Characterization and Properties of Petroleum Fractions

Characterization
and Properties of
Petroleum Fractions
Characterization
and Properties of
Petroleum Fractions
First Edition
M. R. Riazi
Professor of Chemical Engi neeri ng
Kuwai t University
P.O. Box 5969
Safat 13060, Kuwait
riazi@kuc01 .kuniv.edu.kw
ASTM Stock Number: MNL50
ASTM
100 Barr Harbor
West Conshohocken, PA 19428-2959
Printed in the U.S.A.
Library of Congress Catal ogi ng- i n- Publ i cati on Data
Riazi, M.-R.
Characterization and properties of pet rol eum fractions / M.-R. Riazi--1 st ed.
p. cm. --( ASTM manual series: MNL50)
ASTM stock number: MNL50
I ncludes bibliographical references and index.
I SBN 0-8031-3361-8
1. Characterization. 2. Physical property estimation. 3. Pet rol eum fract i ons--crude oils.
TP691.R64 2005
666.5---dc22
2004059586
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Printed in Philadelphia, PA
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To
Shiva, Touraj, and Nazly
Contents
Foreword
Pref ace
Chapter 1--I nt roduct i on
Nomenclature
1.1 Nature of Petroleum Fluids
1.1.1 Hydrocarbons
1.1.2 Reservoir Fluids and Crude Oil
1.1.3 Petroleum Fractions and Products
1.2 Types and I mportance of Physical Properties
1.3 I mportance of Petroleum Fluids Characterization
1.4 Organization of the Book
1.5 Specific Features of this Manual
1.5.1 Introduction of Some Existing Books
1.5.2 Special Features of the Book
1.6 Applications of the Book
1.6.1 Applications in Petroleum Processing
(Downstream)
1.6.2 Applications in Petroleum Production
(Upstream)
1.6.3 Applications in Academia
1.6.4 Other Applications
1.7 Definition of Units and the Conversion Factors
1.7.1 I mportance and Types of Units
1.7.2 Fundamental Units and Prefixes
1.7.3 Units of Mass
1.7.4 Units of Length
1.7.5 Units of Time
1.7.6 Units of Force
1.7.7 Units of Moles
1.7.8 Units of Molecular Weight
1.7.9 Units of Pressure
1.7.10 Units of Temperature
1.7.11 Units of Volume, Specific Volume, and
Molar Volume---The Standard Conditions
1.7.12 Units of Volumetric and Mass Flow Rates
1.7.13 Units of Density and Molar Density
1.7.14 Units of Specific Gravity
1.7.15 Units of Composition
1.7.16 Units of Energy and Specific Energy
1.7.17 Units of Specific Energy per Degrees
1.7.18 Units of Viscosity and Kinematic Viscosity
1.7.19 Units of Thermal Conductivity
1.7.20 Units of Diffusion Coefficients
1.7.21 Units of Surface Tension
1.7.22 Units of Solubility Parameter
1.7.23 Units of Gas-to-Oil Ratio
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viii CONTENTS
1.7.24 Values of Universal Constants
1.7.24.1 Gas Constant
1.7.24.2 Other Numerical Constants
1.7.25 Special Units for the Rates and Amounts of
Oil and Gas
1.8 Problems
References
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Chapter 2 - - Ch aract eri z at i on and Properti es of Pure
Hydrocarbons
Nomenclature
2.1 Definition of Basic Properties
2.1.1
2.1.2
2.1.3
2.1.4
2.1.5
2.1.6
2.1.7
2.1.8
2.1.9
2.1.10
2.1.11
2.1.12
2.1.13
2.1.14
2.1.15
2.1.16
2.1.17
Molecular Weight
Boiling Point
Density, Specific Gravity, and API
Gravity
Refractive Index
Critical Constants (Tc, Pc, Vc, Zc)
Acentric Factor
Vapor Pressure
Kinematic Viscosity
Freezing and Melting Points
Flash Point
Autoignition Temperature
Flammability Range
Octane Number
Aniline Point
Watson K
Refractivity Intercept
Viscosity Gravity Constant
2.1.18 Carbon-to-Hydrogen Weight Ratio
2.2 Data on Basic Properties of Selected Pure
Hydrocarbons
2.2.1 Sources of Data
2.2.2 Properties of Selected Pure Compounds
2.2.3 Additional Data on Properties of Heavy
Hydrocarbons
2.3 Characterization of Hydrocarbons
2.3.1 Development of a Generalized Correlation
for Hydrocarbon Properties
2.3.2 Various Characterization Parameters for
Hydrocarbon Systems
2.3.3 Prediction of Properties of Heavy Pure
Hydrocarbons
2.3.4 Extension of Proposed Correlations to
Nonhydrocarbon Systems
2.4 Prediction of Molecular Weight, Boiling Point, and
Specific Gravity
2.4.1 Prediction of Molecular Weight
2.4.1.1 Riazi-Daubert Methods
2.4.1.2 ASTM Method
2.4.1.3 API Methods
2.4.1.4 Lee--Kesler Method
2.4.1.5 Goossens Correlation
2.4.1.6 Other Methods
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CONTENTS
2.4.2 Prediction of Normal Boiling Point
2.4.2.1 Riazi-Daubert Correlations
2.4.2.2 Soreide Correlation
2.4.3 Prediction of Specific Gravity/API Gravity
2.5 Prediction of Critical Properties and Acentric
Factor
2.5.1 Prediction of Critical Temperature and
Pressure
2.5.1.2 API Methods
2.5.1.3 Lee-Kesler Method
2.5.1.4 Cavett Method
2.5.1.5 Twu Method for To, Pc, Vc, and M
2.5.1.6 Winn-Mobil Method
2.5.1.7 Tsonopoulos Correlations
2.5.2 Prediction of Critical Volume
2.5.2.2 Hall-Yarborough Method
2.5.2.3 API Method
2.5.3 Prediction of Critical Compressibility Factor
2.5.4 Prediction of Acentric Factor
2.5.4.2 Edmister Method
2.5.4.3 Korsten Method
2.6 Prediction of Density, Refractive Index, CH Weight
Ratio, and Freezing Point
2.6.1 Prediction of Density at 20~C
2.6.2 Prediction of Refractive Index
2.6.3 Prediction of CH Weight Ratio
2.6.4 Prediction of Freezing/Melting Point
2.7 Prediction of Kinematic Viscosity at 38
and 99~
2.8 The Winn Nomogram
2.9 Analysis and Comparison of Various
Characterization Methods
2.9.1 Criteria for Evaluation of a Characterization
Method
2.9.2 Evaluation of Methods of Estimation of
Molecular Weight
Critical Properties
Acentric Factor and Other Properties
2.10 Conclusions and Recommendations
2.11 Problems
References
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Chapter 3- - Characterization of Petrol eum Fractions
Nomenclature
3.1 Experimental Data on Basic Properties of
Petroleum Fractions
3.1.1 Boiling Point and Distillation Curves
3.1.1.1 ASTM D86
3.1.1.2 True Boiling Point
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x CONTENTS
3.1.1.3 Simulated Distillation by Gas
Chromatography
3.1.1.4 Equilibrium Flash Vaporization
3.1.1.5 Distillation at Reduced Pressures
3.1.2 Density, Specific Gravity, and API Gravity
3.1.3 Molecular Weight
3.1.4 Refractive Index
3.1.5 Compositional Analysis
3.1.5.1 Types of Composition
3.1.5.2 Analytical I nstruments
3.1.5.3 PNA Analysis
3.1.5.4 Elemental Analysis
3.1.6 Viscosity
3.2 Prediction and Conversion of Distillation Data
3.2.1 Average Boiling Points
3.2.2 Interconversion of Various Distillation Data
3.2.2.1 Riazi-Daubert Method
3.2.2.2 Daubert's Method
3.2.2.3 Interconverion of Distillation Curves
at Reduced Pressures
3.2.2.4 Summary Chart for Interconverion
of Various Distillation Curves
3.2.3 Prediction of Complete Distillation Curves
3.3 Prediction of Properties of Petroleum Fractions
3.3.1 Matrix of Pseudocomponents Table
3.3.2 Narrow Versus Wide Boiling Range
Fractions
3.3.3 Use of Bulk Parameters (Undefined
Mixtures)
3.3.4 Method of Pseudocomponent (Defined
Mixtures)
3.3.5 Estimation of Molecular Weight, Critical
Properties, and Acentric Factor
3.3.6 Estimation of Density, Specific Gravity,
Refractive Index, and Kinematic Viscosity
3.4 General Procedure for Properties of Mixtures
3.4.1 Liquid Mixtures
3.4.2 Gas Mixtures
3.5 Prediction of the Composition of Petroleum
Fractions
3.5.1 Prediction of PNA Composition
3.5.1.1 Characterization Parameters for
Molecular Type Analysis
3.5.1.2 API Riazi-Daubert Methods
3.5.1.3 API Method
3.5.1.4 n-d-M Method
3.5.2 Prediction of Elemental Composition
3.5.2.1 Prediction of Carbon and Hydrogen
Contents
3.5.2.2 Prediction of Sulfur and Nitrogen
Contents
3.6 Prediction of Other Properties
3.6.1 Properties Related to Volatility
3.6.1.1 Reid Vapor Pressure
3.6.1.2 WL Ratio and Volatility Index
3.6.1.3 Flash Point
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CONTENTS
3.6.2 Pour Point
3.6.3 Cloud Point
3.6.4 Freezing Point
3.6.5 Aniline Point
3.6.5.1 Winn Method
3.6.5.2 Walsh-Mortimer
3.6.5.3 Linden Method
3.6.5.4 Albahri et al. Method
3.6.6 Cetane Number and Diesel Index
3.6.7 Octane Number
3.6.8 Carbon Residue
3.6.9 Smoke Point
3.7 Quality of Petroleum Products
3.8 Minimum Laboratory Data
3.9 Analysis of Laboratory Data and Development
of Predictive Methods
3.11 Problems
References
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Chapter A Characteri z ati on o f Reserv oi r Fl ui ds and
Crude Oils
Nomenclature
4.1 Specifications of Reservoir Fluids and Crude
Assays
4.1.1 Laboratory Data for Reservoir Fluids
4.1.2 Crude Oil Assays
4.2 Generalized Correlations for Pseudocritical
Properties of Natural Gases and Gas Condensate
Systems
4.3 Characterization and Properties of Single Carbon
Number Groups
4.4 Characterization Approaches for C7+ Fractions
4.5 Distribution functions for Properties of
Hydrocarbon-plus Fractions
4.5.1 General Characteristics
4.5.2 Exponential Model
4.5.3 Gamma Distribution Model
4.5.4 Generalized Distribution Model
4.5.4.1 Versatile Correlation
4.5.4.2 Probability Density Function for the
Proposed Generalized Distribution
Model
4.5.4.3 Calculation of Average Properties of
Hydrocarbon-Plus Fractions
4.5.4.4 Calculation of Average Properties of
Subfractions
4.5.4.5 Model Evaluations
4.5.4.6 Prediction of Property Distributions
Using Bulk Properties
4.6 Pseudoization and Lumping Approaches
4.6.1 Splitting Scheme
4.6.1.1 The Gaussian Quadrature Approach
4.6.1.2 Carbon Number Range Approach
4.6.2 Lumping Scheme
4.7 Continuous Mixture Characterization Approach
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xii CONTENTS
4.8 Calculation of Properties of Crude Oils and
Reservoir Fluids
4.8.1 General Approach
4.8.2 Estimation of Sulfur Content of a Crude Oil
4.10 Problems
References
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Chapter 5mPVT Relations and Equations of State
Nomenclature
5.1 Basic Definitions and the Phase Rule
5.2 PVT Relations
5.3 I ntermolecular Forces
5.4 Equations of State
5.4.1 Ideal Gas Law
5.4.2 Real Gases--Liquids
5.5 Cubic Equations of State
5.5.1 Four Common Cubic Equations (vdW, RK,
SRK, and PR)
5.5.2 Solution of Cubic Equations of State
5.5.3 Volume Translation
5.5.4 Other Types of Cubic Equations of State
5.5.5 Application to Mixtures
5.6 Noncubic Equations of State
5.6.1 Virial Equation of State
5.6.2 Modified Benedict-Webb-Rubin Equation
of State
5.6.3 Carnahan-Starling Equation of State and Its
Modifications
5.7 Corresponding State Correlations
5.8 Generalized Correlation for PVT Properties of
Liquids--Rackett Equation
5.8.1 Rackett Equation for Pure Component
Saturated Liquids
5.8.2 Defined Liquid Mixtures and Petroleum
Fractions
5.8.3 Effect of Pressure on Liquid Density
5.9 Refractive Index Based Equation of State
5.10 Summary and Conclusions
5.11 Problems
References
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Chapter 6- - - Thermodynamic Relations for Property Estimations
Nomenclature
6.1 Definitions and Fundamental Thermodynamic
Relations
6.1.1 Thermodynamic Properties and
Fundamental Relations
6.1.2 Measurable Properties
6.1.3 Residual Properties and Departure
Functions
6.1.4 Fugacity and Fugacity Coefficient for Pure
Components
6.1.5 General Approach for Property Estimation
6.2 Generalized Correlations for Calculation of
Thermodynamic Properties
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CONTENTS xiii
6.3 Properties of Ideal Gases
6.4 Thermodynamic Properties of Mixtures
6.4.1 Partial Molar Properties
6.4.2 Properties of Mixtures--Property Change
Due to Mixing
6.4.3 Volume of Petroleum Blends
6.5 Phase Equilibria of Pure Components--Concept
of Saturation Pressure
6.6 Phase Equilibria of Mixtures--Calculation
of Basic Properties
6.6.1 Definition of Fugacity, Fugacity Coefficient,
Activity, Activity Coefficient, and Chemical
Potential
6.6.2 Calculation of Fugacity Coefficients from
Equations of State
6.6.3 Calculation of Fugacity from Lewis Rule
6.6.4 Calculation of Fugacity of Pure Gases and
Liquids
6.6.5 Calculation of Activity Coefficients
6.6.6 Calculation of Fugacity of Solids
6.7 General Method for Calculation of Properties of
Real mixtures
6.8 Formulation of Phase Equilibria Problems for
Mixtures
6.8. I Criteria for Mixture Phase Equilibria
6.8.2 Vapor-Liquid Equilibria--Gas Solubility in
Liquids
6.8.2.1 Formulation of Vapor-Liquid
Equilibria Relations
6.8.2.2 Solubility of Gases in
Liquids--Henry' s Law
6.8.2.3 Equilibrium Ratios (K/Values)
6.8.3 Solid-Liquid Equilibria--Solid Solubility
6.8.4 Freezing Point Depression and Boiling Point
Elevation
6.9 Use of Velocity of Sound in Prediction of Fluid
Properties
6.9.1 Velocity of Sound Based Equation
of State
6.9.2 Equation of State Parameters from Velocity
of Sound Data
6.9.2.1 Virial Coefficients
6.9.2.2 Lennard-Jones and van der Waals
Parameters
6.9.2.3 RK and PR EOS Paramet ers--
Property Estimation
6.10 Summary and Recommendations
6.11 Problems
References
Chapter 7--Applications: Estimation of Thermophysical
Properties
Nomenclature
7.1 General Approach for Prediction of
Thermophysical Properties of Petroleum Fractions
and Defined Hydrocarbon Mixtures
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7.2 Density
7.2.1 Density of Gases
7.2.2 Density of Liquids
7.2.3 Density of Solids
7.3 Vapor Pressure
7.3.1 Pure Components
7.3.2 Predictive Methods--Generalized
Correlations
7.3.3 Vapor Pressure of Petroleum Fractions
7.3.3.1 Analytical Methods
7.3.3.2 Graphical Methods for Vapor
Pressure of Petroleum Products
and Crude Oils
7.3.4 Vapor Pressure of Solids
7.4 Thermal Properties
7.4.1 Enthalpy
7.4.2 Heat Capacity
7.4.3 Heats of Phase Changes--Heat of
Vaporization
7.4.4 Heat of Combustion--Heating Value
7.6 Problems
References
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Chapter 8mAppHcations: Estimation of Transport Properties
Nomenclature
8.1 Estimation of Viscosity
8.1.1 Viscosity of Gases
8.1.2 Viscosity of Liquids
8.2 Estimation of Thermal Conductivity
8.2.1 Thermal Conductivity of Gases
8.2.2 Thermal Conductivity of Liquids
8.3 Diffusion Coefficients
8.3.1 Diffusivity of Gases at Low Pressures
8.3.2 Diffusivity of Liquids at Low Pressures
8.3.3 Diffusivity of Gases and Liquids at High
Pressures
8.3.4 Diffusion Coefficients in Mutlicomponent
Systems
8.3.5 Diffusion Coefficient in Porous Media
8.4 Interrelationship Among Transport Properties
8.5 Measurement of Diffusion Coefficients in Reservoir
Fluids
8.6 Surface/Interracial Tension
8.6.1 Theory and Definition
8.6.2 Predictive Methods
8.8 Problems
References
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Chapter 9--Applications: Phase Equilibrium Calculations
Nomenclature
9.1 Types of Phase Equilibrium Calculations
9.2 Vapor-Liquid Equilibrium Calculations
9.2.1 Flash Calculations--Gas-to-Oil Ratio
9.2.2 Bubble and Dew Points Calculations
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CONTENTS x v
9.2.3 Generation of P-T Diagrams--True Critical
Properties
9.3 Vapor-Liquid-Solid Equi l i bri um--Sol i d
Precipitation
9.3.1 Nature of Heavy Compounds, Mechanism of
their Precipitation, and Prevention Methods
9.3.2 Wax Precipitation--Solid Solution Model
9.3.3 Wax Precipitation: Multisolid-Phase
Model ~Cal cul at i on of Cloud Point
9.4 Asphakene Precipitation: Solid-Liquid Equilibrium
9.5 Vapor-Solid Equi l i bri um--Hydrat e Format i on
9.6 Applications: Enhanced Oil Recovery--Evaluation
of Gas I njection Projects
9.7 Summary and Recommendat i ons
9.8 Final Words
9.9 Problems
References
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Appendix
I nde x
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Foreword
THIS PUBLICATION, Characterization and Properties of Petroleum Fractions, was sponsored
by ASTM Committee D02 on Petroleum Fuels and Lubricants. The aut hor is M. R. Riazi,
Professor of Chemical Engineering, Kuwait University, Safat, Kuwait. This publication
is Manual 50 of ASTM's manual series.
xvii
Pref ace
Scientists do not belong to any particular country, ideology, or religion, they belong
to the world community
THE FIELD OF Petroleum Characterization and Physical Properties has received significant
attention in recent decades with the expansion of comput er simulators and advanced
analytical tools and the availability of more accurate experimental data. As a result of
globalization, structural changes are taking place in the chemical and pet rol eum indus-
try. Engineers working in these industries are involved with process simulators to design
and operate various units and equipment. Nowadays, a large number of process simula-
tors are being produced that are equipped with a variety of t hermodynami c models and
choice of predictive met hods for the physical properties. A person familiar with devel-
opment of such met hods can make appropri at e use of these simulators saving billions
of dollars in costs in investment, design, manufact ure, and operation of various units
in these industries. Petroleum is a complex mixture of thousands of hydrocarbon com-
pounds and it is produced from an oil well in a form of reservoir fluid. A reservoir fluid is
converted to a crude oil through surface separation units and then the crude is sent to a
refinery to produce various pet rol eum fractions and hydrocarbon fuels such as kerosene,
gasoline, and fuel oil. Some of the refinery products are the feed to petrochemical plants.
More t han half of world energy sources are from pet rol eum and probabl y hydrocarbons
will remai n the most convenient and i mport ant source of energy and as a raw material
for the petrochemical plants at least t hroughout the 21 st century. Other fossil type fu-
els such as coal liquids are also mixtures of hydrocarbons although they differ in type
with pet rol eum oils. From 1970 to 2000, the share of Middle East in the world crude oil
reserves raised from 55 to 65%, but this share is expected to rise even further by 2010-
2020 when we near the point where half of oil reserves have been produced. The world
is not running out of oil yet but the era of cheap oil is perhaps near the end. Therefore,
economical use of the remaining oil and t reat ment of heavy oils become increasingly
important. As it is discussed in Chapter 1, use of more accurate physical properties for
pet rol eum fractions has a direct and significant i mpact on economical operation and
design of pet rol eum processing and product i on units which in turn would result in a
significant saving of existing pet rol eum reserves.
One of the most i mport ant tasks in pet rol eum refining and related processes is the
need for reliable values of the volumetric and t hermodynami c properties for pure hy-
drocarbons and their mixtures. They are i mport ant in the design and operation of al most
every piece of processing equipment. Reservoir engineers analyze PVT and phase behav-
ior of reservoir fluids to estimate the amount of oil or gas in a reservoir, to determine an
opt i mum operating condition in a separat or unit, or to develop a recovery process for
an oil or gas field. However, the most advanced design approaches or the most sophisti-
cated simulators cannot guarantee the opt i mum design or operation of a unit if required
input physical properties are not accurate. A process to experimentally determine the
volumetric, t hermodynami c, and t ransport properties for all the industrially i mport ant
materials would be prohibitive in bot h cost and time; indeed it could probabl y never
be completed. For these reasons accurate estimations of these properties are becomi ng
increasingly important.
Characterization factors of many types permeat e the entire field of physical, ther-
modynami c, and t ransport property prediction. Average boiling points, specific gravity,
molecular weight, critical temperature, critical pressure, acentric factor, refractive index,
and certain molecular type analysis are basic paramet ers necessary to utilize met hods
of correlation and prediction of the thermophysical properties. For correlating physi-
cal and t hermodynami c properties, met hods of characterizing undefined mixtures are
xi x
x x PREFACE
necessary to provide input data. It could be imagined that the best method of character-
izing a mixture is a complete analysis. However, because of the complexity of undefined
mixtures, complete analyses are usually impossible and, at best, inconvenient. A predic-
tive method to determine the composition or amount of sulfur in a hydrocarbon fuel is
vital to see if a product meets specifications set by the government or other authorities
to protect the environment.
My first interaction with physical properties of petroleum fluids was at the time that
I was a graduate student at Penn State in the late 70s working on a project related to
enhanced oil recovery for my M.S. thesis when I was looking for methods of estimation
of properties of petroleum fluids. It was such a need and my personal interest that
later I joined the ongoing API project on thermodynamic and physical properties of
petroleum fractions to work for my doctoral thesis. Since that time, property estimation
and characterization of various petroleum fluids has remained one of my main areas of
research. Later in the mid-80s I rejoined Penn State as a faculty member and I continued
my work with the API which resulted in development of methods for several chapters
of the API Technical Data Book. Several years later in late 80s, I continued the work
while I was working at the Norwegian Institute of Technology (NTH) at Trondheim
where I developed some characterization techniques for heavy petroleum fractions as
well as measuring methods for some physical properties. I n the 90s while at Kuwait
University I got the opportunity to be in direct contact with the oil companies in the
region through research, consultation, and conducting special courses for the industry.
My association with the University of Illinois at Chicago in early 90s was helpful in the
development of equations of state based on velocity of sound. The final revision of the
book was completed when I was associated with the University of Texas at Austin and
McGill University in Montreal during my leave from Kuwait University.
Characterization methods and estimating techniques presented in this book have been
published in various international journals or technical handbooks and included in
many commercial softwares and process simulators. They have also been presented as
seminars in different oil companies, universities, and research centers worldwide. The
major characteristics of these methods are simplicity, generality, accuracy, and avail-
ability of input parameters. Many of these methods have been developed by utilizing
some scientific fundamentals and combining them with a broad experimental data set
to generate semi-theoretical or semi-empirical correlations. Some of these methods have
been in use by the petroleum industry and research centers worldwide for the past two
decades.
Part of the materials in this book were prepared when I was teaching a graduate course
in applied thermodynamics in 1988 while at NTH. The materials, mainly a collection of
technical papers, have been continuously updated and rearranged to the present time.
These notes have also been used to conduct industrial courses as well as a course on fluid
properties in chemical and petroleum engineering. This book is an expansion with com-
plete revision and rewriting of these notes. The main objective of this book is to present
the fundamentals and practice of estimating the physical and thermodynamic proper-
ties as well as characterization methods for hydrocarbons, petroleum fractions, crude
oils, reservoir fluids, and natural gases, as well as coal liquids. However, the emphasis is
on the liquid petroleum fractions, as properties of gases are generally calculated more
accurately. The book will emphasize manual calculations with practical problems and
examples but also wilI provide good understanding of techniques used in commercial
software packages for property estimations. Various methods and correlations developed
by different researchers commonl y used in the literature are presented with necessary
discussions and recommendations.
My original goal and objective in writing this book was to provide a reference for the
petroleum industry in both processing and production. It is everyone's experience that in
using thermodynamic simulators for process design and equipment, a large number of
options is provided to the user for selection of a method to characterize the oil or to get
an estimate of a physical property. This is a difficult choice for a user of a simulator, as
the results of design calculations significantly rely on the method chosen to estimate the
properties. One of my goals in writing this book was to help users of simulators overcome
this burden. However, the book is written in a way that it can also be used as a textbook
for graduate or senior undergraduate students in chemical, petroleum, or mechanical
engineering to understand the significance of characterization, property estimation and
PREFACE xxi
met hods of their development. For this purpose a set of probl ems is presented at the
end of each chapter. The book covers characterization as well as met hods of estimation
of t hermodynami c and t ransport properties of various pet rol eum fluids and products. A
great emphasi s is given to t reat ment of heavy fractions t hroughout the book. An effort
was made to write the book in a way that not only would be useful for the profession-
als in the field, but would also be easily understandable to those non-engineers such as
chemists, physicists, or mat hemat i ci ans who get involved with the pet rol eum industry.
The word properties in the title refers to t hermodynami c, physical, and t ransport proper-
ties. Properties related to the quality and safety of pet rol eum products are also discussed.
Organization of the book, its uses, and i mport ance of the met hods are discussed in detail
in Chapter 1. I nt roduct i on of similar books and the need for the present book as well as
its application in the industry and academi a are also discussed in Chapter 1. Each chap-
ter begins with nomencl at ure and ends with the references used in that chapter. Exercise
problems in each chapt er contain additional i nformat i on and methods. More specific
information about each chapt er and its contents are given in Chapter 1. As Goethe
said, "Things which mat t er most must never be at the mercy of things which mat t er
least."
I am indebted to many people especially teachers, colleagues, friends, students, and,
above all, my parents, who have been so helpful t hroughout my academic life. I am partic-
ularly thankful to Thomas E. Daubert of Pennsylvania State University who introduced
to me the field of physical properties and pet rol eum characterization in a very clear and
understandable way. Likewise, I am thankful to Farhang Shadman of the University of
Arizona who for the first time introduced me to the field of chemical engineering re-
search during my undergraduat e studies. These two individuals have been so influential
in shaping my academic life and I am so indebted to t hem for their human characters
and their scientific skills. I have been fortunate to meet and talk with many scientists and
researchers from both the oil industry and academi a from around the world during the
last two decades whose thoughts and ideas have in many ways been helpful in shaping
the book.
I am also grateful to the institutions, research centers, and oil compani es that I have
been associated with or that have invited me for lecturing and consultation. Thanks to
Kuwait University as well as Kuwait Petroleum Corporation (KPC) and KNPC, many of
whose engineers I developed working relations with and have been helpful in evaluation
of many of the estimating met hods t hroughout the years. I am thankful to all scientists
and researchers whose works have been used in this book and I hope that all have
been correctly and appropriately cited. I would be happy to receive their comment s and
suggestions on the book. Financial support from organizations such as API, NSF, GPA,
GRI, SI NTEE Petrofina Exploration Norway, NSERC Canada, Kuwait University, and
KFAS that was used for my research work over the past two decades is also appreciated.
I am grateful to ASTM for publishing this work and particularly to Geroge Totten who
was the first to encourage me to begin writing this book. His advice, interest, support,
and suggestions t hrough the long years of writing the book have been extremely helpful
in completing this project. The introductory comment s from hi m as well as those from
Philip T. Eubank and Jos6 Luis Pefia Diez for the back cover are appreciated. I am also
grateful to the four unani mous reviewers who tirelessly reviewed the entire and lengthy
manuscri pt with their constructive comment s and suggestions which have been reflected
in the book. Thanks also to Kathy Dernoga, the publishing manager at ASTM, who was
always cooperative and ready to answer my questions and provided me with necessary
information and tools during the preparat i on of this manuscript. Her encouragement s
and assistance were quite useful in pursuing this work. She also was helpful in the de-
sign of the front and back covers of the book as well as providing editorial suggestions.
I am thankful to Robert a Storer and Joe Ermigiotti for their excellent job of editing and
updating the manuscript. Cooperation of other ASTM staff, especially Monica Siperko,
Carla J. Falco, and Marsha Fi rman is highly appreciated. The art work and most of
the graphs and figures were prepared by Khaled Damyar of Kuwait University and his
efforts for the book are appreciated. I also sincerely appreciate the publishers and the
organizations that gave their permissions to include some published materials, in partic-
ular API, ACS, AIChE, GPA, Elsevier (U.K.), editor of Oil & Gas J., McGraw-Hill, Marcel
and Dekker, Wiley, SPE, and Taylor and Francis. Thanks to the manager and personnel
of KI SR for allowing the use of phot os of their instruments in the book. Finally and
xxi i PREFACE
most importantly, I must express my appreciation and thanks to my family who have
been helpful and patient during all these years and without whose cooperation, moral
support, understanding, and encouragement this project could never have been under-
taken. This book is dedicated to my family, parents, teachers, and the world scientific
community.
M. R. Riazi
August 2004
MNL50-EB/Jan. 2005
Introduction
NOMENCLATURE
API API gravity
A% Percent of aromatics in a pet rol eum
fraction
D Diffusion coefficient
CH Carbon-to-hydrogen weight ratio
d Liquid density at 20~ and 1 at m
Kw Watson K factor
k Thermal conductivity
Ki Equilibrium ratio of component i in
a mixture
log10 Logarithm of base l0
In Logarithm of base e
M Molecular weight
Nmin Mi ni mum number of theoretical plates
in a distillation col umn
N% Percent of napht henes in a pet rol eum
fraction
n Sodi um D line refractive index of liquid
at 20~ and 1 atrn, dimensionless
n Number of moles
P Pressure
Pc Critical pressure
psat Vapor ( saturation) pressure
P% Percent of paraffins in a pet rol eum
fraction
R Universal gas constant
Ri Refractivity intercept
SG Specific gravity at 15.5~ (60~
SUS Saybolt Universal Seconds (unit of
viscosity)
S% Weight % of sulfur in a pet rol eum
fraction
T Temperat ure
Tb Boiling point
Tc Critical t emperat ure
TF Flash point
Tp Pour point
TM Melting (freezing point) point
V Volume
Xm Mole fraction of a component in
a mixture
Xv Volume fraction of a component in
a mixture
Xw Weight fraction of a component in a
mixture
y Mole fraction of a component in a vapor
phase
1
Greek Letters
Relative volatility
~0 Fugacity coefficient
a~ Acentric factor
Surface tension
p Density at t emperat ure T and pressure P
/~ Viscosity
v Kinematic viscosity
Acronyms
API-TDB American Petroleum I nstitute-Technical Data
Book
bbl Barrel
GOR Gas-to-oil ratio
IUPAC I nternational Union of Pure and Applied Chem-
istry
PNA Paraffin, naphthene, aromat i c content of a
pet rol eum fraction
SC Standard conditions
scf Standard cubic feet
stb Stock t ank barrel
STO Stock t ank oil
STP St andard t emperat ure and pressure
I N THIS INTRODUCTORY CHAPTER, f i r s t the nature of pet rol eum
fluids, hydrocarbon types, reservoir fluids, crude oils, natural
gases, and pet rol eum fractions are introduced and then types
and i mport ance of characterization and physical properties
are discussed. Application of materials covered in the book in
various parts of the pet rol eum industry or academi a as well
as organization of the book are then reviewed followed by
specific features of the book and introduction of some other
related books. Finally, units and the conversion factors for
those paramet ers used in this book are given at the end of the
chapter.
1.1 NATURE OF PETROLEUM FLUI DS
Petroleum is one of the most i mport ant substances consumed
by man at present time. I t is used as a mai n source of energy
for industry, heating, and t ransport at i on and it also pro-
vides the raw materials for the petrochemical plants to pro-
duce polymers, plastics, and many other products. The word
petroleum, derived from the Latin words petra and oleum,
means literally rock oil and a special type of oil called oleum
[1]. Petroleum is a complex mixture of hydrocarbons that
occur in the sedi ment ary rocks in the form of gases ( natural
Copyright 9 2005 by ASTM International www.astm.org
2 CHARACTERI ZATI ON AND PROPERT I ES OF PET ROL EUM FRACTI ONS
gas), liquids (crude oil), semi sol i ds (bi t umen), or solids (wax
or asphaltite). Li qui d fuels are nor mal l y pr oduced f r om liq-
ui d hydr ocar bons, al t hough conver si on of nonl i qui d hydro-
car bons such as coal, oil shale, and nat ur al gas to liquid fuels
is bei ng investigated. I n t hi s book, onl y pet r ol eum hydrocar-
bons in t he f or m of gas or liquid, si mpl y called petroleum flu-
ids, are consi dered. Li qui d pet r ol eum is also si mpl y called oil.
Hydr ocar bon gases in a r eser voi r are called a natural gas or
si mpl y a gas. An under gr ound reservoi r t hat cont ai ns hydro-
car bons is called petroleum reservoir and its hydr ocar bon con-
t ent s t hat can be r ecover ed t hr ough a pr oduci ng well is called
reservoir fluid. Reservoi r fluids in t he reservoi rs are usual l y in
cont act wi t h wat er in por ous medi a condi t i ons and because
t hey are l i ght er t han water, t hey st ay above t he wat er level
under nat ur al condi t i ons.
Al t hough pet r ol eum has been known for ma ny cent uri es,
t he first oi l -produci ng well was di scovered in 1859 by E.L.
Drake in t he st at e of Pennsyl vani a and t hat mar ked t he
bi r t h of mode r n pet r ol eum t echnol ogy and refining. The
mai n el ement s of pet r ol eum are car bon (C) and hydr ogen
(H) and s ome smal l quant i t i es of sul fur (S), ni t rogen (N),
and oxygen (O). There are several t heori es on t he f or mat i on
of pet r ol eum. I t is general l y believed t hat pet r ol eum is de-
ri ved f r om aquat i c pl ant s and ani mal s t hr ough conver si on of
organi c compounds i nt o hydr ocar bons. These ani mal s and
pl ant s under aquat i c condi t i ons have conver t ed i norgani c
compounds dissolved in wat er (such as car bon dioxide) to
organi c compounds t hr ough t he energy provi ded by t he sun.
An exampl e of such r eact i ons is shown below:
(1.1) 6CO2 + 6H20 d- energy --~ 602 + C6H1206
in whi ch C6H1206 is an organi c compound called car bohy-
drat e. I n some cases organi c compounds exist in an aquat i c
envi r onment . For exampl e, t he Nile ri ver in Egypt and t he
Ur uguay ri ver cont ai n consi derabl e amount s of organi c ma-
terials. This mi ght be t he r eason t hat mos t oil reservoi rs are
l ocat ed near t he sea. The organi c compounds f or med ma y be
decomposed i nt o hydr ocar bons under cert ai n condi t i ons.
(1.2) (CHEO)n --~ xCO2 d-yCH4
in whi ch n, x, y, and z are i nt eger number s and yCHz is t he
cl osed f or mul a for t he pr oduced hydr ocar bon compound.
Anot her t heor y suggests t hat t he i nor gani c compound cal-
ci um car bonat e (CaCO3) wi t h alkali met al can be conver t ed to
cal ci um carbi de (CaC2), and t hen cal ci um carbi de wi t h wat er
(H20) can be conver t ed t o acet yl ene (C2H2). Finally, acet yl ene
can be conver t ed to pet r ol eum [ 1]. Conversi on of organi c mat -
t ers i nt o pet r ol eum is called maturation. The mos t i mpor t ant
fact ors in t he conver si on of organi c compounds t o pet r ol eum
hydr ocar bons are (1) heat and pressure, (2) radi oact i ve rays,
such as g a mma rays, and (3) cat al yt i c react i ons. Vanadi um-
and ni ckel -t ype cat al yst s are t he mos t effective cat al yst s in
t he f or mat i on of pet r ol eum. For this r eason some of t hese
met al s ma y be f ound in smal l quant i t i es in pet r ol eum fluids.
The rol e of radi oact i ve mat er i al s in t he f or mat i on of hydro-
car bons can be best observed t hr ough radi oact i ve bombar d-
ing of fat t y aci ds (RCOOH) t hat f or m paraffi n hydr ocar bons.
Occasi onal l y t races of radi oact i ve mat er i al s such as ur ani um
and pot as s i um can also be f ound in pet r ol eum. I n summar y,
t he following st eps ar e requi red f or t he f or mat i on of hydrocar-
bons: (1) a sour ce of organi c mat er i al , (2) a process t o convert
organi c compounds i nt o pet r ol eum, and (3) a seal ed r eser voi r
space to st ore t he hydr ocar bons pr oduced. The condi t i ons re-
qui red for t he process of conver si on of organi c compounds
i nt o pet r ol eum (as shown t hr ough Eq. (1.2) ar e (1) geologic
t i me of about 1 mi l l i on years, (2) ma x i mu m pr essur e of
about 17 MPa (2500 psi), and (3) t emper at ur e not exceed-
ing 100-120~ (~210-250~ I f a l eak occur r ed s omet i me
in t he past , t he expl orat i on well will encount er onl y smal l
amount s of resi dual hydr ocar bons. I n s ome cases bact er i a
may have bi odegr aded t he oil, dest royi ng light hydr ocar bons.
An exampl e of such a case woul d be t he l arge heavy oil accu-
mul at i ons in Venezuela. The hydr ocar bons gener at ed grad-
ually mi gr at e f r om t he ori gi nal beds t o mor e por ous rocks,
such as sandst one, and f or m a pet r ol eum reservoir. A series
of reservoi rs wi t hi n a c ommon r ock is called an oil field.
Pet r ol eum is a mi xt ur e of hundr eds of di fferent identifiable
hydr ocar bons, whi ch are di scussed in t he next section. Once
pet r ol eum is accumul at ed in a r eser voi r or in vari ous sedi-
ment s, hydr ocar bon compounds ma y be conver t ed f r om one
f or m to anot her wi t h t i me and varyi ng geological condi t i ons.
Thi s process is called in-situ alteration, and exampl es of chem-
ical al t erat i on are t her mal mat ur at i on and mi cr obi al degra-
dat i on of t he r eser voi r oil. Exampl es of physi cal al t erat i on of
pet r ol eum are t he preferent i al loss of low-boiling const i t uent s
by t he di ffusi on or addi t i on of new mat er i al s to t he oil in
pl ace f r om a source out si de t he r eser voi r [1]. The mai n dif-
ference bet ween vari ous oils f r om di fferent fields ar ound t he
worl d is t he di fference in t hei r composi t i on of hydr ocar bon
compounds. Two oils wi t h exactly t he s ame composi t i on have
i dent i cal physi cal pr oper t i es under t he s ame condi t i ons [2].
A good revi ew of statistical dat a on t he a mount of oil and
gas reservoi rs, t hei r product i on, processi ng, and cons ump-
t i on is usual l y r epor t ed yearl y by t he Oil and Gas Journal
(OGJ). An annual refi nery survey by OGJ is usual l y publ i shed
in December of each year. OGJ al so publ i shes a forecast and
revi ew r epor t in J anuar y and a mi dyear forecast r epor t in
Jul y of each year. I n 2000 it was r epor t ed t hat t ot al pr oven oil
reserves is est i mat ed at 1016 billion bbl (1.016 x 10 tz bbl),
whi ch for a t ypi cal oil is equi val ent to appr oxi mat el y 1.39 x
1011 tons. The r at e of oil pr oduct i on was about 64.6 mi l l i on
bbl / d (~3. 23 billion t on/ year) t hr ough mor e t han 900 000 pro-
duci ng wells and some 750 refineries [3, 4]. These number s
vary f r om one source to another. For exampl e, Ener gy Infor-
mat i on Admi ni st r at i on of US Depar t ment of Ener gy r epor t s
worl d oil reserves as of J anuar y 1, 2003 as 1213.112 billion
bbl accor di ng to OGJ and 1034.673 billion bbl accor di ng to
World Oil (www. eia. doe. gov/emeu/iea). Accordi ng to t he OGJ
worl dwi de pr oduct i on r epor t s (Oil and Gas Journal, Dec. 22,
2003, p. 44), wor l d oil reserves est i mat es changed f r om 999.78
in 1995 to 1265.811 billion bbl on J anuar y 1, 2004. For t he
s ame per i od wor l d gas reserves est i mat es changed f r om 4.98 x
1015 scf to 6.0683 x 1015 scf. I n 2003 oil cons umpt i on was
about 75 billion bbl/day, and it is expect ed t hat it will in-
crease t o mor e t han 110 mi l l i on bbl / day by t he year 2020.
Thi s means t hat wi t h existing pr oduct i on rat es and reserves,
it will t ake nearl y 40 years for t he world' s oil to end. Oil
reserves life (reserves-t o-product i on rat i o) in s ome sel ect ed
count ri es is given by OGJ (Dec. 22, 2004, p. 45). Accordi ng
t o 2003 pr oduct i on rates, reserves life is 6.1 years in UK,
10.9 years in US, 20 years in Russia, 5.5 years in Canada,
84 years in Saudi Arabia, 143 years in Kuwai t , and 247 years
1. I NTRODUCTI ON 3
in Iraq. As in January l, 2002, the total number of world oil
wells was 830 689, excluding shut or service wells (OGJ, Dec.
22, 2004). Estimates of world oil reserves in 1967 were at
418 billion and in 1987 were at 896 billion bbl, which shows
an increase of 114% in this period [5]. Two-thirds of these
reserves are in the Middle East, although this portion de-
pends on the type of oil considered. Although some people
believe the Middle East has a little more than half of world
oil reserves, it is believed that many undiscovered oil reser-
voirs exist offshore under the sea, and with increase in use
of the other sources of energy, such as natural gas or coal,
and through energy conservation, oil production may well
continue to the end of the century. January 2000, the total
amount of gas reserves was about 5.15 1015 scf, and
its production in 1999 was about 200 x 109 scf/d (5.66 x
109 sm3/d) through some 1500 gas plants [3]. In January
2004, according to OGJ (Dec. 22, 2004, p. 44), world natu-
ral gas reserves stood at 6.068 1015 scf (6068.302 trillion
scf). This shows that existing gas reserves may last for some
70 years. Estimated natural gas reserves in 1975 were at
2.5 x 1015 scf (7.08 x 1013 sm3), that is, about 50% of current
reserves [6]. In the United States, consumption of oil and gas
in 1998 was about 65% of total energy consumption. Crude
oil demand in the United State in 1998 was about 15 million
bbl/d, that is, about 23% of total world crude production [3].
Worldwide consumption of natural gas as a clean fuel is on
the rise, and attempts are underway to expand the trans-
fer of natural gas through pipelines as well as its conver-
sion to liquid fuels such as gasoline. The world energy con-
sumption is distributed as 35% through oil, 31% through
coal, and 23% through natural gas. Nearly 11% of total
world energy is produced through nuclear and hydroelectric
sources [ 1].
1.1.1 Hydrocarbons
In early days of chemistry science, chemical compounds were
divided into two groups: inorganic and organic, depending
on their original source. Inorganic compounds were obtained
from minerals, while organic compounds were obtained from
living organisms and contained carbon. However, now or-
ganic compounds can be produced in the laboratory. Those
organic compounds that contain only elements of carbon (C)
and hydrogen (H) are called hydrocarbons, and they form
the largest group of organic compounds. There might be as
many as several thousand different hydrocarbon compounds
in petroleum reservoir fluids. Hydrocarbon compounds have
a general closed formula of CxHy, where x and y are integer
numbers. The lightest hydrocarbon is methane (CH4), which
is the main component in a natural gas. Methane is from a
group of hydrocarbons called paraffins. Generally, hydrocar-
bons are divided into four groups: (1) paraffins, (2) olefins,
(3) naphthenes, and (4) aromatics. Paraffins, olefins, and
naphthenes are sometime called aliphatic versus aromatic
compounds. The International Union of Pure and Applied
Chemistry (IUPAC) is a nongovernment organization that
provides standard names, nomenclature, and symbols for dif-
ferent chemical compounds that are widely used [7]. The
relationship between the various hydrocarbon constituents
of crude oils is hydrogen addition or hydrogen loss. Such
interconversion schemes may occur during the formation,
maturation, and in-situ alteration of petroleum.
Paraffins are also called alkanes and have the general for-
mula of C, Han+a, where n is the number of carbon atoms.
Paraffins are divided into two groups of normal and isoparaf-
fins. Normal paraffins or normal alkanes are simply written
as n-paraffins or n-alkanes and they are open, straight-chain
saturated hydrocarbons. Paraffins are the largest series of hy-
drocarbons and begin with methane (CH4), which is also rep-
resented by C1. Three n-alkanes, methane (C1), ethane (C2),
and n-butane (C4), are shown below:
H H H H H H H
I I I I I I I
H- - C - - H H- - C - - C - - H H- - C - - C - - C - - C - - H
I I I I I I I
H H H H H H H
Methane Ethane n-Butane
(CH4) (C2H6) (C4H1~
The open formula for n-C4 can also be shown as CH3--
CH2--CH2--CH3 and for simplicity in drawing, usually the
CH3 and CH2 groups are not written and only the carbon-
carbon bonds are drawn. For example, for a n-alkane com-
pound of n-heptadecane with the formula of C17H36, the
structure can also be shown as follows:
n-Heptadecane (C17H36)
The second group of paraffins is called isoparaffins; these
are branched-type hydrocarbons and begin with isobutane
(methylpropane), which has the same closed formula as n-
butane (Call10). Compounds of different structures but the
same closed formula are called isomers. Three branched or
isoparaffin compounds are shown below:
CH3 CH3 CH3
CH3--CH--CH3 CH3--CH~CH2--CH3 CH3--CH--CH2--CH2--CH2--CH2--CH3
isobutane isopen~ane (methylbutane) isooctane (2-methylheptane)
(C4HIo) (C5H12) (C8HI8)
In the case of isooctane, if the methyl group (CH3) is at-
tached to another carbon, then we have another compound
(i.e., 3-methylheptane). It is also possible to have more than
one branch of CH3 group, for example, 2,3-dimethylhexane
and 2-methylheptane, which are simply shown as following:
2-Met hyl hept ane (CsHls) 2, 3-Di met hyl hexane (C8H18)
Numbers refer to carbon numbers where the methyl group
is attached. For example, 1 refers to the first carbon either
1.0E+15
I.OE+IO
1 . 0 E ~ 5
1.0E+O0
0 10 20 30 40
Number of Carbon Atoms
FIG. 1.1reNumber of possible alkane isomers.
4 CHARACTERI ZATI ON AND PROPERTI ES OF PETROLEUM FRACTI ONS
50
from the right or from the left. There are 2 isomers for bu-
tane and 3 for pentane, but there are 5 isomers for hexane, 9
for heptane, 18 for octane (C8H18), and 35 for nonane. Sim-
ilarly, dodecane (C12H26) has 355, while octadecane (C18H38)
has 60523 and C40 has 62 x 1012 isomers [1, 8, 9]. The num-
ber of isomers rapidly increases with the number of carbon
atoms in a molecule because of the rapidly rising number of
their possible structural arrangements as shown in Fig. 1.1.
For the paraffins in the range of Cs-C12, the number of iso-
mers is more than 600 although only about 200-400 of them
have been identified in petroleum mixtures [ 10]. Isomers have
different physical properties. The same increase in number
of isomers with molecular weight applies to other hydro-
carbon series. As an example, the total number of hydrocar-
bons (from different groups) having 20 carbon atoms is more
than 300000 [10]!
Under standard conditions (SC) of 20~ and 1 atm, the
first four members of the alkane series (methane, ethane,
propane, and butane) are in gaseous form, while from C5Hl1
(pentane) to n-hexadecane (C16H36) they are liquids, and from
n-heptadecane (C17 H38) the compounds exist as waxlike solids
at this standard temperature and pressure. Paraffins from C1
to C40 usually appear in crude oil and represent up to 20% of
crude by volume. Since paraffins are fully saturated (no dou-
ble bond), they are stable and remain unchanged over long
periods of geological time.
Olefms are another series of noncyclic hydrocarbons but
they are unsaturated and have at least one double bond
between carbon-carbon atoms. Compounds with one dou-
ble bond are called monoolefins or alkenes, such as ethene
(also named ethylene: CH2=CH2) and propene or propylene
(CH2=CH--CH3). Besides structural isomerism connected
with the location of double bond, there is another type of iso-
merism called geometric isomerism, which indicates the way
atoms are oriented in space. The configurations are differen-
tiated in their names by the prefixes cis- and trans- such as
cis- and trans-2-butene. Monoolefins have a general formula
of CnH2n. If there are two double bonds, the olefin is called
diolefin (or diene), such as butadiene (CH2=CH--CH=CH2).
Unsaturated compounds are more reactive than saturated hy-
drocarbons (without double bond). Olefins are uncommon in
crude oils due to their reactivity with hydrogen that makes
them saturated; however, they can be produced in refiner-
ies through cracking reactions. Olefins are valuable prod-
ucts of refineries and are used as the feed for petrochemical
plants to produce polymers such as polyethylene. Similarly
compounds with triple bonds such as acetylene (CH------CH) are
not found in crude oils because of their tendency to become
saturated [2].
Napht henes or cycloalkanes are ring or cyclic saturated hy-
drocarbons with the general formula of CnH2n. Cyclopentane
(C5H10), cyclohexane (C6H12), and their derivatives such as
n-alkylcyclopentanes are normally found in crude oils. Three
types of naphthenic compounds are shown below:
Cyclopentane Methylcyclopentane Ethylcyclohexane
(CsHIo) (C6HI2) (C8H~6)
If there is only one alkyl group from n-paraffins (i.e., methyl,
ethyl, propyl, n-butyl . . . . ) attached to a cyclopentane hydro-
carbon, the series is called n-alkylcyclopentanes, such as the
two hydrocarbons shown above where on each junction of the
ring there is a CH2 group except on the alkyl group juncture
where there is only a CH group. For simplicity in drawing,
these groups are not shown. Similarly there is a homologous
napthenic series of n-alkylcyclohexanes with only one satu-
rated ring of cyclohexane, such as ethylcyclohexane shown
above. Napthenic hydrocarbons with only one ring are also
called monocycloparaffins or mononaphthenes. In heavier oils,
saturated multirings attached to each other called polycy-
cloparaffins orpolynaphthenes may also be available. Thermo-
dynamic studies show that naphthene rings with five and six
carbon atoms are the most stable naphthenic hydrocarbons.
The content of cycloparaffins in petroleum may vary up to
60%. Generally, any petroleum mixture that has hydrocarbon
compounds with five carbon atoms also contains naphthenic
compounds.
A r o m a t i c s are an important series of hydrocarbons found
in almost every petroleum mixture from any part of the world.
Aromatics are cyclic but unsaturated hydrocarbons that begin
with benzene molecule (C6H6) and contain carbon-carbon
double bonds. The name aromatic refers to the fact that such
hydrocarbons commonly have fragrant odors. Four different
aromatic compounds are shown below:
\
9
(C6H6) (C7H8) (C8H1o) (C1o~8)
Benzene Toluene O-xylene Naphthalene
(Methylbenzene) ( 1,2-Dimethylbenzene)
In the above structures, on each junction on the benzene
ring where there are three bonds, there is only a group of CH,
while at the junction with an alkylgroup (i.e., toluene) there
is only a C atom. Although benzene has three carbon-carbon
double bonds, it has a unique arrangement of electrons that
allows benzene to be relatively unreactive. Benzene is, how-
ever, known to be a cancer-inducing compound [2]. For this
reason, the amount of benzene allowed in petroleum prod-
ucts such as gasoline or fuel oil is limited by government
regulations in many countries. Under SC, benzene, toluene,
and xylene are in liquid form while naphthalene is in a solid
state. Some of the common aromatics found in petroleum
and crude oils are benzene and its derivatives with attached
methyl, ethyl, propyl, or higher alkyl groups. This series of
aromatics is called alkylbenzenes and compounds in this ho-
mologous group of hydrocarbons have a general formula
of CnH2n-6 (where n _> 6). Generally, aromatic series with
only one benzene ring are also called monoaromatics (MA)
or mononuclear aromatics. Naphthalene and its derivatives,
which have only two unsaturated rings, are sometime called
diaromatics. Crude oils and reservoir fluids all contain aro-
matic compounds. However, heavy petroleum fractions and
residues contain multi-unsaturated rings with many benzene
and naphthene rings attached to each other. Such aromatics
(which under SC are in solid form) are also calledpolyaromat-
ics (PA) or polynuclear aromatics (PNA). In this book terms of
mono and polyaromatics are used. Usually, heavy crude oils
contain more aromatics than do light crudes. The amount of
aromatics in coal liquids is usually high and could reach as
high as 98% by volume. It is common to have compounds
with napthenic and aromatic rings side by side, especially
in heavy fractions. Monoaromatics with one napthenic ring
have the formula of CnH2n-8 and with two naphthenic rings
the formula is C~Hzn-8. There are many combinations of alkyl-
naphthenoaromatics [ 1, 7].
Normally, high-molecular-weight polyaromatics contain
several heteroatoms such as sulfur (S), nitrogen (N), or oxygen
(O) hut the compound is still called an aromatic hydrocarbon.
Two types of these compounds are shown below [1 ]:
H
Dibenzothiophene Benzocarbazole (CI6H1 IN)
Except for the atoms S and N, which are specified in the above
structures, on other junctions on each ring there is either a
CH group or a carbon atom. Such heteroatoms in multiring
aromatics are commonly found in asphaltene compounds as
shown in Fig. 1.2, where for simplicity, C and H atoms are not
shown on the rings.
Sulfur is the most important heteroatom in petroleum and
it can be found in cyclic as well as noncyclic compounds such
as mercaptanes (R--S--H) and sulfides (R--S--W), where R
and R' are alkyl groups. Sulfur in natural gas is usually found
in the form of hydrogen sulfide (H2S). Some natural gases
C: 8 3 . 1 %
H: 8 . 9 %
N: 1 . 0 %
O: 0%
S: 7 . 0 %
H/C: i.28
Mol ecul ar Wei ghh 1 3 7 0
1. I NTRODUCTI ON 5
FIG. 1. 2mAn exampl e of asphal tene mol ecul e. Reprinted from
Ref. [1], p. 463, by court esy of Marcel Dekker, Inc.
contain HzS as high as 30% by volume. The amount of sulfur
in a crude may vary from 0.05 to 6% by weight. In Chapter 3,
further discussion on the sulfur contents of petroleum frac-
tions and crude oils will be presented. The presence of sulfur
in finished petroleum products is harmful, for example, the
presence of sulfur in gasoline can promote corrosion of en-
gine parts. Amounts of nitrogen and oxygen in crude oils are
usually less than the amount of sulfur by weight. In general
for petroleum oils, it appears that the compositions of ele-
ments vary within fairly narrow limits; on a weight basis they
are [1]
Carbon (C), 83.0-87.0%
Hydrogen (H), 10.0-14.0%
Nitrogen (N), 0.1-2.0%
Oxygen (O), 0.05-1.5%
Sulfur (S), 0.05-6.0%
Metals (Nickel, Vanadium, and Copper), < 1000 ppm (0.1%)
Generally, in heavier oils (lower API gravity, defined by
Eq. (2.4)) proportions of carbon, sulfur, nitrogen, and oxygen
elements increase but the amount of hydrogen and the overall
quality decrease. Further information and discussion about
the chemistry of petroleum and the type of compounds found
in petroleum fractions are given by Speight [ 1]. Physical prop-
erties of some selected pure hydrocarbons from different ho-
mologous groups commonl y found in petroleum fluids are
given in Chapter 2. Vanadium concentrations of above 2 ppm
in fuel oils can lead to severe corrosion in turbine blades and
deterioration of refractory in furnaces. Ni, Va, and Cu can also
severely affect the activities of catalysts and result in lower
products. The metallic content may be reduced by solvent
extraction with organic solvents. Organometallic compounds
are precipitated with the asphaltenes and residues.
1. 1. 2 Res ervoi r Fl ui ds and Crude Oil
The word fluid refers to a pure substance or a mixture of com-
pounds that are in the form of gas, liquid, or both a mixture
of liquid and gas (vapor). Reservoir fluid is a term used for the
mixture of hydrocarbons in the reservoir or the stream leaving
a producing well. Three factors determine if a reservoir fluid is
in the form of gas, liquid, or a mixture of gas and liquid. These
factors are (1) composition of reservoir fluid, (2) temperature,
and (3) pressure. The most important characteristic of a reser-
voir fluid in addition to specific gravity (or API gravity) is its
gas-to-oil ratio (GOR), which represents the amount of gas
6 CHARACT ERI Z AT I ON AND P R OP E R T I E S OF P E T R OL E UM FRACT I ONS
TABLE 1.1--Types and characteristics of various reservoir fluids.
Reservoir fluid type GOR, scf/sth CH4, mol% C6+, tool% API gravity of STO a
Black oil <1000 _<50 >_30 <40
Volatile oil 1000-3000 50-70 10-30 40-45
Gas condensate 3000-50 000 70-85 3-10 _>45
Wet gas _>50 000 >-75 <3 >50
Dry gas >- 10 0000 >_90 < 1 No liquid
"API gravity of stock tank oil (STO) produced at the surface facilities at standard conditions (289 K and 1 atm).
pr oduced at SC i n st andar d cubi c feet (scf) to the amount of
l i qui d oil produced at the SC i n stock t ank barrel (stb). Ot her
uni t s of GOR are di scussed i n Sect i on 1.7.23 and its calcula-
t i on is di scussed i n Chapt er 9. Generally, reservoir fluids are
cat egori zed i nt o four or five types (their charact eri st i cs are
given i n Table 1.1). These five fluids i n t he di rect i on of i n-
creasi ng GOR are bl ack oil, volatile oil, gas condensat e, wet
gas, and dry gas.
If a gas after surface separator, under SC, does not pro-
duce any l i qui d oil, it is called dry gas. A nat ur al gas t hat after
pr oduct i on at the surface facilities can produce a little l i qui d
oil is called wet gas. The word wet does not mean t hat t he
gas is wet wi t h water, but refers to the hydr ocar bon l i qui ds
t hat condense at surface condi t i ons. I n dry gases no l i qui d
hydr ocar bon is formed at t he surface condi t i ons. However,
bot h dry and wet gases are i n t he category of nat ur al gases.
Volatile oils have also been called high-shrinkage crude oil and
near-critical oils, since t he reservoi r t emper at ur e and pressure
are very close to the critical poi nt of such oils, but t he critical
t emper at ur e is always great er t han the reservoi r t emper at ur e
[i 1]. Gases and gas condensat e fluids have critical t empera-
t ures less t han the reservoi r t emperat ure. Black oils cont ai n
heavi er compounds and t herefore t he API gravity of stock
t ank oil is general l y l ower t han 40 and the GOR is less t han
1000 scf/stb. The specifications given i n Table 1.1 for vari ous
reservoi r fluids, especially at the boundar i es bet ween differ-
ent types, are arbi t rary and vary from one source to anot her
[9, 11]. It is possible to have a reservoi r fluid type t hat has
propert i es out si de t he correspondi ng l i mi t s ment i oned ear-
lier. Det er mi nat i on of a type of reservoir fluid by the above
rul e of t humb based on the GOR, API gravity of stock t ank
oil, or its color is not possible for all fluids. A more accu-
rat e met hod of det er mi ni ng the type of a reservoi r fluid is
based on the phase behavi or cal cul at i ons, its critical poi nt ,
and shape of t he phase di agr am whi ch will be di scussed i n
Chapt ers 5 and 9. I n general, oils produced from wet gas,
gas condensat e, volatile oil, and bl ack oil i ncrease i n spe-
cific gravity (decrease i n API gravity and quality) i n t he same
order. Here qual i t y of oil i ndi cat es lower carbon, sulfur, ni t ro-
gen, and met al cont ent s whi ch correspond to hi gher heat i ng
value. Liquids from bl ack oils are viscous and bl ack i n color,
while t he l i qui ds from gas condensat es or wet gases are clear
and colorless. Volatile oils produce fluids br own wi t h some
red/ green color liquid. Wet gas cont ai ns less met hane t han a
dry gas does, but a l arger fract i on of C2-C 6 component s. Ob-
vi ousl y the mai n difference bet ween these reservoi r fluids is
t hei r respective composi t i on. An exampl e of composi t i on of
different reservoir fluids is given i n Table 1.2.
I n Table 1.2, C7+ refers to all hydr ocar bons havi ng seven
or hi gher car bon at oms and is called hept ane-pl us fraction,
whi l e C6 refers to a group of all hydr ocar bons wi t h six car-
bon at oms (hexanes) t hat exist i n the fluid. MT+ and SG7+ are
t he mol ecul ar wei ght and specific gravity at 15.5~ (60~ for
the C7+ fract i on of the mi xt ure, respectively. It shoul d be re-
alized t hat mol ecul ar wei ght and specific gravity of the whol e
reservoir fluid are less t han the correspondi ng values for the
TABLE 1.2---Composition (mol%) and properties of various reservoir fluids and a crude oil
Component Dry gas ~ Wet gas b Gas condensate C Volatile oil d Black oil e Crude oil f
CO2 3.70 0.00 0.18 1.19 0.09 0.00
N2 0.30 0.00 0.13 0.51 2.09 0.00
H2S 0.00 0.00 0.00 0.00 1.89 0.00
C1 96.00 82.28 61.92 45.21 29.18 0.00
C2 0.00 9.52 14.08 7.09 13.60 0.19
C3 0.00 4.64 8.35 4.61 9.20 1.88
iC4 0.00 0.64 0.97 1.69 0.95 0.62
nC4 0.00 0.96 3.41 2.81 4.30 3.92
iC5 0.00 0.35 0.84 1.55 1.38 2.11
nC5 0.00 0.29 1.48 2.01 2.60 4.46
C6 0.00 0.29 1.79 4.42 4.32 8.59
C7+ 0.00 1.01 6.85 28.91 30.40 78.23
Total 100.00 100.00 100.00 100.00 100.00 100.00
GOR (scf/stb) ... 69917 4428 1011 855
M7+ ... 113 143 190 209.8 266
SG7+ (at 15.5~ ... 0.794 0.795 0.8142 0.844 0.895
API7+ 46.7 46.5 42.1 36.1 26.6
"Gas sample from Salt Lake, Utah [12].
bWet gas data from McCaln [11].
CGas condensate sample from Samson County, Texas (M. B. Standing, personal notes, Department of Petroleum
Engineering, Norwegian Institute of Technology, Trondheim, Norway, 1974).
dVolatile oil sample from Raleigh Field, Smith County, Mississipi (M. B. Standing, personal notes, Department of
Petroleum Engineering, Norwegian Institute of Technology, Trondheim, Norway, 1974).
eBlack oil sample from M. Ghuraiba, M.Sc. Thesis, Kuwait University, Kuwait, 2000.
fA crude oil sample produced at stock tank conditions.
1. I NT RODUCT I ON 7
heptane-plus fraction. For example, for the crude oil sample
in Table 1.2, the specific gravity of the whole crude oil is 0.871
or API gravity of 31. Details of such calculations are discussed
in Chapter 4. These compositions have been determined from
recombination of the compositions of corresponding sepa-
rator gas and stock tank liquid, which have been measured
through analytical tools (i.e., gas chromatography, mass spec-
trometry, etc.). Composition of reservoir fluids varies with the
reservoir pressure and reservoir depth. Generally in a produc-
ing oil field, the sulfur and amount of heavy compounds in-
crease versus production time [10]. However, it is important
to note that within an oil field, the concentration of light hy-
drocarbons and the API gravity of the reservoir fluid increase
with the reservoir depth, while its sulfur and C7+ contents de-
crease with the depth [ 1 ]. The lumped C7+ fraction in fact is
a mixture of a very large number of hydrocarbons, up to C40
or higher. As an example the number of pure hydrocarbons
from C5 to C9 detected by chromatography tools in a crude oil
from North Sea reservoir fluids was 70 compounds. Detailed
composition of various reservoir fluids from the North Sea
fields is provided by Pedersen et al. [13]. As shown in Chapter
9, using the knowledge of the composition of a reservoir fluid,
one can determine a pressure-temperature (PT) diagram of
the fluid. And on the basis of the temperature and pressure
of the reservoir, the exact type of the reservoir fluid can be
determined from the PT diagram.
Reservoir fluids from a producing well are conducted to
two- or three-stage separators which reduce the pressure and
temperature of the stream to atmospheric pressure and tem-
perature. The liquid leaving the last stage is called stock tank
oil (STO) and the gas released in various stages is called as-
sociated gas. The liquid oil after necessary field processing is
called crude oil. The main factor in operation and design of an
oil-gas separator is to find the optimum operating conditions
of temperature and pressure so that the amount of produced
liquid (oil) is maximized. Such conditions can be determined
through phase behavior calculations, which are discussed in
detail in Chapter 9. Reservoir fluids from producing wells are
mixed with free water. The water is separated through gravi-
tational separators based on the difference between densities
of water and oil. Remaining water from the crude can be re-
moved through dehydration processes. Another surface oper-
ation is the desalting process that is necessary to remove the
salt content of crude oils. Separation of oil, gas, and water
from each other and removal of water and salt from oil and
any other process that occurs at the surface are called surface
production operations [14].
The crude oil produced from the atmospheric separator has
a composition different from the reservoir fluid from a pro-
ducing well. The light gases are separated and usually crude
oils have almost no methane and a small C2-C3 content while
the C7+ content is higher than the original reservoir fluid. As
an example, the composition of a crude oil produced through
a three-stage separator from a reservoir fluid is also given in
Table 1.2. Actually this crude is produced from a black oil
reservoir fluid (composition given in Table 1.2). Two impor-
tant characterisitcs of a crude that determine its quality are
the API gravity (specific gravity) and the sulfur content. Gen-
erally, a crude with the API gravity of less than 20 (SG > 0.934)
is called heavy crude and with API gravity of greater than 40
(SG < 0.825) is called light crude [1, 9]. Similarly, if the sulfur
content of a crude is less than 0.5 wt% it is called a sweet
oil. It should be realized that these ranges for the gravity and
sulfur content are relative and may vary from one source to
another. For example, Favennec [15] classifies heavy crude as
those with API less than 22 and light crude having API above
33. Further classification of crude oils will be discussed in
Chapter 4.
1. 1. 3 Pe t r ol e um Fract i ons and Pr oduc t s
A crude oil produced after necessary field processing and
surface operations is transferred to a refinery where it is
processed and converted into various useful products. The
refining process has evolved from simple batch distillation
in the late nineteenth century to today's complex processes
through modern refineries. Refining processes can be gener-
ally divided into three major types: (1) separation, (2) con-
version, and (3) finishing. Separation is a physical process
where compounds are separated by different techniques. The
most important separation process is distillation that occurs
in a distillation column; compounds are separated based on
the difference in their boiling points. Other major physical
separation processes are absorption, stripping, and extrac-
tion. In a gas plant of a refinery that produces light gases,
the heavy hydrocarbons (Cs and heavier) in the gas mixture
are separated through their absorption by a liquid oil sol-
vent. The solvent is then regenerated in a stripping unit. The
conversion process consists of chemical changes that occur
with hydrocarbons in reactors. The purpose of such reactions
is to convert hydrocarbon compounds from one type to an-
other. The most important reaction in modem refineries is
the cracking in which heavy hydrocarbons are converted to
lighter and more valuable hydrocarbons. Catalytic cracking
and thermal cracking are commonl y used for this purpose.
Other types of reactions such as isomerization or alkylation
are used to produce high octane number gasoline. Finishing is
the purification of various product streams by processes such
as desulfurization or acid treatment of petroleum fractions to
remove impurities from the product or to stabilize it.
After the desalting process in a refinery, the crude oil en-
ters the atmospheric distillation column, where compounds
are separated according to their boiling points. Hydrocarbons
in a crude have boiling points ranging from -160~ (boil-
ing point of methane) to more than 600~ (ll00~ which
is the boiling point of heavy compounds in the crude oil.
However, the carbon-carbon bond in hydrocarbons breaks
down at temperatures around 350~ (660~ This process is
called cracking and it is undesirable during the distillation
process since it changes the structure of hydrocarbons. For
this reason, compounds having boiling points above 350~
(660+~ called residuum are removed from the bottom of
atmospheric distillation column and sent to a vacuum dis-
tillation column. The pressure in a vacuum distillation col-
umn is about 50-100 mm Hg, where hydrocarbons are boiled
at much lower temperatures. Since distillation cannot com-
pletely separate the compounds, there is no pure hydrocarbon
as a product of a distillation column. A group of hydrocarbons
can be separated through distillation according to the boiling
point of the lightest and heaviest compounds in the mixtures,
The lightest product of an atmospheric column is a mixture of
methane and ethane (but mainly ethane) that has the boiling
8 CHARACTERIZATION AND PROPERTIES OF PETROLEUM FRACTIONS
TABLE 1.3---Some petroleum fractions produced from distillation columns.
Approximate boiling range
Petroleum fraction Approximate hydrocarbon range ~ ~
Light gases C2 - C4 -90 to 1 -130-30
Gasoline (light and heavy) C4- C10 - 1 - 2 0 0 30-390
Naphthas (light and heavy) C 4 - C l l - 1 - 2 0 5 30-400
Jet fuel C9- C14 150-255 300-490
Kerosene C11-C14 205-255 400-490
Diesel fuel C] 1-C16 205-290 400-550
Light gas oil C14-C18 255-315 490-600
Heavy gas oil C18-C28 315-425 600-800
Wax Cls-Ca6 315-500 600-930
Lubricating oil >C25 >400 >750
Vacuum gas oil C28-C55 425-600 800-1100
Residuum > C55 > 600 > 1100
I n f o r ma t i o n gi ve n i n t h i s t a bl e i s o b t a i n e d f r om di f f er ent s our c e s [ 1, 18, 19] .
range of -180 to -80~ ( - 260 to -40~ which corresponds
to the boiling point of methane and ethane. This mixture,
which is in the form of gas and is known as fuel gas, is actu-
ally a petroleum fraction. In fact, during distillation a crude
is converted into a series of petroleum fractions where each
one is a mixture of a limited number of hydrocarbons with a
specific range of boiling point. Fractions with a wider range
of boiling points contain greater numbers of hydrocarbons.
All fractions from a distillation column have a known boiling
range, except the residuum for which the upper boiling point
is usually not known. The boiling point of the heaviest com-
ponent in a crude oil is not really known, but it is quite high. 60
The problem of the nature and properties of the heaviest com-
pounds in crude oils and petroleum residuum is still under
investigation by researchers [i 6, 17]. Theoretically, it can be
assumed that the boiling point of the heaviest component in a
crude oil is infinity. Atmospheric residue has compounds with 50
carbon number greater than 25, while vacuum residue has
compounds with carbon number greater than 50 (M > 800).
Some of the petroleum fractions produced from distillation
columns with their boiling point ranges and applications are 40
given in Table 1.3. The boiling point and equivalent carbon
number ranges given in this table are approximate and they
may vary according to the desired specific product. For ex-
ample, the light gases fraction is mainly a mixture of ethane, E
propane, and butane; however, some heavier compounds z 30
t-
(C5+) may exist in this fraction. The fraction is further frac- o
tionated to obtain ethane (a fuel gas) and propane and butane
(petroleum gases). The petroleum gases are liquefied to get
liquefied petroleum gas (LPG) used for home cooking pur-
poses. In addition the isobutane may be separated for the gas 20
mixture to be used for improving vapor pressure characteris-
tics (volatility) of gasoline in cold weathers. These fractions
may go through further processes to produce desired prod-
ucts. For example, gas oil may go through a cracking process 10-
to obtain more gasoline. Since distillation is not a perfect sep-
aration process, the initial and final boiling points for each
fraction are not exact and especially the end points are ap-
proximate values. Fractions may be classified as narrow or
wide depending on their boiling point range. As an example, 0
the composition of an Alaska crude oil for various products 0
is given in Table 1.4 and is graphically shown in Fig. 1.3.
The weight and volume percentages for the products are
near each other. More than 50% of the crude is processed
in vacuum distillation unit. The vacuum residuum is mainly
resin and asphaltenes-type compounds composed of high
molecular weight multiring aromatics. The vacuum residuum
may be mixed with lighter products to produce a more valu-
able blend.
Distillation of a crude oil can also be performed in the lab-
oratory to divide the mixture into many narrow boiling point
range fractions with a boiling range of about 10~ Such nar-
row range fractions are sometimes referred to as petroleum
cuts. When boiling points of all the cuts in a crude are known,
then the boiling point distribution (distillation curve) of the
A t m o s p h e r i c D i s t i l l a t i o n 46.1%
Light
Kerosene Gas Oil
, q
t
H e a v y
Gas O i l
V a c u u m
Gas Oil
Residuum
o
V a c u u m D i s t i l l a t i o n 53.9 '/o
i i
20 40 60 80
.655
9
t -
fl_
t -
o
345
,205
90
100
V o l u m e P e r c e n t
Light Gases ~ Light Gaserine
FI G. 1 . 3 - - P r o d u c t s and composi t i on of Al aska cr ude oi l .
Petroleum fraction
Atmospheric distillation
Light gases
Light gasoline
Naphthas
Kerosene
Light gas oil (LGO)
Sum
Vacuum distillation (VD)
Heavy gas oil (HGO)
Vacuum gas oil (VGO)
Residuum
Sum
Total Crude
1. I NTRODUCTI ON 9
TABLE 1.4--Products and composition of alaska crude oil.
Approximate boiling range a
Approximate hydrocarbon range ~ ~ vol% wt%
C2-C4 --90 to 1 --130-30 1.2 0.7
C4-C7 -1-83 30-180 4.3 3.5
C7-Cll 83--205 180--400 16.0 14.1
Cll-C16 205-275 400-525 12.1 11.4
C16-C21 275-345 525-650 12.5 12.2
C2-C21 -90-345 -130-650 46.1 41.9
C21-C31 345-455 650-850
C3l-C48 455-655 850-1050
>C48 655+ 1050+
C21-C48+ 345-655+ 650-1050
C2-C48+ -9(P655+ -130-650+
Information given in this table has been extracted from Ref. [ 19].
aBoiling ranges are interconverted to the nearest 5~ (~
20.4 21.0
15.5 16.8
18.0 20.3
53.9 58.1
100.0 100.0
whole crude can be obtained. Such distillation data and their
uses will be discussed in Chapters 3 and 4. In a petroleum
cut, hydrocarbons of various types are lumped together in
four groups of paraffins (P), olefins (O), naphthenes (N), and
aromatics (A). For olefin-free petroleum cuts the composi-
tion is represented by the PNA content. If the composition
of a hydrocarbon mixture is known the mixture is called a
defined mixture, while a petroleum fraction that has an un-
known composition is called an undefined fraction.
As mentioned earlier, the petroleum fractions presented
in Table 1.3 are not the final products of a refinery. They
go through further physicochemical and finishing processes
to get the characteristics set by the market and government
regulations. After these processes, the petroleum fractions
presented in Table 1.3 are converted to petroleum products.
The terms petroleum fraction, petroleum cut, and petroleum
product are usually used incorrectly, while one should re-
alize that petroleum fractions are products of distillation
columns in a refinery before being converted to final prod-
ucts. Petroleum cuts may have very narrow boiling range
which may be produced in a laboratory during distillation
of a crude. In general the petroleum products can be divided
into two groups: (1) fuel products and (2) nonfuel products.
The major fuel petroleum products are as follows:
1. Liquefied petroleum gases (LPG) that are mainly used for
domestic heating and cooking (50%), industrial fuel (clean
fuel requirement) (15%), steam cracking feed stock (25%),
and as a mot or fuel for spark ignition engines (10%). The
world production in 1995 was 160 million ton per year
(---5 million bbl/d) [20]. LPG is basically a mixture of
propane and butane.
2. Gasoline is perhaps one of the most important products of
a refinery. It contains hydrocarbons from C4 to Cll (molec-
ular weight of about 100-110). It is used as a fuel for cars.
Its main characteristics are antiknock (octane number),
volatility (distillation data and vapor pressure), stability,
and density. The main evolution in gasoline production has
been the use of unleaded gasoline in the world and the use
of reformulated gasoline (RFG) in the United States. The
RFG has less butane, less aromatics, and more oxygenates.
The sulfur content of gasoline should not exceed 0.03% by
weight. Further properties and characteristics of gasoline
will be discussed in Chapter 3. The U.S. gasoline demand
in 1964 was 4,4 million bbl/d and has increased from 7.2 to
8.0 million bbl/d in a period of 7 years from 1991 to 1998
[6, 20]. In 1990, gasoline was about a third of refinery prod-
ucts in the United States.
3. Kerosene and jet fuel are mainly used for lighting and jet
engines, respectively. The main characteristics are sulfur
content, cold resistance (for jet fuel), density, and ignition
quality,
4. Diesel and heating oil are used for mot or fuel and domestic
purposes. The main characteristics are ignition (for diesel
oil), volatility, viscosity, cold resistance, density, sulfur con-
tent (corrosion effects), and flash point (safety factor).
5. Residual fuel oil is used for industrial fuel, for thermal pro-
duction of electricity, and as mot or fuel (low speed diesel
engines). Its main characteristics are viscosity (good at-
omization for burners), sulfur content (corrosion), stabil-
ity (no decantation separation), cold resistance, and flash
point for safety.
The major nonfuel petroleum products are [18] as follows:
i. Solvents are light petroleum cuts in the C4-C14 range and
have numerous applications in industry and agriculture.
As an example of solvents, white spirits which have boiling
points between 135 and 205~ are used as paint thinners.
The main characteristics of solvents are volatility, purity,
odor, and toxicity. Benzene, toluene, and xylenes are used
as solvents for glues and adhesives and as a chemical for
petrochemical industries.
2. Naphthas constitute a special category of petroleum sol-
vents whose boiling points correspond to the class of white
spirits. They can be classified beside solvents since they are
mainly used as raw materials for petrochemicals and as
the feeds to steam crackers. Naphthas are thus industrial
intermediates and not consumer products. Consequently,
naphthas are not subject to government specifications but
only to commercial specifications.
3. Lubricants are composed of a main base stock and addi-
tives to give proper characteristics. One of the most im-
portant characteristics of lubricants is their viscosity and
viscosity index (change of viscosity with temperature). Usu-
ally aromatics are eliminated from lubricants to improve
10 CHARACTERIZATION AND PROPERTI ES OF PETROLEUM FRACTIONS
t hei r vi scosi t y index. Lubr i cant s have st r uct ur e si mi l ar
to i soparaffi ni c compounds. Additives used for l ubr i cant s
are vi scosi t y i ndex addi t i ves such as pol yacryl at es and
olefin pol ymer s, ant i wear addi t i ves (i.e., fat t y esters), an-
t i oxi dant s (i.e., al kyl at ed ar omat i c ami nes), cor r osi on in-
hi bi t ors (i.e., fat t y acids), and ant i f oami ng agent s (i.e., poly-
di met hyl si l oxanes). Lubri cat i ng greases are anot her class
of l ubri cant s t hat are semisolid. The pr oper t i es of lubri-
cant s t hat shoul d be known are vi scosi t y index, ani l i ne
poi nt (i ndi cat i on of ar omat i c cont ent ), volatility, and car-
bon residue.
4. Pet r ol eum waxes are of t wo types: t he paraffi n waxes in
pet r ol eum distillates and t he mi crocryst al l i ne waxes in pe-
t r ol eum resi dua. In s ome count ri es such as France, paraf-
fin waxes are si mpl y called paraffins. Paraffin waxes are
hi gh mel t i ng poi nt mat er i al s used to i mpr ove t he oil's pour
poi nt and are pr oduced dur i ng dewaxi ng of vacuum dis-
tillates. Paraffi n waxes are mai nl y st rai ght chai n al kanes
(C18-C36) wi t h a very smal l pr opor t i on of i soal kanes and
cycloalkanes. Thei r freezi ng poi nt is bet ween 30 and 70~
and t he average mol ecul ar wei ght is ar ound 350. When
present , ar omat i cs appear onl y in t race quant i t i es. Waxes
f r om pet r ol eum r esi dua (mi crocryst al l i ne f or m) are less
defined al i phat i c mi xt ur es of n-alkanes, i soal kanes, and cy-
cl oal kanes in vari ous pr opor t i ons. Thei r average mol ecul ar
wei ght s are bet ween 600 and 800, car bon numbe r r ange is
a l k a n e s C30- C60 , and t he freezi ng poi nt r ange is 60-90~
[ 13]. Paraffin waxes (when compl et el y dear omat i zed) have
appl i cat i ons in t he food i ndust ry and food packagi ng. They
are also used in t he pr oduct i on of candles, polishes, cos-
metics, and coat i ngs [ 18]. Waxes at or di nar y t emper at ur e of
25~ are in solid st at es al t hough t hey cont ai n some hydro-
car bons in liquid form. When mel t ed t hey have relatively
l ow viscosity.
5. Asphalt is anot her maj or pet r ol eum pr oduct t hat is pro-
duced f r om va c uum di st i l l at i on resi dues. Asphalts cont ai n
nonvol at i l e hi gh mol ecul ar wei ght pol ar ar omat i c com-
pounds, such as asphal t enes ( mol ecul ar wei ght s of several
t housands) and cannot be distilled even under very hi gh
vacuum condi t i ons. I n s ome count ri es asphal t is called
bi t umen, al t hough s ome suggest t hese t wo are di fferent
pet r ol eum product s. Li qui d asphal t i c mat er i al s are in-
t ended for easy appl i cat i ons t o roads. Asphal t and bi t u-
men are f r om a cat egory of pr oduct s called hydr ocar bon
bi nders. Maj or pr oper t i es t o det er mi ne t he qual i t y of as-
phal t are flash poi nt (for safety), composi t i on (wax con-
tent), vi scosi t y and soft eni ng poi nt , weat heri ng, densi t y or
specific gravity, and st abi l i t y or chemi cal resi st ance.
6. There are s ome ot her pr oduct s such as whi t e oils (used in
phar maceut i cal s or in t he food i ndust ry), ar omat i c ext ract s
(used in t he pai nt i ndust ry or t he manuf act ur e of plastics),
and coke (as a fuel or t o pr oduce car bon el ecrodes for alu-
mi num refining). Pet r ol eum cokes general l y have boi l i ng
poi nt s above 1100+~ (~2000+~ mol ecul ar wei ght of
above 2500+, and car bon numbe r of above 200+. Aromat i c
ext ract s are bl ack mat eri al s, compos ed essentially of con-
densed pol ynucl ear ar omat i cs and of het erocycl i c ni t rogen
and/ or sul fur compounds. Because of t hi s hi ghl y ar omat i c
st ruct ure, t he ext ract s have good solvent power.
Fur t her i nf or mat i on on technology, propert i es, and test-
ing met hods of fuels and l ubri cant s is given in Ref. [21].
I n general, mor e t han 2000 pet r ol eum pr oduct s wi t hi n s ome
20 cat egori es are pr oduced in refineries in t he Uni t ed St at es
[ 1, 19]. Bl endi ng t echni ques are used to pr oduce some of t hese
pr oduct s or t o i mpr ove t hei r quality. The pr oduct specifica-
t i ons mus t satisfy cust omer s' r equi r ement s f or good perfor-
mance and gover nment regul at i ons for safet y and envi ron-
ment prot ect i on. To be able to pl an refi nery operat i ons, t he
availability of a set of pr oduct quality pr edi ct i on met hods is
t her ef or e very i mpor t ant .
There are a numbe r of i nt ernat i onal or gani zat i ons t hat are
known as st andar d organi zat i ons t hat r e c omme nd specific
charact eri st i cs or st andar d measur i ng t echni ques for vari ous
pet r ol eum pr oduct s t hr ough t hei r regul ar publ i cat i ons. Some
of t hese or gani zat i ons in di fferent count ri es t hat are known
wi t h t hei r abbr evi at i ons are as follows:
1. ASTM (Ameri can Society for Testing and Mat eri al s) in t he
Uni t ed St at es
2. ISO (Int ernat i onal Or gani zat i on for St andardi zat i on),
whi ch is at t he i nt ernat i onal level
3. IP (Inst i t ut e of Pet r ol eum) in t he Uni t ed Ki ngdom
4. API (Ameri can Pet r ol eum Inst i t ut e) in t he Uni t ed St at es
5. AFNOR (Association Fr ancai se de Normal i sat i on), an offi-
cial st andar d or gani zat i on in France
6. Deut sche I nst i t ut f ur Nor r nung (DIN) in Ger many
7. J apan Inst i t ut e of St andar ds (J-IS) in J apan
ASTM is compos ed of several commi t t ees i n whi ch t he D-02
committee is responsi bl e for pet r ol eum pr oduct s and lubri-
cant s, and for t hi s r eason its t est met hods for pet r ol eum ma-
terials are desi gnat ed by t he prefix D. For exampl e, t he t est
met hod ASTM D 2267 provi des a st andar d pr ocedur e t o de-
t er mi ne t he benzene cont ent of gasol i ne [22]. I n Fr ance t hi s
t est met hod is desi gnat ed by EN 238, whi ch are document ed
in AFNOR i nf or mat i on document M 15-023. Most st andar d
t est met hods in di fferent count ri es are very si mi l ar in prac-
tice and follow ASTM met hods but t hey ar e desi gnat ed by
di fferent codes. For exampl e t he i nt ernat i onal st andar d ISO
6743/0, accept ed as t he Fr ench st andar d NF T 60-162, t reat s
all t he pet r ol eum l ubri cant s, i ndust ri al oils, and rel at ed prod-
ucts. The abbr evi at i on NF is used for t he Fr ench st andar d,
while EN is used for Eur opean st andar d met hods [ 18].
Gover nment regul at i ons to pr ot ect t he envi r onment or t o
save energy, in ma ny cases, rel y on t he r ecommendat i ons
of official st andar d organi zat i ons. For exampl e, in France,
AFNOR gives speci fi cat i ons and r equi r ement s for vari ous
pet r ol eum product s. For diesel fuels it r ecommends (aft er
1996) t hat t he sul fur cont ent shoul d not exceed 0.05 wt % and
t he flash poi nt shoul d not be less t han 55~ [18].
1. 2 TYPES AND I MPORTANCE
OF PHYSI CAL PROPERTI ES
On t he basi s of t he pr oduct i on and refining pr ocesses de-
scri bed above it may be sai d t hat t he pet r ol eum i ndust r y
is involved wi t h ma ny t ypes of equi pment f or pr oduct i on,
t r anspor t at i on, and st orage of i nt er medi at e or final pet r ol eum
product s. Some of t he mos t i mpor t ant uni t s ar e listed below.
i. Gravi t y decant er (to separ at e oil and wat er)
2. Separ at or s to separ at e oil and gas
3. Pumps, compr essor s, pipes, and valves
1. I N T R O D U C T I O N 11
4. St orage t anks
5. Distillation, absorpt i on, and st ri ppi ng col umns
6. Boilers, evaporat ors, condensers, and heat exchangers
7. Fl ashers (to separ at e l i ght gases f r om a liquid)
8. Mixers and agi t at ors
9. React or s (fixed and fluidized beds)
10. Online anal yzers (to moni t or t he composi t i on)
11. Fl ow and liquid level meas ur ement devices
12. Cont rol uni t s and cont rol valves
The above list shows some, but not all, of t he uni t s i nvol ved
in t he pet r ol eum industry. Opt i mum desi gn and oper at i on
of such uni t s as well as manuf act ur e of pr oduct s t o meet
mar ket demands and gover nment regul at i ons requi re a com-
pl et e knowl edge of pr oper t i es and charact eri st i cs for hydro-
carbons, pet r ol eum fract i ons/ product s, crude oils, and reser-
voi r fluids. Some of t he mos t i mpor t ant charact eri st i cs and
pr oper t i es of t hese fluids are listed bel ow wi t h some exam-
ples for t hei r appl i cat i ons. They are divided i nt o t wo gr oups
of t emper at ur e- i ndependent par amet er s and t emper at ur e-
dependent propert i es. The t emperat ure-i ndependent properties
and par amet er s are as follows:
1. Specific gravity (SG) or densi t y (d) at SC. These par a-
met er s are t emper at ur e- dependent ; however, specific
gravi t y at 15.5~ and 1 a t m and densi t y at 20~ and 1
at m used in pet r ol eum char act er i zat i on are i ncl uded in
t hi s cat egory of t emper at ur e- i ndependent propert i es. The
specific gravi t y is also pr esent ed in t er ms of API gravity.
I t is a useful pa r a me t e r t o char act er i ze pet r ol eum fluids,
to det er mi ne composi t i on (PNA) and t he qual i t y of a fuel
(i.e., sul fur cont ent ), and to est i mat e ot her pr oper t i es such
as critical const ant s, densi t y at vari ous t emper at ur es, vis-
cosity, or t her mal conduct i vi t y [23, 24]. I n addi t i on to its
di rect use for size cal cul at i ons (i.e., pumps , valves, t anks,
and pipes), it is also needed in desi gn and oper at i on of
equi pment s such as gravi t y decant ers.
2. Boi l i ng poi nt (Tb) or distillation curves such as t he t rue
boi l i ng poi nt curve of pet r ol eum fractions. I t is used to
det er mi ne volatility and to est i mat e char act er i zat i on pa-
r amet er s such as average boi l i ng poi nt , mol ecul ar weight,
composi t i on, and ma ny physi cal pr oper t i es (i.e., critical
const ant s, vapor pressure, t her mal propert i es, t r ans por t
pr oper t i es) [23-25].
3. Mol ecul ar wei ght ( M) is used to convert mol ar quant i t i es
i nt o mass basi s needed for pract i cal appl i cat i ons. Ther-
modynami c rel at i ons al ways pr oduce mol ar quant i t i es
(i.e., mol ar density), while in pract i ce mas s specific val-
ues (i.e., absol ut e density) ar e needed. Mol ecul ar wei ght
is also used to charact eri ze oils, to predi ct composi t i on
and qual i t y of oils, and to predi ct physi cal pr oper t i es such
as vi scosi t y [26-30].
4. Refractive i ndex (n) at s ome reference condi t i ons (i.e., 20~
and 1 at m) is anot her useful char act er i zat i on pa r a me t e r
t o est i mat e t he composi t i on and qual i t y of pet r ol eum frac-
tions. I t is al so used t o est i mat e ot her physi cal pr oper t i es
such as mol ecul ar weight, equat i on of st at e par amet er s,
t he critical const ant s, or t r ans por t pr oper t i es of hydrocar-
bon syst ems [30, 31].
5. Defined char act er i zat i on par amet er s such as Wat son K,
carbon-to-hydrogen wei ght ratio, (CH wei ght ratio), refrac-
tivity intercept (Ri), and vi scosi t y gravity cons t ant (VGC)
to det er mi ne t he qual i t y and composi t i on of pet r ol eum
fract i ons [27-29].
6. Composi t i on of pet r ol eum fract i ons in t er ms of wt % of
paraffi ns (P%), napht henes (N%), ar omat i cs (A%), and
sul fur cont ent (S%) are i mpor t ant t o det er mi ne t he qual-
ity of a pet r ol eum fract i on as well as to est i mat e physi cal
pr oper t i es t hr ough ps eudocomponent met hods [31-34].
Composi t i on of ot her const i t uent s such as asphal t ene and
resin component s are qui t e i mpor t ant for heavy oils t o
det er mi ne possi bi l i t y of sol i d-phase deposi t i on, a maj or
pr obl em in t he product i on, refining, and t r anspor t at i on
of oil [35].
7. Pour poi nt (Tp), and mel t i ng poi nt (TM) have l i mi t ed uses
in wax and paraffi ni c heavy oils t o det er mi ne t he degree
of solidification and t he wax cont ent as well as mi ni mum
t emper at ur e requi red to ensur e fluidity of t he oil.
8. Ani l i ne poi nt to det er mi ne a r ough est i mat e of ar omat i c
cont ent of oils.
9. Flash poi nt (TF) is a very useful pr oper t y for t he safet y of
handl i ng volatile fuels and pet r ol eum pr oduct s especially
in s umme r seasons.
10. Critical t emperat ure (To), critical pressure (Pc), and critical
v ol ume (Vc) k n o wn as critical const ant s or critical pr op-
ert i es are used to est i mat e vari ous physi cal and t her mo-
dynami c pr oper t i es t hr ough equat i ons of st at e or gener-
alized correl at i ons [36].
11. Acentric f act or (w) is anot her par amet er t hat is needed
t oget her wi t h critical pr oper t i es to est i mat e physi cal and
t her modynami c pr oper t i es t hr ough equat i ons of st at e
[36].
The above pr oper t i es are mai nl y used to char act er i ze t he
oil or to est i mat e t he physi cal and t her modynami c proper-
ties whi ch are all t emperat ure-dependent . Some of t he mos t
i mpor t ant pr oper t i es are listed as follows:
1. Densi t y (p) as a funct i on of t emper at ur e and pr essur e
is per haps t he mos t i mpor t ant physi cal pr oper t y for
pet r ol eum fluids (vapor or liquid forms). I t has great ap-
pl i cat i on in bot h pet r ol eum pr oduct i on and processi ng as
well as its t r anspor t at i on and st orage. It is used in t he
cal cul at i ons rel at ed to sizing of pi pes, valves, and st orage
t anks, power requi red by pumps and compr essor s, and
fl ow-measuri ng devices. I t is also used in r eser voi r si mul a-
t i on t o est i mat e t he amount of oil and gas in a reservoir, as
well as t he amount of t hei r pr oduct i on at vari ous r eser voi r
condi t i ons. I n addi t i on densi t y is used in t he cal cul at i on
of equi l i br i um rat i os (for phase behavi or cal cul at i ons) as
well as ot her propert i es, such as t r ans por t propert i es.
2. Vapor pressure (pv~p) is a meas ur e of volatility and it is
used in phase equi l i bri um calculations, such as flash, bub-
ble poi nt , or dew poi nt pr essur e calculations, in or der to
det er mi ne t he st at e of t he fluid in a r eser voi r or to sep-
ar at e vapor f r om liquid. I t is needed in cal cul at i on of
equi l i bri um rat i os for oper at i on and desi gn of distilla-
tion, absorber, and st ri ppi ng col umns in refineries. I t is
al so needed in det er mi nat i on of t he a mount of hydrocar-
bon losses f r om st orage facilities and t hei r pr esence in
air. Vapor pr essur e is t he pr oper t y t hat r epr esent s igni-
t i on charact eri st i cs of fuels. For exampl e, t he Rei d vapor
pressure (RVP) and boiling r ange of gasol i ne govern ease
of st art i ng engine, engi ne war m- up, r at e of accel erat i on,
mi l eage economy, and t endency t owar d vapor l ock [ 19].
3. Heat capacity (Cp) of a fluid is needed in design and oper-
ation of heat transfer units such as heat exchangers.
4. Enthalpy (H) of a fluid is needed in energy balance cal-
culations, heat requirements needed in design and oper-
ation of distillation, absorption, stripping columns, and
reactors.
5. Heat of vaporization (AHvap) is needed in calculation of
heat requirements in design and operation of reboilers or
condensers.
6. Heats of formation (hHf), combustion (AHc), and reaction
(AHr) are used in calculation of heating values of fuels
and the heat required/generated in reactors and furnaces
in refineries. Such information is essential in design and
operations of burners, furnaces, and chemical reactors.
These properties together with the Gibbs free energy are
used in calculation of equilibrium constants in chemical
reactions to determine the optimum operating conditions
in reactors for best conversion of feed stocks into the prod-
ucts.
7. Viscosity (t*) is another useful property in petroleum pro-
duction, refining, and transportation. It is used in reser-
voir simulators to estimate the rate of oil or gas flow
and their production. It is needed in calculation of power
required in mixers or to transfer a fluid, the amount of
pressure drop in a pipe or column, flow measurement de-
vices, and design and operation of oil/water separators
[37, 38].
8. Thermal conductivity (k) is needed for design and opera-
tion of heat transfer units such as condensers, heat ex-
changers, as well as chemical reactors [39].
9. Diffusivity or diffusion coefficient (D) is used in calcula-
tion of mass transfer rates and it is a useful property in
design and operation of reactors in refineries where feed
and products diffuse in catalyst pores. In petroleum pro-
duction, a gas injection technique is used in improved oil
recovery where a gas diffuses into oil under reservoir con-
ditions; therefore, diffusion coefficient is also required in
reservoir simulation and modeling [37, 40-42].
10. Surface tension (a) or interfacial tension (IFT) is used
mainly by the reservoir engineers in calculation of cap-
illary pressure and rate of oil production and is needed
in reservoir simulators [37]. In refineries, IFT is a use-
ful parameter to determine foaming characteristics of oils
and the possibility of having such problems in distillation,
absorption, or stripping columns [43]. It is also needed
in calculation of the rate of oil dispersion on seawater
surface polluted by an oil spill [44].
11. Equilibrium ratios (Ki) and fugacity coefficients (~Pi) are
the most important thermodynamic properties in all
phase behavior calculations. These calculations include
vapor-liquid equilibria, bubble and dew point pressure,
pressure-temperature phase diagram, and GOR. Such cal-
culations are important in design and operation of distilla-
tion, absorption and stripping units, gas-processing units,
gas-oil separators at production fields, and to determine
the type of a reservoir fluid [45, 46].
Generally, the first set of properties introduced above
(temperature-independent) are the basic parameters that are
used to estimate physical and thermodynamic properties
given in the second set (temperature-dependent). Properties
such as density, boiling point, molecular weight, and refrac-
tive index are called physical properties. Properties such as en-
thalpy, heat capacity, heat of vaporization, equilibrium ratios,
and fugacity are called thermodynamic properties. Viscosity,
thermal conductivity, diffusion coefficient, and surface ten-
sion are in the category of physical properties but they are also
called transport properties. In general all the thermodynamic
and physical properties are called thermophysical properties.
But they are commonl y referred to as physical properties or
simply properties, which is used in the title of this book.
A property of a system depends on the thermodynamic state
of the system that is determined by its temperature, pressure,
and composition. A process to experimentally determine var-
ious properties for all the industrially important materials,
especially complex mixtures such as crude oils or petroleum
products, would be prohibitive in both cost and time, indeed
it could probably never be completed. For these reasons ac-
curate methods for the estimation of these properties are be-
coming increasingly important. In some references the term
property prediction is used instead of property estimation;
however, in this book as generally adopted by most scientists
both terms are used for the same purpose.
1 . 3 I MP OR T A N C E OF P E T R O L E U M F L U I D S
C HA R A C T E R I Z A T I ON
In the previous section, various basic characteristic para-
meters for petroleum fractions and crude oils were intro-
duced. These properties are important in design and oper-
ation of almost every piece of equipment in the petroleum in-
dustry. Thermodynamic and physical properties of fluids are
generally calculated through standard methods such as cor-
responding state correlations or equations of state and other
pressure-volume-temperature (PVT) relations. These corre-
lations and methods have a generally acceptable degree of ac-
curacy provided accurate input parameters are used. When
using cubic equation of state to estimate a thermodynamic
property such as absolute density for a fluid at a known tem-
perature and pressure, the critical temperature (Tc), critical
pressure (Pc), acentric factor (~0), and molecular weight (M)
of the system are required. For most pure compounds and hy-
drocarbons these properties are known and reported in var-
ious handbooks [36, 47-50]. If the system is a mixture such
as a crude oil or a petroleum fraction then the pseudocritical
properties are needed for the calculation of physical proper-
ties. The pseudocritical properties cannot be measured but
have to be calculated through the composition of the mix-
ture. Laboratory reports usually contain certain measured
properties such as distillation curve (i.e., ASTM D 2887) and
the API gravity or specific gravity of the fraction. However,
in some cases viscosity at a certain temperature, the per-
cent of paraffin, olefin, naphthene, and aromatic hydrocar-
bon groups, and sulfur content of the fraction are measured
and reported. Petroleum fractions are mixtures of many com-
pounds in which the specific gravity can be directly measured
for the mixture, but the average boiling point cannot be mea-
sured. Calculation of average boiling point from distillation
data, conversion of various distillation curves from one type
to another, estimation of molecular weight, and the PNA com-
position of fractions are the initial steps in characterization of
1. INTRODUCTION 13
petroleum fractions [25, 46, 47]. Estimation of other basic pa-
rameters introduced in Section 1.2, such as asphaltenes and
sulfur contents, CH, flash and pour points, aniline point, re-
fractive index and density at SC, pseudocrtitical properties,
and acentric factor, are also considered as parts of charac-
terization of petroleum fractions [24, 28, 29, 51-53]. Some of
these properties such as the critical constants and acentric
factor are not even known for some heavy pure hydrocarbons
and should be estimated from available properties. Therefore
characterization methods also apply to pure hydrocarbons
[33]. Through characterization, one can estimate the basic
parameters needed for the estimation of various physical and
thermodynamic properties as well as to determine the com-
position and quality of petroleum fractions from available
properties easily measurable in a laboratory.
For crude oils and reservoir fluids, the basic laboratory
data are usually presented in the form of the composition
of hydrocarbons up to hexanes and the heptane-plus frac-
tion (C7+), with its molecular weight and specific gravity
as shown in Table 1.2. In some cases laboratory data on a
reservoir fluid is presented in terms of the composition of
single carbon numbers or simulated distillation data where
weight fraction of cuts with known boiling point ranges are
given. Certainly because of the wide range of compounds ex-
isting in a crude oil or a reservoir fluid (i.e., black oil), an
average value for a physical property such as boiling point
for the whole mixture has little significant application and
meaning. Characterization of a crude oil deals with use of
such laboratory data to present the mixture in terms of a
defined or a continuous mixture. One commonl y used char-
acterization technique for the crudes or reservoir fluids is
to represent the hydrocarbon-plus fraction (C7+) in terms of
several narrow-boiling-range cuts called psuedocomponents
(or pseudofractions) with known composition and character-
ization parameters such as, boiling point, molecular weight,
and specific gravity [45, 54, 55]. Each pseudocomponent is
treated as a petroleum fraction. Therefore, characterization
of crude oils and reservoir fluids require characterization of
petroleum fractions, which in turn require pure hydrocarbon
characterization and properties [56]. It is for this reason that
properties of pure hydrocarbon compounds and hydrocarbon
characterization methods are first presented in Chapter 2,
the characterization of petroleum fractions is discussed in
Chapter 3, and finally methods of characterization of crude
oils are presented in Chapter 4. Once characterization of a
petroleum fraction or a crude oil is done, then a physical
property of the fluid can be estimated through an appropri-
ate procedure. In summary, characterization of a petroleum
fraction or a crude oil is a technique that through available
laboratory data one can calculate basic parameters necessary
to determine the quality and properties of the fluid.
Characterization of petroleum fractions, crude oils, and
reservoir fluids is a state-of-the-art calculation and plays an
important role in accurate estimation of physical properties
of these complex mixtures. Watson, Nelson, and Murphy of
Universal Oi1 Products (UOP) in the mid 1930s proposed ini-
tial characterization methods for petroleum fractions [57].
They introduced a characterization parameter known as
Watson or UOP characterization factor, Kw, which has been
used extensively in characterization methods developed in the
following years. There are many characterization methods
r~
,<
<C
0
"0
3 0
20
10
. ~ e%Q
\
_ ~ . , ~ - . : . , . . - - 9 . . . . . . . . . . i
: , , , - . , . . . . . . , .
" , , ~ x e ~ ..," o~"*~" ~ 6e ~'*l'~ 9
I , I I I i ,
- 6 - 4 - 2 0 2 4
D e v i a t i o n f r o m E x p e r i m e n t a l V a l u e , %
F I G . 1 . ~ ! n f l u e n c e o f e r r o r i n c r i t i c a l t e m p e r a t u r e o n e r r o r s
i n p r e d i c t e d p h y s i c a l p r o p e r t i e s o f t o l u e n e . T a k e n f r o m R e f . [ 5 8 ]
w i t h p e r m i s s i o n .
suggested in the literature or in process simulators and each
method generates different characterization parameters that
in turn would result different estimated final physical prop-
erty with subsequent impact in design and operation of re-
lated units. To decide which method of characterization and
what input parameters (where there is a choice) should be
chosen depends very much on the user's knowledge and ex-
perience in this important area.
To show how important the role of characterization is in
the design and operation of units, errors in the prediction
of various physical properties of toluene through a modified
BWR equation of state versus errors introduced to actual crit-
ical temperature (To) are shown in Fig. 1.4 [58]. In this figure,
errors in the prediction of vapor pressure, liquid viscosity,
vapor viscosity, enthalpy, heat of vaporization, and liquid den-
sity are calculated versus different values of critical tempera-
ture while other input parameters (i.e., critical pressure, acen-
tric factor, etc.) were kept constant. In the use of the equation
of state if the actual (experimental) value of the critical tem-
perature is used, errors in values of predicted properties are
generally within 1-3% of experimental values; however, as
higher error is introduced to the critical temperature the error
in the calculated property increases to a much higher magni-
tude. For example, when the error in the value of the critical
temperature is zero (actual value of Tc), predicted vapor pres-
sure has about 3% error from the experimental value, but
when the error in Tc increases to 1, 3, or 5%, error in the pre-
dicted vapor pressure increases approximately to 8, 20, and
40%, respectively. Therefore, one can realize that 5% error in
an input property for an equation of state does not necessar-
ily reflect the same error in a calculated physical property but
can be propagated into much higher errors, while the pre-
dictive equation is relatively accurate if actual input parame-
ters are used. Similar results are observed for other physical
100
. . . . . n-Hexane
80
x ~ ( , ~ . - - - - n-Octane
60 "~ \ ' X . . . . . n-Eieosane
~ n-Tetracosane
40
20 "%\
"~~' ~' ~-20.400 " ~ " ~
-60
-6 -4 -2 0 2 4 6
% Deviation in Critical Temperature
F I G . 1 . 5 ~ l n f l u e n c e o f e r r o r i n c r i t i c a l t e m p e r a t u r e o n e r r o r s
o f p r e d i c t e d v a p o r p r e s s u r e f r o m L e e - K e s l e r m e t h o d .
pr oper t i es and wi t h ot her correl at i ons for t he est i mat i on of
physi cal pr oper t i es [59]. Effect of t he er r or in t he critical
t emper at ur e on t he vapor pr essur e of di fferent compounds
pr edi ct ed f r om t he Lee- Kesl er met hod (see Sect i on 7.3.2) is
shown in Fig. 1.5. When t he act ual critical t emper at ur e is
used, t he er r or in t he pr edi ct ed vapor pr essur e is al most neg-
ligible; however, if t he critical t emper at ur e is under-predi ct ed
by 5%, t he er r or in t he vapor pr essur e i ncreases by 60-80%
for t he vari ous compounds evaluated.
As shown in Chapt er 6, vapor pr essur e is one of t he key
par amet er s in t he cal cul at i on of equi l i bri um rat i os (Ki) and
subsequent relative volatility (a12), whi ch is defined in a bi-
nar y syst em of component s 1 and 2 as follows:
(1.3) K1 = y~
X1
X2
Ka Yl x -
(1.4) oq2 = ~ = x--~ Y2
wher e xl and x2 are t he mol e fract i ons of component s 1 and
2 in t he liquid phase, respectively. Si mi l arl y yt and y2 are t he
mol e fract i ons in t he vapor phase for component s 1 and 2,
respectively. For an ideal bi nar y syst em at l ow pressure, t he
equi l i br i um rat i o Ki is di rect l y pr opor t i onal to t he vapor pres-
sure as will be seen in Chapt er 6.
The mos t i mpor t ant aspect in t he desi gn and oper at i on of
distillation col umns is t he numbe r or t rays needed t o make a
specific separ at i on for specific feed and product s. I t has been
shown t hat a smal l er r or in t he val ue of relative volatility coul d
l ead to a muc h great er er r or in t he cal cul at i on of numbe r of
t r ays and t he l engt h of a distillation col umn [60]. The mi ni -
mu m numbe r of t rays requi red in a distillation col umn can be
cal cul at ed f r om t he knowl edge of relative volatility t hr ough
t he Fenske Equation given bel ow [61].
( 1 . 5 ) N m i n = l n [ x D ( 1 - - XB) / XB( 1 - - XD) ] - - 1
l n ( u l z )
wher e Nmin is t he mi ni mum numbe r of plates, and xD and xB
are t he mol e fract i on of t he light component in t he distillate
(top) and bot t om product s, respectively. Equat i on (1.5) is de-
vel oped for a bi nar y mi xt ure; however, a si mi l ar equat i on has
been devel oped for mul t i component mi xt ur es [61]. For differ-
ent val ues of or, er r or s cal cul at ed for t he mi ni mum numbe r
of t rays versus errors i nt r oduced in t he val ue of ~ t hr ough
Eq. (1.5) are shown in Fig. 1.6. As is shown in t hi s figure,
a - 5 % er r or in t he val ue of a when its val ue is 1.1 can gen-
erat e an er r or of mor e t han 100% in the cal cul at i on of mi n-
i mum numbe r of t rays. I t can be i magi ned t hat t he er r or in
t he act ual numbe r of t rays woul d be even hi gher t han 100%.
I n addi t i on, t he cal cul at ed number s of t rays are t heoret i cal
and when conver t ed to real numbe r of t rays t hr ough overall
col umn efficiency, t he er r or ma y i ncrease to several hundr ed
percent . The appr oach of bui l di ng t he col umn hi gher to have
a safe desi gn is quite expensive.
As an exampl e, a distillation col umn of di amet er 4.5 m
and hei ght 85 m has an i nvest ment cost of appr oxi mat el y
$4 mi l l i on (~4.5 million) as st at ed by Dohr n and Pfohl [60].
Er r or in t he cal cul at i on of relative volatility, a, coul d have
been caused by t he er r or in cal cul at i on of vapor pressure,
whi ch itself coul d have been caused by a smal l er r or in an
i nput pa r a me t e r such as critical t emper at ur e [58, 59]. There-
fore, f r om t hi s si mpl e anal ysi s one can realize t he ext r eme
cost and loss in t he i nvest ment t hat can be caused by a smal l
er r or in t he est i mat i on of critical t emper at ur e. Si mi l ar ot her
exampl es have been given in t he l i t erat ure [62]. Nowadays,
i nvest ment in refineries or t hei r upgr adi ng cost s billions of
dollars. For exampl e, for a t ypi cal refi nery of 160000 bbl/d
(8 mi l l i on t ons/ year) capacity, t he cost of const r uct i on in
Eur ope is about $2 billion [18]. This is equi val ent to refining
cost of $7.5/bbl while this numbe r for refineries of 1980s
was about $2/bbl. I n addi t i on to t he ext ra cost of i nvest ment ,
i nappr opr i at e desi gn of uni t s can cause ext ra oper at i ng costs
and short en t he pl ant life as well as pr oduce pr oduct s t hat
do not mat ch t he ori gi nal desi gn specifications. The use of a
pr oper char act er i zat i on met hod to cal cul at e mor e accur at e
100
80
60
~ ) 4O
20
z
o
Ze -2o
-4o
4 o
-80
-100
a =l . l
. . . . . a=l . 2
a = 1.3
-10 -8 -6 -4 -2 0 2 4 6 g 10
% Error in Relative Volatility
F I G . 1 . & - E f f e c t o f e r r o r i n t h e r e l a t i v e v o l a t i l -
i t y o n t h e e r r o r o f m i n i m u m n u m b e r o f p l a t e s
o f a d i s t i l l a t i o n c o l u m n .
1. I NT RODUCT I ON 15
pr oper t i es of pet r ol eum fract i ons can save a large por t i on of
such huge addi t i onal i nvest ment and oper at i ng costs.
1 . 4 OR GA N I Z A T I ON OF T H E B O O K
As t he title of t he book por t r ays and was di scussed in Sec-
t i ons 1.2 and 1.3, t he book pr esent s met hods of charact eri za-
t i on and est i mat i on of t her mophysi cal pr oper t i es of hydrocar-
bons, defined mi xt ures, undefi ned pet r ol eum fract i ons, cr ude
oils, and r eser voi r fluids. The ent i re book is wri t t en in ni ne
chapt er s in a way such t hat in general every chapt er requi res
mat er i al s pr esent ed in previ ous chapt ers. I n addi t i on t her e is
an appendi x and an index. Chapt er 1 gives a general i nt ro-
duct i on to t he subj ect f r om basi c definition of vari ous t erms,
t he nat ur e of pet r ol eum, its f or mat i on and composi t i on, t ypes
of pet r ol eum mi xt ures, and t he i mpor t ance of charact eri za-
t i on and pr oper t y pr edi ct i on to specific feat ures of t he book
and its appl i cat i on in t he pet r ol eum i ndust ry and academi a.
Because of t he i mpor t ance of uni t s in pr oper t y calculations,
t he l ast sect i on of Chapt er 1 deals wi t h uni t conver si on fac-
t ors especially bet ween SI and Engl i sh uni t s for the par ame-
t ers used in t he book. Chapt er 2 is devot ed to pr oper t i es and
char act er i zat i on of pur e hydr ocar bons f r om C1 to C22 f r om
di fferent hydr ocar bon groups, especi al l y f r om homol ogous
gr oups commonl y found in pet r ol eum fluids. Propert i es of
s ome nonhydr ocar bons f ound wi t h pet r ol eum fluids such as
H20, H2S, CO2, and Nz are also given. Basi c par amet er s ar e
defined at t he begi nni ng of t he chapter, fol l owed by charac-
t eri zat i on of pur e hydr ocar bons. Predictive met hods for vari-
ous pr oper t i es of pur e hydr ocar bons are pr esent ed and com-
par ed wi t h each other. A di scussi on is given on t he state-of-
t he-art s devel opment of predi ct i ve met hods. The pr ocedur es
pr esent ed in this chapt er are essent i al for char act er i zat i on of
pet r ol eum fract i ons and crude oils di scussed in Chapt ers 3
and 4.
Chapt er 3 di scusses vari ous char act er i zat i on met hods for
pet r ol eum fract i ons and pet r ol eum product s. Charact eri za-
t i on par amet er s are i nt r oduced and anal yt i cal i nst r ument s in
l abor at or y are discussed. I n t hi s chapt er one can use mi n-
i mu m l abor at or y dat a to char act er i ze pet r ol eum fract i ons
and to det er mi ne t he qual i t y of pet r ol eum product s. Esti-
mat i on of some basi c pr oper t i es such as mol ecul ar weight,
mol ecul ar-t ype composi t i on, sul fur cont ent , flash, pour poi nt
and freezi ng poi nt s, critical const ant s, and acent ri c fact or for
pet r ol eum fract i ons are pr esent ed in t hi s chapter. A t heoret -
ical di scussi on on devel opment of char act er i zat i on met hods
and gener at i on of predi ct i ve correl at i ons f r om exper i ment al
dat a is also present ed. Met hods of Chapt er 3 are ext ended t o
Chapt er 4 for t he char act er i zat i on of vari ous reservoi r fluids
and crude oils. Chapt ers 2- 4 are per haps the mos t i mpor-
t ant chapt ers in t hi s book, as t he met hods pr esent ed in t hese
chapt er s influence t he ent i re field of physi cal pr oper t i es in t he
r emai ni ng chapt ers.
I n Chapt er 5, PVT relations, equat i ons of state, and
cor r espondi ng st at e correl at i ons are pr esent ed [31, 63-65].
The use of t he velocity of light and sound in devel opi ng
equat i ons of st at e is also pr esent ed [31, 66-68]. Equat i ons of
st at e and cor r espondi ng st at e correl at i ons are powerful tools
in t he est i mat i on of vol umet ri c, physi cal , t r anspor t , and
t her modynami c pr oper t i es [64, 65, 69]. Pr ocedur es out l i ned
in Chapt er 5 will be used in t he pr edi ct i on of physi cal
pr oper t i es di scussed in t he fol l ow-up chapt ers. Fundament al
t her modynami c rel at i ons for cal cul at i on of t her modynami c
pr oper t i es are pr esent ed in Chapt er 6. The l ast t hr ee chapt er s
of t he book show appl i cat i ons of met hods pr esent ed in Chap-
t ers 2- 6 for cal cul at i on of vari ous physical, t her modynami c,
and t r ans por t propert i es. Met hods of cal cul at i on and esti-
mat i on of densi t y and vapor pr essur e are gi ven in Chapt er 7.
Ther mal pr oper t i es such as heat capacity, enthalpy, heat
of vapori zat i on, heat s of combus t i on and react i on, and
t he heat i ng val ue of fuels are also di scussed in Chapt er 7.
Predictive met hods for t r ans por t pr oper t i es namel y viscosity,
t her mal conductivity, diffusixdty, and surface t ensi on are
given in Chapt er 8 [ 30, 31, 42, 43, 69, 70] . Finally, phase
equi l i bri um calculations, est i mat i on of equi l i bri um ratios,
GOR, cal cul at i on of pr es s ur e- t emper at ur e (PT) di agr ams,
solid format i ons, t he condi t i ons at whi ch asphal t ene, wax,
and hydr at e are formed, as well as t hei r prevent i ve met hods
are di scussed in Chapt er 9.
The book is wri t t en accor di ng to t he st andar ds set by ASTM
for its publ i cat i on. Every chapt er begi ns wi t h a general i nt ro-
duct i on t o t he chapter. Since in t he following chapt er s for
mos t pr oper t i es several predi ct i ve met hods are present ed, a
sect i on on concl usi on and r ecommendat i ons is added at t he
end of t he chapter. Pract i cal pr obl ems as exampl es are pre-
sent ed and solved for each pr oper t y di scussed in each chap-
ter. Finally, t he chapt er ends by a set of exercise pr obl ems
followed by a ci t at i on sect i on for t he references used in t he
chapter.
The Appendi x gives a s umma r y of definitions of t er ms and
pr oper t i es used in t hi s manual accor di ng to t he ASTM dictio-
nar y as well as t he Greek letters used in t hi s manual . Finally
t he book ends wi t h an i ndex to provi de a qui ck gui de to find
specific subjects.
1 . 5 S P E C I F I C F E A T U R E S OF
T H I S MA N U A L
I n t hi s par t several existing books in t he ar ea of charact er-
i zat i on and physi cal pr oper t i es of pet r ol eum fract i ons are
i nt r oduced and t hei r di fferences wi t h t he cur r ent book are
discussed. Then some speci al feat ures of this book are pre-
sented.
1. 5. 1 I n t r o d u c t i o n o f S o me Ex i s t i n g Bo o k s
There are several books avai l abl e t hat deal wi t h physi cal pr op-
ert i es of pet r ol eum fract i ons and hydr ocar bon syst ems. The
mos t compr ehensi ve and wi del y used book is t he API Tech-
nical Data Book--Pet rol eum Refining [47]. I t is a book wi t h
15 chapt er s in t hree vol umes, and t he first edi t i on appear ed
in mi d 1960s. Every 5 years since, some chapt er s of t he book
have been revi sed and updat ed. The proj ect has been con-
duct ed at t he Pennsyl vani a St at e University and t he sixth
edi t i on was publ i shed in 1997. I t cont ai ns a dat a bank on
pr oper t i es of pur e hydr ocar bons, chapt ers on charact eri za-
t i on of pet r ol eum fract i ons, t her modynami c and t r ans por t
pr oper t i es of liquid and gaseous hydr ocar bons, t hei r mi x-
t ures, and undefi ned pet r ol eum fract i ons. For each propert y,
one predi ct i ve met hod t hat has been appr oved and selected
by t he API-TDB commi t t ees as t he best avai l abl e met hod is
present ed. This book will be r ef er r ed as API-TDB t hr oughout
t hi s book.
Anot her i mpor t ant book in this ar ea is The Properties of
Gases and Liquids t hat was originally wri t t en by Rei d and
Sher wood in 1950s and it has been revised and updat ed nearl y
every decade. The fifth and l at est edi t i on was publ i shed in
November 2000 [36] by t hree aut hor s di fferent f r om t he orig-
inal t wo aut hors. The book has been an excellent r ef er ence f or
st udent s and pract i cal engi neers in t he i ndust r y over t he past
five decades. I t di scusses vari ous met hods for pr edi ct i on of
pr oper t i es of pur e hydr ocar bons as well as nonhydr ocar bons
and t hei r defined mi xt ures. However, it does not t r eat un-
defined pet r ol eum fract i ons, cr ude oils, and r eser voi r fluids.
Most of t he met hods for pr oper t i es of pur e compounds re-
qui re t he chemi cal st r uct ur e of compounds (i.e., gr oup con-
t r i but i on t echni ques). The book compar es vari ous met hods
and gives its r ecommendat i ons for each met hod.
Ther e are several ot her books in t he ar ea of pr oper t i es of
oils t hat document empi ri cal l y devel oped predi ct i ve met hods,
among t hem is t he book Properties of Oils and Natural Gases,
by Pedersen et al. [ 13]. The book mai nl y t reat s reservoi r fluids,
especi al l y gas condensat es f r om Nor t h Sea, and it is mai nl y
a useful r ef er ence for r eser voi r engineers. Books by McCai n
[11], Ahmed [71], Whi t son [45], and Danesh [72] are all writ-
t en by reservoi r engi neers and cont ai n i nf or mat i on mai nl y f or
phase behavi or cal cul at i ons needed in pet r ol eum pr oduct i on
and r eser voi r si mul at ors. However, t hey cont ai n s ome useful
i nf or mat i on on met hods of pr edi ct i on of some physi cal pr op-
ert i es of pet r ol eum fract i ons. Anot her good r ef er ence book
was wri t t en by Tsonopoul os et al. [73] on t her modynami c
and t r anspor t pr oper t i es of coal liquids in t he mi d 1980s.
Al t hough t here are ma ny si mi l ari t i es bet ween coal liquids
and pet r ol eum fractions, t he book does not consi der cr ude
oils and r eser voi r fluids. But it provi des some useful correl a-
t i ons for pr oper t i es of coal liquids. The book by Wauqui er [ 18]
on pet r ol eum refining has several useful chapt er s on charac-
t eri zat i on and physi cal pr oper t i es of pet r ol eum fract i ons and
finished product s. I t also provi des t he t est met hods accord-
i ng t o Eur opean st andards. Some or gani zat i ons' Web sites
also provi de i nf or mat i on on fluid physi cal propert i es. A good
exampl e of such online i nf or mat i on is pr ovi ded by Nat i onal
Inst i t ut e of St andar ds (http://webbook. nist. gov) whi ch gives
mol ecul ar weight, names, formul as, st ruct ure, and some dat a
on vari ous compounds [74].
1. 5. 2 S p e c i a l Fe a t u r e s o f t h e Bo o k
Thi s book has objectives and ai ms t hat are di fferent f r om
t he books ment i oned in Sect i on 1.5.1. The mai n obj ect i ve
of this book has been to provi de a qui ck reference in t he
ar ea of pet r ol eum char act er i zat i on and pr oper t i es of vari ous
pet r ol eum fluids for t he peopl e who wor k in t he pet r ol eum
i ndust r y and r esear ch cent ers, especially in pet r ol eum pro-
cessing (downstream), pet r ol eum pr oduct i on (upstream), and
rel at ed industries. One speci al charact eri st i c of t he book is its
di scussi on on devel opment of vari ous met hods whi ch woul d
hel p t he users of process/ reservoi r si mul at or s to become fa-
mi l i ar wi t h the nat ur e of char act er i zat i on and pr oper t y esti-
mat i on met hods for pet r ol eum fractions. Thi s woul d in t ur n
hel p t hem to choose t he pr oper predi ct i ve met hod a mong t he
ma ny met hods avai l abl e in a process simulator. However, t he
book has been wri t t en in a l anguage t hat is under st andabl e
to under gr aduat e and gr aduat e st udent s in all areas of engi-
neeri ng and science. I t cont ai ns pract i cal solved pr obl ems as
well as exercise pr obl ems so t hat t he book woul d be sui t abl e
as a t ext for educat i onal purposes.
Special feat ures of this book are Chapt ers 2, 3, and 4 t hat
deal wi t h t he char act er i zat i on of hydr ocar bons, pet r ol eum
fract i ons, and cr ude oils and t hei r i mpact on t he ent i re field
of pr oper t y pr edi ct i on met hods. I t di scusses bot h light as
well as heavy fract i ons and pr esent s met hods of pr edi ct i on
of t he i mpor t ant charact eri st i cs of pet r ol eum pr oduct s f r om
mi ni mum l abor at or y dat a and easily measur abl e par ame-
ters. I t present s several char act er i zat i on met hods devel oped
in r ecent years and not document ed in existing references.
The book also present s vari ous predi ct i ve met hods, i ncl udi ng
t he mos t accur at e and wi del y used met hod for each pr oper t y
and di scusses poi nt s of st rengt h, weaknesses, and l i mi t at i ons.
Recommended met hods are based on t he generality, si mpl i c-
ity, accuracy, and availability of i nput par amet er s. Thi s is
anot her special feat ure of t he book. I n Chapt ers 5 and 6 it
di scusses equat i ons of st at e based on t he vel oci t y of sound
and light and how t hese t wo measur abl e pr oper t i es can be
used t o predi ct t her modynami c and vol umet r i c pr oper t i es of
fluids, especially heavy compounds and t hei r mi xt ur es [31,63,
66-68]. Significant at t ent i on is given t hr oughout t he book on
how to est i mat e pr oper t i es of heavy hydr ocar bons, pet r ol eum
fract i ons, crude oils, and r eser voi r fluids. Most of t he met hods
devel oped by Riazi and cowor ker s [23, 24, 26-33, 51-56, 63,
65-70], whi ch have been in use by t he pet r ol eum i ndust r y
[47, 75-82], are document ed in t hi s book. I n addi t i on, a new
exper i ment al t echni que to meas ur e di ffusi on coefficients in
r eser voi r fluids under r eser voi r condi t i ons is pr esent ed in
Chapt er 8 [42]. I n Chapt er 9 s ome new met hods for det ermi -
nat i on of onset of solid f or mat i on are i nt roduced. Repor t ed
exper i ment al dat a on charact eri st i cs and pr oper t i es of var-
i ous oils f r om di fferent par t s of t he worl d are i ncl uded in
vari ous chapt er s for di rect eval uat i ons and t est i ng of met h-
ods. Although bot h gases and liquids are t r eat ed i n t he book,
emphasi s is on t he liquid fractions. Generally, t he met hods
of est i mat i on of pr oper t i es of gases are mor e accur at e t han
t hose f or liquid syst ems. Most of t he met hods pr esent ed in t he
book are suppor t ed by s ome scientific basi s and t hey are not
si mpl y empi r i cal correl at i ons derived f r om a cert ai n gr oup of
data. This wi dens t he appl i cat i on of t he met hods pr esent ed
in t he book to di fferent t ypes of oils. However, all basi c pa-
r amet er s and necessar y engi neeri ng concept s are defined in
a way t hat is under st andabl e for t hose nonengi neer scientists
who are worki ng in t he pet r ol eum or rel at ed industry. Nearl y
all met hods are expressed t hr ough mat hemat i cal rel at i ons so
t hey are conveni ent f or comput er appl i cat i ons; however, mos t
of t hem are si mpl e such t hat t he pr oper t i es can be cal cul at ed
by hand cal cul at ors for a qui ck est i mat e whenever appl i ca-
ble special met hods are given for coal liquid fract i ons. Thi s is
anot her uni que feat ure of t hi s book.
1 . 6 A P P L I C A T I ON S OF T H E B O O K
The i nf or mat i on t hat is pr esent ed in t he book ma y be appl i ed
and used in all areas of t he pet r ol eum industries: pr oduct i on,
processing, and transportation. It can also be used as a
textbook for educational purposes. Some of the applica-
tions of the materials covered in the book were discussed in
Sections 1.2 and 1.3. The applications and uses of the book
may he summarized as follows.
1.6.1 Appl i c a t i ons i n Petrol eum Processing
(Downstream)
Engineers, scientists, and operators working in various sec-
tors of petroleum processing and refining or related industries
can use the entire material discussed in the book. It helps
laboratory people in refineries to measure useful properties
and to test the reliability of their measurements. The book
should be useful for engineers and researchers to analyze ex-
perimental data and develop their own predictive methods.
It is also intended to help people who are involved with de-
velopment of computer softwares and process simulators for
design and operation of units and equipments in petroleum
refineries. Another objective was to help users of such simu-
lators to be able to select an appropriate predictive method
for a particular application based on available data on the
fraction.
1.6.2 Applications in Petrol eum Production
(Upstream)
Reservoir, chemical, and mechanical engineers may use the
book in reservoir simulators, design and operation of surface
separators in production fields, and feasibility studies for en-
hanced oil recovery projects, such as gas injection projects.
Another application of the book by reservoir engineers is to
simulate laboratory data on PVT experiments for the reser-
voir fluids, determination of the nature and type of reservoir
fluids, and calculation of the initial amounts of oil and gas in
the reservoir. Reservoir engineers may also use Chapter 9 to
determine the conditions that a solid may form, amount of
solid formation, and method of its prevention during produc-
tion. Practically all chapters of the book should be useful for
reservoir engineers.
1.6.3 Applications in Academia
Although the original goal and aim in writing this book was
to prepare a reference manual for the industry, laboratories,
and research institutions in the area of petroleum, it has been
written in a way such that it can also be used as a textbook
for educational purposes. It can be used as a text for an elec-
tive course for either undergraduate (senior level) or graduate
level. Students from chemical, petroleum, and mechanical en-
gineering fields as well as from chemistry and physics can take
the course and understand the contents of the book. However,
it should not be hard for students from other fields of engi-
neering and science to use this book. The book may also be
used to conduct short courses in the petroleum industry.
1.6.4 Other Applications
There are several other areas in which the book can be used.
One may use this book to determine the quality of crude oils,
petroleum fuels, and products for marketing and government
1. I NTRODUCTI ON 17
organizations that set the standards for such materials. As
an example, the amount of sulfur or aromatic contents of a
fuel can be estimated through mi ni mum laboratory data to
check if they meet the market demand or government regu-
lations for environmental protection. This book can be used
to determine properties of crude oil, its products, and natural
gases that are needed for transportation and storage. Exam-
ples of such properties are density, boiling point, flash and
pour points, sulfur content, vapor pressure, and viscosity.
The book can also be used to determine the properties of
oils for clean-up operations where there is an oil spill on sea-
water. To simulate the fate of an oil spill and the rate of its
disappearance at least the following properties are needed in
order to use appropriate simulators [44, 83-85]:
9 Characterization of petroleum fractions (Chapter 3)
9 Pour point (Chapter 3)
9 Characterization of crude oil (Chapter 4)
9 Solubility parameter (Chapters 4, 6, and 9)
9 Density (Chapters 5 and 7)
9 Vapor pressure (Chapter 7)
9 Viscosity, diffusion coefficient, and surface tension
(Chapter 8)
Accurate prediction of the fate of a crude oil spill depends on
the characterization technique used to estimate the physical
properties. For example, to estimate how much of the ini-
tial oil would be vaporized after a certain time, accurate val-
ues of the diffusion coefficient, vapor pressure, and molecular
weight are needed in addition to an appropriate characteriza-
tion method to split the crude into several pseudocomponents
E833.
1.7 DEFINITION OF UNITS AND
THE CONVERSION FACTORS
An estimated physical property is valuable only if it is ex-
pressed in an appropriate unit. The most advanced process
simulators and the most sophisticated design approaches
fail to perform properly if appropriate units are not used.
This is particularly important for the case of estimation
of physical properties through various correlations or re-
porting the experimental data. Much of the confusion with
reported experimental data arises from ambiguity in their
units. If a density is reported without indicating the tem-
perature at which the density has been measured, this value
has no use. In this part basic units for properties used in
the book are defined and conversion factors between the
most commonly used units are given for each property.
Finally some units specifically used in the petroleum indus-
try are introduced. Interested readers may also find other
information on units from online sources (for example,
http://physics.nist.gov/cuu/contents/index.html).
1.7.1 Importance and Types of Uni t s
The petroleum industry and its research began and grew
mainly in the United States during the last century. The rela-
tions developed in the 1930s, 1940s, and 1950s were mainly
graphical. The best example of such methods is the Winn
nomogram developed in the late 1950s [86]. However, with the
b i r t h of t he c o mp u t e r a n d i t s e xpa ns i on, mo r e a na l yt i c a l
me t h o d s i n t he f o r m of e q u a t i o n s we r e d e v e l o p e d i n t he 1960s
a n d ma i n l y i n t he 1970s a n d 1980s. Ne a r l y al l c o r r e l a t i o n s
a n d g r a p h i c a l me t h o d s t ha t we r e d e v e l o p e d unt i l t he e a r l y
1980s a r e i n Engl i s h uni t s . However , s t a r t i ng f r om t he 1980s
ma n y b o o k s a n d h a n d b o o k s a p p e a r e d i n t he SI uni t s ( f r om Le
Sy s t e me I n t e r n a t i o n a l d' Uni t es ) . The ge ne r a l t r e n d i s t o uni f y
al l e ngi ne e r i ng b o o k s a n d d o c u me n t s i n SI uni t s t o be us e d
b y t he i n t e r n a t i o n a l c o mmu n i t y . However , ma n y books , r e-
por t s , h a n d b o o k s , a n d e q u a t i o n s a n d f i gur es i n v a r i o u s publ i -
c a t i ons a r e st i l l i n Engl i s h uni t s . The Uni t e d St a t e s a n d Uni t e d
Ki n g d o m b o t h of f i ci al l y us e t he Engl i s h s ys t e m of uni t s .
Ther ef or e, i t i s e s s e nt i a l t h a t e ngi ne e r s be f a mi l i a r wi t h b o t h
uni t s ys t e ms of Engl i s h a n d SI. The o t h e r uni t s y s t e m t ha t
i s s o me t i me s u s e d f or s o me p r o p e r t i e s i s t he cgs ( cent i met er ,
gr a m, s e c ond) uni t , wh i c h i s de r i ve d f r o m t he SI uni t .
Si nc e t he b o o k i s p r e p a r e d f or a n i n t e r n a t i o n a l a udi e nc e ,
t he p r i ma r y uni t s ys t e m u s e d f or e qua t i ons , t abl es , a n d fig-
ur e s i s t he SI; however , i t ha s b e e n t r i e d t o p r e s e n t e qui va l e nt
of n u mb e r s a n d val ues of p r o p e r t i e s i n b o t h SI a n d Engl i s h
uni t s . The r e a r e s o me f i gur es t h a t a r e t a k e n f r o m o t h e r r ef -
e r e nc e s i n t he l i t e r a t u r e a n d a r e i n Engl i s h uni t s a n d t h e y
have be e n p r e s e n t e d i n t h e i r or i gi na l f or m. The r e a r e s o me
s pe c i a l uni t s t ha t a r e c o mmo n l y u s e d t o e xpr e s s s o me spe-
ci al pr ope r t i e s . Fo r e xa mpl e , vi s c os i t y i s u s u a l l y e xpr e s s e d i n
c e nt i poi s e (cp), k i n e ma t i c vi s c os i t y i n c e nt i s t oke ( cSt ) , de n-
s i t y i n g/ cm 3, speci f i c gr a vi t y ( SG) a t s t a n d a r d t e mp e r a t u r e
of 60~ o r t he GOR i n scf / st b. Fo r s uc h pr ope r t i e s , t he s e pr i -
ma r y uni t s have b e e n u s e d t h r o u g h o u t t he book, whi l e t h e i r
r e s pe c t i ve e qui va l e nt val ues i n SI a r e al s o pr e s e nt e d.
1. 7. 2 Fundame nt al Uni t s and Pref i xes
Ge ne r a l l y t he r e a r e f our f u n d a me n t a l qua nt i t i e s of l e ngt h (L),
ma s s (M), t i me (t), a n d t e mp e r a t u r e ( T) a n d wh e n t h e i r uni t s
a r e known, uni t s of al l o t h e r de r i ve d qua nt i t i e s c a n be det er -
mi ne d. I n t he SI s ys t em, uni t s of l engt h, ma s s , a n d t e mp e r a -
t ur e a r e me t e r ( m) , k i l o g r a m (kg), a n d Kel vi n (K), r espect i vel y.
I n Engl i s h uni t s t he s e d i me n s i o n s have t he uni t s of f oot (ft),
p o u n d ma s s Ohm), a n d de gr e e s Ra n k i n e (~ r espect i vel y. The
u n i t of t i me i n al l uni t s ys t e ms i s t he s e c o n d (s), a l t h o u g h
i n En g l i s h uni t , h o u r (h) i s al s o us e d f or t he uni t of t i me .
F r o m t he s e uni t s , uni t of a ny o t h e r q u a n t i t y i n SI i s known.
F o r e x a mp l e t he uni t of f or ce i s SI i s kg- m/ s 2 wh i c h i s c a l l e d
Newt on (N) a n d as a r e s ul t t he uni t of p r e s s u r e mu s t be
N/ m 2 o r Pascal (Pa). Si nc e 1 Pa i s a ve r y s ma l l quant i t y,
l a r g e r uni t s s uc h as kPa (1000 Pa) o r me g a Pa s c a l ( MPa)
a r e c o mmo n l y us ed. The s t a n d a r d pr i f i xes i n SI uni t s a r e as
f ol l ows:
Gi ga (G) = 109
Me ga (M) = 106
Ki l o (k) = 103
He c t o (h) = 102
De ka ( da) = 101
Deci (d) = 10 -1
Cent i (c) = 10 -2
Mi l l i ( m) = 10 -3
Mi c r o (/z) = i 0 6
Na n o (n) = 10 -9
As a n e x a mp l e 1 000 000 Pa c a n be e xpr e s s e d as 1 MPa. The s e
pr ef i xes a r e n o t us e d i n c o n j u n c t i o n wi t h t he Engl i s h uni t s .
However , i n t he Engl i s h s y s t e m of uni t s wh e n v o l u me t r i c
qua nt i t i e s of gas es a r e p r e s e n t e d i n l ar ge n u mb e r s , u s u a l l y
ever y 1000 uni t s i s e xpr e s s e d by one pr ef i x of M. F o r ex-
a mpl e , 2000 scf of gas i s e xpr e s s e d as 2 Ms c f a n d s i mi l a r l y
2 000 000 scf i s wr i t t e n as 2 MMscf . Ot he r s y mb o l s us u-
al l y us e d t o e xpr e s s l a r ge qua nt i t i e s a r e b f or bi l l i on
(1000 mi l l i o n or 109) a n d t r f or t r i l l i on ( one mi l l i o n mi l l i ons
or 1012).
1. 7. 3 Uni t s o f Mass
The ma s s i s s h o wn b y m a n d i t s u n i t i n SI i s kg ( ki l ogr a m) , i n
cgs i s g ( gr am) , a n d i n t he Engl i s h uni t s y s t e m i s Ibm ( p o u n d -
mas s ) . On ma n y oc c a s i ons t he s u b s c r i p t m i s d r o p p e d f or l b
wh e n i t i s r e f e r r e d t o ma s s . I n t he Engl i s h u n i t s ys t em, uni t s
of o u n c e ( oz) a n d gr a i ns a r e al s o us e d f or ma s s uni t s s ma l l e r
t h a n a p o u n d . F o r l a r g e r va l ue s of ma s s , uni t of ton i s us ed,
wh i c h i s def i ned i n t hr e e f or ms of l ong, s hor t , a n d me t r i c .
Ge ne r a l l y t he t e r m t o n i s a p p l i e d t o t he me t r i c t o n (1000 kg).
The c o n v e r s i o n f a c t or s a r e as f ol l ows:
1 kg = 1000g = 2. 204634 l b --- 35. 27392 oz
11b = 0. 45359kg = 453. 59g = 16oz = 7000 g r a i n
1 g = 0.001 kg = 0. 002204634 l b = 15. 4324 g r a i n
i t on ( me t r i c ) = 1000kg = 2204. 6341b
1 t o n ( s hor t ) = 2000 l b = 907. 18 kg
I t on ( l ong) = 22401b = 1016kg = 1. 12t on ( s hor t )
= 1. 1016 t o n ( me t r i c )
1. 7. 4 Uni t s o f Lengt h
The u n i t of l e ngt h i n SI i s me t e r ( m) , i n cgs i s c e n t i me t e r
( cm) , a n d i n Engl i s h uni t s ys t e m i s f oot (ft). Sma l l e r val ues
of l e ngt h i n Engl i s h s ys t e m a r e p r e s e n t e d i n i n c h (i n. ). The
c o n v e r s i o n f a c t or s a r e as f ol l ows:
1 m = 1 0 0 c m = 10-3 km= 1000 mm= 106 mi c r o n s (i xm)
= 101~ a n g s t r o ms (A) = 3. 28084 ft = 39. 37008 i n.
= 1. 0936yd ( yar d)
1 ft = 12i n. = 0. 3048 m = 30. 48 c m = 304. 8 mm
= 3. 048 10- 4ki n = 1 / 3 y d
1 c m = 1 0 - 2 m = 10-5 k m = 1 0 mm = 0. 0328084 ft
= 0. 393701 i n.
l k m = 1 0 0 0 m = 3280. 48 ft = 3. 93658 x 104i n.
1 i n. = 2. 54 c m = 0. 0833333 ft = 0. 0254 m = 2. 54x 10-5 kr n
1 mi l e = 1609. 3m = 1. 609km = 5279. 8 ft
1 . 7 . 5 Un i t s o f T i me
The uni t of t i me i n al l ma j o r s ys t e ms i s t he s e c o n d (s); how-
ever, f or l ar ge val ues of t i me o t h e r uni t s s uc h as mi n u t e ( r ai n) ,
h o u r (h), d a y (d), a n d s o me t i me s even y e a r ( year ) a r e u s e d
appropriately. The conversi on fact ors among t hese uni t s are
as follows:
1 year = 365 d = 8760h = 5.256 105 mi n = 3.1536 x 107 s
1 d = 2.743973 x 10 3 year = 24 h = 1440 mi n = 8.64 x 104 s
l h = 1.14155 10-4year = 4.16667 10-2d
= 60 mi n = 3600 s
i mi n = 1.89934 10-6year = 6.94444 10 -4 d
= 1.66667 x 10-2h = 60s
1 s = 3.17098 x 10 -8 year = 1.157407 10 -5 d
= 2.77777 x 10 -4 h = 1.66667 x 10 -2 mi n
1. 7. 6 Uni t s o f Fo r c e
As ment i oned above, t he uni t of force i n t he SI syst em is
Newt on (N) and i n t he Engl i sh uni t syst em is pound- f or ce
(lbf). 1 lbf is equi val ent to t he wei ght of a mass of 1 lbm at
t he sea level wher e t he accel erat i on of gravity is 32.174 ft/s 2
(9.807 m/s2). In t he cgs system, t he uni t of force is dyne (dyn).
Anot her uni t for t he force i n t he met r i c syst em is kgf, whi ch
is equi val ent t o t he wei ght of a mass of 1 kg at t he sea level.
The conversi on factors are as follows:
1N = 1 kg. m/ s 2 = 105 dyn = 0.2248 lbf = 1.01968 x 10 -1 kgf
1 lbf = 4.4482 N = 0.45359 kgf
1 kgf = 9.807 N = 2.204634 lbf
1 dyn = 10 5 N = 2.248 x 10 -6 lbf
1. 7. 7 Uni t s o f Mo l e s
Anot her uni t to present amount of mat t er especi al l y i n en-
gi neeri ng cal cul at i ons is mole (mol), whi ch is defined as t he
rat i o of mass (m) to mol ecul ar wei ght (M).
m
(1.6) n = - -
M
In SI syst em t he uni t of mol e is kmol , wher e m in t he above
equat i on is in kg. In t he Engl i sh system, t he uni t of mol is
l bmol . In t he cgs system, t he uni t of mol is gmol, whi ch is
usual l y wr i t t en as mol. For exampl e, for met hane (mol ecul ar
wei ght 16.04) i mol of t he gas has mass of 16.04 g. One mol e
of any subst ance cont ai ns 6.02 x 1023 number of mol ecul es
(Avogadro' s number). The conver si on fact ors bet ween vari-
ous uni t s of mol es are t he same as gi ven for t he mass i n
Sect i on 1.7.3.
1 krnol = l O00mol = 2. 2046341bmol
1 l bmol = 0.45359 kmol = 453.59 mol
1 tool = 0.001 krnol = 0.002204634 l bmol
1. 7. 8 Uni t s o f Mo l e c u l a r We i g ht
Mol ecul ar wei ght or mol ar mass shown by M is a number
t hat 1 mol of any subst ance has equi val ent mass of M g. In
t he SI syst em t he uni t of M is kg/ kmol and i n t he Engl i sh
syst em t he uni t is lb/lbmol, whi l e in t he cgs syst em t he uni t
of M is g/mol. Mol ecul ar wei ght is r epr esent ed by t he same
number in all uni t syst ems regardl ess of t he syst em used. As
an exampl e, met hane has t he mol ecul ar wei ght of 16 g/mol,
16 lb/lbmol, and 16 kg/krnol i n t he uni t syst ems of cgs, SI,
and English, respectively. For this reason, i n many cases t he
uni t for t he mol ecul ar wei ght is not ment i oned; however, one
must real i ze t hat it is not a di mensi onl ess paramet er. Most
recent compi l at i ons of mol ar masses are pr ovi ded by Copl en
[87].
1. 7. 9 Uni t s o f Pr e s s u r e
Pressure is t he force exert ed by a fluid per uni t area; t herefore,
in t he SI syst em it has t he uni t of N/ m 2, whi ch is called Pascal
(Pa), and in t he Engl i sh syst em has t he uni t of lbf/ft 2 (psf) or
lbf/in. 2 (psi). Ot her uni t s commonl y used for t he pressure are
t he bar (bar) and standard atmosphere (atm). Pressure may
also be expressed in t er ms of mm Hg. In this book uni t s of
MPa, kPa, bar, at m, or psi are commonl y used for pressure.
The conver si on fact ors are gi ven as follows:
1 at m = 1.01325 bar = 101 325Pa = 101. 325kPa
= 0.101325 MPa = 14. 696psi
i at m = 1.0322 kgf / cm 2 = 760 mm Hg (torr) = 29.921 in. Hg
= 10.333 mH20 (4~
1 bar = 0.98692 at m = 1 105 Pa = 100kPa
= 0.1 MPa = 14. 5038psi
1 Pa = i x 10 - 3kPa = 1 x 10 - 6MPa = 9.8692 x 10 - 6at m
= 1 10- 5bar = 1.45037 x 10-4psi
1 psi = 6.804573 10 -2 at m = 6.89474 x 10- 2bar
= 6.89474 x 10 -3 MPa
1 psf = 144 psi = 9.79858 at m = 9.92843 bar = 0.99285 MPa
1 kgf / cm 2 = 0.96784 at m = 0.98067 bar = 14.223 psi
The act ual pressure of a fluid is t he absolute pressure, whi ch
is measur ed rel at i ve to vacuum. However, some pressure
meas ur ement devices are cal i brat ed t o read zero i n t he at-
mospher e and t hey show t he di fference bet ween t he abso-
lute and at mospher i c pressure. This di fference is cal l ed gage
pressure. Nor mal l y "a" is used to i ndi cat e t he absol ut e val ue
(i.e., psia, bara) and "g" is used to show t he gage pressure
(i.e., psig). However, for absol ut e pressure very oft en "a" is
dr opped f r om t he uni t (i.e., psi, at m, bar). Anot her uni t for
t he pressure is vacuum pressure t hat is defi ned for pressure
bel ow at mospher i c pressure. Rel at i ons bet ween t hese uni t s
are as follows:
( 1 . 7 ) Pg a g e = P a b s - - P a t m
( 1 . 8 ) P a b s = Pat na - - P v a c
General l y gage pressure uni t is used to express pressures
above t he at mospher i c pressures and vacuum pressure uni t is
used for pressures bel ow at mospher i c and may be expressed
i n vari ous uni t s (i.e., mm Hg, psi).
1. 7. 10 Uni t s o f Te mp e r a t u r e
Temper at ur e (T) is t he most i mpor t ant par amet er affect i ng
propert i es of fluids and it is r epr esent ed in Cent i grade (~
and Kel vi n (K) i n t he SI syst em and i n Fahr enhei t (~ and
degrees Ranki ne (~ in t he Engl i sh uni t system. Temper at ur e
in most equat i ons is i n absol ut e degrees of Kel vi n or Rank-
ine. However, accor di ng to t he defi ni t i on of Kel vi n and de-
grees Ranki ne wher e t here is a t emper at ur e di fference (AT),
uni t of ~ is t he same as K and ~ is t he same as ~ These
t emper at ur e uni t s are rel at ed t hr ough the following rel at i ons:
(1.9) T(K) = T(~ + 273.15
(1.10) T(~ = T(~ + 459.67
(1.11) AT(K) = AT(~
(1.12) AT(~ = AT(~
(1.13) T(~ = 1.8T(K)
(1.14) T(~ = 1.8T(~ + 32
As an example, absol ut e t emper at ur e of 100 K is equi val ent to
1.8 x 100 or 180~ Therefore, the conversi on factors bet ween
K and ~ are as follows:
1 K = 1.8~ (for absol ut e t emper at ur e T and the t emper at ur e
difference, AT)
1 ~ C -- 1.8 ~ F (only for t he t emper at ur e difference, A T)
1. 7. 11 Uni t s o f Vo l u me , Spe c i f i c Vo l u me ,
a n d Mol ar V o l u me - - T h e S t a n d a r d Co n d i t i o n s
Volume (V) has t he di mensi on of cubi c l engt h (L 3) and t hus
i n SI has the uni t of m 3 and i n Engl i sh its uni t is cubi c feet
(cf or ft3). Some uni t s part i cul arl y used for l i qui ds i n t he SI
syst em are liter (L), cm 3 (cc), or mi l l i l i t er (mL) and i n Engl i sh
uni t s are gal l on (in U.S. or Imperi al ) and barrel (bbl). Volume
of one uni t mass of a fluid is called specific vol ume and the
vol ume of 1 mol of a fluid is called mol ar vol ume. Some of
the conversi on factors are as follows.
1 m 3 = 106 cm 3 = 1000 L = 35.315 f t 3 ~- 264.18 gal l on (U.S,)
= 35.316 ft 3 = 6. 29bbl
I ft 3 = 2.8316 x 10- 2m 3 = 28. 316L = 7.4805 gal l on(U. S. )
1 bbl = 42 gallon(U.S.) = 158.98 L = 34.973 gal l on (Imperi al )
1 gal l on (U.S.) = 0.8327 gal l on (Imperial)
= 0.023809 bbl = 3.7853 L
1 mL = i cm 3 = 10-3L = 10- 6m 3 ---0. 061024in. 3
For the mol ar vol umes some of t he conversi on factors are
given as follows:
1 m3/ kmol = 1 L/ mol = 0.001 m3/ mol = 1000 cm3/ mol
= 16.019 ft 3/ l bmol
1 ft 3/ l bmol = 6.24259 x 10 -2 m3/ kmol
= 6.24259 x 10 -5 ma/ mol = 62.4259 cm3/ mol
1 cma/ mol = i mL/ mol = 1 L/ kmol = 0.001 ma/ kmol
= 1.6019 x 10 -2 ft a/ l bmol
It shoul d be not ed t hat the same conversi on factors appl y
to specific vol umes. For example,
1 ft3/lb = 6.24259 10 -2 m3/ kg = 62. 4259cm3/ g
Si nce vol ume and specific or mol ar vol umes depend on t em-
perat ure and pressure of t he system, values of vol ume i n any
uni t syst em are meani ngl ess if t he condi t i ons are not spec-
ified. This is part i cul arl y i mpor t ant for gases i n whi ch bot h
t emper at ur e and pressure st rongl y i nfl uence t he vol ume. For
this reason, to express amount of gases i n t erms of vol ume,
nor mal l y some SC are defined. The SC i n the met ri c SI uni t s
are 0~ and 1 at m and i n the Engl i sh syst em are 60~ and
1 at m. Under these condi t i ons mol ar vol ume of any gas is
equi val ent to 22.4 L/mol (in SI) and 379 scf/lbmol (in Engl i sh
uni t s). I n reservoir engi neeri ng cal cul at i ons and pet r ol eum
i ndust r y i n general, t he SC i n the SI uni t s are also set at 60~
(15.5~ or 289 K) and 1 at m. The choice of st andar d t emper-
at ure and pressure (STP) varies from one source to another.
I n this book when the st andar d T and P are not specified the
STP refers to 289 K and I at m, whi ch is equi val ent to t he STP
i n Engl i sh uni t syst em rat her t han SI syst em (273 K and 1
atm). However, for l i qui d syst ems the vol ume is less affected
by pressure and for this reason specification of t emper at ur e
al one is sufficient.
1. 7. 12 Uni t s o f Vo l u me t r i c a n d Ma s s F l o w Ra t e s
Most processes i n the pet rol eum i ndust r y are cont i nuous and
usual l y the vol ume or mass quant i t i es are expressed i n the
form of rate defined as vol ume or mass per uni t time. One
part i cul ar vol umet ri c flow rate used for l i qui ds i n t he Engl i sh
syst em is gal l on (U.S.) per mi nut e and is known as GPM.
Some of the conversi on factors for these quant i t i es are
l m3/ s = 1 103 L/ s = 1.5851 x 104 GPM
= 5.4345 x I0 s bbl / d = 1.27133 x 105 ft a/ h
1 fta/h = 7.86558 x 10-4m3/ s = 0. 12468GPM
= 4.27466 bbl / d
1 GPM = 2.228 x 10 -3 ft3/s = 8.0205 ma/ h = 34.285 bbl / d
I bbl / d = 2.9167 x 10 -2 GPM = 1.8401 10 -4 ma/ s
= 0.23394 ft3/h
The conversi on factors for the mass rates are as follows:
1 kg/s -- 7.93656 x 103 lb/h --- 3.5136 x 107 t on/ year
1 lb/s = 1.63295 x 103 kg/h = 39. 1908t on/ d
The same conversi on factors appl y to mol ar rates.
1. 7. 13 Uni t s o f De n s i t y a n d Mol ar De n s i t y
Densi t y shown by d or p is defined as mass per uni t vol ume
and it is reci procal of specific vol ume. The conversi on factors
can be obt ai ned from reversi ng those of specific vol ume i n
Sect i on 1.7.11.
1 kg/m3 = 6.24259 x 10-21b/ft3 = 1 x 10-3 g/cm 3
= 8.3455 x 10 -3 lb/gal
1 lb/ft 3 = 16. 019kg/m 3 = 1.6019 10-2 g/cm 3
= 0.13368 lb/gal
1 g/cm 3 = 1 kg/L = 103 kg/m 3 = 62.42591b/ft 3
-= 8.3455 lb/gal
1 lb/gal = 1.19825 x 102 kg/m 3 = 7.48031b/ft 3
= 0.119825 g/cm 3
Densi t y may also be present ed i n t erms of numbe r of mol es
per uni t vol ume, whi ch is called molar density and is recipro-
cal of mol ar vol ume. It can be obt ai ned by di vi di ng absol ut e
densi t y to mol ecul ar wei ght . The conversi on factors for mol ar
densi t y are exactly the same as t hose for the absol ut e densi t y
(i.e., I mol / cm 3 = 62.4259 lbmol/ft3). I n pract i cal cal cul at i ons
1. I N T R ODUC T I ON 21
the conversion factors may be simplified without major
error in the calculations. For example, 62.4 instead of 62.4259
or 7.48 instead of 7.4803 are used in practical calculations.
In expressing values of densities, similar to specific volumes,
the SC must be specified. Generally densities of liquid hydro-
carbons are reported either in the form of specific gravity at
15.5~ (60~ or the absolute density at 20~ and 1 atm in
g/cm 3 .
1.7.14 Units of Specific Gravity
For liquid systems, the specific gravity (SG) is defined as the
ratio of density of a liquid to that of water, and therefore, it is
a dimensionless quantity. However, the temperature at which
specific gravity is reported should be specified. The specific
gravity is also called relative densi t y versus absolute density.
For liquid petroleum fractions and crude oils, densities of
both the oil and water are expressed at the SC of 60~ (15.5~
and 1 atm, and they are usually indicated as SG at 60~176
or simply SG at 60~ Another unit for the specific gravity of
liquid hydrocarbons is defined by the American Petroleum
Institute (API) and is called API degree and is defined in terms
of SG at 60~ (API = 141.5/SG-131.5). For gases, the spe-
cific gravity is defined as the ratio of density of the gas to
that of the air at the SC, which is equivalent to the ratio of
molecular weights. Further discussion on specific gravity, def-
initions, and methods of calculation are given in Chapter 2
(Section 2.1.3).
1.7.15 Units of Composition
Composition is the most important characteristic of homoge-
nous mixtures in which two or more components are uni-
formly mixed in a single phase. Because of the nature of
petroleum fluids, accurate knowledge of composition is im-
portant. Generally composition is expressed as percent-
age (%) or as fraction (percent/100) in terms of weight, mole,
and volume. Density of the components (or pseudocompo-
nents) constituting a mixture is required to convert composi-
tion from weight basis to volume basis or vice versa. Similarly
conversion of composition from mole basis to weight basis
or vice versa requires molecular weight of the constituting
components (or pseudocomponents). Mole, weight, and vol-
ume fractions are shown by Xm, X~, and xv, respectively. Mole,
weight, and volume percentages are shown by mol%, wt%,
and vol%, respectively. Some references use mol/mol, wt/wt,
and vol/vol to express fractional compositions. For normal-
ized compositions, the sum of fractions for all components
in a mixture is 1 (Y~xi = 1) and the sum of all percentages is
100. If the molecular weights of all components in a mixture
are the same, then the mole fraction and weight fraction are
identical. Similarly, if the density (or specific gravity) of all
components is the same, the weight and volume fractions are
identical. The formula to calculate weight fraction from mole
fraction is given as
Xmi Mi
(1.15) Xwi - - ~ N 1 xmiMi
where N is the total number of components, Mi is the molec-
ular weight, and Xwi and Xmi are the weight and mole fractions
of component i, respectively. The conversion from weight to
volume fraction can be obtained from the following equation:
Xwi / SGi
(1.16) xvi -- ~N=I Xwi/SGi
in which x~ is the volume fraction and SGi is the specific grav-
ity of component i. In Eq. (1.16) density (d) can also be used
instead of specific gravity. If mole and weight fractions are
multiplied by 100, then composition is calculated on the per-
centage basis. In a similar way the conversion of composition
from volume to weight and then to mole fraction can be ob-
tained by reversing the above equations. The composition of
a component in a liquid mixture may also be presented by its
molar density, units of which were discussed in Section 1.7.13.
Generally, a solution with solute molarity of 1 has 1 mol of
solute per 1 L of solution (1 mol/L). Through use of both
molecular weight of solute and density of solution one can
obtain weight fraction from molarity. Another unit to express
concentration of a solute in a liquid solution is mol al i t y. A so-
lution with molality of 1 has 1 mol of solute per i kg of liquid
solvent.
Another unit for the composition in small quantities is the
ppm (part per mi l l i on), which is defined as the ratio of unit
weight (or volume) of a component to 106 units of weight or
volume for the whole mixture. Therefore, ppm can be pre-
sented in terms of both volume or weight. Usually in gases
the ppm is presented in terms of volume and in liquids it is
expressed in terms of weight. When ppm is presented in terms
of weight, its relation with wt% is 1 ppm = 10 .4 wt%. For ex-
ample, the maxi mum allowable concentration of H2 S in air
for prolonged exposure is 10 ppm or 0.001 wt%. There is an-
other smaller unit definedas part per billion known as ppb
(1 ppm = 1000 ppb). In the United States a gas is considered
"sweet" if the amount of its H2S content is no more than one
quarter grain per i00 scf of gas. This is almost equivalent to
4 x 10 .4 mol fraction [88]. This is in turn equivalent to 4 ppm
on the gas volume basis. Gas composition may also be rep-
resented in terms of partial pressure where sum of all partial
pressures is equivalent to the total pressure.
In general, the composition of gases is presented in volume
or mole fractions, while the liquid composition may be pre-
sented in any form of weight, mole, or volume. For gases at
low pressures (< 1 atm where a gas may be considered an ideal
gas) mole fraction and volume fractions are the same. How-
ever, generally under any conditions, volume and mole frac-
tions are considered the same for gases and vapor mixtures.
For narrow boiling range petroleum fractions with composi-
tions presented in terms of PNA percentages, it is assumed
that densities and molecular weights for all three representa-
tive pseudocompoents are nearly the same. Therefore, with
a good degree of approximation, it is assumed that the PNA
composition in all three unit systems are the same and for
this reason on many occasions the PNA composition is repre-
sented only in terms of percentage (%) or fraction without in-
dicating their weight or volume basis. However, this is not the
case for the crude or reservoir fluid compositions where the
composition is presented in terms of boiling point (or carbon
number) and not in the form of molecular type. The following
example shows conversion of composition from one type to
another for a crude sample.
22 CHARACT ERI Z AT I ON AND P R OP E R T I E S OF P E T R OL E UM FRACT I ONS
TABLE 1.5--Conversion of composition of a crude oil sample from mole to weight and volume percent.
Component t ool % Molecular weight (M) Specific gravity (SG) wt% vol%
C2 0.19 30.07 0.356 0.03 0.06
C 3 1.88 44.10 0.508 0.37 0.64
iC4 0.62 58.12 0.563 0.16 0.25
nC4 3.92 58.12 0.584 1.02 1.52
iC5 2.11 72.15 0.625 0.68 0.95
nC5 4.46 72.15 0.631 1.44 1.98
C6 (fraction) 8.59 82.00 a 0.690 3.15 3.97
C7+ (fraction) 78.23 266.00 0.895 93.15 90.63
Sum 100.00 100.00 100.00
~Thi s i s mo l e c u l a r we i g h t o f C6 h y d r o c a r b o n g r o u p a n d s h o u l d n o t b e mi s t a k e n wi t h M o f nC6 wh i c h i s 86. 2.
Ex ampl e / . / - - The composi t i on of a Middle East crude
oil is given i n Table 1.5 i n t erms of mol % wi t h known
mol ecul ar wei ght and specific gravity for each component /
pseudocomponent . Calculate t he composi t i on of the crude i n
bot h wt% and vo1%.
Sol ut i on- - I n this t abl e values of mol ecul ar wei ght and spe-
cific gravity for pure compounds are obt ai ned from Chapt er 2
(Table 2.1), whi l e for t he C6 group, val ues are t aken from
Chapt er 4 and for the C7 ~ fraction, val ues are given by the
laboratory. Conversi on cal cul at i ons are based on Eqs. (1.15)
and (1.16) on t he percent age basis and the results are also
given i n Table 1.5. I n this cal cul at i on it is seen t hat i n t erms
of wt % and vo1%, heavi er compounds (i.e., C7+) have hi gher
val ues t han i n t erms of mol%. t
1. 7. 16 Uni t s o f Energy a nd Speci f i c Energy
Energy i n vari ous forms (i.e., heat, work) has the uni t of Joule
(1 J -- 1 N- m) i n the SI and ft -l bf i n the Engl i sh system. Val-
ues of heat are also present ed i n t erms of calorie (in SI) and
BTU (British Ther mal Unit) i n the Engl i sh system. There are
t wo types of joules: absol ut e j oul es and i nt er nat i onal joules,
where i Joule (int.) =1. 0002 Joul e (abs.). In this book onl y ab-
solute j oul es is used and it is desi gnat ed by J. There are also
t wo types of calories: t her mochemi cal and I nt er nat i onat i onal
St eam Tables, where I cal (i nt ernat i onal st eam tables) --
1.0007 cal (t hermochemi cal ) as defined i n the API-TDB [47].
I n this book cal refers to the i nt er nat i onal st eam tables unl ess
ot herwi se is specified. I n the cgs syst em the uni t of energy is
dyn-cm, whi ch is also called erg. The uni t of power i n t he SI
system is J/s or wat t (W). Therefore, kW. h equi val ent to 3600
kJ is also a uni t for the energy. The pr oduct of pressure and
vol ume (PV) may also present the uni t of energy. Some of t he
conversi on factors for the uni t s of energy are given as follows:
1 J = 1 N. m -- 10 -3 kJ = 107erg -- 0.23885 cal
= 9.4783 x 10-4 Bt u = 2.778 x 10- 7kW. h
1 J = 3.725 x 10 -7 hp. h ---= 0.73756 ft.lbf = 9.869 L. at m
I cal ( I nt er nat i onal Tables) -- 3.9683 x 10 -3 Bt u = 4.187 J
= 3.088 ft . l bf = 1.1630 x 10- 6kW. h
1 cal (t hermochemi cal ) = 1 cal = 3.9657 x 10 .3 Bt u
= 4. 184J = 3.086 ft -l bf = 1.1622 x 10- 6kW. h
1Btu = 1055 J = 251.99 cal = 778.16 ft. lbf
= 2.9307 x 10- 4kW. h
1 ft -l bf = 1. 3558J = 0. 32384cal = 1.2851 x 10-3 Bt u
= 3.766 x 10- 7kW. h
1 kW. h = 3600kJ = 3412.2 Bt u = 2.655 106 ft -l bf
Energy per uni t mass is called specific energy t hat may be
used to present propert i es such as specific enthalpy, specific
i nt er nal energy, specific heats of react i on, and combust i on or
the heat i ng values of fuels. Some of the conversi on factors are
given below.
1 J/g = 103 J/kg = 1 kJ/kg -- 0.42993 Bt u/ l b
1 Btu/lb = 2.326 J/g = 0.55556 cal/g
The same conversi on factors apply to the uni t s of mol ar en-
ergy such as mol ar enthalpy.
1. 7. 17 Uni t s o f Speci f i c Energy per De gr e e s
Propert i es such as heat capaci t y have the uni t of specific en-
ergy per degrees. The conversi on factors are as follows:
J J
1 ~ = 1 x 10 .3 kgOC = 1 = 0.23885 Bt u
cal Bt u J
1 ~ = 1 ~ = 4.1867 go--~
As ment i oned i n Sect i on 1.7.13, for the difference i n t em-
perat ure (AT), uni t s of ~ and K are the same. There-
fore, t he uni t s of heat capaci t y may also be represent ed
i n t erms of specific energy per Kel vi n or degrees Ranki ne
(i.e., 1 ~ = 1 Bm = 1 ~ = 1 ~) The same conversi on fac-
l b. ~ g. ~ ~-.~ "
tors appl y to uni t s of mol ar energy per degrees such as mol ar
heat capacity.
Anot her par amet er whi ch has the uni t of mol ar energy per
degrees is the uni versal gas const ant (R) used i n t hermody-
nami c rel at i ons and equat i ons of state. However, the uni t of
t emper at ur e for this par amet er is the absol ut e t emper at ur e
(K or ~ and ~ or ~ may never be used i n this case. Si mi l ar
conversi on factors as t hose used for the heat capaci t y given
above also appl y to t he uni t s of gas const ant s i n t erms of mo-
l ar energy per absol ut e degrees.
Bt u cal cal (t hermochemi cal )
1 - 1 - 1.0007
l bmol 9 ~ mol . K mol . K
J
= 4.1867 x 1 0 3 _
kmol . K
of t he gas const ant are gi ven i n Sec- Numeri cal val ues
t i on 1.7.24.
1 . 7 . 1 8 Un i t s o f Viscosity and Kinematic Viscosity
Vi s cos i t y ( a bs ol ut e vi s cos i t y) s h o wn b y / ~ i s a p r o p e r t y t h a t
c h a r a c t e r i z e s t he f l ui di t y of f l ui ds a n d i t ha s t he d i me n s i o n
of ma s s p e r l e ngt h p e r t i me ( M/ L . t ) . I f t he r e l a t i o n b e t we e n
d i me n s i o n s of f or ce ( F) a n d ma s s (M) i s u s e d ( F = M. L. t - 2 ) ,
t h e n a b s o l u t e vi s c os i t y f i nds t he d i me n s i o n of F . t . L -2 wh i c h
i s t he s a me as d i me n s i o n f or t he p r o d u c t of p r e s s u r e a n d
t i me . Ther ef or e, i n t he SI s y s t e m t he uni t of vi s cos i t y i s Pa - s
( N. m- 2. s ) . I n t he cgs s ys t e m t he uni t of vi s c os i t y i s i n g/ c m- s
t ha t i s cal l ed poi s e (p) a n d i t s h u n d r e d t h i s c a l l e d c e nt i poi s e
(cp), wh i c h i s e qui va l e nt t o mi ] l i - Pa , s ( mP a . s). The conver -
s i on f a c t or s i n v a r i o u s uni t s a r e gi ven bel ow.
1 cp = 1.02 x 10 -4 kgf . s / m 2 : 1 10 -3 Pa - s = 1 mP a . s
= 10- 2p = 2. 089 x 10 -5 l bf . s / f t 2 = 2. 4191b/ h- ft
= 3.6 kg/ h. m
1 Pa. s = 1 k g / m. s = 1000 cp = 0. 67194 l b/ f t - s
1 l b/ h. f t = 8. 634 x 10 -6 l bf . s/ ft 2 = 0. 4134 cp = 1.488 kg/ h. m
i kgf. s / m 2 = 9. 804 x 103cp = 9. 804 Pa . s = 0. 20476 l bf 9 s/ ft 2
1 l bf . s/ ft 2 = 4. 788 104 cp = 4. 884 kgf. s / m 2
The r a t i o of vi s c os i t y t o d e n s i t y i s k n o wn as k i n e ma t i c vi s -
c os i t y ( v) a n d ha s t he d i me n s i o n of L/ t 2. I n t he cgs s ys t em,
t he uni t of k i n e ma t i c vs i c os i t y i s cm2/ s al s o c a l l e d s t o k e ( St )
a n d i t s h u n d r e d t h i s c e nt i s t oke ( cSt ) . The c o n v e r s i o n f a c t or s
a r e gi ven bel ow.
1 ft2/h = 2. 778 10-4 ft2/s --- 0. 0929m2/ h = 25.81 c St
1 f t 2/ s = 9. 29 104 c St = 334. 5 m2/ h
1 c St = 10 -2 S t - - 10- 6m2/ s = 1 mmZ/ s = 3. 875 x 10 -2 f t Z/ h
= 1.076 10 - s ft2/s
1 m2/ s = 104 St = 106 c St = 3. 875 x 104 f t 2/ h
An o t h e r u n i t t o e xpr e s s k i n e ma t i c vi s c os i t y of l i qui ds i s
S a y b o l t u n i v e r s a l s e c o n d s ( SUS) , wh i c h i s t he uni t f or t he
Sa y b o l t uni ve r s a l vi s c os i t y ( ASTM D 88). Def i ni t i on of vi scos-
i t y gr a vi t y c o n s t a n t (VGC) i s b a s e d on SUS uni t f or t he vi scos-
i t y at t wo r e f e r e nc e t e mp e r a t u r e s of 100 a n d 210~ (37. 8 a n d
98. 9 ~ The VGC i s u s e d i n Ch a p t e r 3 t o e s t i ma t e t he c om-
p o s i t i o n of he a vy p e t r o l e u m f r a c t i ons . The r e l a t i o n b e t we e n
SUS a n d c St i s a f u n c t i o n of t e mp e r a t u r e a n d i t i s gi ven i n t he
API TDB [47]. The a na l yt i c a l r e l a t i ons t o c onve r t c St t o SUS
a r e gi ven b e l o w [47].
SUSeq = 4.6324VT
[1. 0 + 0.03264VT]
+
[(3930. 2 + 262. 7vr + 23.97v~- + 1.646v 3) x 10 -5]
(1. 17)
wh e r e vr i s t he k i n e ma t i c vi s c os i t y a t t e mp e r a t u r e T i n cSt .
The SUSeq c a l c u l a t e d f r om t hi s r e l a t i on i s c o n v e r t e d t o t he
SUSr at t he d e s i r e d t e mp e r a t u r e of T t h r o u g h t he f ol l owi ng
r e l a t i on.
(1. 18) SUST = [1 + 1.098 x 10- 4( T - 311)]SUSeq
wh e r e T i s t he t e mp e r a t u r e i n kel vi n (K). F o r c o n v e r s i o n of
c ST t o SUS at t he r e f e r e nc e t e mp e r a t u r e of 311 K (100~
onl y Eq. (1. 17) i s ne e de d. Eq u a t i o n (1. 18) i s t he c o r r e c t i o n
t e r m f or t e mp e r a t u r e s o t h e r t h a n 100~ Fo r k i n e ma t i c vi s-
c os i t i e s g r e a t e r t h a n 70 cSt , Eqs. (1. 17) a n d (1. 18) c a n be
1. I N T R O D U C T I O N 2 3
s i mpl i f i e d t o t he f ol l owi ng f or m at t he t e mp e r a t u r e s of 311
(100~ a n d 372 K, ( 210~ r e s pe c t i ve l y [1].
(1. 19) SUS100F = 4.632v100F vl00F > 7 5 c St
(1. 20) SUS210F = 4.664VZ10F V210F > 75 c St
wh e r e Vl00F i s t he k i n e ma t i c vi s cos i t y at 100~ (311 K) i n cSt .
As a n e xa mpl e , a p e t r o l e u m f r a c t i on wi t h k i n e ma t i c vi s cos i t y
of 5 c St at 311 K ha s a n e qui va l e nt S a y b o h Uni ve r s a l Vi s cos i t y
of 42. 4 SUS as c a l c u l a t e d f r o m Eq. (1. 17).
An o t h e r uni t f or t he vi s cos i t y i s SFS ( Sa ybol t f our a l sec-
onds ) e xpr e s s e d f or Sa y b o l t f our a l vi scosi t y, wh i c h i s me a -
s u r e d i n a wa y s i mi l a r t o Sa y b o l t uni ve r s a l vi s cos i t y b u t
me a s u r e d b y a l a r ge r or i f i ce ( ASTM D 88). The c o n v e r s i o n
f r o m c St t o SFS i s e xpr e s s e d t h r o u g h t he f ol l owi ng e q u a t i o n s
at t wo r e f e r e nc e t e mp e r a t u r e s of 122~ (323 K) a n d 210~
(372 K) [47].
13924
(1. 21) SFS122F = 0.4717Vm22F +
U 1 2 2 F 2 - - 72.59VI22F + 6816
5610
(1. 22) SFS210F = 0.4792v2mF + v2210F + 2130
Fo r c onve r s i on of Sa y b o l t f our a l vi s cos i t y ( SFS) t o k i n e ma t i c
vi s c os i t y ( cSt . ) , t he a bove e q u a t i o n s s h o u l d be u s e d i n r e ve r s e
o r t o us e t a b u l a t e d val ues gi ven b y API - TDB [47]. As a n e xa m-
pl e, a n oi l wi t h Sa y b o l t f our a l vi s c os i t y of 450 SFS at 210~
ha s a k i n e ma t i c vi s cos i t y of 940 cSt . Gener al l y, vi s c os i t y of
hi ghl y vi s c ous oi l s i s p r e s e n t e d by SUS or SFS uni t s .
1.7.19 Units of Thermal Conductivity
Th e r ma l c onduc t i vi t y (k) as d i s c u s s e d i n Ch a p t e r 8 r e p r e s e n t s
a mo u n t of h e a t p a s s i n g t h r o u g h a u n i t a r e a of a me d i u m f or
one uni t of t e mp e r a t u r e g r a d i e n t ( t e mp e r a t u r e di f f e r e nc e p e r
uni t l engt h) . Ther ef or e, i t ha s t he d i me n s i o n of e ne r gy p e r
t i me p e r a r e a p e r t e mp e r a t u r e gr a di e nt . I n t he SI uni t s i t i s
e xpr e s s e d i n J/ s- m. K. Si nc e t h e r ma l c onduc t i vi t y i s def i ned
b a s e d on a t e mp e r a t u r e di f f e r e nc e ( AT) , t he u n i t of ~ ma y
al s o be us e d i n s t e a d of K. Be c a us e J/ s i s de f i ne d as wa t t (W),
t he u n i t of t h e r ma l c onduc t i vi t y i n t he SI s ys t e m i s u s u a l l y
wr i t t e n as W/ r e . K. I n t he Engl i s h s ys t em, t he uni t of t her -
Btu a n d i n s o me r e f e r e nc e s i s wr i t t e n ma l c onduc t i vi t y i s
as ~ ' h ~ t " wh i c h i s t he r a t i o of h e a t f l ux t o t he t e mp e r a t u r e
9 . o / . . .
gr a di e nt . The c o n v e r s i o n f a c t or s b e t we e n v a r i o u s u mt s a r e
gi ven bel ow.
1 W/ r e . K ( J/ s. m. ~ = 0. 5778 Bt u/ f l . h. ~
= 1.605 10 -4 Bt u/ f t . s. ~
= 0. 8593 kcal / h- m-~
1 Bt u/ f t - h- ~ = 1. 7307 W/ m. K
1 c a l / c m 9 s.~ = 242. 07 Bt u/ f t 9 h- ~ = 418. 95 W/ m. K
1 . 7 . 2 0 Un i t s o f Di f f u s i o n C o e f f i c i e n t s
Di f f us i on coef f i ci ent o r di f f us i vi t y r e p r e s e n t s t he a mo u n t of
ma s s di f f us e d i n a me d i u m p e r uni t a r e a p e r uni t t i me p e r
uni t c o n c e n t r a t i o n gr a di e nt . As s h o wn i n Ch a p t e r 8, i t ha s
t he s a me d i me n s i o n as t he k i n e ma t i c vi scosi t y, wh i c h i s
squared l engt h per t i me (L2]t). Usually it is expressed i n
cm2/s.
i cm2/s = 10 -4 m2/s : 9.29 x 10 -6 f t 2 / s : 3.3445 x 10 -4 f t 2 / h
1. 7. 21 Uni t s o f Surf ace Tens i on
Surface t ensi on or i nt erfaci al t ensi on (a) as descri bed i n Sec-
t i on 8.6 (Chapt er 8) has t he uni t of energy (work) per uni t area
and t he SI uni t of surface t ensi on is J/ m 2 = N/re. Si nce N/ m
is a large uni t the values of surface t ensi on are expressed i n
mi l l i -N/ m (mN/ m) whi ch is t he same as t he cgs uni t of surface
t ensi on (dyn/cm). The conversi on factors for this propert y are
as follows:
1 dyn/ cm = 1 erg/cm 2 = 10 - 3 J/ m 2 = 1 mJ/ m 2
= 10 -3 N/ m -- I mN/ m
1. 7. 22 Uni t s o f Sol ubi l i t y Paramet er
Predi ct i on of sol ubi l i t y par amet er (~) for pet r ol eum fract i ons
and crude oil is di scussed i n Chapters 4 and 10 and it has
t he uni t of (energy/volume) ~ The t radi t i onal uni t of ~ is i n
(cal/cm3) ~ Anot her form of the uni t for the sol ubi l i t y pa-
r amet er is (pressure) ~ Some conversi on factors are given
below.
1 (calth/cm3) ~ = 2.0455 (J/cm3) ~ = 2.0455 (MPa) ~
= 2.0455 x 103 (j/m3) ~
= 2.0455 x 103 (Pa) ~ = 10.6004 (Btu/ft3) ~
= 31.6228 (kcalth/m 3)~
6.4259 (atm) ~ = 2.05283 (ft-lbf/ft3) ~
1 (MPa) ~ = 0.4889 (calth/cm3) ~ = l(J/cm3) ~ = 103 (Pa) ~
Values of surface t ensi on i n the l i t erat ure are usual l y ex-
pressed i n (cal/cm3) ~ where cal represent s t her mochemi cal
uni t of calories.
1. 7. 23 Uni t s o f Gas-to-Oi l Rat i o
Gas-to-oil rat i o is an i mpor t ant par amet er i n det er mi ni ng the
type of a reservoi r fluid and i n set t i ng the opt i mum operat i ng
condi t i ons i n t he surface separat ors at t he pr oduct i on field
(Chapt er 9, Sect i on 9.2.1). I n some references such as the
API-TDB [47], this par amet er is called gas-to-liquid ratio and
is shown by GLR. GOR represent s the rat i o of vol ume of gas to
t he vol ume of l i qui d oil from a separat or under the S C of 289 K
and 101.3 kPa (60~ and 14.7 psia) for bot h the gas and liquid.
Uni t s of vol ume were di scussed i n Sect i on i. 7.13. Three types
of uni t s are commonl y used: the oilfield, t he metric, and t he
Engl i sh uni t s.
9 Oilfield uni t s: st andar d cubi c feet (scf) is used for the vol ume
of gas, and stock tank barrels (stb) is used for t he vol ume of
oil. Therefore, GOR has the uni t of scf/stb.
9 Metric uni t s: st andar d cubi c met ers (sm 3) is used for the
gas, and stock t ank cubi c met ers (st m 3) uni t is used for the
oil. The vol ume of l i qui d oil pr oduced is usual l y present ed
under t he stock t ank condi t i ons, whi ch are 60~ (15.5~
and 1 at m. Therefore, GOR uni t i n this syst em is sma/ st m 3.
9 Engl i sh uni t : scf is used for the gas, and sock tank cubic feet
(stft 3) is used for the l i qui d vol ume. Thus t he GOR has t he
uni t s of scf/ st ft 3. This uni t is exactly t he same as sm3/stm 3
i n the SI uni t .
The conversi on factors bet ween these t hree uni t s for the GOR
(GLR) are given as follows:
1 scf/stb = 0.1781 scf/stft 3 -- 0.1781 sm3/stm 3
I sm3/stm 3 : I scf/stft 3 = 5.615 scf/stb
1. 7. 24 Val ues o f Uni versal Cons t ant s
I. 7.24.1 Gas Cons t ant
The uni versal gas const ant shown by R is used i n equat i ons
of state and t her modynami c rel at i ons i n Chapt ers 5, 6, 8, and
10. It has the uni t of energy per mol e per absol ut e degrees. As
di scussed i n Sect i on 1.7.17, its di mensi on is si mi l ar to t hat of
mol ar heat capacity. The val ue of R i n the SI uni t is 8314 J/
kmol - K. The energy di mensi on may also be expressed as the
pr oduct of pressure and vol ume (PV), whi ch is useful for ap-
pl i cat i on i n the equat i ons of state. Value of R i n t erms of en-
ergy uni t is more useful i n the cal cul at i on of t her modynami c
propert i es such as heat capaci t y or enthalpy. Values of this
par amet er i n several ot her uni t s are given as follows.
R -- 8.314 J/mol 9 K = 8314 J/kmol 9 K = 8.314 mapa/ mol - K
= 83.14 cm3bar/ mol 9 K
= 82.06 cm 3.atm/mO1 9 K = 1.987 calth/mol - K
= 1.986 cal/mol 9 K = 1.986 Bt u/ l bmol 9 R
-- 0.7302 ft 3-atm/lbmOl' R = 10.73 ft 3. psi a/ l bmol 9 R
= 1545 ft. lbf/lbmol 9 R
1. 7. 24. 2 Ot her Nume r i c al Cons t ant s
The Avogadro numbe r is the numbe r of mol ecul es i n 1 mol of
a subst ance.
NA = Avogadro numbe r -- 6.022 x 1023 mo1-1
For exampl e 1 mol of met hane (16 g) consi st s of 6.022 1023
molecules. Ot her const ant s are
Bol t zman const ant = kB -- R/NA = 1.381 x 10 -23 J/K.
Pl anck const ant = h = 6.626 1 0 - 3 4 J. s.
Speed of light i n vacuum = c = 2.998 x 108 m/s.
Numer i cal const ant s
~r = 3.14159265
e = 2.718281 828
lnx = lOgl0 x/logl0e = 2.30258509 log10 x.
1. 7. 25 Speci al Uni t s f or t he Rat es and Amount s
o f Off and Gas
Amount s of oil and gas are usual l y expressed i n vol umet ri c
quant i t i es. I n t he pet r ol eum i ndust r y the common uni t for
vol ume of oil is barrel (bbl) and for the gas is st andar d cubi c
feet (scf) bot h at the condi t i ons of 60~ (15.5~ and 1 at m.
The pr oduct i on rat e for t he crude is expressed i n bbl / d and
for the gas i n scf/d.
I n some cases, amount of crude oil is expressed i n t he met -
ric ton. Conversi on from vol ume to wei ght or vice versa re-
qui res densi t y or specific gravity (API) of the oil. For a light
Saudi Ar abi an cr ude of 35.5 API (SG = 0. 847), t he fol l ow-
i ng conver s i on f act or s appl y bet ween wei ght a nd vol ume of
cr udes a nd t he rat es:
1 t on ~ 7.33 bbl = 308gal l on (U.S.) 1 bbl ~- 0. 136t on
1 bbl / d ~ 50 t on/ year
For a Mi ddl e Eas t cr ude of API 30, 1 t on - 7. 19 bbl (1 bbl
0. 139 t on).
Anot her wa y of expr essi ng quant i t i es of var i ous s our ces of
e ne r gy is t h r o u g h t hei r heat i ng val ues. For exampl e, by bur n-
i ng 1 x 106 t ons of a cr ude oil, t he s ame a mo u n t of ener gy
c a n be p r o d u c e d t ha t is p r o d u c e d t h r o u g h bur ni ng 1.5 x 109
t ons of coal . Of cour s e t hi s val ue ver y mu c h depends on t he
t ype of cr ude a nd t he coal. Ther ef or e, s uch eval uat i ons and
c ompa r i s ons ar e appr oxi mat e. I n s ummar y, 1 mi l l i on t ons of
a t ypi cal cr ude oil is equi val ent t o ot her f or ms of ener gy:
1 x 106t ons of cr ude oil ~ 1.111 x 1 0 9 s m 3 (39.2 x 109s cf )
of nat ur al gas
1.5 x 109t ons o f c o a l
---- 12 x 109 kW. h of el ect r i ci t y
The --- si gn i ndi cat es t he a ppr oxi ma t e val ues, as t hey de pe nd
on t he t ype of oil or gas. For a t ypi cal cr ude, t he heat i ng val ue
is appr oxi mat el y 10 500 cal/g (18 900 Bt u/ l b) and f or t he nat -
ur al gas is a bout 1000 Bt u/ scf (37. 235 x 103 kJ / s m3) . Appr ox-
i mat el y 1 mi l l i on t ons of a t ypi cal cr ude oil c a n pr oduc e an
ener gy equi val ent t o 4 x 109 kW- h of el ect r i ci t y t h r o u g h a t yp-
ical powe r pl ant . I n 1987 t he t ot al nucl ear ener gy p r o d u c e d
i n t he wor l d was equi val ent t o 404 x 106 t ons of cr ude oil
bas ed on t he ener gy p r o d u c e d [5]. I n t he s a me ye a r t he t o-
t al hydr oel ect r i c ener gy was equi val ent t o 523. 9 x 106 t ons of
cr ude oil. I n 1987 t he t ot al coal r eser ves i n t he wor l d wer e
es t i mat ed at 1026 x 109 t ons, whi l e t he t ot al oil r eser ves wer e
a bout 122 x 109 t ons. However , f r o m t he ener gy poi nt of vi ew
t he t ot al coal r eser ves ar e equi val ent t o onl y 0.68 x 109 t ons
of cr ude oil. The subj ect of heat i ng val ues will be di scussed
f ur t he r i n Chapt er 7 (see Sect i on 7.4.4).
Uni t conver s i on is an i mp o r t a n t ar t i n engi neer i ng cal cu-
l at i ons and as was st at ed bef or e wi t h t he knowl edge of t he
def i ni t i on of s ome bas i c uni t s f or onl y a few f unda me nt a l
quant i t i es (energy, l engt h, mas s , t i me, a nd t emper at ur e) , t he
uni t f or ever y ot he r pr ope r t y c a n be obt ai ned. The basi c i dea
i n t he uni t conver s i on is t ha t a val ue of a p a r a me t e r r e ma i ns
t he s a me wh e n it is mul t i pl i ed by a f act or of uni t y i n a wa y
t ha t t he i ni t i al uni t s ar e el i mi nat ed and t he desi r ed uni t s ar e
kept . The f ol l owi ng exampl es de mons t r a t e h o w a uni t c a n be
conver t ed t o a not he r uni t s ys t em wi t hout t he use of t abul at ed
conver s i on f act or s.
Example 1 . 2 - - Th e mo l a r heat i ng val ue of me t h a n e is 802 kJ/
mol . Cal cul at e t he heat i ng val ue of me t h a n e i n t he uni t s of
cal / g and Btu/lb. The mol e c ul a r wei ght of me t h a n e is 16.0,
Solution--In t hi s cal cul at i on a pr act i ci ng engi neer has t o re-
me mb e r t he f ol l owi ng bas i c conver s i on f act or s: 1 l b = 453. 6 g,
1 cal = 4. 187 J, and 1 Bt u = 252 cal. The val ue of mol e c ul a r
wei ght i ndi cat es t ha t 1 t ool = 16 g. I n t he c onve r s i on pr oces s
t he i ni t i al uni t is mul t i pl i ed by a seri es of k n o wn conver s i on
1. INTRODUCTION 2 5
f act or s wi t h r at i os of uni t y as fol l ows:
8 0 2 ~ o l = ( 8 0 2 ~ o l ) x m~ Xl 6 g 1 0 ~ ~ 0 ~ J x
/ 8 0 2 x 1 0 0 0 \
- ~ ~- x- - 4. - i - ~) [cal/g] = l 1971. 58cal / g
The conver s i on t o t he Engl i s h uni t is pe r f or me d i n a si mi l ar
way:
453. 6 g Bt u
1 1 9 7 1 . 5 8 c a l / g =( l 1 9 7 1 . 5 8 c a l / g ) x ~ x 2 5 2 c a l
( 11971. 58 x 4 5 3 . 6 )
= ~ [Bt u/ l b]
= 21549. 2 Bt u/ l b
I n t he above cal cul at i ons all t he r at i o of t er ms i nsi de t he II
si gn have val ues of unity.
Example 1 . 3 - - Th e r ma l conduct i vi t y of a ker os ene s ampl e at
60~ is 0.07 Bt u/ h - ft-~ Wh a t is t he val ue of t he r ma l c onduc -
t i vi t y i n mW/ mK f r om t he f ol l owi ng pr ocedur es :
1. Use of a ppr opr i a t e conver s i on f a c t or i n Sect i on 1.7.19.
2. Di r ect cal cul at i on wi t h use of conver s i on f act or s f or f un-
da me nt a l di mens i ons .
Solution--
1 . I n Sect i on 1.7.19 t he conver s i on f act or be t we e n SI a nd
Engl i s h uni t s is gi ven as:
1 W/mK = 0. 5778 Bt u/ f t . h. ~ Wi t h t he knowl edge t ha t
W = 1000 rnW, t he conver s i on is car r i ed as:
0 . 0 7 B t u / h . f t . ~ \.(007 h. ft . oF]xBt U 1~ 100OmW
W/ mK ~
x 0 . 5 7 7 8 Bt u / h - ft - =121. 1 mW/ mK
2. The conver s i on can be car r i ed out wi t hout use of t he con-
ver s i on t abl es if a pr act i ci ng engi neer is f ami l i ar wi t h t he
bas i c def i ni t i ons a nd conver s i on fact ors. These ar e 1 W =
1 J/s, 1 W = 1000 mW, 1 cal = 4. 187 J, I Bt u = 251. 99 cal,
1 h = 3600 s, 1 ft = 0. 3048 m, 1 K = I~ = 1.8~ (for t he
t e mpe r a t ur e di fference). I t s houl d be not ed t ha t t he r ma l
conduct i vi t y is def i ned ba s e d on t e mpe r a t ur e di f f er ence.
0. 07 Bt u/ h - f t . ~
[ 0 0 7 [ Bt u 1 251. 99c a l 4. 187J 3 6 @0 s I
= [ " ] h . f t . ~ x Bt u x cal x
x ~/s x 1 0 0 ~ W x 0. 3~48m x 1 ' ~ x
[0. 07 x 251. 99 x 4. 187 x 1000 x 1.8 m_~.~
= 121.18 mW/ mK
Exampl es 1.2 a nd 1.3 s how t hat wi t h t he knowl edge of onl y
ver y f ew conver s i on f act or s a nd bas i c def i ni t i ons of f unda-
ment al uni t s, one can obt ai n t he conver s i on f act or bet ween
any t wo uni t s ys t ems f or a ny pr ope r t y wi t hout use of a refer-
ence conver s i on table.
1 . 8 P R O B L E M S
1.1. State one theory for the formation of petroleum and give
names of the hydrocarbon groups in a crude oil. What
are the most important heteroatoms and their concen-
tration level in a crude oil?
1.2. The following compounds are generally found in the
analysis of a crude oil: ethane, propane, isobutane,
n-butane, isopentane, n-pentane, 2,2-dimethylbutane,
cyclopentane, cyclohexane, n-hexane, 2-methylpentane,
3-methylpentane, benzene, methylcyclopentane, 1,1-
dimethylcyclopentane, and hydrocarbons from C7 and
heavier grouped as C7+.
a. For each compound, draw the chemical structure and
give the formula. Also indicate the name of hydrocar-
bon group that each compound belongs to.
b. From the above list give the compounds that possibly
exist in a gasoline fraction.
1.3. Give the names of n-C20, n-C3o, n-C40, and three isomers
of n-heptane according to the IUPAC system.
1.4. List the 10 most important physical properties of crude
and its products that are required in both the design and
operation of an atmospheric distillation column.
1.5. What thermodynamic and physical properties of gas
and/or liquid fluids are required for the following two
cases?
a. Design and operation of an absorption column with
chemical reaction [40, 89].
b. Reservoir simulation [37].
1.6. What is the characterization of petroleum fractions,
crude oils, and reservoir fluids? Explain their differ-
ences.
1.7. Give the names of the following compounds according
to the IUPAC system.
a
(b) (c) (d)
e.
CH2-~-~ CH-- CH~CH- - CHz---CH3
1.8. From an appropriate reference find the following data
in recent years.
a. What is the distribution of refineries in different parts
of the world (North America, South America, Western
Europe, Africa, Middle East, Eastern Europe and
Former Soviet Union, and Asia Pacific)?
b. Where is the location of the biggest refinery in the
world and what is its capacity in bbl/d?
c. What is the history of the rate of production of gaso-
line, distillate, and residual from refineries in the
world and the United States for the last decade?
1.9. Characteristics of three reservoir fluids are given below.
For each case determine the type of the reservoir fluid
using the rule of thumb.
a. GOR = 20 scf/stb
b. GOR = 150 000 scf/stb
c. CH4 mol% -- 70, API gravity of STO = 40
1.10. GOR of a reservoir fluid is 800 scf/stb. Assume the molec-
ular weight of the stock tank oil is 260 and its specific
gravity is 0.87.
a. Calculate the GOR in sma/stm a and the mole fraction
of gases in the fluid.
b. Derive a general mathematical relation to calculate
GOR from mole fraction of dissolved gas (XA) through
STO gravity (SG) and oil molecular weight (M).
Calculate XA using the developed relation.
1.11.The total LPG production in 1995 was 160 million
tons/year. If the specific gravity of the liquid is assumed
to be 0.55, what is the production rate in bbl/d?
1.12. A C7+ fraction of a crude oil has the following composi-
tion in wt%. The molecular weight and specific gravity of
each pseudocomponent are also given below. Calculate
the composition of crude in terms of vol% and mol%.
Pseudocomponent wt% M SG
C7+ (1) 17.3 110 0.750
C7+ (2) 23.6 168 0.810
C7+ (3) 31.8 263 0.862
C7+ (4) 16.0 402 0.903
C7+ (5) 11.3 608 0.949
Total C7+ 100
1.13. It is assumed that a practicing engineer remembers the
following fundamental unit conversion factors without
a reference.
1 ft = 0.3048 m -- 12 in.
1 atm = 101.3 kPa = 14.7 psi
1 K = 1.8~
1 Btu = 252 cal
1 c a l = 4 . 1 8 J
1 k g = 2 . 2 l b
g = 9.8 m/s 2
1 lbmol = 379 scf
Molecular weight of methane -- 16 g/tool
Calculate the following conversion factors using the
above fundamental units.
a. The value of gas constant is 1.987 cal/mol 9 K. What is
its value in psi. ft3/lbmol 9 R?
b. Pressure of 5000 psig to atm
c. 1 kgf/cm 2 to kPa
d. 1 Btu/lb.~ to J/kg. K
e. 1 Btu/lbmol to cal/g
f. 1000 scf of methane gas to lbmol
g. 1 MMM scf of methane to kg
h. 1 cp to lb/ft.h
i. 1 Pa. s to cp
j. 1 g/cm 3 to lb/ft a
1.14. A crude oil has API gravity of 24. What is its densi t y i n
g/cm 3, lb/ft 3, kg/L, kg/m3 ?
1.15. Convert the following uni t s for t he viscosity.
a. Crude viscosity of 45 SUS (or SSU) at 60~ (140~ to
cSt.
b. Viscosity of 50 SFS at 99~ (210~ to cSt.
c. Viscosity of 100 cp at 38~ (100~ to SUS
d. Viscosity of 10 cp at 99~ (210~ to SUS
1.16. For each t on of a typical crude oil give the equi val ent
est i mat es i n the following t erms:
a. Vol ume of crude i n bbl.
b. Tons of equi val ent coal.
c. St andar d cubi c feet (scf) and sm 3 of nat ur al gas.
1.17. I n t erms of equi val ent energy values, compare exist-
i ng reserves for t hree maj or fossil types and nonr enew-
able sources of energy: oil, nat ur al gas, and coal by
cal cul at i ng
a. the rat i o of existing worl d total gas reserves to the
worl d total oil reserves.
b. t he rat i o of existing worl d total coal reserves to t he
worl d total oil reserves.
c. t he per cent share of amount of each energy source i n
total reserves of all t hree sources.
REFERENCES
[1] Speight, J. G, The Chemistry and Technology of Petroleum, 3rd
ed., Marcel Dekker, New York, 1998.
[2] Jahn, F., Cook, M., and Graham, M., Hydrocarbon Exploration
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Elsevier, Amsterdam, 1998.
[3] Oil and Gas Journal Data Book, 2000 edition, PennWell
Publishing, Tulsa, OK, 2000.
[4] International Petroleum Encyclopedia, Volume 32, PennWell
Publishing, Tulsa, OK, 1999.
[5] BP Statistical Review of World Energy, British Petroleum,
United Kingdom, June 1988.
[6] Basic Petroleum Data Book, Petroleum Industry Statistics,
Volume XVIII, Number i, American Petroleum Institute (API),
Washington, DC, 1988.
[7] The International Union of Pure and Applied Chemistry
(IUPAC), 2003, Research Triangle Park, NC, URL:
http://www.iupac.org.
[8] Nelson, W. L., Petroleum Refinery Engineering, McGraw-Hill,
New York, 1958.
[9] Manning, F. S. and Thompson, R. R., Oilfield Processing,
PennWell Publishing, Tulsa, OK, 1995.
[10] Altagelt, K. H. and Boduszynski, M. M., Composition and
Analysis of Heavy Petroleum Fractions, Marcel Dekker, New
York, 1994.
[11] McCain, Jr., W. D., The Properties of Petroleum Fluids, 2nd ed.,
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Estimation of Thermophysical Properties of Hydrocarbons and
Their Mixtures," Industrial and Engineering Chemistry
Research, Vol. 40, No. 8, 2001, pp. 1975-1984.
[32] Riazi, M. R., Prediction of Thermophysical Properties of
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[33] Riazi, M. R. and Al-Sahhaf, T., "Physical Properties of
n-Alkanes and n-Alkyl Hydrocarbons: Application to Petroleum
Mixtures," Industrial and Engineering Chemistry Research,
Vol. 34, 1995, pp. 4145-4148.
[34] Daubert, T. E., "Property Predictions," Hydrocarbon Processing,
Vol. 59, No. 3, 1980, pp. 107-112.
[35] Riazi, M. R. and Roomi, Y. A., "Compositional Analysis of
Petroleum Fractions," Preprints of Division of Petroleum
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[37] Mattax, C. C. and Dalton, R. L., Reservoir Simulation, SPE
Nomograph 13, Society of Petroleum Engineers, Richardson,
TX, 1990.
[38] Riazi, M. R. and Faghri, A., "Effect of the Interfacial Drag on
Gas Absorption with Chemical Reaction in a Vertical Tube,"
American Institute of Chemical Engineers Journal,
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[39] Dabir, B., Riazi, M. R., and Davoudirad, H. R., "Modeling of
Falling Film Reactors," Chemical Engineering Science, Vol. 51,
No. 11, 1996, pp. 2553-2558.
[40] Riazi, M. R., Whitson, C. H., and da Silva, E, "Modelling of
Diffusional Mass Transfer in Naturally Fractured Reservoirs,"
Journal of Petroleum Science and Engineering, Vol. 10, 1994,
pp. 239-253.
[41] Riazi, M, R. and Whitson, C. H., "Estimating Diffusion
Coefficients of Dense Fluids," Industrial and Engineering
Chemistry Research, Vol. 32, No. 12, 1993, pp. 3081-3088.
[42] Riazi, M. R., "A New Method for Experimental Measurement of
Diffusion Coefficient in Reservoir Fluids," Journal of Petroleum
Science and Engineering, Vol. 14, 1996, pp. 235-250.
[43] Kister, H. Z., Distillation Operations, McGraw-Hill, New York,
1990.
[44] Villoria, C. M., Anselmi, A. E., Intevep, S. A., and Garcia, E R.,
"An Oil Spill Fate Model," SPE23371, Society of Petroleum
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Volume 20, Society of Petroleum Engineers, Richardson, TX,
2000.
[46] Firoozabadi, A., Thermodynamics of Hydrocarbon Reservoirs,
McGraw-Hill, New York, 1999.
[47] Daubert, T. E. and Danner, R. E, Eds., API Technical Data
Book--Petroleum Refining, 6th ed., American Petroleum
Institute (API), Washington, DC, 1997.
[48] AIChE DIPPR| Design Institute for Physical
Property Data (DIPPR), EPCON International, Houston, TX,
1996.
[49] Rowley, R. L., Wilding, W. V., Oscarson, J. L., Zundel, N. A.,
Marshall, T. L., Daubert, T. E., and Danner, R. P., DIPPR Data
Compilation of Pure Compound Properties, Design Institute for
Physical Properties (DIPPR), Taylor & Francis, New York, 2002
(http://dippr.byu.edu).
[50] Thermodynamic Research Center, National Institute of
Standards and Technology (NIST), Boulder, CO, URL:
http://www.nist.gov/.
[51] Riazi, M. R. and Daubert, T. E., "Analytical Correlations
Interconvert Distillation Curve Types," Oil and Gas Journal,
August 25, 1986, pp. 50-57.
[52] Riazi, M. R. and Daubert, T. E., "Improved Characterization of
Wide Boiling Range Undefined Petroleum Fractions," Industrial
and Engineering Chemistry Research, Vol. 26, 1987, pp. 629-632.
[53] Riazi, M. R. and Daubert, T. E., "Predicting Flash Points and
Pour Points of Petroleum Fractions," Hydrocarbon Processing,
September 1987, pp. 81-84.
[54] Riazi, M. R., "A Distribution Model for C7+ Fractions
Characterization of Petroleum Fluids," Industrial and
Engineering Chemistry Research, Vol. 36, 1997, pp. 4299-4307.
[55] Riazi, M. R., "Distribution Model for Properties of
Hydrocarbon-Plus Fractions," Industrial and Engineering
Chemistry Research, Vol. 28, 1989, pp. 1731-1735.
[56] Riazi, M. R. and Al-Sahhaf, T. A., "Physical Properties of Heavy
Petroleum Fractions and Crude Oils," Fluid Phase Equilibria,
Vol. 117, 1996, pp. 217-224.
[57] Watson, IC M., Nelson, E. E, and Murphy, G. B.,
"Characterization of Petroleum Fractions," Industrial and
Engineering Chemistry, Vol. 27, 1935, pp. 1460-1464.
[58] Brule, M. R., Kumar, K. H., and Watanasiri, S.,
"Characterization Methods Improve Phase Behavior
Predictions," Oil & Gas Journal, February 11, 1985, pp. 87-93.
[59] Riazi, M. R., A1-Sahhaf, T. A., and Al-Shammari, M. A.,
"A Generalized Method for Estimation of Critical Constants,"
Fluid Phase Equilibria, Vol. 147, 1998, pp. 1-6.
[60] Dohrn, R. and Pfohl, O. "Thermophysical
Properties--Industrial Directions," Paper presented at the
Ninth International Conference on Properties and Phase
Equilibria for Product and Process Design (PPEPPD 2001),
Kurashiki, Japan, May 20-25, 2001.
[61] McCabe, W. L., Smith, J. C., and Harriot, P., Unit Operations of
Chemical Engineering, 4th ed., McGraw-Hill, New York, 1985.
[62] Peridis, S., Magoulas, K., and Tassios, D., "Sensitivity of
Distillation Column Design to Uncertainties in Vapor-Liquid
Equilibrium Information," Separation Science and Technology,
Vol. 28, No. 9, 1993, pp. 1753-1767.
[63] Riazi, M. R. and Mansoori, G. A., "Simple Equation of State
Accurately Predicts Hydrocarbon Densities," Oil and Gas
Journal, July 12, 1993, pp. 108-111.
[64] Lee, B. I. and Kesler, M. G., "A Generalized Thermodynamic
Correlation Based on Three- Parameter Corresponding States,"
American Institute of Chemical Engineers Journal, Vol. 21, No. 5,
1975, pp. 510-527.
[65] Riazi, M. R. and Daubert, T. E., "Application of Corresponding
States Principles for Prediction of Self-Diffusion Coefficients in
Liquids," American Institute of Chemical Engineers Journal,
Vol. 26, No. 3, 1980, pp. 386-391.
[66] Riazi, M. R. and Mansoori, G. A., "Use of the Velocity of Sound
in Predicting the PVT Relations," Fluid Phase Equilibria, Vol. 90,
1993, pp. 251-264.
[67] Shabani, M. R., Riazi, M. R., and Shabau, H. I., "Use of Velocity
of Sound in Predicting Thermodynamic Properties from Cubic
Equations of State," Canadian Journal of Chemical Engineering,
Vol. 76, 1998, pp. 281-289.
[68] Riazi, M. R. and Roomi, Y. A., "Use of Velocity of Sound in
Estimating Thermodynamic Properties of Petroleum
Fractions,"Preprints of Division of Petroleum Chemistry
Symposia, American Chemical Society (ACS), August 2000,
Vol. 45, No. 4, pp. 661-664.
[69] Riazi, M. R. and Faghri, A., "Prediction of Thermal
Conductivity of Gases at High Pressures," American Institute of
Chemical Engineers Journal, Vol.31, No.l, 1985, pp. 164-167.
[70] Riazi, M. R. and Faghri, A., "Thermal Conductivity of Liquid
and Vapor Hydrocarbon Systems: Pentanes and Heavier at Low
Pressures," Industrial and Engineering Chemistry, Process
Design and Development, Vol. 24, No. 2, 1985, pp. 398-401.
[71] Ahmed, T., Hydrocarbon Phase Behavior, Gulf Publishing,
Houston, TX, 1989.
[72] Danesh, A., PVT and Phase Behavior of Petroleum Reservoir
Fluids, Elsevier, Amsterdam, 1998.
[73] Tsonopoulos, C., Heidman, J. L., and Hwang, S.-C.,
Thermodynamic and Transport Properties of Coal Liquids, An
Exxon Monograph, Wiley, New York, 1986.
[74] National Institute of Standards and Technology (NIST),
Boulder, CO, 2003 (http://webbook.nist.gov/chemistry/).
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Houston, TX, 2000.
[76] Edmister, W. C. and Lee, B. I., Applied Hydrocarbon
Thermodynamics, 2nd ed., Gulf Publishing, Houston, TX, 1985.
[77] HYSYS, "Reference Volume 1., Version 1.1", HYSYS Reference
Manual for Computer Software, HYSYS Conceptual Design,
Hyprotech Ltd., Calgary, Alberta, Canada, 1996.
1. I NT R ODUCT I ON 29
[78] EPCON, "API Tech Database Software," EPCON International,
Houston, TX, 2000 (www.epcon.com).
[79] Stange, E. and Johannesen, S. O., "HYPO*S, A Program for
Heavy End Characterization," An Internal Program from the
Norsk Hydro Oil and Gas Group (Norway), Paper presented at
the 65th Annual Convection, Gas Processors Association, San
Antonio, TX, March 10, 1986.
[80] Aspen Plus, "Introductory Manual Software Version," Aspen
Technology, Inc., Cambridge, MA, December 1986.
[81] PRO/II, "Keyword Manual," Simulation Sciences Inc.,
Fullerton, CA, October 1992.
[82] Riazi, M. R., "Estimation of Physical Properties and
Composition of Hydrocarbon Mixures," Analytical Advartces
for Hydrocarbon Research, edited by Hsu, C. Samuel, Kluwer
Academic/Plenum Publishers, New York, NY, 2003, pp. 1-26.
[83] Riazi, M. R. and A1-Enezi, G., "A Mathematical Model for the
Rate of Oil Spill Disappearance from Seawater for Kuwaiti
Crude and Its Products," Chemical Engineering Journal, Vol. 73,
1999, pp. 161-172.
[84] Riazi, M. R. and Edalat, M., "Prediction of the Rate of Oil
Removal from Seawater by Evaporation and Dissolution,"
pp. 291-300.
[85] Fingas, M. E, "A Literature Review of the Physics
and Predictive Modeling of Oil Spill Evaporation,"
Journal of Hazardous Materials, Vol. 42, 1995, pp. 157-
175.
[86] Winn, E W., "Physical Properties by Nomogram," Petroleum
Refiners, Vol. 36, No. 21, 1957, pp. 157.
[87] Coplen, T. B., "Atomic Weights of the Elements 1999," Pure and
Applied Chemistry, Vol. 73, No. 4, 2001, pp. 667-683.
[88] Maddox, R. N., Gas Conditioning and Processing, Volume 4,
Campbell Petroleum Series, John M. Campbell, Norman, OK,
1985.
[89] Riazi, M. R., "Estimation of Rates and Enhancement Factors in
Gas Absorption with Zero-Order Reaction and Gas Phase Mass
Transfer Resistances," Chemical Engineering Science, Vol. 41,
No. 11, 1986, pp. 2925-2929.
MNL50-EB/Jan. 2005
Characterization and
Properties of Pure
Hydrocarbons
N OMEN C L A T UR E
API API gravity defined in Eq. (2.4)
A, B Paramet ers in a potential energy relation
A, B Paramet ers in a t wo-paramet er cubic
equation of state
a, b, . . . i Correlation constants in various
equations
CH Carbon-to-hydrogen weight ratio
d20 Liquid density at 20~ and 1 atm, g/ cm 3
dc Critical density defined by Eq. (2.9), g/ cm 3
F I nt ermol ecul ar force
I Refractive index paramet er defined in
Eq. (2.36)
Kw Watson (UOP) K factor defined by
Eq. (2.13)
In Natural logarithm (base e)
lOgl0 Logari t hm to the base 10
M Molecular weight, g/rnol (kg/krnol)
n Sodi um D line refractive index of
liquid at 20~ and l at m, dimensionless
NA Avogadro's number
Nc Carbon number ( number of carbon
at oms in a hydrocarbon molecule)
Pc Critical pressure, bar
pvap Vapor (saturation) pressure, bar
R Universal gas constant, 8.314 J/ mol.K
/~ Refractivity intercept defined in Eq. (2.14)
Rm Molar refraction defined in Eq. (2.34),
crna/g
R 2 R squared, defined in Eq. (2.136)
r Distance between molecules
SG Specific gravity of liquid substance at
15.5~ (60~ defined by Eq. (2.2),
dimensionless
SGg Specific gravity of gas substance at 15.5~
(60~ defined by Eq. (2.6), dimensionless
Tb Boiling point, K
Tc Critical t emperat ure, K
TF Flash point, K
TM Melting (freezing point) point, K
V Molar volume, cm3/ gmol
V Saybolt universal viscosity, SUS
Vc Critical volume (molar), cm3/ mol
(or critical specific volume, cm3/g)
VGC Viscosity gravity constant defined by
Eq. (2.15)
Zc Critical compressibility factor defined by
Eq. (2.8), dimensionless
Greek Letters
P Potential energy defined in Eq. (2.19)
Polarizability defined by Eq. (2.33), cm3/ mol
e Energy par amet er in a potential energy relation
# Absolute ( dynamic) viscosity, cp [mPa.s]. Also
used for dipole moment
u Kinematic viscosity defined by Eq. (2.12), cSt
[mm2/s]
0 A propert y of hydrocarbon such as M , To, Pc, Vc, I,
d, Tb . . . .
p Density at a given t emperat ure and pressure,
g]cm 3
a Surface tension, dyn/ cm ( = raN/ m)
a Size paramet er in potential energy relation
w Acentric factor defined by Eq. (2.10),
dimensionless
Superscript
~ Properties of n-alkanes from Twu correlations
Subscripts
A Aromatic
N Naphthenic
P Paraffinic
T Value of a propert y at t emperat ure T
o A reference state for T and P
ec Value of a propert y at M -~ e~
20 Value of a propert y at 20~
38(100) Value of kinematic viscosity at 38~ (100~
99(210) Value of kinematic viscosity at 99~ (210~
Ac r onyms
%AAD Average absolute deviation percentage defined by
Eq. (2.135)
API-TDB American Petroleum I nstitute--Technical Data
Book
%D Absolute deviation percentage defined by
Eq. (2.134)
EOS Equation of state
IUPAC I nternational Union of Pure and Applied
Chemistry
%MAD Maxi mum absolute deviation percentage
NI ST National I nstitute of Standards and Technology
RK Redlich-Kwong
vdW van der Waals
R 2 R squared, Defined in Eq. (2.136)
30
2. CHARACTERI ZATI ON AND PROPERTI ES OF PURE HYDROCARBONS 31
As DISCUSSED IN CHAPTER 1, the characterization of petroleum
fractions and crude oils depends on the characterization and
properties of pure hydrocarbons. Calculation of the prop-
erties of a mixture depends on the properties of its con-
stituents. In this chapter, first basic parameters and properties
of pure compounds are defined. These properties are either
temperature-independent or values of some basic properties
at a fixed temperature. These parameters are the basis for
calculation of various physical properties discussed in this
book. Reported values of these parameters for more than
100 selected pure compounds are given in Section 2.2. These
values will be used extensively in the following chapters, es-
pecially in Chapters 3 to determine the quality and properties
of petroleum fractions. In Section 2.3, the characterization of
hydrocarbons is introduced, followed by the development of
a generalized correlation for property estimation that is a
unique feature of this chapter. Various correlations and meth-
ods for the estimation of these basic parameters for pure hy-
drocarbons and narrow boiling range petroleum fractions are
presented in different sections. Finally, necessary discussion
and recommendations for the selection of appropriate pre-
dictive methods for various properties are presented.
2.1 DEFINITION OF BASIC PROPERTIES
In this section, all properties of pure hydrocarbons presented
in Section 2.2 are defined. Some specific characteristics of
petroleum products, such as cetane index and pour point, are
defined in Chapter 3. Definitions of general physical proper-
ties such as thermal and transport properties are discussed in
corresponding chapters where their estimation methods are
presented.
The units and definition of molecular weight or molar mass,
M, was discussed in Section 1.7.8. The molecular weight of a
pure compound is determined from its chemical formula and
the atomic weights of its elements. The atomic weights of the
elements found in a petroleum fluid are C = 12.011, H = 1.008,
S = 32.065, O = 16.0, and N = 14.01, as given by the IUPAC
standard [ 1 ]. As an example, the molecular weight of methane
(CH4) is calculated as 12.011 + 4 x 1.008 = 16.043 kg/kmol or
16.043 g/tool (0.01604 kg/mol) or 16.043 lb/lbmol. Molecular
weight is one of the characterization parameters for hydro-
carbons.
2.1.2 Boiling Point
The boiling point of a pure compound at a given pressure
is the temperature at which vapor and liquid exist together
at equilibrium. If the pressure is 1 atm, the boiling point is
called the normal boiling point. However, usually the term
boiling point, Tb, is used instead of normal boiling point and
for other pressures the term saturation temperature is used.
In some cases, especially for heavy hydrocarbons in which
thermal cracking may occur at high temperatures, boiling
points at pressures other than atmospheric is specified. Boil-
ing points of heavy hydrocarbons are usually measured at 1,
10, or 50 mm Hg. The conversion of boiling point from low
pressure to normal boiling point requires a vapor pressure re-
lation and methods for its calculation for petroleum fractions
are discussed in Chapter 3. The boiling point, when available,
is one of the most important characterization parameters for
hydrocarbons and is frequently used in property estimation
methods.
Density is defined as mass per unit volume of a fluid. Density
is a state function and for a pure compound depends on both
temperature and pressure and is shown by p. Liquid densities
decrease as temperature increases but the effect of pressure
on liquid densities at moderate pressures is usually negligible.
At low and moderate pressures (less than a few bars), satu-
rated liquid density is nearly the same as actual density at the
same temperature. Methods of the estimation of densities of
fluids at various conditions are discussed in Chapters 5 and 7.
However, liquid density at the reference conditions of 20~
(293 K) and 1 atm is shown byd and it is used as a characteri-
zation parameter in this chapter as well as Chapter 3. Parame-
ter d is also called absolute density to distinguish from relative
density. Other parameters that represent density are specific
volume (l/d), molar volume (M/d), and molar density (d/M).
Generally, absolute density is used in this book as the charac-
teristic parameter to classify properties of hydrocarbons.
Liquid density for hydrocarbons is usually reported in
terms of specific gravity (SG) or relative density defined as
density of liquid at temperature T
(2.1) SG =
density of water at temperature T
Since the standard conditions adopted by the petroleum in-
dustry are 60 ~ F (15.5 ~ C) and 1 atm, specific gravities of liquid
hydrocarbons are normally reported at these conditions. At a
reference temperature of 60~ (15.5~ the density of liquid
water is 0.999 g/cm 3 (999 kg/m 3) or 8.337 lb/gal(U.S.). There-
fore, for a hydrocarbon or a petroleum fraction, the specific
gravity is defined as
(2.2) SG (60~176 -- density of liquid at 60~ in g/cm 3
0.999 g/cm 3
Water density at 60~ is 0.999 or almost 1 g/cm3; therefore,
values of specific gravities are nearly the same as the density
of liquid at 15.5~ (289 K) in g/cm 3 . The Society of Petroleum
Engineers usually uses y for the specific gravity and in some
references it is designated by S. However, in this book SG de-
notes the specific gravity. Since most of hydrocarbons found
in reservoir fluids have densities less than that of water, spe-
cific gravities of hydrocarbons are generally less than 1. Spe-
cific gravity defined by Eq. (2.2) is slightly different from the
specific gravity defined in the SI system as the ratio of the den-
sity of hydrocarbon at 15~ to that of water at 4~ designated
byd 15. Note that density of water at 4~ is exactly I g/cm 3 and
therefore d 15 is equal to the density of hydrocarbon at 15~
in g/cm 3 . The relation between these two specific gravities is
approximately given as follows:
(2.3) SG = 1.001 d4 is
In this book specific gravity refers to SG at 60~176 (15.5~
In the early years of the petroleum industry, the American
32 C H A R A C T E R I Z A T I O N A N D P R O P E R T I E S OF P E T R O L E U M F R A C T I O N S
Petroleum Institute (API) defined the API gravity (degrees
API) to quantify the quality of petroleum products and crude
oils. The API gravity is defined as [2]
141.5
(2.4) API gravity - SG (at 60~ - 131.5
Degrees API was derived from the degrees Baum6 in which
it is defined in terms of specific gravity similar to Eq. (2.4)
except numerical values of 140 and 130 were used instead
of 141.5 and 131.5, respectively. Liquid hydrocarbons with
lower specific gravities have higher API gravity. Aromatic hy-
drocarbons have higher specific gravity (lower API gravity)
than do paraffinic hydrocarbons. For example, benzene has
SG of 0.8832 (API of 28.72) while n-hexane with the same car-
bon number has SG of 0.6651 (API gravity of 81.25). A liquid
with SG of i has API gravity of 10. Once Eq. (2.4) is reversed it
can be used to calculate specific gravity from the API gravity.
141.5
(2.5) SG =
API gravity + 131.5
The definition of specific gravity for gases is somewhat dif-
ferent. It is defined as relative density of gas to density of air at
standard conditions. In addition, density of gases is a strong
function of pressure. Since at the standard conditions (15.5 ~ C
and 1 arm) the density of gases are estimated from the ideal
gas law (see Chapter 5), the specific gravity of a gas is pro-
portional to the ratio of molecular weight of gas (Mg) to the
molecular weight of air (28.97).
(2.6) SGg-- Mg
28.97
Therefore, to obtain the specific gravity of a gas, only its
molecular weight is needed. For a mixture, Mg can be deter-
mined from the gas composition, as discussed in Chapter 3.
2. 1. 4 Re f r a c t i ve I n d e x
Refractive index or refractivity for a substance is defined as
the ratio of velocity of light in a vacuum to the velocity of
light in the substance (fluid) and is a dimensionless quantity
shown by n:
velocity of light in the vacuum
(2.7) n =
velocity of light in the substance
In other words, when a light beam passes from one substance
(air) to another (a liquid), it is bent or refracted because of
the difference in speed between the two substances. In fact,
refractive index indicates the degree of this refraction. Refrac-
tive index is a state function and depends on the temperature
and pressure of a fluid. Since the velocity of light in a fluid is
less than the velocity of light in a vacuum, its value for a fluid
is greater than unity. Liquids have higher values of refractive
index than that of gases. For gases the values of refractive
index are very close to unity.
All frequencies of electromagnetic radiation (light) travel
at the same speed in vacuum (2.998 x l0 s m/s); however, in a
substance the velocity of light depends on the nature of the
substance (molecular structure) as well as the frequency of
the light. For this reason, standard values of refractive index
must be measured at a standard frequency. Usually the refrac-
tive index of hydrocarbons is measured by the sodium D line
at 20~ and 1 atm. The instrument to measure the refractive
index is called a refractometer and is discussed in Chapter 3.
In some references the values of refractive index are reported
at 25~ however, in this book the refractive index at 20~ and
1 arm is used as a characterization parameter for hydrocar-
bons and petroleum fractions. As is shown in this chapter and
Chapter 3, refractive index is a very useful characterization
parameter for pure hydrocarbons and petroleum fractions,
especially in relation with molecular type composition. Val-
ues of n vary from about 1.3 for propane to 1.6 for some
aromatics. Aromatic hydrocarbons have generally higher
n values than paraffinic compounds as shown in Table 2.1.
2 . 1 . 5 Cr i t i c a l C o n s t a n t s (T~, Pc, V~, Zc)
The critical point is a point on the pressure-volume-
temperature diagram where the saturated liquid and satu-
rated vapor are identical and indistinguishable. The temper-
ature, pressure, and volume of a pure substance at the critical
point are called critical t emperat ure (To), critical pressure (Pc),
and critical v ol ume (Vc), respectively. In other words, the crit-
ical temperature and pressure for a pure compound are the
highest temperature and pressure at which the vapor and liq-
uid phase can coexist at equilibrium. In fact, for a pure com-
pound at temperatures above the critical temperature, it is
impossible to liquefy a vapor no matter how high the pres-
sure is. A fluid whose temperature and pressure are above
the critical point is called supercritical fluid. For pure com-
pounds, critical temperature and pressure are also called true
critical temperature and true critical pressure. However, as
will be discussed in Chapter 3, pseudocri t i cal properties are
defined for mixtures and petroleum fractions, which are dif-
ferent from true critical properties. Pseudocritical properties
are important in process calculations for the estimation of
thermophysical properties of mixtures.
The critical compressibility factor, Z~, is defined from To, Pc,
and Vc according to the general definition of compressibility
factor.
PcVc
(2.8) Zc =
RTc
where R is the universal gas constant. According to Eq. (2.8),
Zc is dimensionless and Vc must be in terms of molar vol-
ume (i.e., cm3/mol) to be consistent with R values given in
Section 1.7.24. Critical temperature, pressure, and volume
( Tc, Pc, Vc) are called the critical const ant s or critical properties.
Critical constants are important characteristics of pure com-
pounds and mixtures and are used in corresponding states
correlations and equations of state (EOS) to calculate PVT and
many other thermodynamic, physical, and transport proper-
ties. Further discussion on the critical point of a substance
is given in Chapter 5. As was discussed in Section 1.3, the re-
sults of EOS calculations very much depend on the values of
critical properties used. Critical volume may be expressed in
terms of specific critical volume (i.e., ma/kg), molar critical
volume (i.e., ma/kinol), or critical density dc (i.e., kg/m 3) or
critical molar density (i.e., kmol/m3). Critical density is re-
lated to the critical molar volume as
M
(2.9) d~ = - -
vo
2. CHARACT ERI Z AT I ON AND P R OP E R T I E S OF PURE HY DR OCA R B ONS 33
Experimental values of critical properties have been re-
ported for a large number of pure substances. However, for
hydrocarbon compounds, because of thermal cracking that
occurs at higher temperatures, critical properties have been
measured up to C18 [2]. Recently some data on critical proper-
ties of n-alkanes from C19 to C36 have been reported [3]. How-
ever, such data have not yet been universally confirmed and
they are not included in major data sources. Reported data
on critical properties of such heavy compounds are generally
predicted values and vary from one source to another. For ex-
ample, the API-TDB [2] reports values of 768 K and 11.6 bar
for the critical temperature and pressure of n-eicosane, while
these values are reported as 767 K and 11.1 bar by Poling
et al. [4]. Generally, as boiling point increases (toward heav-
ier compounds), critical temperature increases while critical
pressure decreases. As shown in Section 2.2, aromatics have
higher Tc and Pc relative to those of paraffinic compounds
with the same carbon atoms.
2. 1. 6 Acentri c Fact or
Acentric factor is a parameter that was originally defined by
Pitzer to improve accuracy of corresponding state correla-
tions for heavier and more complex compounds [5, 6]. Acen-
tric factor is a defined parameter and not a measurable quan-
tity. It is a dimensionless parameter represented by w and is
defined as
w = - log10 (PI vap) -- 1.0 (2.10)
where
pvap
pvap
= reduced vapor pressure, pvap/Pc, dimensionless
= vapor pressure at T = 0.7 Tc (reduced temperature
of 0.7), bar
Pc -- critical pressure, bar
T = absolute temperature, K
Tc = critical temperature, K
Acentric factor is defined in a way that for simple fluids such
as argon and xenon it is zero and its value increases as the
size and shape of molecule changes. For methane w -- 0.001
and for decane it is 0.489. Values reported for acentric fac-
tor of pure compounds are calculated based on Eq. (2.10),
which depends on the values of vapor pressure. For this rea-
son values reported for the acentric factor of a compound may
slightly vary from one source to another depending on the re-
lation used to estimate the vapor pressure. In addition, since
calculation of the acentric factor requires values of critical
temperature and pressure, reported values for w also depend
on the values of Tc and Pc used.
2. 1. 7 Vapor Pressure
In a closed container, the vapor pressure of a pure compound
is the force exerted per unit area of walls by the vaporized
portion of the liquid. Vapor pressure, pvap, can also be de-
fined as a pressure at which vapor and liquid phases of a
pure substance are in equilibrium with each other. The vapor
pressure is also called saturation pressure, psat, and the cor-
responding temperature is called saturation temperature. In
an open air under atmospheric pressure, a liquid at any tem-
perature below its boiling point has its own vapor pressure
that is less than 1 atm. When vapor pressure reaches 1 atm,
the saturation temperature becomes the normal boiling point.
Vapor pressure increases with temperature and the highest
value of vapor pressure for a substance is its critical pressure
(Pc) in which the corresponding temperature is the critical
temperature (To). When a liquid is open to the atmosphere at
a temperature T in which the vapor pressure of liquid is pvap,
vol% of the compound vapors in the air is
(2.11) vol% = 100 \ Pa ]
where Pa is the atmospheric pressure. Derivation of Eq. (2.11)
is based on the fact that vapor pressure is equivalent to partial
pressure (mole fraction total pressure) and in gases under
low-pressure conditions, mole fraction and volume fraction
are the same. At sea level, where P~ = 1 atm, calculation of
vol% of hydrocarbon vapor in the air from Eq. (2.11) is simply
100 pvap, if pwp is in atm.
Vapor pressure is a very important thermodynamic prop-
erty of any substance and it is a measure of the volatility of
a fluid. Compounds with a higher tendency to vaporize have
higher vapor pressures. More volatile compounds are those
that have lower boiling points and are called light compounds.
For example, propane (Ca) has boiling point less than that of
n-butane (nCa) and as a result it is more volatile. At a fixed
temperature, vapor pressure of propane is higher than that
of butane. In this case, propane is called the light compound
(more volatile) and butane the heavy compound. Generally,
more volatile compounds have higher critical pressure and
lower critical temperature, and lower density and lower boil-
ing point than those of less volatile (heavier) compounds, al-
though this is not true for the case of some isomeric com-
pounds. Vapor pressure is a useful parameter in calculations
related to hydrocarbon losses and flammability of hydrocar-
bon vapor in the air (through Eq. 2.11). More volatile com-
pounds are more ignitable than heavier compounds. For ex-
ample, n-butane is added to gasoline to improve its ignition
characteristics. Low-vapor-pressure compounds reduce evap-
oration losses and chance of vapor lock. Therefore, for a fuel
there should be a compromise between low and high vapor
pressure. However, as will be seen in Chapter 6, one of the
major applications of vapor pressure is in calculation of equi-
librium ratios (Ki values) for phase equilibrium calculations.
Methods of calculation of vapor pressure are given in detail in
Chapter 7. For pure hydrocarbons, values of vapor pressure at
the reference temperature of 100~ (38~ are provided by the
API [2] and are given in Section 2.2. For petroleum fractions,
as will be discussed in Chapter 3, method of Reid is used to
measure vapor pressure at 100~ Reid vapor pressure (RVP)
is measured by the ASTM test method D 323 and it is approx-
imately equivalent to vapor pressure at 100~ (38~ RVP is
a major characteristic of gasoline fuel and its prediction is
discussed in Chapter 3.
2. 1. 8 Ki nemat i c Vi scosi ty
Kinematic viscosity is defined as the ratio of absolute (dy-
namic) viscosity/z to absolute density p at the same temper-
ature in the following form:
(2.12) v = ~-
P
As discussed in Section 1.7.18, kinematic viscosity is ex-
pressed in cSt, SUS, and SFS units. Values of kinematic vis-
cosity for pure liquid hydrocarbons are usually measured and
reported at two reference temperatures of 38~ (100~ and
99~ (210~ in cSt. However, other reference temperatures
of 40~ (104~ 50~ (122~ and 60~ (140~ are also used
to report kinematic viscosities of petroleum fractions. Liq-
uid viscosity decreases with an increase in temperature (see
Section 2.7). Kinematic viscosity, as it is shown in Chapter 3,
is a useful characterization parameter, especially for heavy
fractions in which the boiling point may not be available.
2 . 1 . 9 Fr e e z i n g a n d Me l t i n g P o i n t s
Petroleum and most petroleum products are in the form of a
liquid or gas at ambient temperatures. However, for oils con-
taining heavy compounds such as waxes or asphaltinic oils,
problems may arise from solidification, which cause the oil
to lose its fluidity characteristics. For this reason knowledge
of the freezing point is important and it is one of the ma-
jor specifications of jet fuels and kerosenes. For a pure com-
pound the freezing poi nt is the temperature at which liquid
solidifies at 1 atm pressure. Similarly the melting point, TM,
is the temperature that a solid substance liquefies at 1 atm.
A pure substance has the same freezing and melting points;
however, for petroleum mixtures, there are ranges of melting
and freezing points versus percent of the mixture melted or
frozen. For a mixture, the initial melting point is close to the
melting point of the lightest compound in the mixture, while
the initial freezing point is close to the freezing point (or melt-
ing point) of the heaviest compound in the mixture. Since the
melting point increases with molecular weight, for petroleum
mixtures the initial freezing point is greater than the initial
melting point. For petroleum mixtures an equivalent term of
pour point instead of initial melting point is defined, which
will be discussed in Chapter 3. Melting point is an important
characteristic parameter for petroleum and paraffinic waxes.
2 . 1 . 1 0 Fl a s h Po i n t
Flash point, TF, for a hydrocarbon or a fuel is the mi ni mum
temperature at which vapor pressure of the hydrocarbon is
sufficient to produce the vapor needed for spontaneous igni-
tion of the hydrocarbon with the air with the presence of an
external source, i.e., spark or flame. From this definition, it is
clear that hydrocarbons with higher vapor pressures (lighter
compounds) have lower flash points. Generally flash point
increases with an increase in boiling point. Flash point is
an important parameter for safety considerations, especially
during storage and transportation of volatile petroleum prod-
ucts (i.e., LPG, light naphtha, gasoline) in a high-temperature
environment. The surrounding temperature around a storage
tank should always be less than the flash point of the fuel
to avoid possibility of ignition. Flash point is used as an in-
dication of the fire and explosion potential of a petroleum
product. Estimation of the flash point of petroleum fractions
is discussed in Chapter 3, and data for flash points of some
pure hydrocarbons are given in Table 2.2. These data were ob-
tained using the closed cup apparatus as described in ASTM
D 93 (ISO 2719) test method. There is another method of
measuring flash point known as open cup for those oils with
flash point greater than 80~ (ASTM D 92 or ISO 2592 test
methods). Flash point should not be mistaken with fire point,
which is defined as the mi ni mum temperature at which the
hydrocarbon will continue to burn for at least 5 s after being
ignited by a flame.
2. 1. 11 Au t o i g n i t i o n Te mpe r a t ur e
This is the mi ni mum temperature at which hydrocarbon va-
por when mixed with air can spontaneously ignite without
the presence of any external source. Values of autoignition
temperature are generally higher than flash point, as given
in Table 2.2 for some pure hydrocarbons. Values of autoigni-
tion temperature for oils obtained from mineral sources are
in the range of 150-320~ (300-500~ for gasoline it is about
350~ (660~ and for alcohol is about 500~ (930~ [7].
With an increase in pressure the autoignition temperature
decreases. This is particularly important from a safety point
of view when hydrocarbons are compressed.
2. 1. 12 Fl a mma bi l i t y Ra nge
To have a combustion, three elements are required: fuel (hy-
drocarbon vapor), oxygen (i.e., air), and a spark to initiate the
combustion. One important parameter to have a good com-
bustion is the ratio of air to hydrocarbon fuel. The combustion
does not occur if there is too much air (little fuel) or too lit-
tle air (too much fuel). This suggests that combustion occurs
when hydrocarbon concentration in the air is within a certain
range. This range is called flammability range and is usually
expressed in terms of lower and upper volume percent in the
mixture of hydrocarbon vapor and air. The actual volume per-
cent of hydrocarbon vapor in the air may be calculated from
Eq. (2.1 i) using vapor pressure of the hydrocarbon. If the cal-
culated vol% of hydrocarbon in the air is within the flamma-
bility range then the mixture is flammable by a spark or flame.
2. 1. 13 Oc t a ne Nu mb e r
Octane number is a parameter defined to characterize an-
tiknock characteristic of a fuel (gasoline) for spark ignition
engines. Octane number is a measure of fuel's ability to re-
sist auto-ignition during compression and prior to ignition.
Higher octane number fuels have better engine performance.
The octane number of a fuel is measured based on two ref-
erence hydrocarbons of n-heptane with an assigned octane
number of zero and isooctane (2,2,4-trimethylpentane) with
assigned octane number of 100. A mixture of 70 vol% isooc-
tane and 30 vol% n-heptane has an octane number of 70.
There are two methods of measuring octane number of a fuel
in the laboratory. The methods are known as mot or octane
number (MON) and research octane number (RON). The MON
is indicative of high-speed performance (900 rpm) and is mea-
sured under heavy road conditions (ASTM D 357). The RON
is indicative of normal road performance under low engine
speed (600 rpm) city driving conditions (ASTM D 908). The
third type of octane number is defined as posted octane num-
ber (PON), which is the arithmetic average of the MON and
RON [PON = (MON + RON)/2]. Generally isoparaffins have
higher octane number than do normal paraffins. Naphthenes
have relatively higher octane number than do corresponding
2. C HA R A C T E R I Z A T I ON A N D P R OP E R T I E S OF P UR E HY DR OC A R B ON S 35
paraffins and aromatics have very high octane numbers. The
octane number of a fuel can be improved by adding tetra-
ethyl-lead (TEL) or methyl-tertiary-butyl-ether (MTBE). Use
of lead (Pb) to improve octane number of fuels is limited in
many industrial countries. In these countries MTBE is used
for octane number improvement. However, there are prob-
lems of groundwater contamination with MTBE. MTBE has
MON and RON of 99 and 115, respectively [8]. Lead gener-
ally improves octane number of fuels better than MTBE. The
addition of 0.15 g Pb/L to a fuel of RON around 92 can im-
prove its octane number by 2-3 points. With 0.6 g Pb/L one
may improve the octane number by 10 points [8]. However,
as mentioned above, because of environmental hazards use
of lead is restricted in many North American and West Euro-
pean countries. Values of the octane number measured with-
out any additives are called clear octane number. For pure
hydrocarbons values of clear MON and RON are given in Sec-
tion 2.2. Estimation of the octane number of fuels is discussed
in Chapter 3.
2. 1. 14 Ani l i ne Poi nt
The aniline point for a hydrocarbon or a petroleum fraction is
defined as the mi ni mum temperature at which equal volumes
of liquid hydrocarbon and aniline are miscible. Aniline is an
aromatic compound with a structure of a benzene molecule
where one atom of hydrogen is replaced by the -NH2 group
(C6Hs-NH2). The aniline point is important in characteriza-
tion of petroleum fractions and analysis of molecular type.
As discussed in Chapter 3, the aniline point is also used as a
characterization parameter for the ignition quality of diesel
fuels. It is measured by the ASTM D 611 test method. Within
a hydrocarbon group, aniline point increases with molecu-
lar weight or carbon number, but for the same carbon num-
ber it increases from aromatics to paraffinic hydrocarbons.
Aromatics have very low aniline points in comparison with
paraffins, since aniline itself is an aromatic compound and it
has better miscibility with aromatic hydrocarbons. Generally,
oils with higher aniline points have lower aromatic content.
Values of the aniline point for pure hydrocarbons are given
in Table 2.2, and its prediction for petroleum fractions is dis-
cussed in Chapter 3.
2. 1. 15 Wat s on K
Since the early years of the petroleum industry it was desired
to define a characterization parameter based on measurable
parameters to classify petroleum and identify hydrocarbon
molecular types. The Watson characterization factor denoted
by Kw is one of the oldest characterization factors originally
defined by Watson et al. of the Universal Oil Products (UOP)
in mid 1930s [9]. For this reason the parameter is sometimes
called UOP characterization factor and is defined as
(1.8Tb) 1/3
(2.13) Kw =
SG
where
Tb = normal boiling point K
SG = specific gravity at 15.5~
In the original definition of Kw, boiling point is in degrees
Rankine and for this reason the conversion factor of 1.8
is used to have Tb in the SI unit. For petroleum fractions
Tb is the mean average boiling point (also see Chapter 3). The
purpose of definition of this factor was to classify the type
of hydrocarbons in petroleum mixtures. The naphthenic hy-
drocarbons have Kw values between paraffinic and aromatic
compounds. In general, aromatics have low Kw values while
paraffins have high values. However, as will be discussed in
Chapter 3 there is an overlap between values of Kw from dif-
ferent hydrocarbon groups. The Watson K was developed in
1930s by using data for the crude and products available in
that time. Now the base petroleum stocks in general vary sig-
nificantly from those of 1930s [10, 11]. However, because it
combines two characterization parameters of boiling point
and specific gravity it has been used extensively in the devel-
opment of many physical properties for hydrocarbons and
petroleum fractions [2, 11, 12].
2. 1. 16 Ref ract i vi t y I nt e r c e pt
Kurtz and Ward [13] showed that a plot of refractive index
against density for any homologous hydrocarbon group is
linear. For example, plot of refractive index of n-paraffins ver-
sus density (d20) in the carbon number range of C5-C45 is a
straight line represented by equation n = 1.0335 + 0.516d20,
with R E value of 0.9998 (R 2 = I, for an exact linear relation).
Other hydrocarbon groups show similar performance with
an exact linear relation between n and d. However, the inter-
cept for various groups varies and based on this observation
they defined a characterization parameter called ref ract i vi t y
i nt ercept , R~, in the following form:
d
(2.14) Ri = n - -
2
where n and d are refractive index and density of liquid hy-
drocarbon at the reference state of 20~ and I atm in which
density must be in g/cm 3 . Ri is high for aromatics and low for
naphthenic compounds, while paraffins have intermediate/~
values.
2. 1. 17 Vi s cos i t y Gravi ty Cons t ant
Another parameter defined in the early years of petroleum
characterization is the v i s c os i t y gr avi t y c o n s t a n t (VGC). This
parameter is defined based on an empirical relation developed
between Saybolt viscosity (SUS) and specific gravity through
a constant. VGC is defined at two reference temperatures of
38~ (100~ and 99~ (210~ as [14]
10SG - 1.0752 log10(V38 - 38)
(2.15) VGC =
10 - logl0(V3s - 38)
SG - 0.24 - 0.022 log10(V99 - 35.5)
(2.16) VGC =
0.755
where
V38 = viscosity at 38~ (100~ in SUS (Saybolt Universal
Seconds)
V99 = Saylbolt viscosity (SUS) at 99~ (210~
Conversion factors between cSt and SUS are given in Sec-
tion 1.7.18. Equations (2.15) and (2.16) do not give identical
values for a given compound but calculated values are close to
each other, except for very low viscosity oils. Equation (2.16)
is recommended only when viscosity at 38~ (100~ is not
available. VGC varies for paraffinic hydrocarbons from 0.74 to
0.75, for naphthenic from 0.89 to 0.94, and for aromatics from
0.95 to 1.13 [15]. In Chapter 3, VGC along with other param-
eters has been used to estimate the composition of petroleum
fractions. Values of VGC for some hydrocarbons are given in
Table 2.3. The main limitation in use of VGC is that it cannot
be defined for compounds or fractions with viscosities less
than 38 SUS (--~3.6 cSt) at 38~ A graphical method to esti-
mate VGC of petroleum fractions is presented in Chapter 3.
ASTM D 2501 suggests calculation of VGC using specific
gravity and viscosity in mm2/s (cSt) at 40~ v40 in the follow-
ing form:
SG - 0.0664 - 0.1154 logl0(v40 - 5.5)
(2.17) VGC--
0.94 - 0.109 logl0(v40 - 5.5)
Values of VGC calculated from Eq. (2.17) are usually very
close to values obtained from Eq. (2.15). If viscosity at 40~
is available, use of Eq. (2.17) is recommended for calculation
of VGC. Another relation to calculate VGC in metric units was
proposed by Kurtz et al. [16] in terms of kinematic viscosity
and density at 20~ which is also reported in other sources
[17].
(2.18) VGC= d- 0.1384 log10(v20-20) +0. 0579
0.1526[7.14 - logl0(V20 - 20)]
in which d is density at 20~ and 1 atm in g/cm 3 and v20 is the
kinematic viscosity at 20~ in cSt. In this method viscosity
of oil at 20~ must be greater than 20 cSt. However, when
there is a choice Eq. (2.15) should be used for the procedures
described in Chapter 3.
Example 2.1--API RP-42 [ 18] reports viscosity of some heavy
hydrocarbons, 1,1-Di-(alphadecalyl)hendecane (C31H56) is a
naphthenic compound with molecular weight of 428.8 and
specific gravity of 0.9451. The kinematic viscosity at 38~
(100~ is 20.25 cSt. Calculate the viscosity gravity constant
for this compound.
Sol ut i on--Usi ng Eq. (1.17), the viscosity is converted from
cSt to SUS: V3s = 99.5 SUS. Substituting values of V3s and
SG = 0.9451 into Eq. (2.15) gives VGC = 0.917. The VGC may
be calculated from Eq. (2.17) with direct substitution of vis-
cosity in the cSt unit. Assuming there is a slight change in
viscosity from 38 to 40~ the same value of viscosity at 38~
is used for v40. Thus v40 ~ 20.25 cSt (mm2/s) and Eq. (2.17)
gives VGC = 0.915. The small difference between calculated
values of VGC because in Eq. (2.17) viscosity at 40~ must
be used, which is less than the viscosity at 38~ Calculated
VGC is within the range of 0.89-0.94 and thus the hydrocar-
bon must be a naphthenic compound (also see Fig. 3.22 in
Chapter 3). #
general formula of CnHan+2, alkylcyclopentanes or alkylcy-
clohexanes (naphthenes) have formula of C, H2,, and alkyl-
benzenes (aromatics) have formula of CnH2n-6 (n ~ 6). This
shows that at the same carbon number, the atomic ratio of
number of carbon (C) atoms to number of hydrogen (H)
atoms increases from paraffins to naphthenes and aromat-
ics. For example, n-hexane (C6H14), cyclohexane (C6H12), and
benzene (C6H6) from three different hydrocarbon groups all
have six carbon atoms, but have different CH atomic ratios
of 6/14, 6/12, and 6/6, respectively. If CH atomic ratio is mul-
tiplied by the ratio of atomic weights of carbon (12.011) to
hydrogen (1.008), then CH weight ratio is obtained. For ex-
ample, for n-hexane, the CH weight ratio is calculated as
(6/14)x(12.011/1.008) = 5.107, This number for benzene is
11.92. Therefore, CH weight ratio is a parameter that is ca-
pable of characterizing the hydrocarbon type. In addition,
within the same hydrocarbon group, the CH value changes
from low to high carbon number. For example, methane has
CH value of 2.98, while pentane has CH value of 4.96. For
extremely large molecules (M ~ o0), the CH value of all hy-
drocarbons regardless of their molecular type approaches the
limiting value of 5.96. This parameter is used in Section 2.3
to estimate hydrocarbon properties, and in Chapter 3 it is
used to estimate the composition of petroleum fractions. In
some references HC atomic ratio is used as the character-
izing parameter. According to the definition, the CH weight
ratio and HC atomic ratio are inversely proportional. The
limiting value of HC atomic ratio for all hydrocarbon types
is 2.
Another use of CH weight ratio is to determine the quality of
a fossil-type fuel. Quality and the value of a fuel is determined
from its heat of combustion and heating value. Heating value
of a fuel is the amount of heat generated by complete com-
bustion of 1 unit mass of the fuel. For example, n-hexane has
the heating value of 44734 kJ/kg (19232 Btu/lb) and benzene
has the heating value of 40142 kJ/kg (17258 Btu/lb). Calcula-
tion of heating values are discussed in Chapter 7. From this
analysis it is clear that as CH value increases the heating value
decreases. Hydrogen (H2), which has a CH value of zero, has a
heating value more than that of methane (CH4) and methane
has a heating value more than that of any other hydrocarbon.
Heavy aromatic hydrocarbons that have high CH values have
lower heating values. In general, by moving toward lower CH
value fuel, not only do we have better heating value but also
better and cleaner combustion of the fuel. It is for this reason
that the use of natural gas is preferable to any other type of
fuel, and hydrogen is an example of a perfect fuel with zero
CH weight ratio (CH = 0), while black carbon is an example
of the worst fuel with a CH value of infinity. Values of CH for
pure hydrocarbons are given in Section 2.2 and its estimation
methods are given in Section 2.6.3.
2. 1. 18 Carbon-to-Hydrogen Wei ght Rati o
Carbon-to-hydrogen weight ratio, CH weight ratio, is defined
as the ratio of total weight of carbon atoms to the total weight
of hydrogen in a compound or a mixture and is used to
characterize a hydrocarbon compound. As was discussed in
Section 1.1.1, hydrocarbons from different groups have dif-
ferent formulas. For example, alkanes (paraffins) have the
2 . 2 DATA ON B AS I C P R O P E R T I E S
OF S E L E C T E D P U R E H Y D R O C A R B O N S
2.2.1 Sources of Data
There are several sources that provide data for physical prop-
erties of pure compounds. Some of these sources are listed
below.
1. API: Technical Data Book~Petroleum Refining [2]. The first
chapt er of API-TDB compi l es basi c pr oper t i es of mor e t han
400 pur e hydr ocar bons and some nonhydr ocar bons t hat
are i mpor t ant in pet r ol eum refining. For s ome compounds
wher e exper i ment al dat a ar e not available, pr edi ct ed val ues
f r om t he met hods r ecommended by t he API are given.
2. DIPPR: Desi gn Inst i t ut e for Physi cal and Pr oper t y Dat a
[ 19]. The proj ect initially suppor t ed by t he AIChE began in
earl y 1980s and gives vari ous physi cal pr oper t i es for bot h
hydr ocar bon and nonhydr ocar bon compounds i mpor t ant
in t he industry. A comput er i zed versi on of t hi s dat a bank is
pr ovi ded by EPCON [20].
3. TRC Ther modynami c Tabl es - - Hydr ocar bons [21]. The
Ther modynami c Resear ch Cent er ( f or mer l y at t he Texas
A&M University) current l y at t he Nat i onal Inst i t ut e of Stan-
dar ds and Technol ogy (NIST) at Boulder, CO, (http://www.
trc.nist.gov/) in conj unct i on wi t h t he API Resear ch Proj ect
44 [22] has regul arl y publ i shed physi cal and basi c proper-
ties of l arge numbe r of pur e hydr ocar bons.
4. API Resear ch Proj ect 44 [22]. This proj ect sponsor ed by t he
API was conduct ed at Texas A&M University and provi des
physi cal pr oper t i es of sel ect ed hydr ocar bons.
5. API Resear ch Proj ect 42 [18]. This dat a compi l at i on com-
pl et ed in t he 1960s provi des exper i ment al dat a on den-
sity, refract i ve index, viscosity, and vapor pr essur e for
mor e t han 300 hydr ocar bons wi t h car bon numbe r gr eat er
t han Cu.
6. Dor t mund Dat a Bank (DDB) [23]. Thi s proj ect on physi cal
pr oper t i es has been conduct ed at t he University of Olden-
bur g in Germany. DDB cont ai ns exper i ment al dat a f r om
open l i t erat ure on vari ous t her modynami c pr oper t i es of
pur e compounds and some defined mi xt ures. Dat a have
been pr ogr a mme d in a comput er soft ware conveni ent for
ext ract i ng data. Maj ori t y of dat a are on t her modynami c
propert i es, such as vapor - l i qui d equi l i bri um (VLE), activ-
ity coefficients, and excess propert i es. However, dat a on
viscosity, density, vapor pr essur e, t her mal conductivity, and
surface t ensi on have also been compl i ed as ment i oned in
t hei r Web site. Unf or t unat el y t hey have not compi l ed char-
act eri st i c dat a on hydr ocar bons and pet r ol eum fract i ons
i mpor t ant in t he pet r ol eum industry. Also t he dat a on t rans-
por t pr oper t i es are mai nl y for pur e compounds at at mo-
spheri c pressures.
7. The f our t h and fifth edi t i ons of The Properties of Gases
and Liquids [4] also provi de vari ous pr oper t i es for mor e
t han 400 pur e compounds ( hydr ocar bons and nonhydr o-
carbons). However, dat a in t hi s book have been mai nl y
t aken f r om t he TRC Tables [21 ].
8. There are also some free onl i ne sources t hat one may use t o
obt ai n some physi cal pr oper t y dat a. The best exampl e is t he
one provi ded by NIST (http://webbook. nist. gov). Various
uni versi t i es and r esear cher s have also devel oped special on-
line sources for free access to some dat a on physi cal proper-
ties. For exampl e t he Cent er for Resear ch in Comput at i onal
Ther mochemi s t r y (CRCT) of Ecol e Pl oyt echni que Mon-
t real provi des onl i ne cal cul at i onal soft ware at http://www.
crct. polymfl. caJfact/index. php/. The Cent er for Applied
Ther modynami c Studies (CATS) at t he University of I daho
al so provi des soft wares for pr oper t y cal cul at i ons at its web-
site (ht t p: / / www. webpages. ui daho. edu/ ~cat s/ ). G. A. Man-
soori in his per sonal Web site also provi des some onl i ne
sources for physi cal pr oper t y dat a (http://tigger.uic.edu/
~mansoor i / Ther modynami c. Dat a. and. Pr oper t y- ht ml / ) .
I n ma ny occasi ons di fferent sources provi de di fferent val ues
for a par t i cul ar pr oper t y dependi ng on t he ori gi nal sour ce
of data. Cal cul at ed pr oper t i es such as critical const ant s and
acent ri c f act or for compounds heavi er t han C18 shoul d be
t aken wi t h care as in di fferent sources di fferent met hods have
been used to predi ct t hese par amet er s.
2. 2. 2 Propert i es of Sel ect ed Pure Compounds
The basi c pr oper t i es of pur e hydr ocar bons f r om di fferent
gr oups t hat will be used in t he predi ct i ve met hods pr esent ed
in t he following chapt er s are t abul at ed in Tables 2.1 and 2.2.
The basi c pr oper t i es of M, TM, Tb, SG, d20, n20, To, Pc, Vc, Zc,
and ~0 are pr esent ed in Table 2.1. Secondar y pr oper t i es of kine-
mat i c viscosity, API gravity, Kw, vapor pressure, aniline poi nt ,
flash and aut oi gni t i on poi nt s, f l ammabi l i t y range, and oc-
t ane numbe r are given in Table 2.2. Compounds sel ect ed are
mai nl y hydr ocar bons f r om paraffins, napht henes, and aro-
mat i cs t hat const i t ut e crude oil and its product s. However,
some olefinic and nonhydr ocar bons found wi t h pet r ol eum
fluids are also included. Most of t he compounds are f r om ho-
mol ogous hydr ocar bon gr oups t hat are used as model com-
pounds for char act er i zat i on of pet r ol eum fract i ons di scussed
in Chapt er 3. The pr oper t i es t abul at ed are t he basi c prop-
ert i es needed in char act er i zat i on t echni ques and t her mody-
nami c correl at i ons for physi cal pr oper t i es of pet r ol eum frac-
tions. Al t hough t her e ar e separ at e chapt er s for est i mat i on of
density, viscosity, or vapor pressure, t hese pr oper t i es at s ome
reference t emper at ur es are pr ovi ded because of t hei r use in
t he char act er i zat i on met hods given in Chapt er 3 and 4. Ot her
physi cal pr oper t i es such as heat capaci t y or t r ans por t pr op-
ert i es are given in cor r espondi ng chapt ers wher e t he predi c-
tive met hods are discussed. Dat a for mor e t han 100 sel ect ed
compounds are pr esent ed in t hi s sect i on and are l i mi t ed to
C22 mai nl y due to t he l ack of sufficient exper i ment al dat a for
heavi er compounds. Dat a pr esent ed in Tables 2.1 and 2.2 are
t aken f r om t he API-TDB [2, 22]. St andar d met hods of mea-
s ur ement of t hese pr oper t i es are pr esent ed in Chapt er 3.
Exampl e 2. 2- - As s ume l arge a mount of t ol uene is pour ed on
t he gr ound in an open envi r onment at whi ch t he t emper at ur e
is 38~ (100~ Det er mi ne if t he ar ea sur r oundi ng t he liquid
surface is wi t hi n t he f l ammabi l i t y range.
Sol ut i on--From Table 2.2, t he f l ammabi l i t y r ange is 1.2-7.1
vol% of t ol uene vapor. Fr om this table, t he vapor pr essur e of
t ol uene at 38~ is 0.071 bar (0.07 at m). Subst i t ut i ng this vapor
pr essur e value in Eq. (2.11) gives t he value of vol % = 100 x
0.07/1.0 = 7% of t ol uene in t he ai r mi xt ure. Thi s numbe r is
wi t hi n t he f l ammabi l i t y r ange (1.2 < 7 < 7.1) and t her ef or e
t he sur r oundi ng ai r is combust i bl e. r
2. 2. 3 Addi ti onal Dat a on Propert i es
o f He a v y Hydrocarbons
Some dat a on density, refract i ve index, and vi scosi t y of s ome
heavy hydr ocar bons are given in Table 2.3. These dat a are
t aken f r om API RP 42 [18]. Values of P~ and VGC in t he t abl e
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5

