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CHEMICAL ENGINEERING MONOGRAPHS 17
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Gas Transport in
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The Dusty-Gas
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Gas Transport in Porous Media
The Dusty-Gas Model
CHEMICAL ENGINEERING MONOGRAPHS
Advisory Editor: Professor S.W. CHURCHILL, Department of Chemical Engineering,
University of Philadelphia, PA 19104, U.S.A.
Vol. 1 Polymer Engineering (Williams)
Vol. 2 Filtration Post-Treatment Processes (Wakeman)
Vol. 3 Multicomponent Diffusion (Cussler)
Vol. 4 Transport in Porous Catalysts (Jackson)
Vol. 5 Calculation of Properties Using Corresponding State Methods
(Sterbacek et al.)
Vol. 6 Industrial Separators for Gas Cleaning (Storch et al.)
Vol. 7 Twin Screw Extrusion (Janssen)
Vol. S Fault Detection and Diagnosis in Chemical and Petrochemical
Processes (Himmelblau)
Vol. 9 Electrochemical Reactor Design (Pickett)
Vol. 10 Large Chemical Plants (Froment, editor)
Vol. 11 Design of Industrial Catalysts (Trimm)
Vol. 12 Steady-state Flow-sheeting of Chemical Plants (Benedek, editor)
Vol. 13 Chemical Reactor Design in Practice (Rose)
Vol. 14 Electrostatic Precipitators (Bohm)
Vol. 15 Toluene, the Xylenes and their Industrial Derivatives
(Hancock, editor)
Vol. 16 Dense Gas Dispersion (Britter and Griffiths, editors)
Vol. 17 Gas Transport in Porous Media: The Dusty-Gas Model
(Mason and Malinauskas)
TTErDce © O Q S S ^ C S aS
E.A. MASON
Brown University, Providence, Rhode Island, U.S.A.
and
A.P. MALINAUSKAS
Oak Ridge National Laboratory, Oak Ridge, Tennessee, U.S.A.
ELSEVIER - Amsterdam - Oxford - New York 1983
ELSEVIER SCIENCE PUBLISHERS B.V.
Molenwerf 1
P.O. Box 211, 1000 AE Amsterdam, The Netherlands
Distributors for the United States and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY INC.
52, Vanderbilt Avenue
New York, NY 10017
Library of Congress Cataloging in Publication Data
Mason, Ε. Α., 1926-
Gas transport in porous media.
(Chemical engineering monographs ; 17)
Bibliography: p.
Includes index.
1. Gases. 2. Gas dynamics. 3. Porous materials.
I. Malinauskas, A. P. IT. Title. III. Series.
TP2li2.M325 1983 66θ.2?81ι2 83-5588
ISBN 0-W-l*2190-l* (U.S.)
ISBN 0-444-42190-4 (Vol. 17)
ISBN 0-444-41295-6 (Series)
© Elsevier Science Publishers B.V., 1983
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or
transmitted in any form or by any means, electronic, mechanical, photocopying, recording or other-
wise, without the prior written permission of the publisher, Elsevier Science Publishers B.V., P.O.
Box 330, 1000 AH Amsterdam, The Netherlands
Printed in The Netherlands
ν
PREFACE
The development of a theory for the description of gas transport in
porous media has had a long and tangled history· Over the past two decades
or so, however, the main problems appear to have been sorted out, and a
rather comprehensive and consistent understanding of the problem now
exists. The more recent advances are due in large measure to the develop-
ment of a mathematical model which is currently known as the "dusty-gas
model," in which the porous medium is viewed as one component of a gas mix-
ture, but consisting of giant molecules (or dust particles) that are
constrained in space.
One of the purposes of this monograph is to present an historical
account of the development of the dusty-gas model. Our primary intent,
however, is to describe the model and some of its applications in suf-
ficient detail that it can be employed in engineering practice and that it
may also serve as a guide for further advances in this interesting though
not fully understood field. The model has a much broader scope than the
simple physical picture behind it would seem to suggest, and several exten-
sions of the model are also described. These include slip and creep pheno-
mena exhibited by gases over solid surfaces, radiometer effects, aerosol
motion, and membrane transport. Areas where gaps in knowledge remain are
also indicated.
Our thanks go to Mrs. G. Goditt in Providence and to Beverly Varnadore
in Oak Ridge for their careful work and patient good humor during the pre-
paration of the manuscript. It is also a pleasure to acknowledge the sup-
port of our own research during the development of the dusty-gas model by
the U.S. Department of Energy, Office of Basic Energy Sciences, by the U.S.
Army Research Office, by the U.S. National Aeronautics and Space
Administration, and by the U.S. National Science Foundation.
