brodkey, r. s. and hershey, h. c. - transport phenomena - a unified approach

_I
.T

TRANSPORT PHENOMENA
A Unified. Approach
.
.
McGraw-Hill Chemical Engineering Series
Editorial Advisory Board
James J. Carbeny, Professor of Chemical Engineering, University of Notre Dame
James R. Fair, Professor
of
Chemical Engineering, University of Texas, Austin
William P. Schowalter, Professor of Chemical Engineering, Princeton University
Matthew
‘IkreU,
Professor of Chemical Engineering, University of Minnesota
James Wei, Professor of Chemical Engineering, Massachusetts Institute of Technology
I&xx
S. Peters, Emeritus Professor of Chemical Engineering, University of Colorado
BUILDING THE LITERATURE OF A PROFESSION
Fifteen prominent chemical engineers first met in New York more than
60
years ago to
plan a continuing literature for their rapidly growing profession. From industry came
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sequently by S. D. Kirkpatrick as consulting editor.

After several meetings, this committee submitted its report to the McGraw-Hill Book
Company in September 1925. In the report were detailed specifications for a correlated
series of more than a dozen texts and reference books which have since become the
McGraw-Hill Series in Chemical Engineering and which became the cornerstone of the
chemical engineering curriculum.
From this beginning there has evolved a series of texts surpassing by far the scope and
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Chemical engineering is a dynamic profession, and its literature continues to evolve.
McGraw-Hill and its consulting editors remain committed to a publishing policy that
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ars to come.
THE SERIES
Bailey and Ouip: Biochemical Engineering Fundamentals
Bennett and Myers: Momentum, Heat, and Mass Transfer
Beveridge and Schechter: Optimization: Theory and Practice
Brodkey
and
Hershey: Transport Phenomena:
4
iif
fied Approach
cprberry:
Chemical and Catalytic Reaction
En&
&ig

1

)
_,
Coughanowr
and
Koppel:
Process Systems Analysis and Control
Edgar and Himmelbhm: Optimization of Chemical Processes
Fabien: Fundamentals of Transport Phenomena
FInlayson: Nonlinear Analysis in Chemical Engineering
Gates, Katzer, and
!3chuit:
Chemistry of Catalytic Processes
Holland: Fundamentals of Multicomponent Distillation
Holland and Liipis: Computer Methods for Solving Dynamic Separation Problems
Katz, Cornell, Kobayashi, Poettmann, Vary, Elenbaas, and Weinang:
Handbook of Natural Gas Engineering
King: Separation Processes
Loyben:
Process Modeling, Simulation, and Control for Chemical Engineers
McCabe, Smith, J. C., and Harriott: Unit Operations of Chemical Engineering
Mickley, Sherwood, and Reed: Applied Mathematics in Chemical Engineering
Nelson: Petroleum Refinery Engineering
Perry and Cbilton (Editors): Chemical Engineers’ Handbook
Peters: Elementary Chemical Engineering
Peters and Timmerhaus: Plant Design and Economics for Chemical Engineers
Probstein and Hicks: Synthetic Fuels
Reid, Prausnitz, and Shenvood: The Properties of Gases and Liquids
Resnick Process Analysis and Design for Chemical Engineers

Sattertield:
Heterogeneous
C?talysis
in Practice
Sherwood, Pigford, ind Wiie: Mass Transfer
Smith, B. D.: Design of Equilibrium Stage Processes
Smith, J. M.: Chemical Engineering Kinetics
Smith, J. M., and Van Ness: Introduction to Chemical Engineering Thermodynamics
Treybal:
Mass Transfer Operations
Valle-Riestraz Project Evolution in the Chemical Process Industries
Van Ness and Abbott: Classical Thermodynamics of Nonelectrolyte Solutions:
With Applications to Phase Equilibria
Van Wile: Distillation
Volk: Applied Statistics for Engineers
Wdas: Reaction Kinetics for Chemical Engineers
Wei, Russell, and Swartzlander: The Structure of the Chemical Processing Industries
Whitwell and Toner: Conservation of Mass and Energy
“f
TRANSPORT
PHENOMENA
A Unified Approach
Robert S. Brodkey
The Ohio State
Universi@
Harry C. Hershey
The Ohio State University
McGraw-Hill Book Company
New York St. Louis San Francisco Auckland BogotP Hamburg
London Madrid Mexico Milan Montreal New Delhi Panama

