PI-21 Data correlation in homogeneous azeotropic mixtures using NRTL model and stochastic optimization methods. A. Briones-Ramírez

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1 ESAT 009 PI- Data correlaton n homogeneous azeotropc mxtures usng NRTL model and stochastc optmzaton methods C. Gutérrez-Antono (, A. Bonlla-Petrcolet (, J. G. Segova-Hernández (3, (,( 4 A. Brones-Ramírez ( CIATEQ, A.C., Av. del Retablo # 50 Col. Fovssste, 7650, Querétaro, Querétaro, Méxco, Correspondng author: clauda.guterrez@cateq.mx ( Departamento de Ingenería Químca, Insttuto Tecnológco de Aguascalentes, Av. Adolfo López Mateos #80 Ote. Fracc. Bonagens, 056, Aguascalentes, Aguascalentes, Méxco (3 Unversdad de Guanajuato, Campus Guanajuato, Dvsón de Cencas Naturales y Exactas, Departamento de Ingenería Químca, Nora Alta S/N, 36050, Guanajuato, Guanajuato, Méxco. (4 Innovacón Integral de Sstemas S.A. de C.V., Calle Número # 5 Interor 3, Parque Industral Jurca, 760, Querétaro, Querétaro, Méxco. Introducton Azeotropy s a specal case of phase equlbrum phenomenon that occurs n many ndustral applcatons, and ts presence restrcts the separaton amount of a multcomponent mxture that can be acheved by dstllaton. Azeotropes can be classfed as homogeneous and heterogeneous, dependng on the number of lqud phases nvolved n the phase equlbrum condton. In the context of process system engneerng, the descrpton of homogeneous azeotropy s essental for the selecton of strateges n synthess, desgn and operaton of separaton unts. Usually, excess Gbbs energy models are used for modelng vapor-lqud equlbrum (VLE n azeotropc mxtures, due to ts smplcty and predctve capabltes. However, proper parameters of these models are necessary for the relable desgn of separaton systems. Usually, expermental data are used to determne the adjustable parameters of local composton models for vapor-lqud equlbrum modelng. The parameter estmaton problem for vapor-lqud equlbrum modelng n azeotropc mxtures s a challengng numercal task. Ths problem nvolves fttng the thermodynamc model to expermental VLE data by mnmzng a sutable objectve functon. However, ths objectve functon may become non-convex due to nonlnear form of the thermodynamc equatons. Therefore, the parameter estmaton requres the soluton of a nonlnear and dffcult optmzaton problem even for relatvely smple thermodynamc equatons such as Wlson, UNIQUAC and NRTL [, ]. Recently, some studes have shown that tradtonal approaches used for parameter estmaton n local composton models have several lmtatons [-3]. Most of the exstng methods for solvng optmzaton problems are local n nature, and, at best, yeld only local solutons [4]. It s mportant to note that local optmal parameters Santago de Compostela, Span 435 ESAT 009

2 ESAT 009 (.e. local solutons of the parameter estmaton problem affect the predctve capablty of thermodynamc models causng qualtatve and quanttatve dscrepances of phase behavor [3]. In partcular for NRTL model, falures n the data correlaton process may result n ncorrect predctons of the azeotropc states, and n qualtatve dscrepances of the phase behavor such as predcton of spurous phase splt and modelng of homogeneous azeotrope as heterogeneous [,]. Undoubtedly, these falures mply errors and uncertantes n process desgn and erroneous conclusons about model performance. Relable optmzaton strateges must be used for determnng the adjustable parameters of NRTL equaton, and other local composton models for VLE modelng n azeotropc mxtures. In ths context, stochastc optmzaton methods are robust numercal strateges that offer several advantages wth respect to conventonal optmzaton methods, and are very attractve for ts applcaton n vapor-lqud equlbrum modelng wth homogeneous azeotropy. Based on ths fact, n ths study two stochastc methods are used and compared for parameter estmaton n vapor-lqud equlbrum modelng usng NRTL equaton. Specfcally, the performance of Smulated Annealng and Partcle Swarm Optmzaton s tested and compared n ths thermodynamc applcaton. In addton, we llustrate the effect of NRTL adjusted parameters by consderng the possble qualtatvely errors n the predcton of homogeneous azeotropes. Parameter estmaton for VLE modelng Accordng to classcal Thermodynamcs, the equlbrum between vapor and lqud phases n a c multcomponent system mples that temperature (T, pressure (P and the chemcal potental of each component must be the same n both phases. At low pressure, the VLE condtons can be smplfed because the fugacty coeffcent of pure components nearly cancels each other, and Poyntng correctons usually are very close to unty. Neglectng these correctons, now yelds: 0 xp yp for,..., c ( 0 where s the actvty coeffcent of component, P s the vapor pressure of pure component, x and y are the equlbrum mole fracton at lqud and vapor phase, respectvely. Usng Eq. (, the non-deal behavor for non-deal systems s descrbed solely by lqud-phase actvty coeffcent. Therefore, the objectve functon commonly used for VLE data modelng s gven as: obj ndat c j exp calc j j F ( exp j exp calc where and are the expermental and calculated values for the actvty coeffcent of component and ndat s the number of expermental data used for parameter estmaton, respectvely. For the case of complete VLE data (.e., x y P at constant T, or x y T at constant P, excess Gbbs energy equatons are wdely appled for phase equlbrum ESAT Santago de Compostela, Span

