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PowderTechnolog~87 ( 19961 13-20

Process modelling tools and their application to particulate processes C o n s t a n t i n o s C . P a n t e l i d e s *, M i n O h Centre for Process Systems Enginee~#~g. Imperial College of Science, Technology and Medicine, London SW7 2BF, UK

Received19October 1994; revised 1 September1995

Abstract This paper considers the special problems posed by the modelling and simulation of particulate processes and analyses the masons which obstructed the application of general purpose process modelling tools to these processes in the past. It argues that the latest generation of these tools is now capable of directly representing and simulating particulate processes. It outlines the features of one ~uch tool and demonstrates its capabilities by applying it to the study of the start-up of a continuous mixed suspension mixed product removal (CMSMPR) crystallization unit. KeF,cords: Particulateprocesses;Processmotioning;Dynamicsimulalion

1, Introduction The modelling of particulate processes poses special problems out encountered in more conventional process operations The state of such systems is usually characterized by particle size distribution functions instead of, or in addition to, standard point properties such as concentrations. Moreover, the steady-state and dynamic behaviour of these systems is described by population balance equations rather than simple mass balances. Finally, the physical properties of solids encountered in particulate processes are generally much less well characterized than those of fluids. Traditionally, most process modalling and simulation tools have been aimed primarily at the mainstream chemical and petrochemical industry. Commercial steady-state simulation packages have now reached a high degree of sophistication, encompassing extensive libraries of unit operation models, as well as large compilations of physical property data and calculation techniques. However, given the differences outlined above, it is hardly surprising that the area of perticulate processes has not been served well by the tools now u~ed routinely by process engineers in other areas. Most of the established steady-state simulation packages are of the sequential-modular type (see Westerberg et al. [ I ] ). In these, each unit operation is represented by a module which, given the properties (flowrate, composition, temperatum, pressure, etc.) of the unit input streams and the associated equipment parameters, calculates the output stream * Correspondingauthor, e-mail:[email protected]~.ul~ 0032-5910/96/$15.00 © 1996ElsevierS¢iesc¢S.A.All fightsrescaved $8DI0032-5910(95)03079-0

properties. Entire processes are modelled by using streams to link a number of such modules together in a flowsheet. Several factors limit the flexibility of sequential modular packages in modelling non-standard unit operations. FLrSt, each module encompasses both t ~ modelling equations and the techniques used for their solution, all described within procedural computer code (usually in the form of a subroutine). Consequently, creating new modules is a far from trivial task, sometimes involving a significant degree of computer programming and mathematical skill. Secondly, the type of information that is carried by streams, while entirely apuropriam for standard petrochemical operations, may not be suitable for applications in other areas. The desire m ad~eas the above limitations was one of the prime motivations for the development of eqnation-oriented process modelling tools. In these, each umt operation is described in terms of a set of variables and th'. equations that relate them. The flowsheet is still describ:J in a modular fashion, but it is the responsibility of the modelling system to assemble the mathematical descriptions of all modules in it into one large system of equations and then to solve it using a general purpose solution method. In this context, the definition of models of novel unit operations becomes relatively easy as it is no longer necessary for the user to be concerned with the actual solution of the modelling equations. Furthermore, streams arc simply aser-defined snbsets oftbe variables in each model, and can therefore be configured to match the needs of individual applications. Another advantage of equation-oriented modelling systems is that the model is defined and maintained quite dis-

C C Pantelides, M. Oh/P~nvder Technology 87 (1996) 13-20

tinctly from the solution method. This allows the same model to be us,~d for a variety of model-based applications, such as steady-state and dynamic simulation and optimization, parameter estimation, data reconciliation, control system design, etc. Therefore, instead of just a simulation tool, we now have a general purpose process modelling environment. A typical equation-oriented package is SpeedUp [2,3]. The system provides a high level declarative language for describing unit operation models in terms of mixed sets of ordinary differential and algebraic equations. It also representsa practical implementation of the conocpt of a modelling environment in the sense described here. The package is now widely used in industry [4]. It should be recognized, however, that even with relatively sophisticated general purpose process modelling tools, the modelling and simulation of particulate processes still presents serious difficulties. One key problem is the mathematical complexity of the models: population balances invariably lead to partial differential equations, and these are often coupled with other equations describing the evolution of properties in the fluid surrounding the particles through integral terms. This results in systems of integro-partial dift~:reutial equations which may be difficult to solve. In fact, most current equation-oriented packages cannot even describe directly such distributed parameter systems. Our recent research has been concerned with the development of gPROMS, a process modelling tool aimed specifically at addressing the above limitations. In the next section, we review the major concepts and features of this package. We then proceed to illustrate its applicability to particulate processes using crystallization as an example, before drawing some general conclusions.

