Analysis and Prediction of Input Multiplicity for the Reactive Flash Separation using Reaction- Invariant Composition Variables

Transcription

1 Insttuto Tecnologco de Aguascalentes From the SelectedWors of Adran Bonlla-Petrcolet 2012 Analyss and Predcton of Input Multplcty for the Reactve Flash Separaton usng Reacton- Invarant Composton Varables Jose E. Jame-Leal, Unversdad de Guanajuato Adran Bonlla-Petrcolet Juan Gabrel Segova-Hernandez, Unversdad de Guanajuato Salvador Hernandez Hector Hernandez-Escoto, Unversdad de Guanajuato Avalable at:

2 chemcal engneerng research and desgn 9 0 ( ) Contents lsts avalable at ScVerse ScenceDrect Chemcal Engneerng Research and Desgn journal homepage: Analyss and predcton of nput multplcty for the reactve flash separaton usng reacton-nvarant composton varables José Enrque Jame-Leal a, Adrán Bonlla-Petrcolet b,, Juan Gabrel Segova-Hernández a, Salvador Hernández a, Héctor Hernández-Escoto a a Unversdad de Guanajuato, Guanajuato, Gto., Mexco b Insttuto Tecnológco de Aguascalentes, Aguascalentes, Aguascalentes, Mexco abstract In ths study, we ntroduce a new approach for predctng and analyzng the nput multplcty n reactve flash separaton processes. Specfcally, we have dentfed necessary condtons to detect these multple states n reactve flash separatons usng reacton-nvarant composton varables. The presence of the nput multplcty s studed for the reactve systems of MTBE and TAME producton to llustrate the capabltes of our methodology. For these reactve systems, we report the exstence of multple states for dfferent operatng condtons. In summary, our strategy can be appled wth any reactve system and thermodynamc model, assumng that all reactons are reversble and n thermodynamc equlbrum and the operatng condtons are away from the retrograde regon. In general, our method s a robust procedure for the multplcty analyss n flash separaton of mult-reactve and mult-component systems The Insttuton of Chemcal Engneers. Publshed by Elsever B.V. All rghts reserved. Keywords: Reactve flash separaton; Input multplcty; Reacton-nvarant composton varables 1. Introducton Reactve separaton schemes (e.g., reactve dstllaton, extracton and crystallzaton) are ntegrated unt operatons wdely used n the current chemcal ndustry due to ther well-nown economcal and operatonal advantages. Specfcally, these separaton systems may mprove the process performance va the reducton of captal cost, the ncrement of selectvty and converson, the decrement of heat demand, the suppresson of sde reactons and the avodance of undesrable phase equlbrum condtons such as homogeneous azeotropy (Taylor and Krshna, 2000). However, the relable modelng of reactve separaton process s dffcult due to the multcomponent nature of the reactve systems, the nonlnearty of the thermodynamc models caused by the nteracton of smultaneous chemcal and physcal equlbrum, and also by the type of varables nvolved n defnng the mathematcal model, whch are generally composton varables n molar unts and extents of reacton. In partcular, reactve separaton processes exhbt a hgh non-lnear behavor and, as a consequence, the multplcty of solutons s often possble durng the desgn and modelng of these separaton schemes (Taylor and Krshna, 2000; Chen et al., 2002). Multplcty of solutons s an mportant feature of ndustral processes and plays an mportant role n desgn, smulaton and control of separaton unts (Monroy-Loperena, 2001). In process system engneerng, t s mportant to predct all multple states wthn the practcal doman of operatng varables, to now whether they are desrable, and to understand how the separaton scheme responses to changes n the operatng condtons (Tscareño et al., 1998; Yang et al., 2006). Accordng to the lterature, reactve separaton systems can exhbt two types of multplcty: nput and output multplcty (Sngh et al., 2005a; Malnen and Tansanen, 2010). Input multplcty occurs when two or more sets of nput varables produce the same output condtons, whle output multplcty occurs when one set of nput varables results n two or more ndependent sets of output varables (Sngh et al., 2005a,b; Correspondng author. Tel.: x127. E-mal address: petrcolet@hotmal.com (A. Bonlla-Petrcolet). Receved 27 August 2011; Receved n revsed form 13 December 2011; Accepted 29 February /$ see front matter 2012 The Insttuton of Chemcal Engneers. Publshed by Elsever B.V. All rghts reserved. do: /j.cherd

