Cbe 324 Manual Uw Madison Cbe Assessment Home Page - Compress
Cbe 324 Manual Uw Madison Cbe Assessment Home Page - Compress
Cbe 324 Manual Uw Madison Cbe Assessment Home Page - Compress
Transport Phenomena
By E.J. Crosby
Revised by Thomas W. Chapman
Updated by Rafael Chavez, 2002
Chemical Engineering Department
University of Wisconsin-Madison
Madison, Wisconsin 53706
Copyright © 1999 by T.W. Chapman
ChE 324 Lab Manual
Experiments in
Transport Phenomena
by E.J. Crosby
revised by T.W. Chapman
updated by Rafael Chavez
Preface
Chemical Engineering 324, Transport Phenomena Laboratory, is an important
course in the chemical engineering curriculum. It is intended to accomplish three
objectives:
The textbook Transport Phenomena by Bird, Stewart, and Lightfoot (2002) is the
main source for the theoretical aspects of most of the topics treated in the laboratory.
Generally the notation used in this manual will be the same as that used in that book.
In this revision of the manual, the references were updated to the Second Edition
of the Transport Phenomena book. Some content was also added or modified to make the
manual more self-contained and easier to use.
Preface
Page I-1
ChE 324 Lab Manual
During the semester the students work in small groups, performing weekly
experiments. Individual reports are prepared and submitted at the subsequent class
session. Each week students will be asked to prepare either a formal technical report or a
shorter technical memo. Each student will also make one oral presentation.
This lab manual provides general guidelines regarding the operation of the course
as well as descriptions of each of the laboratory experiments. Students are expected to
review the subject of each week's laboratory prior to the class in order to understand
better the significance of the lab exercises. Also, a plan for data collection and analysis
should be prepared ahead of time. Planning prepares the students to complete many of the
necessary calculations during the lab period. Short quizzes may be given at the
beginning of the lab sessions to confirm such preparation. The course will be much less
time consuming for students who can complete most of the data analysis during the lab
session.
Each week the assigned experiment is put into context by a hypothetical memo
written by a fictitious industrial supervisor to his engineering staff. These memos,
included in Appendix 15, are intended to give the students a practical motivation for
conducting the assigned study. With a concrete context, the students should find it easier
to write a realistic and relevant report rather than simply commenting on whether their
data agreed with “theory”, that is, what they perceive as the “right” answer because it
comes from a textbook. Thus, student reports should be written in response to these
assignment memos.
References
Alley, M., The Craft of Scientific Writing, 3rd edition, Springer-Verlag, New York (1996)
Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, 2nd.Edition, John
Wiley & Sons, New York (2002)
Beer, D., and D. McMurrey, A Guide to Writing as an Engineer, John Wiley & Sons,
New York (1997)
Pfeiffer, William S., Pocket Guide to Technical Writing, 2nd Edition, Prentice Hall, Upper
Saddle River, New Jersey (2001)
Strunk, W., Jr., and E.B. White, The Elements of Style, 3rd edition, Macmillan, New York
(1972)
Preface
Page I-2
ChE 324 Lab Manual
Experiments in
Transport Phenomena
by E.J. Crosby
revised by T.W. Chapman
updated by R. Chavez
Contents
Preface I-1
Introduction I-5
Table of Contents
Page I-3
ChE 324 Lab Manual
Appendices
Table of Contents
Page I-4
Introduction
ChE 324 Lab Manual
With complex geometries or with turbulent fluid flow, the macroscopic balances
of transport phenomena are still relevant, but empirical models must be used to
characterize interfacial rates. Thus, one defines fluid-film transport coefficients such as
the friction factor, the heat-transfer coefficient, and the mass-transfer coefficient.
Although dimensional analysis of the differential conservation equations can identify
what independent groups of variables should appear in the functional dependence of rate
quantities on operating conditions, the actual relationships must be determined
The purpose of this course is that students recognize that the physical quantities
discussed in their transport phenomena course can indeed be measured. The experiments
are grouped into four categories: measurement of transport properties, observation of
profiles, measurements of transport coefficients, and analyses of macroscopic systems.
In each section there are experiments that deal with fluid flow, with heat transfer, and
with mass transfer.
Too often undergraduate students approach laboratory courses with the idea that
their objective is to prove basic theories or to obtain results that agree with published
information. Laboratory reports then focus on whether the experimental results agree
with the "right" answer and on explanations of why the agreement is not perfect.
Although there may be a correct value for an intrinsic property value of a material, such
as density or thermal conductivity, the same can not be said of transport characteristics.
Transport coefficients and similar efficiency factors of chemical-process systems depend
on many variables. Thus, no generalized correlation given in the literature can be
expected to predict the behavior of a particular experimental system perfectly.
The course meets weekly for one four-hour period. The first hour is normally
used for instruction, discussion, oral presentations, and occasional quizzes. The
remaining three hours are devoted to measurements and calculations in the laboratory.
2. GRADING PROCEDURE
The grade for this course will be based approximately upon the following
distribution of credit:
Category Weight
Class Quizzes and Exercises 10 %
Oral reports 5%
Laboratory Reports 80 %
Professionalism 5%
Reports are due at the next class following the experimental session. Late reports
will be penalized 10%/day. Weekends count as two days.
3. DAMAGE TO EQUIPMENT
Fee cards with charges for costs will be issued to anyone who, in the opinion of
the instructional staff, damages or destroys equipment because of carelessness or
negligence.
4. HOUSEKEEPING
group of students is responsible for its work area. All utilities are to be turned off, and all
spills are to be cleaned up before any member of the group leaves the laboratory.
5. SAFETY
All personnel are required to wear safety glasses and proper clothing when in the
laboratory in accordance with the specified safety procedures. Each individual student
must acquire his or her own safety glasses and wear them at all times in the laboratory.
Substantial footware, other than sandals, and proper clothing that provides protection
from accidental spills and burns should be worn.
7. COMPUTATIONAL AIDS
8. LABORATORY REPORTS
The laboratory instructor will present examples of proper formats for reports and
memos and will indicate how the reports should be bound.
9. WRITING SKILLS
Course Guidelines
Page I-10
ChE 324 Lab Manual
Software
The majority of students in ChE 324 use Microsoft Word for their reports and
Microsoft Excel or Mathcad for the data analysis and preparation of graphs. Use of these
particular programs is not required, but they are readily available through the Computer-
Aided Engineering (CAE) facilities in the College. Also, in general, the lab instructors
are familiar with these common programs and thus are prepared to offer assistance to
students performing specific tasks with their data or reports. It will be assumed that the
students already have a working knowledge of computers and programs of this type so
detailed instruction will not be presented. Those people who need help should see the
teaching assistants during their office hours and make use of the handouts and manuals
and of the consultants at CAE.
Computer Resources
Students are encouraged to use the many CAE computers provided in the College
to work on their reports. During the laboratory, a lab-group member or two should be
able to do much of the data analysis while others are collecting more data. There is a
satellite CAE computer room adjacent to the transport lab that now contains Windows-
based machines. These machines are connected to the CAE network so that students can
save their files directly. Of course, one can also save documents on Diskettes.
Course Homepages
Most ChE324 instructors take advantage of internet resources to distribute materials and
to communicate with students. Materials related to ChE 324 are posted on a course
homepage at http://courses.engr.wisc.edu/ecow/get/che/324/. Students are expected to
Use of Computers in ChE324
Page I-11
ChE 324 Lab Manual
make use of this method to obtain course materials. The Appendix of this manual
contains several Excel and Mathcad programs that have been set up to facilitate the data
analysis in particular experiments. Those files are also available on the course homepage.
The materials can be accessed from off-campus computers as in the CAE labs.
1. Safety Glasses: All persons in the laboratory will wear safety glasses whenever they are
whithin the restricted area of the laboratory.
2. Shoes: Footware must provide protection. Sandals and open-toed shoes will not be
permitted.
4. Chemical Hazards: Review the hazards, toxicity, and proper disposal of any chemicals
used in the lab. Such information is available on the Cornell web site mentioned in
Section III.
7. Clothing: The wearing of shorts and sleeveless shirts in the laboratory will not be
permitted. Trousers and skirts must extend below the knees. The use of a laboratory coat
is recommended.
8. Medical Insurance: The University does not carry blanket accident insurance to cover
medical costs in case of accidents involving students. Students should obtain adequate
medical or accident insurance for themselves.
9. Housekeeping: Housekeeping and safety are closely related. Sloppy housekeeping and
poor safety practices in the laboratory and workroom/classroom should be reported to the
instructor. We do not intend to allow any unsafe practices to develop. This will help
insure the safety of all.
10. Paying attention: Most accidents can be avoided by diligence, awareness, and caution. No
horseplay or practical jokes are allowed.
V. Technical Communication
A major objective of this course is to provide instruction and practice in technical
communication. Students will report on the laboratory experiments with complete,
formal technical reports, which are appropriate for archival purposes, with shorter
memoranda, which are commonly used to convey progress reports in industry, and with
oral summaries, which are also used quite often for providing overviews to colleagues
and supervisors.
A. Formal Reports
1. DEADLINE FOR SUBMISSION
Reports are due at the beginning of the period following that period during which
the experiment was performed. Late reports will be penalized 10% of the grade for each
day that submission is delayed.
2. MECHANICS OF PRESENTATION
Reports are to be word-processed (or typed) with double spacing on one side only
of standard 8 1/2" by 11" white paper. A 12-point font size is required and the Time New
Roman font is suggested. The left-hand margin should be 1-1/4". All pages, including
those with accompanying graphs and figures as well as appendices, are to be assembled
in proper order, numbered, and stapled. Figures and tables cited in the body of the report
should appear immediately following the citation, either at the end of the paragraph in the
case of tables or on the next page following. (The body of the report should be printed on
good-quality paper. If reports are printed at CAE, students may find it necessary to
purchase an additional paper allowance. Appendices may be printed, written, or drawn
on lower-quality or used paper, provided that the face side is clean.)
3. CONTENTS
There are few if any absolute rules governing the style or format of technical
reports, other than the basic requirements of clarity and neatness. The writer must adopt a
style that is appropriate for the purposes of the report and recognizes the interests and
background of the likely audience. Different formats may be appropriate for different
reports, but often a standardized format is imposed on the writer by an organization or
publication editor. For the sake of consistency in ChE 324, reports will comprise the
following designated sections in the order given. Some reports will be submitted in a
Technical Communication
Formal Reports Page I-14
ChE 324 Lab Manual
memo format. In that case, some for the following sections may be condensed,
combined, or even eliminated. Standards for memos are summarized in Section V.B.
3.2 ABSTRACT
The abstract is a simple, clear, and concise paragraph or two covering (i) what
was done, (ii) how it was done, and (iii) what was accomplished. This differs from a
Summary of Results in that it is written as an advertisement, to interest the passing reader
in the rest of the report, or as a very general summary of the nature and scope of the
report. It should be self-contained, i.e., no references to figures, tables, or equations
should be included. The abstract should contain text only.
3.3 INTRODUCTION
Reports normally begin with an introduction that sets the context for the report.
(The format at some companies and laboratories might include an executive summary
before the introduction, either as an alternative to the abstract or in addition to it.) In
general, the introduction specifies the topic of the report and states the motivation for the
reported study, giving the objectives or purpose. Some background is given, such as
references to earlier related work or to the theoretical basis for subsequent analysis. For a
Technical Communication
Formal Reports Page I-15
ChE 324 Lab Manual
longer report, it is helpful to the reader to give a paragraph or two outlining the structure
and content of the report and even the nature of the results.
For a research study, there may be a separate section devoted to the development
of the relevant theoretical analysis, which provides the basis for the subsequent data
analysis and interpretation of results. In ChE 324, the theory is available in common
textbooks and references so no separate theory section is required. Reference to relevant
theory and its source should be given in the introduction or in a later section when it is
used. Equations that will be used in presenting the results might be given in the
Introduction in order to define terms. Use the textbook Transport Phenomena as a guide
to the proper format for presenting equations within the text. Equations should appear as
part of a sentence.
This part of the report should include a summary form the specific results of the
experiment, presented in neat, clear-cut tabular and graphical form, and a thoughtful
analysis of their significance, implications, or possible applications. Pertinent discussion
of the experimental procedure (especially deviations from that suggested or possible
sources of error), theoretical analysis of the system studied as needed to interpret the
results, analysis of the data (including estimates of error), and conclusions are all given in
this section. While the earlier sections should be quite objective, this is the place where
the author can present his or her observations and interpretations. As a basis for the
analysis or to support specific conclusions, one should refer freely to relevant citations in
the literature.
In writing both the Introduction and the Discussion, the author should keep in
mind both the presumed motivation for the study and the likely interests of the reader.
For each experiment this manual presents a hypothetical context for the experiment, other
than an academic exercise. Thus, the report should be written in response to the
presumed motivation. If specific questions are posed in the manual or by the instructors,
they should be used as suggestions for topics to be discussed in the overall context of the
report.
Technical Communication
Formal Reports Page I-16
ChE 324 Lab Manual
Most reports lead to specific conclusions based on the reported results and the
analysis given. Recommendations for future work or applications and other implications
might be stated in a final section. In shorter reports this section may be omitted with the
relevant conclusions and recommendations given at the end of the discussion section. In
ChE 324 reports a separate conclusions section is usually not warranted.
3.8 APPENDICES
This section should include one example of each calculation made in connection
with the analysis of the experimental data. Each calculation is to be accompanied by the
formula involved, written in terms of the clearly defined variables, with the item being
calculated appearing on the left-hand side of the equation. The numerical values and
units of each quantity appearing on the right-hand side of the equation are to be given
when substituted into same. The calculations must be presented in neat, logical order
with the answers either underlined or set off by blocks. Extensive arithmetic
manipulations need not be shown. Identify the part of the experiment to which each
calculation applies. Simply presenting a spreadsheet with the calculated values is not
sufficient.
Any derivations performed in connection with the data analysis and discussion are
to be demonstrated in this section. This includes analysis of errors based on equations
appearing elsewhere.
A neat, orderly data sheet, for recording the original data, is to be prepared prior
to beginning the experiment and preferably before coming to the lab. All data taken in
the laboratory are to be included in their original form. Lab partners may make
photocopies for inclusion in individual reports. For the case of computer-acquired data
files, hard copies of the files are usually not required in the report.
Technical Communication
Formal Reports Page I-17
ChE 324 Lab Manual
Key graphs and tables should be presented in the body of the reports, either in the
Summary of Results or in the Discussion. Less important ones, or those used in
intermediate calculations, should be placed in the Appendix. Graphs and tables in the
body of the report should be placed on or immediately following the page where they are
first mentioned, as is done in textbooks. Graphs, as well as drawings and diagrams, are
numbered sequentially and called “Figure x.” Tables are also numbered in a separate
series. Graphs require complete, self-explanatory captions, placed below the figure but
within the margins. Tables should include a descriptive heading at the top of the table.
The main text should say what is presented in each figure and describe the significance of
each graph and table.
5. GRADING
Reports will be graded on the basis of neatness, grammar, spelling, and clarity as
well as technical validity. Strive to be clear and concise.
Technical Communication
Formal Reports Page I-18
ChE 324 Lab Manual
B. Memorandum Reports
Much communication within a technical or industrial organization is accomplished
with memos. Memos are short documents that can be read quickly and easily. They are
designed to convey their message clearly and concisely. Memos may be used to give a
notice, to make a request, or to provide a report. In Chemical Engineering 324 memos are
written for the latter purpose. The memos used here resemble those that an engineer in
industry would use to provide a progress report or summary of a project to supervisors and
colleagues. Typical formats for memos and guidelines for their content are provided in
technical writing textbooks and at the EPD Technical Communication web site.
The difference between a memo and a complete formal report is the greater amount
of detail contained in the latter. A report is intended for the purposes of communicating
with a wide, varied audience and of establishing a relatively permanent record of what was
done in a project. A memo is generally written for more immediate needs, such as
conveying recent results, e.g. a weekly progress report, to a knowledgeable reader such as
a supervisor. A memo is also designed to be read and understood quickly.
Thus, a memo does not need to present all the background material that goes into a
formal report. The format is a bit different, and it is obviously shorter. The emphasis is on
the results and your interpretation. Recommendations for additional work are usually
appropriate.
You may abbreviate the formal title page of a report into a memo heading such as
that used in each weekly memo to students from the ChE 324 instructor, and one may omit
an abstract. Separate sections headings may or may not be required, depending on the
length and complexity of the memo. Headings are probably appropriate and helpful if a
memo is longer than a page or two. Whether there are headings or not, several sections
appearing in a report may be combined logically within a memo. For example, the
introduction, theory, apparatus, and procedure may all be combined into several coherent
paragraphs, and the discussion and conclusions might flow together. It is a good idea to
give a summary of the most important results and conclusions in the first paragraph, which
then serves as a type of abstract or summary. This is the information that a busy reader is
most eager to obtain.
Depending on the complexity of the material in the memo, you may or may not
need an appendix. In any case, you should make liberal use of figures and tables, but make
sure that the headings and captions are thorough and descriptive. Generally, you should
include a list of references. For the purposes of this course, sample calculations are always
required as an appendix with memos.
Technical Communication
Memorandum Reports Page I-19
ChE 324 Lab Manual
C. Oral Reports
Engineers are frequently called upon to present oral reports. These may be brief
summaries to a team of colleagues on a project in process, a proposal to senior
management for a major investment, a tutorial to other engineers on a specialized subject,
or a paper at a technical conference reporting on a completed project. Oral reports may
be presented to a small group around a conference table or delivered in an auditorium to
an audience of hundreds. The duration may be only five or ten minutes or as long as an
hour or even more. In all cases, the purpose of an oral report is to convey information to
the audience rapidly and efficiently, preferably with a sense of the speaker's attitudes and
personality that is no so readily conveyed in a written report. Although an oral report
may not be able to cover a topic in as much detail as a written document can, it allows the
speaker an opportunity to emphasize and communicate his more important points.
Another advantage of an oral report is that the audience is often able to ask questions to
clarify the speaker's intent.
Probably everyone who delivers an oral report feels some nervousness about
standing up and talking before an audience. Such nervousness should not be a cause for
concern but a source of energy for the presentation. Nervousness diminishes with
experience, but for inexperienced speakers as well as old hands, preparation is the key to
avoiding any feared awkwardness or embarrassment during the oral presentation.
Beer and McMurrey (1997) present a very sensible discussion about giving oral
reports. Some of their primary points are summarized below. Pfeiffer (2001) presents a
similar discussion.
Preparation
After a speaker has identified her primary purpose and the key points to be
conveyed, she next must select a structure for the talk. That is, a logical sequence must
be selected as the path by which she will lead the audience through the subject at hand.
Every talk must have a beginning, a central part, and an end. The beginning is an
introduction and a preview that prepares the audience and sets the stage for what follows.
The end is the summary of what has been covered, with conclusions and perhaps
recommendations. The end should reiterate key points just as the beginning might
suggest what key issues are to be covered.
The central part of a talk is the technical development of the specific subject.
This part of the talk, just like a written report, should be organized to make the trip from
the original objective and premises to the conclusions as effortless as possible for the
audience. Designing such a path requires selecting a logical structure. As indicated by
Beer and McMurray, there are a number of alternative strategies that may be selected,
depending on the topic and the audience. One may proceed chronologically or spatially;
one may go from simple to complex or vice versa, one may organize the points in order
of decreasing or increasing importance, familiarity, difficulty, etc. Regardless of what
logical sequence is selected, the speaker should be consistent so that the audience does
not get confused. Also, it is imperative that the degree of detail presented be adjusted to
fit the time allotment for the talk and the technical level of the audience.
After the overall structure of the talk has been designed and the content selected,
the speaker should design visual aids and graphics to enhance the clarity and efficiency of
his presentation. Slides or overhead transparencies should be used to reinforce what the
speaker is saying, helping to convey the overall logic of the presentation. As a picture is
worth many words, the same is true of well-designed graphs and diagrams. Each graphic
should have a descriptive heading, summarizing the significance of the illustration. On
all sheets one should use large letters that are easy to read and avoid cluttering it up with
too much information. Each page to be displayed should be kept quite simple and
contain lots of blank space so that the observer does not get overloaded and can focus on
the key point.
In preparation for the presentation, the speaker should give special thought to
what will be said in the introduction and in the conclusion. These portions of a talk
should appear to be ad lib, but they should be quite polished to make a good impression
on the audience. A speaker may want to make some notes as an aid in the presentation,
but for most of the talk, the visuals themselves should be sufficient reminders of what
needs to be said and in what order.
Finally, in preparation for an oral presentation, a speaker should practice the talk.
If possible, some friends or colleagues should be asked to listen to trial runs. Such
practice is needed, first of all to ensure that the talk will not be too long but also to check
the quality of the visual aids, to practice speaking on one's feet, and to test the planned
wording of the introduction and the conclusion.
Presentation
Giving a speech in front of an audience is always stressful, even for the most
accomplished speakers. You can reduce the stress by following the guidelines given by
Pfeiffer (2001).
Technical Communication
Oral Reports Page I-21
ChE 324 Lab Manual
When the time comes for the actual presentation, there are a few other issues to
keep in mind. Think about the many bad talks (or lectures) that you have attended and
think of all of the mistakes that the speaker made. These are mistakes that you wish to
avoid.
With respect to delivery, remember to speak at a sufficiently audible level that
those in the back of the room can easily hear what you say. The graphics should be
designed so that the same people can easily read them. Look at your own projected
graphics from the same distance to see how they work. Be careful to speak at a
comfortable pace, neither too rapidly nor too slowly, and inject some dynamics into your
delivery. Maintain eye contact with your audience to sense whether they are following
you. If you see a puzzled face, you might ask whether there is a question.
When displaying projected graphics, use a pointer to help keep the audience with
you. Use the hand closer to the screen to avoid blocking the view or turning your back to
the audience. Leave the graphics up on the screen long enough for the audience to absorb
the content. Although it is not desirable to read one's entire talk from the screen, some
reading is helpful for the audience. That is, one should not be expecting the audience to
be reading an outline or a statement on the screen and at the same time listening to the
speaker make a separate point.
Speak naturally, not too stiffly, but avoid also being too informal. That is, use
proper English and avoid slang and clichés. Also, try to eliminate nervous gestures and
hemming and hawing that will distract or annoy the audience.
Finally, try to make the planned logical structure of the talk transparent to the
audience. Orally and with visual aids, emphasize clear transitions as you step through the
presentation. Also, it is very helpful to the audience when the speaker repeats the key
points of the talk. One old recommendation, “The Preacher’s Maxim” is:
First tell them what you are going to tell them, then tell them, and finally tell them
what you told them.
People are generally not very good listeners. They remember only a portion of
what they hear and a bit more of what they read. They do remember the most when they
both hear and see the information. It is the speaker’s obligation to help the audience to
absorb and to remember the most important information from a talk. This can be done
through planning, preparation, and practice of the presentation.
References
Beer, David, and David McMurrey, A Guide to Writing as an Engineer, John Wiley &
Sons, Inc., New York, 1997, Chapter 8.
