Bubble and dew point calculations in multicomponent and multireactive mixtures

Insttuto Tecnologco de Aguascalentes From the SelectedWorks of Adran Bonlla-Petrcolet 2006 Bubble and dew pont calculatons n multcomponent and multreactve mtures Adran Bonlla-Petrcolet Azucena Acosta-Martínez Ulses I. Bravo-Sánchez Juan Gabrel Segova-Hernández, Unversdad de Guanauato Avalable at: http://works.bepress.com/adran_bonlla_petrcolet/29/

A. BONILLA-PETICIOLET et al., Bubble and Dew Pont Calculatons n, Chem. Bochem. Eng. Q. 20 (2) 8 (2006) BubbleandDewPontCalculatonsnMultcomponentandMultreactveMtures A. Bonlla-Petrcolet,* A. Acosta-Martínez, U. I. Bravo-Sánchez, andj. G. Segova-Hernández** Insttuto Tecnológco de Aguascalentes, Departamento de Ingenería Químca, Av. López Mateos 80, CP 20256, Aguascalentes, Ags. Méco Orgnal scentfc paper **Unversdad de Guanauato, Facultad de Cencas Químcas, eceved: August 28, 2005 Nora Alta s/n, C.P. 6050, Guanauato, Gto. Méco Accepted: February, 2006 Bubble and dew pont calculatons are useful n chemcal engneerng and play an mportant role n the study of separaton equpments for non-reactve and reactve mtures. To the best of the authors s knowledge, few methods have been proposed for these calculatons n systems wth several chemcal reactons. The obectve of ths paper s to ntroduce new condtons for performng bubble and dew pont calculatons n reactve mtures. We have developed these condtons based on the applcaton of transformed varables of Ung and Doherty (995). Usng these transformed varables, the soluton space s restrcted to compostons that are already at chemcal equlbrum and by consequence the problem dmenson s also reduced. The relablty and effcency of three equaton-solvng methods are tested and compared usng our equlbrum condtons: a) a smultaneous equaton-solvng approach usng Newton method (SESN), b) an equaton-decouplng approach usng successve substtuton method (EDSS) and c) an optmzaton approach usng the stochastc optmzaton method Smulated Annealng (OSA). Our results ndcated that even for smple reactve systems, bubble and dew pont calculatons are challengng for classcal equaton-solvng methods and requre robust strateges. We conclude that OSA and EDSS methods are relable to locate bubble and dew ponts n reactve systems. EDSS s more effcent than OSA; however, OSA does not need ntal guesses and s more sutable for dffcult problems. Key words: Chemcal equlbrum, phase equlbrum, bubble pont, dew pont, global optmzaton, equaton-solvng method comparson Introducton The descrpton of phase equlbrum n multreacton mtures s an mportant topc nvolved n several ndustral applcatons and t s the bass for desgn, analyss, and synthess of separaton processes. The phase equlbrum problem wth several chemcal reactons s hghly non lnear and multvarable. 2 Durng the last years, there has been a growng nterest for developng new methods to treatment the thermodynamc behavor of mtures under physcal and chemcal equlbrum. 2 Almost all developed methods use the mole fractons and reacton etents as ndependent varables and unknowns of the reactve phase equlbrum problem. Untl now, only a few methods have used theores of varable transformaton wth the am of reducng problem dmensonalty and mprovng the numercal behavor (effcency and relablty) of soluton methods.,9 These knds of algorthms are very attractve for the smulaton of separaton process and favor the study of comple multreactve multcomponent systems. *Correspondng author: (52) 449905002 et. 27, petrcolet@hotmal.com A specal case of phase equlbrum problems are the bubble and dew pont calculatons. These calculatons are useful n chemcal engneerng and play an mportant role n the study of separaton equpments. 