Paris
SHo
Paula Singapore Sydney Tokyo -Toronto
TRANSPORT PHENOMENA
A Unified Approach
INTERNATIONAL EDITION
\
Copyright
@
1988
Exclusive rights by McGraw-Hill Book Co
T
Singapore
for manufacture and export. This book cannot be
re-exported from the country to which it is consigned by
McGraw-Hill.
34567CM0943210
Copyright
0
1988 by McGraw-Hill Inc. All rights reserved.
Except as permitted under the United States Copyright Act of 1976,
no part of this publication may be reproduced or distributed in any
form or by any means, or stored in a data base or retrieval system,
without the prior written permission of the publisher.
This book was set in Times Roman.
The editor was B.J. Clark;
the production supervisor was Denise L. Puryear; ’
Project supervision was done by Universities Press, Belfast.
Library of Congress Cataloging-in-Publication Data
Brodkey, Robert S.

Transport phenomena.
(McGraw-Hill chemical engineering series)
Bibliography: p.
Includes index.
1. Transport theory. I. Hershey Harry C.
II. Title. III. Series
TPI
56.T7B76 1988 660.2’842 86-34414
ISBN
0-07-007963-3
When ordering this title use
ISBN
0-07-100152-2
Printed in Singapore
“{
CONTENTS
I
i
Preface
To the Instructor
xv
xvii
Part I Basic Concepts in Transport Phknomena
1
1.1
1.2
1.3
1.4
1.5
2

2.1
2.2
2.3
2.4
2.5
Introduction to Transport Phenomena
Transport Phenomena and Unit Operations
Equilibrium and Rate Processes
Fundamental Variables and Units
The Role of Intermolecular Forces
Simple Balances
Problems
References
Molecular Transport Mechanisms
The Analogy
2.1.1
The Case for Heat Transfer
2.1.2 The Case for Mass Transfer
2.1.3 The Case for Momentum Transfer
2.1.4 The Analogous Forms
Heat Transfer
Mass Transfer
Momentum Transfer
Heat, Mass and Momentum Diffusivities
2.51 Thermal Conductivity
2.5.2 Diffusion Coefficient
2.5.3 Viscosity
3
4
4

5
9
9
11
13
14
18
18
21
22
25
30
32
40
46
47
50
51
vii
.

.

.
vul

CONTENTS
2.6
A Comparison of the Transports
53

Problems
55
References
59
3.4
The Continuity Equation
3.5
The General Property Balance for an Incompressible Fluid
3.6 Summary
Problems
4
pI$ek&ar Transport and the General Property
4.1
Steady Transport in One Direction Involving Input-Output
with no Generation
4.1.1 Constant-area Transport
4.1.2 Variable-area Transport
4.2
Steady State Transport With Generation
4.2.1
Heat or Mass Transport with Constant Generation
4.2.2 Momentum Transfer with Generation at Steady-State
4.2.3 Laminar Flow in a Tube
4.2.4 Laminar Flow Between Parallel Plates
4.2.5 Variable Generation
4.3 Concluding Remarks
Problems
References
5
Transport with a Net Convective Flux

5.1
Convective Flux Caused by Forced Convection
5.1.1 The Balance Equation
51.2 Coordinate Systems
51.3 Relationship Between Shear Stress and Shear Rate
51.4 The Continuity Equation
51.5 The Energy Balance
5.1.6 The Navier-Stokes Equation
5.1.7 The Boundary Layer
5.2 Convected Coordinates
5.3
Mass
Diffusion
Phenomena
5.3.1
Mass Flwes in Stationary and Convected Coordinates
60
62
64
65
66
67
72
72
72
77
82
85
87
87

90
93
95
95
103
104
108
113
119
124
125
126
128
129
132
134
134
135
138
142
146
157
160
161
161
5.4
5.5
6
6.1
6.2

6.3
6.4
6.5
6.6
7
7.1
7.2
7.3
CONTENTS
5.3.2
Total Flux and
Fick’s
Law
5.3.3 Binary Mass Diffusion in Gases
5.3.4 Binary Mass Diffusion in Liquids
5.3.5