3 chemcal engneerng research and desgn 9 0 ( ) Nomenclature F V L c r a K eq n N P T v x X z Z K ˆ V V T bub T dew feed vapor phase lqud phase number of components number of ndependent chemcal reactons actvty of component reacton equlbrum constant mole number of component nvertble matrx of stochometrc coeffcents of reference components pressure temperature stochometrc coeffcent of component mole fracton transformed mole fracton feed mole fracton transformed feed mole fracton actvty coeffcent of component n the mxture phase equlbrum constant of component transformed amount fracton for vapor phase mole fracton of vapor phase bubble-pont temperature dew-pont temperature Kumar and Kastha, 2008; Malnen and Tansanen, 2010); Fg. 1 llustrates these types of multplcty. It s convenent to remar that the nput varables of separaton processes are those that can be manpulated by controllers. These varables nclude the reflux rato, reboler duty, and feed flow, among others. On the other hand, the output varables can be also controlled or used to descrbe the process condtons, e.g., the stage temperatures and compostons (Sngh et al., 2005b). In partcular, the nput multplcty (IM) s relevant and mportant because of t mposes sgnfcant control and operaton problems than those obtaned for output multplcty (Vaca et al., 2006; Kumar and Kastha, 2008). Theoretcal and expermental studes have shown the exstence of multple states n reactve separaton processes especally for reactve dstllaton columns (Hauan et al., 1997; Eldars and Douglas, 1998; Güttnger and Morar, 1999a,b; Taylor and Krshna, 2000; Chen et al., 2002; Baur et al., 2003; Sngh et al., 2005a,b; Yang et al., 2006; Kumar and Kastha, 2008; Svandová et al., 2009; Ramzan et al., 2010). Several authors have studed and analyzed the presence of multplcty n reactve dstllaton columns usng several operatve varables and desgn parameters such as the system confguraton, the selecton of both thermodynamc and netc models, the reflux rato, and the locaton of feed nlets, among others (Taylor and Krshna, 2000). Lterature ndcates that the exstence and possble explanaton of multplcty n reactve dstllaton columns have been nvestgated by several authors for well-nown reactve systems nvolved n the producton of fuel ethers le MTBE and TAME (Mohl et al., 1999). Dfferent numercal methods have been used to detect and predct the presence of multple states n these separaton systems where homotopy contnuaton approaches have been wdely appled for predctng and analyzng the multplcty va bfurcaton dagrams (Güttnger and Morar, 1999a,b; Taylor and Krshna, 2000; Rodríguez et al., 2001, 2004; Chen et al., 2002; Malnen Fg. 1 Types of multplcty n reactve separaton systems: (a) nput multplcty and (b) output multplcty. and Tansanen, 2010). However, to the best of our nowledge, only few studes have been conducted on multplcty n reactve flash separatons (Rodríguez et al., 2001, 2004; Laerveld et al., 2005; Ruz et al., 2006). Note that the flash separaton problem s one of the most mportant chemcal engneerng problems and s recurrent n the modelng and desgn of separaton systems based on vapor lqud phase equlbrum. Despte sgnfcant progress on desgn and modelng of reactve dstllaton columns have been acheved, more general results are needed concernng the presence of multplcty n reactve flash operaton. It s convenent to remar that the relable determnaton and analyss of the exstence of multple states n reactve separaton schemes s a challengng tas and t s more complex than those nvolved for conventonal non-reactve separaton schemes because of the presence of chemcal reactons ncreases the complexty and dmensonalty of flash problems. Therefore, proper and relable numercal strateges are requred for modelng reactve systems and predctng the presence of multple solutons n reactve separaton processes. Therefore, n ths study we ntroduce a new approach to predct the exstence of nput multplcty n reactve flash separatons. Ths approach s based on the applcaton of reacton-nvarant composton varables of Ung and Doherty (1995a,b). Note that these varables allow that the classcal procedures for analyzng and modelng non-reactve mxtures can be extended to systems wth chemcal reactons. The presence of the nput multplcty s studed for the

4 1858 chemcal engneerng research and desgn 9 0 ( ) Fg. 2 Schematc dagram of a reactve flash separaton. reactve systems of MTBE and TAME producton to llustrate the capabltes of our methodology. For these reactve systems, we report the exstence of multple states for dfferent operatng condtons usng our approach. Fnally, we show that necessary K-based condtons for the exstence of multple solutons n non-reactve flash operaton, prevously reported by Tscareño et al. (1998) and Monroy-Loperena (2001), can be useful for multplcty analyss of reactve flash separaton n tested reactve systems. 