4 Under ths contet, robust, and effcent methods for these calculatons are desrable. To the best of the authors s knowledge, few methods have been developed to calculate bubble and dew ponts n multreactve mtures. 9,2 Ths paper ntroduces a new method for performng dew and bubble pont calculatons n multreactve and multcomponent systems. We have proposed new condtons for these equlbrum problems based on the theory of reactve varables of Ung and Doherty. Also, we have tested and compared three equaton-solvng methods for performng those calculatons, usng our equlbrum condtons. Formulaton of new condtons for bubble and dew pont calculatons n multreactve mtures Ung and Doherty showed that the Gbbs energy n a reactve system behaves as n a non-reac-

2 A. BONILLA-PETICIOLET et al., Bubble and Dew Pont Calculatons n, Chem. Bochem. Eng. Q. 20 (2) 8 (2006) tve system f transformed composton varables are used nstead of the conventonal composton varables. Usng transformed compostons, we restrct the soluton space to the compostons that are already n chemcal equlbrum. Ths reduces the problem dmenson by the number of ndependent reactons and makes t possble to represent phase equlbrum n reactve systems n a smlar way as n non-reactve mtures. So, the reactve phase dagrams look smlar to the non-reactve ones and the non-reactve flash algorthms can be easly modfed to account for the equlbrum reactons. 2 Also, Ung and Doherty showed that the chemcal potental follows all the thermodynamc relatonshps of a non-reactve system as long as all the thermodynamc propertes are functons of the transformed composton varables. Consderng ths, equlbrum condtons among two phases of a reactve mture, wth n components and n ndependent chemcal reactons, are met when where T,, nn () T {} 0 (2) T T wth the unversal gas constant, 0 the chemcal potental of the pure component and the transformed chemcal potental of component at phase whch s a functon of the transformed mole fracton. Eq (2) can be epressed n terms of fugacty or actvty coeffcents as follows {} ln ln ( y{}),, nn () T where s the mole fracton of component, s the fugacty coeffcent of component n the mture, s the fugacty coeffcent of the pure component, and s the actvty coeffcent of component, respectvely. Usng Eq () and (), we can deduce the followng relaton for mole fractons at phase equlbrum n n,, (4) Then, we defne the transformed phase equlbrum constant K as {, K },, nn (5) and the transformed mole fractons by 8,9, are gven N v 0,, nn (6) v N where s the column vector of n erence component mole fractons, v s the row vector of stochometrc number of component for each reacton v, s a row vector where each element corresponds to reacton r and t s the sum of the stochometrc number for all components that partcpate n reacton r, and N s a square matr formed from the stochometrc number of the erence components n the n reactons. 8,9, The erence mole fractons are calculated usng Eq (6) and from the equlbrum constants for each reacton K eq by solvng a system of n nonlnear equatons gven by n K a r,, n (7) eq v r where a s the actvty of component and v r s the stochometrc number of component n reacton r, respectvely. When we know the erence mole fractons, the remanng mole fractons are calculated usng Eq (6). We wrte the phase equlbrum condton n terms of transformed varables, usng Eq (4) (6), as follows and K,, nn (8) v v vn ( K ) v N N N (9),, nn (0) The materal balance on component s F. ( ) 0,, nn () and the transformed mole phase fracton s defned as ( v N ) ( v N ) F, (2) where s the conventonal mole fracton of phase whose feasble doman s (0, ) and F s the column vector of n erence component mole