2. The gPROMS process modelling system gPROMS (general PROcess Modelling System) is a new equation-oriented package under development at Imperial College, London. It differs significantly from earlier modelling software in that it is designed to deal effectively with a much wider range of processes, including those with combined discrete and continuous characteristics and lumped and distributed parameter systems. As such, it is especially suited to the modelling and simulation of particulate processes. gPROMS is designed on the premise that a process encompasses not only the physical plant, but also tbe operating procedures and control actions that are employed to operate it [ 5 ]. Two distinct types of fundamental entity are therefore recognised: MODELs, describing the physical, chemical and biological behaviour of the plant; and TASKs representing the external actions and disturbances imposed on the plant. A third type of entity, the PROCESS, generally comprises a TASK driving a MODEL. Like earlier equation-oriented process modelling tools, gPROMS models plant bebaviour in terms of sets of variables and equations. However, much more generality is allowed.

Thus models can be expressed in terms of combinations of partial and ordinary differential, algebraic and integral equations. Model variables may be distributed over an arbitrary number of domains, and the degree of distribution may be different for different variables in the same model. These considerations are also reflected in the structure of partial and integral equations and the associated bolmdary conditions. A more detailed description of the distributed sy stem modelling capability of gPROMS is given in Ref. [6]. Another novel feature ofgPROMS is its ability for describing and handling model discontinuities of a very general nature. Thus one can define equations that take different forms under different conditions. Discontinuities that exhibit hysteresis phenomena and irreversible discontinuities are also allowed. One important characteristic of process modelling tools is the extent to which they can handle the complexity of realistic processes. Such complexity may arise in both the intrinsic plant behaviour and in the plant operating procedures, gPROMS allows any MODEL and TASK to be built from instances of lower level MODELs and TASKs. thus establishing hierarchies of arbitrary depth. Thus. for instance, the model of a plant may be described at the highest level as comprising a reaction section and a separation section. The latter may involve a number of separation units, and each of these may comprise a number of distinct items such as stirred tanks, valves, local controllers, heating and stirring equiputent, etc. An analogous hierarchical approach can be employed in describing, for instance, a start-up operating procedure for this plant. This may involve starting the reaction and the separation sections in sequence. Starting the separation section may entail starting up a number of units in parallel and starting up a unit typically involves a more complex sequence of operations on its constituent components. The important thing to note is that, by properly exploiting this hierarchical approach, the model builder needs to bc concerned with describing each level o f the hierarchy in terms of a relatively small number of entities at the immediately lower level, instead o f large numbers of elementary plant components such as individual valves, pumps, etc. and operations, for example, opening and closing valves, starting and stopping pumps. This reduces the probability of errors being made, and also renders the structure of the model much clearer, thus significantly improving its future maintainability. gPROMS employs a combination of symbolic, structural and numerical techniques for the solution of the underlying mathematical problem. For instance, symbolic manipulation is used to generate analytical partial derivatives of al.~ equations with respect to all unknowns in the system, thereby enhancing the reliability and efficiency of the numerical codes. A structural analysis of the equations allows their sparsity to be exploited, thus permitting the solution of systems involving tens of thousands of equations and variables. It is important to point nut that all of these mathematical

C.C. Pantelide,s M. Oh/Powder Technology87 (1996) 13-20 manipulations are carried out in a manner totally transparent to the user who is therefore leR free to concentrate on the physics of the problem. The main numerical code in gPROMS handles the solution of large sets of mixed systems of ordinary differential (with respect to time) and algebraic equations (DAEs), with facilities for initialization and automatic detection and handling of di~ontinuities [7]. Partial differential and integral equations are converted automatically to DAEs by discretizing all non-temporal dimensions. A number of different techniques, including methods used on finite difference approximations and orthogonal collocation on finite elements, are provided for this purpose [6]. In the next section, we illustrate some of the capabilities of gPROMS by applying it to a crystallization process.

