Gibbs Energy Minimization Using Simulated Annealing for Two-phase Equilibrium Calculations in Reactive Systems

Transcription

1 A. BONILLA-PETRICIOLET et al., Gbbs Enery Mnmzaton Usn Smulated, Chem. Bochem. En. Q. (3) (008) 85 Gbbs Enery Mnmzaton Usn Smulated Annealn for Two-phase Equlbrum Calculatons n Reactve Systems A. Bonlla-Petrcolet, a,* U. I. Bravo-Sánchez, a F. Castllo-Bora, a S. Frausto-Hernández, a and J. G. Seova-Hernández b a Insttuto Tecnolóco de Auascalentes, Departamento de Inenería Químca Av. López Mateos 80, CP 056, Auascalentes, As. Méco b Unversdad de Guanauato, Facultad de Cencas Químcas, Nora Alta s/n, C.P , Guanauato, Gto. Méco Ornal scentfc paper Receved: Aprl, 007 Accepted: February 8, 008 Phase equlbrum calculatons n systems subect to chemcal reactons are nvolved n the desn, synthess and optmzaton of reactve separaton processes. Untl now, several methods have been developed to perform smultaneously physcal and chemcal equlbrum calculatons. However, publshed methods may face numercal dffcultes such as varable ntalzaton dependence, dverence and converence to trval solutons or unstable equlbrum states. Besdes, these methods enerally use conventonal composton varables and reactons etents as unknowns whch drectly affect the numercal mplementaton, relablty and effcency of solvn stratees. The obectve of ths work s to ntroduce and test an alternatve approach to perform Gbbs enery mnmzaton n phase equlbrum problems for reactve systems. Specfcally, we have employed the transformed composton varables of Un and Doherty and the stochastc optmzaton method Smulated Annealn for two-phase equlbrum calculatons n reactn systems. Performance of ths stratey has been tested usn several benchmark problems and results show that proposed approach s enerally sutable for the lobal mnmzaton of transformed Gbbs enery n reactve systems wth two-phase equlbrum. Key words: Global optmzaton, Gbbs enery mnmzaton, smulated annealn, chemcal equlbrum, phase equlbrum Introducton Phase equlbrum calculatons n reactn systems are often requred for desnn processes of the chemcal, petrochemcal and metallurcal ndustres. Durn the last years, there has been a rown nterest for developn new numercal tools to model the thermodynamc behavor of mtures under physcal and chemcal equlbrum. Untl now, several methods have been proposed to perform reactve phase equlbrum calculatons. Prncpally, these methods have been developed to model reactve dstllaton process and they are based on equlbrum constant (K-value) method or Gbbs enery mnmzaton technques. Also, they can be classed as ether stochometrc or nonstochometrc, dependn on the way n whch the elemental abundance constrants are used n the alorthm. 3 Stron nteractons amon components, phases and reactons may cause that ths thermodynamc problem ehbts hhly nonlnear behavor. By * Correspondn author: tel: (5) et. 7, e-mal: petrcolet@hotmal.com consequence, there are frequently computatonal dffcultes n these calculatons and publshed methods could not be relable enerally. Specfcally, there are ntalzaton troubles, the presence of trval solutons or local mnmums of Gbbs enery and numercal methods may also present slow converence or dverence. It s mportant to note that few lobal solvn methods have been proposed for ths area.,7,9,0 Specfcally, McDonald and Floudas proposed a determnstc lobal optmzaton method whch uarantees the lobal mnmzaton of Gbbs enery n reactn mtures usn soluton models. Also, Jalal and Seader 7 have successfully appled a nonlnear optmzaton based on homotopy contnuaton and Larane functon whle Buros-Solórzano et al. 0 have used a relable stablty analyss, based on nterval mathematcs, to valdate the results of phase and chemcal equlbrum calculatons. The stochastc optmzaton methods such as Smulated Annealn (SA), Tabu Search (TS) or Genetc Alorthm (GA) have not been etensvely studed n thermodynamc calculatons for reactn mtures. These methods have a reat potental n

