fundamentals of momentum, heat, and mass transfer 5th edition

Fundamentals of Momentum,
Heat, and Mass Transfer
5th Edition
Fundamentals of Momentum,
Heat, and Mass Transfer
5
th
Edition
James R. Welty
Department of Mechanical Engineering
Charles E. Wicks
Department of Chemical Engineering
Robert E. Wilson
Department of Mechanical Engineering
Gregory L. Rorrer
Department of Chemical Engineering
Oregon State University
John Wiley & Sons, Inc.
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Printed in the United States of America
10987654321
Preface to the 5
th
Edition
The first edition of Fundamentals of Momentum, Heat, and Mass Transfer, published in
1969, was written to become a part of what was then known as the ‘‘engineering science
core’’ of most engineering curricula. Indeed, requirements for ABET accreditation have
stipulated that a significant part of all curricula must be devoted to fundamental subjects.
The emphasis on engineering science has continued over the intervening years, but the
degree of emphasis has diminished as new subjects and technologies have entered the
world of engineering education. Nonetheless, the subjects of momentum transfer (fluid
mechanics), heat transfer, and mass transfer remain, at least in part, important components
of all engineering curricula. It is in this context that we now present the fifth edition.
Advances in computing capability have been astonishing since 1969. At that time, the
pocket calculator was quite new and not generally in the hands of engineering students.
Subsequent editions of this book included increasingly sophisticated solution techniques as
technology advanced. Now, more than 30 years since the first edition, computer competency

among students is a fait accompli and many homework assignments are completed using
computer software that takes care of most mathematical complexity, and a good deal of
physical insight. We do not judge the appropriateness of such approaches, but they surely
occur and will do so more frequently as software becomes more readily available, more
sophisticated, and easier to use.
In this edition, we still include some examples and problems that are posed in English
units, but a large portion of the quantitative work presented is now in SI units. This is
consistent with most of the current generation of engineering textbooks. Th ere are still some
subdisciplines in the thermal/fluid sciences that use English units conventionally, so it
remains necessary for students to have some familiarity with pounds, mass, slugs, feet, psi,
and so forth. Perhaps a fifth edition, if it materializes, will finally be entirely SI.
We, the original three authors (W
3
), welcome Dr. Greg Rorrer to our team. Greg is a
member of the faculty of the Chemical Engineering Department at Oregon State University
with expertise in biochemical engineering. He has had a significant influence on this
edition’s sections on mass transfer, both in the text and in the problem sets at the end of
Chapters 24 through 31. This edition is unquestionably strengthened by his contributions,
and we anticipate his continued presence on our writing team.
We are gratified that the use of this book has continued at a significant level since the
first edition appeared some 30 years ago. It is our continuing belief that the transport
phenomena remain essential parts of the foundation of engineering education and practice.
With the modifications and modernization of this fourth edition, it is our hope that
Fundamentals of Momentum, Heat, and Mass Transfer will continue to be an essential
part of students’ educational experiences.
Corvallis, Oregon J.R. Welty
March 2000 C.E. Wicks
R.E. Wilson
G.L. Rorrer
v

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Contents
1. Introduction to Mom entum Transfer 1
1.1 Fluids and the Continuum 1
1.2 Properties at a Point 2
1.3 Point-to-Point Variation of Properties in a Fluid 5
1.4 Units 8
1.5 Compressibility 9
1.6 Surface Tension 11
2. Fluid Statics 16
2.1 Pressure Variation in a Static Fluid 16
2.2 Uniform Rectilinear Acceleration 19
2.3 Forces on Submerged Surfaces 20
2.4 Buoyancy 23
2.5 Closure 25
3. Description of a Fluid in Motion 29
3.1 Fundamental Physical Laws 29
3.2 Fluid-Flow Fields: Lagrangian and Eulerian Representations 29
3.3 Steady and Unsteady Flows 30
3.4 Streamlines 31
3.5 Systems and Control Volumes 32
4. Conservation of Mass: Control-Volume Approach 34
4.1 Integral Relation 34
4.2 Specific Forms of the Integral Expression 35
4.3 Closure 39
5. Newton’s Second Law of Motion: Control-Volume Approach 43
5.1 Integral Relation for Linear Momentum 43
5.2 Applications of the Integral Expression for Linear Momentum 46
5.3 Integral Relation for Moment of Momentum 52
5.4 Applications to Pumps and Turbines 53

