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ApplicationofNumericalMetho dsinChemicalPro cessEngineeringFrerich J? KeilTechnical University of Hamburg?Harburg? Dept? of Chemical Engineering? Ei?en?dorfer Str? 38? D?21073 Hamburg? GermanyIntro ductionNumerical metho ds in chemical engineering deal with a broad range of prob?lems starting from calculations on atomic or molecular level to the optimiza?tion of complete chemical plants? From an engineer?sp oint of view? we willexp ound the following sub jects:?quantum mechanical calculations of atoms and molecules?numerical treatment of chemical reaction kinetics?transp ort pro cesses?mathematical description of unit op erations?stationary and instationary simulation and optimization of chemical plantsBecause of this extensive ?eld we will have to refer to other overview pap ers?Quantum mechanics of atoms and molecules? Monte CarloandMolecularDynamics Metho dsExactsolutionsformany?electron systemsareessentiallyunattainable? ex?ceptforsometrivialcases?sinceanexactsolutionwouldimplyanexactHamiltonian?whichisunavailable?Inpracticemo delHamiltoniansareinuse? b oth on nonrelativistic and relativistic level? In short? one can say thatthe energyof an n?electronsystem can b epartioned into theenergy of oneelectronmoving in theaverage ?eldof theremaining ?n?1? ?HF?SCF?elec?trons and nuclei? This is the Hartree?Fo ckmo del? The next more advancedlevel takes the electron correlation into account? The Hartree?Fo ck approachallows that two electrons with di?erent spins can b e found at the same spa?tial p oint?indicating thattheCoulomb holecannotb erepresentedontheHF?level? Various metho ds have b eenprop osedfor calculating electron cor?relation e?ects?In common use is the M?ller?Plesset p erturbation theory ofsecond order?MP2?? In this approach? the total Hamiltonian of thesystemis partioned into two pieces:a zeroth?orderpart?H0

? which is the Hartree?Fo ck Hamiltonian and a p ertubation?V? The exact energy is then expressedasanin?nite sum ofcontributionsof increasingcomplexity? Going b eyond

2KeilHF? SCF or MP2 is di?cult and matter for sp ecialists in quantum chemistryresearch ?for details see ?1?4? ??TheenergyER

ofamolecularsystemisobtainedasasolutionoftheelectronic part of the Schr?odinger equation for a ?xed con?gurationRof thenuclei ?Born?Opp enheimer approximation?:H?r?R???r? ?ER

?R ?r??1?The n?electron wave function?R

?r? describ es the motion of the electronsinthe?eldofthenuclei?Duetotheelectron?electroninteractiontermintheHamiltonian? thisequationcannotb esolvedwithoutapproximations?The HF approximation assumes that then?particle wave function??r? canb ewrittenasanantisymmetrized pro ductofone?electronfunctions?i

?r1 ??so?called orbitals?:??r1 ? r2 ? ???? rn 1 p n 1 ?r1 ??2 ?r1 ?? ? ??n ?r1 1 ?r2 ??2 ?r2 ?? ? ??n ?r2 1 ?rn ??2 ?rn ?? ? ??n ?rn

?2?Eq? ?2? is called a Slater determinant? The set of orbitals that yields thelowest energy of a molecular system in the sense of a variational principle isgiven by the following set of HF integro di?erential equations:F?r1

??i ?r1 ? ?Ei ?i ?r1 ??3?The orbitals?i ?r1 ? are called molecular orbitals ?MOs? andF?r1

? is theFo ck op erator which comprises the di?erential op erator of the kinetic energyand the electron?electron interaction term? The expansion of the MOs?i

intoa ?nite series of basic functions?i ?r?? i ?r1 ? ???r1 ?ci ? ??1 ?r1 ?? ?2 ?r1 ?? ? ? ? ? ?m ?r1 0 B BB? c 1ic 2i? ?c mi 1 C CCA