3. Modelling and simulation of crystallization processes We shall consider a continuous mixed suspension mixed product removal (CMSMPR) crystallization unit operating in cooling mode to produce potassium sulphate crystals from aqueous solutions. 3. I. Modelling equations We assume that the temperature in the crystallizer is constant and ignore any crystal breakage and aggregation. A mass balance on the solute in both the liquid and the solid phase yields the equation t: B~

vo~t =Wfc~.-e)

(1)

where E denotes the combined solid and liquid phase concentration of the solute, defined as:

~ = c + M-'2 P

(2)

(4b)

The boundary condition for this equation is of the form: n ( 0 , t) = Be G~

(5a)

which can also be expressed in logarithmicterms: In n(0, t) = I n ( ~

(Sb)

The residence time ~"in the system is related to the mass flowrate W through: Vp ¢= - W

(6)

The crystal growth rate G(L), the nucleation rate Be and the nuclei growth rate Go arc all functions of the relative supersaturation ar of the solution given by: {r = max( e - ccq, 0)

(7)

where the equilibrium saturation concentration of the solute, c~q, is generally a function of temperature. For the purposes of this paper, we use the following cmrelations obtained by Chianese et al. [8] from experimental data on the crystallization of potassium sulphate from aqueous solutions 2: G(L) = 892o'-'( l + 5.87L)

(8)

Bo=4.12 x 10JaMto-3'=

(9)

and Go= 847o =

(lo)

The crystal volumetric shape factor, Kv(L) is given by the following discontinuous correlation [9]: Kv(L) = 0.898 exp[0A68(lfl00L) 1/2_ g.234L]

The magma density M, is given by: = Mt = p~IKv(L)Lan(L, t) dL o

i) l n n ( L , t ) . . . . i)ln n(L) ~3~(L) + _1 at +~(L,) - - - - - ~ - + ~ I" = 0

15

for Lg0.1 mm (31

4.460 exp[ - 0.0797(1000L) t/2 + 6.76 × l0 - ~L] for L>0.1 mm

(ll)

A population balance on the crystals yields the partial diffarenfial equation:

Finally the saturation concentration of potassium sulphate in water is taken as [ 10]:

~ ( L , t) O(G(L)n(L, 1)1 n(L, t) - -=" + =0 i}t i)L "r

cm= 0.0735 + 1.675× 10-3T

(4a)

However, this equation may pose numerical problems because of the potentially large magnitude of the crystal number concentrations n ( L, t), and it may be preferable to express it in terms of the natural logarithm of these quantities: * The mathematicalnotationusedis explainedin the Listof symbolsat the end of the paper.

(121

Overall, the model of the crystallizer comprises the partial differential equation (Eq. (4b)), the ordinary differential equation (Eq. ( I11, the integral equation (Eq. (31) and the algebraic relations E,qs. (2), (5b), and (61-( 111. It is worth zThe unitsof measurementof the vmlo4usquantiiiesappeanngin these and subsequentcorrelationsate givenin the Listof symbolsat the end of thispager.

C C. Pantelides. M. Oh/Powder Teclmology 87 ¢i996) 13-20

16

I

MODEL

Crystalliser

2 3 4 5

PA/~ETER MaxE~ze, SmallSize rho, rhoS V

6 7

DISTRIBUTION_DOMAIN size

8 9 I0 II 12 13

AS AS AS

AS

REAL REAL REAL

( C

: MaxSize

18

VARIABLE logn G Kv c, cc, cin. ceq Mt sigma tau temp W

19 20

BOUNDARY logn(0)

= log

21 22 23

EQUATION # Tetal V * rho

mass balance on solute * $cc = W * ( t i n - cc}

]4 15 ]6 17

AS AS AS

DISTRIBUTION DISTRIBUTION DISTRIBUTION

1

(size) (size) (size)

OF OF OF

AS AS AS AS AS AS

lognumber_density growthrate shape factor concentration mass_density supersaturation resldencetime temperature flow_rate

(4.12e14

of c o m b i n e d / rho

* Mt

* si~a~3.4

24 25

# Definition cc = c + Mt

26 27 28

# Definition of magma density Mt - rhoS * IE-12 * INTEGRAL ( L := 0 : M a x S i z ~

29

# Crystal

/

(847

* siva^2

concentration

; Kv(L)

* L~3

* EXP(logn(L)))

30 31 32 33

population balance F O R L := 0 ] + T O M a x S i z e DO $ 1 o g n (L) + G (L) * P A R T I A L { I o g n + i I tau = 0 END

