Numerical Methods for Chemical Engineering: Applications in MATLAB

486 Pages·2006·8.08 MB·English

by  Beers

Numerical Methods for Chemical Engineering
Suitableforafirst-yeargraduatecourse,thistextbookunitestheapplicationsofnumerical
mathematicsandscientificcomputingtothepracticeofchemicalengineering.Writtenin
apedagogicstyle,thebookdescribesbasiclinearandnonlinearalgebraicsystemsallthe
way through to stochastic methods, Bayesian statistics, and parameter estimation. These
subjectsaredevelopedatanominalleveloftheoreticalmathematicssuitableforgraduate
engineers. The implementation of numerical methods in M® is integrated within
eachchapterandnumerousexamplesinchemicalengineeringareprovided,togetherwitha
libraryofcorrespondingM programs.Althoughtheapplicationsfocusonchemical
engineering,thetreatmentofthetopicsshouldalsobeofinteresttonon-chemicalengineers
andotherappliedscientiststhatworkwithscientificcomputing.Thisbookwillprovidethe
graduatestudentwiththeessentialtoolsrequiredbyindustryandresearchalike.
Supplementary material includes solutions to homework problems set in the text,
M programs and tutorial, lecture slides, and complicated derivations for the more
advancedreader.Theseareavailableonlineatwww.cambridge.org/9780521859714.
K JB hasbeenAssistantProfessoratMITsince2000.Hehastaughtexten-
sivelyacrosstheengineeringdisciplineatboththeundergraduateandgraduatelevel.This
bookisaresultofthesuccessfulcoursetheauthordevisedatMITfornumericalmethods
appliedtochemicalengineering.

Numerical Methods for
Chemical Engineering
Applications in M ®
ATLAB
KENNETH J. BEERS
MassachusettsInstituteofTechnology
  
