HYDRODYNAMICS, MASS AND
HEAT TRANSFER IN CHEMICAL
ENGINEERING
Topics in Chemical Engineering
A
series edited by
R.
Hughes, University of Salford,
UK
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Engineering series.
HYDRODYNAMICS, MASS AND
HEAT TRANSFER IN CHEMICAL
ENGINEERING
A.D. Polyanin
lnstitute for Problems in Mechanics, Moscow, Russia
A.M. Kutepov
Moscow State Academy of Chemical Engineering, Moscow, Russia
A.V. Vyazm in
Karpov lnstitute of Physical Chemistry, Moscow, Russia
D.A. Kazenin
Moscow State Academy of Chemical Engineering, Moscow, Russia
CRC PRESS
Boca Raton London
New
York Washington,
D.C.
First published 2002 by Taylor
&
Francis
11 New Fetter Lane, London EC4P
4EE
Simultaneously published in the USA and Canada
by Taylor
&
Francis Inc.
29 West 35th Street, New York, NY 10001
Taylor
&

ISBN 0-415-27237-8
CONTENTS
Introduction to the Series
Preface
Basic Notation
xiii
xv
xvii
1
.
Fluid Flows in Films. Jets. nbes. and Boundary Layers 1.1. Hydrodynamic Equations and Boundary Conditions
.

1.1- 1
.
Laminar Flows Navier-Stokes Equations

1.1-2. Initial Conditions and the Simplest Boundary Conditions

1.1-3. Translational and Shear Flows
1.1-4. Turbulent Flows 1.2. Flows Caused by a Rotating Disk

1.2-1. Infinite Plane Disk
1.2-2. Disk of Finite Radius

1.5-4. Tube of Elliptic Cross-Section

1.5-5. Tube of Rectangular Cross-Section

1.5-6. Tube of Triangular Cross-Section

1.5-7. Tube of Arbitrary Cross-Section

1.6. Turbulent Flows in Tubes

1.6-1. Tangential Stress
.
Turbulent Viscosity

1.6.2. Structure of the Flow
.
Velocity Profile in a Circular Tube

1.6-3. Drag Coefficient of a Circular Tube

1.6-4. Turbulent Flow in a Plane Channel

1.6-5. Drag Coefficient for Tubes of Other Shape

1.7. Hydrodynamic Boundary Layer on a Flat Plate

1.7- 1
.
Preliminary Remarks


.
.
2.1 1 Stokes Equations

2.1-2. General Solution for the Axisymmetric Case
2.2.
Spherical Particles. Drops. and Bubbles in Translational Stokes Flow