2 86 A. BONILLA-PETRICIOLET et al., Gbbs Enery Mnmzaton Usn Smulated, Chem. Bochem. En. Q. (3) (008) ths area because they are relable numercal tools that can be appled for multvarable and nonconve problems and, by consequence, they are sutable for performn phase equlbrum calculatons n reactve systems. For eample, Reynolds et al. 4 have outlned a eneral procedure for the lobal mnmzaton of Gbbs enery n non-reactve and reactve mtures usn SA method and a nonstochometrc approach. Unfortunately, these authors have reported results for only non-reactve systems. On the other hand, we consder convenent to remark that most of publshed methods use reacton etents and mole fractons as ndependent varables. For eample, n ths cateory the methods proposed by McDonald and Floudas and Jalal and Seader 7 are classed. A sutable choce to reduce the problem dmensonalty, and to favor the numercal performance of soluton alorthms, conssts n usn technques of varable transformaton. 4,5 Only a few methods have used these transformed varables wth the am of mprovn the numercal behavor (effcency and relablty) of solvn stratees. 3,4,8,9 These knds of alorthms are very attractve for the smulaton of separaton process and favor the study of comple multreactve multcomponent systems. Moreover, these approaches, n combnaton wth stochastc optmzaton technques, can be used to develop relable stratees for phase equlbrum calculatons n reactve systems. The obectve of ths work s to ntroduce an alternatve approach for performn two-phase equlbrum calculatons n reactve systems. We use the transformed varables of Un and Doherty 3,5,6 and the stochastc optmzaton method Smulated Annealn for the lobal mnmzaton of Gbbs enery n reactn mtures wth two-phase equlbrum. Numercal performance of ths stratey s tested usn several multcomponent reactve mtures and our results show that t s enerally a sutable method to perform ths knd of thermodynamc calculatons. Thermodynamc problem statement Un and Doherty 3,5,6 proposed the use of transformed composton varables wth the obectve of developn a smpler thermodynamc framework for modeln reactve systems. These transformed varables depend only on the ntal composton of each ndependent chemcal speces and are constant as the reactons proceed. They also restrct the soluton space to the compostons that satsfy stochometry requrements and reduce the dmenson of the composton space by the number of ndependent reactons. These characterstcs allow that all of the procedures used to obtan thermodynamc propertes of non reactn mtures can be etended to reactn systems and, by consequence, the non-reactve phase equlbrum alorthms can be easly modfed to account for the equlbrum reactons. 9 Also, reactve phase darams n transformed coordnates look very smlar to the non-reactve ones n the standard mole fracton coordnates. The man dfference s n the shape of the composton space, whch depends on the type of reacton n the system and could chane f dfferent reference components are selected. 9 Transformed amount fractons are defned as ref v N v N TOT ref,, cr () where c s the number of components, R s the number of ndependent reactons, ref s the column vector of R reference component mole fractons, v s the row vector of stochometrc number of component for each reacton, v TOT s a row vector where each element corresponds to the sum of the stochometrc coeffcents for all components that partcpate n reacton r, and N s a square matr formed from the stochometrc coeffcents of the reference components n the R reactons. 3,5 It s mportant to note that eq. () provdes constant values of tranformed mole fractons n snle-phase reactons, or constant overall mole fractons n heteroeneous reactons: Z ref z v N z v N z TOT ref,, cr. In multple-phase reactons, -values n coestent phases are varable and subect to phase equlbra requrements. The reference mole fractons ref are calculated usn eq. () and the equlbrum constants for each r reacton K eq by solvn a system of R nonlnear equatons ven by r eq c v r K a r,, R () where a s the actvty of component and v r s the stochometrc number of component n the reacton r, respectvely. When we know the R reference mole fractons, for a set of c R specfed transformed varables, the correspondn mole fractons of c R non-reference components are calculated usn eq. (). In ths work, we used the bsecton method to fnd the mole fracton of reference component n all eamples wth only one reacton. Our eperence ndcates that bsecton method works very well for ths purpose. 7 On the other

3 A. BONILLA-PETRICIOLET et al., Gbbs Enery Mnmzaton Usn Smulated, Chem. Bochem. En. Q. (3) (008) 87 hand, for multreactve mtures, the Newton method can be used to fnd the mole fractons of reference components. 3,5 Classcal thermodynamcs ndcates that mnmzaton of Gbbs enery s a natural course for calculatn the equlbrum state of a system. Gbbs enery mnmzaton alorthm was ntroduced by Whte et al. 8 and, for reactve mtures, mnmzn Gbbs enery s equvalent to mnmzn the transformed Gbbs enery. 9 In a multcomponent multreacton mture, wth c components and R ndependent chemcal reactons, the dmensonless transformed molar Gbbs free enery of mn can be defned as 9 c R { } 0 RT RT (3) cr { } ln( { }) ln cr where R s the as unversal constant, T s the temperature, s the transformed { } 0 RT RT chemcal potental of component, 0 s the pure component free enery, s the transformed molar Gbbs free enery, s the fuacty coeffcent of pure component, s the fuacty coeffcent of component n the mture and s the actvty coeffcent of component, respectvely. Note that and are functons of the transformed composton varables. Also, calculatons of pure component free eneres are avoded, whch do not nfluence the equlbrum and stablty calculatons, f nstead of we use the transformed molar Gbbs enery of mn. RT 9 For a feed wth a lobal transformed composton Z that splts n phases, s ven by F RT cr RT (4) The equlbrum transformed mole fractons for all phases must satsfy the materal balance ZF 0,, cr (5) where F s the transformed amount fracton for phase. All transformed varables are subect to the follown restrctons cr F (6),, (7) We note that F s defned as v F TOT N ref vtot N zref ( ),, (8) ( ) ben the mole fracton of phase whose feasble doman s (0, ) and, n accordance wth, 6 s subect to where s the column vector v TOT of R etents of reacton and z ref s the column vector of R reference component mole fractons assocated to the transformed feed Z, respectvely. At equlbrum, must be at the lobal mnmum whch s a necessary and suffcent condton for a thermodynamcally stable state. 3,9,5,9 However, the lobal mnmzaton of transformed Gbbs enery s a challenn optmzaton problem due to the obectve functon s nonconve and may have multple local optmums even for two-phase reactn systems modeled wth smple thermodynamc equatons. Consdern ths fact, local optmzaton methods are not sutable to solve ths problem and a robust optmzaton stratey must be used. In the follown secton we descrbe the optmzaton procedure used to perform the lobal mnmzaton of transformed Gbbs enery for two-phase equlbrum calculatons n reactve systems. Optmzaton approach In ths paper we have etended the optmzaton stratey proposed by Ranaah, 0 whch was ornally developed to non-reactve mtures, to perform an unconstraned Gbbs enery mnmzaton n two-phase reactve systems. Usn ths approach, the transformed Gbbs enery mnmzaton problem can be smplfed by usn a set of new varables nstead of the transformed mole numbers or fractons as decson varables n the optmzaton stratey. The ntroducton of these varables elmnates the restrctons mposed by materal balances, reduces problem dmensonalty and the optmzaton problem s transformed to an unconstraned one. Ths approach s more sutable than the Larane multpler formulaton due to the snfcant reducton of problem dmensonalty. So, for two-phase reactn systems where all trans-