34 35

~ Residence time tau = V * the / W

36 37 38 39 40

41

# Supersaturation IF c < ceq + IE-6 THEN siva IE-~ ELSE sigma = C - ceq END

42 43 44 45

# Crystal growth rate correlation (Chianese F O R L := 0 T O M a x S i z e DO G(L) = 892 * sigma*2 * (I+5.87"L) END

46 47 48 49 50 51 52

# Vol~etric shape factor correlation ( B u t z e t a l . , 1987) F O R L := 0 T O S m a l l S i z e DO Kv(L) = 0.898 * exp(O.168*(1000*L)*0.5 - 8.234"L ) END F O R L := s m a l l S i z e l + TO MaxSize DO Kv{L) = 4.46 " exp(-0.0797*(1000*L)^0.5 + 6.?6E-I*L ) END

53 54

# Equilibrium ceg = 0.0735

55

END

# Model

* IE-3})

(L), s i z e ) ;

concentration (Perry + 1.675E-3 * temp

and

+ PARTIAL(

et

Chilton,

Crystalliser

Fig. I. Crystallizer model in gPROMS.

al.,

G (L),

1987~

1973)

size

)

CC Panlelides,M. Oh~PowderTechnoloRy87(1996)13-20 Table I Inputstreamspecificationsand parametervaluesforsimulationstudy Parameter

Value

c,, (kg solid/kgsolvent) !~'(kg/h) T('C) V(m~)

0.15 100 25 0,05

p (kg/m ~)

I000

p~(kg/m3)

2660

noting that Eq, (7) introduces an implicit discontinuity in the system as, in general, it is not known a priori if and when the liquid will reach its saturation point. 3.2, Model implementation in gPROMS The model of the crystallization unit expressed in the gPROMS language is shown in Fig. 1. It can be seen that the model defines a distribution domain called size (lines 6-7 ~). Several model variables are declared as being distributed over this domain (lines 9-11 ). The boundary conditions (lines 19-20) and other equations (lines 21-54) correspond closely to the mathematical model described in the previous section. It should be noted that the $ operator denotes partial differentiation with respect to time, while differentiation with respect to crystal size is done using the PARTIAL operator (see lines 31-32). The definition of the magma density (3) requires the use of the INTEGRAL operator (see lines 27-28). Finally, a conditional equation is introduced to define the relative supersab uration o" (linns 37-41 4). Overall, the model comprises a number of cuuplod partial and ordinary differential, algebraic and integral equations, it is worth pointing oat that the gPROMS language is purely declarative: the order or precise form in which the equations are written is immaterial. 3.3, Dynamic simulalion in gPROMS The values of the various parameters used for the simulation are shown in Table 1. Crystals of sizes up to 1.5 nun (corresponding to parameter MaxSize in the gPROMS model) were considered, A first-order backward finite difference method with 151 nodes was used for the discretization of the crystal size domain. Simulations with different numbers of nodes were carried out to verify that this scheme leads to sufficiently accurate results without incurring excessive computation. The dynamic simulation considers the start-up of the crystallizer, with the unit initially being full of pure water. Thus 3Notethailinenumbersare not pallofthe gPROMSlanguage.Theyarc used hem for easeof mfefence,

,LThesmalltergflof tO-6is addedtOthisequationfo¢numericalreasons sincesomeequationsate notdefinedfor o"Ueingexactlyzcm,

the initial liquid concentration of solute in the unit is set to zero, and so is the logarithmic number density for all nonzero crystal sizes. The latter specification actually corresponds to a population density of I crystal/m 4, but this is equivalent to zero for all practical purposes. The values of the various model parameters and the input stream specifications are specified in the gPROMS PROCESS shown in Fig. 2. The discrotization method to be applied to the particle size distribution is also specified here (see line 12). In this particular example, no separate TASK specification is necessary. The required plant operation is described directly in the SCHEDULE section of the PROCESS (lines 27-28 ). It simply involvesa dynamic simulation starting from the specified initial condition and continuing until it reaches steady state. Here the latter is defined m occur when the rate of change of the combined solid/liquid conceutration of the solute in the crystallizer falls below 10 -6 kg/(kg h) in absolute value. The manner in which gPROMS performs a dynamic simulation of the type considered here is outlined in Fig. 3. The input file, containing the MODEL, TASK and PROCESS descriptions is first translated into an internal representation; some syntactical and semantic validation of the input is also carried out at this stage. It is worth noting that the same input file may contain several PROCESS descriptions, each defining a different simulation ( for example, atarbup, normal shotdown, emergency shut-down, etc.) that can be peffonnnd using the same MODELs and TASKs. The user specifies the simulation that is actually to be performed by executing the corresponding PROCESS. The execution of a PROCESS has several effects. First of all, the integral, partial differential and algebraic equations (IPDAEs) describing the system are discrefized with r~pect to all non-tamporal dimensions (for instance, particle size) using the method and discretization grid specified in the PROCESS (such as first-order backward finite differences). This results in a large set of ordinary differential (with respect to time) and algebraic equations (DAEs). Most sophisticated numerical methods for the solution of DAEs require the partial derivatives of the equations with respect to the unknowns occurring in them. In gPROMS, these are determined using symbolic manipulation. Finally, other information regarding the mathematical model and its initialconditions, for example its sparsity structure, is also determined. This is used to check further whether the problem is well posed, and is also required by the nmnefical solution methods. Once the DAEs representing the system have been deft red and analysed as described before, they are solvednumerically using a suitable algorithm. The many complications that this entails (such as the derivation of cunsistent initial values for all problem variables and the detection, location and handling of discontinuities) are outside the scope of the present paper but the interested reader should see Ref. [ I 1]. Here, it suffices to emphasize that all of the symbolic, structural and numerical manipulations shown within the dashed box in Fig. 3 are

