Numerical Techniques for Chemical and Biological Engineers Using MATLAB® Numerical Techniques for Chemical and Biological Engineers Using MATLAB® A Simple Bifurcation Approach Said Elnashaie Frank Uhlig with the assistance of Chadia Affane MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. Library of Congress Control Number: 2006930111 ISBN-10: 0-387-34433-0 ISBN-13: 978-0-387-34433-1 © 2007 Springer Science + Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science + Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 987654321 springer.com Professor Said S.E.H. Elnashaie Pennsylvania State University at Harrisburg Room TL 176 Capital College 777 W. Harrisburg Pike Middletown, PA 17057-4898 sse10@psu.edu Professor Frank Uhlig Department of Mathematics and Statistics Auburn University 312 Parker Hall Auburn, AL 36849 uhligfd@auburn.edu Chadia Affane Department of Mathematics and Statistics Auburn University Auburn, AL 36849 affanac@auburn.edu SAID ELNASAIE Born 1947, Cairo/Egypt; grew up in Egypt; married, two children, three grandchildren. Chemical Engineering student at Cairo University, University of Waterloo, and University of Edinburgh. B. S., Cairo University, 1968; M.S., U of Waterloo, Canada, 1970; Ph.D., U Edinburgh, UK, 1973. Postdoc, University of Toronto and McGill U, 1973 – 1974. Professor, Cairo University, 1974 – 1992; King Saud University, Saudi Arabia, 1986 – 1996; University of British Columbia, Canada, 1990; University Putra Malaysia, 1996/97; King Fahad University, Saudi Arabia, 1999; Auburn University, 1999 – 2005; Visiting Professor, U British Columbia, Vancouver, 2004; U British Columbia, Vancouver, 2005 – Vice President, Environmental Energy Systems and Services (EESS), Egypt, 1996 – 1998. Research Areas: Modeling, Simulation and Optimization of Chemical and Biological Processes, Clean Fuels (Hydrogen, Biodiesel and Ethanol), Fixed and Fluidized Bed Catalytic Reactors, Nonlinear Dynamics, Bifurcation and Chaos, Clean Technology, Utilization of Renewable Materials, Sustainable Engineering. 300+ papers, 3+ books. nashaie@chml.ubc.ca FRANK UHLIG Born 1945, M¨agdesprung, Germany; grew up in M¨ulheim/Ruhr, Germany; married, two sons. Mathematics student at University of Cologne, California Institute of Technology. Ph.D. CalTech 1972; Assistant, University of W¨urzburg, RWTH Aachen, Germany, 1972 – 1982. Two Habilitations (Mathematics), University of W¨urzburg 1977, RWTH Aachen 1978. Visiting Professor, Oregon State University 1979/1980; Professor of Mathematics, Auburn University 1982 – Two Fulbright Grants; (Co-)organizer of eight research conferences. Research Areas: Linear Algebra, Matrix Theory, Numerical Analysis, Numerical Algebra, Geometry, Krein Spaces, Graph Theory, Mechanics, Inverse Problems, Mathematical Education, Applied Mathematics, Geometric Computing. 50+ papers, 3+ books. uhligfd@auburn.edu www.auburn.edu/ ~ uhligfd v CHADIA AFFANE Born 1968, Fes/Morocco; grew up in Morocco; married, two children. Engineering student at ´ Ecole Sup´erieure de Technologie, E.S.T., Fes, Morocco, 1989 – 1991. Chemical engineering student at Texas A&M University, 1997 – 1999; B.S. Texas A&M University, 1999. MS in Applied Mathematics, Auburn University, 2003. Ph. D. student, Mathematics, Auburn University, 2004 – Research Areas: Numerical Analysis, Applied Mathematics. affanac@auburn.edu vi Preface This book has come about by chance. The first author, Said Elnashaie, and his wife, Shadia Elshishini, moved next door to the second author, Frank Uhlig, and his family in 2000. The two families became good neighbors and friends. Their chats covered the usual topics and occasionally included random teaching, departmental, and university matters. One summer day in 2003, Said showed Frank a numerical engineering book that he had been asked to review. Neither of them liked what they saw. Frank eventually brought over his “Numerical Algorithms” book and Said liked it. Then Said brought over his latest Modeling book and Frank liked it, too. And almost immediately this Numerical Chemical and Biological Engineering book project started to take shape. Said had always felt more mathematically inclined in his work on modeling problems and