5.5 Closure 57
6. Conservation of Energy: Control-Volume Approach 63
6.1 Integral Relation for the Conservation of Energy 63
6.2 Applications of the Integral Expression 69
vii
6.3 The Bernoulli Equation 72
6.4 Closure 76
7. Shear Stress in Laminar Flow 81
7.1 Newton’s Viscosity Relation 81
7.2 Non-Newtonian Fluids 82
7.3 Viscosity 83
7.4 Shear Stress in Multidimensional Laminar Flows of a Newtonian Fluid 88
7.5 Closure 90
8. Analysis of a Differential Fluid Element in Laminar Flow 92
8.1 Fully Developed Laminar Flow in a Circular Conduit of Constant
Cross Section
92
8.2 Laminar Flow of a Newtonian Fluid Down an Inclined-Plane Surface 95
8.3 Closure 97
9. Differential Equations of Fluid Flow 99
9.1 The Differential Continuity Equat ion 99
9.2 Navier-Stokes Equations 101
9.3 Bernoulli’s Equation 110
9.4 Closure 111
10. Inviscid Fluid Flow 113
10.1 Fluid Rotation at a Point 113
10.2 The Stream Function 114
10.3 Inviscid, Irrotational Flow about an Infinite Cylinder 116
10.4 Irrotational Flow, the Velocity Potential 117
10.5 Total Head in Irrotational Flow 119

10.6 Utilization of Potential Flow 119
10.7 Potential Flow Analysis—Simple Plane Flow Cases 120
10.8 Potential Flow Analysis—Superposition 121
10.9 Closure 123
11. Dimensional Analysis and Similitude 125
11.1 Dimensions 125
11.2 Dimensional Analysis of Governing Differential Equations 126
11.3 The Buckingham Method 128
11.4 Geometric, Kinematic, and Dynamic Similarity 131
11.5 Model Theory 132
11.6 Closure 134
12. Viscous Flow 137
12.1 Reynolds’s Experiment 137
12.2 Drag 138
viii Contents
12.3 The Boundary-Layer Concept 144
12.4 The Boundary-Layer Equations 145
12.5 Blasius’s Solution for the Laminar Boundary Layer on a Flat Plate 146
12.6 Flow with a Pressure Gradient 150
12.7 von Ka
´
rma
´
n Momentum Integral Analysis 152
12.8 Description of Turbulence 155
12.9 Turbulent Shearing Stresses 157
12.10 The Mixing-Length Hypothesis 158
12.11 Velocity Distribution from the Mixing-Length Theory 160
12.12 The Universal Velocity Distribution 161
12.13 Further Empirical Relations for Turbulent Flow 162

12.14 The Turbulent Boundary Layer on a Flat Plate 163
12.15 Factors Affecting the Transition From Laminar to Turbulent Flow 165
12.16 Closure 165
13. Flow in Closed Conduits 168
13.1 Dimensional Analysis of Conduit Flow 168
13.2 Friction Factors for Fully Developed Laminar, Turbulent,
and Transition Flow in Circular Conduits
170
13.3 Friction Factor and Head-Loss Determination for Pipe Flow 173
13.4 Pipe-Flow Analysis 176
13.5 Friction Factors for Flow in the Entrance to a Circular Conduit 179
13.6 Closure 182
14. Fluid Machinery 185
14.1 Centrifugal Pumps 186
14.2 Scaling Laws for Pumps and Fans 194
14.3 Axial and Mixed Flow Pump Configurations 197
14.4 Turbines 197
14.5 Closure 197
15. Fundamentals of Heat Transfer 201
15.1 Conduction 201
15.2 Thermal Conductivity 202
15.3 Convection 207
15.4 Radiation 209
15.5 Combined Mechanisms of Heat Transfer 209
15.6 Closure 213
16. Differential Equations of Heat Transfer 217
16.1 The General Differential Equation for Energy Transfer 217
16.2 Special Forms of the Differential Energy Equation 220
16.3 Commonly Encountered Boundary Conditions 221
16.4 Closure 222