2. Problem formulaton of nput multplcty n reactve flash separaton 2.1. Model descrpton In general, a flash problem conssts of fndng the correct number and types of phases and ther correspondng equlbrum compostons such that the Gbbs free energy of the system s at the global mnmum. The chemcal equlbrum constrants must be consdered to determne the phase dstrbuton and compostons of reactve systems. In ths study, we wll analyze the case of two-phase flash separaton wth a reactve vapor lqud equlbrum. Gven a temperature T and pressure P, consder a reactve flash separaton process for a system of c components wth an ntal composton n 0 =(n 1,0,...,n c,0 ) that undergoes r ndependent chemcal reactons (see Fg. 2). Usng ths formulaton for the PT flash problem, we can dscard the energy balance from the analyss and ths thermodynamc problem s posed as a Gbbs free energy mnmzaton problem. Under these condtons, the problem for modelng the reactve vapor lqud equlbrum s to solve the system of non-lnear equatons obtaned from the statonary condtons of the Gbbs free energy mnmzaton problem, whch nvolves the materal balance, phase and chemcal equlbrum, and consstency equatons. Usually, ths problem s formulated usng conventonal composton varables (.e., mole numbers) as ndependent varables and unnowns (Laerveld et al., 2005; Ruz et al., 2006). However, as ndcated by Ung and Doherty (1995a,b), the numbers of moles are not proper varables to use n the modelng of reactve systems because they do not have the same dmensonalty as the number of degrees of freedom (.e., they are nconsstent wth respect to the Gbbs phase rule). To obtan a convenent descrpton of reactve vapor lqud equlbrum problem and to smplfy the analyss of multplcty n flash systems subject to chemcal reactons, we have appled the reacton-nvarant composton varables proposed by Ung and Doherty (1995a,b). These varables are based on transformaton of the physcal compostons and ts prncpal beneft s that the chemcal and physcal equlbrum approach n the reactve mxture s dentcal to a strctly physcal equlbrum model. The dmenson of reacton-nvarant composton space s equal to the number of degrees of freedom obtaned from the Gbbs phase rule. Thus, these varables depend only on the ntal composton of each ndependent chemcal speces, restrct the soluton space to the compostons that satsfy stochometry requrements and also reduce the dmenson of the composton space by the number of ndependent reactons (Ung and Doherty, 1995a,b). These features allow all of the procedures used to model non-reactve mxtures to be easly extended to systems subject to chemcal equlbrum and, consequently, non-reactve algorthms can be modfed to account for chemcal reactons. Note that several approaches for modelng and desgnng reactve separaton processes have been developed usng these reacton-nvarant composton varables and they nclude phase equlbrum calculatons, phase stablty analyss, azeotropy predcton, calculaton of resdues curves, among other calculatons (e.g., Güttnger and Morar, 1999a,b; Bonlla-Petrcolet et al., 2006a,b, 2008a,b; Carrera- Rodrguez et al., 2011a,b). For a reactve system of c components and r chemcal reactons, transformed mole varables X are defned as X = x v N 1 x ref 1 v TOT N 1 x ref = 1,...,c r (1) where x s the mole fracton of component, x ref s a column vector of mole fractons for r reference components, v s the row vector of stochometrc coeffcents of component for each of the r reactons, N s an nvertble and square matrx formed from the stochometrc coeffcents of the reference components n the r reactons, and v TOT s a row vector where each element corresponds to the sum of stochometrc coeffcents for all components that partcpate n each of the r reactons, respectvely. These transformed mole fractons (X) n reactve systems are smlar to the mole fractons (x) n non-reactve mxtures, and the sum of all transformed mole fractons s equal to unty (.e., c r =1 X = 1), but a transformed mole fracton can be negatve or postve dependng on the reference components, number and type of reactons. Therefore, for a reactve mxture, mnmzng Gbbs free energy wth respect to conventonal compostons varables (.e., mole numbers or fractons) s equvalent to mnmzng the Gbbs free energy formulated wth respect to the reacton-nvarant composton varables (Ung and Doherty, 1995a,b). It s mportant to remar that X has the desrable property of tang the same numercal values before and after the reactons. Ths s n contrast to conventonal mole varables (x and n), whch have dfferent values for the components n the unmxed and mxed (.e., reactng) states (Ung and Doherty, 1995a,b). Note that the transformed varables X are related to x va the reacton equlbrum constants K eq, : K eq, = c a v = 1,...,r (2) =1 where v s the stochometrc coeffcent of component n reacton, and a s the actvty of component. To evaluate thermodynamc propertes n reactve systems usng