Pfeiffer, William S., Pocket Guide to Technical Writing, 2nd Edition, Prentice Hall, Upper
Saddle River, New Jersey (2001), Chapter 3.
Technical Communication
Oral Reports Page I-22
ChE 324 Lab Manual
There are generally two types of quantities that must be measured. There are the
material properties, and there are macroscopic characteristics of a certain type of system or
process. The unknown quantity may be a single constant, or it may be an unknown
function that varies with changes in local conditions. In the case of basic materials
properties, thermodynamics usually reveals a set of independent variables upon which a
quantity should depend. In the more general case, dimensional analysis often helps one to
identify an appropriate set of independent variables to be considered. Based on theoretical
analysis, one may know the functional form that the unknown quantity should follow.
When a theoretical form is known, the task of the experimenter is to find the specific
parameter values that enable the function to fit the behavior of the particular material or
process of interest. When there is no theoretical guidance or experience that provides a
functional form for the expected dependence of the measured quantity on its independent
variables, the experiment has to seek an empirical functional form that represents the
Experimental Design and Statistical Analysis of Data
Page I-23
ChE 324 Lab Manual
system behavior. One must also determine the associated parameter values that provide a
quantitative description of the phenomenon.
Thus, in the conduct of practical chemical engineering there are two related
activities that are crucial to the effectiveness of an experimental program. First, one must
consider the matter of experimental design. The other issue is statistical analysis of the
data.
When one has available a deterministic model for a system that is based on a
rigorous theoretical analysis, the purpose of experimentation might be simply to confirm
the validity of the theory. In this case one might use statistical analysis, combined with
replicated measures, to discrimiate experimental error from shortcomings in the theoretical
model. More often, the chemical engineer is working with a deterministic process model
that contains some unknown parameters. These unknown parameters may be
thermodynamic or transport properties of the material or they may be parameters such as
transport coefficients that depend on the detailed geometry or flow conditions in the
equipment. In this case, the objective is to estimate the value of the unknown parameter or
parameters from the experimental tests.
In the case where one does not know initially what functional form should
represent the magnitude and variation of a quantity of interest, it is appropriate to adopt a
strategy of experimental design. That is, one has to decide which independent variables
might have an effect on the outcome of an experiment. Then one must choose values of
those variables to use in setting up the experiment. A number of experiments must be
conducted at different settings of the suspected independent variables to see what effect
each actually has. Because experiments are usually costly and time-consuming, one hopes
to answer this question with a minimum number of tests.
For example, suppose one were interested in maximizing the yield of a particular
reaction in a certain type of reactor. The independent variables that might be relevant
could include temperature, pressure, reactant concentrations, residence time in the reactor,
as well as mixing characteristics. Another variable might be the concentration of a
possible catalyst. To find out under what conditions the amount of product produced from
the reactants is maximized, one could do many experiments at different settings of the
various variables, but an exhaustive study might be prohibitively expensive.
To make the experimental study of a problem like this most efficient, statisticians
have developed techniques known as factorial design. The first objective of statistical
design is to determine which variables have a large effect and which have little or no effect
on the outcome of the experimental system. More advanced analysis considers whether the
effects of variables are independent or whether there are interactions among the variables.
Then there is the question of how the dependence on the variables can be represented by
quantitative formula.
Box, Hunter, and Hunter (1978) provide a good treatment of experimental design
and factorial analysis. Many of their examples are taken from the field of chemistry. The
fourth and last part of this book deals with model building, that is, identification of
quantitative functions that can successfully describe observed experimental behavior.
Although models based on statistics and empiricism, rather than a rigorous underlying
theory, are limited in their predictive capacity to the actual range of variables studied, they
are nevertheless quite useful for practical purposes.
Reference
Box, G.E.P., W.G. Hunter, and J.S. Hunter, Statistics for Experimenters, An Introduction
to Design, Data Analysis, and Model Building, John Wiley & Sons, New York (1978
Part A
Measurement of Transport Properties
ChE 324 Lab Manual
Experiment A.1
where (-∆℘) is the net driving force for the flow, Q is the volumetric flow rate of fluid,
and R is the tube radius. The quantity ℘ is defined as (p+ρgh) where p is static, or
thermodynamic, pressure, ρ is fluid density, g is the acceleration of gravity, and h is
vertical elevation above a datum plane. Thus, ℘ represents the combined effects of
pressure and gravity in causing the fluid motion. (The notation in this manual follows
that used in Transport Phenomena by Bird et al., 2002, which presents a summary table
on pp. 757-764. For example, ∆x ≡ x2-x1 where the subscript 1 indicates the value of a
quantity x at the fluid entrance and 2 the value at the exit.)
There are a number of assumptions involved in the development of the Hagen-
Poiseuille law. Among other conditions, the flow must be laminar and free from end
effects. If the construction and operation of an experimental apparatus can conform
accurately to the key assumptions, it is possible to use Equation A.1.1 to measure the
viscosity of Newtonian fluids.
A simple experimental arrangement which could yield a viscosity determination
based on Equation A.1.1 is the steady flow Q of a fluid in a long, straight tube that is
maintained at a constant temperature and is equipped with a device to measure the
pressure gradient ∆p/L at some distance from the ends of the tube; a capillary manometer,
for example. In most instances the control of the operating conditions over the entire
length of the tube, the cleaning difficulties, and the need for a large sample of liquid to
fill the length of the tube prohibit or make very difficult the use of such a device. Other
more convenient and compact types of viscometers to which the Hagen-Poiseuille
Experiment A.1
Viscosity of Newtonian Fluids Page A.1-1
ChE 324 Lab Manual
equation may be applied have been developed. The Cannon-Fenske viscometer and other
modifications of the Ostwald pipette are examples (ASTM, 1955).
When the total change in the driving force ℘ associated with a flow rate Q
through a tube is due to hydrostatic head alone, Equation A.1.1 may be written as
4
π g ( − ∆h ) R
ν = (A.1-2)
8Q L
where (-∆h) is upstream elevation minus downstream elevation, called the hydrostatic
head difference, and the quantity ν is defined as
ν= µ/ρ (A.1-3)
and is called the kinematic viscosity. The kinematic viscosity is often expressed in units
of cm2/sec, which is called stoke.
Consider steady fluid flow through a straight capillary tube of fixed length L for
which the hydrostatic head differential ∆h is constant. If one measures the time for a
fixed volume of fluid V to pass through a particular tube, the kinematic viscosity should
be related to the observed efflux time te as follows:
ν = C te (A.1-4)
where C is called the viscometer constant. If C is evaluated by observing te with a liquid
of known viscosity, C may be calculated for the apparatus. Then measurements of te for
the same V in the same cell with other fluids allows the kinematic viscosities of the latter
to be calculated from Equation A.1.4. Dynamic viscosity value is then obtained by
multiplication with the density of the liquid.
Q = V/te (A.1-5)
into Equation A.1.2 and combining all constant factors into one term. The viscometer
constant C is thus identified to be
π g ( − ∆h) R 4
C= (A.1-6)
8VL
Although the preceding equations are derived for constant ∆h and constant Q,
they may be applied with reasonable success to a pipette-type viscometer in which a
liquid drains under a slowly varying hydrostatic head. In that case, one may use average
values of Q and ∆h in Equations A.1.2 and 6, and the constant C should still be a property
of only the viscometer geometry and not depend on the properties of the fluid.
Apparatus
Experiment A.1
Viscosity of Newtonian Fluids Page A.1-2
ChE 324 Lab Manual
Experiment A.1
Viscosity of Newtonian Fluids Page A.1-3
ChE 324 Lab Manual
Working capillary
Procedure
1. Ensure that the thermostat baths have attained the predetermined temperatures at
which the viscosity measurements are to be made, 30, 45, and 60oC.
2. Clean the viscometer thoroughly before using. In the case where aqueous solutions of
organic materials are involved, clean with cleaning solution, rinse with distilled water
followed by acetone, and dry with filtered air. NOTE: In order for the viscometer to
operate properly, it must be absolutely clean.
Experiment A.1
Viscosity of Newtonian Fluids Page A.1-4
ChE 324 Lab Manual
3. Calibrate the viscometer using the 60-weight-percent aqueous sucrose solution that is
provided. Measurements are to be taken at three temperatures, 30, 45, and 60oC.
4. With the viscometer in a vertical position, use the 10-ml graduated pipette to
introduce exactly 6.5 ml of the sucrose solution into the wider leg of the viscometer.
NOTE: All liquids are to be introduced into the viscometer at room temperature.
5. Place the viscometer in a constant temperature bath. It should be submerged such
that the bath-water is at least one centimeter above the upper of the two small
reservoirs. Allow at least ten minutes for the viscometer and its contents to reach
thermal equilibrium with the bath, particularly at the higher temperatures. The filled
viscometer will be moved from bath to bath to obtain data at various temperatures.
6. Before measuring any efflux times, align the viscometer vertically in the constant
temperature bath, in the orientation shown in Figure A.1-1.
7. Apply suction to the narrow leg of the viscometer until the liquid level is about 0.5
cm above the etched mark between the two small reservoirs.
8. Place a thumb over open end of narrow leg to maintain the liquid level. At this point
an unbroken column of liquid should extend from the large bulb at the bottom to a
level near the bottom of the upper small reservoir.
9. Remove thumb and measure with the stopwatch the time required for the liquid
meniscus to pass from the upper etched mark to the lower etched mark.
10. Repeat Steps 7. through 9 to obtain replicate data points. The runs go faster at the
higher temperatures so it is more convenient to take replicates in the warmer baths.
11. Clean the viscometer thoroughly and dry it, as described in Step 2, both when a new
liquid is to be introduced into the cell and when no further measurements are to be
made.
12. Repeat Steps 4 through 11, replacing the sucrose solution first with the 85-weight-
percent aqueous glycerol solution, then with distilled water. For glycerol use the
same temperatures as in the calibration process. With pure water, it is sufficient to
make a measurement only at 30oC; this measurement will be used to test the
applicability of the method to less viscous fluids.
13. Collect the following data needed for the determination of the density of the sucrose
and glycerol solutions, using water as a standard:
a) Weight of the empty, dry pycnometer
b) Weight of the pycnometer plus distilled water
c) Weight of pycnometer plus sucrose solution
d) Weight of pycnometer plus glycerol solution
e) Temperatures of all solutions weighed
These measurements may be done at room temperature or in the 30oC bath. Calculate
densities in order to determine the actual solution concentrations from the density
tables given in Appendix 4.
14. Note also approximate values of the quantities appearing in Equation A.1.6. These can
be used to estimate the expected value of C.
Experiment A.1
Viscosity of Newtonian Fluids Page A.1-5
ChE 324 Lab Manual
15. Always pour the solutions slowly. Otherwise, they will entrain air bubbles that are
very slow to escape and can affect the experimental results.
16. Be especially careful while cleaning and drying your viscometer. Return test solutions
to their containers, and wipe up any spills. Rinse out the glassware as thoroughly as
possible with distilled water and with Alconox if necessary. Rinse with a minimal
amount of acetone, disposing of the waste acetone in the waste-solvent container
provided, and dry very gingerly with compressed air. The greatest risk of breaking the
glass occurs during the drying.
Data Analysis
1. For the 60-weight-percent aqueous sucrose solution plot:
a. density vs. temperature
b. absolute viscosity vs. temperature
c. kinematic viscosity vs. temperature
These properties are given in Appendix 4 and 4a.
2. Use these sucrose-solution data with measured efflux times to determine the
viscometer constant C. Consider whether the data indicate any dependence of C on
temperature.
3. Determine the experimental kinematic viscosity of the glycerol solution as a function
of temperature. Plot the results, and for comparison include in the plot a literature
value for the kinematic viscosity of an 85-weight-percent aqueous glycerol solution at
20°C. Properties of glycerol solutions are given in Appendix 4.
4. Compare the experimentally determined viscosity of water with published values.
Properties of water are given in Appendix 5.
5. As a check on the validity of the viscometer calibration, estimate the geometrical
parameters of the cell, and calculate the expected value of C from Equation A.1.6.
6. For laminar-flow conditions, the entrance length Le , i.e., the distance in the tube
required for the flow patterns to become fully developed, has been found to be a
function of the Reynolds number:
Le ≅ (0.05)(2R)(Re) (A.1-7)
where Re ≡ (2R<vz>ρ)/µ is the Reynolds number , and <vz> is the average velocity
in the tube. Estimate the volume V and the mean velocity <vz> in order to estimate
Re and the entrance length for both the glycerol solution and the water at 45oC.
Compare the estimated entrance-length values with the actual capillary length to
check the validity of neglecting end effects in the data analysis.
References
Bird, R.B., W.E. Stewart and E.N. Lightfoot, Transport Phenomena, 2nd Edition, John
Wiley and Sons, Inc., New York (2002).
Experiment A.1
Viscosity of Newtonian Fluids Page A.1-6
ChE 324 Lab Manual
American Society for Testing Materials, Book of ASTM Standards, Part 5, Fuels,
Petroleum, Aromatic Hydrocarbons, Engines Antifreezes. Philadelphia (1955). Tentative
Method of Test for Kinematic Viscosity, ASTM Designation. D 445-53T, pp. 197, 200-
224.
Viscometers, Bulletin 19, Cannon Instrument Company, Box 812, State College,
Pennsylvania.
Prandtl, L. and O.G. Tietjens, Applied Hydro- and Aeromechanics, Dover Publications,
Inc., New York (1934), pp. 26-27.
Cannon, M.R., R.E. Manning, and J.D. Bell, Anal. Chem., 32, 355-358 (1960).
Cannon, M.R., and M.R. Fenske, Ind. Eng. Chem. (Anal. Ed.), 10, 297-301 (1938).
Experiment A.1
Viscosity of Newtonian Fluids Page A.1-7
Experiment A.2
The minus sign in Fourier's law indicates that heat always flows from regions of high
temperature to regions of lower temperature.
The thermal conductivity of solids can exhibit values that range over many orders
of magnitude. Good conductors such as metals have high conductivity, while good
insulators, like wool, have much smaller values. It is necessary to measure the
conductivity of a material experimentally in order to ascertain the correct value of k to use
in quantitative heat-transfer calculations. This experiment demonstrates one method for
measuring the thermal conductivity of solids.
This experiment is based on measurement of transient temperature changes in a
sample of an initially cool solid material after it is immersed in a hot fluid bath. The
experiment is modeled by use of Fourier's law, combined with the principle of
conservation of energy, in order to obtain a theoretical relation for temperature as a
function of time in the unsteady-state heating process. Comparison between experimental
data for temperature as a function of time and the theoretical prediction allows calculation
of the thermal conductivity.
Unfortunately, there is some uncertainty concerning the effect of the fluid mixing
in the bath on the rate of heating in the solid. Therefore the apparatus must be calibrated
with solids of known conductivities in order to determine the efficiency of heat transfer
from the stirred fluid to the outer surface of the solid samples.
Theory
A microscopic energy balance in a homogeneous solid, where the physical
properties are assumed to be constant yields
ChE 324 Lab Manual
~ ∂T
ρ Cp
= k ∇2T (A.2-2)
∂t
In solids of simple geometry, Equation A.2-2 can be used along with appropriate boundary
conditions to solve for the temperature T within the solid body as a function of time t as
well as position. Spatial derivatives ∇ and ∇2 are given for various coordinate systems in
Bird et al. (2002, §A.7).
As an example consider a thin, wide slab of solid material with thickness 2b that is
initially at a uniform temperature To. At time t=0 the slab is exposed on its surfaces to a
fluid held at a different temperature T∞. The temperature profiles in the solid can be
calculated from Equation 2 if the heat conduction from the edges of the slab is neglected
and the temperature profile is taken to be a function only of time and distance y. The
position coordinate y is measured from the center plane of the slab, and Equation A.2-2
becomes,
~ ∂ T ∂ 2T
ρC p =k (A.2-3)
∂ t ∂y 2
T∞ − T ( y, t )
Θ= , (A.2-4)
T∞ − To
where T∞ is the surrounding fluid temperature and To is the solid’s initial temperature. A
dimensionless position η is defined as
y
η= , (A.2-5)
b
where b is the distance from the center to the surface of the slab. The dimensionless time,
τ, is defined as
αt
τ= , (A.2-6)
b2
k
α= ~ . (A.2-7)
ρ Cp
Observe that the magnitude of the thermal diffusivity is proportional to the value of the
thermal conductivity, k. Equation A.2-3 becomes
∂ Θ ∂ 2Θ
= (A.2-8)
∂τ ∂η 2
Experiment A.2
Thermal Conductivity of Solids Page A.2-2
ChE 324 Lab Manual
Two boundary conditions and an initial conditions are needed to solve the problem.
At t=0, T=To, which in dimensionless from becomes,
Θ = 1 at τ = 0 (A.2-9)
In general, heat transfer from a stirred fluid to a solid surface is not perfectly efficient. In
that case T1 is not equal to T∞ at all times, and the boundary condition for solving Equation
2 must be established accordingly. Although the flow patterns and convective heat transfer
in the fluid phase surrounding the solid may be quite complex, it is common to represent
the heat-transfer efficiency by use of Newton's "law" of cooling:
q1 = h (T1 − T∞ ) (A.2-10)
where q1 is the flux of heat crossing the solid-fluid interface into the fluid. Equation A.2-
10 is not a fundamental law; it is merely a convenient approximation used to describe the
efficiency of the fluid-side heat-transfer process. It defines the proportionality factor h, the
fluid-film heat transfer coefficient, the value of which depends on the flow conditions and
geometry as well as the properties of the fluid. Better mixing and more efficient heat
transfer give larger values of h.
Assuming that in a given situation, one can estimate the value of h, Equation A.2-
10 can be used as a more realistic boundary condition on the solid surface instead of the
condition of a constant temperature. That is, the appropriate boundary condition for
solving Equation A.2-2 for a slab becomes
∂T
q1 = − k = h (T − T∞ ) at y = ±b (A.2-11)
∂y
where the first relation expresses Fourier's law for the heat flux on the solid side of the
interface and the second gives the flux on the fluid side.
∂Θ
+ Bi Θ = 0 at η=±1 (A.2-12)
∂η
bh
Bi = (A.2-13)
k
and called the Biot number. The magnitude of the Biot number indicates the resistance to
heat flow of the solid body relative to that in the surrounding fluid.
This problem has been solved and is given in many textbooks on heat conduction
(Carslaw and Jaeger, 1959, Jacob, 1949). The result is an infinite series solution of the
form
Experiment A.2
Thermal Conductivity of Solids Page A.2-3
ChE 324 Lab Manual
Bi sec( β n )cos( β nη )e ( − β nτ )
2
∞
Θ=2∑ (A.2-14)
n =1 Bi( Bi + 1)+ β n2
where the βn quantities are called eigenvalues and are identified as the positive roots of the
relation
β tan( β ) − Bi = 0 (A.2-15)
The solution given in Equation A.2-14 converges slowly at short times. On the
other hand, at longer times, as τ gets large, the exponential factors in each term get smaller,
particularly those of higher order with large values of βn. At sufficiently long times the
first term, with the smallest βn value, dominates, and the approach to the equilibrium
temperature is everywhere in the solid a pure exponential decay.
For example, at times sufficiently long that Θ has fallen below 0.8 everywhere in
the solid, the dimensionless temperature at the center of the slab may be approximated
accurately by
ln Θ (τ , 0; Bi ) ≈ − β 12 τ + ln A1 (A.2-16)
Bi J o ( β n η ) e − β n τ
2
∞
Θ=2∑
[β ]
(A.2-17)
n =1
2
n + Bi 2 J o ( β n )
where Jo(x) is a Bessel function of the first kind and zero order. The eigenvalues in this
case are determined as the roots of
β J 1 ( β ) − Bi J o ( β ) = 0 (A.2-18)
where J1(x) is the Bessel function of the first kind and first order.
Bi sin( β n η) e − β n τ
2
∞
Θ=2 ∑η
[ Bi ] sin(β
(A.2-19)
n =1
2
− Bi + β 2
n n)
Experiment A.2
Thermal Conductivity of Solids Page A.2-4
ChE 324 Lab Manual
β cot( β ) + Bi − 1 = 0 (A.2.20)
For the cylinder and sphere the characteristic distance b used to define the
dimensionless groups in Equations A.2-5, 6, and 13 is the radius of the body, and the
dimensionless distance η is the fractional distance from the center of the body to the
surface. For these two cases as well as the slab, the behavior of the temperature at the
center of the body at longer times takes on the form of Equation A.2-16.
Values of the first (smallest) eigenvalue, β1, calculated from Equations A.2-15, 18,
and 20, are given for the three geometrical cases as functions of Bi in Figure A.2-1.
Tabulated values are given in Appendix 6. These values can be used in Equation A.2-16,
which is valid for all three cases.
20
18
Sphere
16
Cylinder
14 Slab
12
1/Bi=k/hb
10
0
0 0.5 1 1.5 2 2.5 3 3.5
β1
Figure A.2-1. The first eigenvalue β1 for heat conduction in a slab, cylinder, or sphere,
given in relation to the Biot number. At a given value of Bi, the magnitude of the first
eigenvalue is largest for the sphere and smallest for the slab.
For very large values of Bi, it is also possible to derive approximate asymptotic
forms for the β1 factor that appears in Equation A.2-16. The following approximations,
which are accurate within about 1% for the range indicated, may be more convenient that
tabulated values for use in data analysis:
Experiment A.2
Thermal Conductivity of Solids Page A.2-5
ChE 324 Lab Manual
π
β1 =
2
[1− ε + ε 2
− 01775
. ]
ε 3 + ... for ε < 0.3 (A.2-21)
for a slab,
ε2
β 1 = 2.405 1 − ε + + ..., for ε < 0.2
2
(A.2-22)
for a cylinder, and
[ ]
β 1 = π 1 − ε + 3.29 ε 3 +..., for ε < 0.2 (A.2-23)
Apparatus
The apparatus for this experiment consists of (1) a relatively large constant-
temperature bath with automatic temperature control; (2) a circulation chamber for
contacting a solid specimen with the bath fluid under controlled flow conditions;(3) a
pump to circulate the bath fluid from the thermostat through the circulation chamber; (4) a
mercury thermometer; (5) copper-constantan thermocouples connected to a digital
thermometer; (6) a stop watch; and (7) solid test specimens of various shapes and
materials, each with a copper-constantan thermocouple inserted at its center and mounting
brackets attached for suspending it in the circulation chamber.
The test specimens are shown schematically in Figure A.2-2. Physical properties
and dimensions of the materials used are given in Table A.2-1. A diagram of the apparatus
is given in Figure A.2-3.
Experiment A.2
Thermal Conductivity of Solids Page A.2-6
ChE 324 Lab Manual
Table A.2-1. Physical properties and dimensions of the specimens used in Experiment A.2
Material of Density, ρ Heat Thermal Specimen Shape
Construction (lbm/in3) Capacity, CP Conductivity, k and size
(Btu/°F·lbm) (Btu/hr·ft·°F)
Aluminum Bronze 0.274 0.170 41 Sphere: D = 3.0 in.