bifurcation and chaos in chemical/biological engineering; Frank had lately turned more numerical in his perception and efforts as a mathematician. This book is the outcome of Said’s move to Auburn University and his chance moving in next door to Frank. It was born by a wonderful coincidence! Said and Frank’s long evening walks through Cary Woods contributed considerably to- wards most of the new ideas and the educational approach in this book. We have both learned much about numerics, chemical/biological engineering, book writing, and think- ing in our effort to present undergraduates with state of the art chemical/biological engineering models and state of the art numerics for modern chemical/biological engi- neering problems. Chadia is a chemical engineer who has turned towards applied mathematics in her gradu- ate studies at Auburn University and has helped us bridge the gap between our individual perspectives. The result is an interdisciplinary, totally modern book, in contents, treatment, and spirit. We hope that the readers and students will enjoy the book and benefit from it. For help with our computers and computer software issues we are indebted to A. J., to Saad, and to Darrell. Auburn and Vancouver, 2006 vii Contents Introduction 1 HowtoUsethisBook 5 1 Computations and MATLAB 11 1.1 MATLABSoftwareandProgramming 12 1.1.1 TheBasicsofMATLAB 12 1.2 NumericalMethodsandMATLABTechniques 19 1.2.1 SolvingScalarEquations 20 Exercises 33 1.2.2 Differential Equations; the Basic Reduction to First Order Systems . . . . 34 1.2.3 SolvingInitialValueProblems 37 1.2.4 Solving Boundary Value Problems 42 1.2.5 MATLABmandOtherFilesandBuilt-inMATLABFunctions 43 2 Modeling, Simulation, and Design 55 2.1 SystemTheoryanditsApplications 55 2.1.1 Systems 55 2.1.2 Steady State, Unsteady State, and Thermodynamic Equilibrium 57 2.2 Basic Principles for Modeling Chemical and Biological Engineering Systems . . . 58 2.3 ClassificationofChemicalandBiologicalEngineeringSystems 59 2.4 Physico-ChemicalSourcesofNonlinearity 61 2.5 SourcesofMultiplicityandBifurcation 65 3 Some Models with Scalar Equations 69 3.1 ContinuousStirredTankReactor:TheAdiabaticCase 69 Exercises 89 3.2 ContinuousStirredTankReactor:TheNonadiabaticCase 92 Exercises 114 3.3 ABiochemicalEnzymeReactor 115 3.4 ScalarStaticEquations 118 3.4.1 Simple Examples of Reactions with No Possible Multiple Steady States . 119 3.4.2 Solving Some Static Transcendental and Algebraic Equations from the ChemicalandBiologicalEngineeringFields 121 ix Exercises 129 ProblemsforChapter3 130 4 Initial Value Problems 135 4.1 ANonisothermalDistributedSystem 135 4.1.1 Vapor-PhaseCrackingofAcetone 138 4.1.2 PreludetotheSolutionoftheProblem 138 4.1.3 MaterialBalanceDesignEquationinTermsofVolume 139 4.1.4 HeatBalanceDesignEquationinTermsofVolume 141 4.1.5 NumericalSolutionoftheResultingInitialValueProblem 142 Exercises 154 4.2 AnaerobicDigester 155 4.2.1 ProcessDescriptionandRateEquations 155 4.2.2 MathematicalModelingforaContinuousAnaerobicDigester 156 4.2.3 SolutionoftheSteady-StateEquations 157 4.2.4 Steady-StateVolumeintermsoftheFeedRate 157 4.2.5 Steady-StateConversioninTermsoftheFeedConcentration 159 4.2.6 The Unsteady-State Behavior of the Digester and the Solution of the IVP 165 Exercises 168 4.3 HeterogeneousFluidizedBedCatalyticReactors 169 4.3.1 MathematicalModelingandSimulationofFluidizedBeds 169 4.3.2 AnalyticalManipulationoftheJointIntegrodifferentialEquations 174 4.3.3 Bifurcation and Dynamic Behavior of Fluidized Bed Catalytic Reactors . 177 4.3.4 DynamicModelsandChemisorptionMechanisms 177 4.3.5 FluidizedBedCatalyticReactorwithConsecutiveReactions 181 4.3.6 Numerical Treatment of the Steady-State and Dynamical Cases of the Bubbling Fluidized Bed Catalytic Reactor with Consecutive Reactions . . 184 Exercises 221 4.4 ABiomedicalExample:TheNeurocycleEnzymeSystem 222 4.4.1 Fundamentals 223 4.4.2 The Simplified Diffusion-Reaction Two Enzymes/Two Compartments Model223 4.4.3 DynamicModelDevelopment 225 4.4.4 NormalizedFormoftheModelEquations 229 4.4.5 IdentificationofParameterValues 231 4.4.6 NumericalConsiderations 232 Exercises 249 ProblemsforChapter4 250 5 Boundary Value Problems 255 5.1 TheAxialDispersionModel 255 5.1.1 FormulationoftheAxialDispersionModel 257 5.1.2 Example of an Axial Dispersion Model. Linear and Non-linear Two-point Boundary Value Problems (BVPs) 262 TheLinearCase 262 AnalyticSolutionoftheLinearCase 265 x TheNonlinearCase 272 5.1.3 Numerical Solution of Nonlinear BVPs. The Non-Isothermal Case . . . . 