Contents
ix
17. Steady-State Conduction 224
17.1 One-Dimensional Conduction 224
17.2 One-Dimensional Conduction with Internal Generation of Energy 230
17.3 Heat Transfer from Extended Surfaces 233
17.4 Two- and Three-Dimensional Systems 240
17.5 Closure 246
18. Unsteady-State Conduction 252
18.1 Analytical Solutions 252
18.2 Temperature-Time Charts for Simple Geometric Shapes 261
18.3 Numerical Methods for Transient Conduction Analysis 263
18.4 An Integral Method for One-Dimensional Unsteady Conduction 266
18.5 Closure 270
19. Convective Heat Transfer 274
19.1 Fundamental Considerations in Convective Heat Transfer 274
19.2 Significant Parameters in Convective Heat Transfer 275
19.3 Dimensional Analysis of Convective Energy Transfer 276
19.4 Exact Analysis of the Laminar Boundary Layer 279
19.5 Approximate Integral Analysis of the Thermal Boundary Layer 283
19.6 Energy- and Momentum-Transfer Analogies 285
19.7 Turbulent Flow Considerations 287
19.8 Closure 293
20. Convective Heat-Transfer Correlations 297
20.1 Natural Convection 297
20.2 Forced Convection for Internal Flow 305
20.3 Forced Convection for External Flow 311
20.4 Closure 318
21. Boiling and Condensation 323
21.1 Boiling 323

21.2 Condensation 328
21.3 Closure 334
22. Heat-Transfer Equipment 336
22.1 Types of Heat Exchangers 336
22.2 Single-Pass Heat-Exchanger Analysis: The Log-Mean Temperature
Difference
339
22.3 Crossflow and Shell-and-Tube Heat-Exchanger Analysis 343
22.4 The Number-of-Transfer-Units (NTU) Method of Heat-Exchanger
Analysis and Design
347
22.5 Additional Considerations in Heat-Exchanger Design 354
22.6 Closure 356
x Contents
23. Radiation Heat Transfer 359
23.1 Nature of Radiation 359
23.2 Thermal Radiation 360
23.3 The Intensity of Radiation 361
23.4 Planck’s Law of Radiation 363
23.5 Stefan-Boltzmann Law 365
23.6 Emissivity and Absorptivity of Solid Surfaces 367
23.7 Radiant Heat Transfer Between Black Bodies 370
23.8 Radiant Exchange in Black Enclosures 379
23.9 Radiant Exchange in Reradiating Surfaces Present 380
23.10 Radiant Heat Transfer Between Gray Surfaces 381
23.11 Radiation from Gases 388
23.12 The Radiation Heat-Transfer Coefficient 392
23.13 Closure 393
24. Fundamentals of Mass Transfer 398
24.1 Molecular Mass Transfer 399

24.2 The Diffusion Coefficient 407
24.3 Convective Mass Transfer 428
24.4 Closure 429
25. Differential Equations of Mass Transfer 433
25.1 The Differential Equation for Mass Transfer 433
25.2 Special Forms of the Differential Mass-Transfer Equation 436
25.3 Commonly Encountered Boundary Conditions 438
25.4 Steps for Modeling Processes Involving Molecular
Diffusion
441
25.5 Closure 448
26. Steady-State Molecular Diffusion 452
26.1 One-Dimensional Mass Transfer Independent of Chemical Reaction 452
26.2 One-Dimensional Systems Associated with Chemical Reaction 463
26.3 Two- and Three-Dimensional Systems 474
26.4 Simultaneous Momentum, Heat, and Mass Transfer 479
26.5 Closure 488
27. Unsteady-State Molecular Diffusion 496
27.1 Unsteady-State Diffusion and Fick’s Second Law 496
27.2 Transient Diffusion in a Semi-Infinite Medium 497
27.3 Transient Diffusion in a Finite-Dimensional Medium Under Conditions of
Negligible Surface Resistance
500
27.4 Concentration-Time Charts for Simple Geometric Shapes 509
27.5 Closure 512
Contents
xi
28. Convective Mass Transf er 517
28.1 Fundamental Considerations in Convective Mass Transfer 517
28.2 Significant Parameters in Convective Mass Transfer 519

