Distributed Models in Plantwide Dynamic Simulation

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DistributedModels inPlantwideDynamicSimulatorsW. S.Martinson and P. I. Barton

Dept. of Chemical E ngineering, Massachusetts Institute of Technology, Cambridge, MA 02139

Modelingsupport for dynamic simulation of chemical-process flowsheets, which is ofsignificant alue for plantwide dynamic simulation using differential algebraic modelformulations, is to date ery limited when one or more unit models include partialdifferential equations. Seeral new techniques that proide modeling support for suchsimulations are presented. These techniques are based on a generalized characteristic

analysis and a differentiation index analysis of partial differential algebraic models.Theycan beused to uncoer systemsthat cannot besoled aspart of a dynamic simula-tion, and to determine whether or not the initial and boundary conditions supplied bythe modeler forma well-posed problem. In a network flow context, they can further beused to select, enforce, and adapt the boundary conditions as required to maintainautomaticallya mathematicallywell-posed problem. Each of these proidestime-saingsupport to thesystem modeler.

Introduction

Significant research over the past twenty years has pro-duced several highly developed software packages that aredesigned specifically for plantwide dynamic simulation. Ex-

.amples include SpeedUp Perkins and Sargent, 1982 , DIVA . Marquardt et al., 1987 , gPROM S Barton and Pantelides,

. .1994 , and A BACU SS Allgor et al., 1996 . These packagesfacilitate large-scale system simulation by isolating the engi-neer from numerical algorithms, code generation, and debug-ging, therebyleavinghimor her freeto concentrate on modelformulation and application. The feasibility of dynamic simu-lation-based activities such as optimal batch policy synthesis,parametric sensitivity studies, safety interlock design verifica-tion, control system design, and start-uprchangeoverrshut-down studies in an industrial setting often depends on the

productivity gains provided by this modeling support Long-.well, 1993 .

These existing packages deal very effectively with the dif-

.ferentialalgebraic equation DA E formulation of a plantmodel, which arises when spatial variations of dependentvariables in the unit operations are ignored. This formulationadmits a fairly general dynamic description of a chemicalprocessing system. Given such a model, a modern simulation

Correspondence concerning this article should be addressed to P. I. Barton.Present address of W. S. M artinson: Cargill Central Research, Cargill I nc., Mi n-

neapolis, M N 55440.

package typically advances the solution to the entire flow-wsheet using a single integration method such as DA SSL

. .Petzold, 1982 , DA SOLV J arvis, 1992 , or the DA SSL vari-

.ant DSL48S Feehery et al., 1997 , which employ a Gear-type.xvariable stepsize, variable-order BDF method . Modern soft-

ware packages also automatically perform calculations to, forexample, locate state eventsrimplicit discontinuities Park and

.Barton, 1996 , reinitialize the system after a state event .Mayer et al., 1995 , and integrate high-index systems .Feehery and Barton, 1996.

Sometimes a finer degree of detail is required than can beprovided by the DAE, or lumped, formulation. F or example,when the spatial variations of some dependent variablesacross a processing unit are important, that unit must be de-scribed bypartial differential equations. Similarly, populationbalances or polymer chain-length distributions in an other-wise lumped system also give rise to partial differential equa-

tions. A model that includes partial differential equations, andpossibly ordinary differential and algebraic equations as well,is referred to as a partial differentialalgebraic equation .PDAE , or distributed model.

Efforts to provide modeling support for distributed modelsin network simulations have so far focused on generatingandanalyzing a discretization of the partial differential equa-tions. Oh and Pantelides have addressed semidiscretizationin the context of network flow simulations by developing aninput language for generating discrete equationson rectangu-

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.lar domains Oh and Pantelides, 1996 . This input languageforms part of the larger gPR OM S dynamic simulation soft-ware mentioned earlier. The language allows the user to de-fine spatial coordinate axes and represent derivatives in eachcoordinate direction using either finite differences or colloca-tion on a monospaced grid along each axis. gPR OM S thenuses the specified grid and method to substitute an approxi-mating set of DA Es for the original PDE s, couples them to

the rest of the flow-sheet equations, and solves the entiresystem in the same manner as a purely lumped model.

TR IF IT is a mesh-generation language van der Wijngaart,.1994 . It consists of a set of operators for several common

manipulations of unstructured triangular two-dimensionalgrids. T hese operators can be used to refine or smooth suchgrids. Discrete approximations to partial differential equa-tions can also be expressed compactly using the operators.

The language, like the mesh-generation syntax in gPROMs,is designed to reduce the time required to generate a discretescheme for solving systems of PDEs. H owever, it is not inte-grated into a large simulation environment.

The GRI DOP package Liska and Shashkov, 1991; Liska.et al., 1994 provides similar tools for generation of conserva-

tive finite difference schemes on logically rectangular do-mains in an arbitrary number of independent variables. Thepackage takes as input a user-supplied definition of functionspaces and associated scalar products, together with user-supplied definitions of grid operators as finite differenceschemes. The user may then provide partial differentialequations in terms of the defined grid operators or the ad-

joints of those operators, and the package returns the finitedifference equations.

These tools are all designed to generate or evaluate a dis-cretization scheme for solving a distributed model. The even-tual goal is for these tools and others like them to advance tothe point where an engineer simply provides a distributedmodel, and the simulator will generate a numerical solution

along with rigorous guarantees of its accuracy. This wouldcorrespond to the current capabilities of process simulatorsfor dealing with lumped models.

One step on the path toward such full support of dis-tributed models by dynamic process simulators is automatedscreening for models that the simulator cannot solve. It issomewhat unreasonable to expect a simulator to solvea modelthat is mathematically ill-posed, for example. Similarly, DA Esthat are high index are not amenable to numeric integrationby standard integration codes. This article will focus on waysfor a process simulator to examine models generated by anengineer and identify ones that are mathematically ill-posedor may be expected to lead to high-index DA Es in method-of-lines solution techniques.