4 88 A. BONILLA-PETRICIOLET et al., Gbbs Enery Mnmzaton Usn Smulated, Chem. Bochem. En. Q. (3) (008) formed mole fractons have values n the rane ( 0, ) or ts phase equlbrum reon satsfes ths restrcton, real varables n the rane [0, ] are defned and employed as optmzaton varables usn the net equatons T n n,, cr (9) T n n n,, cr (0) T where n n n s the transformed mole number T of component n the feed, n nv N nref s the transformed mole number of component at phase and s the optmzaton varable of the unconstraned optmzaton problem, respectvely. The beneft of ths modfcaton s that all tral compostons wll satsfy the materal balance whch allows the easy applcaton of optmzaton stratees. It s mportant to note that ths formulaton can be used f n T 0 for all, cr. Transformed Gbbs enery functon s mnmzed usn the smulated annealn (SA) stochastc optmzaton method. SA s a robust numercal tool that presents a reasonable computatonal effort n the optmzaton of multvarable functons; t s applcable to ll-structure or unknown structure problems, t requres only calculatons of the obectve functon and can be used wth all thermodynamc models., In fact, SA has the attrbutes of a ood numercal optmzaton method f s properly mplemented: eneralty, acceptable computatonal tme, relablty and ease of use. 3 5 It s consdered as lobal optmzaton stratey and s one of the most used stochastc methods n enneern applcatons. 6 Specfcally, n the feld of thermodynamcs, ths method has been successfully used n phase stablty and equlbrum calculatons 4,7,0,7 and nonlnear parameter estmaton. 8,9 Ths work ntroduces the applcaton of ths optmzaton stratey n the mnmzaton of transformed Gbbs enery n reactve mtures. SA s a eneralzaton of a Monte Carlo method and ts concept s based on the thermodynamc process of cooln of molten metals to acheve the lowest enery state. Generally, t can locate the lobal optmum ndependently of ntal uesses f the values for ts alorthm parameters are properly selected. We have used the alorthm proposed by Corana et al. 30 because of ts proved relablty n thermodynamc calculatons. 7,0,9 In ths alorthm, a tral pont, randomly chosen wthn the step lenth VM (a vector of lenth n optmzaton varables) s the startn pont. The functon evaluated at ths tral pont s compared to ts value at the ntal pont. The Metropols crteron, 3 wth a parameter termed annealn temperature T SA, s used to accept or reect the tral pont. If the tral pont s accepted, the alorthm moves on from that pont. If the tral s reected, another pont s chosen for a tral evaluaton. After havn adusted each element of VM perodcally, half of all functon evaluatons n that drecton are acceptable. The temperature reducton factor RT s used to decrease T SA employn T SA RTTSA where s the teraton counter. Thus, as T SA declnes, downhll moves are less lkely to be accepted and the percentae of reectons rses. As T SA declnes, VM falls and the method focuses upon the most promsn area for optmzaton. If the fnal functon values from the last temperatures (NEPS = 4) dffer from the correspondn value at the current temperature by less than a tolerance value (EPS) and the fnal functon value at the current temperature dffers from the current optmal functon value by less than ths tolerance value, alorthm eecuton termnates. Corana et al. 30 provde a full descrpton of ths alorthm and we used the subroutne mplemented by Goffe et al. 3 F. shows the flow daram of ths alorthm. F. Flow daram of SA optmzaton alorthm 9

5 A. BONILLA-PETRICIOLET et al., Gbbs Enery Mnmzaton Usn Smulated, Chem. Bochem. En. Q. (3) (008) 89 Numercal performance (relablty and effcency) of SA method s snfcantly affected by the cooln schedule whch s lnked to the parameters T 0 SA, RT and NT. These parameters requre pre-calbraton for new problems. We have tuned these parameters by performn several flash calculatons for the reactve mture: acetc acd + n-butanol water + n-butyl acetate at 5 C. Thermodynamc propertes of ths mture were modeled wth UNIQUAC equaton usn the model parameters reported by Wasylkewcz and Un. 9 Based on these calculatons (not reported n ths paper) and our numercal eperence n phase stablty analyss for several reactve mtures, 7 we have defned that SA works acceptably well usn the follown condtons: T 0 SA = 0, RT = 0.85, NT = 5 (c R) where NT s the teraton number before annealn temperature reducton. Results and dscusson We have tested the numercal performance of proposed method usn several eamples wth dfferent dmensonalty and thermodynamc models. Most of our eamples are standard benchmark n the lterature of reactve dstllaton process, predcton of reactve homoeneous and heteroeneous azeotropes, multple steady states n reactve separaton unts and phase stablty analyss. All eamples were solved several tmes usn dfferent feed condtons (temperature, pressure, chemcal equlbrum constants or feed compostons). We have assumed that all chemcal reactons are reversble and they occur n both phases. For varable transformaton, n all eamples we have selected arbtrarly the reference components. Also, tested condtons were arbtrary but we consder that they are suffcent to demonstrate the numercal performance of proposed stratey. To determne the computatonal behavor of our alorthm, 00 runs are performed for each eample usn random ntal values for decson varables and random number seeds for SA. We have defned a tolerance value of ESP = 0 6 for the converence of SA method. Snce stochastc optmzaton methods do not provde an accurate soluton of the lobal optmum, 5,33 we have consdered that the lobal mnmzaton of transformed Gbbs enery s successful upon satsfyn the condton 4 OBJ OBJ OBJ () calc mn where OBJ mn s the lobal mnmum of the transformed Gbbs enery and OBJ calc s the calculated value for ths thermodynamc functon wth the optmzaton method, respectvely. Eq. () has been mn used n several applcatons of other stochastc optmzaton methods We must remark that the lobal optmum OBJ mn for all eamples was determned by solvn the equalty of transformed chemcal potentals usn several ntal estmatons. The stable two-phase solutons obtaned by ths procedure were consdered as the correspondn lobal optmum OBJ mn values. On the other hand, the numercal performance of our optmzaton stratey s characterzed usn wdespread crterons reported n the lterature of stochastc optmzaton methods; 5,33,36,37 a) success rate (SR) of fndn the lobal mnmum ven as percent of calculatons performed that satsfes eq. (), b) mean total number of functon evaluatons (NFEV) durn the optmzaton procedure, c) mean computatonal tme and d) mean absolute percentae devaton of the calculated compostons from the known compostons at lobal mnmum Calc cr 00 AAD c mn () ( R) where mn s the lobal optmum value for the transformed composton of component at phase Calc and s the calculated value for the transformed composton of component at phase usn the stochastc method, respectvely. Crtera a) and d) are used to characterze the relablty of optmzaton stratey whle remann ones are ndcators of alorthm effcency. 5 Mean values of NFEV, AAD and computatonal tme are calculated consdern only the successful calculatons. All calculatons were performed on a Processor Intel Pentum M.73 GHz wth 504 MB of RAM usn Fortran 4.0 software. Phase stablty analyss of calculated equlbrum compostons was performed usn the Reactve Tanent Plane Dstance Functon (RTPDF) whch s ven by 9,9 cr mn RTPDF ( Z ) (3) where the lobal mnmum of RTPDF s 0 for stable mtures. Ths functon was lobally mnmzed also usn SA optmzaton method. Bonlla-Petrcolet et al. 7 have reported that SA s robust to perform phase stablty analyss n reactn and non-reactn mtures. All reported solutons n ths work are thermodynamcally stable. Fnally, the overall method s outlned as follows for a flash calculaton n a multcomponent reactve system:

6 90 A. BONILLA-PETRICIOLET et al., Gbbs Enery Mnmzaton Usn Smulated, Chem. Bochem. En. Q. (3) (008). Input n 0, equlbrum constants and model parameters. 0 n. Calculate z by z and Z usn eq. c (). Do a phase stablty analyss usn Z and eq. (3). Contnue wth step 3 f the lobal optmum of RTPDF < 0, otherwse propose other n 0 and repeat step. T n 0 c 0 0 TOT ref 3. Calculate n n v N n. 4. Guess ( 0, ) for,, cr. 5. Calculate n and n usn eq. (9) and (0). 6. Calculate n n n T n n and n n. cr n and F n T where 7. Defne as functon of and ref. 8. Introduce n the equatons of equlbrum constants for each reacton K r eq. 9. Determne ref by solvn the system of R nonlnear equatons formed wth K r eq. Ths step s performed for both phases to determne ther composton n conventonal mole fractons. 0. Calculate transformed Gbbs enery usn eq. (4).. Check for mnmum; proceed wth SA method wth new ( 0, ) and repeat steps 5 untl satsfy the converence crteron of SA method. Ths alorthm has been appled for all calculatons performed n ths paper. In the follown tet, we descrbe the results obtaned for several reactn systems usn ths stratey. Eample. Our frst eample s a hypothetcal reactn ternary mture A A A 3 wth lqud lqud equlbrum. Ths system was ntroduced by Un and Doherty 5 n ther seres of publcatons related to ther transformed varables for reactn mtures. Transformed mole fractons, consdern thrd component as reference component, are ven by (4) (5) where ( 0, ). Marules soluton model wth the data reported by Un and Doherty 5 s used to calculate thermodynamc propertes G R e A (6) wth A 4786.,A and A ,respectvely. We have studed ths system at 33.5 K and phase equlbrum calculatons are performed for dfferent values of the reacton equlbrum constant K eq (.5, 30) and feed compostons. Based on our formulaton, we have only two optmzaton varables for ths eample. Table shows the results of equlbrum calculatons for ths reactn mture. For all calculatons performed, the proposed optmzaton stratey s very relable to fnd the equlbrum compostons correspondn to the lobal mnmum of transformed Gbbs enery. Wth respect to effcency of SA method, mean NFEV raned from to where ths numercal effort s equvalent to 5 s of computatonal tme. For ths mture, AAD s around %. Table Numercal performance of smulated annealn n the unconstraned lobal mnmzaton of Transformed Gbbs enery for the reactn mture A A A 3 at 33.5 K (Marules soluton model) Feed condtons Z K eq Equlbrum condtons transformed Gbbs enery Numercal performance ± ± ± ± ± ± ± ± ± ± NFEV s the mean total number of functon evaluatons nvolved n the lobal mnmzaton of transformed Gbbs enery and SR s the success rate to fnd the transformed phase equlbrum compostons. 00 trals performed wth random ntal values for optmzaton varables and random number seeds.