2. CHARACTERI ZATI ON AND PROPERT I ES OF PURE HY DROCARBONS 45
are cal cul at ed by Eqs. (2.14) and (2.15) and will be used t o
devel op predi ct i ve met hods for t he composi t i on of heavy frac-
t i ons di scussed in Chapt er 3 (Sect i on 3.5).
2. 3 CHARACTERIZATION
OF HYDROCARBONS
The wor k on char act er i zat i on of pur e hydr ocar bons began in
1933 when Wat son and Nel son for t he first t i me devel oped
t wo empi r i cal chart s rel at i ng mol ecul ar wei ght to ei t her boil-
ing poi nt and Kw or boiling poi nt and API gravi t y [24]. I n
t hese chart s boi l i ng poi nt and specific gravi t y (or API grav-
ity) are used as t he t wo i ndependent i nput par amet er s. Since
t hen t he wor k on char act er i zat i on and met hods of est i mat i on
of basi c pr oper t i es of pur e hydr ocar bons and pet r ol eum frac-
t i ons has cont i nued to t he pr esent time. Met hods devel oped
in 1930s till 1960s wer e mai nl y graphi cal , while wi t h t he use
of comput er, met hods devel oped in 1970s till pr esent t i me are
in t he f or ms of anal yt i cal correl at i ons. The best exampl e of
char t - t ype correl at i ons, whi ch has been in use by t he industry,
is t he Wi nn nomogr a m t hat rel at es mol ecul ar weight, critical
pressure, aniline poi nt , and CH wei ght rat i o t o boiling poi nt
and specific gravi t y [25]. A versi on of Wi nn nomogr aph as
used by t he API [2] is pr esent ed in Fig. 2.12. Some of t he an-
alytical correl at i ons t hat are used in t he i ndust ry are Cavett
[26], Kesl er-Lee [12], Lee- Kesl er [27], Ri azi - Dauber t [28, 29],
Twu [30], and Ri azi - Sahhaf [31]. Most of t hese correl at i ons
use boiling poi nt and specific gravi t y as t he i nput par ame-
t ers to est i mat e par amet er s such as mol ecul ar weight, critical
const ant s, and acent ri c factor. Most recent l y Kor st en [32] has
devel oped a char act er i zat i on scheme t hat uses boiling poi nt
and a par amet er called doubl e- bond equi val ent (DBE) as t he
i nput par amet er s. DBE can be est i mat ed f r om H/C at omi c
ratio. I n anot her paper Kor st en [33] lists and eval uat es vari-
ous correl at i ons for est i mat i on of critical pr oper t i es of pur e
hydr ocar bons. Tsonopol us et al. [34] give t he list of correl a-
t i ons devel oped for char act er i zat i on of coal liquids in t er ms
of boi l i ng poi nt and specific gravity. There are some met h-
ods of est i mat i on of pr oper t i es of pur e compounds t hat are
based on vari ous gr oup cont r i but i on t echni ques. The mos t
accur at e met hods of gr oup cont r i but on for vari ous proper-
ties wi t h necessar y r ecommendat i ons are given in t he fifth
edi t i on of Properties of Gases and Liquids [4]. Even some of
t hese gr oup cont r i but i on met hods requi re pr oper t i es such as
mol ecul ar wei ght or boiling poi nt . Exampl es of such proce-
dures are t he Lydersen and Ambr ose met hods [4]. The pr ob-
l em wi t h gr oup cont r i but i on met hods is t hat t he st r uct ur e of
t he compound mus t be known. For this r eason t hey are not
appr opr i at e for undefi ned pet r ol eum fractions. However, t hey
can be used to pr edi ct pr oper t i es of pur e compounds when
exper i ment al dat a are not avai l abl e (i.e., critical pr oper t i es
of heavy pur e hydr ocar bons) . I n fact, on t hi s basi s t he pr op-
ert i es of hydr ocar bons heavi er t han C18 have been pr edi ct ed
and r epor t ed by t he API [2].
As discussed, duri ng t he past 70 year s ma ny met hods in
t he f or ms of chart s and equat i ons wer e pr oposed to esti-
mat e t he basi c pr oper t i es of hydr ocar bons f r om t he knowl -
edge of t he boiling poi nt and specific gravi t y or t he mol ecul ar
weight. Nearl y all of t hese correl at i ons are empi r i cal in nat ur e
wi t hout any t heoret i cal expl anat i on. Boiling poi nt and spe-
cific gravi t y were used in mos t correl at i ons based on expe-
ri ence and t hei r availability. However, the char act er i zat i on
met hods pr oposed by Riazi and Dauber t [28, 29, 35] are based
on t he t heor y of i nt er mol ecul ar forces and EOS par amet er s
[36]. Al t hough EOS are di scussed in Chapt er 5, t hei r appl i ca-
t i on in t he devel opment of anal yt i cal correl at i ons t o charac-
t eri ze hydr ocar bons are di scussed here. I n t he following par t s
in t hi s sect i on several char act er i zat i on schemes devel oped by
Riazi et al. [28, 29, 31, 35, 37] are pr esent ed al ong wi t h ot her
met hods.
2.3.1 Devel opment of a Generalized Correlation
for Hydrocarbon Properties
Propert i es of a fluid depend on t he i nt er mol ecul ar forces t hat
exist bet ween mol ecul es of t hat fluid [38, 39]. As s ummar i zed
by Pr ausni t z et al. [39] t hese forces are gr ouped i nt o f our cat -
egories. (1) El ect rost at i c forces bet ween char ged mol ecul es
(ions) and bet ween per manent dipoles or hi gher mul t i pol es.
These forces resul t f r om t he chemi cal st r uct ur e of mol ecul es
and are i mpor t ant in pol ar compounds (i.e., water, met hanol ,
et hanol , etc.). (2) I nduct i on forces on mol ecul es t hat are po-
l ari zabl e when subj ect ed t o an electric field f r om pol ar com-
pounds. These forces are also called dipole forces and are de-
t er mi ned by di pol e mome nt of mol ecul es 0z), whi ch is propor-
t i onal to pol ari zabi l i t y factor, ~, and the field st rengt h. These
forces are pr opor t i onal t o/ z 2 x ~. (3) The t hi rd t ype of forces
are at t r act i on (di spersi on forces) and r epul si on bet ween non-
pol ar mol ecul es. These forces, al so called London forces, are
static in nat ur e and ar e pr opor t i onal to u2. (4) The l ast are spe-
cial (chemi cal ) forces l eadi ng to associ at i on or compl ex for-
mat i on such as chemi cal bonds. According to London t hese
forces are additive and except for very pol ar compounds, t he
st rongest forces are of t he London type. For light and me di um
hydr ocar bon compounds, London forces are t he domi nant
force bet ween t he mol ecul es.
The i nt er mol ecul ar force, F, is rel at ed t o t he pot ent i al en-
ergy, F, accor di ng to t he following relation:
dF
(2.19) F --
dr
wher e r is t he di st ance bet ween molecules. The negat i ve of
t he pot ent i al energy, - F( r ) , is t he wor k requi red to sepa-
r at e t wo mol ecul es f r om t he i nt er mol ecul ar di st ance r t o in-
finite separat i on. Equat i on of st at e par amet er s can be esti-
mat ed f r om t he knowl edge of t he pot ent i al energy rel at i on
[39]. Most hydr ocar bon compounds, especially t he light and
me di um mol ecul ar wei ght hydr ocar bons, are consi der ed as
nonpol ar subst ances. There are t wo forces of at t r act i on (dis-
persi on forces) and r epul si on bet ween nonpol ar mol ecul es.
The c ommon convent i on is t hat t he force of at t r act i on is neg-
ative and t hat of r epul si on is positive. As an exampl e, when
mol ecul es of met hane are 1 nm apart , t he force of at t r act i on
bet ween t hem is 2 x 10 -s dyne [39]. The following rel at i on
was first pr oposed by Mie f or t he pot ent i al energy of nonpol ar
mol ecul es [39]:
Ao /3o
(2.20) F -
r n r m
4 6 C HA R A C T E R I Z A T I ON A N D P R OP E R T I E S OF P E T R OL E UM F R A C T I ON S
wher e
N o
Prep -- - -
- - r n
Bo
P a t t = - - -
r m
Prep = r epul si ve pot ent i al
Fat t = at t r act i ve pot ent i al
Ao = p a r a me t e r char act er i zi ng t he r epul si ve f or ce ( >0)
/3o = p a r a me t e r char act er i zi ng t he at t r act i ve f or ce ( >0)
r --- di st ance bet ween mol ecul es
n, m = posi t i ve numbe r s , n > m
The ma i n char act er i s t i cs of a t wo- pa r a me t e r pot ent i al ener gy
f unct i on is t he mi n i mu m val ue of pot ent i al energy, P m i n , desi g-
nat ed by e = - Pmin a nd t he di s t ance bet ween mol ecul es wher e
t he pot ent i al ener gy is zer o (F = 0) whi c h is des i gnat ed by a.
Lo n d o n st udi ed t he t he or y of di sper si on ( at t r act i on) f or ces
a nd has s hown t hat m = 6 a nd it is f r equent l y conveni ent f or
ma t he ma t i c a l cal cul at i ons t o let n = 12. I t can t hen be s hown
t ha t Eq. (2.20) r educes t o t he f ol l owi ng r el at i on k n o wn as
Le n n a r d - J o n e s pot ent i al [39]:
(2.21) P = 48 [ ( ~) 1 2 - ( ~ ) 6 1
I n t he above r el at i on, e is a p a r a me t e r r epr es ent i ng mol ec-
ul ar ener gy a nd a is a pa r a me t e r r epr es ent i ng mol e c ul a r
size. Fur t he r di s cus s i on on i nt er mol ecul ar f or ces is gi ven i n
Sect i on 5.3.
Accor di ng t o t he pr i nci pl e of st at i st i cal t h e r mo d y n a mi c s
t her e exists a uni ver sal EOS t hat is val i d f or all fl ui ds t ha t
follow a t wo- pa r a me t e r pot ent i al ener gy r el at i on s uch as
Eq. (2.21) [40].
(2.22) Z = fl(g, a, T, P)
(2.23) Z - PVr,~
RT
whe r e
Z = di mens i onl es s compr es s i bi l i t y f act or
Vr, e = mo l a r vol ume at abs ol ut e t emper at ur e, T, a nd pr es-
sur e, P
/'1 = uni ver sal f unc t i on s a me f or all fl ui ds t ha t f ol l ow
Eq. (2.21).
By c ombi ni ng Eqs. (2. 20)-(2. 23) we obt ai n
(2.24) Vr, v = f2(Ao, ]3o, T, P)
wher e Ao and/ 3o ar e t he t wo pa r a me t e r s i n t he pot ent i al en-
er gy r el at i on, whi c h di ffer f r o m one fl ui d t o anot her . Equa t i on
(2.24) is cal l ed a t wo- pa r a me t e r EOS. Ear l i er EOS s uc h as van
der Waal s (vdW) a nd Re dl i c h- Kwong (RK) devel oped f or si m-
pl e fl ui ds all have t wo pa r a me t e r s A a nd B [4] as di s cus s ed i n
Chapt er 5. Ther ef or e, Eq. (2.24) can al so be wr i t t en i n t er ms
of t hese t wo par amet er s :
(2.25) Vr, p = f 3(A, B, T, P)
The t hr ee f unct i ons f l , f2, a nd f3 i n t he above equat i ons var y
i n t he f o r m a nd style. The condi t i ons at t he cri t i cal poi nt f or
a ny PVT r el at i on ar e [41 ]
(2.26) ( 0 ~ ) = 0
re.Pc
(2.27) k OV2} = 0
Appl i cat i on of Eqs. (2.26) a nd (2.27) t o any t wo- pa r a me t e r
EOS woul d r esul t i n r el at i ons f or cal cul at i on of pa r a me t e r s
A a nd B i n t er ms of Tc and Pc, as s hown i n Chapt er 5. I t s houl d
be not e d t ha t EOS pa r a me t e r s ar e gener al l y des i gnat ed by
l ower case a a nd b, but her e t hey ar e s hown by A a nd B. No-
t at i on a, b, c . . . . ar e us ed f or cor r el at i on pa r a me t e r s i n var i ous
equat i ons i n t hi s chapt er . Appl yi ng Eqs. (2. 26) a n d (2. 27) t o
Eq. (2.25) r esul t s i nt o t he f ol l owi ng t hr ee r el at i ons f or To, Pc,
a nd Vc:
(2.28) Tc = f4(A, B)
(2.29) Pc = f s( A, B)
(2.30) Vc = f6(A, B)
Func t i ons f4, f5, a nd f6 ar e uni ver sal f unct i ons a nd ar e t he
s a me f or all fl ui ds t ha t obey t he pot ent i al ener gy r el at i on ex-
pr es s ed by Eq. (2.20) or Eq. (2.21). I n fact, i f pa r a me t e r s A
a nd B i n a t wo- pa r a me t e r EOS i n t er ms of Tc a nd Pc ar e rear-
r a nge d one can obt a i n r el at i ons f or Tc and Pc i n t er ms of t hese
t wo par amet er s . For exampl e, f or van der Waal s a nd Re dl i c h-
Kwo n g EOS t he t wo pa r a me t e r s A a nd B ar e gi ven i n t er ms
of Tc a nd Pc [21] as s hown i n Chapt er 5. By r e a r r a nge me nt of
t he v d W EOS pa r a me t e r s we get
To= AB-1 Pc = AB -2 Vc = 3B
and f or t he Re dl i c h- Kwong EOS we have
[ (0"0867)5R] 2/3 A2/3B-2/3
r C= k ~ J
[ (0. 0867) 5 R 1 '/3 A 2/3 B -s/3
Pc = L ~ J vc = 3. 847B
Si mi l ar r el at i ons c a n be obt a i ne d f or t he pa r a me t e r s of ot he r
t wo- pa r a me t e r EOS. A gener al i zat i on c a n be ma d e f or t he
r el at i ons bet ween To, Pc, a nd V~ i n t er ms o f EOS pa r a me t e r s
A a nd B i n t he f ol l owi ng f or m:
(2.31) [Tc, Pc, Vc] = aAbB c
wher e pa r a me t e r s a, b, and c ar e t he cons t ant s whi c h di ffer
f or r el at i ons f or To, Pc, and Vc. However, t hese cons t ant s ar e
t he s a me f or each cri t i cal pr ope r t y f or all fl ui ds t ha t f ol l ow
t he s a me t wo- pa r a me t e r pot ent i al ener gy r el at i on. I n a t wo-
pa r a me t e r EOS s uch as vdW or RK, Vc is r el at ed t o onl y one
pa r a me t e r B so t ha t Vc/B is a c ons t a nt f or all c ompounds .
However, f or mul a t i on of Vc t h r o u g h Eq. (2.30) s hows t ha t
Vc mu s t be a f unc t i on of t wo pa r a me t e r s A a nd B. Thi s is
one of t he r eas ons t ha t t wo- pa r a me t e r s EOS ar e not accur at e
ne a r t he cri t i cal r egi on. Fur t he r di s cus s i on on EOS is gi ven
i n Chapt er 5.
To fi nd t he na t ur e of t hese t wo char act er i zi ng pa r a me t e r s
one s houl d r eal i ze t ha t A a nd B i n Eq. (2.31 ) r epr es ent t he t wo
pa r a me t e r s i n t he pot ent i al ener gy rel at i on, s uc h as e a nd c~ i n
Eq. (2.22). These pa r a me t e r s r epr es ent ener gy a nd si ze char -
act er i st i cs of mol ecul es. The t wo pa r a me t e r s t ha t ar e r eadi l y
me a s ur a bl e f or h y d r o c a r b o n s ys t ems ar e t he boi l i ng poi nt ,
Tb, a nd speci fi c gravity, SG; i n fact, Tb r epr es ent s t he ener gy
2. C H A R A C T E R I Z A T I O N A N D P R O P E R T I E S OF P U R E H Y D R O C A R B O N S 47
parameter and SG represents the size parameter. Therefore,
in Eq. (2.31) one can replace parameters A and B by Tb and
SG. However, it should be noted that Tb is not the same as
parameter A and SG is not the same as parameter B, but it
is their combination that can be replaced. There are many
other parameters that may represent A and B in Eq. (2.31).
For example, if Eq. (2.25) is applied at a reference state of To
and P0, it can be written as
(2.32) Vr0,P0 = f3(A, B, To, Po)
where Vro,eo is the molar volume of the fluid at the reference
state. The most convenient reference conditions are tempera-
ture of 20~ and pressure of 1 atm. By rearranging Eq. (2.32)
one can easily see that one of the parameters A or B can be
molar volume at 20~ and pressure of 1 atm [28, 36, 42].
To find another characterization parameter we may con-
sider that for nonpolar compounds the only attractive force
is the London dispersion force and it is characterized by fac-
tor polarizability, a, defined as [38, 39]
(2.33) ct = ~ x x ~,r/2 q- 2/
where
NA = Avogadro's number
M = molecular weight
p = absolute density
n = refractive index
In fact, polarizability is proportional to molar refraction, Rm,
defined as
(2.34) Rrn----- (-~-) X {n2-l'~,tz 2 q- 2`/
M
(2.35) V = - -
p
n2- 1
(2.36) I = - -
n2+2
in which V is the molar volume and I is a characterization
parameter that was first used by Huang to correlate hydrocar-
bon properties in this way [ 10, 42]. By combining Eqs. (2.34)-
(2.36) we get
Rm actual molar volume of molecules
(2.37) I - -
V apparent molar volume of molecules
Rm, the molar refraction, represents the actual molar volume
of molecules, V represents the apparent molar volume and
their ratio, and parameter I represents the fraction of total
volume occupied by molecules. Rm has the unit of molar vol-
ume and I is a dimensionless parameter. Rm/M is the specific
refraction and has the same unit as specific volume. Parame-
ter I is proportional to the volume occupied by the molecules
and it is close to unity for gases (Ig ~ 0), while for liquids it
is greater than zero but less than 1 (0 < /liq <Z 1 ) . Parameter
I can represent molecular size, but the molar volume, V, is a
parameter that characterizes the energy associated with the
molecules. In fact as the molecular energy increases so does
the molar volume. Therefore, both V and I can be used as two
independent parameters to characterize hydrocarbon proper-
ties. Further use of molar refraction and its relation with EOS
parameters and transport properties are discussed in Chap-
ters 5, 6, and 8. It is shown by various investigators that the
ratio of Tb/Tc is a characteristic of each substance, which is
related to either Tr or Tb [36, 43]. This ratio will be used to
correlate properties of pure hydrocarbons in Section 2.3.3.
Equation (2.31) can be written once for Tr in terms of V and I
and once for parameter Tb/Tr Upon elimination of parameter
V between these two relations, a correlation can be obtained
to estimate T~ from Tb and I. Similarly through elimination
of Tc between the two relations, a correlation can be derived
to estimate V in terms of Tb and I [42].
It should be noted that although both density and refrac-
tive index are functions of temperature, both theory and ex-
periment have shown that the molar refraction (Rm = VI ) is
nearly independent of temperature, especially over a narrow
range of temperature [38]. Since V at the reference tempera-
ture of 20~ and pressure of 1 atm is one of the characteriza-
tion parameters, I at 20~ and 1 arm must be the other char-
acterization parameter. We chose the reference state of 20~
and pressure of 1 atm because of availability of data. Simi-
larly, any reference temperature, e.g. 25~ at which data are
available can be used for this purpose. Liquid density and re-
fractive index of hydrocarbons at 20~ and 1 atm are indicated
by d20 and n20, respectively, where for simplicity the subscript
20 is dropped in most cases. Further discussion on refractive
index and its methods of estimation are given elsewhere [35].
From this analysis it is clear that parameter I can be used
as one of the parameters A or B in Eq. (2.31) to represent the
size parameter, while Tb may be used to represent the energy
parameter. Other characterization parameters are discussed
in Section 2.3.2. In terms of boiling point and specific gravity,
Eq. (2.31) can be generalized as following:
(2.38) 0 = aT~SG c
where Tb is the normal boiling point in absolute de-
grees (kelvin or rankine) and SG is the specific gravity at
60 ~ F(15.5 ~ C). Parameter 0 is a characteristic property such as
molecular weight, M, critical temperature, To, critical pres-
sure, Pc, critical molar volume, Vc, liquid density at 20~
d20, liquid molar volume at 20~ and I atm, V20, or refrac-
tive index parameter, I, at 20~ It should be noted that
0 must be a temperature-independent property. As mentioned
before, I at 20~ and 1 atm is considered as a character-
istic parameter and not a temperature-dependent property.
Based on reported data in the 1977 edition of API-TDB, con-
stants a, b, and c were determined for different properties
and have been reported by Riazi and Daubert [28]. The con-
stants were obtained through linear regression of the loga-
rithmic form of Eq. (2.38). Equation (2.38) in its numerical
form is presented in Sections 2.4-2.6 for basic characteri-
zation parameters. In other chapters, the form of Eq. (2.38)
will be used to estimate the heat of vaporization and trans-
port properties as well as interconversion of various distil-
lation curves. The form of Eq. (2.38) for T~ is the same as
the form Nokay [44] and Spencer and Daubert [45] used to
correlate the critical temperature of some hydrocarbon com-
pounds. Equation (2.38) or its modified versions (Eq. 2.42),
especially for the critical properties and molecular weight,
have been in use by industry for many years [2, 8, 34, 46-56].
Further application of this equation will be discussed in
Section 2.9.
One should realize that Eq. (2.38) was developed based on
a two-parameter potential energy relation applicable to non-
polar compounds. For this reason, this equation cannot be
used for systems containing polar compounds such as alco-
hol, water, or even some complex aromatic compounds that
are considered polar. In fact constants a, b, and c given in
Eq. (2.38) were obtained based on properties of hydrocar-
bons with carbon number ranging from Cs to C20. This is
almost equivalent to the molecular weight range of 70-300.
In fact molecular weight of n-C20 is 282, but considering the
extrapolation power of Eq. (2.38) one can use this equation
up to molecular weight of 300, which is roughly equivalent to
boiling point of 370~ (700~ Moreover, experimental data
on the critical properties of hydrocarbons above C~8 were not
available at the time of development of Eq. (2.38). For heav-
ier hydrocarbons additional parameters are required as will
be shown in Section 2.3.3 and Chapter 3. The lower limit for
the hydrocarbon range is C5, because lighter compounds of
C1-C4 are mainly paraffinic and in the gaseous phase at nor-
mal conditions. Equation (2.38) is mainly applied for unde-
fined petroleum fractions that have average boiling points
higher than the boiling point of C5 as will be seen in Chapter
3. Methods of calculation of properties of natural gases are
Exampl e 2. 3--Show that the molecular weight of hydrocar-
bons, M, can be correlated with the boiling point, Tb, and
specific gravity, SG, in the form of Eq. (2.38).
Sol ut i on- - I t was already shown that molar volume at 20~
and l atm, V20, can be correlated to parameters A and B of a
potential energy function through Eq. (2.32) as follows: V20 =
gl(A, B). In fact, parameter V2o is similar in nature to the
critical molar volume, Vr and can be correlated with Tb and
SG as Eq. (2.38): V20 = aTbbSG c. But V20 = M/d2o, where d20
is the liquid density at 20~ and 1 atm and is considered a
size parameter. Since Tb is chosen as an energy parameter
and SG is selected as a size parameter, then both d20 and SG
represent the same parameter and can be combined with the
energy parameter as M = aT~SG r which has the same form
of Eq. (2.38) for 0 = M.
2. 3. 2 Va r i o us Cha r a c t e r i z a t i o n Pa r a me t e r s
f o r Hy d r o c a r b o n S y s t e ms
Riazi and Daubert [29] did further study on the expansion
of the application of Eq. (2.38) by considering various in-
put parameters. In fact instead of Tb and SG we may con-
sider any pair of parameters 0a and 02, which are capable of
characterizing molecular energy and the molecular size. This
means that Eq. (2.31) can be expressed in terms of two pa-
rameters 01 and 02:
(2.39) 0 = aOblO~
However, one should realize that while these two parame-
ters are independent, they should represent molecular energy
and molecular size. For example, the pairs such as (Tb, M)
or (SG, I) cannot be used as the pair of input parameters
(01, 02). Both SG and I represent size characteristics of
molecules and they are not a suitable characterization pair.
In the development of a correlation to estimate the proper-
ties of hydrocarbons, all compounds from various hydrocar-
bon groups are considered. Properties of hydrocarbons vary
from one hydrocarbon type to another and from one carbon
number to another. Hydrocarbons and their properties can
be tabulated as a matrix of four columns with many rows.
Columns represent hydrocarbon families (paraffins, olefins,
naphthenes, aromatics) while rows represent carbon num-
ber. Parameters such as Tb, M, or kinematic viscosity at 38~
(100~ v38(100), vary in the vertical direction with carbon
number, while parameters such as SG, I, and CH vary signif-
icantly with hydrocarbon type. This analysis is clearly shown
in Table 2.4 for C8 in paraffin and aromatic groups and prop-
erties of C7 and C8 within the same group of paraffin family.
As is clearly shown by relative changes in various properties,
parameters SG, I, and CH clearly characterize hydrocarbon
type, while Tb, M, and 1)38(100 ) are good parameters to charac-
terize the carbon number within the same family. Therefore,
a correlating pair should be selected in a way that character-
izes both the hydrocarbon group and the compound carbon
number. A list of properties that may be used as pairs of cor-
relating parameters (01,02) in Eq. (2.39) are given below [29].
(01, 02)
Pairs:
Tc (K), Pc (bar), Vc (cm3/g), M, T b (K), SG, I (20~ CH
(T b, SG), (Tb, I), (Tb, CH), (M, SG), (M, I), (M, CH),
( v 3 8 ( 1 0 0 ) , SG), (v38(100) ,I), ( v 3 8 ( 1 0 0 ) , CH)
The accuracy of Eq. (2.39) was improved by modification of
its a parameter in the following form:
(2.40) 0 = a exp[b0a + c02 " [ - d O l O 2 ] O ~ O f 2
Values of constants a - f in Eq. (2.40) for various param-
eters of 0 and pairs of (01, Oz) listed above are given in
Table 2.5. It should be noted that the constants reported by
Riazi-Daubert have a follow-up correction that was reported
later in the same volume [29]. These constants are obtained
from properties of hydrocarbons in the range of C5-C20 in
TABLE 2.4--Comparison of properties of adjacent members of paraffin family and two families
of C8 hydrocarbons.
Hydrocarbon group Tb, K M v38000), cSt SG 1 CH
Paraffin family
C7Hi6(n-heptane) 371.6 100.2 0.5214 0.6882 0.236 5.21
C8H18(n-octane) 398.8 114.2 0.6476 0.7070 0.241 5.30
% Difference in property +7.3 +14.2 +24.2 +2.7 +2.1 +1.7
Two Families (C8)
Paraffin (n-octane) 398.8 114.2 0.6476 0.7070 0.241 5.30
Aromatic (ethylbenzene) 409.4 106.2 0.6828 0.8744 0.292 9.53
% Difference in property +2.7 -7.0 5.4 +23.7 +21.2 +79.8
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50 C HA R A C T E R I Z A T I ON A N D P R OP E R T I E S OF P E T R OL E UM F R A C T I ON S
a si mi l ar appr oach as t he const ant s of Eq. (2.38) were ob-
tained. Similarly, Eq. (2.40) can be appl i ed t o hydr ocar bon
syst ems in t he mol ecul ar wei ght range of 70-280, whi ch is
appr oxi mat el y equi val ent to t he boi l i ng r ange of 30-350~
("-80-650~ However, t hey ma y be used up t o C22 or mol ec-
ul ar wei ght of 300 ( ~boi l i ng poi nt 370~ wi t h good accuracy.
I n obt ai ni ng t he const ant s, Eq. (2.40) was first conver t ed i nt o
l i near f or m by t aki ng l ogar i t hm f r om bot h side of t he equat i on
and t hen usi ng a l i near regressi on pr ogr am in a spr eadsheet
t he const ant s were det er mi ned. The val ue of R 2 (index of cor-
rel at i on) is general l y above 0.99 and in some cases near 0.999.
However, when vi scosi t y or CH par amet er s are used t he R 2
val ues are lower. For t hi s r eason use of ki nemat i c vi scosi t y or
CH wei ght rat i o shoul d be used as a l ast opt i on when ot her pa-
r amet er s are not available. Propert i es of heavy hydr ocar bons
are di scussed in t he next section. When Eq. (2.40) is appl i ed
t o pet r ol eum fract i ons, t he choi ce of i nput par amet er s is de-
t er mi ned by t he availability of exper i ment al data; however,
when a choi ce exists t he following t rends det er mi ne t he char-
act eri zi ng power of i nput par amet er s used in Eq. (2.39) or
(2.40): The first choi ce for 01 is Tb, fol l owed by M, and t hen
v38(100), while for t he par amet er 02 t he first choi ce is SG, fol-
l owed by par amet er s I and CH. Ther ef or e t he pai r of (M, SG)
is pr ef er abl e t o (M, CH) when t he choi ce exists.
2. 3. 3 Pr e di c t i on o f Propert i es o f He a v y
Pure Hydrocarbons
One of t he maj or pr obl ems in char act er i zat i on of heavy
pet r ol eum f r act i ons is t he l ack of sufficient met hods to pr edi ct
basi c charact eri st i cs of heavy hydr ocar bons. As ment i oned in
t he previ ous section, Eqs. (2.38) or (2.40) can be appl i ed to
hydr ocar bons up t o mol ecul ar wei ght of about 300. Crude oils
and reservoi r fluids cont ai n fract i ons wi t h mol ecul ar wei ght s
hi gher t han t hi s limit. For exampl e, pr oduct s f r om va c uum
distillation have mol ecul ar wei ght above t hi s range. For such
fract i ons appl i cat i on of ei t her Eq. (2.38) or (2.40) leads t o
s ome er r or s t hat will affect t he overall pr oper t y of t he whol e
cr ude or fluid. While si mi l ar correl at i ons may be devel oped
f or hi gher mol ecul ar wei ght syst ems, exper i ment al dat a are
l i mi t ed and mos t dat a (especially for critical pr oper t i es for
such compounds) are pr edi ct ed values. As ment i oned in t he
previ ous section, t he heavy hydr ocar bons are mor e compl ex
and t wo par amet er s ma y not be sufficient to correl at e prop-
ert i es of t hese compounds.
One way t o char act er i ze heavy fract i ons, as is di scussed in
t he next chapter, is to model t he fract i on as a mi xt ur e of pseu-
docompounds f r om vari ous homol ogous hydr ocar bon fami -
lies. I n fact, wi t hi n a single homol ogous hydr ocar bon group,
such as n-alkanes, onl y one char act er i zat i on pa r a me t e r is suf-
ficient to correl at e t he propert i es. Thi s single char act er i zat i on
pa r a me t e r shoul d be one of t hose par amet er s t hat best char-
act eri zes pr oper t i es in t he vert i cal di rect i on such as car bon
numbe r (Nc), Tb, or M. As shown in Table 2.4, par amet er s
SG, I20, and CH wei ght r at i o are not sui t abl e for t hi s purpose.
Kregl ewi ski and Zwol i nski [57] used t he following rel at i on to
correl at e critical t emper at ur e of n-alkanes:
(2.41) ln(0~ - O) = a - bN2c/3
wher e 0~ r epr esent s val ue of a pr oper t y such as Tc at Nc -~ ~ ,
and 0 is t he val ue of Tc for t he n-al kane wi t h car bon numbe r of
Nc. Later, t hi s t ype of cor r el at i on was used by ot her investiga-
t ors t o correl at e Tc and Pc for n-al kanes and al kanol s [58-60].
Based on t he above discussion, M or Tb may also be used in-
st ead of Nc. Equat i on (2.41) suggests t hat for ext r emel y hi gh
mol ecul ar wei ght hydr ocar bons (M -+ or critical t emper a-
t ur e or pr essur e appr oaches a finite val ue (Tcoo, Pc~). While
t here is no pr oof of t he validity of t hi s cl ai m, t he above equa-
t i on shows a good capabi l i t y for correl at i ng pr oper t i es of
n-al kanes for t he mol ecul ar wei ght range of i nt erest i n prac-
tical appl i cat i ons.
Based on Eq. (2.41), t he following general i zed cor r el at i on
was used to char act er i ze hydr ocar bons wi t hi n each homol o-
gous hydr ocar bon group:
(2.42) ln(0~ - O) = a - b M c
The r eason for t he use of mol ecul ar wei ght was its avail-
ability for heavy fract i ons in whi ch boi l i ng poi nt dat a may
not be avai l abl e due to t her mal cracki ng. For f our gr oups
of n-alkanes, n-al kyl cycopent anes, n-alkylcyclohexanes, and
n-alkylbenzenes, const ant s in Eq. (2.42) were det er mi ned us-
ing exper i ment al dat a r epor t ed in t he 1988 edi t i on of API-TDB
[2] and 1986 edi t i on of TRC [21]. The const ant s for TM, Tb, SG,
dE0, I , Tbr ( Tb/ Tc) , Pc, dc, w, and cr are given in Table 2.6 [31].
Car bon numbe r r ange and absol ut e and average absol ut e de-
vi at i ons (AAD) for each pr oper t y are also given in Table 2.6.
Er r or s are general l y l ow and wi t hi n t he accur acy of t he ex-
per i ment al data. Equat i on (2.42) can be easily reversed t o
est i mat e M f r om Tb for di fferent fami l i es if Tb is chosen as t he
char act er i zi ng paramet er. Then est i mat ed M f r om Tb can be
used t o predi ct ot her pr oper t i es wi t hi n t he s ame gr oup (fam-
ily), as is shown l at er in t hi s chapter. Si mi l arl y if Nc is chosen
as t he char act er i zat i on par amet er , M f or each fami l y can be
est i mat ed f r om Nc before usi ng Eq. (2.42) t o est i mat e vari ous
propert i es. Appl i cat i on and definition of surface t ensi on are
di scussed in Chapt er 8 (Sec 8.6).
Const ant s given in Table 2.6 have been obt ai ned f r om t he
pr oper t i es of pur e hydr ocar bons in t he car bon numbe r r anges
specified. For TM, Tb, SG, d, and I , pr oper t i es of compounds
up to C40 wer e available, but for t he critical pr oper t i es val ues
up t o C20 were used to obt ai n t he numer i cal const ant s. One
condi t i on i mposed in obt ai ni ng t he const ant s of Eq. (2.42)
for t he critical pr oper t i es was t he cri t eri a of i nt ernal con-
si st ency at at mospher i c pressure. For light compounds crit-
ical t emper at ur e is great er t han t he boi l i ng poi nt (Tbr < 1)
and t he critical pr essur e is gr eat er t han 1 at m (Pc > 1.01325
bar). However, this t r end changes for very heavy compounds
wher e t he critical pr essur e appr oaches 1 at m. Actual dat a
for t he critical pr oper t i es of such compounds are not avail-
able. However, t heor y suggests t hat when Pc --~ 1.01325 bar,
Tc ~ Tb or Tb~ ~ 1. And for infinitely l arge hydr ocar bons
when Nc --~ ~ (M --~ oo), Pc --~ 0. Some met hods devel oped
for pr edi ct i on of critical pr oper t i es of hydr ocar bons l ead to
Tbr =- 1 as Nc -~ oo[43]. Thi s can be t rue onl y if bot h Tc and
Tb appr oach infinity as Nc ~ oo. The val ue of car bon num-
ber for t he c ompound whose Pc = 1 at m is desi gnat ed by N~.
Equat i on (2.42) predi ct s val ues of Tb~ = 1 at N~ for di fferent
homol ogous hydr ocar bon groups. Values of N~ for di fferent
hydr ocar bon gr oups are given in Table 2.7. I n pract i cal ap-
pl i cat i ons, usual l y val ues of critical pr oper t i es of hydrocar-
bons and fract i ons up to C45 or C50 are needed. However,
accurat e pr edi ct i on of critical pr oper t i es at N~* ensures t hat
2. CHARACT ERI Z AT I ON AND P R OP E R T I E S OF PURE HY DR OCA R B ONS 5 1
TABLE 2.6---Constants of Eq. (2.42) for various parameters.
Constants in Eq. (2.42)
0 C No . Range 00r a
Const ant s f or physi cal pr oper t i es of n-al kanes [3 1]a
b c AAD b %AAD b
TM Cs-C40 397 6.5096 0.14187 0.470 1.5 0.71
T b C5-C40 1070 6.98291 0.02013 2/3 0.23 0.04
SG C5-C19 0.85 92.22793 89.82301 0.01 0.0009 0.12
d20 C5-C40 0.859 88.01379 85.7446 0.01 0.0003 0.04
I C5-C4o 0.2833 87.6593 86.62167 0.01 0.00003 0.002
Tbr ---- Tb/ Tc C5-C20 1.15 - 0. 41966 0.02436 0.58 0.14 0.027
- P c C5-C20 0 4.65757 0.13423 0.5 0.14 0.78
de C5-C20 0.26 - 3. 50532 1.5 x 10 -8 2.38 0.002 0.83
- t o C5-C2o 0.3 - 3. 06826 - 1. 04987 0.2 0.008 1.2
a C5-C20 33.2 5.29577 0.61653 0.32 0.05 0.25
Const ant s f or physi cal pr oper t i es of n-al kyl cycl opent anes
TM C7-C41 370 6.52504 0.04945 2/3 1.2 0.5
Tb C6-C41 1028 6.95649 0.02239 2/3 0.3 0.05
SG C7-C25 0.853 97.72532 95.73589 0.01 0.0001 0.02
d20 C5-C41 0.857 85.1824 83.65758 0.01 0.0003 0.04
I C5-C4I 0.283 87.55238 86.97556 0.01 0.00004 0.003
Tbr ---- Tb/ Tc C5-C18 1.2 0.06765 0.13763 0.35 1.7 0.25
- P c C6-C18 0 7.25857 1.13139 0.26 0.4 0.9
- de C6-C20 - 0. 255 - 3. 18846 0.1658 0.5 0.0004 0.11
- t o C6-C20 0.3 - 8. 25682 - 5. 33934 0.08 0.002 0.54
o" C6-C25 30.6 14.17595 7.02549 0.12 0.08 0.3
Const ant s f or physi cal pr oper t i es of n-al kyl cycl ohexane
TM C7-C20 360 6.55942 0.04681 0.7 1.3 0.7
Tb C6-C20 1100 7.00275 0.01977 2/3 1.2 0.29
SG C6-C20 0.845 - 1. 51518 0.05182 0.7 0.0014 0.07
d20 C6-C21 0.84 - 1.58489 0.05096 0.7 0.0005 0.07
I C6-C20 0.277 - 2. 45512 0.05636 0.7 0.0008 0.06
Tbr = Tbf r c C6-C20 1.032 - 0. 11095 0.1363 0.4 2 0.3
- Pc C6-C20 0 12.3107 5.53366 0.1 0.15 0.5
- dc C6-C20 - 0. 15 - 1. 86106 0.00662 0.8 0.0018 0.7
--CO C7-C20 0.6 - 5. 00861 - 3. 04868 0.1 0.005 1.4
a C6-C20 31 2.54826 0.00759 1.0 0.17 0,6
Const ant s f or physi cal pr oper t i es of n-al kyl benzenes
TM C9-C42 375 6.53599 0.04912 2/3 0.88 0.38
T b C6-C42 1015 6.91062 0.02247 2/3 0.69 0.14
- S G C6-C20 - 0. 8562 224.7257 218.518 0.01 0.0008 0.1
-d20 C6- C42 - 0. 854 238.791 232.315 0.01 0.0003 0.037
- I C6-C42 - 0. 2829 137.0918 135.433 0.01 0.0001 0.008
Tbr = Tb/ Tc C6-C20 1.03 - 0. 29875 0.06814 0.5 0.83 0.12
- P c C6-C20 0 9.77968 3.07555 0.15 0.22 0.7
- de C6-C20 - 0. 22 - 1. 43083 0.12744 0.5 0.002 0.8
-co C6-C20 0 - 14. 97 - 9. 48345 0.08 0.003 0.68
tr C6-C20 30.4 1.98292 - 0. 0142 1.0 0.4 1.7
With permission from Ref. [31].
aData sources: TM Tb, and d are taken from TRC [21]. All other properties are taken from API-TDB-1988 [2].
TM, Tb, and Tc are in K; d20 and d~ are in g/cm3; Pc is in bar; a is in dyn/cm.
b AD and AAD% given by Eqs. (2.134) and (2.135).
Units:
t h e e s t i ma t e d c r i t i c a l p r o p e r t i e s b y Eq. ( 2. 42) a r e r e a l i s t i c f o r
h y d r o c a r b o n s b e y o n d Cla. Th i s a n a l y s i s i s c a l l e d internal con-
sistency f o r c o r r e l a t i o n s o f c r i t i c a l p r o p e r t i e s .
I n t h e c h a r a c t e r i z a t i o n me t h o d p r o p o s e d b y Ko r s t e n [32,
33] i t i s a s s u me d t h a t f o r e x t r e me l y l a r g e h y d r o c a r b o n s
( Nc ~ o~), t h e b o i l i n g p o i n t a n d c r i t i c a l t e mp e r a t u r e a l s o a p-
p r o a c h i nf i ni t y. Ho we v e r , a c c o r d i n g t o Eq. ( 2. 42) as Nc --~ er
o r ( M- - ~ o~), p r o p e r t i e s s u c h as Tb, SG, d, I , Tbr, Pc, de,
co, a n d a al l h a v e f i ni t e va l ue s . F r o m a p h y s i c a l p o i n t o f
v i e w t hi s ma y b e t r u e f or mo s t o f t h e s e p r o p e r t i e s . Ho we v e r ,
Ko r s t e n [ 33] s u g g e s t s t h a t as Nc --~ o~, Pc a n d dc a p p r o a c h
z e r o wh i l e Tb, To, a n d mo s t o t h e r p r o p e r t i e s a p p r o a c h i nf i ni t y.
Go o s s e n [61] d e v e l o p e d a c o r r e l a t i o n f o r mo l e c u l a r we i g h t o f
h e a v y f r a c t i o n s t h a t s u g g e s t s b o i l i n g p o i n t f o r e x t r e me l y l a r g e
mo l e c u l e s a p p r o c h e s a f i ni t e v a l u e of Tbo~ = 1078. I n a n o t h e r
p a p e r [ 62] h e s h o ws t h a t f o r i nf i ni t e p a r a f f i n i c c h a i n l e n g t h ,
TABLE 2.7--Prediction of atmospheric critical pressure from Eq. (2.42).
Nc* calculated at N* calculated at Predicted Pc (bar) at
Hydrocarbon type Tb = Tc Pc ~ 1.01325 Tb = Tc
n-Mkanes 84.4 85 1.036
n-Mkyl cycl opent anes 90.1 90.1 1.01
n-Mtcylcyc]ohexanes 210.5 209.5 1.007
n-Mkyl benzenes 158.4 158.4 1,013
With permission from Ref. [31 ].
5 2 CHARACTERI ZATI ON AND PROPERTI ES OF PETROLEUM FRACTI ONS
1.2
1.1
1.0
b.
o 0 . 9
~ 0 . 8
o
o
o
0 . 7
0.6
~n- al kanes
. . . . n-all~,lcyclope~t aaes
; ~ . . . . . ,-alkylbenzenes ~ / ~ . . . _ .
. /
5 L i i ~ i j J ~ l i i i I i i i l l K i
10 100 1000
J , , J , I
10000
Molecular Weight
F I G. 2 . 1 - - R e d u c e d b o i l i n g p o i n t o f h o mo l o g o u s h y d r o c a r b o n g r o u p s f r o m
Eq. (2.42).
do~ = 0.8541 a nd no~ = 1.478 (Ioo = 0.283), whi l e t he val ues
obt a i ne d t h r o u g h Eq. (2.42) (see Table 2.6) ar e Tboo = 1070,
doo = 0.859, and Ioo = 0. 2833. One can see h o w cl ose t he val-
ues ar e a l t hough t hey have been der i ved by t wo di f f er ent
met hods . However , t hese val ues ar e of little pr act i cal appl i ca-
t i on as l ong as a pr opos e d cor r el at i on satisfies t he condi t i on
of Tbr = 1 at Pc = 1.0133 bar. Equa t i on (2.42) will be us ed l at er
i n Chapt er 4 t o devel op phys i cal pr oper t i es of si ngl e c a r b o n
n u mb e r (SCN) cut s up t o C50 f or t he es t i mat i on of pr oper t i es
of heavy cr ude oils a nd r eser voi r oils. Gr aphi cal pr e s e nt a t i on
of Eq. (2.42) f or Tbr and Tc ver sus mol e c ul a r wei ght of dif-
f er ent h y d r o c a r b o n f ami l i es is s hown i n Figs. 2.1 and 2.2 f or
mol e c ul a r wei ght s up t o 3000 (Nc ~ 214).
One di r ect appl i cat i on of cr i t i cal pr oper t i es of h o mo l o g o u s
h y d r o c a r b o n s is t o cal cul at e pha s e equi l i br i um cal cul at i ons
f or wax pr eci pi t at i on and cl oud poi nt of r eser voi r fl ui ds a nd
cr ude oils as s hown by Pan et al. [63, 64]. These i nvest i gat or s
eval uat ed pr oper t i es cal cul at ed t h r o u g h Eq. (2.42) and mod-
ified t hi s e qua t i on f or t he cri t i cal pr es s ur e of PNA hydr ocar -
bons wi t h mol e c ul a r wei ght above 300 t h r o u g h t he f ol l owi ng
r el at i on:
(2.43) Pc = a - b e x p ( - c M)
wher e a, b, a nd c ar e gi ven f or t he t hr ee h y d r o c a r b o n gr oups
i n Table 2. 8 [64]. However, Eq. (2.43) does not hol d t he inter-
nal cons i s t ency at Pc of I at m, whi c h was i mpos e d i n der i vi ng
t he cons t ant s of Eq. (2.42). But t hi s ma y not affect r esul t s f or
pr act i cal cal cul at i ons as cr i t i cal pr es s ur es o f even t he heavi est
c o mp o u n d s do not r each t o a t mos phe r i c pr essur e. A compar -
i s on bet ween Eq. (2.42) and (2.43) f or t he cri t i cal pr es s ur e
of paraffi ns, napht henes , a nd a r oma t i c s is s h o wn i n Fig. 2.3.
Pan et al. [63, 64] al so r e c o mme n d use of t he f ol l owi ng re-
l at i on f or t he acent r i c f act or of a r oma t i c s f or h y d r o c a r b o n s
wi t h M < 800:
(2.44) I n to = - 3 6 . 1 5 4 4 + 30. 94M 0"026261
and wh e n M > 800, to = 2.0. Equa t i on (2.42) is r e c o mme n d e d
f or cal cul at i on of ot he r t h e r mo d y n a mi c pr oper t i es bas ed on
t he eval uat i on ma d e on t h e r mo d y n a mi c pr oper t i es of waxes
and as phal t enes [63, 64].
For h o mo l o g o u s h y d r o c a r b o n gr oups, var i ous cor r el at i ons
ma y be f ound sui t abl e f or t he cr i t i cal pr oper t i es. For exampl e,
a not he r r el at i on t hat was f ound t o be appl i cabl e t o cri t i cal
pr es s ur e of n-al kyl f ami l i es is i n t he f ol l owi ng f or m:
(2.45) Pc = (a + bM)-"
wher e Pc is i n b a r and M is t he mol ecul ar wei ght of pur e hy-
d r o c a r b o n f r o m a h o mo l o g o u s gr oup. Cons t a nt n is gr eat er
t h a n uni t y and as a r esul t as M ~ oo we have Pc ~ 0, whi c h
satisfies t he gener al cr i t er i a f or a Pc cor r el at i on. Bas ed on dat a
on Pc of n- al kanes f r o m C2 t o C22, as gi ven i n Tabl e 2.1, it was
f ound t hat n = 1.25, a = 0. 032688, and b = 0. 000385, whi c h
gives R 2 = 0. 9995 wi t h aver age devi at i on of 0. 75% f or 21 c om-
pounds . To s how t he degr ee of ext r apol at i on of t hi s equat i on,
if dat a f r o m C2 t o C10 ( onl y ni ne c o mp o u n d s ) ar e us ed t o
TABLE 2.8---Coefficients of Eq. (2.43).
Coefficient Paraffins Naphthenes Aromatics
a 0.679091 2.58854 4.85196
b -22. 1796 -27. 6292 -42.9311
c 0.00284174 0.00449506 0.00561927
Taken from PanetaL[63,64].
2. CHARACTERI ZATI ON AND PROPERTI ES OF PURE HYDROCARBONS 5 3
700
500
?
o)
#
-~ 300
100
* API-TDB
- - n - a h k a n e s
. . . . . . . n-alkylcyclopentanes
. . . . n-alkylbenzenes
lO 100 1000
Mol ecul ar Weight
FIG. 2. 2- - Cr i t i cal temperature of homol ogous hydrocarbon groups from
Eq. (2,42).
50
40
m 30
o ~
~ 20
10
o
",x
*" t . DIPPR Data
. ~ R-S: Eq. 2.42
., ~ - - n - a l k ~ e s
.: ~ . . . . . . . n-alkyleyclopentanes
~ . . . . n-alkylb~zenes
~ ~ ~ P-F: Eq. (2.43)
- - n - a l k a n e s
. . . . n-all~,lcyclopentanes
X - - - - n-alkylbeazenes
I 0 I00 1000 10000
Mol ecul ar Weight
FIG. 2. 3- - Pr edi ct i on of critical pressure of homol ogous hydrocarbon groups from
Eqs. (2.42) and (2.43).
TABLE 2.9--Constants in Eqs. (2.46a and 2.46b) 0 = al exp(bl01 + Cl SG + dl01SG) 0~SG f for various properties of heavy hydrocarbons.
o o1 al bi cl dl el fl AAD%
Tc Tb 35.9413 -6. 9 10 -4 -1.4442 4.91 10 -4 0.7293 1.2771 0.3
Pc Tb 6.9575 -0.0135 -0.3129 9.174 10 3 0.6791 -0.6807 5.7
Vc T b 6.1677 x 101~ -7.583 10 -3 -28.5524 0.01172 1.20493 17.2074 2.5
I T b 3.2709 10 -3 8.4377 10 -4 4.59487 -1.0617 10 -3 0.03201 -2.34887 0.l
d20 Tb 0.997 2.9 10 -4 5.0425 -3.1 10 -4 -0.00929 1.01772 0.07
0 01 32 b2 c2 d2 e2 ~ AAD%
T b M 9.3369 1.65 10 -4 1.4103 -7.5152 x 10 -4 0.5369 -0.7276 0.3
Tc M 218.9592 -3. 4 10 -4 -0.40852 -2.5 10 -5 0.331 0.8136 0.2
Pc M 8.2365 104 -9.04 10 -3 -3.3304 0.01006 -0.9366 3.1353 6.2
Vc M 9.703 106 -9.512 10 -3 -15.8092 0.01111 1.08283 10.5118 1.6
I M 1.2419 x 10 -2 7.27 10 -4 3.3323 -8.87 x 10 -4 6.438 10 -3 -1.61166 0.2
d20 M 1.04908 2.9 10 4 -7.339 10 -2 -3. 4 1 0 - 4 3.484 10 -3 1.05015 0.09
Data generated from Eq. (2.42) have been used to obtain these constants. Units: Vc in cm3/mol; To, and Tb in K;
Pc in bar; d20 in g/cm 3 at 20~ Equations are recommended for the carbon range of C20-C50; however, they may be used for the C5-C20 with lesser degree of
accuracy.
obt ai n a and b i n t he above equat i on we get a = 0.032795 and
b = 0.000381. These coefficients give R 2 ---- 0.9998 but when it
is used to est i mat e Pc from C2 to C22 AAD of 0.9% is obt ai ned.
These coefficients est i mat e Pc of t / - C3 6 a s 6.45 bar versus val ue
given i n DIPPR as 6.8. This is a good ext rapol at i on power. I n
Eq. (2.45) one may replace M by Tb or Nc and obt ai n new
coefficients for cases t hat these paramet ers are known.
Properties of pure compounds predi ct ed t hr ough Eqs.
(2.42) and (2.43) have been used to develop the following gen-
eralized correl at i ons i n t erms of (Tb, SG) or (M, SG) for the
basi c propert i es of heavy hydr ocar bons from all hydr ocar bon
groups i n the C6-C50 range [65].
Tc, Pc, Vc, I, d20- - al [exp(blTb +c l S G+d l Tb S G) ] T~ 1 SG fi
(2.46a)
Tb, Tc, Pc, Vc, I, d20 =a2[ exp( bEM+CESG
(2.46b) +d2MSG)] M e2 SG f2
where Vc i n these rel at i ons is i n cm3/mol. Const ant s a l - f l
and a2-f2 i n these rel at i ons are given i n Table 2.9. These cor-
rel at i ons are r ecommended for hydr ocar bons and pet r ol eum
fract i ons i n t he car bon numbe r range of C20-C50. Al t hough
these equat i ons may be used to predi ct physi cal propert i es
of hydr ocar bons i n the range of C6-C20, if the syst em does
not cont ai n heavy hydr ocar bons Eqs. (2.38) and (2.40) are
r ecommended.
2. 3. 4 Ext e ns i on o f Pr opos e d Correl at i ons
t o Nonhydr oc ar bon Sys t e ms
Equat i ons (2.38) and (2.40) cannot be appl i ed to systems con-
t ai ni ng hydrocarbons, such as met hane and et hane, or hydro-
gen sulfide. These equat i ons are useful for hydr ocar bons wi t h
car bon number s above Cs and are not appl i cabl e to nat ur al
gases or refinery gases. Est i mat i on of t he propert i es of nonhy-
dr ocar bon systems is beyond t he objective of this book. But
i n reservoi r fluids, compounds such as light hydr ocar bons
or H2S and CO2 may be present . To develop a general i zed
correl at i on i n t he form of Eq. (2.40) t hat i ncl udes nonhydr o-
carbons, usual l y a t hi rd par amet er is needed to consi der the
effects of polarity. I n fact Vetere [66] has defined a pol ari t y
factor i n t erms of the mol ecul ar wei ght and boi l i ng poi nt to
predi ct propert i es of pol ar compounds. Equat i on (2.40) was
ext ended i n t erms of t hree paramet ers, Tb, d20, and M, to es-
t i mat e the critical propert i es of bot h hydr ocar bons and non-
hydr ocar bons [37].
Tc, Pc, Vc = exp[a + bM + cTb + dd20 + eTbd2o] MfT~+hMdi2o
(2.47)
Based on the critical propert i es of more t han 170 hydrocar-
bons from C1 to C18 and more t han 80 nonhydr ocar bons, such
as acids, sul fur compounds, nitriles, oxide gases, alcohols,
hal ogenat ed compounds, ethers, ami nes, and water, t he ni ne
paramet ers i n Eq. (2.47) were det er mi ned and are given i n
Table 2.10. I n usi ng Eq. (2.47), the const ant d shoul d not be
mi st aken wi t h par amet er d20 used for l i qui d densi t y at 20~
As i n the ot her equat i ons i n this chapter, values of Tb and Tc
are i n kelvin, Pc is i n bar, and Vc is i n cma/g. Par amet er d20 is
the l i qui d densi t y at 20~ and 1.0133 bar i n g/cm 3. For light
gases such as met hane (C1) or et hane (C2) i n whi ch t hey are i n
the gaseous state at t he reference condi t i ons, a fictitious value
of dE0 was obt ai ned t hr ough the ext rapol at i on of densi t y val-
ues at l ower t emper at ur e given by Reid et al. [4]. The values
of d20 for some gases f ound i n this manner are as follows:
ammoni a, NH3 (0.61); ni t r ous oxide, N20 (0.79); met hane,
C1 (0.18); et hane, C2 (0.343); propane, C3 (0.5); n-but ane, nC4
(0.579); i sobut ane, iC4 (0.557); ni t rogen, N2 (0.135); oxygen,
02 (0.22); hydrogen sulfide, H2S (0.829); and hydrogen chlo-
ride, HC1 (0.837). I n some references different values for liq-
ui d densities of some of these compounds have been reported.
For example, a val ue of 0.809 g/cm 3 is report ed as t he den-
sity of N2 at 15.5~ and 1 at m by several aut hors i n reservoi r
engi neeri ng [48, 51]. This value is very close to the densi t y of
N2 at 78 K [4]. The critical t emper at ur e of N2 is 126.1 K and
TABLE 2.10--Constants for Eq. (2.47).
o ~ Vc
Constants Tc. K Pc, MPa cm3'/g
a 1.60193 10.74145 -8.84800
b 0.00558 0.07434 -0.03632
c -0.00112 -0.00047 -0.00547
d -0.52398 -2.10482 0.16629
e 0.00104 0.00508 -0.00028
f -0.06403 -1.18869 0.04660
g 0.93857 -0.66773 2.00241
h -0.00085 -0.01154 0.00587
i 0.28290 1.53161 -0.96608
t her ef or e at t emper at ur e 288 or 293 K it cannot be a liquid
and val ues r epor t ed for densi t y at t hese t emper at ur es are fic-
titious. I n any case t he val ues given here for densi t y of N2,
CO2, C1, Ca, and H2S shoul d not be t aken as real val ues and
t hey are onl y r ecommended for use in Eq. (2.47). I t shoul d
be not ed t hat dE0 is t he s ame as t he specific gravi t y at 20~
in t he SI syst em (d4a~ Thi s equat i on was devel oped based on
t he fact t hat nonhydr ocar bons are mai nl y pol ar compounds
and a t wo- par amet er pot ent i al energy rel at i on cannot rep-
resent t he i nt er mol ecul ar forces bet ween mol ecul es, t here-
fore a t hi rd pa r a me t e r is needed to char act er i ze t he syst em.
Thi s met hod woul d be part i cul arl y useful t o est i mat e t he bul k
pr oper t i es of pet r ol eum fluids cont ai ni ng light hydr ocar bons
as well as nonhydr ocar bon gases. Eval uat i on of this met hod
is pr esent ed in Sect i on 2.9.
2. 4 PREDICTION OF MOLECULAR WEIGHT,
BOILING POINT, AND SPECIFIC GRAVITY
Mol ecul ar weight, M, boiling poi nt , Tb, and specific gravity,
SG, are per haps t he mos t i mpor t ant char act er i zat i on par am-
et ers for pet r ol eum fract i ons and ma ny physi cal pr oper t i es
ma y be cal cul at ed f r om t hese par amet er s. Various met hods
commonl y used to cal cul at e t hese pr oper t i es are pr esent ed
here. As ment i oned before, t he mai n appl i cat i on of t hese cor-
rel at i ons is for pet r ol eum fract i ons when exper i ment al dat a
are not available. For pur e hydr ocar bons ei t her exper i ment al
dat a are avai l abl e or gr oup cont r i but i on met hods are used to
est i mat e t hese par amet er s [4]. However, met hods suggest ed
in Chapt er 3 to est i mat e pr oper t i es of pet r ol eum fract i ons are
based on t he met hod devel oped f r om t he pr oper t i es of pur e
hydr ocar bons in t hi s chapter.
2.4.1 Prediction of Molecular Weight
For pur e hydr ocar bons f r om homol ogous groups, Eq. (2.42)
can be reversed to obt ai n t he mol ecul ar wei ght f r om ot her
propert i es. For exampl e, if Tb is available, M can be est i mat ed
f r om t he following equat i on:
(2.48) M = [a - ln(Tb~ -- Tb)] /
wher e values of a, b, c, and Tboo are t he s ame const ant s as
t hose given in Table 2.6 for t he boi l i ng poi nt . For exampl e,
for n-alkanes, M can be est i mat ed as follows:
(2.49) Mp - ~ / ~ l [6 98291 - ln(1070 - Tb)]} 3/2
[0. 02013 '
in whi ch Mp is mol ecul ar wei ght of n-al kane (n-paraffins)
whose nor mal boi l i ng poi nt is Tb. Values obt ai ned f r om
Eq. (2.49) are very close to mol ecul ar wei ght of n-alkanes.
Si mi l ar equat i ons can be obt ai ned for ot her hydr ocar bon
gr oups by use of val ues given in Table 2.6. Once M is deter-
mi ned f r om Tb, t hen it can be used wi t h Eq. (2.42) to obt ai n
ot her pr oper t i es such as specific gravi t y and critical const ant s.
2. 4. 1. 1 Ri az i - Daube r t Me t hods
The met hods devel oped in t he previ ous sect i on are commonl y
used to cal cul at e mol ecul ar wei ght f r om boi l i ng poi nt and
specific gravity. Equat i on (2.38) for mol ecul ar wei ght is [28]
(2.50)
M = 1.6607 x 10- 4T21962SG -1"0164
This equat i on fails to pr oper l y predi ct pr oper t i es for hydro-
car bons above C2s. This equat i on was extensively eval uat ed
for vari ous coal liquid sampl es al ong wi t h ot her correl at i ons
by Tsonopoul os et al. [34]. They r ecommended t hi s equat i on
for t he est i mat i on of t he mol ecul ar wei ght of coal liquid frac-
tions. Const ant s in Eq. (2.40) for mol ecul ar weight, as given
in Table 2.5, were modi fi ed t o i ncl ude heavy hydr ocar bons up
to mol ecul ar wei ght of 700. The equat i on in t er ms of Tb and
SG becomes
M = 42.965[exp(2.097 x 10-4Tb -- 7.78712SG
(2.51) + 2.08476 x 10-3TbSG)]Tbl26~176 4"98308
This equat i on can be appl i ed to hydr ocar bons wi t h mol ecul ar
wei ght rangi ng f r om 70 to 700, whi ch is nearl y equi val ent to
boiling poi nt r ange of 300-850 K (90-1050~ and t he API
gravi t y r ange of 14.4-93. These equat i ons can be easily con-
vert ed in t er ms of Wat son K f act or (Kw) and API degrees
usi ng t hei r definitions t hr ough Eqs. (2.13) and (2.4). A graph-
ical pr esent at i on of Eq. (2.51) is shown in Fig. 2.4. ( Equat i on
(2.51) has been r ecommended by t he API as it will be dis-
cussed later.) Equat i on (2.51) is mor e accur at e for light frac-
t i ons (M < 300) wi t h an %AAD of about 3.5, but f or heavi er
fract i ons t he %AAD is about 4.7. Thi s equat i on is i ncl uded in
t he API-TDB [2] and is recogni zed as t he st andar d met hod
of est i mat i ng mol ecul ar wei ght of pet r ol eum fract i ons in t he
industry.
For heavy pet r ol eum fract i ons boiling poi nt ma y not be
available. For this r eason Riazi and Dauber t [67] devel oped
a t hr ee- par amet er correl at i on in t er ms of ki nemat i c vi scosi t y
based on t he mol ecul ar wei ght of heavy fract i ons in t he r ange
of 200-800:
[. (-1.2435+1.1228SG) .(3.4758-3.038SG)'1 SG-0.6665
M = 223.56 l_v38(loo) u99(21o ) j
(2.52)
The t hree i nput par amet er s ar e ki nemat i c viscosities (in cSt)
at 38 and 98.9~ (100 and 210~ shown by v38000) and 1)99(210),
respectively, and t he specific gravity, SG, at 15.5~ I t shoul d
be not ed t hat viscosities at t wo di fferent t emper at ur es repre-
sent t wo i ndependent par amet er s: one t he value of vi scosi t y
and t he ot her t he effect of t emper at ur e on viscosity, whi ch is
anot her charact eri st i c of a compound as di scussed in Chap-
t er 3. The use of a t hi rd par amet er is needed to charact er-
ize compl exi t y of heavy hydr ocar bons t hat follow a t hree-
pa r a me t e r pot ent i al energy rel at i on. Equat i on (2.52) is onl y
r ecommended when t he boiling poi nt is not available. I n
a case wher e specific gravi t y is not available, a met hod is
pr oposed in Sect i on 2.4.3 to est i mat e it f r om vi scosi t y data.
Gr aphi cal pr esent at i on of Eq. (2.52) is shown in Fig. 2.5 in
t er ms of API gravity. To use this figure, based on t he val ue of
v3m00) a poi nt is det er mi ned on t he vert i cal line, t hen f r om
v a l u e s of I)99(210 ) and SG, anot her poi nt on t he char t is speci-
fied. A line t hat connect s t hese t wo poi nt s i nt ersect s wi t h t he
line of mol ecul ar wei ght where it ma y be r ead as t he est i mat ed
value.
56 CHARACTERI ZATI ON AND PROPERTI ES OF PETROLEUM FRACTIONS
800
700
600
500
400
300
200
100
0 !
0. 60 0. 70 0. 80 0. 90 1.00
o n-al kane
y
_. 200~
~ _ _ 150~
30oC 100~
i t , I }
Specific Gravity
FIG. 2. 4---Esti mati on of mol ecul ar wei ght from Eq. (2.51).
i
1.10
2.4.1.2 ASTM Method
ASTM D 2502 me t h o d [68] pr ovi de s a c h a r t t o c a l c ul a t e t he
mo l e c u l a r we i ght of vi s c ous oi l s us i ng t he k i n e ma t i c vi s cos i -
t i es me a s u r e d at 100~ (38~ a n d 210~ (99~ The me t h o d
wa s e mp i r i c a l l y de ve l ope d by Hi r s c h l e r i n 1946 [69] a n d i s
p r e s e n t e d b y t he f ol l owi ng e qua t i on.
(2. 53) M = 180 + K(Ha80oo) + 60)
wh e r e
K = 4. 145 - 1.733 l o g t o ( VS F - 145)
VSF = / - / 3 8 ( 1 0 0 ) - H 9 9 ( 2 1 0 )
H = 870 lOglo[lOglo(v + 0. 6)] + 154
i n wh i c h v i s t he k i n e ma t i c vi s cos i t y i n cSt . Thi s e q u a t i o n
wa s de ve l ope d s o me 60 ye a r s a go a n d r e qui r e s k i n e ma t i c
vi s cos i t i es at 38 a n d 99~ i n c St as t he onl y i n p u t p a r a m-
et er s. The Hi r s c h l e r me t h o d wa s i n c l u d e d i n t he API - TDB
i n 1964 [2], b u t i n t he 1987 r e vi s i on of API - TDB i t wa s r e-
p l a c e d b y Eq. (2. 52). Ri a z i a n d Da u b e r t [67] ext ens i vel y c om-
p a r e d Eq. (2. 52) wi t h t he Hi r s c h l e r me t h o d a n d t he y f o u n d
t h a t f or s o me 160 f r a c t i ons i n t he mo l e c u l a r we i ght r a nge of
200- 800 t he p e r c e n t a ve r a ge a b s o l u t e de vi a t i on (%AAD) f or
t he s e me t h o d s we r e 2. 7% a n d 6. 9%, r espect i vel y. Eve n i f t he
c ons t a nt s of t he Hi r s c h l e r c o r r e l a t i o n we r e r e o b t a i n e d f r o m
t he d a t a b a n k u s e d f or t he e va l ua t i ons , t he a c c u r a c y of t he
me t h o d i mp r o v e d onl y s l i ght l y f r om 6. 9 t o 6. 1% [67].
Example 2 . 4 - - Th e vi s c os i t y a n d o t h e r p r o p e r t i e s of 5-n-
but yl docos ane, C26H54, as gi ven i n API RP- 42 [18] a r e M =
366. 7, SG - - 0. 8099, v38000) = 11.44, a n d 1)99(210 ) = 3. 02 cSt .
Cal cul at e t he mo l e c u l a r we i ght wi t h %AD f r o m t he API
me t h o d , Eq. (2. 52), a n d t he Hi r s c h l e r me t h o d ( ASTM 2502),
Eq. (2. 53).
Sol ut i on--In us i ng Eq. (2. 52) t hr e e p a r a me t e r s of v38o00),
P99(210), a n d SG a r e ne e de d.
M = 223. 56 x [11. 44 (-1"2435+1"1228x0"8099) 3. 02 (3"475a-a'038x0"8099) ]
%AD= 4 . 5 %. F r o m Eq. (2. 53):
H38 = 183.3, H99 ~--- - 6 5 . 8 7 , VSF- - - 249. 17, K = 0. 6483, M =
337. 7, %AD = 7. 9%. #
2.4.1.3 API Methods
The API - TDB [2] a d o p t e d me t h o d s d e v e l o p e d b y Ri a z i a n d
Da u b e r t f or t he e s t i ma t i o n of t he mo l e c u l a r we i ght of h y d r o -
c a r b o n s ys t ems . I n t he 1982 e d i t i o n of API - TDB, a mo d i f i e d
ve r s i on of Eq. (2. 38) wa s i nc l ude d, b u t i n i t s l a t e s t e di t i ons
( f r om 1987 t o 1997) Eqs. (2. 51) a n d (2. 52) a r e i n c l u d e d a f t e r
r e c o mme n d a t i o n s ma d e b y t he API - TDB Commi t t e e .
2.4.1.4 Lee--Kesl er Method
The mo l e c u l a r we i ght i s r e l a t e d t o boi l i ng p o i n t a n d s peci f i c
gr a vi t y t h r o u g h a n e mp i r i c a l r e l a t i o n as f ol l ows [13]:
M = - 1 2 2 7 2 . 6 + 9486. 4SG + (8. 3741 - 5. 9917SG )Tb
+ (1 - 0. 77084SG - 0. 02058SG 2)
(2. 54) x (0. 7465 - 222.466/Tb)lO7/Tb
+ (1 -- 0. 80882SG + 0. 02226SG 2)
x (0. 3228 - 17.335/Tb)1012/T~
Hi g h - mo l e c u l a r - we i g h t d a t a we r e al s o u s e d i n o b t a i n i n g t he
c ons t a nt s . The c o r r e l a t i o n i s r e c o mme n d e d f or us e u p t o a
boi l i ng p o i n t o f a b o u t 750 K ( ~850~ I t s e v a l u a t i o n i s s h o wn
i n Se c t i o n 2.9.
2. CHARACTERIZATION AND PROPERTI ES OF PURE HYDROCARBONS 5 7
_ 1 0 0 0 0
"~ooo
- 8 0 0 0
_~7ooo
- 6 0 0 0
= 5 0 0 0
~ ' 4 0 0 0
~ 3 0 0 0
--- 2 0 0 0
m
- ' 1 0 0 0
9 0 0
8 0 0
7OO
6 0 0
5 0 0
4 0 0
3OO
- 2 0 0
100
9 0
8 0
7 0
6 0
5 0
4 0
3 0
- 2 0
! 0
g
8
7
6
5
4
3
_=
E ,
o
o
T - -
~ 0
o
o
. ( ~
>
E
c
API Gr a v i t y
2 0
4 BOO
%
o
5
10
7 0 0
6 0 0
._~ 5 0 0
(D
0
o
4OO
3 0 0
2 0 0
1 0 0
F I G . 2 . 5 - - - E s t i m a t i o n o f m o l e c u l a r w e i g h t f r o m E q . ( 2 . 5 2 ) . T a k e n f r o m R e f . [ 6 7 ] w i t h p e r -
m i s s i o n .
2.4.1.5 Goossens Correlation
Mos t r e c e nt l y Goos s e ns [61] c o r r e l a t e d M t o Tb a n d d20 i n t he
f ol l owi ng f o r m us i ng t he d a t a on 40 p u r e h y d r o c a r b o n s a n d
23 p e t r o l e u m f r a c t i ons :
(2. 55) M = 0.01077Tb~/d] ~
wh e r e /3 = 1. 52869 + 0. 06486 ]n[Tb/ (1078 -- Tb)]. I n s p e c t i o n
of t hi s e q u a t i o n s hows t h a t i t ha s t he s a me s t r u c t u r e as
Eq. (2. 38) b u t wi t h a va r i a bl e b a n d c = - 1 . P a r a me t e r b i s
c o n s i d e r e d as a f u n c t i o n of Tb, whi l e SG i n Eq. (2. 38) i s r e-
p l a c e d b y d~ ~ t he speci f i c gr a vi t y at 20/4~ (d~ ~ i s t he s a me
as d20 i n g/ cm3). The d a t a b a n k u s e d t o de ve l op t hi s equa-
t i on cover s t he c a r b o n r a n g e of C5-C120 ( M ~ 70- 1700, Tb
300- 1000 K, a n d d ~ 0. 63- 1. 08) . F o r t he s a me 63 d a t a p o i n t s
u s e d t o o b t a i n t he c o n s t a n t s of Eq. (2. 55), t he a ve r a ge e r r o r
wa s 2. 1%. However , p r a c t i c a l a p p l i c a t i o n of Eq. (2. 55) i s l i m-
i t ed t o mu c h l owe r mo l e c u l a r we i ght f r a c t i ons b e c a u s e he a vy
fract i on distillation dat a is not usual l y available. When d20 is
not avai l abl e it may be est i mat ed f r om SG usi ng t he met hod
given in Sect i on 2.6.1.
2.4.1.6 Other Methods
Twu [30] pr opos ed a set of correl at i ons for t he cal cul at i on of
M, Tc, Pc, and Vc of hydr ocar bons. Because t hese correl at i ons
are i nt errel at ed, t hey are all given in Sect i on 2.5.1. The com-
put er i zed Wi nn met hod is gi ven by Eq. (2.93) in Sect i on 2.5.1
and in t he f or m of char t in Fig. 2.12.
Example 2. 5- - For n-But yl benzene est i mat e t he mol ecul ar
wei ght f r om Eqs. (2.50), (2.51), (2.54), and (2.55) usi ng t he
i nput dat a f r om Table 2.1
Solution--From Table 2.1, n-but yl benzene has Tb = 183.3~
SG- - 0.8660, d = 0.8610, and M = 134.2. Applying vari ous
equat i ons we obt ai n t he following: f r om Eq. (2.50), M = 133.2
wi t h AD-- 0.8%, Eq. (2.51) gives M = 139.2 wi t h AD = 3.7%,
Eq. (2.54) gives M = 143.4 wi t h AD-- 6.9%, and Eq. (2.55)
gives M = 128.7 wi t h AD = 4.1%. For t hi s pur e and light hy-
dr ocar bon, Eq. (2.50) gives t he l owest er r or because it was
mai nl y devel oped f r om t he mol ecul ar wei ght of pur e hydro-
car bons whi l e t he ot her equat i ons cover wi der r ange of mol ec-
ul ar wei ght because dat a f r om pet r ol eum fract i ons were also
used in t hei r devel opment , t
2. 4. 2 Pr e di c t i on o f Nor mal Boi l i ng Poi nt
2.4.2.1 Ri azi -Daubert Correlations
These correl at i ons are devel oped in Sect i on 2.3. The best in-
put pai r of par amet er s to predi ct boiling poi nt are (M, SG) or
(M, 1). For light hydr ocar bons and pet r ol eum fract i ons wi t h
mol ecul ar wei ght in t he r ange of 70-300, Eq. (2.40) ma y be
used f or boiling point:
Tb = 3.76587[exp(3.7741 x 10-3M + 2.98404SG
(2.56) - 4. 25288 10-3MSG)]M~176 -1'58262
For hydr ocar bons or pet r ol eum fract i ons wi t h mol ecul ar
wei ght in t he r ange of 300-700, Eq. (2.46b) is r ecommended:
Tb = 9.3369[exp(1.6514 x 10-4M + 1.4103SG
(2.57) - 7.5152 x 10-4MSG)]M~ -0"7276
Equat i on (2.57) is also appl i cabl e to hydr ocar bons havi ng
mol ecul ar wei ght r ange of 70-300, wi t h less accuracy. Esti-
mat i on of t he boi l i ng poi nt f r om t he mol ecul ar wei ght and
refract i ve i ndex par amet er ( I ) is given by Eq. (2.40) wi t h con-
st ant s in Table 2.5. The boi l i ng poi nt may also be cal cul at ed
t hr ough Kw and API gravi t y by usi ng definitions of t hese pa-
r amet er s gi ven in Eqs. (2.13) and (2.4).
2.4.2.2 Soreide Correlation
Based on ext ensi on of Eq. (2.56) and dat a on t he boiling poi nt
of some C7+ fract i ons, Sorei de [51, 52] devel oped t he follow-
ing correl at i on for t he nor mal boi l i ng poi nt of fract i ons in t he
r ange of 90-560~
Tb = 1071.28 -- 9.417 x 104 exp(--4.922 x 10-aM
(2.58) - 4.7685SG + 3.462 x 10-3MSG) M-~176 TM
This rel at i on is based on t he assumpt i on t hat t he boi l i ng poi nt
of ext r emel y l arge mol ecul es (M --~ c~) appr oaches a finite
val ue of 1071.28 K. Sorei de [52] compar ed f our met hods f or
t he pr edi ct i on of t he boiling poi nt of pet r ol eum fractions:
(1) Eq. (2.56), (2) Eq. (2.58), (3) Eq. (2.50), and (4) Twu
met hod given by Eqs. (2.89)-(2.92). For his dat a bank on
boiling poi nt of pet r ol eum fract i ons in t he mol ecul ar wei ght
r ange of 70-450, he f ound t hat Eq. (2.50) and t he Twn cor-
rel at i ons over est i mat e t he boiling poi nt while Eqs. (2.56)
and (2.58) are al most i dent i cal wi t h AAD of about 1%. Si nce
Eq. (2.56) was ori gi nal l y based on hydr ocar bons wi t h a mol ec-
ul ar wei ght r ange of 70-300, its appl i cat i on to heavi er com-
pounds shoul d be t aken wi t h care. I n addi t i on, t he dat abase
f or eval uat i ons by Sorei de was t he s ame as t he dat a used to
deri ve const ant s in his correl at i on, Eq. (2.58). For heavi er hy-
dr ocar bons (M > 300) Eq. (2.57) ma y be used.
For pur e hydr ocar bons f r om di fferent homol ogous fami l i es
Eq. (2.42) shoul d be used wi t h const ant s given in Table 2.6 for
Tb to est i mat e boi l i ng poi nt f r om mol ecul ar weight. A graph-
ical compar i s on of Eqs. (2.42), (2.56), (2.57), and (2.58) for
n-al kanes f r om C5 to C36 wi t h dat a f r om API-TDB [2] is shown
in Fig. 2.6.
2. 4. 3 Pr e di c t i on o f Speci f i c Gravity/API Gravity
Specific gravi t y of hydr ocar bons and pet r ol eum fract i ons is
nor mal l y avai l abl e because it is easily measur abl e. Specific
gravi t y and t he API gravi t y are rel at ed to each ot her t hr ough
Eq. (2.4). Therefore, when one of t hese par amet er s is known
t he ot her one can be cal cul at ed f r om t he definition of t he API
gravity. Several correl at i ons are pr esent ed in t hi s sect i on for
t he est i mat i on of specific gravi t y usi ng boi l i ng poi nt , mol ec-
ul ar weight, or ki nemat i c vi scosi t y as t he i nput par amet er s.
2.4.3.1 Ri azi -Daubert Methods
These correl at i ons for t he est i mat i on of specific gravi t y re-
qui re Tb and I or vi scosi t y and CH wei ght rat i o as t he i nput
par amet er s (Eq. 2.40). For light hydr ocar bons, Eq. (2.40) and
Table 2.5 can be used to est i mat e SG f r om di fferent i nput
par amet er s such as Tb and I.
SG = 2.4381 x 107 exp( - 4. 194 x 10-4Tb -- 23.55351
(2.59) + 3.9874 x lO-3Tbl)Tb~
wher e Tb is in kelvin. For heavy hydr ocar bons wi t h mol ecul ar
wei ght in t he r ange 300-700, t he following equat i on in t er ms
of M and I can be used [65]:
SG = 3.3131 x 104 exp( - 8. 77 x 1 0 - a M- 15.0496I
(2.60) + 3.247 x lO-3MI)M-~176 4"9557
Usually for heavy fractions, Tb is not avai l abl e and for t hi s rea-
son, M and I are used as t he i nput par amet er s. Thi s equat i on
also ma y be used f or hydr ocar bons bel ow mol ecul ar wei ght
of 300, if necessary. The accur acy of this equat i on is about 0.4
%AAD for 130 hydr ocar bons in t he car bon numbe r r ange of
C7-C50 (M ~ 70-700).
For heavi er fract i ons ( mol ecul ar wei ght f r om 200 t o 800)
and especi al l y when t he boiling poi nt is not avai l abl e t he fol-
l owi ng rel at i on in t er ms of ki nemat i c viscosities devel oped by
600
2. CHARACTERIZATION AND PROPERTI ES OF PURE HYDROCARBONS 5 9
500
o
o API Data 9 r
. . . . . . . Eq. 2.42 ~ , , ~, ~
. . . . Eq. 2.56
400
300
200
I00
0 1 i I i
0 5 10 15 20 25 30 35
i i I i I I L J I I I I I I I i I I I
Carbon Number
i
4O
F I G. 2.6--Estimation of boiling point of n-alkanes from various methods.
P
50 0.78
40
0,88
30
20
0.98
l 0
o
o
o
0 1.08
10 100 1000
Kinematic Viscosity at 37.8 ~ cSt
FIG. 2.7--API gravity and viscosity of heavy hydrocarbon fractions by Eq. (2.61).
Riazi and Dauber t may be used [67]:
(2.61) SG ---~ 0. 1157 r - 0. 16161
0.7717 [v38000) ] [1299(210 ) ]
in whi ch v38~100~ and ~d99(210 ) are ki nemat i c viscosities in cSt at
i 00 and 210~ (37.8 and 98.9~ respectively. Equat i on (2.61)
is shown in Fig. 2.7 and has also been adopt ed by t he API and
is i ncl uded in 1987 versi on of API-TDB [2]. This equat i on
gives an AAD of about 1.5% for 158 fract i ons in t he mol ecul ar
wei ght r ange of 200-500 ( ~SG r ange of 0.8-1.1).
For coal liquids and heavy resi dues t hat are hi ghl y aro-
mat i c, Tsonopoul os et al. [58] suggest t he following rel at i on
in t er ms of nor mal boiling poi nt (Tb) for t he est i mat i on of
specific gravity.
SG = 0.553461 + 1.15156To - 0.708142To 2 + 0.196237T 3
(2.62)
wher e To = ( 1. 8Tb- 459.67) in whi ch Tb is in kelvin. This
equat i on is not r ecommended for pur e hydr ocar bons or
pet r ol eum fract i ons and has an average relative devi at i on of
about 2.5% f or coal liquid fract i ons [58]. For pur e homol o-
gous hydr ocar bon groups, Eq. (2.42) wi t h const ant s given in
Table 2.6 for SG can be used. Anot her appr oach to est i mat e
specific gravi t y is to use t he Racket t equat i on and a known
densi t y dat a poi nt at any t emper at ur e as di scussed in Chap-
t er 5 (Section 5.8). A very si mpl e and pract i cal met hod of
est i mat i ng SG f r om densi t y at 20~ d, is given by Eq. (2.110),
whi ch will be di scussed in Sect i on 2.6.1. Once SG is esti-
mat ed t he API gravi t y can be cal cul at ed f r om its definition,
i.e., Eq. (2.4).
2 . 5 P R E D I C T I O N OF CRI TI CAL
P R O P E R T I E S A N D A C E N T R I C F ACT OR
Critical propert i es, especi al l y t he critical t emper at ur e and
pressure, and t he acent ri c f act or are i mpor t ant i nput par am-
et ers for EOS and general i zed correl at i ons t o est i mat e phys-
ical and t her modynami c pr oper t i es of fluids. As shown in
Chapt er 1 even smal l errors in pr edi ct i on of t hese proper-
ties great l y affect cal cul at ed physi cal propert i es. Some of t he
met hods wi del y used in t he pet r ol eum i ndust ry are given in
t hi s section. These procedures, as ment i oned in t he previ ous
sections, are mai nl y devel oped based on critical pr oper t i es of
pur e hydr ocar bons in whi ch val i dat ed exper i ment al dat a are
avai l abl e onl y up to C18. The following correl at i ons are given
in t er ms of boi l i ng poi nt and specific gravity. For ot her in-
put par amet er s, appr opr i at e correl at i ons given in Sect i on 2.3
shoul d be used.
2. 5. 1 Pr e d i c t i o n o f Cri t i cal Te mpe r a t ur e
a n d Pr e s s u r e
Simplified equat i ons to cal cul at e T~ and Pc of hydr ocar bons
in t he r ange of C5-C20 are given by Eq. (2.38) as follows [28].
(2.63) Tc = 19.06232Tb~ ~
(2.64) Pc = 5.53027 107Tb2"3125SG 2"3201
wher e Tc and Tb are in kelvin and Pc is in bar. In t he litera-
t ure, Eqs. (2.50), (2.63), and (2.64) are usual l y referred to as
Ri azi -Daubert or Riazi met hods. These equat i ons are r ecom-
mended onl y for hydr ocar bons in t he mol ecul ar wei ght r ange
of 70-300 and have been wi del y used in i ndust r y [2, 47, 49, 51,
54, 70]. However, t hese correl at i ons wer e repl aced wi t h mor e
accur at e correl at i ons pr esent ed by Eq. (2.40) and Table 2.5 in
t er ms of Tb and SG as given below:
Tc = 9. 5233[exp(-9. 314 x 1 0 - 4 Tb - - 0 . 5 4 4 4 4 2 S G
(2.65) +6. 4791 1 0 - 4 T b S G) ] T ~ 1 7 6 0"53691
Pc = 3.1958 x 105[exp(-8. 505 x 1 0 - 3 Tb - - 4 . 8 0 1 4 S G
(2.66) + 5.749 x 10-3TbSG)]Tb~ 4"0846
These correl at i ons were also adopt ed by t he API and have
been used in ma ny i ndust ri al comput er soft wares under t he
API met hod. The same l i mi t at i ons and uni t s as t hose for Eqs.
(2.63) and (2.64) appl y to t hese equat i ons. For heavy hydro-
car bons (>C20) t he following equat i ons are obt ai ned f r om
Eq. (2.46a) and const ant s in Table 2.9:
Tc = 35. 9413[exp(-6. 9 x 1 0 - 4 Tb - - 1.4442SG
(2.67) +4. 91 X 10-4TbSG)]Tb~ 1"2771
Pc = 6. 9575[exp(-1. 35 1 0 - 2 Zb - - 0.3129SG
(2.68) + 9.174 x 1 0 - 3 T b S G) ] T ~ -0"6807
I f necessary t hese equat i ons can also be used for hydrocar-
bons in t he r ange of C5-C20 wi t h good accuracy. Equat i on
(2.67) predi ct s val ues of Tc f r om C5 t o Cs0 wi t h %AAD of 0.4%,
but Eq. (2.68) predi ct s Pc wi t h AAD of 5.8%. The r eason for
t hi s hi gh average er r or is l ow values of Pc (i.e, a few bars) at
hi gher car bon number s whi ch even a smal l absol ut e devi at i on
shows a large val ue in t er ms of relative deviation.
2.5.1.2 API Methods
The API-TDB [2] adopt ed met hods devel oped by Riazi and
Dauber t for t he est i mat i on of pseudocri t i cal pr oper t i es of
pet r ol eum fract i ons. I n t he 1982 edi t i on of API-TDB, Eqs.
(2.63) and (2.64) were r ecommended for critical t emper at ur e
and pr essur e of pet r ol eum fractions, respectively, but in its
edi t i ons f r om 1987 to 1997, Eqs. (2.65) and (2.66) are i ncl uded
aft er eval uat i ons by t he API-TDB Commi t t ee. For pur e hy-
dr ocar bons, t he met hods r ecommended by API are based on
gr oup cont r i but i on met hods such as Ambrose, whi ch requi res
t he st r uct ur e of t he compound to be known. These met hods
are of mi nor pract i cal use in this book since pr oper t i es of
pur e compounds of i nt erest are given in Sect i on 2.3 and for
pet r ol eum fract i ons t he bul k pr oper t i es are used r at her t han
t he chemi cal st r uct ur e of individual compounds.
Kesl er and Lee [12] pr oposed correl at i ons for est i mat i on of
Tc and Pc si mi l ar t o t hei r cor r el at i on for mol ecul ar weight.
Tc = 189.8 + 450.6 SG + (0.4244 + 0.1174 SG)Tb
(2.69) +(0. 1441 - 1.0069 SG)105/Tb
In Pc = 5.689 - 0. 0566/ SG
- (0.43639 + 4. 1216/ SG + 0. 21343/ SG 2) x 1 0 - 3 Tb
(2.70) + (0.47579 + 1.182/SG + 0. 15302/ SG 2) x 10 - 6 T~
- (2.4505 + 9. 9099/ SG 2) 10 -1~ x T~
wh e r e Tb a n d Tc a r e i n kel vi n a n d Pc i s i n bar. I n t he s e equa-
t i ons a t t e mp t s we r e ma d e t o ke e p i n t e r n a l c ons i s t e nc y a mo n g
Tc a n d Pc t ha t at Pc e qua l t o 1 a t m, Tc i s c o i n c i d e d wi t h nor -
ma l boi l i ng poi nt , Tb. The c or r e l a t i ons we r e r e c o mme n d e d
by t he a u t h o r s f or t he mo l e c u l a r r a nge of 70- 700 (~C5-C50).
However , t he val ues of Tc a n d Pc f or c o mp o u n d s wi t h c a r b o n
n u mb e r s g r e a t e r t h a n Cl s us e d t o de ve l op t he a bove c or r e l a -
t i ons we r e not b a s e d on e x p e r i me n t a l evi dence.
2. 5. 1. 4 Cavet t Met hod
Cavet t [26] d e v e l o p e d e mp i r i c a l c or r e l a t i ons f or Tc a n d Pc i n
t e r ms of boi l i ng p o i n t a n d API gr avi t y, wh i c h a r e st i l l a va i l a bl e
i n s o me p r o c e s s s i mu l a t o r s as a n o p t i o n a n d i n s o me c a s e s
gi ve g o o d e s t i ma t e s of Tc a n d Pc f or l i ght t o mi d d l e di s t i l l a t e
p e t r o l e u m f r a c t i ons .
Tc = 426. 7062278 + (9. 5187183 x 10-~)(1.8Tb -- 459. 67)
- (6. 01889 x 10-4)(1.8Tb -- 459. 67) 2
-- (4. 95625 x 10-3)(API)(1. 8Tb - 459. 67)
(2. 71) +( 2. 160588 x 10-7)(1.8Tb - 459. 67) 3
+ (2. 949718 10-6)(API)(1. 8Tb - 459. 67) 2
+( 1. 817311 x 10- 8) ( API Z) ( 1. 8Tb- 459. 67) 2
l og(Pc) = 1. 6675956 + (9. 412011 x 10-4)(l . 8Tb - 459. 67)
- (3. 047475 10-6)(1.8Tb -- 459. 67) 2
- ( 2 . 0 8 7 6 1 1 x 10- 5) ( API ) ( 1. 8Tb- 459. 67)
(2. 72) + (1. 5184103 x 10-9)(1.8Tb -- 459. 67) 3
+ ( 1. 1047899 x 10-8)(API)(1. 8Tb - 459. 67) 2
- ( 4. 8271599 x 10-8)(API2)(1. 8Tb - 459. 67)
+ ( 1. 3949619 x 10-1~ - 459. 67) 2
I n t he s e r e l a t i ons Pc i s i n b a r whi l e Tc a n d Tb a r e i n kel vi n a n d
t he API gr a vi t y i s def i ned i n t e r ms of speci f i c gr a vi t y t h r o u g h
Eq. (2. 4). Ter ms (1.8Tb - 459. 67) c o me f r om t he f act t h a t t he
uni t of Tb i n t he or i gi na l r e l a t i ons wa s i n de gr e e s f a hr e nhe i t .
2. 5. 1. 5 Twu Met hod f or Tc, Pc, Vc, and M
Twu [30] i ni t i a l l y c o r r e l a t e d c r i t i c a l p r o p e r t i e s (To, Pc, Vc),
speci f i c gr avi t y ( SG) , a nd mo l e c u l a r we i ght (M) of n- a l ka ne s
t o t he boi l i ng p o i n t (Tb). The n t he di f f e r e nc e b e t we e n s pe-
ci fi c gr a vi t y of a h y d r o c a r b o n f r om o t h e r g r o u p s ( SG) a n d
speci f i c gr avi t y of n- a l ka ne ( SG ~ wa s us e d as t he s e c o n d pa -
r a me t e r t o c or r e l a t e p r o p e r t i e s of h y d r o c a r b o n s f r o m di f f er -
e nt gr oups . Thi s t ype of c or r e l a t i on, k n o wn as a p e r t u r b a t i o n
e xpa ns i on, wa s f i r st i n t r o d u c e d by Ke s l e r - Le e - S a n d l e r ( KLS)
[71] a n d l a t e r u s e d by Li n a n d Chao [72] t o c or r e l a t e c r i t i c a l
p r o p e r t i e s of h y d r o c a r b o n s us i ng n- a l ka ne as a r e f e r e nc e f l ui d
a n d t he speci f i c gr avi t y di f f e r e nc e as t he c o r r e l a t i n g p a r a m-
et er. However , KLS c or r e l a t i ons d i d not f i nd p r a c t i c a l a ppl i -
c a t i o n b e c a u s e t he y def i ned a n e w t h i r d p a r a me t e r s i mi l a r t o
t he a c e nt r i c f a c t or wh i c h i s not a va i l a bl e f or p e t r o l e u m mi x-
t ur es . Li n a n d Chao (LC) c o r r e l a t e d Tc, l n(Pc), w, SG, a n d Tb
of n- a l ka ne s f r om CI t o C20 t o mo l e c u l a r wei ght , M. The s e
p r o p e r t i e s f or al l o t h e r h y d r o c a r b o n s i n t he s a me mo l e c u l a r
we i ght we r e c o r r e l a t e d t o t he di f f e r e nc e i n Tb a n d SG of t he
s u b s t a n c e of i nt e r e s t wi t h t ha t of n- al kane. Ther ef or e, LC cor-
r e l a t i o n s r e q u i r e t hr e e i n p u t p a r a me t e r s of Tb, SG, a n d M f or
e a c h pr ope r t y. Ea c h c o r r e l a t i o n f or e a c h p r o p e r t y c o n t a i n e d
as ma n y as 33 n u me r i c a l c ons t a nt s . Thes e c o r r e l a t i o n s a r e
i n c l u d e d i n s o me r e f e r e nc e s [49Z1. However , t he Twu c or r e l a -
t i ons a l t h o u g h b a s e d on t he s a me f o r ma t as t he KLS o r LC
r e q u i r e i n p u t p a r a me t e r s of Tb a n d SG a n d a r e a p p l i c a b l e t o
h y d r o c a r b o n s b e y o n d C20. Fo r he a vy h y d r o c a r b o n s s i mi l a r
t o t he a p p r o a c h of Le e - Ke s l e r [12], Twu [30] us e d t he cr i t -
i cal p r o p e r t i e s b a c k c a l c u l a t e d f r o m v a p o r p r e s s u r e d a t a t o
e x p a n d hi s d a t a b a n k on t he c r i t i c a l c o n s t a n t s of p u r e hydr o-
c a r b o n c o mp o u n d s . F o r t hi s r e a s o n t he Twu c o r r e l a t i o n s have
f o u n d a wi d e r r a nge of a ppl i c a t i on. The Twu c o r r e l a t i o n s f or
t he c r i t i c a l pr ope r t i e s , s peci f i c gr avi t y, a nd mo l e c u l a r we i ght
of n- a l ka ne s a r e as f ol l ows:
T~ ~ = Tb(0. 533272 + 0. 34383 x 10 -3 Tb
+2 . 5 2 6 1 7 x 10 -7 T~ - 1.658481 10 -1~ x T 3
(2. 73) +4 . 6 0 7 7 3 x 1024 x Tb-13) -1
(2. 74) ot = 1 - Tt,/T~
(2. 75)
P~ = (1. 00661 + 0.31412ot 1/2 + 9. 161063
+ 9. 504132 + 27.358860t4) 2
V~ ~ = (0. 34602 + 0.301710~ + 0.93307ot 3 + 5655.414314) - s
(2. 76)
SG ~ = 0. 843593 - 0 . 1 2 8 6 2 4 o t - 3 . 3 6 1 5 9 0 t 3 - 13749.5312
(2. 77)
Tb = exp( 5. 12640 + 2. 71579fl -- 0. 286590fl 2 -- 39. 8544/ f l
(2. 78) --0.122488/fl 2) -- 13. 7512fl + 19. 6197fl 2
wh e r e Tb i s t he boi l i ng p o i n t of h y d r o c a r b o n s i n kel vi n a n d
3 = l n ( M ~ i n wh i c h M ~ i s t he mo l e c u l a r we i ght n- a l ka ne r ef -
e r e nc e c o mp o u n d . Cr i t i cal p r e s s u r e is i n b a r a n d c r i t i c a l vol -
u me i s i n cm3/ mol . Da t a on t he p r o p e r t i e s of n- a l ka ne s f r om
C1 t o C100 we r e u s e d t o o b t a i n t he c o n s t a n t s i n t he a b o v e r el a-
t i ons . Fo r he a vy h y d r o c a r b o n s b e y o n d C20, t he va l ue s of t h e
c r i t i c a l p r o p e r t i e s o b t a i n e d f r o m v a p o r p r e s s u r e d a t a we r e
us e d t o o b t a i n t he c ons t a nt s . The a u t h o r o f t he s e c or r e l a -
t i ons al s o i ndi c a t e s t ha t t he r e i s i me r n a l c ons i s t e nc y b e t we e n
Tc a n d Pc as t he cr i t i cal t e mp e r a t u r e a p p r o a c h e s t he boi l i ng
poi nt . Eq u a t i o n (2. 78) i s i mp l i c i t irt c a l c ul a t i ng M ~ f r o m Tb. To
sol ve t hi s e q u a t i o n by i t e r a t i o n a s t a r t i ng val ue c a n be f o u n d
f r om t he f ol l owi ng r e l a t i on:
(2. 79) M ~ = Tb/(5. 8 -- 13'.0052Tb)
F o r o t h e r h y d r o c a r b o n s a n d pet r o] [ eum f r a c t i ons t he r e l a t i o n
f or t he e s t i ma t i o n of To, Pc, Vc, a nd M a r e as f ol l ows:
Critical temperature
(2. 80) Tc = Tc[(1 + 2f r)/ (1 - 2fr)] 2
f r = AS GT [ - 0.27016/T~/2
(2. 81) + (0. 0398285 - 0.706691/T~/2)ASGr]
(2. 82)
Critical volume
(2. 83)
AS Gr = e xp[ 5( SG ~ - SG) ] - 1
Vc = vf [ ( 1 + 2 f v) / ( 1 - 2 f v ) ] 2
62 CHARACTERIZATION AND PROPERTI ES OF PETROLEUM FRACTI ONS
fv = ASGv[O.347776/T~/2
(2.84) + ( - 0.182421 + 2.248896/T~/2)ASGv]
(2.85) ASGv = exp[4(SG ~ - SG2)] - 1
Critical pressure
(2.86) Pc = P~(TclT~) x (Vc/Vc)[(1 + 2fv)l(1 - 2f p) ] 2
fe = ASGe[(2.53262 - 34.4321/T~/2 - 2.30193Tb/1000)
+( -- 11.4277 + 187.934/Tb 1/2 + 4.11963Tb/lOOO)ASGv]
(2.87)
(2.88) ASGv = exp[0.5(SG ~ - SG)] - 1
Molecular weight
(2.89) ln(M) = (ln M~ + 2 f~)/(1 - 2fM)] 2
f?a = ASGm[x + (-0. 0175691 + 0.143979/T~/2)ASGM]
(2.90)
(2.91) X --- ]0.012342 - 0.244541/T1/2[
(2.92) ASGM = exp[5(SG ~ - SG)] - 1
In the above relations Tb and Tc are in kelvin, Vc is in cm3/mol,
and Pc is in bar. One can see that these correlations should
be solved simultaneously because they are highly interrelated
to each other and for this reason relations for estimation of
M and Vc based on this method are also presented in this part.
Example 2.6---Estimate the molecular weight of n-eicosane
(C20H42) from its normal boiling point using Eq. (2.49) and
the Twu correlations.
Solution--n-Eicosane is a normal paraffin whose molecular
weight and boiling point are given in Table 2. I as M = 282.55
and Tb = 616.93 K. Substituting Tb in Eq. (2.49) gives M =
282.59 (%AD = 0.01%). Using the Twu method, first an initial
guess is calculated through Eq. (2.79) as M ~ = 238 and from
iteration the final value of M ~ calculated from Eq. (2.78) is
281.2 (%AD = 0.48%). Twu method for estimation of proper-
ties of hydrocarbons from other groups is shown later in the
next example. r
2.5.1.6 Wi nn-Mobi l Met hod
Winn [25] developed a convenient nomograph to estimate var-
ious physical properties including molecular weight and the
pseudocritical pressure for petroleum fractions. Mobil [73]
proposed a similar nomograph for the estimation of pseudo-
critical temperature. The input data in both nomographs are
boiling point (or Kw) and the specific gravity (or API gravity).
As part of the API project to computerize the graphical meth-
ods for estimation of physical properties, these nomographs
were reduced to equation forms for computer applications by
Riazi [36] and were later reported by Sire and Daubert [74].
These empirically developed correlations have forms similar
to Eq. (2.38) and for M, To and Pc are as follows.
(2.93) M = 2.70579 x 10-5T~4966SG -1'174
(2.94) In Tc = -0. 58779 + 4.2009T~176 ~176
(2.95) Pc = 6.148341 x 107Tb23177SG 2"4853
where Tb and Tc are in kelvin and Pc is in bar. Comparing
values estimated from these correlations with the values from
the original figures gives AAD of 2, 1, and 1.5% for M, To, and
Pc, respectively, as reported in Ref. [36]. In the literature these
equations are usually referred as Winn or Sim-Daubert and
are included in some process simulators. The original Winn
nomograph for molecular weight and some other properties
is given in Section 2.8.
2.5.1.7 Tsonopoulos Correlations
Based on the critical properties of aromatic compounds,
Tsonopoulos et al. [34] proposed the following correlations
for estimation of Tc and Pc for coal liquids and aromatic-rich
fractions.
lOgl0 Tc = 1.20016 + 0.61954(log10 Tb)
(2.96) + 0.48262(1og10 SG) + 0.67365(log10 SG) 2
log10 Pc = 7.37498 - 2.15833(log10 Tb)
(2.97) + 3.35417(log10 SG) + 5.64019(log10 SG) 2
where Tb and Tc are in kelvin and Pc is in bar. These correla-
tions are mainly recommended for coal liquid fractions and
they give average errors of 0.7 and 3.5% for the estimation of
critical temperature and pressure of aromatic hydrocarbons.
2. 5. 2 Predi ct i on o f Critical Vol ume
Critical volume, Vo is the third critical property that is not
directly used in EOS calculations, but is indirectly used to
estimate interaction parameters (kii) needed for calculation
of mixture pseudocritical properties or EOS parameters as
will be discussed in Chapter 5. In some corresponding state
correlations developed to estimate transport properties of flu-
ids at elevated pressure, reduced density (Vc/V) is used as the
correlating parameter and values of Vc are required as shown
in Chapter 8. Critical volume is also used to calculate critical
compressibility factor, Zc, as shown by Eq. (2.8).
2.5.2.1 Ri azi -Daubert Methods
A simplified equation to calculate Vc of hydrocarbons in the
range of C5-C20 is given by Eq. (2.38) as follows.
(2.98) Vc = 1.7842 x 10-4T2"a829SG -1"683
in which Vc is in cma/mol and Tb is in kelvin. When evalu-
ated against more than 100 pure hydrocarbons in the carbon
range of C5--C20 an average error of 2.9% was observed. This
equation may be used up to Css with reasonable accuracy. For
heavier hydrocarbons, Vc is given by Eq. (2.46a) and in terms
of Tb and SG is given as
Vr = 6.2 x 101~ x 10-3Tb -- 28.5524SG
(2.99) + 1.172 x aO-2TbSG)]Tl2~ 17"2074
where Vc is in cm3/mol. Although this equation is recom-
mended for hydrocarbons heavier than C20 it may be used, if
necessary, for the range of C5-C50 in which the AAD is about
2.5%. To calculate Vc from other input parameters, Eqs. (2.40)
and (2.46b) with Tables 2.5 and 2.9 may be used.
2. CHARACTERI ZATI ON AND PROPERTI ES OF PUAE HYDROCARBONS 63
2.5.2.2 Hall-Yarborough Met hod
Thi s met hod for est i mat i on of critical vol ume follows t he gen-
eral f or m of Eq. (2.39) in t er ms of M and SG and is given as
[75]:
(2.100) Vc = 1.56 Ml l SSG -0"7935
Predi ct i ve met hods in t er ms of M and SG are usual l y useful
for heavy fract i ons where distillation dat a may not be avail-
able.
2.5.2.3 API Met hod
I n t he mos t recent API-TDB [2], t he Reidel met hod is rec-
omme nde d t o be used for t he critical vol ume of pur e hydro-
car bons given in t er ms of To, Pc, and t he acent ri c f act or as
follows:
RTc
(2.101) Vc =
P~[3.72 + 0.26(t~R - 7.00)]
in whi ch R is t he gas const ant and OR is t he Riedel f act or given
in t er ms of acent ri c factor, o3.
(2.102) Ot R = 5.811 + 4.919o3
I n Eq. (2.101), t he uni t of Vc mai nl y depends on t he uni t s of
To, Pc, and R used as t he i nput par amet er s. Values of R in
di fferent uni t syst ems are given in Sect i on 1.7.24. To have Vc
in t he uni t of cma/ mol , Tc mus t be in kelvin and if Pc is in
bar, t hen t he value of R mus t be 83.14. The API met hod for
cal cul at i on of critical vol ume of mi xt ures is based on a mi xi ng
rul e and pr oper t i es of pur e compounds, as will be di scussed
in Chapt er 5. Twu's met hod for est i mat i on of critical vol ume
is given in Sect i on 2.5.1.
2. 5. 3 Pr e d i c t i o n o f Cri t i cal Co mpr e s s i bi l i t y Fa c t o r
Critical compressi bi l i t y factor, Zc, is defined by Eq. (2.8) and
is a di mensi onl ess paramet er. Values of Zc given in Table 2.1
show t hat t hi s pa r a me t e r is a charact eri st i c of each com-
pound, whi ch vari es f r om 0.2 to 0.3 for hydr ocar bons in t he
r ange of C1-C20. General l y it decreases wi t h i ncreasi ng car-
bon numbe r wi t hi n a homol ogous hydr ocar bon group. Zc is
in fact value of compressi bi l i t y factor, Z, at t he critical poi nt
and t her ef or e it can be est i mat ed f r om an EOS. As it will be
seen in Chapt er 5, t wo- par amet er EOS such as van der Waals
or Peng- Robi nson give a single val ue of Zc for all compounds
and f or t hi s r eason t hey are not accur at e at t he critical re-
gion. Thr ee- par amet er EOS or general i zed correl at i ons gen-
eral l y give mor e accur at e val ues for Zc. On t hi s basi s s ome
r esear cher s cor r el at ed Zc t o t he acent ri c factor. An exampl e
of such correl at i ons is given by Lee- Kesl er [27]:
(2.103) Z~ = 0.2905 - 0.085w
Ot her references give vari ous versi ons of Eq. (2.103) wi t h
slight di fferences in t he numer i cal const ant s [6]. Anot her ver-
si on of t hi s equat i on is given in Chapt er 5. However, such
equat i ons are onl y appr oxi mat e and no single par amet er is
capabl e of predi ct i ng Zc as its nat ur e is di fferent f r om t hat of
acent ri c factor.
Anot her met hod t o est i mat e Z~ is to combi ne Eqs. (2.101)
and (2.102) and usi ng t he definition of Zc t hr ough Eq. (2.8)
to devel op t he following rel at i on f or Zc in t er ms of acent ri c
factor, m:
1.1088
(2.104) Zc --
o3 + 3.883
Usually for light hydr ocar bons Eq. (2.103) is mor e accur at e
t han is Eq. (2.104), whi l e for heavy compounds it is t he op-
posite; however, no compr ehensi ve eval uat i on has been made
on t he accur acy of t hese correl at i ons.
Based on t he met hods pr esent ed in this chapter, t he mos t
appr opr i at e met hod t o est i mat e Zc is first to est i mat e To, Pc,
and Vc t hr ough met hods given in Sect i ons 2.5.1 and 2.5.2 and
t hen to cal cul at e Zc t hr ough its definition given in Eq. (2.8).
However, for consi st ency in est i mat i ng To, Pc, and Vc, one
met hod shoul d be chosen for cal cul at i on of all t hese t hree
par amet er s. Fi gure 2.8 shows pr edi ct i on of Zc f r om vari ous
correl at i ons for n-al kanes f r om C.5 to C36 and compar i ng wi t h
dat a r epor t ed by API-TDB [2].
Example 2.7--The critical pr oper t i es and acent ri c f act or of
n-hexatriacontane (C36H74) are given as follows [20]: Tb =
770.2 K, SG = 0.8172, M = 506.98, Tc = 874.0 K, Pc = 6.8 bar,
Vc = 2090 cm3/mol, Zc = 0.196, and w = 1.52596. Cal cul at e
M, To, Pc, Vc, and Zc f r om t he following met hods and for each
pr oper t y cal cul at e t he per cent age relative devi at i on (%D) be-
t ween est i mat ed value and ot her act ual value.
a. Ri azi - Dauber t met hod: Eq. (2.38)
b. API met hods
c. Ri azi - Dauber t ext ended met hod: Eq. (2.46a)
d. Ri azi - Sahhaf met hod for hon~tologous groups, Eq. (2.42),
Pc f r om Eq. (2.43)
e. Lee- Kesl er met hods
f. Cavett met hod (only Tc and Pc), Zc f r om Eq. (2.104)
g. Twu met hod
h. Wi nn met hod (M, Tc, Pc) and t t al l - Yar bor ough f or Vc
i. Tabul at e %D for vari ous pr oper t i es and met hods.
Solution--(a) Ri azi - Dauber t met hod by Eq. (2.38) f or M, To,
Pc, and Vc are given by Eqs. (2.50), (2.63), (2.64), and (2.98).
(b) The API met hods for pr edi ct i on of M, To, Pc, Vc, and
Zc are expressed by Eqs. (2.51), (2.65), (2.66), (2.101), and
(2.104), respectively. (c) The ext ended Ri azi - Dauber t met hod
expressed by Eq. (2.46a) for hydr ocar bons heavi er t han C20
and const ant s for t he critical pr oper t i es are given in Table 9.
For Tc, Pc, and Vc t hi s met hod is pr esent ed by Eqs. (2.67),
(2.68), and (2.99), respectively. The rel at i on for mol ecul ar
wei ght is t he s ame as t he API met hod, Eq. (2.51). (d) Ri azi -
Sahhaf met hod is given by Eq. (42) in whi ch t he const ant s
for n-al kanes given in Table 2.6 shoul d be used. I n usi ng this
met hod, if t he gi ven val ue is boiling poi nt , Eq. (2.49) shoul d be
used to cal cul at e M f r om Tb. Then t he pr edi ct ed M will be used
to est i mat e ot her propert i es. I n t hi s met hod Pc is cal cul at ed
f r om Eq. (2.43). For par t s a, b, c, g, and h, Zc is cal cul at ed f r om
its definition by Eq. (2.8). (e) Lee- Kesl er met hod for M, To,
Pc, and Zc are given in Eqs. (2.54), (2.69), (2.70), and (2.103),
respectively. Vc shoul d be back cal cul at ed t hr ough Eq. (2.8)
usi ng Tc, Pc, and Zc. (f) Si mi l arl y for t he Cavett met hod, Tc and
Pc are cal cul at ed f r om Eqs. (2.71) and (2.72), whi l e Vc is back
cal cul at ed f r om Eq. (2.8) wi t h Zc cal cul at ed f r om Eq. (2.104).
(g) The Twu met hods are expressed by Eqs. (2.73)-(2.92) for
M, To, Pc, and Vc. Zc is cal cul at ed f r om Eq. (2.8). (h) The Wi nn
6 4 CHARACT ERI Z AT I ON AND P R OP E R T I E S OF P E T R OL E UM FRACT I ONS
0.30
84
@
r.)
9 DIPPR Data
. . . . . . . Eq. 2. 103
. . . . . Eq. 2. 104
~ Eq. 2. 42
- - , . - . - : ,
9 @ ~ 1 7 6
9 " . .
020. ~ i t s I i , , . i I , , , i i , , , ,
5 l 0 1 5 2 0 25
Carbon Number
FI G. 2. 8- - - Est i mat i on of cri t i cal compr essi bi l i t y f act or of n- al kanes f r om var i ous
met hods.
met hod for M, To, and Pc are given by Eqs. (2.93)-(2.95). I n
part h, Vc is cal cul at ed from t he Hal l -Yarborough t hr ough
Eq. (2.100) and Zc is cal cul at ed t hr ough Eq. (2.8). Summar y
of results is gi ven i n Table 2.11. No j udgement can be made
on accuracy of these different met hods t hr ough this single-
poi nt eval uat i on. However, met hods of Ri azi - Sahhaf (Part d)
and Twu (Part g) give t he most accurat e results for this par-
t i cul ar case. The reason is t hat these met hods have specific
rel at i ons for n-al kanes fami l y and n-hexat ri acont ane is hydro-
car bon from this family. I n addi t i on, the values for the critical
propert i es from DIPPR [20] are est i mat ed val ues rat her t han
t rue experi ment al values. #
2 . 5 . 4 P r e d i c t i o n o f A c e n t r i c F a c t o r
Acentric factor, w, is a defined par amet er t hat is not directly
measurabl e. Accurate values of t he acent ri c factor can be ob-
t ai ned t hr ough accurat e values of T~, Pc, and vapor pressure
wi t h use of Eq. (2.10). At t empt s to correlate co wi t h par ame-
ters such as Tb and SG all have failed. However, for homol o-
gous hydr ocar bon groups t he acent ri c factor can be rel at ed to
mol ecul ar wei ght as given by Eqs. (2.42) or (2.44). For ot her
compounds t he acent ri c factor shoul d be cal cul at ed t hr ough
its definition, i.e., Eq. (2.10), wi t h the use of a correl at i on to
est i mat e vapor pressure. Use of an accurat e correl at i on for
vapor pressure woul d resul t i n a more accurat e correl at i on
for t he acent ri c factor. Met hods of t he cal cul at i on of t he vapor
pressure are di scussed i n Chapt er 7. There are t hree si mpl e
correl at i ons for the est i mat i on of vapor pressure t hat can be
used i n Eq. (2.10) to derive correspondi ng correl at i ons for the
acent ri c factor. These t hree met hods are present ed here.
2. 5. 4. 1 Lee- Kes l er Me t h o d
They proposed t he following rel at i ons for t he est i mat i on of
acent ri c factor based on t hei r proposed correl at i on for vapor
pressure [27].
TABLE 2.11--Prediction of critical properties of n-hexatriacontane from different methods a (Example 2.7).
M Tc, K APe,bar Vc, cma/mol Zc
Part Method(s) Est.** %D Est. %D Est. %D Est. %D Est. %D
Data from DIPPR [20] 507.0 --. 874.0 ... 6.8 .-- 2090.0 --. 0.196 -.-
a R-D: Eq. (2.38) 445.6 -12.1 885.8 1.3 7.3 7.4 1894.4 -9.3 0.188 -4. 2
b API Methods 512.7 1.1 879.3 0.6 7.37 8.4 1849. 7 -11.5 0.205 4.6
c R- D (ext.): Eq. (2.46a) . . . . . . 870.3 -0. 4 5.54 -18.5 1964.7 -6. 0 0.150 -23.3
d R-S: Eqs. 2.42 &2.43 506.9 0 871.8 -0. 3 5.93 -12.8 1952.5 -6. 6 0.16 -18.4
e L- K Methods 508.1 0.2 935.1 7.0 5. 15 -24.3 2425.9 16. 0 0. 161 -18.0
f Cavett & Eq. (2.104) . . . . . . 915.5 4.7 7.84 15. 3 . . . . . . . . . . . .
g Twu 513.8 1.3 882.1 0.9 6.02 -11.4 2010.0 -3. 8 0.165 -15.8
h Winn and H- Y 552.0 8.9 889.5 1.77 7.6 11. 8 2362.9 13.1 0.243 24.0
aThe references for the methods are (a) R-D: Riazi-Daubert [28]; (b) API: Methods in the API-TDB [2]; (c) Extended Riazi-Dubert [65]; (d) Riazi-Sahhaf [31];
(e) Kesler-Lee [12] and Lee-Kesler [27]; (f) Cavett [26]; Twu [31]; (h) Winn [25] and Hall-Yarborough [75]. Est.: Estimated value. %D: % relative deviation defined
in Eq. (2.134).
F o r Tbr < 0.8 (<C20 ~ M < 280)
- In Pc/1.01325 - 5.92714 + 6.09648/Tb~ + 1.28862 In Tb~ -- 0.169347T~
15.2518 - 15.6875/Tbr - 13.4721 In Tbr + 0.43577Tb6r
(2. 105)
wh e r e Pc i s i n b a r a n d Tbr i s t he r e d u c e d boi l i ng p o i n t wh i c h
i s def i ned as
(2. 106) Tbr = Tb/Tc
a n d Ke s l e r - Le e [12] p r o p o s e d t he f ol l owi ng r e l a t i o n f or Tbr >
0. 8 ( ~>C20 ~ M > 280):
0) = - 7 . 9 0 4 + 0. 1352Kw - 0. 007465K 2 + 8.359Tb~
(2. 107) + (1. 408 -- O.Ol063Kw)/Tbr
i n wh i c h Kw i s t he Wa t s o n c h a r a c t e r i z a t i o n f a c t or def i ned by
Eq. (2. 13). Eq u a t i o n (2. 105) ma y al s o be us e d f or c o mp o u n d s
h e a v i e r t h a n C20 (Tbr > 0. 8) wi t h o u t ma j o r e r r o r as s h o wn i n
t he e x a mp l e be l ow
2. 5. 4. 2 Edmi st er Met hod
The Ed mi s t e r c or r e l a t i on [76] i s de ve l ope d o n t he s a me ba s i s
as Eq. (2. 105) b u t us i ng a s i mp l e r t wo - p a r a me t e r e q u a t i o n
f or t he v a p o r p r e s s u r e de r i ve d f r om Cl a pe yr on e q u a t i o n ( see
Eq. 7.15 i n Ch a p t e r 7).
(2. 108) o ) = ( 3 ) x ( T b r ~ X I1Ogl0 ( Pc
'
wh e r e logm0 i s t he l o g a r i t h m b a s e 10, Tbr i s t he r e d u c e d boi l i ng
poi nt , a n d Pc i s t he c r i t i c a l p r e s s u r e i n bar. As i s c l e a r f r om
Eqs. (2. 105) a n d (2. 108), t he s e t wo me t h o d s r e q u i r e t he s a me
t h r e e i n p u t p a r a me t e r s , namel y, boi l i ng poi nt , c r i t i c a l t e mpe r -
at ur e, a n d c r i t i c a l pr e s s ur e . Eq u a t i o n s (2. 105) a n d (2. 108) a r e
di r e c t l y de r i ve d f r o m v a p o r p r e s s u r e c o r r e l a t i o n s di s c us s e d i n
Ch a p t e r 7.
2.5.4.3 Korsten Met hod
The Ed mi s t e r me t h o d u n d e r e s t i ma t e s a c e nt r i c f a c t or f or
he a vy c o mp o u n d s a n d t he e r r o r t e nds t o i n c r e a s e wi t h i n-
c r e a s i ng mo l e c u l a r we i ght of c o mp o u n d s b e c a u s e t he v a p o r
p r e s s u r e r a p i d l y de c r e a s e s . Mo s t r e c e nt l y Ko r s t e n [77] mo d -
i f i ed t he Cl a pe yr on e q u a t i o n f or v a p o r p r e s s u r e of h y d r o -
c a r b o n s ys t e ms a n d de r i ve d a n e q u a t i o n ver y s i mi l a r t o t he
Ed mi s t e r me t h o d :
(2. 109) o ) = 0. 5899 [ ~ x l og - 1
\ 1 - Tr r / 1.0~25
To c o mp a r e t hi s e q u a t i o n wi t h t he Ed mi s t e r e qua t i on, t he
f a c t or (3/7), wh i c h i s e qui va l e nt t o 0. 42857 i n Eq. (2. 108), ha s
be e n r e p l a c e d by 0. 58990 a n d t he e x p o n e n t of Tb~ ha s be e n
c h a n g e d f r om 1 t o 1.3 i n Eq. (2. 109).
One c a n r e a l i z e t h a t a c c u r a c y of t he s e me t h o d s ma i n l y de-
p e n d s on t he a c c u r a c y of t he i n p u t p a r a me t e r s . However , f or
p u r e c o mp o u n d s i n wh i c h e x p e r i me n t a l d a t a on p u r e hydr o-
c a r b o n s a r e a va i l a bl e t he Le e - Ke s l e r me t h o d , Eq. (2. 105),
gi ves a n AAD of 1- 1. 3%, whi l e t he Ed mi s t e r me t h o d gi ves
h i g h e r e r r o r of a b o u t 3- 3. 5%. The Ko r s t e n me t h o d i s n e w
a n d i t ha s not b e e n ext ens i vel y e va l ua t e d f or p e t r o l e u m f r ac-
t i ons , b u t f or p u r e h y d r o c a r b o n s i t s e e ms t h a t i t i s mo r e ac-
c u r a t e t h a n t he Ed mi s t e r me t h o d b u t l ess a c c u r a t e t h a n t he
Le e - Ke s l e r me t h o d . Gener al l y, t he Ed mi s t e r me t h o d i s not
r e c o mme n d e d f or p u r e h y d r o c a r b o n s a n d i s u s e d t o cal cu-
l at e a c e nt r i c f act or s of u n d e f i n e d p e t r o l e u m f r act i ons . F o r
p e t r o l e u m f r act i ons , t he p s e u d o c r i t i c a l t e mp e r a t u r e a n d pr e s -
s ur e n e e d e d i n Eqs. (2. 105) a n d (2. 108) mu s t be e s t i ma t e d
f r om me t h o d s d i s c u s s e d i n t hi s s ect i on. Usual l y, wh e n t he
Cavet t o r Wi n n me t h o d s a r e us e d t o e s t i ma t e Tc a n d Pc,
t he a c e nt r i c f a c t or i s c a l c u l a t e d b y t he Ed mi s t e r me t h o d .
Al l o t h e r me t h o d s f or t he e s t i ma t i o n of c r i t i c a l p r o p e r t i e s
us e Eq. (2. 105) f or c a l c ul a t i on of t he a c e nt r i c f act or . Equa -
t i on (2. 107) i s a p p l i c a b l e f or he a vy f r a c t i ons a n d a de t a i l e d
e va l ua t i on of i t s a c c u r a c y i s n o t a va i l a bl e i n t he l i t e r a t ur e . Fur -
t he r e va l ua t i on of t he s e me t h o d s is gi ven i n Se c t i on 2.9. The
me t h o d s of c a l c ul a t i on of t he a c e nt r i c f a c t or f or p e t r o l e u m
f r a c t i ons a r e di s c us s e d i n t he next chapt er .
Exampl e 2. 8- - - Cr i t i cal p r o p e r t i e s a n d a c e nt r i c f a c t or of
n - h e x a t r i a c o n t a n e (C36H74) are gi ven as b y DI PPR [20] as
Tb ---- 770. 2K, SG = 0. 8172, Tc = 874. 0 K, Pc = 6.8 bal ; and~o =
1. 52596. Es t i ma t e t he a c e n t r i c f a c t o r o f n - h e x a t r i a c o n t a n e us -
i ng t he f ol l owi ng me t h o d s :
a. Ke s l e r - Le e me t h o d wi t h Tc, Pc f r om DI PPR
b. Le e - Ke s l e r me t h o d wi t h To Pc f r om DI PPR
c. Ed r n i s t e r me t h o d wi t h Tc, Pc f r om DI PPR
d. Ko r s t e n me t h o d wi t h To Pc f r om DI PPR
e. Ri a z i - S a h h a f c or r e l a t i on, Eq. (2. 42)
TABLE 2. 12--Prediction of acentric factor of n-hexatriacontane from different
methods (Example 2.8).
Method for %
Part Met hod for o) Tc & pa To. K Pc, bar Calc. o) Rel. de~
a Kesler-Lee DIPPR 874.0 6.8 1.351 - 11. 5
b Lee-Kesl er DIPPR 874.0 6.8 1.869 22.4
c Edmi st er DIPPR 874.0 6.8 1.63 6.8
d Korst en DIPPR 874.0 6.8 1.731 13.5
e Ri azi - Sahhaf not needed . . . . . . 1.487 - 2. 6
f Korsten R-D-80 885.8 7.3 1.539 0.9
g Lee-Kesl er API 879.3 7.4 1.846 21.0
h Korsten Ext. RD 870.3 5.54 1.529 0.2
i Lee-Kesl er R-S 871.8 5.93 1.487 - 2. 6
j Edmi st er Winn 889.5 7.6 1.422 - 6. 8
k Kesler-Lee L-K 935.1 5.15 0.970 - 36. 4
1 Lee-Kesl er Twu 882.1 6.03 1.475 - 3. 3
aR-D-80: Eqs. (2.63) and (2.64); API: Eqs. (2.65) and (2.66); Ext. RD: Eqs. (2.67) and (2.68);
R-S: Eqs. (2.42) and (2.43); Winn: Eqs. (2.94) and (2.95); L-K: Eqs. (2.69) and (2.70);
Twu: Eqs. (2.80) and (2.86).
f. Lee- Kesl er met hod wi t h To Pc obt ai ned f r om Part a in
Exampl e 2.6
g. Lee-Kesl er met hod wi t h Tc, Pc obt ai ned f r om Part b in
Exampl e 2.6
h. Lee- Kesl er met hod wi t h To Pc obt ai ned f r om Par t c in
Exampl e 2.6
i. Lee- Kesl er met hod wi t h To Pc obt ai ned f r om Part d in
Exampl e 2.6
j. Edmi st er met hod wi t h Tc, Pc obt ai ned f r om Part h in Ex-
ampl e 2.6
k. Lee- Kesl er met hod wi t h Tc, Pc obt ai ned f r om Part e in
Exampl e 2.6
1. Lee- Kesl er met hod wi t h T~, Pc obt ai ned f r om Par t g in
Exampl e 2.6
m. Tabul at e %D for est i mat ed value of acent ri c f act or in each
met hod.
Lee- Kesl er met hod refers to Eq. (2.105) and Kesl er-Lee to
Eq. (2.107).
SolutionkAll t hree met hods of Lee-Kesler, Edmister, and
Kor st en requi re Tb, To, and Pc as i nput par amet er s. The
met hod of Kesl er-Lee requi res Kw in addi t i on to Tbr. Fr om
definition of Wat son K, we get Kw --- 13.64. Subst i t ut i ng t hese
val ues f r om vari ous met hods one cal cul at es t he acent ri c fac-
tor. A s umma r y of t he resul t s is given in Table 2.12. The l east
accur at e met hod is t he Kesl er-Lee correl at i ons while t he mos t
accur at e met hod is Kor st en combi ned wi t h Eqs. (2.67) and
(2.68) for t he critical const ant s, t
2 . 6 P R E D I C T I O N OF DE NS I T Y,
R E F R A C T I V E I N D E X , CH WE I G H T
RATI O, A N D F R E E Z I N G P O I N T
Est i mat i on of densi t y at di fferent condi t i ons of t emper at ur e,
pressure, and composi t i on (p) is di scussed in detail in Chap-
t er 5. However, liquid densi t y at 20~ and 1 at m desi gnat ed by
d in t he uni t of g/ cm 3 is a useful char act er i zat i on pa r a me t e r
whi ch will be used in Chapt er 3 for t he composi t i onal analy-
sis of pet r ol eum fract i ons especially in conj unct i on wi t h t he
definition of refract i vi t y i nt er cept by Eq. (2.14). The sodi um
D line refract i ve index of liquid pet r ol eum fract i ons at 20~
and 1 at m, n, is anot her useful char act er i zat i on par amet er .
Refract i ve index is needed in cal cul at i on of refract i vi t y inter-
cept and is used in Eq. (2.40) for t he est i mat i on of vari ous
pr oper t i es t hr ough pa r a me t e r I defined by Eq. (2.36). More-
over refract i ve index is useful in t he cal cul at i on of densi t y and
t r anspor t pr oper t i es as di scussed in Chapt ers 5 and 8. Carbon-
t o- hydr ogen wei ght rat i o is needed in Chapt er 3 for t he esti-
mat i on of t he composi t i on of pet r ol eum fractions. Freezi ng
poi nt , TF, is useful for anal yzi ng solidification of heavy com-
ponent s in pet r ol eum oils and t o det er mi ne t he cl oud poi nt
t emper at ur e of crude oils and reservoi r fluids as di scussed in
Chapt er 9 (Section 9.3.3).
2. 6. 1 Pr e d i c t i o n o f De n s i t y at 20~
Numer i cal val ues of d20 for a given compound is very close
to t he value of SG, whi ch r epr esent s densi t y at 15.5~ in t he
uni t of g/ cm 3 as can be seen f r om Tables 2.1 and 2.3. Li qui d
densi t y general l y decreases wi t h t emper at ur e. Vari at i on of
densi t y wi t h t emper at ur e is di scussed in Chapt er 6. However,
in t hi s sect i on met hods of est i mat i on of densi t y at 20~ d20,
are pr esent ed to be used for t he char act er i zat i on met hods dis-
cussed in Chapt er 3. The mos t conveni ent way t o est i mat e d20
is t hr ough specific gravity. As a rul e of t humb d20 = 0.995 SG.
However, a bet t er appr oxi mat i on is provi ded t hr ough calcu-
l at i on of change of densi t y wi t h t emper at ur e (Ad/AT), whi ch
is negat i ve and for hydr ocar bon syst ems is given as [7]
(2.110) Ad/AT = - 1 0 -3 x (2.34 - 1.9dr)
wher e dr is densi t y at t emper at ur e T in g/ cm 3. Thi s equat i on
may be used to obt ai n densi t y at any t emper at ur e once a val ue
of densi t y at one t emper at ur e is known. This equat i on is qui t e
accur at e wi t hi n a nar r ow t emper at ur e r ange limit. One can
use t he above equat i on t o obt ai n a val ue of density, d20, at
20~ (g/cm 3) f r om t he specific gravi t y at 15.5~ as
(2.111) d20 = SG - 4.5 x 10-3(2.34 - 1.9SG)
Equat i on (2.111) ma y also be used to obt ai n SG f r om densi t y
at 20 or 25~
SG -- 0.9915d20 + 0.01044
(2.112)
SG = 0.9823d25 + 0.02184
Si mi l arl y densi t y at any ot her t emper at ur e may be cal cul at ed
t hr ough Eq. (2.110). Finally, Eq. (2.38) ma y also be used to
est i mat e d20 f r om TB and SG in t he following form:
(2.113) d20 = 0.983719Tb~176176 1~176
This equat i on was devel oped for hydr ocar bons f r om C5 to C20;
however, it can be safely used up to C40 wi t h AAD of less t han
0.1%. A compar i s on is made bet ween t he above t hree met h-
ods of est i mat i ng d for some n-paraffins wi t h act ual dat a t aken
f r om t he API-TDB [2]. Results of eval uat i ons are given in
Table 2.13. Thi s s umma r y eval uat i on shows t hat Eqs. (2.111)
and (2.113) are al most equivalent, while as expect ed t he rul e
of t humb is less accurat e. Equat i on (2.111) is r ecommended
for pract i cal calculations.
2. 6. 2 Pr e d i c t i o n o f Re f r ac t i ve I n d e x
The refract i ve i ndex of liquid hydr ocar bons at 20~ is corre-
l at ed t hr ough par amet er I defined by Eq. (2.14). I f par amet er
I is known, by r ear r angi ng Eq. (2.14), t he refract i ve index, n,
can be cal cul at ed as follows:
(2.114) n = (11+~2//) 1/2
For pur e and four di fferent homol ogous hydr ocar bon com-
pounds, pa r a me t e r I is pr edi ct ed f r om Eq. (2.42) usi ng mol ec-
ul ar weight, M, wi t h const ant s in Table 2.6. I f boiling poi nt is
available, M is first cal cul at ed by Eq. (2.48) and t hen I is cal-
culated. Predi ct i on of I t hr ough Eq. (2.42) for vari ous hydro-
car bon gr oups is shown in Fig. 2.9. Actual val ues of refract i ve
i ndex f r om API-TDB [2] are also shown in t hi s figure.
For all t ypes of hydr ocar bons and narrow-boi l i ng r ange
pet r ol eum fract i ons t he si mpl est met hod to est i mat e par am-
et er I is given by Riazi and Dauber t [28] in t he f or m of
Eq. (2.38) for t he mol ecul ar wei ght r ange of 70-300 as follows:
(2.115) I = 0.3773Tb-~176 0~9182
1 . 6
2. CHARACTERIZATION AND PROPERTIES OF PURE HYDROCARBONS 67
TABLE 2.13--Prediction of density (at 20~ of pure hydrocarbons.
Estimated density, g/cm 3
SG d, g/cm 3 Eq. (2.113) %AD Eq. (2.111) %AD 0. 995SG %AD
0.6317 0.6267 0.6271 0.06 0.6266 0.02 0.6285 0.29
0.7342 0.7303 0.7299 0.05 0.7299 0.05 0.7305 0.03
0.7717 0.768 0.7677 0.03 0.7678 0.03 0.7678 0.02
0.7890 0.7871 0.7852 0.24 0.7852 0.24 0.7851 0.26
0.8048 0.7996 0.8012 0.20 0.8012 0.19 0.8008 0.15
0.8123 0.8086 0.8088 0.03 0.8087 0.01 0.8082 0.04
0.8172 0.8146 0.8138 0.09 0.8137 0.12 0.8131 0.18
0.10 0.10 0.14
wh e r e Tb i s i n Kel vi n. Thi s e q u a t i o n pr e di c t s n wi t h a n aver -
age e r r o r of a b o u t 1% f or p u r e h y d r o c a r b o n s f r om C5 t o C20.
Mor e a c c u r a t e r e l a t i ons a r e gi ven b y Eq. (2. 40) a n d Tabl e 2. 5
i n t e r ms of va r i ous i n p u t p a r a me t e r s . The f ol l owi ng me t h o d
de ve l ope d b y Ri a z i a n d Da u b e r t [29] a n d i n c l u d e d i n t he API -
TDB [2] have a c c u r a c y of a b o u t 0. 5% on n i n t he mo l e c u l a r
we i ght r a nge of 70- 300.
I = 2. 34348 10 -2 [ exp (7. 029 10-4Tb + 2. 468SG
(2. 116) - 1. 0267 x 10-3TbSG)] Tb~176 -0"720
wh e r e Tb i s i n kel vi n. Fo r h e a v i e r h y d r o c a r b o n s (>C20) t he
f ol l owi ng e q u a t i o n de r i ve d f r o m Eq. ( 2. 46b) i n t e r ms of M
a n d SG c a n be us ed.
I = 1. 2419 x 10 -2 [ exp (7. 272 x 10- 4M + 3. 3223SG
(2. 117) - 8 . 8 6 7 10- 4MSG) ] M~176176 -1"6117
Eq u a t i o n (2. 117) i s ge ne r a l l y a p p l i c a b l e t o h y d r o c a r b o n s wi t h
a mo l e c u l a r we i ght r a nge of 70- 700 wi t h a n a c c u r a c y of l ess
t h a n 0. 5%; however , i t i s ma i n l y r e c o mme n d e d f or c a r b o n
n u mb e r s g r e a t e r t h a n C20. I f o t h e r p a r a me t e r s a r e a va i l a bl e
Eqs. (2. 40) ma y be u s e d wi t h c o n s t a n t s gi ven i n Tabl es 2. 5
a n d 2.9. The API me t h o d t o e s t i ma t e I f or h y d r o c a r b o n s wi t h
M > 300 i s s i mi l a r t o Eq. (2. 116) wi t h di f f er ent n u me r i c a l
c ons t a nt s . Si nc e f or he a vy f r a c t i ons t he boi l i ng p o i n t i s us u-
al l y n o t avai l abl e, Eq. (2. 117) i s p r e s e n t e d her e. An o t h e r r el a-
t i on f or e s t i ma t i o n of I f or he a vy i hydr oc a r bons i n t e r ms of Tb
a n d SG i s gi ven by Eq. ( 2. 46a) wi t h p a r a me t e r s i n Tabl e 2.9,
wh i c h c a n be u s e d f or he a vy h y d r o c a r b o n s i f di s t i l l a t i on d a t a
i s avai l abl e.
Once r e f r a c t i ve i nde x at 20~ i s e s t i ma t e d, t he r e f r a c t i ve
i nde x at o t h e r t e mp e r a t u r e s ma y be p r e d i c t e d f r om t he
f ol l owi ng e mp i r i c a l r e l a t i o n [37].
(2. 118) nr = n20 - 0. 0004( T - 293. 15)
whe r e n20 i s r e f r a c t i ve i nde x at 20~ (293 K) a n d n r i s t he
r e f r a c t i ve i nde x at t he t e mp e r a t u r e T i n wh i c h T i s i n kel vi n.
Al t hough t hi s e q u a t i o n i s s i mpl e, b u t i t gi ves s uf f i ci ent
a c c u r a c y f or p r a c t i c a l a ppl i c a t i ons . A mo r e a c c u r a t e r e l a t i o n
c a n be de ve l ope d b y c o n s i d e r i n g t he s l ope of dnr / dT ( val ue
4~
1 . 4
1.5
n-alkylcyclopentanes
n-alkylbenzenes
n-Paraffin Tb, K
n-C5 309.2
n-Clo 447.3
n-Cl5 543.8
n-C2o 616.9
n-C25 683.2
n-C3o 729.3
n-C36 770.1
Overall
3 , , , , , , , , I , , , , , , , , , , , , , , , ,
10 100 1000 10000
Molecular Weight
FIG. 2.9--Prediction of refractive indices of pure hydrocarbons from Eq. (2.42).
of - 0. 0004 in Eq. (2.118)) as a funct i on of n20 r at her t han
a const ant . Anot her appr oach to est i mat e refract i ve i ndex
at t emper at ur es ot her t han 20~ is t o as s ume t hat specific
refract i on is const ant for a given hydr ocar bon:
lr I2o
(2.119) Specific r ef r act i on - dr - d2o - const ant
wher e I2o is t he refract i ve i ndex pa r a me t e r at 20~ and I r is its
val ue at t emper at ur e T. Si mi l arl y dr is densi t y at t emper at ur e
T. I n fact t he val ue of specific r ef r act i on is t he s ame at all t em-
per at ur es [38]. If/20, d2o, and dT are known, t hen IT can be esti-
mat ed f r om t he above equat i on. Value of nr can be cal cul at ed
f r om IT and Eq. (2.114). Equat i on (2.119) has t he s ame accu-
r acy as Eq. (2.118), but at t he t emper at ur es f ar f r om t he refer-
ence t emper at ur e of 20 ~ C accur acy of bot h met hods decrease.
Because of simplicity, Eq. (2.118) is r ecommended f or calcu-
l at i on of refract i ve i ndex at di fferent t emper at ur es. I t is obvi-
ous t hat t he reference t emper at ur e in bot h Eqs. (2. I 18) and
(2.119) can be changed to any desi red t emper at ur e in whi ch
refract i ve i ndex is available. Refract i ve i ndex is also rel at ed to
anot her pr oper t y called di el ect ri c cons t ant , e, whi ch for non-
pol ar compounds at any t emper at ur e is e -- n 2. For exampl e,
at t emper at ur e of 20~ a paraffi ni c oil has dielectric const ant
of 2.195 and refract i ve i ndex of 1.481 (n 2 -- 2.193). Dielectric
const ant s of pet r ol eum pr oduct s ma y be used to i ndi cat e t he
pr esence of vari ous const i t uent s such as asphal t enes, resins,
etc. [11]. However, for mor e compl ex and pol ar mol ecul es
such as muhi r i ng ar omat i cs, t hi s si mpl e rel at i on bet ween e
and n 2 is not valid and t hey are rel at ed t hr ough di pol e mo-
ment . Fur t her di scussi on on t he met hods of est i mat i on of
refract i ve i ndex is given by Ri azi and Roomi [37].
2. 6. 3 Pr e d i c t i o n o f CH We i ght Ra t i o
Car bon- t o- hydr ogen wei ght rat i o as defined in Sect i on 2.1.18
is i ndi cat i ve of t he qual i t y and t ype of hydr ocar bons pr esent
in a fuel. As will be shown in Chapt er 3 f r om t he knowl edge
of CH value, composi t i on of pet r ol eum fract i ons ma y be es-
t i mat ed. CH value is also rel at ed to car bon resi dues as it is
di scussed in t he next chapter. For hydr ocar bons wi t h mol ec-
ul ar wei ght in t he r ange of 70-300, t he rel at i ons to est i mat e
CH values are given t hr ough Eq. (2.40) and Table 2.5. I n t er ms
of TD and SG t he rel at i on is al so given by Eq. (2.120) whi ch
is al so r ecommended f or use in pr edi ct i on of composi t i on of
pet r ol eum fract i ons [78].
CH = 3.4707 [exp (1.485 x 10-2Tb + 16.94SG
(2.120) - 1. 2492 x 10-2TbSG)] Tb2'725SG -6'798
wher e Tb is in kelvin. The above equat i on was used to ext end
its appl i cat i on for hydr ocar bons f r om C6 t o C50.
CH = 8.7743 x 10 -l ~ [exp (7.176 10-3Tb + 30.06242SG
(2.121) - 7. 35 x 10-3TbSG)] T b ~ -18"2753
wher e Tb is in kelvin. Al t hough t hi s equat i on was devel oped
based on dat a in t he range of C20-C50, it can also be used
f or l ower hydr ocar bons and it gives AAD of 2% f or hydrocar-
bons f r om C20 to C50. Most of t he dat a used in t he develop-
ment of t hi s equat i on are f r om n-al kanes and n-alkyl mono-
cyclic napht heni c and ar omat i c compounds. Est i mat i on of
CH wei ght r at i o f r om ot her i nput par amet er s is possi bl e
t hr ough Eq. (2.40) and Table 2.5. Once CH wei ght r at i o is
det er mi ned t he at omi c HC rat i o can be cal cul at ed f r om t hei r
definitions as descri bed in Sect i on 2.1.18:
11.9147
(2.122) HC ( at omi c rat i o) =
CH(wei ght rat i o)
E x a mp l e 2. 9- - Es t i mat e t he val ues of CH (weight) and HC
(at omi c) rat i os for n-t et radecyl benzene (C20H34) f r om Eqs.
(2.120) and (2.121) and compar e wi t h t he act ual value. Also
dr aw a gr aph of CH values f r om C6 to C50 f or t he t hree homol -
ogous hydr ocar bon gr oups f r om paraffins, napht henes, and
ar omat i cs based on Eq. (2.121) and act ual values.
S o l u t i o n - - T h e act ual val ues of CH wei ght and HC
at omi c rat i os are cal cul at ed f r om t he chemi cal f or mul a
and Eq. (2.122) as CH --- (20 x 12.011)/(34 x 1.008) -- 7.01,
HC(atomic) = 34/ 20 = 1.7. Fr om Table 2.1, for n-t et radecyl -
benzene (C20H34), Tb ~ 627 K and S G= 0.8587. Subst i t ut -
ing t hese val ues i nt o Eq. (2.120) gives CH- - 7.000, and f r om
Eq. (2.122) at omi c HC r at i o- - 1. 702. The er r or f r om
Eq. (2.134) is %D = 0.12%. Equat i on (2.121) gives CH- -
6.998, whi ch is nearl y t he s ame as t he val ue obt ai ned f r om
Eq. (2.120) wi t h t he s ame error. Si mi l arl y CH val ues are cal-
cul at ed by Eq. (2.121) for hydr ocar bons r angi ng f r om C6 t o
C50 in t hree homol ogous hydr ocar bon gr oups and are shown
wi t h act ual val ues in Fig. 2.10. #
2 . 6 . 4 Pr e d i c t i o n o f Fr e e z i n g / Me l t i n g P o i n t
For pur e compounds, t he nor mal freezi ng poi nt is t he s ame
as t he mel t i ng poi nt , TM. Melting poi nt is mai nl y a pa r a m-
et er t hat is needed for predi ct i ng sol i d-l i qui d phase behav-
ior, especi al l y for t he waxy oils as shown in Chapt er 9. All
at t empt s to devel op a general i zed correl at i on for TM in t he
f or m of Eq. (2.38) have failed. However, Eq. (2.42) devel oped
by Riazi and Sahhaf for vari ous homol ogous hydr ocar bon
gr oups can be used to est i mat e mel t i ng or freezi ng poi nt of
pur e hydr ocar bons f r om C7 to C40 wi t h good accur acy (error
of 1-1.5%) f or pr act i cal cal cul at i ons [31]. Using t hi s equat i on
wi t h appr opr i at e const ant s in Table 2.6 gives t he following
equat i ons for predi ct i ng t he freezi ng poi nt of n-al kanes (P),
n-al kycycl opent anes (N), and n-al kybenzenes (A) f r om mol ec-
ul ar weight.
(2.123) TMp = 397 -- exp(6.5096 -- 0.14187M ~
(2.124) TMN = 370 exp(6. 52504-- 0.04945M 2/3)
(2.125) TMA = 395 -- exp(6.53599 -- 0.04912M 2/3)
wher e TM is in kelvin. These equat i ons are valid in t he car-
bon r anges of C5-C40, C7-C40, and C9-C40 f or t he P, N, and
A groups, respectively. I n fact in wax pr eci pi t at i on l i near
hydr ocar bons f r om CI t o CI5 as well as ar omat i cs are ab-
sent, t her ef or e t here is no need f or t he mel t i ng poi nt of very
light hydr ocar bons [64]. Equat i on (2.124) is for t he mel t -
ing poi nt of n-al kyl cycl opent anes. A si mi l ar cor r el at i on for
n-al kyl cycl ohexanes is given by Eq. (2.42) wi t h const ant s in
Table 2.6. I n Chapt er 3, t hese correl at i ons will be used to es-
t i mat e freezi ng poi nt of pet r ol eum fractions.
Won [79] and Pan et al. [63] also pr oposed correl at i ons
f or t he freezi ng poi nt s of hydr ocar bon groups. The Won
0
. = ~
O8
O8
i
0
2. CHARACTERIZATION AND PROPERTIES OF PURE HYDROCARBONS 6 9
10.0
9. 0
8. 0
7.0
6, 0
5. 0
t * Actual Values forn-alkanes
9 Actual Values forn-alkyicyclopentanes
A Actual Values forn-alkybcnzenes
. ~; ~. . o e l , ~ e o ~ ooe o o l e o o e o , I . Q , ooo~ o o e o o e ~ 9
5 15 25 35 45 55
Carbon Number
FIG. 2. 10- - Est i mat i on of CH wei ght Ratio from Eq. (2.121) f or vari ous fami l i es.
50
0
-50
N
~ -100
-150
-200
,, K
i I
~
I t
. Y . . P -
o Data for n-alkanes
Predicted: R-S Method
Predicted: P-F Method
A Data for n-alkyl~'clopenlanes
. . . . . . . Predicted: R-S Method
. . . . Predicted: P-F Method
I Data for n-alkylbenzenes
- - - - - Predicted: R-S Method
m Predicted: P-F Method
5 10 15 20 25
Car bon Number
FI G. 2. 11- - Est i mat i on of f r eezi ng poi nt of pur e hydr ocar bons f or var i ous f ami -
l i es. [ R- S r ef er s t o Eqs. (2.123)-(2.125); P-F ref ers t o Eqs. (2.126) and (2.127).
correlation for n-alkanes is
(2.126) Trap = 374.5 + 0.02617M - 20172/M
where TMe is in kelvin. For naphthenes, aromatics, and
isoparaffins the melting point temperature may be esti-
mated from the following relation given by Pan-Fi rrozabadi -
Fotland [63].
(2.127) Tra(iP,N,A) = 333.45 -- 419 exp(-0.00855M)
where TM is in kelvin. Subscripts iP, N, and A indicate iso-
paraffins, naphthenes, and aromatics, respectively.
Exampl e 2. 10--Estimate the freezing point of n-hexa-
triacontane ( C36 H7 4 ) from Eqs. (2.123 ) and (2.126 ) and com-
pare with the actual value of 348.19 K [20]. Also draw a graph
of predicted Tra from Eqs. (2. i23) to (2.127) for hydrocarbons
from C7 to C40 for the three homologous hydrocarbon groups
from paraffins, naphthenes, and aromatics based on the above
two methods and compare with actual values given up to C20
given in Table 2.2.
Solution--For n-C36 , we have M = 36 x 12.011 + 74 x 1.008 =
508.98 and T~a = 348.19 K. From Eq. (2.123), TM = 397-
exp(6. 5096- 0.14187 508.980'47) ---- 349.78 K. The percent
absolute relative deviation (%AD) is 0.2%. Using Eq. (2.126),
T~a = 348.19 K with %AD of 0.24%. A complete evaluation is
demonstrated in Fig. 2.1 I. On an overall basis for n-alkanes
Eq. (2.126) is more accurate than Eq. (2.123) while for naph-
thenes and aromatics, Eqs. (2.124) and (2.125) are more ac-
curate than Eq. (2.127).
2 . 7 P R E D I C T I O N OF KI NE MAT I C
V I S C O S I T Y AT 3 8 A N D 9 9 ~
Detailed prediction of the viscosities of petroleum fractions
will be discussed in Chapter 8. However, kinematic viscos-
ity defined by Eq. (2.12) is a characterization parameter
needed to calculate parameters such as VGC (Section 2.1.17),
which will be used in Chapter 3 to determine the compo-
sition of petroleum fractions. Kinematic viscosity at two
reference temperatures of 100~ (37.78 ~ 38~ and 210~
(98.89 ~ 99~ are generally used as basic characterization
parameters and are designated by 1)38(100) and 1)99(210), respec-
tively. For simplicity in writing, the reference temperatures of
100 and 210~ are presented as 38 and 99~ rather than accu-
rate values of 37.78 and 98.89. Kinematic viscosity decreases
with temperature and for highly viscous oils values of 1)99(210)
are reported rather than u38~00). The temperature dependency
of viscosity is discussed in Chapter 8 and as will be seen, the
viscosity of petroleum fractions is one of the most complex
physical properties to predict, especially for very heavy frac-
tions and multiring aromatic/naphthenic compounds. Heavy
oils with API gravities less than 10 could have kinematic vis-
cosities of several millions cSt at 99~ (210~ These viscos-
ity values would be almost impossible to predict from bulk
properties such as boiling point and specific gravity. How-
ever, there are some relations proposed in the literature for
the estimation of these kinematic viscosities from Tb and SG
or their equivalent parameters Kw and API gravity. Relations
developed by Abbott et al. [80] are commonl y used for the
estimation of reference kinematic viscosities and are also in-
cluded in the API-TDB [2]:
l og v38000) =
4. 39371 - 1. 94733Kw + 0.12769K~v
+3 , 2 6 2 9 x 10- 4API 2 - 1. 18246 x 10-ZKwAPI
0.171617K2w + 10. 9943(API) + 9. 50663 x 10-2(API) 2 - 0. 860218Kw( API )
+
(API) + 50. 3642 - 4. 78231 Kw
(2.128)
log 1)99(210) =
- 0 . 4 6 3 6 3 4 - 0.166532(API) + 5.13447
10-4(API) 2 - 8.48995 x 10-3KwAPI
8.0325 10 -2 Kw + 1.24899(API) + 0.19768(API) 2
+
(API) + 26.786 - 2.6296Kw
(2.129)
Kw and API are defined by Eqs. (2.13) and (2.4). In these
relations the kinematic viscosities are in cSt (mm2/s). These
correlations are also shown by a nomograph in Fig. 2.12. The
above relations cannot be applied to heavy oils and should
be used with special care when Kw < 10 or Kw > 12.5 and
API < 0 or API > 80. Average error for these equations is in
the range of 15-20%. They are best applicable for the viscos-
ity ranges of 0.5 < 1)38(100 ) < 20 mm2/s and 0.3 < 1)99(210) < 40
mm2/s [8]. There are some other methods available in the lit-
erature for the estimation of kinematic viscosities at 38 and
99~ For example Twu [81 ] proposed two correlations for the
kinematic viscosities of n-alkanes from C1 to C100 in a similar
fashion as his correlations for the critical properties discussed
in Section 2.5.1. Errors of 4-100% are common for prediction
of viscosities of typical oils through this method [ 17].
Once kinematic viscosities at two temperatures are known,
ASTM charts (ASTM D 341-93) may be used to obtain viscos-
ity at other temperatures. The ASTM chart is an empirical
relation between kinematic viscosity and temperature and it
is given in Fig. 2.13 [68]. In using this chart two points whose
their viscosity and temperature are known are located and
a straight line should connect these two points. At any other
temperature viscosity can be read from the chart. Estimated
values are more accurate within a smaller temperature range.
This graph can be represented by the following correlation
[ 8 ] :
(2.130) log[log(1)r + 0.7 + c:~)] = A1 + B1 log T
where vr is in cSt, T is the absolute temperature in kelvin,
and log is the logarithm to base 10. Parameter Cr varies with
value of Vr as follows [8]:
0.085(1)r - 1.5) 2 if vr < 1.5 cSt [mm2/s]
(2.131) Cr =
0.0 if 1)~ >_ 1.5 cSt [mmZ/s]
If the reference temperatures are 100 and 210~ (38 and
99~ then A1 and B1 are given by the following relations:
Al ----- 12.8356 x (2.57059D1 - 2.49268D2)
B1 = 12.8356(D2 - D1)
(2.132)
D1 = log[log(v38000) + 0.7 + c38000))]
/)2 ----- log[log(1)99(210) + 0.7 + C99(210)) ]
Various forms of Eq. (2.130) are given in other sources
[2, 11, 17]. Errors arising from use of Eq. (2.130) are better or
1 . 0 ~
1.2
1.6-
1 . 8
u~
O)
0
u ~
0
I/I
0
u2
0
g
0
a E
9 ~
.~o
4 3o
0
s--t L- 2o0
I soo
1 7oo
70~
1 2 . 5 - -
5 0
4 0
3 0
2 0
! 0
0_
<(
I Z O-
x /
~" 11..5"
O
t -
O
N
t -
O
t -
O
~ .! 1.0"
10.5 -
lO.O-J--
165
F I G . 2 . 1 2 - - P r e d i c t i o n o f k i n e m a t i c v i s c o s i t y f r o m Kw a n d t h e A P I g r a v i t y . W i t h
p e r m i s s i o n f r o m R e f . [ 2 ] ,
at least in the same range of errors for the prediction of viscos-
ity from Eqs. (2.128) and (2.129). Similarly constants AI and
B1 in Eq. (2.130) can be determined when values of viscosity
at two temperatures other than 100 and 210~ are known.
When vr is being calculated from Eq. (2.130) at temperature
T, a trial and error procedure is required to determine param-
eter cr. The first estimate is calculated by assuming vr > 1.5
cSt and thus Cr = 0. If calculated value is less than 1.5 cSt,
then Cr is calculated from Eq. (2.131). Extrapolated values
from Fig. 2.12 or Eq. (2.130) should be taken with caution.
An application of this method to estimate kinematic viscosity
of petroleum fractions is demonstrated in Chapter 3. Further
discussion on the estimation of viscosity is given in Chapter 8.
Consi st ency Text - - One way to check reliability of a predicted
physical property is to perform a consistency test through
different procedures. For example, laboratory reports may
consist of viscosity data at a temperature other than 38 or
T
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99~ In many cases kinematic viscosity at 40~ (122~ or
60~ (140~ is reported. One may estimate the kinematic
viscosities at 38 and 99~ through Eqs. (2.128) and (2.129)
and then use the ASTM chart (or Eq. 2.128) to obtain the
value of viscosity at 40 or 60~ If the interpolated value is far
from laboratory data then the estimation method cannot be
trusted and other methods should be considered.
Another consistency test can be made through estimation
of the molecular weight by Eq. (2.52) using estimated vis-
cosities by Eqs. (2.128) and (2.129). If value of M calculated
through Eq. (2.52) is near the value of M estimated from Tb
and SG by Eqs. (2.51) or (2.50), then all estimated values can
be trusted. Such consistency tests can be extended to all other
physical properties. The following example demonstrates the
test method.
Example 2.11--The viscosity of a pure multiring hydrocar-
bon from an aromatic group (naphthecene type compound)
with formula C26H40 has been i~eported in the API RP-42
[ 18]. Data available are M = 352.6, SG = 0.9845, and 1299(210) =
13.09 cSt. Estimate the kinematic viscosity of this hydrocar-
bon at 38 and 99~ (100 and 210~ by Eqs. (2.128) and
(2.129). How can you assess the validity of your estimated
kinematic viscosity at 38~
Solution--To estimate the viscosity through the Abbott cor-
relations, Kw and API gravity are needed. However, Tb is not
available and should be estimated from M and SG. Since
M > 300, we use Eq. (2.57) in terms of M and SG to esti-
mate Tb as follows: Tb = 720.7 K. Using Eqs. (2.13) and (2.4),
Kw and API are calculated as Kw = 11.08 and API = 12.23.
Using Eqs. (2.128) and (2.129) the viscosities are calculated
as 1)38(100 ) = 299.76 cSt. 1399(210 ) ~ I 1.35 cSt. At 99~ the esti-
mated value can be directly evaluated against the experimen-
tal data: %D = [(11.08 - 13.09)/13.09)] x 100 = -15.4%. To
evaluate accuracy of estimated viscosity at 38~ a consistency
test is required. Since the actual value of molecular weight,
M, is given, one can estimate M through Eq. (2.52) using
estimated values of 1)38(100), 1)99(210), and SG as the input pa-
rameter. The estimated M is 333.8 which in comparison with
actual value of 352.6 gives %AD of 11.36%. This error is ac-
ceptable considering that Eq. (2.52) has been developed based
on data of petroleum fractions and the fact that input param-
eters are estimated rather than actual values. Therefore, we
can conclude that the consistency test has been successful
and the value of 299.8 cSt as viscosity of this hydrocarbon at
38~ is acceptable. The error on estimated viscosity at 99~
is 15.4%, which is within the range of errors reported for the
method. It should be realized that the equations for prediction
of kinematic viscosity and estimation of molecular weight by
Eq. (2.52) were originally recommended for petroleum frac-
tions rather than pure compounds. #
2 . 8 T H E WI N N N O MO G R A M
Development of estimation techniques through graphical
methods was quite common in the 1930s through the 1950s
when computational tools were not available. Nomogram or
homograph usually refers to a graphical correlation between
different input parameters and desired property when more
than two input parameters are involved. By drawing a straight
line between values of input parameters, a reading can be
made where the straight line intersects with the line (or curve)
of the desired property. The best example and widely used
nomogram is the one developed by Winn in 1957 [25]. This
nomogram, which is also included in the API-TDB [2], re-
lates molecular weight (M), CH weight ratio, aniline point,
and Watson K to boiling point and specific gravity (or API
gravity) on a single chart and is shown in Fig. 2.14.
Application of this figure is mainly for petroleum fractions
and the mean average boiling point defined in Chapter 3
is used as the boiling point, Tb. If any two parameters are
available, all other characterization parameters can be deter-
mined. However, on the figure, the best two input parameters
are Tb and SG that are on the opposite side of the figure.
Obviously use of only M and Tb as input parameters is not
suitable since they are near each other on the figure and an
accurate reading for other parameters would not be possible.
Similarly CH and SG are not suitable as the only two input
parameters. Previously the computerized form of the Winn
nomogram for molecular weight was given by Eq. (2.95).
Use of the nomogram is not common at the present time
especially with availability of personal computers (PCs) and
simulators, but still some process engineers prefer to use a
nomogram to have a quick estimate of a property or to check
their calculations from analytical correlations and computer
programs.
If the boiling point is not available, methods discussed in
Section 2.4.2 may be used to estimate the boiling point be-
fore using the figure. Equation (2.50) for molecular weight
may be combined with Eq. (2.13) to obtain a relation for the
estimation of Kw from M and SG [51].
(2.133) Kw = 4. 5579M~ -0"84573
This equation gives an approximate value for Kw and should
be used with care for hydrocarbons heavier than C30. A more
accurate correlation for estimation of Kw can be obtained if
the boiling point is calculated from Eq. (2.56) or (2.57) and
used in Eq. (2.13) to calculate Kw.
Example 2.12--Basic properties of n-tridecylcyclohexane
(CIgH38) a r e given in Table 2.1. Use M and SG as available
input parameters to calculate
a. Kw from Eq. (2.133).
b. Kw from most accurate method.
c. Kw from Winn Nomogram.
d. CH weight ratio from M and SG.
e. CH weight ratio from Winn method.
f. %D for each method in comparison with the actual values.
Solution--From Table 2.1, M = 266.5, Tb = 614.7 K,
and SG = 0.8277. HC atomic ratio -- 38/19 = 2.0. Using
Eq. (2.122), CH weight ratio = 11.9147/2.0 = 5.957. From
definition of Kw, i.e., Eq. (2.13), the actual value of Kw is
calculated as Kw = (1.8 x 614.7)1/3/0.8277 = 12.496.
a. From Eq. (2.133), Kw = 4.5579 x [(266.5) ~ x
[(0.8277) -~ = 12.485. %D = -0.09%.
b. The compound is from the n-alkylcyclohexane family and
the most accurate way of predicting its boiling point is
through Eq. (2.42) with constants given in Table 2.6 which
90-i_
s s - - - - ~s
8o-
75-2
70-_ -0.70
6 5 "
6 0 -
- -0.75
55- -
50-
4 s - - ' o s o
._.g,
4 0 -
0
E
< 3 5 - -0.85
.._
3 o - 2
_--1200
L
: l 10O
- ~ o o o
~00 r
L800
500
" F -
O
u. ~too 240~ ~ ~ 00 70O
. ~ 2 2 o . . 2 r .
, 3- 5- 300
c~ ' - 6 0 0 ~ .
' - ' v 200- ~- #_
~ . . 09 ~ .-
~ , 2 . . " ~ ~ Lsoo ~~
o 150--~ ~ "- 200 ,~
ID
c -
. ~ 150
-- ~ - ~
2 s - -o.9o ~a- ~ : _
:I: - - 3 0 0
-
- o9- 2 '
--0.95 - 9 o
15-
- 8 0 -20O
10 tOO
5 -
~t 05
- -100
0-
FIG. 2 . 1 4 ~ Wi n n nomogram f or characterization of petroleum fractions. With
p e r m i s s i o n f r o m Re f . [ 2] .
gives Tb = 1100 -- exp[7.00275 -- 0.01977 x (266.5) 2/3] =
615.08 K. Usi ng Tb = 615.08 and SG = 0.8277 in Eq. (2.13)
gives Kw = 12.498. The %D is +0. 016%.
c. When usi ng Wi nn nomogr am (Fig. 2.14) it is easi er to con-
vert SG to API gravity, whi ch t hr ough Eq. (2.4) is 39.46.
A st rai ght line bet ween poi nt s 266.5 on t he M l i ne and
39.5 on t he API gravity l i ne i nt ersect s t he Wat son K l i ne
at Kw = 12.27 and t he i nt ersect i on wi t h t he CH l i ne is at
CH = 6.1. The %D for Kw is - 1. 8%.
d. CH wei ght rat i o can be est i mat ed f r om Eq. (2.40) usi ng
M and SG as i nput par amet er s wi t h const ant s in Table 2.5,
whi ch resul t in CH val ue of 6.2 wi t h %D = + 4% ( C H a c t u a i =
5.96).
CH = 2.35475 x [exp(9.3485 10 -3 266.5 + 4.74695
x 0.8277 - 8.01719 x 10 -3 x 266.5 x 0.8277)]
x [(266,5) -0'68418] x [(0.8277) -0'7682] = 6.2
e, CH wei ght rat i o f r om t he Wi nn met hod was obt ai ned i n
Part c as CH=6. 1, whi ch gives %D = +2. 3%.
f. For all part s %D is cal cul at ed f r om Eq. (2.134). Part b
gives t he most accur at e Kw val ue because Tb was cal cul at ed
accurately. However, Part b in this case is also accurate. Es-
timation of the CH value is less accurate than prediction of
boiling point and gives errors higher than Kw.
2 . 9 ANALYSI S AND COMP ARI S ON OF
VARI OUS CHARACTERI ZATI ON METHODS
Generally there are a large number of pure hydrocarbons
and their properties can be used for evaluation purposes.
However, hydrocarbons from certain groups (i.e., paraffins,
naphthenes, and aromatics) are more abundant in petroleum
fractions and can be used as a database for evaluation pur-
poses. Molecular weight, critical properties, and acentric fac-
tor are important properties and their predictive methods
are presented in this chapter. Errors in any of these proper-
ties greatly influence the accuracy of the estimated physical
property. Methods of estimation of these properties from bulk
properties such as boiling point and specific gravity that are
presented in this chapter have been in use in the petroleum
industry for many years. In some process simulators a user
should select a characterization method out of more than a
dozen methods included in the simulator [56]. In each ap-
plication, the choice of characterization method by the user
strongly influences the simulation results. Although there has
not been a general and comprehensive evaluation of various
characterization methods, a conclusion can be made from
individual's experiences reported in the literature. In this sec-
tion first we discuss criteria for evaluation of various methods
and then different predictive methods for molecular weight,
critical constants and acentric factor are compared and eval-
uated.
2. 9. 1 Cri teri a f or Eval uat i on
o f a Charact eri zat i on Me t hod
Methods of characterization and correlations presented in
this chapter are mainly based on properties of pure hydrocar-
bons. However, some of these correlations such as Eq. (2.52)
for estimation of the molecular weight of heavy fractions or
the correlations presented for prediction of the kinematic vis-
cosity are based mainly on the properties of fractions rather
than pure compounds. The main application of these correla-
tions is for basic properties of undefined petroleum fractions
in which bulk properties of a fraction are used to estimate
a desired parameter. Therefore, the true evaluation of these
characterization methods should be made through properties
of petroleum fractions as will be discussed in upcoming chap-
ters. However, evaluation of these methods with properties of
pure hydrocarbons can be used as a preliminary criteria to
judge the accuracy of various correlations. A method, which
is more accurate than other methods for pure hydrocarbons,
is not necessarily the best method for petroleum mixtures. A
database for pure hydrocarbons consists of many compounds
from different families. However, evaluations made by some
researchers are based primarily on properties of limited pure
hydrocarbons (e.g., n-alkanes). The conclusions through such
evaluations cannot be generalized to all hydrocarbons and
petroleum fractions. Perhaps it is not a fair comparison if
a data set used to develop a method is also used to evalu-
ate the other methods that have used other databases. Type
of compounds selected, the source of data, number of data
points, and the basis for the evaluation all affect evaluation
outcome. The number of numerical constants in a correla-
tion and number of input parameters also affect the accu-
racy. Usually older methods are based on a fewer and less
accurate data than newer methods. It would be always useful
to test different methods on a set of data that have not been
used in obtaining the correlation coefficients. The roost ap-
propriate procedure would be to compare various methods
with an independent data set not used in the development of
any methods considered in the evaluation process. Another
fair comparison of two different correlation would be to use
the same database and reobtain the numerical constants in
each correlation from a single database. This was done when
Eqs. (2.52) and (2.53) were compared, as discussed in Sec-
tion 2.4.1. These are the bases that have been used to compare
some of the correlations presented in this chapter.
Basically there are two parameters for the evaluation of
a correlation. One parameter is the percent average absolute
deviation (%AAD). Average errors reported in this chapter and
throughout the book are based on percent relative deviation
(%D). These errors are defined as following:
estimated value - actual val ue)
(2.134) %D = \ ac~u~ualva~ue x 100
where N is the total number of data points and summation is
made on all the points. I%D[ is called percent absolute devi-
ation and it is shown by %AD. The maximum value of I%D[
in a data set is referred as %MAD. The second parameter is
called R squared (R 2) that is considered as an index of the cor-
relation when parameters of a correlation are obtained from
a data set. A value of 1 means perfect fit while values above
0.99 generally give good correlation. For a set of data with
X column (independent variable) and Y column (dependent
variable) the parameter is defined as
(2.136) R2= [ N( y~xY) - ( ~X) ( Y~Y) ] 2
[ Ny~X 2 - ( ~X) 2] x [ N~Y 2 - (y~y)2]
where X and Y are values of the independent and correspond-
ing dependent variables and N is the number of data points.
The ~ is the summation over all N values of X, X 2, Y, y2, and
XY as indicated in the above equation. The R 2 value can be
interpreted as the proportion of the variance in y attributable
to the variance in x and it varies from 0 to maxi mum value
of 1.
For most of the correlations presented in this chapter such
as Eqs. (2.40), (2.42), or (2.46a) the %AAD for various prop-
erties is usually given in the corresponding tables where the
constants are shown. Most of these properties have been cor-
related with an R 2 value of mi ni mum 0.99. Some of these
properties such as kinematic viscosity or CH weight ratio
showed lower values for R 2. Evaluation of some of the other
correlations is made through various examples presented in
this chapter.
Nowadays with access to sophisticated mathematical tools,
it is possible to obtain a very accurate correlation from any
dat a set. For exampl e, when t he met hod of neural network is
used t o obt ai n correl at i ons for est i mat i on of critical proper-
ties, a very accur at e correl at i on can be obt ai ned for a l arge
numbe r of compounds [82]. However, such correl at i ons con-
t ai n as ma ny as 30 numer i cal values, whi ch l i mi t t hei r power
of extrapolatability. I t is our experi ence t hat when a corre-
l at i on is based on some t heoret i cal foundat i on, it has fewer
const ant s wi t h a wi der r ange of appl i cat i on and bet t er ex-
trapolatability. Thi s is par t i cul ar l y evi dent for t he case of
Eq. (2.38) devel oped based on t he t heor y of i nt er mol ecul ar
forces and EOS par amet er s. Equat i on (2.38) has onl y t hree
par amet er s t hat are obt ai ned f r om dat a on pr oper t i es of pur e
hydr ocar bons f r om C5 t o C20. This equat i on f or vari ous pr op-
ert i es can be safely used up to C30. Tsonopoul os et aL [34]
and Li n et al. [83] have extensively eval uat ed Eq. (2.50) for
est i mat i on of t he mol ecul ar wei ght of di fferent sampl es of
coal liquids, whi ch are mai nl y ar omat i cs, and compar ed wi t h
ot her sophi st i cat ed mul t i par amet er correl at i ons specifically
devel oped for t he mol ecul ar wei ght of coal liquids. Thei r con-
cl usi on was t hat Eq. (2.50) gave t he l owest er r or even t hough
onl y pur e component dat a wer e used to devel op this equat i on.
Fur t her eval uat i on of char act er i zat i on met hods for mol ecu-
l ar wei ght and critical pr oper t i es are given in t he following
part s.
2. 9. 2 Eval uati on of Met hods of Est i mat i on
of Mol ecul ar Wei ght
As ment i oned above mos t of t he eval uat i ons made on
Eq. (2.50) for t he mol ecul ar wei ght of pet r ol eum fract i ons
bel ow 300 suggest t hat it predi ct s qui t e well for vari ous
TABLE 2.14--Evaluation of methods for estimation of molecular
weight of petroleum fractions, a
Abs Dev %**
Method Equation(s) AAD% MAD%
API (Riazi-Daubert) (2.51 ) 3.9 18.7
Twu (2.89)-(2.92) 5.0 16.1
Kesler-Lee (2.54) 8.2 28.2
Winn (2.93) 5.4 25.9
~ Nu r n b e r o f d a t a p o i n t s : 625; R a n g e s o f d a t a : M ~ 7 0 - 7 0 0 , Tb ~ 3 0 0 - 8 5 0 , S G
0 . 6 3 - 0 . 9 7
b De f i n e d b y Eq s . ( 2 . 1 3 4 ) a n d ( 135) . R e f e r e n c e [ 29] .
fractions. This equat i on has been i ncl uded in mos t pr ocess
si mul at or s [54-56]. Whi t son [51, 53] has used t hi s equat i on
and its conver si on t o Kw (Eq. 2.133) for fract i ons up t o C25
in his char act er i zat i on met hods of r eser voi r fluids. A mor e
general f or m of t hi s equat i on is given by Eq. (2.51) for t he
mol ecul ar wei ght r ange of 70-700. This equat i on gives an
average er r or of 3.4% for fract i ons wi t h M < 300 and 4.7%
for fract i ons wi t h M > 300 for 625 fract i ons f r om Penn St at e
dat abase on pet r ol eum fract i ons. An advant age of Eq. (2.51)
over Eq. (2.50) is t hat it is appl i cabl e to bot h light and heavy
fractions. A compar at i ve eval uat i on of vari ous correl at i ons
f or est i mat i on of mol ecul ar wei ght is given in Table 2.14 [29].
Process si mul at or s [55] usual l y have referred to Eq. (2.50)
as Ri azi - Dauber t met hod and Eq. (2.51) as t he API met hod.
The Wi nn met hod, Eq. (2.93), has been al so r ef er r ed as Si m-
Dauber t met hod in some sources [55, 84].
For pur e hydr ocar bons t he mol ecul ar wei ght of t hree ho-
mol ogous hydr ocar bon gr oups pr edi ct ed f r om Eq. (2.51) is
dr awn versus car bon numbe r in Fig. 2.15. For a given car-
bon numbe r t he di fference bet ween mol ecul ar wei ght s of
Y~
e~
o
3000
2500
2000
1500
1 0 0 0
500
0
200
o A P I D a t a
. . . . . n - A l k a n e s
N a p h t h e n e s
- - n-Alk3,1benzenes
/
/
I i f i I i f
400 600 800 1000 1200
Boiling Point, K
FIG. 2.15--Evaluation of Eq. (2.51) for molecular weight of pure compounds.
2. CHARACTERIZATION AND PROPERTI ES OF PURE HYDROCARBONS 77
400
o API D a t a ..M ~ " ,
-- R-D: Eq. 2.50 ~ A~. "~
350
. . . . A P I : E q . 2 . 5 1 ~ r
. . . . . . R.s: 248 / . G . . ? : :
- - - - - T w u : E q . 2 . 8 9 - 2 . 9 2 ~ 7
300 Lee-Kcsler: E q ~
250
200 " ' "
O
E
150
100
5 0 , , . . , ~ , , , , r , , , , f , , , t , , , ,
5 10 15 20 25 30
Carbon Number
FIG. 2. 16- - Eval uat i on of vari ous methods for prediction o1 molecular wei ght
of n-alkylcycohexanes. Ri azi - Daubert : Eq. (2,50); API: Eq. (2.51); Ri azi - Sahhaf :
Eq. (2.48); Lee- Kesl er: Eq. (2.54); Twu: Eqs, ( 2, 89) - ( 2, 92) .
hydr ocar bons f r om di fferent gr oups is small. Actual val ues of
mol ecul ar wei ght of n-al kyl benzenes up to C20 as r epor t ed by
API-TDB [2] are also shown on t he figure. Equat i on (2.51) is
not t he best met hod for t he pr edi ct i on of mol ecul ar wei ght of
pur e compounds as it was pr i mar i l y devel oped for pet r ol eum
fract i ons. Various met hods for t he est i mat i on of mol ecul ar
wei ght for n-al kyl cyl ohexanes wi t h t he API dat a (up t o C26)
are shown in Fig. 2.16 for t he r ange of C6-Cs0. At hi gher car-
bon number s t he devi at i on bet ween t he met hods i ncreases.
The Twu met hod accur at el y est i mat es mol ecul ar wei ght of
l ow-mol ecul ar-wei ght pur e hydr ocar bons; however, at hi gher
mol ecul ar wei ght s it deviates f r om act ual data. A compar i -
son bet ween eval uat i ons pr esent ed in Fig. 2.16 and Table 2.14
shows t hat a met hod t hat is accur at e f or predi ct i on of proper-
ties of pur e hydr ocar bons is not necessari l y t he best met hod
for pet r ol eum fractions. Eval uat i on of met hod of pr edi ct i on
of mol ecul ar wei ght f r om vi scosi t y (Eqs. (2.52) and (2.53))
has been di scussed in Sect i on 2.4.1.
2. 9. 3 Eval uat i on o f Me t hods of Es t i mat i on
of Cri t i cal Pr ope r t i e s
Eval uat i on of correl at i ons for est i mat i on of critical proper-
ties of pur e compounds can be made di rect l y wi t h t he act ual
val ues for hydr ocar bons up to C18. However, when t hey are
appl i ed to pet r ol eum fractions, pseudocri t i cal pr oper t i es are
cal cul at ed whi ch are not di rect l y measur abl e. These val ues
shoul d be eval uat ed t hr ough ot her pr oper t i es t hat ar e mea-
surabl e but requi re critical pr oper t i es for t hei r calculations.
For exampl e, ent hal pi es of pet r ol eum fract i ons are calcu-
l at ed t hr ough general i zed correl at i ons whi ch requi re critical
pr oper t i es as shown in Chapt ers 6 and 7. The phase behav-
i or pr edi ct i on of reservoi r fluids also requi res critical prop-
erties of pet r ol eum cut s t hat make up t he fluid as di scussed
in Chapt er 9. These t wo i ndi rect met hods are t he basi s f or
t he eval uat i on of correl at i ons for est i mat i on of critical pr op-
erties. These eval uat i ons very muc h depend on t he t ype of
fract i ons evaluated. For exampl e, Eqs. (2.63)-(2.66) for esti-
mat i on of Tc and Pc have been devel oped based on t he critical
dat a f r om C5 to Cls; t herefore, t hei r appl i cat i on to heavy frac-
t i ons is not reliable al t hough t hey can be safely ext r apol at ed to
C25-C30 hydr ocar bons. I n t he devel opment of t hese equat i ons,
t he i nt ernal consi st ency bet ween Tc and Pc was not i mposed as
t he correl at i ons were devel oped f or fract i ons wi t h M < 300.
These correl at i ons wer e pr i mar i l y devel oped for light frac-
t i ons and me di um distillates t hat are pr oduced f r om at mo-
spheri c distillation col umns.
For pur e hydr ocar bons f r om homol ogous families,
Eq. (2.42) wi t h const ant s in Table (2.6) provi de accur at e val-
ues for Tc, Pc, and Vc. Predi ct i on of Tc and Pc f r om this equa-
t i on and compar i s on wi t h t he API-TDB dat a are shown in
Figs. 2.2 and 2.3, respectively. Eval uat i on of vari ous met hods
for critical t emper at ur e, pressure, and vol ume of di fferent hy-
dr ocar bon fami l i es is demons t r at ed in Figs. 2. 17-2. 19 respec-
tively. A s umma r y of eval uat i ons for Tc and Pc of hydr ocar bons
f r om different gr oups of all t ypes is pr esent ed in Table 2.15
[29]. Di scont i nui t y of API dat a on Pc of n-al kyl cycl opent anes,
as seen in Fig. 2.18, is due to predi ct i on of Pc f or heav-
i er hydr ocar bons (>C20) t hr ough a gr oup cont r i but i on
met hod.
~Z
E
900
o A P I D a t a
~ W i n n
- - R - D _ , ~ . ~ , ~ . . . . ~ 9
- - - T w t l . . . ~ ~
750
600
450
1050
l I I I I L I L L e i i i i i i i l i i i i i i i i i l
10 20 30 40 50
Carbon Number
FIG. 2. 17- - Compar i son of var i ous met hods f or est i mat i on of cri t i cal t emperat ure of n-
alkanes. API Data: API-TDB [2]; Wi nn: Eq. (2.94); R-D: Ri azi -Daubert , Eq. (2.63); Twu:
Eq. (2.80)-(2.82); Ext. R-D: Extended Riazi-Daubert, Eq. (2.67); L-K: Lee-Kesl er: Eq. (2.69);
API : Eq. (2.55); R-S: Ri azi -Sahhaf, Eq. (2.42); and Table 2.6.
60
50
40
30
~ 20
(,9
10
o API Data
\ ~ - - - Winn
* ~ - - - - R-D
- - - - API
Ext. R-D
- - - - L - K
- - - P - F
- R - S
r ~ i I i t L i ~ I I I I I I I I I ~ I i I I I I I
5 10 15 20 25 30 35 40 45
Carbon Number
FIG. 2. 18~Compar i son of var i ous met hods f or est i mat i on of cri ti cal pressure of n-
al kyl cycl opent anes. API Data: API-TDB [2]; Wi nn: Eq. (2.95); R-D: Ri azi -Daubert , Eq. (2.64);
APh Eq. (2.56); Ext. R-D: Extended Ri azi -Daubert , Eq. (2.68); L-K: Lee-Kesl er, Eq. (2.70);
P-F: Pl an-Fi roozabadi , Eq. (2.43); and Table 2.8; R-S: Ri azi -Sahhaf, Eq. (2.42); and Table 2.6.
i
50
2. CHARACTERIZATION AND PROPERTI ES OF PURE HYDROCARBONS 79
3000
2500
2 0 0 0
1500
= 1000
500
S
o API Data / .
/ .
TWO / ' i /
. . . . Ext. R-D / . " ~
. . . . . R-S ,//~d - ~ / - ' / ~
- - - --R-D ~ Z . . / /
i i I I r i i I i i i i i i i l i i
0 10 20 30 40 50
Carbon Number
F I G . 2 . 1 9 - - C o m p a r i s o n o f v a r i o u s m e t h o d s f o r e s t i m a t i o n o f c r i t i c a l v o l u m e o f
n - a l k y l b e n z e n e s . A P I D a t a : A P I - T D B [ 2 ] ; T w u : E q s . ( 2 . 8 3 ) - ( 2 . 8 5 ) ; A P h E q . ( 2 . 1 0 1 ) ;
R - S : R i a z i - S a h h a f , E q . ( 2 . 4 2 ) a n d T a b l e 2 . 6 ; R - D : R i a z i - D a u b e r t , E q . ( 2 . 9 8 ) ; H - Y :
H a l I - Y a r b o r o u g h , E q . ( 2 . 1 0 0 ) .
Ev a l u a t i o n of t he s e me t h o d s f or c r i t i c a l p r o p e r t i e s of hy-
d r o c a r b o n s h e a v i e r t h a n C20 wa s n o t p o s s i b l e due t o t he l a c k
of c o n f i r me d e x p e r i me n t a l dat a. Ap p l i c a t i o n of t he s e me t h o d s
f or c r i t i c a l p r o p e r t i e s of p e t r o l e u m f r a c t i ons a n d r e s e r voi r
f l ui ds i s b a s e d on t he a c c u r a c y of p r e d i c t e d phys i c a l p r o p -
ert y. Thes e e va l ua t i ons a r e d i s c u s s e d i n Ch a p t e r 3, whe r e t he
me t h o d of p s e u d o c o mp o n e n t i s i n t r o d u c e d f or t he e s t i ma -
t i on of p r o p e r t i e s of p e t r o l e u m f r act i ons . Gener al l y, a mo r e
a c c u r a t e c o r r e l a t i o n f or p r o p e r t i e s of p u r e h y d r o c a r b o n s doe s
n o t ne c e s s a r i l y gi ve b e t t e r p r e d i c t i o n f or p e t r o l e u m f r a c t i ons
e s pe c i a l l y t hos e c o n t a i n i n g he a vy c o mp o u n d s .
Ev a l u a t i o n of me t h o d s of e s t i ma t i o n of c r i t i c a l p r o p e r t i e s
f or p e t r o l e u m f r a c t i ons i s a di f f i cul t t a s k as t he r e s ul t s d e p e n d
on t he t ype of p e t r o l e u m f r a c t i on u s e d f or t he e va l ua t i on. The
Ri a z i a n d Da u b e r t c or r e l a t i ons p r e s e n t e d b y Eq. (2. 63) a n d
(2. 64) or t he API me t h o d s p r e s e n t e d by Eqs. (2. 65) a n d (2. 66)
we r e d e v e l o p e d b a s e d on c r i t i c a l p r o p e r t y d a t a f r om C5 t o C18;
t he r e f or e , t he i r a p p l i c a t i o n t o p e t r o l e u m f r a c t i ons c o n t a i n i n g
ver y he a vy c o mp o u n d s wo u l d be l ess a c c ur a t e . The Ke s l e r -
Lee a n d t he Twu me t h o d we r e or i gi na l l y d e v e l o p e d b a s e d on
s o me c a l c u l a t e d d a t a f or c r i t i c a l p r o p e r t i e s of he a vy hydr oc a r -
bons a n d t he c ons i s t e nc y of To a n d Pc we r e o b s e r v e d at Pc = 1
a t m at wh i c h T6 wa s set e qua l t o To. Twu u s e d s o me val ues of
Tc a n d Pc b a c k - c a l c u l a t e d f r om v a p o r p r e s s u r e d a t a f or hydr o-
c a r b o n s h e a v i e r t h a n C20 t o e xt e nd a p p l i c a t i o n of hi s cor r e-
l a t i ons t o he a vy h y d r o c a r b o n s . Ther ef or e, i t i s e xpe c t e d t h a t
f or he a vy f r a c t i ons or r e s e r voi r f l ui ds c o n t a i n i n g he a vy c om-
p o u n d s t he s e me t h o d s p e r f o r m b e t t e r t h a n Eqs. ( 2. 63) - ( 2. 66)
TABLE 2. 15--Eval uat i on of various methods for prediction of critical temperature and pressure
of pure hydrocarbons from C5 to C20.
Abs Dev%**
T~ Pc
Method Equation(s) AD% MAD% AD% MAD%
API (2.65)-(2.66) 0.5 2.2 2.7 13.2
Twu (2.73)-(2.88) 0.6 2.4 3.9 16.5
Kesler-Lee (2.69)-(2.70) 0.7 3.2 4 12.4
Cavett (2.71 )-(2.73) 3.0 5.9 5.5 31.2
Winn (Si m-Daubert ) (2.94)-(2.95) 1.0 3.8 4.5 22.8
Ri azi -Daubert (2.63)-(2.64) 1.1 8.6 3.1 9.3
Lin & Chao Reference [72] 1.0 3.8 4.5 22.8
aData on Tc and Pc of 138 hydrocarbons from different families reported in API-TDB were used for the evalua-
tion process [29].
bDefined by Eqs. (2.134) and (2.135).
for Tc and Pc as observed by s ome r esear cher s [51, 85]. How-
ever, Eq. (2.42) and subsequent l y deri ved Eqs. (2.67) and
(2.68) have t he i nt ernal consi st ency and can be used f r om
C5 t o C50 al t hough t hey are devel oped for hydr ocar bons f r om
C20 to C50
The 1980 Ri azi - Dauber t correl at i ons for Tc and Pc were
general l y used and r ecommended by ma ny r esear cher s for
light fract i ons (M < 300, car bon numbe r < C22). Yu et al. [84]
used 12 di fferent correl at i ons to char act er i ze t he C7+ pl us
fract i on of several sampl es of heavy r eser voi r fluids and bi-
t umens. Based on t he resul t s pr esent ed on gas- phase compo-
sition, GOR, and sat ur at i on pressure, Eqs. (2.63) and (2.64)
showed bet t er or equi val ent predi ct i ons to ot her met hods.
Whi t son [53] made a good anal ysi s of correl at i ons for t he crit-
ical pr oper t i es and t hei r effects on char act er i zat i on of reser-
voi r fluids and suggest ed t he use of Eqs. (2.63) and (2.64) for
pet r ol eum cut s up to C25. But l at er [51] based on his observa-
t i on for phase behavi or pr edi ct i on of heavy r eser voi r fluids,
he r ecommended t he use of Kesl er-Lee or Twu for est i mat i on
of Tc and Pc of such fluids, while for est i mat i on of critical
vol ume he uses Eq. (2.98). Sorei de [52] in an extensive eval-
uat i on of vari ous correl at i ons for t he est i mat i on of critical
pr oper t i es r ecommends use of t he API-TDB [2] met hod for
est i mat i on Tc and Pc (Eqs. (2.65) and (2.66)) but he r ecom-
mends Twu met hod for t he critical vol ume. Hi s r ecommenda-
t i ons are based on phase behavi or cal cul at i ons for 68 sampl es
of Nor t h Sea reservoi r fluids. I n a recent l y publ i shed Hand-
book o f Reservoi r Engi neeri ng [48], and cal cul at i ons made on
phase behavi or of r eser voi r fluids [86], Eqs. (2.65), (2.66) have
been sel ect ed for t he est i mat i on of critical pr oper t i es of unde-
fined pet r ol eum fractions. Anot her possi bi l i t y to r educe t he
er r or associ at ed wi t h critical pr oper t i es of heavy fract i ons is
to back-cal cul at e t he critical pr oper t i es of t he heavi est end of
t he reservoi r fluid f r om an EOS based on a meas ur ed physi cal
pr oper t y such as densi t y or sat ur at i on pr essur e [51, 52, 70].
Fi r oozabadi et al. [63, 64] have st udi ed extensively t he wax
and asphal t ene pr eci pi t at i on in r eser voi r fluids. They ana-
lyzed vari ous met hods of cal cul at i ng critical pr oper t i es of
heavy pet r ol eum fract i ons and used Eq. (2.42) for t he critical
pr oper t i es and acent ri c f act or of paraffins, napht henes, and
ar omat i cs, but t hey used Eq. (2.43) for t he critical pr essur e of
vari ous hydr ocar bon gr oups wi t h M > 300. Thei r eval uat i on
was based on t he cal cul at i on of t he cl oud poi nt of di fferent
oils. It is believed t hat fract i ons wi t h mol ecul ar wei ght gr eat er
t han 800 (NC57) mai nl y cont ai n ar omat i c hydr ocar bons [63]
and t her ef or e Eq. (2.42) wi t h const ant s given in Table 2.6 for
ar omat i cs is an appr opr i at e correl at i on to est i mat e t he pr op-
erties of such fractions.
Mor e recent l y Ji anzhong et al. [87] revi ewed and evalu-
at ed vari ous met hods of est i mat i on of critical pr oper t i es of
pet r ol eum and coal liquid fract i ons. Thei r wor k fol l owed
t he wor k of Voulgaris et al. [88], who r ecommended use of
Eq. (2.38) for est i mat i on of critical pr oper t i es for t he pur pose
of predi ct i on of physi cal pr oper t i es of pet r ol eum fract i ons
and coal liquids. They correct l y concl uded t hat compl exi t y
of correl at i ons does not necessari l y i ncrease t hei r accuracy.
They eval uat ed Lee-Kesler, Ri azi - Dauber t , and Twu met hods
wi t h mor e t han 318 compounds (> C5) i ncl udi ng t hose f ound
in coal liquids wi t h boi l i ng poi nt up to 418~ (785 ~ F) and spe-
cific gravi t y up to 1.175 [87]. They suggest ed t hat Eq. (2.38)
is t he mos t sui t abl e and accur at e rel at i on especi al l y when
t he coefficients are modified. Based on t hei r dat abase, t hey
~ 9
<
3.00
2.50
2.03
1.50
1.03
(150
(103
A P I Da t a : n- Al k a ne s /
/
- - - - Predicted: n-Alkanes
9 A P I D auc n- Al kyl cycl opent anes:
. . . . Pr edi ct ed: n - Al k y l c y c l o p e n t a n e s
9 A P I Dat a: n - A l k y l b e n z enes
Preclicted~ n-~dkylbeazenes ~ j . . / /
: f5
J
I t I I I r I I r
0 10 ~!0 30 40 5o
03
Carbon Number
F I G . 2 . 2 0 - - - P r e d i c t i o n o f a c e n t r i c f a c t o r o f p u r e h y d r o c a r b o n s f r o m E q . ( 2 . 4 2 ) .
obtained the coefficients for Tc, Pc, and Vo in Eq. (2.38) with
use of d20 (liquid density at 20~ and 1 atm in g/cm 3) instead
of SG (Tc, Pc, Vc = aTbbd~0). They reported the coefficients as
[87] T j K (a = 18.2394, b = 0.595251, c = 0.347420), Pc/bar
(a = 2.95152, b = -2.2082, c = 2.22086), and Vc/cma/mol (a =
8.22382 x 10 -5, b = 2.51217, c = -1.62214). Equation (2.38)
with these coefficients have not been extensively tested
against data on properties of petroleum fractions as yet but
for more than 300 pure hydrocarbons gives average errors of
0.7, 3.8, and 2.9% for To Pc, and Vc, respectively [87].
2. 9. 4 Eva l ua t i on o f M e t h o d s o f E s t i m a t i o n
o f A c e n t r i c F a c t o r a n d O t h e r Pr ope r t i e s
For the calculation of the acentric factor of pure hydrocar-
bons Eq. (2.42) is quite accurate and will be used in Chapter 3
for the pseudocomponent method. Firoozabadi suggests that
for aromatics with M > 800, co = 2. Generally there are three
methods for the estimation of the acentric factor of undefined
petroleum fractions. Perhaps the most accurate method is to
estimate the acentric factor through its definition, Eq. (2.10),
and vapor pressure estimated from a reliable method [86].
This method will be further discussed in Chapter 7 along with
methods of calculation of vapor pressure. For pure hydrocar-
bons the Lee-Kesler method is more accurate than the Edmis-
ter method [36]. The Korsten method for estimating acentric
factor is new and has not yet been evaluated extensively. For
three different homologous hydrocarbon families from C6 to
C50, values of acentric factor calculated from Eq. (2.42) are
compared with values reported in the API-TDB [2] and they
are shown in Fig. 2.20. Prediction of acentric factors from dif-
ferent methods for n-allcylcyclopentanes and n-alkylbenzenes
2.0
O
O
<
1.5
1.0
0.5
- 0 . 0
0 5 0
o API Data
Riazi- 5ahhaf /
....... Lee-Kesler / /
Kesler-Lee ~ , , " . . ~. -~
. . . . ~ ' = . ~ 7 : - : ' ~ . " "
- -
@ ~ 1 7 6 ~
1 0 2 0 3 0 4 0
Carbon Number
F I G . 2 . 2 1 - - P r e d i c t i o n o f a c e n t r i c f a c t o r o f n - a l k y l c y c l o -
p e n t a n e s f r o m v a r i o u s m e t h o d s . A P I D a t a : A P I - T D B [ 2 ] ; R - S :
R i a z i - S a h h a f , E q . ( 2 . 4 2 ) a n d T a b l e 2 . 6 ; L - K : L e e - K e s l e r ,
E q . ( 2 . 1 0 5 ) ; K - L : K e s l e r - L e e , E q . ( 2 . 1 0 7 ) ; E d m i s t e r : E q . ( 2 . 1 0 8 ) ;
K o r s t e n : E q . ( 2 . 1 0 9 ) .
3.000
o DIPPR Data
Riazi-Sahhaf
2.500 P ~ et al. /
9 - Korstea /
2.O00
/
m ~ Lee-Kesler /
/.:.--:...---
< 1 . 0 0 0
0 . 5 0 0
0.000
0 1 0 2 0 3 0 4 0 5 0
Carbon Number
F I G . 2 . 2 2 - - P r e d i c t i o n o f a c e n t r i c f a c t o r o f n - a l k y l b e n z e n e s
f r o m v a r i o u s m e t h o d s . D I P P R D a t a : D I P P R [ 2 0 ] ; R i a z i - S a h h a f :
E q . ( 2 . 4 2 ) a n d T a b l e 2 . 6 ; P a n e t a l . : R e f . [ 6 3 , 6 4 ] , E q . ( 2 . 4 4 ) ;
K o r s t e n : E q . ( 2 . 1 0 9 ) ; L e e - K e s l e r : E q . ( 2 . 1 0 5 ) ; K e s l e r - L e e :
E q . ( 2 . 1 0 7 ) .
are presented in Figs. 2.21 and 2.22, respectively. The Riazi-
Sahhaf method refers to Eq. (2.42) and coefficients given
in Table 2.6 for different hydrocarbon families. In Fig. 2.22
the Pan et al. [63, 64] method refers to Eq. (2.44), which
has been recommended for n-alkylbenzenes (aromatics). The
Lee-Kesler method, Eq. (2.105), has been generally used for
the estimation of accentric factor of undefined petroleum
fractions [27]. The Kesler-Lee method refers to Eq. (2.107),
which was recommended by Kesler-Lee [12] for estimation
of the acentric factor of hydrocarbons with Tbr > 0.8, which
is nearly equivalent to hydrocarbons with molecular weights
greater than 300. However, our experience shows that this
equation is accurate for pure compounds when true critical
temperatures are used and high errors can occur when the
predicted critical temperature is used in the equation. For
heavy hydrocarbons and petroleum fractions (M > 300) with
estimated critical properties, either the method of pseudo-
component discussed in Chapter 3 or the Lee-Kesler may be
the most appropriate method. The accuracy of a method to
estimate acentric factor also depends on the values of Tc and
Pc used to calculate co as was shown in Example 2.7. Usually
the Cavett correlations for Tc and Pc are used together with
the Edmister method. Evaluation of these methods for the
prediction of properties of undefined petroleum fractions is
The accuracy of correlations presented for estimation of
other properties such as density, refractive index, boiling
point, and CH has been discussed in the previous section
where these methods are presented. Prediction of the refrac-
tive index for pure hydrocarbons is shown in Fig. 2.9. Predic-
tion of viscosity at 38~ (100~ I ) 3 8 , through Eq. (2.128) for
pure hydrocarbons from three hydrocarbon groups is shown
in Fig. 2.23. Further assessment of accuracy of these methods
o
30
25
20
15
10
i
Z~ A P I D a t a : n-Al kanes /
P r e d i c t e d : n - A l k a n e s /
o A P I D a t a : n - a l k y l e y c l o p e n t a n e s /
. . . . . P r e d i c t e d: n - a l k y l c y c l o p e n t a n e s /
9 A P I D a t a : n - a l k y l c b e n z e n e s / /
- - - - P r e d i e t e d : n - a l l q , l c b e n z e n e s 1 / . ,
/
/
/
/ . . /
/ , ' /
/ . ' o
/ / o
. . . . ~ . i . . . . i . . . . J . . . .
0 5 10 15 20 25 30 35
Carbon Number
FI G. 2 . 2 3 - - P r e f l i c t i o n of k i n e ma t i c v i s c o s i t y o f p u r e h y d r o c a r b o n s a t 3 7 . 8 ~
f r o m Eq. ( 2 . 1 2 8 ) .
is di scussed in t he following chapt ers wher e pr oper t i es of
pet r ol eum fract i ons are cal cul at ed.
2 . 1 0 C O N C L U S I O N S A N D
R E C O M M E N D A T I O N S
I n t hi s chapt er met hods of char act er i zat i on of pur e hydro-
car bons have been present ed. These met hods will be used
in Chapt ers 3 and 4 for t he char act er i zat i on of pet r ol eum
fract i ons and crude oils, respectively. Thi s chapt er is an i m-
por t ant chapt er in this book as t he met hod selected for t he
char act er i zat i on of hydr ocar bons affects t he accur acy of es-
t i mat i on of every physi cal pr oper t y t hr oughout t he book. I n
t hi s chapt er t he basi c char act er i zat i on par amet er s have been
i nt r oduced and based on t he t heor y of i nt er mol ecul ar forces,
a general i zed correl at i on for t he char act er i zat i on of hydro-
car bon syst ems has been developed. I t is shown t hat funda-
ment al l y devel oped correl at i ons are si mpl er wi t h a wi der field
of appl i cat i on and accuracy. For light fract i ons (M < 300),
general l y t wo- par amet er correl at i ons are sufficient for prac-
tical calculations, while for heavi er hydr ocar bons or nonhy-
dr ocar bons t he use of a t hi rd pa r a me t e r is needed. The t wo
char act er i zat i on par amet er s shoul d r epr esent t he energy and
size charact eri st i cs of molecules. Charact eri zat i on par ame-
t ers such as Tb, M, and v3s000) may be used as energy, par am-
et ers while SG, I , and CH coul d be used as size par amet er s.
Boiling poi nt and specific gravi t y are t he mos t easily measur -
abl e and appr opr i at e char act er i zat i on par amet er s fol l owed
by mol ecul ar wei ght and refract i ve index. Viscosity and CH
par amet er s may be used as t he l ast opt i on for pr edi ct i on of
pr oper t i es of hydr ocar bons. Various met hods of est i mat i on
of t hese par amet er s as well as critical pr oper t i es and acen-
t ri c f act or used in cor r espondi ng st at e correl at i ons and a de-
t ai l ed revi ew of t hei r appl i cat i on for di fferent pur poses and
r ecommendat i ons made in t he l i t erat ure are present ed. Basi c
pr oper t i es of mor e t han 100 sel ect ed compounds are given in
Tables 2.1 and 2.2 and will be used frequent l y t hr oughout t he
book.
The mos t i mpor t ant i nf or mat i on pr esent ed in this chap-
t er is t he met hods of est i mat i on of mol ecul ar weight, critical
const ant s, and acent ri c f act or for pur e hydr ocar bons. These
met hods are al so r ecommended t o est i mat e pr oper t i es of nar-
r ow boiling r ange pet r ol eum fract i ons as di scussed in Chapt er
3. A s umma r y of eval uat i ons made by vari ous r esear cher s was
revi ewed in Sect i on 2.9. Based on t hese eval uat i ons it is cl ear
t hat t heoret i cal l y based correl at i ons such as Eq. (2.38) or its
modi fi ed versi on Eq. (2.40), while si mpl er t han ot her empi ri -
cally devel oped correl at i ons, have a wi de r ange of appl i cat i on
wi t h r easonabl e accuracy. Based on t hese eval uat i ons a list
of r ecommended met hods f or di fferent pr oper t i es of vari ous
t ypes of hydr ocar bons and nar r ow boiling r ange fract i ons is
given in Table 2.16. Est i mat i on of wide boi l i ng r ange frac-
t i ons is di scussed in t he next chapter. The choi ce for met h-
ods of cal cul at i on of pr oper t i es not pr esent ed in Table 2.16
is general l y nar r ow and comment s have been made wher e
t he met hods are i nt r oduced in each section. The i nf or mat i on
pr esent ed in t hi s chapt er shoul d hel p pract i cal engi neers t o
devel op new correl at i ons or t o select an appr opr i at e charac-
t eri zat i on scheme when usi ng a process simulator.
TABLE 2. 16--Recommended methods for the prediction of the basic properties of pure hydrocarbons
and narrow boiling range petroleum fractions a.
Property Range of M Method Equation
M 70-700 API [2] 2.51
70-300 Ri azi - Dabuber t [28] 2.50
200-800 API [2] 2.52(b)
70-700 Twu [30] 2.89-2.92(c)
Tc 70-300 API [2] 2.65
70-700 Lee-Kesl er [ 12] 2.69
70-700 Ext ended API [65] 2.67
70-800 Ri azi - Sahhaf [31 ] 2.42 d
<70 e Ri azi et al. [37] 2.47 e
Pc 70-300 API [2] 2.66
70-700 Lee- Kesl er [12] or Twu [30] 2.70
70-700 Ext ended API [65] 2.68
70-300 Ri azi - Sahhaf [31 ] 2.42 d
300-800 Pan- Fi r oozabadi - Fot l and [63] 2.43 d
<70 e Ri azi et al. [37] 2.47 e
Vc 70-350 Ri azi - Dauber t [28] 2.98
300-700 Ext ended R- D [65] 2.99
70-700 Ri azi - Sahhaf [31] 2.42 d
<70 e Ri azi et al. [37] 2.47 e
Zc 70-700 By defi ni t i on of Zc 2.8
w 70-300 Lee- Kesl er [27] 2.105
300-700 Kor st en [77] 2.109
70-700 Ri azi - Sahhaf [31 ] 2.42 f
300-700 Pan- Fi r oozabadi - Fof l and [63] 2.44g
Tb 70-300 Ri azi - Dauber t [29] 2.56
300-700 Ext ended R- D [65] 2.57
70-700 Ri azi - Sahhaf [31 ] 2.42 d
SG All r ange Denis et al. [8] 2.112
70-300 Ri azi - Dauber t [29] 2.59
70-700 Ext ended R- D [65] 2.60
200-800 API [2] 2.61 d
I 70-300 Ri azi - Dauber t [29] 2.116
300-700 Ext ended R- D [65] 2.117
70-700 Ri azi - Sahhaf [31 ] 2.42 d
d All r ange Deni s et al. [8] 2.111
70-350 Ri azi - Dauber t [28] 2.113
T M 70-700 Pan- Fi r oozabadi - Fot l and [63] 2.126 h
Ri azi - Sahhaf [31] 2.124 and 2.125 i
Methods recommended for pure homologous hydrocarbons (designated by c-i) are also recommended for the pseu-
docomponent method discussed in Chapter 3 for petroleum fractions. The 300 boundaryis approximate and methods
recommended for the range of 70-300 may be used safely up to molecular weight of 350 and similarly methods rec-
ommended for the range 300-700 may be used for molecular weight as low as 250.
a For narrow boiling range fractions a midpoint distillation temperature can be used as Tb.
bOnly when Tb is not available.
CRecommended for pure hydrocarbons from all types.
'/Recommended for pure homologous hydrocarbon groups.
e For compounds and fractions with molecular weight less than 70 and those containing nonhydrocarbon compounds
(H2S,CO2, N2, etc.) Eq. (2.47) is recommended.
fEquation (2.42) is applicable to acentric factor of n-alkylbenzenes up to molecular weight of 300.
gEquation (2.44) is applicable to acentric factor of aromatics for 300 < M < 800 and for M > 800, w - 2 should be
used.
hFor pure hydrocarbons from n-alkanes family.
i For pure hydrocarbons from n-alkylcylopentanes (naphthenes) and n-alkylbenzenes (aromatics) families.
2. 11 PROBLEMS
2. 1. F o r l i g h t h y d r o c a r b o n s a n d n a r r o w b o i l i n g r a n g e f r a c -
t i o n s u s u a l l y a f e w me a s u r e d p a r a me t e r s a r e a va i l -
abl e. F o r e a c h o n e o f t h e f o l l o wi n g c a s e s d e t e r mi n e
t h e b e s t t wo p a r a me t e r s f r o m t h e s e t o f a v a i l a b l e
d a t a t h a t a r e s u i t a b l e t o b e u s e d f o r p r o p e r t y p r e d i -
c t i ons :
a. Tb, M, SG
b. CH, 1)38(100), n20
c. CH, n20, SG
d. Tb, M, n20, CH
e. 1338(100), Tb, CH, M
2. 2. F o r h e a v y a n d c o mp l e x h y d r o c a r b o n s o r p e t r o l e u m f r a c -
t i ons , b a s i c p r o p e r t i e s c a n b e b e s t d e t e r mi n e d f r o m
t h r e e p a r a me t e r s . De t e r mi n e t h e b e s t t h r e e p a r a me t e r s
f o r e a c h o f t h e f o l l o wi n g cas es :
a. Tb, M, SG, I)38(100 )
b. CH, 1)38(100), n20, 1)99(210), API Gr a v i t y
c. CH, n20, SG, M, 1)99(210)
d. Tb, M, n20 , CH, Kw
2. 3. You wi s h t o d e v e l o p a p r e d i c t i v e c o r r e l a t i o n f o r p r e d i c -
t i o n o f mo l a r v o l u me , Vr, i n t e r ms o f v38(100), SG, a n d
t e mp e r a t u r e T. Ho w d o y o u p r o p o s e a s i mp l e r e l a t i o n
wi t h t e mp e r a t u r e d e p e n d e n t p a r a me t e r s f o r e s t i ma t i o n
o f mo l a r v o l u me ?
2.4. A t a nk cont ai ns pur e h y d r o c a r b o n l i qui d f r om t he
n- par af f i n gr oup. Det er mi ne t he l i ght est h y d r o c a r b o n
f r o m t he n- al kane f ami l y t hat can exi st i n an ope n ves-
sel at t he e nvi r onme nt of 38~ (100~ and 1 a t m wi t h-
out da nge r of f l ammabi l i t y i n t he va por phas e ne a r t he
vessel.
2.5. Devel op t hr ee r el at i ons f or es t i mat i on of CH wei ght
r at i o of n-paraffi ns, n- al kyl cycl opent anes, a nd n-alkyl-
be nz e ne i n t er ms of t hei r r espect i ve mol e c ul a r wei ght .
For each gr oup cal cul at e CHo~ ( and HCo~). Show gr aph-
i cal pr e s e nt a t i on of t he pr edi ct ed val ues ver sus act ual
val ues of CH f or t he t hr ee f ami l i es on a si ngl e gr aph.
2.6. Pr edi ct t he r ef r act i ve i ndex of n-paraffi ns, n-al kyl -
cycl opent anes , and n- al kyl benzene ver s us c a r b o n n u m-
ber f r o m C6 t o Cs0 us i ng Eq. (2. 46a) a nd c o mp a r e gr aph-
i cal l y wi t h val ues f r o m Eq. (2.42). I n us i ng Eq. (2. 46a)
it is neces s ar y t o obt ai n M f r om Nc i n each family, and
t hen f r o m Eq. (2.42) Tb and SG ma y be es t i mat ed f or
each c a r b o n n u mb e r i n each family.
2.7. A pur e h y d r o c a r b o n has mol e c ul a r wei ght of 338. 6 a nd
speci fi c gr avi t y of 0. 8028. Usi ng a ppr opr i a t e me t hods
cal cul at e
a. boi l i ng poi nt , Tb.
b. r ef r act i vi t y i nt er cept , Ri .
c. ki nemat i c vi scosi t y at 38 a nd 99~
d. VGC f r o m t hr ee di f f er ent met hods .
2.8. For n- but yl cycl ohexane, cri t i cal pr oper t i es and mol ecu-
l ar wei ght ar e give i n Table 2.1. Use Tb and SG as t he
i nput pa r t a me t e r s a nd cal cul at e
a. M, To, Pc, de, a nd Zc f r om t he API-TDB-87 met hods .
b. M, To, Pc, dc, and Zc t he Lee- Kes l er cor r el at i ons.
c. M, To, Pc, de, a n d Zc f r om t he Ri a z i - Da ube r t cor r el a-
t i ons (Eq. 2.38).
d. M, Tc, Pc, de, and Zc f r o m t he Twu cor r el at i ons.
e. Compa r e val ues f r o m each me t h o d wi t h act ual val ues
and t abul at e t he %D.
2.9. Use cal cul at ed val ues of Tc and Pc i n Pr obl em 2.8 t o cal-
cul at e acent r i c f act or f r om t he Lee- Kes l er a nd Kor s t en
cor r el at i ons f or each par t , t hen obt ai n t he er r or s (%D)
f or each met hod.
2.10. Es t i ma t e t he acent r i c f act or of i s ooct ane f r o m Le e -
Kesler, Edmi st er , a nd Kor s t en cor r el at i ons us i ng i nput
dat a f r o m Table 2.1. Cal cul at e t he %D f or each met hod.
2.11. Es t i ma t e t he ki nemat i c vi scosi t y of n- hept ane at 38
and 99~ a nd c o mp a r e wi t h t he exper i ment al val ues
r e por t e d by t he API - TDB [2]. Also es t i mat e vi scosi t y
of n- hept ane at 50~ f r o m Eq. (2.130) and t he ASTM
vi s cos i t y- t emper at ur e char t .
2.12. For n- al kyl cyl opent anes f r o m C5 t o C10, est i mat e d20
f r o m SG us i ng t he r ul e of t h u mb s a nd a mor e a c c ur a t e
met hod. Compa r e t he r esul t s wi t h act ual val ues f r o m
Table 2.1. For t hese c o mp o u n d s al so es t i mat e r ef r act i ve
i ndex at 25~ us i ng M as t he onl y i nput dat a avai l abl e.
Use bot h me t hods f or t he effect of t e mpe r a t ur e on re-
f r act i ve i ndex as di scussed i n Sect i on 2.6.2 and c o mp a r e
y o u r r esul t s wi t h t he val ues r e por t e d b y t he API - TDB [2].
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Equilibria: Wax Formation from Heavy Hydrocarbon
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pp. 379-384.
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MNL50-EB/Jan. 2005
Characteri zati on of Petroleum
Fractions
NOMENCLATURE
AP
API
A,B, . . . . F
a,b. . i
CABP
CH
d
Kw
I
M
MABP
MeABP
n
Nc
SG
SGg
SL
S%
Tb
Tc
TF
TM
T~o
Ts0
V
V
v~
VGC
VABP
WABP
Aniline point, ~ (unless specified otherwise)
API gravity defined in Eq. (2.4)
Correlation coefficients in various equations
Correlation coefficients in various equations
Cubic average boiling point, K
Carbon-to-hydrogen weight ratio
Liquid density at 20~ and 1 at m, g/ cm 3
Watson (UOP) K factor defined by Eq. (2.13)
Refractive index par amet er defined in Eq. (2.36)
Molecular weight, g/ mol [kg/kmol]
Molal average boiling point, K
Mean average boiling point, K
Sodi um D line refractive index of liquid at 20~
and 1 atm, dimensionless
Carbon number ( number of carbon at oms in a
hydrocarbon molecule)
Critical pressure, bar
Refractivity intercept in Eq. (2.14)
Specific gravity of liquid substance at 15.5~
Specific gravity of gas substance at 15.5~ (60~
defined by Eq. (2.6), dimensionless
ASTM D 86 slope between 10 and 90% points,
~ (K)/vol%
Weight percent of sulfur in a fraction
Boiling point, K
Critical t emperat ure, K
Flash point, K
Melting (freezing point) point, K
Temperature on distillation curve at 10% volume
vaporized, K
Temperature on distillation curve at 50% volume
vaporized, K
Molar volume, cma/ gmol
Saybolt universal viscosity, SUS
Critical volume (molar), cm3/ gmol
Viscosity gravity constant defined by Eqs. (2.15)-
(2.18)
Volume average boiling point, K
Weight average boiling point, K
0 A property of hydrocarbon such as: M , To, Pc, Vc, I, d,
Tb, 9 9 9
p Density at a given t emperat ure and pressure, g/ cm 3
a Surface tension, dyn/ cm [ =mN/ m]
o~ Acentric factor defined by Eq. (2.10), dimensionless
Superscript
~ Properties of n-alkanes from Twu correlations
Subscripts
A Aromatic
N Naphthenic
P Paraffinic
T Value of a propert y at t emperat ure T
~ A reference state for T and P
c~ Value of a propert y at M --~ c~
20 Value of a propert y at 20~
39(100) Value of kinematic viscosity at 37.8~ (100~
99(210) Value of kinematic viscosity at 98.9~ (210~
Acronyms
API-TDB
ASTM D
%AD
%AAD
EFV
EOS
FBP
GC
GPC
HPLC
KI SR
I BP
I R
MA
MS
PA
PIONA
Greek Letters
RVP
F Gamma function RS
# Absolute (dynamic) viscosity, cp [mPa.s]. Also used for SD
dipole moment TBP
v Kinematic viscosity defined by Eq. (2.12), cSt [mm2/s] UV
87
American Petroleum I nstitute--Technical Data
Book
ASTM I nternational (test met hods by D com-
mittee)
Absolute deviation percentage defined by
Eq. (2.134)
Average absolute deviation percentage defined by
Eq. (2.135)
Equilibrium flash vaporization
Equation of state
Final boiling point (end point)
Gas chromat ography
Gel permeat i on chromat ography
High performance liquid chromat ography
Kuwait I nstitute for Scientific Research
Initial boiling point
I nfrared
Monoaromat i c
Mass spectroscometry
Poly (di- tri-, and higher) aromat i c
Paraffin, isoparaffin, olefin, naphthene,
and aromat i c
Reid vapor pressure
R squared (R2), defined by Eq. (2.136)
Simulated distillation
True boiling point
Ultraviolet
8 8 CHARACTERI ZATI ON AND PROPERTI ES OF PETROLEUM FRACTI ONS
IN THIS CHAPTER methods of characterization of petroleum
fractions and products are discussed. Petroleum fractions are 330
mixtures of hydrocarbon compounds with a limited boiling
point range. Experimental methods on measurement of ba- 320
sic properties that can be obtained from laboratory testing are
first presented and then methods of prediction of properties 310
that are not available will be discussed. Two general meth-
ods are presented: one for defined mixtures and another for ~ 300
2
undefined mixtures in which the composition is not known ~ 290
but some bulk properties are available. Petroleum fractions =
are also divided into light and heavy as well as narrow and
wide boiling range mixtures in which different characteriza- 280
tion methods are proposed. In addition to methods of estima-
tion of characterization parameters discussed in Chapter 2 270
for pure hydrocarbons, predictive methods for some char-
acteristics specifically applicable to petroleum fractions are 260
presented in this chapter. These characteristic parameters in-
clude distillation curve types and their interconversions, hy- 250
drocarbon type composition, sulfur content, carbon residue,
octane number, pour, cloud, aniline, and smoke points that af-
fect the quality of a fuel. Standard test methods recommended
by ASTM are given for various properties. Finally, mi ni mum
laboratory data needed to characterize various fractions as
well as analysis of laboratory data and criteria for develop-
ment of a predictive method are discussed at the end of this
chapter. Most of methods presented in this chapter will also
be used in Chapter 4 to characterize crude oils and reservoir
fluids.
3 . 1 E X P E R I M E N T A L D A T A O N B A S I C
P R O P E R T I E S O F P E T R O L E U M F R A C T I O N S
In this section characterization parameters that are usually
measured in the laboratory as well as methods of their mea-
surements are discussed. Generally not all of these parame-
ters are reported in a laboratory report, but at least from the
knowledge of some of these properties, all other basic prop-
erties for the fraction can be determined from the methods
presented in this chapter.
3 . 1 . 1 B o i l i n g P o i n t a n d D i s t i l l a t i o n C u r v e s
Pure compounds have a single value for the boiling point;
however, for mixtures the temperature at which vaporization
occurs varies from the boiling point of the most volatile com-
ponent to the boiling point of the least volatile component.
Therefore, boiling point of a defined mixture can be repre-
sented by a number of boiling points for the components ex-
isting in the mixture with respect to their composition. For
a petroleum fraction of unknown composition, the boiling
point may be presented by a curve of temperature versus
vol% (or fraction) of mixture vaporized. Different mixtures
have different boiling point curves as shown in Fig. 3.1 for a
gas oil petroleum product [1]. The curves indicate the vapor-
ization temperature after a certain amount of liquid mixture
vaporized based on 100 units of volume, The boiling point of
the lightest component in a petroleum mixture is called ini-
tial boiling point (IBP) and the boiling point of the heaviest
compound is called the final boiling point (FBP). In some ref-
erences the FBP is also called the end point. The difference
Gas Oil
. . . . n-Tetradecane
n-Hexadecane
m ~ n-Nonadecane
L t i I
20 40 60 80 0 100
Vol% Vaporized
FIG. 3. 1- - Di st i l l at i on curve for a gas oi l and t hree pure
hydrocarbons,
between FBP and IBP is called boiling point range or simply
boiling range. For petroleum fractions derived from a crude
oil, those with wider boiling range contain more compounds
than fractions with narrower boiling range. This is due to
the continuity of hydrocarbon compounds in a fraction. Ob-
viously, in general, for defined mixtures this is not the case.
For a pure component the boiling range is zero and it has
a horizontal distillation curve as shown in Fig. 3.1 for three
n-alkane compounds of C14, C16, and C19. For the gas oil sam-
ple shown in Fig. 3.1 the IBP is 248~ (477~ and the FBP is
328~ (62 I~ Therefore its boiling range is 80~ (144~ and
compounds in the mixture have approximate carbon number
range of C14-C19. Crude oils have boiling ranges of more than
550~ (~1000~ but the FBPs are not accurate. For heavy
residues and crude oils the FBPs may be very large or even
infinite as the heaviest components may never vaporize at
all. Generally, values reported as the IBP and FBP are less
reliable than other points. Ft3P is in fact the maximum tem-
perature during the test and its measurement is especially
difficult and inaccurate. For heavy fractions it is possible
that some heavy compounds do not vaporize and the high-
est temperature measured does not correspond to the boiling
point of heaviest component present in the mixture. If the
temperature is measured until, i.e. 60% vaporized, then the
remaining 40% of the fraction is called residue. The boiling
point curve of petroleum fractions provides an insight into the
composition of feedstocks and products related to petroleum
refining processes. There are several methods of measuring
and reporting boiling points of petroleum fractions that are
described below.
3.1.1.1 ASTM D 86
ASTM D 86 is one of the simplest and oldest methods of mea-
suring and reporting boiling points of petroleum fractions
and is conducted mainly for products such as naphthas,
gasolines, kerosenes, gas oils, fuel oils, and other similar
petroleum products. However, this test cannot be conducted
3. CHARACTERIZATION OF PETROLEUM FRACTIONS 89
FIG. 3.2--Experimental apparatus for measurement of boiling point of
petroleum fractions by ASTM D 86 method (courtesy of Kuwait Institute
for Scientific Research).
for mixtures containing very light gases or very heavy com-
pounds that cannot be vaporized. The test is conducted at
atmospheric pressure with 100 mL of sample and the result
is shown as a distillation curve with temperatures at 0, 5, 10,
20, 30, 40, 50, 60, 70, 80, 90, 95, and 100% volume vapor-
ized. The final boiling point (at 100%) is the least accurate
value and it is usually less than the true final boiling point. In
many cases only a few temperatures are reported. An exposed
thermometer is used and temperatures are reported without
stem corrections. For heavy products, temperatures are re-
ported at maximum of 90, 70, or even 50% volume vaporized.
This is due to the cracking of heavy hydrocarbons at high tem-
peratures in which vaporization temperatures do not repre-
sent boiling points of the original compounds in the mixture.
The cracking effect is significant at temperatures above 350 ~ C
(660~ however, ASTM D 86 temperatures reported above
250~ (480~ should be used with caution. Corrections ap-
plied to consider the effects of cracking are applicable from
250 to 500~ however, these procedures have not been widely
used and generally have not been confirmed. In the new revi-
sions of API-TDB-97 no correction for cracking in ASTM D 86
temperatures has been recommended [2]. An apparatus to
measure distillation of petroleum fractions by ASTM D 86
method is shown in Fig. 3.2.
3.1.1.2 True Boiling Point
ASTM D 86 distillation data do not represent actual boiling
point of components in a petroleum fraction. Process engi-
neers are more interested in actual or true boiling point (TBP)
of cuts in a petroleum mixture. Atmospheric TBP data are
obtained through distillation of a petroleum mixture using a
distillation column with 15-100 theoretical plates at relatively
high reflux ratios (i.e., 1-5 or greater). The high degree of
fractionation in these distillations gives accurate component
distributions for mixtures. The lack of standardized appara-
tus and operational procedure is a disadvantage, but vari-
ations between TBP data reported by different laboratories
for the same sample are small because a close approach to
complete component separation is usually achieved. Mea-
surement of TBP data is more difficult than ASTM D 86 data
in terms of both time and cost. TBP and ASTM D 86 curves
for a kerosene sample are shown in Fig. 3.3 based on data
provided by Lenoir and Hipkin [ 1 ]. As shown in this figure
the IBP from TBP curve is less than the IBP from ASTM D
86 curve, while the FBP of TBP curve is higher than that of
ASTM curve. Therefore, the boiling range based on ASTM D
86 is less than the actual true boiling range. In TBP, the IBP
is the vapor temperature that is observed at the instant that
the first drop of condensate falls from the condenser.
3.1.1.3 Simulated Distillation by Gas Chromatography
Although ASTM D 86 test method is very simple and conve-
nient, it is not a consistent and reproducible method. For this
reason another method by gas chromatography (GC) is be-
ing recommended to present distillation data. A distillation
curve produced by GC is called a simulated distillation (SD)
and the method is described in ASTM D 2887 test method.
Simulated distillation method is simple, consistent, and
3 0 0 ...........................................................................................................................
. . . . . . . ASTM D86
250 TBP .-""
~'~ . . . . - " ~176 *" - ' ~
150
100
0 20 40 60 80 100
Vol% Vaporized
FIG. 3. 3--ASTM D 86 and TBP curves for a
kerosene sample.
250 ......................................................................................................................................................................
200
150
100
50
....... ASTMD86
SD(ASTMD ~
_ _ i i i i
0 20 40 60 80 100
Percent Vaporized
FIG. 3. 4---Si mul ated and ASTM D 86 di sti l l a-
t i on cur ves for a petrol eum fracti on. ( The per-
cent i s in vol % for ASTM D 86 and is in wt % for
ASTM D 2887. )
reproducible and can represent the boiling range of a
petroleum mixture without any ambiguity. This method is ap-
plicable to petroleum fractions with a FBP of less than 538~
(1000~ and a boiling range of greater than 55~ (100~ and
having a vapor pressure sufficiently low to permit sampling at
ambient temperature. The ASTM D 2887 method is not appli-
cable to gasoline samples and the ASTM D 3710 test method
is recommended for such fractions. Distillation curves by SD
are presented in terms of boiling point versus wt% of mixture
vaporized because as described below in gas chromatogra-
phy composition is measured in terms of wt% or weight frac-
tion. Simulated distillation curves represent boiling points of
compounds in a petroleum mixture at atmospheric pressure;
however, as will be shown later SD curves are very close to ac-
tual boiling points shown by TBP curves. But these two types
of distillation data are not identical and conversion methods
should be used to convert SD to TBP curves. In comparison
with ASTM D 86, the IBP from a SD curve of a petroleum mix-
ture is less than IBP from ASTM D 86 curve, while the FBP
from SD curve is higher than the FBP from ASTM D 86 of the
same mixture (see Fig. 3.4). This is the same trend as that of
TBP curves in comparison with ASTM curves as was shown
in Fig. 3.3. A typical SD curve for a gas oil sample is shown
in Fig. 3.4. Note that in this figure the percent vaporized for
ASTM D 2887 (SD) is in wt% while for the ASTM D 86 curve
is in vol%.
The gas chromatography technique is a separation method
based on the volatility of the compounds in a mixture. The GC
is used for both generation of distillation curves as well as to
determine the composition of hydrocarbon gas or liquid mix-
tures, as will be discussed later in this chapter. For this reason
in this part we discuss the basic function of chromatography
techniques and elements of GC. In an analysis of a mixture by
a GC, the mixture is separated into its individual compounds
according to the relative attraction of the components for
a stationary and a mobile phase. Recent advances in chro-
matography make it possible to identify and separate com-
pounds with boiling points up to 750~ (1380~ A small fluid
sample (few microliters for liquid and 5 mL for gas samples)
is injected by a needle injector into a heated zone in which
the sample is vaporized and carried by a high-purity carrier
gas such as helium or nitrogen. The stationary phase is either
solid or liquid. A component that is more strongly attracted to
the mobile phase than to the stationary phase is swept along
with the mobile phase more rapidly than a component that is
more strongly attracted to the stationary phase. The mobile
phase can be a liquid phase as well; in this case the chro-
matography method is called liquid chromatography (LC).
The basic elements of a GC are a cylinder of carrier gas,
flow controller and pressure regulator, sample injector, col-
umn, detector, recorder, and thermostats for cylinder, col-
umn, and detector. The sample after injection enters a heated
oven where it enters the GC column (stationary phase). The
eluted components by the carrier gas called effluents enter
a detector where the concentration of each component may
be determined. The presence of a component in the carrier
gas leaving the column is continuously monitored through
an electric signal, which eventually is fed to a recorder for a
visual readout.
There are two types of columns, packed or capillary
columns, and two types of detectors, flame ionization detec-
tor or thermal conductivity detector. Packed columns have
inner diameters of 5-8 mm and length of 1-5 m. Column
and detector types depend on the nature of samples being
analyzed by the GC. The capillary columns are equivalent to
hundreds of theoretical equilibrium stages and can be used in
preference to packed columns. The inner diameter of capri-
lary columns is about 0.25-0.53 mm and their length is about
10-150 m. The stationary phase is coated on the inside wall
of columns. The flame ionization detector (FID) is highly sen-
sitive to all organic compounds (10 -12 g) but is not sensitive
to inorganic compounds and gases such as H20, CO2, N2,
and 02. The FID response is almost proportional to the mass
concentration of the ionized compound. Hydrogen of high
purity is used as the fuel for the FID. The thermal conduc-
tivity detector (TCD) is sensitive to almost all the compounds
but its sensitivity is less than that of FID. TCD is often used
for analysis of hydrocarbon gas mixtures containing nonhy-
drocarbon gases. The retention time is the amount of time re-
quired for a given component spent inside the column from
its entrance until its emergence from the column in the efflu-
ent. Each component has a certain retention time depending
on the structure of compound, type of column and station-
ary phase, flow rate of mobile phase, length, and tempera-
ture of column. More volatile compounds with lower boil-
ing points have lower retention times. Detector response is
measured in millivolts by electric devices. The written record
obtained from a chromatographic analysis is called a chro-
matograph. Usually the time is the abscissa (x axis) and mV
is the ordinate (y axis). A typical chromatograph obtained
to analyze a naphtha sample from a Kuwaiti crude is shown
in Fig. 3.5. Each peak corresponds to a specific compound.
Qualitative analysis with GC is done by comparing retention
times of sample components with retention times of reference
compounds (standard sample) developed under identical ex-
perimental conditions. With proper flow rate and tempera-
ture, the retention time can he reproduced within I%. Ev-
ely component has only one retention time; however, compo-
nents having the same boiling point or volatility but different
molecular structure cannot be identified through GC analysis.
In Fig. 3.5, compounds with higher retention time (x coordi-
nate) have higher boiling points and the actual boiling point
or the compound can be determined by comparing the peak
3. CHARACTERI ZATI ON OF PET ROL EUM FRACTI ONS 91
2~Q
!
r ' o
J
q ~
i
i
i
+
z~ ~ ~ - .................. / ~ ~ ,45
FIG. 3 . 5 ~ A typi cal chromat ograph f or a Kuwaiti naphtha sampl e.
J
+ ti
wi t h t he s i mi l a r p e a k of a k n o wn c o mp o u n d wi t h a k n o wn
boi l i ng poi nt . I n t he qua nt i t a t i ve a na l ys i s of a mi xt ur e , i t c a n
be s h o wn t h a t t he a r e a u n d e r a p a r t i c u l a r c o mp o n e n t p e a k ( as
s h o wn i n Fi g. 3. 5) i s di r e c t l y p r o p o r t i o n a l t o t he t ot a l a mo u n t
( ma s s ) of t he c o mp o n e n t r e a c h i n g t he det ect or . The a mo u n t
r e a c h i n g t he d e t e c t o r i s al s o p r o p o r t i o n a l t o t he c o n c e n t r a -
t i on ( wei ght p e r c e n t or we i ght f r act i on) of t he c o mp o n e n t
i n t he s a mp l e i nj ect ed. The p r o p o r t i o n a l i t y c o n s t a n t i s det er -
mi n e d wi t h t he a i d of s t a n d a r d s c o n t a i n i n g a k n o wn a mo u n t
of t he s a mp l e c o mp o n e n t . Mo d e m GCs a r e e q u i p p e d wi t h a
c o mp u t e r t h a t di r e c t l y me a s u r e s t he a r e a s u n d e r e a c h p e a k
a n d c o mp o s i t i o n c a n be di r e c t l y d e t e r mi n e d f r o m t he c om-
p u t e r pr i nt out . A p r i n t o u t f or t he c h r o ma t o g r a p h of Fi g. 3.5
i s s hown i n Tabl e 3.1 f or t he n a p h t h a s a mpl e . The a r e a per -
c e nt i s t he s a me as c o mp o s i t i o n i n wt % wi t h boi l i ng p o i n t s of
c o r r e s p o n d i n g c o mp o n e n t s . I n a na l ys i s of s a mp l e s b y a GC,
t he c o mp o s i t i o n i s a l wa ys d e t e r mi n e d i n wt % a n d n o t i n vol %
o r f r a c t i on. Fo r t hi s r e a s o n t he o u t p u t of a GC a na l ys i s f or a
s i mu l a t e d di s t i l l a t i on i s a cur ve of t e mp e r a t u r e ( boi l i ng p o i n t )
ver s us wt % of mi x t u r e va por i z e d, as c a n be s e e n i n Fi g. 3.4.
F u r t h e r i n f o r ma t i o n f or us e of GC f or s i mu l a t e d di s t i l l a t i on
up t o 750~ i s p r o v i d e d b y Cur ver s a n d va n d e n Enge l [3].
A t ypi c a l GC f or me a s u r e me n t of boi l i ng p o i n t of p e t r o l e u m
p r o d u c t s i s s h o wn i n Fi g. 3.6.
3. 1. 1. 4 Equi l i br i um Fl ash Vapori zat i on
Eq u i l i b r i u m f l ash v a p o r i z a t i o n ( EFV) i s t he l e a s t i mp o r t a n t
t ype of di s t i l l a t i on cur ve a n d i s ve r y di f f i cul t t o me a s u r e . I t i s
TABLE 3. 1--Calculation of composition of a naphtha sample with GC chromatograph shown in Fig. 3.4.
No.
1
2
3
4
5
6
7
8
9
Name Time, min Area Area % Tb, ~ C
n-Hexane 11.16 1442160 3.92 68.7
Benzene 14.64 675785 1.84 80.1
Cyclohexane 15.55 3827351 10.40 80.7
n-Heptane 18.90 5936159 16.14 98.4
2,2,3-Trimethyl-pentane 20.38 8160051 22.18 109.8
Toluene 27.53 8955969 24.34 110.6
Ethylbenzene 42.79 1678625 4.56 136.2
p-Xylene 45.02 4714426 12.82 138.4
o-Xylene 49.21 1397253 3.8 144.4
Total 36787780 100
FIG. 3.6--A GC for measurement of boiling point of products (courtesy of KISR).
pr esent ed in t er ms of t he t emper at ur e versus vol % vapor i zed.
It involves a seri es of exper i ment s at const ant at mos pher i c
pr essur e wi t h t ot al vapor in equi l i br i um wi t h t he unvapor -
i zed l i qui d. In fact t o det er mi ne each poi nt on t he EFV curve
one exper i ment is r equi r ed. To have a full shape of an EFV
curve at l east five t emper at ur es at 10, 30, 50, 70, and 90 vol %
vapor i zed ar e r equi r ed. EFV di st i l l at i on curves ar e useful in
t he desi gn and oper at i on of over head par t i al condenser s and
bot t om r eboi l er s si nce t he EFV t emper at ur es r epr esent act ual
equi l i br i um t emper at ur es. In cont r ast wi t h TBR t he EFV ini-
t i al t emper at ur e of a mi xt ur e is gr eat er t han t he IBP of ASTM
D 86 curve, whi l e t he FBP f r om a EFV curve is l ower t han t he
FBP f r om t he TBP curve for t he same mi xt ure. EFV curves at
pr essur es above at mos pher i c up t o pr essur es of 15 bar may
al so be useful for desi gn and oper at i on of vapor i zi ng or con-
densi ng vessel s under pr essur e.
3.1.1.5 Distillation at Reduced Pressures
At mospher i c di st i l l at i on curves pr esent boi l i ng poi nt s of
pr oduct s f r om an at mos pher i c di st i l l at i on col umn. For pr od-
uct s such as heavy gas oils t hat cont ai n heavy compounds and
may under go a cr acki ng pr ocess dur i ng vapor i zat i on at at mo-
spher i c pr essur e, di st i l l at i on dat a are meas ur ed at r educed
pr essur es f r om I t o 760 mm Hg. The exper i ment al pr ocedur e
is descr i bed in ASTM D 1160 t est met hod (see Fig. 3.7). Distil-
l at i on of heavy pet r ol eum f r act i ons is nor mal l y pr esent ed at
1, 2, 10, or 50 mmHg. Bot h a manual and an aut omat i c
met hod ar e specified. The t emper at ur e of t he vapor shoul d
not exceed 400~ (750~ ASTM D 1160 di st i l l at i on dat a ar e
meas ur ed mor e accur at el y t han ASTM D 86 si nce i t is con-
duct ed at l ow pressure. For t hi s r eason ASTM D 1 I60 curves
ar e cl oser to TBP curves at t he same pr essur e base. Con-
ver si on of di st i l l at i on dat a f r om l ow pr essur e to equi val ent
FIG. 3.7--An apparatus for experimental measurement of boiling point
at reduced pressures by ASTM D 1160 test method (courtesy of KISR).
3. CHARACTERI ZATI ON OF PETROLEUM FRACTI ONS 93
at mospheri c boiling points are given in the ASTM Manual [4]
and will be discussed later in this chapter.
Specific gravity (SG) or relative density and the API gravity are
defined in Section 2.1.3 and for pure hydrocarbons are given
in Tables 2.1 and 2.2. Aromatic oils are denser t han paraffinic
oils. Once specific gravity is known, the API gravity can be de-
termined by Eq. (2.4), which corresponds to the ASTM D 287
method. The standard t emperat ure to measure the specific
gravity is 15.56~ (60~ however, absolute density is usually
reported at 20~ Specific gravity or density for a pet rol eum
mixture is a bulk property that can be directly measured for
the mixture. Specific gravity is a property that indicates the
quality of a pet rol eum product and, as was shown in Chapter
2, is a useful propert y to estimate various physical proper-
ties of pet rol eum fluids. A standard test met hod for density
and specific gravity of liquid pet rol eum products and distil-
lates in the range of 15-35~ through use of a digital density
met er is described in ASTM D 4052 met hod [4]. The appa-
ratus must be calibrated at each t emperat ure and this test
met hod is equivalent to ISO 12185 and IP 365 methods. An-
other met hod using a hydromet er is described under ASTM D
1298 test method. Hydromet er is a glass float with lead ballast
that is floated in the liquid. The level at which hydromet er is
floating in the liquid is proportional to the specific gravity
of the liquid. Through graduation of the hydromet er the spe-
cific gravity can be read directly from the stalk of hydrometer.
This met hod is simpler t han the ASTM 4052 met hod but is
less accurate. The French standard procedure for measuring
density by hydromet er is described under NFT 60-101 test
method. With some hydromet ers densities with accuracy of
0.0005 g/mL can be measured. A digital density met er model
DMA 48 from PARA (Austria) is shown in Fig. 3.8.
Molecular weight is anot her bulk propert y that is indicative of
molecular size and structure. This is an i mport ant propert y
that usually laboratories do not measure and fail to report
when reporting various properties of pet rol eum fractions.
This is perhaps due to the low accuracy in the measur ement
of the molecular weight of pet rol eum fractions, especially for
heavy fractions. However, it should be realized that experi-
mental uncertainty in reported values of molecular weight
is less t han the errors associated with predictive met hods for
this very useful parameter. Since pet rol eum fractions are mix-
tures of hydrocarbon compounds, mixture molecular weight
is defined as an average value called number average molec-
ular weight or simply molecular weight of the mixture and it
is calculated as follows:
(3.1) M = ~x4Mi
i
where xi and M/are the mole fraction and mol ecul ar weight of
component i, respectively. Molecular weight of the mixture,
M, represents the ratio of total mass of the mixture to the total
moles in the mixture. Exact knowledge of molecular weight
of a mixture requires exact composi t i on of all compounds in
the mixture. For pet rol eum fractions such exact knowledge is
not available due to the large number of component s present
in the mixture. Therefore, experimental measurement of mix-
ture molecular weight is needed in lieu of exact composi t i on
of all compounds in the mixture.
There are three met hods that are widely used to measure
the molecular weight of various pet rol eum fractions. These
are cryoscopy, the vapor pressure method, and the size ex-
clusion chromatography (SEe) method. For heavy pet rol eum
fractions and asphaltenic compounds the SEC met hod is
commonl y used to measure distribution of mol ecul ar weight
FIG. 3.8--PARA model DMA 48 digital density meter (courtesy of Chemical Engineering
Department at Kuwait University).
FIG. 3.9--A GPC from waters model 150-C plus (courtesy of Chemical Engineering
Department at Kuwait University).
in the fraction. The SEC method is mainly used to determine
molecular weights of polymers in the range of 2000 to 2 106.
This method is also called gel permeation chromatography
(GPC) and is described in the ASTM D 5296 test method. In
the GPC method, by comparing the elution time of a sample
with that of a reference solution the molecular weight of the
sample can be determined. A GPC instrument is shown in
Fig. 3.9. The SEC experiment is usually performed for heavy
residues and asphaltenes in crude oils and gives the wt% of
various constituents versus molecular weight as will be dis-
cussed in Chapter 4.
The vapor pressure method is based on the measurement of
the difference between vapor pressure of sample and that of
a known reference solvent with a vapor pressure greater than
that of the sample. A solution of about 1 g of sample in 25 mL
of the reference solvent is prepared. This solution, which has
vapor pressure less than that of the solvent, tends to condense
the vapors of solvent on the sample thus creating a temper-
ature difference which is measured by two thermistors. The
molarity of the solution is calculated using calibrated curves.
This method is described by the ASTM D 2503 test method
and is applicable to oils with an initial boiling point greater
than 220~ [5]. A typical experimental error and uncertainty
in measuring the molecular weight is about 5%.
The third and most widely used method of determining
the molecular weight of an unknown petroleum mixture is
by the cryoscopy method, which is based on freezing point
depression. The freezing point of a solution is a measure of
the solution's concentration. As the concentration of the so-
lute increases, the freezing point of the solution will be lower.
The relation between freezing point depression and concen-
tration is linear. For organic hydrocarbons, benzene is usually
used as the solvent. Special care should be taken when work-
ing with benzene [6]. Calibration curves can be prepared by
measuring the freezing points of different solute concentra-
tions with a known solute and a known solvent. A cryoscope
can measure the freezing point depression with an accuracy
of about 0.001 ~ The relation to obtain molecular weight of
a sample is [6]
1000 x Kf x ml
(3.2) M =
AT x m2
where Kf is molal freezing point depression constant of the
solvent and is about 5.12~ AT is the freezing point de-
pression and the reading from the cryoscope, rnl is the mass
of solute and m2 is the mass of solvent both in grams. It gener-
ally consists of refrigerator, thermometer and the apparatus
to hold the sample. A cryoscope is shown in Fig. 3.10.
3.1.4 Refractive Index
Refractive index or refractivity is defined in Section 2.1.4 and
its values at 20~ for pure hydrocarbons are given in Table
2.1. Refractive indexes of hydrocarbons vary from 1.35 to 1.6;
however, aromatics have refractive index values greater than
naphthenes, which in turn have refractive indexes greater
than paraffins. Paraffinic oils have lower refractive index val-
ues. It was shown in Chapter 2 that refractive index is a useful
parameter to characterize hydrocarbon systems and, as will
be seen later in this chapter, it is needed to estimate the com-
position of undefined petroleum fractions. Refractive index is
the ratio of the speed of light in a vacuum to that of a medium.
In a medium, the speed of light depends on the wavelength
and temperature. For this reason refractive index is usually
measured and reported at 20~ with the D line sodium light.
For mixtures, refractive index is a bulk property that can
be easily and accurately measured by an instrument called
a refractometer Refractive index can be measured by digital
refractometers with a precision of 4-0.0001 and temperature
precision of 4-0. I~ The amount of sample required to mea-
sure refractive index is very small and ASTM D 1218 provides
a test method for clear hydrocarbons with values of refractive
indexes in the range of 1.33-1.5 and the temperature range
of 20-30~ In the ASTM D 1218 test method the Bausch and
FIG. 3.10~Model 5009 wide range cryoscope to measure molecular weight (courtesy of
Chemical Engineering Department at Kuwait University).
Lomb refract omet er is used. Refractive index of viscous oils
with values up to 1.6 can be measured by the ASTM D 1747
test method. Samples must have clear color to measure their
refractive index; however, for darker and more viscous sam-
ples in which the actual refractive index value is outside the
range of application of refractometer, samples can be diluted
by a light solvent and refractive index of the solution should be
measured. From the composi t i on of the solution and refrac-
tive index of pure solvent and that of the solution, refractive
index of viscous samples can be determined. A Model Abbe re-
fract omet er (Leica) is shown in Fig. 3.11. This refract omet er
measures refractive index of liquids within the t emperat ure
range of - 20 to 100~ with t emperat ure accuracy of 176
Because of simplicity and i mport ance of refractive index it
would be extremely useful if laboratories measure and report
its value at 20~ for a pet rol eum product, especially if the
composi t i on of the mixture is not reported.
3.1.5 Compositional Analysis
Petroleum fractions are mixtures of many different types of
hydrocarbon compounds. A pet rol eum mixture is well defined
if the composition and structure of all compounds present in
the mixture are known. Because of the diversity and number
of constituents of a pet rol eum mixture, the determination of
such exact composi t i on is nearly impossible. Generally, hy-
drocarbons can be identified by their carbon number or by
their molecular type. Carbon number distribution may be
determined from fractionation by distillation or by molec-
ular weight distribution as discussed earlier in this section.
However, for narrow boiling range pet rol eum products and
pet rol eum cuts in which the carbon number range is quite
limited, knowledge of molecular type of compounds is very
important. As will be seen later, properties of pet rol eum frac-
tions with detailed compositional analysis can be estimated
with a higher degree of accuracy t han for undefined fractions.
After distillation data, mol ecul ar type composition is the
most i mport ant characteristic of pet rol eum fractions. In this
FIG. 3.11--Leica made Abbe refractometer (courtesy of
Chemical Engineering Department at Kuwait University).
sect i on vari ous t ypes of composi t i on of pet r ol eum fract i ons
and di fferent met hods of t hei r meas ur ement are present ed.
3.1.5.1 Types of Composition 1
1
Based on t he nat ur e of pet r ol eum mi xt ure, t here are sev-
eral ways to express t he composi t i on of a pet r ol eum mi xt ure.
Some of t he mos t i mpor t ant t ypes of composi t i on are given
below: Asphaltenes
9 PONA (paraffins, olefins, napht henes, and ar omat i cs)
9 PNA (paraffins, napht henes, and ar omat i cs)
/
9 PIONA (paraffins, isoparaffins, olefins, napht henes, and aro- ]
mat i cs)
1 9 SARA (sat urat es, ar omat i cs, resins, and asphal t henes)
9 El ement al anal ysi s (C, H, S, N, O) Resins
Since mos t pet r ol eum fract i ons are free of olefins, t he hydro-
car bon t ypes can be expressed in t er ms of onl y PINA and if
paraffi ns and i soparaffi ns are combi ned a fract i on is si mpl y
expressed in t er ms of PNA composi t i on. Thi s t ype of anal -
ysis is useful for light and nar r ow boiling r ange pet r ol eum
pr oduct s such as distillates f r om at mospher i c cr ude dis-
tillation units. But t he SARA analysis is useful for heavy
pet r ol eum fract i ons, resi dues, and fossil fuels (i.e., coal liq-
uids), whi ch have hi gh cont ent s of ar omat i cs, resins, and as-
phal t enes. The el ement al anal ysi s gives i nf or mat i on on hy-
dr ogen and sul fur cont ent s as well as C/H ratio, whi ch are
i ndi cat i ve of t he qual i t y of pet r ol eum product s.
3.1.5.2 Analytical Inst rument s
General l y t hree met hods may be used to anal yze pet r ol eum
fractions. These are
9 separ at i on by solvents
9 chr omat ogr aphy met hods
9 spect r oscopi c met hods
The met hod of separ at i on by solvents is based on solubil-
ity of s ome compounds in a mi xt ur e in a par t i cul ar solvent.
The r emai ni ng i nsol ubl e compounds may be in a solid or an-
ot her i mmi sci bl e liquid phase. This met hod is par t i cul ar l y Feedstock/
useful for heavy pet r ol eum fract i ons and resi dues cont ai n- [ n-Pentane
ing asphahenes, resins, and sat ur at e hydr ocar bons. The de-
/
gree of solubility of a c ompound in a solvent depends on t he |
chemi cal st r uct ur e of bot h t he solute and t he solvent. I f t he
t wo st r uct ur es are si mi l ar t here is a gr eat er degree of solubil-
ity. For exampl e, hi gh-mol ecul ar-wei ght asphal t enes are not Asphaltenes
soluble in a l ow-mol ecul ar-wei ght paraffi ni c solvent such as
n-hept ane. Therefore, if n- hept ane is added t o a heavy oil, as-
phal t enes preci pi t at e while t he ot her const i t uent s f or m a solu-
/
ble sol ut i on wi t h t he solvent. I f solvent is changed t o pr opane, |
because of t he gr eat er di fference bet ween t he st r uct ur e of
t he solvent and t he hi gh-mol ecul ar-wei ght asphal t enes, mor e
asphal t eni c compounds preci pi t at e. Si mi l arl y if acet one is Resins
added t o a deasphal t ed oil (DAO), resi ns preci pi t at e whi l e
l ow-mol ecul ar-wei ght hydr ocar bons r emai n soluble in ace-
tone. I n Fig. 3.12 an all-solvent fract i ons pr ocedur e is shown
for SARA anal ysi s [7].
One of t he di sadvant ages of t he all-solvent separ at i on tech-
ni que is t hat in some i nst ances a very l ow t emper at ur e (0 to
- 10 ~ C) is requi red, whi ch causes i nconveni ence in l abor at or y
operat i on. Anot her difficulty is t hat in ma ny cases large vol-
umes of sol vent may be requi red and solvents mus t have suffi-
ciently l ow boi l i ng poi nt so t hat t he sol vent can be compl et el y
Feedstock
n-Heptane (or n-Pentane)
t
Deasphaltened Oil
Acetone
1
Oils
Dimethylforrnamide
1
Saturates
Deasphaltened 0il
Clay
Oils
~l Silica Gel
l 1
Aromatics Saturates
FIG. 3. 13---The ASTM D 2007 procedure. Repri nted from
Ref. [7], p. 280, by cour t esy of Marcel Dekker, inc.
1
Aromatics
FIG. 3 . 1 2 - - An al l -sol vent fracti onati on procedure. Repri nted
from Ref. [7], p. 267, by cour t esy of Marcel Dekker, Inc.
r emoved f r om t he pr oduct [7]. ASTM [4] provi des several
met hods based on solvent separ at i on t o det er mi ne amount s
of asphal t enes. I n ASTM D 2007 t est met hod n- pent ane is used
as t he solvent, while in ASTM D 4124 asphal t ene is separ at ed
by n-hept ane. Schemat i cs of t hese t est met hods are shown in
Figs. 3.13 and 3.14, respectively, as given by Spei ght [7]. As-
phal t enes are soluble in liquids wi t h a surface t ensi on above
25 dyne/ cm such as pyri di ne, car bon disulfide, car bon t et ra-
chloride, and benzene [7].
The pri nci pl e of separ at i on by chr omat ogr aphy t echni que
was descri bed in Sect i on 3.1.1.3. I f t he mobi l e phase is gas
t he i ns t r ument is called a gas chr omat ogr aph (GC), while for
l
Asphaltenes
Feedstock
n-Heptane
1
Deasphaltened Oil
~ Alumina
Polar Aromatics Oils
(Resins) ~ Alumina
l
Naphthene Aromatics Saturates
(Aromatics)
FIG. 3.14.--Separation of asphaltenes and resins from
petroleum fractions. Reprinted from Ref. [7], p. 281, by cour-
tesy of Marcel Dekker, Inc.
liquid mobile phase it is called a liquid chromatograph (LC).
As discussed earlier, component s can be separated by their
boiling points t hrough GC analysis. In advanced pet rol eum
refineries aut omat i c online GCs are used for continuous anal-
ysis of various st reams to control the quality of products. A
st ream may be analyzed every 20 mi n and aut omat i c adjust-
ment can be made to the refinery unit. In crude assay anal-
ysis distillation is being replaced by chromat ography tech-
niques. The LC met hod is used for less volatile mixtures such
as heavy pet rol eum fractions and residues. Use of LC for sepa-
ration of saturated and aromat i c hydrocarbons is described in
ASTM D 2549 test method. Various forms of chromat ography
techniques have been applied to a wide range of pet rol eum
products for analysis, such as PONA, PIONA, PNA, and SARA.
One of the most useful types of liquid chromat ography is
high performance liquid chromatography (HPLC), which can
be used to identify different types of hydrocarbon groups.
One particular application of HPLC is to identify asphaltene
and resin type constituents in nonvolatile feedstocks such as
residua. Total time required to analyze a sample by HPLC is
just a few minutes. One of the mai n advantages of HPLC is
that the boiling range of sample is immaterial. A HPLC ana-
lyzer is shown in Fig. 3.15.
The accuracy of chromat ography techniques mainly
depends on the type of detector used [7]. In Section 3.1.1.3,
flame ionization (FID) and t hermal conductivity (TCD) detec-
tors are described, which are widely used in GC. For LC the
most common detectors are refractive index detector (RID)
and wavelength UV (ultraviolet) detector. UV spectroscopy is
particularly useful to identify the types of aromatics in asphal-
tenic fractions. Another spectroscopy met hod is conventional
infrared (IR) spectroscopy, which yields i nformat i on about
the functional features of various pet rol eum constituents.
For example, IR spectroscopy will aid in the identification of
N--H and O--H functions and the nature of polymethylene
FIG. 3.15--A HPLC instrument (courtesy of Chemical Engineering Department at Kuwait
University).
FIG. 3, 16--HP made GC- MS model 5890 Series II. (courtesy of Chemical Engineering
Department at Kuwait University),
chains (C--H) and the nature of any polynuclear aromat i c
systems [7].
Another type of analysis of pet rol eum fractions to iden-
tify molecular groups is by spectrometric met hods such as
mass spectrometry (MS). In general, there is a difference be-
tween spectroscopy and spect romet ry met hods although in
some references this difference is not acknowledged. Spec-
troscopy refers to the techniques where the molecules are ex-
cited by various sources, such as UV and IR, to return to their
normal state. Spect romet ry refers to the techniques where
the molecules are actually ionized and fragmented. Evolution
of spectroscopic met hods comes after chromat ography tech-
niques, nonetheless, and in recent decades they have received
considerable attention. While volatile and light pet rol eum
products can be analyzed by gas chromatography, heavier
and nonvolatile compounds can be analyzed and identified by
spectrometric methods. One of the most i mport ant types of
spect romet ry techniques in analysis of petroleum fractions is
mass spectrometry (MS). In this method, masses of mol ecul ar
and at omi c component s that are ionized in the gaseous state
by collision with electrons are measured. The advantage of
MS over ot her spectrometric met hods is its high reproducibil-
ity of quantitative analysis and i nformat i on on mol ecul ar type
in complex mixtures. Mass spect romet ry can provide the most
detailed quantitative and qualitative i nformat i on about the
at omi c and molecular composition of organic and inorganic
compounds. However, use of MS is limited to organic com-
pounds that are stable up to 300~ (570~ At higher tem-
peratures t hermal decomposition may occur and the anal-
ysis will be biased [7]. Through MS analysis, hydrocarbons
of similar boiling points can be identified. In the MS analy-
sis, molecular weight, chemical formul a of hydrocarbons, and
their amount s can be determined. The most powerful instru-
ment to analyze pet rol eum distillates is the combi nat i on of
a GC and an MS called GC-MS instrument, which separates
compounds bot h t hrough boiling point and molecular weight.
For heavy pet rol eum fractions containing high-boiling-point
compounds an integrated LC-MS unit may be suitable for
analysis of mixtures; however, use of LC-MS is more difficult
t han GC-MS because in LC-MS solvent must be removed
from the elute before it can be analyzed by MS. A GC-MS
i nst rument from Hewlett Packard (HP) is shown in Fig. 3.16
Another type of separation is by SEC or GPC, which can
be used to determine molecular weight distribution of heavy
pet rol eum fractions as discussed in Section 3.1.3. Fractions
are separated according to their size and molecular weight
and the met hod is particularly useful to determine the amount
of asphaltenes. Asphaltenes are polar multiring aromat i c
compounds with molecular weight above 1000 (see Fig. 1.2).
It is assumed that in this molecular weight range only aro-
matics are present in a pet rol eum fraction [8].
3.1.5.3 PNA Analysis
As determination of the exact composi t i on of a pet rol eum
fraction is nearly an impossible task, for narrow boiling range
pet rol eum fractions and products a useful type of compo-
sitional analysis is to determine the amount s of paraffins
(P), napht henes (N), and aromatics (A). As ment i oned be-
fore, most pet rol eum products are olefin free and PNA anal-
ysis provides a good knowledge of mol ecul ar type of mixture
constituents. However, some analyzers give the amount of
isoparaffins and olefins as well. These analyzers are called
PIONA analyzer, and a Chrompack Model 940 PIONA ana-
lyzer is shown in Fig. 3.17. An output of this type of analyzer
is similar to the GC output; however, it directly gives wt% of
n-paraffins, isoparaffins, olefins, naphthenes, and aromatics.
The composition is expressed in wt%, which can be converted
to mole, weight, and volume fractions as will be shown later
in this chapter.
3.1.5.4 Elemental Analysis
The mai n elements present in a pet rol eum fraction are car-
bon (C), hydrogen (H), nitrogen (N), oxygen (O), and sulfur
(S). The most valuable i nformat i on from elemental analysis
FIG. 3,17mA chrompack model 940 PIONA analyzer (courtesy of Chemical Engineering
Department at Kuwait University),
that can be obtained is on the C/H ratio and sulfur content
of a petroleum mixture from which one can determine the
quality of oil. As boiling points of fractions increase or their
API gravity decrease the amount of C/H ratio, sulfur content,
nitrogen content, and the metallic constituents increase, sig-
nifying a reduction in the quality of an oil. Sulfur content of
very heavy fractions can reach 6-8% and the nitrogen content
can reach 2.0-2.5 wt%. There are specific methods to measure
these elements individually. However, instruments do exist
that measure these elements all together; these are called ele-
mental analyzers. One of these apparatuses is CHN analyzers
in which there is a simultaneous combustion in pure oxygen
at 1000~ Carbon is reduced to CO2, H is reduced to H20,
and N is converted to nitrogen oxides. Nitrogen oxides are
then reduced over copper at 650~ to nitrogen by eliminating
oxygen. A mixture of CO2, H/O, and N2 is separated by gas
chromatography with TCD. In a similar fashion, sulfur is ox-
idized to SO2 and is detected by TCD after detection of CO2,
Na, and H20. Oxygen is determined by passing it over carbon
at high temperature and converted to CO, which is measured
by a GC [5]. ASTM test methods for elemental analysis of
petroleum products and crude oils include hydrogen content
(ASTM D 1018, D 3178, D 3343), nitrogen content (ASTM
D 3179, D 3228, D 3431), and sulfur content (ASTM D 129/IP
61, D 1266/IP 107, D 1552, D 4045). An elemental analyzer
Model CHNS-932 (Leco Corp., St. Joseph, MI, USA) is shown
in Fig. 3.18. In this analyzer, the CO2, H20, and SO2 are
detected by infrared detector (IRD) while N2 is determined
by the TCD method.
Another group of heteroatoms found in petroleum mix-
tures are metallic constituents. The amount of these met-
als are in the range of few hundreds to thousand ppm and
their amounts increase with increase in boiling points or de-
crease in the API gravity of oil. Even a small amount of these
metals, particularly nickel, vanadium, iron, and copper, in
the feedstocks for catalytic cracking have negative effects on
the activity of catalysts and result in increased coke forma-
tion. Metallic constituents are associated with heavy com-
pounds and mainly appear in residues. There is no general
method to determine the composition of all metals at once
but ASTM [4] provides test methods for determination of
various metallic constituents (i.e., ASTM D 1026, D 1262,
D 1318, D 1368, D 1548). Another method is to burn the oil
sample in which metallic compounds appear in inorganic
ashes. The ash should be digested by an acid and the so-
lution is examined for metal species by atomic absorption
spectroscopy [7].
3.1.6 Viscosity
Absolute and kinematic viscosities are defined in Section
2.1.8 and experimental data for the kinematic viscosity of
some pure hydrocarbons are given in Table 2.2. Viscosity of
petroleum fractions increase with a decrease in the API grav-
ity and for residues and heavy oils with the API gravity of
less than 10 (specific gravity of above 1), viscosity varies from
several thousands to several million poises. Viscosity is a bulk
property that can be measured for all types of petroleum frac-
tions in liquid form. Kinematic viscosity is a useful character-
ization parameter for heavy fractions in which boiling point
data are not available due to thermal decomposition during
distillation. Not only is viscosity an important physical prop-
erty, but it is a parameter that can be used to estimate other
physical properties as well as the composition and quality of
undefined petroleum fractions as shown later in this chap-
ter. Since viscosity varies with temperature, values of viscos-
ity must be reported with specified temperature. Generally,
kinematic viscosity of petroleum fractions are measured at
standard temperatures of 37.8~ (100~ and 98.9~ (210~
However, for very heavy fractions viscosity is reported at tem-
peratures above 38~ i.e., 50~ (122~ or 60~ (140~ When
viscosity at two temperatures are reported from the method
of Section 2.7 one can obtain the viscosity at other tempera-
tures. Measurement of viscosity is easy but the method and
FIG. 3.18--Leco made CHNS-932 model elemental analyzer (courtesy of Chemical Engi-
neering Department at Kuwait University).
the instrument depend on the type of sample. For Newto-
nian and high-shear fluids such as engine oils, viscosity can
be measured by a capillary U-type viscometer. An example of
such viscometers is the Cannon-Fenske viscometer. The test
method is described in ASTM D 445, which is equivalent to
ISO 3104 method, and kinematic viscosity is measured at tem-
peratures from 15 to 100~ (~60 to 210~ In this method,
repeatability and reproducibility are 0.35 and 0.7%, respec-
tively [5]. Another type of viscometer is a rotary viscometer,
which is used for a wide range of shear rates, especially for
low shear rate and viscous fluids such as lubricants and heavy
petroleum fractions. In these viscometers, fluid is placed be-
tween two surfaces, one is fixed and the other one is rotating.
In these viscometers absolute viscosity can be measured and
an example of such viscometers is the Brookfield viscome-
ter. Details of measurement and prediction of viscosity of
petroleum fractions are given in Chapter 8. As the viscosity
of petroleum fractions, especially the heavy oils, is one of the
most difficult properties to estimate, its experimental value is
highly useful and desirable.
3. 2 PREDICTION AND CONVERSION
OF DISTILLATION DATA
Various distillation curves are introduced in Section 3.1.1.
For simplicity ASTM is used to refer to ASTM D 86 distilla-
tion curve, similarly TBP, SD, and EFV refer to true boiling
point, simulated distillation (ASTM D 2887), and equilibrium
flash vaporization, respectively. Petroleum fractions have a
range of boiling points. To use the correlations introduced
in Chapter 2, a single value for boiling point is required. For
this reason there is a need for the definition of an average
boiling point or a characteristic boiling point based on a dis-
tillation curve. Availability of one type of distillation curve
for simplicity in experimental measurement and the need for
another type for its application requires conversion methods
between various distillation curves. The tedious procedures
necessary to obtain experimental EFV data have given im-
petus to the development of correlations for predicting EFV
data from the analytical ASTM and TBP distillations. Sim-
ulated distillation by gas chromatography appears to be the
most simple, reproducible, and consistent method to describe
the boiling range of a hydrocarbon fraction unambiguously.
TBP is the most useful distillation curve, while available data
might be ASTM D 86, ASTM D 2887, or ASTM D 1160 distilla-
tion curves. ASTM [4] has accepted this technique as a tenta-
tive method for the "Determination of Boiling Range Distribu-
tion of Petroleum Fractions by Gas Chromatography" (ASTM
D 2887). In most cases distillation data are reported in terms
of ASTM D 86 or SD. In this section methods of calculation of
average boiling points, interconversion of various distillation
curves, and prediction of complete distillation curves from a
limited data are presented.
3.2.1 Average Boiling Points
Boiling points of petroleum fractions are presented by distil-
lation curves such as ASTM or TBR However, in prediction of
physical properties and characterization of hydrocarbon mix-
tures a single characteristic boiling point is required. Gener-
ally an average boiling point for a fraction is defined to de-
termine the single characterizing boiling point. There are five
average boiling points defined by the following equations [9].
Three of these average boiling points are VABP (volume aver-
age boiling point), MABP (molal average boiling point) and
WABP (weight average boiling point), defined for a mixture
of n components as
tt
(3.3) ABP = ~-~ x4Tb~
i =1
3. CHARACTERI Z ATI ON OF PETROL EUM FRACTI ONS 101
wher e ABP is t he VABP, MABP, or WABP and xi is t he corre-
sp ondi ng vol ume, mol e, or wei ght fract i on of comp onent i.
Tbi is t he nor mal boi l i ng p oi nt of comp onent i i n kelvin. Two
ot her average boi l i ng p oi nt s ar e CABP ( cubi c average boi l i ng
poi nt ) and MeABP ( mean average boi l i ng p oi nt ) defi ned as
(3.4) CABP = x~ (1.8Tbi -- 459. 67) 1/3 + 255.37
MABP + CABP
(3.5) MeABP =
2
wher e T~ in Eq. (3.4) is in kelvin. The conver si on fact ors in
Eq. (3.4) come f r om t he fact t hat t he or i gi nal defi ni t i on of
CABP is in degrees Fahr enhei t . F or p et r ol eum f r act i ons i n
whi ch vol ume, wei ght , or mol e f r act i ons of comp onent s are
not known, t he average boi l i ng p oi nt s ar e cal cul at ed t hr ough
ASTM D 86 di st i l l at i on curve as
T10 + T30 + T50 + T70 + T90
(3.6) VABP =
5
wher e 1"10, T30, Ts0, I"70, and T90 ar e ASTM t emp er at ur es at
10, 30, 50, 70, and 90 vol % di st i l l ed. ASTM di st i l l at i on curves
can be char act er i zed by t he magni t udes of t emp er at ur es and
overal l sl ope of t he curve. A p ar amet er t hat ap p r oxi mat el y
char act er i zes sl ope of a di st i l l at i on curve is t he sl ope of a
l i near l i ne bet ween 10 and 90% poi nt s. This sl ope shown by
SL is defi ned as
T90 - Tl0
(3.7) SL -
80
wher e T10 and T90 ar e t he ASTM D 86 t emp er at ur es at 10 and
90% of vol ume vapori zed. The 10-90 slope, SL, in some refer-
ences is r ef er r ed to as t he Engl er sl ope and is i ndi cat i ve of a
var i et y of comp ounds i n a p et r ol eum fract i on. When t he boi l -
i ng p oi nt s of comp ounds are near each ot her t he val ue of SL
and t he boi l i ng r ange of t he f r act i on are low. F or p et r ol eum
fract i ons, WABP, MABP, CABP, and MeABP ar e cor r el at ed
t hr ough an emp i r i cal pl ot to VAPB and SL in Chap t er 2 of t he
API -TDB [2]. Anal yt i cal cor r el at i ons based on t he API pl ot
wer e devel oped by Zhou [10] for use in a di gi t al comput er.
For heavy fract i ons and vacuum di st i l l at es in whi ch di st i l l a-
t i on dat a by ASTM D i 160 are avai l abl e, t hey shoul d first be
conver t ed to ASTM D 86 and t hen average boi l i ng p oi nt s ar e
cal cul at ed. Anal yt i cal cor r el at i ons for est i mat i on of average
boi l i ng p oi nt s ar e given by t he fol l owi ng equat i ons in t er ms
of VABP and SL [2, 10].
(3.8) ABP = VABP - AT
l n( - ATw) = - 3. 64991 - 0.02706(VABP - 273.15) 0.6667
(3.9) + 5. 163875SL ~
ln(ATM) = - 1.15158 - 0.01181 (VABP - 273.15) 0"6667
(3.10) + 3. 70612SL 0'333
ln( ATc) = - 0. 82368 - 0.08997(VABP - 273.15) 0.45
(3.1 I ) + 2. 456791SL ~
ln( ATue) = - 1.53181 - 0.0128(VABP - 273.15) 0.6667
(3.12) + 3. 646064SL ~
wher e ABP is an average boi l i ng p oi nt such as WABP, MABP,
CABP, or MeABP and AT is t he cor r esp ondi ng cor r ect i on
t emp er at ur e for each ABP. All t emp er at ur es ar e i n kelvin.
VABP and SL ar e defi ned in Eqs. (3.6) and (3.7). Once AT
is cal cul at ed for each case, cor r esp ondi ng ABP is cal cul at ed
f r om Eq. (3.8). Equat i ons ( 3. 8) -( 3. 12) cal cul at e val ues of var-
i ous ABP very cl ose to t hose obt ai ned f r om emp i r i cal pl ot in
t he API -TDB [2]. The fol l owi ng examp l e shows ap p l i cat i on of
t hese equat i ons in cal cul at i on of var i ous ABE
Ex ampl e 3 . 1 - - A l ow boi l i ng nap ht ha has t he ASTM D 86 t em-
p er at ur es of 77.8, 107.8, 126.7, 155, and 184.4~ at 10, 30, 50,
70, and 90 vol % di st i l l ed [11]. Cal cul at e VABE WABE MABP,
CABP, and MeABP for t hi s fract i on.
Sol ut i on- - Fr om Eqs. (3.6) and (3.7) VABP and SL are cal-
cul at ed as follows: VABP = (77.8 + 107.8 + 126.7 + 155 +
184.4)/5 = 130.3~ = 403.5 K, and SL = ( 184. 4- 77. 8) /
80 = 1.333~ (K)/%. F r om Eqs. ( 3. 9) -( 3. 12) var i ous cor r ect i on
t emp er at ur es ar e cal cul at ed: ATw = -3. 3~ ATu = 13.8~
ATc = 3.2~ and ATMe = 8.6~ F r om Eq. (3.8) var i ous av-
erage boi l i ng p oi nt s ar e cal cul at ed: WABP = 133.7, MABP =
116.5, CABP = 127.1, and MeABP = 121.7~
Appl i cat i on and est i mat i on of var i ous boi l i ng p oi nt s ar e
di scussed by Van Wi nkl e [12]. Si nce t he mat er i al s boi l over a
r ange of t emp er at ur e, any one average boi l i ng p oi nt fails t o be
useful for cor r el at i on of all pr oper t i es. The most useful t ype
of ABP is MeABP, whi ch is r ecommended for cor r el at i on of
most p hysi cal p r op er t i es as well as cal cul at i on of Wat son K
as will be di scussed l at er i n t hi s chapt er. However, for cal cul a-
t i on of specific heat , VABP is r ecommended [ 12]. I n Examp l e
3.1, MeABP is 121.7~ whi ch vari es f r om 126.7 for t he ASTM
D 86 t emp er at ur e at 50 vol % di st i l l ed (T50). However, based on
our experi ence, for nar r ow boi l i ng r ange fract i ons wi t h SL <
0.8~ t he MeABP is very cl ose to 50% ASTM t emp er at ur e.
As an exampl e, for a gas oil samp l e [11] wi t h ASTM t emp er a-
t ures of 261.7, 270, 279.4, 289.4, and 307.2~ at I 0, 30, 50, 70,
and 90 vol%, t he MeABP is cal cul at ed as 279, whi ch is very
cl ose to 50% ASTM t emp er at ur e of 279.4~ F or t hi s fract i on
t he val ue of SL is 0.57~ whi ch i ndi cat es t he boi l i ng r ange
is qui t e narrow. Si nce none of t he average boi l i ng p oi nt s de-
fi ned here r ep r esent t he t rue boi l i ng p oi nt of a fract i on, t he
50% ASTM t emp er at ur e may be used as a char act er i st i c boi l -
i ng p oi nt i nst ead of average boi l i ng poi nt . I n t hi s case it is
as s umed t hat t he di fference bet ween t hese t emp er at ur es is
wi t hi n t he r ange of exp er i ment al uncer t ai nt y for t he r ep or t ed
di st i l l at i on dat a as wel l as t he cor r el at i on used to est i mat e a
p hysi cal propert y.
3 . 2 . 2 I nt e r c onv e r s i on o f Vari ous Di s t i l l at i on Dat a
Wor k to devel op emp i r i cal met hods for convert i ng ASTM dis-
t i l l at i ons to TBP and EFV di st i l l at i ons began in t he l at e 1920s
and cont i nued t hr ough t he 1950s and 1960s by a l arge number
of r esear cher s [ 13-18] . All of t he cor r el at i ons were based on
di scor dant exp er i ment al dat a f r om t he l i t erat ure. Exp er i men-
tal ASTM, TBP, and EFV dat a on whi ch t he emp i r i cal correl a-
t i ons are based suffer a l ack of r ep r oduci bi l i t y because t her e
were no st andar di zed p r ocedur es or ap p ar at us avai l abl e. All
of t hese cor r el at i ons were eval uat ed and comp ar ed t o each
ot her by House et al. [ 19] to sel ect most ap p r op r i at e met hods
for i ncl usi on in t he API -TDB. As a resul t of t hei r eval uat i ons,
t he fol l owi ng met hods were adop t ed in t he API Dat a Book
102 CHARACTERI Z ATI ON AND PROPERTI ES OF PETROL EUM FRACTI ONS
as t he best met hod: Edmi st er - Pol l ock [14] for ASTM to TBP,
Edmi s t er - Okamot o [ 15-17] for ASTM to EFV, and Maxwel l
for conver si on of TBP to EFV [19]. Most of t hese cor r el at i ons
wer e in gr ap hi cal f or ms and i nconveni ent for comp ut er appl i -
cat i ons. Later, Ar nol d comp ut er i zed t hese gr ap hi cal met hods
t hr ough a set of nt h or der p ol ynomi al s [20]. Cor r el at i on to
convert ASTM D 2887 (SD) to ASTM D 86 were first devel op ed
by F or d usi ng mul t i p l i er r egr essi on anal ysi s [21]. I n t he mi d
1980s Ri azi and Dauber t [22] devel op ed anal yt i cal met hods
for t he conver si on of di st i l l at i on curves based on t he general -
i zed cor r el at i on for hydr ocar bon p r op er t i es gi ven by Eq. (2.2).
These met hods were adop t ed by t he API in t he fifth edi t i on of
API -TDB-88 [2] t o r ep l ace t he pr evi ous met hods. Cont i nued
i nt erest s f r om t he p et r ol eum i ndust r y for t hese conver si on
met hods l ed t o devel op ment of f ur t her met hods. The l at est
met hods for t he conver si on of di st i l l at i on curves were devel-
op ed by Dauber t in mi d 1990s [23] t hr ough modi f yi ng Ri azi -
Dauber t correl at i ons. I n t hi s sect i on t he API met hods ( Ri azi -
Dauber t and Dauber t ) for conver si on of di st i l l at i on dat a are
pr esent ed, whi ch ar e al so r ecommended and used in ot her
references and i ndust r i al soft ware [24, 25].
3. 2. 2. 1 Ri azi - Daubert M et hod
Ri azi and Dauber t met hods for t he i nt er conver si on of vari -
ous di st i l l at i on dat a are based on t he gener al i zed cor r el at i on
for p r op er t y est i mat i on of hydr ocar bons in t he f or m of Eq.
(2.38). Avai l abl e di st i l l at i on t emp er at ur e and specific gravi t y
of t he fract i on are used as t he i np ut p ar amet er s t o est i mat e
t he desi r ed di st i l l at i on dat a in t he fol l owi ng f or m [22]:
(3.13) T/ ( desi red) = a [T/ ( avai l abl e) ] b SG c
wher e T/ ( avai l abl e) is t he avai l abl e di st i l l at i on t emp er at ur e
at a specific vol % di st i l l ed and T/ ( desired) is t he desi r ed di st i l -
l at i on dat a for t he same vol % di st i l l ed, bot h ar e i n kelvin. SG
is t he specific gravi t y of f r act i on at 15.5~ and a, b, and c are
cor r el at i on p ar amet er s specific for each conver si on t ype and
each vol % p oi nt on t he di st i l l at i on curve. F or exampl e, if t hi s
equat i on is used to convert ASTM to EFV at 10%, T,. ( avai l abl e)
is ASTM t emp er at ur e at 10% and T/ ( desired) is t he EFV t em-
p er at ur e at 10% and const ant s a, b, and c are specific for t hi s
conver si on t ype at 10% of vol ume vapor i zed.
3. 2. 2. 1. 1 ASTM D 86 and TBP Conversi on- - I f di st i l l at i on
dat a avai l abl e ar e i n t he f or m of ASTM D 86 and desi r ed dis-
t i l l at i on is TBP, Eq. (3.13) can be used, but for t hi s p ar t i cul ar
t ype of conver si on val ue of const ant c for all p oi nt s is zero
and t he equat i on r educes to
(3.14) TBP = a( ASTM D 86) b
wher e bot h TBP and ASTM t emp er at ur es are for t he same
vol % di st i l l ed and are i n kelvin. Const ant s a and b at var i ous
p oi nt s al ong t he di st i l l at i on curve wi t h t he r ange of appl i ca-
t i on are gi ven i n Table 3.2.
For a t ot al of 559 dat a p oi nt s for 80 di fferent sampl es, Eq.
(3.14) gives an average absol ut e devi at i on (AAD) of about
5~ whi l e t he Edmi st er - Pol l ock met hod [14] gives an AAD
of about 7~ Gener al l y p r edi ct i ons at 0% give hi gher er r or s
and ar e less rel i abl e. Det ai l s of eval uat i ons ar e gi ven in our
pr evi ous p ubl i cat i ons [22, 26]. Equat i on (3.14) can be easi l y
reversed to p r edi ct ASTM f r om TBP dat a, but t hi s is a r ar e
ap p l i cat i on as usual l y ASTM dat a are avai l abl e. I f TBP di st i l -
TABLE 3.2---Correlation constants for Eq. (3.14).
ASTM D 86
Vol% a b range, a ~ C
0 0.9177 1.0019 20-320
10 0.5564 1.0900 35-305
30 0.7617 1.0425 50-315
50 0.9013 1.0176 55-320
70 0.8821 1.0226 65-330
90 0.9552 1.0110 75-345
95 0.8177 1.0355 75-400
Source: Ref. [22].
aTemperatures are approximated to nearest 5.
l at i on curve is avai l abl e t hen ASTM curve can be est i mat ed as
(3.15) ASTM D 86 = (TBP) 1/b
wher e const ant s a and b ar e gi ven i n Table 3.2 as for Eq. (3.14).
3. 2. 2. 1. 2 ASTM D 86 and EFV Conversi ons- - Appl i cat i on
of Eq. (2.13) to t hi s t ype of conver si on gives
(3.16) EFV = a( ASTM D 86)b(SG) c
wher e const ant s a, b, and c were obt ai ned f r om mor e t han 300
dat a p oi nt s and ar e gi ven in Table 3.3. Equat i on (3.16) was
eval uat ed wi t h mor e t han 300 dat a p oi nt s f r om 43 di fferent
samp l es and gave AAD of 6~ whi l e t he met hod of Edmi s t er -
Okamot o [15] gave an AAD of 10~ [22, 26].
I n usi ng t hese equat i ons if specific gravi t y of a fract i on is
not avai l abl e, it may be est i mat ed f r om avai l abl e di st i l l at i on
curves at 10 and 50% p oi nt s as gi ven by t he fol l owi ng equa-
t i on:
(3.17) SG = aTbmoT(o
wher e const ant s a, b, and c for t he t hr ee t ypes of di st i l l at i on
dat a, namel y, ASTM D 86, TBP, and EFV, ar e gi ven in Table
3.4. Temp er at ur es at 10 and 50% are bot h i n kelvin.
3. 2. 2. 1. 3 SD to ASTM D 86 Conversi ons- - The equat i on
deri ved f r om Eq. (3.13) for t he conver si on of si mul at ed dis-
t i l l at i on ( ASTM D 2887) to ASTM D 86 di st i l l at i on curve has
t he fol l owi ng form:
(3.18) ASTM D 86 = a(SD)b(F) C
wher e const ant F is a p ar amet er speci fi cal l y used for t hi s t ype
of conver si on and is given by t he fol l owi ng equat i on:
(3.19) F = 0. 01411( SD 10~ 50~
in whi ch SD 10% and SD 50% ar e t he SD t emp er at ur es in
kel vi n at 10 and 50 wt % di st i l l ed, respectively. Par amet er F
cal cul at ed f r om Eq. (3.19) mus t be subst i t ut ed in Eq. (3.18) to
est i mat e ASTM D 86 t emp er at ur e at cor r esp ondi ng p er cent
p oi nt expr essed in vol ume basi s. Equat i on (3.18) cannot be
ASTM D 86
Vol% a b c range, a ~ C
0 2.9747 0.8466 0.4209 10-265
10 1.4459 0.9511 0.1287 60-320
30 0.8506 1.0315 0.0817 90-340
50 3.2680 0.8274 0.6214 110-355
70 8.2873 0.6871 0.9340 130-400
90 10.6266 0.6529 1.1025 160-520
100 7.9952 0.6949 1.0737 190-430
Source: Ref. [22].
3. CHARACTERI Z ATI ON OF PETROL EUM FRACTI ONS
Distillation Tl0 7"50 SG No. of AAD
type range,a~ range,a ~ range a b c data points %
ASTM D 86 35-295 60-365 0.70-1.00 0.08342 0.10731 0.26288 120 2.2
TBP 10-295 55-320 0.67-0.97 0.10431 0.12550 0.20862 83 2.6
EFV 79-350 105-365 0. 74--0. 91 0.09138 -0.0153 0.36844 57 57
Source: Ref. [22].
1 0 3
used i n a reverse form to predi ct SD from ASTM D 86, but this
type of conversi on is usual l y not desired as most predictive
met hods use ASTM D 86 dat a while l aborat ori es report SD
data. Const ant s a, b, and c i n Eq. (3.18) were obt ai ned from
81 different sampl es and 567 dat a poi nt s and are given i n
Table 3.5 wi t h the range of SD dat a at each percent age al ong
the di st i l l at i on curve.
Equat i on (3.18) and t he met hod of Ford publ i shed by
ASTM, i ncl uded i n the earlier edi t i ons of API-TDB [21], were
eval uat ed by some 570 dat a poi nt s and gave AAD of 5 and
5.5~ respectively [22, 26]. Larger errors were observed at
t he i ni t i al and final boi l i ng poi nt s (0 and 100%) but excl udi ng
these poi nt s the AAD reduces to about 3~ for conversi ons
wi t hi n the range of 10-90% distilled.
The procedures given i n this sect i on shoul d be used wi t h the
range of dat a specified i n Tables 3.1-3.4. Use of these equa-
t i ons out si de the specified ranges coul d cause large errors.
Graphi cal forms of these equat i ons for conversi on of vari ous
di st i l l at i on curves are given i n Reference [22] as well as i n t he
fourt h edi t i on of the API-TDB-88 [2]. One of the advant ages of
these equat i ons is t hat they can be used i n reversed form. This
means one may est i mat e EFV from TBP dat a t hr ough conver-
si on of TBP to ASTM by Eq. (3.15) and t hen usi ng Eq. (3.16)
to est i mat e EFV from cal cul at ed ASTM curve. The exampl e
bel ow shows this conversi on process.
Exampl e 3. 2- - F or a bl end of nap ht ha- ker osene sample,
ASTM, TBP, and EFV di st i l l at i on curves are given i n the API-
TDB [2]. These dat a are represent ed i n Table 3.6. Use the
Ri azi - Dauber t met hods to predi ct EFV curve from TBP curve.
$ ol ut i on- - TBP dat a are used as available i np ut data. Equa-
t i on (3.15) shoul d be used to est i mat e ASTM D 86 from TBP.
For the i ni t i al poi nt at 0%, the cal cul at i ons are as follows.
ASTM D 86 = (1/0.9177) 1/1~176 (10 + 273) 1/1~176 = 305 K =
poi nt s:
305 - 273 = 32~ The act ual dat a for the i ni t i al ASTM t em-
perat ure is 35~ whi ch is close to the cal cul at ed value. (3.21)
Now to est i mat e EFV from Eq. (3.16), specific gravity, is re-
where
qui red whi ch is not given by the probl em. SG can be esti-
mat ed from Eq. (3.17) and const ant s given i n Table 3.3 for Y/ =
the TBP. Fr om Table 3.6, T10(TBP) = 71.1 and Ts0(TBP) =
204.4~ Usi ng these values i n Eq. (3.17) gives SG = 0.10431 Xi --
(71.1 + 273) 0"1255 (204.4 + 273)020862= 0.7862. Now from
A, B =
SD
Vol% a b c rangefl ~ C
0 5.1764 0.7445 0.2879 -20-200
10 3.7452 0.7944 0. 2671 25-230
30 4.2749 0.7719 0.3450 35-255
50 1. 8445 0.5425 0.7132 55-285
70 1. 0751 0.9867 0.0486 65-305
90 1. 0849 0.9834 0.0354 80-345
100 1.799l 0.9007 0.0625 95--405
Source:Ref.[22].
~Ternperatures are approximated to nearest 5.
cal cul at ed ASTM and SG, the EFV t emper at ur es can be es-
t i mat ed from Eq. (3.16) wi t h const ant s given i n Table 3.2.
EFV = 2.9747 (32 + 273) 0.8466 (0.7862)~176 : 340.9 K ---
340.9 - 273 = 67.9~ The cal cul at ed value is very close to the
act ual val ue of 68.3~ (see Table 3.5). Si mi l arl y EFV values at
ot her poi nt s are cal cul at ed and results are shown i n Fig. 3.19.
Predi ct ed EFV curve from TBP are very close to the act ual
EFV curve. The AAD bet ween predi ct ed EFV and experi men-
tal dat a is 2.6 K. It shoul d be not ed t hat if experi ment al ASTM
dat a and specific gravity were used, the predi ct ed values of
EFV woul d be even closer to t he experi ment al values. #
3. 2. 2. 2 Daubert ' s M e t hod
Daubert and his group developed a different set of equat i ons
to convert ASTM to TBP, SD to ASTM, and SD to TBP [23].
These met hods have been i ncl uded i n the sixth edi t i on of API-
TDB [2] and are given i n this section. I n these met hods, first
conversi on shoul d be made at 50% poi nt and t hen the differ-
ence bet ween two cut poi nt s are correl at ed i n a form si mi l ar
to Eq. (3.14). I n this met hod SD dat a can be convert ed di-
rectly to TBP wi t hout cal cul at i ng ASTM as was needed i n t he
Ri azi - Dauber t met hod.
3. 2. 2. 2. 1 ASTM and TBP Conv e r s i on- - The following
equat i on is used to convert an ASTM D 86 di st i l l at i on at 50%
poi nt t emper at ur e to a TBP di st i l l at i on 50% poi nt t empera-
ture.
TBP(50%) = 255.4 + 0.8851[ASTM D 86(50%) - 255.4] 1~
(3.20)
where ASTM (50%) and TBP (50%) are t emper at ur es at 50%
vol ume distilled i n kelvin. Equat i on (3.20) can also be used
i n a reverse form to est i mat e ASTM from TBP. The following
equat i on is used to det er mi ne the difference bet ween two cut
Y/ = AX/B
difference i n TBP t emper at ur e bet ween two cut
points, K (or ~
observed difference i n ASTM D 86 t emper at ur e be-
t ween two cut poi nt s, K (or ~
const ant s varyi ng for each cut poi nt and are gi ven
i n Table 3.7
TABLE 3.6---Data on various distillation curves
for a naphtha-kerosene blend [2].
Vol% ASTM D 86, TBP, EFV,
distilled ~ C ~ C ~ C
0 35.0 10.0 68.3
10 79.4 71.1 107.2
30 145.6 143.3 151.1
50 201.7 204.4 182.2
70 235.6 250.6 207.2
90 270.6 291.7 228.3
300
~ 20o
[~ 100
~
~ 1 7 6
. . . . . . . TBP ( exp ) . . * " " 9
EF V ( exp ) ~176 ~ O
o EF V ( cal c) - " " ~
/
~176
0 i i i i
0 20 40 60 80 100
Vol% Distilled
FIG. 3 . 1 9 ~ Pr edi ct i on of EFV from T BP curv e
for a napht ha- k erosene blend ( Ex ampl e 3.2) .
To det er mi ne t he t rue boi l i ng p oi nt t emp er at ur e at any
p er cent di st i l l ed, cal cul at i on shoul d begi n wi t h 50% TBP
t emp er at ur e and addi t i on or subt r act i on of t he p r op er t em-
p er at ur e di fference Y/.
TBP (0%) = TBP (50%) - Y4 - Y5 - I16
TBP (10%) = TBP (50%) - Y4 - Y5
TBP (30%) - TBP (50%) - Y4
(3.22)
TBP (70%) = TBP (50%) + Y3
TBP (90%) = TBP (50%) + Y3 + Y2
TBP (100%) = TBP (50%) + Y3 + Y2 + Y1
Thi s met hod was devel oped bas ed on samp l es wi t h ASTM
50% p oi nt t emp er at ur e of less t han 250~ (480~ but it is
r ecommended for ext r ap ol at i on up to fract i ons wi t h ASTM
50% t emp er at ur e of 315~ (600~ as suggest ed by t he API [2].
Average absol ut e devi at i on for t hi s met hod as r ep or t ed by t he
API -TDB [2] is about 4.6~ for some 70 sampl es. Pr edi ct ed
TBP at 0 and 100% are t he l east accur at e val ues fol l owed by
val ues at i 0 and 90% p oi nt s as it is shown i n t he fol l owi ng
exampl e.
Exampl e 3. 3- - ASTM D 86 and TBP di st i l l at i on dat a for a
ker osene samp l e [1] are gi ven i n Table 3.7. Pr edi ct t he TBP
curve f r om ASTM dat a usi ng Ri azi - Dauber t and Daubert ' s
met hods and cal cul at e AAD for each met hod.
Sol ut i on- - The Ri azi - Dauber t met hod for conver si on of
ASTM to TBP dat a is p r esent ed by Eq. (3.14) and const ant s in
Table 3.2. The Daubert ' s met hod is expr essed by Eqs. ( 30. 20) -
TABLE 3.7--Correlation constants for Eq. (3.21).
Cut point Maximum allowable
range, % A B xi,a~
1 100-90 0.1403 1.6606
2 90-70 2.6339 0.7550 "55--
3 70-50 2.2744 0.8200 85
4 50-30 2.6956 0.8008 140
5 30-10 4.1481 0.7164 140
6 10-0 5.8589 0.6024 55
Source: Refs. [2, 23].
(3.22). The s ummar y of resul t s is gi ven i n Table 3.8. The
overal l average absol ut e devi at i ons (AAD) for Eqs. (3.14) and
(3.20) ar e cal cul at ed as 2.2 and 3.8~ respectively. As it is
seen i n Table 3,8, Eqs. ( 3. 20) -( 3. 22) are mor e accur at e at
30, 50, and 70% p oi nt s t han at t he l ower or hi gher ends of
t he di st i l l at i on curve. 0
3. 2. 2. 2. 2 SD to TBP Conversi on- - As descr i bed before,
si mul at ed di st i l l at i on by gas chr omat ogr ap hy ( ASTM D 2887)
is now commonl y used as a means of measur i ng boi l i ng p oi nt s
of l i ght p et r ol eum fract i ons. SD curves ar e expr essed i n t er ms
of t emp er at ur e versus wt % di st i l l ed, whi l e TBP curves ar e ex-
p r essed in t er ms of t emp er at ur e versus vol % di st i l l ed. I n t he
Daubert ' s met hod of conver si on of SD to TBP it is as s umed
t hat TBP at 50 vol % di st i l l ed is equal to SD t emp er at ur e at
50 wt % di st i l l ed. Equat i ons for conver si on of SD to TBP ar e
si mi l ar t o equat i ons devel oped for conver si on of ASTM to
TBE
(3.23) TBP( 50 vol%) = SD( 50 wt %)
wher e SD (50 wt %) and TBP (50 vol%) are t emp er at ur es at
50% di st i l l ed i n kel vi n ( or ~ The di fference bet ween adj a-
cent cut p oi nt s is cal cul at ed f r om t he fol l owi ng equat i on as
given by t he API -TDB [2].
(3.24) Vii = CWi n
wher e
V/ = di fference in TBP t emp er at ur e bet ween t wo cut
poi nt s, K ( or ~
W/ = obser ved di fference in SD t emp er at ur e bet ween
t wo cut poi nt s, K ( or ~
C, D = const ant s var yi ng for each cut p oi nt and are gi ven
in Table 3.9
To det er mi ne t he t r ue boi l i ng p oi nt t emp er at ur e at any per-
cent di st i l l ed, cal cul at i on shoul d begi n wi t h 50% TBP t emper-
at ur e and addi t i on or subt r act i on of t he p r op er t emp er at ur e
di fference V/.
TBP( 5%) = TBP( 50%) - Vs - V6 - - V7
TBP( 10%) = TBP( 50%) - Vs - V6
TBP( 30%) = TBP( 50%) - Vs
(3.25) TBP( 70%) = TBP( 50%) + V4
TBP( 90%) = TBP( 50%) + V4 + 173
TBP( 95%) = TBP( 50%) + V4 + V3 + V2
TBP( 100%) = TBP( 50%) + V4 + V3 + V2 + V1
Thi s met hod is ap p l i cabl e to fract i ons wi t h TBP 50% p oi nt s
in t he range of 120-370~ ( 250- 700~ Average absol ut e de-
vi at i on for t hi s met hod as r ep or t ed by t he API -TDB [2] is
about 7.5~ for about 21 sampl es. Based on 19 dat aset s it
was obser ved t hat er r or s i n di r ect conver si on of SD to TBP is
sl i ght l y hi gher t han if SD is conver t ed first t o ASTM and t hen
est i mat ed ASTM is conver t ed to TBP by Eqs. ( 3.20) -( 3.22) .
Det ai l s of t hese eval uat i ons are gi ven by t he API [2]. Pr edi ct ed
TBP at 5, 95, and 100% ar e t he l east accur at e val ues fol l owed
by val ues at 10 and 90% p oi nt s as is shown i n t he fol l owi ng
exampl e.
Exampl e 3. 4- - Exp er i ment al ASTM D 2887 (SD) and TBP dis-
t i l l at i on dat a for a p et r ol eum f r act i on are gi ven in Table 3.9 as
t aken f r om API [2]. Pr edi ct t he TBP curve f r om SD dat a usi ng
Vol%
di st i l l ed
TABLE 3.8--Prediction of TBP from ASTM for a kerosene sample of Example 3.3.
ASTM D 86 TBP Eq. (3.14) Eqs. (3.20)-(3.22)
exp,~ exp,~ TBP calc,~ AD,~ TBP calc,~ AD,~
0 165.6 146.1 134.1 12.0 133.1 13.0
10 176.7 160.6 160.6 0.0 158.1 2.5
30 193.3 188.3 188.2 0.1 189.2 0.9
50 206.7 209.4 208.9 0.5 210.6 1.2
70 222.8 230.6 230.2 0.4 232.9 2.3
90 242.8 255.0 254.7 0.3 258.1 3.1
Overall AAD,~ 2.2 3.8
Ri azi - Dauber t and Daubert ' s met hods and cal cul at e AAD for
each met hod.
Sol ut i on- - The Ri azi - Dauber t met hods do not pr ovi de a di-
rect conver si on f r om SD to TBP, but one can use Eqs. (3.18)
and (3.19) to convert SD t o ASTM D 86 and t hen Eq. (3.14)
shoul d be used t o convert ASTM to TBP dat a. F r om Eq. (3.19)
and use of SD at 10% and 50% poi nt s, t he val ue of p ar am-
et er F is cal cul at ed as 0.8287. Value of SD t emp er at ur e at
50 wt % is 168.9~ f r om Eq. (3.18) wi t h ap p r op r i at e const ant s
in Table 3.5 one can obt ai n ASTM D 86 (50%) = 166.3~
Subst i t ut i ng t hi s val ue for ASTM i nt o Eq. (3.14) gives TBP
(50 vol%) = 167.7~ whi l e t he exp er i ment al val ue as gi ven
i n Table 3.10 is 166.7~ The AD is t hen cal cul at ed as
167. 7-166. 7 = 1 ~ C. Daubert ' s met hod for conver si on of SD
t o TBP is di r ect and is p r esent ed t hr ough Eqs. ( 3.23) -( 3.25) .
Accor di ng to Eq. (3.23), TBP (50%) = SD (50%) = 168.9~
whi ch gives an AD of 2.2~ for t hi s poi nt . A s ummar y of com-
pl et e cal cul at i on resul t s is gi ven in Table 3.10. The overal l AAD
for Eqs. (3.14) and (3.18) is 4.8, whi l e for Eqs. ( 3. 23) -( 3. 25)
is 2.2~
Resul t s p r esent ed i n Examp l e 3.4 show t hat Eqs. ( 3. 23) -
(3.25) are mor e accur at e t han Eqs. (3.14) and (3.18) for t he
conver si on of SD t o TBE One of t he r easons for such a r esul t
is t hat t he samp l e p r esent ed in Table 3.10 to eval uat e t hese
met hods is t aken f r om t he same dat a bank used to devel op
cor r el at i ons of Eqs. ( 3.23) -( 3.25) . I n addi t i on t hese equat i ons
p r ovi de a di r ect conver si on of SD to TBE However, one shoul d
real i ze t hat Eqs. ( 3. 23) -( 3. 25) ar e based on onl y 19 dat aset s
and t hi s l i mi t s t he ap p l i cat i on of t hese equat i ons. Whi l e Eqs.
(3.14), (3.18), and (3.19) ar e based on much l ar ger dat a banks
wi t h wi der r ange of appl i cat i on. As avai l abl e dat a on bot h
SD and TBP are very l i mi t ed, a concr et e r ecommendat i on on
sup er i or i t y of t hese t wo ap p r oaches cannot be made at t hi s
t i me.
3,2. 2. 2. 3 SD to ASTM D 86 Conversi on- - Equat i ons to
convert SD ( ASTM D 2887) di st i l l at i on dat a to ASTM D 86
TABLE 3.9--Correlation constants for Eq. (3.24).
Cut p oi nt Maxi mum al l owabl e
i range, % C D W/,~
1 100-95 0.03849 1.9733 15
2 95-90 0.90427 0.8723 20
3 90-70 0.37475 1.2938 40
4 70-50 0.25088 1.3975 40
5 50-30 0.08055 1.6988 40
6 30-10 0.02175 2.0253 40
7 10-0 0.20312 1.4296 20
Source: Taken wi t h p er mi ssi on from Refs. [2, 23].
aTemperatures are ap p r oxi mat ed to nearest 5.
dat a ar e si mi l ar to t he equat i ons devel oped by Dauber t [2, 23]
to conver t ASTM or SD to TBP as gi ven in t hi s sect i on. The
equat i ons are s ummar i zed as fol l owi ng:
ASTM D 86(50vo1%) = 255.4 + 0.79424
(3.26) [ SD( 50 wt %) - 255.4] 1.0395
wher e SD (50 wt %) and ASTM D 86 (50 vo1%) ar e t emp er a-
t ures at 50% di st i l l ed in kelvin. The di fference bet ween adj a-
cent cut p oi nt s is cal cul at ed f r om t he fol l owi ng equat i on as
gi ven by t he API -TDB [2].
(3.27) Ui = ETf
wher e
Ui = di fference i n ASTM D 86 t emp er at ur es bet ween
T/ = obser ved di fference in SD t emp er at ur es bet ween
E, F = const ant s varyi ng for each cut p oi nt and are gi ven
in Table 3.11
To det er mi ne t he ASTM D 86 t emp er at ur e at any p er cent
di st i l l ed, cal cul at i ons shoul d begi n wi t h 50% ASTM D 86 t em-
p er at ur e and addi t i on or subt r act i on of t he p r op er t emp er a-
t ur e di fference Ui.
ASTM D 86(0%) = ASTM D 86( 50%)
- U4 - Us - U6
ASTM D 86( 10%) -- ASTM D 86( 50%)
- u4 - u5
ASTM D 86( 30%) -- ASTM D 86( 50%) - U4
(3.28)
ASTM D 86( 70%) = ASTM D 86( 50%) + U3
ASTM D 86( 90%) = ASTM D 86( 50%)
.-~ U3 -~ U 2
ASTM D 86( 100%) = ASTM D 86( 50%)
+ U3 + U2 + Ul
Thi s met hod is ap p l i cabl e to f r act i ons wi t h ASTM D 86 50%
p oi nt s in t he r ange of 65-315~ ( 150-600~ The average ab-
sol ut e devi at i on for t hi s met hod as r ep or t ed by t he API -TDB
[2] is about 6~ for some 125 samp l es and ap p r oxi mat el y 850
dat a poi nt s. Pr edi ct ed ASTM t emp er at ur es at 0 and 100% ar e
t he l east accur at e val ues fol l owed by val ues at 10 and 90%
p oi nt s as is shown in t he fol l owi ng exampl e.
Exampl e 3. 5- - Exp er i ment al ASTM D 2887 (SD) and ASTM
