momentum, heat, and mass transfer fundamentals

Momentum, Heat, and
Mass Transfer
Fundamentals
David P. Kessler
Robert A. Greenkorn
MARCEL DEKKER, INC.
Momentum,
Heat,
and
Mass
Transfer
David
I?
Kessler
Robert
A.
Greenkorn
Purdue
University
West La fayette,
Indiana
MARCEL
.
.
.
.
.
.
-
-
-

MARCEL
DEKKER,
INC.
DEKKER
NEW
YORK
BASEL
Library
of
Congress Cataloging-in-Publication Data
Kessler, David P.
Greenkorn.
Momentum, heat, and mass transfer fundamentals
/
David P. Kessler, Robert A.
p. cm.
Includes bibliographical references and index.
ISBN 0-8247-1972-7 (alk. paper)
1.
Transport theory. 2. Heat-Transmission.
3.
Chemical engineering.
I.
Greenkorn, Robert Albert.
TP 156.T7K48 1999
11.
Title.
66W.284242
I
99- 10432

CIP
This book is printed on acid-free paper.
Headquarters
Marcel Dekker, Inc.
270
Madison Avenue, New York, NY
100
16
tel: 2 12-696-9000; fax: 2 12-685-4540
Eastern Hemisphere Distribution
Marcel Dekker
AG
Hutgasse 4, Postfach 8 12, CH-400
1
Basel, Switzerland
tel: 4 1-6 1-26 1-8482; fa: 4 1-6 1-26 1-8896
World Wide Web
http://www .dekker.com
The publisher offers discounts
on
this book when ordered in bulk quantities. For more infor-
mation, write to Special SalesProfessional Marketing at the headquarters address above.
Copyright
0
19W
by Marcel Dekker, Inc.
All
Rights Reserved.
Neither this book nor any part may be reproduced
or

transmitted in any form or by any
means, electronic or mechanical, including photocopying, microfilming, and recording,
or
by any information storage and retrieval system, without permission in writing from the
publisher.
Current printing
(last
digit):
10987654321
PRINTED IN THE UNITED STATES
OF
AMERICA
PREFACE
This text springs from our experience over the past
30+
years teaching the
momentum, heat, and mass transferhansport sequence in the School of
Chemical Engineering at Purdue University.
As
faculty members
in
a state land-
grant institution, we encounter students with
a
wide variety
of
backgrounds
planning for a wide variety
of
ultimate careers. We believe that

with
a
fm
grasp
of engineering fundamentals,
our
graduates can readily progress to careers that
involve either highly technical functions or broader responsibilities
in
management.
Our
objective with this volume is to provide a foundation in basic
momentum, heat, and mass transfer/transport sufficient to permit the student to
do elementary design and analysis, and adequate
as
a base from which to learn
more advanced concepts.
We
present the fundamentals of both microscopic and
macroscopic processes. The text is built around a large number of examples
which are worked
in
detail. Many of the examples are,
of
course, idealized,
because their objective is to illustrate elementary principles, but we have kept
them
as
realistic
as

possible.
Since the
book
is intended
as
a textbook, we have incorporated a
high
level
of detail and fedundancy
in
an
effort
to
make the text readable for those just being
introduced to the area. At the expense
of
conciseness
in
many
places, we have
attempted
to
avoid those gaps
in
derivations that
are
obvious to
one
familiar
to

the area, but utterly opaque to the novice.
We have included
many
references
to
more advanced material, or simply
to
other approaches
to
the material herein. Age
and,
we
hope,
wisdom has disabused
us
of
any conceit that we possesses the only valid approach to the subject. To
make access to other material easier for the student, we have included page
numbers for most references to avoid the necessity
of
an index search
in
the
citation.
In
the same vein, we have attempted to make our nomenclature conform
to the most common usage
in
the
area and have incorporated

an
extensive
nomenclature table.
An
abbreviated thumb index permits rapid access to chapters
and the more commonly accessed tables and figures.
Problems
in
momentum, heat,
and
mass transfer
in
fluids are profoundly
difficult because the most practical application
of
the area
is
to the turbulent
flow regime.
To
date, very little
in
the way
of
rigorous
solution
to
even
the
most

elementary turbulent flow problems is possible, although the exponential
increase
in
computing power
with
time holds great promise for the future.
V
vi
Pre
fuce
We feel that dimensional analysis is, and for the foreseeable future
will
be,
still crucial for design of experiments, scale-up of equipment, and simplifying
differential equations and associated boundary conditions.
Closed-form analytical solutions are still important for the insight that they
bring; however, most applied problems seem clearly destined to
be
solved
numerically. For this reason we have included a fair amount of numerical
solution techniques.
Emphasis is mostly on finite difference algorithms, which are often easily
implemented on a spreadsheet. Finite element analysis
is
heavily reliant on
software, which requires too heavy a time investment
to
permit other than the
abbreviated treatment here, although the area
is

certainly
of
growing importance.
We have attempted to give
an
overview sufficient to enable the student to read
further on hidher
own.
At
Purdue we cover the material
in
this text
in
two three-credit required
undergraduate courses. In general, depending on the background of the students,
the material in Chapter
5
on systems of units will receive more or less
emphasis. Similarly, the transfer of heat
by
radiation stands alone and can be
tailored
to
the desires of the particular professor. Perry’s
ChernicuZ Engineer’s
Hundbookl
remains
a
useful supplemental reference for physical properties
arid

empirical correlations. One of the commercial design packages can give help
with
physical property estimation, pumps and pipe networks, and more detailed
heat exchanger design.
David
P.
Kessler
Robert
A.
Greenkorn
1
Perry,
R.
H.
and
D.
W.
Green, Eds.
(1984).
Clzernicul
Engineer’s
Hwuibook.
New York, NY, McGraw-Hill.
THUMB
INDEX
CONTENTS
1
ESSENTIALS
1.6.2
Types

of
derivatives
2
THE MASS BALANCES
2.1.1 The macroscopic total mass balance
2.1.2 The macroscopic species mass balance
2.2.1 The microscopic total mass balance (continuity equation)
2.2.2 The microscopic species mass balance
3
THE ENERGY BALANCES
3.1.2 The macroscopic total energy balance
3.1.3 The macroscopic mechanical energy balance
3.1.4 The macroscopic thermal energy balance
3.2.1 The microscopic total energy balance
3.2.2 The microscopic mechanical energy balance
3.2.3 The microscopic thermal energy balance
4
THE MOMENTUM BALANCES
4.1 The Macroscopic Momentum Balance
4.2
The Microscopic Momentum Balance
4.3 Summary
of Balance Equations
and
Constitutive Relationships
5
APPLICATION
OF
DIMENSIONAL ANALYSIS
6

