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1 Ths artle appeared n a journal publshed by Elsever. The attahed opy s furnshed to the author for nternal non-ommeral researh and eduaton use, nludng for nstruton at the authors nsttuton and sharng wth olleagues. Other uses, nludng reproduton and dstrbuton, or sellng or lensng opes, or postng to personal, nsttutonal or thrd party webstes are prohbted. In most ases authors are permtted to post ther verson of the artle e.g. n Word or Tex form to ther personal webste or nsttutonal repostory. Authors requrng further nformaton regardng Elsever s arhvng and manusrpt poles are enouraged to vst:

2 Flud Phase Equlbra Contents lsts avalable at SeneDret Flud Phase Equlbra journal homepage: Constraned and unonstraned Gbbs free energy mnmzaton n reatve systems usng genet algorthm and dfferental evoluton wth tabu lst Adrán Bonlla-Petrolet a, Gade Pandu Rangaah b,, Juan Gabrel Segova-Hernández a Department of Chemal Engneerng, Insttuto Tenológo de Aguasalentes, 256, Mexo b Department of Chemal & Bomoleular Engneerng, Natonal Unversty of Sngapore, Sngapore, , Sngapore Department of Chemal Engneerng, Unversdad de Guanajuato, 350, Mexo artle nfo abstrat Artle hstory: Reeved 12 July 10 Reeved n revsed form 18 Otober 10 Aepted 24 Otober 10 Avalable onlne 3 November 10 Keywords: Phase and hemal equlbrum Smulated annealng Dfferental evoluton wth tabu lst Genet algorthm Global optmzaton Phase equlbrum modelng plays an mportant role n desgn, optmzaton and ontrol of separaton proesses. The global optmzaton problem nvolved n phase equlbrum alulatons s very hallengng due to the hgh non-lnearty of thermodynam models espeally for mult-omponent systems subjet to hemal reatons. To date, a few attempts have been made n the applaton of stohast methods for reatve phase equlbrum alulatons ompared to those reported for non-reatve systems. In partular, the populaton-based stohast methods are known for ther good exploraton abltes and, when optmal balane between the exploraton and explotaton s found, they an be relable and effent global optmzers. Genet algorthms GAs and dfferental evoluton wth tabu lst DETL have been very suessful for performng phase equlbrum alulatons n non-reatve systems. However, there are no prevous studes on the performane of both these strateges to solve the Gbbs free energy mnmzaton problem for systems subjet to hemal equlbrum. In ths study, the onstraned and unonstraned Gbbs free energy mnmzaton n reatve systems have been analyzed and used to assess the performane of GA and DETL. Spefally, the numeral performane of these stohast methods have been tested usng both onventonal and transformed omposton varables as the deson vetor for free energy mnmzaton n reatve systems, and ther relatve strengths are dsussed. The results of these strateges are ompared wth those obtaned usng SA, whh has shown ompettve performane n reatve phase equlbrum alulatons. To the best of our knowledge, there are no studes n the lterature on the omparson of reatve phase equlbrum usng both the formulatons wth stohast global optmzaton methods. Our results show that the effetveness of the stohast methods tested depends on the stoppng rteron, the type of deson varables, and the use of loal optmzaton for ntensfaton stage. Overall, unonstraned Gbbs free energy mnmzaton nvolvng transformed omposton varables requres more omputatonal tme ompared to onstraned mnmzaton, and DETL has better performane for both onstraned and unonstraned Gbbs free energy mnmzaton n reatve systems. 10 Elsever B.V. All rghts reserved. 1. Introduton The aurate modelng of phase equlbrum plays a major role n the desgn, development, operaton, optmzaton and ontrol of hemal proesses. For example, phase behavor has sgnfant mpat on equpment and energy osts of separaton and purfaton proesses n hemal ndustry. Further, solvng phase equlbrum problems s a domnant task n the proess smulaton software. The development of relable methods has long been a hallenge and s stll a researh top of ontnual nterest [1]. The determnaton of the number of phases, ther dentty, and ompos- Correspondng author. Tel.: ; fax: E-mal address: hegpr@nus.edu.sg G.P. Rangaah. ton at equlbrum of mult-omponent systems s a omplex ssue and presents several numeral dffultes [1]. Chemal reatons, f present, nrease the omplexty and dmensonalty of phase equlbrum problems, and so phase splt alulatons n reatve systems are more hallengng due to non-lnear nteratons among phases and reatons [2]. Ths fat has prompted growng nterest n relable and effent methods for the smultaneous omputaton of physal and hemal equlbrum. The phase dstrbuton and omposton at equlbrum of a reatve mxture are determned by the global mnmzaton of Gbbs free energy G subjet to mass balane and hemal equlbrum onstrants [3]. Spefally, the global optmzaton problem for reatve phase equlbrum alulatons follows the form: mnmze F obj u subjet to h j u = 0 for j =1, 2,..., m and u where u s a vetor of ontnuous varables n the doman R n, m s /$ see front matter 10 Elsever B.V. All rghts reserved. do: /j.flud

3 A. Bonlla-Petrolet et al. / Flud Phase Equlbra the number of equalty onstrants related to materal balanes and hemal equlbrum, and F obj u=g: R s a real-valued funton. The doman s defned by the upper and lower lmts of eah deson varable, whh are omposton varables. Ths optmzaton problem an be formulated usng ether onventonal omposton varables.e., mole fratons or numbers, or transformed omposton varables [3 5]. Based on the problem formulaton and the numeral strategy used for ths mnmzaton, the methods an be grouped nto two man ategores: equatonsolvng methods and dret optmzaton strateges. In addton, dependng on the handlng of materal balane onstrants, these strateges an also be lassfed as ether stohometr or nonstohometr [3,6]. In general, lassal strateges for determnng the phase equlbrum of non-reatve systems have been extended and appled to systems subjet to hemal reatons [3]. Equaton-solvng methods are based on the soluton of non-lnear equatons obtaned from