Diffusion
in Solids
53.6 Diffusion due to a Pressure Gradient
53.7 Diffusion with Three or More Components
Less Common Types of Mass and Thermal Transport
5.4.1 Heat Transport
5.4.2 Mass Transport
Summary
Problems
References
Flow Turbulence
Transitional and Turbulent Flow
6.1.1 The Reynolds Experiment

6.1.2 Transitional Flow
6.1.3 Fully Developed Turbulent Flow
The Equations for Transport under Turbulent Conditions
6.2.1
Reynolds Rules of Averaging
6.2.2 Reynolds Equation for Incompressible Turbulent Flow
6.2.3 Reynolds Stresses
6.2.4 Turbulent Flow in Channels and Pipes
Turbulence Models
6.3.1 The Boussinesq Theory
6.3.2 The Prandtl Mixing Length Theory
6.3.3 Analogies
6.3.4
Film and Penetration Theories
The Velocity Distribution
Friction Factor
Summary
Problems
References
Ink
8
ral Methods of Analysis
The eneral Integral Balance
7.1.1
The Integral Mass Balance
7.1.2
The Integral Balance on an Individual Species
7.1.3 The Integral Momentum Balance
7.1.4 The Integral Energy Balance
7.1.5 The Mechanical Energy Equation and the Engineering

Bernoulli Equation
Fluid Statics
7.2.1 Manometers
:
7.2.2 Buoyant Forces
7.2.3
Variation of Pressure with Depth
Recapitulation
ix
168
172
179
180
182
186
186
187
187
188
189
193
195
198
198
201
206
210
214
220
223

225
227
227
229
234
236
240
257
260
261
263
265
268
270
273
275
286
295
305
305
316
319
321
x
coN-rENTs
Problems
323
References
326
8 Methods of Analysis

8.1
Inspection of the Basic Differential Equations
8.2 Dimensional Analysis
8.2.1
Rayleigh
Method of Analysis
8.2.2 Buckingham Method
8.2.3 Completeness of Sets
8.3 Modeling
Problems
References
327
330
335
339
346
350
353
355
356
Part II Applications of Transport Phenomena
9
9.1
9.2
9.3
9.4
9.5
9.6
9.7
10

10.1
10.2
Agitation
Introduction to Agitation
Equipment
Geometric Similarity and Scale-up
Design Variables
Dimensionless Numbers
Scale-up
9.6.1
Scale-up Procedures for Turbulent Flow with Three or
More Test Volumes
9.6.2 Scale-up Procedures for Turbulent Flow with Two Test
Volumes
9.6.3 Scale-up Procedures for Turbulent Flow with a Single
Test Volume
9.6.4 Scale-up Procedure for Laminar Flow
9.6.5
Scale-up Without Geometric Similarity
Summary
Problems
References
Transport in Ducts
Review
10.1.1 Laminar Pipe Flow
10.1.2 Turbulent Pipe Flow
Piping Systems
10.2.1 Roughness
10.2.2
Pressure Drop in Rough Pipes

10.2.3 von Karman Number
10.2.4
Solutions of Large Molecules
10.2.5
The Velocity Head Concept
10.2.6 Curved Tubes
10.2.7
Expansion and Contraction Losses
10.2.8
Pipe Fittings and Valves
359
362
364
371
372
374
383
384
385
386
395
396
396
397
398
400
403
403
406
409

409
413
417
420
421
422
424
430
CoNreNTs
xi
10.3
10.4
10.5
10.6
11
11.1
11.2
11.3
11.4
11.5
11.6
I2
12.1
10.2.9 Gases
10.2.10 Complex Fluid Flow Systems
Noncircular Conduits
Measurement of Fluid Flow
10.4.1 Orifice Meter
10.4.2 Venturi and Nozzle
10.4.3 Rotameter

10.4.4 Pitot Tube
10.45 Other Flow Metering Devices
Measurement of Pressure
Measurement of Temperature and Concentration
Problems
References
Heat and Mass Transfer in Duct Flow
Review and Extensions
11.1.1 Radiation
11.1.2 Convection
11.1.3 Conduction
11.1.4 The Resistance Concept
11.15 Slope at the Wall
11.1.6
Bulk and Film Temperatures
Laminar Pipe Flow
11.2.1 Fully Developed Transfer
11.2.2 Entry Region
Heat and Mass Transfer During Turbulent Flow
11.3.1
Review of Turbulence Models
11.3.2
Correlations for Fully Developed HOW
11.3.3 The Analogies
11.3.4 Other Methods
Double-pipe Heat Exchangers
11.4.1
The Overall Heat Transfer Coefficient
11.4.2
Contact Resistance and Fouling Factors