3 ESAT 009 modelng. Gven VLE measurements and assumng an deal vapor phase, the expermental values for the actvty coeffcents can be calculated from experments and usng Eq. (. There are several local composton models for the calculaton of lqud-phase actvty coeffcents. Partcularly, NRTL equaton s a flexble local composton model that can be used for the correlaton of, and for representng complex VLE behavors n multcomponent systems. For a bnary mxture, the actvty coeffcents usng the NRTL model are gven by ln ln x x G G G G (3 beng G j g g RT g g RT exp( j j (4 where three adjustable parameters are necessary for each bnary par: = g g, = g g, and the nonrandomness factor =. Parameters of the lqud-phase model are estmated by mnmzng Eq. (. Note that the thrd parameter can be treated as adjustable, but several authors have suggested that better t should be fxed between 0. and We wll show that ths choce s very mportant due to t may cause qualtatve errors n the predcton of azeotropc states. As ndcated, several studes have shown that the hghly non-lnear form of local composton models makes that F obj s non-lnear, potentally non-convex wth several local mnma ponts wthn the specfed bounds. Prevous studes have shown that parameter estmaton for VLE data modelng nvolves the solvng of a global optmzaton problem [ - 4]. In partcular for NRTL model, due to ts flexblty, ths model can predct more phases that actually exst n the system f the parameter estmaton procedure s not performed adequately. Ths qualtatve dscrepancy occurs more frequently n the VLE modelng of azeotropc mxtures [4]. Therefore, t s necessary to apply a sutable numercal strategy for relably solvng the parameter estmaton n VLE modelng. In ths study we have tested the numercal performance of two stochastc optmzaton methods for VLE data fttng of bnary systems usng NRTL equaton. Descrpton of optmzaton strateges: Partcle Swarm Optmzaton and Smulated Annealng We have used the Smulated Annealng (SA and Partcle Swarm Optmzaton (PSO for solvng the VLE parameter estmaton problem wth NRTL model. Both stochastc methods are consdered global optmzaton strateges, and they have Santago de Compostela, Span 437 ESAT 009

4 ESAT 009 found several applcatons n scence and engneerng, ncludng thermodynamc calculatons. Specfcally, SA smulates the process of slow coolng of metals to acheve the mnmum functon value n a mnmzaton problem. The coolng phenomenon s modeled by controllng a temperature lke parameter ntroduced wth the concept of Boltzmann probablty dstrbuton. By a controlled temperature reducton as the algorthm proceeds, the convergence of the algorthm can be controlled. We have used the SA code developed by Goffe et al. [5], whch s based on the algorthm developed by Corana et al. [6]. On the other hand, PSO s a novel and promsng populaton based method that belongs to the class of swarm ntellgence algorthms. Kennedy and Eberhart [7] ntroduced ths strategy for global optmzaton, whch s nspred by the socal behavor of flockng swarms of brds and fsh schools. It explots a populaton of potental solutons to dentfy promsng areas for optmzaton. In ths context, the populaton of potental solutons s called the swarm, and each soluton s called partcle. Partcles are conceptual enttes, whch fly through the mult-dmensonal search space. The success hstores of the partcles nfluence both ther own search patterns and those of ther peers. Each partcle has two state varables: ts current poston and ts current velocty. In the local verson of PSO, whch s used n ths study, the search s focused on promsng regons by basng each partcles velocty toward both the partcles own remembered best poston and the communcated best ever neghborhood locaton. The relatve weghts of these two postons are scaled by the socal and cogntve parameters. Once the partcles are all ntalzed, the postons and veloctes of all the partcles are modfed. After calculatng the veloctes and poston for the next teraton, the current teraton s completed. The best partcle s only updated when a new one s found yeldng a decrease n the objectve functon value. Ths process s performed untl satsfes the stoppng crteron. Results and dscusson Wth llustratve purposes, we have tested the performance of SA and PSO for the parameter estmaton n bnary VLE data of the system water +, ethanedol at 430 mmhg. Ths system has no azeotropes but ts VLE parameter estmaton problem s very challengng [,]. Thus, the global optmum of selected example s well dentfed. Equaton ( s optmzed wth respect three parameters of NRTL nsde followng ntervals:, (-000, 5000 and (0.0, 0.0. Expermental data are taken from Dechema collecton. We note that the global mnmzaton of objectve functon can be done as an unconstraned optmzaton problem usng NRTL equaton. Ths VLE problem was solved 00 tmes usng random ntal values, va dfferent random number seed for both SA and PSO. Fgure shows the plot of success rate (SR versus the number of teratons for both SA and PSO. Note that SR s defned as the number of runs out of 00 that satsfy the condton f opt f calc 0-04 ; where the known global optmum of the objectve functon s f opt, and f calc s the value of objectve functon calculated by stochastc methods. In frst nstance, to drectly compare the performance of algorthms, we keep constant the numercal effort va the number of teratons, and compare the results obtaned n terms of SR. Fgure shows that the relablty of the stochastc methods hghly depends on the maxmum number of teratons allowed. It s notced that as the number of teratons ncreased, the success rate to fnd the global optmum of all methods also ncreased. Based on our numercal practce usng PSO and SA, we can ESAT Santago de Compostela, Span

