MINLP Model for Synthesis of Paraxylene Separation Processes Based on Crystallization Technology

Transcription

1 MNLP Model for Synthesis of Paraxylene Separation Proesses Based on Crystallization Tehnology Carlos A. Mendez a John Myers b Sott Roberts b Jeffery Logsdon b Anastasia Vaia b and gnaio E. Grossmann a* a Department of Chemial Engineering Carnegie Mellon University Pittsbrgh PA15213; b BP Researh Center Naperville L60563 Abstrat n this paper we address the development of an MNLP optimization model for the synthesis of paraxylene separation proesses based on rystallization tehnology. A sperstrtre is proposed that onsiders different alternative designs and operating onditions for the reovery of a high prity paraxylene prodt from a feed stream ontaining other xylene isomers and ethylbenzene. The goal is to maximize the overall annal profitability of the proess inlding apital and operating ost as well as prodt revenes. De to the large size of the MNLP model as well as the great diffilty in onverging the orresponding eqations a soltion strategy based on initialization models for the topology and the operating onditions is desribed to enhane the robstness of the optimization method. Appliation of the proposed model yields flowsheet designs with improved savings in apital and tilities ost. Keywords: Synthesis rystallization paraxylene separation proess optimization 1. ntrodtion The C8 aromati isomers are very diffilt to separate by onventional distillation proesses bease of their lose boiling points whih in the partilar ase of paraxylene and metaxylene differ by less than 1 C. However paraxylene an be readily separated from other xylene omponents taking advantage of its higher freezing point. n this way separation proesses based on rystallization and liqid/solid separation eqipment are an alternative tehnology allowing paraxylene to be solidified and reovered in rystal form at a temperatre at whih the other isomers are present in liqid phase. This proess generally onsists of mltiple proessing stages in series and parallel for reating and separating solids where rystallizers liqid/solid separation devies sh as different types of entrifges drms heat exhangers and refrigeration systems are the most important eqipment that is sed. Althogh separation proesses based on rystallization tehnology have beome inreasingly important in hemial indstry they have reeived mh less attention in the optimization area where most efforts have been fosed on distillation systems (Barttfeld M. et al. 2002). We address the synthesis and optimization of separation proesses based on rystallization where * Athor to who orrespondene shold be addressed: grossmann@m.ed

2 major deisions involve seleting: (a) nmber of rystallization stages (b) type nmber and size of eqipment at eah stage and interonnetions between nit operations () temperatre levels for rystallizers heat exhangers refrigeration system and flows in the system and (d) alternatives for the retrn of reyles. 2. Crystallization sperstrtre Figre 1 illstrates a general topology for a n-stage rystallization proess. The feed stream ontains the following 4 omponents in the liqid phase: paraxylene metaxylene orthoxylene and ethylbenzene. However sine paraxylene is rystallized dring the proess we treat paraxylene in the liqid and solid phase as two different omponents that an be present in eah proess stream. n order to garantee that only paraxylene be solidified the eteti point (temperatre at whih a seond solid phase will form) is sed as a lower bond on the proess temperatre. n addition the paraxylene inipient rystallization temperate (CT) is sed to satisfy ertain operating restritions. Sine rystallizers represent the most important eqipment and involve signifiant apital and operating ost a sperstrtre is embedded into the model to onsider all the alternatives for series/parallel arrangements for a given nmber of rystallizers. Figre 2 depits the sperstrtre for a set of rystallizers. REFRGERATN SYSTEM Feed STAGE 1 STAGE N PX Prod to isomerization setion reyles Figre 1. A general n-stage separation proess rystallization inlet stream 1 1 rystallization otlet stream N Figre 2. A general sperstrtre for a rystallization proess 3. MNLP formlation This setion desribes the proposed MNLP model for the synthesis problem. Sine this proess involves mltiomponent streams we define individal omponent flows f s for eah stream. As a reslt bilinear terms are only reqired for the material balanes in the splitters.