?4?leads to a transformation of the Eq? ?3? into the so?called Ro othaan matrixequations:FC?SCE?5?withF? h?jF?iFo ck matrix?6?S? h?j?ioverlap matrix?7?The HF solution is given as m column vectorsci

of co e?cients related toa chosen basis set?: Application of Numerical Metho ds in Pro cess Engineering3C? ?c1 ?c2 ? ? ? ? ?cm ?;Eij ?Ei ?ij ?8?The one?electron density function???r?? is given for this basis set asR? n Xk c k c ?k ?9?The Fo ck matrix elements are:F ?h?jh?i?G?R??? ?10?withG?R??? X? X? R ?h??j??i ? h??j? ?i??11?and h??j??i? Z? ?r1 ?r1 1 r12 ?r2 ?r2 ?dr1 dr2

?12?In practice billions of integrals of typ e ?12? have to b e calculated? storedand reread in each iteration? Eq? ?5? is a pseudo?eigenvalue equation b ecauseChas to b e known in b eforehand? Therefore? one starts with an assumption ofCand solves Eq? ?5? iteratively until convergence has b een achieved? Matrixeigenvalues aremostly computed according tothemetho d of Davidson ?5??In its bruteforce form the Hartree?Fo ckSCF metho d needsa computationof?N

4?N?Numb eroffunctionsinitsbasisset??r1???Duetosometricks one can reducethis numb er to?N

3? With the MP2 approach ab out?N

5integralshaveto b ecalculated?Moresophisticatedmetho ds need?N

7integral computations? Quantum chemical computations are a standardto ol?evenforexp erimentalists?seee?g??2?6?7???Blackb oxprogramsarecommercially available ?e?g? GAUSSIAN; TURBOMOLE??With thedevelopment of material science??nechemistry? molecular bi?ology and many branchesof condensed?matter physics? the problem of howtodealwiththequantummechanicsofmany?particlesystemsformedbythousands of electrons and hundreds of nuclei has attained relevance? An al?ternative of ab?initio metho ds is the density functional approach ?8?10? whichgives resultsof anaccuracycomparable to ab?initio metho ds? Thedensity?functional metho d bypassesthecalculation of then?electronwave?functionby using the electron density??r?? The energy of a many?electron system is aunique function of electron density? The computational work grows like?N

3instead of?N

4in HF?Unlike in classical physics? in quantum mechanics relativistic e?ects mayb e quite striking? esp ecially for heavy atoms in connection of quantum chem?istry? Pyykk?o ?11? gives a review of relativistic quantum chemistry?Theparallelization of a molecular quantum chemical co deis at leastinparts trivial? Esp ecially thecomputation of the integrals ?matrix elements?

4Keilcan b e done indep endently? Dep ending on the exact structure of the integralpackage? one easily develops a strategy for the typ e of granularity b est suitedfor a certain parallel computer?Quantum chemical computations are in use for the determination of ther?mophysical data? arrangement of molecules on solid catalyst surfaces? relativestability of zeolites etc?Monte Carlo metho ds are a class of techniques that can b e used to simu?late the b ehaviour of a physical system ?12?16?? They are distinguished fromother simulation metho ds such as molecular dynamics? by b eing sto chastic?that is? nondetermistic in some manner? This sto chastic b ehaviour in MonteCarlo metho ds generally resultsfrom the use of random numb er sequences?MonteCarlometho dsareusedfortheevaluationofmultidimensional in?tegrals with complicated b oundary conditions? wheregrid metho ds b ecomeine?cient? solution of di?erential equations ?e?g? Schr?odinger equation?? studyof systems with a large numb er of strongly coupled degrees of freedom? such asliquids? phase transitions? disordered materials? and strongly coupled solids?A striking application of Monte Carlo metho ds is the computation of di?usionand reaction pro cesses in zeolites ?17??For an N?particle system? the average of a functionF?q