MOMENTUM TRANSFER IN FLUIDS
6.7
Drag Coefficients
Table 6.3-1 Elementary plane
flows
Table 6.7.2-1 Properties of
pipe
Figure 6.7.2-3 Moody friction factor chart
xi
1
63
73
74
86
96
103
113
114
141
149
150
157
158
169
169
196
199
211
281
302

371
386
399
vii
Thumb
Index
ix
7
HEAT TRANSFER MODELS
Table 7.2.1
-
1 Components of Fourier Equation
7.2.4 One-dimensional steady-state conduction in rectangular
coordinates
7.2.5 One-dimensional steady-state conduction in cylindrical
coordinates
7.2.6 One-dimensional steady-state conduction in spherical
coordinates
7.2.8 One-dimensional unsteady-state conduction
Semi-infinite slab
Finite slab
Infinite cylinder and sphere
7.2.9 Multi-dimensional unsteady-state conduction
7.3.2 Heat transfer coefficients
7.4 Conduction and Convection in Series
7.5 Radiation Heat Transfer Models
Heisler charts
Reciprocity relation
Summation rule
7.7.3

NTU
methd for design of heat exchangers
7.7.4 F-factor method for design of heat exchangers
8
MASS TRANSFER MODELS
Table 8.2.3-1 Equivalent forms
of
Fick's Law
8.3 Convective Mass Transfer Models
Height of transfer unit models
8.7 Design of
Mass
Transfer Columns
8.8
Mass Transfer with Chemical Reaction
APPENDIX A: VECTOR AND
TENSOR
OPERATIONS
APPENDIX C: NOMENCLATURE
INDEX
517
=
530
=
536
=
574
=
581
=

629
630
638
=
651
667
-
675
=
719
=
734
=
741
=
759
-
760
=
806
=
822
=
843
=
860
882
=
903
=

923
-
963
=
989
=
997
-
1009
=
TABLE
OF
CONTENTS
Preface
Thumb
Index
1
ESSENTIALS
1.1 Models
Figure 1.1- 1 Modeling the weather
Figure 1.1-2
A
poor model of the weather
1.1.1 Mathematical models and the
real
world
1.1.2 Scale of the model
1.2 The Entity Balance
Example
1.2-1

An
entity
balance
1.2.1 Conserved quantities
1.2.2
S
teady-state processes
1.3 The Continuum Assumption
1.4 Fluid Behavior
Figure 1.3-1 Breakdown
of
continuum assumption
1.4.1 Laminar and turbulent flow
1.4.2 Newtonian fluids
Figure 1.4.1-1 Injection
of
dye in pipe flow
Figure 1.4.2- 1
Shear
between layers
of
fluid
Figure 1.4.2-2 Momentum transfer between layers of fluid
Figure 1.4.2-3 Sign convention for momentum
flux
between
layers of fluid
Figure 1.4.2-4 Sign convention for shear stress on surface
layers of fluid
Table 1.4.2- 1 Summary of sign convention for

stresslmomentum flux tensor
Figure 1.4.2-5 Migration
of
momentum by molecular motion
Figure 1.4.2-6 Viscosity of common fluids
Example
I
.4.2-1
Flow
offluids
between frxed parallel
plates
1.4.3 Complex fluids
Figure 1.4.3-1 Complex fluids
Figure 1.4.3-2 Mechanical analog of viscoelasticity
V
vii
1
1
1
2
5
8
12
14
15
16
17
18
19

19
20
21
21
24
24
25
26
26
28
28
29
30
31
xi
xii Tuble
of
Contents
1.4.4
Compressible vs. incompressible flows
1.5.1
General cone
t
of
average
Figure
1.5.1-1
Time-average
speed
for travel between two

points
Figure
1.5.1-2
Distance-average speed for travel between two
points
1.5
Averages
Example
I.
P
.1
-I
Time-average vs distance-average speed
1.5.2
Velocity averages
Area-averaged velocity
Example 1.5.2-1 Area-averuged velocity
for
luminar pipe
POW
Figure
1.5.2-1
Velocity profile
Time-averaged velocity
Exumple 1.5.2-2 Time-uveraged velocity
for
turbulent
POW
Example 1.5.2-3 Area-averuge
of

time-averaged velocity
for
turbulent pipe
flow
1.5.3
Temperature averages
Example 1.5.3-1 Area-uverage temperuture vs. bulk
temperature
Example 1.5.3-2 Bulk temperuture
for
quadratic
temperuture profile, laminar pipe
flow
Example 1.5.4-1 Bulk concentration
Example
I
S.5-1
Case examples
of
logarithmic mean
Example
1.5.5-2
Approximation
of
logarithmic mean
by
urithmetic mean
1.5.4
Concentration averages
1.5.5

Arithmetic, logarithmic, and geometric means
1.6
Scalars, Vectors, Tensors and Coordinate Systems
1.6.1
The viscous stress tensor
Components
of
the viscous
stress
tensor
Figure
1.6.1-1
(a) Vectors associated by a particukv viscous
stress
tensor
with
the direction
of
the rectangular Cartesian
axes
Figure
1.6.1-1
(b)
Vector associated
with
the 3-direction
decomposed into
its
components
1.6.2

Types of derivatives
Partial
derivative
Total derivative
Substantial derivative, material derivative, derivative following the
motion
Example 1.6.2-1 Rute
of
change
of
pollen density
1.6.3
Transport theorem
32
33
33
34
35
36
37
37
39
39
41
42
42
44
46
50
52

54
57
59
59
60
60
61
62
63
63
63
64
64
65
66
Table of Contents xiii
Figure 1.6.3-1 Motion
of
continuum
Chapter 1 Problems
2
THE
MASS BALANCES
2.1
The
Macroscopic Mass Balances
Figure 2.1
-
1 System for
mass

balances
2.1.1 The macroscopic total
mass
balance
Accumulation of
mass
Input and output of mass
Simplified forms of the macroscopic total
mass
balance
Example
2.1.1-1
Mass balance on a surge tank
Figure 2.1.1
-
1
Surge
tank
Example
2.1.1 -2
Volumetricjlow rate offluid in laminar
flow in circular pipe
Example
2.
I.
1-3
Air storage tank
Example
2.
I.