28.3 Dimensional Analysis of Convective Mass Transfer 521
28.4 Exact Analysis of the Laminar Concentration Boundary Layer 524
28.5 Approximate Analysis of the Concentration Boundary Layer 531
28.6 Mass, Energy, and Momentum-Transfer Analogies 533
28.7 Models for Convective Mass-Transfer Coefficients 542
28.8 Closure 545
29. Convective Mass Transf er Between Phases 551
29.1 Equilibrium 551
29.2 Two-Resistance Theory 554
29.3 Closure 563
30. Convective Mass-Transfer Correlations 569
30.1 Mass Transfer to Plates, Spheres, and Cylinders 569
30.2 Mass Transfer Involving Flow Through Pipes 580
30.3 Mass Transfer in Wetted-Wall Columns 581
30.4 Mass Transfer in Packed and Fluidized Beds 584
30.5 Gas-Liquid Mass Transfer in Stirred Tanks 585
30.6 Capacity Coefficients for Packed Towers 587
30.7 Steps for Modeling Mass-Transfer Processes Involving Convection 588
30.8 Closure 595
31. Mass-Transfer Equipment 603
31.1 Types of Mass-Transfer Equipment 603
31.2 Gas-Liquid Mass-Transfer Operations in Well-Mixed Tanks 605
31.3 Mass Balances for Continuous Contact Towers: Operating-Line Equations 611
31.4 Enthalpy Balances for Continuous-Contact Towers 620
31.5 Mass-Transfer Capacity Coefficients 621
31.6 Continuous-Contact Equipment Analysis 622
31.7 Closure 636
Nomenclature 641
APPENDIXES
A. Transformations of the Oper ators = and =

2
to Cylindrical Coordinates 648
B. Summary of Differential Vector Operations in Various Coordinate Systems 651
C. Symmetry of the Stress Tensor 654
D. The Viscous Contribution to the Normal Stress 655
E. The Navier–Stokes Equations for Constant r and m in Cartesian,
Cylindrical, and Spherical Coordinates
657
F. Charts for Solution of Unsteady Transport Problems 659
xii Contents
G. Properties of the Standard Atmosphere 672
H. Physical Properties of Solids 675
I. Physical Properties of Gases and Liquids 678
J. Mass-Transfer Diffusion Coefficients in Binary Systems 691
K. Lennard–Jones Constants 694
L. The Error Function 697
M. Standard Pipe Sizes 698
N. Standard Tubing Gages 700
Author Index 703
Subject Index 705
Contents
xiii
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Fundamentals of Momentum,
Heat, and Mass Transfer
5th Edition
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Chapter 1
Introduction to Momentum
Transfer

Momentum transfer in a fluid involves the study of the motion of fluids and the
forces that produce these motions. From Newton’s second law of motion it is known
that force is directly related to the time rate of change of mom entum of a system.
Excluding action-at-a-distance forces such as gravity, the forces acting on a fluid, such
as those resulting from pressure and shear stress, may be shown to be the result of
microscopic (molecular) transfer of mom entum. Thus the subject under consideration,
which is historically fluid mechanics, may equally be termed momentum transfer.
The history of fluid mechanics shows the skillful blending of the nineteenth- and
twentieth century analytical work in hydrodynamics with the empirical knowledge in
hydraulics that man has collected over the ages. The mating of these separately
developed disciplines was started by Ludwig Prandtl in 1904 with his boundary-layer
theory, which was verified by experiment. Modern fluid mechanics, or momentum
transfer, is both analytical and experimental.
Each area of study has its phraseology and nomenclature. Momentum transfer
being typical, the basic definitions and concepts will be introduced in order to provide a
basis for communication.
1.1 FLUIDS AND THE CONTINUUM
A fluid is defined as a substance that deforms continuously under the action of a shear stress.
An important consequence of this definition is that when a fluid is at rest, there can be no
shear stresses. Both liquids and gases are fluids. Some substances such as glass are
technically classified as fluids. However, the rate of deformation in glass at normal
temperatures is so small as to make its consideration as a fluid impractical.
Concept of a Continuum. Fluids, like all matter, are composed of molecules whose
numbers stagger the imagination. In a cubic inch of air at room conditions there are some
10
20
molecules. Any theory that would predict the individual motions of these many
molecules would be extremely complex, far beyond our present abilities.
Most engineering work is concerned with the macroscopic or bulk behavior of a fluid
rather than with the microscopic or molecular behavior. In most cases it is convenient to

