Transport Phenomena I: Department of Chemical Engineering Cairo University

Transport Phenomena I
Department of Chemical Engineering

Cairo University
2016/2017
Transport Phenomena – Lectures

TRANSPORT PHENOMENA I
INSTRUCTOR:
Ahmed Sherif Eissa, aseissa@gmail.com 0100-5806060 (Mob)
Ahmed Refaat, aarefaat@hotmail.com 0122-2277897 (Mob)
Hours:
Saturday 10:45 AM to 12:45 PM
Textbook:
1. R.B. Bird, W.E. Stewart and E.N.
Lightfoot, Transport Phenomena, J. Wiley
and Sons, Inc., NY, 2003

Course Activities
• Lectures
• Home Works
• Project
• Exams
Homework
• There will be ~ 2 homework assignments
• Homework should be worked on in groups of 5. Only one
solution per group needs to be turned in.
Project
There will be a group project (5 students per group)
Project Topic: Using Comsol Software for Modeling of

Momentum Transport Problem.
Project report: to be submitted sometimes during the term (date to

be announced later)

Exams& Grading
• There will be a mid term exam
The course grades will be determined as follows:
Final Examination 70%

Homeworks: 5%
Project: 10 %
Exams (2 Mid Terms): 15 %
(1st Exam on 5th of Nov, 2nd Exam on 10th of Dec,
Inshallah)

ATTENDANCE
•As per the university law:
You have to attend at least 75% of lectures to be allowed

to sit for the final exam.

Course Outline
Applied Mathematics and Momentum Transport:
1. Mathematical Preliminaries (Vector/tensor notation)

Index notation, scalar and cross products, Kroenecker delta, permutation
symbol, determinants, tensors, dyadics, symmetric and skew symmetric
tensor.
2. Conservation of Mass & Momentum

Continuity Equation, Equations of Motion, N-S Equation
3. Constitutive Equations & Stress Tensor

Newtonian and Generalized Newtonian Fluids (GNF), Boundary Conditions

Course Outline-cont’d
Complex Flows & Approximate Methods:
4. Lubrication Theory
Order of magnitude analysis for low Reynolds # creeping flows and
simplification of NS equations
5. Stream Functions
Analysis of flow around objects
6. Boundary Layer Approximations

Flow near boundaries
7. Turbulent Flows

Today’s Agenda
Transport Phenomena: (Chapter 0 in BSL)
Overview and Perspective
Course Objectives
Applications & Examples
Mathematical Preliminaries: (Appendix A in BSL)

Vectors and Scalars
Index Notation
Dot Products
Operators

Transport Phenomena, Why?

Levels of Studying Transport Phenomena
(a) Macroscopic
(b) Microscopic
(c) Molecular

How is our Book Organized?

Momentum Transport

Momentum Transport Phenomenon
Fluids Deformation
tied together via Stresses and
or Flow constitutive eqns
(Kinematics)
Forces
Solids:
Fluids:
log τ
.
ie: Newtonian Fluid: τyx = µγ
τyx [=] shear stress µ

.
µ [=] viscosity
γ [=] rate of strain .
log γ
Applications:
a) Flow through a pipeline

Need to know what
is happening in
bulk, boundary
conditions
b) Falling film problems
gas-liquid
interface
θ

Applications:
c) Coating
d) Flow around objects, i.e. sphere

Overall Objectives
Math Preliminaries Conservation law

-Mass
-Momentum Static Problems
Constitutive eqn’s
& BC’s
Exact solns to Complex flows

flow problems; Approx solns
Non-Newtonian
Newtonian -BL
flows/fluids -Turbulence
-Lubrication
Math
Preliminaries

Math preliminaries: Vectors
Index Notation:
x3 a
a3
a2
a1 x2
a = a e + a e +a e
1 1 2 2 3 3
x1
Where e1, e2 and e3 are unit vectors
Unit vector: vector of unit length along axis

e3
e2 |ei| = 1
e1
Vectors: Index notation
Can also write a as:
i =3
a = ∑ae
i =1
i i
Components of a are ai, where i=1, 2 or 3
For simplification, we omit the summation symbol:
a = a e
i i