3 Solbility. The non-linear solbility orrelation fsol predits the maximm liqid onentration of paraxylene in eah stream s at a temperatre T s. Conseqently the amont of paraxylene in the liqid and solid phases is expliitly defined for most of the proess streams whih is a entral aspet of the problem. However we have to onsider two possible alternatives: (i) the solbility predits the exat amont of paraxylene in the liqid phase or (ii) the solbility predits a onentration of liqid paraxylene that annot be met with the rrent stream omposition. To deal with the seond ase we define onstraints (2) and (3) and the variable f s whih beomes positive and allows eqation (4) to be always satisfied as eqality. Eqation (3) represents a smooth approximation of the max[0x] fntion (Biegler et al. 1987) whih defines the exat amont of paraxylene that wold be reqired to meet the solbility predition. sols = fsol( Ts ) s (1) ss = sols fs f L f S s (2) s s f s = C s 2 2 s + 2 s s f L = sol f f s s L fs s s + s (4) L C Material balanes. Eqation (5) defines the mass balane for those omponents that an only be in liqid phase whereas eqation (6) fores the mass balane of liqid and solid material for the omponent to be separated. L fs = fs C (5) s S s S s S f L + f S = f L f S U s s + (6) s s s S Additionally the ondition that the streams leaving a splitter have the same ratios in flow for eah omponent is enfored by onstraint (7). f s (7) SP s = fs' e s C s' Stream density. A linear fntion fd of the stream temperatre is sed to predit the density of eah stream whih is then tilized to size the drms sed in the separation proess. denss = fd( Ts ) s (8) Heat balanes. Speifi non-linear fntions fe are sed to ompte the heat transfer taking plae in eah proess nit. fe( f s Ts Q ) = 0 (9) Eqipment sizing. The area or the volme of the eqipment ( size ) is determined throgh a nonlinear fntion fv. Minimm and maximm apaities are enfored for those nits belonging to the proess i.e. Y = 1. size = fv( f T dens ) (10) V min s s Y size V max Y s (3) (11)

4 Liqid/solid separation devies. The performane of the liqid/solid separation devies is determined throgh Constraint (12). n this way the amont of omponent leaving the nit in eah otlet stream will depend on ertain eqipment parameters and on the stream feeding the nit whih is onsidered in the non-linear fntion fs. f ' (12) SL s = fs( fs ) s s' Unit operating limits. Certain operating onditions over minimm and maximm liqid and solid flowrates that some nits are able to handle are modeled in onstraint (13). n addition this onstraint fores all the streams related to partilar nit to be zero when the piee of eqipment is not inlded in the proess. min max F Y fs F Y s (13) Prodtion target. The total amont of prodt and the minimm omponent onentration to be reovered is imposed throgh eqations (14) (15). f PT (14) C s prod f prod L + f prod S p f prod s s s C (15) bjetive fntion. The nonlinear fntion f omptes the annalized profit onsidering prodt revenes feed ost reyled material prie (to the isomerization setion) tilities ost (steam and eletriity) and apital investment. n order to reflet eonomies of sale we se onave ost fntions for apital osts (Doglas 1988; Biegler et al. 1997). Min Z = f f feed f prod f reyle Q size ) (16) ( s s s 4. Soltion strategies Taking into aont the omplexity of the separation proess the large nmber of eqipment and flows and the highly nonlinear eqations involved the general mathematial formlation introded in the previos setion generates a large and omplex MNLP model where even feasible soltions are diffilt to find. n order to enhane the model robstness withot ompromising the optimality of the soltion we se different strategies that signifiantly improve the model behavior and at the same time generate better initial soltions for the nonlinear optimization method. The most important soltion strategies are desribed below. 1. Seqential proedre for solving the entire optimization problem. The main idea here is to gradally inorporate aspets of inreasing omplexity throgh the following seqential optimization steps. (i) Solve the NLP model where all disrete variables are fixed to 1 and linear solbility orrelations desribed below are sed (ii) Corret solbility preditions by sing exat non-linear orrelations (NLP model) (iii) Solve the MNLP model by only fixing to 1 the disrete variables related to liqid/solid separation devies (iv) Solve the MNLP model onsidering all the disrete deisions. n this way different deisions are seqentially made and a signifiant improvement in the onvergene is obtained. t shold be noted that