N?? which dep endsonly on the con?gurational variablesq

N? ?q1

?q2 ? ? ? ? ?qN ?? is given byhFi? ZF?q N?p?q

N?dq?13?wherep?q

N?is theprobability density? Becauseof themany thousandsof degreesof freedom of thesystem? it is practically imp ossible to evaluateaverages from this integral? A solution to the problem is to limit the samplingto a con?ned region of the con?guration space? An algorithm which providesthisapproachistheMetrop olisalgorithm ?18?? Thisalgorithm generatesarandom walk in the con?gurational spacewith theconstraint that the con??gurations have to b estatesof an irreducible Markovian chain ?19?? Let ussupp ose to have N molecules in a given con?guration and con?ned in a cubicb ox? In the Monte Carlo pro cedurethe following stepsare p erformed manytimes: ?1? a molecule is chosen randomly; ?2? a trial move is attempted: themolecule is displaced and rotated around the three rotational axis? so that anew set of co ordinates is generated from the old one; ?3? the change in energyof the system ??E? due to the trial move is calculated; if the energyis de?creased? the move is accepted with probability equal to one? If the energy isincreased the move is accepted with probabilityexp???E ?k T? and rejectedwith probability ?1?exp???E ?k T???wherekistheBoltzmann constant?and Tisthetemp eratureof thesystem? Inshort?aMonteCarlo programrunslikethis:?1?input:T?Nandinitialcon?guration;?2?calculationofthe system interaction energy; ?3? generation of trial move co?ordinates; ?4?computation oftrialmove energydi?erence;?5?acceptanceorrejectionofthe move; ?6? up date of the co ordinates and energy arrays; ?7? after numb erof moves has b een completed: output of ?nal results? Steps ?1? and ?7? o ccur

Application of Numerical Metho ds in Pro cess Engineering5once? steps ?3? to ?6? take place inside a lo op over a given numb er of moves?In step?6?thetotal energyof thesystemis evaluatedaftereachmove? Inorder to avoid accumulation of round errors? step ?2? must b e done from timeto time? For a single move steps ?2? and ?4? take by far most of the computingtime? Therefore? these steps are worth mo difying to execute in parallel?Forthecomputationofdynamic prop ertiesofphysicalsystemswithavery high degreeof freedom the Molecular dynamics ?MD? approach is themost widely used simulation technique? Monte Carlo and Molecular dynamicsmetho dsarelinkedviastatisticalmechanics andtheassumption of ergo d?icity? Thus?if equilibrium prop ertiesof anergo dicsystem aredesiredtheneither of thesemetho ds may b e employed? However? if time dep endent?dy?namical? prop ertiesarerequiredthenMDsimulations arethesoleoption?An intro duction to MD is given in ?12? 20? 21?? MD calculates the motion ofan ensemble of atoms and?or molecules by integrating Newton?s equations?From themotion oftheensembleofatoms andmolecules microscopicandmacroscopic information can b e extracted?e?g? transp ortco e?cients? phasediagrams? and structural prop erties? The physics of the mo del is contained ina p otential energy functional for the system from which force equations foreach atom and molecule are derived? MD simulations are large in resp ecttothe numb er of time steps and the numb er of atoms and?or molecules?Let usconsideran ensemble of N molecules in a ?xedvolume V with a?xedtotalenergyE?Thisisamicro canonicalensembleofclassicalstatis?tical mechanics? Typical values forN usedin thesesimulations of chemicalinterest is of the order of hundreds to a few thousands? In order to simulatean?in?nite? system? p erio dicb oundaryconditionsareinvariably imp osed?Thus a typical MD system would consist of N molecules enclosed in a cubicb ox with each side equal to length L? MD solves the equations of motion fora molecule i:f

i Xj f ij

Xj?6?i?