the statonary ondtons of the optmzaton rteron. Loal searh methods wth and wthout deouplng strateges are frequently used to solve these equatons n onjunton wth the mass balane and hemal equlbrum restrtons [7]. However, they are prone to severe omputatonal dffultes and may fal to onverge to the orret soluton when ntal estmates are not sutable, espeally for non-deal mult-omponent and mult-reatve systems [1,3,6]. Note that the mnmzaton of G n reatve systems nvolves many omplextes beause t s generally non-onvex, onstraned, hghly non-lnear wth many deson varables, and often has unfavorable attrbutes suh as dsontnuty and non-dfferentablty e.g., when ub equatons of state or asymmetr models are used for modelng thermodynam propertes. Addtonal omplextes arse near the phase boundares, n the vnty of rtal ponts or saturaton ondtons, and when the same model s used for determnng the thermodynam propertes of the mxture [1,3]. As onsequene, G may have several loal mnma nludng trval and non-physal solutons. In these ondtons, onventonal numeral methods are not sutable for performng reatve phase equlbrum alulatons. On the other hand, a number of optmzaton strateges for performng the mnmzaton of G n reatve systems have been proposed, and they omprse loal and global methods e.g., [2,3,5,6,8 23]. The use of Lagrange multplers s usually the preferred approah for G mnmzaton but ts performane s hghly dependent on ntal estmates of Lagrange multplers [17]. There has been sgnfant and nreasng nterest n the development of determnst and stohast global strateges for relably solvng reatve phase equlbrum problems. Studes on determnst reatve phase equlbrum alulatons have been foused on the applaton of the lnear programmng [9,21], branh and bound global optmzaton [11], homotopy ontnuaton methods [13,15,], and nterval analyss usng an nterval-newton/generalzed bseton algorthm [16]. Although these methods have proven to be promsng, some of them are model-dependent, may requre problem reformulaton or sgnfant omputatonal tme for mult-omponent systems [1,24]. Alternatvely, stohast optmzaton tehnques have often been found to be as relable and effetve as determnst methods. Further, they offer more advantages for the global optmzaton of G. These methods are robust, requre a reasonable omputatonal effort for the optmzaton of multvarable funtons generally less tme than determnst approahes, applable to ll-struture or unknown struture problems, requre only objetve funton alulatons and an be used wth all thermodynam models. In fat, t appears that they may fulfll the requrements of an deal algorthm: relablty, generalty and effeny. To date, a few attempts have been made n the applaton of stohast methods for reatve phase equlbrum alulatons, ompared to those reported for non-reatve systems [14,19,22,23,25]. Spefally, Lee et al. [14] ntrodued the applaton of the random searh method of Luus and Jaakola for the global mnmzaton of G usng a nonstohometr formulaton. On the other hand, Bonlla-Petrolet et al. [19] formulated the unonstraned optmzaton problem for G mnmzaton usng smulated annealng SA and transformed omposton varables. In another study, partle swarm optmzaton PSO and several of ts varants have been appled for reatve phase equlbrum alulatons usng transformed omposton varables [22]. Reently, our group [23] has tested and ompared the performane of dfferental evoluton DE and tabu searh TS for the global mnmzaton of G usng reaton-nvarant omposton varables. Fnally, Reynolds et al. [25] outlned a general proedure for the global optmzaton of G n reatve systems usng SA and a non-stohometr approah. Results of these studes have shown the potental of stohast optmzaton solvers for phase equlbrum alulatons subjet to hemal reatons. In partular, the populaton-based stohast methods are known for ther good exploraton abltes; when optmal balane between the exploraton and explotaton s found, they an be relable and effent global optmzers. Ths s beause at eah generaton/teraton a whole populaton of potental solutons s mproved rather than a sngle soluton. A varety of populaton-based stohast methods have been proposed for hemal engneerng applatons nludng the modelng of phase equlbrum, e.g. [26 29]. Spefally, genet algorthms GAs and dfferental evoluton wth tabu lst DETL have been very suessful for performng phase equlbrum alulatons n non-reatve systems [26,28]. However, to the best of our knowledge, there are no studes on the performane of both these strateges for G mnmzaton n systems subjet to hemal equlbrum. These methods are sutable and promsng for overomng the numeral dffultes of ths global optmzaton problem. In ths study, the onstraned and unonstraned Gbbs free energy mnmzaton n reatve systems have been analyzed and used to assess the performane of GA and DETL. Spefally, the numeral performane of these stohast methods have been tested usng both onventonal and transformed omposton varables as the deson vetor for G mnmzaton, and ther relatve strengths are dsussed. The results of GA and DETL are ompared wth those obtaned usng SA, whh has shown a ompettve performane n reatve phase equlbrum alulatons [19]. Our results on a varety of reatve systems ndate that DETL s superor to SA and GA for both the onstraned and unonstraned Gbbs free energy mnmzaton n reatve systems. 2. Formulaton of the Gbbs free energy mnmzaton n reatve systems 2.1. Gbbs free energy funton Classal thermodynams ndates that, at onstant temperature T and pressure P, the equlbrum for a mult-omponent and mult-phase system s aheved when the G funton s at the global mnmum [1]. Ths thermodynam funton s expressed as a lnear ombnaton of the hemal potental of eah omponent n eah phase, then G = n j j 1 where n j s the number of moles of omponent present n phase j and j s the hemal potental of omponent n phase j, respetvely. For reatve phase equlbrum, the mass balane restrtons and non-negatvty requrements are usually formulated usng the