2.2-1. Flow Past
a
Spherical Particle

2.2-2. Flow Past a Spherical Drop or Bubble

2.2-3. Steady-State Motion of Particles and Drops in a Fluid

2.2-4. Flow Past Drops With a Membrane Phase

2.2-5. Flow Past a Porous Spherical Particle

2.3.
Spherical Particles in Translational Flow at Various Reynolds Numbers

2.3-1. Oseen's and Higher Approximations as
Re
+
0

2.3-2. Flow Past Spherical Particles in a Wide Range of
Re

2.6-3. Translational Stokes Flow Past Particles of Arbitrary Shape

2.6-4. Sedimentation of Isotropic Particles

2.6-5. Sedimentation of Nonisotropic Particles

2.6-6. Mean Velocity of Nonisotropic Particles Falling in a Fluid

2.6-7. Flow Past Nonspherical Particles at Higher Reynolds Numbers

2.7. Flow Past a Cylinder (the Plane Problem)

2.7-1. Translational Flow Past a Cylinder

2.7-2. Shear Flow Around a Circular Cylinder

2.8. Flow Past Deformed Drops and Bubbles

2.8-1. Weak Deformations of Drops at Low Reynolds Numbers

2.8-2. Rise of an Ellipsoidal Bubble at High Reynolds Numbers

2.8-3. Rise of a Large Bubble of Spherical Segment Shape
vii

2.8-4. Drops Moving in Gas at High Reynolds Numbers

2.9. Constrained Motion of Particles

2.9- 1


3.1-4. Mass Transfer Complicated by a Volume Chemical Reaction

3.1-5. Diffusion Fluxes and the Sherwood Number

.
3.1.6. Heat Transfer The Equation and Boundary Conditions

3.1-7. Some Methods of Theory of Mass and Heat Transfer

3.1-8. Mass and Heat Transfer in Turbulent Flows

3.2. Diffusion to a Rotating Disk
.

3.2. 1 Infinite Plane Disk

3.2-2. Disk of Finite Radius

3.3. Heat Transfer to a Flat Plate

.
3.3- 1 Heat Transfer in Laminar Flow

3.3-2. Heat Transfer in Turbulent Flow

3.4. Mass Transfer in Liquid Films

3.4-1. Mass Exchange Between Gases and Liquid Films


4
.
Mass and Heat Exchange Between Flow and Particles. Drops. or Bubbles
4.1. The Method of Asymptotic Analogies in Theory of Mass and Heat Transfer
.
.
4.1 1 Preliminary Remarks

4.1-2. Transition to Asymptotic Coordinates

4.1.3. Description of the Method

viii
CONTENTS

4.2. Interiors Heat Exchange Problems for Bodies of Various Shapes

4.2-1. Statement of the Problem

4.2-2. General Formulas for the Bulk Temperature of the Body

4.2-3. Bulk Temperature for Bodies of Various Shapes
4.3. Mass and Heat Exchange Between Particles of Various Shapes and a Stagnant
Medium 4.3-1. Stationary Mass and Heat Exchange

4.3-2. Transient Mass and Heat Exchange


4.7- 1
.
Mass Transfer at Low Reynolds Numbers

4.7-2. Mass Transfer at Moderate and High Reynolds Numbers

4.7-3. General Correlations for the Sherwood Number

4.8. Particles, Drops, and Bubbles in Linear Shear Flows
.
Arbitrary Peclet
Numbers

4.8-1. Linear Straining Shear Flow
.
High Peclet Numbers

4.8-2. Linear Straining Shear Flow
.
Arbitrary Peclet Numbers

4.8-3. Simple Shear and Arbitrary Plane Shear Flows

4.9.
Mass Transfer in a Translational-Shear Flow and in a Flow with Parabolic
Profile

4.9-1. Diffusion to a Sphere in a Translational-Shear Flow

4.9-2. Diffusion to a Sphere in a Flow with Parabolic Profile


4.12-1. Statement of the Problem 197

4.12-2. Spherical Particles and Drops at High Peclet Numbers
198

4.13-3. Spherical Particles and Drops at Arbitrary Peclet Numbers
199
4.12-4. Nonspherical Particles, Drops, and Bubbles

200
4.13. Qualitative Features of Mass Transfer Inside a Drop at High Peclet Numbers
201
4.13-1. Limiting Diffusion Resistance of the Disperse Phase

201
4.13-2. Comparable Diffusion Phase Resistances

205
4.14. Diffusion Wake
.
Mass Exchange of Liquid with Particles or Drops Arranged
inLines

206
4.14-1. Diffusion Wake at High Peclet Numbers

206
4.14-2. Diffusion Interaction of Two Particles or Drops


5.1-2. Rotating Disk and a Flat Plate
5.1-3. Circular Tube

Diffusion to a Rotating Disk and a Flat Plate Complicated by a Volume
Reaction 5.2-1. Mass Transfer to the Surface of a Disk Rotating in a Fluid

5.2-2. Mass Transfer to a Flat Plate in a Translational Flow
Mass Transfer Between Particles, Drops, or Bubbles and Flows, with Volume
Reaction 5.3-1. Statement of the Problem

5.3-2. Particles in a Stagnant Medium
.

5.3.3. Particles, Drops, and Bubbles First-Order Reaction
.

5.3.4. Particles, Drops, and Bubbles Arbitrary Rate of Reaction
Mass Transfer Inside a Drop (Cavity) Complicated by a Volume Reaction
.

5.4-1. Spherical Cavity Filled by a Stagnant Medium

5.4-2. Nonspherical Cavity Filled by a Stagnant Medium



5.7-2. Equation for the Thickness of the Film
.
Nusselt Solution 5.7-3. Some Generalizations

5.8. Nonisothermal Flows in Channels and Tubes

.
5.8-1. Heat Transfer in Channel Account of Dissipation
.

