Solution of differential equation models by polynomial approximation

231 Pages·1978·43.73 MB·English

by  John Villadsen

Solution of
Differential Equation
Models by
Polynomial Approximation
JOHN VILLADSEN
MICHAEL L. MICHELSEN
Instituttet for Kemiteknik
Denmark
PRENTICE-HALL, INC.
Englewood Cliffs, New Jersey 07632
Library of Congress Cataloging in Publication Data
VILLADSEN, JOHN.
Solution of differential equatiOn models by
polynomial approximation.
(Prentice-Hall international senes in the physical
and chemical engineering sc1ences)
Includes bibliographies and index.
Contents
1. Differential equations-Numerical solutions.
2. ApproximatiOn theory. 3. Polynomials. 4. Chem1cal
engmeering-Mathematical models. I. Michelsen, M. L.,
joint author. II. Title.
QA.371.V52 515'.35 77-4331
ISBN 0-13-822205-3
© 197 8 by Prentice-Hall, Inc. Preface xi
Englewood Cliffs, N.J. 07632
All rights reserved. No part of this book 1. A Preliminary Study
may be reproduced in any form or by any means of Some Important Mathematical Models
without permission in writing from the publisher. from Chemical Engineering 1
Introduction, 1.
10 9 8 7 6 5 4 3 2 1
1.1 The General Mathematical Model, 3.
1.2 Steady State Homogeneous Flow Model for a Reacting
Printed in the United States of America
System, 10.
1.3 Steady State and Transient Models for Solids, 23.
1.4 Heterogenous Model for a Reacting System, 34.
1.5 A Model for Hollow-Fiber Reverse-Osmosis Systems, 40.
1.6 Flow of Polymer Melts in Extruders, 45.
1.7 Solution of Linear Differential Equations, 50.
Exercises, 59.
References, 61.
Prentice-Hall International, Inc., London
Prentice-Hall of Australia Pty. Limited, Sydney
2. Polynomial Approximation-
Prentice-Hall of Canada, Ltd., Toronto
Prentice-Hall of India Private Limited, New Delhi A First View of Construction Principles 67
Prentice-Hall of Japan, Inc., Tokyo
Prentice-Hall of Southeast Asia Pte. Ltd., Singapore Introduction, 67.
Whitehall Books Limited, Wellington, New Zealand 2.1 A Taylor Series Approximation, 68.
vii
viii
Contents
Contents
IX
2.2 The Lowest-Order MWR Approximations Y1(x), 73.
5.2 Existence and Uniqueness Theorems, 204.
2.3 Higher-Order MWR Approximations y~x), 77.
5.3 Application of Comparison Differential Equations, 207.
2.4 Nonlinear Problems, 92.
5.4 Global Collocation Solution of a Nonlinear Differential
2.5 Reformulations of the Nth-Order Approximation YN(x ), 98.
Equation, 212.
Exercises, 108.
5.5 Concentration Profiles for Nonisothermal Reactions, 215.
References, 109.
Exercises, 228.
References, 230.
3. Some Important Properties
of Orthogonal Polynomials
6. One-Point Collocation 232
Formulation of
a Standard Collocation Procedure 111
Introduction, 232.
6.1 Application of One-Point Collocation
Introduction, 111.
to Ordinary Differential Equations, 233.
3.1 The Power Series Representation of PK:·13)(x ), 112.
6.2 Application of One-Point Collocation
3.2 Zeros of Orthogonal Polynomials, 115.
to Partial Differential Equations, 253
3.3 Differentiation and Integration of Lagrange Interpolation
6.3 One-Point Collocation for Initial Value Problems, 266.
Polynomials, 118.
Exercises, 269.
3.4 Program Description, 131.
References, 271.
3.5 Discretization of Differential Equations in Terms of
Ordinates, 135.
Exercises, 138.
References, 141.
7. Global Spline Collocation 273
4. Solution of Linear Differential Equations
Introduction, 273.
By Collocation 143 7.1 Global Spline Collocation for Two-Point Boundary Value
Problems, 274.
