Mathematical Modeling in Chemical Engineering

196 Pages·2014·3.14 MB·English

by  Rasmuson A.

Trim:247mm×174mm Top:12.653mm Gutter:16.871mm
CUUK2531-FM CUUK2531/Rasmuson ISBN:9781107049697 November13,2013 18:34
MathematicalModelinginChemicalEngineering
A solid introduction to mathematical modeling for a range of chemical engineering
applications,coveringmodelformulation,simplification,andvalidation.Itexplainshow
todescribeaphysical/chemicalrealityinmathematicallanguageandhowtoselectthe
typeanddegreeofsophisticationforamodel.Modelreductionandapproximatemethods
are presented, including dimensional analysis, time constant analysis, and asymptotic
methods.Anoverviewofsolutionmethodsfortypicalclassesofmodelsisgiven.Asfinal
stepsinmodelbuilding,parameterestimationandmodelvalidationandassessmentare
discussed.Thereaderisgivenhands-onexperienceofformulatingnewmodels,reducing
themodels,andvalidatingthemodels.
Theauthorsassumeaknowledgeofbasicchemicalengineering,inparticulartransport
phenomena,aswellasbasicmathematics,statistics,andprogramming.Theaccompa-
nying problems, tutorials, and projects include model formulation at different levels,
analysis,parameterestimation,andnumericalsolution.
Anders Rasmuson is a Professor in Chemical Engineering at Chalmers University of
Technology, Gothenburg, Sweden. He obtained his PhD in chemical engineering at
the Royal Institute of Technology, Stockholm, in 1978. His research has focused on
mathematical modeling combined with experimental work in many areas of chemical
engineering,forexampleparticulateprocesses,multiphaseflowsandtransportphenom-
ena,mixing,andseparationprocesses.
Bengt Andersson is a Professor in Chemical Engineering at Chalmers University of
Technology. He obtained his PhD in chemical engineering at Chalmers in 1977. His
research has focused on experimental studies and modeling of mass and heat transfer
invarious chemical reactors,rangingfromautomotive catalysistothree-phase flowin
chemicalreactors.
Louise Olsson is a Professor in Chemical Engineering at Chalmers University of
Technology. She obtained her PhD in chemical engineering at Chalmers in 2002. Her
researchhasfocusedonexperimentsandkineticmodelingofheterogeneouscatalysis.
Ronnie Andersson is an Assistant professor in Chemical Engineering at Chalmers
University of Technology. He obtained his PhD at Chalmers in 2005, and from 2005
until2010heworkedasconsultantatEpsilonHighTechasaspecialistincomputational
fluid dynamic simulations of combustion and multiphase flows. His research projects
involvephysicalmodeling,fluiddynamicsimulations,andexperimentalmethods.
Trim:247mm×174mm Top:12.653mm Gutter:16.871mm
CUUK2531-FM CUUK2531/Rasmuson ISBN:9781107049697 November13,2013 18:34
Trim:247mm×174mm Top:12.653mm Gutter:16.871mm
CUUK2531-FM CUUK2531/Rasmuson ISBN:9781107049697 November13,2013 18:34
Mathematical Modeling in
Chemical Engineering
ANDERS RASMUSON
ChalmersUniversityofTechnology,Gothenberg
BENGT ANDERSSON
ChalmersUniversityofTechnology,Gothenberg
LOUISE OLSSON
ChalmersUniversityofTechnology,Gothenberg
RONNIE ANDERSSON
ChalmersUniversityofTechnology,Gothenberg
Trim:247mm×174mm Top:12.653mm Gutter:16.871mm
CUUK2531-FM CUUK2531/Rasmuson ISBN:9781107049697 November13,2013 18:34
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107049697
(cid:2)C CambridgeUniversityPress2014
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2014
PrintedintheUnitedKingdombyMPGPrintgroupLtd,Cambridge
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloginginPublicationdata
ISBN978-1-107-04969-7Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication,
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
Trim:247mm×174mm Top:12.653mm Gutter:16.871mm
CUUK2531-FM CUUK2531/Rasmuson ISBN:9781107049697 November13,2013 18:34
Contents
Preface pageix
1 Introduction 1
1.1 Whydomathematicalmodeling? 1
1.2 Themodelingprocedure 5
1.3 Questions 9
2 Classification 10