Possibly the first step toward development of a tool of thisnature was taken by Marquardt and coworkers, who havedemonstrated PDEDI S, a software package for rapid con-struction and evaluation of method-of-lines semidiscretiza-

tions for one-dimensional PDE systems Pfeiffer and M ar-.quardt, 1993. The software accepts as input a system of at

most second-order spatial derivatives and first-order timederivatives. It can symbolically discretize the spatial deriva-tive terms using either a finite difference or a weighted resid-ual method, retaining grid spacings or function weightsasun-

knowns. This symbolic form can be easily evaluated, given agrid and values for any parameters and dependent variablesrequired to calculate the coefficient matrices. This informa-tion isthen output toa fileand submitted toMA TL AB, wheretemporal eigenvaluesare calculated. Anypositive real part ofan eigenvalue indicates an unstable discretization.

As part of the preprocessing capabilities, PDEDI S is ableto characterize the system as hyperbolic, parabolic, or ellip-

tic; provide characteristic directions for purely hyperbolic sys-tems; identify self-adjoint spatial operators; and perform sev-eral other classifications of the equations. This informationallows consistencyof themodel to beevaluated, although only

basic consistency checks which are not detailed in the arti-.cle are implemented. However, this automated analysis of

the model equations as provided by the engineer is preciselythe type of technology that will be explored and developed inthis article.

The question of well-posedness of systems of parabolic andhyperbolic type is very well understood Courant and Hilbert,

.1962. Recent work has extended some of the classic analysisto more general linear PDAEs of neither hyperbolic norparabolic type, that may include purely algebraic equations

.Martinson and Barton, 2001 . The notion of the index ofdifferentialalgebraic systems has evolved steadily over the

past decade, and is also well understood Campbell, 1982;.Brenan et al., 1989 . T he concept of the index of a system of

partial differential equations has been the subject of a grow-ing amount of research Campbell and Marszalek, 1997; Mar-.tinson and Barton, 2000 .

The article will begin with several motivating examples ofdifficulties with particular dynamic simulation problems, fol-lowed by a very brief review of some current work in theareas of index analysis and well-posedness of PDAEs. Calcu-lations that can be performed bya simulator and the implica-tions of the results will then be presented. These calculationswill be applied to the examples in the following section, and

the results examined. The article will conclude with a discus-sion of directions for future work.

Motivating Examples

The following examples illustrate some of the difficultiesthat may arise when trying to perform a dynamic simulationusing models that involve partial differential equations. Theflowsheet sections are simple and are chosen to illustratespecific problems; they are not intended to belarge-scale casestudies.

Pressure-swing adsorption

Consider greenhouse gas removal from a nitrogen gas

stream by a two-column pressure-swing adsorption process.Part of the process flowsheet appears in Figure 1. A continu-ous high-pressure feed to the system is directed through oneof the columns, where greenhouse gases are removed fromthe nitrogen stream by adsorption onto a zeolite packing. Atthe same time, a low-pressure nitrogen stream is blownthrough the other columns to remove the adsorbed speciesand carry them to another treatment unit. When the packingin the high-pressure column approaches saturation, thehigh-pressure feed is switched over to the second column,

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and the low-pressure stream is switched to the first column.The process in repeated.

Here the engineersoverall task is to improve the operatingpolicy for the process, using dynamic simulation for as muchpreliminary work as possible, because the system cannot betaken off-line without major expense. Laboratory experi-ments have provided good values for the parameters in theKikkinides and Y ang model of pressure-swing adsorption

.processes Kikkinides and Y ang, 1991 , which describes col-

umn behavior under the assumptions of isothermal opera-tion, negligible axial dispersion and pressure drop, plug flow,instantaneous solidgas phase equilibrium, and perfect gasbehavior, all of which are judged to be reasonable for thisprocess.

Under this model, the adsorbate concentration on the solidq , mole fractions in the gas phase of adsorbate yis1...3 is1...3and inert y , and flow velocity u are related by the following4system of equations over time t and axial position in the ab-sorber z. Pressure P, pressurization rate P , temperature T,tbed void fraction , bed density , gas constant R, satura-Btion loadings qsat , and load-relation correlation constantsis1...3n and B are parameters. The values of these pa-is1...3 is1...3rameters have been experimentally validated for this process

3 RT Bq q P q u s0 i t ztP Pis 1

RT yB iy q q q P q uy s0, is13 .i i t i zt tP P

4

y s1 iis 1

1rnsat iq B y P .i i iq y s0, is13. 1 .i 1rnj31q B y P .js1 j j

The first equation is the total material balance. The secondequation is the material balance for each adsorbed species.

The third equation forces the mole fractions in the gas phaseto sumto unity. The fourth equation is the loading ratio cor-relation that gives the equilibrium loading of each adsorbedcomponent.

The project requires dynamic simulation of the system froma cold start. Initial conditions for the six differential variablesare

y 0, z s1.0=10y6, is13 .i

q 0, z s 0, is13, 2 . .i

while boundary conditions at startup are given by the feedcompositions y and velocity u s0f,is1...3 f

y t,0 s y , is13 .i f,i

u t,0 s u . 3 . .f

Partial derivatives with respect to z are discretized using afirst-order upwind finite difference scheme, and an implicit

Figure 1. PSA flowsheet.

BDF integration method is used to advance the solution for-ward in t. T he disappointing results appear in Figure 2. T hesimulation fails after a simulated time of 30 s, when the reini-

Figure 2. Simulationresults forthe PSA process.



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Figure 3. Vessel depressurization flowsheet.

tialization calculation required after the first valve positionchange does not converge.

What is wrong? The task facing the engineer is to figureout what is wrong, and do it as quickly as possible.

Compressible flow

The second example involves simulation of a vessel depres-surization. The simplified flow sheet for this process consistsof two pressure vessels, two valves, and the process piping,and appears in Figure 3. T he gas is compressible, and if fric-tion losses, gravity, and radial variations are ignored, and thegas is assumed ideal, flow is described by the Euler equations .J effrey, 1976; R oe, 1986

q u s0 . xt

12u q pq u s0 . t /2 x

h q upy uh s0 . .t x

ps y1 i .

12hs iq u . 4 .