A. BONILLA-PETICIOLET et al., Bubble and Dew Pont Calculatons n, Chem. Bochem. Eng. Q. 20 (2) 8 (2006) fractons n the ntal overall composton. Usng Eq (8) (), s gven by nn, F, ( K ),, nn () From the transformed mole fracton restrcton, we have nn K( F, ) 0 (4) K nn F, ( K 0 (5) By applyng the condtons 0 and, we obtan the followng new functons for bubble and dew pont calculatons n multreactve mtures nn f ( K ) 0 (6) bubble f dew F, nn F 0 K (7) In contrast wth other formulatons,,5 our functons are ndependent of reacton etents and have fewer unknowns. Eq (6) and (7) are a functon of temperature or pressure and n n transformed mole fracton of the lqud or vapor phase, respectvely. On the other hand, these equlbrum problems are nonlnear and multvarable so that conventonal numercal methods may present dffcultes, such as poor ntalzaton or dvergence behavor. We have performed a comparson of three approaches to solve our bubble and dew pont functons. In the net secton we descrbe the strateges appled n ths work. Soluton approachs Smultaneous equaton-solvng wth Newton method (SESN) Ths s the classcal approach used for performng flash calculatons n non-reactve and reactve systems snce t s conceptually smple and straghtforward. The popular Newton method s used because t provdes quadratc convergence when the ntal estmates are close to the soluton and t s readly avalable n computer programs. 6 Bubble and dew ponts are calculated by solvng smultaneously the followng n n equatons F K,, nn (8) Fnn f (9) where f s the bubble or dew pont functon, Eq (6) or (7), respectvely. The unknowns are n n transformed mole fractons of vapor or lqud phase and the temperature or pressure. In ths work, the above equatons are solved usng the Numercal ecpes Fortran subroutne NEWT. Equaton-decouplng wth Successve Substtuton method (EDSS) Ths approach s often descrbed n tetbooks on thermodynamcs for flash calculatons n non-reactve mtures and s based on the achford-ce formulaton. Ths method s relable for flash calculatons n non-reactve systems and has a lnear rate of convergence. 6 The procedure used for bubble and dew pont calculatons n reactve systems s as follows: n the nner loop, we have appled a Newton method, usng the Numercal ecpes Fortran subroutne NEWT, to update the temperature or pressure by solvng Eq (6) or (7), whle n the outer loop the transformed mole fractons are updated wth a successve substtuton method usng Eq (8) or (). In ths work, we have not appled an acceleraton technque for ths approach. Optmzaton approach usng Smulated Annealng method (OSA) We have tested an optmzaton technque for solvng our equlbrum condtons. As ndcated by Henderson et al., 7 the formulaton of thermodynamc calculatons for optmzaton problems offers some advantages: a) the use of a robust optmzaton method, b) the possblty of usng a drect optmzaton method whch requres only calculatons of the obectve functon and c) the use of an teratve procedure whose convergence s ndependent on the ntal guess. Thus, the calculaton of bubble and dew pont condtons can be performed by mnmzng the net obectve functon nn 2 2 f ob f F (20) where F s descrbed by Eq (8) for =,, n n and f s the bubble or dew pont functon, respectvely. At the bubble and dew pont condtons, the global mnmum of the obectve functon must be zero, but we assume that a soluton s found when we fnd the transformed mole fractons and temperature or pressure that make the value of the