5 ESAT 009 conclude the relablty of SA s better compared to PSO n terms of both success rate and computatonal tme SA PSO 60 SR, % Iteratons Fgure. Success rate (SR versus teratons of Smulated Annealng and Partcle Swarm Optmzaton for parameter estmaton n vapor-lqud equlbrum modelng. To llustrate the qualtatve dscrepances that may occur n azeotropc predctons by fxng parameter to an unsutable value, we have consdered the parameter estmaton for VLE data of the bnary mxture,3-dmethyl--butane + methanol at atmospherc pressure [7]. Expermentally, a homogeneous azeotrope s observed for ths mxture. Data correlaton was performed usng NRTL model and SA method. For parameter estmaton, three scenaros were consdered: a VLE data modelng usng as an adjustable parameter, b VLE data modelng usng = 0. and c VLE data modelng usng = 0.4. Results of parameter estmaton are gven n Table. In ths table, we also report the predcted azeotropes (homogeneous and heterogeneous usng adjusted parameters of NRTL model. We can observe that, f s fxed to 0. or 0.4 nsde parameter estmaton procedure, NRTL model predcts smultaneously stable heterogeneous azeotropes (.e., a vapor-lqud-lqud equlbrum, and unstable homogeneous azeotropes. These dscrepances between model and experment are due to the fact that parameter s not used as adjustable parameter, and, under these condtons, NRTL model predcts a lqud-lqud splt whle expermentally ths phase behavor does not occur. However, when three parameters of NRTL are used as optmzaton varables, ths local composton model closely match the expermental data, ncludng the presence of a homogeneous azeotrope. These results hghlght the use of a sutable strategy for parameter estmaton n VLE modelng n azeotropc systems usng NRTL model. Table. Results of VLE data modelng of bnary system,3-dmethyl--butane + methanol at atmospherc pressure usng NRTL and Smulated Annealng NRTL parameters Predcton of azeotropes Homogeneous Heterogeneous Yes No Yes, Unstable Yes, Stable Yes, Unstable Yes, Stable Santago de Compostela, Span 439 ESAT 009

6 ESAT 009 Concludng remarks Parameter estmaton for VLE modelng usng local composton models s a dffcult global optmzaton problem, and ther characterstcs pose a challenge to any optmzaton technque. Our numercal experence ndcates that both PSO and SA are relable, f properly mplemented, for solvng ths thermodynamc problem. However, t appears that SA s the most sutable method for parameter estmaton for VLE modelng. Fnally, the mproper applcaton of NRTL model for data fttng of azeotropc mxtures may cause severe qualtatve and quanttatve dscrepances n predctng phase behavor. Acknowledgements Fnancal support of ths work provded by CONACyT (Méxco through the Project 8455 s gratefully acknowledged. References [] C.Y. Gau, J.F. Brennecke, and M.A. Stadtherr. Flud Phase Equlbra (000, 68: -8. [] A. Bonlla-Petrcolet, U.I. Bravo-Sanchez, F. Castllo-Borja, J.G. Zapan- Salnas, and J.J. Soto-Bernal, Brazlan Journal of Chemcal Engneerng (007, 4: 5-6. [3] G.M. Bollas, P.I. Barton, and A. Mtsos, Chemcal Engneerng Scence (009, 64: [4] V.H. Alvarez, R. Larco, Y. Ianos and M. Aznar. Brazlan Journal of Chemcal Engneerng (008, 5: [5] B. Goffe, G. Ferrer, and J. Rogers. Journal of Econometrcs (994, 60: [6] A. Corana, M. Marches, C. Martn, and S. Rdella. ACM Transactons on Mathematcal Software (987, 3: [7] J. Kennedy, and R. Eberhart. Partcle swarm optmzaton. In Proceedngs of IEEE Internatonal Conference on Neural Networks (995: [8] P. Uus-Kyyny, J.P. Pokk, Y. Km, and J. Attamaa. Journal of Chemcal Engneerng Data (004, 49: ESAT Santago de Compostela, Span