5 chemcal engneerng research and desgn 9 0 ( ) ths approach, mole fractons are obtaned from the transformaton procedure X x usng Eqs. (1) and (2). Ths stage requres soluton of one or more nonlnear equatons dependng on the number of chemcal reactons. In partcular, any solver such as the bsecton method (for systems wth only one reacton) or the Newton method (for mult-reacton systems) can be used for ths varable transformaton. The resultng mole fracton values (x) satsfy the stochometry requrements and are chemcally equlbrated. It s mportant to remar that multple solutons are not possble for x ref durng varable transformaton X x because only one soluton set of x smultaneously satsfes the chemcal equlbrum equatons and corresponds to the specfed values of the transformed composton varables (Ung and Doherty, 1995a). Therefore, the presence of multple states n the modelng of reactve flash separaton s not caused by the use of these varables. The reactve vapor lqud equlbrum problem for a multcomponent and multreactve system can be defned as (Bonlla-Petrcolet et al., 2006a,2008a): X L (Z = ı ˆ V ) ( ˆ V (K 1) + 1) X V = X L K + ı = (Z K + ı (1 ˆ V )) ( ˆ V (K 1) + 1) where (1 v TOT N 1 x L ref ) = (1 v TOT N 1 x V ref ) (v N 1 (K x L ref xv ref )) ı = (1 v TOT N 1 x V ref ) = 1,...,c r (3) = 1,...,c r (4) subject to the materal balance gven also n transformed composton space Z (1 ˆ V )X L X V ˆ V = 0 = 1,...,c r (6) V (1 v TOT N 1 x V ˆ V ref ) = (1 v TOT N 1 z ref ) and equalty constrants c r X L = 1 =1 c r X V = 1 =1 ˆ V + ˆ L = 1 where X L and X V are the transformed mole fracton of component at lqud and vapor phase at equlbrum, Z s the global transformed composton of component n the feed, K s the phase equlbrum constant of component, ˆ V s the transformed amount fracton for vapor phase whle V s the conventonal mole fracton of vapor phase whose feasble doman s (0, 1). Note that the feasble doman of ˆ V s not the same as that gven for V due to stochometrc constrants of the reactve system, see Eq. (7). However, ths fact s not a lmtaton to robustly solve the reactve phase equlbrum problem. (5) (7) (8) The mplct functon to evaluate the transformed amount fracton for vapor phase s based on an alternatve Rachford Rce formulaton usng X (Bonlla-Petrcolet et al., 2006a,2008a). Thus, for the reactve flash problem we have to solve the followng non-lnear equaton c r [ ] f ( ˆ V (Z (K ) = 1) + ı ) = 0 (9) ( ˆ V (K 1) + 1) =1 Eq. (9) s employed to evaluate ˆ V (.e., V ) and determne the vapor lqud equlbrum compostons subject to chemcal equlbrum. Heren, we restrct our analyss of reactve flash equatons on the appearance of multple solutons n the regon of physcal sgnfcance for flash separatons,.e. phase equlbrum calculatons are bounded between reactve bubble and dew pont condtons where V (0, 1). Therefore, bubble and dew pont calculatons are performed usng the next functons (Bonlla-Petrcolet et al., 2006a,2008a): c r f bubble = 1 (K Z + ı ) = 0 (10) =1 =1 c r [ Z ] ı f dew = 1 = 0 (11) K Note that the phase equlbrum constant K are calculated from K {X L,X V }= ˆϕL ˆϕ V = L V = 1,...,c r (12) where ˆϕ s the fugacty coeffcent of component n the mxture and s the actvty coeffcent of component, respectvely. These thermodynamc propertes are determned usng a proper model (e.g., equaton of state or local composton model) and employng the results of the transformaton procedure X x (.e., we use mole fractons that satsfy the chemcal equlbrum to evaluate the system thermodynamc propertes). Note that the general steps nvolved n the varable transformaton between X and x are ndependent of both the type of reactve system and the model used n the calculaton of thermodynamc propertes for the dfferent phases (e.g., equatons of state or local composton models). In partcular, the thermodynamc model only affects the approach used for calculatng the fugacty or actvty coeffcents, whch are nvolved n the evaluaton of Eqs. (2) and (12); whle the characterstcs of the reactve system (.e., the number of components and reactons) determne the dmensonalty of the non-lnear equaton system to be solved durng the transformaton of composton varables. Therefore, n ths study no restrctons are made wth regard to these parameters. On the other hand, dfferent numercal strateges can be used for performng flash calculatons n reactve systems wth reacton-nvarant composton varables and they nclude, for example, smultaneous equaton-solvng methods, equaton decouplng approaches and global optmzaton technques (Bonlla-Petrcolet et al., 2006a,b, 2008a,b). In ths study, proper numercal methods have been appled for relably modelng the phase behavor of reactve systems used as cases of study and these calculatons do not requre a sgnfcant numercal effort and computer tme.