Experiment A.2
Thermal Conductivity of Solids Page A.2-7
ChE 324 Lab Manual
Procedure
1. Turn on the thermostat, if it has not been done, and set the bath temperature at 60o C.
2. Turn on the circulation pump, and check that there is flow through the test chamber.
3. Ensure that the bath has reached the pre-determined temperature and that it is holding
constant.
4. Inspect the digital thermometers and make sure that they are all reading room
temperature properly. Note any offset observed with a particular thermocouple.
5. Choose a particular geometry for your experiment, slab, cylinder, or sphere. Check the
dimensions, and note the materials of the solid test specimens that will be used. See
Appendix 7.
6. Just prior to placing a test specimen in the circulation chamber, measure the bath
temperature with the mercury thermometer, and record the temperature at the center of the
test specimen.
7. Place the test specimen in the circulation chamber and record the temperature history at
the center of the specimen by recording both time and temperature as the solid heats up.
Experiment A.2
Thermal Conductivity of Solids Page A.2-8
ChE 324 Lab Manual
This may be done most conveniently by choosing a temperature, using the stopwatch to
record when that temperature is reached, and then selecting the next temperature to be
recorded. Be careful not to turn off the stopwatch when interval readings are taken. Take
as many readings as possible at the beginning of a run while the temperature is changing
most rapidly. Later the data may be taken in a more leisurely manner. Take data until the
solid centerline temperature reaches at least 95% of its ultimate change, that is, until Θ ≤
0.05 at η=0.
8. Just after the measurement of the temperature history at the center of the test specimen
has been completed, once again measure the bath temperature with the mercury
thermometer.
9. Repeat this procedure with two known materials (iron or steel and copper or brass) and
with the unknown (nylon), all of the same shape and size.
Note: Initial filling and any make-up of water lost from the bath should be done with hot
tap water to minimize bath-temperature recovery time.
Data Analysis
The logic of this experiment is that the thermal conductivity of nylon, presumably
unknown, can be determined in an apparatus in which the rate of heating of a nylon object
can be observed. We have a theoretical model for the rate of heating of simple solid shapes
that relates the changing temperature of the solid body to the thermal conductivity of the
solid. There is a complication, however, in that the rate of heating also depends on the
efficiency of the fluid in transferring heat to the surface of the solid. This efficiency
depends on the properties of the bath fluid, the intensity of fluid mixing, and the geometry.
These factors are characterized by the parameter h.
In order to calibrate the apparatus, that is, to determine the value of h for the bath
being used here, one observes heating rates with one or two materials of the same shape
and size but with known thermal properties.
Calculations:
1. For each of the known materials tested, plot the temperature-time data in
dimensionless form on a semi-log graph, according to Equation A.2-16. Obtain the value
of β1 from the slope of the linear region and calculate the Biot number, hL/k, from the
appropriate equation (A.2-15,18, or 20) depending on the shape. Calculate the value of h.
The fluid-film heat-transfer coefficient, h, is characteristic of the water bath, its flow rate,
and of the shape of the solid body, but it is independent of the thermal properties of the
solid. Therefore, the solid specimens should yield similar values of h.
2. After you have fit the semi-log plots of your temperature data for the two known
materials and estimated the corresponding values of β1 (and Bi) from the slopes of the
linear regions, estimate the values of the intercept ln A1 in Equation A.2-16 calculated
from β1 and the truncated theoretical model. This theoretical intercept may be compared
Experiment A.2
Thermal Conductivity of Solids Page A.2-9
ChE 324 Lab Manual
with the intercepts of your linear fit of the data for each known material in order to check
for consistency.
2 Bi sec β 1
A1 = (A.2-24)
Bi ( Bi + 1) + β 12
for the slab,
2 Bi
A1 = (A.2-25)
[β 2
1 ]
+ Bi 2 J 0 ( β 1 )
for a cylinder, and
2 Bi β 1
A1 = (A.2-26)
( β + Bi 2 − Bi ) sin( β 1 )
2
1
for a sphere. Values of the Bessel function Jo(β1) are tabulated in Appendix 6.
3. From the temperature measurements on the nylon object, make an initial estimate of k
by comparing the temperature-versus-time data with the theoretical form. Again, use only
the data falling in the linear region of the semi-log plot. Because we can guess that the
thermal conductivity of this polymer is relatively small, we can get an initial estimate of k
by assuming that the Biot number is very large, that is, k/hb ≅ 0, and use the corresponding
theoretical value of β1. (If this approximation turns out to be a good assumption, we would
not need to determine the actual value of h from measurements with known materials.)
4. Refine your estimate of k of nylon by accounting for the effect of a finite Bi on β1, that
is, the effect of the finite resistance to heat transfer in the water. This could be done by
iterative calculations, starting with the initial estimate of k obtained in the previous step
and successively revising the values of β1, α, and k until a good fit of the data is obtained.
More conveniently, you may solve for k directly by noting that you have two independent
expressions that may be solved simultaneously for β1 and k with nylon. The procedure is
the following.
First, you have in Appendix 6 or Figure A.2-1 a relation between β1 and Bi into
which you can substitute the definition of Bi from Equation A.2-13 and the known value of
h to obtain β1 as a function of k. Also, you can plot the dimensionless temperature
calculated from the experimental data with nylon versus real time. According to Equation
A.2-16 and the definition of τ in Equation A.2-6, the slope of the linear region of such a
plot, called m, will be
m = − β 12 (α / b 2 ) (A.2-27)
Substituting the definition of α into Equation A.2-27 yields a second relation between β1
and k. Simultaneous solution of the two relations gives the values of k and β1 for the nylon.
Experiment A.2
Thermal Conductivity of Solids Page A.2-10
ChE 324 Lab Manual
The two relations may be solved graphically by plotting them both as curves of β1 versus k
and determining the location of their intersection.
References
Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Second Edition, John
Wiley and Sons, Inc., New York (2002).
Carslaw, H.S., and J.C. Jaeger, Conduction of Heat in Solids, 2nd edition, Oxford
University Press, London (1959), pp. 121-124
Jacob, M., Heat Transfer, Volume 1, John Wiley and Sons, Inc., New York (1949), pp.
270-287
Kreith, F., Principles of Heat Transfer, International Textbook Company, Scranton (1958),
pp. 137-145
Experiment A.2
Thermal Conductivity of Solids Page A.2-11
ChE 324 Lab Manual
Part B
Measurement of Profiles of Velocity, Temperature, and Concentration
ChE 324 Lab Manual
Experiment B.1
VELOCITY PROFILES IN STEADY TURBULENT FLOW
When the velocity is fast relative to the fluid viscosity, fluid flow becomes
unstable with respect to various disturbances, and the streamlines experience
instantaneous fluctuations in both magnitude and direction called turbulent eddies. For
pipe flow this usually occurs when the Reynolds number, Re = D<vz>ρ/µ, exceeds 2100.
The fluctuations in velocity provide an additional mechanism for momentum transport
across the time-averaged streamlines, with the result that the rate of momentum transport
to a solid wall is increased. Also, the enhancement of the momentum flux by turbulent
eddies modifies the time-smoothed velocity profile of the fluid; in regions of effective
turbulent transport steep velocity gradients are not needed to drive momentum transport
by viscous forces. Thus, the time-smoothed velocity profile can be flatter than the
corresponding laminar flow field. On the other hand, near a solid wall, where the
velocity fluctuations are blocked or damped out, viscous forces must carry the
momentum flux into the wall. A laminar sublayer exists near the wall in which the
velocity gradient becomes steep according to Newton's law of viscosity.
The principle of conservation of momentum and the associated equations of
motion, are still valid in the case of turbulent flow. When the equations are averaged
Experiment B.1
Velocity Profiles in Steady Turbulent Flow Page B.1-1
ChE 324 Lab Manual
over a time period that is long compared with the frequency of the turbulent fluctuations
one obtains equations for the time-smoothed velocity profiles. As shown by Bird et al.
(2002, §5.2),
Time-smoothed (∇ • v) = 0 (B.1-1)
equation of
continuity
Time-smoothed Dv (B.1.2)
equation of motion ρ = −∇p − [∇ • τ ( v ) ] − [∇ • τ ( t ) ] + ρg
Dt
These equations contain, in addition to the usual viscous transport and inertial
terms, extra terms that arise from the mixing effects of the eddies. The over bars indicate
time-smoothed quantities. The extra terms are identified as the turbulent momentum flux,
τ (t ) , and called the Reynolds stresses. If one could identify a general relation between
the turbulent momentum flux and the time-smoothed velocity gradient for turbulent
flows, in a form analogous to Newton's law of viscosity, then one could solve the time-
smoothed equations of motion to obtain the averaged velocity profile and the shear stress
on the walls.
A number of empirical relationships have been proposed to describe the turbulent
momentum flux; Bird et al. (2002, §5.4) summarizes some of them. The empirical
relations of Prandtl and Diessler were combined with the time-smoothed equation of
motion and experimental data to yield the so-called Universal Velocity Profile, which
agrees closely with experimental data for the time-smoothed velocity distributions in
pipes at Reynolds numbers greater than 20,000. The profile has three identifiable regions:
the laminar sublayer, the buffer layer, and the turbulent core. The three following semi-
empirical expressions and ranges given by McCabe et al.(2001) describe the profile very
closely:
vz
v+ = (B.1-6)
v*
Experiment B.1
Velocity Profiles in Steady Turbulent Flow Page B.1-2
ChE 324 Lab Manual
where v * = τ o ρ and τ 0 is the normal shear stress (or momentum flux) at the wall.
Also,
yv * ρ
y+ = (B.1-7)
µ
The Universal velocity profile is plotted, along with numerous experimental data
for turbulent flow in pipes, in Figure 5.5-3 of Bird et al. (2002). Although these
equations and the corresponding plot fit data on the turbulent velocity profile in pipes at
high Reynolds numbers, an awkward aspect of this approach is that Equation B.1-5 (and
the corresponding graph) do not recognize the existence of the centerline of the pipe,
where the velocity profile should be flat. The pipe radius R does not appear in these
correlations because they focus on the effect of the wall, namely the shear stress τ0, on
the structure of the turbulent boundary layer.
The average shear stress at the wall can be determined from a macroscopic force
balance on the pipe. For steady flow in a horizontal pipe the shear stress on the wall
balances the net pressure force acting axially on the fluid. That is,
2 π RL τ 0 = π R 2 ( p0 − p L ) (B.1-8)
τ0 =
( p0 − p L ) R (B.1-9)
2L
17
vz r
= 1 − (B.1-10)
v z ,max R
Schlichting (1951) has broadened the applicability of Equation B.1-10 by letting the
exponent be an empirical function of Reynolds number. That is, he proposed the
following empirical equation to describe the velocity distribution for steady flow in round
tubes:
1/ n
r
v z = v z,max 1 − (B.1-11)
R
Experiment B.1
Velocity Profiles in Steady Turbulent Flow Page B.1-3
ChE 324 Lab Manual
Re 4 x 103 7.3 x 104 1.1 x 105 1.1 x 106 2.0 x 106 3.2 x 106
n 6.0 6.6 7.0 8.8 10 10
One of the simplest methods of measuring point velocities within a flowing fluid
is with an impact tube, also called a pitot tube, which is described by McCabe et al.
(2001). By conversion of kinetic energy head to static pressure head at the mouth of a
tubular probe, the undisturbed velocity in an impinging streamline can be related to the
rise in pressure within the impact tube above the static pressure in the fluid at the point of
impact. When this pressure difference is measured by a manometer, the local velocity of
the fluid impacting the mouth of the tube vn is related to the manometer reading by the
relation
(ρ − ρ b )
1/ 2
2∆pg c
v n = 2 g ∆h sinθ a = (B.1-12)
ρ H 2O ρH O
2
where vn is fluid velocity normal to the mouth of the tube, ∆h is the differential length
reading on the manometer scale, θ is the angle of the manometer relative to the horizon,
and the subscripts in the density-difference term refer to the heavier manometer fluid (a)
and to the lighter fluid (b) above it. The second expression can be used when the
pressure drop is measured directly using an electronic transducer.
When the manometer is damped, as it is in this experiment, and the impact tube is
aligned with the pipe axis, Equation B.1.12 may be used to relate the time-averaged value
of the differential reading of the manometer, ∆ h , to the time-averaged axial point
velocity, v z (r ) . Be careful that the calculated velocities are dimensionally correct.
Apparatus
The apparatus for this experiment is illustrated in Figures B1-1 and 2. The
equipment consists of
1. A test section of cylindrical pipe that is equipped with two piezometer rings for
measuring the local static pressure and a traversing impact tube with a static pressure
tap. (Inside diameter of test section = 1-1/16 inches; distance between piezometer
rings = 3 ft; length of test section before impact tube = 5 ft; length of test section
Experiment B.1
Velocity Profiles in Steady Turbulent Flow Page B.1-4
ChE 324 Lab Manual
after impact tube = 1 ft.) The configuration of the impact tube is shown in Figure
B.1-1. A scale on the probe-positioning mechanism is graduated in tenths of an inch.
2. Two manometers (24-inch air-over-water, and 15-inch water-over-mercury). The
manometer board can be oriented at several angles relative to the horizontal in order
to amplify the ∆h reading.
3. A source of clean water and a 55-gallon galvanized steel drum supply reservoir.
4. A scale, collection container, and stopwatch for measuring the mass flow rate of
water.
5. A thermometer
6. Auxiliary piping (1.5 inch, Schedule 40, galvanized iron pipe) as shown in Figure
B.1-2. Valve V-10 controls the flow rate through the pipe.
7. A centrifugal pump driven by a 1.5 horsepower, 60 cycle, 220 volt, 3-phase electric
motor at 1800 rpm. The pump is rated to deliver 45 gal/min under 25 ft of liquid
head at 1750 rpm.
Figure B.1-1. Diagram of the test section and traversing impact tube.
Experiment B.1
Velocity Profiles in Steady Turbulent Flow Page B.1-5
ChE 324 Lab Manual
Procedure
Measure the time-smoothed velocity profile in the pipe at three Reynolds numbers
according to the following procedure.
1. Locate all the valves and become familiar with the operation of the equipment,
particularly that concerning functions of the manometers and the pressure taps on the
test section. Check and record dimensions of the apparatus, and measure the water
temperature.
a) Start with all valves closed except V-10, which is to be fully opened. Let
water flow through the test section.
b) Turn valves V-1 and V-2 to the “piezometer” position.
c) Open valves V-3, V-4, and V-5. When no more air bubbles are visible in the
manometer, close V-3, V-4, and V-5.
Experiment B.1
Velocity Profiles in Steady Turbulent Flow Page B.1-6
ChE 324 Lab Manual
d) Open V-8 and V-9. When no more air bubbles are visible in the
transparent plastic tubing, close V-8 and V-9.
e) Turn valves V-1 and V-2 to the “impact tube” position.
f) Open valves V-3, V-4, and V-5. When no more air bubbles are visible in the
manometer, close V-3, V-4, and V-5.
g) Open V-8 and V-9. When no more air bubbles are visible in the
transparent tubing, close V-8 and V-9.
3. Adjust the height of the water columns in the air-over-water manometer as follows:
a) Start with all valves closed. There is to be no water flowing through the test
section, but the test section is to be full of water.
b) Open valves V-8 and V-9.
c) If the water columns are too high on the manometer scale, open V-6 and turn
V-1 and V-2 to the “piezometer” position. Inject air into the top of the
manometer by means of the pressurizing bulb. When the heights of the water
columns have been adjusted to the desired level, close valves V-1, V-2, and
V-6.
d) If the water columns are too low on the manometer scale, turn on the pump,
open V-7 and turn V-1 and V-2 to the “piezometer” position. Slowly open V-
10. When the heights of the water columns have been adjusted to the desired
level, close valves V-1, V-2, and V-7.
e) Close valves V-8 and V-9.
4. Turn on the pump, and adjust the flow rate through the test section to the approximate
flow rate desired. Determine the actual mass flow rate by collecting and weighing the
discharge from the test section. It is suggested that you make runs with valve V-10
fully open and with it set approximately at two-thirds and at one-half of fully open.
The by-pass valve on the pump should remain closed. Also, the water level in the
reservoir should be kept fairly constant and above the entrance of the outlet line.
5. Determine the rates of momentum transfer in the test section by measuring the
pressure drop between the piezometer rings. Valves V-1 and V-2 should be set to the
“piezometer” position. Record no manometer readings when the manometer
differential is less than two inches. In order to get reasonable accuracy with a
significant ∆h value, either adjust the angle of the manometer or switch from the
water-over-mercury to the air-over-water manometer by closing valves V-3 and V-5
and opening V-8 and V-9. On the other hand, be careful not to blow mercury from the
manometer with a large pressure difference.
6. Determine the fluid velocity at a radial position in the test section by measuring the
pressure differential between the impact tube and the static pressure tap for various
positions of the probe. (Turn valves V-1 and V-2 to the “impact tube” position).
Again, take no manometer readings when the differential is less than two inches.
Also, be certain to allow ample time for the flow and manometer reading to become
stable. Record the position of the probe for each pressure measurement.
Experiment B.1
Velocity Profiles in Steady Turbulent Flow Page B.1-7
ChE 324 Lab Manual
7. Perform Step 6 for several radial distances from the center of the pipe. Make
measurements on both sides of the centerline of the pipe to obtain a complete velocity
profile. In particular, take readings at the extreme positions of the probe, close to the
walls, in order to identify the relationship between the scale on the probe-positioning
device and the coordinates within the pipe. Do not, however, use excessive force in
positioning the probe against the pipe wall.
9. Perform Steps 4 through 8 for at least three different flow rates. (It is suggested that
the first measurement be made at the highest attainable water rate and that subsequent
measurements be made at successive reductions of the flow rate.)
Data Analysis
1. For each volumetric flow rate:
o Calculate τo from the pressure drop measurement using Equation B1-9.
o Calculate the mass flow rate and average water velocity from the timed
collection and/or the electronic flow meter data.
o Calculate the velocity profile and plot it as v z (r ) v z ,max versus r/R. Plot
the profiles for all flow rates on the same graph.
o Integrate the velocity profile to compute the flow rate of water. Calculate
the corresponding average velocity < v z > and the Reynolds number.
Tabulate the mass flow rate obtained from weighing and from this
integration and discuss the differences.
o Fit the experimental local velocity data to Equation B.1-11 and determine
the apparent value of n; compare the results with values reported in Table
B.1-1. Calculate the average velocity in the pipe for each flow rate from
the resulting equations, and compare with those determined by weighing.
2. For the largest flow rate only, plot the velocity profile in the form
v + versus y + using the calculated value of τo. In the same graph, plot the
universal velocity profile using Equations B.1-3, 4, and/or 5, as appropriate, or
Figure 5.5-3 of Bird et al. (2002). Discuss the results.
3. Check whether the shear stress and pressure drop can be calculated from
Equation B.1-11.
Experiment B.1
Velocity Profiles in Steady Turbulent Flow Page B.1-8
ChE 324 Lab Manual
References
Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Second Edition,
John Wiley and Sons, Inc., New York (2002).
McCabe, W.L., J.C. Smith, and P. Harriott, Unit Operations of Chemical Engineering,
6th Edition, McGraw-Hill, Inc., New York (2001), pp. 105-106, 232-234
Prengle, R.S., and R.R. Rothfus, Ind. Eng. Chem., 47, 379-386 (1955).
Schlichting, H., Boundary Layer Theory, Fourth Edition, McGraw-Hill Book Co., Inc.,
New York (1960), pp. 504-506.
Experiment B.1
Velocity Profiles in Steady Turbulent Flow Page B.1-9
ChE 324 Lab Manual
Experiment B.2
TEMPERATURE PROFILES IN SOLID RODS
When heat flows through a solid body by conduction, Fourier's law gives the
local rate of heat flow. As shown Experiment A.2, it is often possible to predict
temperature profiles within the body for both steady and unsteady-state conditions.
Solution of the differential equation that expresses conservation of energy within the
solid to obtain the temperature profiles depends, however, on the thermal boundary
conditions imposed upon the body. At a surface, possible boundary conditions include
specification of the temperature, specification of the heat flux, or knowledge of some
fixed relation between temperature and the flux, such as Newton's law of cooling.
In Experiment A.2, the flow of heat was essentially one dimensional, and
Newton's law of cooling was used to represent the resistance of heat flow in the fluid
phase, which lay in series with the resistance of the solid itself. By calibrating the
apparatus to obtain the magnitude of the heat-transfer coefficient in the fluid adjacent to
the solid surface, one was able to calculate the thermal conductivity of a solid material
from the observed temperature changes. Experiment B.2 is similar in nature except that
the thermal conductivities of the solid materials used are all known, and the heat transfer
to the fluid is by free--not forced--convection. Although the flow of heat here is actually
two-dimensional, a one-dimensional approximation is used to describe the temperature
profiles.
In this experiment, long, solid metal rods of different materials and diameters are
heated by steam at one end. Heat flows down the rod and it is transferred to the
surrounding air by free convection. Initially, the heat flow is in an unsteady state, while
the rods are heating up. Eventually, when the rate of input of heat from the steam is
balanced by the loss of heat to the air, the temperature profiles in the rods reach a steady
state. A simplified theoretical model of this process is presented. Measurements of the
axial temperature profiles allow comparison with the theoretical model and estimation of
the fluid-film heat-transfer coefficient for the air surrounding the horizontal cylinders.
The situation studied in this experiment is representative of heat-exchanger fins,
that is, masses of metal attached to heat-exchanger surfaces to provide increased surface
area for transfer of heat from a solid to a fluid. Such structures enhance the overall rate
of heat transfer if the resistance to heat flow in the solid is less than that in the fluid film.
Theory
A long, solid rod of uniform circular cross section with radius R, and initially at a
uniform temperature equal to that of the surrounding air, is suddenly heated at one end by
a constant-temperature heat source. Assume that the following approximations are
acceptable: The physical properties of the solid are independent of temperature, the rod is
very long, a temperature gradient ∇T exists only in the axial direction, and the fluid-film
heat-transfer coefficient, h, of the air on the cylindrical surface is independent of position.
Experiment B.2
Temperature profiles in solid rods Page B.2-1
ChE 324 Lab Manual
The β factor has dimensions of (time)-1 and should no be confused with the eigenvalues
used in Experiment A.2.
Note that treating the problem as a one-dimensional conduction problem has put
the heat-transfer coefficient, which normally appears in a boundary condition, into the
differential equation. Thus, the last term in Equation B.2-1 accounts for the local loss of
energy from the metal to the surrounding air.
The appropriate boundary conditions for this problem are the following:
T = Ta for t ≤ 0 and all z (B.2-4)
T = Ts for z = 0 and t > 0 (B.2-5)
T = Ta for z → ∞ (B.2-6)
where Ts is the temperature of the energy source at the base of the rod. The last condition
applies if the rod is long enough that its temperature far from the source remains
unaffected and in equilibrium with the air.
One may verify that the solution to this problem, given by Carslaw and Jaeger
(1959), is
T (t , z ) − Ta 1 z β /α
z β /α
z
= e erfc + β t + e − z erfc − β t (B.2-7)
Ts − Ta 2 4α t 4α t
Experiment B.2
Temperature profiles in solid rods Page B.2-2
ChE 324 Lab Manual
Figure B.2-1 shows a plot of the error function and the complementary error function.