4 A. BONILLA-PETICIOLET et al., Bubble and Dew Pont Calculatons n, Chem. Bochem. Eng. Q. 20 (2) 8 (2006) obectve functon less than 0 6. The stochastc global optmzaton method Smulated Annealng (SA) s used to mnmze our obectve functon. SA s nspred n the thermodynamc process of coolng of molten metals to acheve the lowest free energy state. 8 The SA algorthm presents ease of computatonal mplementaton and, f the values for ts algorthm parameters are properly selected, t can converge to the global optmum ndependently of ntal guesses. Ths optmzaton method has been successfully used n the resoluton of several thermodynamc problems. 7,9 24 It s mportant to note that other phase equlbrum problems (e.g. calculatons of crtcal ponts and homogeneous azeotropes) have been solved usng an optmzaton approach coupled wth the SA method. 7,24 In our calculatons, we have used the algorthm proposed by Corana et al. 25 because of ts hgh relablty n thermodynamc calculatons. In ths algorthm, a tral pont s randomly chosen wthn the step length VM (a vector of length n varables) of a startng pont defned n the feasble doman of optmzaton varables. The functon s evaluated at ths tral pont and ts value s compared to ts value at the ntal pont. The Metropols crteron, 26 wth a parameter called annealng temperature T, s used to accept or reect the tral pont. If the tral pont s accepted, the algorthm moves on from that pont. If t s reected, another pont s chosen nstead for a tral evaluaton. Each element of VM s perodcally adusted so that half of all functon evaluatons n that drecton are accepted. A fall n T s mposed upon the system wth the Tvarable by T + = T T, where s the teraton counter and Ts the temperature reducton factor. Thus, as T declnes, downhll moves are less lkely to be accepted and the percentage of reectons rses. Gven the scheme for the selecton for VM, VM falls. Thus, as T declnes, VM falls and SA focuses upon the most promsng area for optmzaton. A full descrpton of ths algorthm s found n Corana et al., 25 and the FOTAN subroutne mplemented by Goffe et al. 27 s used n ths work. The choce of the coolng schedule s a crucal aspect n the mplementaton of SA because t affects the numercal performance of the optmzaton procedure. Based on prelmnary calculatons, we propose the followng values for the quanttes of SA: T 0 = 0.0, T=0.85andN T =5,whereN T s the teraton number before temperature reducton. esults and dscusson For the present study, four eamples are used to compare the numercal performance of the three equaton-solvng methods. In all eamples, we have consdered a tolerance of 0 6 for the convergence of all methods. All eamples are solved 25 tmes to evaluate the relablty and effcency of solvng methods, usng n each tme dfferent random ntal values for unknowns, T or p. All calculatons were performed on a Processor Intel Centrno 600 MHz wth.00 GB of AM. The performance of the three equaton-solvng methods s tested usng the followng crterons: a) success rate of fndng the bubble or dew pont n the performed calculatons, b) average number of K-value evaluatons and c) computaton tme. Phase stablty of all calculated bubble and dew ponts s evaluated by mnmzng the tangent plane dstance functon for reactve mtures 2 nn TPDF ( {} { F }) (2) The TPDF functon s globally mnmzed usng the Smulated Annealng method. Bonlla-Petrcolet 24 has tested the SA method wth several reactve systems and found that ths method s robust for reactve phase stablty analyss. All calculated bubble and dew ponts are stable. Eample. sobutene + methanol methyl ter-butyl ether Our frst eample s the equlbrum reacton of sobutene () and methanol (2) to produce methyl ter-butyl ether (), A + A 2 A. Ths system has