1-4
Water
manifold
2.1.2 The macroscopic species
mass
balance
Generation
of
mass
of
a
species
Accumulation of mass of a
species
Input
and
output of mass of
a
species
Example
2.1.2-1
Macroscopic species
mass
balance with
zero
-0
rde
r
irreversible reaction
Example

2.
I
.2-2
Macroscopic species
mass
balance with
.first-order irreversible reaction
Figure 2.1.2-1 Perfectly mixed
tank
with reaction
2.2.1 The microscopic
total
mass
balance (continuity equation)
Special
cases
of
the continuity equation
Continuity equation
in
different coordinate systems
2.2 The Microscopic Mass Balances
Table
2.2.1-1
Continuity
equation
(microscopic
total
mass
balance) in rectangular, cylindrical, and spherical coordinate

tiames
Example
2.2.
I
-I
Velocity components
in
two-dimensional
steady incompressible
jlow,
rectangular coordinates
Example
2.2.1-2
Velocity components in two-dimensional
steady incompressible jlow, cylindrical coordinates
Example
2.2.1-3
Compression of air
Figure 2.2.1-1
Air
compression by pisiston
2.2.2 The microscopic species mass balance
Diffusion
Chapter 2 Problems
3
THE ENERGY
BALANCES
3.1 The Macroscopic Energy Balances
67
69

73
73
73
74
74
75
77
78
78
79
81
82
86
87
87
88
90
94
94
96
96
98
99
99
99
101
102
102
103
105

105
113
113
xiv
Table
of
Contents
3.1.1
Forms of energy
3.1.2
The macroscopic total energy balance
Rate of accumulation
of
energy
Rates of
input
and
output
of
energy
Figure
3.1.2-1
Flow work
Simplified
forms
of the macroscopic total energy balance
The
potential energy term
The kinetic energy
term

The enthalpy term
Averages and the macroscopic energy equations
Energy balance approximation
-
turbulent
flow
Energy balance approximation
-
laminar
flow
Figure
3.1.2-2
Gravitational field
of
earth
S
teady-state cases
of
the macroscopic total energy balance
Table
3.1.2-
1
Qualitative comparison
of
ranges
of
enthalpy
changes (kcal/mol) for processes involving organic
compounds
Example

3.1.2-1
Relative magnitudes
of
mechmical and
thermal energy terms with phase change
Figure
3.1.2-3
Mechanical energy and thermal energy
terms
compared for a boiler
(I)
Figure
3.1.24
Mechanical and thermal energy terms compared
for a boiler
(U)
Example 3.1.2-2 Steam production
in
a boiler
Exunzple 3.1.2-3 Temperuture rise from conversion
of
mechanical to thermal energy
Figure
3.1.2-5
Water supply system
Emmple 3.1.2-4 Heuted
tank,
steudy state
in
mss

und
unsteady state
in
energy
Figure
3.1.2-6
Heated
tank
3.13
The macroscopic mechanical energy balance
Exumple 3.1.3-1 Mechunical energy and pole vaulting
Exumple 3.1.3-2 Culculution
of
lost work
in
pipe
Figure
3.1.3-1
Pipe system
3.1.4
The macroscopic thermal energy balance
3.2.1
The microscopic total energy balance
Eulerian forms of the microscopic total energy balance
Lagr'angian forms of the microscopic total energy balance
3.2.2
The microscopic mechanical energy balance
3.2.3
The microscopic thermal energy balance
3.2

The Microscopic Energy Balances
Chapter
3
Problems
4
THE
MOMENTUM
BALANCES
113
114
115
116
117
121
121
122
125
127
128
128
129
130
132
132
133
134
134
136
136
138

141
144
147
147
140
150
150
153
156
157
158
162
I69
Table
of
Contents
xv
4.1 The Macroscopic Momentum Balance
Example
4.1
-1
Momentum
flux
offluid in laminar
flow
in
circular pipe
4.1.1 Types
of
forces

4.1.2 Influence of uniform pressure over entire surface of irregular
objects
Figure 4.1.2- 1 Approximation
of
solid by prisms
Figure 4.1.2-2 Detail of prism
4.1.3 Averages
and
the
momentum equation
Momentum balance approximation
-
turbulent
flow
Momentum balance approximation
-
laminar
flow
Example
4.1.3-1
Force
on
a
nozzle
Example
4.1.3-2
Thrust
of
aircru? engine
Example

4.1.3-3
Piping support
Example
4.1.3-4
Jet bout
Exclmple
4.1.3-5
Horizontal force
on
tunk
4.2 The Microscopic Momentum Balance
4.3
Summary
of
Balance
Equations
and
Constitutive Relationships
4.4 The Momentum Equation in Non-Inertial Reference Frames
Chapter 4 Problems
Table 4.3-1 Tabulation of balance equations
Table 4.3-2 Tabulation
of
common constitutive relationships
5
APPLICATION
OF
DIMENSIONAL
ANALYSIS
5.1 Systems of Measurement

Example
5.
I
-I
Weight
vs.
mass;
g
vs.
g,
Table 5.1
-
la
Systems
of
Units
Table 5.1
-
1 b Systems of Units
Table 5.1-2
SI
Prefixes
5.2 Buckingham's Theorem
Example
5.2-1
Dimensionless variables
for
pipe
flow
5.2.1 Friction factors and drag coefficients