2.1 Groupingofmodelsintooppositepairs 10
2.2 Classificationbasedonmathematicalcomplexity 14
2.3 Classificationaccordingtoscale(degreeofphysicaldetail) 16
2.4 Questions 18
3 Modelformulation 20
3.1 Balancesandconservationprinciples 20
3.2 Transportphenomenamodels 22
3.3 Boundaryconditions 26
3.4 Populationbalancemodels 28
3.4.1 ApplicationtoRTDs 32
3.5 Questions 34
3.6 Practiceproblems 35
4 Empiricalmodelbuilding 40
4.1 Dimensionalsystems 40
4.2 Dimensionlessequations 41
4.3 Empiricalmodels 46
4.4 Scalingup 48
4.5 Practiceproblems 52
5 Strategiesforsimplifyingmathematicalmodels 53
5.1 Reducingmathematicalmodels 54
Trim:247mm×174mm Top:12.653mm Gutter:16.871mm
CUUK2531-FM CUUK2531/Rasmuson ISBN:9781107049697 November13,2013 18:34
vi Contents
5.1.1 Decouplingequations 55
5.1.2 Reducingthenumberofindependentvariables 55
5.1.3 Lumping 56
5.1.4 Simplifiedgeometry 56
5.1.5 Steadystateortransient 58
5.1.6 Linearizing 61
5.1.7 Limitingcases 63
5.1.8 Neglectingterms 64
5.1.9 Changingtheboundaryconditions 67
5.2 Casestudy:Modelingflow,heat,andreactioninatubularreactor 68
5.2.1 Generalequationforacylindricalreactor 68
5.2.2 Reducingthenumberofindependentvariables 69
5.2.3 Steadystateortransient? 70
5.2.4 Decouplingequations 72
5.2.5 Simplifiedgeometry 72
5.2.6 Limitingcases 75
5.2.7 Conclusions 76
5.3 Errorestimations 76
5.3.1 Sensitivityanalysis 77
5.3.2 Over-andunderestimations 77
5.4 Questions 77
5.5 Practiceproblems 78
6 Numericalmethods 81
6.1 Ordinarydifferentialequations 81
6.1.1 ODEclassification 81
6.1.2 Solvinginitial-valueproblems 82
6.1.3 Numericalaccuracy 87
6.1.4 Adaptivestepsizemethodsanderrorcontrol 88
6.1.5 Implicitmethodsandstability 90
6.1.6 Multistepmethodsandpredictor–correctorpairs 93
6.1.7 SystemsofODEs 94
6.1.8 Transforminghigher-orderODEs 96
6.1.9 StiffnessofODEs 97
6.2 Boundary-valueproblems 99
6.2.1 Shootingmethod 99
6.2.2 FinitedifferencemethodforBVPs 102
6.2.3 Collocationandfiniteelementmethods 107
6.3 Partialdifferentialequations 108
6.3.1 ClassificationofPDEs 109
6.3.2 Finitedifferencesolutionofparabolicequations 110
6.3.3 Forwarddifferencemethod 110
6.3.4 Backwarddifferencemethod 113
Trim:247mm×174mm Top:12.653mm Gutter:16.871mm
CUUK2531-FM CUUK2531/Rasmuson ISBN:9781107049697 November13,2013 18:34
Contents vii
6.3.5 Crank–Nicolsonmethod 114
6.4 Simulationsoftware 114
6.4.1 MATLAB 114
6.4.2 MiscelleanusMATLABalgorithms 115
6.4.3 AnexampleofMATLABcode 116
6.4.4 GNUOctave 117
6.5 Summary 117
6.6 Questions 117
6.7 Practiceproblems 118
7 Statisticalanalysisofmathematicalmodels 121
7.1 Introduction 121
7.2 Linearregression 121
7.2.1 Leastsquaremethod 123
7.3 Linearregressioninitsgeneralizedform 125
7.3.1 Leastsquaremethod 126
7.4 Weightedleastsquares 127
7.4.1 Stabilizationofthevariance 127
7.4.2 Placinggreater/lessweightoncertainexperimentalparts 128
7.5 Confidenceintervalsandregions 130
7.5.1 Confidenceintervals 130
7.5.2 Student’st-testsofindividualparameters 133
7.5.3 Confidenceregionsandbands 134
7.6 Correlationbetweenparameters 136
7.6.1 Varianceandco-variance 136
7.6.2 Correlationmatrix 137
7.7 Non-linearregression 138
7.7.1 Intrinsicallylinearmodels 138
7.7.2 Non-linearmodels 139
7.7.3 Approximateconfidencelevelsandregionsfornon-linearmodels 140
7.7.4 Correlationbetweenparametersfornon-linearmodels 142
7.8 Modelassessments 142
7.8.1 Residualplots 142
7.8.2 Analysisofvariance(ANOVA)table 145
7.8.3 R2statistic 149
7.9 Casestudy7.1:Statisticalanalysisofalinearmodel 149
7.9.1 Solution 150
7.10 Casestudy7.2:Multipleregression 153
7.10.1 Solution 154
7.11 Casestudy7.3:Non-linearmodelwithonepredictor 158
7.11.1 Solution 159
7.12 Questions 163
7.13 Practiceproblems 163
Trim:247mm×174mm Top:12.653mm Gutter:16.871mm
CUUK2531-FM CUUK2531/Rasmuson ISBN:9781107049697 November13,2013 18:34
viii Contents
AppendixAMicroscopictransportequations 168
AppendixBDimensionlessvariables 170
AppendixCStudent’st-distribution 173
Bibliography 180
Index 181