Table Of ContentULLMANN’S
Chemical Engineering
and Plant Design
Volume 1
Mathematics and Physics
in Chemical Engineering
Fundamentals of
Chemical Engineering
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V
Preface
Sincetheunabridged40-volumeUllmann’sEncyclopediaisinaccessibletomanyreaders–particu-
larlyindividuals,smallercompaniesorinstitutes–alltheinformationonchemicalengineeringand
plantdesignhasbeencondensedintothisconvenienttwo-volumeset.
BasedontheverylatestprinteditionofUllmann’s,thisreadyreferenceistheone-stopresourcefor
theplantdesignengineeringcommunity.Startingwiththequantitativetreatmentandfundamentals
ofchemicalengineering,itcombinesallaspectsofprocessdevelopmentandreactortechnology,as
wellasdetailingtheirpracticalapplicationsinsectionsdevotedtoplantdesign,scale-upandplant
safety.Eachofthedetailedandcarefullyeditedarticlesiswrittenbyrelevantexpertsfromindustry
oracademia.Akeywordandanauthorindexcompletethecontentsofthishandbook.
Throughout,readersbenefitfromtherigorousandcross-indexednatureoftheparentreference,and
willfindbothbroadintroductoryinformationaswellasin-depthdetailsofsignificancetoindustrial
andacademicenvironments.
ThePublisher
Contents VII
Contents
Volume1 NondestructiveTesting................ 785
On-LineMonitoringofChemical
SymbolsandUnits.................... IX Reactions........................ 821
ConversionFactors................... XI
Abbreviations........................ XII
Volume2
CountryCodes....................... XVII
PeriodicTableofElements............XVIII
PlantandProcessDesign
ProcessDevelopment................. 873
MathematicsandPhysicsin
ChemicalEngineering SeparationProcesses,Introduction...... 915
Solid–LiquidSeparation,Introduction.. 923
MathematicsinChemicalEngineering.. 3 Solid–SolidSeparation,Introduction... 931
MathematicalModeling............... 165 Mixing,Introduction.................. 939
TransportPhenomena................. 271 SolidsTechnology,Introduction........ 941
FluidMechanics...................... 371 ReactorTypesandTheirIndustrial
Applications...................... 953
DesignofExperiments................ 423
ChemicalPlantDesignandConstruction 987
ComputationalFluidDynamics........ 463
ModelReactorsandTheirDesign PinchTechnology..................... 1075
Equations........................ 487 PilotPlants........................... 1083
Scale-UpinChemicalEngineering..... 1093
BiochemicalEngineering.............. 1117
Fundamentals EnergyManagementinChemical
ofChemicalEngineering Industry.......................... 1187
PlantandProcessSafety............... 1205
EstimationofPhysicalProperties....... 537 EnvironmentalManagementinthe
ConstructionMaterialsinChemical ChemicalIndustry................ 1331
Industry.......................... 605
Corrosion............................ 653
AbrasionandErosion................. 735 AuthorIndex....................... 1359
MechanicalPropertiesandTestingof
MetallicMaterials................ 761 SubjectIndex....................... 1363
Mathematics and Physics in Chemical Engineering
MathematicsinChemicalEngineering 3
Mathematics in Chemical Engineering
BruceA.Finlayson,DepartmentofChemicalEngineering,UniversityofWashington,Seattle,Washington
98195,UnitedStates(Chaps.1–9,11,12)
LorenzT.Biegler,CarnegieMellonUniversity,Pittsburgh,Pennsylvania15231,UnitedStates(Chap.10)
IgnacioE.Grossmann,CarnegieMellonUniversity,Pittsburgh,Pennsylvania15231,UnitedStates(Chap.10)
ArthurW.Westerberg,CarnegieMellonUniversity,Pittsburgh,Pennsylvania15231,UnitedStates(Chap.10)
1. SolutionofEquations . . . . . . . . 6 7. OrdinaryDifferentialEquationsas
1.1. LinearAlgebraicEquations . . . . 6 BoundaryValueProblems . . . . . 76
1.2. NonlinearAlgebraicEquations . . 12 7.1. SolutionbyQuadrature . . . . . . . 76
1.3. LinearDifferenceEquations . . . . 14 7.2. ShootingMethods . . . . . . . . . . . 77
1.4. Eigenvalues . . . . . . . . . . . . . . . 15 7.3. FiniteDifferenceMethod . . . . . . 79
2. ApproximationandIntegration . . 16 7.4. OrthogonalCollocation . . . . . . . 82
2.1. Introduction . . . . . . . . . . . . . . 16 7.5. Orthogonal Collocation on Finite
2.2. GlobalPolynomialApproximation 16 Elements . . . . . . . . . . . . . . . . . 87