5.2.2 Shape factors
Example 5.2.2-1 Drag force
on
ship hull
Exumple
5.2.2-2
Deceleration
of
compressible fluid
5.3
Systematic Analysis
of
Variables
Example
5.3-1
Drag
force
on
a
sphere
Example 5.3-2 Dimensionless groups
foraflow
over
u
flut
plate
Example
5.3-3
Consistency
of

dimensionless
groups
across
system
of
dimensions
Example
5.3-4
Capillary interface height via dimensional
analysis
169
173
1 74
175
176
177
178
178
181
182
186
188
192
194
196
199
199
200
200
203

211
21 1
215
220
221
22 1
222
223
227
229
230
232
234
235
236
238
243
xvi Table
of
Contents
5.4
Dimensionless groups
and
differential
models
Example
5.4-1
Pipe jlow
of
incompressible fluid with

constant viscosity
Example
5.4-2
One-dimensional energy transport
Example
5.4-3
Mass
transport in
a
binary mixture
Example
5.4-4
Extrapolating mdel results j?om one
category
of
momentum, heat, or
mss
transport to another
Table
5.4-
1
Dimensionless variables
Table
5.4-2
Dedimensionalized balance
equations
Table
5.4-3
Dimensionless numbers
5.5

Similarity, Models and Scaling
Example
5.5-1
Drag on immersed body
Example 5.5-2 Scale effects
Chapter
5
Problems
6
MOMENTUM TRANSFER IN FLUIDS
6.1
Fluid Statics
6.1.1
Manometers
Example
6.1.1
-1
Pressure difference using
a
manometer
Figure
6.1.1
-
1
Measurement of pressure difference with
manometer
Example
6.1.1-2
Pressure dinerence between tanks
Figure

6.1.1-2
Pressure difference between
tanks
Example 6.1.1-3 Differential manometer
Figure
6.1.1-3
Differential manometer
Figure
6.2-1
Paths
between streamlines
6.2
Description of Flow Fields
6.2.1 Irrotational flow
6.3
Potential
Flow
Table 6.3-1 Elementary plane flows
Table
6.3-2
Superposition of elementary plane flows
Example
6.3-1
Flow around
a
circular cylinder
Example 6.3-2 Flow
of
an ideal
jluid

through a corner
Example 6.3-3 Flow around
a
rotating cylinder
Figure
6.3-
1 Flow around circular cylinder
Figure
6.3-2
Flow through a corner
6.4
Laminar Flow
6.4.1
Laminar flow between infinite parallel plates
Figure
6.4.1-1
Steady flow between infinite stationary parallel
plates
Example
6.4.
I
-
I
Steady flow between infinite parallel
plates
Figure
6.4.1-2
Flow between infinite parallel plates, top plate
moving at
vo

245
249
2.50
252
2.53
260
26
1
262
264
266
268
272
281
281
284
284
284
285
285
286
286
288
29
1
292
295
302
304
307

308
309
310
31
I
314
315
315
31
8
3
19
Table
of
Contents
xvii
Figure
6.4.1-3
Velocity profiles for laminar flow
of
Newtonian fluid between parallel plates
with
imposed
pressure
drop,
top plate moving at steady velocity
Example 6.4.1-2 Flow between infinite rotating concentric
cylinders
6.4.2
Laminar flow in a circular pipe

Figure
6.4.2-1
Control volume for force balance on fluid in
Figwe
6.4.2-2
Velocity profile
for
laminar
flow
of
a
Newtonian fluid
in
a pipe or duct of circular cross-section
Figure
6.4.2-3
Shear
stress profile for laminar flow
of
a
Newtonian fluid in
a
pipe
or
duct of circular cross-section
Pipe
Example 6.4.2-1 Flow in a capillary viscometer
Example 6.4.2-2 Flow between
two
concentric cylinders

Example 6.4.2-3 Film jlow
down
U
wall
Example 6.4.2-4 Flow adjacent to ujlat plate
instantuneously set
in
motion
Figure
6.4.2-4
Viscometric
flow
between cylinders
Figure
6.4.2-5
Film flow down wall
Figure
6.4.2-6
Flow adjacent
to
flat plate instantaneously set
in motion
6.5
Turbulent Flow
Figure
6.5-1
Local
velocity
in
turbulent flow

as
a
function of
time
Figure
6.5-2
Laminar and
time-smoothed
turbulent
(1/7
power
model) velocity profiles in steady pipe flow
6.5.1
Time
averaging the equations of change
Exumple 6.5-2 Time averaging
of
velocity product
6.5.2
The mixing length
model
Figure
6.5.2-1
Mixing length model
Figure
6.5.2-2
Universal velocity distribution
Example 6.5.2-1 Size
of
sublayer

and
bu$er tone
in
turbulent jlow
6.6
The Boundary Layer Model
Figure
6.6-1
Boundaq layer development
on
flat plate
6.6.1
Momentum
balance
-
integral equations
Figure
6.6.1-1
Element in boundary layer
Figure
6.6.1-2
Velocity profile development
in
the entrance
region
to
a
pipe
6.6.2
De-dimensionalhation of tbe

boundary
layer equations
6.6.3
Exact solution of the momentum boundary layer equations via
similarity variables
Example
6.6-1
Displacement
thickness
321
321
322
322
325
325
326
327
327
330
331
332
332
338
339
341
341
347
347
348
351

351
353
353
354
356
356
359
360
362
xviii Tuble
of
Contents
Example
6.6.3-1
Similarity vuriuble developed
from
dimensional analysis
Example
6.6.3-2
Runge-Kuttu solution
of
Blusius problem
Figure
6.6.3-1
Solution to Blasius boundary layer equation
6.7
Drag Coefficients
Figure
6.7-1
Flow around

an
airfoil (a) without and (b) with
separation
6.7.1
Drag on immersed bodies (external flow)
Figure
6.7.1-1
Drag coefficient for smooth flat plate oriented
parallel
to
flow
stream
Example
6.7.1-1
Drag
on
u
flat
plute
Figure
6.7.1-2
Flow past circular cylinder
Figure
6.7.1-3
Drag coefficient for circular cylinder
Figure
6.7.1-4
Drag coefficient for sphere
Exumple 6.7.1-2 Wind force
on