D 86 di st i l l at i on dat a for a p et r ol eum f r act i on are gi ven i n
Table 3.11 as t aken f r om t he API -TDB [2]. Pr edi ct t he ASTM
TABLE 3.10~Prediction of TBP from SD for a petroleum fraction of Example 3.4,
Wt% or vol % ASTM D 2887 TBP Eqs. (3.18) and (3.14) Eq. (3.23)-(3.25)
distilled (SD) exp,~ exp,~ TBP calc,~ AD,~ TBP calc,~ AD,~
10 151.7 161.1 146.1 15.0 164.3 3.2
30 162.2 163.3 157.1 6.2 166.9 3.5
50 168.9 166.7 167.7 1.0 168.9 2.2
70 173.3 169.4 170.7 1.3 170.9 1.5
90 181.7 173.9 179.1 5.3 176.7 2.8
Overall AAD, ~ 4.8 2.2
D 86 curve f r om SD dat a usi ng Ri azi - Dauber t and Daubert ' s
met hods and cal cul at e AAD for each met hod.
Sol ut i on- - Bot h met hods p r ovi de di rect met hods for conver-
si on of SD to ASTM D 86 and cal cul at i ons ar e si mi l ar to t hose
p r esent ed in Examp l es 3.3. and 3.4. Equat i ons ( 3. 26) -( 3. 28)
ar e used for Daubert ' s met hod, whi l e Eqs. (3.18) and (3.19)
ar e used for Ri azi - Dauber t met hod. A s umma r y of comp l et e
cal cul at i on resul t s is given in Table 3.12. The overal l AAD for
Eq. (3.18) is 1.5, whi l e for Eqs. ( 3. 26) -( 3. 28) is 2.0~ #
Resul t s p r esent ed in Examp l e 3.5 show t hat Eq. (3.18) is
sl i ght l y mor e accur at e t han Eqs. ( 3. 26) -( 3. 28) for t he conver-
si on of SD to ASTM D 86. This is consi st ent wi t h AAD r ep or t ed
for t hese met hods. However, Eqs. ( 3. 26) -( 3. 28) are based on
a l ar ger dat a set t han is Eq. (3.18). I n general , Ri azi - Dauber t
met hods ar e si mp l er and easi l y reversi bl e, whi l e t he exi st i ng
API met hods ar e sl i ght l y mor e compl ex. The advant age of
Daubert ' s met hods is t hat t he p r edi ct ed curve is s moot h and
uni f or m, whi l e in t he Ri azi - Dauber t met hods every p oi nt is
p r edi ct ed i ndep endent of adj acent p oi nt and l ack of smoot h-
ness i n p r edi ct ed curve is possi bl e, al t hough t hi s is r ar el y ob-
served in our experi ence. Si nce i n t he Daubert ' s met hods t em-
p er at ur es at 0 and 10% p oi nt s are cal cul at ed f r om p r edi ct ed
val ues at 30 and 50% poi nt s, l ar ger er r or s ar e obser ved at t he
l ower (0, 5, and 10% di st i l l ed) or up p er ends (90, 95, and 100%
di st i l l ed) of p r edi ct ed di st i l l at i on curves. I n general t he accu-
r acy of bot h met hods in t he p r edi ct i on of di st i l l at i on curves
at 0 and 100% p oi nt s ar e l i mi t ed. This is mai nl y due to t he
exp er i ment al uncer t ai nt y in meas ur ement of t emp er at ur es at
t he end poi nt s.
3. 2. 2. 3 Int erconveri on of DistiUation Curves
at Reduced Pressures
Nor mal boi l i ng p oi nt s of heavy p et r ol eum fract i ons such as
p r oduct s of a vacuum di st i l l at i on col umn cannot be meas ur ed
due to t he t her mal decomp osi t i on of heavy hydr ocar bons at
hi gh t emp er at ur es. For t hi s r eason di st i l l at i on dat a ar e re-
p or t ed at r educed p r essur es of 1-50 mmHg, as descr i bed
Cut point Maximum allowable
i range, % E F T/,~
1 100-90 2.13092 0.6596 55
2 90-70 0.35326 1,2341 55
3 70-50 0.19121 1.4287 55
4 50-30 0.10949 1.5386 55
5 30-10 0.08227 1.5176 85
6 10-0 0.32810 1.1259 85
Source: Taken with permission from Ref. [2].
ear l i er in t hi s chap t er under ASTM D 1160 t est met hod. F or
p r edi ct i on of p hysi cal and t her modynami c p r op er t i es nor mal
boi l i ng p oi nt s are requi red. F or t hi s r eason met hods of cal-
cul at i on of equi val ent at mosp her i c boi l i ng p oi nt ( EABP) ar e
i mp or t ant . One has t o r ecogni ze t hat EABP is not a real boi l -
i ng p oi nt as for such heavy fract i ons t here is no act ual and real
exp er i ment al val ue for t he nor mal boi l i ng poi nt . This p ar am-
et er can be obt ai ned f r om conver si on of di st i l l at i on curves at
l ow p r essur es to equi val ent di st i l l at i on curves at at mos p her i c
p r essur es and it is j ust an ap p ar ent nor mal boi l i ng poi nt . The
basi s of such conver si on is vap or p r essur e cor r el at i on for t he
f r act i on of i nt erest , whi ch will be di scussed in Chap t er 6. I n
t hi s p ar t we p r esent cal cul at i on met hods for t he conver si on
of ASTM D 1160 to at mos p her i c di st i l l at i on curve and for t he
p r edi ct i on of at mosp her i c TBP curves f r om ASTM D 1160.
I t shoul d be not ed t hat ASTM D 1160 does not refer to any
specific pr essur e. The p r essur e may vary f r om 1 to 50 mm Hg.
When D 1160 curve is conver t ed t o a di st i l l at i on curve at at-
mosp her i c p r essur e t hr ough a vap or p r essur e cor r el at i on t he
r esul t i ng di st i l l at i on curve is not equi val ent t o ASTM D 86 or
to TBP curve. The r esul t i ng di st i l l at i on curve is r ef er r ed to
as equi val ent at mos p her i c ASTM D 1160. Anot her l ow pres-
sure di st i l l at i on dat a is TBP di st i l l at i on curve at 1, 10, or
50 mm Hg. Thr ough vap or p r essur e cor r el at i ons TBP at re-
duced p r essur es can be conver t ed to at mos p her i c TBP. There
is a p r ocedur e for t he conver si on of ASTM D 1160 to TBP
at 10 mm Hg whi ch is p r esent ed in t hi s sect i on. Therefore,
to convert ASTM D 1160 to TBP at at mos p her i c p r essur e
one has to convert D 1160 at any p r essur e to D 1160 at 10
mmHg and t hen to convert resul t i ng D 1160 t o TBP at 10
mm Hg. This means if ASTM D 1160 at 1 mm Hg is avail-
able, it mus t be first conver t ed t o D 1160 at 760 mmHg,
t hen to D 1160 at 10 mm Hg fol l owed by conver si on to TBP
at 10 mmHg and fi nal l y t o TBP at 760 mmHg. A s umma r y
char t for var i ous conver si ons is p r esent ed at t he end of t hi s
sect i on.
3,2. 2. 3. 1 Conversion of a Boi l i ng Poi nt at Sub- or Super-
At mospheri c Pressures to the Normal Boi l i ng Poi nt or
V ice V ersa- - The conver si on of boi l i ng p oi nt or sat ur at i on
t emp er at ur e at s ubat mos p her i c ( P < 760 mm Hg) or super-
at mos p her i c ( P > 760 mm Hg) condi t i ons to nor mal boi l i ng
p oi nt is based on a vap or p r essur e correl at i on. The met hod
wi del y used in t he i ndust r y is t he cor r el at i on devel oped for
p et r ol eum fract i ons by Maxwel l and Bonnel l [27], whi ch is
also used by t he API -TDB [2] and ot her sour ces [24] and is
p r esent ed here. This cor r el at i on is gi ven for several p r essur e
r anges as follows:
748.1 QT
(3.29) 2r~ = 1 + T( 0. 3861Q - 0.00051606)
TABLE 3.12--Prediction of ASTM D 86 from SD for a petroleum fraction of Example 3.5.
Eqs. (3.18) and (3.19) Eqs. (3.25)-(3.28)
Vol% ASTM D 2887 ASTM D 86 ASTM D 86 ASTM D 86
distilled (SD) exp,~ exp,~ calc,~ AD,~ calc,~ AD,~
10 33.9 56.7 53.2 3.4 53.5 3.2
30 64.4 72.8 70.9 1.9 68.2 4.5
50 101.7 97.8 96.0 1.8 96.8 1.0
70 140.6 131.7 131.3 0.4 132.5 0.9
90 182.2 168.3 168.3 0.0 167.8 0.6
Overall AAD, ~ 1.5 2.0
Q=
6.761560 - 0.987672 lOglo P
3000.538 - 43 loglo P
5.994296 - 0.972546 loglo P
2663.129 - 95.76 loglo P
6.412631 - 0.989679 loglo P
2770.085 - 36 loglo P
( P < 2mmHg)
Q=
(2 < P < 760 mm Hg)
Q=
( P > 760mmHg)
P
Tb T,'
= b + 1. 3889F( Kw - 12)logao 760
F = 0
F = - 3. 2985 + 0.009 Tb
F = - 3. 2985 + 0.009 Tu
(Tu < 367 K) or when Kw
is not avai l abl e
(367 K _< Tb < 478 K)
(Tb > 478 K)
wher e
P = p r essur e at whi ch boi l i ng p oi nt or di st i l l at i on dat a
is avai l abl e, mm Hg
T = boi l i ng p oi nt or i gi nal l y avai l abl e at p r essur e P, in
kel vi n
T~ = nor mal boi l i ng p oi nt cor r ect ed to Kw = 12, in
kel vi n
Tb = nor mal boi l i ng poi nt , in kel vi n
Kw = Wat son (UOP) char act er i zat i on f act or [ = (1.8Tb) 1/3
/SG]
F = cor r ect i on f act or f or t he fract i ons wi t h Kw di fferent
f r om I 2
logl0 = c ommon l ogar i t hm ( base 10)
The ori gi nal eval uat i on of t hi s equat i on is on p r edi ct i on of va-
p or p r essur e of p ur e hydr ocar bons. Rel i abi l i t y of t hi s met hod
for nor mal boi l i ng p oi nt of p et r ol eum f r act i ons is unknown.
When t hi s equat i on is ap p l i ed to p et r ol eum fract i ons, gener-
al l y Kw is not known. For t hese si t uat i ons, T~ is cal cul at ed
wi t h t he as s ump t i on t hat Kw is 12 and Tb = T~. This is to
equi val ent to t he as s ump t i on of F = 0 for l ow- boi l i ng- poi nt
comp ounds or fract i ons. To i mp r ove t he r esul t a second r ound
of cal cul at i ons can be made wi t h Kw cal cul at ed f r om esti-
mat ed val ue of T~. When t hi s equat i on is ap p l i ed to di st i l l a-
t i on curves of cr ude oils it shoul d be r eal i zed t hat val ue of
Kw may change al ong t he di st i l l at i on curve as bot h Tb and
specific gravi t y change.
Equat i on (3.29) can be easi l y used in its reverse form to
cal cul at e boi l i ng p oi nt s (T) at l ow or el evat ed p r essur es f r om
nor mal boi l i ng p oi nt (Tb) as follows:
wher e
P
7"~ = Tb -- 1.3889 F (Kw - 12) logl0 760
wher e all t he p ar amet er s are defi ned i n Eq. (3.29). The mai n
ap p l i cat i on of t hi s equat i on is to est i mat e boi l i ng p oi nt s at
I 0 mmHg f r om at mos p her i c boi l i ng poi nt s. At P = 10
mmHg, Q = 0.001956 and as a r esul t Eq. (3.30) r educes t o
t he fol l owi ng si mp l e form:
0.683398T~
(3.31) T( 10mmHg) = 1 - 1.63434 x I0-4T~
i n whi ch T~ is cal cul at ed f r om Tb as given in Eq. (3.30) and
bot h ar e in kelvin. Temp er at ur e T (10 mm Hg) is t he boi l i ng
p oi nt at r educed p r essur e of 10 mm Hg in kelvin. By assum-
i ng Kw = 12 ( or F = 0) and for l ow-boi l i ng f r act i ons val ue
of nor mal boi l i ng poi nt , Tb, can be used i nst ead of T~ i n Eq.
(3.31). To use t hese equat i ons for t he conver si on of boi l i ng
p oi nt f r om one l ow p r essur e to anot her l ow p r essur e (i.e.,
f r om 1 to 10 mm Hg) , t wo st eps are requi red. I n t he first step,
nor mal boi l i ng p oi nt or T (760 mm Hg) is cal cul at ed f r om
T (1 mmHg) by Eq. (3.29) and in t he second st ep T (10
mm Hg) is cal cul at ed f r om T (760 mm Hg) or Tb t hr ough Eqs.
(3.30) and (3.31).
I n t he mi d 1950s, anot her gr ap hi cal cor r el at i ons for t he
est i mat i on of vap or p r essur e of hi gh boi l i ng hydr ocar bons
were p r op os ed by Myers and Fenske [28]. Lat er t wo si mp l e
l i near r el at i ons were der i ved f r om t hese char t s to est i mat e
T ( I 0 mm Hg) from t he nor mal boi l i ng p oi nt (Tb) or boi l i ng
p oi nt at 1 mm Hg as fol l ows [29]:
T(10 mm Hg) = 0.8547T(760 rnm Hg) - 57.7 500 K < T(760 mm) < 800K
T(10mmHg) = 1.07T(1 rnmHg) + 19 300K < T(1 ram) < 600K
(3.32)
wher e all t emp er at ur es are in kelvin. These equat i ons r epr o-
duce t he ori gi nal figures wi t hi n 1%; however, t hey shoul d
be used wi t hi n t he t emp er at ur es r anges specified. Equa-
t i ons (3.30) and (3.31) ar e mor e accur at e t han Eq. (3.32) but
for qui ck hand est i mat es t he l at t er is mor e conveni ent . An-
ot her si mpl e r el at i on for qui ck conver si on of boi l i ng p oi nt at
var i ous p r essur es is t hr ough t he fol l owi ng correct i on, whi ch
was p r op os ed by Van Kr anen and Van Nes, as gi ven by Van
Nes and Van West en [30].
Tb - 41 1393 - T
log10 PT = 3.2041 1 - 0.998 ~ 1393 - Tb]
(3.33)
wher e T is t he boi l i ng p oi nt at p r essur e Pr and Tb is t he nor mal
boi l i ng poi nt . Pr is in bar and T and Tb are in K. Accur acy of
t hi s equat i on is about I %.
(3.30) T =
748.1 O - T[(0.3861 O - 0.00051606)
3. 2. 2. 3. 2 Conversion of a Distillation Curve from Sub- or
Super- Atmospheric Pressures to a Distillation Curve at
Atmospheric Pressure--The met hod of conver si on of boi l -
i ng p oi nt s t hr ough Eqs. ( 3. 29) -( 3. 32) can be used to every
p oi nt on a di st i l l at i on curve under ei t her sub- or sup er at mo-
sp her i c p r essur e condi t i ons. I n t hese equat i ons Tb or T (760
mmHg) r ep r esent a p oi nt al ong t he di st i l l at i on curve at at-
mos p her i c pr essur e. I t can be ap p l i ed to any of TBP, EFV,
or ASTM D 1160 di st i l l at i on curves. However, it shoul d be
not ed t hat t hese equat i ons convert di st i l l at i on curves f r om
one p r essur e t o anot her wi t hi n t he same type. For exampl e, it
is not possi bl e to use t hese equat i ons t o di r ect l y convert ASTM
D 1160 at 10 mmHg to TBP at 760 mmHg. Such conver si ons
r equi r e t wo st eps t hat are di scussed i n t he fol l owi ng sect i on.
The onl y di st i l l at i on curve t ype t hat mi ght be r ep or t ed un-
der s up er at mos p her i c p r essur e ( P > 1.01325 bar) condi t i on
is t he EFV di st i l l at i on curve. TBP curve may be at I , 10, 100,
or 760 mm Hg pr essur e. Exp er i ment al dat a on ASTM D 1160
ar e usual l y r ep or t ed at i , 10, or 50 mm Hg. ASTM D 86 di st i l -
l at i on is al ways r ep or t ed at at mos p her i c pr essur e. I t shoul d
be not ed t hat when ASTM D 1160 di st i l l at i on curve is con-
ver t ed to or r ep or t ed at at mos p her i c p r essur e (760 mm Hg)
it is not equi val ent to or t he same as ASTM D 86 di st i l l at i on
dat a. They ar e di fferent t ypes of di st i l l at i on curves and t here
is no di r ect conver si on bet ween t hese t wo curves.
in t he above r el at i ons all t emp er at ur es are ei t her in ~ or i n
kelvin.
3. 2. 2. 4 Summary Chart for Interconverion of Various
Distillation Curves
A s ummar y of all conver si on met hods is shown in Fig. 3.20.
I t shoul d be not ed t hat any di st i l l at i on curve at l ow p r essur e
(i.e., ASTM D 1160 or EFV at 1, 10, 50, mmHg or TBP at
1 mm Hg) shoul d be first conver t ed to TBP di st i l l at i on curve
at 10 mm Hg bef or e t hey are conver t ed to TBP at at mos p her i c
pr essur e.
Example 3. 6- - F or a p et r ol eum f r act i on t he ASTM D 1160
di st i l l at i on dat a at 10 mm Hg are gi ven i n Table 3.13. Pr edi ct
t he TBP curve at at mos p her i c pr essur e.
Solution--ASTM D 1160 dat a have been conver t ed to TBP
at 10 mmHg by Eq. (3.34). Then Eq. (3.29) wi t h P- - 10
mmHg and Q = 0.001956 is used to conver t TBP f r om 10
to 760 mmHg. A s ummar y of resul t s is gi ven i n Table 3.13.
The second and less accur at e met hod to conver t TBP f r om 10
to 760 mm Hg is t hr ough Eq. (3.32), whi ch in its reverse f or m
becomes T (760 mmHg) = 1.17T (10 mmHg) + 67.51. Esti-
mat ed TBP at 760 mm Hg t hr ough t hi s r el at i on is p r esent ed
in t he l ast col umn of Table 3.13. #
3. 2. 2. 3. 3 Conversion of ASTM D 1160 at 10 mmHg to
TBP Distillation Curve at 10 mm Hg- - The onl y met hod
wi del y used under s ubat mos p her i c p r essur e condi t i on for
conver si on of di st i l l at i on curves is t he one devel op ed by
Edmi s t er - Okamot o [17], whi ch is used to conver t ASTM
D 1160 t o TBP, bot h at 10 mmHg. Thi s met hod is gr aphi -
cal and it is al so r ecommended by t he API -DTB [2]. I n t hi s
met hod it is as s umed t he at 50% p oi nt s ASTM D 1160 and
TBP t emp er at ur es ar e equal . The Edmi s t er - Okamot o char t
is conver t ed i nt o equat i on f or m t hr ough r egr essi on of val ues
r ead f r om t he figure i n t he fol l owi ng f or m [2]:
TBP( 100%)
TBP( 90%)
TBP( 70%)
(3.34) TBP( 50%)
TBP( 30%)
TBP( 10%)
TBP( 0%)
= ASTM D 1160(100%)
= ASTM D 1160(90%)
= ASTM D 1160(70%)
= ASTM D 1160(50%)
= ASTM D 1160(50%) - F1
= ASTM D 1160(30%) - F2
= ASTM D 1160(10%) - F3
wher e f unct i ons F1, F2, and/ : 3 ar e given i n t er ms of t emper -
at ur e di fference i n t he ASTM D 1160:
F1 = 0.3 + 1.2775(AT1) - 5.539 10-3(AT1) 2 @ 2.7486
X 10-5(AT1) 3
F2 = 0.3 + 1.2775(AT2) - 5.539 x 10-3(AT2) 2 + 2.7486
10-5(AT2) 3
F3 = 2.2566(AT3) - 266.2 x 10-4(AT3) 2 + 1.4093
10-4(AT3) 3
AT1 = ASTM D 1160(50%) - ASTM D 1160(30%)
AT2 = ASTM D 1160(30%) - ASTM D 1160(10%)
AT3 = ASTM D 1160(10%) - ASTM D 1160(0%)
3. 2. 3 Predi ct i on of Compl et e Di st i l l at i on Curves
I n many cases di st i l l at i on dat a for t he ent i re r ange of p er cent
di st i l l ed ar e not avai l abl e. Thi s is p ar t i cul ar l y t he case when a
f r act i on cont ai ns heavy comp ounds t owar d t he end of di st i l l a-
t i on curve. For such fract i ons di st i l l at i on can be p er f or med to
a cer t ai n t emp er at ur e. For exampl e, in a TBP or ASTM curve,
di st i l l at i on dat a may be avai l abl e at 10, 30, 50, and 70% p oi nt s
hut not at 90 or 95% poi nt s, whi ch are i mp or t ant for p r ocess
engi neers and are char act er i st i cs of a p et r ol eum p r oduct . For
heavi er fract i ons t he di st i l l at i on curves may even end at 50%
poi nt . F or such fract i ons it is i mp or t ant t hat val ues of t emper -
at ur es at t hese hi gh p er cent age p oi nt s to be est i mat ed f r om
avai l abl e dat a. I n t hi s sect i on a di st r i but i on f unct i on for bot h
boi l i ng p oi nt and densi t y of p et r ol eum f r act i ons is p r esent ed
so t hat its p ar amet er s can be det er mi ned f r om as few as t hree
dat a p oi nt s on t he curve. The f unct i on can p r edi ct t he boi l i ng
p oi nt for t he ent i re r ange f r om i ni t i al p oi nt to 95% poi nt . This
f unct i on was p r op os ed by Ri azi [31] based on a p r obabi l i t y
di st r i but i on model for t he p r op er t i es of hep t ane pl us fract i ons
in cr ude oils and r eser voi r fluids and its det ai l ed char act er i s-
t i cs ar e di scussed in Sect i on 4.5.4. The di st r i but i on model is
p r esent ed by t he fol l owi ng equat i on ( see Eq. 4.56):
in whi ch T is t he t emp er at ur e on t he di st i l l at i on curve i n
kel vi n and x is t he vol ume or wei ght fract i on of t he mi xt ur e
di st i l l ed. A, B, and To ar e t he t hr ee p ar amet er s t o be det er-
mi ned f r om avai l abl e dat a on t he di st i l l at i on curve t hr ough a
l i near regressi on. To is in fact t he i ni t i al boi l i ng p oi nt (T at x =
0) but has to be det er mi ned f r om act ual dat a wi t h x > 0. The
exp er i ment al val ue of To shoul d not be i ncl uded in t he regres-
si on pr ocess si nce i t is not a rel i abl e poi nt . Equat i on (3.35)
I
TBP ! --'-I
1, 50, 100mm 6
I v
3. CHARACTERI Z ATI ON OF PETROL EUM FRACTI ONS 1 0 9
ASTM D2887
Simulated
Distillation
(SD)
ASTM D86
760 mmHg
TBP
760 mmHg
Superatmospheric
EFV
(P >760 mmHg)
EFV
760 mmHg
TBP
10 mmHg
EFV
10 mrnHg
ASTM D- 1160
10 mmHg
4 V
ASTM D-1160 ASTM D-1160
6 Reported
at
1, 30, 50 mmHg ~" 760 mmHg
STEP METHOD A METHOD B
Eqs. (3.14)or (3.15)
Eq. (3.34)
Eqs. (3.18) & (3.19)
Eq. (3.31)
Eq. (3.16)
Eq. (3.29)
Eqs. (3.23) - (3.25)
Eqs. (3.20) - (3.22)
Eqs. (3.26) - (3.28)
Eq. (3.32)
FIG. 3 . 20- - Summar y of methods for the i nterconv ersi on of v ari ous distillation curv es.
does not give a finite value for T at x = 1 ( end poi nt at 100%
distilled). Accordi ng to this model the final boiling poi nt is
infinite (oo), whi ch is t rue for heavy residues. Theoretically,
even for light product s wi t h a limited boiling range t here is a
very small amount of heavy comp ound since all comp ounds
in a mi xt ure cannot be compl et el y separat ed by distillation.
For this reason predi ct ed values f r om Eq. (3.35) are reliable
up to x = 0.99, but not at the end point. Paramet ers A, B, and
To in Eq. (3.35) can be directly det ermi ned by usi ng Solver
(in Tools) in Excel spreadsheets. Anot her way to det ermi ne
the const ant s in Eq. (3.35) is t hr ough its conversi on i nt o t he
following linear form:
(3.36) Y = Cz + C2X
where Y= I n [(T - To) / To] and X- - I n I n [1 / (1 - x) ] . Con-
stants C1 and Ca are det ermi ned from linear regressi on of Y
versus X wi t h an initial guess for To. Const ant s A and B are
det ermi ned from C1 and C2 as B = 1/C2 and A = B exp (C1B).
Paramet er To can be det ermi ned by several estimates t o maxi-
mi ze the R squared (RS) value for Eq. (3.36) and mi ni mi ze the
AAD for predi ct i on of T form Eq. (3.35). I f the initial boiling
poi nt in a distillation curve is available it can be used as the
TABLE 3.13---Conversion of ASTM D 1160 to TBP at 760 mm Hg
for the petroleum fraction of Example 3.6.
XZol% ASTM
distilled D 1160, ~ TBP%, ~ TBl~76O, ~ TBP~76o, ~
10 150 142.5 280.1 280.7
30 205 200.9 349.9 349.0
50 250 250 407.2 406.5
70 290 290 453.1 453.3
90 350 350 520.4 523.5
aEq. (3.34).
bEq. (3.29).
eEq. (3.32).
i ni t i al guess, but val ue of To shoul d al ways be less t han val ue
of T for t he first dat a poi nt . For fract i ons wi t h final boi l i ng
p oi nt very hi gh and uncer t ai n, such as at mos p her i c or vac-
uum r esi dues and hep t ane- p l us fract i on of cr ude oils, val ue of
B can be set as 1.5 and Eq. (3.35) r educes to a t wo- p ar amet er
equat i on. However, for var i ous p et r ol eum fract i ons wi t h fi-
ni t e boi l i ng r ange p ar amet er B shoul d be det er mi ned f r om
t he r egr essi on anal ysi s and val ue of B for l i ght fract i ons is
hi gher t han t hat of heavi er fract i ons and is nor mal l y gr eat er
t han 1.5. Equat i on (3.35) can be ap p l i ed to any t ype of distil-
l at i on dat a, ASTM D 86, ASTM D 2887 (SD), TBP, EFV, and
ASTM D 1160 as wel l as TBP at r educed p r essur es or EFV
at el evat ed pr essur es. I n t he case of SD curve, x is cumul a-
tive wei ght f r act i on di st i l l ed. The average boi l i ng p oi nt of t he
f r act i on can be det er mi ned f r om t he fol l owi ng rel at i on:
Tav -= To(1 + Tar )
( 3. 37' ( A) : ~ ( 1)
T~*v= F 1+
i n whi ch F is t he ga mma f unct i on and may be det er mi ned
f r om t he fol l owi ng r el at i on when val ue of p ar amet er B is
gr eat er t han 0.5.
F ( l +l ) =o. 992814- O. 504242B- l +O. 696215B -z
- 0. 272936B -3 + 0. 088362B -4
(3.38)
Devel op ment of t hese r el at i ons is di scussed i n Chap t er 4.
I n Eq. (3.35), i f x is vol ume fract i on, t hen Tar cal cul at ed f r om
Eq. (3.37) woul d be vol ume average boi l i ng p oi nt (VABP)
and if x is t he wei ght f r act i on t hen Tar woul d be equi val ent
to t he wei ght average boi l i ng p oi nt (WABP). Si mi l ar l y mol e
average boi l i ng p oi nt can be est i mat ed f r om t hi s equat i on
if x is i n mol e fract i on. However, t he mai n ap p l i cat i on of
Eq. (3.35) is t o p r edi ct comp l et e di st i l l at i on curve f r om a l i m-
i t ed dat a avai l abl e. I t can al so be used to p r edi ct boi l i ng p oi nt
of r esi dues in a cr ude oil as will be shown in Chap t er 4. Equa-
t i on (3.35) is al so perfect l y ap p l i cabl e to densi t y or specific
gravi t y di st r i but i on al ong a di st i l l at i on curve for a p et r ol eum
fract i on and cr ude oils. F or t he case of density, p ar amet er T
is r ep l aced by d or SG and densi t y of t he mi xt ur e may be cal-
cul at ed f r om Eq. (3.37). When Eqs. ( 3. 35) -( 3. 38) ar e used for
p r edi ct i on of densi t y of p et r ol eum fract i ons, t he val ue of RS
is less t han t hat of di st i l l at i on dat a. Whi l e t he val ue of B for
t he case of densi t y is gr eat er t han t hat of boi l i ng p oi nt and is
usual l y 3 for very heavy fract i ons (C7+) and hi gher for l i ght er
mi xt ures. I t shoul d be not ed t hat when Eqs. ( 3. 35) -( 3. 38) ar e
ap p l i ed to specific gravi t y or density, x shoul d be cumul at i ve
vol ume fract i on. F ur t her p r op er t i es and ap p l i cat i on of t hi s
di st r i but i on f unct i on as well as met hods of cal cul at i on of av-
erage p r op er t i es for t he mi xt ur e ar e gi ven in Chap t er 4. Here-
in we demons t r at e use for t hi s met hod for p r edi ct i on of dis-
t i l l at i on curves of p et r ol eum fract i ons t hr ough t he fol l owi ng
exampl e.
Exampl e 3. 7- - ASTM D 86 di st i l l at i on dat a f r om i ni t i al to
final boi l i ng p oi nt for a gas oil samp l e [ i ] are gi ven in t he first
t wo col umns of Table 3.14. Pr edi ct t he di st i l l at i on curve for
t he fol l owi ng four cases:
a. Use dat a p oi nt s at 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, and
95 vo1% di st i l l ed.
b. Use all dat a p oi nt s f r om 5 to 70 vol% di st i l l ed.
c. Use t hree dat a p oi nt s at 10, 30, and 50%.
d. Use t hr ee dat a p oi nt s at 30, 50, and 70%.
TABLE 3.14--Prediction of ASTM D 86 distillation curve
Data Set A Data Set B
Vol% distilled Temp. exp, K Pred, K AD, K Pred, K
0 520.4 526.0 5.6 525.0
5 531.5 531.6 0.1 531.5
10 534.8 534.6 0.2 534.7
20 539.8 539.5 0.4 539.6
30 543.2 543.8 0.7 544.0
40 548.2 548.1 0.1 548.1
50 552.6 552.5 0.1 552.4
60 557.0 557.3 0.3 557.0
70 562.6 562.9 0.3 562.2
80 570.4 569.9 0.5 568.7
90 580.4 580.4 0.0 578.2
95 589.8 589.6 0.2 586.6
100 600.4 608.3 7.9 603.1
AAD (total), K 1.3
No. of data used 11
To 526
A 0.01634
B 1.67171
RS 0.9994
AAD (data used), K 0.25
VABP, K 554.7 555.5
for gas oil sample of Example 3.7.
Data Set C Data Set D
AD, K Pred, K AD, K Pred, K AD, K
4.6 530.0 9.6 512.0 8.4
0.0 532.7 1.2 526.4 5.1
0.1 534.8 0.0 531.2 3.6
0.2 538.9 0.9 537.8 2.0
0.8 543.1 0.0 543.1 0.0
0.0 547.6 0.5 547.9 0.3
0.2 552.6 0.0 552.6 0.0
0.1 558.5 1.4 557.4 0.4
0.4 565.6 3.0 562.6 0.0
1.7 575.2 4.8 568.8 1.6
2.2 590.6 10.3 577.5 2.9
3.3 605.2 15.4 584.8 5.1
2.7 637.1 36.7 598.4 2.0
1. 3 6. 5 2 . 4
8 3 3
525 530 512
0.0125 0.03771 0.00627
1.80881 1.21825 2.50825
0.999 1 1
0.23 0 0
555 557 554
For each case give parameters To, A, and B in Eq. (3.35) as
well as value of RS and AAD based on all data points and
based on data used for the regression. Also calculate VABP
from Eq. (3.37) and compare with actual VABP calculated
from Eq. (3.6).
Solution--Summary of calculation results for all four cases
are given in Table 3.14. For Case A all experimental data given
on the distillation curve (second column in Table 3.14) from
5 to 95% points are used for the regression analysis by Eq.
(3.36). Volume percentages given in the first column should be
converted to cumulative volume fraction, x, (percent values
divided by 100) and data are converted to X and Y defined in
Eq. (3.36). The first data point used in the regression process
is at x = 0.05 with T = 531.5 K; therefore, the initial guess
(To) should be less than 531. With a few changes in To values,
the maximum RS value of 0.9994 is obtained with mi ni mum
AAD of 0.25 K (for the 11 data points used in the regression
process). The AAD for the entire data set, including the IBP
and FBP, is 1.3 K. As mentioned earlier the experimentally re-
ported IBP and especially the value of FBP are not accurate.
Therefore, larger errors for prediction of IBP and FBP are ex-
pected from Eq. (3.35). Since values of FBP at x = 1 are not
finite, the value of T at x = 0.99 may be used as an approx-
imate predicted value of FBP from the model. These values
are given in Table 3.14 as predicted values for each case at
100 vol% vaporized. Estimated VABP from Eq. (3.37) for Case
A is 555.5 versus value of 554.7 from actual experimental data
and definition of VABP by Eq. (3.6).
For Case B, data from 5 to 70 vol% distilled are used for
the regression process and as a result the predicted values
up to 70% are more accurate than values above 70% point.
However, the overall error (total AAD) is the same as for Case
A at 1.3 K. For Case C only three data points at 10, 30, and
50% are used and as a result much larger errors especially
for points above 50% are observed. I n Case D, data at 30, 50,
~9
650
600
550
o Exp. data "l
-- Pred. (data set A) '1 /
. . . . . . . Pred. (data set B) ; "t
. . . . . Pred. (data set C) 1' ~'
m ~ Perd. (data set D) ~ o4'(.~q?
500 . . . . . . . .
0 20 40 60 80 100
Vol % Distilled
FIG. 3 . 21 1 Pr edi ct i on of distillation curv es f or the g as oil
sampl e of Ex ampl e 3.7.
and 70% points are used and the predicted values are more
accurate than values obtained in Case C. However, for this last
case the highest error for the IBP is obtained because the first
data point used to obtain the constants is at 30%, which is far
from 0% point. Summary of results for predicted distillation
curves versus experimental data are also shown in Fig. 3.2 I.
As can be seen from the results presented in both Table 3.14
and Fig. 3.21, a good prediction of the entire distillation curve
is possible through use of only three data points at 30, 50, and
70%. r
3 . 3 P R ED I C T I ON OF P R OP ER T I ES
OF P ET R OL EUM FR A C T I ON S
As discussed in Chapter 1, petroleum fractions are mixtures of
many hydrocarbon compounds from different families. The
most accurate method to determine a property of a mixture
is through experimental measurement of that property. How-
ever, as this is not possible for every petroleum mixture, meth-
ods of estimation of various properties are needed by process
or operation engineers. The most accurate method of esti-
mating a property of a mixture is through knowledge of the
exact composition of all components existing in the mixture.
Then properties of pure components such as those given in
Tables 2.1 and 2.2 can be used together with the composi-
tion of the mixture and appropriate mixing rules to determine
properties of the mixture. I f experimental data on properties
of pure compounds are not available, such properties should
be estimated through the methods presented in Chapter 2.
Application of this approach to defined mixtures with very
few constituents is practical; however, for petroleum mixtures
with many constituents this approach is not feasible as the
determination of the exact composition of all components
in the mixture is not possible. For this reason appropriate
models should be used to represent petroleum mixtures by
some limited number of compounds that can best represent
the mixture. These limited compounds are different from the
real compounds in the mixture and each is called a "pseudo-
component" or a "pseudocompound". Determination of these
pseudocompounds and use of an appropriate model to de-
scribe a mixture by a certain number of pseudocompounds
is an engineering art in prediction of properties of petroleum
mixtures and are discussed in this section.
3.3.1 Mat r i x of Ps e u d o c o mp o n e n t s Tabl e
As discussed in Chapter 2, properties of hydrocarbons vary
by both carbon number and molecular type. Hydrocarbon
properties for compounds of the same carbon number vary
from paraffins to naphthenes and aromatics. Very few frac-
tions may contain olefins as well. Even within paraffins fam-
ily properties of n-paraffins differ from those of isoparaffins.
Boiling points of hydrocarbons vary strongly with carbon
number as was shown in Table 2.1; therefore, identification of
hydrocarbons by carbon number is useful in property predic-
tions. As discussed in Section 3,1.5.2, a combination of GS-
MS in series best separate hydrocarbons by carbon number
and molecular type. If a mixture is separated by a distillation
column or simulated distillation, each hydrocarbon cut with a
single carbon number contains hydrocarbons from different
TABLE 3.15--Presentation of a petroleum fraction (diesel fuel) by a matrix of 30 pseudocomponents.
Carbon number n-Paraffins Isoparaffins Olefins Naphthenes Aromatics
C11 1 2 3 4 5
C12 6 7 8 9 10
C13 11 12 13 14 15
C14 16 17 18 19 20
C15 21 22 23 24 25
C16 26 27 28 29 30
groups, which can be identified by a PIONA analyzer. As an
example in Table 1.3 (Chapter 1), carbon number ranges for
different petroleum products are specified. For a diesel fuel
sample, carbon number varies from Cll to C16 with a boiling
range of 400-550~ I f each single carbon number hydrocar-
bon cut is further separated into five pseudocomponents from
different groups, the whole mixture may be represented by a
group of 30 pseudocomponents as shown in Table 3.15. Al-
though each pseudocomponent is not a pure hydrocarbon but
their properties are very close to pure compounds from the
same family with the same carbon number. If the amounts of
all these 30 components are known then properties of the mix-
ture may be estimated quite accurately. This requires exten-
sive analysis of the mixture and a large computation time for
estimation of various properties. The number of pseudocom-
ponents may even increase further if the fraction has wider
boiling point range such as heptane plus fractions as will be
discussed in Chapter 4. However, many petroleum fractions
are olefin free and groups of n-paraffins and isoparaffins may
be combined into a single group of paraffins. Therefore, the
number of different families reduces to three (paraffins, naph-
thenes, and aromatics). I n this case the number of compo-
nents in Table 3.15 reduces to 6 x 3 or 18. If a fraction is nar-
row in boiling range then the number of rows in Table 3.15
decreases indicating lower carbon number range. I n Chapter
4, boiling points of various single carbon number groups are
given and through a TBP curve it would be possible to deter-
mine the range of carbon number in a petroleum fraction. In
Table 3.15, if every two carbon number groups and all paraf-
fins are combined together, then the whole mixture may be
represented by 3 x 3 or 9 components for an olefin-free frac-
tion. Similarly if all carbon numbers are grouped into a single
carbon number group, the mixture can be represented by only
three pseudocomponents from paraffins (P), naphthenes (N),
and aromatics (A) groups all having the same carbon number.
This approach is called pseudocomponent technique.
Finally the ultimate simplicity is to ignore the difference in
properties of various hydrocarbon types and to present the
whole mixture by just a single pseudocomponent, which is
the mixture itself. The simplicity in this case is that there is
no need for the composition of the mixture. Obviously the
accuracy of estimated properties decreases as the number of
pseudocomponents decreases. However, for narrow boiling
range fractions such as a light naphtha approximating the
mixture with a single pseudocomponent is more realistic and
more accurate than a wide boiling range fraction such as an
atmospheric residuum or the C7+ fraction in a crude oil sam-
ple. As discussed in Chapter 2, the differences between prop-
erties of various hydrocarbon families increase with boiling
point (or carbon number). Therefore, assumption of a single
pseudocomponent for a heavy fraction (M > 300) is less ac-
curate than for the case of light fractions. For fractions that
are rich in one hydrocarbon type such as coal liquids that
may have up to 90% aromatics, it would be appropriate to di-
vide the aromatics into further subgroups of monoaromatics
(MA) and polyaromatics (PA). Therefore, creation of a matrix
of pseudocomponents, such as Table 3.15, largely depends on
the nature and characteristics of the petroleum mixture as
well as availability of experimental data.
3. 3. 2 Narrow Versus Wi de Boi l i ng Rang e Fract i ons
I n general, regardless of molecular type, petroleum fractions
may be divided into two major categories: narrow and wide
boiling range fractions. A narrow boiling range fraction was
defined in Section 3.2.1 as a fraction whose ASTM 10-90%
distillation curve slope (SL) is less than 0.8~ although
this definition is arbitrary and may vary from one source to
another. Fractions with higher 10-90% slopes may be consid-
ered as wide boiling range. However, for simplicity the meth-
ods presented in this section for narrow fractions may also
be applied to wider fractions. For narrow fractions, only one
carbon number is considered and the whole fraction may be
characterized by a single value of boiling point or molecu-
lar weight. For such fractions, if molecular type is known
(PNA composition), then the number of pseudocomponents
in Table 3.15 reduces to three and if the composition is not
known the whole mixture may be considered as a single pseu-
docomponent. For this single pseudocomponent, properties
of a pure component whose characteristics, such as boiling
point and specific gravity, are the same as that of the fraction
can be considered as the mixture properties. For mixtures the
best characterizing boiling point is the mean average boiling
point (MeABP); however, as mentioned in Section 3.2.1, for
narrow fractions the boiling point at 50 vol% distilled may
be considered as the characteristic boiling point instead of
MeABP. The specific gravity of a fraction is considered as the
second characteristic parameter for a fraction represented by
a single pseudocomponent. Therefore, the whole mixture may
be characterized by its boiling point (Tb) and specific gravity
(SG). I n lieu of these properties other characterization pa-
rameters discussed in Chapter 2 may be used.
Treatment of wide boiling range fractions is more compli-
cated than narrow fractions as a single value for the boiling
point, or molecular weight, or carbon number cannot rep-
resent the whole mixture. For these fractions the number of
constituents in the vertical columns of Table 3.15 cannot be
reduced to one, although it is still possible to combine var-
ious molecular types for each carbon number. This means
that the mi ni mum number of constituents in Table 3.14 for
a wide fraction is six rather than one that was considered
for narrow fractions. The best example of a wide boiling
range fraction is C7+ fraction in a crude oil or a reservoir
fluid. Characterization of such fractions through the use of a
di st ri but i on model t hat reduces t he mi xt ure into a number of
p seudocomp onent s wi t h known charact eri zat i on paramet ers
will be di scussed in detail in Chapt er 4. However, a si mpl er
ap p r oach based on the use of TBP curve is outlined in Ref.
[32]. I n this ap p r oach t he mi xt ure propert y is calculated from
the following relation:
1
(3.39) 0 = [ O(x) dx
1/
0
in whi ch 0 is t he physical propert y of mi xt ure and O(x) is t he
value of propert y at poi nt x on t he distillation curve. This
ap p r oach may be applied to any physical property. The in-
t egrat i on shoul d be carri ed out by a numeri cal met hod. The
fract i on is first divided into a number of p seudocomp onent s
al ong t he entire range of distillation curve wi t h known boiling
points and specific gravity. Then for each comp onent physi-
cal propert i es are calculated f r om met hods of Chapt er 2 and
finally t he mi xt ure propert i es are calculated t hr ough a sim-
ple mixing rule. The procedure is outlined in t he following
example.
Example 3. 8- - F or a low boiling napht ha, TBP curve is pro-
vided al ong wi t h the density at 20~ as t abul at ed bel ow [32].
Est i mat e specific gravity and mol ecul ar weight of this fract i on
usi ng the wide boiling range approach. Compare the calcu-
lated results wi t h the experi ment al values report ed by Leni or
and Hi pki n and ot hers [1, 11, 32] as SG = 0.74 and M = 120.
vo1% 0 (IBP) 5 10 20 30 50 70 90 95
TBP, K 283.2 324.8 348.7 369.3 380.9 410.4 436.5 467.6 478.7
d20, g/ cm 3 . . . 0.654 0.689 0.719 0.739 0.765 0.775 0.775 0.785
Solution--For this fract i on the 10-90% slope based on TPB
curve is about 1.49~ This value is slightly above the slope
based on t he ASTM D 86 curve but still indicates how wi de
t he fraction is. For this sampl e based on the ASTM distilla-
t i on dat a [1], t he 10-90% slope is 1.35~ whi ch is above
the value of 0.8 specified for nar r ow fractions. To use the
met hod by Ri azi -Daubert [32] for this relatively wide frac-
tion, first di st ri but i on funct i ons for bot h boiling poi nt and
specific gravity shoul d be det ermi ned. We use Eqs. ( 3. 35) -
(3.38) to det ermi ne t he di st ri but i on funct i ons for bot h prop-
erties. The mol ecul ar weight, M, is est i mat ed for all points on
the curve t hr ough appropri at e relations in Chapt er 2 devel-
oped for pure hydrocarbons. The value of M for the mi xt ure
t hen may be est i mat ed from a simple i nt egrat i on over the en-
tire range of x as given by Eq. (3.39): May = fd M(x)dx, where
M(x) is t he value of M at poi nt x det ermi ned f r om Tb(X) and
SG(x). May is the average mol ecul ar wei ght of t he mixture.
For this fract i on values of densities given al ong the distilla-
t i on curve are at 20~ and shoul d be convert ed to specific
gravity at 15.5~ (60~ t hr ough use of Eq. (2.112) in Chap-
ter 2: SG = 0.9915d20 + 0.01044. Paramet ers of Eq. (3.35) for
bot h t emperat ure and specific gravity have been det ermi ned
and are given as following.
Parameters in Eq. (3.35) To, K SGo A B RS
TBP curve 240 1.41285 3.9927 0.996
SG curve 0.5 0.07161 7.1957 0.911
The values of To and SGo det ermi ned f r om regression of
dat a t hr ough Eq. (3.35) do not mat ch well wi t h the exper-
i ment al initial values. This is due to the maxi mi zi ng value
of RS wi t h dat a used in the regressi on analysis. Actually
one can i magi ne t hat t he actual initial values are l ower t han
experimentally measur ed values due to the difficulty in such
measurement s. However, these initial values do not affect
subsequent calculations. Predi ct ed values at all ot her poi nt s
f r om 5 up to 95% are consi st ent wi t h t he experi ment al values.
Fr om calculated values of SGo, A, and B for t he SG curve,
one can det ermi ne the mixture SG for t he whol e fract i on
t hr ough use of Eqs. (3.37) and (3.38). For SG, B = 7.1957 and
f r om Eq. (3.38), F ( I + 1 / B) = 0.9355. Fr om Eq. (3.37) we
get SG~v = ( ~) 1/ 71957F ( 1 + 7.1~57) ---- 0.5269 x 0.9355 =
0.493. Therefore, for t he mixture: SGav -- 0.5(1 + 0. 493) =
0.746. Compari ng wi t h experi ment al value of 0.74, the
percent relative deviation (%D) wi t h experi ment al value is
0.8%. I n Chapt er 4 anot her met hod based on a di st ri but i on
funct i on is i nt roduced t hat gives slightly bet t er predi ct i on
for t he density of wi de boiling range fract i ons and crude
oils.
To calculate a mi xt ure propert y such as mol ecul ar weight,
the mi xt ure is divided to some nar r ow pseudocomponent s,
Np. I f the mi xt ure is not very wi de such as in this exam-
ple, even Nv = 5 is sufficient, but for wi der fract i ons the mix-
t ure may be divided to even larger number of p seudocomp o-
nent s (10, 20, etc.). I f Nv = 5, t hen values of T and SG at x =
0, 0.2, 0.4, 0.6, 0.8, and 0.99 are evaluated t hr ough Eq. (3.35)
and paramet ers det ermi ned above. Value of x = 0.99 is used
instead of x = 1 for the end poi nt as Eq. (3.35) is not defined
at x = 1. At every point, mol ecul ar weight, M, is det ermi ned
from met hods of Chapt er 2. I n this example, Eq. (2.50) is quite
accurat e and may be used to calculate M since all compo-
nent s in the mi xt ure have M < 300 ( ~Nc < 20) and are wi t hi n
the range of appl i cat i on of this met hod. Equat i on (2.50) is
M = 1.6604 x 10-4Tb 2"1962 SG -1-~ Calculations are summa-
ri zed in t he following table.
x Tb, K SG M
0 240.0 0.500 56.7
0.2 367.1 0.718 99.8
0.4 396.4 0.744 114.0
0.6 421.0 0.764 126.7
0.8 448.4 0.785 141.6
0.99 511.2 0.828 178.7
The t rapezoi dal rule for i nt egrat i on is quite accurat e to
estimate the mol ecul ar weight of the mixture. May -- ( 1/ 5) x
[ ( 56. 7+ 178. 7) / 2+( 99. 8 + 114 + 126. 7+ 141. 6+ 178.7)] =
119.96 - 120. This is exactly t he same as the experi ment al
value of mol ecul ar wei ght for this fract i on [ 1, 11].
I f t he whol e mi xt ure is consi dered as a single pseudocom-
ponent , Eq. (2.50) shoul d be applied directly t o t he mi xt ure
usi ng the MeABP and SG of the mixture. For this fract i on
the Wat son K is given as 12.1 [1]. Fr om Eq. (3.13) usi ng
experi ment al value of SG, average boiling poi nt is calcu-
lated as Tb ---- (12.1 x 0.74)3/ 1.8 ---- 398.8 K. Fr om Eq. (2.50),
the mi xt ure mol ecul ar wei ght is 116.1, whi ch is equivalent to
%D = -3. 25%. For this sample, the difference bet ween 1 and
5 p seudocomp onent s is not significant, but for wi der fract i on
the i mp r ovement of the proposed met hod is much larger.
Example 3. 9--I t is assumed for the same fraction of Ex-
ample 3.8, the only information available is ASTM D 86 data:
temperatures of 350.9, 380.9, 399.8, 428.2, and 457.6 K at 10,
30, 50, 70 and 90 vol% distilled, respectively. State how can
you apply the proposed approach for wide boiling range frac-
tions to calculate the molecular weight of the fraction.
Solution- - Since distillation data are in terms of ASTM
D 86, the first step is to convert ASTM to TBP through
Eq. (3.14). The second step is to determine the TBP distri-
bution function through Eqs. (3.35) and (3.36). The third
step is to generate values of T at x -- 0, 0.2, 0.4, 0.6, 0.8, and
0.99 from Eq. (3.35) and parameters determined from TBP
distillation curve. Since the specific gravity for this frac-
tion is not known it may be estimated from Eq. (3.17) and
constants in Table 3.4 for the ASTM D 86 data as follows:
SG = 0.08342 (350.9)~176176 = 0.756. The Wat-
son K is calculated as Kw = (1.8 x 399.8)1/3/0.756 = 11.85.
Now we assume that Kw is constant for the entire range of
distillation curve and on this basis distribution of SG can be
calculated through distribution of true boiling point. At every
point that T is determined from Eq. (3.35) the specific grav-
ity can be calculated as SG : (1.8 x T)I/a/Kw, where T is the
temperature on the TBP curve. Once TBP temperatures and
SG are determined at x = 0, 0.2, 0.4, 0.6, 0.8, and 0.99 points,
molecular weight may be estimated from Eq. (2.50). Numer-
ical integration of Eq. (3.39) can be carried out similar to the
calculations made in Example 3.8 to estimate the molecular
weight. I n this approach the result may be less accurate than
the result in Example 3.8, as ASTM distillation curve is used
as the only available data. #
Although the method outlined in this section improves
the accuracy of prediction of properties of wide boiling
range fraction, generally for simplicity in calculations most
petroleum products are characterized by a single value of boil-
ing point, molecular weight, or carbon number regardless of
their boiling range. The proposed method is mainly applied
to crude oils and C7+ fraction of reservoir fluids with an ap-
propriate splitting technique as is shown in the next chapter.
However, as shown in the above example, for very wide boiling
range petroleum products the method presented in this sec-
tion may significantly improve the accuracy of the estimated
physical properties.
3. 3. 3 Use of Bul k Parameters
( Undefined Mixtures)
An undefined petroleum fraction is a fraction whose com-
position (i.e., PNA) is not known. For such fractions infor-
mation on distillation data (boiling point), specific gravity,
or other bulk properties such as viscosity, refractive index,
CH ratio, or molecular weight are needed. I f the fraction is
considered narrow boiling range then it is assumed as a sin-
gle component and correlations suggested in Chapter 2 for
pure hydrocarbons may be applied directly to such fractions.
All limitations for the methods suggested in Chapter 2 should
be considered when they are used for petroleum fractions.
As mentioned in Chapter 2, the correlations in terms of Tb
and SG are the most accurate methods for the estimation of
various properties (molecular weight, critical constants, etc.).
For narrow boiling range fractions, ASTM D 86 temperature
at 50 vol% vaporized may be used as the characterizing boil-
ing point for the whole mixture. However, for a wide boiling
range fraction if it is treated as a single pesudocomponent
the MeABP should be calculated and used as the character-
izing parameter for Tb in the correlations of Chapter 2. I f
for a fraction TBP distillation data are available the average
boiling point calculated through Eq. (3.37) with parameters
determined from TBP curve would be more appropriate than
MeABP determined from ASTM D 86 curve for use as the
characterizing boiling point. For cases where only two points
on the distillation curve are known the interpolated value at
50% point may be used as the characterizing boiling point of
the fraction.
For heavy fractions (M > 300) in which atmospheric dis-
tillation data (ASTM D 86, SD, or TBP) are not available, if
ASTM D 1160 distillation curve is available, it should be con-
verted to ASTM D 86 or TBP through methods outlined in
Section 3.2. I n lieu of any distillation data, molecular weight
or viscosity may be used together with specific gravity to esti-
mate basic parameters from correlations proposed in Chap-
ter 2. I f specific gravity is not available, refractive index or
carbon-to-hydrogen weight ratio (CH) may be used as the
second characterization parameter.
3. 3. 4 Method of Pseudocomponent
( Defined Mixtures)
A defined mixture is a mixture whose composition is known.
For a petroleum fraction if at least the PNA composition
is known it is called a defined fraction. Huang [11,33] used
the pseudocompounds approach to estimate enthalpies of
narrow and defined petroleum fractions. This technique has
been also used to calculate other physical properties by
other researchers [34, 35]. According to this method all com-
pounds within each family are grouped together as a sin-
gle pseudocomponent. An olefin-free fraction is modeled
into three pseudocomponents from three homologous groups
of n-alkanes (representing paraffins), n-alkylcyclopentanes
or n-alkylcyclohexanes (representing naphthenes), and n-
alkylbenzenes (representing aromatics) having the same boil-
ing point as that of ASTM D 86 temperature at 50% point.
Physical properties of a mixture can be calculated from prop-
erties of the model components by the following mixing rule:
(3.40) 0 : XpOp @ XNON + XAOA
where 0 is a physical property for the mixture and 0p, ON, and
0A are the values of 0 for the model pseudocomponents from
the three groups. I n this equation the composition presented
by xA, XN, and XAshould be in mole fraction, but because the
molecular weights of different hydrocarbon groups having
the same boiling point are close to each other, the compo-
sition in weight or even volume fractions may also be used
with mi nor difference in the results. I f the fraction contains
olefinic compounds a fourth term for contribution of this
group should be added to Eq. (3.40). Accuracy of Eq. (3.40)
can be increased if composition of paraffinic group is known
in terms of n-paraffins and isoparaffins. Then another pseudo-
component contributing the isoparaffinic hydrocarbons may
be added to the equation. Similarly, the aromatic part may be
split into monoaromatics and polyaromatics provided their
3. CHARACTERI ZATI ON OF PETROLEUM FRACTIONS 115
amount in the fraction is known. However, based on our expe-
rience the PNA t hree-pseudocomponent model is sufficiently
accurate for olefin-free pet rol eum fractions. For coal liquids
with a high percentage of aromat i c content, splitting aromat -
ics into two subgroups may greatly increase the accuracy of
model predictions. I n using this met hod the mi ni mum data
needed are at least one characterizing par amet er (Tb or M)
and the PNA composition.
Properties of pseudocomponent s may be obtained from in-
terpolation of values in Tables 2.1 and 2.2 to mat ch boiling
point to that of the mixture. As shown in Section 2.3.3, prop-
erties of homologous groups can be well correlated to only
one characterization paramet er such as boiling point, molec-
ular weight, or carbon number, depending on the availability
of the paramet er for the mixture. Since various properties of
pure homol ogous hydrocarbon groups are given in t erms of
molecular weight by Eq. (2.42) with constants in Table 2.6,
if molecular weight of a fraction is known it can be used di-
rectly as the characterizing parameter. But if the boiling point
is used as the characterizing parameter, molecular weights of
the three model component s may be estimated t hrough rear-
rangement of Eq. (2.42) in t erms of boiling point as following:
(3.41) Mp = [ 1_____2___[6.98291 _ ln(1070 - Tb)]/3/2/
| 0.02013
(3.42) MN = { ~[ 6. 95649 -- ln(1028 - Tb)]} 3/2
(3.43) MA = / 1- - - - ~[ 6. 91062 -- ln(1015 -- Tb)]} 3/2
| 0.02247
where Mp, MN, and MA are molecular weights of paraffinic,
naphthenic, and aromat i c groups, respectively. Tb is the char-
acteristic boiling point of the fraction. Predicted values of Mp,
MN, and MA versus Tb were presented in Fig. 2.15 in Chapter 2.
As shown in this figure the difference between these molecu-
lar weights increase as boiling point increases. Therefore, the
pseudocomponent approach is more effective for heavy frac-
tions. I f ASTM D 86 distillation curve is known the t empera-
ture at 50% point should be used for Tb, but if TBP distillation
data are available an average TBP would be more suitable to
be used for Tb. Once Mr, MN, and MA are determined, they
should be used in Eq. (2.42) to determine properties from cor-
responding group to calculate other properties. The met hod
is demonst rat ed in the following example.
Example 3. 10---A pet rol eum fraction has ASTM D 86 50%
t emperat ure of 327.6 K, specific gravity of 0.658, molecular
weight of 78, and PNA composi t i on of 82, 15.5, and 2.5 in
vol% [36]. Est i mat e molecular weight of this fraction using
bulk properties of Tb and SG and compare with the value esti-
mat ed from the pseudocomponent method. Also estimate the
mixture specific gravity of the mixture t hrough the pseudo-
component technique and compare the result with the exper-
imental value.
Solution- - For this fraction the characterizing paramet ers
are Tb ---- 327.6 K and SG = 0.672. To estimate M from these
bulk properties, Eq. (2.50) can be applied since the boiling
point of the fraction is within the range of 40-360~ ( ~C5-
C22). The results of calculation is M = 85.0, with relative
deviation of 9%. From the pseudocomponent approach, Mp,
MN, and MA are calculated from Eqs. (3.41)-(3.43) as 79.8,
76.9, and 68.9, respectively. The mi xt ure molecular weight is
calculated t hrough Eq. (3.40) as M = 0.82 x 79.8 + 0.155 x
76.9 + 0.025 68.9 = 79, with relative deviation of 1.3%. I f
values of Alp, MN, and MA are substituted in Eq. (2.42) for
the specific gravity, we get SGp = 0.651, SGN = 0.749, and
SGA ----- 0.895. From Eq. (3.40) the mixture specific gravity is
SG = 0.673, with AD of 2.3%. It should be noted t hat when
Eq. (3.40) is applied to molecular weight, it would be more
appropri at e to use composition in t erms of mole fraction
rat her t han volume fraction. The composi t i on can be con-
verted to weight fraction t hrough specific gravity of the three
component s and then to mole fraction t hrough molecular
weight of the component s by equations given in Section
1.7.15. The mole fractions are Xmp = 0.785, XmN = 0.177, and
Xmn=0.038 and Eq. (3.39) yields M = 78.8 for the mixture
with deviation of 1%. The difference between the use of
volume fraction and mole fraction in Eq. (3.40) is mi nor and
within the range of experimental uncertainty. Therefore, use
of any form of composi t i on in t erms of volume, weight, or
mole fraction in the pseudocomponent met hod is reasonable
without significant effect in the results. For this reason, in
most cases the PNA composi t i on of pet rol eum fractions are
simply expressed as fraction or percentage and they may
considered as weight, mole, or volume.
I n the above example the met hod of pseudocomponent pre-
dicts molecular weight of the fraction with much bet t er ac-
curacy t han the use of Eq. (2.50) with bulk properties (%AD
of 1.3% versus 9%). This is the case for fractions that are
highly rich in one of the hydrocarbon types. For this frac-
tion paraffinic content is nearly 80%, but for pet rol eum frac-
tions with normal distribution of paraffins, naphthenes, and
aromat i cs bot h met hods give nearly similar results and the
advantage of use of three pseudocomponent s from different
groups over the use of single pseudocomponent with mixture
bulk properties is minimal. For example, for a pet rol eum frac-
tion the available experimental data are [36] M = 170, Tb =
487 K, SG = 0.802, xp --- 0.42, XN = 0.41, andxn = 0.17. Equa-
tion (2.50) gives M = 166, while Eq. (3.40) gives M = 163 and
SG = 0.792. Equation (3.40) is particularly useful when only
one bulk property (i.e., Tb) with the composi t i on of a frac-
tion is available. For highly aromat i c (coal liquids) or highly
paraffinic mixtures the met hod of pseudocomponent is rec-
ommended over the use of bulk properties.
3. 3. 5 Es t i mat i on of Mol ecul ar Wei ght, Critical
Propert i es, and Acent ri c Fact or
Most physical properties of pet rol eum fluids are calculated
t hrough corresponding state correlations that require pseu-
docritical properties (Tpc, Pr~, and Vpc) and acentric factor
as a third parameter. I n addition molecular weight (M) is
needed to convert calculated mole-based propert y to mass-
based property. As ment i oned in Section 1.3, the accuracy
of these properties significantly affects the accuracy of es-
t i mat ed properties. Generally for pet rol eum fractions these
basic characterization paramet ers are calculated t hrough ei-
ther the use of bulk properties and correlations of Chapter 2
TABLE 3.16---Comparison of various methods of predicting pseudocritical properties and
acentric factor through enthalpy calculation of eight petroleum fractions [37].
Method of estimating input AAD, kJ/kg
P arameters*~ Liquid Vapor
Item To, Pc to (437 data points) (273 data points)
1 Pseudocomp. Pseudocomp. 5.3 7.9
2 RD (80) LK 5.9 7.7
3 KL LK 5.8 7.4
4 Winn LK 9.9 12.8
~* Pseudocomp.: The pseudocomponent method by Eqs. (3.40)-(3.43) and (2.42) for T~, P~ and co; RD: Riazi-
Daubert [38] by Eqs. (2.63) and (2.64); LK: Lee,-Kesler [39] by Eq. (2.103); KL: Kester-Lee [40] by Eqs. (2.69)
and (2.70); Winn method [41] by Eqs. (2.94) and (2.95),
or by the pseudocomponent approach as discussed in Sec-
tions 3.3.2-3.3.4.
For pet rol eum fractions, pseudocritical properties are not
directly measurabl e and therefore it is not possible to make
a direct evaluation of different met hods with experimental
data. However, these met hods can be evaluated indirectly
t hrough prediction of other measurabl e properties (i.e., en-
thalpy) t hrough corresponding state correlations. These cor-
relations are discussed in Chapters 6-8. Based on more t han
700 data points for enthalpies of eight pet rol eum fractions
over a wide range of t emperat ure and pressure [1], different
met hods of estimation of pseudocritical temperature, pres-
sure (Tpo Pp~), and acentric factor (w) have been evaluated
and compared [37]. These pet rol eum fractions ranging from
napht ha to gas oil all have molecular weights of less t han 250.
Details of these enthalpy calculations are given in Chapter 7.
Summar y of evaluation of different met hods is given in Ta-
ble 3.16. As shown in Table 3.16, the met hods of pseudocom-
ponent, Lee-Kesler, and Ri azi -Daubert have nearly similar
accuracy for estimating the critical properties of these light
pet rol eum fractions. However, for heavier fractions as it is
shown in Example 3.11, the met hods of pseudocomponent
provide more accurate results.
Example 3. 1 l kExp er i ment al data on molecular weight and
composi t i on of five heavy pet rol eum fractions are given in
Table 3.17, I n addition, normal boiling point, specific grav-
ity, density, and refractive index at 20~ are also given [36].
Calculate the molecular weight of these fractions from the fol-
lowing five methods: (1) API met hod [2, 42] using Eq. (2.51),
(2) Twu met hod [42] using Eqs. (2.89)-(2.92), (3) Goossens
met hod [43] using Eq. (2.55), (4) Lee-Kesler met hod [40] us-
ing Eq. (2.54), and (5) the pseudocomponent met hod using
Eqs. (3.40)-(3.43). Calculate the %AAD for each method.
Solution- - M ethods 1, 2, and 4 require bulk properties of Tb
and SG, while the met hod of pseudocomponent requires Tb
and the PNA composition as it is shown in Exampl e 3.10.
Method 2 requires Tb and density at 20~ (d20). Results of
calculations are given in Table 3.18.
The Twu met hod gives the highest error (AAD of 14.3%) fol-
lowed by the Goossens with average deviation of 11.4%. The
Twu and Goossens met hods both underest i mat e the molecu-
lar weight of these heavy fractions. The Lee-Kesler met hod is
more accurate for lighter fractions, while the API met hod is
more accurate for heavier fractions. The pseudocomponent
met hod gives generally a consistent error for all fractions and
the lowest AAD%. Errors generated by the API, Lee-Kesler,
and the pseudocomponent met hods are within the experi-
ment al uncertainty in the measurement of molecular weight
of pet rol eum fractions. r
I n summary, for light fractions (M < 300) met hods recom-
mended by the API for Tr and PC (Eqs. 2.65 and 2.66) [2] or the
simple met hod of Ri azi -Daubert (Eqs. 2.63 and 2.64) [38] are
suitable, while for heavier fractions the Lee-Kesler met hod
(Eqs. 2.69 and 2.70) [40] may be used. The pseudocompo-
nent met hod may also be used for bot h Tc and Pc when the
composition is available. For all fractions met hods of calcu-
lation of acentric factor from the pseudocomponent or the
met hod of Lee-Kesler [39] presented by Eq. (2.105) may be
used. Molecular weight can be estimated from the API met hod
[2] by Eq. (2.51) from the bulk properties; however, if the PNA
composition is available the met hod of pseudocomponent is
preferable especially for heavier fractions.
3. 3. 6 Es t i ma t i on of Densi t y, Sp eci f i c Gr avi t y,
Ref ract i v e I ndex, and Ki nemat i c Vi scosi t y
Density (d), specific gravity (SG), and refractive index (n) are
all bulk properties directly measurabl e for a pet rol eum mix-
ture with relatively high accuracy. Kinematic viscosity at 37.8
or 98.9~ (1)38(100), 1,'99(210)) are usually reported for heavy frac-
tions for which distillation data are not available. But, for
light fractions if kinematic viscosity is not available it should
be estimated t hrough measurabl e properties. Methods of es-
timation of viscosity are discussed in Chapter 8; however, in
this chapt er kinematic viscosity at a reference of t emperat ure
of 37.8 or 98.9~ (100~ or 210~ is needed for estimation
of viscosity gravity constant (VGC), a paramet er required for
prediction of composi t i on of pet rol eum fractions. Generally,
TABLE 3.17--Molecular weight and composition of five heavy petroleum fractions of Example 3.11 [36].
No. M Tb,~ SG d20, g/ml n20 P% N% A%
1 233 298.7 0.9119 0.9082 1.5016 34.1 45.9 20.0
2 267 344.7 0.9605 0.9568 1.5366 30.9 37.0 32.1
3 325 380.7 0.8883 0.8845 1.4919 58.4 28.9 12.7
4 403 425.7 0.9046 0.9001 1.5002 59.0 28.0 13.0
5 523 502.8 0.8760 0.8750 1.4865 78.4 13.3 8.3
TABLE 3.18---Comparison of various methods of predicting molecular weight of petroleum fractions of Table 3. 17 (Example 3.12).
(1) API (2) Twu, (3) Goossens, (4) Lee-Kesler, (5) Pseudocomp.,
Eq. (2.51) Eqs. (2.89)-(2.92) Eq. (2.55) Eq. (2.54) Eqs. (3.40)-(3.43)
No. M, exp, M, calc AD% M, catc AD% M, calc AD% M, calc AD% M, calc AD%
1 233 223.1 4.2 201.3 13.6 204.6 12.2 231.7 0.5 229.1 1.7
2 267 255.9 4.2 224.0 16.1 235.0 12.0 266.7 0.1 273.2 2.3
3 325 320.6 1.4 253.6 16.8 271.3 11.0 304.3 0.2 321.9 1,0
4 403 377.6 6.3 332.2 17.6 345.8 14.2 374.7 7.0 382.4 5.1
5 523 515.0 1.5 485.1 7.2 483.8 7.5 491,7 6.0 516.4 1.3
Total, AAD% 3.5 14.3 11.4 2.8 2.3
density, which is required in various predictive methods mea-
sured at 20~ is shown by d or d20 in g/mL. These properties
can be directly estimated through bulk properties of mixtures
using the correlations provided in Chapter 2 with good accu-
racy so that there is no need to use the pseudocomponent
approach for their estimation.
Specific gravity (SG) of petroleum fractions may be es-
timated from methods presented in Sections 2.4.3 and 2.6.
I f API gravity is known, the specific gravity should be di-
rectly calculated from definition of API gravity using Eq. (2.5).
I f density at one temperature is available, then Eq. (2.110)
should be used to estimate the increase in density with de-
crease in temperature and therefore density at 15.5~ (60~
may be calculated from the available density. A simpler re-
lation between SG and da0 based on the rule of t humb is
SG = 1.005d20. If density at 20~ (d20) is available, the fol-
lowing relation developed in Section 2.6.1 can be used:
(3.44) SG -- 0.01044 + 0.9915d20
where d20 is in g/mL and SG is the specific gravity at 15.5~
This equation is quite accurate for estimating density of
petroleum fractions. If no density data are available, then
SG may be estimated from normal boiling point and refrac-
tive index or from molecular weight and refractive index for
heavy fractions in which boiling point may not be available.
Equation (2.59) in Section 2.4.3 gives SG from Tb and I for
fractions with molecular weights of less than 300, while for
heavier fractions Eq. (2.60) can be used to estimate specific
gravity from M and I. Parameter I is defined in terms of re-
fractive index at 20~ n20, by Eq. (2.36). If viscosity data are
available Eq. (2.61) should be used to estimate specific grav-
ity, and finally, if only one type of distillation curves such as
ASTM D 86, TBP, or EFV data are available Eq. (3.17) may be
used to obtain the specific gravity.
Density (d) of liquid petroleum fractions at any temper-
ature and atmospheric pressure may be estimated from the
methods discussed in Section 2.6. Details of estimation of
density of petroleum fractions are discussed in Chapter 6;
however, for the characterization methods discussed in this
chapter, at least density at 20~ is needed. I f specific gravity
is available then the rule of t humb with d = 0.995SG is the
simplest way of estimating density at 20~ For temperatures
other than 20~ Eq. (2.110) can be used. Equation (2.113)
may also be used to estimate d20 from Tb and SG for light
petroleum fractions (M < 300) provided that estimated den-
sity is less than the value of SG used in the equation. This
is an accurate way of estimating density at 20~ especially
for light fractions. However, the simplest and most accurate
method of estimating d20 from SG for all types of petroleum
fractions is the reverse form of Eq. (3.44), which is equivalent
to Eq. (2.111). I f the specific gravity is not available, then it is
necessary to estimate the SG at first step and then to estimate
the density at 20 ~ C. The liquid density decreases with increase
in temperature according to the following relationship [24].
d = dlss - k( T - 288.7)
where d15.5 is density at 15.5~ which may be replaced by
0.999SG. T is absolute temperature in kelvin and k is a con-
stant for a specific compound. This equation is accurate
within a narrow range of temperature and it may be applied
to any other reference temperature instead of 15.5~ Value
of k varies with hydrocarbon type; however, for gasolines it is
close to 0.00085 [24].
Refract i ve i ndex at 20~ n20, is an important character-
ization parameter for petroleum fractions. It is needed for
prediction of the composition as well as estimation of other
properties of petroleum fractions. I f it is not available, it may
be determined from correlations presented in Section 2.6 by
calculation of parameter I. Once I is estimated, n20 can be cal-
culated from Eq. (2.114). For petroleum fractions with molec-
ular weights of less than 300, Eq. (2.115) can be used to esti-
mate I from Tb and SG [38]. A more accurate relation, which
is also included in the API-TDB [2], is given by Eq. (2.116).
For heavier fraction in which Tb may not be available, Eq.
(2.117) in terms of M and SG may be used [44]. Most re-
cently Riazi and Roomi [45] made an extensive analysis of
predictive methods and application of refractive index in pre-
diction of other physical properties of hydrocarbon systems.
An evaluation of these methods for some petroleum fractions
is demonstrated in the following example.
Exampl e 3 . 1 2- - Experi ment al data on M , Tb, SG, d2o, and n20
for five heavy petroleum fractions are given in Table 3.17, Es-
timate SG, dzo, and n2o from available methods and calculate
%AAD for each method with necessary discussion of results.
Sol ut i on- - The first two fractions in Table 3.17 may be consid-
ered light (M < 300) and the last two fractions are considered
as heavy (M > 300). The third fraction can be in either cat-
egory. Data available on d20, Tb, n20, and M may be used to
estimate specific gravity. As discussed in this section, SG can
be calculated from d20 or from Tb and / or from M and I.
I n this example specific gravity may be calculated from four
methods: (1) rule of t humb using d20 as the input parameter;
(2) from d20by Eq. (3.44); (3) from Tb and n20 by Eq. (2.59); (4)
from M and n20 by Eq. (2.60). Summary of results is given in
Table 3.19. Methods 1 and 2, which use density as the input
parameter, give the best results. Method 4 is basically devel-
oped for heavy fractions with M > 300 and therefore for the
last three fractions density is predicted with better accuracy.
1 18 CHARACTERI Z ATI ON AND PROPERTI ES OF PETROL EUM FRACTI ONS
TABLE 3.19--Comparison of various methods of predicting specific gravity of petroleum fractions of Table 3.17 (Example 3.12).
(1) Rule of thumb, (2) Use of d20, (3) Use of Tb & n20, (4) Use of M & n20,
SG = 1.005 d20 Eq. (3.44) Eq. (2.59) for M < 300 Eq. (2.60) for M >300
No. M, exp SG, exp SG, calc AD% SG, calc AD% SG, calc AD% SG, calc AD%
1 233 0.9119 0.9127 0.09 0.9109 0.11 0.8838 3.09 0.8821 3.27
2 267 0.9605 0.9616 0.11 0.9591 0.15 0.9178 4.44 0.9164 4.59
3 325 0.8883 0.8889 0.07 0.8874 0.10 0.8865 0.20 0.8727 1.76
4 403 0.9046 0.9046 0.00 0.9029 0.19 0.9067 0.23 0.8867 1.98
5 523 0.8760 0.8794 0.39 0.8780 0.23 0.9062 3.45 0.8701 0.67
Total, AAD% 0.13 0.15 2.28 2.45
Met hod 3, whi ch is r ecommended for l i ght fract i ons, gives
bet t er resul t s for t he specific gravi t y of heavy fract i ons. I t
shoul d he not ed t hat t he boundar y of 300 for l i ght and heavy
f r act i ons is ap p r oxi mat e and met hods p r op os ed for l i ght frac-
t i ons can be used well above t hi s boundar y l i mi t as shown i n
Met hod 3.
Est i mat i on of densi t y is si mi l ar to est i mat i on of specific
gravity. When bot h Tb and SG are avai l abl e Eq. (2.113) is t he
most accur at e met hod for est i mat i on of densi t y of p et r ol eum
fract i ons. Thi s met hod gives AAD of 0.09% for t he five frac-
t i ons of Table 3.17 wi t h hi gher er r or s for t he l ast t wo fract i ons.
This equat i on may be used safel y up to mol ecul ar wei ght of
500 but for heavi er fract i ons Eq. (3.44) or t he rul e of t humb
shoul d be used. Pr edi ct ed val ue of densi t y at 20~ f r om Eq.
(2.113) is not rel i abl e if it is gr eat er t han t he val ue of speci fi c
gravi t y used i n t he equat i on. The met hod of rul e of t humb
wi t h d = 0.995 SG gives an AAD of 0.13% and Eq. (3.44) gives
an AAD of 0.15%.
Refract i ve i ndex is est i mat ed f r om t hree di fferent met hods
and resul t s are gi ven in Table 3.20. I n t he first met hod, Tb
and SG ar e used as t he i np ut p ar amet er s wi t h Eq. (2.115) to
est i mat e I and n is cal cul at ed f r om Eq. (2.114). I n t he sec-
ond met hod Eq. (2.116) is used wi t h t he same i np ut dat a.
Equat i ons (2.115) and (2.116) ar e bot h devel oped wi t h dat a
on refract i ve i ndex of p ur e hydr ocar bons wi t h M < 300. How-
ever, Eq. (2.116) in t hi s range of ap p l i cat i on is mor e accur at e
t han Eq. (2.115). But for heavi er fract i ons as shown i n Table
3.20, Eq. (3.115) gives bet t er resul t . This is due to t he si mpl e
nat ur e of Eq. (2.115) whi ch al l ows its ap p l i cat i on to heavi er
fract i ons. Equat i on (2.116) does not give very accur at e refrac-
tive i ndex for f r act i on wi t h mol ecul ar wei ght s of 500 or above.
Equat i on (2.117) in t er ms of M and SG is devel oped basi cal l y
for heavy fract i ons and for t hi s r eason it does not give accu-
rat e resul t s for f r act i ons wi t h mol ecul ar wei ght s of less t han
300. This met hod is p ar t i cul ar l y useful when boi l i ng p oi nt is
not avai l abl e but mol ecul ar wei ght is avai l abl e or est i mabl e.
However, if boi l i ng p oi nt is avai l abl e, even for heavy f r act i ons
Eq. (2.115) gives mor e accur at e resul t s t han does Eq. (2.117)
as shown in Table 3.20. r
Ki ne ma t i c vi s cos i t y of p et r ol eum fract i ons can be esti-
mat ed f r om met hods p r esent ed i n Sect i on 2.7 of t he pr evi ous
chapt er. At r ef er ence t emp er at ur es of 37.8 and 98.9~ (100
and 210~ v3m00) and 1)99(210 ) can be det er mi ned f r om Eqs.
(2.128) and (2.129) or t hr ough Fig. 2.12 usi ng API gravi t y and
Kw as t he i np ut p ar amet er s. I n use of t hese equat i ons at t en-
t i on shoul d be p ai d to t he l i mi t at i ons and t o check if API and
Kw ar e wi t hi n t he r anges speci fi ed for t he met hod. To cal cu-
l at e ki nemat i c vi scosi t y at any ot her t emp er at ur e, Eq. (2.130)
or Fig. 2.13 may be used. The p r ocedur e is best demons t r at ed
t hr ough t he fol l owi ng exampl e.
Exampl e 3. 13- - A p et r ol eum f r act i on is p r oduced t hr ough
di st i l l at i on of a Venezuel an cr ude oil and has t he specific grav-
ity of 0.8309 and t he fol l owi ng ASTM D 86 di st i l l at i on dat a:
vol% distilled 10 30 50 70 90
ASTM D 86 temperature,~ 423 428 433 442 455
Est i mat e ki nemat i c vi scosi t y of t hi s fract i on at 100 and 140~
(37.8 and 60~ Comp ar e t he cal cul at ed val ues wi t h t he ex-
p er i ment al val ues of 1.66 and 1.23 cSt [46].
Sol ut i on- - Ki nemat i c vi scosi t i es at 100 and 210~ v3m00) and
v99(210), ar e cal cul at ed f r om Eqs. (2.128) and (2.129), respec-
tively. The API gravi t y is cal cul at ed f r om Eq. (2.4): API = 38,8.
To cal cul at e Kw f r om Eq. (2.13), MeABP is r equi r ed, For t hi s
f r act i on si nce it is a nar r ow boi l i ng r ange t he MeABP is near l y
t he same as t he mi d boi l i ng p oi nt or ASTM 50% t emp er at ur e.
However, si nce comp l et e ASTM D 86 curve is avai l abl e we use
Eqs. ( 3. 6) -( 3. 12) to est i mat e t hi s average boi l i ng poi nt . Cal-
cul at ed p ar amet er s ar e VABP = 435.6~ and SL = 0.4~
F r om Eqs. (3.8) and (3.12) we get MeABP = 434~ (223.3~
As expect ed this t emp er at ur e is very cl ose to ASTM 50%
t emp er at ur e of 433~ F r om Eq. (2.13), Kw = 11.59. Si nce
0 < API < 80 and 10 < Kw < 11, we can use Eqs. (2.128)
and (2.129) for cal cul at i on of ki nemat i c vi scosi t y and we
get 1238(100 ) = 1. 8, 1299(210 ) = 0.82 cSt. To cal cul at e vi scosi t y at
140 ~ v60(140), we use Eqs. ( 2, 130) -( 2. 132) . F r om Eq. (2.131)
TABLE 3.20--Comparison of various methods of predicting refractive index of petroleum fractions of Table 3.17
(Example 3.12).
(1) Use of Tb & SG, (2) Use of T b & SG, (4) Use of M & SG,
Eq. ( 2. 115) for M < 300 Eq. ( 2.116) for M < 300 Eq. ( 2. 117) f or M > 300
No. M, exp n20 exp n2o exp. AD% n20 cal c AD% n2o cal c AD%
1 233 1.5016 1.5122 0.70 1.5101 0.57 1.5179 1.08
2 267 1.5366 1.5411 0.29 1.5385 0.13 1.5595 1.49
3 325 1.4919 1.4960 0.28 1.4895 0.16 1.4970 0.34
4 403 1.5002 1.5050 0.32 1.4952 0.34 1.5063 0.41
5 523 1.4865 1.4864 0.01 1.4746 0.80 1.4846 0.13
Total, AAD% 0.32 0.40 0.69
s i nc e 1)38(100) > 1. 5 a nd 1)99(210 ) < 1. 5 c St we have C38(100 ) -~- 0
and c99(210)= 0.0392. From Eq. (2.132), A1 = 10.4611, B1 =
-4.3573, D1 = -0.4002, DE = --0.7397, and from Eq. (2.130)
at T -- 140~ (60~ we calculate the kinematic viscosity. It
should be noted that in calculation of v60(140) from Eq. (2.130)
trial and error is required for calculation of parameter c. At
first it is assumed that c = 0 and after calculation of 1)60(140) if
it is less than 1.5 cSt, parameter c should be calculated from
Eq. (2.131) and substituted in Eq. (2.130). Results of calcula-
tions are as follows: 1)38(100 ) = 1.8 and 1)60(140) = 1.27 cSt. Com-
paring with the experimental values, the percent relative de-
viations for kinematic viscosities at 100 and 140~ are 8.4 and
3.3%, respectively. The result is very good, but usually higher
errors are observed for estimation of kinematic viscosity of
petroleum fractions from this method.
3 . 4 GEN ER A L P R OC ED UR E FOR
P R OP ER T I ES OF M I X T UR ES
Petroleum fluids are mixtures of hydrocarbon compounds,
which in the reservoirs or during processing could he in the
form of liquid, gas, or vapor. Some heavy products such as
asphalts and waxes are in solid forms. But in petroleum proc-
essing most products are in the form of liquid under atmo-
spheric conditions. The same liquid products during proc-
essing might be in a vapor form before they are stored as a
product. Certain properties such as critical constants, acen-
tric factor, and molecular weight are specifications of a com-
pound regardless of being vapor of liquid. However, physical
properties such as density, transport, or thermal properties
depend on the state of the system and in many cases sepa-
rate methods are used to estimate properties of liquid and
gases as will be discussed in the following chapters. I n this
section a general approach toward calculation of such prop-
erties for liquids and gases with known compositions is pre-
sented. Since density and refractive index are important phys-
ical properties in characterization or petroleum fractions they
are used in this section to demonstrate our approach for mix-
ture properties. The same approach will be applied to other
properties throughout the book.
3. 4. 1 Li qui d Mi xt ur es
I n liquid systems the distance between molecules is much
smaller than in the case of gases and for this reason the inter-
action between molecules is stronger in liquids. Therefore, the
knowledge of types of molecules in the liquid mixtures is more
desirable than in gas mixtures, especially when the mixture
constituents differ significantly in size and type. For example,
consider two liquid mixtures, one a mixture of a paraffinic hy-
drocarbon such as n-eicosane (n-C20) with an aromatic com-
pound such as benzene (C6) and the second one a mixture of
benzene and toluene, which are both aromatic with close
molecular weight and size. The role of composition in the
n-C20-benzene mixture is much more important than the
role of composition in the benzene-toluene mixture. Simi-
larly the role of type of composition (weight, mole, or volume
fraction) is more effective in mixtures of dissimilar con-
stituents than mixtures of similar compounds. It is for this
reason that for narrow-range petroleum fractions, use of the
PNA composition in terms of weight, volume, or mole does
not seriously affect the predicted mixture properties. Use of
bulk properties such as Tb and SG to calculate mixture prop-
erties as described for petroleum fractions cannot be used for
a synthetic and ternary mixture of C5-C10-C25. Another exam-
ple of a mixture that bulk properties directly cannot be used to
calculate its properties is a crude oil or a reservoir fluid. For
such mixtures exact knowledge of composition is required
and based on an appropriate mixing rule a certain physical
property for the mixture may be estimated. The most simple
and practical mixing rule that is applicable to most physical
properties is as follows:
N
(3.45) 0m = E XiOi
i=1
where xi is the fraction of component i in the mixture, Oi is a
property for pure component i, and 0m is property of the mix-
ture with N component. This mixing rule is known as Kay
mixing rule after W. B. Kay of Ohio State, who studied mix-
ture properties, especially the pseudocritical properties in the
1930s and following several decades. Other forms of mixing
rules for critical constants will be discussed in Chapter 5 and
more accurate methods of calculation of mixture properties
are presented in Chapter 6.
Equation (3.45) can be applied to any property such as crit-
ical properties, molecular weight, density, refractive index,
heat capacity, etc. There are various modified version of Eq.
(3.45) when it is applied to different properties. Type of com-
position used forxi depends on the type of property. For exam-
ple, to calculate molecular weight of the mixture (0 = M) the
most appropriate type of composition is mole fraction. Sim-
ilarly mole fraction is used for many other properties such
as critical properties, acentric factor, and molar properties
(i.e., molar heat capacity). However, when Eq. (3.45) is ap-
plied to density, specific gravity, or refractive index parameter
[ I = (n 2 - 1)/(n 2 + 2)], volume fraction should be used for xi.
For these properties the following mixing rule may also be
applied instead of Eq. (3.45) if weight fraction is used:
N
(3.46) 1/0m = EXwi/Oi
i =l
where Xwi is the weight fraction and the equation can be ap-
plied to d, SG, or parameter I. I n calculation of these proper-
ties for a mixture, using Eq. (3.45) with volume fraction and
Eq. (3.46) with weight fraction gives similar results. Applica-
tion of these equations in calculation of mixture properties
will be demonstrated in the next chapter to calculate proper-
ties of crude oils and reservoir fluids.
For liquid mixtures the mixing rule should be applied to the
final desired property rather than to the input parameters. For
example, a property such as viscosity is calculated through
a generalized correlation that requires critical properties as
the input parameters. Equation (3.45) may be applied first
to calculate mixture pseudocritical properties and then mix-
ture viscosity is calculated from the generalized correlation.
An alternative approach is to calculate viscosity of individual
components in the mixture for the generalized correlation
and then the mixing rule is directly applied to viscosity. As
it is shown in the following chapters the second approach
gives more accurate results for properties of liquid mixtures,
while for gaseous mixtures there is no significant difference
between these two methods.
3. 4. 2 Gas Mi xtures
As discussed earlier the gases at atmospheric pressure condi-
tion have much larger free space between molecules than do
liquids. As a result the interaction between various like and
unlike molecules in a gaseous state is less than the molecular
interactions in similar liquid mixtures. Therefore, the role of
composition on properties of gas mixtures is not as strong as
in the case of liquids. Of course the effect of composition on
properties of gas mixtures increases as pressure increases and
free space between molecules decreases. The role of compo-
sition on properties of dense gases cannot be ignored. Under
low-pressure conditions where most gases behave like ideal
gases all gas mixtures regardless of their composition have the
same molar density at the same temperature and pressure. As
it will be discussed in Chapter 5, at the standard conditions
(SC) of 1.01325 bar and 298 K (14.7 psia and 60~ most gases
behave like ideal gas and RT/P represents the molar volume
of a pure gas or a gas mixture. However, the absolute density
varies from one gas to another as following:
Mmix P
(3.47) PmLx -- - -
83.14T
where Pmix is the absolute density of gas mixture in g/cm 3,
Mmi~ is the molecular weight of the mixture in g/tool, P is
pressure in bar, and T is the temperature in kelvin. Equation
(3.1) can be used to calculate molecular weight of a gas mix-
ture, MmL~. However, the mole fraction of component i in a
gas mixture is usually shown as Yi to distinguish from com-
position of liquid mixtures designated by x~. From definition
of mole and volume fractions in Section 1.7.15 and use of
Eq. (3.47) it can be shown that for ideal gas mixtures the
mole and volume fractions are identical. Generally volume
and mole fractions are used interchangeably for all types of
gas mixtures. Composition of gas mixtures is rarely expressed
in terms of weight fraction and this type of composition has
very limited application for gas systems. Whenever composi-
tion in a gas mixture is expressed only in percentage it should
be considered as tool% or vol%. Gas mixtures that are mainly
composed of very few components, such as natural gases, it is
possible to consider them as a single pseudocomponent and
to predict properties form specific gravity as the sole param-
eter available. This method of predicting properties of nat-
ural gases is presented in Chapter 4 where characterization
of reservoir fluids is discussed. The following example shows
derivation of the relation between gas phase specific gravity
and molecular weight of gas mixture.
Example 3. 14 Specific gravity of gases is defined as the ra-
tio of density of gas to density of dry air both measured at
the standard temperature and pressure (STP). Composition
of dry air in tool% is 78% nitrogen, 21% oxygen, and 1% ar-
gon. Derive Eq. (2.6) for the specific gravity of a gas mixture.
where Mg is the gas molecular weight. Density of both a gas
mixture and air at STP can be calculated from Eq. (3.47).
Mgas Psc
(3.48) SGg - Pga~ _ 83.14Ts~ _ Mg~
Pair M~irPsc Mair
83.14Tsc
where sc indicates the standard condition. Molecular weight
of air can be calculated from Eq. (3.48) with molecular
weight of its constituents obtained from Table 2.1 as MN2 =
28.01, 3/lo2 = 32.00, and MAr = 39.94. With composition given
as YN2 = 0.78, Yo2 = 0.21, and YA~= 0.01, from Eq. (3.1) we
get Mair = 28.97 g/mol. Equation (2.6) can be derived from
substituting this value for Mair in Eq. (3.49). I n practical cal-
culations molecular weight of air is rounded to 29. I f for a gas
mixture, specific gravity is known its molecular weight can be
calculated as
(3.49) Mg = 29SGg
where SGg is the gas specific gravity. It should be noted that
values of specific gravity given for certain gases in Table 2.1
are relative to density of water for a liquefied gas and are
different in definition with gas specific gravity defined from
Eq. (2.6). #
3 . 5 PR ED I C T I ON OF T HE C OMPOS I T I ON
OF PET R OL EUM FR A C T I ON S
As discussed earlier the quality and properties of a petroleum
fraction or a petroleum product depend mainly on the
mixture composition. As experimental measurement of the
composition is time-consuming and costly the predictive
methods play an important role in determining the quality
of a petroleum product. I n addition the pseudocomponent
method to predict properties of a petroleum fraction requires
the knowledge of PNA composition. Exact prediction of all
components available in a petroleum mixture is nearly im-
possible. I n fact there are very few methods available in the
literature that are used to predict the composition. These
methods are mainly capable of predicting the amounts (in
percentages) of paraffins, naphthenes, and aromatic as the
main hydrocarbon groups in all types of petroleum fractions.
These methods assume that the mixture is free of olefinic hy-
drocarbons, which is true for most fractions and petroleum
products as olefins are unstable compounds. I n addition to the
PNA composition, elemental composition provides some vital
information on the quality of a petroleum fraction or crude
oil. Quality of a fuel is directly related to the hydrogen and
sulfur contents. A fuel with higher hydrogen or lower carbon
content is more valuable and has higher heating value. High
sulfur content fuels and crude oils require more processing
cost and are less valuable and desirable. Methods of predict-
ing amounts of C, H, and S% are presented in the following
section.
Solution--Equation (2.6) gives the gas specific gravity as
(2.6) SGg- Mg
28.97
3. 5. 1 Predi ct i on of PNA Composi t i on
Parameters that are capable of identifying hydrocarbon types
are called characterization parameters. The best example of
such a paramet er is the Watson characterization factor, which
along with other paramet ers is introduced and discussed in
this section. However, the first known met hod to predict the
PNA composition is the n-d-M met hod proposed by Van Nes
and Van Westen [30] in the 1950s. The n-d-M met hod is
also included in the ASTM manual under ASTM D 3238 test
method. The mai n limitation of this met hod is that it can-
not be applied to light fractions. Later in the 1980s Riazi and
Daubert [36, 47] proposed a series of correlations based on
careful analysis of various characterization parameters. The
unique feature of these correlations is that they are appli-
cable to bot h light and heavy fractions and identify various
types of aromatics in the mixture. I n addition various meth-
ods are proposed based on different bulk properties of the
mixture that mi ght be available. The Ri azi -Daubert met hods
have been adopted by the API Committee on characteriza-
tion of pet rol eum fractions and are included in the fourth
and subsequent editions of the API-TDB [2] since the early
1980s. The other met hod that is reported in some literature
sources is the Bergamn' s met hod developed in the 1970s [48].
This met hod is based on the Watson K and specific gravity
of the fraction as two mai n characterization parameters. One
common deficiency for all of these met hods is that they do
not identify n-paraffins and isoparaffins from each other. I n
fact compositional types of PIONA, PONA, and PINA can-
not be determined from any of the met hods available in the
literature. These met hods provide mi ni mum information on
the composi t i on that is predictive of the PNA content. This is
mai nl y due to the complexity of pet rol eum mixtures and dif-
ficulty of predicting the composi t i on from measurabl e bulk
properties. The met hod of Riazi-Daubert, however, is capable
of predicting the monoaromat i c (MA) and pol yaromat i c (PA)
content of pet rol eum fractions.
I n general low boiling point fractions have higher paraffinic
and lower aromat i c contents while as boiling point of the frac-
tion increases the amount of aromat i c content also increases.
I n the direction of increase in boiling point, in addition to
aromat i c content, amount s of sulfur, nitrogen, and other het-
eroat oms also increase as shown in Fig. 3.22.
3. 5. 1. 1 Characterization Parameters for Molecular
Type Analysis
A characterization paramet er that is useful for molecular type
prediction purposes should vary significantly from one hy-
drocarbon type to another. I n addition, its range of varia-
tion within a single hydrocarbon family should be minimal.
With such specifications an ideal paramet er for character-
izing mo] ecular type should have a constant value within a
single family but different values in different families. Some
of these characterization paramet ers (i.e., SG, I , VGC, CH,
and Kw), which are useful for mol ecul ar type analysis, have
been introduced and defined in Section 2. i. As shown in Table
2.4, specific gravity is a paramet er that varies with chemical
structure particularly from one hydrocarbon family to an-
other. Since it also varies within a single family, it is not a
perfect characterizing par amet er for molecular type analysis
but it is more suitable t han boiling point that varies within a
single family but its variation from one family to anot her is
not significant. One of the earliest paramet ers to characterize
hydrocarbon molecular type was defined by Hill and Coats in
1928 [49], who derived an empirical relation between viscos-
ity and specific gravity in t erms of viscosity gravity constant
(VGC), which is defined by Eq. (2.15) in Section 2.1.17. Def-
inition of VGC by Eqs. (2.15) or (2.16) limits its application
to viscous oils or fractions with kinematic viscosity at 38~
(100~ above 38 SUS (~3.8 cSt.). For quick hand estima-
tion of VGC from viscosity and specific gravity, ASTM [4] has
provided a homograph, shown in Fig. 3.23, that gives VGC
values close to those calculated from Eq. (2.15). Paraffinic
oils have low VGC, while napthenic oils have high VGC val-
ues. Watson K defined by Eq. (2.13) in t erms of MeABP and
SG was originally introduced to identify hydrocarbon type
[9, 50, 51], but as is shown later, this is not a very suitable
paramet er to indicate composi t i on of pet rol eum fractions.
FIG. 3.22mVariation of composition of petroleum fractions with boiling point. Reprinted
from Ref. [7], p. 469, by courtesy of Marcel Dekker, Inc.
1 2 2 CHARACTERIZATION AND PROPERTIES OF PETROLEUM FRACTIONS
- - 41
4. 6
I
42
5- ---' 13
- - 44-
- - - 45 L02O
6 - 1.02
. LOoo .~
7 -7-5o
6 1.00
. 980
IO , 6 0 .$ 8
- 7 0 ,960 1
-9 40 . 9 6
45, - a o
8 - o~ ~ _ .goo 20
o 8 8 9 o . 1 _~ . gz ,-" - C 3 , P -
3 0- r .~ > . .
- o: i <
.880= E 25
40- CO < 0
-200 _~ .~ =~-.9 0 ,-r
5 0 -840 ~ ~
CO
-_ -z s o
60 - . aco ,e2o -: " ~ .as
7 0 ~_: 50
8 0 -~ , 800
9 0 -- 4tO0
I 0 0 -~ - - 7 8 0 - . 8 6
120 ~ - 500
55
~. - 6 00 , ; " 60 --
-- -- - 7 00 ~ _ ~ .84
140
1 60 ~ - 7 40
1 8 0 . - 800
z oo - - 9 oo
L I 0 0 0 . 7 2 0 4 0
- .82
-- 1200 .ZOO
5 0 0 "-- t 400
1600
i -
"-- 1 8 00
400
FI G . 3 . 2 3 ~ E s t i ma t i o n of V G C f r om k i nemat i c v i s cos i t y a nd
s p eci f i c g r av i t y [ 4].
Aromatic oils have low Kw values while paraffinic oils have
high Kw values. Kurtz and Ward in 1935 [52] defined refrac-
tivity intercept, Ri, in terms of refractive index (n) and density
(d) at 20~ which is presented by Eq. (2.14). The definition
is based on this observation that a plot of refractive index
against density for any homologous hydrocarbon group is lin-
ear. Ri is high for aromatics and low for naphthenic stocks.
The most recent characterization parameter was introduced
in 1977 by Huang [53] in terms of refractive index and it
is defined by Eq. (2.36). Paraffinic oils have low I values while
aromatics have high I values. Carbon-to-hydrogen weight
ratio defined in Section 2.1.18 is also a useful parameter
that indicates degree of hydrocarbon saturation and its value
increases from paraffinic to naphthenic and aromatic oils.
Methods of prediction of CH was discussed in Section 2.6.3.
Application of the hydrogen-to-carbon ratio in characteriza-
tion of different types of petroleum products is demonstrated
by Fryback [54]. An extensive analysis and comparison of
various characterization parameters useful for prediction of
the composition of petroleum fractions is presented by Ri-
azi and Daubert [36,47]. Comparison of parameters P~, VGC,
Kw, and I is presented in Table 3.21 and Fig. 3.24. From this
analysis it is clear that paramet ers/ ~ and VHC best separate
hydrocarbon types, while parameters Kw and I show large
variations for aromatic and naphthenic compounds making
them less suitable for prediction of composition of petroleum
fractions.
Another very useful parameter that not only separates
paraffins and aromatics but also identifies various hydrocar-
bon types is defined through molecular weight and refractive
index as [36]:
(3.50) m = M ( n- 1.475)
where n is the refractive index at 20~ Parameter m was de-
fined based on the observation that refractive index varies lin-
early with 1/M with slope of m for each hydrocarbon group
[55]. Values of parameter m for different hydrocarbon groups
calculated from Eq. (3.50) are given in Table 3.22.
As shown in Table 3.22, paraffins have low m values while
aromatics have high m values. I n addition, paraffinic and
naphthenic oils have negative m values while aromatic oils
have positive m values. Parameter m nicely identifies vari-
ous aromatic types and its value increases as the number of
tings increases in an aromatic compound. A pure hydrocar-
bon whose m value is calculated as - 9 has to be paraffinic,
it cannot be naphthenic or aromatic. This parameter is par-
ticularly useful in characterizing various aromatic types in
aromatic-rich fractions such as coal liquids or heavy residues.
Besides the parameters introduced above there are a num-
ber of other parameters that have been defined for the purpose
of characterizing hydrocarbon type. Among these parameters
viscosity index (VI) and correlation index (CI) are worth defin-
ing. The viscosity index was introduced in 1929 by Dean and
Davis and uses the variation of viscosity with temperature
as an indication of composition of viscous fractions. It is an
empirical number indicating variation of viscosity of oil with
temperature. A low VI value indicates large variation of vis-
cosity with temperature that is a characteristic of aromatic
oils. Similarly, paraffinic hydrocarbons have high VI values.
The method is described under ASTM D 2270-64 [4] and in
TABLE 3.2t--Values of characterization factors.
Value Range
Hydrocarbon type M Ri VGC K I
Paraffin 337-535 1.048-l.05 0. 74- 0, 75 13. 1- 1. 35 0.26-0.273
Naphthene 248--429 1. 03- 1. 046 0. 89- 0. 94 10. 5- 13. 2 0.278-0.308
Aromatic 180-395 1. 07- 1. 105 0. 95- 1. 13 9. 5- 12. 53 0.298-0.362
Taken with Permission from Ref. [47].
Ri
I
1 . 0 2
N
I
1 . 0 4
3. CHARACTERI Z ATI ON OF PETROL EUM FRACTI ONS
I[] I A I
I I I I I I
1 . 0 6 1 . 0 8 1 . 0 0
123
VGC
[ ]
I
0.76
I - - - ' 7- ] 1 A
I I I I I I I
0.84 0.92 1.00 1.08
K
10 14
N I
A I
I I I I I I
11 12 13
N I
I A I
I I I I I I I I I I I
0.26 0.28 0.30 0.32 0.34 0.36
FIG. 3 .24--Comparison of different characterization factors for prediction
of composition of petroleum fractions (see Table 3.20).
the API-TDB [2]. The VI is defined as [2]
(3.51) VI - L - U 100
D
where
L -- ki nemat i c viscosity of reference oil at 40~ wi t h 0
VI oil, cSt.
U -- ki nemat i c viscosity of oil at 40~ whose VI is to be
calculated, cSt.
D = L - H, i n cSt.
H = ki nemat i c viscosity of reference oil at 40 ~ C wi t h 100
VI oil, cSt.
The reference oils wi t h 0 and 100 VI and the oil whose
VI is to be cal cul at ed have t he same ki nemat i c viscosity at
100~ I n Engl i sh uni t s of system, VI is defined i n t erms of
viscosity at 37.8 and 98.9~ whi ch correspond to 100 and
210~ respectively [2]. However, i n the SI uni t s, viscosity at
reference t emper at ur es of 40 and 100~ have been used to
define the VI [5]. For oils wi t h a ki nemat i c viscosity of 100~
of less t han 70 mm2/ s (cSt.), the values of L and D are gi ven
TABLE 3.22--Values of parameter m for different types
of hydrocarbons [36].
Hydrocarbon type m
Paraffins -8.79
Cyclopentanes - 5.41
Cyclohexanes -4.43
Benzenes 2.64
Naphthalenes 19.5
Condensed Tricyclics 43.6
i n tables i n the st andar d met hods (ASTM D 2270, ISO 2909)
as well as i n t he API-TDB [2]. However, for viscous oils wi t h
a ki nemat i c viscosity at 100~ of great er t han 70 mm2/ s, the
values of L and D are given by t he following rel at i onshi ps [5]:
L = 0.8353Y 2 + 14.67Y - 216
(3.52) D = 0.6669Y 2 + 2.82Y - 119
where Y is t he ki nemat i c viscosity of oil whose VI is to be cal-
cul at ed at 100~ i n cSt. I n Engl i sh uni t s i n whi ch reference
t emper at ur es of 37.8 and 98.9~ (100 and 210~ are used,
values of numer i cal coefficients i n Eq. (3.52) are slightly dif-
ferent and are given i n the API-TDB [2].
L = 1.01523Y 2 + 12. 154Y- 155.61
(3.53)
D ---- 0.8236Y 2 - 0.5015Y - 53.03
Viscosity i ndex defined by Dean and Davis i n t he form of
Eq. (3.51) does not work very well for oils wi t h VI values of
great er t han 100. For such oils ASTM D 2270 describes t he
cal cul at i on met hod and it is summar i zed bel ow for viscosity
at reference t emper at ur es of 40 and 100~ [4, 5]:
10 N -- 1
(3.54) VI - - - + 100
0.00715
where N is gi ven by t he following rel at i on:
(3.55) N = log H - log U
l ogY
i n whi ch U and Y are ki nemat i c viscosity of oil i n cSt whose
VI is to be cal cul at ed at 40 and 100~ respectively. Values of
TABLE 3.23---Comparison of VI, VGC, Kw, and CI for several groups of oils [29].
Type of oil VI VGC Kw CI
High VI distillate 100 0.800-0.805 12.2-12.5 < 15
Medium VI distillate 80 0.830-0.840 11.8-12.0 15-50
Low VI distillate 0 0.865-0.890 11.0-11.5 >50
Solvent Extracts ... 0.880-0.950 10.0-11.0 ...
Recycle Stock ... 0.900-0.950 10.0-11.0 ...
Cracked Residues ... 0.950-1.000 9.8-11.0 ...
H versus ki nemat i c vi scosi t y of oil at 100 ~ (Y) when Y is less
t han 70 cSt ar e t abul at ed i n t he ASTM D 2270 met hod [4]. F or
oils wi t h val ues of Y gr eat er t han 70 cSt H mus t be cal cul at ed
f r om t he r el at i on given bel ow [5].
(3.56) H = 0.1684Y 2 + 11. 85Y- 97
I f kinematic vi scosi t i es ar e avai l abl e i n Engl i sh uni t s at 37.8
and 98.9~ (100 and 210~ t hen Eqs. (3.54) and (3.56) shoul d
be r ep l aced wi t h t he fol l owi ng r el at i ons as gi ven i n t he API -
TDB [2]:
10 N - I
(3.57) VI - - - + 100
0.0075
(3.58) H = 0.19176Y 2 + 12. 6559Y- 102.58
in whi ch Eq. (3.58) must be used for fract i ons wi t h ki nemat i c
vi scosi t y at 99~ (210~ gr eat er t han 75 cSt [2].
Fi nal l y t he cor r el at i on i ndex (CI) defi ned by t he U.S. Bur eau
of Mi nes is expr essed by t he fol l owi ng equat i on [7]:
48640
(3.59) CI - + 473. 7SG - 456.8
rb
in whi ch Tb is t he vol ume average boi l i ng p oi nt (VABP) in
kelvin. Values of CI bet ween 0 and 15 i ndi cat e a p r edomi -
nant l y paraffi ni c oil. A val ue of CI gr eat er t han 50 i ndi cat es a
p r edomi nance of ar omat i c comp ounds [7]. I t has a t endency
to i ncr ease wi t h i ncr easi ng boi l i ng p oi nt in a gi ven cr ude oil. A
comp ar i s on bet ween val ues ofVI , VGC, Kw, and CI for several
t ypes of p et r ol eum fract i ons and p r oduct s is p r esent ed in Ta-
bl e 3.23. A comp l et e comp ar i s on of var i ous char act er i zat i on
p ar amet er s i ndi cat i ng comp osi t i on of p et r ol eum fract i ons for
t he t hr ee hydr ocar bon gr oup s is p r esent ed in Table 3.24. All
p ar amet er s except Ri, Kw, and VI i ncr ease in t he di r ect i on
f r om paraffi ni c to nap ht heni c and ar omat i c oils.
Exampl e 3. 15--Calculate vi scosi t y i ndex of an oil havi ng
ki nemat i c vi scosi t i es of 1000 and 100 cSt at 37.8 and 98.9~
(100 and 210~ respectively.
Sol ut i on- - For t hi s oil 1)99(210) > 70 cSt, t hus we can use Eqs.
( 3. 51) -( 3. 57) for cal cul at i on of VI. Si nce t he VI is not known
we assume it is gr eat er t han 100 and we use Eqs. ( 3. 53) -( 3. 57)
to cal cul at e t he VI. However, si nce ki nemat i c vi scosi t i es ar e
gi ven at 38 and 99~ Eqs. (3.57) and (3.58) shoul d be used.
F r om t he i nf or mat i on gi ven U = i 000 and Y = 100. Si nce Y
is gr eat er t han 75 cSt, Eq. (3.58) mus t be used to cal cul at e
p ar amet er H, whi ch gives H -- 3080.6. F r om Eqs. (3.55) and
(3.57) we get N = 0.2443 and VI--- 200.7. Si nce cal cul at ed VI is
gr eat er t han 100, t he i ni t i al as s ump t i on is correct . Ot herwi se,
Eq. (3.51) mus t be used. Si nce t he val ue of VI is qui t e hi gh
t he oil is paraffi ni c as shown in Table 3.24. I f i)99(210 ) was less
t han 70 cSt t hen t abl es p r ovi ded by ASTM [4] or API -TDB [2]
shoul d be used to cal cul at e p ar amet er s L and D.
3. 5. 1. 2 API ( Ri azi - Daubert ) M et hods
To devel op a met hod for p r edi ct i ng t he comp os i t i on of olefin-
free p et r ol eum fract i ons t hr ee equat i ons ar e r equi r ed to ob-
t ai n fract i ons of paraffi ns (xp), nap ht henes (XN), and ar omat -
ics (XA). The first and most obvi ous equat i on is known f r om
t he mat er i al bal ance:
(3.60) Xp + XN + XA = 1
Two ot her equat i ons can be est abl i shed by ap p l yi ng Eq. (3.40)
for t wo p ar amet er s t hat can char act er i ze hydr ocar bon t ypes
and are easi l y measur abl e. Anal ysi s of var i ous char act er i za-
t i on fact ors shown i n Table 3.21 and Fig. 3.24 i ndi cat es t hat / ~
and VGC are t he most sui t abl e p ar amet er s to i dent i fy hydr o-
car bon type. F or exampl e, if for a p ur e hydr ocar bon/ ~ = 1.04,
it has to be a nap ht heni c hydr ocar bon, it cannot be paraffi ni c
or ar omat i c si nce onl y for t he nap ht heni c gr oup /~ vari es
f r om 1.03 to 1.046. Refract i vi t y i nt er cept has been r el at ed
to t he p er cent of nap ht heni c car bon at oms (%CN) as /~ =
1.05-0.0002 %CN [7]. Ri azi and Dauber t [47] used bot h/ ~
and VGC to devel op a pr edi ct i ve met hod for t he comp os i t i on
of vi scous p et r ol eum fract i ons. Pr oper t i es of p ur e hydr ocar -
bons f r om t he API RP-42 [56] have been used to cal cul at e/ ~
and VGC for a number of heavy hydr ocar bons wi t h mol ecu-
l ar wei ght s gr eat er t han 200 as shown in Table 2.3 and Table
3.21. Based on t he val ues of / ~ and VGC for all hydr ocar bons,
average val ues of t hese p ar amet er s were det er mi ned for t he
t hr ee gr oup s of paraffi ns, nap ht henes, and ar omat i cs. These
average val ues f or / ~ are as follows: 1.0482 (P), 1.0138 (N),
and 1.081 (A). Si mi l ar average val ues for VGC ar e 0.744 (P),
TABLE 3.24--Comparison of various characterization parameters for molecular type analysis.
Parameter Defined by Eq. (s) Paraffins Naphthenes Aromatics
/~ (2.14) medium low high
VGC (2.15) or (2.16) low medium high
m (3.50) low medium high
SG (2.2) low medium high
I (2.3) (2.36) low medium high
CI (3.59) low medium high
Kw (2.13) high medium low
VI (3.51)-(3.58) high medium low
0.915 (N), and 1.04 (A). Appl yi ng Eq. (3.40) for Ri and VGC
gives t he fol l owi ng t wo rel at i ons.
(3.61) /~ = 1.0482xp + 1.038XN + 1.081XA
(3.62) VGC = 0.744Xp + 0.915XN + 1.04XA
A r egr essi on of 33 defi ned hydr ocar bon mi xt ur es changes
t he numer i cal const ant s in t he above equat i ons by less t han
2% as fol l ows [29, 47]:
(3.63) Ri = 1.0486xv + 1.022XN + 1.1 lXA
(3.64) VGC = 0.7426xv + 0.9XN + 1.112XA
Si mul t aneous sol ut i on of Eqs. (3.60), (3.63), and (3.64)
gives t he fol l owi ng equat i ons for est i mat i on of t he PNA com-
p osi t i on of f r act i ons wi t h M > 200.
(3.65) Xp = - 9. 0 + 12.53/ ~ - 4. 228VGC
(3.66) xN = 18.66 - 19.9/~ + 2. 973VGC
(3.67) XA = --8.66 + 7.37/ ~ + 1. 255VGC
These equat i ons can be ap p l i ed to fract i ons wi t h mol ecul ar
wei ght s i n t he r ange of 200-600. As ment i oned earlier, Xp,
Xn, and XA cal cul at ed f r om t he above r el at i ons may r ep r esent
vol ume, mol e, or wei ght fract i ons. Equat i ons ( 3. 65) -( 3. 67)
cannot be ap p l i ed to l i ght f r act i ons havi ng ki nemat i c vi scosi t y
at 38~ of less t han 38 SUS ( ~3. 6 cSt.) . This is because VGC
cannot be det er mi ned as defi ned by Eqs. (2.15) and (2.16). For
such fract i ons Ri azi and Dauber t [47] defi ned a p ar amet er
si mi l ar to VGC cal l ed vi scosi t y gravi t y funct i on, VGF, by t he
fol l owi ng rel at i ons:
(3.68) VGF = - 1.816 + 3. 484SG - 0.1156 I n v38(100)
(3.69) VGF = - 1. 948 + 3. 535SG - 0.1613 I n i)99(210 )
wher e 1)38(100) and 1)99(210) ar e ki nemat i c vi scosi t y i n mm2/ s
( cSt) at 38 and 99~ (100 and 210~ respectively. F or a
p et r ol eum fract i on, bot h Eqs. (3.68) and (3.69) give near l y t he
same val ue for VGF; however, if ki nemat i c vi scosi t y at 38~ is
avai l abl e Eq. (3.68) is p r ef er abl e for cal cul at i on of VGE These
equat i ons have been der i ved bas ed on t he obser vat i on t hat at
a fixed t emp er at ur e, pl ot of SG versus I n 1) is a l i near l i ne for
each homol ogous hydr ocar bon group, but each gr oup has
a specific slope. F ur t her i nf or mat i on on der i vat i on of t hese
equat i ons is p r ovi ded by Ri azi and Dauber t [47]. Par amet er
VGF is basi cal l y defi ned for f r act i ons wi t h mol ecul ar wei ght s
of less t han 200. Based on t he comp os i t i on of 45 defi ned
mi xt ur es ( synt het i c) and wi t h an ap p r oach si mi l ar t o t he
one used to devel op Eqs. ( 3.65) -( 3.67) , t hr ee r el at i onshi p s i n
t er ms of / ~ and VGF have been obt ai ned to est i mat e t he PNA
comp os i t i on (Xp, XN, XA) of l i ght ( M < 200) fract i ons [47].
These equat i ons were l at er modi f i ed wi t h addi t i onal dat a for
bot h l i ght and heavy fract i ons and ar e gi ven bel ow [36].
For fract i ons wi t h M <_ 200
(3.70) xv -- - 13. 359 + 14.4591R~ - 1. 41344VGF
(3.71) xn = 23.9825 - 23.33304Ri + 0. 81517VGF
(3.72) Xg = 1 -- (Xp + XN)
F or fract i ons wi t h M > 200
(3.73) Xv = 2.5737 + 1.0133/ ~ - 3. 573VGC
(3.74) Xs = 2.464 -- 3.6701/ ~ + 1. 96312VGC
I n t hese set of equat i ons XA mus t be cal cul at ed f r om Eq.
(3.72). F or cases t hat cal cul at ed XA is negat i ve it shoul d be
set equal to zero and val ues of Xp and XN mus t be nor mal i zed
i n a way t hat Xp + xr~ = 1. The same p r ocedur e shoul d be
ap p l i ed t o Xp or Xn if one of t hem cal cul at ed f r om t he
above equat i ons is negat i ve. F or 85 samp l es Eqs. (3.70) and
(3.72) give average devi at i on of 0.04 and 0.06 for xv and
XN, respectively. For 72 heavy fract i ons, Eqs. ( 3. 72) -( 3. 74)
p r edi ct Xp, XN, and XA wi t h average devi at i ons of 0.03, 0.04,
and 0.02, respect i vel y [36]. These devi at i ons ar e wi t hi n t he
r ange of exp er i ment al uncer t ai nt y for t he PNA comp osi t i on.
Equat i ons ( 3. 70) -( 3. 74) ar e r ecommended to be used if
exp er i ment al dat a on vi scosi t y are avai l abl e. F or cases t hat
n2o and d2o are not avai l abl e, t hey can be accur at el y est i mat ed
f r om t he met hods p r esent ed in Chap t er 2.
For fract i ons t hat ki nemat i c vi scosi t y is not avai l abl e, Ri azi
and Dauber t [36] devel oped a seri es of cor r el at i ons i n t er ms of
ot her char act er i zat i on p ar amet er s whi ch are r eadi l y avai l abl e
or pr edi ct abl e. These p ar amet er s are SG, m, and CH and t he
pr edi ct i ve equat i ons for PNA comp os i t i on ar e as follows:
For f r act i ons wi t h M < 200
Xp = 2.57 - 2. 877SG + 0. 02876CH
XN = 0.52641 -- 0.7494Xp -- 0. 021811m
(3.75)
(3.76)
or
(3.77)
(3.78)
Xp = 3.7387 - 4. 0829SG + 0. 014772m
XN = - 1. 5027 + 2. 10152SG - 0. 02388m
F or fract i ons wi t h M > 200
xp = 1.9842 - 0.27722Ri - 0. 15643CH
XN = 0.5977 -- 0.761745Ri + 0. 068048CH
(3.79)
(3.80)
or
(3.81)
(3.82)
xv -- 1.9382 + 0. 074855m- 0. 19966CH
XN ----- --0.4226 -- 0. 00777m+ 0. 107625CH
I n all of t hese cases XA mus t be cal cul at ed f r om Eq. (3.72).
Equat i ons (3.75) and (3.76) have been eval uat ed wi t h PNA
comp os i t i on of 85 fract i ons in t he mol ecul ar wei ght r ange
of 78-214 and give average devi at i ons of 0.05, 0.08, and
0.07 for xv, XN, and xA, respectively. For t he same dat a set
Eqs. (3.77) and (3.78) give AAD of 0.05, 0.086, and 0.055 for
Xp, XN, and xa, respectively. For 72 fract i ons wi t h mol ecu-
l ar wei ght range of 230-570, Eqs. ( 3. 79) -( 3. 82) give near l y
t he same AAD of 0.06, 0.06, and 0.02 for Xp, xN, and XA,
respectively. I n cases t hat i np ut p ar amet er s for t he above
met hods ar e not avai l abl e Eqs. (3.77) and (3.78) i n t er ms of
SG and m ar e mor e sui t abl e t han ot her equat i ons si nce re-
fract i ve i ndex and mol ecul ar wei ght can be est i mat ed mor e
accur at el y t han CH. Al t hough Eqs. (3.77) and (3.78) have
been der i ved f r om a dat a set on fract i ons wi t h mol ecul ar
wei ght s up to 200, t hey can be safel y used up t o mol ecu-
l ar wei ght of 300 wi t hout ser i ous errors. Most recently, Eqs.
(3.77) and (3.78) have been modi f i ed to exp and t he r ange
of ap p l i cat i on of t hese equat i ons for heavi er fract i ons, but
in gener al t hei r accur acy is not si gni fi cant l y di fferent f r om
t he equat i ons p r esent ed her e [45]. For exampl e, for frac-
t i ons wi t h 70 < M < 250, Ri azi and Roomi [45] modi f i ed Eqs.
(3.77) and (3.78) as: Xp -- 3. 2574- 3.48148 SG + 0.011666m
andxN = --1.9571 + 2.63853 SG - 0.03992m. Most of the data
used in development of Eqs. (3.70)-(3.82) were in terms of
volume fractions for xp, XN, and Xg. Therefore, generally esti-
mated values represent volume fractions; however, they can
be used as mole fractions without serious errors.
I n all the above equations total aromatic content is cal-
culated from Eq. (3.72). As discussed earlier for cases that
aromatic content is high it should be split into two parts for
a more accurate representation of hydrocarbon types in a
petroleum mixture. Aromatics are divided into monoaromat-
ics (XMA) and polyaromatics (XPA) and the following relations
have been derived for fractions with molecular weights of less
than 250 [36]:
(3.83) XMA= --62.8245 + 59.90816R~ - 0.0248335m
(3.84) XpA = 11.88175 -- 11.2213R~ + 0.023745m
(3.85) XA -~- XM A -~- XpA
The equations may be applied to fractions with total aromatic
content in the range of 0.05-0.96 and molecular weight range
of 80-250. Based on a data set for aromatic contents of 75
coal liquid sample, Eqs. (3.83)-(3.85) give AAD of about 0.055,
0.065, and 0.063 for XMA, XPA, and XA, respectively. Maximum
AD is about 0.24 for Xpg. Equations (3.83) and (3.84) have not
been evaluated against petroleum fractions. For heavier frac-
tions no detailed composition on aromatics of fractions were
available; however, if such data become available expressions
similar to Eqs. (3.83) and (3.84) may be developed for heavier
fractions.
Exampl e 3. 16- - A gasoline sample produced from an Aus-
tralian crude oil has the boiling range of C5-65~ specific
gravity of 0.646, and PNA composition of 91, 9, and 0 vol%
(Ref. [46], p. 302). Calculate the PNA composition from a suit-
able method and compare the results with the experimental
values.
Sol ut i on- - For this fraction the only information available
are boiling point and specific gravity. From Table 2.1 the boil-
ing point of n-C5 is 36~ Therefore, for the fraction the
characteristic boiling point is Tb ---- (36 + 65)/2 ---- 50.5~ The
other characteristic of the fraction is SG -- 0.646. This is
a light fraction (low Tb) so we use Eq. (2.51) to calculate
molecular weight as 84.3. Since viscosity is not known, the
most suitable method to estimate composition is through
Eqs. (3.77) and (3.78). They require parameter m, which
in turn requires refractive index, n. From Eq. (2.115), I =
0.2216 and from Eq. (2.114), n = 1.3616. With use of M and
n and Eq. (3.50), m= -9.562. From Eq. (3.77) and (3.78),
xp = 0.96 and xN -- 0.083. From Eq. (3.72), ;cA = -4.3. Since
XA is negative thus ;CA---- 0 and Xp, XN should be normalized
as xp = 0.96/(0.96 + 0.083) -- 0.92 and XN = l -- 0.92 ---- 0.08.
Therefore, the predicted PNA composition is 92, 8, 0% versus
the experimental values of 91, 9, and 0%.
The aromatic content for this fraction is zero and there
is no need to estimate XMAand XpA from Eqs. (3.83) and
(3.84); however, to see the performance of these equations
for this sample we calculate ;CA from Eq. (3.85). From
Eq. (2.113), d = 0.6414 and from Eq. (2.14), R4 -- 1.0409. Us-
ing these values of / ~ and m in Eqs. (3.83) and (3.84), we
get XMA= --0.228 and XpA = -0.025. Since both numbers are
negative the actual estimated values are XMA= 0 and XpA = 0.
From Eq. (3.85), ;cA = 0, which is consistent with the previous
result from Eqs. (3.77), (3.78), and (3.72). t
3. 5. 1. 3 API M et hod
Since 1982 API has adopted the methods developed by Ri-
azi and Daubert [36, 47] for prediction of the composition
of petroleum fractions. Equations (3.65)-(3.67) and similar
equations developed for light fractions in terms of Ri and
VGF by Riazi and Daubert in 1980 [47] were included in
the fourth edition of the API-TDB-82. However, after devel-
opment of Eqs. (3.70)-(3.74) in terms of viscosity and Eqs.
(3.83)-(3.84) for prediction of the amount of different types
of aromatics in 1986 [36], they were included in the fifth and
subsequent editions of the API-TDB [2]. The API methods for
prediction of the composition of petroleum fractions require
kinematic viscosity at 38 or 99~ and if not available, it should
be estimated from Eq. (2.128) or (2.129) in Chapter 2.
3. 5. 1. 4 n- doM M et hod
This method requires three physical properties of refractive
index (n20), density (d20), and molecular weight (M). For this
reason the method is called n-d-M method and is the oldest
method for prediction of the composition of petroleum frac-
tions. The method is described in the book by Van Nes and
Van Westen in 1951 [30] and it is included in the ASTM man-
ual [4] under ASTM D 3238 test method. The method does
not directly give the PNA composition, but it calculates the
distribution of carbon in paraffins (%Cp), naphthenes (%CN),
and aromatics (%CA). However, since carbon is the dominant
element in a petroleum mixture it is assumed that the %Cp,
%CN, and %CA distribution is proportional to %P, %N, and %A
distribution. I n this assumption the ratio of carbon to hydro-
gen is considered constant in various hydrocarbon families.
Errors caused due to this assumption are within the range of
uncertainty in experimental data reported on the PNA com-
position. Another input data required for this method is sulfur
content of the fraction in wt% (%S) and should be known if
it exceeds 0.206 wt%. The method should not be applied to
fractions with sulfur content of greater than 2%. This method
is applicable to fractions with boiling points above gasoline.
I n addition this method should be applied to fractions with
ring percent, %CR (%CN + %CA) up to 75% provided that %CA
(as determined from the n-d-M method) is not higher than
1.5 times %CN [7]. The n-d-M method also provides equations
for calculation of total number of rings (RT), number of
aromatic rings (Ra), and number of naphthenic rings (RN)
in an average molecule in the fractions. The method is
expressed in two sets of equations, one for n2o, d20 (20~
and another set for n70 and d7o (70~ as input data. I n this
section correlation in terms of n20 and d20 are presented. The
other set of correlations for measurement of n and d at 70~
is given in the literature [7, 24].
%CA = av + 3660/ M
%CN = %CR -- %CA
(3.86) %Cp = 100 - %CR
Rn = 0.44 + bvM
RN = RT-- R,
3. CHARACTERIZATION OF PETROLEUM FRACTIONS 1 2 7
where
v = 2.51(n - 1.475) - (d - 0.851)
430 i fv > 0
a = 670 if v< 0
b= [ 0" 055 i f v> 0
! 0.080 if v< 0
w = ( d- 0.851) - 1. 11( n- 1.475)
%CR = [ 820w- - 3%S+10000/ M if w> 0
1440w-- 3%S + 10600/M if w < 0
1. 33+0. 146M( w- 0. 005x%S) i f w> 0
RT= 1. 33+0. 180M( w- 0. 005x%S) if w< 0
Once carbon distribution is calculated from Eq. (3.86), the
PNA composition can be determined as follows:
xv = %Cv / l O0
(3.87) XN ----- %C~/100
XA = %CA/100
As mentioned above the n-d-M method cannot be applied
to light fractions with molecular weights of less than 200.
However, when it was evaluated against PNA composition
of 70 fractions for the molecular weight range of 230-570,
AAD of 0.064, 0.086, and 0.059 were obtained for xv, XN, and
xh, respectively. Accuracy of the n-d-M method for prediction
of composition of fractions with M > 200 is similar to the
accuracy of Eqs. (3.79)-(3.82). But accuracy of Eqs. (3.73)
and (3.74) in terms of viscosity (API method) is more than
the n-d-M method [30, 36].
I n addition to the above methods there are some other
procedures reported in the literature for estimation of the
PNA composition of petroleum fractions. Among these
methods the Bergman' s method is included in some refer-
ences [48]. This method calculates the PNA composition in
weight fraction using the boiling point and specific gravity
of the fraction as input data. The weight fraction of aromatic
content is linearly related to Kw. The xv and XN are calculated
through simultaneous solution of Eqs. (3.72) and (3.46)
when they are applied to specific gravity. Specific gravity
of paraffinic, naphthenic, and aromatic pseudocomponents
(SGp, SGN, and SGA) are calculated from boiling point of
the fraction. Equation (2.42) may be used to calculate SG for
different groups from Tb of the fraction. Except in reference
[48] this method is not reported elsewhere. There are some
other specific methods reported in various sources for
each hydrocarbon group. For example, ASTM D 2759 gives a
graphical method to estimate naphthene content of saturated
hydrocarbons (paraffins and naphthenes only) from refrac-
tivity intercept and density at 20~ I n some sources aromatic
content of fractions are related to aniline point, hydrogen
content, or to hydrogen-to-carbon (HC) atomic ratio [57]. An
example of these methods is shown in the next section.
3 . 5 . 2 P r e d i c t i o n o f El e me n t a l C o mp o s i t i o n
As discussed earlier, knowledge of elemental composition
especially of carbon (%C), hydrogen (%H), and sulfur con-
tent (%S) directly gives information on the quality of a fuel.
Knowledge of hydrogen content of a petroleum fraction helps
to determine the amount of hydrogen needed if it has to go
through a reforming process. Petroleum mixtures with higher
hydrogen content or lower carbon content have higher heat-
ing value and contain more saturated hydrocarbons. Predic-
tive methods for such elements are rare and limited so there
is no possibility of comparison of various methods but the
presented procedures are evaluated directly against experi-
mental data.
3. 5. 2. 1 Prediction of Carbon and Hydrogen Contents
The amount of hydrogen content of a petroleum mixture is
directly related to its carbon-to-hydrogen weight ratio, CH.
Higher carbon-to-hydrogen weight ratio is equivalent to lower
hydrogen content. I n addition aromatics have lower hydrogen
content than paraffinic compounds and in some references
hydrogen content of a fraction is related to the aromatic con-
tent [57] although such relations are approximate and have
low degrees of accuracy. The reason for such low accuracy
is that the hydrogen content of various types of aromatics
varies with molecular type. Within the aromatic family, dif-
ferent compounds may have different numbers of rings, car-
bon atoms, and hydrogen content. In general more accurate
prediction can be obtained from the CH weight ratio method.
Several methods of estimation of hydrogen and carbon con-
tents are presented here.
3. 5. 2. 1. 1 Riazi M ethod--This method is based on calcula-
tion of CH ratio from the method of Riazi and Daubert given
in Section 2.6.3 and estimation of %S from Riazi method in
Section 3.5.2.2. The main elements in a petroleum fraction
are carbon, hydrogen, and sulfur. Other elements such as ni-
trogen, oxygen, or metals are in such small quantities that
on a wt% basis their presence may be neglected without se-
rious error on the composition of C, H, and S. This is not to
say that the knowledge of the amounts of these elements is
not important but their weight percentages are negligible in
comparison with weight percentages of C, H, and S. Based
on this assumption and from the material balance on these
three main elements we have
(3.88) %C + %H + %S = 100
%C
(3.89) = CH
%H
From simultaneous solution of these two equations, assum-
ing %S is known, the following relations can be obtained for
%H and %C:
100 - %S
(3.90) %H -
1 +CH
(3.91) %C= ( CI ~CH ) x( 100- %S)
where %S is the wt% of sulfur in the mixture, which should
be determined from the method presented in Section 3.5.2.2
if the experimental value is not available. Value of CH may
be determined from the methods presented in Section 2.6.3.
I n the following methods in which calculation of only %H is
presented, %C can be calculated from Eq. (3.88) if the sulfur
content is available.
3. 5. 2. 1. 2 Goossens' M ethod--M ost recently a simple re-
lation was proposed by Goossens to estimate the hydrogen
content of a petroleum fraction based on the assumption of
mol ar additivity of structural contributions of carbon types
[58]. The correlation is derived from data on 61 oil fractions
with a squared correlation coefficient of 0.999 and average
deviation of 3% and has the following form:
82.952 - 65.341n 306
(3.92) %H = 30.346 +
d M
where M is the molecular weight and n and d are refractive
index and density at 20~ respectively. This met hod should
be applied to fractions with mol ecul ar weight range of 84-
459, boiling point range of 60-480~ refractive index range
of 1.38-1.51, and hydrogen content of 12.2-15.6 wt%. I n
cases that M is not available it should be estimated from the
Goossens correlation given by Eq. (2.55).
3. 5. 2. 1. 3 ASTM M ethod---ASTM describes a met hod to es-
t i mat e the hydrogen content of aviation fuels under ASTM
D 3343 test met hod based on the aromat i c content and
distillation data [4]:
%H = (5.2407 + 0.01448Tb -- 7.018XA)/SG - 0.901XA
(3.93) + 0.01298XATb -- 0.01345Tb + 5.6879
where xA is the fraction of aromatics in the mixture and
Tb is an average value of boiling points at 10, 50, and 90
vol% vaporized in kelvin [Tb = (/'10 + Ts0 + %0)/3]. This cor-
relation was developed based on 247 aviation fuels and 84
pure hydrocarbons. This met hod is quite accurate if all the
input data are available from experimental measurement .
3. 5. 2. 1. 4 Jenkins- Walsh M et hod- They developed a sim-
ple relation in t erms of specific gravity and aniline point in
the following form [59]:
(3.94) %H = 11.17 - 12.89SG + 0.0389AP
where AP is the aniline point in kelvin and it may be deter-
mined from the Winn nomogr aph (Fig. 2.14) presented in
:1:: 20
6
10
~0
5
(a)
0
........................................................... i ........................................ ............... ; ................... ........................................................... ~ ..............
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . 7 . . . . . . . . . . . . . .
~iiiiiiiiiiiiiil 4+ 1. 27274%H) 1 IIIIIIIIIIIII
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :: 4
, J ; r i
5 10 15
Hydr ogen Wei ght Percent , %H
.< 70
6O
s o
o 40
.2 30
20
o
0
11 12 13 14 15
(b) Hydr ogen Wei ght Percent , %H
FIG. 3.25--Relationships between fuel hydrogen content,
(a) CH weight ratio and (b) aromatic content.
Section 2.8. The correlation is specifically developed for jet
fuels with aniline points in the range of 56-77~
There are a number of other methods reported in the liter-
ature. The Winn nomograph may be used to estimate the CH
ratio and then %H can be estimated from Eq. (3.90). Fein-
Wilson-Sherman also related %H to aniline point through
API gravity [60]. The oldest and simplest method was pro-
posed by Bureau of Standards in terms of specific gravity as
given in reference [61 ]:
(3.95) %H = 26 - 15SG
The other simple correlation is derived from data on jet fuels
and is in terms of aromatic content (XA) in the following form
[57]:
(3.96) %H = 14.9 - 6.38xg
Finally Fig. 3.25 is based on data from Ref. [57]. Analytical
correlation is also presented in Fig. 3.25(a), which represents
data with an average deviation of 0.5%. Equation (3.96) is pre-
sented in Fig. 3.25(b). When CH ratio is available, %H can be
determined from Fig. 3.25(a) and then %A can be determined
from Fig. 3.25(b).
3.5.2.2 Prediction of Sulfur and Nitrogen Contents
Sulfur is the most important heteroatom that may be present
in a crude oil or petroleum products as dissolved free sulfur
and hydrogen sulfide (H2S). It may also be present as organic
compounds such as thiophenes, mercaptanes, alkyl sulfates,
sulfides (R--S--W), disulfides (R--S--S--R'), or sulfoxides
(R--SO--R'), where R and R' refer to any aliphatic or aromatic
group. Its presence is undesirable for the reasons of corrosion,
catalysts poisoning, bad odor, poor burning, and air pollu-
tion. I n addition presence of sulfur in lubricating oils lowers
resistance to oxidation and increases solid deposition on en-
gine parts [62]. New standards and specifications imposed
by governments and environmental authorities in industrial
countries require very low sulfur content in all petroleum
products. For example, reformulated gasolines (RFG) require
sulfur content of less than 300 ppm (<0.03 wt%) [63]. Re-
cently a federal court has upheld an Environmental Protec-
tion Agency (EPA) rule to cut pollution from tractor-trailers
and other large trucks and buses. The rule is expected to re-
duce tailpipe emissions from tractors, buses, and other trucks
up to 90%. The EPA also calls on refineries to reduce the sul-
fur content in diesel oils to 15 ppm by 2007 from the current
level of 500 ppm. The American Lung Association claims that
low-sulfur fuel will reduce the amount of soot from larger
trucks by 90%. This is expected to prevent 8300 premature
deaths, 5500 cases of chronic bronchitis, and another 17600
cases of acute bronchitis in children as provided by the EPA
[64]. Products with high-sulfur contents have low quality and
heating values. Generally, sulfur is associated with heavy and
aromatic compounds [7]. Therefore, high aromatic content
or high boiling point fractions (i.e., residues and coal liquids)
have naturally higher sulfur contents. Distribution of sulfur in
straight-run products of several crude oils and the world aver-
age crude with 2 and 5% sulfur contents is shown in Fig. 3.26.
Data used to generate this figure are taken from Ref. [61]. As
the boiling point of products increases the sulfur content in
the products also increases. However, the distribution of sul-
fur in products may vary from one crude source to another.
10
. . . . Venezuel an ( 5%) . . - "~ ""
8 . . . . . Mi ddl e East (5%) . - ' "
. . . . . . . West Afri can (5%) . ' "
Wor l d Aver age (5%) . "~
9 " S"
. . . . Wor l d Average ( 2%) . . ~/ , - " . j . s
6 : ,- . - -
, , , . . j ' ~ -
~ . : r ~ l "
. . " 9 9 ot
0 100 200 300 400 500 600
Mid Boiling Point, ~
FIG. 3 . 26~ Di st r i but i on of sul fur in strai g ht-run prod-
ucts for several crude oils. Numbers in the parentheses
i ndi cate sul fur content of crudes.
As boiling point, specific gravity, or aromatic content of
a fraction increases the sulfur content also increases [7].
Parameters Ri, m, and SG have been successfully used to
predict the PNA composition especially aromatic content of
petroleum fractions as shown in Section 3.5.2.1. On this ba-
sis the same parameters have been used for the estimation of
sulfur content of petroleum fractions in the following form
for two ranges of molecular weight.
For fractions with M < 200
(3.97) %S = 177.448 - 170.946Ri + 0.2258m+ 4.054SG
and for fractions with M > 200
(3.98) %S --- -58. 02 + 38.463R~ - 0.023m+ 22.4SG
For light fractions in which Eq. (3.96) may give very small
negative values, %S would be considered as zero. Squared cor-
relation coefficients (R 2) for these equations are above 0.99.
A summary of evaluation of these equations is presented in
Table 3.25 as given in Ref. [62].
I n using these equations parameters n2o, d2o, M, and SG
are required. For samples in which any of these parameters
are not known they can be estimated from the methods dis-
cussed earlier in this chapter. I n Chapter 4, it is shown how
this method can be used to estimate sulfur content of whole
crudes. The author is not familiar with any other analytical
method for estimation of sulfur content of petroleum frac-
tions reported in the literature so a comparison with other
methods is not presented. Generally amount of sulfur in var-
ious products is tabulated for various crudes based on the
sulfur content of each crude [61 ].
Another heteroatom whose presence has adverse effect on
the stability of the finished product and processing catalysts
is nitrogen. High nitrogen content fractions require high hy-
drogen consumption in hydro processes. Nitrogen content of
crudes varies from 0.01 to 0.9 wt%. Most of the compounds
having nitrogen have boiling points above 400~ (~750~
and are generally within the aromatic group. Crudes with
higher asphaltene contents have higher nitrogen content as
TABLE 3.25--Prediction of sulfur content of petroleum fractions [62].
Erroff
Fraction type No. of point Mol% range SG range Sulfur wt% range AAD% MAD%
Light 76 76-247 0.57-0.86 0.01-1.6 0.09 0.7
Heavy 56 230-1500 0.80-1.05 0.07-6.2 0.24 1.6
Overall 132 76-1500 0.57-1.05 0.01-6.2 0.15 1.6
~AAD% = Absolute average deviation, %; MAD% = maximum average deviation, %.
well. Si mi l ar t o sulfui, ni t r ogen cont ent of var i ous p et r ol eum
fract i ons is p r esent ed in t er ms of ni t r ogen cont ent of t he
cr ude oil [61]. Bal l et al. [65] have shown t hat ni t r ogen con-
t ent of cr ude oils for each geol ogi cal p er i od is l i near l y r el at ed
t o car bon r esi due of t he crude. However, t he cor r el at i on does
not p r ovi de i nf or mat i on on ni t r ogen cont ent of p et r ol eum
pr oduct s. I n gener al ni t r ogen cont ent of f r act i ons whose mi d
boi l i ng p oi nt is less t han 450~ have ni t r ogen cont ent s less
t han t hat of cr ude and for heavi er cut s t he ni t r ogen wt % in t he
f r act i on is gr eat er t han t hat of cr ude [61]. However, t he val ue
of 450~ at whi ch ni t r ogen cont ent of t he f r act i on is near l y t he
same as t hat of cr ude is ap p r oxi mat e and it may vary sl i ght l y
wi t h t he t ype of t he crude. Dat a r ep or t ed in Ref. [61] for dis-
t r i but i on of ni t r ogen cont ent of st r ai ght r un di st i l l at es have
been cor r el at ed i n t he fol l owi ng form:
%N2 i n fract i on
= - 0. 4639 + 8. 47267T - 28. 9448T 2
%N2 in cr ude
(3.99) + 27. 8155T 3
wher e T = Tb/1000 in whi ch Tb is t he mi d boi l i ng p oi nt of t he
cut in kelvin. This equat i on is val i d for cut s wi t h mi d boi l i ng
p oi nt s gr eat er t han 220~ and is not ap p l i cabl e to fi ni shed
p et r ol eum p r oduct s. Amount of ni t r ogen in at mos p her i c
di st i l l at es is qui t e smal l on p er cent basi s. The wt % r at i o in
Eq. (3.99) can be r ep l aced by p p m wei ght r at i o for smal l quan-
t i t i es of ni t rogen. Est i mat i on of comp osi t i on of el ement s is
demons t r at ed i n Examp l es 3.17 and 3.18.
Exampl e 3 . 1 7 ~A p et r ol eum fract i on wi t h a boi l i ng r ange of
250-300~ is p r oduced f r om a Venezuel an cr ude oil (Ref. [46],
p. 360). Exp er i ment al l y meas ur ed p r op er t i es are as follows:
ASTM di st i l l at i on 262. 2, 268. 3, and 278.9~ at 10, 50, and 90
vol % recovered, respect i vel y; specific gravi t y 0.8597; car bon-
t o- hydr ogen wei ght r at i o 6.69; ani l i ne p oi nt 62~ ar omat i c
cont ent 34.9%; and sul fur wt % 0.8. Est i mat e sul fur cont ent
of t he f r act i on f r om t he met hod p r esent ed in Sect i on 3.5.2.2.
Also cal cul at e %C and %H f r om t he fol l owi ng met hods: exper-
i ment al dat a, Ri azi , Goossens, ASTM, J enki ns- Wat sh, Bur eau
of Mi nes and Eq. (3.96).
Sol ut i on- - To est i mat e t he sul fur cont ent , p ar amet er s M, n2o,
and d20 ar e r equi r ed as t he i np ut dat a. The f r act i on is a nar-
r ow f r act i on and t he boi l i ng p oi nt at 50% di st i l l ed can be
consi der ed as t he char act er i st i c average boi l i ng poi nt , Tb =
268.3~ = 541.5 K. This is a l i ght fract i on wi t h M < 300;
t herefore, M, d2o, and n20 ar e cal cul at ed f r om Eqs. (2.50) and
( 2. 112) -( 2. 114) as 195.4, 0.8557, and 1.481, respectively. F r om
Eqs. (2.15) and (3.50) we get Ri -- 1.0534, m- - 1.2195. Si nce
M < 250, Eq. (3.97) is used to est i mat e t he sul fur cont ent as
%S = 1. i % versus t he exp er i ment al val ue of 0.8%. Therefore,
t he er r or is cal cul at ed as follows: 1. 1%-0. 8% --- 0.3%.
To cal cul at e %C and %H f r om exp er i ment al dat a, val ues of
CH = 6.69 wi t h %S = 0.8 ar e used i n Eqs. (3.90) and (3.91).
This woul d resul t in %C -- 86.3 and %H -- 12.9. Accor di ng to
t he gener al met hod p r esent ed in t hi s book ( aut hor' s p r op os ed
met hod) , CH is cal cul at ed f r om Eq. (2.120) as CH = 6.75 and
wi t h est i mat ed val ue of sul fur cont ent as %S -- 1.1, %C and
%H ar e cal cul at ed f r om Eqs. (3.90) and (3.91) as %C = 86.1
and %H = 12.8. I n use of Goossens met hod t hr ough Eq. (3.92),
est i mat ed val ues of n, d, and M ar e r equi r ed wher e M shoul d
be est i mat ed f r om Eq. (2.55) as M = 190. F or t hi s met hod %C
may be cal cul at ed f r om Eq. (3.88) if %S is known. A s umma r y
or resul t s for cal cul at i on of %H wi t h AD for var i ous met hods
is gi ven in Table 3.26. The Goossens met hod gives t he hi gh-
est er r or because all i np ut dat a r equi r ed ar e p r edi ct ed val ues.
The ASTM met hod gives t he same val ue as exp er i ment al val ue
because t he exp er i ment al val ues on all t he i np ut p ar amet er s
r equi r ed in Eq. (3.93) are avai l abl e in t hi s p ar t i cul ar exampl e.
However, in many cases ar omat i c cont ent or comp l et e di st i l l a-
t i on curve as r equi r ed by t he ASTM met hod ar e not avai l abl e.
The gener al met hod of aut hor p r esent ed in t hi s sect i on based
on cal cul at i on of CH and %S gives good resul t s al t hough 50%
ASTM di st i l l at i on t emp er at ur e and specific gravi t y have been
used as t he onl y avai l abl e dat a. t
Exampl e 3. 18- - A p et r ol eum cut has t he boi l i ng r ange of 370-
565~ and is p r oduced f r om a cr ude oil f r om Dani sh Nor t h
Sea fields (Ref. [46], p. 353). The ni t r ogen cont ent of cr ude
is 1235 p p m. Cal cul at e ni t r ogen cont ent of t he f r act i on and
comp ar e wi t h t he exp er i ment al val ue of 1625 p p m.
Solution- - Tb = (370 + 565) / 2 = 467.5~ = 740.6 K. T = Tb/
1000 = 0.7406. Subst i t ut i ng T i n Eq. (3.99) gives %N2 in cut ----
1.23 x 1235 = 1525. The p er cent rel at i ve devi at i on wi t h t he
exp er i ment al val ue is - 6%. Thi s is rel at i vel y a good p r edi c-
tion, but nor mal l y l ar ger er r or s ar e obt ai ned especi al l y for
l i ght er cuts. t
3 . 6 PR ED I C T I ON OF OT HER PR OPER T I ES
I n t hi s sect i on, pr edi ct i ve met hods for some i mp or t ant p r op -
ert i es t hat are useful to det er mi ne t he qual i t y of cer t ai n
p et r ol eum p r oduct s ar e pr esent ed. Some of t hese pr oper -
t i es such as flash p oi nt or p our p oi nt are useful for safet y
TABLE 3. 26--Estimation of hydrogen content of petroleum fraction in Example 3.17.
Method Riazi Goossens ASTM D 3343 Jenkins-Walsh Bureau of Mines Eq. (3.97)
%H, calc. 12.8 12.6 12.9 13.1 13.1 12.7
AD,% 0.1 0.3 0 0.2 0.2 0.2
consideration or storage and transportation of products. One
of the most important properties of petroleum products re-
lated to volatility after the boiling point is vapor pressure.
For petroleum fractions, vapor pressure is measured by the
method of Reid. Methods of prediction of true vapor pressure
of petroleum fractions are discussed in Chapter 7. However,
Reid vapor pressure and other properties related to volatility
are discussed in this section. The specific characteristics of
petroleum products that are considered in this part are flash,
pour, cloud, freezing, aniline, and smoke points as well as car-
bon residue and octane number. Not all these properties apply
to every petroleum fraction or product. For example, octane
number applies to gasoline and engine type fuels, while car-
bon residue is a characteristic of heavy fractions, residues,
and crude oils. Freezing, cloud, and pour points are related
to the presence of heavy hydrocarbons and are characteristics
of heavy products. They are also important properties under
very cold conditions. Predictive methods for some of these
properties are rare and scatter. Some of these methods are
developed based on a limited data and should be used with
care and caution.
3. 6. 1 Propert i es Rel at ed t o Vol ati l i ty
Properties that are related to volatility of petroleum fraction
are boiling point range, density, Reid vapor pressure, and flash
point. Prediction of boiling point and density of petroleum
fractions have been discussed earlier in this chapter. I n this
FIG. 3 .27 --A pparatus to measure RVP of
petroleum products by A STM D 323 test method
( courtesy of KISR) .
part, methods of prediction of vapor pressure, fuel vapor liq-
uid (V/L) ratio, fuel volatility index, and flash points are pre-
sented.
3. 6. 1. i Reid V apor Pressure
Reid vapor pressure is the absolute pressure exerted by a mix-
ture at 37.8~ (311 K or 100~ at a vapor-to-liquid volume
ratio of 4 [4]. The RVP is one of the important properties of
gasolines and jet fuels and it is used as a criterion for blend-
ing of products. RVP is also a useful parameter for estimation
of losses from storage tanks during filling or draining. For
example, according to Nelson method losses can be approxi-
mately calculated as follows: losses in vol% = (14.5 RVP - 1)/6,
where RVP is in bar [24, 66]. The apparatus and procedures
for standard measurement of RVP are specified in ASTM
D 323 or IP 402 test methods (see Fig. 3.27). I n general, true
vapor pressure is higher than RVP because of light gases dis-
solved in liquid fuel. Prediction of true vapor pressure of pure
hydrocarbons and mixtures is discussed in detail in Chapter 7
(Section 7.3). The RVP and boiling range of gasoline governs
ease of starting, engine warm-up, mileage economy, and ten-
dency toward vapor lock [63]. Vapor lock tendency is directly
related to RVP and at ambient temperature of 21~ (70~
the maximum allowable RVP is 75.8 kPa (11 psia), while this
limit at 32~ (90~ reduces to 55.2 kPa (8 psia) [63]. RVP
can also be used to estimate true vapor pressure of petroleum
fractions at various temperatures as shown in Section 7.3.
True vapor pressure is important in the calculations related
to losses and rate of evaporation of liquid petroleum prod-
ucts. Because RVP does not represent true vapor pressure,
the current tendency is to substitute RVP with more modern
and meaningful techniques [24]. The more sophisticated in-
struments for measurement of TVP at various temperatures
are discussed in ASTM D 4953 test method. This method
can be used to measure RVP of gasolines with oxygenates
and measured values are closer to actual vapor pressures
E4, 24].
As will be discussed in Chapters 6 and 7, accurate calcu-
lation of true vapor pressure requires rigorous vapor liquid
equilibrium (VLE) calculations through equations of state.
The API-TDB [2] method for calculation of RVP requires a
tedious procedure with a series of flash calculations through
Soave cubic equation of state. Simple relations for estima-
tion of RVP have been proposed by Jenkins and White and
are given in Ref. [61]. These relations are in terms of tem-
peratures along ASTM D 86 distillation curve. An example of
these relations in terms of temperatures at 5, i0, 30, and 50
vol% distilled is given below:
RVP = 3.3922 - 0.02537(T5) - 0.070739(T10) + 0.00917(T30)
- 0.0393(Ts0) + 6.8257 x 10- 4( T10) 2
(3.100)
where all temperatures are in~ and RVP is in bar. The diffi-
culty with this equation is that it requires distillation data up
to 50% point and frequently large errors with negative RVP
values for heavier fuels have been observed. Another method
for prediction of RVP was proposed by Bird and Kimball [61 ].
I n this method the gasoline is divided into a number (i.e.,
28) of cuts characterized by their average boiling points. A
blending RVP of each cut is then calculated by the following
equation:
7.641
Bi=
exp (0.03402Tbi + 0.6048)
i = 28
(3.101) Pa = ~ BiXvi
i = l
f -- 1.0 + 0.003744 (VAPB - 93.3)
RVP = fPa
where B~ = RVP is the blending number for cut i and Tbi =
normal boiling point of cut i in ~ Xvi is the volume fraction
of cut i, VABP is the volume average boiling point in~ and
RVP is the Reid vapor pressure in bars. The constants were ob-
tained from the original constants given in the English units.
The average error for this method for 51 samples was 0.12
bar orl.8 psi [67, 68].
Recently some data on RVP of gasoline samples have been
reported by Hatzioznnidis et al. [69]. They measured vapor
pressure according to ASTM D 5191 method and related to
RVP. They also related their measured vapor pressure data to
TVP thus one can obtain RVP from TVP, but their relations
have not been evaluated against a wide range of petroleum
fractions. Other relations for calculation of TVP from RVP
for petroleum fractions and crude oils are given in Section
7.3.3. TVP at 100~ (311 K) can be estimated from Eq. (3.33)
a s
(3.102) logl0(TVP)100=3.204x 1- 4x l ~_ - ~r b
where Tb is the normal boiling point in K and TVP100 is the
true vapor pressure at 100~ (311 K). Once TVP is calculated
it may be used instead of RVP in the case of lack of sufficient
data. When this equation is used to estimate RVP of more than
50 petroleum products an average error of 0.13 bar ( ~ 1.9 psi)
and a maxi mum error of 5.9 psi were obtained [67, 68].
RVP data on 52 different petroleum products (light and
heavy naphthas, gasolines, and kerosenes) from the Oil and
Gas Journal data bank [46] have been used to develop a sim-
ple relation for prediction of RVP in terms of boiling point
and specific gravity in the following form [67]:
RVP = Pc exp(Y)
X (TbSG'~ (1 - Tr) s
r = - ',--g--~ /
X = -276.7445 + 0.06444Tb + 10.0245SG - 0.129TbSG
9968.8675
+ + 44.6778 In Tb + 63.6683 In SG
TbSG
T~ = 311/Tc
(3.103)
where Tb is the mid boiling point and Tc is the pseudocritical
temperature of the fraction in kelvin. Pc is the pseudocritical
pressure and RVP is the Reid vapor pressure in bars. The basis
for development of this equation was to use Miller equation
for TVP and its application at 311 K (100~ The Miller equa-
tion (Eq. 7.13) is presented in Section 7.3.1. The constants
of vapor pressure correlation were related to boiling point
and specific gravity of the fraction. Critical temperature and
pressure may be estimated from Tb and SG using methods
presented in Chapter 2. This equation is based on data with
RVP in the range of 0.0007-1.207 bar (0.01-17.5 psia), normal
boiling point range of 305-494 K, and specific gravity range
of 0.65-1.08. The average absolute deviation for 52 samples is
0.061 bar (0.88 psia). The above equation may be used for cal-
culation of RVP to determine quality characteristics of a fuel.
The calculated RVP value should not be used for calculation
of TVP when very accurate values are needed. (Appropriate
methods for direct estimation of TVP of petroleum fractions
are discussed in Section 7.3.3.) Vapor pressure of a petroleum
mixture depends on the type of its constituents and with use
of only two bulk properties to predict RVP is a difficult task.
This equation is recommended for a quick and convenient
estimation of RVP, but occasionally large errors may be ob-
tained in use of this equation. For more accurate estimation
of RVP the sophisticated method suggested in the API-TDB
[2] may be used. I n this method RVP is calculated through a
series of vapor-liquid-equilibrium calculations.
RVP is one of the main characteristics that is usually used
to blend a fuel with desired specifications. The desired RVP
of a gasoline is obtained by blending naphtha with n-butane
(M = 58, RVP = 3.58 bar or 52 psia) or another pure hydro-
carbon with higher RVPs than the original fuel. For condi-
tions where RVP should be lowered (hot weather), heavier
hydrocarbons with lower RVP are used for blending purposes.
RVP of several pure hydrocarbons are given as follows: i-C4:
4.896 bar (71 psia); n-C4:3.585 (52); i-Cs:1.338 (19.4); n-Cs:
1.0135 (14.7); i-C6:0.441 (6.4); n-C6:0.34 (5.0); benzene: 0.207
(3.0); and toluene: 0.03 (0.5), where all the numbers inside the
parentheses are in psia as given in Ref. [63]. However in the
same reference in various chapters different values of RVP
for a same compound have been used. For example, values of
4.14 bar (60 psi) for n-C4, 1.1 bar (16 psi) for n-Cs, and 0.48
bar (7 psi) for i-C6 are also reported by Gary and Handwerk
[63]. They also suggested two methods for calculation of RVP
of a blend when several components with different RVPs are
blended. The first method is based on the simple Kay's mixing
rule using mole fraction (Xr~) of each component [63]:
(3.104) RVP(blend) = E Xmi (RVP)/
i
where (RVP)i is the RVP of component i in bar or psia. The
second approach is to use blending index for RVP as [63]:
(RVPBI)i = (RVP)] 2s
(3.105) RVPBI (blend) = EXvi(RV PBI)i
i
RVP (blend) = [RVPBI (blend)] ~
where (RVPBI)/ is the blending index for (RVP)i and Xvi is
the volume fraction of component i. Both units of bar or
psia may be used in the above equation. This relation was
originally developed by Chevron and is also recommended
in other industrial manuals under Chevron blending number
[61]. Equations (3.104) and (3.105) may also be applied to
TVP; however, methods of calculation of TVP of mixtures are
discussed in Section 7.3 through thermodynamic relations.
Example 3. 19--Estimate RVP of a gasoline sample has
molecular weight of 86 and API gravity of 86.
Sol ut i on- - API = 86 and M = 86. From Eq. (2.4), SG = 0.65
and from Eq. (2.56), Tb = 338 K. Since only Tb and SG are
known, Eq. (3.103) is used to calculate the RVP. From Eqs.
(2.55) and (2.56) we get Tc -- 501.2 Kand Pc = 28.82 bar. From
Eq. (3.103), Tr = 0.6205, X = 1.3364, and Y = -3.7235. Thus
we calculate RVP = 0.696 bar or 10.1 psia. The experimental
value is I 1.1 psia [63]. #
3. 6. 1. 2 V / L Rat i o and V olatility Index
Once RVP is known it can be used to determine two other
volatility characteristics, namely vapor liquid ratio (V /L) and
fuel volatility index (FVI), which are specific characteristics
of spark-ignition engine fuels such as gasolines. V /L ratio is a
volatility criterion that is mainly used in the United States and
Japan, while FVI is used in France and Europe [24]. The V /L
ratio at a given temperature represents the volume of vapor
formed per unit volume of liquid initially at 0~ The proce-
dure of measuring V /L ratio is standardized as ASTM D 2533.
The volatility of a fuel is expressed as the temperature levels at
which V /L ratio is equal to certain values. Usually V /L values
of 12, 20, and 36 are of interest. The corresponding tempera-
tures may be calculated from the following relations [24]:
T(V /L)12 m_ 88. 5 - - 0. 19E70 - 42.5 RVP
(3.106) T(v/L)2o = 90.6 - 0.25E70 - 39.2 RVP
T~v/L)36 = 94.7 -- 0.36E70 - 32.3 RVP
where T(v/L)x is the temperature in ~ at which V /L = x.
Parameter E70 is the percentage of volume distilled at 70~
E70 and RVP are expressed in percent distilled and bar,
respectively. Through Lagrange interpolation formula it is
possible to derive a general relation to determine temper-
ature for any V /L ratio. E70 can be calculated through a
distribution function for distillation curve such as Eq. (3.35)
in which by rearrangement of this equation we get
E ]
(3.107) E70 = 100- 100 exp - \ To
where To is the initial boiling point in kelvin and together
with parameters A and B can be determined from the
method discussed in Section 3.2.3. Another simple relation
to calculate T(V /L)20 is given in terms of RVP and distillation
temperatures at 10 and 50% [61]:
(3.108) T(V/L)2O = 52.5 + 0.2T10 + 0.17T50 - 33 RVP
where T10 and Ts0 are temperatures at 10 and 50 vol% distilled
on the ASTM D 86 distillation curve. All temperatures are
in~ and RVP is in bar. For cases that T10 is not available it
may be estimated through reversed form of Eq. (3.17) with T50
and SG. Several petroleum refining companies in the United
States such as Exxon and Mobil use the critical vapor locking
index (CVLI), which is also related to the volatility index [61 ].
(3.109) CVLI -- 4.27 + 0.24E70 + 0.069 RVP
The fuel volatility index is expressed by the following relation
[24]:
(3.110) FVI = 1000 RVP + 7E70
where RVP is in bar. FV/ is a characteristic of a fuel for its per-
formance during hot operation of the engine. I n France, spec-
ifications require that its value be limited to 900 in summer,
1000 in fall/spring, and 1150 in the winter season. Automobile
manufacturers in France require their own specifications
that the value of FVI not be exceeded by 850 in summer [24].
3. 6. 1. 3 Flash Poi nt
Flash point of petroleum fractions is the lowest temperature
at which vapors arising from the oil will ignite, i.e. flash,
when exposed to a flame under specified conditions. There-
fore, the flash point of a fuel indicates the maximum tem-
perature that it can be stored without serious fire hazard.
Flash point is related to volatility of a fuel and presence of
light and volatile components, the higher vapor pressure cor-
responds to lower flash points. Generally for crude oils with
RVP greater than 0.2 bar the flash point is less than 20~ [24].
Flash point is an important characteristics of light petroleum
fractions and products under high temperature environment
and is directly related to the safe storage and handling of
such petroleum products. There are several methods of de-
termining flash points of petroleum fractions. The Closed
Tag method (ASTM D 56) is used for petroleum stocks with
flash points below 80 ~ C (175 ~ The Pensky-Martens method
(ASTM D 93) is used for all petroleum products except waxes,
solvents, and asphalts. Equipment to measure flash point ac-
cording to ASTM D 93 test method is shown in Fig. 3.28.
The Cleveland Open Cup method (ASTM D 92) is used for
petroleum fractions with flash points above 80~ (175~ ex-
cluding fuel oil. This method usually gives flash points 3-6~
higher than the above two methods [61]. There are a number
of correlations to estimate flash point of hydrocarbons and
petroleum fractions.
Buffer et al. [70] noticed that there is a linear relationship
between flash point and normal boiling point of hydrocar-
bons. They also found that at the flash point temperatures, the
product of molecular weight (M) and vapor pressure (pvap) for
pure hydrocarbons is almost constant and equal to 1.096 bar
(15.19 psia).
(3.111) M P vap = 1.096
Another simple relation for estimation of flash point of hydro-
carbon mixtures from vapor pressure was proposed by Walsh
and Mortimer [71].
(3.112) TF = 231.2 -- 40 l ogP vav
where pvap is the vapor pressure at 37.8~ (100~ in bar and
TF is the flash point in kelvin. For simplicity RVP may be used
for pwp. Methods of calculation of vapor pressure are dis-
cussed in Chapter 7. Various oil companies have developed
special relations for estimation of flash points of petroleum
fractions. Lenoir [72] extended Eq. (3.100) to defined mix-
tures through use of equilibrium ratios.
The most widely used relation for estimation of flash point
is the API method [2], which was developed by Riazi and
Daubert [73]. They used vapor pressure relation from Clasius-
Clapeyron (Chapter 6) together with the molecular weight
relation form Eq. (2.50) in Eq. (3.111) to develop the following
relation between flash point and boiling point:
(3.113) 1 / TF=a+b / Tb +c l nTb +dl nSG
where Tb is the normal boiling point of pure hydrocarbons.
It was observed that the coefficient d is very small and TF is
FIG, 3.28---Equipment for measurement of flash point of petroleum fractions by A STM
D 93 test method (courtesy of Chemical Engineering Department at Kuwait University),
nearly independent of specific gravity. Based on data from
pure hydrocarbons and some pet rol eum fractions, the con-
stants in Eq. (3.113) were determined as
1 2.84947
(3.114) ~FF = --0. 024209+ T1----~ +3. 4254 10 -3 lnTm
where for pure hydrocarbons T10 is normal boiling point,
while for pet rol eum fractions it is distillation t emperat ure at
10 vol% vaporized (ASTM D 86 at 10%) and it is in kelvin. TF is
the flash point in kelvin determined from the ASTM D 93 test
met hod ( Pensky-Martens closed cup tester). This equation is
presented in Fig. 3.29 for a quick and convenient estimate
of flash point. For 18 pure hydrocarbons and 39 fractions,
Eq. (3.114) predicts flash points with an average absolute de-
viation (AD) of 6.8~ (12~ while Eq. (3.111) predicts the
flash points with AD of 18.3~
Equation (3.114) should be applied to fractions with nor-
mal bor i ng points from 65 to 590~ 150-1100~ Equation
150
o ~176 ILILL
-50
0 100 200 300 400
ASTM 10% Temperature, ~
FIG. 3.29--Prediction of flash
point of petroleum fractions from
Eq. (3.114).
(3.114) is adopted by the API as the standard met hod to esti-
mat e flash point of pet rol eum fractions [2]. I t was shown that
Eq. (3.114) can be simplified into the following linear form
[73]:
(3.115) TF = 15.48 + 0.70704/'1o
where both Tx0 and TF are in kelvin. This equation is applica-
ble to fractions with normal bor i ng points (i.e., ASTM D 86
t emperat ure at 50%) less t han 260~ (500~ For such light
fractions, Eq. (3.115) is slightly more accurate t han Eq.
(3.t14). For heavier fractions Eq. (3.114) should be used.
There are some relations in the literature that correlate flash
points to either the initial boiling point (T10) or the distillation
t emperat ure at 50% point (T50). Such correlations are not ac-
curate over a wide range of fractions, especially when they are
applied to fractions not used in obtaining their coefficients.
Generally reported initial boiling points for pet rol eum frac-
tions are not reliable and if mi d boiling point t emperat ure is
used as the characteristics boiling point it does not truly rep-
resent the boiling point of light component s that are initially
being vaporized. For this reason the correlations in t erms of
distillation t emperat ure at 10% point (7"10) are more accurate
t han the other correlations for estimation of flash points of
pet rol eum fractions. Flash points of pet rol eum fractions may
also be estimated from the pseudocomponent met hod using
the PNA composi t i on and values of flash points of pure hy-
drocarbons from Table 2.2. However, volumetric averaging
of component flash point t hrough Eq. (3.40) generally over-
predicts the flash point of the blend and the blending index
approach described below should be used to estimate flash
point of defined mixtures.
I f the flash point of a pet rol eum fraction or a pet rol eum
product does not meet the required specification, it can be
adjusted by blending the fraction with other compounds hav-
ing different flash points. For example in hot regions where
the t emperat ure is high, heavy hydrocarbons may be added
to a fraction to increase its flash point. The flash point of the
blend should be det ermi ned from the flash point indexes of
the component s as given below [74]:
(3.116) lOgl0 BIF = -6. 1188 + - -
2414
TF - 42.6
where log is the logarithm of base 10, BIF is the flash point
blending index, and TF is the flash point in kelvin. Once BIF
is determined for all component s of a blend, the blend flash
point index (BIB) is determined from the following relation:
(3.117) BIB = ~_ x,,iBIi
where xvi is the volume fraction and BIi is the flash point
blending index of component i. As it will be shown later, the
blending formul a by Eq. (3.117) will be used for several other
properties. Once Bi t s is calculated it should be used in Eq.
(3.116) to calculate the flash point of the blend, TFB. Another
relation for the blending index is given by Hu-Burns [75]:
Tl/X
(3.118) BI F = ~F
where TF is the flash point in kelvin and the best value of x
is -0. 06. However, they suggest that the exponent x be cus-
t omi zed for each refinery to give the best results [61]. The
following example shows application of these methods.
Example 3.20---A kerosene product with boiling range of
175-260~ from Mexican crude oil has the API gravity of 43.6
(Ref. [46], p. 304). (a) Estimate its flash point and compare
with the experimental value of 59~ (b) For safety reasons it
is required to have a mi ni mum flash point of 65~ to be able
to store it in a hot summer. How much n-tetradecane should
be added to this kerosene for a safe storage?
Solutions(a) To estimate flash point we use either Eq.
(3.114) or its simplified form Eq. (3.15), which require ASTM
10% t emperat ure, T10. This t emperat ure may be estimated
from Eq. (3.17) with use of specific gravity, SG = 0.8081, and
ASTM 50% temperature, Ts0. Since complete ASTM curve is
not available it is assumed that the mi d boiling point is the
same as Ts0; therefore, T50 = 217.5~ and from Eq. (3.17) with
coefficients in Table 3.4,/'10 = 449.9 K. Since Ts0 is less t han
260~ Eq. (3.115) can be used for simplicity. The result is
TF = 60.4~ which is in good agreement with the experimen-
tal value of 59~ considering the fact that an estimated value
of ASTM 10% t emperat ure was used.
(b) To increase the flash point from 59 to 65~ n-Ct4 with
flash point of 1000C (Table 2.2) is used. I f the vol ume frac-
tion of ~/-C14 needed is shown by Xadd, then using Eq. (3.117)
we have BIFB = (1 --Xadd) X BIvK + Xadd X BIVadd where BI ~,
BIFK, and BIradd are the blending indexes for flash points
of final blend, kerosene sample, and the additive (n-C14),
respectively. The blending indexes can be estimated from
Eq. (3.116) as 111.9, 165.3, and 15.3, respectively, which re-
sult in xaaa = 0.356. This means that 35.6% in volume of rt-C14
is required to increase the flash point to 65~ I f the blending
indexes are calculated from Eq. (3.118), the amount of r/-C14
required is 30.1%. ,
3 . 6. 2 Po ur Po i nt
The pour point of a pet rol eum fraction is the lowest t empera-
ture at which the oil will pour or flow when it is cooled without
stirring under standard cooling conditions. Pour point repre-
sents the lowest t emperat ure at which an oil can be stored
and still capable of flowing under gravity. Pour point is one
of low t emperat ure characteristics of heavy fractions. When
t emperat ure is less t han pour point of a pet rol eum product it
cannot be stored or transferred t hrough a pipeline. Test pro-
cedures for measuri ng pour points of pet rol eum fractions are
given under ASTM D 97 (ISO 3016 or IP 15) and ASTM D 5985
methods. For commerci al formulation of engine oils the pour
point can be lowered to the limit of - 25 and -40~ This is
achieved by using pour point depressant additives t hat inhibit
the growth of wax crystals in the oil [5]. Presence of wax and
heavy compounds increase the pour point of pet rol eum frac-
tions. Heavier and more viscous oils have higher pour points
and on this basis Riazi and Daubert [73] used a modified ver-
sion of generalized correlation developed in Chapter 2 (Eq.
2.39) to estimate the pour point of pet rol eum fractions from
viscosity, molecular weight, and specific gravity in the follow-
ing form:
Tp = 130.47[SG 297~ x [M (~176
F (0.310331-0.32834SG) 1
(3.119) x lP38(100) J
where Tp is the pour point (ASTM D 97) in kelvin, M is the
molecular weight, and v38o00) is the kinematic viscosity at
37.8~ (100~ in eSt. This equation was developed with data
on pour points of more t han 300 pet rol eum fractions with
molecular weights ranging from 140 to 800 and API gravities
from 13 to 50 with the AAD of 3.9~ [73]. This met hod is also
accepted by the API and it is included in the API-TDB since
1988 [2] as the standard met hod to estimate pour point of
pet rol eum fractions. As suggested by Hu and Burns [75, 76],
Eqs. (3.117) and (3.118) used for blending index of flash point
can also be used for pour point blending index (TpB) with
x = 0.08 :
T1/ 0. 08
(3.120) BIp = ~p
where Tp is the pour point of fraction or blend in kelvin. The
AAD of 2.8~ is reported for use of Eqs. (3.117) and (3.120)
to estimate pour points of 47 blends [76].
3 . 6. 3 Cl oud Poi nt
The cloud point is the lowest t emperat ure at which wax crys-
tals begin to form by a gradual cooling under standard con-
ditions. At this t emperat ure the oil becomes cloudy and the
first particles of wax crystals are observed. The standard pro-
cedure to measure the cloud point is described under ASTM
D 2500, IP 219, and ISO 3015 test methods. Cloud point
is anot her cold characteristic of pet rol eum oils under low-
t emperat ure conditions and increases as molecular weight of
oil increases. Cloud points are measured for oils that con-
tain paraffins in the form of wax and therefore for light frac-
tions, such as napht ha or gasoline, no cloud point data are
reported. Cloud points usually occur at 4-5~ (7 to 9~ above
the pour point although the t emperat ure differential could be
in the range of 0-10~ (0-18~ as shown in Table 3.27. The
TABLE 3.27--Cloud and pour points and their differences for some petroleum products.
Fraction API gravity T~,, ~ C TCL, ~ C Tp -TcL,
Indonesian Dist. 33.0 -43.3 -53.9 10.6
Australian GO 24.7 -26.0 -30.0 4.0
Australian HGO 22.0 -8. 0 -9. 0 1.0
Abu Dhabi LGO 37.6 -19.0 -27.0 8.0
Abu Dhabi HGO 30.3 7.0 2.0 5.0
Abu Dhabi Disst. 21.4 28.0 26.0 2.0
Abu Dhabi Diesel 37.4 - 12.0 - 12.0 0.0
Kuwaiti Kerosene 44.5 -45.0 -45.0 0.0
Iranian Kerosene 44.3 -46.7 -46.7 0.0
Iranian Kerosene 42.5 -40.6 -48.3 7.8
Iranian GO 33.0 - 11.7 - 14.4 2.8
North Sea GO 35.0 6.0 6.0 0.0
Nigerian GO 27.7 -32.0 -33.0 1.0
Middle East Kerosene 47.2 -63.3 -65.0 1.7
Middle East Disst. 38.9 -17.8 -20.6 2.8
Source: Ref. [46].
Tp: pour point; TcL: cloud point.
~
difference bet ween cl oud and p our poi nt depends on t he na-
t ure of oil and there is no simplified correl at i on to predi ct this
difference. Cl oud poi nt is one of the i mpor t ant characteris-
tics of crude oils under l ow-t emperat ure conditions. As t em-
perat ure decreases bel ow the cl oud point, format i on of wax
crystals is accelerated. Therefore, low cl oud poi nt product s
are desirable under l ow-t emperat ure conditions. Wax crys-
tals can pl ug the fuel syst em lines and filters, whi ch coul d
lead to stalling aircraft and diesel engines under cold condi-
tions. Since cl oud poi nt is hi gher t han p our point, it can be
consi dered t hat the knowl edge of cl oud poi nt is mor e impor-
t ant t han the p our poi nt in establishing distillate fuel oil spec-
ifications for cold weat her usage [61]. Table 3.27 shows the
difference bet ween cl oud and p our points for some pet rol eum
product s. Cloud and p our poi nt s are also useful for predict-
ing the t emperat ure at whi ch t he observed viscosity of an oil
deviates from t he t rue ( newt oni an) viscosity in the low tem-
perat ure range [7]. The amount of n-paraffins in pet rol eum
oil has direct effect on the cl oud poi nt of a fract i on [8]. Pres-
ence of gases dissolved in oil reduces the cl oud poi nt whi ch is
desirable. The exact cal cul at i on of cl oud poi nt requires sol i d-
liquid equi l i bri um calculations, whi ch is discussed in Chap-
ter 9. The bl endi ng index for cl oud poi nt is cal cul at ed from
the same relation as for p our poi nt t hr ough Eq. (3.118) wi t h
x = 0.05 :
(3.121) BIcL = T~/~176
where TCL is t he cl oud poi nt of fract i on or bl end in kelvin.
Accuracy of this met hod of calculating cl oud poi nt of blends is
t he same as for the p our poi nt (AAD of 2.8~ Once the cl oud
poi nt index for each comp onent of blend, BIcLi, is det ermi ned
t hr ough Eq. (3.12 t), the cl oud poi nt index of the blend, Blczs,
is cal cul at ed t hr ough Eq. (3.117). Then Eq. (3.121) is used in
its reverse f or m to calculate cl oud poi nt of the blend f r om its
cl oud poi nt index [76].
3 . 6. 4 Freez i ng Poi nt
Freezi ng poi nt is defined in Section 2.1.9 and freezing poi nt s
of pure hydr ocar bons are given in Table 2.2. For a pet rol eum
fraction, freezing poi nt test involves cooling the sampl e until a
sl urry of crystals form t hr oughout the sampl e or it is t he tem-
perat ure at whi ch all wax crystals di sappear on r ewar mi ng
the oil [61]. Freezi ng poi nt is one of t he i mpor t ant charac-
teristics of aviation fuels where it is det ermi ned by t he pro-
cedures descri bed in ASTM D 2386 (U.S.), I P 16 ( England) ,
and NF M 07-048 ( France) test met hods. Maxi mum freezing
poi nt of jet fuels is an i nt ernat i onal specification whi ch is re-
qui red to be at - 47~ ( - 53~ as specified in the "Aviation
Fuel Quant i t y Requi rement s for Jointly Operat ed Systems"
[24]. This maxi mum freezing poi nt indicates the lowest t em-
perat ure t hat t he fuel can be used wi t hout risk of separat i on
of solidified hydr ocar bons (wax). Such separat i on can result
in the blockage in fuel tank, pipelines, nozzles, and filters [61 ].
Wal sh-Mort i mer suggest a t her modynami c model based on
the solubility of n-paraffin hydr ocar bons in a pet rol eum mix-
t ure to det ermi ne the freezing poi nt [71]. Accurat e determi-
nat i on of freezing poi nt requires accurat e knowl edge of t he
composi t i on of a fuel whi ch is normal l y not known. However,
the met hod of det ermi nat i on of car bon number di st ri but i on
al ong wi t h solid-liquid equi l i bri um can be used t o det ermi ne
freezing poi nt s of pet rol eum fract i ons and crude oils as will be
di scussed in Chapt er 9. A si mpl er but less accurat e met hod to
det ermi ne freezing points of pet rol eum fract i ons is t hr ough
the p seudocomp onent ap p r oach as shown in t he following
example.
Exampl e 3. 21- - A kerosene sampl e pr oduced from a crude
oil from Nor t h Sea Ekofisk field has t he boiling range of 150-
204.4~ (302--400~ and API gravity of 48.7. Est i mat e t he
freezing poi nt of this kerosene and comp ar e wi t h the experi-
ment al value of - 65~ ( -85~
Sol ut i on- - The mi d boiling poi nt is Tb = 177.2~ and t he spe-
cific gravity is SG = 0.785. We use the met hod of pseudo-
comp onent usi ng predi ct ed composi t i on. Fr om Eq. (2.50),
M = 143 and since M > 143, we use Eqs. (3.77), (3.78), and
(3.72) to predi ct Xp, XN, and XA, respectively. Fr om Eqs. (2.114)
and (2.115), n = 1.439 and f r om Eq. (3.50), m = -5. 1515. Us-
ing SG and m, we calculate t he PNA composi t i on as xp =
0.457,xN = 0.27, and XA = 0.273. Fr om Eqs. (3.41)-(3-43),
Mp = 144.3, MN = 132.9, and MA = 129.3. Using Eq. (2.42)
for predi ct i on of the freezing poi nt for different families we
get TFp = 242.3, TFN = 187.8, and TrA = 178.6 K. Using Eq.
(3.40) we get TF = 210.2 K or -63. 1~ versus t he measur ed
FIG. 3 .3 0~ A pparatus to measure aniline point of petroleum fuels by
A STM D 611 test method (courtesy of KISR).
value of -65~ The result is quite satisfactory considering
mi ni mum data on Tb and SG are used as the only available
parameters.
3 . 6. 5 Ani l i ne Po i nt
Aniline point of a petroleum fraction is defined as the mini-
mum temperature at which equal volumes of aniline and the
oil are completely miscible. Method of determining aniline
point of petroleum products is described under ASTM D 611
test method and the apparatus is shown in Fig. 3.30. Aniline
point indicates the degree of aromaticity of the fraction. The
higher the aniline point the lower aromatic content. For this
reason aromatic content of kerosene and jet fuel samples may
be calculated from aniline point [59]:
(3.122) %A = 692.4 + 12.15(SG)(AP) - 794(SG) - 10.4(AP)
where %A is the percent aromatic content, SG is the specific
gravity, and AP is the aniline point in~ There are a number
of methods to estimate aniline point of petroleum fractions.
We discuss four methods in this section,
3. 6. 5. 1 Winn M ethod
Aniline point can be estimated from Winn nomograph
(Fig. 2.14) using Tb and SG or M and SG as the input pa-
rameters.
3. 6. 5. 2 Walsh-M ortimer
The aniline point can be calculated from the following rela-
tion [61,71]:
100.5c /03
(3.123) AP = -204. 9 - 1.498Cs0 +
SG
where AP is the aniline point in ~ and C5o is the carbon
number of n-paraffin whose boiling point is the same as the
mid boiling point of the fraction. C50 may be calculated from
Mp - 14
(3.124) C50 - - -
2
in which Me is the molecular weight of n-paraffin whose boil-
ing point is the same as mid boiling point of the fraction which
can be determined from Eq. (3.41).
3. 6. 5. 3 Linden M ethod
This relation is a mathematical representation of an earlier
graphical method and is given as [73]
(3.125) AP = -183. 3 + 0.27(API)T~/3 + 0.317Tb
where AP is in ~ Tb is the mid boiling point in kelvin and
API is API gravity. The blending index for aniline point may be
calculated from the following relation developed by Chevron
Research [61 ]:
(3.126) BIAp = 1.124[exp (0.00657AP)]
where AP is in ~ and BIAp is the blending index for the aniline
point. Once the blending indexes of components of a blend are
determined, Eq. (3.117) should be used to calculate blending
index for aniline point of the blend.
3. 6. 5. 4 Albahri et al. M ethod
Most recently Mbahri et al. [68] developed predictive meth-
ods for determination of quality of petroleum fuels. Based on
the idea that aniline point is mainly related to the aromatic
content of a fuel, the following relation was proposed:
(3.127) AP = -9805.269(Ri) + 711.85761(SG) + 9778.7069
where AP is in ~ and Ri is defined by Eq. (2.14). Equations
(3.123), (3.125), and (3.127) were evaluated against data on
aniline points of 300 fuels with aniline point range: 45-107~
boiling range: 115-545~ and API gravity range of 14-56. The
average absolute deviation (AAD) for Eq. (3.127) was 2.5~
while for Eqs. (3.123) and (3.125) the errors were 4.6 and
6.5~ respectively [68]. Error distribution for Eq. (3.127) is
shown in Fig. 3.31.
3 . 6. 6 Ce t ane N u mb e r a nd D i e s e l I nde x
For diesel engines, the fuel must have a characteristic that fa-
vors auto-ignition. The ignition delay period can be evaluated
1 3 8 CHARACTERI Z ATI ON AND PROPERTI ES OF PETROL EUM FRACTI ONS
120.0
110.0
100.0
90.0
fl 80.0 . 9 9
70.0 , ,
60.0 . - ~ . ~"
"%Z "
~o.o " / :
40.0 /
30.0
30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0 120.0
Experimental Aniline Point, ~
FIG. 3 . 3 1 - - Er r or distribution for prediction of ani l i ne point
from Eq. (3.127). Taken with permission from Ref. [ 68] .
by the fuel characterization factor called cetane number (CN).
The behavior of a diesel fuel is measured by comparing
its performance with two pure hydrocarbons: n-cetane or
n-hexadecane (n-C16H34) which is given the number 100 and
a-methylnaphthalene which is given the cetane number of
0. A diesel fuel has a cetane number of 60 if it behaves
like a binary mixture of 60 vol% cetane and 40 vol% ~-
methylnaphthalene. I n practice heptamethylnonane (HMN)
a branched isomer of n-cetane with cetane number of 15 is
used instead of a-methylnaphthalene [24, 61]. Therefore, in
practice the cetane number is defined as:
(3.128) CN = vol%(n-cetane) + 0.15(vo1% HMN)
The cetane number of a diesel fuel can be measured by the
ASTM D 613 test method. The shorter the ignition delay pe-
riod the higher CN value. Higher cetane number fuels re-
duce combustion noise and permit improved control of com-
bustion resulting in increased engine efficiency and power
output. Higher cetane number fuels tend to result in easier
starting and faster warm-up in cold weather. Cetane num-
ber requirement of fuels vary with their uses. For high speed
city buses in which kerosene is used as fuel the required CN
is 50. For premium diesel fuel for use in high speed buses
and light marine engines the required number is 47 while for
marine distillate diesel for low speed buses and heavy marine
engines the required cetane number is 38 [61]. I n France the
mi ni mum required CN of fuels by automotive manufacturers
is 50. The product distributed in France and Europe have CN
in the range of 48-55. I n most Scandinavian countries, the
United States and Canada the cetane number of diesel fuels
are most often less than 50. Higher cetane number fuels in
addition to better starting condition can cause reduction in
air pollution [24].
Since determination of cetane number is difficult and
costly, ASTM D 976 (IP 218) proposed a method of calcu-
lation. Calculated number is called calculated cetane index
(CCI) and can be determined from the following relation:
CCI = 454.74 - 1641.416SG + 774.74SG 2
(3.129) - 0.554Ts0 + 97.083(log10 T50) 2
where/ ' so is the ASTM D 86 temperature at 50% point in ~
Another characteristic of diesel fuels is called diesel index (DI)
defined as:
(API)(1.8AF + 32)
(3.130) OI =
100
which is a function of API gravity and aniline point in ~
Cetane index is empirically correlated to DI and AP in the
following form [24]:
(3.131) CI = 0.72DI + 10
(3.132) CI = AP - 15.5
where AP is in ~ Calculated cetane index (CI) is also related
to n-paraffin content (%NP) of diesel fuels in the following
from [87].
(3.133) %NP = 1.45CI - 57.5
The relation for calculation of cetane number blending in-
dex is more complicated than those for pour and cloud point.
Blending indexes for cetane number are tabulated in various
sources [61, 75]. Cetane number of diesel fuels can be im-
proved by adding additives such as 2-ethyl-hexyl nitrate or
other types of alkyl nitrates. Cetane number is usually im-
proved by 3-5 points once 300-1000 ppm by weight of such
additives is added [24]. Equation (3.129) suggested for cal-
culating cetane number does not consider presence of addi-
tives and for this reason calculated cetane index for some
fuels differ with measured cetane index. Generally, CCI is less
than measured CN and for this reason in France automobile
manufacturers have established mi ni mum CN for both the
calculated CI (49) and the measured CN (50) for the quality
requirement of the fuels [24].
3 . 6. 7 Oc t a ne N u mb e r
Octane number is an important characteristic of spark en-
gine fuels such as gasoline and jet fuel or fractions that
are used to produce these fuels (i.e., naphthas) and it rep-
resents antiknock characteristic of a fuel. Isooctane (2,2,4-
trimethylpentane) has octane number of 100 and n-heptane
has octane number of 0 on both scales of RON and MON.
Octane number of their mixtures is determined by the vol%
of isooctane used. As discussed in Section 2.1.13, isoparaffins
and aromatics have high octane numbers while n-paraffins
and olefins have low octane numbers. Therefore, octane num-
ber of a gasoline depends on its molecular type composition
especially the amount of isoparaffins. There are two types of
octane number: research octane number (RON) is measured
under city conditions while motor octane number (MON)
is measured under road conditions. The arithmetic average
value of RON and MON is known as posted octane number
(PON). RON is generally greater than MON by 6-12 points,
although at low octane numbers MON might be greater than
RON by a few points. The difference between RON and MON
is known as sensitivity of fuel. RON of fuels is determined
by ASTM D 908 and MON is measured by ASTM D 357 test
methods. Generally there are three kinds of gasolines: regu-
lar, intermediate, and premium with PON of 87, 90, and 93,
respectively. I n France the mi ni mum required RON for su-
perplus gasoline is 98 [24]. Required RON of gasolines vary
with parameters such as air temperature, altitude, humidity,
engine speed, and coolant temperature. Generally for every
300 m altitude RON required decreases by 3 points and for
every I I~ rise in temperature RON required increases by 1.5
points [63]. I mproving the octane number of fuel would result
in reducing power loss of the engine, improving fuel economy,
and a reduction in environmental pollutants and engine dam-
age. For these reasons, octane number is one of the important
properties related to the quality of gasolines. There are a num-
ber of additives that can improve octane number of gasoline
or jet fuels. These additives are tetra-ethyl lead (TEL), alco-
hols, and ethers such as ethanol, methyl-tertiary-butyl ether
(MTBE), ethyl-tertiary-butyl ether (ETBE), or tertiary-amyl
methyl ether (TAME). Use of lead in fuels is prohibited in
nearly all industrialized countries due to its hazardous nature
in the environment, but is still being used in many third world
and underdeveloped countries. For a fuel with octane number
(ON) of 100, increase in the ON depends on the concentra-
tion of TEL added. The following correlations are developed
based on the data provided by Speight [7]:
TEL = -871. 05 +2507.81 ~ - 2415.94 \ 100/
(oN 3
(3.134) +779.12 \ 100]
ON = 100.35 + 11.06(TEL) - 3.406(TEL) 2
(3.135) + 0.577(TEL) 3 - 0.038(TEL) 4
where ON is the octane number and TEL is milliliter TEL
added to one U.S. gallon of fuel. These relations nearly repro-
duce the exact data given by Speight and valid for ON above
100. I n these equations when clear octane number (without
TEL) is 100, TEL concentration is zero. By subtracting the cal-
culated ON from 100, the increase in the octane number due
to the addition of TEL can be estimated, which may be used
to calculate the increase in ON of fuels with clear ON different
from 100. Equation (3.134) is useful to calculate amount of
TEL required for a certain ON while Eq. (3.135) gives ON of
fuel after a certain amount of TEL is added. For example, if
0.3 mL of TEL is added to each U.S. gallon of a gasoline with
RON of 95, Eq. (3.135) gives ON of 104.4, which indicates an
increase of 4.4 in the ON. This increase is based on the refer-
ence ON of 100 which can be used for ON different from 100.
Therefore, the ON of gasoline in this example will be 95 + 4.4
or 99.4. Different relations for octane number of various fuels
(naphthas, gasolines, and reformates) in terms of TEL con-
centration are given elsewhere [88].
Octane numbers of some oxygenates (alcohols and ethers)
are given in Table 3.28 [24]. Once these oxygenates are added
to a fuel with volume fraction of :Cox the octane number of
product blend is [24]
(3.136) ON = Xox(ON)ox + (1 - Xox)(ON)clear
where ONclear is the clear octane number (RON or MON) of
a fuel and ON is the corresponding octane number of blend
T A B L E 3. 28--Octane numbers of some alcohols and
ethers (oxygenates).
Compound RON MON
Methanol 125-135 100-105
MTBE 113-I17 95-101
Ethanol 120-130 98-103
ETBE 1 1 8 - 1 2 2 100-102
TBA 105-110 95-100
TAME 1 1 0 - 1 1 4 96-100
MTBE: met hyl - t er t i ar y- but yl et her; ETBE: et hyl - t er t i ar y- but yl et her ;
TBA: t er t i ar y- but yl al cohol ; TAME: t er t i ar y- amyl - met hyl et her.
Sour ce: Ref. [24],
after addition of an additive. ONox is the corresponding octane
number of oxygenate, which can be taken as the average val-
ues for the ranges of RON and MON as given in Table 3.28. For
example for MTBE, the range of RONox is 113-117; therefore,
for this oxygenate the value of RONox for use in Eq. (3.136)
is 115. Similarly the value MONox for this oxygenate is are
98. Equation (3.136) represents a simple linear relation for
octane number blending without considering the interaction
between the components. This relation is valid for addition
of additives in small quantities (low values of Xox, i.e., < 0.15).
However, when large quantities of two components are added
(i.e., two types of gasolines on 25:75 volume basis), linear mix-
ing rule as given by Eq. (3.136) is not valid and the interac-
tion between components should be taken into account [61 ].
Du Pont has introduced interaction parameters between two
or three components for blending indexes of octane number
which are presented in graphical forms [89]. Several other
blending approaches are provided in the literature [61]. The
simplest form of their tabulated blending indexes have been
converted into the following analytical relations:
BIRoN =
2 3 4 5
36. 01+38. 33X- 99, 8X +341. 3X - 507. 2X +268. 64X 11_ < RON< 76
- 299. 5 + 1272X - 1552.9X 2 + 651X 3 76 _< RON _< 103
2206. 3- 4313. 64X+ 2178. 57X 2 103 _< RON < 106
/ x = RON/IO0
(3.137)
where BIRoN is the blending index for RON and should be used
together with Eq. (3.117) to calculate RON of a blend. Equa-
tion (3.137) reproduce the tabulated values of RON blending
indexes with AAD of 0.06%.
Estimation of octane number of a fuel from its bulk proper-
ties is a challenging task, since ON very much depends on the
chemical structure of components of the mixture. Figure 3.32
shows variation of RON with boiling point of pure hydrocar-
bons from different families as produced from data given in
Table 2.2. If PIONA composition of a fuel is known, RON of a
fuel may be estimated from the pseudocomponent techniques
in the following form:
RON = x~p(RON)Np + xI~(RON)Ip + xo(RON)o
(3.138) + xN (RON)N + XA(RON)A
where x is the volume fraction of different hydrocarbon
families i.e., n-paraffins (NP), isoparaffins (IP), olefins (O),
naphthenes (N), and aromatics (A). RONNp, RONIp, RONo,
RONN, and RONA are the values of RON of pseudocompo-
nents from n-paraffin, isoparaffins, olefins, naphthenes, and
aromatics families whose boiling points are the same as the
,2o ; 7 - : : .......... ...... . " ................. --omZ; 5l
, o o .
~\ l : ~ ~phthenes
.~ 40 n-paraft~s \ k
0 A n-alky Icy clop ent anes : : x
O n-alkylbenzenes
-20 . . . . . ' . . . . ' . . . . ' . . . . i . . . . , , i ;
-100 -50 0 50 100 150 200
Boiling Point, ~
FIG. 3 . 3 2- - Resear ch octane number of di fferent
fami l i es of hydrocarbons. Taken wi th permi ssi on
from Ref. [ 68 ],
mi d boi l i ng p oi nt or t he ASTM D 86 t emp er at ur e at 50%
p oi nt of t he fract i on and can be det er mi ned f r om Fig. 3.32
or Table 2.2. Gener al l y p et r ol eum p r oduct s ar e free of olefins
and t he mai n gr oup s p r esent i n a p et r ol eum p r oduct s are
n-paraffi ns, i soparaffi ns, nap ht henes, and ar omat i cs. The rol e
of i soparaffi ns on oct ane number is si gni fi cant as t hey have
ON val ues gr eat er t han n-paraffi ns. I n addi t i on di fferent t ypes
of i soparaffi ns have di fferent oct ane number s at t he same
boi l i ng poi nt . As t he number of br anches in an i so-paraffi n
comp ound i ncr eases t he oct ane number al so i ncreases. F or
t hi s r eason it woul d be mor e ap p r op r i at e if RONIp in Eq.
(3.138) is an average val ue of oct ane number s of var i ous t ypes
of i soparaffi ns. For conveni ence and comp ut er cal cul at i ons,
val ues of RON for t hese var i ous homol ogous hydr ocar bon
gr oup s have been cor r el at ed t o nor mal boi l i ng poi nt , Tb in
t he fol l owi ng form:
(3.139) RON =a +b T +cT2 +dT3 +eT 4
wher e RON is t he cl ear r esear ch oct ane number and T =
( Tb- - 273. 15) / 100 in whi ch Tb is t he boi l i ng p oi nt i n
kelvin. Based on t he dat a t aken f r om t he API -TDB [2],
t he coeffi ci ent s a - e were det er mi ned and are gi ven i n
Table 3.29 [68, 78]. I t shoul d be not ed t hat for i sopar af -
fins t he coeffi ci ent s are gi ven for f our di fferent gr oup s of
2- met hyl pent anes, 3- met hyl pent anes, 2, 2- di met hyl pent anes,
and 2, 3- di met hyl pent anes. Oct ane number s of var i ous
i soparaffi ns var y si gni fi cant l y and for t hi s r eason an average
val ue of RON for t hese f our di fferent i so-paraffi ni c gr oup s is
consi der ed as t he val ue of RONIp for use in Eq. (3.138).
8o
! i I I I
6 0 i ' - ' I , , o ~ ,
~ so ,
~ 40 .
$ 03~ t J'l " ol' oox: "< ~176 r' ""
j I . 1
I I . 3 11.4 1T.5 11.6 11.7 11.8 11,9 12.0 12.1 12.2 12.3
Characteri zati on Factor
FIG. 3 . 3 3 - - Resear ch octane number of naphthas ( ~ = 1 .8 x
~ + 3 2) . Taken wi th permi ssi on from Ref. [ 79].
Nor mal l y when det ai l ed PI ONA comp os i t i on is not avail-
able, PNA comp os i t i on is p r edi ct ed f r om t he met hods pre-
sent ed in Sect i on 3.5.1. For such cases Eq. (3.138) may he
si mpl i fi ed by consi der i ng xo = 0 and XNp = Xip = xp/2. Be-
cause RON of n-paraffi ns and i soparaffi ns di ffers si gni fi cant l y
(Fig. 3.32), t he as s ump t i on of equal amount s of n-paraffi ns
and i soparaffi ns can l ead to subst ant i al er r or s in cal cul at i on
of RON for fuels whose nor mal and iso paraffi ns cont ent s dif-
fer significantly. For such cases t hi s met hod est i mat es RON of
a fuel wi t h a hi gher er r or but requi res mi ni mum i nf or mat i on
on di st i l l at i on and specific gravity.
Nel son [79] gives gr ap hi cal r el at i on for est i mat i on of RON
of nap ht has in t er ms of Kw char act er i zat i on f act or or paraffi n
cont ent ( wt%) and mi d boi l i ng p oi nt as given i n Figs. 3.33
and 3.34, respectively.
As ment i oned ear l i er if amount of paraffi ns in wt % is not
avai l abl e, vol % may be used i nst ead of wt % if necessary. Once
RON is det er mi ned, MON can be cal cul at ed f r om t he fol l ow-
i ng r el at i on p r op os ed by J enki ns [80]:
MON = 22.5 + 0.83 RON - 20.0 SG - 0.12 ( %0)
(3.140) + 0.5 ( TML) + 0.2 ( TEL)
wher e SG is t he specific gravity, TML and TEL ar e t he con-
cent r at i ons of t et r a met hyl l ead and t et ra et hyl l ead in mL/ UK
gal l on, and %0 is t he vol % of olefins in t he gasol i ne. F or olefin-
and l ead-free fuels ( %0 = TML = TEL = 0) and Eq. (3.140)
r educes to a si mp l e f or m i n t er ms of RON and SG. F r om t hi s
TABLE 3.29--Coefficients for Eq. (3.139) for estimation of RON [68, 78].
Hydrocarbon family a b c d
n-Paraffins 92.809 -70.97 - 53 20 10
isoparaffins
2-Methyl-pentanes 95.927 -157.53 561 -600 200
3-Methyl-pentanes 92.069 57.63 - 65 0 0
2,2-Dimethyl-pentanes 109.38 -38.83 - 26 0 0
2,3-Dirnethyl-pentanes 97.652 -20. 8 58 -200 100
Naphthenes -77.536 471.59 -418 100 0
Aromatics 145.668 -54.336 16.276 0 0
Taken with permission from Ref. [68].
9 i i !
6C ~ tm
Z 50 " ' " 1 7 5~ P"
,
%' 200 ~ F.
4~ J \ \ "
rr 30 \ .-
\ ,
i \
X \ ~ !oo ~ F.
20 I i x
\ " ? 50 ~ P. I
I
10 I i
10 20 30 40 50 60 70 80 90 100
W t% Paraffin Hydrocarbons
FI G . 3 . 3 4- - F i e s e a r c h oc t a ne n u mb e r v e r s u s par af f i n c ont e nt
( ~ ---- 1 . 8 x ~ + 3 2 ) . T ak en wi t h p e r mi s s i o n f r om Fief. [ 7 g ].
equat i on sensi t i vi t y of gasol i ne can be det er mi ned. Gasol i nes
wi t h l ower sensi t i vi t y ar e desi rabl e.
Example 3. 22~A nap ht ha samp l e f r om an Aust r al i an cr ude
oil has t he fol l owi ng char act er i st i cs: boi l i ng p oi nt r ange
15. 5- 70~ specific gravi t y 0.6501, n-paraffi ns 49.33%,
i soparaffi ns 41.45%, nap ht henes 9.14%, ar omat i cs 0.08%,
cl ear RON 69.6, and MON 66.2 [Ref. [46], p. 359).
a. Est i mat e RON f r om t he p s eudocomp onent met hod usi ng
exp er i ment al comp osi t i on.
b. Est i mat e RON f r om t he p s eudocomp onent met hod usi ng
p r edi ct ed PNA comp osi t i on.
c. Est i mat e RON f r om Fig. 3.33.
d. Est i mat e RON f r om Fig. 3.34.
e. Est i mat e MON f r om act ual r ep or t ed RON.
f. Est i mat e MON f r om p r edi ct ed RON f r om Par t a.
g. F or each case cal cul at e t he er r or ( devi at i on bet ween esti-
mat ed and r ep or t ed values) .
Solution- - For t hi s fract i on: Tb = (15.5 + 70) / 2 = 42.8~
SG = 0.6501,Xp = 0.4933, xw = 0.4145, XN = 0.0914, XA :
0.008.
a. RON can be est i mat ed f r om Eq. (3.138) t hr ough pseu-
docomp onent met hod usi ng RON val ues for p ur e hydr o-
car bons cal cul at ed f r om Eq. (3.139) and Table 3.29
wi t h Tb = 315.9 K. Resul t s of cal cul at i on ar e (RON)np =
54.63, (RON)Ip = (90.94 + 104.83 + 88 + 87. 05) / 4 = 92.7,
(RON)N -- 55.57, and (RON)A ---- 125.39. I n cal cul at i on of
( RON) w, an average val ue for RON of f our fami l i es i n
Table 3.29 is cal cul at ed. F r om Eq. (3.139), cl ear RON can
be cal cul at ed as :
RON = 0.4933 54.63 + 0.4145 x 92.7 + 0.0914 x 55.57
+0. 0008 125.39 = 70.55.
I n comp ar i s on wi t h t he r ep or t ed val ue of 69.6 t he er r or is
70.55 - 69.6 = 0.95.
b. To p r edi ct PNA, we cal cul at e M f r om Eq. (2.50) as M =
79.54. Si nce M < 200 and vi scosi t y is not avai l abl e we use
Eqs. (3.77) and (3.78) and (3.72) to p r edi ct t he comp osi -
t i on. F r om Eqs. (2.126) and (2.127), n20 = 1.3642 and f r om
Eq. (3.50), m= - 8. 8159. The p r edi ct ed comp os i t i on for
%P, %N, and %A is 95.4%, 7.4%, and - 2. 8%, respect i vel y.
Si nce p r edi ct ed %A is negat i ve, it is set equal t o zero and
t he nor mal i zed comp os i t i on is xp -- 0.928, xN -- 0.072, and
XA = 0.0. To use Eq. (3.139) we spl i t t he paraffi n cont ent
equal l y bet ween n-paraffi ns and i soparaffi ns as xNp = Xip =
0. 928/ 2 = 0.464. I n t hi s case RON = 72.36. The er r or on
cal cul at i on of RON is 2.76.
c. To use Fig. 3.33 we need Tb = 42.8~ -- 109~ and Kw,
whi ch f r om Eq. (2.13) is cal cul at ed as Kw -- 12.75. Si nce
t he Kw is out si de t he r ange of val ues in Fig. 3.33, accur at e
r eadi ng is not possi bl e, but f r om val ue of t he boi l i ng p oi nt
it is obvi ous t hat t he RON f r om ext r ap ol at i on of t he curves
is above 70.
d. To use Fig. 3.34 we need t ot al paraffi ns whi ch is %- -
49.33 + 41.45 = 90.78 and Tb = 109~ I n t hi s case Tb is out -
si de t he range of val ues on t he curves, but wi t h ext r apol a-
t i on a val ue of about 66 can be read. The er r or is about
- 3. 6.
e. To cal cul at e MON we use Eq. (3.140) wi t h R ON=
69.6, SG = 0.6501, and %0 -- TML = TEL ----- 0. The esti-
mat ed val ue is (MON)e~t. = 67.3, whi ch is in good agree-
ment wi t h t he r ep or t ed val ue of 66.2 [46] wi t h er r or of + 1.1.
f. I f est i mat ed RON val ue of 70.55 ( from Par t a) is used i n
Eq. (3.140), t he p r edi ct ed MON is 68 wi t h devi at i on of + 1.8.
g. Er r or s ar e cal cul at ed and gi ven i n each part . Equat i on
(3.138) gives t he l owest er r or wi t h devi at i on of less t han 1
when exp er i ment al PI ONA comp osi t i on is used. For sam-
pl es in whi ch t he di fference bet ween amount s of n-paraffi ns
and i soparaffi ns is smal l , Eq. (3.138) gives even bet t er re-
sults. I n t he cases t hat t he comp os i t i on is not avai l abl e t he
p r ocedur e used i n Par t b usi ng p r edi ct ed comp os i t i on wi t h
mi ni mum dat a on boi l i ng p oi nt and specific gravi t y gives
an accep t abl e val ue for RON. *
3 . 6 . 8 C a r b o n R e s i d u e
When a p et r ol eum f r act i on is vap or i zed in t he absence of ai r
at at mos p her i c pr essur e, t he nonvol at i l e comp ounds have a
car bonaceous r esi due known as carbon residue, whi ch is des-
i gnat ed by CR. Therefore, heavi er f r act i ons wi t h mor e aro-
mat i c cont ent s have hi gher car bon r esi dues whi l e vol at i l e and
l i ght f r act i ons such as nap ht has and gasol i nes have no car-
bon resi dues. CR is p ar t i cul ar l y an i mp or t ant char act er i st i c
of cr ude oils and p et r ol eum resi dues. Hi gher CR val ues in-
di cat e l ow- qual i t y fuel and less hydr ogen cont ent . There are
t wo ol der di fferent t est met hods to measur e car bon resi dues,
Rams bot t om ( ASTM D 524) and t he Conr adson ( ASTM
D 189). The r el at i onshi p bet ween t hese met hods ar e al so gi ven
by t he ASTM D 189 met hod. Oils t hat have ash f or mi ng com-
p ounds have er r oneousl y hi gh car bon r esi dues by bot h met h-
ods. F or such oils ash shoul d be r emoved bef or e t he measur e-
ment . There is a mor e r ecent t est met hod ( ASTM D 4530) t hat
r equi r es smal l er samp l e amount s and is oft en r ef er r ed as mi-
crocarbon residue ( MCR) and as a resul t it is less pr eci se i n
p r act i cal t echni que [7]. I n most cases car bon r esi dues ar e
reported in wt% by Conradson method, which is designated
by %CCR.
Carbon residue can be correlated to a number of other prop-
erties. It increases with an increase in carbon-to-hydrogen
ratio (CH), sulfur content, nitrogen content, asphaltenes con-
tent, or viscosity of the oil. The most precise relation is
between CR and hydrogen content in which as hydrogen con-
tent increases the carbon residues decreases [7]. The hydro-
gen content is expressed in terms of H/C atomic ratio and
the following relation may be used to estimate CCR from H/C
[81].
(3.141) %CCR = 148.7 - 86.96 H/C
if H/C _> 1.71 (%CCR < 0), set %CCR = 0.0 and if H/C < 0.5
(%CCR > 100), set %CCR = 100. H/C ratio can be estimated
from CH ratio methods given in Section 2.6.3.
The carbon residue is nearly a direct function of high boil-
ing asphaltic materials and Nelson has reported a linear re-
lation between carbon residue and asphalt yield [82]. One
of the main characteristic of residuum is its asphaltene con-
tent. Asphaltenes are insoluble in low molecular weight n-
alkanes including n-pentane. Knowledge of n-pentane insol-
ubles in residual oils is quite important in determining yields
and products qualities for deasphalting, thermal visbreaking,
and hydrodesulfurization processing. The relation between
the normal pentane insolubles and carbon residue is as fol-
lows [61]:
(3.142) %NCs = 0.74195 (%CCR) + 0.01272 (%CCR) 2
where %NC5 is the wt% of n-pentane insolubles and %CCR is
the wt% of Conradson carbon residue. Once %NCs is known,
the asphaltene content (asphaltene wt%) of a residue can be
determined from the following empirical relation:
(3.143) %Asphaltene = a(%NCs)
where a is 0.385 for atmospheric residue and 0.455 for vac-
uum residues [61,66]. These equations are approximate and
do not provide accurate predictions.
Example 3. 23--A vacuum residue of an Australian crude oil
has carbon-to-hydrogen weight ratio of 7.83. Estimate its car-
bon residue and asphaltene contents and compare the results
with the experimental values of 15.1 and 4.6%, respectively
[46].
Solution--With CH=7.83, from Eq. (2.122), HC atomic ratio
is calculated as HC --- 1.52. From Eq. (3.141), %CCR = 16.4%
and from Eq. (3.142), %NC5 = 15.6%. From Eq. (3.143) with
a = 0.455 (for vacuum residue) we calculate %Asphaltene =
7.1. The results show that while Eq. (3.141) provides a
good prediction for %CCR, prediction of %Asphaltene from
Eq. (3.143) is approximate.
3 . 6. 9 S mo k e Po i nt
Smoke point is a characteristic of aviation turbine fuels and
kerosenes and indicates the tendency of a fuel to burn with
a smoky flame. Higher amount of aromatics in a fuel causes
a smoky characteristic for the flame and energy loss due to
thermal radiation. The smoke point (SP) is a maximum flame
height at which a fuel can be burned in a standard wick-fed
lamp without smoking. It is expressed in millimeters and a
high smoke point indicates a fuel with low smoke-producing
tendency [61]. Measurement of smoke point is described un-
der ASTM D t322 (U.S.) or IP 57 (UK) and ISO 3014 test
methods. For a same fuel measured smoke point by IP test
method is higher than ASTM method by 0.5-1 mm for smoke
point values in the range of 20-30 mm [61].
Smoke point may be estimated from either the PNA com-
position or from the aniline point. The SP of kerosenes from
IP test method may be estimated from the following relation
[90]:
SP = 1.65X - 0.0112X 2 - 8.7
(3.144) 100
X =
0.61Xp + 3.392xN + 13.518xA
where SP is the smoke point by IP test method in mm and Xv,
xN, and :cA are the fraction of paraffin, naphthene, and aro-
matic content of kerosenes. The second method is proposed
by Jenkins and Walsh as follows [83]:
SP = -255. 26 + 2.04AP - 240.8 ln(SG) + 7727(SG/AP)
(3.145)
where AP is the aniline point in ~ and SG is the specific grav-
ity at 15.5~ Both Eqs. (3.144) and (3.145) estimate SP ac-
cording to the IP test method. To estimate SP from the ASTM
D 1322 test method, 0.7 mm should be subtracted from the
calculated I P smoke point. Equations (3.144) and (3.145) are
based on data with specific gravity in the range of 0.76-0.82,
and smoke points in the range of 17-39 mm. Based on some
preliminary evaluations, Eq. (3.133) is expected to perform
better than Eq. (3.144), because smoke point is very much re-
lated to the aromatic content of the fuel which is expressed in
terms of aniline point in the Jenkins-Walsh method. I n addi-
tion the specific gravity, which is an indication of molecular
type, is also used in the equation. Equation (3.144) may be
used for cases that the aniline point is not available but ex-
perimental PNA composition is available. Albahri et al [68]
also proposed the following relation for prediction of smoke
point using API gravity and boiling point:
(3.146) SP = 0.839(API) + 0.0182634(Tb) -- 22.97
where SP is in mm (ASTM method) and Tb is the average
boiling point in kelvin. This equation when tested for 136
petroleum fractions gave an average error of about 2 mm [68].
Example 3. 24--A Nigerian kerosene has an API gravity of
41.2, aniline point of 55.6~ and the PNA composition of
36.4, 49.3, and 14.3%. Estimate the smoke point of this fuel
from. Equations (3.144)-(3.146) and compare with the exper-
imental value of 20 mm (Ref. [46], p. 342).
Solution--To estimate SP from Eq. (3.144), we have Xp =
0.364, XN = 0.493, and XA = 0.143 which give X = 26.13. Cal-
culated SP is 26.8 mm according to the IP method or 26.1
mm according to the ASTM method. To use Eq. (3.145) we
have from API gravity, SG = 0.819, AP -- 55.6~ the calcu-
lated SP is SP = 20 mm. The ASTM smoke point is then 19.3
mm which is in very good agreement with the experimental
value of 20 with deviation of -0. 7 mm. Predicted value from
Eq. (3.146) is 17.6 mm.
3 . 7 QUAL I T Y OF PET R OL EUM PR OD UC T S
Met hods p r esent ed i n t hi s chap t er can be used to eval uat e
t he qual i t y of p et r ol eum p r oduct s f r om avai l abl e p ar amet er s.
The qual i t y of a p et r ol eum p r oduct dep ends on cer t ai n
speci fi cat i ons or p r op er t i es of t he fuel t o sat i sfy r equi r ed
cr i t er i a set by t he mar ket demand. These char act er i st i cs ar e
speci fi ed for best use of a fuel (i.e., hi ghest engi ne perfor-
mance) or for cl eaner envi r onment whi l e t he fuel is i n use.
These speci fi cat i ons var y f r om one p r oduct t o anot her and
f r om one count r y t o anot her. F or exampl e, for gasol i ne t he
qual i t y is det er mi ned by a seri es of p r op er t i es such as sul fur
and ar omat i c cont ent s, oct ane number, vap or pr essur e,
hydr ogen cont ent , and boi l i ng range. Engi ne war m- up t i me
is affect ed by t he p er cent di st i l l ed at 70~ and t he ASTM 90%
t emp er at ur e. For t he ambi ent t emp er at ur e of 26.7~ (80~
a gasol i ne mus t have ASTM 90% t emp er at ur e of 188~ and
3% di st i l l ed at 70~ t o give accep t abl e war m- up t i me [63].
St andar d or gani zat i ons such as ASTM give such specifica-
t i ons for var i ous pr oduct s. For r ef or mul at ed gasol i ne sul fur
cont ent of less t han 300 p p m (0.03 wt %) is r equi r ed [63].
Amount of p ar t i cul at e emi ssi ons is di rect l y r el at ed t o t he
ar omat i c and sul fur cont ent of a fuel. Fi gur e 3.35 shows t he
i nfl uence of sul fur r educt i on in gasol i ne f r om 500 to 50 p p m
in t he r educt i on of p ol l ut ant emi ssi ons [24].
Vapor p r essur e of gasol i ne of j et fuel det er mi nes t hei r ig-
ni t i on charact eri st i cs. Whi l e freezi ng p oi nt is i mp or t ant for
j et fuels i t is not a maj or char act er i st i c for gas oils. F or l ubri -
cat i ng oil p r op er t i es such as vi scosi t y and vi scosi t y i ndex ar e
i mp or t ant in addi t i on t o sul fur and PNA comp osi t i on. Ani l i ne
p oi nt is a useful char act er i st i c t o i ndi cat e p ower of sol ubi l -
i t y of sol vent s as wel l as ar omat i c cont ent s of cer t ai n fuels.
For heavy p et r ol eum p r oduct s knowl edge of p r op er t i es such
as car bon resi due, p our poi nt , and cl oud p oi nt are of inter-
est. Some i mp or t ant speci fi cat i ons of j et fuels ar e gi ven i n
Table 3.30.
One of t he t echni ques used i n refi ni ng t echnol ogy to pr o-
duce a p et r ol eum p r oduct wi t h a cer t ai n char act er i st i c is t he
bl endi ng met hod. Once a cer t ai n val ue for a p r op er t y (i.e.,
viscosity, oct ane number, p our poi nt , etc.) of a p et r ol eum
p r oduct is r equi r ed, t he mi xt ur e may be bl ended wi t h a cer-
t ai n comp onent , addi t i ve or anot her p et r ol eum f r act i on to
-5]
-10
-15
-20
% Reduction
HC CO NOx
f
FIG. 3.35--Influence of sulfur content in gasoline (from
500 to 50 ppm) in reduction of pollutant gases. Taken with
permission from Ref. [24].
p r oduce t he r equi r ed final p r oduct . I n Examp l e 3.20, it was
shown t hat t o have a p r oduct wi t h cer t ai n flash poi nt , one can
det er mi ne t he vol ume of var i ous comp onent s in t he bl end.
The s ame ap p r oach can be ext ended to any ot her propert y. For
exampl e, to i ncr ease vap or p r essur e of gasol i ne n- but ane may
be added dur i ng wi nt er season to i mp r ove engi ne st ar t i ng
char act er i st i cs of t he fuel [63]. The amount of r equi r ed but ane
to r each a cer t ai n vap or p r essur e val ue can be det er mi ned
t hr ough cal cul at i on of vap or p r essur e bl endi ng i ndex for t he
comp onent s and t he p r oduct as di scussed in Sect i on 3.6.1.1.
3 . 8 MI N I MUM L A B OR A T OR Y DATA
As di scussed ear l i er meas ur ement of all p r op er t i es of var i ous
p et r ol eum fract i ons and p r oduct s in t he l abor at or y is an
i mp ossi bl e t ask due t o t he r equi r ed cost and t i me. However,
Characteristics
TABLE 3. 30---Some general characteristics of three fuels [24].
Specifications a
Gasoline b Jet fuel Diesel fuel
Max. total sulfur, wt% 0.05 0.2
Max aromatics content, vol% 20
Max olefins content, vol% 5
Distillation at 10 vol%,~ 204
Max final boiling point, ~ 215 300
Range of % evaporation at 70~ (E70) 15-47
Min research octane number (RON) 95
Min flash point, ~ 38
Specific gravity range 0.7254).78 0.775-0.84
Min smoke point, ~ 25
Max freezing point, ~ - 47
Range of Reid vapor pressure, bar 0.35-0.90 d
Min cetane number
aEuropean standards in the mid 1990s.
bEuropean unleaded Super 98 premium gasoline.
CAt this temperature minimum of 95 vol% should be evaporated.
dVaries with season according to the class of gasoline.
0.05
370 c
55
0.82-0.86
49
t here are a numbe r of basi c par amet er s t hat must be known
for a fract i on to det ermi ne vari ous propert i es from t he
met hods present ed i n this chapter. As mor e experi ment al
dat a are available for fract i on a bet t er charact eri zat i on of
the fract i on is possible. For example, to est i mat e sul fur
cont ent of a fract i on from Eqs. (3.97) and (3.98), t he i np ut
par amet er s of specific gravity, mol ecul ar weight, densi t y and
refractive i ndex at 20~ are needed. If experi ment al values of
all these par amet er s are available a good est i mat e of sul fur
cont ent can be obt ai ned. However, since nor mal l y all these
dat a are not available, M, n, and dE0 shoul d be est i mat ed from
SG and Tb. Therefore, a mi ni mum of two par amet er s t hat
are boi l i ng poi nt and specific gravity are needed to est i mat e
t he sul fur cont ent . However, for heavy fract i ons i n whi ch
di st i l l at i on dat a are not reported, M shoul d be est i mat ed
from ki nemat i c viscosity at 38 and 99~ (1)38 and 1)99) and
specific gravity t hr ough Eq. (2.52). Once M is est i mat ed,
n can be est i mat ed from M and SG t hr ough Eq. (2.127) and
d is cal cul at ed from SG t hr ough Eq. (2.123). Wi t h t he knowl -
edge of M and SG all ot her par amet er s can be est i mat ed from
met hods present ed i n Chapt er 2. Therefore, at least t hree
paramet ers of I)38 , v99, and SG must be known to det er mi ne
sul fur cont ent or ot her characteristics. I n a case t hat oul y one
viscosity dat a is known, i.e., vas, ki nemat i c viscosity at 99~
1)99, can be est i mat ed from Eq. (2.61). I n this way est i mat ed
val ue of M is less accurat e t han the case t hat t hree val ues of
1)38, I)99 and SG are known from experi ment al measur ement s.
We see t hat agai n for heavy fract i ons wi t h knowl edge of onl y
t wo paramet ers (i.e., 1)3a and SG or 1)99 and SG) all basi c
propert i es of the fract i on can be est i mat ed. Therefore, to
obt ai n t he basi c charact eri zat i on paramet ers of a pet r ol eum
fract i on a mi ni mum of two paramet ers are needed.
I f the onl y i nf or mat i on is the di st i l l at i on curve, t hen spe-
cific gravity can be est i mat ed from T10 and T50 t hr ough Eq.
(3.17) and Table 3.4. Havi ng T50 and SG, all ot her par amet er s
can be est i mat ed as di scussed above. When onl y a por t i on of
di st i l l at i on curve (i.e., T2o, T40, and T60) is available, t hr ough
Eq. (3.35) the compl et e curve can be predi ct ed and from this
equat i on T10 and Ts0 can be det ermi ned. Therefore, a por t i on
of di st i l l at i on curve can also be used to generat e all par ame-
ters rel at ed to propert i es and qual i t y of pet rol eum fractions.
We showed t hat wi t h t he knowl edge of PNA composi t i on a
bet t er charact eri zat i on of a fract i on is possible t hr ough pseu-
docomp onent t echni que. Therefore, if the composi t i on al ong
boi l i ng poi nt is available, nearl y all ot her par amet er s can be
det er mi ned t hr ough mi d boi l i ng poi nt and PNA composi t i on
wi t h bet t er accuracy t han usi ng onl y Tb and SG. For heavy
fract i ons i n whi ch Tb may not be available, the pseudocom-
p onent t echni que can be appl i ed t hr ough use of M and PNA
composi t i on where M may be est i mat ed from viscosity dat a
if it is not available. As t here are many scenari os to est i mat e
basi c propert i es of pet r ol eum fractions, use of available dat a
to predi ct the most accurat e charact eri zat i on par amet er s is
an engi neeri ng art whi ch has a direct i mpact on subsequent
predi ct i on of physi cal propert i es and event ual l y on desi gn cal-
cul at i ons. The basi c l aborat ory dat a t hat are useful i n charac-
t eri zat i on met hods based on t hei r significance and si mpl i ci t y
are given below:
1. di st i l l at i on data, boi l i ng poi nt
2. specific gravity
3. composi t i on (i.e, PNA cont ent )
4. mol ecul ar wei ght
5. refractive i ndex
6. el ement al anal ysi s (i.e., CHS composi t i on)
7. ki nemat i c viscosity at 37.8 and 98.9~ (100 and 210~
One can best charact eri ze a p et r ol eum fract i on if all t he
above par amet er s are known from l aborat ory measur ement s.
However, among these seven i t ems at least two i t ems must be
known for charact eri zat i on purposes. I n any case when exper-
i ment al value for a charact eri zat i on p ar amet er is available it
shoul d be used i nst ead of predi ct ed value. Among these seven
i t ems t hat can be measur ed i n laboratory, refractive i ndex and
specific gravity are the most conveni ent propert i es to mea-
sure. Mol ecul ar wei ght especially for heavy fract i ons is also
very useful to predi ct ot her properties. As di scussed i n Chap-
ter 2, for light fract i ons (M < 300; Nc < 22, Tb < 350~ t he
best two pai rs of paramet ers i n the order of t hei r character-
i zi ng power are (Tb, SG), (Tb, n), (M, SG), (M, n), (v, SG), (Tb,
CH), (M, CH), (v, CH). The most sui t abl e pai r is (Tb, SG) and
t he least one is (v, CH). As it is expl ai ned i n t he next section,
for heavy fract i ons t hree p ar amet er correl at i ons are more ac-
curate. Therefore, for heavy fract i ons i n whi ch boi l i ng poi nt
cannot be measur ed a mi ni mum of t hree par amet er s such
as viscosity at two different t emperat ures and specific gravity
(i.e., v38, I)99, SG) are needed. For heavy fract i ons the pseudo-
comp onent met hod is much more accurat e t han use of bul k
propert i es for the est i mat i on of vari ous properties. Therefore,
TABLE 3.31---Standard test methods for measurement of some properties of liquid petroleum products.
Property ASTM D IP ISO Property ASTM D IP ISO
Aniline Point 611 2/98 2977 Flash Point 93 34/97 2719
Carbon Residue (Ramsbottom) 524 14/ 94 4262 Freezing Point 2386 16/98 3013
Carbon Residue (Conradson) 189 13/ 94 6615 Hydrocarbon Types 1319 156/ 95 3837
Centane Number 4737 380/98 4264 Heating Value 240 12
Cloud Point 2500 219/94 3016 Kinematic Viscosity 445 71/97 3104
Color 1500 196/ 97 2049 Octane Number (Motor) 2700 236 5163
Density/Sp. Gr. 4052 365/97 2185 Refractive Index 1218
Distillation at Atm. Pressure 86 123/ 99" 3405 Pour Point 97 15/95 3015
Distillation at Reduced Pressures 1160 6616" Sulfur Content 1266 107/ 86 2192
Distillation by Gas Chromatography 2887 406/99* Thermal Conductivity 2717-95
Distillation of Crude Oils 2892 8708* Vapor Pressure (Reid) 323 69/94 3007
Viscosity (Viscous Oils) 2983 370/85*
ASTM has test methods for certain properties for which other test methods do not suggest equivalent procedures. Some of these methods
include heat of combustion: D 4809; smoke point: D 1322; surface tension: D 3825; vapor-liquid ratio: D 2533; viscosity temperature chart: D
341; autoignition: D 2155 (ISO 3988). Further test methods for some specific properties are given in the text where the property is discussed.
ASTM methods are taken from Ref. [4]. IP methods are taken from Ref. [85]. Methods specified by * are similar but not identical to other
standard methods. Most IP methods are also used as British Standard under BS2000 methods [85]. The number after IP indicates the year of
]ast approval. ISO methods are taken from Refs. [24] and [85].
the knowledge of PNA composi t i on for prediction of prop-
erties of heavy fractions is more useful t han for light frac-
tions. For wide boiling range fractions knowledge of complete
distillation curve is quite useful to consider nature of different
compounds and their effects on the properties of the mixture.
As it is shown in Chapter 4, for wide and heavy fractions such
as C7+ fractions, distribution of carbon number in the frac-
tion is the most useful i nformat i on besides specific gravity.
Furt her analysis of mi ni mum l aborat ory data for characteri-
zation of pet rol eum fractions is provided in our previous work
[84].
Predictive met hods of characterization must be used when
experimental data are not available. I f possible, one should
make maxi mum use of available experimental data, A sum-
mary of standard test met hods for some specifications of
liquid pet rol eum products is given in Table 3.31. For some
properties equivalent test met hods according to the interna-
tional standards organization (ISO) are also specified in this
table [24, 85].
3 . 9 A N A L Y S I S OF L A B OR A T OR Y DATA A N D
D EVEL OP M EN T OF PR ED I C T I VE M ET H OD S
I n Chapter 2 and this chapter the predictive met hods in t erms
of readily available paramet ers are presented for estimation of
various properties related to basic characteristics and quality
of pet rol eum fractions. Generally these met hods fall within
two categories of empirical and semiempirical correlations.
I n an empirical correlation the structure of the correlation
is determined t hrough fitting the data and the type of input
paramet ers in each correlation are determined t hrough anal-
ysis of experimental data. While in a semi-empirical corre-
lation, the structure and functionality of the relation is de-
t ermi ned from a theoretical analysis of paramet ers involved
and t hrough analysis of existing theoretical relations. Once
the mai n functionality and nature of a correlation between
various physical properties is determined, the correlation co-
efficients can be determined from experimental data. The best
example of such a predictive met hod is development of Eq.
(2.38) in Chapter 2, which was developed based on the un-
derstanding of the intermolecular forces in hydrocarbon sys-
tems. This generalized correlation has been successfully used
to develop predictive met hods for a variety of physical prop-
erties. An example of an empirical correlation is Eq. (2.54)
developed for estimation of molecular weight of pet rol eum
fractions. Many other correlations presented in this chapt er
for estimation of properties such as aniline and smoke points
or met hods presented for calculation of octane numbers for a
blend are also purely empirical in nature. I n development of
an empirical relation, knowledge of the nature of properties
involved in the correlation is necessary. For example, aniline
point is a characteristic that depends on the molecular type
of hydrocarbons in the fraction. Therefore, it is appropri at e
to relate aniline point to the paramet ers that characterize hy-
drocarbon types (i.e., Ri) rat her t han boiling point that char-
acterizes carbon number in a hydrocarbon series.
Mathematical functions can be expressed in the form of
polynomial series; therefore, it is practically possible to de-
velop correlations in the forms of polynomial of various de-
grees. With powerful comput at i onal tools available at present
it is possible to find an empirical correlation for any set of
laboratory data for any physical propert y in t erms of some
other paramet ers. As the complexity and the number of pa-
ramet ers increases the accuracy of the correlation also in-
creases with respect to the data used in the development of
the correlation. However, the mai n probl em with empirical
correlations is their limited power of extrapolation and the
large number of numeri cal constants involved in the correla-
tion. For example, in Chapter 2, several correlations are pro-
vided to estimate molecular weight of hydrocarbons in t erms
of boiling point and specific gravity. Equation (2.50) derived
from Eq. (2.38) has only three numerical constants, which
are developed from molecular weight of pure hydrocarbons.
Tsonopoulos et al. [86] made an extensive analysis of various
met hods of estimation of molecular weights of coal liquids.
Equation (2.50) was compared with several empirical corre-
lations specifically developed for coal liquids having as many
as 16 numerical constants. They concluded that Eq. (2.50)
is the most accurate met hod for the estimation of mol ecul ar
weight of coal liquids. No data on coal liquid were used in
development of constants in Eq. (2.50). However, since it was
developed with some physical basis and properties of pure
hydrocarbons were used to obtain the numerical constants
the equation has a wide range of applications from pure hy-
drocarbons to pet rol eum fractions and coal liquids, which are
mainly aromatics. This indicates the significance of develop-
ment of correlations based on the physical understanding of
the nature of the system and its properties. The mai n advan-
tage of such correlations is their generality and simplicity.
The mai n characteristics of an ideal predictive met hod for a
certain propert y are accuracy, simplicity, generality, and avail-
ability of input parameters. The best approach t oward the
development of such correlations would be to combi ne phys-
ical and theoretical fundament al s with some modifications.
An example of such type of correlations is Eq. (2.39), which is
an extension of Eq. (2.38) derived from physical basis. A pure
empirical correlation might be quite accurate to represent the
data used in its development but when it is applied to other
systems the accuracy is quite low. I n addition characterizing
the systems according to their degree of complexity is helpful
to develop more accurate correlations. For example, heavy
fractions contain heavy and nonpol ar compounds, which dif-
fer with low-molecular-weight hydrocarbons present in light
pet rol eum fractions. Therefore, in order to increase the de-
gree of accuracy of a predictive met hod for a certain prop-
erty it is quite appropri at e to develop one correlation for
light and one correlation for heavy fractions. For heavy frac-
tions because of the nature of complex compounds in the
mixture three input paramet ers are required. As variation in
properties of pure hydrocarbons from one family to anot her
increases with increase in carbon numbers (i.e., see Figs. 2.15
and 2.23), the role of composition on estimation of such prop-
erties for heavy fractions is more t han its effect on the prop-
erties of light fractions. Therefore, including molecular type
in the development of predictive met hods for such properties
of heavy fractions is quite reasonable and useful and would
enhance the accuracy of the method.
Once the structure of a correlation is det ermi ned from
theoretical developments between various properties, ex-
perimental data should be used to determine the numeri cal
constants in the correlation. I f the data on properties of
pure hydrocarbons from different families are used to
determine the constants, the resulting correlation would be
more general and applicable to various types of petroleum
fractions within the same boiling point range. The accuracy
of a correlation for a specific group of fractions could be
increased if the coefficients in the correlation are obtained
from the same group of fractions. I n obtaining the constants,
many equations may be convertible into linear forms and
spreadsheets such as Lotus or Excel programs may be used to
obtain the constants by means of least squared method. Non-
linear regression of correlations through these spreadsheets
is also possible. I n analyzing the suitability of a correlation,
the best criteria would be the R 2 or correlation parameter
defined by Eq. (2.136) in which values of above 0. 99 indicate
an equation is capable of correlating data. Use of a larger data
bank and most recent published data in obtaining the nu-
merical constants would enhance accuracy and applicability
of the correlation. A fair way of evaluations and comparison
of various correlations to estimate a certain property from
the same input parameters would be through a data set not
used in obtaining the coefficients in the correlations. I n such
evaluations AAD or %AAD can be used as the criterion to
compare different methods. Average absolute deviation may
be used when the range of variation in the property is very
large and small values are estimated. For example, in predic-
tion of PNA composition, amount of aromatics varies from
1% in petroleum fractions to more than 90% in coal liquids.
AAD of 2 (in terms of percentage) in estimating aromatic
content is quite reasonable. This error corresponds to 200%
in terms of %AD for fractions with t% aromatic content.
Experience has shown that correlations that have fewer nu-
merical constants and are based on theoretical and physical
grounds with constants obtained from a wide range of data
set are more general and have higher power of extrapolation.
3 . 1 0 CONCLUS I ONS AND
RECOMMENDAT I ONS
based on a mi ni mum of three data points along the distillation
curve. For prediction of each characteristic of a petroleum
fraction, several methods are provided that have been in use
in the petroleum industry. Limitations, advantages, and dis-
advantages of each method are discussed.
Basically two approaches are proposed in characterization
of petroleum fractions. One technique is based on the use
of bulk properties (i.e., Tb and SG) considering the whole
mixture as a single pseudocomponent. The second approach
called pseudocomponent technique considers the fraction as
a mixture of three pseudocomponents from the three fami-
lies. This technique is particularly useful for heavy fractions.
A third approach is also provided for wide boiling range frac-
tions. However, since behavior of such fractions is similar to
that of crude oils the technique is mainly presented in the next
chapter. Fractions are generally divided into light and heavy
fractions. For heavy fractions a mi ni mum of three character-
ization parameters best describe the mixture. Recommenda-
tions on the use of various input parameters and advantages
of different methods were discussed in Sections 3.8 and 3.9.
For light fractions (M < 300, Nc < 22) and products of atmo-
spheric distillation unit, Eq. (2.38) for M, I, and d is quite ac-
curate. For such light fractions, To, Pc, and co can be estimated
from Eqs. (2.65), (2.66), and (2.105), respectively. For pre-
diction of the PNA composition for fractions with M < 200,
Eqs. (3.77) and (3.78) in terms of m and SG are suitable. For
fractions with M > 200, Eqs. (3.71)-(3.74) in terms of P~ and
VGC are the most accurate relations; however, in cases that
viscosity data are not available, Eqs. (3.79) and (3.80) in terms
of P~ an CH are recommended. Special recommendations on
use of various correlations for estimation of different prop-
erties of petroleum fractions from their bulk properties have
been given in Section 2.10 and Table 2.16 in Chapter 2.
3 . 1 1 PR OB L EMS
I n this chapter various characterization methods for differ-
ent petroleum fractions and mixtures have been presented.
This is perhaps one of the most important chapters in the
book. As the method selected for characterization of a frac-
tion would affect prediction of various properties discussed in
the remaining part of the book. As it is discussed in Chapter 4,
characterization and estimation of properties of crude oils de-
pend on the characterization of petroleum fractions discussed
in this chapter. Through methods presented in this chapter
one can estimate basic input data needed for estimation of
thermodynamic and physical properties. These input param-
eters include critical properties, molecular weight, and acen-
tric factor. I n addition methods of estimation of properties
related to the quality of a petroleum product such as distil-
lation curves, PNA composition, elemental composition, vis-
cosity index, carbon residue, flash, pour, cloud, smoke, and
freezing points as well as octane and cetane numbers are pre-
sented. Such methods can be used to determine the quality
of a fuel or a petroleum product based on the mi ni mum labo-
ratory data available for a fraction. Methods of conversion of
various types of distillation curves help to determine neces-
sary information for process design on complete true boiling
point distillation curve when it is not available. I n addition a
method is provided to determine complete distillation curve
3.1. List four different types of analytical tools used for com-
positional analysis of petroleum fractions.
3.2. What are the advantages/disadvantages and the differ-
ences between GC, MS, GC-MS, GPC and HPLC instru-
ments?
3.3. A jet naphtha has the following ASTM D 86 distillation
curve [ 1 ]:
3.4.
ASTM D 86 temperature, ~ 151.1 156.1 160.6 165.0 171.7
a. Calculate VABP, WABR MABP, CABP, and MeABP for
this fraction. Comment on your calculated MeABR
b. Estimate the specific gravity of this fraction and com-
pare with reported value of 0.8046.
c. Calculate the Kw for this fraction and compare with
reported value of 11.48.
A kerosene sample has the following ASTM D 86, TBP,
and SG distribution along distillation curve [1]. Convert
ASTM D 86 distillation curve to TBP by Riazi-Daubert
and Daubert's (API) methods. Draw actual TBP and pre-
dicted TBP curves on a single graph in ~ Calculate the
average specific gravity of fraction form SG distribution
and compare with reported value of 0.8086.
Vol % I BP 5 10 20 30 40 50 60 70 80 90 95 FBP
ASTM, ~ 330 342 350 366 380 390 404 417 433 450 469 482 500
TBP, ~ 258 295 312 337 358 . . . 402 ... 442 ... 486 499 . . .
SG ... 0.772 0.778 0.787 0.795 ... 0.817 ... 0.824 ... 0.829 0.830 ...
3.5. For t he ker osene samp l e of Pr obl em 3.4, assume ASTM
t emp er at ur es at 30, 50, and 70% p oi nt s are t he onl y
known i nf or mat i on. Based on t hese dat a p oi nt s on t he
di st i l l at i on curve p r edi ct t he ent i re di st i l l at i on curve and
comp ar e wi t h t he act ual val ues in bot h t abul at ed and
gr ap hi cal forms.
3.6. F or t he ker osene samp l e of Pr obl em 3.4 cal cul at e t he
Wat son K f act or and gener at e t he SG di st r i but i on by
consi der i ng const ant Kw al ong t he curve. Cal cul at e
%AAD for t he devi at i ons bet ween p r edi ct ed SG and ac-
t ual SG for t he 8 dat a poi nt s. Show a gr ap hi cal eval ua-
t i on of p r edi ct ed di st r i but i on.
3.7. F or t he ker osene samp l e of Pr obl em 3.4 gener at e SG
di st r i but i on usi ng Eq. (3.35) and dr aw p r edi ct ed SG dis-
t r i but i on wi t h act ual val ues. Use Eq. (3.37) to cal cul at e
specific gravi t y of t he mi xt ur e and comp ar e wi t h t he ac-
t ual val ue of 0.8086.
3.8. F or t he ker osene samp l e of Pr obl em 3.4 cal cul at e den-
si t y at 75~ and comp ar e wi t h t he r ep or t ed val ue of
0.8019 g/ cm 3 [1].
3.9. For t he ker osene samp l e of Pr obl em 3.4, assume t he onl y
dat a avai l abl e are ASTM D 86 t emp er at ur es at 30, 50,
and 70% poi nt s. Based on t hese dat a cal cul at e t he spe-
cific gravi t y at 10, 30, 50, 70, and 90 % and comp ar e wi t h
t he val ues given in Pr obl em 3.4.
3.10. A p et r ol eum fract i on has t he fol l owi ng ASTM D 1160
di st i l l at i on curve at 1 mmHg:
ASTMD 1160 temperature, ~ 104 143 174 202 244
Pr edi ct TBP, ASTM D 86, and EFV di st i l l at i on curves all
at 760 mm Hg.
3.11. A gas oil samp l e has t he fol l owi ng TBP and densi t y di s-
t r i but i on [32]. Exp er i ment al val ues of M, n20, and SG
ar e 214, 1.4694, and 0.8475, respectively.
a. Use t he met hod out l i ned for wi de boi l i ng r ange frac-
t i ons and est i mat e M and n20.
b. Use Eqs. (2.50) and (2.115) to est i mat e M and n
c. Comp ar e %AD f r om met hods a and b.
3.13. F or t he f r act i on of Pr obl em 3.12 cal cul at e acent r i c fac-
tor, w f r om t he fol l owi ng met hods:
a. Lee- Kesl er met hod wi t h Tc and Pc f r om Part s (a), (b),
(c), and (d) of Pr obl em 3.12.
b. Edmi s t er met hod wi t h Tc and Pc f r om Part s and (a)
and (e) of Pr obl em 3.12.
c. Kor st en met hod wi t h Tc and Pc f r om Par t (a) of Prob-
l em 3.12.
d. Ps eudocomp onent met hod
3.14. A p ur e hydr ocar bon has a boi l i ng p oi nt of 110.6~ and
a specific gravi t y of 0.8718. What is t he t ype of t hi s hy-
dr ocar bon (P, N, MA, or PA)? How can you check your
answer ? Can you guess t he comp ound? I n your anal ysi s
it is as s umed t hat you do not have access to t he t abl e of
p r op er t i es of p ur e hydr ocar bons.
3.15. Exp er i ment al dat a on sul fur cont ent of some p et r ol eum
p r oduct s al ong wi t h ot her basi c p ar amet er s ar e gi ven in
Table 3.32. I t is as s umed t hat t he onl y dat a avai l abl e for
a p et r ol eum p r oduct is its mi d boi l i ng poi nt , Tb, ( col umn
1) and refract i ve i ndex at 20~ n20 ( col umn 2). Use ap-
p r op r i at e met hods to comp l et e col umns 4,6,8,9,10, and
12 i n t hi s t abl e.
3.16. ASTM 50% t emp er at ur e (Tb), specific gravi t y (SG), and
t he PNA comp os i t i on of 12 p et r ol eum f r act i ons ar e given
i n Table 3.33. Compl et e col umns of t hi s t abl e by cal cul at -
i ng M, n, m, and t he PNA comp os i t i on for each f r act i on
by usi ng ap p r op r i at e met hods.
3.17. A gasol i ne p r oduct f r om Nor t h Sea cr ude oil has boi l i ng
r ange of C5-85~ and specific gravi t y of 0.6771. Pr edi ct
t he PNA comp os i t i on and comp ar e wi t h exp er i ment al l y
det er mi ned comp os i t i on of 64, 25, and 11 in wt % [46].
3.18. A r esi due f r om a Nor t h Sea cr ude has t he fol l owi ng ex-
p er i ment al l y det er mi ned charact eri st i cs: 1)99(210 ) = 14.77
cSt., SG ---- 0.9217. F or t hi s fract i on est i mat e t he fol l ow-
i ng p r op er t i es and comp ar e wi t h t he exp er i ment al val-
ues wher e t hey ar e avai l abl e [46]:
a. Ki nemat i c vi scosi t y at 38~ ( 100~
b. Mol ecul ar wei ght and average boi l i ng p oi nt
Vol% IBP 5 10 20 30 40 50 60 70 80 90 95
TBP, ~ 420 451 470 495 514 ... 544 ... 580 ... 604 621
d20 ... 0.828 0.835 0.843 0.847 ... 0.851 ... 0.855 ... 0.856 0.859
3.12. A j et nap ht ha has ASTM 50% t emp er at ur e of 321 ~ and
specific gravi t y of 0.8046 [ i ] . The PNA comp osi t i on of
t hi s fract i on is 19, 70, and 11%, respectively. Est i mat e M,
n20, dzo, To, Pc, and Vc f r om Tb and SG usi ng t he fol l owi ng
met hods:
a. Ri azi - Dauber t (1980) met hods [38].
b. API -TDB met hods [2].
c. Twu met hods for M, Tc, Pc, and Vc.
d. Kesl er - Lee met hods for M, Tc, and Pc.
e. Si m- Dauber t ( comp ut er i zed Wi nn nomogr ap h)
met hod for M, Tc, and Pc.
f. Ps eudocomp onent met hod.
3.19.
c. Densi t y and refract i ve i ndex at 20~
d. Pour p oi nt ( exp er i ment al val ue is 39~
e. Sul f ur cont ent ( exp er i ment al val ue is 0.63 wt %)
f. Conr adson car bon r esi due ( exp er i ment al val ue is 4.6
wt %)
A cr ude oil f r om Nor t hwest Aust r al i an field has t ot al ni-
t r ogen cont ent of 310 p p m. One of t he p r oduct s of at mo-
sp her i c di st i l l at i on col umn for t hi s cr ude has t rue boi l -
i ng p oi nt r ange of 190-230~ and t he API gravi t y of 45.5.
For t hi s p r oduct det er mi ne t he fol l owi ng p r op er t i es and
TABLE 3. 32- - Basic parameters and sulfur content of some undefined petroleum products (Problem 3. 15). a
(i) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
Tb, K n2o SG, SG, M, M, d20, d20, rr~ RI Sulfur %,
Fraction exp. exp. exp. calc. exp. calc. exp. calc. calc. calc. exp.
Kuwaiti kerosene 468 1. 441 0.791 . . . . . . 0.01
Kuwaiti diesel oil 583 1.480 0.860 1.3
US jet naphtha 434 1.444 0.805 144 01861 1.0
US high boiling naphtha 435 1.426 0.762 142.4 0.759 0.0
US kerosene 480 1.444 0.808 162.3 0.804 0.0
US fuel off 559 1.478 0.862 227.5 0.858 1.3
(12)
Sulfur %,
calc.
aExperimental data for the first two Kuwaiti fractions are taken from Riazi and Roomi [62] and for the US fractions are taken from Lenoir
and Hipkin data set [ 1 ].
comp ar e wi t h t he exp er i ment al val ues as gi ven bel ow
[46].
a. What is t he t ype of t hi s p r oduct ?
b. Mol ecul ar wei ght ( exp er i ment al val ue is 160.1)
c. Ki nemat i c vi scosi t i es at 20 and 40~ ( exp er i ment al
val ues ar e 1.892 and 1.28)
d. Fl ash p oi nt ( exp er i ment al val ue is 69~
e. PNA comp osi t i on ( the exp er i ment al PNA comp osi -
t i on f r om GC- MS anal ysi s in vol % ar e 46.7, 40.5, and
12.8, respect i vel y)
f. Smoke p oi nt ( exp er i ment al val ue is 26 mm)
g. Ani l i ne p oi nt ( exp er i ment al val ue is 64.9~
h. Pour p oi nt ( exp er i ment al val ue is - 45~
i. Fr eezi ng p oi nt ( exp er i ment al val ue is - 44~
j. Hydr ogen cont ent ( exp er i ment al val ue is 14.11 wt %)
3.20. An at mos p her i c r esi due p r oduced f r om t he same cr ude
of Pr obl em 3.19 has API gravi t y of 25 and UOP K f act or
of 12.0. Pr edi ct t he fol l owi ng p r op er t i es and comp ar e
wi t h t he exp er i ment al val ues [46].
a. Mol ecul ar wei ght ( exp er i ment al val ue is 399.8)
b. Total ni t r ogen cont ent ( exp er i ment al val ue is 0.21
wt%)
c. Ki nemat i c vi scosi t y at 100~ ( exp er i ment al val ue is
8.082)
d. Ki nemat i c vi scosi t y at 70~ ( exp er i ment al val ue is
17.89)
e. Sul f ur cont ent ( exp er i ment al val ue is 0.17 wt %)
f. Conr adson car bon r esi due ( exp er i ment al val ue is 2.2
wt%)
g. Car bon cont ent ( exp er i ment al val ue is 86.7 wt %)
h. Hydr ogen cont ent ( exper i ment al val ue is 13 wt %)
i. Ani l i ne p oi nt ( exper i ment al val ue is 95.2~
j. Pour p oi nt ( exp er i ment al val ue is 39~
3.21. A gas off p r oduced f r om a cr ude f r om Sor oosh field
( I ran) has boi l i ng r ange of 520-650~ and t he API gravi t y
of 33 [46]. Rep or t ed Wat son K fact or is 11.72. Cal cul at e
t he fol l owi ng p r op er t i es and comp ar e wi t h exp er i men-
t al val ues.
a. Average boi l i ng p oi nt f r om Kw and comp ar e wi t h mi d
boi l i ng p oi nt
b. Cet ane i ndex ( r ep or t ed val ue is 50.5)
c. Ani l i ne p oi nt ( r ep or t ed val ue is 152.9~
3.22. A heavy nap ht ha samp l e f r om Aust r al i an cr ude oil
has t he boi l i ng r ange of 140-190~ speci fi c gravi t y of
0.7736 and mol ecul ar wei ght of 131.4. The exp er i men-
t al l y det er mi ned comp os i t i on for n-paraffi ns, i soparaf-
fins, nap ht henes, and ar omat i cs are 29.97, 20.31, 38.72,
and 13%, respectively. For t hi s samp l e t he exp er i ment al
val ues of RON and MON are 26 and 28, respect i vel y as
r ep or t ed in Ref. [46], p. 359. Est i mat e t he RON f r om t he
p s eudocomp onent met hod ( Eqs. 3.138 and 3.139) and
t he Nel son met hods ( Fig. 3.33 and 3.34). Also cal cul at e
t he MON f r om J enki ns met hod (Eq. 3.140). For each
case cal cul at e t he er r or and comment on your resul t s.
3.23. A l i ght nap ht ha f r om Abu Dhabi field (UAE) has boi l -
i ng r ange of C5-80~ t he API gravi t y of 83.1 and Kw of
12.73 [46]. Est i mat e t he fol l owi ng oct ane number s and
comp ar e wi t h t he exp er i ment al val ues.
a. Cl ear RON f r om t wo di fferent met hods ( exp er i ment al
val ue is 65)
b. RON + 1.5 mL of TEL / U.S. Gal l on ( exp er i ment al
val ue is 74.5)
c. Cl ear MON ( exp er i ment al val ue is 61)
d. How much MTBE shoul d be added to t hi s nap ht ha to
i ncr ease t he RON f r om 65 to 75.
3.24. A p et r ol eum f r act i on p r oduced f r om a Venezuel an cr ude
has ASTM D 86 di st i l l at i on curve as:
ASTM D 86 temperature, ~ 504 509 515 523 534
TABLE 3. 33- - Estimation of composition of petroleum fractions (Problem 3. 16). a
Exp. composition
No, Fraction Tb, K exp. SG exp.
1 China Heavy Naphtha 444.1 0.791
2 Malaysia Light Naphtha 326.6 0.666
3 Indonesia Heavy Naphtha 405.2 0.738
4 Venezuela Kerosene 463.6 0.806
5 Heavy I ranian Gasoline 323.5 0.647
6 Qatar Gasoline 309.4 0.649
7 Sharjah Gasoline 337.7 0.693
8 American Gasoline 317.2 0.653
9 Libya Kerosene 465.5 0.794
10 U.K. North Sea Kerosene 464.9 0.798
11 U.K. North Sea Gas Oil 574.7 0.855
12 Mexico Naphtha 324.7 0.677
Estimated composition
aExperimental data on Tb, S, and the composition are taken from the Oil & Gas Journal Data Book [46].
n20 M m Xp XN XA Xp XN XA
48.9 30.9 20.2
83.0 17.0 0.0
62.0 30.0 8.0
39.8 41.1 19.0
93.5 5.7 0.8
95.0 3.9 1.1
78.4 14.4 7.2
92.0 7.3 0.7
51.2 34.7 14.1
42.5 36.4 21.1
34.3 39.8 25.9
81.9 13.9 4.2
The specific gravi t y ( at 60/ 60~ ) of f r act i on is 0.8597
[46]. Pr edi ct t he fol l owi ng p r op er t i es for t he p r oduct
and comp ar e wi t h r ep or t ed val ues.
a. Ki nemat i c vi scosi t i es at 100, 140, and 210~ ( experi -
ment al val ues are 3.26, 2.04, and 1.12 cSt.)
b. Mol ecul ar wei ght f r om vi scosi t y and comp ar e wi t h
mol ecul ar wei ght est i mat ed f r om t he boi l i ng poi nt .
c. Boi l i ng p oi nt and specific gravi t y f r om exp er i men-
t al val ues of ki nemat i c vi scosi t i es at 100 and 210~
( Part a) and comp ar e wi t h act ual val ues.
d. Ani l i ne p oi nt ( exp er i ment al val ue is 143.5~
e. Cet ane number ( exp er i ment al val ue is 43.2)
f. Fr eezi ng p oi nt ( exp er i ment al val ue is 21 ~
g. Fl ash p oi nt ( exp er i ment al val ue is 230~
h. Car bon- t o- hydr ogen wei ght r at i o ( exp er i ment al val ue
is 6.69)
i. Ar omat i c cont ent f r om exp er i ment al vi scosi t y at
100~ ( exp er i ment al val ue is 34.9)
j. Ar omat i c cont ent f r om exp er i ment al ani l i ne p oi nt
( exp er i ment al val ue is 34.9)
k. Refract i ve i ndex at 75~ ( exper i ment al val ue is
1.4759)
3.25. A vacuum r esi due has ki nemat i c vi scosi t y of 4.5 mm2/ s
at 100~ and specific gravi t y of 0.854. Est i mat e vi scosi t y
i ndex (VI) of t hi s f r act i on and comp ar e wi t h r ep or t ed
val ue of 119 [24].
3.26. A ker osene samp l e has boi l i ng r ange of 180-225~ and
specific gravi t y of 0.793. Thi s p r oduct has ar omat i c
cont ent of 20.5%. Pr edi ct smoke p oi nt and freezi ng
p oi nt of t hi s p r oduct and comp ar e wi t h t he experi -
ment al val ues of 19 mm and - 50~ [24]. How much
2- met hyl nonane (C10Hzz) wi t h freezi ng p oi nt of - 74~
shoul d be added to t hi s ker osene t o r educe t he freezi ng
p oi nt to - 60~
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"Prediction of Flash Points of Middle Distillates," Industrial
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[71] Walsh, R. E and Mortimer, J. V., "New Way to Test Product
Quality," Hydrocarbon Processing, Vol. 50, No. 9, 1971,
pp. 153-158.
[72] Lenoir, J. M., "Predict Flash Points Accurately," Hydrocarbon
Processing, Vol. 54, No. 1, 1975, pp. 153-158.
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Points," Hydrocarbon Processing, Vol. 66, No. 9, 1987,
pp. 81-83.
[74] Wickey, R. O. and Chittenden, D. I-I., "Flash Points of Blends
Correlated," Hydrocarbon Processing, Vol. 42, No. 6, 1963,
pp. 157-158.
[75] Hu, J. and Bums, A. M., "New Method Predicts Cloud, Pour
and Flash Points," Hydrocarbon Processing, Vol. 49, No. 11,
1970, pp. 213-216.
[76] Hu, J. and Bums, A. M., "Index Predicts Cloud, Pour and Flash
Points of Distillate Fuel Blends," Oil & Gas Journal, Vol. 68,
No. 45, 1970, pp. 66-69.
[77] Linden, H. R., "The Relationship of Physical Properties and
Ultimate Analysis of Liquid Hydrocarbon," Oil & Gas Journal,
Vol. 48, No. 9, 1949, pp. 60-65.
[78] Albahri, T. A., Riazi, M. R., and Qattan, A., "Octane Number
and Aniline Point of Petroleum Fractions," Preprints, Division
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18-22, 2002.
[79] Nelson, W. L., "Octane Numbers of Naphthas," Oil and Gas
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[80] Jenkins, G. I., "Calculation of the Motor Octane Number from
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[82] Nelson, W. L., "Estimating Yields of Asphalt in Cmde
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[83] Jenkins, G. I. and Walsh, R. E, "Quick Measure of Jet Fuel
Properties," Hydrocarbon Processing, Vol. 47, No. 5, 1968,
pp. 161-164.
[84] Riazi, M. R. and Roomi, Y. A., "Minimum Laboratory Data for
Properties of Petroleum Fractions and Crude Oils," Preprints of
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1999.
[87] Knepper, J. I. and Hutton, R. E, "Blend for Lower Pour
Point," Hydrocarbon Processing, Vol. 54, No. 9, 1975,
pp. 129-136.
[88] Auckland, M. H. T. and Charnock, D. J., "The Development
of Linear Blending indexes for Petroleum Products,"
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pp. 322-329.
[89] Morris, W. E., "The Interaction Approach To Gasoline
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[90] Modern Petroleum Technology, 3rd ed., The Institute of
Petroleum, London, 1962.
MNL50-EB/Jan. 2005
Characteri zati on of Reservoir
Fluids and Crude Oils
NOMENCLATURE
API
A, B
a, b . . . . i
CH
40
E
F( P)
h
API Gravity defined in Eq. (2.4)
Coefficients in Eq. (4.56) and other equations
Correlation constants in various equations
Liquid density at 20~ and 1 atm, g/ cm 3
Critical density defined by Eq. (2.9), g/ cm 3
Error function defined in Eqs. (4.41) or (4.42)
Probability density function in t erms of propert y P
Difference in molecular weight of successive SCN
groups
KT Equilibrium ratio for a component whose boiling
point is T
Kw Watson (UOP) K factor defined by Eq. (2.13)
I Refractive index paramet er defined in Eq. (2.36)
J I ntegration paramet er defined in Eq. (4.79),
dimensionless
M Molecular weight, g/ mol [kg/kmol]
Mb Variable defined by Eq. (4.36)
Nc Carbon number ( number of carbon at oms in
a single carbon number hydrocarbon group)
Np Number of pseudocomponent s
N Carbon number of the residue or plus fraction
in a mixture
n Refers to carbon number in a SCN group
n20 Sodi um D line refractive index of liquid at 20~ and
1 atm, dimensionless
P A property such as Tb, M, SG, or I used
in a probability density function
P* Dimensionless paramet er defined by Eq. (4.56) as
[ =( P - Po)/Po].
Ppc Pseudocritical pressure, bar
pvap Vapor (saturation) pressure, bar
R 2 Rsquar ed (R2), defined in Eq. (2.136)
SG Specific gravity of]iquid substance at 15.5~ (60~
SGg Specific gravity of gas substance at 15.5~ (60~
Tb Boiling point, K
Tbr Reduced boiling point (=Tb/Tc in which bot h Tb and
Tc are in K), dimensionless
Tpc Pseudocritical temperature, K
Vr Critical volume (molar), cma/ mol (or critical
specific volume, cma/g)
x~ Cumulative mole, weight, or volume fraction
x~ Cumulative mole fraction
Xr~ Discrete mole fraction of component i
x* Defined in Eq. (4.56) [=1 - xc]
X Paramet er defined in Eq. (4.57) [ = In In(I/x* )]
Y Paramet er defined in Eq. (4.57) [ =l n P* ]
Yi A Gaussian quadrat ure point in Section 4.6.1.1
Zc Critical compressibility factor defined by Eq. (2.8),
dimensionless
zj Predicted mole fraction of pseudocomponent i in a C7+
fraction
wi A weighting factor in Gaussian quadrat ure splitting
scheme
S
r(x)
r ( a, q)
F
0
#i
0
Greek Letters
Paramet er for gamma distribution model, Eq. (4.31)
fl Composite paramet er for gamma distribution model,
Eq. (4.31)
Solubility paramet er [ =cal/ cm 3 ]1/2
Error par amet er
Gamma function defined by Eq. (4.43)
I ncompl et e gamma function defined by Eq. (4.89)
Activity coefficient
Coefficient in gamma distribution model, Eq. (4.31)
Fugacity coefficient
Chemical potential of component i in a mixture
A propert y of hydrocarbon such as M, Tc, Pc, Vc, I , d,
Tb~ 9 9 9
a Surface tension, dyn/ cm [ =mN/ m]
co Acentric factor
Superscript
c Adjusted pseudocritical properties for the effects of
nonhydrocarbon compounds in nat ural gas system
as given by Eqs. (4.5) and (4.6).
cal Calculated value
exp Experimental value
i A component in a mixture.
j A pseudocomponent in a C7+ fraction
L Liquid phase
V Vapor phase
- Lower value of a property for a SCN group
+ Upper value of a propert y for a SCN group
Subscripts
A Aromatic
av Average value for a propert y
c Cumulative fraction
M Molecular weight
152
4. CHARACTERIZATION OF RESERV OI R FLUIDS AND CRUDE OILS 153
m Mole fraction
N Naphthenic
n Refers to SCN group with n carbon number
P Paraffinic
pc Pseudo-Critical
T A distribution coefficient in Eq. (4.56) for
boiling point
v Volume fraction
w Weight fraction
o Value of a property at xc = 0 in Eq. (4.56)
oo Value of a property at M --~ c~
20 Value of a property at 20~
Acrony ms
%AAD Average absolute deviation percentage defined
by Eq. (2.135)
API-TDB American Petroleum Institute--Technical
Data Book
%D Absolute deviation percentage defined by
Eq. (2.134)
GC Gas chromatography
KISR Kuwait Institute for Scientific Research
%MAD Maximum absolute deviation percentage
OGJ Oil & Gas Journal
PDF Probability density function
PNA Paraffins naphthenes aromatics
RMS Root mean squares defined by Eq. (4.59)
RVP Reid vapor pressure
RS R squared (R2), defined in Eq. (2.136)
SCN Single carbon number
TBP True boiling point
VLE Vapor-liquid equilibrium
As DISCUSSED IN CHAPTER 1, reservoir fluids are in the forms of
natural gases, gas condensates, volatile oils, and black oils. As
shown in Table 1.1, these fluids contain hydrocarbons from C1
to compounds with carbon number greater than 50. Composi-
tion of a reservoir fluid is generally expressed in tool% of non-
hydrocarbon compounds (i.e., H2S, CO2, N2), C1, C2, C3, nC4,
i Ca, nCs, i C5, C6, and C7 The boiling range of reservoir fluids
can be greater than 550~ (~ > 1000~ Crude oil is produced
by separating light gases from a reservoir fluid and bringing
its condition to surface atmospheric pressure and tempera-
ture. Therefore, crude oils are generally free from methane
and contain little ethane. The main difference between vari-
ous reservoir fluid and produced crude oil is in their composi-
tion, as shown in Table 1.1. Amount of methane reduces from
natural gas to gas condensate, volatile oil, black oil, and crude
oil while amount of heavier compounds (i.e., C7+) increase in
the same direction. Characterization of reservoir fluids and
crude oils mainly involves characterization of hydrocarbon-
plus fractions generally expressed in terms of C7+ fractions.
These fractions are completely different from petroleum frac-
tions discussed in Chapter 3. A C7 fraction of a crude oil has
a very wide boiling range in comparison with a petroleum
product and contains more complex and heavy compounds.
Usually the only information available for a C7+ fraction is
the mole fraction, molecular weight, and specific gravity.
The characterization procedure involves how to present this
mixture in terms of arbitrary number of subfractions (pseu-
docomponents) with known mole fraction, boiling point, spe-
cific gravity, and molecular weight. This approach is called
pseudoization. The main objective of this chapter is to present
methods Of characterization of hydrocarbon-plus fractions,
which involves prediction of distribution of hydrocarbons in
the mixture and to represent the fluid in terms of several nar-
row range subfractions. However, for natural gases and gas
condensate fluids that are rich in low-molecular-weight hy-
drocarbons simple relations have been proposed in the lit-
erature. I n this chapter types of data available for reservoir
fluids and crude oils are discussed followed by characteriza-
tion of natural gases. Then physical properties of single car-
bon number (SCN) groups are presented. Three distribution
models for properties of hydrocarbon plus fractions are intro-
duced and their application in characterization of reservoir
fluids is examined. Finally, the proposed methods are used to
calculate some properties of crude oils. Accuracy of charac-
terization of reservoir fluids largely depends on the distribu-
tion model used to express component distribution as well as
characterization methods of petroleum fractions discussed in
Chapter 2 to estimate properties of the narrow boiling range
pseudocomponents.
4 . 1 S PEC I FI C A T I ON S OF R ES ER VOI R
FL UI D S A N D C R UD E A S S A Y S
Characterization of a petroleum fluid requires input para-
meters that are determined from laboratory measurements.
I n this section types of data available for a reservoir fluid or
a crude oil are presented. Availability of proper data leads to
appropriate characterization of a reservoir fluid or a crude oil.
4. 1. 1 Laborat ory Dat a for Reserv oi r Fl ui ds
Data on composition of various reservoir fluids and a crude oil
were shown in Table i. 1. Further data on composition of four
reservoir fluids from North Sea and South West Texas fields
are given in Table 4.1. Data are produced from analysis of the
fluid by gas chromatography columns capable of separating
hydrocarbons up to C40 or C4s. Composition of the mixture is
usually expressed in terms of tool% for pure hydrocarbons up
to Cs and for heavier hydrocarbons by single carbon number
(SCN) groups up to C30 or C40. However, detailed composition
is available for lower carbon numbers while all heavy hydro-
carbons are lumped into a single group called hydrocarbon-
plus fraction. For example in Table 4.1, data are given up to
C9 for each SCN group while heavier compounds are grouped
into a Ci0+ fraction. It is customary in the petroleum indus-
try to lump the hydrocarbons heavier than heptane into a C7+
fraction. For this reason the tool% of C7+ for the four mixtures
is also presented in Table 4.1. For hydrocarbon-plus fractions
it is important to report a mi ni mum of two characteristics.
These two specifications are generally molecular weight and
specific gravity (or API gravity) shown by M7+ and SG7+, re-
spectively. I n some cases a reservoir fluid is presented in terms
of true boiling point (TBP) of each SCN group except for the
plus fraction in which boiling point is not available. The plus
fractions contain heavy compounds and for this reason their
Component
T A B L E 4. 1---Composi t i on of several reservoir fluids.
North Sea gas condensate North Sea oil Texas gas condensate
mol% SG M tool% SG M mol% SG M
Texas oil
mol% SG M
N2 0.85 0.69 0 0
CO2 0.65 3.14 0 0
Ct 83.58 52.8I 91.35 52.00
C2 5.95 8.87 4.03 3.81
C3 2.91 6,28 1.53 2.37
IC4 0.45 1.06 0,39 0.76
nC4 1.11 2.48 0.43 0.96
IC5 0.36 0.87 0.15 0.69
nC5 0.48 1.17 0.19 0.51
C6 0.60 1.45 0.39 2.06
C7 0.80 0.7243 95 2.39 0.741 91.7 0.361 0.745 100 2.63
C8 0.76 0.7476 103 2.67 0.767 104.7 0.285 0.753 114 2.34
C9 0.47 0.7764 116 1.83 0.787 119.2 0.222 0.773 128 2.35
C10+ 1.03 0.8120 167 14.29 0.869 259.0 0.672 0.814 179 29.52
C7+ 3.06 0.7745 124 21.18 0.850 208.6 1.54 0.787 141. 1 36.84
0.749 99
0.758 110
0.779 121
0.852 221
0.841 198.9
Source: North Sea gas condensate and oil samples are taken from Ref. [1]. South West Texas gas condensate and oil samples are
taken from Ref. [2]. Data for Cv+ have been obtained from data on C7, Cs, C9, and C10+ components.
boi l i ng p oi nt cannot be measur ed; onl y mol ecul ar wei ght and
speci fi c gravi t y ar e avai l abl e for t he pl us fract i ons. Charact er-
i st i cs and p r op er t i es of SCN gr oups are gi ven l at er in t hi s
chap t er ( Sect i on 4.3).
Gener at i on of such dat a for mol ecul ar wei ght and den-
si t y di st r i but i on f r om gas chr omat ogr ap hy (GC) anal ysi s for
cr ude oils is shown by Osj ord et al. [3]. Det ai l ed comp osi -
t i on of SCN gr oup s for C6+ or C7+ fract i ons can al so be ob-
t ai ned by TBP di st i l l at i on. Exp er i ment al dat a obt ai ned f r om
di st i l l at i on are t he most accur at e way of anal yzi ng a reser-
voi r fluid or cr ude oil, especi al l y when it is combi ned wi t h
measur i ng specific gravi t y of each cut. However, GC anal ysi s
r equi r es smal l er samp l e quant i t y, less t i me, and less cost t han
does TBP anal ysi s. The ASTM D 2892 p r ocedur e is a s t andar d
met hod for TBP anal ysi s of cr ude oils [4]. The ap p ar at us used
in ASTM D 2892, is shown in Fig. 4.1 [5]. A GC for det er mi ni ng
SCN di st r i but i on in cr ude oils is shown in Fig. 4.2. The out -
p ut f r om t hi s GC for a Kuwai t i cr ude oil samp l e is shown in
Fig. 4.3. I n t hi s fi gure var i ous SCN f r om Cs up to C40 are i den-
tified and t he r et ent i on t i mes for each car bon gr oup ar e gi ven
on each pick. A comp ar i s on of mol ecul ar wei ght and specific
gravi t y di st r i but i on of SCN gr oup s obt ai ned f r om TBP di st i l -
l at i on and GC anal ysi s for t he same cr ude oil is al so shown by
Osj ord et al. [3]. Peder sen et al. [6] have al so p r esent ed com-
p osi t i onal dat a for many gas condensat e samp l es f r om t he
Nor t h Sea. An ext ended comp osi t i on of a l i ght waxy cr ude
oil is gi ven i n Table 4.2 [7]. Di st r i but i on of SCN gr oup s for
t he Kuwai t cr ude det er mi ned f r om Fig. 4.3 is al so gi ven i n
Table 4,2, Ot her p r op er t i es of SCN gr oup s ar e gi ven i n
Sect i on 4.3. One of t he i mp or t ant char act er i st i cs of cr ude oi l s
is t he cl oud p oi nt (CPT). Thi s t emp er at ur e i ndi cat es when t he
p r eci p i t at i on of wax comp onent s i n a cr ude begi ns. Cal cu-
l at i on of CPT r equi r es l i qui d- sol i d equi l i br i um cal cul at i ons,
whi ch ar e di scussed in Chap t er 9 ( Sect i on 9.3.3).
FIG. 4. 1 mA pparat us to conduct T BP anal ysi s of crude oi l s
and reserv oir fl ui ds ( courtesy of KISR [5]).
4 . 1 . 2 Crude Oil Assay s
Comp osi t i on of a cr ude may be expr essed si mi l ar to a reser-
voi r fl ui d as shown in Table 1.1. A cr ude is p r oduced t hr ough
r educi ng t he p r essur e of a r eser voi r fl ui d to at mos p her i c
p r essur e and sep ar at i ng l i ght gases. Therefore, a cr ude oil
is usual l y free of met hane gas and has a hi gher amount
of C7+ t han t he ori gi nal r eser voi r fluid. However, in many
cases i nf or mat i on on char act er i st i cs of cr ude oils are gi ven
t hr ough cr ude assay. A comp l et e dat a on cr ude assay cont ai n
i nf or mat i on on speci fi cat i on of t he whol e cr ude oil as wel l
FI G. 4.2--A GC for measuri ng SCN di st ri but i on in crude oi l s and reservoi r f l ui ds (courtesy
of KI SR [5]),
| ? w
~~]i " '''' ~" I !' ~ ' " i ~ ' ,
r 0. r i ~ a 0. ~ G0~ 0 "
=1 ; 3 : 3 : 1 ~ - _ _ O , 0. Q. ~ nO .
i t q i i ~ r I i
' I
T ime, Mi n
FI G. 4.3 --A sample of out put from the GC of Fig. 4.2 for a Kuwai t crude oil.
TABLE 4. 2--Extended compositional data for a light waxy crude oil and a Middle East crude.
Waxy crude oil Middle East crude oil
Component mol% M, g/tool wt% M Normalized mol%
C2 0.0041 30 0.0076 30 0.0917
C3 0.0375 44 0.1208 44 0.9940
iC4 0.0752 58 0.0921 58 0.5749
NC4 0.1245 58 0.4341 58 2.7099
iC5 0.3270 72 0.4318 72 2.1714
NC5 0.2831 72 0.7384 72 3.7132
C6 0.3637 86 1.6943 82 7.4812
C7 3.2913 100 2.2346 95 8.5166
C8 8.2920 114 2.7519 107 9.3120
C9 10.6557 128 2.8330 121 8.4772
Clo 11.3986 142 2.8282 136 7.5294
Cll 10.1595 156 2.3846 149 5.7946
Ct2 8.7254 170 2.0684 163 4.5945
C13 8.5434 184 2.1589 176 4.4413
C14 6.7661 198 1.9437 191 3.6846
Cls 5.4968 212 1.9370 207 3.3881
C16 3.5481 226 1.5888 221 2.6030
C17 3.2366 240 1.5580 237 2.3802
C18 2.1652 254 1.5006 249 2.1820
C19 1.8098 268 1.5355 261 2.1301
C20 1.4525 282 1.5441 275 2.0330
C21 1.2406 296 1.1415 289 1.4301
C22 1.1081 310 1.4003 303 1.6733
C23 0.9890 324 0.9338 317 1.0666
C24 0.7886 338 1.0742 331 1.1750
C25 0.7625 352 1.0481 345 1.1000
C26 0.6506 366 0.9840 359 0.9924
C27 0.5625 380 0.8499 373 0.8250
C28 0.5203 394 0.9468 387 0.8858
C29 0.4891 408 0.8315 400 0.7527
C30 0.3918 422 0.8141 415 0.7103
C31 0.3173 436 0.7836 429 0.6613
C32 0.2598 450 0.7450 443 0.6089
C33 0.2251 464 0.7099 457 0.5624
C34 0.2029 478 0.6528 471 0.5018
C35 0.1570 492 0.6302 485 0.4705
C36 0.1461 506 0.5400 499 0.3918
C37 0.t230 520 0.5524 513 0.3899
C38 0.1093 534 0.5300 528 0.3634
C39 0.1007 548 0.4703 542 0.3142
C40+ 3.0994 700 C40:0.4942 C40:556 0.3217
C41+ 51.481
Source: waxy oil data from Ref. [7] and Middle East crude from Ref. [5]. For the Middle East crude the
GC output is shown in Fig. 4.3. Normalized mole% for the Middle East crude excludes C41+ fraction.
Based on calculated value of M41+ = 865, mole% of C41 + is 17.73%.
as its product s from at mospheri c or vacuum di st i l l at i on
col umns. The Oil & Gas Journal Data Book has publ i shed a
comprehensi ve set of dat a on vari ous crude oils from ar ound
the worl d [8]. Characteristics of seven crude oils from ar ound
the worl d and t hei r product s are given i n Table 4.3. A crude
assay dat aset cont ai ns i nf or mat i on on API gravity, sul fur and
met al cont ent s, ki nemat i c viscosity, p our poi nt , and Rei d
vapor pressure (RVP). I n addi t i on to boi l i ng poi nt range, API
gravity, viscosity, sul fur cont ent , PNA composi t i on, and ot her
charact eri st i cs of vari ous product s obt ai ned from each crude
are given. Fr om i nf or mat i on gi ven for vari ous fractions, boil-
i ng poi nt curve can be obt ai ned.
Qual i t y of crude oils are mai nl y eval uat ed based on hi gher
val ue for t he API gravity ( lower specific gravity), l ower aro-
mat i c, sulfur, ni t rogen and met al cont ent s, l ower p our poi nt ,
carbon-t o-hydrogen (CH) wei ght ratio, viscosity, car bon res-
idue, and salt and wat er cont ent s. Hi gher API crudes usual l y
cont ai n hi gher amount of paraffins, l ower CH wei ght ratio,
less sul fur and metals, and have l ower car bon resi dues and
viscosity. For t hi s reason API gravity is used as t he p r i mar y
p ar amet er to quant i fy qual i t y of a crude. API gravity of
crudes varies from 10 to 50; however, most crudes have API
gravity bet ween 20 and 45 [9]. A crude oil havi ng API gravity
great er t han 40 (SG < 0.825) is consi dered as light crude,
whi l e a crude wi t h API gravity less t han 20 (SG > 0.934) is
consi dered as a heavy oil. Crudes wi t h API gravity bet ween 20
and 40 are called intermediate crudes. However, this di vi si on
may vary from one source to anot her and usual l y there is
no sharp di vi si on bet ween light and heavy crude oils. Crude
oils havi ng API gravity of 10 or l ower (SG > 1) are referred
as very heavy crudes and oft en have more t han 50 wt %
residues. Some of Venezuel an crude oils are from this
category. Anot her p ar amet er t hat charact eri zes qual i t y of a
crude oil is t he total sul fur cont ent . The total sul fur cont ent
is expressed i n wt % and it varies from less t han 0.1% to more
t han 5% [9]. Crude oils wi t h total sul fur cont ent of great er
t han 0.5% are t er med as sour crudes while when the sul fur
cont ent is less t han 0.5% they are referred as "sweet" crude
[9]. After sul fur cont ent , lower ni t r ogen and met al cont ent s
signify qual i t y of a crude oil.
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V
/
N
i
,