? V ij ?q N? ?qi ?14?whereVij

is the pair p otential function b etween molecules i and j? A re?strictiontobinary molecular interactionsisnotnecessary?Thesummationover j is usually con?ned to the molecules within a spherical cuto? radius Rof particlei? Theradiusis usually chosento b ehalf thelengthof theb ox?For long rangeforces? suchasCoulombic or dip ol?dip ol forces?interactionsb eyond the cuto? are usually calculated according to the Ewald summationtechniqueorthereactionmetho d ?21?? For systemsconsistingof many dif?ferent typ esof complex molecules? many di?erenttyp esof p otential energyfunctions must b e evaluated at each time step? For a given pair of interact?ing particles? the appropriate p otential energy function and parameters mustb e identi?ed according to particle typ e? The evaluation of thetotal force isby far the most time consuming step? It requires the computation of a widerange of expressionswhich consistof terms containing fractional p owersor

6Keiltranscendental functions? The metho d of table lo ok?up and the so?called La?grangian particle tracking are imp ortant for calculating many di?erent typ esofinteractionsorarbitrarycomplexityinlargeMDsimulations?Optimalprogramming structures of these techniques have recently b een published byDunnandLambrakos?22??Theintegrationoftheequationsofmotionisdone by the Verlet? Verlet leap frog? Gear ?xed or variable time step or e?g?Gauss?Radau algorithm? A comparison of di?erent algorithms in MD simula?tions has b een published by Bolton and Nordhohn ?23?? The Gauss?Raudaualgorithm turned out to b e the most e?cient and accurate one?MD computations areinherently parallel ?24? and sp ecial?purp osehard?ware for MD calculations is also available?25??Numerical treatment ofchemicalreaction?kineticsInchemical reactionengineeringnumerical simulation andidenti?cation ofreactionsystems isof an outstanding imp ortance? Evaluating reactionrateparameters isa common problem for thechemical engineer?Basedonpro?p osed chemical mechanisms and carefully done measurements of ?ow rates?pressures?temp eratures and comp ositions the rate constants have to b e de?termined? Detailsofnumerical metho dstotacklethisproblemisgivenbyBo ck ?26? or Deu?hard and Nowak ?27? 28?? In general a system of chemicalreactions is describ ed by a set of di?erential equations which corresp onds to aprop osed chemical reaction mechanism? The set of di?erential equations eval?uates thenC

concentrationsCof the involved sp ecies? They may b e describ edas:?

C?f?C?k? t??15?The rate equations on the right?hand sides are p olynomial inC? In generalthey are nonlinear inCand linear in the rate parametersk?constant for T ?const??? The parameterskare to b e determined? They follow the Arrheniusequation:k

ij ?T? ?Aij T ijexp??Eij ?T??16?If the parametersAij ? ?ij andEij

are all known? the initial concentrationsandatemp eraturepro?learegiven?therateequationswouldpredicttheb ehaviour of the reaction? For very large systems a program LARKIN thatintegratesthe?ingeneralsti??systemofequations?27??Theinitialvalueproblems may b e solved by routines like METAN1 ?29? or SODEX ?30? 31??Both metho ds are based on a semi?implicit midp oint rule?Themathematical problem ofidenti?cationofrateparametersisaso?called inverse problem? which may b e stated like this ?26?:Find parameterskand a solutionC?k? t? of the di?erential equations?

C?f?C?k? t??17?

Application of Numerical Metho ds in Pro cess Engineering7such thatkr1 ?C0 ? ? ? ? ?Ck ?k?k 22
?M I N??18?constraints:r 2 ?C0 ? ? ? ? ?Ck ?k? ? 0?19?r 3 ?C0 ? ? ? ? ?Ck ?k??0?20?whereC?C?k? tj ?? withtj

known?This problem is a constrained overdetermined multip oint b oundary valueproblem? Bo ck ?26? has taken a multiple sho oting technique to solve this prob?lem? This time interval is devided into pieces ?e?g? according to measurementp oints?:t

0 ??0 ? ?1 ? ? ? ? ? ?m ?tk ?21?TheconcentrationsCvi at thesep ointsareadditional parameters? suchthat an augmented variable vector?Cv0 ? ? ? ? ?Cv0 ?k??22?is obtained? For a given estimate of this vector the solutionsC?Cv i?k? t?of m initial value problems on the time subintervals are computed:?23??