Vectors: Index Notation
Example:
F =a b c expanded
j i j i
F j = aib j ci ∴ repeated index " i ", so s u m,
F1 = a b c + a b c + a b c
111 2 1 2 3 1 3
Similarly, F2 = ? = ai b2 ci
F =a b c =a b c +a b c +a b c
2 i 2i 121 2 2 2 323
F =a b c +a b c +a b c
3 131 232 333
Vectors: Scalar (dot) Products
Scalar (dot) Products:
a a ⋅ b = a b cos ( θab ) = ai ei ⋅ b j e j
θab
b ( )
= ai b j ei ⋅ e j = ?ai b j δi j
Define δij = Kronecker delta Named after

ei ⋅ e j = δij German
Mathematician:
Leopold
1 0 0 Kronecker
1 if i = j  
δij =  = 0 1 0 so e1 ⋅ e1 = cos 0° = 1
0 if i ≠ j  0 0 1  e1 ⋅ e2 = cos 90° = 0
 
Vectors: Scalar (dot) Products
a⋅a =
(
ai ei ⋅ a j e j = ai a j ei ⋅ e j )
2
ai a j δij = ai ai = a Magnitude of a
So a•b =?
a ⋅ b = (ai ei ) ⋅ (b j e j ) = ai b j ( ei ⋅ e j )
= ai b j δij = ai bi = a1b1 + a2b2 + a3b3

Vectors
The components of a vector can be obtained by dotting
the vector with its corresponding unit vector
ak = ek ⋅ a
i .e.,
e1 ⋅ a = e1 ⋅ ai ei
= ( e1 ) ⋅ ( a1 e1 + a2 e2 + a3 e3 )

Vectors
Another format:
a=aλ
wh ere λ = un it vect or in di rect io n of a
λ = λ1 e1 + λ 2 e 2 + λ3 e3 , λ =1
How to get λι: a ⋅ e j = a λi ei ⋅ e j = a λi δij

ai = a λi
a
λi = i
a
Operators
∂ ∂ ∂
“del” or ∇≡ e1 + e2 + e3
“nabla” ∂x1 ∂x2 ∂x3
∂ψ ∂ψ ∂ψ ∂ψ
Example: ∇ψ ≡ e1 + e2 + e3 = ei
∂x1 ∂x2 ∂x3 ∂xi
∂T ∂T
V ⋅∇T = νi ei ⋅ e j = νi ei ⋅ e j
∂x j ∂x j
∂T
= νi δij
∂x j
∂T ∂T ∂T ∂T
= νi = V1 + V2 + V3
∂xi ∂x1 ∂x2 ∂x3
In Class Exercise
?
∇ ⋅ (V ψ ) = ψ ( ∇ ⋅ V ) + (V ⋅∇ψ )
LHS :
∂
= ei ⋅ e jV j ψ
∂ xi
∂
(
= ei ⋅ e j
∂xi
) (
V jψ )
∂ ∂ψ ∂Vi
=
∂xi
( Vi ψ ) = Vi
∂xi
+ψ
∂xi
= V ⋅∇ψ + ψ ( ∇ ⋅ V )

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Transport Phenomena I

Department of Chemical Engineering

Transport Phenomena – Lectures

Transport Phenomena – Lectures

Project Topic: Using Comsol Software for Modeling of

Project report: to be submitted sometimes during the term (date to

Transport Phenomena – Lectures

• There will be a mid term exam

The course grades will be determined as follows:

Final Examination 70%

Transport Phenomena – Lectures

•As per the university law:

You have to attend at least 75% of lectures to be allowed

Transport Phenomena – Lectures

1. Mathematical Preliminaries (Vector/tensor notation)

2. Conservation of Mass & Momentum

3. Constitutive Equations & Stress Tensor

Transport Phenomena – Lectures

6. Boundary Layer Approximations

Transport Phenomena – Lectures

Mathematical Preliminaries: (Appendix A in BSL)

Transport Phenomena – Lectures

Transport Phenomena – Lectures

Transport Phenomena – Lectures

Transport Phenomena – Lectures

Transport Phenomena – Lectures

τyx [=] shear stress µ

a) Flow through a pipeline

Transport Phenomena – Lectures

d) Flow around objects, i.e. sphere

Transport Phenomena – Lectures

Math Preliminaries Conservation law

Exact solns to Complex flows

Transport Phenomena – Lectures

Unit vector: vector of unit length along axis

Components of a are ai, where i=1, 2 or 3

For simplification, we omit the summation symbol:

Transport Phenomena – Lectures

Define δij = Kronecker delta Named after

Transport Phenomena – Lectures

Transport Phenomena – Lectures

How to get λι: a ⋅ e j = a λi ei ⋅ e j = a λi δij

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