p
p
m
:

6
/
1

H
2
S
,

w
t
%
:

<
0
.
0
0
0
1

S
a
l
t

(
a
s

N
a
C
I
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w
t
%
:

0
.
0
1
5

W
a
t
e
r

c
o
n
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e
n
t
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w
t
%
:

0
.
3
8

L
i
g
h
t
e
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d
s

(
C
I
-
C
5
)

Y
i
e
l
d
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w
t
%
:

5
.
8
7

R
a
n
g
e
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~

I
B
P
-
9
5

Y
i
e
l
d
,

v
d
%
:

1
2
.
3

Y
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l
d
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t
%
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1
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3

D
e
n
s
i
t
y
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t

1
5
~

k
g
/
L
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0
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6
9
2
4

S
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l
f
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r
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w
t
%
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0
.
0
0
0
6

M
e
r
c
a
p
t
a
n

S
,

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p
m
:

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1

P
a
r
a
f
f
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n
s
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w
t
%
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8
2
.
8

N
a
p
h
t
h
e
n
e
s
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t
%
:

1
1
.
9

A
r
o
m
a
t
i
c
s
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w
t
%
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5
.
3

n
-
P
a
r
a
f
f
i
n
s
,

w
t
%
:

3
0
.
9

R
a
n
g
e
,

~

9
5
-
1
7
5

Y
i
e
l
d
,

v
o
l
%
:

1
6
.
7

Y
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l
d
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w
t
%
:

1
5
.
4

D
e
n
s
i
t
y
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t

1
5
~

k
g
/
L
;

0
.
7
6
9
3

S
u
l
f
u
r
,

w
t
%
:

0
.
0
0
1
2

M
e
r
c
a
p
t
a
n

S
,

p
p
m
:

1
4

P
a
r
a
f
f
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n
s
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w
t
%
:

4
4
.
9

N
a
p
h
t
h
e
n
e
s
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w
t
%
:

3
6
.
7

A
r
o
m
a
t
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c
s
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w
t
%
:

1
8
.
4

n
-

P
a
r
a
f
f
i
n
s
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w
t
%
:

2
0
.
1

B
r
e
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t

B
l
e
n
d
,

U
.
K
.