2

Here is the fluid density, u is the flow velocity, p is pres-sure, h is the specific total energy, and i is the specific inter-nal energy. The first three model equations are conservationof mass, momentum, and energy, respectively. The fourth isthe ideal gas law, with a constant fluid heat capacity ratio of. The final equation relates total, internal, and kinetic en-ergy.

The pipe segment under consideration is 10 m in length, so0F xF10, and also let tG0. The initial and boundary condi-

tions are

0, x s79.6kgrm3 .

u 0, x s0.0 mrs .

p 0, x s2.76 MPa .

p t,0 s f t . .valve1

p t,10 s f t . 5 . . .valve2

Figure 4. Pipepressureprofile.

The first scenario of interest is a case where the pressure inthe pipe is initially slightly higher than the pressure in both

vessels. The pressure in one vessel is significantly higher thanthe other.

Again, the problem will be solved using a first-order up- .wind finite difference scheme Strikwerda, 1989 . Initially,

flow out of both ends of the pipe is expected, followed byestablishment of a steady pressure gradient and flow fromthe high-pressure vessel to the low-pressure vessel.

Simulation results, specifically the pressure profile alongthe pipe, appear in Figure 4. Clearly, something is wrong.

The calculated pressure profile blows up at the right end-point. One would expect a rarefaction to enter the pipe fromboth ends, followed by establishment of a steady pressuregradient between the two ends. I nstead, the calculated solu-tion blows up after less than 0.3 simulated seconds.

Possible problems include improper boundary conditions,an improper discretization scheme, a time step or mesh spac-ingthat is toolarge, and simplecode bugs. The engineer againfaces the task of uncovering the root of the problem and cor-recting it.

Electric power transmission

Next, consider simulations of 420-kV power transmissionlines in an electric power distribution grid. Current flow Iand voltage with respect to ground u over a transmission lineare described by the following simple system of two equa-tions, which are known as the telegraphersequations

0 L u 1 0 u 0 R uq q s0. 6 .

C 0 I 0 1 I G 0 It x

Here L , C, R, and G are the inductance, capacitance, re-sistance, and conductance of the line per unit length.

The scenario of interest is a 1% increase in current de-mand occurring over 0.5 s, to be delivered over a 10-km line.For this particular line, L s0.0046 srkm, Cs6.5 nFrkm,G s33.3 1r km, and Rs0.030 rkm.



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Measured values at the substation are 380 kV at 50 Hz,with a typical current demand of 3,160 A. These values willbe used for boundary conditions. The current demand will begiven as a sinusoid increase from 3,160 to 3,192 over 0.5 s

u 0,t s190,000)sin 50t . .

w xI 0,t s 1.0q0.005 1.0qsin 2tq1.5 3,160. 7 4 . . . .

The domain is a 10-Km line, and the simulation will cover

the surge in demand, so 0F xF10 and 0F tF0.5.Here, the engineer wants to build the complexity of the

simulation slowly, and therefore begins with a simplified form .Massobrio and A ntognetti, 1993 of the telegraphers equa-tions that neglects the line inductance, resistance, and con-ductance

0 0 u 1 0 uq s0. 8 .

C 0 I 0 1 It x

While these assumptions behind this simplification are notvalid for this system, experience with chemical process simu-lations has taught the engineer to start with simplified mod-els, and move to simulations based on more rigorous models

once the simulation based on a simplified model is working.The partial derivative terms in x are discretized using cen-tered finite differences, and the line voltage is initialized to190 kV. Simulation results for the simplified model appear inFigure 5. The results look good, so the engineer proceeds tothe full model.

The partial derivative of current with respect to time, whileabsent fromthe simplified model, is present in the full model.

The engineer initializes the current in the line to its nominaldemand of 3160 A. Resultsfor the full current delivery modelappear in Figure 6. The simulation blows up immediately.Once again, the task is to determine what is causing the sim-ulation to fail.

ReviewWe consider a first-order PDAE system with u the depen-

dent variables, and two independent variables x and t.The differentiation index with respect to t, , or simply thet

.indexwith respectto t, is defined M artinson and Barton, 2000as the minimumnumber of times some or all of the equationsmust be differentiated in order to determine u as a continu-tous function of u, x, and t. The index with respect to x isdefined in an analogous manner.

wThis particular definition of the index as opposed to per-turbation, modal, or algebraic indices Campbell and Marsza-

.xlek, 1997 is a natural generalization of the differentiation .index of a DA E Brenan et al., 1989 . As such, the index with

respect to t provides insight into the expected index of any

DAE that is generated by a method of lines semidiscretiza-tion. It also allows algorithms based on the index of a DAE

to be applied with only minor modification to PDE s Martin-.son and Barton, 2001 . The index is important because high

.index 2 or greater DA Es cannot be solved accurately by .standard numeric integration codes Brenan et al., 1989 ; the

calculated solution may fail or, worse, drift away from thetrue solution with no indication that specified error toler-ances had been violated. Index analysis may also be used toassist with the task of proper Cauchy data formulation

Figure 5. Simulation results for simplified elec-trical-current model.

.Martinson and Barton, 2000 . No classic analysis exists forthis problem, because it is trivial in the case of a strictly hy-perbolic or parabolic system; however, it can become signifi-cantly more complex for the general PDAE models consid-ered in this article.

A PDE system is said to be well-posed if it has a uniquesolution that depends continuously on its data Lieberstein,

.1972 . T he model equations typically admit a family of solu-tions, and proper initial and boundary conditions must beprovided in order to specify a unique member of that family.If nosolution exists, or it is not possible to determinea uniquesolution, one cannot expect a standard numerical code to

generate meaningful results. If the solution does not dependcontinuously on its data, tiny errors in initial or boundaryconditions may govern the computed solution. Models thatdo not depend continuously on their data are therefore notsuitable for dynamic simulation.

