Calculation of homogeneous azeotropes in reactive and nonreactive mixtures using a stochastic optimization approach

Transcription

1 Insttuto Tecnologco de Aguascalentes From the SelectedWorks of Adran Bonlla-Petrcolet 2009 Calculaton of homogeneous azeotropes n reactve and nonreactve mxtures usng a stochastc optmzaton approach Adran Bonlla-Petrcolet Gustavo A. Iglesas-Slva Kenneth R. Hall, Texas A & M Unversty - College Staton Avalable at:

2 Ths artcle appeared n a journal publshed by Elsever. The attached copy s furnshed to the author for nternal non-commercal research and educaton use, ncludng for nstructon at the authors nsttuton and sharng wth colleagues. Other uses, ncludng reproducton and dstrbuton, or sellng or lcensng copes, or postng to personal, nsttutonal or thrd party webstes are prohbted. In most cases authors are permtted to post ther verson of the artcle (e.g. n Word or Tex form) to ther personal webste or nsttutonal repostory. Authors requrng further nformaton regardng Elsever s archvng and manuscrpt polces are encouraged to vst:

3 Flud Phase Equlbra 281 (2009) Contents lsts avalable at ScenceDrect Flud Phase Equlbra journal homepage: Short communcaton Calculaton of homogeneous azeotropes n reactve and non-reactve mxtures usng a stochastc optmzaton approach Adran Bonlla-Petrcolet a,, Gustavo A. Iglesas-Slva b, Kenneth R. Hall c a Insttuto Tecnologco de Aguascalentes, Depto. de Ing. Qumca, Aguascalentes, Mexco b Insttuto Tecnologco de Celaya, Depto. de Ing. Qumca, Celaya, Mexco c Texas A&M Unversty, Department of Chemcal Engneerng, College Staton, TX, USA artcle nfo abstract Artcle hstory: Receved 3 December 2008 Receved n revsed form 4 March 2009 Accepted 5 March 2009 Avalable onlne 21 March 2009 Keywords: Homogeneous azeotrope Reactve homogeneous azeotrope Smulated Annealng Global optmzaton Phase equlbrum In ths paper we ntroduce an alternatve strategy to fnd homogeneous azeotropes n reactve and nonreactve mxtures whch s based upon the Smulated Annealng optmzaton technque. Ths stochastc optmzaton method s used to robustly solve a system of non-lnear equatons that results from the equaltes of the orthogonal dervatves of the Gbbs energy and the Gbbs energy of mxng n the vapor and the lqud phases. For non-reactve systems, ths equaton system s solved by consderng conventonal composton varables whle for reactve cases we use the transformed composton varables proposed by Ung and Doherty. Numercal performance of our approach s llustrated usng several examples prevously reported n the lterature and results show that t s a sutable and robust strategy for the calculaton of homogeneous azeotropes n mxtures wth or wthout chemcal reactons Elsever B.V. All rghts reserved. 1. Introducton Azeotropy s a phase equlbrum phenomenon that occurs n many ndustral applcatons and ts presence alters product dstrbuton and restrcts the separaton amount of a multcomponent mxture that can be acheved by dstllaton [1]. Azeotropes can occur n reactve and non-reactve mxtures and they can be classfed as homogeneous and heterogeneous dependng on the number of lqud phases nvolved n the equlbrum condton. For non-reactve mxtures, homogeneous azeotropes occur when the composton of vapor and lqud phases at equlbrum s dentcal. The same defnton apples for homogeneous azeotropes n reactve systems but usng reacton-nvarant composton space [1]. The descrpton of reactve and non-reactve homogeneous azeotropy s essental for the selecton of strateges n synthess, desgn and operaton of separaton unts [1,2]. Ths descrpton conssts of establshng the temperature, pressure and composton as well as the component number of the azeotrope. Tradtonal methods for the calculaton of non-reactve homogeneous azeotropes use local procedures and dfferent objectve functons employng fugacty coeffcents or relatve volatltes [3,4]. However, several relable technques have evolved recently to calculate homogeneous azeotropes n non-reactve systems [5 10]. Correspondng author. Tel.: ; fax: E-mal address: petrcolet@hotmal.com (A. Bonlla-Petrcolet). For example, Fdkowsk et al. [5] developed a globally convergent method to locate all the homogeneous azeotropes usng a homotopy technque together wth a robust phase stablty analyss. Hardng et al. [6] used a determnstc global optmzaton procedure wth convex underestmatng functons for the thermodynamc equatons to perform a global mnmzaton. The success or falure of ths technque depends upon proper constructon of the convex functons. Maer et al. [7] used an nterval- Newton/generalzed-bsecton algorthm to locate all the solutons of the thermodynamc condtons for homogeneous azeotropy usng soluton models and deal gas behavor. Later, Salomone and Espnosa [8] combned ths nterval-newton/generalzed-bsecton algorthm wth Zharov Serafmov topologcal ndex theory to reduce the total computaton tme for the calculaton of homogeneous azeotropes. Fnally, Aslam and Sunol [9] used a homotopy contnuaton method to study the senstvty of azeotropc states to actvty coeffcent model parameters and operatng condtons. Also, ths method has been successfully appled wth EoS models [10]. On the other hand, for systems subject to chemcal reactons, only a few robust methods have been developed to address the problem of locatng all reactve homogeneous azeotropes [1,11 13]. Okasnsk and Doherty [1] reported a homotopy contnuaton approach to locate multcomponent reactve azeotropes. They have analyzed the effect of reacton equlbrum constant on the exstence of reactve azeotropes. Later, Hardng and Floudas [11] appled ther determnstc optmzaton method n the /$ see front matter 2009 Elsever B.V. All rghts reserved. do: /j.flud