C.C Pamelides. M Oh/Powder Technology 87 (1996) 13-20 PROCESS StartUp

2 3

UNIT CMSMPE

4 5 6 7

SET # S p e c i f i c a t i o n of m o d e l p a r a m e t e r v a l u e s WITHIN CMSMPR DO SmallSize := 0.i MaxSlze := 1.5 rho := i 0 0 0 . 0 ; rhoS := 2660 V := 0.050 ;

8

9 10 It 12 13

AS

Crystalliser

# D i s c r e t i z a t i o n m e t h o d for p a r t i c i e size d i s t ~ i b u t t ~ p d ~ m a i n size := [ bfdm, I, 150 ] ; END

14 15 16 17 18 t9

A S S I G N # D e q r e e of f r e e d o m s p e c i f i c a t l o n s W I T H I N C M S M P R DO tin := 0.15 temp := 25.0 w := i 0 0 . 0 ; END

20 21 22 23 24 25 26

I N I T I A L # S p e c i f i c a t i o n of i n i t i a l c o n d i t i o n s W I T H I N C M S M P R DO

27 28 29

FOR L logn(L) SND END

(inlet s t r e a m p r o p e r t i e s )

TO M a x S t z ~ DO - 0 :

SCHEDULE CONTINUE UNTIL ABS(CMSMPR.$CC)

< IE-6

END

~g,2, Dynamicsimulationofe~slallizerslaix~upin gPROMS. carried out automatically in a manner that is totally transparent to the user. 3.4. Dynamic simulation results

Because of the potential complexity of the dynamic simulation results in g P R O M S , their archiving and display are

Input Tm nElation/V alidalion

1

• A~,~maticdisCrelis~.ionof PDAEsystem * Symbolicdiltete.tiatio. • Det=rnunatio,~t systemstructuralproperlles

handled by a separate software system called g R M S (gP R O M S Results Management S y s t e m ) . At the start of the execution of a P R O C E S S , g P R O M S registers with g R M S the names and other properties o f the variables to be monitored (see Fig. 3). Then, during the simulation, g P R O M S periodically transmits the values for these variables to g R M S which has a sole responsibility for archiving them and dis-

'*

"-

gRMS ResultsAtchlvingantiDisplay

. . . . . . . . . . . . . . . . . . . . . Fig, 3. Rowchall of gPROMS operalion.

~I~MSlt~uJt~~tes

C.C Pantelldes. M. Oh/Powder Technology 87 (19¢J6l 13-20

0.1E

19

26 24

0.14

22 20 18

+o~ 0.101

E

008 I

~ t2 m

~m 0.o6. 0.04 0,02

J

0.00 1

2

3

4

5

6

1

7

2

3

"lime (hr) Fig, 4. Variation of solule cont.'eat ~ t i o n with time. t - - l . liquid and solid; ( . . . . . ) equilibrium.