been analyzed by Okasnsk and Doherty, 28 Maer et al. 29 and Hardng and Floudas 0 n the contet of calculaton of reactve homogeneous azeotropes. We have consdered non-dealty n the lqud phase and deal gas n the vapor phase. We use the Wlson equaton for the actvty coeffcent n the lqud phase and the Antone equaton to calculate the propertes of vapor phase. Maer et al. 29 report the parameters for both equatons. The reacton equlbrum constant s assumed to be ndependent of temperature K eq = 0.04, and methyl ter-butyl ether (MTBE) s the erence component so that the transformed mole fractons are and 2 2. We use and temperature as unknowns for all equaton-solvng methods. The ntal values are randomly generated n the feasble domans of (0, ) for and (0.0, 00) C for temperature. These domans are also used n the mplementaton of the OSA method. Bubble and dew pont calculatons are performed at dfferent pressures for a feed of F (0.5, 0.5) and the results are reported n Tab. The numercal performance of tested methods appears n Tab 2. In all calculatons,

A. BONILLA-PETICIOLET et al., Bubble and Dew Pont Calculatons n, Chem. Bochem. Eng. Q. 20 (2) 8 (2006) 5 Table Bubble and dew pont calculatons for sobutene + methanol methyl ter-butyl ether. Wlson equaton and deal gas, K eq = 0.04 and F (0.5,0.5). p /atm Dew pont Bubble pont T / C T / C 2 64.94 (0.096, 0.9804).944 (0.96245, 0.0755) 4 8.894 (0.0274, 0.97259) 6.425 (0.94082, 0.0598) 6 96.56 (0.05, 0.96647) 5.56 (0.9224, 0.07676) 8 05.462 (0.0882, 0.968) 62.859 (0.90804, 0.0996) Table 2 Performance of equaton-solvng methods tested n bubble and dew pont calculatons n reactve systems Success rate n percent (No. of K-values evaluatons) Eample No. T or p Bubble pont Dew pont SESN DESS OSA SESN DESS OSA 2atm 6 (7) 00 (56) 00 (2089) 6 (0) 00 (42) 00 (245) 4atm 44 (72) 00 (52) 00 (2409) 6 (54) 00 (59) 00 (229) 6atm 44 (84) 00 (48) 00 (2265) 20 () 00 (02) 00 (245) 8atm 60 (84) 00 (4) 00 (2289) 24 (44) 00 (06) 00 (227) 50 C 68 (55) 00 (7) 96 (42977) 4 (22) 00 (79) 00 (4297) 2 70 C 76 (5) 00 (247) 00 (445) 20 (2) 00 (275) 00 (477) 90 C 68 (59) 00 (208) 00 (4297) 40 (56) 00 (220) 00 (4505) atm 76 (86) 00 (270) 00 (24) 4 (22) 00 (59) 00 (2677) 5atm 28 (50) 00 (29) 00 (86) 2 (29) 00 (420) 00 (22) 0atm 64 (64) 00 (70) 00 (2005) 8 (22) 00 (00) 00 (2449) atm 28 (20) 88 (92) 00 (27) 28 (2) 00 (49) 00 (227) 4 4atm 6 (6) 80 (46) 00 (2257) 28 (24) 00 () 00 (224) 6atm 28 (5) 56 (40) 00 (205) 56 (2) 00 (27) 00 (247) 8atm 28 () 48 (5) 00 (269) 72 (24) 00 (5) 00 (24) Both parameters are calculated based on 25 calculatons wth random ntal values. DESS and OSA methods fnd the bubble and dew ponts wthout problems whle SESN dverged several tmes. We observe that SESN shows more falures n dew pont calculatons. These results ndcated, that even for smple reactve systems, bubble and dew pont calculatons are challengng for classcal equaton-solvng methods and requre robust strateges. The number of K-value evaluatons of solvng-methods s gven by: OSA >>> DESS > SESN. Computaton tme mantans around of 6 s for OSA, 0.0 s for DESS and 0.02 s for SESN, respectvely. Eample 2. sobutene + methanol methyl ter-butyl ether wth n-butane as nert In the second eample, we use the same reacton as before but nclude n-butane (4) as an nert. Ung and Doherty, 9 Hardng and Floudas 0 and Maer et al. 29 have also studed ths reactve equlbrum problem. The transformed mole fractons for ths mture are 2. 2 and