4 A. Bonlla-Petrcolet et al. / Flud Phase Equlbra 281 (2009) calculaton of all reactve azeotropes of multcomponent systems. Ths method also offers a theoretcal guarantee for fndng all azeotropes, but dependng on the thermodynamc model, t s necessary to reformulate the problem. Maer et al. [12] have also extended ther nterval-newton/generalzed-bsecton algorthm to locate all homogeneous reactve azeotropes usng soluton models and deal gas behavor. These authors provde a computatonal guarantee for the convergence of ther method. More recently, Q and Sundmacher [13] proposed a geometrcal approach, based on a contnuaton method, to locate all knd of azeotropes (reactve, non-reactve and knetc). It s mportant to note that these methods and those descrbed for non-reactve mxtures are useful for predcton of polyazeotropy,.e. two or more azeotropes at sobarc or sothermal condtons. Although these methods have demonstrated to be very promsng, some of them are model dependent, may requre problem reformulatons or a sgnfcant computatonal tme for multcomponent systems. Under ths context, the stochastc optmzaton methods offer some advantages for the calculaton of homogeneous azeotropes and may overcome the dffcultes descrbed above. Specfcally, these methods are robust numercal tools that present a reasonable computatonal effort n the optmzaton of multvarable functons; they are applcable to ll-structure or unknown structure problems, are effcent f properly mplemented, requre only calculatons of the objectve functon and can be used wth all thermodynamc models [14]. Many thermodynamc problems that are very dffcult to solve by conventonal technques can be solved by stochastc methods. In thermodynamcs, Smulated Annealng, genetc algorthm and Tabu Search have been successfully appled n phase equlbrum and stablty problems, crtcal pont calculatons and parameter estmaton [14 19]. To the best of our knowledge, stochastc optmzaton methods have not been appled n the calculaton of homogeneous azeotropes n both reactve and non-reactve mxtures. In ths paper we ntroduce an alternatve technque to fnd homogeneous azeotropes n reactve and non-reactve mxtures whch s based upon the Smulated Annealng optmzaton technque. Ths stochastc optmzaton method s used to robustly solve a system of non-lnear equatons that results from the equaltes of the orthogonal dervatves of the Gbbs energy and the Gbbs energy of mxng n the vapor and the lqud phases. For non-reactve systems, ths equatons system s solved by consderng conventonal composton varables whle for reactve cases we use the transformed composton varables proposed by Ung and Doherty [20]. Numercal performance of our approach s llustrated usng several examples prevously reported n the lterature and results show that t s a smple and relable strategy for the calculaton of homogeneous azeotropes n mxtures wth or wthout chemcal reactons. 2. Thermodynamc problem formulaton Recently, Iglesas-Slva et al. [21] have used the orthogonal dervatves, the tangent plane crteron and mass balances to fnd the equlbrum compostons of a multcomponent mxture. For a non-reactve mxture wth two phases at equlbrum, they solve a system of 2c 1 non-lnear equatons ( g x ) T,P,x j /= c 1 (gˇ g ) j=1 z = x + xˇ ˇ ( g x )ˇ (xˇ j T,P,x j /= = 0 = 1, 2,...,c 1 (1) x j ) ( g x j ) T,P,x l /= j = 0 (2) = 1, 2,...,c 1 (3) beng ( ) g x T,P,x j /= = c = 1, 2,...,c 1 (4) where g s the Gbbs energy of mxng, z s the mole fracton of component n the feed, x k s the mole fracton of component n the phase k, k s the fracton of moles n phase k where + ˇ =1, c s the number of components, s the chemcal potental of component at the mxture, and ˇ denote the vapor and lqud phases, respectvely. Note that the sum of all mole fractons n each c phase must equal unty: x c = 1, xˇ c = 1 and =1 =1 =1 z = 1, respectvely. If a mxture presents homogeneous azeotropy (z = x = xˇ ), the above equatons smplfy to F 1 = g gˇ = 0 (5) F +1 = ( g x ) T,P,x k /= ( g x )ˇ T,P,x k /= = 0 = 1, 2,...,c 1 where Eqs. (5) and (6) are a system of c non-lnear equatons wth c 1 unknown compostons (mole fractons) plus the unknown temperature or pressure of the azeotrope. Eqs. (1) (4) can be extended for the calculaton of phase equlbrum n reactve systems f we consder the transformed composton varables proposed by Ung and Doherty [20]. Gbbs energy n a reactve system behaves as n a non-reactve system f transformed composton varables are used nstead of the conventonal composton varables. Also, these authors showed that the chemcal potental follows all the thermodynamc relatonshps of a non-reactve system as long as all the thermodynamc propertes are functon of the transformed composton varables. Based on ths dea, two-phase equlbrum n a reactve system can be calculated by usng [22] ( ĝ X ) T,P,X j /=,j {1,...,c R 1} ( ĝ X )ˇ T,P,Xˇ j /=,j {1,...,c R 1} = 0 = 1,...,c R 1 (7) c R 1 ( ) ĝ ĝˇ ĝ (Xˇ X ) = 0 (8) X =1 T,P,X j /=, j {1,...,c R 1} Z = X + Xˇ ˇ and ( ĝ (6) = 1,...,c R 1 (9) = ˆ ˆ c R = 1,...,c R 1 (10) X )T,P,X j /=,j {1,...,c R 1} subject to c Keq r = a vr r = 1,,R (11) =1 where R s the ndependent reacton number, ĝ and ĝˇ are the dmensonless transformed molar Gbbs energy of mxng for reactng system of the phases at equlbrum, Z s the transformed overall mole fracton composton of component, Keq r s the reacton equlbrum constant n reacton r, a s the actvty of component, v r s the stochometrc coeffcent of component n reacton r, ˆ s the transformed chemcal potental of component at the mxture