U
distillation column
Exumpk
6.7.1-3
Ternziniil velocity
of
a polymer sphere in
water
6.7.2
Drag
in
conduits
-
pipes (internal flow)
Table
6.7.2-
1
Properties of pipe
Figure
6.7.2-1
Momentum balance on cylindrical fluid element
in
horizontal pipe
Figure
6.7.2-2
Momentum balance on cylindrical fluid element
in
non-horizontal
pipe
Figure

6.7.2-3
Moody friction factor chart
Figure
6.7.2-4
Relative roughness for clean new pipes
Example
6.7.2-1
Expunsion losses
Figure
6.7.2-5
Equivalent lengths for losses
in
pipes
Example 6.7.2-2 Direction offow between tunks at
differing pressures
und
heights
Exmzple 6.7.2-3 Friction loss
in
(I
piping system
Case
1:
Pressure
drop
unknown
Exumple 6.7.2-4 Pressure loss
for
flow
between tunks

Case
2:
Diameter unknown
Example 6.7.2-5 Transfer line
from
tank to column
Example
6.7.2-6
Minimum pipe diameter
Exumple 6.7.2-7 Air supply through hose
Emrnple 6.7.2-8 Flow rute unknown
Example 6.7.2-9 Culculution offow rute via
Kurmun
number when pressure drop is
known
Friction factor calculations
-
serial paths
Case
3:
Length unknown
Case
4:
Flow rate unknown
Figure
6.7.2-6
Friction factor
vs.
Karman
number

Non-circular conduits
362
366
366
37
1
372
375
375
378
3
79
3
80
380
381
382
385
386
395
396
399
400
402
403
405
406
408
400
409

41
1
41
1
41 4
416
41
6
417
418
422
423
424
Table
of
Contents
xin
Example
6.7.2-10
Flow
in
a smooth annulus
Example
6.7.2-1
1
Pressure
drop
in a pipe annulus
Example
6.7.2-12

Pipe network with imposed pressure
drop
Example
6.7.2-13
Flow
in
a parallel piping system
Example
6.7.2-14
Input
of
additional fluid to
an
existing
pipe network
6.8
Non-Newtonian Flow
6.8.1
Bingham plastics
6.8.2
Power-law fluids
6.9
Flow
in
Porous Media
Friction factor calculations
-
parallel paths
Table
6.7.2-2

Convergence of Newton's Method
Figure
6.8.1-1
Tube
flow
of
Bingham plastic
Example
6.8-1
Flow
of
polymer melt
6.9.1
Darcy's law
Figure
6.9.1-1
Permeability
as
a function of porosity for a bed
of spheres
Table
6.9.1
-
1
Porosities (void fractions)
for
dumped
packings
Table
6.9.1-2

Porosity
and
permeability for typical
materials
Example
6.9.1-1
Flow
of
water
in
sandstone
6.9.2
Packed
beds
Example
6.9.2-1
Pressure
drop.for
air flowing though bed
of
spheres
Example
6.9.2-2
Pressure drop
for
waterjlowing though
bed
of
cylinders
Example

6.9.3-1
Production scale filter performance
prediction from pilot plant data
Example
6.9.3-2
Filter performance fiom data
Example
6.9.3-3
Upting existingjilter to new product
6.10
Flow Measurement
6.9.3
Filters
6.10.1
Pitot tube
Figure
6.10.1-1
Pitot tube
schematic
Figure
6.10.1-2
Flow at mouth of pitot
tube
Example
6.10.1-1
Pitot tube traverse
6.10.2
Venturi meter
Figure
6.10.2-1

Venturi schematic
Figure
6.10.2-2
Venturi meter coefficient
Example
6.10.2-1
Flow measurement with venturi meter
6.10.3
Orifice meter and flow nozzle
Figure
6.10.3-1
Orifice meter, flow nozzle
Figure
6.10.3-2
Orifice coefficient
425
426
427
427
43
1
433
435
438
438
439
440
443
444
445

448
452
453
454
455
4.58
460
462
465
467
469
472
472
473
473
474
478
478
480
480
48
1
482
483
xx
Table
of
Contents
Example
6.10.3-1

Metering
of
crude oil with orifice
Chapter
6
Problems
7
HEAT
TRANSFER
MODELS
7.1
The Nature
of
Heat
7.1.1
Forced convection heat transfer
7.1.2
Free convection heat transfer
Table
7.1.2-
1
Dimensionless
Fonns:
Mass,
Energy, and
Momentum Equations for Natural and
Forced
Convection
7.2.1
Three-dimensional conduction

in
isotropic media
Table
7.2.1-1
Components
of
Fourier Equation
in
Various
Coordinate Systems
7.2.2
Boundary conditions at solid surfaces
7.2.3
Thermal conductivity
Table
7.2.3-1
Relative Values of Thermal Conductivity
7.2.4
One-dimensional steady-state conduction
in
rectangular
coordinates
7.2
Conduction Heat Transfer Models
Analytical solution
Figure
7.2.4-1
Homogeneous solid
Figure
7.2.4-2

Temperature profile
in
1-D
heat transfer
by
conduction, rectangular coordinates
Interface condition between
solids
-
series
conduction
Figure
7.2.4-3
Conduction
with
two solids
in
contact
assuming no temperature drop at interface
Figure
7.2.4-4
Temperature profile
with
interfacial resistance
Figure
7.2.4-5
Contact resistance
treated
as
an

intermediate
solid
Figure
7.2.4-6
Development of equivalent conductance for
series conduction
Figure
7.2.4-7
Development
of
equivalent conductance for
parallel conduction
Figure
7.2.4-8
Equivalent circuit for parallel conduction
Excrmple 7.2.4-1 Series conduction
through
layers
-
constant temperature at external surfaces
Equivalent thermal resistance
-
series conduction
Equivalent thermal resistance
-
parallel conduction
Figure
7.2.4-9
Series conduction through layers
with

constant
temperature at external surfaces
Exumple 7.2.4-2 Series conduction through luyers
-
constant convective heat trunsfer coeficient at external
surfaces
483
485
517
5
17
520
525
528
528
529
530
532
533
533
536
537
537
539
539
540
540
541
542
542