4

5

6

qqme (hi') liquid; ( - - - ) , Fig. 5, Variation of m a g m a density with lime,

playing them to the user either while the simulation is running or at a later stage. The results of the CMSMPR simulation are shown in Figs. 4-6 which also provide some examples of typical gRMS output. Fig. 4 shows the actual liquid phase concentration of the solute, as well as the corresponding equilibrium value. The system stays below the saturation line for the first 0.735 h of its operation. Even after crossing the saturation line, the liquid phase eoneenlration continues to rise towards the feed value (0.15 kg/kg) until the rate of crystallization increases

sufficiently to reverse this trend. This is also evident in Fig. 5 as the point at which the magma density starts becoming appreciable, The combined (liquid and solid phase) solute concentration is also shown in Fig. 4. It can be seen that this rises monotonically from zero to the inlet liquid concentration value ofO.15 kg/kg. Fig. 6 shows the variation of crystal population density (in natural logarithm terms) with respect to crystal size and time.

31.0 24.! 211.6 17.2

6

t

I03 13.7 6.9 3.4 -O+O -6.9

t/ Fig. 6. Var~tion of crystalpopulationdensitywilh cryslalsize andtime.

20

C C Pantelides. M. Oh/Powder Technology 87 (1996) 13-20

4. Conclusions

Ccq

This paper has considered the special problems posed by the modelling of particulate processes. It has also attempted to analyse the reasons why, in the past, these peculiarities have severely linfited the applicability , f general purpose process modelling packages to such operatlt,ns. An implicitassumption throughout this paper has been that the application of general purpose process modelling tools to particulate processes is preferable to the development of process specific tools. We believe this to be true for a number of reasons. General purpose packages have now reached a high degree of sophistication regarding, for instance, the reliability of the solution methods and the ability to describe the intrinsic behaviour and the operating procedures of complex plants. Obviously all of these features are very important for both particulate and more conventional processes. However, the development effort tltat would be required to incorporate such capabilities within packages aimed at a narrow range of applications may not be economically justifiable. Furthermore, general purpose tools are more suitable for modelling processes that incorporate a combination of particulate and conventional operations. Finally. we have argued that the latest generation of process modelling tools can indeed be applied to particulate processes. A key element in this development is the provision of facilities for the description of di~:.ributed parameter processes, and the ability of solving wid~ classes of systems of mixed integro-parfial differential equations. These factors, in fact, make these tools good platforms for the construction of even more sophisticated (for example, spatially as well as size distributed) models of particulate processes than the ones commonly used at present.

c~,

5. List of symbols Bo c

nucleation rate solute concentration in liquid (kg solid/kg solvent) combined solute concentration in both liquid and solid phases (kg solid/kg solvent)

G Go Kv L Mt n t T V W

equilibrium solute concentration (kg solid/kg solvent) solute concentration in feed stream (kg solid/kg solvent) crystal growth rote (ram/h) nuclei growth rate (ram/h) crystal volumetric shape factor crystal size (ram) magma density (kg solid/m3) crystal population density (crystals/m4) time (h) temperature (QC) volume of crystallizer (m 3) mass flowrate of liquid through crystallizer (kg/h)

Greek letters

p p~ cr "r

density af solvent (kg/m 3) density of solid (kg/m 3) relative supersaturation (kg solid/kg solvent) mean residence time ( h )

References

[ 1] A.W.Westerberg,H.P.Hutchison.R.L.MotardandP. Winter.Process Flowsheefng, CambridgeUniversityPress,Cambridge,UK, 1973. [2] LD. Perkinsand R.W.H.Sargent,AIChESymp. Set,. 214 (1978) I. [3] C.C.Pantelides,Comput. Chem. Eng., 12 (1988) .45. [4] Aspetl Technology.Inc.. SpeedUp User Manual. Cambridge,MA, 1994. [5] P.I. Bartonand C.C. Pantelides,AICh£ J.. 40 (1994) 966. [6l M. Oh and C.C. PanteUdes, Prec. 5th Int. Syrup. Prec. System Engineering, Korean InstituteofChemicalEngineers, 1994,p. 37. [7] RB. J~rvisand C,C. Pantelides,DASOLV:A differentiat-algebl~ic equation solver.Tech. Rep,. Centrefor ProcessSystemsEngineering, ImperialCollege,London,1992. [8l A. Chianese,S. di Caveand B. Mazza~tta, Ing. Chim. (Milan). 23 (1987) 8, [9] L Budz,AG. JonesandJ.W,MuHin,Ind. Eng. Chem. Res.. 26 (|987) 820. [10] R.H. Peny and C.H. Chilton, Chemical Engineer's Handbook, McGraw-Hill,NewYork,5th edn., 1973. till C.C. Pantelidesand P.l. Barton, Comput, Chem, Eng., I?S (1993) $263.