5.8-2. Heat Transfer in Circular Tube Account of Dissipation

5.8-3. Qualitative Features of Heat Transfer in Highly Viscous Liquids

5.8-4. Nonisothermal Turbulent Flows in Tubes

5.9.
Thermogravitational and Thermocapillary Convection in a Fluid Layer
5.9-1. Thermogravitational Convection 5.9-2. Joint Thermocapillary and Thermogravitational Convection
.

5.9-3. Thermocapillary Motion Nonlinear Problems
5.10. Thermocapillary Drift of a Drop
6.1-6. Viscoelastic Fluids
6.2. Motion of Non-Newtonian Fluid Films

.

6.2-1. Statement of the Problem Formula for the Friction Stress
.
6.2-2. Nonlinearly Viscous Fluids Power-Law Fluids

.
6.2-3. Viscoplastic Media The Shvedov-Bingham Fluid

6.3. Mass Transfer in Films of Rheologically Complex Fluids

6.3-1. Mass Exchange Between a Film and a Gas

6.3-2. Dissolution of a Plate by a Fluid Film

6.4. Motion of Non-Newtonian Fluids in Tubes and Channels

.
6.4-1. Circular Tube Formula for the Friction Stress

.
6.4-2. Circular Tube Nonlinearly Viscous Fluids

.
6.4-3. Circular Tube Viscoplastic Media


6.8-2. Exact Solutions

6.8-3. Jet Width and Volume Rate of Flow

Motion and Mass Exchange of Particles, Drops, and Bubbles in
Non-Newtonian Fluids

6.9- 1
.
Drag Coefficients

6.9-2. Sherwood Numbers

6.10. Transient and Oscillatory Motion of Non-Newtonian Fluids

296
6.10-1. Transient Motion of an Infinite Flat Plate

296
6.10-2. Oscillating Flat-Plate Flow for Maxwellian Fluids

299

6.10-3. Transient Simple Shear Flow of Shvedov-Bingham Fluids 299
7
.
Foams: Structure and Some Properties

301

313
7.3.3. Kinetics of Surfactant Adsorption in a Transient Foam Body

314
Internal Hydrodynamics of Foams
.
Syneresis and Stability

315
7.4-1. Internal Hydrodynamics of Foams

316
7.4-2. Generalized Equation of Syneresis

317
7.4.3. Gravitational and Centrifugal Syneresis

318
7.4-4. Barosyneresis

319
7.4.5. Stability, Evolution, and Rupture of Foams

320
Rheological Properties of Foams

322
7.5. 1
.
Macrorheological Models of Foams

.
1-4
.
Heat Equation in Spherical Coordinates

S.2. Formulas for Constructing Exact Solutions S.2-1. Duhamel Integrals

S.2-2. Problems with Volume Reaction

S.3. Orthogonal Curvilinear Coordinates

S.3-1. Arbitrary Orthogonal Coordinates
S.3-2. Cylindrical Coordinates
R,
p,
Z

S.3-3. Spherical Coordinates
R,
0,
cp S.3-4. Coordinates of a Prolate Ellipsoid of Revolution
a,
r,
cp


S.6- 1
.
Equations in Rectangular Cartesian Coordinates
S.6-2. Equations in Cylindrical Coordinates

S.6-3. Equations in Spherical Coordinates

References

Index

Introduction to the Series
The subject matter of chemical engineering covers a very wide spectrum of
learning and the number of subject areas encompassed in both undergraduate
and graduate courses is inevitably increasing each year. This wide variety of
subjects makes it difficult to cover the whole subject matter of chemical
engineering in a single book. The present series is therefore planned as a
number of books covering areas of chemical engineering which, although
important, are not treated at any length in graduate and postgraduate standard
texts. Additionally, the series will incorporate recent research material which
has reached the stage where an overall survey is appropriate, and where
sufficient information is available to merit publication in book form for the
benefit of the profession as a whole.
Inevitably, with a series such as this, constant revision is necessary if the
value of the texts for both teaching and research purposes is to be maintained.
I
would be grateful to individuals for criticisms and for suggestions for future
editions.
R.