think of a fluid as a continuous distribution of matter or a continuum. There are, of course,
certain instances in which the concept of a continuum is not valid. Consider, for example, the
number of molecules in a small volume of a gas at rest. If the volume were taken small
enough, the number of molecules per unit volume would be time-dependent for the
microscopic volume even though the macroscopic volume had a constant number of
1
molecules in it. The concept of a continuum would be valid only for the latter case. The
validity of the continuum approach is seen to be dependent upon the type of information
desired rather than the nature of the fluid. The treatment of fluids as continua is valid
whenever the smallest fluid volume of interest contains a sufficient numb er of molecules to
make statistical averages meaningful. The macroscopic properties of a continuum are
considered to vary smoothly (conti nuously) from point to point in the fluid. Our immediate
task is to define thes e properties at a point.
1.2 PROPERTIES AT A POINT
When a fluid is in motion, the quantities associated with the state and the motion of the fluid
will vary from point to point. The definition of some fluid variables at a point is presented
below.
Density at a Point. The density of a fluid is defined as the mass per unit volume. Under
flow conditions, particularly in gases, the density may vary greatly throughout the fluid. The
density, r, at a particular point in the fluid is defined as
r ¼ lim
DV!dV
Dm
DV
where Dm is the mass contained in a volume DV, and dV is the smallest volume surrounding
the point for which statistical averages are meaningful. The limit is shown in Figure 1.1.
The concept of the density at a mathematical point, that is, at DV ¼ 0 is seen to be
fictitious; however, taking r ¼ lim
DV!dV
(Dm/DV) is extremely useful, as it allows us to

describe fluid flow in terms of continuous functions. The density, in general, may vary
from point to point in a fluid and may also vary with respect to time as in a punctured
automobile tire.
∆V
δV
∆m
∆V
Molecular domain Continuum domain
Figure 1.1 Density at a point.
2 Chapter 1 Introduction to Momentum Transfer
Fluid Properties and Flow Properties. Some fluids, particularly liquids, have densities
that remain almost constant over wide ranges of pressure and temperature. Fluids which
exhibit this quality are usually treated as being incompressible. The effects of compres-
sibility, however, are more a property of the situation than of the fluid itself. For example, the
flow of air at low velocities is described by the same equations that describe the flow of
water. From a static viewpoint, air is a compressible fluid and water incompressible. Instead
of being classified according to the fluid, compressibility effects are considered a property of
the flow. A distinction, often subtle, is made between the properties of the fluid and the
properties of the flow, and the student is hereby alerted to the importance of this concept.
Stress at a Point. Consider the force DF acting on an element DA of the body shown in
Figure 1.2. The force DF is resolved into components normal and parallel to the surface of
the element. The force per unit area or stress at a point is defined as the limit of DF/DA as
DA !dA where dA is the smallest area for which statistical averages are meaningful
lim
DA!dA
DF
n
DA
¼ s
ii

lim
DA!dA
DF
s
DA
¼ t
ij
Here s
ii
is the normal str ess and t
ij
the shear stress. In this text, the double-subscript
stress notation as used in solid mechanics will be employed. The student will recall that
normal stress is positive in tension. The limiting process for the normal stress is illustrated in
Figure 1.3.
Forces acting on a fluid are divided into two general groups: body forces and surface
forces. Body forces are those which act witho ut physical contact, for example, gravity and
electrostatic forces. On the contrary, pressure and frictional forces requi re physical contact
for transmission. As a surface is required for the action of these forces they are called surface
forces. Stress is therefore a surface force per unit area.
1
∆A
∆F
n
∆F
s
∆F
Figure 1.2 Force on an
element of fluid.
∆A