11.4.3 Design Equations
11.4.4 Simple Solutions
Multipass Heat Exchangers
11.51 Equipment
11.5.2 Design Equations
Other Topics
Problems
References
Transport Past Immersed Bodies
The Boundary Layer and the Entry Region
12.1.1
The Laminar Boundary Layer
12.1.2
The Turbulent Boundary Layer
12.1.3
Heat and Mass Transfer During Boundary Layer Flow
Past a Flat Plate
442
443
455
459
460
469
471
476
479
481
482
484
487

489
493
493
493
494
494
504
505
506
506
510
512
512
512
516
520
526
528
530
532
535.
539
539
541
546
547
549
551
556
557

566
571
xii

CONrENTs
12.2
Flow Over Cylinders and Spheres
578
12.2.1
Ideal Flow (Nonviscous Fluids)
578
12.2.2
Stokes Flow Past a Sphere
587
12.2.3 Drag Coefficient Correlations
591
12.3
Flow Phenomena with Solids and Fluids
600
12.3.1 Introduction to Fluidization
601
12.3.2 Gas-Solid
Fhridization 606
12.3.3 Liquid-Solid Fluidization 613
12.3.4 Packed Beds
619
12.35 Single-Cylinder Heat Transfer
623
12.3.6
Banks of Tubes

626
12.4
Flow Phenomena with Gas-Liquid and Liquid-Liquid Mixtures
634
Problems
63.5
References
637
I.3
Unsteady-state Transport
13.1 Basic Equations
13.1.1 Heat Transfer Equation
13.1.2 Mass Transfer
13.1.3 Error Function
13.1.4
Heat Transfer with Negligible Internal Resistance
13.2
Finite Slab and Cylinder
13.2.1 Fourier Series Solution
13.2.2 Lapiace Transform Solution
13.2.3 Generalized Chart Solution
13.2.4 Numerical Solution
13.3 Other Geometries
13.3.1 Infinite Slab
13.3.2 Semi-infinite Slab
13.3.3 Cylinder
13.3.4 Sphere
Problems
References
640

644
645
646
646
647
652
654
665
669
685
696
696
698
701
702
703
706
Part III Transport Property
14
Estimation of Transport Coefficients
711
14.1 Cases
714
14.1.1 Kinetic Theory of Gases
714
14.1.2 Nonuniform Gas Theory
721
14.1.3 Empirical Correlations for Gases
731
14.2

Liquids
733
14.2.1 Viscosity
733
14.2.2 Thermal Conductivity
736
14.2.3 Diffusion Coefficient
738
14.3 Solids
745
14.4
Measurement of the Transport Properties
745
15
15.1
15.2
15.3
15.4
15.5
15.6
coNl?wrs
14.4.1 Viscosity Measurements
745
14.4.2 Thermal Conductivity
746
14.4.3 Diffusion Coefficient Measurements
746
Problems
747
References

749
Non-Newtonial Phenomena
Rheological Characteristics of Materials
15.1.1 Time-Independent Behavior
151.2 Time-Dependent Behavior
15.1.3 Viscoelastic Behavior
Rheological Measurements
15.2.1 Capillary Viscometer
15.2.2 Rotational Viscometers
Turbulent Flow
Agitation of Non-Newtonian Fluids
Heat Transfer in Pipe Flow
Summary
Problems
References
Appendixes
A Properties of Materials
A.1
Properties of Water and Air
Table A.1
Thermophysical Properties of Saturated
Water
Table A.2
Thermophysical Properties of Dry Air
A.2 Prediction of Transport Properties
Table A.3 Constants in the Lennard-Jones 12-6
Potential as Determined from Viscosity Data
Table A.4
Le Bas Atomic and Molar Volumes at the
Normal Boiling Point

B
Mechanical Characteristics of Pipe and Tubing
Table B.l Standard Steel Pipe Dimensions, Capacities,
and Weights
Table B.2 Condenser and Heat-Exchanger Tube Data
C
Physical Constants, Units, and Conversion
Tables
Table C.l Physical Constants
Table C.2 SI Base and Supplementary Quantities and
Units
Table C.3
Derived Units of SI Which Have Special
Names
.