1.2
0.8
erf(x), erfc(x)
erf
0.6
0.4 erfc
0.2
0
0 0.5 1 1.5 2 2.5 3
x
Figure B.2-1. Error function and complementary error function. The error function
starts at 0 and asymptotically approaches 1, whereas the complementary error
function starts at 1 and asymptotically approaches 0.
At steady state, Equation B.2-7 becomes simply
T ( z ) − Ta β /α
= e−z (B.2-9)
Ts − Ta
The instantaneous rate of total heat discharge by the rod to the air Q may be
computed either by applying Fourier's law at the base of the rod,
∂T
Q = −π R2 k (B.2-10)
∂z z =0
Q = 2π R ∫ h (T − Ta ) dz (B.2-11)
0
The solution for T(t,z) from Equation 7 may be used in either of the latter two equations.
In general, the result is
e −β t β
Q (t ) = π R 2 k ( Ts − Ta ) + erf ( β t ) (B.2-12)
π α t α
Experiment B.2
Temperature profiles in solid rods Page B.2-3
ChE 324 Lab Manual
Figure B.2-2 illustrates the apparatus for this experiment. The equipment consists
of:
1. A steam chest that serves as a constant-temperature heat source. The steam chest is
connected to the low-pressure steam line by a quick-release flexible hose. The steam
pressure should remain constant at about 8-10 psig with the main valve completely
open and the steam trap operating properly.
2. Three metal rods mounted horizontally with one end inserted into the steam chest.
Two of the rods are made of aluminum (Alloy 2011-T3), and one is made of steel
(Type 304 SS). The aluminum rods have diameters of 0.5 and 1.0 inch; the steel rod
has a 1.0-inch diameter.
3. Copper-constantan thermocouples installed on the centerlines of the rods at specific
axial distances from the steam chest. The locations of the thermocouples measured
from the first thermocouple are 0.1, 0.3, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5, and 3.0 feet.
4. Either a portable precision potentiometer with manual switches to record
thermocouple readings or a computer system interfaced with the equipment to log
temperature measurements from the array of thermocouples as functions of time.
5. A copper-constantan thermocouple probe designed to measure surface temperatures.
6. A stopwatch.
The properties of the metal rods are given in Table B.2-1.
Experiment B.2
Temperature profiles in solid rods Page B.2-4
ChE 324 Lab Manual
Figure B.2-2. Diagram of the Apparatus for Measuring Temperature Profiles in Solid
Rods
Procedure
Measure the transient and steady-state temperature profiles in all three rods
according to the following procedure.
Experiment B.2
Temperature profiles in solid rods Page B.2-5
ChE 324 Lab Manual
Only two switches on the virtual instrument panel are used to record the data:
a) The "Run" arrow ( ⇒ ) on the left side of the tool bar starts data acquisition.
b) The "Enable/Off" green switch stops acquisition and saves the data to a file.
The default sampling-time interval is 30 seconds, which should be adequate for this
experiment.
To become familiar with the data acquisition system, check the rod temperatures
(ambient) sampled by the computer before the steam is turned on. Start acquisition
by clicking on the Run arrow, and stop acquisition after a few minutes by clicking on
the green Enable/Off switch. DO NOT CLICK ON THE STOP ICON AS THIS
WILL PREVENT THE DATA FILE FROM CLOSING PROPERLY AND ALL
YOUR DATA WILL BE LOST. A prompt for a filename will appear within 2
cycles of the sampling interval (default is 2 x 30 sec). Save the initial thermocouple
readings to a file on a diskette. Temporary data storage in the Labview directory on
the C: drive is permissible, but data will be erased weekly.
After the ambient temperature profile has been recorded in a file, it is safest to
close and re-open the LabVIEW program file (“324 Expt. B.2 - Temp Profiles”) so
that all settings are re-initialized.
4. Measure and record the lengths of the rods and the distance of the closest
thermocouple position from the outside of the steam chest. Measure the positions of
the other thermocouples to check the values given above.
5. Open the steam valve and monitor the pressure in the steam chest. Start the computer
program by clicking on the Run arrow just after the steam is introduced. Wait until
steady state has been reached in all of the rods before stopping the data acquisition.
(This may take ~ 2 hours for the stainless steel rod.)
During the data-acquisition period it is very important that the computer program
is not interrupted or stopped because the time scale will subsequently be incorrect.
When steady state has been reached, the data may be saved to a spreadsheet-
compatible file by clicking on the Enable/Off switch. It may take up to 2 cycles of
the sampling time for a prompt to appear. DO NOT "CANCEL" THE FILENAME
PROMPT WHILE WAITING FOR THE FILE TO BE SAVED, OR ALL
DATA WILL BE LOST.
During data acquisition it is possible to change the scaling on the graphical
display or to scan through the data appearing in spreadsheet format on the screen. Be
careful not to accidentally interrupt the acquisition process.
6. Record the steam pressure in the steam chest, and look up the corresponding
saturation temperature for steam. This is the value to be used as Ts.
7. With the surface-temperature probe, attempt to measure the surface temperatures on
the rods at several locations to see if they differ significantly from the interior
temperatures.
8. Observe and record once again the air temperature in the room and the steam pressure
to confirm that the boundary-condition temperatures have not changed significantly
Experiment B.2
Temperature profiles in solid rods Page B.2-6
ChE 324 Lab Manual
Data Analysis
From the steady-state temperature profiles, one can estimate average values for
the air-film heat-transfer coefficient on each rod. Plot the steady-state profiles on a semi-
log graph, according to Equation B.2-9, to estimate the values of h. If the measured rod
temperatures do not seem to approach the steam temperature Ts at the edge of the steam
chest, adjust the nominal location of the origin of z by choosing z=0 to be the position
where the steady-state profile extrapolates to Ts. This adjustment may be a bit different
for each rod because of different conduction effects within the steam chest itself.
Estimate the steady-state flow rate of heat Qss from the steam chest to each rod by
dT
estimating from the data the temperature derivative and using Equation B.2-10.
dz z =0
Compare these values with the results calculated from Equation B.2-11 and the estimated
values of h.
Using the temperatures of the rod and the air at locations 2, 5, and 8, calculate the
predicted local heat transfer coefficient. Equation B.2-14 shows the correlation between
the Nusselt,Grashof, and Prandtl numbers for free convection (Bird et al., 2002, §14.6).
0.671
Nu = 0.772 (Gr Pr)1 / 4 (B.2.14)
[
1 + (0.492 / Pr) 9 / 16 ]
4/9
Cpµ f
Pr = (B.2-16)
kf
D 3 ρ 2f gβ f ∆T
Gr = (B.2-14)
µ 2f
Experiment B.2
Temperature profiles in solid rods Page B.2-7
ChE 324 Lab Manual
Note that βf in Equation B.2-14 is the coefficient of thermal expansion for air (1/Tf
if ideal gas is assumed), not the convection factor defined in Equation B.2-3. The
subscript f indicates that the properties are evaluated at the “film” temperature, i.e., the
arithmetic average of the metal surface and air bulk temperatures.
Test the consistency of your data and the validity of the theoretical model by
comparing the transient temperature data for each of the three rods with values predicted
by Equation B.2-7 and the values of h obtained from the steady-state temperature
profiles. Specifically, plot the experimental dimensionless temperatures at the location of
the third thermocouple, counting from the steam chest, as a function of β t , and show
the theoretically predicted curve on the same plot.
The equations developed in the Theory section have been programmed in a
Mathcad sheet, and it is presented as Appendix 8. Students may copy this Mathcad
program from the Appendix on the course web page and use it as a template to calculate
the theoretical temperature profiles and heat flows for various values of h.
As an optional exercise, one might estimate the local heat flux from the rod
surfaces by radiation. Especially at higher temperatures radiation can be a significant
mechanism for heat transfer that acts in parallel with conduction and convection.
Approximate calculations can indicate whether this mechanism is significant a particular
situation and should be considered in the process model; see for example Bird et al.
(2002, §16.5).
References
Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Second Edition,
John Wiley and Sons, Inc., New York (2002)
Carslaw, H.S., and J.C. Jaeger, Conduction of Heat in Solids, Second Edition, Oxford
University Press, London (1959), pp.133-135.
Experiment B.2
Temperature profiles in solid rods Page B.2-8
ChE 324 Lab Manual
Experiment B.3
CONCENTRATION PROFILES IN A STAGNANT GAS
Experiment B.3
Concentration profiles in a stagnant gas Page B.3-1
ChE 324 Lab Manual
maintains a concentration xA0=0. The column of gas is thus viewed as a stagnant film of
B across which A diffuses.
As Bird et al. (2002, p. 543) show, Equation B.3-2 gives Fick’s law for a binary
mixture
dx A
N Az = − c DAB + x A ( N Az + N Bz ) (B.3-2)
dz
where c is the molar density of the gas mixture.
Because component B is insoluble, its flux at the gas-liquid interface must be
zero, and since the flux is a constant (steady state), it must therefore be zero everywhere:
N Bz = 0 (B.3-3)
Therefore, Equation B.3-2 becomes
c DAB dx A
N A0 = − (B.3-4)
(1 − x A ) dz
Experiment B.3
Concentration profiles in a stagnant gas Page B.3-2
ChE 324 Lab Manual
= (B.3-5)
1 − x A0 1 − x A0
which becomes linear as xA0 gets small, thus
z
lim x A ( z )= x A0 1 − (B.3-6)
x A →0 L
Equation B.3-6 is analogous to the temperature profile in a similar situation.
From combination of Equations B.3-4 and 5 the flux of vapor is given by
c DAB 1 c DAB
N A0 = ln = ( x A0 ) (B.3-7)
L 1 − x A0 L ( x B ) ln
where (xB)ln, the log mean of the mole fraction of B across the film, is defined as
x BL − x B 0
( x B ) ln = (B.3-8)
ln BL x
x
B0
In the limit of very small liquid volatility, (xB)ln→1.0, and the flux becomes
cD AB
lim N A0 = x A0 (B.3-9)
x A →0 L
which is analogous to the comparable heat-conduction result because the convection term
in Equation B.3-2 becomes negligible as the solute concentration becomes very small.
The unsteady-state diffusion in this geometry, for the case where the
concentration of vapor A in the gas is initially zero and the time is short enough that the
concentration xA at z=L is not yet significantly different from zero, is analyzed in Bird et
al. (2002, §20.1). As a simplifying mathematical approximation, the top of the column is
taken to be at z=∞. The result for the concentration profile is given in Equation B.3-10:
x A ( z , t ) 1 − erf ( Z − ϕ )
= (B.3-10)
x A0 1 + erf (ϕ )
where erf(y) is the error function and
z
Z= (B.3-11)
4 DAB t
Experiment B.3
Concentration profiles in a stagnant gas Page B.3-3
ChE 324 Lab Manual
2. A liquid reservoir in the form of a glass cup fitted with a vertical side arm to permit
observation of the liquid level in the cup.
3. A gas-storage cylinder providing pure nitrogen gas to be used as the inert gas phase
and a rotameter for monitoring the gas flow rate;
4. Pure liquid acetone for use as the evaporating material;
5. A gas chromatograph with an integrating chart recorder and gas-sampling syringes
for measuring the gas-phase concentration of acetone vapor;
Experiment B.3
Concentration profiles in a stagnant gas Page B.3-4
ChE 324 Lab Manual
Figure B.3-1. Diagram of the Gas Diffusion Cell used in Experiment B.3
Procedure
Diffusion of acetone vapor in nitrogen gas is observed by means of the following
procedure:
1. Locate all components of the equipment and become familiar with their operation,
including the gas chromatograph.
Experiment B.3
Concentration profiles in a stagnant gas Page B.3-5
ChE 324 Lab Manual
Experiment B.3
Concentration profiles in a stagnant gas Page B.3-6
ChE 324 Lab Manual
7. Stop the diffusion process by rotating, smoothly and slowly but not too slowly, the
movable section of the cell to displace and isolate the alternate sections of the gas
column. Be careful not to obstruct the outlet stream.
8. After the cell compartments have been isolated, determine the average composition
of acetone vapor in each section of the cell in the following manner:
a) Open the valve on the gas-sampling port.
b) Insert a syringe through the septum and mix the gas in the chamber by a gentle
pumping action with the syringe.
c) Slowly draw a gas sample, of about 3.0 ml, and run it in the chromatograph.
d) Run duplicate samples from several cell compartments to check the accuracy of
the GC data.
9. Measure at least one unsteady-state concentration profile. Flush the cell and set it up
for diffusion by following the same procedure as above, but close the cell and stop the
diffusion process about five minutes after introducing the liquid into the reservoir.
Data Analysis
From the chromatography data calculate the concentration profiles in the cell for
each run. Tabulate the average mole fraction of acetone in each compartment as a
function of position (and time in the unsteady-state case) by using the distance from the
gas-liquid interface to the mid-point of the cell chamber. Note that xA0 will not
correspond exactly to that based on the vapor pressure of acetone at room temperature
because the liquid surface will be cooled a bit by the evaporation process. Also, although
xAL should be close to zero because of the flushing flow of pure nitrogen through the gas
manifold, it is possible that the actual value may be small but finite.
. Extrapolate the concentration profile to the position of the interface, z=0, to
estimate xA0, and extrapolate to the screen at z=L to check xAL, by comparing the
experimental steady-state concentration profile with Equations B.3-5 and 6. This may be
done by plotting experimental values of xA versus z on rectangular coordinates along with
theoretical curves computed for various values of xA0 (with xAL = 0 as assumed in the
Theory section). Identify the value of xA0 that agrees best with your data. Calculate the
interfacial temperature that would give such a mole fraction at equilibrium.
Calculate (xB)ln to assess the magnitude of its influence in Equation B.3-7. Plot
the ratio of the acetone flux given by Equation B.3-7 to that given by Equation B.3-9 as a
function of xA0. This plot shows the relative deviation from the simple analogy with heat
conduction across a stagnant film.
With the unsteady-state data, use the concentration measured in each
compartment and Equation B.3-10 to estimate the binary diffusivity of acetone vapor in
nitrogen DAB at the prevailing temperature and pressure. This calculation is set up as a
Mathcad program given in Appendix 9. Average the values of DAB calculated, and
compare the result with experimental values found in the literature and with a theoretical
estimate calculated from, Equation B.3-13 (Bird et al., 2002, §17.3).
Experiment B.3
Concentration profiles in a stagnant gas Page B.3-7
ChE 324 Lab Manual
1 1
T +
MA MB
cD AB = 2.2646 × 10 −5 (B.3-13)
σ AB
2
Ω D , AB
In Equation B.3-13, c is the molar concentration in mol cm-3, DAB the diffusivity
in cm2 sec-1, T is the temperature in Kelvin, MA and MB are the molecular weights of the
two species, and σAB and ΩD,AB are the binary Lennard-Jones parameters given by
Equations B.3-14 and 15.
1
σ AB = (σ A + σ B ) (B.3-14)
2
ε AB εA εB
= (B.3-15)
κ κ κ
And the collision integral, ΩD,AB is a function of κT/εAB . The tabulated collision
function and the Chapman-Enskog calculation are given in Appendix 9.
For the calculation of the parameters in Equations B.3-14 and 15 one may use the
molecular parameters given in Table B.3-1 (McCabe et al.,2001).
σ, Å ε/κ, K
Nitrogen 3.798 71.4
Acetone 4.600 560.2
References
Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Second Edition,
John Wiley and Sons, Inc., New York, 2002.
McCabe, W.L., J.C. Smith, and P. Harriott, Unit Operations of Chemical Engineering,
McGraw-Hill Book Co., New York, 2001, p. 1090.
Experiment B.3
Concentration profiles in a stagnant gas Page B.3-8
ChE 324 Lab Manual
Part C
Measurement of Transport Coefficients
ChE 324 Lab Manual
Experiment C.1
FRICTION FACTORS FOR FLOW IN CIRCULAR TUBES
When a fluid flows through a pipe, the fluid exerts a drag force on the pipe wall.
This force is the manifestation of axial momentum being transmitted from the fluid to the
wall. One may view it as fluid friction because the equal and opposite force of the wall
on the fluid tends to retard the flow. The tangential drag force per unit area of fluid-solid
interface is called the shear stress, τ0. At steady state, a macroscopic force balance on a
section of straight, horizontal, circular pipe of length L and radius R indicates that the
fluid friction causes a pressure drop in the direction of flow:
( p0 − p L ) π R 2 = τ 0 2π RL , (C.1-1)
where p0 is the fluid pressure at the entrance of the length of pipe, and pL is the pressure
at the exit. Equation C.1-1 means that the force due to the upstream pressure, acting on
the fluid in the direction of the flow, is balanced by the tangential drag force and by the
downstream pressure acting in the opposite direction.
For practical purposes, it is desirable to relate τ0 to the flow rate of the fluid so
that one may use Equation C.1-1 to calculate the relationship between flow rate and
pressure drop in pipes. For Newtonian fluids, Newton's law of viscosity gives the shear
stress at the wall:
∂ vz
τ0 =−µ (C.1-2)
∂r r=R
where µ is the fluid viscosity, vz is the fluid velocity in the axial direction, and r is the
radial coordinate in the pipe. For laminar flow, one can solve for the steady-state
velocity profile vz(r) as shown in Section 2.3 of Bird et al. (2002). With that profile, one
can then calculate the total volumetric flow rate of fluid Q as well as the derivative in
Equation C.1-2. These two results are combined to obtain the Hagen-Poiseuille law for a
horizontal pipe,
π ( p0 − p L ) R 4
Q= , (C.1-3)
8µ L
which may be used to calculate either Q from (p0-pL) or vice versa.1
With turbulent flow, however, the chaotic nature of the flow and the radial
transport of momentum by random eddies make it impossible to calculate the velocity
profile from first principles. In Experiment B.1 it was seen that it is even difficult to
measure the velocity profile near the pipe wall very accurately. In this case Equation
1
For a non-horizontal pipe, one must substitute the pressure p with the expression p-ρgz to include the
effect of the static head. This quantity is designated by a script P in Bird et al.,2002.
Experiment C.1
Friction factors for flow in circular tubes Page C.1-1
ChE 324 Lab Manual
C.1-2 is of limited utility, and one must resort to empiricism in order to relate pressure
drop to fluid flow rate and to the properties of the system.
Based on the physical principles that govern fluid flow, even when the related
differential equations cannot be solved, one can define a useful quantity for correlating
experimental data on fluid flow behavior in laminar or turbulent flow and in more
complex geometries. This quantity, called the friction factor, is calculated from
measured values of pressure drops and flow rates for various geometries and fluids.
According to dimensional analysis, friction factor values may be presented in fairly
general dimensionless correlations that can be used for practical predictions of flow
behavior. In this experiment, one measures pressure drop as a function of flow rate for
water passing through horizontal pipes of several sizes. From the data a general
correlation of the friction factor is constructed.
Theory
Pressure po Pressure
r
R
D z
z= z=L
Figure C.1-1. Flow inside a pipe.
For fully developed, steady-state flow the local, time-smoothed shear stress at the
wall τ0, given by Equation C.1-2, is independent of position on the wall because the time-
smoothed velocity profile is independent of the axial distance and of the angular position.
A dimensionless friction factor f for this case may be defined as the proportionality factor
in the relation
1
τ 0 = ρ < vz > 2 f (C.1-4)
2
where <vz> is the average velocity in the pipe, defined by
Q
< v z >= (C.1-5)
π R2
This f is called the Fanning friction factor for pipe flow.
Combination of Equation C.1-4 with Equation C.1-1 yields the relation
π 2 R 5 p0 − p L
f= (C.1-6)
ρ Q2 L
Experiment C.1
Friction factors for flow in circular tubes Page C.1-2
ChE 324 Lab Manual
which allows calculation of f from experimental data on pressure and flow rate. At the
same time, combination of Equation C.1-4 with Equation C.1-2 gives another expression
for f:
∂ vz
− 2µ
∂r
f= r=R
(C.1-7)
ρ< v z > 2
which provides a theoretical form for computing f if the detailed velocity profile is
known.
Even if the velocity profile cannot be computed from basic principles, as in
turbulent flow for example, the underlying differential equations may be subjected to
dimensional analysis as shown in Section 3.7 of Bird et al. (2002). The conclusion of
such a dimensional analysis for steady, fully developed Newtonian flow in a straight pipe
is that form and that the solution of the basic equations must be a general function of the
form
v z ( r , ρ ,µ , R ) r
= v *z ( ; Re) (C.1-8)
< vz > R
Where the left hand side of equation C.1-8 is the dimensionless velocity profile. The
Reynolds number is defined as
D < vz > ρ
Re = (C.1-9)
µ
where D is the pipe diameter, D=2R.
Substituting Equation C.1-8 into Equation C.1-7 leads to the conclusion that the
friction factor function for incompressible, Newtonian fluids in a straight, smooth pipes
should be simply
f = f (Re) (C.1-10)
More generally, the friction factor can depend on dimensionless geometrical ratios. For
example, for shorter pipes in which the velocity profile is developing and τ0 depends on
the distance downstream from an entrance, the expected relationship for the average
friction factor is of the form
L
f = f (Re, ) (C.1-11)
D
Or if the pipe wall is rough with a scale of roughness of average magnitude k, the
correlation for f in fully developed flow is expected to be
k
f = f (Re, ) (C.1-12)
D
One can calculate values of f from flow data on selected fluids in straight pipes
using Equation C.1-6. Those values may then be fitted into an empirical dimensionless
correlation to obtain the dimensionless function indicated in Equation C.1-10 (or C.1-11
Experiment C.1
Friction factors for flow in circular tubes Page C.1-3
ChE 324 Lab Manual
for short pipes). According to the principle of dimensional analysis, the resulting
dimensionless function should be valid for all Newtonian fluids for pipes of all sizes.
Thus, when the flow rate is specified for a different fluid in a pipe of any size, one may
calculate the Reynolds number and estimate f from the general correlation. Finally, the
expected pressure drop in the pipe may be predicted with Equation C.1-6. A similar
approach is applied in the definition and use of heat or mass transfer coefficients, as
explained in Chapters 14 and 22 of Bird et al. (2002).
A well-known general correlation of f values has been constructed from a large
body of data and is shown in Figure C.1-2. A larger diagram is presented in Figure 5.10
of McCabe et al. (2001) and in Figure 6.2-2 of Bird et al. (2002).
0.100
0.010
f
0.001
100 1000 10000 100000 1000000
Re
On this plot is included the theoretical relationship obtained from Equation C.1-7 for
fully developed laminar flow, for which the velocity profile is given by
( po − p L ) R 2 r 2
vz = 1 − (C.1-13)
4 µL R
Also shown are empirical curves for turbulent flow in pipes of various degrees of
roughness, characterized by the scale dimension k. Although there are two possible
solutions for f at any value of Re, corresponding to both laminar and turbulent flow, the
former is stable only at Reynolds numbers lower than 2100. The turbulent-flow friction
factor is always greater than that for laminar flow because turbulent mixing enhances
momentum transport to the pipe wall. Similarly, finite values of L/D or k/D tend to
increase the value of f.