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,Sa˜oPaulo
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB22RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521859714
(cid:1)C K.J.Beers2007
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithout
thewrittenpermissionofCambridgeUniversityPress.
Firstpublished2007
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary
ISBN-13 978-0-521-85971-4hardback
ISBN-10 0-521-85971-9hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofURLsfor
externalorthird-partyinternetwebsitesreferredtointhispublication,anddoesnotguaranteethat
anycontentonsuchwebsitesis,orwillremain,accurateorappropriate.
Contents
Preface pageix
1 Linearalgebra 1
Linearsystemsofalgebraicequations 1
Reviewofscalar,vector,andmatrixoperations 3
Eliminationmethodsforsolvinglinearsystems 10
Existenceanduniquenessofsolutions 23
Thedeterminant 32
Matrixinversion 36
Matrixfactorization 38
Matrixnormandrank 44
Submatricesandmatrixpartitions 44
Example.Modelingaseparationsystem 45
Sparseandbandedmatrices 46
MATLABsummary 56
Problems 57
2 Nonlinearalgebraicsystems 61
Existenceanduniquenessofsolutionstoanonlinearalgebraicequation 61
IterativemethodsandtheuseofTaylorseries 62
Newton’smethodforasingleequation 63
Thesecantmethod 69
Bracketingandbisectionmethods 70
Findingcomplexsolutions 70
Systemsofmultiplenonlinearalgebraicequations 71
Newton’smethodformultiplenonlinearequations 72
EstimatingtheJacobianandquasi-Newtonmethods 77
Robustreduced-stepNewtonmethod 79
Thetrust-regionNewtonmethod 81
SolvingnonlinearalgebraicsystemsinMATLAB 83
Example.1-Dlaminarflowofashear-thinningpolymermelt 85
Homotopy 88
Example.Steady-statemodelingofacondensation
polymerizationreactor 89
v
vi Contents
Bifurcationanalysis 94
MATLABsummary 98
Problems 99
3 Matrixeigenvalueanalysis 104
Orthogonalmatrices 104
Aspecificexampleofanorthogonalmatrix 105
Eigenvaluesandeigenvectorsdefined 106
Eigenvalues/eigenvectorsofa2×2realmatrix 107
Multiplicityandformulasforthetraceanddeterminant 109
Eigenvaluesandtheexistence/uniquenesspropertiesoflinear
systems 110
Estimatingeigenvalues;Gershgorin’stheorem 111
ApplyingGershgorin’stheoremtostudytheconvergenceofiterative
linearsolvers 114
Eigenvectormatrixdecompositionandbasissets 117
NumericalcalculationofeigenvaluesandeigenvectorsinMATLAB 123
Computingextremaleigenvalues 126
TheQRmethodforcomputingalleigenvalues 129
Normalmodeanalysis 134
Relaxingtheassumptionofequalmasses 136
Eigenvalueproblemsinquantummechanics 137
SinglevaluedecompositionSVD 141
Computingtherootsofapolynomial 148
MATLABsummary 149
Problems 149
4 Initialvalueproblems 154
Initialvalueproblemsofordinarydifferentialequations
(ODE-IVPs) 155
Polynomialinterpolation 156
Newton–Cotesintegration 162
Gaussianquadrature 163
Multidimensionalintegrals 167
LinearODEsystemsanddynamicstability 169
OverviewofODE-IVPsolversinMATLAB 176
Accuracyandstabilityofsingle-stepmethods 185
StiffstabilityofBDFmethods 192
Symplecticmethodsforclassicalmechanics 194
Differential-algebraicequation(DAE)systems 195
Parametriccontinuation 203
MATLABsummary 207
Problems 208
Contents vii
5 Numericaloptimization 212
Localmethodsforunconstrainedoptimizationproblems 212
Thesimplexmethod 213
Gradientmethods 213
Newtonlinesearchmethods 223
Trust-regionNewtonmethod 225
Newtonmethodsforlargeproblems 227
UnconstrainedminimizerfminuncinMATLAB 228
Example.Fittingakineticratelawtotime-dependentdata 230
Lagrangianmethodsforconstrainedoptimization 231
ConstrainedminimizerfminconinMATLAB 242
Optimalcontrol 246
MATLABsummary 252
Problems 252
6 Boundaryvalueproblems 258
BVPsfromconservationprinciples 258
Real-spacevs.function-spaceBVPmethods 260
Thefinitedifferencemethodappliedtoa2-DBVP 260
Extendingthefinitedifferencemethod 264
Chemicalreactionanddiffusioninasphericalcatalystpellet 265
Finitedifferencesforaconvection/diffusionequation 270
Modelingatubularchemicalreactorwithdispersion;treating
multiplefields 279
NumericalissuesfordiscretizedPDEswithmorethantwo
spatialdimensions 282
TheMATLAB1-Dparabolicandellipticsolverpdepe 294
Finitedifferencesincomplexgeometries 294
Thefinitevolumemethod 297
Thefiniteelementmethod(FEM) 299
FEMinMATLAB 309
FurtherstudyinthenumericalsolutionofBVPs 311
MATLABsummary 311
Problems 312
7 Probabilitytheoryandstochasticsimulation 317
Thetheoryofprobability 317
Importantprobabilitydistributions 325
Randomvectorsandmultivariatedistributions 336
Browniandynamicsandstochasticdifferentialequations
(SDEs) 338
Markovchainsandprocesses;MonteCarlomethods 353
Geneticprogramming 362
viii Contents
MATLABsummary 364
Problems 365
8 Bayesianstatisticsandparameterestimation 372
Generalproblemformulation 372
Example.Fittingkineticparametersofachemicalreaction 373
Single-responselinearregression 377
Linearleast-squaresregression 378
TheBayesianviewofstatisticalinference 381
Theleast-squaresmethodreconsidered 388
Selectingapriorforsingle-responsedata 389
Confidenceintervalsfromtheapproximateposteriordensity 395
MCMCtechniquesinBayesiananalysis 403
MCMCcomputationofposteriorpredictions 404
Applyingeigenvalueanalysistoexperimentaldesign 412
Bayesianmultiresponseregression 414
Analysisofcompositedatasets 421
Bayesiantestingandmodelcriticism 426
Furtherreading 431
MATLABsummary 431
Problems 432
9 Fourieranalysis 436
Fourierseriesandtransformsinonedimension 436
1-DFouriertransformsinMATLAB 445
Convolutionandcorrelation 447
Fouriertransformsinmultipledimensions 450
Scatteringtheory 452
MATLABsummary 459
Problems 459
References 461
Index 464
Preface
Thistextfocusesontheapplicationofquantitativeanalysistothefieldofchemicalengi-
neering. Modern engineering practice is becoming increasingly more quantitative, as the