Consider a first-order linear PDE system over two inde-pendent variables t and x of the form

Au q Bu qCus f t, x 9 . .t x



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Figure 6. Simulation results for full electrical-current

model.

on the semi-infinite domain aF xF b, tG 0, with A, B, Cg

n=n, ugn, and f:2n.If A is invertible, multiplication on the left by Ay1 pro-

duces

u q Bu qCus f t, x . 10 . .t x

The solution to such a system depends continuously on itsw .xdata and is hyperbolic Courant and Hilbert, 1962 if B is

diagonalizable with strictly real eigenvalues Kreiss and.Lorenz, 1989 .

Let L be a matrix that consists of the left eigenvectors ofB, and let be a diagonal matrix that contains the corre-

y1sponding eigenvalues, so that LBL s . If Cs0, multipli-cation of the system on the left by L and introduction of new

variables s Lu produces a system of decoupled for C/0,the equations are coupled, but the implications for boundary

.condition placement are unchanged equations of the form

q s f t, x , 11 . .i i i it x

which is equivalent to an ODE along dxrdts . A s such, aniinitial condition on determines a unique solution. This isithe characteristic form of the hyperbolic system.

Let there be n eigenvalues of B greater than zero, and n1 2

less than zero. The solution to a hyperbolic system exists andis unique if n independent initial conditions are provided atts0, n independent boundary conditions are provided on1xs a, and n independent boundary conditions are specified2on xs b.

This analysis can provide information on all three compo-nents of well-posedness, and furthermore because it reliesonly on calculation of eigenvalues and eigenvectors of a ma-trix, it is amenable to implementation in a chemical-processsimulator. However, the analysis applies only to systems withA invertible and B diagonalizable. This precludes analysis ofmore general, nonhyperbolic systems. In particular, algebraicequations or equations that only involve partial derivativeswith respect to x make a system nonhyperbolic.

.More recent work Martinson and Barton, 2001 has ex-tended this analysis to a much broader class of nonhyperbolicsystems. First, it has been proven that a system of the form

.given earlier Eq. 9 depends continuously on its data if all .generalized eigenvalues of the matrix pair A, B are strictly

real and have geometric multiplicity 1. The proof also allowslocal consideration of quasi-linear systems byfreezing the co-efficient matrices. I f some generalized eigenvalues of the ma-trix pair are strictly real, but infinite and of geometric multi-plicity 2, the solution has also been shown to depend continu-ously on its data.

This work also developed a canonical form that may bethought of as a generalization of the characteristic form of ahyperbolic system. If the coefficient matrix pair forms a regu-

lar pencil, then the system is equivalent to one of the form

1 1J IN I q1 2 2

N2I 3 3t x

f t, x .1U f t, x .qC us , 12 .2

f t, x .3

where J is a lower J ordan matrix, and N and N are lower1 2J ordan matrices of nilpotencies and , respectively.1 2

Each subblock in one of the three block rows has beenshown to be equivalent to an ODE system along a particular

.direction in the t, x plane. The direction is given by the .generalized eigenvalue , that corresponds to the sub-i i

block. For the first block row, called the hyperbolic part, thenumber of boundary conditions that are required on xs aand xs b are again equal to the number of positive and neg-ative generalized eigenvalues, respectively, that are associ-ated with the hyperbolic part. The number of initial condi-tions is equal to the dimension of the hyperbolic part. If any



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associated generalized eigenvalues have nonzero degeneracy,which is defined as one less than the geometric multiplicitythe geometric multiplicity of a generalized eigenvalue is de-fined as the dimension of the associated J ordan block; also,the overall degeneracy of the system is defined as the maxi-

.mum degeneracy of any generalized eigenvalue , the systemdoes not depend continuously on its data.

No boundary conditions are required for the third blockrow, which is called the differential part. The number of initial

conditions is equal to the dimension of the differential part.If any associated generalized eigenvalues have nonzero de-generacy, the system again fails to depend continuously on itsdata.

The total number of boundary conditions that are requiredfor the second block row, which is the parabolic part, is equalto its dimension. The boundary condition associated with ageneralized eigenvalue of degeneracy zero in the parabolicpart can be enforced at either xs a or xs b. No initial con-dition is required for a subblock associated with a general-ized eigenvalue of degeneracy zero. F or a generalized eigen-value of degeneracy one in the parabolic part, if the index ofthe entire system with respect to t is less than two, the solu-tion can depend continuously on its data and the problem

can be well posed only if the two associated boundary condi-tions for the subblock are not enforced at the same point,and an initial condition is also specified.

.A system for which the coefficient matrix pair A, B doesnot form a regular pencil such as in the presence of alge-

.braic equations , but that is equivalent to an algebraic systemcoupled to a PDE with a regular coefficient matrix pencil

A 0 u B 0 u C 0 u11 1 11 1 11 1q q

A 0 u B 0 u C C u21 2 21 2 21 22 2t x

f1s , 13 .

f2

. ..where dim C s ny r, r s max rank Aq B , with22 g .A , B regular and C invertible, can be handled in the11 11 22same manner as one with a regular pencil. Because the firstblock row involves only u , it can be considered independ-1ently of the second block row. Again assuming a dynamicsimulation based on a time-evolution method, the first blockrow provides the same information regarding dependence onand location of data given bygeneralized characteristic analy-sis in the regular coefficient matrix pencil case. Once the firstblock rowis solved for u , no additional data are required to1uniquely determine u . The second block row is therefore2called the algebraic part of the system.

A differential system that is equivalent to an algebraic sys-tem can also be coupled to a regular PDE and handled in thesame way. Let N and N be two conforming nonzero nilpo-1 2tent matrices, both either strictly upper triangular or strictlylower triangular

A 0 u B 0 u C 0 u11 1 11 1 11 1q q

A N u B N u C I u21 1 2 21 2 2 21 2t x

f1s . 14 .

f2

An important special case is systems that contain one ormore strictly algebraic equations. An algebraic equation con-strains the dependent variables on eery surface in the inde-pendent variable space, so a system that contains an alge-braic equation can be viewed as one for which every surfaceis characteristic. This corresponds to A s B s0 in the21 21

.form just considered Eq. 13 .If an algebraic equation is differentiated once with respect

to time, it becomes an ordinary differential equation. This is

an interior partial differential equation on surfaces of theform xs constant, such as the domain boundaries. In otherwords, differentiation with respect to t transforms an alge-braic equation, which constrains the solution on all surfaces,to one that constrains the solution on domain boundaries ofthe form xs constant. If one is interested in the equationsthat partially determine the solution u on a domain bound-ary, the original and differentiated algebraic equations areequivalent. Because it may be difficult to identify what vari-ables belong to the algebraic part, differentiating all alge-braic equations once and analyzing the resulting system is aviable alternative if it produces a regular coefficient matrix

.pencil. It has been proven Martinson and Barton, 2001 thatif the differentiation index with respect to either t or x is

zero, or equal to one subject to a mild rank condition, thatdifferentiating the algebraic equations indeed regularizes thecoefficient matrix pencil.