4 122 A. Bonlla-Petrolet et al. / Flud Phase Equlbra onservaton of hemal elements n the omponents [3,14] d l n j = b l l = 1,...,m e 2 where d l represents the number of gram-atoms of element l n omponent, b l s the total number of gram-atoms of element l n the system, and m e s the number of elements, respetvely. So, the bounds on n j are gven by 0 d l n j b l = 1,...,; j = 1,...,; l = 1,...,m e 3 Therefore, to determne the phase equlbrum ompostons n reatve systems, t s neessary to fnd the global mnmum of Eq. 1 wth respet to n j subjet to onstrants gven by Eq. 2 and n the regon bounded by Eq. 3. The expressons for G and ts mathematal propertes depend ompletely on the struture of the thermodynam equatons hosen to model eah of the phases that may exst at equlbrum [26]. Alternatvely, the G funton n reatve systems an be expressed n terms of transformed omposton varables that have been ntrodued by dfferent researh groups [4,5] to provde a smpler framework for treatng reatve systems. As stated by Ung and Doherty [4], mole numbers are not the natural omposton varables to use n the modelng of reatve systems beause they do not have the same dmensonalty as the number of degrees of freedom.e., they are nonsstent wth respet to the Gbbs phase rule. Therefore, n ths study, we have appled the reatonnvarant omposton varables proposed by Ung and Doherty [4]. These varables are based on the transformaton of physal ompostons, restrt the soluton spae to the ompostons that satsfy stohometry requrements, and also redue the dmenson of the omposton spae by the number of ndependent reatons satsfyng the Gbbs phase rule. These features allow all the proedures and algorthms used to model non-reatve mxtures to be easly modfed and extended to systems subjet to hemal reatons [4]. For a system of omponents that undergo r ndependent hemal reatons, the transformed mole numbers ˆn are defned by seletng r referene omponents ˆn = n v N 1 n ref for = 1,..., r 4 where n s the number of moles of omponent, v s the row vetor of dmenson r of stohometr oeffents of omponent n r reatons, N s an nvertble, square matrx formed from the stohometr oeffents of r referene omponents n r reatons, and n ref s a olumn vetor of dmenson r of moles of eah of the referene omponents. The transformed mole fratons X are gven by X = ˆn = x v N 1 x ref ˆn T 1 v TOT N 1 for = 1,..., r 5 x ref where x s the mole fraton of omponent, x ref s a olumn vetor r of mole fratons of r referene omponents, ˆn T = ˆn, and v TOT s a row vetor of dmenson r where eah element orresponds to the sum of stohometr oeffents of all omponents n eah of the r reatons. The transformed mole fratons X n reatve systems are smlar to the mole fratons x n non-reatve mxtures, and the sum r of all transformed mole fratons s equal to unty.e., X = 1, but a transformed mole fraton an be negatve or postve dependng on the referene omponents, number and type of reatons. It s mportant to note that the set of X and ˆn has the desrable property of takng the same numeral values before and after the reatons. Ths s n ontrast to onventonal mole varables x and n, whh have dfferent values for the omponents n the unmxed and mxed.e., reatng states [4]. The transformed varables X are related to x va the reaton equlbrum onstants K eq,k : K eq,k = a v k k = 1,...,r 6 where v k s the stohometr oeffent of omponent n reaton k, and a s the atvty of omponent. To evaluate thermodynam propertes n reatve systems usng ths approah, mole fratons are obtaned from the transformaton proedure X x usng Eqs. 5 and 6, whh requres soluton of one or more nonlnear equatons. The resultng mole fraton values x satsfy the stohometry requrements and are hemally equlbrated [4].In our study, bseton method s used to perform the omposton transformaton. Note that multple solutons are not possble for x ref durng varable transformaton X x beause only one soluton set of x smultaneously satsfes the hemal equlbrum equatons and orresponds to the spefed values of the transformed omposton varables [4]. For more detals on ths transformaton proedure, see our reent work [23]. For a reatve mxture, mnmzng the Gbbs free energy wth respet to n j s equvalent to mnmzng the transformed Gbbs free energy Ĝ wth respet to ˆn j [4]. For a mult-phase reatve system, Ĝ s defned as r Ĝ = ˆn j j 7 where ˆn j s the transformed mole numbers of omponent n phase j. In transformed omposton spae, the materal balanes are gven by ˆn j = Z ˆn F = 1,..., r 8 j = 1 0 ˆn j Z ˆn F = 1,..., r; j = 1,..., 9 where ˆn F s the total amount of transformed moles n the feed, and Z s the orrespondng transformed mole fraton of omponent. So, the transformed phase ompostons at equlbrum are determned by the global mnmzaton of Eq. 7 subjet to onstrants mposed by Eq. 8 n the feasble regon defned by Eq. 9. Note that ths formulaton requres the transformaton proedure X x for evaluatng the objetve funton value. The global mnmzaton of G and Ĝ s dffult and requres robust numeral methods sne these funtons are multvarable, non-onvex and hghly non-lnear. In ths study, two dfferent optmzaton approahes.e., onstraned and unonstraned, usng the onventonal and transformed omposton varables, are adopted for global optmzaton of Gbbs free energy. To the best of our knowledge, there are no studes n the lterature on the omparson and modelng of phase equlbrum n reatve systems usng both formulatons wth ether determnst or stohast global optmzaton methods. In the followng seton, formulatons for both onstraned and unonstraned optmzaton problems are desrbed Constraned mnmzaton approah For modelng reatve systems, the hemal equlbrum ondton an be evaluated from ether Gbbs free-energy data or hemal equlbrum onstants determned expermentally [30]. In suh ases, we an use dfferent objetve funtons for the onstraned mnmzaton of Gbbs energy funton. In prate, Gbbs

