Transport Phenomena - LEC 5

8/9/2019 Transport Phenomena - LEC 5

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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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11

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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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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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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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

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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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Transport Phenomena - LEC 5

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