7 A. BONILLA-PETRICIOLET et al., Gbbs Enery Mnmzaton Usn Smulated, Chem. Bochem. En. Q. (3) (008) 9 Eample. Butyl acetate s wdely used as solvent n coatn, adhesves and pant ndustres and t can be produced va the lqud-phase reacton of butanol and acetc acd n the presence of a sutable acdc catalyst. Ths reacton s ven by acetc acd () + n-butanol () water (3) + n-butyl acetate (4). Khaled and Bshno have modeled the producton of butyl acetate va reactve dstllaton process. In ths work, we have calculated several te-lnes for ths mture at 5 C. UNIQUAC model s used to predct thermodynamc propertes wth the parameters reported by Wasylkewcz and Un. 9 The chemcal equlbrum constant s calculated usn ln K eq 450. T 08 where T s ven n K. Transformed compostons are defned usn n-butyl acetate as reference component and are ven by 4 (7) 4 (8) (9) where, ( 0, ) and 3 (, ). Transformed Gbbs enery s mnmzed consdern three optmzaton varables. Calculated te-lnes for ths system are shown n F. and the numercal performance of optmzaton stratey appears n Table. In ths table, we also report the slopes of te-lnes whch are calculated usn cr cr,, (0) where. Our results show that proposed alorthm s relable to fnd the equlbrum compostons and t enerally ehbts a 00 % success rate. Only for two feeds, t shows a poor performance. In these cases, SA method convered to trval solutons (local optmums) where ts value of transformed Gbbs enery s very near to the lobal one. For eample, at Z(0.394, 0.74, 0.33), the trval soluton ( Z ) has a transformed Gbbs enery value of.5 67 whle the lobal optmum s equal to.5 7. We note that these local optmums satsfy the condton ven by eq. (). However, the phase stablty analyss helped us to dentfy these falures of the proposed optmzaton method. As ndcated by Buros-Solozarno et al., 0 phase stablty analyss s a fundamental procedure whch must be used to valdate the results of any phase equlbrum calculaton. On the other hand, for these dffcult cases, t s convenent to modfy the cooln schedule of SA alorthm to favor the converence to the lobal optmum of F. Calculated te-lnes n transformed mole fractons for the reacton () acetc acd + () n-butanol (3) water + (4) n-butyl acetate at 5 C. UNIQUAC soluton model. transformed Gbbs enery. Specfcally, an ncrement on T SA 0 and NT provdes a better performance, of course, at the cost of a reater computatonal effort. Also, we can mprove the performance of SA method n these dffcult problems usn a proper ntalzaton stratey. For eample, we can use the results of phase stablty as ntal values for Gbbs enery mnmzaton. 4 Fnally, wth respect to eff- Table Numercal performance of smulated annealn n the unconstraned lobal mnmzaton of transformed Gbbs enery for the reactn mture () acetc acd + () n-butanol (3) water + (4) n-butyl acetate at 5 C (UNIQUAC soluton model) Z Equlbrum condtons transformed Gbbs enery Numercal performance (0.0, 0.4, 0.59) ± (0., 0., 0.7) ± (0.5, 0.5, 0.35) ± (0., 0.3, 0.5) ± (0.3, 0.3, 0.4) ± (0.3, 0.4, 0.3) ± (0.397, 0.94, 0.309) (0.394, 0.74, 0.33) ± ± (0.3, 0.5, 0.55) ± (0.7, 0., 0.63) ± NFEV s the mean total number of functon evaluatons nvolved n the lobal mnmzaton of transformed Gbbs enery and SR s the success rate to fnd the transformed phase equlbrum compostons. 00 trals performed wth random ntal values for optmzaton varables and random number seeds.

8 9 A. BONILLA-PETRICIOLET et al., Gbbs Enery Mnmzaton Usn Smulated, Chem. Bochem. En. Q. (3) (008) cency, the computatonal effort n terms of mean functon evaluatons raned from 9 97 to whle mean computaton tme s around 40 seconds and AAD s lower than % for all tested condtons. Eample 3. Thrd eample s the reacton of sobutene (), methanol () and methyl tert-butyl ether (3) wth n-butane (4) as an nert. MTBE s an mportant ndustral chemcal and has been used an ant-knock aent to replace tetra-ethyl lead n asolne. Several smulaton and epermental researches have been performed to study the MTBE reactve dstllaton process. 38 For eample, Okansnsk and Doherty 39 have studed the effect of the reacton equlbrum constant on the estence and locaton of reactve homoeneous azeotropes n ths mture. In ths work, we have consdered the vapor lqud equlbrum for ths reacton at p = 0.35 bar and = 00 C. Transformed mole fractons are defned usn MTBE as reference component and are ven by () () 4 (3) 3 where all transformed mole fractons raned from 0 to. Wlson soluton model and Antone equaton are used to calculated thermodynamc propertes. Model parameters are taken from Maer et al. 40 and the reacton equlbrum constant s calculated usn 0 rs G / R T0667. T lnt where T s ven n K. For deal as behavor, transformed Gbbs enery s ven by 4 RT cr p ln sat p (4) F. 3 Calculated te-lnes n transformed mole fractons for the reacton () sobutene + () methanol (3) methyl tert-butyl ether and (4) n-butane as nert at 00 C and 0.3 bar. Wlson model and Antone equaton. sat where p s the vapor pressure of pure component. For ths reactn mture, we also have three optmzaton varables. Calculated te-lnes for ths mture appear n F. 3 and detals of equlbrum calculatons are reported n Table 3. For all calculatons performed, SA method shows a 00 % relablty where phase equlbrum compostons are located wthout numercal problems. In the other hand, mean value of NFEV raned from to for all calculatons performed whle mean computaton tme s around of 85 s. For ths reactn mture, ADD raned from to %. It s mportant to note that some publshed methods cannot handle the presence of nert components. 3 The nert components do not partcpate n any of the reactons, but have an nfluence on the phase equlbrum. Based on these results, t appears that proposed stratey s robust even when nert components are consdered. Table 3 Numercal performance of smulated annealn n the unconstraned lobal mnmzaton of transformed Gbbs enery for the reactn mture () sobutene + () methanol (3) methyl tert-butyl ether and (4) n-butane as nert at 00 C and 0.3 bar. (Wlson soluton model and deal as). Z (0.78, 0.365, 0.357) Equlbrum condtons Numercal performance transformed Gbbs enery ± (0.3, 0.3, 0.4) ± (0.35, 0.5, 0.4) ± (0.4, 0.5, 0.35) ± (0.5, 0.3, 0.) ± 7 00 (0.7, 0.5, 0.05) ± (0.673, 0.36, 0.07) ± (0.5, 0.6, 0.5) ± (0.005, , ± ) (0.6, 0.3, 0.) ± NFEV s the mean total number of functon evaluatons nvolved n the mnmzaton of transformed Gbbs enery and SR s the success rate to fnd the transformed phase equlbrum compostons. 00 trals performed wth random ntal values for optmzaton varables and random number seeds.