C?f?C?k? t?;t2??i

? ?i?1 ??C??i ? ?Cv

i?23??24?which leads to a discontinuous tra jectory? This approach reduces the in?u?ence of p o or estimates of the rate constants k? Information ab out C?t? takenfrom measurementscanb ebroughtin? Thus?theinitial tra jectoryiscloseto theobserveddata? A large constrainednonlinear least?squaresproblemsobtained ?26?:Find parameterskand concentrationsCvo

? ? ? ? ?Cvm ? such thatkR1 ?Cv0 ? ? ? ? ?Cvm ?k?k 22
?M I N??25?with constraints:R 2 ?Cv0 ? ? ? ? ?Cvm ?k? ? 0?26?R 3 ?Cv0 ? ? ? ? ?Cvm

?k??0?27?In order to ensure continuity of the ?nal solution? the additional smo oth?ing conditions have to b e metk

i ?Cvi ?Cvi?1 ?k? :?C??i?1 ?Cvi ?k??Cvi?1 ? 0?28?

8KeilThe least?squares problem has b een solved by a generalized Gau? ?Newtonmetho d ?26?53?? Thealgorithm of theinverseproblem of kineticparameteridenti?cation is available as a co de called PARFIT? Nowak and Deu?hard ?27?have develop ed a software package PARKIN for the identi?cation of kineticparameters?Inprinciple?theseprogramscanb ecombinedwithanyintegratorforsti? problems? Available co des are ?30? 31? e?g? EULEX? ODEX? DIFEX1 orSTIFF3 ?32?? Further information is available from E?mail address eZib?zib?b erlin?de and hairer ? uni 2a?unige?ch?Transp ortpro cessesMulticomp onent mass transfer combined with heat and momentum transp ortareomnipresentinchemical pro cessengineering?42?44?? Di?usion?reactionpro cesses have attracted many mathematicians and is a ?eld of mathematicalresearch of its own ?45?? Exemplary? di?usion and reaction in catalyst parti?cles will b e discussed in more detail? Catalytic gas?solid reactions take placewithin p oroussolid supp ortsin whichcatalytic activesitesarefound? Thereacting gases have to di?use into thesep orous solids? adsorb on the activesites? react? and the pro ducts have to di?use back to the outer surface? Firstof all thep orousmaterial has tob edescrib ed?Inthepastthis hasmostlyb een done by continuum mo dels? In a continuum mo del? the p orous materialis treated as a continuum within which temp erature? ?uid sp ecies concentra?tions are de?ned as smo oth functions of time and p osition? These p ointwisefunctions are governed by suitable energy and sp ecies concentration balances?Continuum mo dels are sets of ordinary or partial di?erential equations whichinclude ?e?ective? parameters like transp ort co e?cients ?di?usivity? reactiv?ity co e?cients etc??? A spatial averaging technique is used which is meaningfulwhenthecharacteristiclengthforthevariationinmacroscopicconcentra?tionsismuchlargerthanthelineardimensionofastatisticallyadequatematerial region? In many practical systems? the length scale over which theinhomogeneities are imp ortant is small enough to fully justify the continuumtreatment? Continuum mo dels cannot satisfactorily describ e the connectivityof the material from a global standp oint? The ma jority of continuum mo delsare capillary mo dels that treat the p ore space as a collection of capillaries ofoneormore radii eachof whichsatis?esthe?uxrelations foran in?nitelylong cylinder? The mo dels di?er from each other by the way that the ?uxesin capillaries of di?erent sizes are combined with each other? The multicom?p onentdi?usionismostlydescrib edbytheso?calleddustygasmo del ?46?which is based on the Stefan?Maxwell equations?In practice the active sites are not homogeneously distributed within thesolidp ellet?Furthermore?duetodep osits?e?g?coke?withinthep oresthestructureof the p oreschangeswith time? This demands a lo cal descriptionof the p ore space? At presentnetwork mo dels are preferredfor this purp ose