5 24 A. Bonlla-Petrcolet et al. / Flud Phase Equlbra 281 (2009) whch s a functon of X, and and ˇ are the transformed fractons of moles at the vapor and lqud phases, respectvely. Due to mass balance restrctons, the sum of transformed varables must c R equal unty,.e. X c R = 1, Xˇ = 1 and + ˇ =1. =1 =1 Transformed mole fractons X are defned as [20] X = x v N 1 x ref 1 v TOT N 1 x = 1,...,c R (12) ref where x ref s the column vector of R reference component mole fractons, v s the row vector of stochometrc coeffcents 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 matrx formed from the stochometrc coeffcents of the reference components n the R reactons [20]. For varable transformaton X x, we note that the reference mole fractons are calculated usng Eq. (12) and from the equlbrum constants for each reacton Keq r by solvng a system of R non-lnear equatons. When the reference mole fractons are known, the remanng mole fractons are calculated usng Eq. (12). We have used the bsecton method for fndng the mole fracton of reference components n smple reactve systems. Ung and Doherty [23] showed that Z = X = Xˇ s a necessary and suffcent condton for the presence of a homogeneous reactve azeotrope. In the transformed composton space X, ths condton has the same functonal form as the condton for non-reactve homogeneous azeotropy n mole fracton space x [23]. Therefore, f we replace ths equalty crteron nto Eqs. (7) (9), now yelds F 1 = ĝˇ ĝ = 0 (13) F +1 = ( ĝ X ) T,P,X j /=,j {1,...,c R 1} ( ĝ X )ˇ T,P,Xˇ j /=,j {1,...,c R 1} = 0 = 1,...,c R 1 (14) These equatons can be used to fnd c R 1 transformed composton varables and the reactve azeotrope temperature or pressure. The materal balance s meanngless for both reactve and non-reactve azeotropes. If the Gbbs energy, or transformed Gbbs energy, s plotted for both lqud and vapor phases aganst the correspondng composton varable at the homogeneous azeotrope pressure and temperature, the Gbbs curves of each phase wll ntersect at one pont satsfyng the equalty crteron of the tangent plane and the equalty of chemcal potentals. Fg. 1a shows the Gbbs energy surface for a stable homogeneous azeotrope. Note that t s possble to fnd unstable solutons (.e. unstable phases) for the thermodynamc condtons of homogeneous azeotropy. Ths case s llustrated n Fg. 1b and generally suggests the presence of a heterogeneous azeotrope [4]. Therefore, any soluton of homogeneous azeotropy problem, Eqs. (5) and (6), and (13) and (14), should next be checked for phase stablty [7]. The systems of Eqs. (5) and (6) and (13) and (14) are an alternatve set of thermodynamc condtons to locate all homogeneous non-reactve and reactve azeotropes n multcomponent systems whch can be appled wth any thermodynamc model. Also, they are non-lnear and may have one, more than one or no solutons (one azeotrope, several azeotropes or the non-exstence of a homogeneous azeotrope). As ndcated by several authors [5 13,24], the hghly non-lnear form of the thermodynamc condtons makes the computaton of reactve and non-reactve azeotropes a challengng numercal problem. Therefore, conventonal root-fnders cannot be used robustly to solve the systems of Eqs. (5) and (6) and (13) and (14). So, a relable strategy must be consdered for the calculaton Fg. 1. Gbbs energy of mxng of the lqud and vapor phases for a homogeneous azeotrope (reactve or non-reactve): (a) stable azeotrope and (b) unstable azeotrope. of homogeneous azeotropes. In the next secton, we descrbe our proposed approach to solve ths thermodynamc problem. 3. Global optmzaton approach for the calculaton of homogeneous azeotropes We use an optmzaton approach to locate all solutons of the system of Eqs. (5) and (6) and (13) and (14). We formulate the system of equatons as a sngle objectve functon that should mnmze by a sutable optmzaton technque. The objectve functon s gven by nec F obj = F F 2 (15) =2 where nec s equal to c 1 for non-reactve mxtures whle n reactve mxtures t s gven by c R 1. Due to the non-lnear nature of thermodynamc models, Eq. (15) s potentally non-convex and may have multple local optmums. At the homogeneous azeotrope condton, the global mnmum of ths functon must be zero, but we assume that a azeotrope s found when we fnd the composton varables (x or X ) and temperature or pressure that make the value of the functon less than or equal to ; otherwse, we consder that an azeotrope does not exst n the mxture at the gven condtons. For polyazeotropy systems, our objectve functon wll show several optmums where F obj Note that ths objectve functon value s equvalent to F where ε s around Ths tolerance value s suffcently small to accurately determne the azeotropc condtons snce very accurate results are requred for process desgn. Smlar tolerance values have been used n other studes for solvng non-lnear equatons systems related to

6 A. Bonlla-Petrcolet et al. / Flud Phase Equlbra 281 (2009) Fg. 2. Flow chart for the calculaton of homogeneous azeotropes n non-reactve and reactve systems usng Smulated Annealng method. phase equlbrum calculatons [6,11,15]. In addton, our numercal experence ndcates that the non-exstence of azeotropes s gven by F obj > In Secton 4 of ths paper, we provde results that support the choce of ths crteron for determnng the presence of a homogeneous azeotrope. The followng procedure s used to locate all homogeneous reactve or non-reactve azeotropes. Frst, we defne the component number (azeotrope order) and ntal search ntervals for the composton varables and the temperature or pressure. In these ntervals, the objectve functon s mnmzed usng the Smulated Annealng optmzaton method and, once a mnmum s found, we create new search ntervals usng the found soluton for the temperature or pressure. In these new ntervals, we agan mnmze the objectve functon. Ths procedure s repeated untl the convergence crteron cannot be satsfed. Fg. 2 shows a flowchart of ths algorthm. We penalze the objectve functon n the lmts of the ntervals obtaned from segmentaton to avod convergence to the last soluton when system shows polyazeotropy. Usually, the value of the objectve functon s greater than f one nterval does not contan an azeotrope. We note that a smlar approach has been also appled by Fretas et al. [15] and Justo-García et al. [18] n the calculaton of crtcal ponts usng an optmzaton