δA
∆F
n
∆A
Molecular domain
Continuum domain
Figure 1.3 Normal stress at a point.
1
Mathematically, stress is classed as a tensor of second order, as it requires magnitude, direction, and
orientation with respect to a plane for its determination.
1.2 Properties at a Point 3
Pressure at a point in a Static Fluid. For a static fluid, the normal stress at a point may be
determined from the application of Newton’s laws to a fluid element as the fluid element
approaches zero size. It may be recalled that there can be no shearing stress in a static fluid.
Thus, the only surface forces present will be those due to normal stresses. Consider the
element shown in Figure 1.4. This element, while at rest, is acted upon by gravity and normal
stresses. The weight of the fluid elemen t is rg(Dx Dy Dz/2).
For a body at rest, SF ¼ 0. In the x direction
DF
x
À DF
s
sin u ¼ 0
Since sin u ¼ Dy/Ds, the above equation becomes
DF
x
À DF
s
Dy
Ds

¼ 0
Dividing through by Dy Dz and taking the limit as the volume of the element approaches
zero, we obtain
lim
DV!0
DF
x
Dy Dz
À
DF
s
Ds Dz
!
¼ 0
Recalling that normal stress is positive in tension, we obtain, by evaluating the above
equation
s
xx
¼ s
ss
(1-1)
In the y direction, applying SF ¼ 0 yields
DF
y
À DF
s
cos u Àrg
Dx Dy Dz
2
¼ 0

Since cos u ¼ Dx/Ds, one has
DF
y
À DF
s
Dx
Ds
À rg
Dx Dy Dz
2
¼ 0
y
x
z
∆y
∆F
x
∆F
s
∆F
y
∆x
q
∆z
∆s
Figure 1.4 Element in a static fluid.
Dividing through by Dx Dz and taking the limit as before, we obtain
lim
DV!0
DF

y
Dx Dz
À
DF
s
Ds Dz
À
rgDy
2
!
¼ 0
4
Chapter 1 Introduction to Momentum Transfer
which becomes
Às
yy
þ s
ss
À
rg
2
ð0Þ¼0
or
s
yy
¼ s
ss
(1-2)
It may be noted that the angle u does not appear in equation (1-1) or (1-2), thus the
normal stress at a point in a static fluid is independent of direction, and is therefore a scalar

quantity.
As the element is at rest, the only surface forces acting are those due to the normal stress.
If we were to measure the force p er unit area acting on a submerged element, we would
observe that it acts inward or places the element in compression. The quantity measured is,
of course, pressure, which in light of the preceding development, must be the negative of the
normal stress. This important simplification, the reduction of stress, a tensor, to pressure, a
scalar, may also be shown for the case of zero shear stress in a flowing fluid. When shearing
stresses are present, the normal stress components at a point may not be equal; however, the
pressure is still equal to the average normal stress; that is
P ¼À
1
3
ðs
xx
þ s
yy
þ s
zz
Þ
with very few exceptions, one being flow in shock waves.
Now that certain properties at a point have been discussed, let us investigate the manner
in which fluid properties vary from point to point.
1.3 POINT-TO-POINT VARIATION OF PROPERTIES IN A FLUID
In the continuum approach to momentum transfer, use will be made of pressure, temperature,
density, velocity, and stress fields. In previous studies, the concept of a gravitational field has
been introduced. Gravity, of course, is avector, and thus a gravitational field is a vector field. In
this book, vectors will be written in boldfaced type. Weather maps illustrating the pressure
variation over this country are published daily in our newspapers. As pressure is a scalar
quantity, such maps are an illustration of a scalar field. Scalars in this book will be set in
regular type.