In many cases of pipe flow the flow rate is not known a priori so the Reynolds
number and friction factor cannot be calculated directly. For example, in the case where
the pressure drop is imposed by external conditions, the flow rate and hence the Reynolds
Experiment C.1
Friction factors for flow in circular tubes Page C.1-4
ChE 324 Lab Manual
number are dependent variables to be determined from the f correlations. In this case one
may rearrange the relation given by Equation C.1-10 to solve for the quantity
Dρ ( p0 − p L ) D
Re f = (C.1-14)
µ 2 Lρ
which is independent of <vz> in order to obtain a new correlation
(
f = f Re f ) (C.1-15)
For example, Figure C.1-2 may be replotted in the form of Equation C.1-15. The
quantity Re f can be calculated from a known pressure difference by means of
Equation C.1-14 and used to estimate the value of f. Then, Equation C.1-6 may be used
to predict the volumetric flow rate Q.
In the present experiment, one measures pressure drops and
flow rates for water passing through horizontal pipes of three different diameters. From
the data one may calculate values of both the Reynolds number and the friction factor f,
from which a correlation corresponding to Figure C.1-2 may be constructed.
It is instructive to compare regions of the empirical
friction-factor correlation with relations predicted from various forms of the velocity
profile according to Equation C.1-7. For example, the parabolic laminar velocity profile
yields the theoretical relation, which can be confirmed experimentally for Re<2100,
16
f= (C.1-16)
Re
If the time-smoothed velocity profile in fully developed turbulent flow is fit by the
power-law form given in Equation B.1-10 the corresponding result from Equation C.1-7
is
0.0791
f= (C.1-17)
Re1/ 4
On the other hand, if one fits the velocity profile in turbulent flow with Equation B.1-5, a
specific form of Equation C.1-15 is obtained:
1
f
[ ]
= 4.0 log 10 Re f − 0.40 (C.1-18)
Equation C.1-17 is generally valid up to Re = 105, while Equation C.1-18 fits smooth-
pipe turbulent-flow data up to Re=4x106 (Bird et al, 2002, §6.2).
Apparatus
The apparatus for this experiment is a rig mounted with three straight horizontal
pipes of two different diameters. The dimensions of the pipes are given in Table C.1-1.
The test rig is illustrated in Figure C.1-3. Each pipe has affixed a pair of piezometer rings
for measuring pressure drops between the locations of the two rings. Tubing connects the
Experiment C.1
Friction factors for flow in circular tubes Page C.1-5
ChE 324 Lab Manual
piezometer rings to a pair of 60-inch manometers, one containing air over water and the
other containing water over mercury.
Clean water is supplied to the inlet ends of the three pipes by a manifold fed by a
roller-type, positive-displacement pump, and there are valves on each line to allow flow
to pass, to connect with the manometers, and to control the flow rate, as indicated in
Figure C.1-2. The following materials and equipment should be available: containers to
collect water leaving the system outlet, a platform scale for weighing the amount of water
collected, a stop watch for measuring collection time, and a thermometer for measuring
the temperature of the water.
Experiment C.1
Friction factors for flow in circular tubes Page C.1-6
ChE 324 Lab Manual
Figure C.1-3. Diagram of the rig with three test pipes and manometers for measuring
flow rates and pressure drops in the pipes. The inlet flow comes from a positive-
displacement pump fed from a water reservoir. The outlet flow is returned to the
reservoir through a flexible rubber hose, which can also be directed into the weighing
bucket.
Procedure
The relation between pressure drop and flow rate in the pipes is observed in the following
manner.
1. Locate all valves and the pump controls. Become familiar with the operation of the
equipment, particularly the manometers but also the platform scales and stop watch.
Check the dimensions of the system, particularly the distance between the manometer
taps. Weigh the empty water-collection bucket, and record room temperature.
2. Purge the manometer lines of air by the following procedure:
a. Open valves V-2, V-3, V-4, and V-7, and let water flow through Tube A.
Experiment C.1
Friction factors for flow in circular tubes Page C.1-7
ChE 324 Lab Manual
b. Open valves V-18, V-19, and V-20 and then open valves V-12 and V-15. When
no more air bubbles are visible in the transparent sections of the tubing, close
valves V-12 and V-15.
c. Direct the flow through Tube B by opening valves V-5 and V-8 and closing V-4
and V-7.
d. Open valves V-11 and V-14. When no more air bubbles are visible, close
valves V-11 and V-14.
e. Repeat with Tube C by opening valves V-6 and V-9 and closing valves V-5 and
V-8.
f. Open valves V-10 and V-13. When no more air bubbles can be detected in the
tubing to the manometers, close valves V-10 and V-13 as well as V-20.
3. Adjust the height of the water columns in the air-over-water manometer as follows:
a. Start with all valves closed except V-3 and the valves associated with the pump.
b. Open valve V-9 on Tube C.
c. If the water levels are too high on the manometer scale,
i. Open valves V-16, V-17, V-10, and V-13.
ii. Open valve V-22 and V-6, and let the water columns fall to a desired level
near the mid-point on the scale.
iii. Close valves V-22, V-16, V-17, V-6, and V-9.
d. If the water levels are too low on the manometer scale,
i. Start the pump with the inlet-line and by-pass valves open.
ii. Open valves V-16, V-17, V-10, and V-13.
iii. Crack valve V-6 open only slightly.
iv. Open valve V-22, and let the water columns rise in both columns to a level
somewhat higher than that desired.
v. Close valve V-22.
vi. Close valve V-6.
vii. Stop the pump.
viii. Adjust the levels following the procedure given above in Step 3.c.
4. Set the approximate water flow rate desired in the following manner:
a. Start with all valves closed except the inlet and by-pass valves associated with
the pump.
b. With valve V-3 closed, open completely the valves that will allow water to flow
through one of the pipes, except for the respective control valve, V-4, V-5, or
V-6.
c. Open the valves needed to connect the appropriate manometer to the test pipe.
Experiment C.1
Friction factors for flow in circular tubes Page C.1-8
ChE 324 Lab Manual
d. Slowly open valve V-4, V-5, or V-6 and adjust it in order to set the pressure
drop in the test pipe to the level desired. Table C.1-2 contains suggested
manometer readings for which data might be taken, in order to cover a wide
range of flow rates. These values are given only as a guide; they need not be set
exactly. Set a steady flow rate in the test pipe that yields a manometer reading
corresponding approximately to a value suggested in the table. Record the
manometer’s actual, steady state reading.
5. Determine the flow rate by collecting a mass of water from the system outlet in the
weighing receptacle during a timed period. Hold the outlet hose in a fixed position,
and direct some flow into the floor drain. Then place the collection bucket under the
hose outlet without changing its position or elevation. Weigh the mass collected, and
record its weight and the corresponding collection time. Record also the water
temperature for each run.
6. Adjust the control valve (V-4, V-5, or V-6) and, if necessary, the valves in the
manometer-connection lines, to attain flow rates corresponding to the six or seven
levels indicated in Table C.1-2.
7. Close valve V-3 after the flow rate and pressure drop values have been recorded.
Close all other valves that were opened in Step 4.
8. Repeat Steps 4 through 7 for the other two pipes.
9. Turn off the pump and close the valve on the pump inlet.
An Excel spreadsheet program for data entry and calculations of flow rates and
friction factors is presented in Appendix 11. Data should be entered during the
laboratory period and a preliminary log-log plot of friction factor vs. Reynolds number
should be prepared. If there are any obvious gaps in the graph, make additional runs to
complete the correlation of f with Re.
Data Analysis
Experiment C.1
Friction factors for flow in circular tubes Page C.1-9
ChE 324 Lab Manual
References
Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Second Edition,
John Wiley and Sons, Inc., New York, 2002.
McCabe, W.L., J.M. Smith, and P. Harriott, Unit Operations of Chemical Engineering,
6th ed., McGraw-Hill Book Co., New York, 2001.
Perry, R.H., and D. Green (eds.), Perry's Chemical Engineers' Handbook, 6th ed.,
McGraw-Hill Book Co., New York, 1984.
Experiment C.1
Friction factors for flow in circular tubes Page C.1-10
ChE 324 Lab Manual
Experiment C.2
HEAT-TRANSFER COEFFICIENTS IN CIRCULAR TUBES
When a fluid flows through a pipe whose wall temperature is different from that
of the fluid, heat flows between the pipe wall and the flowing fluid, raising or lowering
the fluid's temperature. This is the situation in most practical heat exchangers. The
transport of heat in the flowing fluid occurs by both conduction and convection. If the
flow is turbulent, the dominant mechanism of transport in the bulk fluid is turbulent
mixing, that is, the convection provided by turbulent eddies. Near the pipe wall the
turbulence is damped out, and the flow of heat is ultimately carried across the solid-fluid
interface by molecular conduction. In laminar flow conduction is the mechanism of radial
transport throughout the pipe's cross section, but axial convection also affects the
temperature profiles.
In principle, theoretical methods based on Fourier’s Law and Newton’s viscosity
law could be used to calculate temperature profiles and heat fluxes within the flowing
fluid. In practice, however, especially with turbulent flow, the mathematical problem
becomes intractable, and an alternative approach must be used to calculate the
performance of heat-transfer equipment.
From a practical point of view, one usually wants to know how much the
temperature of a flowing fluid will change as it passes through a heat-transfer pipe. That
problem can be formulated in terms of a macroscopic energy balance. In many cases,
specifically at steady state when kinetic energy, potential energy, and work effects are
negligible, the macroscopic energy balance for a flowing fluid is
w ∆H = Q (C.2-1)
where w is the mass flow rate of fluid, H is its enthalpy per unit mass, and Q is the rate of
heat transfer from the wall into the fluid. If thermal effects are dominant in the enthalpy
change, if the fluid’s heat capacity, Cp, is assumed constant, and if no latent heat effects
occur, Equation C.2-1 may be approximated by
w C p ∆T = Q (C.2-2)
where T is the average temperature of the fluid. In both of these equations the symbol ∆
indicates the change in a quantity from the entrance to the exit of the system. If the value
of Q in Equation C.2-2 is known or can be estimated, one can calculate the change in the
fluid temperature as it passes through a heat-exchanger tube.
The heat transfer rate Q is defined as the integral of the heat flux over the surface
area of the solid-fluid interface, and the flux is determined by the shape of the
temperature profile at the wall according to Fourier's law (Equation A.2-1). The
difficulty in calculating the detailed temperature profile in most flow situations, however,
precludes the systematic calculation of Q from first principles. As an alternative one can
Experiment C.2
Heat-Transfer Coefficients in Circular Tubes Page C.2-1
ChE 324 Lab Manual
Theory
Cold stream in d
Tc1
d Cold stream out Tc2
z
Hot stream out Th2
Figure C.2-1. Heat transfer between two streams
Experiment C.2
Heat-Transfer Coefficients in Circular Tubes Page C.2-2
ChE 324 Lab Manual
directly with the total heat rate, Q, rather than the local rate, qw, there is necessarily an
averaging that takes place:
( )
Q = ∫∫ q w dA = ∫∫ h Tw − T f dA = have Tw − T f ( ) ave
A (C.2-4)
where A is the total interfacial area for heat transfer. The identity of have and its relation
to the local value defined in Equation C.2-3 depend on what convention one chooses in
calculating the average driving force (Tw-Tf)ave. If the local value of h is constant over
the entire surface area A, a common approximation, it may be taken out of the integral in
Equation C.2-4, and the appropriate choice for the average driving force is the
logarithmic mean (Tw-Tf)ln, defined as
(T − Tf ) − (T − T )
(T w − Tf ) ln
=
w in
(T − T )
w f out
(C.2-5)
ln
w f in
(T − T )
w f out
The other complication that arises in the definition and use of heat-transfer
coefficients is that the wall temperature is often not known. In most cases, heat transfer
occurs from one fluid to another through a solid wall. Only the bulk fluid temperatures
are readily measured. In this case it is appropriate to use an overall heat-transfer
coefficient U. Locally, at a position at some distance z downstream in the pipe the flux
across the inner wall may be written as
1 dQ
qw = = U w (Th − Tc ) (C.2-6)
2π rw dz
where rw is the inner radius of the pipe, and the subscripts h and c refer to the hot and
cold fluids.
As shown in Bird et al. (2002, Example 10.6-1), the overall coefficient, which
represents the total resistance to heat flow from the hot fluid to the cold fluid, is given by
( )
−1
1 ln rout , j / rin , j 1
U w = rw−1 +∑ + (C.2-7)
rw hi kj ro ho
j
where hi is the local film heat-transfer coefficient of the fluid inside the pipe, ho is that for
the fluid on the outside with its interface at radial position ro, and the summation accounts
for solid cylindrical layers between the fluids, each with thermal conductivity kj. The
subscript w on U indicates that it is based on the cylindrical surface area at rw. Other
choices for the reference radius are possible. Because of the cylindrical geometry, the
local radial flux and hence U vary according to the radial position where they are
evaluated. The value of total heat flow from one side of the cylindrical wall to the other
side is, however, fixed and invariant with this choice.
As with the film coefficient, there is a question of how U should be averaged
when Equation C.2-6 is integrated to get Q. Normally, one assumes that U is constant
with z. In that case, there is no distinction between local and average values of U if the
integrated rate expression is written as
Experiment C.2
Heat-Transfer Coefficients in Circular Tubes Page C.2-3
ChE 324 Lab Manual
Experiment C.2
Heat-Transfer Coefficients in Circular Tubes Page C.2-4
ChE 324 Lab Manual
Dh
Nu = (C.2-12)
k
Re is the Reynolds number
Dvρ
Re = (C.2-13)
µ
and Pr is the Prandtl number
Cp µ
Pr = (C.2-14)
k
where D is the pipe diameter, v is the average fluid velocity, k is the thermal conductivity
of the fluid, ρ is its density, µ the viscosity, and L is the length of the pipe. For long
pipes, the effect of L/D usually becomes unimportant. On the other hand, the fact that the
viscosity may be a strong function of temperature leads one to add another dimensionless
group, the ratio of the viscosity at the pipe wall to that of the bulk fluid, into some
correlations.
For laminar flow exact solutions for the temperature profiles and heat flux have
been derived for some relatively simple situations. The results can be used to obtain a
theoretical form for h, and they may be put into dimensionless form. For example, for a
short heat-transfer section in a fully developed laminar pipe flow with constant wall
temperature and constant fluid properties, the specific form of Equation C.2-11 is found
to be
. [ Re Pr ( D / L)]
1/ 3
Nuln =162 (C.2-15)
where the subscript on Nu indicates that the log-mean temperature difference is used to
define have. Other theoretical forms have been obtained for other boundary conditions
and limiting cases.
For turbulent flow correlations for h must be empirical and based on experimental
data. It is often assumed that the general form of Equation C.2-11 for relatively long
pipes can be represented by a function
Nu = a Re b Pr c ( µ / µ w )
d
(C.2-16)
where a, b, c, and d are constants, and µw is the viscosity at the wall temperature. One
important result for pipe flow with Re>20,000 is
Nu = 0.026 Re 0.8 Pr 1/ 3 ( µ / µ w )
0.14
(C.2-17)
Bird et al. (2002, p. 436) present a plot of this equation and other correlations covering a
range of Reynolds numbers from 103 to 105 for smooth pipes.
Thus, for geometries where individual film heat-transfer coefficients have been
measured and correlated in generalized form, predicted values may be used to estimate
the overall coefficient U from Equation C.2-7, and Equation C.2-8 may then be used for
heat-exchanger design or simulation. Conversely, for unknown situations, experiments
may be run to measure U. When estimates can be made for most of the components of U,
Experiment C.2
Heat-Transfer Coefficients in Circular Tubes Page C.2-5
ChE 324 Lab Manual
either from empirical correlations or theoretical considerations, then one can calculate
one unknown term, such as the h of one fluid or a fouling factor, from Equation C.2-7.
Experiments using steam simplify these calculations a bit. First of all, during
condensation at a fixed pressure, the temperature of steam remains constant. Its enthalpy
balance, Equation C.2-1, becomes
ws λ s = Q (C.2-18)
where ws is the rate of steam consumption and λs is its latent heat of condensation. Also,
as long as the condensate drains from the heat-transfer surface readily, the film heat-
transfer coefficient for steam is quite large, and its contribution in Equation C.2-7 usually
becomes negligible.
Apparatus
The apparatus for this experiment is a rig mounted with three straight horizontal
double-pipe heat exchangers. On each rig, all of the heated copper tubes have a length of
41 inches, but each has a different diameter. The dimensions of the tubes are given in
Table C.2-1. The outside pipes, which are insulated and into which steam is introduced,
all have a two-inch inside diameter, and they are fitted with a steam-pressure gauge.
Table C.2-1. Dimensions of the Heat-exchanger Tubes. All have length 41 inches.
Inside Pipe Outside Pipe
Tube Location on Test Diameter (in) Diameter
Rig (in)
Unit No. 1
A top 0.1425 0.190
B middle 0.314 0.374
C bottom 0.788 0.874
Unit No.2
A top 0.1862 0.250
B middle 0.4325 0.501
C bottom 1.025 1.125
Unit No. 3
A top 0.250 0.3125
B middle 0.666 0.750
C bottom 0.913 1.050
Experiment C.2
Heat-Transfer Coefficients in Circular Tubes Page C.2-6
ChE 324 Lab Manual
The test rigs are illustrated in Figure C.2-1. Each heat exchanger has affixed
copper-constantan thermocouples for measuring inlet and outlet temperatures of steam
and water as well as a temperature on the outside wall of the copper tubing.
Also available are digital thermometers, collection buckets, stopwatches, and
scales for determining water flow rate, a supply of low-pressure steam, and an auxiliary
system for supplying water.
The construction of the heat exchangers is shown in Figure C.2-2.
Figure C.2-1. Diagram of the Rig Showing Three Double-pipe Heat Exchangers with
Different Size Tubes for Heating Water with Steam.
Experiment C.2
Heat-Transfer Coefficients in Circular Tubes Page C.2-7
ChE 324 Lab Manual
Procedure
Measure overall heat-transfer coefficients in the three heat exchangers by means
of the following procedure:
1. Locate all valves and the pump controls. Become familiar with the operation of the
equipment. Confirm the dimensions of the system. Weigh the empty water-
collection bucket.
2. Purge air from the water lines and from the tubes of the heat exchangers by the
following procedure:
a. Open valves V-2A, V-2B, and V-2C, turn on the pump, and let water flow
through all three tubes. Control water flow rate by means of these three valves.
Keep the pump inlet valve wide open to avoid cavitation. The centrifugal pumps
are robust and can tolerate any backpressure, but the piping manifold may leak
when exposed to the maximum pump head. Keep the pump head below 100 psi
by adjusting the pump by-pass valve. At the same time do not open the by-pass
valve too wide, or the pump motor may overheat and trip the circuit breaker.
b. When there is no air remaining in the water lines, close the three water valves.
4. Open valve V-4 to clear the steam lines and jackets of condensate.
5. Test one heat exchanger at a time, in the order C, B, A, doing two runs with each.
Start up each of the heat exchangers, A, B, or C, in the following manner:
a. Open valve V-2 A, B, or C, and allow water to run through one of the tubes at a
maximum flow rate. The water flow is returned to the water storage reservoir.
Experiment C.2
Heat-Transfer Coefficients in Circular Tubes Page C.2-8
ChE 324 Lab Manual
Check the flow rate by collecting and weighing a portion of water leaving the
tube. Calculate the value of the Reynolds number. (See Appendix 12 for a
spreadsheet that facilitates this calculation.) Cover a range of Re values, greater
than 10,000, by making two steady-state runs with each tube. That is, in the
second run with each exchanger reduce the water flow rate by 25 to 50% to
obtain a lower value of Re. A lower-flow-rate run may be omitted in the
smallest tube if two different values of Re greater than 10,000 can not be
attained.
b. Open completely the steam-condensate drain valve, V-1 A, B, or C, for the heat
exchanger to be tested. Then open the corresponding steam-supply valve, V-3
A, B, or C, to allow steam to enter. Control the steam pressure, if necessary,
with the latter valve. Make sure that the purge valve V-4 is closed.
5. Determine the steady-state heat-transfer rate Q in each run with the following steps:
a. Adjust the water flow rate through the tube in order to obtain a Reynolds
number of the desired magnitude.
b. Adjust the steam pressure in the shell of the exchanger to the lowest level that
gives only a slight (2-4 oF) temperature difference between the steam inlet and
the condensate outlet. Check the measured steam pressure and temperatures
against the steam tables to confirm that all instruments are operating properly.
c. Adjust the water flow rate or the steam pressure such that the rise in the water
temperature across the exchanger is between 10 and 40oF. This range allows
reasonable accuracy in calculating the energy balance but does not introduce
great variations in fluid properties. Take care especially to avoid boiling of
water within the exchanger. That is, keep the outlet water temperature well
below 212oF.
d. When all temperatures attain steady state, record their values.
e. Measure the water flow rate again at the final steady-state conditions.
6. Avoid cavitation in the pump by keeping the inlet water temperature below 160oF as
well as keeping the valve on the water-intake line completely open. The water in the
feed tanks heats up gradually as the experiments proceed. It may be cooled by city
water in a double-pipe heat exchanger that is mounted on the back of the
experimental rig. Crack open the cold-water inlet valve to keep the recycled water
temperature in the storage tank from rising too much.
7. Shut down the operation of an exchanger by first closing the steam supply valve, V-3
A, B, or C, and then closing the condensate drain valve. Never allow steam to pass
into an exchanger unless there is water flowing through the tube. Keep all valves
closed except those for the exchanger being tested. Do not operate two heat
exchangers at once.
Data Analysis
Calculate the heat load Q and the overall coefficient Uw using Equations C.2-2
and C.2-8 with the measured temperatures and water flow rates for each steady-state run.
Calculate also the Reynolds number attained by the water in each run. An Excel
Experiment C.2
Heat-Transfer Coefficients in Circular Tubes Page C.2-9
ChE 324 Lab Manual
spreadsheet is presented in Appendix 12, and available on the web page, to facilitate the
calculations.
References
Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Second Edition,
John Wiley and Sons, New York, 2002.
McCabe, W.L., J.M. Smith, and P. Harriott, Unit Operations of Chemical Engineering,
5th ed., McGraw-Hill Book Co., New York, 2001.
Perry, R.H., and D. Green (eds.), Perry's Chemical Engineers' Handbook, 6th ed.,
McGraw-Hill Book Co., New York, 1984.
Experiment C.2
Heat-Transfer Coefficients in Circular Tubes Page C.2-10
ChE 324 Lab Manual
ChE 324 Lab Manual
Part D
Analysis of Macroscopic Systems
ChE 324 Lab Manual
Experiment D.2
EFFLUX TIME FOR A TANK WITH AN EXIT PIPE
Although there exists a theoretical framework for describing fluid flow behavior
by calculating complete profiles of velocity and pressure (Bird et al., 2002), many
practical problems cannot be solved with such a fundamental approach. One difficulty
that arises is that the geometry and boundary conditions are too complex to allow a
precise mathematical formulation and solution. The other obstacle is that the flow may
become unstable and turbulent, introducing a random component in local fluctuations that
cannot be predicted in any precise way. In such situations the fundamental laws of
physics still apply, but an alternative approach is needed to obtain descriptions with
engineering utility.