Introduction, 143. 7.2 Eigenvalues and Entry Length Problems by Spline
4.1 Solution of a Linear Boundary Value Problem, 144. Collocation, 285.
4.2 Integration of Initial Value Problems, 148. Exercises, 293.
4.3 Linear Parabolic Differential Equations, 166. References, 295.
4.4 Collocation Solution of a Linear PDE Compared to Exact
Solution, 175
4.5 Construction of Eigenfunctions by Forward Integration, 183.
4.6 An Ordinary Differential Equation at the Boundary of a 8. Coupled Differential Equations 297
PDE, 188.
Exercises, 191.
Introduction, 297.
References, 196.
8.1 On the Numerical Structuring of Coupled Differential
Equations, 299.
5. Noklinear Ordinary Differential Equations 198 8.2 The General Initial Value Problem-
Accuracy, Convergence, and Stability Considerations, 305.
8.3 Sensitivity Functions, 328.
Introduction, 198.
8.4 Partial Differential Equations, 332.
5.1 Multiple Solutions of a Trivial Boundary Value Problem,
Exercises, 335.
200.
References, 344.
X Contents
9. Selected Research Problems 347
Introduction, 347
p ,
9.1 The Graetz Problem with Axial Conduction, 348. .f
fej ace
9.2 Asymptotic Stability of a Catalyst Particle, 365.
9.3 Fixed Bed Reactor Dynamics-Transfer Functions and State
Space Formulation by Collocation, 394.
Exercises, 405.
References, 413.
Appendix A:
Computer Programs with Test Examples 417
Subject Index 441
Author Index 445
The principal aim of this book is to support the engineer-specifically the
chemical engineer-who is interested in quantitative treatment of physical
models. Engineering models often appear in the guise of differential
equations and we present some tools for solving these models by polyno
mial approximation.
Many years of teaching experience have made it abundantly clear that
.chemical engineering students and graduates are by no means less
mathematically gifted than their colleagues working in other engineering
sciences. Once they understand that even very complex models can be
attacked with the aid of only a few, basic methods and that these methods
are available as computer codes, many of the students develop a vora
cious appetite for mathematical studies and may have to be pushed gently
away from their newly acquired hobby of mathematical modeling before
they become estranged from their true vocation of chemical engineering:
inventiveness in the field of chemical processes, the actual large- or
small-scale synthesis of chemical compounds, and the interplay between
the chemical industry and society as a whole.
Scandinavian technical universities have a long tradition of giving their
students a solid background in classical mathematics-at least with
respect to the number of required courses. Still it is an undeniable fact
that their lack of preparedness for numerical manipulations with models
is often lamentable. Since the same observation is also made by col
leagues in American universities, it may express some rather fundamental
misconception in our way of teaching mathematics to young people even
before the university level.
xi
xii Preface Preface XIII
The graduate student or engineer working on a mathematical project honorable for this preoccupancy with chemical reactions and transport
is interested in the solution of the model and in the physical insight that phenomena, but hopefully the last two examples of chapter 1 will give
the solution can give him. Unless he feels that the mathematics course some inspiration to the reader who is interested in fluid mechanics and
helps him understand the physical background of his own problem, he rheology. The numerical methods of the following chapters can certainly
will approach it as just another required course that stands between him also be applied to these subjects.
and his true interests. Not only may a successful course in mathematics Chapters 2 to 4 give a systematic treatment of weighted residual
improve the student's understanding of physical sciences, but it may also methods applied to linear problems. These chapters may be regarded as
prompt him at some later stage of his development to study specific the core of the book, and the student who becomes familiar with the
subjects of mathematics in depth. application of the algorithms in chapters 3 and 4 will be able to solve
Countless books exist on applied mathematics and quite a few on many models of practical interest. The orthogonal collocation method is
numerical methods. Hildebrand's series is an example of excellent publi our preferred choice of weighted residual method, for reasons that are
cations from the first category, while Lapidus' Digital Computation for stated clearly in chapters 2 and 4. We are not convinced, however, that
Chemical Engineers has taught many generations of students how to talk
this very popular method is a vademecum for the solution of differential
sense with a computer. equations. We believe that collocation is a convenient, mechanical
Our text tries to incorporate model understanding and numerical
method that may give even more desirable results in unison with quite
model analysis into a single book. It does not deal with numerical different techniques such as perturbation methods. There are numerous
methods per se (classical methods such as finite differences are hardly
examples of this combination of methods in chapters 2, 4, 6, 8, and 9.
