Art of Modelling in Science and Engineering.pdf

The Art of

MODELING

SCIENCE

and

ENGINEERING

Diran Basmadjian

CHAPMAN & HALL/CRC

Boca Raton London New York Washington, D.C.

Library of Congress Cataloging-in-Publication Data

Basmadjian, Diran

The art of modeling in science and engineering / Diran Basmadjian.

p. cm.

Includes bibliographical references and index.

ISBN 1-58488-012-0

1. Mathematical models. 2. Science—Mathematical models. 3. Engineering—

Mathematical models. I. Title.

QA401.B38 1999

511'.8—dc21

99-11443

CIP

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Preface

The term

as used in this text, is understood to refer to the ensemble of

equations which describe and interrelate the variables and parameters of a physical

system or process. The term

model,

in turn refers to the derivation of appropriate

equations that are solved for a set of system or process variables and parameters.

These solutions are often referred to as simulations, i.e., they simulate or reproduce

the behavior of physical systems and processes.

Modeling is practiced with uncommon frequency in the engineering disciplines

and indeed in all physical sciences where it is often known as “Applied Mathemat-

ics.” It has made its appearance in other disciplines as well which do not involve

physical processes per se, such as economics, ﬁnance, and banking. The reader will

note a chemical engineering slant to the contents of the book, but that discipline

now reaches out, some would say with tentacles, far beyond its immediate narrow

conﬁnes to encompass topics of interest to both scientists and engineers. We address

the book in particular to those in the disciplines of chemical, mechanical, civil, and

environmental engineering, to applied chemists and physicists in general, and to

students of applied mathematics.

The text covers a wide range of physical processes and phenomena which

generally call for the use of mass, energy, and momentum or force balances, together

with auxiliary relations drawn from such subdisciplines as thermodynamics and

chemical kinetics. Both static and dynamic systems are covered as well as processes

which are at a steady state. Thus, transport phenomena play an important but not

exclusive role in the subject matter covered. This amalgam of topics is held together

by the common thread of applied mathematics.

A plethora of related specialized tests exist. Mass and energy balances which

arise from their respective conservation laws have been addressed by Reklaitis

(1983), Felder and Rousseau (1986) and Himmelblau (1996). The books by Reklaitis

and Himmelblau in particular are written at a high level. Force and momentum

balances are best studied in texts on ﬂuid mechanics, among many of which are by

Streeter, Wylie, and Bedford (1998) and White (1986) stand out. For a comprehensive

and sophisticated treatment of transport phenomena, the text by Bird, Stewart, and

Lightfoot (1960) remains unsurpassed. Much can be gleaned on dynamic or unsteady

systems from process control texts, foremost among which are those by Stephano-

poulos (1984), Luyben (1990) and Ogunnaike and Ray (1996).

In spite of this wealth of information, students and even professionals often

experience difﬁculties in setting up and solving even the simplest models. This can

be attributed to the following factors:

modeling

•

A major stumbling block is the proper choice of model. How complex

should it be? One can always choose to work at the highest and most

rigorous level of partial differential equations (PDE), but this often leads

to models of unmanageable complexity and dimensionality. Physical

parameters may be unknown and there is a rapid loss of physical insight

caused by the multidimensional nature of the solution. Constraints of time

and resources often make it impossible to embark on elaborate exercises

of this type, or the answer sought may simply not be worth the effort. It

is surprising how often the solution is needed the next day, or not at all.

Still, there are many occasions where PDEs are unavoidable or advantage

may be taken of existing solutions. This is particularly the case with PDEs

of the “classical” type, such as those which describe diffusion or conduc-

tion processes. Solutions to such problems are amply documented in the

deﬁnitive monographs by Carslaw and Jaeger (1959) and by Crank (1978).

Even here, however, one often encounters solutions which reduce to PDEs

of lower dimensionality, to ordinary differential equations (ODEs) or even

algebraic equations (AEs). The motto must therefore be “PDEs if neces-

sary, but not necessarily PDEs.”

• The second difﬁculty lies in the absence of precise solutions, even with

the use of the most sophisticated models and computational tools. Some

systems are simply too complex to yield exact answers. One must resort

here to what we term

i.e., establishing upper or

lower bounds to the answer being sought. This is a perfectly respectable

exercise, much practiced by mathematicians and theoretical scientists and

engineers.

bracketing the solution,

•

The third difﬁculty lies in making suitable simplifying assumptions and

approximations. This requires considerable physical insight and engineer-

ing skill. Not infrequently, a certain boldness and leap in imagination is

called for. These are not easy attributes to satisfy.

Overcoming these three difﬁculties constitute the core of

The

Art of Modeling.

Although we will not make this aspect the exclusive domain of our effort, a large

number of examples and illustrations will be presented to provide the reader with

some practice in this difﬁcult craft.

Our approach will be to proceed slowly and over various stages from the

mathematically simple to the more complex, ultimately looking at some sophisti-

cated models. In other words, we propose to model “from the bottom up” rather

than “from the top down,” which is the normal approach particularly in treatments

of transport phenomena. We found this to be pedagogically more effective although

not necessarily in keeping with academic tradition and rigor.

As an introduction, we establish in Chapter 1 a link between the physical system

and the mathematical expressions that result. This provides the reader with a sense

of the type and degree of mathematical complexity to be expected. Some simple

classical models such as the

stirred tank

and what we term the

one-dimensional pipe

and

are introduced. We examine as well the types of balances,

i.e., the equations which result from the application of various conservation laws to

different physical entities and the information to be derived from them.

These introductory remarks lead, in Chapter 2, to a ﬁrst detailed examination

of practical problems and the skills required in the setting up of equations arising

quenched steel billet

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