N
o
r
t
h

S
e
a

R
a
n
g
e
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~

I
B
P
-
I
5
0

Y
i
e
l
d
,

v
o
l
%
:
2
3
.
8

Y
i
e
l
d
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w
t
%
:

2
0
.
8

D
e
n
s
i
t
y

a
t

1
5
~

k
g
/
L
:
0
.
7
2
7
9

S
u
l
f
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r
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w
t
%
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0
.
0
0
0
7

M
e
r
c
a
p
t
a
n

S
,

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m
:

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1

P
a
r
a
f
f
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n
s
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w
t
%
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6
5
.
3

N
a
p
h
t
h
e
n
e
s
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w
t
%
:

2
3
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8

A
r
o
m
a
t
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c
s
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w
t
%
:

1
0
.
9

n
-
P
a
r
a
f
f
i
n
s
,

w
t
%
:

2
5
.
9

R
a
n
g
e
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~

1
5
0
-
2
3
0

Y
i
e
l
d
,

v
o
l
%
:

1
3
.
9

Y
i
e
l
d
,

w
t
%
:

1
3
.
3

D
e
n
s
i
t
y
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t

1
5
~

k
g
/
L
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0
.
8
0
0
1

S
u
l
f
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r
,
w
t
%
:

0
.
0
0
4
8

M
e
r
c
a
p
t
a
n

S
,

p
p
m
:

4
2

K
i
n
e
m
a
t
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c

v
i
s
c
o
s
i
t
y
a
t

4
0
~

c
S
t
:

1
.
1
3
5

K
i
n
e
m
a
t
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c

v
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s
c
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s
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t
y
a
t

6
0
~

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S
t
:

0
.
9
4
0
8

A
c
i
d
i
t
y
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m
g

K
O
H
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g
:

0
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0
3
6

S
m
o
k
e

p
o
i
n
t
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n
m
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2

F
r
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z
e

p
o
i
n
t
,

-
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1
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5

A
n
i
l
i
n
e
p
o
i
n
t
,

~

5
5
.
0

C
e
t
a
n
e

i
n
d
e
x
:

3
6
.
6

H
y
d
r
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n
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w
t
%
:

1
3
.
4
5

C
o
l
o
r
s
t
a
b
i
l
i
t
y
:
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t
a
b
l
e

N
a
p
h
t
h
a
l
e
n
e
s
,

v
o
l
%
:
2
.
3
0

R
a
n
g
e
,

~

2
3
0
-
3
5
0

Y
i
e
l
d
,

v
d
%
:

2
2
.
2

Y
i
e
l
d
,

w
t
%

2
2
.
5

D
e
n
s
i
t
y

a
t

1
5
~

k
g
/
L
:
0
.
8
4
6
1

S
u
l
f
u
r
,

w
t
%
:

0
.
2
4

K
i
n
e
m
a
t
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c
v
i
s
c
o
s
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t
y
a
t

5
0
~

c
S
t
:

2
.
8
3
1

K
i
n
e
m
a
t
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c
v
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s
c
o
s
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t
y
a
t

1
0
0
~
C
,
c
S
t
:

1
.
3
3
9

C
l
o
u
d

p
o
i
n
t
,

~

-
9

P
o
u
r

p
o
i
n
t
,

~

-
9

W
a
x

c
o
n
t
e
n
t
,

w
t
%
:

6
.
2

T
o
t
a
l

N
,

m
g
/
k
g
:

1
0
7

A
c
i
d
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t
y
,

m
g

K
O
I
-
U
g
:
0
.
0
2
8

A
n
i
l
i
n
e
p
o
i
n
t
,

~

7
0
.
6

C
e
t
a
n
e

i
n
d
e
x
:

5
1

C
o
l
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r
s
t
a
b
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l
i
t
y
:
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t
a
b
l
e

R
a
n
g
e
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~

3
5
0
-
3
7
5

Y
i
e
l
d
.

v
d
%
:

3
.
8

Y
i
e
l
d
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w
t
%
:

4
.
0

D
e
n
s
i
t
y
a
t

1
5
~

k
g
/
L
:
0
.
8
7
9
5

S
u
l
f
u
r
,
w
t
%
:

0
.
4
9

K
i
n
e
m
a
t
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c
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s
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t
y
a
t

5
0
~

c
S
t
:

7
.
8
0
7

K
i
n
e
m
a
t
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c
v
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s
c
o
s
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t
y
a
t

1
0
0
~
C
,
c
S
t
:
2
.
7
1
4

C
l
o
u
d
p
o
i
n
t
,

~

1
9

P
o
u
r

p
o
i
n
t
,

~

1
5

W
a
x

c
o
n
t
e
n
t
,

w
t
%
:

1
7
.
3

T
o
t
a
l

N
,

m
g
/
k
g
:

5
3
2

A
n
i
l
i
n
e
p
o
i
n
t
,

~

8
0
.
5

C
e
t
a
n
e

i
n
d
e
x
:

4
9
.
5

R
a
n
g
e
,

~

3
7
5
-
5
5
0

Y
i
e
l
d
,

v
d
%
:

2
0
.
7

Y
i
e
l
d
,

w
t
%
:

2
2
.
5

D
e
n
s
i
t
y
a
t

1
5
~

k
g
/
L
:
0
.
9
0
5
9

S
u
l
f
u
r
,
w
t
%
:

0
.
6
1

K
i
n
e
m
a
t
i
c
v
i
s
c
o
s
i
t
y
a
t

6
0
~

c
S
t
:

2
6
.
2
3

K
i
n
e
m
a
t
i
c

v
i
s
c
o
s
i
t
y
a
t

8
0
~

c
S
t
:

1
2
.
6
9

K
i
n
e
m
a
t
i
c
v
i
s
c
o
s
i
t
y
a
t

1
0
0
~
C
,
c
S
t
:

7
.
7
1
1

W
a
x

c
o
n
t
e
n
t
,

w
t
%
:

2
0
.
7

T
o
t
a
l

N
,

m
g
/
k
g
:

1
4
4
7

B
a
s
i
c

N
,

m
g
/
k
g
:

3
5
4

A
c
i
d
i
t
y
,

m
g

K
O
H
/
g
:

0
.
0
5
2

A
n
i
l
i
n
e
p
o
i
n
t
,

~

9
0
.
5

R
e
f
r
a
c
t
i
v
e
i
n
d
e
x

a
t

6
0
~

1
.
4
8
8
4

R
a
n
g
e
,

~

3
5
0
+

Y
i
e
l
d
,

v
o
l
%
:
3
6
.
7

Y
i
e
l
d
,

w
t
%
:

4
0
.
9

D
e
n
s
i
t
y
a
t

1
5
~

k
g
/
L
:
0
.
9
2
8
5

S
u
l
f
u
r
,

w
t
%
:

0
.
8
6

K
i
n
e
m
a
t
i
c
v
i
s
c
o
s
i
t
y
a
t

5
0
~

c
S
t
:

1
2
6
.
8

K
i
n
e
m
a
t
i
c
v
i
s
c
o
s
i
t
y
a
t

6
0
~
C
,

c
S
t
:

7
7
.
3
8

K
i
n
e
m
a
t
i
c
v
i
s
c
o
s
i
t
y
a
t

8
0
~
C
,

c
S
t
:

3
1
.
8
7

K
i
n
e
m
a
t
i
c
v
i
s
c
o
s
i
t
y
a
t

1
0
0
~
C
,
c
S
t
:

1
6
.
8
0

K
i
n
e
m
a
t
i
c
v
i
s
c
o
s
i
t
y
a
t

1
2
0
~

c
S
t
:

9
.
5
*

K
i
n
e
m
a
t
i
c
v
i
s
c
o
s
i
t
y
a
t

1
5
0
~

c
S
t
:

5
.
5
*

P
o
u
r

p
o
i
n
t
,

~

3
6

W
a
x

c
o
n
t
e
n
t
,

w
t
%
:

1
4
.
6

T
o
t
a
l

N
,

m
g
/
k
g
:
2
6
6
7

A
c
i
d
i
t
y
,

m
g

K
O
H
/
g
:

0
.
1
8

M
i
c
r
o

c
a
r
b
o
n

r
e
s
.
,

w
t
%
:

5
.
0

A
s
p
h
a
l
t
e
n
e
s
,

w
t
%
:

0
.
9
5

V
/
N
i
,

p
p
m
:

1
2
/
1

X
y
l
e
n
e

e
q
u
i
v
a
l
e
n
t
,
v
o
l
%
:
5
-
1
0
+

R
a
n
g
e
,

~

5
5
0
+

Y
i
e
l
d
,

v
o
l
%
:

1
2
.
2

Y
i
e
l
d
,

w
t
%
:
1
4
.
4

D
e
n
s
i
t
y

a
t

1
5
~

k
g
/
L
:
0
.
9
8
7
6

S
u
l
f
u
r
,

w
t
%
:

1
.
2
1

K
i
n
e
m
a
t
i
c
v
i
s
c
o
s
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t
y
a
t

8
0
~

c
S
t
:

9
7
5
.
4

K
i
n
e
m
a
t
i
c
v
i
s
c
o
s
i
t
y
a
t

1
0
0
~
C
,
c
S
t
:

3
0
3
.
1

K
i
n
e
m
a
t
i
c
v
i
s
c
o
s
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t
y
a
t

1
2
0
~

c
S
t
:

1
2
0
"

K
i
n
e
m
a
t
i
c
v
i
s
c
o
s
i
t
y
a
t

1
5
0
~
C
,

c
S
t
:

4
0
*

P
e
n
e
t
r
a
t
i
o
n

a
t

2
5
~

>
4
0
0

S
o
f
t
e
n
i
n
g
p
o
i
n
t
,

~

3
1
.
8

T
o
t
a
l

N
,

m
g
/
k
g
:

5
2
4
2

M
i
c
r
o

c
a
r
b
o
n

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e
s
.
,

w
t
%
:

2
.
0
5

A
s
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h
a
l
t
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n
e
s
,

w
t
%
:

2
.
0
5

V
/
N
i
,

p
p
m
:

3
5
/
3

X
y
l
e
n
e

e
q
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i
v
a
l
e
n
t
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v
o
l
%
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1
0

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e
w
a
x
e
d

o
i
l

D
e
n
s
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t
y
a
t

1
5
~

k
g
/
L
:
0
.
9
2
4
4

K
i
n
e
m
a
t
i
c
v
i
s
c
o
s
i
t
y
a
t

4
0
~

c
S
t
:

1
0
4
.
2

K
i
n
e
m
a
t
i
c
v
i
s
c
o
s
i
t
y
a
t

6
0
~
C
,

c
S
t
:

3
7
.
1
7

K
i
n
e
m
a
t
i
c
v
i
s
c
o
s
i
t
y
a
t

1
0
0
~
C
,
c
S
t
:
9
.
5
3
9

V
i
s
c
o
s
i
t
y
i
n
d
e
x
:

5
4

P
o
u
r

p
o
i
n
t
,

~
C
:

-

1
8

*
E
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1

K
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R
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1
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2
0

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l
d
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o
l
%
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.
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3

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r
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~

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5
.
9

P
o
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r

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t
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4
5

S
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l
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r
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w
t
%
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0
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3
9

A
n
i
l
i
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2
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9

C
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t
a
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e

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n
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e
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5
3
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8

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e

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H
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l

R
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n
g
e
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~

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2
0
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6
8
0

Y
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l
d
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o
l
%
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4
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9

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r
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v
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t
y
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2
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9

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s
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e
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t

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3

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o
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t
%
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o
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6
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2

C
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4
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3

G
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R
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v
o
l
%
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r
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4

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r

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t
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t

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3
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R
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t
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t

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m
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r
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t
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3

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4

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r
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r

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t
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t
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t

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m

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t
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3

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r

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t
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8
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t
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t

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6
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/
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m
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s
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m

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d
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v
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l
%
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4

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r
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t
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r
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t
%
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6

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r

p
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8

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/
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p
p
m
:

1
4
3
/
4
4

C
r
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d
e

6

H
o
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o

M
o
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e
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7

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o
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r
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v
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t
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~

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9
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4

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G
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0
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0
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9
3
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7

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7
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t
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e
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o
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6
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r

p
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t
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0

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i
g
h
t

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d
s

s
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m
s

(
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d
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v
o
l
%
:
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6
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R
a
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e
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~

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8
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7

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d
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v
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l
%
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1
2

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N
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r
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4

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r
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t
%
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6
6

R
a
n
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e
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5
6
3

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v
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l
%
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4
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t
%
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1
4

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l
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e
p
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n
t
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2
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.
0

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i
s
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o
s
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t
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t

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t

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6

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r

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n
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8

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m
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k
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n
t
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m
m
:

1
8
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2

R
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6
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5
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l
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v
o
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%
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6
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1
6

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l
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t
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8
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n
t
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~

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8

V
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s
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o
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t
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t

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t

1
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.
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r
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o
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P
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r

p
o
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n
t
,

~

2
9

R
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n
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4
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l
%
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t
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t
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4
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m
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l
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t
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t

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t

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4
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o
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r

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o
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n
t
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/
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p
m
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r
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n
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p
m
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0

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7
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5
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l
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o
l
%
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5
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n
e
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r
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t
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o
n

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t

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m
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s
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t

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t

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4 . 2 GEN ER A L I Z ED C OR R EL A T I ON S
FOR PS EUD OC R I T I C A L P R OP ER T I ES
OF N A T UR A L GA S ES A N D GAS
C ON D EN S A T E S Y S T EM S
Natural gas is a mixture of light hydrocarbon gases rich in
methane. Methane content of natural gases is usually above
75 % with C7+ fraction less than 1%. I f mole fraction of H2 S in
a natural gas is less than 4 x I 0 -6 (4 ppm on gas volume basis)
it is called "sweet" gas (Section 1.7.15). A sample composition
of a natural gas is given in Table 1.2. Dry gases contain no
C7+ and have more than 90 mol% methane. The main differ-
ence between natural gas and other reservoir fluids is that the
amount of C7+ or even C6+ in the mixture is quite low and the
main components are light paraffinic hydrocarbons. Proper-
ties of pure light hydrocarbons are given in Tables 2.1 and 2.2.
The C6+ or C7+ fraction of a mixture should be treated as an
undefined fraction and its properties may be determined from
the correlations given in Chapter 2. I f the detailed composi-
tion of a natural gas is known the best method of characteriza-
tion is through Eq. (3.44) with composition in terms of mole
fraction (xi) for calculation of pseudocritical properties, acen-
tric factor, and molecular weight of the mixture. Although
the Key's mixing rule is not the most accurate mixing rule
for pseudocritical properties of mixtures, but for natural gas
systems that mainly contain methane it can be used with rea-
sonable accuracy. More advanced mixing rules are discussed
in Chapter 5. Once the basic characterization parameters for
the mixture are determined various physical properties can
be estimated from appropriate methods.
The second approach is to consider the mixture as a single
pseudocomponent with known specific gravity. This method
is particularly useful when the exact composition of the mix-
ture is not known. There are a number of empirical correla-
tions in the literature to estimate basic properties of natural
gases from their specific gravity. Some of these methods are
summarized below.
I n cases that the composition of a natural gas is unknown
Brown presented a simple graphical method to estimate pseu-
docritical temperature and pressure from gas specific gravity
(SGg) as shown by Ahmed [10]. Standing [11] converted the
graphical methods into the following correlations for estima-
tion of Tpc and Ppc of natural gases free of CO2 and H2S:
(4.1) Tpc = 93.3 180.6SGg - 6.94SG~
(4.2) Ppc = 46.66 1.03SGg - 2.58SG~
where Too and Ppc are the pseudocritical temperature and
pressure in kelvin and bar, respectively. SGg is defined in
Eq. (2.6). This method is particularly useful when the exact
composition of the mixture is not available. This method pro-
vides acceptable results since nearly 90% of the mixture is
methane and the mixture is close to a pure component. There-
fore, assumption of a single pseudocomponent is quite rea-
sonable without significant difference with detailed composi-
tional analysis. Application of these equations to wet gases is
less accurate.
Another type of reservoir fluids that are in gaseous phase
under reservoir conditions are gas condensate systems. Com-
position of a gas condensate sample is given in Table 1.2. Its
C7+ content is more than that of natural gases and it is about
few percent, while its methane content is less than that of
natural gas. However, for gas condensate systems simple cor-
relations in terms of specific gravity have been proposed in
the following forms similar to the above correlations and are
usually used by reservoir engineers [ 10]:
(4.3) Tpc -- 103.9 + 183.3SGg - 39.7SGg 2
(4.4) Ppc = 48.66 + 3.56SGg - 0.77SG~
These equations give higher critical temperature and lower
critical pressure than do the equations for natural gases since
gas condensate samples contain heavier compounds. Because
of the greater variation in carbon number, the equations for
gas condensate systems are much less accurate than those
for natural gas systems. For this reason properties of gas con-
densate systems may be estimated more accurately from the
distribution models presented in Section 4.5.
Equations (4.1)-(4.4) proposed for pseudocritical proper-
ties of natural gas and gas condensate systems are based
on the assumption that mixtures contain only hydrocarbon
compounds. However, these reservoir fluids generally contain
components such as carbon dioxide (CO2), hydrogen sulfide
(H2S), or nitrogen (N2). Presence of such compounds affects
the properties of the gas mixture. For such cases, correc-
tions are added to the calculated pseudocritical properties
from Eqs. (4.1)-(4.4). Corrections proposed by Wichert and
Aziz [12] and Carr et al. [13] are recommended for the effects
of nonhydrocarbons on properties of natural gases [8]. The
method of Carr et al. for adjustment of calculated Tpc and Pvc
is given as follows:
(4.5) Tp c = Tpc -44.44yco2 + 72.22yH2S -- 138.89yN2
(4.6) Ppc = Ppc + 30.3369yco2 + 41.368yH2S -- 11.721yN2
where T~c and Ppc are the adjusted (corrected) pseudocriti-
cal temperature and pressure in kelvin and bar, respectively.
Yco2, YH2S and YN2are the mole fractions of CO2, H2S, and N2,
respectively. Tpc and Ppc are unadjusted pseudocritical tem-
perature and pressure in kelvin and bar, respectively. These
unadjusted properties may be calculated from Eqs. (4.1) and
(4.2) for a natural gas. The following example shows calcula-
tion of pseudocritical properties for a natural gas sample.
Example 4. 1--A natural gas has the following composition in
mol%: H2S 1.2%, N2 0.2%; CO2 1%, C1 90%, C2 4.8%, Ca 1.7%,
iC4 0.4, F/C4 0.5%, iC5 0.1, nC5 0.1%.
a. Calculate Tc, Pc, o9, and M using properties of pure com-
pounds.
b. Calculate the gas specific gravity.
c. Calculate Tc and Pc using Eqs. (4.1) and (4.2) and SG cal-
culated from Part (b).
d. Adjust Tc and Pc for the effects of nonhydrocarbon com-
pounds present in the gas.
Solution--V alues of M, Tc, Pc, and co for pure components
present in the gas mixture can be obtained from Table 2.1.
These values as well as calculated values of M, Tc, Pc, and co
for the mixture based on Eq. (3.44) are given in Table 4.4. The
calculated values of M, To, Pc, and w as shown in Table 4.4 are:
M = 18.17, Tpc = -68.24~ Ppc = 46.74 bar, and w = 0.0234.
This method should be used for gases with SGg > 0.75 [10].
a. Equation (2.6) can be used to calculate gas specific gravity:
SGg = 18.17/28.96 = 0.6274.
No.
4. CHARACTERI Z ATI ON OF RESERV OI R FL UI DS AND CRUDE OI L S 161
TABLE 4. 4--Calculation of pseudoproperties of the natural gas system of Example 4.1.
Component x/ Mi Tci, ~ Pci, bar o)/ x/ x Mi xi x Tci xi x Pcl xi x o)i
1 HeS 0.012 34.1 100.38 89.63 0.0942 0.41 1.20 1.08 0.0011
2 N2 0.002 28.0 -146.95 34.00 0.0377 0.06 -0. 29 0.07 0.0001
3 CO2 0.01 44.0 31.06 73.83 0.2236 0.44 0.31 0.74 0.0022
4 C1 0.90 16.0 -82.59 45.99 0.0115 14.40 -74.33 41.39 0.0104
5 C2 0.048 30.1 32.17 48.72 0.0995 1.44 1.54 2.34 0.0048
6 Ca 0.017 44.1 96.68 42.48 0.1523 0.75 1.64 0.72 0.0026
7 i C4 0.004 58.1 134.99 36.48 0.1808 0.23 0.54 0.15 0.0007
8 nC4 0.005 58.1 151.97 37.96 0.2002 0.29 0.76 0.19 0.0010
9 iCs 0.001 72.2 187.28 33.81 0.2275 0.07 0.19 0.03 0.0002
10 nC5 0.001 72.2 196.55 33.70 0.2515 0.07 0.20 0.03 0.0003
Sum Mixture 1.00 18.17 -68.24 46.74 0.0234
b. The syst em is a nat ur al gas so Eqs. (4.1) and (4.2) are used
to est i mat e p seudocr i t i cal p r op er t i es usi ng SGg as t he onl y
avai l abl e i np ut dat a.
Tpc = 93.3 + 180.6 0.6274 - 6.9 x 0.62742
= 203.9 K = - 69. 26~
Ppc = 46.66 + 1.03 x 0.6274 2.58 x 0.62742 = 46.46 bar.
c. To cal cul at e t he effects of nonhydr ocar bons p r esent in
t he syst em Eqs. (4.5) and (4.6) ar e used to cal cul at e ad-
j ust ed p seudocr i t i cal propert i es. These equat i ons r equi r e
mol e fract i ons of H2S, CO2, and N2, whi ch are gi ven i n
Table 4.4 as: Yrh s = 0.012, yco2 = 0.01, and YN2 = 0.002. Un-
adj ust ed Tpc and Ppc ar e gi ven in Part c as Tpc = - 69. 26~
and Ppc = - 46. 46 bar.
Tpc = - 69. 26 - 44.44 0.01 + 72.22 x 0.012
- 138. 89 0.002 = -69. 12~
P~ = 46.46 + 30.3369 x 0.01 + 41.368 x 0.012
- 11.721 x 0.002 = 47.24 bar.
Al t hough use of Eqs. (4.1) and (4.2) t oget her wi t h Eqs. (4.5)
and (4.6) gives rel i abl e resul t s for nat ur al gas syst ems, use
of Eqs. (4.3) and (4.4) for gas condensat e syst ems or gases
t hat cont ai n C7+ fract i ons is not rel i abl e. For such syst ems
p r op er t i es of C7+ fract i ons shoul d be est i mat ed accor di ng to
t he met hods di scussed l at er in t hi s chapt er. r
4 . 3 CHARACT ERI ZAT I ON A N D PR OPER T I ES
OF S I N GL E C A R B ON N UM B ER GR OUPS
As shown in Tables 4.1 and 4.2, comp osi t i onal dat a on reser-
voi r fluids and cr ude oils ar e gener al l y expr essed in t er ms of
t ool ( or wt) p er cent of p ur e comp onent s ( up to C5) and SCN
gr oup s for hexanes and heavi er comp ounds (C6, C7, C8 . . . . .
CN+), wher e N is t he car bon number of pl us fract i on. I n
Table 4.1, N is i 0 and for t he cr ude oil of Table 4.2, N is
40. Pr op er t i es of a cr ude oil or a r eser voi r fluid can be ac-
cur at el y est i mat ed t hr ough knowl edge of accur at e p r op er t i es
of i ndi vi dual comp onent s in t he mi xt ure. Pr oper t i es of p ur e
comp onent s up to Cs can be t aken f r om Tables 2.1 and 2.2. By
anal yzi ng t he p hysi cal p r op er t i es of some 26 condensat es and
cr ude oils, Kat z and Fi r oozabadi [14] have r ep or t ed boi l i ng
poi nt , specific gravity, and mol ecul ar wei ght of SCN gr oup s
f r om C6 up to C45. Lat er Whi t son [15] i ndi cat ed t hat t here is
i nconsi st ency for t he p r op er t i es of SCN gr oup s r ep or t ed by
Kat z and F i r oozabadi for Nc > 22. Whi t son modi f i ed proper-
t i es of SCN gr oup s and r ep or t ed val ues of Tb, SG, M, Tc, Pc, and
02 for SCN gr oup s f r om C 6 to C45. Whi t son used Ri azi - Dauber t
cor r el at i ons ( Eqs. (2.38), (2.50), (2.63), and (2.64)) to gener at e
cri t i cal p r op er t i es and mol ecul ar wei ght f r om Tb and SG. He
al so used Edmi s t er met hod (Eq. (2.108)) to gener at e val ues of
acent ri c factor. As di scussed in Chap t er 2 ( see Sect i on 2.10),
t hese ar e not t he best met hods for cal cul at i on of p r op er t i es of
hydr ocar bons heavi er t han C22 ( M > 300). However, p hysi cal
p r op er t i es r ep or t ed by Kat z and F i r oozabadi have been used
in r eser voi r engi neer i ng cal cul at i ons and based on t hei r t ab-
ul at ed dat a, anal yt i cal cor r el at i ons for cal cul at i on of M , To
Pc, Vc, Tb, and SG of SCN gr oup s f r om C6 to C45 i n t er ms of
Nc have been devel op ed [ I 0] .
Ri azi and Ms ahhaf [16] r ep or t ed a new set of dat a on pr op-
ert i es of SCN f r om C6 t o C50. They used boi l i ng p oi nt and
specific gravi t y dat a for SCN gr oup s p r op os ed by Kat z and
F i r oozabadi f r om C6 to C22 to est i mat e PNA comp os i t i on of
each gr oup usi ng t he met hods di scussed i n Sect i on 3.5.1.2.
Then Eq. (2.42) was used to gener at e p hysi cal p r op er t i es of
paraffi ni c, nap ht heni c, and ar omat i c groups. Pr oper t i es of
SCN f r om C6 to C22 have been cal cul at ed t hr ough Eq. (3.39)
usi ng t he p s eudocomp onent ap p r oach. These dat a have been
used to obt ai n coeffi ci ent s of Eq. (2.42) for p r op er t i es of
SCN groups. The p s eudocomp onent met hod p r oduced boi l -
i ng p oi nt s for SCN gr oup s f r om C6 to C22 near l y i dent i cal to
t hose r ep or t ed by Kat z and F i r oozabadi [ 14]. Devel op ment of
Eq. (2.42) was di scussed i n Chap t er 2 and i t is gi ven as
(4.7) 0 = 0o~ - exp( a - b M c)
wher e 0 is val ue of a p hysi cal p r op er t y and 0~ is val ue of 0
as M --~ oo. Coefficients 0~, a, b, and c are specific for each
propert y. Equat i on (4.7) can al so be expressed in t er ms of
car bon number, Nc. Values of Tb and SG f r om C6 to C22 have
been cor r el at ed i n t er ms of Nc as
(4.8) Tb ---- 1090 -- exp( 6. 9955 -- 0.11193Nc 2/3)
(4.9) SG = 1.07 - exp( 3. 65097 - 3. 8864N ~
wher e Tb is in kelvin. Equat i on (4.8) r ep r oduces Ka t z -
F i r oozabadi dat a f r om C6 to C22 wi t h an AAD of 0. 2% ( ~1 K)
and Eq. (4.9) r ep r oduces t he or i gi nal dat a wi t h AAD of
0.1%. These equat i ons were used to gener at e Tb and SG for
SCN gr oup s heavi er t han C22. Physi cal p r op er t i es f r om C6
to C22 were used to obt ai n t he coeffi ci ent s of Eq. (4.7) for
SCN groups. I n doi ng so t he i nt er nal consi st ency bet ween
Tc and Pc were obser ved so t hat as Tb = To Pc becomes
1 at m (1.013 bar) . Thi s occur s for t he SCN gr oup of C99
( ~M = 1382). Coefficients of Eq. (4.7) for Nc > C~0 ar e gi ven
in Table 4.5, whi ch may be used well beyond C50. Dat a on
sol ubi l i t y p ar amet er of SCN gr oup s r ep or t ed by Won were
us e d t o obt a i n t he coef f i ci ent s f or t hi s p r op e r t y [6]. Val ues of
p hys i cal p r op e r t i e s of SCN gr oup s f or C6 t o C50 ar e t a bul a t e d
i n Tabl e 4. 6 [16]. Val ues of Pc cal cul at ed f r om Eq. ( 4.7)
ar e l ower t ha n t hos e r e p or t e d i n ot he r s our ces [10, 17, 18].
As di s c us s e d i n Cha p t e r 2, t he Le e - Ke s l e r cor r el at i ons Tb 1080 6.97996 0.01964 2/3 0.4
SG 1.07 3.56073 2.93886 0.1 0.07
ar e s ui t abl e f or p r e di c t i on of cr i t i cal p r op e r t i e s of heavy d20 1.05 3.80258 3.12287 0.1 0.1
hydr oc a r bons . F or t hi s r e a s on Tc a nd Pc ar e cal cul at ed f r om I 0.34 2.30884 2.96508 0.1 0.1
Eqs. ( 2. 69) -( 2. 70) whi l e dc is cal cul at ed t hr ough Eq. ( 2. 98) Tbr = Tb/Tc 1.2 -0. 34742 0.02327 0.55 0.15
a nd val ues. F or heavy hydr oc a r bons Eq. ( 2. 105) is us e d t o es- - Pc 0 6.34492 0.7239 0.3 1.0
t i ma t e to. R e p or t e d val ues of to f or heavy SCN ar e l ower t ha n -d~ -0. 22 -3. 2201 0.0009 1.0 0.05
-r 0.3 -6. 252 -3. 64457 0.1 1.4
t hos e e s t i ma t e d t hr ough Eq. (4.7). As di s c us s e d i n Cha p t e r a 30.3 17.45018 9.70188 0.1 1.0
2 a nd r e c omme nde d by Pa n et al. [7], f or hydr oc a r bons 3 8.6 2.29195 0.54907 0.3 0.1
wi t h M > 800 it is s ugges t ed t ha t to = 2.0. Ot her p r op e r t i e s
ar e cal cul at ed t hr ough Eq. ( 4. 7) or t aken f r om Ref. [16].
TABLE 4.5---Coefficients of Eq. (4. 7) for physical properties of SCN
groups (>Clo) in reservoir fluids and crude oils.
Constants in Eq. (4.7)
0 0oo A b c %AAD
Taken with permission from Ref. [ 16].
Units: Tb, Tc in K; Pc in bar; d20 and dc in g/cm3; ~ in dyne/cm; 8 in (cal/cm s)1/2
TABLE 4.6---Physical properties of SCN groups.
Carbon
number M Tb SG n20 d20 Tc Pc dc Zc ~o cr 3
6 84 337 0.690 1.395 0.686 510.3 34.4 0.241 0.275 0.255 18.6 7.25
7 95 365 0.727 1.407 0.723 542.6 31.6 0.245 0.272 0.303 21.2 7.41
8 107 390 0.749 1.417 0.743 570.2 29.3 0.246 0.269 0.346 23.0 7.53
9 121 416 0.768 1.426 0.762 599.0 26.9 0.247 0.265 0.394 24.4 7.63
10 136 440 0.782 1.435 0.777 623.7 25.0 0.251 0.261 0.444 25.4 7.71
11 149 461 0.793 1.442 0.790 645.1 23.5 0.254 0.257 0.486 26.0 7.78
12 163 482 0.804 1.448 0.802 665.5 21.9 0.256 0.253 0.530 26.6 7.83
13 176 500 0.815 1.453 0.812 683.7 20.6 0.257 0.249 0.570 27.0 7.88
14 191 520 0.826 1.458 0.822 700.9 19.6 0.262 0.245 0.614 27.5 7.92
15 207 539 0.836 1.464 0.831 716.5 18.5 0.267 0.241 0.661 27.8 7.96
16 221 556 0.843 1.468 0.839 732.1 17.6 0.269 0.237 0.701 28.1 7.99
17 237 573 0.851 1.472 0.847 745.6 16.7 0.274 0.233 0.746 28.3 8.02
18 249 586 0.856 1.475 0.852 758.8 15.9 0.274 0.229 0.779 28.5 8.05
19 261 598 0.861 1.478 0.857 771.1 15.2 0.275 0.226 0.812 28.6 8.07
20 275 611 0.866 1.481 0.862 782.7 14.7 0.278 0.222 0.849 28.8 8.09
21 289 624 0.871 1.484 0.867 793.8 14.0 0.281 0.219 0.880 28.9 8.11
22 303 637 0.876 1.486 0.872 804.9 13.5 0.283 0.215 0.914 29.0 8.13
23 317 648 0.881 1.489 0.877 814.2 13.0 0.287 0.212 0.944 29.1 8.15
24 331 660 0.885 1.491 0.880 824.1 12.5 0.289 0.209 0.977 29.2 8.17
25 345 671 0.888 1.493 0.884 833.3 12.0 0.291 0.206 1.007 29.3 8.18
26 359 681 0.892 1.495 0.888 841.7 11.7 0.295 0.203 1.034 29.3 8.20
27 373 691 0.896 1.497 0.891 850.2 11.3 0.298 0.200 1.061 29.4 8.21
28 387 701 0.899 1.499 0.895 858.2 10.9 0.301 0.197 1.091 29.4 8.22
29 400 710 0.902 1.501 0.898 865.5 10.6 0.303 0.194 1.116 29.5 8.24
30 415 720 0.905 1.503 0.901 873.5 10.2 0.306 0.191 1.146 29.5 8.25
31 429 728 0.909 1.504 0.904 880.1 10.0 0.310 0.189 1.169 29.6 8.26
32 443 737 0.912 1.506 0.906 887.4 9.7 0.312 0.187 1.195 29.6 8.27
33 457 745 0.915 1.507 0.909 894.0 9.5 0.316 0.184 1.218 29.7 8.28
34 471 753 0.917 1.509 0.912 900.2 9.2 0.319 0.182 1.244 29.7 8.29
35 485 760 0.920 1.510 0.914 906.1 9.0 0.323 0.180 1.263 29.7 8.30
36 499 768 0.922 1.511 0.916 912.2 8.8 0.325 0.177 1.289 29.8 8.31
37 513 775 0.925 1.512 0.918 917.7 8.6 0.328 0.175 1.311 29.8 8.32
38 528 782 0.927 1.514 0.920 923.1 8.3 0.332 0.173 1.333 29.8 8.33
39 542 789 0.929 1.515 0.922 928.6 8.2 0.335 0.171 1.355 29.8 8.34
40 556 795 0.931 1.516 0.924 933.4 8.0 0.338 0.169 1.374 29.9 8.35
41 570 802 0.933 1.517 0.926 938.8 7.8 0.341 0.167 1.396 29.9 8.35
42 584 808 0.934 1.518 0.928 943.6 7.7 0.344 0.165 1.415 29.9 8.36
43 599 814 0.936 1.519 0.930 948.4 7.5 0.348 0.164 1.434 29.9 8.36
44 614 820 0.938 1.520 0.932 952.5 7.4 0.353 0.163 1.448 29.9 8.37
45 629 826 0.940 1.521 0.933 956.9 7.2 0.356 0.160 1.470 29.9 8.38
46 641 831 0.941 1.522 0.935 961.6 7.1 0.358 0.159 1.489 30.0 8.38
47 656 836 0.943 1.523 0.936 965.7 7.0 0.362 0.158 1.504 30.0 8.39
48 670 841 0.944 1.524 0.938 969.4 6.9 0.366 0.156 1.522 30.0 8.39
49 684 846 0.946 1.524 0.939 973.5 6.8 0.369 0.155 1.537 30.0 8.40
50 698 851 0.947 1.525 0.940 977.2 6.6 0.372 0.153 1.555 30.0 8.40
Tc and Pc are calculated from Lee--Kesler correlations (Eqs. 2.69-2.70); d~ is calculated from Eq. (2.98) and Zc from Eq. (2.8). For
Nc > 20, ~ is calculated from Lee-Kesler, Eq. (2.105). All other properties are taken from Ref. [16] or calculated through Eq. (4.7)
for Nc > 10. Units: T b, Tc in K; Pc in bar; d2o and d~ in g/cmS; a in dyne/cm; 8 in (cal/cm3) 1/2.
Taken with permission from Ref. [ 16].
Values of critical properties and acent ri c fact or for SCN
groups great er t han C30 est i mat ed by different met hods vary
significantly especially for hi gher SCN groups. Reliability of
these values is subject t o furt her research and no concret e
r ecommendat i on is given in t he literature.
Values of M given in Table 4.6 are mor e consi st ent wi t h
values of M r ecommended by Pedersen et al. [6] t han t hose
suggested by Whi t son [15] as di scussed by Riazi et al. [16].
Mol ecul ar weights of SCN groups r ecommended by Whi t son
[15] are based on t he Kat z-Fi roozabadi values. However,
mor e recently Whi t son [ 17] r ecommends values of M for SCN
groups, whi ch are very close t o t hose suggest ed by Riazi [16].
Al t hough he refers t o Kat z-Fi rrozabadi mol ecul ar weights,
his report ed values are much hi gher t han t hose given in Ref.
[14]. I n addi t i on t he Wat son charact eri zat i on fact or (Kw) re-
port ed in Ref. [15] from C6 to C4s shoul d be consi dered wi t h
caut i on as t hey are nearly const ant at 12 while t owar d heavier
hydr ocar bon factions as the amount of aromat i cs increases,
t he Kw values shoul d decrease. Values of propert i es of SCN
groups given in Table 4.6 or t hose given in Ref. [15] are ap-
proxi mat e as these propert i es may vary for one fluid mi xt ure
to another. As shown in Table 4.1, values of M for the SCN
of C7 in four different reservoir fluids vary from 92 to 100.
I n Table 4.2, mol ecul ar wei ght of each SCN fract i on is deter-
mi ned by addi ng 14 to mol ecul ar wei ght of precedi ng SCN
gr oup [7]. Thus M for C6 is specified as 86, whi ch is deter-
mi ned by addi ng 14 to mol ecul ar weight nC5 t hat is 72. The
Pedersen mol ecul ar wei ght of SCN groups is given in t erms
of Nc by t he following relation [6]:
(4.10) M = 14Nc - 4
I n very few references the value of 2 is used instead of 4
in Eq. (4.10). To obt ai n propert i es of heavier SCN groups
(Nc > 50), Eq. (4.7) shoul d be used wi t h coefficients given in
Table 4.5. However, to use this equation, it is necessary t o
calculate the mol ecul ar wei ght from boiling poi nt t hr ough
reversed f or m of Eq. (4.7). Tb is calculated by Eq. (4.8) usi ng
t he car bon number. The cal cul at i on met hod is demonst r at ed
in Exampl e 4.2.
Example 4. 2--Cal cul at e Tb, SG, d20, n2o, To, Pc, Vc, a, and 8
for C60, C70, and C80 SCN groups.
Sol ut i on- - The onl y dat a needed for cal cul at i on of physical
propert i es of SCN groups is the car bon number. For Nc = 60,
from Eq. (4.8): Tb = 1090- exp(6.9955 --0. 11193 x 602/3) =
894 K. Equat i on (4.7) in a reversed form can be used to
est i mat e M f r om Tb wi t h coefficients given in Table 4.5:
M = [ ~ (6.97996 - l n( I 080 - Tb)] 3/2. For Tb = 894 K we
get M = 844. This value of M shoul d be used to calculate ot her
properties. For example SG is calculated as: SG = 1. 07-
exp(3.56073 - 2. 93886 x 844 ~ -- 0.96. Similarly ot her prop-
erties can be est i mat ed and t he results are given in Table 4.7.
Values of To Pc, and o~ are calculated from Lee-Kesl er
correl at i ons and dc is calculated f r om Eq. (2.98). Actual val-
ues of w are probabl y great er t han t hose given in this table.
Values of Zc cal cul at ed f r om its definition by Eq. (2.8) for C60,
C70, and Cs0 are 0.141, 0.132, and 0.125, respectively. ,
4 . 4 CHARACTERI ZATI ON APPROACHES
FOR C7+ FRACTI ONS
I n descri pt i on of composi t i on of a reservoir fluid, C6 is a very
nar r ow boiling range fract i on and charact eri zat i on met hods
di scussed in Chapt er 2 (i.e., Eq. 2.38 [17]) can be used to esti-
mat e various propert i es of this group. Cont rary to C6 group,
the C7+ fract i on has a very wide boiling range especially for
crude oils. Therefore, met hods of Chapt er 2 or 3 cannot be di-
rectly applied to a C7+ fraction. However, for a nat ural gas t hat
its C7+ fract i on has a nar r ow boiling range and t he amount
of C7+ is quite small, equat i ons such as Eq. (2.40) in t erms
of M and SG may be used to estimate vari ous properties. I n
some references t here are specific correl at i ons for propert i es
of C7+ fractions. For example, Pedersen et al. [6] suggest ed
use of the following relation for cal cul at i on of critical vol ume
of C7+ fractions in t erms of MT+ and SG7+:
Vc7+ = 0.3456 + 2.4224 x 10-4M7+ - 0.443SG7+
(4.11) + 1.131 x 10-3MT+SG7+
where Vc7+ is the critical vol ume of C7+ fract i on in cma/ gmol.
St andi ng represent ed the graphi cal correl at i on of Kat z for the
pseudocri t i cal t emperat ure and pressure of C7+ fractions into
t he following analytical correl at i ons [18]:
To7+ = 338 + 202 x log(MT+ - 71.2)
(4.12) +( 1361 x log M7 - 2111) logSG7+
Po7+ = 81.91 - 29.7 x log(M7+ - 61.1) + (SG7 - 0.8)
(4.13) x [159.9 - 58.7 x log(MT+ - 53.7)]
where To7+ and Pc7+ are in K and bar, respectively. The origi-
nal devel opment of these correl at i ons goes back to t he earl y
1940s and there is no i nformat i on on the reliability of these
equations. Use of such relations to a C7+ fract i on as a sin-
gle p seudocomp onent leads to serious errors especially for
mi xt ures wi t h consi derabl e amount of C7+. Properties of C7+
fract i ons have significant effects on est i mat ed propert i es of
the reservoir fluid even when t hey are present in small quan-
tities [20, 21].
Choru and Mansoori [21 ] have document ed vari ous met h-
ods of charact eri zat i on of C7+ fractions. Generally t here are
t wo t echni ques t o charact eri ze a hydr ocar bon plus fraction:
(i) The p seudocomp onent and (ii) the cont i nuous mi xt ure ap-
proaches. I n t he p seudocomp onent appr oach the C7+ is split
into a number of subfract i ons wi t h known mol e fraction, Tb,
SG, and M [22-26]. I n this met hod t he TBP curve can also
be used to split t he mi xt ure into a number of p seudocomp o-
nents. Moreover, each subfract i on may furt her be split i nt o
T AB L E 4.7--Calculated values for physical properties of C60, C70, and Ca0 of Example 4.2.
NC M Tb SG n2o d2o Tc Pc dc o~ a ?J
60 844 894.0 0.960 1.532 0.952 1010.5 5.7 0.410 1.699 30.1 8.4
70 989 927,0 0.969 1.538 0.961 1035.6 5.1 0.447 1.817 30.1 8.5
80 1134 953.0 0.977 1.542 0.969 1055.7 4.7 0.487 1.909 30.2 8.5
Units: Tb, Tc in K; Pc in bar; d2o and dc in g/cm3; a in dyne/cm; * in (cal/crn3) 1/2.
three pseudocomponents from paraffinic, naphthenic, and
aromatic groups. Although a higher number of pseudocom-
ponents leads to more accurate results, the increase in the
number of components complicates the calculations as the
number of input data required increases significantly. For
example, the application of a two-parameter equation of state
(such as Peng-Robinson EOS) requires four input parame-
ters for each component: Tc, Pc, co, and a binary interaction
coefficient (ki i), which is a correction factor for a mixture of
dissimilar components. The number of variables needed for
a 20-component mixture in two-parameter EOS calculations
is 290! [27].
The second approach is the continuous mixture character-
ization method. I n this method instead of mole fractions, a
distribution function is introduced to describe the composi-
tion of many component mixtures [24, 25, 28-32]. Since com-
position of a reservoir fluid up to C5 is given in terms of dis-
crete mole fractions, application of this approach to reservoir
fluids is also referred as semicontinuous approach in which
the distribution function is applied to C6+ part of the mixture.
Distribution of components in mixtures that consist of many
species is presented by a distribution function F(P) whose
independent property P is defined in terms of a measurable
property such as molecular weight (M), boiling point (Tb), or
carbon number (Nc) and varies from a value for the light-
est component (Po) to the value for the heaviest component
(P0o) present in the mixture. Generally the value of Poo for M
or Tb for a plus fraction is assumed as infinity (oo). Classical
thermodynamics for vapor-liquid equilibrium (VLE) calcula-
tions of multicomponent systems require equality of temper-
ature, pressure, as well as equality of chemical potential of
each component in both phases:
(4.14) #L=/ zV i = 1, 2 . . . . . N
where #/L and/ z v are chemical potential of component i in
liquid and vapor phase, respectively. Equation (4.14) should
be valid for all N components in the mixture. For VLE calcu-
lations of continuous mixtures Eq. (4.12) becomes
(4.15) / z L( p ) = #V(p) Po < P < oo
where P is an independent variable such as molecular weight
or boiling point. Similarly in calculation of all other thermo-
dynamic properties for the mixture, distribution function is
used instead of mole fraction for application of a mixing rule.
It should be noted that even when composition of a mixture
is expressed in terms of a distribution function, the mixture
may be presented in terms of a number of pseudocompo-
nents. Further characteristics of distribution functions and
their application to petroleum mixtures are discussed in the
next section.
4 . 5 D I S T R I B UT I ON F UN C T I ON S F OR
P R OP ER T I ES OF H Y D R OC A R B ON -
P L US F R A C T I ON S
As mentioned before, accurate characterization of a reservoir
fluid or a crude oil requires a complete analysis of the mix-
ture with known mole fraction and carbon number such as
those shown in Table 4.2. For mixtures that the composition
of heavy hydrocarbons is presented by a single hydrocarbon-
plus fraction, it is important to know distribution of carbon
numbers to describe the mixture properly. A mathematical
function that describes intensity of amount of a carbon num-
ber, or value of molecular weight, or boiling point for com-
pounds with Nc > 6 is referred as probability density function
(PDF). The PDF can be obtained from a distribution function
that describes how various components or their properties
are distributed in a mixture. I n this section, general char-
acteristics of density functions are discussed and then three
different distribution models used to describe properties of
hydrocarbon-plus fractions are presented.
4 . 5 . 1 Ge n e r a l C h a r a c t e r i s t i c s
Distribution functions can be applied to determine distribu-
tion of compounds from hexane or heavier in a reservoir fluid.
However, since the mole fraction of C6 fraction in reservoir
fluid is usually known and heavier hydrocarbons are grouped
in a C7+ group, distribution functions are generally used to de-
scribe properties of C7+ fractions. Mole fraction versus molec-
ular weight for SCN groups heavier than C6 in the West Texas
gas condensate sample in Table 4.1 and the waxy and Kuwaiti
crude oils of Table 4.2 are shown in Fig. 4.4. Such graphs are
known as molar distribution for the hydrocarbon plus (in this
case C6+) fraction of reservoir fluids. As can be seen from this
figure the molar distribution of gas condensates is usually
exponential while for the black oil or crude oil samples it is
left-skewed distribution.
For the same three samples shown in Fig. 4.4, the proba-
bility density functions (PDF) in terms of molecular weight
are shown in Fig. 4.5. Functionality of molecular weight ver-
sus cumulative mole fraction, M(x), for the three samples is
shown in Fig. 4.6.
15
10
5
0
0
Carbon Number, N c
10 20 30 40
i I r I i I r
. . . . . W. Texas Gas Conddensate
- - - Kuwaiti Crude
200 400 600
Molecular Weight, M
FI G. 4, 4~ Mo l a r di st ri but i on f or a g as condensat e and a
cr ude oil sampl e,
4. CHARACTERI Z ATI ON OF RESERV OI R FLUI DS AND CRUDE OI LS 165
Carbon Number, Nc based on normalized mole fraction becomes
oo 1
0.01 , 1,0 2,0 , 30 , 4 (4.17) fF(P)dP=fdx =l
i . . . . . W. Texas Gas Condensate eo o
[ - - -- Kuwaiti Crude If the upper limit of the integral in Eq. (4.17) is at prop-
0.008 i i / ~ Waxy Oil erty P, then the upper limit of the right-hand side should be
; I ~ cumulative Xc as shown in the following relation:
0 0 0 6
(4.18) Xc = F( P) dP
0.004 Integration of Eq. (4.16) between limits of P1 and P2 gives
i'k~.~_._~ ~ (4.Pr~ mole fraction of all components in the mixture w h o s e 1 9 ) P is ine~theF(P)dprange of P1 _ < P < P2:
0.002 P2
/ = x~2 x~l = xp~__,p~
0 " P1
0 200 400 600
Molecular Weight, M
FIG. 4. 5---Probabi l i ty densi ty functi ons for the g as conden-
sate and crude oil sampl es of Fig. 4.4.
The continuous distribution for a property P can be ex-
pressed in terms of a function such that
(4.16) F( P) dP = dx~
where P is a property such as M, Tb, Nc, SG, or I (defined
by Eq. 2.36) and F is the probability density function. If the
original distribution of P is in terms of cumulative mole frac-
tion (Xcm), then xc in Eq. (4.16) is the cumulative mole fraction.
As mentioned before, paramet er P for a continuous mixture
varies from the initial value of Po to infinity. Therefore, for the
whole continuous mixture (i.e., C7+), integration of Eq. (4.16)
600 45
. . . . . W. Texas Gas Condensate
- - Kuwaiti Crude /
. 400
~2
200 , ,
O
0 , ~ ~ ~ - ' , - - - i - - , - ' - , ' ' | 0
0 0.2 0.4 0.6 0.8 1
Cumulative Mole Fraction, xcm
FIG. 4. 6- - V ari at i on of mol ecul ar wei g ht with cumul ati v e
mol e fracti on for the g as condensate and crude oil sampl es
of Fig. 4.4.
where xcl and xc2 are the values Of Xc at P1 and P2, respectively.
xp~_~p2 is sum of the mole fractions for all components having
P1 _< P _< PE. Equation (4.19) can also be obtained by apply-
ing Eq. (4.18) at Xcz and xci and subtracting from each other.
Obviously if the PDF is defined in terms of cumulative weight
or volume fractions x represents weight or volume fraction,
respectively. The average value of parameter P for the whole
continuous mixture, Pay, is
1 oo
(4.20) Pay : / P(xc)dxc : / PF( P) dP
d
, /
0 Po
where P (x) is the distribution function for property P in terms
of cumulative mole, weight, or volume fraction, Xc. For all
the components whose parameters varies from/ ~ to P2 the
average value of property P, Pav(t,l_ _ ,e2), is determined as
f~2 PF( P) dP
(4.21) Pav(Pl-~P2)- fle2 F( P) dP
This is shown in Fig. 4.7 where the total area under the curve
from Po to c~ is equal to unity (Eq. 4.17) and the area under
curve from P1 to Pa represents the fraction of components
whose property P is greater than P1 but less than P2- Fur-
ther properties of distribution functions are discussed when
different models are introduced in the following sections.
4. 5. 2 Exponent i al Model
The exponential model is the simplest form of expressing dis-
tribution of SCN groups in a reservoir fluid. Several forms of
exponential models proposed by Lohrenz (1964), Katz (1983),
and Pedersen (1984) have been reviewed and evaluated by
Ahmed [26]. The Katz model [33] suggested for condensate
systems gives an easy met hod of breaking a C7+ fraction into
various SCN groups as [19, 26, 33]:
(4.22) xn = 1.38205 exp(--0.25903CN)
where x~ is the normalized mole fraction of SCN in a C7+
fraction and CN is the corresponding carbon number of the
SCN group. For normalized mole fractions of C7+ fraction, the
mole fraction of C7+ (x7+) is set equal to unity. In splitting a
/ / / / / / /
/ / / / / / /
I I / l l l l
/ / / / / / /
I / i i l / J
/ / / / / / J
F( P) , MHH
/ / f f / f /
/ / / / / / /
/ / / / / / /
i i i i i r l
/ i / / l i /
/ / / f / / /
/ / / / / / /
I / / / / / /
l / r / / r /
l i t / i l l
f i l l / l /
I / / / / / /
l / r i l l 1
PO P1 ~ ~
P
FIG. 4 . 7 - - Goneml sch emat i c ~ a ptobabiliN don$ iN function ~ r a
pr ol ~ r ~
hydr ocar bon pl us fract i on i nt o SCN groups by Eq. (4.22), the
last hydr ocar bon group is shown by CN+ fract i on (i.e., 40+
i n t he waxy oil sampl e of Table 4.2). Mol ecul ar wei ght and
specific gravity of the last fract i on can be det er mi ned from
t he following equat i ons
N
(4.23) Ex, Mn ----- /1//7+
n=7
~ x.M~ mT+
(4.24)
,=7 SG, SG7+
where MT+ and SG7+ are known i nf or mat i on for t he C7+ frac-
tion. M. and SG, are mol ecul ar wei ght and specific gravity of
SCN group t hat may be t aken from Table 4.6 or est i mat ed
from Eq. (4.7) and coefficients given i n Table 4.5. Equa-
t i on (4.24) is i n fact equi val ent to Eq. (3.45) when it is ap-
pl i ed to SG. The following exampl e shows appl i cat i on of t hi s
met hod to generat e SCN groups.
Exampl e 4. 3- - Use the M7+ and SG7+ for the Nort h Sea gas
condensat e mi xt ure of Table 4.1 to generat e t he composi t i on
of SCN groups up to C10+ and compar e wi t h act ual data.
Sol ut i on- - The C7+ has/147+ = 124, SG7+ = 0.7745, andx7+ =
0.0306. At first the mol e fract i ons of SCN groups based on
nor mal i zed composi t i on are cal cul at ed from Eq. (4.22) and
t hen based on mol e fract i on of C7+ t hey are convert ed i nt o
mol e fract i ons i n the ori gi nal mi xt ure. For this probl em, N =
i0+; therefore, mol e fract i ons of C7, C8, and C9 shoul d be esti-
mat ed. The results are x7 = 0.225, x8 = 0.174, and x9 = 0.134.
Mole fract i on of Ct0+ is cal cul at ed from mat eri al bal ance
as xl0+ = 1.0 - (0.225 + 0.174 + 0.134) = 0.467. The tool% of
these comp onent s i n the ori gi nal mi xt ure can be obt ai ned
t hr ough mul t i pl yi ng nor mal i zed mol e fract i ons by mol % of
C7+, 3.06; t hat is mol % of C7 = 0.225 x 3.06 = 0.69. Sum-
mar y of results is gi ven i n Table 4.8. The M10+ and SG10+
are cal cul at ed t hr ough Eqs. (4.23) and (4.24) usi ng M, and
SG, from Table 4.5 as M10+ = [124 - (0.225 x 95 + 0.174 x
107 + 0.134 x 121)]/0.467 = 145.2. Si mi l arl y from Eq. (4.24),
M10+/SG10+ = 181.5. Therefore, SGl0+ = 145.2/ 181.5 = 0.80.
Compar i son wi t h the act ual values from Table 4.1 is present ed
also i n Table 4.8. As is shown i n this table this met hod does
not provi de a good est i mat e of SCN di st ri but i on. The errors
will be much l arger for oil systems. r
Yarborough [34] and Pedersen et al. [35] have suggested for
gas condensat e systems to assume a l ogari t hmi c di st r i but i on
of the mol e fract i on x~ versus the car bon numbe r CN as
(4.25) lnx,~ = A + B x CN
where A and B are const ant s specific for each mi xt ure. This
equat i on is i n fact si mi l ar to Eq. (4.22), except i n this case
the coefficients A and B are det errni ned for each mi xt ure. I f
CN
TABLE 4.8--Prediction of SCN groups from Eq. (4.22) for a gas condensate system
of Example 4.3.
Actual data from Table 4.1 Predicted values from Eqs. (4.22)-(4.24) a
mol% b tool% b
Nor. b M SG Nor? M SG
7 0.80 26 95 0.7243 0.69 23 95 0.727
8 0.76 25 103 0.7476 0.53 17 107 0.749
9 0.47 15 116 0.7764 0.41 13 121 0.768
10+ 1.03 34 167 0.8120 1.43 47 145.2 0.800
~Values of M and SG for SCN groups are taken from Table 4.6.
bValues of mol% in the first column represent composition in the whole original fluid while in the second
column under Nor. represent normalized composition ( ~ = 100) for the C7+ fraction.
4. CHARACTERI Z ATI ON OF RESERV OI R FL UI DS AND CRUDE OI LS 167
TABLE 4.9~Prediction of SCN groups from Eq. (4.27) for a gas condensate system
of Example 4.4.
Actual data from Table 4.1 Predicted values from Eqs. (4.27)-(4.30) a
mol% b mol% b
CN Nor. b M SG NorP M SG
7 0.80 26 95 0.7243 0.95 31 95 0.727
8 0.76 25 103 0.7476 0.69 22 107 0.749
9 0.47 15 116 0.7764 0.45 15 121 0.768
10+ 1.03 34 167 0.8120 0.97 32 166 0.819
~Values of M and SG for SCN groups up to C50 are taken from Table 4.6. C10+ fraction represents SCN
from 10 to 50. ~roups
Values of mola~ in the first column represent composition in the whole original fluid while in the second
column under Nor. represent normalized composition for the C7+ fraction.
Eq. (4.10) is combi ned wi t h Eq. (4.25), an equat i on for mol ar
di st r i but i on of hydr ocar bon- p l us fract i ons can be obt ai ned as
(4.26) l nx~ = A1 + Bi x M~
whi ch may al so be wr i t t en i n t he fol l owi ng exponent i al form.
(4.27) x~ = a exp( B x M~)
wher e B = B1 and A = exp( A: ) . I t shoul d be not ed t hat A and
B i n Eq. (2.27) are di fferent f r om A and B i n Eq. (2.25). Para-
met er s A1 and Bi can be obt ai ned by r egr essi on of dat a be-
t ween ln(x~) and MR such as t hose given in Table 4.2 for t he
waxy oil. Most c ommon case is t hat t he det ai l ed comp osi -
t i onal anal ysi s of t he mi xt ur e is not avai l abl e and onl y M7+
is known. For such cases coeffi ci ent s A1 and BI ( or A and B)
can be det er mi ned by appl yi ng Eqs. (4.23) and (4.24) assum-
i ng t hat t he mi xt ur e cont ai ns hydr ocar bons to a cer t ai n gr oup
(i.e., 45, 50, 60, 80, or even hi gher) wi t hout a pl us fract i on.
Thi s is demons t r at ed i n Examp l e 4.4.
Exampl e 4. 4- - Rep eat Examp l e 4.3 usi ng t he Peder sen expo-
nent i al di st r i but i on model , Eq. (4.27).
Sol ut i on- - For t hi s case t he onl y i nf or mat i on needed are
/147+ = 124 and x7+ = 0.0306. We cal cul at e nor mal i zed mol e
f r act i ons f r om Eq. (4.27) aft er obt ai ni ng coeffi ci ent s A and B.
Si nce t he mi xt ur e is a gas condensat e we assume ma xi mum
SCN gr oup in t he mi xt ur e is C50. Mol ecul ar wei ght s of CN
f r om 7 t o 50 ar e gi ven in Table 4.6. I f Eq. (4.26) is ap p l i ed to
all SCN gr oup s f r om 7 to 50, si nce it is as s umed t her e is no
N50+, for t he whol e C7+ we have
50 50
(4.28) Z xn ---- Z a exp(BM~) = 1
n=7 n=7
wher e x, is nor mal i zed mol e f r act i on of SCN gr oup n in t he
C7+ fract i on. F r om t hi s equat i on A is f ound as
(4.29) A = exp(BM~
Par amet er B can be obt ai ned f r om MT+ as
t )
( 4. 30) M7+= (n~=Texp(BM n)) [exp(BMn)]Mn
wher e t he onl y unknown p ar amet er is B. F or bet t er accu-
r acy t he l ast SCN can be as s umed gr eat er t han 50 wi t h M,
cal cul at ed as di scussed i n Examp l e 4.2. F or t hi s examp l e we
use val ues of Mn f r om Table 4.6 whi ch yi el ds B = - 0. 0276.
Par amet er A is cal cul at ed f r om Eq. (4.29) as: A = - 4. 2943.
Usi ng p ar amet er s A and B, nor mal i zed mol e f r act i ons for
SCN f r om 7 to 50 ar e cal cul at ed f r om Eq. (4.27). Mol e f r act i on
of C10+ can be est i mat ed f r om sum of mol e f r act i ons of C10
to C50. M10+ and SG10+ ar e cal cul at ed as in Examp l e 4.3 and
t he s ummar y of resul t s ar e given i n Table 4.9. I n t hi s met hod
M10+ and SG10+ are cal cul at ed as 166 and 0.819, whi ch are
cl ose t o t he act ual val ues of 167 and 0.812. r
As shown i n Examp l e 4.4, t he exponent i al model wor ks wel l
for p r edi ct i on of SCN di st r i but i on of some gas condensat e
syst ems, but gener al l y shows weak p er f or mance for cr ude oils
and heavy r eser voi r fluids. As an examp l e for t he Nor t h Sea
oil descr i bed in Table 4. i , based on t he p r ocedur e descr i bed
in Examp l e 4.4, M10+ is cal cul at ed as 248 versus act ual val ue
of 259. I n t hi s met hod we have used up to C50 and t he coef-
ficients of Eq. (4.27) ar e A = 0.2215, B = - 0. 0079. I f we use
SCN gr oup s up to C40, we get A = 0.1989 and B = - 0. 0073
wi t h M10+ = 246.5, but if we i ncl ude SCN hi gher t han C50
sl i ght i mp r ovement will be observed. The exp onent i al di st ri -
but i on model as expr essed in t er ms of Eq. (4.27) is i n fact a
di scret e funct i on, whi ch gives mol e fract i on of SCN groups.
The cont i nuous f or m of t he exponent i al model will be shown
l at er in t hi s sect i on.
4 . 5 . 3 Ga mma Di s t ri b ut i on Mode l
The ga mma di st r i but i on model has been used to express
mol ar di st r i but i on of wi der r ange of r eser voi r fl ui ds i ncl udi ng
bl ack oils. Charact eri st i cs, speci fi cat i ons, and ap p l i cat i on of
t hi s di st r i but i on model to mol ecul ar wei ght and boi l i ng p oi nt
have been di scussed by Whi t son i n det ai l s [15, 17, 22, 23,
36, 37]. The PDF in t er ms of mol ecul ar wei ght for t hi s dis-
t r i but i on model as suggest ed by Whi t s on has t he fol l owi ng
form:
(4.31) F(M ) =
( M- ~)~-1 exp ( - ~)
wher e or, r, and 17are t hr ee p ar amet er s t hat shoul d be deter-
mi ned for each mi xt ur e and F( u) is t he gamma f unct i on to
be defi ned later. Par amet er 0 r ep r esent s t he l owest val ue of M
in t he mi xt ure. Subst i t ut i on of F( M ) i nt o Eq. (4.20) gives t he
average mol ecul ar wei ght of t he mi xt ur e (i.e., M7+) as:
(4.32) MT+ = 0 + aft
whi ch can be used to estimate paramet er/ 3 in t he following
form.
M7+ -
(4.33) fl - - -
0t
Whi t son et al. [37] suggest an approxi mat e relation bet ween
t / and a as follows:
( ' )
(4.34) t / = 110 1 - 1 + 4.0zi3cr ~
By subst i t ut i ng Eq. (4.31) into Eq. (4.18), cumul at i ve mol e
fraction, Xcm versus mol ecul ar weight, M can be obt ai ned
whi ch in t erms of an infinite series is given as:
(4.35) Xcm = [exp(--Mb)] ~ r ( a + 1 + j )
1=0
where p ar amet er Mb is a variable defined in t erms of M as
M- t /
(4.36) Mb --
Since M varies f r om r/ to c~, par amet er Mb varies from 0
t o oo. The summat i on in Eq. (4.35) shoul d be di scont i nued
~--, J+l - - J
when z-4=0 Y~-i=0 -< 10-8. For a subfract i on i with mol ecu-
lar wei ght bounds Mi-1 and Mi, the discrete mol e fraction, x~i
is calculated f r om the difference in cumul at i ve mol e fractions
calculated f r om Eq. (4.35) as follows:
(4.37) Xm,i = Xcm, i - - / c m, i - 1
where Mbi is calculated f r om Eq. (4.36) at Mi. The average
mol ecul ar wei ght of this subfract i on, Mav, i is t hen calculated
from the following formul a:
--( 1 _ _ Xcm, )
Xcm,i -- Xcm,i_ 1
(4.38) Mav, i = rl + err x \ X--~m,i
where X~m,i is evaluated f r om Eq. (4.35) by st art i ng t he sum-
mat i on at J = 1 instead of j = 0, whi ch is used to evaluate
Xcrn, i .
Equat i on (4.37) can be used t o estimate mol e fract i ons of
SCN groups in a C7+ fract i on if l ower and up p er mol ecul ar
wei ght boundari es (M2, M +) for the group are used instead
of Mi-1 and Mi. The l ower mol ecul ar wei ght boundar y for a
SCN group n, M 2 is the same as t he up p er mol ecul ar wei ght
boundar y for the precedi ng SCN group, t hat is
(4.39) Ms = m+_l
For a SCN gr oup n, the up p er mol ecul ar weight boundar y
M + may be calculated f r om t he mi dpoi nt mol ecul ar weights
of SCN groups n and n + 1 as following:
(4.40) M~ + _ M~ + M~+I
2
where M~ and M~+a are mol ecul ar weight of SCN groups n
and n + 1 as given in Table 4.6. For example, in this table val-
ues of mol ecul ar weight for M6, M7, Ms, and M9 are given as
82, 95, 107, and 121, respectively. For C6 the upper mol ec-
ul ar wei ght boundar y is M~- = (82 + 95)/2 = 88.5, whi ch can
be approxi mat ed as 88. Similarly, M~- = (95 + 107)/2 = 101
and M + = 114. The l ower mol ecul ar wei ght boundari es are
calculated f r om Eq. (4.39) as My = M~ = 88 and similarly,
M~- = 101. Therefore for t he SCN group of C8, the l ower
mol ecul ar wei ght is 101 and the upper boundar y is 114. For
250
o
200
150
100
t
{
{
50 ~ i i i J i i
7 9 11 13 15
Carbon Number, C N
FIG. 4.8 ---The lower and upper molecular weight
boundaries for SCN groups.
SCN groups f r om C7 to C15 the mol ecul ar wei ght boundari es
are shown in Fig. 4.8.
I n this di st ri but i on model, p ar amet er 0t can be det ermi ned
by mi ni mi zi ng onl y one of t he error funct i ons El ( a) or E2(a)
whi ch for a C7+ fract i on are defined as follows:
N- I
(4.41) El ( 0t ) = Z ( Mc a ! - Me xp ) 2
~---av, t ---av, i ]
i =7
N- 1
(4.42) E2( ~) = E ( Xca,li _ Xm, ijexp'~2
i=7
whe r e MC~i is calculated t hr ough Eq. (4.38) and M2~ p is
experi ment al value of average mol ecul ar wei ght for the sub-
fract i on i. x~ is the calculated mol e fract i on of subfract i on
( or SCN group) from Eq. (4.37). N is the last hydr ocar bon
gr oup in the C7+ fract i on and is normal l y expressed in t erms
of a plus fraction. Paramet er ot det ermi nes t he shape of PDF in
Eq. (4.31). For C7+ fract i on of several reservoir fluids the PDF
expressed by Eq. (4.31) is shown in Fig. 4.9. Values of para-
met ers or, r, and 0 for each sampl e are given in the figure. As
is shown in this figure, when 01 < 1, Eq. (4.35) or (4.31) re-
duces to an exponential di st ri but i on model, whi ch is suitable
for gas condensat e systems. For values of ot > 1, the syst em
shows left-skewed di st ri but i on and demonst rat es a maxi mum
in concent rat i on. This peak shifts t oward heavier comp onent s
as the value of a increases. As values of t/ increase, the whol e
curve shifts t o the right. Paramet er 0 represents t he mol ecu-
lar wei ght of the lightest comp onent in the C7 fract i on and it
varies f r om 86 t o 95 [23]. However, this p ar amet er is mai nl y
an adjustable mat hemat i cal const ant rat her t han a physical
propert y and it may be det ermi ned from Eq. (4.34). Whi t son
[17] suggests t hat for mixtures t hat detailed composi t i onal
analysis is not available, r ecommended values for t/ and
are 90 and 1, respectively, while p ar amet er fl shoul d always
be calculated f r om Eq. (4.33). A detailed step-by-step calcula-
t i on met hod to det ermi ne paramet ers a, 0, and fl is given by
Whi t son [17].
4. CHARACTERI ZATI ON OF RESERV OI R FLUIDS AND CRUDE OILS 169
0.01
Carbon Number, CN
0 20 40 60 80
/ t 1
0.008
0.006
g~
0.004
0.002
0 200 400 600 800 1000 1200
Molecular Weight, M
FIG. 4. 9 ~ Mol ar distribution by gamma density function
(Eq. 4.31).
In evaluation of the summation in Eq. (4.35), the gamma
function is defined as:
oo
(4.43) F(x) = I tX-le-tdt
q]
0
where t is the integration variable. As suggested by Whitson
[16], the gamma function can be estimated by the following
equation provided in reference [37]:
8
(4.44) P(x + 1) = 1 + C Aixi
i = l
where for 0 < x < 1, A1 = -0.577191652, A2 = 0.988205891,
A3 - -0.897056937, A4 = 0.918206857, As = -0.756704078,
A6 = 0.482199394, A7 --- -0.193527818, and As =
0.035868343. And for x > 1, the recurrence formula may be
used:
(4.45) F(x + 1) = xF(x)
where from Eq. (4.44), F(1) = 1 and thus from the above equa-
tion P(2) = 1.
Equation (4.31) with ~ = 1 reduces to an exponential dis-
tribution form. From Eq. (4.33) with ot = 1, fl = M7+ - ~ and
substituting these coefficients into Eq. (4.31) the following
density function can be obtained:
1 M- ~
(4.46) F( M ) =( M 7 +_ O) e x p( - ~7 7 +- - )
For a SCN group n, with molecular weight boundaries of Mff
and M +, substitution of Eq. (4.46) into Eq. (4.19) will result:
xm, ~= - e xP( M7; _ 0)
(4.47) x [exp ( - M7+M+ r T) -- e xp ( - - - - M7+ M2- 7) ]
where x,~n is the mole fraction of SCN group n. Substitut-
ing Eq. (4.46) in Eq. (4.21) for molecular weight gives the
following relation for the average molecular weight of the
SCN group n:
M,v~, = - ( MT+- - ~) e xp ( E_ 0 ) \ Xm, n
x r, " ." , + l~ex p(./t,. M~ --r/) M+
Lt MT + - rl
(4.48) (M~.+ - 17 + 1 )ex p(_ M.
M7+-
where M .... is the average molecular weight of SCN group n.
Equations (4.47) and (4.48) can also be applied to any group
with known lower and upper molecular weight boundaries in
a C7+ fraction that follows an exponential distribution.
Example 4. 5--Show that distribution model expressed by
Eq. (4.46) leads to Eq. (4.27) for exponential distribution of
SCN groups.
Solution- - Equation (4.46) can be written in the following
exponential form:
(4.49) F(M ) = a exp(bM)
where parameters a and b are given as
1 exp( ~ )
a = M7+ - 0 MT+ - r/
(4.50)
1
b-
M7+ - r/
Substituting Eq. (4.49) into Eq. (4.18) gives the following
relation for the cumulative mole fraction, Xcm at molecular
weight M:
M
X c m : [aexp(bM)aM = ( b ) [ e x p ( b M) -
P
(4.51) exp(br/)]
For a SCN group n with lower and upper molecular weights
of M~- and M~ and use of Eq. (4.19) we get mole fraction of
the group, x,:
(4.52) x~ = ( ; ) [ e xp ( bM~) - exp(b/ ~-)]
From Eqs. (4.39) and (4.40) we have
M, + M,,+I Mn-i + Mn
~- 2 ~- = 2
Now if we assume the difference between M, and M,-1 is a
constant number such as h we have Mn+l = M, + h and M,_I =
M~- h, thus M~ + = M~ + h/ 2 and M~- = M~ - h/2. A typical
value for h is usually 14. Substituting for M~ and M~- in
Eq. (4.52) gives
= ( ; ) {exp[ b( Mn + h/ Z) ] - exp[b(M~ - h/2)]}
Xn
(4.53) = ( ; ) [ exp( ~f f ) - e xp ( - ~) ] exp(bM~)
This equation can be written as
(4.54) x~ = A exp(BMn)
TABLE 4. I O~Prediction of molecular weight of SCN groups
from Eq. (4.48) for Example 4.6.
SCN, n M~ M n - M~ x .... Mn, cMc. Mn.calc-Mn
7 95 88 101 0.0314 94.5 --0.5
8 107 101 114 0.0304 107.5 0.5
9 121 114 128.5 0.0328 121.2 0.2
10 136 128.5 142.5 0.0306 135.5 --0.5
11 149 142.5 156 0.0285 149.2 0.2
12 163 156 169.5 0.0276 162.7 --0.3
13 176 169.5 184 0.0286 176.7 0.7
14 191 184 199 0.0286 191.5 0.5
15 207 199 214 0.0275 206.5 --0.5
16 221 214 229 0.0265 221.5 0.5
17 237 229 243 0.0239 236.0 --1.0
18 249 243 255 0.0199 249.0 0.0
19 261 255 268 0.0209 261.5 0.5
20 275 268 282 0.0217 275.0 0.0
Equat i on (4.54) is identical to Eq. (4.27) wi t h paramet ers A
and B defined in t erms of paramet ers a and b in the exponen-
tial di st ri but i on model (Eq. (4.49)) as following:
(4.55) A = ( ; ) I exp ( ~) - exp ( - ~) 1
B =b
where a and b are defined in t erms of di st ri but i on paramet ers
by Eq. (4.50). ,
Example 4. 6- - Use the exponential model to estimate average (4.56)
mol ecul ar weights of SCN groups from C7 t o C20 and comp ar e
wi t h values in Table 4.6. where
Solution--Average mol ecul ar weight of a mi xt ure t hat fol-
lows t he exponential di st ri but i on model is given by Eq. (4.48).
I n usi ng this equation, x,~,~ is needed whi ch shoul d be cal-
culated from Eq. (4.47). Two paramet ers of ~ and M7+ are
needed. Arbi t rary values for these paramet ers may be chosen.
Paramet er ~ has no effect on t he cal cul at i on as l ong as it is less
t han M2 and M7+ does not affect the results as l ong as is well
above M~. Change in the chosen value for M7+ does change
value of x~, but not calculated M~. For our calculations since
we need to estimate M20 we choose MT+ = 500 and ~ = 90.
Values of M~- and M ff for each SCN gr oup are calculated f r om
Eqs. (4.39) and (4.40). Summar y or results for cal cul at i on of
M~ and comp ar i son wi t h values from Table 4.6 is given in
Table 4.10. The maxi mum difference bet ween calculated Mn
and values from Table 4.6 is 1, while for most cases bot h val-
ues are identical.
4. 5. 4 General i z ed Di st ri but i on Model
The exponential model is t he simplest f or m of expressing dis-
t ri but i on of SCN groups in a reservoir fluid but it is mai nl y
applicable to gas condensat e systems or at most to volatile
oils. For this reason the gamma di st ri but i on model has been
used to express mol ar di st ri but i on of heavier oils. Al t hough
this model also has been applied to express di st ri but i on of
boiling poi nt but it is not suitable for specific gravity distri-
bution. For this reason the idea of const ant Wat son K for the
whol e C7+ subfract i ons has been used [17]. I n this approach,
based on calculated Kw for C7+ from 11//7+ and SG7+, values
to SG can be est i mat ed for each subfract i on usi ng their cor-
respondi ng boiling point. As it will be shown, this ap p r oach
dos not provi de an accurat e di st ri but i on of specific gravity in
a wide and heavy hydrocarbon-pl us fraction. As was shown in
Chapt er 2, specific gravity is an i mpor t ant p ar amet er in char-
act eri zat i on of pet rol eum fract i ons and errors in its value
cause errors in est i mat i on of physical propert i es of t he sys-
tem. However, when these model s are applied to very heavy
fract i ons especially for mi xt ures in whi ch the density func-
t i on F(M) sharpl y decreases for the heaviest component s,
t hei r per f or mance decreases [24, 25]. I n fact these distribu-
t i on funct i ons are among many st andard PDF model s t hat
has been selected for appl i cat i on to pet rol eum mi xt ures for
expression of their mol ar distributions because of its mat h-
emat i cal convenience. For these reasons, Riazi at t empt ed to
develop a general di st ri but i on model for vari ous propert i es
and applicable t o different types of pet rol eum mi xt ures espe-
cially heavy oils and residues [24, 25].
4. 5. 4. 1 Versatile Correlation
An extensive analysis was made on basic charact eri zat i on pa-
ramet ers for C7+ fractions of wide range of gas condensat e
systems and crude oils, light and heavy as well as nar r ow and
wide pet rol eum fractions. Based on such analysis the follow-
ing versatile equat i on was f ound to be t he most suitable fit
for vari ous propert i es of mor e t han 100 mi xt ures [24]:
1
(l)].
P* = I n ~-~
P- Po
P* - - - x* = l - xc
Po
P is a propert y such as absolute boiling poi nt (Tb), mol ecul ar
wei ght (M), specific gravity (SG) or refractive index p ar am-
eter ( I ) defined by Eq. (2.36). xc is cumul at i ve weight, mole,
or vol ume fraction. Po is a p ar amet er specific for each prop-
ert y (To, Mo, and SGo) and each sample. Usually cumul at i ve
mol e fraction, Xcm is used for mol ecul ar wei ght and cumu-
lative wei ght fraction, Xcw is used to express di st ri but i on of
boiling point. Ei t her cumul at i ve vol ume fraction, Xcv or cu-
mulative weight fract i on X~w can be used for present i ng distri-
but i on of specific gravity, density, or refractive index p ar am-
eter, I . I n Eq. (4.56), P* is a di mensi onl ess parameter. Equa-
t i on (4.56) is not defined at x~ -- 1 (x* = 0). I n fact accordi ng
to this model, it is theoretically assumed t hat the last compo-
nent in the mi xt ure is extremely heavy wi t h P --~ oo as x~ -~ 1.
A and B are t wo ot her paramet ers whi ch are specific for each
propert y and may vary f r om one sampl e t o another. Equat i on
(4.56) has t hree paramet ers (Po, A, B); however, for mor e t han
100 mixtures investigated it was observed t hat p ar amet er B
for each propert y is the same for most sampl es [24] reduc-
ing the equat i on into a t wo-paramet er correlation. Parame-
ter Po corresponds to the value of P at x~ = 0, where x* = 1
and P* = 0. Physically Po represent s value of propert y P for
the lightest comp onent in the mixture; however, it is mai nl y
a mat hemat i cal const ant in Eq. (5.56) t hat shoul d be deter-
mi ned for each mi xt ure and each property. I n fact Eq. (4.56)
has been already used in Section (2.2.3) by Eq. (3.34) for pre-
diction of compl et e distillation curves of pet rol eum fraction.
The mai n idea behi nd Eq. (4.56) is to assume every pet rol eum
mi xt ur e cont ai ns all comp ounds i ncl udi ng ext r emel y heavy
comp ounds up to M --~ oo. However, what di ffers f r om one
mi xt ur e to anot her is t he amount of i ndi vi dual comp onent s.
For l ow and me di um mol ecul ar wei ght r ange fract i ons t hat
do not cont ai n hi gh mol ecul ar wei ght comp ounds, t he model
expr essed by Eq. (4.56) assumes t hat ext r emel y heavy com-
p ounds do exist i n t he mi xt ur e but t hei r amount is i nfi ni t el y
smal l , whi ch in mat hemat i cal cal cul at i ons do not affect mi x-
t ure pr oper t i es.
When suffi ci ent dat a on p r op er t y P versus cumul at i ve mol e,
wei ght , or vol ume fract i on, Xo are avai l abl e const ant s in
Eq. (4.56) can be easi l y det er mi ned by conver t i ng t he equa-
t i on i nt o t he fol l owi ng l i near form:
(4.57) Y = Ca + CzX
wher e Y= l nP* and X=l n[ l n( 1 / x* ) ] . By combi ni ng
Eqs. (4.56) and (4.57) we have
1
B= - -
(4.58) C2
A = B exp( C1B)
I t is r ecommended t hat for samp l es wi t h amount of r esi dues
( last hydr ocar bon group) gr eat er t han 30%, t he r esi due dat a
shoul d not be i ncl uded i n t he r egr essi on anal ysi s to obt ai n
t he coeffi ci ent s i n Eq. (4.57). I f a fixed val ue of B is used for
a cer t ai n propert y, t hen onl y p ar amet er Ca shoul d be used t o
obt ai n coeffi ci ent A f r om Eq. (4.58).
To est i mat e Po in Eq. (4.56), a t r i al - and- er r or p r ocedur e
can be used. By choosi ng a val ue for Po, whi ch mus t be l ower
t han t he first dat a p oi nt i n t he dat aset , p ar amet er s A and B
can be det er mi ned f r om l i ner r egr essi on of dat a. Par amet er
Po can be det er mi ned by mi ni mi zi ng t he er r or f unct i on E( Po)
equi val ent to t he r oot mean squar es ( RMS) defi ned as
(4.59) E(Po) = (p21 c _ pi~xp)2
i=1
wher e N is t he t ot al number of dat a p oi nt used in t he regres-
si on p r ocess and p/~al~ is t he cal cul at ed val ue of p r op er t y P for
t he subf r act i on i f r om Eq. (4.56) usi ng est i mat ed p ar amet er s
Po, A, and B. As an al t er nat i ve obj ect i ve funct i on, best val ue
of Po can be obt ai ned by maxi mi zi ng t he val ue of R 2 defi ned
by Eq. (2.136). Wi t h sp r eadsheet s such as Mi cr osof t Excel,
p ar amet er Po can be di r ect l y est i mat ed f r om t he Sol ver t ool
wi t hout t r i al - and- er r or p r ocedur e. However, an i ni t i al guess
for t he val ue Po is al ways needed. F or a C7+ f r act i on val ue of
p r op er t y P for C7 or C6 hydr ocar bon gr oup f r om Table 4.6
may be used as t he i ni t i al guess. Al t hough l i near r egr essi on
can be p er f or med wi t h sp r eadsheet s such as Excel or Lot us,
coeffi ci ent s C1 and C2 in Eq. (4.57) can be det er mi ned by ha nd
cal cul at or s usi ng t he fol l owi ng r el at i on der i ved f r om t he l east
squar es l i near r egr essi on met hod:
Z x , ~ - N~( X i ~)
C2 =
(4.60) ( ~ X~)z - N Z (X~/)
ZY~ - c2 z x~
C1 =
N
wher e each sum ap p l i es t o al l dat a p oi nt s used i n t he regres-
si on and N is t he t ot al number of p oi nt s used. The l east
squar es l i near r egr essi on met hod is a s t andar d met hod for
obt ai ni ng t he equat i on of a st r ai ght line, such as Eq. (4.57),
f r om a set of dat a on Xi and Y/.
Ex ampl e 4 . 7 - - The nor mal i zed comp osi t i on of a C7+ f r act i on
der i ved f r om a Nor t h Sea gas condensat e samp l e (GC) i n
t er ms of wei ght f r act i ons of SCN gr oups up t o C17 is gi ven
in Table 4.11. M and SG of Cas+ fract i on ar e 264 and 0.857,
respectively. F or t he whol e C7+ f r act i on t he M7+ and SG7+ ar e
118.9 and 0.7597, respectively. Obt ai n p ar amet er s Po, A, and
B in Eq. (4.56) for M, Tb, and SG and comp ar e cal cul at ed
val ues of t hese p r op er t i es wi t h dat a shown in Table 4.6.
Sol ut i on- - For SCN gr oup s f r om C 7 to C17 val ues of M , Tb, and
SG can be t aken f r om Table 4.6 and ar e gi ven i n Table 4.11.
An al t er nat i ve to t hi s t abl e woul d be val ues r ecommended by
Whi t son [ 16] for SCN gr oup s less t han Cz5. Di scret e mol e frac-
t i ons, Xr~ can be cal cul at ed f r om di scret e wei ght fract i ons, Xwi
and Mi by a reversed f or m of Eq. (1.15) as follows:
Xwi / Mi
(4.61) X mi - - N
Ei=m xwd Mi
wher e N is t he t ot al number of comp onent s ( i ncl udi ng t he
l ast pl us fract i on) and for t hi s examp l e it is 12. Di scret e vol-
ume fract i ons x~/ can be cal cul at ed f r om Xwi and SGi t hr ough
Eq. (1.16). Values of x~ and x,~ ar e gi ven in Table 4.11. To ob-
t ai n p ar amet er s i n Eq. (4.56), cumul at i ve mol e (xcm), wei ght
(Xcw), or vol ume (XCv) f r act i ons ar e needed. A samp l e cal cu-
l at i on for t he est i mat i on of mol ecul ar wei ght versus Xcm is
shown here. A si mi l ar ap p r oach can be t aken to est i mat e cu-
mul at i ve wei ght or vol ume fract i ons.
TABLE 4.11---Sample data on characteristics of a C7+ fraction for a gas condensate
Fraction Carbon
No. No. Xw M Tb, K SG Xm xv
system in Example 4. 7.
Xcm Xcw Xcv
1 7 0.261 95 365 0.727 0.321 0.273 0.161 0.130 0.137
2 8 0.254 107 390 0.749 0.278 0.259 0.460 0.388 0.403
3 9 0.183 121 416 0.768 0.176 0.181 0.687 0.607 0.622
4 10 0.140 136 440 0.782 0.121 0.137 0.836 0.768 0.781
5 11 0.010 149 461 0.793 0.008 0.009 0.900 0.843 0.854
6 12 0.046 163 482 0.804 0.033 0.043 0.920 0.871 0.880
7 13 0.042 176 500 0.815 0.028 0.040 0.951 0.915 0.922
8 14 0.024 191 520 0.826 0.015 0.022 0.972 0.948 0.953
9 15 0.015 207 539 0.836 0.009 0.014 0.984 0.967 0.971
10 16 0.009 221 556 0.843 0.005 0.008 0.990 0.979 0.982
11 17 0.007 237 573 0.851 0.003 0.006 0.994 0.987 0.988
12 18+ 0.010 264 - - 0.857 0.004 0.009 0.998 0.995 0.996
xw, xm, andxvare w~ght, m~e, and volume ~actions, respectivet~ V~uesofM, Tb, and SGaretaken ~omTab~ 4.6. xcm,Xcw, and
x~ arecumulative mole, weight, andvolume ~a~ionsc~culated from Eq.(4.62).
40
30
E
O
~
20
10
0
0 25 50 75 100
Cumulative Mole Percent
FIG. 4.1 O --Rel ati on between di screte and cumul a-
ti v e mole fractions for t he system of Ex ample 4.7.
For the mi xt ure shown i n Table 4.11 t here are 12 compo-
nent s each havi ng mol ecul ar wei ght of Mi and mol e fract i on
of a~ (i = 1 . . . . . 12). Values of cumul at i ve mol e fraction, Xcmi
correspondi ng to each val ue of Mi can be est i mat ed as:
Xmi - 1 -~- Xmi
(4.62) x~ = xcmi-1 + i = 1, 2 . . . . . N
2
where bot h Xcm0 and Xm0 (i = 0) are equal to zero. Accordi ng to
this equat i on, for the last fract i on (i = N), XcmN = 1 -- XmN/2.
Equat i on (4.62) can be appl i ed to wei ght and vol ume fract i ons
as well by repl aci ng the subscri pt s m wi t h w or v, respectively.
Values of X~m/, X~,,~, and xc~i are cal cul at ed from Eq. (4.62)
and are given i n the last t hree col umns of Table 4.11. Si nce
amount of the last fract i on ( residue) is very small, x~N is very
close to unity. However, i n most cases especially for heavy oils
the amount of residues may exceed 50% and value of Xc for
the last dat a poi nt is far from unity. The rel at i on bet ween Xcm
and xm is shown i n Fig. 4.10.
To obt ai n mol ar di st r i but i on for t hi s system, par amet er s
Mo, AM, and BM for Eq. (4.56) shoul d be cal cul at ed from the
l i near rel at i on of Eq. (4.57). Based on the values of Mi and
X~mi i n Table 4.11, values of Y/ and Xi are cal cul at ed from M*
and x* as defined by Eq. (4.57). I n cal cul at i on M* a value of
Mo is needed. The first i ni t i al guess for Mo shoul d be less t han
M1 ( mol ecul ar wei ght of the first comp onent i n t he mi xt ure) .
The best val ue for Mo is the l ower mol ecul ar wei ght boundar y
for C7 group t hat is M 7 i n Table 4.10, whi ch is 88. Si mi l arl y
the best i ni t i al guess for Tbo and SGo are 351 K and 0.709,
respectively. These number s can be simplified to 90, 350, and
0.7 for the i ni t i al guesses of Mo, Tbo, and SGo, respectively.
Si mi l arl y for a C6+ fraction, the i ni t i al guess for its Mo can be
t aken as t he l ower mol ecul ar wei ght boundar y for C6 (M~).
For t hi s example, based on t he val ue of Mo = 90, paramet ers
Y/ and Xi are cal cul at ed and are given i n Table 4.12. A l i near
regressi on gives val ues of C1 and C2 and from Eq. (4.58) pa-
ramet ers A and B are cal cul at ed whi ch are given i n Table 4.12.
For these values of Mo, A, and B, values of Mi are cal cul at ed
from Eq. (4.56) and the error f unct i on E(M o) and AAD% are
cal cul at ed as 2.7 and 1.32, respectively. Value of Mo shoul d
be changed so t hat E(M o) cal cul at ed from Eq. (4.59) is mi ni -
mi zed. As shown i n Table 4.12, the best val ue for this sam-
ple is Mo = 91 wi t h A = 0.2854 and B = 0.9429. These co-
efficients gives RMS or E(Mo) of 2.139 and AAD of 0.99%,
whi ch are at mi ni mum. At Mo = 91.1 the value of E(M o) is
cal cul at ed as 2.167. The same values for coefficients Mo, AM,
and BM can be obt ai ned by usi ng Solver tool i n Microsoft
Excel spreadsheets. Experi ence has shown t hat for gas con-
densat e systems and light fract i ons value of B~a is very close
to one like i n this case. For such cases BM can be set equal
to uni t y whi ch is equi val ent to C2 = 1. I n this exampl e at
Mo = 89.856, we get C1 = - 1. 1694 and Cz = 1 whi ch from
Eq. (4.58) yields AM = 0.3105 and BM = 1. Use of these co-
efficients i n Eq. (4.56) gives E(M o) of 2.83 and AAD of 1.39%,
whi ch is slightly hi gher t han the error for the op t i mum val ue
of Mo at 91. Therefore, the final values of coefficients of
Eq. (4.56) for M i n t erms of cumul at i ve mol e fract i on are de-
t er mi ned as: Mo = 91, Ata = 0.2854, Bu = 0.9429. The mol ar
di st r i but i on can be est i mat ed from Eq. (4.56) as
/ 0. 2854 1 ) t ( I n I / 1. 06056
M* = [ ~l n~- ~ 0- ~r ~=0. 28155\ 1 - X c m/
Fr om defi ni t i on of M* i n Eq. (4.56) we can cal cul at e M as
(4.63) M = Mo x (1 + M*)
and for this exampl e we get:
M= 89.86[I +~ ( ln J - - - L- - / ' ~176 1 - Xcm/ J
TABLE 4. 12--Determination of coefficients of Eq. (4.56) for molecular weight from data of Table 4.11.
Mo = 90, C1 = -1.1809, C2 = 1.0069, A = 0.3074,
B = 0.9932, R 2 = 0.998, RMS = 2.70, AAD = 1.32%
Mo = 91, C1 = -1.2674, C2 = 1,0606, A = 0.2854,
B = 0.9429, R z = 0.999, RMS = 2.139, AAD = 0.99%
-3.125 95.0 0.0 0.0
-1.738 106.3 0.4 0.6
-1.110 121.1 0.0 0.0
-0.704 139.0 8.8 2.2
-0.450 153.0 16.1 2.7
-0.234 159.5 12.4 2.2
-0.068 173.3 7.1 1.5
0.094 189.9 1.2 0.6
0.243 205.6 1.8 0.7
0.357 220.9 0.0 0.1
0.473 236.0 1.0 0.4
0.642 266.4 5.9 0.9
95 0.839 -1.743 0.056 -2.89 94.8 0.0 0.2 0.044
107 0.54 -0.484 0.189 -1.667 107.0 0.0 0.0 0.176
121 0.313 0.15 0.344 -1.066 122.1 1.3 0.9 0.330
136 0.164 0.591 0. 511 -0.671 140.1 16.9 3.0 0.495
149 0.1 0.833 0.656 -0.422 153.9 24.5 3.3 0.637
163 0,08 0.927 0. 811 -0.209 160.3 7.5 1.7 0.791
176 0.049 1.101 0.956 -0.045 173.7 5.3 1.3 0.934
191 0.028 1.273 1.122 0.115 189.6 2.0 0.7 1.099
207 0.016 1.413 1.3 0.262 204.6 5.8 1.2 1.275
221 0.01 1.53 1.456 0.375 219.0 4.0 0.9 1.429
237 0.006 1.634 1.633 0.491 233.2 14.3 1.6 1.604
264 0.002 1.814 1.933 0.659 261.6 5.6 0.9 1.901
A Mi 2 = (M calc- M~)2, %AD = Percent absolute relative deviation.
4. CHARACTERIZATION OF RESERV OIR FLUIDS AND CRUDE OILS 173
300 800 0.9
200
O0
"3
100
o Actual data
- - Optimum B
. . . . . . B=I
o-------------
f z I T T Z r r
0.2 0.4 0.6 0.8 1
Cumulative Mole Fraction, Xcm
FIG. 4.11--Prediction of molar distribution from
Eq. (4.56) for the GC system of Example 4.7.
600
b~
o 400
..=
200
0
0
o Actual Data for Tb
Predicted Tb
zx Actual data for SG
[ r I T T r l r P r
0.2 0.4 0.6 0.8
L~
0.8
C. )
Z , )
0J
0.7
Cumul at i ve Wei ght Fraction, Xcm
FIG. 4.12--Prediction of boiling point and specific gravity dis-
tributions from Eq. (4.56) for the GC system of Example 4.7.
I f t he fixed val ue of BM = 1 is used wi t h Mo = 89.86 and
AM = 0.3105 t hen t he mol ar di st r i but i on is gi ven by a si m-
p l er r el at i on
M = 89. 86( 1 + 0.3105 I n 1 _ ~l xcm)
Pr edi ct i on of mol ar di st r i but i on based on t hese t wo r el at i ons
(BM = 0.9429 and B~ = 1) ar e shown in Fig. 4.11. The t wo
curves are al most i dent i cal except t owar d t he end of t he curve
wher e Xcm ~ I and t he di fference is not vi si bl e i n t he figure.
Usi ng a si mi l ar ap p r oach, coeffi ci ent s in Eq. (4.56) for Tb
and SG ar e det er mi ned. For SG bot h cumul at i ve wei ght and
vol ume f r act i ons can be used. The val ue of Tb for t he r esi due
(C~s+) is not known, for t hi s r eason onl y 11 dat a p oi nt s
ar e used for t he r egr essi on anal ysi s. Summa r y of resul t s
for coeffi ci ent s of Eq. (4.56) for M , Tb, and SG in t er ms of
var i ous xc is gi ven in Table 4.13. Based on t hese coeffi ci ent s
Tb and SG di st r i but i ons p r edi ct ed f r om Eq. (4.56) are shown
in Fig. 4.12. The l i near r el at i on bet ween p ar amet er s X and Y
defi ned i n Eq. (4.57) for SG is demons t r at ed i n Fig. 4.13. Pre-
di ct i on of PDF for Tb and SG are shown i n Figs. 4.14 and 4. l 5,
respectively. Bot h Eqs. (4.66) and (4.70) have been used to
i l l ust rat e densi t y f unct i on for bot h Tb and SG. As shown i n
Table 4.13, t he best val ues of Mo, Tbo, and SGo are 91, 350 K,
and 0.705, whi ch ar e very cl ose to t he val ues of l ower bound-
ar y p r op er t i es for t he C7+ group. ( M 7 = 88, Tb~ = 350 K,
SG 7 = 0.709). For GC and l i ght oils val ues of B for M ar e
very cl ose to one, for Tb are cl ose to 1.5, and for SG ar e cl ose
b
LI
-1
-2
o Data
Predicted j
- 3 I I I I I I I
-2 -1 0 1 2
X = In In ( l/ x* )
FIG. 4.13--Linearity of parameters Y and X defined in
Eq. (4.57) for specific gravity of the system in Example 4.7.
TABLE 4.13---Coefficients of Eq. (4.56) for M, Tb, and SG for data of Table 4.11.
Property Type of Xc Po A B RMS %AAD R 2
M Mole 91 0.2854 0.9429 2.139 0.99 0.999
Tb Weight 350 (K) 0.1679 1.2586 3.794 0.62 0.998
SG Volume 0.705 0.0232 1.8110 0.004 0.32 0.997
SG Weight 0.705 0.0235 1.8248 0.004 0.33 0.997
Coefficients of Po and A with fixed value of B for each property
M Mole 89.86 0.3105 1 2.83 1.39 0.998
T b Weight 340 (K) 0.1875 1.5 5.834 1.15 0.993
SG Volume 0.665 0.0132 3 0.005 0.54 0.984
SG Weight 0.6661 0.0132 3 0.005 0.53 0.985
4 10
3
X__J
2
0.009
0.2 0.4 0.6 0.8
Tb*
0.006
0.003
0 I A
300 700 400 500 600
Boiling Point, Tb, K
800
0.012
0.8 0.9
Specific Gravity, SG
FIG. 4.14--Prediction of PDF for boiling point by
Eqs. (4.66) and (4.70) for the system of Example 4.7.
t o 3. I f these fix values are used errors for predi ct i on of
di st ri but i on of M, Tb, and SG t hr ough Eq. (4.56) increases
slightly as shown in Table 4.13. ,
I n Exampl e 4.7, met hod of det ermi nat i on of t hree coef-
ficients of Eq. (4.56) was demonst rat ed. As shown in this
exampl e fixed values of B (BM = I, Br ---- 1.5, BsG = 3, Bt = 3)
may be used for certain mi xt ures especially for gas conden-
sate systems and light oils, whi ch reduce the di st ri but i on
model into a t wo-paramet er correlation. I t has been observed
t hat even for most oil sampl es t he fixed values of Br -- 1.5 and
Bs6 = Bx = 3 are also valid. Fur t her eval uat i on of Eq. (4.56)
as a t hree-paramet er or a t wo-paramet er correl at i on and a
comp ar i son wi t h the gamma di st ri but i on model are shown
in Section 4.5.4.5.
4. 5. 4. 2 Probability Density Function for the Proposed
Generalized Distribution M odel
The di st ri but i on model expressed by Eq. (4.56) can be trans-
formed i nt o a probabi l i t y density funct i on by use of Eq. (4.16).
Equat i on (4.56) can be r ear r anged as
f A , ~\
(4.64) 1 - Xc = e xp ~- ~P )
6
r,r
a~
i
0.1 0.2 0.3
SG*
0.4
14
12
10
--- 8
2
0
0.7
FIG. 4.1 5--Prediction of PDF for specific gravity by
Eqs. (4.66) and (4.70) for the system of Example 4.7.
Fr om Eq. (4.16) and in t erms P*, the PDF is given as
dxc
(4.65) F(P*) = - -
dP*
where F(P*) is the PDF in t erms of di mensi onl ess par amet er
P* whi ch can be det ermi ned by differentiation of Eq. (4.64)
wi t h respect to P* accordi ng to the above equat i on:
(4.66) F(P*) = B2 ,B-1
:
Equat i on (4.66) is in fact the probabi l i t y density funct i on for
the generalized di st ri but i on model of Eq. (4.56) in t erms of
par amet er P* . I n a hydr ocar bon plus fraction, p ar amet er P*
varies from 0 to ~. Application of Eq. (4.17) in t erms of P*
gives:
OO
(4.67) / F(P*)dP* = 1
0
and xr at P* can be det ermi ned from Eq. (4.18) in t erms
of P* :
p *
/ i
(4.68) xc = / F(P*)dP*
. i
0
4. CHARACTERIZATION OF RESERV OIR FLUIDS AND CRUDE OILS 175
I t is much easier to work in t erms of P* rat her t han P, since for
any mi xt ure P* starts at 0. However, based on the definition
of P* in Eq. (4.56), t he PDF expressed by Eq. (4.66) can be
wri t t en in t erms of original propert y P. Since dx = F(P)dP =
F(P*)dP* and dP = PodP* , t herefore we have
(4.69) F(P) = -~-~ F(P*)
Subst i t ut i ng F(P*) from Eq. (4.66) into t he above equat i on
and use of definition of P* we get
F ( p ) = ( ~) x( ~_ ~2) ) [p_ po. ~S lexp[_ B(P_ po,~ Po .] J
(4.70)
wi t h this form of PDE Eq. (4.18) shoul d be used to calcu-
late cumulative, Xc at P. Obviously it is mor e conveni ent to
wor k in t erms of P* t hr ough Eq. (4.68) and at the end P* can
be convert ed to P. This ap p r oach is used for cal cul at i on of
average propert i es in the next section.
A simple comp ar i son of Eq. (4.70) or (4.66) wi t h the gamma
di st ri but i on funct i on, Eq. (4.31), indicates t hat par amet er Po
is equivalent to par amet er ~ and p ar amet er B is equivalent to
p ar amet er u. Paramet er A can be related t o 0t and r; however,
the biggest difference bet ween these two model s is t hat inside
the exponential t erm in Eq. (4.66), P* is raised to the expo-
nent B, while in the gamma di st ri but i on model, Eq. (4.31),
such exponent is always unity. At B = 1, t he exponent i al t erm
in Eq. (4.66) becomes similar to t hat of Eq. (4.31). I n fact
at B = 1, Eq. (4.66) reduces to the exponential di st ri but i on
model as was t he case for the gamma di st ri but i on model when
0~ = 1. For this reason for gas condensat e systems, the mol ar
di st ri but i on can be present ed by an exponential model as t he
behavi or of t wo model s is t he same. However, for mol ar dis-
t ri but i on of heavy oils or for propert i es ot her t han mol ecul ar
wei ght in whi ch p ar amet er B is great er t han 1, the differ-
ence bet ween t wo model s become mor e apparent. As it is
shown in Section 4.5.4.5, the gamma di st ri but i on model fails
t o present properl y t he mol ar di st ri but i on of very heavy oils
and residues. For the same reason Eq. (4.66) is applicable for
present at i on of ot her propert i es such as specific gravity or
refractive index as it is shown in Section 4.5.4.4. A compar-
ison bet ween the gamma di st ri but i on model (Eq. 4.31) and
general i zed model (4.70) when Ado = ~ and B = a is shown in
Fig. 4.16. As shown in this figure the difference bet ween the
pr oposed model and the gamma model increases as value of
p ar amet er B or a ( keeping t hem equal) increases. Effect of
p ar amet er B on the form and shape of di st ri but i on model by
Eq. (4.70) is shown in Fig. 4.17. For bot h Figs. 4.16 and 4.17,
it is assumed t hat the mi xt ure is a C7+ fract i on wi t h Mo = 90
and M7+ = 150.
4.5.4.3 Calculation of Average Properties
of Hydrocarbon-Plus Fractions
Once the PDF for a propert y is known, the average propert y
for the whol e mi xt ure can be det ermi ned t hr ough appl i cat i on
of Eq. (4.20). I f the PDF in t erms of P* is used, t hen Eq. (4.20)
becomes
oo
P~ = ] (4.71) P*F(P*)dP*
v
0
Carbon Number, Nc
5 10 15 20
0.018 , , ,
25
- - Generalized Model
B=2
....... Gamma Model
B= 3
0.012
"~ 0.006
.=
80 160 240 320
Molecular Weight, M
FIG. 4. 1 6---Compari son of Eqs. ( 4.31) and Eq. ( 4.66)
for Mo = O = 90, B = ~ , and Mr+ = 150.
where Pa*v is the average value of P* for the mixture. Substi-
t ut i ng Eq. (4.66) into Eq. (4.71) gives the following rel at i on
for P'v:
(4.72) P~*~ = F 1 +
where F( I + 1/B) is the gamma funct i on defined by Eq. (4.43)
and may be evaluated by Eq. (4.44) wi t h x = 1/B. A si mpl er
version of Eq. (4.44) was given in Chapt er 3 by Eq. (3.37) as
F ( I + I ) = 0. 992814- 0. 504242B- l +0. 696215B -2
(4.73) - 0.272936B -3 + 0.088362B -4
0.018
Carbon Number, N C
5 10 15 20 25
0.012
O
gr.
0.006
0~
80 160 240 320
Molecular Weight, M
FIG. 4. 1 7 mEffect of parameter B on t he shape of
Eq. ( 4.70) for Mo = 9 0 and M7+ - - 150.
in which F(1 + 1/B) can be evaluated directly from parame-
ter B. This equation was developed empirically for mat hemat -
ical convenience. Values estimated from this equation vary by
a maxi mum of 0.02% (at B = 1) with those from Eq. (4.43).
Therefore, for simplicity we use Eq. (4.73) for calculation of
average values t hrough Eq. (4.72).
As ment i oned earlier for many systems fixed values of B
for different properties may be used. These values are BM = 1
for M, BT = 1.5 for Tb, and BsG = t31 = Ba = 3 for SG,/ 20 or
d20. For these values of B, F(1 1/B) has been evaluated by
Eq. (4.73) and substituted in Eq. (4.72), which yields the fol-
lowing simplified relations for calculation of average proper-
ties of whole C7+ fraction in t erms of coefficient A for each
property:
(4.74) M~*v = AM
(4.75) T~,~ = 0.689A~ 3
(4.76) SG* v = 0. 619A~
I t should be noted that Eq. (4.76) can be used when SG
is expressed in t erms of cumulative volume fraction. Equa-
tion (4.76) is based on Eq. (4.72), which has been derived
from Eq. (4.71). As it was discussed in Chapter 3 (Section 3.4),
for SG, d (absolute density), and I (defined by Eq. 2.36) two
types of mixing rules may be used to calculate mixture prop-
erties. Linear Kay mixing rule in the form of Eq. (3.45) can
be used if composition of the mixture is expressed in volume
fractions, but when composi t i on if given in t erms of weight
fractions, Eq. (4.46) must be used. Both equations give sim-
ilar accuracy; however, for mixtures defined in t erms of very
few compounds that have SG values with great differences,
Eq. (3.46) is superior to Eq. (3.45). Equation (4.46) can be
applied to SG in a continuous form as follows:
1
1 f dxcw
(4.77) SG~ - SG(xcw)
0
where SG~ is the average specific gravity of C7+ and SG(x~w) is
the continuous distribution function for SG in t erms of cumu-
lative weight fraction. SG(xcw) can be expressed by Eq. (4.56).
Equation (4.77) in a dimensionless form in t erms of SG*
becomes
oo
(4.78) - - 1 - f F(SG* ) dSG*
SG* v + 1 SG* + 1
0
I n this equation integration is carried on the variable SG*
and F(SG* ) is the PDF for SG* in t erms of Xcw. I ntegration
in Eq. (4.78) has been evaluated numerically and has been
correlated to paramet er Aso in the following form [24]:
oo
r F G* dSG*
J =J ( S) SG* +I
0
1
= 1.3818 + 0.3503AsG - 0.1932A~G for AsG > 0.05
1
= 1.25355 + 1.44886AsG -- 5.9777A2G + 0.02951 lnAsc
(4.79) for As6 < 0.05
where J is just an integration par amet er defined in Eq. (4.79).
As6 is the coefficient in Eq. (4.56) when SG is expressed
in t erms of cumulative weight fraction, Xcw. For most sam-
ples evaluated, par amet er Asa is between 0.05 and 0.4; how-
ever, for no system a value greater t han 0.4 was observed.
By combining Eqs. (4.78) and (4.79) with definition of SG* by
Eq. (4.56), SG~ can be calculated from the following relation:
SOa ( )SOo
this equation should be used when SG is expressed in t erms
of Xcw by Eq. (4.56). For analytical integration of Eq. (4.78)
see Problem 4.4.
In general, once P*~ is determined from Eq. (4.72), P~ can
be det ermi ned from the definition of P* by the following
relation:
(4.81) Pay = Po(1 + Pay)
Average properties determined by Eqs. (4.74)-(4.76) can be
converted to M~v, Tbav, and SGa~ by Eq. (4.81). Equation (4.76)
derived for SG~, can also be used for refractive index para-
met er I or absolute density (d) when they are expressed in
t erms ofx~. Similarly Eqs. (4.78)-(4.80) can be applied to/2o
or d20 when they are expressed in t erms of Xcw. The following
example shows application of these equations.
Example 4. 8- - For the gas condensate system of Exampl e 4.7
calculate mixture molecular weight, boiling point, and spe-
cific gravity using the coefficients given in Table 4.13. The ex-
perimental values are MT+ = 118.9 and SG7+ --0. 7569 [24].
Also calculate the boiling point of the residue ( component
no. 12 in Table 4.11).
Solution--For mol ecul ar weight the coefficients of PDF in
t erms Of Xcm for Eq. (4.66) as given in Table 4.13 are: Mo = 91,
AM ---- 0.2854, and BM = 0.9429. From Eq. (4.73), F( I + 1/B) =
1.02733 and from Eq. (4.72), M* v = 0.2892. Finally May is cal-
culated from Eq. (4.81) as 117.3. For this system BM is very
close to unity and we can use the coefficients in Table 4.13 for
Mo = 89.86, AM = 0.3105, and BM = 1. From Eqs. (4.74) and
(4.81) we get May -- 89.86 x (1 + 0.3105) = 117.8. Comparing
with the experimental value of 118.9, the relative deviation
is - 1%.
For specific gravity, the coefficients in t erms of Xcv are:
SGo -- 0.705, ASG = 0.0232, and Bsa -- 1.811. From Eq. (4.76),
SG~v = 0.0801 and from Eq. (4.81), SGav = 0.7615. Compar-
ing with experimental value of 0.7597, the relative devia-
tion is 0.24%. I f the coefficients in t erms of Xcw are used,
SGo = 0.6661, AsG = 0.0132, and from Eq. (4.79) we get 1/J =
1.1439 using appropri at e range for As6. From Eq. (4.80),
SG~v = 0.7619 which is nearly the same as using cumulative
volume fraction.
For Tb the coefficients in t erms Xcw with fixed value of BT are
To = 340 K, AT = 0.1875, and Br = 1.5. From Eq. (4.75) and
(4.81) we get: Ta~ = 416.7 K. To calculate Tb for the residue we
use the following relation:
Ta v __ N- 1
~- ~i = 1 XwiTbi
(4.82) TbN =
XwN
where TbN is the boiling point of the residue. For this exam-
ple from Table 4.11, N- - 12 and XwN = 0.01. Using values
Of Xwi and Tbi for i = l t o N - 1 from Table 4 . 1 1 , we get
TbN = 787.9 K. r
Exampl e 4. 9- - Show how Eq. (4.78) can be derived from
Eq. (4.77).
Sol ut i on- - From Eq. (4.16): dxw = F( SG) dSG and from defi-
ni t i on of P* in Eq. (4.56) we have SG = SGoS* + SGo, whi ch
after differentiation we get dSG = SGodSG* . I n addition,
f r om Eq. (4.69), F ( SG* ) = SGoF( SG) and f r om Eq. (4.56),
when Xcw = 0, we have SG* = 0 and at Xcw = 1, we have
SG* = o~. By combi ni ng these basic relations and substitut-
ing t hem into Eq. (4.77) we get
1 = j F(SG* )SGodSG*
SGoSG~v + SGo SGoSG* + SGo
0
whi ch after simplification reduces t o Eq. (4.78). r
4. 5. 4. 4 Calculation of Average Properties
of Sub f ract i ons
I n cases t hat the whol e mi xt ure is divided into several pseu-
docomp onent s (i.e., SCN groups) , it is necessary to calculate
average propert i es of a subfract i on i whose propert y P varies
f r om Pi-~ to Pi. Mole, weight, or vol ume fract i on of the groups
shown by zi can be calculated t hr ough Eq. (4.19), whi ch in
t erms of P* becomes
P:
P
zi = [ (4.83) F( P* ) dP*
Subst i t ut i ng F(P*) f r om Eq. (4.66) into t he above equat i on
gives
(4.84) zi = exp ( - Bp i * _ ~) - e xp ( - Bp i *B)
Average propert i es of this subfract i on shown by P/* ~ can be
calculated from Eq. (4.21), whi ch can be wri t t en as
1/
* - - - P* F( P* ) dP*
(4.85) Pi, , v- zi
eL~
by subst i t ut i ng F(P*) from Eq. (4.66) and carryi ng the inte-
grat i on we get
( 4 . 8 6 ) P. * 1 1 - ( 1 +
.... =~' / ( A) I / B[ F( I + ~ , qi - , ) F 1 ,q i ) ]
where
(4.87) qi = B pi,B
A
zi shoul d he calculated from Eq. (4.84). Pi,av is cal cul at ed f r om
P,* t hr ough Eq. (4.81) as
t , av
(4.88) Pi,~v = Po(1 + Pi*av)
I n Eq. (4.86), F(1 + I/B, qi) is the i ncompl et e gamma funct i on
defined as [38]
oo
(4.89) F(a, q) = f t~- le- tdt
q
7
1.5 I
1.2
0.9
0.6
0.3 I
0
0
- - B =0 . 7
, ~ . . . . B = 1 . 0
- - - - B = l . 5
. . . . B=2.0
",, \ . . . . B=3.0
2 4 6
qi
FIG. 4. 1 8 ~ l ncompl et e g amma function
I' ( 1 + l I B , @) for different v alues of B. Taken
with permission from Ref. [ 40].
where for t he case of Eq. (4.86), a = 1 + 1/B. Values of
F(1 + 1/B, qi) can be det ermi ned from vari ous numeri cal
handbooks (e.g., Press et al. [38]) or t hr ough mat hemat i -
cal comp ut er soft ware such as M ATHEM ATICA. Values of
F(1 + 1/B, qi) for B = 1, 1.5, 2, 2.5, and 4 versus qi are shown
in Fig. 4.18 [39]. As qi --~ oo, F(1 + 1/B, qi) ~ 0 for any value
of B. At B = 1, Eq. (4.89) gives t he following rel at i on for
F(1 + 1/B, qi):
oo
(4.90) r( 2, q) = te- tdt = - ( 1 + t)e -t]q = (1 +q) e -q
q
Furt her propert i es of i ncompl et e gamma funct i ons are given
in Ref. [39]. Subst i t ut i on of Eq. (4.90) i nt o Eq. (4.86) we
get the following relation to estimate Pi*av for the case of
B= I :
P~,*av =
i ex i
Pi-1 "~ ex { Pi-l'~ (1 p*'x / P*'x']
( A) [ ( 1 + ~ - ) pt - - ~ - - ) - + ~ - )
(4.91)
where z i is obtained from Eq. (4.84) which for the case of
B = 1 becomes:
(4.92) zi = exp - - exp -
I n these relations, Pi* and P/*-I are the upper and l ower bound-
aries of the subfract i on i. One can see t hat if we set P/* = M~*
and Pi*-i = M~-*, t hen Eq. (4.91) is equivalent t o Eq. (4.48)
for est i mat ed mol ecul ar wei ght of SCN groups t hr ough t he
exponent i al model.
Exampl e 4. 10- - For t he C7+ fract i on of Exampl e 4.7, com-
posi t i on and mol ecul ar wei ght of SCN groups are given in
Table 4.11. Coefficients of Eq. (4.56) for the mol ar distribu-
tion of this syst em are given in Table 4.13 as Mo = 89.86,
A --- 0.3105, and B = 1. Calculate average mol ecul ar weights
of C~2-C13 gr oup and its mol e fraction. Compare calculated
values f r om t hose given in Table 4.11.
Sol ut i on- - I n Table 4.11 for Ct2 and C13 the mole fractions are
0.033 and 0.028, respectively. The molecular weights of these
component s are 163 and 176. Therefore the average molec-
ular weight of the C12-C13 group for this mi xt ure is M~ =
(0.033 163 + 0.028 x 176)/(0.033 + 0.028) = 169. The mole
fraction of these component s is 0.033 + 0.028 or 0.061. Group
of C12-C13 iS referred to as subfraction i with average molec-
ular weight of Mi,a~ and mole fraction of zi.
Equations (4.84) and (4.86) should be used to calculate zi
and Mi .... respectively. However, to use these equations, P/-1
and Pi represent the lower and upper molecular weights of
the subfraction. I n this case, the lower molecular weight is
M12 and the upper limit is M~3. These values are given in
Table 4.10 as 156 and 184, respectively. Pi*-~ = (156 - 89.9)/
89.9 = 0.7353 and P~* = 1047. Substituting in Eq. (4.84) we
get: z~ = 0.059.
For this system, A = 0.3501 and B = 1; therefore, from
Eq. (4.87), qi-1 = 2.37048 and qi = 3.37401, which gives [39]
F(1 + 1/B, q~_~) = 0.3149 and F(1 + 1/B, q~) = 0.1498. Substi-
tuting these values in Eq. (4.86) gives M~ = 0.8662 which
yields Mn = 167.7. Therefore the predicted values for zi and
Mi,av for group of Clz-C13 are 0.059 and 167.7, respectively,
versus actual values of 0.061 and 169. r
4. 5. 4,5 M odel Eval uat i ons
The distribution model expressed by Eq. (4.56) can be used for
M, Tb, SG, d, and refractive index par amet er I . The exponen-
tia] model expressed by Eq. (4.27) or Eq. (4.31) with a = 1 can
only be used for molecular weight of light otis and gas conden-
sate systems. The gamma distribution model can be applied
to both M and Tb, but for SG, the met hod of constant Watson
K is recommended by Whitson [20]. I n this met hod Kw for
the whole C7+ is calculated from its M7+ and SG7+ (Eq. 2.133)
and it is assumed to be constant for all components. For each
component, SG is calculated from Eq. (2.133) using the Kw
of the mixture and M for the component.
As ment i oned earlier the mai n advantage of generalized
model is its capability to predict distribution of properties
of heavy oils. This is demonst rat ed in Fig. 4.19, for molecu-
lar weight distribution of a heavy residue [25]. Experimental
data on the mol ar distribution are taken from Rodgers et al.
[41 ]. The experimentally determined mixture weight averaged
molecular weight is 630 [41 ]. For this sample, paramet ers Mo,
Am, and Bra for Eq. (4.56) in t erms of cumulative weight frac-
tion are calculated as 144, 71.64, and 2.5, respectively. For
this heavy oil sample both Mo and par amet er B are higher
t han their typical values for oil mixtures. Predicted mixture
molecular weight from Eqs. (4.72) and (4.81) is 632, which is
in good agreement with the experimental data. I n Fig. 4.19
prediction of mol ar distribution from the exponential and
gamma models are also illustrated. I t is obvious that the ex-
ponential model cannot be applied to heavy oils. The gamma
distribution model tends to predict higher values for M to-
ward heavier components.
Evaluation of these models for boiling point of a North Sea
black oil with MT+ and SG7+ of 177.5 and 0.8067 is shown
in Fig. 4.20. This is sample No. 8 in Ref. [25] in which the
experimental data on boiling points of 14 subfractions are
available. By applying Eq. (4.56), it was found that To = 346 K,
Ar = 0.5299, and Br = 1.3, which yields an average error of
15
12
9
o
" ~ 6
o Experimental
- - Generalized
. ~ . . . . Gam2:ntia 1
+ I I I I
500 1000 1500
Molecular Weight, M
FIG. 4.1 ~ Compari son of v ari ous distribution
model s for mol ecul ar wei g ht of a heav y petrol eum
mi x ture. Taken wi th permi ssi on from Ref. [ 25].
I~ Applying the gamma distribution model by Eq. (4.31)
gives Or = 349.9 K, ar = 1.6, and fir = 112.4 K. Use of these
coefficients in Eq. (4.31) for prediction of Tb distribution gives
average error of 1.6~ The exponential model (Eq. (4.56) with
B = 1) gives an average error of 4~ For this mixture with
intermediate mol ecul ar weight, the generalized and gamma
distribution models both are predicting boiling point with a
good accuracy. However, the exponential model is the least
accurate model for the boiling point distribution since Br in
Eq. (4.56) is greater t han unity.
Distribution of specific gravity for the C7+ fraction of a black
oil system from Ekofisk field of North Sea fields is shown in
Fig. 4.21. The generalized model, exponential model, and the
700
O
~0
O
600
500
400
o Experimental
- - Generalized Model
. . . . . . . Gamma Dist. Model /
300 ~ + l I t l J I
0 0.2 0.4 0.6 0.8
Cumulative Weight Fraction, Xcw
FIG. 4. 20- - Compar i son of v ari ous distribution model s for
prediction of boiling point of C7+ of a North Sea Black oil.
Taken with permi ssi on from Ref, [ 25].
4. CHARACTERI Z ATI ON OF RESERV OI R FLUI DS AND CRUDE OILS 179
1.2
1.1
1.0
0.9
0.8
0.7
0
o Experimental
Generalized Model
. . . . Constant Kw /
/
- - - Exponential / . , ~
I I I I I I I I I
0.2 0.4 0.6 0.8
Cumulative Volume Fraction, Xcv
FIG. 4.21--Comparison of three models for prediction of
specific gravity distribution of C7+ of an oil system.
const ant Kw met hod for est i mat i on of SG di st r i but i on are
compar ed i n this figure. For t he mi xt ure M7 and SG7 are
232.9 and 0.8534, respectively. For the general i zed model t he
coefficients are SGo = 0.666, AsG = 0.1453, and Bs6 -- 2.5528
whi ch yields an average error of 0.31% for predi ct i on of SG
di st ri but i on. I n the const ant Kw approach [15, 23], Kw is cal-
cul at ed from Eq. (2.133) usi ng M7+ and SG7+ as i np ut data:
Kw7+ = 11.923. I t is assumed t hat all comp onent s have t he
same Kw as t hat of the mi xt ure. Then for each comp onent
SGi is cal cul at ed from its Mi and Kw7+ for t he mi xt ure usi ng
the same equat i on. For this syst em predi ct ed SG di st ri bu-
t i on gives an average error of 1.7%. The exponent i al model is
t he same as Eq. (4.56) assumi ng B = 1, whi ch yields average
error of 3.3%. Fr om this figure it is clear t hat t he general i zed
model of Eq. (4.56) wi t h the fixed val ue of B ( ~3) generat es
t he best SG di st ri but i on. For very light gas condensat e sys-
t ems the Wat son K approach and general i zed model predi ct
nearl y si mi l ar SG di st ri but i on.
Fur t her eval uat i on of Eq. (4.56) and gamma di st r i but i on
model for 45 bl ack oil and 23 gas condensat e syst ems is re-
port ed i n Ref. [25]. Equat i on (4.56) can be used as ei t her a
two- or a t hr ee- par amet er model. Summar y of eval uat i ons is
given i n Tables 4.14 and 4.15. As ment i oned earlier Eq. (4.56)
is more or less equi val ent to t he gamma di st ri but i on model for
mol ar di st r i but i on of gas condensat e systems and light oils.
Mol ecul ar wei ght range for sampl es eval uat ed i n Table 4.14 is
from 120 to 290 and for this reason bot h model s give si mi l ar
errors for predi ct i on of M di st r i but i on (1.2%). However, two-
p ar amet er form of Eq. (4.56) is equi val ent to the exponent i al
model (B = 1) and gives hi gher error of 2.2%. For these sys-
tems, the exponent i al model does not give hi gh errors si nce
the systems are not qui t e heavy. For heavy oils exponent i al
model is not appl i cabl e for predi ct i on of mol ar di st ri but i on.
For Tb di st r i but i on bot h t hree-paramet er form of Eq. (4.56)
and t he gamma model are equi val ent wi t h error of about 0.6%
( ~6 K), while t he l at t er gives slightly hi gher error. The two-
par amet er general i zed model (B = 1.5 i n Eq. 4.56) gives an
average error of 0.7% for predi ct i on of Tb di st ri but i on. For
SG di st r i but i on t hr ough Eq. (4.56), t he best val ue of B is 3
and t here is no need for t hree-paramet er model. However,
the met hod is much more accurat e t han the const ant Kw
met hod whi ch gives an error more t han twice of the error
from t he general i zed model. Summar y of resul t s for predic-
t i on of M7 and SG7+ for the same systems of Table 4.14 is
shown i n Table 4.15. The gamma di st r i but i on model predi ct s
MT+ more accurat e t han Eq. (4.66) mai nl y because most of
the systems st udi ed are light oil or gas condensat e. It is not
possi bl e to eval uat e predi ct ed Tb of the whol e mi xt ure si nce
the experi ment al dat a were not available. Refractive i ndex can
be accurat el y predi ct ed by Eq. (4.56) wi t h B -- 3 as shown i n
Tables 4.14 and 4.15. Results shown for eval uat i on of refrac-
tive i n Table 4.14 are based on about 160 dat a poi nt s for 13
oil samples. Average error for cal cul at i on of refractive i ndi ces
of 13 oils is 0.2% as shown i n Table 4.15. Fur t her eval uat i on
of appl i cat i on of the general i zed model is demonst r at ed i n
Exampl e 4.11.
The general i zed di st r i but i on model expressed by Eqs. (4.56)
and (4.66) can be appl i ed to ot her physical propert i es and to
pet r ol eum fract i ons ot her t han C7+ fractions. I n general t hey
are appl i cabl e to any wide boi l i ng range and hydr ocar bon-
pl us fraction. The following exampl e demonst r at es how this
T A B L E 4 . 1 4 - - Ev a l u a t i o n of various distribution models for estimation of properties
of C7+ fractions for 68 mixtures, a
Generalized model, Eq. (4.56)
Two-parameter model Three-parameter model
Gamma distribution
model, Eq. (4.31)
Property AAD b %AAD c AAD %AAD AAD %AAD
M 4.09 d 2.2 d 2.28 1.2 2.31 1.2
T b, ~ 7.26/1.8 0.7 5.78/1.8 0.56 6.53/1.8 0.63
SG 0.005 0.6 0.005 0.6 0.01 e 1.24 e
n20 0.0025 0.18 f 0.0025 0.18 g g
aMost of samples are from North Sea reservoirs with M ranging from 120 to 290. Systems include 43
black oil and 23 gas condensate systems with total of 941 data points. Full list of systems and r e f e r e nc e
for data are given in Ref. [24].
bAAD = absolute average deviation = (1/N)E ]estimated property - experimental property].
C%AAD = Percent absolute average deviation= (1/N)Z[](estimated property- experimental property)/
experimental property I x 100].
dSame as exponential model for molar distribution.
~Method of constant Watson K.
fRefractive index was evaluated for 13 oils [42] and total of 161 data points.
gThe gamma model is not applicable to refractive index.
TABLE 4.15--Evaluation of various distribution models for estimation of mixture
average properties of C7+ fractions for 68 mixtures of Table 4.14.a
Generalized model, Eq. (4.56) Gamma distribution
Two-parameter model Three-parameter model model, Eq. (4.31)
Property AAD b %AAD b AAD %AAD AAD %AAD
M 12.58 6.8 11.9 6.4 5.4 2.9
SG 0.003 0.35 0.003 0.35 c c
n20 0.003 d 0.2 0.003 0.2 c c
aM7+ range: 120-290. SG7+ range: 0.76-0.905.
bDeflned in Table 4.14.
CThe gamma model cannot be applied to SG or n20.
aFor 13 oil samples.
TABLE 4.16---Prediction of distribution of refractive index of a C6+ fraction from Eq. (4.56).
Nc Vol% FiE0 Xcv I I, pred. n2o, pred. %AD
6 2.50 1.3866 0.013 0.2352 0.2357 1.3875 0.06
7 5.47 1.4102 0.056 0.2479 0.2467 1.4080 0.16
8 4.53 1.4191 0.109 0.2526 0.2542 1.4222 0,22
9 5.06 1.4327 0.161 0.2597 0.2596 1.4324 0.02
10 2.55 1.4407 0.201 0.2639 0.2632 1.4393 0.09
11 3.62 1.4389 0.234 0.2630 0.2659 t.4445 0.39
I2 3.70 1.4472 0.274 0.2673 0.2688 1.4502 0.20
13 4.19 1.4556 0.316 0.2716 0.2718 1.4560 0.03
14 3.73 1,4615 0.358 0.2747 0.2747 1.4615 0.00
15 3.96 1.4694 0.399 0.2787 0.2774 1.4668 0.18
16 3.03 1.4737 0.437 0.2809 0.2798 1.4715 0.15
17 3.40 1.4745 0.471 0.2813 0.2819 1.4758 0.09
18 3.13 1.4755 0.506 0.2818 0.2842 1.4802 0.31
19 2.94 1.4808 0.538 0.2845 0.2862 1.4842 0.23
20+ 41.70 1.5224 0.777 0.3052 0.3031 1.5182 0.28
Mixture 93.51 0.14
tExperimental data on n20 are taken from Berge [42].
met hod can be used to predi ct di st r i but i on of refractive i ndex
of a C6+ fraction.
Exampl e 4. 11- - For a C6+ of an oil sampl e experi ment al dat a
on refractive i ndex at 20~ are given versus vol% of SCN
groups from C6 to C20+ i n Table 4.16. Refractive i ndex of the
whol e fract i on is 1.483. Use Eq. (4.56) to predi ct refractive
i ndex di st r i but i on and obt ai n the AAD% for the model pre-
diction. Also graphi cal l y compar e the model predi ct i on wi t h
the experi ment al dat a and calculate the mi xt ure refractive
index.
Sol ut i onwSi mi l ar to Exampl e 4.7, vol% shoul d be first con-
verted to nor mal i zed vol ume fract i ons and t hen to cumul at i ve
vol ume fract i on (xcv). For refractive i ndex the charact eri za-
t i on p ar amet er is/20 i nst ead of n20. Therefore, i n Eq. (4.56) we
use p ar amet er I ( defined by Eq. 2.36) for propert y P. Values
of I versus Xcv are also given i n Table 4.16. Upon regressi on of
dat a t hr ough Eq. (4.58), we get: Io = 0.218, A/ = 0.1189, and
BI = 3.0. For these coefficients the RMS is 0.001 and %AAD
is 0.14%. Value of Io for this sampl e is close to t he l ower I
val ue of C6 group and p ar amet er B is same as t hat of spe-
cific gravity. A graphi cal eval uat i on of predi ct ed di st r i but i on
is shown i n Fig. 4.22. Si nce B = 3, Eq. (4.76) shoul d be used
to cal cul at e I* v and t hen from Eq. (4.81) Iav is cal cul at ed as
Iav = 0.2844. Fr om Eq. (2.114), t he mi xt ure refractive i ndex
is cal cul at ed as nav = 1.48 i, whi ch differs from experi ment al
val ue of 1.483 by- 0. 15%.
Fur t her eval uat i on of Eq. (4.56) for predi ct i on of di st ri bu-
t i on of refractive i ndex shows t hat refractive i ndex can be
predi ct ed wi t h B -- 3 wi t h an accuracy of 0.2% as shown i n
Table 4.14. As di scussed i n Chapt er 2, p ar amet er I is a size pa-
r amet er si mi l ar to densi t y or specific gravity and t herefore t he
average for a mi xt ure shoul d be cal cul at ed t hr ough Eq. (4.76)
or (4.80). #
1.6
~ 1.5
1.4
o Experimental
- - Predicted
O
1 . 3 l I I I l I I I
0 0.2 0.4 0.6 0.8
Cumulative Volume Fraction, xcv
FIG. 4. 22- - Predi cUon of di stri buti on of refractive i ndex of
C6+ of a North Sea oil from Eq. ( 4.56) .
4. 5. 4. 6 Prediction of Property Distributions
Using Bulk Properties
As discussed above, Eq. (4.56) can be used as a two-parameter
relation with fixed values of B for each property (BM = 1,
Br = 1.5, and Bsa = Bz = 3). I n this case Eq. (4.56) is referred
as a two-parameter distribution model. I n such cases only
parameters Po and A must be known for each property to ex-
press its distribution in a hydrocarbon plus fraction. The two-
parameter model is sufficient to express property distribu-
tion of light oils and gas condensate systems. For very heavy
oils two-parameter model can be used as the initial guess to
begin calculations for determination of the three parameters
in Eq. (4.56). I n some cases detailed composition of a C7+
fraction in a reservoir fluid is not available and the only infor-
mation known are MT+ and SG7+, while in some other cases
in addition to these properties, a third parameter such as re-
fractive index of the mixture or the true boiling point (TBP)
curve are also known. For these two scenarios we show how
parameters Po and A can be determined for M, Tb, SG, and
/20.
Met hod A: MT+, SGT+, and nT+ are known--Three bulk
properties are the mi ni mum data that are required to predict
complete distribution of various properties [24, 43]. In ad-
dition to/147+ and SGT+, refractive index, n7+, can be easily
measured and they are known for some 48 C7+ fractions [24].
As shown by Eqs. (4.74)-(4.76), if P*v is known, parameter A
can be determined for each property. For example, if Mo is
known, Ma* v can be determined from definition of M* as:
May- Mo
(4.93) M~*v- /14o
where Mav is the mixture molecular weight of the C7+ fraction,
which is known from experimental measurement. Similarly,
SG* v and I~*~ can be determined from $7+ and n7+ (or I7+).
Parameters AM, AsG, and AI are then calculated from Eqs.
(4.72) and (4.81). For fixed values of B, Eqs. (4.74)-(4.76)
and (4.79) and (4.80) may be used. One should realize that
Eq. (4.74) was developed based on cumulative mole fraction,
while Eqs. (4.79) and (4.80) are based on cumulative weight
fraction. Once distribution of M and SG are known, distribu-
tion of Tb can be determined using equations given in Chapter
2, such as Eqs. (2.56) or (2.57), for estimation of Tb from M
and SG. Based on data for 48 C7+ samples, the following re-
lation has been developed to estimate Io from Mo and SGo
[23]:
/o ~ 0.7454exp(-0.01151Mo - 2.37842SGo
(4.94) + 0.01225 Mo SGo)M~ SG~ "53147
This equation can reproduce values of Io with an average
deviation of 0.3%. Furthermore, methods of estimation of
parameter I from either Tb and SG or M and SG are given in
Section 2.6.2 by Eqs. (2.115)-(2.117). Equation (2.117) may
be applied to the molecular weight range of 70-700. However,
a more accurate relation for prediction of parameter I from
M and SG is Eq. (2.40) with coefficients from Table 2.5 as
follows:
I = 0.12399 exp(3.4622 x 10-* M + 0.90389SG
(4.95) -6. 0955 10-4MSG)M~176 0'22423
This equation can be used for narrow boiling range fractions
with M between 70 and 350. I n this molecular weight range
this equation is slightly more accurate than Eq. (2.117). Once
distribution of I is determined from these equations, if the
initial values of Mn and SGo are correct then 1" v calculated
from the distribution coefficients and Eq. (4.72) should be
close to the experimental value obtained from n7+. For cases
that experimental data on n7+ is not available it can be esti-
mated from M7+ and SG7+ using Eq. (4.95) or (2.117). Equa-
tion (2.117) estimates values of nT+ for 48 systems [23] with
an average error of 0.4%. Steps to predict M, Tb, SG, and I
(or n) distributions can be summarized as follows [23]:
1. Read values of M7+, SG7+, and 17+ for a given C7+ sample.
I f 17+ is not available Eq. (2.117) may be used to estimate
this parameter.
2. Guess an initial value for Mo (assume Mo = 72) and cal-
culate M* v from Eq. (4.93).
3. Calculate AM from Eq. (4.72) or Eq. (4.74) when B = 1.
4. Choose 20 (or more) arbitrary cuts for the mixture with
equal mole fractions (Xmi) of 0.05 (or less). Then calculate
Mi for each cut from Eq. (4.56).
5. Convert mole fractions (x~/) to weight fractions (x~)
through Eq. (1.15) using Mi from step 4.
6. Guess an initial value for So (assume So = 0.59 as a starting
value).
7. Calculate 1/J from Eq. (4.80) using SGo and SG7+. Then
calculate As6 from Eq. (4.79) using Newton's method.
8. Using Eq. (4.56) with AsG and So from steps 6 and 7 and
B = 3, SG distribution in terms of Xcw is determined and
for each cut SGi is calculated.
9. Convert Xwi to Xvi using Eq. (1.16) and SG/ values from
step 8.
i0. For each cut calculate Tbi from M/ and SGi through
Eq. (2.56) or (2.57).
11. For each cut calculate // from Mi and SGi through
Eq. (2.95).
12. From distribution of I versus Xcv find parameters Io, At
and BI through Eqs. (4.56)-(4.57). Then calculate Iav from
Eq. (4.72) and (4.81).
13. Calculate el = [(Iav, calc.- I7+)/I7+[.
14. I f el < 0.005, continue from step 15, otherwise go back
to step 6 with SGo,new = SGo,old + 0.005 and repeat steps
7-13.
15. Calculate Io from Eq. (4.94).
16. Calculate 82 = [(lo,step15 -- Io,step12)/Io.step15[ 9
17. GO back to step 2 with a new guess for Mo (higher than the
previous guess). Repeat steps 2-16 until either e2 < 0.005
or 82 becomes minimum.
18. For heavy oils large value of 82 may be obtained, because
value of BM is greater than 1. For such cases values of
BM = 1.5, 2.0, and 2.5 should be tried successively and
calculations from step 2 to 17 should be repeated to min-
imize 82.
19. Using data for Tb versus Xcw, determine parameters To, At,
and Br from Eqs. (4.56) and (4.57).
20. Print Mo, AM, BM, SGo, AsG, To, At, Br, Io, AI, and Bz.
21. Generate distributions for M, Tb, SG, and n20 from
Eq. (4.56).
TABLE 4.17--Sample calculations for prediction of distribution of properties of the C7+ fraction
in Example 4.12.
Xm Xcm M Xw Xcw SG x~ Xc~ Tb, K 1
No. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
1 0.05 0.025 90.7 0.038 0.019 0.719 0.002 0.001 353.9 0.242
2 0.05 0.075 92.3 0.039 0.058 0.727 0.007 0.006 357.9 0.245
3 0.05 0.125 93.9 0.040 0.097 0.732 0.012 0.015 361.6 0.246
I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The fol l owi ng examp l e shows ap p l i cat i on of t hi s met hod to
fi nd p r op er t y di st r i but i on of a C7+ when onl y mi ni mum dat a
of MT+ and SG7+ ar e avai l abl e.
Exampl e 4. 12- - For t he C7+ of Examp l e 4.7, M, Tb, and SG
di st r i but i ons ar e gi ven i n Table 4.11. Di st r i but i on model co-
efficients ar e gi ven in Table 4.13. For t hi s gas condensat e
syst em, assume t he onl y dat a avai l abl e ar e M7+ = 118.9 and
SG7+ = 0.7597. Usi ng t he met hod descr i bed above gener at e
M, Tb, SG, and I di st r i but i ons.
Sol ut i on- - Si nce rt7+ is not avai l abl e we cal cul at e 17+ f r om
Eq. (4.95) usi ng M7+ and SG7+ as /7+ = 0.2546 ( equi val ent
t o nT+ = 1.4229). St ep- by- st ep cal cul at i ons ar e fol l owed and
resul t s of first few p oi nt s as samp l e cal cul at i ons are gi ven i n
Table 4.17 wher e cal cul at i ons ar e cont i nued up to i = 20.
1. M7+ = 118.9, SG7+ = 0.7597, I7+ = 0.2546.
2. F or t he i ni t i al guess of Mo t he mi ni mum val ue of 72 can be
used for comp ut er p r ogr ams. However, t he act ual val ue of
Mo is very cl ose t o M7, whi ch is 88. F or t hi s gas condensat e
syst em we assume Mo = 90. I f t he cal cul at ed er r or is hi gh
t hen st ar t f r om 72. By Eq. (4.93), M~* v = 0.3211.
3. Assumi ng BM = I, f r om Eq. (4.72) AM = M~* = 0.3211.
4. The C7+ fract i on is di vi ded i nt o 20 cut s wi t h equal mol e
fract i ons: Xmi = 0.05 ( col umn 1 i n Table 4.17). Now Xcm is
cal cul at ed f r om x~ as gi ven i n col umn 2. Mi for each cut is
est i mat ed t hr ough Eq. (4.56) wi t h Mo = 90, AM = 0.3211,
and BM = 1, and val ue of x~. Cal cul at ed val ues of M/ a r e
gi ven i n col umn 3.
5. Wei ght f r act i ons (xwi) ar e cal cul at ed usi ng Xmi and Mi
t hr ough Eq. ( I .15) and are gi ven i n col umn 4.
6. The l owest val ue of SGo sui t abl e for comp ut er cal cul at i ons
is 0.59; however, it is usual l y cl ose t o val ue of t he l ower
l i mi t of SG for C7 ( SG7) , whi ch is 0.709. Her e i t is as s umed
SGo = 0.7.
7. Wi t h SG7+ = 0.7579 and SGo = 0.7, f r om Eq. (4.80) we
get 1/J = 1.0853. Usi ng Eq. (4.79) AsG is cal cul at ed f r om
1/J as Asc = 0.0029 ( the second equat i on is used si nce
As~ < 0.05).
8. Cumul at i ve Xcw is cal cul at ed f r om Xwi and ar e gi ven i n
col umn 5. Usi ng SGo, AsG, and BsG = 3, SG di st r i but i on
is cal cul at ed t hr ough use of Xcw and Eq. (4.56). Values of
SGi ar e gi ven in col umn 6.
9. Vol ume f r act i ons (x,~) ar e cal cul at ed f r om Xw/ and SGi
usi ng Eq. (1.16) and ar e gi ven i n col umn 7. Cumul at i ve
vol ume f r act i on is given i n col umn 8.
10. F or each cut, T~ is cal cul at ed f r om Eq. (2.56) usi ng Mi
and SGi and is gi ven i n col umn 9.
11. For each c ut , / / i s cal cul at ed f r om Eq. (4.95) usi ng Mi and
SGi and is gi ven in col umn 10.
12. F r om col umns 9 and 10, di st r i but i on coeffi ci ent s of
Eq. (4.56) for I ar e cal cul at ed as lo = 0.236, AI = 5.3 x
10 -5, and B = 4.94 ( R 2 = 0.99 and %AAD = 0.16%).
13. F r om Eqs. (4.72) and (4.81), lay = 0.2574 whi ch gives
el = 0.01.
14. el i n st ep 13 is gr eat er t han 0.005; however, a change i n
SGo causes a sl i ght change in t he er r or p ar amet er so t hi s
val ue of el is accept abl e.
15. / o is cal cul at ed f r om Eq. (4.94) usi ng Mo and SGo as
0.2364.
16. ez is cal cul at ed f r om Io i n st eps 15 and 12 as 0.0018, whi ch
is less t han 0.005.
17. Go to st ep 18 si nce e2 < 0.005.
18. Si nce val ues of el and e2 ar e accep t abl e t he as s umed val ue
of BM = i is OK.
19. F r om col umns 5 and 9, di st r i but i on coeffi ci ent s for Tb
are cal cul at ed as To = 350 K, Ar = 0.161, and Br = 1.3
( R 2 = 0.998 and %AAD = 0.3).
20. Fi nal p r edi ct ed di st r i but i on coeffi ci ent s for M, Tb, SG, and
I ar e gi ven i n Table 4.18.
21. Pr edi ct ed di st r i but i ons for M, Tb, SG, and I ar e shown in
Figs. 4.20--4.23, respectively. r
Me t h od B: MT+, SG7+, a nd TBP ar e known- - I n some
cases t rue boi l i ng p oi nt (TBP) di st i l l at i on curve for a cr ude
or C7+ f r act i on is known t hr ough si mul at ed di st i l l at i on or
ot her met hods descr i bed in Sect i on 4.1.1. Gener al l y TBP is
avai l abl e in t er ms of boi l i ng p oi nt versus vol ume or wei ght
fract i on. I f in addi t i on to TBP, t wo bul k p r op er t i es such as
M7+ and SG7+ or MT+ and nT+ ar e known, t hen a bet t er pre-
di ct i on of comp l et e di st r i but i on of var i ous p r op er t i es is possi -
bl e by ap p l yi ng t he gener al i zed di st r i but i on model . F or t hese
cases an i ni t i al guess on SGo gives comp l et e di st r i but i on of SG
t hr ough Eq. (4.56) al ong t he Tb di st r i but i on, whi ch is avail-
abl e f r om dat a. Havi ng Tb and SG for each cut, Eq. (2.51) can
be used to p r edi ct M for each subf r act i on. Usi ng Eq. (2.115)
or (2.116), 12o can be est i mat ed for cut s wi t h M val ues up to
350. F or heavi er cut s Eq. (2.117) may be used. The p r ocedur e
can be s ummar i zed as fol l ows [24]:
1. Read val ues of M7+, SG7+, and t he TBP di st r i but i on (i.e.,
SD curve) for a gi ven cr ude oil sampl e.
2. F r om TBP det er mi ne di st r i but i on coeffi ci ent s i n Eq. (4.56)
for Tb in t er ms of Xcw or Xcv. I f si mul at ed di st i l l at i on is
avai l abl e, Xcw shoul d be used.
TABLE 4.18---Estimated coefficients of Eq. (4.56) for the C7+
of Example 4.12.
Property Po A B Type of xc
M 90 0.3211 1.0 Xcm
T b 350 0.1610 1.3 Xcw
SG 0.7 0.0029 3.0 Xcw
I 0.236 5.4 x 10 -5 4.94 Xcv
TABLE 4.19--Sample calculations for prediction of distribution of properties of the C7+
fraction in Example 4.13.
Xw/ Xcw Tbi, K SGi Xvl x ~ M i Xmi li
No. (1) (2) (3) (4) (5) (6) (7) (8) (9)
1 0.05 0.025 353.8 0.720 0.053 0.026 91.4 0.065 0.244
2 0.05 0.075 359.3 0.730 0.052 0.079 93.9 0.063 0.247
3 0.05 0.125 364.3 0.735 0.052 0.131 96.3 0.062 0.249
i . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Choose 20 ( or mor e) ar bi t r ar y cut s for t he mi xt ur e wi t h
equal wei ght ( or vol ume) f r act i ons of 0.05 ( or less) . Then
det er mi ne Tbi for each cut f r om Eq. (4.56) and coeffi ci ent s
f r om st ep 2.
4. Guess an i ni t i al val ue for SGo ( l owest val ue is 0.59).
5. Cal cul at e 1/J f r om Eq. (4.80) usi ng SGo and SG7+. Then
cal cul at e AsG f r om Eq. (4.79) usi ng Newt on' s met hod or
ot her ap p r op r i at e p r ocedur es. I f ori gi nal TBP is i n t er ms
of X~v, t hen Eq. (4.76) shoul d be used t o det er mi ne ASG in
t er ms of x~.
6. I f ori gi nal TBP is in t er ms of xov, find SG di st r i but i on f r om
Eq. (4.56) in t er ms ofx~v. Then use SG to convert xv t o xw
t hr ough Eq. (1.16).
7. Usi ng val ues of SG and Tb for each cut det er mi ne val ues
of M f r om Eq. (2.56).
8. Use val ues of M f r om st ep 7 t o convert Xw i nt o Xr, t hr ough
Eq. (1.15).
9. F r om dat a cal cul at ed in st ep 8, find mol ar di st r i but i on by
est i mat i ng coeffi ci ent s Mo, AM, and BM i n Eq. (2.56).
10. Cal cul at e val ue of I for each cut f r om Tb and SG t hr ough
Eq. (2.115) or (2.116).
11. Fi nd coeffi ci ent s Io, AI, and BI ( set Bt = 3) f r om dat a ob-
t ai ned in st ep 10. Set / 1 = Io.
12. F r om SGo as s umed in st ep 4 and Mo det er mi ned in st ep 9,
est i mat e Io t hr ough Eq. (4.94). Set 12 = Io.
13. Cal cul at e el = 1(12 - 11)/111.
14. I f el _> 0.005 go back to st ep 4 by guessi ng a new val ue for
SGo. I f el < 0.005 or it is mi ni mum go to st ep 15.
15. Pri nt Mo, AM, BM, SGo, As6, To, At , By, Io, At, and Bz.
16. Gener at e di st r i but i ons for M, Tb, SG, and n20 f r om
Eq. (4.56).
The fol l owi ng examp l e shows ap p l i cat i on of Met hod B to find
p r op er t y di st r i but i on of a C7+ when dat a on TBP di st i l l at i on,
MT+ and SG7+ ar e avai l abl e.
Exampl e 4. 13- - For t he C7+ of Examp l e 4.7, assume t hat
Tb di st i l l at i on curve is avai l abl e, as given i n col umns 3 and
5 i n Table 4.11. I n addi t i on assume t hat for t hi s samp l e
M7+ = 118.9 and SG7+ -- 0.7597 ar e al so avai l abl e. Usi ng t he
met hod descr i bed above ( Met hod B) gener at e M, Tb, SG, and
I di st r i but i ons. Gr ap hi cal l y comp ar e p r edi ct i on of var i ous
di st r i but i ons by Met hods A and B wi t h act ual dat a gi ven i n
Table 4.11.
Sol ut i on- - Si mi l ar to Examp l e 4.12, st ep-by-st ep p r ocedur e
descr i bed under Met hod B shoul d be fol l owed. Si nce dat a on
di st i l l at i on ar e gi ven i n t er ms of wei ght fract i ons ( col umn 3 in
Table 4.11) we choose wei ght f r act i on as t he r ef er ence for t he
comp osi t i on. F r om dat a on Tbi versus Xwi di st r i but i on coeffi-
ci ent s i n Eq. (4.56) can be det er mi ned. Thi s was al r eady done
i n Examp l e 4.7 and t he coeffi ci ent s ar e gi ven i n Table 4.13 as:
To = 350 K, Ar = 0.1679, and Br = 1.2586. An i ni t i al guess
val ue of SGo = 0.7 is used to cal cul at e ASG and SG di st r i bu-
t i on coefficients. Now we di vi de t he whol e f r act i on i nt o 20
cut s wi t h equal wei ght fract i ons as x~ = 0.05. Si mi l ar t o cal-
cul at i ons shown i n Table 4.17, Xcw is cal cul at ed and t hen for
each cut val ues of Tbi and SGi are cal cul at ed. F r om t hese t wo
p ar amet er s Mi and Ii ar e cal cul at ed by Eqs. (2.56) and (2.115),
respectively. F r om Xwi and Mi mol e fract i ons ( Xmi ) a r e cal cu-
l at ed. Samp l e cal cul at i on for t he first few p oi nt s is gi ven in
Table 4.19 wher e cal cul at i on cont i nues up t o i = 20. The co-
efficients of Eq. (4.56) det er mi ned f r om dat a in Tabl e 4.19
ar e gi ven i n Table 4.20. I n this met hod p ar amet er el = 0.0018
( st ep 13), whi ch is less t han 0.005 and t her e is no need to
re-guess SGo. I n t hi s set of cal cul at i ons si nce i ni t i al guess for
SGo is t he same as t he act ual val ue onl y one r ound of cal-
cul at i ons was needed. Coefficients gi ven i n Table 4.20 have
been used to gener at e di st r i but i on for var i ous p r op er t i es and
t hey ar e comp ar ed wi t h p r edi ct ed val ues f r om Met hod A as
well as act ual val ues gi ven in Table 4.11. Resul t s are shown i n
Figs. 4.23 and 4.24 for p r edi ct i on of M and Tb di st r i but i ons.
Met hods A and B p r edi ct si mi l ar di st r i but i on curves mai nl y
TABLE 4.20---Estimated coefficients of Eq. (4.56)
for the C7+ of Example 4.13.
Property Po A B Type of Xc
M 90 0.3324 1.096 Xcm
Tb 350 0.1679 1.2586 Xcw
SG 0.7 0.0029 3.0 Xcw
I 0.236 7.4 x 10 -4 3.6035 Xcv
300
~0
.,..4
0)
(9
o
250
200
150
100
o Experimental
Method A
. . . . . Method B
i
f
J
o--------"
50 i i i i i i I i i
0 0.2 0.4 0.6 0.8
Cumul at i ve Mol e Fraction, Xcm
FIG. 4. 23 - - Predi ct i on of molar di stri buti on in Ex ampl es
4.1 2 and 4.13.
600
O
cx0
550
500
450
400
350
o Experimental
_ . - - . MethodA J
I I I I l I I I I
0.2 0.4 0.6 0.8
Cumulative Mole Fraction, Xcw
FIG. 4.24--Prediction of boiling point distribution in Exam-
ples 4.12 and 4.13.
because the system is gas condensate and value of BM is one.
For very heavy oils Method B predicts better prediction. As
shown in Fig. 4.24 for Tb, Method B gives better prediction
mainly because information on at least one type of distribu-
tion was available. #
Method C: M7+ and SG7+ are known- - An alternative to
method A when only/I//7+ and SG7+ are known is to predict M
distribution by assuming BM = 1 and a value for Mo as steps
1-5 in method A. For every value of M, SG is estimated from
Eq. (4.7) using coefficients given in Table 4.5 for SG. Then
parameters SGo, AsG, and BsG are calculated. From these co-
efficients SGav is estimated and compared with experimental
value of SG7+. The initial guessed values for Mo and BM are ad-
justed until error parameter for calculated SGav is minimized.
I n this approach, refractive index is not needed.
Method D: Distribution of only one property (i. e. , Na,
Mi, Tbi, SGi, or Ni) is known- - I n this case distribution of
only one parameter is known from experimental data. As an
example in Table 4.2, distribution of only Mi for the waxy
crude oil is originally known versus weight or mole fraction.
I n this case from values of Mi, boiling point and specific grav-
ity are calculated through Eq. (4.7) and coefficients given in
Table 4.5 for Tb and SG. Once distributions of Tb, SG, and
M are known the distribution coefficients can be determined.
Similarly if instead of Mi, another distribution such as Tbi,
SGi, or / / i s known, Eq. (4.7) can be used in its reversed form
to determine M/ distribution as well as other properties.
Method E: Only one bulk property (MT+, TbT+, SG7+,
or nT+) is known- - One bulk property is the mi ni mum in-
formation that can be known for a mixture. I n this case if
M7 is known, parameter Mo is fixed at 90 and B~ = 1. Para-
meter Ara is calculated from Eq. (4.74). Once distribution of
M is found, SG distribution can be estimated through use of
Eq. (4.7) and coefficients in Table 4.5 for SG. Similarly if only
SG7+ is known, assume SGo = 0.7 and BsG = 3. Coefficient
AsG is calculated from Eq. (4.76) and then distribution of SG
versus x~, can be obtained through Eq. (4.56). Once SG dis-
tribution is known, the reversed form of Eq. (4.7) should be
used to estimate M distribution. I n a similar approach if n7+
is known distributions of M, Tb, and SG can be determined by
assuming Io = 0.22 and BI = 3 and use of Eq. (4.7). Obviously
this method gives the least accurate distribution since min-
i mum information is used to obtain the distributions. How-
ever, this method surprisingly well predicts boiling point dis-
tribution from specific gravity (as the only information avail-
able) for some crude oils as shown by Riazi et al. [40].
4 . 6 P S EUD OI Z A T I ON A N D
L UM P I N G A P P R OA C H ES
Generally analytical data for reservoir fluids and crude oils are
available from C1 to C5 as pure components, group C6, and all
remaining and heavier components are grouped as a C7+ frac-
tion as shown in Tables 1.2 and 4. I. As discussed earlier for
wide C7+ and other petroleum fractions assumption of a sin-
gle pseudocomponent leads to significant errors in the char-
acterization scheme. I n such cases, distribution functions for
various characterization parameters are determined through
Methods A or B discussed in Section 4.5.4.6. Once the molar
distribution is known through an equation such as Eq. (4.56),
the mixture (i.e., C7+) can be split into a number of pseudo-
components with known xi, Mi, Tbi, and SGi. This technique is
called pseudoization or splitting and is widely used to charac-
terize hydrocarbon plus fractions, reservoir fluids, and wide
boiling range petroleum fractions [ 15, 17, 18, 23, 24, 26, 36].
I n some other cases detailed analytical data on the compo-
sition of a reservoir fluid are available for SCN groups such
as those shown in Table 4.2. Properties of these SCN groups
are determined from methods discussed in Section 4.3. How-
ever, when the numbers of SCN components are large (i.e.,
see Table 4.2) computational methods specially those related
to phase equilibrium would be lengthy and cumbersome. I n
such cases it is necessary to lump some of these components
into single groups in order to reduce the number of com-
ponents in such a way that calculations can be performed
smoothly and efficiently. This technique is called lumping or
grouping [24, 26]. I n both approaches the mixture is expressed
by a number of pseudocomponents with known mole frac-
tions and characterization parameters which effectively de-
scribe characteristics of the mixture. These two schemes are
discussed in this section in conjunction with the generalized
distribution model expressed by Eqs. (4.56) and (4.66).
4. 6. 1 S pl i t t i ng S c h e me
Generally a C7+ fraction is split into 3, 5, or 7 pseudocompo-
nents. For light oils and gas condensate systems C7+ is split
into 3 components and for black oils it is split into 5 or 7 com-
ponents. For very heavy oils the C7+ may be split to even 10
components. But splitting into 3 for gas condensate and 5 for
oils is very common. When the number of pseudocomponents
reaches oo, behavior of defined mixture will be the same as
continuous mixture expressed by a distribution model such
as Eq. (4.56). Two methods are presented here to generate
the pseudocomponents. The first approach is based on the
application of Gaussian quadrature technique as discussed
by Stroud and Secrest [44]. The second method is based on
carbon number range approach in which for each pseudo-
component the lower and higher carbon numbers are speci-
fied.
4. 6. 1. 1 The Gaussi an Quadrat ure Approach
The Gaussian quadrature approach is used to provide a dis-
crete representation of continuous functions using different
numbers of quadrature points and has been applied to define
pseudocomponents in a petroleum mixture [23, 24, 28]. The
number of pseudocomponents is the same as the number of
quadrature points. Integration of a continuous function such
as F( P) can be approximated by a numerical integration as
in the following form [44]:
Np
(4.96) f f ( y) exp( -y) dy = ~wi f(Yi) = 1
i=1
0
where Np is the number of quadrature points, wi are weighting
factors, yi are the quadrature points, and f ( y) is a continuous
function. Sets of values of Yi and wi are given in various mathe-
matical handbooks [38]. Equation (4.96) can be applied to a
probability density function such as Eq. (4.66) used to express
molar distribution of a hydrocarbon plus fraction. The left
side of Eq. (4.96) should be set equal to Eq. (4.67). I n this
application we should find f ( y) in a way that
(4.97) [ F(P*)dP* = [ f ( y) exp( -y) dy = 1
0 0
where F(P*) is given by Eq. (4.66). Assuming
(4.98) y = BP *B
and integrating both sides
B 2
(4.99) dy = - xP* B- l dp *
Using Eq. (4.66) we have
F(P*)dp* = B 2 p* B - l e x p( - B P* n)
(4.100) = 1 x exp( -y) dy
By comparing Eqs. (4.97) and (4.100) one can see that
(4.101) f ( y) = 1
and from Eq. (4.96) we get
(4.102) zi = wi
where zi is the mole fraction of pseudocornponent i. Equa-
tion (4.102) indicates that mole fraction of component i is
the same as the value of quadrature point wi. Substituting
definition of P* as (P - Po)/Po in Eq. (4.98) gives the follow-
ing relation for property Pi:
I(at ]
(4.103) Pi = Po 1 + y]l~
Coefficients Po, A and B for a specific property are known
from the methods discussed in Section 4.5.4.6. Table 4.21
TABLE 4. 21--Gaussian quadrature points
and weights for 3 and 5 points [38].
Root yi Wei ght wi
Np=3
1 0. 41577 7.11093 x 10 -1
2 2. 29428 2. 78518 10 -1
3 6.28995 1.03893 x 10 -2
Np=5
1 0. 26356 5. 21756 x 10 -1
2 1.41340 3. 98667 x 10 -1
3 3. 59643 7. 59424 x 10 -2
4 7.08581 3.61176 x 10 -3
5 12.64080 2. 33700 x 10 -5
gives a set of values for roots Yi and weights wi as given in
Ref. [37].
Similarly it can be shown that for the gamma distribution
model, Eq. (4.31), f( y) in Eq. (4.96) becomes
y~- I
(4.104) f ( Y) - F(a)
and mole fraction of each pseudocomponent, zi, is calculated
a s
y~,-i
(4.105) zi = wi f(Yi) = wi - -
r(~)
Molecular weight Mi for each pseudocomponent is calculated
from
(4.106) Mi = y~r + 0
where u, fl, and ~ are parameters defined in Eq. (4.31). It
should be noted that values of zi in Eq. (4.102) or (4.105)
is based on normalized composition for the C7+ fraction
(i.e., z7+ = 1) at which the sum of zi for all the defined pseudo-
components is equal to unity. For both cases in Eqs. (4.102)
and (4.106) we have
Np
(4.107) Zz ~ = 1
i=1
To find mole fraction of pseudocomponent i in the original
reservoir fluid these mole fractions should be multiplied by
the mole fraction of C7+. Application of this method is demon-
strated in Example 4.14.
Exampl e 4. 14--For the gas condensate system described in
Example 4.13 assume the information available on the C7+
are M7+ -- 118.9 and SG7+ --- 0.7597. Based on these data, find
three pseudocomponents by applying the Gaussian quadra-
ture method to PDF expressed by Eq. (4.66). Find the mixture
M7+ based on the defined pseudocomponents and compare
with the experimental value. Also determine three pseudo-
components by application of Gaussian quadrature method
to the gamma distribution model.
Sol ut i on- - For Eq. (4.56), the coefficients found for M in Ex-
ample 4.13 may be used. As given in Table 4.20 we have
Mo = 90, AM = 0.3324, and BM = 1.096. Values of quadra-
ture points and weights for three components are given in
Table 4.21. For each root, yi, corresponding value of Mi is de-
termined from Eq. (4.103). Mole fractions are equal to the
TABLE 4.22---Generation of pseudocomponents from Gaussain
quadrature method for the C7+ sample in Example 4.14.
Yi wi Zi Mi zi A/~
1 0.416 0.711 0.711 103.6 73.7
2 2.294 0.279 0.279 154.6 43.1
3 6.290 0.010 0.010 252.2 2.6
Mixture ... 1.000 1.000 ... 119.4
weighting factors according to Eq. (4.102). Summary of cal-
culations and mole fractions and molecular weights of the
components are given in Table 4.22. As shown in this table
average molecular weight of C7+ calculated from the 3 pseu-
docomponents is 119.4, which varies with experimental value
of 118.9 by 0.4%. To apply the Gaussian method to the gamma
distribution model first we must determine coeff or, r,
and 0. Since detailed compositional data are not available as
discussed in Section 4.5.3, we assume ~ = 90 and ~ = 1. Then
fl is calculated from Eq. (4.33) as fl = (i 18.9 - 90)/1.0 = 28.9.
Substituting ~ = 1 in Eq. (4.105), considering that F(a) = 1
we get zi ---- w/. Values of M/ are calculated from Eq. (4.106)
with yi taken from Table 4.21. Three components have molec-
ular weights of 102, 156.3, and 271.6, respectively. M7+ calcu-
lated from these values and mole fractions given in Table 4.22
is exactly 118.9 the same as the experimental data. The rea-
son is that this value was used to obtain parameter ft. For
this sample zi calculated from the gamma PDF is the same as
those obtained from Eq. (4.102) since o~= 1 and Eq. (2.105)
reduces to Eq. (2.102). But for values of u different from unity,
the two models generate different mole fractions and dif-
ferent Mi values, t
4. 6. 1. 2 Carbon Number Range Approach
I n this approach we divide the whole C7+ fraction into a num-
ber of groups with known carbon number boundaries. As an
example if five pseudocomponents are chosen to describe the
mixture, then five carbon number ranges must be specified. It
was found that for gas condensate systems and light oils the
carbon number ranges of C7-C10, CH-C15, C16-C2s, C25-C36,
and C36 well describe the mixture [23]. It should be noted
that the heaviest component in the first group is C+0, which
is the same as the lightest component in the second group is
C~- 1. Values of the lower and upper limit molecular weights
for each SCN group can be calculated from Eqs. (4.39) and
(4.40) and SCN up to C20 were calculated and are given in
Table 4.10. For example for the C7-C10, the molecular weight
range is from Mo (initial molecular weight of a C7+ fraction) to
M~0 or Mo - 142.5 and for the CH-CI5, the molecular weight
range is M~-M~5 or 142.5-214. Similarly molecular weight
range of other groups can be determined as: 214-352 for C16-
C25,352--492 for C26-C35, and 492-oo for C36+-cx~. For the last
group; i.e., C36+ the molecular weight range is from M36 to
to. Once the lower and upper values of M are known mole
fraction and molecular weight for each group can be deter-
mined from appropriate equations developed for each dis-
tribution model. Mole fraction and molecular weight of each
group for the gamma distribution model are determined from
Eqs. (4.37) and (4.38), respectively. For the generating distri-
bution model these equations are Eq. (4.84) and (4.86) and
for the exponential model mole fractions are calculated from
Eq. (4.92) and molecular weights from Eq. (4.91). Step-by-
step calculations for both of these methods with an example
are given in the next section.
4 . 6 . 2 L u mp i n g S c h e me
The lumping scheme is applied when composition of a reser-
voir fluid or crude oil is given in terms of SCN groups such
as those given in Table 4.2. Whitson [15, 17] suggests that the
C7+ fraction can be grouped into Alp pseudocomponents given
by
(4.108) ATe= 1 + 3. 3 log~ 0( N+ - 7)
where Np is the number of pseudocomponents and N+ is the
carbon number of heaviest fraction in the original fluid de-
scription. Obviously Np is the nearest integer number calcu-
lated from the above equation. The groups are separated by
molecular weight M i given by
(4.109) M i = M7+(MN+/M7+) 1/Nv
where j = 1 . . . . . Ale. SCN groups in the original fluid descrip-
tion that have molecular weights between boundaries Mi_ l
and M i are included in the group j. This method can be ap-
plied only to those C7+ fractions that are originally separated
by SCN groups and Ne >__20 [17].
The lumping scheme is very similar to the pseudoization
method, except the distribution coefficients are determined
for data on distribution of carbon number. For this reason
the lumping scheme generates better and more accurate pseu-
docomponents than does the splitting method when distribu-
tion coefficients are determined from only two bulk properties
such as M7+ and SG7+. Method of lumping is very similar to
the calculations made in Example 4.10 in which SCN groups
of C12 and C13 for the C7+ sample in Table 4.11 were lumped
together and the mole fraction and molecular weight of the
group were estimated. Here the two methods that can be used
for lumping and splitting schemes are summarized to show
the calculations [24]. I n these methods the generalized distri-
bution model is used; however, other models (i.e., gamma or
exponential) can be used in a similar way.
M e t h o d I: Ga us s i a n Qua dr a t ur e A p p r o a c h
1. Read properties of SCN groups and properties of plus frac-
tions (e.g., M30+ and SG30+). Normalize the mole fractions
( ~xmi = 1).
2. I f M and SG for each SCN group are not available, obtain
these properties from Table 4.6.
3. Determine distribution parameters for molecular weight
(Mo, AM, and BM) in terms of cumulative mole fraction and
for specific gravity (SGo, AsG, and BsG) in terms of cumu-
lative weight fraction.
4. Choose the number of pseudocomponents (i.e., 5) and cal-
culate their mole fractions (zi) and molecular weight (M/)
from Eqs. (4.102) and (4.103) or from Eqs. (4.105) and
(4.106) for the case of gamma distribution model.
5. Using Mi and zi in step 4, determine discrete weight frac-
tions, zwi, through Eq. (1.15).
6. Calculate cumulative weight fraction, Z~w, from z~ and es-
timate SGi for each pseudocomponent through Eq. (4.56)
with coefficients determined for SG in step 3. For example,
SG1 can be determined from Eq. (4.56) at z~1 = Zwl/2.
TABLE 4. 23--Lumping of SCN groups by two methods for the C7+ sample in Example 4.15 [24].
Method I: Gaussian quadrature approach Method II: Carbon number range approach
Component Mole Weight Mole Weight
i fraction fraction Mi SGi fract i on fraction Mi SGi
1 0.5218 0.3493 102.1 0.7436 0.532 0.372 106.7 0.7457
2 0.3987 0.4726 180.8 0.8023 0.302 0.328 165.5 0.7957
3 0.0759 0.1645 330.4 0.8591 0.144 0.240 254.4 0.8389
4 0.0036 0.0134 569.5 0.9174 0.019 0.049 392.7 0.8847
5 2.3 x 10 -5 1.4 x 10 -4 950.1 0.9809 0.003 0.011 553.5 0.9214
Mixture 1.0000 1.0000 152.5 0.7905 1.000 1.0000 152. 5 0.7908
7. Obt ai n May and SGav for t he mi xt ure from May = ~,~/P=I Zi Mi
and 1/SGav = ~7~1Zwi/ SGj.
Method II: Carbon Number Range Approach
1. Same as Met hod I.
4. Choose number of p seudocomp onent s (i.e., 5) and corres-
p ondi ng car bon number ranges, e.g., gr oup i: C7-C10,
group 2: Cll-C15, group 3: C16-C25, group 4:C25-C36 and
C36+.
5. Obtain mol ecul ar wei ght boundari es f r om Eqs. (4.39) and
(4.40). For exampl e for the groups suggested in step 4
t he mol ecul ar wei ght ranges are: ( Mo-142. 5) , ( 142. 5-214) ,
( 214-352) , ( 352- 492) and ( 492-oo) . The number of pseu-
docomp onent s (Np) and mol ecul ar wei ght boundari es may
also be det ermi ned by Eqs. (4.108) and (4.109).
6. Using the mol ecul ar wei ght boundari es det ermi ned in
step 5, calculate mol e fractions (zi) and mol ecul ar wei ght
(Mi) of these p seudocomp onent s f r om Eqs. (4.84) and
(4.86) or from Eqs. (4.92) and (4.91) when BM in Eq. (4.56)
is equal to unity.
7. Same as step 5 in Met hod I.
I n this met hod if the calculated mol e fract i on for a pseudo-
comp onent in step 6 is t oo hi gh or t oo low, we may reduce or
increase the correspondi ng car bon number range chosen for
t hat p seudocomp onent in step 4. Application of these met h-
ods is shown in t he following example.
Exampl e 4. 15- - Fluid descri pt i on of a C7+ f r om Nort h Sea
fields ( sample 42 in Ref. [24]) is given in t erms of mol e frac-
tions of SCN groups from C7 to C20+ as
N c 7 8 9 10 11 12 13 14 15 16 17 18 19 20+
xi 0.178 0.210 0.160 0.111 0.076 0,032 0.035 0.029 0.022 0.020 0.020 0.016 0,013 0.078
where Nc represents car bon number gr oup and x~ is its cor-
respondi ng normal i zed mol e fraction. For this mi xt ure t he
/147+ = 151.6 and SG7+ = 0.7917. Lump these comp onent s
i nt o an appropri at e number of p seudocomp onent s and give
t hei r mol ecul ar wei ght and specific gravity usi ng the above
t wo met hods.
Sol ut i on- - For this sampl e N+ = 20 and we may use
Eq. (4.108) t o det ermi ne t he number of pseudocomponent s.
Np = 1 + 3.31 log(20 - 7) = 4.7. The nearest integer number
is 5, t herefore Np = 5, whi ch is the same number as suggest ed
in step 4 of t he above met hods. For car bon number s f r om C7
t o C19, values of M and SG are t aken from Table 4.6 and mol e
fractions are convert ed into wei ght fract i on (x~-). Distribu-
t i on coefficients for M in t erms of Xcm and SG in t erms of Xcw
are t hen det ermi ned f r om Eqs. (4.56) and (4.57). The results
for M are Mo = 84, AM = 0.7157, and BM = 1 and for SG t he
coefficients are SGo = 0.655, Asc = 0.038, and Bso = 3. For
the 5 pseudocomponent s, Met hods I and I I have been applied
step by step and for each group j values of Zr~, Z~, Mi, and
SGi are given in Table 4.23. Specific gravity and mol ecul ar
wei ght of C7+ calculated from p seudocomp onent s generat ed
by Met hod I are 0.7905 and 152.5, whi ch are very close to ex-
peri ment al values of 0.7917 and 151.6. Met hod I I gives similar
results as shown in Table 4.23. Specific gravity differs f r om
the experi ment al value by 0.1%. Obviously comp onent s
1, 2 . . . . generat ed in Met hod I are not the same compo-
nent s generat ed by Met hod II, but combi nat i on of all 5 com-
ponent s by t wo met hods represent the same mixture. That is
why Mi and SGi for t he 5 p seudocomp onent s generat ed by
Met hods I and I I are not t he same.
4 . 7 C ON T I N UOUS M I X T UR E
C HA R A C T ER I Z A T I ON A PPR OA C H
A mor e compl i cat ed but mor e accurat e t reat ment of a C7+
fract i on is t o consi der it as a cont i nuous mixture. I n this ap-
p r oach the mi xt ure is not expressed in t erms of a finite num-
ber of p seudocomp onent but its propert i es are given by a con-
t i nuous funct i on such as Eq. (4.56). This met hod is in fact
equivalent to t he p seudocomp onent ap p r oach but wi t h infi-
nite number of comp onent s (Np = oo). Mansoori and Chor n
[27] di scussed a general ap p r oach t oward charact eri zat i on of
cont i nuous mixtures. I n this ap p r oach instead of specifying
a comp onent by i, it is expressed by one of its charact eri s-
tic paramet ers such as Tb or M. For mul at i on of cont i nuous
mi xt ures for phase equi l i bri um calculations is best expressed
by Eq. (4.15), while for t he p seudocomp onent ap p r oach for a
defined discrete mi xt ure it is formul at ed t hr ough Eq. (4.14).
To show appl i cat i on of a PDF in charact eri zat i on of a crude
oil by t he cont i nuous mi xt ure approach, we use Eq. (4.15)
to formul at e vapor-l i qui d equi l i bri um (VLE) and to obt ai n
species di st ri but i on of vapor and liquid product s once such
di st ri but i on is known for the feed duri ng a flash distillation
process. Theory of VLE is di scussed in Chapt er 6 and its appli-
cat i on is shown in Chapt er 9. I n Eq. (4.15), if we take boiling
poi nt as t he charact eri zat i on p ar amet er for P t he equi l i bri um
relation in t erms of fugacity is (see Eq. 6.173)
(4.110) fV(T) = fL( T) To < T < oo
where fV(T) is the fugacity of a specie in the vapor phase
whose boiling point is T. When T = To the above equation is
applied to the lightest component in the mixture and when
T = o0 it is applied to the last and heaviest component whose
boiling point may be considered as infinity. For simplicity it is
assumed that the vapor phase is ideal gas and the liquid phase
is an ideal solution. Under such conditions Eq. (4.110) for a
component with boiling point T in a mixture can be written
as (4.115)
(4.111) dyrp = dxrpl's
where dyr is the mole fraction of a component having boil-
ing point T in the vapor phase and dxr is the mole fraction
of the same component in the liquid phase, prs is the satura-
tion pressure (or vapor pressure) of components with boiling
point T at temperature T s and p is the total pressure at which
vapor and liquid are in equilibrium. T s is in fact the tempera-
ture at which separation occurs and Prs is a function of T s and
type of component that is characterized by boiling point (see
Problem 4.16). This relation is known as the Raoult's law and
its derivation will be discussed in Chapter 6. I n Eq. (4.111),
dyTp is the fugacity of components with boiling point T in an
ideal gas vapor phase while dxTprs is the fugacity of compo-
nents with boiling point T in an ideal liquid solution. To apply
Eq. (4.111) for a continuous mixture, we can use Eq. (4.16)
to express dxr and dyr by a PDF in each phase:
(4.112) dxr = FLdT
(4.113) dyr = F~dr
where F L and F v are the PDF in terms of boiling point T for
the liquid and vapor phases, respectively. Equations (4.70)
or (4.31) may be used to express F v or F L. Substituting (4.117)
Eqs. (4.112) and (4.113) into Eq. (4.111) we get
(4.114) FV p = F~.p
Equation (4.114) is the Raouh' s law in terms of a PDF appli-
cable to a continuous mixture. I f the liquid phase is nonideal,
the right-hand side of above equation should be multiplied by
activity coefficient V (T, T s) for those components with boiling (4.118)
point T at temperature T ~. And if the vapor phase is nonideal
gas the left-hand side of Eq. (4.114) should be multiplied by
fugacity coefficient ~0(T, T s, p) for components with boiling
point T at temperature T s and pressure p. These thermody-
namic properties are defined in Chapter 6 and can be obtained
from an equation of state for hydrocarbon systems. A more
general form of Eq. (4.114) for high-pressure VLE calcula-
tions is in terms equilibrium ratio can be written as
where Kr is the equilibrium ratio for a component with boil-
ing point T at temperature T s and pressure p. As it will be
shown in Chapter 6, Kr depends on vapor pressure pS.
Now we apply the above equations for design and operation
of a separation unit for flash distillation of reservoir fluids
and crude oils. As shown in Fig. 4.25 we assume 1 mol of
feed enters the unit that is operating at temperature T s and
pressure p. The products are r moles of vapor and 1 - ~b moles
of liquid in which ~ is the fraction of the feed vaporized in
a single-stage flash distillation unit. Material balance on the
distillation unit for a component whose boiling point is T can
be written as
(4.116) dzr x 1 = dxr x (I - ~) +dyr x
where dzT is the mole fraction of all components having boil-
ing point T and can be expressed in terms of a PDF similar to
Eq. (4.112). Substituting Eqs. (4.112) and (4.113) for dxr and
dyT and similarly for dzr into the above equation gives
FT F = (1 - ~)FT v + 4)FL4)
where F F is the density function for the feed in terms of boil-
ing point T. For all three probability density functions, F v,
F v, and F L we have
/ / / FFdT= FV dT= FLdT = 1
TO To To
~ Feed
1 mole
Tb
Vapor
C mo l e
|
~ Ts,P
~ L i qu i d
I-~ mole
T b
e~
T b
FIG. 4. 25- - Schemat i c of a si ng l e-stag e flash distillation unit.
From Eqs. (4.114), (4.117), and (4.118) one can derive the
following relation for calculation of par amet er r
f FFdT = 0
P P~
(4.119) (1 -- ~-)-p-~r
To
where the integration should be carried numerically and
may be determined by trial-and-error procedure. As will be
shown in Chapter 7, combi nat i on of Trouton' s rule for the heat
of vaporization and the Clasius-Clapeyron equation leads to
the following relation for the vapor pressure:
(4.120) prs = pa exp [10.58 (1 - T) ]
where T is the boiling point of each cut in the distribution
model, T s is the saturation temperature, and Pa is the atmo-
spheric pressure. Both T and T S must be in K. By combining
Eqs. (4.114) and (4.117) we get
P F F
(4.121) FTL = ( 1--r r
P~ F v
(4.122) FV = (1 - ~b)p + Cp~
After finding r from Eq. (4.119), it can be substituted in the
above equations to find density functions for the vapor and
liquid products.
For evaluation and application of these equations, data on
boiling point distribution of a Russian crude oil as given by
Ratzch et al. [31] were used. I n this case TBP distributions for
feed, vapor, and liquid st reams during flash distillation of the
crude are available. Molecular weight, specific gravity, and
refractive index of the mixture are 200, 0.8334 and 1.4626,
respectively. Applying Method A discussed in Section 4.5.4.6,
we obtain distribution coefficients for boiling point of feed
as: To = 241.7 K, Ar = 1.96, and Br = 1.5 and Fr F was deter-
mi ned from Eq. (4.70). Fraction of feed vaporized, q~, was
det ermi ned from Eq. (4.119) as 0.7766. Boiling point distri-
butions for the liquid (F L) and vapor (F v) products were de-
t ermi ned from Eqs. (4.121) and (4.122), respectively. Results
of calculations for F~, F v, and F L for this crude are shown
in Fig. 4.26 and compared with the experimental values pro-
vided in Ref. [31]. Since heavier component s appear in the
liquid product, therefore, the curve for F L is in the right side
of bot h F F and F v corresponding to higher values of boiling
points.
Part of errors for predicted distributions of F L and Fr v is
due to assumpt i on of an ideal solution for VLE calculations
as well as an approxi mat e relation for the estimation of vapor
pressures. For more accurate calculations Eq. (4.115) can be
used which would result in the following relations:
(4.123) f ( i - u,, -r ~, -1277 'K---r-r'~ Kr FrFdT = 0
To
1
F v
(4.124) FL = ( 1- ~b) +~Kr r
Kr FF
(4.125/ FrY = (1 - r +r r
0.005
"" -. - - Crude Feed
0.004 , ' Liquid Product
. . . . . . . Vapor Product
"-~ 0.003 "
~ .002
0.001
0
-200 0 200 400 600 800 1000
Boiling Point, Tb, ~
FIG. 4. 26~ Pr edi ct ed probabi l i ty densi ty f uncti ons
of feed, liquid, and v apor at 3 00~ f or flash v apori za-
tion of a Russi an crude oil. A ctual data are tak en f rom
Ref. [ 31].
where Eqs. (4.123)-(4.125) are equivalent to Eqs. (4.119),
(4.121), and (4.122) for ideal systems, respectively. Calcula-
tion of equilibrium ratios from equations of state will be dis-
cussed in Chapters 5 and 10. Probability density functions in
these equations may be expressed in t erms of other character-
ization paramet ers such as molecular weight or carbon num-
ber. However, as discussed in Chapter 2, boiling point is the
most powerful characterization paramet er and it is preferable
to be used once it is available. Similarly the same approach
can be used to obtain distribution of any other propert y (see
Probl em 4.16).
I n t reat ment of a reservoir fluid, the mixture may be pre-
sented in t erms of composi t i on of pure hydrocarbon com-
pounds from Ct to C5 and nonhydrocarbon compounds such
as HES and CO2 as well as grouped C6+ or a SCN group of C6
and C7+. For these mixtures the continuous mixture approach
discussed in this section can be applied to the hydrocarbon-
plus portion, while the discrete approach can be applied
to the lower portion of the mixture containing compounds
with known composition. This approach is known as semi-
continuous approach and calculation of different properties
of reservoir fluids by this approach has been discussed by
various researchers [27, 28, 43].
4 . 8 CAL CUL AT I ON OF P R OP ER T I ES
OF C R UD E OI L S A N D R ES ER VOI R F L UI D S
As discussed in Chapter 2, properties of a hydrocarbon com-
pound depend on its carbon number and mol ecul ar type.
Accurate calculation of properties of a pet rol eum mixture
rely on accurate knowledge of the composi t i on of the mixture
by individual constituents, their properties, and an appropri-
ate mixing rule to estimate the mixture properties. I n this
part based on the met hods outlined in this chapt er a crude
oil or a reservoir fluid is presented by a number of pseudo-
component s and a general approach is outlined to estimate
p r op er t i es of such mi xt ures. Ap p l i cat i on of t hi s ap p r oach is
shown t hr ough est i mat i on of sul fur cont ent of cr ude oils.
4. 8. 1 General Approach
F or a r eser voi r fluid accur at e p r op er t i es can be cal cul at ed
t hr ough det ai l ed comp osi t i onal anal ysi s of p ur e comp ounds
f r om C1 to C5 and SCN gr oup s f r om C6 and heavi er gr oup s
up t o at l east C50. The r emai ni ng p ar t can be gr oup ed as C50+.
F or very heavy oils, SCN gr oup sep ar at i on may be ext ended
up to C80 and t he r esi due gr oup ed as C80+. To est i mat e var i ous
p r op er t i es of t hese SCN gr oup s at l east t wo char act er i zi ng pa-
r amet er s such as Tb and SG or M and SG shoul d be known.
Thi s is shown in Table 4.24, wher e known dat a ar e i ndi cat ed
by + sign. Met hods out l i ned i n Sect i ons 4.5 and 4.6 can l ead
to gener at e such i nf or mat i on for a r eser voi r fluid. F or SCN
gr oup s of C6 and heavier, met hods i n Chap t er 2 can be used
t o est i mat e var i ous p r op er t i es (Tb, To Pc . . . . ) usi ng M and
SG as avai l abl e i nput p ar amet er s. For p ur e comp ounds up
to C5, all basi c p r op er t i es ar e gi ven in Tables 2.1 and 2.2 and
no est i mat i on met hod is r equi r ed. For mor e accur at e pre-
di ct i on of p r op er t i es of a r eser voi r fluid, each SCN gr oup s
f r om C6 up to C50+ may be di vi ded i nt o f ur t her t hree p seudo-
comp onent s as paraffi ni c, nap ht heni c, and ar omat i c. Met h-
ods of Sect i on 3.5 can be used to det er mi ne PNA comp os i t i on
of each SCN group. I n t hi s way number of comp onent s i n
Table 4.24 i ncr eases t o 152. For heavy oils t he numbe r of com-
p onent s woul d be even higher. For each homol ogous gr oup,
di fferent p r op er t i es may be est i mat ed f r om mol ecul ar wei ght
of i ndi vi dual SCN gr oup t hr ough t he r el at i ons gi ven in Sec-
t i on 2.3.3. Obvi ousl y cal cul at i on of mi xt ur e p r op er t i es when
it is expr essed in t er ms of l arge number of comp onent s is not
an easy task. For t hi s r eason t he number of comp onent s i n
Table 4.24 may be r educed by gr oup i ng to SCN comp onent s
or spl i t t i ng t he C7 f r act i on i nt o j ust 3 or 5 p s eudocomp onent s .
Fur t her mor e, iC4 and nC4 may be gr oup ed as C4 and iC5 and
nC5 coul d be gr oup ed as C5. I n t hi s way t he mi xt ur e can be pre-
sent ed by 10-15 comp onent s wi t h known speci fi cat i ons. The
fol l owi ng examp l e shows how a cr ude oil can be p r esent ed
TABLE 4. 24--Matrix table of components for estimation
of properties of reservoir fluids.
No. Compound Mole fraction M SG
1 H2S q-
2 CO2 +
3 N2 +
4 H20 +
5 C1 +
6 C2 +
7 C3 +
8 iC4 q-
9 nC4 q-
10 iC5 +
11 nC5 +
12 C6 +
13 C7 +
14 C8 +
15-5 5 a C9-C49 q-
56 C50 +
57 C5o+ +
-I- +
+ +
+ +
+ +
+ +
+ +
aCompounds from 15 to 55 represent SCN groups from C9 to C49.
For compounds 1-11, properties are given in Tables 2.1 and 2.2.
TABLE 4. 25--Pseudoization of the C7+ for the Kuwaiti crude
in Example 4.16.
Pseudocomponent 1 2 3 4 5
Weight fraction 0.097 0.162 0.281 0.197 0.264
Mole fraction 0.230 0.255 0.280 0,129 0,106
Molecular weight 112.0 169.1 267.1 405,8 660.9
Specific gravity 0.753 0.810 0.864 0,904 0.943
by an adequat e number of p s eudocomp onent s wi t h known
p ar amet er s.
Exampl e 4. 16- - Comp os i t i onal dat a on a Kuwai t i cr ude oil is
given as follows:
Component C2 C3 iC4 nC4 iC5 nC5 C6 C7+
Wt% 0.03 0.39 0.62 1. 08 0.77 1. 31 1.93 93.87
The char act er i st i cs of t he C7+ fract i on ar e M7+ = 266.6 and
SG7+ =-- 0.891 [44]. Divide t he C7 f r act i on i nt o 5 p s eudocom-
p onent s and p r esent t he cr ude in t er ms of mol e and wei ght
f r act i ons of r ep r esent at i ve const i t uent s wi t h known M, SG,
and Tb. Est i mat e M and SG for t he whol e crude.
Sol ut i on- - For t he C 6 gr oup f r om Table 4.6 we have M6 = 82,
SG6 --- 0.69, and Tb6 : - 337 K. For p ur e comp onent s f r om C2
to C5, M and SG can be t aken f r om Table 2.1. Usi ng M and
x~, mol e f r act i on Xm can be est i mat ed t hr ough Eq. (4.61).
Usi ng Met hod A in Sect i on 4.5.4.6 di st r i but i on coeffi ci ent s
in Eq. (4.56) for t he C7+ f r act i on are f ound as Mo = 90,
AM = 1.957, and BM = 1.0. F r om Met hod I I out l i ned in Sec-
t i on 4.6.1.2 and speci fyi ng 5 car bon number r anges t he C7+
can be spl i t i nt o 5 p s eudocomp onent s wi t h known mol e frac-
t i on ( nor mal i zed) , M and SG as gi ven i n Table 4.25. I n t hi s
t abl e t he wei ght f r act i ons ar e cal cul at ed t hr ough Eq. (1.15)
usi ng mol e f r act i on and mol ecul ar wei ght . Values of wei ght
f r act i ons in Table 4.25 shoul d be mul t i p l i ed by wt % of C7+ in
t he whol e cr ude to est i mat e wt % of each p s eudocomp onent
i n t he crude. Values of mol % in t he or i gi nal fl ui d ar e cal-
cul at ed f r om wt % and mol ecul ar wei ght of all comp onent s
p r esent in t he mi xt ur e as shown in Table 4.26. F or t he 5 pseu-
docomp onent s gener at ed by spl i t t i ng t he C7+, boi l i ng p oi nt s
are cal cul at ed f r om M and SG usi ng Eq. (2.56). F r om Tb and
SG of p s eudocomp onent s gi ven in Table 4.26, one may esti-
mat e basi c char act er i zat i on p ar amet er s to est i mat e var i ous
TABLE 4.26--Characterization of the Kuwait crude oil
in Example 4.16.
Component Wt% Mol% M SG Tb ,; C
C2 0.03 0.22 30.1 0.356
C 3 0.39 1.99 44.1 0.507
iC4 0.62 2.40 58.1 0.563
nC4 1.08 4.18 58.1 0.584
iC5 0.77 2.40 72.2 0.625
nC5 1.31 4,08 72.2 0.631
C6 1.93 5.29 82 0.690
C7+(1) 9.1 18,28 112,0 0.753
C7+(2) 15.2 20.22 169.1 0.810
C7+(3) 26.4 22.23 267.1 0.864
C7+(4) 18.5 10.26 405.8 0.904
C7+(5) 24.8 8.44 660.9 0.943
Total 100 100 225.2 0.8469
64
123
216
333
438
527
properties, For example, for the whole crude the mol ecul ar
weight is calculated from Xmi and Mi using Eq. (3.1) and spe-
cific gravity of the whole crude is calculated from Xwi and SGi
using Eq. 3.44. Calculated M and SG for the crude are 225
and 0.85, respectively. These values are lower t han those for
the C7+ as the crude contains component s lighter t han C7. I n
the next example estimation of sulfur content of this crude is
demonstrated.
4 . 8 . 2 Es t i mat i on o f S ul f ur Cont ent o f a Crude Oi l
Est i mat i on of sulfur content of crude oils is based on the gen-
eral a p p r oa c h f or e s t i ma t i on of var i ous p r op e r t i e s of cr ude
oi l s a nd r es er voi r fl ui ds de s c r i be d i n Sect i on 4.8.1. Once a
cr ude is p r e s e nt e d by a numbe r of p ur e c omp one nt s a nd s ome
na r r ow boi l i ng r a nge p s e udoc omp one nt s wi t h known mol e
f r act i on, Tb, and SG, any p hys i cal p r op e r t y ma y be e s t i ma t e d
t hr ough me t hods di s c us s e d i n Chap t er s 2 a nd 3. Pr op er t i es
No. wt%
TABLE 4. 27--Estimation of sulfur content of crude oil in Example 4.18.
Experimental data Calculated parameters
Tb, K d20, g/cm 3 S% exp. SG M Rt m S%, pred. wt% x S% pre
1 1. 7 20
2 0.26 25
3 0.29 30
4 0.31 35
5 0.33 40
6 0.37 45
7 0.38 50
8 0.42 55
9 0.44 60
10 0.47 65
11 0.49 70
12 0.52 75
13 0,55 80
14 0,58 85
15 0.6 90
16 0.63 95
17 0.66 100
18 0.4 105
19 0.52 110
20 0.59 115
21 0.66 120
22 0.71 125
23 0.73 130
24 0.76 135
25 0.76 140
26 0.77 145
27 0.76 150
28 0.75 155
29 0.75 160
30 0.74 165
31 0.73 170
32 0.72 175
33 0.71 180
34 0.71 185
35 0.71 190
36 0.71 195
37 0.71 200
38 1.45 210
39 1.47 220
40 1.51 230
41 1.56 240
42 1.58 250
43 1.6 260
44 1.59 270
45 1.56 280
46 1.49 290
47 1.42 300
48 1.32 310
49 1.23 320
50 1.18 330
51 1.18 340
52 1.29 350
53 1.56 360
54 26 449
55 28.1 678
Sum 100.0
0.566 0.006 0.570 76.9 1.037 - 11.970 0.000 0.00
0,583 0.006 0.587 77.4 1.038 - 11.310 0.000 0.00
0.597 0.006 0.602 78.4 1.038 -10. 835 0.000 0.00
0.610 0.006 0.615 79.5 1.039 -10. 395 0.000 0.00
0.623 0.006 0.627 80.7 1.040 -10. 000 0.000 0.00
0.634 0.007 0.639 82.1 1.041 -9. 644 0.000 0.00
0.645 0.007 0.649 83.5 1.041 -9. 313 0.000 0.00
0.655 0.007 0.659 85.0 1.042 -9. 009 0.002 0.00
0.664 0.007 0.669 86.7 1.042 -8. 731 0.008 0.00
0.673 0.008 0.677 88.4 1.043 -8. 480 0.013 0.01
0.681 0.008 0,685 90.2 1.043 -8. 244 0.017 0.01
0.688 0.008 0.693 92.1 1.044 -8. 032 0.021 0.01
0.695 0.008 0,700 94.0 1.044 -7. 829 0.025 0.01
0.702 0.008 0.706 96.1 1.045 -7. 645 0.028 0.02
0.708 0.009 0.713 98.2 1.045 -7. 473 0,032 0.02
0.714 0.009 0.718 100.3 1.045 -7. 314 0.036 0.02
0.719 0.009 0.724 102.6 1.045 -7. 170 0.040 0.03
0.724 0.01 0.729 104.8 1.046 -7. 027 0.045 0.02
0.729 0.011 0.734 107.2 1.046 -6. 893 0.050 0.03
0.734 0.016 0.738 109.6 1.046 -6. 769 0.055 0.03
0.738 0.019 0.743 112.0 1.047 -6. 649 0.061 0.04
0.743 0.022 0.747 114.5 1.047 -6. 532 0.068 0.05
0.747 0.026 0.751 117.1 1.047 -6. 420 0.076 0.06
0.750 0.031 0.755 119.7 1.047 -6. 305 0.085 0.06
0.754 0.036 0.758 122.3 1.047 -6. 196 0.094 0.07
0,758 0.041 0.762 125.0 1,047 -6. 086 0.104 0.08
0.761 0.047 0.766 127.7 1.048 -5. 973 0.115 0.09
0.765 0.054 0.769 130.5 1.048 -5. 859 0.128 0.10
0.768 0.061 0.772 133.2 1.048 -5. 745 0.141 0.11
0.771 0.068 0.775 136.1 1.048 -5. 629 0.155 0.11
0.774 0.077 0.779 138.9 1.048 -5. 505 0.171 0.12
0.777 0.086 0.782 141.8 1.048 -5. 380 0.188 0.14
0,781 0.095 0.785 144.7 1.049 -5. 246 0.206 0.15
0.784 0.106 0.788 147.7 1.049 -5. 102 0.227 0.16
0.787 O. 117 0.791 150.7 1.049 -4, 949 0.249 O. 18
0.790 0.129 0.794 153.7 1.049 -4. 796 0.271 0.19
0.793 0.142 0.797 156.7 1.049 -4. 633 0.296 0.21
0.799 0.17 0.803 162.8 1.049 -4. 290 0.350 0.51
0.805 0.201 0.809 169.1 1.050 -3. 907 0.411 0.60
0.810 0.31 0.814 175.4 1.050 -3. 484 0.481 0.73
0.816 0.46 0.820 181.9 1.050 -3. 032 0.556 0.87
0.822 0.64 0.826 188.4 1.051 -2. 550 0.639 1.01
0.828 0.83 0.832 195.1 1.051 -2. 027 0.730 1.17
0.833 1.03 0.837 201.8 1.051 -1. 485 1.191 1.89
0.839 1.21 0.842 208.7 1.051 -0. 901 1.312 2.05
0.844 1.37 0.848 215.7 1.052 -0. 313 1.424 2.12
0.849 1.5 0.853 222.9 1.052 0.316 1.536 2.18
0.854 1.6 0.858 230.2 1.052 0.948 1.640 2.16
0.859 1.67 0.862 237.6 1.052 1.593 1.739 2.14
0.863 1.75 0.867 245.2 1.053 2.249 1.832 2.16
0.868 1.86 0.871 252.9 1.053 2.930 1.922 2.27
0.872 2.05 0.875 260.8 1.053 3.606 2.005 2.59
0.875 2.4 0.879 269.1 1.053 4.210 2.072 3.23
0.915 2.81 0.918 343.4 1.055 13.025 2.847 74.02
1.026 5.2 1.028 561.1 1.064 57.206 4.631 130.13
233.94
of pure components can be directly obtained from Tables 2.1
and 2.2. For each physical property an appropriate mixing
rule should be applied. For example, SG of the mixture should
be calculated from Eq. (3.44) as was shown in Example 4.16.
For the sulfur content of a crude oil the appropriate mixing
rule is [45]:
(4.126) sulfur wt% of crude = ~ xwi(sutfur wt%)i
i
in which x~ is the weight fraction of pseudocomponent i in
the crude. The method is well demonstrated in the following
examples for calculation of sulfur content of crude oils.
Example 4. 17--For the crude oil of Example 4.16 estimate
the total sulfur content in wt%. The whole crude has API grav-
ity of 31 and sulfur content of 2.4 wt% [45].
Solution- - The crude is presented in terms of 12 compo-
nents (6 pure compounds and 6 pseudocompounds) in Ta-
ble 4.26. Sulfur content of the crude should be estimated
through Eq. (4.126). For pure hydrocarbons from C2 to nC5
the sulfur content is zero; however, sulfur content of pseu-
docomponents from C6 to C7 should be calculated from
Eqs. (3.96) and (3.97). Parameters n20 and d20 needed in these
equations have been estimated from methods discussed in
Sections 2.6.1 and 2.6.2. Estimated sulfur content of C6 and
the 5 C7+ pseudocomponents are 0.2, 0.1, 0.7, 1.9, 2.9, and 3.8,
respectively. Substituting these values into Eq. (4.126) would
estimate sulfur content of the whole crude as 2.1% versus
experimental value of 2.4% with -0. 3 wt% error.
Example 4. 18--For the crude oil of Example 4.17 a complete
TBP, SG and sulfur wt% curves versus weight fraction are
available as given in the first five columns of Table 4.27. Esti-
mate the sulfur content curve and graphically compare with
the experimental values. Also estimate the sulfur content of
whole crude from predicted sulfur content curve and com-
pare with the experimental value of 2.4 wt%.
Solution- - A complete characterization dataset on a crude oil
include two suitable characterization parameters such as Tb
and SG versus cumulative weight or volume fraction with low
residue. When such data are available, properties of the crude
may be estimated quite accurately. I n Table 4.27, Tb, d2o, and
sulfur wt% of 55 cuts are given with known wt%. The boil-
ing point of last cut (residue) was not originally known from
experimental data. Based on the fractions with Tb greater than
100~ (cuts 21-54), weight fractions were normalized and cu-
mulative weight fractions were calculated. Temperature of
100~ is near the boiling point of nC7. For the C7+ portion
of the crude distribution, coefficients in Eq. (4.56) were de-
termined as To = 360 K, Ar = 1.6578, and Br = 1.485. Using
these values, boiling point of the residue (cut 55) was deter-
mined from Eq. (4.56) as 678.3~ Specific gravity of cuts were
determined from d20 and parameters M, t~, and m were de-
termined using the methods discussed in Chapters 2 and 3.
Sulfur content of each cut were determined from Eq. (3.96)
for cuts with M < 200 and from Eq. (3.97) for cuts 44-55 with
M > 200. For cuts 1-7 calculated values of S% from Eq. (3.96)
were slightly less than zero and they are set as zero as dis-
cussed in Section 3.5.2.2. Finally sulfur content of the whole
o Experimental
5
5 4
3
2
= 1
0 ~ E m ' ~ ' - J ' ~ ' I
0 200 400 600 800
Boiling Point, ~
FIG. 4. 27 - - Di st ri but i on of sul fur content in the
crude oil of Ex ampl e 4.1 8 . Taken wi th permi ssi on
from Ref. [ 45],
crude is calculated from Eq. (4.126) as shown in the last col-
umn of Table 4.27. The estimated sulfur content of the crude
is 2.34 wt%, which is near the experimental value of 2.4%.
A graphical comparison between predicted and experimen-
tal sulfur distribution along distillation curve is presented in
Fig. 4.27.
Calculations made in Examples 4.17 and 4.18 show that as
more characterization data for a crude are available better
property prediction is possible. I n many cases characteriza-
tion data on a crude contain only the TBP curve without SG
distribution. I n such cases M and SG distributions can be de-
termined from Eq. (4.7) and coefficients given in Table 4.5.
Equation (4.7) can be used in its reversed form using Tb as in-
put instead of M. Once M is determined it can be used to esti-
mate SG, n20, and d20 from Eq. (4.7) with corresponding coef-
ficients in Table 4.5. This approach has been used to estimate
sulfur content of 7 crudes with API gravity in the range of 31-
40. An average deviation of about 0.3 wt% was observed [45].
4 . 9 C ON C L US I ON S A N D
R EC OM M EN D A T I ON S
I n this chapter methods of characterization of reservoir fluids,
crude oils, natural gases and wide boiling range fractions have
been presented. Crude assay data for seven different crudes
from around the world are given in Section 4.1.2. Charac-
terization of reservoir fluids mainly depends on the charac-
terization of their C7+ fractions. For natural gases and gas
condensate samples with little C7+ content, correlations de-
veloped directly for C7+, such as Eqs. (4.11)-(4.13), or the
correlations suggested in Chapters 2 and 3 for narrow-boiling
range fractions may be used. However, this approach is not
applicable to reservoir fluids with considerable amount of C7+
such as volatile or black oil samples. The best way of charac-
terizing a reservoir fluid or a crude oil is to apply a distribu-
tion model to its C6+ or C7+ portion and generate a distribu-
tion of SCN groups or a number of pseudocomponents that
represent the C7+ fraction. Various characterization param-
eters and basic properties of SCN groups from C6 to C50 are
given in Table 4.6 and in the form of Eq. (4.7) for computer
applications.
Characterization of C7+ fraction is presented through appli-
cation of a distribution model and its parameters may be de-
termined from bulk properties with mi ni mum required data
on M7+ and SG7+. Three types of distribution models have
been presented in this chapter: exponential, gamma, and a
generalized model. The exponential model can be used only to
molecular weight and is suitable for light reservoir fluids such
as gas condensate systems and wet natural gases. The gamma
distribution model can be applied to both molecular weight
and boiling point of gas condensate systems. However, the
model does not accurately predict molar distribution of very
heavy oils and residues. This model also cannot be applied
to other properties such as specific gravity or refractive in-
dex. The third model is the most versatile distribution model
that can be applied to all major characterization parameters
of M, Tb, SG, and refractive index parameter I. Furthermore,
the generalized distribution model predicts molar distribu-
tion of heavy oils and residues with reasonable accuracy. Ap-
plication of the generalized distribution model (Eq. 4.56) to
phase behavior prediction of complex petroleum fluids has
been reported in the literature [46]. Both the gamma and the
generalized distribution models can be reduced to exponen-
tial in the form of a two-parameter model.
Once a distribution model is known for a C7+ fraction, the
mixture can be considered as a continuous mixture or it could
be split into a number of pseudocomponents. Examples for
both cases are presented in this chapter. The method of con-
tinuous distribution approach has been applied to flash dis-
tillation of a crude oil and the method of pseudocomponent
approach has been applied to predict sulfur content of an oil.
Several characterization schemes have been outlined for dif-
ferent cases when different types of data are available. Meth-
ods of splitting and grouping have been presented to represent
a crude by a number of representative pseudocomponents.
A good characterization of a crude oil or a reservoir fluid is
possible when TBP distillation curve is available in addition
to M7+ and SG7+. The most complete and best characteriza-
tion data on a crude oil or a C7+ fraction would be TBP and SG
distribution in terms of cumulative weight or volume fraction
such as those shown in Table 4.27. Knowledge of carbon num-
ber distribution up to C40 and specification of residue as C40+
fraction is quite useful and would result in accurate property
prediction provided the amount of the residue ( hydrocarbon
plus) is not more than a few percent. For heavy oils separation
up to C60+ or C80+ may be needed. When the boiling point of
the residue in a crude or a C7+ fraction is not known, a method
is proposed to predict this boiling point from the generalized
distribution model. When data on characterization of a crude
are available in terms of distribution of carbon number such
as those shown in Table 4.2, the method of grouping should be
used to characterize the mixture in terms of a number of sub-
fractions with known mole fraction, M, Tb and SG. Further in-
formation on options available for crude oil characterization
from mi ni mum data is given by Riazi et al. [40]. Properties
of subfractions or pseudocomponents can be estimated from
Tb and SG using methods presented in Chapters 2 and 3. For
light portion of a crude or a reservoir fluid whose composition
is presented in terms weight, volume, or mole fraction of pure
compounds, the basic characterization parameters and prop-
erties may be taken from Tables 2.1 and 2.2. Once a crude
is expressed in term of a number of components with known
properties, a mixture property can be determined through ap-
plication of an appropriate mixing rule for the property as it
will be shown in the next chapter.
4 . 1 0 P R OB L EM S
4.1. Consider the dry natural gas, wet natural gas, and gas
condensate systems in Table 1.2. For each reservoir fluid
estimate the following properties:
a. SGg for and the API gravity.
b. Estimate Tp~ and/ ' pc from methods of Section 4.2.
c. Estimate Tpc, Ppo and Vp~ from Eq. (3.44) using pure
components properties from Table 2.1 and C7+ prop-
erties from Eqs. (4.12) and (4.13).
d. Compare the calculated values for Tp~ and Pp~ in parts
b and c and comment on the results.
4.2. Calculate Tb, SG, d20, n20, Tc, Pc, Vo a, and ~ for C55, C65,
and C75 SCN groups.
4.3. Predict SCN distribution for the West Texas oil sample
in Table 4.1, using Eq. (4.27) and M7+ and x7+ (mole
fraction of C7+) as the available data.
4.4. Derive an analytical expression for Eq. (4.78), and show
that when SG is presented in terms of X~w we have
SG.v+~-~]- - J0 = ~ ( - 1)k+l ! ~) F 1 + ---~-
k=0
4.5. Basic characterization data, including M, Tb, and SG,
versus weight fraction for seven subfractions of a C7+
fluid are given in Table 4.28. Available experimental bulk
properties are MT+ = 142.79, and SG7+ -- 0.7717 [47].
Make the following calculations:
a. Calculate Xr~ and x~.
b. Estimate distribution parameter I from Tb and SG
using methods of Chapter 2.
c. Using experimental data on M, Tb, SG and I distribu-
tions calculate distribution coefficients Po, A and B in
Eq. (4.56) for these properties. Present M in terms of
x~ and Tb, SG and I in terms of Xcw.
d. Calculate PDF from Eq. (4.66) and show graphical
presentation of F(M ), F(T), F(SG), and F(I) .
e. Find refractive index distribution
f. Calculate mixture M, Tb, SG, and n20 based on the
coefficients obtained in part c.
g. For parts b and f calculate errors for M, Tb, and SG in
terms of AAD.
TABLE 4.28---Characterization parameters for the C7+
fraction of the oil.system in Problem 4.5 [47].
x~ M~ Tb~, K SG
0.1269 98 366.5 0.7181
0.0884 110 394.3 0.7403
0.0673 121 422.1 0.7542
0.1216 131 449.8 0.7628
0.1335 144 477.6 0.7749
0.2466 165 505.4 0.7859
0.2157 216 519.3 0.8140
4.6. Repeat Problem 4.5 for the gamma and exponential dis-
tribution models to
a. find the coefficients of Eq. (4.31): 0, a, and fl in
Eq. (4.31) for M and Tb.
b. estimate SG distribution based on exponential model
and constant Kw approach.
c. calculate mixture M, Tb, and SG and compare with
experimental data.
d. make a graphical comparison between predicted dis-
tributions for M, Tb, and SG from Eq. (4.56) as ob-
tained in Problem 4.5, gamma and exponential mod-
els with each other and experimental data.
4.7. For the C7+ of Problem 4.5 find distributions of M, Tb,
and SG assuming:
a. Only information available are MT+ = 142.79 and
SG7+ = 0.7717.
b. Only information available is MT+ = 142.79.
c. Only information available is SGT+ = 0.7717.
d. Graphically compare predicted distributions from
parts a, b and c with data given in Table 4.28.
e. Estimate MT+ and SG7+ form distribution parameters
obtained in parts a, b and c and compare with the
experimental data.
4.8. Using the Guassian Quadrature approach, split the C7+
fraction of Problem 4.7 into three pseudocomponents.
Determine, xm, M, Tb, and SG for each component. Cal-
culate the mixture M and SG from the three pseudocom-
ponents. Repeat using carbon number range approach
with t 5 pseudocomponents and appropriate boundary
values of Mi.
For the C7+ fraction of Problem 4.5 estimate total sulfur
content in wt%.
For the waxy oil in Table 4.2 present the oil in six
groups as C2-C3, C4-C6, C7-Cl0, Cll-C20, C21-C30, and
C31+. Determine M and SG for each group and calculate
M and the API gravity of the oil. Compare estimated
M from the six groups with M calculated for the crude
based on the detailed data given in Table 4.2.
Use the crude assay data for crude number 7 in Table 4.3
to
a. determine Tb and SG distributions.
b. estimate Tb for the residue based on the distribution
found in Part a.
c. estimate M for the residue from Tb in Part b and
SG.
d. estimate M for the residue from viscosity and SG and
compare with value from c.
e. Determine distribution of sulfur for the crude and
graphically evaluate variation of S% versus cumula-
tive wt%.
f. Estimate sulfur content of the crude based on the pre-
dicted S% distribution.
For the crude sample in Problem 4.8 find distribution of
melting point and estimate average melting point of the
whole crude.
Estimate molecular weights of SCN groups from 7 to
20 using Eqs. (4.91) and (4.92) and compare your re-
sults with those calculated in Example 4.6 as given in
Table 4.10.
Construct the boiling point and specific gravity curves
for the California crude based on data given in Table 4.3
4.9.
4.10.
4.11.
4.12.
4.13.
4.14.
4.15.
4.16.
4.17.
(crude number 6). I n constructing this figure the mid-
volume points may be used for the specific gravity. De-
termine the distribution coefficients in Eq. (4.56) for Tb
and SG in terms Of Xcv and compare with the experimen-
tal values. Also estimate crude sulfur content.
Show how Eqs. (4.104), (4.105), and (4.106) have been
derived.
As it will be shown in Chapter 7, Lee and Kesler have
proposed the following relation for estimation of vapor
pressure (Pvap) of pure compounds, which may be ap-
plied to narrow boiling range fractions (Eq. 7.18).
lnP~ ap = 5.92714 - 6.09648/Tbr -- 1.28862 In Tbr
+ 0.169347T6r + o)(15.2518 -- 15.6875/Tbr
-- 13.4721 In Tbr + 0.43577T~r)
where p~ap = pv,p/pc and Tb~ = Tb/Tc in which both Tb
and Tc must be in K. Use the continuous mixture ap-
proach (Section 4.7) to predict distribution of vapor
pressure at 311 and 600 K for the waxy crude oil in
Table 4.2 and graphically show the vapor pressure dis-
tribution versus cumulative mol% and carbon number.
Minimum information that can be available for a crude
oil is its API or specific gravity. A Saudi light crude has
API gravity of 33.4 (SG = 0.8581), and experimental data
on boiling point and specific gravity of its various cuts
are given in the following table as given in the Oil and
Gas Journal Data Book (2000) (p. 318 in Ref. [8]).
Vol% SG Tb, K SG (calc) Tb, K (calc)
23.1 _ 370.8 ? ?
23.1 0.8"1"31 508.3 ? ?
8.5 0.8599 592.5 ? ?
30.2 0.9231 727.5 ? ?
15.1 1.0217 ... ?
a. Using the mi ni mum available data (API gravity), es-
timate values of Tb and SG in the above table and
compare with given experimental data graphically.
b. Similar data exist for a Saharan crude oil from Algeria
(page 320 in Ref. [8]) with API gravity of 43.7. Con-
struct Tb and SG distribution diagram in terms of cu-
mulative volume fraction.
4.18. Similar to the continuous mixture approach introduced
in Section 4.7, calculate vapor and liquid product distri-
butions for flash distillation of the same crude at 1 atm
and 400~ Present the results in a fashion similar to
Fig. 4.26 and calculate the vapor to feed ratio (~b).
4.19. Repeat Problem 4.18 but instead of Eq. (4.120) for the
vapor pressure, use the Lee-Kesler correlation given in
Problem 4.16. Compare the results with those obtained
in Problem 4.17 and discuss the results.
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pp. 406-411.
MNL50-EB/Jan. 2005
PVT Relations and Equations
of State
N OMEN C L A T UR E
API
A, B, C, . . .
ac
b, C, . . .
B
C
C
d2o
&
e
exp
F
f~
Ul, U2
SG
API Gravity defined in Eq. (2.4)
Coefficients in various equations
Paramet er defined in Eq. (5.41) and given in
Table 5.1
Constants in various equations
Second virial coefficient
Third virial coefficient
Volume translation for use in Eq. (5.50), cma/ mol
Liquid density of liquid at 20~ and i atm, g/ cm s
Critical density defined by Eq. (2.9), g/ cm 3
Correlation parameter, exponential function
Exponential function
Degrees of freedom in Eq. (5.4)
A function defined in t erms of co for paramet er
a in PR and SRK equations as given in Table 5.1
and Eq. (5.53)
h Paramet er defined in Eq. (5.99), dimensionless
ka Bohzman constant (---- R/NA = 1.2 x 10 -2~ J/K)
k/i Binary interaction par amet er (BIP),
dimensionless
I Refractive index paramet er defined in Eq. (2.36)
M Molecular weight, g/ mol [kg/kmol]
m Mass of system, g
NA Avogadro number = number of molecules in one
mole (6.022 x 1023 mo1-1)
N Number of component s in a mixture
n Number of moles
n20 Sodi um D line refractive index of liquid at 20~
and 1 atm, dimensionless
P Pressure, bar
psat Saturation pressure, bar
Pr Reduced pressure defined by Eq. (5.100)
( = P/Po), dimensionless
R Gas constant = 8.314 J/ mol. K (values are given
in Section 1.7.24)
Rm Molar refraction defined by Eq. (5.133), cm3/ mol
r Reduced mol ar refraction defined by Eq. (5.129),
dimensionless
r I nt ermol ecul ar distance in Eqs. (5.10)-(5.12),
A (10 -m m)
r A par amet er specific for each substance in
Paramet ers in Eqs. (5.40) and (5.42)
T Absolute t emperat ure, K
197
ZRA
zi
Z1, Z2, and Z3
Tc Critical temperature, K
Tcdc Cri condent herm t emperat ure, K
Tr Reduced t emperat ure defined by Eq. (5.100)
(-- T/Tc), dimensionless
V L Saturated liquid mol ar volume, cma/ gmol
V sat Saturation mol ar volume, cma/ gmol
V v Saturated vapor mol ar volume, cma/ gmol
Vc Critical volume (molar), cma/ mol (or critical
specific volume, cma/g)
Vr Reduced volume (-- V/Vc)
x4 Mole fraction of i in a mixture (usually used
for liquids)
yi Mole fraction of i in a mixture (usually used
for gases)
Z Compressibility factor defined by Eq. (5.15)
Z~ Critical compressibility factor defined by
Rackett parameter, dimensionless
Mole fraction of i in a mixture
Roots of a cubic equation of state
Greek Letters
Paramet er defined by Eq. (5.41), dimension-
less
Polarizability factor defined by Eq. (5.134),
c m 3
a, V Paramet ers in BWR EOS defined by Eq.
(5.89)
/3 A correction factor for b par amet er in an EOS
A Difference between two values of a par amet er
8ii Paramet er defined in Eq. (5.70), dimension-
less
e Energy par amet er in a potential energy func-
tion
F Potential energy function defined by Eq.
(5.10)
r Volume fraction of i in a liquid mixture de-
fined by Eq. (5.125)
H Number of phases defined in Eq. (5.4)
# Dipole moment in Eq. (5.134)
0 A propert y in Eq. (5.1), such as volume, en-
thalpy, etc.
0 Degrees in Eq. (5.47)
g/ cm 3 ( mol ar density unit: cm3/ mol)
pO Value of density at low pressure ( at mospheri c
pressure), g/ cm a
a Size paramet er in a potential energy func-
tion, A ( I 0 -~~ m)
~0 Acentric factor defined by Eq. (2.10)
Packing fraction defined by Eq. (5.91), di-
mensionless
Superscript
bp Value of a propert y for a defined mixture at
its bubble point
c Value of a propert y at the critical point
cal Calculated value
g Value of a propert y for gas phase
HS Value of a propert y for hard sphere molecules
ig Value of a propert y for an ideal gas
L Saturated liquid
1 Value of a propert y for liquid phase
V Saturated vapor
sat Value of a propert y at saturation pressure
(0) A dimensionless t erm in a generalized corre-
lation for a propert y of simple fluids
(1) A dimensionless t erm in a generalized corre-
lation for a propert y of acentric fluids
Subscripts
C
i
J
i , j
m
P
P
P, N, A
Acronyms
API-TDB
BI P
BWRS
COSTALD
CS
EOS
GC
HC
HS
HSP
KI SR
~PWS
LJ
LJ EOS
Value of a propert y at the critical point
A component in a mixture
A component in a mixture
Effect of bi nary interaction on a propert y
Value of a propert y for a mixture
Value of a property at pressure P
Pseudoproperty for a mixture
Value of par amet er c in Eq. (5.52) for paraf-
fins, naphthenes, and aromatics
Value of a propert y for the whole (total) sys-
t em
American Petroleum I nstitute--Technical
Data Book
Binary interaction paramet er
Starling modification of Benedi ct -Webb-
Rubin EOS (see Eq. 5.89)
correspondi ng st at e liquid density (given by
Eq. 5.130)
Carnahan-St arl i ng EOS (see Eq. 5.93)
Equations of state
Generalized correlation
Hydrocarbon
Hard sphere
Hard sphere potential given by Eq. (5.13)
Kuwait I nstitute for Scientific Research
I nternational Association for the Properties
of Water and St eam
Lennard-J ones potential given by Eq. (5.11)
Lennard-J ones EOS given by Eq. (5.96)
LK GC Lee-Kesler generalized correlation for Z
(Eqs. 5.107-5.113)
LK EOS Lee-Kesler EOS given by Eq. (5,109)
MRK Modified Redlich-Kwong EOS given by Eqs,
(5.38) and (5.137)-(5.140)
NI ST National I nstitute of Standards and Technol-
ogy
OGJ Oil and Gas Journal
PHCT Perturbed Hard Chain Theory (see Eq, 5.97)
PR Peng-Robi nson EOS (see Eq. 5.39)
RHS Right-hand side of an equation
RK Redlich-Kwong EOS (see Eq. 5.38)
RS R squared (R2), defined in Eq. (2.136)
SRK Soave-Redl i ch-Kwong EOS given by Eq.
(5.38) and paramet ers in Table 5.1
SAFT Statistical associating fluid theory (see Eq.
5.98)
SW Square-Well potential given by Eq. (5.12).
vdW van der Waals (see Eq. 5.21)
%AAD Average absolute deviation percentage de-
fined by Eq. (2.135)
%AD Absolute deviation percentage defined by
Eq. (2.134)
%MAD Maxi mum absolute deviation percentage
AS DISCUSSED I N CHAPTER 1, the mai n application of charac-
terization met hods presented in Chapters 2-4 is to provide
basic data for estimation of various thermophysical proper-
ties of pet rol eum fractions and crude oils. These properties
are calculated t hrough t hermodynami c relations. Although
some of these correlations are empirically developed, most
of t hem are based on sound t hermodynami c and physical
principles. The most i mport ant t hermodynami c relation is
pressure-vol ume-t emperat ure (PVT) relation. Mathematical
PVT relations are known as equations of state. Once the PVT
relation for a fluid is known various physical and thermody-
nami c properties can be obtained through appropri at e rela-
tions that will be discussed in Chapter 6. I n this chapt er we
review principles and theory of propert y estimation met hods
and equations of states that are needed to calculate various
thermophysical properties.
5 . 1 B A S I C D EF I N I T I ON S A N D T H E
P H A S E R UL E
The state of a system is fixed when it is in a t hermodynami c or
phase equilibrium. A system is in equilibrium when it has no
tendency to change. For example, pure liquid wat er at 1 at m
and 20~ is at stable equilibrium condition and its state is
perfectly known and fixed. For a mixture of vapor and liquid
wat er at 1 at m and 20~ the system is not stable and has a
tendency to reach an equilibrium state at anot her t empera-
ture or pressure. For a system with two phases at equilibrium
only t emperat ure or pressure (but not both) is sufficient to
determine its state. The state of a system can be det ermi ned
by its properties. A propert y that is independent of size or
mass of the system is called intensive property. For example,
t emperat ure, pressure, density, or mol ar volume are inten-
sive properties, while total volume of a system is an extensive
property. All mol ar propert i es are intensive propert i es and are
related t o total propert y as
0 t
(5.1) 0 = - -
n
where n is t he number of moles, 0 t is a total propert y such as
vol ume, V t, and 0 is a mol ar propert y such as mol ar volume,
V. The number of mol es is related t o t he mass of t he system,
m, t hr ough mol ecul ar wei ght by Eq. (1.6) as
m
( 5 . 2 ) n = - -
M
I f total propert y is divided by mass of t he syst em (m), instead
of n, t hen 0 is called specific property. Bot h mol ar and spe-
cific propert i es are intensive propert i es and t hey are related
t o each ot her t hr ough mol ecul ar weight.
(5.3) Mol ar Propert y = Specific Propert y x M
Generally t her modynami c relations are developed among
mol ar propert i es or intensive properties. However, once a mo-
lar propert y is calculated, t he total propert y can be cal cul at ed
f r om Eq. (5.1).
The phase rule gives the mi ni mum number of i ndependent
variables t hat must be specified in order to det ermi ne ther-
modynami c state of a syst em and various t her modynami c
properties. This number is called degrees of freedom and is
shown by F. The phase rule was stated and formul at ed by the
Ameri can physicist J. Willard Gibbs in 1875 in t he following
f or m [1]:
(5.4) F = 2 + N - Fl
where rI is t he number of phases and N is the number of
comp onent s in t he system. For exampl e for a pure compo-
nent ( N = 1) and a single phase (Fl = 1) syst em the degrees
of freedom is calculated as 2. This means when t wo intensive
propert i es are fixed, t he state of the syst ems is fixed and its
propert i es can be det ermi ned f r om the t wo known parame-
ters. Equat i on (5.4) is valid for nonreact i ve systems. I f t here
are some react i ons among t he comp onent s of t he systems, de-
grees of freedom is reduced by t he number of react i ons wi t hi n
the system. I f we consi der a pure gas such as met hane, at least
t wo intensive propert i es are needed to det ermi ne its t hermo-
dynami c properties. The most easily measurabl e propert i es
are t emperat ure (T) and pressure (P). Now consi der a mix-
t ure of t wo gases such as met hane and et hane wi t h mol e frac-
t i ons xl and x2 (x2 = 1 - xl). Accordi ng t o the phase rule t hree
propert i es must be known t o fix the state of t he system. I n ad-
dition to T and P, the t hi rd variable coul d be mol e fract i on of
one of t he comp onent s (xl or x2). Similarly, for a mi xt ure wi t h
single phase and N comp onent s t he number of propert i es t hat
must be known is N + 1 (i.e., T, P, xl, x2 . . . . . XN-1). When the
number of phases is i ncreased the degrees of freedom is de-
creased. For example, for a mi xt ure of cert ai n amount of ice
and liquid wat er (H = 2, N = 1) f r om Eq. (5.4) we have F = 1.
This means when onl y a single variable such as t emperat ure
is known t he state of the syst em is fixed and its propert i es
can be det ermi ned. Mi ni mum value of F is zero. A syst em of
pure comp onent wi t h t hree phases in equi l i bri um wi t h each
other, such as liquid water, solid ice, and vapor, has zero de-
grees of freedom. This means t he t emperat ure and pressure
of t he syst em are fixed and onl y under uni que condi t i ons of
5. PV T RELATI ONS AND EQUATIONS OF STATE 199
T and P t hree phases of a pure comp onent can coexist all t he
time. This t emperat ure and pressure are known as triple point
temperature and triple point pressure and are charact eri st i cs
of any pure comp ound and their values are given for many
comp ounds [2, 3]. For example, for wat er t he triple poi nt t em-
perat ure and pressure are 0.01~ and 0.6117 kPa ( ~0. 006 bar) ,
respectively [3]. The most recent t abul at i on and formul at i on
of propert i es of wat er r ecommended by I nt ernat i onal Asso-
ci at i on for the Properties of Wat er and St eam (IAPWS) are
given by Wagner and Pruss [4].
A t her modynami c propert y t hat is defined t o formul at e the
first law of t her modynami cs is called internal energy shown
by U and has t he uni t of energy per mass or energy per mol e
(i.e., J/ mol). I nt ernal energy represent s bot h kinetic and po-
tential energies t hat are associ at ed wi t h t he mol ecul es and for
any pure subst ance it depends on t wo propert i es such as T and
V. When T increases the kinetic energy increases and when
V increases the pot ent i al energy of molecules also increases
and as a result U increases. Anot her useful t her modynami c
propert y t hat includes PV energy in addi t i on to t he internal
energy is enthalpy and is defined as
( 5 . 5 ) H = U + P V
where H is t he mol ar ent hal py and has the same uni t as U.
Furt her definition of t her modynami c propert i es and basi c re-
lations are present ed in Chapt er 6.
5 . 2 P VT R EL A T I ON S
For a pure comp onent syst em after t emperat ure and pressure,
a propert y t hat can be easily det ermi ned is t he vol ume or
mol ar volume. Accordi ng to the phase rule for single phase
and pure comp onent systems V can be det ermi ned f r om T
and P:
(5.6) V = f~(T, P)
where V is the mol ar vol ume and fl represent s funct i onal
relation bet ween V, T, and P for a given system. This equat i on
can be rearranged to find P as
(5.7) P = f2(T, V )
where t he forms of funct i ons fl and f2 in the above t wo rela-
t i ons are different. Equat i on (5.6) for a mi xt ure of N compo-
nent s wi t h known composi t i on is wri t t en as
( 5 . 8 ) P = f 3 ( T, V , . ~l , X 2 . . . . . XN_I)
where x4 is the mol e fract i on of comp onent i. Any mat hemat -
ical rel at i on bet ween P, V, and T is called an equation of state
(EOS). As will be seen in the next chapter, once t he PVT rela-
t i on is known for a syst em all t her modynami c propert i es can
be calculated. This indicates the i mpor t ance of such relations.
I n general the PVT relations or any ot her t her modynami c re-
lation may be expressed in t hree forms of (1) mat hemat i cal
equations, (2) graphs, and (3) tables. The graphi cal ap p r oach
is tedious and requires sufficient dat a on each subst ance t o
const ruct the graph. Mat hemat i cal or analytical forms are
t he most i mpor t ant and conveni ent relations as t hey can be
. . . . . . . . 1- . . . . .
S~ t~ O~ 0%/ 'Liquid/vapor'lTOO)o s ' ~: ~: ~ ~ <Tc
vc
V
FIG. 5. 1 - - T ypi cal PV di ag ram for a pure substance.
Gas
Fusion / C
Curve ',,.~ / Liquid Region / i
P SolidRegion ~ / :
%
, atm . . . . . . . . . . . . F . . . . - f :
Trip,e Point i
Vapor ',
I I
I I
I I
Tf Tb Tc
(a) T
used in comput er programs for accurate estimation of vari-
ous properties. Graphical and tabulated relations require in-
terpolation with hand calculations, while graphical relations
were quite popul ar in the 1950s and 1960s. With the growth
of comput ers in recent decades mat hemat i cal equations are
now the most popul ar relations.
A typical PVT relation in the form of PV and PT diagrams
for a pure substance is shown in Figs. 5.1 and 5.2a, respec-
tively. The solid, liquid, and vapor phases are clearly specified
in the PT diagram. The two-phase region of vapor and liquid is
best shown in the PV diagram. I n Fig. 5.1, three isotherms of
T1 < T2 < T3 are shown where isotherm T2 passes t hrough the
critical point, that is Tc = T2. I n the PV di agram lines of satu-
rated liquid (solid line) and saturated vapor (dotted line) meet
each other at the critical point. At this point properties of va-
por phase and liquid phase become identical and two phases
are indistinguishable. Since the critical isotherm exhibits a
horizontal inflection at the critical point we may impose the
following mat hemat i cal conditions at this point:
(5.9) ~ n, Pc \ b- ~ / I n, po
The first and second partial derivatives of P with respect to
V (at constant T) may be applied to any EOS in the form
of Eq. (5.7) and at the critical point they should be equal to
zero. Simultaneous solution of resulting two equations will
give relations for calculation of EOS paramet ers in t erms of
critical constants as will be seen later in Section 5.5.1.
The two-phase region in the PV di agram of Fig. 5.1 is
under the envelope. As is seen from this figure the slope of an
i sot herm in the liquid region is much greater t han its slope in
the vapor phase. This is due to the greater change of volume of
a gas with pressure in compari son with liquids that show less
dependency of volume change with pressure under constant
t emperat ure condition. The dotted lines inside the envelope
indicate percentage of vapor in a mixture of liquid and vapor,
which is called quality of vapor. On the saturated vapor curve
(right side) this percentage is 100% and on the saturated liq-
uid curve (left side) this percentage is zero. Vapor region is
part of a greater region called gas phase. Vapor is usually re-
ferred to a gas that can be liquefied under pressure. A vapor
at a t emperat ure above Tc cannot be liquefied no mat t er how
Bubble Point
9 Curve C Gas
Liquid Region ~~" ~" ~' ~ ~' ~l
!
Liquid + Vapor I
Region t
I
I
Dew Point
s s Curve
S
s
s
s
s
s s Vapor Region
(b) T
FIG. 5. 2- - T ypi cal PT di ag rams for a pure sub-
stances and mi x tures.
high the pressure is and it is usually referred as a gas. When
T and P of a substance are greater than its Tc and Pc the sub-
stance is neither liquid nor vapor and it is called supercriti-
cal fluid or simply fluid. However, the word fluid is generally
used for either a liquid or a vapor because of many similari-
ties that exist between these two phases to distinguish t hem
from solids.
As is seen in Fig. 5.1, lines of saturated liquid and vapor
are identical in the PT diagram. This line is also called vapor
pressure (or vaporization) curve where it begins from the
triple point and ends at the critical point. The saturation line
between solid and liquid phase is called fusion curve while
between solid and vapor is called sublimation curve. I n Fig.
5.2 typical PT diagrams for pure substances (a) and mixtures
(b) are shown.
I n Fig. 5.2a the freezing point t emperat ure is al most the
same as triple point t emperat ure but they have different
corresponding pressures. The normal boiling point and crit-
ical point both are on the vaporization line. A compari son
between PV and PT diagrams for pure substances (Figs. 5.1
and 5.2a) shows that the two-phase region, which is an area in
the PV diagram, becomes a line in the PT diagram. Similarly
5. PV T RELATIONS AND EQUATIONS OF STATE 201
triple point, whi ch is a poi nt on the PT di agram, becomes a
line on the PV di agram. For a mixture, as shown in Fig. 5.2b,
t he t wo-phase regi on is under the envelope and bubbl e poi nt
and dew points curves meet each ot her at the critical point.
The mai n appl i cat i on of PT di agr am is to det ermi ne the phase
of a syst em under certain condi t i ons of t emperat ure and pres-
sure as will be discussed in Chapt er 9 ( Section 9.2.3). Figure
5.1 shows t hat as t emperat ure of a pure subst ance increases,
at const ant pressure, the following phase changes occur:
Subcool ed solid (1) ~ Sat urat ed solid at subl i mat i on
t emperat ure (2) ~ Sat urat ed liquid at subl i mat i on
t emperat ure (3) --~ Subcoot ed liquid (4) ~ Sat urat ed
liquid at vapori zat i on t emperat ure (5) ~ Sat urat ed
vapor at vapori zat i on t emperat ure (6)---> Superheat ed
vapor (7)
The process from (2) to (3) is called fusion or mel t i ng and
t he heat requi red is called heat of fusion. The process f r om
(5) to (6) is called vapori zat i on or boiling and the heat re-
qui red is called heat of vaporization. Fusi on and vapori zat i on
are t wo-phase change processes at whi ch bot h t emperat ure
and pressure r emai n const ant while volume, internal energy,
and ent hal py woul d increase. A gas whose t emperat ure is
great er t han Tc cannot be liquefied no mat t er how hi gh the
pressure is. The t erm vapor usually refers to a gas whose tem-
perat ure is less t han Tr and it can be convert ed to liquid as
pressure exceeds t he vapor pressure or sat urat i on pressure at
t emperat ure T.
A C
Critical Point .
/ ! ' t ,<
~++ t t t t ~ i ' ~ i f i
" + , ; {o !+
+ # / ? ' + +P+~, = u
" I .,."_+ IL5
E I
I !
I I
+ i .
T c Tcri c
Temperatm-e, T
FIG. 5. 3 - - T ypi cal PT di ag ram for a reserv oi r
fluid mi x ture.
An ext ended versi on of Fig. 5.2b is shown in Fig. 5.3 for
a typical PT di agram of a reservoir fluid mixture. Lines of
const ant quality in the t wo-phase region converge at the crit-
ical point. The sat urat ed vapor line is called dewpoint curve
( dotted line) and t he line of sat urat ed liquid is usually called
bubblepoint curve (solid line) as i ndi cat ed in Figs. 5.2b and
5.3. I n Fig. 5.3 when pressure of liquid is reduced at con-
st ant t emperat ure (A to B), vapori zat i on begins at t he bubbl e
poi nt pressure. The bubbl epoi nt curve is locus of all these
bubbl e points. Similarly for t emperat ures above Tc when gas
FIG. 5. 4---A DB Robi nson computeri zed PVT cell ( courtesy of KISR) [ 5].
pressure is r educed (C to D) or i ncreased ( E to F) at const ant
T, the first drop of liquid appears at the dew poi nt pressure.
The dewpoi nt curve is l ocus of all these dewpoi nt s (dotted).
The dot t ed lines under the envelope in this figure indicate
const ant percent vapor in a mi xt ure of liquid and vapor. The
100% vapor line corresponds t o sat urat ed vapor ( dewpoint)
curve. The PT di agram for reservoir fluids has a t emperat ure
called cricondentherm t emperat ure (Tcnc) as shown in Fig. 5.3.
When t emperat ure of a mi xt ure is great er t han Tcno a gas can-
not be liquefied when pressuri zed at const ant t emperat ure.
However, as is seen in Fig. 5.3, at Tc < T < Tcn~ a gas can be
convert ed to liquid by either increase or decrease in pressure
at const ant t emperat ure dependi ng on its pressure. This phe-
nomenon is called retrograde condensation. Every mi xt ure has
a uni que PT or PV di agram and varies in shape f r om one mix-
t ure to another. Such di agrams can be developed from phase
equi l i bri um calculations t hat require composi t i on of the mix-
t ure and is discussed in Chapt er 9.
Accurat e measur ement of fluid phase behavi or and related
physical propert i es can be obt ai ned from a PVT apparat us.
The central part of this equi pment is a t ransparent cylindrical
cell of about 2.0-2.5 cm di amet er and 20 cm length sealed by
a pi st on t hat can be moved to adjust desired volume. A typical
moder n and mercury-free PVT syst em made by D B Robi n-
son, court esy of KI SR [5], is shown in Fig. 5.4. Variation of
P and V can be det ermi ned at various i sot herms for differ-
ent syst ems of pure comp ounds and fluid mixtures. The PVT
cell is part i cul arl y useful in t he st udy of phase behavi or of
reservoir fluids and const ruct i on of PT di agrams as will be
discussed in Chapt er 9.
5 . 3 I N T ER MOL EC UL A R FOR C ES
As di scussed in Section 2.3.1, propert i es of a subst ance de-
p end on t he i nt ermol ecul ar forces t hat exist bet ween its
molecules. The type of PVT relation for a specific fluid also
depends on t he i nt ermol ecul ar forces. These forces are de-
fined in t erms of potential energy funct i on (F) t hr ough Eq.
(2.19). Potential energy at the i nt ermol ecul ar distance of r
is defined as the work requi red to separat e t wo molecules
f r om di st ance r t o di st ance ~ where the interrnotecular force
is zero and mat hemat i cal l y F is defined in the following
forms:
dF = - Fdr
oo
(5.10)
r(r) = t F(r)dr
r
where the first equat i on is t he same as Eq. (2.19) and the sec-
ond one is derived from i nt egrat i on of the first equat i on con-
sidering the fact t hat F(oo) = 0. F is comp osed of repulsive
and attractive t erms where t he latter is negative. For ideal
gases where the distance bet ween the molecules is large, it
is assumed t hat P = 0 as shown in Fig. 5.5 [6]. For nonp ot ar
comp ounds such as hydr ocar bon systems for whi ch t he dom-
i nant force is London di spersi on force, the potential energy
may be expressed by Lennar d- J ones (L J) model given by
Eq. (2.21) as
(5.11) P= e s [ ( ~) 12- - ( r ) 6]
0
r
FIG. 5.5--Potential energy
for ideal gases.
where e and a are energy and size paramet ers, whi ch are char-
acteristics of each subst ance. The significance of this func-
t i on is t hat (a) at r = a, F ---- 0 (i.e., at r = cr repul si on and
at t ract i on forces are just bal anced) and (b) F = - dF / dr = 0
at F = - e. I n fact F = - e is the mi ni mum potential energy,
whi ch defines equi l i bri um separat i on where force of attrac-
t i on is zero. The potential model is illustrated in Fig. 5.6.
Since the LJ potential is not mat hemat i cal l y conveni ent
to use, the following potential model called Square-WeU
potential (SWP) is proposed to represent the LJ model for
nonp ol ar systems:
{ oo r<_ a
(5.12) F ( r ) = - e a < r < r * a
0 r >_ r*a
where in the regi on 1 < r/ a < r* we have Square-We/ / ( SW) .
This model is also shown in Fig. 5.6. The SW model has t hree
paramet ers (a, e, r*), whi ch shoul d be known for each sub-
stance f r om mol ecul ar properties. As will be seen later in this
chapter, this model conveni ent l y can be used to estimate the
second virial coefficients for hydr ocar bon systems.
Anot her pot ent i al model t hat has been useful in develop-
ment of EOS is hard-sphere potential (HSP). This model as-
sumes t hat t here is no i nt eract i on until t he mol ecul es collide.
At the t i me of collision t here is an infinite interaction. I n this
model attractive forces are neglected and mol ecul es are like
rigid billiard balls. I f the mol ecul ar di amet er is a, at the time
of collision, t he di st ance bet ween centers of t wo mol ecul es is
r = a and it is shown in Fig. 5.7. As shown in this figure, the
HSP can be expressed in t he following form:
oo a t r < a
(5.13) P= 0 a t r > a
I t is assumed t hat as T -+ oo all gases behave like hard
sphere molecules. Application of this model will be di scussed
in Chapt er 6 for t he devel opment of EOS based on velocity
of sound. I n all model s accordi ng to definition of potential
! Square Well
i
Lennard-Jones
,I
r * ( ~
FIG. 5. 6--Lennard-Jones and Square-
W ell potential models.
F 0 ~* ff:l
r
FIG. 5.7mHard sphere po-
tential model.
energy we have, as r --> c~, F ~ 0. For example, in the
Sut herl and model it is assumed t hat the repul si on force is
but the at t ract i on force is proport i onal wi t h Ur n , t hat is
for r > ~, F = - D/ r 6, where D is the model p ar amet er [6].
Potential energy model s present ed in this section do not de-
scribe mol ecul ar forces for heavy hydr ocar bons and pol ar
compounds. For such molecules, additional paramet ers must
be i ncl uded in the model. For example, dipole moment is a
par amet er t hat charact eri zes degree of pol ari t y of molecules
and its knowl edge for very heavy mol ecul es is quite useful for
bet t er propert y predi ct i on of such compounds. Furt her dis-
cussi on and ot her potential energy funct i ons and i nt ermol ec-
ul ar forces are di scussed in various sources [6, 7].
5. 4 EQUATI ONS OF STATE
An EOS is a mat hemat i cal equat i on t hat relates pressure, vol-
ume, and t emperat ure. The simplest form of these equat i ons
is t he ideal gas law t hat is onl y applicable to gases. I n 1873,
van der Waals proposed t he first cubi c EOS t hat was based
on the t heory of cont i nui t y of liquids and gases. Since t hen
many modi fi cat i ons of cubi c equat i ons have been developed
and have f ound great industrial appl i cat i on especially in the
pet rol eum i ndust ry because of their mat hemat i cal simplic-
ity. More sophi st i cat ed equat i ons are also proposed in re-
cent decades t hat are useful for cert ai n syst ems [8]. Some of
these equat i ons part i cul arl y useful for pet rol eum fluids are
reviewed and di scussed in this chapter.
5. 4. 1 I deal Gas Law
As di scussed in the previ ous section the i nt ermol ecul ar forces
depend on the distance bet ween the molecules. Wi t h an
increase in mol ar vol ume or a decrease in pressure the
i nt ermol ecul ar distance increases and t he i nt ermol ecul ar
forces decrease. Under very l ow-pressure conditions, the in-
t ermol ecul ar forces are so small t hat t hey can be neglected
(F = 0). I n addi t i on since the empt y space bet ween the
mol ecul es is so large the vol ume of mol ecul es may be ne-
glected in comp ar i son wi t h the gas volume. Under these con-
ditions any gas is consi dered as an ideal gas. Properties of
ideal gases can be accurat el y est i mat ed based on t he kinetic
t heory of gases [9, 10]. The universal form of t he EOS for ideal
gases is
(5.14) PV ig = RT
where T is absolute t emperat ure, P is t he gas absolute pres-
sure, V ig is t he mol ar vol ume of an ideal gas, and R is the
5. PV T REL ATI ONS AND EQUATI ONS OF STATE 203
universal gas const ant in whi ch its values in different uni t s
are given in Section 1.7.24. The condi t i ons t hat Eq. (5.14)
can be used depend on the subst ance and its critical proper-
ties. But approxi mat el y this equat i on may be applied to any
gas under at mospheri c or subat mospheri c pressures wi t h an
accept abl e degree of accuracy. An EOS can be nondi mensi on-
alized t hr ough a par amet er called compressi bi l i t y factor, Z,
defined as
V PV
(5.15) Z - - - -
V ig RT
where for an ideal gas Z = 1 and for a real gas it can be great er
or less t han uni t y as will be di scussed later in this chapter. Z in
fact represent s the ratio of vol ume of real gas t o t hat of ideal
gas under the same condi t i ons of T and P. As the deviation
of a gas from ideality increases, so does deviation of its Z
fact or f r om unity. The appl i cat i on of Z is in cal cul at i on of
physical propert i es once it is known for a fluid. For example,
i f Z is known at T and P, vol ume of gas can be cal cul at ed f r om
Eq. (5.15). Application of Eq. (5.15) at the critical poi nt gives
critical compressi bi l i t y factor, Zc, whi ch was initially defined
by Eq. (2.8).
I n ideal gases, molecules have mass but no vol ume and
t hey are i ndependent f r om each ot her wi t h no interaction.
An ideal gas is mat hemat i cal l y defined by Eq. (5.14) wi t h t he
following relation, whi ch indicates t hat the i nt ernal energy is
onl y a funct i on of t emperat ure.
(5.16) U i g= f4(T )
Subst i t ut i on of Eqs. (5.14) and (5.16) into Eq. (5.5) gives
(5.17) H ig = f s( T)
where H ig is the ideal gas ent hal py and it is onl y a funct i on
of t emperat ure. Equat i ons (5.14), (5.16), and (5.17) si mpl y
define ideal gases.
5. 4. 2 Real Gas e s - - L i qui ds
Gases t hat do not follow ideal gas condi t i ons are called real
gases. At a t emperat ure bel ow critical t emperat ure as pressure
increases a gas can be convert ed to a liquid. I n real gases, vol-
ume of mol ecul es as well as the force bet ween mol ecul es are
not zero. A comp ar i son among an ideal gas, a real gas, and
a liquid is demonst r at ed in Fig. 5.8. As pressure increases
behavi or of real gases approaches t hose of t hei r liquids. The
space bet ween t he mol ecul es in liquids is less t han real gases
and in real gases is less t han ideal gases. Therefore, t he in-
t ermol ecul ar forces in liquids are much st ronger t han t hose
in real gases. Similarly t he mol ecul ar forces in real gases are
hi gher t han t hose in ideal gases, whi ch are nearl y zero. I t is
for this reason t hat predi ct i on of propert i es of liquids is mor e
difficult t han propert i es of gases.
Most gases are actually real and do not obey t he ideal gas
law as expressed by Eqs. (5.14) and (5.16). Under limiting
condi t i ons of P -~ 0 (T > 0) or at T and V --> o~ (finite P) we
can obt ai n a set of const rai nt s for any real gas EOS. When
T -~ oo t ransl at i onal energy becomes very large and ot her
energies are negligible. Any valid EOS for a real gas shoul d
obey t he following const rai nt s:
(5.18) lira ( PV ) = RT
P-~0
0
o
o
o
o o o
O O
(a)
1 Mole of Ideal Gas
( b) (c)
1 Mole of Real Gas 1 Mole of Liquid
pc>p sat
Pa (atmospheric pressure) Pb>Pa
1 mol e of a fluid consi st of 6.02x1023 mol ecul es, b = vol ume of 1 mol e of har d mol ecul es.
V=volume of 1 mole of fluid
vi g=Vi gspac e vg=Vgspac e + b vl=Vlspace + b
" vg .~" vi g
b < < Vlgspac e b< Vg Vlspace < < - - space-- - - space
FI G . 5 . 8 - - Di f f e r e nc e b et ween a n i deal g as, a real g as, and a l i qui d.
(5.19) lim ( O~) = _ R
T--~oo p p
(5.20) lim ( 32V'~
~ \~-~] ~ = 0
I n general for any gas as P ~ 0 ( or V ~ oo) it becomes an
ideal gas; however, as T --~ oo it is usually assumed t hat gas
behavi or approaches t hose of hard sphere gases. Const rai nt s
set by the above equat i ons as well as Eq. (5.9) may be used to
exami ne validity of an EOS for real fluids.
5 . 5 C UB I C EQUA T I ON S OF S T A T E
The ideal gas law expressed by Eq. (5.14) is nei t her applicable
to real gases ( high pressure) nor to liquids where t he vol ume
of molecules cannot be i gnored in comp ar i son wi t h the vol-
ume of gas (see Fig. 5.8). Cubic EOS are desi gned t o overcome
these t wo short comi ngs of ideal gas law wi t h mat hemat i cal
convenience. Several commonl y used equations, t hei r solu-
tion, and characteristics are di scussed in this section.
5. 5. 1 Four C o mmo n Cubi c Equat i ons
(vdW, RK, SRK, and PR)
The behavi or of hi gh-pressure gases approaches the behavi or
of liquids until t he critical poi nt where bot h gas and liquid
behavi or become identical, van der Waal (vdW) proposed t he
idea of cont i nui t y of gases and liquids and suggested t hat a
single equat i on may represent the PVT behavi or of bot h gases
and liquids. He modified Eq. (5.14) by repl aci ng P and V wi t h
appropri at e modi fi cat i ons to consi der real gas effects in the
following form [1]:
(5.21) P+~ ( V- b) = R T
where a and b are t wo const ant s specific for each subst ance
but i ndependent of T and P. The above equat i on is usually
wri t t en as
RT a
(5.22) P -
V - b V 2
To find V from T and P, the above equat i on maybe rearranged
a s
(5.23) (;)
V 3- b+ V2+ V- ~= 0
where it is a cubi c equat i on in t erms of V. For this reason the
vdW EOS, Eq. (5.22), is known as a cubi c EOS. As a mat t er
of fact any EOS t hat can be convert ed into a cubi c f or m is
called a cubic EOS. I n Eq. (5.22), paramet ers a and b have
physical meani ngs. Paramet er b also called co-vol ume or re-
pulsive p ar amet er represent s vol ume of 1 mol of hard cores
of molecules and has t he same uni t as the mol ar vol ume (V).
Paramet er a is also referred t o as at t ract i on p ar amet er and
has the same uni t as t hat of PV 2 (i.e., bar. cm6/mol2). I n Eq.
(5.22), the t erm RT/ ( V - b) represents the repulsive t erm of a
molecule, while a/ V 2 represents attractive t erm and account s
for noni deal behavi or of gas. V - b is in fact the space bet ween
mol ecul es (Figs. 5.8b and 5.8c). When paramet ers a and b are
zero Eq. (5.22) reduces to ideal gas law. Mat hemat i cal l y it can
be shown from Eq. (5.22) t hat as P ~ cr V --* b and the free
vol ume bet ween molecules disappears.
Since Eq. (5.21) has onl y two paramet ers it is also known
as a t wo-paramet er EOS. Paramet ers a and b in the vdW EOS
can be best det ermi ned from experi ment al dat a on PVT. How-
ever, mat hemat i cal l y these const ant s can be det ermi ned by
i mposi ng Eq. (5.9) as shown in the following example.
Exampl e 5. ~- - Obt ai n vdW paramet ers in t erms of Tc and Pc
usi ng Eq. (5.9) and (5.21). Also det ermi ne Zc for fluids t hat
obey vdW EOS.
Sol ut i on- - OP/ OV and 02p/ OV 2 a r e calculated f r om Eq. (5.22)
by keepi ng T const ant and set equal to zero at T = To, P = Pc,
and V = V~ as
(5.24) a~_ rc RTc 2a
- (Vc_ b)~ +W =~
(5.25)
82P 2RTc 6a
- - o
By taking the second terms to the right-hand side in each
equation and dividing Eq. (5.24) by Eq. (5.25) we get
(5.26) b = Vc
3
By substituting Eq. (5.26) into Eq. (5.24) we get
9 E
(5.27) a = ~Tc c
Since Tc and Pc are usually available, it is common to express
parameters a and b in terms of Tc and Pc rather than Tc and
Vc. For this reason Vc can be found from Eq. (5.21) in terms
of Tc and Pc and replaced in the above equations. Similar
results can be obtained by a more straightforward approach.
At the critical point we have V = Vc or V - Vc = 0, which can
be written as follows:
( 5 . 2 8 ) ( v - v c ) 3 = 0
Application of Eq. (5.23) at Tc and Pc gives
(5.29) V 3- b+ Pc ] V2 + Pc V- ~= 0
Expansion of Eq. (5.28) gives
(5.30) (V - Vc) 3 = V 3 - 3VcV 2 + 3Vc2V - V 3 = 0
Equations (5.29) and (5.30) are equivalent and the corre-
sponding coefficients for V 3, V 2, V 1, and V ~ must be equal
in two equations. This gives the following set of equations for
the coefficients:
(5.31) - ( b + RTc~ = -3Vc coefficients of V 2
\ Pc/
a 3Vc2 coefficients of V (5.32) ~ =
ab
(5.33) - - V 3 coefficients of V ~
Pc
By dividing Eq. (5.33) by (5.32), Eq. (5.26) can be obtained. By
substituting Vc = 3b (Eq. 5.26) to the right-hand side (RHS)
of Eq. (5.31) the following relation for b is found:
RTc
(5.34) b = - -
sic
Combining Eqs. (5.26) and (5.34) gives
3 RTc
(5.35) Vc -
8 Pc
Substituting Eq. (5.35) into Eq. (5.27) gives
9 ( 3RTc' ~ _ 27R2Tc 2
/2
RTc \ 8Pc ] 64Pc
Therefore, the final relation for parameter a in terms of Tc and
Pc is as follows:
27R2T 2
(5.36) a - - -
64Pc
I n calculation of parameters a and b unit of R should be con-
sistent with the units chosen for Tc and Pc. Another useful
result from this analysis is estimation of critical compress-
ibility factor through Eq. (5.35). Rearranging this equation
and using definition of Zc from Eq. (2.8) gives
PcVc 3
(5.37) Zc - - - 0.375
RTc 8
Equation (5.37) indicates that value of Zc is the same for all
compounds. Values of Zc given in Table 2.1 varies from 0.28
to 0.21 for most hydrocarbons. Therefore, vdW EOS signifi-
cantly overpredicts values of Zc (or Vc) and its performance
in the critical region is quite weak. Similar approaches can
be used to determine EOS parameters and Zc for any other
EOS. r
Since the introduction of the vdW EOS as the first cubic
equation 130 years ago, dozens of cubic EOSs have been pro-
posed, many of them developed in recent decades. The math-
ematical simplicity of a cubic EOS in calculation of thermo-
dynamic properties has made it the most attractive type of
EOS. When van der Waals introduced Eq. (5.21) he indicated
that parameter a is temperature-dependent. It was in 1949
when Redlich and Kwong (RK) made the first modification
to vdW EOS as [11]
RT a
(5.38) P -- - -
V - b V ( V + b)
where parameter a depends on temperature as ac/ T ~ in
which ac is related to Tc and Pc. Parameters a and b in Eq.
(5.38) are different from those in Eq. (5.22) but they can be
obtained in a similar fashion as in Example 5.1 (as shown
later). The repulsive terms in Eqs. (5.38) and (5.22) are iden-
tical. Performance of RK EOS is much better than vdW EOS;
however, it is mainly applicable to simple fluids and rare gases
s uch as Kr, CH4, or O2, but for heavier and complex com-
pounds it is not a suitable PVT relation.
The RK EOS is a source of many modifications that began
in 1972 by Soave [12]. The Soave modification of Redlich-
Kwong equation known as SRK EOS is actually a modifica-
tion of parameter a in terms of temperature. Soave obtained
parameter a in Eq. (5.38) for a number of pure compounds us-
ing saturated liquid density and vapor pressure data. Then he
correlated parameter a to reduced temperature and acentric
factor. Acentric factor, co, defined by Eq. (2.10) is a parameter
that characterizes complexity of a molecule. For more com-
plex and heavy compounds value of o~ is higher than simple
molecules as given in Table 2.1. SRK EOS has been widely
used in the petroleum industry especially by reservoir engi-
neers for phase equilibria calculations and by process engi-
neers for design calculations. While RK EOS requires Tc and
Pc to estimate its parameters, SRK EOS requires an additional
parameter, namely a third parameter, which in this case is oJ.
As it will be seen later that while SRK EOS is well capable of
calculating vapor-liquid equilibrium properties, it seriously
underestimates liquid densities.
Another popular EOS for estimation of phase behavior
and properties of reservoir fluids and hydrocarbon systems
is Peng-Robinson (PR) proposed in the following form [13]:
RT a
(5.39) P - - -
V - b V ( V + b) + b( V - b)
where a and b are the two parameters for PR EOS and are
calculated similar to SRK parameters. Parameter a was cor-
related in terms of temperature and acentric factor and later it
was modified for properties of heavy hydrocarbons [14]. The
original idea behind development of PR EOS was to improve
liquid density predictions. The repulsive term in all four cubic
equations introduced here is the same. I n all these equations
TABLE 5. 1--Constants in Eq. (5.40) for four common cubic EOS (with permission from Ref. [15]).
Equat i on /A 1 b/2 ac ~ b Zc
vdW 0 0 27 RzT~ 1 RZ
Pe ~ 0.375
RI~ 1 0 0"42748R2 T2 Tr 1/2 ~ 0.333
Pc c
, 0 04274s.2 pc 0.333
f,0 = 0. 48 + 1.574c0 - 0. 176w 2
PR 2 - 1 0"45724R2T~pc [1 + f w( l - Trl / 2) ] 2 0"07780RTcpc 0. 307
f,o ---- 0. 37464 + 1. 5422&0 -- 0. 2699
when a -- b = 0, t he equat i on r educes to i deal gas law, Eq.
(5.14). I n addi t i on, all equat i ons sat i sfy t he cri t eri a set by Eqs.
( 5. 18) -( 5. 20) as wel l as Eq. (5.9). For exampl e, consi der t he
PR EOS expr essed by Eq. (5.39). To show t hat cr i t er i a set
by Eq. (5.18) ar e satisfied, t he l i mi t s of all t er ms as V -* oo
( equi val ent t o P -~ 0) shoul d be cal cul at ed. I f bot h si des of
Eq. (5.39) are mul t i p l i ed by V / RT and t aki ng t he l i mi t s of all
t er ms as V ~ oo ( or P --~ 0), t he first t er m i n t he RHS ap-
p r oaches uni t y whi l e t he second t er m ap p r oaches zero and
we get Z ~ 1, whi ch is t he EOS for t he i deal gases.
Rei d et al. [15] have p ut vdW, RK, SRK, and PR t wo-
p ar amet er cubi c EOS i nt o a p r act i cal and uni fi ed fol l owi ng
form:
RT a
(5.40) P - - -
V - b V 2 + ul bV + u2b 2
wher e Ua and u2 ar e t wo i nt eger val ues speci fi c for each cubi c
equat i on and ar e gi ven in Table 5.1. Par amet er a is i n gener al
t emp er at ur e- dep endent and can be expr essed as
(5.41) a = aco~
wher e ot is a di mensi onl ess t emp er at ur e- dep endent p ar ame-
t er and usual l y is expr essed in t er ms of r educed t emp er at ur e
(Tr = T/ Tc) and acent ri c f act or as gi ven in Table 5.1. For bot h
vdW and RK equat i ons t hi s p ar amet er is unity. Par amet er s
ua and u2 i n Eq. (5.40) ar e t he same for bot h RK and SRK
equat i ons, as can be seen in Table 5.1, but vdW and PR equa-
t i ons have di fferent val ues for t hese p ar amet er s. Equat i on
(5.40) can be conver t ed i nt o a cubi c f or m equat i on si mi l ar to
Eq. (5.23) but in t er m of Z r at her t han V:
Z 3- (1 + B - Ul B) g 2 + ( A + u2 B 2 - t t l B -- UlB2)Z
(5.42) - AB - uz B 2 - u2B 3 = 0
aP b P
wher e A- - and B = - -
R2T z RT
in whi ch p ar amet er s A and B as wel l as all t er ms in Eq. (5.42)
ar e di mensi onl ess. Par amet er s a and b and Zc have been de-
t er mi ned in a way si mi l ar to t he met hods shown in Examp l e
5. I . Z~ for bot h RK and SRK is t he same as 1/3 or 0.333 whi l e
for t he PR it is l ower and equal to 0.307 for all comp ounds.
As it will be shown l at er p er f or mance of all t hese equat i ons
near t he cri t i cal r egi on is weak and l eads to l arge er r or s for
cal cul at i on of Zc. Pr edi ct i on of an i sot her m by a cubi c EOS is
shown i n Fig. 5.9. As is seen i n t hi s figure, p r essur e p r edi ct i on
for an i sot her m by a cubi c EOS i n t he t wo- p hase r egi on is not
rel i abl e. However, i sot her ms out si de t he t wo- p hase envel ope
ma y be p r edi ct ed by a cubi c EOS wi t h a r easonabl e accuracy.
I n cal cul at i on of Z for sat ur at ed l i qui d and sat ur at ed vap or at
t he same T and P, Eq. (5.42) shoul d be sol ved at once, whi ch
Actual I sot herm
k ~ Predicted by Cubi c
Voltune, V
FI G. 5. 9 ---Predi cti ofl of i sot herms by a cu-
bic EO S,
gives t hr ee r oot s for Z. The l owest val ue of Z cor r esp onds
to sat ur at ed l i qui d, t he hi ghest r oot gives Z for t he sat ur at ed
vapor, and t he mi ddl e r oot has no physi cal meani ng.
5. 5. 2 S ol ut i on of Cubic Equat i ons o f St at e
Equat i on (5.42) can be sol ved t hr ough sol ut i on of t he follow-
i ng gener al cubi c equat i on [16, 17]:
( 5. 43) Z 3 +a l Z2 +a2Z +a 3 -~ 0
Let' s defi ne p ar amet er s Q, L, D, $1, and $2 as
O _ 3a2 - a l 2
9
9a l a 2 -- 27a3 - 2a~
L=
54
(5.44)
D = Q3 + L 2
S 1 : ( L --}- %ffO)l/3
$2 = ( L - ~/ O) l / 3
The t ype and number of r oot s of Eq. (5.43) dep ends on t he
val ue of D. I n cal cul at i on of X 1/3 if X < 0, one may use
X1/ 3 = - ( - X) 1/3.
I f D > 0 Eq. (5.43) has one real r oot and t wo comp l ex con-
j ugat e root s. The real r oot is gi ven by
(5.45) Z1 = $1 + Sz - al / 3
I f D = 0 all r oot s ar e real and at l east t wo ar e equal . The
unequal r oot is gi ven by Eq. (5.45) wi t h $1 = $2 = L 1/3. The
5. PV T REL ATI ONS AND EQUATI ONS OF STATE 2 0 7
t wo equal r oot s ar e
( 5.46) Z2 = Z3 : - L 1/3 - al / 3
I f D < 0 all r oot s ar e r eal and unequal . I n t hi s case $1 and
Sz ( Eq. 5.44) c a nnot be cal cul at ed and t he c omp ut a t i on is
si mp l i f i ed by use of t r i gonome t r y as
Z1 = 2- ~ZQ Cos (-130 + 120 ~ a13
)al
Z2 = 2~/ - ~ Cos 70 + 240 ~ 3
( 5. 47)
Z3 = 2 - , r Cos (31-0) a13
L
whe r e Cos 0 -
whe r e 0 is i n degr ees. To check val i di t y of t he sol ut i on, t he
t hr ee r oot s mus t sat i sfy t he f ol l owi ng r el at i ons
Z 1 + Z2 + Z3 = - a l
( 5. 48) Z1 Z2 + 22 Z3 + Z3 Zl = a2
Z 1 Z 2 Z 3 = - a 3
A c omp a r i s on of Eq. ( 5.42) and ( 5. 43) i ndi cat es t hat t he
f ol l owi ng r el at i ons exi st bet ween coef f i ci ent s ai ( s) and EOS
p a r a me t e r s
al = - ( I + B - ul B )
( 5. 49) a2 = A + uz B 2 - ul B - uaB 2
a3 = - AB - u2 B2 - u2 B3
F or t he case t hat t her e ar e t hr ee di f f er ent r eal r oot s ( D < 0),
Z liq is equal t o t he l owes t r oot (Z1) whi l e Z Vap is equal t o t he
hi ghes t r oot (Z3). The mi ddl e r oot (Z2) is di s r egar ded as p hys-
i cal l y meani ngl es s . Equa t i on ( 5.42) ma y al so be sol ved by suc-
cessi ve s ubs t i t ut i on met hods ; however , ap p r op r i at e f or ms of
t he e qua t i on and i ni t i al val ues ar e di f f er ent f or va p or and l i q-
ui d cases. F or examp l e, f or gases t he best i ni t i al val ue f or Z is
1 whi l e f or l i qui ds a good i ni t i al guess is b P/ RT [ 1 ]. Sol ut i on of
cubi c equat i ons t hr ough Eq. ( 5.42) is s hown i n t he f ol l owi ng
examp l e.
Ex ampl e 5. 2- - Es t i ma t e mol a r vol ume of s at ur at ed l i qui d and
va p or f or n- oct ane at 279. 5~ and p r es s ur e of 19.9 ba r f r om
t he RK, SRK, and PR cubi c EOS. Val ues of V L and V v ex-
t r act ed f r om t he exp er i ment al dat a ar e 304 and 1216 cm3/ mol ,
r esp ect i vel y [18]. Al so es t i mat e t he cr i t i cal vol ume.
Sol ut i on- - To us e SRK and PR EOS p ur e c omp one nt dat a
f or n-C8 ar e t aken f r om Tabl e 2.1 as Tc = 295. 55~ ( 568. 7 K) ,
Pc = 24. 9 bar, co = 0. 3996, and Vc = 486. 35 cm3/ mol . Whe n T is
i n K, P is i n bar, and V is i n cm3/ mol , val ue of R f r om Sect i on
1.7.24 is 83. 14 c m 3 9 ba r / mol . K. Sa mp l e cal cul at i on is s hown
her e f or SR K EOS. Tr = ( 279. 5 +273. 15) / 568. 7 = 0. 972.
F r om Tabl e 5.1, /A1 = 1, u2 = 0 and aSRK and bSRK ar e cal cu-
l at ed as
f~ = 0. 48 + 1.574 x 0. 3996 - 0. 176 x ( 0. 3996) 2 = 1. 08087
0. 42748 ( 83. 14) 2 x ( 568. 7) 2
asRx = 24. 9
x [1 + 1. 08087 x (1 -- 0.97171/ 2)] 2
= 3. 957 x 107 cm6/ mol 2.
bsRx = 0. 08664 x 83. 14 568. 7 = 164.52 cm3/ mol .
24. 9
Par amet er s A and B ar e cal cul at ed f r om Eq. ( 5.42) :
3. 957 x 107 x 19.9
A = = 0. 373
( 83. 14) 2 ( 552. 65) 2
and
164.52 x 19.9
B = = 0. 07126
83. 14 552. 65
Coef f i ci ent s al , a2, and a3 ar e cal cul at ed f r om Eq. ( 5. 49) as
al = - ( 1 + 0. 07126 - 1 x 0. 07126) -- - 1
a2 = 0. 373 + 0 0.071262 - 1 0. 07126 - 1 x 0. 071262
= 0. 29664
a3 = - 0. 37305 0. 07126 - 0 x 0.071262 - 0 x 0.071263
---- - 0. 026584
F r om Eq. ( 5.44) , Q = - 0. 01223, L --- 8.84 10 -4, and D =
- 1.048 x t 0 -6. Si nce D < 0, t he s ol ut i on is gi ven by Eq. ( 5.47) .
0 = Cos-1( 8. 84 x 10- 4/ x/ - ( - 0. 01223) 3) = 492 ~ and t he r oot s
ar e Z1 = 0. 17314, Z2 = 0. 28128, and Z3 = 0. 54553. Accep t -
abl e r esul t s ar e t he l owes t and hi ghes t r oot s whi l e t he i n-
t er medi at e r oot is not usef ul : Z L = Z1 = 0. 17314 and Z v =
Z2 = 0. 54553. Mol ar vol ume, V , can be cal cul at ed f r om Eq.
( 5.15) : V = ZRT/ P i n whi c h T = 552. 65 K, P = 19.9 bar, and
R = 83. 14 c m3- ba r / mol . K; t her ef or e, V L= 399. 9 cm3/ mol
and V v ---- 1259.6 cm3/ mol . F r om Tabl e 5.1, Zc = 0. 333 and
Vc is cal cul at ed f r om Eq. (2.8) as Vc = ( 0. 333 x 83. 14 x
568. 7) / 24. 9 = 632. 3 cm3/ mol . Er r or s f or V L, V v, and Vc ar e
31.5, 3.6, and 30%, respect i vel y. I t s houl d be not ed t hat Zc
can al so be f ound f r om t he s ol ut i on of cubi c e qua t i on wi t h
T = Tc and P = Pc. However , f or t hi s case D > 0 and t her e is
onl y one s ol ut i on whi c h is obt ai ned by Eq. ( 5.45) wi t h s i mi l ar
answer. As is seen i n t hi s examp l e, l i qui d and cr i t i cal vol ume s
ar e gr eat l y over es t i mat ed. Summa r y of r esul t s f or al l f our cu-
bi c equat i ons ar e gi ven i n Tabl e 5.2. t
5 . 5 . 3 Vo l u me T r a n s l a t i o n
I n p r act i ce t he SR K and PR equat i ons ar e wi del y us ed f or VLE
cal cul at i ons i n i ndus t r i al ap p l i cat i ons [ 19-21 ]. However , t hei r
abi l i t y t o p r edi ct vol ume t r i c dat a esp eci al l y f or l i qui d syst ems
TABLE 5. 2- - Prediction of saturated liquid, vapor and critical molar vol umes for n-octane in
Example 5. 2.
Equation V L, cma/mol %D V v, cma/mol %D Vc, cm3/mol %D
Data* 304.0 . . . 1216.0 . . . 486.3
RK 465.9 53.2 1319.4 8.5 632.3 i6"
SRK 399.9 31.5 1259.6 3.6 632.3 30
PR 356.2 17.2 1196.2 - 1. 6 583.0 19.9
Source: V L and V v from Ref. [18]; Vc from Table 2.1.
is weak. Usually the SRK equation predicts densities more ac-
curately for compounds with low acentric values, while the
PR predicts better densities for compounds with acentric fac-
tors near 0.33 [21]. For this reason a correction term, known
as volume translation, has been proposed for improving vol-
umetric prediction of these equations [8, 15, 19, 22]:
(5.50) V = V E~ - c
where c is the volume translation parameter and has the same
unit as the molar volume. Equation (5.50) can be applied to
both vapor and liquid volumes. Parameter c mainly improves
liquid volume predictions and it has no effect on vapor pres-
sure and VLE calculations. Its effect on vapor volume is negli-
gible since V v is very large in comparison with c, but it greatly
improves prediction of liquid phase molar volumes. Values of
c have been determined for a number of pure components up
to C10 for both SRK and PR equations and have been included
in references in the petroleum industry [19]. Peneloux et al.
[22] originally obtained values of c for some compounds for
use with the SRK equation. They also suggested the following
correlation for estimation of c for SRK equation:
Rrc
(5.51) c = 0.40768 (0.29441 - ZRA) - -
Pc
where ZRA is the Rackett parameter, which will be discussed
in Section 5.8.1. Similarly Jhaveri and Yougren [23] obtained
parameter c for a number of pure substances for use with PR
EOS and for hydrocarbon systems have been correlated to
molecular weight for different families as follows:
Cp = bpR (1 -- 2.258M~ -~
(5.52) CN = bva (1 -- 3.004M~ ~
CA = bpR (1 -- 2.516M~ ~176176
where bpR refers to parameter b for the PR equation as given
in Table 5.1. Subscripts P, N, and A refer to paraffinic, naph-
thenic, and aromatic hydrocarbon groups. The ratio of c/b
is also called shift parameter. The following example shows
application of this method.
Exampl e 5. 3--For the system of Example 5.2, estimate V L
and V v for the PR EOS using the volume translation method.
Sol ut i on- - For n-Cs, from Table 2.1, M= 114 and beR are
calculated from Table 5.1 as 147.73 cm3/mol. Since the hy-
drocarbon is paraffinic Eq. (5.51) for Cp should be used,
c = 7.1 cma/mol. From Table 5.2, V L(PR) = 356.2 and V v(vR) =
1196.2 cma/mol. From Eq. (5.50) the corrected molar volumes
are V L = 356.2 - 7.1 = 3491 and V v = 1196.2 - 7.1 = 1189.1
cm3/mol. By the volume translation correction, error for V L
decreases from 17.2 to 14.8% while for V v it has lesser effect
and it increases error from -1.6% to -2.2%. r
As is seen in this example improvement of liquid volume
by volume translation method is limited. Moreover, estima-
tion of c by Eq. (5.51) is limited to those compounds whose
ZRA is known. With this modification at least four parameters
namely Tc, Pc, co, and c must be known for a compound to
determine its volumetric properties.
5. 5. 4 Ot her Types of Cubi c Equat i ons of St at e
I n 1972 Saove for the first time correlated parameter a in a
cubic EOS to both Tr and co as given in Table 5.1. Since then
this approach has been used by many researchers who tried
to improve performance of cubic equations. Many modifica-
tions have been made on the form of f~ for either SRK or
PR equations. Graboski and Daubert modified the constants
in the f~ relation for the SRK to improve prediction of va-
por pressure of hydrocarbons [24]. Robinson and Peng [14]
also proposed a modification to their fo~ equation given in
Table 5.1 to improve performance of their equation for heav-
ier compounds. They suggested that for the PR EOS and for
compounds with ~o > 0.49 the following relation should be
used to calculate f~:
(5.53) fo, = 0.3796 + 1.485co - 0.1644o92 + 0.01667co 3
Some other modifications give different functions for param-
eter ot in Eq. (5.41). For example, Twu et al. [25] developed
the following relation for the PR equation.
ct = Tr 0"171813 exp [0.125283 (I - Tr1"77634)]
(5.54) + (.0 {Tr -0"607352 exp [0.511614 (I - V2~
- Tr ~ exp [0.125283 (1 - T)77634)] }
Other modifications of cubic equations have been derived by
suggesting different integer values for parameters ui and uz in
Eq. (5.40). One can imagine that by changing values of ul and
u2 in Eq. (5.40) various forms of cubic equations can be ob-
tained. For example, most recently a modified two-parameter
cubic equation has been proposed by Moshfeghian that cor-
responds to ut = 2 and u2 = - 2 and considers both parame-
ters a and b as temperature-dependent [26]. Poling et al. [8]
have summarized more than two dozens types of cubic equa-
tions into a generalized equation similar to Eq. (5.42). Some
of these modifications have been proposed for special sys-
tems. However, for hydrocarbons systems the original forms
of SRK and PR are still being used in the petroleum industry.
The most successful modification was proposed by Zudke-
fitch and Joffe [27] to improve volumetric prediction of RK
EOS without sacrificing VLE capabilities. They suggested that
parameter b in the RK EOS may be modified similar to Eq.
(5.41) for parameter a as following:
(5.55) b = bRKfl
where fl is a dimensionless correction factor for parameter
b and it is a function of temperature. Later Joffe et al. [28]
determined parameter 0~in Eq. (5.41) and fl in Eq. (5.55) by
matching saturated liquid density and vapor pressure data
over a range of temperature for various pure compounds. I n
this approach for every case parameters a and/ ~ should be
determined and a single dataset is not suitable for use in all
cases. SRK and ZJRK are perhaps the most widespread cu-
bic equations being used in the petroleum industry, especially
for phase behavior studies of reservoir fluids [19]. Other re-
searchers have also tried to correlate parameters oe and fl in
Eqs. (5.41) and (5.55) with temperature. Feyzi et al. [29] cor-
related a 1/2 and fl1/2 for PR EOS in terms Tr and co for heavy
reservoir fluids and near the critical region. Their correla-
tions particularly improve liquid density prediction in com-
parison with SRK and PR equations while it has similar VLE
5. PV T REL ATI ONS AND EQUATI ONS OF STATE 2 0 9
N
1~0
0.8 ~
0 . 6
0. 4'
0 . 2 "
0 . 0 '
L4
A ~ A A ~
o.s E6 0:8 0.9
-'. ~ =~rJa~' ~ 6 ~ 4~ ~ ~h,,~, ~ ~, ~ ~ d k , ~ ~ , , ~ ~k ~
O Z~ ,exp ~
& Zv,exp ~ & . &
o Zc, e)q:, ~
Q z ~ . ~ ~ ,
Z1, calc
---" Z%Ca|C
Re d u c e d T e mp e r a t u r e , T r
FI G . 5 . 1 0 - - Pr e d i c t i on of s at ur at i on c ur v e s f or e t ha ne us i ng a modi f i ed PR E O S [ 29 ].
prediction capabilities. Another improvement in their corre-
lation was prediction of saturation curves near the critical
region. This is shown for prediction of compressibility fac-
tor of saturated liquid and vapor curves as well as the critical
point for methane and ethane in Fig. 5.10.
Prediction of isotherms by a cubic EOS is shown on PV di-
agram in Fig. 5.9. As shown in this figure in the two-phase
region the prediction of isotherm is not consistent with true
behavior of the isotherm. I n addition, performance of these
cubic equations in calculation of liquid densities and derived
thermodynamic properties such as heat capacity is weak. This
indicates the need for development of other EOS. Further in-
formation on various types of cubic EOS and their character-
istics are available in different sources [30-34].
5 . 5 . 5 A ppl i c a t i o n t o Mi xt ur e s
Generally when a PVT relation is available for a pure sub-
stance, the mixture property may be calculated in three ways
when the mixture composition (mole fraction, x~) is known.
The first approach is to use the same equation developed for
pure substances but the input parameters (To, Pc, and to) are
estimated for the mixture. Estimation of these pseudocritical
properties for petroleum fractions and defined hydrocarbon
mixtures were discussed in Chapter 3. The second approach
is to estimate desired physical property (i.e., molar volume
or density) for all pure compounds using the above equations
and then to calculate the mixture property using the mixture
composition through an appropriate mixing rule for the prop-
erty (i.e., Eq. (3.44) for density). This approach in some cases
gives good estimate of the property but requires large calcu-
lation time especially for mixtures containing many compo-
nents. The third and most widely used approach is to calculate
EOS parameters (parameters a and b) for the mixture using
their values for pure components and mixture composition.
The simplest EOS for gases is the ideal gas law given by Eq.
(5.14). When this is applied to component i with n~ moles in
a mixture we have
(5.56) PV//t = n~RT
where V/t is the total volume occupied by i at T and P of the
mixture. For the whole mixture this equation becomes
(5.57) PV t = nt RT
where V t is the total volume of mixture (V t = EV/t) and n t is
the total number of moles (n t -- En/). By dividing Eq. (5.56)
by Eq. (5.57) we get
r~ Vii t
(5.58) Yi -- --
n V t
where yi is the mole fraction of i in the gas mixture. The
above equation indicates that in an ideal gas mixture the mole
fractions and volume fractions are the same (or mol% of i =
vol% of i). This is an assumption that is usually used for gas
mixtures even when they are not ideal.
For nonideal gas mixtures, various types of mixing rules
for determining EOS parameters have been developed and
presented in different sources [6, 8]. The mixing rule that is
commonly used for hydrocarbon and petroleum mixtures is
called quadratic mi xi ng rule. For mixtures (vapor or liquid)
with composition xi and total of N components the following
equations are used to calculate a and b for various types of
cubic EOS:
N N
i =1 j = l
N
(5.60) bmix = ~xqbi
i =1
where aij is given by the following equation:
(5.61) aq = (aiai)l/2(1 - kq)
For the volume translation c, the mixing rule is the same as
for parameter b:
N
(5.62) Cmi~ = y~xi ci
i =1
I n Eq. (5.61), kq is a dimensionless parameter called binary in-
teractionparameter (BIP), where kqi = 0 andkq = kji. For most
(5.59) amix = ~] ~ xixjai j
Comp.
TABLE 5. 3--Recommended BIPs for SRK and PR EOS [19].
PR EOS SRK EOS
N2 CO2 H2S N2 CO2 H2S
N2 0.000 0.000 0.130 0.000 0.000 0.120
CO2 0.000 0.000 0.135 0.000 0.000 0.135
H2S 0.130 0.135 0.000 0.120 0.120 0.000
C1 0.025 0.105 0.070 0.020 0.120 0.080
C2 0.010 0.130 0.085 0.060 0.150 0.070
C3 0.090 0.125 0.080 0.080 0,150 0.070
iC4 0.095 0.120 0.075 0.080 0.150 0.060
nC4 0.090 0.115 0.075 0.800 0.150 0.060
iC5 0.100 0.115 0.070 0.800 0.150 0.060
nC5 0.110 0.115 0.070 0.800 0.150 0.060
C6 0.110 0.115 0.055 0.800 0.150 0.050
C7+ 0.110 0.115 0.050 0.800 0.150 0.030
Values recommended for PR EOS by Ref. [6] are as follows: N2/CO2: -0.013; N2/CI: 0.038; C1/CO2: 0.095; N2/C2:
0.08; C1/C2: 0.021.
hydrocarbon systems, k/l = 0; however, for the key hydrocar-
bon compounds in a mixture where they differ in size value
of k/i is nonzero. For example, for a reservoir fluid that con-
tains a considerable amount of methane and C7+ the BIP for
C1 and C7 fractions cannot he ignored. For nonhydrocarbon-
hydrocarbon pairs k/i values are nonzero and have a signif-
icant impact on VLE calculations [20, 35]. Values of k/i for
a particular pair may be determined from matching exper-
imental data with predicted data on a property such as va-
por pressure. Values of k/i are specific to the particular EOS
being used. Some researchers have determined k/j for SRK
or PR equations. Values of BIP for N2, CO2, and methane
with components in reservoir fluids from C1 to C6 and three
subfractions of C7+ for PR and SRK are tabulated by Whit-
son [19]. Values that he has recommended for use with SRK
and PR equations are given in Table 5.3. There are some gen-
eral correlations to estimate BIPs for any equation [36, 37].
The most commonl y used correlation for estimating BIPs
of hydrocarbon-hydrocarbon (HC-HC) systems is given by
Chueh and Prausnitz [37]:
(5.63) kq ----- A - (V~i)l/3 + (V)/3)J /
where Vci and V~j are critical molar volume of components
i and j in cm3/mol. Originally A = 1 and B = 3; however, in
practical cases B is set equal to 6 and A is adjusted to mat ch
saturation pressure and other variable VI E data [20, 38]. For
most reservoir fluids, A is within 0.2-0.25; however, as is seen
in Chapter 9 for a Kuwaiti oil value of A was found as 0.18.
As discussed by Poling et al. [8], Tsonopoulos recommends
the original Chueh-Prausnitz relation (A = 1 and B = 3) for
nonpolar compounds. Pedersen et al. [39] proposed another
relation for calculation of BIPs for HC-HC systems. Their cor-
relation is based on data obtained from North Sea reservoir
fluids and it is related to molecular weights of components
i and ] as k4i ~ O.O01MilMi where Mi > Mj. Another corre-
lation was proposed by Whitson [40] for estimation of BIPs
of methane and C7+ fraction components based on the data
presented by Katz and Firoozabadi [36] for use with PR EOS.
His correlation is as: k U = 0.14 SGi - 0.0688, where 1 refers
to methane and j refers to the CT+(j) fraction, respectively.
Equations (5.59)-(5.62) can be applied to either liquid or
vapor mixtures. However, for the case of vapor mixtures with
N components, mole fraction yi should be used. Expansion
of Eq. (5.59) for a ternary gas mixture (N = 3) becomes
amix "--- y2all + y2a22 + y2a33 + 2ylY2a12 + 2ylx3a13 + 2y2Yaa23
(5.64)
where all = al, a22 ----- a2, and a33 ----- a3. I nteraction coefficients
such as a12 can be found from Eq. (5.61): a12 = a~d]-~(1 - k12)
whe r e k12 may be taken from Table 5.3 or estimated from Eq.
(5.63). a13 and a23 can be calculated in a similar way.
5 . 6 N ON C UB I C EQUA T I ON S OF STATE
The main reason for wide range application of cubic EOS
is their application to both phases of liquids and vapors,
mathematical simplicity and convenience, as well as pos-
sibility of calculation of their parameters through critical
constants and acentric factor. However, these equations are
mainly useful for density and phase equilibrium calculations.
For other thermodynamic properties such as heat capacity
and enthalpy, noncubic equations such as those based on
statistical associating fluid theory (SAFT) or perturbed hard
chain theory (PHCT). Some of these equations have been
particularly developed for special mixtures, polar molecules,
hard sphere molecules, and near critical regions. Summary
of these equations is given by Poling et al. [8]. I n this sec-
tion three important types of noncubic EOS are presented:
(I) virial, (2) Carnahan-Starling, and (3) modified Benedict-
Webb-Rubin.
5. 6. 1 Virial Equat i on o f State
The most widely used noncubic EOS is the virial equation or
its modifications. The original virial equation was proposed
in 1901 by Kammerlingh-Onnes and it may be written either
in the form of polynomial series in inverse volume (pressure
explicit) or pressure expanded (volume explicit) as follows:
B C D
(5.65) Z = I + V + ~ + V ~ +"
Z - - l +
( 5. 6 6 ) + ( D - 3 B C + 2B3\
~- ~ )m3 +.
\
TABLE 5.4--Second virial coefficients for several gases [41~
Temperature, K
Compound 200 300 400 500
N2 -35. 2 -4. 2 9 16.9
CO2 - 122.7 60.5 -29. 8
CH4 - i 05 - 42 - 15 -0. 5
C2H6 -410 -182 - 96 - 52
C3H8 . . . -382 -208 - 124
Note: Values of B are given in cma/mol.
wher e B, C, D . . . . ar e cal l ed second, t hi rd, and f our t h vi ri al co-
efficients and t hey ar e all t emp er at ur e- dep endent . The above
t wo forms of vi ri al equat i on are t he same and t he second
equat i on can be der i ved f r om t he first equat i on ( see Pr obl em
5.7). The second f or m is mor e p r act i cal to use si nce usual l y T
and P are avai l abl e and V shoul d be est i mat ed. The number
of t er ms in a vi ri al EOS can be ext ended t o i nfi ni t e t er ms
but cont r i but i on of hi gher t er ms r educes wi t h i ncr ease i n
p ower of P. Virial equat i on is p er hap s t he most accur at e PVT
r el at i on for gases. However, t he di ffi cul t y wi t h use of vi ri al
equat i on is avai l abi l i t y of its coeffi ci ent s especi al l y for hi gher
t erms. A l arge numbe r of dat a ar e avai l abl e for t he second
vi ri al coeffi ci ent B, but less dat a ar e avai l abl e for coeffi ci ent
C and very few dat a are r ep or t ed for t he f our t h coeffi ci ent
D. Dat a on val ues of vi ri al coeffi ci ent s for several comp ounds
ar e gi ven in Tables 5.4 and 5.5. The vi ri al coeffi ci ent has fi rm
basi s in t heor y and t he met hods of st at i st i cal mechani cs al l ow
der i vat i on of its coefficients.
B r ep r esent s t wo- body i nt er act i ons and C r ep r esent ed
t hr ee- body i nt er act i ons. Si nce t he chance of t hr ee- body in-
t er act i on is less t han t wo- body i nt er act i on, t herefore, t he i m-
p or t ance and cont r i but i on of B is much gr eat er t han C. F r om
quant um mechani cs it can be shown t hat t he second vi ri al
coeffi ci ent can be cal cul at ed f r om t he knowl edge of p ot ent i al
f unct i on ( F) for i nt er mol ecul ar forces [6]:
oo
(5.67) S = 2zrNA ] (1 -- e-r(r)/ kr)r2dr
0
wher e NA is t he Avogadro' s number (6.022 x 1023 mo1-1) and
k is t he Bol t zman' s const ant (k = R/NA). Once t he r el at i on for
F is known, B can be est i mat ed. For exampl e, if t he fl ui d
fol l ows har d sp her e p ot ent i al funct i on, one by subst i t ut i ng
Eq. (5.13) for F i nt o t he above equat i on gives B = (2/3)zr NA 33.
Vice versa, const ant s in a p ot ent i al r el at i on (e and a) may
be est i mat ed f r om t he knowl edge of vi ri al coefficients. F or
mi xt ur es, Bmi can be cal cul at ed f r om Eq. (5.59) wi t h a bei ng
r ep l aced by B. For a t er nar y syst em, B can be cal cul at ed f r om
Eq. (5.64). Bii is cal cul at ed f r om Eq. (5.67) usi ng Fii wi t h r
and eli gi ven as [6]
1
(5.68) a# = ~(ai + a/ )
(5.69) eli = (eiSi) 1/2
TABLE 5. 5--Sample values of different virial coefficients for
several compounds [1].
Compound T, ~ B, cma/mol C, cm6/mol 2
Methane a 0 -53. 4 2620
Ethane 50 - 156.7 9650
Steam (H20) 250 -152.5 -5800
Sulfur dioxide (SO2) 157.5 -159 9000
aFor met hane at 0~ t he fourt h virial coefficient D is 5000 cmg/ mol 3,
Anot her f or m of Eq. (5.59) for cal cul at i on of Bm~ can be wri t -
t en as fol l owi ng:
N I NN
Bmix = Ey i B i i + "~ Z Ey i Yj Si / wher e 8i/~-2Bij - Bi i - B i i
i=l i=l j=l
(5.70)
Ther e ar e several cor r el at i ons devel op ed based on t he the-
or y of cor r esp ondi ng st at e p r i nci p l es t o est i mat e t he second
vi ri al coeffi ci ent s in t er ms of t emp er at ur e. Some of t hese rel a-
t i ons cor r el at e B/V c t o Tr and o9. Pr ausni t z et al. [6] r evi ewed
some of t hese r el at i ons for est i mat i on of t he second vi ri al
coefficients. The r el at i on devel oped by Tsonopoul os [42] is
useful to est i mat e B f r om To, Pc, and o9.
BPc = B(0) + OgB(1)
RTc
0.330 0.1385 0.0121 0.000607
B (~ = 0.1445
r/ r3 T/
0.331 0.423 0.008
B (1) = 0.0637 + - -
Tr 2 Tr 3 T r s
( 5 . 7 1 )
wher e Tr = T/Tc. There ar e si mp l er r el at i ons t hat can be used
for nor mal fluids [ 1 ].
BPc = B(O) + ogBO )
RTc
0.422
(5.72) B (~ = 0.083 -
Tr 1-6
0.172
B ( 1 ) = 0 . 1 3 9 - - -
Tr4.2
Anot her r el at i on for p r edi ct i on of second vi ri al coeffi ci ent s of
si mpl e fluids is gi ven by McGl ashan [43]:
(5.73) BPc = 0.597 - 0.462e ~176176
RTc
A gr ap hi cal comp ar i s on of Eqs. ( 5. 71) -( 5. 73) for p r edi ct i on
of second vi ri al coeffi ci ent of et hane is shown i n Fig. 5.11.
Coefficient B at l ow and moder at e t emp er at ur es is negat i ve
and i ncr eases wi t h i ncr ease in t emp er at ur e; however, as is
seen f r om t he above cor r el at i ons as T --+ ~, B ap p r oaches a
posi t i ve number.
To p r edi ct Bm~, for a mi xt ur e of known comp osi t i on, t he in-
t er act i on coeffi ci ent Bi / i s needed. This coeffi ci ent can be cal-
cul at ed f r om/ ~ and Bi / usi ng t he fol l owi ng r el at i ons [1, 15]:
B~/ = ~RTcii (B(O) + ogiiB(1) )
B (~ and B (1) are cal cul at ed t hr ough T~i = T/Tcij
(5.74)
o9~ + w/
o9i/-- 2
Tcij = (TdTcj)l/2(1 - kii)
Pii - Zij RTc~i
Vc#
z~ + zc/
Zci/ - 2
2
50
.. ~176 . ~ . . . . _. -~
-50
-150
-250 /4 . . . . Normal Fluids
u
"~ r Tsonopoulos _
~ o Experimental Data
f
"~ -350
o3
- 4 5 0 ? ~ ' ~ ' P ~ '
100 300 500 700 900 1100
Temperature, K
FIG. 5. 1 1 - - Predi ct i on of second virial coefficient for
ethane from different methods. Ex perimental data
from Table 5.4: McGl ashan, Eq. ( 5.73) ; Normal fluids,
Eq. ( 5.72) ; T sopoul os, Eq. ( 5.71) .
where k/i is the i nt eract i on coefficient and for hydr ocar bons of
similar size it is zero. B ~~ and B ~x) must be calculated from the
same relations used to calculate Bu and Bii. Anot her si mpl er
met hod t hat is fairly accurat e for light, nonp ol ar gases is the
geomet ri c mean:
Bij = ( ni nj ) 1/2
Bmixm(i~=lYiB1/ 2)2
The i mport ance of these relations is t hat at moderat e pres-
sures, Eq. (5.66) may be t runcat ed after the second t erm as
follows:
BP
(5.75) z = 1 + - -
RT
This equat i on is usually referred to as the t runcat ed virial
equat i on and may be used wi t h a reasonabl e degree of accu-
racy in cert ai n ranges of reduced t emperat ure and pressure:
Vr > 2.0 [i.e., (Pr < 0.5, Tr > 1), (0.5 < Pr < 1, Tr > 1.2), (1 <
Pr < 1.7, Tr > 1.5)]. At l ow-pressure range (Pr < 0.3), Eq.
(5.72) provides good predi ct i on for t he second virial coeffi-
cients for use in Eq. (5.75) [1].
A more accurat e form of virial equat i on for gases is ob-
t ai ned when Eq. (5.65) or (5.66) are t runcat ed after t he t hi rd
t erm:
B C
(5.76) Z = 1 + ff + V- 5
An equivalent form of this equat i on in t erms of P can be ob-
t ai ned by Eq. (5.66) wi t h three t erms excluding fourt h virial
coefficient and hi gher terms. Because of lack of sufficient
data, a general i zed correl at i on to predi ct the t hi rd virial co-
efficient, C, is less accurat e and is based on fewer data. The
general i zed correl at i on has t he following form [6]:
C _ (0.232Tr_0.25 + 0.468Tr5 ) x [1 - e(1-1"S9T2)]
(5.77) + de - ( 2"49-2' 30Tr )
where Vc is the mol ar critical vol ume in cm3/ mol and d is
a p ar amet er t hat is det ermi ned for several compounds, i.e.,
d = 0.6 for met hane, 1 for ethane, 1.8 for neopent ane, 2.5 for
benzene, and 4.25 for n-octane. I n general when Tr > 1.5 the
second t erm in the above equat i on is insignificant. A mor e
pract i cal and general i zed correl at i on for t hi rd virial coeffi-
cient was proposed by Orbey and Vera [44] for nonp ol ar com-
p ounds in a f or m similar to Eq. (5.71), whi ch was pr oposed
for the second virial coefficients:
C p2 _ C ~o) + o~C (I)
(RTr 2
0.02432 0.00313
(5.78) C ~~ = 0.01407 + - -
r/8 Ty5
0.0177 0.040 0.003 0.00228
C (1) = - 0. 02676 + ~ + T~ Tr 6 Tr l~
where C (~ and C (~ are di mensi onl ess paramet ers for simple
and correct i on t erms in the generalized correlation. Est i ma-
tion of the t hi rd virial coefficients for mi xt ures is quite diffi-
cult as there are t hree-way i nt eract i ons for C and it shoul d be
calculated f r om [6]:
(5.79) Cmi x- EEEy i y j y k Cq k
Met hods of est i mat i on of cross coefficients Ciik are not re-
liable [6]. For simplicity, generally it is assumed t hat Ciii =
Ciii = Ciii but still for a bi nary syst em at least t wo cross coef-
ficients of Cl12 and C~22 must be estimated. I n a bi nary system,
Cl12 expresses i nt eract i on of two molecules of comp onent 1
wi t h one mol ecul e of comp onent 2. Orbey and Vera [44] sug-
gest t he following rel at i on for cal cul at i on of Ciik as
( 5. 80) Cij k : (CijCikCjk) 1/3
where C# is evaluated f r om Eq. (5.78) usi ng Tr Pcij and ogii
obt ai ned from Eq. (5.74). This ap p r oach gives sat i sfact ory
estimates for bi nary systems.
There are cert ai n specific correl at i ons for the virial coeffi-
cients of some specific gases. For example, for hydrogen the
following correl at i ons for B and C are suggested [6]:
g
B = ~ bi X(2i-1)/4
1
C = 1310.5x 1/2 (1 + 2.1486x 3) [ i - exp (1 - x-3) ]
(5.81)
109.83 b
where x - ~- , a = 42.464, bz = -37. 1172,
b3 = -2. 2982, and b4 = - 3. 0484
where T is in K, B is in cm3/ mol, and C is in cm6/ mol 2. The
range of t emperat ure is 15-423 K and the average deviations
for B and C are 0.07 cm3/ mol and 17.4 cm6/ mo] 2, respectively
[6].
As det ermi nat i on of hi gher virial coefficients is difficult,
appl i cat i on of t runcat ed virial EOS is mai nl y limited t o gases
and for this reason t hey have little appl i cat i on in reservoir
fluid studies where a single equat i on is needed for bot h liquid
and vapor phases. However, t hey have wi de appl i cat i ons in
est i mat i on of propert i es of gases at low and moderat e pres-
sures. I n addition, special modi fi cat i ons of virial equat i on has
industrial applications, as di scussed in t he next section. Fr om
mat hemat i cal relations it can be shown t hat any EOS can be
convert ed i nt o a virial form. This is shown by t he following
example.
Exampl e 5. 4- - Conver t RK EOS i nt o the virial form and ob-
t ai n coefficients B and C i n t erms of EOS paramet ers.
Sol ut i on- - The RK EOS is given by Eq. (5.38). If bot h sides
of this equat i on are mul t i pl i ed by V / RT we get
PV V a
(5.82) Z -
RT V - b RT( V + b)
Assume x = b/ V and A = alRT, t hen the above equat i on can
be wri t t en as
1 1 1
(5.83) Z- - - l _ x A~x l +x
Si nce b < V , therefore, x < 1 and the t erms i n the RHS of
the above equat i on can be expanded t hr ough Taylor series
[16, 17]:
(5.84) f ( x) = s f(~)(x~ (x
n! - x~
n=0
where f(')(Xo) is the nt h order derivative d" f ( x) / dx" eval uat ed
at x = Xo. The zerot h derivative of f is defined to be f itself
and bot h 0! and 1! are equal to 1. Applying this expansi on rul e
at Xo = 0 we get:
i
- - 1 + x + x 2 ~ - X 3 + X 4 - ~- . . .
1- - x
(5.85)
1
- 1 - - X Ar X 2 - - x a+x 4 . . . .
l +x
I t shoul d be not ed t hat the above rel at i ons are valid when
Ixl < 1. Subst i t ut i ng t he above two rel at i ons i n Eq. (5.83) we
get
Z = (1 +x +x 2 " ~- X 3 "~- 9 " ) - - A 1- - x ( 1 - x + x 2 - - X 3 -~-' " ' )
V
(5.86)
I f x is repl aced by its defi ni t i on b/ V and A by a/ RT we have
b - a/ RT b 2 + ab / RT +b 3 - ab2/ RT
Z= I + - - + + +
V V 2 V 3
(5.87)
A comp ar i son wi t h Eq. (5.65) we get the virial coefficients i n
t erms of RK EOS par amet er s as follows:
a ab ab 2
(5.88) B= b- R ~ C= b 2+~ D= b 3- - R T
Consi deri ng the fact t hat a is a t emp er at ur e- dep endent
p ar amet er one can see t hat the virial coefficients are all
t emp er at ur e- dep endent paramet ers. Wi t h use of SRK EOS,
si mi l ar coefficients are obt ai ned but p ar amet er a also depends
on t he acent ri c factor as given i n Table 5.1. This gives bet-
ter est i mat i on of the second and t hi rd virial coefficients (see
Probl em 5.10) r
The following exampl e shows appl i cat i on of t r uncat ed
virial equat i on for cal cul at i on of vapor mol ar vol umes.
Exampl e 5. 5- - Pr op ane has vapor pressure of 9.974 bar at
300 K. Sat urat ed vapor mol ar vol ume is V v = 2036.5 cma/ mol
[Ref. 8, p. 4.24]. Calculate (a) second virial coefficient from
Eqs. (5.71)-(5.73), (b) t hi rd virial coefficient from Eq. (5.78),
(c) V v from virial EOS t r uncat ed after second t erm usi ng Eqs.
(5.65) and (5.66), (d) V v from virial EOS t r uncat ed after t hi rd
t erm usi ng Eqs. (5.65) and (5.66), and (e) V v from ideal gas
l aw.
Sol ut i on- - ( a) and (b): For pr opane from Table 2.1 we
get Tc = 96.7~ (369.83 K), Pc 42.48 bar, and c0 = 0.1523.
Tr = 0.811, Pr = 0.23, and R = 83.14 cm 3 9 bar/ mol - K. Second
virial coefficient, B, can be est i mat ed from Eqs. (5.71) or
(5.72) or (5.73) and the t hi rd virial coefficient from Eq. (5.78).
Results are given i n Table 5.6. (c) Truncat ed virial equat i on
after second t er m from Eq. (5.65) is Z = 1 + B/ V , whi ch is
referred to as V expansi on form, and from Eq. (5.66) is Z =
1 + BP/ RT, whi ch is the same as Eq. (5.75) and it is referred
to as P expansi on form. For the V expansi on (Eq. 5.65), V
shoul d be cal cul at ed t hr ough successive subst i t ut i on met hod
or from mat hemat i cal sol ut i on of the equat i on, whi l e i n P
expansi on form (Eq. 5.66) Z can be directly cal cul at ed from
T and P. Once Z is det ermi ned, V is cal cul at ed from Eq.
(5.15): V = ZRT/ P. I n part (d) virial equat i on is t r uncat ed
after t he t hi rd t erm. The V expansi on form reduces to Eq.
(5.76). Summar y of cal cul at i ons for mol ar vol ume is given
i n Table 5.6. The resul t s from V expansi on (Eq. 5.65) and P
expansi on (5.66) do not agree wi t h each other; however, t he
difference bet ween these two forms of virial equat i on reduces
as the numbe r of t erms increases. When the numbe r of t erms
becomes i nfi ni t y ( compl et e equat i on) , t hen t he two forms
of virial equat i on give i dent i cal results for V. Obvi ousl y for
t r uncat ed virial equat i on, the V expansi on form, Eq. (5.65),
gives more accurat e resul t for V as the virial coefficients
are ori gi nal l y det er mi ned from this equat i on. As can be seen
from Table 5.6, when B is cal cul at ed from Eq. (5.71) bet t er
TABLE 5.6--Prediction of molar volume of propane at 300 K and 9. 974 bar from virial equation with different methods for second virial
coefficient (Example 5.5).
Virial equation with two terms Virial equation with three terms a
Method of estimation of P expansion b V expansion c P expansion d V expansion e
second virial coefficient (B) B, cma/mol V, cma/mol %D V, cma/mol %D V, cm3/mol %D V, cm3/mol %D
Tsonopoulos (Eq. 5.71) -390.623 2110.1 3.6 2016.2 -1. 0 2056.8 1.0 2031.6 -0. 2
Normal fluids (Eq. 5.72) -397.254 2103.5 3.3 2005.3 -1. 5 2048.1 0.6 2021.0 -0. 7
McGlashan (Eq. 5.73) -360.705 2140.0 5.1 2077.8 2.0 2095.7 2.9 2063.6 1.3
The experimental value of vapor molar volume is: V = 2036.5 cm3/mol (Ref. [8], p. 4.24).
aIn all calculations with three terms, the third virial coefficient C is calculated from Eq. (5.78) as C = 19406.21 cm6/mol 2.
bTruncated two terms (P expansion) refers to pressure expansion virial equation (Eq. 5.66) truncated after second term (Eq. 5.75): Z = 1 + BP/RT.
CTruncated two terms (V expansion) refers to volume expansion virial equation (Eq. 5.65) truncated after second term: Z = 1 + B/V.
aTruncated three terms (P expansion) refers to pressure expansion virial equation (Eq. 5.66) truncated after third term: Z = 1 + BP/RT + (C - B 2) P2/(RT)2.
eTruncated three terms (V expansion) refers to volume expansion virial equation (Eq. 5.65) truncated after third term (Eq. 5.76): Z = 1 + B/V + C/V 2.
TABLE 5.7---Coefficients for the BWRS EOS--Eq. (5.89) [21].
Bo/Vc -- 0.44369 + 0.115449w Eo/(RTScVc) = 0.00645 - 0.022143w x exp(-3.8w)
Ao/(RTcVc) = 1.28438 - 0.920731~o
Co/(RT3cVc) = 0.356306 + 1.7087w b/(V 2) = 0.528629 + 0.349261w
Do/(RT4Vc) = 0.0307452 + 0.179433w a/(RTcV 2) = 0.484011 + 0.75413~0
d/(RT~V~) = 0.0732828 + 0.463492w
~/V~=0.0705233 - 0.044448w
d(RT~V~) = 0.504087 + 1.32245w
v/V~=0. 544979 - 0.270896w
predi ct i ons are obtained. Equat i on (5.72) also gives reason-
able results but Eq. (5.73) gives a less accurat e estimate of B.
The best result is obt ai ned from Eq. (5.76) with Eqs. (5.71)
and (5.78), whi ch give a deviation of 0.2%. (e) The ideal gas
l aw (Z - i) gives V v -- 2500.7 cm3/ mol with a deviation of
+22. 8%. #
5 . 6. 2 Modi f i e d B e ne di c t - We b b - - R ub i n
Equa t i o n o f S t at e
Anot her i mpor t ant EOS t hat has industrial appl i cat i on is the
Benedi ct - Webb- Rubi n ( BWR) EOS [45]. This equat i on is in
fact an empi ri cal expansi on of virial equation. A modi fi cat i on
of this equat i on by Starling [46] has f ound successful applica-
t i ons in pet rol eum and nat ural gas industries for propert i es
of light hydr ocar bons and it is given as
( co Oo
P = RT + BoRT- Ao- ~-~ + T3 ~-~ V2
1 ( a + 1
(5.89) +( b RT- a- : ) ~- ~+a_ : ) ~6
+T-T-V-~v 3 I +~- ~ exp
where the 11 const ant s Ao, Bo, . . . , a, b . . . . . a and y are given
in Table 5.7 in t erms of Vc, Tc, and w as report ed in Ref. [21].
This equat i on is known as BWRS EOS and may be used for
cal cul at i on of density of light hydr ocar bons and reservoir flu-
ids. I n the original BWR EOS, const ant s Do, Eo, and d were all
zero and the ot her const ant s were det ermi ned for each spe-
cific comp ound separately. Al t hough bet t er vol umet ri c dat a
can be obt ai ned f r om BWRS t han from cubic-type equations,
but predi ct i on of phase equi l i bri um f r om cubi c equat i ons are
quite compar abl e in some cases ( dependi ng on t he mixing
rules used) or bet t er t han this equat i on in some ot her cases.
Anot her probl em wi t h the BWRS equat i on is large comput a-
t i on t i me and mat hemat i cal i nconveni ence to predi ct vari ous
physical properties. To find mol ar vol ume V f r om Eq. (5.89),
a successive substitutive met hod is required. However, as it
will be discussed in t he next section, this type of equat i ons can
be used to develop general i zed correl at i ons in the graphi cal
or t abul at ed forms for predi ct i on of various t hermophysi cal
properties.
5 . 6. 3 C a r na h a n- S t a r l i ng Equ a t i o n o f S t at e
a nd I t s Modi f i c at i ons
Equat i ons of state are mai nl y developed based on t he un-
derst andi ng of i nt ermol ecul ar forces and potential energy
funct i ons t hat cert ai n fluids follow. For example, for hard
sphere fluids where the pot ent i al energy funct i on is given by
Eq. (5.13) it is assumed t hat t here are no attractive forces. For
such fluids, Car nahan and Starling proposed an EOS t hat has
been used extensively by researchers for devel opment of mor e
accurat e EOS [6]. For hard sphere fluids, t he smallest possible
vol ume t hat be can occupi ed by N mol ecul es of di amet er a is
V oN=N( V~
\NAJ
(5.90)
Vo -~ (-~2 or3) NA
where NA is t he Avogadro' s number and Vo is t he vol ume of
1 mol (NA molecules) of hard spheres as packed mol ecul es
wi t hout empt y space bet ween the molecules. VoN is t he total
vol ume of packed N molecules. I f the mol ar vol ume of fluid is
V, t hen a di mensi onl ess reduced density, ~, is defined in t he
following form:
Par amet er g is also known as packing fraction and indicates
fract i on of total vol ume occupi ed by hard molecules. Substi-
t ut i ng Vo from Eq. (5.90) into Eq. (5.91) gives the following
relation for packi ng fraction:
The Carnahan-St arl i ng EOS is t hen given as [6]
PV 1+~+~2- ~ 3
(5.93) Z Hs = - - =
RT (1 _~) 3
where Z us is t he compressi bi l i t y fact or for har d sphere
molecules. For this EOS there is no bi nary const ant and
t he onl y p ar amet er needed is mol ecul ar di amet er a for each
molecule. I t is clear t hat as V -+ oo ( P --~ 0) from Eq. (5.93)
( ~ 0 and Z ns ~ 1, whi ch is in fact identical to t he ideal
gas law. Car nahan and Starling extended the HS equat i on to
fluids whose spherical molecules exert attractive forces and
suggest ed t wo equat i ons based on t wo different attractive
t erms [6]:
(5.94) Z = Z Hs a
RTV
or
(5.95) Z = Z ~s a - ~ (V - b) -1 T -t/2
where Z ns is t he hard sphere cont ri but i on given by Eq.
(5.93). Obviously Eq. (5.94) is a t wo-paramet er EOS (a, a)
and Eq. (5.95) is a t hree-paramet er EOS (a, b, a). Bot h Eqs.
(5.94) and (5.95) reduce to ideal gas law (Z -+ Z Hs --* 1) as
V -~ cc ( or P ~ 0), whi ch satisfies Eq. (5.18). For mixtures,
the quadrat i c mi xi ng rule can be used for p ar amet er a while
a linear rule can be applied to p ar amet er b. Application of
these equat i ons for mi xt ures has been discussed in recent ref-
erences [8, 47]. Anot her modi fi cat i on of CS EOS is t hr ough
LJ EOS in the following form [48, 49]:
(5.96) Z --- Z Hs 32e~
3kBT
where e is the molecular energy paramet er and ( (see Eq. 5.92)
is related to a the size parameter, e and a are two paramet ers
in the LJ potential (Eq. 5.11) and ks is the Boltzman constant.
One advanced noncubic EOS, which has received significant
attention for propert y calculations specially derived proper-
ties (i.e., heat capacity, sonic velocity, etc.), is that of SAFT
originally proposed by Chapman et al. [50] and it is given in
(5.97) Z sAFT = 1 + Z Hs + Z cHAIN + Z DIsc + Z Ass~
where HS, CHAIN, DI SE and ASSOC refer to contributions
from hard sphere, chain format i on molecule, dispersion, and
association terms. The relations for Z Hs and Z cH~aN are simple
and are given in the following form [47]:
zSAFr= l +r ~_ ~) 33+( 1- - r )
(5.98) + Z DISP -[- Z AssOC
where r is a specific paramet er characteristic of the substance
of interest. ( in the above relation is segment packing fraction
and is equal to ( from Eq. (5.92) multiplied by r. The relations
for Z DIsP and Z gss~ are more complex and are in t erms of
summat i ons with adjusting paramet ers for the effects of asso-
ciation. There are other forms of SAFT EOS. A more practical,
but much more complex, form of SAFT equation is given by Li
and Englezos [51]. They show application of SAFT EOS to cal-
culate phase behavior of systems containing associating fluids
such as alcohol and water. SAFT EOS does not require criti-
cal constants and is particularly useful for complex molecules
such as very heavy hydrocarbons, complex pet rol eum fluids,
water, alcohol, ionic, and polymeric systems. Paramet ers can
be determined by use of vapor pressure and liquid density
data. Further characteristics and application of these equa-
tions are given by Prausnitz et al. [8, 47]. I n the next chapter,
the CS EOS will be used to develop an EOS based on the
velocity of sound.
5 . 7 C OR R ES PON D I N G STATE
CORRELATI ONS
One of the simplest forms of an EOS is the t wo-paramet er RK
EOS expressed by Eq. (5.38). This equation can be used for
fluids that obey a t wo-paramet er potential energy relation. I n
fact this equation is quite accurate for simple fluids such as
methane. Rearrangement of Eq. (5.38) t hrough multiplying
bot h sides of the equation by V / RT and substituting param-
eters a and b from Table 5.1 gives the following relation in
t erms of dimensionless variables [ 1]:
1 4. 934( h ) 0.08664Pr
(5.99) Z -- 1 - ~ TrL~ ~ where h =- ZTr
where T~ and P~ are called reduced temperature and reduced
pressure and are defined as:
T P
(5.100) Tr ~ Pr =
Tc Pc
where T and Tc must be in absolute degrees (K), similarly P
and Pc must be in absolute pressure (bar). Both T~ and Pr are
dimensionless and can be used to express t emperat ure and
pressure variations from the critical point. By substituting pa-
ramet er h into the first equation in Eq. (5.99) one can see that
(5.101) Z = f(Tr, Pr)
This equation indicates that for all fluids that obey a two-
paramet er EOS, such as RK, the compressibility factor, Z,
is the only function of Tr and Pr. This means that at the
critical point where Tr = Pr = 1, the critical compressibility
factor, Zc, is constant and same for all fluids (0.333 for RK
EOS). As can be seen from Table 2.1, Zc is constant only for
simple fluids such as N2, CH4, O2, or Ar, which have Zc of
0.29, 0.286, 0.288, and 0.289, respectively. For this reason RK
EOS is relatively accurate for such fluids. Equation (5.101)
is the fundament al of corresponding states principle (CSP) in
classical t hermodynami cs. A correlation such as Eq. (5.101)
is also called generalized correlation. I n this equation only
two paramet ers (To and Pc) for a substance are needed to
determine its PVT relation. These types of relations are usu-
ally called t wo-paramet er corresponding states correlations
(CSC). The functionality of function f in Eq. (5.101) can be
determined from experimental data on PVT and is usually
expressed in graphical forms rat her t han mat hemat i cal
equations. The most widely used t wo-paramet er CSC in a
graphical form is the Standing-Katz generalized chart that
is developed for natural gases [52]. This chart is shown in
Fig. 5.12 and is widely used in the pet rol eum industry [19, 21,
53, 54]. Obviously this chart is valid for light hydrocarbons
whose acentric factor is very small such as met hane and
ethane, which are the mai n component s of natural gases.
Hall and Yarborough [55] presented an EOS that was based
on data obtained from the Standing and Katz Z-factor chart.
The equation was based on the Carnahan-Stafling equation
(Eq. 5.93), and it is useful only for calculation of Z-factor of
light hydrocarbons and natural gases. The equation is in the
following form:
(5.102) Z=O, O6125PrT~- ay- ' exp[ - 1. 2( 1- Trl) 2]
where Tr and Pr are reduced t emperat ure and pressure and y
is a dimensionless par amet er similar to par amet er ~ defined
in Eq. (5.91). Paramet er y should be obtained from solution
of the following equation:
F(y) = - 0.06125PrTr 1 exp [ -1. 2 (1 - Tr ' ) 2]
+ y + y2 + y3 _ y4 (14"76Tfl-9"76Tr-E +4"58T~-3)Y 2
( 1 - y)3
+ (90.7T~ -1 - 242.2T~ -2 + 42.4Tr 3) y(2aS+2"82r~-~) = 0
(5.103)
The above equation can be solved by the Newt on-Raphson
method. To find y an initial guess is required. An approxi mat e
relation to find the initial guess is obtained at Z = 1 in Eq.
(5.102):
(5.104) y(k) = 0.06125PrTr-1 exp [ -1. 2 (1 - T~-t) 2]
Substituting y(k) in Eq. (5.103) gives F (k), which must be used
in the following relation to obtain a new value of y:
F(k)
(5.105) y(k+l) = y(k) dF (k)
dy
Reduced Pressure, Pr
s 6
! , ! 'L1
! . 0 I . e
*t7
0 . 9 5
o-~
l
O.| ~.7
0, 7 1 . 6
N N
O O
I t . LL
a ~
"~ ~ - - ~.i " : ' ' ~ . . . . . . : 1.4 t~
E ~ . . . . . . . . . . . . . . . ~ E
G ,3 , ; ~ : ~ - : : ~ i t . 2
-.+- }
i ~ "+"
FI G . 5 . 1 2 ~ S t a n d i n g - Ka t z g ener al i z ed char t f or comp r es s i b i l i t y f act or of nat ur al g as es
( c o ur t e s y of GPS A and GPA [ 53] ) .
wher e dF(k)/dy is t he deri vat i ve of F wi t h r esp ect to y at y =
y(k) and it is gi ven by t he fol l owi ng rel at i on:
dF
dy
1 + 4y + 4y 2 - 4y 3 + y4
(1 - y)4
- ( 29. 52Tr I - 19.52Tr -2 + 9. 16Tr 3) y
+ ( 2 . 1 8 + 2.82T~ -1) ( 90. 7Tr I - 242.2T~ -2 + 42.4T,73)
y(2"l 8+2"82T~ -1 )
(5.106)
Cal cul at i ons mus t be cont i nued unt i l t he di fference bet ween
y(k _ y(k) becomes smal l er t han a t ol er ance (e.g., 10-1~
As ment i oned before, t he St andi ng- Kat z char t or its equiv-
al ent Hal l - Yar bor ough cor r el at i on is ap p l i cabl e onl y t o l i ght
hydr ocar bons and t hey ar e not sui t abl e to heavi er fl ui ds such
as gas condensat es, o) of whi ch is not near zero. F or t hi s rea-
son a modi f i ed ver si on of t wo- p ar amet er CSC is needed. As
it can be seen f r om Table 2.1, for mor e comp l ex comp ounds,
val ue of Zc decr eases from t hose for si mpl e fluids and Eq.
(5.101) wi t h const ant Zc is no l onger valid. A p ar amet er t hat
i ndi cat es comp l exi t y of mol ecul es is acent r i c f act or t hat was
defined by Eq. (2.10). Acentric factor, co, is defined in a way
t hat for simple fluids it is zero or very small. For example, N2,
CH4, 02, or Ar have acent ri c factors of 0.025, 0.011, 0.022,
and 0.03, respectively. Values of w increase wi t h compl exi t y
of molecules. I n fact as shown in Section 2.5.3, Z~ can be cor-
related to w and bot h indicate deviation from simple fluids.
Acentric fact or was originally i nt roduced by Pitzer [56, 57] to
extend appl i cat i on of t wo- par amet er CSC t o mor e compl ex
fluids. Pitzer and his coworkers realized t he linear relation
bet ween Zc and w (i.e., see Eq. (2.103)) and assumed t hat
such linearity exists bet ween w and Z at t emperat ures ot her
t han To. They i nt roduced the concept of t hree-paramet er cor-
respondi ng states correl at i ons in t he following form:
(5.107) Z = Z (~ + wZ O)
where bot h Z (~ and Z O) are funct i ons of Tr and Pr- For
simple fluids (w ~ 0), this equat i on reduces to Eq. (5.101).
Z (~ is t he cont ri but i on of simple fluids and Z (1) is the cor-
rect i on t erm for compl ex fluids, tt can be shown t hat as
P --~ 0, Z (~ --+ 1 while Z O) --+ 0, therefore, Z --+ 1. The origi-
nal t hree-paramet er CSC developed by Pitzer was in the f or m
of t wo graphs similar to Fig. (5.12): one for Z (~ and the ot her
for Z (~), bot h in t erms of Tr and Pr. Pitzer correl at i ons f ound
wide appl i cat i on and were ext ended to ot her t her modynami c
properties. They were in use for mor e t han t wo decades; how-
ever, t hey were f ound to be i naccurat e in the critical regi on
and for liquids at low t emperat ures [58].
The most advanced and accurat e t hree-paramet er corre-
spondi ng states correl at i ons were developed by Lee and
Kesler [58] in 1975. They expressed Z in t erms of values of Z
for t wo fluids: simple and a reference fluid assumi ng l i near
rel at i on bet ween Z and w as follows:
(5.108) Z = Z (~ + -~r~(Z (r) - Z (~
where Z (r) and w (r) represent compressi bi l i t y fact or and acen-
tric fact or of the reference fluid. A comp ar i son bet ween Eqs.
(5.107) and (5.108) indicates t hat [Z (r) -Z( ~ or) is equiva-
lent t o Z (1). The simple fluid has acent ri c fact or of zero, but
t he reference fluid shoul d have t he highest value of o) t o cover
a wi der range for appl i cat i on of t he correlation. However,
t he choice of reference fluid is also limited by availability of
PVT and ot her t her modynami c data. Lee and Kesler chose
n-oct ane wi t h ~o of 0.3978 (this number is slightly different
f r om the most recent value of 0.3996 given in Table 2.1) as
the reference fluid. The same EOS was used for bot h the sim-
ple and reference fluid, whi ch is a modified version of BWR
EOS as given in t he following reduced form:
o
(5. 109) exp
where Vr is t he reduced vol ume defined as
V
(5.110) Vr = - -
v~
Coefficients B, C, and D are t emperat ure-dependent as
B = bl bE b3 b4 c2 c3 d2
Tr T~ 2 Tr 3 C = c l - "Tr + Tr3 D = dl + "~r
(5.111)
TABLE 5.8----Constants for the Lee-Kesler modification of BWR
EOS---Eq. (5. 109) [581.
Constant Simple fluid Re~rencefluid
bl 0.1181193 0.2026579
b2 0.265728 0.331511
b3 0.154790 0.027655
b4 0.030323 0.203488
cl 0.0236744 0.0313385
c2 0.0186984 0.0503618
c3 0.0 0.016901
c4 0.042724 0.041577
dl x 104 0.155488 0.48736
d2 x 104 0.623689 0.0740336
fl 0.65392 1.226
y 0.060167 0.03754
I n det ermi ni ng the const ant s in these equat i ons t he con-
straints by Eq. (5.9) and equality of chemi cal potentials or
fugacity (Eq. 6.104) bet ween vapor and liquid at sat urat ed
condi t i ons were imposed. These coefficients for bot h simple
and reference fluids are given in Table 5.8.
I n usi ng Eq. (5.108), bot h Z (~ and Z (r) shoul d be calculated
f r om Eq. (5.109). Lee and Kesler also t abul at ed values of Z (~
and Z 0) versus Tr and Pr for use in Eq. (5.107). The original
Lee-Kesl er (LK) tables cover reduced pressure f r om 0.01 t o
10. These tables have been widely used in maj or texts and ref-
erences [1, 8, 59]. However, the API -TDB [59] gives ext ended
tables for Z (~ and Z O) for t he Pr range up t o 14. Lee-Kesl er
tables and their ext ensi on by the API -TDB are perhaps t he
most accurat e met hod of est i mat i ng PVT rel at i on for gases
and liquids. Values of Z (~ and Z (I) as given by LK and t hei r
extension by API -TDB are given in Tables 5.9-5.11. Table 5.11
give values of Z (~ and Z (1) for Pr > 10 as provi ded in t he API-
TDB [59]. I n Tables 5.9 and 5.10 the dot t ed lines separat e liq-
ui d and vapor phases from each ot her up to t he critical point.
Values above and t o the ri ght are for liquids and bel ow and t o
the left are gases. The values for liquid phase are hi ghl i ght ed
wi t h bol d numbers. Graphi cal represent at i ons of these tables
are given in the API -TDB [59]. For comp ut er applications,
Eqs. (5.108)-(5.111) shoul d be used wi t h coefficients given in
Table 5.8. Graphi cal present at i on of Z (~ and Z (1) versus Pr
and Tr wi t h specified liquid and vapor regions is shown in
Fig. 5.13. The t wo-phase regi on as well as sat urat ed curves
are also shown in this figure. For gases, as Pr -* 0, Z (~ ~ 1
and Z (1) ~ 0. I t is interesting to not e t hat at t he critical poi nt
(T~ = Pr = 1), Z (~ = 0.2901, and Z (I) = -0. 0879, whi ch after
subst i t ut i on i nt o Eq. (5.107) gives t he following relation for
Zc:
(5.112) Zr -- 0.2901 - 0.0879w
This equat i on is slightly different f r om Eq. (2.93) and gives
different values of Zr for different compounds. Therefore, in
the critical regi on the LK correl at i ons per f or m bet t er t han
cubi c equations, whi ch give a const ant value for Z~ of all
compounds. Graphi cal present at i ons of bot h Z (~ and Z ~ for
cal cul at i on of Z f r om Eq. (5.107) are given in ot her sources
[60].
For t he l ow-pressure regi on where t he t runcat ed virial
equat i on can be used, Eq. (5.75) may be wri t t en in a gen-
eralized di mensi onl ess f or m as
B y
(5.113) Z = 1 + ~-~ = 1 + \ RTc ]
~-7(o) or
TABLE S.9---Values oT ~ fi use in Eq. (5.107) from the Lee-Kesler modification of BWR EOS (Eq. 5.109) [58].
Pr ~ 0.01 0.05 0.1 0.2 0.4 0.6 0.8 1 1.2 1.5 2 3 5 7 10
0.30 0. 0029 0. 0145 0. 0290 0. 0579 0. 1158 0. 1737 0. 2315 0.2892 0.3479 0.4335 0.5775 0.8648 1.4366 2.0048 2.8507
0.35 0. 0026 0. 0130 0. 0261 0. 0522 0. 1043 0. 1564 0. 2084 0.2604 0.3123 0.3901 0.5195 0.7775 1.2902 1.7987 2.5539
0.40 0. 0024 0. 0119 0. 0239 0. 0477 0. 0953 0. 1429 0. 1904 0.2379 0.2853 0.3563 0.4744 0.7095 1.1758 1.6373 2.3211
0.45 0. 0022 0. 0110 0. 0221 0. 0442 0. 0882 0. 1322 0. 1762 0.2200 0.2638 0.3294 0.4384 0.6551 1.0841 1.5077 2.1338
0.50 0. 0021 0. 0103 0. 0207 0. 0413 0. 0825 0. 1236 0. 1647 0.2056 0.2465 0.3077 0,4092 0.6110 1.0094 1.4017 1.9801
0.55 0.9804 0. 0098 0. 0195 0. 0390 0, 0778 0. 1166 0. 1553 0.1939 0.2323 0.2899 0.3853 0.5747 0.9475 1.3137 1.8520
0.60 0.9849 0. 0093 0. 0186 0. 0371 0, 0741 0. 1109 0. 1476 0.1842 0.2207 0.2753 0.3657 0.5446 0.8959 1.2398 1.7440
0.65 0,9881 0.9377 0. 0178 0. 0356 0. 0710 0. 1063 0. 1415 0.1765 0.2113 0.2634 0.3495 0.5197 0.8526 1.1773 1.6519
0.70 0.9904 0.9504 0.8958 0. 0344 0. 0687 0. 1027 0. 1366 0.1703 0.2038 0.2538 0.3364 0.4991 0.8161 1.1341 1.5729
0.75 0.9922 0.9598 0.9165 0. 0336 0. 0670 0. 1001 0. 1330 0.1656 0.1981 0.2464 0.3260 0.4823 0.7854 1.0787 1.5047
0,80 0.9935 0.9669 0.9319 0.8539 0. 0661 0. 0985 0. 1307 0.1626 0.1942 0,2411 0.3182 0.4690 0.7598 1.0400 1.4456
0.85 0.9946 0.9725 0.9436 0.8810 0. 0661 0. 0983 0. 1301 0.1614 0.1924 0.2382 0.3132 0.4591 0.7388 1.0071 1.3943
0.90 0.9954 0.9768 0.9528 0.9015 0.7800 0. 1006 0. 1321 0.1630 0.1935 0.2383 0.3114 0.4527 0.7220 0.9793 1.3496
0.93 0.9959 0.9790 0.9573 0.9115 0.8059 0.6635 0. 1359 0.1664 0.1963 0.2405 0.3122 0.4507 0.7138 0.9648 1.3257
0.95 0.9961 0.9803 0.9600 0.9174 0.8206 0.6967 0, 1410 0. i 705 0,1998 0.2432 0.3138 0.4501 0.7092 0.9561 1.3108
0.97 0.9963 0.9815 0.9625 0.9227 0.8338 0.7240 0.5580 0, t 779 0,2055 0.2474 0.3164 0.4504 0.7052 0.9480 1.2968
0.98 0.9965 0.9821 0.9637 0.9253 0.8398 0.7360 0.5887 0.1844 0.2097 0.2503 0.3182 0.4508 0.7035 0.9442 1.2901
0.99 0.9966 0.9826 0.9648 0.9277 0.8455 0.7471 0,6138 0.1959 0.2154 0.2538 0.3204 0.4514 0.7018 0.9406 1.2835
1.00 0.9967 0.9832 0.9659 0.9300 0.8509 0.7574 0,6355 0.2901 0.2237 0.2583 0.3229 0.4522 0.7004 0.9372 1.2772
1.01 0.9968 0.9837 0.9669 0.9322 0.8561 0.7671 0.6542 0.4648 0.2370 0.2640 0.3260 0.4533 0.6991 0.9339 1.2710
1.02 0.9969 0.9842 0.9679 0.9343 0.8610 0.7761 0.6710 0.5146 0.2629 0.2715 0.3297 0.4547 0.6980 0.9307 1.2650
1.05 0.9971 0.9855 0.9707 0.9401 0.8743 0.8002 0.7130 0.6026 0.4437 0.3131 0.3452 0.4604 0.6956 0.9222 1.2481
1.10 0.9975 0.9874 0.9747 0.9485 0.8930 0.8323 0.7649 0.6880 0.5984 0.4580 0.3953 0.4770 0.6950 0.9110 1.2232
1.15 0.9978 0.9891 0.9780 0.9554 0.9081 0.8576 0.8032 0.7443 0.6803 0.5798 0.4760 0.5042 0.6987 0.9033 1.2021
1.20 0.9981 0.9904 0.9808 0.9611 0.9205 0.8779 0.8330 0.7858 0.7363 0.6605 0.5605 0.5425 0.7069 0.8990 1.1844
1.30 0.9985 0.9926 0.9852 0.9702 0.9396 0.9083 0.8764 0.8438 0.8111 0.7624 0.6908 0.6344 0.7358 0.8998 1.1580
1.40 0.9988 0.9942 0.9884 0.9768 0.9534 0.9298 0.9062 0.8827 0.8595 0.8256 0.7753 0.7202 0.7761 0.9112 1.1419
1.50 0.9991 0.9954 0.9909 0.9818 0.9636 0.9456 0.9278 0.9103 0.8933 0.8689 0.8328 0.7887 0.8200 0.9297 1.1339
1.60 0.9993 0.9964 0.9928 0.9856 0.9714 0.9575 0.9439 0.9308 0.9180 0.9000 0.8738 0.8410 0.8617 0.9518 1.1320
1.70 0.9994 0.9971 0.9943 0.9886 0.9775 0.9667 0.9563 0.9463 0.9367 0.9234 0.9043 0.8809 0.8984 0.9745 1.1343
1.80 0.9995 0.9977 0.9955 0.9910 0.9823 0.9739 0.9659 0.9583 0.9511 0.9413 0.9275 0.9118 0.9297 0.9961 1.1391
1.90 0.9996 0.9982 0.9964 0.9929 0.9861 0.9796 0.9735 0.9678 0.9624 0.9552 0.9456 0.9359 0.9557 1,0157 1.1452
2.00 0.9997 0.9986 0.9972 0.9944 0.9892 0.9842 0.9796 0.9754 0,9715 0.9664 0.9599 0.9550 0.9772 1,0328 1.1516
2.20 0,9998 0,9992 0.9983 0.9967 0.9937 0.9910 0.9886 0.9865 0.9847 0.9826 0.9806 0.9827 1.0094 1.0600 1.1635
2.40 0.9999 0.9996 0.9991 0.9983 0.9969 0.9957 0.9948 0.9941 0.9936 0.9935 0.9945 1.0011 1.0313 1.0793 1.1728
2.60 1.0000 0.9998 0.9997 0.9994 0.9991 0.9990 0.9990 0.9993 0.9998 1.0010 1.0040 1.0137 1.0463 1,0926 1.1792
2,80 1.0000 1.0000 1.0001 1.0002 1.0007 1.0013 1.0021 1.0031 1.0042 1.0063 1.0106 1.0223 1.0565 1.1016 1.1830
3.00 1.0000 1.0002 1.0004 1.0008 1.0018 1.0030 1.0043 1.0057 1.0074 1.0101 1.0153 1.0284 1.0635 1.1075 1.1848
3.50 1.0001 1.0004 1.0008 1.0017 1.0035 1.0055 1.0075 1,0097 1.0120 1.0156 1.0221 1.0368 1.0723 1.1138 1. t 834
4.00 1.0001 1.0005 1.0010 1.0021 1.0043 1.0066 1.0090 1.0115 1.0140 1.0179 1.0249 1.0401 1.0747 1.1136 1.1773
whe r e BPc/ RTc c a n be e s t i ma t e d f r om Eq. ( 5. 71) or ( 5. 72)
t hr ough T~ a nd 0). Equa t i on ( 5. 114) ma y be us e d at l ow Pr
a nd Vr > 2 or Tr > 0, 686 + 0. 439Pr [ 60] i ns t e a d of c omp l e x
Eqs. ( 5. 108) - ( 5. 111) . The API - TDB [ 59] al so r e c omme nds
t he f ol l owi ng r el at i on, p r op os e d by Pi t z e r et al. [ 56] , f or
c a l c ul a t i on of Z f or gas es at Pr --< 0.2.
Z = 1 + - ~[ ( 0. 1445 + 0. 0730) ) - ( 0. 33 - 0.460))T~ -1
- ( 0. 1385 + 0.50))Tr 2 - ( 0. 0121 + 0.0970) ) Tr -3
( 5. 114) - 0.00730)T~ - s]
Obvi ous l y ne i t he r Eq. ( 5. 113) nor ( 5. 114) c a n be ap p l i ed t o
l i qui ds.
The LK c or r e s p ondi ng s t at es c or r e l a t i ons e xp r e s s e d by
Eq. ( 5. 107) a nd Tabl es 5. 9- 5. 11 c a n al so be a p p l i e d t o mi x-
t ur es , Suc h c or r e l a t i ons ar e s ens i t i ve t o t he i np ut da t a f or
t he p s e udoc r i t i c a l p r op e r t i e s . The mi xi ng r ul es us e d t o cal -
cul at e mi xt ur e cr i t i cal t e mp e r a t ur e a nd p r e s s ur e ma y gr eat l y
af f ect c a l c ul a t e d p r op e r t i e s s p eci al l y whe n t he mi xt ur e con-
t a i ns di s s i mi l a r c omp ounds , Lee a nd Kes l er p r op os e d s p eci al
set of e qua t i ons f or mi xt ur e s f or us e wi t h t he i r cor r el at i ons ,
Thes e e qua t i ons ar e e qui va l e nt t o t he f ol l owi ng e qua t i ons as
p r ovi de d by t he API - TDB [ 59] .
Vu = ZaRTcg/P,~
Za = 0. 2905 - 0.085w~
Vmc = ~ x iVci + 3 Xi 2/3 xi /3
\ i = 1 1 3
Trnc ~ ~ xi Vci Tc i + 3 xi V 2/3 ~"~d xi v2/3x/~T-~
mc ki=l
N
0)m = EX/ 0) i
i=1
Pmc = ZrncRTmc/Vmc c = ( 0. 2905 - O.0850)m)RTrnc/grnc
(5.115)
whe r e xi i s t he mol e f r a c t i on of c omp one nt i, N i s t he numbe r
of c omp ounds i n t he mi xt ur e, a nd Tmo Pmc, a nd Vmc ar e t he
mi xt ur e p s e udoc r i t i c a l t e mp e r a t ur e , p r es s ur e, a nd vol ume re-
spect i vel y. 0)m is t he mi xt ur e a c e nt r i c f a c t or a nd i t i s c a l c ul a t e d
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1
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1
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1
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2
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4

