Applied Linear Analysis for Chemical Engineers: A Multi-scale Approach with Mathematica® PDF
Preview Applied Linear Analysis for Chemical Engineers: A Multi-scale Approach with Mathematica®
VemuriBalakotaiah,RamR.Ratnakar AppliedLinearAnalysisforChemicalEngineers Also of Interest ChemicalReactionTechnology DmitryYu.Murzin,2022 ISBN978-3-11-071252-0,e-ISBN(PDF)978-3-11-071255-1, e-ISBN(EPUB)978-3-11-071260-5 Non-equilibriumThermodynamicsandPhysicalKinetics HalidBikkin,IgorI.Lyapilin,2021 ISBN978-3-11-072706-7,e-ISBN(PDF)978-3-11-072719-7, e-ISBN(EPUB)978-3-11-072738-8 ProcessTechnology.AnIntroduction AndréB.deHaan,JohanT.Padding,2022 ISBN978-3-11-071243-8,e-ISBN(PDF)978-3-11-071244-5, e-ISBN(EPUB)978-3-11-071246-9 Multi-levelMixed-IntegerOptimization.ParametricProgramming Approach StylianiAvraamidou,EfstratiosPistikopoulos,2022 ISBN978-3-11-076030-9,e-ISBN(PDF)978-3-11-076031-6, e-ISBN(EPUB)978-3-11-076038-5 DataScience.TimeComplexity,InferentialUncertainty,andSpacekime Analytics IvoD.Dinov,MilenVelchevVelev,2021 ISBN978-3-11-069780-3,e-ISBN(PDF)978-3-11-069782-7, e-ISBN(EPUB)978-3-11-069797-1 OutliersinControlEngineering.FractionalCalculusPerspective PawełD.Domański,YangQuanChen,MaciejŁawryńczuk,2022 ISBN978-3-11-072907-8,e-ISBN(PDF)978-3-11-072912-2, e-ISBN(EPUB)978-3-11-072913-9 Vemuri Balakotaiah, Ram R. Ratnakar Applied Linear Analysis for Chemical Engineers | ® A Multi-scale Approach with Mathematica Authors Prof.VemuriBalakotaiah Dr.RamR.Ratnakar UniversityofHouston ShellInternationalExploration&Production DeptofChemicalandBiomolecularEngineering Houston,TX77082 4800CalhounRoad USA Houston,TX77204-4004 [email protected] USA [email protected] Thecitationofregisterednames,tradenames,trademarks,etc.inthisworkdoesnotimply,evenin theabsenceofaspecificstatement,thatsuchnamesareexemptfromlawsandregulations protectingtrademarksetc.andthereforefreeforgeneraluse. ISBN978-3-11-073969-5 e-ISBN(PDF)978-3-11-073970-1 e-ISBN(EPUB)978-3-11-073978-7 LibraryofCongressControlNumber:2022944897 BibliographicinformationpublishedbytheDeutscheNationalbibliothek TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de. ©2023WalterdeGruyterGmbH,Berlin/Boston Coverimage:akinbostanci/iStock/GettyImagesPlus Typesetting:VTeXUAB,Lithuania Printingandbinding:CPIbooksGmbH,Leck www.degruyter.com Preface ThisbookisbasedonacoursethatthefirstauthortaughtattheUniversityofHous- tonforabout30years.Thiscoursewasarequirementforallfirst-yeargraduatestu- dentsandwasaprerequisitefortwootheroptionalcourses,takenmostlybygraduate studentswhoseresearchinvolvedmodeling,computationalandnonlinearanalysis. AswestateintheIntroduction,whilethisbookdealsonlywiththesolutionoflin- earequations,linearanalysisisthefoundationofallnumericalandnonlineartech- niques. Sincetherearemanybooksalreadyavailableonappliedmathematicsforchem- icalengineers,itisfairtoaskthequestionwhyanotherbook?Forthis,ourresponse isthateveryauthorhasauniqueperspectivethatmaybeappealingtoothers.Fur- ther,theauthorsarenotawareofanybookthatdealsexclusivelywiththesolution ofvariouslinearequationsthatariseinengineeringinaunifiedmanner,andwith examples. The senior author had the pleasure of taking the applied mathematics course from Professor Neal R. Amundson and later teaching the same course when Pro- fessor Amundson retired. Both authors have used the material extensively