5 A. Bonlla-Petrolet et al. / Flud Phase Equlbra free-energy data are not avalable at tested ondtons and the use of hemal equlbrum onstants obtaned from expermental measurements s more onvenent. Based on ths and to perform a dret omparson of results obtaned employng both onventonal and transformed varables as the deson vetor, we have used a G funton defned n terms of hemal equlbrum onstants K eq,k. Ths objetve funton s derved from the relatonshp between G and Ĝ [31]. However, n our analyss, Gbbs free energy of mxng s used to avod the alulaton of pure omponent free energes, whh do not nfluene the equlbrum and stablty results [4]. For a mult-phase and mult-omponent reatve system, the transformed Gbbs free energy of mxng s defned as [4,15,31] r ĝ = ˆn j lnx j j 10 where j and x j are respetvely the atvty oeffent and mole fraton of omponent n phase j. Usng the defnton of transformed omposton varables Eq. 4, ĝ an be wrtten as [31]: r ĝ = n j v N 1 n ref,j lnx j j = [ n j lnx j j = r+1 v N 1 n ref,j lnx j j n j v N 1 n ref,j lnx j j ] 11 Defne an unt, row vetor e = 0,...,1,..., 0 of length r wth all elements zeros exept one n the th poston. For all omponents seleted as referene for the transformed omposton varables, the vetor v an be alulated as follows v = e N = r + 1,..., 12 Ths expresson s appled to the matrx produt v N 1 n ref,j =e NN 1 n ref,j =n j = r + 1,...,; j = 1,..., 13 Ths result gves [n j v N 1 n ref,j lnx j j ] = 0 j = 1,..., 14 = r+1 Sne ln j x j s a salar quantty, we have v N 1 n ref,j lnx j j = v lnx j j N 1 n ref,j [ ] = v lnx j j N 1 n ref,j 15 Reall that v s the row vetor of dmenson r of stohometr oeffents of omponent n r reatons. So, element k of the row vetor v lnx j j an be re-wrtten as v k lnx j j = lnx j j v k =ln x j j v k =ln K eq,k 16 Substtutng Eq. 16 nto Eq. 15, yelds v N 1 n ref,j lnx j j = ln K eq N 1 n ref,j 17 where ln K eq s a row vetor of logarthms of hemal equlbrum onstants for all r ndependent hemal reatons. Then, ĝ Eqs. 10 and 11 beomes F G = ĝ = g ln K eq N 1 n ref,j 18 where g = n j lnx j j s the Gbbs free energy of mxng. Eq. 18 s an alternatve objetve funton nvolvng reaton equlbrum onstants, for performng phase equlbrum alulatons n reatve systems. Ths objetve funton must be globally mnmzed subjet to mass balane restrtons. In ths ontext, mass balane equatons an be rearranged to redue the number of deson varables of the optmzaton problem [14] and to elmnate equalty onstrants whh are hallengng for stohast algorthms. The hange n the number of moles of eah reatng omponent, whle the set of reatons s proeedng, s gven by n j = n F + v ε for =1,..., where n,f s the ntal moles of omponent n the feed and ε s the vetor of the r extents of reatons for eah of the r reatons. Under these ondtons, a set of r referene omponents an be hosen to fnd the r extents of reaton n the followng way [4]: ε = N 1 n ref n ref,f 19 Reall that N s an nvertble, square matrx formed from the stohometr oeffents of the referene omponents n the r reatons, and n ref s a olumn vetor of moles of eah of the referene omponents. Therefore, we an establsh that n j v N 1 n ref,j = n F v N 1 n ref,f = 1,..., r Thus, the mass balane restrtons an be used as follows to redue the number of deson varables. 1 n = n F v N 1 n ref,f n ref, n j v N 1 n ref,j = 1,..., r 21 Usng Eq. 21, the deson varables are 1 + r mole numbers n j. Then, the global optmzaton problem an be solved by mnmzng F G wth respet to 1 + r deson varables n j and the remanng r mole numbers n are determned from Eq. 21 and subjet to the nequalty onstrants n > 0. Note that the bounds on deson varables are gven by Eq. 3. The global mnmzaton of F G s a onstraned optmzaton problem. The searh spae n onstraned optmzaton problems onssts of both feasble and nfeasble ponts. In reatve phase equlbrum alulatons, feasble ponts satsfy all the mass balane onstrants, Eq. 21 and bounds, Eq. 3, whle nfeasble ponts volate at least one of them.e., n < 0 where =1,..., r. In ths study, the penalty funton approah [32,33] s used to solve the onstraned Gbbs free energy mnmzaton n reatve systems. It s one of the popular tehnques for handlng onstrants n the stohast methods. Ths method s easy to mplement and s onsdered effent. It transforms the onstraned problem nto an unonstraned problem by penalzng nfeasble solutons. In our alulatons, an absolute value of onstrant volaton s multpled wth a hgh penalty weght and then added to the orrespondng Gbbs free energy funton. In ase of more than one onstrant volaton, all onstrant volatons are frst multpled wth the penalty weght, and all of them are added to the objetve funton value.

6 124 A. Bonlla-Petrolet et al. / Flud Phase Equlbra Spefally, the penalty funton s gven by { FG f n j > 0 = 1,...,; j = 1,...,, F 1 = 22 F G + p otherwse, where p s the penalty term whose value s postve. For phase equlbrum alulatons, nfeasble solutons.e., n < 0 mply that the Gbbs free energy funton of phase annot be determned due to the logarthm terms of the atvty or fugaty oeffents. Takng nto aount the nequalty onstrants n > 0, the penalty term s defned as n unf p = 10 n 23 where n s obtaned from Eq. 21 and n unf s the number of nfeasble mole fratons.e., n < 0 where =1,..., r. Ths penalty term s straghtforward and prelmnary alulatons ndate that ts value s approprate for handlng nfeasble solutons n the onstraned Gbbs free energy mnmzaton n systems subjet to hemal reatons Unonstraned mnmzaton approah Alternatvely, Gbbs free energy funton an be wrtten n terms of reaton-nvarant omposton varables ˆn j to transform the problem nto an unonstraned form. Spefally, to perform an unonstraned mnmzaton of F 2 = ĝ Eq. 10, we an use a set of new varables, namely, ˇj as deson varables. The ntroduton of these varables elmnates the restrtons mposed by materal balanes, redues problem dmensonalty, and the optmzaton problem s transformed nto an unonstraned one [19,26 28]. However, ths wll requre the soluton of nonlnear equatons for evaluatng the objetve funton due to the transformaton proedure X x usng Eqs. 5 and 6. For mult-phase reatve systems, real varables ˇj 0, 1 are defned and employed as deson varables by usng the followng expressons ˆn 1 = ˇ1 Z ˆn F = 1,..., r 24 j 1 ˆn j = ˇj Z ˆn F ˆn m m=1 = 1,..., r; j = 2,..., ˆn = Z ˆn F ˆn j = 1,..., r 26 Usng ths formulaton, equalty onstrants, Eqs. 8 and 9 are elmnated and all tral transformed ompostons satsfy the materal balanes allowng the easy applaton of optmzaton strateges. For the unonstraned mnmzaton of ĝ, Eq. 10, the overall number of deson varables.e., ˇj s r 1. Fnally, t s mportant to note that the global optma of F 1 and F 2 are equal but ther deson varables are dfferent n j and ˇj, respetvely. 