277 Exercises 297 5.2 ThePorousCatalystPelletBVP 298 5.2.1 DiffusionandReactioninaPorousStructure 298 5.2.2 NumericalSolutionfortheCatalyticPelletBVP 303 TheHeatBalanceModel 304 TheMassandHeatBalanceModel 314 Exercises 323 ProblemsforChapter5 324 6 Heterogeneous and Multistage Systems 327 6.1 HeterogeneousSystems 327 6.1.1 Material Balance and Design Equations for Heterogeneous Systems . . . . 328 GeneralizedMassBalanceandDesignEquations 328 OverallHeatBalanceandDesignEquations 333 TwoPhaseSystems 335 TheCo-andCountercurrentCases 337 The Equilibrium Case 338 StageEfficiency 339 GeneralizedMassBalanceforTwoPhaseSystems 339 6.1.2 Steady State Models for Isothermal Heterogeneous Lumped Systems . . . 340 6.1.3 Steady State Models for Isothermal Heterogeneous Distributed Systems . 344 MultipleReactionsinTwoPhaseSystems 346 6.1.4 NonisothermalHeterogeneousSystems 348 LumpedSystems 348 HeterogeneousLumpedSystems 349 DistributedSystems 351 Exercises 353 6.2 NonreactingMultistageIsothermalSystems 353 6.2.1 Absorption Columns or High Dimensional Lumped, Steady State and Equilibrium Stages Systems . . 353 The Case of a Linear Equilibrium Relation . 354 MultistageAbsorption 361 6.2.2 Nonequilibrium Multistages with Nonlinear Equilibrium Relations . . . . 373 Exercises 381 6.3 IsothermalPackedBedAbsorptionTowers 382 6.3.1 ModelDevelopment 383 6.3.2 ManipulationoftheModelEquations 384 6.3.3 Discussion and Results for both the Simulation and the Design Problem . 384 Exercises 399 6.4 TheNonisothermalCase:aBatteryofCSTRs 399 6.4.1 ModelDevelopment 399 6.4.2 NumericalSolutions 402 6.4.3 TheSteadyStateEquations 419 xi Exercises 421 ProblemsforChapter6 422 7 Industrial Problems 425 7.1 ASimpleIllustrativeExample 426 7.1.1 MassBalancefortheReactor 427 7.1.2 HeatBalancefortheReactor 428 7.1.3 ReactorModelSummary 429 7.1.4 The Catalyst Pellet Design Equations and the Computation of the Effec- tiveness Factor η 430 7.1.5 PelletModelSummary 431 7.1.6 ManipulationandReductionoftheEquations 432 Exercises 436 7.2 Industrial Fluid Catalytic Cracking FCC Units 436 7.2.1 Model Development for Industrial FCC Units . . 437 7.2.2 Static Bifurcation in Industrial FCC Units 442 TheSteadyStateModel 443 SolutionoftheSteadyStateEquations 445 Steady State Simulation Results for an Industrial Scale FCC Unit 446 7.2.3 Industrial Verification of the Steady State Model and Static Bifurcation of Industrial Units . . 451 SimulationProcedure;VerificationandCrossVerification 453 Simulation and Bifurcation Results; Discussion for two Industrial FCC Units453 7.2.4 Preliminary Dynamic Modeling and Characteristics of Industrial FCC Units459 TheDynamicModel 459 Results for the Dynamic Behavior of FCC Units and their Relation to the StaticBifurcationCharacteristics 461 7.2.5 Combined Static and Dynamic Bifurcation Behavior of Industrial FCC Units 469 TheDynamicModel 470 Exercises 472 7.3 TheUNIPOLProcessforPolyethyleneandPolypropyleneProduction 473 7.3.1 ADynamicMathematicalModel 475 GeneralAssumptions 475 HydrodynamicRelations 476 TheModelEquations 478 7.3.2 NumericalTreatment 482 Exercises 483 7.4 Industrial Steam Reformers and Methanators 484 7.4.1 RateExpressions 484 7.4.2 ModelDevelopmentforSteamReformers 488 7.4.3 ModelingofSide-FiredFurnaces 490 7.4.4 ModelforaTop-FiredFurnace 491 7.4.5 ModelingofMethanators 491 7.4.6 Dusty Gas Model for Catalyst Pellets in Steam Reformers and Methanators492 xii [...]... 