6 A. BONILLA-PETICIOLET et al., Bubble and Dew Pont Calculatons n, Chem. Bochem. Eng. Q. 20 (2) 8 (2006) Table Bubble and dew pont calculatons for sobutene + methanol methyl ter-butyl ether wth n-butane as nert. Wlson equaton and deal gas. z (0.,0.,0.4) T / C Dew pont Bubble pont p /atm p / atm 50.950 (0.400, 0.45659, 0.4).9 (0.2692, 0.2009, 0.75299) 70.84 (0.4602, 0.44792, 0.606) 5.7 (0.4796, 0.4425, 0.70779) 90 7.006 (0.4077, 0.4648, 0.676) 9.02 (0.6826, 0.7076, 0.66099) 4 4 2, where the MTBE s the erence component. Agan, the Wlson and Antone equatons are used for the calculaton of thermodynamc propertes. Maer et al. 29 report the parameters for both equatons. The reacton equlbrum constant s determned from G rs / = 0 4205.05 + 0.0982 T 0.2667 T ln T, where T s n K. For SESN and DESS methods, we use, 2 and T as unknowns whle n the optmzaton approach, the obectve functon s mnmzed wth respect to, 2, 4 and T. We set a feasble doman of (0, ) for all the transformed mole fractons and (0.000, 20) bar for pressure. The ntals values are randomly generated n these domans. For OSA method, at each evaluaton of the obectve functon, the transformed mole fractons are normalzed to unty. We have analyzed an ntal global composton of F (0., 0., 0.4). Tab shows the results of bubble and dew pont calculatons at dfferent temperatures and Tab 2 shows the performance of the three equaton-solvng methods. For ths case, DESS shows 00 % relablty for the locaton of bubble and dew ponts at all tested temperatures, whle OSA method fals only one tme n all calculatons performed. Agan, SESN shows several falures n the locaton of reactve saturaton condtons. For dew pont calculatons, the performance of SESN method s very poor. OSA method shows the hghest computaton tme (22 s) whle SESN and DESS showed a computaton tme of 0.08 and 0. s, respectvely. Eample. acetc acd + sopropanol sopropyl acetate + water Thrd eample s the formaton of sopropyl acetate through esterfcaton reacton of sopropanol and acetc acd: acetc acd () + sopropanol (2) sopropyl acetate () + water (4). Hardng and Floudas 0 and Maer et al. 29 have studed ths system. For the lqud phase, we use actvty coeffcents from the NTL equaton and consder the assocaton of acetc acd n the vapor phase, usng a correcton factor as suggested by Maer et al. 29 The correcton factor for the assocaton s gven by and sat 2 / ( 4kp ) 2 / [ 4kp( 2)] 2 / 2 { [ 42 kp( )] } 2 / ( 2 ){ [ 4kP ( 2 )] } (22) for = 2,, 4(2) sat where p s the saturaton pressure of pure component and the dmerzaton constant s calculated from log 0 k = log 0 k = 2.5454 + 66/T wth T n Kelvn and k and k n Pa. Maer et al. 29 gves the Antone and NTL equaton parameters. The reacton equlbrum constant s ndependent of temperature, K eq = 8.7, and water s the erence component. The transformed mole fractons are gven by 2 4, 2 and 4 4 4 4 2. We use, 2 and T as 4 unknowns for all methods. The ntal values are randomly generated usng the feasble doman (0, ) for transformed mole fractons and (50, 200) C for temperature. Bubble and dew pont calculatons are performed for a feed of F (0.5, 0., 0.2) at dfferent pressures (see Tab 4and Tab 2) shows the numercal performance of all tested methods. DESS and OSA methods are very relable to fnd the bubble and dew pont condtons and they dd not show any falures n all the calculatons performed. Agan, SESN showed the worst performance and ts numercal behavor s hghly dependent on ntal guesses. The mean number of K-value evaluatons s gven by: SESN < DESS << OSA, whle the computaton tme mantans around of 7 s for OSA, 0.2 s for DESS and 0.0 s for SESN.