touched); neither is it a book about applied (or engineering) mathematics.
Nonlinear problems are treated in chapters 5 and 8. Here simplifica
Classical methods for solving linear differential equations are reviewed in
tions of the numerical procedure-similar to the model simplifications of
chapter 1 and well-known techniques such as approximation by perturba
chapter 1-are pursued whenever possible. To reduce the numerical
tion series occur frequently. Yet there are much better and more com work, many different tricks have to be played. In chapters 5 and 8, many
plete treatises on these subjects. of these tricks are advocated, most notably the sensitivity analysis that
On reflection, the book seems to owe much to Lanczos' beautifully
makes tracing of the solution as a function of model parameters much
written Applied Analysis. He tried to enamor the student with the easier.
philosophy of numerical work rather than presenting the subject as either
Specific difficulties of the problem may require special techniques.
a string of theorems or a stack of digested subroutines on punched cards.
Chapter 7, which describes global spline collocation as a method of
Each physical problem must be treated on its own merits-at least if it is
solving entry length problems, is specifically noteworthy. Part of chapter
worth any research effort-and a careful analysis of the problem is more
8 treats an efficient procedure for integration of sets of stiff equations, a
than half its solution. Once the ephemeral "feeling" for numerical work
numerical problem of tremendous interest in current research. The
has been instilled into the student, he can produce remarkable results
procedure is based on semi-implicit Runge-Kutta methods that are
with only a few tools. related to the collocation methods of chapter 4 but are easier to apply.
The first chapter presents what we regard as the basic units of The techniques of chapter 6-one-point collocation methods-are
mathematical models in chemical engineering: fluid flow, diffusion, and eminently suited for qualitative study of the problem. The one-point
chemical reaction. In developing the model for a fixed bed reactor, the collocation method is mentioned in numerous papers from the last few
intimate coupling between transport processes that leads to an almost
years. Different variants of the method are reported, and usually the
untractable mathematical complex is clearly seen. Much emphasis is put object is to study the influence of model parameters on the solution.
on the simplifications that are possible by judicious model approximation.
In chapter 9 we report results on three larger research projects. The
The student who understands how the model can be simplified without
problems are all from very active areas of research, and their solution
sacrificing t~ desired essential features will save himself many hours of might be interesting in itself, but the intention of the chapter is to show
fruitless computer work, and he is likely to emerge from his analysis with
how closely all the methods of the previous chapters work together.
a far better understanding of the physical process.
Perhaps this will give the reader determination to analyze his own
The personal interests of the authors in chemical reactor simulation as problems in a similar fashion.
an academic research subject and in industrial practice is apparent not
Finally, in the appendix we have collected a number of subroutines
only in the first chapter but in the entire book. We make no amende
that should help the reader to solve a majority of computational problems
xiv Preface Preface XV
part of the material; most recently a full-semester course was offered at
in differential equations. The routines have been used for over 6 years at
the University of Houston, where the graduate students helped to draw
this university and in some large Danish engineering companies. The
attention to weak passages.
routines are available in the libraries of computer centers at the Univer
Drs. Bruce A. Finlayson and Bruno van den Bosch read the manu
sity of Houston and at the University of California at Berkeley and are
script and suggested many valuable corrections. The authors graciously
used at a number of other American universities; in Salford, Great
acknowledge their contributions.