.

.
xlu
752
755
756
761
762
770
771
777
778
783

784
786
786
788
791
791
792
794
796
797
799
802
803
805
806
807
807
808
xiv

CONTENTS
Table C.4 SI Prefixes
808
Table
C.5
Density (or Specific Volume)
809
Table C.6
Diffusivity
809

Table C.7 Force
809
Table C.8 Gravitational Conversion Constant
809
Table C.9 Heat Capacity
809
Table
C.10
Heat Transfer Coefficient
810
Table C.ll Length
810
Table C.12 Mass
810
Table C. 13 Mass Transfer Coefficient
810
Table C.14 Power
811
Table C. 15
Pressure or Momentum Flux or Shear Stress
811
Table C.16 Thermal Conductivity
811
Table C. 17 Viscosity
812
Table C.18 Volume
812
Table C.19. Work, Energy, and Torque
812
Table

C.20
Miscellaneous
813
D Vector Mathematics
814
D. 1 Introduction
814
D.2 Scalar Quantities and Vectors
814
D.3 Tensors
816
E Computer Programs
E. 1 Table E. 1
Index of Computer Programs
817
817
Index
818
PREFACE
After publication of the pioneering book Transport Phenomena by Bird,
Stewart, and Lightfoot in 1960, educators everywhere recognized that the
previous “unit operations-unit processes” organization of material for the
curricula of chemical engineers was inadequate for modern engineering
education. Many schools found that the 1960 book was suitable for graduate
courses and an excellent reference, but too difficult for most undergraduates,
especially if the course was offered early in the curriculum. Others followed
this pioneering effort by writing simpler versions.
This book was designed to provide an integrated treatment of the three
areas of transport: momentum, heat, and mass. The similarities and the
differences of the three transports are clearly stated at a level suitable for

second-semester sophomores and first-semester juniors in engineering or the
other sciences where the mathematics requirement is similar. Many of the
basic equations are mathematically identical, when expressed in terms of the
generalized flux and property variables. This identity helps the student
understand transport phenomena and forms the basis for the organization of
the material here. A typical curriculum teaches momentum transfer before
heat and mass because a complete treatment of these latter two is not possible
without a prior discussion of fluid dynamics. This text allows heat transfer,
which is encountered daily by everyone and easily visualized, to explain by
analogy momentum transfer, which is not easily visualized or understood by
neophytes. Transport is rapidly becoming more widely used in most branches
of engineering, and this text provides all engineering disciplines with a
readable and otherwise useful treatment of this difficult subject. In most of the
other books on this subject, these topics are covered separately.
We believe that this text provides a solid foundation for engineering
design and research. At the same time, some interesting and important
problems are solved. A study of transport phenomena does not replace unit
operations, but understanding of transport phenomena provides deeper insight
xv
XVi PREFACE
into the fundamental processes occurring in the unit operations. The engineer
who masters the material in this text will be better able to analyze the unit
operations he or she encounters.
McGraw-Hill and the authors would like to express their thanks for the
many useful comments and suggestions provided by colleagues who reviewed
this text during the course of its development, especially Charles E. Hamrin,
Jr., University of Kentucky; Richard W. Mead, University of New Mexico;
Robert Powell, University of California-Davis; and James Wei, Massachusetts
Institute of Technology.
Finally, the authors owe much thanks to many who have helped over the