2.3. PiecewiseApproximation . . . . . . 18 7.6. GalerkinFiniteElementMethod . 88
2.4. Quadrature . . . . . . . . . . . . . . . 21 7.7. CubicB-Splines . . . . . . . . . . . . 90
2.5. LinearLeastSquares . . . . . . . . . 24 7.8. AdaptiveMeshStrategies . . . . . . 91
2.6. NonlinearLeastSquares . . . . . . 26 7.9. Comparison . . . . . . . . . . . . . . . 92
2.7. FourierTransforms 7.10. SingularProblemsandInfiniteDo-
ofDiscreteData . . . . . . . . . . . . 27 mains . . . . . . . . . . . . . . . . . . . 93
2.8. Two-DimensionalInterpolation 8. PartialDifferentialEquations . . . 94
andQuadrature . . . . . . . . . . . . 29 8.1. ClassificationofEquations . . . . . 94
3. ComplexVariables . . . . . . . . . . 30 8.2. HyperbolicEquations . . . . . . . . 96
3.1. IntroductiontotheComplexPlane 30 8.3. ParabolicEquations
3.2. ElementaryFunctions . . . . . . . . 31 inOneDimension . . . . . . . . . . . 98
3.3. Analytic Functions of a Complex 8.4. EllipticEquations . . . . . . . . . . . 104
Variable . . . . . . . . . . . . . . . . . 33 8.5. ParabolicEquationsinTwo
3.4. IntegrationintheComplexPlane 34 orThreeDimensions . . . . . . . . . 108
3.5. OtherResults . . . . . . . . . . . . . . 37 9. IntegralEquations . . . . . . . . . . 109
4. IntegralTransforms . . . . . . . . . 37 9.1. Classification . . . . . . . . . . . . . . 109
4.1. FourierTransforms . . . . . . . . . . 37 9.2. Numerical Methods for Volterra
4.2. LaplaceTransforms . . . . . . . . . 42 EquationsoftheSecondKind . . . 111
4.3. SolutionofPartialDifferential 9.3. NumericalMethodsforFredholm,
EquationsbyUsingTransforms . . 48 Urysohn,andHammersteinEqua-
5. VectorAnalysis . . . . . . . . . . . . 51 tionsoftheSecondKind . . . . . . 113
6. OrdinaryDifferentialEquationsas 9.4. NumericalMethodsforEigenvalue
InitialValueProblems . . . . . . . . 61 Problems . . . . . . . . . . . . . . . . . 115
6.1. SolutionbyQuadrature . . . . . . . 62 9.5. Green’sFunctions . . . . . . . . . . . 115
6.2. ExplicitMethods . . . . . . . . . . . 63 9.6. Boundary Integral Equations and
6.3. ImplicitMethods . . . . . . . . . . . 67 BoundaryElementMethod . . . . . 117
6.4. Stiffness . . . . . . . . . . . . . . . . . 68 10. Optimization . . . . . . . . . . . . . . 118
6.5. Differential–AlgebraicSystems . 69 10.1. Introduction . . . . . . . . . . . . . . 118
6.6. ComputerSoftware . . . . . . . . . 71 10.2. ConditionsforOptimality . . . . . 119
6.7. Stability, Bifurcations, Limit Cy- 10.3. StrategiesofOptimization . . . . . 123
cles . . . . . . . . . . . . . . . . . . . . . 72 10.4. SuccessiveQuadratic
6.8. SensitivityAnalysis . . . . . . . . . . 74 Programming(SQP) . . . . . . . . . 127
6.9. Eigenvalues and Roots by Initial 10.5. LinearProgramming . . . . . . . . . 130
ValueTechniques . . . . . . . . . . . 75 10.5.1. BasicProperties . . . . . . . . . . . . . 131
4 MathematicsinChemicalEngineering
10.5.2. SimplexAlgorithm . . . . . . . . . . . 132 11.4. Factorial Design of Experiments
10.6. Mixed-IntegerProgramming . . . 133 andAnalysisofVariance . . . . . . 150
12. MultivariableCalculusAppliedto
10.7. SolutionofDynamicOptimization
Problems . . . . . . . . . . . . . . . . . 136 Thermodynamics . . . . . . . . . . . 153
12.1. StateFunctions . . . . . . . . . . . . . 153
11. ProbabilityandStatistics . . . . . . 142
12.2. ApplicationstoThermodynamics 154
11.1. Concepts . . . . . . . . . . . . . . . . . 143
12.3. PartialDerivativesofAllThermo-
11.2. SamplingandStatisticalDecisions 146 dynamicFunctions . . . . . . . . . . 155
11.3. ErrorAnalysisinExperiments . . 150 13. References . . . . . . . . . . . . . . . . 156
Symbols h setofm equalityconstraintsforanopti-
mizationproblem
Variables
H the inverse of matrix Q. Also the scalar
a scalar constant in quadratic approxima-
Hamiltonian function for a dynamic op-
tionforF,theobjectivefunction
timizationproblem
A m×n matrix of constant coefficients for
I identitymatrix
equality constraints in linear program-
k iterationmatrix
mingmodel
L scalarLagrangefunction
b vector of constants premultiplying the r
m numberofequalityconstraintsforanop-
independent variables u in the quadratic
timizationproblem
approximationforF,theobjectivefunc-