Britain; at the Technion in Haifa; and in various Scandinavian univer
Finally, the secretaries at Instituttet for Kemiteknik have labored over
sities. We believe our long experience with the programs have made
several years to produce the editions of notes that have now finally
them almost failproof.
reached a measure of completeness. Mrs. Bente Hansen and Mrs.
The readers of this text are usually from various fields of engineering
Johanne Nielsen bore the brunt of this work. For their patience we owe
or science. The problems may be stated in chemical engineering terms
them our sincere thanks.
but they are usually of a quite general nature. In our university (with a
5-year curriculum for the degree of MSc in engineering) it is used in the
fourth year as a textbook in a one-semester course {4 credit hours). In Lyngby, Denmark
JOHN VILLADSEN
American universities it is probably best suited for a first-year graduate
MICHAEL L. MICHELSEN
course, although some of the material can certainly be taught at an
undergraduate level. Sections 1.1 to 1.4, 1. 7, and 8.2, and chapters 2,
3, 4, and 6 contain material for approximately one semester. Very brief
continuing-education courses, supplemented by one week of intense
problem solving, have been given from the appendix alone with good
results.
Exercises are given in each chapter. Many of these have been selected
to support the understanding of a specific point in the text. Another
group of exercises points to applications of the methods that are not
immediately obvious from the text. Finally, a number of exercises are
formulated as small research problems-either as an analysis of a recent
paper from the literature or as an extension of previously reported
results. This last group is the most important to us because we wish to
equip the student for individual research in numerical treatment of
mathematical models. Naturally some basic computer experience should
be required as a prerequisite to the course, but in many of the computer
exercises, no more than a straightforward application of the algorithms of
the appendix is needed.
The introductions to each chapter should help focus the attention on
the major ideas of the chapter and after each chapter there is a brief list
of references (with comments about the most important ones). No
attempt has b~en made to present a complete set of references for each
subject.
In finishing a project of this size the authors have incurred debts to
many persof.ts. J. Villadsen wishes to thank Professor Warren E. Stewart
of the University of Wisconsin for an invigorating experience more than
10 years ago. It was in cooperation with this dedicated scientist that the
first version of orthogonal collocation was conceived.
Thanks are also due to the many chemical engineering departments in
different parts of the world who have provided funds for short courses on
Solution of
Differential Equation
Models by
Polynomial Approximation
A Preliminary Study 1
of Some Important
Mathematical Models
from Chemical Engineering
Introduction
In this chapter, we present the main items of the book:
1. Define the models that are to be solved by approximate methods
and, in a purely qualitative way, discuss the expected behavior of
the mathematical solution.
2. Present a few important mathematical tools that have been used to
obtain "exact" or "closed-form" solutions of various differential
equation models.
The physical systems to be considered are composed of one or several
phases, each of which is supposed to be continuous on the microscopic
scale. Differential mass balances and heat balances for each phase and
equilibrium relations between the phases are our main modeling tools. If
the system is a flowing fluid, both mass and heat balance are influenced by
the velocity field of the fluid, and this is described by a momentum
balance. Our starting point is equations (1) to (3) of section 1.1, and
these equations are used as far as possible in an attempt to give a unified
treatment of the model formulation. The equations are certainly not an
exact description of the physical systems, and the reader should note the
restrictions that the application of a simple set of basic equations imposes
on the treatment. Model formulation must necessarily be based on actual
physical systems, and sections 1.2 to 1.6 are a catalog of models that we
have chosen to use as examples. The choice of examples is directed by
2 Mathematical Models from Chemical Engineering Chap. 1 Sec. 1.1 The General Mathematical Model
3
the interests of the authors, and the reader who finds that too much
Two subjects, however, are given separate treatments. These are the
emphasis has been put on models for chemical reactors can hopefully be solution of coupled linear differential equations with constant coefficients
convinced that models for these systems exhibit most of the important and the Fourier series solution of linear partial differential equations.
features of the vaguely defined model building concept. The examples As fur.ther ar?ued in the introduction to section 1. 7, these specific
are organized as shown in table 1.1. mathematical topics are used in almost any type of numerical treatment
of much more complicated nonlinear differential equations.