in their own research and would like to point out the following highlights of the material presented:(i)useofsymbolicsoftware(Mathematica®)forillustratingandenhanc- ing the impact of physical parameter changes on solutions, (ii) multiscale analysis of chemical engineering problems with physical interpretation of time and length scales in terms of eigenvalues and eigenvectors/eigenfunctions, (iii) detailed dis- cussion of compartment models for various finite- dimensional problems and their solution in phase spaces, (iv) evaluation and illustration of functions of matrices (anduse of symbolicmanipulation)tosolvemulticomponentdiffusion-convection- reactionproblems,(v)illustrationofthetechniquesandinterpretationofsolutions to several classical chemical engineering and related problems, (vi) emphasis on the connection between discrete (matrix algebra) and continuum models (initial, boundary and initial-boundary value problems), (vii) physical interpretation of ad- jointoperatorandadjointsystemsandtheirapplicationinsolvinginverseproblems and(viii)useofcomplexanalysisandalgebrainthesolutionofpracticalengineering problems. TheseniorauthorhastaughtmostofcontentsofPartsI,III,IVandVinasingle semester(14weeksor28lecturesof90minutesduration).However,theentirecon- tentsofthebookcanbetaughtinatwo-semestercourse.Forasinglesemestercourse, werecommendcoveringChapters1to5,14,17,selectedsectionsofChapters18to21 and23to25. Wewishtoacknowledgemanycolleagues,formerstudentsandourmentorswho overtheyearscontributedtoourunderstandingandorganizationofthesubject. https://doi.org/10.1515/9783110739701-201 VI | Preface WealsowanttothankKarinSora,NadjaSchedensackandVilmaVaičeliūnienėof DeGruyterfortheirhelpduringproduction. Thesecondauthorwishestoacknowledgetheconstantencouragementandsup- portofhisfamilty,especiallyhiseldestbrotherSiddheshSatyakar. Finally,thefirstauthorwishestoacknowledgethepatienceandunderstandingof hiswife,NaliniVemuri,anddedicateittoherwithaffectionandgratitude. Introduction Thisbookdealswiththesolutionoflinearequations.Wediscussthesolutionsoflin- ear algebraic equations, linear initial value problems, linear boundary value prob- lems, linear integral equations and linear partial differential equations along with theirapplicationtovariouschemicalengineeringproblems. Itshouldbepointedoutthatmostpracticalproblemsencounteredbyengineers arenonlinearandareoftensolvedonacomputerusingnumericaltechniques.Inmost cases,thenonlinearproblemislinearizedaroundaknownorapproximatesolution andamoreaccuratesolutionisobtainedbysolvingasequenceoflinearproblems.The nonlinearmethodsofanalysisaswellasthenumericaltechniquesusedbyengineers drawheavilyfromthelinearanalysis.Inotherwords,linearanalysisisthefoundation ofallnonlinearandnumericaltechniques. Generallyspeaking,mostlinearproblemsthatariseinapplicationsmaybeclas- sifiedintotwogroups:(i)problemsdescribingthesteadystateorequilibriumstateof aphysicalsystemand(ii)problemsdescribingthedynamicortransientbehaviorof aphysicalsystem.Thefirsttypeofproblemsaredescribedbylinearequationsofthe form Lu=f (1) whereLisalinearoperator,uisastatevectorandfisasourcefunction.Forexample, in finite dimensions, equation (1) may be a set of n linear algebraic equations in n unknowns, Au=b (2) whereAisan×nmatrix,uandbaren×1vectors.Whenthestatevectorubelongstoan