For semilinear and quasi-linear systems, boundary condi-tion requirements, dependence on data, and the index arelocal properties that maychange with different values of x, t,or u. The boundary-condition requirements can be deter-

.mined by evaluating or freezing the coefficient matrices at .some nominal value u , x , t of interest, and examining0 0 0

the canonical form of the resulting linear system. If the sys-tem has a parabolic part, additional assumptions are re-quired, because the boundary-condition analysis is inherently

.nonlocal. It has been proven Martinson and Barton, 2001that the local dependence on data also can be determined by

the generalized eigenvalues of the frozen coefficient system.

Implementation

The goal of this work is to automate the analyses of theprevious sections as much as possible. In particular, determi-nation of the index, degeneracy, characteristic directions, andvariables associated with the subsystems of the canonical formwill allow a simulator to verify initial and boundary condi-tions, identify systems of high index with respect to the evolu-tion variable t, and detect some ill-posed systems.

Difficulties with direct calculation of the canonical form of .a DA E Bujakiewicz, 1994 and a desire to develop methods

that can be used for nonlinear problems have led to the de-

velopment of structural index algorithms Kroner et al., 1992;.Pantelides, 1988 . These algorithms work with the occurrence

information to determine the minimum number of differenti- .ations required to produce a low-index zero or one system.

It is well known that DA Es of high index due to numericalsingularities may escape detection by structural algorithms.

.Recent work Reisszig et al., 2000 has highlighted the factthat structural algorithms may also oerestimate the numberof differentiations required to produce a low-index system.However, the low computational cost of these algorithms and



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their applicability to nonlinear and large, sparse systems al-lows them to be used with considerable success in practical

applications if new algorithms emerge that provably performthis analysis properly, then they can be applied directly and

.the answer will be unambiguous .A second algorithm, called the method of dummy deriva-

.tives Mattsson and Soderlind, 1993 , has been used success-fully in conjunction with Pantelides algorithm to generateautomatically a low index system that is mathematically

.equivalent at least locally to the original system and explic-itly preserves all constraints. From this dummy reformulationof the original system, one can obtain the dynamic degrees offreedom, which is equal to the number of differential vari-ables. Note that this number may be correct even in the casewhere the number of differentiations has been overstated bythe structural algorithm.

Both algorithms can beapplied in an extremely straightfor-ward manner to PDEs. The index with respect to t, for exam-ple, is determined by considering all interior partial differen-tial operators together with algebraic operators. The inci-dence matrix for t-algebraic occurrences of the dependentvariables is formed by simply merging the incidence matricesfor u and u. Once this has been done, the two algorithmsx

.will in the absence of numerical singularities produce anequivalent system of index 0 or 1 with respect to t that re-flects the true number of t-differential variables. The numberof initial conditions required in order to determine a uniquesolution is equal to the number of t-differential variables inthe t-dummy reformulation. An analogous x-dummy refor-mation is evident.

The most basic necessary condition for well-posedness of alinear system is the regularity condition of Campbell and

.Marszalek 1997 . A system that does not possess an outputset assignment will not satisfy this necessary condition. Pan-telides algorithm also identifies systems that fail to meet thisnecessary condition as a result of structural singularity, as apreprocessing step that guarantees the algorithm has finite

termination. If the system is linear and has the general form .considered earlier Eq. 9 with Cs0, and if any generalized

eigenvalues of the coefficient matrix pair are given by 0r0,the system also fails the regularity condition.

Routines that calculate the generalized eigenvalues andtheir degeneracies for regular coefficient matrix pairs are

readily available Golub and van L oan, 1989; Demmel and.Kagstrom, 1993 . If any generalized eigenvalues are complex,

the system is ill-posed. Otherwise, if the degeneracy of thesystem is zero, the solution depends continuously on its data.If the degeneracy of the system is nonzero but the forcing issimple, the system is weakly well-posed. For linear forcingand nonzero degeneracy, it is not in general possible atpresent to distinguish between weakly well-posed and strongly

ill-posed systems.Index analysis can be used to identify the total number of

boundary conditions required to determine a unique solu-tion. J ust as index analysis with respect to t gives the numberof dynamic degrees of freedom on surfaces of the form tsconst., index analysis with respect to x gives the number ofdynamic degrees of freedom on surfaces of the form xsconst. In a dynamic simulation with t as the evolution vari-able, all such degrees of freedom on surfaces of the formtsconst. must be specified as initial conditions, while dy-

namic degrees of freedom on surfaces of the form xsconst.can be specified on either xs a or xs b.

The distribution of these boundary conditions between theboundaries xs a and xs b can be ascertained from the gen-eralized eigenvalues. Each block in the hyperbolic subsystemwas shown to be equivalent to an ODE along a particular

.direction in the x, t plane, given by dxrdtsr. Because ai idynamic simulation in t is assumed, data provided at t may2not be used to specify a unique solution at t -t , so initial1 2

conditions for these ODEs must be provided as boundaryconditions on xs a for ODE s along dxrdt)0, and asboundary conditions on xs b for ODEs along dxrdt-0.

Blocks in the parabolic subsystem are equivalent to ODEsin x, or alongthe direction dtrdxs0. An initial condition forsuch an ODE can in general be given at either domainboundary in x. In particular, a parabolic block of dimension1 requires a boundary condition at either xs a or xs b.