TABLE 1.1
ORGANIZATION OF MODELS DESCRIBED IN CHAPTER 1
1.1 The General Mathematical Model
Section Description of model
1.1 General model: momentum-energy-and mass balances. Explicit models for A differential mass balance for component A in a dilute binary mixture
velocity distribution in unidirectional flow. takes the form
1.2 Models for a homogeneous fluid phase with or without reaction. No interac idCiAt + (v . v cA ) = (V . D vc A ) - RA (1)
tion fluid-bed packing.
1.3 Steady state and transient models for a solid phase on which a chemical A corresponding thermal energy balance is almost identical to (1):
reaction occurs.
pc.[~~
1.4 The interplay between the flowing fluid phase and the stationary solid phase. + (v • VT)] = (V · kVT) + Q (2)
Steady state and transient fixed bed reactor models.
1.5 Purification of a fluid by reverse osmosis in hollow fibers. The example shows The velocity field v, which must be known in order to solve these two
how complicated models can sometimes be treated gradually by suitable equations, is given by a momentum balance for the fluid.
assumptions.
For a constant-density, constant-viscosity Newtonian fluid one obtains
1.6 Two-dimensional flow and heat transfer to a non-Newtonian fluid. Material the f~llowing vector differential equation, the so-called N~vier-Stokes
beyond the level of section 1.1 is used in the model formulation. The final equatiOn:
model has an unusual mathematical appearance.
p[: pG Vp
+ (v • Vv)J = J.LV2v + ~ (3)
The models of sections 1.2 to 1.4 are all characterized by rather
simple flow patterns: The convective fluid flow is one-dimensional in the Equation (3) int~oduces a new variable, the pressure gradient Vp. In the
axial direction of a cylinder. Diffusion gradients inside solids are con examples of sect10~s 1.2 ~o 1.4, the pressure is known at every point of
sidered only across a thin plate, in the axial and radial directions of a the system. EquatiOn (3) IS supplemented by the equation of continuity
cylinder and radially in a sphere. With these restrictions added to the
dp
assumptions of the basic model (1) to (3), an apparently complete at = -[V . (pv)]
treatment of these frequently occurring systems, all of which are treated
in subsequent chapters by numerical methods, can be given. or for a flow at constant density, simply
The layout of the chapter would give the reader too bright a picture of
v. v = 0
model formulation as an easily systematized subject if no more examples (4)
were included. Some feeling for the complexity of the subject may be . The reader _who is familiar with the subject of transport phenomena
obtained tJy"ough a study of the last two examples in sections 1.5 and 1.6. will have n.o difficulty in recognizing this set of four equations as the
These examples are not very complicated, although they include a few source of mnumerable mathematical models of interest to chemical
features such as a two-dimensional velocity field that was not present in engineers. He will also know that these leave some physical phenomena
the previous examples. and .syst~ms ~ntouched. If he is in doubt about their applicability in a
Results from classical mathematical analysis occur throughout the specific SituatiOn, he should consult a standard text such as Bird (1960,
book interspersed with the approximative methods that will be our main chapters 3, 10, and 18) for an almost complete guide to the model
tools for solving models. building process.
4 Mathematical Models from Chemical Engineering Chap. 1 Sec. 1.1 The General Mathematical Model 5
We shall never need a more general basis for the models solved in the Finally, pG and -Vp in equation (3) describe the production of
present text. If any phenomenon that is not cov.ered by the ~ssumptions momentum by body forces proportional to the fluid density (the action of
of (1) to (3) appears, the system will be very simple othe~wi.se and the gravity is an example) or by a pressure gradient Vp in the flowing system.
Equations (1) and (2) are scalar equations, and their solutions are the
appropriate differential balances can be set up from firs~ pnnciple~.