A. BONILLA-PETICIOLET et al., Bubble and Dew Pont Calculatons n, Chem. Bochem. Eng. Q. 20 (2) 8 (2006) 7 Table 4 Bubble and dew pont calculatons for acetc acd + sopropanol sopropyl acetate + water. NTL equaton and deal gas,k eq = 8.7 and F (0.5,0.,0.2). p / atm Dew pont Bubble pont T / C T / C 94.5 (0.70299, 0.786, 0.842) 85.55 (0.79, 0.4569, 0.20522) 5 49.68 (0.6046, 0.496, 0.22558) 8.272 (0.42525, 0.4785, 0.09625) 0 79.78 (0.59782, 0.8, 0.2708) 66.45 (0.442, 0.5047, 0.05260) Eample 4. A +A 2 A and 2A A 4 +A 2 Our last eample s a hypothetcal quaternary system that follows the reactons: A + A 2 A and 2A A 4 + A 2. Ung and Doherty 9 have studed the phase equlbrum behavor of ths mture. We have consdered deal behavor for both lqud and vapor phases. The Antone equaton s used to calculate the vapor pressures of pure components wth the parameters reported by Ung and Doherty. 9 The transformed mole fractons are defned usng A and A 4 as erence components, so they are gven by 24 24 and 2. 24 24 The chemcal equlbrum constants are calculated usng K eq ep 2257. and 768 T 2 6844. 9790 K eq ep 7. 0265, where T T 2 T s gven n Kelvn. We use and temperature as unknowns and ther feasble domans are (0, ) and (27.5, 400) K, respectvely. andom ntal values are generated nsde these domans. Bubble and dew pont calculatons are performed for a feed F (0.5, 0.5) at several pressures. The results of phase equlbrum calculatons appear n Tab 5 and performance of all tested methods are reported n Tab 2. Only, the OSA method showed 00 % relablty to locate the bubble and dew ponts n ths reactve system, however, t showed the hghest computaton tme (. seconds) and the hghest mean number of K-values evaluatons. On the other hand, several falures are reported for the DESS and SESN methods. For non-reactve systems, Mchelsen has ndcated that bubble and dew pont calculatons usng hybrd models rarely present convergence problems. However, our results show that ths doesn t apply for phase equlbrum calculatons n reactve systems. Even for smple reactng systems, SESN and DESS methods present convergence problems and depend on the ntal guesses. The OSA method s very relable to fnd the bubble and dew ponts and ts performance s almost ndependent of ntal values. However, we can not offer a theoretcal guaranty for the locaton of bubble and dew ponts usng ths optmzaton approach. Fnally, we have tested the performance of these equaton-solvng methods wth other reactve systems reported by Okasnsk and Doherty, 28 Hardng and Floudas 0 and Maer et al.; 29 and the results, not reported n ths paper, ndcated that OSA and DESS methods are relable strateges for the calculaton of dew and bubble ponts n multreactve mtures. Conclusons New condtons for bubble and dew pont calculatons n multreactve mtures have been derved n terms of the varable transformaton theory of Ung and Doherty. We have tested and compared the numercal performance of three equaton-solvng meth- Table 5 Bubble and dew pont calculatons for a hypothetcal quaternary mture A +A 2 A and 2A A 4 +A 2. Ideal soluton and deal gas. F (0.5,0.5) p /atm Dew pont Bubble pont T /K T /K 55.488 (0.02678, 0.9722) 06.95 (0.96045, 0.0955) 4 95.265 (0.0452, 0.9548) 28.7 (0.97759, 0.0224) 6 409.04 (0.0599, 0.9460) 7.9 (0.979, 0.02089) 8 49.525 (0.06, 0.9867) 45.829 (0.97829, 0.027)