Application of Numerical Metho ds in Pro cess Engineering9?seee?g?Rieckmann andKeil?47??? Thesemo dels leadtoverylargelinearsystems of equations of ab out 10

6unknowns? For thesimple caseof a ?rstorder reaction A??B one gets a symmetric matrixAin the systemAx?b?29?whereA2?

?n?n?isadiagonal?dominant sparsematrix?Aandbaregiven? The unknown vectorxrepresentsthe concentrationsin the no desofnetwork? This problem can b esolved by e?g? preconditionedconjugate gra?dient?PCG?metho ds?48?50??Ingeneral?thenetworkmatrixAisnon?symmetric? In this case one has to refer to routines like Generalized MinimalResidual ?GMRES?? the BiConjugate Gradient ?BiCG?? BiConjugate Gradi?ent Stabilized ?Bi? CGSTAB?? Conjugate Gradient Squared Metho d?CGS?or Quasi?Minimal Residual ?QMR? or others ?51??Catalystdeactivation?e?g?dep ositionofcokeasafunctionoftime?changes the p orous structure with time? At a certain time the p ore networkcannot b e passed by the reactant gas? This is the so?called p ercolation thresh?old ?52?? Before this p ercolation threshold app earsp ore clustersof di?erentsizemay b efound within thenetworkand atleastoneclusterrangesoverthe entire p ellet? Percolation theory deals with the numb er and prop erties ofthese clusters?In the future network mo dels will b e completed by random p ore mo delsinwhichdi?usionandreactionproblemscanb etreatedbyMonteCarlometho ds only?Mathematical descriptionofunitop erationsMo delling of unit op erations like chemical reactors? multipass heat exchang?ersordistillation columnsleadstosetsof ordinaryandpartialdi?erentialequations sometimes augmented by algebraic equations? Under transient con?ditions one gets mixed systems of di?erential and algebraic equations ?DAEs??Numerical solutions of initial and b oundary problems of ordinary di?erentialequationsaregiveninmanyb o oks?seee?g??30?31?33?35???Numericalsolutions of partial di?erential equations ?PDEs? are e?g? given in b o oks byLapidus and Pinder ?36? or Ames ?37?? A recent compilation of literature onDAEs is given by Unger et al? ?38?? Mo dels for unit op erations are scatteredover many journals? Bo oks that refer to sp ecial typ es of unit op erations aree?g? for chemical reactors ?39? or separation pro cesses?40? 41??Phase distribution at equilibrium is an imp ortant problem in pro cess de?sign? Vap our?liquid equilibria can b e computed in a rather simple way? Butsingularities often arise as phase app ear and disapp ear for nonideal solutionsandatelevatedpressuresinthecriticalregion?Atagiventemp eratureTand pressure P? phase equilibria are calculated by minimizing Gibbs free en?ergy sub ject to the mass balance constraints? This is a nonlinear programing

10Keil?NLP? problem? Near two? and three?phase critical p oints? near phase b ound?aries? and with chemical reaction? when the phase distribution is uncertain?numericaldi?cultiesmayarise?Themostobvioussingularityapp earsata critical p oint? wheretwo or more phasescoalesce?For two? phasecriticalp oints? the singularity is expressed explicitly using the tangent plane distancefunction:g?y? ?