strategy. Eq. (15) must be solved for each combnaton of k components to fnd all k-ary homogeneous azeotropes n a multcomponent system where k =2,..., c for non-reactve mxtures and k =2,..., c R for reactve mxtures, respectvely. It should be noted that our problem formulaton avods the trval solutons representng the pure components. The stochastc optmzaton method Smulated Annealng (SA) s used to globally mnmze Eq. (15). In general, SA method overcomes most of the numercal dffcultes that local optmzaton methods show. Due to the statstcal nature of the SA, a local mnmum can be hopped much more easly than n conventonal methods [25]. Specfcally, SA smulates the process of slow coolng of metals to acheve the mnmum functon value n a mnmzaton problem. The coolng phenomenon s modeled by controllng a temperature lke parameter ntroduced wth the concept of Boltzmann probablty dstrbuton. By a controlled temperature reducton as the algorthm proceeds, the convergence of the algorthm can be controlled. Krkpatrck et al. [26] suggested the use of SA as a technque for dscrete optmzaton, and several researchers have successfully extended the method to problems nvolvng contnuous varables nsde the thermodynamcs and others scence s felds [14 16,18,19,25]. In the present study, the SA code developed by Goffe et al. [27] was used, whch s based on the algorthm developed by Corana et al. [28]. Ths SA algorthm has been successfully appled n several thermodynamc problems [14 16,18,19]. Itsanteratve random search procedure wth adaptve moves along the coordnate drectons. The step szes for the optmzaton varables are adjusted n an attempt to mantan approxmately 1:1 rate between accepted and rejected confguratons. For mnmzng a functon, the algorthm starts wth a hgh ntal annealng temperature T SA and a startng value set of m optmzaton varables. Tral ponts are then generated usng random numbers and ntal step length VM for each optmzaton varable. The tral pont s accepted f ts functon value s lower than that for the prevous one. If the tral pont has a hgher functon value, Metropols crteron [29] s used to accept o reject t, where T SA s consdered on ths crteron. The step lengths of the m varables are adjusted after NS steps. After carryng these cycles NT tmes, the annealng temperature s gradually reduced by a factor RT. These cycles contnue untl t s observed that the reducton n the functon value n NE successve cycles s less than a user-defned small number. In ths study, the tolerance value used for the convergence of SA method s Numercal performance (relablty and effcency) of SA method s affected by the coolng schedule whch s lnked to the parameters T SA, RT and NT. However, SA s a robust optmzaton method f an approprate coolng schedule s used [27]. So, the parameters of coolng schedule requre pre-calbraton for new problems. Dfferent combnatons for parameters T SA (10, 1000), NT (5, 20) and RT (0.1, 0.85) were tred to test the performance of SA method usng random ntal values and random number seeds for the calculaton of homogeneous azeotropes n several reactve and non-reactve examples. We have found that n calculatons performed SA generally converges to the global mnmum of Eq. (15), ndependently of the ntal values for azeotropc condtons, usng the dfferent coolng schedules. Also, we fnd that the optmzaton method works acceptably well and converges usng fewer functon evaluatons (or computatonal tme) wth RT=0.1,NT=5,andT SA = 10 for both non-reactve and reactve mxtures. Fnally, phase stablty of all calculated reactve and non-reactve azeotropes must be tested by mnmzng the correspondng tangent plane dstance functon [30,31] c TPDF = x [ (x ) (z )] (16) or =1 c R RTPDF = X [ˆ (X ) ˆ (Z )] (17) =1 where TPDF s the phase stablty crteron for non-reactve mxtures whle RTPDF s the correspondng one for reactve mxtures, respectvely. If the global mnmum of TPDF or RTPDF < 0; the azeotrope under analyss s unstable. We mnmze both functons also usng SA method. Bonlla-Petrcolet et al. [32] has tested the

7 26 A. Bonlla-Petrcolet et al. / Flud Phase Equlbra 281 (2009) Table 1 Test examples for the calculaton of homogeneous azeotropes n non-reactve mxtures usng Smulated Annealng method. No. System Thermodynamc models Optmzaton varables 1 Benzene + sopropanol at kpa NRTL model and deal gas. x benzene (0,1) Model parameters reported by Maer et al. [7] T (283,373) K 2 Ethanol + methyl ethyl ketone + water at kpa 3 Acetone + chloroform + methanol + benzene at kpa NRTL model and deal gas. n (0,1) Model parameters reported by Maer et al. [7] T (283,373) K NRTL model and deal gas. n (0,1) Model parameters reported by Maer et al. [7] T (283,373) K 4 CO 2 + ethane at kpa SRK EoS wth conventonal mxng rules. x CO2 (0, 1) k 12 = Model parameters reported by Gow et al. [33] T (200,270) K 5 Chloroform + ethanol at K G E /R gt = x 1x 2(1.42x x 2) x Chlor (0, 1) B (P P sat )+ 1 2 x j x l (2ı j ı jl ) j l ϑ = exp Rg T P (0,200) kpa ı j =2B j B B jj, B 11 = 963, B 22 = 1523, B 12 =52cm 3 mol 1. P sat = and 1 P sat = kpa 2 6 R23 + R13 at K P = Rg T V b a(t) g Tc 2 (V+c) 2 64PcTr n x R23 (0,1) ( ) ( ) b = V 4Zc c c = V 8Zc c T (2000,3000) kpa Model parameters reported by Gow [34] 7 Benzene + hexafluorbenzene at kpa NRTL model and deal gas. x benzene (0,1) Model parameters reported by Maer et al. [7] T (283,373) K 8 Methanol + -propanol at kpa NRTL model and deal gas. x methanol (0,1) Model parameters reported by Maer et al. [7] T (283,373) K 9 Acetone + chloroform + methanol at kpa 10 Acetone + chloroform + ethanol + benzene at kpa 11 Acetone + chloroform + methanol + ethanol + benzene at kpa NRTL model and deal gas. n (0,1) Model parameters reported by Maer et al. [7] T (373,473) K NRTL model and deal gas. n (0,1) Model parameters reported by Maer et al. [7] T (283, 373) K NRTL model and deal gas. n (0,1) Model parameters reported by Maer et al. [7] T (283,373) K SA method n phase stablty analyss of several reactve and nonreactve systems and they found that ths method s very relable for ths purpose. In ths study, all reported azeotropes are thermodynamcally stable. whle for an deal gas behavor we have c ( ) x P gˇ = x ln P sat =1 (20) 4. Results and dscusson We have calculated the homogeneous azeotropes for bnary, ternary, quaternary and qunary reactve and non-reactve mxtures that appear n the lterature for testng the numercal performance of our strategy. We use soluton models and equatons of state to represent the lqud phase, and also deal gas behavor and equatons of state to represent the vapor phase. The dmensonless molar Gbbs free energy of mxng for a non-reactve mxture usng an EoS model s gven by ( ) c x g = x ln ϕ (18) ϕ =1 where ϕ and ϕ are the fugacty coeffcent of component at the mxture and of the pure component, respectvely. If a soluton model s consdered for a lqud phase, g s defned as g = c x ln(x ) (19) =1 beng s the actvty coeffcent of component at the mxture and P sat s the saturaton pressure of pure component. For reactve mxtures, these thermodynamc functons are c R ( ) x ˆϕ ĝ = X ln (21) ϕ =1 ĝ = ĝˇ = c R X ln(x ) (22) =1 and c R ( ) x P X ln P sat =1 (23) where the dmensonless transformed molar Gbbs free energy of mxng s evaluated at the transformed mole fractons X. To evaluate Eqs. (21) (23) n reactve systems, conventonal mole fractons x are obtaned from the transformaton procedure X x. These mole fractons satsfy the stochometry requrements and are chemcally equlbrated.

8 A. Bonlla-Petrcolet et al. / Flud Phase Equlbra 281 (2009) Table 2 Test examples for the calculaton of homogeneous azeotropes n reactve mxtures usng Smulated Annealng method. No. Reacton Thermodynamc models Optmzaton varables 1 A+B C at kpa Ideal gas and deal soluton X 1 (0,1) G rxs= 8314 J/mol T (283,473) K X 1 = x 1 +x 3, 1+x 3 X 2 = x 2 +x 3 = 1 X 1+x 1 3 Model parameters reported by Maer et al. [12] 2 A+B C + D at kpa Ideal gas and deal soluton X 1, X 2 (0,1) G rxs = J/mol T (300,400) K X 1 = x 1 + x 4, X 2 = x 2 + x 4 X 3 = x 3 x 4 =1 X 1 X 2 Model parameters reported by Maer et al. [12] 3 Isobutene (1) + methanol (2) methyl tert-butyl ether (3) at kpa Wlson model and deal gas X 1 (0,1) K eq = 0.04 and K eq = 20.0 T (283,473) K 4 Isobutene (1) + methanol (2) methyl tert-butyl ether (3) wth n-butane (4) as nert at kpa X 1 = x 1 +x 3 1+x 3, X 2 = x 2 +x 3 1+x 3 = 1 X 1 Model parameters reported by Maer et al. [12] Wlson model and Ideal gas G rxs/r g = T T ln T X 1 = x 1 +x 3, X 1+x 2 = x 2 +x x 3 X 4 = x 4 1+x 3 = 1 X 1 X 2 Model parameters reported by Maer et al. [12] n 1, n 2, n 4 (0,1) where X = T (323,473) K n = 1, 2 and 4 n 1+ n2+ n4 5 Acetc acd (1) + sopropanol (2) sopropyl acetate (3) + water (4) at kpa and K eq = 8.7 NRTL model and Ideal gas wth assocaton for Acetc acd. X 1, X 2 (0,1) log 10 k = /T, k [=] Pa 1 1 = 1+(1+4kPsat 1 )1/2 1+[1+4kPx 1 (2 x 1 )] 1/2 T (293,473) K = 2{1 x 1 +[1+4kPx 1 (2 x 1 )]1/2 } (2 x 1 ){1+[1+4kPx 1 (2 x 1 )] 1/2 } X 1 = x 1 + x 4, X 2 = x 2 + x 4 X 3 = x 3 x 4 =1 X 1 X 2 Model parameters reported by Maer et al.[12] 6 2A B + C at kpa Wlson model and Ideal gas X 1 (0,2) K eq = 1000 T (350,500) K X 1 = x 1 + 2x 3, X 2 = x 2 x 3 =1 X 1 Model parameters reported by Okasnsk and Doherty [1] Tables 1 and 2 show our examples, thermodynamc models, test condtons and optmzaton varables used for the calculaton of homogeneous azeotropes. Note that non-reactve examples 8 11 and reactve example 3 (wth K eq = 20.0) are used to llustrate the performance of our approach for verfyng the non-exstence of homogeneous azeotropes. By convenence n some examples, dependng upon the problem characterstcs, we used mole numbers or mole fractons (conventonal or transformed) as optmzaton varables. Due to the stochastc nature of SA method, all examples were solved 100 tmes usng random ntal values for optmzaton varables and random number seeds. Our calculatons were performed on a Processor Intel Pentum M 1.73 GHz wth 504 MB of RAM. Calculated azeotrope condtons for test examples are reported n Tables 3 7. In all calculatons performed, our method fnds all azeotropes, or t determnes the non-exstence of azeotropes, wthout any numercal problem ndependently of the ntal values used for optmzaton varables. In fact, our method s very relable for fndng homogeneous azeotropes n both reactve and non-reactve mxtures. Wth respect to effcency, Fg. 3 shows the mean number of functon evaluatons (NFEV) requred for fndng the homogeneous azeotropes n reactve and non-reactve examples usng SA method. In general, mean NFEV ranged from 2831 to 7356 where ths numercal effort s equvalent to 0.03 and 2.39 s of computatonal tme (see Tables 3 and 4). Even though the NFEV appears to be sgnfcant, the computaton tme nvolved for the calculaton of azeotropc states s very reasonable. On the other hand, the computatonal tme n reactve examples s hgher due to the varable transformaton procedure X x nvolved n the calculaton of Table 3 Results of azeotrope calculatons n non-reactve test examples usng Smulated Annealng method. No. Homogeneous azeotrope CPU tme (s) 1 x aze ( , ) and K x aze ( , , ) and K 3 x aze ( , , , ) and K 4 x aze ( , ) and K x aze ( , ) and kpa x aze ( , ) and kpa x aze1 ( , ) and K 0.03 x aze2 ( , ) and K No azeotrope No azeotrope No azeotrope No azeotrope 0.17