E. A. MASON
A. P. MALINAUSKAS
vii
CONTENTS
Page
PREFACE ν
I. INTRODUCTION AND HISTORICAL BACKGROUND 1
II. THEORY 11
A. ELEMENTARY ARGUMENTS 11
1. General Procedure 11
2. Modes of Gas Transport 15
a. Free-Molecule or Knudsen Flow 15
b. Viscous Flow 17
c. Continuum, or Ordinary, Diffusion 19
d. Surface Flow or Diffusion 23
3. Combined Transport 24
4. Multicomponent Mixtures 28
B. DUSTY-GAS MODEL 30
1. Description of the Model and Critique of Assumptions . 32
2. Diffusion Equations 37
3. Viscous-Flow Equations 42
4. Dusty-Gas Limit 42
C. STRUCTURE OF THE POROUS MEDIUM 50
1. Transport Parameters from Structural Models 51
2. Effect of Heteroporosity on Flux Equations 54
a. Parallel Pores 56
b. Pores in Series 63
c. Interconnection of Pores 65
3. Remarks 69
D. SUMMARY OF EQUATIONS 69
III. EXPERIMENTAL TESTS OF THE THEORY 73
A. REPRESENTATION OF DATA 74
B. ISOTHERMAL MEASUREMENTS 83
1. Single Gases 84
2. Binary Mixtures 87
a. Augmented Diffusion and Permeability Coefficients . 87
b. Uniform-Pressure Diffusion 89
c. Equal Countercurrent Diffusion 92
d. Diffusion Pressure Effect (Diffusive Slip) . . .. 93
e. Effect of Pressure Gradients on Fluxes 96
3. Multicomponent Mixtures 97
C. NONISOTHERMAL MEASUREMENTS 99
1. Single Gases 100
2. Binary Mixtures 108
a. Pressure and Composition Dependence of Thermal
Transpiration 108
b. Pressure Dependence of Thermal Diffusion Ill
vi ii
Page
D. REMARKS 114
IV. EXTENSIONS AND GENERALIZATIONS OF THE THEORY 117
A. SLIP AND CREEP PHENOMENA IN RAREFIED GASES 117
B. RADIOMETER EFFECTS 122
C. AEROSOL MOTION 125
1. General Formulation 126
2. Drag on a Particle 132
3. Diffusiophoresis 136
4. Thermophoresis 139
D. MEMBRANE TRANSPORT 142
1. Generalization of the Dusty-Gas Theory 144
2. Statistical-Mechanical Derivation 146
3. Comparison with Phenomenological Equations 147
a." Onsager (Kedem-Katchalsky) Linear Laws 147
b. Frictional Model 152
c. Diffusion Model 153
4. Semipermeable Membranes 155
V. CONCLUDING REMARKS 159
REFERENCES 161
LIST OF SYMBOLS 175
AUTHOR INDEX 183
SUBJECT INDEX 189
1
Chapter I.
INTRODUCTION AND HISTORICAL BACKGROUND
The primary purpose of this monograph is to give a detailed presentation
of a theory that had been developed to describe the transport of gases through
porous media. This theory is often known as the "dusty-gas model," because
the porous medium is treated as one component of the gas mixture, consisting
of giant molecules held fixed in space, and the highly developed kinetic
theory of gases is applied to this supermixture. The physical picture
involved is both simple and appealing — so much so that the model has been
invented independently at least four times, starting with James Clerk Maxwell
in 1860 [M25]. However, it is only in recent years that the consequences and
ramifications of the model have been worked out in sufficient detail to
constitute a comprehensive and consistent body of theory. This theory is now
mature and developed enough to be a useful tool for engineers. It has, in
fact, already been used to some extent for modeling chemical reactions in
porous catalysts, as has been reviewed in a monograph by Jackson [Jl].
Nevertheless, the theory does not seem to be widely known among engineers,
probably because much of it has been published in journals primarily devoted
to physics or physical chemistry. Furthermore, the language of kinetic theory
used in the basic papers is not always familiar to engineers.
That diffusion can be an important limitation of reactivity in porous cat-
alysts has been recognized for a long time [SI, S8, S19, T3, W4], so that one
area of application of the dusty-gas model to engineering problems is fairly
obvious. It is less obvious that the model has a much wider scope than just
gas transport in porous media. Two examples may be mentioned at this point.
First, if the forces holding the giant molecules ("dust") fixed in space are
removed, so that these giant molecules are free to move in the gas, the result
is an aerosol system. Thus a small change in the dusty-gas equations converts
them from a description of gas transport in a porous medium to a description
of aerosol motion in nonuniform gas mixtures. Put another way, in one case we
rest on the particles and account for the motion of the gas, and in the other
case, we sit out in the gas and account for the motion of the particles. It
2
is thus not so surprising that the two cases can be described by similar sets
of equations.
As a second example, the model can be applied to transport in membranes by
regarding the gas mixture as a solution, and the porous medium as a membrane.
The model then serves as a test and a diagnostic probe of supposedly very
general phenomenological theories of transport in membranes. Any difficulties
or deficiencies identified in this way are usually easily remedied, either by
modifying the original phenomenological equations, or by replacing specifi-
cally gas-like terms of the dusty-gas equations with terms appropriate for
arbitrary fluids (e.g., partial pressures are replaced by activities).
Ultimately, one may hope to learn enough about the crucial points of the
problem to construct a general statistical-mechanical theory that is valid for
general fluids and not just perfect gases.
The scope of this monograph is limited to a description of the dusty-gas
model, with emphasis on its application to gas transport in porous media. In
addition, a brief discussion of its applications in other fields is presented,
and some indication of specific applications in chemical engineering is given.
However, we do not try to give a complete survey of the subject of gas
transport in porous media and its applications in chemical engineering; but a
number of books and review articles are already available for the reader who
seeks such a survey [B2, B3, CI, D12, H7, Jl, SI, Yl].
In the remainder of this chapter, we present a brief review of the
historical background of the dusty-gas model. The model has had an unusually
long history, and its origins are intertwined with the origins of the kinetic
theory of gases itself. Many crucial points, both experimental and theoreti-
cal, have been elucidated, then forgotten or misunderstood, and later rediscov-
ered independently. In Chapter II, a systematic critical account of the theory
behind the model is given, beginning with simple momentum-transfer arguments,
and then proceeding to the full machinery of modern gas kinetic theory. The
chapter is concluded with a summary of the final transport equations developed
from the model. Chapter III is devoted to a selection of experimental tests
of the model as applied to gas transport, whereas in Chapter IV it is shown
how the fundamental ideas behind the model can be used to develop extensions
and generalizations of the results for porous media; four examples are given —
slip and creep phenomena exhibited by gases over solid surfaces, radiometer
effects, aerosol motion, and membrane transport. Finally, some specific
applications to chemical engineering are mentioned in Chapter V.