9 A. BONILLA-PETRICIOLET et al., Gbbs Enery Mnmzaton Usn Smulated, Chem. Bochem. En. Q. (3) (008) 93 Eample 4. Ths eample s also a hypothetcal reactn ternary mture A A A 3 wth lqud lqud equlbrum. We have consdered a reacton equlbrum constant ndependent of temperature and thermodynamc propertes are calculated usn the Marules soluton model where E (5) RT Consdern thrd component as reference component, transformed compostons are ven by eqs. (4) and (5). Ilesas-Slva et al. 4 used ths system to ntroduce the concept of equal area rule for multphase equlbrum n reactn mtures based on transformed varables. Based on ther paper, ths mture shows three-phase equlbrum at K eq and Z (0.8336, 0.85). We have performed phase equlbrum calculatons for dfferent values of K eq and feed compostons that show a transformed Gbbs functon wth multple local mnma. Wth llustratve purposes, -surface for some tested condtons appears n F. 4 where the tanent planes for stable and unstable equlbrums are ndcated. F. 4 shows that ths mture can present unstable two-phase equlbrum states where a local optmzaton method can be easly trapped. For eample, at Z(0.6, 0.4) and K eq = ths system has two unstable phase equlbrum states: ( ,. 0633), (. 0804,. 0896) Table 4 Numercal performance of smulated annealn n the unconstraned lobal mnmzaton of transformed Gbbs enery for the reactn mture A A A 3 (Marules soluton model 36. E ) RT Feed condtons Equlbrum condtons Numercal performance Z K eq transformed Gbbs enery ± ± ± ± ± ± NFEV s the mean total number of functon evaluatons nvolved n the mnmzaton of transformed Gbbs enery and SR s the success rate to fnd the transformed phase equlbrum compostons. 00 trals performed wth random ntal values for optmzaton varables and random number seeds. F. 4 Transformed Gbbs enery surface of a hypothetcal reactn mture A A A 3 (Marules soluton model E )a)K RT eq = 0.95, b) K eq = and c) K eq =.0 wth = 0.45 and (. 097, ), (. 0843,. 0857) wth = So, ths eample s a ood choce to test the relablty of the proposed optmzaton stratey. Two optmzaton varables are consdered n the mnmzaton of transformed Gbbs enery and results of phase equlbrum calculatons are reported n Table 4. Aan, for all tested condtons, SA method s capa-

10 94 A. BONILLA-PETRICIOLET et al., Gbbs Enery Mnmzaton Usn Smulated, Chem. Bochem. En. Q. (3) (008) ble of fndn the lobal mnmum of transformed Gbbs enery wthout numercal problems. On the other hand, NFEV raned from to whle computaton tme s around.5 s. ADD s lower than 0.0 % for all cases. Eample 5. Tert-amyl methyl ether (TAME) s an mportant chemcal for asolne and s produced by lqud-phase synthess from methanol and so-amylenes catalyzed by a sulfonc acd on echane resn. 38 In ths reacton, fve components take part: methanol, -methyl--butene, -methyl- --butene, TAME and n-pentane as nert component. In ths work, we have consdered the lumped snle reacton whch can be wrtten as: -methyl- --butene () + -methyl--butene () + methanol (3) TAME (4) wth n-pentane (5) as an nert solvent.,38 In frst nstance, VLE of ths reacton wthout n-pentane as nert s studed. Transformed mole fractons, consdern TAME as reference component, are ven by 054. (6) (7) (8) All transformed fractons raned from 0 to. Wlson and deal as models have been used to calculate thermodynamc propertes of ths mture where thermodynamc parameters are taken from Chen et al. 38 Reacton equlbrum constant s calculated usn T Keq / e where T s ven n K. Phase equlbrum calculatons are performed for several feeds at 335 K and.5 bar. Detals of calculatons are reported n Table 5 and calculated te-lnes reported n transformed mole fractons appear n F. 5. Only for one feed, the SA method shows a success rate lower than 00 %. For remann calculatons, ths method s very relable to fnd the lobal mnmum of transformed Gbbs enery. Wth respect to effcency, NFEV raned from to where mean computatonal tme s around 83 s and AAD s lower than % for all tested condtons. In our second scenaro for ths eample, we have consdered the presence of n-pentane as nert. Transformed mole fractons are defned by eqs. (6) (7) and (9) (30) 4 F. 5 Calculated te-lnes n transformed mole fractons for the reacton () -methyl--butene + () -methyl--butene + + (3) methanol (4) tert-amyl methyl ether at 335 K and.5 bar. Wlson model and deal as. Transformed Gbbs enery s mnmzed consdern four optmzaton varables. Phase equlbrum calculatons are performed at 335 K and.5 bar where the detals of calculatons and numercal Table 5 Numercal performance of smulated annealn n the unconstraned lobal mnmzaton of transformed Gbbs enery for the reactn mture () -methyl--butene + () -methyl--butene + (3) methanol (4) tert-amyl methyl ether at 335 K and.5 bar (Wlson model and deal as) Z Equlbrum condtons Numercal performance transformed Gbbs enery (0.3, 0.5, 0.55) ± (0.3, 0., 0.48) ± (0.354, 0.83, 0.463) ± (0., 0.07, 0.73) ± (0.5, 0.0, 0.83) ± (0.7, 0.3, 0.43) ± (0., 0.35, 0.45) ± (0., 0.35, 0.5) ± (0.05, 0.3, 0.65) ± (0.05, 0.3, 0.675) ± NFEV s the mean total number of functon evaluatons nvolved n the lobal mnmzaton of transformed Gbbs enery and SR s the success rate to fnd the transformed phase equlbrum compostons. 00 trals performed wth random ntal values for optmzaton varables and random number seeds.