9 28 A. Bonlla-Petrcolet et al. / Flud Phase Equlbra 281 (2009) Table 4 Results of azeotrope calculatons n reactve test examples usng Smulated Annealng method. No. Homogeneous azeotrope Transform mole fractons Mole fractons CPU tme (s) 1 X aze (0.3515, ) x ( , , ) 0.39 T = K xˇ ( , , ) 2 X aze (0.4266, , ) x ( , , , ) 0.91 T = K xˇ ( , , , ) 3a a X aze (0.9358, ) x ( , , ) 0.69 T = K xˇ ( , , ) 3b b No azeotrope X aze (0.0413, , ) x ( , , , ) 2.39 T = K xˇ ( , , , ) 5 X aze (0.2611, , ) x ( , , , ), xˇ ( , , , ) 2.23 T = K 6 X aze (0.0008, ) x ( , , ) 0.53 T = K xˇ ( , , ) a K eq = b K eq = Table 5 Results of azeotrope calculatons n non-reactve test example no.7 usng Smulated Annealng method. Intal value Optmzaton nterval Azeotrope condton x benzene T (K) x benzene T (K) x benzene T (K) No azeotrope No azeotrope thermodynamc propertes. In some cases, CPU tmes requred for verfyng the non-exstence of homogeneous azeotropes are slghtly hgher than those reported to compute an azeotrope. For comparson purposes, prelmnary calculatons wth local solvers for fndng homogeneous azeotropes ndcates that n general these methods showed a success rate from 70% to 99% also usng random ntal values. When the unknown number ncreases, the performance of local solvers s worst. Of course, they are more effcent than SA method, however, are less relable and more dependent on the ntal guesses. As example of polyazeotropy, Table 5 shows the procedure used to fnd both azeotropes of the mxture benzene + hexafluorbenzene at kpa. The frst azeotrope s located at K, and we then use ths value to create two dfferent search ntervals for temperature: K and K where the objectve functon s mnmzed. In the frst nterval, a second azeotrope occurs and, n the second nterval, no set of values satsfes the convergence crtera. For the second azeotrope, new search ntervals do not provde other soluton. We note that the sequence for locatng both azeotropes depends on the ntal values of composton varables and temperature. However, the relablty of our method s ndependent of ntal guesses and of the values of algorthm parameters used for SA method. For llustratve purposes, Fg. 4 shows the numercal performance of SA method by plottng F obj versus NFEV for selected non-reactve examples, whch have no azeotropes. Even though the objectve functon value decreases n the ntal teratons, t does not mprove sgnfcantly over further teratons and s hgher than These results and the numercal experence obtaned wth other examples, not reported n ths paper, ndcated that F obj > s a sutable crteron to dentfy the non-exstence of homogeneous azeotropes usng SA method. However, we cannot Table 6 Results of azeotrope calculatons n reactve and non-reactve test examples usng a hybrd approach based on Smulated Annealng and Newton method. No. Example Non-reactve NFEV CPU tme (s) No. Reactve NFEV CPU tme (s) SA Newton SA Newton a a b b < < a K eq = b K eq = 20.0.

10 A. Bonlla-Petrcolet et al. / Flud Phase Equlbra 281 (2009) Table 7 Results of the calculaton of all k-ary homogeneous azeotropes for the qunary system acetone + chloroform + methanol + ethanol + benzene at kpa usng NRTL model and deal gas. Model parameters taken from Maer et al. [7]. Components Homogeneous azeotrope SA-Newton method x (A,C,M,E,B) a T (K) NFEV CPU tme (s) AC (0.3609, , 0.0, 0.0, 0.0) <0.01 AM (0.7844, 0.0, , 0.0, 0.0) <0.01 AE No azeotrope AB No azeotrope CM (0.0, , , 0.0, 0.0) <0.01 CE (0.0, , 0.0, , 0.0) <0.01 CB No azeotrope ME No azeotrope MB (0.0, 0.0, , , 0.0) <0.01 EB (0.0, 0.0, 0.0, , ) <0.01 ACM (0.3768, , , 0.0, 0.0) ACE (0.3604, , 0.0, , 0.0) AME No azeotrope AMB No azeotrope AEB No azeotrope CME No azeotrope CMB No azeotrope CEB No azeotrope MEB No azeotrope ACME No azeotrope ACMB (0.3569, , , 0.0, ) ACEB No azeotrope AMEB No azeotrope CMEB No azeotrope ACMEB No azeotrope a A = acetone, C = chloroform, M = methanol, E = ethanol, B = benzene. offer a theoretcally based guarantee for ths convergence crteron. Fg. 5a and b shows the convergence curves of F obj durng the calculaton of homogeneous azeotropes for selected non-reactve and reactve examples wth dfferent dmensonalty. We observe that SA method requres a sgnfcant computatonal effort to mprove the accuracy of azeotrope condtons after F obj has reached a value lower than Ths performance prevals n all test examples. The man reason for ths slow convergence s that SA explores the search space of decson varables (.e. azeotropc condtons) by creatng random movements nstead of determnng a logcal optmzaton trajectory. Based on these results, we tred a hybrd approach startng wth SA method as a robust algorthm to ensure a certan progress from poor ntalzatons, and endng wth a more effcent local method to accelerate convergence. Specfcally, we used the Newton method as local strategy to solve the system of equatons defned by Eqs. (5) and (6) or (13) and (14) usng the soluton obtaned by SA method. The swtch to the Newton method takes place once F obj s lower or equal than Therefore, the local method s used to mprove the computatonal effcency and accuracy of fnal soluton. All examples were solved usng ths approach (also performng 100 runs wth random ntal values) and obtaned results are reported n Table 6. For all calculatons performed usng our hybrd method, the azeotropes were found wthout any numercal problem. The results ndcate that a sutable ntalzaton for Newton method s avalable after applcaton of SA method, and convergence s easly acheved. Furthermore, computatonal tme s sgnfcantly reduced for both reactve and non-reactve examples (see Fg. 5). Generally, there s a 3:1 rate between the computatonal tmes of SA method and SA-Newton hybrd method. It appears that the tme reducton s more sgnfcant n non-reactve test examples. Note that the task of verfyng the non-exstence of an azeotrope s performed by SA wthout the swtch to Newton method due to F obj s hgher than for ths condton. Fg. 3. Mean number of functon evaluatons (NFEV) for the calculaton of homogeneous azeotropes n both reactve and non-reactve test examples usng Smulated Annealng method. Fg. 4. Convergence profles for verfyng the non-exstence of homogeneous azeotropes n selected non-reactve examples by Smulated Annealng.