If the only blocks with nonzero degeneracy are part of theparabolic subsystem and of dimension 2, and the index of thesystem with respect to t is 1, the solution to the parabolicblocks will not depend continuously on their data if thosedata are enforced at a single side of the domain in x. Such aproblem may still be well-posed as an evolution problem in t

if one boundary condition is enforced at each side of the do-main in x for every parabolic block of degeneracy 1.

By the same approach but with the roles of t and x re-versed, if the only blocks with nonzero degeneracy are part ofthe differential subsystem and the index of the system withrespect to x is 1, the solution will not depend continuouslyon its data if those data are enforced at a single surface. Asan evolution problem in t, the problem is therefore ill-posed.

It is possible to move beyond simply counting the numberof required boundary conditions and to identify the informa-tion that those boundary conditions must provide. The matri-ces P and Q that transform the system to its generalizedcharacteristic formcan becomputed stably only when the de-generacy of the system is zero; when the degeneracy is

nonzero, stable similarity transforms exist that take both Aand B to upper triangular matrices Demmel and K agstrom,

.1993 . While not the characteristic form of the system, thisgeneralized upper triangular form can be used in the samemanner as the characteristic form for a more detailed bound-ary-condition analysis.

Consider nowa linear system in generalized upper triangu-lar form the generalized characteristic form can be used in-

.stead if available

PAQ q PBQ sy PCQq Pf t, x . 15 . .t x

. .Let s PAQ and s PBQ . Because the coefficienti ii i iimatrix pencil is assumed regular, it is not possible for si is0, and thus an output set assignment of to equation i isiimplied. Given this output set assignment, each dependent

.variable is given as the solution to a possibly degenerateone-way wave.

A dynamic simulation implies advancing a solution forwardin t. The values of the dependent variables , for which theiassociated characteristic direction r is nonpositive, arei idetermined at xs a by the outward-directed characteristics.Similarly, values associated with characteristics that have



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6. If the number of boundary conditions at xs a is lessthan the number of positive generalized eigenvalues, or thenumber of boundaryconditions at xs b is less than the num-ber of negative generalized eigenvalues, the solution is notunique, and the problem is ill-posed.

7. If any eigenvalue given by r, /0 has degeneracy 1,and -2, the solution does not depend continuously on itsxdata, and the problem is ill-posed.

8. If any generalized eigenvalue given by r, s0 has

degeneracy 2, and -2, the solution may depend continu-tously on itsdata only if one boundary condition for the corre-sponding block is enforced at each domain endpoint, and aninitial condition is also specified.

Again note that this analysis applies rigorously only to lin-ear systems. Extensions based on local linearization can bemade to semilinear and quasi-linear systems, but very fewgeneral statements can be made about truly nonlinear dis-tributed unit models.

Demonstrations

Pressure-swing adsorption

Could the analyses outlined in the previous section enable

a simulator to provide some insight into the cause of the dif-ficulties with the pressure-swing adsorption simulations? Thefirst step is estimation of the index. Pantelides algorithm dif-ferentiates the isotherm once before terminating, indicatingthat the index of the system with respect to t is 2, and thusimmediately pointing to the underlying cause of the simula-tion failure. The original system had a high index with re-spect to t, which was preserved by the method-of-linessemidiscretization, and thus produced a high-index DAE.

The simulator could provide an equivalent dummy refor-mulation of the original PDE that had index 1 with respect tot. There are two possible dummy reformulations; one is

3 RT BXq q P q u s0 i t zP Pi s1

RT yB iXy q q q P q uy s0, is13 .i i t i zt P P

4

y s1 iis 1

1rnsat iq B y P .i i iq y s0, is13 19 .i 1rnj31q B y P .js1 j j

3 31rn Xj 1rn sat 1rnj i1q B y P q q q B P y q B P . j j i i j i i

/ /js1 js 1

1w1r n y1.xi= y y s0, is1...3.i it /ni

By item 3 in the analysis of the equations, only three initialconditions should be enforced.

Discretizing this system using the same upwind finite dif-ference scheme and employing the same BDF integrator in

.time produces a low-index DA E. Once the redundant initial

Figure 7. Simulation results for reformulated problem.

conditions on q are eliminated, the solution proceedsis1...3normally. Results for the first few operating cycles appear inFigure 7.

In this case automated model analysis is able to immedi-ately identify the root cause of the simulation failure. Fur-thermore, a simulator would be able to correct the underly-ing problem automatically, with no intervention on the engi-neers part.

Compressible flow

What about the difficulties with the compressible flow sim-

ulation? Can a process simulator use these tools to help getthis simulation working?

In quasi-linear form, the model equations are

1 0 0 0 0 u 0 0 0 uh 0 0 0 p

0 0 0 0 0 h0 0 0 0 0 i t

u 0 0 0 2u 2u 1 0 0 u

q py uh py h u y u 0h0 0 0 0 0i0 0 0 0 0 x

000

s . 20 .py y1 i .1

2iy hq u2



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Pantelides algorithm, applied to determine the index withrespect to t, locates no structurally singular subsets of equa-tions. The index with respect to t is in fact 1. No dummyreformulation is necessary.

Differentiating the algebraic equations with respect to tproduces

1 0 0 0 0 u 0 0 0 uh 0 0 0 p

1y i 0 1 0 1y . . hi0 u 0 y1 1 t

u 0 0 0 02u 2u 1 0 0 u 0

q s .p 0y uh py h u y u 00h0 0 0 0 00i0 0 0 0 0 x

21 .

The system is quasi-linear, so the coefficient matrices must

be frozen at a point of interest. Consider the domain bound-ary at xs10, and let conditions at xs10 be s79.6 kgrm3,us0.00 mrs, ps2.76 MPa, hs86.6kJ, and is86.6kJ . Thefrozen-coefficient matrices are submitted to an eigensolver,such as the LAPACK routine dgegv. The result is three char-acteristic directions parallel to the t coordinate axis and twocomplex characteristic directions.