n numberoftotalvariablesinanoptimiza-
tion. Also vector of m right-hand-sides
tionproblem
forequalityconstraintsinlinearprogram-
N (n-m)×m matrix corresponding to the
mingproblem
non-basis variables in a linear program-
B approximationofthen×nHessianmatrix
mingproblem
fortheLagrangefunctionforthesucces-
p number of inequality constraints for an
sive quadratic programming algorithm.
optimizationproblem.Alsovectorofpa-
Also m×m non-singular matrix corre-
rameters(donotvarywithtime)fordy-
spondingtothebasisvariablesinalinear
namicoptimizationproblem
programming problem. Also coefficient
Pe Pecletnumber
matrixforbinaryvariablesyinthesetof
Q anr×rmatrixofconstantsusedindefin-
equality constraints for a mixed integer
ingthequadratictermforaquadraticap-
programmingproblem
proximationforF,theobjectivefunction
Bi Biotnumber
r number of independent variables for an
c vectorofnconstantcostcoefficientsfor
optimizationproblem(r=n-m)
all variables z in linear programming
Re thespaceofrealnumbers
problem
R matrixdefinedinthequadraticprogram-
Co Courantnumber
mingalgorithm.Alsothevectorofresid-
d searchdirectioninthespaceofallnvari-
ual equations in the finite element ap-
ablesz(bothdependentandindependent
proach to solving dynamic optimization
variables)
problems
D diffusionconstant
Ren realnumbervectorspaceofdimensionn
Da Damko¨hlernumber
s changeintheindependentvariablesu(a
f nonlinearscalarcontributiontoobjective
vector with r elements in it). Also sen-
function which is a function only of the
sitivityofthefunctionsf withrespectto
continuousvariables,y,foramixedinte-
theparameterpforadynamicoptimiza-
gerprogrammingproblem
tionproblem
F scalarobjectivefunctionforanoptimiza-
Sh Sherwoodnumber
tionproblem
u vector of r independent variables for an
g setofpinequalityconstraintsforanop-
optimizationproblem.Foratime-varying
timizationproblem
MathematicsinChemicalEngineering 5
problem, the independent time-varying optimizationproblem,theLagrangemul-
controlvariables tiplierforthestatevariableconstraints.
W r×rweightingmatrixusedincomputing µ vectorofpKuhn–Tuckermultipliers
Frobeniusnorm ν vectorofLagrangemultipliers(varywith
x vector of m dependent variables for an time) for the algebraic equality con-
optimizationproblem(thestatevariables straintsforadynamicoptimizationprob-
foradynamicoptimizationproblem) lem
y variables used in linear programming τ rescaledtimesoitliesinrange[0,1]
problem.Alsobinaryvariablesinthefor- Y vector of polynomial approximation
mulationofamixedintegerprogramming function for the control variables for fi-
problem nite element method for dynamic opti-
Y n×m matrix whose columns are vectors mizationproblem
thatspantherangespaceofthelinearized
constraints (see section on successive
Specialsymbols
quadraticprogrammingalgorithm)
z vectorofallnvariables–boththem de- | subjectto
pendentvariablesxandtherindependent : mapping. For example, h : Rn→Rm,
variablesu–foranoptimizationproblem statesthatfunctionshmaprealnumbers
z¯ interior point for linear programming into m real numbers. There are m func-
problem tionshwrittenintermsofnvariables
Z n×(n-m)matrixwhosecolumnsarevec- ∈ memberof
tors that span the null space of the → mapsinto
linearized constraints (see section on
successivequadraticprogrammingalgo-
Subscripts
rithm)
A g are the active constraints (i.e., pre-
A
ciselyequaltozero)amongtheinequality
Greeksymbols
constraints
α scalar for setting the step size in a
B basis variables in a linear programming
line search algorithm for the successive
problem
quadratic programming algorithm. Also
f valueatthefinaltimeforadynamicop-
avectorofcoefficientstoformaconvex
timizationproblem
combinationofextremepointsforalinear
i i-thelementofavectorofvariables
programmingproblem.Alsoscalarvari-
N non-basis variables in a linear program-
ableusedastheobjectiveforthemixedin-
mingproblem
teger programming problem in equation
NE number of elements in a finite element
76.Alsolengthofanelementinthefinite
model
elementapproachtosolvingdynamicop-
OA outerapproximation
timizationproblems
o valueattheinitialtimeforadynamicop-
γ changeingradientoftheobjectivefunc-
timizationproblem
tioninmovingfrompointktopointk+1.