TABLE 5. 11--Values of Z (0) and Z (1) for use in Eq. (5.107) from the Lee-Kesler modification of BWR EOS---Eq. (5.109) [59].
Pr ~ Tr Z (~ Z (1)
~, 10 11 12 14 10 11 12 14
0.30 2.851 3.131 3.411 3.967 - 0. 792 - 0. 869 - 0. 946 - 1.100
0.35 2.554 2.804 3.053 3.548 - 0. 886 - 0. 791 - 1.056 - 1.223
0.40 2.321 2.547 2.772 3.219 - 0. 894 - 0. 978 - 1.061 - 1.225
0.45 2.134 2.340 2.546 2.954 -0. 861 - 0. 940 - 1.019 - 1.173
0.50 1.980 2.171 2.360 2.735 - 0. 810 - 0. 883 - 0. 955 - 1. 097
0.55 1.852 2.029 2.205 2.553 - 0. 752 - 0. 819 - 0. 885 - 1.013
0.60 1.744 1.909 2.073 2.398 - 0. 693 - 0. 753 - 0. 812 - 0. 928
0.65 1.652 1.807 1.961 2.266 - 0. 635 - 0. 689 - 0. 742 - 0. 845
0.70 1.573 1.720 1.865 2.152 - 0. 579 - 0. 627 - 0. 674 - 0. 766
0.75 1.505 1.644 1.781 2.053 - 0. 525 - 0. 568 - 0. 610 -0. 691
0.80 1.446 1.578 1.708 1.966 - 0. 474 - 0. 512 - 0. 549 -0. 621
0.85 1.394 1.520 1.645 1.890 - 0. 425 - 0. 459 - 0. 491 - 0. 555
0.90 1.350 1.470 1.589 1.823 - 0. 379 - 0. 408 - 0. 437 - 0. 493
0.95 1.311 1.426 1.540 1.763 - 0. 334 - 0. 360 - 0. 385 - 0. 434
0.98 1.290 1.402 1.513 1.731 - 0. 308 - 0. 331 - 0. 355 - 0. 40I
0.99 1.284 1.395 1.504 1.721 - 0. 299 - 0. 322 - 0. 345 - 0. 390
1.00 1.277 1.387 1.496 1.710 - 0. 290 - 0. 313 - 0. 335 - 0. 379
1.01 1.271 1.380 1.488 1.701 - 0. 282 - 0. 304 - 0. 326 - 0. 368
1.02 1.265 1.373 1.480 1.691 - 0. 273 - 0. 295 - 0. 316 - 0. 357
1.03 1.259 1.367 1.473 1.682 - 0. 265 - 0. 286 - 0. 307 - 0. 347
1.04 1.254 1.360 1.465 1.672 - 0. 256 - 0. 277 - 0. 297 - 0. 337
1.05 1.248 1.354 1.458 1.664 - 0. 248 - 0. 268 - 0. 288 - 0. 326
1.06 1.243 1.348 1.451 1.655 - 0. 239 - 0. 259 - 0. 278 - 0. 316
1.07 1.238 1.342 1.444 1.646 -0. 231 - 0. 250 - 0. 269 - 0. 306
1.08 1.233 1.336 1.438 1.638 - 0. 222 -0. 241 - 0. 260 - 0. 296
1.09 1.228 1.330 1.431 1.630 - 0. 214 - 0. 233 -0. 251 - 0. 286
1.10 1.223 1.325 1.425 1.622 - 0. 206 - 0. 224 - 0. 242 - 0. 276
1.11 1.219 t. 319 1.419 1.614 - 0. 197 - 0. 215 - 0. 233 - 0. 267
1.12 1.214 1.314 1.413 1.606 - 0. 189 - 0. 207 - 0. 224 - 0. 257
1.13 1.210 1.309 1.407 1.599 -0. 181 - 0. 198 - 0. 215 - 0. 247
1.15 1.202 1.299 1.395 1.585 - 0. 164 -0. 181 - 0. 197 - 0. 228
1.20 1.184 1.278 1.370 1.552 - 0. 123 - 0. 139 - 0. 154 - 0. 183
1.25 1.170 1.259 1.348 1.522 - 0. 082 - 0, 098 - 0. 112 - 0. 139
1.30 1.158 1.244 1.328 1.496 - 0. 042 - 0. 058 - 0. 072 - 0. 097
1.40 1.142 1.220 1.298 1.453 0,035 0.019 0.005 - 0. 019
1.50 1.134 1.205 1.276 1.419 O. 106 0.090 0.076 0.052
1.60 1.132 1.197 1.262 1.394 0.167 0.152 0.138 0.116
1.70 1.134 1.193 1,253 1.374 0.218 0.204 0.192 0.171
1.80 1.139 1.192 1,247 1.359 0.258 0.247 0.237 0.218
2.00 1.152 1.196 1.243 1.339 0.310 0.305 0.300 0.290
2.50 1.176 1.210 1.244 1.316 0.348 0.356 0.362 0.37I
3.00 1.185 1.213 1.241 1.300 0.338 0.353 0.365 0.385
3.50 1.183 1.208 1.233 1.284 0.319 0.336 0.350 0.376
4.00 1.177 1.200 1.222 1.268 0.299 0.316 0.332 0.360
Hi gh Pressure Range: Value of Z (~ and Z (1) for 10 < Pr < 14.
s i mi l a r t o t he Kay' s mi xi ng r ul e. Ap p l i c a t i on of Kay' s mi xi ng
r ul e, e xp r e s s e d by Eq. ( 3. 39) , gi ves t he f ol l owi ng r e l a t i ons f or
c a l c ul a t i on of p s e udoc r i t i c a l t e mp e r a t ur e a nd p r es s ur e:
N N
( 5. 116) Tpc = Y~. x4T~- Ppc -- )-~ x4 P~
i=1 i=1
whe r e Tpc a nd Pp~ ar e t he p s e udoc r i t i c a l t e mp e r a t ur e a nd
p r es s ur e, r esp ect i vel y. Ge ne r a l l y f or s i mp l i c i t y p s e udoc r i t i -
cal p r op e r t i e s ar e c a l c ul a t e d f r om Eqs. ( 5. 116) ; however , us e
of Eqs . ( 5. 115) f or t he LK c or r e l a t i ons gi ves be t t e r p r op e r t y
p r e di c t i ons [ 59] .
Ex ampl e 5. 6- - - Rep eat Exa mp l e 5. 2 us i ng LK ge ne r a l i z e d cor-
r e l a t i ons t o e s t i ma t e V v a nd V L f or n- oc t a ne at 279. 5~ a nd
19. 9 bar.
Sol ut i on- - For n- oct ane, f r om Exa mp l e 5.2, Tc = 295. 55~
( 568. 7 K) , Pc = 24. 9 bar, w = 0. 3996. T~ = 0. 972, a nd Pr = 0. 8.
F r om Tabl e 5. 9 i t c a n be s een t ha t t he p oi nt ( 0. 972 a nd 0. 8) i s
on t he s a t ur a t i on l i ne; t her ef or e, t he r e ar e bot h l i qui d a nd va-
p or p ha s e s at t hi s c ondi t i on a nd val ues of Z ~~ a nd Z~l ) are sep -
a r a t e d by dot t e d l i nes. F or t he l i qui d p ha s e at Pr = 0.8, ext r ap -
ol a t i on of va l ue s of Z ~~ at Tr = 0. 90 a nd Tr -- 0. 95 t o T~ = 0. 972
gi ves Z C~ = 0. 141 + [ ( 0. 972 - 0. 93) / ( 0. 95 - 0. 93) ] ( 0. 141 -
0. 1359) = 0. 1466, s i mi l ar l y we get Z ~1~= - 0. 056. Subs t i t ut -
i ng Z (~ a nd Z (~) i nt o Eq. ( 5. 107) gi ves Z L = 0. 1466 + 0. 3996
( - 0. 056) = 0. 1242. Si mi l ar l y f or t he va p or p ha s e , val ues of
Z ~~ a nd Z 0~ be l ow t he dot t e d l i ne s houl d be used. F or t hi s
cas e l i ne a r i nt e r p ol a t i ons be t we e n t he val ues f or Z (~ a nd
Z C1~at Tr = 0. 97 a nd T~ -- 0. 98 f or t he gas p ha s e gi ve Z O~ =
0. 5642, Z ~1~= - 0. 1538. F r om Eq. ( 5. 107) we get Z v = 0. 503.
F r om Eq. ( 5. 15) c or r e s p ondi ng vol ume s ar e V L -- 286. 8 a nd
I t i i i l i i i : i ! i i : i I i ! i ~i l ! l l
I ~~i
1.2 '~ . . . . . . . . . . ........ ! . . . . : , , i . , ~ , ~ . : i ~ - ................ : ........... i ----~ ---~ : ~ + " ............... ....... - i ~..,,~/,..t-./J
........ T 7 ~ 0 6 , "~ . . . . . . : 9" T : '" . . . . . . . . . i . . . . . . . . . i - i q - ! : i . . . . . . . . . i ............ i ....
, !:
0.8 : ~ : ~15
0.8 ................. i ............ ~ ! .......... ' ::~' ~i ........................... ~-i-' -i ~-i~b~ ................... i ..... i .......
~ .......... ........ 11i:! ] iiiiiii; ................. ..... i .................. ..........
0.0
0.01 0.1 1 10
( a) Reduced Pressure, Pr
0 4 i i : ' :
i i~ iil i ! ! i i ~ ! i
Z d ) - 0 2 .................... i - - - - i ~ ~ ......... -~ ...........
- 0 . 4 ................. : ............ i - i ': : ! : : ....................... : .............. ? : : : , : ............. i ............ ,
- u . o .............. : ..... ? - ? : - 1 . . . . . . . . . . . .
i i ''~ i i ~ i ! i } i :
0.01 0.1 1 10
(b) Re d u c e d Pr essur e, Pr
FIG. 5. 1 3 ~ Compressi bi l i t y factor (a) Z 1~ and ( b) ~ 1 ) from Tables 5.9 and 5.10.
V v -- 1161.5 cm3/ mol, whi ch give er r or s of - 5. 6 and - 4. 5% for
t he l i qui d and vap or vol umes, respectively.
The cor r esp ondi ng st at es cor r el at i on expr essed by Eq.
(5.107) is deri ved f r om p r i nci p l es of cl assi cal t her mody-
nami cs. However, t he same t heor y can be der i ved f r om
mi cr oscop i c t her modynami cs. Previ ousl y t he r el at i on be-
t ween vi ri al coeffi ci ent s and i nt er mol ecul ar forces was shown
t hr ough Eq. (5.67). F r om Eq. (5.11), F can be wr i t t en m a
di mensi onl ess f or m as
r
/ r
whi ch is t he basi s for t he devel op ment of mi cr oscop i c ( mol ec-
ul ar) t heor y of cor r esp ondi ng st at es. Subst i t ut i on of Eq.
(5.117) i nt o Eq. (5.67) woul d r esul t i nt o a gener al i zed cor-
r el at i on for t he second vi ri al coeffi ci ent [6].
5 . 8 GEN ER A L I Z ED CORREL AT I ON
FOR PVT PR OPER T I ES OF
L I QUI D S - - R A C KET T EQUA T I ON
Al t hough cubi c EOS and gener al i zed cor r el at i ons di scussed
above can be used for bot h l i qui d and vap or phases, it was
ment i oned t hat t hei r p er f or mance for t he l i qui d p hase is weak
especi al l y when t hey are used for l i qui d densi t y pr edi ct i ons.
F or t hi s r eason in many cases sep ar at e cor r el at i ons have been
devel op ed for p r op er t i es of l i qui ds. As can be seen f r om Fig.
5.1, t he var i at i on of P wi t h V for an i sot her m in t he l i qui d
p hase is very st eep and a smal l change in vol ume of l i qui d, a
bi g change in p r essur e is needed. I n addi t i on it is seen f r om
t hi s fi gure t hat when t he p r essur e is near t he sat ur at i on pres-
sure, l i qui d vol ume is very cl ose to sat ur at i on vol ume. I n t hi s
sect i on t he Racket t equat i on, whi ch is wi del y used for pre-
di ct i on of sat ur at ed l i qui d densi t i es, is i nt r oduced for p ur e
subst ances and defi ned mi xt ures. Then t he met hod of pr edi c-
t i on of l i qui d densi t i es at hi gh p r essur es is pr esent ed.
5. 8. 1 Racket t Equat i on f or Pure Component
Sat urat ed Li qui ds
I f Eq. (5.6) is ap p l i ed at t he sat ur at i on pr essur e, Pr sat we have
( 5. 118) V sat = f~ ( r , ps. t )
Si nce for any subst ance, psat dep ends onl y on t emp er at ur e
t hus t he above equat i on can be r ear r anged i n a r educed f or m
as
( 5. 119) VrSat= f2 ( r r )
wher e V~ sat is t he r educed sat ur at i on vol ume (VSWVc) and Tr is
t he r educed t emp er at ur e. To i mp r ove t hi s gener al i zed corre-
l at i on a t hi r d p ar amet er such as Zc can be used and Racket t
[61] suggest ed t he fol l owi ng si mp l e f or m for V2 at versus Tr:
V sat _ Z(1-Tr)2/7
( 5. I 20) Vr s a t - Vc
This equat i on is i n fact a gener al i zed cor r el at i on for sat ur at ed
l i qui ds and it is in di mensi onl ess form. Lat er Sp encer and
Danner [62] modi f i ed t hi s equat i on and r ep l aced p ar amet er
Zc wi t h anot her p ar amet er cal l ed R a c ke r p ar amet er shown
by ZRA:
( 5. 121) vsat = ( p ~) Z~ A n = 1.0 + ( 1. 0- Tr) 2/7
Values of Z~ ar e cl ose to t he val ues of Zr and t hey ar e re-
p or t ed by Sp encer and Adl er [63]. For some sel ect ed com-
p ounds, val ues of Zv, A ar e given i n Table 5.12 as r ep or t ed by
t he API -TDB [59]. A l i near r el at i on bet ween ZV, A and o) si mi -
l ar to Eq. (5.112) was p r op os ed based on t he i ni t i al val ues of
Racket t p ar amet er [64].
(5.122) Z~ = 0.29056 - 0.08775o)
I t shoul d be not ed t hat t he API -TDB [59] r ecommends val ues
of Zw di fferent f r om t hose obt ai ned f r om t he above equat i on.
Usual l y when t he val ue of Z~ is not avai l abl e, it may be re-
p l aced by Zc. I n t hi s case Eq. (5.121) r educes t o t he or i gi nal
Racket t equat i on ( Eq. 5.120). The most accur at e way of pre-
di ct i ng Zga is t hr ough a known val ue of density. I f densi t y of
a l i qui d at t emp er at ur e T is known and is shown by dx, t hen
No.
TABLE 5.12---Values of Rackett parameter for selected compounds [59].
ZRA No. ZRA
Paraffins
1 Methane
2 Ethane
3 Propane
4 n-Butane
5 2-Methylpropane (isobutane)
6 n-Pentane
7 2-Methylbutane (isopentane)
8 2,2-Dimethylpropane (neopentane)
9 n-Hexane
10 2-Methylpentane
11 n-Heptane
12 2-Methylhexane
13 n-Octane
14 2-Methylheptane
15 2.3,4-Trimethylpentane
16 n-Nonane
17 n-Decane
18 n-Undecane
19 n-Dodecane
20 n-Tridecane
21 n-Tetradecane
22 n-Pentadecane
23 n-Hexadecane
24 n-Heptadecane
25 n-Octadecane
26 n-Nonadecane
27 n-Eicosane
Naphthenes
28 Cyclopentane
29 Methylcyclopentane
30 Cyclohexane
31 Methylcyclohexane
a Calculated from Eq. (5.123) using specific gravity.
0.2880 32
0.2819 33
0.2763 34
0.2730 35
0.2760 36
0.2685 37
0.2718
0.2763 38
0.2637
0.2673
0.2610 39
0.2637 40
0.2569 41
0.2581 42
0.2656 43
0.2555 44
0.2527 45
0.2500 46
0.2471 47
0.2468 48
0.2270 49
0,2420
0,2386 59
0.2343 51
0.2292 52
0.2173 a 53
0.2281 54
55
0.2709 56
0.2712 57
0.2729 58
0.2702 59
Olefms
Ethene (ethylene)
Propene (propylene)
1-Butene
i-Pentene
i -Hexene
1-Heptene
Di-olefin
Ethyne (acetylene)
Aromatics
Benzene
Methylbenzene (toluene)
Ethylbenzene
1,2-Dimethylbenzene (o-xylene)
1.3-Dimethylbenzene (m-xylene)
1.4-Dimethylbenzene (p-xylene)
n-Propylbenzene
Isopropylbenzene (cumene)
n-Butylbenzene
Naphthalene
Aniline
Nonhydrocarbons
Ammonia
Carbon dioxide
Hydrogen
Hydrogen sulfide
Nitrogen
Oxygen
Water
Methanol
Ethanol
Diethylamine (DEA)
0.2813
0.2783
0.2735
0.2692
0.2654
0.2614
0.2707
0.2696
0.2645
0.2619
0.2626
0.2594
0.2590
0.2599
0.2616
0.2578
0.2611
0.2607
0.2466
0.2729
0.3218
0.2818
0.2893
0.2890
0.2374
0.2334
0.2502
0.2568
Eq. (5.121) can be rearranged to get ZRA:
( M P~ .~l/n
(5.123) ZRA = \ RTcdy:
where n is calculated from Eq. (5.121) at temperature T
at which density is known. For hydrocarbon systems and
petroleum fractions usually specific gravity (SG) at 15.5~ is
known and value of 288.7 K should be used for T. Then dT (in
g/cm 3) is equal to 0.999SG according to the definition of SG
by Eq. (2.2). I n this way predicted values of density are quite
accurate at temperatures near the reference temperature at
which density data are used. The following example shows
the procedure.
Example 5. 7~F or n-octane of Example 5.2, calculate satu-
rated liquid molar volume at 279.5~ from Rackett equation
using predicted ZRA.
Solution- - From Example 5.2, M = 114.2, SG = 0.707, T~ =
295.55~ (568.7 K), Pc = 24.9 bar, R = 83.14 c m 3. bar/tool 9 K,
and T~ = 0.972. Equation (5.123) should be used to predict
ZRA from SG. The reference temperature is 288.7 K, which
gives Tr = 0.5076. This gives n = 1.8168 and from Eq. (5.123)
we calculate ZRA = 0.2577. ( Z~ = 0.2569 from Table 5.12).
From Eq. (5.121), V sat is calculated: n = 1 + (1 - 0.972) 2/7 =
1.36, V ~at = (83.14 x 568.7/24.9) x 0. 2577 TM ----- 300 cm3/mol.
Comparing with actual value of 304 cm3/mol gives the er-
ror of -1.3%. Calculated density is p = 114.2/300 = 0.381
g/cm 3 . #
5 . 8 . 2 De f i ne d Li qui d Mi xt ures and
Pe t r o l e um Fract i ons
Saturation pressure for a mixture is also called bubble point
pressure and saturation molar volume is shown by V bp. Liquid
density at the bubble point is shown by pbp, which is related
to V bp by the following relation:
M
(5.124) p bp = V bp
where pbp is absolute density in g/cm 3 and M is the molec-
ular weight. V bp c a n be calculated from the following set of
equations recommended by Spencer and Danner [65]:
g bp = R x/ ZRA m
n = 1 + (1 - Tr ) 2/ 7
N
ZRA~ = ~--~ X4ZR~
i=1
Tr = T/ Tcm
N N
(5.125) Tcm= E E ~)i4)jTcij
i=1 j=l
xi V~i
~i - E~=I xi vci
~j 1. O-
L (vl,, + d
5. PV T RELATIONS AND EQUATIONS OF STATE 223
This method is also included in the API-TDB [59]. Another
approach to estimate density of defined liquid mixtures at its
bubble point pressure is through the following mixing rule:
1 x-~N x~
(5.126)
pb'--p -- L psa---t
where x~ is weight fraction of / i n the mixture, p~at ( = M~ V/sat)
is density of pure saturated liquid i and should be calculated
from Eq. (5.121) using Tci and Ze,~.
For petroleum fractions in which detailed composition is
not available Eq. (5.121) developed for pure liquids may be
used. However, ZRA should be calculated from specific grav-
ity using Eq. (5.123) while Tc and Pc can be calculated from
methods given in Chapter 2 through Tb and SG.
5. 8. 3 Ef f ect o f Pr e s s ur e on Li qui d De ns i t y
As shown in Fig. 5.1, effect of pressure on volume of liquids
is quite small specially when change in pressure is small.
When temperature is less than normal boiling point of a liq-
uid, its saturation pressure is less than 1.0133 bar and den-
sity of liquid at atmospheric pressure can be assumed to be
the same as its density at saturation pressure. For temper-
atures above boiling point where saturation pressure is not
greatly more than 1 atm, calculated saturated liquid density
may be considered as liquid density at atmospheric pressure.
Another simple way of calculating liquid densities at atmo-
spheric pressures is through Eq. (2.115) for the slope of den-
sity with temperature. If the only information available is spe-
cific gravity, SG, the reference temperature would be 15.5~
(288.7 K) and Eq. (2.115) gives the following relation:
p~ = 0.999SG - 10 -3 X (2.34 -- 1.898SG) (T - 288.7)
(5.127)
where SG is the specific gravity at 15.5~ (60~176 and T
is absolute temperature in K. p~ is liquid density in g/cmaat
temperature T and atmospheric pressure. I f instead of SG at
15.5~ (288.7 K), density at another temperature is available
a similar equation can be derived from Eq. (2.115). Equation
(5.127) is not accurate if T is very far from the reference tem-
perature of 288.7 K.
The effect of pressure on liquid density or volume becomes
important when the pressure is significantly higher than 1
atm. For instance, volume of methanol at 1000 bar and 100~
is about 12% less than it is at atmospheric pressure. I n gen-
eral, when pressure exceeds 50 bar, the effect of pressure on
liquid volume cannot be ignored. Knowledge of the effect of
pressure on liquid volume is particularly important in the de-
sign of high-pressure pumps in the process industries. The
following relation is recommended by the API-TDB [59] to
calculate density of liquid petroleum fractions at high pres-
sures:
pO p
(5.128) - - = 1. 0- - -
p By
where pO is the liquid density at low pressures (atmospheric
pressure) and p is density at high pressure P (in bar). Br
is called isothermal secant bulk modulus and is defined
as -(1/ p~ Parameter By indicates the slope of
change of pressure with unit volume and has the unit of pres-
sure. Steps to calculate By are summarized in the following
set of equat i ons:
Br = r e X+ Bz
m = 1492.1 + 0. 0734P + 2.0983 x 10- 6p 2
X = (/320 - 105) / 23170
(5.129)
logB2o = - 1. 098 10- 3T + 5.2351 + 0. 7133p ~
/3i = 1.0478 x 103+ 4.704 P- 3.744 IO- 4p 2
+2. 2331 x l O- Sp 3
wher e Br is i n bar and po is t he l i qui d densi t y at at mos p her i c
p r essur e i n g/ cm 3. I n t he above equat i on T is absol ut e t em-
p er at ur e i n kel vi n and P is t he p r essur e i n bar. The average
er r or f r om t hi s met hod is about 1.7% except near t he cri t i cal
p oi nt wher e er r or i ncr eases to 5% [59]. Thi s met hod is not
r ecommended for l i qui ds at Tr > 0.95. I n cases t hat po is not
avai l abl e it may be est i mat ed f r om Eq. (5.121) or (5.127). Al-
t hough t hi s met hod is r ecommended for p et r ol eum f r act i ons
but it gives r easonabl e resul t s for p ur e hydr ocar bons (>C5)
as well.
F or l i ght and medi um hydr ocar bons as wel l as l i ght
p et r ol eum f r act i ons t he Tait-COSTALD ( cor r esp ondi ng s t at e
l i qui d densi t y) cor r el at i on ori gi nal l y p r op os ed by Hanki ns on
and Thoms on may be used for t he effect of p r essur e on l i qui d
densi t y [66]:
[ Cl n[ B +P) I - '
(5.130) Pe = peo 1 - \ f f - ~- ~] j
wher e pe is densi t y at p r essur e P and Ovo is l i qui d densi t y at
r ef er ence p r essur e of po at whi ch densi t y is known. When peo
is cal cul at ed f r om t he Racket t equat i on, po = psat wher e p~at
is t he sat ur at i on ( vapor) pr essur e, whi ch may be est i mat ed
f r om met hods of Chap t er 7. Par amet er C is a di mensi onl ess
const ant and B is a p ar amet er t hat has t he same uni t as pres-
sure. These const ant s can be cal cul at ed f r om t he fol l owi ng
equat i ons:
B
- - = - 1 - 9.0702 (1 - Tr) ~/3 + 62.45326 (1 - T~) 2/3
Pc
- 135.1102 (1 - Tr) +e (1 - T~) 4/3
(5.131)
e = exp (4.79594 + 0.250047o) + 1.14188o) 2)
C = 0.0861488 + 0.0344483o9
wher e Tr is t he r educed t emp er at ur e and o) is t he acent r i c
factor. All t he above r el at i ons ar e in di mensi onl ess forms.
Obvi ousl y Eq. (5.130) gives very accur at e r esul t when P is
cl ose to P~ however, it shoul d not be used at Tr > 0.95. The
COSTALD cor r el at i on has been r ecommended for i ndust r i al
ap p l i cat i ons [59, 67]. However, in t he API -TDB [59] it is rec-
omme nde d t hat speci al val ues of acent r i c f act or obt ai ned
f r om vap or p r essur e dat a shoul d be used for o). These val ues
for some hydr ocar bons are gi ven by t he API -TDB [59]. The
fol l owi ng examp l e demons t r at es ap p l i cat i on of t hese met h-
ods. The most r ecent modi f i cat i on of t he Thoms on met hod
for p ol ar and associ at i ng fluids was p r op os ed by Garvi n i n
t he fol l owi ng f or m [68]:
, [ B+P/ Pc e "~]
v- - v j
K - -
vs. t 5-~ r ( BP~ e + P)
(5.132) B --- - I - 9. 070217r I/3 + 62. 45326r 2/3
- 135. 1102r + er 4/3
C = 0.0861488 + 0.034448309
e = exp (4.79594 + 0.25004709 + 1.14188o) z)
r = 1 - T/ Tc
wher e V s~t is t he sat ur at i on mol ar vol ume and psat is t he sat-
ur at i on p r essur e at T. V is l i qui d mol ar vol ume at T and P
and x is t he i sot her mal bul k comp r essi bi l i t y defi ned in t he
above equat i on ( also see Eq. 6.24). T~ is t he cri t i cal t emper-
at ur e and o) is t he acent r i c factor. Poe is equi val ent cri t i cal
pressure, whi ch for all al cohol s was near t he mean val ue of
27.0 bar. Thi s val ue for di ol s is about 8.4 bar. F or ot her seri es
of comp ounds Pce woul d be di fferent . Garvi n f ound t hat use of
Pc e si gni fi cant l y i mp r oves p r edi ct i on of V and x for al cohol s.
F or exampl e, for est i mat i on of r of met hanol at 1000 bar and
100~ Eq. (5.132) pr edi ct s x val ue of 7.1 x 10 -7 bar -1, whi ch
gives an er r or of 4.7% versus exp er i ment al val ue of 6.8 x 10 -7
bar -1, whi l e usi ng Pc t he er r or i ncr eases to 36.6%. However,
one shoul d not e t hat t he numer i cal coeffi ci ent s for B, C, and e
i n Eq. (5.132) may var y for ot her t ypes of p ol ar l i qui ds such
as coal l i qui ds.
Anot her cor r el at i on for cal cul at i on of effect of p r essur e on
l i qui d densi t y was p r op os ed by Chueh and Pr ausni t z [69] and
is bas ed on t he est i mat i on of i sot her mal compr essi bi l i t y:
Pe = peo[I + 9fl ( P - po) ] U9
fl = a (1 - 0. 894' ~ ) exp (6.9547 - 76.2853Tr + 191.306T 2
- 203.5472T~ + 82. 7631T 4)
V~ Zc
RTc Pc
(5.133)
The p ar amet er s ar e defi ned t he same as were defi ned in Eqs.
(5.132) and (5.133). V~ is t he mol ar cri t i cal vol ume and t he
uni t s of P, V~, R, and Tr mus t be consi st ent in a way t hat
PVc/RT~ becomes di mensi onl ess. Thi s equat i on is ap p l i cabl e
for Tr r angi ng f r om 0.4 to 0.98 and accur acy of Eq. (5.134) is
j ust mar gi nal l y less accur at e t han t he COSTALD cor r el at i on
[67].
Exampl e 5 . 8mPropane has vap or p r essur e of 9.974 bar at 300
K. Sat ur at ed l i qui d and vap or vol umes ar e V L = 90.077 and
V v = 2036.5 cma/ mol [Ref. 8, p. 4.24]. Cal cul at e sat ur at ed liq-
ui d mol ar vol ume usi ng (a) Racket t equat i on, (b) Eqs. ( 5. 127) -
(5.129), (c) Eqs. (5.127) and (5.130), and (d) Eq. (5.133).
$olution- - ( a) Obviously the most accurate method to esti-
mate V L is through Eq. (5.121). From Table 2.1, M = 44.I,
SG = 0.507, T~ = 96.7~ (369.83 K), Pc 42.48 bar, and w =
0.1523. From Table 5.12, ZgA ---- 0.2763. Tr ---- 0.811 so from
Eq. (5.121), V sat = 89.961 cma/mol ( -0. 1% error). (b) Use of
Eqs. (5.127)-(5.129) is not suitable for this case that Rack-
ett equation can be directly applied. However, to show the
application of method V sat is calculated to see their perfor-
mance. From Eq. (5.127) and use of SG = 0.507 gives pO __
0.491 g/cm 3. From Eq. (5.129), m = 1492.832, B20 = 180250.6,
X = 3.46356, BI -- 1094.68, and Br = 6265.188 bar. Using Eq.
(5.128), 0.491/p = 1-9. 974/ 6265. 188. This equation gives
density at T (300 K) and P (9.974 bar) as p = 0.492 g/cm 3.
V sat = M/p = 44.1/0.492 -- 89.69 cma/mol (error of -0.4%).
(c) Use of Eqs. (5.127) and (5.130) is not a suitable method
for density of propane, but to show its performance, satu-
rated liquid volume is calculated in a way similar to part
(b): From Eq. (5.131), B = 161.5154 bar and C = 0.091395.
For Eq. (5.130) we have ppo --- 0.491 g/cm 3, po = 1.01325 bar,
P -- 9.974 bar, and calculated density is pp = 0.4934 g/ cm 3.
Calculated V sat is 89.4 cm3/mol, which gives a deviation of
-0. 8% from experimental value of 90.077 cma/mol. (d) Us-
ing the Chueh-Prausnitz correlation (Eq. 5.133) we have
Z~ = 0.276, 0t = 0.006497, fl = 0.000381, pp = 0.49266 g]cm 3,
and VSa]tc = 89.5149 cm3/mol, which gives an error of -0. 62%
from the actual value. )
5 . 9 R EFR A C T I VE I N D EX B A S ED EQUA T I ON
OF S TATE
From the various PVT relations and EOS discussed in this
chapter, cubic equations are the most convenient equations
that can be used for volumetric and phase equilibrium cal-
culations. The main deficiency of cubic equations is their (5.134)
inability to predict liquid density accurately. Use of volume
translation improves accuracy of SRK and PR equations for
liquid density but a fourth parameter specific of each equa-
tion is required. The shift parameter is not known for heavy
compounds and petroleum mixtures. For this reason some
specific equations for liquid density calculations are used. As
an example Alani-Kennedy EOS is specifically developed for
calculation of liquid density of oils and reservoir fluids and
is used by some reservoir engineers [19, 21]. The equation
is in van der Waals cubic EOS form but it requires four nu-
merical constants for each pure compound, which are given
from Ca to Ca0. For the C7+ fractions the constants should be (5.135)
estimated from M7+ and SG7+. The method performs well for
light reservoir fluids and gas condensate samples. However,
as discussed in Chapter 4, for oils with significant amount of
heavy hydrocarbons, which requires splitting of C7+ fraction,
the method cannot be applied to C7+ subfractions. I n addition
the method is not applicable to undefined petroleum fractions
with a limited boiling range.
Generally constants of cubic equations are determined
based on data for hydrocarbons up to C8 or C9. As an ex-
ample, the LK generalized correlations is based on the data
for the reference fluid of n-C8. The parameter that indicates
complexity of a compound is acentric factor. I n SRK and PR
EOS parameter a is related to w in a polynomial form of at
least second order (see f~ in Table 5.1). This indicates that
extrapolation of such equations for compounds having acen-
tric factors greater than those used in development of EOS
parameters is not accurate. And it is for this reason that most
cubic equations such as SRK and PR equations break down
when they are applied for calculation of liquid densities for
C10 and heavier hydrocarbons. For this reason Riazi and Man-
soori [70] attempted to improve capability of cubic equations
for liquid density prediction, especially for heavy hydrocar-
bons.
Most modifications on cubic equations is on parameter a
and its functionality with temperature and w. However, a pa-
rameter that is inherent to volume is the co-volume parameter
b. RK EOS presented by Eq. (5.38) is the simplest and most
widely used cubic equation that predicts reasonably well for
prediction of density of gases. I n fact as shown in Table 5.13
for simple fluids such as oxygen or methane (with small oJ)
RK EOS works better than both SRK and PR regarding liquid
densities.
For liquid systems in which the free space between
molecules reduces, the role of parameter b becomes more
important than that of parameter a. For low-pressure gases,
however, the role of parameter b becomes less important than
a because the spacing between molecules increases and as
a result the attraction energy prevails. Molar refraction was
defined by Eq. (2.34) as
M ( n2 - 1 ~
Rm = V I = ~ \ n 2 +2 /
where Rm is the molar refraction and V is the molar volume
both in cma/mol. Rm is nearly independent of temperature
but is normally calculated from density and refractive index
at 20~ (d20 and n20). Rm represents the actual molar volume
of molecules and since b is also proportional to molar volume
of molecules (excluding the free space); therefore, one can
conclude that parameter b must be proportional to Rm. I n
fact the polarizability is related to Rm in the following form:
3
a' = "7"'77-~. Rm -- #(T)
~Tr/VA
where NA is the Avogadro's number and/ z( T) is the dipole
moment, which for light hydrocarbons is zero [7]. Values of
Rm calculated from Eq. (5.134) are reported by Riazi et al.
[70, 71] for a number of hydrocarbons and are given in Table
5.14. Since the original RK EOS is satisfactory for methane we
choose this compound as the reference substance. Parameter
Compound
Methane
Oxygen
T A B L E 5.13--Evaluation of RK, SRK, and PR EOS for prediction of density of simple fluids.
%AAD
No. of data points Temperature range, K Pressure range, bar RK SRK PR
135 90-500 0.7-700 0.88 1.0 4.5
120 80-1000 1-500 1,1 1.4 4.0
Data source
Goodwin [72]
TRC [73]
2 2 6 CHARACTERI ZATI ON AND PROPERTI ES OF PETROLEUM FRACTIONS
TABLE 5.14---Data source for development of Eq. (5.139), values of parameter r and predicted Zc from MRK EOS [70].
Rm, at 20~ No. of Temp. Pressure
No. Compound cm3/mol R data points range, K range, bar
1 Methane (C1) 6.987 1.000 135 90-500 0.5-700
2 Et hane (C2) 11.319 1.620 157 90-700 0.1-700
3 Ethylene 10.508 1.504 90 100-500 1--400
5 Propane (Ca) 15.784 2.259 130 85-700 0.1-700
6 I sobut ane 20.647 2.955 115 110-700 0.1-700
7 n-Butane (C4) 20.465 2.929 183 130-700 0.1-700
8 n-Pentane (C5) 25.265 3.616 . . . . . .
9 n-Hexane (C6) 29.911 4.281 "1"()0 298-1000 1-500
10 Cyclohexane 27.710 3.966 140 320-1000 t-500
11 Benzene 26.187 3.748 110 310-1000 1-500
12 Toluene 31.092 4.450 110 330-1000 1-500
13 n-Hept ane (C7) 34.551 4.945 100 300-1000 1-500
14 n-Octane (C8) 39.183 5.608 80 320-1000 1-500
15 / -Octane 39.260 5.619 70 340-1000 1-500
16 n-Heptane (C7) a 34.551 4.945 35 303-373 50-500
17 n-Nonane (C9) a 43,836 6.274 35 303-373 50-500
18 n-Decane (C10) a 48.497 6. 941 . . . . . . .
19 n-Undecane (CH) a 53.136 7.605 "35 303-373 50-500
20 n-Dodecane (Clz) a 57.803 8.273 . . . . . . .
21 n-Tridecane (C13) a 62.478 8.942 "30 303-373 50-500
22 n-Tetradecane (C14) a 67.054 9.597 . . . . . . . . .
23 n-Pentadecane (C15) a 71.708 10. 263 . . . . . . . . .
24 n-Hexadecane (C16) ~ 76.389 10. 933 . . . . . . .
25 n- Hep t adecane (C17) a 81.000 11.593 '30 323-573 50-500
26 n-Eicosane (C20) a 95.414 13.656 20 373-573 50-500
27 n-Triacosane (C30) a 141.30 20.223 20 373-573 50-500
28 n-Tetracontane (C40) a 187.69 26.862 20 423-573 50-500
Critical compressibility, Zc
Ref. Table 2.1 Pred. MRI ( %AD
Goodwin [72] 0.288 0.333 15.6
Goodwin et al. [74] 0.284 0.300 5.6
McCarty and 0.276 0.295 6.9
Jacobsen [75]
Goodwin and 0.280 0.282 0.7
Haynes [76]
Goodwin and 0.282 0.280 0.7
Haynes [76]
Haynes and 0.274 0.278 1,5
Goodwin [77]
0.269 0.271 0,7
TI~C Tables [73] 0.264 0.266 0.7
TRC Tables [73] 0.273 0.269 1.5
TRC Tables [73] 0.271 0.270 0.4
TRC Tables [73] 0.264 0.265 0.4
TRC Tables [73] 0.263 0.262 0.4
TRC Tables [73] 0.259 0.258 0.4
TRC Tables [73] 0.266 0.256 3.8
Doolittle [78] . . . . . . . . .
Doolittle [78] 0.255 0.254 0.4
0.249 0.250 0.4
Doolittle [78] 0.243 0.247 1.6
0.238 0.245 2.9
Doolittle [78] 0.236 0.242 2.6
.., 0.234 0.240 2.5
... 0.228 0.238 4.3
0.225 0.235 4.2
l)oolittle [78] 0.217 0.233 7.4
Doolitfle [78] 0.213 0.227 6.6
Doolittle [78] ... 0.213 ...
Doolittle [78] . . . . . .
Overall . . . . . . 1745 90-1000 0.1-700 3.0
Density data for compounds 16-28 are all only for liquids [78]. Compounds specified by bold are used in development of Eq. (5.139). Calculated values of Zc from
SRK and PR E OSs for all compounds are 0.333 and 0.307, respectively. These give average errors of 28.2 and 18.2 %, respectively.
aPVT data for the following compounds were not used in development of Eq. (5.I39).
fl is defined as
bactual
(5.136) 3 = - -
b~
wher e bact~al is t he op t i mum val ue of b and bRK is t he val ue of
b obt ai ned for RK EOS and is cal cul at ed t hr ough t he rel at i on
gi ven i n Table 5.1. For t he r ef er ence fluid, 3~ef. = 1. We now
assume t hat
(5.137) fl-~- = ~--~-- = f ( ~, ~, T~)
~ref C/ref
Par amet er r is defined as
Rm Rm
(5.138) r . . . .
Rm, ref. 6.987
r is a di mensi onl ess p ar amet er and represent s r educed mol ec-
ul ar size. Values of r cal cul at ed from Eq. (5.138) are al so
gi ven in Table 5.14. By combi ni ng Eqs. (5.137) and (5.138)
and based on dat a for densi t i es of hydr ocar bons f r om C2 t o
C8 ,the fol l owi ng rel at i on was f ound for cal cul at i on of p ar am-
et er b i n t he RK EOS:
1
- = 1 + {0.0211 - 0.92 exp ( - 1000 IT~ - 11)] - 0.035 (Tr - 1)}
(r - 1)
(5.139)
Once 3 is det er mi ned f r om t he above rel at i on, t he co- vol ume
p ar amet er b for t he RK can be cal cul at ed by subst i t ut i ng bRK
f r om Table 5.1 i nt o Eq. (5.136) as
( 0. 08664RTc ~
(5.140) b= \ -~ ] 3
Par amet er a for t he RK EOS is given in Table 5.1 as
0.42748R2T~
(5.141) a =
Pc
Therefore, t he modi fi ed RK EOS is comp osed of Eq. (5.38)
and Eqs. (5.138)-(5.141) for cal cul at i on of t he p ar amet er s a
and b. Equat i on (5.39) for t he PVT rel at i on and Eq. (5.141)
for p ar amet er a are t he same as t he ori gi nal RK EOS. This
modi fi ed versi on of R K EOS is referred as MRK. I n fact when
3 = 1 t he MRK EOS reduces to RK EOS. The exponent i al
t er m i n Eq. (5.139) is t he cor r ect i on for t he critical region. At
T~ = 1 this equat i on reduces to
(5.142) batrc = I + 0.0016(r - 1)
This equat i on i ndi cat es t hat t he MRK EOS does not give a
const ant Zc for all comp ounds but di fferent val ues for dif-
ferent comp ounds. For this r eason this EOS does not satisfy
t he const rai nt s set by Eq. (5.9). But cal cul at i ons show t hat
(OP/~V)rc and (O2p/oV2)r c are very small. For hydr ocar bons
f r om C1 to C20 t he average val ues for t hese deri vat i ves are
0.0189 and 0.001, respect i vel y [70]. I n s ummar y 1383 dat a
poi nt s on densi t i es of l i qui ds and gases for hydr ocar bons f r om
C2 to C8 wi t h pressure range of 0. 1-700 bar and t emp er at ur e
up to 1000 K wer e used in devel op ment of Eq. (5.139). The
TABLE 5.15--Evaluation of various EOS for prediction of liquid
density of heavy hydrocarbons [70].
%AAD
No. of
Compound data points MRK RK SRK PR
n-Heptane (n-C7) 35 0.6 12.1 10.5 1.4
n-Nonane (n-C9) 35 0.6 15.5 13.4 3.4
n-Undecane (n-C11) 35 1.7 18.0 15.5 5.4
n- Tr i decane (n-C13) 30 2.8 20.3 17.7 7.9
n-Heptadecane (n-C17) 30 1.2 27.3 24.8 16.0
n-Eicosane (n-C20) 20 2.8 29.5 26.7 18.2
n-Triacontane (n-C30) 20 0.6 41.4 39.4 32.5
n-Tetracontane (n-C40) 20 4.1 50.9 49.4 44.4
Total 225 1.6 24.3 22.1 13.3
MRK: Eqs. (5.38), (5.138), and (5.141). Note none of these data were used
in development of Eq. (5.139).
i nt er est i ng p oi nt about t hi s equat i on is t hat it can be used up
t o C40 for densi t y est i mat i ons. Obvi ousl y t hi s equat i on is not
desi gned for VLE cal cul at i ons as no VLE dat a were used to
devel op Eq. (5.139). Pr edi ct i on of Zc f r om MRK EOS is shown
i n Table 5.14. Eval uat i on of MRK wi t h PR and SRK equat i ons
for p r edi ct i on of l i qui d densi t y of heavy hydr ocar bons is gi ven
i n Table 5.15. Dat a sources for t hese comp ounds ar e gi ven in
Table 5.14. Overal l resul t s for p r edi ct i on of densi t y for bot h
l i qui d and gaseous hydr ocar bon comp ounds f r om C1 to C40
is shown in Table 5.15. The overal l er r or for t he MRK EOS
for mor e t han 1700 dat a p oi nt s is about 1.3% in comp ar i s on
wi t h 4.6 for PR and 7.3 for SRK equat i ons.
To ap p l y t hi s EOS to defi ned mi xt ur es a set of mi xi ng rul es
ar e gi ven in Table 5.17 [70]. For p et r ol eum f r act i ons p ar ame-
t ers can be di r ect l y cal cul at ed for t he mi xt ure. F or bi nar y and
t er nar y l i qui d mi xt ur es cont ai ni ng comp ounds f r om C1 to C20
an average er r or of 1.8% was obt ai ned for 200 dat a p oi nt s [70].
F or t he same dat aset RK, SRK, and PR equat i ons gave er r or s
of 15, 13, and 6%, respectively. F ur t her char act er i st i cs and
eval uat i ons of t hi s modi f i ed RK EOS ar e di scussed by Ri azi
and Roomi [71]. Appl i cat i on of t hi s met hod i n cal cul at i on of
densi t y is shown in t he fol l owi ng exampl e.
Exampl e 5. 9- - Rep eat Examp l e 5.2 for p r edi ct i on of l i qui d
and vap or densi t y of n-oct ane usi ng MRK EOS.
Sol ut i on- - The MRK EOS is to use Eq. (5.38) wi t h p ar amet er s
obt ai ned f r om Eqs. ( 5. 139) -( 5. 141) . The i np ut dat a needed to
use MRK EOS are Tc, Pc, a ndr . F r om Examp l e 5.2, Tc = 568.7
K, Pc = 24.9 bar, and Tr = 0.9718 K. F r om Table 5.14 for
n-Cs, r = 5.608. F r om Eq. (5.139), fl = 1.5001 x 10 -4. F r om
Eq. (5.139), b- - 150.01 cma/ mol and f r om Eq. (5.141), a =
3.837982 x 107 cm6/ mol 2. Sol vi ng Eq. (5.42) wi t h ul = 1 and
u2 = 0 ( Table 5.1) and in a way si mi l ar to t hat p er f or med i n
Examp l e 5.2 we get V L = 295.8 and V v -- 1151.7 cma/ mol . De-
vi at i ons of p r edi ct ed val ues f r om exp er i ment al dat a ar e - 2. 7%
and - 5. 3% for l i qui d and vap or mol ar vol ume, respect i vel y.
TABLE 5.16---Comparison of various EOSs for prediction of
density of liquid and gaseous hydrocarbons.
%AAD
No. of
Compound data points MRK RK SRK PR
C1 C~ 1520 1.3 4.9 5.1 3.3
CT-Cb0 225 1.6 24.3 22.1 13.3
Total 1745 1.33 7.38 7.28 4.59
aThese are the compounds that have been marked as bold in Table 5.14 and
are used in development of Eq. (5.139).
bThese are the same compounds as in Table 5.15.
TABLE 5. 17nMixing rules for MRK EOS parameters (Eqs. (5.38)
and (5.137)-(5.140)).
Tcm ~i ~,j xixi Tc2ij/Pci]
= ~,i ~j xixjTcij/Pcij
Pcm ( Ei v/;~j xixjTcij/Pcij) 2
Rm = ~i )-~-j ~x j r i j
r c . = (rd rcj) '/~(i - ~j)
8r~q
P~q = [ (r~i/p~i)l/3+(rc./pc.),/3] 3
1/3_ 1/3X 3
rii +rjj )
rij - - 8
Pr edi ct ed l i qui d densi t i es f r om SRK and PR equat i ons ( Exam-
pl e 5.2) devi at e f r om exp er i ment al dat a by +31. 5 and 17.2%,
respectively. Advant age of MRK over ot her cubi c equat i ons
for l i qui d densi t y is gr eat er for heavi er comp ounds as shown
in Table 5.15. t
Thi s modi f i ed ver si on of RK EOS is devel op ed onl y for den-
si t y cal cul at i on of hydr ocar bon syst ems and t hei r mi xt ures.
I t can be used di r ect l y t o cal cul at e densi t y of p et r ol eum frac-
t i ons, once M, d20,/'/20, Tc, and Pc ar e cal cul at ed f r om met h-
ods di scussed i n Chapt er s 2 and 3. Mor eover p ar amet er r can
be accur at el y est i mat ed for heavy fract i ons, whi l e p r edi ct i on
of acent r i c f act or for heavy comp ounds is not rel i abl e ( see
Figs. 2. 20-2. 22) . The mai n char act er i st i c of t hi s equat i on is its
ap p l i cat i on to heavy hydr ocar bons and undef i ned p et r ol eum
fract i ons. The fact t hat Eq. (5.139) was devel op ed bas ed on
dat a for hydr ocar bons f r om C2 t o C8 and it can wel l be used up
t o C40 shows its ext r ap ol at i on capabi l i t y. The l i near r el at i on
t hat exists bet ween 1/fl and p ar amet er r makes its ext rapo-
l at i on to heavi er hydr ocar bons possi bl e. I n fact i t was f ound
t hat by changi ng t he f unct i onal i t y of 1/fi wi t h r, bet t er pre-
di ct i on of densi t y is possi bl e but t he r el at i on woul d no l onger
be l i near and its ext r ap ol at i on t o heavi er comp ounds woul d
be less accur at e. For exampl e, for C17 and C18, i f t he const ant
0.02 i n Eq. (5.139) is r ep l aced by 0.018, t he %AAD for t hese
comp ounds r educes f r om 2 to 0.5%. The fol l owi ng examp l e
shows ap p l i cat i on of t hi s met hod.
Anal ysi s of var i ous EOS shows t hat use of refract i ve i ndex
i n obt ai ni ng const ant s of an EOS is a p r omi s i ng ap p r oach.
F ur t her wor k in t hi s ar ea shoul d involve use of sat ur at i on
p r essur e in addi t i on t o l i qui d densi t y dat a t o obt ai n r el at i ons
for EOS p ar amet er s t hat woul d be sui t abl e for bot h l i qui d
densi t y and VLE cal cul at i ons.
5 . 1 0 S UM M A R Y A N D C ON C L US I ON S
I n t hi s chap t er t he f undament al of PVT r el at i ons and mat h-
emat i cal EOS ar e pr esent ed. Once t he PVT r el at i on for a
fluid is known var i ous p hysi cal and t her modynami c pr oper -
t i es can be det er mi ned as di scussed i n Chapt er s 6 and 7. I n-
t er mol ecul ar forces and t hei r i mp or t ance i n p r op er t y pr edi c-
t i ons were di scussed i n t hi s chapt er. For l i ght hydr ocar bons
t wo- p ar amet er p ot ent i al ener gy r el at i ons such as LJ descr i bes
t he i nt er mol ecul ar forces and as a resul t t wo- p ar amet er EOS
ar e suffi ci ent to descr i be t he PVT r el at i on for such fluids. I t is
shown t hat EOS p ar amet er s can be di rect l y cal cul at ed f r om
t he p ot ent i al ener gy rel at i ons. Cri t eri a for cor r ect EOS ar e
gi ven so t hat val i di t y of any EOS can be anal yzed. Three cat-
egory of EOSs ar e p r esent ed in t hi s chapt er : ( I ) cubi c type,
(2) noncubi c type, and (3) gener al i zed cor r el at i ons.
Four types of cubic equations vdW, RK, SRK, and PR and
their modifications have been reviewed. The mai n advantage
of cubic equations is simplicity, mat hemat i cal convenience,
and their application for bot h vapor and liquid phases. The
mai n application of cubic equations is in VLE calculations
as will be discussed in Chapters 6 and 9. However, their abil-
ity to predict liquid phase density is limited and this is the
mai n weakness of cubic equations. PR and SRK equations are
widely used in the pet rol eum industry. PR equation gives bet-
ter liquid density predictions, while SRK is used in VLE calcu-
lations. Use of volume translation improves capability of liq-
uid density prediction for both PR and SRK equations; how-
ever, the met hod of calculation of this par amet er for heavy
pet rol eum fractions is not available and generally these equa-
tions break down at about C10. Values of input paramet ers
greatly affect EOS predictions. For heavy hydrocarbons, ac-
curate prediction of acentric factor is difficult and for this rea-
son an alternative EOS based on modified RK equation is pre-
sented in Section 5.9. The MRK equation uses refractive index
paramet er instead of acentric factor and it is recommended
for density calculation of heavy hydrocarbons and undefined
pet rol eum fraction. This equation is not suitable for VLE and
vapor pressure calculations. In Chapter 6, use of velocity of
sound data to obtain EOS paramet ers is discussed [79].
Among noncubic equations, virial equations provide more
accurate PVT relations; however, prediction of fourth and
higher virial coefficients is not possible. Any EOS can be con-
verted into a virial form. For gases at moderat e pressures,
truncated virial equation after third t erm (Eq. 5.75) is recom-
mended. Equation (5.71) is recommended for estimation of
the second virial coefficient and Eq. (5.78) is recommended
for prediction of the third virial coefficients. For specific com-
pounds in which virial coefficients are available, these should
be used for more accurate prediction of PVT data at certain
moderat e conditions such as those provided by Gupta and
Eubank [80].
Several ot her noncubic EOS such as BWRS, CS, L J, SPHC,
and SAFT are presented in this chapter. As will be discussed
in the next chapter, recent studies show that cubic equations
are also weak in predicting derivative properties such as en-
thalpy, Joule Thomson coefficient, or heat capacity. For this
reason, noncubic equations such as simplified pert urbed hard
chain (SPHC) or statistical associating fluid theory (SAFT)
are being investigated for prediction of such derived prop-
erties [81]. For heavy hydrocarbons in which t wo-paramet er
potential energy functions are not sufficient to describe the in-
termolecular forces, three- and perhaps four-paramet er EOS
must be used. The most recent reference on the theory and
application of EOSs for pure fluids and fluid mixtures is pro-
vided by Sengers et al. [82]. I n addition, for a limited number
of fluids there are highly accurate EOS that generally take on a
modified MBWR form or a Helmholtz energy representation
like the IAPWS wat er standard [4]. Some of these equations
are even available free on the webs [83].
The theory of corresponding state provides a good PVT rela-
tion between Z-factor and reduced t emperat ure and pressure.
The LK correlation presented by Eqs. (5.107)-(5.11 I) is based
on BWR EOS and gives the most accurate PVT relation if ac-
curate input data on To, Pc, and co are known. While the cubic
equations are useful for phase behavior calculations, the
LK corresponding states correlations are r ecommended for
calculation of density, enthalpy, entropy, and heat capacity of
hydrocarbons and pet rol eum fractions. Analytical form of LK
correlation is provided for comput er applications, while the
tabulated form is given for hand calculations. Simpler two-
par amet er empirical correlation for calculation of Z-factor of
gases, especially for light hydrocarbons and natural gases, is
given in a graphical form in Fig. 5.12 and Hall-Yarborough
equation can be used for comput er applications..
For calculation of liquid densities use of Rackett equa-
tion (Eq. 5.121) is recommended. For pet rol eum fractions
in which Racket paramet er is not available it should be de-
t ermi ned from specific gravity t hrough Eq. (5.123). For the
effect of pressure on liquid density of light pure hydrocar-
bons, defined hydrocarbon mixtures and light pet rol eum frac-
tions, the COSTALD correlation (Eq. 5.130) may be used. For
pet rol eum fractions effect of pressure on liquid density can
be calculated t hrough Eq. (5.128).
For defined mixtures the simplest approach is to use Kay' s
mixing rule (Eqs. 3.39 and 5.116) to calculate pseudocriti-
cal properties and acentric factor of the mixture. However,
when molecules in a mixture are greatly different in size (i.e.,
Cs and C20), more accurate results can be obtained by using
appropri at e mixing rules given in this chapt er for different
EOS. For defined mixtures liquid density can be best calcu-
lated through Eq. (5.126) when pure component densities are
known at a given t emperat ure and pressure. For undefined
narrow boiling range pet rol eum fractions Tc, Pc, and co should
be estimated according to the met hods described in Chapters
2 and 3. Then the mixture may be treated as a single pseudo-
component and pure component EOS can be directly applied
to such systems. Some other graphical and empirical meth-
ods for the effect of t emperat ure and pressure on density and
specific gravity of hydrocarbons and pet rol eum fractions are
given in Chapter 7. Furt her application of met hods presented
in this chapt er for calculation of density of gases and liquids
especially for wide boiling range fractions and reservoir fluids
will be presented in Chapter 7. Theory of prediction of ther-
modynami c properties and their relation with PVT behavi or
of a fluid are discussed in the next chapter.
5 . 1 1 PR OB L EMS
5.1. Consider three phases of water, oil, and gas are in equi-
librium. Also assume the oil is expressed in t erms of
10 component s (excluding water) with known specifica-
tions. The gas contains the same compounds as the oil.
Based on the phase rule determine what is the mi ni mum
i nformat i on that must be known in order to det ermi ne
oil and gas properties.
5.2. Obtain coefficients a and b for the PR EOS as given in
Table 5.1. Also obtain Zc = 0.307 for this EOS.
5.3. Show that the Dieterici EOS exhibits the correct limiting
behavi or at P ~ 0 (finite T) and T --~ o0 (finite P)
RTb e-a/n:rv
P- V -
where a and b are constants.
5.4. The Lorentz EOS is given as
a bV
where a and b are the EOS constants. Is this a valid
EOS?
5.5. A graduate student has come up with a cubic EOS in the
following form:
I aV2 ] (V - b) = RT
P + (V +b ) ( V - b )
Is this equation a correct EOS?
5.6. Derive a relation for the second virial coefficient of a
fluid that obeys the SWP relation. Use data on B for
methane in Table 5.4 to obtain the potential energy
parameters, a and s. Compare your calculated values
with those obtained from LJ Potential as a = 4.01 A and
elk = 142.87 K [6, 79].
5.7. Derive Eq. (5.66) from Eq. (5.65) and discuss about your
derivation.
5.8. Show that for the second virial coefficient, Eq. (5.70) can
be reduced to a form similar to Eq. (5.59). Also show that
these two forms are identical for a binary system.
5.9. Derive the virial form of PR EOS and obtain the virial
coefficients B, C, and D in terms of PR EOS parameters.
5.10. With results obtained in Example 5.4 and Problem 5.9
for the virial coefficients derived from RK, SRK, and PR
equations estimate the following:
a. The second virial coefficient for propane at temper-
atures 300, 400 and 500 K and compare the results
with those given in Table 5.4. Also predict B from
Eqs. (5.71)-(5.73).
b. The third virial coefficients for methane and ethane
and compare with those given in Table 5.5.
c. Compare predicted third virial coefficients from (b)
with those predicted from Eq. (5.78).
5.11. Specific volume of steam at 250~ and 3 bar is 796.44
cm3/g [1]. The virial coefficients (B and C) are given in
Table 5.5. Estimate specific volume of this gas from the
following methods:
a. RK, SRK, and PR equations.
b. Both virial forms by Eqs. (5.65) and (5.66). Explain
why the two results are not the same.
c. Virial equation with coefficients estimated from Eqs.
(5.71), (5.72), and (5.78)
5.12. Estimate molar volume of n-decane at 373 K and 151.98
bar from LK generalized correlations. Also estimate the
critical compressibility factor. The actual molar volume
is 206.5 cma/mol.
5.13. For several compounds liquid density at one tempera-
ture is given in the table below.
5.14.
Componen# N2 H20 C1 C2 C3 n-C4
T, K 78 293 112 183 231 293
p, g/cm 3 0.804 0. 998 0. 425 0. 548 0. 582 0.579
~Source: Reid et al. [15].
For each compound calculate the Rackett parameter
from reference density and compare with those given
in Table 5.12. Use estimated Rackett parameter to cal-
culate specific gravity of Ca and n-Ca at 15.5~ and com-
pare with values of SG given in Table 2.1.
For a petroleum fraction having API gravity of 31.4 and
Watson characterization factor of 12.28 estimate liquid
5. PV T RELATI ONS AND EQUATI ONS OF STATE 229
density at 68~ and pressure of 5400 psig using the fol-
lowing methods. The experimental value is 0.8838 g/ cm 3
(Ref. [59] Ch 6)
a. SRK EOS
b. SRK using volume translation
c. MRK EOS
d. Eq. (5.128)
e. COSTALD correlation (Eq. 5.130)
f. LK generalized correlation
g. Compare errors from different methods
5.15. Estimate liquid density of n-decane at 423 K and 506.6
bar from the following methods:
a. PR EOS
b. PR EOS with volume translation
c. PR EOS with Twu correlation for parameter a (Eq.
5.54)
d. MRK EOS
e. Racket equation with COSTALD correlation
f. Compare the values with the experimental value of
0.691 g/cm 3
5.16. Estimate compressibility factor of saturated liquid and
vapor (Z L and Z v) methane at 160 K (saturation pres-
sure of 15.9 bar) from the following methods:
a. Z L from Racket equation and Z v from Standing-Katz
chart
b . PR EOS
c. PR EOS with Twu correlation for parameter a (Eq.
5.54)
d. MRK EOS
e. LK generalized correlation
f. Compare estimated values with the values from Fig.
6.12 in Chapter 6.
5.17. Estimate Z v of saturated methane in Problem 5.16 from
virial EOS and evaluate the result.
5.18. A liquid mixture of C1 and n-C5 exists in a PVT cell at
311.1 K and 69.5 bar. The volume of liquid is 36.64 cm 3 .
Mole fraction of C1 is 0.33. Calculate mass of liquid in
grams using the following methods:
a. PR EOS with and without volume translation
b. Rackett equation and COSTALD correlation
c. MRK EOS
5.19. A natural gas has the following composition:
Component CO2 H2S N2 C1 C2 C3
mol% 8 16 4 65 4 3
5.20.
Determine the density of the gas at 70 bar and 40~ in
g/ cm 3 using the following methods:
a. Standing-Katz chart
b. Hall-Yarborough EOS
c. LK generalized correlation
Estimate Z L and Z v of saturated liquid and vapor ethane
at Tr = 0.8 from MRK and virial EOSs. Compare calcu-
lated values with values obtained from Fig. 5.10.
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Boulder, CO, 2003, http://webbook.nist.gov/chemistry/fluid/.
MNL50-EB/Jan. 2005
Thermodynamic Relations
for Property Estimations
N OM EN C L A T UR E
A
API
A, B, C . . . .
a,b
ac
ai
a, b , c , . . .
b
B
B', B"
C
C', C"
C
CpR
Cp
Cv
,t2 o
F(x, y)
f
f?L t
f?
foL
fr L
fo,
G
G R
H
CH
Helmholtz free energy defined in Eq. (6.7), J/ mol
Cubic EOS paramet ers given in Table 5.1
Paramet er defined in Eq. (5.41) and given in
Table 5.1
Activity of component i defined in Eq. (6.111),
dimensionless
A paramet er defined in the Standing correlation,
Eq. (6.202), K
Second virial coefficient, cm3/ mol
First- and second-order derivatives of second
virial coefficient with respect to t emperat ure
Third virial coefficient, (cm3/mol) 2
First- and second-order derivatives of third virial
coefficient with respect to t emperat ure
Velocity of sound, m/ s
Velocity of sound calculated from PR EOS
Heat capacity at constant pressure defined by
Eq. (6.17), J/ mol. K
Heat capacity at constant volume defined by
Eq. (6.18), J/ mol 9 K
Liquid density at 20~ and 1 atm, g/ cm 3
A mat hemat i cal function of independent
variables x and y.
Fugacity of a pure component defined by
Eq. (6.45), bar
Fugacity of component i in a mixture defined by
Eq. (6.109), bar
Fugacity of pure liquid i at standard pressure
(1.01 bar) and t emperat ure T, bar
Fugacity of pure solid i at P and T (Eq. 6.155),
bar
Fugacity of pure hypothetical liquid at
t emperat ure T (T > Tc), bar
Reduced fugacity of pure hypothetical liquid at
t emperat ure T ( = f~ dimensionless
A function defined in t erms of oJ for par amet er a
in the PR and SRK equations as given in
Table 5.1 and Eq. (5.53)
Molar Gibbs free energy defined in Eq. (6.6),
J/ mol
Molar residual Gibbs energy ( = G - Gig),
J/ mol
Molar enthalpy defined in Eq. (6.1), J/ tool
2 3 2
psub
/ ' 1, / ' 2, / ' 3
Qrev
RC
U
Ul , IA2
S
S
Ki Equilibrium ratio in vapor-liquid equilibria
(Ki -=- yi/xi) defined in Eq. (6.196), dimensionless
K~ sL Equilibrium ratio in solid-liquid equilibria
( K sL = xS/xi L) defined in Eq. (6.208),
dimensionless
Kw Watson characterization factor defined by
Eq. (2.13)
kB Boltzman constant ( = R/NA = 1.381 x 10 -23 J/K)
/~ Henry' s law constant defined by Eq. (6.184),
bar
k/.u Henry' s law constant of component i in a
muhi component solvent, bar
/~i Binary interaction par amet er (BIP),
dimensionless
M A mol ar propert y of system (i.e., S, V, H, S, G . . . . )
M E Excess propert y ( = M - M ~a)
M t Total propert y of system ( = ntM)
Mi Partial mol ar propert y for M defined by
Eq. (6.78)
M Molecular weight, g/ mol [kg/ kmol]
M g Gas mol ecul ar weight, g/tool [kg/ kmol]
NA Avogadro number = number of molecules in
I mol (6.022 x 10 23 tool -1)
Arc Number of carbon at oms in a hydrocarbon
compound
n Number of moles (g/ molecular wt), tool
r~ Number of moles of component i in a mixture,
tool
P Pressure, bar
psat Saturation pressure, bar
Pa Atmospheric pressure, bar
Pr Reduced pressure defined by Eq. (5. 100)
( = P/P~), dimensionless
Sublimation pressure, bar
Derivative paramet ers defined in Table 6.1
Heat transferred to the system by a reversible
process, J/ mol
R Gas constant = 8.314 J/ mol. K (values in
different units are given in Section 1.7.24)
An objective function defined in Eq. (6.237)
Molar internal energy, J/tool
Paramet ers in Eqs. (5.40) and (5.42) as given in
Table 5.1 for a cubic EOS
Liquid mol ar volume, cm3/ mol
Molar ent ropy defined by Eq. (6.2), J/ mol- K
Shrinkage factor defined by Eq. (6.95),
dimensionless
6. THERM ODYNAM I C REL ATI ONS FOR PROPERTY ESTI M ATI ONS 233
SG g Specific gravity of gas fluid (pure or mixture)
[ = Mg/29], dimensionless
SG Specific gravity of liquid substance at 15.5~ (60~
T Absolute temperature, K
Tr Reduced temperature defined by Eq. (5.100)
( = T/Tc), dimensionless
TB A parameter in the Standing correlation
(Eq. 6.202), K
TM Freezing (melting) point for a pure component at
1.013 bar, K
Ttp Triple point temperature, K
V Molar volume, cm3/gmol
V t Saturated liquid molar volume, cm3/gmol
V sat Saturation molar volume, cm3/gmol
V v Saturated vapor molar volume, cm3/gmol
Vc Critical volume (molar), cm3/Inol (or critical specific
volume, cm3/g)
Vr Reduced volume (= V/Vc)
V25 Liquid molar volume at 25~ cm3/mol
x4 Mole fraction of component i in a mixture (usually
used for liquids), dimensionless
Xwi Weight fraction of component i in a mixture (usually
yi Mole fraction of i in mixture (usually used for gases),
dimensionless
Z Compressibility factor defined by Eq. (5.15),
dimensionless
Z L Compressibility factor of liquid phase,
dimensionless
Z v Compressibility factor of vapor phase, dimensionless
Greek Letters
Ors, ]~S
Parameter defined by Eq. (5.41), dimensionless
Parameters defined based on velocity of sound for
correction of EOS parameters a and b defined by
fl Coefficient of thermal expansion defined by
Eq. (6.24), K -1.
A Difference between two values of a parameter
~i Solubility parameter for i defined in Eq. (6.147),
( J / cm3) l / 2
8i Parameter used in Eq. (6.126), dimensionless
3ii Parameter defined in Eq. (5.70)
e Energy paramet er in a potential energy function
e Error parameter defined by Eq. (106), dimensionless
qI) i Volume fraction of i in a liquid mixture defined by
Eq. (6.146)
Fugacity coefficient of pure i at T and P defined by
q~i Fugacity coefficient of component i at T and P in an
ideal solution mixture, dimensionless
6i Fugacity coefficient of component i in a mixture at T
and P defined by Eq. (6.110)
0 A paramet er defined in Eq. (6.203), dimensionless
p Density at a given temperature and pressure, g/cm 3
(molar density unit: cma/mol)
a Diameter of hard sphere molecule,/ ~ (10 -l~ rn)
o"
(9
K
Y
Y/
ACpi
A H yap
AHmix
AM
AS/
ASv~p
ATb2
ATM 2
AVm~x
Molecular size parameter,/ ~ (10 -1~ m)
Acentric factor defined by Eq. (2.10)
Packing fraction defined by Eq. (5.86),
dimensionless
I sothermal compressibility defined by Eq. (6.25),
bar -I
J oule-Thomson coefficient defined by Eq. (6.27),
K/ bar
Heat capacity ratio (-- Ce/Cv), dimensionless
Activity coefficient of component i in liquid solution
Activity coefficient of a solid solute ( component 1) in
the liquid solution defined by Eq. (6.161), dimension-
less
Activity coefficient of component i in liquid solution
at infinite dilution (x4 ~ 0), dimensionless
Chemical potential of component i defined in
Eq. (6.115)
Difference between heat capacity of liquid and solid
for pure component i (= cLi -- CSi), J/ mol. K
Heat of fusion (or latent heat of melting) for pure
component i at the freezing point and 1.013 bar,
J/tool
Heat of vaporization (or latent heat of melting) at
1.013 bar defined by Eq. (6.98), J/mol
Heat of mixing. J/mol
Property change for M due to mixing defined by
Eq. (6.84)
Ent ropy of fusion for pure component i at the freezing
point and 1.013 bar, J / mol -K
Entropy of vaporization at 1.013 bar defined by
Eq. (6.97), J/ mol
Boiling point elevation for solvent 2 (Eq. 6.214), K
Freezing point depression for solvent 2
(Eq. 6.213), K
Volume change due to mixing defined by Eq. (6.86)
Superscript
E Excess property defined for mixtures (with respect to
ideal solution)
HS Value of a property for hard sphere molecules
ig Value of a property for a component as ideal gas at
temperature T and P --~ 0
id Value of a property for an ideal solution
L Value of a property for liquid phase
R A residual property (with respect to ideal gas
property)
V Value of a property for vapor phase
vap Change in value of a property due to vaporization
S Value of a property for solid phase
sat Value of a property at saturation pressure
t Value of a property for the whole (total) system
[](0) A dimensionless t erm in a generalized correlation for
a property of simple fluids
[](1) A dimensionless term in a generalized correlation for
a property of acentric fluids
[](r) A dimensionless term in a generalized correlation for
a property of reference fluids
or, fl Value of a propert y for phase a or phase fl
oo Value of a propert y for i in the liquid solution at
infinite dilution as x~ --~ 0
o Value of a propert y at standard state, usually the
standard state is chosen at pure component at T
and P of the mi xt ure according to the
Lewis/ Randall rule
A Value of mol ar propert y of a component in the
mixture
Subscripts
PR
SRK
c Value of a propert y at the critical point
i A component in a mi xt ure
j A component in a mixture
i, [ Effect of binary interaction on a property
m Value of a propert y for a mi xt ure
mix Change in value of a propert y due to mixing at
constant T and P
Value of a propert y determined from PR EOS
Value of a propert y determined from SRK EOS
Acronyms
API-TDB American Petroleum I nstitute--Technical Data
Book
BI P Binary interaction par amet er
bbl Barrel, unit of volume of liquid as given in Section
1.7.11
CS Carnahan-St arl i ng EOS (see Eq. 5.93)
DI PPR Design I nstitute for Physical Property Data
GC Generalized correlation
GD Gi bbs-Duhem equation (see Eq. 6.81)
HS Hard sphere
HSP Hard sphere potential given by Eq. (5.13)
IAPWS I nternational Association for the Properties of
Water and St eam
LJ Lennard-J ones potential given by Eq. (5.1 I)
LJ EOS Lennard-J ones EOS given by Eq. (5.96)
LK EOS Lee-Kesler EOS given by Eq. (5.104)
LLE Liquid-liquid equilibria
NI ST National I nstitute of Standards and Technology
PVT Pressure-vol ume-t emperat ure
RHS Right-hand side of an equation
RK Redlich-Kwong EOS (see Eq. 5.38)
SRK Soave-Redl i ch-Kwong EOS given by Eq. (5.38)
and paramet ers in Table 5.1
SAFT Statistical associating fluid theory (see
Eq. 5.98)
SLE Solid-liquid equilibrium
SLVE Solid-liquid-vapor equilibrium
VLS Vapor-liquid-solid equilibrium
VS Vapor-solid equilibrium
%AAD Average absolute deviation percentage defined by
Eq. (2.135)
Eq. (2.134)
IN CHAPTER 5 THE PVT relations and theory of intermolecu-
lar forces were discussed. The PVT relations and equations of
states are the basis of propert y calculations as all physical and
t hermodynami c properties can be related to PVT properties.
I n this chapt er we review principles and theory of propert y
estimation met hods and basic t hermodynami c relations that
will be used to calculate physical and t hermodynami c prop-
erties.
The PVT relations and equations of state are perhaps the
most i mport ant t hermodynami c relations for pure fluids and
their mixtures. Once the PVT relation is known, various phys-
ical and t hermodynami c properties needed for design and op-
eration of units in the pet rol eum and related industries can be
calculated. Density can be directly calculated from knowledge
of mol ar volume or compressibility factor t hrough Eq. (5.15).
Various t hermodynami c properties such heat capacity, en-
thalpy, vapor pressure, phase behavior and vapor liquid equi-
librium (VLE), equilibrium ratios, intermolecular parame-
ters, and transport properties all can be calculated t hrough
accurate knowledge of PVT relation for the fluid. Some of
these relations are developed in this chapt er t hrough funda-
ment al t hermodynami c relations. Once a propert y is related
to PVT, using an appropri at e EOS, it can be estimated at any
t emperat ure and pressure for pure fluids and fluid mixtures.
Development of such i mport ant relations is discussed in this
chapter, while their use to estimate thermophysical properties
for pet rol eum mixtures are discussed in the next chapter.
6. 1 DEFI NI TI ONS AND FUNDAMENTAL
THERMODY NAMI C RELATI ONS
I n this section, t hermodynami c properties such as entropy,
Gibbs energy, heat capacity, residual properties, and fugacity
are defined. Thermodynami c relations that relate these prop-
erties to PVT relation of pure fluids are developed.
6.1.1 Thermodynami c Properties and
Fundamental Relations
Previously two t hermodynami c properties, namel y internal
energy (U) and enthalpy ( H) , were defined in Section 5.1.
The enthalpy is defined in t erms of U and PV (Eq. 5.5) as
(6.1) H = U + PV
Another t hermodynami c propert y that is used to formulate
the second law of t hermodynami cs is called entropy and it is
defined as
(6.2) dS = 8O~v
T
where S is the ent ropy and ~Q~v is the amount of heat trans-
ferred to the system at t emperat ure T t hrough a reversible
process. The symbol ~ is used for the differential heat Q to
indicate that heat is not a t hermodynami c propert y such as
H or S. The unit of ent ropy is energy per absolute degrees, e.g.
J/K, or on a mol ar basis it has the unit of J/ tool. K in the SI
unit system. The first law of t hermodynami cs is derived based
on the law of conservation of energy and for a closed system
( constant composi t i on and mass) is given as follows [1, 2]:
(6.3) dU = 8Q - PdV
6. THERM ODYNAM I C RELATI ONS FOR PROPERTY ESTI M ATI ONS 2 3 5
Combining Eqs. (6.2) and (6.3) gives the following relation:
(6.4) dU = TdS- PdV
This relation is one of the fundament al t hermodynami c rela-
tions. Differentiating Eq. (6.1) and combining with Eq. (6.4)
gives
(6.5) dH = TdS+ V dP
Two other t hermodynami c properties known as auxiliary
functions are Gibbs free energy (G) and Helmholtz free energy
(A) that are defined as
(6.6) G - H - TS
(6.7) A =- U - TS
G and A are mai nl y defined for convenience and formulation
of useful t hermodynami c properties and are not measurabl e
properties. Gibbs free energy also known as Gibbs energy is
particularly a useful property in phase equilibrium calcula-
tions. These two paramet ers bot h have units of energy simi-
lar to units of U, H, or PV. Differentiating Eqs. (6.6) and (6.7)
and combining with Eqs. (6.4) and (6.5) lead to the following
relations:
(6.8) dG = V dP - Sdr
(6.9) dA = - PdV - SdT
Equations (6.4), (6.5), (6.8), and (6.9) are the four fundamen-
tal t hermodynami c relations that will be used for propert y
calculations for a homogenous fluid of constant composition.
I n these relations either mol ar or total properties can be used.
Another set of equations can be obtained from mat hemat i -
cal relations. I f F = F (x, y) where x and y are two independent
variables, the total differential of F is defined as
( OF']dx + OF
(6.10) dF = \ ~xj y ( - ~y ) ?y
which may also be written as
(6.11) dE = M (x, y)dx + g( x, y) dy
where M(x, y ) = (OF/Ox)y and N(x, y ) = (OF/Oy)x. Consider-
ing the fact that 02F/OxOy = 02F/OyOx, the following relation
exists between M and N:
(6.12) (O-~y)x = (0~xN)y
Applying Eq. (6.12) to Eqs. (6.4), (6.5), (6.8), and (6.9) leads to
the following set of equations known as Maxwell's equations
[1, 2]:
(6.13) ( ~) S = -(~---~)V
(6.14) ( ~- ~) S= ( ~- ~) / ,
(6.15) ( ~- ~) p = - ( 0~- ) r
(6.16) (O0---~)v=(~-~)r
Maxwell's relations are the basis of propert y calculations by
relating a propert y to PVT relation. Before showing appli-
cation of these equations, several measurabl e properties are
defined.
6. 1 . 2 Meas urab l e Propert i es
I n this section some t hermodynami c properties that are di-
rectly measurabl e are defined and introduced. Heat capacity
at constant pressure (Cp) and heat capacity at constant vol-
ume (Cv) are defined as:
(6.17) Cp = -d-f p
~Q
(6.18) Cv= ( ~f ) v
Molar heat capacity is a t hermodynami c property that indi-
cates amount of heat needed for 1 mol of a fluid to increase its
t emperat ure by 1 degree and it has unit of J/ mol - K ( same as
J/ tool. ~ in the SI unit system. Since t emperat ure units of K
or ~ represent the t emperat ure difference they are bot h used
in the units of heat capacity. Similarly specific heat is defined
as heat required to increase t emperat ure of one unit mass of
fluid by 1 ~ and in the SI unit systems has the unit of kJ/kg 9 K
(or J/ g. ~ I n all t hermodynami c relations mol ar properties
are used and when necessary they are converted to specific
property using molecular weight and Eq. (5.3). Since heat is
a pat h function and not a t hermodynami c property, amount
of heat transferred to a system in a constant pressure process
differs from the amount of heat transferred to the same sys-
t em under constant volume process for the same amount of
t emperat ure increase. Combining Eq. (6.3) with (6.18) gives
(6.19) Cv = ~ v
similarly Cp can be defined in t erms of enthalpy t hrough Eqs.
(6.2), (6.5), and (6.17):
(6.20) Cv = ~-~ P
For ideal gases since U and H are functions of only t empera-
ture (Eqs. 5.16 and 5.17), from Eqs. (6.20) and (6.19) we have
(6.21) dH ig --- C~dT
ig
(6.22) dU ig = C v dT
where superscript ig indicates ideal gas properties. I n some
references ideal gas properties are specified by superscript ~
or * (i.e., C~ or C~ for ideal gas heat capacity). As will be seen
later, usually C~ is correlated to absolute t emperat ure T in
the form of polynomial of degrees 3 or 5 and the correlation
coefficients are given for each compound [1-5]. Combining
Eqs. (6.1), (5.14), (6.21), and (6.22) gives the following rela-
tion between Cie g and C~ t hrough universal gas constant R:
(6.23) C~ - C~ = R
For ideal gases C~ and C~ are bot h functions of only temper-
ature, while for a real gas Cp is a function of bot h T and P as
it is clear from Eqs. (6.20) and (6.28). The ratio of Cp/ Cv is
called heat capacity ratio and usually in t hermodynami c texts
is shown by y and it is greater t han unity. For monoat omi c
gases (i.e., helium, argon, etc.) it can be assumed that y = 5/3,
and for diatomic gases (nitrogen, oxygen, air, etc.) it is as-
sumed that y = 7/ 5 = 1.4.
2 3 6 CHARACTERI ZATI ON AND PROPERTI ES OF PETROLEUM FRACTI ONS
There are two other measurable properties: coefficient of
thermal expansion, r, and the bulk isothermal compressibility,
g. These are defined as
(6.24) f l = V ~ p
(6.25) x- - - - V V r
since OV/OP is negative, the minus sign in the definition of
x is used to make it a positive number. The units of fl and
x in SI system are K -1 and Pa -1, respectively. Values of fl
and r can be calculated from these equations with use of an
equation of state. For example, with use of Lee-Kesler EOS
(Eq. 5.104), the value of ~ is 0.84 x 10 -9 Pa -1 for liquid ben-
zene at temperature of 17~ and pressure of 6 bar, while the
actual measured value is 0.89 x 10 -9 Pa -1 [6]. Once fl and
are known for a fluid, the PVT relation can be established
for that fluid (see Problem 6.1). Through the above thermo-
dynamic relations and definitions one can show that
TV B 2
(6.26) Ce - Cv = - -
g
Applying Eqs. (6.24) and (6.25) for ideal gases (Eq. 5.14)
gives B ig = I / T and gig ___ 1/P. Substituting B ig and gig into
Eq. (6.26) gives Eq. (6.23). From Eq. (6.26) it is clear that
Cp > Cv; however, for liquids the difference between Cp and
Cv is quite small and most thermodynamic texts neglect this
difference and assume Cp ~- Cv. Most recently Garvin [6] has
reviewed values of constant volume specific heats for liquids
and concludes that in some cases Cp - Cv for liquids is signif-
icant and must not be neglected. For example, for saturated
liquid benzene when temperature varies from 300 to 450 K,
the calculated heat capacity ratio, Cp/Cv, varies from 1.58
to 1.41 [6]. Although these values are not yet confirmed as
they have been calculated from Lee-Kesler equation of state,
but one should be careful that assumption of Cp ~- Cv for liq-
uids in general may not be true in all cases. I n fact for ideal
incompressible liquids, B ~ 0 and x- ~ 0 and according to
Eq. (6.26), (Cp - Cv) --~ O, which leads to y = Cp/ Cv --~ 1.
There is an EOS with high accuracy for benzene [7]. I t gives
Cp/C~ for saturated liquids having a calculated heat capacity
ratio of 1.43-1.38 over a temperature range of 300-450 K.
Another useful property is Joule-Thomson coefficient that
is defined as
This property is useful in throttling processes where a fluid
passes through an expansion valve at which enthalpy is nearly
constant. Such devices are useful in reducing the fluid pres-
sure, such as gas flow in a pipeline, tj expresses the change of
temperature with pressure in a throttling process and can be
related to C~ and may be calculated from an equation of state
(see Problem 6.10).
6. 1. 3 Res i dual Propert i es and
Depart ure Func t i ons
Properties of ideal gases can be determined accurately
through kinetic theory. I n fact all properties of ideal gases are
known or they can be estimated through the ideal gas law.
Values of C~ are known for many compounds and they are
given in terms of temperature in various industrial handbooks
[5]. Once C~ is known, C~, U ig, H ig, and S ig can also be de-
termined from thermodynamic relations discussed above. To
calculate properties of a real gas an auxiliary function called
residual property is defined as the difference between prop-
erty of real gas and its ideal gas property (i.e., H - Hig). The
difference between property of a real fluid and ideal gas is
also called departure from ideal gas. All fundamental relations
also apply to residual properties. By applying basic thermo-
dynamic and mathematical relations, a residual property can
be calculated through a PVT relation of an equation of state.
I f only two properties such as H and G or H and S are known
in addition to values of V at a given T and P, all other prop-
erties can be easily determined from basic relations given in
this section. For example from H and G, entropy can be calcu-
lated from Eq. (6.6). Development of relations for calculation
of enthalpy departure is shown here. Other properties may be
calculated through a similar approach.
Assume that we are interested to relate residual enthalpy
(H - HiE) into PVT at a given T and P. For a homogenous
fluid of constant composition (or pure substance), H can be
considered as a function of T and P:
(6.28) H = H(T, P)
Applying Eq. (6.10) gives
OH
(6.29) dH=( - ~- ~) e dT+( ~p) r dP
Dividing both sides of Eq. (6.5) to OP at constant T gives
( )
OH = V + T
(6.30) ~ r r
Substituting for (OS/OP)T from Eq. (6.15) into Eq. (6.30)
and substitute resulting (OH/OP)T into Eq. (6.29) with use of
Eq. (6.20) for (OH/OT)p, Eq. (6.29) becomes
(6.31) dH=CpdT+[ V-T(OV~kOT/Pj]dP
where the right-hand side (RHS) of this equation involves
measurable quantities of Cp and PVT, which can be deter-
mined from an equation of state. Similarly it can be shown
that
(6.32)
dS=C~l - - ~ e
Equations (6.31) and (6.32) are the basis of calculation of en-
thalpy and entropy and all other thermodynamic properties
of a fluid from its PVT relation and knowledge of Cp. As an ex-
ample, integration of Eq. (6.31) from (T~, PL) to (T2, P2) gives
change of enthalpy (AH) for the process. The same equation
can be used to calculate departure functions or residual prop-
erties from PVT data or an equation of state at a given T and
P. For an ideal gas the second term in the RHS of Eq. (6.32)
is zero. Since any gas as P ---> 0 behaves like an ideal gas, at a
fixed temperature of T, integration of Eq. (6.31) from P --+ 0
to a desired pressure of P gives
P
0V
(6.33) ( H- H' g ) r =f [ V - T( ~- f ) p] dP (at constant T)
0
6. THERMODYNAMIC RELATIONS FOR PROPERTY ESTIMATIONS 2 3 7
For practical applications the above equation is converted
into dimensionless form in t erms of paramet ers Z defined by
Eq. (5.15). Differentiating Z with respect to T at constant P,
from Eq. (5.15) we get
(6.34) ~ e = ~ kS--T)e + ~ -
Dividing bot h sides of Eq. (6.33) by RT and combining with
Eq. (6.34) gives
P
(6.35) H- H ig f ( OZ) dP
R ~ - T 8-T e --P- (at constant T)
0
It can be easily seen that for an ideal gas where Z- - 1,
Eq. (6.35) gives the expected result of H - H i g = 0. Similarly
for any equation of state the residual enthalpy can be calcu-
lated. Using definitions of Tr and Pr by Eq. (5.100), the above
equation may be written as
H - H ig f ( OZ ) dPr
(6.36) R ~ -- Tr2 ~ P~ ~ (at constant T)
0
where the t erm in the left-hand side and all paramet ers in the
RHS of the above equation are in dimensionless forms. Once
the residual enthalpy is calculated, real gas enthalpy can be
determined as follows:
(6.37)
H= Hig_]_ RTc ( H- Hig~
\ RT~ ]
I n general, absolute values of enthalpy are of little interest
and normally the difference between enthalpies in two differ-
ent conditions is useful. Absolute enthalpy has meani ng only
with respect to a reference state when the value of enthalpy
is assigned as zero. For example, tabulated values of enthalpy
in st eam tables are with respect to the reference state of satu-
rated liquid wat er at 0~ [1]. As the choice of reference state
changes so do the values of absolute enthalpy; however, this
change in the reference state does not affect change in en-
thalpy of systems from one state to another.
A relation similar to Eq. (6.33) can be derived in t erms of
volume where the gas behavior becomes as an ideal gas as
V- ~ oo:
(6.38)
V
( H- Hig)T,V = f
V---~ oo
[ T( ~T) v- P] dV +PV - RT
Similar relation for the ent ropy departure is
V
(6.39) (S-~g)r,v : f [ ( ~T) v- R] dv
Once H is known, U can be calculated from Eq. (6.1). Sim-
ilarly all other t hermodynami c properties can be calculated
from basic relations and definitions.
Example 6. 1--Deri ve a relation for calculation of Cp from
PVT relation of a real fluid at T and P.
Solution--By substituting the Maxwelrs relation of Eq. (6.15)
into Eq. (6.30) we get
(6.40) ~ r \ST ]e
differentiating this equation with respect to T at constant P
gives
(6.41) = - T \ 8T 2 ,iv
From mat hemat i cal identity we have
0H 3H
(6.42) [ ~---~ (-~-ff )r ] l = [ 8 (-ff~ )p]r
Using definition of Cv t hrough Eq. (6.20) and combining the
above two equations we get
(6.43) \ OP Jr =- T \ ~T~ je
Upon integration from P = 0 to the desired pressure of P at
constant T we get
P
(6.44) f ]
\ 8T 2 ] e J r dP
P=O
Once C~ is known, Ce can be determined at T and P of in-
terest from an EOS, PVT data, or generalized corresponding
states correlations, t
6 . 1 . 4 F u g a c i t y a n d F u g a c i t y C o e f f i c i e n t
f o r P u r e C o mp o n e n t s
Another i mport ant auxiliary function that is defined for cal-
culation of t hermodynami c properties, especially Gibbs free
energy, is called fugacity and it is shown by f. This par amet er
is particularly useful in calculation of mixture properties and
formulation of phase equilibrium problems. Fugacity is a pa-
ramet er similar to pressure, which indicates deviation from
ideal gas behavior. It is defined to calculate properties of real
gases and it may be defined in the following form:
(6.45) P-~0 lim ( f ) = 1
With this definition fugacity of an ideal gas is the same as
its pressure. One mai n application of fugacity is to calculate
Gibbs free energy. Application of Eq. (6.8) at constant T to an
ideal gas gives
(6.46) dG ig -- RTd In P
For a real fluid a similar relation can be written in t erms of
fugacity
(6.47) dG -- RTdl n f
where for an ideal gas fig = p. Subtracting Eq. (6.46) from
(6.47), the residual Gibbs energy, G R, can be det ermi ned
t hrough fugacity:
G R
G
Gi g
- In f = In ~b
(6.48) RT -- g ~
/ -,
Equation of state
Definition of dimensionless
parameters
Definition of various
parameters
Residual heat capacity and
heat capacity ratio
H_ Hig
RT
TABLE 6.1--Calculation of thermodynamic properties from cubic equations of state [8].
RK EOS I SRK EOS ] PR EOS
Z 3 - Z 2 +( A- B - B 2 ) Z - AB = 0 [ Z 3 - (1 - B)Z 2 + ( A- 2 B - 3B2) Z- AB+B 2 +B 3 = 0
al ~T a2 d~- P1 ( ~ ) V P2 . . . . (~V)T P3 -- ~T y dv
oo
ig C_/L
Cp- Cf =TP3 - w- R Cv - C v =TP3 Y= Cv
(Z - 1) + b~( a - Tal)ln z+~ (Z - 1) + ~ ( a - Tal) x In z+B(l+4~)z+8(1-v' ~)
( L) z+B. - ~)
In z - 1 - ln(Z - B) + ~- In ~z Z - 1 - ln(Z - B) + 2-~--~2 n In z+B0+,/~)
P1
e2
-gr ~(2 v+b~
-F V 2(V +b)2 (V-Z~b) + V 2(2bV +b2)2
1)3 1 V
--/;a2 In
ctI/2 1 a _ a~g 1/2 f
- - ~ y rcTrl/2 I ~
3 a a ~ I
r--~ 2 r~r)/ 2 f o,( + f ~)
SRK, and PR EOS are given in Table 5.1.
d2
a = aca, where ac, a, f~, and b for RK
1 - In v+b--,/~
o~ G( t + f~)
2r2r) n
The ratio of f / P is a di mensi onl ess p ar amet er called fugacity
coefficient and it is shown by r
f
(6.49) ~b = --
P
where for an ideal gas, r = 1. Once q~ is known, G can be
cal cul at ed t hr ough Eq. (6.48) and f r om G and H, S may be
calculated from Eq. (6.6). Vice versa when H and S are known,
G and eventually f can be det ermi ned.
6 . 1 . 5 Ge n e r a l A p p r o a c h f o r P r o p e r t y Es t i ma t i o n
Similar to the met hod used in Exampl e 6.1, every t hermody-
nami c propert y can be related to PVT rel at i on either at a given
T and P or at a given T and V. These relations for ( H -/ _/ ig)
are given by Eqs. (6.33) and (6.39). For t he residual ent ropy an
equivalent rel at i on in t erms of pressure is (see Probl em 6.3)
P
(6.50) ( S- Si g) r, , = f [ R - (~----f ) e] dP
0
This equat i on can be wri t t en in t erms of Z as
P P
( S- ~g ) r , P: - Rf ~- - - ~dP- RT f d_P_Pp
0 0
(6.51) (at const ant T)
Once residual ent hal py and ent ropy are calculated, residual
Gi bbs energy is cal cul at ed f r om t he following relation based
on Eq. (6.6):
(6.52) (G - - Gi g ) T, p = ( H - Hi g ) T, p - - T( S- ~g ) T, P
Subst i t ut i ng Eqs. (6.33) and (6.50) into Eq. (6.52) and com-
bi ni ng wi t h Eq. (6.48) give s the following equat i on whi ch can
be used to calculate fugaci t y coefficient for a pure component :
P
(6.53) ln =f (z-1)
0
For t he residual heat capaci t y (Ce - C~), the relation at a
fixed T and P is given by Eq. (6.44). I n general when fugac-
ity coefficient is cal cul at ed t hr ough Eq. (6.53), residual Gibbs
energy can be cal cul at ed f r om Eq. (6.48). Properties of ideal
gases can be cal cul at ed accurat el y as will be di scussed later
in this chapter. Once H and G are known, S can be calcu-
lated f r om Eq. (6.6). Therefore, either H and S or H and r
are needed to calculate vari ous properties. I n this chapter,
met hods of cal cul at i on of H, Ce, and r are presented.
When residual propert i es are related t o PVT, any equat i on
of state may be used to calculate propert i es of real fluids and
depart ure functions. Calculation of ( H - Hig), (Cp - C~), and
I n r f r om RK, SRK, and PR equat i ons of state are given in Ta-
ble 6.1. RK and SRK give similar results while t he onl y differ-
ence is in p ar amet er a, as given in Table 5. i. EOS paramet ers
needed for use in Table 6.1 are given in Table 5.1. Relations
present ed in Table 6.1 are applicable to bot h vapor and liq-
ui d phases whenever the EOS can be applied. However, when
t hey are used for t he liquid phase, values of Z and V must
be obt ai ned for the same phase as discussed in Chapt er 5.
I t shoul d be not ed t hat relations given in Table 6.1 for var-
ious propert i es are based on assumi ng t hat p ar amet er b in
t he correspondi ng EOS is i ndependent of t emperat ure as for
RK, SRK, and PR equations. However, when p ar amet er b is
consi dered t emperat ure-dependent , t hen its derivative with
respect t o t emper at ur e is not zero and derived relations for
residual propert i es are significantly mor e compl i cat ed t han
t hose given in Table 6. I. As will be di scussed in t he next sec-
tion, cubi c equat i ons do not provi de accurat e values for en-
t hal py and heat capaci t y of fluids unless their const ant s are
adjusted for such calculations.
6 . 2 GEN ER A L I Z ED C OR R EL A T I ON S
FOR CAL CUL AT I ON OF
T H ER M OD Y N A M I C P R OP ER T I ES
I t is generally believed t hat cubi c equat i ons of state are not
suitable for cal cul at i on of heat capaci t y and ent hal py and
6. THERM ODYNAM I C RELATI ONS FOR PROPERTY ESTI M ATI ONS 239
in some cases give negative heat capacities. Cubic equations
are widely used for calculation of molar volume (or density)
and fugacity coefficients, Usually BWR or its various modi-
fied versions are used to calculate enthalpy and heat capacity.
The Lee-Kesler (LK) modification of BWR EOS is given by
Eq. (5.109). Upon use of this PVT relation, residual proper-
ties can be calculated. For example, by substituting Z from
Eq. (5,109) into Eqs. (6.53) and (6.36) the relations for the
fugacity coefficient and enthalpy departure are obtained and
are given by the following equations [9]:
(6.54)
o
In = Z- l - l n( Z) +~r r +~ +~5 +E
H - H ig ( bz + 2b3/Tr + 3b4/T f
R~ - T~ \ Z - I - T~Vr
c2 - 3c3/T 2 d2 )
(6.55)
2TrV 2 5rrVr5 + 3E
where parameter E in these equations is given by:
c4 [ ( ~/ ) ( _ ~r2 ) 1 E- 2Tr3~, r f +l +~ exp Y
The coefficients in the above equations for the simple fluid
and reference fluid of n-octane are given in Table 5.8. Sim-
ilar equations for estimation of (Cv -Cipg), (Cv ig - Cv), and
(S - S ~g) are given by Lee and Kesler [9]. To make use of these
equations for calculation of properties of all fluids a similar
approach as used to calculate Z through Eq. (5.108) is rec-
ommended. For practical calculations Eq. (6.55) and other
equations for fugacity and heat capacity can be converted
into the following corresponding states correlations:
H -/-/1 -H - Hig] (1)
(6.56) L RTr J =
(6.58) [ln ( ~- ) ] = [ln ( ~) ] ( ~ + co [ln ( ~- ) ] (1)
where for convenience Eq. (6.58) may also be written as [1]
(6.59) ~b = (~b (~ (~bO)) ~
Simple fluid terms such as [ ( H- Hig)/RTc] (~ can be esti-
mated from Eq. (6.55) using coefficients given in Table 5.8
for simple fluid. A graphical presentation of [(H - I-Pg)/RTc] (~
and [(H - Hig)/RTc] O) is demonstrated in Fig. 6.1 [2].
The correction term [(H - Hig)/RTc] (1) is calculated from
(6.60) L~. ] = ~r ] / L RTc J [H ~ T cH I A I
where [(H - Hig)IRTc] (r) should be calculated from Eq. (6.55)
using coefficients in Table 5.8 for the reference fluid (n-
octane). O~ris the acentric factor of reference fluid in which
for n-C8 the value of 0.3978 was originally used. A simi-
lar approach can be used to calculate other thermodynamic
4
j
3
=:
2
0
O. Ol o. 1 1 l O
(a)
8 ........ ~ ~ : : : , ~ ~ ii ........................
7 . . . . . . . . . . . . . .......
I
~ 5
1 - . . 0 _ 6: : . . : :: : : : : : : . . . . . : . . . . . . . . . . i 5 i
_l
0.01 0.1 1 10
R e d u c e d Pressure, Pr
(b)
FI G . 6. 1 - - T he Lee- Kes l er cor r el at i on f or ( a) [ ( H - H i g ) /
RTc] (~ and ( b) [ ( H - - Hi g ) / RT c] (1) in t er ms of Tr and P r ,
properties. While this method is useful for computer calcula-
tions, it is of little use for practical and quick hand calcula-
tions. For this reason tabulated values similar to Z (~ and Z O)
are needed. Values of residual enthalpy, heat capacity, and
fugacity in dimensionless forms for both [](0) and [](a) terms
are given by Lee and Kesler [9] and have been included in
the API-TDB [5] and other references [1, 2, 10]. These values
f o r enthalpy, heat capacity, and fugacity coefficient are given
in Tables 6.2-6.7. In use of values for enthalpy departure it
should be noted that for simplicity all values in Tables 6.2
and 6.3 have been multiplied by the negative sign and this is
indicated in the tires of these tables. In Tables 6.4 and 6.5,
for heat capacity departure there are certain regions of max-
i mum uncertainty that have been specified by the API-TDB
[5]. In Table 6.4, when Pr > 0.9 and values of [(Cv - C~)/R] (~
are greater than 1.6 there is uncertainty as recommended
by the API-TDB. In Table 6.5, when Pr > 0.72 and values of
[(Cp - C~) / R] O) are greater than 2.1 the uncertainty exists
as recommended by the API-TDB. In these regions values of
heat capacity departure are less accurate. Tables 6.6 and 6.7
give values of r and r that are calculated from (lnr (~ as
given by Smith et al. [1].
(o)
TABLE 6. 2--V alues of --[ B~--T~ ] for use in Eq. (6.56).
L ~ J
0.01 0.05 0.1 0.2 0,4 0.6 0.8 1 1.2 1.5 2 3 5 7 I 0
0.30 6. 045 6. 043 6. 040 6. 034 6. 022 6.011 5, 999 5.987 5.975 5.957 5.927 5.868 5,748 5.628 5.446
0.35 5. 906 5, 904 5,901 5. 895 5. 882 5, 870 5, 858 5.845 5.833 5.814 5,783 5,721 5.595 5.469 5,278
0.40 5 . 7 63 5 . 7 61 5 . 7 5 7 5 . 7 5 1 5 . 7 3 8 5 . 7 2 6 5 . 7 1 3 5.700 5.687 5.668 5.636 5.572 5,442 5.311 5.113
0.45 5 . 61 5 5 . 61 2 5 . 6 0 9 5 . 60 3 5 . 5 9 0 5 . 5 7 7 5 . 5 6 4 5,551 5.538 5.519 5.486 5.421 5.288 5.154 5.950
0,50 5. 465 5. 469 5. 459 5. 453 5. 440 5. 427 5. 414 5.401 5.388 5.369 5.336 5.270 5.135 4.999 4.791
0.55 0.032 5 . 3 1 2 5 . 3 0 9 5 . 3 0 3 5 . 2 9 0 5 . 2 7 8 5 . 2 65 5.252 5.239 5.220 5.187 5.121 4.986 4.849 4.638
0.60 0.027 5. 162 5. 159 5. 153 5.141 5. 129 5. 116 5.104 5.091 5.073 5.041 4.976 4.842 4.704 4.492
0.65 0.023 0.118 5, 008 5. 002 4. 991 4. 980 4. 968 4.956 4.945 4.927 4.896 4.833 4.702 4.565 4.353
0.70 0.020 0.101 0,213 4. 848 4. 838 4. 828 4. 818 4.808 4.797 4,781 4,752 4.693 4.566 4.432 4.221
0.75 0.017 0,088 0.183 4. 687 4. 679 4. 672 4. 664 4.655 4.646 4,632 4.607 4.554 4,434 4.303 4.095
0.80 0,015 0,078 0.160 0.345 4, 507 4. 504 4. 499 4,494 4.488 4.478 4.459 4.413 4.303 4.178 3.974
0.85 0.014 0.069 0,141 0.300 4. 309 4. 313 4. 316 4.316 4.316 4,312 4.302 4.269 4.173 4.056 3.857
0.90 0.012 0.062 0,126 0.264 0.596 4. 074 4. 094 4.108 4.118 4.127 4.132 4.119 4.043 3.935 3.744
0.93 0.011 0.058 0.118 0.246 0.545 0.960 3. 920 3.953 3.976 4.000 4.020 4.024 3,963 3.863 3.678
0.95 0.011 0.056 0.113 0.235 0.516 0.885 3. 763 3.825 3.865 3.904 3.940 3.958 3.910 3.815 3.634
0,97 0.011 0.054 0.109 0.225 0.490 0,824 1.356 3.658 3.732 3.796 3.853 3.890 3,856 3.767 3.591
0.98 0.010 0.053 0,107 0.221 0.478 0,797 1.273 3.544 3.652 3.736 3.806 3.854 3.829 3.743 3.569
0.99 0.010 0,052 0.105 0.216 0.466 0.773 1.206 3.376 3.558 3,670 3.758 3.818 3.801 3.719 3,548
1.00 0,010 0,052 0,105 0.216 0,466 0,773 1,206 2.593 3.558 3.670 3.758 3.818 3,801 3,719 3.548
1.01 0,010 0,051 0,103 0.212 0.455 0.750 1,151 1.796 3.441 3,598 3.706 3.782 3,774 3.695 3.526
1,02 0.010 0,049 0,099 0.203 0.434 0.708 1.060 1.627 3.039 3.422 3.595 3.705 3.718 3.647 3.484
1.05 0.009 0.046 0.094 0.192 0.407 0.654 0.955 1,359 2.034 3.030 3.398 3.583 3.632 3.575 3.420
1,10 0.008 0.042 0.086 0.175 0.367 0.581 0.827 1.120 1.487 2.203 2.965 3.353 3,484 3.453 3.315
1.15 0.008 0.039 0.079 0.160 0.334 0.523 0.732 0.968 1.239 1.719 2.479 3.091 3.329 3.329 3.211
1.20 0.007 0.036 0.073 0.148 0.305 0.474 0.657 0.857 1.076 1.443 2.079 2.807 3,166 3.202 3.107
1.30 0.006 0.031 0.063 0.127 0.259 0.399 0.545 0.698 0.860 1.116 1.560 2.274 2,825 2.942 2.899
1.40 0.005 0.027 0.055 0.110 0.224 0.341 0.463 0,588 0.716 0.915 1.253 1.857 2.486 2.679 2.692
1.50 0.005 0.024 0.048 0.097 0.196 0,297 0.400 0.505 0.611 0.774 1.046 1.549 2.175 2.421 2.486
1.60 0.004 0,021 0.043 0.086 0.173 0.261 0.350 0.440 0.531 0.667 0,894 1,318 1.904 2.177 2.285
1.70 0.004 0.019 0.038 0.076 0,153 0.231 0.309 0.387 0.446 0,583 0.777 1.139 1.672 1.953 2,091
1.80 0.003 0.017 0.034 0.068 0.137 0.206 0.275 0.344 0.413 0.515 0.683 0.996 1.476 1.751 1.908
1.90 0.003 0.015 0.031 0.062 0.123 0.185 0.246 0.307 0.368 0.458 0.606 0.880 1.309 1.571 1.736
2.00 0.003 0.014 0.028 0.056 0.111 0.167 0.222 0.276 0.330 0.411 0.541 0.782 1.167 1.411 1.577
2.20 0.002 0.012 0.023 0.046 0.092 0.137 0.182 0.226 0.269 0.334 0.437 0.629 0,937 1.143 1.295
2.40 0.002 0.010 0,019 0.038 0,076 0.114 0.150 0,187 0,222 0.275 0.359 0.513 0.761 0.929 1.058
2.60 0.002 0.008 0.016 0.032 0,064 0.095 0.125 0.155 0,185 0.228 0,297 0.422 0.621 0.756 0,858
2.80 0.001 0.007 0.014 0.027 0.054 0.080 0.105 0.130 0,154 0.190 0.246 0,348 0.508 0.614 0.689
3.00 0,001 0,006 0.011 0.023 0.045 0,067 0.088 0.109 0.129 0.159 0.205 0,288 0,415 0.495 0.545
3.50 0.001 0.004 0,007 0.015 0.029 0.043 0.056 0,069 0,081 0,099 0.127 0.174 0.239 0.270 0,264
4.00 0.000 0.002 0.005 0.009 0.017 0.026 0.033 0,041 0.048 0.058 0.072 0,095 0,116 0.110 0.061
Taken with permission from Ref. [9]. The value at the critical point (Tr = Pr = 1) is taken from the API-TDB [5]. Bold numbers indicate liquid region.
F or l ow a nd mode r a t e p r e s s ur e s whe r e t r unc a t e d vi r i al
e qua t i on i n t he f or m of Eq. ( 5. 113) i s val i d, t he r e l a t i on f or
f ugaci t y coef f i ci ent c a n be der i ved f r om Eq. ( 6. 53) as
B P
( 6. 61) l n( ~) -
RT
Thi s r e l a t i on ma y al so be wr i t t e n as
[Pr(SPc]l
( 6. 62) ~ -~ exp Z \ R - ~} J
whe r e ( BPc/ RTc) c a n he c a l c ul a t e d f r om Eq. ( 5. 71) or ( 5. 72) .
Si mi l ar l y e nt ha l p y de p a r t ur e ba s e d on t he t r unc a t e d vi r i al
e qua t i on i s gi ven as [ 1 ]
( 6. 63) H - H ig _ [B( 0 ) dB (~ (
dB (1)
RTo Pr Tr-d -~ + o~\B (1) -
whe r e B (~ a nd B (1) ar e gi ven by Eq. ( 5. 72) wi t h dB(~ =
0. 675/ Tr 26 a nd dB0) / dTr = 0. 722/ Tr s2. Obvi ous l y B (~ a nd B (1)
ma y be us e d f r om Eq. ( 5. 71) , but c or r e s p ondi ng der i vat i ves
mus t be us ed, The a bove e qua t i on ma y be a p p l i e d at t he s a me
r e gi on t ha t Eq. ( 5. 75) or ( 5. 114) we r e ap p l i cabl e, t ha t is, V~ >
2. 0 or T~ > 0. 686 + 0, 439Pr [ 2] .
F or r eal gas es t ha t f ol l ow t r unc a t e d vi r i al e qua t i on wi t h
t hr e e t e r ms ( coef f i ci ent s D a nd hi ghe r a s s ume d zer o i n
Eq. 5. 76) , t he r e l a t i ons f or Cp a nd Cv ar e gi vens as
E: ]
Cv - C~ _ T " ( B - TB" ) z - C + TC' - TzC" / 2
R V 2
( 6. 64)
Cv - C~ [ 2TB' + T2B" TC' + T2 C' / 2 ]
( 6. 65)
- - W - = [ v v-~ " j
whe r e B' a nd C' ar e t he f i r s t - or der der i vat i ves of B a nd C wi t h
r e s p e c t t o t e mp e r a t ur e , whi l e B" a nd C" ar e t he s e c ond- or de r
der i vat i ves of B a nd C wi t h r e s p e c t t o t e mp e r a t ur e .
6. THERM ODYNAM I C RELATI ONS FOR PROPERTY ESTI M ATI ONS
of - [ ~] (1) for use in Eq. (6.56). TABLE 6. 3--V alues
Pr
241
0.01 0.05 0.1 0.2 0.4 0.6 0.8 1 1.2 1.5 2 3 5 7 10
0.30 11. 098 11. 096 11. 095 11.091 11. 083 11. 076 11. 069 11.062 11.055 11.044 11.027 10.992 10,935 10.872 10.781
0.35 10. 656 10.655 10. 654 10. 653 10. 650 10. 646 10. 643 10.640 10.637 10.632 10.624 10.609 t0.581 10.554 10.529
0.40 10.121 10.121 10.121 10.121 10.121 10.121 10.121 10.121 10.121 10.121 10.122 10.123 10.128 10.135 10.150
0,45 9 . 5 1 5 9 . 5 1 5 9. 516 9. 516 9 . 5 1 9 9 . 5 2 1 9 . 5 2 3 9,525 9.527 9.531 9.537 9.549 9.576 9.611 9.663
0.50 8 . 8 68 8 . 8 69 8 . 8 7 0 8 . 8 7 0 8 . 8 7 6 8 . 8 8 0 8 . 8 8 4 8.888 8.892 8.899 8.909 8.932 8,978 9.030 9.111
0.55 0.080 8.211 8 . 2 1 2 8 . 2 1 5 8 . 2 2 1 8 . 2 2 6 8 . 2 3 2 8.238 8.243 8.252 8.267 8.298 8.360 8,425 8.531
0.60 0,059 7 . 5 68 7 . 5 7 0 7 . 5 7 3 7 . 5 7 9 7 . 5 8 5 7 . 5 9 1 7.596 7.603 7.614 7.632 7.669 7.745 7.824 7.950
0.65 0.045 0.247 6. 949 6. 9 5 2 6. 9 5 9 6, 966 6. 973 6.980 6.987 6.997 7.017 7.059 7.147 7.239 7.381
0.70 0.034 0.185 0.415 6. 360 6. 367 6. 373 6. 381 6.388 6.395 6.407 6.429 6.475 6.574 6.677 6.837
0.75 0.027 0.142 0.306 5 . 7 9 6 5 . 8 0 2 5 . 8 0 9 5, 816 5.824 5.832 5.845 5.868 5.918 6.027 6.142 6. 3t 8
0.80 0. 02t 0,110 0.234 0.542 5, 266 5 . 2 7 1 5 . 2 7 8 5.285 5.293 5.306 5.330 5.385 5.506 5.632 5.824
0.85 0.017 0.087 0.182 0.401 4. 753 4 . 7 5 4 4 . 7 5 8 4.763 4.771 4.784 4.810 4.872 5.008 5.149 5.358
0.90 0.014 0.070 0.144 0.308 0.751 4. 254 4. 248 4.249 4.255 4.268 4.298 4.371 4.530 4.688 4.916
0.93 0.012 0.061 0.126 0.265 0.612 1.236 3. 942 3.934 3.937 3.951 3.987 4.073 4.251 4.422 4.662
0.95 0.011 0.056 0.115 0.241 0.542 0.994 3. 737 3.712 3.713 3.730 3.773 3.873 4.068 4.248 4.497
0.97 0.010 0.052 0.105 0.219 0.483 0.837 1.616 3.470 3.467 3.492 3.551 3.670 3.885 4,077 4.336
0.98 0.010 0.050 0.101 0.209 0.457 0.776 1.324 3.332 3.327 3.363 3.434 3.568 3.795 3.992 4.257
0.99 0.009 0.048 0.097 0.200 0.433 0.722 1.154 3.164 3.164 3.223 3.313 3.464 3.705 3.909 4.178
1.00 0.009 0.046 0.093 0.191 0.410 0.675 1.034 2.348 2.952 3.065 3.186 3.358 3.615 3.825 4.100
1.01 0.009 0.044 0.089 0.183 0.389 0.632 0.940 1.375 2.595 2.880 3.051 3.251 3.525 3.742 4.023
1.02 0.008 0.042 0.085 0.175 0.370 0.594 0.863 1.180 1.723 2.650 2.906 3.142 3,435 3,661 3.947
1.05 0.007 0.037 0.075 0.153 0.318 0.498 0.691 0.877 0.878 1.496 2.381 2.800 3.167 3.418 3.722
1.10 0.006 0.030 0.061 0.123 0.251 0.381 0.507 0.617 0.673 0.617 1.261 2.167 2.720 3.023 3.362
1.15 0.005 0.025 0.050 0.099 0.199 0.296 0.385 0.459 0.503 0.487 0.604 1.497 2.275 2.641 3.019
1.20 0.004 00.020 0.040 0.080 0.158 0.232 0.297 0.349 0,381 0.381 0.361 0.934 1.840 2,273 2.692
1.30 0.003 0.013 0.026 0.052 0,100 0.142 0.177 0.203 0.218 0.218 0.178 0.300 1.066 1.592 2.086
1.40 0.002 0.008 0.016 0.032 0,060 0.083 0.100 0.111 0.115 0.108 0.070 0.044 0.504 1.012 1.547
1.50 0.001 0.005 0.009 0.018 0,032 0.042 0.048 0.049 0.046 0.032 - 0. 008 - 0. 078 0.142 0.556 1.080
1.60 0.000 0.002 0.004 0.007 0.012 0.013 0.011 0.005 - 0. 004 - 0. 023 - 0. 065 - 0. 151 - 0. 082 0.217 0.689
1.70 0.000 0.000 0.000 0.000 - 0. 003 - 0. 009 - 0. 017 - 0. 027 - 0. 040 - 0. 063 - 0. 109 - 0. 202 - 0. 223 - 0. 028 0.369
1.80 - 0. 000 - 0. 001 - 0. 003 - 0. 006 - 0. 015 - 0. 025 - 0. 037 -0. 051 - 0. 067 - 0. 094 - 0. 143 - 0. 241 - 0. 317 - 0, 203 0.112
1.90 -0. 001 - 0. 003 - 0. 005 - 0. 011 - 0, 023 - 0. 037 - 0. 053 - 0. 070 - 0. 088 - 0. 117 - 0. 169 - 0. 271 -0. 381 - 0. 330 - 0. 092
2.00 - 0. 001 - 0. 003 - 0. 007 - 0. 015 - 0, 030 - 0. 047 - 0. 065 - 0. 085 - 0. 105 - 0. 136 - 0. 190 - 0. 295 - 0. 428 - 0. 424 - 0. 255
2.20 - 0. 001 - 0. 005 - 0. 010 - 0. 020 - 0, 040 - 0. 062 - 0. 083 - 0. 106 - 0. 128 - 0. 163 - 0. 221 - 0. 331 - 0. 493 - 0. 551 - 0. 489
2.40 -0. 001 - 0. 006 - 0. 012 - 0. 023 - 0. 047 - 0. 071 - 0. 095 - 0. 120 - 0. 144 -0. 181 - 0. 242 - 0. 356 - 0. 535 -0. 631 - 0. 645
2.60 -0. 001 - 0. 006 - 0. 013 - 0. 026 - 0. 052 - 0. 078 - 0. 104 - 0. 130 - 0. 156 - 0. 194 - 0. 257 - 0. 376 - 0. 567 - 0. 687 - 0. 754
2.80 - 0. 001 - 0. 007 - 0. 014 - 0. 028 - 0, 055 - 0. 082 - 0. 110 - 0. 137 - 0. 164 - 0. 204 - 0. 269 - 0. 391 - 0. 591 - 0. 729 - 0. 836
3.00 -0. 001 - 0. 007 - 0. 014 - 0. 029 - 0. 058 - 0. 086 - 0. 114 - 0. 142 - 0. 170 - 0. 211 - 0. 278 - 0. 403 -0. 611 - 0. 763 - 0. 899
3.50 - 0. 002 - 0. 008 - 0. 016 -0. 031 - 0. 062 - 0. 092 - 0. 122 - 0. 152 - 0. 181 - 0. 224 - 0. 294 - 0. 425 - 0. 650 - 0. 827 - 1. 015
4.00 - 0. 002 - 0. 008 - 0. 016 - 0. 032 - 0. 064 - 0. 096 - 0. 127 - 0. 158 - 0. 188 - 0. 233 - 0. 306 - 0. 442 - 0. 680 - 0. 874 - 1. 097
6. 3 PR OPER T I ES OF I DEAL GA S ES
Ca l c ul a t i on of t he r modyna mi c p r op e r t i e s t hr ough t he me t h-
ods out l i ne d a bove r e qui r e s p r op e r t i e s of i deal gases. Ba s e d
on def i ni t i on of i deal gases, U ig, H ig, a nd C~ ar e f unc t i ons
of onl y t e mp e r a t ur e . Ki net i c t he or y s hows t ha t t he mo-
l ar t r a ns l a t i ona l e ne r gy of a monoa t omi c i deal gas i s 3 RT,
whe r e R is t he uni ve r s a l gas c ons t a nt a nd T is t he abs o-
l ut e t e mp e r a t ur e [ 10]. Si nce f or i deal gas es t he i nt e r na l en-
er gy i s i nde p e nde nt of p r e s s ur e t hus U i g = 2 3- I RT. Thi s l eads
9 . . "g
l g 5 l g 5 l g 3 Cp 5
t o H = ~R T, Cp = ~R , C v =~R, andF=7 ~- - - - - ~=1 . 6 6 7 .
C
I deal gas he a t c a p a c i t y of monoa t omi c gas es s uc h as ar-
gon, he l i um, et c. ar e c ons t a nt wi t h r es p ect t o t e mp e r a t ur e
[ 10] . Si mi l a r l y f or di a t omi c gas es s uc h as N2, 02, air, et c. ,
7 whi c h l eads t o C~ -- 7 i g = 5 a nd 7/ 5 H ig = ~R, ~R, C V 2R, y -- =
1.4. I n f act va r i a t i on of he a t c a p a c i t i e s of i deal di a t omi c
gas es wi t h t e mp e r a t ur e i s ver y mode r a t e . F or mul t i a t omi c
mol e c ul e s s uc h as hydr oc a r bons , i deal gas p r op e r t i e s do
c ha nge wi t h t e mp e r a t ur e ap p r eci abl y. As t he numbe r of
a t oms i n a mol e c ul e i ncr eas es , de p e nde nc y of i deal gas p r op -
er t i es t o t e mp e r a t ur e al so i ncr eas es . Da t a on p r op e r t i e s of
i deal gas es f or a l ar ge numbe r of hydr oc a r bons ha ve be e n
r e p or t e d by t he API - TDB [ 5] . Thes e da t a f or i deal gas he a t ca-
p aci t y ha ve be e n c or r e l a t e d t o t e mp e r a t ur e i n t he f ol l owi ng
form [5]:
( 6. 66) C~Pg = A -b B T -k CT 2 + DT 3 q- ET 4
R
whe r e R is t he gas c ons t a nt ( Se c t i on 1. 7. 24) , C~ i s t he mol a r
he a t c a p a c i t y i n t he s a me uni t as R, a nd T i s t he a bs ol ut e
t e mp e r a t ur e i n kel vi n. Val ues of t he c ons t a nt s f or a numbe r
of nonhydr oc a r bon gas es as wel l as s ome s el ect ed hydr oc a r -
bons ar e gi ven i n Tabl e 6.8. The t e mp e r a t ur e r a nge at whi c h
t he s e c ons t a nt s c a n be us e d is al so gi ven f or e a c h c omp ound
i n Tabl e 6.8. F or a c omp ound wi t h known c he mi c a l s t r uc-
t ur e, i deal gas he a t cap aci t y is us ual l y p r e di c t e d f r om gr oup
T. ~B] . E 6.4---ValbleS of r r r i g [ ~P---~-R ~ ~j(~ for use in Eq. (6.57).
Pr
0.01 0.05 0.1 0.2 0.4 0.6 0.8 1 1.2 1.5 2 3 5 7 10
0.30 2 . 8 0 5 2 . 8 0 7 2 . 8 0 9 2 . 8 1 4 2 . 8 3 0 2 . 8 4 2 2 . 8 5 4 2,866 2.878 2.896 2,927 2,989 3.122 3.257 3.466
0.35 2 . 8 0 8 2 . 8 1 0 2 . 8 1 2 2 . 8 1 5 2 . 8 2 3 2 . 8 3 5 2.844 2.853 2,861 2.875 2,897 2.944 3.042 3.145 3,313
0.40 2 . 9 2 5 2 . 9 2 6 2 . 9 2 8 2 . 9 3 3 2 . 9 3 5 2,940 2,945 2.951 2,956 2.965 2.979 3,014 3,085 3,164 3.293
0.45 2.989 2 . 9 9 0 2 . 9 9 0 2 . 9 9 1 2 . 9 9 3 2 . 9 9 5 2.997 2.999 3,002 3.006 3.014 3,032 3,079 3.135 3.232
0.50 3 . 0 0 6 3 . 0 0 5 3 . 0 0 4 3 . 0 0 3 3 . 0 0 1 3 . 0 0 0 2.998 2,997 2,996 2.995 2.995 2,999 3.019 3.054 3.122
0.55 0.118 3 . 0 0 2 3 . 0 0 0 2 . 9 9 7 2 . 9 9 0 2 . 9 8 4 2 . 9 7 8 2,973 2,968 2.961 2.951 2,938 2,934 2.947 2.988
0.60 0,089 3 . 0 0 9 3 . 0 0 6 2 . 9 9 9 2 . 9 8 6 2,974 2.963 2,952 2.942 2.927 2.907 2.874 2,840 2.831 2.847
0.65 0.069 0.387 3.047 3.036 3.014 2.993 2,973 2,955 2,938 2.914 2.878 2.822 2,753 2.720 2.709
0.70 0,054 0.298 0.687 3.138 3 . 0 9 9 3 . 0 65 3.033 3.003 2.975 2.937 2.881 2.792 2,681 2.621 2.582
0.75 0.044 0.236 0.526 3.351 3.284 3.225 3.171 3.122 3.076 3.015 2.928 2.795 2.629 2.537 2.469
0.80 0.036 0.191 0.415 1.032 3.647 3.537 3.440 3.354 3.277 3.176 3.038 2.838 2.601 2.473 2.373
0.85 0.030 0.157 0.336 0.794 4.404 4.158 3.957 3.790 3.647 3.470 3.240 2.931 2.599 2.427 2.292
0.90 0,025 0.131 0.277 0.633 1.858 5.679 5.095 4.677 4.359 4.000 3.585 3.096 2.626 2.399 2.227
0.93 0,023 0.118 0.249 0.560 1.538 4.208 6.720 5.766 5.149 4.533 3.902 3.236 2.657 2.392 2.195
0.95 0.021 0.111 0.232 0.518 1.375 3.341 9.316 7.127 6.010 5.050 4.180 3.351 2.684 2.391 2.175
0.97 0,020 0.104 0.217 0.480 1.240 2.778 9.585 10.011 7.451 5.785 4.531 3.486 2.716 2.393 2.159
0,98 0.019 0.101 0.210 0.463 1.181 2.563 7.350 13,270 8.611 6.279 4.743 3,560 2.733 2.395 2.151
0.99 0.019 0,098 0.204 0.447 1.126 2.378 6,038 21,948 10.362 6.897 4.983 3.641 2.752 2,398 2.144
1.00 0.018 0.095 0.197 0.431 1.076 2,218 5,156 oo 13,182 7.686 5,255 3.729 2.773 2.401 2.138
1.01 0,018 0,092 0.191 0.417 1.029 2.076 4.516 22,295 18,967 8.708 5,569 3.82t 2,794 2.405 2.131
1.02 0.017 0.089 0.185 0.403 0.986 1.951 4.025 13.183 31.353 10.062 5.923 3.920 2.816 2.408 2.125
1.05 0.016 0 . 0 82 0.169 0.365 0.872 1.648 3.047 6.458 20.234 16.457 7.296 4.259 2.891 2.425 2.110
1.10 0.014 0.071 0.147 0.313 0.724 1.297 2.168 3.649 6.510 13.256 9.787 4.927 3.033 2.462 2.093
1.15 0.012 0.063 0.128 0.271 0.612 1.058 1.670 2,553 3.885 6.985 9.094 5.535 3.186 2,508 2.083
1.20 0. 011 0.055 0.113 0 . 23 7 0.525 0.885 1.345 1.951 2.758 4.430 6.911 5. 710 3.326 2.555 2.079
1.30 0.009 0.044 0.089 0. 185 0 . 4 0 0 0. 651 0.946 1.297 1.711 2.458 3.850 4.793 3.452 2.628 2.077
1.40 0.007 0.036 0.072 0.149 0.315 0.502 0.711 0.946 1.208 1.650 2.462 3.573 3.282 2.626 2.068
1.50 0.006 0.029 0.060 0.122 0.255 0.399 0.557 0.728 0.912 1.211 1.747 2.647 2.917 2.525 2.038
1.60 0.005 0. 025 0 . 0 5 0 0. 101 0.210 0.326 0.449 0.580 0.719 0.938 1.321 2.016 2,508 2.347 1.978
1.70 0.004 0.021 0.042 0.086 0.176 0.271 0. 371 0.475 0.583 0.752 1.043 1.586 2. 128 2.130 1.889
1.80 0.004 0.018 0.036 0.073 0.150 0.229 0.311 0.397 0.484 0.619 0.848 1.282 1.805 1.907 1.778
1.90 0.003 0.016 0.031 0.063 0.129 0.196 0.265 0.336 0.409 0.519 0.706 1.060 1.538 1.696 1.656
2.00 0. 003 0.014 0.027 0.055 0 . 1 1 2 0.170 0.229 0.289 0.350 0.443 0.598 0.893 1. 320 1.505 1.531
2.20 0.002 0. 011 0.021 0.043 0.086 0.131 0.175 0.220 0.265 0.334 0.446 0.661 0.998 1.191 1.292
2.40 0.002 0.009 0.017 0.034 0.069 0.104 0.138 0.173 0. 208 0.261 0.347 0.510 0. 779 0.956 1.086
2.60 0.001 0.007 0.014 0.028 0.056 0.084 0.112 0.140 0.168 0.210 0.278 0.407 0.624 0.780 0.917
2.80 0. 001 0 . 0 0 6 0.012 0.023 0.046 0.070 0.093 0.116 0.138 0.172 0.227 0.332 0.512 0.647 0.779
3.00 0.001 0.005 0 . 0 1 0 0.020 0.039 0,058 0.078 0.097 0.116 0.144 0.190 0.277 0.427 0.545 0.668
3.50 0.001 0.003 0.007 0. 013 0.027 0.040 0.053 0.066 0.079 0.098 0.128 0.187 0.289 0.374 0.472
4. 00 0. 000 0.002 0.005 0.010 0.019 0.029 0.038 0.048 0.057 0.071 0.093 0.135 0.209 0.272 0.350
cont r i but i on me t hods [4, 11]. Once C~" ig is known, Cv, H ig, and
S ig can be de t e r mi ne d f r om t he f ol l owi ng r el at i ons:
( 6. 67) C~ _ C~ 1 = A - 1 + BT + CT 2 + DT 3 + ET 4
R R
( 6. 68) Hi g =At t +AT+B T2 +CT3 +DT4 +ET 5
C 2 D E 4
( 6. 69) ~g =As +AI nT+B T+~T + Ta+~T
The r el at i on f or C~ is obt ai ned t hr ough Eqs. ( 6. 23) and ( 6.66) ,
whi l e r el at i ons f or H ig and S ig have been obt ai ned f r om Eqs.
( 6. 21) and (6.32), respect i vel y. Cons t ant s An and As ar e ob-
t ai ned f r om i nt egr at i on of r el at i ons f or dH ig and dS ig and can
be de t e r mi ne d bas ed on t he r ef er ence st at e f or i deal gas en-
t hal p y and ent ropy. These p a r a me t e r s ar e not neces s ar y f or
cal cul at i on of H and S as t hey ar e omi t t e d dur i ng cal cul at i ons
wi t h r es p ect t o t he ar bi t r ar y r ef er ence st at e chos en f or H and
S. Usual l y t he choi ce of r ef er ence st at e is on H and not on H ig.
F or examp l e, Lenoi r and Hi p ki n [12] r e p or t e d exp er i ment al
dat a on ent hal p y of s ome p e t r ol e um f r act i ons wi t h r ef er ence
st at e of s at ur at ed l i qui d at 75~ at whi c h H = 0. I n s t e a m t a-
bl es whe r e p r op er t i es of l i qui d wa t e r and s t eam ar e r e p or t e d
[1] t he r ef er ence st at e at whi c h H - S = 0 is s at ur at ed l i qui d
at 0. 01~ Ther ef or e t her e is no need f or t he val ues of i nt e-
gr at i on cons t ant s An and As i n Eqs. ( 6.68) and ( 6. 69) as t hey
cancel i n t he cour s e of cal cul at i ons. Ther e ar e sever al ot he r
f or ms of Eq. ( 6. 66) f or C~, as an examp l e t he f ol l owi ng s i mp l e
f or m is gi ven f or i deal gas heat cap aci t y of wa t e r [1]:
( 6.70) - ~- = 3. 47 + 1.45 x 10- 3T + 0.121 x 105T -2
a not he r r el at i on f or wa t e r is gi ven by D I PPR [ 13]:
C~ 4.0129+3.222[ 2610.5/r ]2
R Lsi nh ( 2610. 5/ T) J
[ 1169/ T 12
( 6. 71) + 1.07 L c os h( 11- ~/ T) J
wher e i n bot h equat i ons T is i n kel vi n and t hey ar e val i d up t o
2000~ A gr ap hi cal c omp a r i s on of C~/ R f or wa t e r f r om Eqs.
( 6.66) , (6.70), a nd ( 6. 71) is s hown i n Fig. 6.2. Equa t i ons ( 6. 66)
6. THERM ODYNAM I C REL ATI ONS FOR PROPERTY ESTI M ATI ONS
TABLE 6. 5--V alues of [ ~ ]O) for use in Eq. (6.57).
er
2 4 3
0.01 0. 05 0. I 0. 2 0. 4 0. 6 0. 8 1 1.2 1.5 2 3 5 7 10
0.30 8 . 4 62 8 . 4 4 5 8 . 4 2 4 8 . 3 8 1 8 . 2 8 1 8 . 1 9 2 8 . 1 0 2 8.011 7.921 7.785 7.558 7.103 6.270 5.372 4.020
0.35 9. 775 9. 762 9. 746 9. 713 9. 646 9. 568 9. 499 9.430 9.360 9.256 9.080 8.728 8.013 7.290 6.285
0.40 11. 494 11. 484 11.471 11. 438 11, 394 11.343 11.291 11.240 11.188 11.110 10.980 10.709 10.170 9.625 8.803
0.45 12.651 12. 643 12. 633 12. 613 12. 573 12, 532 12. 492 12.451 12.409 12.347 12.243 12.029 11.592 11.183 10.533
0.50 13.111 13. 106 13. 099 1 3 . 0 8 4 1 3 . 0 5 5 1 3 . 0 2 5 12. 995 12.964 12.933 12.886 12.805 12.639 12.288 11.946 11.419
0.55 0.511 1 3 . 0 3 5 1 3 . 0 3 0 1 3 . 0 2 1 1 3 . 0 0 2 25. 981 12.961 12.939 12.917 12.882 12.823 12.695 12.407 12.103 11.673
0.60 0.345 12. 679 12.675 12. 668 12. 653 12. 637 12. 620 12.589 12.574 12.550 12.506 12.407 12.165 11.905 11.526
0.65 0.242 1.518 12. 148 12. 145 12. 137 12. 128 12. 117 12.105 12.092 12.060 12.026 11.943 11.728 11.494 11.141
0.70 0.174 1.026 2.698 11. 557 11. 564 11. 563 11. 559 11.553 11.536 11.524 11.495 11.416 11.208 10.985 10.661
0.75 0.129 0.726 1.747 10. 967 10. 995 11.011 11. 019 11.024 11.022 11.013 10.986 10.898 10.677 10.448 10.132
0.80 0.097 0.532 1.212 3.511 10. 490 10. 536 10. 566 10.583 10.590 10.587 10.556 10.446 10.176 9.917 9.591
0.85 0.075 0.399 0.879 2.247 9. 999 10.153 10. 245 10.297 10.321 10.324 10.278 10.111 9.740 9.433 9.075
0.90 0.058 0.306 0.658 1.563 5.486 9. 793 10. 180 10.349 10.409 10.401 10.279 9.940 9.389 8.999 8.592
0.93 0.050 0.263 0.560 1.289 3.890 1 0 . 2 8 5 10.769 10.875 10.801 10.523 9.965 9.225 8.766 8.322
0.95 0.046 0.239 0.505 1.142 3.215 9".3"89 9. 993 11.420 11.607 11.387 10.865 10.055 9.136 8.621 8.152
0.97 0.042 0.217 0.456 1.018 2.712 6.588 13.001 . . . 12.498 11.445 10.215 9.061 8.485 7.986
0.98 0.040 0.207 0.434 0.962 2.506 5.711 2(Z'9"18 14.884 14.882 13. 420 11.856 10.323 9.037 8.420 7.905
0.99 0.038 0.198 0.414 0.863 2.324 5.027 ... . . . . . . 12.388 10.457 9.011 8.359 7.826
1.00 0.037 0.189 0.394 0.863 2.162 4.477 10".5"11 oo 25. 650 16.895 13.081 10.617 8.990 8.293 7.747
1.01 0.035 0.181 0.376 0.819 2.016 4.026 8.437 . . . . . . . . . . . . 10.805 8.973 8.236 7.670
1.02 0.034 0.173 0.359 0.778 1.884 3.648 7.044 15.109 115.101 269 15.095 11.024 8.960 8.182 7.595
1.05 0.30 0.152 0.313 0.669 1.559 2.812 4.679 7.173 2.277 ... ... 11.852 8.939 8.018 7.377
1.10 0.024 0.123 0.252 0.528 1.174 1.968 2.919 3.877 4.002 3.927 . . . . . . 8.933 7.759 7.031
1.15 0.020 0.101 0.205 0.424 0.910 1.460 2.048 2.587 2.844 2.236 7.716 12.812 8.849 7.504 6.702
1.20 0.016 0.083 0.168 0.345 0.722 1.123 1.527 1.881 2.095 1.962 2.965 9.494 8.508 7.206 6.384
1.30 0.012 0.058 0.118 0.235 0.476 0.715 0.938 1.129 1.264 1.327 1.288 3.855 6.758 6.365 5.735
1.40 0.008 0.042 0.083 0.166 0.329 0.484 0.624 0.743 0.833 0.904 0.905 1.652 4.524 5.193 5.035
1.50 0.006 0.030 0.061 0.120 0.235 0.342 0.437 0.517 0.580 0.639 0.666 0.907 2.823 3.944 4.289
1.60 0.005 0.023 0.045 0.089 0.173 0.249 0.317 0.374 0.419 0.466 0.499 0.601 1.755 2.871 3.545
1.70 0.003 0.017 0.034 0.068 0.130 0.187 0.236 0.278 0.312 0.349 0.380 0.439 1.129 2.060 2.867
1.80 0.003 0.013 0.027 0.052 0.100 0.143 0.180 0.212 0.238 0.267 0.296 0.337 0.764 1.483 2.287
1.90 0.002 0.011 0.021 0. 04t 0.078 0. I 11 0.140 0.164 0.185 0.209 0.234 0.267 0.545 1.085 1.817
2.00 0.002 0.008 0.017 0.032 0.062 0.088 0.110 0.130 0.146 0.166 0.187 0.217 0.407 0.812 1.446
2.20 0.001 0.005 0.011 0.021 0.042 0.057 0.072 0.085 0.096 0.110 0.126 0.150 0.256 0.492 0.941
2.40 0.001 0.004 0.007 0.014 0.028 0.039 0.049 0.058 0.066 0.076 0.089 0.109 0.180 0.329 0.644
2.60 0.001 0.003 0.005 0.010 0.020 0.028 0.035 0.042 0.048 0.056 0.066 0.084 0.137 0.239 0.466
2.80 0.000 0.002 0.004 0.008 0.014 0.021 0.026 0.031 0.036 0.042 0.051 0.067 0.110 0.187 0.356
3.00 0.000 0.001 0.003 0.006 0.011 0.016 0.020 0.024 0.028 0.033 0.041 0.055 0.092 0.153 0.285
3.50 0.000 0.001 0.002 0.003 0.006 0.009 0.012 0.015 0.017 0.021 0.026 0.038 0.067 0.108 0.190
4.00 0.000 0.001 0.001 0.002 0.004 0.006 0.008 0.010 0.012 0.015 0.019 0.029 0.054 0.085 0.146
Taken with permission from Ref. [9]9 The value at the critical point (Tr = Pr = 1) is taken from the API-TDB [5]. Bol d numbe r s i ndi cat e l i qui d regi on.
a nd ( 6. 71) ar e a l mos t i dent i cal a nd Eq. ( 6. 70) is not val i d at
ver y l ow t e mp e r a t ur e s . The mos t a c c ur a t e f or mul a t i on a nd
t a bul a t i on of p r op e r t i e s of wa t e r a nd s t e a m i s ma de by I APWS
[ 14] .
To cal cul at e i deal gas p r op e r t i e s of p e t r ol e um f r act i ons , t he
p s e udoc omp one nt me t hod di s c us s e d i n Se c t i on 3. 3. 4 ma y be
us ed. Kes l er a nd Lee [ 15] p r ovi de a n e qua t i on f or di r ect cal -
c ul a t i on of i deal gas he a t c a p a c i t y of p e t r ol e um f r a c t i ons i n
t e r ms of Wa t s on Kw, a nd a c e nt r i c fact or, w:
C~ -- M [Ao + A, T + A2T 2 - C(Bo + B1T + B2T2) ]
Ao = - 1. 41779 + 0. 11828Kw
A1 = - ( 6. 99724 - 8. 69326Kw + 0. 27715K 2) x 10 -4
A2 = - 2. 2582 10 -6
( 6. 72) Bo = 1. 09223 - 2. 48245w
B1 = - ( 3. 434 - 7. 14w) l 0 -3
B2 = - ( 7. 2661 - 9. 2561w) x 10 7
C = [ (12 " 8 - Kw ) x ( I O - Kw ) ]
whe r e C~ i s i n J / t ool - K, M i s t he mol e c ul a r we i ght ( g/ mol ) , T
i s i n kel vi n, Kw i s def i ned by Eq. ( 2. 13) , a nd w ma y be det er -
mi ne d f r om Eq. ( 2. 10) . Ts onop oul os et al. [ 16] s ugge s t e d t ha t
t he c or r e c t i on t e r m C i n t he a bove e qua t i on s houl d e qua l t o
zer o whe n Kw is l ess t ha n 10 or gr e a t e r t ha n 12,8. But our
e va l ua t i ons s how t ha t t he e qua t i on i n i t s or i gi na l f or m p r e-
lg
di ct s val ues of C~ f or hydr oc a r bons i n t hi s r a nge of Kw ver y
cl ose t o t hos e r e p or t e d by DI PPR [ 13] . Thi s e qua t i on ma y
al so be a p p l i e d t o p ur e hydr oc a r bons wi t h c a r bon numbe r
gr e a t e r t ha n or e qua l t o C5. I deal gas he a t c a p a c i t i e s of sev-
er al hydr oc a r bons f r om p ar af f i ni c gr oup p r e di c t e d f r om Eqs.
( 6. 66) a nd ( 6. 72) ar e s hown i n Fig. 6.3. As e xp e c t e d he a t cap ac-
i t y a nd e nt ha l p y i nc r e a s e wi t h c a r bon numbe r or mol e c ul a r
wei ght . Equa t i on ( 6. 72) gener al l y p r e di c t s C~ of p ur e hydr o-
c a r bons wi t h e r r or s of 1- 2% as e va l ua t e d by Kes l er a nd Lee
[ 15] a nd c a n be us e d i n t he t e mp e r a t ur e r a nge of 255- 922 K
( 0- 1200~ The r e ar e s i mi l a r ot he r c or r e l a t i ons f or es t i ma-
t i on of i deal gas he a t c a p a c i t y of na t ur a l gas es a nd p e t r ol e um
f r a c t i ons [ 17, 18]. The r e l a t i on r e p or t e d by F i r ooz a ba di [ 17]
f or c a l c ul a t i on of he a t c a p a c i t y of na t ur a l gas es i s i n t he f or m
TABLE ~ ( 0 ) Eq. 6.6---Values o1~, for use in (6.59).
t'r
Tr 0.01 0.05 0.1 0.2 0.4 0.6 0.8 1 1.2 1.5 2 3 5 7 10
0.30 0 . 0 0 0 2 0 . 0 0 0 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.35 0 . 0 0 9 4 0 . 0 0 0 7 0 . 0 0 0 9 0 . 0 0 0 2 0 . 0 0 0 1 0 . 0 0 0 1 0 . 0 0 0 1 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.40 0 . 0 2 7 2 0 . 0 0 5 5 0 . 0 0 2 8 0 . 0 0 1 4 0 . 0 0 0 7 0 . 0 0 0 5 0 . 0 0 0 4 0.0003 0.0003 0.0003 0.0002 0.0002 0.0002 0.0002 0.0003
0.45 0 . 1 9 2 1 0 . 0 2 6 6 0 . 0 1 9 5 0 . 0 0 6 9 0 . 0 0 9 6 0 . 0 0 2 5 0 . 0 0 2 0 0.0016 0.0014 0.0012 0.0010 0.0008 0.0008 0.0009 0.0012
0.50 0 . 4 5 2 9 0 . 0 9 1 2 0 . 0 4 61 0 . 0 2 9 5 0 . 0 1 2 2 0 . 0 0 8 5 0 . 0 0 6 7 0.0055 0.0048 0.0041 0.0034 0.0028 0.0025 0.0027 0.0034
0.55 0.9817 0 . 2 4 9 2 0 . 1 2 2 7 0 . 0 6 2 5 0 . 0 9 2 5 0 . 0 2 2 5 0 . 0 1 7 6 0.0146 0.0127 0.0107 0.0089 0.0072 0.0063 0.0066 0.0080
0,60 0.9840 0 . 5 9 8 9 0 . 2 7 1 6 0 . 1 9 8 4 0 . 0 7 1 8 0 . 0 4 9 7 0 . 0 9 8 6 0.0321 0.0277 0.0234 0.0193 0.0154 0.0132 0.0135 0.0160
0.65 0.9886 0,9419 0 . 5 2 1 2 0 . 2 6 5 5 0 . 1 9 7 4 0 . 0 9 4 8 0 . 0 7 9 8 0.0611 0.0527 0.0445 0.0364 0.0289 0.0244 0.0245 0,0282
0.70 0.9908 0.9528 0.9057 0.4560 0 . 2 9 6 0 0 . 1 6 2 6 0 . 1 2 6 2 0.1045 0.0902 0.0759 0.0619 0.0488 0.0406 0.0402 0.0453
0.75 0. 9931 0.9616 0.9226 0.7178 0 . 3 7 1 5 0 . 2 5 5 9 0 . 1 9 8 2 0.1641 0.1413 0.1188 0.0966 0.0757 0.0625 0.0610 0.0673
0.80 0.9931 0.9683 0.9354 0.8730 0 . 5 4 4 5 0 . 9 7 5 0 0 . 2 9 0 4 0.2404 0.2065 0.1738 0.1409 0.1102 0.0899 0.0867 0.0942
0.85 0.9954 0.9727 0.9462 0.8933 0 . 7 5 9 4 0 . 5 1 8 8 0 . 4 0 1 8 0.3319 0.2858 0.2399 0.1945 0.1517 0.1227 0.1175 0.1256
0,90 0.9954 0.9772 0.9550 0.9099 0.8204 0 . 6 8 2 9 0 . 5 2 9 7 0.4375 0.3767 0.3162 0.2564 0.1995 0.1607 0.1524 0.1611
0.95 0.9954 0.9817 0.9616 0.9226 0.8472 0.7709 0.6668 0. 5521 0.4764 0,3999 0.3251 0.2523 0.2028 0.1910 0.2000
1.00 0.9977 0.9840 0.9661 0.9333 0.8690 0.8035 0.7379 0.6668 0.5781 0.8750 0.3972 0.3097 0.2483 0.2328 0.2415
1.05 0.9977 0.9863 0.9705 0. 9441 0.8872 0.8318 0.7762 0.7194 0.6607 0.5728 0.4710 0.3690 0.2958 0.2773 0.2844
1.10 0.9977 0.9886 0.9750 0.9506 0.9016 0.8531 0.8072 0.7586 0.7112 0.6412 0.5408 0.4285 0.3451 0.3228 0.3296
1.15 0.9977 0.9886 0.9795 0.9572 0.9141 0.8730 0.8318 0.7907 0.7499 0.6918 0.6026 0.4875 0.3954 0.3690 0.3750
1.20 0.9977 0.9908 0.9817 0.9616 0.9247 0.8892 0.8531 0.8166 0.7834 0.7328 0.6546 0.5420 0.4446 0.4150 0.4198
1.30 0.9977 0. 9931 0,9863 0.9705 0.9419 0. 9141 0.8872 0.8590 0.8318 0.7943 0.7345 0.6383 0.5383 0.5058 0.5093
1.40 0.9977 0.9931 0.9886 0.9772 0.9550 0.9333 0.9120 0.8892 0.8690 0.8395 0.7925 0.7145 0.6237 0.5902 0.5943
1.50 1.0000 0.9954 0.9908 0.9817 0.9638 0.9462 0.9290 0. 9141 0.8974 0.8730 0.8375 0.7745 0.6966 0.6668 0.6714
1.60 1.0000 0.9954 0,993l 0.9863 0.9727 0.9572 0.9441 0.9311 0.9183 0.8995 0.8710 0.8222 0.7586 0.7328 0.7430
1,70 1.0000 0.9977 0.9954 0.9886 0.9772 0. 9661 0.9550 0.9462 0.9354 0.9204 0.8995 0.8610 0.8091 0.7907 0.8054
1.80 1.0000 0.9977 0.9954 0.9908 0.9817 0.9727 0.9661 0.9572 0.9484 0.9376 0.9204 0.8913 0.8531 0.8414 0.8590
1.90 1.0000 0.9977 0.9954 0. 9931 0.9863 0.9795 0,9727 0.9661 0.9594 0.9506 0.9376 0.9162 0.8872 0.8831 0.9057
2.00 1.0000 0.9977 0.9977 0.9954 0.9886 0.9840 0.9795 0.9727 0.9683 0.9616 0.9528 0.9354 0.9183 0.9183 0.9462
2.20 1.0000 1.0000 0.9977 0.9977 0.9931 0.9908 0.9886 0.9840 0.9817 0.9795 0.9727 0.9661 0.9616 0.9727 1.0093
2.40 1.0000 1.0000 1.0000 0.9977 0.9977 0.9954 0.9931 0.9931 0.9908 0.9908 0.9886 0.9863 0.9931 1.0116 1.0568
2.60 1. 0000 1. 0000 1.0000 1. 0000 1.0000 0.9977 0.9977 0.9977 0.9977 0.9977 0.9977 1.0023 1. 0162 1.0399 1.0889
2.80 1. 0000 1.0000 1.0000 1.0000 1.0000 1. 0000 1. 0023 1. 0023 1.0023 1.0046 1.0069 1. 0116 1. 0328 1.0593 1.1117
3.00 1. 0000 1, 0000 1.0000 1.0000 1. 0023 1. 0023 1,0046 1.0046 1.0069 1. 0069 1. 0116 1. 0209 1. 0423 1.0740 1.1298
3.50 1. 0000 1.0000 1.0000 1. 0023 1. 0023 1. 0046 1.0069 1.0093 1. 0116 1.0139 1. 0186 1.0304 1.0593 1. 0914 1.1508
4.00 1.0000 1. 0000 1.0000 1, 0023 1.0046 1. 0069 1. 0093 1.0116 1. 0139 1.0162 1. 0233 1.0375 1.0666 1.0990 1.1588
Taken with permission from Ref.[9].Thev~ue ~ thecfitic~ Point(~ = ~ = 1)istaken from the API-TDB[5].Bdd numbe~ indicateliquidre~on.
of Cf f = A +B T+ C( SG g) + D(SGg) 2 + E[T( SGg) ], wher e T is
t emp er at ur e and SG g is gas specific gravi t y (Mg/29), Al t hough
t he equat i on is very useful for cal cul at i on of C~ of undefi ned
nat ur al gases but usi ng t he r ep or t ed coefficients we coul d not
obt ai n rel i abl e val ues for C~. I n anot her correl at i on, ideal
heat capaci t i es of hydr ocar bons (Nc > C5) were rel at ed to
boi l i ng poi nt and specific gravi t y in t he f or m of Eq. (2.38)
at t hr ee t emp er at ur es of 0~ ( ~255 K), 600~ ( ~589 K),
and 1200~ (922 K) [18]. For light gases based on t he dat a
gener at ed t hr ough Eq. (6.66) for comp ounds f r om C1 to
C5 wi t h H2S, CO2, and N2 t he fol l owi ng rel at i on has been
det er mi ned:
(6.73) C~ 2
-~- = ~ ( A i + ~ M ) ~ i
i =0
for nat ural and l i ght gases wi t h 16 < M < 60
wher e
A0 = 3.3224 B0 = - 2. 5379 x 10 -2
A1 = - 7. 3308 x 10 -3 B1 = 7.5939 x 10 -4
A2 = 4.3235 x 10 -6 B2 = - 2. 6565 10 -7
i n whi ch T is t he absol ut e t emp er at ur e in kelvin. This equa-
t i on is based on mor e t han 500 dat a poi nt s gener at ed i n
t he t emp er at ur e r ange of 50-1500 K and mol ecul ar wei ght
range of 16-60. The average devi at i on for this equat i on for
t hese ranges is 5%; however, when it is appl i ed in t he t em-
p er at ur e range of 200-1000 K and mol ecul ar wei ght r ange
of 16-50, t he er r or reduces to 2.5%. Use of this equat i on is
r ecommended for undefi ned l i ght hydr ocar bon gas mi xt ures
when gas specific gravity (SG g) is known (M = 29 SGg), For
defi ned hydr ocar bon mi xt ur es of known comp osi t i on t he fol-
l owi ng rel at i on may be used to cal cul at e mi xt ur e ideal gas
heat capacity:
ig ig
(6.74) Cp'mixR = EYi-RCpi
i
wher e Cp g is t he mol ar ideal heat capaci t y of comp onent i
( with mol e fract i on yi) and may be cal cul at ed f r om Eq. (6.66).
Exampl e 6. 2- - Cal cul at e Cp for sat urat ed l i qui d benzene at
450 K and 9.69 bar usi ng general i zed cor r el at i on and SRK
EOS and comp ar e wi t h t he val ue of 2.2 kJ / kg. K as gi ven
in Ref. [6]. Also cal cul at e heat capaci t y at const ant vol ume,
heat capaci t y ratio, and resi dual ent hal py ( H - H ig) f r om bot h
SRK EOS and general i zed correl at i ons of LK.
Sol ut i on- - Fr om Table 2.1 for benzene we have Tc = 562 K,
Pc = 49 bar, ~o = 0.21, M = 78.1, and Kw = 9.72 and f r om
t- ( l) r
TABLE 6. 7--V alues ol q~ fo use in Eq. (6.59).
Pr
2 4 5
Tr 0.01 0.05 0.1 0.2 0.4 0.6 0.8 1 1.2 1.5 2 3 5 7 10
0.30 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.35 0. 0000 0. 0000 0. 0000 0. 0000 0. 0000 0. 0000 0. 0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.40 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.45 0 . 0 0 0 2 0 . 0 0 0 2 0 . 0 0 0 2 0 . 0 0 0 2 0 . 0 0 0 2 0 . 0 0 0 2 0 . 0 0 0 2 0.0002 0.0002 0.0002 0.0002 0.0001 0.0001 0.0001 0.0001
0.50 0 . 0 0 1 4 0 . 0 0 1 4 0 . 0 0 1 4 0 . 0 0 1 4 0 . 0 0 1 4 0 . 0 0 1 4 0. 0013 0.0013 0.0013 0.0013 0.0012 0.0011 0.0009 0.0008 0.0006
0.55 0.9705 0 . 0 0 69 0 . 0 0 68 0 . 0 0 68 0 . 0 0 66 0 . 0 0 65 0 . 0 0 64 0.0063 0.0062 0.0061 0.0058 0.0053 0.0045 0.0039 0.0031
0.60 0.9795 0 . 0 2 2 7 0 . 0 2 2 6 0 . 0 2 2 3 0 . 0 2 2 0 0. 0216 0. 0213 0.0210 0.0207 0.0202 0.0194 0.0179 0.0154 0.0133 0.0108
0.65 0.9863 0.9311 0 . 0 5 7 2 0 . 0 5 68 0 . 0 5 5 9 0 . 0 5 5 1 0 . 0 5 4 3 0.0535 0.0527 0.0516 0.0497 0.0461 0.0401 0.0350 0.0289
0.70 0.9908 0.9528 0.9036 0. 1182 0. 1163 0. 1147 0. 1131 0.1116 0.1102 0.1079 0.1040 0.0970 0.0851 0.0752 0.0629
0.75 0.9931 0.9683 0.9332 0. 2112 0. 2078 0 . 2 0 5 0 0 . 2 0 2 2 0.1994 0.1972 0.1932 0.1871 0.1754 0.1552 0.1387 0.1178
0.80 0.9954 0.9772 0.9550 0.9057 0. 3302 0. 3257 0. 3212 0.3168 0.3133 0.3076 0.2978 0.2812 0.2512 0.2265 0.1954
0.85 0.9977 0.9863 0.9705 0.9375 0. 4774 0. 4708 0. 4654 0.4590 0.4539 0.4457 0.4325 0.4093 0.3698 0.3365 0.2951
0.90 0.9977 0.9908 0.9795 0.9594 0.9141 0. 6323 0. 6250 0.6165 0.6095 0.5998 0.5834 0.5546 0.5058 0.4645 0.4130
0.95 0.9977 0.9931 0.9885 0.9750 0.9484 0.9183 0. 7888 0.7797 0.7691 0.7568 0.7379 0.7063 0.6501 0.6026 0.5432
1.00 1.0000 0.9977 0.9931 0.9863 0.9727 0.9594 0.9440 0.9311 0.9204 0.9078 0.8872 0.8531 0.7962 0.7464 0.6823
1.05 1.0000 0.9977 0.9977 0.9954 0.9885 0.9863 0.9840 0.9840 0.9954 1.0186 1.0162 0.9886 0.9354 0.8872 0.8222
1.10 1.0000 1.0000 1.0000 1.0000 1.0023 1.0046 1.0093 1.0163 1.0280 1.0593 1.0990 1.1015 1.0617 1.0186 0.9572
1.15 1.0000 1.0000 1.0023 1.0046 1.0116 1.0186 1.0257 1.0375 1.0520 1.0814 1.1376 1.1858 1.1722 1.1403 1.0864
1.20 1.0000 1.0023 1.0046 1.0069 1.0163 1.0280 1.0399 1.0544 1.0691 1.0990 1.1588 1.2388 1.2647 1.2411 1.2050
1.30 1.0000 1.0023 1.0069 1.0116 1.0257 1.0399 1.0544 1.0716 1.0914 1.1194 1.1776 1.2853 1.3868 1.4125 1.4061
1.40 1.0000 1.0046 1.0069 1.0139 1.0304 1,0471 1.0642 1.0815 1.0990 1.1298 1.1858 1.2942 1.4488 1.5171 1.5524
1.50 1.0000 1.0046 1.0069 1.0163 1.0328 1.0496 1.0666 1.0865 1.1041 1.1350 1.1858 1.2942 1.4689 1.5740 1.6520
1.60 1.0000 1.0046 1.0069 1.0163 1.0328 1.0496 1.0691 1.0865 1.1041 1.1350 1.1858 1.2883 1.4689 1.5996 1.7140
1.70 1.0000 1.0046 1.0093 1.0163 1.0328 1.0496 1.0691 1.0865 1.1041 1.1324 1.1803 1.2794 1.4622 1.6033 1.7458
1.80 1.0000 1.0046 1.0069 1.0163 1.0328 1.0496 1.0666 1.0840 1.1015 1.1298 1.1749 1.2706 1.4488 1.5959 1.7620
1.90 1.0000 1.0046 1.0069 1.0163 1.0328 1.0496 1.0666 1.0815 1.0990 1.1272 1.1695 1.2618 1.4355 1.5849 1.7620
2.00 1.0000 1.0046 1.0069 1.0163 1.0304 1.0471 1.0642 1.0815 1.0965 1.1220 1.1641 1.2503 1.4191 1.5704 1.7539
2.20 1.0000 1.0046 1.0069 1.0139 1.0304 1.0447 1.0593 1.0765 1.0914 1.1143 1.1535 1.2331 1.3900 1.5346 1.7219
2.40 1.0000 1.0046 1.0069 1.0139 1.0280 1.0423 1.0568 1.0716 1.0864 1.1066 1.1429 1.2190 1.3614 1.4997 1.6866
2.60 1.0000 1.0023 1.0069 1.0139 1.0257 1.0399 1.0544 1.0666 1.0814 1.1015 1.1350 1.2023 1.3397 1.4689 1.6482
2.80 1.0000 1.0023 1.0069 1.0116 1.0257 1.0375 1.0496 1.0642 1.0765 1.0940 1.1272 1.1912 1.3183 1.4388 1.6144
3.00 1.0000 1.0023 1.0069 1.0116 1.0233 1.0352 1.0471 1.0593 1.0715 1.0889 1.1194 1.1803 1.3002 1.4158 1.5813
3.50 1.0000 1.0023 1.0046 1.0023 1.0209 1.0304 1.0423 1.0520 1.0617 1.0789 1.1041 1.1561 1.2618 1.3614 1.5101
4.00 1.0000 1.0023 1.0046 1.0093 1.0186 1.0280 1.0375 1.0471 1.0544 1.0691 1.0914 1.1403 1.2303 1.3213 1.4555
Tabl e 5. 12, ZRA = 0. 23. F r om Eq. ( 6. 72) , C~ = 127. 7 J / mol . K,
T~ = T/Tc = 0. 8, a nd Pr = 0. 198- 0.2. F r om Tabl es 6. 4 a nd
6. 5, [(Ce - C~) / R] (~ = 3. 564, [(Cp - C~) / R] (1) = 10. 377. I t i s
i mp or t a nt t o not e t ha t t he s ys t e m is s a t ur a t e d l i qui d a nd
e xt r a p ol a t i on of val ues f or t he l i qui d r e gi on f r om Tr = 0. 7 a nd
0. 75- 0. 8 at Pr = 0. 2 is r e qui r e d f or bot h ( 0) a nd ( 1) t e r ms .
Di r ect da t a gi ven i n t he t abl es at Tr = 0. 8 a nd Pr = 0. 2 cor-
r e s p ond t o s a t ur a t e d va p or a nd s p eci al car e s houl d be t a ke n
whe n t he s ys t e m i s at s a t ur a t e d c ondi t i ons . F r om Eq. ( 6. 57) ,
[ ( Cv - C~) / R] = 5. 74317, Cp - C~ = 5. 74317 x 8. 314 = 47. 8
J / moI - K, a nd Cv = 47. 8 + 127. 7 = 175. 5 J / t ool - K. The sp e-
ci fi c he a t is c a l c ul a t e d t hr ough mol e c ul a r we i ght us i ng
Eq. ( 5. 3) as Cv = 175. 5/ 78. 1 = 2. 25 J / g. ~ Thi s va l ue is
bas i cal l y t he s a me as t he r e p or t e d val ue. F or SR K EOS,
a = 1. 907 x 107 ba r ( c m3/ mol ) 2, b= 82. 69 cm3/ mol , z L=
0. 033, V L = 126.1 cm3/ mol , c = 9. 6 cm3/ mol , a nd V L = 116. 4
( cor r ect ed) , whi c h gi ves Z L ( cor r ect ed) = 0. 0305, whe r e
c i s c a l c ul a t e d f r om Eq. ( 5. 51) . F r om ge ne r a l i z e d cor r e-
l at i ons Z( ~ 0. 0328 a nd Z ( ~ = - 0. 0138, whi c h gi ves Z =
0. 0299 t ha t i s ver y cl ose t o t he val ue c a l c ul a t e d f r om
SR K EOS. Us i ng uni t s of kel vi n f or t e mp e r a t ur e , ba r f or
p r e s s ur e , a nd c m 3 f or vol ume , R = 83. 14 c m3. ba r / mol . K
a nd V = 116. 4 cr n3/ mol . F r om r e l a t i ons gi ven i n Tabl e 6.1
we cal cul at e al = - 33017 a nd a2 = 60. 9759. P1 = 4. 13169,
P2 = - 38. 2557, a nd P3 -- 0. 402287. F r om Tabl e 6.1 f or SR K
EOS we ha ve Cp - C~ = 450 0. 39565 - 450 ( 3. 887) 2/
( - 32. 8406) - 83. 14 = 301. 936 c m 3 . ba r / mol . K. Si nce 1 J =
10 c m 3 . bar , t hus CF - C~ = 301. 936/ 10- - 30. 194 J / t ool . K.
Cv = (Ce - C~) + C~ = 30. 194 + 127. 7 = 157. 9 J / mol . K or
Cv = 157. 9/ 78. 1 = 2. 02 J / g. ~ The de vi a t i on wi t h gener -
al i zed c or r e l a t i on is - 8. 1%. Ef f ect of c ons i de r i ng vol ume
t r a ns l a t i on c on vol ume i n c a l c ul a t i on of Cv i s mi nor
a nd i n t hi s p r obl e m i f V L di r ect l y c a l c ul a t e d f r om SR K
e qua t i on ( 126. 1 c m3/ mol ) is us ed, val ue of c a l c ul a t e d Ce
woul d be st i l l t he s a me as 2. 02 J / g- ~ F or c a l c ul a t i on
of Cv, SR K e qua t i on i s us e d wi t h r e l a t i ons gi ven i n Ta-
bl e 6.1. Cv - Cv g = TP3 = 450 x 0. 402287 = 178. 04 c m 3 . ba r /
mol - K = 178. 04/ 10 = 17. 8 J / mol . K. C~ = Cie g - R = 127. 7 -
8. 314 = 119. 4 J / mol . K. Thus , Cv = 119. 4+ 17. 8 = 137. 2
J / mol . K = 137. 2/ 78. 1 = 1. 75 J / g. . K. The he a t c a p a c i t y r at i o
is g = Cp/ Cv = 2. 02/ 1. 75 = 1. 151.
To cal cul at e H- H ig f r om ge ne r a l i z e d c or r e l a t i ons we
get f r om Tabl es 6. 2 a nd 6. 3 as [ ( H- Hig)/RTc] (0) = - 4. 518
a nd [ ( H - Hig)/RTc] (1) = - 5. 232. F r om Eq. ( 6. 56) , [ ( H - Hig) /
RTc] = - 5. 6167. Agai n i t s houl d be not e d t ha t t he val ues of
[](0) a nd [](1) t e r ms ar e t a ke n f or s a t ur a t e d l i qui d by ext r ap -
ol a t i on of Tr f r om 0. 7 a nd 0. 75 t o 0.8. Val ues i n t he t a bl e s
f or s a t ur a t e d va p or ( at Tr = 0. 8, Pr = 0. 2) s houl d be avoi ded.
TABLE 6.8---Constants for Eqs. (6.66)-(6.69) for ideal gas heat capacity, enthaIpy, and entropy.
No. Comp ound na me F or mul a M A B x 10 3 C x 10 6 D x 1010 E 10 I4 Trrdn, K Tmax, K
Pa r a t ~l i l $
1 Met hane CH4 16.043 4.34610 - 6. 14488 26.62607 -219. 2998 588.89965
2 Et hane C2H6 30.070 4.00447 - 1. 33847 42.86416 - 452. 2446 1440.4853
3 Propane C3H8 44.096 3.55751 10.07312 39.13602 - 475. 7220 1578.1656
4 n-But ane C4Hlo 58.123 2.91601 28.06907 15.37435 -292. 9255 1028.0462
5 I sobut ane C4H1o 58,123 2.89796 25.14031 26.04226 -405. 3691 1396.6324
6 n-Pent ane C5H12 72,150 4.06063 29.87141 30.46993 -461. 3523 1559.8971
7 I sopent ane C5H12 72,150 0.61533 49.99361 - 9. 72605 - 121. 1597 563.52870
8 Neopent ane C5H12 72.150 6.60029 24.43268 32.52759 -402. 96336 1258.46299
9 n-Hexane C6I-I14 86.177 3.89054 41.42970 24.35860 -457. 5222 1599.4100
10 n-Hept ane C7H12 100.204 4.52739 47.36877 31.09932 -570. 22085 1999.68224
11 n-Oct ane C8Ht8 114.231 4.47277 57.81747 29.07465 -621. 09106 2265.33690
12 n-Nonane C9H20 128.258 3.96754 68.72207 31.85998 -758. 47191 2875.17975
13 n-Decane CIOH22 142.285 14.56771 - 9. 12133 283.5241 -3854. 9259 16158.7933
14 n-Undecane CllH24 156.312 15.72269 -8. 39015 308.0195 -4205. 1509 17634.1470
15 n-Dodecane C12H26 170.338 16.87761 - 7. 65919 332.5144 -4555. 3529 19109.3961
16 n-Tridecane C13H28 184.365 30.63938 - 107. 2144 632.4036 -8053. 8502 33377.9390
17 n-Tetradecane C14H30 198.392 -2. 95801 - 6. 19822 381.5291 -5256. 0878 22061.2353
18 n-Pent adecane
19 n-Hexadecane
20 n-Hept adecane
21 n-Oct adecane
22 n-Nonadecane
23 n-Ei cosane
24 2-Met hyl pent ane
25 3-Met hyl pent ane
26 2, 2-Di met hyl but ane
27 2- Met h~hexane
28 3-Met hyl hexane
29 2, 4- Di met h~p ent ane
30 2- Met h~hep t ane
C15H32 212.419 - 2. 65315
C16H34 226.446 -36. 57941
C17H36 240.473 23.25896
C18H38 254.500 - 2. 20866
C19H40 268.527 25.68345
C20H42 282.553 26.82718
C6H14 86.177 0.44073
C6H14 86.177 - 0. 07902
C6H14 86.177 1.00342
C7H16 100.204 0.57808
C7H16 100.204 - 0. 37490
C7H16 100.204 - 3. 20582
C8H18 114.231 0.92650
31 2, 2, 4-Trimethyl-pentane C8H18 114.231 - 1. 85230
Ol ef ms
32 Et hyl ene
33 Pr op ~ene
34 1-Butene
35 1-Pentene
36 1-Hexene
37 1-Heptene
38 1-Octene
39 1-Nonene
40 1-Decene
41 1-Undecene
42 1-Dodecene
43 1-Tfidecene
44 1-Tetradecene
45 1-Pent adecene
46 1-Hexadecene
47 1-Hept adecene
C2H4 28.054 2.11112
C3H6 42.081 2.15234
C4H8 56.107 4.25402
C5HIo 70.135 2.04789
C6H12 84.162 0.00610
C7H14 98.189 3.47887
C8H16 112.216 3.98703
C9H18 126.243 4.54519
CloH20 140.270 4.95682
CllH22 154.219 5.68918
C12H24 168.310 5.94633
C13H26 182.337 - 0. 32099
C14H28 196.364 - 0. 29904
C15H3o 210.391 0.09974
C16H32 224.418 0.54495
C17H34 238.445 0.41533
-5. 09511 404.3408 -5576. 7343 -23366. 9827
- 4. 73820 430.5450 -5956. 8328
- 4. 00829 431.5163 -6307. 0717
- 3. 27840 479.5420 -6657. 3161
- 2. 54834 504.0402 -7007. 5535
- 1. 81886 528.5571 -7358. 0352
60.77573 - 10. 93570 -180. 70573
63.31181 - 18. 82562 - 90. 58759
56.10078 -1. 05011 -237. 84301
70.71556 - 15. 00679 -187. 48705
75.26096 -22. 63052 -131. 65268
98.77224 - 72. 48550 293.10189
78.42561 -11, 24742 -281. 97592
96.08105 - 47. 77416 68.93159
25013.0768
26488.4354
27963.8136
29439.1464
30915.5067
833.40865
510,28364
956.53142
899.92106
744.03635
-500. 62465
1265.61161
137.05449
8.32103 11.24746 -155. 42099
17.76767 8.26700 -166. 33525
10.78298 47.84869 -597. 94406
37.52066 5.51442 -254. 11404
63.24725 - 35. 49665 88.16710
48.09877
54.62745
60.98631
68.28457
73.39300
81.98315
129.0630
138.5541
145.9734
152.863
163.244
23.25712 -531. 31261
29.77494 -652. 07494
36.26160 -769. 45635
40.89381 -873. 20972
49.96821 -1012. 1464
51.37713 -1083. 4987
- 25. 20577 -505. 45836
-25. 09881 -568. 90191
- 20. 68363 -670. 89438
-15. 10253 -781. 81088
- 16. 44658 -838. 48469
50 1500
50 1500
50 1500
200 1500
50 1500
200 1500
200 1500
220 1500
200 1500
200 1500
200 1500
200 1000
200 1000
200 1000
2O0 1000
200 1000
200 1000
200 1000
200 1000
200 1000
200 1000
20O 1000
200 1000
200 1500
200 1500
200 1500
200 1500
200 1500
200 1500
200 1500
200 1500
48 1-Octadecene
49 1-Nonadecene
50 1-Eicosene
Naphthenes
51 Cycl opent ane
52 Met hyl cycl opent ane
53 Et hyl cycl opent ane
54 n-Propyl cycl opent ane
55 n-But yl cycl opent ane
56 n-Pent yl cycl opent ane
57 n-Hexyl cycl opent ane
58 n-Hept yl cycl opent ane
59 n-Oct yl cycl opent ane
60 n-Nonyl cycl opent ane
61 n-Decylcyclopentane
62 n-Unoecyl cycl opent ane Ct6H32 224.420 - 8. 71319
63 n-Dodecyl cycl opent ane C17H34 238.440 - 8. 81568
C18H36 252.472 31.69585 -73. 95796 647.4299 -8000. 0231
CI9H38 266.490 0.77613 180.596 -13. 04267 -993. 43903
C20H40 280.517 - 0. 20146 196.7237 -27. 69713 -923. 83278
C5H10 70.134 -7. 43795
C6H12 84.161 - 6. 81073
C7H14 98.188 - 7. 51027
C8H16 112.216 -7. 61363
C9Hla 126.243 -7. 58208
C10H20 140.270 - 8. 03062
CllH22 154.290 - 5. 33508
C12H24 168.310 -8. 17951
C13H26 182.340 - 8. 20466
C14H28 196.360 - 8. 27104
C15H30 210.390 - 8. 70424
69.82174 - 43. 64337 122.59611
80.58175 - 50. 42977 141.93915
93.72668 - 58. 81706 167.05996
105.1051 - 65. 42900 183.75014
115.9123 - 71. 30847 197.13396
128.8771 -80. 20325 225.20999
122.4990 -48. 15565 -101. 53331
151.3485 - 93. 06028 256.93490
162.3974 -99. 23258 271.5150
173.6250 - 105. 7060 287.84321
186.5104 - 114. 4444 314.93805
197.4859 -120. 5348 329.23444
208.8487 - 127. 1364 345.90292
- 92. 59304 300 1500
-107. 77270 300 1500
-127. 54803 300 1500
-138. 61493 300 1500
-146. 30090 300 1500
-169. 73008 300 1500
871.78359 300 1500
-190. 08988 300 1500
-199. 08614 300 1500
-209. 81363 300 1500
-232. 04645 300 1500
-240. 60241 300 1500
-251. 65896 300 1500
(Con~nued)
516.14291 200 1500
582.37236 200 1500
2039.55839 50 1500
993.58809 300 1200
- 57. 32517 200 1500
1997.9072 200 1500
2453.5780 200 1500
2891.5695 200 1500
3296.6252 200 1500
3796.2346 200 1500
4090.6826 200 1500
2487.2522 200 1500
2764.1344 200 1500
3163.1228 200 1500
3585.1609 200 1500
3862.2068 200 1500
29069.9319 200 1500
4506.8347 200 1500
4366.8186 200 1500
No. Compound name Formula M
64 n-Tridecylcyclopentane C18H36 252.470
65 n-Tetradecylcyclopentane C19H38 266.490
66 n-Pent adecyl cycl opent ane C20H40 280.520
67 n-Hexadecyl cycl opent ane C21H42 294.550
68 Cyclohexane C6H12 84.161
69 Met hyl cycl ohexane C7H14 98.188
70 Et hyl cycl ohexane C8H16 112.215
71 n-Propylcyclohexane C9H18 126.243
72 n-Butylcyclohexane C10H20 140.270
73 n-Pentylcyclohexane CllH22 154.290
74 n-Hexylcyclohexane C12H24 168.310
75 n-Heptylcyclohexane C13H26 182.340
76 n-Octylcyclohexane C14H28 196.360
77 n-Nonylcycloi-iexane C15H30 210.390
78 n-Decylcyclohexane C16Ha2 224.420
79 n-Undecylcyclohexane C17H34 238.440
80 n-Dodecylcyclohexane C18H36 252.470
81 n-Tridecylcyclohexane C19H38 266.490
82 n-Tetradecylcyclohexane C20H40 280.520
83 n-Hexadecylcyclohexane C22H44 308.570
Ar omat i cs
84 Benzene C6H6 78.114
85 Toluene C7H8 92.141
86 Et hyl benzene C8H10 106.167
87 m-Xylene C8H10 106.167
88 o-Xylene C8H10 106.167
89 p-Xylene C8Hlo 106.167
90 n-Propyl benzene C9H12 120.195
91 n-But yl benzene CloH14 134.222
92 m-Cymene CloH14 134.222
93 o-Cymene CloH14 134.222
94 p-Cymene CloH14 134.222
95 n-Pent yl beuzene CIIH16 148.240
96 n-Hexyl benzene C12H18 162.260
97 n-Hept yl benzene C13H2o 176.290
98 n-Oct yl benzene C14H22 190.320
99 St yrene C8Hs 104.152
Di enes and acetylenes
100 Propadi ene C3H4 40.065
101 1, 2-Butadiene C4H6 54.092
102 Acetylene C2H2 26.038
Diaromatics
103 Nap ht hal ene CloH8 128.174
Nonhydr oc a r bons
104 Wat er H20 18.015
105 Carbon dioxide CO2 44.01
106 Hydrogen sulfide H2S 34.08
107 Ni t rogen N2 28.014
108 Oxygen 02 32
109 Ammoni a NH3 17.03
110 Car bon monoxi de CO 28.01
111 Hydrogen H2 2.016
112 Ni t rogen dioxide NO2 46.01
113 Ni t rous oxide NO 30.01
TABLE 6, 8- - ( Cont i nued)
A B xl03 C 106 D xl 0 l~ E Tmln, K Tm~, K
- 8. 82057 219.8119 - 133. 2056 360.15605 -260. 35782 300 1500
- 8. 81992 227.8540 - 139. 2730 374.44617 -268. 88152 300 1500
- 9. 29147 243.8142 -148. 2795 402.88398 -292. 64837 300 1500
- 9. 34807 254.9133 -154. 7201 419.05364 -303. 20007 300 1500
-7. 66115 77.46123 -31. 65303 - 45. 48807 456.29714 300 1500
-8. 75751 100.2054 - 62. 47659 169.33320 -123. 27361 300 1500
- 5. 50074 91.59292 - 26. 04906 -192. 84542 1021.80248 300 1500
- 8. 87526 124.6789 -76. 99183 180.70008 20.22888 300 1500
- 7. 38694 127.4674 - 67. 63120 73.28814 355.51905 300 1500
- 10. 16016 152.5757 - 98. 38009 265.14011 -106. 39559 300 1500
-9. 58825 161.8750 -104. 4133 302.29623 -236. 75537 300 1500
- 12. 53870 188.4588 -138. 5801 523.83412 -881. 68097 300 1500
-7. 88711 178.2886 -112. 8765 330.77533 -271. 09270 300 1500
-8. 48961 187.0067 - 105. 2157 192.47573 192.71532 300 1500
-10. 58196 209.8953 - 134. 2136 385.00443 -297. 79779 300 1500
- 9. 25980 214.8824 -131. 9175 357.79740 -261. 20128 300 1500
- 9. 94518 228.7293 -141. 7915 389.80571 -289. 05527 300 1500
-10. 06895 240.3258 -148. 9432 410.23509 -304. 86051 300 1500
- 10. 98687 255.5423 -161. 2184 455.82197 -347. 77976 300 1500
-8. 96825 268.1151 -159. 9818 417.54247 -292. 26560 300 1500
- 7. 29786 75.33056 - 69. 66390 336.46848 -660. 39655 300 1500
- 2. 46286 57.69575 - 19. 66557 -106. 61110 654.52596 200 1500
4.72510 9,02760 141.1887 -1989. 2347 8167.1805 50 1000
- 4. 00149 76,37388 -44. 21568 82.57499 90.13866 260 1500
- 1. 51679 68.03181 - 33. 61164 24.37900 206.82729 260 1500
- 4. 77265 80.94644 -51. 89215 136.1966 -45. 64845 260 1500
4.42447 33.21919 74.42459 -1045. 5561 3656.7834 50 1500
- 6. 24190 110.6923 -74. 17854 221.3160 -178. 64701 300 1500
- 4. 41825 103.1174 - 65. 46564 182.7512 -138. 15307 300 1500
- 2. 40242 96.87475 - 58. 63517 154.5568 -109. 45170 300 1500
- 4. 47668 102.5377 - 64. 61930 179.2371 -134. 70678 300 1500
- 6. 89760 124.5723 -84. 11348 251.1513 -201. 56517 300 1500
-7. 66975 139.1540 -95. 04913 285.7856 -230. 69678 300 1500
- 8. 36450 153.2807 -105. 2641 316.7510 -255. 85057 300 1500
-9. 35221 168.8057 -117. 4996 357.7099 -292. 04881 300 1500
-6. 20755 91.11255 - 83. 45606 411.3630 -842. 07179 300 1500
1.30128 23.37745 --13.57151 26.91489 26.81000 200 1500
3.43878 19.01555 11.36858 --212.98223 751.33700 50 1500
1.04693 21.20409 --29.08273 203.04028 -533. 31364 50 1500
--5.74112 86.70543 --46.55922 --1.47621 531.58512 200 1500
4.05852 --0.71473 2.68748 --11.97480 13.19231 50 1500
3.51821 --2.68807 31.88523 --499.2285 2410.9439 50 1000
4.07259 --1.43459 6.47044 --45.32724 103.38528 50 1500
3.58244 -0. 84375 2.09697 --10.19404 11.22372 50 1500
3.57079 -1. 18951 4.79615 --40.80219 110.40157 50 1500
0.98882 --0.68636 3.61604 --32.60481 96.53173 50 1500
3.56423 --0.78152 2.20313 --11.29291 13.00233 50 1500
3.24631 1.43467 --2.89398 25.8003 --73.9095 160 1220
3.38418 3.13875 3.98534 --58.69776 197.35202 200 1500
4.18495 --4.19791 9.45630 --72.74068 192.33738 50 1500
Tmin and Tm~x are approximated to nearest 10. Data have been determined from Method 7A1.2 given in the API-TDB [5].
( H - H ig) = - 5. 6167 x 8. 314 x ( 1/ 78. 1) x 562 = - 336 kJ / kg.
To us e SR K EOS, Eq. ( 5. 40) s houl d be us ed, whi c h gi ves
Z L = 0. 0304 a nd B = 0. 02142. F r om Tabl e 6. 1 ( H - Hi g ) / RT =
- 7. 438, whi c h gi ves ( H- H ig) = - 7. 438 x 8. 314 ( 1/ 78. 1) x
450 = - 356 kJ / kg. The di f f er ence wi t h t he ge ne r a l i z e d cor r e-
l a t i on i s a bout 6%. The ge ne r a l i z e d c or r e l a t i on gi ves mor e
a c c ur a t e r e s ul t t ha n a c ubi c EOS f or c a l c ul a t i on of e nt ha l p y
a nd he a t cap aci t y. #
6. 4 T HER MOD Y N A MI C PR OPER T I ES
OF MI X T UR ES
Thermodynami cs of mixtures also known as solution ther-
modynami cs is p a r t i c ul a r l y i mp or t a nt i n e s t i ma t i on of p r op -
er t i es of p e t r ol e um mi xt ur e s es p eci al l y i n r e l a t i on wi t h p ha s e
e qui l i br i um cal cul at i ons . I n t hi s s e c t i on we di s cus s p ar t i al
mol a r quant i t i es , c a l c ul a t i on of p r op e r t i e s of i deal a nd r eal
6 . 0
m
5.5
5.0
4.5
4.0
3.5
Eq. (6.66)
....... Eq. (6.70) J "
. . . . . . . , -'"
i i
0 5 0 0 1 0 0 0 1 5 0 0
T e mp e r a t u r e , K
FIG. 6. 2~ Pr edi ct i on of ideal g as heat capac-
ity of wat er from v ari ous methods.
solutions, and volume change due to mixing and blending of
petroleum mixtures.
6. 4. 1 Part i al Mol ar Pr ope r t i e s
Consider a homogeneous phase mixture of N components
at T and P with number of moles of nl, rt2,..., nN. A total
property is shown by M t where superscript t indicates total
(extensive) property and M can be any intensive thermody-
namic property (i.e., V, H, S, G). I n general from the phase
rule discussed in Chapter 5 we have
(6.75) M t = Mr(T, P, nl, n2, n3 . . . . . nN)
N
(6.76) n = ) ~ r~
i=1
M t
(6.77) M = - -
where n is the total number of moles and M is the molar
property of the mixture. Partial molar property of component
45
. . . . Lee- Kesler Met hod for Pentane
Pentane
Propane / -
30 " -
15
~ .. ........ -... _
Z' . ~ ...................
0 I I
0 5 0 0 1 0 0 0 1 5 0 0
T e mp e r a t u r e , K
FIG. 6. 3 ~ Pr edi ct i on of ideal g as heat capaci ty
of some hydrocarbons from Eq. (6.66) and Lee-
Kesl er met hod ( Eq. 6.7 2) .
i in a mixture is shown by ~/i and is defined as
(6.78) M~ = (O-~ )r,e,ni~ i
~/i indicates change in property M t per infinitesimal addi-
tion of component i at constant T, P, and amount of all other
species. This definition applies to any thermodynamic prop-
erty and ~/i is a function of T, P, and composition. Partial
molar volume (l?i) is useful in calculation of volume change
due to mixing for nonideal solutions, partial molar enthalpy
(/7i) is useful in calculation of heat of mixing, and Gi is par-
ticularly useful in calculation of fugacity and formulation of
phase equilibrium problems. The main application of partial
molar quantities is to calculate mixture property from the
following relation:
N
(6.79) Mt = E r//~/i
i=1
or on the molar basis we have
N
(6.80) M = E X4~/~i
i=1
where x4 is mole fraction of component i. Similar equations
apply to specific properties (quantity per unit mass) with re-
placing mole fraction by mass or weight fraction. I n such
cases/I)/i is called partial specific property.
Partial molar properties can be calculated from the knowl-
edge of relation between M and mole fraction at a given T
and P. One relation that is useful for calculation of Mi is
the Gibbs-Duhem (GD) equation. This equation is also a use-
fu] relation for obtaining a property of one component in a
mixture from properties of other components. This equation
can be derived by total differentiation of M t in Eq. (6.75) and
equating with total differential of M t from Eq. (6.79), which at
constant T and P can be reduced to the following simplified
form [11:
(6.81) Ex4d/ f/ i = 0 (at constant T, P)
i
This equation is the constant T and P version of the GD equa-
tion. As an example for a binary system (x2 = 1 - xl) we can
show that Eqs. (6.80) and (6.81) give the following relations
for calculation of/ f/ i:
dM
M1 = M +x 2 - -
(6.82) dxl
dM
M2 = M - xl dx~
Based on these relations it can be shown that when graphical
presentation of M versus xl is available, partial molar proper-
ties can be determined from the interceptions of the tangent
line (at xl) with the Y axis. As shown in Fig. 6.4 the intercep-
tion of tangent line at xl = 0 gives/f/2 and at xl = 1 gives/f/1
according to Eq. (6.82).
Example 6. 3--Based on the graphical data available on en-
thalpy of aqueous solution of sulfuric acid (H2SO4) [1], the
following relation for molar enthalpy of acid solution at 25~
is obtained:
H = 123.7 - 1084.4x~ + 1004.5x~1 - 1323.2x~3~ + 1273.7x41
e ~
2
"6
MI
M2
X 1
I
0 0.5 1.0
Mol e Fraction, Xl
FIG. 6,4~ Graphi cal method for calculation of partial
molar properties,
where H is the specific enthalpy of solution in kJ/kg and xwl
is the weight fraction of H2504. Calculate/ )1 and/ ) 2 for a
solution of 66.7 wt% sulfuric acid. Also calculate H for the
mixture from Eq. (6.78) and compare with the value from the
above empirical correlation.
Sol ut i on- - Eq uat i on (6.82) is used to calculate /)1 and/ ) 2.
By direct differentiation of H with respect to xwl we have
dH/ dxwl = -1084. 4 + 2009Xwl - 3969.6x21 + 5094.8x3~1 . At
xwl = 0.667 we calculate H = - 293. 3 kJ/kg and dH/ dxwl =
-1. 4075 kJ/kg. From Eq. (6.82) we have /)1 = - 293. 3+
(0.333) (1.4075) = -292. 8 and/ ) 2 -- -294. 2 kJ/kg. Substi-
tuting the values in Eq. (6.80) we get H(at xwl = 0.667) =
0.667 x ( -293.3) + 0.333 x ( -294.2) = -293. 3 kJ/kg, which
is the same value as obtained from the original relation for
H. Graphical calculation of partial specific enthalpies /)1
and/ ) 2 is shown in Fig. 6.5. The tangent line at xl = 0.667 is
almost horizontal and it gives equal values for/ ) 1 and/ ) 2 as
-295 kJ/kg.
6. 4. 2 Properties of Mixtures- - Property Change
Due to Mi xi ng
Calculation of properties of a mixture from properties of its
pure components really depends on the nature of the mixture.
100
0
-~ -100
-200
-300
-400
H2
I I I I I I I I I
0 0.2 0.4 0.6 0.8 1
H1
200
Weight Fraction H2SO 4
FIG. 6.5--Specific enthalpy of sulfuric acid solution
at 25~ (part of Example 6,3).
I n general the mixtures are divided into two groups of ideal
solutions and real solutions. An ideal solution is a homoge-
nous mixture in which all components (like and unlike) have
the same molecular size and intermolecular forces, while real
solutions have different molecular size and intermolecular
forces. This definition applies to both gas mixtures and liq-
uid mixtures likewise; however, the terms normally are ap-
plied to liquid solutions. Obviously all ideal gas mixtures are
ideal solutions but not all ideal solutions are ideal gas mix-
tures. Mixtures composed of similar components especially
with similar molecular size and chemical structure are gener-
ally ideal solutions. For example, benzene and toluene form
an ideal solution since both are aromatic hydrocarbons with
nearly similar molecular sizes. A mixture of polar component
with a nonpolar component (i.e., alcohol and hydrocarbon)
obviously forms a nonideal solution. Mixtures of hydrocar-
bons of low-molecular-weight hydrocarbons with very heavy
hydrocarbons (polar aromatics) cannot be considered ideal
solutions. If molar property of an ideal solution is shown by
M id and real solution by M the difference is called excess prop-
erty shown by M E
(6.83) M E = M - M id
M E is a property that shows nonideality of the solution and it
is zero for ideal solutions. All thermodynamic relations that
are developed for M also apply to M E as well. Another im-
portant quantity is property change due to mixing which is
defined as
(6.84) AMmix = M - Ex i M i = Ex i ( f f / l i - M i )
i i
During mixing it is assumed that temperature and pressure
remain constant. From the first law it is clear that at constant
T and P, the heat of mixing is equal to AHmix, therefore
(6.85) Heat of mixing = AHmix = ~x/ ( / r t i -/ -/ / )
i
Similarly the volume change due to mixing is given by the
following relation:
Volume change due to mixing = AVmix ---- ~ x4 ( ~ - V/)
i
(6.86)
where H/ and V/are molar enthalpy and volume of pure com-
ponents at T and P of the mixture. For ideal solutions both
the heat of mixing and the volume change due to the mixing
are zero [19]. This means that in an ideal solution, partial
molar volume of component i in the mixture is the same as
pure component specific volume (~k~ri = V/ ) and 17i nor/ ) i vary
with composition. Figure 6.6 shows variation of molar vol-
ume of binary mixture with mole fraction for both a real and
an ideal solution (dotted line) for two cases. I n Fig. 6.6a the
real solution shows positive deviation, while in Fig. 6.6b the
solution shows negative deviation from ideal solution. Sys-
tems with positive deviation from ideality have an increase in
volume due to mixing, while systems with negative deviation
have decrease in volume upon mixing.
Equations (6.85) and (6.86) are useful when pure compo-
nents are mixed to form a solution. If two solutions are mixed
then the volume change due to mixing can be calculated from
>
V2
ideal
Xl
0 Mole Fraction, xl
(a) Systems with increase in volume due to mixing
Vl
Ex ampl e 6. 4 For the mixture of Exampl e 6.3 calculate the
heat of mixing at 25~
Sol ut i on- - Heat of mixing is calculated from Eq. (6.85) using
values of/ ~i and/ 42 calculated in Exampl e 6.3 as -292. 8 and
-294. 2 kJ/kg, respectively. Pure component s/ / 1 and H2 are
calculated from the correlation given for H in Exampl e 6.3 at
xl = 1 (for H1) andxt = 0( for H2)as H1 = - 5. 7kg/ kJ and H2 =
123.7 kJ/kg. From Eq. (6.85), AHr,~x = (0.667) [ ( -292. 8) -
( -5. 7) ] + (1 - 0.667) x [ ( -294. 2) - (123.7)] = -330. 7 kJ/kg.
This means that to make 1 kg of solution of 66.7 wt% sul-
furic acid at 25~ 330.7 kJ heat will be released, t
>
>
I
I
X I
Mole Fraction, xl
Vl
(b) Systems with decrease in volume due to mixing
FI G. 6. 6- - V ari at i on of mol ar v ol ume of a bi nary mi x ture wi th
composi ti on.
the following relation [ 17]:
/~Vtmixing = ~ F//,afterV/(r, P,/q-/,after) -- E Ft/'bef~ (T, P, t't/,before)
i i
(6.87)
where r~,before is the moles of i before mixing and r~,aner rep-
resents moles of i in the solution after the mixing. Obviously
since the mi xt ure composition before and after the mixing are
not the same, 17i for i in the solution before the mixing and its
value for i in the solution after the mixing are not the same.
The same equation may be applied to enthalpy by replacing
H with V to calculate heat of mixing when two solutions are
mixed at constant T and P. Partial mol ar volume and enthalpy
may be calculated from their definition, Eq. (6.78) t hrough an
EOS. For example in deriving the relation for l?i, derivative
[O(nV)/~Yli]r,p,nj~ i should be det ermi ned from the EOS. For
the PR EOS the partial mol ar vol ume is given as [20]
(6.88) 17,i _ X~ + X2
X3+X4
where
Xt = ( RT + biV ) x (V 2 + 2bV - b 2)
= [2bi RT - 2 ~i xiaii - 2bi P ( V - b)] x (V - b) + X2 bia
Xa = n (V 2 + 2bV - b 2) + 2P (V - b) (V + b)
X 4 = - 2 RT ( V + b) + a
where V is the mixture mol ar volume calculated from PR
EOS. For more accurate calculation of ffi, corrected V
t hrough use of volume translation concept (Eq. 5.50) may
be used. Similar relation f or / }i can be obtained (see Prob-
l em 6.5).
( 6. 90) V id
(6.91)
(6.92)
For the ideal solutions, H, V, G, and S of the mixture may
be calculated from pure component properties t hrough the
following relations [1, 21 ]:
(6.89) H id = ~x/ H/
i
= ~-~ x~ V/
i
G ia = ~_ xi Gi + RT ~_ xi lnxi
i i
sid --- ~-~ xiSi - R E xi l nx i
i i
where H ia, V id, Gig, and ~g can be either mol ar or specific
enthalpy, volume, Gibbs energy, and ent ropy of mixture. I n
case of specific property, x4 is weight fraction. For example, if
V is specific volume ( = 1/ p) , Eq. (6.90) can be written in the
following form for density:
(6.93) P
where x~/ i s the weight fraction of i and pi is the density of
pure i. This equation was previously introduced in Chapter 3
(Eq. 3.45). Although all hydrocarbon mixtures do not really
behave like ideal solutions, mixtures that do not contain non-
hydrocarbons or very heavy hydrocarbons, may be assumed
as ideal solutions. For simplicity, application of Eqs. (6.89)
and (6.90) is extended to many t hermodynami c properties as
it was shown in Chapters 3 and 4. Mixture heat capacity, for
example, is calculated similar to enthalpy as:
(6.94) Cv = E xiCpi
i
where xi is either mole or mass fraction depending on the unit
of Cp. I f Cp is the specific heat (i.e., J/ g. ~ weight fraction
should be used for x~. Obviously the mai n application of these
equations is when values of properties of pure component s
are available. For cases that these properties are predicted
from equations of state or other correlations, the mixing rules
are usually applied to critical properties and the input param-
eters of an EOS rat her t han to calculated values of a t hermo-
dynami c propert y in order to reduce the time and complexity
of calculations. For hy drocarbon mixtures that contain very
light and very heavy hydrocarbons the assumpt i on of ideal so-
lution and application of Eqs. (6.89)-(6.93) will not give accu-
rate results. For such mixtures some correction t erms to con-
sider the effects of nonideality of the system and the change in
molecular behavior in presence of unlike molecules should be
added to the RHS of such equations. The following empirical
6. THERM ODYNAM IC RELATIONS FOR PROPERTY ESTIM ATIONS 251
met hod r ecommended by the API for cal cul at i on of vol ume
of pet rol eum blends is based on such t heory [5].
6. 5 P H A S E EQUI L I B R I A OF P UR E
C OMPON EN T S - - - C ON C EPT
OF S A T UR A T I ON P R ES S UR E
6. 4. 3 Vol ume of Pe t r ol e um Bl e nds
One of t he appl i cat i ons of partial mol ar vol ume is to calculate
vol ume change due to mixing as shown by Eq. (6.86). How-
ever, for pract i cal applications a si mpl er empi ri cal met hod
has been developed for cal cul at i on of vol ume change when
pet rol eum product s are blended.
Consi der t wo liquid hydr ocar bons or t wo different petrol-
eum fract i ons ( products) whi ch are bei ng mi xed to pr oduce
a bl end of desired characteristics. I f the mi xt ure is an ideal
solution, vol ume of the mi xt ure is simply the sum of vol-
umes of t he comp onent s before t he mixing. This is equiva-
lent t o "no vol ume change due t o mixing." Experi ence shows
t hat when a l ow-mol ecul ar-wei ght hydr ocar bon is added to a
heavy mol ecul ar wei ght crude oil t here is a shri nkage in vol-
ume. This is part i cul arl y t he case when a crude oil API grav-
ity is i mproved by addi t i on of light product s such as gasoline
or lighter hydr ocar bons (i.e., but ane, propane) . Assume the
vol ume of light and heavy hydr ocar bons before mixing are
Vlight and Vh~,,y, respectively. The vol ume of the bl end is t hen
calculated from the following relation [5]:
Vblend = Vheavy - I re~ight(1 -- S)
S = 2.14 10-5C-~176176
(6.95) G = APIlight -- APIh~vy
C = vol% of light comp onent in t he mi xt ure
where S is called shri nkage fact or and G is t he API gravity
difference bet ween light and heavy component . The amount
of shri nkage of light comp onent due to mixing is Vlight ( 1 - S).
The following example shows appl i cat i on of this met hod.
As di scussed in Sect i on 5.2, a pure subst ance may exist in
a solid, liquid or vapor phases (i.e., see Fig. 5.1). For pure
subst ances four types of equi l i bri um exist: vapor-l i qui d (VL),
vapor-sol i d (VS), l i qui d-sol i d (LS) and vapor-l i qui d-sol i d
(VLS) phases. As shown in Fig. 5.2 the VLS equi l i bri um oc-
curs onl y at the triple point, while VL, VS, and LS equi l i bri um
exist over a range of t emperat ure and pressure. One i mpor t ant
type of phase equilibria in t he t her modynami cs of pet rol eum
fluids is vapor-liquid equilibria (VLE). The VLE line also called
vapor pressure curve for a pure subst ance begins from triple
poi nt and ends at t he critical poi nt (Fig. 5.2a). The equilib-
r i um curves bet ween solid and liquid is called fusion line and
bet ween vapor and solid is called sublimation line. Now we
formul at e VLE; however, the same ap p r oach may be used to
formul at e any type of mul t i phase equilibria for single com-
p onent systems.
Consi der vapor and liquid phases of a subst ance coexist in
equi l i bri um at T and P (Fig. 6.7a). The pressure is called sat-
urat i on pressure or vapor pressure and is shown by psat. AS
shown in Fig. 5.2a, vapor pressure increases wi t h t emperat ure
and the critical point, normal boiling poi nt and triple poi nt
are all located on the vapor pressure curve. As was shown in
Fig. 2.1, for hydr ocar bons t he rat i o Tb/Tc known as reduced
boiling point varies from 0.6 t o mor e t han one for very heavy
compounds. While the triple poi nt t emperat ure is al most the
same as t he freezing poi nt t emperat ure, but the triple poi nt
pressure is much l ower t han at mospheri c pressure at whi ch
Example 6. 5--Cal cul at e vol ume of a bl end and its API gravity
p r oduced by addi t i on of 10000 bbl of light nap ht ha wi t h API
gravity of 90 t o 90000 bbl of a crude oil wi t h API gravity of 30.
Solution--Equation (6.95) is used to calculate vol ume of
blend. The vol% of light comp onent is 10% so C = 10. G =
90 - 30 = 60. S = 2.14 10 -5 x (10 -0"0704) 60176 = 0.025.
VBlend = 90000 + 10000(1 -- 0.025) = 99750 bbl. The amount
of shri nkage of nap ht ha is 10000 x 0.025 = 250 bbl. As can
be seen from Eq. (6.95) as t he difference bet ween densities
of t wo comp onent s reduces t he amount of shri nkage also
decreases and for t wo oils wi t h the same density t here is
no shrinkage. The percent shri nkage is 100S or 2.5% in this
example. I t shoul d be not ed t hat for cal cul at i on of density
of mi xt ures a new composi t i on shoul d be calculated as:
x~ = 9750/ 99750 = 0.0977 whi ch is equivalent t o 9.77% in-
st ead of 10% originally assumed. For this exampl e the mi xt ure
API gravity is calculated as: SGL = 0.6388 and SGH = 0.8762
where L and H refer to light and heavy component s. Now
usi ng Eq. (3.45): SGBlend = (1 -- 0.0977) 0.8762 + 0.0977 x
0.6388 = 0.853 whi ch gives API gravity of bl end as 34.4 while
direct appl i cat i on of mixing rule to the API gravity wi t h orig-
inal composi t i on gives APIBl~nd = (1 -- 0.1) X 30 + 0.1 90 =
36. Obviously t he mor e accurat e value for the API gravity of
bl end is 34.4. #
Vapor (V)
at T, psat
fV(T psat)= fL(T psat)
a. Pure Component System
Vapor
at T, p~t, Yi
fi v (T, psat, Yi) = fi L (T, psat, Xi )
i!iii!iii!iiiiiiililililililililiiiiiiiiiiiiiii
iiiiiiiiii!iii!i!ili i iiiililiiiiii!iii!i
!: !: !: !: !: !: !: !: ~ i ~ !l L~ .: ~ : i i !: i : !: i : ~ : i : i : ~
b. Multi Component System
FIG. 6. 7 --General criteria for
vapor-liquid equilibrium.
freezi ng occurs. When a syst em is in equi l i br i um its ener gy is
in mi ni mum level ( dG = 0), whi ch for a syst em wi t h onl y va-
p or and l i qui d is d( G v - G L) = 0, whi ch can be wr i t t en as [1]:
(6.96) dGV(T, psat) = dGL(T, psat)
wher e psat i ndi cat es t hat t he r el at i on is val i d at t he sat ur at i on
t emp er at ur e and pr essur e. Si mi l ar equat i on appl i es t o sol i d-
l i qui d or s ol i d- vap or phases.
Dur i ng a p hase change (i.e., vap or to l i qui d or vice versa),
t emp er at ur e and p r essur e of t he syst em r emai n const ant and
t her ef or e f r om Eq. (6.5) we have:
A/-yap
(6.97) AS yap - -
T
wher e AH v~p is heat of vaporization and AS yap is t he entropy
of vaporization. A/-/yap is defi ned as:
AH yap ---- H(T, psat, sat ur at ed vapor )
(6.98) - H(T, psat, sat ur at ed l i qui d)
Si mi l ar l y AS wp and AV v~p ar e defi ned. F or a p hase change
f r om sol i d to l i qui d i nst ead of heat of vap or i zat i on A/ -F ap,
heat of fusi on or mel t i ng A H fu~ is defi ned by t he di fference be-
t ween ent hal p y of sat ur at ed l i qui d and solid. Si nce ps~t is onl y
a f unct i on of t emp er at ur e, AS yap and Avvap ar e al so f unct i ons
of t emp er at ur e onl y for any p ur e subst ance. AH yap and AW ap
decr ease wi t h i ncr ease i n t emp er at ur e and at t he cri t i cal p oi nt
t hey ap p r oach zero as vap or and l i qui d p hases become i den-
tical. Whi l e AV yap can be cal cul at ed f r om an equat i on of st at e
as was di scussed i n Chap t er 5, met hods of cal cul at i on A/-/vap
will be di scussed i n Chap t er 7. By ap p l yi ng Eq. (6.8) to bot h
dG v and dG L and use of Eqs. (6.97) and (6.98) t he fol l owi ng
r el at i on known as Clapeyron equation can be deri ved:
dpsat A/-/yaP
(6.99)
dT TAV yap
This equat i on is t he basi s of devel op ment of pr edi ct i ve met h-
ods for vap or p r essur e versus t emp er at ur e. Now t hr ee si m-
pl i fyi ng as s ump t i ons ar e made: ( I ) over a nar r ow r ange of
t emp er at ur e, A/-/vap is const ant , (2) vol ume of l i qui d is smal l
in comp ar i s on wi t h vap or vol ume ( AV yap -~-- V v -- V L ~'~ Vv) ,
and (3) vol ume of vap or can be cal cul at ed f r om i deal gas l aw
( Eq. 5.14). These assump t i ons ar e not t rue in gener al but at
a nar r ow r ange of t emp er at ur e and l ow p r essur e condi t i ons
t hey can be used for si mpl i ci t y. Upon ap p l i cat i on of assump -
t i ons 2 and 3, Eq. (6.99) can be wr i t t en i n t he fol l owi ng f or m
known as Clausius-Clapeyron equation:
d I n psat aHvap
(6.100)
d( 1/ T) R
wher e R is t he uni ver sal gas const ant . This equat i on is t he
basi s of devel op ment of si mp l e cor r el at i ons for est i mat i on
of vap or p r essur e versus t emp er at ur e or cal cul at i on of heat
of vap or i zat i on f r om vap or p r essur e dat a. F or exampl e, by
usi ng t he first as s ump t i on ( const ant AH v~p) and i nt egr at i ng
t he above equat i on we get
B
(6.101) I n psat ~ A - --
T
wher e T is absol ut e t emp er at ur e and A and B ar e t wo posi t i ve
const ant s specific for each p ur e subst ance. Thi s equat i on sug-
gest s t hat I n pv~p versus 1/T is a st r ai ght l i ne wi t h sl ope of - B.
Const ant B is in fact same as AI-FaO/R. Because of t hr ee ma-
j or si mpl i f yi ng as s ump t i ons made above, Eq. (6.101) is very
ap p r oxi mat e and it may be used over a nar r ow t emp er at ur e
r ange when mi ni mum dat a ar e avai l abl e. Const ant s A and
B can be det er mi ned f r om mi ni mum t wo dat a p oi nt s on t he
vap or p r essur e curve. Usual l y t he cri t i cal p oi nt (To Pc) and
nor mal boi l i ng p oi nt (1.01325 bar and Tb) ar e used to obt ai n
t he const ant s. I f A/-/yap is known, t hen onl y one dat a p oi nt (Tb)
woul d be suffi ci ent to obt ai n t he vap or p r essur e cor r el at i on.
A mor e accur at e vap or p r essur e cor r el at i on is t he fol l owi ng
t hr ee- const ant cor r el at i on known as Ant oi ne equat i on:
(6.102) I n psat = A - - -
B
T+C
A, B, and C, known as Ant oi ne const ant s, have been det er-
mi ned for a l arge number of comp ounds. Ant oi ne p r op os ed
t hi s si mp l e modi f i cat i on of t he Cl asi us- Cl ap eyr on equat i on
i n 1888. Vari ous modi f i cat i ons of t hi s equat i on and ot her cor-
r el at i ons for est i mat i on of vap or p r essur e ar e di scussed in t he
next chapt er.
Exampl e 6. 6- - F or p ur e water, est i mat e vap or p r essur e of wa-
t er at 151.84~ What is its heat of vap or i zat i on? The act ual
val ues as gi ven in t he st eam t abl es are 5 bar and 2101.6 kJ/ kg,
respect i vel y [1]. Assume t hat onl y Tb, To, and Pc ar e known.
Sol ut i on- - From Table 2.1 for wat er we have Tc = 647.3 K,
Pc = 220.55 bar, and Tb = I00~ Appl yi ng Eq. (6.101) at t he
cri t i cal p oi nt and nor mal boi l i ng p oi nt gives lnPc = A - B~ Tc
and I n (1.01325) ~- A - B/Tb. Si mul t aneous sol ut i on of t hese
equat i ons gives t he fol l owi ng r el at i ons to cal cul at e A and B
f r om Tb, To, and Pc.
l n( J ~- )
1.01325
B= 1 1
(6.103) rb r~
B
A = 0.013163 +
wher e Tc and Tb mus t be in kel vi n and Pc mus t be i n bar.
The same uni t s mus t be used in Eq. (6.101). I n cases t hat
a val ue of vap or p r essur e at one t emp er at ur e is known it
shoul d be used i nst ead of Tc and Pc so t he r esul t i ng equa-
t i on will be mor e accur at e bet ween t hat p oi nt and t he boi l i ng
poi nt . As t he di fference bet ween t emp er at ur es of t wo refer-
ence p oi nt s used to obt ai n const ant s in Eq. (6.101) reduces,
t he accur acy of r esul t i ng equat i on for t he vap or p r essur e be-
t ween t wo r ef er ence t emp er at ur es i ncreases. F or wat er f r om
Eq. (6.103), A = 12.7276 bar and B = 4745.66 bar . K. Subst i -
t ut i ng A and B in Eq. (6.101) at T = 151.84 + 273.15 = 425 K
gives I n P = 1.5611 or P = 4.764 bar. Comp ar i ng p r edi ct ed
val ue wi t h t he act ual val ue of 5 bar gives an er r or of - 4. 9%,
whi ch is accep t abl e consi der i ng si mpl e r el at i on and mi n-
i mum dat a used. Heat of vap or i zat i on is cal cul at ed as
follows: AH yap = R B = 8.314 x 4745.66 = 39455.4 J / t ool =
39455.4/ 18 = 2192 kJ/ kg. This val ue gives an er r or of +4. 3%.
Obvi ousl y mor e accur at e met hod of est i mat i on of heat of va-
p or i zat i on is t hr ough A/ -F ap = H sat,yap - n sat,liq, wher e H sat,yap
a nd H sat'liq can be cal cul at ed t hr ough gener al i zed correl a-
t i ons. Emp i r i cal met hods of cal cul at i on of heat of vap or i za-
t i on ar e gi ven i n Chap t er 7.
An al t er nat i ve met hod for f or mul at i on of VLE of p ur e sub-
st ances is to combi ne Eqs. (6.47) and (6.96), whi ch gives t he
fol l owi ng r el at i on i n t er ms of fugaci t y:
(6.104) f v = fL
wher e f v and fL ar e fugaci t y of a p ur e subst ance i n vap or
and l i qui d p hases at T and psat. Obvi ousl y for sol i d- l i qui d
equi l i br i um, sup er scr i p t V i n t he above r el at i on is r ep l aced
by S i ndi cat i ng fugaci t y of sol i d is t he same as fugaci t y of
l i qui d. Si nce at VLE p r essur e of bot h p hases is t he same, an
al t ernat i ve f or m of Eq. (6.104) is
(6.105) ~bV(T, psat) = ~bL(T, psat)
An equat i on of st at e or gener al i zed cor r el at i on may be used
to cal cul at e bot h s v and ~L if T and ps~t ar e known. To cal-
cul at e vap or p r essur e (ps~t) f r om t he above equat i on a t ri al -
and- er r or p r ocedur e is r equi r ed. Value of ps~t cal cul at ed f r om
Eq. (6.101) may be used as an i ni t i al guess. To t er mi nat e cal-
cul at i ons an er r or p ar amet er can be defi ned as
e = 1 - (6.106)
when e is less t han a smal l val ue (i.e., 10 -6) cal cul at i ons may
be st opped. I n each r ound of cal cul at i ons a new guess for pres-
sure may be cal cul at ed as follows: pnew = pold(~bL/~bv). The
fol l owi ng examp l e shows t he p r ocedur e.
Ex ampl e 6 . 7 - - Repeat Examp l e 6.6 usi ng Eq. (6.105) and t he
SRK EOS t o est i mat e vap or p r essur e of wat er at 151.84~
Also cal cul at e V L and V v at t hi s t emp er at ur e.
Sol ut i on- - For wat er Tc = 647.3 K, Pc = 220.55 bar, ~o =
0.3449, and ZRX = 0.2338. Usi ng t he uni t s of bar, cm3/ mol,
and kel vi n for P, V, and T wi t h R = 83.14 cm 3 . ba r / mol - K
and T = 423 K, SRK p ar amet er s ar e cal cul at ed usi ng
r el at i ons gi ven i n Tables 5.1 and 6.1 as follows: ac =
5.6136 x 10 -6 bar ( cma/ mol ) 2, a = 1.4163, a = 7.9504 x 10 -6
bar ( cma/ mol ) 2, b = 21.1 cma/ mol , A = 0.030971, and B=
0.00291. Rel at i on for cal cul at i on of ~b for SRK is gi ven i n
Table 6.1 as follows: l n$= Z- 1- I n( Z- B) + Aln(z--~B),
wher e Z for bot h sat ur at ed l i qui d and vap or is cal cul at ed
f r om sol ut i on of cubi c equat i on ( SRK EOS) : Z 3 - Z 2 + (A -
B - B2) Z - AB = 0. The first i ni t i al guess is to use t he val ue
of P cal cul at ed i n Examp l e 6.6 f r om Eq. (6.101): P = 4.8
bar, whi ch resul t s in e = 1.28 x 10 -2 ( from Eq. (6.106)) as
shown in Table 6.9. The second guess for P is cal cul at ed
as P = 4.86 x ( 0.9848/ 0.97235) = 4.86, whi ch gives a l ower
val ue for e. Summa r y of resul t s is shown i n Table 6.9. The fi-
nal answer is psat = 4.8637 bar, whi ch di ffers by - 2. 7% f r om
t he act ual val ue of 5 bar. Val ues of specific vol umes of l i qui d
and vap or ar e cal cul at ed f r om Z L and ZV: Z L = 0.003699 and
Z v = 0.971211. Mol ar vol ume is cal cul at ed f r om V = ZRT/ P,
wher e R = 83.14, T = 425 K, and P = 4.86 bar. V L = 26.9
and V v = 7055.8 cm3/ mol. The vol ume t r ansl at i on p ar ame-
t er c is cal cul at ed f r om Eq. (5.51) as c = 6.03 cma/ mol , whi ch
t hr ough use of Eq. (6.50) gives V L = 20.84 and V v = 7049.77
cm3/ mol. The specific vol ume is cal cul at ed as V ( mol ar) / M ,
wher e for wat er M = 18. Thus, V L = 1.158 and V v = 391.653
cma/ g. Act ual val ues of V L and V v ar e 1.093 and 374.7 cm3/ g,
respect i vel y [1]. The er r or s for cal cul at ed V L and V v ar e +5. 9
and +4. 6%, respectively. For a cubi c EOS t hese er r or s ar e ac-
cept abl e, al t hough wi t hout cor r ect i on f act or by vol ume t r ans-
l at i on t he er r or for V L is 36.7%. However, for cal cul at i on of
vol ume t r ansl at i on a f our t h par amet er , namel y Racket pa-
r amet er is requi red. I t is i mp or t ant t o not e t hat i n cal cul at i on
of fugaci t y coeffi ci ent s t hr ough a cubi c EOS use of vol ume
t r ansl at i on, c, for bot h vap or and l i qui d does not affect re-
sul t s of vap or p r essur e cal cul at i on f r om Eq. (6.105). Thi s has
been shown in var i ous sour ces [20]. r
Equat i on (6.105) is t he basi s of det er mi nat i on of EOS pa-
r amet er s f r om vap or p r essur e dat a. For exampl e, coeffi ci ent s
gi ven in Table 5.8 for t he LK EOS ( Eqs. 5. 109-5. 111) or t he
f~ r el at i ons for var i ous cubi c EOSs gi ven in Table 5.1 wer e
f ound by mat chi ng p r edi ct ed psat and sat ur at ed l i qui d den-
si t y wi t h t he exp er i ment al dat a for p ur e subst ances for each
equat i on.
The s ame pr i nci pl e may be ap p l i ed to any t wo- p hase sys-
t em in equi l i br i um, such as VSE or SLE, i n or der t o deri ve
a r el at i on bet ween sat ur at i on p r essur e and t emp er at ur e. F or
exampl e, by ap p l yi ng Eq. (6.96) for sol i d and vap or phases,
a r el at i on for vap or p r essur e curve for subl i mat i on (i.e., see
Fig. 5.2a) can be deri ved. The final r esul t i ng equat i on is si m-
i l ar to Eq. (6.101), wher e p ar amet er B is equal t o AHsub/ R
in whi ch AH sub is t he heat of subl i mat i on in J / mol as shown
by Eq. (7.27). Then A and B can be det er mi ned by havi ng
t wo p oi nt s on t he subl i mat i on curve. One of t hese p oi nt s is
t he t r i pl e p oi nt (Fig. 5.2a) as di scussed in Sect i on 7.3.4. The
same ap p r oach can be ap p l i ed to SLE and deri ve a r el at i on
for mel t i ng ( or freezi ng) p oi nt l i ne ( see Fig. 5.2a) of p ur e com-
ponent s. Thi s is shown in t he fol l owi ng exampl e.
Ex ampl e 6. 8- - Ef f ect of pressure on the melting poi nt : Deri ve a
gener al r el at i on for mel t i ng p oi nt of p ur e comp onent s versus
p r essur e in t er ms of heat of mel t i ng ( or fusi on) , AH u, and
vol ume change due t o mel t i ng AV M, assumi ng bot h of t hese
p r op er t i es are const ant wi t h r esp ect to t emp er at ur e. Use t hi s
equat i on t o p r edi ct
a. mel t i ng p oi nt of n- oct adecane (n-C18) at 300 bar and
b. t ri pl e p oi nt t emp er at ur e.
The fol l owi ng dat a ar e avai l abl e f r om DI PPR dat a bank [13]:
Nor mal mel t i ng poi nt , TMO= 28.2~ heat of mel t i ng at nor-
mal mel t i ng poi nt , AHM= 242.4597 kJ/ kg; l i qui d densi t y
at Tuo, p L= 0.7755 g/cm3; sol i d densi t y at TM, pS = 0.8634
g/cm3; and t ri pl e p oi nt pr essur e, Pt p = 3.39 x 10 -5 kPa.
TABLE 6. 9--Estimation of vapor pressure of water at 151.8~ from
SRK EOS (Example 6.7).
P, bar Z L Z v ~L ~bV E
4.8 0.00365 0.9716 0.9848 0.97235 1.2 x 10 -2
4.86 0.003696 0.9712 0.9727 0.972 7.4 x 10 -4
4.8637 0.003699 0.971211 0.971981 0.971982 1.1 x 10 -6
Sol ut i on- - To deri ve a gener al r el at i on for sat ur at i on pres-
sure versus t emp er at ur e for mel t i ng/ f r eezi ng p oi nt of p ur e
comp ounds we st ar t by ap p l yi ng Eq. (6.96) bet ween sol i d and
l i qui d. Then Eq. (6.99) can be wr i t t en as
dP AH M
dTM - TMAV M
where TM is the melting point temperature at pressure P. I f
atmospheric pressure is shown by po (1.01325 bar) and the
melting point at P~ is shown by TMo (normal melting point),
integration of the above equation from po to pressure P gives
[ AV M x( P- Po)]
(6.107) TM = TMo exp [_ AH M
where in deriving this equation it is assumed that both AV M
and AH M are constants with respect to temperature (melting
point). This is a reasonable assumption since variation of TM
with pressure is small (see Fig. 5.2a). Since this equation is
derived for pure substances, TM is the same as freezing point
(Tf) and AH M is the same as heat of fusion (AHf).
(a) To calculate melting point of n-C18 at 300 bar, we have
P = 300 bar, Po = 1.01325 bar, TMo = 301.4 K, and AV M =
1/p v - Up s = 0.I313 cm3/g. AH M = 242.4597 J/g, I/J = 10
bar. cm 3, thus from Eq. (6.107) we have TM = 301.4 x exp
[0.1313 x (300 -- 1.013)/10 x 242459.7] = 301.4 x 1.0163 =
301.4 x 1.0163 = 306.3 K or TM = 33.2~ This indicates
that when pressure increases to 300 bar, the melting point
of n-C18 increases only by 5~ I n this temperature range as-
sumption of constant AV M and AH M is quite reasonable.
(b) To calculate the triple point temperature, Eq. (6.107)
must be applied at P= Pt p = 3. 39x10 -5 kPa = 3. 39x
10 -7 bar. This is a very low number in comparison
with Po = 1 bar, thus TM = 301.4 x exp(--0.1313 x 1.013/10 x
242459.7) = 301.4 x 0.99995 ~ 301.4 K. Thus, we get triple
point temperature same as melting point. This is true for most
of pure substances as Ptp is very small. It should be noted that
Eq. (6.107) is not reliable to calcnlate pressure at which melt-
ing point is known because a small change in temperature
causes significant change in pressure. This example explains
why melting point of water decreases while for n-octadecane
it increases with increase in pressure. As it is shown in Sec-
tion 7.2 density of ice is less than water, thus AV i for water
is negative and from Eq. (6.107), TM is less than TMo at high
pressures. #
6. 6 PHA S E EQUI LI BRI A OF
MI XTURES - - CALCULATI ON
OF BASI C PR OPER T I ES
Perhaps one of the biggest applications of equations of state
and thermodynamics of mixtures in the petroleum science
is formulation of phase equilibrium problems. I n petroleum
production phase equilibria calculations lead to the determi-
nation of the composition and amount of oil and gas produced
at the surface facilities in the production sites, PT diagrams to
determine type of hydrocarbon phases in the reservoirs, solu-
bility of oil in water and water in oils, compositions of oil and
gas where they are in equilibrium, solubility of solids in oils,
and solid deposition (wax and asphaltene) or hydrate forma-
tion due to change in composition or T and P. I n petroleum
processing phase equilibria calculations lead to the determi-
nation of vapor pressure and equilibrium curves needed for
design and operation of distillation, absorption, and stripping
columns.
A system is at equilibrium when there is no tendency to
change. I n fact for a multicomponent system of single phase
to be in equilibrium, there must be no change in T, P, and
xl, x2 . . . . . x~c-t. When several phases exist together while at
equilibrium similar criteria must apply to every phase. I n this
case every phase has different composition but all have the
same T and P. We know for mechanical equilibrium, total
energy (i.e., kinetic and potential) of the system must be min-
imum. The best example is oscillation of hanging object that
it comes to rest when its potential and kinetic energies are
mi ni mum at the lowest level. For thermodynamic equilib-
rium the criterion is mi ni mum Gibbs free energy. As shown
by Eq. (6.73) a mixture molar property such as G varies with
T and P and composition. A mathematical function is mini-
mum when its total derivative is zero:
(6.108) dG(T, P,x~) = 0
Schematic and criteria for VLE of multicomponent systems
are shown in Fig. 6.7b. Phase equilibria calculations lead to
determination of the conditions of T, P, and composition at
which the above criteria are satisfied. I n this section general
formulas for phase equilibria calculations of mixtures are pre-
sented. These are required to define new parameters such as
activity, activity coefficient, and fugacity coefficient of a com-
ponent in a mixture. Two main references for thermodynam-
ics of mixtures in relation with equilibrium are Denbigh [19]
and Prausnitz et al. [21 ].
6. 6. 1 Defi ni ti on of Fugacity, Fugaci ty
Coefficient, Activity, Activity Coefficient,
and Chemi cal Pot ent i al
I n this section important properties of fugacity, activity, and
chemical potential needed for formulation of solution ther-
modynamics are defined and methods of their calculation are
presented. Consider a mixture of N components at T and P
and composition yi. Fugacity of component i in the mixture
is shown bye- and defined as
(6.109) lim ( f-~p ~ ~I
\Yi }P-~O
where sign ^ indicates the fact that component i is in a mix-
ture. When yi --~ 1 we have f --~ f , where f is fugacity of
pure i as defined in Eq. (6.45). The fugacity coefficient of i in
a mixture is defined as
(6.110) q~i ~ f ~/
YiP
where for an ideal gas, ~i = 1 or f = Yi P. I n a gas mixture
yi P is the same as partial pressure of component i. Activity of
component i, di, is defined as
f
(6.111) ai = ~/o
where ff is fugacity of i at a standard state. One common
standard state for fugacity is pure component i at the same
T and P of mixture, that is to assume f/~ = f , where f. is the
fugacity of pure i at T and P of mixture. This is usually known
as standard state base on Lewis rule. Choice of standard state
for fugacity and chemical potential is best discussed by Den-
bigh [I 9]. Activity is a parameter that indicates the degree of
nonideality in the system. The activity coefficient of compo-
nent i in a mixture is shown by yi and is defined as
ai
(6.112) Fi = --
Yq
6. THERM ODYNAM I C RELATI ONS FOR PROPERTY ESTI M ATI ONS 255
where x/ i s mol e fract i on of comp onent i in the mixture. Bot h
di and yi are di mensi onl ess paramet ers. With t he above defini-
t i ons one may cal cul at e~ f r om one of the following relations:
(6.113) ~ = ~/yiP
(6.114) ~ = xi)'i fi
Al t hough generally ~i and ~,/ are defined for any phase, but
usually ~i is used t o calculate fugacity of i in a gas mi xt ure
and Yi is used to calculate fugaci t y of comp onent i in a liq-
ui d or solid solution. However, for liquid mi xt ures at hi gh
pressures, i.e., hi gh pressure VLE calculations, ~ is calculated
from q~/ through Eq. (6.113). I n such calculations as it will be
shown later in this section, for t he sake of simplicity and con-
venience, ~/ f or bot h phases are calculated t hr ough an equa-
t i on of state. Bot h q~/and )'i indicate degree of noni deal i t y for a
system. I n a gas mixture, q~i indicates deviation f r om an ideal
gas and in a liquid solution, Vi indicates deviation f r om an
ideal solution. To formul at e phase equi l i bri um of mi xt ures a
new par amet er called chemical potential must be defined.
= (OGt~
(6.115) /2i \-O-n~ni l r, v,n/~i = (~i
where/ ~i is the chemi cal potential of comp onent i in a mi xt ure
and Gi is the partial mol ar Gibbs energy. General definition
of partial mol ar propert i es was given by Eq. (6.78). For a pure
comp onent bot h partial mol ar Gibbs energy and mol ar Gibbs
energy are the same: G/ = Gi. For a pure ideal gas and an ideal
gas mi xt ure f r om t her modynami c relations we have
(6.116) dGi = RTdl n P
(6.117) d~/ = RTd I n (Yi P)
where in Eq. (6.117) if Yi = 1, it reduces to Eq, (6.116) for pure
comp onent systems. For real gases these equat i ons become
(6.118) dGi = d/ z i = RTd I n f~-
(6.119) dGi = d/ 2/ = RTd I n
Equat i on (6.118) is t he same as Eq. (6.46) derived for pure
component s. Equat i on (6.119) reduces to Eq. (6.118) at
Yi -- I. Subt ract i ng Eq. (6.118) f r om Eq. (6.119) and usi ng
Eq. (6.114) for]~ one can derive t he following relation for/ 2i
in a solution:
(6.120) / 2/ -/ z~ = RTln )'iN
where/ z[ is t he pure comp onent chemi cal pot ent i al at T and
P of mi xt ure and x4 is the mol e fract i on of i in liquid solu-
tion. For ideal solutions where Yi = 1, Eq. (6.120) reduces t o
/2i - tz~ = RTln xi. I n fact this is anot her way t o define an ideal
solution. A sol ut i on t hat is ideal over t he entire range of com-
posi t i on is called perfect solution and follows this relation.
6. 6. 2 Cal cul ati on o f Fug aci ty Coef f i ci ent s
f rom Equat i ons of State
Thr ough t her modynami c relations and definition of G one
can derive the following relation for the mi xt ure mol ar Gibbs
free energy [21].
P
( 6 . 1 21 ) c = f (v- ) d e +R T y il n (r / P ) + y / C,
0
where G 7 is the mol ar Gibbs energy of pure i at T of t he
syst em and pressure of 1 arm (ideal gas state). By replac-
ing G = Y'~.Yi/2i, and V = ~y i V i in the above equat i on and
removi ng the summat i on sign we get
P
(6.122) /2i = f (ff'i - R--~)dP + RTln( yiP) +G~
0
I nt egrat i on of Eq. (6.119) f r om pure ideal gas at T and P = 1
at m to real gas at T and P gives
/ .
^
(6.123) /2i - / x 7 = RTln 1
wher e/ x T is the chemi cal potential of pure comp onent i at
T and pressure of 1 at m (ideal gas as a st andard state). For
a pure comp onent at t he same T and P we have: /z~ = G~.
Combi ni ng Eqs. (6.122) and (6.123) gives
P
( f~ ) =RTl n~/ =Z ( r
(6.124) RTln ~p o
where 17i is the partial mol ar vol ume of comp onent i in the
mixture. I t can be seen t hat for a pure comp onent (17i = V/ and
y~ = i) this equat i on reduces to Eq. (6.53) previ ousl y derived
for cal cul at i on of fugaci t y coefficient of pure component s.
There are ot her forms of this equat i on in whi ch i nt egrat i on
is carri ed over vol ume in the following form [21]:
(6.125) RTl n~i = OP
V t T, V , nir ~ dV t - I n Z
where V t is the total vol ume (V t = nV). I n usi ng these equa-
tions one shoul d not e t hat n is t he sum of r~ and is not const ant
when derivative wi t h respect t o r~ is carried. These equat i ons
are the basis of cal cul at i on of fugaci t y of a comp onent in a
mixture. Exampl es of such derivations are available in vari-
ous texts [ I , 4, 11, 20-22]. One can use an EOS to obt ai n ITi
and up on subst i t ut i on in Eq. (6.124) a relation for cal cul at i on
of ~i can be obtained. For the general f or m of cubi c equat i ons
given by Eqs. (5.40)-(5.42), q~i is given as [11]
ln /= ( z - 1 ) - I n ( Z - B ) +
(6.126) x I n 2z +B( . ~+~
b~ "&/P~
where -
b Ejyjrcj/ P~j
and - 2ay
a i
i fal l k/ i = 0 t hen ~i = 2( - ~) 1/2
M1 paramet ers in t he above equat i on for vdW, RK, SRK, and
PR equat i ons of state are defined in Tables 5.1 and 6. I. Pa-
ramet ers a and b for the mi xt ure shoul d he cal cul at ed f r om
Eqs. (5.59)-(5.61). Equat i on (6.126) can be used for calcula-
t i on of fugacity of i in bot h liquid and vapor phases provi ded
appropri at e Z values are used as for the case of pure compo-
nent syst ems t hat was shown in Exampl e 6.7. For cal cul at i on
of ~/ f r om PR and SRK equat i ons t hr ough the above relation,
use of vol ume t ransl at i on is not required.
256 CHARACTERI ZATI ON AND PROPERTI ES OF PETROL EUM FRACTI ONS
I f truncated virial equation (Eq. 5.75) is used, ~i is calcu-
lated from the following relation as derived from Eq. (6.124):
(6.127) ln~)i = ( 2 Ey i B i i - B) P-~
j RT
where B (for whole mixture) and Bii (interaction coefficients)
should be calculated from Eqs. (5.70) and (5.74), respectively.
As discussed earlier Eq. (5.70) is useful for gases at moderat e
pressures. Equation (6.127) is not valid for liquids.
Example 6. 9--Suppose that fugacity coefficient of the whole
mixture, CmJx, is defined similar to that of pure components.
Through mixture Gibbs energy, derive a relation between f ~
and/ ~ for mixtures.
Solution- - Applying Eq. (6.80) to residual mol ar Gibbs free
energy ( G R = G - Gig) gives G R = ~-~yiG R and since t~i = Gi
from Eq. (6.119) dGi =RTdl n/ ~ and for ideal gases from
Eq. (6.117) we have dGi g = RTdlnyiP. Subtracting these two
relations from each other gives dG/R = RTd In ~i, which after
integration gives ~R = RTl n ~i. Therefore for the whole mix-
ture we have
(6.128) G ~ = RT E xi ln~,
where after compari ng with Eq. (6.48) for the whole mixture
we get
(6.129) lnCmi~ = Ex/ l nSi
or in t erms of fugacity for the whole mixture, fmi~, it can be
written as
^
(6.130) In fmix = Exi In f--/
x~
This relation can be applied to bot h liquid and gases, fmix is
useful for calculation of properties of only real mixtures but
is not useful for phase equilibrium calculation of mixtures
except under certain conditions (see Probl em 6.19). #
6. 6. 3 Calculation of Fugacity from Lewis Rul e
Lewis rule is a simple met hod of calculation of fugacity of a
component in mixtures and it can be used if the assumpt i ons
made are valid for the system of interest. The mai n assump-
tion in deriving the Lewis fugacity rule is t hat the mol ar vol-
ume of the mixture at constant t emperat ure and pressure is
a linear function of the mole fraction (this means Vi = V/ =
constant). This assumpt i on leads to the following simple rule
for~ known as Lewis~Randall or simply Lewis rule [21, 22]:
(6.131) /~(T, P) = Yi fi(T, P)
where f~.(T, P) is the fugacity of pure i at T and P of mixture.
Lewis rule simply says that in a mixture ~i is only a function of
T and P and not a function of composition. Direct conclusion
of Lewis rule is
(6.132) ~i(T, P) = ~i(T, P)
which can be obtained by dividing both sides of Eq. (6.131) by
Yi P. The Lewis nile may be applied to bot h gases and liquids
with the following considerations [21]:
- - Good approxi mat i on for gases at low pressure where the
gas phase is nearly ideal.
- - Good approxi mat i on at any pressure whenever i is present
in large excess (say, yi > 0.9). The Lewis rule becomes exact
in the limit of Yi ~ 1.
- - Good approxi mat i on over all range of pressure and com-
position whenever physical properties of all component s
present in the mixture are the same as (i.e., benzene and
toluene mixture).
- - Good approxi mat i on for liquid mixtures whose behavi or is
like an ideal solution.
- - A poor approxi mat i on at moderat e and high pressures
whenever the molecular properties of component s in the
mixture are significantly different from each other (i.e., a
mixture of met hane and a heavy hydrocarbon) .
Lewis rule is attractive because of its simplicity and is usu-
ally used when the limiting conditions are applied in certain
situations. Therefore when the Lewis rule is used, fugacity of
i in a mixture is calculated directly from its fugacity as pure
component. When Lewis rule is applied to liquid solutions,
Eq. (6.114) can be combi ned with Eq. (6.131) to get Yi = 1
(for all components) .
6. 6. 4 Calculation of Fugacity of Pure Gases
and Liquids
Calculation of fugacity of pure component s using equations
of state was discussed in Section 6.5. Generally fugacity of
pure gases and liquids at moderat e and high pressures may
be estimated from equations given in Table 6.1 or t hrough
generalized correlations of LK as given by EQ. (6.59). For pure
gases at moderat e and low pressures Eq. (6.62) derived from
virial equation can be used.
To calculate fugacity of i in a liquid mixture through
Eq. (6.114) one needs fugacity of pure liquid i in addition to
the activity coefficient. To calculate fugacity of a pure liquid i
at T and P, first its fugacity is calculated at T and correspond-
ing saturation psat. Under the conditions of T and psat bot h
vapor and liquid phases of pure i are in equilibrium and thus
(6.133) f/ r(T ' psat) = f/V(T ' psat) = r
where r is the fugacity coefficient of pure vapor at T and
psat. Effect of pressure on liquid fugacity should be consid-
ered to calculate fiL(T, P) from fiL(T, p s a t ) . This is obtained
by combi ni ng Eq. (6.8) (at constant T) and Eq. (6.47):
(6.134) dGi = Rr dl n/ ~ = V/dP
I ntegration of this equation from psat to desired pressure of
P for the liquid phase gives
P
(6.135) In fi~(T, psat) = dP
Combining Eqs. (6.133) and (6.135) leads to the following
relation for fugacity of pure i in liquid phase.
P L
:~" i exp~Jt RT )
p~at is the sat urat i on pressure or vapor pressure of pure i at T
a nd met hods of its cal cul at i on are di scussed in t he next chap-
ter. ~b sat is t he vapor phase fugaci t y coefficient of pure compo-
nent i at p/sat and can be cal cul at ed from met hods di scussed
i n Section 6.2. The exponential t erm in the above equat i on is
called Poy nt i ng correct i on and is calculated from liquid mol ar
volume. Since vari at i on of V/L wi t h pressure is small, usually it
is assumed const ant versus pressure and t he Poynt i ng fact or
is simplified a s e xp [ V/ L( P -- P/ sat ) / RT] . I n such cases V/L may
be t aken as mol ar vol ume of sat urat ed liquid at t emperat ure
T and it may be calculated f r om Racket equat i on ( Section
5.8). At very low pressures or when ( P - psat) is very small,
t he Poynt i ng fact or approaches uni t y and it coul d be r emoved
from Eq. (6.136). I n addition, when p/ sat is very small ( ~ 1 at m
or less), ~b~ ~t may be consi dered as uni t y and fi r is simply equal
t o p/sat. Obviously this simplification can be used onl y in spe-
cial situations when t he above assumpt i ons can be justified.
For cal cul at i on of Poynt i ng fact or when V/L is in cm3/ mol, P
in bar, and T in kelvin, t hen the value of R is 83.14.
6 . 6 . 5 C a l c u l a t i o n o f A c t i v i t y C o e f f i c i e n t s
Activity coefficient Yi is needed in cal cul at i on of fugacity of i
in a liquid mi xt ure t hr ough Eq. (6.114). Activity coefficients
are related to excess mol ar Gibbs energy, G E, t hr ough ther-
modynami c relations as [21 ]
(6.137) RT l nF i = C, = L o,~ J l" ,e,n,~,
where Gi is t he partial mol ar excess Gibbs energy as defined
by Eq. (6.78) and may be cal cul at ed by Eq. (6.82). This equa-
t i on leads to anot her equally i mpor t ant rel at i on for the act i v -
i t y coefficient in t erms of excess Gibbs energy, GE:
(6.138) G E = RT E xi l n yi
i
where this equat i on is obt ai ned by subst i t ut i on of Eq. (6.137)
i nt o Eq. (6.79). Therefore, once the relation for G E is known
it can be used to det ermi ne Fi. Similarly, when Fi is known
G E can be calculated. Various model s have been proposed
for G E of bi nary systems. Any model for G E must satisfy t he
c o n d i t i o n t hat when xl = 0 or 1 (x2 = 0), G E must be equal t o
zero; therefore, it must be a fact or of xlx2. One general model
for G E of bi nary systems is called Redl i ch-Ki st er expansi on
and is given by the following power series form [1, 21]:
G E
(6.139) ~ = X l X 2 [ A -}- B( Xl - x2) -4- C( X l - x2) 2 q- . . . ]
where A, B . . . . are empi ri cal t emperat ure-dependent coeffi-
cients. I f all these coefficients are zero t hen the sol ut i on is
ideal. The simplest noni deal solution is when onl y coeffi-
cient A is not zero but all ot her coefficients are zero. This
is known as two-suffix Margules equat i on and up on applica-
t i on of Eq. (6.137) t he following relations can be obt ai ned for
~,1 and yz:
I n yt = --~-A x2
RT
(6.140)
A x2
I n Y2 = RT 1
Accordi ng to t he definition of yi when x4 = 1 ( pure i) t hen
Yi = 1. Generally for bi nary syst ems when a rel at i on for ac-
t i v i t y coefficient of one comp onent is known t he rel at i on for
activity coefficient of ot her comp onent s can be det ermi ned
from t he following relation:
dlnF1 dl nF2
(6.141) xx dx 1 - x2 dx 2
whi ch is derived f r om Gi bbs- Duhem equation. One can ob-
t ai n y2 f r om yl by appl yi ng t he above equat i on wi t h use of
x2 = 1 - xl and dx2 = - dxl . Const ant A in Eq. (6.140) can be
obt ai ned f r om dat a on the activity coefficient at infinite di-
lution (Fi~ whi ch is defined as limxi-~0(Yi). This will result
in A = RTl n y ~ = RTl n y ~. This simple model applies well
to simple mi xt ures such as benzene-cycl ohexane; however,
for mor e compl ex mi xt ures ot her activity coefficient model s
must be used. A mor e general f or m of activity coefficients for
bi nary systems t hat follow Redl i ch-Ki st er model for G E a r e
g i v e n as
(6.142)
RTl n y, = a, x 2 + a2x32 + a3x 4 + a4 X5 -~-''"
RTl n 2 = b , x 2 + b2x~ + b3x 4 + b4 x5 --~- . . .
I f in Eq. (6.139) coefficient C and hi gher order coefficients
are zero t hen resulting activity coefficients cor r espond to onl y
the first t wo t erms of t he above equation. This model is called
four-suffix Margules equation. Since dat a on yi ~ are useful
i n obt ai ni ng the const ant s for an activity coefficient model,
many researchers have measur ed such dat a for vari ous sys-
tems. Figure 6.8 shows values of yi ~ for n-C4 and n-C8 in var-
ious n-alkane solvents f r om C15 to C40 at 100~ based on dat a
available f r om C20 t o C36 [21]. As can be seen f r om this fig-
ure, as the size of solvent mol ecul e increases t he deviation of
activity coefficients f r om uni t y also increases.
Anot her popul ar model for activity coefficient of bi nary
systems is the van Laar model proposed by van Laar dur-
ing 1910-1913. This model is part i cul arl y useful for binaries
whose mol ecul ar sizes vary significantly. Van Laar model is
g
2
r~
r~
L~
._>
0.9
0.8
0.7
0.6
0.5
9 C4: Y=1.337-O.0269X + O.OOO241X ~
. . . n-Butane
n- Oct ane
10 20 30 40 50
Carbon Number of n-Alkane Solvent
FIG. 6.8---Values of 7 T for n-butane and n-octane in
r~ paraffi n s o l v e n t s at 1 0 0 ~
based on the Wohl' s model for the excess Gibbs energy [21 ].
The G ~ relation for the van Laar model is given by
G E
(6.143) - - = XlX2[A + B(x1 - x2)] -1
RT
Upon appl i cat i on of Eq. (6.137), the activity coefficients are
obt ai ned as
A,2x, -2
lnya --- A12 1 + A21x2]
( 6.144) ,
lnyz -~ A21( l + A21x2 /
where coefficients AlE and A21 are related to A and B in Eq.
(6.143) as A- B = 1/A12 and A + B = 1/A21. Coefficients A12
and A21 can be det ermi ned from t he activity coefficients at
infinite dilutions (Alz = In y~, Azl = I n y~) . Once for a given
syst em VLE dat a are available, t hey can be used to calculate
activity coefficients t hr ough Eqs. (6.179) or (6.181) and t hen
GE/RT is calculated from Eq. (6.138). Fr om the knowl edge of
G~/RT versus (xl - x2) the best model for G ~ can be found.
Once the rel at i on for G ~ has been det ermi ned the activity
coefficient model will be found.
For regular solutions where different comp onent s have the
same i nt ermol ecul ar forces it is generally assumed t hat V ~ =
S E = 0. Obviously syst ems cont ai ni ng pol ar comp ounds gen-
erally do not fall into the cat egory of regul ar solutions. Hy-
dr ocar bon mi xt ures may be consi dered as regul ar solutions.
The activity coefficient of comp onent i in a bi nary liquid solu-
t i on accordi ng to t he regul ar sol ut i on t heory can be cal cul at ed
from the Scat char d- Hf l debr and rel at i on [21, 22]:
(8, - h )
I n Yl = RT
(6.145)
I n Yz = VL (81 - 82) 2 ev~
RT
where V L is t he liquid mol ar vol ume of pure comp onent s (1
or 2) at T and P and 8 are the solubility p ar amet er of pure
comp onent s 1 or 2. qb~ is the vol ume fract i on of comp onent
1 and for a bi nary syst em it is given by
x~ V~
(6.146) Ol =
where xl and xz are mol e fract i ons of comp onent s 1 and 2.
The solubility p ar amet er for comp onent i can be calculated
f r om the following rel at i on [21, 22]:
(6.147) 8i = \ - - - ~/ L ] = \ v/L
where AU yap and AH/ yap a r e t he mol ar internal energy and
heat of vapori zat i on of comp onent i, respectively. The tradi-
tional uni t for ~ is (cal/cm3)l/z; however, in this chapt er the
uni t of (J/cm3) V2 is used and its conversi on to ot her uni t s is
given in Section 1.7.22. Solubility p ar amet er originally pro-
posed by Hi l debrand has exact physical meani ng. Two par am-
eters t hat are used to define 8 are energy of vapori zat i on and
mol ar volume. I n Chapt er 5 it was di scussed t hat for nonp ol ar
comp ounds t wo paramet ers, namel y energy p ar amet er and
size p ar amet er describe t he i nt ermol ecul ar forces. Ener gy of
vapori zat i on is directly related to the energy requi red to over-
come forces bet ween mol ecul es in the liquid phase and mol ar
vol ume is proport i onal to the mol ecul ar size. Therefore, when
t wo comp onent s have similar values of 8 their mol ecul ar size
and forces are very similar. Molecules wi t h similar size and
interrnolecular forces easily can dissolve in each other. The
i mpor t ance of solubility p ar amet er is t hat when t wo compo-
nent s have 8 values close to each ot her t hey can dissolve in
each ot her appreciably. I t is possible to use an EOS to cal-
culate 8 from Eq. (6.147) (see Probl em 6.20). Accordi ng to
t he t heory of regul ar solutions, excess ent ropy is zero and it
can be shown t hat for such solutions RT lnF/ is const ant at
const ant composi t i on and does not change wi t h t emperat ure
[11]. Values of V~ L and 8i at a reference t emperat ure of 298
K is sufficient t o calculate ~'i at ot her t emperat ures t hr ough
Eq. (145). Values of solubility p ar amet er for single car bon
number comp onent s are given in Table 4.6. Values of V/L and
~i at 25~ for a number of pure subst ances are given in Ta-
ble 6.10 as provi ded by DI PPR [13]. I n this table values of 8
have t he uni t of (J/cm3) 1/2. I n Table 6.10 values of freezing
poi nt and heat of fusion at the freezing poi nt are also given.
These values are needed in cal cul at i on of fugaci t y of solids as
will be seen in the next section.
Based on the dat a given in Table 6.10 the following rela-
t i ons are developed for est i mat i on of liquid mol ar vol ume of
n-alkanes (P), n-alkylcyclohexanes (N), and n-alkylhenzenes
(A) at 25~ V25 [23]: I t gives Cp/Cv for sat urat ed liquids hav-
ing a cal cul at ed heat capaci t y ratio of 1.43 to 1.38 over a tem-
perat ure range of 300-450 K.
(6.148)
I n V25 = - 0. 51589 + 2. 75092M ~
for n-alkanes (C1 - C36)
V2s = 10. 969+ 1.1784M
for n-al kyl cycl ohexanes (C6 - C16)
I n V25 = - 96. 3437 + 96. 54607M ~176
for n-al kyl benzenes (C6 - C24)
where V25 is in cm3/ mol. These correl at i ons can reproduce
dat a in Table 6.10 wi t h average deviations of 0.9, 0.4, and
0.2% for n-alkanes, n-alkylcyclohexanes, and n-alkylbenzens,
respectively. Similarly the following relations are developed
for est i mat i on of solubility p ar amet er at 25~ [23]:
(6.149)
6 = 16.22609 [ I + exp (0.65263 - 0.02318M)] -~176176
for n-alkanes (C1 - C36)
8 = 16.7538 + 7.2535 10-5M
for n-alkylcyclohexanes (C6 - C16)
8 = 26. 8557- 0. 18667M+ 1.36926 x 10-3M 2
- 4 . 3 4 6 4 10-6M3 + 4.89667 10-9M 4
for n-alkylbenzenes (C6 - Cz4)
where 8 is in (J/cm3) U2. The conversi on fact or from this
uni t to the t radi t i onal units is given in Section 1. 7. 22: 1
(cal/ cm3) 1/2 = 2.0455 (J/cm3) V2. Values predi ct ed f r om these
equat i ons give average deviation of 0.2% for n-alkanes, 0.5%
for n-alkylcyclohexanes, and 1.4% for n-alkylbenzenes. I t
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
6O
61
TABLE 6. 10--Freezi ng point, heat of fusion, molar volume, and solubility parameters for some selected compounds [DIPPR].
Compound Formula Nc M TM, K AHf/ RTu at TM V25, cm3/mol 82s,
n-ParmTms
Met hane CH4 1 16.04 90.69 1.2484 37.969(52) a
Et hane C2H6 2 30.07 90.35 3.8059 55.229(68) a
Propane C3H8 3 44.09 85.47 4.9589 75.700(84) a
n-But ane C4H10 4 58.12 134.86 4.1568 96.48(99.5) a
n- Pent ane C5H12 5 72.15 143.42 7.0455 116.05
n-Hexane C6H14 6 86.17 177.83 8.8464 131.362
n-Hept ane C7HI6 7 100.20 182.57 9.2557 147.024
n-Oct ane C8H18 8 114.22 216.38 11.5280 163.374
n-Nonane C9H20 9 128.25 219.66 8.4704 179.559
n-Decane C10H22 10 142.28 243.51 14.1801 195.827
n-Undecane CllH24 11 156.30 247.57 10.7752 212.243
n-Dodecane C12H26 12 170.33 263.57 16.8109 228.605
n-Tridecane C13H28 13 184.35 267.76 12.8015 244.631
n-Tet radecane C14H30 14 198.38 279.01 19.4282 261.271
n-Pent adecane C15H32 15 212.41 283.07 14.6966 277.783
n-Hexadecane C16H34 16 226.43 291.31 22.0298 294.213
n-Hept adecane C17H36 17 240.46 295.13 16.3674 310.939
n-Oct adecane C18H38 18 254.48 301.31 24.6307 328.233
n-Nonadecane C19H40 19 268.51 305.04 18.0620 345.621
n-Ei cosane C20H42 20 282.54 309.58 27.1445 363.69
n-Henei cosane C21H44 21 296.56 313.35 18.3077 381.214
n-Docosane C22H46 22 310.59 317.15 18.5643 399.078
n-Triacosane C23H48 23 324.61 320.65 20.2449 416.872
n-Tetracosane C24H50 24 338.64 323.75 20.3929 434.942
n-Hexacosane C26H54 26 366.69 329.25 22.1731 469.975
n-Hept acosane C27H56 27 380.72 332.15 21.8770 488.150
n-Oct acosane C28H58 28 394.74 334.35 23.2532 506.321
n-Nonacosane C29H60 29 408.77 336.85 23.6034 523.824
n-Tri acont ane C30H62 30 422.80 338.65 24.4439 540.500
n-Docont ane C32H66 32 450.85 342.35 26.8989 576.606
n-Hexacont ane C36H74 36 506.95 349.05 30.6066 648.426
Isoparaffins
I sobut ane C4H10 4 58.12 113.54 4.8092 105.238
I sopent ane C5H12 5 72.15 113.25 5.4702 117.098
I sooct ane ( 2, 2, 4-t ri met hyl pent ane) C8H18 8 114.23 165.78 6.6720 165.452
n-Alkylcyelopentanes (naphthenes)
Cycl opent ane C5H10 5 70.14 179.31 0.4084 94.6075
Met hy] cycl opent ane C6H12 6 84.16 146.58 4.7482 128.1920
Et hyl cycl opent ane C7H14 7 98.19 134.71 6.1339 128.7490
n-Propyl cycl opent ane C8H16 8 112.22 155.81 7.7431 145.1930
n-But yl cycl opent ane C9H18 9 126.24 165.18 8.2355 161.5720
n.Alkylcyclohexanes (naphthenes)
Cyclohexane C6H12 6 84.16 279.69 1.1782 108.860
Met hyl cycl ohexane C7H14 7 98.18 146.58 5.5393 128.192
Et hyl cycl ohexane C8H16 8 112.21 161.839 6.1935 143.036
n-Propylcyclohexane C9H18 9 126.23 178.25 6.9970 159.758
n-Butylcyclohexane CIoH20 10 140.26 198.42 8.5830 176.266
n-Decylcyclohexane C16H32 16 224.42 271.42 17.1044 275.287
n-Alkylbenzenes (aromatics)
Benzene C6H 6 6 78.11 278.65 4.2585 89.480
Met hyl benzene (Toluene) C7H8 7 92.14 178.15 4.4803 106.650
Et hyl benzene Call10 8 106.17 178.15 6.1983 122.937
Propyl benzene C9H12 9 120.20 173.55 6.4235 139.969
n-But yl benzene C10H14 10 134.22 185.25 7.2849 156.609
n-Pent yl benzene CllH16 11 148.25 198.15 9.2510 173.453
n-Hexyl benzene C12H18 12 162.28 211.95 10.4421 189.894
n-Hept yl benzene C13H20 t 3 176.30 225.15 11.6458 206.428
n-Oct yl benzene C14H22 14 190.33 237.15 13.1869 223.183
n-Nonyl benzene C15H24 15 204.36 248.95 13.9487 239.795
n-Decylbenzene C16H26 16 218.38 258.77 15.1527 256.413
n-Undecyl benzene C17H28 17 232.41 268.00 16.1570 272.961
n-Dodecyl benzene C18H30 18 246.44 275.93 17.5238 289.173
n-Tridecylbenzene C19H32 19 260.47 283.15 18.6487 306.009
n-Tetradecylbenzene C20H34 20 274.49 289.15 19.8420 322.197
n-Pent adecyl benzene C21H36 21 288.52 295.15 20.9874 339.135
(J/cm3)l/2
11.6
12.4
13.1
13.7
14.4
14.9
15.2
15.4
15.6
15.7
15.9
16.0
16.0
16.1
16.1
16.2
16.2
16.2
16.2
16.2
16.2
16.2
16.3
16.3
16.3
16.2
16.2
16.2
16.2
16.2
16.2
12.57
13.86
14.08
16.55
16.06
16.25
16.36
16.39
16.76
16.06
16.34
16.35
16.40
16.65
18.70
18.25
17.98
17.67
17.51
17.47
17.43
17.37
17.37
17.39
17.28
17.21
17.03
16.87
16.64
16.49
(Continued)
TABLE 6.10---(Continued)
No. Compound Formula Nc M TM, K AHf/RTM at T~ V2s, cm3/mol 82s, (J/cm3) 1/2
62 n-Hexadecylbenzene C22H38 22 302.55 300.15 22.1207 356.160 16.39
63 n-Heptadecylbenzene C23H40 23 316.55 305.15 22.9782 373.731 16.30
64 n-Octadecylbenzene C24H42 24 330.58 309.00 23.7040 390.634 16.24
1 -rvAlkylnaphthalenes (aromatics)
65 Napht hal ene C10H8 10 128.16 353.43 6.4588 123.000 19.49
66 1-Methylnaphthalene Cl l Hm 11 142.19 242.67 3.4420 139.899 19.89
67 1-Ethylnaphthalene C12H12 12 156.22 259.34 7.5592 155.579 19.85
68 1-n-Propylnaphthalene C13H14 13 170.24 264.55 7.9943 172.533 19.09
69 1-n-Butylnaphthalene C14H16 14 184.27 253.43 11.9117 189.358 19.10
70 1-n-Pentylnaphthalene ClsHt8 15 198.29 248.79 11.3121 205.950 18.85
71 1-n-Hexylnaphthalene C16H20 16 212.32 255.15 . . . 224.155 18.72
72 1-n-Nonylnaphthalene C19H26 19 254.40 284,15 . . . 272.495 17.41
73 l-n-Decylnaphthalene C20H2s 20 268.42 288.15 . . . 289.211 17.20
Other organic compounds
74 Benzoic acid C7H602 7 122.12 395.52 5.4952 112.442 24.59
75 Di phenyl met hane C13H12 13 168.24 298.39 7.3363 167.908 19.52
76 Ant heracene C14H10 14 190.32 488.93 7.7150 182.900 17.75
Nonhydrocarbons
77 Water H20 . . . 18.02 273.15 2.6428 18.0691 47.81
78 Methanol CH3OH 1 32.04 -97. 68 0.2204 40.58 29.59
79 Et hanol C2HsOH 2 46.07 -114. 1 0,3729 58.62 26.13
80 I sobutano] C4H9OH 4 74.12 -108. 0 0.4634 . . . 22.92
81 Carbon dioxide CO 2 1 44.01 216.58 5.0088 37.2744 14.56
82 Hydrogen sulfide H2S . . . 34.08 187.68 1.5134 35.8600 18.00
83 Nitrogen N2 . . . 28.01 63.15 1.3712 34.6723 9.082
84 Hydrogen H2 . . . 2.02 13.95 1.0097 28.5681 6.648
85 Oxygen 02 . . . 32.00 54.36 0.9824 28.0225 8.182
86 Ammoni a NH3 17.03 195.41 3.4819 24.9800 29.22
87 Carbon monoxi de CO "1' 28.01 68.15 1.4842 35.4400 6.402
aAPI-TDB [111 gives different values for V2s of light hydrocarbons. These values are given in parentheses and seem more accurate, as also given in Table 6.11.
Values in this table are obtained from a program in Ref. [13].
should be noted that the polynomial correlation given for n-
alkylbenzenes cannot be used for compounds heavier t han
C24. The other two equations may be extrapolated to heavier
compounds. Equations (6.148) and (6.149) may be used to-
gether with the pseudocomponent met hod described in Chap-
ter 3 to estimate V2s and 8 for pet rol eum fractions whose
molecular weights are in the range of application of these
equations. Values of V/L and 8 given in Table 6.10 are taken
from Ref. [13] at t emperat ure of 25~ It seems that for some
light gases (i.e., CH4), there are some discrepancies with re-
ported values in ot her references. Values of these properties
for some compounds as recommended by Pruasnitz et al. [21 ]
are given in Table 6.11. Obviously at 25~ for light gases such
as CH4 or N2 values of liquid" properties represent extrapo-
lated values and for this reason they vary from one source to
another. It seems that values given in Table 6.10 for light gases
correspond to t emperat ures lower t han 25~ For this reason
for compounds such as C1, C2, H2S, CO2, N2, and 02 values
of V/L and 8 at 25~ as given in Table 6.11 are recommended
t o be used.
F or mul t i c omp one nt s ol ut i ons , Eqs. ( 6. 145) a nd ( 6. 146) ar e
r e p l a c e d by t he f ol l owi ng r el at i on:
I n Yi -- V/L (8i - 8mix)2
RT
(6.150) 6mlx = Y~ e;i& i
i
x; vj r
| - Ek x~V ~
where the summat i on applies to all component s in the mix-
ture. Regular solution theory is in fact equivalent to van
Laar theory since by replacing A12 = ( V ~/ RT) ( SI - 82) 2 and
A21 = ( V? / R T) ( ~1 - ~2) 2 into Eq. (6.144), it becomes identical
to Eq. (6.145). However, the mai n advantage of Eq. (6.145)
over Eq. (6.144) is that paramet ers Vii L and ~i a r e calcula-
ble from t hermodynami c relations. Riazi and Vera [23] have
shown that predicted values of solubility are sensitive to the
values of Vii L and 8i and they have recommended some specific
values for 8i of various light gases in pet rol eum fractions.
Other commonl y used activity coefficient models include
Wilson and NRTL ( nonrandom two-liquid) models, which
are applicable to systems of heavy hydrocarbons, water, and
TABLE 6.1 l--V alues of liquid molar volume and solubility
parameters for some pure compounds at 90 and 298 K.
Compound V/L, (cm3/mol) 8i, (J/cm3) 1/2
N2 (at 90 K) 38.1 10.84
N2 (at 298 K) 32.4 5.28
CO (at 90 K) 37.1 11.66
CO (at 298 K) 32.1 6.40
02 (at 90 K) 28.0 14.73
02 (at 298 K) 33.0 8.18
CO2 (at 298 K) 55.0 12.27
CH4 (at 90 K) 35.3 15.14
CH4 (at 298 K) 52.0 11.62
C2H6 (at 90 K) 45.7 19.43
C2H6 (at 298 K) 70.0 13.50
Taken from Ref. [21]. Components N2, CO, 02, CO2, CH4, and CzH6 at
298 K are in gaseous phase (To < 298 K) and values of ~L and t~i are
hypothetical liquid values which are recommended to be used. Values
given at 90 K are for real liquids. All other components are in liquid form
at 298 K. Values reported for hydrocarbons heavier than Cs are similar
to the values given in Table 6.10. For example, for n-Cl6 it provides
values of 294 and 16.34 for ~L and ~i, respectively. Similarly for benzene
values of 89 and 18.8 were provided in comparison with 89.48 and 18.7
given in Table 6.10.
alcohol mixtures [21]. For hydrocarbon systems, UNIQUAC
(universal quasi chemical) model that is based on a group
contribution model is often used for calculation of activity
coefficient of compounds with known structure. More details
on activity coefficient models and their applications are dis-
cussed in available references [4, 21]. The maj or application
of activity coefficient models is in liquid-liquid and solid-
liquid equilibria as well as low pressure VLE calculations
when cubic equations of state do not accurately estimate liq-
uid fugacity coefficients.
6. 6. 6 Cal cul ati on of Fugaci ty of Sol i ds
I n the pet rol eum industry solid fugacity is used for SLE cal-
culations. Solids are generally heavy organics such as waxes
and asphaltenes that are formed under certain conditions.
Solid-liquid equilibria follows the same principles as VLE.
Generally fugacity of solids are calculated similar to the meth-
ods that fugacity of liquids are calculated. I n the study of
solubility of solids in liquid solvents usually solute (solid) is
shown by component 1 and solvent (liquid) is shown by com-
ponent 2. Mole fraction of solute in the solution is xl, which is
the mai n paramet er that must be estimated in calculation of
solubility of solids in liquids. We assume that the solid phase
is pure component 1. I n such a case fugacity of solid in the
solution is shown byf s, which is given by
(6.151) f~ (solid in liquid solution) = xl~,sf~
where f~ is the fugacity of solute at a standard state but tem-
perat ure T of solution. )/s is the activity coefficient of solid
component in the solution. Obviously for ideal solutions yl s
is unity. Model to calculate ys is similar to liquid activity co-
efficients, such as two-suffix Margules equation:
A
(6.152) l nF s = ~ (1 - - Xl ) 2
A more accurate activity coefficient model for nonpol ar so-
lutes and solvents is given by the Scat chard-Hi l debrand rela-
tion (Eq. 6.145):
(6.153) I n y1S : vL( 81 -- 82)2(I)2
RT
where VIE is the liquid mol ar volume of pure component 1 at
T and P, 82 is solubility of solvent, 81 is the solubility parame-
ter of subcooled component 1, and ep2 is the volume fraction
of solvent and is given by Eq. (6.146). Methods of calculation
of 81 and apl have been discussed in Section 6.6.5.81 can be
calculated from Eq. (6.147) from the knowledge of heat of
vaporization of solute, AH~ ap. Values of the solubility param-
eter for heavy single carbon number component s are given in
Table 4.6. When the liquid solvent is a mixture 82 is replaced
by 8mu and ys is calculated t hrough Eq. (6.150). I t should be
noted that when Eq. (6.153) is used to calculate fugacity of a
solid in a liquid solution value of 8 can be obtained from Table
6.10 from liquid solubility data. However, when this equation
is applied for calculation of fugacity of a solid component i
in a homogeneous solid phase mixture (i.e., wax) then solid
solubility, 8 s, should be used for value of 8 as recommended
by Won [24]. I f a value of liquid solubility given in Tables 6.10
and 6.11 is shown by 8 L, then 8 s may be calculated from the
following relation [24]:
An
(6.154) (ss) 2 = (8/L) 2 +
in which 8 is in (J/cm3) 1/2, A/~/ is in J/ mol, and V/ is in
cm3/mol.
Calculation of fugacity of solids t hrough Eq. (6.151) re-
quires calculation of f/~ For convenience the standard state
for calculation of f~ is considered subcooled liquid at temper-
ature T and for this reason we show it by fi E. I n the following
discussion solute component 1 is replaced by component i
to generalize the equation for any component. Based on the
SLE for pure i at t emperat ure T it can be shown that [21, 25]
~s( r, p) = ~( i r, p)
eXP L RTMi Q -- ~- ) - ACvi
)< -----~ (1-- ~ ) - ACPi "
(6.155)
where f/S(T, P) is the fugacity of pure solid at T and P, A/ ff is
the mol ar heat of fusion of solute, TMi is the melting or freezing
point t emperat ure, and ACpi = cL i - - CSi , which is the differ-
ence between heat capacity of liquid and solid solute at av-
erage t emperat ure of (T + TMi)/2. Derivation of Eq. (6.155)
is similar to the derivation of Eq. (6.136) for calculation of
fugacity of pure liquids but in this case equilibrium between
solid and liquid is used to develop the above relation. Firooz-
abadi [17] clearly describes calculation of fugacity of solids.
Since met hods of calculation of f/L we r e discussed in the pre-
vious section, f/s can be calculated from the above equation.
Values of TM and A/ ~/ for some selected compounds are given
in Table 6.10 along with liquid mol ar volume and solubility
parameter. Est i mat i on of freezing point TM for pure hydro-
carbons was discussed in Section 2.6.4. From Eq. (2.42) and
coefficients given in Table 2.6 we have
TM = 397 -- exp (6.5096 -- 0.1487M ~
n-alkanes (C5 - C40)
TM ---- 370 -- exp (6.52504 -- 0.04945M 2/3)
(6.156)
n-alkylcyclopentanes (C7 - C40)
TM = 375 -- exp (6.53599 -- 0.04912M 2/3)
n-alkanes (C9 - C42)
where TM is in kelvin. Average deviation for these equations
are 1.5, 1.2, and 0.9%, for n-alkanes, n-alkylcyclopentanes,
and n-alkylbenzenes, respectively. Similarly based on the data
given for A ~ in Table 6.10 the following relations are devel-
oped for estimation of heat of fusion of pure hydrocarbons
for the PNA homol ogous families.
In AHf _ -71. 9215 + 70.7847M ~176
RTM
for n-alkanes (C2 - Ca6)
AHf
n R~M ---- 0.8325 + 0.009M
for n-alkylcyclohexanes (C7 - C16)
AHf _ 1.1556 + 0.009M + 0.000396M 2 - 6.544 10-7M 3
RTM
for n-alkylbenzenes(C6 - C24)
(6.157)
where TMi is the melting point in kelvin and R is the gas con-
stant. The ratio AI-Iif./RTta is dimensionless and represents en-
t ropy of fusion. Unit of A/ if/ depends on the unit of R. A/ ff
may also be calculated from ent ropy change of fusion, AS/f ,
whenever it is available.
(6.158) A/ ~/ = TMiAS/f
The above equation may also be used to estimate AS/f from
A/~. calculated t hrough Eq. (6.157). Equation (6.157) can
reproduce data with average deviations of 12.5, 5.4, and
3.8% for n-alkanes, n-alkylcyclohexanes, and n-alkylbenzens,
respectively. Firoozabadi and co-workers [17, 24-26] have
provided the following equations for calculation of A/-~:
A/~f =0.07177M~ for paraffins
RT~ai
(6.159) A/ ~ = 0.02652M/ for napht henes and isoparaffins
RTM~
A/-ff = 5.63664 for aromat i cs
RTM~
where AI-~/RTM is dimensionless. The relation given for cal-
culation of A/ if/ of aromatics (Eq. 6.159) suggests that the en-
tropy of fusion is constant for all aromatics. While this may
be true for some multiring aromatics, it certainly is not true
for n-alkylbenzenes. Graphical compari sons of Eqs. (6.157)
and (6.159) for calculation of ent ropy of fusion of n-alkanes
and n-alkylbenzenes and evaluation with data given in Ta-
ble 6.10 are shown in Figs. 6.9 and 6.10. As is seen from Fig.
6.10, the ent ropy of n-alkylbenzenes does change with carbon
number.
Calculation of fugacity of solids also requires ACre. The fol-
lowing relation developed for all types of hydrocarbons (P, N,
and A) by Pedersen et al. [26] is recommended by Firoozabadi
for calculation of ACpi [17]:
(6.160) ACe~ = R(0.1526M~ - 2.3327 10-4M~T)
where T is the absolute t emperat ure in kelvin and Mi is molec-
ular weight of i. The unit of ACr~ is the same as the unit of R.
Evaluation of this equation with data from DI PPR [13] for
30
n-Alkylbenzenes
o Dat a ^o~
Pr op osed Equat i on
20 .......
, 0
0 I i i I I
0 5 10 15 20 25 30
Carbon Number, Nc
FIG. 6.10--Prediction of entropy of fusion of
n-alkylbenzens. Proposed equation: Eq. (6.157);
W on method: Eq. (6.159); data from DIPPR [13],
n-alkanes at two different t emperat ure of 298 K and freez-
ing point is shown in Fig. 6.11. As is seen from this figure,
Eq. (6.160) gives values higher t han actual values of ACpi.
Generally, actual values of ACr~ are small and as will be seen
later they may be neglected in the calculation of f/s from
Eq. (6.155) with good approximation.
Another type of SLE that is i mport ant in the pet rol eum in-
dustry is precipitation of heavy organics, such as asphaltenes
and waxes, that occurs under certain conditions. Wax and
asphaltene precipitation can plug the well bore formations
and it can restrict or plug the tubing and facilities, such
as flowlines and product i on handling facilities, which can
lead to maj or economic problems. For this reason, knowl-
edge of the conditions at which precipitation occurs is impor-
tant. I n formulation of this phase transition, the solid phase
is considered as a solution of mixtures of component s that
fugacity of i is shown by ~ s and can be calculated from the
following relation:
^ S
(6.161) fi (solid i in solid mixture) = xSyi s fi s
40
n-Alkanes
o Dat a .-' "
30
Proposed Equat i on . - " " ' f
. . . . . . . WonMe t h od o . : * / o
~ o o
o .. Oo
"9 lO ~ ~176
20 o .." oo
o ~ 1 7 6
0 l 0 20 30 40
Carbon Number, Nc
FIG. 6.9--Prediction of entropy of fusion
of n-alkanes. Proposed equation: Eq. (6.157);
W on method: Eq. (6.159); data from DIPPR
[13].
40
3O
20
10
Won Method at 25 C # ,
9 Data from DIPPR at 25 C J O
/
- - - Won Method at Freezing Point / o
o Data from " " . *
t 9 9
. / -
9 o
@
o
0
0
82 o ,
10 20 30 40
Carbon Number, Nc
FIG. 6.11--Values of ACp~ ( = c L j - C S) for
n-alkanes. W on method: Eq. (6.160); data from
DIPPR [13].
where f/s is the fugacity of pure i at T and P of the system.
I n wax precipitation usually the solid solution is considered
ideal and yi s is assumed as unity [17]. x s is the mole fraction
of solid i in the solid phase solution. Here the term solution
means homogeneous mixture of solid phase. As it will be seen
in the next chapter these relations can also be used to deter-
mine the conditions at which hydrates are formed.
Calculation of fugacity of pure solids through Eq. (6.155)
is useful for SLE calculations where the temperature is above
the triple-point temperature (Ttp). When temperature is less
than Ttp we have solid-vapor equilibrium as shown in Fig.
5.2a. For such cases the relation for calculation of fugacity of
pure solids can be derived from fugacity of pure vapor and
effect of pressure on vapor phase fugacity similar to deriva-
tion of Eq. 6.136, where f/L, plat, and V/L should be replaced
by f/s, p/sub, and V/s, respectively. However at T < Ttp, Pi sub or
solid-vapor pressure is very low and &at is unity. Furthermore
molar volume of solid, V/s is constant with respect to pressure
(see Problem 6.15).
6. 7 GEN ER A L M ET H OD FOR CAL CUL AT I ON
OF P R OP ER T I ES OF R EA L M I X T UR ES
Two parameters have been defined to express nonideality of
a system, fugacity coefficient and activity coefficient. Fugac-
ity coefficient indicates deviation from ideal gas behavior and
activity coefficient indicates deviation from ideal solution be-
havior for liquid solutions. Once residual properties (devia-
tion from ideal gas behavior) and excess properties (deviation
from ideal solution behavior) are known, properties of real
mixtures can be calculated from properties of ideal gases or
real solutions. Properties of real gas mixtures can be calcu-
lated through residual properties. For example, applying the
definition of residual property to G we get
( 6. 162) G = G ig + G R
whe r e G R is the residual Gibbs energy (defined as G - Gig).
G R is related to q~i by Eq. (6.128), which when combined with
the above equation gives
(6.163) G = G ig + RT Ey i ln~i
Furthermore from thermodynamic relations one can show
that [ 1]
H=Y~. y i Hi i g - RT2 ~y i ( Ol n~) i ~
\ or ] , , ,
(6.164)
V= y~yiv/ig + RT~] y i I~
\ - Sg - / T,, '
Calculation of properties of ideal gases have been discussed
in Section 6.3, therefore, from the knowledge of fugacity co-
efficients one can calculate properties of real gases.
Similarly for real liquid solutions a property can be cal-
culated from the knowledge of excess property. Properties of
ideal solutions are given by Eqs. (6.89)-(6.92). Property of
a real solution can be calculated from knowledge of excess
property and ideal solution property using Eq. (6.83):
(6.165) M ~--- M ig "4- M E
where M E is the excess property and can be calculated from
activity coefficients. For example, G ~ can be calculated from
Eq. (6.138). Similarly V E and H E can be calculated from yi
and H and V of the solution may be calculated from the fol-
lowing relations:
( O ln yi "]
H = H id - - RT 2 Y'~xi \ ~T~j l , ~i
(6.166)
(01n),i ~
V = V ia + RT ~_ ,xi \ - - ~- - , / Tm
Once G, H, and V are known, all other properties can be cal-
culated from appropriate thermodynamic relations discussed
in Section 6.1.
Another common way of determining thermophysical pro-
perties is through thermodynamic diagrams. I n these dia-
grams various properties such as H, S, V , T, and P for both
liquid and vapor phases of a pure substance are graphically
shown. One type of these di agrams is the P- H diagram that
is shown in Fig. 6.12 for methane as given by the GPA [28].
Such diagrams are available for many industrially important
pure compounds [28]. Most of these thermodynamic charts
and computer programs were developed by NIST [29]. Val-
ues used to construct such diagrams are calculated through
thermodynamic models discussed in this chapter. While these
diagrams are easy to use, but it is hard to determine an accu-
rate value from the graph because of difficulty in reading the
values. I n addition they are not suitable for computer appli-
cations. However, these figures are useful for the purpose of
evaluation of an estimated property from a thermodynamic
model. Other types of these diagrams are also available. The
H- S diagram known as Mollier diagram is usually used to
graphically correlate properties of refrigerant fluids.
6. 8 FOR MUL A T I ON OF P H A S E EQUI L I B R I A
P R OB L EM S FOR M I X T UR ES
I n this section equations needed for various phase equi-
librium calculations for mixtures are presented. Two cases
of vapor-liquid equilibria (VLE) and liquid-solid equilibria
(LSE) are considered due to their wide application in the
petroleum industry, as will be seen in Chapter 9.
6. 8. 1 Criteria f or Mi xt ure Ph as e Equi l i b ri a
The criteria for phase equilibrium is set by mi ni mum Gibbs
free energy, which requires derivative of G to be zero at the
conditions where the system is in thermodynamic equilib-
rium as shown by Eq. (6.108). Gibbs energy varies with T, P,
and x4. At fixed T and P, one can determine x~ that is when G is
minimized or at a fixed T (or P) and x~, equilibrium pressure
(or temperature) can be found by minimizing G. At different
pressures functionality of G with x~ at a fixed temperature
varies. Baker et al. [29] have discussed variation of Gibbs en-
ergy with composition. A typical curve is shown in Fig. 6.13.
To avoid a false solution to find equilibrium conditions, there
is a second constraint set by the second derivative of G as
[17, 20, 30]
(0G)r,p = 0
(6.167) (02G)r,e > 0
This discussion is known as stability criteria and it has re-
ceived significant attention by reservoir engineers in analysis
-~ l l 0 -I ]1~0 -I ~ 00 -]tl70 -1 840 -I I I 10 -1 7 1 ~ -0~ 0 -ff~ 0
100OO
-~ 80 -NS0 -1 420 -1 5~ 0 -1 56 0 - I ~ -/, ~ 00 -I ]PT0 -a40
- - - 10OO0
100E
lOC
~O00
I 00
1t11 . . . . . . . .
-1580 -I S30 -lg00 -1 87 0 -181,0 -1 81 0 -1 7 80 -1 " /~ 0 -1 7 2(I -l ! 1 0 -i6t)0 -1 1 3 0 -1100 -1S70 -1~140 ~1510 -1 4~ -1 450 -14ZO -1350 -1 3 6 0 -1 3 3 0 -1300 -1 27 0 -1240
SC. ewn = ~ ~.~. ~ En t h a l p y, Bt u / I b Me t h a n e
FIG. 6. 1 2- - T he P - H di ag r am f or met hane. Uni t conv ersi on: ~ = ~ 1.8 + 3 2 psi a - -
1 4. 504 bar. T ak en wi th permi ssi on f rom Ref. [ 27].
of fluid phase equilibrium of pet rol eum mixtures. Furt her dis-
cussion regarding phase stability is given in a number of re-
cent references [17, 20, 31].
Derivation of the general formul a for equilibrium condi-
tions in t erms of chemical potential and fugacity for multi-
component systems is shown here. Consider a mixture of N
t5
0 1.0
Composition, xi
FI G. 6. 1 3 - - A sampl e v ari at i on of G i bbs en-
erg y v er sus composi t i on for a bi nary syst em
at const ant T and P.
component s with two phases of 0t and ft. Applying Eq. (6.79)
to total Gibbs energy, G t, and taking the derivative of G t with
respect to r~ at constant T and P and combi ng with the Gi bbs-
Duhem equation (Eq. 6.81) gives the following relation:
(6.168) dG t = ~ / 2i dn/
i
where /2i is the chemical potential defined by Eq. (6.115).
Combining Eqs. (6.167) and (6.168) for all phases of the sys-
t em at equilibrium gives
(6.169) E ~ dn~ d- ~/~/~dn/~ - - 0
i i
Since ni = n~ + nf and r~ is constant (closed system without
chemical reaction), therefore, dn~ -- - dnf , which by substi-
tuting into the above equation leads to the following conclu-
sion:
(6.170) /27 --/2/~ (at constant T and P)
This relation must apply to all component s when the system
is in equilibrium. I f there are more than two phases (i.e., ~, r ,
y . . . . ) the same approach leads to the following conclusion:
(6.17 I) /2~ =/~/~ =/ 21 . . . . for every i at constant T and P
Using the relation between fii and~ given by Eq. (6.119) we
get
(6.172) / ~ --/~ --/~Y . . . . for ever y/ at constant T and P
Equations (6.171) or (6.172) are the basis for formulation
of mixture phase equilibrium calculations. Application of
Eq. (6.172) to VLE gives
(6.173) /~V(T, P, Yi) =~L( T, P, xi)
for all i components at constant T and P
For SLE, Eq. (6.172) becomes
(6.174) ~S(T, P,x s) =/~ L(T, P,x L)
for all i components at constant T and P
where x s is the mole fraction of i in the solid phase. Simi-
larly Eq. (6.172) can be used in liquid-liquid equilibria (LLE),
solid-liquid-vapor equilibria (SLVE), or vapor-liquid-liquid
equilibria (VLLE).
6. 8 . 2 Vapor- Li qui d Equi l i bri a- - - Gas S ol ub i l i t y
i n Li qui ds
In this section general relations for VLE and specific relations
developed for certain systems such as Raoult's and Henry' s
laws are presented. For high pressure VLE calculations equi-
librium ratio (Ki) is defined and its methods of estimation for
hydrocarbon systems are presented.
6. 8. 2. 1 Formulation of V apor-Liquid
Equilibria Relations
Formulation of VLE calculations requires substitution of rela-
tions for/~ v and/~ L from Eqs. (6.113) and (6.114). Combining
Eqs. (6.114) and (6.173) gives the following relation:
(6.175) yiqbV P = xi Yi fi L
where f/L is the fugacity of pure liquid i at T and P of the mix-
ture and it may be calculated through Eq. (6.136). The activity
coefficient ~/i is alSO a temperature-dependent parameter in
addition to x4. Another general VLE relation may be obtained
when bot h~ v and]~ L are expressed in terms of fugacity coef-
ficients ~v and ~L through Eq. (6.113) and are substituted in
Eq. (6.173):
(6.176) Yi~pV(T, P, Yi) = xi~L(T, P, xi)
where pressure P from both sides of the equation is dropped.
Equation (6.176) is essentially the same as Eq. (6.175) and
the activity coefficient can be related to fugacity coefficient
as [17]
(6.177) In Fi = ln~i(T, P, x4) - ln@(T, P)
where ~i(T, P, xi) is the fugacity coefficient of i in the liquid
mixture and r P) is fugacity coefficient of pure liquid i at
T and P of mixture. In fact one may use an EOS to calcu-
late Fi by calculating ~i and @ for the liquid phase through
Eq. (6.126). Application of PR EOS in the above equation, at
x~ ~ 0, will result in the following relation for calculation of
activity coefficient at infinite dilution for component 1 (F~)
in a binary system of components 1 and 2 at T and P [17]:
(6.178)
bl , /' Zl -- B1 ~l
l n) , ~ = ~2(Z2 - 1) - ( Z1 - 1) + m ~ )
A1
+ 2_ _ ~i in ( Z~+ 2.414B1
- 0.414B1 )
_ ( al 2P~ 1 l n( z~+2. 414B2 ~
blA2 ln( Z2 +2. 414Bz']
-~ 2~B2b~2 \ Z2 ~ 1
where a12 = a~/2a~/2(I -- k12) in which k12 is the binary inter-
action parameter. Parameters a, b, A, and B for PR EOS are
given in Table 5.1. Z1 and Z2 are the compressibility factor
for components 1 and 2 calculated from the PR EOS.
The main difference between Eqs. (6.175) and (6.176) for
VLE calculations is in their applications. Equation (6.176)
is particularly useful when bot h ~v and q~L are calculated
from equations of state. Cubic EOSs generally work well in
the VLE calculation of petroleum systems at high pressures
through this equation. ~v and ~/L may be calculated through
Eq. (6.126) with use of appropriate composition and Z; that
is, x L and Z L must be used in calculation of ~, while yV
and ZVare used in calculation of ~v. Binary interaction coeffi-
cients (BIPs) given in Table 5.3 must be used when dissimilar
(very light and very heavy or nonhydrocarhon and hydrocar-
bon) molecules exist in a mixture. However, as mentioned
earlier there is no need for use of volume translation or shift
parameter in calculation of ~v and ~/L for use in Eq. (6.176).
At low and moderate pressures use of Eq. (6.175) with
activity coefficient models is more accurate t han use of
Eq. (6.176) with an EOS. Assuming const ant V/L and substi-
tuting Eq. (6.136) into Eq. (6.175) we have
(6.179)
Yi~ vP xiFi~ bsatP?at [ V/L(p _ p/sat)]
= ~ ~ e x p L ) @ J
where the effect of pressure on the liquid molar volume is
neglected and saturated liquid molar volume V/sat may be used
for V/t. As discussed in Section 6.5, the vapor pressure p/sat
is a function of temperature and the highest temperature at
which p/sat can be calculated is Tc, where p/sat = Pc. Therefore,
Eq. (6.179) cannot be applied to a component in a mixture at
which T > Tc. For ideal liquid solutions or those systems that
follow Lewis rule (Section 6.6.3), the activity coefficient for
all components is unity (yi = 1). If pressure P and saturation
pressure p/sat are low and the gas phase can be considered
as an ideal gas, t hen 6v and q~at are unity and the Poynting
factor is also unity; therefore, the above relation reduces to
the following simple form:
(6.180) Yi P = ~ p~at
This is the simplest VLE relation and is known as the Raoult's
law. This rule only applies to ideal solutions such as benzene-
toluene mixture at pressures near or below 1 atm. If the gas
phase is ideal gas, but the liquid is not ideal solution then
Eq. (6.180) reduces to
(6.181) yiP - ~- X./yi P/sat
This rel at i on also known as modified Raouh' s l aw is valid for
noni deal syst ems but at pressures of I at m or less where t he
gas phase is consi dered ideal gas. We know t hat as x/--~ 1 (to-
ward a pure component ) t hus yi --~ 1 and t herefore Eq. (6.181)
reduces to Raoult' s law even for a real solution. Noni deal sys-
t ems wi t h ~/i > 1 show positive deviation while wi t h ~'i < 1
show negative deviation f r om the Raoult' s law. One direct ap-
pl i cat i on of modified Raouh' s law is to calculate composi t i on
of a comp ound in the air when it is vapori zed f r om its pure
liquid phase (xi = 1, Yi = I).
(6.182) YiP = psat
Since for ideal gas mi xt ures vol ume and mol e fract i ons are
t he same t herefore we have
(6.183) vol% of i in air = p/sat
Pa
(for vapori zat i on of pure liquid i)
where Pa is at mospheri c pressure. This is the same as Eq.
(2.11) t hat was used to calculate amount of a gas in the air for
fl ammabi l i t y test. Behavi or of ideal and noni deal systems is
shown in Fig. 6.14 t hr ough Txy and Pxy di agrams. Calculation
of bubbl e and dew poi nt pressures and generat i on of such
di agrams will be di scussed in Chapt er 9.
6. 8. 2. 2 Solubility of Gases in Liquids- - Henry' s Law
Anot her i mp or t ant VLE relation is t he rel at i on for gas solubil-
ity in liquids. Many years ago it has been observed t hat solu-
bility of gases in liquids (x/) is proport i onal to partial pressure
of comp onent in the gas phase (Yi P), whi ch can be formul at ed
as [21]
(6.184) yi P = k~x~
This relation is known as Henry's law and the proport i on-
ality const ant k~ is called Henry' s constant. ]q-solvent has t he
uni t of pressure per mol e ( or weight) fract i on and for any
given solute and solvent syst em is a funct i on of t emperat ure.
Henry' s law is a good approxi mat i on when pressure is l ow
( not exceeding 5-10 bar) and the solute concent rat i on in the
solvent, x/, is low ( not exceeding 0~03) and t he t emperat ure
is well bel ow the critical t emperat ure of solvent [21]. Henry' s
law is exact as x/--~ 0. I n fact t hr ough appl i cat i on of Gi bbs-
Duhem equat i on in t erms of Yi (Eq. 6.141), it can be shown
t hat for a bi nary syst em when Henry' s law is valid for one
comp onent t he Raoult' s law is valid for the ot her comp onent
(see Probl em 6.32). Equat i on (6.184) may be applied t o gases
at hi gher pressures by mul t i pl yi ng the left side of equat i on
by~ v.
sat
T:
T
P-Const.
V
Dew point
W-Xl ~
L Bubble point
T~ t p~at
T-Const.
L
Dew point V
p~at
0 1. 0
xbyl
(a)Txy diagram for an ideal binary system
Xl,Yl
(b)Pxy diagram for an ideal binary system
L I
I az az
~,xl = yl
0 1.0
xj~yl
(c)Txy diagram for a real binary system
azeotropi {
x =yl J
0 1.
Xl,Yl
(d)Pxy diagram for a real binary system
FIG. 6.14--Txy and Pxy diagrams for ideal and nonideal systems.
/ " kl
Constant T and P ,'
,
t t
i
t 1
t t
t
f
s
Henry's Law /
t t
AL t "
t'1 = klx] /
i t
~L / f~ (Pure l)
S 9 sss sj. SS~S~SJS'~
/ . - ~ L =x,fL
1.0
XI
FIG. 6.15~Variation o f ~ with x~ in a binary liquid solu-
tion and comparison with its values from Henry' s law and
Lewis rule.
The RHS of Eq. (6.184) is~ L and in fact the exact definition
of Henry' s constant is [1, 21]
(6.185) /q - lim it~_~0 ( ~)
Therefore, k/ is in fact the slope of ]~L versus x~ at x~ = 0.
This is demonst rat ed in Fig. 6.15 for a binary system. The
Henry' s law is valid at low values of xl ( ~< 0.03) while as
Xl ~ 1, the system follows Raoult' s law. Henry' s constant gen-
erally decreases with increase in t emperat ure and increases
with increase in pressure. However, there are cases that that k4
increases with increase in t emperat ure such as Henry' s con-
stant for H2S and NH3 in wat er [21]. Generally with good
approximation, effect of pressure on Henry' s constant is ne-
glected and ki is considered only as a function of t emperat ure.
Henry' s law constant for a solute ( component i) in a solvent
can be estimated from an EOS t hrough liquid phase fugacity
coefficient at infinite dilution (q~L,~ = lim~_~0 q~L) [21].
(6.186) k~ = ~L,~p
P16cker et al. [33] calculated/ q using Lee-Kesler EOS t hrough
calculation of ~/L'~ the above equation for solute hydro-
gen ( component 1) in various solvents versus in t emperat ure
range of 295-475 K. Their calculated values of/ q for HE in
n-C16 are presented in Fig. 6.16 for the t emperat ure range of
0-200 ~ These calculated values are in good agreement with
the measured values. The equation used for extrapolation of
data is also given in the same figure that reproduce original
data with an average deviation of 1%. Another useful rela-
tion for the Henry' s constant is obtained by combining Eqs.
(6.177) and (6.186):
(6.187) /q = ),/~ f L
1300
1200
1100
1000
o
900
800
m
700
600
FIG.
0 50 100 150 200
Temperature,~
6.16--Henry' s constant for hydrogen in n-
hex adecane( ~ C1 6H~ ) .
where yi ~ is the activity coefficient at infinite dilution and f/L
is the fugacity of pure liquid i at T and P of the system. Where
if yi ~ is calculated t hrough Eq. (6.178) and the PR EOS is used
to calculate liquid fugacity coefficient ()rE = ~bLp), Henry' s
constant can be calculated from the PR EOS.
The general mixing rule for calculation of Henry' s constant
for a solute in a mixed solvent is given by Prausnitz [21]. For
t ernary systems, Henry' s law constant for component 1 into
a mixed solvent (2 and 3) is given by the following relation:
(6.188)
lnkl,M = x2 Ink1,2 + X3 lnk],3 - o/ 23x2x3
- +
O/23
2RT
where ~ is the solubility parameter, V is mol ar volume, and
x is the mole fraction. This relation may be used to calcu-
late activity coefficient of component 1 in a t ernary mixture.
Herein we assume that the mixture is a binary system of com-
ponent s 1 and M, where M represents component s 2 and 3
together (xu = 1 - x0. Activity coefficient at infinite dilution
is calculated t hrough Eq. (6.187) as Yl,M~176 = kx,M / f L. Once Yl,g~176
is known, it can be used to calculate paramet ers in an activity
coefficient model as discussed earlier.
The mai n application of Henry' s law is to calculate solu-
bility of gases in liquids where the solubility is limited (small
xl). For example, solubilities of hydrocarbons in wat er or light
hydrocarbons in heavy oils are very limited and Henry' s law
may be used to estimate the solubility of a solute in a solvent.
The general relation for calculation of solubility is t hrough
Eq. (6.147). For various homol ogous groups, Eq. (6.149) may
be used to estimate solubility par amet er at 25~ One ma-
j or probl em in using Eq. (6.179) occurs when it is used to
calculate solubility of light gases (C1, C2, or C3) in oils at tem-
peratures greater t han Tc of these components. I n such cases
calculation of p~at is not possible since the component is not
in a liquid form. For such situations Eq. (6.175) must be used
and f/L represents fugacity of component i in a hypothetical
liquid state. I f solute (light gas) is indicated as component 1,
the following equation should be used to calculate fugacity
10.0
t~ ~ 1.0
O.1
o Data
0.5 1.0 1.5 2.0 2.5 3.0
T r
FIG. 6.1 7 --Fug acity of hypothetical liquid at 1.013 bar. Data taken from
Ref. [21] for CH4, C2H4, C2Hs, and Na.
of pure comp onent 1 as a hypot het i cal liquid when Tr > 1
[21]:
(6.189) fL= fr~ ]
where f~L is the reduced hypot het i cal liquid fugaci t y at pres-
sure of 1 at m (fr ~ = f~ and it shoul d be cal cul at ed f r om
Fig. 6.17 as explained in Ref. [21]. Dat a on foe Of C1, C2, N2,
CO, and CO2 have been used t o const ruct this figure. For con-
venience, values obt ai ned f r om Fig. 6.17 are represent ed by
t he following equat i on [23]:
(6.190) f ; L= e xp ( 7. 902 8.19643Tr 3. 081nTr)
where Tr is the reduced t emperat ure. Dat a obt ai ned f r om Fig.
6.17 in the range of 0.95 < T~ < 2.6 are used to generat e t he
above correl at i on and it reproduces the gr aph wi t h %AAD of
1.3. This equat i on is not valid for T~ > 3 and for comp ounds
such as H2.
I f the vapor phase is pure comp onent 1 and is in cont act
with solvent 2 at pressure P and t emperat ure T, t hen its solu-
bility in t erms of mol e fraction, xl, is f ound f r om Eq. (6.168)
a s
~' P
(6.191) xl - Yl fL
where ~b v is the fugaci t y coefficient of pure gas ( comp onent
1) at T and P. Yi is the activity coefficient of solute 1 in sol-
vent 2, whi ch is a funct i on of xl. f~ is the fugaci t y of pure
comp onent 1 as liquid at T and P and it may be calculated
f r om Eq. (6.189) for light gases when T > 0.95Tcl. I t is clear
t hat to find xl f r om Eq. (6.191) a t ri al -and-error procedure is
requi red since ~/1 is a funct i on of xl. To start t he calculations
an initial value of xl is normal l y obt ai ned from Eq. (6.191) by
assumi ng 7/1 = 1. As an alternative met hod, since values of Xl
are normal l y small, initial value ofxl can be assumed as zero.
For hydr ocar bon syst ems Y1 may be calculated f r om regul ar
sol ut i on theory. The following exampl e shows t he met hod.
Example 6./O--Estimate solubility of met hane in n-pent ane
at 100~ when the pressure of met hane is 0.01 bar.
SolutionnMethane is consi dered as t he solute ( comp onent
1) and n-pent ane is t he solvent ( component 2). Properties of
met hane are t aken f r om Table 2.1 as M = 16, Tc = 190.4 K,
and Pc = 46 bar. T = 373.15 K (Tr = 1.9598) and P = 0.01
bar. Since the pressure is quite l ow the gas phase is ideal
gas, t hus ~b v = 1.0. I n Eq. (6.191) onl y Y1 and f~ must be
calculated. For C1-C5 system, the regul ar sol ut i on t heory
can be used to calculate Yx t hr ough Eq. (6.145). Fr om Ta-
ble 6.11, at 298 K, V( = 52 and V L = 116 cm3/ mol, ~i -- 11.6,
and ~2 = 14.52 (J/cm3) 1/z. Assumi ng ~a ~ 0 (~2 -~ 1), from
Eq. (6.145), I n 7/1 = 0.143 or Yl = 1.154. Since Tr > 1, fL is
calculated f r om Eq. (6.189). Fr om Eq. (6.190), f7 L = 5.107
and f r om Eq. (6.189): fL = 234.8 bar. Therefore, t he solu-
bility is xl = 0.01/ ( 1.1519 x 234.8) = 3.7 x 10 -5. Since xl is
very small, the initial guess for @t = 0 is accept abl e and t here
is no need for recal cul at i on of y~. Therefore, t he answer is
3.7 x 10 -5, whi ch is close to value of 4 10 -5 as given in Ref.
[21]. #
One type of useful dat a is correl at i on of mol e fract i on sol-
ubility of gases in wat er at 1.013 bar (1 atm) . Once this infor-
mat i on is available, it can be used to det ermi ne solubility at
ot her elevated pressures t hr ough Henry' s law. Mole fract i on
solubility is given in the following correl at i ons for a number
of gases in wat er versus t emperat ure as given by Sandi er [22]:
met hane ( 275-328) l nx = - 416. 159289 + 15557. 5631/ T + 65.25525911n T - 0. 0616975729T
(6.192)
et hane ( 275-323) l nx = -11268. 4007 + 221617. 099/ T + 2158.4217911n T - 7.18779402T + 4.0501192 x 10-3T 2
propane ( 273-347) l nx = - 316. 46 + 15921. 2/ T + 44.32431 I n T
n-but ane ( 276-349) l nx = - 290. 238 + 15055. 5/ T + 40.19491n T
/ -but ane ( 278-343) l nx = 96.1066 - 2472. 33/ T - 17. 36631nT
H2S ( 273-333) l nx = - 149. 537 + 8226. 54/ T + 20.2308 I n T + 0. 00129405T
CO2 ( 273-373) l nx ---- - 4957. 824 + 105, 288. 4/ T + 933.17 I n T - 2. 854886T + 1.480857 10-aT 2
N2 ( 273-348) l nx -- - 181. 587 + 8632. 129/ T + 24.79808 I n T
H2 ( 274-339) l nx = - 180. 054 + 6993. 54/ T + 26.31211n T - 0.0150432T
For each gas the range of t emperat ure (in kelvin) at whi ch
t he correl at i on is applicable is given in parenthesis. T is t he
absol ut e t emperat ure in kelvin and x is the mol e fract i on of
dissolved gas in wat er at 1.013 bar. Henry' s const ant of light
hydr ocar bon gases (C1, C2, C3, C4, and i - C4) in wat er may be
est i mat ed f r om t he following correl at i on as suggested by t he
API -TDB [5]:
(6.193) lnkgas-water = A1 + A2T + - ~ + A4 l nT
where kgas-water is t he Henry' s const ant of a light hydr ocar bon
gas in wat er in the uni t of bar per mol e fract i on and T is t he
absol ut e t emperat ure in kelvin. The coefficients A1-A4 and
the range of T and P are given in Table 6.12.
To calculate solubility of a hydr ocar bon liquid mi xt ure in
t he aqueous phase, t he following relation may be used:
where ~ is the solubility of comp onent i in the wat er when it
is in a liquid mixture, xi is t he solubility of pure i in t he water.
/~L is the fugacity of i in t he mi xt ure of liquid hydr ocar bon
phase and f/L is t he fugacity of pure i in t he liquid phase. More
accurat e calculations can be performed t hr ough l i qui d-l i qui d
phase equi l i bri um calculations.
For cal cul at i on of solubility of wat er in hydr ocar bons t he
following correl at i on is proposed by t he API -TDB [5]:
1ogI0xH20 = -- ( CH weight4200 ratio + 1050) x (--1T - 0"0016)
(6.194)
where T is in kelvin and xH2O is t he mol e fract i on of wat er
in liquid hydr ocar bon at 1.013 bar. CH wei ght ratio is t he
carbon-t o-hydrogen wei ght ratio. This equat i on is known as
Hi bbard correl at i on and shoul d be used for pent anes and
heavier hydr ocar bons (C5+). The reliability of this met hod is
4-20% [5]. I f this equat i on is applied to undefi ned hydrocar-
bon fractions, t he CH wei ght ratio may be est i mat ed from t he
met hods di scussed in Section 2.6.3 of Chapt er 2. However,
API -TDB [5] r ecommends the following equat i on for calcula-
t i on of solubility of wat er in some undefi ned pet rol eum frac-
tions:
1841.3
nap ht ha log10 xn2o = 2.94
T
2387.3
kerosene log10 XH20 = 2.74
T
(6.195) 1708.3
paraffinic oil log10 xu2o = 2.69
T
1766.8
gasoline log10 xa~o = 2.63
T
I n the above equat i ons T is in kelvin and XH20 is t he mol e
fract i on of wat er in t he pet rol eum fraction. Obviously these
correl at i ons give approxi mat e values of wat er solubility as
composi t i on of each fract i on vary f r om one source t o another.
6. 8. 2. 3 Eq ui l i b r i um Rat i os (Ki V al ues)
The general formul a for VLE cal cul at i on is obt ai ned t hr ough
definition of a new par amet er called equilibrium ratio shown
by Ki :
(6.196) Ki - yi
xi
Ki is a di mensi onl ess p ar amet er and in general varies wi t h
T, P, and composi t i on of bot h liquid and vapor phases.
I n many references, equi l i bri um ratios are referred as Ki
value and can be calculated from combi ni ng Eq. (6.176) wi t h
Eq. (6.196) as in t he following form:
(6.197) Ki = ?bE(T' P, xi)
~aV(T, P, Yi)
I n hi gh-pressure VLE calculations, Ki values are cal cul at ed
from Eq. (6.197) t hr ough Eq. (6.126) for cal cul at i on of
fugacity coefficients wi t h use of cubi c equat i ons ( SRK or
PR). I n cal cul at i on of Ki values from a cubi c EOS use of
bi nary i nt eract i on paramet ers (BIPs) i nt roduced in Chapt er 5
is requi red specially when comp onent s such as N2, H2S, and
CO2 exist in the hydr ocar bon mixture. Also in mi xt ures when
the difference in mol ecul ar size of comp onent s is appreci abl e
TABLE 6. 12--Constants for Eq. (6.193) for estimation of Henry's constant for light gases in water [5].
Gas T range, K Pressure range, bar A1 A2 A3 A4 %AAD
Methane 274-444 1-31 569.29 0.107305 - 19537 -92.17 3.6
Ethane 279--444 1-28 109.42 -0.023090 - 8006.3 - 11.467 7.5
Propane 278-428 1-28 1114.68 0.205942 -39162.2 - 181.505 5.3
n-Butane 277-444 1-28 182. 41 -0.018160 - 11418.06 -22.455 6.2
/-Butane 278-378 1-10 1731.13 0.429534 -52318.06 -293.567 5.3
~ z f !
. . . . . . . . . . . . . . . . . . ~. o. . J _
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I I . . . . . . . . . .
_ 1
0,0~ ~ a m
3~ 300
P(BARs)
0,1
I 0
1
0,~
0.0~
3O 3OO
(a) (b)
FIG. 6. 1 8 - - Compari son of predicted equilibrium ratios (K~ values)
from PR EO S wi thout (a) and with ( b) use of interaction parameters, 9
Experimental data for a crude oil. Taken with permission from Ref. [32],
(i.e., C1 and some heavy compounds) use of BIPs is required.
Effect of BIPs in calculation of Ki values is demonstrated in
Fig. 6.18. If bot h the vapor and liquid phases are assumed as
ideal solutions, then by applying Eq. (6.132) the Lewis rule,
Eq. (6.197) becomes
~(~, P)
(6.198) Ki - 4~V(T, p)
where q~v and ~0/L are pure component fugacity coefficients
and Ki is independent of composition and depends only on
T and P. The main application of this equation is for light
hydrocarbons where their mixtures may be assumed as ideal
solution. For systems following Raoult's law (Eq. 6.180) the
Ki values can be calculated from the relation:
(6.199) Ki(T, P) = p/sat(T)
P
Equilibrium ratios may also be calculated from Eq. (6.181)
through calculation of activity coefficients for the liquid
phase.
Another met hod for calculation of Ki values of nonpolar
systems was developed by Chao and Seader in 1961 [34]. They
suggested a modification of Eq. (6.197) by replacing q~/L with
(Yi4J~), where q~]- is the fugacity coefficient of pure liquid i and
Fi is the activity coefficient of component i.
(6.200) K~ = Y~ = ),~ck~(Ta, Pa, r
x,
yi must be evaluated from Eq. (6.150)
~v must be evaluated from the Redlich-Kwong EOS
~bi L empirically developed correlation in terms of T~/, P~/, COl
must be evaluated with the Scatchard-Hildebrand regu-
lar solution relationship (Eq. 6.150). q~v must be evaluated
with the original Redlich-Kwong equation of state. Further-
more, Chao and Seader developed a generalized correlation
for calculation of ~0~ in terms of reduced temperature, pres-
sure, and acentric factor of pure component i (T,~, P~i, and
wi). Later Grayson and Streed [35] reformulated the corre-
lation for o/L to temperatures about 430~ ( ~ 800~ Some
process simulators (i.e., PRI/II [36]) use the Greyson-Streed
expression for q~/L. This met hod found wide industrial appli-
cations in the 1960s and 1970s; however, it should not be
used for systems containing polar compounds or compounds
with close boiling points (i.e., i-C4/ n-C4) . It should not be
used for temperatures below -17~ (0~ nor near the criti-
cal region where it does not recognize x~ = yi at the critical
point [37]. For systems composed of complex molecules such
as very heavy hydrocarbons, water, alcohol, ionic (i.e., salt,
surfactant), and polymeric systems, SAFT EOS may be used
for phase equilibrium calculations. Relations for convenient
calculation of fugacity coefficients and compressibility factor
are given by Li and Englezos [38].
Once Ki values for all components are known, various VLE
calculations can be made from the following general relation-
ship between xi and Yi:
(6.201) Yi = Kix~
Assuming ideal solution for hydrocarbons, Ki values at var-
ious temperature and pressure have been calculated for n-
paraffins from C1 to C10 and are presented graphically for
quick estimation. These charts as given by Gas Processor As-
sociation (GPA) [28] are given in Figs. 6.19-6.31 for various
6. THERM ODYNAM I C REL ATI ONS FOR PROPERTY ESTI M ATI ONS 2 7 1
1 0 2
1 ,0001
K = Y/x
PRESSURE, PSIA
3 0 4 5 0 6 7 8 9 1 0 0 2 3 0 0 4 5 0 0 6 7 8 9 1 , 0 0 0 2 3 , 0 0 0 4 6 7 8 9 1 0 , 0 0 0
1,000
X X x X X
\ X \ X
~X " X X
" -,
%%
x " ~ \ x x x
x x ,
"~% x x ~ x x x
x x , " x x " x
x x
x \ ~ x x x ~ x ~
x x x ~ x
\ x x x ,
x
x
\ x
x
x
\
K= Y/ x
0/ ] [ 01
1 0 2 3 0 4 5 0 6 7 8 9 1 0 0 2 3 0 0 4 5 0 0 6 7 8 9 1 , 0 0 0 2 3 , 0 0 0 4 6 7 8 9 1 0 , 0 0 0
PRESSURE, PSIA " ME T H A N E
CONV. PRESS. 5000 PSIA
FIG. 6. 1 9 ~ K~ v al ues of methane. Unit conv ersion: ~ 1 7 6 x 1 . 8 + 3 2 p s i a =
1 4. 504 x bar, Taken wi th permi ssi on f rom Ref. [ 28 ].
c o mpo ne nt s f r om me t h a ne t o de c ane and h y dr og e n sul fi de.
Equi l i b r i um rat i os are perh aps t h e mo s t i mpor t ant par ame t e r
f or h i g h - pres s ure VLE cal cul at i ons as de s c r i b e d i n Ch apt er 9.
For h y drocarb on s y s t e ms and res erv oi r fl ui ds t h ere are
s o me empi ri cal correl at i ons f or cal cul at i on of Ki v al ues. Th e
correl at i on pr opos e d b y Ho f f ma n et al. [ 3 9 ] i s w i de l y us e d i n
t h e i ndustry. Lat er S t andi ng [ 40] us ed v al ues of Ki report ed
b y Kat z and Ha c h mut h [ 4 1 ] on crude oi l and nat ural g as
s y s t e ms t o ob t ai n t h e f ol l owi ng equat i ons b as ed on t h e
Ho f f ma n ori g i nal correl at i on:
Ki = ( 1 ) 10 (a+c/r)
b 1
a = 0 . 0 3 8 5 + 6. 5 2 7 x 1 0 - 3 P + 3 . 1 5 5 x 1 0 - 5 p 2
c = 0 . 8 9 - 2 . 4 65 6 x 1 0 - 3 P - 7 . 3 62 61 x 1 0 - 6p 2
1 0
PRESSURE, PSI A ~"
2 3 0 4 50 6 7 8 9 1 00 2 3 00 4 500 6 z 8 9 1 , 000 2 3 , 000 4 8 7 S 81 0, 000
K = Y/x K = Y/x
7
' 6
~5
14
! 3
I
i
1
0.01 0.C
1 0 2 3 0 4 50 6 7 8@1 00 2 3 00 45006 7 89 1 , 000 it 3 , 000 4 6 7 89 1 0, 000
PRESSURE, PSI A ~" ETHANE
CONV. PRESS. 10,000 PSI A
FI G . 8 . 20- - - / ( i v al ues of et hane. Uni t conv er s i on: ~ = ~ x 1 . 8 + 3 2 p s i a =
1 4. 504 x bar. T ak en wi t h p er mi s s i on f r om Ref . [ 28 ].
where P is the pressure in bar and T is the temperature in
kelvin. These equations are restricted to pressures below 69
bar (-~1000 psia) and temperatures between 278-366 K (40-
200~ Values ofb and TB for these T and P ranges are given in
Tables 6.13 for some pure compounds and lumped C6 group.
These equations reproduce original data within 3.5% error.
For C7+ fractions the following equations are provided by
Standing [40]:
0 = 3.85 + 0.0135T + 0.02321P
(6.203) bT+ = 562.78 + 1800 - 2.36402
Ta,7+ = 167.22 + 33.250 - 0.539402
where T is in kelvin and P is in bar. It should be noted that
all the original equations and constants in Table 6.11 were
given in the English units and have been converted to the SI
units as presented here. As it can be seen in these equations
Ki is related only to T and P and they are independent of
composition and are based on the assumption that mixtures
behave like ideal solutions. These equations are referred as
Standing method and they are recommended for gas conden-
sate systems and are useful in calculations for surface sep-
arators. Katz and Hachmut h [41] originally recommended
that K7+ = 0.15Kn-c7, which has been used by Glaso [42] with
satisfactory results. As will be seen in Chapter 9, in VLE
1 0 2 3 0
6. THERM ODYNAM IC ~L ATI ONS FOR PROPERTY ESTIM ATIONS
PRESSURE, PSI A
4 50 6 7 89 1 00 2 3 00 45006 7 89 1 , 0C0 2 3 , 000 4 6 7 89 1 0, 000
2 7 3
,t,,
0.0(
1 0
001
2 3 0 4 50 6 7 8 9 1 00 2 3 00 4 500 6 7 8 9 1,0(X ] 2 3 , 000 4 8 7 8 9 1 0, 000
PRESSURE, PSI A " PRO PANE
FI G . 6. 2 1 - - Ki v a l ue s of p r op ane. Uni t conv er s i on: ~ = ~ x 1 . 8 -1- 3 2 psi a =
1 4. 504 bar, T ak en wi t h p er mi s s i on f r om Ref . [ 28 ].
calculations some initial Ki values are needed. Whitson [31]
suggests use of Wilson correlation for calculation of initial Ki
Values:
(6.204) Ki = exp [5.37 (1 + wi) (1 - Tffl ) ]
P~/
where Tu and Pu are the reduced temperature and pressure
as defined in Eq. (5.100) and wi is the acentric factor. It
can be shown that Wilson equation reduces to Hoffman-type
equation when the Edmister equation (Eq. 2.108) is used for
the acentric factor (see Problem 6.39).
Example 6. 11--Pure propane is in contact with a nonvolatile
oil (M = 550) at 134~ and pressure of 10 bar. Calculate Ki
value using the regular solution theory and Standing correla-
tion.
Solution--Consider the system as a binary system of com-
ponent 1 (propane) and component 2 (oil). Component 2 is
PRESSURE, PSI A ~'
10 2 3 0 4 50 6 7 8 9100 2 3 00 4 500 6 7 8 9 1 , 000 2 3 , 000 4 . . . . ~o
,%
O,OI 001
10 2 3 0 4 50 6 7 89 1 00 2 300 4 5008 7 89 1 , 000 2 3 , 0~ 4 6 7 89 1 0, 000
PRESSURE, PSI A - BUTANE
FI G. 6. 22mKz v al ues of / - but ane. Uni t conv er si on: ~ = ~ x 1 .8 + 3 2 psi a =
1 4. 504 x bar. T ak en wi t h per mi ssi on f rom Ref. [ 28 ].
in fact sol vent for comp onent 1, whi ch can be consi der ed as
sol ut e. Also for si mpl i ci t y consi der oil as a si ngl e car bon num-
ber wi t h mol ecul ar wei ght of 550. Thi s as s ump t i on does not
cause maj or er r or i n t he cal cul at i ons as p r op er t i es of p r op ane
ar e needed for t he cal cul at i on. Ki is defi ned by Eq. (6.196) as
K1 -- yl / xl . Si nce t he oil is nonvol at i l e t hus t he vap or p hase
is p ur e p r op ane and Yl = 1, t herefore, KI = 1/x1. TO cal cu-
l at e xl , a si mi l ar met hod as used in Examp l e 6.10 is fol l owed.
I n t hi s examp l e si nce P = 10 bar t he gas p hase is not i deal
and t o use Eq. (6.179), ~v mus t be cal cul at ed for p r op ane
at 10 bar and T = 134~ (407 K). F r om Table 2.1 for C3 we
have M = 44.1, SG = 0.5063, Tc = 369.83 K, Pc = 42.48 bar,
and to = 0.1523. T~ = 1.125, Pr = 0.235. Si nce Pr is low, ~ can
be conveni ent l y cal cul at ed f r om vi ri al EOS by Eq. (6.62) to-
get her wi t h Eq. (5.72) for cal cul at i on of t he second vi ri al co-
efficient. The r esul t is ~v __ 0.94. Cal cul at i on of ~'1 is si mi l ar
to Examp l e 6.10 wi t h use of Eq. (6.145). I t r equi r es p ar am-
et ers V L and ~ for bot h C3 and t he oil. Value of V L for Ca
as gi ven i n Table 6.10 seems to be l ower t han ext r ap ol at ed
val ue at 298 K. The mol ar vol ume of p r op ane at 298 K can
10 2 3 0
6. THERM ODYNAM IC RELATIONS FOR PROPERTY ESTIM ATIONS
PRESSURE, PSI A J'
4 50 6 7 89 1 00 2 3 00 4 5006 7 89 1 , 000 2 3,000 4 6 7 89 1 0. 000
= 100
2 7 5
K = Y/x~ I ~ ~ \ \ \ \ \ "~ ~ ~ ~ J J ~ . / / / / / / / / / I~ K = Ylx
0.0(
10
001
2 3 0 4 50 6 7 8 9100 8 3 00 4 500 6 7 8 9 1,C00 8 3,000 4 6 7 8 9 1 0, 000
PRESSURE, PSl A , n - BUT ANE
FI G. 6.23 - - - K i v al ues of n- but ane. Uni t conv er si on: ~ = ~ x 1 . 8 + 3 2 psi a =
be cal cul at ed from its density. Subst i t ut i ng SG = 0.5063 and
T = 298 K i n Eq. (5.127) gives densi t y at 25~ as 0.493 g/ cm 3
and t he mol ar vol ume is V L = 44.1/ 0.493 = 89.45 cma/ mol.
Si mi l arl y at 134~ we get V1L= 128.7 cm3/ mol. Value of
for Ca is given i n Table 6.10 as 8 = 13.9 (J/cm3) 1/2. Fr om
Eq. (4.7) and coefficients i n Table 4.5 for oil of M = 550, we
get d20 = 0.9234 g/ cm 3 and 82 = 8.342 (cal/cm3) 1/2. These val-
ues are very approxi mat e as oil is assumed as a single car bon
number . Densi t y is correct ed to 25~ t hr ough Eq. (2.115) as
d25 = 0.9123 g/ cm 3. Thus at 298 K for comp onent 2 ( solvent)
we have V f = 550/ 0.9123 = 602.9 cm3/ mol. To cal cul at e Yx
from Eq. (6.145), Xl is required. The i ni t i al val ue of Xl is
cal cul at ed t hr ough Eq. (6.191) assumi ng Yl = 1. Si nce Tr >
1, t he val ue of fL is cal cul at ed t hr ough Eqs. (6.189) and
(6.190) as fl L = 51.13 bar. Finally, t he val ue of yl is cal cul at ed
as 1.285, whi ch gives Xl = 0.94 x 10/ (1.285 x 51.13) = 0.144.
Thus, K1 = 1/ 0.144 = 6.9. To cal cul at e K1 from the St and-
i ng met hod, Eq. (6.202) shoul d he used. F r om Table 6.13 for
propane, b = 999.4 K and TB = 231.1 K, and from Eq. (6.202)
at 407 K and 10 bar, a = 0.1069, c = 0.8646, and K1 = 5.3. r
PRESSURE, PSI A
1 0 ~ 3 0 4 50 6 7 8 9 1 00 2 3 00 4 56 0 6 7 a 9 1 , 000 2 3 , 000 4 6 7 8 9 1 0~ 00,
K=Y/x K=Y/x
l I 0.001 0.001
1 0 2 3 0 4 50 s 7 a 9 1 00 2 3 00 4 500 s 7 a 9 1 , 000 2 3 , 000 4 6 7 8 9 1 0, 000
PRESSURE, PSI A ~ i - PENT ANE
CONY. PRESS. 10,000 PSI A
FI G . 6. 2 4~ Ki v al ues of i - p ent ane. Uni t conv er s i on: ~ = ~ x 1 . 8 + 3 2 p s i a =
1 4. 504 bar. T ak en wi t h p er mi s s i on f r om Ref . [ 28 ].
To summarize methods of VLE calculations, recommended
methods for some special cases are given in Table 6.14 [37].
6.8.3 Solid-Liquid Equi l i br i a - - Sol i d Sol ubi l i t y
Formulation of SLE is similar to that of VLE and it is made
through Eq. (6.174) with equality of fugacity of i in solid and
liquid phases, where the relations for calculation of]~ s and
E L are given in Section 6.6. To formulate solubility of a solid
in a liquid, the solid phase is assumed pure,]~ s = f/s, and the
above relation becomes
(6.205) f/s = x~yi f/L
by substituting f/s from Eq. (6.155) we get
In 1 A///f ( 1_ ~) + ACpi ( ~)
Yi ~- RTui - - R- 1 -
ACpi, Tui
(6.206) + - ~ m T
10 2 3 0
PRESSURE, PSI A "
4 50 6 7 89 1 00 2 3 00 4 5006 7 89 1 , 000 2 3 , 000 4 6 7 89 1 0, 000
277
K= Y/x K= Y/x
10 2 3 0 4 50 8 7 8 9100 2 3 00 4 500 6 7 8 81 , 000 2 3 , 000 4 6 7 8 9 1 0, 000
PRESSURE, PSI A ) n - PENTANE
FI G . 6. 2 5 - - / ( / v al ues of n- p ent ane. Uni t conv er s i on: ~ = ~ x 1 . 8 + 3 2 psi a =
1 4. 504 x bar. T ak en wi t h p e r mi s s i on f r om Ref . [ 28 ].
I t should be noted that this equation can be used to calculate
solubility of a pure solid into a solvent, yi (a function ofx~) can
be calculated from met hods given in Section 6.6.6 and x~ must
be found by trial-and-error procedure with initial value of xl
calculated at yi = 1.0. However, for ideal solutions where yi
is equal to unity the above equation can be used to calculate
solubility directly. Since actual values of ACpi/R are gener-
ally small (see Fig. 6.11) with a fair approxi mat i on the above
relation for ideal solutions can be simplified as [17, 21, 43]
(6.207) x/ L L<IMi
= exp ~ 1 - -
The above equation provides a quick way of calculating sol-
ubility of a solid into a solvent where chemical nature of so-
lute is similar to that of solvent; therefore, only properties of
solute are needed. Calculation of heat of fusion (AH/f) was
discussed in Section 6.6.6 and calculation of freezing point
(TMi) was discussed in Section 2.6.4. Solubility of naphthalene
in several hydrocarbons is given in Ref. [43]. At 20~ mole
fraction of solid napht hal ene in solvents hexane, benzene,
and toluene is 0.09, 0.241, and 0.224, respectively. Naphtha-
lene ( aromatics) has higher solubility in benzene and toluene
(also aromatics) t han in hexane (a paraffinic). Naphthalene
PRESSURE, PSI A
K=Yj = Y/x
000( .0001
1 0 2 3 0 4 50 6 7 89 1 00 2 3 00 4 5006 7 89 1 , 000 2 3 , 000 4 6 7 89 1 0, 000
PRESSURE, PSI A - H EX ANE
FI G. 6. 2 6~ KI v al ues of hex ane. Uni t conv er si on: ~ = ~ x 1 . 8 + 3 2 psi a =
1 4. 504 x bar. T ak en wi t h per mi ssi on f rom Ref. [ 28 ],
has A/-//f of 18.58 kJ/ mol and its melting point is 80~ [43].
Solubility of napht hal ene calctflated t hrough Eq. (6.207) is
xl = 0.27. Better prediction can be obtained by calculating yl
for each system. For example, t hrough regular solution theory
for napht hal ene ( 1) -toluene (2) system at 20 ~ C yl is calculated
as yl = 1.17 ( met hod of calculation was shown in Exampl e
6.10). The corrected solubility is 0.27/1.17 = 0.23, which is in
good agreement with the experimental value of 0.224. As dis-
cussed before, compounds with similar structures have bet t er
solubility.
For solid precipitation such as wax precipitation, the solid
phase is a mixture and the general relation for SLE is given by
Eq. (6.174). A SLE ratio, K sL, can he defined similar to VLE
ratio as
(6.208) x s ---- KiSLx L
By combining Eqs. (6.114) and (6.161) with Eq. (6.174) and
use of the above definition we get [17]
(6.209) K sL = f~L(T' P)yiL(T' P' xL)
f/S(T, P )yiS(T, P, x s)
10 2 3 0
6. THERM ODYNAM IC RELATIONS FOR PROPERTY ESTIM ATIONS
PRESSURE, PSI A ~"
4 50 6 7 8 9 1 O0 2 3 00 4 500 6 7 8 9 1 , 000 2 3 , 000 4 6 9 8 91=%000
2 7 9
%
K = Y/x K = Y/x
" ~ 0
,0001
1 0 2 3 0 4 50 6 7 89 1 00 2 3 00 45006 7 89 1 , 000 2 3 , 000 4 6 7 89 1 0, 000
PRESSURE, PSI A 9 H EP' ] " A~
FI G. 6, 2 7 - - KI v al ues of hept ane. Uni t conv er si on: ~ = ~ x 1 . 8 + 3 2 p s i a =
1 4. 504 x bar, T ak en wi t h per mi ssi on f rom Ref. [ 28 ],
where for ideal liquid and solid solutions yi L and yi s are
unity and the pure component fugacity ratio f/L/f/S can be
calculated from Eq. (6.155). For nonideal solutions, yi s may
be calculated from Eq. (6.153). However, 8 s calculated from
Eq. (6.153) must be used for 8 in Eq. (6.154). Equations
(6.208) and (6.209) can be used to construct freezing/melting
points or liquid-solid phase diagram ( Tx Sx L) based on SLE
calculations as shown in Chapter 9. For an ideal binary sys-
tem the Tx Sx L diagram is shown in Fig. 6.32. Such figures
are useful to determine the temperature at which freezing
begins for a mixture (see Problem 6.29). Multicomponent
SLE calculations become very easy once the stability analysis
is made. From stability analysis consideration a component
in a liquid mixture with mole fraction zi may exist as a pure
solid if the following inequality holds [ 17]:
(6.210) f i ( T, P, zi ) - ~S(T, P) > 0
where ~s is fugacity of pure solid i. This equation is the ba-
sis of judgment to see if a component in a liquid mixture
will precipitate as solid or not. The answer is yes if the above
PRESSURE, PSI A
1 0 2 3 0 4 506 7 89 1 00 2 3 00 45006 7 89 1 , 000 2 3 , 000 4 6 7 89110(~000
Plotted ftom19471al~ulalk~s ~f 9
B G.G. Brown, University of Mich-
Egan.Ex lmpo/ated and drawn by
The Fruor Cotp, Ltd, ~n1957,
K=Y/x31,,. " ~ " ~ ~ ~ ~ ~oo... ~ ~ J / / / / l l l l / / / I I~ K=Y/x
1 0 2 3 0 4 50 6 7 8 9 1 00 2 3 00 4 500 6 7 8 9 1 , 000 2 3 , 000 4 6 7 8 9 1 0, 000
PRESSURE, PSI A ~" O CT ~ d~ I -E
CONV PRESS. 10,000 PSI A
FIG. 6. 28 --Ki values of octane. Unit conversion: ~ 1 7 6 x 1 . 8 - t - 3 2p s i a=
14.504 x bar. Taken with permission from Ref. [28].
inequality holds for that component. Equality in the above
equation is equivalent to Eq. (6.206). The same criteria apply
to precipitation of a component from a gas mixture, where in
the above inequality fi(T, P, zi) would refer to fugacity of / i n
the gas phase with mole fraction zi. Similar principle applies
in format i on of liquid i from a gas mi xt ure when the tem-
perat ure decreases. One mai n application of this inequality is
to determine the t emperat ure at which solid begins to form
from an oil. This t emperat ure is equivalent to cloud point
of the oil. Applications of Eqs. (6.208) and (6.209) for calcu-
lation of cloud point and wax formation are demonst rat ed
in Chapter 9. Full description of a t hermodynami c model for
wax precipitation is provided in Ref. [ 17]. Application of these
relations to calculate cloud point of crude oils and reservoir
fluids are given in Chapt er 9.
Example 6. 12--How much (in grams) n-hexacontane (n-C36)
can be dissolved in 100 g of n-heptane, so that when the
1 0
1%
2 3 0
PRESSURE, PSIA
4 50 6 7 8 91 O0 2 3 00 4 500 6 7 8 9 1,000 2 3,000 4 6 7 8 9 1 0, 000
I 0
[ ......................... ]!
G. G Brown, university of Mich-
igan. Ex trapolated and drawn by
The Fluor Corp. Lid. in 1957.
2 8 1
\
\ 2
K=Y/x3I~. ~ ~ ~ ~ ~ ~ ~ ~ J / / / / / l l l l l l l / I 19 K=Y/x
0.0001
10 2 3 0 4 50 6 7 9 9100 2 3 00 r 500 6 7 9 9 1,000 2 3 , 000 4 6 7 8 9 1 0, 000
PRESSURE, PSI A NONANE
CONV. PRESS. 10,000 PSIA
FI G. 6. 29 ~ K~ v al ues of nonane. Uni t conv er si on: ~ = ~ x 1 . 8 + 3 2 psi a =
t emp er at ur e of t he sol ut i on is r educed to 15 ~ C, t he sol i d p hase
begi ns to form.
Sol ut i on- - We have a bi nar y syst em of comp onent 1 (C36)
and comp onent 2 (n-CT). Comp onent 1 is sol ut e and com-
p onent 2 is consi der ed as solvent. F r om Table 6.10, Mz =
100.2, TM2 = 182.57 K, AH~2/RTM 2 =9. 2557, M1 = 506.95,
Tun = 349 K, and AH~/RTM1 = 30.6066. Assumi ng i deal so-
l ut i on we use Eq. (6.207) for cal cul at i on of xl at T = 288.2 K:
xl = exp[30.6066( 1 - 349/ 288. 2) ] = 0.0016. Wi t h r esp ect to
M1 = 506.9 and M2 = 100.2, f r om Eq. (1.15) we ge t x~ = 0.08,
whi ch is equi val ent to 0.807 g of n-C36 in 100 g of n-C7. 0
6. 8. 4 F r e e z i ng Poi nt De p r e s s i on a nd Boi l i ng
Poi nt El e va t i on
When a smal l amount of a p ur e sol i d ( sol ut e) is added t o
a solvent, t he freezi ng p oi nt of sol vent decr eases whi l e its
boi l i ng p oi nt i ncreases. Up on addi t i on of a sol ut e ( comp o-
nent 1) to a sol vent ( comp onent 2) mol e f r act i on of sol vent
PRESSURE, PSI A
10 2 3 0 4 50 6 7 8 9 1 00 2 3 00 4 500 6 7 8 9 1,000 2 3 , 000 4 6 7 8 9110(~000
PlOtted froth 1947 tabulations of 9
G. G. Brown, University of Mich-
igan, Ex trapolated and drawn by
The FlUO(Corp. Ltd. in 1957
K = Y/x 31 ~ ~ ~ ~ ~ ~ \,~.~_ ~ ~ ~ J / / / / l l l i l 13 K = Ylx
10 2 3 0 4 50 6 7 89 1 00 2 3 00 45006 7 89 1 , 000 2 3 , 000 4 6 7 89 1 0, 000
PRESSURE, PSI A DECANE
FI G . 6. 3 0- - - Ki v al ues of d ecane. Uni t conv er s i on: ~ = ~ x 1 . 8 + 3 2 p s i a =
1 4. 504 bar. T ak en wi t h p er mi s s i on f r om Ref . [ 28 ],
(x2) reduces from unity. This slight reduction in mole fraction
of x2 causes slight reduction in chemical potential according
to Eq. (6.120). Therefore, at the freezing point when liquid
solvent is in equilibrium with its solid, the activity of pure
solid must be lower than its value that corresponds to nor-
mal freezing point. This decrease in freezing point is called
freezing point depression. At freezing point temperature, liq-
uid and solid phases are in equilibrium and Eq. (6.206) ap-
plies. If the solution is assumed ideal, Eq. (6.206) can be
written for the solvent ( component 2) in the following form
neglecting ACpi:
where A/-/~2 is the molar heat of fusion for pure solvent and
TM2 is the solvent melting point. T is the temperature at which
solid and liquid phases are in equilibrium and is the same
as the freezing point of solution after addition of solute. The
amount of decrease in freezing point is shown by ATe2, which
10
1,000
K= Y/x
PRESSURE, PSI A ~'
2 3 0 4 50 6 7 8 9 1 00 2 3 00 4 500 6 7 8 9 1,000
,%
2
]
References:
1 Petrol.Tm~ AI ME196226 (1953)
2, Petrol Trans.AI ME19599 (1952)
3~C.E.F~ PhaseEquilibde Symlposium
Val ,~SNO 2E 121 ~952)
4 I . & EJ~.30534(1946)
3,000 4 6 7 8 9 1 0, 000
1,000
K = Y/x
0.0
10 2 3 0 4 50 6 7 89 1 00 2 3 00 45006 7 89 1 , 000 2 3,000 4 6 7 89 1 0, 000
PRESSURE, PSI A J. HYDROGEN SULFIDE
FIG. 6. 3 1 - - K/ v al ues of hydrog en sulfide. Unit conv ersi on: ~ = ~ x 1 . 8 +
32 psia = 1 4,504 x bar, Taken with permi ssi on from Ref. [ 28 ].
is equal to (TM2 - T). The amount of solute in the solution is
x~ ( = 1 - x2), which is very small (xl (< 1), and from a mat h-
ematical approxi mat i on we have
(6.212) In(1 - xl) ~ - xl
Therefore, Eq. (6.211) can be solved to find ATM2 [43]:
(6.213) ATM2 ----- x1RT22
I n deriving this equation since the change in the freezing point
is small, TTM2 is approxi mat ed by T~2. Equation (6.213) is
approxi mat e but it is quite useful for calculation of freezing
point depression for hydrocarbon systems. For nonideal so-
lutions, Eq. (6.211) must be used by replacing x2 with y2x2,
where )'2 is a function of x2.
By similar analysis from VLE relation for ideal solutions it
can be shown that boiling point elevation may be estimated
from the following simplified relation [43]:
(6.214) ATb2 ~ xlRT~2
AH~ p
TABLE 6.13--Values orb and TB for use in
computing Ki values from Eq. (6.202) [Ref. 41].
Compound b,K TB, K
N2 261.1 60.6
CO2 362.2 107.8
H2S 631.1 183.9
C1 166.7 52.2
C2 636.1 168.3
C3 999.4 231.1
i-C4 1131.7 261.7
n-C4 1196.1 272.8
i-Cs 1315.6 301.1
n-C5 1377.8 309.4
i - C 6 ( al l ) 1497. 8 335. 0
n-C6 1544.4 342.2
n-C7 1704.4 371.7
n-Ca 1852.8 398.9
n-C9 1994.4 423.9
n-Cm 2126.7 447.2
C6 (lumped) 1521.1 338.9
C7+ Use Eq. (6.203)
where AH; ap is t he mol ar heat of vapori zat i on for the solvent
and ATb2 is t he increase in boiling poi nt when mol e fract i on
yap
of solute in t he solution is xl. Met hods of est i mat i on of AH 2
are di scussed in the next chapter.
Exampl e 6. 13--Calculate t he freezing poi nt depressi on of
t ol uene when 5 g of benzoi c acid is dissolved in 100 g of ben-
zene at 20~
Sol ut i on- - For this syst em t he solute is benzoi c acid ( compo-
nent 1) and the solvent is benzene ( comp onent 2). Fr om Table
6.10 for benzoi c acid, M~ = 122.1 and TM1 = 395.5 K, and for
toluene, M2 = 78.1, TM2 = 278.6 K, and A/-/~2/RTM2 = 4.26. For
5 g benzoi c acid and 100 g benzene f r om a reverse f or m of
Eq. (1.15) we get xl = 0.031. To calculate freezing poi nt de-
pression we can use Eq. (6.213):
ATM2 ~ x1RT22
where xl = 0.031, TM2 = 278.6 K, and RTM2/A/-/~2 = i / 4. 26 =
0.2347. Thus ATM2 = 0.031 x 278.6 x 0.2347 = 2 K. A mor e
accurat e result can be obt ai ned by use of Eq. (6.211) for non-
ideal syst ems as
In
1 AH~2 1___~2 ~ _ _
~2X2 = -- RTM2 RT22
For this syst em since x2 is near unity, F2 = 1.0 and same value
for ATM2 is obtained; however, for cases t hat x2 is substantially
l ower t han uni t y this equat i on gives different result, t
6. 9 US E OF VEL OCI T Y OF S OUN D I N
PR ED I C T I ON OF FL UI D PR OPER T I ES
One appl i cat i on of f undament al relations di scussed in this
chapt er is to develop an equat i on of state based on the ve-
locity of sound. The i mport ance of PVT relations and equa-
t i ons of state in est i mat i on of physical and t her modynami c
propert i es and phase equi l i bri um were shown in Chapt er 5
as well as in this chapter. Cubic equat i ons of state and gen-
eralized cor r espondi ng states correl at i ons are powerful tools
for predi ct i ng t her modynami c propert i es and phase equilib-
ria calculations. I n general most of these correl at i ons pro-
vide reliable dat a if accurat e i nput paramet ers are used (see
Figs. 1.4 and 1.5). Accuracy of t her modynami c PVT mod-
els largely depends on the accur acy of t hei r i nput parame-
ters (To, Pc, and co) part i cul arl y for mi xt ures where no mea-
sured dat a are available on the pseudocri t i cal propert i es and
acent ri c factor. While values of these par amet er s are avail-
able for pure and light hydr ocar bons or t hey may be esti-
mat ed accurat el y for light pet rol eum fract i ons ( Chapt er 2),
for heavy fract i ons and heavy comp ounds f ound in reservoir
fluids such dat a are not available. Various met hods of predict-
ing these paramet ers give significantly different values espe-
cially for hi gh-mol ecul ar-wei ght comp ounds (see Figs. 2.18
and 2.20).
One way to tackle this difficulty is to use a measur ed prop-
erty such as density or vapor pressure t o calculate critical
properties. I t is i mpract i cal t o do this for reservoir fluids un-
der reservoir conditions, as it requires sampl i ng and l abora-
t ory measurement s. Since any t her modynami c propert y can
be related to PVT relations, if accurat el y measur ed values of a
t her modynami c propert y exist, t hey can be used to extract pa-
ramet ers in a PVT relation. I n this way t here is no need to use
vari ous mixing rules or predictive met hods for cal cul at i on of
To, Pc, and ~0 of mi xt ures and EOS paramet ers can be directly
calculated f r om a set of t her modynami c data. One t hermody-
nami c propert y t hat can be used to estimate EOS p ar amet er is
velocity of sound t hat may be measur ed directly in a reservoi r
fluid under reservoi r condi t i ons wi t hout sampling. Such dat a
can be used t o obt ai n an accurat e PVT rel at i on for t he reser-
voi r fluids. For this reason Riazi and Mansoori [44] used ther-
modynami c relations t o develop an equat i on of state based
on velocity of sound and t hen sonic velocity dat a have been
used t o obt ai n t her modynami c propert i es [8, 44, 45]. Colgate
et al. [45, 46] used velocity of sound dat a t o det ermi ne critical
propert i es of substances. Most recently, Ball et al. [48] have
const ruct ed an ul t rasoni c apparat us for measur i ng the speed
of sound in liquids and compr essed gases. They also report ed
speed of sound dat a for an oil sampl e up to pressure of 700
bars (see Fig. 6.34) and di scussed prospect s for use of velocity
of sound in det ermi ni ng bubbl e point, density, and viscosity
of oils.
Pressure
TABLE 6.14--Recommended methods for VLE calculations.
Mixtures of similar substances Mixtures of dissimilar substances
<3.45 bar (50 psia)
<13.8 bar (200 psia)
P < 5-10 bar,
Any P, 255 < T < 645 K
> 13.8 bar (200 psia)
P < 69 bar (1000 psia)
Raoult's law (Eq. 6.180) Modified Raoult's law (Eq. 6.181)
Lewis rule (Eq. 6.198) Activity coefficients (Eq. 6.179)
Henry's law (Eq. 6.184) for dilute liquid systems (xi < ~0.03)
Chao-Seader (Eq. 6.200) for nonpolar systems and outside critical region
Eq. (6.197) with SRK or PR EOS using appropriate BIPs
Standing correlation (Eq. 6.202) for natural gases, gas condensate reservoir
fluids and light hydrocarbon systems with little C7+
TM
TM2
P-Const.
L
~ ~T~xS Melting line
L Freezing l i ne ~
I ~ Melting line
0 1.0 0 1.0
X~ 1 , yl S x s] , yl s
T , l
FI G. 6 .3 2~ Schemat i c of freezi ng-mel ti ng diagram for ideal and nonideal bi nary
systems.
FI G. 6 .3 3 ~ Schemat i c of i nterferometer for measuring vel oci ty of sound in liquids.
(a) General vi ew of ul trasoni c i nterferometer and (b) cross secti on of ul trasoni c cell.
T aken wi th permi ssi on from Ref. [49 ].
| 400 i i i i i i i i i i i i i i i i i H. I I I I I ,,11111ii i
1100
1000
9OO
20 30 40 50 60 70
p~M~
FIG. 6. 3 4~ Sp eed of sound in oil sampl e. 9 335.1 K, 9
3 7 0. 7 K, and 9 402.1 K. T he lines are quadrati c fits. Tak en
wi th permi ssi on from Ref. [ 48 ].
Method of measurement of velocity of sound in liquids
t hrough ultrasonic interferometer is presented in Ref. [49].
I n this met hod the measuri ng cell is connected to the out put
terminal of a high-frequency generator t hrough a shielded
cable. The cell is filled with the experimental liquid before
switching on the generator. Schematic of this i nt erferomet er
is shown in Fig. 6.33. The principle used in the measur ement
of velocity (c) is based on the accurate det ermi nat i on of the
wavelength. Ultrasonic waves of known frequency are pro-
duced by a quartz plate fixed at the bot t om of the cell. The
waves are reflected by a movabl e metallic plate kept paral-
lel to the quartz plate. The sonic velocity in the liquid can
be det ermi ned from the following relation: c = wavelength x
frequency. This simple measuri ng device is useful to deter-
mi ne velocity of sound in liquids under normal at mospheri c
pressure. From velocity of sound measurement it would be
possible to directly determine isothermal or adiabatic com-
pressibilities, excess enthalpy, heat capacity, surface tension,
miscibility, van der Waal's constants, free vol ume between
molecules, mean free pat h of molecules, molecular radius,
etc. [49].
The purpose of this section is to use t hermodynami c rela-
tions discussed in this chapt er to develop an equation of state
based on the velocity of sound and then to use velocity of
sound data to estimate t hermal and volumetric properties of
fluids and fluid mixtures.
6. 9 . 1 Vel oci t y o f S o und B a s e d Equat i on o f S t at e
Sound waves in a fluid are longitudinal contractions and