infinite-dimensionalspace,equation(1)maybeatwo-pointboundaryvalueproblem suchas d du − (p(x) )+q(x)u=f(x), a<x<b (3) dx dx u(a)=u(b)=0 (4) oranintegralequationsuchastheFredholmintegralequationofthefirstkindgiven by b ∫K(x,s)u(s)ds=f(x) (5) a orapartialdifferentialequationsuchasthePoisson’sequation https://doi.org/10.1515/9783110739701-202 VIII | Introduction 𝜕2u 𝜕2u −( + )=f(x,y) inΩ (6) 𝜕x2 𝜕y2 u=0 on𝜕Ω (7) whereΩissomedomaininthex-yplaneand𝜕Ωisitsboundary. Thesecondclassofproblemsareoftheform du =Lu, t >0 (8) dt u=u0 att =0 (9) wheretisthetimeandtheevolutionequation(8)describesthesystembehaviorfor t > 0,whileequation(9)givestheinitialcondition.Inthesimplercaseofthefinite- dimensionalproblems,equations(8)–(9)maybeoftheform du =Au (10) dt u=u0 att =0 (11) whereAisaconstantcoefficientn×nmatrix,uisan×1vectorofstatevariablesandu0 isan×1vectorofinitialconditions.Anexampleofaninitialvalueproblemininfinite dimensionsistheheatequationinonespatialcoordinateandtime: 𝜕u 𝜕2u = ; 0<x<1, t >0 (12) 𝜕t 𝜕x2 u(0,t)=u(1,t)=0 (Boundaryconditions) (13) u(x,0)=f(x) (Initialcondition) (14) Weshallseethatmanyoftheconceptsinvolvedinthesolutionoflinearordinary andpartialdifferentialequationsaregeneralizationsoftheideasinvolvedinthesolu- tionofthefinite-dimensionalproblemsrepresentedbyequations(2)and(10).There- fore,weshallfocusfirstonthefinite-dimensionalcase. Propertiesofsolutionstolinearequations WhenthematrixAisinvertible,thesolutionofequation(2)maybeexpressedas n 1 u=∑ cx (15) λ j j j=1 j wherethescalarsλ (eigenvalues)andthe(eigen)vectorsx dependonlyonthema- j j trixA,whiletheconstantsc aregivenby j ⟨b,y⟩ c = j , (16) j ⟨x,y⟩ j j Introduction | IX wherey areknownasthelefteigenvectorsofA.Here,⟨x,y⟩denotesthedotorinner j product of vectors. When the matrix A is symmetric (self-adjoint), x = y and the j j eigenvectorsarenormalizedtohaveunitlength(⟨x,x⟩ = 1),theexpressionforthe j j constantsc simplifiesto j c =⟨b,x⟩. (17) j j Theaboveformofthesolutionhasadvantagesoverthedirectsolution(e.g.,byGaus- sianelimination)whennislarge.Forexample,whenAissymmetricandtheeigenval- uesarewellseparated(0<|λ |≪|λ |≪|λ | ≪⋅⋅⋅≪|λ |),thefirstfewtermsmaybe 1 2 3 n sufficienttocomputethesolutionifthedesiredaccuracyisnothigh.Asecondadvan- tageisthatthesolutionhasthesameformforalllinearequationsoftheformgiven by(1).Forexample,whenthelinear(differential/integral)operatorLissymmetric, thesamesolutionisapplicablewithaslightmodification: ∞ 1 u=∑ cϕ; c =⟨f,ϕ⟩, (18) λ j j j j j=1 j whereλ aretheeigenvaluesandϕ arethenormalizedeigenfunctionsoftheopera- j j torL. Thesolutionoftheinitialvalueproblem,equations(10)–(11)maybeexpressedas n ⟨u0,y⟩ u(t)=∑ceλjtx; c = j (19) j j j ⟨x,y⟩ j=1 j j whichforthecaseofsymmetricmatrixsimplifiestoc = ⟨u0,x⟩.Thegeneralization j j ofthisresultforthecaseofasymmetricdifferentialoperatoris ∞ u(t)=∑ceλjtϕ; c =⟨f,ϕ⟩. (20) j j j j j=1 Animportantobservationregardingthevarioussolutionstothelinearequationsis thattheyareallexpressedintermsoftheeigenvaluesandeigenfunctionsoftheoper- atorappearingintheequation.Theseeigenvaluesandeigenfunctionscorrespondto varioustimeandlengthscalesthatareofinterestinthephysicalsystem.Animportant taskoflinearanalysisistheidentificationoftheselengthandtimescalesandrelating themtotheparametersofthephysicalsystem.Wehopetoillustratethisforvarious chemicalengineeringproblems.
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