1.2 Numerical Methods and MATLAB Techniques for Chemical and Biological Engineering Problems In this section we give an overview of numerical analysis in general, and of the aspects of numerical analysis that are needed for problems encountered specifically in chemical and biological engineering2 This overview will, by necessity, be rather brief and it cannot substitute for a full semester course on Numerical. .. of format long e, while for most engineering output format short g will give sufficient information format short g limits the output to 9 digits and writes numbers 0.0001 ≤ |x| ≤ 9.99999899 · 108 in standard decimal form and 1.2 Numerical Methods and MATLAB Techniques 19 numbers outside of this range in exponential form be±a In this book we shall use MATLAB codes and explain more involved features of MATLAB. .. concepts of numerical analysis and in its fundamentals In this book we aim to place general and specific chemical/ biological problems in the context of standard numerical techniques and we try to exhibit and explain special numerical considerations that are needed to solve these chemical/ biological problems using MATLAB These concerns are due in part to the special nonlinearities that occur in chemical/ biological. .. suited software for engineering and numerical computations is MATLAB 1 This acronym stands for “Matrix Laboratory” Its operating units and principle are vectors and matrices By their very nature, matrices express linear maps And in all modern and practical numerical computations, the methods and algorithms generally rely on some form of linear approximation for nonlinear problems, equations, and phenomena... will be developed and programmed using MATLAB 2 This is a sophisticated numerical software package MATLAB is powerful numerically through its built-in functions and it allows us to easily develop and evaluate complicated numerical codes that fulfill very specialized tasks Our solution techniques will be developed and discussed from both the chemical/ biological point of view and the numerical point of... Equations are usually solved numerically with the help of computers and suitable software Almost all problems faced by chemical and biological engineers are nonlinear Most if not all of the models have no known closed form solutions Thus the model equations generally require numerical techniques to solve them One central task of chemical/ biological engineers is to identify the chemical/ biological processes... the student/reader has a background in chemical and biological modeling, calculus, matrix notation and possibly MATLAB before attempting to study this book As with any book on science and mathematics that contains mathematical equations, formulas and model derivations, it is important for our readers to take out paper and pencil and try to replicate the equations and their derivations from first principles... encounter them MATLAB has a built-in help menu; typing help format at the >> prompt, or help \, for example, will show the syntax and variations of these two commands “format” and ‘backslash’ Whenever a student encounters a MATLAB command that is not self explanatory, we suggest using this built-in help function of MATLAB There are many MATLAB tutorials available on the web; please enter MATLAB tutorial”... steady states and we introduce several transcendental and algebraic equations of chemical and biological engineering import As always, the students and readers should find their own MATLAB codes for the various problems first before relying on those that are supplied and before solving the included exercises Chapter 4 studies problems that involve change over time or location and that therefore are modeled... efficient solution methods for the model equations Some of the models are very simple and easy to solve, even by hand, but most require medium to high-powered numerical techniques From chapter to chapter, we introduce increasingly more complex chemical/ biological processes and describe methods and develop MATLAB codes for their numerical solution The problem of validating a solution and comparing between . Numerical Techniques for Chemical and Biological Engineers Using MATLAB Numerical Techniques for Chemical and Biological Engineers Using MATLAB A Simple Bifurcation. 2006 vii Contents Introduction 1 HowtoUsethisBook 5 1 Computations and MATLAB 11 1.1 MATLABSoftwareandProgramming 12 1.1.1 TheBasicsofMATLAB 12 1.2 NumericalMethodsandMATLABTechniques 19 1.2.1 SolvingScalarEquations. number of chemical/ biological processes will be presented, modeled, and effi- cient numerical techniques will be developed and programmed using MATLAB ❤ R 2 .This is a sophisticated numerical software