γ isavectorwithr elementsinit
δ Kroneckerdelta Superscript
∆ samplingrate
k iterationindex
ε smallscalar
L lowerbound
φ vector of polynomial approximation
T transposeofavectorormatrix
functionsforthestatevariablesforfinite
U upperbound
element method for dynamic optimiza- ˆ basepointforaTaylorseriesexpansion
tionproblem,alsoThielemodulus
κ conditionnumber
λ vector of m Lagrange multipliers for an
optimizationproblem.Alsoforadynamic
6 MathematicsinChemicalEngineering
1. SolutionofEquations 1.1. LinearAlgebraicEquations
Mathematical models of chemical engineering Considerthen×nlinearsystem
systemscantakemanyforms:theycanbesets
ofalgebraicequations,differentialequations,or a11x1 +a12x2 +...+ a1nxn=f1
integralequations.Massandenergybalancesof
chemical processes typically lead to large sets a21x1 +a22x2 +...+a2nxn = f2
ofalgebraicequations:
...
a11x1+a12x2= b1
an1x1 +an2x2 +...+annxn = fn
a21x1+a22x2=b2
Inthisequationa ,...,a areknownparam-
11 nn
Massbalancesofstirredtankreactorsmaylead eters, f ,...,f are known, and the unknowns
1 n
toordinarydifferentialequations: arex ,...,x .Thevaluesofallunknownsthat
1 n
satisfyeveryequationmustbefound.Thissetof
dy
= f [y(t)] equationscanberepresentedasfollows:
dt
Radiative heat transfer may lead to integral (cid:2)n
equations: aijxj = fj or Ax=f
j=1
(cid:1)1
The most efficient method for solving a set
y(x)=g(x)+λ K(x,s)f (s)ds
of linear algebraic equations is to perform a
0
lower–upper(LU)decompositionofthecorre-
Evenwhenthemodelisadifferentialequation
sponding matrix A. This decomposition is es-
orintegralequation,themostbasicstepintheal-
sentially a Gaussian elimination, arranged for
gorithmisthesolutionofsetsofalgebraicequa-
maximumefficiency.
tions.Thesolutionofsetsofalgebraicequations
TheLUdecompositionisdonebycalculating
isthefocusofChapter1.
inturn
A single linear equation is easy to solve for
eitherxory:
fori= 1,ndo
y=ax+b
forj = 1,ndo
Iftheequationisnonlinear,
a(ij1) = aij
f(x)= 0
itmaybemoredifficulttofindthex satisfying enddo
thisequation.Theseproblemsarecompounded
when there are more unknowns, leading to si- enddo
multaneous equations. If the unknowns appear
fork= 2,ndo
inalinearfashion,thenanimportantconsidera-
tionisthestructureofthematrixrepresentingthe
fori =k +1,ndo
equations; special methods are presented here
for special structures. They are useful because
forj =k +1,ndo
they increase the speed of solution. If the un-
klmenmuoswtisnbmseaupucpsheemdaro(irnie.ea.d,nifomfincalukinlete.aaIrtefgrauastehisvisoento,eftchhtehnpeiqrousbeos-- a(ijk)=ai(jk−1) − a(kak−(i−,k1k−1,−k)1−1)1ak(k−−1,1j)
lution and try to improve the guess). An im-
enddo
portant question then is whether such an iter-
ative scheme converges. Other important types
enddo
ofequationsarelineardifferenceequationsand
eigenvalueproblems,whicharealsodiscussed.
enddo