11 A. BONILLA-PETRICIOLET et al., Gbbs Enery Mnmzaton Usn Smulated, Chem. Bochem. En. Q. (3) (008) 95 Table 6 Numercal performance of smulated annealn n the unconstraned lobal mnmzaton of transformed Gbbs enery for the reactn mture () -methyl--butene + () -methyl--butene + (3) methanol (4) tert-amyl methyl ether wth (5) n-pentane as nert at 335 K and.5 bar (Wlson model and deal as) Equlbrum condtons Numercal performance Z (, 3 ) transformed Gbbs enery (0.5, 0.0, 0.8, 0.83) (0.0099,.48) ± (0., 0., 0.6, 0.) (0.9406, ) ± (0.05, 0.05, 0.85, 0.05) (0.8069, 6.438) ± (0., 0.5, 0.7, 0.05) (6.043, ) ± (0.5, 0.5, 0.6, 0.) (0.8465, ) ± (0.07, 0.7, 0.64, 0.) (7.930, 8.05) ± NFEV s the mean total number of functon evaluatons nvolved n the lobal mnmzaton of transformed Gbbs enery and SR s the success rate to fnd the transformed phase equlbrum compostons. 00 trals performed wth random ntal values for optmzaton varables and random number seeds. performance of SA method for some feeds are reported n Table 6. For all tested condtons, ths method s capable of fndn the transformed equlbrum compostons wth 00 % relablty. We note aan that the presence of nert component does not affect the performance of proposed method. Wth respect to the effcency NFEV raned from to 70 4, whch s equvalent to 99 s of computatonal tme. Fnally, AAD s lower than % for all feeds. Eample 6. In ths eample we have consdered the reactn mture of propene () + water () -propanol (3) whch has been studed by Caster et al. 4 and Stateva and Wakeham. 5 Equlbrum constant s consdered ndependent of temperature and flash calculatons are performed usn K eq 3 at K. 5 We have used the SRK EoS wth conventonal mn rules and all nteracton parameters equal to zero. Parameters of pure components are taken from Smth and Van Ness. 43 For ths reacton, transformed mole fractons are defned usn -propanol as reference component and are ven by eqs. (4) (5). Equlbrum calculatons are performed for dfferent pressures and feed compostons. Numercal results of calculatons performed are reported n Table 7. Wth the ecepton of one feed, the SA method showed a 00 % Table 7 Numercal performance of smulated annealn n the unconstraned lobal mnmzaton of transformed Gbbs enery for the reactn mture () propene + () water (3) -propanol at K (SRK EoS) Equlbrum condtons Numercal performance p/kpa Z transformed Gbbs enery ± ± ± ± ± ± ± ± ± NFEV s the mean total number of functon evaluatons nvolved n the lobal mnmzaton of transformed Gbbs enery and SR s the success rate to fnd the transformed phase equlbrum compostons. 00 trals performed wth random ntal values for optmzaton varables and random number seeds.

12 96 A. BONILLA-PETRICIOLET et al., Gbbs Enery Mnmzaton Usn Smulated, Chem. Bochem. En. Q. (3) (008) success rate to fnd the lobal optmum of transformed Gbbs enery. For all calculatons performed, mean value of NFEV raned from to and computatonal tme s around 30 s. Ths computatonal tme s very reasonable consdern that an EoS model s used. Wth respect to soluton accuracy, AAD s lower than 0.4 % for all cases. Eample 7. Our fnal eample s a hypothetcal quaternary system that follows the two reactons: A +A A 3 and A 3 A 4 +A. We have consdered deal behavor for both lqud and vapor phases, where the Antone equaton s used to calculate the vapor pressures of pure components wth the parameters reported by Un and Doherty. 3 The transformed mole fractons are defned usn A 3 and A 4 as reference components and they are ven by (3) (3) where and ( 0, ).The chemcal equlbrum constants are calculated usn K eq ep. T and K eq ep where T T T s ven n K. Two optmzaton varables are consdered for the lobal mnmzaton of transformed Gbbs enery. So, flash calculatons are performed for fve arbtrary feeds at.035 bar and 30 K, and our results are reported n Table 8. In eneral, the Smulated Annealn s robust to mnmze the transformed Gbbs enery for the performed calculatons. Specfcally, for only one feed, the success rate of SA method s lower than 00 %. On the other hand, the computatonal tme s around s and the mean value of NFEV raned from to 4 04, respectvely. We note that ths computatonal tme s lower than that obtaned for other eamples wth the same dmensonalty. Ths result s due to the use of alebrac approach to fnd the reference mole fractons nsde the stae of varable transformaton. Fnally, ADD s lower than 0.03 % for all calculatons performed. For all systems wth only one reacton, we note that the computatonal tme nvolved n the stae of varable transformaton can be reduced snfcantly by usn Newton method for fndn the mole fracton of reference component. Unfortunately, the Newton method s hhly senstve to ntal values whle bsecton method s more relable for ths purpose. Conclusons In ths paper, we have ntroduced and tested an alternatve approach for performn two-phase equlbrum calculaton n reactve systems based on transformed Gbbs enery mnmzaton usn Smulated Annealn optmzaton method. In fact, ths work ntroduces the use of a stochastc optmzaton stratey for the lobal mnmzaton of Gbbs enery n reactve systems usn transformed varables. Our results show that the SA method enerally can locate the lobal optmum of Gbbs enery n two-phase systems; t can be appled wth multcomponent reactve systems (wth or wthout nert components) usn many thermodynamc model wthout problem reformulatons and requres a reasonable computatonal tme. Althouh we can not offer a theoretcal uarantee for the lobal Table 8 Numercal performance of smulated annealn n the unconstraned lobal mnmzaton of transformed Gbbs enery for the reactn mture A +A A 3 and A 3 A 4 +A at 30 K and.035 bar (Ideal Gas and Ideal Soluton) Equlbrum condtons Numercal performance Z transformed Gbbs enery ± ± ± ± ± NFEV s the mean total number of functon evaluatons nvolved n the lobal mnmzaton of transformed Gbbs enery and SR s the success rate to fnd the transformed phase equlbrum compostons. 00 trals performed wth random ntal values for optmzaton varables and random number seeds.

