Mathematical Modelling Techniques

285 Pages·1995·13.83 MB·English

by  Rutherford Aris

MATHEMATICAL
M ODELLING
TECHNIQUES
Rutherford Aris
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(continued on back flap)
Mathematical
Modelling Techniques

Mathematical
Modelling Techniques
Rutherford Aris
University of Minnesota
DOVER PUBLICATIONS, INC.
New York
Acknowledgments
The material on pages 193-269 of this Dover edition originally appeared in the following sources:
—“Re, k and /tt: A Conversation on Some Aspects of Mathematical Modelling,“ Appl. Math. Modelling,
Vol. 1 (1977), pp. 386-394. Reprinted with the permission of the publisher.
—“The Jail of Shape,” Chem. Eng. Commun., Vol. 24 (1983), pp. 167-181. Copyright © 1983 by Gordon
and Breach, Science Publishers, Inc. Reprinted with the permission of the publisher.
—“The Mere Notion of a Model,” Mathematical Modelling, Vol. 1 (1980), pp. 1-12. Copyright © 1980
Pergamon Press Ltd. Reprinted with kind permission from Pergamon Press Ltd, Headington Hill Hall,
Oxford 0X3 OBW, United Kingdom.
—“Ut Simulacrum, Poesis,” New Literary History, Vol. 20 (1988-89), pp. 323-340. Reprinted with the
permission of the publisher.
—“Manners Makyth Modellers,” Chemical Engineering Science, Vol. 46, No. 7 (1991), pp. 1535-1544.
Copyright © 1991 Pergamon Press pic. Reprinted with kind permission from Pergamon Press Ltd,
Headington Hill Hall, Oxford OX3 OBW, United Kingdom.
—“How to Get the Most Out of an Equation without Really Trying,” Chemical Engineering Education,
Vol. 10 (1976), pp. 114-124. Reprinted with the permission of the publisher.
Copyright
Copyright © 1978, 1994 by Rutherford Aris.
All rights reserved under Pan American and International Copyright Conventions.
Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario.
Published in the United Kingdom by Constable and Company, Ltd., 3 The Lanchesters, 162-164 Fulham Palace
Road, London W6 9ER.
Bibliographical Note
This Dover edition, first published in 1994, is an unabridged, slightly corrected republication of the work
originally published by Pitman Publishing Limited, London, in 1978. The Dover edition has been expanded by the
addition of six journal articles not included in the original edition.
Library of Congress Cataloging-in-Publication Data
Aris, Rutherford.
Mathematical modelling techniques / Rutherford Aris.
p. cm.
Originally published: London : Pitman, 1978.
“The Dover edition has been expanded by the addition of six journal articles not included in the original
edition”—T.p. verso.
Includes bibliographical references and indexes.
ISBN 0-486-68131-9 (pbk.)
1. Mathematical models. I. Title.
QA401.A68 1994
5 IP. 8—dc20 94-23196
CIP
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
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Preface
"Par ma foi! il y a plus de quarante ans que
je dis de la prose sans que jfen susse rien,
et je vous suis le plus oblige du monde de
m1avoir appris cela."
M. Jourdain in Moliere's "Le Bourgeois
Gentilhomme" (Act. II Sc. IV)
The original title under which these notes were written— "Notes toward the
definition of the craft of mathematical modelling"— was somewhat long-winded
and perhaps, by reason of its allusion, a shade pretentious. It had however
the merit of greater precision and conveyed the tentative spirit in which
these notes are put forward for the criticism of a larger public. The whole
activity of mathematical modelling has blossomed forth into such a multitude
of areas in the last few years (witness a 1st International Conference with 2646
pages of Proceedings [199]) that there is indeed a need to define it in the
sense of seeking out its boundary and exploring its interior as well as of
discovering its structure and essential nature. The time is not yet ripe for
a magisterial survey, which would in any case demand an abler pen than mine,
but I believe it can be approached from the angle of craftsmanship. It is a
commonplace in educational circles that it is comparatively easy to teach the
method of solution of a standard mathematical equations, but much harder to
communicate the ability to formulate the equations adequately and economi­
cally. With the notable exception of Lin and Segelfs [223], and Haberman's book
[213] and the papers of Hammersley [80,81,82] few publications pay much
x Mathematical Modelling Techniques
attention to the little things that the experienced mathematical modeller
does, almost by instinct. It would therefore seem to be worthwhile to try
and set down some of these notions in the interests of the craft and with
the hope that it will stimulate further discussion and development. There is
manifestly a danger here, for it may be only the MM. Jourdains who will be
vastly excited to learn that they have been talking prose all their lives.
Nevertheless I hope that some of my peers and betters will find the subject
worthy of their attention.
Later iterations of this effort will demand a wealth of examples drawn
from all branches of the physical and social sciences. In this first attempt
I have chosen three physical examples to serve the illustration of many
points. These examples— the packed bed, the chromatographic column and the
stirred tank— are given in detail in the appendices. They are in some sense
fold-out maps to the text though they cannot be presented as such. (Each
has its own nomenclature which is listed at the end of its discussion; the
nomenclature for other examples is introduced in situ.) These examples and
those introduced at various points of the text are often connected with the
mathematical theory of chemical reactors. I make no apology for this; the
field is a rich one that has stimulated some of the work of the best applied
mathematicians who have used a reactor like a stalking-horse under cover of
which to shoot their wit. Its problems are challenging, yet from the
modelling point of view they do not demand any great knowledge of chemistry
or of engineering and so are accessible to all.
Many of the notions I have advanced here and the order I have tried to
impose on the subject are quite tentative and I shall appreciate any comments
and criticism. I have already benefitted from interaction with colleagues,
both faculty and students, at Caltech and it is one of the virtues of