years with this project. A partial list (in alphabetical order) includes F.
Bavarian, A. M. Cameron, J. F. Davis, L. Economikos, L. S. Fan, L. Fishler,
K. S. Knaebel, S. G. Nychas, J. Y. Oldshue, C. E. Patch, A. Syverson, G. B.
Tatterson, J. L. Zakin, and the many typists who have helped with this effort,
especially Pat Osborn.
Robert S. Brodkey
Harry C. Hershey
$2
TO THE INSTRUCTOR
This text covers transport phenomena in an integrated manner. It is our
opinion that a solid understanding of fluid mechanics is essential to
understanding and solving problems in heat and mass transport. Hence, all
topics suitable for a course in undergraduate fluid mechanics are covered in
detail. This text introduces the basic equations of heat and mass transfer as
well. This text also covers heat and mass transport applications that are in the
transport phenomena area. It does not cover topics that are traditionally
taught as unit operations. It is expected that the students will purchase a unit
operations book for course work and reference beyond our coverage.
After the introductory chapter, the basic equations of molecular trans-
port are covered first, then the general property balance, followed by the
combination of the balance and molecular transport. The topic of convection is
in Chapter 5. Our treatment is especially strong in discerning the differences
between transport problems with flow but no net convective flux, and those .
with a net convective flux. Chapter 5 also contains a lengthy section on the
fundamentals of mass transport phenomena. Chapter 6 on turbulent flow
provides a thorough discussion of modern turbulence theory. There are also
two chapters (7 and 8) on methods of analysis-integral methods and
dimensional and modeling approaches. Dimensional analysis is applied to
agitation in Chapter 9. The remaining chapters contain advanced applications.
Chapters 10 and 11 cover transport in ducts. Flow past immersed bodies and

fluidization are discussed in Chapter 12. Chapter 13 covers unsteady-state
transport phenomena. Chapter 14 covers the estimation of the transport
properties
p,
k, and
DAB;
this chapter can be covered in conjunction with
Chapter 2, or anything thereafter, as the instructor wishes. Chapter 15 on
non-Newtonian phenomena is unique in that this important topic is largely
ignored in other texts. This chapter also can be covered whenever the
instructor wishes; at Ohio State, we have found that our students cannot really
appreciate non-Newtonian flow until they understand Newtonian flow. Hence,
we teach Chapter 15 in conjunction with Chapter 10. If the instructor desires
to cover only the laminar aspects of non-Newtonianism, the appropriate
material from Chapter 15 could be taught much earlier in the course. The
Xvii
xviii
TO THE INSTRUCTOR
$
appendix is in five parts: properties of materials, mechanical characteristics of
piping and tubing, conversion tables, vector mathematics, and a list of
computer programs.
Taken as a whole, this text covers the area of fluid mechanics thoroughly.
The basic equations of change are in the early chapters. Laminar flow solutions
are found in both Chapters 4 and 5 on molecular and convective transport. The
sections on agitation, turbulence, and fluidization contain the most modem
concepts and procedures. Fluid statics is covered in Chapter 7 on integral
methods, where it arises naturally from the general integral balance equations.
Advanced topics include discussions of design of complex piping systems, the
boundary layer, ideal flow, flow past immersed bodies such as spheres and

cylinders, fluidization, packed beds, banks of tubes, and non-Newtonian
fluids.
The inclusion of an entire chapter on non-Newtonian transport phenomena is
an indication of the importance of this subject to chemical engineers;
non-Newtonian fluids are encountered daily in our lives, as well as being
common in industry. The engineer needs some familiarity with this area.
The topics of heat and mass transfer are covered only to the extent that
transport phenomena can be applied. Excellent books exist for these topics,
especially for heat transfer, for which our colleagues in mechanical engineering
have written well-conceived textbooks. In 1984 and 1985, three major books
on mass transfer were published. Because heat and momentum transport are
so closely linked, a weakness of many heat transfer texts lies in their limited
treatment of the fluid mechanics topics needed for heat-exchanger design. Our
integrated approach is intended to explain fully the coupled nature of heat and
momentum transport. The heat transport and momentum transport equations
are presented together for laminar applications, turbulent flow, flow past
immersed bodies, fluidized beds, etc. Similarly, the basic equations for mass
transport are integrated with those for heat and momentum. Chapter 5
discusses mass transport phenomena in detail, including the additional
complexities inherent in mass diffusion. The presentation is in a clear fashion
that undergraduates can understand, especially the reasons why the mass
diffusion equations as simplified for
gases
do not strictly apply to liquid
systems. The basic principle of diffusion in solids is also covered; this topic is
important in catalysis, and other areas as well. The unsteady-state chapter
combines heat and mass transfer and includes the modern numerical methods
as the Crank-Nicolson.
Our text is expected to serve widely as a reference. Hence, more material
and more detail have been included than undergraduates can usually assimi-