For complex geometries or flow situations it is common to adopt a macroscopic
approach rather than trying to solve the differential equations of change. Required
equations include the mass balance, the momentum balance, and the mechanical energy
balance, all applied to a general macroscopic system. A macroscopic system is defined by
the boundaries that contain a phase and, for a flow system, by control planes at the
entrance and exit points. Conservation principles are then applied to the entire volume of
the system rather than to a differential element.
The macroscopic balances are written as algebraic equations for steady-state
systems or as ordinary differential equation when an unsteady-state process is considered.
The terms that appear in the macroscopic balances are of three types: First, the input and
output terms that are evaluated at the entrance and exit planes of the system; second,
terms that express the total-transport rate across the boundaries of the system; and finally,
terms that involve a summation, i.e. an integral, of some quantity over the entire volume
of the system.
Because the purpose of the macroscopic balances is to avoid calculating in detail
the profiles of material properties, such as velocity and pressure, the calculation of the
summation terms must be done approximately. Such terms may be analyzed by
dimensional analysis, and the results combined with experimental data to develop
generalized correlations that can be used for estimates in practical engineering
calculations.
In this experiment one observes the rate of drainage of liquids from a tank through
exit tubes of various sizes. This system is analyzed with a macroscopic mass balance
combined with a macroscopic mechanical energy balance. In the later equation it is
necessary to use dimensionless correlations to estimate the loss of mechanical energy
caused by viscous dissipation within the fluid.
Experiment D.2
Efflux time for a tank with an exit pipe Page D.2-1
ChE 324 Lab Manual
Theory
Consider a vertical, cylindrical tank such as that shown in Figure D.2-1 that
contains a Newtonian liquid up to the level h above the bottom of the tank. The top of
the tank as well as the bottom of the drain pipe are exposed to the atmosphere. Thus, the
liquid drains from the tank under the influence of gravity. A model of the draining
process is sought in order to predict the rate at which the tank empties.
R L
First, one may write a macroscopic mass balance for the liquid in the system,
which is defined as the entire liquid volume from the top surface to the outlet of the pipe.
The total mass of liquid in the system is
mtot = ρ π ( R02 L + R 2 h) (D.2-1)
where ρ is the density of the liquid, R is the radius of the tank, R0 is the pipe, and L is its
length. A mass balance for the liquid contained in the system yields
d
mtot = − ρ π R02 v (D.2-2)
dt
where v is the average velocity out of the bottom of the pipe. There is no velocity term at
the top because there is no flow into the top of the tank.
Taking the density to be constant and substituting Equation D.2-1 into D.2-2, one
obtains
2
dh R
=− 0 v (D.2-3)
dt R
This equation may be integrated from the initial liquid level h1 to a subsequent level h2 to
obtain the corresponding efflux time te. That is,
2
te R h2 dh
te = ∫ dt = − ∫ (D.2-4)
0 R0 h1 v
Experiment D.2
Efflux time for a tank with an exit pipe Page D.2-2
ChE 324 Lab Manual
In order to perform the desired integration, one must know the velocity v as a function of
liquid level h. That relationship comes from the macroscopic mechanical energy balance.
If one neglects the rate of change of the total mechanical energy in the system, by
the so-called pseudo-steady-state approximation, then one can use the macroscopic
mechanical energy balance, also called the Bernoulli equation (Bird et al., 2002, p.204).
1 < v3 > p
∆ + g (h + L) + w = Wm − E c − E v (D.2-5)
2 <v > ρ
. In addition to the pseudo-steady-state approximation, one can assume that there is no
work term, that there is no overall pressure differential, that the fluid is incompressible,
and that the velocity profile in the tube is essentially flat. With those assumptions
Equation D.2-5 becomes
1 2 E
v − g ( L + h) = − v (D.2-6)
2 w
where g is the acceleration of gravity, w is the mass flow rate through the pipe, and Ev is
the rate of viscous dissipation in the liquid.
From the derivation of the Bernoulli equation, one can identify the source of the
viscous dissipation term. For a Newtonian fluid it is possible to apply dimensional
analysis to arrive at the conclusion that
Ev 1 2
= v ev (D.2-7)
w 2
where ev is a dimensionless function of Reynolds number and geometrical ratios and is
called the friction loss factor (Bird et al., 2002, p.206).
The contributions to Ev or ev from various regions of the system are additive. For
regions where most of the viscous dissipation arises from friction and steep velocity
gradients, such as flow through valves or around elbows, one can identify a value of ev
that is relatively constant. Some representative values for obstacles, valves, and fittings
are tabulated in Bird et al. (2002, p 207). A more complete listing is given in Perry's
Handbook (Perry and Green, 1984).
For the geometry considered in this experiment, the only ev value of interest is
that associated with the entrance into the pipe. According to Bird et al.(2002), this value
should be ev = 0.45(1 − β ) , where β is the ratio of the area of the pipe to that of the tank.
There is no exit loss because the liquid leaves the tube in a streamline manner, without
any extra turbulence generated by its contact with air. For the orifice, the value of ev is
( )
2.7(1 − β) 1 − β 2 / β 2 .
For regions of straight pipe, one can replace the friction loss factor with the
Fanning friction factor f:
L
ev = 4 f (D.2-8)
D
Experiment D.2
Efflux time for a tank with an exit pipe Page D.2-3
ChE 324 Lab Manual
where D is the diameter of the pipe, L is its length, and f is a function of Reynolds
number, Re (Bird et al., 2002, p.206)
For the case under consideration, the Bernoulli equation may now be written as
1 2 1 L
v − g ( L + h) = v 2 2 f + ∑ e v,i (D.2-9)
2 2 R0 i
where the summation of friction loss factors includes all obstacles and disturbances that
the fluid encounters. All terms in Equation D.2-9 may be viewed as energy per unit mass
of fluid flowing. The first term on the left-hand side represents the kinetic energy carried
out of the pipe by the fluid. Rearrangement yields an explicit expression for the
instantaneous velocity in the pipe
1
2
2g ( L + h)
v= (D.2-10)
1 + 2f L + e
∑i v,i
R0
Before Equation D.2-10 is substituted into Equation D.2-4, it is necessary to assess the
relative magnitudes of the terms in the denominator. In particular, if the term containing
the friction factor is significant and f varies with Re, is it necessary to deal with the
velocity dependence of that term.
First, suppose that the liquid is in laminar flow, then
16 µ
f= (D.2-11)
2 R 0 vρ
as given in Equation C.1-16. If one assumes in addition that the kinetic energy and other
friction terms are negligible, then
ρgR 20 ( L + h)
v= (D.2-12)
8µL
which is the result of the Hagen-Poiseuille analysis. If this expression for v is substituted
into Equation D.2-4, one obtains the efflux time to be
8µLR 2 L + h 1
te = ln (D.2-13)
ρgR 40 L + h 2
At the other extreme, if the flow is sufficiently turbulent or the pipe is sufficiently
rough that f is a constant (see Figure 6.2-2 in Bird et al., 2002), then the result of the
integration is
1
L 2
2 1 + 2f + ∑ e v ,i
R R0 ( L + h ) 12 − ( L + h ) 12
t e = 2 i (D.2-14)
R0 2g 1 2
Experiment D.2
Efflux time for a tank with an exit pipe Page D.2-4
ChE 324 Lab Manual
v= (D.2-16)
( 0.0791) L µ
4 4
0
It is possible that, as a liquid drains from a tank and the level h drops, the
Reynolds number within the pipe changes from one regime to another. In that case, the
integral of Equation D.2-4 must be evaluated in several parts, using the appropriate
expression for the velocity in each region of h.
Apparatus
The apparatus for this experiment is a cylindrical tank fitted with a level gauge, as
shown in Figure D.2-2. The tank has a 6-inch diameter, and its total depth is 10 inches.
Various interchangeable exit pipes can be screwed into the bottom of the tank. The
available exit pipes are listed in Table D.2-1. Also available are a blank plug and a
simple plug containing an orifice with diameter of 0.188 inches.
Additional equipment includes weighing containers, platform scales, a stopwatch, a
thermometer, and a graduated cylinder. In addition to water, a 60-weight-per cent
aqueous sucrose solution is available to be used as a working fluid. Properties of aqueous
sucrose solutions are given in Appendices 4 and 4a.
Experiment D.2
Efflux time for a tank with an exit pipe Page D.2-5
ChE 324 Lab Manual
Figure D.2-2. Sketch of the draining-tank apparatus with an exit pipe in place
Procedure
Experiment D.2
Efflux time for a tank with an exit pipe Page D.2-6
ChE 324 Lab Manual
the appropriate equation from the Theory section to calculate te, and compare these
values with the experimental values.
Plot efflux time te values from all water runs versus (L/R0) on a single graph as
individual data points. Show on the same graph, as lines, the two theoretical
relationships given by Equations D.2-13 and D.2-17.
In a table compare the experimental efflux times for the sucrose solution
compared with the theoretical values calculated with the appropriate model, depending on
the Reynolds number. In the table show also L/R0 and the average value of Re for each
run.
By reference to the theoretical models, investigate the expected accuracy of the te
predictions and compare the uncertainty in those values with the deviation from
experimental values. As part of this exercise calculate the relative magnitudes of the
contributions to the viscous dissipation term for the various experimental conditions
studied.
References
Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Second Edition,
John Wiley and Sons, New York (2002).
Perry, R.H., and D. Green (eds.), Perry's Chemical Engineers' Handbook, 6th ed.,
McGraw-Hill Book Co., New York, 1984.
Experiment D.2
Efflux time for a tank with an exit pipe Page D.2-8
ChE 324 Lab Manual
Experiment D.3
HEATING LIQUIDS IN TANK STORAGE
dU tot
=w1 Hˆ 1 −w2 Hˆ 2 +Q (D.3-2)
dt
If the individual terms in Equation D.3-2 can be expressed in terms of
temperatures, fluid properties, system geometry, and flow conditions, this approximate
macroscopic balance provides a useful mathematical model of the thermal process. In
this experiment, application of Equation D.3-2 is demonstrated by observing the heating
of water in a storage tank with several modes of heating. The rate of temperature rise of
the water is measured and compared with models derived from Equation D.3-2.
Theory
Consider a stirred storage tank that either contains a mass of cool liquid or is
being filled with a flow of liquid. The liquid in the tank may be heated with steam either
by a steam coil mounted in the tank or by circulating a stream of liquid through an
external heat exchanger. The general case, showing both possibilities, is represented by
Figure D.3-1.
Figure D.3-1. Diagram of a storage tank equipped for either internal or external heating.
tank or through the external heat exchanger, as well as the initial and inlet-stream
temperatures and the steam temperature, are constant in each case. It is assumed that the
heat losses to the surroundings are negligible. Finally, it is assumed that perfect mixing
occurs in the tank, that is, the liquid temperature is uniform, if the agitator is employed,
but that plug flow occurs in the heat exchanger and, if the agitator is not used, in the tank
as well.
For the heat-transfer rate Q, appearing in Equation D.3-2, we shall use an overall
heat transfer coefficient, as defined in Equation C.2-7. The overall coefficient is assumed
to be constant over the heat-transfer surface area A. Thus, for the steam coil in the well-
mixed tank the heat transfer rate is expressed as
Q = UA (Ts − T) (D.3-3)
where Ts is the steam temperature and T is the temperature of the liquid in the tank. For
an external tubular heat exchanger the total rate of heat transfer is given by
Q = UA (Ts − T) ln (D.3-4)
and the reference temperature Tref is chosen as an arbitrary datum at which Utot=0.
The enthalpy per unit mass of a constant-density, constant-heat capacity liquid is
similarly based on the chosen reference state. It may be written as
Experiment D.3.
Heating Liquids in Tank Storage Page D.3-3
ChE 324 Lab Manual
below. It is assumed in the transient problems that the heat capacity of the tank walls and
of the external heat-exchanger loop are zero so that only the volume of liquid in the tank
is included in the accumulation term. The instructor will select the cases that a particular
group should examine during the laboratory session.
Case I. The liquid in the tank is heated by an internal steam coil.
Case I.a. The tank is initially full at room temperature. There is no flow. The tank is
stirred.
In this case one normally assumes that the temperature in the tank is uniform; that
is, the local liquid temperature is everywhere equal to Tave. Equation D.3-2 becomes in
this case
dT
ρVCˆ p =UA(Ts − T ) (D.3-8)
dt
because there are no flow terms. The initial condition is
T = T1 at t = 0 (D.3-9)
where t is time measured from the instant that steam is introduced into the coil, and T1 is
the initial temperature of the liquid. Equation D.3-8 may be integrated to obtain
T −T UA
ln s =− t (D.3-10)
Ts − T1 ρC p V
Case I.b. The tank is initially empty, but liquid flows in at a constant rate.
The mass flow rate of liquid into the tank is w. The temperature of the inlet
stream is T1. There is no flow of liquid out of the tank, so this case is applicable only up
to the time when the tank is full. With agitation, one may again assume that T is uniform
throughout the liquid in the tank.
For convenience one may choose the reference temperature to be the same as the
inlet temperature. Then the enthalpy of the liquid feed is defined as zero and does not
contribute to the energy balance. Thus, Equation D.3-2 becomes
d [V (T − T0 )]
ρCˆ p =UA(Ts − T ) (D.3-11)
dt
In this case it is necessary to recognize that both V and A vary with time as the liquid
level in the tank rises. The volume of liquid in the tank varies linearly with time and is
proportional to the mass flow rate w. The variation of the heat-transfer-surface area
depends on the geometry of the tank and the coil. As an example, one might assume that
the submerged area of the coil is linearly proportional to the liquid volume. This case is
examined by Bird et al. (2002, pp. 466-468). The result is
T − T1
=1−
(
1 − exp − UA f t / ρV f Cˆ p )
Ts − T1 (
UA f t / ρV f Cˆ p) (D.3-12)
Experiment D.3.
Heating Liquids in Tank Storage Page D.3-4
ChE 324 Lab Manual
where Af and Vf are the respective values of area and volume when the tank is full. This
function starts at zero and approaches unity at infinite time. The temperature, Tf, attained
when the tank is full, that is, when time reaches the value tf, identified as
ρVf
tf = (D.3-13)
w
is
T f − T1
=1−
(
1 − exp − UA f / wCˆ p )
Ts − T1 (
UA f / wCˆ p ) (D.3-14)
Case I.c. The tank is initially full. Flow rates in and out are constant and equal. The
tank is stirred
Consider the case where the initial temperature in the tank and the temperature of
the inlet stream are equal at T1. It is convenient to choose T1 as the reference temperature
for calculating enthalpy and internal energy. The flow rate in and out of the tank is w.
Equation D.3-2 then becomes
=− wCˆ p (T − T1 )+UA(Ts − T )
dT
ρVCˆ p (D.3-15)
dt
with the initial condition
T = T1 at t = 0 (D.3-16)
This case is posed in Problem 15.B.8 of Bird et al .(2002)
The solution to this equation is
T − T∞ UA w
=exp− + t (D.3-17)
T1 − T∞
wC p
ˆ ρV
where T∞ is the steady-state temperature in the tank, given by
UA
T1 + T
wCˆ s
T∞ = p
(D.3-18)
UA
1+
wCˆ
p
which is obtained by solving Equation D.3-16 with the time derivative set equal to zero.
Case II. The liquid is heated by pumping a side stream through an external tubular
heat exchanger.
The mass flow rate through the heat exchanger is constant and equal to wh. Although the
tank plus the heat-exchanger loop could be selected as the complete macroscopic system,
it is more convenient to divide the process in two subsystems: the tank and the heat-
exchanger loop. Such a separation allows one to calculate both the temperature of the
Experiment D.3.
Heating Liquids in Tank Storage Page D.3-5
ChE 324 Lab Manual
stream returning from the heat exchanger Th and the average temperature in the tank T. It
is assumed that the tank itself is insulated, so the heat-transfer rate Q for the tank is zero.
Equation D.3-4 yields the rate, Qh, in the heat exchanger.
Case II.a. The tank is initially full when the heating is started, and the tank is perfectly
mixed.
Once again, the assumption of perfect mixing implies that the temperature in the
tank is always uniform at T. Thus, the temperature of the stream entering the heat
exchanger is T. On the other hand, the temperature of the stream returning from the heat
exchanger is Th, thus, the log-mean temperature-difference driving force in the heat
exchanger is
(Ts − T) − ( Ts − Th )
(Ts − T) ln = (D.3-19)
T −T
ln s
Ts − Th
Although both T and Th will be changing with time during the process, the total
volume of liquid in the heat-exchanger loop may be assumed to be small compared with
the volume in the tank. Under this condition, one can neglect the accumulation term in
Equation D.3-2 for the heat exchanger and apply what is called the "pseudo-steady-state"
approximation. That is, one uses a steady-state energy balance on the heat exchanger to
obtain an equation relating Th to T at any given instant:
wh Cˆ p (Th − T )=UA(Ts − T )ln (D.3-20)
= wh Cˆ p (Th − T )
dT
ρVCˆ p (D.3-21)
dt
Combination of Equations D.3-19, 20, and 21 and integration yields the temperature in
the tank T(t) as
T − T1 w h K − 1
ln s =
t
(D.3-22)
s
T − T ρV K
where K is an abbreviation for
UA
K =exp (D.3-23)
w Cˆ
h p
In this case the product UA refers to the overall heat-transfer coefficient and the heat-
transfer surface area inside the tubular heat exchanger.
Experiment D.3.
Heating Liquids in Tank Storage Page D.3-6
ChE 324 Lab Manual
Case II.b. The tank is initially only partly full. There is a constant flow rate into the
tank, but no flow out. The tank is stirred.
For this case, Equations D.3-19 and 20 still describe the performance of the heat
exchanger, which is taken to be at pseudo-steady state. If the initial temperature in the
tank and that of the feed are equal at T1, and the reference temperature for enthalpy and
internal energy calculation is T1, the macroscopic energy balance for the tank is
d [V (T − T1 )]
ρCˆ p = wh Cˆ p (Th − T ) (D.3-24)
dt
There is an inlet stream with flow rate w, but it does not appear in the energy balance
because its enthalpy is by definition zero at T1. The right-hand side of Equation D.3-24
represents the net enthalpy input from the external heat exchanger. Equation D.3-20 can
be used to obtain the temperature difference. Because the steam temperature Ts is
constant, a simpler form of Equation D.3-20, relating Th to T, is
Ts − T UA
=exp ≡K (D.3-25)
Ts − Th wh Cˆ p
Experiment D.3.
Heating Liquids in Tank Storage Page D.3-7
ChE 324 Lab Manual
ρV
tc = (D.3-27)
wh
If the inlet and outlet pipes are placed essentially at the bottom and top of the tank, tc is
the period of one cycle, that is, the time for the inlet to the heat exchanger to experience a
step change in temperature from T1 to Th. Thus, for the initial cycle, Equation D.3-25
predicts that the temperature in the upper region of the tank Tu will be
1
Tu (1) = Ts − (Ts − T1 ) for 0 < t ≤ t c (D.3-28)
K
Subsequently, the inlet temperature to the heat exchanger will be Tu(1) until t=2tc. By
repeated application of Equation D.3-25, taking into account the step changes in the inlet
and outlet temperatures in the heat exchanger loop, leads to the result
Ts − Tu ( N ) t
= K − N for ( N − 1) < ≤ N (D.3-29)
Ts − T1 tc
where N is an integer that counts the number of cycles that the liquid at the top of the
tank has passed through the heat exchanger. Thus, if the liquid in the tank is truly
stratified with no vertical mixing, one should observe successive step changes in Tu with
time.
For all of these cases, the performance characteristic of the equipment that
expresses the efficiency of the heating process is the product UA. The area A can be
measured directly from the geometry of the system. Thus, measurements of the
temperature response for any of the modes of heating considered allow experimental
determination of the overall heat transfer coefficient U. Conversely, if U can be
estimated from its basic definition and from correlations for the contributing film heat-
transfer coefficients, such as those given in Bird et al.(2002, §§14.3, 14.7) , one could use
the models given here to predict the process performance. Note that the precise
definition of U depends on specifying which area is used for expressing A.
Apparatus
The apparatus for this experiment is a cylindrical tank fitted with an variable-
speed agitator and a steam coil, as shown in Figure D.3-1. The geometry of the coil is
designed to make the submerged area available for heat transfer directly proportional to
the volume of liquid in the tank. Mounted on the side of the tank is a loop of piping with
a pump and a tubular heat exchanger. A water line is available to provide the test liquid.
Steam may be introduced into either the coil or the heat exchanger, and the steam
pressure may be regulated with a manual control valve.
The tank has depth of 33.5 inches and inside diameter of 22.25 inches. The steam
coil is fabricated from ¾-inch galvanized iron pipe, 10.5 feet long, with a core diameter
of 14.5 inches and 2.5 turns. The S-40 pipe has inside diameter of 0.824 inch and outside
diameter of 1.05 inch. The same type of pipe is used for the external heating loop.
Experiment D.3.
Heating Liquids in Tank Storage Page D.3-8
ChE 324 Lab Manual
The heat exchanger contains 10 copper tubes, ¼ inch in inside diameter and 18
inches long.
Data Analysis
From the dimensions of the equipment, calculate the nominal heat-transfer area A
upon which reported U values will be based. Use the appropriate model from the theory
section to analyze the time-temperature data for the various heating processes. Generally,
the data should be fit to determine an average value of the overall heat-transfer
coefficient U. For the coil heating, report U as a function of agitator speed. For the heat
exchanger, report U as a function of water flow rate.
For the case of the external heat exchanger, check the accuracy of the data and the
validity of the model used by calculating the heat-input rates from both the temperature in
the tank and the temperature change of the stream passing through the exchanger.
Compare the experimental values of U for the two heaters with one another, with
representative values in the literature, and with general correlations. Consider both the
absolute magnitudes and the variations with operating conditions, such as agitation rate
or flow rate.
Discuss the validity of the assumptions made in the analysis.
References
Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Second Edition,
John Wiley and Sons, New York, 2002.
Kern, D.Q., Process Heat Transfer, McGraw-Hill Book Co., Inc., New York, 1950.
Perry, R.H., and D. Green (eds.), Perry's Chemical Engineers' Handbook, 6th ed.,
McGraw-Hill Book Co., New York, 1984.
Experiment D.3.
Heating Liquids in Tank Storage Page D.3-10
ChE 324 Lab Manual
Appendices
Appendix 1. A Sample Report
The following report was submitted by a ChE 324 student in the fall semester of
1996. It is generally well written so it is offered as a model to 324 students, particularly
as an example of proper report format. It has been rewritten a bit, and editorial comments
have been inserted to indicate where improvements are needed. Editorial comments are
highlighted in bold type.