13 A. BONILLA-PETRICIOLET et al., Gbbs Enery Mnmzaton Usn Smulated, Chem. Bochem. En. Q. (3) (008) 97 mnmzaton of transformed Gbbs enery usn SA method, t appears that ths stratey s a sutable numercal tool for the analyss and study of phase equlbrum behavor of multcomponent reactn systems. In future work, we wll test and compare the numercal performance of SA method wth respect to others stochastc optmzaton methods, as Genetc Alorthm or Tabu search, n ths knd of thermodynamc calculatons. ACKNOWLEDGEMENTS Ths research was supported by CONACYT, DGEST, Insttuto Tecnolóco de Auascalentes and Unversdad de Guanauato. Authors are also rateful for the techncal support of Azucena Acosta-Martnez n some numercal eperments. Nomenclature a actvty, mol dm 3 A parameter of Marules soluton model AAD mean absolute percentae devaton c number of components transformed molar Gbbs free enery, J mol K eq chemcal equlbrum constant n mole number N square matr formed for all components that partcpate n reacton r NT teraton number of Smulated Annealn optmzaton method OBJ obectve functon value p pressure, bar ref reference component R reacton number R unversal as constant, J mol K RT temperature reducton factor of Smulated Annealn optmzaton method T thermodynamc temperature, K T SA temperature of Smulated Annealn optmzaton method r v stochometrc number of component transformed mole fracton slope of te-lne mole fracton Z transformed feed composton z feed mole composton Greek letters, phase at equlbrum phase mole fracton actvty coeffcent fuacty coeffcent optmzaton varable chemcal potental, J mol admensonal etent of reacton F transformed phase mole fracton temperature, C References. Seder, W. D., Flud Phase Equlbr. 3 (996) 83.. McDonald, C. M., Floudas, C. A., Comp. Chem. En. (996). 3. Un, S., Doherty, M. F., Chem. En. Sc. 50 (995) Pérez-Csneros, E. S., Gan, R., Mchelsen, M. L., Chem. En. Sc. 5 (997) Stateva, R. P., Wakeham, W. A., Ind. En. Chem. Res. 36 (997) Phoen, A. V., Hedemann, R. A., Flud Phase Equlbr (998) Jalal, F., Seader, J. D., Comp. Chem. En. 3 (999) Platt, G. M., de Mederos, J. L., Braz. J. Chem. En. 6 (999) Wasylkewcz, S. K., Un, S., Flud Phase Equlbr. 75 (000) Buros-Solorzano, G. I., Brennecke, J. F., Stadtherr, M. A., Flud Phase Equlbr. 9 (004) 45.. Khaled, R., Bshno, P. R., Ind. En. Chem. Res. 45 (006) Ruz, G., Srdhar, L. N., Renaswamy, R., Ind. En. Chem. Res. 45 (006) Smth, W. R., Ind. En. Chem. Fundam. 9 (980). 4. Reynolds, D., Mulholland, A. J., Gomatam, J., J. Math. Chem. (998) Un, S., Doherty, M. F., Chem. En. Sc. 50 (995) Un, S., Doherty, M. F., AIChE Journal 4 (995) Bonlla-Petrcolet, A., Vázquez-Román, R., Ilesas-Slva, G. A., Hall, K. R., Ind. En. Chem. Res. 45 (006) Whte, W. B., Johnson, S. M., Dantzn, G. B., J. Chem. Phys. 8 (958) Wasylkewcz, S. K., Chem. En. Sc. 55 (000) Ranaah, G. P., Flud Phase Equlbr (00) 83.. Henderson, N., de Olvera, J. R., Souto, H. P. A., Marques, R. P., Ind. En. Chem. Res. 40 (00) Fretas, L., Platt, G., Henderson, N., Flud Phase Equlbr. 5 (004) Arora, J. S., Struct. Saf. 7 (990) Bazaraa, M. S., Sheral, H. D., Shetty, C. M., Nonlnear Prorammn: Theory and alorthms, nd ed., John Wley and Sons, Khompatraporn, C., Pnter, J. D., Zabnsky, Z. B., J. Global Optm. 3 (005) Mchalewcz, Z., Foel, D. B. How to solve t? Modern heurstcs. Sprner, München, Zhu, Y., u, Z., Flud Phase Equlbr. 54 (999) Costa, A. L. H., da Slva, F. P. T., Pessoa, F. L. P., Braz. J. Chem. En. 7 (000) Bonlla-Petrcolet, A., Bravo-Sanchez, U. I., Castllo-Bora, F., Zapan-Salnas, J. G., Soto-Bernal, J. J., Braz. J. Chem. En. 4 (007) 5.

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