11 30 A. Bonlla-Petrcolet et al. / Flud Phase Equlbra 281 (2009) We present an alternatve set of thermodynamc condtons to locate all homogeneous non-reactve and reactve azeotropes of multcomponent mxtures. Specfcally, we showed that a homogeneous azeotrope (reactve or non-reactve) must satsfy an equalty of the Gbbs free energy of mxng between vapor and lqud phases and an equalty of the orthogonal dervatves of Gbbs energy wth respect to composton varables n each phase. The resoluton of these thermodynamc condtons can be done wth confdence usng the stochastc optmzaton method Smulated Annealng. Ths optmzaton method s robust for fndng all homogeneous non-reactve and reactve azeotropes and s almost ndependent of the ntal values and ts algorthm parameters. Also, we can sgnfcantly reduce the computatonal tme for the locaton of homogeneous azeotropes by combnng the Smulated Annealng strategy wth the Newton method. Ths hybrd approach s robust and more effcent to locate both reactve and non-reactve azeotropes. Fnally, our method can be used wth any thermodynamc model wthout the problem reformulaton and t s sutable for multcomponent calculatons. Fg. 5. Convergence profles for the calculaton of homogeneous azeotropes n selected examples by Smulated Annealng: (a) non-reactve mxtures and (b) reactve mxtures. Fnally, Table 7 reports the results for the calculaton of all k-ary azeotropes (k 5) n the non-reactve qunary system of acetone + chloroform + methanol + ethanol + benzene at kpa. Ths system has been studed by Fdkowsk et al. [5], Hardng et al. [6], Maer et al. [7] and Salomone and Espnosa [8] for testng the performance of global solvng methods. Our calculatons are performed usng NRTL model and deal gas wth model parameters reported by Maer et al. [7]. All azeotropes reported by these authors are found n ths study wthout any numercal problem usng SA-Newton method. The CPU tme requred to fnd all k- ary azeotropes usng SA-Newton method s only 1.06 s. It seems that our hybrd strategy s effcent and very relable for the calculaton of homogeneous azeotropes n reactve and non-reactve systems. 5. Conclusons Lst of symbols a actvty of component c number of components F obj objectve functon g Gbbs energy of mxng ĝ dmensonless transformed molar Gbbs energy of mxng for reactng system k assocaton equlbrum constant Keq r reacton equlbrum constant m number of optmzaton varables ˆn transformed mole number N square matrx formed from the stochometrc coeffcents of the reference components n the R reactons NT number of teratons before reducton of annealng temperature NE successve cycles n Smulated Annealng algorthm NFEV mean number of functon evaluatons P pressure P c crtcal pressure P sat saturaton pressure of pure component R g unversal gas constant RT reducton factor for annealng temperature RTPDF reactve tangent plane dstance functon TPDF tangent plane dstance functon T temperature T c crtcal temperature T SA annealng temperature v row vector of stochometrc coeffcents of component for each reacton v TOT row vector where each element corresponds to the sum of the stochometrc coeffcents for all components that partcpate n reacton r v r stochometrc coeffcent of component n reacton r V volume VM ntal step length x k mole fracton of component n the phase k x ref column vector of R reference component mole fractons X k transformed mole fracton of component n the phase k z mole fracton of component n the feed Z transformed overall mole fracton composton of component Greek letters k transformed fracton of moles at phase k, ˇ lqud and vapor phases ˆϕ fugacty coeffcent of component at the mxture actvty coeffcent of component at the mxture ϕ fugacty coeffcent of the pure component chemcal potental of component at the mxture ˆ transformed chemcal potental of component at the mxture k fracton of moles n phase k Acknowledgements The authors acknowledge the fnancal support from CONACYT, Insttuto Tecnológco de Aguascalentes, Insttuto Tecnológco de Celaya and Texas A&M Unversty.