rarefactions, which are adiabatic (no heat transfer) and re-
versible (no energy loss) and which travel at a speed c given
by the t hermodynami c quantity [10]:
(6.215) c 2 = - - -
M S- ~ s
where c is the velocity of sound, V is the mol ar volume, M
is the mol ecul ar weight, and constant S refers to the fact
the wave transmission is a constant ent ropy process (adia-
batic and reversible). I t should be noted that if in the above
equation V is the specific volume then par amet er M must be
removed from the relation. Equivalent forms of this equation
in t erms of specific vol ume or mass density are also commonl y
used in various sources. From t hermodynami c relations the
above relation can be converted to the following form:
(6.216) c 2 - YV2 ( OP) = y (O~p)
M -OV r --M r
where V is the mol ar volume, p is the mol ar density ( l / V) ,
and Z is the heat capacity ratio (Ce/Cv). Using definition of
isothermal compressibility, K, (Eq. 6.25), the velocity of sound
can be calculated from the following relation:
(6.217) c 2 = Y
Mpr
From this relation it is apparent that the velocity of sound in a
fluid depends on the fluid properties and it is somewhat less
t han mean velocity of molecules as shown from the kinetic
theory of gases [10]. Since speed of sound is a state function
property, an equation of state can be developed for the velocity
of sound in t erms of t emperat ure and density as independent
variables [43]. Similarly velocity of sound can be calculated
from an EOS t hrough Eq. (6.216) [8]. For example, for ideal
gases Eq. (6.216) reduces to (yRT/ M ) 1/2. I n general velocity
of sound decreases with molecular weight of the fluid. Veloc-
ity of sound at the same condition of T and P is higher in
liquids t han in gases. With increases in t emperat ure, velocity
of sound in gases increases while in liquids decreases. Ve-
locity of sound increases with pressure for bot h gases and
liquids. Some experimental and calculated data on veloc-
ity of sound for several hydrocarbons in gaseous and liquid
phases are reported by Firoozabadi [17]. As an example, ve-
locity of sound in met hane gas increases from 450 to 750 m/ s
when pressure increases from low pressures ( < 1 bar) to
about 400 bars at 16~ Effect of t emperat ure on velocity of
sound at low pressures is much greater t han at high pres-
sures. Velocity of sound in met hane at 50 bar increases from
430 at 16~ to about 540 m/ s at 167~ For liquid n-hexane
velocity of sound decreases from 1200 to about 860 rn/s when
t emperat ure increases from - 10 to 70~ [ 17]. Experimentally
measured velocity of sound in oil sample at various pressures
and t emperat ures is shown in Fig. 6.34 as det ermi ned by Ball
et al. [48]. I n this figure effect of t emperat ure and pressure
on the velocity of sound in liquid phase for a live oil is well
demonstrated. The oil composi t i on is given as follows: CO2
(1), C1 (34), C2-C6 (26), and C7+ (39), where the numbers in-
side parentheses represent mol%. The mol ecul ar weight of oil
is 102 and that of C7+ is 212. Detail of oil composi t i on is given
by Ball et al. [48]. They also showed that velocity of sound in
oils increases linearly with density at a fixed t emperat ure [48].
I t has been shown by Alem and Mansoori [50] that the
expression for the ent ropy departure of a hard-sphere fluid
can be used for ent ropy depart ure of a real fluid provided
that the hard sphere di amet er is taken as t emperat ure- and
density-dependent. By substituting Carnahan-St arl i ng EOS,
Eq. (5.93) into Eq. (6.50), the following relation is obtained
for the ent ropy departure of hard-sphere fluids:
(6.218) ( S- sig) us - R ( ( 4- 3()
(1 - 02
6. THERM ODYNAM IC RELATIONS FOR PROPERTY ESTIM ATIONS 2 8 7
in which ~ is dimensionless packing factor defined by
Eq. (5.86) in terms of hard-sphere diameter a. I n general
for nonassociating fluids, a can be taken as a linear func-
t i onof 1/ T andp [50], i.e., a = do +dip +d2/ T + d3p/T. From
Eq. (6.218) we have
(6.219) S = S(T, V )
Since c is a state function we can write it as a function of only
two independent properties for a pure fluid or fluid mixtures
of constant composition in a single phase (see the phase rule
in Chapter 5):
(6.220) c = c(S, V )
Differentiating Eq. (6.220) with respect to S at constant V
gives
V2 { ~[ ( OP) ] } as ( at const ant V)
(6.221) 2cdc = - - ~ ~ s v
Applying Eqs. (6.10) to (6.219) gives
aS
(6.222) dS=( ~- ~) v dT+( ~V ) r dV
which at constant V becomes
( aS) dT( at c ons t ant V )
(6.223) dS= ~-~ r
From mathematical relations we know
(6.224) O--S ~ s v ~ ~ v s
The Maxwell's relation given by Eq. (6.10) gives (SP/SS)v =
-(OT/SV )s, where -(OT/SV )s can be determined from divid-
ing both sides of Eq. (6.222) by 8V at constant S as
( aT) _ (OS/OV)r
(6.225) - ~ s (OS/OT)v
Substituting Eqs. (6.223)-(6.224) into Eq. (6.221) and inte-
grating from T to T ~ oo, where c ---> c ns gives the following
relation for c in terms of T and V:
oo
f (o2 r (os dr
(6.226) c2 = (cnS)2 - --M J \-O-~ ]s k OT Jv
r
c Hs can be calculated from Eq. (6.216) using the CS EOS,
Eq. (5.93). Derivative (SS/ST)v can be determined from
Eq. (6.218) or (6.219) as a function of T and V only. (02 T/O V2)s
can be determined from Eq. (6.225) as a function of T and V.
Therefore, the RHS of above equation is in terms of only T
and V, which can be written as
(6.227) c = c(T, V )
Equation (6.226) or (6.227) is a cVT relation and is called ve-
locity of sound based equation of state [44]. One direct appli-
cation of this equation is that when a set of experimental data
on cVT or cPT for a fluid or a fluid mixture of constant compo-
sition are available they can be used with the above relations
to obtain the PVT relation of the fluid. This is the essence of
use of velocity of sound in obtaining PVT relations. This is
demonstrated in the next section by use of velocity of sound
data to obtain EOS parameters. Once the PVT relation for a
fluid is determined all other thermodynamic properties can
be calculated from various methods presented in this chapter.
6. 9 . 2 Equat i on o f S t at e Par ame t e r s f r o m
Vel oci t y o f S o und Dat a
I n this section the relations developed in the previous section
for the velocity of sound are used to obtain EOS parameters.
These parameters have been compared with those obtained
from critical constants or other properties in the form of pre-
diction of volumetric and thermodynamic properties. Trun-
cated virial (Eq. 5.76), Carnahan-Starling-Lennard-Jones
(Eq. 5.96), and common cubic equations (Eq. 5.40) have been
used for the evaluations and testing of the suggested method.
Although the idea of the proposed method is for heavy hy-
drocarbon mixtures and reservoir fluids, but because of lack
of data on the sonic velocity of such mixtures applicability of
the method is demonstrated with use of acoustic data on light
and pure hydrocarbons [8, 44].
6. 9. 2. 1 V irial Coefficients
Since any equation of state can be converted into virial form,
in this stage second and third virial coefficients have been
obtained from sonic velocity for a number of pure substances.
Assuming that the entropy departure for a real fluid is the
same as for a hard sphere and by rearranging Eq. (6.218) the
packing fraction of hard sphere can be calculated from real
fluid entropy departure:
(
2 s-sig 4 -
(6.228) ( = ( 6) pNAa3 R
s_ sig
3 - -~
Calculated values of a from the above equation indicate that
there is a simple relation between hard-sphere diameter as in
(6.229) a = do + dA
T
Application of the virial equation truncated after the third
term, Eq. (5.76), to hard sphere fluids gives
BHS C HS
(6.230) Z Ms = 1 + ~- - + V- ~-
By converting the HS EOS, Eq. (5.93), into the above virial
form one gets [51]
2 3
BHS= ~7r NAt7
(6.231)
5 2 2 6
C ns = ~z r N~o
[ ' S- ~ g'] I" P B+ TB' C + TC"~
(6.232) / - - 1 = - [ l n + ~V- - - q
L R Area/fluid k 2-V--~ ' ,]
[" S - S ig "] [ B Ms C IJs \
_ _ = _ l nP +- - + q77~
(6.233) [ R J hard sphere ~ V 2V )
Since it is assumed that the left sides of the above two equa-
tions are equal, so the RHSs must also be equal, which result
in the following relations:
TB' + B = B Hs
(6.234)
TC' + C = C HS
Substituting Eqs. (5.76) and (6.230) for real and hard-sphere
fluids virial equations into Eq. (6.39) one can calculate en-
tropy departures for real and hard-sphere fluids as
where B' = dB/ dT and C' = dC/ dT. Substituting for B ns and
C Hs from Eq. (6.231) into Eq. (6.234) and, combining with
Eq. (6.229), gives two nonhomogeneous differential equa-
tions that after their solutions we get:
B(T) = qo + r-~
(6.235)
In T 3 L,~
c ( r ) = ql - - f - +
T-~
tz=0-
Parameters Pl and L1 are constants of integration while
all other constants are related to parameters do and dl in
Eq. (6.229) [44]. For example, parameters q0 and L0 are
related to do and dl as follows: q0 = 2zrNgdodl and L0 =
(5/18)rrEN2d 6, where NA is the Avogadro's number. Substitu-
tion of the truncated virial EOS, Eq. (5.76), into Eq. (6.216)
gives the following relation for the velocity of sound in terms
of virial coefficients:
(6.236) c 2 = - ~ - [ 1 + p(2 B + 3Cp)]
where Y is the heat capacity ratio (Cp/ Cv) and p is the molar
density ( l/ V) . Once B and C are determined from Eq. (6.235),
Cp and Cv can be calculated from Eqs. (6.64) and (6.65) and
upon substitution into Eq. (6.236) one can calculate velocity
of sound. Vice versa the sonic velocity data can be used to
obtain virial coefficients and consequently constants Pl and
L1 in Eqs. (6.235) by minimizing the following objective
function:
N
(6.237) RC = ~ (ci,~al~. - ci,~xp.) 2
i=1
where N is the number of data points on the velocity of sound.
Thermodynamic data, including velocity of sound for
methane, ethane, and propane, are given by Goodwin eta] .
[52-54]. Entropy data on methane [52] were used to ob-
tain constants do and dl by substituting Eq. (6.235) into
Eq. (6.232). Values ofdo = 2.516 x 10 -1~ manddl -- 554.15 x
10-1~ m. K have been obtained for methane from entropy data
[44]. With knowledge of do and dl all constants in Eq. (6.235)
were determined except Pl and L1. For simplicity, truncated
virial equation after the second term (Eq. 5.75) was used to
obtain constant Pl for the second viria] coefficient, B, by min-
imizing RC in Eq. (6.237). For methane in the temperature
range of 90-500 K and pressures up to 100 bar, it was found
that Pl = -8. 1 x 103 cm 3. K/mol. Using this value into con-
stants for B in Eq. (6.235) the following relation was found
[44]:
~_T 81000
B(T) -- 13274 + 20.1 -
2.924 x 106 1.073 x 101~
(6.238)
T 2 T 3
TABLE 6.1$--Constants in Eq. (6.239) for calculation of second
virial coefficient.
Compound a B c %AAD for Z
Methane 0.02854 19.4 1.6582 0.5
Ethane 0.16 250 0.88 1.1
Propane 0.22 230 1.29 1.4
Taken with permissionfrom Ref. [44].
Number of data points for each compound: 150; pressure range: 0.1-200 bar;
temperature range: 90-500 K for C1, 90-600 K for C2, and 90-700 K for Ca.
0.05 I
O~
-0.0S
-O.1
-0.15
-0.2
-0.25
-0.3
-0.35
-0.4
-O.4, =
~ D ymar~l & Smith Data
Temperature, K
FI G . 6. 3 5 ~ P r e d i c t i o n of s e c ond v i r i al coef f i ci ent of me t h a n e
f r om v el oci t y of s ound dat a ( Eq. 6,23 9 ) . T ak en wi t h p er mi s s i on
f r om Ref . [ 44],
where B is in cm3/mol and T is in kelvin. This equation can
be fairly approximated by the following simpler form for the
second virial coefficient:
106 x l nT i06 x c
(6.239) B(T) = a b
T T
where B is in cma/mo] and T is in kelvin. All three constants
a, b, and c have been directly determined from velocity of
sound data for methane, ethane, and propane and are given
in Table 6.15. When this equation is used to calculate c from
Eq. (6.236) with C = 0, an error of 0.5% was obtained for 150
data points for methane [44]. I f virial equation with coeffi-
dent s B and C (Eq. 5.76) were used obviously lower error
could be obtained. Errors for prediction of compressibility
factor of each compound using Eq. (5.75) with coefficient B
estimated from Eq. (6.239) are also given in Table 6.15. Graph-
ical evaluation of predicted coefficient B for methane from
Eq. (6.239) is shown in Fig. 6.35. Predicted compressibility
factor (Z) for methane at 30 bar, using B determined from
velocity of sound and truncated virial equation (Eq. 5.75), is
shown in Fig. 6.36. Further development in relation between
sonic velocity and virial coefficient is discussed in Ref. [55].
6. 9. 2. 2 Lennard- J ones and van der Waals Paramet ers
I n a similar way Lennard-Jones potential parameters, e and
a have been determined from velocity of sound data using
CSLJ EOS (Eq. 5.96). Calculated parameters have been com-
pared with those determined for other methods and are given
1.1
1 Sonic Vel,. . . . . . . .---* " . . . . . "
O .9 / . , ~ _ ~ ~
i~ 0,8 / Ex per i ment al
N 0.7'
0. 6 ~/Dymond & Smirch Data
/
..... 2 o s6o
Temper~dure, K
FI G . 6. 3 6- - Pr e d i c t i on of Z f act or of me t ha ne at 3 0 bar f r om
t r uncat ed v i r i al E O S wi t h s e c ond coef f i ci ent f r om v el oci t y of
s ound dat a. T ak en wi t h p er mi s s i on f r om Ref . [ 44].
o9,1
= 0"9 t
:~ o, as
i o.8
0.751
8 0.r~
0.65
o,~
/"
Original Parameter
~/Ex perimental . . . . . . . . . . . .
2OO 25O 3OO 350 400 45O 5O0
T emperature, K
FIG. 6. 3 7 - - Predi ct i on of Z factor for methane at 3 0 bar from
v dW EO S using parameters from v elocity of sound data. Taken
i n Tabl e 6.17. Er r or s f or cal cul at ed Z val ues wi t h use of LJ
p a r a me t e r s f r om di f f er ent me t hods ar e al so gi ven i n t hi s t a-
bl e. Van der Waal s EOS p a r a me t e r s de t e r mi ne d f r om vel oci t y
of s ound ar e gi ven i n Tabl e 6. 18 and p r edi ct ed Z val ues f or
me t ha ne and et hane ar e s hown i n Fi gs. 6. 37 and 6. 38, re-
spect i vel y. Pr edi ct ed Z f act or f or p r op a ne f r om CSLJ EOS
( Eq. 5. 96) is s hown i n Fig. 6.39. Resul t s p r es ent ed i n Ta-
bl es 6. 15- 6. 17 and Fi gs. 6. 36- 6. 39 s how t hat EOS p a r a me -
t er s de t e r mi ne d f r om vel oci t y of s ound p r ovi de r el i abl e PVT
dat a and ma y be us ed t o cal cul at e ot her t he r modyna mi c
p r op er t i es.
ooo . 08,
.~ 0. 9
"~ .., ., ~" O tiginal Parameter
cL 0.8
E
8 o.75i , "
r4
~ ~ o 5 ~ 66o 6so 7o0
Temperature, K
FIG. 6. 3 8 --Predi cti on of Z factor for ethane at 100 bar from
v dW EO S using parameters from v elocity of sound data. Taken
1.05
0.96.
0.75,
~ 0.7'
0.65"
1,4 0.6.
~
~ - ~ E x p e d m e n t ~ l
~ o ~ o ~ ~ o ~ ~ o Too
Temperature, K
FIG. 6. 3 9 - - Predi ct i on of Z factor of propane at 30 bar from LJ
EO S with parameters from different methods. Taken with per-
mission from Ref. [44].
6. 9. 2. 3 RK and PR EOS Par amet er s - - Pr oper t y
Es t i mat i on
To f ur t her i nvest i gat e t he p ossi bi l i t y of us i ng vel oci t y of s ound
f or cal cul at i on of PVT and t he r modyna mi c dat a, R K and PR
EOS p a r a me t e r s wer e de t e r mi ne d f or bot h gases and l i qui ds
t hr ough vel oci t y of s ound dat a. Usi ng p a r a me t e r s def i ned i n
Tabl e 6.1 f or cal cul at i on of y, V, and (aP/aV)r a nd subst i -
t ut i ng t he m i nt o Eq. ( 6. 216) , vel oci t y of sound, c, can be es-
t i mat ed. F or bot h R K and SR K equat i ons t he r el at i on f or c
be c ome s
RT a( 2V + b)
- - +
( V - b) 2 V 2(V + b) 2
2 v2 [
CRK, SRK = - - - ~
- T V - b
( 6. 240)
al 9 Ta2 , V .~-lq
V2+bv)2x(C~-R---K-'nv-~J J
and f or PR EOS t he r el at i on f or c be c ome s
V 2 [ RT 2a( V + b)
C2R = - - - M ( V - b) ~ + ( V 2 + 2bV - b2) 2
L
al
- T V b V 2 + 2bV - b 2
( 6. 241)
TABLE 6.16---The Lennard-Jones parameters from the velocity of sound data and other sources.
Velocity of sound Second virial coefficient a Viscosity data a
Compound e/ka, K o, A %AAD for Z e/k~,K a,A %AAD for Z e/kB, K cr, A %AAD for Z
Methane 178.1 3.97 0.8 148.2 3.817 4.0 144.0 3.796 4.7
Ethane 300.0 4.25 0.5 243.0 3.594 3.0 230.0 4.418 3.4
Propane 350.0 5.0 1.1 242.0 5.637 11.5 254.0 5.061 8.0
aThe LJ parameters are used with Eq. (5.96) to calculate Z. The LJ parameters from the second virial coefficient and viscosity are taken from
Hirschfelder et al. [56]. kB is the Boltzman constant (1.381 x 10 -23 J/K) and 1 A = 10 -1~ m.
TABLE 6. 17--The van der Waals constants from the velocity of sound data.
Velocity of sound Original constants a
Compounds a x 10 -~ b %AAD for Z a x 10 _6 b %AAD for Z
Methane 1.88583 44.78 1.0 2.27209 43.05 0.8
Ethane 3.84613 57.18 1.8 5.49447 51.98 2.4
Propane 8.34060 90.51 1.4 9.26734 90.51 1.5
Taken with permission from Ref. [44]. a is in cmT/mol 2. bar and b is in cm3/mol.
=From Table 5.1.
2 9 0 CHARACTERIZATION AND PROPERTI ES OF PETROLEUM FRACTIONS
where parameters al and a2 are first and second derivatives of
EOS parameter a with respect to temperature and for both RK
and PR equations are given in Table 6.1. I n terms of parame-
ters a and b, velocity of sound equation for both RK and SRK
are the same. Their difference lies in calculation of parameter
a through Eq. (5.41), where for RK EOS, a = 1. Now we de-
fine EOS parameters determined from cVT data in terms of
original EOS parameters (aEOS and bEos as given in Table 5.1)
in the following forms:
(6.242) as = asaF.os
bs = ~,b~os
Parameters c~s and/gs can be determined for each compound
or mixtures of constant composition from velocity of sound
data. Parameters aEOS and bEos can be calculated from their
definition and use of critical constants. I n fact values of the
critical constants used in the calculations do not affect the
outcome of results but they affect calculated values of as and
/~s. For this reason as and/gs must be used with the same aEOS
and bEos that were used originally to determine these param-
eters. As an alternative approach and especially for petroleum
TABLE 6. 18. --RK and PR EOS parameters (Eq. 6.242) from velocity
of sound in gases and liquids.
Compound No. of RK EOS a PR EOS ~
(gas) points as fls as fls
Me t ha ne 77 1.025 1.111 0. 936 1.093
Et ha ne 119 1.043 1.123 0. 956 0. 993
Pr op a ne 63 0. 992 1.026 1.013 1.031
I s obut a ne 80 0. 993 1.019 0. 983 0. 983
n- But a ne 86 0. 987 1.015 0. 941 0. 912
n- Pent ane ( l i qui d) 1.04 0. 9
n- Decane ( l i qui d) 1.06 0. 99
~ parameters must be used for gaseous phase with Eq. (5.40) and param-
eters aEOS and baos from Tables 5.1.
mixtures it would be appropriate to determine as and bs from
cVT data and directly use them in the corresponding EOS
without calculation of aEOS and bEos through critical proper-
ties. Therefore, for both RK and SRK we get same values of
as and bs since the original form of EOS is the same. For a
number of light gases, parameters as and fls have been deter-
mined from sonic velocity data for both RK and PR EOSs and
they are given in Table 6.18. Once these parameters are used
TABLE 6. 19--Prediction of thermodynamic properties of light gases from RK and PR
equations with use of velocity of sound and original parameters, a
%AAD for RK EOS %AAD for PR EOS
No. of data Sonic Sonic
Gas system points Property velocity Original velocity Original
Pur e gas 425 C 0. 82 0. 58 0. 62 0. 82
c omp ounds 425 Z 0, 76 0. 92 0. 5 0. 77
Cl , C2, C3, 341 Cp 1.9 1.8 1.3 1.2
/ C4, nC4 341 H 0. 66 0. 53 0. 42 0. 48
Gas mi xt ur e 61 C 9. 2 0. 84 1.47 0. 89
69 Mol % C1, 66 Z 4.1 2. 0 4. 0 1.9
31 Mol % C2 66 Cv 8.2 2. 85 6.5 7. 0
aFor the velocity of sound parameters, values of as and fls from Table 6.18 have been used. For the original
parameters these corrections factors are taken as unity.
I t
I o 0
~ %F. xpetimen~ l . . . + ' ~
Original Parameter
290 310 330 3 ~ 4) 3" 10 390 410 430 450 4/ 0 490 510
Temperat ure, K
FIG. 6. 40- - Predi cti on of const ant pressure heat capaci ty of
ethane g as at 30 bar from RK EOS usi ng parameters from v el oc-
ity o f s o u n d d a t a .
6. THERM ODYNAM IC RELATIONS FOR PROPERTY ESTIM ATIONS 2 9 1
39
= " . . . . . . . . . . . . . .
g .
Original P~rameter
6 7 $ 9 IO
Pressure, M P a
FIG. 6. 41 - - Predi ct i on of constant v ol ume heat capacity
of 69 mol% methane and 31 mol% ethane g as at 260 K from
RK EO S usi ng parameters from v el oci ty of sound data.
t o calculate vari ous physical propert i es errors very similar
t o t hose obt ai ned from original paramet ers are obt ai ned as
shown in Table 6.19 [8]. Predi ct ed const ant pressure heat ca-
paci t y f r om RK EOS wi t h paramet ers det ermi ned f r om veloc-
ity of sound for et hane a gas mi xt ure of met hane and et hane
is shown in Figs, 6.40 and 6.41, respectively.
Similarly soni c velocity dat a for some liquids from Cs t o C t0
were used t o calculate EOS paramet ers. Calculated as and fls
paramet ers for use wi t h PR EOS t hr ough Eq. (6.242) are also
given in Table 6.18. When EOS paramet ers f r om velocity of
sound are used to calculate Cp of liquids rangi ng f r om C5
t o C10 an average error of 6.4% is obt ai ned in comp ar i son
wi t h 7.6% error obt ai ned f r om original paramet ers. Velocity
=. *
. r
+ , , ................. . . . . . . . . . . . . . . . . . . . . . . . . . . ++ . - . . . .
Ex perimental e ~ ~ eee / ~ " ~
270 ~, , ~' ~ . ~., ..~.* ~* " ~' * "
** ,, * * * ~f Sonic Vet.; ~ " * ~ , ~ ~ ' j~
260 " ' ' " ~ ~ . _ ~ ~ ~
~ _ ~ -= ~ ---- Original Parameter
250'
320 330 340 350 360 37O
T e mpe r at ur e , K
FIG. 6. 43 ~ Pr edi ct i on of liquid heat capacity of n-octane at
100 bar from PR EO S usi ng parameters from v el oci ty of sound
data.
of sound for liquids can be est i mat ed from original PR par am-
eters wi t h AAD of 9.7%; while usi ng paramet ers cal cul at ed
from soni c velocity, an error of 3.9% was obt ai ned for 569
dat a poi nt s [8]. Graphi cal eval uat i ons for predi ct i on of liq-
ui d density of a mi xt ure and const ant vol ume heat capaci t y
of n-oct ane are shown in Figs, 6.42 and 6.43. Results shown
in these figures and Table 6,18 indicate t hat EOS paramet ers
det ermi ned f r om velocity of sound are capabl e of predi ct i ng
t her modynami c properties. I t shoul d be not ed t hat dat a on ve-
locity of sound were obt ai ned either for comp ounds as gases
or liquids but not for a single comp ound dat a on sonic veloc-
ity of bot h liquids and gases were available in this eval uat i on
,3
IS
17
15
14
t3
12
Experimen~ ....
tO 2O 30 4 0
Pressm,'e+ MP'=
FIG. 6. 42- - Predi ct i on of l i qui d densi ty of 10 mol% n-hex adecane
and 9 0 mol% carbon diox ide at 20~ from PR EO S usi ng parameters
from v el oci ty of sound data.
process. For this reason there is no continuity in use of pa-
ramet ers at and fls for use in bot h phases. For the same reason
when paramet ers obtained from gas sonic velocity were used
to calculate vapor pressure errors larger t han original EOS
paramet ers were obtained [8].
Research on using the velocity of sound to obtain t hermo-
dynami c properties of fluids are underway, and as more data
on the speed of sound in heavy pet rol eum mixtures become
available usefulness of this technique of calculating proper-
ties of undefined and heavy mixtures becomes more clear.
From the analysis shown here, one may conclude that use
of sonic velocity is a promi si ng met hod for prediction and
calculation of t hermodynami c properties of fluids and fluid
mixtures.
6 . 1 0 S UMMA R Y A N D R EC OM M EN D A T I ON S
I n this chapt er fundament al t hermodynami c relations that
are needed in calculation of various physical and thermody-
nami c properties are presented. Through these relations var-
ious properties can be calculated from knowledge of a PVT
relation or an equation of state. Methods of calculation of
vapor pressure, enthalpy, heat capacity, entropy, fugacity, ac-
tivity coefficient, and equilibrium ratios suitable for hydro-
carbon systems and pet rol eum fractions are presented in this
chapter. These met hods should be used in conjunction with
equations of states or generalized correlations presented in
Chapter 5. I n use of cubic equations of state for phase equilib-
ri um calculations and calculation of Ki values, binary interac-
tion paramet ers recommended in Chapter 5 should be used.
Cubic equations are recommended for high-pressure phase
equilibrium calculations while activity coefficient models are
recommended for low-pressure systems. Methods of calcu-
lation of activity coefficient and Henry' s law constants from
a cubic EOS are presented. Recent studies show that cubic
equations are not the best type of PVT relation for prediction
of derivative properties such as enthalpy, J oul e-Thomson co~
efficient, or heat capacity. For this reason noncubic equations
such as statistical associating fluid theory (SAFT) are being
investigated for prediction of such properties [38, 57]. The
mai n purpose of this chapter was to demonst rat e the role that
theory plays in estimation of physical properties of pet rol eum
fluids. However, among the met hods presented in this chapter,
the LK generalized correlations are the most suitable meth-
ods for calculation of enthalpy, heat capacity, and fugacity for
bot h liquid and gas phases at elevated pressures.
While the cubic equations (i.e., SRK or PR) are useful
for phase behavior calculations, the LK corresponding state
correlations are recommended for calculation of density,
enthalpy, entropy, and heat capacity of hydrocarbons and
pet rol eum fractions. Partial mol ar properties and their meth-
ods of calculation have been presented for estimation of mix-
ture properties. Calculation of volume change due to mixing
or heat of mixing is shown. Fundament al phase equilibria re-
lations especially for vapor-liquid and solid-liquid systems
are developed. Through these relations calculation of vapor
pressure of pure substances, solubility of gases and solids
in liquids are demonstrated. Solubility paramet ers for pure
compounds are given for calculation of activity coefficients
without use of any VLE data. Correlations are presented for
calculation of heat of fusion, mol ar volume, and solubility
paramet ers for paraffinic, naphthenic, and aromat i c groups.
These relations are useful in VLE and SLE calculations for
pet rol eum fractions through the pseudocomponent met hod
of Chapter 3. Data on the enthalpy of fusion and freezing
pointd can be used to calculate freezing point of a mixture or
the t emperat ure at which first solid particles begin to form.
Application of met hods presented in this chapt er require in-
put paramet ers (critical properties, molecular weight, and
acentric factor) that for defined mixtures should be calcu-
lated from mixing rules given in Chapter 5. For undefined
pet rol eum fractions these paramet ers should he calculated
from met hods given in Chapters 2-4. Main application of
met hods presented in this chapt er will be shown in the next
chapt er for calculation of t hermodynami c and physical prop-
erties of hydrocarbons and undefined pet rol eum fractions.
The mai n characteristic of relations shown in this chapt er
is that they can be used for prediction of properties of bot h
gases and liquids t hrough an equation of state. However, as
it will be seen in the next chapt er there are some empirically
developed correlations that are mainly used for liquids with
higher degree of accuracy. Generally properties of liquids are
calculated with lesser accuracy t han properties of gases.
With the help of fundament al relations presented in this
chapt er a generalized cVT relation based on the velocity of
sound is developed. It has been shown that when EOS param-
eters are calculated through a measurabl e propert y such as
velocity of sound, thermophysical properties such as density,
enthalpy, heat capacity, and vapor pressure have been calcu-
lated with better accuracy for bot h liquid and vapor phases
t hrough the use of velocity of sound data. This technique
is particularly useful for mixtures of unknown composi t i on
and reservoir fluids and it is a promising approach for esti-
mat i on of t hermodynami c properties of complex undefined
mixtures.
6. 1 1 P R OB L EM S
6.1. Develop an equation of state in t erms of paramet ers fl
and K.
6.2. I n storage of hydrocarbons in cylinders always a mix-
tures of bot h vapor and liquid (but not a single phase)
are stored. Can you justify this?
6.3. Derive a relation for calculation of (G - Gig)/ RTin t erms
of PVT and then combi ne with Eq. (6.33) to derive
Eq. (6.50).
6.4. Derive Edmi st er equation for acentric factor (Eq. 2.108)
from Eq. (6.101).
6.5. a. Derive a relation for mol ar enthalpy from PR EOS.
b. Use the result from part a to derive a relation for par-
tial mol ar enthalpy from PR EOS.
c. Repeat part a assumi ng par amet er b is a t emperat ure-
dependent parameter.
6.6. Derive a relation for partial mol ar volume from PR EOS
(Eq. 6.88).
6.7. Derive fugacity coefficient relation from SRK EOS for
a pure substance and compare it with results from
Eq. (6.126).
6.8. Derive Eq. (6.26) for the relation between Cv and Cv.
6.9. Show that
V
f l Cv- C~ = L \ or2 ]vJr dV
V =oo
Use this relation with truncated virial equation to derive
Eq. (6.65).
6.10. The Joule-Thomson coefficient is defined as
a. Show that it can be related to PVT in the following
form:
8V ) _ V
TS- ~p
T1-- C~
b. Calculate 0 for methane at 320 K and 10 bar from the
SRK EOS.
6.1 I. Similar to derivation of Eq. (6.38) for enthalpy departure
at T and V, derive the following relation for the heat
capacity departure and use it to calculate residual heat
capacity from RK EOS. How do you judge validity of
your result?
V
(cp ig f
- Ce) r,v = 7" \ OT2]vdV
V---~ oo
OP 2
R
6.12. Show that Eqs. (6.50) and (6.51) for calculation of resid-
ual entropy are equivalent.
6.13. Prove Eq. (6.81) for the Gibbs-Duhem equation.
6.14. Derive Eq. (6.126) for fugacity coefficient of i in a mix-
ture using SRK EOS.
6.15. Derive the following relation for calculation of fugacity
of pure solids at T <Ttp.
]
6.16.
6.17.
where Pi sub is the vapor pressure of pure solid i at tem-
perature T.
Derive Eq. (6.216) for the velocity of sound.
A mixture of C1 and C5 exists at 311 K and 69.5 bar in
both gas and liquid phases in equilibrium in a closed
vessel. The mole fraction of C 1 in the mixture is Zl =
0.541. I n the gas Yl = 0.953 and in the liquid Xl = 0.33.
Calculate K~ and K5 from the following methods:
a. Regular solution theory
b. Standing correlation
c. GPA/NIST graphs
d. PR EOS
I n using PR EOS, use shift parameters of -0. 2044 and
-0. 045 for C1 and n-C5, respectively. Also use BIP value
of kl-5 = 0.054.
6.18. a. For a gas mixture that follows truncated virial EOS,
show that
1 I 2 1 2 y ,
V E =
i j
P Z Z yiyj8ij
G E= -~
i j
P
j ~7
where &i is defined in Eq. (5.70) in Chapter 5.
b. Derive a relation for heat of mixing of a binary gas
that obeys truncated virial EOS.
6.19. I n general for mixtures, equality of mixture fugacity be-
tween two phases is not valid in VLE calculations:
fmV,x= f ~
However, only under a certain condition this relation is
true. What is that condition?
6.20. With the use of PR EOS and definition of solubility pa-
rameter (8) by Eq. (6.147) one can derive the following
relation for calculation of 8 [17]:
I ( ~ ) V L+(l+~/-i) b l
8 =
where da/dT for PR EOS can be obtained from Table 6. I.
With use of volume translation for V L estimate values
of V L and 8 at 25~ for hydrocarbons C5 and C10 and
compare with values given in Table 6.10.
6.21. Calculate freezing point depression of toluene when it
is saturated with solid naphthalene at 20~
6.22. Derive Eq. (6.240) for calculation of velocity of sound
from RK/SRK EOS.
6.23. Consider the dry natural gas (fluid 1) and black oil (fluid
5) samples whose compositions are given in Table 1.2.
Assume there are two reservoirs, one containing the nat-
ural gas and the other containing black oil, both at 400 K
and 300 bar. Calculate velocity of sound in these two flu-
ids using SRK EOS.
6.24. Calculate (U - uig), ( H - Hig), and (S - S ig) for steam at
500~ and I00 bar from SRK and PR EOS. How do you
evaluate your results?
6.25. Calculate the increase in enthalpy of n-pentane when
its pressure increases from 600 to 2000 psia at 190.6~
using the following methods:
a. SRK EOS.
b. LK method.
c. Compare the results with the measured value of 188
Btu/lbmol [17].
6.26. Estimate Cp, Cv, and the speed sound in liquid hexane
at atmospheric pressure and 269 and 300K from the
following methods:
a. SRK EOS
b. PR EOS
c. Compare calculated sonic velocities with reported
values of 1200 (at 269 K) and 1071 m/s (at 300 K)
(Fig. 3.33, Ref. [17]).
6.27. Estimate Fi ~ for the system of n-C4 and n-C32 at
100~ from PR EOS and compare with the value from
Fig. 6.8.
6.28. Using results from Probl em 6.27, estimate t he Henry' s
law const ant for the syst em of n-C4 and n-C32 at 100~
f r om PR EOS.
6.29. For the mi xt ure of benzene and t ol uene const ruct the
freezi ng/ meki ng di agr am similar to Fig. 6.32.
6.30. One of t he advanced liquid solution theories is known as
the quasi chemi cal approxi mat i on, whi ch is part i cul arl y
useful for mi xt ures cont ai ni ng mol ecul es quantitatively
different in size and shape. Accordi ng to this t heory t he
mol ar excess Gibbs energy for a bi nary syst em is given
by [21]:
G E ( w) [ l ( 2w) 1
RT- ~ XlX2 1 - ~ ~ XlX2-~-''"
where the hi gher t erms are neglected. Xl and x2 are mol e
fract i ons of 1 and 2 and k is t he Bol t zman' s constant. W
and z are model paramet ers t hat must be det ermi ned for
each system. W/ zkT is less t han uni t y and z is called co-
ordination number and varies f r om 6 t o 12 [21]. Typical
value of z is 10. Use Eq. (6.137) to derive the relations
for F1 and Y2.
6.31. Show t hat at const ant T and P, t he Gi bbs- Duhem equa-
tion in a mul t i comp onent mi xt ure can be wri t t en in t he
following forms:
}--~x4dln y~ = 0
i
or
~x~dl n] ~ = 0
i
6.32. Consi der a bi nary sol ut i on of comp onent s 1 and 2. Show
t hat in the regi on t hat Raoult' s law is valid for compo-
nent I , Henry' s law must be valid for comp onent 2.
6.33. I n a bi nary liquid, mi xt ures of 1 and 2, fugacity of com-
ponent 1 at 20~ can be approxi mat el y present ed by t he
equation:
fl~ = 30xl - 2 0 x ~
where fl % is the fugacity of 1 in the mi xt ure in bar. At
20~ and 30 bar det ermi ne:
a. The fugaci t y of pure comp onent 1, fL.
b. The fugaci t y coefficient of pure comp onent 1, qh.
c. The Henry' s law const ant for comp onent 1, kl.
d. Rel at i on for the activity coefficient yl in t erms of xl
( based on t he st andard state of Lewis rule).
e. Rel at i on for f2 ~.
6.34. Consi der t hree hydr ocar bon comp onent s benzene, cy-
clohexane, and n-hexane all havi ng six car bon atoms.
Bot h quantitatively and qualitatively state t hat solubility
of benzene in met hyl cyl opent ane is hi gher or benzene
in n-hexane.
6.35. A sol ut i on is made at t he t emperat ure of 298 K by addi ng
5 g of napht hal ene to a mi xt ure of 50 g benzene and 50 g
n-heptane. The t emperat ure is gradual l y l owered until
the particles of solid are observed. What is the temper-
at ure at this point? What is t he t emper at ur e if 10 g of
napht hal ene is added?
6.36. Est i mat e vapor pressure of i sobut ane at 50~ f r om t he
following met hods:
a. aRK EOS.
b. PR EOS.
c. Cl ausi us-Cl apeyron equat i on
d. Compare the results wi t h a report ed value.
6.37. A nat ural gas is comp osed of 85% met hane, 10% ethane,
and 5% propane. What are t he mol e fract i ons of each of
t he comp onent s in wat er (gas solubility) at 300 bar and
298 K.
6.38. Ni net y barrels of n-C36 are diluted wi t h addi t i on of 10
bbl of n-C5 at 25~ Calculate vol ume of the sol ut i on at
1 bar f r om the following met hods:
a. Using partial mol ar vol ume from PR EOS.
b. Using API procedure.
6.39. Show t hat if Edmi st er equat i on (Eq. 2.108) is used
for acent ri c factor, Wilson correl at i on for Ki-values
(Eq. 6.204) reduces to Hof f man type correl at i on (Eq.
6.202).
6.40. Solubility of wat er in a gasoline sampl e at 1 at m can
be det ermi ned approxi mat el y by Eq. (6.195). However,
accurat e solubility of wat er can be est i mat ed t hr ough
a t her modynami c model wi t h activity coefficient calcu-
lations. A gasoline f r om California oil has mi d boiling
poi nt of 404~ and API gravity of 43.5 wi t h PNA compo-
sition of 30.9, 64.3, and 4.8% as report ed by Lenoi r and
Hipkin [ 12]. Est i mat e solubility of wat er in this gasoline
sampl e at 100~ and 1 at m from appropri at e t hermody-
nami c model and comp ar e the predi ct ed value wi t h t he
value est i mat ed from Eq. (6.195).
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[40] Standing, M. B., "A Set of Equations for Computing
Equilibrium Ratios of a Crude Oil/Natural Gas System at
Pressures Below 1000 psia," Journal of Petroleum Technology,
Vol. 31, 1979, pp. 1193-1195.
[41] Katz, D. L. and Hachmuth, K. H., "Vaporization Equilibrium
Constants in a Crude-Oil-Natural-Gas System," Industrial
and Engineering Chemistry Research, Vol. 29, 1937, pp. 1072-
1077.
[42] Glaso, O. S., "The Accuracy of PVT Parameters Calculated from
Computer Flash Separation at Pressures Less than 1000 psia,"
Journal of Petroleum Technology, Vol. 34, 1982, pp. 1811-1813.
[43] Winnick, J., Chemical Engineering Thermodynamics, Wiley, New
York, 1997.
in Predicting the PVT Relations," Fluid Phase Equilibria,
Vol. 90, 1993, pp. 251-264.
[45] Riazi, M. R. and Roomi, Y. A., "Use of Velocity of Sound in
Estimating Thermodynamic Properties of Petroleum
Fractions," Preprints of Division of Petroleum Chemistry
Symposia, American Chemical Society (ACS), Vol. 45, No. 4,
2000, pp. 661-664.
[46] Colgate, S. O., Silvarman, A., and Desjupa, C., "Acoustic
Resonance Determination of Sonic Speed and the Critical
Point," Paper presented at the AIChE Spring Meeting, Paper
No. 29c, New Orleans, 1992.
[47] Colgate, S. O., Silvarman, A., and Desjupa, C., "Sonic Speed
and Critical Point Measurements in Ethane by the Acoustic
Resonance Method," Fluid Phase Equilibria, Vol. 76, 1992,
pp. 175-185.
[48] Ball, S. J., Goodwin, A. R. H., and Trusler, J. P. M., "Phase
Behavior and Physical Properties of Petroleum Reservoir
Fluids from Acoustic Measurements," Journal of Petroleum
Science and Engineering, Vol. 34, No. 1-4, 2002, pp. 1-11.
[49] Mittal Enterprises, New Delhi, India, 2004
(http://www.mittalenterprises.com).
[50] Alem, A. H. and Mansoori, G. A., "The VIM Theory of
Molecular Thermodynamics," American Institute of Chemical
Engineers Journal, VoL 30, 1984, pp. 468-480.
[51] Walas, S. M., Phase Equilibria in Chemical Engineering,
Butterworth, London, 1985.
[52] Goodwin, R. D., The Thermophysical Properties of Methane from
90 to 500 K at Pressures to 700 Bar, NBS Technical Note 653,
National Bureau of Standards, April 1974.
[53] Goodwin, R. D., Roder, H. M., and Starry, G. C., The
Thermophysical Properties of Ethane from 90 to 600 K at
Pressures to 700 Bar, NBS Technical Note 684, National Bureau
of Standards, August 1976.
[54] Goodwin, R. D. and Hayens, W. M., The Thermophysical
NBS Monograph 170, National Bureau of Standards, April
1982.
[55] Mossaad, E. and Eubank, E T., "New Method for Conversion
of Gas Sonic Velocities to Density Second Virial Coefficients,"
No. 1, 2001, pp. 222-229.
[57] Konttorp, K., Peters, C. J., and O'Connell, J. P., "Derivative
Properties from Model Equations of State," Paper presented at
the Ninth International Conference on Properties and Phase
Equilibria for Product and Process Design ( PPEPPD 2001),
MNL50-EB/Jan. 2005
A pplications: Estimation of
Thermophysical Properties
NOMENCL AT URE
API
A, B , C, D, E
a, b
a, b , c , d, e
Cp
dr
f(o), f(1)
/Ogl0
In
M
RV P
S%
SG
Coetficients in various equations
Cubic EOS paramet ers given in Table 5.1
Eq. (6.17), J/ mol 9 K
Cv Heat capacity at constant volume defined by
Eq. (6.18), J / t ool -K
Liquid density at t emperat ure T and 1 atm,
g/ cm 3
Dimensionless functions for vapor pressure
generalized correlation (Eq. 7.17)
H Enthalpy defined in Eq. (6.1), mol ar unit:
J/tool; specific unit: kJ/kg
H% Hydrogen wt% in a pet rol eum fraction
I Refractive index par amet er at t emperat ure T,
defined in Eq. (2.36) [ =( n 2- 1)/(n 2 +2) ] , di-
mensionless
Kw Watson characterization factor defined by
Eq. (2.13)
Common logarithm (base 10)
Natural logarithm (base e)
Molecular weight ( mol ar mass), g/tool [kg/
kmol]
N Number of component s in a mi xt ure
Nc Number of carbon at oms
N% Nitrogen wt% in a pet rol eum fraction
n Liquid refractive index at t emperat ure T and
1 arm, dimensionless
0% Oxygen wt% in a pet rol eum fraction
P Pressure, bar
( ~- P / Pc ), dimensionless
Ptp Triple point pressure, bar
pvap Vapor pressure at a given temperature, bar
p~ub Sublimation pressure ( vapor pressure of a
solid) at a given t emperat ure, bar
Pr ~ap Reduced vapor pressure at a given t emperat ure
(~-pvap/Pc) , dimensionless
Q A paramet er defined in Eq. (7.21)
R Gas constant = 8.314 J / mol - K (values in dif-
ferent units are given in Section 1.7.24)
Reid vapor pressure, bar
Sulfur wt% in a pet rol eum fraction
Tb Normal boiling point, K
( = T /Tc) , dimensionless
To A reference t emperat ure for use in Eq. (7.5), K
TM Freezing (melting) point for a pure component
at 1.013 bar, K
Tbr Reduced boiling point (=Tb/Tc), dimensionless
T~ Pseudocritical temperature, K
V Molar volume at T and P, cma/ mol
Vc Critical mol ar volume, cm3/ mol
V s Molar vol ume of solid, cm3/ mol
x/ Mole fraction of component i in a mixture, di-
mensionless
xwi Weight fraction of component i in a mixture
(usually used for liquids), dimensionless
xp, XN, XA Fractions (i.e., mole) of paraffins, naphthenes,
and aromatics in a pet rol eum fraction, dimen-
sionless
Z Compressibility factor defined by Eq. (5.15),
dimensionless
Greek Letters
A Difference between two values of a par amet er
0m A mol ar propert y (i.e., mol ar enthalpy, mol ar
volume, et c. . . )
0s A specific propert y (i.e., specific enthalpy,
et c. . . )
g/ cm 3 ( molar density unit: cm3/mol)
Pm Molar density at a given t emperat ure and pres-
sure, mol / cm 3
Pr Reduced density ( = P/Pc = V c/ V ), dimension-
less
pW Water density at a given t emperat ure, g/ cm 3
a~ Acentric factor defined by Eq. (2.10), dimen-
sionless
AH~298 Heat of format i on at 298 K, kJ/ mol
AH ~ap Heat of vaporization (or latent heat) at temper-
ature T, J/ mol
S upers cri pt
ig Value of a propert y for component i as ideal
gas at t emperat ure T and P -> 0
L Value of a propert y at liquid phase
V Value of a propert y at vapor phase
2 9 7
vap Change in value of a propert y due to vaporiza-
tion
S Value of a propert y at solid phase
sat Value of a propert y at saturation pressure
sub Value of a propert y at sublimation pressure
[](0) A dimensionless t erm in a generalized correla-
tion for a propert y of simple fluids
[](1) A dimensionless t erm in a generalized correla-
tion for a propert y of acentric fluids
o Value of a propert y at low pressure (ideal gas
state) condition at a given t emperat ure
Subscripts
A
B
b
C
i , j
Value of a propert y for component A
Value of a propert y for component B
Value of a propert y at the normal boiling point
Value of a propert y for component i or j in a
mixture
L Value of a propert y for liquid phase
m Molar propert y (quantity per unit mote)
m Mixture propert y
mix Value of a propert y for a mixture
nbp Value of a liquid phase propert y at the normal
boiling point of a substance
pc Pseudocritical propert y
r Reduced propert y
ref Value of a propert y at the reference state
S Value of a propert y at the solid phase
S Value of a propert y for solvent (LMP)
s Specific propert y (quantity per unit mass)
T Values of propert y at t emperat ure T
tp Value of a propert y at the triple point
W Values of a propert y for wat er
20 Values of propert y at 20~
7+ Values of a propert y for C7+ fraction of an oil
Acronyms
API-TDB
BI P
COSTALD
DI PPR
EOS
GC
HHV
LHV
MB
RVP
PR
PNA
PVT
SRK
scf
Book (see Ref. [9])
Binary interaction par amet er
Corresponding State Liquid Density (given by
Eq. 5.130)
Design I nstitute for Physical Property Data (see
Ref. [i0])
Equation of state
Generalized correlation
Hi gher heating value
Lower heating value
Maxwell and Bonnell (see Eqs. (3.29), (3.30),
and (7.20)-(7.22))
Reid vapor pressure
Peng-Robi nson EOS (see Eq. 5.39)
Paraffins, naphthenes, aromat i cs content of a
pet rol eum fraction
Pressure-vol ume-t emperat ure
Soave-Redl i ch-Kwong EOS given by Eq.
(5.38) and paramet ers in Table 5.1
Standard cubic foot (unit for volume of gas at
1 at m and 60~
stb
TVP
VABP
%AAD
%AD
wt%
Stock tank barrel (unit for volume of liquid oil
at 1 at m and 60~
True vapor pressure
Volume average boiling point defined by Eq.
(3.3)
Average absolute deviation percentage defined
by Eq. (2.135)
Absolute deviation percentage defined by Eq.
(2.134)
Weight percent
THE LAST THREE CHAPTERS of this book deal with application
of met hods presented in previous chapters to estimate var-
ious t hermodynami c, physical, and transport properties of
pet rol eum fractions. I n this chapter, various met hods for pre-
diction of physical and t hermodynami c properties of pure
hydrocarbons and their mixtures, pet rol eum fractions, crude
oils, natural gases, and reservoir fluids are presented. As it was
discussed in Chapters 5 and 6, properties of gases may be esti-
mat ed more accurately t han properties of liquids. Theoretical
met hods of Chapters 5 and 6 for estimation of thermophysical
properties generally can be applied to bot h liquids and gases;
however, more accurate properties can be predicted t hrough
empirical correlations particularly developed for liquids.
When these correlations are developed with some theoretical
basis, they are more accurate and have wider range of appli-
cations. I n this chapt er some of these semitheoretical corre-
lations are presented. Methods presented in Chapters 5 and 6
can be used to estimate properties such as density, enthalpy,
heat capacity, heat of vaporization, and vapor pressure.
Characterization met hods of Chapters 2-4 are used to de-
t ermi ne the input paramet ers needed for various predictive
methods. One i mport ant part of this chapt er is prediction of
vapor pressure that is needed for vapor-liquid equilibrium
calculations of Chapter 9.
7. 1 GENERAL APPROACH FOR
PREDI CTI ON OF THERMOPHY S I CAL
PROPERTI ES OF PETROLEUM FRACTI ONS
AND D EFI N ED HY DROCARBON MI XTURES
Finding reliable values for inadequate or missing physical
properties is the key to a successful simulation, which de-
pends on the selection of correct estimation met hod [1]. I n
Chapters 5 and 6 theoretically developed met hods for calcu-
lation of physical and t hermodynami c properties of hydro-
carbon fluids were presented. Paramet ers involved in these
met hods were mainly based on properties of pure com-
pounds. Methods developed based on corresponding states
approaches or complex equations of state usually predict
the properties more accurately t han those based on cubic
EOSs. For the purpose of propert y calculations, fluids can
be divided into gases and liquids and each group is fur-
ther divided into two categories of pure component s and
mixtures. Furthermore, fluid mixtures are divided into two
categories of defined and undefined mixtures. Examples of
defined mixtures are hydrocarbon mixtures with a known
composition, reservoir fluids with known compositions up to
C6, and pseudocompounds of the C7 fraction. Also pet rol eum
fractions expressed in t erms of several pseudocomponent s
7. APPLICATIONS: ESTIM ATION OF THERM OPHYSICAL PROPERTIES 299
can be considered as defined mixtures. Examples of unde-
fined mixtures are pet rol eum fractions and reservoir fluids
whose compositions are not known. For such mixtures, some
bulk properties are usually known.
Theoretically developed met hods are generally more ac-
curate for gases t han for liquids. Kinetic theory provides
sound predictive met hods for physical properties of ideal
gases [2, 3]. For this reason, empirical correlations for calcu-
lation of physical properties of liquids have been proposed.
Similarly, theoretical met hods provide a more accurate es-
timation of physical properties of pure compounds t han of
their mixtures. This is mai nl y due to the complexity of inter-
action of component s in the mixtures especially in the liquid
phase. For undefined mixtures such as pet rol eum fractions,
properties can be calculated in three ways. One met hod is to
consider t hem as a single pseudocomponent and to use the
met hods developed for pure component s. The second met hod
is to develop empirical correlations for pet rol eum fractions.
Such empirically developed met hods usually have limited ap-
plications and should be used with caution. They are accurate
for those data for which correlation coefficients have been
obtained but may not provide reliable values for properties
of other fractions. These two approaches cannot be applied
to mixtures with wide boiling range, such as wide fractions,
crude oils, or reservoir fluids. The third approach is used
for available data on the mixture to express the mixture in
t erms of several pseudocomponent s, such as those met hods
discussed in Chapters 3 and 4. Then, met hods available for
prediction of properties of defined mixtures can be used for
such pet rol eum fluids. This approach should particularly be
used for wide boiling range fractions and reservoir fluids.
Fluid properties generally depend on t emperat ure (T), pres-
sure (P), and composition (xi). Temperature has a significant
effect on properties of bot h gases and liquids. Effect of pres-
sure on properties of gases is much larger t han effect of pres-
sure on properties of liquids. The magni t ude of this effect
decreases for fluids at higher pressures. For the liquid flu-
ids, generally at low pressures, effect of pressure on prop-
erties is neglected in empirically developed correlations. As
pressure increases, properties of gases approach properties
of liquids. Effect of composi t i on on the properties of liquid
is stronger t han the effect of composi t i on on properties of
gases. Moreover, when component s vary in size and proper-
ties the role of composition on propert y estimation becomes
more important. For gases, the effect of composi t i on on prop-
erties increases with increase in pressure. At higher pressures
molecules are closer to each ot her and the effect of interac-
tion between dissimilar species increases. For gases at at mo-
spheric or lower pressures where the gas may be considered
ideal, composi t i on has no role on mol ar density of the mixture
as seen from Eq. (5.14).
There are two approaches to calculate properties of defined
mixtures. The first and more commonl y used approach is to
apply the mixing rules introduced in Chapter 5 for the in-
put paramet ers (To, Pc, oJ) of an EOS or generalized correla-
tions and then to calculate the properties for the entire mix-
ture. The second approach is to calculate desired propert y
for each component in the mixture and then to apply an ap-
propri at e mixing rule on the property. This second approach
usually provides more accurate results; however, calculations
are more tedious and time-consuming, especially when the
number of component s in the mixture is large since each
propert y must be calculated for each component in the mix-
ture. I n applying a mixing rule, the role of binary interaction
paramet ers (BIPs) is i mport ant when the mixture contains
component s of different size and structure. For example, in a
reservoir fluid containing C1 and a heavy component such as
C30 the role of BI P between these two component s cannot be
ignored. Similarly when nonhydrocarbon component s such
as H2S, N:, H20, and CO2 exist in the mixture, the BIPs of
these compounds with hydrocarbons must be considered. For
some empirically developed correlations specific interaction
paramet ers are recommended that should be used.
Theoretically developed t hermodynami c relations of Chap-
ters 5 and 6 give t hermodynami c properties in mol ar quan-
tifies. They should be converted into specific properties by
using Eq. (5.3) and mol ecul ar weight. I n cases that no spe-
cific mixing rule is available for a specific propert y the simple
Kay' s mixing rule (Section 3.4.1) may be used to calculate
mixture properties from pure component properties at the
same conditions of T and P. I f mol ar properties for all com-
ponent s (0r~) are known, the mixture mol ar propert y (0m)
may be calculated as
( 7 . 1 ) 0m = ZXmiOmi
i
where Xmi is mole fraction of component i and the summa-
tion is on all component s present in the mixture. Subscript
m indicates that the propert y is a mol ar quantity (value of
propert y per unit mole). For gases especially at low pressures
(< 1 bar), the volume fraction, Xvi may be used instead of mole
fraction. Similarly for specific properties this equation can be
written as
(7.2) 0s : ZXwiOsi
where Xwi is weight fraction of i in the mixture and subscript
s indicates that the propert y is a specific quantity (per unit
mass). I n the above two equations, 0 is a t hermodynami c
propert y such as volume (V), internal energy (U), enthalpy
(H), heat capacity (Cp), ent ropy (S), Helmholtz free energy
(A), or Gibbs free energy (G). Usually Eq. (7. I) is used to cal-
culate mol ar propert y of the mixture as well as its mol ecul ar
weight and then Eq. (5.3) is used to calculate specific propert y
wherever is required. I n fact Eqs. (7.1) and (7.2) are equiva-
lent and one may combi ne Eqs. (5.3) and (1.15) with Eq. (7. I)
to derive Eq. (7.2). These equations provide a good estimate
of mixture properties for ideal solutions and mixtures of sim-
ilar compounds where the interaction between species may
be ignored.
Empirically developed correlations for properties of un-
defined or defined mixtures are based on a certain group of
data on mixtures. Correlations specifically developed based
on data of pet rol eum fractions usually cannot be used for
estimation of properties of pure hydrocarbons. However, if
in development of correlations for properties of undefined
pet rol eum fractions pure component data are also used, then
the resulting correlation will be more general. Such correla-
tions can be applied to bot h pure component s and undefined
mixtures and they can be used more safely to fractions
that have not been used in development of the correlation.
However, one i mport ant limitation in use of such correlations
is boiling point range, carbon number, or mol ecul ar weight
of fractions and compounds used in the development of
the correlations. For example, correlations that are based
on properties of pet rol eum fractions and pure component s
with carbon number range of C5-C20 cannot be used for
estimation of properties of light gases ( natural gases or LPG),
heavy residues, or crude oils. Another limitation of empir-
ically developed correlations is the met hod of calculation
of input paramet ers, For example, a generalized correlation
developed for properties of heavy fractions requires critical
properties as input parameter. For such correlations the same
met hod of estimation of input paramet ers as the one used
in the development of correlation should be used. The most
reliable correlations are those that have some theoretical
background, but the coefficients have been det ermi ned
empirically from data on pet rol eum fractions as well as pure
compounds. One technique that is often used in recent years
to develop correlations for physical properties of bot h pure
compounds and complex mixtures is the artificial neural
network met hod [4-6]. These met hods are called neural net-
works met hods because artificial neural networks mi mi c the
behavior of biological neurons. Although neural nets can be
used to correlate data accurately and to identify correlative
patterns between input and target values and the i mpact of
each input par amet er on the correlation, they lack necessary
theoretical basis needed in physical propert y predictions.
The resulting correlations from neural nets are complex and
involve a large number of coefficients. For this reason corre-
lations are inconvenient for practical applications and they
have very limited power of extrapolation outside the ranges
of data used in their developments. However, the neural
net model can be used to identify correlating paramet ers in
order to simplify theoretically developed correlations.
book. Equation (7.3) is valid for bot h liquids and gases once
their Z values are calculated from an equation of state or a
generalized correlation. I f Z is known for all component s in
a mixture, then Zm can be calculated from Eq. (7.1) and Pm
from Eq. (7.3). Specific met hods and recommendat i ons for
calculation of density of gases and liquids are given in the
following sections.
7.2.1 Density of Gases
Generally both equations of state and the Lee-Kesler gener-
alized correlation (Section 5.7) provide reliable prediction of
gaseous densities. For high-pressure gases, cubic EOS such
as PR or SRK EOS give acceptable values of density for bot h
pure and mixtures and no volume translation (Section 5.5.3)
is needed. For practical calculations, properties of gases can
be calculated from simple equations of state. For example,
Press [7] has shown that the original simple t wo-paramet er
Redlich-Kowng equation of state (RK EOS) gives reasonably
acceptable results for predicting gas compressibility factors
needed for calculation of valve sizes. For moderat e pressures
truncated virial equation (Eq. 5.76) can be used with coeffi-
cients (B and C) calculated from Eqs. (5.71) and (5.78). For
low-pressure gases ( <5 bar), virial equation truncated after
the second t erm (Eq. 5.75) with predicted second virial coef-
ficient from Tsonopoulos correlation (Eq. 5.71) is sufficient
to predict gas densities. For light hydrocarbons and natural
gases, the Hall-Yarborough correlation (Eq. 5.102) gives a
good estimate of density. For defined gas mixtures the mixing
rule may be applied to the input paramet ers (Tc, Pc, and o)) and
the mixture Z value can be directly calculated from an EOS.
For undefined natural gases, the input paramet ers may be
calculated from gas-specific gravity using correlations given
in Chapter 4 (see Section 4.2).
7 . 2 DENS I TY
Density is perhaps one of the most i mport ant physical proper-
ties of a fluid, since in addition to its direct use in size calcula-
tions it is needed to predict other t hermodynami c properties
as shown in Chapter 6. As seen in Section 7.5, met hods to
estimate t ransport properties of dense fluids also require re-
duced density. Therefore once an accurate value of density
is used as an input par amet er for a correlation to estimate
a physical property, a more reliable value of that propert y
can be calculated. Methods of calculation of density of fluids
have been discussed in Chapter 5. Density may be expressed
in the form of absolute density (p, g/cm3), mol ar density
(Pm, mol/ cm 3), specific volume (V, cm3/g), mol ar volume (Vm,
cm3/mol), reduced density (Pr ----- P/Pc = Vc/V, dimensionless),
or compressibility factor (Z = PV /RT = V / V i g , dimensionless).
Equations of states or generalized correlations discussed in
Chapter 5 predict Vm or Z at a given T and P. Once Z is known,
the absolute density can be calculated from
M P
(7.3) P = ZRT
where M is the molecular weight, R is the gas constant, and
T is the absolute t emperat ure. I f M is in g/mol, P in bars,
T in kelvin, and R = 83.14 bar. cm3/ mol. K, then p is calcu-
lated in g/ cm 3, which is the standard unit for density in this
7. 2. 2 Density of Liquids
For high-pressure liquids, density may be estimated from cu-
bic EOS such as PR or SRK equations. However, these equa-
tions break at carbon number of about C10 for liquid density
calculations. They provide reasonable values of liquid density
when appropri at e volume translation introduced in Section
5.5.3 is used. The error of liquid density calculations from
cubic equations of states increases at low and at mospheri c
pressures. For saturated liquids, special care should be taken
to take the right Z value (the lowest root of a cubic equation).
Once a cubic equation is used to calculate various t hermody-
nami c properties (i.e., fugacity coefficient) at high pressures,
it is appropri at e to use a cubic equation such as SRK or PR
with volume translation for bot h liquid and gases. However,
when density of a liquid alone is required, PR or SRK are not
the most appropri at e met hod for calculation of liquid density.
For heavy hydrocarbons and pet rol eum fractions, the modi-
fied RK equation of state based on refractive index proposed
in Section 5.9 is appropri at e for calculation of liquid densi-
ties. The refractive index of heavy pet rol eum fractions can be
estimated accurately with met hods outlined in Chapter 2. One
should be carefl,d that this met hod is not applicable to non-
hydrocarbons (i.e., water, alcohols, or acids) or highly polar
aromat i c compounds.
For the range that Lee-Kesler generalized correlation (Eq.
(5.107) and Table 5.9) can be used for liquids, it gives density
32 ~
7. APPLICATIONS: ESTI M ATI ON OF THERM OPHYSICAL PROPERTI ES 301
~ 30G ~C~' 5~0 60O
LO
~ 9
0,9
0,8
0~
0.7
06
0.4
300 400 500
T~em~ra~re. "F
60O
FIG. 7 . 1 - - Ef f ect of t emperat ure on the liquid speci fi c g rav i ty of hydrocarbons. Units
conv ersion: ~ ---- ( ~ 1.8 -I- 32. Taken with permi ssi on from Ref. [ 8 ].
values more accurate t han SRK or PR equations without vol-
ume translation. The Lee-Kesler correlation is particularly
useful for rapi d-hand calculations for a single data point.
The most accurate met hod for prediction of saturated liq-
uid densities is t hrough Rackett equation introduced in Sec-
tions 5.8. However, for high-pressure liquids the met hod of
API (Eq. 5.129) or the COSTALD correlation (Eq. 5.130)
may be used combi ned with the Rackett equation to pro-
vide very accurate density values for both pure component s
and pet rol eum fractions. These met hods are also applicable to
nonhydrocarbons as well. At low pressures or when the pres-
sure is near saturation pressure, no correction on the effect of
pressure is required and saturated liquid density calculated
from Rackett equation may be directly used as the density of
compressed (subcooled) liquid at pressure of interest.
For liquid mixtures with known composition, density can
be accurately calculated from density of each component
(or pseudocomponent s) through Eq. (7.2) when it is applied
8
E
ea
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I 1 ....... I
I l f l
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I I
I !
I i l l I I I
f i l l I I I
1 I I l : l l l
i l I l i l l l I I 11t t t I I I I 111f } [ t i l l t l I I I I I I I I l l
= l t I l I MI ] I I l l ] l I ] l l l I I I 1 i t I [ I I I I I I I ! 11 I l l
9 I l l d_ l J l l l I I I i 1 i t 1 l . l 11 11 I t t 1 t i I I I I I1 l ] ! I I I [ Xl
l l i I l l i l l l I t 1 , ~ 1 t 1 1 1 1 1 1 1 [ 1 l I I 1 l I I i l 1 I I 1 1 1 1 1
l r %l ] l l l l I I 1 1 t t i t l ] l I I I I I I 1 ! 1 1 I 1 I I I I I ! f I l l
~ ] _ l _ 1 2 i t t i l l l l l l l 1 i i l i 1 1 i l l ..... | . . . i l l I l l
, ~ r i 1 f I I i 1 1 I I 1 1 1 i l l 1 ! 1 1 I f i
i i I Pd ! I r %~ ; l q I 1 t I 1 1 t 1 ! I I l l , , i , l l t l '
[ ] l i / l l - l l l l i i I 1 1 1 1 1 1 1 l i t t 1 I I I t l i t I i l f l l i
r l i l l l l l I I 1 t l I I ! 1 t i l l l l I 1 1 f 1 1 1 I I I I I I i l l
l l l l I l i l l l l I 1 Mt I I t ! 1 f I I I I l I I l i 1 1 1 ! 1 I I 1 1 I I !
l ~ I l kt 1 1 [ i l L I l ! i 1 I I I i 1 l l I i
~ 1 I i l l t t I I P l [ l l ] " ~ t [ I ! 1 I I I t i l l
~ F 1 K ~ J I % I I l i t t i l l l I ~ f ! I 1 1 1 ! 1 l I I 1 I I I
! I %1 1 1 1 1 1 t ' ~ ] ! t %1 ~ I I t I 1 1 1 t i I l i t i J i t I I f I i l i
5l i J_ _ l i I 3 t _ l _ ] T l ' ~ l ] i t ~ t I I 1 1 1 i t l 1 1 I I 1 1 1 I t I NI I l l
~ ' k ~ L l l I l i l I i 1 1 I l l 1 ~ I J I I I 1 1 1 I i I 1 1 1 1
i - I KI T N- [ I " I - I TM ~ t . _ ~ kLLJ_ L~ L[ ] I I i 1 t I 1 ! 1 1 1 1 I 1 1 1 I I I f 1 I
~ K I - P t L ~ T 1 1 i l t I t I 1 1 1 I ! 1 1 I I I I I I I
~ ~ t - : ~ ' ! 1 i I t I I I I I I I I I ! I I I I I
4 ~ ~ R ~ ~ ~ I I t t ! 1 1 I t ! 1 t I ! I 1 1 l l
~ ) ~ . 1 i l l r % L I I I I % - % " 1 i I ! I i "[ I J l l I I I t l I I
i ~ j _ _ ~ j _ ~ : ~ i i _ l _ ~ i ~ l _ ~ kl 1 ~ t l 1 1 1 ~ l : ~ / e l ] ! I I / I ! 1 t I I t I l l i
g4-4-3d~2~,~.ll~ i ~ t l i i ~ ~ i l i i i i = i i i l l z : ] : l ,
i ' 7 - I - ] KI - I ~ Lr . I %I I ~ I I I %i 1 %1 1 1 1 F~ kl I l l I I I I I I I 1 1 I l l
. I k l I I ~ = l I ~ I ~ i ~ % t I f N t l I ~ 1 i l I I % J l I I I ! 1 t I I ! I 1 i l l
~ ~ " l ~ ~ I I ~ f ~ t i .I t ~ ~ . ki ! ! 1 i I 1 I I t I I !
~ ~ . . ~ i i ~ t ~ i I ] ~ d 1 gr ~ LI I I I l l ~ l ~ d l l ] t i I l l I l l
~ ~ ] ~ I ! N: l ~ I kl t 1~1~11 r " - ~ l i l t I I ~ ! f I I l l
i i t. I ~ i I l - ~ i ~ : i ~ % l ~ i l l - l i ~ l i T ~ I i I ! 1 I I i ~ I t i
! I t I I i % ~ i A ~ I ~ k I P*t~A l-_'~l~L I 1 " ~ t 4J 1 F - - ~ ~
I I I ! 1 1 %l 1 ! 7%~1 1--11~Cid l ' ~ l l i l ~ T i -k~ .JL i I ] ~ ' -] -4~ . l J 1 I i ~ I 1 !
~ . 1 1 1 ! ~ _ 1 1 1 1 ~ 1 1 1 ~ 7 i l ~ l _ F ' ~ l ~ d . _ l I I / ~ , _ " 1 1 I I !
~ - I L ~ I I I ~ 1 I t l r ~ d ~ l r ' l ~ k. . L. ~ i ] - l ~ ' ~ i [ - ~ ! I I I 1- I 1 1
0 Ill ! | - 9 9 II 9 ! 9
25 30 35 40 45 50 55 60 65
Density at NY"F ancf 14.7 ~ . Ib Per cu I1
F I G . 7 . 2 - - E f f e c t of p r e s s u r e o n t h e l i qu i d d e n s i t y a t 60 ~ ( 1 5 . 5 ~ Un i t c o n v e r -
s i o n : p [ g / c m 3 ] = p [ I b / f t 3 ] / 62 . 4; ~ = ( ~ x 1 . 8 + 3 2 ; p s i a = b a r x 1 4. 5 0 4. T a k e n
wi t h p e r mi s s i o n f r o m Re f . [ 8 ].
to specific vol ume (Vs = 1/p) as
(7.4) 1 ? x~
Pmi x 9 Pi
where/ 9mi x is t he mi xt ur e l i qui d densi t y (i.e., g/ cm 3) and Xwi
is wei ght fract i on of comp onent i in t he l i qui d mi xt ure, pi
shoul d be known f r om dat abase, exper i ment , or may be cal-
cul at ed f r om Racket t equat i on. Equat i on (7.4) can be ap p l i ed
t o specific gravi t y but not to mol ar density. Thi s equat i on is
p ar t i cul ar l y useful for cal cul at i on of speci fi c gravi t y and den-
si t y of cr ude oi l s wi t h known comp osi t i on at at mos p her i c
pr essur e, as it is shown l at er i n t hi s chapt er.
When onl y a mi ni mum of one dat a p oi nt for l i qui d densi t y
of a p et r ol eum fract i on at at mos p her i c p r essur e is known (i.e.,
SG, d20, or d25), t hen Eq. (2.1 10) may be used to cal cul at e liq-
ui d densi t y at at mos p her i c p r essur e and ot her t emp er at ur es.
For exampl e, Eq. (2.111) can be used to est i mat e densi t y at
t emp er at ur e T f r om SG. The gener al f or mul a for cal cul at i on
of dx ( densi t y at T and 1 at m) f r om known densi t y at a refer-
ence t emp er at ur e To can be der i ved f r om Eq. (2.110) as
(7.5) dT = dro - 10 -3 x (2.34 - 1.9dT) x ( T - To)
wher e bot h T and To ar e in kel vi n or in ~ and dr and dTo
are in g/ cm 3 . This met hod pr ovi des rel i abl e densi t y val ues at
t emp er at ur es near t he r ef er ence t emp er at ur e at whi ch densi t y
is known ( when T is near To). When t he act ual t emp er at ur e is
far f r om t he r ef er ence t emp er at ur e, t hi s equat i on shoul d be
used wi t h caut i on.
F or qui ck densi t y cal cul at i ons several gr ap hi cal met hods
have been devel oped, whi ch ar e less accur at e t han t he met h-
ods out l i ned above. Gr ap hi cal met hods for l i qui d densi t y cal -
cul at i ons r ecommended by GPA [8] ar e shown i n Figs. 7. 1-7. 3.
30
7. APPLICATIONS: ESTI M ATI ON OF THERM OPHYSICAL PROPERTIES
3 5 40 ~5 ~0 55 60
3 03
8
E
E
30 35 40 45 50 55 6O
C,r at 60~F and P~essute, I b p~ c~ (I
FI G . 7 . 3 - - E f f e c t of t emp er at ur e on l i qui d dens i t y at p r es s ur e P. Uni t conv er s i on: p [ g / cm 3] =
p[ I b/ f t a]/ 62. 4; ~ = ( ~ 1 . 8 -I- 3 2. T ak en wi t h p er mi s s i on f r om Ref . [ 8 ].
Figure 7.1 gives effect of temperature on the liquid specific
gravity (SG). Once SG at 60~ (15.5~ is known, with the use
of this figure specific gravity at temperature T (SGT) can be
determined. Then density at T and 1 atm can be determined
through multiplying specific gravity by density of water. This
figure may be used for calculation of density of liquids at low
pressures where the effect of pressure on liquid density can be
neglected (pressures less than ~50-70 bar). Figure 7.2 shows
effect of pressure on density at 60~ and Fig. 7.3 shows effect
of T on liquid density at any pressure P (P > 1000 psia or
TABLE 7. 1--Properties of saturated liquid and solid at the freezing point for some hydrocarbons [10].
No. Compound Formula M T~,t/Ttp, K Ptp, bar pL, g/cm 3 pS, g/cm 3
n -Paraffins
1 n-Pentane C5H12 72.15 143. 42 6.8642 x 10 -7 0.7557 0.9137
2 n-Hexane C6H14 86.17 177.83 9.011 10 -6 0.7538 0.8471
3 n-Heptane C7H16 100.20 182.57 1.8269 10 -6 0.7715 0.8636
4 n-Octane C8H18 114. 22 216.38 2.108 x 10 -5 0.7603 0.8749
5 n-Nonane C9H20 128.25 219.66 4.3058 x 10 -6 0.7705 0.8860
6 n-Decane C10H22 142. 28 243.51 1. 39297 10 -5 0.7656 0.8962
7 n-Tetradecane C14H30 198. 38 279. 01 2.5269 10 -6 0.7722 0.9140
8 n-Pentadecane C15H32 212.41 283.07 1. 2887 x 10 6 0.7752 0.9134
9 n-Eicosane C20H42 282.54 309.58 9.2574 x 10 -8 0.7769 0.8732
10 n-Hexacosane C26H54 366.69 329.25 5.1582 10 -9 0.7803 0.9254
11 n-Nonacosane C29H60 408.77 336.85 6.8462 10 -10 0.7804 0.9116
12 n-Triacontane C30H62 422.80 338.65 2.0985 x 10 -1~ 0.7823 0.9133
13 n-Hexacontane C36H74 506.95 349.05 2.8975 10 -12 0.7819 0.9610
n-Alkylcyclohexanes (naphthenes)
14 Cyclohexane C6H12 84.16 279.69 5.3802 x 10 .2 0.7894 0.8561
15 n-Decylcyclohexane C16H32 224.42 271.42 4.5202 x 10 -8 0.8327 0.9740
n-Alkylbenzenes (aromatics)
16 Benzene C6H6 78. 11 278.65 4.764 x 10 .4 0.8922 1.0125
17 n-Butylbenzene C10H14 134. 22 185. 25 1. 5439 x 10 -9 0.9431 1.1033
18 n-Nonylbenzene Ct5H24 204.36 248.95 6.603 x 10 -9 0.8857 1.0361
19 n-Tetradecylbenzene C20H34 274.49 289.15 9.8069 x 10 -9 0.858 1.0046
1 -n -Alkylnaphthalenes (aromatics)
20 Naphthalene C10H8 128. 16 353.43 9.913 x 10 -3 0.9783
21 1-Methylnaphthalene CllH10 142. 19 242.67 4.3382 x 10 -7 1.0555
22 1-n-Decylnaphthalene C20H28 268.42 288.15 8.4212 x 10 -9 0.9348
Other organic compounds
23 Benzoic acid C7H602 122. 12 395.52 7.955x 10 -3 1.0861
24 Diphenylmethane C13H12 168. 24 298.39 1. 9529 x 10 -5 1.0020
25 Antheracene C14H10 190.32 488.93 4.951 x 10 .2 0.9745
Nonhydrocarbons
26 Water H20 18. 02 273.15/ 6.117 x 10 .3 1.0013
273.16
27 Carbon dioxide CO2 44. 01 216.58 5.187 1.1807
C~, J/g-K C~, J/g. K
1.9509 1.4035
1.9437 1.4386
1.9949 1.4628
2.0077 1.5699
2.0543 1.6276
2.0669 1.6995
2.1589 1.8136
2.1713
2.2049 2.2656
2.3094 2.2653
2.2553 1.8811
2.2632
2.3960 2.4443
1.7627 1.6124
1.9291 1.5398
1.6964 1.6793
1.5268 1.1309
1.7270 1.6882
1.8799 1.7305
1.157 1.687 1.6183
1.2343 1.4237 1.0796
1.0952 1.7289 1.5601
1.2946 2.0506 1.5684
1.0900 1.5727 1.3816
1.2167 2.0339 2.0182
0.9168 4.227 2.1161
1.5140 1.698 1.3844
70 bar) . Wi t h use of Figs. 7.2 and 7.3, one may cal cul at e den-
sity of a l i qui d p et r ol eum fract i on wi t h mi ni mum i nf or mat i on
on specific gravity as shown i n t he fol l owi ng exampl e. These
figures are mai nl y useful for densi t y of undefi ned p et r ol eum
fract i ons by hand cal cul at i on.
Exampl e 7. I - - A p et r ol eum fract i on has API gravi t y of 31.4.
Cal cul at e densi t y of this fract i on at 20~ (68~ and 372.3 bar
(5400 psia). Comp ar e t he est i mat ed val ue wi t h t he exper i men-
tal val ue of 0.8838 g/ cm 3 as gi ven in Chapt er 6 of Ref. [9].
Sol ut i on- - For this fraction, t he mi ni mum i nf or mat i on of
SG is avai l abl e f r om API gravi t y (SG = 0.8686); t her ef or e
Figs. 7.2. and 7.3 can be used to get est i mat e of densi t y at
T and P of interest. Densi t y at 60~ and 1 at m is cal cul at ed as
0.999 x 0.8686 x 62.4 = 54.2 lb/ ft 3. F r om Fig. 7.2 for pressure
of 5400 psi a we r ead f r om t he y axis t he val ue of 1.2, whi ch
shoul d be added to 54.2 to get densi t y at 60~ and 5400 psi a
as 54.2 + 1.2 = 55.4 lb/ ft 3. To consi der t he effect of t emp er a-
ture, use Fig. 7.3. For t emp er at ur e of 68~ and at densi t y of
55.4 lb/ ft 3 t he di fference bet ween densi t y at 60 and 68~ is
read as 0.25 lb/ ft 3. This smal l val ue is due to smal l t emper a-
t ure di fference of 8~ Fi nal l y densi t y at 68~ and 55.4 lb/ ft 3
is cal cul at ed as 55. 4- 0. 25 = 55.15 lb/ ft 3. This densi t y is
equi val ent to 55. 15/ 62. 4 - 0.8838 g/ cm 3, whi ch is exactly t he
same as t he exper i ment al value. #
Once specific gravi t y of a hydr ocar bon at a t emp er at ur e
is known, densi t y of hydr ocar bons at t he same t emp er at ur e
can be cal cul at ed usi ng Eq. (2.1), whi ch requi res t he densi t y
of wat er at t he same t emp er at ur e (i.e., 0.999 g/ cm 3 at 60~ A
cor r el at i on for cal cul at i on of densi t y of l i qui d wat er at 1 at m
for t emp er at ur es in t he range of 0-60~ is gi ven by DI PPR-
EPCON [10] as
(7.6) dr = A x B- [ 1+O-tIc)D]
wher e T is in kelvin and dr is t he densi t y of wat er at t em-
per at ur e T in g/ cm 3. The coefficients are A = 9.83455 x 10 -2,
B ---- 0.30542, C -- 647.13, and D --- 0.081. This equat i on gives
an average er r or of 0.1% [10].
7 . 2 . 3 D e ns i t y o f S o l i ds
Al t hough t he subj ect of solid propert i es is out si de of t he dis-
cussi on of t hi s book, as shown in Chapt er 6, such dat a are
needed in sol i d-l i qui d equi l i bri a ( SLE) cal cul at i ons. Densi-
ties of solids are less affect ed by pressure t han are propert i es
of l i qui ds and can be assumed i ndep endent of pressure (see
Fig. 5.2a). I n addi t i on t o density, solid heat capaci t y and t ri pl e
7. APPL I CATI ONS: ESTI M ATI ON OF THERM OPHYSI CAL PROPERTI ES 3 0 5
point temperature and pressure (Ttp, Ptp) are also needed in
SLE calculations. Values of density and heat capacity of liquid
and solid phases for some compounds at their melting points
are given in Table 7.1, as obtained from DIPPR [ 10]. The triple
point temperature (Ttp) is exactly the same as the melting or
freezing point temperature (TM). As seen from Fig. 5.2a and
from calculations in Example 6.5, the effect of pressure on the
melting point of a substance is very small and for a pressure
change of a few bars no change in TM is observed. Normal
freezing point TM represents melting point at pressure of 1
atm. Ptp for a pure substance is very small and the maxi mum
difference between atmospheric pressure and Ptp is less than
1 atm. For this reason as it is seen in Table 7.1 values of TM
and Ttp are identical (except for water).
Effect of temperature on solid density in a limited temper-
ature range can be expressed in the following linear form:
(7.7) pS m = A- (10 -6 B) T
where ps m is the solid molar density at T in mol/ cm 3. A and
B are constants specific for each compound, and T is the
absolute temperature in kelvin. Values of B for some com-
pounds as given by DIPPR [10] are n-Cs: 6.0608; n-C10: 2.46; n-
C20: 2.663; benzene: 0.3571; naphthalene: 2.276; benzoic acid:
2.32; and water (ice): 7.841. These values with Eq. (7.7) and
values of solid density at the melting point given in Table 7.1
can be used to obtain density at any temperature as shown in
the following example.
Ex ampl e 7. 2--Estimate density of ice at -50~
Sol ut i on- - Fr om Table 7.1 the values for water are obtained
as M= 18.02, TM ---- 273.15 K, p S= 0.9168 g/cm 3 (at TM).
I n Eq. (7.7) for water (ice) B ---- 7.841 and pS is the molar
density. At 273.15 K, pS = 0.050877 mol/ cm 3. Substituting
in Eq. (7.7) we get A = 0.053019. With use of A and B in
Eq. (7.7) at 223.15 K (-50~ we get pS = 0.051269 mol/ cm 3
or ps = 0.051269 x 18.02 = 0.9238 g/ cm 3. #
7 . 3 VA POR P R ES S UR E
As shown in Chapters 2, 3, and 6, vapor pressure is required
in many calculations related to safety as well as design and
operation of various units. In Chapter 3, vapor pressure rela-
tions were introduced to convert distillation data at reduced
pressures to normal boiling point at atmospheric pressure. I n
Chapter 2, vapor pressure was used for calculation of flamma-
bility potential of a fuel. Major applications of vapor pres-
sure were shown in Chapter 6 for VLE and calculation of
equilibrium ratios. As it was shown in Fig. 1.5, prediction
of vapor pressure is very sensitive to the input data, partic-
ularly the critical temperature. Also it was shown in Fig. 1.7
that small errors in calculation of vapor pressure (or relative
volatility) could lead to large errors in calculation of the height
of absorption/distillation columns. Methods of calculation of
vapor pressure of pure compounds and estimation methods
using generalized correlations and calculation of vapor pres-
sure of petroleum fractions are presented hereafter.
7. 3. 1 Pur e C o mpo ne nt s
Experimental data for vapor pressure of pure hydrocarbons
are given in the TRC Thermodynamic Tables [11]. Figures 7.4
and 7.5 show vapor pressure of some pure hydrocarbons from
praffinic and aromatic groups as given in the API-TDB [9].
Further data on vapor pressure of pure compounds at 37.8~
(100~ were given earlier in Table 2.2. For pure compounds
the following dimensionless equation can be used to estimate
vapor pressure [9]:
(7.8) l nPr "p = ( Tr' ) x (ar +br 15 +c r 26 +dr s)
where r = 1 - Tr and p~ap is the reduced vapor pressure
(pvap/Pc) , and Tr is the reduced temperature. Coefficients a-
d with corresponding temperature ranges are given in Ta-
ble 7.2 for a number of pure compounds. Equation (7.8) is
a linearized form of Wagner equation. I n the original Wagner
equation, exponents 3 and 6 are used instead of 2.6 and 5 [ 12].
The primary correlation recommended in the API-TDB [9]
for vapor pressure of pure compounds is given as
B E
(7.9) In pv,p = A + ~ + C In T + DT 2 + T- ~
where coefficients A- E are given in the API-TDB for some 300
compounds (hydrocarbons and nonhydrocarbons) with spec-
ified temperature range. This equation is a modified version
of correlation originally developed by Abrams and Prausnitz
based on the kinetic theory of gases. Note that performance
of these correlations outside the temperature ranges specified
is quite weak. I n DIPPR [10], vapor pressure of pure hydro-
carbons is correlated by the following equation:
B
(7.10) l nP vap= A+ -~ +Cl nT + DT E
where coefficients A- E are given for various compounds in
Ref. [ 10]. I n this equation, when E = 6, it reduces to the Riedel
equation [12]. Another simple and commonl y used relation
to estimate vapor pressure of pure compounds is the Antoine
equation given by Eq. (6.102). Antoine parameters for some
700 pure compounds are given by Yaws and Yang [13]. An-
toine equation can be written as
B
(7.11) In PVap(bar) = A - - -
T+C
where T is in kelvin. Antoine proposed this simple modifi-
cation of the Clasius-Clapeyron equation in 1888. The lower
temperature range gives the higher accuracy. For some com-
pounds, coefficients of Eq. (7.11) are given in Table 7.3. Equa-
tion (7.11) is convenient for hand calculations. Coefficients
may vary from one source to another depending on the tem-
perature range at which data have been used in the regres-
sion process. Antoine equation is reliable from about 10 to
1500 mmHg (0.013-2 bars); however, the accuracy deteri-
orates rapidly beyond this range. It usually underpredicts
vapor pressure at high pressures and overpredicts vapor pres-
sures at low pressures. One of the convenient features of this
equation is that either vapor pressure or the temperature can
be directly calculated without iterative calculations. No gener-
alized correlation has been reported on the Antoine constants
and they should be determined from regression of experimen-
tal data.
1,000
cO
t r
CO
G0
UJ
t r
Q.
n-
O
Q.
X
100
10
0.1 L
-50
VAPOR PRESSURE O F
NO RMALPARAFFI N
H YDROCARBONS
j I )
TECH NICAL DATABOOK
February 1994
-25 0 25 50 7 5 100 150 200 250
T EMPERAT URE, F
FI G. 7 . 4~ V a p o r pressure of some n- al k ane hydrocarbons. Uni t conv ersi on: ~ = ( ~
1 .8 + 3 2; psi a = bar 1 4.504. T ak en wi th permi ssi on f rom Ref. [ 9].
3 00 400 500
An expanded form of Antoine equation, which covers a
wider temperature range by including two additional terms
and a fourth parameter, is given in the following form as sug-
gested by Cox [12]:
B
(7.12) In pvap : A + ~ + CT + DT 2
Another correlation is the Miller equation, which has the fol-
lowing form [12]:
(7.13) In pvap = - ~[ 1 - Tf +B ( 3 +Tr ) ( 1 -T~) 3]
where A and B are two constants specific for each compound.
These coefficients have been correlated to the reduced boiling
point Tbr (=Tb/Tc) and Pc of pure hydrocarbon vapor pressure
in the following form:
Tbrln(PJl. O1325)
AI =
1 - Tbr
(7.14) A ----- 0.4835 + 0.4605A1
A/A~ - (1 + Tbr)
B =
( 3 + Tbr)(1 - Tbr) 2
where Pc is in bar. Equations (7.13) and (7.14) work better
at superatmospheric pressures (T > Tb) rather than at sub-
atmospheric pressures. The main advantage of this equation
is that it has only two constants. This was the reason that
it was used to develop Eq. (3.102) in Section (3.6.1.1) for
calculation of Reid vapor pressure (RVP) of petroleum fuels.
For RVP prediction, a vapor pressure correlation is applied at
a single temperature (100~ or 311 K) and a two-parameter
correlation should be sufficient. Some other forms of equa-
tions used to correlate vapor pressure data are given in
Ref. [12].
7. 3. 2 Pr edi ct i ve Me t hods - - Ge ne r a l i z e d
C o r r e l a t i o n s
I n Section 6.5, estimation of vapor pressure from an equa-
tion of state (EOS) through Eq. (6.105) was shown. When an
appropriate EOS with accurate input parameters is used, ac-
curate vapor pressure can be estimated through Eq. (6.105)
or Eq. (7.65) [see Problem 7.13]. As an example, prediction of
vapor pressure of p-xylene from a modified PR EOS is shown
in Fig. 7.6 [14].
1 00
7. APPLICATIONS: ESTIM ATION OF THERM OPHYSICAL PROPERTIES 307
rt"
E~
o9
cO
LM
n"
Q..
t r
o
t ) . .
X
10
0.1
VAPOR PRESSURE
O F ALKYLBENZENE
H YDROCARBONS
I I TECH NI CAL DATA BOOK
February 1994
25 50 75 1 O0 150 200 250 3 00
T EMPERAT URE, F
FI G . 7 . 5 - - V a p o r p r es s ur e of s o me n- al k y l b enz ene hy d r ocar b ons . Uni t conv er s i on: ~ =
( ~ x 1 . 8 + 3 2; psi a = bar 1 4,504. T ak en wi t h p er mi s s i on f r om Ref , [ 9 ].
400 500
Generally, vapor pressure is predicted through correlations
similar to those presented in Section 7.3.2. These correlations
require coefficients for individual components. A more use-
ful correlation for vapor pressure is a generalized correlation
for all compounds that use component basic properties (i.e.,
Tb) as an input parameter. A perfect relation for prediction
of vapor pressure of compounds should be valid from triple
point to the critical point of the substance. Generally no sin-
gle correlation is valid for all compounds in this wide tem-
perature range. As the number of coefficients in a correlation
increases it is expected that it can be applied to a wider tem-
perature range. However, a correct correlation for the vapor
pressure in terms of reduced temperature and pressure is ex-
pected to satisfy the conditions that at T = To, pvap = Pc and at
T = Tb, pvap = 1.0133 bar. The temperature range Tb _< T _< Tc
is usually needed in practical engineering calculations. How-
ever, when a correlation is used for calculation of vapor pres-
sure at T < Tb (PvaP _< 1.0133 bar), it is necessary to satisfy
the following conditions: at T = Ttp, pvap =Ptp and at T = Tb,
pwp = 1.0133 bar, where Ttp and Ptp are the triple point tem-
perature and pressure of the substance of interest.
The origin of most of predictive methods for vapor pres-
sure calculations is the Clapeyron equation (Eq. 6.99). The
simplest method of prediction of vapor pressure is through
Eq. (6.101), which is derived from the Clapeyron equation.
Two parameters of this equation can be determined from two
data points on the vapor pressure. This equation is very ap-
proximate due to the assumptions made (ideal gas law, ne-
glecting liquid volume, and constant heat of vaporization) in
its derivation and is usually useful when two reference points
on the vapor pressure curve are near each other. However,
the two points that are usually known are the critical point
(Tc, Pc) and normal boiling point (Tb and 1.013 bar) as demon-
strated by Eq. (6.103). Equations (6.101) and (6.103) may be
combined to yield the following re]ation in a dimensionless
forln:
Pc
(7.15) l nP~a P= [ l n( ~) ] x \ ~] ( Tbr ~x ( 1- ~)
where Pc is the critical pressure in bar and Tbr is the reduced
normal boiling point (Tbr = Tb/Tc). The main advantage of
Eq. (7.15) is simplicity and availability of input parameters
3 0 8 CHARACTERI ZATI ON AND PROPERTI ES OF PETROL EUM FRACTI ONS
TABLE 7. 2---Coefficients of Eq. (7. 8) for vapor pressure of pure compounds [9].
(7.8) lnpV ap = (T~-l ) x (ar + b~l'5 + c~2"6 + dr 5)
Compound name a b c d Ttp, K Train, K Tmax, K
Nonhydrocarbon
1 Oxygen -6. 0896 1.3376 -0. 8462 - 1.2860 54 54 154
2 Hydrogen -4. 7322 0.5547 1.5353 -1. 1391 14 14 33
3 Water -7. 8310 1.7399 -2. 2505 -1. 9828 273 273 647
4 Hydrogen chloride -6. 2600 0.1021 1.0793 -4. 8162 159 159 324
5 Hydrogen sulfide -5. 7185 -0. 4928 1.0044 -4. 5547 188 188 373
6 Carbon monoxi de -6. 2604 1.5811 - 1.5740 -0. 9427 68 68 133
7 Carbon dioxide -6. 9903 1.3912 -2. 2046 -3. 3649 217 217 304
8 Sulfur dioxlde -6. 8929 1.3119 -3. 5225 0.6865 198 203 431
Par af ms
9 Met hane -5, 9999 1.2027 -0, 5310 -1, 3447 91 91 191
10 Et hane -6. 4812 1.4042 -1. 2166 -1. 7143 91 91 306
11 Propane -6. 8092 1.6377 -1. 8173 -1. 8094 86 86 370
12 n-Butane -7. 0524 1.6799 -2. 0398 -2. 0630 135 135 425
13 I sobut ane -6. 7710 1.0669 -0. 9201 -3. 8903 113 125 408
14 n-Pentane -7. 2048 1.3503 -1. 5540 -4. 2828 143 157 469
15 I sopent ane -7. 1383 1.5320 -1. 8896 -2. 7290 113 178 461
16 Neopent ane -6. 9677 1.5464 -1. 9563 -2. 6057 257 257 434
17 n-Hexane -7. 3505 0.9275 -0. 7303 -6. 7135 178 178 507
18 n-Heptane -7. 4103 0.7296 -1. 3081 -5. 9021 183 183 540
19 2-Methylhexane -7. 6340 1.6113 -2. 4895 -3. 7538 155 222 531
20 n-Octane -8. 0092 1.8442 -3. 2907 -3. 5457 216 286 569
21 2,2-Dimethylhexane -7. 5996 1.4415 -2. 3822 -4. 2077 152 236 550
22 n-Nonane -9. 5734 5.7040 -8. 9745 3.3386 219 233 596
23 2-Methyloctane -9. 4111 5.6082 -9. 1179 3.9544 193 303 587
23a 2,2,4-Trimethylpentane -7. 4717 1.5074 -2. 2532 -3.5291 166 225 544
24 n-Decane -8. 4734 2.0043 -3. 9338 -4. 5270 243 286 618
25 n-Undecane -8. 6767 1.8339 -3. 6173 -6. 5674 248 322 639
26 n-Dodecane -9. 1638 2.8127 -5. 5268 -4. 1240 263 294 658
27 n-Tridecane -11. 5580 9,5675 -17, 8080 23.9100 268 333 676
28 n-Tetradecane -9. 5592 2,6739 -5. 3261 -7. 2218 279 369 692
29 n-Pent adecane -9. 8836 2.9809 -5. 8999 -7. 3690 283 383 707
30 n-Hexadecane -10. 1580 3.4349 -7. 2350 -4. 7220 291 294 721
31 n-Hept adecane -8. 7518 -1. 2524 0.6392 -21. 3230 295 311 733
32 n-Octadecane -11. 30200 6.3651 -12. 4510 0.2790 301 322 745
33 n-Nonadecane -10. 0790 2,7305 -7. 8556 -5. 3836 306 400 756
34 n-Eicosane -9. 2912 0.7364 -8. 1737 -0. 4546 309 353 767
35 n-Tetracosane -14. 4290 12.0240 -21. 5550 11.2160 324 447 810
36 n-Octacosane -11. 4490 2.0664 -7. 4138 -15. 4770 334 417 843
Naphthenes
37 Cyclopentane -7. 2042 2,2227 -2. 8579 -1. 2980 179 203 512
38 Methylcyclopentane -7. 1157 1.5063 -2. 0252 -2. 9670 131 178 533
39 Ethylcyclopentane -7. 2608 1.3487 -1. 8800 -3. 7286 134 183 569
40 n-Propylcyclopentane -1. 3961 0.2383 -5. 7723 -6. 0536 156 244 603
41 Cyclohexane -7. 0118 1.5792 -2. 2610 -2. 4077 279 279 553
42 Methylcyclohexane -7. 1204 t.4340 -1. 9015 -3. 3273 147 217 572
43 Ethylcyclohexane -5. 9783 -1. 2708 0.2099 -5. 3117 162 228 609
44 n-Propylcyclohexane -5. 6364 -2. 1313 0.6054 -6. 0405 178 228 639
45 I sopropylcyclohexane -7. 8041 2.0024 -2. 8297 -3. 4032 184 208 627
46 n-Butylcyclohexane -4. 9386 -3. 9025 2.0300 -7. 8420 198 286 667
47 n-Decylcyclohexane -9. 5188 2.4189 -4. 5835 -7. 7062 272 322 751
48 Cycloheptane -7. 3231 1.8407 -2. 2637 -3. 4498 265 265 604
Olefms
49 Ethylene -6. 3778 1.3298 -1. 1667 -2. 0209 104 104 282
50 Propylene -6. 7920 1.7836 -2.0451 -1. 5370 88 88 366
51 1-Butene -6. 9041 1.3587 -1. 3839 -3. 7388 88 125 420
52 1-Pentene -6. 6117 0.0720 0.0003 -5. 4313 108 167 465
53 1,3-Butadiene -5. 6060 -0. 9772 -0. 3358 -3. 1876 127 156 484
Diolefins and Acetylenes
54 Acetylene -7. 3515 2.8334 -4. 5075 6.8797 192 192 308
Aromatics
55 Benzene -7. 0200 1.5156 -1. 9176 -3. 5572 279 279 562
56 Toluene -7. 2827 1.5031 -2. 0743 -3. 1867 178 244 592
57 Et hyl benzene -7. 5640 1.7919 -2. 7040 -2. 8573 178 236 1
58 m-Xylene -7. 6212 1.6059 -2. 4451 -3. 0594 226 250 617
59 o-Xylene -7. 5579 1.5648 -2. 1826 -3. 7093 248 248 631
60 p-Xylene -7. 6935 1.8093 -2. 5583 -3. 0662 287 287 616
T~,K
154
33
647
324
373
133
304
431
191
306
370
425
408
469
461
434
507
54O
531
569
550
596
587
544
618
639
658
676
692
707
721
733
745
756
767
810
843
512
533
569
603
553
572
609
639
627
667
751
604
282
366
420
465
484
308
562
592
617
617
631
616
Max% err
2.0
6.0
0.4
4.6
7.3
10.1
0.8
3.8
0.1
0.1
2.7
6.5
1.6
6.5
0.3
3.2
5.8
4.6
1.9
3.1
0.7
3.9
1.4
5.0
3.8
4.2
5.4
3.2
0.4
0.2
5.9
4.9
6.6
2.8
8.0
4.0
5.7
3.9
2.0
1.5
0.1
4.3
1.2
0.2
0.4
3.5
0.2
9.2
0.7
8.9
5.2
2.4
6.5
0.3
2.4
1.6
3.0
0.8
3.9
2.2
7.5
Ave% err
0.2
0.8
0.0
0.6
1.0
0.9
0.1
1.7
0.0
0.0
0.9
0.4
1.2
0.8
0.1
0.2
1.0
0.4
0.2
0.3
0.1
0.6
0.3
0.3
0.5
0.2
0.2
1.5
0.1
0.1
0.3
0.7
1.9
0.5
1.2
0.8
1.2
0.6
0.1
0.1
0.0
0.7
0.2
0.0
0.0
0.1
0.0
0.4
0.1
0.6
0.3
0.5
1.5
0.1
0.4
0.3
0.3
0.1
0.2
0.2
0.4
(Continued)
7. APPL I CATI ONS: ESTI M ATI ON OF THERM OPHYSI CAL PROPERTI ES 309
TABLE 7. 2--(Continued).
Compound name a b c d Ttp, K Tmin, K Tmax, K To K Max% err Ave% err
61 i-PropyIbenzene -8.1015 2.6607 -3.8585 -2.2594 173 236 638 638 5.4 0.4
62 n-Butylbenzene -7.8413 1. 3055 -2.1437 -5.3415 186 233 661 661 5.6 0.7
63 n-Pentylbenzene -8.7573 3.1808 -4.7169 -2.7442 198 311 680 680 2.8 0.2
64 n-Hexylbenzene -8.0460 0.6792 -1.4190 -8.1068 212 333 698 698 1.8 0.2
65 n-Heptylbenzene -9.1822 3.1454 -4.8927 -4.5218 225 356 714 714 2.0 0.2
66 n-Octylbenzene -10.7760 7.0482 -10.5930 1.7304 237 311 729 729 8.0 0.8
67 Styrene -6.3281 -1.2630 0.9920 -7.1282 243 243 636 636 0.6 0.1
68 n-Nonylbenzene -10.7760 7.0038 -10.4060 1.1027 249 311 741 741 1.4 0.4
69 n-Decylbenzene -10.5490 4.7502 -7.2424 -4.8469 259 333 753 753 0.1 0.0
70 n-Undecylbenzene -11.8950 8. 0001 -12.7000 4.6027 268 383 764 764 1.1 0.2
71 n-Dodecylbenzene -10.6650 3.9860 -7.6855 -1.7721 276 333 659 774 9.6 1.8
72 n-Tridecylbenzene -11.995 6.5968 -10.1880 -5.2923 283 417 783 783 2.1 0.4
73 Cumene -7.4655 1. 2449 -2.0897 -4.5973 177 228 631 631 2.6 0.3
Diaromatics
74 Naphthalene -7.6159 1. 8626 -2.6125 -3.1470 353 353 748 748 17.5 0.8
75 1-Methylnaphthalene -7.4654 1. 3322 -3.4401 -0.8854 243 261 772 772 7.1 1.7
76 2-Methylnaphthalene -7.6745 1.0179 -1.3791 -5.6038 308 308 761 761 7.8 0.9
77 2,6-Dimethylnaphthalene -7.8198 -2.5419 9.2934 -24.3130 383 383 777 777 0.1 0.0
78 i -Ethylnaphthalene -6.7968 -0.5546 -1.2844 -5.4126 259 322 776 776 11.1 0.6
89 Anthracene -8.4533 1.3409 -1.5302 -3.9310 489 489 873 873 5.6 0.5
80 Phenanthrene -11.6620 9.2590 -10.0050 1.2110 372 372 869 869 1.0 0.2
Oxygenated compounds
81 Methanol -8.6413 1. 0671 -2.3184 -1.6780 176 176 513 513 5.9 0.7
82 Ethanol -8.6857 1. 0212 -4.9694 1.8866 159 194 514 514 4.9 0.4
83 Isopropanol -7.9087 -0.6226 -4.8301 0.3828 186 200 508 508 8.4 1.6
84 Methyl-tert-hutyl ether -7.8925 3.3001 -4.9399 0.2242 164 172 497 497 8.0 1.3
85 tert-Butyl ethyl ether -6.1886 -1.0802 -0.9282 -2.9318 179 179 514 514 8.7 4.8
86 Diisopropyl ether -7.2695 0.4489 -0.9475 -5.2803 188 188 500 500 22.7 2.7
87 Methyl tert-pentyl ether -7.8502 2.8081 -4.5318 -0.3252 158 534 534 1.3 0.4
Ttp is the triple point temperature and Tc is the critical temperature. Train and Tmax indicate the range at which Eq. (7.8) can be used with these coefficients. For
quick and more convenient method use Antoine equation with coefficients given in Table 7.3.
(Tb, T0, and Pc) for pure compounds. However, one shoul d
realize t hat si nce the base poi nt s i n deri vi ng t he const ant s
gi ven by Eq. (6.103) are Tb and To t hi s equat i on shoul d be
used i n the t emper at ur e range of Tb _< T _< To Theoretically,
a vapor pressure rel at i on shoul d be valid from triple poi nt
t emper at ur e to the critical t emperat ure. But most vapor pres-
sure correl at i ons are very poor at t emper at ur es near the triple
p oi nt t emperat ure. Usi ng Eq. (7.15) at t emper at ur es bel ow Tb
usual l y leads to unaccept abl e predi ct ed values. For bet t er pre-
di ct i on of vapor pressure near the triple poi nt , the t wo base
poi nt s shoul d be nor mal boi l i ng poi nt (T = Tb, P = 1.01325
bar) and triple poi nt (Ttp, Ptp). Values of Ttp and Ptp for some
comp ounds are given i n Table 7.1. Si mi l arl y if vapor pressure
predi ct i on near 37.8~ (100~ is requi red t he vapor pressure
dat a given i n Table 2 shoul d be used as one of t he reference
poi nt s al ong wi t h Tb, To, or Ttp to obt ai n t he const ant s A and
B i n Eq. (6.101).
One of the latest devel opment s for correl at i on of vapor pres-
sure of pure hydr ocar bons was proposed by Korst en [ 15]. He
i nvest i gat ed modi fi cat i on of Eq. (6.101) wi t h vapor pressure
dat a of hydr ocar bons and he f ound t hat l nP vap varies l i nearl y
wi t h 1/T 1.3 for all hydrocarbons.
B
(7.16) I n pvap = A - - -
TI.3
where T is absol ut e t emper at ur e i n kel vi n and pvap is the va-
por pressure i n bar. I n fact t he mai n difference bet ween t hi s
equat i on and Eq. (6.101) is the exponent of T, whi ch i n this
case is 1.3 ( rat her t han 1 i n the Cl apeyron type equat i ons) . Pa-
ramet ers A and B can be det er mi ned from boi l i ng and critical
poi nt s as it was shown i n Exampl e 6.6. Paramet ers A and B i n
Eq. (7.16) can be det er mi ned from Eq. (6.103) wi t h repl aci ng
Tb and To by T~ 3 and T~ 3. The l i near rel at i onshi p bet ween
I n pvap and 1/T 13 for large numbe r of pure hydr ocar bons is
shown i n Fig. 7.7.
Prel i mi nary eval uat i on of Eq. (7.16) shows no maj or advan-
tage over Eq. (7.15). A compar i son of Eqs. (7.15) and (7.16)
for n-hexane is shown i n Fig. 7.8. Predi ct ed vapor pressure
from the met hod r ecommended i n the API-TDB is also shown
i n Fig. 7.8. Cl apeyron met hod refers to Eq. (7.15), whi l e t he
Korst en met hod refers to Eq. (7.16), wi t h par amet er s A and
B det er mi ned from Tb, To and Pc. Equat i on (7.15) agrees bet-
t er t han Eq. (7.16) wi t h the API-TDB met hod. Subst i t ut i on of
Eq. (6.16) i nt o Eq. (2.10) leads to Eq. (2.109) for predi ct i on of
acent ri c factor by Korst en met hod. Eval uat i on of met hods of
predi ct i on of acent ri c factor present ed i n Sect i on 2.9.4 also
gives some i dea on accuracy of vapor pressure correl at i ons
for pure hydrocarbons.
Korst en det er mi ned t hat all hydr ocar bons exhi bi t a vapor
pressure of 1867.68 bar at 1994.49 K as shown i n Fig. 7.7.
This dat a poi nt for all hydr ocar bons and t he boi l i ng p oi nt
dat a can be used to det er mi ne par amet er s A and B i n Eq.
(7.16). I n this way, t he resul t i ng equat i on requi res onl y one
i np ut p ar amet er (Tb) si mi l ar to Eq. (3.33), whi ch is also shown
i n Sect i on 7.3.3.1 (Eq. 7.25). Eval uat i on of Eqs. (7.25) and
(7.16) wi t h use of Tb as sole i np ut p ar amet er i ndi cat es t hat Eq.
(7.25) is mor e accurat e t han Eq. (7.16) as shown i n Fig. 7.8.
However, not e t hat Eq. (7.25) was developed for pet r ol eum
fract i ons and it may be used for pure hydr ocar bons wi t h
Nc_>5.
Perhaps the most successful general i zed correl at i on for
predi ct i on of vapor pressure was based on the t heory of
TABLE 7. 3--Antione coefficients for calculation of vapor pressure from Eq. (7.11).
B
In pvap = A - - - Units: bar and K
T+C
No. Compound ~, K A B C
n- Al kanes
1 Met hane (C1) 111.66 8.677752 911.2342 - 6. 340
2 Et hane (C2) 184.55 9.104537 1528.272 - 16. 469
3 Propane (Ca) 231.02 9.045199 1851.272 - 26. 110
4 But ane (n-C4) 272.66 9.055284 2154.697 -34. 361
5 I sobut ane (i-C4 ) 261.34 9.216603 2181.791 - 24. 280
6 Pent ane (n-Cs) 309.22 9.159361 2451.885 - 41. 136
7 Hexane (n-C6) 341.88 9.213541 2696.039 - 48. 833
8 Hept ane (n-C7) 371.57 9.256922 2910.258 - 56. 718
9 Oct ane (n-Ca) 398.82 9.327197 3123.134 - 63. 515
10 Nonane (n-C9) 423.97 9.379719 3311.186 - 70. 456
11 Decane (n-C10) 447.3 9.368137 3442.756 - 79. 292
12 Undecane (n-C11) 469.08 9.433921 3614.068 - 85. 450
13 Dodecane (n-C12) 489.48 9.493213 3774.559 - 91. 310
14 Tridecane (n-C13) 508.63 9.515341 3892.912 - 98. 930
15 Tet radecane (n-C14) 526.76 9.527867 4008.524 - 105.430
16 Pent adecane (n-C15) 543.83 9.552251 4121.512 - 111.770
17 Hexadecane (n-C16) 559.98 9.563948 4214.905 - 118. 700
18 Hept adecane (n-C17) 574.56 9.53086 4294.551 - 123.950
19 Oct adecane (n-C18) 588.3 9.502999 4361.787 - 129.850
20 Nonadecane (n-C19) 602.34 9.533163 4450.436 - 135.550
21 Ei cosane (n-C20) 616.84 9.848387 4680.465 - 141.050
1-Alkenes
22 Et hyl ene (C2H4) 169.42 9.011904 1373.561 - 16. 780
23 Propyl ene (Ca H6 ) 225.46 9.109165 1818. t 76 - 25. 570
24 1 -but ane (C4H8) 266.92 9.021068 2092.589 - 34. 610
Naphthenes
25 Cycl opent ane 322.38 9.366525 2653.900 - 38.640
26 Met hyl cycl opent ane 344.98 9.629388 2983.098 - 34. 760
27 Et hyl cycl opent ane 376.59 9.219735 2978.882 - 53. 030
28 Cycl ohexane 353.93 9.049205 2723.438 - 52. 532
29 Met hyl cycl ohexane 374.09 9.169631 2972.564 - 49. 449
Ar omat i cs
30 Benzene (C6H6) 353.24 9.176331 2726.813 - 55. 578
31 Toluene (C7H8) 383.79 9.32646 3056.958 - 55. 525
32 Et hy] benzene 409.36 9.368321 3259.931 - 60. 850
33 Propyl benzene 432.35 9.38681 3434.996 - 65. 900
34 But yl benzene 456.42 9.448543 3627.654 - 71. 950
35 o-Xylene (C8H10) 417.59 9.43574 3358.795 - 61. 109
36 m-Xylene (C8H10) 412.34 9.533877 3381.814 - 57. 030
37 p-Xylene (Call10) 411.53 9.451974 3331.454 - 58. 523
Other hydrocarbons
38 I sooct ane 372.39 9.064034 2896.307 - 52. 383
39 Acetylene (C2H2) 188.40 8.459099 1217.308 - 44. 360
40 Napht hal ene 491.16 9.522456 3992.015 -71. 291
Organics
41 Acetone (C3H60) 329.22 9.713225 2756.217 - 45. 090
42 Pyri di ne (C5H5N) 388.37 9.59600 3161.509 - 58. 460
43 Aniline (C6HTN) 457.17 10.15141 3897.747 - 72. 710
44 Met hanol 337.69 11.97982 3638.269 - 33. 650
45 Et hanol 351.80 12.28832 3795.167 - 42. 232
46 Propanol 370.93 11.51272 3483.673 - 67. 343
Nonhydrocarbons
47 Hydr ogen (H2) 20.38 6.768541 153.8021 2.500
48 Oxygen (O2) 90.17 8.787448 734.5546 - 6. 450
49 Ni t rogen (N2) 77.35 8.334138 588.7250 - 6. 600
50 Hel i um (He) 4.30 3.876632 18.77712 0.560
51 CO 81.66 8.793849 671.7631 - 5. 154
52 CO2 194.65 10.77163 1956.250 - 2. 1117
53 Ammoni a (NH3) 239.82 10.32802 2132.498 - 32. 980
54 H2 S 212.84 9.737218 1858.032 - 21. 760
55 Sul fur (S) 717.75 9.137878 5756.739 - 86. 850
56 CC14 349.79 9.450845 2914.225 - 41. 002
57 Wat er (HE O) 373.15 11.77920 3885.698 - 42. 980
The above coefficients may be used for pressure range of 0.02-2.0 bar except for water for which the pressure range is 0.01-16
bar as reported in Ref. [ 12]. These coefficents can generate vapor pressure near atmospheric pressure with error of less than
0.1%. There are other reported coefficients that give slightly more accurate results near the boiling point. For example, some
other reported values for A, B, and C are given here. For water: 11.6568, 3799.89, and -46.8000; for acetone: 9.7864, 2795.82,
and -43.15 or 10.11193, 2975.95, and-34.5228; for ethanol: 12.0706, 3674.49, and-46.70.
3 1 0
7. APPLICATIONS: ESTI M ATI ON OF THERM OPHYSICAL PROPERTIES 311
40-
3O
<~ ~ P, c~ t l c
= 20
Reduced T emperature, T r
FIG, 7 ,6~ Pr edi ct i on of v apor pressure of p-x yl ene from modi fi ed
PR EO S, A dopted wi th permi ssi on from Ref. [ 14],
cor r es p ondi ng st at es p r i nci p l e as descr i bed i n Sect i on 5.7 ( see
Eq. 5.107) , whi ch was p r op os ed or i gi nal l y by Pi t zer i n t he
f ol l owi ng f or m:
( 7. 17) I n pvap ---- f(0)(Tr) + cofO)(T~)
wher e co is t he acent r i c factor. Lee and Kesl er ( 1975) deve-
l op ed anal yt i cal cor r el at i on for f(0) a nd f(1) i n t he f ol l owi ng
Locz~ of acentric factors .
I l i r
I l i. / " I " ~ # . , / ~ 1 / i ~ / / / 1/ / / , ~
> o-o5i- ....,- /- . . , ' ~ ~ / / ~
/ , ~ / . * . ' / , r
0. 001 L. _ ~ _ ~ ~ , . L- ~ - - - , J . . . . , .
180 200 250 300 400 500 1000 3000
T emperature T in K
3 000 r - ~ ~ ' . ~ ~ . . . . ~ - - . I
'%i
i n-alk341:~r~.
~, u
c
& 1 Locus
o.1!
~ 0, 05 , /
6 ~ t ' t
001 7
0.005 ~- Nt
. . . . .
400'
Temperature T in K
~ 49
FIG. 7 ,7 - - V apor pressure of pure hydrocarbons accordi ng to
Eq. ( 7 .1 6) . A dopted wi th permi ssi on from Ref, [ 15].
f or ms [16]:
6. 09648
I n Pr yap = 5. 92714 Tr 1. 28862 I n Tr -I- 0. 169347T 6
( 7. 18) + aj( 15. 2518 15.6875Tr 13.4721 l nTr + 0. 43577T 6)
wher e pvap = pvap/ p c a nd Tbr ~-~ Tb/ Z c. I n 1989, Ambr os e a nd
Wal t on added a t hi r d t er m i n Eq. ( 7. 17) a nd p r op os ed t he
f ol l owi ng cor r el at i on for es t i mat i on of vap or p r essur e [ 12]:
Tr(ln p~ap)
= - - 5. 97616r + 1. 29874r 15 -- 0. 60394r 2s -- 1. 06841r 5
+w( - - 5. 03365r + 1. 11505r 15 -- 5. 41217r 2"5 -- 7. 46628r 5)
+ w2( --0. 64771r + 2. 41539r 15- 4. 26979r2. 5+ 3. 25259r 5)
( 7. 19)
30
o API
25 Clapeyron
. . . . . . . Korsten
,~ 20 - - - - Modified Riedel
~ . . . . . Miller
> 10
5
0 . . . . . . - ~ ' ~
200 250 300 350 400 450 500
Temperature, K
FIG. 7 . 8 ~ Ev al uat i on of v ari ous methods of cal cul a-
tion of v apor pressure of n-hexane. Methods: a. A Ph
Eq. ( 7 .8 ) wi th coeffi ci ents from Table 7.2; b. Clapeyron:
Eq, ( 7.15) ; c, Korsten: Eq. ( 7.16) ; d. Modi fi ed Riedel:
Eq, ( 7,24) ; e. Miller: Eqs. ( 7 ,1 3 ) and ( 7.14) .
8
>
2000
200
10
2
0.2
0.02
0.002
0
. . . . Ant oi ne
, , , i . . . . , . . . . J . . . . i . . . . i . . . . i . . . . i . . . .
50 100 150 200 250 300 350 400
Temperature, C
FIG. 7 . 9 - - Predi ct i on of v apor pressure of water
from Lee- Kesl er ( Eq. 7.18 ) , A mbrose ( Eq. 7.19) , and
A ntoine ( Eq. 7.11) correlations.
where Tr = T/ To P~P = Pv~P/Pc, and r = 1 - T~. A graphical
comparison between the Antoine equation (Eq. 7.11 with
coefficients from Table 7.3), Lee-Kesler correlation, and
Ambrose correlation for water from triple point to the crit-
ical point is shown in Fig. 7.9. Although Eq. (7.19) is more ac-
curate than Eq. (7.18), the Lee-Kesler correlation (Eq. 7.18)
generally provides reliable value for the vapor pressure and it
is recommended by the API-TDB [9] for estimation of vapor
pressure of pure hydrocarbons.
7 . 3 . 3 Vapor Pr e s s ur e o f Pe t r o l e um Fract i ons
Both analytical as well as graphical methods are presented
here for calculation of vapor pressure of petroleum fractions,
coal liquids, and crude oils.
7. 3. 3. 1 Anal yt i cal M et hods
The generalized correlations of Eqs. (7.18) and (7.19) have
been developed from vapor pressure data of pure hydro-
carbons and they may be applied to narrow boiling range
petroleum fractions using pseudocritical temperature and
pressure calculated from methods of Chapter 2. When using
these equations for petroleum fractions, acentric factor (o~)
should be calculated from Lee-Kesler method (Eq. 2.105).
Simpler but less accurate method of calculation of vapor
pressure is through the Clapeyron method by Eqs. (6.101)
and (6.103) or Eq. (7.15) using Tb, To and Pc of the fraction.
For very heavy fractions, the pseudocomponent method of
Chapter 3 (Eq. 3.39) may be used by applying Eq. (7.18) or
(7.19) for each homologous groups of paraffins, naphthenes,
and aromatics using To, Pc, and w calculated from Eq. (2.42).
There are some methods that were specifically developed
for the vapor pressure of petroleum fractions. These correla-
tions are not suitable for vapor pressure of light hydrocar-
bons (i.e., C1-C4). One of the most commonl y used meth-
ods for vapor pressure of petroleum fractions is the Maxwell
and Bonnell (MB) correlation [17] presented by Eqs. (3.29)-
(3.30). Usually Eq. (3.29) can be used at subatmospheric pres-
sures (P < 1 atm.) for calculation of normal boiling point
(Tb) from boiling points at low pressures (T). Equation (3.30)
is normally used at superatmospheric pressures where nor-
real boiling point (Tb) is known and boiling point at higher
pressures (T) is required. When calculation of vapor pressure
(pvap) at a given temperature (T) is required, Eq. (3.29) can
be rearranged in the following form:
3000.538Q - 6.761560
log10 pvap = 43Q - 0.987672
for Q > 0.0022 (pvap < 2mmHg)
2663.129Q - 5.994296
logl0 pvap = 95. 76Q- 0.972546
for 0.0013 _< Q < 0.0022(2 mmHg < pvap _< 760mmHg)
2770.085Q - 6.412631
logl0 pvap = 36Q - 0.989679
(7.20) for Q < 0.0013 ( p va p > 760mmHg)
Parameter Q is defined as
rs 0.00051606T~
(7.21) Q= r
748.1 - 0.3861T~
where T~ can be calculated from the following relations:
r~ = Tu- ar b
p va p
ATb = 1.3889F(Kw - 12) log10 760
(7.22) F = 0 (Tb < 367 K) or when Kw is not available
F = -3. 2985 + 0.009Tb (367 K _< Tb _< 478 K)
F = -3. 2985 + 0.009Tb (Tb > 478K)
where
p va p =
desired vapor pressure at temperature T, mm Hg
( =bar x 750)
T -- temperature at which pvap is needed, in kelvin
T~ -- normal boiling point corrected to Kw -- 12, in kelvin
Tb ---- normal boiling point, in kelvin
Kw = Watson (UOP) characterization factor [=(1. 8Tb)l/3/
SG]
F = correction factor for the fractions with Kw different
from 12
logl0 = common logarithm (base 10)
It is recommended that when this method is applied to light
hydrocarbons (Nc < 5), F in Eq. (7.22) must be zero and there
is no need for value of Kw(i.e., T~ = Tb). Calculation of pvap
from Eqs. (7.20)-(7.22) requires a trial-and-error procedure.
The first initial value of pvap can be obtained from Eqs. (7.20)
and (7.2t) by assuming Kw = 12 (or T~ = Tb). I f calculation of
T is required at a certain pressure, reverse form of Eqs. (7.20)
and (7.21) as given in Eqs. (3.29) and (3.30) should be used.
Tsonopoulos et al. [18, 19] stated that the original MB cor-
relation is accurate for subatmospheric pressures. They mod-
ified the relation for calculation of ATb in Eq. (7.22) for frac-
tions with Kw < 12. Coal liquids have mainly Kw values of
less than 12 and the modified MB correlation is suggested for
vapor pressure of coal liquids. The relation for ATb of coal
7. APPLICATIONS: ESTI M ATI ON OF THERM OPHYSI CAL PROPERTI ES 313
TABLE 7.4---Prediction of vapor pressure of benzene at 400 K (260~ from different methods in Example 7.3.
API [9] Miller Eqs. (7.13) Lee-Kesler Ambrose Riedel Clapeyron Korsten Maxwell API
Fig. 7.5 and (7.14) Eq. (7.18) Eq. (7.19) Eq. (7.24) Eq. (7.25) Eq. (7.15) Eq. (7.16) Eqs. (7.20-7.22) Eq. (7.8)
pvap, bar 3.45 3.74 3.48 3.44 3.50 3.53 3.43 3.11 3.44 3.53
%Error ... 8.4 0.9 -0.3 1.4 2.3 -0. 6 -9. 9 -0. 3 2.3
liquids is [18, 19]:
T~=Tb - ATb
ATb = FIFzF3
{O Tb < 366. 5K
F1 = 1 + 0.009(Tb -- 255.37) Tb > 366.5 K
F2 = (Kw - 12) - 0.01304(Kw - 12) 2
1.47422 log10 pvap pvap < 1 at m
F3 = [1.47422 log10 pvap + 1.190833 (loga0 pvap)Z pvap > 1 at m
(7.23)
where T~ and Tb are in kelvin and pvap is in at mospheres
( =bar/ 1.01325) . This equat i on was derived based on mor e
t han 900 dat a poi nt s for some model comp ounds in coal liq-
uids i ncl udi ng n-alkylbenzenes. Equat i on (7.23) may be used
instead of Eq. (7.22) onl y for coal liquids and calculated T~
shoul d be used in Eq. (7.21).
Anot her rel at i on t hat is proposed for est i mat i on of vapor
pressure of coal liquids is a modi fi cat i on of Riedel equat i on
(Eq. 7.10) given in t he following f or m by Tsonopoul os et al.
[18, 19]:
B
l nPr yap = A - ~ - C ln Tr + DT 6
A = 5.671485 + 12.439604w
(7.24)
B = 5.809839 + 12.755971w
C = 0.867513 + 9.65416909
D = 0.1383536 + 0.316367w
This equat i on performs well for coal liquids if accurat e i nput
dat a on To Pc, and 09 are available. For coal fractions where
these paramet ers cannot be det ermi ned accurately, modified
MB (Eqs. 7.20-7.23) shoul d be used. When evaluated wi t h
mor e t han 200 dat a poi nt s for some 18 coal liquid fract i ons
modi fi ed BR equat i ons gives an average error of 4.6%, while
t he modified Riedel (Eq. 7.24) gives an error of 4.9% [18].
The simplest met hod for est i mat i on of vapor pressure of
pet rol eum fract i ons is given by Eq. (3.33) as
Tb --41 1393-- T ~
log10 pvap = 3.2041 1 - 0.998 x ~ x 1393 - Tb]
(7.25)
where Tb is the nor mal boiling poi nt and T is t he t emperat ure
at whi ch vapor pressure pvap is required. The correspondi ng
uni t s for T and P are kelvin and bar, respectively. Accuracy of
this equat i on for vapor pressure of pure comp ounds is about
1%. Eval uat i on of this single p ar amet er correl at i on is shown
in Fig. 7.8. I t is a useful relation for qui ck calculations or
when onl y Tb is available as a sole parameter. This equat i on
is hi ghl y accurat e at t emperat ures near Tb.
Exampl e 7. 3- - Est i mat e vapor pressure of benzene at 400 K
from the following met hods:
a. Miller (Eqs. (7.13) and (7.14))
b. Lee-Kesl er (Eq. 7.18)
c. Ambrose (Eq. 7.19)
d. Modified Riedel (Eq. 7.24)
e. Equat i on (7.25)
f. Equat i ons (6.101)-(6.103) or Eq. (7.15)
g. Korst en (Eq. 7.16)
h. Maxwel l -Bonnel l (Eqs. (7.20)-(7.22))
i. API met hod (Eq. 7.8)
j. Compare predi ct ed values f r om different met hods wi t h t he
value f r om Fig. 7.5.
Sol ut i on- - For benzene f r om Table 2.1 we have Tb = 353.3 K,
SG= 0.8832, Tr = 562.1 K, Pc = 48.95 bar, andw = 0.212. T =
400 K, Tr = 0.7116, and Tbr = 0.6285. The cal cul at i on met hods
are st rai ght forward, and the results are summar i zed in Table
7.4. The highest error corresponds to t he Korst en met hod. A
prel i mi nary eval uat i on wi t h some ot her dat a also indicates
t hat t he simple Cl apeyron equat i on (Eq. 7.15) is mor e accu-
rate t han t he Korst en met hod (Eq. 7.16). The Antoine equa-
t i on (Eq. 7.11 ) wi t h coefficients given in Table 7.3 gives a value
of 3.523 bar wi t h accur acy nearl y the same as Eq. (7.8).
7. 3. 3. 2 Graphical M et hods for V apor Pressure
of Pet rol eum Product s and Crude Oils
For pet rol eum fractions, especially gasolines and napht has,
l aborat ori es usually report RVP as a charact eri st i c related t o
quality of the fuel (see Table 4.3). As di scussed in Section
3.6.1.1, t he RVP is slightly less t han t rue vapor pressure (TVP)
at 100~ (37.8~ and for this reason Eq. (7.25) or (3.33) was
used to get an approxi mat e value of RVP from a TVP corre-
lation. However, once RVP is available from l aborat ory mea-
surement s, one may use this value as a basis for cal cul at i on of
TVP at ot her t emperat ures. Two graphi cal met hods for calcu-
lation of vapor pressure of pet rol eum finished product s and
crude oils from RVP are provi ded by the API -TDB [9]. These
figures are present ed in Figs. 7.10 and 7.11, for t he finished
product s and crude oils, respectively. When usi ng Fig. 7.10
the ASTM 10% slope is defined as SL10 = (T15 - T5)/10, where
T5 and T15 are t emperat ures on the ASTM D 86 distillation
curve at 5 and 15 vol% distilled bot h in degrees fahrenheit.
I n cases where ASTM t emperat ures at these poi nt s are not
available, values of 3 (for mot or gasoline), 2 ( aviation gaso-
line), 3.5 (for light napht has wi t h RVP of 9-14 psi), and 2.5
(for napht has wi t h RVP of 2-8 psi) are r ecommended [9]. To
use these figures, the first step is t o locate a poi nt on t he RVP
line and t hen a straight line is dr awn bet ween this poi nt and
the t emperat ure of interest. The i nt ercept i on wi t h t he vertical
line of TVP gives the reading. Values of TVP est i mat ed from
these figures are approxi mat e especially at t emperat ures far
f r om 100~ (37.8~ but useful when onl y RVP is available
f r om experi ment al measurement s. Values of RVP for use in
.m _
0 9
[3 .
U)
O -
B
a3
J
I . -
-O.2
O3
O.4
0.5
~- 0. 6
0,.8
: - 1
2
~3
-:-4
5
6
8
L
- lO
L
:-- 12
:-- 14
=" 16
- - 1 8
~. 2 0
-- 22
~- 2 4
AST M 1 0% Slope
a2 ! 0
2 "}.
N-P
T rue Vapor Pressure of
Gasolines and Finished
Petroleum Products
I I TECH NI CAL DATA BOOK
June 1993
Uni t conv er s i on: ~ = ( ~ x 1 . 8 ~-
wi t h p er mi s s i on f r om Ref . [ 9 ].
1 1 0-3
1 00 -
9 0-
8 0 -
7o-_-
60 -
50"
30"
20"
o-
LL
oJ
CL
E
FI G . 7 , 1 0 - - T r u e v ap or p r es s ur e of p et r ol eum p r oduct s f r om RVP.
3 2; p s i a = bar 1 4. 504, T ak en
Fig. 7.10 should be experimental rather than estimated from
methods of Section 3.6.1.1. I f no experimental data on RVP
are available the TVP should be calculated directly from meth-
ods discussed in Sections 7.3.2 and 7.3.3.1.
For computer applications, analytical correlations have
been developed from these two figures for calculation of vapor
pressure of petroleum products and crude oils from RVP data
[9]. For petroleum products, Fig. 7.10 has been presented by
a complex correlation with 15 constants in terms of RVP and
slope of ASTM D 86 curve at 10%. Similarly for crude oils the
mathematical relation developed based on Fig. 7.11 is given
as [9]
In pvap = A1 + A2 In(RVP) + A3(RVP) + A4T
(7.26) + [B1 + BE ln(RVP) + Ba(RVP) 4]
T
where pvap and RVP are in psia, T is in ~ Ranges of appli-
cation are OF < T(~ < 140F and 2 psi < RVP < 15 psi. The
coefficients are given as A1 = 7.78511307, A2 = - 1.08100387,
A3 = 0.05319502, A4 =0.00451316, B1 = - 5756.8562305, BE=
1104.41248797, and B3 =-0. 00068023. There is no in-
formation on reliability of these methods. Figures 7.10 and
7.11 or Eq. (7.26) are particularly useful in obtaining values
of vapor pressure of products and crude oils needed in esti-
mation of hydrocarbon losses from storage tanks [20].
7 . 3 . 4 Va p o r P r e s s u r e o f S o l i d s
Figure 5. 2a shows the equilibrium curve between solid and
vapor phases, which is known as a sublimation curve. I n fact,
at pressures below triple point pressure (P <Ptp), a solid di-
rectly vaporizes without going through the liquid phase. This
type of vaporization is called sublimation and the enthalpy
change is called heat of sublimation (AHSUb). For ice, heat
of sublimation is about 50.97 k J/tool. Through phase equilib-
rium analysis similar to the analysis made for VLE of pure
substances in Section 6.5 and beginning with Eq. (6.96) for
t~
t~
g
p -
- o
- - 1
- 2
- - 3
- - 4
2- 5
mb
~7
- - 8
- - - - 9
- - - l O
_ _
.z_
15
~ 2 o
- 2
._m _ 3
(/)
o - - a
r
m
n
>
~ - l o
w
15
Figure 5B1 .2
T rue Vapor Pressure
of
Crude O ils
l I TECHNICA L DA TA BO O K
Ju n e 1993
u0 -
130
1 2 0
100-
9o=_:
70
6O
4O
3 o 4
2 0 -
1 0-
0--
LL
03
o -
E
0)
FI G. 7 . 1 1 - - T r ue v apor pr essur e of cr ude oi l s f rom RVP. Uni t conv ersi on:
~ = ( ~ 1 . 8 + 3 2; psi a = bar x 1 4. 504. T ak en wi t h per mi ssi on f rom
Ref, [ 9 ].
vap or - sol i d equi l i br i a ( VSE) one can deri ve a r el at i on si mi l ar
to Eq. (6.101) for est i mat i on of vap or p r essur e of sol i ds:
B
(7.27) I n psub = A - --
T
wher e p~ub is t he vap or p r essur e of a p ur e sol i d al so known as
subl i mat i on pressure and A and B are t wo const ant s specific
for each comp ound. Values of psub are less t han Ptp and one
base p oi nt to obt ai n const ant A is t he t ri pl e p oi nt (Ttp, Ptp).
Values of Ttp and Ptp for some sel ect ed comp ounds are gi ven i n
Table 7.1. I f a val ue of sat ur at i on p r essur e (p~ub) at a r ef er ence
t emp er at ur e of T1 is known it can be used al ong wi t h t he t r i pl e
p oi nt to obt ai n A and B in Eq. (7.27) as follows:
[ P, p ~
I nk e[ub}
B- - - -
1 1
(7.28) r, r,p
= I n (p~ub) + ~1 A
wher e Ttp and T1 ar e i n kelvin. Par amet er B is equi val ent to
AH~ub/R. I n deri vi ng t hi s equat i on, it is as s umed t hat AH ~ub is
const ant wi t h t emp er at ur e. Thi s as s ump t i on can be j ust i fi ed
as t he t emp er at ur e var i at i on al ong t he subl i mat i on curve is
l i mi t ed, I n addi t i on it is as s umed t hat AV sub = V yap - V s ~-
RT/ P sub. Thi s as s ump t i on is r easonabl e as V s << V v and psub
is very smal l so t hat t he vap or is consi der ed as an i deal gas.
I n fact accur acy of Eq. (7.27) is mor e t han Eq. (6.101) be-
cause t he as s ump t i ons made in der i vat i on of t hi s equat i on
ar e mor e real i st i c. The r el at i ons for subl i mat i on p r essur e of
nap ht hal ene is gi ven as [21 ]
3783
I n PSUb(bar) = 8.722 - - -
T
(7.29) (T i n kelvin) for sol i d nap ht hal ene
Vapor p r essur es of sol i d CO2 ar e gi ven at several t emp er a-
t ures as: 9.81 t or t ( at - 120~ 34.63 ( - 110) , 104.81 ( - 100) ,
279.5 ( - 90) wher e t he p r essur es ar e i n mmHg ( t ort ) (1 bar =
750.06 mm Hg) and t he number s i n t he p ar ent heses ar e t he
cor r esp ondi ng t emp er at ur es in ~ as given by Levi ne [22].
Li near r egr essi on of I n psub versus 1/T gives const ant s A and
B i n Eq. (7.27) as
3131.97
I n Psub(bar) = 16.117 (T in kelvin) for sol i d CO2
T
(7.30)
The R 2 for t hi s equat i on is 0.99991 and t he equat i on r ep r o-
duces dat a wi t h an average er r or of 0.37%. Triple p oi nt t em-
p er at ur e of CO2 as gi ven in Table 7.1 is 216.58 K. Subst i t u-
t i on of t hi s t emp er at ur e in t he above equat i on pr edi ct s t ri pl e
p oi nt p r essur e of 5.238 ba r wi t h 1% er r or ver sus act ual val ue
of 5.189 bar as gi ven in Table 7.1.
Example 7. 4- - Vap or p r essur e of ice at - 10~ is 1.95 mm Hg
[21]. Derive a r el at i on for subl i mat i on p r essur e of ice and
est i mat e t he fol l owi ng:
a. Subl i mat i on p r essur es at - 2 a nd- 4 ~ C. Comp ar e cal cul at ed
val ues wi t h exp er i ment al dat a of 3.88 and 3.28 mm Hg [21].
b. The heat of subl i mat i on of ice.
Solution--(a) Dat a avai l abl e on sol i d vap or p r essur e ar e
p~ub = 1.95 mmHg = 0.0026 bar at T1 = -10~ = 263.15 K.
F r om Table 7.1 for wat er Ttp = 273.16 K and Ptp = 6.117 x
10 -3 bar. Subst i t ut i ng t hese val ues i nt o Eq. (7.28) gives A =
17.397 and B = 6144.3741. Thus t he r el at i on for subl i mat i on
p r essur e of ice is det er mi ned f r om Eq. (7.27) as follows:
I n PSUU(bar) = 17.397
(7.31)
6144.3741
T
(T in kelvin) for ice
At T1 = - 2~ = 271.15 Kwe get psub = 0.005871 = 3.88 mm
Hg. Si mi l ar l y at - 4~ t he vap or p r essur e is cal cul at ed as
0.004375 ba r or 3.28 mm Hg. Bot h val ues ar e i dent i cal t o
t he exp er i ment al val ues. (b) Si nce coeffi ci ent B is equi va-
l ent t o AHsUb/R t hus we have AH sub = R B. AH sub = 8.314
6144.3741/ 18 = 2.84 kJ/ g, wher e 18 is mol ecul ar wei ght of
water. r
7 . 4 THERMAL PROPERTI ES
I n t hi s sect i on, met hods of est i mat i on of t her mal p r op er t i es
such as ent hal py, heat capaci t y, heat of vap or i zat i on, and heat -
i ng val ues for p et r ol eum f r act i ons are pr esent ed. These p r op -
ert i es are r equi r ed i n cal cul at i ons r el at ed to energy bal ances
ar ound var i ous pr ocess uni t s as wel l as desi gn and op er at i on
of heat t r ansf er r el at ed equi pment . The f undament al equa-
t i ons for cal cul at i on of ent hal p y and heat cap aci t y wer e dis-
cussed i n Chap t er 6. I n t hi s sect i on ap p l i cat i on of t hose met h-
ods and some emp i r i cal cor r el at i ons devel oped for p r edi ct i on
of such p r op er t i es ar e pr esent ed. Heat capaci t y, heat s of va-
por i zat i on, and combus t i on can be eval uat ed f r om ent hal p y
dat a, but i ndep endent met hods ar e p r esent ed for conveni ence
and bet t er accuracy.
7.4.1 Enthalpy
Ent hal p y ( H) is defi ned by Eq. (6.1) and has t he uni t of en-
ergy p er uni t mas s or mol e (i.e., J/ g or J/ mol) . This p r op er t y
r ep r esent s t he t ot al ener gy associ at ed wi t h a fl ui d and can
be meas ur ed by a cal ori met er. Ent hal p y i ncr eases wi t h t em-
p er at ur e for bot h vap or and l i qui ds. Accor di ng t o Eq. (6.1),
ent hal p y of l i qui ds i ncr eases wi t h pr essur e, but for vap or s
ent hal p y decr eases wi t h i ncr ease in p r essur e because of de-
crease in vol ume. Effect of P on l i qui d ent hal p y is smal l and
can be negl ect ed for moder at e p r essur e changes ( AP ~ 10
bar) . However, effect of P on ent hal p y of vap or s is gr eat er
and cannot be negl ect ed, Effect of T and P on ent hal p y of
gases is best shown i n Fig. 6.12 for met hane.
I n engi neer i ng cal cul at i ons what is needed is t he di fference
bet ween ent hal pi es of a syst em at t wo di fferent condi t i ons of
T and P. Thi s di fference is usual l y shown by AH =/ -/ 2 - / / 1
wher e H1 is t he ent hal p y at 7"1 and P1 and/ / 2 is t he ent hal p y
at T2 and P2. Rep or t ed val ues of absol ut e ent hal p y have a
r ef er ence p oi nt at whi ch ent hal p y is zero. For exampl e, i n t he
st eam t abl es val ues of bot h H and S are gi ven wi t h r esp ect to
a r ef er ence st at e of sat ur at ed l i qui d wat er at its t r i pl e p oi nt of
0.01~ At t he r ef er ence p oi nt bot h ent hal p y and ent r op y are
set equal to zero. The choi ce of r ef er ence st at e is ar bi t r ar y
but usual l y sat ur at ed l i qui d at some reference t emp er at ur e
is chosen. For exampl e, Lenoi r and Hi p ki n i n a p r oj ect for
t he API meas ur ed and r ep or t ed ent hal p i es of ei ght p et r ol eum
fract i ons for bot h l i qui d and vap or p hases [23].
Thi s dat abase is one of t he mai n sour ces of exp er i men-
t al dat a on ent hal p y of p et r ol eum f r act i ons f r om nap ht ha t o
ker osene and gas oil. The dat aset i ncl udes 729 for l i qui d, 331
for vapor, and 277 dat a p oi nt s for t wo- p hase r egi on wi t h t ot al
of 1337 dat a p oi nt s i n t he t emp er at ur e r ange of 75-600~ and
p r essur e r ange of 20-1400 psi a. The reference st at e is sat u-
r at ed l i qui d at 75~ (23.9~ wi t h cor r esp ondi ng sat ur at i on
p r essur e of about 20-40 psi a. Some val ues of ent hal p y f r om
t hi s dat abase ar e gi ven i n Table 7.5. For all t hr ee f r act i ons t he
r ef er ence st at e is sat ur at ed l i qui d at 75~ and 20 psi a. One
shoul d be careful in r eadi ng absol ut e val ues of ent hal py, en-
tropy, or i nt er nal ener gy si nce r ep or t ed val ues dep end on t he
choi ce of r ef er ence state. However, no mat t er what is choi ce
of r ef er ence st at e cal cul at i on of AH is i ndep endent of refer-
ence state.
Heavy p et r ol eum fract i ons possess l ower ent hal p y ( per uni t
mass) t han do l i ght f r act i ons at t he same condi t i ons of T and
P. F or exampl e, for fract i ons wi t h Kw = i 0 and at 530 K,
when t he API gravi t y i ncr eases f r om 0 t o 30, l i qui d ent hal p y
i ncr eases f r om 628 to 721 kJ/ kg. Under t he same condi -
t i ons, for t he vap or p hase ent hal p y i ncr eases f r om 884 t o 988
kJ/ kg as shown by Kesl er and Lee [24]. Based on dat a mea-
sur ed by Lenoi r and Hi p ki n [23], var i at i on of ent hal p y of t wo
p et r ol eum f r act i ons ( jet fuel and gas oil) versus t emp er at ur e
and t wo di fferent p r essur es is shown in Fig. 7.12. Gas oil is
TABLE 7.5--Enthatpies of some petroleum fractions from Lenoir-Hipkin dataset [23].
Petroleum 20 psi and 1400 psi and 600~ and 20 psi and
fraction Kw API 300~ liquid 500~ liquid (P, psi), vapor (T, ~ vapor
Jet fuel 11.48 44.4 117.6 245.4 401.9 (100) 311 (440)
Kerosene 11.80 43.5 120 250.9 404.1 (80) 358.6 (520)
Fuel oil 11.68 33.0 115.8 243.1 346.0 (25) 378.4 (600)
Reference enthalpy (H = 0): saturated liquid at 75~ and 20 psia for all samples. H values are in Btudb.
7. APPLICATIONS: ESTIM ATION OF THERM OPHYSICAL PROPERTIES 3 1 7
Temperature, F
32 212 392 572
800 350
~J e t Fuel at 1.4 bar ~. 0
(20 psia) r
" " " Gas Oil Liquid at 2.8 ~ j r 280
600 bar (40 psia) [ . ~
Gas Oil Liquid at [ ~
400 --
140 ~"
gO
200
70
0 ---O , ~ , ~ , 0
0 100 200 300
Temperature, C
FI G. 7 . 1 2- - Ent hal p y of t wo pet r ol eum f ract i ons.
Ref er ence state: H ----- 0 for sat ur at ed l i qui d at 23 . 9 ~
( 7 5~ and 1 . 3 8 bar for j et f uel , and 23 . 9 ~ ( 7 5~
and 2. 7 6 bar ( 40 psi a) f or g as oil. Speci f i cat i ons:
Jet f uel , M = 1 44, Tb = 1 60. 5~ SG ----- 0. 8 04; g as oil,
M ---- 21 4, Tb ---- 27 9 . 4~ SG = 0. 8 48 . G as oi l i s i n l i q-
ui d st at e for ent i r e t emper at ur e r ang e. Jet f uel has
bubbl e poi nt t emper at ur e of 1 66. 8 ~ and dew poi nt
t emper at ur e of 1 8 3 . 1 ~ at 1 .4 bar ( 20 psi a) . Dat a
sour ce Ref. [ 23 ].
heavier t han jet fuel and its ent hal py as liquid is just slightly
less t han ent hal py of liquid jet fuel. However, there is a sharp
increase in the ent hal py of jet fuel duri ng vaporization. Pres-
sure has little effect on liquid ent hal py of gas oil.
As it was di scussed in Chapt er 6, to calculate H one shoul d
first calculate ent hal py depart ure or the residual ent hal py
from ideal gas state shown by H g = H - H ig. General met h-
ods for cal cul at i on of H a were present ed in Section 6.2. H R is
related to PVT relation t hr ough Eqs. (6.33) or (6.38). For gases
t hat follow t runcat ed virial equat i on of state (T~ > 0.686 +
0.439Pr or V~ > 2.0), Eq. (6.63) can be used to calculate H R.
Calculation of H R from cubi c equat i ons of state was shown in
Table 6.1. However, the most accurat e met hod of calculation
of H g is t hr ough generalized correl at i on of Lee-Kesl er given
by Eq. (6.56) in t he form of di mensi onl ess gr oup HR/RT~.
Then H may be calculated from the following relation:
(7.32)
Rrc
/-/=M-L RTc J+
where bot h H and H ig are in kJ/kg, Tc in kelvin, R = 8.314
J/ mol- K, and M is the mol ecul ar wei ght in g/ mol. The ideal
gas ent hal py H ig is a funct i on of onl y t emperat ure and must be
cal cul at ed at the same t emperat ure at whi ch H is t o be calcu-
lated. For pure hydrocarbons/ _/ ig may be calculated t hr ough
Eq. (6.68). I n this equat i on t he const ant An depends on t he
choi ce of reference state and in cal cul at i on of A H it will be
eliminated. I f the reference state is known, AH can be deter-
mi ned f r om H = 0, at the reference state of T and P. As it is
seen shortly, it is the AH ig t hat must be calculated in calcu-
lation of AH. This t erm can he calculated from the following
rel at i on based on ideal gas heat capacity, Cp g.
r2
(7.33) s ig = f Cpg(T)dT
rl
where T1 and Tz are t he same t emperat ure points t hat AH ig
must be calculated. For pure comp ounds Cp g can be calcu-
lated f r om Eq. (6.66) and combi ni ng wi t h the above equat i on
AH ig can be calculated. For pet rol eum fractions, Eq. (6.72)
is r ecommended for cal cul at i on of C~ g and when it is com-
bi ned wi t h Eq. (7.33) the following equat i on is obt ai ned for
cal cul at i on of AH ~g from/ ' 1 to T2:
A/-Pg = M Ao (T2 - / ' 1) + T (T2 - T'2) + - 3 (T23 - T3)
- C Bo( T2- TO+- ~ ( T2- T2) +- - f (T 3
(7.34)
where T1 and Tz are in kelvin, A/-Pg is in J/ mol, M is t he mol ec-
ul ar weight, coefficients A, B, and C are given in Eq. (6.72)
in t erms of Wat son Kw and co. This equat i on shoul d not be
applied to light hydr ocar bons (Nc < 5) as stated in t he appli-
cat i on of Eq. (6.72). H ig or C~ g of a pet rol eum fract i on may
also be calculated from the p seudocomp ound ap p r oach dis-
cussed in Chapt er 3 (Eq. 3.39). I n this way H ig or Cp g must
be calculated from Eqs. (6.68) or (6.66) for t hree pseudocom-
p ounds f r om groups of n-alkane, n-alkylcyclopentane, and n-
alkylbenzene havi ng boiling poi nt s the same as t hat of the
fraction. Then H ig is calculated from t he following equation:
(7.35) / f g = xpHp g + XNHN g + XAHA g
where xe, XN, and XA refer to the fractions of paraffins (P),
napht henes (N), and aromat i cs (A) in the mixture, whi ch is
known f r om PNA composi t i on or may be det ermi ned f r om
met hods given in Section 3.5. Cp g of a pet rol eum fract i on may
be calculated from t he same equat i on but Eq. (6.66) is used
to calculate C~ g of the P, N, and A comp ounds havi ng boiling
points t he same as t hat of t he fraction.
A summar y of the cal cul at i on procedure for A H f r om an
initial state at / ' 1 and P1 (state 1) to a final state at I"2 and P2
(state 6), for a general case t hat the initial state is a subcool ed
( compressed) liquid and the final state is a superheat ed vapor,
is shown in Fig. 7.13. The t echni que involves step-by-step cal-
cul at i on of AH in a way t hat in each step the cal cul at i on pro-
cedure is available. The subcool ed liquid is t ransferred to a
sat urat ed liquid at T1 and p~at where p~at is the vapor pres-
sure of liquid at t emperat ure T1. For this step (1 t o 2), AH1
represent s t he change in ent hal py of liquid phase at const ant
t emperat ure of T1 from pressure /'1 to pressure p~at. Meth-
ods of est i mat i on of p~at are di scussed in Section 7.3. I n most
cases, the difference bet ween P1 and plat is not significant
and t he effect of pressure on liquid ent hal py can be neglected
wi t hout serious error. This means t hat AH1 ~ 0. However, for
cases t hat this difference is large it may be calculated t hr ough
a cubi c EOS or general i zed correl at i on of Lee-Kesl er as
discussed in Chapt er 6. However, a mor e conveni ent ap p r oach
is t o calculate T( ~t at pressure Pb where T( ~t is t he sat urat i on
t emperat ure correspondi ng to pressure P1 and it may be cal-
cul at ed from vapor pressure correl at i ons present ed in Sect i on
An =/ ~kn 1 + An 2 + AH3 + AH4 + AH5 = H6 - Hj
- 1 -" S ubcoole-d ] iqui-d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' 6- Superheat ed
at T1, P2 vapor
AHI =H2- H l
Ans=H6- Hs=H~
2- Sat urat ed l i qui d
at T1, p~at
An2=H3-H2=AHv~p
3- Sat urat ed vapor
at TI , p~at
AH3=H4- H2=- H ~
4- I deal gas /~drI4=Hs-H4=An ig 5- I deal gas
at Tl , Pl sat at T2, P2
FIG. 7.13--Diagram of enthalpy calculation.
7.3. Then st at e 2 will be sat ur at ed l i qui d at T~ at and/ ~ and
A Hx r ep r esent s const ant p r essur e ent hal p y change of a l i qui d
f r om t emp er at ur e 7"1 to T1 sat and can be cal cul at ed f r om t he
fol l owi ng rel at i on:
T~t
(7.36) An 1 = f CLp(T)dT
rl
wher e C L is t he heat cap aci t y of l i qui d and it may be cal cul at ed
f r om met hods p r esent ed in Sect i on 7.4.2.
Si nce i n most cases t he i ni t i al st at e is l ow- p r essur e l i qui d,
t he ap p r oach p r esent ed in Fig. 7.13 to show t he cal cul at i on
p r ocedur es is used. St ep 2 is vap or i zat i on of l i qui d at const ant
T and P. AH2 r ep r esent s heat of vap or i zat i on at T1 and it can
be cal cul at ed f r om t he met hods di scussed i n Sect i on 7.4.3.
St ep 3 is t r ansf er of sat ur at ed vap or to i deal gas vap or at
const ant T1 and p~at ( or T1 sat and P~). AH3 = - HI R in whi ch H~
is t he r esi dual ent hal p y at T1 and p~at and its cal cul at i on was
di scussed earlier. St ep 4 is conver t i ng i deal gas at T1 and p~at t o
i deal gas at T2 and P2. Thus, AH4 = AH ig, wher e AH ig can be
cal cul at ed f r om Eq. (7.33). The final st ep is t o convert i deal gas
at T2 and P2 to a real gas at t he same T2 and P2 and AH5 = H~,
wher e H~ is t he r esi dual ent hal p y at T2 and P2. Once AH for
each st ep is cal cul at ed, t he overal l AH can be cal cul at ed f r om
s um of t hese AHs as shown i n Fig. 7.13. Si mi l ar di agr ams
can be const r uct ed for ot her cases. F or exampl e, if t he i ni t i al
st at e is a gas at at mos p her i c pr essur e, one may assume t he
i ni t i al st at e as an i deal gas and onl y st eps 4 and 5 i n Fig. 7.13
ar e necessar y for cal cul at i on of AH. I f t he i ni t i al st at e is t he
chosen r ef er ence state, t hen cal cul at ed overal l AH r ep r esent s
absol ut e ent hal p y at T2 and P2. Thi s is demons t r at ed in t he
fol l owi ng exampl e.
Example 7. 5- - Cal cul at e ent hal p y of j et fuel of Table 7.5 at
600~ and 100 psi a. Comp ar e your r esul t wi t h t he exp er i men-
t al val ue of 401.9 Btu/ lb. The r ef er ence st at e is sat ur at ed l i qui d
at 75~ and 20 psi a.
Solution--Calculation char t shown in Fig. 7.13 can be used
for cal cul at i on of ent hal py. The i ni t i al st at e is t he r ef er ence
st at e at T1 = 75~ (297 K) and P1 = 20 psi a (1.38 bar ) and
t he final st at e is Tz = 600~ (588.7 K) and P2 = 100 psi a (6.89
bar) . Si nce P1 = p~at, t herefore, AH1 = 0. p~at is given and
t her e is no need to cal cul at e it. Cal cul at i on of An yap and H R
r equi r es knowl edge of Tc, Pc, o9, and M. The API met hods of
Chap t er 2 ( Sect i on 2.5) are used t o cal cul at e t hese p ar ame-
ters. Tb and SG needed to cal cul at e t hese p ar amet er s can be
cal cul at ed f r om Kw and API gi ven i n Table 7.5. Tb = 437.55 K
and SG = 0.8044. F r om Sect i on 2.5.1 usi ng t he API met hods,
Tc = 632. 2K, Pc = 26.571 bar. Usi ng t he Lee- Kesl er met hod
f r om Eq. (2.105) o9 = 0.3645. M is cal cul at ed f r om t he API
met hod, Equat i on 2.51 as M = 134.3. Trl = 297/ 632. 2 = 0.47,
Prl = 1.38/ 26.571 = 0.052, Tr2 = 0.93, and Pr2 = 0.26. The
ent hal p y dep ar t ur e H- H ig can be est i mat ed t hr ough Eq.
(6.56) and Tables 6.2 and 6.3 fol l owi ng a p r ocedur e si mi l ar
to t hat shown in Examp l e 6.2. At Trl and Prl (0.47, 0.05) as i t
is cl ear f r om Table 6.2, t he syst em is i n l i qui d r egi on whi l e
t he r esi dual ent hal p y for sat ur at ed vap or is needed. The
r eason for t hi s di fference is t hat t he syst em is a p et r ol eum
mi xt ur e wi t h est i mat ed Tc and Pc di fferent f r om t r ue cri t i cal
p r op er t i es as needed for p hase det er mi nat i on. For t hi s
r eason, one shoul d be careful to use ext r ap ol at ed val ues
for cal cul at i on of [ ( H- Hig)/RTc] (~ and [ ( H- Hig)/RTc] (1)
at Trl and Prl- Therefore, wi t h ext r ap ol at i on of val ues at
Tr-- 0.65 and Tr = 0.7 to Tr = 0.47 for Pr = 0.05 we get
[ ( H - Hig)/RTc]i = - 0. 179 + 0.3645 ( - 0. 83) = - 0. 4815. At
Tr2 and /~ (0.93, 0.26) t he syst em is as sup er heat ed vapor:
[ ( H - Hig)/RTc]ii = - 0. 3357 + 0.3645 x ( - 0. 3691) = - 0.47 or
( n - Hi g) i = - 2530. 8 J / mol and ( H - Hi g) i i = - 2470. 4 J/ tool.
Thus, f r om Fig. 7.13 AH3 = - ( H - Hi g) i = +2530.8 J/ tool and
AH5 = +( H - Hig)n = - 2470. 4 J/ mol. AH ig c a n be cal cul at ed
f r om Eq. (7.34) wi t h coeffi ci ent s gi ven i n Eq. (6.72). The
i np ut p ar amet er s ar e Kw = 11.48, o9 = 0.3645, M = 134.3,
T1 ---- 297, and T2 = 588.7 K. The cal cul at i on r esul t is AH4 =
AH ig = 78412 J/ mol. AH vap can be cal cul at ed f r om met h-
ods of Sect i on 7.4.3. ( Eqs. 7.54 and 7.57), whi ch gives
AH2 = An yap = 46612 J/ mol. Thus, AH = AH1 + AH2 +
AH3 + AH4 + AH5 = 0 + 46612 + 2530.8 + 78412 - 2470.4 =
125084.4 J / mol = 125084.4/ 134.3 = 930.7 J/ g = 930.7 kJ/ kg.
F r om Sect i on 1.7.17 we get 1 J/ g = 0.42993 Btu/ lb. Therefore,
AH = 930.7 0.42993 = 400.1 Btu/ lb. Si nce t he i ni t i al st at e
is t he chosen r ef er ence st at e, at t he final T and P ( 600~ and
100 psi a) t he cal cul at ed absol ut e ent hal p y is 400.1 Btu/ lb,
whi ch differs by 1.8 Bt u/ l b or 0. 4% f r om t he exp er i ment al
val ue of 401.9 Btuflb. Thi s is a good p r edi ct i on of ent hal p y
consi der i ng t he fact t hat mi ni mum i nf or mat i on on boi l i ng
p oi nt and specific gravi t y has been used for est i mat i on of
var i ous basi c p ar amet er s. ,
I n addi t i on t o t he anal yt i cal met hods for cal cul at i on of en-
t hal p y of p et r ol eum fract i ons, t her e are some gr ap hi cal met h-
ods for qui ck est i mat i on of t hi s propert y. For exampl e, Kesl er
and Lee [24] devel op ed gr ap hi cal cor r el at i ons for cal cul at i on
of ent hal p y of vap or and l i qui d p et r ol eum fract i ons. They
p r op os ed a seri es of gr ap hs wher e Kw and API gravi t y were
used as t he i np ut p ar amet er s for cal cul at i on of H at a gi ven
T and P. F ur t her di scussi on on heat cap aci t y and ent hal p y
is p r ovi ded i n t he next sect i on. Once H and V ar e cal cul at ed,
t he i nt er nal ener gy (U) can be cal cul at ed f r om Eq. (6.1).
7 . 4 . 2 Heat Capaci t y
Heat capaci t y is one of the most i mp or t ant t hermal propert i es
and is defined at bot h const ant pressure (Cp) and const ant vol-
ume (Cv) by Eqs. (6.17) and (6.18). I t can be measur ed usi ng
a calorimeter. For const ant pressure processes, Cp and in con-
st ant vol ume processes, Cv is needed. Cp can be obt ai ned f r om
ent hal py usi ng Eq. (6.20).
Experi ment al dat a on liquid heat capaci t y of some pure
hydr ocar bons are given in Table 7.6 as report ed by Poling
et al. [12]. For defined mi xt ures where specific heat capaci t y
for each comp ound in the mi xt ure is known, the mixing rule
given by Eq. (7.2) may be used to calculate mi xt ure heat ca-
pacity of liquids cL.. Heat capacities of gases are l ower t han
liquid heat capacities under t he same condi t i ons of T and P.
For example, for pr opane at low pressures (ideal gas state)
the value of C; g is 1.677 J / g- K at 298 K and 3.52 J/ g. K at (7.38)
800 K. Values of Cp g of n-hept ane are 1.658 J/ g. K at 298 K
and 3.403 J / g- K at 800 K. However, for liquid state and at
300 K, C L of C3 is 3.04 and t hat of n-C5 is 2.71 J/ g. K as re-
port ed by Reid et al. [12]. While mol ar heat capaci t y increases
wi t h M, specific heat capaci t y decreases wi t h increase in M.
Heat capaci t y increases wi t h t emperat ure.
The general ap p r oach to calculate Cp is t o estimate heat
capaci t y depart ure from ideal gas [Cp - C~ g] and combi ne it
wi t h ideal gas heat capaci t y (cpg)9 A similar ap p r oach can be
used to calculate Cv. The relation for calculation of C~ g of
pet rol eum fractions was given by Eq. (6.72), whi ch requires. (7.39)
9 l g
Kw and ~0 as i nput paramet ers. C v can be calculated from Cp g
i g i g
t hr ough Eq. (6.23). Bot h Cp and C v are funct i ons of onl y t em-
perature. For pet rol eum fractions, Cp g can also be calculated
from the p seudocomp ound met hod of Chapt er 3 (Eq. 3.39) by
usi ng Eq. (6.66) for pure hydr ocar bons f r om different fami-
lies similar to cal cul at i on of ideal gas ent hal py (Eq" 7.35). The
most accurat e met hod for cal cul at i on of [Cp - Cp g] is t hr ough
generalized correl at i on of Lee-Kesl er (Eq. 6.57). Relations
for cal cul at i on of [Cp - Cp g] and [Cv - C~] f r om cubi c equa-
t i ons of state are given in Table 6.1. For gases at moderat e
pressures the depart ure funct i ons for heat capaci t y can be es-
t i mat ed t hr ough virial equat i on of state (Eqs. 6.64 and 6.65).
Once heat capaci t y depart ure and ideal gas propert i es are de-
t ermi ned, Cr is calculated f r om t he following relation:
(7.37) Cp = [Cp - Cp g] + Cp g
Relations given in Chapt er 6 for the cal cul at i on of [Cp - Cp g]
and Cp g are in mol ar units. I f specific uni t of J/ g. ~ for heat
capaci t y is needed, calculated values from Eq. (7.37) shoul d
be divided by mol ecul ar wei ght of the substance. Generalized (7.40)
correl at i on of Lee-Kesl er normal l y provi de reliable values of
Cp for gases, but for liquids mor e specific correl at i ons espe-
cially at low pressures have been proposed in the literature.
Est i mat i on of Cr and Cv from equat i ons of state was demon-
strated in Exampl e 6.2.
For solids t he effect of pressure on heat capaci t y is ne-
glected and it varies onl y wi t h t emperat ure: C s -- C s = f(T).
At moderat e and low pressures the effect of pressure on liq-
uids may also be neglected as Cp L ~- Cv L = f ( T) . However, this
assumpt i on is not valid for liquids at high pressures. Some
specific correl at i ons are given in the literature for cal cul at i on
of heat capaci t y of hydr ocar bon liquids and solids at at mo-
spheric pressures. At low pressures a generalized expression
in a pol ynomi al form of up to fourt h orders is used to correlate
Cp wi t h t emperat ure:
CL / R = CL / R = A1 + A2T + A3T 2 + A4T 3 -1- A5T 4
CSl R = CSvlR = B1 + BzT + B3T 2 + B4T 3 + B5T 4
where T is in kelvin. Coefficients A1- As and B1- Bs for a num-
ber of comp ounds are given in Table 7.7 as given by DI PPR
[10]. Some of t he coefficients are zero for some comp ounds
and for most solids t he pol ynomi al up to T 3 is needed. I n
fact Debye' s st at i st i cal -mechani cal t heory of solids and exper-
imental dat a show t hat specific heats of nonmet al l i c solids at
very low t emperat ures obey the following [22]:
CSp = aT 3
where T is t he absolute t emperat ure in kelvin. I n this relation
there is onl y one coefficient t hat can be det ermi ned from one
dat a poi nt on solid heat capacity. Values of heat capaci t y of
solids at mel t i ng poi nt given in Table 7.1 may be used as t he
reference poi nt to find coefficient a in Eq. (7.39). Equat i on
(7.39) can be used for a very nar r ow t emperat ure range near
the poi nt where coefficient a is det ermi ned.
Cubic equat i ons of states or the general i zed correl at i on of
Lee-Kesl er for calculation of the residual heat capaci t y of
liquids [C L - C~ g] do not provide very accurat e values espe-
cially at l ow pressures. For this reason, at t empt s have been
made to develop separate correl at i ons for liquid heat capac-
ity. Based on principle of correspondi ng states and usi ng pure
comp ounds' liquid heat capaci t y data, Bondi modified previ-
ous correl at i ons into the following form [12]:
CL - CPg - 1.586 + 0. 49
R 1- Tr
[4.2775 + 6.3 (1 - Tr) ~/3 0.4355 ]
+ 0)
rr
L
TABLE 7. 6---Some experimental values of liquid heat capacity of hydrocarbons, C L [12].
Compound T, K C L, J/g. K Compound T, K C L, J/g- K Compound T, K
Methane 100 3.372 n-Pentane 250 2.129 n-Decane 460
Methane 180 6.769 n-Pentane 350 2.583 Cyclohexane 280
Propane 100 1.932 n-Heptane 190 2.014 Cyclohexane 400
Propane 200 2.120 n-Heptane 300 2.251 Cyclohexane 500
Propane 300 2.767 n-Heptane 400 2.703 Benzene 290
/-Butane 300 2.467 n-Heptane 480 3.236 Benzene 400
n-Pentane 150 1.963 n-Decane 250 2.091 Benzene 490
C L, J/g. K
2.905
1.774
2.410
3.220
1.719
2.069
2.618
TABLE 7. 7--Coefficients of Eq. (7. 38) for liquid (Ai, s) and solid (Bi, s) heat capacity for some selected compounds [10].
cL / R = CL/ R = A1 + A2T + A3 T2 + A4T 3 + A5T 4
(7.38) CS / R = CSv / R = B1 + B2 T + BaT 2 + B4 T 3 + Bs T 4
Compound. M Ai A~ A3 A4 A5 Train, K Tmax, K
Liquid heat capacity, C~
n- Pent ane 72.2 19.134 - 3. 254 x 10 -2 1.197 x 10 . 4 0 0 143 390
n-Hexane 86.2 20.702 - 2, 210 x 10 -2 1.067 x 10 -4 0 0 178 460
n-Decane 142.3 33.512 - 2, 380 x 10 -2 1.291 x 10 -4 0 0 243 460
n-Pent adecane 212.4 41.726 2,641 x 10 -2 7.894 x 10 -5 0 0 283 544
n-Ei cosane 282,5 42.425 9.710 x 10 -2 2.552 x 10 -5 0 0 309 617
n-Hexat ri acont ane (C36) 507.0 84.311 1.771 x 10 -1 0 0 0 353 770
Cyclohexane 84.2 - 26. 534 3.751 x 10 -1 - 1. 13 x 10 -3 1. 285x10 -6 0 280 400
Met hyl cycl ohexane 98.2 15.797 - 7. 590 x 10 -3 9.773 x 10 -5 0 0 146 320
Benzene 78.1 19.598 - 4. 149 x 10 -2 1.029 x 10 -4 0 0 279 500
Toluene 92.1 16.856 - 1. 832 x 10 -2 8.359 x 10 . 5 0 0 178 500
Napht hal ene 128.2 3.584 6.345 x 10 -2 0 0 0 353 491
Ant hracene 178.2 9.203 7.325 x 10 . 2 - 5. 93 x 10 -6 0 0 489 655
Car bon dioxide 44.0 - 998. 833 1.255 x 10 -5. 21 x 10 -2 7.223 x 10 -5 0 220 290
Wat er 18.0 33.242 - 2. 514 x 10 -1 9.77 x 10 -4 - 1. 698 x 10 -6 1.127 x 10 . 9 273 533
B1 /32 /33 /34 B5 Train, K Tmax, K
Solid heat capacity, Cp s
n-Pent ane - 1.209 O. 1215
n-Hexane - 2. 330 0.1992
n-Decane - 4. 198 0.3041
n-Pent adecane - 311. 823 1.3822
n-Ei cosane - 0. 650 0.3877
n-Hexat ri acont ane (C36) --200.000 1.0000
Cyclohexane 15.763 - 0. 0469
Met hyl cycl ohexane - 1.471 0.1597
Benzene 0.890 0.0752
Toluene - 0. 433 0.1557
Napht hal ene 0.341 0,0949
Ant hracene 2.436 0.0531
Car bon dioxide - 2. 199 0.1636
Wat er - 3. 157 10 -2 0.0169
5.136 x 10 -4 - 1. 22 x 10 -5 5.08 x 10 -8 12 134
-1. 01 x 10 -3 2.43 x 10 -6 0 20 178
- 1. 52 x 10 -3 3.43 10 -6 0 20 240
0 0 0 271 283
- 1. 57 x 10 -3 3.65 x 10 -6 0 93 268
0 0 0 300 325
1.747 10 -4 0 0 191 271
- 9. 55 x 10 -4 3.06 x 10 -6 0 12 146
- 3. 23 x 10 -4 8.80 x 10 -7 0 40 279
- 1. 05 x 10 -3 2.97 x 10 -6 0 40 274
- 3. 79 x 10 -4 1.34 x 10 -6 - 1. 34 x 10 -9 30 353
1.04 x 10 -4 - 8. 82 x 10 . 8 3.69 10 -12 40 489
- 1. 46 x 10 .3 6.20 x 10 . 6 - 9. 26 x 10 -9 25 216
0 0 0 3 273
whe r e C~ g is t he i deal gas mol a r he a t cap aci t y. Li qui d he a t ca-
p aci t y i nc r e a s e s wi t h t e mp e r a t ur e . Thi s e qua t i on c a n al so be
a p p l i e d t o nonhydr oc a r bons as wel l . Thi s e qua t i on is r e c om-
me nde d f or Tr < 0. 8 a nd a n aver age e r r or of a bout 2. 5% was
obt a i ne d f or e s t i ma t i on of C L of s ome 200 c omp ounds at 25~
[ 12] . F or 0. 8 < Tr < 0. 99 val ues obt a i ne d f r om Eq. ( 7. 40) ma y
be c or r e c t e d i f he a t cap aci t y of s a t ur a t e d l i qui d is r equi r ed:
C L C L
( 7. 41) - r -sat _ exp ( 2. 1Tr - 17. 9) + exp ( 8. 655 Tr - 8. 385)
R
whe r e C } s houl d be c a l c ul a t e d f r om Eq. ( 7. 40) . Whe n Tr <
0, 8, i t c a n be a s s ume d t ha t C} ~- CsLt a nd t he c or r e c t i on t e r m
ma y be negl ect ed, csLt r e p r e s e nt s t he e ne r gy r e qui r e d whi l e
ma i nt a i ni ng t he l i qui d i n a s a t ur a t e d st at e, Mos t of t e n csLt
is me a s ur e d e xp e r i me nt a l l y whi l e mos t p r edi ct i ve me t hods
e s t i ma t e C } [ 12] .
F or p e t r ol e um f r a c t i ons t he p s e udoc omp one nt me t hod
s i mi l a r t o Eq, ( 7. 35) c a n be us e d wi t h M or Tu of t he f r a c t i on
as a c ha r a c t e r i s t i c p a r a me t e r . However , t he r e ar e s ome gen-
er al i zed c or r e l a t i ons de ve l op e d p a r t i c ul a r l y f or e s t i ma t i on of
he a t c a p a c i t y of l i qui d p e t r ol e um f r act i ons . Kes l er a nd Lee
[ 24] de ve l op e d t he f ol l owi ng c or r e l a t i on f or C } of p e t r ol e um
f r a c t i ons at l ow p r es s ur es :
CLp = a (b + cT)
F or l i qui d p e t r ol e um f r a c t i ons i n t he t e mp e r a t ur e r ange:
145 < T < 0.8Tr ( T a nd Tc bot h i n kel vi n)
( 7. 42)
a = 1. 4651 + 0. 2302 Kw
b= 0. 306469- 0. 16734SG
c = 0. 001467 - 0. 000551SG
whe r e Kw i s t he Wa t s on c ha r a c t e r i z a t i on f a c t or def i ned i n
Eq. ( 2. 13) . Pr e l i mi na r y c a l c ul a t i ons s how t ha t t hi s e qua t i on
ove r p r e di c t s val ues of C L of p ur e hydr oc a r bons a nd a c c ur a c y
of t hi s e qua t i on i s a bout 5%. Equa t i on ( 7. 42) i s r e c omme nde d
i n t he ASTM D 2890 t es t me t hod f or c a l c ul a t i on of he a t ca-
p aci t y of p e t r ol e um di s t i l l at e f uel s [ 25] . The r e ar e ot he r f or ms
s i mi l a r t o Eq. ( 7. 42) c or r e l a t i ng Cp L of p e t r ol e um f r a c t i ons t o
SG, Kw, a nd T us i ng hi ghe r t e r ms a nd or de r s f or t e mp e r a t ur e
but gener al l y gi ve s i mi l a r r es ul t s as t ha t of Eq. ( 7. 42) . Si mp l e r
f or ms of r e l a t i ons f or e s t i ma t i on of Ce L of l i qui d p e t r ol e um
f r a c t i ons i n t e r ms of SG a nd T ar e al so avai l abl e i n t he lit-
e r a t ur e [ 26] . But t he i r abi l i t y t o p r e di c t Cp L i s ver y p oor a nd
i n s ome cas es l ack i nf or ma t i on on t he uni t s or i nvol ve wi t h
s ome e r r or s i n t he coef f i ci ent s r ep or t ed. The c or r e s p ondi ng
s t at es c or r e l a t i on of Eq. ( 7. 40) ma y al so be us e d f or cal cul a-
t i on of he a t c a p a c i t y of l i qui d p e t r ol e um f r a c t i ons us i ng To,
w, a nd C~ g of t he f r act i on. The API me t hod [ 9] f or c a l c ul a t i on
of C L of l i qui d p e t r ol e um f r a c t i ons is gi ven i n t he f ol l owi ng
7. APPLICATIONS: ESTI M ATI ON OF THERM OPHYSI CAL PROPERTI ES 321
form for Tr _< 0.85:
C~ = A1 + AzT + A3 T2
Ax = -4. 90383 + (0.099319 + 0.104281SG)Kw
( 4. 81407 - 0.194833 K~
+
\
A2 = (7.53624 + 6.214610Kw) x (1.12172 -
0. 27634)
] 10 -4
( 0.70958]
A3 = -( 1. 35652 + 1.11863Kw)x 2.9027 ~ ] x 10 - 7
(7.43)
where C~ is in kJ/kg 9 K and T is in kelvin. This equation was
developed by Lee and Kesler of Mobile Oil Corporation in
1975. From this relation, the following equation for estima-
tion of enthalpy of liquid pet rol eum fractions can be obtained.
T
H L = f CLdT + H~Lf = AI(T - Tref)+ 2( T - Tref) 2
Tref
(7.44) + ~-~3(T - T~f) 3 + H;Lf
where HrL~ is usually zero at the reference t emperat ure of T~f.
Equation (7.43) is not recommended for pure hydrocarbons.
The following modified form of Watson and Nelson correla-
tion is recommended by Tsonopoulos et al. [18] for calcula-
tion of liquid heat capacity of coal liquids and aromatics:
C L = (0.28299 + 0.23605Kw)
x I0.645 - 0.05959 SG + (2.32056 - 0.94752 SG)
(1-~00 - 0-25537) ]
(7.45)
where C~ is in kJ/ kg- K and T is in kelvin. This equation pre-
dicts heat capacity of coal liquids with an average error of
about 3.7% for about 400 data points [18]. The following ex-
ampl e shows various met hods of calculation of heat capacity
of liquids.
Exampl e 7. 6--Calculate C L of 1,4-pentadiene at 20~ using
the following met hods and compare with the value of 1.994
J/ g. ~ reported by Reid et al. [12].
a. SRK EOS
b. DI PPR correlation [10]
c. Lee-Kesler generalized corresponding states correlation
(Eq. 6.57)
d. Bondi' s correlation (Eq. 7.40)
e. Kesler-Lee correlation (Eq. 7.42)--ASTM D 2890 met hod
f. Tsonopoulos et al. correlation (Eq. 7.45)
Solution- - Basic properties of 1,4-pentadiene are not given in
Table 2.1. Its properties obtained from other sources such as
DI PPR [10] are as follows: M = 68.1185, Tb = 25.96~ SG =
0.6633, Tc = 205.85~ Pc = 37.4 bar, Zc = 0.285, w = 0.08365,
and Cp g --- 1.419 J/g. ~ (at 20~ From Tb and SG, Kw =
12.264.
(a) To use SRK EOS use equations given in Table 6.1 and
follow similar calculations as in Exampl e 6.2: A = 0.039685,
B = 0.003835, Z L = 0.00492, V L = 118.3 cm3/mol, the volume
translation is c = 13.5 cm3/mol, VL(corrected) = 104.8 cm3/
mol, and ZL( correc. ) =0. 00436. From Table 6.1, P1 =
6.9868, P2 = -103.976, P3 = 0.5445, and [Cp - C~ g] = 21.41
J / mot - K =21.41/ 68.12 = 0.3144 J / g. K. C L = 0. 3144+
1.419 = 1.733 J / g. K ( error of -13%) . (b) DI PPR [10] gives
the value of C L = 2.138 J / g- K ( error of +7%). (c) From the
Lee-Kesler correlation of Eq. (6.57), Tr = 0.612 and Pr =
0.0271. From Tables 6.4 and 6.5, using interpolation (for Pr)
and extrapolation (for Tr, extrapolation from the .liquid re-
gion) we get [(Cp - c~g)/R] (~ = 1.291 and [(Cv - c~g)/R] (1) =
5.558. I n obtaining these values special care should be made
not to use values in the gas regions. From Eqs. (6.57) and
(7.38) using paramet ers R, M, and Cp g we get Cp L = 1.633
( error of-18%) . (d) From Eq. (7.40), [(C L - Cpg)/R] = 3.9287,
Cp L = 1.899 ( error of -4. 8%) . (e) From Eq. (7.42), a = 4.2884,
b = 0.19547, c = 0.0011, C L = 2.223 J/ g. K ( error of +11.5%).
This is the same as ASTM D 2890 test method. (f) From
Tsonopoulos correlation, Eq. (7.45), C L = 2.127 J / g- K ( error
of +6.6%). The generalized Lee-Kesler correlation (Eq.
6.57) gives very high error because this met hod is mai nl y
accurate for gases. For liquids, Eq. (7.40) is more accurate
t han is Eq. (6.57). Equation (7.45) although recommended
for coal liquids predicts liquid heat capacity of hydrocarbons
relatively with relative good accuracy. r
There are some other met hods developed for calculation of
C~. I n general heat capacity of a substance is proport i onal to
mol ar volume and can be related to the free space between
molecules. As this space increases the heat capacity decreases.
Since paramet er I (defined by Eq. 2.36) also represents mo-
lar volume occupied by the molecules Riazi et al. [27] showed
that C L varies linearly with 1/(1 - I). They obtained the fol-
lowing relation for heat capacity of homologous hydrocarbon
groups:
(7.46) CA=( al M +b l ) I ~- I ) + c l M + d l R
I n the above relation M is molecular weight, R is the gas con-
stant, and coefficients al -dl are specific for each hydrocarbon
family. Paramet ers I is calculated t hroughout Eqs. (2.36) and
(2.118) at the same t emperat ure at which C L is being cal-
culated. Paramet ers al-dl for different hydrocarbon families
and solid phase are given in Table 7.8.
7. 4. 3 Heats of Phase Changes- - Heat
of Vaporization
Generally there are three types of phase changes: solid to
liquid known as fusion (or melting), liquid to vapor (vapor-
ization), and solid to vapor (sublimation), which occurs at
pressures below triple point pressure as shown in Fig. 5.2a.
During phase change for a pure substance or mixtures of con-
stant composition, the t emperat ure and pressure remai n con-
stant. According to the first law of t hermodynami cs, the heat
TABLE 7.8--Constants for estimation of heat capacity from refractive index (Eq. 7.46).
Cp/R = (aiM + bl)[I/(1 - 1)] + cl M + dl
No. of
Gr oup St at e Car bon r ange Temp. r ange, ~ al b~ cl d~ dat a p oi nt s AAD% MAD%
n-Alkanes Liquid C5-C20 - 15-344 -0.9861 -43.692 0.6509 5.457 225 0.89 1.36
1-Alkenes Liquid C5- C20 -60-330 -1.533 40.357 0.836 -21.683 210 1.5 5.93
n-Alkyl-cyclopentane Liquid Cs-C20 -75-340 -1.815 56.671 0.941 -28.884 225 1.05 2.7
n-Alkyl-cyclohexane Liquid C6- C20 -100-290 -2.725 165.644 1.270 -68.186 225 1.93 2.3
n-Alkyl-benzene Liquid C6- C20 -250-354 -1.149 4.357 0.692 -3.065 225 1.06 4.71
n-Alkanes Solid C5-C20 -180-3 -1.288 -66.33 0.704 14.678 195 2.3 5.84
AAD%: Average absol ut e devi at i on per cent . MAD%: Ma xi mum absol ut e devi at i on p er cent . Coefficients ar e t aken f r om Ref. [ 27] . Dat a source: DI PPR [10].
t r ansf er r ed t o a syst em at const ant p r essur e is t he same as t he
ent hal p y change. Thi s amount of heat ( Q) is cal l ed ( l at ent )
heat of p hase change.
Q ( l at ent heat ) = AH ( phase t r ansi t i on)
(7.47) at const ant T and P
The t er m l at ent is nor mal l y not used. Si nce dur i ng p hase t ran-
si t i on, t emp er at ur e is also const ant , t hus t he ent r op y change
is gi ven as
AH ( phase change)
AS ( phase change) =
T ( phase change)
(7.48) at const ant T and P
Heat of fusi on was di scussed i n Sect i on 6.6.5 (Eq. 6.157) and
is usual l y needed in cal cul at i ons r el at ed to cl oud p oi nt and
p r eci p i t at i on of sol i ds i n p et r ol eum fluids ( Sect i on 9.3.3). I n
t hi s sect i on cal cul at i on met hods for heat of vap or i zat i on of
p et r ol eum f r act i ons ar e di scussed.
Heat of vap or i zat i on ( A/ P ap ) c a n be cal cul at ed i n t he t em-
p er at ur e r ange f r om t r i pl e p oi nt to t he cri t i cal poi nt . Ther mo-
dynami cal l y, A/ P ap is defi ned by Eq. (6.98), whi ch can be
r ear r anged as
(7.49) AH vap= ( H v - Hig) sat - - ( H L - Hig) sat
wher e ( H v - Hi g) sat and ( H E - Hig) s~t can be bot h cal cul at ed
f r om a gener al i zed cor r el at i on or a cubi c equat i on of st at e at
T and cor r esp ondi ng p~at (i.e., see Examp l e 7.7). At t he cri t i -
cal p oi nt wher e H v and H E become i dent i cal , ~xH yap becomes
zero. For several comp ounds, var i at i on of A/ P ap versus t em-
p er at ur e is shown in Fig. 7.14. The fi gure is const r uct ed based
500
400
2~
300
200
100
0
-200
~ . . . ~ n-Pentane
. . . . . . . n-Decane
- - - - n-Butyibenzene
. . -, ,
' . %
' , ~x
" . x
I i i i I
-100 0 100 200 300 400
Temperature, C
500
FIG. 7.14--Enthalpy of vaporization of several
hydrocarbons versus temperature.
on dat a gener at ed f r om cor r el at i ons p r ovi ded in Ref. [10].
Speci fi c val ue of AHn'~ p (kJ/g) decr eases as car bon number of
hydr ocar bon ( or mol ecul ar wei ght ) i ncreases, whi l e t he mol ar
val ues ( kJ / mol ) i ncr eases wi t h i ncr ease i n t he car bon number
or mol ecul ar wei ght . I n t he API -TDB [9], AHT p for p ur e com-
p ounds is cor r el at ed t o t emp er at ur e in t he fol l owi ng form:
(7.50) AHT ~p = A (1 - Tr) B+CTr
wher e coeffi ci ent s A, B, and C for a l arge number of com-
p ounds ar e p r ovi ded [9]. F or most hydr ocar bons coeffi ci ent
C is zero [9]. F or some comp ounds val ues of A, B, and C ar e
gi ven i n Table 7.9 as p r ovi ded in t he API -TDB [9].
The most ap p r oxi mat e and si mp l e rul e to cal cul at e A/-/yap is
t he Trout on' s rule, whi ch assumes AS ~ap at t he nor mal bor i ng
p oi nt (Tb) is r oughl y 10.5R ( ~87. 5 J / t ool . K) [22]. I n some
references val ue of 87 or 88 is used i nst ead or 87.5. Thus,
f r om Eq. (7.48)
H yap = 87.5Tb (7.51) nbp
wher e A f4vav is t he heat of vap or i zat i on at t he nor mal boi l -
" * nbp
i ng p oi nt in J / mol and Tb is i n K. Thi s equat i on is not val i d
for cer t ai n comp ounds and t emp er at ur e ranges. The accu-
r acy of t hi s equat i on can be i mp r oved subst ant i al l y by t aki ng
AS~ p as a f unct i on of Tb, whi ch gives t he fol l owi ng r el at i on for
AHnV~gp [22]:
(7.52) T4vap = RTb (4.5 + lnTb)
* * nbp
wher e R is 8.314 J / mol 9 K. Thi s equat i on at Tb = 400 K r educes
t o Eq. (7.51). I n general , A/ P ap can be det er mi ned f r om a
vap or p r essur e cor r el at i on t hr ough Eq. (6.99).
AHva p [ dl n PrSat ]
(7.53) RTc -- Azvap ' d( 1/ Tr )
wher e p~at is t he r educed vap or ( sat ur at i on) p r essur e at re-
duced t emp er at ur e of Tr. AZ vap is t he di fference bet ween Z v
and Z L wher e at l ow p r essur es Z L << Z v and AZ vap can be
ap p r oxi mat ed as Z v. Fur t her mor e, at l ow p r essur e if t he gas
is as s umed i deal , t hen AZ yap = Z v = 1. Under t hese condi -
t i ons, use of Eq. (6.101) in t he above equat i on woul d r esul t i n
A/ P ap = RB, wher e B is t he coeffi ci ent in Eq. (6.101). Obvi-
ously, because of t he assump t i ons made t o deri ve Eq. (6.101),
t hi s met hod of cal cul at i on of A/_pap is very ap p r oxi mat e. Mor e
accur at e pr edi ct i ve cor r el at i ons for A/ P ap can be obt ai ned by
usi ng a mor e accur at e r el at i on for t he vap or p r essur e such as
Eqs. (7.17) and (7.18).
There ar e a number of gener al i zed cor r el at i ons for pre-
di ct i on of A/ P ap based on t he p r i nci p l e of cor r esp ondi ng
st at es theory. Pi t zer cor r el at ed AHVap/RTc t o Tr t hr ough acen-
t ri c f act or w si mi l ar to Eq. (7.17). I n such cor r el at i ons,
7. APPLICATIONS: ESTI M ATI ON OF THERM OPHYSI CAL PROPERTI ES
TABLE 7. 9~Coefficients of Eq. 7.50 for calculation of enthalpy of vaporization of pure compounds versus temperature f9].
AHTaPEkJ/kg] = A(l - Tr) B+CTr
323
Compound A B C Compound A B C
Water 2612.982 -0.0577 0.3870 Methylcyclopentane 527.6931 0.3967 0.0000
Ammonia 1644.157 - 0.017 0.3739 Ethylcyclopentane 502.1246 0.3912 0.0000
HaS 754.073 0.3736 0.0000 Pentylcyclopentane 442.0789 0.3800 0.0000
CO2 346.1986 -0.6692 0.9386 Decylcyclopentane 397.8670 0.3800 0.0000
N2 228.9177 - 0.1137 0.4281 Pentadecylcyclopentane 372.6050 0.3800 0.0000
CH4 570.8220 -0. l 119 0.4127 Cyclohexane 534.5225 0.3974 0.0000
C2H6 588.1554 0.0045 0.3236 Methylcyclohexane 503.9656 0.4152 0.0000
C3H8 610.2175 0.3649 0.0000 Ethylene 679.2083 0.3746 0.0000
n-C4Hl0 568.6540 0.3769 0.0000 Propylene 539.9479 0.0169 0.0000
n-CsH12 540.6440 0.3838 0.0000 Benzene 651.8210 0.6775 -0.2695
n-C6H14 515.2685 0.3861 0.0000 Toluene 544.7929 0.3859 0.0000
n-C7H16 497.0039 0.3834 0.0000 Ethylbenzene 515.2839 0.3922 0.0000
n-Call18 489.0450 0.4004 0.0000 o-Xylene 521.7788 0.3771 0.0000
n-C10H22 461.4396 0.3909 0.0000 Propylbenzene 500.4582 0.3967 0.0000
n-C15H32 431.6786 0.4185 0.0000 n-Butylbenzene 470.0009 0.3808 0.0000
n-C20H42 407.3617 0.4089 0.0000 n-Octylbenzene 456.0581 0.4281 0.0000
Cyclopentane 517.7318 0.1808 0.1706 Naphthalene 371.4852 -0.3910 0.0000
f(~ and f(1)(T~) are cor r el at ed t o ( I - Tr), wher e as T~ ~ 1,
AHvap/RTc ~ O. However, mor e accur at e pr edi ct i ve met hods
ar e devel oped in t wo steps. I n t he first st ep heat of vapor i za-
t i on at nor mal boi l i ng poi nt , A/-/n~b p, is cal cul at ed and t hen
cor r ect ed to t he desi r ed t emp er at ur e by a second correl a-
t i on. One of t he most successful cor r el at i ons for p r edi ct i on
of AHnV~ p was p r op os ed by Ri edel [12]:
(7.54) AHn~p p l neo - 1.013
-- 1.093RTcTbr - 0- ~- _ - ~
wher e Tb~ is t he r educed boi l i ng p oi nt (Tb/T~) and Pc is t he
cri t i cal p r essur e in bars. The uni t of AH yap dep ends on t he
nbp
uni t s of R and Tc. Lat er Chen and Vetere devel op ed si mi l ar
cor r el at i ons for cal cul at i on of AH~'~ p i n t er ms of Pc and Tb~
[12]. For exampl e, Chen cor r el at i on is in t he fol l owi ng form:
gTcTb~ 3.978Tbr - 3.958 + 1.555 I n Pc
(7.55) AH~b~p p = 1.07 -- Tb~
Al t hough for cer t ai n p ur e comp ounds t he Chen cor r el at i on
is sl i ght l y sup er i or to t he Ri edel met hod, but for p r act i cal
ap p l i cat i ons especi al l y for p et r ol eum f r act i ons in whi ch T~
and Pc ar e cal cul at ed val ues, t he Ri edel equat i on is r easonabl y
accur at e. A mor e di r ect cal cul at i on of A/ -/ ~p p for p et r ol eum
f r act i ons is use of fract i on' s bul k p r op er t i es such as Tb and
SG or ot her avai l abl e p ar amet er s in an equat i on si mi l ar to
Eq. (2.38) [28]:
(7.56) AHnb ~p = aObO~
wher e A r4v~p is in J / mol ( or kJ/ krnol) and const ant s a, b, and
"~nbp
c ar e gi ven i n Table 7.10 for a numbe r of di fferent i np ut p ar a-
T A B L E 7.10---Coefficients of Eq. (7.56) for estimation of heat of
vaporization of petroleum fractions at the normal boiling point [28].
yap a0b0~
(7.56) A//~b p =
A//~p p, J/mo] 01 02 A b c
A g/yap Tb, K SG 37.32315 1.14086 9.77089 x 10 -3
"~l,nbp
AH;?n~ p T b, K I 39.7655 t.13529 0.024139
AH3afbp M I 5238.3846 0.5379 0.48021
met ers. Once t he val ue of A ~rvap "~bp is cal cul at ed, it shoul d be
di vi ded by M to convert its uni t f r om kJ/ krnol to kJ/ kg.
Equat i on (7.56) wi t h coeffi ci ent s gi ven in Table 7.10 can be
used for f r act i ons wi t h mol ecul ar wei ght r ange of 70-300 (~Tb
of 300-600 K) wi t h accur acy of about 2% when t est ed agai nst
138 p ur e hydr ocar bons. Appl i cat i on of t he equat i on can be
ext ended up to 700 K wi t h r easonabl e accuracy. Once AH~b~p v
is det er mi ned, t he Wat son r el at i on can be used t o cal cul at e
A/ -F ap at t he desi r ed t emp er at ur e (T).
(7.57) A/ -F a~ = AHn~ p \l--Z~br ]
wher e Tr and Tbr a r e t he r educed t emp er at ur e and r educed
boi l i ng poi nt , respectively. The same equat i on can be used t o
cal cul at e An yap at any t emp er at ur e when its val ue at one t em-
p er at ur e is avai l abl e. As it was shown i n Examp l e 7.5, use of
Eqs. (7.54) and (7.57) p r edi ct s AH v~p of p et r ol eum fract i ons
wi t h good accuracy. Tsonopoul os et al. [ 18] modi f i ed t he orig-
i nal Lee- Kesl er cor r el at i on for cal cul at i on of heat of vapor-
i zat i on of coal l i qui ds and ar omat i cs i n t he fol l owi ng form:
(7.58) ( Anva P) Tr = 0. 8 = RTr (4.0439 + 5.38260))
wher e R is 8.314 J / t ool . K, To is t he cri t i cal t emp er at ur e i n
kelvin, c0 is t he acent r i c factor, and A/-/yap is t he heat of va-
p or i zat i on at T = 0.8To in J/ tool. Equat i on (7.57) can be used
to cal cul at e An vap at t emp er at ur es ot her t han Tr = 0.8. An
eval uat i on of var i ous met hods for est i mat i on of AHVb p of sev-
eral coal l i qui d samp l es is shown i n Tables 7.11 and 7.12.
Basi c cal cul at ed p ar amet er s are gi ven in Table 7.11, whi l e es-
_f A r r vap
t i mat ed val ues o "n~bp f r om Ri edel , Vetere, Ri azi - Dauber t ,
and Lee- Kesl er ar e gi ven in Table 7.12. I n Table 7.11, M is
cal cul at ed f r om Eq. (2.51), whi ch is r ecommended for heavy
fract i ons. I f Eq. (2.50) were used to est i mat e M, t he %AAD
for t he f our met hods i ncr ease to 4.5, 3.2, 4.9, and 2.3, re-
spectively. Equat i on (2.50) is not ap p l i cabl e to heavy f r act i ons
( M > 300), whi ch shows t he i mp or t ance of t he char act er i za-
t i on met hod used to cal cul at e mol ecul ar wei ght of hydr ocar -
bon fract i ons. Eval uat i ons shown in Table 7.12 i ndi cat e t hat
bot h t he Ri edel met hod and Eq. (7.56) p r edi ct heat s of va-
p or i zat i on wi t h good accur acy despi t e t hei r si mpl i ci t y. For a
coal l i qui d samp l e 5HC in Table 7.1 l , exp er i ment al dat a on
TABLE 7. 11--Experimental data on heat of vaporization of some coal liquid fractions with calculated
basic parameters [28].
Fraction (a) Tb, K SG AH~ p, kJ/kg M Tr K Pc, bar co
5HC 433.2 0.8827 309.4 121.8 649.1 33.1 0.302
8HC 519,8 0.9718 281,4 162.7 748.1 27.1 0,394
11HC 612,6 1.0359 269,6 223,1 843.1 21.5 0,512
16HC 658.7 1,0910 245,4 247.9 896.4 20.5 0.552
17HC 692.6 1.1204 239,3 272.0 932.2 19.4 0,590
M from Eq. (2.51), Tr and Pc from Eqs. (2.63) and (2.64), ~ofrom Eq. (2.108). Experimental value on Tb, SG, and A Hnbt~
are taken from J. A. Gray, Report DOEfET/10104-7, April 1981; Department of Energy, Washington, DC and are also
given in Ref. [28].
TABLE 7. 12--Evaluation of various methods of prediction of heat of vaporization of petroleum
fractions with data of Table 7.11.
Riedel, Eq. (7.54) Chen, Eq. (7.55) RD, Eq. (7.56) MLK, Eq. (7.58)
Fraction AHb ap exp. Calc. %Dev. Calc. %Dev. Calc. %Dev. Calc. %Dev.
5HC 309.4 305.9 - 1. 1 303.9 - 1. 8 311.8 0.8 304.7 - 1. 5
8HC 281.4 282.5 0.4 278.9 -0. 9 287.7 2.2 276.6 -1.7
11HC 269.6 252.2 -6. 4 246.3 -8. 6 253.2 -6.1 240.5 -10.8
16HC 245.4 248.5 1.3 241.5 -1, 6 247.6 0.9 234.7 -4. 4
17HC 239.3 241.8 1.0 233.8 -2, 3 239.0 -0.1 226.2 -5.5
%AAD 2.0 3,0 2.0 4.8
Values of M, Tc, Pc, and w from Table 7.10 have been used for the calculations. RD refers to Riazi-Daubert method or
Eq. (7.56) in terms of Tb and SG as given in Table 7.10. MLK refers to modified Lee-Kesler correlation or Eq. (7.58).
ya p - . .
In use of Eq. (7.58), values of AHnb p have been obtained by correcting estimated values at Tr = 0.8 to Tr = Trb, using
Eq. (7.57).
AHT ~p in t he t emperat ure range of 350-550 K are given in
Ref. [28]. Predi ct ed values f r om Eq. (7.57) wi t h use of differ-
ent met hods for cal cul at i on of AHV~p p as given in Table 7.12
are comp ar ed graphi cal l y in Fig. 7.15. The average deviations
for t he Riedel, Vetre, Ri azi -Daubert , and Lee-Kesl er are 1.5,
1.8, 1,9 and 1.7%, respectively. The dat a show t hat the Riedel
Hvap vap
met hod gives the best result for bot h A ~bp and AH T when
the latter is calculated f r om t he Wat son met hod.
As a final met hod, A/ -F ~p can be calculated from Eq. (7.49)
by calculating residual ent hal py for bot h sat urat ed vapor and
liquid from an equat i on of state. This is demonst r at ed in t he
following example for cal cul at i on of An yap from SRK EOS.
Exampl e 7. 7--Deri ve a relation for t he heat of vapori zat i on
from SRK equat i on of state.
[-,
g
> <
350
300
250
o ExD Data - ~, ~
Riedel x~. . ,
. . . . Vetre "~o
. . . . . R D
. . . . . . . LK
200 ' ' ' ' '
350 400 450 500 550
Temperature, T, K
FI G. 7 . 1 5- - Ev al uat i on of v ari ous met hods f or es-
t i mat i on of heat of v apori zat i on of coal l i qui d 5HC.
Charact eri st i cs of 5HC f ract i on and descri pt i on of
met hods are g i v en in T abl es 7.11 and 7 . 1 2,
Sol ut i on- - The ent hal py depart ure from SRK is given in Table
6.1. I f it is applied to bot h sat urat ed vapor and sat urat ed liquid
at t he same t emperat ure and pressure and subt ract ed f r om
each ot her based on Eq, (7.49) we get:
H v - H L = AH yap = RT (Z v - Z L)
a _ Z v Z L
(7.59) +( ~ Tb) [ln ( ~ ) - I n ( Z~- - ~+ B) ]
where al is da/ dT as given in Table 6.1 for t he SRK EOS.
Repl aci ng for Z = PV / RT and B = bP/RT and consi deri ng t hat
t he ratio of V V / ( v v + b) is nearly uni t y (since b << VV):
at low t emperat ures where Z e << Z v, the first t erm in the right-
hand side can be repl aced by RTZ v. At hi gher t emperat ures
where t he difference bet ween Z v and Z L decreases Z e can-
not be neglected in comp ar i son with ZV; however, t he t erm
(Z v - Z L) becomes zero at the critical point. I n cal cul at i on
of AH yap f r om t he above equat i on one shoul d be careful of
t he uni t s of a, b and V. I f a is in bar ( cm6/ mol 2) and b is in
cm3/ mol, t hen the second t erm in t he ri ght -hand side of t he
above equat i on shoul d be divided by fact or 10 to have t he
uni t of J/ mol and R in t he first t erm shoul d have the value of
8.314 J/ tool. K. Eubank and Wang [29] also developed a new
identity to derive heat of vapori zat i on f r om a cubi c equat i on
of state (see Eq. (7.65) in probl em 7.13). #
7 . 4 . 4 Heat of C omb us t i on- - He at i ng Val ue
Combust i on is a chemi cal react i on wherei n the product s of
t he react i on a r e H20( g) , CO2( g) , SO2( g) , and N2(g), where (g)
refers t o t he gaseous state. The mai n react ant s in the react i on
are a fuel (i.e., hydrocarbon, H2, SO, CO, C . . . . ) and oxygen
(02). I n case of combust i on of H2 or CO, t he p r oduct is onl y
7. APPLI CATI ONS: ESTI M ATI ON OF THERM OPHYSI CAL PROPERTI ES 325
one comp ound (i.e., H20 or CO2). However, when a hydro-
car bon (CxHy) is bur ned the onl y product s are H20 and CO2.
Combust i on is a react i on in whi ch the ent hal py of product s is
less t han ent hal py of react ant s and as a result the heat of reac-
t i on ( ent hal py of product s - ent hal py of reactants) is always
negative. This heat of react i on is called heat of combustion
and is shown by zXH c. Heat of combust i on depends on the
t emperat ure at whi ch the combust i on takes place. The stan-
dard t emperat ure at whi ch usually values of A H c are report ed
is 25~ (298 K).
Amount of heat released by bur ni ng one uni t mass (i.e., kg,
g, or lb) of a fuel is called heating value or calorific value and
has the uni t of l o/ kg or Btu/ lb (1 l o/ kg = 0.42993 Btu/lb). I n
some cases for liquid fuels t he heat i ng values are given per
uni t vol ume (i.e., lOlL of fuel), whi ch differs from specific
( mass unit) heat i ng values by liquid density. I f in the combus-
t i on process p r oduced H20 is consi dered as liquid, t hen the
heat pr oduced is called gross heat of combust i on or higher
heating value (HHV). When p r oduced H20 is consi dered as
vapor (as in the actual cases), t hen the heat pr oduced is called
lower heating value (LHV). The LHV is also known as t he net
heating value (NHV). The difference bet ween HHV and LHV is
due to the heat requi red to vapori ze pr oduced wat er from liq-
ui d to vapor f or m at the st andard t emperat ure (43.97 kJ / mol
or 2.443 l o/ g of H20) . The amount of H20 f or med depends on
t he hydrogen cont ent of fuel. I f the hydrogen wt% of fuel is
H% t hen the relation bet ween HHV and LHV is given as [30]:
(7.61) LHV = HHV - 0. 22H%
where bot h LHV and HHV are in kJ/g. The heat i ng values can
also be det ermi ned from st andard heat s of format i on (A H2f98).
Values of AHaf98 for any el ement (i.e., H2, 02, C, S, etc.) is
zero and for f or med molecules such as H20 are given in
most t her modynami cs references [12, 21, 31]. For example,
for H20( g) , CO2(g), CO(g), SO2, CH4(g), C2H6(g), C3H8(g),
and n-CmH22 the respective values of AHf98 are -241. 81,
-393. 51, -110. 53, -296. 81, -74. 52, -83. 82, -104. 68, and
- 249. 46 lo/ mol. The following example shows cal cul at i on of
heat i ng values f r om heat s of format i ons.
Exampl e 7. 8--Cal cul at e HHV and LHV of hydrogen, me-
thane, propane, carbon, and sulfur from heat s of format i on.
Sol ut i on- - Here the cal cul at i on of heat i ng value of CH4
is demonst r at ed and a similar ap p r oach can be used for
ot her fuels. The chemi cal react i on of combust i on of CH4 is
CH4(g) + 202( g) -+ 2H20( g) + CO2(g) + AH c, where AH c =
2AH2f98( H2 O) + AH2f98(CO2) -- z~sf 98( CH4) - AS2f98( O2) ~--- 2x
( -241. 835) + (-393.51) - ( -74. 8936) - ( 0) = - 802. 286kJ /
mol. Since the produced wat er is assumed to be in gas phase
so the LHV is calculated as 802.286/ 16.04 = 50.01 lo/ g.
This is equivalent to 11953 cal/ g or 21500 Btu/ lb. The
HHV can be calculated by addi ng heat of vapori zat i on of
wat er (2 x 43.97 = 87.94 kJ/ mol) to the mol ar LHV. HHV =
802.286 + 87.94 = 890.2 kJ/ mol or 55.5 kJ/ g of CH4. Equa-
t i on (7.61) to convert LHV to HI-IV or vice versa usi ng
H% of fuel may also be used. I n this case, H% of CH4 =
( 4/ 16) x 100 = 25 wt%. Thus HHV = 50 + 0.22 x 25 = 55.5
kJ/g. Similarly for H2, LHV = 241.81/ 2.0 = 121 kJ/g. The
HHV = 121 +0. 22 x 100- - 143 l o/ g or 61000 Btu/ lb. The
heat i ng values of ot her fuels are calculated as follows:
Hydrogen, Methane, Propane, Carbon, Sulfur,
Fuel H2 CH4 C3 I-I8 C S
LI-I~, kJ/g 121 50 46.4 32.8 9.3
HHV, kJ/g 143 55.5 50.4 32.8 9.3
As it can be seen from these calculations, hydr ogen has the
highest heat i ng value and car bon has t he lowest heat i ng value.
Thus hydrogen is t he best, while car bon is consi dered as the
worst fuel. Sulfur heat i ng value is even less t han car bon but
sulfur is not really consi dered as a fuel. Some values of HHV
for several ot her fuels as report ed by Felder and Rousseau
[32] are given in Table 7.13. The calculated value of HHV of
C is near the HHV of hard coal (i.e., solid form) as given in
Table 7.13. I n nat ural gases since t here are some hydrocar-
bons heavier t han met hane, its heat i ng value is somewhat
l ower t han t hat of pure met hane, t
Exampl e 7.8 shows t hat t he heat i ng value generally in-
creases as t he hydrogen cont ent of fuel increases and car-
bon cont ent decreases. I n ot her words, as CH wei ght rat i o
increases the heat i ng value decreases. Furt hermore, presence
of sulfur furt her reduces the heat i ng value. For this reason,
some researchers have correl at ed HHV t o wt % of C, H, S, N,
and O cont ent of fuel. For example, Tsonopoul os et al. [18]
proposed the following relation for est i mat i on of HHV of coal
liquids:
HHV [lo/ g] = 0.3506 (C%) + 1.1453 (H%) + 0.2054 (S%)
(7.62) + 0.0617 (N%) - 0.0873 ( 0%)
S, N, and O are usually f ound in heavy fuels and aromat i c ri ch
fuels such as coal liquids. This equat i on predicts HHV of coal
liquids wi t h %AAD of 0.55 for some 130 fuels. This equat i on
predicts HHV of pure C as 35 lo/ g. However, this equat i on is
not r ecommended for light fuels, pet rol eum fraction, or pure
compounds. There is a si mpl er relation for cal cul at i on of LHV
of heavy fuels and pet rol eum fractions [30]:
(7.63) LHV[ l o/ g] = 55.5 - 14.4 SG - 0.32S%
where S% is the sulfur wt% in the fuel. A very simple but
approxi mat e formul a for cal cul at i on of HHV of crude oils is
[26]:
(7.64) HHV = 51.9 - 8.8 SG 2
where HHV is in l o/ g ( or MJ/kg) and SG is the specific gravity
of crude and S% is the sulfur wt % of the crude. Accuracy of
these equat i ons is usually about 1%. A typical crude oil has
heat i ng value of about 10 500 cal/ g ( ~44 lo/ g). I ncrease in hy-
drogen cont ent of a fuel not only increases the heat i ng value
TABLE 7.13--Heating values of some fitels.
Higher heating value
Fuel kJ/g Btu/ lb
Wood 18 7700
Hard coal 35 15 000
Crude oil 44 19 000
Natural gas 54 23 000
Hydrogen 143 61 000
of t he fuel but also decreases amount of unbur ned hydrocar-
bon and CO/CO2 product i on. For these reasons, nat ural gas
is consi dered to be a clean fuel but the cleanest and most
valuable fuel is hydrogen. This is the reason for global ac-
cel erat i on of devel opment of hydrogen fuel cells as a clean
energy, al t hough still pr oduct i on of energy from hydr ogen is
very costly [33].
HHV can be measur ed in t he l aborat ory t hr ough combus-
t i on of t he fuel in a bomb cal ori met er sur r ounded by water.
Heat pr oduced can be calculated from the rise in t he t empera-
t ure of water. The experi ment al procedure to measure heat i ng
value is explained in ASTM D 240 test met hod. The heat i ng
value of a fuel is one of the charact eri st i cs t hat det ermi nes
pri ce of a fuel.
7 . 5 S UMMA R Y A N D R EC OM M EN D A T I ON S
I n this chapter, appl i cat i on of met hods and procedures pre-
sented in t he book for cal cul at i on and est i mat i on of vari ous
t hermophysi cal propert i es are shown for pure hydr ocar bons
and t hei r defined mixtures, nat ural gases and nonhydrocar-
bon gases associ at ed wi t h t hem (i.e. H2S, CO2, N2, H20) ,
defined and undefi ned pet rol eum fractions, crude oils, coal
liquids, and reservoir fluids. Charact eri zat i on met hods of
Chapt ers 2-4 and t her modynami c relations of Chapters 5 and
6 are essential for such propert y calculations. Basically, ther-
mophysi cal propert i es can be est i mat ed t hr ough equat i ons of
state or general i zed correlations. However, for some special
cases empi ri cal met hods in t he forms of graphi cal or analyt-
ical correl at i ons have been present ed for qui ck est i mat i on of
cert ai n properties.
Met hods of predi ct i on of propert i es i nt roduced in t he pre-
vious chapt ers such as density, enthalpy, heat s of vapori zat i on
and melting, heat capaci t y at const ant pressure and volume,
vapor pressure, and fuels' heat i ng values are presented.
For cal cul at i on of propert i es of pure comp onent s when a
correl at i on for a specific comp ound is available it must be
used wherever applicable. Generalized correl at i ons shoul d
be used for cal cul at i on of propert i es of pure hydr ocar bons
when specific correl at i on ( analytical or graphical) for t he
given comp ound is not available. For defined mi xt ures the
best way of cal cul at i on of mi xt ure propert i es when experi-
ment al dat a on propert i es of individual comp onent s of t he
mi xt ure are available is t hr ough appropri at e mixing rules for
a given propert y usi ng pure comp onent s propert i es and mix-
t ure composi t i on. For defined mixtures wherei n propert i es
of pure comp onent s are not available, t he basi c i nput pa-
ramet ers for equat i ons of states or generalized correl at i ons
shoul d be calculated f r om appropri at e mixing rules given in
Chapt er 5. These basi c propert i es are generally Tr Pc, Vc, w,
M, and C~ g, whi ch are known for pure component s. For
pet rol eum fract i ons these paramet ers shoul d be est i mat ed
and t he met hod of their est i mat i ons has a great i mpact on
accur acy of predi ct ed physi cal properties. I n fact the i mpact
of est i mat i on of basic i nput propert i es is great er t han the im-
pact of selected t her modynami c met hod on the accur acy of
propert y predictions.
For predi ct i on of propert i es of pet rol eum fractions, spe-
cial met hods are provi ded for undefi ned mixtures. For bot h
pure comp ounds and pet rol eum mixtures, propert i es of gases
can be est i mat ed wi t h greater accuracy t han for propert i es of
liquids. This is mai nl y due to bet t er underst andi ng of inter-
mol ecul ar forces in gaseous systems. Similarly properties of
gases at low pressures can be est i mat ed with bet t er accuracy
in comp ar i son wi t h gases at hi gh pressures. Effect of pres-
sure on propert i es of gases is much great er t han t he effect
of pressure on propert i es of liquids. At hi gh pressures as we
ap p r oach the critical regi on propert i es of gases and liquids
ap p r oach each ot her and under such condi t i ons a uni que gen-
eralized correl at i on for bot h gases and liquids t er med dense
fluids may be used for predi ct i on of propert i es of bot h gases
and liquids. For wi de boiling range fract i ons or crude oils t he
mi xt ure shoul d be split into a number of p seudocomp onent s
and treat the fluid as a defined mixture. A mor e accurat e ap-
p r oach woul d be to consi der t he fluid as a cont i nuous mi xt ure
When usi ng a t her modynami c model, cubi c equat i ons of
state (i.e., PR or SRK) shoul d be used for cal cul at i on of
PVT and equi l i bri um propert i es at pressures great er t han
about 13 bars ( ~200 psia). At low pressures and especially
for liquids, propert i es calculated from a cubi c EOS are not
reliable. For liquid systems specific generalized correl at i ons
developed based on liquid propert i es are mor e accurat e t han
ot her met hods. I t is on this basis t hat Racket t equat i on pro-
ri des mor e accurat e dat a on liquid density t han any ot her
correlation. I n appl i cat i on of EOS to pet rol eum mi xt ures t he
BI Ps especially for t he key comp onent s have significant im-
pact on accur acy of predi ct ed results. Wherever possible BI P
of key comp onent s (i.e., C1-C7+ in a reservoir fluid) can be
t uned wi t h available experi ment al dat a (i.e., density or satu-
rat i on pressure) t o i mprove predi ct i on by an EOS model [34].
TABLE 7.14---Summary of recommended methods ]or various
properties.
Property Methods of estimation for various fluids
Density
Vapor pressure
Enthalpy
Liquid heat capacity, C L
AHvap
9 Eq. (7.3) for gases with Lee-Kesler
generalized correlation for calculation
of Z (Ch 5), also see Section 7.2.1.
9 For pure liquid hydrocarbons, Table
2.1 and Eq. (7.5) or Rackett equation.
9 Eq. (7.4) for defined liquid mixture.
9 For petroleum fractions use Rackett
equation, Eq. (7.5), or Figs. 7.1-7.3.
9 See Section 7.2.2 for other cases.
9 Eq. (7.8) for pure compounds and
if coefficients are not known use
Eqs. (7.18) or (7.19).
9 Use Eqs. (7.20) and (7.22) for
petroleum fractions and Eqs. (7.21)
and (7.23) for coal liquids.
9 For crude oils use Eq. (7.26) or
Fig. 7.11.
9 Use Eq (7.32) and Fig. 7.13 with
Lee-Kesler correlations of Ch 6 for
petroleum fractions.
9 Use Eq. (7.34) for AH ig.
9 For special cases see Section 7.4.1.
9 Eq. (7.40) for pure compounds.
9 Eq. (7.43) for petroleum fractions.
9 Eq. (7.45) for coal liquids.
9 Eq. (7.50) or Eqs. (7.54) and (7.57) for
pure compounds.
9 Eqs. (7.54) and (7.57) for petroleum
fractions.
9 Eqs. (7.58) and (7.57) for coal liquids.
Heating value See Section 7.4.4.
These recommendations are not general and for special cases one should see
specific recommendations in each section.
7. APPLICATIONS: ESTI M ATI ON OF THERM OPHYSICAL PROPERTI ES 327
I n general when model paramet ers are tuned with available
experimental data especially for complex mixtures and heavy
fractions, accuracy of model prediction for the given systems
can be greatly improved. A summar y of some recommended
met hods for different physical and t hermodynami c proper-
ties is given in Table 7.14.
7 . 6 P R OB L EM S
7.1. For storage of light hydrocarbons and their mixtures in
sealed tanks, always a mixture of liquid and vapor are
stored. Why is this practiced, rat her t han storing 100%
gas or 100% liquid phase?
7.2. Figure 7.16 shows reported l aborat ory data on variation
of P with V for a fluid mixture at constant T and com-
position. Can you comment on the data?
7.3. Figure 7.17 shows reported l aborat ory data on variation
of H with T for a fluid mi xt ure at constant P and com-
position. Can you comment on the data?
7.4. A kerosene sample has specific gravity and molecular
weight of 0.784 and 167, respectively. Methane is dis-
solved in this liquid at 333 K and 20.7 bar. The mole frac-
tion of met hane is 0.08. Use the graphical met hod sug-
gested in this chapt er to calculate mol ar density of the
mixture and compare it with the value of 5.224 kmol / m 3
as given in Ref. [35]. What is the predicted value from
an EOS?
7.5. Derive Eq. (4.120) for vapor pressure.
7.6. A pet rol eum product has mi d boiling point of 385 K and
specific gravity of 0.746. Estimate its vapor pressure at
323 K from three most suitable methods.
7.7. For the pet rol eum product of Problem 7.6 calculate RVP
from an appropri at e met hod in Chapter 3 and then use
Fig. 7.10 to calculate TVE Compare the result with those
estimated in Probl em 7.6.
7.8. Sublimation pressure of benzene at -36.7~ is 1.333 Pa
[21]. Derive a relation for sublimation pressure of ben-
zene. Calculate sublimation pressure of benzene at -11. 5
and-2. 6 ~ C. Compare estimated values with reported val-
ues of 13.33 and 26.67 Pa [21]. Also estimate heat of
sublimation of benzene.
Constant Temp. & Comp.
Volume
FIG. 7 . 1 6- - Pr essur e- v ol ume data for
Problem 7.2.
Constant Press. & Comp.
Temperature
FIG. 7 . 1 7 mEnt hal py- t emperat ure data for
Problem 7,3.
7.9. When n-pentane is heated from 190.6~ and 600 psia to
370.7~ and 2000 psia, the enthalpy increases by 8655
Btu/ lbmol [36]. Calculate this enthalpy change from
Lee-Kesler and PR EOSs and compare calculated val-
ues with the experimental value.
7.10. I n the previous probl em consider the initial pressure is
2000 psia. I n this case the process becomes heating at
constant pressure. Calculate the enthalpy change and
compare with the experimental value of 8467 Btu/ lbmol
[36].
7.11. Calculate enthalpy of kerosene of Table 7.5 in liquid
phase at 500~ and 1400 psia using Lee-Kesler general-
ized correlation and SRK EOS. Compare the results with
experimental value given in Table 7.5. Use the API meth-
ods for prediction of M, Tc, Pc, and the Lee-Kesler corre-
lation for calculation of acentric factor. Repeat the cal-
culations using Lee-Kesler generalized correlation for
the enthalpy departure and Twu correlations for M, To,
and Pc.
7.12. The purpose of this probl em is to show the i mpact of
bot h the selected predictive met hod and the selected
characterization met hod on the estimation of t hermal
properties of hydrocarbon fractions. A coal liquid has a
boiling point of 476~ and specific gravity of 0.9718. Es-
timate its heat of vaporization at 600~ using the meth-
ods proposed by Riedel, Chen, and Riazi-Daubert. For
each met hod use API, Ri azi -Daubert (I 980), Lee-Kesler,
and Twu met hods for estimation of input paramet ers.
The experimental value is 110.9 Btu/lb [28].
7.13. For pure components, the Maxwell Equal Area Rule
(MEAR) is a t hermodynami c identity for vapor-liquid
equilibria [29]:
(7.65)
V V
p sat ( vV -- vL) ~ fvL pEOS (dV) T
Use this equation to calculate vapor pressure of benzene
at 25, 100, 140, and 220~ from SRK EOS and compare
with actual data.
7.14. Derive relations for heat of vaporization based on RK
and PR EOS.
R EF ER EN C ES
[1] Carlson, E. C., "Don't Gamble with Physical Properties for
Simulations," Chemical Engineering Progress, Vol. 92, No. 10,
October 1996, pp. 35-46.
[2] Alberty, R. A. and Silbey, R. J., "Physical Chemistry," 2nd ed.,
[4] Bhagat, Ph., "An Introduction to Neural Nets," Chemical
Engineering Progress, Vol. 86, No. 8, August 1990, pp. 55-60.
[5] Riazi, M. R. and Elkamel, A., "Using Neural Network Models to
Estimate Critical Constants," Simulators International, Vol. 29,
No. 3, 1997, pp. 172-177.
[6] Lee, M.-J., Hwang,, S.-M., and Chen, J.-T., "Density and
Viscosity Calculations for Polar Solutions via Neural
Networks," Journal of Chemical Engineering of Japan, Vol. 27,
No. 6, 1994, pp. 749-754.
[7] Press, J., "Working with Non-Ideal Gases," Chemical
Engineering Progress, VoL 99, No. 3, 2003, pp. 39-41.
[8] Engineering Data Book, Vol. H, Sections 17-26, 10th ed., Gas
Processors (Suppliers) Association, Tulsa, OK, 1987.
[10] AIChE DIPPR| Database, Design Institute for Physical
1996.
[11] Hall, K. R., Ed., TRC Thermodynamic Tables--Hydrocarbons,
Thermodynamic Research Center, The Texas A&M University
System, College Station, TX, 1993.
Gases and Liquids, 5th ed., McGraw-Hill, New York, 2000. Reid,
R. C., Prausnitz, J. M., Poling, B. E., Eds., Properties of Gases
and Liquids,4th ed., McGraw-Hill, New York, 1987.
[13] Yaws, C. L. and Yang, H.-C., "To Estimate Vapor Pressure
Easily," Hydrocarbon Processing, Vol. 68, 1989, pp. 65-68.
[14] Feyzi, E, Riazi, M. R., Shaban, H. I., and Ghotbi, S.,
"Improving Cubic Equations of State for Heavy Reservoir
Fluids and Critical Region," Chemical Engineering
[15] Korsten, H., "Internally Consistent Prediction of Vapor
Pressure and Related Properties," Industrial and Engineering
pp. 510-527.
[17] Maxwell, J. B. and Bonnell, L. S., Vapor Pressure Charts for
Petroleum Engineers, Exxon Research and Engineering
Company, Florham Park, NJ, 1955. Reprinted in 1974.
"Deviation and Precision of a New Vapor Pressure Correlation
for Petroleum Hydrocarbons," Industrial and Engineering
Chemistry, Vol. 49, 1957, pp. 1187-1196.
[19] Wilson, G. M., Johnston, R. H., Hwang, S. C., and Tsonopoulos,
C., "Volatility of Coal Liquids at High Temperatures and
Pressures," Industrial Engineering Chemistry, Process Design
and Development, Vol. 20, No. 1, 1981, pp. 94-104.
[20] Marks, A., Petroleum Storage Principles, PennWell, Tulsa, OK,
1983.
[21] Sandier, S. I., Chemical and Engineering Thermodynamics, 3rd
ed., Wiley, New York, 1999.
[22] Levine, I. N., Physical Chemistry, 4th ed., McGraw-Hill, New
York, 1995.
Data, Vol. 18, No. 2, 1973, pp. 195-202.
Fractions," Hydrocarbon Processing, Vol. 55, No. 3, 1976,
pp. 153-158.
Products, Lubricants, and Fossil Fuels (in 5 Vols.), ASTM
Research, Vol. 26, 1987, pp. 755-759. Corrections, p. 1268.
[29] Eubank, E T. and Wang, X., "Saturation Properties from
Equations of State," Industrial and Engineering Chemistry
[30] Wauquier, J.-E, Petroleum Refining, Vol. 1: Crude Oil,
Petroleum Products, Process Flowsheets, Editions Technip,
Pads, 1995.
[32] Felder, R. M. and Rousseau, R. W., Elementary Principles of
Chemical Processes, 2nd ed., Wiley, New York, 1986.
[33] Mang, R. A., "Clean Energy for the Hydrogen Planet," The
Globe and Mall, Canada's National Newspaper, June 9, 2003,
Page H1.
Characterization Data," Journal of Petroleum Science &
Engineering, 1999, Vol. 22, pp. 67-93.
[35] Riazi, M. R. and Whitson, C. H., "Estimating Diffusion
MNL50-EB/Jan. 2005
A pplications: Estimation
of Transport Properties
N OM EN C L A T UR E
AT
API
A,B,C,D,E
a, b , c, d, e
BI,~s
DA-mix
D ~ L
d
E
F
g
I
J Ay
Kw
k
kB
log10
In
M
m
m
A Helmholtz free energy defined in Eq. (6.7),
J/ mol
Adhesion tension (Eq. 8.85), N or dyne
Blending index for viscosity of liquid hydrocar-
bons (see Eq. 8.20), dimensionless
cs Velocity of sound, m/ s
DA Self diffusion coefficient of component A,
cm2/ s ( = 10 -4 m2/ s)
DAB Binary ( mutual) diffusion coefficient (diffusiv-
ity) of component A in B, cm2/s
Effective diffusion coefficient (diffusivity) of
component A in a mixture, cm2/s
Liquid binary diffusion coefficient (diffusivity)
of component A in B at infinite dilution
(X A -'+ 0), cm2/ s ( = 10 -4 m2/ s)
Molecular diameter, m (1 A = 10 -1~ m)
Activation energy (see Eq. 8.55), kcal/ mol
Format i on resistivity factor in a porous medi a
(see Eq. 8.72), dimensionless
Acceleration of gravity ( = 9.8 m2/s)
Refractive index par amet er defined in
Eq. (2.36) [ = (n 2 - 1)/(n 2 + 2)], dimensionless
Mass diffusion flux of component A in the y
direction (i.e., g/ cm 2. s)
Watson characterization factor defined by
Eq. (2.13)
Thermal conductivity, W/ m. K
Thermal conductivity of liquid at normal boil-
ing point, W/ re. K
Thermal conductivity of liquid at normal melt-
ing point, W/ re. K
Boltzman constant (=R/NA = 1.381 x 10 -23
J/K)
Common logarithm (base 10)
Natural logarithm (base e)
Molecular weight ( molar mass), g/ mol
(kg/kmol)
Mass of one molecule ( = M/NA), kg
Cementation factor, a paramet er characteristic
of a porous media, dimensionless (i.e., see Eqs.
(8.73) and (8.74))
NA Avogadro number = number of molecules in
one mole (6.022 x 1023 mo1-1)
Xp, XN, X A
3 2 9
Npr Prandtl number defined in Eq. (8.29), dimension-
less
n A paramet er in various equations
n Liquid refractive index at t emperat ure T and
1 atm, dimensionless
P Pressure, bar
Pa Parachor number (see Eq. 8.86)
( = P/Pc), dimensionless
p A dimensionless paramet er for use in Eq. (8.78)
Q A paramet er for use in Eq. (8.88)
qy Heat flux in the y direction, W/ m 2
R Gas constant = 8.31.4 J / mol - K (values in different
units are given in Section 1.7.24)
r Radius of capillary tube (see Eqs. (8.80)-(8.82))
SG Specific gravity of liquid substance at 15.5~
t Temperature related paramet er (i.e., see Eq. (8.37)
or (8.44))
( = T/Tc), dimensionless
TM Freezing (melting) point for a pure component at
1.013 bar, K
Tbr Reduced boiling point ( = Tb/Tc), dimensionless
V Molar volume, cma/ mol
VA Liquid mol ar volume of pure component A at nor-
mal boiling point, cm3/ mol
Vc Critical vol ume (molar), cm3/ mol (or critical spe-
cific volume, cm3/g)
Vi Molar vol ume of pure component i at T and P,
cm3/ mol
V~ Reduced volume ( = V/Vc), dimensionless
Vx Velocity of fluid in the x direction, m/ s
V~ Molar volume of liquid mixture, cma/ mol
x~ Mole fraction of component i in a mixture (usually
Xwi Weight fraction of component i in a mixture (usu-
ally used for liquids), dimensionless
Fractions (i.e., mole) of paraffins, naphthenes,
and aromatics in a pet rol eum fraction, dimension-
less
Yi Mole fraction of i in a mixture (usually used for
gases), dimensionless
Zc Critical compressibility factor [Z = PcVffRTc], di-
mensionless
Greek
0/
t~AB
O'so
O)
#
/Za
#c
#v
#r
p
v38000)
Letters
Thermal diffusivity ( = k/pCp), m2/s or cm2/s
A t her modynami c p ar amet er for noni deal i t y of a
liquid mi xt ure defined by Eq. (8.63), di mensi on-
less
ei Energy par amet er for comp onent i (see Eq. 8.57)
cI'ii Di mensi onl ess p ar amet er defined in Eq. (8.7)
Porosi t y of a p or ous medi a (Eq. 8.73), di mensi on-
less
Owo Oi l -wat er cont act angle, in degrees as used in Eq.
(8.84)
( mol ar density unit: cm3/ mol)
PM Mol ar density at a given t emperat ure and pressure,
mol / c m 3
Pr Reduced density ( = p/Pc = Vc/V), di mensi onl ess
po Oil density at a given t emperat ure, g/ cm 3
pw Wat er density at a given t emperat ure, g/ cm 3
a Mol ecul ar size paramet er, / ~ [1/ ~ = 10 -1~ m]
a Surface t ensi on of a liquid at a given t emperat ure,
dyn/ cm
an Surface t ensi on of a hydr ocar bon at a given temper-
ature, dyrdcm
awo I nterracial t ensi on of oil and wat er at a given temper-
ature, dyn/ cm
Surface t ensi on of wat er wi t h rock surface, dyn/ cm
Acentric fact or defined by Eq. (2.10), di mensi onl ess
Viscosity par amet er defined by Eq. (8.5), (cp) -1
Absolute viscosity, mPa- s (cp)
Viscosity at at mospheri c pressure, mPa. s (cp)
Critical viscosity, mPa. s (cp)
Viscosity at pressure P, mPa. s
Reduced viscosity (=/ z/ / zc), di mensi onl ess
Ki nemat i c viscosity ( = tz/p), cSt (10 -2 cm2/s)
Ki nemat i c viscosity of a liquid at 37.8~ (100~ cSt
( 10 -2 cm2/ s )
Mol ecul ar energy par amet er (i.e., see Eqs. 8.31 or
8.57)
F Paramet er defined in Eq. (8.38)
Paramet er defined in Eq. (8.34), m. s- mo1-1.
YA Activity coefficient of comp onent A in liquid solution
defined by Eq. (6.112), di mensi onl ess
~gs Association p ar amet er defined in Eq. (8.60), di men-
sionless
r Tortuosity, di mensi onl ess p ar amet er defined for pore
connect i on st ruct ure in a por ous medi a syst em (see
Eq. 8.71)
zyx x comp onent of moment um flux in the y direction,
N/ m 2
Jr A numeri cal const ant = 3,14159265
Superscri pt
ig Value of a propert y for comp onent "i" as ideal gas at tem-
perat ure T and P -~ 0
o Value of a propert y at low pressure (ideal gas state) con-
dition at a given t emper at ur e
Subscripts
A Value of a propert y for comp onent A
B Value of a propert y for comp onent B
b Value of a propert y at the nor mal boiling poi nt
c Value of a propert y at the critical poi nt
i,j Value of a propert y for comp onent i or j in a mi xt ure
L Value of a propert y for liquid phase
m Mixture propert y
od Value of a propert y for dead oil ( crude oil) at at mospheri c
pressure
r Reduced propert y
w Values of a propert y for wat er
Acronyms
API -TDB Ameri can Pet rol eum I nst i t ut e--Techni cal Dat a
Book (see Ref. [5])
BI P Bi nary i nt eract i on par amet er
bbl Barrel, uni t of vol ume of liquid as given in Sect i on
1.7.11.
cp Centipoise, uni t of viscosity, (1 cp = 0.01 p = 0.01
g. cm. s = 1 mPa- s = 10 -3 kg/ m. s)
cSt Centistoke, uni t of ki nemat i c viscosity, (1 cSt =
0.01 St = 0.01 cm2/s)
DI PPR Design I nstitute for Physical Propert y Dat a (see
Ref. [10])
EOS Equat i on of state
GLR Gas-to-liquid ratio
I FT I nterracial t ensi on
PNA Paraffins, napht henes, aromat i cs cont ent of a
pet rol eum fract i on
scf St andard cubi c foot ( unit for vol ume of gas at 1
at m and 60~
stb St ock t ank barrel ( unit for vol ume of lquid oil at 1
at m and 60~
IN THIS CHAFrER, appl i cat i on of various met hods present ed
in previ ous chapt ers is ext ended to estimate anot her t ype of
physical properties, namely, t ransport propert i es for vari ous
pet rol eum fract i ons and hydr ocar bon mixtures. Transport
propert i es generally include viscosity, t hermal conductivity,
and diffusion coefficient (diffusivity). These are mol ecul ar
propert i es of a subst ance t hat indicate the rat e at whi ch
specific ( per uni t vol ume) moment um, heat, or mass are
transferred. Science of the st udy of these processes is called
transport phenomenon. One good text t hat describes these
processes was wri t t en by Bird et al. [1]. The first edition
appeared in 1960 and remai ned a leading source for four
decades until its second publ i cat i on in 1999. A fourt h prop-
erty t hat also det ermi nes t ransport of a fluid is surface or
interracial t ensi on (IFT), whi ch is needed in calculations re-
lated to t he rise of a liquid in capillary t ubes or its rat e of
spreadi ng over a surface. Among these properties, viscosity is
consi dered as one of the most i mpor t ant physical propert i es
for cal cul at i ons related t o fluid flow followed by t hermal con-
ductivity and diffusivity. I nterracial t ensi on is i mpor t ant in
reservoir engi neeri ng calculations to det ermi ne t he rat e of oil
8. APPLICATIONS: ESTI M ATI ON OF TRANSPORT PROPERTIES 331
recovery and for process engineers it can be used to determine
foaming characteristics of hydrocarbons in separation units.
As was discussed in Chapter 7, properties of gases may
be estimated more accurately than can the properties of liq-
uids. Kinetic theory provides a good approach for develop-
ment of predictive methods for transport properties of gases.
However, for liquids more empirically developed methods are
used for accurate prediction of transport properties. Perhaps
combination of both approaches provides most reliable and
general methods for estimation of transport properties of flu-
ids. For petroleum fractions and crude oils, characterization
methods should be used to estimate the input parameters. It
is shown that choice of characterization method may have
a significant impact on the accuracy of predicted transport
property. Use of methods given in Chapters 5 and 6 on the
development of a new experimental technique for measure-
ment of diffusion coefficients in high-pressure fluids is also
demonstrated.
8 . 1 ES T I M A T I ON OF VI S C OS I T Y
Viscosity is defined according to the Newton's law of viscosity:
avx a @vx)
( 8. 1) ~yx=- ~- y =- ~ Oy ~=-p
where Zyx is the x component of flux of moment um in the y di-
rection in which y is perpendicular to the direction of flow x.
Velocity component in the x direction is V~. p is the density
and pV~ is the specific moment um ( moment um per unit vol-
ume). OV~/Oy is the velocity gradient or shear rate (with di-
mension of reciprocal of time, i.e., s -1) and 3(pVx)/Oy is gra-
dient of specific momentum, r represents tangent force to
the fluid layers and is called shear stress with the dimension
of force per unit area (same as pressure). Velocity is a vector
quantity, while shear stress is a tensor quantity. While pres-
sure represents normal force per unit area, r represents tan-
gent (stress) force per unit area, which is in fact the same as
moment um (mass x velocity) per unit area per unit time or
moment um flux. Thus/ x has the dimension of mass per unit
length per unit time. For example, in the cgs unit system it has
unit of g/ cm. s, which is called poise. The most widely used
unit for viscosity is centipoise (1 cp = 0.01 p = 10 -4 micro-
poise). The ratio of Ix/p is called kinematic viscosity and is
usually shown by v with the unit of stoke (cm2/s) in the cgs
unit system. The common unit for v is cSt (0.01 St), which
is equivalent to mm2/s. Liquid water at 20~ exhibits a vis-
cosity of about 1 cp, while its vapor at atmospheric pressure
has viscosity of about 0.01 cp. More viscous fluids (i.e., oils)
have viscosities higher than the viscosity of water at the same
temperature. Fluids that follow linear relation between shear
stress and shear rate (i.e., Eq. 8.1) are called Newtonian. Poly-
mer solutions and many heavy oils with large amount of wax
or asphahene contents are considered non-Newtonian and
follow other relations between shear stress and shear rate.
Viscosity is in fact a measure of resistance to motion and the
reciprocal of viscosity is called fluidity. Fluids with higher vis-
cosity require more power for their transportation. Viscosity
is undoubtedly the most important transport property and it
has been studied both experimentally and theoretically more
than other transport properties. I n addition to its direct use
for fluid-flow calculations, it is needed in calculation of other
properties such as diffusion coefficient. Experimental values
of gas viscosity at 1 atm versus temperature for several hydro-
carbon gases and liquids are shown in Fig. 8.1. As it is seen
from this figure, viscosity of liquids increases with molecular
weight of hydrocarbon while viscosity of gases decreases.
8. 1. 1 Vi scosi t y of Gas es
Viscosity of gases can be predicted more accurately than can
the viscosity of liquids. At low pressures (ideal gas condition)
viscosity can be well predicted from the kinetic theory of gases
[1, 3, 4].
2
(8.2) / ~- 3jr2/3 d2
where m is the mass of one molecule in kg (m = O.O01M/NA),
kB is the Boltzmann constant (= R/NA), and d is the molecular
diameter. I n this relation if m is in kg, kB in J/K, T in K, and d
in m, then/ x would be in kg/ m. s (1000 cp). This relation has
been obtained for hard-sphere molecules. Similar relations
can be derived for viscosity based on other relations for the
interrnolecular forces [ 1 ]. The well-known Chapman-Enskog
equations for transport properties of gases at low densities
(low pressure) are developed on this basis by using Lennard-
Jones potential function (Eq. 5.11). The relation is very sim-
ilar to Eq. (8.2), where/ x is proportional to (MT)I/2/(a2S2) in
which a is the molecular collision diameter and f2 is a func-
tion of kBT/~. Parameters a and E are the size and energy
parameters in the Lennard-Jones potential (Eq. 5.1 i). From
such relations, one may obtain molecular collision diameters
or potential energy parameters from viscosity data. At low
pressures, viscosity of gases changes with temperature. As
shown by the above equation as T increases, gas viscosity also
increases. This is mainly due to increase in the intermolecular
collision that is caused by an increase in molecular friction.
At high pressures, the behavior of the viscosity of gases and
liquids approach each other.
For pure vapor compounds, the following correlation was
developed by the API-TDB group at Penn State and is recom-
mended in the API-TDB for temperature ranges specified for
each compound [5]:
1000AT a
C D (8. 3) " = (1 + ~ + ~)
where correlation coefficients A-D are given for some selected
compounds in Table 8.1. The average error over the entire
temperature range is about 5% but usually errors are less
than 2%. This equation should not be applied at pressures in
which Pr > 0.6.
The following relation developed originally by Yoon and
Thodos [6] is recommended in the previous editions of API-
TDB and DIPPR manuals for estimation of viscosity of hy-
drocarbons as well as nonhydrocarbons and nonpolar gases
at atmospheric pressures:
/z~ x 10 s = 1 + 46.1T ~ - 20.4 exp(-0.449Tr)
(8.4) + 19.4 exp(-4.058Tr)
(8. 5)
1 1 2
= Tc 6 M-2 (0.987Pc)-~
3o0
250 9
u)
0
r
2 200'
u
E
"6
~ lo0
0
>
50'
o
200
YS
_._ .,,,"1 ~
J
f j
J
f
~oise = 0.0672E - 6 I b/ft s ]
[
0 200 400 600 8 00 1 0 0 0
Temperature, C
(a) Gases
O
r l
C
8
10
~2r
."1
"6
8
i
0. 1 L
\
\
X'x,,~
\
13-,
--~Metl~ane; ~.~. AEthanq
, ' I I
i[1 centipoise = 0.000672 I b/ft sj
" ~ i' ~ . N'%" ~., ~ IPr~ IButanq
0,01
-200 -150 -1 oo -50 0 50 1 oo 1 50 200
Temperature, C
(b) Liquids
FI G . 8 . 1 - - V i s cos i t y of sev eral l i g ht hydr ocar bons v er sus t emper at ur e at at mo-
spher i c pressure. T ak en wi t h per mi ssi on f rom Ref. [ 2].
wher e/ z is in cp, T~ is the reduced t emperat ure, and ~ is a
p ar amet er t hat has a di mensi on of inverse viscosity and is ob-
t ai ned from kinetic t heory of gases. The fact or 0.987 comes
from the original definition t hat uni t of at m was used for Pc. I n
t he above equat i on Pc, Tc, and M are in bar, kelvin, and g/reel,
respectively. I n cases where dat a on gas viscosity is available,
it woul d be mor e appropri at e t o det ermi ne ~ from viscosity
dat a rat her t han t o calculate it from the above equation. Reli-
ability of this equat i on is about 3-5%. For some specific com-
pounds such as hydrogen, t he numeri cal coefficients in Eq.
(8.4) are slightly different and in the same order as given in the
DI PPR manual are 47.65, - 20. 0, -0. 858, +19.0, and -3. 995.
I n t he 1997 edition of the API -TDB [5], the mor e commonl y
used correl at i on developed earlier by Stiel and Thodos [7] is
r ecommended:
/*~ = 3.4 x 10-4Tr TM for Tr _< 1.5
(8.6)
/,~ = 1.778 x 10-4(4.58Tr - 1.67) 0.625 f or T r > 1.5
where uni t s of ~ and ~ are the same as in Eq. (8.4). For de-
fined gas mi xt ures at low pressures, Eq. (8.6) may be used
wi t h To, Pc, and M calculated from Kay' s mixing rule (Eq. 7.1).
However, when viscosity of individual gases in a mi xt ure are
known, a mor e accurat e met hod of est i mat i on of mi xt ure vis-
cosity is provi ded by Wilke, whi ch can be applied for pressures
wi t h Pr < 0.6 [1, 5]:
(8.7)
N X/ / ~ i
#m = Y~
i = l ' ~N= I Xj~ij
# ~ij =~ l +M i } I + # \ M i . ] J
This semi empi ri cal met hod is r ecommended by bot h API-
TDB and DI PPR for cal cul at i on of viscosity of gas mi xt ures
of known composi t i on at low pressures. Accuracy of this
equat i on is about 3% [5, 8]. I n the above rel at i on dPii :# dPii.
8. APPL I CATI ONS: ESTI M ATI ON OF TRANSPORT PROPERTI ES 3 3 3
TABLE 8.1---Coefficients of Eq. (8.3) for viscosity of pure vapor compounds. (Taken with permission from Ref. [5].)
1000ArB Units: cp and kelvin (8.3)
/* = (l+Y.+TT_)
API No.
794
781
845
771
786
789
774
775
797
770
1
2
3
4
5
6
7
8
9
10
14
15
23
24
37
41
62
73
74
75
76
77
78
79
80
81
82
86
9O
101
102
103
109
146
147
148
156
157
158
168
192
193
194
198
204
292
322
335
336
337
Compound A B C D Trai n , K Tmax, K
Oxygen 1. 1010E- 06 5. 6340E-01 9. 6278E+0t 0. 0000E+00 54 1500
Hydrogen 1. 7964E-07 6. 8500E-01 - 5. 8889E- 01 1. 4000E+02 14 3000
Wat er 6. 1842E- 07 6. 7780E-01 8. 4722E+02 - 7. 4074E+04 273 1073
Ammoni a 4. 1856E- 08 9. 8060E-01 3. 0800E+01 0. 0000E+00 196 1000
Hydrogen sulfide 5. 8597E- 08 1. 0170E+00 3. 7239E+02 - 6. 4198E+04 250 480
Ni t rogen 6. 5593E- 07 6. 0810E- 01 5. 4711E+01 0. 0000E+00 63 1970
Car bon monoxi de 1. 1131E- 06 5. 3380E-01 9. 4722E+01 0. 0000E+00 68 1250
Car bon dioxide 2. 1479E- 06 4. 6000E- 01 2. 9000E+02 0. 0000E+00 194 1500
Sul fur trioxide 3. 9062E- 06 3. 8450E-01 4. 7011E+02 0. 0000E+00 298 694
Air 1. 4241E-06 5. 0390E-01 1.0828E+02 0. 0000E+00 80 2000
Par af f i ns
Met hane 5. 2553E- 07 5. 9010E-01 1. 0572E+02 0, 0000E+00 91 1000
Et hane 2. 5904E- 07 6. 7990E-01 9. 8889E+01 0. 0000E+00 91 1000
Propane 2. 4995E- 07 6. 8610E-01 1. 7928E+02 - 8, 2407E+03 86 1000
n-But ane 2. 2982E- 07 6. 9440E-01 2. 2772E+02 - 1, 4599E+04 135 1000
I sobut ane 6. 9154E- 07 5. 2140E-01 2. 2900E+02 0. 0000E+00 150 1000
n- Pent ane 6. 3411E- 08 8. 4760E-01 4. 1722E+01 0. 0000E+00 143 1000
I sopent ane 1. 1490E- 06 4. 5720E- 01 3. 6261E+02 - 4. 9691E+03 113 1000
Neopent ane 4. 8643E- 07 5. 6780E-01 2. 1289E+02 0, 0000E+00 257 1000
n-Hexane 1. 7505E-07 7. 0740E-01 1. 5711E+02 0, 0000E+00 178 1000
2-Met hyl pent ane 1. 1160E- 06 4. 5370E- 01 3. 7472E+02 0, 0000E+00 119 1000
n-Hept ane 6. 6719E- 08 8. 2840E-01 8. 5778E+01 0, 0000E+00 183 1000
2-Met hyl hexane 1. 0130E-06 4. 5610E- 01 3. 5978E+02 0. 0000E+00 155 1000
n-Oct ane 3. 1183E- 08 9. 2920E-01 5. 5089E+01 0. 0000E+00 216 1000
2-Met hyl hept ane 4. 4595E- 07 5. 5350E-01 2. 2222E+02 0. 0000E+00 164 1000
2, 2, 4-Tri met hyl pent ane 1. 1070E-07 7. 4600E-01 7. 2389E+01 0. 0000E+00 166 1000
n-Nonane 1. 0339E- 07 7. 7300E-01 2. 2050E+02 0. 0000E+00 219 i 000
n-Decane 2. 6408E- 08 9. 4870E-01 7. 1000E+01 0. 0000E+00 243 1000
n-Undecane 3. 5939E- 08 9. 0520E-01 1.2500E+02 0. 0000E+00 248 1000
n-Dodecane 6. 3443E- 08 8. 2870E-01 2. 1950E+02 0. 0000E+00 263 1000
n-Tridecane 3. 5581E- 08 8. 9870E-01 1.6528E+02 0. 0000E+00 268 1000
n-Tet radecane 4. 4566E- 08 8. 6840E-01 2. 2822E+02 4. 3519E+03 279 1000
n-Pent adecane 4. 0830E- 08 8. 7660E-01 2. 1272E+02 0. 0000E+00 283 1000
n-Hexadecane 1. 2460E-07 7. 3220E-01 3. 9500E+02 6. 0000E+03 291 1000
n-Hept adecane 3. 1340E- 07 6. 2380E-01 6. 9222E+02 0. 0000E+00 295 1000
n-Oct adecane 3. 2089E- 07 6. 1840E-01 7. 0889E+02 0. 0000E+00 301 1000
n-Nonadecane 3. 0460E- 07 6. 2220E-01 7. 0556E+02 0. 0000E+00 305 1000
n-Ei cosane 2. 9247E- 07 6. 2460E-01 7. 0278E+02 0. 0000E+00 309 1000
n-Tet racosane 2. 6674E- 07 6. 2530E-01 7. 0000E+02 0. 0000E+00 324 1000
n-Oct acosane 2. 5864E- 07 6. 1860E-01 6. 9833E+02 0. 0000E+00 334 1000
Naph t h enes
Cycl opent ane 2. 3623E- 07 6. 7460E-01 1. 3900E+02 0. 0000E+00 179 1000
Met hyl cycl opent ane 9. 0803E- 07 4. 9500E- 01 3. 5589E+02 0. 0000E+00 131 1000
Et hyl cycl opent ane 2. 1695E- 06 3. 8120E-01 5. 7778E+02 0. 0000E+00 134 1000
n-Propyl cycl opent ane 2. 6053E- 06 3. 4590E-01 5. 8556E+02 0. 0000E+00 156 1000
Cyclohexane 6. 7700E- 08 8. 3670E-01 3. 6700E+01 0. 0000E+00 279 900
Met hyl cycl ohexane 6. 5276E- 07 5. 2940E-01 3. 1061E+02 0. 0000E+00 147 1000
Et hyl cycl ohexane 4. 1065E- 07 5. 7140E-01 2. 3011E+02 0. 0000E+00 162 1000
n-Propylcyclohexane 9. 7976E- 07 4. 5420E- 01 3. 8589E+02 0. 0000E+00 178 1000
I sopropyl cycl ohexane 5. 7125E- 07 5. 2610E-01 2. 7989E+02 0. 0000E+00 184 1000
n-Butylcyclohexane 5. 3514E- 07 5. 2090E-01 2. 7711E+02 0. 0000E+00 198 1000
n-Oecylcyclohexane 3. 3761E- 07 5. 4480E-01 2. 0728E+02 0. 0000E+00 272 1000
Ol efi ns
Et hyl ene 2. 0793E- 06 4. 1630E- 01 3. 5272E+02 0. 0000E+00 169 1000
Propyl ene 8. 3395E- 07 5. 2700E-01 2. 8339E+02 0. 0000E+00 88 1000
1-Butene 1. 0320E-06 4. 8960E- 01 3. 4739E+02 0. 0000E+00 175 1000
l ~ 1. 6706E-06 4. 1110E- 01 4. 3028E+02 0. 0000E+00 108 1000
1-Hexene 1. 3137E-06 4. 3220E- 01 4. 0211E+02 0. 0000E+00 133 1000
Di ol efi ns and acet y l ene
1, 3-Butadiene 2. 6963E- 07 6. 7150E- 01 1. 3472E+02 0.0000E+O0 164 1000
Acetylene 1. 2019E-06 4. 9520E- 01 2. 9139E+02 0. 0000E+00 192 600
Aromati cs
Benzene 3. 1347E- 08 9. 6760E-01 7. 9000E+00 0.O000E+00 279 1000
Toluene 8. 7274E- 07 4. 9400E- 01 3. 2378E+02 0. 0000E+00 178 1000
Et hyl benzene 3. 8777E- 07 5. 9270E-01 2. 2772E+02 0. 0000E+00 178 1000
(Continued)
TABLE 8. 1- - ( Continued)
API No. Compound A B C D Tmi~, K Tr~ax, K
338 o-Xylene 3.8080E-06 3.1520E-01 7.7444E+02
339 m-Xylene 4.3098E-07 5.7490E-01 2.3861E+02
340 p-Xylene 5.7656E-07 5.3820E-01 2.8700E+02
341 n-Propylbenzene 1.6304E-06 4.1170E-01 5.4722E+02
349 n-Butylbenzene 9.9652E-07 4.6320E-01 4.3278E-02
371 n-Pentylbenzene 4.2643E-07 5.5740E-01 2.5900E+02
372 n-Hexylbenzene 5.5928E-07 5.1090E-01 2.8722E+02
373 n-Heptylbenzene 4.3188E-07 5.3580E-01 2.4561E-02
374 n-Octylbenzene 5.4301 E- 07 4.9890E- 01 2.7711E - 02
375 n-Nonylbenzene 4.8731E-07 5.0900E-01 2.6178E-02
376 n-Decylbenzene 4.6333E-07 5.1060E-01 2.5611E-02
377 n-Undecylbenzene 4.3614E-07 5.1410E-01 2.4761E-02
378 n-Dodecylbenzene 3.7485E-07 5.2390E-01 2.1878E-02
379 n-Tridecylbenzene 3.5290E-07 5.2760E-01 2.1039E-02
384 Styrene 6.3856E-07 5.2540E-01 2.9511E+02
342 Cumene 4.1805E-06 3.0520E-01 8.8000E+02
Di aromat i cs
427 Naphthalene 6.4323E-07 5.3890E-01 4.0022E+02
428 l-Methylnaphthalene 2.6217E-07 6.4260E-01 2.3522E+02
474 Anthracene 7.3176E-08 7.5320E-01 1.0000E+00
475 Phenanthrene 4.3474E-07 5.2720E-01 2.3828E+02
Aromati cs ami nes
746 Pyridine 5.2402E-08 9.0080E-01 6.2722E+01
749 Quinoline 1.3725E-06 4.8350E-01 9.2389E+02
Sul fur
776 Carbonyl sulfide 2.2405E-05 2.0430E-01 1.3728E+03
828 Methyl mercaptan 1.6372E-07 7.6710E-01 1.0800E+02
891 Thiophene 1.0300E-06 5.4970E-01 5.6944E+02
892 Tetrahydrothiophene 1.6446E-07 7.4400E-01 1.4472E+02
Al cohol s
709 Methanol 3.07E-007 6.9650E-001 2.0500E+02
710 Ethanol 1.06E-006 8.0660E-001 5.2700E+02
712 Isopropanol 1.99E-007 7.2330E-001 1.7800E+02
766 Methyl-ten-butyl ether 1.54E-007 7.3600E-001 1.0822E+02
0.0000E+00 248 1000
0.0000E+00 226 1000
0.0000E+00 287 1000
0.0000E+00 173 1000
0.0000E+00 186 1000
0.0000E+00 198 1000
0.0000E+00 212 1000
0.0000E+00 225 1000
0.0000E+00 237 1000
0.0000E+00 249 1000
0.0000E+00 259 1000
0.0000E+00 268 1000
0.0000E+00 276 1000
0.0000E+00 283 1000
0,0000E+00 243 1000
0.0000E+00 177 1000
0.0000E+00 353 1000
0.0000E+00 243 1000
0.0000E+00 489 1000
0.0000E+00 372 1000
0.0000E+00 232 1000
-6.7901E+04 511 1000
0.0000E+00 134 1000
0.0000E+00 150 1000
0.0000E+00 235 1000
0.0000E+00 394 1000
0.0000E+00 240 1000
0.0000E 200 1000
0.0000E+00 186 1000
0.0000E+00 164 1000
A si mp l er ver si on of Eq. (8.7) for a gas mi xt ur e is gi ven as [9]:
( 8. 8) /J~om -- N
~i=1 xi~
wher e N is t he t ot al number of comp ounds i n t he mi x-
t ure, r = MiX/z, and subscr i p t o i ndi cat es l ow p r essur e ( at-
mosp her i c and bel ow) whi l e subscr i p t m i ndi cat es mi xt ur e
propert y. By assumi ng q~i = 1 t hi s equat i on r educes t o Kay' s
mi xi ng rul e (#m = ~ xitzi), whi ch usual l y gives a r easonabl y
accep t abl e r esul t at very l ow pressure.
Pr essur e has a good effect on t he vi scosi t y of real gases and
at a const ant t emp er at ur e wi t h i ncr ease in p r essur e vi scosi t y
al so i ncreases. For si mpl e gases at hi gh pr essur es, r educed
vi scosi t y (/~r) is usual l y cor r el at ed to Tr and Pr bas ed on t he
t heor y of cor r esp ondi ng st at es [1]. /Zr is defi ned as t he r at i o
of tz/tzc, wher e/ zc is cal l ed cri t i cal vi scosi t y and r ep r esent s
vi scosi t y of a gas at its cri t i cal p oi nt (Tc and Pc).
3 1 2
(8.9) /Zc = 6.16 10- (MTc)~(V c)-~
(8.10) /z~ = 7.7 10-4~ -1
I n t he above rel at i ons, / zr is i n cp, Tc i n kelvin, Vc is in cm3/ mol,
and ~ is defi ned by Eq. (8.5). Equat i on (8.10) can be obt ai ned
by combi ni ng Eqs. (8.9) and (8.5) wi t h Eq. (2.8) assumi ng
Z~ = 0.27. I n some pr edi ct i ve met hods, r educed vi scosi t y is
defi ned wi t h r esp ect to vi scosi t y at at mos p her i c p r essur e (i.e.,
/Zr = IZ//Za), where/ ~a is t he vi scosi t y at I at m and t emp er at ur e
T at whi ch/ z mus t be cal cul at ed. Anot her r educed f or m of
vi scosi t y is (/z -/Za)~, whi ch is al so cal l ed as r esi dual vi scosi t y
( si mi l ar to r esi dual heat capaci t y) and is usual l y cor r el at ed t o
t he r educed densi t y (Pr = P/Pc = Vc/V). F or p ur e hydr ocar bon
gases at hi gh p r essur es t he fol l owi ng met hod is r ecommended
in t he API -TDB [5]:
(/s -- # a ) ~ ---- 1. 08 X 10 - 4 [exp (1.439p~) - exp ( - 1. 1 lp1S58)]
(8.11)
The s ame equat i on can he ap p l i ed to mi xt ur es i f To, Pc, M, and
Vc of t he mi xt ur e are cal cul at ed f r om Eq. (7.1). V or p can be
est i mat ed f r om met hods of Chap t er 5. F or mi xt ur es, in cases
t hat t her e is at l east one dat a p oi nt on tz, it can be used to ob-
t ai n IZa r at her t han to use its est i mat ed val ue. Equat i on (8.11)
may al so be used for nonp ol ar nonhydr ocar bons as r ecom-
mended in t he DI PPR manual [ 10]. However, i n t he API -TDB
anot her gener al i zed cor r el at i on for nonhydr ocar bons is gi ven
in t he f or m of tz/tZa versus Tr and Pr wi t h some 22 numer i cal
const ant s. The advant age of t hi s met hod is mai nl y si mpl i ci t y
i n cal cul at i ons si nce t her e is no need to cal cul at e Pr and IZ can
be di rect l y cal cul at ed t hr ough ~a and Tr and Pr.
I n t he p et r ol eum i ndust r y one of t he mos t wi del y used cor-
r el at i ons for est i mat i on of vi scosi t y of dense hydr ocar bons is
p r op os ed by J ossi et al. [1 l] :
[ ( / z - - / Zo) ~ -a t- 10- 4] ~ = 0.1023 + 0.023364pr + 0.058533Pr 2
(8.12) - 0.040758p~ + 0. 0093324p 4
8. APPLICATIONS: ESTIM ATION OF TRANSPORT PROPERTIES 335
This equat i on is, in fact, a modi fi cat i on of Eq. (8.11) and
was originally developed for nonp ol ar gases in t he range of
0.1 < Pr < 3. /Zo is t he viscosity at low pressure and at t he
same t emperat ure at whi ch/ z is to be calculated./ Zo may be
cal cul at ed from Eqs. (8.6)-(8.8). However, this equat i on is
also used by reservoir engineers for the cal cul at i on of t he vis-
cosity of reservoir fluids under reservoir condi t i ons [9, 12].
Lat er Stiel and Thodos [13] pr oposed similar correl at i ons for
the residual viscosity of pol ar gases:
(/z -/ Zo) ~ = 1.656 x 10-4p TM f or p r _< 0.1
(/z - / ~o) ~ = 6.07 10 -6 (9.045pr + 0.63) 1"739
(8.13) for 0.1 < Pr < 0.9
loga0 {4 -- log10 [(/~ -- #o) x 104~]} = 0.6439 -- 0.1005pr
for 0.9 < Pr < 2.2
These equat i ons are mai nl y r ecommended for cal cul at i on of
viscosity of dense pol ar and nonhydr ocar bon gases. At hi gher
reduced densities accur acy of Eqs. (8.1 i) -( 8. 13) reduces.
For undefi ned gas mi xt ures wi t h known mol ecul ar wei ght
M, t he following relation can be used to estimate viscosity at
t emperat ure T [5]:
#go = - 0. 0092696 + ~( 0. 001383 - 5.9712 x 10- s v~)
(8.14) + 1.1249 x lO-5M
where T is in kelvin and ~go is the viscosity of gas at low pres-
sure in cp. Reliability of this equat i on is about 6% [5]. There
are a number of empi ri cal correl at i ons for cal cul at i on of vis-
cosity of nat ural gases at any T and P; one widely used cor-
relation was proposed by Lee et al. [14]:
]~g = 10-4A [exp (B x pC)]
A = [(12.6 + 0.021M) T 15] / (116 + 10.6M + T)
(8.15) 548
B = 3.45 + 0. 01M+ - -
T
C = 2.4 - 0.2B
where/ ~g is the viscosity of nat ural gas in cp, M is t he gas
mol ecul ar weight, T is absolute t emperat ure in kelvin, and p
is t he gas density in g/ cm 3 at the same T and P t hat ~g shoul d
be calculated. This equat i on may be used up to 550 bar and
in the t emperat ure range of 300-450 K. For cases where M is
not known, it may be calculated f r om specific gravity of t he
gas as di scussed in Chapt er 3 ( M = 29 SGg). For sour nat ural
gases, correl at i ons in t erms of H2S cont ent of nat ural gas are
available in handbooks of reservoir engi neeri ng [ 15, 16].
8.1.2 Viscosity of Liquids
Met hods for the predi ct i on of t he viscosity of liquids are less
accurat e t han the met hods for gases, especially for the estima-
t i on of viscosity of undefi ned pet rol eum fractions and crude
oils. Errors of 20-50% or even 100% in predi ct i on of liquid
viscosity are not unusual . Crude oil viscosity at r oom t emper-
at ure varies f r om less t han 10 cp (light oils) t o many t hou-
sands of cp (very heavy oils). Usually convent i onal oils wi t h
API gravities f r om 35 to 20 have viscosities f r om 10 to 100 cp
and heavy crude oils wi t h API gravities from 20 to 10 have
viscosities f r om 100 to 10000 cp [17]. Most of the met hods
developed for est i mat i on of liquid viscosity are empi ri cal in
nature. An approxi mat e t heory for liquid t ransport propert i es
is t he Eyri ng rat e t heory [1, 4]. Effect of pressure on t he liq-
uid viscosity is less t han its effect on viscosity of gases. At low
and moder at e pressure, liquid viscosity may be consi dered as
a funct i on of t emperat ure only. Viscosity of liquids decreases
wi t h increase in t emperat ure. Accordi ng to t he Eyri ng rat e
model t he following relation can be derived on a semi t heo-
retical basis:
Ngh f 3.8Tb'~
(8.16) tt = - ~- exp ~- - )
where/ ~ is t he liquid viscosity in posie at t emperat ure T, NA is
the Avogadro number (6.023 x 1023 gmol-1) , h is t he Planck' s
const ant (6.624 x 10 -27 g. c m2/ s ) , V is the mol ar vol ume at
t emperat ure T in cma/ mol, and Tb is t he nor mal boiling point.
Bot h Tb and T are in kelvin. Equat i on (8.16) suggests t hat
In/~ versus 1/T is linear, whi ch is very similar t o the Cl asi us-
Cl apeyron equat i on (Eq. 7.27) for vapor pressure. More ac-
curat e correl at i ons for t emperat ure dependency of liquid vis-
cosities can be obt ai ned based on a mor e accurat e relation
for vapor pressure. I n t he API-TDB [5] liquid viscosity of pure
comp ounds is correl at ed accordi ng to the following relation:
(8.17) / z= IO00exp(A + B/ T +Cl nT + DT E)
where T is in kelvin and/ z is in cp. Coefficients A-E for a
number of comp ounds are given in Table 8.2 [5]. Liquid vis-
cosity of some n-alkanes versus t emperat ure calculated f r om
Eq. (8.17) is shown in Fig. 8,2. Equat i on (8.17) has uncer-
t ai nt y of bet t er t han over the entire t emperat ure ranges
given in Table 8.2. I n most cases t he errors are less t han 2%
as shown in the API -TDB [5].
For defined liquid mi xt ures the following mixing rules are
r ecommended in t he API -TDB and DI PPR manual s [5, 10]:
(i )3
/Zr n = X/IZ~/3 f or liquid hydr ocar bons
( 8. t8)
N
In/ z m = ~ x~ In/zi for liquid nonhydr ocar bons
i=1
where /~m is the mi xt ure viscosity in cp and x~ is the mol e
fract i on of comp onent i wi t h viscosity /zi. There are some
ot her mi xi ng rules t hat are available in the literature for liquid
viscosity of mi xt ures [ 18].
For liquid pet rol eum fractions ( undefined mixtures) , usu-
ally ki nemat i c viscosity v is ei t her available f r om experi men-
tal measur ement s or can be est i mat ed f r om Eqs. ( 2. 128) -
(2.130), at low pressures and t emperat ures. The following
equat i on developed by Si ngh may also be used to estimate
v at any T as r ecommended in the API -TDB [5]:
log10 (vr) A ( 311) B
= - - - 0.8696
(8.19) A = log10 (v3s000)) + 0.8696
B = 0.28008 x log~0 (v38(100)) + 1.8616
where T is in kelvin and v38000) is the ki nemat i c viscosity at
100~ (37.8~ or 311 K) in cSt, whi ch is usually known from
experiment. The average error for this met hod is about 6%.
For bl endi ng of pet rol eum fractions the simplest met hod is
API No.
TABLE 8.2---Coefficients of Eq. (8.17) for viscosity of pure liquid compounds. (Taken with permission from Ref. [5].)
lz= IO00exp(A + B/ T +C lnT + DT E) (8.17)
Compound A B C D E Train, K Tmax, K
794 Oxygen - 4. 1480E+00 9. 4039E+01 - 1. 2070E+00 0. 0000E+00 0. 0000E+00
781 Hydrogen - 1. 1660E+01 2. 4700E+01 - 2. 6100E- 01 - 4. 1000E- 16 1.0000E+01
845 Wat er - 5, 2840E+01 3. 7040E+03 5. 8660E+00 - 5. 8791E- 29 1.0000E+01
771 Ammoni a - 6. 7430E+00 5. 9828E+02 - 7. 3410E- 01 - 3. 6901E- 27 1.0000E+01
786 Hydrogen sulfide - 1.0900E+01 7. 6211E+02 - 1. 1860E-01 0. 0000E+00 0. 0000E+00
798 Ni t rogen 1.6000E+01 - 1. 8160E+02 - 5. 1550E+00 0, 0000E+00 0. 0000E+00
775 Car bon dioxide 1.8770E+01 - 4. 0290E+02 - 4. 6850E+00 - 6. 9999E- 26 1.0000E+01
Paraffins
1 Met hane - 6. 1570E+00 1. 7810E+02 - 9. 5240E- 01 - 9. 0611E- 24 1.0000E+01
2 Et hane - 3. 4130E+00 1. 9700E+02 - 1. 2190E+00 - 9, 2022E- 26 1.0000E+01
3 Propane - 6. 9280E+00 4. 2080E+02 - 6. 3280E- 01 - 1, 7130E- 26 1,0000E+01
4 n-But ane - 7. 2470E+00 5. 3480E+02 - 5. 7470E- 01 - 4, 6620E- 27 1.0000E+01
5 I sobut ane - 1. 8340E+01 1.0200E+03 1. 0980E+00 - 6. 1001E- 27 1.0000E+01
6 n-Pent ane - 2. 0380E+01 1. 0500E+03 1. 4870E+00 - 2. 0170E- 27 1.0000E+01
7 I sopent ane - 1.2600E+01 8. 8911E+02 2. 0470E- 01 0. 0000E+00 0. 0000E+00
8 Neopent ane - 5. 6060E+01 3. 0290E+03 6. 5860E+00 0. 0000E+00 0. 0000E+00
9 n-Hexame - 2. 0710E+01 1.2080E+03 1. 4990E+00 0. 0000E+00 0. 0000E+00
10 2-Met hyl pent ane - 1. 2860E+01 9. 4689E- 04 2. 6190E-01 0. 0000E+00 0. 0000E+00
14 n-Hept ane - 2. 4450E+01 1. 5330E+03 2. 0090E+00 0. 0000E+00 0. 0000E+00
15 2-Met hyl hexane - 1. 2220E+01 1.0210E+03 1. 5190E-01 0. 0000E+00 0. 0000E+00
23 n-Oct ane - 2. 0460E+01 1, 4970E+03 1. 3790E+00 0. 0000E+00 0. 0000E+00
24 2-Met hyl hept ane - 1. 1340E+01 1.0740E+03 1. 3050E-02 0. 0000E+00 0. 0000E+00
37 2, 2, 4-Tri met hyl pent ane - 1.2770E+01 1. 1300E+03 2. 3460E-01 - 3. 7069E- 28 1.0000E+01
41 n-Nonane - 2. 1150E+01 1.6580E+03 1. 4540E+00 0. 0000E+00 0. 0000E+00
62 n-Decane - 1.6470E+01 1.5340E+03 7. 5110E-01 0. 0000E+00 0. 0000E+00
73 n-Undecane - 1.9320E+01 1. 7930E+03 1. t 430E+00 0. 0000E+00 0. 0000E+00
74 n-Dodecane - 2. 1386E+05 1.9430E+03 1. 3200E+00 0. 0000E+00 0. 0000E+00
75 n-Tridecane - 2. 1010E+01 2. 0430E+03 1. 3690E+00 0. 0000E+00 0. 0000E+00
76 n-Tet radecane - 2. 0490E+01 2. 0880E+03 1. 2850E+00 0. 0000E+00 0. 0000E+00
77 n-Pent adecane - 1.9300E+01 2. 0890E+03 1. 1090E+00 0. 0000E+00 0. 0000E+00
78 n-Hexadecane - 2. 0180E+01 2. 2040E+03 1. 2290E+00 0. 0000E+00 0. 0000E+00
79 n-Hept adecane - 1.9990E+01 2. 2450E+03 1. 1980E+00 0. 0000E+00 0. 0000E+00
80 n-Oct adecane - 2. 2690E+01 2. 4660E+03 1. 5700E+00 0. 0000E+00 0. 0000E+00
81 n-Nonadecane - 1.63995E+01 2. 1200E+03 6. 8810E-01 0. 0000E+00 0. 0000E+00
82 n-Ei cosane - 1. 8310E+01 2. 2840E+03 9. 5480E- 01 0. 0000E+00 0. 0000E+00
86 n-Tetracosane - 2. 0610E+01 2. 5360E+03 1. 2940E+00 - 7. 0442E- 30 1,0000E+01
Naphthenes
101 Cycl opent ane - 3. 2610E+00 6. 1422E+02 - 1. 1560E+00 0. 0000E+00 0. 0000E+00
102 Met hyl cycl opent ane - 1. 8550E+00 6. 1261E+02 - 1. 3770E+00 0. 0000E+00 0. 0000E+00
103 Et hyl cycl opent ane - 6. 8940E+00 8. 1861E+02 - 5. 9410E- 01 0. 0000E+00 0. 0000E+00
109 n-Propyl cycl opent ane - 2. 3300E+01 1.6180E+03 1. 8470E+00 0. 0000E+00 0. 0000E
110 I sopropyl cycl opent ane - 1.0500E+01 1.0840E+03 - 8. 2650E- 02 0. 0000E+00 0. 0000E+00
146 Cyclohexane - 6. 9310E+01 4. 0860E+03 8. 5250E+00 0. 0000E+00 0. 0000E+00
147 Met hyl cycl ohexane - 1. 5920E+01 1.4440E+03 6. 6120E-01 2. 1830E- 27 1.0000E+01
148 Et hyl cycl ohexane - 2. 2110E+01 1.6730E+03 1. 6410E+00 0. 0000E+00 0. 0000E+00
156 n-Propylcyclohexane - 3. 1230E+01 2. 1790E+03 2. 9730E+00 0. 0000E+00 0. 0000E+00
158 n-But yl cydohexane - 3. 9820E+01 2. 6870E+03 4. 2270E+00 0. 0000E+00 0. 0000E+00
168 n-Decylcyclohexane - 2. 7670E+01 2. 9210E+03 2. 1910E+00 0. 0000E+00 0. 0000E+00
Ol efl ns
192 Et hyl ene 1. 8880E+00 7. 8861E+01 - 2. 1550E+00 0. 0000E+00 0. 0000E+00
193 Propyl ene - 9. 1480E+00 5. 0090E+02 - 3. 1740E- 01 0, 0000E+00 0. 0000E+00
Di ol ef ms a nd acet yl enes
322 Acetylene 6. 2240E+00 - 1. 5180E - 2. 6550E 0. 0000E 0. 0000E+00
Aromatics
335 Benzene - 7. 3700E+00 1. 0380E+03 - 6. 1810E- 01 - 1. 1020E-28 1.0000E+01
336 Toluene - 6. 0670E+01 3. 1490E+03 7. 4820E+00 - 5. 7092E- 27 1.0000E+01
337 Et hyl benzene - 1.0450E+01 1. 0480E+03 - 7. 1500E- 02 0. 0000E+00 0. 0000E+00
338 o-Xylene - 1. 5680E+01 1.4040E+03 6. 6410E- 01 0. 0000E+00 0. 0000E+00
341 n-Propyl benzene - 1. 8280E+01 1.5500E+03 1. 0450E+00 0. 0000E+00 0. 0000E+00
349 n-But yl benzene - 2. 3800E+01 1. 8870E+03 1. 8480E+00 0. 0000E+00 0. 0000E+00
371 n-Pent yl benzene - 7. 8290E+01 4. 4840E+03 9. 9270E+00 - 2. 3490E- 27 1.0000E+01
372 n-Hexyl benzene - 8. 8060E+01 5. 0320E+03 1.1360E+01 - 2. 6390E- 27 1.0000E+01
373 n-Hept yl benzene - 9. 5724E+01 5. 4770E+03 1.2480E+01 - 2. 8510E- 27 1.0000E+01
374 n-Oct yl benzene - 9. 4614E+01 5. 5678E+03 1.2260E+01 - 1. 8370E- 27 1.0000E+01
54 150
14 33
273 646
196 393
188 350
63 124
219 304
91 188
91 300
86 360
135 420
190 400
143 465
150 310
257 304
178 343
119 333
183 373
155 363
216 399
164 391
166 541
219 424
243 448
248 469
263 489
268 509
279 528
283 544
291 564
295 576
301 590
305 603
309 617
324 793
225 325
248 353
253 378
200 404
162 399
285 354
200 393
200 405
248 430
253 454
272 420
104 250
88 320
204 384
279 545
178 384
248 413
248 418
200 432
200 457
220 478
220 499
336 519
237 538
(Con~nued)
API No. Comp ound A
8. APPL I CATI ONS: ESTI M ATI ON OF TRANSPORT PROPERTI ES 337
TABLE 8. 2--(Continued)
B C D E Train, K Tmax, K
375 n-Nonylbenzene 1.0510E+02
376 n-Decylbenzene 1.0710E+02
377 n-Undecylbenzene - 1.0260E+02
378 n-Dodecylbenzene 8.8250E+01
379 n-Tridecylbenzene 4.5740E
383 Cyclohexylbenzene -4.3530E+00
386 Styrene -2.2670E+01
342 Cumene -2.4962E+01
Diaromatics and condensed rings
427 Naphthalene 1.9310E+01
472 Acenaphthene 2.0430E+01
473 Fluorene 4.1850E+00
474 Anthracene 2.7430E+02
709 Methanol 1.2135E+04
710 Ethanol 7.8750E+00
6.1272E+03 1.3820E+01 -2. 8910E-27 1.0000E+01 360 555
6.3311E+03 1.4080E -2. 7260E-27 1.0000E+01 253 571
6.2200E+03 1.3380E+01 -2. 4450E-27 1.0000E+01 258 587
5.6472E+03 1.1230E+01 -1. 8200E-27 1.0000E+01 268 601
3.6870E+03 4.9450E+00 -5. 8391E-28 1.0000E 328 614
1.4700E+03 - 1.1600E+00 0.0000E+00 0.0000E+00 280 513
1.7580E+03 1.6700E+00 0.0000E+00 0.0000E+00 243 418
1.8079E+03 2.0556E+00 0.0000E+00 0.0000E+00 200 400
1.8230E+03 1.2180E+00 0.0000E+00 0.0000E+00 353 633
1.0380E+02 -4.6070E+00 0.0000E+00 0.0000E+00 367 551
7.2328E+02 -2.1490E+00 0.0000E+00 0.0000E+00 388 571
2.1060E+04 3.6180E+01 0.0000E+00 0.0000E+00 489 595
1.7890E+03 2.0690E+04 0.0000E+00 0.0000E+00 176 338
7.8200E+02 -3.0420E+00 0.0000E+00 0.0000E+00 200 440
to use Eq. (3.105) by calculating bl endi ng index of t he mix-
ture. The viscosity-blending index can be calculated from t he
following relation proposed by Chevron Research Comp any
[193:
B I , ~ - l og l 0 v
(8.20) 3 + log10 v
BImix = Y~. xviBIi
in whi ch v is t he ki nemat i c viscosity in cSt. Once v is de-
t ermi ned absolute viscosity of a pet rol eum fract i on can be
est i mat ed from density ( # -- p x v). I t shoul d be not ed t hat
Eqs. (2.128)-(2.130) or Eqs. (8.19) and (8.20) are not suitable
for pure hydrocarbons.
To consi der t he effect of pressure on liquid viscosity of
hydrocarbons, the t hree-paramet er correspondi ng states
correl at i ons may be used for predi ct i on of viscosity of high-
pressure liquids [5]:
(8.21) //Jr = /Z ~ [ / ~r] ( 0) "}- co [ / Zr] ( 1)
where [#r] (~ and [/x~] (1) are funct i ons of Tr and Pr. These func-
tions are given in the API-TDB [5] in the form of pol ynomi al s
in t erms of Tr and Pr wi t h mor e t han 70 numeri cal constants.
10
n-Pentane
", - - ......... n-Decane
. . . . . n-Eicosane
\ ~ " " - - - - - Cyclohexane
1
\ . . ~. 4- . - - Benzene
"~ " ' \ ~ ........ . m Water
;_q 0A
0.01 ~ ~
200 300 400 500 600 700
Temperature, K
FIG. 8 .2--Liquid viscosity of several compounds
versus temperature at atmospheric pressure.
More recent l y a correspondi ng state correl at i on similar t o
this equat i on was proposed for est i mat i on of viscosity of hy-
dr ocar bon fluids at elevated pressures in whi ch t he reduced
mol ar refract i on ( paramet er r defined by Eq. 5.129) was used
instead of co [20], Paramet ers [/zr] (~ and [/zr] (1) have been
correl at ed t o Tr and Pr. Results show t hat for hydr ocar bon
systems, par amet er co can be repl aced by r in the correspond-
ing states correlations. Such correl at i ons have hi gher power
of ext rapol at i on to heavier hydrocarbons. Moreover, param-
eter r can be accurat el y calculated for heavy pet rol eum frac-
tions and undefi ned hydr ocar bon mixtures as di scussed in
Section 5.9.
Equat i on (8.21) is r ecommended for l ow-mol ecul ar-wei ght
hydr ocar bons [5]. For such systems, Jossi' s correl at i on
(Eq. 8.12) can also be used for cal cul at i on of viscosity of high-
pressure liquids. However, this appr oach is not appropri at e
for heavy or hi gh-mol ecul ar-wei ght liquid hydr ocar bons and
their mixtures. For such liquids the Kouzel correl at i on is rec-
ommended in the API -TDB [5]:
( / Zp) P- 1. 0133 ( _1. 48+5. 86, 0. 181)
(8.22) log10 ~a -- 10000
where P is pressure in bar and #a is low-pressure (1 atm) vis-
cosity at a given t emperat ure in cp. / zr is the viscosity at pres-
sure P and given t emperat ure in cp. The maxi mum pressure
for use in the above equat i on is about 1380 bar ( ~20000 psi)
and average error is about 10% [5].
When a gas is dissolved in a pure or mi xed liquid hydrocar-
bons viscosity of solution can be calculated from viscosity of
gas-free hydr ocar bon (/za) and gas-to-liquid ratio (GLR) usi ng
the following rel at i on [5]:
P~ _-- / _I.651(GLR) + 137#~/3 ~_ 538.4 / 3
~a | ,~/31137 + 4.891( GLR) ] + 538.4 ]
(8.23)
log 0 ( )
where both/ Z m and/ z a are at 37.8~ (100~ in cp and GLR is in
m 3/m3. # r is the viscosity of solution at t emperat ure T, where
T is in kelvin. This equat i on shoul d not be used for pressures
above 350 bar. If/Za at 37.8~ (100~ is not available, it may
be estimated; however, if/xa at the same t emperat ure at whi ch
/x is to be calculated is available then/ x may be est i mat ed f r om
/g = A(Ix,) B, where A and B are funct i ons of GLR (see Prob-
l em 8.4). GLR were calculated from the following relation:
379xA
(8.24) GLR -----
MB
where XA is t he mol e fract i on of dissolved gas in liquid, MB
is mol ecul ar wei ght of liquid, and SGB is the specific grav-
ity of liquid. I n this rel at i on GLR is cal cul at ed as st m 3 of
gas/ stm 3 of liquid (1 m3/ m 3 = 1 scf/ st, ft 3 = 5.615 scf/bbl).
Units of GLR are di scussed in Section 1.7.23. Predi ct i on of
viscosity of crude oils (gas free dead oils at 1 atm) is quite dif-
ficult due to compl exi t y of mixtures. However, t here are many
empi ri cal correl at i ons developed for cal cul at i on of crude oils
[15, 16]. For example t he Glaso' s correl at i on for viscosity of
crude oils is given as
/god = (3.141 x 1010) [ ( 1. 8T- 460) -3"444] [logl0(API)] ~
n = 10.313 [logl0(1.8T - 460)] - 36.447
(8.25)
where/ zoo is t he viscosity of dead oil (gas free at 1 atm.), T is
t emperat ure in kelvin, and API is t he oil gravity. This equat i on
shoul d be used for crude oils wi t h API gravity in t he range of
20-48 and in t he t emperat ure range of 283-422 K (50-300~
More advanced and accurat e met hods of cal cul at i on of viscos-
ity of crude oils is based on splitting t he oil into several pseu-
docomp onent s and t o use met hods di scussed in Chapt er 4
for cal cul at i on of the mi xt ure properties. Accurat e predi ct i on
of viscosities of heavy crude oils is a difficult task and most
correl at i ons result in large errors and errors of 50-100% are
quite c ommon in such predictions.
As seen f r om Eqs. (8.11) and (8.25), viscosity of liquids
and oils is mai nl y related to density. I n general, heavier oils
( lower API gravity) exhibit hi gher viscosity. Pure hydrocar-
bon paraffins have viscosity of about 0.35 cp (0.5 cSt.), naph-
t henes about 0.6 cp, n-alkylbenzenes ( aromat i cs) about 0.8 cp
(I.1 cSt.), gasoline about 0.6 cp, kerosene about 2 cp, and
residual oils' viscosity is in the range of 10-100 000 cp [17].
The met hods of measur ement of viscosity of oils are given in
ASTM D 445 and D 446. A graphi cal met hod for cal cul at i on
of viscosity of t he bl end is given by ASTM D 341. For light
oils capillary vi scomet ers are suitable for measuri ng liquid
viscosity in whi ch viscosity is proport i onal to t he pressure
difference in t wo tubes.
Most recently Riazi et al. [21] developed a relation for es-
t i mat i on of viscosity of liquid pet rol eum fract i ons by usi ng
refractive index at 20 ~ C as one of the i nput paramet ers in addi-
t i on to mol ecul ar wei ght and boiling poi nt (see Probl em 8.3).
Anot her devel opment on t he predi ct i on of viscosity and ot her
t ransport propert i es for liquid hydr ocar bon systems was t o
use refractive index t o estimate a t ransport propert y at the
same t emper at ur e in whi ch relative index is available. The-
ory of Hi l debrand [22] suggests t hat fluidity (1//g) of a liquid
is proport i onal to t he free space bet ween the molecules.
(8.26) --/gl = E ( ~0V0)
where E is a const ant , V is t he liquid vol ume (i.e., molar) , and
V0 is t he value of V at zero fluidity (/g -+ 0). Paramet ers E and
V0 may be det ermi ned f r om regressi on of experi ment al data.
The t erm (V -V o) represent s t he free space bet ween molecules.
As t emperat ure increases V also increases and/ g decreases.
This t heory is applicable t o liquids at low pressures. I n Chap-
ter 2 it was shown t hat par amet er I (defined by Eq. 2.36) is
proport i onal wi t h fract i on of liquid occupi ed by molecules.
Therefore p ar amet er I is proport i onal t o Vo/V and t hus
(8.27) /g-1 = C (1-1 - 1)
where/ g and I are evaluated at given t emperat ure. Met hods of
cal cul at i on of I were di scussed in Chapt er 2 (see Eqs. (2.36)
and (2.118)). On this basis, one can see t hat 1/# varies lin-
early wi t h 1/I for any subst ance. This rel at i on has been also
confi rmed wi t h experi ment al dat a [23]. Similar correl at i ons
for t hermal conduct i vi t y and diffusivity were developed and
the coefficients were related to hydr ocar bon propert i es such
as mol ecul ar wei ght [23, 24]. Equat i on (8.27) is applicable
onl y to nonpol ar and hydr ocar bon liquid systems in whi ch t he
i nt ermol ecul ar forces can be det ermi ned by London forces.
Ot her devel opment s in the cal cul at i on of liquid viscosity are
report ed by Chung et al. ( generalized correl at i ons for pol ar
and nonpol ar comp ounds) [25] and Qui nones-Ci sneros et al.
( pure hydr ocar bons and t hei r mixtures) [26].
Exampl e &/ - - Cons i der a liquid mi xt ure of 74.2 mol %
acet one and 25.8 mol % car bon t et rachl ori de (CCI4) at 298.2 K
and 1 atm. Est i mat e its viscosity assumi ng the onl y i nforma-
t i on known for this syst em are To Pc, Vo ~o, M, and ZRn of each
comp ound. Compare est i mat ed value wi t h the experi ment al
value of 0.395 mPa. s (cp) [10].
Solution--CC14 and acet one are nonhydr ocar bons whose
critical propert i es are not given in Table 2.1 and for this rea-
son t hey are obt ai ned f r om ot her sources such as DI PPR [ 10]
or any chemi cal engi neeri ng t her modynami cs text as [ 18,27]:
for acetone, Tc = 508.2 K, Pc = 47.01 bar, Vc = 209 cm3/ mol,
w = 0.3065, M = 58.08 g/tool, and ZRA = 0.2477; for CC14,
Tc = 556.4 K, Pc -= 45.6 bar, Vc = 276 cm3/ mol, o) = 0.1926,
M = 153.82 g/ mol, and ZRA = 0.2722 [18]. Using t he Kay' s
mixing rule (Eq. 7.1) wi t h xl = 0.742 and x2 = 0.258: Tc =
520.6 K, Pc = 46.6 bar, Vc = 226.3 cm3/ mol, 09 = 0.2274, M =
82.8, and ZRa = 0.254. Mixture liquid density at 298 K is cal-
culated f r om Racket equat i on (Eq. 5.121): V s = 80.5 cm3/ mol
(P25 = 1.0286 g/cm3). This gives Pr = V JV = 226.3/ 80.5 =
2.8112. For cal cul at i on of residual viscosity a general i zed cor-
rel at i on in t erms of Pr may be used. Al t hough Eq. (8.12) is pro-
posed for hydr ocar bons and nonpol ar fluids, for liquids/ o r is
quite hi gh and t he equat i on can be used up t o Pr of 3.0. Fr om
Eq. (8.5), ~ = 0.02428 and Tr = T/Tc = 0.5724 < 1.5. Fr om
Eq. (8.6), /go = 0.00829 cp. Fr om Eq. (8.12), /g = 0.374 cp,
whi ch in comp ar i son wi t h experi ment al value of 0.395 cp
gives an error of onl y - 5.3%. This is a good predi ct i on consid-
ering the fact t hat the mi xt ure cont ai ns a highly pol ar com-
p ound ( acetone) and predi ct ed density was used i nst ead of
a measur ed value. I f actual values of p2s [18] for pure com-
p ounds were used (p25 = 0.784 for acet one and p2s = 1.584
g/ cm 3 for CC14) and density is calculated f r om Eq. (7.4) we get
p25 = 1.03446 g/ cm 3 (Pr = 2.828), whi ch predicts/gmix = 0.392
cp ( error of onl y -0. 8%) . r
8 . 2 ES T I MA T I ON OF T H ER MA L
C ON D UC T I VI T Y
relation for hard-sphere molecules, the following equation is
developed for monoat omi c gases.
Thermal conductivity is a molecular property that is required
for calculations related to heat transfer and design and opera-
tion of heat exchangers. It is defined according to the Fourier's
law"
aT a (pCpT)
qy : - k : -or
8y Oy
(8.28)
k
0g~
pCp
where qy is the heat flux (heat transferred per unit area per
unit time, i.e., J/m 2. s or W/ m 2) in the y direction, 8T/Oy
is the temperature gradient, and the negative sign indicates
that heat is being transferred in the direction of decreasing
temperature. The proportionality constant is called thermal
conductivity and is shown by k. This equation shows that in
the SI unit systems, k has the unit of W/ m. K, where K may
be replaced by ~ since it represents a temperature differ-
ence. I n English unit system it is usually expressed in terms
of Btu/ft. h. ~ ( = 1.7307 W/ m. K). The unit conversions are
given in Section 1.7.I9. In Eq. (8.28), pCpT represents heat
per unit volume and coefficient k/ pCe is called thermal diffu-
sivity and is shown by 0t. A comparison between Eq. (8.28) and
Eq. (8.1) shows that these two equations are very similar in
nature as one represents flux of moment um and the other flux
of heat. Coefficients v and u have the same unit (i.e., cm2/s)
and their ratio is a dimensionless number called Prandtl num-
ber Npr, which is an important number in calculation of heat
transfer by conduction in flow systems. I n use of correlations
for calculation of heat transfer coefficients, Ner is needed [28].
v IzCp
(8.29) Npr
k
At 15.5~ (60~ values of Npr for n-heptane, n-octane,
benzene, toluene, and water are 6.0, 5.0, 7.3, 6.5, and 7.7,
respectively. These values at 100~ (212~ are 4.2, 3.6, 3.8,
3.8, and 1.5, respectively [28]. Vapors have lower Npr num-
bers, i.e., for water vapor Npr = 1.06. Thermal conductivity
is a molecular property that varies with both temperature
and pressure. Vapors have k values less than those for
liquids. Thermal conductivity of liquids decreases with an
increase in temperature as the space between molecules
increases, while for vapors thermal conductivity increases
with temperature as molecular collision increases. Pressure
increases thermal conductivity of both vapors and liquids.
However, at low pressures k is independent of pressure.
For some light hydrocarbons thermal conductivities of both
gases and liquids versus temperature are shown Fig. 8.3.
Methods of prediction of thermal conductivity are very sim-
ilar to those of viscosity. However, thermal conductivity of
gases can generally be estimated more accurately than can
liquid viscosity. For dense fluids, residual thermal conductiv-
ity is usually correlated to the reduced density similar to that
of viscosity (i.e., see Eqs. (8.11)-(8.13)).
8. 2. 1 Th ermal Conduct i v i t y of Gases
Kinetic theory provides the basis of prediction of thermal
conductivity of gases. For example, based on the potential
(8.30) k = d2 V 7r3m
where the parameters are defined in Eq. (8.2). This equation
is independent of pressure and is valid up to pressure of 10
atm for most gases [1]. The Chapmman-Enskog theory dis-
cussed in Section 8.1.1 provides a more accurate relation in
the following form:
1.9 10 -4 ( T) 1/2
(8.31) k =
t72f2
where k is in cal/cm 9 s. K, cr is in/~, and ~2 is a parameter that is
a weak function of T as given for viscosity or diffusivity. This
function is given later in Section 8.3.1 (Eq. 8.57). From Eq.
(8.31) it is seen that thermal conductivity of gases decreases
with increase in molecular weight. For polyatomic gases the
Eucken formula for Prandtl number is [ 1]
Cp
(8.32) Npr -
Cp + 1.25R
where Ce is the molar heat capacity in the same unit as for gas
constant R. This relation is derived from theory and errors as
high as 20% can be observed.
For pure hydrocarbon gases the following equation is given
in the API-TDB for the estimation of thermal conductivity [5]:
(8.33) k = A + BT + CT 2
where k is in W/ m. K and T is in kelvin. Coefficients A, B, and
C for a number of hydrocarbons with corresponding tem-
perature ranges are given in Table 8.3. This equation can be
used for gases at pressures below 3.45 bar (50 psia) and has
accuracy of 4-5%. A generalized correlation for thermal con-
ductivity of pure hydrocarbon gases for P < 3.45 bar is given
as follows [5]:
k=4. 911 x 10 -4TrCP
(a) only for methane and cyclic compounds at Tr < 1
= [11.04 x 10 -5 ( 14.52Tr- 5.14) 2/3]
Ce
k
-7
(b) for all compounds at any T except (a)
)~ = 1.i 1264 Tcl/6M1/2
p2c/3
(8.34)
Equation (8.34) also applies to methane and cyclic com-
pounds at Tr > 1, but for other compounds can be used at
any temperature. The units are as follows: Cp in J/ mol. K,
ire in K, Pc in bar, and k in W/ m-K. This equation gives an
average error of about 5%.
For gas mixtures the following mixing rule similar to
Eq. (8.7) can be used [18]:
xik4
(8.35) k~ = N
i=I ~j =l xj Aij
0.25
.. 0.2
o
15
1D
t . -
8
0.15
0.1
0.05"
o
f
f
200 400 600
Temperature, C
J
J
w/m K = 0.5788 ~TU,'hr ft R I
l
8 0 0 1 0 0 0
(a)Gases
,,,,,
' 5
o
o
E
0.22
0. 2'
0.18
0.16
0.1 4
0. 1 2
0.1
0.08
0.06
-200
\
, IP,opa~ iB ~
\
1,1 W/rn K = 0.5788 BT U/I lr ft RI
-150 -1 O0 -50 0 50 1 O0 150 200
Temperature, C
(b) Liquids
FI G. 8 . 3 mT her mal conduct i v i t y of sev eral l i g ht hydr ocar bons v ersus t emper at ur e
at at mospher i c pressure. T ak en wi t h per mi ssi on f rom Ref. [ 2].
wher e Aij may be set equal to ~ii gi ven in Eq. (8.7). Anot her
mi xi ng rul e t hat does not r equi r e vi scosi t y of p ur e comp onent
is gi ven by Pol i ng et al. [18]. A mor e advanced mi xi ng rul e
for cal cul at i on of mi xt ur e t her mal conduct i vi t y of gases and
l i qui ds is p r ovi ded by Mat hi as et al. [29]. For vap or s f r om un-
defi ned p et r ol eum fact i on, t he fol l owi ng equat i on has been
deri ved f r om r egr essi on of an ol d figure devel op ed i n t he
1940s [5]:
k = A + B ( T - 255.4)
0.42624 1.9891
(8.36) A = 0.00231 + ~ 4 M2
1.3047 x 10 -4 0.00574
B---- 1.0208 x 10 -4 + + - -
M M 2
wher e k is in W/ m. K and T is in kelvin. The equat i on shoul d
be used for p r essur e bel ow 3.45 bar, for p et r ol eum fract i ons
wi t h M bet ween 50-150 and T in t he r ange of 260-811 K. Thi s
equat i on is oversi mpl i fi ed and shoul d be used when ot her
met hods are not appl i cabl e. Ri azi and Faghr i [30] used t he
general r el at i onshi p bet ween k, T, and P at t he cri t i cal p oi nt
(To, Pc) to devel op an equat i on si mi l ar to Eq. (2.38) for es-
t i mat i on of t her mal conduct i vi t y of p et r ol eum f r act i ons and
p ur e hydr ocar bons.
k = 1 . 7 3 0 7 A( 1 . 8Tb) BSG c
A = exp (21.78 - 8.07986t + 1.12981t 2 - 0.05309t 3)
(8.37) B = - 4. 13948 + 1.29924t - 0.17813t 2 + 0. 00833t 3
C = 0.19876 - 0.0312t - 0.00567t 2
1.8T - 460
t =
100
wher e k is in W/ m. K, Tb and T ar e i n kelvin. Fact or s 1.7307
and 1.8 come f r om t he fact t hat t he ori gi nal uni t s were in
Engl i sh. This equat i on can be ap p l i ed to p ur e hydr ocar bons
TABLE 8.3--Coefficients of Eq. (8.33) for thermal conductivity of pure gases [5].
k =A+B T+CT 2 (8.33)
No. Compound name A x 10 -1 B x 10 4 C x 10 -7 Range, K
1 Methane -0.0076 0.9753 0.7486 97-800
2 Ethane -0.1444 0.9623 0.7649 273-728
3 Propane -0.0649 0.4829 1.1050 233-811
4 n-Butane 0.0000 0.0614 1.5930 273-444
5 n-Pentane 0.0327 -0.0676 1.5580 273-444
6 n-Hexane 0.0147 0.0654 1.2220 273-683
7 n-Heptane -0.0471 0.2788 0.9449 378-694
8 n-Octane -0.1105 0.5077 0.6589 416-672
9 n-Nonane -0.0876 0.4099 0.6937 450-678
10 n-Decane -0.2249 0.8623 0.2636 450-678
11 n-Undecane -0.1245 0.4485 0.6230 472-672
12 n-Dodecane -0.2535 0.8778 0.2271 516-666
13 n-Pentadecane -0.3972 1.3280 -0.2523 566-644
14 Ethene -0.0174 0.3939 1.1990 178-589
15 Propene -0.0844 0.6138 0.8086 294-644
16 Cyclohexane -0.0201 0.0154 1.4420 372-633
17 Benzene -0.2069 0.9620 0.0897 372-666
18 Toluene -0.3124 1.3260 -0.1542 422-661
19 Ethylbenzene -0.3383 1.3240 -0.1295 455-678
20 1,2-Dimethylbenzene(o-Xylene) -0.1430 0.8962 0.0533 461-694
21 n-Propylbenzene -0.3012 0.9695 0.7099 455-616
( C5- C16) or to p et r ol eum f r act i ons wi t h M > 70 ( boi l i ng p oi nt
r ange of 65-300~ in t he t emp er at ur e r ange of 200-370~
( ~400-700~ Accur acy of t hi s equat i on for p ur e comp ounds
wi t hi n t he above r anges is about 3%.
The effect of p r essur e on t he t her mal conduct i vi t y of gases
is usual l y consi der ed t hr ough gener al i zed cor r el at i ons si m-
i l ar to t hose gi ven for gas vi scosi t y at hi gh pressures. The
fol l owi ng r el at i on for cal cul at i on of t her mal conduct i vi t y of
dense gases and nonp ol ar fluids by St i el and Thodos [31] is
wi del y used wi t h accur acy of about 5- 6% as r ep or t ed i n var-
i ous sources [10, 18]:
A
k = k ~ + ~ [exp (Bpr) -t- C]
(8.38)
M3 c' 1' 6 ZcS
r = 4" 642x104\ p4 ]
F or p r < 0. 5: A= 2. 702, B= 0. 535, C= - 1. 000
0. 5< p r < 2. 0: A= 2. 528, B= 0. 670, C= - 1. 069
2. 0< p r < 2. 8: A= 0. 574, B= 1. 155, C= 2. 016
wher e k ~ is t he t her mal conduct i vi t y of l ow- p r essur e ( at mo-
sp her i c pr essur e) gas at gi ven t emp er at ur e and k is t he corre-
sp ondi ng t her mal conduct i vi t y at given t emp er at ur e and pres-
sure of i nt erest . Pr is t he r educed densi t y ( V JV ) , Tc is i n K, Pc is
i n bar, and Zc is t he cri t i cal comp r essi bi l i t y factor. Bot h k and
k ~ ar e i n W/ m. K. I n t he API -DTB [5] a gener al i zed cor r el at i on
devel op ed by Cr ook and Dauber t is r ecommended for cal cu-
l at i on of k of dense hydr ocar bon gases. However, t hi s met hod
r equi r es cal cul at i on of i sochor i c ( const ant vol ume) heat ca-
p aci t y (Cv) at t he T and P of i nt erest . Anot her gener al i zed
cor r el at i on for est i mat i on of t her mal conduct i vi t y of gases at
hi gh p r essur e was devel oped by Ri azi and Faghr i [32]:
k
(8.39) kr = ~ = (0.5 - o))kr 1 + ~ok~
wher e k~ is t he r educed t her mal conduct i vi t y and k is t he ther-
mal conduct i vi t y at T and P of i nt er est i n W/ m. K whi l e/ ~ is
t he t her mal conduct i vi t y at t he cri t i cal p oi nt (Tc and Pc). Pa-
r amet er s k~ 1) and k~ 2) are det er mi ned as a f unct i on of ~ and
Pr. Values of k~ were det er mi ned f r om exp er i ment al dat a for
a number of hydr ocar bons and are given in Table 8.4 [32].
Values of k~ 1) and ~2) ar e gi ven in Table 8.5. For t hose com-
p ounds for whi ch val ues of kr ar e not avai l abl e t hey may be
det er mi ned f r om Eq. (8.39) if onl y one dat a on k is avai l abl e
(k~ = k/kr). I n some cases, as shown i n t he fol l owi ng exampl e,
k~ can be obt ai ned f r om i nt er / ext r ap ol at i on of val ues gi ven i n
Table 8.4.
Values of ~ r ep or t ed in Table 8.4 ar e bas ed on ext r apol a-
t i on f r om exp er i ment al dat a at subcr i t i cal condi t i ons. I t is
bel i eved t hat t her e is a great enl ar gement of t her mal conduc-
t i vi t y at t he cri t i cal p oi nt for fluids. For mi xt ures, t he cri t i cal
enhancement s are si gni fi cant but t he t her mal conduct i vi t y
r emai ns finite [29]. Act ual val ues of cri t i cal t her mal conduc-
t i vi t y may be subst ant i al l y di fferent f r om t he val ues gi ven i n
t hi s t abl e. For exampl e, val ue of kr f r om met hane as shown
by Mat hi as et al. [29] is 0.079 W/ m. K, whi l e t he val ue gi ven
i n Table 8.4 is 0.0312 W/ m- K. However for et hane t he val ue
of kr f r om t hi s t abl e is t he same as obt ai ned f r om met hod of
Mat hi as et al. [29]. Equat i on (8.39) is mai nl y r ecommended
for condi t i ons di fferent f r om t he cri t i cal p oi nt and as l ong as
val ues of kr f r om Table 8.4 ar e used, p r edi ct ed val ues f r om
Eq. (8.39) ar e rel i abl e.
Exampl e 8. 2- - Cons i der n- p ent ane vap or at 300~ and
100 bar. Cal cul at e its t her mal conduct i vi t y f r om St i el - Thodos
and Ri azi - F aghr i met hods.
Sol ut i on- - Fr om Table 2.1, M = 72.2, Tc = 196.55~ =
469.7 K, Pc = 33.7 bar, Vc -- 313.05 cm3/ mol, and Zc -- 0.2702.
F r om Eq. (8.33) and coeffi ci ent s for n-C5 i n Table 8.3 at
TABLE 8. 4--Critical thermal conductivity of some pure
compounds [32].
Compound k~, W/mK Compound ke, W/mK
Methane 0.0312 Ethene 0.0379
Ethane 0.0319 Cyclohexane 0.0533
Propane 0.0433 Benzene 0.0472
n-Butane 0.0478 Toluene 0.0526
n-Heptane 0.0535 Ethylbenzne 0.0526
TABLE 8.S--Vatues ofk<r u and k(r 2) for Eq. (8.39). (Taken with permission from Ref. [32].)
kr = ~ ---- (0.5 -- (.o)k~ + o) k~ ( 8. 101)
Pr
Tr 0.2 0.5 1.0 1.5 2.0 3.0 4.0 6.0 8.0 10.0
Values of k! 1) versus Tr and Pr
1.00 1.1880 1.3307 2.0000 4.15t7 4.4282 4.7900 5.2140 5.7989 6.2080 6.5132
1.05 1.3002 1.3640 1.8922 3.2806 3.7990 4.4915 4.7590 5.2817 5.7710 6.2040
1.10 1.4300 1.4810 1.8660 2.5989 3.3334 4.1068 4.4746 4.9502 5.3740 5.8812
1.15 1.5182 1.5365 1.8356 2.2978 2.9769 3.8583 4.4676 4.9404 5.3734 5.8760
1.20 1.8311 1.8956 2.1200 2.3983 2.8809 3.5626 4.2067 4.9285 5.3731 5.8699
1.40 2.1838 2.2520 2.3589 2.5291 2.7120 3.3000 4.0020 4.6327 5.2404 5.7656
1.60 2.5971 2.6589 2.7305 2.8572 3.0035 3.3760 3.8239 4.4385 4.8967 5.3031
2.00 3.6763 3.6984 3.7418 3.9161 3.9594 4.1370 4.3768 4.7138 5.0462 5.3614
3.00 6.9896 7.0010 7.0310 7.0617 7.1079 7.1452 7.2197 7.4077 7.5915 7.7685
Values of kr (2) versus Tr and Pr
1.00 1.6900 1.6990 2.0000 2.0619 2.3112 2.3140 2.3160 2.3180 2.3210 2.3212
1.05 1.7200 1.7290 1.8100 1.8170 2.1318 2.1912 2.3010 2,8380 2.3398 2.3400
1.10 1.8001 1.8211 1.8300 1.8310 1.9672 2.1384 2.1369 2.3614 2.3988 2.4105
1.15 2.0599 2.0601 2.0661 2.0700 2.0801 2.1269 2.2246 2.3780 2.4618 2.4622
1.20 2.1441 2.1539 2.1629 2.1681 2.1689 2.1901 2.2319 2.3981 2.4640 2.4701
1.40 2.6496 2.6772 2.6865 2,6889 2.6900 2.6911 2.7001 2.7119 2.8079 2.8810
1.60 3.2184 3.2448 3.2559 3.2886 3.3142 8.8292 3.3343 3.3352 3.8869 3.4525
2.00 4.5222 4.5330 4.5465 4.6871 4.6378 4.7108 4.8148 4.8119 4.8850 4.9885
3.00 8.4002 8.4158 8.4234 8.4503 8.4504 8.5038 8.6083 8.6204 8.6732 8.7454
T = 573.2 K (300~ k~ 0.048 W/ m. K. Fr om Lee-Kesl er
correl at i on (Eq. 5.107), t he mol ar vol ume at 573.2 K and
100 bar is cal cul at ed as Z = 0.59 or V = 281 cm3/ mol. Thus
Pr = Vc/V = 313.05/281 = 1.114. Si nce 0.5 < pr < 2, from
Eq. (8.38) F = 151.82, A = 2.702, B = 0.67, C = - 1. 069, and
k = 0.048 + 0.017 = 0.065 W/ m- K.
To cal cul at e k from Eq. (8.39), kr is required. Si nce i n
Table 8.4 val ue of k~ for n-C5 is not given, one can ob-
t ai n it from i nt er pol at i on of values given for C4 and C7
by assumi ng a l i near rel at i on be t we e n/ ~ and To. For C4,
= 0.0478 and Tc = 425.2 K and for C7, k~ = 0.0535 and
ire = 540.2 K. For C5 wi t h T~ = 469.7 by l i near i nt erpol at i on,
/~ = [(0.0535 - 0.0478)/ (540.2 - 425.2)] (469.7 - 425.2) +
0.0478 = 0.05 W/ m. K. Ext rapol at i on bet ween values ofkc for
C3 and Ca to k~ of C5 gives a slightly different value. At T and P
of interest, Tr = 1.22 and Pr = 2.97. Fr om Table 8.5, k~ 1) = 3.5
and k~ 2) = 2.2. Fr om Eq. (8.39), /q = 1.42 and k = 0.05
1.42 = 0.071 W/ m. K. St i el -Thodos met hod varies by 8.5%
from Ri azi -Faghri met hod, whi ch represent s a reasonabl e
deviation. I n this case t he St i el -Thodos met hod is more ac-
curat e since the val ue of k ~ is cal cul at ed mor e accurately. #
8. 2. 2 Thermal Conductivity of Liquids
Theory of t her mal conduct i vi t y of l i qui ds was proposed by
Br i dgman [1]. I n this theory, it is assumed t hat mol ecul es
are arranged as cubi c lattice wi t h cent er-t o-cent er spaci ng
of (V /NA) 1/3, i n whi ch V is t he mol ar vol ume and NA is t he
Avogadro number. Furt hermore, it is assumed t hat energy
is t ransferred from one lattice to anot her at the speed of
sound, cs. This t heory provides the basis of predi ct i on of ther-
mal conduct i vi t y of liquids. For monoat omi c l i qui ds the fol-
l owi ng rel at i on can be obt ai ned from this t heory [ I]:
2
3 f NAykBcs
(8.40) k
= \ v /
where ks is t he Bol t zman' s const ant and met hods of calcula-
t i on of cs have been di scussed i n Sect i on 6.9. For pure l i qui d
hydrocarbons, t her mal conduct i vi t y varies l i nearl y wi t h t em-
perat ure:
(8.41) k = A + BT
Coefficients A and B can be det er mi ned if at least two dat a
poi nt s on t her mal conduct i vi t y are available. Values of ther-
mal conduct i vi t y of some comp ounds at mel t i ng and boi l i ng
poi nt s are given i n Table 8.6, as given i n the API-TDB [5]. Liq-
ui d t her mal conduct i vi t y of several n-paraffins as cal cul at ed
from Eq. (8.41) (or Eq. 8.42) is shown i n Fig. 8.4.
If values of t her mal conduct i vi t y at mel t i ng and boi l i ng
poi nt s are t aken as reference poi nt s, t hen Eq. (8.41) can be
used to obt ai n val ue of t her mal conduct i vi t y at any ot her t em-
perat ure:
T- TM
(8.42) kr r = k~ + (k~ - k~) Tb Z
where TM and Tb are nor mal mel t i ng (or triple) and boi l i ng
points, respectively, k~ and k~ are values of l i qui d t her mal
conduct i vi t y at TM and Tb, respectively, kr L is val ue of l i qui d
t her mal conduct i vi t y at t emper at ur e T. Accordi ng to API-TDB
[5] t hi s equat i on can predi ct values of l i qui d t her mal conduc-
tivity of pure comp ounds up to pressure of 35 bar wi t h an
accuracy of about 5% [5]. There are a number of general i zed
correl at i ons developed for predi ct i on of t her mal conduct i vi t y
of pure hydr ocar bon liquids. The Riedel met hod is i ncl uded
i n t he API-TDB [5]:
C M"
( 8. 43) k L -
3 ~- 2; (1 298"15~2/3
\ - ~7-~ ]
For unbr anched, st rai ght -chai n hydrocarbons,
n = 1.001 and C = 0.1811
For br anched and cyclic hydrocarbons,
n = 0.7717 and C = 0.4407
TABLE 8. 6---Liquid thermal conductivity of some pure compounds
melting and boiling points [5].
at their normal
No. Compound TM, K k at TM, W/InK Tb, K k at Tb, W/mK
1 Methane 90.69 0.2247 111.66 0.1883
2 Propane 85.47 0.2131 231.11 0.1289
3 n-Butane 134.86 0.1869 272.65 0.1176
4 n-Pentane 143.42 0.1783 309.22 0.1086
5 n-Hexane 177.83 0.1623 341.88 0.1042
6 2-Methyipentane I19.55 0.1600 333.41 0.1000
7 3-Methylpentane 110.25 0.1646 336.42 0.1010
8 n-Heptane 182.57 0.1599 371.58 0.1025
9 n-Octane 216.38 0.1520 398.82 0.0981
10 2.24-Trimethylpentane 165. 78 0.1284 372.39 0.0815
11 n-Nonane 219.66 0.1512 423.97 0.0972
12 n-Decane 243.51 0.1456 447.31 0.0946
13 n-Undecane 247.57 0.1461 469.04 0.0930
14 n-Dodecane 263.57 0.1436 489.47 0.0909
15 n-Tridecane 267.76 0.1441 508.62 0.0896
16 n-Tetradecane 279.01 0.1423 526.73 0.0882
17 n-Pentadecane 283.07 0.1446 543.83 0.0874
18 n-Hexadecane 291.31 0.1438 560.02 0.0849
19 n-Heptadecane 295.13 0.1441 575.26 0.0819
20 n-Octadecane 301.31 0.1460 589.86 0.0810
21 n-Nonadecane 305.04 0.1453 603.05 0.0797
22 n-Eicosane 309.58 0.1488 616.94 0.0801
23 n-Heneicosane 313.35 0.1499 629.66 0.0799
24 n-Docosane 317.15 0.1513 641.75 0.0809
25 n-Tricosane 320.65 0.1516 653.35 0.0811
26 n-Tetracosane 323.75 0.1530 664.45 0.0819
27 Cyclopentane 179.31 0.1584 322.40 0,1198
28 Methylcyclopentane 130.73 0.1605 344.96 0.1071
29 Cyclohexane 279.69 0.1282 353.87 0.1096
30 Methylcyclohexane 146.58 0.1449 374.04 0.0935
31 Cyclohexane 169.67 0.1653 356.12 0.1167
32 Benzene 278.68 0.1494 353.24 0.1266
33 Methylbenzene (toluene) 178.18 0.1616 383.78 0.1117
34 Ethylbenzene 178.20 0.1576 409.35 0.1025
35 n-Propylbenzene 173.55 0.1528 432.39 0.1014
36 n-Butylbenzene 185.30 0.1501 456.46 0.0957
wher e V L is t he l i qui d mol ar vol ume at 25~ (298 K) i n
cm3/ mol. For some comp ounds t hese val ues of V~ are gi ven
in Table 6.10. M is t he mol ecul ar wei ght i n g/ mol and k L
is desi red l i qui d t her mal conduct i vi t y at T in W/ m. K. Av-
erage er r or for this equat i on is about 5% as r ep or t ed in t he
0.8
~0. 6
0.4
~ 0.2
- - - - - n- Pe nt a ne
n- De c a ne
f
. . . . . n- Ei c os a ne j
7
- - Benzene
- - -- Water
0
100
i i i I i
200 300 400
Temperature, K
500
FIG. 8 . 4~ Li qui d thermal conducti v i ty of n-
alk anes v ersus temperature at atmospheri c
pressure.
API -TDB [5]. This equat i on can be used for t emp er at ur es at
Tr < 0.8 and pressure bel ow 35 bar. For est i mat i on of k L at
t emp er at ur es above nor mal boi l i ng poi nt ( compr essed or sat-
ur at ed liquids) , t here are a number of met hods t hat use re-
duced densi t y Pr as a correl at i ng p ar amet er [5, 8]. Ri azi and
Faghri [30] also devel oped a met hod si mi l ar to Eq. (8.37) for
predi ct i on of t her mal conduct i vi t y of l i qui d hydr ocar bons for
pent anes and heavier.
k = 1.7307A(1.8Tb)BSG c
A = exp ( - 4. 5093 - 0.6844t - 0.1305t a)
(8.44) B = 0.3003 + 0.0918t + 0.01195t a
C = 0.1029 + 0.0894t + 0.0292t a
t = (1.8T - 460) / 100
wher e k is in W/ m- K, whi l e Tb and T are in kelvin. This
equat i on can be appl i ed to pur e hydr ocar bons (C5-Ca2) or to
p et r ol eum fract i ons wi t h 70 < M < 300 ( boiling poi nt r ange
of 65-360~ in t he t emp er at ur e r ange of - 20- 150~ ( ~0-
300~ and pressures bel ow 30-35 bar. I f Eq. (8.44) is appl i ed
to t her mal conduct i vi t y dat a at t wo reference t emp er at ur es
of 0 and 300~ (256 and 422 K) one can get
k256 = 1.1594 10-3Tb~ 0"5478
(8.45)
kaz2 = 2.2989 10-2Tb~ 0"0094
TABLE 8.7--Comparison of various methods of calculation of liquid thermal conductivity at 20~ (Example 8.3).
Linear, a Eq. (8.42) API, Eq. (8.43) Eq. (8.52) RF, b Eq. (8.44) RF, Eq. (8.46)
Nc k~xp K %Dev k %Dev k %Dev k %Dev k %Dev
5 0.114 0.114 0.4 0.113 --0.8 0.107 --6.6 0.107 --6.5 0.113 --0.6
6 0.121 0.121 0.2 0.119 --1.4 0.111 --8.4 0.112 --7.8 0.118 --2.4
7 0.1262 0.126 --0.1 0.124 --1.7 0.114 --9.7 0.116 --8.3 0.122 --3.3
8 0.1292 0.129 0.0 0.127 --1.5 0.116 --10.0 0.119 --7.6 0.126 --2.8
9 0.1316 0.132 0.0 0.130 --1.2 0.118 -10. 2 0.123 -6. 9 0.129 --2.2
10 0.133 0.133 0.0 0.132 --0.6 0.120 --9.9 0.125 --5.8 0.132 --1.1
Overall 0.1 1.2 9.1 7.1 2.1
aLinear refers to linear relation betweern k and T.
bRF referes to Riazi-Faghri methods.
wher e k256 refers to t he val ue of k at 256 K (0~ and k422 is
t he val ue of k at 422 K (300~ Usi ng Eq. (8.41) and on t he
basi s of l i near i nt er p ol at i on of t her mal conduct i vi t y f r om t he
above equat i ons, t he fol l owi ng r el at i on was al so der i ved for
t he t emp er at ur e r ange and mol ecul ar wei ght r anges speci fi ed
for Eq. (8,44):
k = 10 -2 (0.11594T~176 - 2.2989T~ 0"0094)
x ( 1. 8T- 460] 10 2T~ ~176176
\ 300 / + 2.2989
(8.46)
wher e Tb and T ar e in kel vi n and k is in W/ m. K. Accur acy of
t hi s equat i on for p ur e comp ounds wi t h t he speci fi ed r anges is
about 3.8% [30] and it is r ecommended i nst ead of Eq. (8.44).
Exampl e 8. 3~Es t i ma t e val ues of t her mal conduct i vi t y of
l i qui d nor mal al kanes f r om Cs to C10 at 20~ and 1 at m,
usi ng met hods gi ven i n Eqs. ( 8. 42) -( 8. 44) and (8.46). Com-
p ar e cal cul at ed val ues wi t h exp er i ment al dat a as gi ven in t he
l i t er at ur e [8, 10].
Sol ut i on- - Sampl e cal cul at i ons ar e shown for n-Cs and si mi -
l ar ap p r oach can be used to est i mat e val ues of k L for ot her n-
al kane comp ounds. F r om Table 2. I , for n- p ent ane Tb = 36.1 ~
(309.3 K), SG = 0.6317, TM = - 129. 7~ (143.45 K), and Tc =
196.55~ (469.8 K). F r om r ef er ence [10], k20 = 0.114 W/ m. K.
F r om Table 8.4, / ~= 0. 1758 and k~= 0. 1079 W/ m. K.
Subst i t ut i ng in Eq. (8.42) k L = 0.1758 + (0.1079 - 0.1758) x
(298.15 - 143. 45) / ( 309. 3 - 143.45) -- 0.1758 - 0.06334 =
0.1145 W/ m. K. This gives an er r or of + 0.43%. F r om
Eq. (8.43), n = 1.001 and C = 0. I 811 and it gives k L = 0. I 15,
wi t h 0.7% error. F r om Eq. (8.44), t = 0.68, A = 0.006524,
B = 0.36787, C = 0.17677, and k L = 0.107 ( er r or of - 6.5%).
Equat i on (8.46) gives k L -- 0.1134, wi t h er r or of - 0.57%.
Lat er i n t hi s sect i on several ot her emp i r i cal cor r el at i ons for
est i mat i on of l i qui d t her mal conduct i vi t y ar e pr esent ed. F or
exampl e, Eq. (8.52) is p r op os ed for t her mal conduct i vt y of
coal l i qui ds. Thi s equat i on gives a val ue of 0.107 ( - 6. 6%) .
Summa r y of resul t s are gi ven in Table 8.7 and al so shown
in Fig. 8.5. As expect ed, Eq. (8.44) because of its si mp l i ci t y
and Eq. (8.52) p r op os ed for coal l i qui ds give t he hi ghest
er r or s in est i mat i on of t her mal conduct i vi t y of l i qui d hydr o-
car bons. $
For defi ned mi xt ur es t he fol l owi ng mi xi ng rul e p r op os ed
by Li is r ecommended i n t he APLTDB [5] for cal cul at i on of
l i qui d t her mal conduct i vi t y of hydr ocar bon syst ems:
= EZ~i ~i k L
i J
+ wher e k/i = kii and k/i = ki
q~i - xiVi L
i n whi ch k~ is t he t her mal conduct i vi t y of l i qui d mi xt ure, V/L
is t he l i qui d mol ar vol ume at a r ef er ence t emp er at ur e (20 or
25~ x~ is mol e fract i on, and ~i is t he vol ume f r act i on of
comp onent i in t he mi xt ure. Average er r or for t hi s met hod
is about 5% [5]. Li p r op os ed a si mp l er mi xi ng rule, whi ch is
r ecommended in t he DI PPR manual [ 10] for nonhydr ocar bon
l i qui ds:
F 7 - 1/ 2
wher e x~i is t he wei ght f r act i on of i in t he mi xt ure. This
equat i on gives an average devi at i on of about 4- 6% [10]. The
J ami eson met hod for a bi nar y l i qui d mi xt ur e is suggest ed by
Pol i ng et al. [18]:
(8.49) k~ =Xwl k L +x~2/ ~ -0t12 (/ ~ - k L) (1 - q:-~-2) Xw2
Par amet er a12 is an adj ust abl e p ar amet er t hat can be de-
t er mi ned f r om an exp er i ment al dat a on mi xt ur e t her mal
0.14
~ 0.13
IF 0.12
~ 0.11
0.10
o F~
Method A
Method B ~ .
- - - - - Method C ~/ / w/ / ' /
. . . . Method D ~ - - / ' ~/ / / " / . "
- -Method E / ~/ / / f / / /
J / / " / . . ' " -
/ " / -
/-
/
i
2 6 8 12
Carbon Number
i k
4
r
10
FIG. 8 . 5~ Est i mat i on of liquid thermal con-
ductiv ity of n-alkanes at 20~ and atmospheric
pressure ( Ex ampl e 8 .3) . Method A : Eq. ( 8 .42) ;
Method B: Eq. ( 8 .43) ; Method C: Eq. ( 8 .44) ;
Method D: Eq. ( 8 .46) ; Method E: Eq. ( 6.52) .
8. APPLICATIONS: ESTIM ATION OF TRANSPORT PROPERTIES 345
conductivity and when no data are available it can be con-
sidered as unity [18].
For calculation of thermal conductivity of liquid petroleum
fractions, if the PNA composition is available the pseudocom-
ponent method using Eq. (8.42) and Table 8.4 may be applied.
The simplest method of calculation of k L for petroleum frac-
tions when there is no information on a fraction is provided
in the API-TDB [5]:
(8.50) k~ = 0.164 - 1.277 x 10-4T
where T is in kelvin and k L in W/re. K. I n other references this
equation is reported with slight difference in the coefficients.
For example, Wauquier [8] gives k z = 0.17 - 1.418 x 10-4T.
At 298 K (25~ this relation gives a value of 0.128 W/re. K
(near k of n-C8), while Eq. (8.50) gives a value of 0.126, which
is the same as the value of k for n-heptane. The error for this
equation is high, especially for light and branched hydrocar-
bons. Average error of 10% is reported for this equation [5]
and it may be used in absence of any information on a frac-
tion. A more accurate relation uses average boiling point of
the fraction as an input parameter and was developed by the
API group at Penn State [5]:
(8.51) k L = Tb 0"2904 X (2.551 10 -2 - 1.982 10-ST)
where both Tb and T are in kelvin. This equation gives an av-
erage error of about 6%. For n-C5 of Example 8.3, this equa-
tion gives a value of 0.104 W/ m. K with - 9% error. However,
this equation is not recommended for pure hydrocarbons.
For petroleum fractions Eq. (8.46) can be used with better
accuracy with the specified ranges of boiling point and tem-
perature when both Tb and SG are available. For coal liquids
and heavy fractions, Tsonopoulos et al. [33] developed the
following relation based on the corresponding states method
of Sato and Riedel:
(8.52) k L = 0.05351 + 0.10177 (I - Tr) 2/3
where k L is in W/re. K. This equation is not recommended for
pure hydrocarbons. For some eight coal liquid samples and
74 data points this equation gives an average error of about
3% [33].
For liquid hydrocarbons and petroleum fractions when
pressure exceeds 30-35 atm, effect of pressure on liquid ther-
mal conductivity should be considered. However, this effect
is not significant for pressures up to 70-100 atm. For the re-
duced temperature range of 0.4-0.8 and pressures above 35
atm, the following correction factor for the effect of pressure
on liquid thermal conductivity is recommended in the API-
TDB [5]:
L L C2
( 8. 53) ~ = k~
C = 17.77 + 0.065Pr - 7.764Tr 2054T2
exp (0.2Pr)
To calculate value of k~ at I"2 and P2, value of k L at/' 1 and P1
must be known. I n case of lack of an experimental value, the
value of k~ at T1 and P~ can be calculated from Eqs. (8.41)-
(8.43). There are some other generalized correlations based
on the theory of corresponding states for prediction of both
viscosity and thermal conductivity of dense fluids [25, 34].
However, these methods, although complex, are not widely
recommended for practical applications in the petroleum
industry.
8. 3 DI FFUS I ON COEFFI CI ENTS
Diffusion coefficient or diffusivity is the third transport prop-
erty that is required in calculations related to molecular dif-
fusion and mass transfer in processes such as mixing and
dissolution. I n the petroleum and chemical processing, dif-
fusion coefficients of gases in liquids are needed in design
and operation of gas absorption columns and gas-gas diffu-
sion coefficients are required to determine rate of reactions
in catalytic-gas-phase reactions, where mass transfer is a con-
trolling step. I n the petroleum production, knowledge of dif-
fusion coefficient of a gas in oil is needed in the study of gas
injection projects for improved oil recovery. I f a binary sys-
tem of components A and B is considered, where there is a
gradient of concentration of A in the fluid, then it diffuses
in the direction of decreasing concentration (or density) --a
process similar to heat conduction due to temperature gradi-
ent. I n this case the diffusion coefficient of component A in
the system of A and B is called binary (or mutual) diffusion
coefficient and is usually shown by DAB, which is defined by
the Fick's law [1]:
(8.54) JAy = -- DAB ~ = - p DAB ~dd~Wy A
where JAy is the mass flux of component A in the y direction
(i.e., g/cm 2. s) and dpA/dy is the gradient of mass density in
the y direction, p is the mass density (g/cm 3) of mixture and
XwAis the weight fraction of component A in the mixture. In
the above relation, the second equality holds when p is con-
stant with respect to y. JA represents the rate of transport of
mass in the direction of reducing density of A. It can be shown
that in binary systems DaB is the same as DBA [1]. From the
above equation, it can be seen that the unit of diffusivity in
the cgs unit system is cm2/s (or 1 cm2/s --- 10 -4 m2/s). Diffu-
sion of a component within its own molecules (DAA) is called
self-diffusion coefficient. From thermodynamic equilibrium
point of view, the driving force behind molecular diffusion is
gradient of chemical potential O#A/Oy. Since chemical poten-
tial is a function of T, P, and concentration, for systems with
uniform temperature and pressure,/ za is only a function of
concentration (see Eq. 6.12 i) and Eq. (8.54) is justified. Var-
ious forms of Fick's law can be established in the forms of
gradients of molar concentration, mole, weight, or mass frac-
tions [1]. A comparison between Eqs. (8.1), (8.28), and (8.54)
shows the similarity in momentum, heat, and mass trans-
fer processes. The corresponding molecular properties (i.e.,
kinematic viscosity (v), thermal diffusivity (a), and diffusion
coefficient (D)) that characterize the rate of these processes
have the same unit (i.e., cm2/s or ft2/h). This is the reason that
these physical properties are called transport properties. The
diffusion process may also be termed mass transfer by con-
duction. The ratio of v/D or (Iz/pD) is a dimensionless num-
ber called Schmidt number (Ns~) and is similar to the Prandfl
number (Npr) in heat transfer (see Eq. 8.29). Schmidt number
represents the ratio of mass transfer by convection to mass
transfer by diffusion. Values of Nsc of methane, propane, and
n-octane in the air at 0~ and 1 atm are 0.69, 1.42, and 2.62,
TABLE 8.8--Order of magnitude of binary diffusion coefficient and its concentration dependency
for various systems [35].
Order of magnitude Activation energy (E), Concentration
Type of system of D, cm2/s kcal/mol dependence
Gas-gas (vapor-gas) 0.1-1.0 E < 5 Very weak
Gas-liquid ~10 -5 E _< 5 Weak
Normal liquids 10-5-10 -6 5-10 +100%
Polymer solutions 10 -5-10 -8 10-20 + 1000%
Gas or liquid in polymer or solids ~10-12-10 -15 E _> 40 Factor of 1000%
respectively. The or der of magni t ude of Nsc in liquids such as
wat er is 103 .
Diffusion coefficient like any ot her t her modynami c prop-
ert y is a funct i on of the state of a syst em and depends on
T, P, and concent rat i on (i.e., xi). One t heory t hat describes
mol ecul ar diffusion is based on the assumpt i on t hat mol ecu-
lar diffusion requires a j ump in t hei r energy level. This energy
is called activation energy and is shown by EA. This act i vat i on
energy, al t hough not t he same, is very similar to t he activa-
t i on energy requi red for a chemi cal react i on to occur. Heavier
mol ecul es have hi gher activation energy and as a result l ower
diffusion coefficients. Based on this theory, dependency of D
wi t h T can be expressed by Arrhenius-type equat i on in t he
following form:
D f EA~
(8.55) D = oexp ~- ~- ~)
where Do is a const ant ( with respect t o T), EA is the act i vat i on
energy, R is the gas constant, and T is the absolute t empera-
ture. The order of magni t ude of D and EA in vari ous syst ems
and concent rat i on dependency of D are shown in Table 8.8.
Diffusion coefficients depend on t he ability of molecules to
move. Therefore, larger mol ecul es have mor e difficulty to
move and consequent l y their diffusivity is lower. Similarly in
liquids where t he space bet ween molecules is small, diffusion
coefficients are l ower t han in the gases. I ncrease in T woul d
increase diffusion coefficients, while increase in P decreases
diffusivity. The effect of P on diffusivity of liquids is less t han
its effect on t he diffusivity of gases. At very hi gh pressures,
values of diffusion coefficients of liquids ap p r oach t hei r val-
ues for the gas phase. At the critical point, bot h liquid and
gas phases have the same diffusion coefficient called critical
diffusion coefficient and it is represent ed by Dc.
I n this section, met hods of est i mat i on of diffusion coef-
ficients in gases and liquids as well as in mul t i comp onent
systems and the effect of por ous medi a on diffusivity are pre-
sented. I n t he last part a new met hod different from conven-
tional met hods for experi ment al measur ement of diffusion
coefficients in dense hydr ocar bon fluids ( bot h gases and liq-
uids) is presented.
8. 3. 1 Di f f us i v i t y o f Gas es at L ow Pr e s s ur e s
Similar to viscosity and t hermal conductivity, kinetic t heory
provi des a relatively accurat e relation for diffusivity of rigid
( hard) molecules wi t h different size. Based on this theory, for
gases at low pressures (ideal gas condi t i ons) the following
relation is developed for gas-gas diffusivities [ 1, 3]:
3xn ( k~ ~ 1/2 ( 1 1 ~,/2 T3/2
DAB
- 8 p( )z
(8.56)
where DAB is in cm2/s, kB is the Bol t zman' s const ant (1.381
10 -23 J/K), T is t emperat ure in kelvin, P is t he pressure in
bar, m is the mol ecul ar mass in kg [M/NA, i.e., mA = MA
10-3/(6.022 x 1023)], and d is the hard sphere mol ecul ar di am-
eter in m (1 nm = 1 x 10 -9 m). Values of d may be det ermi ned
f r om measur ed viscosity or t hermal conduct i vi t y dat a by Eqs.
(8.2) and (8.30), respectively. For example, for CH4 value of d