544
544
545
546
547
549
Table
of
Contents
Figure
7.2.4-10
Series conduction through layers
with
constant convective heat transfer coefficient at external
SlllfZMXS
Example 7.2.4-3
conductivity
Conduction with variable thermal
Figure
7.2.4-1
1
Conduction through firebrick with variable
thermal conductivity
Figure
7.2.4- 12
Temperature profile
Example 7.2.4-4 One-dimensional steady-state conduction
'
with parallel path
Figure

7.2.4-13
Conduction with parallel paths
Figure
7.2.4-14
Analogous circuit
Numerical solution
The finite element method
Figure
7.2.4-14
Examples of
1-D
and
2-D
finite elements
Figure
7.2.4-15
Interpolation functions, Shape function
Example 7.2.4-5 Solution by finite elements
of
steady-state
conduction with generation
Figure
7.2.4-16
Bar
with
thermal
energy source
Table
7.2.4-1
Global node numbering scheme

Figure
7.2.4-17
Comparison of finite element and analytic
solution
7.2.5
Onedimensional steady-state conduction
in
cylindrical
Coordinates
Figure
7.2.5-1
Conduction through the
wall
of a composite
Example 7.2.5-1 Conduction in a fuel rod
Example 7.2.5-2 Conduction through
un
insulated pipe
Finite element method
in
higher dimensions
cylinder
7.2.6
One-dimensional steady-state conduction
in
spherical
coordinates
Figure
7.2.6-1
Radial conduction

in
spherical geometry
Example 7.2.6-1 Conduction through shiekiing
7.2.7
Two-dimensional steady-state conduction
Tay lor series
Analytical solution
Orthogonal
functions
Example 7.2.7-2 Convergence
of
steady-state rectangular
coordinate solution
Numerical solution
Finite difference method
Forward difference approximation
to
the first derivative
Backward difference approximation to the
fmt
derivative
Central difference approximation to the first derivative
550
552
553
555
555
555
556
558

558
560
562
563
564
569
572
573
574
574
577
578
581
582
583
586
589
59
1
598
603
605
605
607
607
608
xxii Tuble
of
Contents
Approximation

of
second derivative
Finite difference approximation to the Laplace equation
Example 7.2.7-1 Determination
of
steady-state
temperature distribution in a rectangular slab
Irregular boundaries, Dirichlet boundary conditions
Normal
derivative (Neumann)
boundary
condition at nodal point
Generation
terns
Example 7.2.7-2 Finite difference solution
of
2-0 steady-
state conduction
7.2.8
One-dimensional unsteady
-sta
te conduction
Analytical methods for one-dimensional unsteady-state conduction
Semi-infinite slab
Figure 7.2.8-1 Semi-infinite slab with constant face
temperature
Example 7.2.8-1 Semi-infinite slub: conduction in a brick
wull
Finite slab
Figure 7.2.8-2 Finite slab with constant face temperatures

Figure 7.2.8-3 Unsteady-state heat transfer in
a
finite slab
with
uniform initial temperature and constant, equal surface
temperatures
Exunzple 7.2.8-2 Finite
slab
nwdel
vs.
semi-infinite slab
model for one-dimnsionul unsteady-stute conductive heut
trunsfe r
Infinite cylinder
and
sphere
Figure 7.2.8-4 Unsteady-state heat transfer in
an
infinite
cylinder with uniform initial temperature and constant surface
temperatwe
Figure 7.2.8-5 Unsteady-state heat transfer in a sphere with
uniform initial temperature and constant surface temperature
Numerical Methods for One-Dimensional Unsteady-State
Conduction
Finite difference method
Finite difference explicit form
Figure 7.2.8-6 Finite difference grid for 1D unsteady-state
conduction
Example 7.2.8-3 Unsteady-stute heut transfer by explicit

finite differences
Finite difference implicit form
Example 7.2.8-4 Finite slab unsteady-stute heut transfer
by finite differences
7.2.9
Multi-dimensional unsteady-state conduction
Analytical solution for regular geometries
609
609
61 2
614
618
619
621
629
630
630
630
635
63
8
638
646
646
65
1
652
65
3
65

3
654
654
654
656
658
662
667
667
Table
of
Contents
xxiii
Figure
7.2.9-1
Multidimensional unsteady-state temperature
profiles for conduction
in
regular geometries expressed
as
product of one-dimensional solutions
Numerical solution of two-dimensional unsteady-state conduction
Figure
7.2.9-2
Alternating direction implicit method
Finite difference method
Finite element method
7.3
Convection Heat Transfer Models
7.3.1

The thermal boundary layer
7.3.2
Heat transfer coefficients
Single-phase heat transfer coefficients
Correlations for prediction
of
heat transfer
Average heat transfer coefficients
Example
7.3.2-
I
pipe
flow
Figure
7.3.1-1
Solution of Equation
(7.3.1-1)
Figure
7.3.2-
1
Single-phase heat transfer coefficients
Average heat transfer coefficients
for
Design equations for convective heat transfer
Forced convection in laminar
flow
Table
7.3.2-1
Nusselt number limit for laminar flow
in

ducts
with
various cross-sections
Forced convection in turbulent flow
Example
7.3.2-2
Comparison of the Dittus-Boelter, Colburn,
and Sieder-Tate equations
Heat transfer
in
non-circular conduits and annular flow
External flows, natural and ford convection
Table
7.3.2-2
Values of
b
and
n
for Equation
(7.3.2-104)
Table
7.3.2-3
Values of a
and
m
for
use
with Equation
(7.3.2-
105)

Example
7.3.2-3
Heat transfer
with
flow
normal to
pipes
Heat
transfer with phase change
Boiling
-
mechanism
Condensation
-
mechanism
Boiling coefficients
Condensing coefficients
Figure
7.3.2-2
Boiling Curve
Table
7.3.2-3
Values of
C
for nucleate boiling model
7.4
Conduction
and
Convection in Series
7.4.1