cal hydrodynamics for graduate and postgraduate students.
In Chapters
1
and
2
we study fluid flows, which underlie numerous processes
of chemical engineering science. We present up-to-date results about transla-
tional and shear flows past particles, drops, and bubbles of various shapes at a
wide range of Reynolds numbers. Single particles and systems of particles are
considered. Film and jet flows, fluid flows through tubes and channels of various
shapes, and flow past plates, cylinders, and disks are examined.
In Chapters
3
and
4
we analyze mass and heat transfer in plane channels,
tubes, and fluid films. We consider the mass and heat exchange between parti-
cles, drops, or bubbles and uniform or shear flows at various Peclet and Reynolds
numbers. The results presented are of great importance in obtaining scientifi-
cally justified methods for a number of technological processes such as dissolu-
tion, drying, adsorption, aerosol and colloid sedimentation, heterogeneous catalytic
reactions, absorption, extraction, and rectification.
In Chapter
5
some problems of mass and heat transfer with various complic-
ating factors are discussed. Mass transfer problems are investigated for various
xvi
PREFACE
kinetics of volume and surface chemical reactions. Nonlinear problems of con-
vective mass and heat exchange are considered taking into account the depend-

material and its compact presentation permit the book to be used as a concise
handbook in chemical engineering science and related fields in hydrodynamics,
heat and mass transfer, etc.
The authors are grateful to
A.
E.
Rednikov and Yu.
S.
Ryazantsev, who wrote
Sections
5.8-5.10,
Z.
D.
Zapryanov, who contributed to Sections 1.2,
1.3,
and
2.9,
and
A.
G.
Petrov, who contributed to Subsections 2.4-3,2.8-2, and 2.8-4. We
express our deep gratitude to
V.
E.
Nazaikinskii and
A.
I. Zhurov for fruitful
discussions and valuable remarks.
The work on this book was supported in part by the Russian Foundation for
Basic Research.

=
Fs(C)
kinetic function of volume reaction,
F,
=
Fv(C)
Froude number
dimensionless kinetic function of surface reaction,
fs
=
fs(c)
dimensionless kinetic function of volume reaction,
fv
=
f,(c)
mean value of dimensionless kinetic function,
(f,)
=
A:
f,(c) dc
shear matrix coefficients
Grashof number
acceleration due to gravity
metric tensor components
film thickness; half-width of plane channel
dimensionless total diffusion flux
total diffusion flux
dimensionless total heat flux
unit vectors of Cartesian coordinate system
momentum of jet

R
=
\/x2
+
YZ
+
Z2
cylindrical coordinate system,
R
=
m
Reynolds number,
Re
=
aU/u
Reynolds number based on diameter,
Red
=
dU/v
local Reynolds number,
Rex
=
XU/u
dimensionless radial spherical coordinate,
r
=
R/a
dimensionless area of surface,
S
=

urn,
UT
U*
v
(V)
Vx,
VY,
vz
Vx
VR,
VB,
Vp
VR, v., Vp
v$',
vs"'
characteristic flow velocity
nonperturbed fluid velocity (in incoming flow remote from particle)
maximum fluid velocity at surface of film or on tube axis
thermocapillary drift velocity of drop
friction velocity (for turbulent flows),
U,
=
m
fluid velocity vector
mean flow rate velocity,
(V)
=
Q/S,
fluid velocity components in Cartesian coordinate system
average component of velocity in turbulent flow

p2/p1
Laplace operator
total pressure drop along a tube part of length
L,
AP
>
0
thickness of hydrodynamic boundary layer
thickness of thermal boundary layer
Kronecker delta
friction temperature (for turbulent flows)
angular coordinate
thermal conductivity coefficient
von Karman constant
drag coefficient (for tubes and channels)
eigenvalues
viscosity
plastic viscosity for Shvedov-Bingham fluid
viscosity of continuous phase
viscosity of disperse phase
kinematic viscosity,
V
=
p/p
turbulent viscosity coefficient
shape factor,
IT
=
Sh S, /a;
disjoining pressure