1 RT C Xj?1 y j ?uj ?y???j ?z???30?where g is the dimensionsless Gibbs free energy of mixing at comp osition

yminusthelinearapproximationtotheGibbsfreeenergyofmixingatcomp ositionzwhich is the equation of a tangent hyp erplane? and?j

is thechemical p otential of comp onent j? Ashasb eenshown by Michelsen?54? acritical p oint o ccurs aty?zat a givenz? for a binary mixture? wheng?z? ?

dg dy ???z d 2gdy 2 ???z d 3gdy 3 ???z ? 0; d 4gdy 4 ???z

?0?31?Similar expressions may b e found for multicomp onent mixtures and an al?gorithm is given to ?ndTc

andPc

where the second? and third?order terms arezero? Michelsen ?55? has also formulated nonlinear equations for the calcula?tion of phase envelop es and critical p oints for multicomp onent mixtures? Thenonlinear equations may b e solved by e?g? a di?erential arclength homotopy?continuation algorithm published by Allgower and Georg ?56? that integratesthe ODEs along the arclength using an Euler predictor step followed by New?ton correction steps? This approach was implemented by Wayburn and Seader?57? with attention to the metho ds for step?size adjustment and parameter?variable exchange to avoid singularities near limit p oints? Kovach and Seider?58? extendedtheturning?p oint algorithm tobypassthelimit p ointswhenmultiplesolutionsareencounteredasasecondliquidphaseisintro ducedon thetraysof aheterogeneousazeotropicdistillation tower? Awidely dis?tributed program package for the implementation of Newton and ?xed?p ointhomotopies is HOMPACK ?62?? A similar program called CONSOL has b eendescrib ed in a b o ok written by Morgan ?63?? Seader et al? ?64? have extendedthese metho ds to systems of nonlinear equations with transcendental terms?They have applied a global ?xed?p oint homotopy to ?nd from a single arbi?trary starting guess all solutions to several sets of nonlinear equations? evenwhen transcendental terms are present? However? dep ending up on the start?ing guess selected? the homotopy path may consist of branches that are onlyconnected at in?nities? which traverse the dep endent variables and?or the ho?motopy parameter from?1to ?1? By using mapping functions? the ?xedp oint homotopy path may b e transformed into a ?nite domain? whereinallsolutions lie on the path?

Application of Numerical Metho ds in Pro cess Engineering11Paloschi ?68? has presenteda hybrid algorithm that combines a Newtoniterationpro cesswiththecontinuationco dePITCON?69?? Acollectionofnonlinear mo del problems is given by More ?70??Ashomotopy?continuationalgorithmsfollowthesteady?statesolutionbranches? they bracket limit p oints? where the JacobianjJjis equal to zero?Butnormally theypassover higher?ordersingular p oints? e?g? Hopf bifur?cation p oints? By evaluating the eigenvalues of the Jacobian a lo cal stabilityanalysiscanb ep erformedalongthesolutionbranches?ande?g?theHopfbifurcation p oints can b e lo cated? Alternatively? the necessary and su?cientconditions for thesep oints canb esolved ?65?? At a Hopf bifurcation p oint?twocomplexeigenvaluesb ecomepureimaginary?thesteady?statebranchdestabilizes or remains unstable? and a branch of p erio dic solutions has b eencreated?In orderto tracethe p erio dicbranchesa very comprehensivesoft?ware package called AUTO is available ?66?? Kubicek and Marek ?65? give aprintout of their program DERPAR? Holo dniok and Kubicek have describ eda similar program DERPER ?67??The ab ove mentioned mathematical metho ds may also b e used to mo delunit op erations without phase transitions? In reaction engineering PDEs withtwo spatial variables are dominating? For time dep endent problems the timeis an additional variable? Nonlinear mathematical metho ds arealso gettingmore and more employed in chemical pro cess control? A survey of this sub jectis given by Bequette ?74??Stationaryandinstationarysimulation andoptimizationofchemicalplantsWidely distributed software packages like ASPEN