B.K. Badger
Partners: Xxxx Xxxx
Experiment Completed:
11/6/96
Report Submitted:
11/13/96
ChE 324 Lab Manual
ABSTRACT
factor and Reynolds number for Newtonian flow in smooth, cylindrical pipes. The
Reynolds number was determined by measurement of mass flow rates through pipes of
known dimensions. The friction factor was calculated by measuring the pressure drop
along a finite length of pipe, using mercury/water and air/water manometers connected
by piezometer rings to the pipe. Results show two different correlations for friction
factor versus Reynolds number, one for laminar flow and one for turbulent flow. The
transition region from laminar to turbulent flow was found to occur at a Reynolds number
between 2000 and 4900 . New parameters for both the Blasius formula and the Prandtl
regime.
INTRODUCTION
In the design of industrial flow processes, one factor that must be accounted for is
the pressure drop associated with fluid flow through a pipe. The pressure drop is the
result of the shear stress between the flowing fluid and the stationary solid at the wall of
the pipe. The shear stress is the result of frictional drag and is proportional to the inside
surface area of the pipe and the kinetic energy of the fluid flowing through it. This
pressure drop must be accounted for in the design process in order to properly size pumps
and compressors and to minimize the energy requirements of the given system.
The design of flow processes requires a relationship between flow rate through a
pipe and the corresponding pressure drop along the length of the pipe. This relationship
can be determined for many systems if the velocity and pressure profiles in the fluid are
correlations have been developed to represent the flow characteristics for geometrically
similar systems.
One such correlation for flow of fluids through a conduit is known as the friction
factor correlation. As a fluid flows through a pipe, it exerts a tangential force, F, on the
solid surface of the pipe. This force is proportional to the interfacial area and to a
characteristic flow kinetic energy with the proportionality factor being identified as the
1 P0 − PL D L
f= = f Re, (1)
4 1 2 L D
ρ< v >
2
where D is the diameter of the pipe, L is the length of the pipe, (Po-PL) is the pressure
drop along the length of the pipe, ρ is the density of the fluid, and < v > is the average
velocity of the fluid. For fully developed flow and long, smooth pipes, the friction factor,
f, is independent of L/D and is therefore a function only of the Reynolds number. The
D<v>ρ
Re = (2)
µ
where µ is the viscosity of the fluid flowing through the pipe. A graph of the friction
factor, f, versus the Reynolds number, Re, provides a convenient method to estimate
pressure drops across a length of pipe given the physical characteristics of the pipe and
For circular pipes where the Reynolds number is less than 2100, the flow is
known to be laminar, meaning that all of the velocity streamlines are smooth, steady, and
parallel in the axial direction. At a Reynolds number above 2100, the flow becomes
unstable and enters the turbulent region. In the turbulent region, the velocities throughout
the tube fluctuate randomly, with the average bulk velocity being in the direction of fluid
flow. Because of the qualitative difference between turbulent and laminar flow, it is
expected that there will be two regions of the friction factor correlation, one for laminar
Although the velocity distribution and drag forces in laminar flow can be
calculated theoretically, the chaotic nature of turbulent flow requires that the macroscopic
experimental data. Bird et al. (1960) report that the friction factor, f, can be represented
in the turbulent regime by one of several empirical forms, including the Blasius formula
and the Prandtl resistance law. Each of these equations applies in only a limited range of
Reynolds-number values.
a
f= for 4000 < Re < 100000 (3)
Re b
where the parameters are reported to be a = 0.0791 and b = 0.25 (Bird et al., 1960). On
1
= a log Re − b for 4000 < Re < 50000 (4)
f
with the accepted parameter values appearing here being a = 4.0 and b = 0.40 (Bird et
al., 1960).
The goal of this study was to test the reported relations for f and to develop a
correlation of friction factors with Reynolds number over a range of flow rates for
smooth, cylindrical pipes. Experimental results for pressure drop as a function of flow
rate were used to compute friction-factor values. Experimental f values were compared
with predictions from the standard correlations commonly used in designing flow
processes, namely the Blasius formula and the Prandtl resistance law. This comparison
showed the agreement of the accepted correlations with the experimental data. Revised
values of the a and b parameters were calculated to obtain a better representation of the
[Note: Some more information should be given about the test system, particularly its
dimensions. This could be done by making reference to Experiment C.1 the lab manual.
Also, it would be helpful to give page references in BSL.]
SUMMARY OF RESULTS
Figure 1 shows the relationship observed between the friction factor, f, and
Reynolds number, Re. [Note by editor: The writer should have made clear that the
figure presents values calculated from experimental data. There should be a brief
summarizing all of the experimental conditions and the corresponding results. This
information is needed to make the presentation of results clear and to lay the
improvements in the experimental methods. The missing table might take the
following form.
1
Friction Factor (dimensionless)
0.1
0.01
0.001
Reynolds Number (dimensionless)
from this graph; they just make it look cluttered. The word “dimensionless” could
be omitted from the axes. Finally, the numbers on the x axis should appear at the
bottom of the graph.]
The experimental values for the friction factor and Reynolds number shown in
Figure 1 [and in the missing table summarizing the results] were used to estimate
values of the a and b parameters for both the Blasius formula and the Prandtl resistance
law. [Comment on how the curve fitting was done. Give some estimate of accuracy,
data scatter, or confidence limits.] The results are presented in Table 1 along with
Table 1 - Parameters for the Blasius formula and Prandtl resistance law which correlate
friction factor with Reynolds number for turbulent flow in pipes. The literature values
were taken from Bird et al. (1960). [Omit the boxes in tables. A horizontal line
between the headings and the entries would be sufficient.]
Paramet Blasius formula Blasius formula Prandtl Prandtl
er (literature (experimental resistance law resistance law
value) values) (literature) (experimental)
a 0.0791 0.112 2.0 3.08
function of Reynolds number with predictions of the Blasius formula and of the Prandtl
0.025
0.02
f (dimensionless)
0.015 f(experimental)
f(Blasiusformula)
0.01 f(Prandtl Resistance-law)
0.005
0
0 10000 20000 30000 40000
Re(dimensionless)
[There should be no points given for the model predictions; they are smooth curves.
Also, there should be no box around the figure. It might be helpful to indicate in
some way which data points came from which tubes.]
DISCUSSION
As shown in Figure 1, there are clearly two different relations between the friction
factor f and Reynolds number Re for the laminar and the turbulent flow regions. Figure 1
also shows that the friction factor generally decreases as the Reynolds number increases.
This decrease is less pronounced, however, in the turbulent region. The theoretical form
that can be developed for laminar flow [which should have been given in the
Introduction] predicts that f varies inversely with the second power of the Reynolds
number. According to the form of Equation 1, this means that the pressure drop varies
linearly with the flow rate through a pipe. The turbulent-flow behavior shown in Figure 1
indicates that pressure drop at the higher Reynolds numbers increases more rapidly with
flow rate.
Figure 1 also confirms that the transition from laminar to turbulent flow occurs
somewhere between a Reynolds number of 2000 and 4900. A lack of data in this region
prevents a more exact identification of the transition point from laminar to turbulent flow.
Also, the apparatus did not allow very accurate measurement of flow rate for slow flows.
The exit hose may need to be removed to allow more accurately measure laminar flow
rates. Due to the extremely low flow rate in the laminar region and elevation changes of
the exit hose, pooling of the exit stream occurred within the hose. [Note that reference
to a hose and suggestions for changing the apparatus make no sense here because
the apparatus and procedure have not been described. There should have been at
least a reference to the lab manual to define the system that was used in the lab.]
Figure 2 shows that the Blasius formula and Prandtl resistance law do not agree
exactly, but each provides a fair description of the variation of the friction factor with
increasing Reynolds number for the range of flow rates studied. The Prandtl resistance
law, however, overestimates the measured friction factor by 30-50%, while the Blasius
formula underestimates the measured friction factor by 9-40%. Therefore, the parameters
in each of these formulas should be replaced with those listed in Table 1 to provide a
factor could be due to two sources of systematic error in our measurements. First, both
equations assume that the pipes are hydraulically smooth. Slight roughness on the inside
surfaces of the pipes would yield higher values for the measured friction factors.
Second, air bubbles in the manometer connector lines were a persistent problem that
introduced inaccuracy into the pressure-drop measurement, which would give artificially
low values for the measured friction factor. [Why low values? This is not obvious.]
This problem was most likely due to leaks within the test-system valves.
In this experiment, a correlation was obtained between the friction factor and the
Reynolds number for both turbulent and laminar pipe flow. The relationship obtained,
which is presented in Figure 2, can be used in the design of flow processes in order to
estimate the pressure drop associated with a given flow of fluid through a circular pipe.
The form of the correlation is significantly different for laminar and turbulent flow.
Further measurements should be taken to clarify at what exact Reynolds number the
correlation was compared with the Blasius formula and Prandtl resistance law to illustrate
the validity of these models for predicting friction factors in turbulent flow. New
parameters for both the Blasius and Prandtl formulas were determined to reflect more
REFERENCES
1. Bird, R.B., Stewart, W.E. and Lightfoot, E.N., Transport Phenomena, John Wiley &
Sons, New York, (1960), Chapters 5-6.
2. Crosby, E.J., Experiments in Transport Phenomena, Department of Chemical
Engineering, University of Wisconsin-Madison, August 1961, pp. 1-14, B10-B17.
[No specific reference was made to the Crosby manual in the text of the report.
Such reference should have been given in order to document the equipment and
procedure used. Otherwise, the citation should not appear in the list of references.
A bibliography, on the other hand, lists references that an author has consulted in
preparing an article but does not necessary cite explicitly. Bibliographies are more
common with articles expressing opinion rather than with scientific reports.]
APPENDICES
Appendix 2 (Suppressed).
Appendix 3 (Suppressed).
Temperature (C)
0 10 20 30 40 50 60 70 80 90 100
Weight Percent
Glycerol Density (g/cc)
5 1.01015 1.00735
10 1.0251 1.0241 1.02210 1.01905 1.0155 1.0113 1.0062 1.0006 0.9946 0.9883 0.9818
15 1.03450 1.03130
20 1.0566 1.0495 1.04690 1.04350 1.0395 1.0351 1.0296 1.0239 1.0175 1.0110 1.0039
25 1.05980 1.05605
30 1.0789 1.0762 1.07270 1.06855 1.0637 1.0585 1.0529 1.0473 1.0409 1.0343 1.0273
35 1.08600 1.08165
40 1.1068 1.1035 1.09930 1.09475 1.0893 1.0839 1.0781 1.0719 1.0657 1.0591 1.0523
45 1.11280 1.10795
50 1.1349 1.1311 1.12630 1.12110 1.1154 1.1095 1.1035 1.0973 1.0909 1.0843 1.0774
55 1.14005 1.13470
60 1.1635 1.1591 1.15380 1.14830 1.1425 1.1366 1.1302 1.1239 1.1175 1.1110 1.1039
65 1.16750 1.16195
70 1.1922 1.1872 1.18125 1.17565 1.1694 1.6360 1.1571 1.1506 1.1441 1.1375 1.1304
75 1.19485 1.18900
80 1.2197 1.2144 1.20850 1.20240 1.1961 1.1900 1.1836 1.1768 1.1699 1.6320 1.1561
85 1.22180 1.21565
Appendix 4. Properties of glycerol and sucrose solutions
90 1.2470 1.1424 1.23510 1.22890 1.2223 1.2159 1.2093 1.2028 1.1966 1.1897 1.1828
95 1.24825 1.24190
100 1.2725 1.2668 1.26108 1.25495 1.2487 1.2426 1.2360 1.22940 1.2224 1.2153 1.2089
These densities are relative to that of water at 4C where the mass of 1 cc is taken as 1 gm. These densities have been calculated from measured densities at 20 and 30C
and the data for thermal expansion determined by Gerlach, G, Th., and reported by Fresnius, W., Z. anal. Chem., 24, 99-119 (1885); see Lawrie, J.W.,
"Glycerol and the Glycols," A.C.S. Monograph Series No. 44, The Chemical Catalog Company, Inc., New York (1926), p. 170.
Data for 20 and 30 C: Bosart, L.W, and A.O. Snoddy, Ind. Eng. Chem., 20, 1377-1379 (1928).
Ap.4-1
Viscosity of Aqueous Glycerol Solutions 1
Temperature (C)
0 10 20 30 40 50 60 70 80 90 100
Weight Percent
Glycerol Viscosity (cp)
10 2.44 1.74 1.31 1.03 0.826 0.680 0.575 0.500
20 3.44 2.41 1.76 1.35 1.1 0.879 0.731 0.635
30 5.14 3.49 2.50 1.57 1.5 1.16 0.956 0.816 0.690
40 8.25 5.37 3.72 2.72 2.1 1.62 1.00 1.09 0.918 0.763 0.668
50 14.6 9.01 6.00 4.21 3.1 2.37 1.86 1.53 1.25 1.05 0.91
60 29.9 17.4 10.8 7.19 5.1 3.76 2.85 2.29 1.84 1.52 1.28
65 45.7 25.3 15.2 9.85 6.8 4.89 3.66 2.91 2.28 1.86 1.55
70 76.0 38.8 22.5 14.1 9.4 6.61 4.86 3.78 2.90 2.34 1.93
75 132 65.2 35.5 21.2 13.6 9.25 6.61 5.01 3.80 3.00 2.43
80 255 116 60.1 33.9 20.8 13.6 9.42 6.94 5.13 4.03 3.18
85 540 223 109 58.0 33.5 21.2 14.2 10.0 7.28 5.52 4.24
90 1310 498 219 109 60.0 35.5 22.5 15.5 11.0 7.93 6.00
95 3690 1270 523 237 121 67.0 39.9 26.4 17.5 12.4 9.08
96 4600 1585 624 281 143 77.8 45.4 29.7 19.6 13.6 10.1
97 5770 1950 765 340 166 88.9 51.9 33.6 21.9 15.1 10.9
98 7370 2460 939 409 196 104 59.8 38.5 24.8 17.0 12.2
99 9420 3090 1150 500 235 122 69.1 43.6 27.8 19.0 13.2
100 12070 3900 1412 612 284 142 81.3 50.6 31.9 21.3 14.8
1 Segur, J.B., and H.E. Oberstar, Ind. Eng. Chem., 43, 2117-2120 (1951).
Ap.4-2
1
Density of Aqueous Sucrose Solutions
These densities are relative to that of water at 4C where the mass of 1 cc is taken as 1 gm. These densities have been calculated from measured densities at 20 and 30C
and the data for thermal expansion given by Bates, F.J., et al., "Polarimetry, Saccharimetry and the Sugars," Circular of the National Bureau of Standards C440,
U.S. Dept. of Commerce, U.S. Government Printing Office. Washington D.C. (1942), Table 119, pp. 648-649
Data for 20 C: Bates, F.J. et al., op. cit., Table 113, pp. 626-631.
Ap.4-3
Viscosity of Aqueous Sucrose Solutions 1
1 Swindells, J.F., Snyder, C.F., Hardy, R.C., and Golden P.E., "Viscosities of Sucrose Solutions at Various Temperatures: Tables of Recalculated Values," Supplement to Circular
of the National Bureau of Standards C440, U.S. Dept. of Commerce, U.S Government Printing Office, Washington, D.C., July 31, 1958, Tables 132 and 133.
Data for 10% sucrose at 20 C: Ibid., Table 131.
Data for 85-100 C: Lange, N.A., editor, "Handbook of Chemistry," 9th edition, Handbook Publishers Inc., Sandusky, Ohio (1956), p. 1667.
Ap.4-4
10
15ºC
20ºC
25ºC
30ºC
35ºC
1
40ºC
45ºC
Kinematic viscosity (cm2/s)
50ºC
55ºC
60ºC
65ºC
0.1
0.01
40 45 50 55 60 65 70
Sucrose concentration (%)
Absolute Density 1
Temperature 2 3,4
Absolute Density Viscosity
(°C) (g/cm3) (g/cm3) (cp)
0 0.9998681 0.999841 1.7921
5 0.9999919 0.999965 1.5188
10 0.9997282 0.9997 1.3077
15 0.9991266 0.999099 1.1404
20 0.9982343 0.998203 1.0000
50 0.98807 0.5494
55 0.98573 0.5064
60 0.98324 0.4688
65 0.98059 0.4355
70 0.97781 0.4061
75 0.97489 0.3799
80 0.97183 0.3565
85 0.96865 0.3355
90 0.96534 0.3165
95 0.96192 0.2994
100 0.95838 0.2838
1
These densities are relative to that at 4°C where the mass of 1 cm3 is taken as 1 g; Forsythe, W.E.,
“Smithsonian Physical Tables,”9th edition, The Smithsonian Institution, Washington, D.C. (1954),
Tables 287 and 290, pp. 296 and 298.
2
Hodgman, C.D., editor, Handbook of Chemistry and Physics , 44th edition, The Chemical Rubber
Publishing Company, Cleveland, Ohio (1962), p. 2197.
3
Forsyth, W.E., op. cit., Table 311, p. 319.
4
The viscosity of water is 1.0000 cp at .20°C; Hodgman, C.D. op. cit., p. 2257.
Ap.5-1
Appendix 6. - Calculated values of the first (smallest) eigenvalues for unsteady heat conduction in a
slab, cylinder, or sphere with a Newton's law of cooling boundary condition.
To be used for data analysis in ChE 324, Experiment A.2, Thermal Conductivity of Solids.
T.W. Chapman, 10/7/96
Reviewed by R. Chavez 2002
Relations between the first eigenvalue, Beta1, and the Biot number Bi=hR/k for unsteady heat
conduction into a slab, a cylinder, and a sphere. These tables may be used in the data analysis of
Experiment A.2 of ChE 324
Ap.6-1
20
18
Sphere
16
Cylinder
14 Slab
12
1/Bi=k/hb
10
0
0 0.5 1 1.5 2 2.5 3 3.5
β1
Figure Ap.6-1. Relationship of the Biot number with the first eigenvalue
for heat conduction in solids of simple geometry and convective surface
Ap.6-2
Appendix 8. Calculation of temperature profiles and heat flows into the rod. (Experiment B.2)
β
θ ( z , t , α , β ) := ⋅ z ⋅ 1 − erf + β ⋅ t ...
1 z
⋅ exp
2 α 4⋅ α ⋅ t
β
+ β ⋅ t ⋅ ( −1 )
z
+ exp α ⋅ ( −z) ⋅ 1 − erf
4 ⋅ α ⋅ t
Steady-state temperature profile (Eq. B.2-9):
β
θss( z , α , β ) := exp − ⋅z
α
exp( −β ⋅ t) β D
Q( t , α , β ) := + erf ( β ⋅ t) ⋅ ⋅
π ⋅α ⋅t α 2
Calculations
Conversion. MathCAD will convert to default English units. You may edit any of the units to
make a conversion to a different one.
BTU lb BTU
k = 9.434 ρ = 499.424 C p = 0.12
ft⋅ hr⋅ R 3 lb⋅ R
ft
2
ft −1
α = 0.157 β = 0.4 hr
hr
Ap.8-1
Dimensionless temperature and heat flow at z := .1⋅ ft
t := .1 ⋅ hr , .2⋅ hr .. 2 ⋅ hr
t = θ ( z , t , α , β ) = Q( t , α , β ) =
Steady-state values:
0.1 hr 0.565 0.195
0.2 0.676 0.143 θss( z , α , β ) = 0.853
0.3 0.725 0.121
0.4 0.754 0.108 Qss( α , β ) = 0.066
0.5 0.773 0.1
0.6 0.787 0.094
Heat flow at the base as a function of time:
0.7 0.797 0.09
0.8 0.805 0.086 0.25
0.9 0.812 0.084
1 0.817 0.082 Q( t , α , β )
1.1 0.821 0.08 0.13
Qss( α , β )
1.2 0.825 0.078
1.3 0.828 0.077
1.4 0.83 0.076 0
0 1 2
1.5 0.833 0.075 t
1.6 0.835 0.074 hr
1.7 0.836 0.073 time in hours
1.8 0.838 0.073 For example:
1.9 0.839 0.072 t := 3 ⋅ hr Q( t , α , β ) = 0.069
2 0.84 0.072
t := 10⋅ hr
Q( t , α , β ) = 0.066
Temperature profiles at various values of position and time. One could specify z and t as
range variables to calculate θ(z,t,α,β). In such case, the temperatures for all z values and for
the first time value would be at the top of the array, followed by the values for subsequent
times. Here we show a list for only t=1.0 hr. and z from 0 to 2, at 0.2 intervals.
z := 0 ⋅ ft , .2⋅ ft .. 2 ⋅ ft t := 1 ⋅ hr
z = θ (z , t , α , β ) = θss( z , α , β ) =
0 ft 1 1 2
ft −1 BTU
0.2 0.657 0.727 α = 0.157 β = 0.4 hr h = 0.5
hr 2
0.4 0.404 0.528 hr⋅ ft ⋅ R
0.6 0.23 0.384
0.8 0.119 0.279
1 0.056 0.203
1.2 0.024 0.147
1.4 9.133·10 -3 0.107
1.6 3.115·10 -3 0.078
1.8 9.471·10 -4 0.057
2 2.563·10 -4 0.041
Ap.8-2
The following plot shows how we can plot spatial temperature profiles at various
values of time, including the steady-state profile, without having to generate the
lists for those times:
θ ( z , .01⋅ hr , α , β )
Dimensionless temperature
θ ( z , .1⋅ hr , α , β )
θ ( z , .5⋅ hr , α , β )
0.5
θ ( z , 1⋅ hr , α , β )
θ ( z , 2⋅ hr , α , β )
θss( z , α , β )
0
0 0.5 1 1.5 2
z
Distance from base, ft.
θ i , j := θ ( z( i) , t ( j) , α , β )
Generate the matrix
The resulting matrix is obtained by typing "θ=" (shortcut for θ is q CTRL-G). The rest
of the temperature values may be viewed by clicking on the box and scrolling across
the window.
0 1 2 3 4 5
0 0 1 1 1 1 1
1 0 0.422 0.565 0.634 0.676 0.704
θ= 2 0 0.109 0.254 0.347 0.41 0.457
3 0 0.017 0.088 0.161 0.221 0.27
4 0 1.408·10 -3 0.023 0.063 0.105 0.144
5 0 6.625·10 -5 4.675·10 -3 0.02 0.044 0.07
Ap.8-3
View the corresponding values of z and t by entering the row and column number, respectively,
inside the parentheses in the expressions below. For example z(1)=0.1ft and t(1)=0.05hr:
z( 1 ) = 0.1 ft t( 1 ) = 0.05 hr
2
ft −1 BTU
α = 0.157 β = 0.4 hr h = 0.5
hr 2
hr⋅ ft ⋅ R
1 1
θ1, j
θi , 1
θ2, j
θi , 5
θ5, j 0.5 0.5
θ i , 10
θ 10 , j
θ i , 20
θ 20 , j
0 0
0.5 1 0 1 2
t( j) z( i)
Distance from base, ft
hr
Notes:
- The time is shown divided by hr on the graph at left. This is because MathCAD plots the variables in the default
units, in this case seconds. To change the plot to a different scale one must divide by the conversion factor; hr
means 3600 sec/hr. The graph on the right is expressed in the default units (ft) so there is no need to scale.
- To change the limits of an axis: 1) Click on the plot; you will see four numbers showing the limits of each axis.
Change the appropriate limit to modify the scale.
