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Transport Phenomena
By
Farhan Ahmad
Department of Chemical Engineering,
University of Engineering & Technology Lahorewww.engineering-resource.com
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Course outline:
2
Viscosity and the mechanism of momentum transport
Velocity distributions in laminar flow
The equations of change for isothermal systems
Velocity distribution with more than one independent variable
Thermal conductivity and mechanism of energy transport
Temperature distribution in solids and in laminar flow
The equations of change for non-isothermal systems
Diffusivity and the mechanisms of mass transport
Concentration distribution in solids and in laminar flow
The equations of change for multi-component systems
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Recommended Books
Text Book
Reference
Books
Transport Phenomena2nd EditionR. Byron Bird, Warren E. Stewart and Edwin N. Lightfoot
1. Transport Phenomena Fundamentals, J. Plawsky, CRC Press, 2009.
2. Transport Phenomena: A Unified Approach, R.S. Brodkey, H.C. Hershey, McGraw-
Hill.
3. Analysis of Transport Phenomena, W.M. Deen, Oxford Univ. Press, 1998.
4. Welty, J.R., Wicks, C.E., Wilson, R.E., Fundamentals of Momentum, Heat, and MassTransfer, 3rd edition, John Wiley & Sons, 1984.
5. Slattery, J.C., Advanced Transport Phenomena, Cambridge University Press,1999.
6. Modeling in Transport Phenomena - A Conceptual Approach, Ismail Tosun
7. Transport Phenomena and Unit Operations - A combined approach, Richard G.Griskey
8. Momentum, heat and mass transfer fundamentals, David P. Kessler, Robert A.Greenkorn
9. Transport Processes and Separation Process Principles, Christie John Geankoplis
10. Momentum Heat and Mass Transfer, C.O. Bennett, J.E. Myers11. Incropera, Frank P., and David P. DeWitt. Fundamentals of Heat and Mass Transfer.
5th ed.
12. J.R. Backhurst, J.H. Harker, J.M. Coulson and J.F. Richardson, Chemical Engineering
Vol.1: Fluid Flow, Heat Transfer and Mass Transfer.www.engineering-resource.com
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4
An Introduction
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Subject
Subject code
Contact hours
Credit hours
Transport Phenomena
Ch.E - 407
3
3
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Pre-requisite for this course .. ??????
Objective of this course:
To provide an understanding of fundamental knowledge of heat, mass, and
momentum transport phenomena.
Illustrate how to solve the problems by using fundamental relations.
To master the skills of applying this knowledge to the design of chemical
engineering unit operations.
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Transport Phenomena:
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What .. ???
Why .. ???
How .. ???
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Levels
8
Macroscopic
Microscopic
Molecular
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Problems:
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Class 1
Class 2
Class 3
Class 4
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Suggestions:
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Always read the text with pencil and paper in hand; work through thedetails of the mathematical developments and supply any missingsteps.
Whenever necessary, go back to the mathematics textbooks to brush upon calculus, differential equations, vectors, etc. This is an excellent timeto review the mathematics that was learned earlier.
Make it a point to give a physical interpretation of key results; that is,get in the habit of relating the physical ideas to the equations.
Always ask whether the results seem reasonable. If the results do notagree with intuition, it is important to find out which is incorrect.
Make it a habit to check the dimensions of all results. This is one verygood way of locating errors in derivations.
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Transport Phenomena - An Introduction
Basic Concepts
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Basic Concepts:
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Conserved Quantities
Chemical species
Mass
Momentum
Energy
Law of Conservation of Quantities
Conservation of Chemical species
Conservation of Mass
Conservation of Momentum
Conservation of Energy
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Basic Concepts:
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Rate Equation
It describes the transformation of conserved quantity.
Transformation of conserved quantity is based on specified unit of time (Rate).
Components of Rate Equation
Input
Output
Generation
Consumption
Accumulation
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Basic Concepts - Characteristics
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Independent of the level of application
Independent of the coordinate system to which they are applied
Independent of the substance to which they are applied
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Basic Concepts - Application
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Balances
Control Volume
Control surface
Types of Balances
Overall Balance
Differential Balance
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Basic Concepts - Definitions
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The notation of conserved quantity is
x, y & z = three independent space variables
t= one independent time variable
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Basic Concepts - Definitions
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Steady-state
Uniform Equilibrium
Flux
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Basic Concepts Mathematical formulation
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1. Inlet and Outlet terms
2. Generation and consumption term3. Accumulation term
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Basic Concepts Simplification of Rate equation
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Case I : Steady state transport without regeneration
Case II : Steady state transport with regeneration
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Momentum Transport
Viscosity and Mechanism of Momentum Transport
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Momentum Transport - Introduction
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o Matter
Solid
Fluid
Liquid
Gas
o What is the difference between Solid and Fluid?
