Applied Linear Analysis for Chemical Engineers: A Multi-scale Approach with Mathematica®

783 Pages·2023·8.828 MB·English

by  Balakotaiah V., Ratnakar R.R.

VemuriBalakotaiah,RamR.Ratnakar
AppliedLinearAnalysisforChemicalEngineers
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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
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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.