Ap.8-4
Appendix 9. Concentration Profiles in a Stagnant Film and determination of diffusivity of
acetone in nitrogen. (Experiment B.3)
( ) ( − 1)
Equation 20.1-18 in BSL.
1 + π ⋅ ( 1 + erf ( φ) ) ⋅ φ⋅ exp φ
2
π
ψ ( φ) := φ⋅
xAo( φ) You may click on the tables and scroll to see other values.
φ= xAo( φ) = ψ ( φ) =
0.1 0.166 1.067 Plot of the phi and psi factors as functions of
xAo:
0.2 0.311 1.14
0.3 0.436 1.22
4
0.4 0.543 1.306
0.5 0.634 1.398
0.6 0.71 1.498
0.7 0.773 1.606 φ
0.8 0.824 1.721 2
ψ( φ)
0.9 0.866 1.843
1 0.899 1.972
1.1 0.925 2.108
1.2 0.945 2.251
0
1.3 0.96 2.4 0 0.5 1
1.4 0.972 2.554 xAo( φ )
1.5 0.98 2.712
1.6 0.986 2.875
Ap.9-1
This table shows a narrower range of φ values
φ := 0.1 , .105 .. .2
φ= xAo( φ) = ψ ( φ) =
0.1 0.166 1.067
We calculate the theoretical concentration profile
0.105 0.174 1.071
which can be compared with experimental values to
0.11 0.181 1.074 estimate the diffusivity if xA0 is known. We estimate
0.115 0.189 1.078 xAo (xAovalue) using Antoine's Equation and the ideal
0.12 0.197 1.081 gas law. You must enter P and T from your
experimental data.
0.125 0.204 1.085
0.13 0.212 1.088
1210.595
0.135 0.219 1.092 7.11714−
P := 740 mmHg T := 17 ºC T+229.664
0.14 0.226 1.096 pA := 10
0.145 0.234 1.099 pA
xAovalue := xAovalue = 0.219
0.15 0.241 1.103 P
0.155 0.248 1.107
0.16 0.255 1.11
0.165 0.263 1.114
0.17 0.27 1.118
0.175 0.277 1.121
Guess phi: φ := 0.1 If the given guess for φ doesn't work, use the
plot or tables above to supply a better value.
Given
xAo( φ) = xAovalue
φ := Find( φ) φ = 0.135
Ap.9-2
The following is the theoretical plot of Z vs. X for xAo( φ) = 0.219and φ = 0.135
This is the same type of plot as that shown in Figure 20.1-1 of BSL.
Z := 0 , .05 .. 2
Z= X( φ , Z) = 1
0 1
0.05 0.952
0.1 0.903 X ( φ , Z) 0.5
0.15 0.854
0.2 0.805
0.25 0.756 0
0 0.5 1
0.3 0.708
Z
0.35 0.661
0.4 0.615 Now we are ready to obtain the diffusivity. Enter the following
0.45 0.57 values of the measured xA, position z, in cm, and time in
0.5 0.526 seconds obtained from your experiment.
0.55 0.484
xA := 0.107 z := 5.715cm t := 300 sec
0.6 0.444
xA
0.65 0.405 Thus Xvalue := Xvalue = 0.489
xAo( φ)
0.7 0.368
0.75 0.334
Guess : Z := 0.3 If the given guess for Z doesn't work, use the plot
above to supply a better value
Given
X( φ , Z) = Xvalue
Z := Find( Z) Z = 0.544
z
Since Z := we can solve for D to obtain its value
4Dt
2
z
Z
D :=
( 4 ⋅ t)
2
cm
D = 0.092
s
Ap.9-3
Calculation of molecular diffusivity using Chapman-Enskog Theory
The following scrollable table contains the collision integral values. The first column contains the
κT
values for , the second and third columns contain the collision integral values for the viscosity
ε
and diffusivity, respectively.
Ω :=
0 1 2 Ω D_AB( x ) := linterp Ω ( 〈0〉
,Ω
〈2〉
,x )
0 0.3 2.79 2.66
1 0.35 2.63 2.48 Units definitions:
2 0.4 2.49 2.32 −8 −3
A := 10 ⋅ cm g := 10 ⋅ kg
3 0.45 2.37 2.18
3 3
4 0.5 2.26 2.07 kPa := 10 ⋅ Pa kJ := 10 ⋅ J
J
Rgas := 8.314 ⋅
mol⋅ K
Enter here Lennard-Jones parameters and other data:
Acetone
εA g
σ A := 4.600 ⋅ A := 560.2 ⋅ K MA := 58⋅ T := 293⋅ K
κ mol
Nitrogen
ε B g
σ B := 3.798 ⋅ A := 71.4⋅ K MB := 28⋅ P := 101.3 ⋅ kPa
κ mol
Calculation:
ε AB ε A ε B
:= ⋅ σ AB :=
1
(
⋅ σA + σB )
κ κκ 2
ε AB σ AB = 4.199 A
κT := T κT = 1.465
= 199.996 K
ε ε AB
κ ε
κ
Ω D := Ω D_AB
κT Ω D = 1.21 c :=
P
c = 41.585 mol m
-3
ε Rgas⋅ T
T⋅
1 1
0.5 2 0.5
+
− 5 mol ⋅ A ⋅ g
DAB := 2.26⋅ 10 ⋅ ⋅
MA MB
0.5 2
cm⋅ s⋅ K c⋅ σ AB ⋅ Ω D 2
cm
DAB = 0.1
s
Ap.9-4
Appendix 10. Acetone data for Experiment B.3
-2.0 60
7.7 100
22.7 200
39.5 400
56.5 760
Values in Table Ap.10-1 are plotted in Figure Ap.10-1 in the form log Pvap vs. 1/Tabs. The
values can be approximated by the following Antoine equation:
100
Vapor Pressure (mm Hg)
10
1
0 .0 0 3 0 .0 0 3 2 0 .0 0 3 4 0 .0 0 3 6 0 .0 0 3 8 0 .0 0 4 0 .0 0 4 2 0 .0 0 4 4 0 .0 0 4 6 0 .0 0 4 8 0 .0 0 5
-1
1 /Te m p e r a tu r e (K )
Ap.10-1
Appendix 11. Spreadsheet for preliminary analysis of data in Experiment C.1. Friction Factors for Flow in Circular Tubes
General Information
D, in Length,ft
Tube A 0.250 5.0
Tube B 0.141 5.0
Tube C 0.141 2.5
Run: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Input data: Units
Water temperature (C or F?)
Water density (gm/ml) g/ml
Water viscosity (cP) cP
Room temperature (C or F?)
Manometer-water density g/ml 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Manometer-mercury density g/ml 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13
Tube (A, B, or C) A B C C C C C C C C C C C C C C C C C C
Friction factor
The user should check the formulas that are programmed here to confirm that they are correct.
Note: Actual physical properties corresponding to experimental temperatures should be inserted into this spreadsheet,
along with measured values and apparatus dimensions.
Be sure to check the validity of all formulas and the consistency of all units used in this spreadsheet.
Based on notes prepared for ChE 324 by Prof. D.J. Klingenberg, Fall 1997
Introduction
In this course, you will be measuring a variety of variables, then analyzing the
data by calculating derived quantities, and often fitting the data to equations in order to
estimate some parameters. Engineers must appreciate the fact that all measured data
contain some degree of uncertainty; all reported results are only estimates. To make these
estimates meaningful, we must also somehow report the uncertainty in the results, or how
confident we are about the results.
The purpose of these notes is to provide some guidance for reporting uncertain
results, especially for the cases where measured data are analyzed and fit to equations to
extract useful engineering information.
Experimental Errors
Experimental errors are typically grouped into two categories: systematic errors
and random errors. Systematic errors are usually relatively fixed, and they influence all
the data in a series of measurement in essentially the same way. Examples of this type of
error are pressure measurements with a gauge that is not properly calibrated,
measurement of temperature rise in a system that is supposedly adiabatic but in fact is
not, and incorrect thermometer readings due to thermal expansion of the thermometer
glass. Such errors can often be eliminated by auxiliary experiments and calibrations or
theoretical corrections to the experimental data.
Random errors are usually more difficult to detect and to deal with. Examples of
these errors include mistakes by the experimenter when recording the data, random
electronic fluctuations in measuring equipment, and incidental vibrations that affect a
mechanical system. Even after systematic errors are eliminated from an experiment,
random errors will still be present. It is the experimenter's job to take these persistent
errors into account by assessing the uncertainty in the data, trying to minimize it, and
indicating the degree of uncertainty in the reported results.
Appendix A.13.
Statistical Analysis of Experimental Data Ap.13- 1
indeed inconsistent rather than revealing an important characteristic of the system.
Exceptions to this rule arise in instances where one can explain on physical grounds why
a suspect data point does not reflect the system of interest. For example, if one realizes
that part of the equipment was not operating properly during a test, for example, some
controlled variable such as temperature or flowrate deviated from its set point , it is
reasonable to throw away the data taken during such a disruption. Generally, however,
one should investigate the relative validity of all data points in a set of measurements.
Here, the indicated uncertainty is based more on common sense rather than a systematic
statistical analysis. Such uncertainties in measured values should be estimated
conservatively. In this example, we would state an uncertainty of 0.5 mm H2O only if we
were very confident that the true pressure lies somewhere between 46.7 and 47.7 mm
H2O.
Propagation of Error
Very often the experimenter will measure certain types of data and then obtain
derived quantities from the measured values. For example, one might measure the current
and voltage across a resistor and then calculate the power loss by the formula P = VI. For
purposes of a report and its conclusions, it may be more important to identify the
uncertainty of the derived quantity P than that of the raw data, I and V.
Appendix A.13.
Statistical Analysis of Experimental Data Ap.13- 2
R = R(xl, x2, • • •, xn)
from which the derived quantity is calculated from the measurements. In the previous
example, the function is P = P(V, I) = VI. Now let wi be the uncertainty of variable xi ;
that is, xi = <xi> ± wi where <xi> is the estimated value of xi. The uncertainty in the
derived quantity, wR, may be estimated by the formula
1
∂ R
2
∂R
2
∂R
2 2
Equation 1 accounts for the propagation of the individual uncertainties into the calculated
value of R. The partial derivatives of R with respect to each xi are taken while all other xj
are held constant; these terms arise from a two-term Taylor-series analysis of the function
R. The terms are squared before adding to avoid unjustified cancellation of errors. Thus,
Equation 1 gives a conservative estimate of wR.
The Engineering Equation Solver (EES) program can perform error propagation analysis
for a system of equations. EES is available on CAE Windows stations (CAE Applications
> Engineering > EES) or it can be downloaded from the CAE web site.
Statistical Analysis
1 m
z = ∑ zi
m i =1
(A.13-2)
Appendix A.13.
Statistical Analysis of Experimental Data Ap.13- 3
d i = zi − z (A.13-3)
This deviation is not very useful because the average of all readings is zero.
1
1 m
(z i − z )2
2
σ = ∑
m − 1 i =1
(A.13-4)
The standard deviation σ is an estimate of the average magnitude of the errors that
arise in the measurements. If one assumes that systematic errors have been eliminated,
the deviations are the random errors.
The definition of the sample standard deviation given in Equation 4 is used when
the underlying population is not known, that is, when we don't know what would happen
if there were an infinite number of readings. If we compared a set of readings against a
known distribution such that the "correct" value of z were known, for example, in the
calibration of a pressure gauge with a known, reliable instrument such as a dead-weight
tester, then the factor of 1/(m-1) in Equation 4 would be replaced with 1/m. The
explanation of this difference is that, when the sample comes from an unknown
distribution, one measurement is used in estimating the sample mean, and thus only (m-1)
measurements remain to estimate the standard deviation. If the mean were known ahead
of time, however, then all the measurements could be used to estimate the standard
deviation.
Linear Regression
Engineers often need to analyze sets of data where each data point is represented
by a pair of numbers. For example, one may measure the vapor pressure of a liquid as a
function of temperature. The data in this case will be a collection of paired values of
temperature and pressure. In general, we are typically concerned with a set of n data
points, each represented by a pair of numbers (xi, yi) where i goes from 1 to n. The
objective of the data analysis is to infer a functional relationship between x and y.
Two different cases are common. In the first case, the engineer may simply want
to consolidate a large number of data points into a simple equation describing the
functional relationship between the x and y values, y = f(x). Here the objective is simply
Appendix A.13.
Statistical Analysis of Experimental Data Ap.13- 4
to present the results in a neat, concise form. The function f(x) may be chosen arbitrarily,
and the result is called an empirical correlation.
The other case arises when a functional relationship between the x and y values is
known or specified a priori, usually on theoretical grounds. Then the function f(x) is
called a system model, and the objective of the experimental study is to estimate values of
parameters appearing in the function. In the vapor-pressure example, thermodynamic
analysis suggests that the relationship between pressure and absolute temperature should
have the form ln P = A+B/T. In this case one might want to estimate from the
measurements the parameter B because it would indicate the heat of vaporization of the
material studied.
In both of these problems, one must somehow "fit" the experimental data to a
specified functional form and to determine the parameter values that provide this best fit.
Also, particularly in the former case, several different forms of the function f(x) might be
tried to see which form is most successful in representing the variation of the data. In this
course, we will concern ourselves with fitting data to straight lines, in which case the
curve-fitting procedure is called linear regression. The method by which we fit the data
to a straight line is called least-squares linear regression.
For linear regression, we wish to fit data to a line described by the equation
y = f(x) = a x + b
where the constants a and b are called the parameters of the function. Our job is to
determine the values of a and b from the n sets of paired numbers (xi, yi). This procedure
should always begin by creating plot of y vs. x to determine whether or not the data
indeed follow something close to a straight-line relationship. Such a plot is depicted in
Figure 1. In this figure, the data appear to follow a straight line approximately, but the
points are scattered such that they will not fit a straight line perfectly. There is clearly an
infinite number of straight lines that would go more or less through the data points. We
could also come up with a large number of potential algorithms by which we determine
the "best" line by minimizing some measure of the distance between the line and the data
points. Without additional information, it is not clear that any such method would be
better than any other.
Appendix A.13.
Statistical Analysis of Experimental Data Ap.13- 5
12
10
6
y
0
0 2 4 6 8 10
x
1. that the uncertainty in the x values is much less than the uncertainty in the y values;
then the best fit of the data to a straight line can be determined by finding the values of a
and b that minimize the sum of squared deviations of y values from those predicted by
the straight line. Thus, we must find the values of a and b that minimize the sum-of-
squares objective function
n
S (a, b)=∑ [ y i −(axi + b )]
2
(A.13-5)
i =1
This function S, which is a measure of the total discrepancy between the function f(x) and
the data, depends on a and b, and we want to find the values that make S the smallest.
As long as all the xi values are distinct, minimizing S will give a unique answer. It also
turns out that this procedure typically gives informative results even if the above
assumptions are relaxed.
Appendix A.13.
Statistical Analysis of Experimental Data Ap.13- 6
By taking the partial derivatives of S with respect to a and b and setting both
derivatives to zero, one can show that the values of a and b that minimize S are
∑ xy − n (∑ x )(∑ y )
1
a= (A.13-6)
1
∑x − n (∑ x )
2 2
and
b=
(∑ y)(∑ x ) − (∑ xy)(∑ x) = y − ax
2
(A.13-7)
n( ∑ x ) − (∑ x )
2 2
where the summations are taken over all n pairs of data points, and x and y are mean
values of each of the measured variables. Remember that a and b are the slope and
intercept of the straight line fit through the data.
Estimating Uncertainty
The results given in Equations 6 and 7 are only estimates of the true values of a
and b. Even if the assumed form of f(x) is indeed the actual relation between x and y, if
systematic errors are eliminated, and if the conditions listed as conditions for the least-
square method are satisfied, the inevitable occurrence of random errors in any real
measurements means that a very large number of data points is required before the true
values of a and b can be estimated with great confidence. Therefore, it is necessary for
the experimenter to make some estimate of the reliability of his results.
2
1
(
∑ xy − n ∑ x ∑ y )( )
r2 = (A.13-8)
( ) ( )
1 2 1 2
∑ x − n ∑ x ∑ y − n ∑ y
2 2
Appendix A.13.
Statistical Analysis of Experimental Data Ap.13- 7
It can be shown that the absolute value of r is less than or equal to unity. A perfect fit of
the data to the linear equation y = ax + b gives the limiting value of r2 = 1.0 . Completely
random scatter of x and y with no correlation would yield r = 0.
A much better method for describing the quality of the fit is to report the
uncertainty in the estimated values of the parameters a and b. The estimated values of the
standard error values for the two parameters, sa and sb are given by
se2
sa2 = (A.13-9)
∑ ( xi − x )
2
and
s =s
2 2 ∑x 2
i
(A.13-10)
n∑ ( x − x )
b e 2
i
1
se2 = S ( a , b) (A.13-11)
n−2
The Microsoft Excel spreadsheet program contains a function called LINEST that
performs linear regression on a set of data points. In addition to calculating the values of
a and b, LINEST produces the statistical parameters sa and sb as well as the coefficient of
determination of the fit, r.
Given these statistical quantities, we can now report values of the parameters a
and b along with their confidence limits as
a ± t sa and b ± t sb .
The additional factor t in the estimated uncertainties is a parameter that arises in the so-
called Student's t test and the t distribution. It is a function of the number of data points
available as well as the confidence level that one wishes to put on the parameter
estimates.
Values of the factor t are given in Table 1 as a function of the number of degrees
of freedom ν and a parameter α that expresses the confidence interval of the estimate.
The number of degrees of freedom is the number of data points minus the number of
Appendix A.13.
Statistical Analysis of Experimental Data Ap.13- 8
parameters being fit to the data. For the straight-line equation with finite intercept b,
ν = n - 2, where n is the number of data points, and α is given by
X
α = 1.0 − (A.13-12)
100
Table A.13-1. Tabulated values of t as a function of the degrees of freedom (v) and tail
probability α.
Appendix A.13.
Statistical Analysis of Experimental Data Ap.13- 9
For engineering purposes, it is common to report results at a 95% confidence limit
or higher. If X is chosen to be 95%, then α = 0.05 . Suppose that an experiment yielded
5 data points. Then ν = 3, and Table 1 indicates that the corresponding value of t is 3.182
. If twice as many data points were taken such that ν = 8, then t would be 2.306 . Taking
more data thus tightens up the range of uncertainty in the parameter estimates. Also,
decreasing the specified confidence probability X reduces the range of uncertainty.
Conversely, using more data points and specifying a wider range of uncertainty in the
parameter values increase the probability X that the true parameter values are within the
stated uncertainty limits.
LINEST Command
There are four arguments in the LINEST function: the range of cells containing
the dependent variable values, y, the range of cells containing the independent variables
values, the "const" logic value, and the "stats" logic value. One usually wants the "const"
value to be true; if const is false or zero, the fit equation is forced to pass through the
origin, i.e., a0=0. The "stats" value needs to be "true" for LINEST to give detailed error
information on the regression results.
To fit a polynomial function using LINEST, one has to calculate explicitly the
polynomial variables in order to force it to work. For example, if one were trying to fit a
set of y(x) data to the polynomial function y=a0+a1*x+a2*x2+a3*x3, one would need to
provide a column of y values, a column for the x values, and separate columns for the x2
Appendix A.13.
Statistical Analysis of Experimental Data Ap.13- 10
values and the x3 values. That is, x2 and x3 are treated like additional independent
variables in order to calculate the coefficients a0, a1, a2, and a3.
After one enters the LINEST function, execution requires hitting shift-control-
enter on the Windows machines or shift-applekey-enter on the Macintosh machines.
Then the statistical information and the values of the coefficients in the fitted function
will appear in the output area.
Bibliography
G.E.P. Box, W.G. Hunter, and J.S. Hunter, Statistics for Experimenters, John Wiley &
Sons, New York, 1978
J. P. Holman, Experimental Methods for Engineers, 3rd ed., McGraw-Hill, New York,
1978
Appendix13
Revised 9/17/98; TWC
Revised 1/20/2002; RCC
Appendix A.13.
Statistical Analysis of Experimental Data Ap.13- 11
Expt. D.3 benchmark problem 1
The benchmark problem for Experiment D.3 in Experiments in Transport Phenomena (Crosby,
1999) is as follows: A tank similar to those in the pilot plant initially contains 5.0 gal of
water at 60 F, and then receives a feed stream of water at 60 F at the rate of 2.0 gal/min.
Steam heat is applied at a constant steam pressure of 10.0 psig. Initally, there is no
ow
out of the tank, but once the tank is full, an exit
ow of 2.0 gal/min is started to maintain
a constant level in the tank.
We want to analyze the heating this system with a steam coil or with an external heat ex-
changer of the types used in the pilot plant. This report contains the theoretical development
necessary to analyze this benchmark case.
= V Ft V (3)
1 0
= V0 =V1 (4)
= V=V1 (5)
where Ts is the steam temperature. Note that is dened dierently that in Crosby (1999).
Equation (1) becomes
( ) = (1 ) + : (6)
For the steam coil, assume that the heat transfer area is proportional to the volume in the
tank. Thus, let = A(t)=A1, and
( ) = (1 ) + : (7)
Expt. D.3 benchmark problem 2
Again, we can make the equations dimensionless, this time using the following variables:
= T T0 (25)
T s T0
= VFt (26)
1
N = UA 1: (27)
^
F Cp
Equation (24) becomes
d + (1 + N ) = N (28)
d
with the initial condition = f at = 0. Thus, at steady state,
1 = 1 +N N (29)
and the transient solution is
( ) = N 1 e(1+N ) + f e(1+N )
(30)
1+N
where f is calculated using the end result from the rst part of this development. This
equation is equivalent to equation (D.3-17) in Crosby (1999) when f = 0.
In this case, the tank is full and the water starts at temperature Tf , but the heating comes
from the heat exchanger. The value of Tf for this case will be dierent from that for the
steam coil case, since the steam coil and heat exchanger heat at dierent rates while the
tank is lling with water. Again, the inlet temperature T1 equals T0.
Again, the volume is constant at V1 . The energy balance on the tank is
C^pV1 dT ^ ^
dt = FhCp(Th T ) FCp(T T0) (31)
and on the heat exchanger,
2 3
FhC^p(Th T ) = (UA)h 4 (Ts T ) (Ts Th) 5 : (32)
ln TTssTTh
These energy balances can be made dimensionless using the same denitions for and as
for the steam coil with a lled tank. In addition, we need to dene Nh = (UA)h =(FhC^p),
as before with the heat exchanger.
Expt. D.3 benchmark problem 5
Assuming a pseudo-steady state condition, the dimensionless energy balance on the heat
exchanger becomes
h = 1 Nh
(1 )
e : (33)
For the tank, we get
+ (1 + ) =
d
(34)
d
where is as dened in equation (18). The initial condition for equation (34) is = f at
= 0.
The solution is therefore
( ) = 1 +
1 e
(1+ )
+ f e
(1+ )
(35)
and
1 = 1 +
: (36)
References
Bird, R. B., Stewart, W. E., and Lightfoot, E. N., 1960, Transport Phenomena. New York:
John Wiley & Sons, pp. 712-713.
Crosby, E. J., 1999, Experiments in Transport Phenomena, revised ed. University of Wisconsin{
Madison.