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Momentum Transfer:
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o Fluid Mechanics
Fluid Statics
Fluid Dynamics
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Momentum Transport - Introduction
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o Viscosity
o Newton's Law of Viscosity
o Applications of Newtons Law
o Kinematic Viscosity
o Viscosity in Laminar flow
o Viscosity in Turbulent flow
o Viscosity of gases
o Viscosity of liquids
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Momentum Transport - Introduction
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o Rheology
o Types of fluids
o
Newtonian fluids
o Non-Newtonian fluids
o Classification of Non-Newtonian fluids
o Time Independent
o Time dependent
o Viscoelastic fluids
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Newton's Law of Viscosity
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Non-Newtonian fluids:
Time Independent
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Non-Newtonian fluids:
The functional dependence between the shear stress and the velocity gradient ismore complex.
We can write in the most general format:
Steady state rheological behavior:
Where,
= Apparent Fluid Viscosity, a function of either yx/ dvx/dy/ both
decreases with shear rate
increases with shear rate
independent of shear rate
, , 0xyxdv
f fluid propertiesdy
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Models for Non-Newtonian fluids:
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o The Bingham Model
o The Ostwald-de Waele Model
o The Eyring Model
o The Ellis Model
o The Reiner-Philippoff Model
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These model equations should not be usedoutside their range of validity.
Graphical representation of two-parameter model
These models are empirical, that is theparameters of the models are obtained bycurve fitting
The parameters are function of T, P andcomposition
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Turbulence Model:
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Zero equation model
Baldwin-Lomax model
Cebeci-Smith model
One equation model
Spalart-Allmaras model
Baldwin-Barth model
Two equation model
K-omega model
K-epsilon model
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Prediction of Viscosity of gases & liquids:
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o Extensive data is available
o Estimation by empirical methods
For mixtures:
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Example 1.3-1
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Estimate the viscosity ofN2 at 50
oC and 854 atm,given M = 28.0 g/g-mole,Pc = 33.5 atm, and Tc =126.2 K.
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33
Velocity Distribution in Laminar Flow
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Shell momentum balances and boundary conditions
(differential momentum balances)
Flow of a falling film
Flow through a circular tube
Flow through an annulus
Flow of two adjacent immiscible fluids
Creeping flow around a sphere
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(a) Laminar flow, in which fluid layers movesmoothly over one another in the direction offlow, and
(b) Turbulent flow, in which the flow pattern iscomplex and time-dependent, with considerablemotion perpendicular to the principal flow
direction.
The methods and problems in this chapter apply only to steady flow.
Pressure, density, and velocity components at each point in the stream do not
change with time.
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Momentum
Rate of change of momentum
Momentum flux
Momentum balance or Conservation of momentum
Ways of momentum transfer
Shell momentum balance
Steps in shell momentum balance
Boundary conditions
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Shell Momentum Balance:
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Momentum Balance over a thin shell of fluid
For steady-state flow, where the rate of accumulation = zero
Rate of Rate of All forces acting 0Momentum In Momentum Out on the system
1. A mathematical expression showing the balance of rate of change of momentum and
forces acting on the control volume(NEWTONS SECOND LAW OF MOTION)
2. Balance is made on a small shell of dimensionsx,y,z
3. All quantities are written in terms of fluxes
4. Solution gives velocity distribution leading to maximum velocity, average velocity,
flow rates and stresses at surfaces etc
5. This procedure of analysis is called analysis through first principle
6. Generally can be applied to simple geometries and idealized flow situations
7. A combination of these simple analysis lead to complex geometries and flow systems
8. Simple system analysis help in understanding complex systems
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The procedure for setting up and solving viscous flow problemsusing Shell Momentum Balance:
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Select a shell of finite thickness
Write a momentum balance over a thin shell
Let the thickness of the shell approach zero and make use of the definition
of the first derivative to obtain the corresponding differential equation for
the momentum flux.
Integrate this equation to get the momentum-flux distribution.
Insert Newton's law of viscosity and obtain a differential equation for the
velocity.
Integrate this equation to get the velocity distribution.
Use the velocity distribution to get other quantities, such as the maximum
velocity, average velocity, or force on solid surfaces.
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Boundary conditions:
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statements about the velocity or stress at the boundaries of the system
At solid-fluid interface
o fluid velocity equals the velocity with which the solid surface is moving
At liquid-gas interface
o Momentum flux in liquid phase is very nearly zero.
At liquid-liquid interface
o Momentum flux perpendicular to the interface, and velocity are continuous across the
interface.
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Common Boundary Conditions in Fluid Mechanics
NO-SLIP AT THE WALL
Also called boundary condition of the first kind (Dirichlet BC)
At solid-fluid interface, the fluid velocity equals to the velocity of the solidsurface.
at the wallfluid wallV V
SYMMETRY
At the plane of symmetry in flows the velocity field is the same on eitherside of the plane of symmetry, the velocity must go through a minimum or amaximum at the plane of symmetry.
Thus, the boundary condition to use is that the first derivative of thevelocity is zero at the plane of symmetry
at the plane of symmetry
0fluid
m
V
x
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STRESS CONTINUITY When a fluid forms one of the boundaries of the flow, the stress iscontinuous from one fluid to another, there are two possibilities
1. For a viscous fluid in contact with an inviscid (zero or very low viscosity fluid)
At the boundary, the stress in the viscous fluid is the same as the stress in theinviscid fluid.