3. Desrpton of stohast optmzaton methods In ths study, we used three methods: SA, GA and DETL for the global mnmzaton of onstraned and unonstraned Gbbs free energy funtons usng both onventonal.e., onstraned mnmzaton approah and transformed omposton varables.e., unonstraned mnmzaton approah. SA has reently been used for reatve systems n [19,25], whereas DETL and GA have not yet been tred for phase equlbrum alulatons n reatve systems. SA s a pont-to-pont method whle GA and DETL are populaton-based methods. These algorthms are desrbed brefly n the followng subsetons, and ther detaled desrpton and flowharts s avalable n the ted Referenes 3.1. Smulated annealng SA s a stohast method that mms the thermodynam proess of oolng molten metals to attan the lowest free energy state [34]. In the mnmzaton problems, ths algorthm performs a stohast searh of the spae defned for deson varables where uphll moves may be aepted wth a probablty ontrolled by the parameter alled annealng temperature: T SA. The probablty of aeptane of uphll moves dereases as T SA dereases. At hgh T SA, the searh s almost random, whle at low T SA the searh beomes seletve where good moves are favored. The ore of SA algorthm s the Metropols rteron [35] used to aept or rejet uphll movements wth the aeptane probablty gven by MT SA = mn { 1, exp f T SA } 27 where f s the hange n objetve funton value from the urrent pont to new/tral pont. In ths study, the SA algorthm proposed by Corana et al. [36] has been used beause of ts good performane n thermodynam alulatons, e.g., [19,26,37]. In ths algorthm, a tral pont s randomly hosen wthn the step length VM whh s a vetor of length n var from the urrent pont. The objetve funton s evaluated at ths tral pont, and ts value s ompared to the objetve value at the urrent pont. Eq. 27 s used to aept or rejet the tral pont. If ths tral pont s aepted, the algorthm ontnues the searh usng that pont; otherwse, another tral pont s generated wthn the neghborhood of the urrent pont. Eah element of VM s perodally adjusted so that half of all funton evaluatons n that dreton are aepted. A fall n T SA, after NT NS n var funton evaluatons, s mposed upon the system usng the oolng shedule. Note that NT s the number of teratons before T SA reduton and NS s the number of yles for updatng the deson varables. In our alulatons, oolng shedule for dereasng T SA s defned as 17k T SA,k = 0.5T SA,0 T SA,F 1 tanh 5 + T SA,F 28 Iter max where Iter max s the maxmum number of teratons for SA, T SA,k s the annealng temperature at teraton k, and T SA,0 and T SA,F are respetvely the ntal and fnal values of the annealng temperature. Thus, as T SA delnes, downhll moves are less lkely to be aepted and SA fouses on the most promsng area for optmzaton. The teratve steps are performed untl the spefed stoppng rteron: ether the maxmum number of suessve teratons S max wthout mprovement n the best funton value, or untl the maxmum number of teratons Iter max, s satsfed. The man parameters of SA are T SA,0, T SA,F, NS, NT, S max and Iter max. Detaled desrpton of ths algorthm an be found n Corana et al. [36].We have used, after sutable modfatons, the subroutne developed by Goffe et al. [38], for the present study Genet algorthm GA s a stohast tehnque that smulates natural evoluton on the soluton spae of the optmzaton problem. It operates on a populaton of potental solutons ndvduals n eah teraton.e., generaton. Spefally, the frst step of GA s to reate randomly an ntal populaton of NP solutons n the feasble regon. GA

7 A. Bonlla-Petrolet et al. / Flud Phase Equlbra works on ths populaton, and ombnes rossover and modfes mutaton some hromosomes aordng to spefed genet operatons, to generate a new populaton wth better haratersts. Indvduals for reproduton are seleted based on ther objetve funton values and the Darwnan prnple of the survval of the fttest [39]. Genet operators are used to reate new ndvduals for the next populaton from the seleted ndvduals of the urrent populaton, and they serve as the searhng mehansms n GA. In partular, rossover forms two new ndvduals by frst hoosng two ndvduals from the matng pool ontanng the seleted ndvduals and then swappng dfferent parts of genet nformaton between them. Ths ombnng rossover operaton takes plae wth a userdefned rossover probablty P ros so that some parents reman unhanged even f they are hosen for reproduton. Mutaton s an unary operator that reates a new soluton by a random hange n an ndvdual wth a probablty P mut. It ensures that the probablty of searhng any gven strng wll never be zero and atng as a safety net to reover good genet materal whh may be lost through the aton of seleton and rossover. Seleton, rossover and mutaton proedures are reursvely used to mprove the populaton and to dentfy promsng areas for optmzaton. GA termnates when the user-spefed rteron s satsfed. For omparson purposes, the stoppng ondtons desrbed for SA have been mplemented n all stohast methods tested n ths work. Spefally, GA stops after evolvng for the spefed number of generatons Gen max, or untl performng the maxmum number of suessve generatons S max wthout mprovement n the best objetve value. We have used GA wth floatng-pont enodng, seleton va stohast unversal samplng, modfed arthmet rossover and non-unform mutaton. Detals of ths algorthm are avalable n Rangaah [26]. The key parameters of GA are NP, P ros, P mut, Gen max and S max Dfferental evoluton wth tabu lst Ths reent stohast method developed by Srnvas and Rangaah [28] s a hybrd strategy obtaned from dfferental evoluton DE and tabu searh TS. DETL begns wth the seleton of values for parameters: populaton sze NP, amplfaton fator A, rossover onstant CR, tabu radus tr, tabu lst sze tls, Gen max and S max. The algorthm generates the ntal populaton