12 A. Bonlla-Petrcolet et al. / Flud Phase Equlbra 281 (2009) References [1] M.J. Okasnsk, M.F. Doherty, Thermodynamc behavor of reactve azeotropes, AIChE Journal 43 (1997) [2] J. Gmehlng, J. Menke, J. Krafczyk, K. Fscher, A data bank for azeotropc data status and applcatons, Flud Phase Equlbra 103 (1995) [3] S.H. Wang, W.B. Whtng, New algorthm for calculaton of azeotropes from equaton of state, Industral & Engneerng Chemstry Process Desgn and Development 25 (1986) [4] R.G. Chapman, S.P. Goodwn, A general algorthm for the calculaton of azeotropes n flud mxtures, Flud Phase Equlbra 85 (1993) [5] Z.T. Fdkowsk, M.F. Malone, M.F. Doherty, Computng azeotropes n multcomponent mxtures, Computers and Chemcal Engneerng 17 (1993) [6] S.T. Hardng, C.D. Maranas, C.M. McDonald, C.A. Floudas, Locatng all homogeneous azeotropes n multcomponent mxtures, Industral and Engneerng Chemstry Research 36 (1997) [7] R.W. Maer, J.F. Brennecke, M.A. Stadtherr, Relable computaton of homogeneous azeotropes, AIChE Journal 44 (1998) [8] E. Salomone, J. Espnosa, Predcton of homogeneous azeotropes wth nterval analyss technques explotng topologcal consderatons, Industral and Engneerng Chemstry Research 40 (2001) [9] N. Aslam, A.K. Sunol, Senstvty of azeotropc states to actvty coeffcent model parameters and system varables, Flud Phase Equlbra 240 (2006) [10] N. Aslam, A.K. Sunol, Relable computaton of bnary homogeneous azeotropes of mult-component mxtures at hgher pressures through equaton of states, Chemcal Engneerng Scence 59 (2004) [11] S.T. Hardng, C.A. Floudas, Locatng all heterogeneous and reactve azeotropes n multcomponent mxtures, Industral and Engneerng Chemstry Research 39 (2000) [12] R.W. Maer, J.F. Brennecke, M.A. Stadtherr, Relable computaton of reactve azeotropes, Computers and Chemcal Engneerng 24 (2000) [13] Z. Q, K. Sundmacher, Geometrcally locatng azeotropes n ternary systems, Industral and Engneerng Chemstry Research 44 (2005) [14] N. Henderson, J.R. de Olvera, H.P.A. Souto, R.P. Marques, Modelng and analyss of the sothermal flash problem and ts calculaton wth the smulated annealng algorthm, Industral and Engneerng Chemstry Research 40 (2001) [15] L. Fretas, G. Platt, N. Henderson, Novel approach for the calculaton of crtcal ponts n bnary mxtures usng global optmzaton, Flud Phase Equlbra 225 (2004) [16] G.P. Rangaah, Evaluaton of genetc algorthms and smulated annealng for phase equlbrum and stablty problems, Flud Phase Equlbra (2001) [17] Y.S. Teh, G.P. Rangaah, Tabu search for global optmzaton of contnuous functons wth applcaton to phase equlbrum calculatons, Computers and Chemcal Engneerng 27 (2003) [18] D.N. Justo-García, F. García-Sánchez, J. Águla-Hernández, R. Eustaquo-Rncón, Applcaton of the smulated annealng technque to the calculaton of crtcal ponts of multcomponent mxtures wth cubc equatons of state, Flud Phase Equlbra 264 (2008) [19] M. Kundu, S.S. Bandyopadhyay, Modellng vapor lqud equlbrum of CO 2 n aqueous N-methyldethanolamne through the smulated annealng algorthm, Canadan Journal of Chemcal Engneerng 83 (2005) [20] S. Ung, M.F. Doherty, Theory of phase equlbra n multreacton systems, Chemcal Engneerng Scence 50 (1995) [21] G.A. Iglesas-Slva, A. Bonlla-Petrcolet, P.T. Eubank, J.C. Holste, K.R. Hall, An algebrac method that ncludes Gbbs mnmzaton for performng phase equlbrum calculatons for any number of components or phases, Flud Phase Equlbra 210 (2003) [22] A. Bonlla-Petrcolet, G.A. Iglesas-Slva, K.R. Hall, An effectve calculaton procedure for two-phase equlbra n multreacton systems, Flud Phase Equlbra 269 (2008) [23] S. Ung, M.F. Doherty, Necessary and suffcent condtons for reactve azeotropes n multreacton mxtures, AIChE Journal 41 (1995) [24] S.K. Wasylkewcz, L.C. Kobylka, F.J.L. Castllo, Pressure senstvty analyss of azeotropes, Industral and Engneerng Chemstry Research 42 (2003) [25] Y. Xang, D.Y. Sun, W. Fan, X.G. Gong, Generalzed smulated annealng algorthm and ts applcaton to the Thomson model, Physcs Letters A 233 (1997) [26] S. Krkpatrck, C.D. Gelatt, M.P. Vecch, Optmzaton by smulated annealng, Scence 220 (1983) [27] W.L. Goffe, G.D. Ferrer, J. Rogers, Global optmzaton of statstcal functons wth smulated annealng, Journal of Econometrcs 60 (1994) [28] A. Corana, M. Marches, C. Martn, S. Rdella, Mnmzng multmodal functons of contnuous varables wth the smulated annealng algorthm, ACM Transactons on Mathematcal Software 13 (1987) [29] N. Metropols, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, Equaton of state calculatons by fast computng machnes, The Journal of Chemcal Physcs 21 (1953) [30] M.L. Mchelsen, The sothermal flas problem: part I. Stablty, Flud Phase Equlbra 9 (1982) [31] S.K. Wasylkewcz, S. Ung, Global phase stablty analyss for heterogeneous reactve mxtures and calculaton of reactve lqud lqud and vapor lqud lqud equlbra, Flud Phase Equlbra 175 (2000) [32] A. Bonlla-Petrcolet, R. Vázquez-Román, G.A. Iglesas-Slva, K.R. Hall, Performance of stochastc global optmzaton methods n the calculaton of phase stablty analyses for nonreactve and reactve mxtures, Industral and Engneerng Chemstry Research 45 (2006) [33] A.S. Gow, X. Guo, D. Lu, A. Luca, Smulaton of refrgerant phase equlbra, Industral and Engneerng Chemstry Research 36 (1997) [34] A.S. Gow, A modfed Clausus equaton of state for calculaton of multcomponent refrgerant vapor lqud equlbra, Flud Phase Equlbra 90 (1993)