5 althogh we are onsidering the entire optimization problem in the last step a very good initial soltion is provided by performing the previos steps. 2. Linearization of solbility orrelations. The solbility is a entral aspet of the problem that involves highly non-linear and non-onvex orrelations. Linear approximations are sed in the first step of the seqential proedre in order to find a good initial soltion whih is then orreted in following steps by sing the original nonlinear solbility orrelations. 3. Sper-eqipment definition. Sine eqipment working in parallel have fixed apital ost and apaity they an be treated in the model as a single aggregated eqipment with the disrete variables defining the nmber of individal nits. n this way the nmber of proess streams and onseqently the model size is greatly reded. 4. Minimm eqipment reqirement. The proposed sperstrtre omprises a large nmber of proess nits and 0-1 variables defining whether they are sed or not in the optimal soltion. Taking advantage of the proess knowledge and the nit operating limits it is possible to estimate the minimm nmber of a partilar type of eqipment that wold be reqired in any feasible soltion. n this way some disrete variables an be fixed to 1 a priori whih greatly redes the CPU time. 5. Nmerial reslts The proposed MNLP formlation together with the soltion strategies desribed above were sessflly sed to ondt stdies for the design and synthesis of different shemes for the rystallization proess. Figre 3 shows the generi flowsheet for a rystallization proess design. The MNLP formlation of or atal problem involves 20 binary variables 1381 ontinos variables and 1229 onstraints and was solved on a Pentim V PC sing CNPT for the NLP solver and CPLEX 8.1 for the MLP solver in GAMS 21.2/DCPT++ (Viswanatahn and Grossmann 1990). The total soltion time for the seqential proedre was 33 seonds sing the heristi termination that stops the searh when the objetive fntion of the NLP sbproblems starts to deteriorate. Both variable and annalized apital osts ontribted signifiantly (>10%) to the objetive fntion. The entire proess omprises mltiple rystallizers in series and parallel and many liqid/solid separation devies with different onfigrations most of them working at fll apaity. Figre 3. General flowsheet for a mltiple-stage rystallization design

6 6. Conlsions An effiient MNLP approah for the optimal design and synthesis of paraxylene separation proesses based on rystallization tehnology has been presented. Different designs and operations onditions were evalated and signifiant savings in apital and tilities were obtained with low omptational reqirements. Notation (a) ndies omponent s reyle stream reyled to isomerization setion L liqid omponent that an be solidified s streams onneted to nit S solid omponent s in inlet stream to holanit s stream s ot otlet stream from nit s feed feed stream nit s prod prodt stream (b) Sets C omponents S inlet streams for nit C L omponents that an only be in the liqid Phase S otlet streams for nit C LS omponents that an be in the liqid and U eqipment solid phase (paraxylene) S streams U X nit of type x = SP(splitter) MX(mixer) CR(rystallizer) DR(drm) EX(heat exhanger) SL(solid/liqid separation devie) () Parameters F min minimm flowrate of omponent in nit PT prodtion target F max maximm flowrate of omponent in nit p minimm prodt omposition V min minimm size or apaity for nit V max maximm size or apaity for nit (d) Variables dens s density of stream s sol s solbility predition for stream s e s split fration orresponding to otlet stream s of splitter s s maximm amont of omponent reqired to meet solbility predition f s flowrate of omponent in stream s T s temperatre of stream s f s exat amont of omponent reqired to meet T s temperatre of stream s solbility predition Q heat transfer in nit Y whether or not nit is sed size total area or volme orresponding to nit Aknowledgments The athors wold like to thank BP for finanial spport of this projet Referenes Barttfeld M. Agirre P.A. and Grossmann.E A deomposition method for synthesizing omplex olmn onfigrations sing tray-by-tray GDP models. Compters and Chemial Engineering Brooke A. Kendrik D. Meeras A. and Raman R GAMS langage gide Release 2.25 Version 92. GAMS Development Corporation. Biegler L.T. Grossmann.E. and Westerberg A.W Systemati methods of hemial proess design. Prentie Hall New Jersey. Doglas J.M Coneptal design of hemial proesses. MGraw-Hill New York. Viswanathan J. and Grossmann.E A Combined penalty fntion and oter approximation method for MNLP optimization Compters and Chemial Engineering