of sze NP usng unformly dstrbuted random numbers to over the entre feasble regon. The objetve funton s evaluated at eah ndvdual/pont, and the best one s seleted. The tabu onept of TS s mplemented n the generaton step of DE.e., after rossover and mutaton to mprove the dversty among the ndvduals and onsequently the omputatonal effeny. It employs a tabu lst wth the parameters: tr and tls, to keep trak of the evaluated ponts for avodng revsts to them durng the subsequent searh. The three man steps: mutaton, rossover and seleton of DE along wth tabu hekng are performed on the populaton durng eah generaton. For ths, a mutant ndvdual s generated for eah randomly hosen target ndvdual X,j n the populaton by V,G+1 = X R1,G + AX R2,G X R3,G = 1, 2, 3,...,NP 29 where random numbers R 1, R 2 and R 3 are dstnt and belong to the set {1, 2, 3,..., NP}, and X R1,G, X R2,G and X R3,G represent the three random ndvduals hosen from the urrent generaton, to produe the mutant vetor for the next generaton, V,G+1. The random numbers should be dfferent from the runnng ndex,, and hene NP should be 4 to allow mutaton. Parameter A s a real value between 0 and 2, and t ontrols the amplfaton of the dfferental varaton between the two random ndvduals. In the rossover step, a tral ndvdual/vetor s generated by opyng some elements of the mutant ndvdual to the target ndvdual wth a probablty of CR. A boundary volaton hek s performed to hek the feasblty of the resultng tral ndvdual; f any bound s volated, the tral ndvdual s replaed by generatng a new ndvdual. The tral ndvdual s then ompared to the already evaluated ponts n the tabu lst n terms of the Euldean dstane. If the Euldean dstane s smaller than the tabu radus, whh ndates that the objetve funton value at the tral vetor and at one of the ponts n the tabu lst are omparable, the tral ndvdual s rejeted as t may not gve new nformaton about the objetve funton exept nreasng the number of funton evaluatons. The rejeted pont s replaed by generatng another tral pont by mutaton and rossover operatons, untl the Euldean dstane between the new pont and eah of the ponts n the tabu lst s greater than the tabu radus. Whenever a tral ndvdual s rejeted, the number of rejeted ndvduals N fal at the urrent generaton s updated. The objetve funton s evaluated at the tral ndvdual only f t s away from all the ponts n the tabu lst and f N fal <15n var.in ths algorthm, the parameter N fal s used to avod ndefnte ylng n the generaton step. After eah evaluaton, the tabu lst s updated dynamally to keep the latest ponts n the lst by replang the earlest entered ponts. In the seleton step, objetve funton value s used to selet the better one between the tral and target ndvduals. If the tral ndvdual s seleted, t replaes the target ndvdual n the populaton mmedately and may partpate n the subsequent mutaton and rossover operatons. If the target ndvdual s better, then t remans n the populaton and may partpate n the subsequent mutaton and rossover operatons. The algorthm runs untl the stoppng rteron Gen max or S max s satsfed, and gves the best pont obtaned over all the generatons. More detals on DETL algorthm an be found n Srnvas and Rangaah [28] Implementaton of the methods In the present study, FORTRAN odes developed for the three stohast algorthms were used. All odes are avalable to nterested readers upon request to the orrespondng author. Eah method has been mplemented n ombnaton wth a loal optmzaton tehnque at the end of global searh. Spefally, the best pont dentfed by the stohast algorthm s used as the ntal guess for loal optmzaton. Ths s beause stohast optmzaton methods may requre a sgnfant omputatonal effort to mprove the auray of the soluton sne they explore the searh spae by reatng random movements nstead of usng a logal optmzaton trajetory. Therefore, ntensfaton step usng a loal optmzer s needed for rapd onvergene and for mprovng auray of the best soluton obtaned wth a stohast method. In ths study, performane of SA, DETL and GA s tested for onstraned and unonstraned Gbbs free energy mnmzaton wth and wthout loal optmzaton. The quas-newton method mplemented n the subroutne DBCONF of IMSL lbrary was used for loal optmzaton. Ths subroutne alulates the gradent va fnte dfferenes and approxmates the Hessan matrx aordng to BFGS formula. For more detals on ths loal strategy, see the book by Denns and Shnabel []. The default values of DBCONF parameters n the IMSL lbrary were employed. Prelmnary alulatons ndate that these parameter settngs are a reasonable ompromse between numeral effort and relablty for ntensfaton stage. All alulatons were performed on a HP Workstaton wth Dual-Core AMD Opteron 2.19 GHz proessor wth 1.87 GB of RAM. Ths omputer performs 254 mllon floatng pont operatons per seond for the LINPACK benhmark program avalable at for a matrx of order 500.

8 126 A. Bonlla-Petrolet et al. / Flud Phase Equlbra Table 1 Examples seleted for the onstraned and unonstraned Gbbs free energy mnmzaton n reatve systems. No. System Feed ondtons Thermodynam models Ref. 1 A 1 + A 2 A 3 + A 4 1 Ethanol 2 Aet ad 3 Ethyl aetate 4 Water 2 A 1 + A 2 A 3, and A 4 as an nert omponent 1 Isobutene 2 Methanol 3 Methyl ter-butyl ether 4 n-butane 3 A 1 + A 2 +2A 3 2A Methyl-1-butene 2 2-Methyl-2-butene 3 Methanol 4 Tert-amyl methyl ether 4 A 1 + A 2 A 3 + A 4 1 Aet ad 2 n-butanol 3 Water 4 n-butyl aetate n F = 0.5, 0.5, 0.0, 0.0 at 355 K and kpa n F = 0.3, 0.3, 0.0, 0.4 at K and kpa n F = 0.354, 0.183, 0.463, 0.0 at 335 K and kpa n F = 0.3, 0.4, 0.3, 0.0 at K and kpa NRTL model and deal gas. K eq,1 = Wlson model and deal gas. = T T ln T G 0 rxs R ln K eq,1 = G0 rxs RT where T s n K. Wlson model and deal gas. K eq,1 = e /T where T s n K. UNIQUAC model. ln K eq,1 = 450 T where T s n K. 5 A 1 + A 2 A 3 n F = 0.6, 0.4, 0.0 Margules soluton model. g E = 3.6x1x x1x x2x3 Rg T K eq,1 = A 1 + A 2 +2A 3 2A 4 wth A 5 as nert n F = 0.1, 0.15, 0.7, 0.0, 0.05 at Wlson model and deal gas. omponent 335 K and kpa K eq,1 = e /T where T s n K. 1 2-Methyl-1-butene 2 2-Methyl-2-butene 3 Methanol 4 