useofscientificcomputingbecomesevermorecloselyintegratedintothedailyactivities
ofallengineers.Itisnolongerthedomainofasmallcommunityofspecialistpractitioners.
Whereasinthepast,onehadtohand-craftaprogramtosolveaparticularproblem,carefully
husbandingthelimitedmemoryandCPUcyclesavailable,nowwecanveryquicklysolvefar
morecomplexproblemsusingpowerful,widely-availablesoftware.Thishasintroducedthe
needforresearchengineersandscientiststobecomecomputationallyliterate–toknowthe
possibilitiesthatexistforapplyingcomputationtotheirproblems,tounderstandthebasic
ideasbehindthemostimportantalgorithmssoastomakewisechoiceswhenselectingand
tuningthem,andtohavethefoundationalknowledgenecessarytonavigateindependently
throughtheliterature.
This text meets this need, and is written at the level of a first-year graduate student
inchemicalengineering,aconsequenceofitsdevelopmentforuseatMITforthecourse
10.34, “Numerical methods applied to chemical engineering.” This course was added in
2001tothegraduatecorecurriculumtoprovideallfirst-yearMastersandPh.D.students
withanoverviewofquantitativemethodstoaugmenttheexistingcorecoursesintransport
phenomena,thermodynamics,andchemicalreactionengineering.Carehasbeentakento
develop any necessary material specific to chemical engineering, so this text will prove
usefultootherengineeringandscientificfieldsaswell.Thereaderisassumedtohavetaken
the traditional undergraduate classes in calculus and differential equations, and to have
®
someexperienceincomputerprogramming,althoughnotnecessarilyinMATLAB .
Even a cursory search of the holdings of most university libraries shows there to be a
greatnumberoftextswithtitlesthatarevariationsof“AdvancedEngineeringMathematics”
or“NumericalMethods.”Sowhyaddyetanother?
I find that there are two broad classes of texts in this area. The first focuses on intro-
ducing numerical methods, applied to science and engineering, at the level of a junior
orseniorundergraduateelectivecourse.Thescopeisnecessarilylimitedtorathersimple
techniquesandapplications.Thesecondclassistargetedtoresearch-levelworkers,either
higher graduate-level applied mathematicians or computationally-focused researchers in
scienceandengineering.Thesemaybeeitheradvancedtreatmentsofnumericalmethods
formathematicians,ordetaileddiscussionsofscientificcomputingasappliedtoaspecific
subjectsuchasfluidmechanics.
ix
x Preface
Neitheroftheseclassesoftextisappropriateforteachingthefundamentalsofscientific
computingtobeginningchemicalengineeringgraduatestudents.Examplesshouldbetyp-
ical of those encountered in graduate-level chemical engineering research, and while the
studentsshouldgainanunderstandingofthebasisofeachmethodandanappreciationof
itslimitations,theydonotneedexhaustivetheory-prooftreatmentsofconvergence,error
analysis, etc. It is a challenge for beginning students to identify how their own problems
maybemappedintoonesamenabletoquantitativeanalysis;therefore,anyappropriatetext
shouldhaveanextensivelibraryofworkedexamples,withcodeavailabletoservelateras
templates.Finally,thetextshouldaddresstheimportanttopicsofmodeldevelopmentand
parameterestimation.Thisbookhasbeendevelopedwiththeseneedsinmind.
Thistextfirstpresentsafundamentaldiscussionoflinearalgebra,toprovidethenecessary
foundationtoreadtheappliedmathematicalliteratureandprogressfurtheronone’sown.
Next, a broad array of simulation techniques is presented to solve problems involving
systems of nonlinear algebraic equations, initial value problems of ordinary differential
anddifferential-algebraic(DAE)systems,optimizations,andboundaryvalueproblemsof
ordinary and partial differential equations. A treatment of matrix eigenvalue analysis is
included,asitisfundamentaltoanalyzingthesesimulationtechniques.
Nextfollowsadetaileddiscussionofprobabilitytheory,stochasticsimulation,statistics,
andparameterestimation.Asengineeringbecomesmorefocuseduponthemolecularlevel,
stochasticsimulationtechniquesgaininimportance.ParticularattentionispaidtoBrownian
dynamics,stochasticcalculus,andMonteCarlosimulation.Statisticsandparameteresti-
mationareaddressedfromaBayesianviewpoint,inwhichMonteCarlosimulationprovesa
powerfulandgeneraltoolformakinginferencesandtestinghypothesesfromexperimental
data.
In each of these areas, topically relevant examples are given, along with MATLAB
(www.mathworks.com)programsthatservethestudentsastemplateswhenlaterwriting
their own code. An accompanying website includes a MATLAB tutorial, code listings of
all examples, and a supplemental material section containing further detailed proofs and
optionaltopics.Ofcourse,whilesignificantefforthasgoneintotestingandvalidatingthese
programs,noguaranteeisprovidedandthereadershouldusethemwithcaution.
Theproblemsaregradedbydifficultyandlengthineachchapter.ThoseofgradeAare
simple and can be done easily by hand or with minimal programming. Those of grade B
requiremoreprogrammingbutarestillratherstraightforwardextensionsorimplementations
oftheideasdiscussedinthetext.ThoseofgradeCeitherinvolvesignificantthinkingbeyond
thecontentpresentedinthetextorprogrammingeffortatalevelbeyondthattypicalofthe
examplesandgradeBproblems.
Thesubjectscoveredarebroadinscope,leadingtotheconsiderable(thoughhopefully
notexcessive)lengthofthistext.Thefocusisuponprovidingafundamentalunderstanding
oftheunderlyingnumericalalgorithmswithoutnecessarilyexhaustivelytreatingalloftheir
details,variations,andcomplexitiesofuse.Masteryofthematerialinthistextshouldenable
first-yeargraduatestudentstoperformoriginalworkinapplyingscientificcomputationto
their research, and to read the literature to progress independently to the use of more
sophisticatedtechniques.