6 1860 chemcal engneerng research and desgn 9 0 ( ) Input multplcty analyss Usng transformed varables, the reactve flash problem has c r + 2 degrees of freedom. For the analyss of nput multplcty, they are fxed by specfyng c r transformed mole fractons Z of the feed, the pressure P of the reactve flash separaton process and a product composton (X L or X V ). In ths study, we assume that the operatng condtons of a nonazeotropc mxture are chosen below the crtcal state to avod retrograde effects. If under these condtons the system shows multple solutons (.e., nput multplcty), more than one soluton may exst for the reactve flash problem. It s mportant to note that our analyss also rests on the fact that the soluton (.e., phase equlbrum compostons) obtaned for reactve flash separaton problem s globally stable wth respect to the formaton of addtonal phases (Mchelsen, 1982). In partcular, Tscareño et al. (1998) and Monroy-Loperena (2001) have suggested that f nput multplctes exst n nonreactve mxtures, the correspondng flash equaton for lqud or vapor mole fracton of component must be a concave or convex functon wth respect to temperature or vapor phase fracton. Intutvely, we can expect the same performance for Eq. (3) or (4) n reactve systems usng X. Therefore, analyzng the reactve flash problem of a nonazeotropc mxture n the temperature doman between the bubble-pont temperature T bub and the dew-pont temperature T dew, whch mples a real vapor lqud equlbrum soluton, and assumng that a maxmum or a mnmum exsts for X j T, the condton for a statonary pont s gven by dx j dt = 0 (13) or, equvalently, for X j ˆ V we have dx j dˆ V = 0 (14) where j s the lqud (L) or vapor phase (V), respectvely. Note that the nput multplcty analyss can be performed n the vapor fracton doman or n the temperature doman because there s only one stable soluton n flash calculatons performed at operatng condtons below the crtcal state of the mxture (Luca, 1986; Monroy-Loperena, 2001). However, based on the fact that the transformed composton varables are used only to smplfy and facltate the modelng of thermodynamc behavor of multcomponent and multreactve systems, the multplcty analyss must be performed n the doman of mole fracton composton, whch s the real operaton varable. As stated, transformed composton varables (X) depend on both the molar composton (x)of the reactve system and T at the equlbrum condtons, f P s gven. Therefore, we can state the mathematcal functonalty: X = f(x) and x = f(t) orx = f ( ˆ V ). Then, for reactve phase equlbrum, Eqs. (13) and (14) are the composton of two functons and can be wrtten usng Lebnz notaton as follows dx j dt = dx j dxj dx j dt = 1,...,c r; j = L, V; = 1,...,c (15) or usng the vapor phase fracton: dx j dˆ = dx j dxj = 1,...,c r; j = L, V; = 1,...,c (16) V dx j dˆ V where x j s the mole fracton of component n phase j at the equlbrum condton. Rearrangng these functons, we can defne the statonary condtons for dentfyng nput multplcty n reactve flash separaton usng reacton-nvarant composton varables dx j d = dx j d dx j = 0 (17) dx j where = T or ˆ V dependng on the varable to be analyzed. Note that f Eq. (17) s to be satsfed, then we can have the followng scenaros for fndng a statonary pont (.e., a maxmum or a mnmum exsts for x j ): Condton I : Condton II : Condton III : dx j j d = 0 f dx d dx j j d = 0 f dx d dx j j d = 0 f dx d j dx = 0 and dx j /= 0 j dx /= 0 and = 0 dx j j dx = 0 and = 0 dx j These condtons ndcate that the dervatve of X j wth respect to equals zero or the dervatve of x j wth respect to Xj equals zero, n an ndependent way or smultaneously. If the statonary pont of Eq. (17) s bounded by T [T bub, T dew ] and V [0,1], away from the retrograde regon, we can confrm the presence of nput multplctes for the mole