The two terms of the left-hand side of the equations descnbe the scalar quantities cA and T as functions of the space coordinates and time
accumulation of mass, thermal energy, and momentum per unit time in a t. Equation (3) is a vector equation with three components, one equation
unit volume of the fluid as seen by an observer who travels through the for each of the three unknown scalar components of the fluid velocity v.
system, passively drifting along the streamlines. Thi~ total a~cumulation We denote the coordinates of the normal rectangular coordinate
term-the substantial derivative or D/ Dt in the notatiOn of Bud (1960, P· system as either (x, y, z) or (x1, x2, x3), whichever is more convenient.
73)-may also be conceived as the sum of two perh~ps more ~ell-known The axial coordinate in cylinder geometry is called z, and the radial
terms. The first term is the conventional accumulation term m a volume coordinate in cylinder and spherical geometry is called r. We shall never
that is fixed in space, and the second term is the rate of change of the treat problems in the angular coordinates of these last two geometries
the dependent variables will be constant in these directions. This restric
property by the convective flow through the element.
The right-hand sides of the equations also consist of two terms. The tion makes a complete description of the relevant vectorial quantities of
first term is the change of molecular diffusion flow across the element. equations (1) to (3) for all three geometries quite easy.
Fick's law, Fourier's law, and Newton's viscosity law are used to correlate The three components of v are vxb vx2, vx3 or, in a more convenient
the flux q of an extensive property (mass, heat, ?r mom~ntum) t? the notation, vb v2, v3. The gradient of the scalar quantities cA and T is a
vector with the partial derivatives in the 1-, 2-, and 3-directions as
gradients of concentration, temperature, . and flmd. vel?~Ity by simple
semi-empirical proportionality relations with mass diffusivity D, thermal components. In rectangular coordinates, the scalar products v · V cA and
v • VT of equations (1) and (2) are
heat conductivity k, and viscosity JL as proportionality constant~.
As written in (1) and (2), D and k are molecular properties of the
3 acA 3 aT
fluid, and these quantities are well known either fr~m experiments or I v;- and I v;-
1 ax; 1 ax;
from theoretical calculations. We shall often use equations (1) and (2) for
systems that are much too complex to allow a complete solution of (1) to
In equation (3), the quantity Vv-the gradient of the vector v-is a
(3). An empirical velocity distribution is inserted in (1) and (2), but now second-order tensor, which is a (3 x 3) matrix M with element m;i =
the transport parameters are empirical quantities without relation to k
av/a x; in row i and column j.
and D as they stand in (1) and (2): A turbulent axial mixing coefficient, a The scalar product v · Vv is the product of the (row) vector v =
mass transport coefficient inside the pore system of a catalyst, and a (v11 v2, v3) and M. It is thus a (1 X 3) vector, where each component is
radial heat conductivity in a fixed bed reactor will all be called D and k
the scalar product of v and a column of M.
with only a brief reference to their physical significance. The r~ason. that
this violation of the assumptions of the general model does not mvahdate av1 av2 av3
the results is that the mechanisms of the more complicated transport ax1 ax1 ax1
phenomena are empirically found to be similar to those leading to the
av1 av2 av3
diffusion terms of (1) and (2), and consequently the mathematical models v · Vv = ( v11 v2, v3)
ax2 ax2 ax2
look alike although their foundation is different. (5)
The last terms-RA, Q, and (pG- Vp)-are volumetric production av1 av2 av3
terms. RA is the rate of disappearance of A per unit volume by a
ax3 ax3 ax3
chemical re.action. Q is a corresponding heat of reaction by the chemical f f f
reaction or ttnother continuous heat source in the fluid. Internal friction, = ( V·av1 av2 v·av3)
Vj '
absorption of radiant heat, or dielectric heating all contribute to 0, while i=1 1 axi ' j=l axj i=1 'axi
external heating from a steam jacket surrounding the fluid has to be
Assume for the moment that D and k are constant; i.e., they are the
treated differently in an accurate model, although we shall see that an
approximation of a boundary heat source by an average volumetric heat same in all directions and they are. independent of cA and T. In this case,
the first term on the right-hand sides of (1) and (2) can be written DV2
source is quite often used. cA