Since the inviscid fluid can support no shear stress (zero viscosity) this meansthat the stress is zero at this interface.
The boundary condition between a fluid such as a polymer and air, for example,would be that the shear stress in the polymer at the interface would be zero.
This is also called Boundary Condition of Second Kind
OR
Newmann BC
at the boundary of two fluids0ij
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STRESS CONTINUITY, contd.
Alternatively if two viscous fluids meet and form a flow boundary,
This same boundary condition would require that the stress in one fluidequal the stress in the other at the boundary.
at the boundary at the boundary
fluid 1 fluid 2ij ij
VELOCITY CONTINUITYWhen a fluid forms one of the boundaries of the flow then along withstress at the boundary, the velocity is also continuous from one fluid toanother.
This is also called Boundary Condition of Fourth Kind
fluid 1 fluid 2at the boundary at the boundary
V V
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Flow of falling film
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Inclined flat plate Length = L
Width = W
Assume viscosity and density of the fluid to be constant.
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L
W
x
x+x
Direction
of
Transport
Shellxz
y
Step 1: Draw the physical diagram
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Step 4: Draw a shell
The shell is one dimensional as there is only one transport direction. The shell is drawn such that the surfaces are at x and x+x,
which are perpendicular to the direction of transport.
Step 2: Possible transport mechanism Transport of shear across the thickness of the film.
Step 3: Frame of coordinates and direction of transport Choose a frame of coordinate (x,z) with x across the film and z along thefilm.
The direction of transport is in the x-direction.
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Step 5: Momentum balance
Rate of z-momentum in at the surface at x = x
xz xWL
surface areashear (force/area) at thesurface x
Rate of z-momentum out at surface x = x + x
xz x xWL surface area
shear (force/area) at thesurface x+ x
Gravity force in the z-direction acting on the volume of the shell =
cosWL x g
Volume of Shell gravity force per unit volume in z-direction
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Momentum balance equation
cos 0xz xzx x xWL WL WL x g
Divide the equation by the volume of the shell and make the shell as thin as possible:
The momentum balance equation is finally a differential equation(valid at any point in the fluid film)
0lim cos 0
xz xzx x x
xg
x
cosxzd
gdx
cosxz g x C Linear
Upon deriving this equation nothing has been said about the fluid behavior. Hence, it is
applicable to Newtonian as well as non-Newtonian fluids.
Definition of derivative:
0
( ) ( ) ( )limx
d f x f x x f x
d x x
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cosxzd
gdx
Step 6: Apply the Newtonian lawFrom this step onward, we haveto decide that what kind of fluidwe are using
zvxz
d
dx
2
z2
v cosd gdx
second-order ODE in terms ofvelocity
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Step 7: Impose physical constraint
v
0 0z
xz
d
x dx
v 0zx
Step 8: Solve ODE for velocity distribution
2
z
2
v cosd g
dx
2
z 1 2
cosv
2
gx C x C
22
z
cosv 1
2
g x
x =
x = 0
is zero at the gas-
liquid interface
At solid-fluid interfacethe velocity is zero
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22
z
cosv 1
2
g x
This is the velocity distribution in the film. It is parabolic in shape (only for Newtonian fluids)
Step 9: Useful quantities
22
z
cosv 1
2g x
Maximum velocity Average velocity
volumetric flow rate Shear force actingon solid surface
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Maximum velocity
2
z max, at x = 0
cosv
2
g
Average velocity(defined as the mean velocity when multiplied by the cross section area will give thevolumetric flow rate)
2
z z, max
cos 2v v
3 3
g
Volumetric flow rate
3cos
Q3
gW
z
0 0
z z
0
0 0
v1
v v
W
W
dxdy
dx
dxdy
Shear force acting on solidsurface
coszF g WL www.engineering-resource.com
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What if the fluid is non-Newtonian
Step 6: Apply the Appropriate Non-Newtonian model
Let us try the Ostwald-de Waele model for non-Newtonian fluids
1n
z z
xz
dv dv
m dx dx
cosxz
dg
dx
Momentum Equation
cos
n
zdvd
m g
dx dx
Step 7: physical constraint
A Nonlinear ODE
v0 0zxz
dx
dx
v 0zx www.engineering-resource.com
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Step 8: Solving for Velocity Distribution
1 11
zcosv 1
1
nn nn
n g x
n m
Maximum velocity
11
z max, at x = 0
cosv
1
nnn g
n m
Average velocity
11
z
cosv
2 1
nnn g
n m
Volumetric flow rate
1
2 1cos
Q2 1
nnnW gW
n m
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For example n= 0.566
Compare withNewtonian Fluids
2.77
z
z max
v 1v
x
2
z
z max
v 1v
x
Stronger than parabolicdependence
Implication of this example:
This study of thin film is useful in the analysis of:
1. wetted wall tower
2. evaporation and gas absorption
3. coating
4. drainage from plate0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
0.0
0.2
0.4
0.6
0.8
1.0
V/(V)max
x/
Non-Newtonian fluid
Newtonian fluid
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