The system is thus ill-posed in a neighborhood of thesenominal values, and cannot be solved by a simulator as partof a dynamic simulation. A process simulator could then ad-visethe engineer that the equations, as entered, are ill-posed,at least in the vicinity of the initial conditions. On review of

.the input, the sign error made in the energy balance Eq. 4should be corrected

h q uhq up s0. 22 . . .t x

The analysis can then be repeated for the corrected sys-tem. Now, all generalized eigenvalues are strictly real. Threeare zero: the other two are"220.3. The corrected problemiswell-posed.

Simulation resultsfor the corrected problem appear in Fig-ure 8. A s expected, a rarefaction enters the pipe from bothends. This time, the simulation failure was the result of asimple sign error on the engineers part. This sign error pro-duced a strongly ill-posed system, which can be detected byaprocess simulator through the use of the analyses developedin this article.

Electric power transmission

Could the automatable analyses presented in this articlehelp uncover the cause of the electric power-line simulationfailure? The index of the system with respect to both t and xis zero; Pantelides algorithm would correctly return no dif-ferentiations. Therefore, no reformulation is necessary. Thecoefficient matrices are linear and have two generalizedeigenvalues, "182,879, each of geometric multiplicity 1. The

Figure 8. Corrected pipe pressureprofile.

problem, as the engineer has defined it, is thus ill-posed, be-cause the two boundaryconditions enforced at the substationdo not determine a unique solution. In this case, it means

that the engineer must obtain data from another substationat the other end of the line, in order to provide the requiredboundary condition at that end of the domain.

Also, once these measurements have been taken, the char-acteristic speeds give a time-step size restriction. For a finitedifference scheme, the time step must be limited by a CFL

.condition Strikwerda, 1989. Here, that restriction is tF xr182,879.

Why, then, did the simplified model work so well? Analysisof the simplified model shows that the index with respect to tis 2. No initial conditions may be arbitrarily specified. Initial-izing u at an inconsistent value caused the small initial jumpin current shown in the simulation results. So, there was infact a problem with the simplified model, but it was less seri-

ous than the outright failure that befell the simulation basedon the full model. Also, the canonical form of the simplifiedsystem consists of a single degenerate parabolic block withsimple forcing. Two boundary conditions at the same domainendpoint therefore do determine a unique solution of thesimplified model. Finally, there is no CFL condition limitingthe time step.

The generalized eigenvectors form the transformation ma-trices P and Q

y1.19E y3 1.00P s

1.19E y3 1.00

y4.21E q2 4.21E q2

Qs 23 .5.00E y1 5.00E y1

that take the system to its canonical form

y5.47E y6 1 q t x5.47E y6 1

y1.40E q4 1.40E q4q s0. 24 .

y1.40E q4 1.40E q4



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Several things are apparent from the canonical form. Asnoted from the eigenvalues, one boundary condition must beenforced at each end of the domain. Furthermore, the math-ematical properties of the simplified model are verydifferentfromthose of the full model. The simplifiedmodel is parabolicand equivalent to an ODE in x, while the full model is hy-perbolic. The analyses uncover these differences, and can beused to provide very understandable feedback to the engi-neer; specifically, that one boundary condition at the left do-

main endpoint must be removed, and one boundary condi-tion must be enforced at the right endpoint. T his means go-ing out into the field and obtaining a new set of measure-ments at a new location, or inferring new information fromexisting data.

Compressibleflow reisited: Adaptie boundary conditions

The boundary-condition evaluation method described ear- .lier Eq. 18 can be modified slightly to create a method by

which a simulator could automatically adapt boundary condi-tions as required to form a well-posed problem.

. The Courant-Isaacson-Rees CIR scheme Courant et al.,.1952 solves hyperbolic partial differential equations using a

linear finite difference approximation to the characteristicform of the model equations. Consider a quasi-linear hyper-bolic system in t and x over the domain 0F xF1, tG0:

u q B u, t, x u s f u, t, x . 25 . . .t x

Let the domain be discretized into a set X of equispacedpoints, and let x g X be a particular point in that set. Initiali

.data give the values of the dependent variables u x , 0 .iThis scheme evaluates the coefficient matrix B at each

node. For example, consider the ith node in Figure 9. Thefrozen coefficient system is

u q B u 0, x , 0, x u s f u 0, x , 0, x . 26w x w x . . .t i i x i i

Now, let L and contain the left eigenvectors and thew . xeigenvalues of B u 0, x , 0, x , respectively, so the charac-i i

Figure 9. Stencil for CIR scheme.

teristic form of the frozen coefficient system is

duL s Lf u 0, x , 0, x along diag Idxw x . .i i

dt

sdiag dt . 27 . .

.This system Eq. 27 is then used as an approximation tothe system after a small increment h in time t. Using theexplicit Euler finite difference approximation to the direc-

tional derivative along each characteristic given equations ofthe form

u h, x y uU .i il s l f u 0, x , 0, x , 28w x . .i i i i /h

where uU is the vector of values of u at the foot of the ithicharacteristics of the frozen coefficient system, calculated byinterpolation between values at grid points on ts0. For ex-

U .ample, in Figure 9, u s u x , 0 is the value at the foot ofa acharacteristic a.

U U .Let s l u and g s l f u , 0, x . Then the equationsi i i i i i i i .that give the value of u x , h arei

Lu x , h s q hg. 29 . .i

This is the CIR scheme. For linear systems with simple orlinear forcing, thecoefficientson theleft- and righthand sidesare constant, so calculating new values after a time step ateach node only requires solving the same system with multi-ple righthand sides.

Performing the same approximation at a boundary node,but retaining only the outward-directed characteristics, pro-ducesthe system that partially determines the solution at that

.boundary Eq. 18 . I f the characteristics associated with eachline in that system are traced back from the next time tq hto the current time t, and interpolation is used to determinethe values at the feet of those characteristics, the righthand

side is given in the same manner as in the CI R scheme, andis depicted graphically in Figure 10.

Performing Gauss elimination with row and column pivot- .ing on this possibly underdetermined system gives a number

of pivot variables that are determined by the characteristicinformation. The simulator could take this information, to-

Figure 10. Modified CIR scheme forboundary point.



