Nonlinear analysis in chemical engineering (1980)

McGRAW-HILL INTERNATIONAL eoOK COMPANY N.:wVork Si.L ouis SanfranclSCO Auckland eo801a Gua1emala �j.°'�.:S\u rg Lisbon Loodo• Madrid Mexico Montreal New Delhi Paris San Juan SioPaulo re �;r.,: Tokyo Toronto Bruce A Finlayson Professor of Chemical Engineering and Applied Machenratks University ofWashington, Seattle NONLINEAR ANALYSIS IN CHEMICAL ENGINEERING � CONTENTS Preface I lntroduclion 1-1 Classification of Equations Algebraic Equalions 2-t 2-2 2-3 · Successive Substitution Newton-Raphson Comparison Study Questions Problems Bibliography Ordinary Qifferential Equations-lnilial-Value Problems 3-1 3-2 3-3 3-4 3-5 3-6 3-7 3-8 3-9 Terminology Interpolation and Quadrature Explicit Integration Methods Implicit Integration Methods Predictor-Corrector and Runge-Kuna Methods Extrapolation and Step-Size Control Stabi!ity High-order Schemes that are Stable and not Oscillate Equation Solvers 11 13 15 15 16 17 17 20 25 27 28 32 37 45 47 riil COl'ffENTS 3-IO S lO , Comparison s1udy Questions Problems Bibliography References 57 l8 l9 60 Boundary-Valac Problems Ordinary Diff'erenlial Equalions- MeihodofWcighted Residuals Finite Difference Method 4-3 Regular Perturbation 4-4 Orthogonal Collocation 4-5 Diffusion and Reaction-Exact Results 4-6 Perturbation Method for Diffusion and Reaction 4-7 Orthogonal Collocation for Diffusion and Reaction 4-8 Lower-Upper Decomposition of Matrices 4-9 Or1hogonal Collocation on Finite Elements 4-IO Galerkin Finite Elements 4-11 Initial-Value Techniques 4-12 Quasilinearization 4-13 Comparison 4-14 Adaptive Meshes Study Questions Problems Bibliography References 60 " 70 4-I 4-2 Parabolic Panial Differenlial Equations - Time 11.ad One Spatial Dimension 5-1 Similarity Transformation Separation of Variab les 5-3 Method ofWeigh ted Residuals 5-4 Orthogonal Colloc ation 5-5 Finite DilTerence S-& Orthogonal Colloc ation on Finite S-7 Galerkin Finite Element Method Elements S-8 Conve ctive Diffusion Equation S-9 Flow Thro ugh Porous Medin S·IO Com paris on 5-2 Study Questions Problems Bibliography References 73 79 88 94 104 113 126 136 142 143 164 166 167 170 171 l7l 172 180 184 191 214 228 230 231 "" 261 265 2'6 269 110 CONTENTS Partial Differential Equations in Two Space Dimensions 6-1 Introduction 6-2 Finite Difference 6-3 Orthogonal Collocation 6-4 Galerkin Finite Element Method 6-5 Comparison Problems Bibliography References Appendix Computer Programs ix 271 272 276 286 291 307 312 13 314 316 Author Index 359 Subject Index 361 PREFACE tho�s or a?alysis t�at arise in introduction to many m� This book provides an ns of ordinary and partial ddTerential equatio Many engineering for the solution s is arc oriented towards linear problem , yet it books, and often many courses, ng Here many methods­ engineeri in arise frequenlly that nonlinear problems applied on-are , perturbati finitc difference, finite element, orthogonal collocation of the method and the to nonlinear problems to illustrate the range of applicability arc solved useful results that can be derived from each method The same problems in practical and methods these with different methods so that the reader can assess similar cases The examples are from the author's own experience: fluid ftow f, (including polymers), heat transrer, and chemical reactor modeling The level of the book is introductory, and the treatment is oriented toward the nons�alist Even so the reader is introduced to the latest, most powerful techniques The course is based on a successrul graduate course at the University or W�ington, and most chemical engineers taking the course are experi­ men � ah�ts The reader desiring to delve deeper into a particular technique or app can follow the leads given in the bibliography of each chapter _ peciall y th�nks the class or 1979, who tested the first ��tten version ort _ k, and especially Dan David and Mike Chang, who were dtligelll _ about providing correctio ns The draft was expertly typed by Karen Fincher and Sylvia Swimm The author is also tha8k�u1 10 �1s famdy for supporting him during the pro ject-both fiscall a d psycholog cally Y Writing a book really involves the whole famil y Spe ia1 t anks go Christine, wh� gav to the author's children Mark, Cady, and r father-child time to make this book possible and to the author' _ s e contmued support and encouragement �� �� ��C:O: � : ;r 5;�1: :::/::; Seatlle 1980 n Bruce A Finlayso CHAPTER ONE INTRODUCTION The goal or this book is to bring the reader into contact with the efficient computation tools that are available today to solve differential equations modeling physical phenomena, such as diffusion, reaction, heat transfer, and ftuid flow Arter mastering the material in this book you should be able to apply a variety or methods-finite difference, finite element, collocation, perturbation, etc.