Tert-amyl methyl ether 5 n-pentane 7 A 1 + A 2 A 3 n F = 0.52, 0.48, 0.0 at K and Margules soluton model kpa K eq,1 = A 1 + A 2 A 3 + A 4 n F = 0.048, 0.5, 0.452, 0.0 at 3 K and kpa NRTL model. K eq,1 = 4.0 [2,11,14] [41] [19] [15,43] [42] [19] [4] [43] Table 2 Problem formulaton for the onstraned and unonstraned Gbbs free energy mnmzaton n seleted reatve systems. No. Constraned optmzaton Unonstraned optmzaton F 1 a n var Deson varables F 2 b Referene omponent n var Deson varables 1, 4, 8 g n 4,1 + n 4,2lnK eq,1 5 n,1 for =1,..., 4 and n 4,2 ĝ A 4 3 ˇ,1 for =1,2,3 2 g n 3,1 + n 3,2lnK eq,1 5 n,1 for =1,..., 4 and n 3,2 ĝ A 3 3 ˇ,1 for =1,2,4 3 g 0.5n 4,1 + n 4,2lnK eq,1 5 n,1 for =1,..., 4 and n 4,2 ĝ A 4 3 ˇ,1 for =1,2,3 5, 7 g n 3,1 + n 3,2lnK eq,1 4 n,1 for =1,..., 3 and n 3,2 ĝ A 3 2 ˇ,1 for =1,2 n,1 lnx,1,1 + n,2 lnx,2 P/P,sat for VLE and g =,1 lnx,1.1 + n,2 lnx,2,2 for LLE. r r 6 g 0.5n 4,1 + n 4,2lnK eq,1 6 n,1 for =1,..., 5 and n 4,2 ĝ A 4 4 a Where g = ˇ,1 for =1,2,3,5 b Where ĝ =,1 lnx,1,1 + ˆn,2 lnx,2 P/P,sat for VLE and ĝ = ˆn,1 lnx,1.1 + ˆn,2 lnx,2,2 for LLE. 4. Results and dsusson 4.1. Desrpton of reatve phase equlbrum problems We have tested and ompared the performane of SA, GA and DETL usng a number of reatve systems and dfferent thermodynam models. The test problems nlude systems wth vapor lqud VL and lqud lqud LL equlbrum. Detals.e., feed ondtons, thermodynam models, objetve funton, deson varables and global optmum of all examples are reported n Tables 1 3. Parameters of thermodynam models for these reatve systems are gven n Appendx A. Most of the seleted reatve systems have been used for testng other determnst and stohast optmzaton strateges, e.g., [2,4,11,14,15,18,22,23,41 43]. In all examples, the number of phases exstng at the equlbrum s assumed to be known a pror. In general, the seleted reatve phase equlbrum problems have dfferent dmensonalty and nherent dffultes e.g., dsontnutes n objetve funton for VL equlbrum problems, or feed omposton near phase boundares whh are generally hallengng for any algorthm. Therefore, we onsder that the number and features of the test problems are suffent to demonstrate and ompare the performane of SA, GA and DETL for solvng reatve phase equlbrums problems va unonstraned and onstraned approahes. Table 3 Global mnmum of the reatve examples studed. No. Equlbrum Global mnmum of F 1 and F 2 1 Vapor Lqud Vapor Lqud Vapor Lqud Lqud Lqud Lqud Lqud Vapor Lqud Lqud Lqud Lqud Lqud Note: Global solutons of all problems an be found n Refs. [2,11,14,15,18,19,23].

9 A. Bonlla-Petrolet et al. / Flud Phase Equlbra The performane of all stohast methods s evaluated based on both relablty measured n terms of number of tmes the algorthm loated the global mnmum out of trals wth random ntal values, refereed to as suess rate SR and omputatonal effeny measured n terms of average number of funton evaluatons NFE and CPU tme. Note that NFE nludes both the funton alls for evaluatng the objetve funton usng the stohast method NFE st and the funton alls for the loal optmzaton NFE qn. The average NFE and CPU tme are evaluated usng suessful trals only. A tral s onsdered suessful f the global optmum s obtaned wth an absolute error of 10 5 or less n the objetve funton value. To ompare the performane of stohast algorthms and to analyze ther relatve merts for reatve phase equlbrum alulatons, we have onsdered Eqs. 10 and 22 as the objetve funton for unonstraned and onstraned Gbbs free energy mnmzaton, respetvely Parameter tunng of SA, GA and DETL Reatve examples 1 and 4 have been used to establsh the most sutable parameter values for solvng the onstraned and unonstraned Gbbs free energy mnmzaton problems effently and relably. Parameter tunng was arred out by varyng one parameter at a tme wth the remanng parameters fxed at nomnal values, whh were establshed usng the reported results n the lterature [19,26,28]. The tested and suggested values for parameters of eah stohast method are summarzed n Table 4. Our prelmnary alulatons suggest that these parameter values are a reasonable ompromse between numeral effort and relablty of tested stohast methods Performane of SA, GA and DETL The three stohast methods were studed usng two stoppng rtera: a maxmum number of teratons/generatons Iter max or Gen max referred to as stoppng rteron 1, SC1 and b maxmum Table 4 Tested and suggested values of parameters n the stohast optmzaton methods for the onstraned and unonstraned Gbbs free energy mnmzaton n reatve systems. Method Parameter a Tested values Suggested values SA T SA, T SA,F GA P ros P mut DETL CR A tr 0.001n var 0.01n var 0.001n var tls a NS NT =NP=10n var where n var s the number of deson varables for the Gbbs free energy mnmzaton problems. number of teratons/generatons wthout mprovement n the best funton value S max referred to as stoppng rteron 2, SC2. The methods are ompared n terms of SR and NFE by examnng dfferent levels of algorthm effeny, whh are obtaned by hangng the values of Iter max /Gen max and S max. Note that optmal values of these parameters may be problem dependent, and they also determne the trade-off between effeny and relablty. As a onsequene, seleton of proper values for them s mportant for the omparson. For all alulatons performed n ths study, NS NT =NP=10n var where n var s the number of deson varables used n reatve phase equlbrum alulatons. The performane of SA, DETL and GA mplemented wth and wthout the loal optmzaton method s gven n Fgs. 1 and 2. For the sake of brevty, algorthm relablty results are summarzed as the global suess rate GSR, defned as the average suesses rate on the olleton of reatve phase equlbrum problems tested: GSR = 1 N prob SR N 30 prob where SR s the suess rate on the th problem. SA DETL GA a F 1 GSR - - b F Iter max / Gen max Stohast method ombned wth loal optmzaton Stohast method Fg. 1. Global suess rate GSR versus Iter max/gen max wthout usng S max of SA, DETL and GA for the a onstraned and b unonstraned Gbbs free energy mnmzaton n reatve systems. Algorthm parameters: NS NT = NP =10n var.