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gether with theflowsheet topologyand a specification of whatvariablesrefer to the same quantities in different unit modelsfor example, in the pipe model refers to the same quantity

.as in the model for valve 1 , and attempt to set DirichletAconditions on the remaining variables by equating values atthe boundary to those in the adjacent unit, in order to formafully determined system.

For this problem, consider use of this adaptive boundary-condition scheme at the pipe ends, together with a Godunov

.scheme Godunov et al., 1962 using R oes Riemann solver .Roe, 1986 on the domain interior. Usingthe LAPACK rou-tine rgg to solve the generalized eigenvalue problem, and al-lowing the quantities that appear in both the pipe and thevalve models to be u, , p, and i, the method described isable to adapt the boundary conditions as needed to maintaina well-posed problem.

Possible characteristic directions at the domain endpointsand corresponding boundary-condition regimes appear inFigure 11. Three characteristics directed into the domaincorrespond to supersonic flow into the pipe at that end, andthree boundary conditions are required. Two characteristicsdirected inward and one outward occurs when flow entersthe pipe at subsonic conditions, and two boundary conditions

are required. One characteristic directed inward correspondsto subsonic flow out of the pipe, which requires one bound-ary condition. Finally, no inward characteristics represents

.supersonic or choked flow out of the pipe, and no boundaryconditions are required. The conditions at the two ends ofthe pipe may occur independently in any combination. Be-cause it is based on the characteristics, the modified CI Rscheme at the boundary together with the boundary condi-tion selection method can correctly adapt to any combinationof these flow regimes.

The pressure profile appears in Figure 12. The dual rar-efaction shown earlier in the short-time profile is replacedquickly by the evolving quasi-steady pressure gradient.

.The boundary condition changes at the left end xs0 ap-pear in Figure 13. The short-time results appear in the bot-tomframe, and resultsfor the entire simulation appear in the

.upper frame. The method correctly adapts from one to .two and i boundary conditions after the flow reversal. It

correctly adjusts again when a sonic transition occurs, and .enforces a third p boundary condition.

Boundary-condition changesenforced bythe method at the .right end xs10 appear in Figure 14. No flow reversal oc-

curs, and the method correctly enforces a single boundarycondition on until the sonic transition at approximately 0.1s. The method removes this boundary condition when it is nolonger needed, and obtains the solution at the boundary en-tirely from characteristic information after the sonic transi-tion.

Without any intervention from the engineer, or even anyknowledge of the mathematical changesin the boundary-con-dition requirements for well-posedness that occur at flow re-versals and sonic transitions, a simulator employing thismethodcould successfully adapt theboundary conditions. Theengineer need only provide information regarding what vari-ables refer to the same physical quantities in the differentunit models.

Boundary-condition placement, stability, andcontinuousdependence on data

Boundary-condition placement for partial differentialequation models is typically motivated by the need to formu-late a well-posed problem. Split boundary conditions for or-dinary differential and differentialalgebraic models are of-ten motivated by stability considerations. These two analysesare very different, even when they produce the same results.

For example, consider a material balance for a single reac-tive species in a PFR in the presence of dispersion. Forincompressible flow with a constant superficial velocity,

Figure 11. Characteristics and boundary-condition requirements for Euler equations of compressible flow.



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Figure 12. Pressure profile.

isothermal operation, irreversible reaction, and a first-order

rate law, the material balance written as a first-order sys-.tem is simply

C s D V yUVy kCt a z

0sC yV, 30 .z

where C is the concentration of the species of interest, V isthe first partial derivative of C with respect to position zalong the length of the reactor, U is the superficial velocity,D is the diffusivity, and k is the reaction rate constant.a

Figure 13. Results atleft end of pipe.

Figure 14. Results atrightend of Pipe.

In canonical form, the system is

U U0 0 C 1 0 Cq

1 0 V 0 1 Vt z

0 y1UCUq s0, 31 .k y V

Da

where CU syCrD . It consists of a single degenerateaparabolic block, much like the simplified telegraphers equa-tions. H owever, here s1, so if two boundary conditionstare enforced at the same side of the domain, the problemwill be ill-posed, because it does not depend continuously onits data. One boundary condition must therefore be enforcedat each side of the domain.

At steady state, the system is

0 y11 0 C Ck Uq s0. 32 .y y0 1 V Vz Da Da

The stability of this system as an evolution problem in z isdetermined bytheeigenvaluesof thecoefficient matrix, whichare

2U 1 U 4ksy " q . 33 .( 22D 2 DDa aa

2 2 .Because D ,k)0, U rD q 4krD ) UrD , and thus' . .a a a athere is onepositive and onenegative eigenvalue. This meansthat the system is unstable as an evolution problem in boththe forward and backward z-directions, and should insteadbe formulated as a split boundary-value problem.

Given a reaction zone of length L , and assuming that thereaction zone is fed by and flows into well-mixed vessels, thewell-known closed-closed Danckwerts boundary conditions

DaC sy C qCvessel zU

C s0 34 .z

produce both a well-posed initial-boundaryvalue problemanda stable steady-state problem.



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Conclusions

Modeling support and automated analysis tools haveproven crucial for flow-sheet-scale dynamic simulation. Mostexisting tools are limited to models that consist only of ordi-nary differential and algebraic equations. This article out-lines new model analysis tools that can be applied to modelsthat include partial differential equations. In particular, theyallow a simulator to identify models that are ill-posed due tomodel inconsistency or incorrect initial or boundary condi-

tions. T hese tools also identify some models that do not de-pend continuously on their data, possibly as a result of a sim-ple sign error on the part of the engineer. Finally, they canidentify some models that are high index with respect to t. Ineach case, numerical method-of-lines solution methods can-not be expected to generate meaningful results. The toolstherefore allow a simulator to screen models and identify thetrue cause of a simulation failure that results from the funda-mental properties of the model equations themselves.

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Manuscript receied No., 15, 1999, and reision receied Dec. 19, 2000.


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Distributed Models in Plantwide Dynamic Simulation

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