properties, for example, the widely used classification based on the Reynolds
number Re, which is the most significant state-geometric parameter." This clas-
sification distinguishes flows at low Re
[179], at high Re (boundary layers [427]),
and at supercritical Re (turbulent flows [188]) and is methodologically impor-
tant in that it introduces a small parameter (Re or ~e-l), which permits one
to solve nonlinear hydrodynamic problems reliably by using expansions with
respect to that parameter. Although this classification is undoubtedly fruitful
and convenient for those studying hydrodynamic problems mathematically and
numerically, in the present book we focus our attention on the practical needs of
industrial engineers who deal with specific units of equipment where the type of
flow of the reactive medium is virtually predetermined by the design. Accord-
ingly, our treatment of hydrodynamics consists of two chapters. Chapter
1
deals
with flows of extended fluid media interacting with each other or with containing
walls (flows in films, tubes, channels, jets, and boundary layers near a solid
surface). In Chapter 2 we consider the hydrodynamic interaction of particles of
various nature (solid, liquid, or gaseous) with the ambient continuous phase.
1.1.
Hydrodynamic Equations and Boundary
Conditions
In this section we present equations and boundary conditions used in solving
hydrodynamic problems. Their detailed derivation, as well as an analysis of
*
There are some other flow classifications, for example, with respect to specific properties of
the boundary of the flow region: fluid flow with free boundaries
[385], fluid flow with interface
[226,
5011, and flow along a permeable boundary

and
Z
in physical space;
t
is time;
gx, gy,
and
gz
are the mass force (e.g., the
gravity force) density components;
v
=
p/p
is the kinematic viscosity of the fluid.
The three components of the fluid velocity
VX, Vy, VZ,
and the pressure
P
are
the unknowns.
By introducing the fluid velocity vector V
=
ixVx
+
iyVy
+
iZVZ,
where
ix, iy,
and

l
.3) becomes
The stream function *(X,
Y)
is introduced by the relations
(l.
l
S)
The continuity equation is satisfied identically.
2.
In axisymmetric problems, all variables are independent of the axial
coordinate
Z
in the cylindrical coordinates
R,
8,
Z.
The continuity equation has
the form (both sides are multiplied by
R)
and the stream function is introduced
by
3. In axisymmetric problems, all variables are independent of the coordi-
nate
cp
in the spherical coordinates R, 8,
cp.
The continuity equation has the form
(both sides are multiplied by R)
1

line there corresponds a constant value of the stream function. The fluid velocity
vector is tangent to the streamline. (Note that the streamlines coincide with the
trajectories of fluid particles only in the stationary case.)
4
FLUID FLOWS
IN
FILMS, TUBES,
AND
JETS
Table 1.1 presents equations for the stream function, obtained from the
Navier-Stokes equations (1.1. l), (1.1.2) in various coordinate systems.
Dimensionlessform of equations.
To analyze the hydrodynamic equations
(1.1.3), (1.1.4), it is convenient to introduce dimensionless variables and un-
known functions as follows:
where
a
and
U
are the characteristic length and the characteristic velocity, re-
spectively. As a result, we obtain
dv
1 1
g
-
+
(v.
V)v=-Vp+ -AV+

dt

In nonstationary problems, where the terms with partial derivatives with
respect to time are retained in the equation of motion, the initial velocity field
must be given in the entire flow region and satisfy the continuity equation
(l.
l. 1)
there. The initial pressure field need not be given, since the equations do not
contain the derivative of pressure with respect to time.*
As a rule, the region occupied by a moving reactive mixture is not the entire
space but only a part bounded by some surfaces. According to whether the
point at infinity belongs to the flow region or not, the problem of finding the
unknown functions is called the exterior or interior problem of hydrodynamics,
respectively.
On the surface
S
of a solid body moving in a flow of a viscous fluid, the
no-slip condition is imposed. This condition says that the vector
Vls
of the fluid
*
Obviously, if
an
arbitrary initial pressure field is given, it may happen that the velocity fields
obtained from the equations of motion do not satisfy the continuity equation fort
>
0
[404].
No such
problems arise in the stationary case.
1.1.
HYDRODYNAMIC