-although you will not be an expert in any of them When faced with a problem to solve you will know which methods are suitable and what information can be easily determined by which method The emphasis is on numerical methods, using a computer, although some of the approaches can also yield powerful results analytically The author's philosophy is to use preprogrammed computer packages when available because they allow the reader to sample, �ruse, and solve difficult problems with less effort The reader is, however, introduced to the theory and techniques used in these computer programs 1-1 CLASSIFICATION OF EQUATIONS Equations modeling physical phenomena have different characteristics depending on how they model evolution in time and the influence or boundary condilions When conrronted with a model, expressed in the rorm of a differential equation, the analyst must decide what type or equation is to be solved That characterization determines the methods that are suitable C��sider a closed system (i.e no interchange or mass with the surroundings) contain mg three chemical components whose concentrations are given by c1, ,·�· and c3 The three components can react (say when the system is illuminated with cnonsystcm Figurel·I Rca the concentration of each ncy), and �he go, is to predict light of a specified frcqu� e The rates eaction are known as a function of the of ti� _is0 s own in Fig 1-1, and the differential species as a function reaction system concentrations The system arc equations governing this t � -k1c1+k2c·2c3 �== df (1-1) �f ==k3d initial _ zero, and the Initially the concentrations of components two and lhrce ai·e concentration of component one is given as c0• We thus wish to solve Eqs 0-l) subject to the initial conditions (1-2) Note that the conditions apply only at time zero not to later times t The reaction proceeds in time; if we know where to start we can integrate the equations indefinitely This evolution property yields equations that are called ini1ial-value problems In this case Eqs (1-1) are ordinary differential equations, since there is only one independent variable, time Thus Eqs (I-I) and (1-2) are governed by a system of ordinary differential equations that are initial-value problems In this text this_ is abbreviated to ODE IVP Consider next diffusion and reaction in a porous medium We have a hetcrogcnc�us system (solid material with pores through which the reactants and prod�cts �11Tuse), but here we model the system as simple diffusion using an tvc diffusion coefficient mass balance on a volume of the porous medium ;�:�� A 1\ · _ ?1 - - : (tJ_, + r1J)' + tJ ) r1X -1,)� �iz + R(c) (1-3) �::r; :�� th� r;•te of _reaction per unit volume (solid plus void volume� J is the r l ni f solid and ::�� 31r: one mdependent (1•6) is second-order and the theory of hnear second-order says that we must specify two constants in the t· stating two boundary conditions, one at each side e we co sid r one side of the slab as impermeable ��o Hux) and the is held fixed at the other side These boundary cond1t1ons are de -D��=O x=O (1-7] (1-8) The problem in Eqs (1-6) to (1-8) is an ordinary differential equation and a boundary-value problem ODE-BVP It is also called a two-point boundary-value problem because the two conditions are expressed at different positions IC the� had both been specified at the same point say then the problem would have been an initial-value problem This nature of boundary-v alue problems-having conditions at each end of the domain- complicates the solution characteristic of diffusion, heat transfer, and Huid-flow problems techniques but is Re�racin� our steps back to Eq (l-5) describing diffusion and ction in a l�ree-d�m rea ens1onal this time let us simplify the equation for one space d1mens1on, as beforespace, , but include transient pheno mena, such x x = ,,,, c that ,\· ( ) ;1f = iiX D,,fl.� + R(c) (1-9) This is a partial differential , · ' �ccause the solution depends on tw� i�dependent variables and equ,nion d11Terent, however Onl a ·1 The �ha�act�r of the dependence on and on �s e der an �vo lution phenome:o sin � • vativ� 1�.1 occurs, and the dependence on is n W require pos111on an mll ta l value of the concentration at each x c x r r t'(X,0)::: c·o(X) The dependence on (1-10) necessary Conditions is like boundary-value probl em and two conditions are lik E ou\d ��w be a functio� 7) a nd (1-8) ure feasible, but the concentration ��' centration We 0��i�� to variations in the bulk-stream d�IJ�=�t1a l equationcall th ' t 'com.rrespondmg t l act that one in one•:p�?c: �� e �qs (I- 7) to ( 1-10) a parabolic partial variable is evolu:: :,����� ��er=�e��· The term purubolic refers x a INTIODUCTION !! f"igurel-l Diffusioninalonr.ca1alys1pellc1 If we solve Eq ( l-5) in two or three space dimensions we also have a parabolic partial differential equation, with the I variable being evolutionary and the '¥:: ·'" and z variables being or boundary-value type In two dimensions we have (1-111 Ir we include two space dimensions but allow only steady-state situations then the equation reduces to ,1 o � "' � a ''' ( ) +a;; (D,,fy) + R(c) = o ( l-12) This equation would model both diffusion and reaction in a catalyst particle that is very long in the z direction, so that z varia1ions are negligible (see Fig 1-3) The type of boundary conditions allowed are Dirichlet-type or boundary conditions or the first kind t'= WR I TE ( 61 55 > 27 28 29 50 30 31 32 33 3' 35 36 GD T D • h XX C l h CX C I > J1 XX C J l CX C J I GO T O Ii lt WR I T E ( 61 55 h X X U h C IC U ) C Dh T l N U E 5 F O R tl A T U Z X1 f l F l 91 l O X I Z X f l 6.1 F l ) RETURN END S IJ B R DUT U I E I H I T I A L C C N I D i lt E N S I O N C C N J D O I•l•N CUl • O R E TURN END S U B R O U T I N E B C C C N1 K E T H C l C N T ) O l " E NS I D N C C N l CCC S E T C l • C C X • O I I F ft E T H • l O R CCC S E T C N • C U • l ) F OR A L L K E T H I F C PI E T H E Q G C T O Cl • t CN • O 3e 39 •037 c c c c c D D D D D D • � ·l ' � R E TURN � �� R O U T I N E D I F F U f H N , T1 C C O O T , C D l'! 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