Lumped capacitance models
Criteria for use of lumped capacitance models
Figure
7.4.1-1
RC
circuit
analog
of unsteady-state lumped-
capacitance
heat
transfer
Figure
7.4.1-2
Steady-state conduction through solid with
convection at interface
668
669
669
670
67
1
672
673
675
675
676
68
1
68 1
690

692
694
696
699
700
702
708
708
709
709
71
0
713
713
715
716
7 16
718
718
7 19
721
726
726
727
xriv
Tuble
of
Contents
Figure
7.4.1-3

Steady-state conduction through solid with
convection at interface,
small
vs. large Biot number
Figure
7.4.1-4
Unsteady-state conduction
through
solid with
convection at interface, small vs. large Biot number
Example
7.4.1-1
Lumped capucitance
models
7.4.2
Distributed capacitance models
Figure
7.4.2-1
Mid-plane temperature for unsteady-state heat
transfer in a slab of finite thickness
2L
with uniform initial
temperature and convective resistance at surfaces
Figure
7.4.2-2
Temperature profile for unsteady-state heat
transfer
in
a slab of finite thickness
with

uniform initial
temperature and convective resistance at surfaces
Figure
7.4.2-3
Centerline temperature for unsteady-state heat
transfer
in
an
infinite cylinder of radius
ro
with uniform initial
temperature and convective resistance at surfaces
Figure
7.4.2-4
Temperature profile for unsteady-state heat
transfer in
an
infinite cylinder of radius ro
with
uniform initial
temperature and convective resistance at surfaces
Figure
7.4.2-5
Center temperature for unsteady-state heat
transfer
in
a sphere of radius ro with uniform initial
temperature and convective resistance at surfaces
Figure
7.4.2-6

Temperature profile for unsteady-state heat
transfer
in
a sphere of radius
ro
with
uniform
initial
temperature and convective resistance at surfaces
Example
7.4.2-1
in series
7.5
Radiation Heat Transfer Models
Convective
und
conductive resistunces
Figure
7.5-1
The electromagnetic
spectrum
7.5.1
Interaction
of
radiation and matter
Geometric description of radiation
Figure
7.5.1-1
Directions
in

space
Intensity of radiation
Figure
7.5.1-2
Radiation leaving
AI
Lumping of quantities used
in
modeling radiation
Incident radiation
Figure
7.5.1-3
Extreme modes
of
reflection
Absorptivity
Reflectivity
Transmi ttivi
ty
Emitted radiation
Blackbodies
Blackbody radiation
Emissivity
Radiosity
729
730
731
732
734
735

736
737
738
739
73Y
74 1
742
742
742
743
743
744
745
746
748
748
748
749
749
750
75
1
752
752
Table
of
Contents
xxv
7.5.2 Radiant heat exchange between two opaque
bodies

with
no
intervening
medium
Table 7.5.2-1 History of radiation emitted
7.5.3 Kirchhoffs law
7.5.4 View factors
Reciprocity relation
Summation
rule
Figure 7.5.3-1
Total
emissivity of some surfaces
Figure 7.5.4-
1
Radiation between surfaces
Exumple 7.5.4-1 Integration to obtain view factor
Example 7.5.4-2 Use
of
reciprocity relation and
surnmution
rule
to infer view factor
for
concentric spheres
Example 7.5.5-1 Heat transfer by radiation
-
blackbody
Example 7.5.5-2 Use
of

view factor tables with blackbody
radiation exchange
Table 7.5.4-1 View Factors
7.5.5 Radiant heat exchange between blackbodies
7.5.6 Radiative exchange between gray bodies
Figure 7.5.6-1 Electrical analog of net radiation from a gray
Surface
Figure 7.5.6-2 Network analog
of
radiation exchange
with
gray
surfaces
in
an
enclosure
Example 7.5.6-1 Two-gray-body exchunge in enclosure
Exumple 7.56-2 Heat transfer by rdiution
-
gray
body
7.6 Overall
Heat
Transfer Coefficients
Figure 7.6-1 Insulated pipe
Figure 7.6-1 Overall heat transfer coefficients
Example 7.6-1 Controlling resistance
for
heat transfer
resistances in series

-
spherical container of liquid oxygen
Example
7.6-2
Controlling resistance
in
replacement
qf
section
of
wall of distillation column
Example
7.6-3
Overall heat transfer coefficient with
fouling
7.7 Heat Exchangers
7.7.1 Average overall temperature difference
7.7.2 Countercurrent
vs.
ioncurrent operation
Figure 7.7.2-1
T-H
diagram for countercurrent flow
of
two
streams
Figure 7.7.2-2
T-H
diagram for concurrent flow
of

two
streams
Figure 7.7.2-3
T-H
diagram for countercurrent flow
of
two
streams
Figure 7.7.24
T-H
diagram for concurrent flow of two
streams
753
754
755
757
757
758
759
760
76
I
764
764
765
766
768
770
772
773

773
775
777
777
78 1
783
785
787
788
789
796
797
798
799
799
nvi
Tuble
of
Contents
Example 7.7.2-1 Concurrent vs. countercurrent flow in a
concentric tube exchanger
7.7.3
NTU
method for design of heat exchangers
Figure 7.7.3-1
T-H
diagram for two
streams
between which
sensible

heat
is to
be
exchanged
in
countercurrent flow
Figure 7.7.3-2
T-H
diagram for two
streams
between which
sensible heat is to
be
exchanged in concurrent
flow
Figure 7.7.3-3 Pinch with concurrent operation
Figure 7.7.3-4 Pinch with countercurrent operation
Table 7.7.3- la Effectiveness/NTU relationships
Table 7.7.3-1b NTU/effectiveness relationships
Example 7.7.3-1 Determination
of
effectiveness for a
concurrent flow exchanger
Example 7.7.3-2 Calculation
of
urea using
NTU
und
E
for