fracton of component n phase j n a reactve flash separaton. In partcular, n the followng secton we provde numercal evdence to valdate and support that Condtons II and III derved from Eq. (17) are necessary and suffcent condtons for detectng multple solutons n reactve flash separatons. In summary, Eq. (17) s useful to predct the presence of multple solutons n reactve flash separaton and to easly recognze what component or set of components exhbts the multplcty at the specfed operatng condtons. It s mportant to recall that the applcaton of reacton-nvarant composton varables allows the easy mplementaton of numercal strateges for modelng thermodynamc propertes of reactve systems and they are used n ths study to develop alternatve condtons for predctng nput multplcty n the reactve flash separaton problem. To explore the presence of nput multplcty n all the system, Eq. (17) must be tested for all components x j for =1,..., c n both vapor and lqud phases. However, ths procedure s not tme consumng because ths crteron can be effectvely tested at both the bubble and dew ponts where a change of sgn n the dervatves wll ndcate the presence of a statonary pont. Recall that these dervatves can be evaluated explctly by consderng dx j dt = T=T sat dx j dx j x=f (T sat ) dxj dt (18) T=T sat

7 chemcal engneerng research and desgn 9 0 ( ) Fg. 3 Flowchart of the proposed method for predctng nput multplcty n reactve flash separaton. where T sat s the bubble-pont temperature T bub or the dewpont temperature T dew. So, f the dervate dx j /d T presents a change of sgns at bubble and dew ponts, we can conclude the exstence of multple solutons for the reactve flash problem. Ths condton s equvalent to that reported by Tscareño et al. (1998) and Monroy-Loperena (2001) for the non-reactve flash problem. It s convenent to remar that n the general case wthout any smplfcaton of the model used for calculaton of the thermodynamc propertes; these dervatves are evaluated straghtforward usng fnte dfferences. Alternatvely, a one-dmensonal drect optmzaton strategy can be used to fnd the mnmum (.e., statonary pont for multplcty analyss) of d 2 where d s gven by dx j /d. If the mnmum of d2 =0, we conclude that the reactve system shows multple solutons n flash separaton for the component and phase under analyss. Note that ths optmzaton approach has also been used for nput multplcty analyss n non-reactve systems (Monroy-Loperena, 2001). Our numercal experence ndcates that an optmzaton approach s more effectve than rootfndng methods for the locaton of the statonary pont of these dervatves. In general, our approach s straghtforward, easy to be mplemented and does not requre complex numercal calculatons. Wth llustratve purposes, Fg. 3 shows the steps of our method for predctng nput multplcty. In the followng secton, we llustrate the applcaton of ths approach for predctng nput multple states n reactve flash separaton problems usng well-nown reactve systems. 3. Results and dscusson We use two reactve systems as cases of study to llustrate the applcaton of our approach for predctng nput multplcty n reactve flash separaton. These systems are well-nown n the lterature for the producton of fuel ethers (Mohl et al., 1999) and nclude the synthess of methyl tert-butyl ether (MTBE) and tert-amyl methyl ether (TAME). Detals of reactve systems are reported n Table 1 and all model parameters are reported by Maer et al. (2000), Bonlla-Petrcolet et al. (2008a,b) and Carrera-Rodrguez et al. (2011a,b). We assume that all reactons are reversble and n thermodynamc equlbrum. Phase equlbrum calculatons, ncludng the determnaton of bubble and dew ponts, have been performed accordng to the numercal strateges reported by Bonlla-Petrcolet et al. (2006a,b) and Bonlla-Petrcolet et al. (2008a,b). Phase