from viscosity is 0.414 nm while from t hermal conduct i vi t y is
0.405 nm. For 02, Ha and COa, values of d are 0.36, 0.272, and
0.464 nm, respectively [3]. As an example, the self-diffusion
coefficient of CH4 at 1 bar and 298 K f r om t he kinetic the-
ory is calculated as mA = mB = m = 2.66 x 10 -26 kg, dn = dB =
d = 0.414 x 10 -9 m and from Eq. (8.56) DAB = 0.194 cm2/s.
Thus one can calculate diffusion coefficient f r om viscosity
dat a t hr ough cal cul at i on of mol ecul ar diameter. For gases
at low pressures D varies inversely wi t h pressure, while it is
proport i onal to T 3/2. Furt hermore, DA varies wi t h M~ 1/2, t hat
is, heavier mol ecul es have l ower diffusivity under the same
condi t i ons of T and P. I n practical cases mol ecul ar di ame-
ters can be est i mat ed from liquid mol ar vol umes in whi ch
actual dat a are available, as will be seen in Eq. (8.59). A mor e
accurat e equat i on for est i mat i on of diffusivity of ideal gases
was derived i ndependent l y by Chap man and Enskog f r om t he
kinetic t heory and is known as Chap man- Ens kog equation,
whi ch may be written as [1, 9]
2. 2648 5T~ ( ~ + u~) ~
(pDAB)~ = a~2 ~
aA +orb
O' AB--
2
~/ = 0.1866V 1/3 Z~ 6/5
1.06036
g2AB -- (T~)0.1561 + 0.193 exp ( - 0. 47635T~) + 1.76474
x exp ( - 3. 89411T~) + 1.03587 exp ( - 1. 52996T~)
T~m = T/ e~
e AB= (eAeB) 1/2
ei = 65.3T~i Z~i 8/5
(8.57)
where (pDAB) ~ represent s t he p r oduct of density-diffusivity
of ideal gas at l ow-pressure condi t i ons accordi ng to the
Chap man- Enskog t heory and is in mol / cm 9 s. e and a are the
8. APPLICATIONS: ESTI M ATI ON OF TRANSPORT PROPERTIES 347
energy and size parameters in the potential energy relation
(i.e., Eq. 5.11). a is in ~,, T and Tc are in kelvin, and Vc is in
cma/mol. The correlations for calculation of Lennard-Jones
(L J) parameters (e and a) from critical constants as given in
Eq. (8.57) were developed by Stiel and Thodos [36]. There are
some other correlations given in the literature for calculation
of LJ parameters [18]. Typical values of e and a determined
from various properties are given in Table 6.16. I n the above
relation low-pressure diffusivity can be calculated through di-
viding (pDAB) ~ by p~ ( = 83. 14T/P) in which T is in kelvin and
P is in bar. Calculated DAB would be in cm2/s.
For practical calculations, a more accurate estimation
method is required. Most of these correlations are based on
the modified version of Chapman-Enskog theory [18]. The
empirical correlation of Chen-Othmer for estimation of DAB
of gases at low pressures is in the following form [28]:
( 1 1) 1/2
1.518 x 10 2T181 ~ -F
(8.58) D~a 3 =
P (TcATCB) ~176 (VcO~ 4 -~- VcO~4) 2
where D~d 3 is the diffusivity of A in B at low pressures in cm2/s,
T is in kelvin, P is in bar, M in g/mol, and TeA and VCAare the
critical temperature and volume of A in kelvin and cm3/mol,
respectively. This method can be used safely up to pressure of
about 5 bar. This equation predicts self-diffusion coefficient
of methane at 298 K and 1 bar as 0.248 cm2/s versus the value
of 0.194 from the kinetic theory (Eq. 8.56). For hydrocarbon-
hydrocarbon systems the API-TDB recommends the Gilliland
method in the following form [5]:
( 1 1) 1/2
4.36 x 10-3T 15 MAA+
(8.59) DAB
p (v1/3 q- V1/3) 2
V/ = 0.285V 1"048
where V/is the liquid molar volume of component i at its nor-
mal boiling point and Va is the molar critical volume and both
are in cm3/mol. Other units are the same as in Eq. (8.58). This
equation can be used up to pressure of 35 bar with an accu-
racy of about 4% as reported in the API-TDB [5]. Several other
methods for prediction of gas diffusivity at low pressures are
given by Poling et al. [18].
8 . 3 . 2 Di f f usi v i t y of Li qui ds at L ow Pres s ures
Calculation of diffusion coefficients for liquids is less accurate
than gases as for any other physical property. This is mainly
due to the lack of a perfect theory for liquids. Generally there
are three theories for diffusivity in liquids: (i) hydrodynamic
theory, which usually applies to systems of solids dissolved in
liquids, (2) Eyring rate theory, and (3) the free-volume theory.
I n the hydrodynamic theory, it is assumed that fluid slides
over a particle according to the Stoke's law of motion. The
Eyring theory was presented earlier by Eq. (8.55) in which
molecules require an energy j ump before being able to dif-
fuse. The free-volume theory says that for a molecule to j ump
to a higher energy level (activation energy), it needs a critical
free-volume (V2) and Eq. (8.55) can be modified by multiply-
ing the right-hand side by factor exp (V~/V), where V is the
apparent molar volume of liquid. None of these theories is
perfect; however, it can be shown by both the Eyring rate and
hydrodynamic theories that in liquid systems diffusion co-
efficient is inversely proportional to viscosity of solvent. For
example, based on the hydrodynamic theory and the Stokes-
Einstein equation, Wilke and Chang developed the following
relation for estimation of diffusion coefficient at infinite dilu-
tion [18, 28]:
(8.60) D~B L = 7.4 x 10 -8 (tI/BMB)I/2 T
/ZBV2 "6
where D~B L is the diffusion coefficient (in cm2/s) of solute A in
solvent B, when concentration of A is small (dilute solution).
The superscript oc indicates the system is dilute in solute and
for this reason concentration of solute is not included in this
equation. MB is the molecular weight of solvent (g/mol), T
is absolute temperature in kelvin, and #B is the viscosity of
solvent B (in cp). Because the solution is dilute,/Zs is almost
the same as viscosity of solution. VA is the molar volume of
solute A at its normal boiling point in cm3/mol and it may
be calculated from Vr according to the relation given in Eq.
(8.59). qrB is called association parameter for solvent where for
water the value of 2.6 is recommended [28]. For methanol and
ethanol, qra is 1.9 and 1.5, respectively. For benzene, heptane,
and unassociated solvents (most hydrocarbons) its value is
1.0 [18, 28]. The average error for this equation for some 250
systems is about 10% [18].
Another simple method derived from Tyn and Calus equa-
tion and is given as follows [18]:
(vo. 67 r
(8.61) D~a3 L = 8.93 10 -s \ ~ ] --/ZB
where the parameters and units are the same as those given in
Eq. (8.60). VB is the molar volume of solvent at its boiling point
and can be calculated from VCBsimilar to VA. Equation (8.61)
is suitable for organic and hydrocarbon systems. Because of
higher accuracy, the Wilke-Chang method (Eq. 8.60) is widely
used for calculation of diffusion coefficient of liquids and it
is also recommended in the API-TDB [5].
As shown in Table 8.8 diffusion coefficient of a binary liquid
system depends on the concentration of solute. This is the rea-
son that most experimental data on liquid diffusivity are re-
ported for dilute solutions without concentration dependency
and for the same reason predictive methods (Eqs. (8.60) and
(8.61)) are developed for diffusion coefficients of dilute solu-
tions. There are a number of relations that are proposed to
calculate D L at different concentrations. The Vignes method
suggests calculation of DAB from D~B L and Dff L as follows [35]:
(8.62) D L = (D~,BL) xB (D~L) xAOtAB
where Xg is the mole fraction of solute andxB is equal to 1 -- XA.
Parameter aAB is a dimensionless thermodynamic factor in-
dicating nonideality of a solution defined as
(01nyA~ = l +( ~l n~
(8.63) ~AB = 1 + \ Olnx~]r,p \ 81nxB]r,e
where YAis the activity coefficient of solute A and can be esti-
mated from methods of Chapter 6. For ideal systems or dilute
solutions (XA ------ 0), ags = 1.0. For simplicity in calculations for
hydrocarbon-hydrocarbon systems this parameter is taken as
unity.
Another simple relation is suggested by Caldwell and Babb
and is also recommended in the API-TDB for hydrocarbon
L
DAB
D~
X A
FIG. 8 .6--Dependency of liquid diffusion coeffi-
cients with composition.
syst ems:
(8,64) D L = ( t - XA) x ( D~ L) +xA ( D~ L)
wher e D~ r and D~ L ar e di ffusi vi t i es at i nfi ni t e di l ut i ons
and ar e known f r om exp er i ment s or may be cal cul at ed f r om
Eq. (8.60) or (8.61). F or noni deal syst ems D L cal cul at ed
f r om Eq. (8.64) mus t be mul t i p l i ed by f act or anB defi ned i n
Eq. (8.63). Thi s is demons t r at ed in Fig. 8.6 i n whi ch t he bi nar y
di ffusi on coeffi ci ent of i deal syst ems is shown by a dot t ed l i ne
whi l e t he act ual di ffusi vi t y of noni deal sol ut i ons is shown by a
sol i d line. Ri azi and Dauber t [37] showed t hat cor r esp ondi ng
st at e ap p r oach can al so be used t o cor r el at e di ffusi on coeffi-
ci ent of l i qui ds and devel op ed a gener al i zed char t for r educed
di ffusi vi t y (Dr = D/ Dc) , in a f or m si mi l ar t o Eq. (8.21), for cal-
cul at i on of l i qui d di ffusi vi t y at l ow pressures.
8. 3. 3 Di ffusi vi ty of Gases and Li qui ds at Hi g h
Pressures
Pr essur e has si gni fi cant effect on di ffusi vi t y of gases whi l e it
has l esser effect on l i qui d diffusivity. At very hi gh p r essur es
di ffusi on coeffi ci ent s of gases ap p r oach t hose of l i qui ds. For
cal cul at i on of di ffusi on coeffi ci ent s of gases at hi gh pr essur es,
Sl at t er y and Bi r d [38] devel oped a gener al i zed char t in t er ms
of ( PD) / ( PD) ~ versus Tr and Pr. The char t is in gr ap hi cal f or m
and is based on a very few dat a on sel f-di ffusi on coeffi ci ent
of si mp l e gases, whi ch were avai l abl e six decades ago. Lat er
Takahashi [39] p r op os ed a si mi l ar and i dent i cal char t but us-
i ng mor e dat a on self- as wel l as some bi nar y- di f f usi on coef-
ficients. Obvi ousl y t hese met hods cannot be used wi t h com-
p ut er t ool s and use of t he char t s is i nconveni ent to obt ai n
an accur at e val ue of di ffusi on coefficient. However, Sl at t er y-
Bi r d char t has been i ncl uded i n t he API -TDB [5].
Si gmund [40] meas ur ed and r ep or t ed bi nar y di ffusi on co-
efficient of dense gases for C1, C2, C3, n-C4, and N2 for t he
p r essur e r ange of 200-2500 psi a ( 14-170 bars) , t emp er at ur e
r ange of 38-105~ ( 100-220~ and mol e fract i on r ange of
0. 1-0. 9 for met hane. Samp l e of Si gmund' s dat aset for some
bi nar y syst ems ar e gi ven i n Table 8.9. Si gmund al so r ep or t ed
exp er i ment al dat a on t he densi t y of mi xt ur es and based on
t he ori gi nal wor k of Dawson et al. [41] cor r el at ed r educed
densi t y- di f f usi vi t y p r oduct (pDnB) to t he r educed densi t y in
a p ol ynomi al f or m as follows:
(pDAg---2) -- 0.99589 + 0.096016p~ -- 0. 22035p 2 + 0. 032874p 3
( pD~) ~
(8.65)
wher e (pDAB) ~ is a val ue of (pDAB) at l ow p r essur e for an
i deal gas and shoul d be cal cul at ed f r om Eq. (8.57). F or devel-
opi ng t hi s cor r el at i on, Si gmund used l i qui d di ffusi vi t y dat a
for bi nar y syst ems of CI - n- C, (n var i ed f r om 6 to 16) i n ad-
di t i on to di ffusi vi t y dat a of dense gases. The mai n advant age
of t hi s equat i on is t hat it can be used for bot h gases and liq-
ui ds and for t hi s r eason r eser voi r engi neers usual l y use t hi s
met hod for cal cul at i on of di ffusi on coefficients of r eser voi r
fluids under r eser voi r condi t i ons. However, t he mai n di sad-
vant age of t hi s met hod is its sensi t i vi t y t o r educed densi t y for
l i qui d syst ems wher e r educed densi t y ap p r oaches 3. This is
shown in Fig. 8.7 in whi ch ( pDAB) / ( pDAB) ~ is p l ot t ed versus pr
accor di ng to Eq. (8.65). For gases wher e Pr < I , r educed dif-
fusi vi t y ( pDAB) / ( pDAB) ~ is about unity; however, for l i qui ds
wher e Pr > 2.5 t he curve is near l y vert i cal and smal l er r or
in p woul d resul t in a much l ar ger er r or in di ffusi vi t y cal cu-
l at i on. For t hi s reason, t hi s equat i on gener al l y gives hi gher
er r or s for cal cul at i on of di ffusi on coeffi ci ent of l i qui ds even
No.
TABLE 8. 9--Diffusion coefficient of gases at high pressures [40].
Component A Component B XA T, K P, bar
1 Methane Propane 0.896 311 14.0
6 Me thane Propane 0.9 378 13.7
8 Methane n-Butane 0.946 311 137.2
14 Methane Nitrogen 0.5 313 14.1
105DAB, cm2/s
883
22.5
16.9
1196
21.6
1267
36.5
55.79
1017
62.99
16.34
1275
26.82
1870
164
232
8. APPLICATIONS: ESTI M ATI ON OF TRANSPORT PROPERTI ES 349
1.2
1.0
o.8 Nk~ ;
0.6
0.4 ~
0.2
\
0.0
0.1 1
Reduced Density, p/p(o)
l 0
FIG. 8 ,7 ~ Cor r el at i on of reduced di ffusi v i ty v ersus re-
duced densi ty ( Eq. 8 ,65) ,
t hough some modifications have been proposed for Pr > 3.
For example, it is suggested that the right-hand side of Eq.
(8.65) be replaced by 0.18839 exp(3 - pr), when Pr > 3 [42].
There are some other empirical correlations for estimation of
diffusion coefficient of light gases in reservoir fluids. For ex-
ample, Renner proposed the following empirical correlation
for calculation of Di-oil in gas injection projects [43]:
t ~- 8 -0.4562 9 1.706 r*-l.831"a~4.524
( 8. 66) DA-oil = 7.47 x it) /Zoi 1 lvl A PMA r l
where DA_oi I is the effective diffusivity of light gas A (C1, C2,
Ca, CO2) in an oil (reservoir fluid) in cm2/s./Zoil is the viscosity
of oil (free of gas A) at T and P in cp, MA is molecular weight
of gas A, PMA is mol ar density of gas A at T and P in mol / cm 3,
P is pressure in bar, and T is absolute t emperat ure in kelvin.
Exponent 4.524 on T indicates that estimated value of DA-oil
is quite sensitive to the value of T considering that the value
of T is a large number. This equation was developed based
on 140 data points for the ranges 1 < P < 176 bar, 273 < T <
333 K, and 16 < Mi < 44. As ment i oned earlier such empirical
correlations are mainly accurate for the data used in their
development.
Another generalized correlation for diffusion coefficient of
dense fluids was developed by Riazi [9]. For liquids, accord-
ing to the Stokes-Einstein and Eyring theories [44], diffusion
coefficient is inversely proport i onal to viscosity (D c(1/ #). I f
it is further assumed that the deviation of diffusivity of a gas
from ideal gas diffusivity is proport i onal to the viscosity de-
viation the following correlation can be developed between
reduced diffusivity and reduced viscosity [9]:
(pDAB) --a ( # ~b+cPr
a = 1. 07 b= - 0. 27- 0. 38~o
(8.67) c = -0. 05 + 0.1~o Pr = P/Pc
Tc = xATcA + XBTcB Pc = XAPcA + XBPcB
60 = XA0)A + XBO)B
where (pDga) ~ must be determined from Eq. (8.57)./z ~ must
be calculated from Eqs. (8.6) and (8.8). I f experimental data
on/ z are not available it should be calculated from Eq. (8.12)
for bot h liquids and gases. Coefficients a, b, and c have
been det ermi ned from data on diffusion coefficients of some
300 bi nary systems as shown in Table 8.10. Errors for bot h
Eqs. (8.65) and (8.67) are also shown in this table. I n eval-
uation of Eq. (8.65) the coefficients were reevaluated from
the same data bank as given in Table 8.10. When Eq. (8.67) is
evaluated against 17 diffusivity data points for bi nary systems
that were not used in the development of this equation, an av-
erage error of 9% was observed [9]. Furt hermore Eq. (8.67)
was evaluated with D~tB L of some dilute binary liquids at atmo-
spheric pressure and results show that it is comparabl e with
the Wilke-Chang equation (8.60) specifically developed for
liquids [9].
The mai n objective of development of Eq. (8.67) was to
have a unified predictive met hod for bot h gas and liquid
diffusivities, which can be safely used for diffusivity predic-
tion of heavy hydrocarbon fluids. The extrapolation ability of
Eq. (8.67) can be seen from the linear relationship between
(pDAB)/(pDAB) ~ and (/z//~ ~ on a log-log scale. For this reason,
this equation can be used with good accuracy for heavy oils up
to molecular weight of 350. Equation (8.67) was developed for
dense gases and for this reason data on diffusion coefficient
of gases at at mospheri c pressure were not used in determina-
tion of its coefficients. Theoretically, coefficient a in Eq. (8.67)
must be unity, but value of 1.07 was obtained from regression
of experimental data. This is mainly due to the fact that major-
ity of data used were at high pressure (see Table 8.10). How-
ever, even at low pressure where/ z/ / z ~ = 1, this equation gives
average deviation of 7% from the Stokes-Einstein equation,
which is within the range of errors for calculation of diffu-
sivity at higher pressures. The Stokes-Einstein equation (Eq.
8.57) usually underpredicts diffusivity at at mospheri c pres-
sure and for this reason coefficient of 1.07 improves accuracy
of prediction of diffusivity at low pressures. However, for low-
pressure gases and liquids, met hods proposed in previous sec-
tions may be used. Although this equation was developed for
hydrocarbon systems, but when applied to some nonhydro-
carbon systems, reasonably good results have been obtained
as shown in the following example.
TABLE 8,10--Data used for development of Eq. (8.67).
Binary M range of P range,
Dense fluid systems No. of data barrier a bar
Gas es N2, C1, Cz, Ca, C4 140 16-58 7- 416
Li qui ds C1, C3, C6, C10, Oil 143 44- 340 2- 310
aMolecular weight range of heavier component in the binary systems. Ref. [9].
%AAD
T range, (/z/~ ~ 104D~, Eq. Eq.
K Range cmZ/s (8.67) (8.65)
155- 354 1-15 1. 4-240 8.1 10.2
274- 411 4- 20000 0. 01- 5 15.4 48. 9
Exampl e 8. 4- - Es t i mat e t he di ffusi vi t y of benzene i n a bi nar y
mi xt ur e of 74.2 mol % acet one and 25.8 mol % car bon t et ra-
chl or i de (CC14) at 298 K and 1 at m pr essur e.
Sol ut i on- - The syst em is a t er nar y mi xt ur e of benzene,
acet one, and CC14. Consi der benzene, t he sol ut e, as comp o-
nent A and t he mi xt ur e of acet one and CCI4, t he solvent, as
comp onent B. Because amount of benzene is smal l ( di l ut e
syst em) , XA = 0.0 and XB = 1.0. TcB = 520, PcB = 46.6 bar,
VcB = 226.3 cm3/ mol, co = 0.2274, M = 82.8. These p r op -
ert i es ar e cal cul at ed f r om p r op er t i es of acet one and CC14
as gi ven i n Ref. [45]. Act ual l y t he l i qui d sol vent is t he
same as t he l i qui d i n Examp l e 8.1, cal cul at ed p r op er t i es
of whi ch ar e p = 0.012422 mol / cm 3, /z ~ = 0.00829 a nd/ z =
0.374, t hus/ z/ / z ~ = 45.1677. F r om Eq. (8.67), b = - 0. 356, c =
- 0. 02726, and (pDAB)/(pDAB)~ 0.2745. F r om Eq. (8.57),
(pDAB) ~ = 1.28 10 -6 mol / cm- s . Therefore, DAB = 1.28 x
10 -6 x 0. 2745/ 0. 012422 = 2.83 x 10 -5 cm2/s. I n comp ar i s on
wi t h t he exp er i ment al val ue of 2.84 x 10 -5 cm2/ s [ 10] an er r or
of - 0. 4% is obt ai ned. I n t hi s examp l e bot h/ z and p have been
cal cul at ed, whi l e in many cases t hese val ues may be known
f r om exp er i ment al measur ement s. #
8. 3. 4 Di f f usi on Coef f i ci ent s
i n Mut l i component S y st ems
I n mul t i comp onent syst ems, di ffusi on coeffi ci ent of a com-
p onent (A) in t he mi xt ur e of N comp onent s is cal l ed effective
diffusion coefficient and is shown by DA-mix. Based on t he ma-
t er i al bal ance and i deal gas l aw Wi l ke der i ved t he fol l owi ng
r el at i on for cal cul at i on of DA-mix [46]:
(8.68) Dg-mix ----- 1 -- YA
EiN~A Yi
wher e yi is t he mol e fract i on of i and Dg-i is t he bi nar y dif-
fusi on coeffi ci ent of A in i. Thi s equat i on may be used for
p r essur es up to 35 bar; however, because of l ack of a rel i abl e
met hod, t hi s is al so used for hi gh- p r essur e gases and l i qui ds
as wel l [9]. For cal cul at i on of DA-mt~ in l i qui ds t he met hod
of Leffler and Cul l i nan is r ecommended i n t he API -TDB [5].
This met hod r equi r es bi nar y di ffusi on coeffi ci ent s at i nfi ni t e
di l ut i on D~_ L, mol e fract i on of each comp onent x4, l i qui d
vi scosi t y of each comp onent / zi , and vi scosi t y of l i qui d mi x-
t ure/ zm. However, t hi s met hod is not r ecommended i n ot her
sources and is not wi del y p r act i ced by p et r ol eum engi neers.
Ri azi has p r op os ed cal cul at i on of DA-mix for bot h gases and
l i qui ds at l ow and hi gh- p r essur e syst ems by as s umi ng t hat
t he mi xt ur e can be consi der ed as a bi nar y sol ut i on of A and B
wher e B is a p s eudocomp onent comp os ed of all comp onent s
i n t he mi xt ur e except A. DA-mlx is as s umed to be t he same as bi-
nar y diffusivity, DAB, whi ch can be cal cul at ed f r om Eq. (8.67).
Dg-mix is cal cul at ed f r om t he fol l owi ng r el at i ons [9]:
Dg-mix = DAB
(8.69) ~_ ,i~1 x40i
i~A
OB-- U
wher e 0B is a p r op er t y such as To, Pc, or w for p s eudocomp o-
nent B. Thi s met hod is equi val ent to t he Wilke' s met hod ( Eq.
8.68) for l ow- pr essur e gases at i nfi ni t e di l ut i on (i.e., Xn --~ 0).
F or a t er nar y syst em of C1-C3-N2 at l ow pr essur e, t he effec-
tive di ffusi on coeffi ci ent of C1 in t he mi xt ur e cal cul at ed f r om
Eq. (8.69) differs by 2- 3% f r om Eq. (8.68) for mol e f r act i on
r ange of 0. 0-0. 5 [9]. Ap p l i cat i on of Eq. (8.69) was pr evi ousl y
shown i n Examp l e 8.4.
8. 3. 5 Di f f usi on Coeffi ci ent i n Porous Medi a
The pr edi ct i ve met hods p r esent ed i n t hi s sect i on are appl i -
cabl e t o nor mal medi a fully filled by t he fluid of i nt erest . I n
cat al yt i c r eact i ons and hydr ocar bon reservoi rs, t he fl ui d is
wi t hi n a p or ous medi a and as a resul t for mol ecul es it t akes
l onger t i me to t ravel a specific l engt h in or der to diffuse. Thi s
in t ur n woul d resul t i n l ower i ng di ffusi on coefficient. The
effective di ffusi on coeffi ci ent i n a p or ous medi a, DAB,elf can
be cal cul at ed as
DAB
(8.70) DAB,eft-- rn
wher e DAB is t he di ffusi on coeffi ci ent in absence of p or ous me-
di a and exp onent n is usual l y t aken as one but ot her val ues of
n ar e al so r ecommended for some p or ous medi a syst ems [47].
r is a di mensi onl ess p ar amet er cal l ed tortuosity defi ned to in-
di cat e degree of comp l exi t y in connect i on of free p at hs i n a
p or ous medi a. I ts defi ni t i on is demons t r at ed i n Fig. 8.8 ac-
cor di ng t o t he fol l owi ng rel at i on:
Actual free distance between points a and b in porous media
Distance of a straight line between a and b
(8.71)
Si nce act ual di st ance bet ween a and b is al ways gr eat er t han a
st r ai ght l i ne connect i ng t he t wo poi nt s, r > 1.0. For det er mi -
nat i on of r in an i deal medi a, assumi ng all par t i cl es t hat f or m
a p or ous medi a ar e spheri cal , t hen as shown in Fig. 8.9 t he
ap p r oxi mat e val ue of t or t uosi t y can be cal cul at ed as r ~ 1.4.
I n act ual cases such as for p et r ol eum reservoi rs wher e t he
Free distance between a and b
Distance of straight line between a and b
FIG. 8 . 8 wDi st ance for trav el i ng a mol ecul e from at o b
in a porous media and concept of tortuosity.
8. APPLI CATI ONS: ESTI M ATI ON OF TRANSPORT PROPERTI ES 351
a b
2L _ 1 1 1 = ~- = 1. 4
2LCos0 Cos0 Cos45~ ~r2 J
2
FIG. 8 .9--A pproximate calculation of tortuosity ( r) .
size and shape of par t i cl es ar e all di fferent , val ue of r vari es
f r om 3 t o 5.
I n a p or ous medi a r is r el at ed to t he formation resistivity
factor and porosity as
(8.72) r = ( F r ~
wher e F is t he resi st i vi t y and r is t he porosi t y, bot h are di-
mensi onl ess p ar amet er s. r is t he fract i on of connect ed emp t y
space in a p or ous medi a and F is an i ndi cat i on of el ect ri cal
r esi st ance of mat er i al s t hat f or m t he p or ous medi a and is
al ways gr eat er t han unity, nl is a di mensi onl ess emp i r i cal pa-
r amet er t hat dep ends on t he t ype of p or ous medi a. Theoret i -
cally, val ue of nl in Eq. (8.72) is one; however, in p r act i ce nl is
t aken as 1.2. Vari ous r el at i ons bet ween z and r ar e given by
Amyx et al. [48] and Langness et al. [49]. One gener al r el at i on
is gi ven as fol l ows [48]:
(8.73) r = ar -m
wher e p ar amet er s a and m ar e specific of a p or ous medi a.
Par amet er m is cal l ed cementation factor and it is specifi-
cal l y a char act er i st i c of a p or ous medi a and it usual l y vari es
f r om 1.3 to 2.5. Some r esear cher s have at t emp t ed to cor r el at e
p ar amet er m wi t h p or osi t y and resistivity. F or some reser-
voi rs a = 0.62 and m = 2.15, whi l e for some ot her reservoi rs,
when ~b > 0.15, a = 0.75 and m= 2 and for r < 0.15, a = 1
and m = 2. By combi ni ng Eqs. (8.72) and (8.73) wi t h nl = 1.2
and a = 1:
( 8.74) r = ~i.2-1.2m
Equat i on (8.74) can be combi ned wi t h Eq. (8.70) to est i mat e
effective di ffusi on coefficients in a p or ous medi a. Par amet er
r ai n Eq. (8.74) can be t aken as an adj ust abl e par amet er , whi l e
for si mpl i ci t y, p ar amet er n in Eq. (8.70) can be t aken as unity.
I n p r act i cal appl i cat i ons, engi neer s use si mp l er r el at i ons
bet ween t or t uosi t y and porosi t y. F or exampl e, Font es et al.
[50] suggest t hat for cal cul at i on of di ffusi on coeffi ci ent s of
gases in p or ous sol i ds (i.e., cat al yt i c r eact or s) effective diffu-
si on coeffi ci ent s can be cal cul at ed f r om t he fol l owi ng equa-
tion:
(8.75) Deft = r
This equat i on can be obt ai ned f r om Eq. (8.70) by assumi ng
~n ~___ (bl.5.
8 . 4 I N T ER R EL A T I ON S H I P A MON G
T R A N S POR T P R OP ER T I ES
I n previ ous sect i ons t hr ee t r ansp or t p r op er t i es of/ z, k, and D
were i nt r oduced. I n t he pr edi ct i ve met hods for t hese mol ecu-
l ar pr oper t i es, t her e exist some si mi l ar i t i es among t hese pr op-
erties. Most of t he pr edi ct i ve met hods for t r ansp or t p r op er t i es
of dense fluids are devel op ed t hr ough r educed density, Pr. I n
addi t i on, di ffusi on coeffi ci ent s of dense fluids and l i qui ds ar e
r el at ed to viscosity. Ri azi and Dauber t devel oped several re-
l at i onshi p s bet ween/ z, k, and D bas ed on t he p r i nci p l e of
di mensi onal anal ysi s [37]. F or exampl e, t hey f ound t hat for
l i qui ds I n (# 2/3D/T) versus I n (T/Tb) is l i near and obt ai ned t he
fol l owi ng rel at i ons:
/Z2/3 ( T~ 0"7805
- - D = 6.3 x 10 -8 for l i qui ds except wat er
; / 3 ( r
- - D = 10.03 x 10 -s for l i qui d wat er
(8.76)
wher e/ ~ is l i qui d vi scosi t y in cp ( mPa. s), T is t emp er at ur e
in kelvin, D is l i qui d sel f-di ffusi vi t y in cruZ/s, and Tb is nor-
mal boi l i ng p oi nt i n kelvin. For exampl e, for n-C5 i n whi ch Tb
is 309 K t he vi scosi t y and sel f-di ffusi on coeffi ci ent at 25~
(298.2 K) are 0.215 cp and 5.5 x 10 -4 cm2/s, respectively.
Equation (8.76) gives value of D = 5.1 x 10 -5 cm2/s. This
equation is developed based on very few compounds includ-
ing polar and nonpol ar substances and is not recommended
for accurate estimation of diffusivity. However, it gives a gen-
eral trend between viscosity and diffusivity. Similarly the
following relation was derived between/ ~, k, and D [37]:
(8.77) - - D = 4.2868 x 10 -9 - ~-
\Cs /
where k is the liquid t hermal conductivity in W/ m. s, cs is the
velocity of sound in m/s, and/ z is liquid viscosity in cp. Values
of cs can be calculated from met hods given in Section 6.9.
Again it should be emphasi zed that this equation is based
on very few compounds and data and it is not appropri at e
for accurate prediction of D from k. However, it shows the
interrelationship among the t ransport properties.
Riazi et al. [23] developed a generalized relation for predic-
tion of/ z, k, and D in t erms of refractive index parameter, I .
I n fact, the Hildebrand relation for fluidity (Eq. 8.26) can
be extended to t hermal conductivity and diffusivity and the
following relation can be derived based on Eq. (8.27):
(8.78)
1 1
whe r e 0- ' k ' D
/z
z~ 2
v
in which A, B, and p are constants specific for each prop-
erty and each compound. These constants for a large number
of compounds are given in Ref. [23]. Equation (8.78) is de-
veloped for liquid hydrocarbons. Paramet er I is defined in
t erms of refractive index (n) by Eq. (2.36) and n must be eval-
uat ed from n20 using Eq. (2.114) at the same t emperat ure at
which a t ransport propert y is desired. Methods of estimation
of refractive index were discussed in Section 2.6.2. The linear
relationships between 1//z or D and ( 1/ I- 1) are shown in
Figs. 8.10 and 8.11, respectively. Similar relations are shown
for k of several hydrocarbons in Ref. [23]. Equation (8.78)
can reproduce original data with an average deviation of less
t han 1% for hydrocarbons from Cs to C20.
Equation (8.78) is applicable for calculation of t ransport
properties of liquid hydrocarbons at at mospheri c pressures.
Coefficients A, B, and p for a number of compounds are given
in Table 8.11. As shown in this table, paramet er p for ther-
mal conductivity is the same for all compounds as 0.1. For
n-alkanes coefficients of p, A, and - B/ A have been correlated
to M as given in Table 8.12 [23]. Equation (8.78) with coeffi-
cients given in Table 8.12 give average deviations of 0.7, 2.1,
and 5.2% for prediction of/ z, k, and D, respectively. Exampl e
8.5 shows application of this met hod of prediction of trans-
port properties.
Example 8. 5--Est i mat e the t hermal conductivity of n-decane
at 349 K using Eq. (8.78) with coefficients predicted from
correlations of Table 9.11. The experimental value is 0.119
W/ m- K as given by Reid et al. [18].
Val u~ of p
O n-octane, data
n-octane, predicted
n-octane 0.75
pmpylcyclopentane, data propylcyclopentane 0.86
........ propytcyclopentane, predicted ~ 0,86
A b enz ene, data / o
J
. . . . . benzene, predicted /
/
/ j' ~
; N"
' " 9 ! . . . . . . . 9 . . . . . . . . . ! - ' . . . . 9 ! L
2~0 2 . 2 2 . 4 2 . 6 2 . 8
O / i - I ) p
FIG. 8 .10---Relationship between fluidity (1//~) (/~ is in mPa. s ( cp) ) and refractive index
parameter I from Eq. ( 8 .78 ) . A dopted from Ref. [23].
op
1'
0 " : i ~ . . . . . . ~ I " ~ " " ~ ' i 9 T i . . . . . . . . . . .
t, 5 1,6 t, 7 1.8 1.9 2,0 2, i
tan - t ) P
FIG. 8 . 1 1 - - Rel at i onshi p between diffusivity ( D) ( D is in 10 s cm2/s) and refractive
index parameter I from Eq. (8 .78 ). W ith permission from Ref. [23].
TABLE 8. 11--~oefficients of Eq. (8. 78) for some liquid hydrocarbons with permission
from Ref. [231.
Compound M n2o p A - B/ A T range, K
9
0 n-nonane, data ( p=0,546) /
- - n-flOr~13e, predicted / /
Coefficients for viscosity ( mPa. s or cp)
n-Pent ane 72.2 1.3575 0.747 7.8802 2.2040 144-297
n-Decane 142.3 1.4119 0.709 6.3394 2,0226 256-436
n-Ei cosane 282.6 1.4424 0.649 4.5250 1.8791 311-603
Cycl opent ane 70.1 1.4065 0.525 8.3935 1.6169 250-322
Met hyl cycl opent ane 84.2 1,4097 0.584 7.9856 1.7272 255-345
n-Decylcyclopentane 210.4 1.4487 0.349 8.5664 1.3444 255-378
Cyclohexane 84.2 1.4262 0.567 8.8898 1.7153 288-345
n-Pentylcyclohexane 154.3 1.4434 0.650 6.5114 1.8300 255-378
n-Decylcyclohexane 224. 4 1. 4534 0.443 7.7700 t . 4899 255-378
Benzene 78.1 1.5011 0.863 11.2888 1.9936 278-344
Toluene 92.1 1.4969 0.777 9.9699 1.8321 233-389
n-Pent yl benzene 148.2 1.4882 0.740 7.6244 1.8472 255-411
n-Decylbenzene 218.4 1.4832 0.565 7.0362 1.6117 255-411
Wat er 18 1.3330 0.750 6.3827 2.5979 273-373
Met hanol 32.0 1.3288 0.919 4.8375 3.1701 268-328
Et hanol 46.1 1.3610 0.440 8.2649 1.6273 280-338
Coefficients for t her mal conduct i vi t y (W/ rnK)
n-Pent ane 72.2 1.3575 0.1 2.6357 0.6638 335-513
n-Decane 142.3 1.4119 0.1 2.0358 0.5152 256-436
n-Ei cosane 282.6 1.4424 0.1 1.6308 0.3661 427-672
Cycl opent ane 70.1 1.4065 0.1 2.5246 0.6335 328-551
Met hyl cycl opent ane 84.2 1.4097 0.1 2.3954 0.6010 328-551
Cyclohexane 84.2 1.4262 0.1 1.9327 0.4755 411-544
Benzene 78.1 1.5011 0.I 2.6750 0.6384 410-566
Toluene 92.1 1.4969 0.I 2.1977 0.5358 354-577
Et hyl benzene 106.2 1.4959 0.1 2.0965 0.5072 354-577
Coefficients for self-diffusion coefficients (105 x crn2/s)
n-Pent ane 72.2 1.3575 0.270 10.4596 1.2595 195-309
n-Decane 142.3 1.4119 0.555 10.0126 1.7130 227-417
Benzene 78.1 1.5011 0.481 16.8022 1.4379 288-313
Wat er 18 1.3330 0.633 14.6030 2.2396 273-373
Met hano] 32.0 1.3288 0.241 11.6705 1.2875 268-328
Et hanol 46.1 1.3610 0.220 15. t893 1.2548 280-338
TABLE 8.12--Coefficients of Eq. (8. 78) for estimation of transport properties of liquid n-alkanes
with permission from Ref. [23].
0 Coefficients of Eq. (8.78) for n-alkanes
1//z (cp) -1 p = 0.8036 - 5.8492 x 10-4M
1/k (W/mK) -a
105D, cm2/s
A~
- B/ A =
- B/ A =
p=
A=
B=
p=
A=
- n =
2.638 + 5.2141nM + 0.0458M - 2.408M ~
2.216 - 1.235 x 10-3M - 94(lnM) -5 +2.1809 103M -2"2, if M < 185
5. 9644- 3. 625x l O- 3M +788( l nM ) - 3- 71. 441M - 0. 4, if M> 185
0.1
3. 27857- 0.01174M+ 1.6 x 10 5M2
-2.50942 + 0.0139M - 2.0 x 10-SM 2
-0.99259 + 0.02706M - 1.4936 x 10-4M 2 + 2.5383 x 10-7M 3
10.06464 + 0.02191M - 2.6223 x 10-4M 2 + 6.17943 x 10-7M 3
-9.80924 + 0.518156M - 3.31368 x 10-3M 2 + 5.70209 x 10-6M 3
Sol ut i on- - For n-Clo, f r om Table 2.1, M= 142.3. F r om
Eq. (2.42) wi t h coeffi ci ent s given in Table 2.6 for 120 of
n-al kanes we get 120 =0. 24875. F r om Eq. (2.114), n20 =
1.41185. F r om Eq. (2.118) at T = 349 K, nr = 1.38945 and
f r om Eq. (2.14) we cal cul at e IT = 0.2368. F r om Table 8.12
for 1/k we get p = 0.1, A = 1.93196, and B = - 0. 9364. Sub-
st i t ut i ng t hese val ues i n Eq. (8.78) we get ( l / k) ~ = 1.93196
( 1/ 0. 2368 - 1) ~ - 0.9364 or k = 0.1206 W/ m. K, whi ch dif-
fers f r om t he exp er i ment al val ue by 1.3%. DI PPR gives
val ue of 0.1215 W/ m. K [45]. r
8 . 5 M EA S UR EM EN T OF D I F F US I ON
C OEFFI C I EN T S I N R ES ER VOI R FL UI D S
Mol ecul ar di ffusi on is an i mp or t ant p r op er t y needed in si mu-
l at i on and eval uat i on of several oil recovery processes. Exam-
pl es are vert i cal mi sci bl e gas fl oodi ng, nont her mal recovery of
heavy oil by sol vent i nj ect i on, and sol ut i on- gas- der i ved reser-
voirs. I n t hese cases when p r essur e is r educed bel ow bubbl e
p oi nt of oil, gas bubbl es ar e f or med and t he r at e of t hei r diffu-
si on is t he cont r ol l i ng step. At t emp t s i n meas ur ement of gas
di ffusi vi t y in hydr ocar bons under hi gh- p r essur e condi t i ons
goes back to t he earl y 1930s and has cont i nued to t he r ecent
year s [37, 51-57] . I n general , met hods of measur i ng di ffusi on
coeffi ci ent s in hydr ocar bon syst ems can be di vi ded i nt o t wo
cat egori es. I n t he first category, dur i ng t he exp er i ment sam-
pl es of t he fl ui d are t aken at var i ous t i mes and ar e anal yzed by
gas chr omat ogr ap hy or ot her anal yt i cal t ool s [37, 55]. I n t he
second category, samp l es ar e not anal yzed but sel f-di ffusi on
coeff are meas ur ed by equi p ment such as NMR and
t hen bi nar y di ffusi on coeffi ci ent s are cal cul at ed [41]. Ot her
met hods i nvol ve measur i ng vol ume of gas di ssol ved in oil
ver sus t i me at const ant p r essur e i n or der to det er mi ne gas
di ffusi vi t y in r eser voi r fluids [43].
I n t he earl y 1990s a si mp l e met hod to det er mi ne diffu-
si on coeffi ci ent s i n bot h gas- gas and gas- l i qui d for bi nar y
and mul t i comp onent syst ems at hi gh p r essur es wi t hout com-
p osi t i onal meas ur ement was p r op os ed by Ri azi [56]. I n t hi s
met hod, gas and oil are i ni t i al l y p l aced i n a PVT under con-
st ant t emp er at ur e condi t i on. As t he syst em ap p r oaches its
equi l i br i um t he p r essur e as wel l as gas- l i qui d i nt er p hase po-
si t i on i n t he cell var y and ar e meas ur ed versus t i me. Based on
t he rat e of change of p r essur e or t he l i qui d level, rat e of diffu-
si on in each p hase can be det er mi ned [56]. The mechani s m of
di ffusi on p r ocess is based on t he p r i nci p l e of t her modynami c
equi l i br i um and t he deri vi ng force in mol ecul ar di ffusi on is
t he syst em' s devi at i on f r om equi l i br i um. Therefore, once a
nonequi l i br i um gas is br ought i nt o cont act wi t h a l i qui d, t he
syst em t ends to ap p r oach equi l i br i um so t hat t he Gi bbs en-
ergy, and t her ef or e pr essur e, decr eases wi t h t i me. Once t he
syst em has r eached an equi l i br i um st at e t he p r essur e as wel l
as comp os i t i on of bot h gas and l i qui d p hases r emai ns un-
changed. Schemat i c of t he p r ocess is shown i n Fig. 8.12. I f
t he gas p hase is hydr ocar bon, di ssol ut i on of a hydr ocar bon
gas in an oil causes i ncr ease in oil vol ume and hei ght of l i qui d
(Lo) i ncreases. F or t he case of ni t rogen, t he r esul t is op p osi t e
and di ssol ut i on of N2 causes decr ease i n t he oil vol ume. I n
f or mul at i on of di ffusi on p r ocess in each phase, t he Fick' s l aw
and mat er i al bal ance equat i ons ar e ap p l i ed for each comp o-
nent in t he syst em. At t he i nt er phase, equi l i br i um cr i t er i on
od gas
is i mp os ed on each comp onent ( [ / =/~i ). I n addi t i on, at
t he i nt er p hase t he rat es of di ffusi on i n each p hase are equal
for each comp onent . A semi anal yt i cal model for cal cul at i on
of rat es of di ffusi on pr ocess i n bot h p hases of gas and l i qui d
is gi ven by Ri azi [56]. The model is a combi nat i on of mat e-
ri al bal ance and vap or - l i qui d equi l i br i um cal cul at i ons. When
t he di ffusi on pr ocesses come to an end t he syst em will be at
equi l i br i um. Di ffusi on coeffi ci ent s needed i n t he model ar e
cal cul at ed t hr ough a met hod such as Eq. (8.67). The model
pr edi ct s comp os i t i on of each phase, l ocat i on of t he l i qui d in-
t erface, and p r essur e of t he syst em versus t i me.
To eval uat e t he p r op os ed met hod, p ur e met hane was p l aced
on p ur e n- p ent ane at 311 K (100~ and 102 bar in a PVT cell
of 21.943 cm hei ght and 2.56 cm di amet er. The i ni t i al vol-
ume of l i qui d was 35% of t he cell vol ume. Pr essur es were
meas ur ed and r ecor ded manual l y at sel ect ed t i mes and con-
t i nuousl y on a st ri p chart . The l i qui d level was meas ur ed
manual l y wi t h a p r eci si on of 4-0.02 mm. Measur ement s were
cont i nued unt i l t her e is no change i n bot h p r essur e and liq-
ui d l engt h at whi ch t he syst em r eaches equi l i br i um. Diffu-
si on coeffi ci ent s were cor r ect ed so t hat p r edi ct ed p r essur e
curve versus t i me mat ches t he exp er i ment al dat a as shown
in Fig. 8.13. When di ffusi vi t i es cal cul at ed by Eq. (8.67) ar e
mul t i p l i ed by 1.1 t he model p r edi ct i on per f ect l y mat ches
exp er i ment al dat a. Thi s t echni que measur es di ffusi on coef-
ficient of Ca-C5 in l i qui d p hase at 311 K and 71 bar s as
1.51 x 10 -4 cm2/s, whi l e t he exp er i ment al dat a r ep or t ed by
Reamer et al. [52] is 1.43 10 4 cm2/s. Di ffusi on coeffi ci ent s
of C1-C5, in bot h gas and l i qui d phases, versus pr essur e, and
comp os i t i on ar e shown in Figs. 8.14 and 8.15, respectively.
Diffusivity of met hane i n heavy oils ( bi t umens) at 50 bar and
50~ is wi t hi n t he or der of magni t ude of 5 10 -4 cm2/s, whi l e
et hane di ffusi vi t y i n such oils is about 2 10 -4 cm2/ s [55].
8. APPLICATIONS: ESTIM ATION OF TRANSPORT PROPERTIES 3 5 5
Lo
GAS 1 zg
! !!ill ! ,, i!i!i !! iiiiii!i
L
O_<z_<L
z =Zo i f z ~ L o
Z= Zg + Lo if z >_ Lo
FIG. 8 . 1 2- - Schemat i c and di mensi ons of a constant v ol ume cell. Taken wi th per-
mission from Ref. [ 23].
This met hod can be ext ended t o muki comp onent syst ems
and it has been successfully used to measure gas diffusiv-
ity in heavy oils [57, 58]. I n this met hod, pressure mea-
surement is mor e accurat e t han measur i ng t he i nt erphase
location. Furt hermore, the initial measur ement s are mor e
critical t han measur ement s near the final equi l i bri um condi-
tion. The amount of initial liquid or gas det ermi nes diffusivity
of whi ch phase can be measur ed mor e accurat el y [56]. As it
can be seen from Fig. 8.13 once a correct value of diffusion co-
efficient is used, the model predi ct i on mat ches experi ment al
dat a t hr oughout the curve. This confi rms the validity of the
1 00"
o e x ~ n e n t af
: R~ W P~ s~ n I ~ ; ~ Co e H ~ t s
g O "
k'- 85 ~
7 5 ~
0 50 1 O0 1 50 200 250
Time, hr
FIG. 8 . 1 3 - - V ari at i on of pressure for the Cl - Cs constant vol-
ume di ffusi on ex peri ment at 311 K. - - Diffusion coeffi ci ent from
Eq. ( 8 .67) ; . . . . . . di ffusi on coeffi ci ent from Eq. ( 8 ,67 ) multiplied
by 1.1; . . . . . . diffusion coeffi ci ent from Eq. ( 8 .65) . Tak en with
permi ssi on f rom Ref, [ 56],
model and the assumpt i ons made in its formul at i on. Met hods
t hat use unrealistic assumpt i ons, i.e., neglecting nat ural con-
vection t erms when it exists, or oversimplified boundar y con-
ditions (i.e., semiinfinite assumpt i on) lead to predi ct i ons t hat
do not mat ch the entire curve. I n these cases, report ed diffu-
sion coefficients are based on a port i on of experi ment al dat a
and this is t he reason t hat in such cases differences as large
as ~ 100% are report ed for diffusion coefficients in liquids at
hi gh pressures for the same systems under the same condi-
tions. The t echni que can also be used to measure diffusivity
in p or ous medi a by pl aci ng a reservoir core in t he bot t om of a
PVT cell sat urat ed initially with liquid oil. With such experi-
ment s and availability of mor e data, Eq. (8.67) can be furt her
studied, modified, and improved.
t .5-
O
Q
0
/
/
/
I
. . . . . . . . . . . . : :
i - ~ ----,-4=~ GA S I
j , ' ' " " " | I
/ t .,
-8
50 10Q t ~ 26 0
T i me, hr
FIG. 8 . 1 4---Di ffusi on coeffi ci ent of the methane--n-pentane
system at 311 K for the liquid and g as phases. Tak en with
permi ssi on f rom Ref, [ 56].
E
o
x % 4 -
o
C I
1 ,1 -
""X " - - "
\
\ ' \ .
"x
. . . . . . . . ' ~= I ~ l I ' I 9 ! ' ....
0.3~
-0,3
-0.2 ~
i LI.
0.t
- 0. 05
0.0
70 7~ go 85 go g5 100
Pressure, bar
FI G. 8 . 1 5- - Di f f us i on coef f i ci ent of l i qui d phase met hane- n- p ent ane sys-
t em at 311 K v er sus pr essur e and composi t i on. T ak en wi t h per mi ssi on
f rom Ref. [ 56].
8 . 6 S UR FA C E/ I N T ER FA C I A L T EN S I ON
Surface t ensi on is an i mp or t ant mol ecul ar p r op er t y i n reser-
voi r engi neer i ng cal cul at i ons. I n addi t i on, surface t ensi on is
needed for t he desi gn and op er at i on of gas- l i qui d sep ar a-
t i on uni t s such as di st i l l at i on and absor p t i on col umns. Based
on t he di fference bet ween surface t ensi on of t op and bot t om
p r oduct s, one can det er mi ne whet her or not f oami ng woul d
occur in a di st i l l at i on or abs or p t i on col umn. F oam f or mat i on
is t he cause of maj or p r obl ems i n sep ar at i on of gas and liq-
ui d phases. I n t hi s sect i on surface and i nt er f aci al t ensi ons ar e
defi ned and t hei r ap p l i cat i on i n cal cul at i on of capi l l ar y pres-
sure is demonst r at ed. Capi l l ar y p r essur e can be an i mp or t ant
f act or i n det er mi nat i on of rat e of oil movement and p r oduc-
t i on f r om a reservoi r. F or t hi s reason, surface t ensi on is al so
cat egor i zed as a t r ans p or t p r op er t y al t hough it is di fferent i n
nat ur e f r om ot her mai n t r ans p or t pr oper t i es. Met hods of es-
t i mat i on of surface t ensi on ar e p r esent ed in t he second p ar t
of t hi s sect i on.
8. 6. 1 Theory and Def i ni t i on
Surface t ensi on of a l i qui d is t he force r equi r ed for uni t in-
cr ease in l engt h. A cur ved surface of a l i qui d, or a cur ved in-
t erface bet ween p hases ( l i qui d- vap or or l i qui d- l i qui d) , exert s
a p r essur e so t hat t he p r essur e is hi gher i n t he p hase on t he
concave si de of t he i nt erface. Surface t ensi on is a mol ecul ar
p r op er t y of a subst ance and is a char act er i st i c of t he i nt erface
bet ween t wo phases. I n fact, t her e are unequal forces act i ng
up on t he mol ecul es in t wo si des of t he i nt erface, whi ch is due
t o di fferent i nt er mol ecul ar forces t hat exist in t wo phases. F or
t he case of a vap or and l i qui d ( pure subst ance) , t he forces be-
t ween gas mol ecul es ar e less t han t he force bet ween l i qui d-
l i qui d mol ecul es, whi ch cause t he cur vat ur e on t he l i qui d
surface. I t is due to t hi s p henomenon, t hat l i qui d dr op l et s
f or m sp her i cal shap es on a sol i d surface (i.e., dr op l et of l i qui d
mer cur y as seen in Fig. 8.16). Gener al l y t ensi on for vap or -
l i qui d i nt erface ( pure subst ances) is r ef er r ed t o as surface
tension and t he t ensi on bet ween t wo di fferent l i qui ds (i.e.,
oi l - wat er ) is r ef er r ed as interfacial tension ( I FT) . However,
t hese t wo t er ms ar e used i nt erchangeabl y. Surface t ensi on is
shown by a and in t he SI uni t syst em it has t he uni t of N/ m
but usual l y t he uni t of dyne/ cm (1 dyne/ cm = 10 -3 N/ m --
1 mN/ m) is used.
Based on t he p r i nci p l e of p hase equi l i br i um, one can show
t hat for a dr op l et of p ur e l i qui ds t he di fference bet ween pres-
sure in t he l i qui d and vap or si des is p r op or t i onal t o t he dr op l et
radi us. Consi der a l i qui d dr op l et of r adi us r and t hat its sur-
face is exp anded in a cl osed cont ai ner at const ant t emp er a-
ture. Because of t he ext ensi on of t he surface dropl et , r adi us
changes by dr. Total vol ume ( l i qui d and vapor ) is const ant
(V t~ = V v -I- V L = const ant or dV t~ -- 0) and as a r esul t
we have dV v = - dV L. The surface areas and vol ume of l i qui d
dr op l et (V L) are gi ven as S = 4Jrr 2 and V L -- (4/3)Jrr 3. I n t hi s
pr ocess ( const ant t emp er at ur e and vol ume) , t he p r i nci p l e of
equi l i br i um is f or mul at ed in t er ms of Hel mhol t z energy, A as
follows:
(8.79) dAr, v = 0
wher e A = A L -~ A v. Wi t h r espect t o defi ni t i on of Hel mhol t z
energy, one can have dA L = - - p LdvL -t-/zLdn L q- adS, wher e
adS r ep r esent s t he wor k r equi r ed to exp and l i qui d dr op l et
by dr. Si mi l ar l y for t he vap or p hase dAv = - p vdvv +/ zVdn v
in whi ch at equi l i br i um/ z v = / z L and dn v = - dn L. Subst i t ut -
i ng dA t and dAv i nt o Eq. (8.79) t he fol l owi ng r el at i on is
obt ai ned:
2a
(8.80) pL _ pV = _ _
/ .
I n t he case of a bubbl e in t he l i qui d, wher e p r essur e in t he
gas si de is hi gher t han t hat of l i qui d, t he left si de of t he above
r el at i on becomes p v _ pL. Thi s can be f or mul at ed t hr ough
cont act angl e 0, whi ch is defi ned to det er mi ne degr ee of l i qui d
Hg
H20
/ / / / / / / / / / / / / / / / / / / / / / I
L
- _ _ _ - ______-
-_-_-_ pV, -_-_
FIG. 8 . 1 6- - T he contact ang l e of a
liquid surface and concept of wetta-
bility.
wet t abi l i t y. Consi der dr op l et s of wat er and mer cur y on a sol i d
surface as shown in Fig. 8.16. For mer cur y 0 > 90 ~ and it
is cal l ed a nonwetting fluid, whi l e for wat er wi t h 0 < 90 ~ is
an examp l e of a wetting liquid. For ot her l i qui ds 0 is var yi ng
bet ween 0 and 180 ~ and have di fferent degrees of wet t abi l i t y.
Equat i on (8.80) was deri ved on t he as s ump t i on t hat t he
dr op l et is spheri cal . However, when a l i qui d is i n cont act wi t h
a sol i d surface wher e t he l i qui d cur vat ur e is not fully sp her i cal
t he above equat i on is cor r ect ed as
2a Cos0
(8.81) pV _ pL _ _ _
r
For a fully nonwet t i ng l i qui d 0 = 180 ~ ( or Cos0 = - i ) ,
Eq. (8.81) r educes to Eq. (8.80). I f a wet t i ng l i qui d (i.e., wat er )
and a nonwet t i ng l i qui d (i.e., mer cur y) are p l aced in t wo cap-
i l l ary t ubes of r adi us r ( di amet er 2r) , t he wet t i ng l i qui d ri ses
whi l e nonwet t i ng l i qui d depr esses in t he t ube, as shown i n
Fig. 8.17. The hei ght of l i qui d ri se is det er mi ned f r om t he
p r essur e di fference pV _ pL [ = (pL _ pV)gh] i n whi ch by sub-
st i t ut i ng i nt o Eq. (8.81) one can get:
2a Cos0
(8.82) h -
(pL _ _ pV) gr
wher e pL and pV ar e t he l i qui d and vap or density, respectively,
and g is t he accel er at i on of gravi t y (9.8 m/s2). At l ow or at mo-
sp her i c p r essur es wher e pV << pL, for si mpl i ci t y pV can be ne-
gl ect ed. At hi gh pr essur es, t he p r essur e di fference ( p v _ p t )
causes l i qui d ri se and it is cal l ed capillary pressure shown by
Pcap. For nonwet t i ng l i qui ds, such as mercury, wher e 0 > 90 ~
Cos0 < 0 and accor di ng t o Eq. (8.81) t he l i qui d depr esses in
t he t ube as shown i n Fig. 8.17. F r om t hi s equat i on, when t he
a-Wetting Fluid b-Nonwetting Fluid
FIG. 8 . 1 7 - - W et t i ng and nonwetti ng liquids
in capillary tubes,
r adi us of t ube decr eases t he hei ght of l i qui d ri se i ncreases. I n
t he case of oil and water, Eq. (8.82) becomes
2awo Cos0
(8.83) h -
(pW _ po)g r
wher e a~o is t he i nt er r aci al t ensi on bet ween oil and wat er
phases, pW and pO ar e densi t y of wat er and oil, respectively.
I n t hi s equat i on, if awo is in N/ m and p is in kg/ m 3, t hen h and
r mus t be i n m.
The i ns t r ument t hat measur es surface t ensi on of a l i qui d
is cal l ed tensiometer, whi ch may be manual or di gi t al . Most
commonl y used met hods of meas ur i ng surface t ensi on in-
cl ude cl assi cal r i ng met hod, capi l l ar y rise, p endant drop, and
bubbl e pr essur e. The p endant met hod is most commonl y used
t o meas ur e surface t ensi on of l i qui d oils. Schemat i c of ap p a-
r at us to measur e i nt erraci al t ensi on usi ng t he p endant dr op
met hod is shown i n Fig. 8.18 [59]. Mi l l et t e et al. [60] r ecom-
mends ma xi mum bubbl e p r essur e met hod t o meas ur e surface
t ensi on of hydr ocar bons at hi gh t emp er at ur es and pr essur es.
Most advanced i nst r ument s can meas ur e sur f ace t ensi on wi t h
an accur acy of +0. 001 mN/ m.
Surface t ensi on usual l y decr eases wi t h bot h p r essur e and
t emp er at ur e. Effect of t emp er at ur e is gr eat er t han effect of
p r essur e on surface t ensi on. As p r essur e i ncr eases t he dif-
ference bet ween (pL _ pV) decr eases and as a r esul t surface
t ensi on al so decreases, accor di ng to Eq. (8.82). The effect of
p r essur e on I FT is di scussed later. Surface t ensi on i ncr eases
wi t h i ncr ease in mol ecul ar wei ght of a comp ound wi t hi n a ho-
mol ogous hydr ocar bon group. Some val ues of surface t ensi on
1[
31 ....
1 0 1 1
I
I I
12
FIG. 8 . 1 8 - - Schemat i c of apparatus to measure interfacial tensi on using the
pendant drop method. Tak en with permi ssi on from Ref. [ 59].
( 1 ) anti-v ibration table; ( 2) light source; ( 3) optical rail; ( 4) light diffuser; ( 5)
iris; ( 6) green filter; ( 7 ) thermostated interfacial tensi on cell with optical flats;
( 8 ) syri ng e to f orm pendant drops; ( 9 ) thermostat; ( 10) photomacrog raphi c
T essov ar zoom lens; ( 11) CCD camera; ( 12) computer with digitizing board,
TABLE 8. 13--V alues of surface tension of some
hydrocarbons at 25~ [45].
Compound cr at 25 ~ C, dyne/cm
n-Pentane 15.47
n-Decane 23.39
n-Pentadecane 26.71
n-Eicosane 28.56
n-Hexatriacontane 30.44
Cyclopentane 21.78
Cyclohexane 24.64
Benzene 28.21
Decylbenzene 30.52
Pentadecylbenzene 31.97
Water (at 15~ 74.83
Water (at 25~ 72.82
for p ur e hydr ocar bons and wat er are gi ven in Table 8.13. Sur-
face t ensi on i ncr eases f r om paraffi ns to nap ht henes and to
ar omat i cs for a same car bon number. Wat er has si gni fi cant l y
hi gher surface t ensi on t han hydr ocar bons. Surface t ensi on
of mer cur y is qui t e hi gh and at 20~ it is 476 mN/ m. Li qui d
met al s have even hi gher surface t ensi ons [18].
Example 8. 6- - Cons i der wat er at 15~ in a cap i l l ar y t ube op en
to at mosp her e, as shown i n Fig. 8.17. I f t he di amet er of t he
t ube is 10 -4 cm, cal cul at e t he ri se of wat er i n t he t ube. What
is t he capi l l ar y p r essur e of wat er ?
Sol ut i on- - Fr om Table 8.13 for water, a at 15~ C = 74.83 mN/ m
and l i qui d densi t y of wat er at 15~ is 0.999 g/ cm 3. Equat i on
(8.82) mus t be used to cal cul at e l i qui d rise. For wat er ( assum-
i ng full wet t abi l i t y) , 0 = 0 and Cos(O) = 1, r = 5 x 10 -7 m,
a = 74.83 x 10 -3 N/ m, and pL = 999 kg/ m 3. Subst i t ut i ng t hi s
i n Eq. (8.82) gives h = (2 x 74.83 x 10 -3 1) / ( 999 x 9.8 x
5 10 -7) = 30.57 m. When r i ncr eases t he ri se i n l i qui d hei ght
decreases. The capi l l ar y p r essur e is cal cul at ed f r om Eq. (8.80)
a s Pcap = 2.99 bar. r
One of t he mai n ap p l i cat i ons of I FT bet ween oil and wat er
is t o det er mi ne t he t ype of r ock wet t abi l i t y i n a p et r ol eum
reservoir. Wet t abi l i t y may be defi ned as "the t endency of one
fluid t o sp r ead on or adher e to a sol i d surface in t he p r esence
of ot her i mmi sci bl e fluids" [15]. Consi der oil and wat er i n a
r eser voi r as shown i n Fig. 8.19. Assume t he surface t ensi on of
oil wi t h t he r eser voi r r ock ( sol i d phase) is shown by ~o and
surface t ensi on of wat er wi t h t he r ock is shown by a~w. The
cont act angl e bet ween oil and wat er is shown by Owo, whi ch
vari es f r om 0 to 180 ~ The adhes i on t ensi on (AT) bet ween oil
and wat er AT is cal cul at ed as follows:
( 8 . 8 4 ) AT = aso - - asw = OwoCos ( 0wo)
/ / / / / / / / / /
FIG, 8 ,1 9 ~ W et t abi l i t y of oil and water on a
reserv oi r rock consi sti ng mainly of calcium
carbonate ( CaCO 3) .
i~iiiiiiiiiiiii~i~i~iiii~i~iii~iiiiiiiiiiiii~iiiiii~iii~ii~iiiiiiiiiiiiiiiii
. . . . . . . . . . . . . . . . . . . . . . i" i ~ ~ .' Naphthenic Aci d i" i"
iiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiii
FIG. 8 . 20- - Compari son of wettability of two dif-
ferent fl ui ds on a calcite surface.
wher e awo is t he I FT bet ween oil and water. I f AT > 0, t he
heavi er l i qui d ( in t hi s case wat er) , is t he wet t abl e fluid. The
hi gher val ue of AT i ndi cat es hi gher degr ee of wet t abi l i t y,
whi ch means t he wet t i ng fluid sp r eads bet t er on t he sol i d
surface. I f 0wo is smal l ( l arge AT), t he heavi er fl ui d qui ckl y
sp r eads t he sol i d surface. I f 0wo < 90 ~ t he sol i d surface is
wet t abl e wi t h r esp ect to wat er and if 0wo > 90 ~ t he sol i d sur-
face is wet t abl e wi t h r esp ect to oil. Wet t abi l i t y of i sooct ane
(i-C8) and nap ht heni c aci d on a cal ci t e ( a r ock consi st i ng
mai nl y CaCO3) is shown in Fig. 8.20. For t he case of i-Ca and
water, t he surface of cal ci t e is wet t abl e wi t h water, whi l e for
t he case of nap ht heni c acid, t he cal ci t e surface is wet t abl e
wi t h r esp ect t o aci d si nce 0 > 90 ~ Wet t abi l i t y of r eser voi r
rocks has di r ect effect on t he p er f or mance of mi sci bl e g a s
fl oodi ng in enhanced oil recovery ( EOR) processes. F or exam-
ple, wat er fl oodi ng has bet t er p er f or mance for reservoi rs t hat
ar e st rongl y wat er wet t han t hose whi ch are oil wet. F or oil
wet reservoi rs wat er fl oodi ng mus t be fol l owed by gas flood-
i ng t o have effective i mp r oved oil r ecover y [61].
8 . 6 . 2 P r e d i c t i v e M e t h o d s
The basi s of cal cul at i on and meas ur ement of surface/
i nt er f aci al t ensi on is Eqs. (8.82) and (8.83). For surface ten-
si on a is r el at ed to t he di fference bet ween sat ur at ed l i qui d
and vap or densi t i es of a subst ance at a gi ven t emp er at ur e
(pL _ pV). Macl eod in 1923 suggest ed t hat a 1/4 is di rect l y pro-
p or t i onal to (pL _ pV) and t he p r op or t i onal i t y const ant cal l ed
parachor (Pa) is an i ndep endent p ar amet er [18]. The most
c ommon r el at i on for cal cul at i on of surface t ensi on is
( 8 . 8 5 ) o l / n = Pa ( pL _ pv )
M
wher e M is mol ecul ar wei ght , p is densi t y i n g/ cm 3, and a
is i n mN/ m ( dyn/ cm) . Thi s r el at i on is usual l y r ef er r ed to a s
Macl eod- Sugden cor r el at i on. Par achor is a p ar amet er t hat
is defi ned to cor r el at e surface t ensi on and vari es f r om one
mol ecul e to anot her. Di fferent val ues for p ar amet er n in
TABLE 8. 14--V alues of parachor for
some hydrocarbons for use in Eq. (8.85)
with n = 3.88 [16].
Compound Parachor
Methane 74.05
n-Pentane 236.0
Isopentane 229.37
n-Hexane 276.71
n-Decane 440.69
n-Pentadecane 647.43
n-Eicosane 853.67
Cyclopentane 210.05
Cyclohexane 247.89
Methylcyclohexane 289.00
Benzene 210.96
Toluene 252.33
Ethylbenzene 292.27
Carbon dioxide 82.00
Hydrogen sulfide 85.50
Eq. (8.85) are suggest ed, t he mos t commonl y used val ues ar e
4, 11/3 (-- 3.67), and 3.88. For exampl e, val ues of p ar achor s
r ep or t ed i n t he API -TDB [5] ar e gi ven for n = 4, whi l e in Ref.
[16] p ar amet er s ar e given for t he val ue of n = 3.88. Par achor
number of p ur e comp ounds may be est i mat ed f r om gr oup
cont r i but i on met hods [5, 18]. For exampl e, for n-al kanes t he
fol l owi ng equat i on can be obt ai ned based on a gr oup cont ri -
but i on met hod suggest ed by Pol i ng et al. [18]:
(8.86) P~ = 111 + a( Nc - 2) for n = 4 i n Eq. (8.85)
wher e Nc is t he car bon number of n-al kane hydr ocar bon and
a = 40 if 2 _< Nc _< 14 or a -- 40.3 if Nc > 14. Cal cul at ed val ues
of surface t ensi on by Eq. (8.85) ar e qui t e sensi t i ve t o t he val ue
of parachor. Values of p ar achor for some comp ounds as gi ven
in Ref. [16] for use i n Eq. (8.85) wi t h n = 3.88 ar e gi ven in
Table 8.14. F or defi ned mi xt ur es t he Kay' s mi xi ng rul e
(Eq. 7.1) can be used as amix = Y~xiai for qui ck cal cul at i ons.
F or mor e accur at e cal cul at i ons, t he fol l owi ng equat i on is sug-
gest ed i n t he API -TDB to cal cul at e surface t ensi on of defi ned
mi xt ur es [5]:
__ pV } n
( 8. 87) anaix=[i=~l[Pa, i(~LXi ~y i ) 1
wher e M L and M v are mol ecul ar wei ght of l i qui d and vap or
mi xt ures, respectively, x~ and Yi ar e mol e f r act i ons of l i qui d
and vap or phases, pL and pV ar e densi t i es of sat ur at ed liq-
ui d and vap or mi xt ur es at gi ven t emp er at ur e i n g/ cm 3 . Some
at t emp t s to cor r el at e surface t ensi on to l i qui d vi scosi t y have
been made in t he f or m of a = A e xp ( - B# ) i n whi ch A is re-
l at ed to PNA comp os i t i on and p ar amet er B is cor r el at ed to M
as wel l as PNA di st r i but i on [34]. At hi gher p r essur es wher e
t he di fference bet ween l i qui d and vap or p r op er t i es reduces,
/~ coul d be r ep l aced by A/~ = (#0.5L --/ z~S) 2" Such correl at i ons,
however, ar e not wi del y used i n t he i ndust ry.
Temp er at ur e dep endency of surface t ensi on can be ob-
served f r om t he effect of t emp er at ur e on densi t y as shown
in Eq. (8.85). At t he cri t i cal poi nt , pL _ pV = 0 and surface
t ensi on r educes to zero ( a = 0). I n fact, t her e is a di r ect cor-
r el at i on bet ween (pL _ pV) and (Tc - T), and one can assume
(pL _ pV) = K(1 - Tr) m wher e K and m ar e const ant s t hat de-
p end on t he fluid wher e n is ap p r oxi mat el y equal to 0.3.
Combi nat i on of t hi s r el at i on wi t h Eq. (8.85) gives a corre-
l at i on bet ween a and ( i - T~) ~ i n whi ch n is cl ose t o 4.0.
General l y, cor r esp ondi ng st at e cor r el at i on in t er ms of re-
duced surface t ensi on versus (1 - Tr) are p r op os ed [18]. The
gr oup a/ Pc 2/3 T~/3 is a di mensi onl ess p ar amet er except for t he
numer i cal const ant t hat dep ends on t he uni t s of a, Pc, and
Tc. There ar e a number of gener al i zed cor r el at i ons for cal cu-
l at i on of a. For exampl e, Bl ock and Bi r d cor r el at i on is gi ven
as fol l ows [18]:
(8.88)
cr = p2/3T1/3Q(I - Zr) 11/9
Tbrln(Pc/1. O1325)]
O= 0. 1196 1+ i ~ ~br / - - 0. 279
wher e a is in dyn/ cm, Pc in bar, T~ i n kelvin, and Tbr is t he
r educed boi l i ng p oi nt (Tb/Tc). Thi s equat i on is rel at i vel y accu-
rat e for hydr ocar bons; however, for nonhydr ocar bons er r or s
as hi gh as 40- 50% ar e observed. I n general , t he accur acy of
t hi s equat i on is about 5%. Anot her gener al i zed cor r el at i on
was devel oped by Mi queu et al. [62] based on an ear l i er cor-
r el at i on p r op os ed by Schmi dt and it is given i n t he fol l owi ng
form:
a = kBTc/\[NA| 2/3 x (4.35 + 4.1&o) x (1 + 0.19r ~ - 0.25r) .gl.26
(8.89)
wher e r = 1 - T~,a i s i n dyn/ cm, kB(= 1.381 10 -16 dyn- crn/
K), NA, To, Tr, Vc, and co are t he Bol t zmann const ant , Avogadro
number, t he cri t i cal t emp er at ur e in kelvin, r educed t emper -
at ure, t he cri t i cal vol ume i n cm3/ mol, and acent r i c factor,
respectively. This equat i on was devel oped bas ed on experi -
ment al dat a for surface t ensi ons of N2, 02, Kr, hydr ocar bons
from C1 to n-Cs ( i ncl udi ng i-C4 and i-C5) and 16 hal ogenat ed
hydr ocar bons ( refri gerant s) wi t h an average r ep or t ed er r or of
3.5%.
For undef i ned p et r ol eum fract i ons t he fol l owi ng r el at i on
suggest ed in t he API -TDB [5] can be used for cal cul at i on of
surface t ensi on:
(8.90) a =
673.7 (1 - Tr) 1"232
Kw
wher e Tr is t he r educed t emp er at ur e and Kw is t he Wat son
char act er i zat i on factor. Tsonopoul os et al. [33] have corre-
l at ed p ar achor of hydr ocar bons, p et r ol eum fract i ons, and
coal l i qui ds to boi l i ng p oi nt and specific gravi t y i n a f or m
si mi l ar to t hat of Eq. (2.38):
(8.91)
0.1/4 = Pa (,oL -- pV)
M
Pa = 1.7237TO.05873SG_0.64927
M
wher e Tb is t he boi l i ng p oi nt in kel vi n and SG is t he specific
gravity. Uni t s for t he ot her p ar amet er s are t he same as t hose i n
Eq. (9.85). Thi s equat i on can p r edi ct surface t ensi on of p ur e
hydr ocar bons wi t h an average devi at i on of about 1% [33].
Recently, Mi queu et al. [59] r ep or t ed some exp er i ment al
dat a on I FT of p et r ol eum fract i ons and eval uat ed var i ous pre-
di ct i ve met hods. They r ecommended t he fol l owi ng met hod
TABLE 8. 15--Effect of characterization method on prediction of interracial tension of some petroleum fractions through Eq. (8.91).
% Er r or on pr edi ct i on of IFT ~
Fract i on Tb, K SG P25, g/ cm 3 M cr at 25~ mN/ m Met hod 1 Met hod 2 Met hod 3
1 429 0.769 0.761 130.9 22.3 26.5 2.7 14.8
2 499 0.870 0.863 167.7 30.7 -29.3 -7.5 -2.0
3 433 0.865 0.858 120.2 29.2 -15.4 3.4 22.9
4 505 0.764 0.756 184.4 25.6 -4.7 7.8 -10.9
Overall 19.0 5.4 12.7
aExper i ment al dat a are t aken from Mi queu et al. [59]. Met hod 1: Tc and Pc from Kesl er-Lee (Eqs. (2.69) and (2.70)) and w from Lee-Kesl er (Eq. 2.105). Met hod 2:
Tc and Pc from API-TDB (Eqs. (2.65) and (2.66)) [5] and o) from Lee-Kesl er (Eq. 2.105). Met hod 3: Tc and Pc from Twu (Eqs. (2.80) and (2.86)) and ~0 from Lee-Kesl er
(Eq. 2.105).
for calculation of surface tension of undefined petroleum frac-
tions:
(8.92)
(0.85 - 0.19o>) T 12/11
Pa=
(Pc/ lO) 9/11
I n this method, n in Eq. (8.85) is equal to 11/3 or 3.6667. I n the
above equation, Tc and Pc are in kelvin and bar, respectively,
is in mN/ m (dyn/cm), and p is in g/ cm 3. Predicted values of
surface tension by this method strongly depend on the char-
acterization method used to calculate To, Pc, and M. For four
petroleum fractions predicted values of surface tension by
three different characterization methods described in Chap-
ter 2 are given in Table 8.15. As it is seen from this table, the
API method of calculating Tc, Pc, w, and M (Section 2.5) yields
the lowest error for estimation of surface tension. Miqueu
et al. [59] used the pseudocomponent method (Section 3.3.4,
Eq. 3.39) to develop the following equation for estimation of
parachor and surface tension of defined petroleum fractions
with known PNA composition.
Pa = xp Pa, p + XNPa,N + XAPa,A
(8.93) Pa, e = 27.503 + 2.9963M
Pa,s = 18.384 + 2.7367M
Pa,A = 25.511 + 2.8332M
where xe, XN, and XA are mole fractions of paraffins, naph-
thenes, and aromatics in the fraction. Units are the same as
in Eq. (8.92). Experimental data of Darwish et al. [63] on
surface tension consist PNA distribution of some petroleum
fractions. For undefined fractions, the PNA composition may
be estimated from methods of Chapter 3. For cases where ac-
curate PNA composition data are not available the parachor
number of an undefined petroleum fraction may be directly
calculated from molecular weight of the fraction (M), using
the following correlation originally provided by Fawcett and
recommended by Miqueu et al. [59]:
(8.94) Pa = 81.2+2.448M value of n in Eq. (8.85) = 11/3
I n this method, only M and liquid density are needed to
calculate surface tension at atmospheric pressure. Firooz-
abadi [64] also provided a similar correlation (Pa = 11.4 +
3.23M - 0.0022M2), which is reliable up to C10, but for heav-
ier hydrocarbons it seriously underpredicts values of surface
tension.
An evaluation of various methods for prediction of surface
tension of n-alkanes is shown in Fig. 8.2 I. Data are taken from
DIPPR [45]. The most accurate method for calculation of sur-
face tension of pure hydrocarbons is through Eq. (8.85) with
values of parachor from Table 8.14 or Eq. (8.86). Method of
Block and Bird (Eq. 8.88) or Eq. (8.90) for petroleum fractions
also provide reliable values for surface tension of pure hydro-
carbons with average errors of about 3%. Equation (8.90) is
perhaps the most accurate method as it gives the lowest er-
ror for surface tension of n-alkanes (error of ~2%), while it
is proposed for petroleum fractions. Equations (8.92)-(8.94)
give generally very large errors, especially for hydrocarbons
heavier than C10. Equation (8.93) is developed for petroleum
fractions ranging from Cs to C10 and Eq. (8.94) is not suitable
for heavy hydrocarbons as shown in Fig. 8.21.
Interfacial tension (IFT) between hydrocarbon and water
is important in understanding the calculations related to oil
recovery processes. The following simple relation is suggested
in the API-TDB [5] to calculate anw from surface tension of
hydrocarbon ~H and that of water ~w:
(8.95) aHw = ffH -~ GW -- 1.10 (GHGW) 1/2
Use of this method is also demonstrated in Example 8.7. An-
other relation for IFT of hydrocarbon-water systems under
reservoir conditions was proposed by Firoozabadi and Ramey
[16, 65] in the following form:
(8.96) amv = 111 ( ~, - p H) 1"024 (T/TcH) -1"25
where cq-iw is the hydrocarbon-water IFT in dyn/ cm (mN/m),
Pw and ~ are water and hydrocarbon densities in g/cm 3, T is
temperature in kelvin, and TcH is the pure hydrocarbon critical
temperature in kelvon. Errors as high as 30% are reported for
this correlation [16]. IFT similar to surface tension decreases
with increase in temperature. For liquid-liquids, such as oil-
water systems, IFT usually increases slightly with pressure;
however, for gas-liquid systems, such as methane-water, the
IFT slightly decreases with increase in pressure.
Exampl e 8. 7- - A kerosene sample has boiling point and spe-
cific gravity of 499 K and 0.87, respectively. Calculate the IFT
of this oil with water at 25~ Liquid density of the fraction
at this temperature is 0.863 g/cm 3.
Solution- - Tb = 499 K and SG = 0.87. From Eq. (2.51), M =
167.7. Parachor can be calculated from the Fawcett method as
given in Eq. (8.94): Pa = 491.73. From data p25 = 0.863 g/cm 3 .
Substituting values of M, Pa, and p25 (for pL) in Eq. (8.85)
with n = 11/3 gives a2s = 30.1 mN/m, where in comparison
with the experimental value of 30.74 mN/ m [59] the error is
-2.1%. When using Eq. (8.85), the value of pV is neglected
35
r
30 0 , ..~ 1 7 6
" " ; ' ; ~ ~ i
O ~ _ t Y """ 0 Data (DIPPR)
# 2/ "-., Method,
~ " \ ~ Method 2
~ 20 ~ \ - ~ - Met hod3
\ - . Method 4
q ~ ~.~ ~ " \ 9 Method 5
. 4[
15 l~ ~ 4h. o. - ~ ~- . . . . . Method 6
m ~ Met hod7
m ~. . Method 8
10
0 5 10 15 20 25 30 35 40
Carbon Number
FIG. 8 . 21 mPredi cti on of surface tensi on of n-alkanes from v ari ous
methods. Method 1: Eq. ( 8 .8 5) and Table 8 .14; Method 2: Eqs, ( 8 .8 5)
and ( 8 .8 6) ; Method 3: Eq, ( 8 ,8 8 ) ; Method 4: Eq. ( 8 .90) ; Method 5:
Eq, ( 8 ,91) ; Method 6: Eq, ( 8 .92) ; Method 7: Eq. ( 8 .93) ; Method 8 : Eq,
( 8 .94) .
with respect to pL at at mospheri c pressure. To calculate I FT
of water-oil, Eq. (8.95) can be used. From Table 8.13 for
wat er at 25~ aw = 72.8 mN/ m. From Eq. (8.95), aW-oil =
72.8 + 30.1 - 1.1(72.8 30.1) 1/2 = 51.4 mN/ m. To calculate
aw-oi] from Eq. (8.96), TcH is calculated from the API met hod
(Eq. 2.65) as 705 K and at 25~ Pw--0. 995 g/ cm 3. From
Eq. (8.96), aW-oil = 40.9 mN/ m. This is about 20% less t han
the value calculated from Eq. (8.95). As ment i oned before
large error may be observed from Eq. (8.96) for calculation of
IFT.
8 . 7 S UMMARY AND R EC OMMEN D A T I ON S
I n this chapter, met hods and procedures presented in the pre-
vious chapters are used for estimation of four t ransport prop-
erties: viscosity, t hermal conductivity, diffusion coefficient,
and surface tension. I n general semitheoretical met hods for
estimation of t ransport properties have wider range of ap-
plications t han do pure empirical correlations and their de-
vel opment and applications are discussed in this chapter. A
summar y of recommended met hods is given below.
For calculation of viscosity of pure gases at at mospheri c
pressure, Eq. (8.3) should be used and for compounds for
which the coefficients are not available, Eq. (8.6) may be used.
For defined gas mixture when viscosity of component s are
known Eq. (8.7) or (8.8) can be used. For hydrocarbon gases at
high pressure, viscosity can be calculated from Eq. (8.12) and
for nonhydrocarbons Eq. (8.13) can be used. For estimation
of viscosity of natural gas at at mospheri c pressure, Eq. (8.14)
and at higher pressure Eq. (8.15) are recommended.
To estimate viscosity of pure liquids, Eq. (8.17) is recom-
mended and for a defined hydrocarbon mixture Eq. (8.18)
can be used. For pet rol eum fractions when kinematic vis-
cosity at 100~ (37.8~ is available, Eq. (8.19) can be used.
When two pet rol eum fractions are mixed, Eq. (8.20) is use-
ful. Viscosity of liquid hydrocarbons at high pressure can
be calculated from Eq. (8.22). For crude oil at at mospheri c
pressure Eq. (8.25) is useful; however, for reservoir fluids
Eq. (8.12) can be used for bot h gases and liquids or their
mixtures.
Thermal conductivity of pure hydrocarbon gases at low
pressures should be calculated from Eq. (8.33) and for those
for which the coefficients are not available, Eq. (8.34) should
be used. For defined hydrocarbon gas mixtures Eq. (8.35)
and for undefined pet rol eum vapor fractions Eq. (8.37) is
recommended. For vapor fractions at t emperat ures in which
Eq. (8.37) is not applicable, Eq. (8.36) is recommended. For
hydrocarbon gases at high pressures Eq. (8.39) may be used
and if not possible Eq. (8.38) can be used for bot h pure gases
and undefined gas mixtures.
For pure hydrocarbon liquids at low pressures, Eq. (8.42) is
recommended and for those compounds whose t hermal con-
ductivity at two reference points are not known, Eq. (8.43)
is recommended. For undefined liquid pet rol eum fractions,
Eq. (8.46) and for defined liquid mixtures Eq. (8.48) can
be used. For fractions without any characterization data,
Eq. (8.50) can be used for determination of approxi mat e value
of t hermal conductivity. For fractions with only boiling point
available, Eq. (8.51) should be used and for coal liquid frac-
tions Eq. (8.52) is recommended. For liquid fractions at high
pressures, Eq. (8.53) is recommended.
To estimate binary diffusion coefficients for hydrocarbon
gases at low pressures, Eq. (8.59) and for nonhydrocar-
bons Eq. (8.58) can be used. For liquid hydrocarbons at
low pressure, diffusion coefficients at infinite dilution can
be estimated from Eq. (8.60) or (8.61) and for the effect
of concentration on binary diffusion coefficients Eq. (8.64)
should be used. For both liquids and gases at high pressures,
Eq. (8.67) is highly recommended and Eq. (8.66) can be used
as alternative method for diffusivity of a gas in oil under reser-
voir conditions. When using Eq. (8.67) recommended meth-
ods for calculation of low-pressures properties must be used.
For multicomponent gas mixtures at low pressure, Eq. (8.68)
and for liquids or gases at high pressures Eq. (8.69) is recom-
mended to calculate effective diffusion coefficients. Effect of
porous media on diffusion coefficient can be calculated from
Eqs. (8.70) and (8.74). Self-diffusion coefficients or when re-
fractive index is available, Eq. (8.78) can be used.
Surface tension of pure compounds should be calculated
from Eq. (8.85) and defined mixtures from Eq. (8.86) with
parachors given in Table 8.14 or Eq. (8.86) for n-alkanes.
For undefined petroleum fraction surface tension can be
calculated from Eq. (8.90). For defined petroleum fractions
(known PNA composition), Eq. (8.93) is recommended. For
coal liquid fractions Eq. (8.91) may be used. Equation (8.95)
is recommended for calculation of IFT of water-hydrocarbon
systems. For specific cases, recommended methods are dis-
cussed in Section 8.6.2.
I n addition to predictive methods, two methods for experi-
mental measurement of diffusion coefficient and surface ten-
sion are presented in Sections 8.5 and 8.6.1. Furthermore,
the interrelationship among various transport properties, ef-
fects of porous media and concept of wettability, calculation
of capillary pressure and the role, and importance of interfa-
cial tension in enhanced oil recovery processes are discussed.
It is also shown that choice of characterization method could
have a significant impact on calculation of transport proper-
ties of petroleum fractions.
8 . 8
8.1.
8.2.
8.3.
PR OB L EMS
Pure methane gas is being displaced in a fluid mixture
of C1, n-C4, and n-C10 with composition of 41, 27, and
32 mol%, respectively. Reported measured diffusion co-
efficient of pure methane in the fluid mixture under the
conditions of 344 K and 300 bar is 1.01 x 10 -4 cm2/s [9].
a. Calculate density and viscosity of fluid.
b. Estimate diffusion coefficient of methane from
Sigmund method (Eq. 8.65).
c. Estimate diffusion coefficient of methane from
Eq. 8.67.
Hill and Lacy measured viscosity of a kerosene sample at
333 K and 1 atm as 1.245 mPa. s [51]. For this petroleum
fraction, M = 167 and SG = 0.7837. Estimate the viscos-
ity from two most suitable methods and compare with
given experimental value.
Riazi and Otaibi [21] developed the following relation
for estimation of viscosity of liquid petroleum fractions
based on Eq. (8.78):
I / ~ = A + B / I
where
A = 37.34745 - 0.20611M + 141.1265SG - 637.727120
- 6.757T~ + 6.98(T~) 2 - 0.81(T~) 3,
B = -15.5437 + 0.046603M - 42.8873SG + 211.6542120
+ 1.676T~ - 1.8(T~) 2 + 0.212(T~) 3,
T~ = (1.8Tb -- 459.67)/1.8
in which Tb is the average boiling point in Kelvin, /z
is in cP, and parameter I should be determined at the
same temperature as # is desired. (Parameter I can be
determined as discussed for its use in Eq. (8.78).)
For kerosene sample of Problem 8.2, calculate visco-
sity based on the above method and obtain the error.
8.4. Methane gas is dissolved in the kerosene sample of Prob-
lem 8.2, at 333 K (140~ and 20.7 bar (300 psia). The
mole fraction of methane is 0.08. For this fluid mix-
ture calculate density, viscosity, and thermal conductiv-
ity from appropriate methods. The experimental value
of density is 5.224 kmol/ m 3.
8.5. Estimate diffusion coefficient of methane in kerosene
sample of Problem 8.4 from Eqs. (8.65)-(8.67).
8.6. Estimate thermal conductivity of N2 at 600~ and 3750
and 10 000 psia. Compare the result with values of 0.029
and 0.0365 Btu/ft 9 h. ~ as reported in the API-TDB [5].
8.7. Consider an equimolar mixture of C1, C3, and N2 at 14
bar and 311 K. The binary diffusion coefficient of Dcl-c3
and DCI_N2 are 88.3 10 -4 and 187 10 -4 cm2/s, re-
spectively. The mixture density is 0.551 kmol/ m 3. Esti-
mate the effective diffusion coefficient of methane in the
mixture from Eq. (8.68) and compare it with the value
calculated from Eqs. (8.67) and (8.69).
8.8. A petroleum fraction has boiling point and specific grav-
ity of 429 K and 0.761, respectively. The experimen-
tal value of surface tension at 25~ is 22.3 mN/ m [59].
Calculate the surface tension at this temperature from
the following methods and compare them against the
experimental value.
a. Five different methods presented by Eqs. (8.88)-(8.92)
with estimated input parameters from the API-TDB
methods.
b. Equation (8.93) with predicted PNA distribution.
c. Fawcett's method for parachor (Eq. 8.94).
d. Firoozabadi' s method for parachor.
R EFER EN C ES
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Nonpolar Substances in the Dense Gaseous and Liquid
Regions," American Institute of Chemical Engineers Journal,
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[32] Riazi, M. R. and Faghri, A., "Prediction of Thermal
Conductivity of Gases at High Pressures," American Institute
of Chemical Engineers Journal, Vol. 31, No. 1, 1985,
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Corresponding States Model for the Prediction of Oil and Gas
Viscosities and Thermal Conductivities," Chemical Engineering
Science, Vol. 42, 1987, pp. 182-186.
[35] Duda, J. L., "Personal Notes in Transport Phenomena,"
Instructor's note for ChE 545 Course, Chemical Engineering
Department, Pennsylvania State University, University Park,
PA, 1977.
[36] Stiel, L. I. and Thodos, G., "Lennard-Jones Constants Predicted
from Critical Properties," Journal of Chemical and Engineering
Data, Vol. 7, No. 2, 1962, pp. 234-236.
[37] Riazi, M. R. and Daubert, T. E., "Application of Corresponding
States Principles for Prediction of Self-Diffusion Coefficients in
Liquids," American Institute of Chemical Engineers Journal,
Vol. 26, No. 3, 1980, pp. 386-391.
[38] Slattery, J. C. and Bird, R. B., "Calculation of the Diffusion
Coefficients of Dilute Gases and of the Self-Diffusion
Coefficients of Dense Gases," American Institute of Chemical
Engineers Journal, Vol. 4, 1958, pp. 137 142.
[39] Takahashi, S., "Preparation of a Generalized Chart for the
Diffusion Coefficients of Gases at High Pressures," Journal of
Chemical Engineering of Japan, Vol. 7, 1974, pp. 417-421.
[40] Sigmund, E M., "Prediction of Molecular Diffusion at Reservoir
Conditions. Part I---Measurement and Prediction of Binary
Dense Gas Diffusion Coefficients," Canadian Journal of
[41] Dawson R., Khoury, E, and Kobayashi, R., "Self-Diffusion
Measurements in Methane by Pulsed Nuclear Magnetic
Resonance," American Institute of Chemical Engineers Journal,
Vol. 16, No. 5, 1970, pp. 725-729.
Vol. 20, Society of Petroleum Engineers, Richardson, TX, 2000.
[43] Renner, T. A., "Measurement and Correlation of Diffusion
Coefficients for CO2 and Rich Gas Applications," SPE Reservoir
Engineering, No. 3 (May), 1988, pp. 517-523.
[44] Eyring, H., Significant Liquid Structure, Wiley, New York, 1969.
[45] AIChE DIPPR| Database, Design Institute for Physical
1996.
[46] Wilke, C. R., "Diffusional Properties of Multicomponent Gases,"
Chemical Engineering Progress, Vol. 46, No. 2, 1950, pp. 95-104.
[47] Saidi, A. M., Reservoir Engineering of Fracture Reservoirs,
TOTAL Edition Presse, Paris, 1987, Ch. 8.
[48] Amyx, J. W., Bass, D. M., and Whiting, R. L., Petroleum
Reservoir Engineering, Physical Properties, McGraw-Hill,
New York, pp. 129 (total of 600 pp.), 1960.
[49] Langness, G. L., Robertson, J. O., Jr., and Chilingar, G. V.,
Secondary Recovery and Carbonate Reservoirs, Elsevier,
New York, 304 pp. 1972.
[50] Fontes, E. D., Byrne, E, and Hernell, O., "Put More Punch Into
Catalytic Reactors," Chemical Engineering Progress, Vol. 99,
No. 3, 2003, pp. 48-53.
[51] Hill, E. S. and Lacy, W. N., "Rate of Solution of Methane in
Quiescent Liquid Hydrocarbons," Industrial and Engineering
Chemistry, Vol. 26, 1934, pp. 1324-1327.
[52] Reamer, H. H., Duffy, C. H., and Sage, B. H., "Diffusion
Coefficients in Hydrocarbon Systems: Methane-Pentane in
Liquid Phase," Industrial and Engineering Chemistry, Vol. 48,
1956, pp. 275-282.
[53] Lo, H. Y., "Diffusion Coefficients in Binary Liquid n-Alkanes
Systems," Journal of Chemical and Engineering Data, Vol. 19,
No. 3, 1974, pp. 239-241.
[54] McKay, W. N., "Experiments Concerning Diffusion of
Mulficomponent Systems at Reservoir Conditions," Journal of
Canadian Petroleum Technology, Vol. 10 (April-June), 197 t,
pp. 25-32.
[55] Nguyen, T. A. and Farouq All, S. M., "Effect of Nitrogen on the
Solubility and Diffusivity of Carbon Dioxide into Oil and Oil
Recovery by the Immiscible WAG Process," Journal of Canadian
[56] Riazi, M. R., "A New Method for Experimental Measurement of
Diffusion Coefficient in Reservoir Fluids," Journal of Petroleum
Science and Engineering, Vol. 14, 1996, pp. 235-250.
[57] Zhang, Y. P., Hyndman, C. L., and Maini, B. B., "Measurement
of Gas Diffusivity in Heavy Oils," Journal of Petroleum Science
and Engineering, Vol. 25, 2000, pp. 37-47.
[58] Upreti, S. R. and Mehrotra, A. K., "Diffusivity of CO2, CH4,
C2H 6 and N2 in Athabasca Bitumen," The Canadian Journal of
Chemical Engineering, Vol. 80, 2002, pp. 117-125.
[59] Miqueu, C., Satherley, J., Mendiboure, B., Lachiase, J., and
Graciaa, A., "The Effect of P/N/A Distribution on the Parachors
of Petroleum Fractions," Fluid Phase Equilibria, Vol. 180, 2001,
pp. 327-344.
[60] Millette, J. P., Scott, D. S., Reilly, I. G., Majerski, P., Piskorz, J.,
Radlein, D., and de Bruijin, T. J. W., "An Apparatus for the
Measurement of Surface Tensions at High Pressures and
Temperatures," The Canadian Journal of Chemical Engineering,
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Reservoir Wettability on Waterflood and Miscible Flood
Performance," Journal of Canadian Petroleum Technology,
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[62] Miqueu, C., Broseta, D., Satherley, J., Mendiboure, B.,
Lachiase, J., and Graciaa, A., "An Extended Scaled
Equation for the Temperature Dependence of the Surface
Tension of Pure Compounds Inferred From an Analysis of
Experimental Data," Fluid Phase Equilibria, Vol. 172, 2000,
pp. 169-182.
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and Correlation of Surface Tension of Naphtha Reformate
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1988, pp. 41-48.
MNL50-EB/Jan. 2005
A pplications: Phase
Equilibrium Calculations
NOMENCLATURE
API
a, b , c , d, e
b
Cp
F
F(V F)
FSL
f/L(T, P, x L)
Ki
A paramet er defined in the Standing correla-
tion, Eq. (6.202), K
Eq. (6.17), J/ mol. K
Number of moles for the feed in VLSE unit, mol
(feed rate in mol/s)
Objective function defined in Eq. (9.4) to find
value of V F
Objective function defined in Eq. (9.19) to find
value of SF
Fugacity of component i in a mixture defined
by Eq. (6.109), bar
Fugacity of component i in a liquid mixture of
composition x L at T and P, bar
Equilibrium ratio in vapor-liquid equilibria
(Ki = yi/xi) defined in Eq. (6.196), dimension-
less
K vs Equilibrium ratio in vapor-solid equilibria
( K sL = yi/xS), dimensionless
kAB Binary interaction coefficient of asphaltene and
asphaltene-free crude oil, dimensionless
L Number of moles of liquid formed in VLE pro-
cess, tool (rate in tool/s)
LF Mole of liquid formed in VLSE process for each
mole of feed (F = 1), dimensionless
M Molecular weight ( molar mass), g/ mol
[kg/kmol]
MB Molecular weight ( mol ar mass) of asphaltene-
free crude oil, g/tool
n s Number of moles of component j in the solid
phase, tool
P Pressure, bar
Bubble point pressure, bar
Ptp Triple point pressure, bar
R Gas constant = 8.314 J/ tool. K (values in differ-
ent units are given in Section 1.7.24)
/~ Refractivity intercept [ = n20 - d20/2] defined in
Eq. (2.14)
Rs Dilution ratio of LMP solvent to oil ( cm 3 of sol-
vent added to 1 g of oil), cm3/g
S Number of moles of solid formed in VLSE sep-
aration process, mol (rate in mol/s)
SF Moles of solid formed in VLSE separation
process for each mole of initial feed (F = 1),
dimensionless
SG Specific gravity of liquid substance at 15.5~
T Absolute temperature, K
Tc Critical t emperat ure, K
TM Freezing (melting) point for a pure component
at 1.013 bar, K
Tpc Pseudocritical t emperat ure, K
Tt~ True-critical t emperat ure, K
V Molar volume, cm3/ mol
V Number of moles of vapor formed in VLSE sep-
aration process, mol (rate in mol/ s)
VA Liquid mol ar volume of pure component A at
normal boiling point, cm3/ mol
VF Mole of vapor formed in VLSE separation pro-
cess for each mole of feed (F = 1), dimension-
less
Vo Critical mol ar volume, cm3/ mol (or critical spe-
cific volume, cm3/g)
V/ Molar volume of pure component i at T and P,
cma/ mol
V L Molar volume of liquid mixture, cma/ mol
x/ Mole fraction of component i in a mixture
(usually used for liquids), dimensionless
x s Mole fraction of component i in a solid mix-
ture, dimensionless
yi Mole fraction of i in a mixture (usually used for
gases), dimensionless
Z Compressibility factor defined by Eq. (5.15), di-
mensionless
Zo Critical compressibility factor [Z = PcVc/RTc],
dimensionless
zi Mole fraction of i in the feed mi xt ure (in VLE
or VLSE separation process), dimensionless
Gr e e k L e t t e r s
A Difference between two values of a paramet er
e Convergence tolerance (e.g., 10 -5)
q~i Volume fraction of component i in a mixture
~i Volume fraction of component i in a mixture
~i Fugacity coefficient of component i in a mix-
ture at T and P defined by Eq. (6.110)
3 6 5
P
PM
CO
8i
Yi
ACpi
t , ~ f
Density at a given t emperat ure and pressure,
g/ cm 3 ( molar density unit: cma/ mol)
Molar density at a given t emperat ure and pres-
sure, mol / c m 3
Acentric factor defined by Eq. (2.10), dimen-
sionless
Chemical potential of component i in a mixture
defined by Eq. (6.115)
Solubility par amet er for i defined in Eq. (6.147),
(J/cm3) 1/2 or ( cal / cm3) 1/2
Activity coefficient of component i in liquid so-
lution defined by Eq. (6.112), dimensionless
Difference between heat capacity of liquid and
solid for pure component i at its melting (freez-
ing) point ( = cLi -- CSi), J/ tool - K
Heat of fusion (or latent heat of melting) for
pure component i at the freezing point and
1.013 bar, J/ mol
Superscript
S Value of a propert y at solid phase
Subscripts
A
A
C
i , i
L
M
Value of a propert y for component A
Value of a propert y for asphaltenes
Value of a propert y for component i or j in a
mixture
Value of a propert y for liquid phase
Value of a propert y at the melting point of a
substance
pc Pseudocritical property
S Value of a propert y at the solid phase
S Value of a propert y for solvent (LMP)
s Specific property ( quantity per unit mass)
tc True critical propert y
t r Value of a propert y at the triple point
7+ Values of a propert y for C7+ fraction of an oil
Acronyms
ABSA
API-TDB
BI P
bbl
CPT
cp
cSt
EOR
LOS
FH
Alkyl benzene sulfonic acid
Book (see Ref. [12])
Binary interaction paramet er
Barrel, unit of vol ume of liquid as given in Sec-
tion 1.7.11
Cloud-point t emperat ure
Centipoise, unit of viscosity, (1 cp = 0.01 p =
0.01 g- cm. s = 1 mPa. s = 10 -3 kg/ m. s)
Centistoke, unit of kinematic viscosity, (1 cSt =
0.01 St = 0.01 cm2/s)
Enhanced oil recovery
Equation of state
Flory-Huggins
GC Gas condensate (a type of reservoir fluid defined
in Chapter 1)
GOR Gas-to-oil ratio, scf/labl
HFT Hydrat e format i on t emperat ure
I FT I nterracial tension
LLE Liquid-liquid equilibria
LMP Low molecular weight n-paraffins (i.e., C3, n-Cs,
n-C7)
LVS liquid-vapor-solid
LS Liquid-solid
MeOH Methanol
SRK Soave-Redl i ch-Kwong EOS given by Eq. (5.38)
and paramet ers in Table 5.1
SAFT Statistical associating fluid theory (see
Eq. 5.98)
SLE Solid-liquid equilibrium
scf Standard cubic foot (unit for vol ume of gas at
1 at m and 60~
stb Stock t ank barrel (unit for volume of liquid oil
at 1 at m and 60~
VABP Volume average boiling point defined by
Eq. (3.3).
VLSE Vapor-liquid-solid equilibrium
VS Vapor-solid
VSE Vapor-solid equilibrium
WAT Wax appearance t emperat ure
WPT Wax precipitation t emperat ure
%AAD Average absolute deviation percentage defined
by Eq. (2.135)
Eq. (2.134)
wt% Weight percent
ONE OF THE MAIN APPLICATIONS of science of t hermodynami cs
in the pet rol eum industry is for the prediction of phase behav-
ior of pet rol eum fluids. I n this chapt er calculations related to
vapor liquid and solid-liquid equilibrium in pet rol eum flu-
ids are presented. Their application to calculate gas-oil ratio,
crude oil composition, and the amount of wax or asphaltene
precipitation in oils under certain conditions of t emperat ure,
pressure, and composi t i on is presented. Methods of calcula-
tion of wax format i on t emperat ure, cloud point t emperat ure
of crude oils, determination of onset of asphaltene, hydrate
format i on temperature, and met hods of prevention of solid
format i on are also discussed. Finally application of character-
ization techniques, met hods of prediction of t ransport prop-
erties, equations of state, and phase equilibrium calculations
are demonst rat ed in modeling and evaluation of gas injection
projects.
9. 1 T Y PES OF PHA S E EQUI LI BRI UM
CALCULATI ONS
Three types of phase equilibrium, namely, vapor-liquid
(VLE), solid-liquid (SLE), and liquid-liquid (LLE), are of
particular interest in the pet rol eum industry. Furt hermore,
vapor-solid (VSE), vapor-liquid-solid (VLSE), and vapor-
liquid-liquid (VLLE) equilibrium are also of i mport ance in
9. APPL I CATI ONS: PHASE EQUI L I B RI UM CAL CUL ATI ONS 3 67
i i i i i i i i i i ! i i i i i
: : : : : : : : : : : : : : : : : : : : : : : : : : : : :
iiii Pe(rdleumFiuid iiii
:i:i F moles
i ii i Composition zi
i : i : TF , PF i i i
. . . .
iiiiiiiiiiiiiiiiiiiiiiiiiiiii
i:i:i:i:i:i:i:i:i:i:i:i:i:i:i
: : : : : : : : : : : : : : : : : : : : : : : : : : : : :
iiiiiiiiiiiiiiiiiiiii!iiiiiii
. . . . . . !
" . ' . ' . Vmoles .' .' ."
9 . ' . " ' . ' . ' . "
" . ' . ' . ' Yi . ' . ' . "
i i Liquid Solution i i
i:[: L moles i:i
! i! i x,L ! il
9 . . . . . . . . . . . . . . . . . . . . . . . . .
ili::i::iii 9 "';oTi~~o'l~tion""iiiiii!i 1
[i!i!i!!!! S moles i!!!!i!i]
li i i i i ............ .......x.,. ............ ....iiiiii
t
all phases
at T and P
Non-equilibrium state Equilibrium state
FIG. 9.1--Typical v apor-liquid-solid equilibrium for solid precipitation.
calculations related to pet rol eum and nat ural gas product i on.
VLE calculations are needed in design and operat i on of sepa-
rat i on units such as mul t i st age surface separat ors at the sur-
face facilities of pr oduct i on fields, distillation, and gas absorp-
t i on col umns in pet rol eum and nat ural gas processi ng as well
as phase det ermi nat i on of reservoir fluids. LLE calculations
are useful in det ermi nat i on of amount of wat er dissolved in
oil or amount of oil dissolved in wat er under reservoir con-
ditions. SLE calculations can be used t o det ermi ne amount
and the condi t i ons at whi ch a solid (wax or asphaltene) may
be formed f r om a pet rol eum fluid. Cl oud-poi nt t emperat ure
(CPT) can be accurat el y cal cul at ed t hr ough SLE calculations.
VSE cal cul at i on is used to calculate hydrat e format i on and
the condi t i ons at whi ch it can be prevented.
Schemat i c of a system at vapor-l i qui d-sol i d equi l i bri um
(VLSE) is shown in Fig. 9.1. The syst em at its initial con-
ditions of TF and PF is in a nonequi l i bri um state. When it
reaches to equi l i bri um state, t he condi t i ons change t o T and
P and new phases may be formed. The initial composi t i on of
t he fluid mi xt ure is zi; however, at t he final equi l i bri um con-
ditions, composi t i ons of vapor, liquid, and solid in t erms of
mol e fractions are specified as yi, x L, and x s, respectively. The
amount of feed, vapor, liquid, and solid in t erms of number of
mol es is specified by F, V , L, and S, respectively. Under VLE
conditions, no solid is f or med (S -- 0) and at VSE state no
liquid exists at t he final equi l i bri um state (L = 0). The syst em
variables are F, zi , T, P, V , Yi, L, x L, S, and x s, where in a typi-
cal equi l i bri um calculation, F, zi, T, and P are known, and V,
L, S, Yi, x L, and x s are to be calculated. I n some calculations
such as bubbl e poi nt calculations, T or P may be unknown
and must be calculated f r om given i nformat i on on P or T and
the amount of V, L, or S. Calculations are formul at ed t hr ough
bot h equi l i bri um relations and mat eri al bal ance for all com-
ponent s in the system. Two-phase equi l i bri um such as VLE
or SLE calculations are somewhat si mpl er t han t hree-phase
equi l i bri um such as VLSE calculations.
I n this chapt er various types VLE and SLE calculations
are formul at ed and applied to various pet rol eum fluids. Prin-
ciples of phase equilibria were di scussed in Section 6.8
t hr ough Eqs. (6.171)-(6.174). VLE calculations are formu-
lated t hr ough equi l i bri um ratios (Ki) and Eq. (6.201), while
SLE calculations can be formul at ed t hr ough Eq. (6.208). I n
addi t i on t here are five types of VLE calculations t hat are dis-
cussed in the next section. Fl ash and bubbl e poi nt pressure
cal cul at i ons are t he most widely used VLE calculations by
bot h chemi cal and reservoir engineers in the pet rol eum pro-
cessing and product i on.
9 . 2 VAPOR- LI QUI D EQUI LI BRI UM
CALCULATI ONS
VLE calculations are perhaps t he most i mpor t ant types
of phase behavi or calculations in the pet rol eum industry.
They involve calculations related to equi l i bri um bet ween t wo
phases of liquid and vapor in a mul t i comp onent system. Con-
sider a fluid mi xt ure wi t h mol e fract i on of each comp onent
shown by zi is available in a sealed vessel at T and P. Under
these condi t i ons assume t he fluid can exist as bot h vapor and
liquid in equilibrium. Furt hermore, assume t here are total of
F mol of fluid in t he vessel at initial t emperat ure and pres-
sure of Tp and PF as shown in Fig. 9.1. The condi t i ons of the
vessel change to t emperat ure T and pressure P at whi ch bot h
vapor and liquid can coexist in equilibrium. Assume V mol of
vapor wi t h composi t i on Yi and L( = F - V ) mol of liquid wi t h
composi t i on x/ ar e pr oduced as a result of phase separat i on
due t o equi l i bri um conditions. No solid exists at t he equilib-
r i um state and S -- 0 and for this reason composi t i on of liquid
phase is simply shown by xi. The amount of vapor may be ex-
pressed by the ratio of V / F or VF for each mol e of the mixture.
The paramet ers involved in this equi l i bri um pr obl em are T,
P, zi, x4, Yi, and VF (for the case of F = 1). The VLE calcu-
lations involve cal cul at i on of t hree of these paramet ers f r om
t hree ot her known paramet ers.
Generally t here are five types of VLE calculations: (i) Flash,
(ii) bubbl e-P, (iii) bubble-T, (iv) dew-P, and (v) dew-T. (i) I n
flash calculations, usually zi, T, and P are known while xi, Yi,
and V are the unknown paramet ers. Obviously cal cul at i ons
can be per f or med so t hat P or T can be f ound for a known
value of V. Flash separat i on is also referred as flash distilla-
tion. (ii) I n the bubbl e-P calculations, pressure of a liquid of
known composi t i on is reduced at const ant T until t he first
vapor molecules are formed. The correspondi ng pressure is
called bubble poi nt pressure (Pb) at t emperat ure T and estima-
t i on of this pressure is known as bubbl e-P calculations. For
analysis of VLE properties, consi der the syst em in Fig. 9.1
wi t hout solid phase (S -- 0). Also assume t he feed is a liquid
wi t h composi t i on (xi = zi) at T = TF and PF. Now at const ant
T, pressure is reduced to P at whi ch infinitesimal amount
of vapor is p r oduced ( ~V = 0 or begi nni ng of vaporization) .
Thr ough bubbl e-P calculations this pressure is calculated.
Bubbl e poi nt pressure for a mi xt ure at t emperat ure T is sim-
ilar to the vapor pressure of a pure subst ance at given T. Off)
I n bubbl e-T calculations, liquid of known composi t i on (x4)
at pressure P is heat ed until t emperat ure T at whi ch first
molecules of vapor are formed. The correspondi ng t empera-
t ure is known as bubble poi nt temperature at pressure P and
est i mat i on of this t emperat ure is known as bubbl e-T calcu-
lations. I n this type of calculations, P = PF and t emperat ure
T at whi ch small amount of vapor is f or med can be calcu-
lated. Bubble poi nt t emperat ure or sat urat i on t emperat ure
for a mi xt ure is equivalent to t he boiling poi nt of a pure sub-
stance at pressure P. (iv) I n dew- P cal cul at i ons a vapor of
known composi t i on (Yi = zi) at t emperat ure T = T~ is com-
pressed to pressure P at whi ch infinitesimal amount of liquid
is pr oduced ( ~L = 0 or begi nni ng of condensat i on) . Thr ough
dew-P calculations this pressure known as dew poi nt pressure
(Pd) is calculated. For a pure subst ance t he dew poi nt pressure
at t emperat ure T is equivalent t o its vapor pressure at T. (v) I n
dew-T calculations, a vapor of known composi t i on is cool ed
at const ant P until t emperat ure T at whi ch first mol ecul es of
liquid are formed. The correspondi ng t emperat ure is known
as dew poi nt temperature at pressure P and est i mat i on of this
t emperat ure is known as dew-T calculations. I n these calcu-
lations, P = PF and t emperat ure T at whi ch condensat i on be-
gins is calculated. Flash, bubble, and dew poi nt s calculations
are widely used in the pet rol eum i ndust ry and are di scussed
in the following sections.
9. 2. 1 Fl as h Cal cul at i ons - - Gas - t o- Oi l Rat i o
I n typical flash calculations a feed fluid mi xt ure of compo-
sition zi enters a separat or at T and P. Product s of a flash
separat or for F mol of feed are V mol of vapor wi t h composi -
t i on Yi and L mol of liquid wi t h composi t i on x4. Calculations
can be performed for each mol e of the feed ( F = 1). By calcu-
lating vapor-to-feed mol e ratio (VF ---- V / F) , one can calculate
t he gas-to-oil ratio (GOR) or gas-to-liquid ratio (GLR). This
par amet er is part i cul arl y i mpor t ant in operat i on of surface
separat ors at t he oil pr oduct i on fields in whi ch pr oduct i on of
maxi mum liquid (oil) is desired by having low value of GOR.
Schemat i c of a cont i nuous flash separat or uni t is shown in
Fig. 9.2.
F e e d
1 mol e
zi
TF, PF
: ' i ' i ' : ' : ' : ' i ' : ' i ' : ' i ' . "
:i:i:i:iT:ZZ:i:i:i
::::::::::::::::::::::::
9 ..................:.... 9 ,..,.......
..............................................
:::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::
.:::,.,.,,,.:::,...........,.,............
i!ili!iiiii~ililili~ii!i!~i:i~i:i:i2i~i~i:i:i:i:i
:::::::::::::::::::::::::::::::::::::::::::::::::
Va p or
9 V mol e s
Yi
Liquid
9 L moles
xi
FIG. 9 . 2- - A cont i nuous flash separator.
Since vapor and liquid leaving a flash uni t are in equilib-
r i um from Eq. (6.201) we have
(9.1) Yi = gix4
in whi ch Ki is t he equi l i bri um ratio of comp onent i at T
and P and composi t i ons xi and Yi. Calculations of Ki values
have been di scussed in Section 6.8.2.3. Mole bal ance equat i on
ar ound a separat or uni t (Fig. 9.2) for comp onent i is given by
the following equation:
(9.2) 1 x zi = LF x xi + VF x yi
Subst i t ut i ng for LF = 1 -- VF, replacing for Yi f r om Eq. (9.1),
and solving for xi gives the following:
zi
(9.3) x~ - -
1 + V F(Ki -- 1)
Subst i t ut i ng Eq. (9.3) into Eq. (9.1) gives a relation for cal-
cul at i on of Yi. Since for bot h vapor and liquid product s we
must have ~ x4 = ~ Yi = 1 or ~ (Yi - x4) = 0. Subst i t ut i ng x4
and Yi from the above equat i ons gives the following objective
funct i on for cal cul at i on of VF:
~_ , z ~( K ~ - 1)
(9.4) F(V F) ---- 1 ~ --1) -- 0
i = 1
Reservoir engineers usually refer to this equat i on as
Rachf or d- Ri ce met hod [ 1 ]. When VF = 0, the fluid is a liq-
uid at its bubbl e poi nt ( sat urat ed liquid) and if VF = 1, t he
syst em is a vapor at its dew poi nt ( sat urat ed vapor) . Correct
solution of Eq. (9.4) shoul d give positive values for all x~ and
Yi, whi ch mat ch the condi t i ons ~ xi = ~_, yi = 1. The follow-
ing step-by-step procedure can be used to calculate VF:
1. Consi der the case t hat values of zi (feed composi t i on) , T,
and P (flash condi t i on) are known.
2. Calculate all Ki values assumi ng ideal sol ut i on (i.e., usi ng
Eqs. 6.198, 6.202, or 6.204). I n this way knowl edge of x4
and Yi are not required.
3. Guess an estimate of VF value. A good initial guess may
be calculated from the following rel at i onshi p [2]: VF = A/
(A - B), where A = ~[ z i (Ki - 1)] and B = ~[ z i ( Ki - 1)/
Ki].
4. Calculate F( V ) f r om Eq. (9.4) usi ng assumed value of VF in
Step 3.
5. I f calculated F(V F) is smaller t han a preset tolerance, e
(e.g., 10-15), t hen assumed value of VF is t he desired an-
swer. I f F(V F) > e, t hen a new value of VF must be calcu-
lated from the following relation:
F(V F)
( 9. 5) V/ ~ e w = V F dF(V F)
dVr
I n whi ch dF(V F)/ dV F is t he first-order derivative of F(V F)
wi t h respect to VF.
dF ( VF ) - L{ z i ( Ki - 1 ) 2 ]
(9.6) d~ - - i = 1 [VF~i---ii-+ 1] 2
The procedure is repeat ed until t he correct value of VF is
obtained. Generally, if F (V F) > O, VF must be reduced and if
F(V F) < O, V~ must be i ncreased to appr oach the solution.
6. Calculate liquid composi t i on, xi, from Eq. (9.3) and the
vapor phase composi t i on, Yi, f r om Eq. (9.1).
9. APPL I CATI ONS: PHASE EQUI L I B RI UM CAL CUL ATI ONS 369
FIG. 9 . 3 ~ Schemat i c of a three-stag e separator test in a Mi ddl e East
producti on field,
7. Cal cul at e Ki val ues f r om a mor e accur at e met hod usi ng xi
and yi cal cul at ed i n St ep 6. F or exampl e, Ki can be cal cu-
l at ed f r om Eq. (6.197) by a cubi c equat i on of st at e (i.e., SRK
EOS) t hr ough cal cul at i ng ~/L and ~v usi ng Eq. (6.126). Sub-
sequent l y fL and f/v can be cal cul at ed f r om Eq. (6.113). F or
i sot her mal flash we mus t have
(9.7) ~.~ - 1 < e
i=1 ~ f/
wher e e is a convergence t ol erance, (e.g., 1 10-13).
8. Rep eat a new r ound of cal cul at i ons f r om St ep 4 wi t h cal cu-
l at ed V~ f r om t he previ ous r ound unt i l t her e is no change
in val ues of VF, Xi, and Yi and i nequal i t y (9.7) is satisfied.
Vari ous ot her met hods of flash cal cul at i ons for fast con-
vergence are gi ven i n di fferent r ef er ences [ 14] . For exampl e,
Whi t son [1] suggest s t hat t he i ni t i al guess for VF mus t be
bet ween t wo val ues of VF,mi n and VF,m~x to obt ai n fast conver-
gence. Mi chel sen al so gives a st abi l i t y t est for flash cal cul a-
t i ons [5, 6]. Accur acy of resul t s of VLE cal cul at i ons l argel y
dep ends on t he met hod used for est i mat i on of Ki val ues and
for t hi s r eason r ecommended met hods i n Table 6.15 can be
used as a gui de for sel ect i on of an ap p r op r i at e met hod for
VLE cal cul at i on. Anot her i mp or t ant fact or for t he accur acy
of VLE cal cul at i ons is t he met hod of char act er i zat i on of C7+
f r act i on of t he p et r ol eum fluid. Ap p l i cat i on of cont i nuous
funct i ons, as it was shown in Sect i on 4.5, can i mp r ove resul t s
of cal cul at i ons. The i mp act of char act er i zat i on on p hase be-
havi or of r eser voi r fluids is al so demons t r at ed i n Sect i on 9.2.3.
The above p r ocedur e can be easi l y ext ended t o LLE or vap or -
l i qui d- l i qui d equi l i br i um ( VLLE) in whi ch t wo i mmi sci bl e
l i qui ds ar e in equi l i br i um wi t h t hemsel ves and t hei r vap or
p hase ( see Pr obl em 9.1).
Once val ue of lie is cal cul at ed in a VLE flash cal cul at i on,
t he gas-t o-l i qui d r at i o ( GLR) or gas-t o-oi l r at i o ( GOR) can be
cal cul at ed f r om t he fol l owi ng r el at i on [7]:
(9.8) GOR [ scf / st b] = 1.33 x 105pLVF
( 1 - VF)ML
wher e PL ( in g/ cm 3) and ME ( in g/ mol ) ar e t he densi t y and
mol ecul ar wei ght of a l i qui d p r oduct , respect i vel y ( see Prob-
l em 9.2). The best met hod of cal cul at i on of PL for a l i qui d
mi xt ur e is to cal cul at e it t hr ough Eq. (7.4), usi ng p ur e comp o-
nent l i qui d densi t i es. I f t he l i qui d is at at mos p her i c p r essur e
and t emp er at ur e, t hen PL can be r ep l aced by l i qui d specific
gravity, SG~, whi ch may al so be cal cul at ed f r om Eq. (7.4) and
comp onent s SG val ues. The met hod of cal cul at i ons is demon-
st r at ed i n Examp l e 9.1.
Exampl e 9. 1 (Three-stage surf ace separat or) - - Schemat i c of
a t hr ee- st age sep ar at or for anal ysi s of a r eser voi r fluid to pr o-
duce cr ude oil is shown in Fig. 9.3. The comp osi t i on of reser-
voi r fluid and p r oduct s as wel l as GOR in each st age and t he
overal l GOR are gi ven i n Table 9.1. Cal cul at e final cr ude com-
p osi t i on and t he overal l GOR f r om an ap p r op r i at e model .
Solution--The first st ep in cal cul at i on is to express t h e C7 +
f r act i on i nt o a numbe r of p s eudocomp onent s wi t h known
TABLE 9. 1--Experimental data for a Middle East reservoirfluid in a three-stage separator
test. Taken with permission from Ref. [7].
1st-Stage 2nd-St age 3rd-St age 3rd-Stage
No. Component Feed gas gas gas liquid
1 N2 0.09 0.77 0.16 0.15 0.00
2 CO2 2.09 4.02 3.92 1.41 0.00
3 H2S 1.89 1.35 4.42 5.29 0.00
4 H20 0.00 0.00 0.00 0.00 0.00
5 C1 29.18 63.27 31.78 5.10 0.00
6 C2 13.60 20.15 33.17 26.33 0.19
7 C3 9.20 7.56 18.84 36.02 1.88
8 n-C4 4.30 1.5 4.14 13.6 3.92
9 i-C4 0.95 0.43 1.24 3.62 0.62
10 n-C5 2.60 0.36 0.92 3.50 4.46
11 i-C6 ! .38 0.24 0.63 2.46 2.11
12 C6 4.32 0.24 0.57 2.09 8.59
13 C7+ 30.40 0.11 0.21 0.43 78.23
SG at 60~ 0.8150
Temp, ~ F 245 105 100 90 90
Pressure, psia 2387 315 75 15 15
GOR, scf/stb 850 601 142 107
TABLE 9.2--Characterization parameters of the C7+ fraction of sample of Table 9.1 [7].
Pseudocomponent mol% wt% M SG Tb, K n20 Nc P% N% A%
C7+ ( 1) 10.0 12.5 110 0.750 391.8 1.419 8 58 22 20
C7+ ( 2) 9.0 17.1 168 0.810 487.9 1.450 12.3 32 35 33
C7+ (3) 7.7 23.1 263 0.862 602.1 1.478 19.3 17 37 46
C7+ (4) 2.5 11.6 402 0.903 709.0 1.501 28.9 6 34 60
C7+ (5) 1.2 8.2 608 0.949 777.6 1.538 44 0 45 55
Total C7+ 30.4 72.5 209.8 0.843 576.7 1.469 15.3 25 34 41
Exp er i ment al val ues on M7+ and SG7+. Di st r i but i on p ar amet er s ( for Eq. 4. 56) a nd cal cul at ed val ues: M7+ = 209.8; Mo =
86.8; So = 0.65; $7+ = 0.844; BM = 1; As = 0.119; n7+ = 1.4698; AM = 1.417; Bs = 3; May = 209.8; Say = 0. 847.
charact eri zat i on paramet ers (i.e., M, Tb, SG, n2o, Nc, and PNA
composi t i on) . This is done usi ng the di st ri but i on model de-
scribed in Section 4.5.4 wi t h M7+ and SG7+ as the i nput pa-
rameters. The basic paramet ers (Tb, n20) are calculated from
the met hods descri bed in Chapt er 2, while t he PNA composi -
t i on for each p seudocomp onent is calculated from met hods
given in Section 3.5.1.2 (Eqs. 3.74-3.81). The cal cul at i on re-
sults wi t h di st ri but i on paramet ers for Eq. (4.56) are given in
Table 9.2. Mol ar and specific gravity distributions of t he C7+
fract i on are shown in Fig. 9.4. The PNA composi t i on is needed
for cal cul at i on of propert i es t hr ough p seudocomp onent ap-
p r oach ( Section 3.3.4). Such i nformat i on is also needed when
a si mul at or (i.e., EOR software) is used for phase behavi or
calculations [9].
To generat e the composi t i on of gases and liquids in sepa-
rators, see Fig. 9.3, the feed t o the first stage is consi dered
as a mi xt ure of 17 comp onent s (12 comp onent s listed in
Table 9.1 and 5 comp onent s listed in Table 9.2). For pure com-
ponent s (first 11 comp onent s of Table 9.1), Tc, Pc, Vc, and to are
t aken f r om Table 2.1. For C 6 fract i on (SCN) and C7+ fract i ons
(Table 9.3) critical propert i es can be obt ai ned from met hods
of Chapt er 2 ( Section 2.5) or f r om Table 4.6. For this example,
Lee-Kesl er correl at i ons for cal cul at i on of To, Pc, and to and
Ri azi -Daubert correl at i ons (the API met hods) for cal cul at i on
of Vc and M ( or Tb) have been used. The bi nary i nt eract i on pa-
ramet ers (BIPs) for nonhydr ocar bon- hydr ocar bon are t aken
from Table 5.3 and for hydr ocar bon- hydr ocar bon pairs are
calculated from Eq. (5.63). Paramet er A in this equat i on has
been used as an adjustable p ar amet er so t hat at least one pre-
dicted propert y mat ches the experi ment al data. This propert y
can be sat urat i on pressure or a liquid density data. For this
calculation, par amet er A was det ermi ned so t hat predi ct ed
liquid specific gravity from last stage mat ches experi ment al
value of 0.815. Liquid SG is calculated f r om Eq. (7.4) usi ng
SG of all comp onent s in the mixture. I t was f ound t hat when
A -- 0.18, a good mat ch is obtained. Anot her adjustable pa-
r amet er can be t he BI P of met hane and the first pseudocom-
0.01
0.008
0.006
~" 0. 004
0.002
0
FIG.
0 200 400 600 800
M
9 . 4~ Pr obabi l i t y densi ty
4
3
~ 2
LL
1
0.6 0.8 1 1.2 1.4
SG
functi ons f or mol ecul ar
wei g ht and speci fi c g rav i ty of the C7+ fracti on giv en in
Table 9.2 [ 8 ].
p onent of heptane-plus, C7(1). The value of BI P of this pai r
exhibits a maj or i mpact in t he cal cul at i on results. Ki values
are calculated from SRK EOS and flash calculations are per-
f or med for t hree stages shown in Fig. 9.3. The liquid p r oduct
from the first stage is used as t he feed for the second stage
separat or and flash cal cul at i on for this stage is performed
to calculate composi t i on of feed for the last stage. Similarly,
the final crude oil is p r oduced f r om the t hi rd stage at at mo-
spheric pressure. Composi t i on of C7+ in each st ream can be
calculated from sum of mol e fract i ons of the five pseudocom-
ponent s of C7i. GOR for each stage is calculated f r om Eq. (9.8).
Summar y of results are given in Table 9.3. Overall GOR is cal-
cul at ed as 853 comp ar ed wi t h actual value of 850 scf/stb. This
is a very good predi ct i on mai nl y due to adjusting BI Ps wi t h
liquid density of pr oduced crude oil. The cal cul at ed composi -
tions in Table 9.3 are also in good agreement wi t h act ual dat a
of Table 9.1.
The met hod of charact eri zat i on selected for t reat ment of
C7+ has a maj or i mpact on the results of calculations as shown
by Riazi et al. [7]. Table 9.4 shows results of GOR calcu-
lations for the t hree stages f r om different charact eri zat i on
met hods. I n the St andi ng met hod, Eqs. (6.204) and (6.205)
have been used to estimate K/ values, assumi ng ideal sol ut i on
mixture. As shown in this table, as t he number of pseudocom-
ponent s for the C7+ fract i on increases bet t er results can be
obtained. #
9 . 2 . 2 B ub b l e a nd D e w Po i nt s C al c ul at i ons
Bubbl e poi nt pressure cal cul at i on is performed t hr ough the
following steps:
1. Assume a liquid mi xt ure of known xi and T is available.
2. Calculate plat ( vapor pressure) of all comp onent s at T f r om
met hods descri bed in Section 7.3.
3. Calculate initial values of Yi and Pbub f r om Raoult' s law as
P = ~-~ x/Pi sat and Yi = xi psat / p.
4. Calculate Ki f r om Eq. (6.197) usi ng T, P, xi, and Yi.
5. Check if 1~ xiK,- - 1 [ < e, where e is a convergence toler-
ance, (e.g., 1 x 10 -12) and t hen go to Step 6. I f not, repeat
calculations f r om Step 4 by guessing a new value for pres-
sure P and yi = Kixi. I f ~x i Ki - 1 < 0, reduce P and if
xiKi - 1 > 0, increase value of P.
6. Write P as the bubbl e poi nt pressure and yi as t he com-
posi t i on of vapor phase. Bubble P can also be calculated
t hr ough flash calculations by finding a pressure at whi ch
Vr ~ 0. I n bubbl e T cal cul at i on x4 and P are known. The
cal cul at i on procedure is similar to bubbl e P cal cul at i on
met hod except t hat T must be guessed instead of guess-
ing P.
TABLE 9.3---Calculated values for the data given in Table 9.1 using proposed characterization
method. Taken with permission from Ref [7].
No. Component Feed 1 st-Stage gas 2nd-Stage gas 3rd-Stage gas 3rd-Stage liquid
1 N2 0.09 0.54 0.12 0.05 0.00
2 CO2 2.09 3.91 4.09 1.44 0.02
3 H2S 1.89 1,47 4.38 5.06 0.14
4 H20 0.00 0.00 0.00 0.00 0.00
5 C1 29.18 64.10 32.12 5.68 0.03
6 C2 13.60 19.62 32.65 25.41 0.38
7 Ca 9.20 7.41 18.24 35.47 3.05
8 n-C4 4.30 1.48 4.56 13.92 4.38
9 i-C4 0.95 0.41 1.23 3.47 0.78
10 n-C5 2.60 0.36 1.01 3.98 4.81
1 1 i-C6 1.38 0.24 0.68 2.61 2.37
12 C6 4.32 0.27 0.61 2.22 9.01
13 C7+ 30.40 0.19 0.31 0.69 75.03
SG at 60~ 0.8105
Temp,~ 245 105 100 90 90
Pressure, psia 2197 315 75 15 15
GOR, scf/stb 853 580 156 117
F or vap or s of known comp os i t i on dew P or dew T can be
cal cul at ed as out l i ned bel ow:
1. Assume a vap or mi xt ur e of known Yi and T is avai l abl e.
2. Cal cul at e Py ( vapor pr essur e) of all comp onent s at T f r om
met hods of Sect i on 7.3.
3. Cal cul at e i ni t i al val ues of xi and Pdew f r om Raoul t ' s l aw as
1/ P = ~_, yi / P~ sat and x/ = yi P / P~ sat.
4. Cal cul at e Ki f r om Eq. (6.197), usi ng T, P, xi, and Yi.
5. Check if ] ~] yi / Ki - 11 < e, wher e e is a convergence tol-
erance, (e.g., 1 x 10 -l z) go to St ep 6. I f not, r ep eat cal-
cul at i ons from St ep 4 by guessi ng a new val ue for pres-
sure P and x~ = yi/ Ki. I f ~, y i / Ki - 1 < O, i ncr ease P and i f
~y i / Ki - 1 > O, decr ease val ue of P.
6. Wri t e P as t he dew p oi nt p r essur e and xi as t he comp osi t i on
of f or med l i qui d phase.
Dew P can al so be cal cul at ed t hr ough fl ash cal cul at i ons
by fi ndi ng a p r essur e at whi ch VF = 1. I n dew T cal cul at i on
Yi and P are known. The cal cul at i on p r ocedur e is si mi l ar to
dew P cal cul at i on met hod except t hat T mus t be guessed in-
st ead of guessi ng P. I n t hi s case if ~ 3# / Ki - 1 < O, decr ease
T and if ~y i / Ki - 1 > O, i ncr ease T. Bubbl e and dew p oi nt
cal cul at i ons ar e used to cal cul at e PT di agr ams as shown in
t he next sect i on.
Reser voi r engi neer s usual l y use emp i r i cal l y devel oped cor-
r el at i ons to est i mat e bubbl e and dew p oi nt s for r eser voi r fluid
mi xt ures. F or exampl e, St andi ng, Glaso, and Vazquez and
Beggs cor r el at i ons for p r edi ct i on of bubbl e p oi nt p r essur e
of r eser voi r fluids are gi ven i n t er ms of t emp er at ur e, GOR,
gas specific gravity, and st ock t ank oil specific gravi t y ( or API
gravity) . These cor r el at i ons are wi del y used by r eser voi r en-
gi neers for qui ck and conveni ent cal cul at i on of bubbl e p oi nt
p r essur es [1, 3, 10]. The St andi ng cor r el at i on for p r edi ct i on
of bubbl e p oi nt p r essur e is [1, 3]
Pb( psia) = 18.2(a x l 0 b- 1.4)
a = ( GOR/ SGgas) 0"83
(9.9)
b = 0 . 0 0 0 9 1 T - 0.0125 (APIofl)
T = Temper at ur e, ~
whe r e / ~ is t he bubbl e p oi nt pr essur e, SGgas is t he gas specific
gravi t y ( = Mg/29), APIoil is t he API gravi t y of p r oduced l i qui d
cr ude oil at st ock t ank condi t i on, and GOR is t he sol ut i on gas-
t o-oi l r at i o in scf/ stb. Use of t hi s cor r el at i on is shown in t he
fol l owi ng exampl e. A devi at i on of about 15% is expect ed f r om
t he above cor r el at i on [3]. Mar houn devel op ed t he fol l owi ng
r el at i on for cal cul at i on of Pb based on PVT dat a of 69 oil
samp l es f r om t he Mi ddl e East [10]:
Pb( psia) = a ( GOR) b (SGgas) c (SGoiI) d ( T) e
(9.10) a = 5.38088 10 -3 b = 0.715082 c = - 1. 87784
d = 3.1437 e = 1.32657 T = t emp er at ur e, ~
whe r e SGoi I is t he specific gravi t y of st ock t ank oil and GOR is
in scf/ stb. The average er r or for t hi s equat i on is about 4-4%.
Ex ampl e 9. 2- - Cal cul at e bubbl e p oi nt p r essur e of r eser voi r
fluid of Table 9.1 at 245~ f r om t he fol l owi ng met hods and
comp ar e t he resul t s wi t h an exp er i ment al val ue of 2387 psi a.
TABLE 9. 4--Calculated GOR from different C7+ characterization methods. Taken with permission
Method Input for C7+
Lab data
Proposed M7+ and SG7+
Standing (Eqs. 6.202 MT+ and SG7+
and 6.203)
Simulation 1 a Nc & Tb
Simulation 2 Nc & Tb
Simulation 3 M & PNA
Simulation 4 M & PNA
from Ref [7].
No. of C7+ Overall GOR,
~a~ions scffs~ Stage 1 Stage 2 Stage 3
850 601 142 107
5 853 580 156 117
1 799 534 134 131
1 699 472 141 86
5 750 516 142 92
1 779 542 142 95
5 797 559 143 95
aCalculations have been performed through PR EOS using a PVT simulator [9].
a. Thermodynamic model with use of SRK EOS similar to the
one used in Example 9.1.
b. Standing correlation, Eq. (9.9).
c. Mahroun' s correlation, Eq. (9.10).
Sol ut i ons( a) The saturation pressure of the reservoir fluid
(Feed in Table 9.1) at 245~ can be calculated along flash cal-
culations, using the method outlined above. Through flash
calculations (see Example 9.1) one can find a pressure at
245~ and that the amount of vapor produced is nearly zero
(V~ -~ 0). The pressure is equivalent to bubble (or saturation)
pressure. This is a single-stage flash calculation that gives
psat = 2197 psia, which differs by - 8% from the experimental
value of 2387 psia. (b) A simpler method is given by Eq. (9.9).
This equation requires GOR, APIoi], and SGga~. GOR is given
in Table 9.1 as 850 scf/sth. APIofl is calculated from the specific
gravity of liquid from the third stage (SG = 0.815), which gives
APIo~ = 42.12. SGg~s is calculated from gas molecular weight,
Mg~s, and definition of gas specific gravity by Eq. (2.6). Since
gases are produced in three stages, Mgas for these stages are
calculated from the gas composition and molecular weights of
components as 23.92, 31.74, and 44.00, respectively. Mga~ for
the whole gas produced from the feed may be calculated from
GOR of each stage as Mg~, = (601 x 23.92 + 142 x 31.74 +
107 44.00)/(601 + 142 + 107) = 27.76. SGg~ = 27.76/29 =
0.957. From Eq. (9.9), A = 139.18 and Pb = 2507.6 psia, which
differs by +5.1% from the experimental value. (c) Using
Marhoun' s correlation (Eq. 9.10) with T = 705~ SGoij =
0.815, SGgas = 0.957, and GOR = 850 we get Pb = 2292 psia
(error of -4%) . I n this example, Marhoun' s correlation gives
the best result since it was mainly developed from PVT data
of oils from the Middle East, r
9. 2. 3 Generat i on of P- T Diagrams- - - True
Critical Propert i es
A typical temperature-pressure (TP) diagram of a reservoir
fluid was shown in Fig. 5.3. The critical temperature and pres-
sure (critical point) in a PT diagram are true critical proper-
ties and not the pseudocritical. For pure substances, both the
true and pseudocritical properties are identical. The main ap-
plication of a PT diagram is to determine the phase (liquid,
vapor or solid) of a fluid mixture. For a mixture of known com-
position, pseudocritical temperature and pressure (Tpc, P~c)
may be calculated from the Kay's mixing rule (Eq. 7.1) or
other mixing rules presented in Chapter 5 (i.e., Table 5.17).
Methods of calculation of critical properties of undefined
petroleum fractions presented in Section 2.5 all give pseudo-
critical properties. While pseudocritical properties are useful
for generalized correlations and EOS calculations, they do not
represent the true critical point of a mixture, which indicates
phase behavior of fluids. Calculated true critical temperature
and pressure for the reservoir fluid of Table 9.1 by simula-
tions i and 2 in Table 9.4 are given in Table 9.5. Generated PT
diagrams by these two simulations are shown in Fig. 9.5. The
bubble point curves are shown by solid lines while the dew
point curves are shown by a broken line. This figure shows
the effect of number of pseudocomponents for the C7+ on
the PT diagram. Critical properties given in Table 9.5 are true
critical properties and values calculated with five pseudocom-
ponents for the C7+ are more accurate. Obviously as discussed
TABLE 9.5--Effect of C7+ characterization methods on calculated
mixture critical properties [7].
Charac. Input for C7+ No. of C7+
scheme of Table 9.3 Fractions Tc, K Pc, bars Zc
Simulation 1 Nc & Tb 1 634 98 0.738
Simulation 2 Nc & Tb 5 651 141 0.831
Calculations have been performed through PR EOS using a PVT simulator [9].
in Chapter 4, for lighter reservoir fluids such as gas conden-
sate samples detailed treatment of C7+ has less effect on the
phase equilibrium calculations of the fluid.
The true critical temperature (Ttc) of a defined mixture may
also be calculated from the following simple mixing rule pro-
posed by Li [11]:
Tt c = ~_~ ~i Tci
i
(9,11) xiVc i
Si Xi Vci
where a~, Tci, and Vci are mole fraction, critical temperature,
and volume of component i in the mixture, respectively. The
average error for this method is about 0.6% (~3 K) with max-
i mum deviation of about 1.6% (--8 K) [12]. The Kreglewski-
Kay correlation for calculation of true critical pressure, Ptc,
is given as [13] follows:
Ptc=Pp~[l +( 5 . 8 0 8 +4 , 9 3 w) ( ~- l ) ]
(9.12)
Tpc= Exi Tci Pp~ = Zxi Pci and w= Ex , ooi
i i i
where Tpc and Ppc are pseudocritical temperature and pres-
sure calculated through Kay's mixing rule (Eq. 7.1). The aver-
age deviation for this method is reported as 3.8% (~2 bar) for
nonmethane systems and average deviation of 50% (~48 bar)
may be observed for met hane-hydrocarbon systems [12].
These methods are recommended in the API-TDB [12] as well
as other sources [3].
240
Bubble Point L : Liquid
. . . . . . . Dew Point V : Vapor
200 9 Critical Point
L
160 f7+ \
12o \
80 ":, \
V
/ c7 (1 component) '
40 : I
- " j
0 h h h i a i, - - - " a" " I , ~
200 300 400 500 600 700 800
Temperature, K
FIG. 9 . 5~ T he PT diagram for si mul ati ons 1 and 2 g i v en
in Table 9,5 with use of Nc and Tb. Taken with permi ssi on
from Ref. [ 7].
9. APPLICATIONS: PHASE EQUILIBRIUM CALCULATIONS 373
For undefined petroleum fractions the following correla-
tion may also be used to estimate true critical temperature
and pressure from specific gravity and volume average boil-
ing point (VABP) of the fraction [12]:
Ttc= 358.79 + 1.6667A - 1.2827(10-3)A 2
(9.13) A = SG (VABP - 199.82)
loglo(Ptc/Ppc) = 0.05 + 5.656 x logl0(Ttc/Tpc)
where Tt~, Tpr and VABP are in kelvin and Ppc and Ptc are in
bars. It is important to note that both Tpc and Ppc must be
calculated from the methods given in Section 2.5 for criti-
cal properties of undefined petroleum fractions. The average
error for calculation of Tt~ from the above method is about
0.7% (~3.3 K) with maximum error of 2.6% (~12 K). Relia-
bility of the above method for prediction of true critical pres-
sure of undefined petroleum fractions is about 5% as reported
in the API-TDB [12]. The above equation for calculation of
Ptc is slightly modified from the correlation suggested in the
API-TDB. This correlation is developed based on an empirical
graph of Smith and Watson proposed in the 1930s. For this
reason it should be used with special caution. The following
method is recommended for calculation of true critical vol-
ume in some petroleum-related references [3]:
Zto RTtc
V t c ~ ' - -
P,c
(9.14)
Ztc : ~ X4Z c i
i
Method of calculation of true critical points (Tt~, Ptc, and Vtc)
of defined mixtures through an equation of state (i.e., SRK)
requires rigorous vapor-liquid thermodynamic relationships
as presented in Procedure 4.B4.1 in Chapter 4 of the API-
TDB [12]. At the true critical point, a correct VLE calculation
should show that x~ = Yi. Most cubic EOSs fail to perform
properly at the critical point and for this reason attempts have
been made and are still continuing to improve EOS phase
behavior predictions at this point.
9. 3 VAPOR- LI QUI D- S OLI D EQUI LI BRI UMm
SOLI D PRECI PI TATI ON
I n this section, practical application of three-phase equilib-
rium in the petroleum industry is demonstrated. Upon re-
ducing the temperature, heavy hydrocarbons present in a
petroleum fluid may precipitate as a solid phase and the
liquid becomes in equilibrium with both the solid and the
vapor phase. I n such cases, the solid is at the bottom, liquid
is in the middle, and the vapor phase is on top of the liquid
phase. A general schematic of typical vapor-liquid-solid equi-
librium (VLSE) during solid precipitation in a petroleum fluid
is shown in Fig. 9.1. Solid precipitation is a serious problem
in the petroleum industry and the basic question is: what is
the temperature at which precipitation starts and under cer-
tain temperature, pressure, and composition how much solid
can be precipitated from a petroleum fluid? These two ques-
tions are answered in this section. Since solids are formed
at low temperatures, under these conditions the amount of
vapor produced is low and the problem reduces to SLE such
as the case for asphaltene precipitation. Initially, this section
discusses the nature of heavy compounds that are present
in petroleum residua and heavy oils. Precipitation of these
heavy compounds under certain conditions of temperature
and pressure or composition follow general principles of SLE,
which were discussed in Section 6.8.3. I n this section, the
problems associated with such heavy compounds as well as
methods that can be used to predict the certain conditions at
which they precipitate will be discussed. Based on the princi-
ple of phase equilibrium discussed in Section 6.8.1, a thermo-
dynamic model is presented for accurate calculation of cloud
point of crude oils under various conditions. Methods for cal-
culating the amount of solid precipitation from sophisticated
thermodynamic models as well as readily available parame-
ters for a petroleum fluid are also discussed in this section.
9.3.1 Nature of Heavy Compounds, Mechani sm
of their Precipitation, and Prevention Methods
Petroleum fluids, especially heavy oils and residues, contain
heavy hydrocarbons from paraffinic, naphthenic, and aro-
matic groups. Generally, there are three types of heavy hydro-
carbons that may exist in a heavy petroleum fluid: (1) waxes,
(2) resins, and (3) asphaltenes. As discussed in Section 1.1.3,
the main type of waxes in petroleum fluids are paraffinic
waxes. They are mainly n-paraffins with carbon number range
of C16-C36 and average molecular weight of about 350. Waxes
that exist in petroleum distillates usually have freezing points
between 30 and 70~ Another group of waxes called crys-
talline waxes are primarily isoparaffins and cycloparaffins
(with long-chain alkyl groups) with carbon number range of
30-60 and molecular weight range of 500-800. The melting
points of commercial grade waxes are in the 70-90~ range.
Solvent de-oiling of petroleum or heavy residue results in
dark-colored waxes or a sticky, plastic to hard nature material
[14]. Waxes present in a petroleum fluid may precipitate when
the conditions of temperature and pressure change. When the
temperature falls, heavy hydrocarbons in a crude or even a
gas condensate may precipitate as wax crystals. The temper-
ature at which a wax begins to precipitate is directly related
to the cloud point of the oil [15, 16]. Effects of pressure and
composition on wax precipitation are discussed by Pan et al.
[17].
Wax formation is undesirable and for this reason, different
additives usually polymer-based materials are used to lower
pour points of crude oils. Wax inhibitor materials include
polyalkyl acrylates and methacrylates, low-molecular-weight
polyethylene waxes, and ethyl-vinyl acetate (EVA) copoly-
reefs. The EVA copolymers are probably the most commonly
used wax inhibitors [14]. These inhibitors usually contain
20-40 wt% EVA. Molecular weight of such materials is usu-
ally greater than 10 000. The amount of EVA added to an oil is
important in its effect on lowering pour point. For example,
when 100 ppm of EVA is added to an oil it reduces pour point
from 30 to 9~ while if 200 ppm of same inhibitor is added
to another oil, it causes an increase in the pour point from
21~ to 25~ [14].
Asphaltenes are multiring aromatics (see Fig. 1.2) that are
insoluble in low-molecular-weight n-paraffins (LMP) such as
C3, n-C4, n-C5, or even n-C7 but soluble in benzene, carbon
disulfide (CS2), chloroform, or other chlorinated hydrocar-
bon solvents [15]. They exist in reservoir fluids and heavy
p et r ol eum fract i ons as pel l et s of 34-40 mi cr ons and ar e mai n-
t ai ned in susp ensi on by r esi ns [16, 18, 19]. Pet r ol eum fluids
wi t h l ow- r esi n cont ent s or under specific condi t i ons of t em-
per at ur e, pr essur e, and LMP concent r at i on may demons t r at e
asp hal t ene dep osi t i on in oi l - p r oduci ng wells. Asphal t ene de-
p osi t i on may al so be at t r i but ed to t he r educt i on of p r essur e
i n t he reservoi rs or due to addi t i on of sol vent s as in t he case
of CO2 i nj ect i on in enhanced oil recovery ( EOR) processes.
Resi ns p l ay a cri t i cal rol e in t he sol ubi l i t y of t he asp hal t enes
and mus t be p r esent for t he asp hal t enes to r emai n i n t he so-
l ut i on. Al t hough t he exact mechani s m is unknown, cur r ent
t heor y st at es t hat resi ns act as mut ual sol vent or f or m sta-
bi l i t y p ep t i de bonds wi t h asp hal t enes [16]. Bot h oils and as-
p hal t enes ar e sol ubl e in resi ns. St r uct ur e of resi ns is not wel l
known, but it cont ai ns mol ecul es wi t h ar omat i c as well as
nap ht heni c ri ngs. Resi ns can be sep ar at ed f r om oil by ASTM
D 2006 met hod. Resi ns are sol ubl e in n- p ent ane or n- hep t ane
( whi l e asp hal t enes ar e not ) and can be ads or bed on surface-
act i ve mat er i al such as al umi na. Resi ns when sep ar at ed ar e
r ed to br own semi sol i ds and can be des or bed by a sol vent
such as p yr i di ne or a benzene/ met hanol mi xed sol vent [15].
The amount of sul fur i n asp hal t enes is mor e t han t hat of
resi ns and sul fur cont ent of resi ns is mor e t han t hat of oils
[ 15]. Oils wi t h hi gher sul fur cont ent s have hi gher as p hahene
cont ent . Ap p r oxi mat e val ues of mol ecul ar wei ght , H/ C wei ght
rat i o, mol ar vol ume, and mol ecul ar di amet er of asphal t enes,
resi ns and oils are given i n Table 9.6. I n t he absence of act ual
dat a t ypi cal val ues of M, d25, AH/f, and TM ar e al so given for
monomer i c asp hal t ene sep ar at ed by n- hep t ane as suggest ed
by Pan and F i r oozabadi [20]. I n general Masph. > Mres. > Mwax
and (H/C)wax > (H/e)resi > (H/e)asph. Waxes have H/ C at omi c
r at i o of 2-2. 1 gr eat er t han t hose of resi ns and asp hal t enes
because t hey are mai nl y paraffi ni c.
I n general , cr ude oil asp hal t ene cont ent i ncr eases wi t h de-
cr ease in t he API gravi t y ( or i ncr ease in i t s densi t y) and for t he
r esi dues t he asp hl at ene cont ent i ncr eases wi t h i ncr ease in car-
bon resi due. Approxi mat el y, when Conr adson car bon r esi due
i ncr eases f r om 3 to 20%, asp hal t ene cont ent i ncr eases f r om 5
to 20% by wei ght [15]. For cr ude oils when t he car bon r esi due
i ncr eases f r om 0 to 40 wt%, asphal t ene, sulfur, and ni t r ogen
cont ent s i ncr ease f r om 0 to 40, 10, and 1.0, respect i vel y [15].
Oils wi t h asp hal t ene cont ent s of about 20 and 40 wt % exhi bi t
vi scosi t i es of about 5 x 106 and 10 x 106 poi ses, respectively.
As di scussed in Sect i on 6.8.2.2, gener al l y t wo subst ances
wi t h di fferent st r uct ur es ar e not very sol ubl e in each other.
F or t hi s reason, when a l ow- mol ecul ar - wei ght n-paraffi n
TABLE 9.6--Properties of typical asphaltenes, resins and oils.
Hydrocarbons M H% H/C V d, .~ D
Asphaltene 1000-5000 9.2-10.5 1.0-1.4 900 14.2 4-8
Resin 800-1000 10.5-12.5 1.4-1.7 700 13 2-3
Oil 200-600 12.5-13.1 1.7-1.8 200-500 8-12 0-0.7
M is molecular weight in g/mol. H% is the hydrogen content in wt%. H/C is
the hydrogen-to-carbon atomic ratio. V is the liquid molar volume at 25~
d is molecular diameter calculated from average molar volume in which for
methane molecules is about 4/~ (1/~ = 10 -~~ m). D is the dipole moment in
Debye. These values are approximate and represent properties of typical as-
phaltenes and oils. For practical calculations for resins one can assume M =
800 g/tool and for a typical monomeric asphaltene separated by n-heptane ap-
proximate values of some properties are as follows: M = 1000 g/mol. Density
of liquid ,-~ density of solid ~ 1.1 g/cm 3. Enthalpy of fusion at the melting
point: AHM = 7300 callmol, melting point: TM= 583 K. Data source: Pan and
Firoozabadi [20].
comp ound such as rt-C7 is added to a p et r ol eum mi xt ure, t he
as p hahene comp onent s ( heavy ar omat i cs) begi n to pr eci pi -
tate. I f p r op ane is added t o t he same oil mor e asp hal t enes
p r eci p i t at e as t he di fference in sol ubi l i t i es of C3-asphal t ene
is gr eat er t han t hat of nCy-asphal t ene. Addi t i on of an aro-
mat i c hydr ocar bon such as benzene will not cause preci p-
i t at i on of asp hal t i c comp ounds as bot h ar e ar omat i cs and
si mi l ar in st ruct ure; t her ef or e t hey are mor e sol ubl e in each
ot her in comp ar i s on wi t h LMP hydr ocar bons. When t hr ee
p ar amet er s for a p et r ol eum fluid change, heavy dep osi t i on
may occur. These p ar amet er s ar e t emp er at ur e, pr essur e, and
fluid comp osi t i on t hat det er mi ne l ocat i on of st at e of a sys-
t em on t he PT p hase di agr am of t he fluid mi xt ure. Preci pi t a-
t i on of a sol i d f r om l i qui d p hase is a mat t er of sol i d- l i qui d
equi l i br i um ( SLE) wi t h f undament al r el at i ons i nt r oduced in
Sect i ons 6.6.6 and 6.8.3.
Est i mat i on of t he amount of asp hal t ene and resi ns in cr ude
oils and der i ved f r act i ons is very i mp or t ant i n desi gn and
op er at i on of p et r ol eum- r el at ed i ndust ri es. As exp er i ment al
det er mi nat i on of asp hal t ene or resi n cont ent of var i ous oils
is t i me- consumi ng and costly, rel i abl e met hods t o est i mat e
asp hal t ene and r esi n cont ent s f r om easi l y meas ur abl e or
avai l abl e p ar amet er s ar e useful. Waxes are i nsol ubl e i n 1:2
mi xt ur e of acet one and met hyl ene chl ori de. Resi ns are i nsol -
ubl e i n 80:20 mi xt ur e of i sobut yl al cohol - cycl ohexane and as-
p hal t enes are i nsol ubl e in hexane [ 15]. ASTM D 4124 met hod
uses n- hep t ane to sep ar at e asp hal t enes f r om oils. Ot her ASTM
t est met hods for sep ar at i on of asp hal t enes i ncl ude D 893 for
sep ar at i on of i nsol ubl es in l ubr i cat i ng oils [21]. The most
wi del y used t est met hod for det er mi nat i on of asp hal t ene con-
t ent of cr ude oils is I P 143 [22]. Asphal t ene p r op or t i ons in a
t ypi cal p et r ol eum r esi dua is shown in Fig. 9.6. Si nce t hese
are basi cal l y p ol ar comp ounds wi t h very l arge mol ecul es,
most of cor r el at i ons devel op ed for t ypi cal p et r ol eum fract i ons
and hydr ocar bons fail when ap p l i ed to such mat er i al s. Met h-
ods devel op ed for p ol ymer i c sol ut i ons are mor e ap p l i cabl e to
asp hal t i c oils as shown in Sect i on 7.6.5.4.
Compl exi t y and si gni fi cance of asp hal t enes and resi ns i n
p et r ol eum r esi dua is cl earl y shown in Fig. 9.6. Spei ght [15]
as wel l as Goual and F i r oozabadi [23] consi der ed a p et r ol eum
fluid as a mi xt ur e of p r i mar i l y t hr ee speci es: asphal t enes,
resi ns, and oils. They as s umed t hat whi l e t he oil comp onent
is nonpol ar, resi ns and asp hal t ene comp onent s ar e pol ar. The
degrees of pol ar i t i es of asp hal t enes and resi ns for several
oils were det er mi ned by measur i ng di pol e moment . They re-
p or t ed t hat whi l e di pol e moment of oil comp onent of var i ous
cr udes is usual l y less t han 0.7 debye (D) and for many oils
zero, t he di pol e moment of r esi ns is wi t hi n 2-3 D and for
asp hal t enes ( sep ar at ed by n-C7) is wi t hi n t he r ange of 4-8 D.
Di pol e moment of waxy oils is zero, whi l e for asp hal t i c cr udes
is about 0.7 D. Therefore, one may det er mi ne degree of as-
p hal t ene cont ent of oil t hr ough measur i ng di p ol e moment .
Values of di pol e moment s of some p ur e comp ounds ar e gi ven
in Table 9.7. n-Paraffi ns have di p ol e moment of zero, whi l e
hydr ocar bons wi t h doubl e bonds or br anched hydr ocar bons
have hi gher degr ee of pol ari t y. Pr esence of het er oat oms such
as N or O si gni fi cant l y i ncr eases degrees of pol ari t y.
The p r obl ems associ at ed wi t h asp hal t ene dep osi t i on are
even mor e severe t han t hose associ at ed wi t h wax deposi t i on.
Asphal t ene al so affects t he wet t abi l i t y of r eser voi r fluid on
sol i d surface of reservoir. Asphal t ene may cause wet t abi l i t y
9. APPLICATIONS: PHASE EQUI L I BRI UM CALCULATI ONS 3 7 5
90
8 0 84
7O
. ~ 60 -
" 0
,~ 5o
~ 4 o
30
20
1 0
Volatile
Saturates/
A romatics
0 20 40 60 80 100
W eight Percent
FIG. 9 . 6- - Represent at i on of proporti ons of resi ns and as-
phal tenes in a petroleum residua. Taken wi th permi ssi on from
Ref. [ 15].
TABLE 9.7--Dipole moments of some compounds and oil mixtures.
No. Compound Dipole, debye
1 Methane (Ca) 0.0
2 Eicosane (Cz0) 0.0
3 Tetracosane (C24) 0.0
4 2-methylpentane 0.1
5 2,3-dimethylbutane 0.2
6 Propene 0.4
7 1-butene 0.3
8 Cyclopentane 0.0
9 Methylcyclopentane 0.3
10 Cyclopentene 0.9
11 Benzene 0.0
12 Toluene 0.4
13 Ethylbenzene 0.2
14 o-Xylene 0.5
15 Acetone (C3H60) 2.9
16 Pyridine 2.3
17 Aniline 1.6
18 NH3 1.5
19 H2S 0.9
20 CO2 0.0
21 CC14 0.0
22 Methanol 1.7
23 Ethanol 1.7
24 Water 1.8
Oi l mixtures
25 Crude Oils <0.7
26 Resins 2-3
27 Asphaltenes a 4-8
Data source for pure compounds: Poling, B. E., Prausnitz, J. M., O'Connell,
J. E, Properties of Gases and Liquids, 5th ed., McGraw-Hill, New York, 2000.
aAsphaltenes separated by n-heptane. Data source for oil mixtures: Goul and
Firoozabadi [23].
reversal and its understanding will help plan for more effi-
cient oil recovery processes [24]. Similarly, asphaltene depo-
sition negatively affects the EOR gas flooding projects [25].
Asphaltene precipitation may also occur during oil processing
in refineries or transportation in pipelines and causes major
problems by plugging pipes and catalysts pores [26, 27]. The
problem is more severe for heavy oils. Further information
on problems associated with asphaltene precipitation during
production, especially in the Middle East fields, is given by
Riazi et al. [28].
There are a number of models and theories that are pro-
posed to describe mechanism of asphaltene formation [29].
Understanding of kinetics of asphaltene formation is much
more difficult than wax formation. There is no universally
accepted model for asphaltene formation; however, most re-
searchers agree on two models: (1) colloidal and (2) micellar.
Schematic of colloidal model is shown in Fig. 9.7. The nature
and shape of the resulting aggregates will determine their
effect on the behavior of the petroleum fluid [30, 31]. I n
this model, asphaltene particles come together to form larger
molecules (irreversible aggregation), which grow in size. Ac-
cording to this model the surface of asphaltene molecules
must be fully covered by resin molecules. For this reason
when concentration of resin exceeds from a certain level, rate
of asphaltene deposition decreases even if its concentration
is high. Because of this it is often possible that an oil with
higher asphaltene content results in less precipitation due to
high resin content in comparison with an oil with lower as-
phaltene and resin contents. Knowledge of the concentration
of resin in oil is crucial in determination of the amount of
asphaltene precipitation.
I n the micellar model, it is assumed that asphaltene mol-
ecules exist as micelles in crude and micellar formation is a
reversible process. Furthermore, it is assumed that the micel-
lar shape is spherical, the micellar sizes are monodispersed
(i.e., all having the same size), and the asphaltene micellar
core is surrounded by a solvated shell as shown in Fig. 9.8.
I n this model too, resins may cover asphaltene cores and pre-
vent precipitation. Thermodynamic models to describe phase
behavior of asphaltic oils depend on such models to describe
nature of asphaltene molecules.
Asphaltenes precipitate when conditions of temperature,
pressure, or composition change. The condition under which
precipitation begins is called the onset of asphaltene precipi-
tation. I n general to select a right method for determination
of asphaltene onset, asphaltene content or asphaltene preven-
tion one must know the mechanism of asphaltene precipita-
tion, which as mentioned earlier very much depends on the oil
composition. Asphaltenes flocculate due to excess amounts of
paraffins in the solution and micellization (self-association) of
asphaltene is mainly due to increase in aromaticity (polarity)
of its medium [32].
During the past decade, various techniques have been de-
veloped to determine asphaltene onset from easily measur-
able properties. These methods include measuring refractive
index to obtain the onset [33]. Fotland et al. [34] proposed
measuring electric conductivity to determine the asphaltene
onset. Escobedo and Mansoori [35] proposed a method to
determine the onset of asphaltene by measuring viscosity of
crude oil diluted with a solvent (n-Cs, n-C7, n-C9). They showed
that with a decrease in deposition rates with increasing crude
(a) Colloidal Phenomenon Due to Increase in Concentration of Polar Miscible Solvent (such as polar aromatic hydrocarbons
shown by solid ellipses) in crude oil
(b) Asphaltene Flocculation and Precipitation
(c) Steric Colloid Formation
FIG. 9 . 7 - - Schemat i c of colloidal model for asphal tene formation. Taken with permi ssi on
from Ref. [ 29],
m
Monomer i c
asphaltene
Monomedc
resi n
N Asphal =free
oil
Micelle~" core Solvated shell
FIG. 9 . 8 - - Schemat i c of micellar formation in asphal tene precipita-
tion. Taken with permi ssi on from Ref. [ 20].
3 7 6
100 .................................................................................................
>
10
1
t
,o , o
9
9
o Heptane
9 Toluene
9
AO
0 91 76
9 O 0
0
Onset 9 0
0
9 0
,%
9
01 . i i ~ i i i i i i i i i i Jl
1 10 100
Volume% Solvent in Crude
FI G. 9 . 9 ~ Det er mi nat i on of asphal t ene onset
f rom v i scosi t y. T ak en wi t h per mi ssi on f rom Ref.
[ 3 5].
oil kinematic viscosity and with increase in production rate,
deposition also increases [35]. Determination of asphahene
onset through viscosity measurement is shown in Fig. 9.9
[35]. When a solvent such as toluene is added to a crude oil the
viscosity of crude-solvent solution decreases as concentration
of solvent increases. For this solvent, asphaltene does not pre-
cipitate and the curve of viscosity versus solvent concentra-
tion is smooth. However, when a solvent such as n-heptane is
added before asphaltene onset, viscosity decreases smoothly
with solvent concentration similar to the case of toluene sol-
vent, but as asphaltene molecules begin to aggregate and form
larger particles viscosity does not fall for a short time. This
is due to the fact that the increase in viscosity is due to par-
ticle formation that will offset a decrease in viscosity due to
dilution of the crude. However, as soon as particles become
large enough to precipitate, viscosity of the crude begins to
drop again but more rapidly than before the onset. There-
fore, the onset of asphahene precipitation is at the concentra-
tion level where viscosity curve shows a change in its trend.
For the example that is shown in Fig. 9.9, this point is at 20
vol% solvent addition. Another technique to determine the
onset of asphaltene is through measuring interfacial tension
(IFT) in which when precipitation occurs there is a sudden
change in IFT. Various methods of determination of the onset
of asphaltene are discussed by Mansoori [32].
To remove precipitated asphaltenes, special chemicals
known as inhibitors are used. Asphaltenes can be precipi-
tated when a solvent is added to a crude oil, but once the
asphaltenes are precipitated they are difficult to redissolve
by a diluent. Some aromatics are used to inhibit asphahene
precipitation in crude oils. Because aromatics are similar in
nature with asphaltene (also an aromatic compound) they are
more soluble in each other than in other types of hydrocar-
bons and as a result precipitation is reduced. Benzene and
toluene are not commonly used as an asphaltene inhibitor
because a large concentration is required [16]. The effect
of toluene in reducing amount of asphaltene precipitation
for a reservoir fluid with different level of CO2 concentra-
tion is shown in Fig. 9.10 [36]. Other types of asphaltene
inhibitors include n-dodecyl-benzenesulfonic acid (DBSA),
which has stronger effect than benzene in reducing asphal-
tene precipitation. I n petroleum reservoirs, the main prob-
lem associated with asphaltene deposition is its adsorption
on formation rocks. Adsorbed asphahene negatively affects
well performance and removal of the asphaltene is desired.
Piro et al. [37] made a good study on the evaluation of sev-
eral chemicals for asphaltene removal and related test meth-
ods. Toluene is a typical solvent for asphaltenes and shows a
very high uptake (several tens of a wt%) when the asphaltenes
"5,
. m
P,
4,( I
3.0
2,0
1.0
O.O
05 @.KK 0~68
r
0% Toluene
- - 9 1% Toluene
u 3% Toluene
m . 5% Toluene
- - 6% Tol uene ~ , . ,."
. . , j '
f , , ' ~ /
, . / / .
f / / ,~-"
9 . _ J ~ - 1 / . , - . , J
0.$2 0. 54 0,56 I ~ 4[I.$ @,G2 @.K4
Mole fraction of C0 2
FI G. 9 . 1 0- - Ef f ect of t ol uene i n r educi ng amount of asphal t ene pr eci pi t at i on f or a reser-
v oi r f l ui d. T ak en wi t h per mi ssi on f rom Ref. [ 3 6].
TABLE 9. 8reEvaluation of three types of additives for asphaltene removal on a rock surface [37].
Removal efficiency (wt% of adsorbed asphaltene)
for different additives
Additive wt% in solvent Time, h Additive A Additive B Additive C
0.1 1 40.5 14.0 2.7
0.1 18 48.9 21.4 6.0
2.0 1 49.9 32.9 8.9
2.0 18 51.5 46.0 10.1
are in the bulk state; on the contrary, the asphaltene up-take
by toluene is very low (10-20 wt%) when the same material
is adsorbed on a rock surface (clays, dolomia, quartz, etc.),
as experienced by Piro et al. [37]. For this reason, they used
additives dissolved in toluene for asphaltenes' removal when
they are adsorbed on rock surface. Three types of additives
were evaluated: additive A was based on alkyl benzene sul-
fonic acid (ABSA); additive B was based on complex poly-
mer; additive C was based on another complex polymer. As-
phaltic materials were obtained from a crude oil of 42 API
gravity by precipitation with n-heptane. The rock on which
asphaltenes were adsorbed was powdered dolomite (average
particle size of 60/ zm and surface area of 10 m2/g) and toluene
was employed as solvent for the additives. Experiments were
conducted to study the effect of different types of additives,
concentration level, and time on the amount of asphaltenes
up-take. A summary of results of experiments is given in
Table 9.8. The results show that addition of additive A (0.1
wt% in toluene solution) can remove up to 41% after 1 h and
up to 49% of asphaltene after 18 h [37]. Therefore, at higher
additive concentrations the contact time can be reduced.
Deasphalted oils may also be used as asphaltene inhibitor
since they contain resins that are effective in keeping asphal-
tene molecules soluble in the oil in addition to their potential
for greater solvency. There are some synthetic resins such 2-
hexadecyl naphthalene that can also be used as asphaltene
inhibitor. Most of these inhibitors are expensive and research
on manufacturing of commercially feasible asphaltene in-
hibitors is continuing. Asphaltenes or other heavy organics
are precipitated under certain conditions that can be deter-
mined through phase diagram (i.e., PT or Px diagrams). An
example of such diagrams is the Px diagram at constant tem-
perature of 24~ for an oil-CO2 system as shown in Fig. 9.11.
Some specifications for this oil are given in Table 9.9. I n this
figure, the solid phase is indicated by S and regions of LVS
and LS are the regions that asphaltenes may precipitate and
should be avoided. The best way to prevent asphaltene pre-
cipitation is to avoid the region in the phase diagram where
asphaltene precipitation can occur. It is for this reason that
phase behavior of petroleum fluids containing heavy organics
is important in determining the conditions in which precipi-
tation can be avoided. Construction of such phase diagrams
is extremely useful to determine the conditions where precip-
itation occurs. Unfortunately such diagrams for various oils
and solvents are not cited in the open literature. Figure 9.11
shows that the solid phase is formed at very high concentra-
tion of CO2, that is, the region that is not of practical applica-
tion and should be considered with cauttion. Thermodynamic
models, along with appropriate characterization schemes can
be applied to waxy or asphaltic oils to determine possibility
and amount of precipitation under certain conditions. For
example, Kawanaka et al. [30] used a thermodynamic ap-
proach to study the phase behavior and deposition region in
CO2-crude mixtures at different pressures, temperatures, and
compositions. I n the next few sections, thermodynamic mod-
els for solid formation are presented to calculate the onset
and amount of solid precipitation.
For the same tank oil shown in Table 9.9, Pan and
Firoozabai [20] used their thermodynamic model based on
micellar theory of asphaltene formation to calculate asphal-
tene precipitation for various solvents. Their data are 'shown
in Fig. 9.12, where amount of precipitation is shown versus
dilution ratio. The dilution ratio (shown by Rs) represents
volume (in cm 3) of solvent added to each gram of crude oil.
The amount of precipitated resin under the same conditions
is also shown in this figure. The onset of asphaltene forma-
tion is clearly shown at the point where amount of precipita-
tion does not change with a further increase in solvent-to-oil
ratio. Lighter solvents cause higher precipitation. Generally
value of Rs at the onset for a given oil is a function of sol-
vent molecular weight (Ms) and it increases with increase in
Ms [38]. Effect of temperature on asphaltene precipitation
depends on the type of solvent as shown in Fig. 9.13 [39].
The amount of solid deposition increases with temperature
for propane, while for n-heptane the effect of temperature
is opposite. Effect of pressure on asphaltene precipitation is
shown in Fig. 9.14. Above the bubble point of oil, increase in
pressure decreases the amount of precipitation, while below
bubble point precipitation increases with pressure.
9 . 3 . 2 Wax Preci pi t at i on- - - S ol i d S ol ut i on Model
There are generally two models for wax formation calcula-
tions. The first and more commonly used model is the solid-
solution model. I n this model, the solid phase is treated as a
homogenous solution similar to liquid solutions. Formulation
of SLE calculations according to this model is very similar
to VLE calculations with use of Eq. (6.205) and equilibrium
ratio, K sL, from Eq. (6.209) instead of Ki for the VLE. This
model was first introduced by Won [41] and later was used
to predict wax precipitation from North Sea oils by Pedersen
et al. [ 14, 42]. The second model called multisolid-phase model
was proposed by Lira-Galeana et al. in 1996 [43], which has
also found some industrial applications [16]. I n this model,
the solid mixture is not considered as a solution hut it is de-
scribed as a mixture of pure components; each solid phase
does not mix with other solid phases. The multisolid-phase
model is particularly useful for calculation of CPT of oils.
The temperature at which wax appears is known as wax ap-
pearance (or precipitation) temperature (WAT or WPT), which
theoretically is the same as the CPT. Both models are based on
the following relation expressing equilibrium between vapor,
L
,
LV
i
0.0 25 50 75 100.0
L
b d - -
Mole Percent C02
(a)
o
98. 0 98.5 99.0 99.5 100.0
1380. 0
908. 5
891. 5
Mole Percent CO 2
(b)
FI G. 9 .1 1 --Phase diagram f or oil-CO 2 mi x tures at 24~ Asphal tene
preci pi tati on occurs in the LS and LVS regions. (a) Entire composi ti on
range. (b) Enlarged LS section. O il properti es are given in T able 9.9.
T aken wi th permi ssi on from Ref. [3 0].
Oil specifications
TABLE 9. 9--Data for shell tank oil of Fig. 9.12 [30].
Asphaltene specifications
mol% C1 + C2 0.6 wt% (resin) in oil
mol% C3-C5 10.6 wt% (asph.) in oil
mol% C6 4.3 density, g/cm 3
mol% C7+ 84.5
M 221.5 (MT+ = 250) M (precipitated) b
SG 0.873 (SGT+ = 0.96) a ~.
14.1 a
4.02
1.2
4500
12.66(1- 8.28 x 10-4T)
Data taken from Refs. [20, 49].
bM for asphaltene in oil (monomer) is about 1000 [20].
c g in (cal/cm3) ~ and T in kelvin.
liquid, and solid phases for a multicomponent system shown
in Fig. 9.15.
(9.15) f~V(T,P, yi)=~L(T,P, xL)=f~S(T,P, xS)
This equation can be split into two parts, one for vapor-liquid
equilibrium and the other for liquid-solid equilibrium. These
two equations can be expressed by two relations in terms of
equilibrium ratios as given by Eqs. (6.201) and (6.208). I n
this section the solid-solution model is discussed while the
multisolid-phase model is presented in the next section.
I n the solid-solution model the solid phase (S) is treated as a
homogeneous solution that is in equilibrium with liquid solu-
tion (L) and its vapor. I n Fig. 9.15, assume the initial moles of
nonequilibrium fluid mixture (feed) is 1 mol (F = 1) and the
molar fraction of feed converted to vapor, liquid, and solid
phases are indicated by VF, LF, and SF, respectively, where
LF = 1 -- VF -- Sr. Following the same procedure as that in the
VLE calculations and using the mass balance and equilibrium
relations that exist between vapor, liquid, and solid phases
yields the following set of equations similar to Eqs. (9.3) and
(9.4) for calculation of VF and SF and compositions of three
phases:
N N z i ( K ? - 1)
(9.16) F vL=~_ ~( yi - - x~) =~_ , ( I _ SF ) - +VF - ( Ki w - 1 ) i = l i=1 = 0
N
zi
(9.17) XL = ~i=1 (1 -- SF) + VF (Ki vL - 1)
o
FIG. g.1 2--Preoipitateci amount of asphal tene ( - - ) and
resin ( ----) for the crude oil g i v en in Table 9.g at 1 bar and
29 5 K. Taken with permi ssi on from Ref. [ 20].
N Zi K/VL
(9.18) Yi = Ei=t (1 -- SF) + Ve (g/vL - 1)
- 1)
(9.19) FSL=~--~(xiS--xL)=i=1 z...,i=l 1 ~-S~/ g-L ~ 1) = 0
(9.20) x L zi
= I+SF(K _ 1)
(9.21) x s = xL K sL
where zi, x L, and x s are the compositions in mole fractions
of the crude oil (before precipitation), the equilibrium liquid
oil phase (after precipitation), and precipitated solid phase,
respectively. SF is number of moles of solid formed (wax
precipitated) from each I mol of crude oil or initial fluid
(before precipitation) and must be calculated from solution
of Eq. (9.9), while VF must be calculated from Eq. (9.16).
I n fact in Fig. 9.15, F is assumed to be 1 tool and 100 SF
represents tool% of crude that has precipitated. Equations
(9.16)-(9.18) have been developed based on equilibrium rela-
tions between vapor and liquid, while Eqs. (9.19)-(9.21) have
been derived from equilibrium relations between liquid and
solid phases. Compositions of vapor and solid phases are cal-
culated from Eqs. (9.18) and (9.21). Equation (9.20) is the
prime equation for calculation of liquid composition, x L. To
validate the calculations it must be the same as x L calculated
from Eq. (9.17). For the case of crude oils and heavy residues,
the amount of vapor produced is small (especially at low tem-
peratures) so that VF = 0. This simplifies the calculations and
solution of only Eqs. (9.19)-(9.21) is required. However, for
light oils, gas condensates, and natural gases VF must be cal-
culated and all the above six equations must be solved simul-
taneously. The Newton-Raphson method described in Sec-
tion 9.2.1 may be used to find both V~ and SF from Eqs. (9.16)
and (9.19), respectively. The onset of solid formation or wax
appearance temperature is the temperature at which S--> 0
[44]. This is equivalent to the calculation of dew point tem-
perature (dew T) in VLE calculations that was discussed in
Section 9.2.1.
The main parameter needed in this model is K/sL that may
be calculated through Eq. (6.209). I n the original Won model,
activity coefficients of both liquid and solids become close to
unity and Hansen et al. [45] recommended use of polymer-
solution theory for calculation of activity coefficients through
Eq. (6.150). On this basis the calculation of K sL can be sum-
marized as in the following steps:
a. Assume T, P, and compositions x L and x s for each i in the
mixture are all known.
b. Calculate the ratio of f L/f/s for each pure i at T and P from
Eq. (6.155).
( a)
99.4
99.3
99.2
99.1
99.0
98.9
98.8
9. APPLI CATI ONS: PHASE EQUI L I BRI UM CALCULATI ONS 3 8 1
270 280 290 300 310 320 330 340 350 360
T,K
(a)
70.0
K
.o
65.0
60.0
55.0
50.0
45.0
40.0
240
( b)
250 260 270 280 290 300 310
T, K
(b)
FIG. 9.13---Effect of temperature on asphal tene precipitation. ( a) Propane diluent; ( b)
n-heptane diluent. Taken with permission from Ref. [39].
320
c. I n calculation of f/L/f/S paramet ers TMi, AH/f, and ACei
must be calculated for each component i.
d. Calculate TMi from Eq. (6.156), AH~/from Eq. (6.157), and
ACpi f r om Eq. ( 6. 161) .
e. Calculate bot h ~/L and yi s from Eq. (6.154). I n calculation of
yi s, calculate ~s from Eq. (6.155). V/s and V/L can be obtained
from Table 7.1.
f. Once f/L/f/S, yi L, and yi s have been determined, calculate
K/SL from Eq. (6.209).
This is a typical solid-solution model for calculation of wax
format i on without the use of any adjustable parameter. All pa-
ramet ers can be calculated from the molecular weight of com-
ponent s or pseudocomponent s as described in Sections 6.6.6.
Using PNA composition for calculation of properties of C7+
pseudocomponent s t hrough Eqs. (6.149), (6.156), and (6.157)
improves model predictions.
Pedersen et al. [42], based on their data for North Sea oils,
showed that bot h Won and Hansen procedures significantly
overestimate bot h the amount of wax precipitation and CPT.
For this reason, they suggested a number of adjustable pa-
ramet ers to be used for calculation of various paramet ers.
Chung [44] has used the following empirical set of correla-
tions for calculation of properties of C7+ fractions for the wax
format i on prediction:
A/-//f = 0.9TMiM~ ~
V/L = 3. 8M O-786
(9.22)
~ = 6.743 + 0.938 (In M~) - 0.0395 (In M~) 2
-13. 039 (In Mi) -1
where TMi is the melting point in kelvin, A/~ f is the mol ar
heat of fusion in cal/mol, V/r is the mol ar liquid volume in
5~0
0
J
E
~ . 2. 0
~ P~ edi c~ ed
A Mea~'~s~d
A
i I
2OO 4O0
FIG. 9 . 1 4- - Ef f ect of pressure on asphal tene precipitation. Taken with
permission from Ref. [40],
0o0 9 , . ~
0 6 00 800 1 003 i 200 1 400
Pressure, bar
330
320
310
o Experi ment al
- - Predicted
300 , , , L
0 100 200 300 400 500
Pressure, bar
FIG. 9 . 1 5- - Predi ct i on of W A T for a Chi nese reser-
v oi r fi ui d usi ng method of Mei et al. [ 46]. A bsol ut e er-
ror between calculated and ex peri mental data is 1.6
K. Composi ti on of reserv oi r fluid is g i v en in Problem
9.10 ( Table 9.18 ) .
cm3/ mol , and 8/L is the solubility par amet er in (cal/ cm3) ~
Various researchers have used similar correl at i ons but wi t h
different numeri cal coefficients. Most recently, Mei et al. [46]
have applied t he Pedersen et al. model to calculate wax pre-
ci pi t at i on in a live oil (oil under reservoir condi t i ons) from
Pubei Oil field l ocat ed in t he west ern part of Chi na where gas
injection is used in EOR processes. Composi t i on of this oil is
given in Probl em 9.10 (Table 9.18). Basically, t hey used Won' s
correl at i ons [41] for ACpi, T~, and AH/ f while t he Thomas
et al.'s correl at i on [47] was used for cal cul at i on of 8s:
7.62 + 2.8a {1 - exp [ - 9. 51
~s = x 10-4 (Mi - 48.2)] } forM/ < 450
10.30 + 1.78 x 10-3a (M i - 394.8) for Mi > 450
(9.23)
where a is an adjustable parameter. They used six adjustable
par amet er s for cal cul at i on of ACPi , TMi , A/-//f, and 8s in t erms
of T and M, whi ch were det ermi ned by mat chi ng cal cul at ed
and experi ment al dat a on measur ed WAT values for the oil
[46]. For t he begi nni ng of the flash calculations, t he initial
values of K~ zL may he est i mat ed from t he Wilson' s correl at i on
(Eq. 6.204) assumi ng ideal solution theory. Mei et al. [46] sug-
gested t hat initial/~SL ValUeS can be set equal t o t he reci procal
of K/vL values also calculated from the Wilson' s formula. Pre-
dicted WAT versus pressure is comp ar ed wi t h measur ed val-
ues and is given in Fig. 9.15. One maj or probl em associ at ed
wi t h this model is t hat it requires experi ment al dat a on wax
preci pi t at i on t emperat ure or the amount of wax format i on to
find t he adjustable paramet ers. This gr aph is developed based
on dat a report ed in Ref. [46].
Composi t i on of this reservoir fluid (Table 9.18 in Probl em
9.10) indicates t hat it is a gas condensat e sample and for gases
usually WAT declines wi t h increase in pressure. Lower WAT
values for an oil are always desirable. This indicates t hat pres-
sure behaves as an i nhi bi t or for wax preci pi t at i on for live oils,
gas condensat e, or nat ural gas samples. However, this is not
: : : : : : : : : " ' Vap or l V mol es" " : : : : : : :
i.i-~.i~ y~ . . . 9
: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
iiii~iiii ' Liquid,L'moles i:i:i:i
::i:i:i:!:i:i:i:i Pur e Sol i dLa ye r 1 !:i:i:i:i:i:
' . ' . ' . ' . ' . ' . ' , ' . . : . : ' : ' X . :
:::::::::::: n s :.:.:.:.:.:.
i:!::i:]i:!:!::i: 1 mol es :i:!:i:!:i:i
: . : . . . . . : . b S . , . . . . . . . ' . ' , . . . .
:!:i:i:!:!:i:i:! s ii!i!iiiiiii!
:::::::::::::::: nj mol es :5::::::::::
!:!:i:i:i:!:i:i n s :!:i:i:i:!:i:i
i:!:i:i:!:!:!:i: NS_ 1 mol es !:i:i:!:!:i:i:
! i! ! iiii! ! i! il . . . . . . . . i2dy ;ii; . . . . . . . . ! i! iiii! i! ii
::::::::::::::: s i:i:i:i:i:i:~:
::::::::::::::: n ~, mol es +:-:-:-:+:
:i:i:?:::::::: . . . . ~'~ . . . . . . . ::::i:i:i:::::
i
I
Ns separate
solid layers
J
FIG. 9 . 1 6~ Schemat i c of mul ti phase-sol i d mod-
el for wax precipitation. Courtesy of Lira-Galeana
et al. [ 43].
the case for heavy liquid oils such as crude oils or dead oils.
Pressure causes slight increase in t he amount of wax precipi-
t at i on as will be di scussed in t he next section where the exact
met hod of cal cul at i on of CPT of crude oils is presented.
9. 3. 3 Wax Preci pi tati on: Mul t i sol i d- Phase
Model - - Cal cul at i on of Cl oud Poi nt
One of the probl ems wi t h the solid-solution model in predic-
t i on of wax format i on is t hat wi t hout the use of adjustable
paramet ers it usually overestimates amount of wax precipi-
t at i on and cl oud poi nt of crude oils. I n this section, anot her
model t hat is part i cul arl y accurat e for cal cul at i on of CPT of
crude oils will be presented. I n this model, the solid is consid-
ered as multilayer, each layer represents a pure comp onent
( or p seudocomp onent ) as a solid t hat is insoluble in ot her
solid layers. This model was developed by Li ra-Gal eana et al.
[43] and is used for cal cul at i on of bot h t he amount of wax
preci pi t at ed in t erms of wt% of initial oil as well as CPT. A
schemat i c of the model is demonst rat ed in Fig. 9.16. I n this
model, it is assumed t hat as t emperat ure is reduced onl y a
selected number of preci pi t at i ng comp onent s will coexist in
SLE. The basis of calculations for this model is the stability
criteria expressed by Eq. (6.210), whi ch shoul d be applied to
all N comp onent s ( pure as well as p seudocomp onent s) in t he
following form:
(9.24) fii(T, P, zi) - ~S(T, P) > 0 i = 1, 2 . . . . . N
where fii(T, P, zi) is t he fugacity of comp onent i in t he orig-
inal fluid mi xt ure at T and P, and f/s is the fugacity of pure
solid i at T and P. A comp onent may exist as a pure solid
phase if inequality by Eq. (9.24) is valid. This inequality can
be applied onl y to single-solid phase and is not applicable
t o solid-phase solutions. Assume comp onent 1 is the lightest
(i.e., Ct in a reservoir fluid) and N is t he heaviest comp onent
(i.e., the last pseudocomponent of a C7+ fraction). I f Eq. (9.24)
is applied to all N components in the mixture the number
of components that satisfy this equation is designated as Ns
(<N). If Ns = N it means that the mixture at T and P is initially
in a solid phase (100% solid). All precipitating components
must satisfy the following isofugacity equations:
(9.25) f,L(T, P, x3) = f/S(r, P) i = (N - Ns + 1) . . . . . N
The material balance equation for the nonprecipitating com-
ponents is
I
zi-x~ 1- ni -Ki xi ~=0
j=(N_ Ns+I) - f
(9.26) i = i . . . . . ( N - Ns)
where n s is the moles of solid phase j and F is the number
of moles of feed (initial fluid mixture). For the precipitating
components where all solid phases are pure
z~ - x~ 1 - ~ ~i n i _ Ky ~x ~ v_
i=(N-Ns+l) ~- -- -- ~- ~ ' F = 0
(9.27) [i = (N - Ns + 1) . . . . . N - 1], (Ns > 1)
I n addition, all components must satisfy the following VLE
isofugacity:
(9.28) i v ( T, P, Yi) =f i L( T, P, xi L) i = 1 . . . . . N
There are two constraint equations for component i in the
liquid and vapor phases:
N N
(9.29) y~x L ---- Z y i = 1
i=1 i=1
Equation (9.28) is equivalent to Eq. (6.201) in terms of VLE
ratios (K~VL). There are Ns equations through Eq. (9.24),
( N- Ns) equations through Eq. (9.26), ( Ns - 1) equations
through Eq. (9.27), N equations through Eq. (9.28), and two
equations through Eq. (9.29). Thus the total number of equa-
tions are 2N + Ns + 1. The unknowns are x~ (N unknowns),
S
Yi ( N unknowns), n i (Ns unknowns), and V / F (one unknown),
with the sum of unknown same as the number of equations
( 2N+ Ns + 1). Usually for crude oils and heavy residues,
where under the conditions at which solid is formed, the
amount of vapor is small and V / F can be ignored in the above
equations. For such cases Eq. (9.28) and ~ Yi = 0 in Eq. (9.29)
can be removed from the set of equations. On this assumption,
the number of equations and unknowns reduces by N + 1 and
yi and V / F are omitted from the list of unknowns. Total num-
ber of moles of solid formed (S) is calculated as
N
S
(9.30) S = Z ni
j =( N-N$+I )
The amount of wax precipitated in terms of percent of oil is
calculated as
(9.31) wax wt% in oil = 100
F N
~i = 1 Zi M i
s
where F is the total number of moles of initial oil and nj
is the moles of component i precipitated as solid. Mi is the
molecular weight of component i and zi is its mole fraction
in the initial fluid. The ratio of S/ F is the same as SF used in
Eq. (9.17). The ratio of V / F in Eqs. (9.26) and (9.27) is the
same as VF in Eqs. (9.17) and (9.18).
The above set of equations can be solved by converting them
into equations similar to Eqs. (9.17)-(9.19). For precipitating
components, x/t can be calculated directly from Eq. (9.26),
while for nonprecipitating components they must be calcu-
lated from Eq. (9.27) after finding V / F and S/ F. Moles of
solid formed for each component, n s, must be calculated from
Eq. (9.27). Values of V / F and S/ F must be found by trial-and-
error procedure so that Eq. (9.29) is satisfied.
The CPT of a crude can be calculated directly from
Eq. (9.24) using trial-and-error procedure as follows:
a. Define the mixture and break C7+ into appropriate number
of pseudocomponents as discussed in Chapter 4.
b. P and zi are known for all component/ pseudocomponents.
c. Guess a temperature that is higher than melting point of the
heaviest components in the mixture so that no component
in the mixture satisfies Eq. (9.24).
d. Reduce the temperature stepwise until at least one com-
ponent (it must be the heaviest component) satisfies the
equality in Eq. (9.24).
e. Record the temperature as calculated CPT of the crude oil.
A schematic of CPT and wax precipitation calculation using
this model is illustrated in Fig. 9.17. To simplify and reduce
the size of the calculations, Lira-Galeana et al. [43] suggest
that solid phases can be combined into three or four groups
where each group can be considered as one pesudocompo-
nent. As the temperature decreases, the amount of precip-
itation increases. Compositions of six crude oils as well as
their experimental and calculated values of CPT according
to this model are given in Table 9.10. Calculated values of
CPT very much depend on the properties (especially molecu-
lar weight) of the heaviest component in the mixture. For oils
the C7+ fractions should be divided into several pseudocom-
ponents according to the methods discussed in Chapter 4. I n
such cases, the heaviest component in the mixture is the last
pseudocomponent of the C7+ and the value of its molecular
weight significantly affects the calculated CPT. I n such cases,
the molecular weight of last pseudocomponent C7+ may be
used as one of the adjustable parameters to mat ch calculated
amount of wax precipitation with the experimental values.
Prediction of the amount of wax precipitation for oils 1 and
6 in Table 9.10 are shown in Fig. 9.18 as generated from the
data provided in Ref. [43].
I n the calculation of solid fugacity through Eq. (6.155), ACe
is required. In many calculations it is usually considered as
zero; however, Lira et al. [43] show that without this term,
considerable error may arise in calculation of solute compo-
sition in liquid phase for some oils as shown in Fig. 9.19. Ef-
fects of temperature and pressure according to the multisolid-
phase model are clearly discussed by Pan et al. [17] and for
several oils they have compared predicted CPT with exper-
imental data at various pressures. They conclude that for
heavy oils at low pressure or live oils (where light gases are
dissolved in oil) the increase in pressure will decrease CPT as
shown in Fig. 9.20. However, for heavy liquid oils (dead oils)
O
30
[N-Ns+ 1] components
N ~ and (N-1)th components
9 CPT
N ~ components
I - - ~ 9
200 400
Temperature, K
FIG. 9 . 1 7 - - Demonst r at i on of mul ti sol i d-phase model for cal cul ati on
of cl oud-poi nt t emperat ure and wax precipitation f or a typi cal crude
oil. Courtesy of Li ra-Gal eana et al. [ 43],
a nd at hi gh p r es s ur es , t he ef f ect of p r e s s ur e i s op p os i t e a nd
a n i nc r e a s e i n p r e s s ur e wi l l c a us e a n i nc r e a s e i n t he CPT as
s hown i n Fi g. 9. 21.
Val ues of WAT c a l c ul a t e d f r om t he s ol i d- s ol ut i on mode l ar e
us ual l y cl os er t o p our p oi nt s of a n oil, whi l e val ues f r om
mul t i s ol i d- p ha s e mode l ar e cl os er t o t he c l oud p oi nt of t he
cr ude. As i t was s hown i n Tabl e 3. 30, t he p our p oi nt t e mp e r a -
t ur e i s us ual l y gr e a t e r t ha n t he c l oud p oi nt f or mos t c r ude oi l s.
As di s c us s e d ear l i er t he me t hod of c ha r a c t e r i z a t i on of oi l a nd
C7+ gr eat l y af f ect s r e s ul t of c a l c ul a t i ons us i ng t hi s me t hod.
TABLE 9. l O---Compositions and cloud point temperature of some oils. Taken with permission from Ref. [43].
Oil No.
1 2 3 4 5 6
Comp mol% M Mol% M tool% M mol% M mol% M tool% M
C1 0.056 0.000 0.016 0.000 0.000 0.021
C2 0.368 0.113 0.144 0.100 0.173 0.254
C3 1.171 1.224 1.385 0.118 1.605 1.236
i-C4 0.466 0.645 1.174 0.106 1.150 0.588
n-C4 1.486 2.832 3.073 0.099 3.596 2.512
i-C5 0.961 1.959 2.965 0.162 3.086 1.955
n-C5 1.396 3.335 3.783 0.038 4.171 3.486
C6 2.251 5.633 7.171 0.458 7.841 6.842
C7 6.536 88.8 9.933 92.8 11.27 94.1 2.194 90.8 11.11 94.1 12.86 92.2
C8 8.607 101.0 10.75 106.3 12.41 107.0 2.847 106.5 13.43 105.4 13.99 105.4
C9 4.882 116.0 7.179 120.0 7.745 122.0 1.932 122.3 9.420 119.0 9.195 119.0
C10 2.830 133.0 6.561 134.0 5.288 136.0 5.750 135.0 5.583 135.0 6.438 134.0
Cll 3.019 143.0 5.494 148.0 5.008 147.0 4.874 149.0 4.890 148.0 5.119 148.0
CI2 3. 1t 9 154.0 4.547 161.0 3.969 161.0 5.660 162.0 3.864 162.0 4.111 161.0
C13 3.687 167.0 4.837 175.0 3.850 175.0 6.607 176.0 4.300 t75. 0 4.231 175.0
C14 3.687 181.0 3.700 189.0 3.609 189.0 6.149 189.0 3.272 188.0 3.682 188.0
C15 3.687 195.0 3.520 203.0 3.149 203.0 5.551 202.0 2.274 203.0 3.044 202.0
C16 3.079 207.0 2.922 216.0 2.300 214.0 5.321 213.0 2.791 216.0 2.255 214.0
C17 3.657 225.0 3.072 233.0 2.460 230.0 5.022 230.0 2.311 232.0 2.405 230.0
C18 3.289 242.0 2.214 248.0 2.801 244.0 4.016 244.0 1.960 246.0 2.006 245.0
C19 3.109 253.0 2.493 260.0 2.100 258.0 4.176 256.0 1.821 256.0 1.770 257.0
C20+ 38.4 423.0 17.0 544.0 14.33 418.0 38.80 473.0 11.33 388.0 12.00 399.0
SG20+ 0.893 0.934 0.880 0.963 0.872 0.887
CPT, K 313.15 311.15 314.15 295.15 305.15 308.15
CPT, Calc. 312.4 308.2 316.0 299.3 301.2 309.5
Error, K 0.75 2.95 - 1. 85 - 4. 15 3.95 - 1. 35
9. APPLICATIONS: PHASE EQUILIBRIUM CALCULATIONS 3 8 5
16
14
12
lO
Y:
8
6
4
2
0
220
0 Data
0 0 0 ~ - Model
ok
0 - X~O
~ o O t q i ~ 0 O0 0 0 0
240 260 280 300
Temperature, K
(a) Oil 1
10
320
4
0
220
0 Data
- - Model
240 260 280 300
Temperature, K
320
(b) Oil 6
FIG. 9 . 1 8 - - Predi ct i on of wax precipitation
and cl oud-poi nt temperature for oi l s 1 and 6
g i v en in Table 9.10. Taken with permi ssi on from
Ref. [ 43].
0.01
t~
,..3
. 2
0.001
2
" 6
~9
o
0.0001
3.35
O Data
Calculated with Cp term
. . . . . . . Calculated without Cp term
i i i i i h i i i
3.4 3.45
1000/T, 1/K
i
3 . 5
FIG. 9 . 1 9 - - Ef f ect of ~ Cp term on cal cul ati on of
sol ute sol ubi l i ti es at 1 bar [ 43]. Drawn based on
data from Ref. [ 43].
9 . 0
, ~ 7 . o
,1=
"~ 5, 0
:ff
| 4.o
"u
_= 3,0
2.0
~ 1, o
* ,~, Me Mu md . I ba r
- - . , C a ~ . 1 20ba r
41P
0,0 ~ ~ '= ,
270,0 280,0 290.0 300.0 310,0 320.0
T emper at ur e, K
FIG. 9 . 20- - Ef f ect of pressure on cl oud-poi nt temperature and
wax formation in a syntheti c crude oil at low pressures. Taken
with permi ssi on from Ref. [17].
322
320
~ 318
o
~ 316
U
314
~ 1 7 6 1 7 6
. . . . " ' " " " ~ ~ i n a l ~ ~ ~ ~' : ~' - ~- ~- ' : " Oil
- - 30% C2
. . . . . 30% C5
i i i i i
0 100 200 300 400 500 600
Pressure, bar
FIG. 9 . 21 mEffect of pressure on cl oud-poi nt
temperature at hig h pressures for a crude oil di-
luted by v ari ous light hydrocarbons. Taken with
permi ssi on from Ref. [ 17].
9 . 4 A S PHA L T EN E PRECI PI TATI ON:
S OL I D - L I QUI D EQUI L I B R I UM
Predi ct i on of asphal t ene pr eci pi t at i on is mor e difficult and
compl ex t han pr edi ct i on of wax preci pi t at i on. The r eason for
this compl exi t y is t he compl ex nat ur e of asphal t enes and t he
mechani s m of t hei r preci pi t at i on. The presence of resi ns fur-
t her compl i cat es model i ng of asphal t ene preci pi t at i on. I n ad-
di t i on asphal t ene mol ecul es are pol ar and when aggregat ed
t hey behave si mi l ar tO p ol ymer mol ecul es. Asphal t enes are
heavi er t han wax and t hey preci pi t at e at a hi gher t emper a-
t ure t han WAP. Asphal t enes usual l y exist in heavi er oils and
for t he case of crude oils at at mosp her i c pressures t he amount
of vapor produced in equilibrium with liquid is quite small so
that the probl em of asphaltene precipitation reduces to LSE.
The effect of t emperat ure on asphaltene precipitation is the
same as for wax precipitation, that is, as the t emperat ure de-
creases the amount of precipitation increases. The effect of
pressure on asphaltene precipitation depends on the type of
oil. For crude oils (flee of light gases) and live oils above their
bubble point pressure, as pressure increases the amount of as-
phaltene precipitation decreases, but for live oils at pressures
below the bubble point pressure, as the pressure increases
asphaltene precipitation also increases so that at the bubble
point the amount of precipitation is maxi mum [48].
There are specific t hermodynami c models developed for
prediction of asphaltene precipitation; these are based on
the principles of SLE and the model adopt ed for the mecha-
ni sm of precipitation. These mechani sms were discussed in
Section 9.3.1. Most t hermodynami c models are based on two
models assumed for asphaltene precipitation: colloidal and
micellization models. A molecular t hermodynami c fl ame-
work based on colloid theory and the SAFT model has been
established to describe precipitation of asphaltene from crude
oil by Wu-Prausni t z-Fi roozabadi [39, 49]. Mansoori [29] also
discusses various colloidal models and proposed some ther-
modynami c models. Pan and Firoozabadi have also devel-
oped a successful t hermodynami c micellization models for
asphaltene precipitation [20, 36].
Most t hermodynami c models consider asphaltene as poly-
mer molecules. Furt hermore, it is assumed that the solid
phase is pure asphaltene. The solution is a mixture of oil
(asphaltene-free) specified by component B and asphaltene
component specified by A. Applying the principle of SLE
to asphaltene component of crude oil in t erms of equality of
fugacity
( 9. 32) f L = fA s
where fL is fugacity of asphahene (A) in the liquid solution
(A and B) and f s is fugacity of pure solid asphaltene. One
good theory describing polymer-solution equilibrium is the
Flory-Huggins (FH) theory, which can be used to calculate
solubility of a pol ymer in a solvent. Many investigators who
studied t hermodynami c models for asphaltene precipitation
have used the FH theory of pol ymer solutions for calculation
of chemical potential of asphaltenes dissolved in oil [38, 50].
Nor-Azian and Adewnmi [48] also used FH theory for the as-
phaltic oils. Moreover, they also considered the vapor phase
in their model with VLE calculations between liquid oil and
its vapor. According to the FH theory the chemical potential
of component i (polymer) in the solution is given as
(9.33) g/L _/Z~L v/L V/L
R ~ = l n( %) + 1 - ~ + ~-~ ($i - ~m) 2
where api = ~, 8m = E~, Oigi,/2 n is the chemical potential
r n. . . = .
of component ~ m the hqmd phase, and #~L lS the chemical
potential at reference state, which is normal l y taken as pure
liquid i. api is the volume fraction of i, V/L is the liquid mol ar
volume of pure i at Y and P of the solution, R is the gas
constant, V L is the liquid mol ar volume of mixture, and 3i is
the solubility paramet er for component i at Y of the solution.
The above equation can be conveniently converted into an
activity coefficient form (y/L) as
(9.34) yL= e xp In +I - ~mL+R T
Once ~i L iS knownfi L can be calculated from Eq. (6.114) and
after substituting into Eq. (9.32) we get the following relation
for the volume fraction of asphaltene in the liquid solution
[38]:
= e xp [ EE- - - 1 - - vk ( ~A- - ~m) 2]
(9.35)
ap~
LV~
where apE is the volume fraction of asphaltenes in the oil
(liquid) phase at the time solid has been precipitated. Once
apE is known, amount of asphaltene precipitated can be cal-
culated from the difference between the initial amount of as-
phaltene in liquid and its amount after precipitation as
L L
mAD = mAT - - PAf~AV~
mAT = 0.01 x (initial asphaltenes in liquid, wt%) x pLkVL
asphaltene precipitated wt% = 100 x PmixV~ "
(9.36)
where mAD is the mass of asphaltenes deposited (precipitated)
and mAT is the mass of total asphaltene initially dissolved in
the liquid (before precipitation) bot h in g. PA and pLm~ are
mass densities of asphaltenes and initial liquid oil (before
precipitation) in g/ cm 3. V L is the total volume of liquid oil
before precipitation in cm 3. pLix can be calculated from an
EOS or from Eq. (7.4). I n determination of mAT, the initial wt%
of asphaltenes in oil (before precipitation) is needed. This
paramet er may be known from experimental data or it can be
considered as one of the adjustable paramet ers to mat ch other
experimental data. mAT can also be determined from Eq. (9.35)
from the knowledge of asphahene composi t i on in liquid at
( , ~, L t TL' t onset
tbeonsetwhenmAoiszeroandmgT=pA~,~AvrJ . A more
accurate model for calculation of asphaltene precipitation is
based on Chung' s model for SLE [44]. This model gives the
following relation for asphaltene content of oil at t emperat ure
T and pressure P [51]:
X A = exp / RTMA 1 - - - - ~- ~ ( ~A - - t~rn) 2
I n A Vk (vk
(9.37) - ~ - 1+~+ )/-T-
where subscript A refers to asphaltene component and PMA is
the pressure at melting point TMa. All other t erms are defined
previously. The last t erm can be neglected when assumed
V~ - V s. This model has been i mpl ement ed into some reser-
voir simulators for use in practical engineering calculations
related to pet rol eum product i on [51 ].
As ment i oned earlier (Table 9.6), in absence of actual data,
PA and MA may be assumed as 1.1 cm3/g and I 000 g/tool, re-
spectively. Other values for asphaltene density are also used
by some researchers. Speight [15] has given a simplified
version of Eq. (9.37) in t erms of asphaltene mole fraction
(XA) as
MA ( ~ L - - 8A) 2
(9.38) lnxg =
RT pA
where he assumed PA = 1.28 cm3/g and MA = 1000 g/mol. ~A
and 8L are the solubilities of asphaltene and liquid solvent
(i.e., oil), respectively. I f 8A and 3L are in (cal/cm3) 1/2 and T is
in kelvin then R = 1.987 cal/ mol-K. This equation provides
only a very approximate value of asphaltene solubility in oils.
I n fact one may obtain Eq. (9.38) from Eq. (9.35) by assuming
molar volumes of both oil and asphaltene in liquid phase are
equal: VA L = V L. As this assumption can hardly be justified,
one may realize the approximate nature of Eq. (9.38).
A mixture of asphaltenes and oil may be considered ho-
mogenous or heterogeneous. Kawanaka et al. [30] have de-
veloped a thermodynamic model for asphaltene precipitation
based on the assumption that the oil is a heterogeneous solu-
tion of a polymer (asphaltenes) and oil. The asphaltenes and
the C7+ part of the oil are presented by a continuous model (as
discussed in Chapter 4) and for each asphahenes component
the equilibrium relation has been applied as
( 9 . 3 9 ) f f i .( T , s ^L X L ) i 1 . . . . .
P, XAi) = tzAi(T, P, = NA
where/ 2 s and/ 2~/ are chemical potentials of ith component
of asphaltene in the solid and liquid phase, respectively. Sim-
ilarly x s and x~/ are the composition of asphaltene compo-
nents in the solid and liquid phases. The sum ~ x s is unity but
the sum y~. x~ is equal to x~ the mole fraction of asphaltenes
in the liquid phase after precipitation. NA is the number of
asphaltenes components determined from distribution model
as it was discussed in Chapter 4. I n this model, the solid phase
is a mixture of NA pseudocomponents for asphaltenes.
I n this thermodynamic model, several parameters for as-
phaltenes are needed that include molecular weight (MA),
mass density (PAL binary interaction coefficient between as-
phaltene and asphaltene-free crude ( k ~) , and the asphaltene
solubility parameter in liquid phase (3L). As discussed in Sec-
tion 9.3.1, in lieu of experimental data on MA and PA they can
be assumed as 1000 and 1.1 g/cm 3, respectively. Kawanaka
et al. [30] recommends the following relations for calculation
of 3L and kAB as a function of temperature:
(9.40)
(9.41)
6A L = 12.66(i -- 8.28 X 10-4T)
k~ = -7. 8109 x 10 3 +3. 8852 x 10-5MB
where ~ is the asphaltene solubility parameter in (cal/cm3) ~
and T is temperature in kelvin. MB is the molecular weight of
asphaltene-free crude oil. To calculate/2s., values of 6 s and
pA s are needed. 8s can be calculated from Eq. (6.154) and ps is
assumed the same as p~. It should be noted that in these rela-
tions asphaltene-free crude refers to the liquid phase in equi-
librium with precipitated solid phase, which include added
solvent (i.e., C3, n-Cs, or n-C7) and the original crude. The
phase diagram shown in Fig. 9.11 was developed based on
this compositional model [30]. Equation (9.40) gives value of
9.5 (cal/cm3) ~ at 25~ which is consistent with the value re-
ported by other investigators. Equation (9.40) is named after
Hirschberg who originally proposed the relation [52].
Most of the thermodynamic models discussed in this sec-
tion predict data with good accuracy when the adjustable
parameters in the model are determined from experimen-
tal data on asphaltene precipitation. Results of a thermody-
namic model based on the colloidal model and SAFT theory
for Suffield crude oil are shown in Fig. 9.22. The crude has
1.1
1.0
"r, 0.9
E
0.8
-~ 0.7
0.6
0.5
0.4
0.3
O _ _ . . . . ~ _ O
Q. -
o
9 _ ~ 1 - - - - t l - - - - t - . . . . - ~
/
'~;"A A II @n- C5
l ' I i II On- C6
1~m points: experi ment al dat a 9 n- C7
lit / lines: cal cual t ed z~ n-C8
f;" ~ ~ J , 9 n-Cl O
i
5 10 15 2O 25 30
Solvent Dilution Ratio (mL solvent added/g oil)
35
FIG. 9 . 22reCal cul ated v ersus ex peri mental
amount of asphal tene precipitated by v ari ous
n-al k anes sol v ents added to Suffield crude oil.
Taken with permi ssi on from Ref. [ 39].
specific gravity of 0.952 and average molecular weight of 360
with resin and asphaltene contents of 8 and 13 wt%, respec-
tively. Effects of temperature and pressure on asphaltene and
solid precipitation were discussed in Section 9.3.1.
To avoid complex calculations for quick and simple esti-
mation of asphahene and resin contents of crude oils, at-
tempts were made to develop empirical correlations in terms
of readily available parameters similar to those presented in
Section 3.5.1.2 for composition of petroleum fractions. Be-
cause of the complex nature of asphaltenes and wide range
of compounds available in a crude, such attempts were not
as successful as those developed for narrow-boiling range
petroleum fractions. However, Ghuraiba [53] developed the
following simple correlation based on limited data collected
from the literature for prediction of asphaltene and resin con-
tents of crude oils:
wt% of asphaltene or resin in crude oil = a + b/~ + cSG
(9.42)
where /~ is the refractivity intercept defined in Eq. (2.14)
as Ri = n2o - dzo/ 2. Amounts of asphaltenes and resin in a
crude mainly depend on the composition of the crude. I n Sec-
tion 3.5.1.2, paramet ers/ ~ and SG were used to predict the
composition of petroleum fractions. Calculation of n20 and d20
for a crude is not as accurate as for a fraction since the crude
has a very wide boiling point range. For this reason, the above
equation gives only an approximate value of asphaltene and
resin contents. Coefficients a, b, and c in Eq. (9.42) are given in
Table 9.11. These coefficients have been determined based on
the calculation of nzo and d20 from Eq. (4.7) and Table 4.5. Only
M is required for calculation of these two properties. If M is
not available it may be estimated from other properties such
as viscosity and SG (i.e., Eq. (2.52) or reversed form of Eq.
(4.7) and Table 4.5). The above correlation generally predicts
amount of asphahene and resin contents with absolute devi-
ation of 1.5-2 wt%. Experimental data points for resin con-
tents were very limited and for this reason predicted values
must be taken with caution. Data to develop these correlations
TABLE 9.11---Constants in Eq. 9. 42 for estimation of asphaltene and resin contents of crude oils.
Constants in Eq. (9.42) Absolute Dev.%
Composition type (wt%) Range of AP][ gravity of oil Range of wt% a b c No. of oils Avg Max
Asphaltene in oil 5.9--43.4 0.1-20 -731 674 31 122 1.4 -5.2
Resin in oil 5.5--43.4 5.6--40 -2511.5 2467 -76 41 1.9 -4.9
were mostly obtained from Speight [15] and the Oi l and Gas
J our nal Dat a B ook [54]. For prediction of amount of asphal-
tene precipitation when it is diluted by an n-alkane solvent,
the following correlation was developed based on very limited
data [53]:
Asphaltene predicted, wt% = a + b (/~) + c (SG)
(9.43) + d (Rs) + e (Ms)
where coefficients a-e are determined from experimental
data. Parameters /~ and SG are the same as in Eq. (9.42)
and should be calculated in the same way. Ms is the solvent
molecular weight (n-alkane) and Rs is the solvent-to-oil ratio
in cm3/g. This correlation was developed based on the data
available for three different Kuwaiti oils and 45 data points,
and for this limited database the coefficients were deter-
mined as a = -2332, b = 2325, c = -112.6, d = 0.0737, and
e = -0.0265. With these coefficients the above equation pre-
dicts asphaltene precipitation of Kuwait oils with AD of 0.5%.
The correlation is not appropriate for other crude oils and
to have a generalized correlation for various oils, the coeffi-
cients in Eq. (9.43) must be reevaluated with more data points
for crude oils from around the world. The following example
shows application of these equations.
Ex ampl e 9. 3--For Suffield crude oil the asphaltene precipi-
tation by various solvents is shown in Fig. 9.22. Calculate
a. asphaltene content.
b. resin content.
c. amount of asphaltene
10 cm 3/g n-decane.
(wt%) precipitated by adding
Sol ut i on- - For this oil, M = 360 and SG = 0.952. n20 and d20
should be calculated through M using Eq. (4.7) with coeffi-
cients in Table 4.5. The results are n2o = 1.4954, d20 = 0.888,
and/ ~ = 1.05115. (a) From Eq. (9.42), asphaltene wt% = 7%.
(b) From Eq. (9.42), resin wt% = 9.3%. (c) For calculation of
asphaltene precipitation from Eq. (9.43) we have Ms = 142
and Rs = 10 cm3/g, thus wt% of asphaltene precipitated is cal-
culated as 1.3%. The experimental value as shown in Fig. 9.22
is 0.5%. The experimental values for asphaltene and resin con-
tents are 13 and 8%, respectively [39]. For resin content the
calculated value is in error by 1.3% from the experimental
data. This is considered as a good prediction. For the amount
of asphaltene precipitated, Eq. (9.43) gives %AD of 0.8. The
biggest error is for asphaltene content with %AD of 6. As men-
tioned these correlations are very approximate and based on
limited data mainly from Middle East. However, the coeffi-
cients may be reevaluated for other oils when experimen-
tal data are available. I n this example predicted values are
relatively in good agreement with experimental data; how-
ever, this is very rare. For accurate calculations of asphaltene
precipitation appropriate thermodynamic models as intro-
duced in this section should be used.
9 . 5 VA POR - S OL I D EQUI L I B R I UM ~
HY D R A T E FOR MA T I ON
I n this section, another application of phase equilibrium in
the petroleum industry is demonstrated for prediction of hy-
drate formation from vapor-solid equilibrium (VSE) calcula-
tions. Hydrates are molecules of gas (C~, C2, C3, iC4, nC4,
N2, CO2, or H2S) dissolved in solid crystals of water. Gas
molecules, in fact, occupy the void spaces in water crystal
lattice and the form resembles wet snow. I n the oil fields
hydrates look like grayish snow cone [i]. Gas hydrates are
solid, semistable compounds that can cause plugging in natu-
ral gas transmission pipelines, gas handling equipments, noz-
zles, and gas separation units. Gas hydrates may be formed at
temperatures below 35~ when a gas is in contact with water.
However, at high pressures (> 1000 bar), hydrate formation
has been observed at temperatures above 35~ Figure 9.23
shows temperature and pressure conditions that hydrates are
formed for natural gases. As pressure increases hydrate can
be formed at higher temperatures. Severe conditions in arctic
and deep drilling have encouraged the development of pre-
dictive and preventive methods. It is generally believed that
large amounts of energy is buried in hydrates, which upon
their dissociation can be released.
Hydrates are the best example of the application of
VSE calculations. Whitson [1] discusses various methods of
calculation of the temperature at which a hydrate may form
1000
i~ ~::~/~:~::i~:~ 84 ::~ :::~ :i 84 ~ :~~ :~ ~ : ~! :~i 84184 :: ~ i::~~:~:i:~!):!:~:ii~ ~!~:i:!:~i~!~:~;i~:i:~ 84
loo i
1~i.i iiii
-15 -10 -5 0 5 10 15 20 25 30
Temperature, ~
FIG. 9.23---Hyflrate formation for methane and natural
gases. Drawn based on data prov ided in Ref. [ 1].
9. APPLI CATI ONS: PHASE EQUI L I BRI UM CALCULATI ONS 3 8 9
T A B L E 9.12--Coefficients (Ci) for Eq. (9.46) for estimation of liFT at very high pressures. Taken
with permission from Ref. [1].
Pressure, bar (psia) Methane Ethane Propane /-Butane n-Butane
414(6000) 18933 20806 28382 30696 17340
483(7000) 19096 20848 28709 30913 17358
552(8000) 19246 20932 28764 30935 17491
620(9000) 19367 21094 29182 31109 17868
690(10000) 19489 21105 29200 30935 17868
at given pressure. Calculation of hydrate-formation t empera-
ture (HFT) is very similar to dewpoint t emperat ure calcula-
tion in VLE. The equilibrium ratio for component i between
vapor and solid phase is defined as K vs = yi/x s, where x s is
the mole fraction of i in the solid hydrate phase. Hydrat e is
formed if at given T and P we have
N
Yi
(9.44) ~ ~ >_ 1
i =1
where equality holds at t emperat ure where hydrate forma-
tion begins. I n the vapor phase the amount of wat er is very
small (<0.001 mol%) thus its presence in the vapor phase can
be neglected in the calculations (yW _ 0). To find the temper-
ature at which a hydrate dissociates and hydrocarbons are
released, a calculation similar to bubble point calculations
can be performed so that ~x SK vs > 1. Katz provided charts
for calculation of Ki vs, which later Sloan converted into em-
pirical correlations in t erms of T and P and they are used in
the pet rol eum industry [1]. It should be noted that these Ki
values are not true VSE ratios as the above calculations are
based on water-free phases. This met hod can be applied to
pressures below 70 bar (~1000 psia). For methane, ethane,
propane, n-butane, and HaS the correlations for calculation
of K vs are given as follows [1]:
17.59 3.403
l nKc vs = 0.00173 + ~ p + 1.3863 x 10-4pT
1.0356P 0.78338 in ( P) - 23.9804 ( ~)
+ T
( p 3)
- 1.34136 x 10-6T 3 - 1.8834 x 10 .5
l nK vs = 3.92157 - 161.268 181.267
+ ~ + 1.8933 x 10-5p a
1.04557P 1.19703 in ( P) 402.16
+ T p2
- 8.8157 ( ~ ) + 0.133231 ( 7 ) - 21.2354 (~ -7 )
+46. 13339( T)
26.1422
l nK vs = - 7. 59224+ T 3.0545 10-5pT +2. 315
( P) 79.3379
x10-3Te+0" 123481n ~- + p---T--
+ 0.05209 ( - ~) - 26. 4294 ( ~3) + 3.2076 x 10-ST 3
406.78
K vs -37. 211 + 1. 5582T+ + 1.9711 x 10-3T 2
n n_ C4 -~-
- 8.6748 ( P) - 8.2183 ( T) + 540.976 ( T)
+4. 6897 x 10 .3 ~-7 - 1.3227 x 10-5T 4
45.9039
lnK; vs -6. 051 +0. 11146T+ 1.9293 x 10-4pT
H 2 s = T
+ 1.94087 ( P) - 0. 64405 In ( P) - 56.87 ( ~2)
- 7.5816 x 10-6T 3
where T = given t emperat ure in kelvin - 255.4 and
P = given pressure in bar
(9.45)
For pressures between 400 and 700 bars ( ~6000-10000 psia),
a simple empirical met hod is proposed by McLeod and
Campbell in the following form as given in Ref. [1]:
/ ~' \ ~ /2
(9.46) T= 2. 16 ti~=l YiCi)
where values of Ci for CI - C 4 a r e given in Table 9.12 at several
pressures encountered in deep-gaswell drilling.
This met hod can be used for quick estimation of HFT or
to check the validity of estimated t emperat ures from other
methods. More sophisticated met hods using chemical poten-
tial and equations of state are discussed in other references
[1].
Because of the probl ems associated with hydrate for-
mation, hydrate inhibitors are used to reduce HFT. Com-
monl y used hydrate inhibitors are methanol, ethanol, glycols,
sodi um chloride, and calcium chloride. These are nearly the
same materials that are used as wat er antifreeze inhibitors.
Effect of methanol (CH3OH) on the depression of HFT of
met hane reservoir fluid is shown in Fig. 9.24 [55]. The com-
position of this condensate sample in t erms of mol% is as
follows: 0.64 N2, 3.11 COa, 73.03 C1, 8.04 Ca, 4.28 C3, 0.73
i-C4, 1.5 n-C4, 0.54 i-C5, 0.6 n-Cs, and 7.53 C6+ with mixture
molecular weight of 32.4. The most commonl y used equation
to calculate the degree of decrease in HFT (AT) is given by
Hammerschmi dt , which is in the following form [1, 14]:
Awt%
(9.47) AT =
M(100 - wt%)
where AT is the decrease in HFT in ~ (or in kelvin), wt% is
the weight percent of inhibitor in the aqueous phase, and M
is the molecular weight of the inhibitor. Values of M and A
300
29wt%] 16wt%/ 0wt%/
/ / MEO~ /
250 9 Experimental / / [
200
150
100
5O
0
260 270 280 290 300
Temperature, K
FIG. 9 . 24- - Depr essi on of hydrate f ormati on
temperature in met hane by methanol -cal cul ated
v ersus measured v alues. Lines represent coex -
istence curv es f or methane, hydrate, and aque-
ous sol uti ons of MeO H. Taken wi th permi ssi on
from Ref. [ 55].
and at 285 K, ~y i / K vs = 1.073. Fi nal l y at T = 285.3877 K,
~, yi / K vs = 1.00000, whi ch is t he cor r ect answer. Thus t he
HFT for t hi s gas at 30 bar is 285.4 K or 12.2~ At this
t emp er at ur e f r om Eq. (9.45), K1 = 2.222, K2 = 0.7603, and
K3 = 0.113. Comp osi t i on of hydr ocar bons in a wat er-free
base hydr at e is cal cul at ed as x s = yi / K vs, whi ch gives x s =
0. 36, x s = 0.197, and x s = 0.443. (b) At 414 bar s Eq. (9.46)
wi t h coeffi ci ent s i n Table 9.12 shoul d be used. At t hi s pres-
sur e HFT is cal cul at ed as 303.2 K or HFT = 30~ (c) To de-
cr ease HFT at 30 bar s an i nhi bi t or sol ut i on t hat can cause
dep r essi on of AT = 12. 2- 5 : 7.2~ is needed. Rear r ang-
i ng Eq. (9.47): wt % = IO0[MAT/(A + MAT) ] , wher e wt % is
t he wei ght p er cent of i nhi bi t or i n aqueous sol ut i on. F r om
Table 9.13 for met hanol , A = 1297.2 and M = 32. Thus wi t h
AT = 7.2, wt % = 15.1. Si nce cal cul at ed wt % of met hanol
is less t han 20% use of Eq. (9.47) is j ust i fi ed. F or p r essur e
of 414 bar s Eq. (9.48) shoul d be used for met hanol wher e
up on r ear r angement one can get XM~OH = i -- exp ( - AT/ 72) .
At AT -- 30 - 5 = 25~ we get XM~OH = 0.293. F or an aque-
ous sol ut i on (MH2o = 18) and f r om Eq. (1.15), t he wt % of
met hanol ( M = 32) can be cal cul at ed as: wt % = 42.4. #
for some c ommon i nhi bi t or s ar e given in Table 9.13. Values
of A ar e cor r ect ed val ues as gi ven in Ref. [14].
Equat i on (9.47) is r ecommended for sweet nat ur al gases
( HzS cont ent of less t han 4 p p m on vol ume basi s, al so see
Sect i on 1.7.15) wi t h i nhi bi t or concent r at i ons of less t han
20 t ool %. F or concent r at ed met hanol sol ut i ons, like those
used to free a pl ugged- up t ubi ng st ri ng in a hi gh- p r essur e well,
Whi t son [1] suggest s a modi f i ed f or m of Ha mme r s c hmi dt
equat i on:
(9.48) AT = - 72 I n (1 - XMeOH)
wher e AT is t he decr ease i n HFT i n ~ ( or in kel vi n) and XMeon
is t he mol e f r act i on of met hanol in t he aqueous sol ut i on.
Example 9. 4- - Comp os i t i on of a nat ur al gas i n t er ms of t ool %
is as follows: 85% C~, 10% Ca, and 5% C3. Cal cul at e
a. HFT at 30 bar s and comp osi t i on of hydr at e formed.
b. HFT at 414 bars.
c. wt % of met hanol sol ut i on needed to decr ease HFT to 5~
for each case.
Sol ut i ons( a) At 30 bar p r essur e ( <70 bar ) t he HFT can
be cal cul at ed f r om Eqs. (9.44) and (9.45) by t r i al - and- er r or
met hod. Assumi ng HFT of 280 K, t he sum i n Eq. (9.44)
is ~]yi/Ki vs = 2,848 si nce it is gr eat er t han 1, t emp er at ur e
shoul d be i ncr eased i n or der t o decr ease K vs val ues. At T =
300 K, ~y i / KV S= 0.308;, at T = 290 K, )-~yi/KV S= 0.504;
TABLE 9.13---Constants in Eq. (9.47) for hydrate formation
inhibitors.
Hydrate formation inhibitor Formula M A
Methanol CH3OH 32 1297.2
Ethanol C2 H5 OH 46 1297.2
Ethylene glycol C2H602 62 1500
Diethylene glycol C4H1003 106 2222.2
Triethylene glycol C6H1404 150 3000
9 . 6 A P P L I C A T I ON S : EN H A N C ED OI L
R EC OVER Y mEVA L UA T I ON OF GA S
I N J EC T I ON P R OJ EC T S
I n t hi s sect i on anot her ap p l i cat i on of some of t he met hods
p r esent ed i n t hi s book is shown for t he eval uat i on of gas in-
j ect i on proj ect s. Gas is i nj ect ed i nt o oil reservoi rs for differ-
ent pur poses: st or age of gas, mai nt enance of r eser voi r pres-
sure, and enhanced recovery of hydr ocar bons. I n t he l ast case,
under s t andi ng and model i ng of t he di ffusi on p r ocess is of
i mp or t ance t o t he p l anni ng and eval uat i on of gas i nj ect i on
proj ect s. Gases such as nat ur al gas, met hane, et hane, l i que-
fied p et r ol eum gas ( LPG) , or car bon di oxi de are used as mi sci -
bl e gas fl oodi ng in EOR t echni ques. Upon i nj ect i on of a gas, it
is di ssol ved i nt o oi l under r eser voi r condi t i ons and i ncr eases
t he mobi l i t y of oil due to decr ease in its viscosity. To r each a
cer t ai n mobi l i t y l i mi t a cer t ai n gas concent r at i on is r equi r ed.
F or p l anni ng and eval uat i on of such proj ect s, it is desi r ed t o
p r edi ct t he amount of gas and dur at i on of its i nj ect i on i n an
oil reservoir. I n such cal cul at i ons, p r op er t i es such as density,
viscosity, di ffusi vi t y and p hase behavi or of oil and gas ar e
needed. The p ur p os e of t hi s sect i on is to show how to ap p l y
met hods p r esent ed i n t hi s book t o obt ai n desi r ed i nf or mat i on
for such proj ect s. This ap p l i cat i on is shown t hr ough model i ng
of f r act ur ed reservoi rs for a Nor t h Sea r eser voi r for t he st udy
of ni t r ogen i nj ect i on. Labor at or y exp er i ment al dat a ar e used
to eval uat e model p r edi ct i ons as di scussed by Ri azi et al. [56].
An i deal i zed mat r i x- f r act ur e syst em is shown in Fig. 9.25,
wher e mat r i x bl ocks are as s umed to be r ect angul ar cubes. Di-
mensi ons of mat r i x bl ocks may vary f r om 30 to 300 cm, and
the t hi ckness of f r act ur es is about 10-2-10 -4 cm. When a gas
is i nj ect ed i nt o a f r act ur ed reservoir, t he gas flows t hr ough
t he f r act ur e channel s i n hor i zont al and vert i cal di rect i ons.
Therefore, all surfaces of a mat r i x bl ock come i nt o cont act
wi t h t he s ur r oundi ng gas in t he fract ure. The i nj ect ed gas
comes i nt o cont act wi t h oil i n t he mat r i x bl ock at t he mat r i x-
fract ure i nt erface. The gas begi ns to di ffuse i nt o oil and l i ght
MA TRIX FRA CTURES
FIG. 9 ,25---Idealized fractured
reserv oi rs ( after W arren and
Root [ 57]) ,
components in the oil diffuse in the opposite direction from
matrix to the fracture. This process continues until the gas
in the fracture reaches in equilibrium with the oil in the ma-
trix block when no longer gas diffuses into oil. In such cases,
it is assumed that the oil and gas inside the matrix blocks
are in thermodynamic equilibrium at all times. Moreover, it
is assumed that at the matrix-fracture interface, oil and gas
are in equilibrium at all the times and there is no diffusion
across the interface. To analyze the diffusion process, a lab-
oratory experiment was conducted with a cell containing a
porous core ( from Ekofisk field) as shown in Fig. 9.26. The
free volume in the cell can be considered as the fracture in
real reservoirs. For simplicity in formulation of diffusion pro-
cess and mathematical solutions, the matrix-fracture system
was converted into a one-dimensional model. Details of the
model and mathematical formulation are given in Ref. [56].
0.75cm
l. om 1
I :::::::::::::::::::::::::::::: I
8.3cm i i ! i i i i i i :: ! " '~"?" "~' ' ! ~i :: i i ! i i i i :: i
. . . . . . . . GAS. . . . . . . . .
I ::::::::::::::::::::::::::::::: I
i :3:! :?:?:] :! :?:i:! :?:! :i:i:! :?:i: I I
~P' . . . . . . " ' " ' - ' " " . . . . . . i '
3cm - - ~ F=Fracture
5.1cm
6.6cm
13cm
FIG. 9 . 26~ Schemat i c of ex perimental cell for diffu-
si on of g as in a matrix block. Taken with permi ssi on
from Ref. [ 56].
1.0
0.9 M
0.8 N~Y
0.6 i F~
" 0.5
r x~
0.4
0.3
0.1 y ~ "~--~
0.0 . . . .
0 10 20 30 40 50 60 70 80 90 100
Time, h
FIG. 9 . 27 - - Cal cul at ed composi ti ons for oil in matrix
and g as in fracture v ersus ti me. Taken from Ref. [ 56],
In a particular experiment, the core was saturated with a
live oil at its bubble point pressure of 382.8 bar and temper-
ature of 403 K. The free volume around the core was filled
with pure nitrogen. As nitrogen diffuses to the matrix block
and light gas diffuses in the opposite direction to the free vol-
ume (fracture), composition of the gas in the fracture was
measured versus time. Composition of oil was expressed by
15 components, including five pseudocomponents generated
by methods of Chapter 4. Critical properties and acentric fac-
tor were estimated through methods of Chapter 2. Diffusion
coefficients were calculated through methods presented in
Chapter 8. Cubic equation of state (PR EOS) of Chapter 5
was used for calculation of PVT properties and flash calcula-
tions inside the matrix blocks. Through solution of diffusion
equations concentration of all components in both the matrix
and the fracture were determined. This composition in terms
of mole fraction of key components (Cx and N2) in the matrix
and fracture versus time is shown in Fig. 9.27. The system
reaches final equilibrium conditions after 100 h. As dimen-
sion of matrix blocks increases, the time required to reach
final state increases as well. Applying this model to real reser-
voirs one can determine how long the gas must be injected in
order to reach the desired degree of oil mobility.
9 . 7 S UMMA R Y A N D R EC OMMEN D A T I ON S
In this chapter, applications of methods and procedures pre-
sented in the book were shown in phase equilibria calcula-
tions of petroleum fluid mixtures. Five types of VLE calcu-
lations, namely, flash, bubble T, bubble P, dew P, and dew
T, as well as construction of phase diagrams (i.e., PT or Px)
are presented and their applications to petroleum reservoir
fluids have been demonstrated. Furthermore, the principles
of phase equilibria introduced in Chapter 6 is applied to VLE,
SLE, VLSE, and VSE calculations for prediction of the onset
and amount of solid format i on in pet rol eum fluids. Some
guidelines for quick convergence in the calculation and de-
t ermi nat i on of key interaction paramet ers are given. Various
models available in most recent publications for calculation
of amount s of wax and asphaltene precipitations and their on-
sets are presented. Mechanism of solid formation, their neg-
ative effects in the pet rol eum industry and met hods of their
prevention are also discussed. Results of calculations from
various models when applied to different pet rol eum mixtures
are given. Effects of t emperat ure and pressure on the wax and
asphaltene precipitation for different oils are demonstrated.
Methods of calculation of the conditions at which hydrates
may be formed are shown. The i mpact of characterization
met hods of Chapters 2-4 on propert y and phase behavior
predictions as well as met hods of calculation of true critical
properties are also presented. The chapter ends with anot her
application of met hods presented in the book in evaluation
of gas injection projects for EOR.
I n VLE calculations, accuracy of the results basically de-
pends on the met hod chosen for calculation of equilibrium
ratios. I n this regards suggestions given in Table 6.14 should
be used as a guide. For calculation of CPT and WFT the
multisolid-phase model provides a reliable met hod without
the need for adjustable paramet ers. A good prediction of onset
of asphaltene precipitation is possible t hrough measurement
of kinematic viscosity.
9 . 8 FI N A L WOR D S
Variety of met hods for prediction and calculation of various
thermophysical properties for pet rol eum and related fluids is
much wider t han the met hods presented in this book. How-
ever, attempts were made to include the most accurate and
widely used met hods by the people from industry and re-
searchers. Limitations of application of methods, points of
strength and weaknesses, and their degrees of accuracy have
been discussed for different systems. Furt hermore, the basis
of development of nearly all met hods discussed in this book
have been discussed so the students and new researchers in
this area can underst and the basic concepts and fundamen-
tals of property calculations. I n addition, the approaches pre-
sented in the book should help researchers in expansion of the
existing met hods andbe used as a guide in the development of
new predictive methods. The met hods presented in the book
should also help users of various simulators (process, PVT,
phase behavior, etc.) to be able to select the most appropri at e
met hod for their property prediction purposes.
Empirical correlations should be used with caution and
as a last option in absence of experimental data or accurate
fundamentally based t hermodynami c models. I n use of these
correlations their limitations and sensitivity to the input pa-
ramet ers must be considered. Some of these met hods are reli-
able when the input paramet ers are determined t hrough rec-
ommended methods. Perhaps the most accurate met hods are
those based on fundament al theoretical approach combi ned
with empirically determined coefficients and parameters. I n
development of such relations availability of input parame-
ters and accuracy of their measurement s should be consid-
ered. Furt hermore, predictive met hods can have general ap-
plication for a wide range of pet rol eum fluids if properties
of pure compounds have been used in their development in
addition to data on pet rol eum mixtures from oils around the
world. The weakest predictive met hods are perhaps those em-
pirically developed correlations that are based on a set of data
for oils from a certain part of the world.
As it is shown in this book the mai n difficulty in prediction
of properties of pet rol eum fractions relies on properties of
heavy fractions containing polar multiring compounds with
few experimental data available on their properties. As heavy
compounds are generally polar with high boiling points, data
on specific gravity and molecular weight alone are not suffi-
cient for their propert y predictions. For such compounds it
is not possible to measure critical properties or even boiling
point. Boiling points of such compounds or their mixtures are
not measurabl e and estimated boiling points based on distil-
lation data at low pressures have little practical applications
as they do not represent true boiling points. For such com-
pounds one has to look at other properties that are directly
measurabl e and represent their characteristics.
Reported values of critical properties of heavy compounds
are usually predicted from met hods developed for lighter hy-
drocarbons. For example, in the API-TDB [ 12] reported values
of critical constants for heavy compounds are calculated from
group contribution methods. Kesler-Lee met hod for calcula-
tion of critical properties of heavy hydrocarbons are based
on calculated values from vapor pressure data [58]. Predicted
values of critical constants and boiling point from different
met hods for heavy compounds differ significantly from each
other, especially as carbon number increases. This leads to an
even greater difference in predicted t hermodynami c proper-
ties. Presence of very heavy compounds in a mi xt ure requires
a rigorous mixing rule for calculation of mi xt ure properties.
Attempts in this area should be focused on standardization
of values of critical constants for heavy hydrocarbons and
characterization of heavy oils.
Use of directly measurabl e properties in calculation of ther-
modynami c properties of heavy pet rol eum mixtures is an ap-
propriate approach as it was discussed in Chapters 5 and 6.
Use of velocity of sound to determine EOS paramet ers was
demonst rat ed in Section 6.9 and new developments in this
area are highly desirable [59]. Measurement and reporting
of this t hermodynami c property on heavy pet rol eum frac-
tions and crude oils would help researchers to find met hods
of calculation of EOS paramet ers from measurabl e proper-
ties. Other useful and measurabl e properties for heavy oils
include molecular weight, density, and refractive index. Use of
refractive index in determination of EOS paramet ers has been
shown in Section 5.9. It seems that more advanced equations
of state such as SAFT equations would be more appropri at e
for prediction of t hermodynami c properties of heavy oils such
as those containing heavy residues, asphaltenes, and complex
polar compounds. I nvestigation of this approach should he
continued for more accurate estimation of thermophysical
properties. Newly developed methods for phase equilibrium
calculations and phase determination of many-component
systems are useful tools in formulation and efficient predic-
tion of hydrocarbon phase behavior and should be pursued
[60].
Another appropri at e approach in characterization of heavy
oils was taken by Goual and Firoozabadi [23] to mea-
sure dipole moment s of such complex systems. Attempts in
9. APPL I CATI ONS: PHASE EQUI L I BRI UM CAL CUL ATI ONS 393
measurement and reporting of such data should be contin-
ued to enable us in our understanding of properties of heavy
petroleum fluids. Upon availability of such data it would be
possible to develop more accurate and physically sound meth-
ods for characterization of heavy petroleum fractions and
crude oils based on their degrees of polarity. Use of dipole
moment in correlation of transport properties of polar fluids
was shown by Chung et al. [61]. Measurement and effects of
heteroatoms in such complex compounds on physical proper-
ties should also be considered with great emphasis. Presence
of heteroatoms such as S, N, or O in a hydrocarbon com-
pound can have appreciable impact on the properties of the
compound.
The market for heavy oils and residues are limited; how-
ever, production of light oil in the world is in decline. There-
fore, heavy oil conversion becomes increasingly important.
Theoretically, the resources for heavy oils are infinite, as it
is near to impossible to produce the last barrels of oils from
heavy oil reservoirs. Considering limited information avail-
able on properties of heavy compounds, the focus of future
studies must be on characterization of heavy hydrocarbons
and petroleum fractions. I n the area of solid formation and
prevention methods generation and development of phase
envelope diagrams for different reservoir fluids would be of
importance for designers and operating engineers. I n this
book attempts were made to address some of the difficul-
ties associated with property prediction of heavy and complex
petroleum mixtures and with limited data available appropri-
ate approaches are recommended; however, the challenge in
this area of petroleum research continues.
9 . 9 PR OB L EMS
9.1. Three-Phase Fl ash--Consi der three phases of water,
hydrocarbon, and vapor in equilibrium under reservoir
conditions. Water (L1) and hydrocarbons (L2) in the
liquid phase form two immiscible phases. Develop ap-
propriate equations for three-phase flash calculations
and derive relations for calculation of x L1, x/L2, and
Yi. Measurement and prediction of VLLE in water-
hydrocarbon systems by PR EOS has been presented
by Eubank et al. [62].
9.2. Derive Eq. (9.8) for calculation of GOR.
9.3. Calculate composition of liquid and gas streams from
the third stage in Table 9.1 (also see Fig. 9.3) using Stand-
ing correlations for calculation of K/.
9.4. Consider the PVT cell and the core sample shown in
Fig. 9.26. The free volume is 268 cm 3 and is filled ini-
tially with pure N2. The core (porous media) has porosity
of 0.31 and is filled with saturated oil with the follow-
ing composition in terms of mole fraction (Table 9.14).
The C7+ has molecular weight (MT and specific gravity
(SG7+) of 228 and 0.853, respectively. Nitrogen diffuses
into the core and light gases from matrix into the free
TABLE 9.15--Properties of gas and liquid phases in a constant
volume cell.
Specification Initial state Final state
Temperature, K
Pressure, bar
Volume of the cell, cm 3
Volume of the liquid phase,
cm 3
Volume of the gas phase, cm 3
Moles of liquid, tool
Moles of gas, mol
Molecular weight of liquid
phase
Molecular weight of gas phase
Mass of liquid, g
Mass of gas phase, g
Density of liquid phase, g/cm 3
Density of gas phase, g/cm 3
Molar density of liquid,
mol/cm 3
Molar density of gas, mol/cm 3
Length of the cell, cm
Length of the liquid phase, cm
Length of the gas phase, cm
Volume fraction of the liquid
Mole fraction of the gas
phase in the cell
Equilibrium ratio of methane
Mole fraction of methane in
the liquid
Mole fraction of methane in
the gas
volume. The system reaches to final equilibrium state at
pressure of 270 bar when temperature is kept constant
at 403 K. Determine the bubble point pressure of oil at
403 K. Also determine the final equilibrium composi-
tion of gas in terms of mole fractions of N2, CO2, C~, Ca,
(Ca q- C4), and C5+ in the free volume.
9.5. Consider a constant volume-temperature cylinder as
shown in Fig. 8.13. The volume of cylinder is 96.64 cm 3
and its length is 20.5 cm. Initially the cell is filled with
30 vol% liquid n-pentane at 311.1 K and 100 bar. The
rest of the cylinder is filled with pure methane at the
same initial temperature and pressure. Since the sys-
tem is not in equilibrium it approaches to a final equi-
librium state at a lower pressure keeping temperature of
the cell constant. Through constant volume isothermal
flash calculations using PR EOS and information given
in the problem complete Table 9.15.
9.6. Composition of a reservoir fluid (gas condensate) sep-
arated in a separator at 300 psig and 62~ is given
in Table 9.16. The C7+ properties are given as fol-
lows: SG7+ -- 0.795 and M7+ = 143. Laboratory mea-
sured value of produced stock tank liquid-to-well stream
ratio is 133.9 bbl/MMscf and the gas-to-feed ratio is
801.66 Mscf/MMscf. Associated gas (separator product)
specific gravity is SGgas = 0.735 and the primary stage
T AB L E 9. 14--Composition of oil for Problem 9.4.
N2 CO2 C 1 C2 C3 i-C4 H-C4 i-C5 n-C5 C6 C7
0.00114 0. 02623 0. 58783 0. 06534 0. 03560 0. 00494 0. 01558 0. 00500 0. 00872 0. 01442 0.23519
TABLE 9. 16--Composition of reservoir fluid of Problem 9.6.
WelI stream, Separator liquid, Separator gas,
Component tool% mol% mol%
CO2 0.18 Trace 0.22
N2 0.13 Trace 0.16
C1 61.92 7.78 75.31
C2 14.08 10.02 15.08
C3 8.35 15.08 6.68
i-C4 0.97 2.77 0.52
n-C4 3,41 11.39 1.44
i-C5 0,84 3.52 0.18
n-C5 1.48 6.50 0.24
C6 1.79 8.61 0.11
C7+ 6.85 34.33 0.06
Total 100 100 100
GOR is 4428 scf/ bbl at 60 ~ The API gravi t y of p r oduced
cr ude oil is 58.5. Cal cul at e t he fol l owi ng:
a. Comp osi t i on of s ep ar at or gas and l i qui d usi ng St and-
i ng cor r el at i on for Ki.
b. SGgas for sep ar at or gas.
c. API gravi t y of sep ar at or l i qui d.
d. GOR in scf/ bbl.
e. St ock t ank l i qui d to wel l st r eam r at i o i n bar r el s/
MMscf.
f. Gas-t o-feed r at i o i n Mscf/ MMscf.
g. Comp ar e p r edi ct ed val ues wi t h avai l abl e l abor at or y
val ues.
9.7. For t he gas condensat e samp l e of Pr obl em 9.6 cal cul at e
Z f act or at t he r eser voi r condi t i ons of 186~ and 5713
psi a and comp ar e it wi t h t he r ep or t ed val ue of 1.107.
What is t he val ue of gas condensat e exp ansi on f act or in
Mscf for each bbl at r eser voi r condi t i ons? The meas ur ed
val ue is 1.591 Mscf/ bbl.
9.8. F or t he gas condensat e samp l e of Pr obl em 9.6 cal cul at e
dew p oi nt p r essur e (Pa) at 186~ and comp ar e it wi t h
t he meas ur ed val ue of 4000 psi a.
9.9. The fol l owi ng dat a ( Table 9.17) on t wo t ypes of Chi nese
r ecombi ned cr ude oils ar e gi ven by Hu et al. [63]:
The r eser voi r t emp er at ur e is at 339 K and meas ur ed
bubbl e p oi nt p r essur es for oils I and 2 ar e 102.8 and
74.2 bar, respectively. Densi t i es of Cl1+ f r act i on at 20~
for oils 1 and 2 are 0.91 and 0.921 g/ cm 3, respect i vel y.
Ml1+ for oils 1 and 2 are 428 and 443, respectively. At
t he r eser voi r p r essur e of 150 bar, vi scosi t i es of oils 1
and 2 are 5.8 and 6. I cP, respectively. Est i mat e t he bub-
bl e p oi nt p r essur es f r om an EOS for t hese t wo oils and
comp ar e wi t h avai l abl e dat a.
9.10. Mei et al. [46] have r ep or t ed exp er i ment al dat a on com-
p osi t i on of a wel l st r eam fl ui d f r om West Chi na oil field
wi t h comp osi t i ons of sep ar at or gas and p r oduced oil as
gi ven i n Table 9.18. Reser voi r condi t i ons (T and P) , sat-
ur at i on p r essur e of fluid at r eser voi r T, and t he GOR of
r eser voi r fl ui d ar e al so gi ven i n t hi s t abl e. Densi t y of t he
r eser voi r fl ui d ( well st r eam under r eser voi r T and P) has
been meas ur ed and r ep or t ed. F r om anal ysi s of dat a i t is
obser ved t hat t her e is an er r or in t he comp os i t i on of wel l
st r eam as t he sum of all number s is 90.96 r at her t han
100. I n addi t i on r eser voi r t emp er at ur e of 94 K is not
cor r ect ( t oo low) . Per f or m t he fol l owi ng cal cul at i ons t o
get cor r ect val ues for t he wel l st r eam comp os i t i on and
r eser voi r t emp er at ur e.
a. Recombi ne sep ar at or gas and oil t ank to get t he ori g-
i nal wel l st ream. Make ap p r op r i at e mat er i al bal ance
cal cul at i ons, usi ng mol ecul ar wei ght , t o gener at e wel l
st r eam comp osi t i on. Also det er mi ne if gi ven GOR is
in st m3/ m 3 or m3/ m 3 at sep ar at or condi t i ons.
b. Use bubbl e- T cal cul at i ons to cal cul at e r eser voi r t em-
p er at ur e at whi ch cor r esp ondi ng sat ur at i on p r essur e
is 311.5 bar.
c. Use t r i al - and- er r or p r ocedur e to fi nd a t emp er at ur e
at whi ch cal cul at ed densi t y of r eser voi r fluid mat ches
meas ur ed r ep or t ed val ue at r eser voi r pressure. Thi s
TABLE 9. 17--Data on two Chinese crudes for Problem 9. 9 [63].
Compound Nz CO2 C1 C2 C3 i-C4 F/-C4 i-C5 ~/-C5 C6 C7 C8 C9 C10 CI 1+
Oil 1 1.20 0.20 30.90 3.50 2.87 0.33 1.41 0.40 1.02 1.69 2.46 2.98 2.53 2.15 46.36
Oil 2 0.96 0.16 24.06 0.76 3.26 0.64 2.70 0.52 1.06 0.70 0.580 1.86 2.30 0.82 59.62
TABLE 9. 18--Composition of an oil sample from Western China field [46J.
Component Gas in separator, tool% Off in tank, tool% Well streama,mol%
CO2 0.62 0.52
N2 5.94 4.97
C 1 67.35 56.36
C2 11.51 0.08 0.64
C3 7.22 0.47 6.12
i-C4 2.31 0.55 2.02
n-C4 2.41 1.01 2.18
i-C5 0.89 1.19 0.94
n-C5 0.72 1.38 0.83
C6 0.59 4.06 1.16
C7 0.31 5.65 1.14
C8 0.13 13.50 2,31
C9 8.53 1.39
C10 6.26 1.02
cbl+ 57.32 9.36
Initial reservoir Saturation
pressure, bar Reservoir temp, K GOR, m3/m 3 pressure, bar
410 94 a 440 311.5
aWeU stream composition and reservoir temperature are not correct. Find the correct values.
bCll + fraction: M11+ = 311 and SGlI+ = 0.838.
Density of reservoir
fluid, _g/cm 3
0.5364
9. APPL I CATI ONS: PHASE EQUI L I BRI UM CAL CUL ATI ONS 395
t emper at ur e must be near t he t emper at ur e cal cul at ed
i n part b.
9.11. For the reservoi r fluid of Pr obl em 9.10 calculate t he
amount of wax preci pi t at ed ( in mo1%) at 280, 300, and
320 K and 410 bar. Also est i mat e WAT at 410 bar usi ng
solid sol ut i on mode[.
9. t 2. Calculate the CPT for crude oil 6 i n Table 9.10 usi ng
mul t i sol i d-phase model. Also cal cul at e the amount of
wax preci pi t at i on i n wt% at 240 K.
9.13. A nat ur al gas has the composi t i on of 70 mo] % met hane,
15 mo] % et hane, 7 mol % propane, 5 mol % n-but ane, and
3 mol % H2S. What is t he hydrat e f or mat i on t emp er at ur e
(HFT) for this gas at pressure of 15 bars? What met hanol
sol ut i on ( in t erms of wt%) is needed to reduce HFT of
the gas to 0~
9.14. A gas mi xt ure of 75 mol % C1, 10 mol % C2, 10 tool%
Ca, and 5 mol % n-C4 exists at 690 bars. Calculate hy-
drat e f or mat i on t emper at ur e and t he concent r at i on of
met hanol sol ut i on requi red to reduce it to 10~
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[52] Hirschberg, A., Delong, L. N. J., Schipper, B. A., and Meijers,
J. G., "Influence of Temperature and Pressure on Asphaltene
Flocculation," Society of Petroleum Engineers Journal, Vol. 24,
No. 3, 1984, pp. 283-293.
[53] Ghuraiba, M. M. A., "Characterization of Asphaltenes in Crude
Oil," M.Sc. Thesis, Chemical Engineering Department, Kuwait
University, Kuwait, 2000.
[54] Oil and Gas Journal Data Book, 2000 Edition, PennWell, Tulsa,
OK, 2000.
[55] Munck, J., Skjold-Jorgensen, S., and Rsmussen, E,
"Compositions of the Formation of Gas Hydrates," Chemical
Engineering Science, Vol. 43, No. 10, 1988, pp. 2661-2672.
[56] Riazi, M. R., Whitson, C. H., and da Silva, E, "Modeling of
Diffusional Mass Transfer in Naturally Fractured Reservoirs,"
Journal of Petroleum Science and Engineering, Vol, 10, 1994,
pp. 239-253.
[57] Warren, J. E. and Root, E J., "The Behavior of Naturally
Fractured Reservoirs," Society of Petroleum Engineers Journal,
Vol. 3, 1963, pp. 245-251.
Fractions," Hydrocarbon Processing, March 1976, pp. 153-158.
[59] Mossaad, E. and Eubank, E T., "New Method for Conversion of
Gas Sonic Velocities to Density Second Virial Coefficients,"
American Institute of Chemical Engineers' Journal, Vol. 47,
No. 1, 2001, pp. 222-229.
[60] Iglesias-Silva, G. A., Bonilla-Petriciolet, A., Eubank, E T.,
Holste, J. C., and Hall, K. R., "An Algebraic Method that
Includes Gibbs Minimization for Performing Phase
Equilibrium Calculations for Any Number of Components
and Phases," Fluid Phase Equilibria, Vol. 210, No. 2, 2003,
pp. 229-245.
[61] Chung, T. H., Ajlan, M., Lee, L. L., and Starling, K. E.,
"Generalized Multiparameter Correlation for Nonpolar and
Polar Fluid Transport Properties," Industrial Engineering and
Chemical Research, Vol. 27, 1988, pp. 671-679.
[62] Eubank, E T., Wu, C. H., Alvarado, E J., Forero, A., and Beladi,
M. K., "Measurements and Prediction of Three-Phase Water/
Hydrocarbon Equilibria," Fluid Phase Equilibria, Vol. 102,
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[63] Hu, Y.-E, Li, S., Liu, N., Chu, Y.-E, Park, S. J., Mansoori, G. A.,
and Guo, T.-M., "Measurement and Corresponding States
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pp. 169-182.
MNL50-EB/Jan. 2005
A ppendix
ASTM DEFI NI TI ONS OF TERMS
ASTM DICTIONARY OF SCIENCE AND TECHNOLOGY 1 defi nes vari -
ous engineering t erms in standard terminology. ASTM pro-
vides several definitions for most properties by its different
committees. The closest definitions to the properties used in
the book are given below. The identifier provided includes the
standard designation in which the t erm appears followed by
the commi t t ee having jurisdiction of that standard. For ex-
ample, D02 represents the ASTM Committee on Petroleum
Products and Lubricants.
Additive---Any substance added in small quantities to an-
other substance, usually to improve properties; somet i mes
called a modifier. D 16, D01
Aniline p oi nt mThe mi ni mum equilibrium solution temper-
ature for equal volumes of aniline ( aminobenzene) and
sample. D 4175, D02
API gr avi t y~An arbitrary scale developed by the American
Petroleum I nstitute and frequently used in reference
to pet rol eum insulating oil. The relationship between API
gravity and specific gravity 60/60~ is defined by the fol-
lowing: Degree API gravity at 60~ = 141.5/(SG 60/60~ -
131.5. [Note: For definition see Eq. (2.4) in this book.]
D 2864, D27
Ash--Resi due after the combust i on of a substance under
specified conditions. D 2652, D28
Assay~Anal ysi s of a mixture to determine the presence or
concentration of a particular component. F 1494, F23
Aut oi gni t i on- - The ignition of material caused by the appli-
cation of pressure, heat, or radiation, rat her t han by an
external ignition source, such as a spark, flame, or incan-
descent surface. D 4175, D02
Aut oi gni t i on t emp er at ur e- - The mi ni mum t emperat ure at
which autoignition occurs. D 4175, D02
Average ( for a series of ob serv at i ons) ~ Th e total divided by
the number of observations. D123, D13
Bar--Uni t of pressure; 14.5 lb/in 2, 1.020 kg/ cm 2, 0.987 atm,
0.1 MPa. D 6161, DI 9
Bi t umen- - A class of black or dark-colored (solid, semisolid,
or viscous) cementitious substances, natural or manufac-
tured, composed principally of high-molecular-weight hy-
drocarbons, of which asphalts, tars, pitches, and asphaltites
are typical. D 8, D04
Boi l i ng p oi nt - - The t emperat ure at which the vapor pressure
of an engine coolant reaches at mospheri c pressure under
equilibrium boiling conditions. [Note: This definition is ap-
plicable to all types of liquids.] D 4725, DI S
Boi l i ng p r essur e- - At a specified temperature, the pressure
at which a liquid and its vapor are in equilibrium.
E 7, E04
ASTM Dictionary of Engineering Science and Technology, 9th ed.,
BTU--One British t hermal unit is the amount of heat re-
quired to raise 1 lb of wat er I ~ E 1705, E48
Carbon blackmA material consisting essentially of elemental
carbon in the form of near-spherical colloidal particles and
coalesced particle aggregates of colloidal size, obtained by
partial combust i on or t hermal decomposition of hydrocar-
bons. D 1566, Dl l
Car bon resi due---The residue formed by evaporation and
t hermal degradation of a carbon-containing material.
D 4175, D02
Cat al yst --A substance whose presence initiates or changes
the rate of a chemical reaction, but does not itself enter
into the reaction. C 904, C03
Cetane numb er (cn)----A measure of the ignition perfor-
mance of a diesel fuel obtained by compari ng it to reference
fuels in a standardized engine test. D 4175, D02
Chemi cal p ot ent i al (/~i or r partial mol ar free energy
of component i, that is, the change in the free energy of a
solution upon adding 1 tool of component i to an infinite
amount of solution of given composition, (SG/~n4)r.v,.~ =
Gi =/ zi, where G -- Gibbs free energy and r~ = number of
moles of the ith component. E 7, E04
Cl oud p oi nt - - The t emperat ure at which a defined liquid mix-
ture, under controlled cooling, produces perceptible haze
or cloudiness due to the format i on of fine particles of an
incompatible material. D 6440, D01
Coal --A brown to black combustible sedi ment ary rock (in
the geological sense) composed principally of consolidated
and chemically altered plant remains. D 121, D05
CokemA carbonaceous solid produced from coal, petroleum,
or other materials by t hermal decomposition with passage
t hrough a plastic state. C 709, D02
Comb us t i on~ A chemical process of oxidation that occurs at
a rate fast enough to produce heat and usually light either
as glow or flames. D 123, DI 3
Compressed natural gas ( CNG) ~Nat ur al gas that is typi-
cally pressurized to 3600 psi. CNG is primarily used as a
vehicular fuel. D 4150, D03
Concent r at i on- - Quant i t y of substance in a unit quantity of
sample. E 1605, E06
Critical p oi nt - - I n a phase diagram, that specific value
of composition, t emperat ure, pressure, or combi nat i ons
thereof at which the phases of a heterogeneous equilibrium
become identical. E 7, E04
Critical pressure---Pressure at the critical point.
E 1142, E37
Critical temperature----( 1) Temperature above which the
vapor phase cannot be condensed to liquid by an increase
in pressure. E 7, E04
( 2) Temperature at the critical point. E 1142, E37
Degr adat i on~Damage by weakening or loss of some prop-
erty, quality, or capability. E 1749, E 06
397
Degree Celsius (~ unit of temperature in the In-
ternational System of Units (SI). E 344, E20
Densi t y--The mass per unit volume of a substrate at a spec-
ified temperature and pressure; usually expressed in g/ mE
kg/L, g/cm 3, g/L, kg/m 3, or lb/gal. D 16, D01
Deposi t i on--The chemical, mechanical, or biological pro-
cesses through which sediments accumulate in a resting
place. D 4410, D19
Dew poi nt --The temperature at any given pressure at which
liquid initially condenses from a gas or vapor. It is specifi-
cally applied to the temperature at which water vapor starts
to condense from a gas mixture (water dew point) or at
which hydrocarbons start to condense ( hydrocarbon dew
point). D 4150, D03
Diffusion--( 1) Spreading of a constituent in a gas, liquid, or
solid tending to make the composition of all parts uniform.
(2) The spontaneous movement of atoms or molecules to
new sites within a material. B 374, B08
Distillation--The act of vaporizing and condensing a liq-
uid in sequential steps to effect separation from a liquid
mixture. E 1705, E 48
Distillation t emp er at ur e (in a col unm distillation)---The
temperature of the saturated vapor measured just above
the top of the fractionating column. D 4175, D02
Endot her mi e r eact i on- - A chemical reaction in which heat
is absorbed. C 1145, C 28
Ent hal pymA thermodynamic function defined by the equa-
tion H = U + PV, where H is the enthalpy, U is the internal
energy, P is the pressure, and V the volume of the system.
[Note: Also see Eq. (6.1) of this book.] E 1142, E37
Equi l i br i um~A state of dynamic balance between the op-
posing actions, reactions, or velocities of a reversible
process. E 7, E04
Evaporat i on--Process where a liquid (water) passes from a
liquid to a gaseous state. D 6161, D19
Fire poi nt - - The lowest temperature at which a liquid or solid
specimen will sustain burning for 5 s. D 4175, D02
Fl ammabl e l i qui d--A liquid having a flash point below
37.8~ (100~ and having a vapor pressure not exceed-
ing 40 psi (absolute) at 37.8~ and known as a Class I liquid.
E 772, E44
Fl ash poi nt - - The lowest temperature of a specimen cor-
rected to a pressure of 760 mmHg (101.3 kPa), at which
application of an ignition source causes any vapor from
the specimen to ignite under specified conditions of test.
D 1711, D09
Fl ui di t ymThe reciprocal of viscosity. D 1695, D01
Freezi ng poi nt --The temperature at which the liquid and
solid states of a substance are in equilibrium at a given
pressure (usually atmospheric). For pure substances it is
identical with the melting point of the solid form.
D 4790, D16
Gas--One of the states of matter, having neither independent
shape nor volume and tending to expand indefinitely.
D 1356, D22
Gasification--Any chemical or heat process used to convert
a feedstock to a gaseous fuel. E 1126, E 48
Gasoline---A volatile mixture of liquid hydrocarbons, nor-
mally containing small amounts of additives, suitable for
use as a fuel in spark-ignition internal combustion engines.
D 4175, D02
Gibbs free energy--The maximum useful work that can be
obtained from a chemical system without net change in
temperature or pressure, AF = AH - TAS. [Note: For def-
inition see Eq. (6.6) in this book; the author has used G for
Gibbs free energy.] E 7, E04
Grai nmUni t of weight; 0.648 g, 0.000143 lb. D 6161, D19
Gross calorific val ue ( synonym: hi gher heat i ng value,
HI-IV)raThe energy released by combustion of a unit quan-
tity of refuse-derived fuel at constant volume or constant
pressure in a suitable calorimeter under specified condi-
tions such that all water in the products is in liquid form.
This the measure of calorific value is predominately used
in the United States. E 856, D34
Heat capaci t y--The quantity of heat required to raise a sys-
tem 1 ~ in temperature either at constant volume or constant
pressure. D 5681, D34
Heat flux ( q) ~The heat flow rate through a surface of unit
area perpendicular to the direction of heat flow (q in SI
units: W/ m 2; q in inch-pound units: Btu/h/ft 2 = Btu/h 9 ft 2)
C 168, C16
Henry' s l aw--The principle that the mass of a gas dissolved
in a liquid is proportional to the pressure of the gas above
the liquid. D 4175, D02
Hi gher heat i ng val ue ( HHV) --A synonym for gross calorific
value. D 5681, D34
I nert component s---Those elements or components of nat-
ural gas (fuel gas) that do not contribute to the heating
value. D 4150, D03
I nhi bi t or--A substance added to a material to retard or pre-
vent deterioration. D 4790, D16
I nitial boiling poi nt - - The temperature observed immedi-
ately after the first drop of distillate falls into the receiving
cylinder during a distillation test. D 4790, D 16
I nterface- - A boundary between two phases with different
chemical or physical properties. E 673, E 42
I nt erraci al t ensi on (IF]F)---The force existing in a liquid-
liquid phase interface that tends to diminish the area of
the interface. This force, which is analogous to the surface
tension of liquid-vapor interfaces, acts at each point on the
interface in the plane tangent at that point. D 459, DI 2
I nternati onal Sy st em of Units, SI --A complete coherent
system of units whose base units are the meter, kilogram,
second, ampere, kelvin, mole, and candela. Other units are
derived as combinations of the base units or are supple-
mentary units. A 340, A06
I nterphase---The region between two distinct phases over
which there is a variation of a property. E 673, E42
I SO--Abbreviation for International Organization for Stan-
dards: An organization that develops and publishes inter-
national standards for a variety of technical applications,
including data processing and communications.
E 1457, F05
J et fuel --Any liquid suitable for the generation of power by
combustion in aircraft gas turbine engines. D 4175, D02
J oul e ( J) --The unit of energy in the SI system of units. One
joule is 1 W..-. A 340, A06
Kelvin ( K) --The unit of thermodynamic temperature; the SI
unit of temperature for which an interval of 1 kelvin (K)
equals exactly an interval of 1~ and for which a level of
273.15 K equals exactly 0~ D 123, D13
APPENDI X 399
Li quefi ed pe t r ol e um gas ( LPG) --A mixture of normally
gaseous hydrocarbons, predominantly propane or butane
or both, that has been liquefied by compression or cooling,
or both, to facilitate storage, transport, and handling.
D 4175, D02
Li qui d- - A substance that has a definite volume but no defi-
nite form, except such given by its container. It has a viscos-
ity of 1 x 10 -3 to 1 x 103 St (1 x 10 -7 to 1 x 10 -1 m 2 "S -1)
at 104~ (40~ or an equivalent viscosity at agreed upon
temperature. (This does not include powders and granular
materials.) Liquids are divided into two classes:
(1) Class A, low viscosity--A liquid having a viscosity of
1 x 10 -3 to 25.00 St (1 x 10 -7 to 25.00 x 10 -4 m 2 -s -I ) at
104~ (40~ or an equivalent viscosity at agreed upon
temperature.
(2) Class B, high viscosity--A liquid having a viscosity of
25.01 to 1 x 103 St (25.01 x 10 -4 to 1 x 10 -1 m 2.s 1)
at 104~ (40~ or an equivalent viscosity at agreed upon
temperature.
D 16, D01
Lower h eat i ng v al ue (LHV)mA synonym for net calorific
value. D 5681, D34
Lubri cant --Any material interposed between two surfaces
that reduces the friction or wear between them.
D 4175, D0 2
MassmThe quantity of matter in a body (also see weight).
D 123, D13
Melting p oi nt - - I n a phase diagram, the temperature at
which the liquids and solids coincide at an invariant point.
E 7, E04
Mi cron (/~ m, mi cromet er) - - A metric unit of measurement
equivalent to 10 -6 m, 10 4 cm.
1) 6161, D19
Mol al i t y~Mol es (gram molecular weight) of solute per
1000 g of solvent. 1) 6161, 1)19
Mol ari t y~Mol es (gram molecular weight) of solute per liter
of total solution 1) 6161, 1)19
Mol ecul ar di f f usi on~A process of spontaneous intermixing
of different substances, attributable to molecular motion,
and tending to produce uniformity of concentration.
D1356, D22
Mole fract i on--The ratio of the number of molecules (or
moles) of a compound or element to the total number of
molecules (or moles) present. 1) 4023, 1)22
Napht ha, aromat i c sol vent --A concentrate of aromatic hy-
drocarbons including C8, C9, and C10 homologs.
D 4790, 1) 16
Napt h eni c oilmAn hydrocarbon process oil containing more
than 30%, by mass, of naphthenic hydrocarbons.
1) 1566, Dl l
Nat ural gas---A naturally occurring mixture of hydrocarbon
and nonhydrocarbon gases found in porous geological for-
mations (reservoirs) beneath the earth's surface, often in
association with petroleum. The principal constituent of
natural gas is methane. 1) 4150, D03
Net cal ori fi c value (Net h eat of c omb us t i on at cons t ant
pressure) - - Th e heat produced by combustion of unit
quantity of a solid or liquid fuel when burned, at constant
pressure of 1 atm (0.1 MPa), under the conditions such
that all the water in the products remains in the form of
vapor. D 121, I)05
Net h eat of c omb us t i on- - T h e oxygen bomb (see Test
Method D 3286) value for the heat of combustion, corrected
for gaseous state of product water. E 176, E05
Oct ane numb e r ( for spark i g ni t i on eng i ne fuel) mAny one
of several numerical indicators of resistance to knock ob-
tained by comparison with reference fuels in standardized
engine or vehicle tests. D 4175, D02
Oxygenate- - - An oxygen-containing ashless organic com-
pound, such as an alcohol or ether, which may be used as
a fuel or fuel supplement. D 4175, D02
Paraffinic oilmA petroleum oil (derived from paraffin crude
oil) whose paraffinic carbon type content is typically greater
than 60%. E 1519, E35
Partial pressure---The contribution of one component of a
system to the total pressure of its vapor at a specified tem-
perature and gross composition. E 7, E04
Porosi t y--The percentage of the total volume of a material
occupied by both open and closed pores. [Note: In this book
porosity represented by ~ (see Eq. 8.72) is the fraction of
total volume of a material occupied by open pores and is
not identical to this definition.] C 709, D02
Pour poi nt --The lowest temperature at which a liquid can
be observed to flow under specified conditions.
1) 2864, 1)27
Preci pi t at i on--Separat i on of new phase from solid, liquid,
or gaseous solutions, usually with changing conditions or
temperature or pressure, or both. E 7, E04
Pressure--The internal force per unit area exerted by any
material. Since the pressure is directly dependent on the
temperature, the latter must be specified. 1) 3064, 1)10
Pressure, sat urat i on--The pressure, for a pure substance
at any given temperature, at which vapor and liquid, or
vapor and solid, coexist in stable equilibrium. [Note: This
is the definition of vapor pressure used in this book.]
E 41, G03
QualitymCollection of features and characteristics of a prod-
uct, process, or service that confers its ability to satisfy
stated or implied needs. E 253, E18
Range---The region between the limits within which a quan-
tity is measured and is expressed by stating the lower and
upper range values. E 344, E20
Refracti ve i ndex--The ratio of the velocity of light (of speci-
fied wavelength) in air to its velocity in the substance under
examination. This is relative refractive index of refraction.
I f absolute refractive index (that is, referred to vacuum)
is desired, this value should be multiplied by the factor
1.00027, the absolute refractive index of air. [Note: I n this
book absolute refractive index is used.] 1) 4175, 1)02
S at urat i on- - Th e condition of coexistence in stable equilib-
rium of a vapor and a liquid or a vapor and solid phase of
the same substance at the same temperature. E 41, G03
S mo ke poi nt mThe maximum height of a smokeless flame of
fuel burned in a wick-fed lamp. 1) 4175, 1)02
SolidmA state of matter in which the relative motion of
molecules is restricted and in which molecules tend to
retain a definite fixed position relative to each other. A
solid may be said to have a definite shape and volume.
E 1547, E 15
Sol ubi l i t y--The extent that one material will dissolve in an-
other, generally expressed as mass percent, or as volume
percent, or parts per 100 parts of solvent by mass or vol-
ume. The t emperat ure should be specified. D 3064, D10
Sol ubi l i t y p a r a me t e r ( of l i qui ds) --The square root of the
heat of vaporization mi nus work of vaporization (cohesive
energy density) per unit volume of liquid at 298 K.
D 4175, D02
Sol ut es- - Mat t er dissolved in a solvent. D 6161, D19
Specific gravi t y (deprecated term of l i qui ds) - - The ratio of
density of a substance to that of a reference substance such
as wat er (for solids and liquids) or hydrogen (for gases)
under specified conditions. Also called relative density.
[Note: I n this book the reference substance for definition
of gas specific gravity is air]. D 4175, D02
Sur f ace t ensi on- - Pr op er t y that exists due to molecular
forces in the surface film of all liquids and tends to pre-
vent the liquid from spreading. B 374, B08
Temp er at ur e- - The t hermal state of mat t er as measured on
a definite scale. B 713, B01
Ther mal conduct i vi t y ( X) --Time rate of heat flow, under
steady conditions, t hrough unit area, per unit t emperat ure
gradient in the direction perpendicular to the area.
E 1142, E37
Ther mal di ffusi vi t y--Rat i o of t hermal conductivity of a
substance to the product of its density and specific heat
capacity. El 142, E37
Vapor - - The gaseous phase of mat t er that normally exists in
a liquid or solid state. D 1356, D22
Vapor pressure---The pressure exerted by the vapor of a liq-
uid when in equilibrium with the liquid. D 4175, D02
Viscosity, absol ut e (~/ )--The ratio of shear stress to shear
rate. I t is the propert y of internal resistance of a fluid that
opposes the relative mot i on of adjacent layers [Note: See
Eq. (8.1) in this book.] The unit most commonl y used for
insulating fluids is centipoise. D 2864, D27
Viscosity, ki nemat i c- - The quotient of the absolute (dy-
namic) viscosity divided by the density, O/p bot h at the
same temperature. For insulating liquids, the unit most
commonl y unit is the centistokes (100 cSt = 1 St). [Note:
See Eq. (8.1) in this book.] D 2864, D27
Viscosity, Saybol t Uni ver sal - - The efflux time in seconds of
60 mL of sample flowing t hrough a calibrated Saybolt Uni-
versal orifice under specified conditions. D 2864, D27
Wax ap p ear ance p oi nt - - The t emperat ure at which wax or
other solid substances first begin to separate from the liquid
oil when it is cooled under prescribed conditions (refer to
D 3117, Test Method for Wax Appearance Point of Distillate
Fuels). D 2864, D27
Wei ght ( synonymous wi t h mas s ) - - The mass of a body
is a measure of its inertia, or resistance to change in motion.
E 867, E17
Greek Al phabet
Alpha
fl Beta
F Gamma (Uppercase)
F Gamma
A Delta (Uppercase)
8 Delta
e Epsilon
( Zeta
Et a
| Theta (Uppercase)
0 Theta
K Kappa (Uppercase)
K Kappa
A Lambda (Uppercase)
)~ Lamhda
# Mu
v Nu
Xi
H Pi (Uppercase)
zr Pi
p Rho
N Sigma (Uppercase)
Sigma
r Tau
v Upsilon
q~ Phi (Uppercase)
q~ Phi
Phi
x Chi
Psi (Uppercase)
Omega ( Upper case)
co Omega
MNL50-EB/Jan. 2005
I ndex
A
Absolute density, 120
Academia, 17
Acentric factor, 11
aromatics, 52
definition, 33
estimation, 80-82, 115-116
pure hydrocarbons, prediction, 64-66
Activation energy, 346
Activity coefficients
mixtures, 254-255
calculation, 257-261
Albahri et al. method, 137
Alcohols, octane number, 139
Alkanes
boiling point, 58-59
critical compressibility factor, 64
critical temperature, 50
entropy of fusion, 262
liquid thermal conductivity, 343-344
surface tension, 361
vapor pressure, 306
n-Alkyl, critical pressure, 52
Alkylbenzene
entropy of fusion, 262
vapor pressure, 307
Analytical instruments, 96- 98
ANFOR M 15-023, 10
Aniline point, 11
definition, 35
petroleum fractions, 137
Antoine coefficients, 310
Antoine equation, 305-306
API degree, 21
API gravity, 11
crude oils, 156
definition, 32
prediction, pure hydrocarbons,
58-60
API methods, 124-126
critical temperature and pressure,
prediction, 60
critical volume, prediciton, 63
molecular weight prediction, 56
API RP 42, 37, 56
API Technical Data Book- Petroleum
Refining, 15
Aromatics, 4-5
Arrhenius-type equation, 346
Asphalt, 10
Asphaltene, 373-378
inhibitor, 377-378
precipitation, 375, 377, 379
solid-liquid equilibrium, 385-388
temperature and pressure effects, 381
Association parameter, 347
ASTM, definitions of terms, 397-400
ASTM D 56, 133
ASTM D 86-90, 92, 100-106, 108, 110,
113-115, 118, 131, 134, 140, 144,
313-314
ASTM D 88, 23
ASTM D 92, 34, 133
ASTM D 93, 34, 133-134, 144
ASTM D 97, 135, 144
ASTM D 129, 99
ASTM D 189, 141,144
ASTM D 240, 144
ASTM D 287, 93
ASTM D 323, 33, 144
ASTM D 341, 70, 338
ASTM D 357, 34, 139
ASTM D 445, 100, 144, 338
ASTM D 446, 338
ASTM D 524, 144, 141
ASTM D 611, 35, 137, 144
ASTM D 613, 138
ASTM D 908, 34, 139
ASTM D 976, 138
ASTM D 1018, 99
ASTM D 1160, 92,
114, 144
ASTM D 1218 94, 144
ASTM D 1262 99
ASTM D 1266 99, 144
ASTM D 1298 93
ASTM D 1319 144
ASTM D 1322 142
ASTM D 1368 99
ASTM D 1500 144
ASTM D 1548 99
ASTM D 1552 99
ASTM D 1747 95
ASTM D 2007 96
ASTM D 2267 10
ASTM D 2270, 122-124
ASTM D 2386, 136, 144
ASTM D 2500, 135, 144
ASTM D 2501, 36
ASTM D 2502, 56
ASTM D 2503, 94
ASTM D 2533, 133
ASTM D 2549, 97
ASTM D 2700, 144
ASTM D 2717-95, 144
ASTM D 2759, 127
100-101,106, 108, 110,
ASTM D 2887, 12, 89-90, I00, 104-105,
110, 144
ASTM D 2890, 320-321
ASTM D 2892, 144, 154
ASTM D 2983, 144
ASTM D 3178, 99
ASTM D 3179, 99
ASTM D 3228, 99
ASTM D 3238, 121, 126
ASTM D 3343, 99, 128, 130
ASTM D 3431, 99
ASTM D 3710, 90
ASTM D 4045, 99
ASTM D 4052, 93, 144
ASTM D 4124, 96
ASTM D 4530, 141
ASTM D 4737, 144
ASTM D 4953, 131
ASTM D 5296, 94
ASTM D 5985, 135
401
ASTM method, 128
molecular weight prediction, 56
Atmospheric critical pressure, heavy
hydrocarbons, 51
Autoignition temperature, definition, 34
Avogadro number, 24
B
Benedict-Webb-Rubin equation of state,
modified, 214, 217-220
Benzene, 4-5
vapor pressure, 3 l 3
Binary interaction parameter, 209-210,
269-270
Binary systems, freezing-melting diagram,
285
Block and Bird correlation, 359
Boiling point, 11
n-alkanes, 58-59
definition, 31
elevation, 282-284
heavy hydrocarbons, 50, 52
hydrocarbon-plus fractions, 173
petroleum fractions, 88-93
prediction, pure hydrocarbons, 58-59
reduced, 251
sub- or superatmospheric pressures,
106-107
true, 89
Boiling points
average, 100-101
range, 88
Boiling range fractions, narrow versus
wide, 112-119
Boltzman constant, 24
Boossens correlation, 57-58
Bubble point, calculations, 370-371
Bubble point curve, 201
Bubble point pressure, 223, 367
Bubble point temperature, 368
Bulk parameters, petroleum fractions, 114
Butane, equilibrium ratios, 274-275
C
Capillary pressure, 357
Carbon
prediction in petroleum fractions, 127
see also SCN groups
Carbon number range approach, petroleum
fractions, 186
Carbon residue, petroleum fractions,
141-142
Carbon-to-hydrogen weight ratio, 11
definition, 36
Carnahan-Starling equation of state,
214-215
Cavett method, 61
Cementation factor, 351
Cetane number, petroleum fractions,
137-138
CH weight ratio, pure hydrocarbons,
prediction, 68-69
Chapman-Enskog equation, 346
Chapman-Enskog theory, 339
Characterization method, evaluation
criteria, 75-76
Chemical potential, mixtures, 254-255
Chen correlation, 323
Chen-Othmer correlation, 347
Chromatography, 96-98
Chueh-Prausnitz relation, 210
Chung's model, 386
Clapeyron equation, 252, 307-309
Clausius-Clapeyron equation, 252
Cloud point
Coal liquid fractions, heat of vaporization,
324
Coefficient of thermal expansion, 236
Colloidal model, 375-376
Composition, units, 21-22
Compressibility factor, 203, 215-221,289
Consistency test, predicted physical
properties, 71, 73
Continuous mixture characterization
approach, petroleum fractions,
187-189
Correlation index, 122-124
Corresponding states principle, 215
COSTALD correlation, 224
Cracking, 7
Cricondentherrn temperature, 202
Critical compressibility factor
definition, 32
Critical constants, definition, 32-33
Critical density, definition, 32
Critical point, 200
Critical pressure, 11
n-alkyl, 52
definition, 32
estimation, 78-80
heavy hydrocarbons, 52-53
PNA hydrocarbons, 52
Critical properties
coal liquids, 62
estimation, 115-116
internal consistency, 51
Critical temperature, 11
n-alkanes, 50
definition, 32
estimation, 78-80
heavy hydrocarbons, 52-53
influence, 13-14
60-62
Critical viscosity, 334
Critical volume, 11
estimation, 79
Crude oils, 5-7
API gravity, 156
asphaltene content, 374-378, 387-388
assays, 154, 156-159
cloud point temperature, 383-384
composition and properties, 6-7
from atmospheric separator, 7
lumping scheme, 186
nomenclature, 152-153
products and composition, 9
properties calculation, 189-191
resin content, 374-375, 387-388
single carbon number groups,
characteristics, 161-163
sulfur content estimation, 191-192
vapor pressure, 313-315
viscosity, 338
Cryoscopy, 94
C6+ fraction,subitem refractive index, 180
C7+ fraction
carbon number range approach, 186
comparison of distribution models,
179-180
probability density functions, 370
C8 hydrocarbons, properties, 48
Cubic equations of state, 204-210, 319
application to mixtures, 209-210
other types, 208-209
Peng-Robinson equation, 205-206, 208
Redlich and Kwong equation, 205,
226-227
Soave modification of Redlich and
Kwong equation, 205,208
solution, 206-207
unified form, 206
van der Waal equation, 204-205
volume translation, 207-208
Cycloalkanes, 4
D
Daubert's method, 103-106
Deasphalted oils, 378
Decane, equilibrium ratios, 282
Defined fraction, 114
Defined mixtures, 114-115
Definition of basic properties, 31
Degrees of freedom, 199
Density, 11,300-305
definition, 31
gases, 300
liquid petroleum fractions, 223-224
liquids, 300-304
pure hydrocarbons, prediction, 66
solids, 304-305
units, 20-21
Dew point, 201-202
calculations, 371-372
Diesel fuel, characteristics, 143
Diffusion coefficients, 12, 345-351
measurement in reservoir fluids, 354-356
multicomponent systems, 350
order of magnitude, 346
porous media, 350-351
units, 23-24
Diffusivity, 12
relation to refractive index parameter,
353
Dipole forces, 45
Dipole moments, 375
Distillation, simulated, by gas
chromatography, petroleum
fractions, 89-91
Distillation curves, 11
interconversion, 101-108
at reduced pressures, 106-108
summary chart, 109
prediction, 108-111
at reduced pressures, petroleum
fractions, 92-93
sub- or superatmospheric pressures, 108
Double-bond equivalent, 45
Dry gas, 6
E
Edmister method, 65
Elemental analysis, petroleum ractions,
98-99
EN 238, 10
End point, 88
Energy, units, 22
Enthalpy, 12, 315-318
calculation diagram, 318
ideal gas, constants, 246-247
two petroleum fractions, 316-317
Enthalpy departure, 317
Enthalpy of vaporization, 322
versus temperature, 323
Entropy, 234
Entropy departure, 237
hard-sphere fluids, 286-287
Entropy of vaporization, 252
Equations of state, 199-204
corresponding state correlations,
215-221
fugacity coefficient calculation,
255-256
ideal gas law, 203
intermolecular forces, 202-203
real gasses, 203-204
refractive index based, 225-227
velocity of sound based, 286-287
see also Cubic equations of state;
Noncubic equations of state
Equilibrium flash vaporization, petroleum
fractions, 91-92
Equilibrium ratios, 12, 14, 269-276
Ethane
compressibility factor, 289
equilibrium ratios, 272
saturation curves, 209
Ethers, octane number, 139
Excess property, 249
Exponential model, hydrocarbon-plus
fractions, 165-167
Extensive property, 198-199
Eyring rate theory, 347
F
Fenske Equation, 14
Flame ionization detector, 90
Flammability range, definition, 34
Flash calculations, 368-370
Flash point, 11
definition, 34
Fluid properties, use of sound velocity,
284-292
Fluidity, relation to refractive index
parameter, 352
Fluids
Newtonian and non-Newtonian, 331
wettahility, 358
Force, units, 19
Fractured reservoirs, idealized, 391
Free-volume theory, 347
Freezing point, 259-260
definition, 34
depression, 281-283
saturated liquid and solid properties,
304
temperature, 200
I NDEX 4 0 3
Fugacity, 187-188, 237-238, 253,
382-383
asphaltene, 386
calculation from Lewis rule, 256
coefficient, mixtures, 254-255
liquids, 268
mixtures, 254-255
pure gases and liquids, 256-257, 268
of solids, 261-263
Fugacity coefficients, 12, 238
calculation from equations of state,
255-256
Fusion curve, 200
Fusion line, 251
G
Gamma density function, molar
distribution, 168-169
Gamma distribution model, 167-170
Gas chromatography, 96-97
simulated distillation, petroleum
fractions, 89-91
Gas condensate system
C7+ fraction characteristics, 171
pseudocritical properties, 160-161
SCN group prediction, 166-167
Gas constant, 22, 24
Gas injection projects, 390-391
Gas mixtures
properties, 120
viscosity, 335
Gas phase, 200
Gas solubility, in liquids, 266-269
see al s o Vapor-liquid equilibria
Gas-to-liquid ratio, 337-338
Gas-to-oil ratio, 368-370
units, 24
Gases
density, 300
diffusivity
at high pressures, 348-350
low pressures, 346-347
thermal conductivity, 339-342
Gasoline, characteristics, 143
Gaussian quadrature approach, splitting,
185-186
Gel permeation chromatography, 94
Generalized correlation, 215
Generalized distribution model,
170-184
boiling point, 178
calculation of average properties,
175-177
subfractions, 177-178
C6+ fraction, 180
C7+ fractions, 179-180
model evaluations, 178-180
prediction using bulk properties,
181-184
probability density function,
174-175
specific gravity, 179
versatile correlation, 170-174
Gibbs energy, 263
binary system, 263-264
excess, 257-258
Gibbs free energy, 12, 235
Gilliland method, 347
Glaso's correlation, 338
Glossary, ASTM definitions, 397-400
Goossens method, 127-128
Grouping, 184
H
Hall-Yarborough method, 63
Hammerschmidt equation, modified, 390
Hard-sphere fluids, entropy departure,
286-287
Hard-sphere potential, 202
Heat capacity, 12, 235
estimation from refractive index, 321-322
mixture, 250
thermodynamic properties, 319-321
Heat capacity coefficients, 320
Heat capacity ratio, 235
Heat of combustion, 12, 324-326
Heat of formation, 12
Heat of fusion, 201,259-261
Heat of mixing, 249
Heat of reaction, 12
Heat of sublimation, 314
Heat of vaporization, 12, 201,252, 321-324
at boiling point, 323
Heating value, 25, 324-326
Heats of phase changes, 321-324
Heavy hydrocarbons
API gravity and viscosity, 59-60
atmospheric critical pressure, 51
boiling point, 50, 52
constants, 50-51, 54
critical pressure, 52-53
critical temperature, 52-53
prediction of properties, 50-54
refractive index and viscosity, 44
Heavy petroleum fractions
enthalpy, 316
molecular weight and composition, 116
Helmhohz free energy, 235
Henry's constant, 267, 269
Henry's law, 266-269
Heptane, equilibrium ratios, 279
Hexane
vapor pressure, 311
n-Hexatriacontane
acentric factor, 65
critical properties, 64
High performance liquid chromatography,
97
High-shrinkage crude oil, 6
Hoffman correlation, 271-272
Hydrate inhibitors, 389-390
Hydrates, formation, 388-390
Hydrocarbon-plus fractions, 153, 164-184
boiling point and specific gravity
prediction, 173
calculation of average properties,
175-177
exponential model, 165-167
gamma distribution model, 167-170
general characteristics, 164-165
generalized distribution model, 170-184
molar distribution, 167, 172-173
molecular weight variation, 165
prediction of PDF, 173-174
probability density functions, 164-165
subfractions, calculation of average
properties, 177-178
Hydrocarbons, 3-5
groups, 3
liquid specific gravity, temperature
effect, 301
pure, see Pure hydrocarbons
research octane number, 140
Hydrodynamic theory, 347
Hydrogen, prediction in petroleum
fractions, 127
Hydrogen sulfide, equilibrium ratios, 283
ISO
ISO
ISO
ISO
ISO
ISO
ISO
ISO
ISO
ISO
ISO
ISO
ISO
ISO
ISO
ISO
ISO
ISO
ISO
ISO
Ideal gas
mixture, heat capacity, 244
Ideal gas law, 203, 209
In-situ alteration, 2
Infrared spectroscopy, 97
Intensive property, 198-199
Interracial tension, see Surface/interracial
tension
Intermolecular forces, 43, 202-203
Internal energy, 199
IP 2/98, 144
IP 12, 144
IP 13/94, 144
IP 14/94, 144
IP 15, 135, 144
IP 16, 136, 144
IP 34/97, 144
IP 57, 142
IP 61, 99
IP 69/94, 144
IP 71/97, 144
IP 107, 99, 144
IP 123/99, 144
IP 156/95, 144
IP 196/97, 144
IP 218, 138
IP 219, 135, 144
IP 236, 144
IP 365, 93, 144
IP 370/85, 144
IP 380/98, 144
IP 402, 131
IP 406/99, 144
ISO 2049, 144
2185, 144
2192, 144
2592 34
2719 144
2909 123
2977 144
3007 144
3013 144
3014 142
3015 135, 144
3016 135, 144
3104 100, 144
3405 144
3837 144
4262 144
4264, 144
5163, 144
6615, 144
6616, 144
6743/0, 10
ISO 8708, 144
ISO 12185, 93
Isofugacity equations, 383
Isoparaffins, 3
Isothermal compressibility, 236
Jenkins-Walsh method, 128-129
Jet fuel
characteristics, 143
enthalpy, 318
Jossi's correlation, 337
Joule-Thomson coefficient, 236
K
Kay's mixing rule, 220, 372
Kesler-Lee method, 79, 81
Kinematic viscosity, 331,337
definition, 33-34
estimation, 118-119
units, 23
Korsten method, 65, 81
Kreglweski-Kay correlation, 372
Kuwait crude oil, characterization, 190
L
Lee-Kesler correlation, 239
Lee-Kesler method, 56, 60-61, 64-65,
8 0 - 8 1
Length, units, 18
Lennard-Jones model, 202
Lennard-Jones parameters, velocity of
sound data, 288-289
Lewis rule, fugacity calculation, 256
Linden method, 137
Liquid chromatography, 90, 97
Liquid density
effect of pressure, 223-225
pressure effect, 302
temperature effect, 303
Liquid mixtures, properties, 11%120
Liquids
density, 300-304
diffusivity
at high pressures, 348-350
at low pressure, 347-348
fugacity, 268
gas solubility, 266-269
heat capacity values, 319
viscosity, 335-338
see also Vapor-liquid equilibria
London forces, 45
Lubricants, 9-10
Lumping scheme, 184
M
Margule equation, 261
Mass, units, 18
Mass flow rates, units, 20
Mass spectrometry, 98
Maturation, 2
Maxwell's equations, 235
Melting point, 11
definition, 34
pressure effect, 253-254
Metals, in petroleum fractions, 99
Methane
hydrate formation, 388
P-H diagram, 263-264
speed of sound in, 286
Micellar model, 375-376
Miller equation, 306
Mixtures
phase equilibria, 254-263
activity coefficients, 254-255,
257-261
criteria, 263-265
fugacity and fugacity coefficients,
254-257
fugacity of solids, 261-263
property change due to mixing, 249-251
Molar density, units, 20-21
Molar distribution, gamma density
function, 168-169
Molar refraction, 47, 225
Molar volume, 259-260
units, 20
Molecular types, characterization
parameters, 121-124
Molecular weight, 11
comparison of distribution models, 178
definition, 31
estimation, 115-116
evaluation of methods, 76-77
55-58
units, 19
Moles, units, 19
Motor octane number, 34-35, 138
Multicomponent systems, diffusion
coefficients, 350
Multisolid-phase model, 378,
382-385
N
n-d-M method, 126-127
Naphthalene, solubility, 277-278
Naphthas, 9
GC chromatograph, 91
research octane number, 140
Naphthenes, 4
Natural gas
hydrate formation, 388
pseudocritical properties, 160-161
sulfur in, 5
wet and dry, 6
Near-critical oils, 6
Newton-Raphson method, 380
Newton's law of viscosity, 331
NF M 07-048, 136
NF T 60-162, 10
NF T 60-101, 93
Nitrogen, prediction in petroleum
fractions, 129-130
Nomenclature, 1
Nonane, equilibrium ratios, 281
Noncubic equations of state, 210-215
Carnahan-Starling equation of state,
214-215
modified Benedict-Webb-Rubin equation
of state, 214, 217-220
SAFT, 215
second virial coefficients, 211-212
truncated virial, 212-213
virial equation of state, 210-214
Non fuel petroleum products, 9-10
Nonhydrocarbon systems, extension of
correlations, 54-55
Nonpolar molecules, potential energy,
45-46
Nonwetting fluid, 357
Numerical constants, 24
O
Octane
liquid heat capacity, 291
Octane number
definition, 34-35
Oil, speed of sound in, 286
Oil field, 2
Oil reserves, 2
Oil wells
history, 2
number of, 3
Oils, enhanced recovery, 390-391
Olefins, 4
Oleum, 1
P
P-T diagrams, 372-373
Packing fraction, 214
Parachor, 358-359
Paraffins, 3-4
content and research octane number,
141
properties, 48
Partial molar properties, mixtures, 248-249
Partial specific property, 248
Pedersen exponential distribution model,
167
Peng-Robinson equation of state, 205-206,
208
velocity of sound data, 289-292
Pentane, equilibrium ratios, 276-277
Percent average absolute deviation, 75
Petroleum, formation theories, 2
Petroleum blends, volume, 251
Petroleum cuts, 8
Petroleum fluids
nature of, 1-3
characterization, importance, 12-15
Petroleum fractions, 7-10, 87-146
acentric factor, estimation, 115-116
aniline point, 137
average boiling point, 100-101
boiling point and composition, 121
boiling point and distillation curves, 88-93
bulk parameters, 114
carbon and hydrogen prediction, 127
carbon number range approach, 186
carbon residue, 141-142
cetane number, 137-138
cloud point, 135-136
composition, 11
compositional analysis, 95-99
continuous mixture characterization
approach, 187-189
critical properties, estimation, 115-116
defined mixtures, 114-115
density
estimation, 117
specific gravity, and API gravity, 93
diesel index, 137-138
distillation
at reduced pressures, 92-93
columns, 8
curve prediction, 108-111
elemental analysis, 98-99
elemental composition prediction,
127-130
equilibrium flash vaporization, 91-92
flash point, 133-135
I NDEX 4 0 5
freezing point, 136-137
gas mixtures, properties, 120
Gaussian quadrature approach, 185-186
ideal gas properties, 243-244
interconversion of distillation data,
101-108
kinematic viscosity, estimation, 118-119
laboratory data analysis, 145-146
liquid mixtures, properties, 119-120
lumping scheme, 186-187
matrix of pseudocomponents, 111-112
method of pseudocomponent, 114-115
minimum laboratory data, 143-145
molecular type prediction, 121-124
molecular weight, 93-94
molecular weight estimation, 76
narrow versus wide boiling range
fractions, 1 t2-114
nomenclature, 87
octane number, 138-141
olefin-free, 115
PNA analysis, 98
PNA composition, prediction, 120-127
pour point, 135-136
predictive method development, 145-146
pseudocritical properties, estimation,
115-116
Rackett equation, 223
refractive index, 94-95
estimation, 117
Reid vapor pressure, 131-133
simulated distillation by gas
chromatography, 89-91
smoke point, 142
specific gravity, estimation, 117
splitting scheme, 184-186
sulfur and nitrogen prediction, 129-130
surface/interracial tension, 359-360
thermodynamic properties, general
approach, 298-300
true boiling point, 89
types of composition, 96
undefined mixtures, 114
viscosity, 99-100
using refractive index, 338
V /L ratio and volatility index, 133
Winn nomogram, 74
Petroleum processing, 17
Petroleum production, 17
Petroleum products
nonfuel, 9-10
quality, 143
Petroleum waxes, 10
Phase equilibrium, 365-393
asphaltene, precipitation, solid-liquid
equilibrium, 385-388
enhanced oil recovery, 390-391
mixtures, 254-263
activity coefficients, 254-255, 257-261
criteria, 263-265
fugacity and fugacity coefficient,
254-257
fugacity of solids, 261-263
pure components, 251-254
types of calculations, 366-367
vapor-solid equilibrium, 388-390
viscosity, 367-373
see also Vapor-liquid-solid
equilibrium-solid precipitation
Phase rule, 199
Physical properties, 1 0-12
Planck constant, 24
PNA analysis, 98
PNA composition, prediction, 120-127
PNA three-pseudocomponent model, 115
Polarizability, 47
Porous media, diffusion coefficients,
350-351
Potential energy, nonpolar molecules,
45-46
Potential energy function, 202
Potential energy relation, two-parameter,
46, 48
Pour point, 11
Poynting correction, 257
Prandtl number, 339
Pressure
triple point, 199
units, 19
Propane
The Properties of Gases and Liquids, 16
Properties of Oils and Natural Gases, 16
Pseudocomponent method, 320
Pseudocomponent technique, 112
Pseudocomponents
generation from Gaussian quadrature
method, 185-186
matrix, 11 I-112
Pseudocritical properties, 12, 32
gas condensate, 160-161
natural gas, 160-161
Pseudoization, 184
Psuedocomponents, 13
Pure components, vapor pressure,
305-306
Pure compounds
critical thermal conductivity, 241
liquid thermal conductivity, 343
vapor pressure, coefficients, 308-309
viscosity coefficients, 333-334
Pure gases, fugacity, 268
Pure hydrocarbons, 30-83
acentric factor, prediction, 64-65, 81
boiling point, prediction, 58-59
CH weight ratio, prediction, 68-69
characterization, 45-55
parameters, 48-50
criteria for evaluation of characterization
method, 75-76
critical temperature and pressure,
prediction, 60-62
critical volume, prediction, 62-63
data sources, 36-37
definition of properties, 31-36
density, prediction, 66
estimation of critical properties, 77-81
extension of correlations to
nonhydrocarbon systems, 54-55
freezing/melting point, prediction, 68-70
generalized correlation for properties,
45-48
heavy, properties, 37, 44-45
kinematic viscosity, prediction, pure
hydrocarbons, 70-73
molecular weight prediction, 55-58
nomenclature, 30
prediction of properties, recommended
methods, 83
properties, 37-43
refractive index, prediction, 66-68
secondary properties, 41-43
specific gravity/API gravity
prediction, 58-60
Winn nomogram, 73-75
see also Heavy Hydrocarbons
PVT relations, 199-202
critical point, 46
intermolecular forces, 202-203
Rackett equation, 222-225
Q
Quadratic mixing rule, 209
R
Rachford-Rice method, 368
Rackett equation, 222-225, 301
pressure effect on liquid density, 223-225
pure component saturated liquids,
222-223
Rackett parameter, 222
Raoult's law, 188, 265-267
Real gases, equations of state, 203-204
Redlich-Kister expansion, 257
Redlich-Kwong equation of state, 46, 205,
226-227, 300
velocity of sound data, 289-292
Refining processes, 7
Refractive index, 11
basis for equations of state, 225-227
C6+ fraction, 180
definition, 32
estimation, 117
heat capacity estimation from, 321-322
heavy hydrocarbons, 44
parameter
relation to fluidity, 352
relation to diffusivity, 353
pure hydrocarbons, prediction, 66-68
Refractivity intercept, 11
definition, 35
Reid vapor pressure, 11, 33, 131-133
Reidel method, 63
Relative volatility, 14
effect of error, 14
Research octane number, 34-35, 138
Reservoir fluids, 2, 5-7
composition and properties, 6-7
C7+ fractions, characteristics, 163-164
definition, 5
diffusion coefficients measurement,
354-356
flash calculation, 369
laboratory data, 153-155
lumping scheme, 186
properties calculation, 189-191
single carbon number groups,
characteristics, 161-163
types and characteristics, 6
Residual enthalpy, 237
Residual Gibbs energy, 237-238
Residual heat capacity, 238
Resins, 374-375
Retention time, 90
Retrograde condensation, 202
Riazi-Daubert correlations, 58, 78-80
Riazi-Daubert methods, 55-57, 58-60, 62,
102-103, 124-126
Riazi-Faghri method, 341,343
Riazi method, 127
Riedel equation, 313,323
Riedel method, 342
S
SAFT model, 386
Saturation curves, ethane, 209
Saturation pressure, thermodynamic
properties, 251- 254
Saybolt viscosity, 35
Scatchard-Hildebrand relation, 258, 261
Schmidt number, 345
SCN groups
characteristics, 161
exponential model, 165-167
molecular weight boundaries, 168
Self-diffusion coefficient, 345
Sensitivity of fuel, 138
Separation by solvents, 96
Shear stress, 331
Shift parameter, 208
SI units, 18
Size exclusion chromatography, 93-94
Smoke point, petroleum fractions, 142
Solid-liquid equilibrium, 385-388
Solid solubility, 276-281
Solid solution model, 378, 380-382
Solids
density, 304-305
fugacity calculation, 261-263
vapor pressure, 31 4-316
Solubility, 259-260
Solubility parameter, units, 24
Solvents, 9
Soreide correlation, 58
Sound velocity
equations of state based on, 286-287
Lennard-Jones and van der Waals
parameters, 288-289
prediction of fluid properties,
284-292
RK and PR EOS parameters, 289-292
virial coefficients, 287-288
Specific energy, units, 22
Specific gravity, 11
comparison of distribution models,
178-179
definition, 31
estimation, 117
hydrocarbon-plus fractions, 173
hydrocarbons, temperature effect, 301
units, 21
Specific volume, units, 20
Spectrometric methods, 98
Speed of light in vacuum, 24
Splitting scheme, petroleum fractions,
184-186
Square-Well potential, 202
Standing-Katz chart, 215-216
Stiel-Thodos method, 341
Stokes-Einstein equation, 349
Sublimation, 314
Sublimation curve, 200
Sublimation line, 251
Sublimation pressure, 315
Sulfur
crude oil content, 191-192
in natural gas, 5
prediction in petroleum fractions,
129-130
Supercritical fluid, 200
Surface/interfacial tension, 12,
356-361
predictive methods, 358-361
theory and definition, 356-358
units, 24
Temperature
triple point, 199
units, 19, 19-20
Tensiometer, 357
Thermal conductivity, 12, 339-345
critical, 341
gases, 339-342
liquids, 342-345
units, 23
Thermal conductivity detector, 90
Thermodynamic properties, 232-292
boiling point, elevation, 282-284
calculation for real mixtures, 263
density, 300-305
departure functions, 236-237
enthalpy, 31 6-318
freezing point depression, 281-283
fugacity, 237-238
generalized correlations, 238-241
heat capacity, 319-321
heat of combustion, 324-326
heat of vaporization, 321-324
heats of phase changes, 321-324
ideal gases, 241-247
measurable, 235-236
mixtures, 247-251
property estimation, 238
residual properties, 236-237
saturation pressure, 251-254
solid-liquid equilibria, 276-281
summary of recommended methods,
326
use of sound velocity, 284-292
vapor-liquid equilibria, 265-276
Thermodynamic property, 199
Time, units, 18-19
Toluene, effect on asphaltene precipitation,
377-378
Tortuosity, 350-351
Transport properties, 329-362
diffusion coefficients, 345-351
diffusivity at low pressures
gases, 346-347
liquids, 347-348
diffusivity of gases and liquids at high
pressures, 348-350
interrelationship, 351-354
measurement of diffusion coefficients in
reservoir fluids, 354-356
surface/interfacial tension, 356-361
viscosity, 331-338
Triple point pressure, 199
Triple point temperature, 199
Trouton's rule, 322
True boiling point, distillation curve, 182
True critical properties, 372-373
Tsonopoulos correlations, 62
Two petroleum fractions, enthalpy,
316-317
Twu method, 61-62, 80
U
Units
composition, 21-22
conversion, 25
density, 20-21
diffusion coefficients, 23-24
energy, 22
force, 19
fundamental, 18
gas-to-oil ratio, 24
importance and types, 17-18
kinematic viscosity, 23
length, 18
mass, 18
mass flow rates, 20
molar density, 20--21
molecular weight, 19
moles, 19
prefixes, 18
pressure, 19
rates and amounts of oil and gas,
24-25
solubility parameter, 24
specific energy, 22
specific gravity, 21
surface tension, 24
temperature, 19-20
thermal conductivity, 23
time, 18-19
viscosity, 23
volume, 20
volumetric flow rates, 20
UOP characterization factor, 13
V
Van der Waal equation, 204-205
Van der Waals parameters, velocity of
sound data, 289
Van Laar model, 257-258
Vapor, 200
Vapor-liquid equilibria, 25t-253, 265-276
equilibrium ratios, 269-276
formation of relations, 265-266
Raoult's law, 265-266
solubility of gases in liquids, 266-269
Vapor-liquid equilibrium calculations,
367-373
bubble and dew point calculations,
370-372
gas-to-oil ratio, 368-370
P-T Diagrams, 372-373
Vapor liquid ratio, volatility index and, 133
Vapor-liquid-solid equilibrium-solid
precipitation, 373-385
heavy compounds, 373-378
wax precipitation
multisolid-phase model, 382-385
solid solution model, 378, 380-382
Vapor pressure, 11,200, 305-316
Antoine coefficients, 310
definition, 33
predictive methods, 306-312
pure components, 305-306
pure compounds, coefficients, 308-309
solids, 314-316
Vapor pressure method, 94
Vapor-solid equilibrium, 388-390
Vignes method, 347
Virial coefficients, velocity of sound data,
287-288
I NDEX 407
Virial equation of state, 210-214
truncated, 240
Viscosity, 12, 331-338
gases, 331-335
heavy hydrocarbons, 44
liquids, 335-338
pressure effect, 334
units, 23
Viscosity-blending index, 337
Viscosity coefficients, pure liquid
compounds, 336-337
Viscosity gravity constant, 11
definition, 35-36
Viscosity index, 122-124
Viscosity-temperature relation, 72
Volatility, properties related to, 131-135
Volatility index, and vapor liquid ratio, 133
Volume, units, 20
Volume translation, cubic equations
of state, 207-208
Volumetric flow rates, units, 20
W
Walsh-Mortimer method, 137
Water
ideal gas heat capacity, 242-243
vapor pressure, 312
Watson characterization factor, 320
Watson K, 11, 13, 323
definition, 35
Wax appearance temperature, 378,
382
Wax precipitation
rnultisolid-phase model, 382-385
solid solution model, 378,
380-382
Waxes, 373
Wet gas, 6
Wetting liquid, 357
Wilke-Chang method, 347
Wilson correlation, 273
Winn method, 137
Winn-Mobil method, 62
Winn nomogram, 73-75
Won model, 380
X
Xylene, vapor pressure, 311

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