??75? and SPEEDUP ?76?make stationary and instationary pro cess simulations available for many en?gineers? An extensivesummary of this sub jectis given in a b o ok editedbySchuler?77?? Apioneeringb o okab outpro cess?owsheetinghasb eenwrit?tenbyWesterb ergetal? ?78?? In Germany theDIVA?77? 79? simulator forinstationary simulation of complete chemical plants is in industrial use?Achemicalpro cessplantconsistsofmanyunitop erationsconnectedbypro cessstreams?Eachpro cessunitmay b emo delled byasetofequa?tions ?ODEs? PDEs? DAEs? algebraic equations?? which include material? en?ergy and momentum balances? phase and chemical equilibrium relations? rateequations and physical prop erty correlations? These equations relate the out?let stream variables to the inlet stream variables for a given set of equipmentparameters? At present? there are three approaches of ?owsheet calculations:the sequential mo dular? the equation oriented approach and the simultaneousmo dular strategy?The equation oriented approach is the most straightforward? All pro cessequations are organized according to structure and ease of solution? Very e??cient metho ds exist for partioning and tearing large sets of algebraic equations

12Keil?see e?g? 80?? The most common simulation strategy in industrial environmentsis thesequential mo dular strategy? Here? equations are group ed in mo dulesaccording to the physical pro cesses they present? and the mo dules are solvedsequentially? following thewaymaterial ?owsthroughthepro cess?Duetorecyclesin?owsheets?iterationsarerequiredtoconvergethesteady?stateequationsofthepro cess?Thesimultaneousmo dularstrategyattemptstobridge the gap b etween equation ? solving and sequential mo dular strategies?It retains the mo dularitiy of the pro cess but allows more ?exible sp eci?cationof calculation pro cedures and additional conditions?Instationary simulation of chemical pro cessesleads to very large systemsof di?erential? algebraic?equations ?DAEs??Thesp ecialfeaturesof DAEshave b een outlined by Petzold ?81?? Instructive examples of DAEs are givenby e?g? Pantelides et al? ?82? and Byrne and Ponzi ?83?? An extensive surveyof the numerical solution of DAEs is given by Bo ck et al? ?84?? These authorsgive a list of sp ecial di?culties connected with the solution of DAEs:?very large implicit? resp? linear?implicit mo dels with several ten to hundredthousands of variables?sti? and highly nonlinear di?erential equations which requiresp ecial dis?cretization techniques?highly nonlinear algebraic constraints whose consistent initialization presentsdi?culties?discontinuities and non?di?erentiabilities of the mo del equations?implicitly de?ned delays e?g? due to transp ort in pip es?due to convective ?ow and di?usion spatially distributed reactive systemswhich have to b e suitably discretized?DAEsystemscanb eclassi?edaccordingtotheirso?calledindex?Theindex may b e de?ned as the minimum numb er of di?erentiations with resp ectto time that the system equations have to undergo to convert the system intoa set of ODEs? Thus? by de?nition any ODE system has index zero? Problemsof index onemay b e solved by some software packages suchas LSODI ?85?86? for linearly?implicit DAEs? DASSL ?87? for implicit DAEs? Both programsimplement the backward di?erence formulas ?BDF? of Gear and are availablein ODEPACK? DASSL provides an option for consistent initialization whichisnotveryreliable?Semiexplicitsystemswithindex?1canb esolvedbyLIMEX ?88?? An implicit Runge?Kutta co de is RADAU5 ?89?? Caracotsios andStewart ?90? have develop ed a robust numerical metho d for integration andparametric sensitivity analysis of nonlinear initial?b oundary?value problemsin a timelike dimension t and space dimension x? Mixed systems of PDEs andalgebraic equations can b e treated? Parametric derivatives of the calculatedstatesareobtaineddirectlyviathelo calJacobianofthestateequations?Initial and b oundary conditions are e?ciently reconciled? The metho d is ableto handle jump conditions induced by changes of equation forms at given t?values? or at unknown t?values dep endent on the solution? Transition p oints

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