In Figure 1.5, the lines drawn are the loci of points of equal pressure. The pressure
varies continuously throughout the region, and one may observe the pressure levels and infer
the manner in which the pressure varies by examining such a map.
Figure 1.5 Weather map—an example of a scalar field.
1.3 Point-to-Point Variation of Properties in a Fluid
5
Of specific interest in momentum transfer is the description of the point-to-point
variation in the pressure. Denoting the directions east and north in Figure 1.5 by x and y,
respectively, we may represent the pressure throughout the region by the general function
P(x, y).
The change in P, written as dP, between two points in the region separated by the
distances dx and dy is given by the total differential
dP ¼
@P
@x
dx þ
@P
@y
dy (1-3)
In equation (1-3), the partial derivatives represent the manner in which P changes along
the x and y axes, respectively.
Along an arbitrary path s in the xy plane the total derivative is
dP
ds
¼
@P
@x
dx
ds
þ

@P
@y
dy
ds
(1-4)
In equation (1-4), the term dP/ds is the directional derivative, and its functional relation
describes the rate of change of P in the s direction.
A small portion of the pressure field depicted in Figure 1.5 is shown in Figure 1.6. The
arbitrary path s is shown, and it is easily seen that the terms dx/ds and dy/ds are the cosine and
sine of the path angle, a, with respect to the x axis. The directional derivative, therefore, may
be written as
dP
ds
¼
@P
@x
cos a þ
@P
@y
sin a (1-5)
There are an infinite number of paths to choose from in the xy plane; however, two
particular paths are of special interest: the path for which dP /ds is zero and that for which
dP/ds is maximum.
The path for which the directional derivative is zero is quite simple to find. Setting dP/ds
equal to zero, we have
sin a
cos a





dP/ds¼0
¼ tan a




dP/ds¼0
¼À
@P/@x
@P/@y
y
x
α
dx
ds
dy
dy
ds
= sin α
dx
ds
= cos α
Path s
Figure 1.6 Path s in the xy
plane.
6 Chapter 1 Introduction to Momentum Transfer
or, since tan a ¼ dy/dx, we have
dy
dx





dP/ds¼0
¼À
@P/@x
@P/@y
(1-6)
Along the path whose slope is defined by equation (1-6), we have dP ¼ 0, and thus P is
constant. Paths along which a scalar is constant are called isolines.
In order to find the direction for which dP /ds is a maximum, we must have the derivative
(d/da)(dP/ds) equal to zero, or
d
da
dP
ds
¼Àsin a
@P
@x
þ cos a
@P
@y
¼ 0
or
tan a
dP/ds is max
¼
@P/@y
@P/@x





(1-7)
Comparing relations (1-6) and (1-7), we see that the two directions defined by these
equations are perpendicular. The magnitude of the directional derivative when the direc-
tional derivative is maximum is
dP
ds




max
¼
@P
@x
cos a þ
@P
@y
sin a
where cos a and sin a are evaluated along the path given by equation (1-7). As the cosine is
related to the tangent by
cos a ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ tan
2
a

p
we have
cos a




dP/dsis max
¼
@P=@x
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð@P/@xÞ
2
þð@P/@yÞ
2
q
Evaluating sin a in a similar manner gives
dP
ds




max
¼
ð@P/@xÞ
2
þð@P/@yÞ
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð@P/@xÞ
2
þð@P/@yÞ
2
q
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
@P
@x

2
þ
@P
@y

2
s
(1-8)
Equations (1-7) and (1-8) suggest that the maximum directional derivative is a vector of
the form
@P
@x
e
x
þ
@P
@y
e
y
where e

x
and e
y
are unit vectors in the x and y directions, respectively.
The directional derivative along the path of maximum value is frequently encountered
in the anlaysis of transfer processes and is given a special name, the gradient. Thus the
gradient of P, grad P,is
grad P ¼
@P
@x
e
x
þ
@P
@y
e
y
1.3 Point-to-Point Variation of Properties in a Fluid 7