late. At the Ohio State University, this text is used for a 4-credit-hour,
one-quarter course, in transport phenomena which is offered to sophomores
who have completed differential equations, freshman chemistry, stoichiometry,
and two quarters of physics. In our course, the material in Chapters 1 to 8 is
covered in detail. The topics of agitation (Chapter 9) is covered rapidly. The
design material in Chapters 10 and 12 occupy the last part of the course.
Chapter 15 on non-Newtonian phenomena is covered briefly after Chapter 10
on fluid flow in ducts. Our thermodynamics course, usually taken later or
concurrently, emphasizes the applications of the first law to flow problems.
Note that this material is in Chapter 7 in our text and in Chapter 2 in
Introduction Co Chemical Engineering Thermodynamics by Smith and Van
Ness. Fluid statics is also covered at the start of the thermodynamics course.
At Ohio State, a second course in transport focuses entirely on heat transfer;
that course covers Chapter 11 in detail plus, of course, much more. That
second course requires the students to purchase a specialized heat transfer text
(usually a mechanical engineering series) in order to cover the specialized
topics in heat transfer, such as radiation, boiling, and condensation.
Further topics in mass transfer can be taught in an additional course or
courses that combine discussions of mass transfer with unit operations not
previously covered, such as absorption, distillation, drying, evaporation, and
filtration. The basic material for mass diffusion is in Section 5.3; most of the
material presented is not covered in the traditional unit operations texts.
Again, the analogy with heat and momentum assists the student in under-
standing the difficult concepts in mass diffusion.
In solving the example problems, we used a computer or a hand-held
calculator. Calculations with these retain many digits in order to reduce
truncation errors. The example problems in this text make no serious effort to
ascertain the correct number of
significant
digits for every final answer,

inasmuch as the purpose of the examples is to illustrate the method of
calculation. The instructor should point out from time to time the probable
accuracy of final answers, especially when the physical constants (such as mass
.diffusivity)
are not usually known accurately or approximate methods of
solution are used. Also, many of the example problems and homework
problems in this text have been in use at Ohio State for more than 20 years.
Hence, their origins are
obscure:
We sincerely apologize if we have inadver-
tently used problems that originated with someone else.
A book on transport phenomena always encounters nomenclature
problems, because the three areas of transport developed independently in the
early days. A problem of more recent vintage is the decision of the American
Institute of Physics to switch the viscosity notation from
~1
to
q.
Chemical
Engineers have used p from the beginning; this text will also use
cc,
although
the instructor may wish to point out that the other symbol is also recom-
mended by some.
Finally, there are some excellent films available, which illustrate most of
the import t topics in fluid mechanics. The
Encyclopaedia
Britannica
Education
a?

Corporation, 425 North Michigan Avenue, Chicago, IL, 60611 has
available for purchase or loan twenty-two
16-mm
films and a hundred
thirty-three &mm film loops as a part of their fluid mechanics program. At the
Ohio State University, we use the &mm loops, which are shown with a small
portable projector and lend themselves easily to informal discussion. These
fihn loops are referenced by number at the appropriate-locations in the text.
Robert S. Brodkey
Harry C. Hershey
%
TRANSPORT PHENOMENA
A Unified Approach
PART
I
BASIC
CONCEPTS
IN TRANSPORT
PHENOMENA
CHAPTER
INTRODUCTION
TO TRANSPORT
PHENOMENA
NOMENCLATURE
A
a
CA
F
&

m
n
P
R
T
V
X
Y
z
t
Area
(m*,

ft*)
Acceleration (m
s-*,
ft
s-*)
Concentration of species A (kmol mv3, lb mol
ft-‘)
Force (N, lb3
Gravitational conversion constant (32.174 lb,,, lb;’ ft
s-*)
Ma= (kg,
lb,)
Number of moles of gas (kmol, lb mol)
Pressure (kPa, atm,
lb,
in *)
Gas constant, see Appendix, Table C.l