10 128 A. Bonlla-Petrolet et al. / Flud Phase Equlbra SA DETL GA a F 1 GSR b F n var 12n var 24n var 6n var 12n var 24n var S max S max Stohast method Stohast method ombned wth loal optmzaton Fg. 2. Global suess rate GSR versus S max of SA, DETL and GA for the a onstraned and b unonstraned Gbbs free energy mnmzaton n reatve systems. Algorthm parameters: NS NT = NP =10n var. Our results ndate that the relablty of SA, DETL and GA for Gbbs free energy mnmzaton vares sgnfantly wth the stoppng ondton, the problem formulaton.e., onstraned and unonstraned, and the use of loal optmzaton method see Fgs. 1 and 2. In general, DETL and SA an aheve hgh GSR values, and ther performane s usually better than or omparable to that of GA usng ether SC1 or SC2 as the stoppng ondton, wth and wthout the quas-newton method, n both onstraned and unonstraned mnmzaton approahes. The stohast methods may fal to fnd the global mnmum of Gbbs free energy funton usng both onventonal and transformed omposton varables see Fgs. 1 and 2. These falures may be due to the presene of loal optma and/or flat objetve funton near the global soluton n some reatve problems. As expeted, the relablty and omputatonal effort of the stohast methods nrease wth Iter max /Gen max and S max see Fgs. 1, 2 and 4 and Tables 5 and 6. Even though the stoppng rteron used for all stohast methods s the same, NFE requred by DETL s generally less than that of both SA and GA beause the tabu hekng and the parameter N fal are mplemented n the generaton step of DETL. SR of the methods s affeted when the stoppng ondtons are lmted to low values.e., early teratons espeally for onstraned funton F 1 and wthout usng quas-newton method. Applaton of loal optmzaton method for the ntensfaton stage s mportant to mprove GSR espeally n onstraned Gbbs free energy mnmzaton usng ether SC1 or SC2 as the stoppng ondton. Wthout loal optmzaton, SA outperformed the DETL and GA n solvng reatve phase equlbrum problems espe- Table 5 Perentage reduton n NFE of DETL wth loal optmzaton for solvng the onstraned F 1 and unonstraned F 2 Gbbs free energy problems n reatve systems wth SC1 alone as the stoppng rteron. Problem F obj Perentage reduton n NFE for Iter max or Gen max a F F F F F F F F F F F F F F F F a % reduton = NFE of DETL mean NFE of SA and GA/mean NFE of SA and GA. Note that the NFE of SA and GA s omparable.

11 A. Bonlla-Petrolet et al. / Flud Phase Equlbra Table 6 NFE and SR of SA, DETL and GA wth loal optmzaton for solvng the onstraned F 1 and unonstraned F 2 Gbbs energy mnmzaton problems n reatve systems usng SC2 alone as the stoppng rteron. Problem S max NFE wth SR n brakets for a Constraned F 1 Unonstraned F s SA DETL GA SA DETL GA 1 6n var n var 10, n var 44,083 11,548 13, n var n var 10, n var 35, , n var n var n var 32,630 10,893 12, n var , n var , n var 29, , , n var n var n var n var n var 18, , n var 66, , n var n var n var 13, n var n var 11, n var 31, , Total NFE 3, , ,834 69,749 49,443 65,036 a Iter max/gen max s restrted to a maxmum of 1500; however, ths ondton was not reahed n all alulatons performed. Algorthm parameters: NS NT =NP=10n var. ally for F 1.e., onstraned formulaton and f SC1 s used alone as the stoppng ondton Fg. 1. SR of SA nreases wth Iter max and ts GSR ranges from 51 to 87% for onstraned formulaton usng SC1. But, SA showed the best performane.e., % GSR at Iter max 500 usng transformed omposton varables even wthout quas-newton method. On the other hand, maxmum GSR of DETL s 59 and 94% for onstraned and unonstraned problems, whereas GA showed maxmum GSR of 1 and 68% for these problems usng SC1 as the termnaton rteron Fg. 1. In unonstraned Gbbs free energy mnmzaton, DETL and SA an often fnd solutons very lose to the global optmum even wthout applyng the loal method and usng SC1 as the stoppng ondton Fg. 1. In general, value of the best soluton obtaned by these methods s nearer to the global mnmum as Iter max /Gen max nreases. It s lear that GA wthout loal strategy s the worse performer for both stoppng ondtons and problem formulatons Fg. 1. If SC2 s used as the stoppng ondton, relablty of DETL wthout quas-newton method s hgher ompared to GA and SA for both onstraned and unonstraned funtons Fg. 2. In fat, DETL offers the best performane and an gve hgh relablty of GSR = 90%, f proper values of Smax are used, for Gbbs free energy mnmzaton employng transformed omposton varables see results n Fg. 2. SA performed worse than all other stohast methods tested for unonstraned problems usng SC2 as the stoppng ondton wth and wthout loal optmzaton. In summary, all stohast methods mprove ts performane wth the applaton of ntensfaton step n Gbbs free energy mnmzaton usng both onventonal and transformed omposton varables, rrespetve of the onvergene rteron used.e., SC1 and SC2. In partular, the ntensfaton stage usng the quas-newton method plays a major role for mprovng numeral performane of stohast methods n onstraned approah, whle t appears that ts use has less mpat for nreasng the relablty of stohast methods n the unonstraned problems tested. Usually, t s expeted that the performane of the optmzaton methods for solvng unonstraned problems s better than those obtaned for onstraned problems beause the optmzaton an be exeuted wthout worryng about the feasblty [44]. For onstraned problems, the varane of solutons obtaned by the stohast method wthout loal optmzaton may be large. Ths s manly related to the apablty of eah stohast method for searhng the feasble regon of the total searh spae. For llustraton, % of the nfeasble solutons versus Iter max /Gen max for seleted problems s gven n Fg. 3. The nfeasble solutons generated at dfferent levels of omputatonal effort by the stohast methods, ndate that DETL s very effetve for handlng onstrants and the perentage of nfeasble solutons dereases from 30 to 5% as the optmzaton searh progresses. Ths may be due to the ablty of DE to explot and ntensfy the searh as teratons nrease. It s nterestng to observe that approxmately % of funton evaluatons for SA orrespond to nfeasble solutons and ths proporton pratally remans onstant throughout the tested range of Iter max. Although ths perentage s hgher than those reported for DETL, SA shows hgh relablty for onstraned problems espeally usng SC1 as the onvergene rteron. On the other hand, GA fals frequently to dentfy feasble optmal solutons n the onstraned Gbbs free energy mnmzaton of reatve systems see Fg. 3. Other studes have reognzed ths drawbak of GA for solvng onstraned optmzaton problems [45]. Lterature also ndates that GA performs well n the dversfaton of the searh spae but may fal n the ntensfaton of the solutons found [46]. Overall, the present results suggest that the dversfaton mehansms n DETL and SA are more effetve than those of GA, to esape from the nfeasble regon and to loate the global optmum.