8 A. BONILLA-PETICIOLET et al., Bubble and Dew Pont Calculatons n, Chem. Bochem. Eng. Q. 20 (2) 8 (2006) ods n the calculaton of bubble and dew ponts usng the new equlbrum condtons. Our results ndcate that an optmzaton approach usng Smulated Annealng s very relable for ths knd of phase equlbrum calculatons n multreactve systems. However, t shows the hghest computaton tme. In the other hand, the equaton-decouplng method wth successve substtuton also presents a good numercal performance and s effcent. It s clear that the smultaneous equaton solvng approach, usng Newton method, s not sutable for bubble and dew pont calculatons n reactve systems because t can dverge very frequently and depends sgnfcantly on ntal values. Based on relablty, we consder that the optmzaton approach usng Smulated Annealng s the most sutable method and perable choce for these equlbrum calculatons n reactve systems, due to ts acceptable convergence speed, and that ntal guesses are not requred. We wll etend ths procedure for performng these equlbrum calculatons usng equatons of state. ACKNOWLEDGEMENTS Authors acknowledge the fnancal support from CONACYT,Insttuto Tecnológco de Aguascalentes and POMEP. Notaton a actvty, mol dm n number of components K eq chemcal equlbrum constant K transformed phase equlbrum constant N teraton number p pressure gven n atm, atm = 0 25 Pa n reacton number unversal gas constant, J mol K T temperature gven n C or K r v stochometrc number of component transformed mole fracton, mole fracton, F transformed feed composton, F feed composton, transformed phase mole fracton, VM vector length Greek letters, phase at equlbrum chemcal potental, J mol actvty coeffcent, fugacty coeffcent, Inde F feed composton erence component phase eferences. Pérez-Csneros,E. S.,Gan,.,Mchelsen,M. L.,Chem. Eng. Sc. 52 (997) 527. 2. Seder,W. D.,Wdagdo,S.,Flud Phase Equlb 2 (996) 28.. Castllo,J.,Grossmann,I. E.,Comp. Chem. Eng. 5 (98) 99. 4. Caster,M.,asmussen,P.,Fredenslund,A.,Chem. Eng. Sc. 44 (989) 27. 5. Xao,W.,Zhu,K.,Yuan,W.,Chen,H. H.,An algorthm for smultaneous chemcal and phase equlbrum calculaton. AIChE J. 5 (989) 8. 6. Grener,H.,An effcent mplementaton of newton s method for comple nondeal chemcal equlbra. Comp. Chem. Eng. 5 (99) 5. 7. Smth,J. V.,Mssen,. W.,Smth,W..,General optmalty crtera for multphase multreacton chemcal equlbrum. AIChE J. 9 (99) 707. 8. Ung,S.,Doherty,M. F.,AIChE J. 4 (995) 28. 9. Ung,S.,Doherty,M. F.,Chem. Eng. Sc. 50 (995) 2. 0. McDonald,C. M.,Floudas,C. A.,Glopeq: a new computatonal tool for the phase and chemcal equlbrum problem. Comp. Chem. Eng. 2 (997).. Stateva,. P.,Wakeham,W. A.,Ind. Eng. Chem. es. 6 (997) 5474. 2. Wasylkewcz,S. K.,Ung,S.,Flud Phase Equlb 75 (2000) 25.. Ung,S.,Doherty M. F.,Chem. Eng. Sc. 50 (995) 20. 4. Eckert,E.,Kubcek,M.,Comp. Chem. Eng. 9 (995) 49. 5. Walas,S. M.,Phase equlbra n chemcal engneerng. Butterworth Publshers, USA, 985. 6. Teh,Y. S.,angaah,G. P.,Trans IChemE 80 (2002) 745. 7. Henderson,N.,Fretas,L.,Platt,G. M.,AIChE J. 50 (2004) 00. 8. Krkpatrck,S.,Gelatt,C.,Vecc,M.,Scence 220 (98) 67. 9. Pan,H.,Froozabad,A.,Comple multphase equlbra calculatons by drect mnmzaton of Gbbs free energy by use of smulated annealng. SPE eservor Eva. Eng. Feb (998) 6. 20. Zhu,Y.,Xu,Z.,Flud Phase Equlb 54 (999) 55. 2. Henderson,N.,Olvera,J..,Ind. Eng. Chem. es. 40 (200) 6028. 22. Gomes,J.,Henderson,N.,ocha,M.,Macromol. Theory Smul. 0 (200) 86. 2. angaah,g. P.,Flud Phase Equlb 87-88 (200) 8. 24. Bonlla-Petrcolet,A.,Calculaton of phase equlbrum n reactve and nonreactve systems. Ph. D. Dssertaton, Insttuto Tecnológco de Celaya, Guanauato, Meco, 2005. 25. Corana,A.,Marches,M.,Martn,C.,della,S.,ACM T. Math. Software (987) 262. 26. Metropols,N.,osenbluth,A.,osenbluth,M.,Teller,A., Teller E., J. Chem. Phys. 2 (95) 087. 27. Goffe,B.,Ferrer,G.,ogers,J.,Global optmzaton of statstcal functons wth smulated annealng. J. Econometrcs 60 (994) 65. 28. Okasnsk,M. J.,Doherty,M. F.,AIChE J. 4 (997) 2227. 29. Maer. W.,Brennecke J. F.,Stadtherr M. A.,Comp. Chem. Eng. 24 (2000) 85. 0. Hardng,S. T.,Floudas,C. A.,Ind. Eng. Chem. es. 9 (2000) 576.. Mchelsen,M. L.,Comp. Chem. Eng. 7 (99) 4.