U
concurrent flow exchanger
Example 7.7.3-3 Calculation
of
exit temperatures using
NTU
and
E
for a heat exchanger
of
known area
7.7.4 F-factor method for design
of
heat exchangers
Figure 7.7.4-1 Correction factor to log mean temperature
difference
-
one shell
pass,
2" tube passes
Emmple 7.7.4-1 Use
of
F
Fuctor compured to
effect iveness/NTU met
hod
Chapter 7 Problems
8
MASS
TRANSFER

MODELS
8.1 The Nature
of
Mass Transfer
8.2 Diffusive
Mass
Transfer Models
8.2.1
Velocities of components
in
a mixture
Figure 8.2.1-1 Diffusion
of
vapor from vessel
Figure
8.2.1-2
Evolution
of
concentration
profile
Figure
8.2.1-3
Velocity of molecule
Figure 8.2.1-4 Changing velocity and displacement of single
molecule via collisions
Example
8.2.
I
-I
Average velocity when individual

particles have the same velocity
Figure 8.2.1-5 Identical molecules, identical velocities
Example
8.2.1
-2 Average velocity when individual
particles have different velocities
Figure 8.2.1-6 Identical molecules, differing velocities
Example 8.2.1-3
Number
uveruge velocity, velocities in
two
dimensions
Figure 8.2.1-7 Number average velocity, velocities in two
dimensions
Table 8.2.1- 1 Coordinate frame motion
800
806
807
808
809
809
812
812
81
3
81
7
81
9
822

825
825
828
843
843
845
845
846
846
848
848
850
850
851
85
1
853
8
54
857
Table
of
Contents xxvii
Table
8.2.1-2
Mass
transfer relationships
8.2.2
Mechanisms of mass transfer
8.2.3

Ficks law
Table
8.2.3-1
Equivalent forms of Ficks Law referred
to
coordinate systems in various motions
Figure
8.2.3-1
Flux
of marbles without diffusion
Figure
8.2.3-2
Flux
of marbles
with
diffusion
Figure
8.2.3-3
Fluxes compared
Figure
8.2.4-1
Diffusivities
in
solids
Figure
8.2.4-2
Diffusivities in liquids
Figure
8.2.4-3
Diffusivities in gases

8.2.5
Solutions of the diffusion equation
One-dimensional equimolar counterdiffusion in rectangular
coordinates
Example 8.2.5-1 Equimolar counterdiflusion
One-dimensional diffusion of
A
through stagnant
B
observed
in
rectangular coordinates
8.2.4
Binary diffusivities
Example
8.2.5-2
Diffusion
of
vapor through
a
stagnant
gus
Figure
8.2.5-1
Diffusion through stagnant gas layer
One dimensional unsteady-state diffusion
in
a semi-infinite slab
Figure
8.2.5-2

Semi-infinite slab with constant face
concentration
8.2.6
Diffusion
in
porous solids
8.2.7
Dispersion
Figure
8.2.7-1
Dispersion
and
diffusion
as
a function of Peclet
number
8.3
Convective Mass Transfer Models
8.3.1
The concentration boundary layer
Figure
8.3.1-1
Concentration boundary layer
Figure
8.3.1-2
Boundary layer
Figure
8.3.1-3
Boundary layer solution for a flat plate
8.3.2

Film theory and penetration-renewal theory
8.4
The
Mass
Transfer Coefficient for a Single Phase
Example 8.4-1 Calculation offluxfrom a
mass
transfer
coefficient
Exutizple 8.4-2 Mass transfer
using
partial pressure
as
a
driving force
Exunzple 8.4-3
Mass
transfer
using
species
mss
density
as
driving force
8.4.1
Design equations for single-phase
mass
transfer coefficients
Flat plates
Average

mass
transfer coeficient j?om
Example 8.4.1-1
loca
1
coefJic ient
858
858
858
860
861
862
862
863
864
864
865
865
865
867
868
869
869
871
87
1
877
880
88
1

882
882
883
885
886
887
888
892
893
894
895
895
896
miii Table
of
Contents
Mass transfer
in
flow in pipes
Mass transfer from spheres,
drops,
and
bubbles
Example
8.4.1 -2
Comparison
of
mars transfer coeffxient
models
Example

8.4.1-3
Mass transfer coefficient
for
dissolution
of
a
sphere
Packedbeds
Height of transfer unit models
8.4.2
Dimensional analysis of mass transfer by convection
8.5
Overall Mass Transfer Coefficients
Figure
8.5-1
Mass transfer concentrations
Figure
8.5-2
Interface conditions
8.5.1
Incorporation of overall mass transfer coefficient into height
of transfer
unit
model
8.6
Relationship
of
Overall
and
Single-Phase Mass Transfer

Coefficients
Example
8.5-1
Culculation
of
interface composition
Example
8.5.1-1
Overall transfer units
Figure
8.6-1
Assumption necessary to utilize overall
mass
transfer coefficient
Example
8.6-1
Controlling resistance for
mass
trunsfer
8.7
Design of
Mass
Transfer Columns
Figure
8.7-1
Typical countercumnt gas absorber
8.7.1
Determination of liquid-to-gas ratio
Figure
8.7.1

-
1
Gas
absorption
8.7.2
Calculation
of
tower diameter
Figure
8.7.2-1
Norton Chemical Process Products Corporation
Intalox@
IMTP@ Packing
Figure
8.7.2-2
IMTPO packing pressure
drop
Table
8.7.2-1
Values of coefficient
F
for
IMP@
packings
Table
8.7.3-1
n
integrals
8.7.3
Calculation of packing height

8.7.4
Applications
Example
8.7.4-1
Analytical calculation
of
interfacial
concentration
Example
8.7.4-2
Analytical determination
of
number
of
transfer units: straight operating
and
equilibrium lines
Example
8.7.4-3
Enect
of
change
of
UG
on
outlet
composition
Exumple
8.7.4-4
Design

of
absorber
Example
8.7.4-5
Economic optimization
of
an absorber
8.8
Mass
Transfer with Chemical Reaction
Figure
8.8-1
Diffusion in
a
membrane
898
898
899
900
902
903
907
910
911
015
91
6
919
9I
9

92 1
921
923
923
926
926
927
930
932
933
934
934
937
937
937
939
943
948
9.56
963
963