Temperature (K, “R, “C, “F)
Volume (m3, ft’)
Unknown in algebraic equation
Unknown in algebraic equation
Unknown in algebraic equation
Shear stress (N m-*,
lbf

ft-‘)
This chapter provides a brief introduction to the material to be covered in
detail in subsequent chapters. First, a brief historical perspective of the role of
transport phenomena in the solution of engineering problems is discussed.
3
4 BASIC CONCEPTS IN TRANSPORT PHENOMkNA
?
Then some fundamental concepts from physics, chemistry, and mathematics
are presented.
1.1 TRANSPORT PHENOMENA AND
UNIT OPERATIONS
The pioneering work Principles of Chemical Engineering was published in
1923
under the authorship of Walker, Lewis, and McAdams [Wl]. This book was
the first to emphasize the concept of unit operations as a fundamental
approach to physical separations such as distillation, evaporation, drying, etc.
This was the era when the profession of chemical engineering matured into a
separate area, no longer the province of the industrial chemist. The study of
unit operations such as distillation is predicated on the idea that similarities in
equipment and fundamentals exist regardless of the process. In other words,
the principles of distillation apply equally to the separation of liquid oxygen
from liquid nitrogen as well as to the thousands of other distillations routinely

carried out in industries around the world. The study of transport phenomena
is undertaken because this topic is the basis for most of the unit operations. In
simple terms, transport phenomena comprise three topics: heat transfer, mass
transfer, and momentum transfer (fluid flow) In many of the unit operations
(such as distillation), all three transport phenomena (i.e., fluid flow, heat
transfer, and mass transfer) occur, otten simultaneously. The concepts
presented in transport phenomena underly the empirical procedures that are
used in the design of unit operations. Empiricism is required because the exact
equations cannot be solved.
1.2 EQUILIBRIUM AND RATE
PROCESSES
Many problems can conveniently be divided into two classifications: equi-
librium and nonequilibrium. Under conditions of nonequilibrium, one or more
variables change with time. The rates of these changes are of much interest,
naturally. A typical engineer-scientist reading this book will be involved with
four types of rate processes: rate of heat transfer, rate of mass transfer, rate of
momentum transfer, and rate of reaction. The first three of these are the
subject of this text. The fourth, rate of reaction, will not be covered in any
detail, except for the inclusion of the appropriate terms in the general
equations and in a few elementary examples.
Equilibrium processes. The science of thermodynamics deals mainly with
systems in equilibrium. Consider Fig. 1.1 which shows a gas composed of 50
mole percent nitrogen and 50 mole percent oxygen enclosed in a tank at a
pressure of 2 atm at 300 K. Let this gas be surrounded by ambient air at the
same temperature. After an appropriately long time, the gas inside the tank is
at physical equilibrium. Its temperature is the same as that of the surrounding
MTRODUCnON
TO TRANSPORT PHENOMENA
5
$/

FlGURFa
1.1
1
Gas mixture
Closed valve
(SO
percent
N2,
50
percent
0,)
gas,
3OOK.
Inside the tank, there will be no concentration gradients. The
science of chemistry tells us that this gas will not be in chemical equilibrium.
Although oxygen and nitrogen can form a series of compounds, such as NzO,
NO, NO,, and
N,O,,
in actuality none of these is formed in the present system
because the rate of reaction is essentially zero. If the gas in the tank were pure
nitrogen, then there would be complete equilibrium inside the tank, i.e.,
physical (often called mechanical) and chemical equilibrium.
.
, ,
Rate
proeemes.
When nonequilibrium processes are considered, the system
under consideration progresses in a manner such as to approach equilibrium.
All such rate processes are characterized by a driving force. The rate of
transport is proportional to the driving force. The topic will be discussed

thoroughly in Chapter 2.
1.3

F'IjNJJAMElWAL
VARIABLES AND
UNITS
Temperature. Interestingly, temperature (T) can be defined only in the
empirical sense as a relative measure of “hotness”
[Dl,
Ml]. The temperature
scales in present use are defined with only one fixed point, the triple point of
water, 273.16 K. Temperature scales are based on changes in properties of
materials with temperature. The change of resistivity of a solid such as
platinum or the change of volume of a liquid such as mercury is easily
measured as a function of temperature, and therefore can be used as an
indication of temperature. Temperature units are Kelvin (K), Celsius
(“C),
Fahrenheit
(“F),
and Rankine
(“R).
The reader already knows how to convert
from one to the other. Temperature is one of the most important quantities in
a system. Temperature manifests itself in the motions of the molecules: the
higher the temperature, the higher are the velocities of the molecules. Almost
all properties are strongly dependent on temperature. The rate processes are
likewise functions of temperature.
Pressare.
The pressure in the tank in Fig. 1.1 has units of force
(F)

per unit