stablty of all reactve phase equlbrum calculatons, ncludng bubble and dew ponts, has been Table 1 Examples selected for nput multplcty analyss n a reactve flash separaton. No. System Thermodynamc models 1 A 1 + A 2 A 3, and A 4 as an nert component (1) Isobutene (2) Methanol (3) Methyl-tert-butyl ether (4) n-butane 2 A 1 + A 2 +2A 3 2A 4 (1) 2-Methyl-1-butene (2) 2-Methyl-2-butene (3) Methanol (4) Tert-amyl methyl ether 3 A 1 + A 2 +2A 3 2A 4 wth A 5 as nert component (1) 2-Methyl-1-butene (2) 2-Methyl-2-butene (3) Methanol (4) Tert-amyl methyl ether (5) n-pentane Wlson model and deal gas: G o rxs /R = T T ln T lnk eq,1 = G0 rxs RT where T s n K Wlson model and deal gas: K eq,1 = e /T where T s n K Wlson model and deal gas: K eq,1 = e /T where T s n K

8 1862 chemcal engneerng research and desgn 9 0 ( ) Reactve system for MTBE synthess We have analyzed the presence of nput multplcty n the reactve system nvolved n the synthess of MTBE (x 3 )from Isobutene (x 1 ), Methanol (x 2 ) and wth n-butane (x 4 ) as nert: Isobutene (x 1 ) + Methanol (x 2 ) MTBE (x 3 ) (19) Wlson and Antone models were used for the calculaton of thermodynamc propertes employng the parameters reported by Maer et al. (2000). Ths reactve system s a benchmar problem used n reactve process desgn and has been studed extensvely by Ung and Doherty (1995a), Bonlla-Petrcolet et al. (2006a,b), Ruz et al. (2006), and Carrera-Rodrguez et al. (2011a,b), among other authors. In partcular, Ruz et al. (2006) showed the presence of Hopf bfurcatons and multple solutons n the sothermal reactve flash processes nvolvng MTBE mxture. In our study, MTBE s selected as the reference component (x 3 ) and transformed mole fractons for ths mxture are gven by X 1 = x 1 + x x 3 (20) X 2 = x 2 + x x 3 (21) X 4 = x x 3 = 1 X 1 X 2 (22) Fg. 4 Vapor phase composton behavor of (a) sobutene and (b) butane as a functon of temperature and vaporzaton fracton for MTBE producton. performed by mnmzng the tangent plane dstance functon for reactve mxtures (Bonlla-Petrcolet et al., 2006a,b). In ths study, all calculated reactve phase equlbrum compostons are stable. In summary, the condtons used n ths study meet the requrement for a unque stable soluton for the reactve phase equlbrum calculatons. Fnally, n ths study bsecton method s used to relably perform the composton transformaton X x durng reactve flash calculatons. Multplcty analyss has been performed for a feed composton n 0 = (0.163, 0.005, 0.081, 0.751), or n transformed varables Z = (0.2257, , ), usng dfferent condtons of pressure (P). Note that Ruz et al. (2006) have analyzed the same feed for obtanng the Hopf bfurcaton dagrams. The multple solutons of reactve flash problem are found by solvng Eqs. (3) (9) for ths feed composton and dfferent operatng condtons (.e., T and P). Fg. 4 shows the results of reactve flash calculatons for x1 V and xv 4 wth respect to the vaporzaton fracton and the temperature at dfferent pressures from 1 to 20 atm. Phase equlbrum behavor of sobutene (x 1 ) and butane (x 4 ) ndcates the presence of multplcty n the vapor phase at some operatng condtons and, as a consequence, Eq. (17) s satsfed for these components where ths dervatve has changes of sgn when T = T bub and Fg. 5 Input multplcty (IM) analyss of the vapor phase for the MTBE producton n a flash separaton at 5 atm.

9 chemcal engneerng research and desgn 9 0 ( ) Fg. 6 Input multplcty (IM) analyss of the vapor phase for the MTBE producton n a flash separaton at 20 atm. T = T dew. Note that we can easly establsh that both components have multple solutons n the vapor phase but at dfferent condtons of P (see Fg. 4). As example, Fgs. 5 and 6 show the multplcty analyss for the vapor phase at 5 and 20 atm. In partcular, these fgures show the dependence of x wth T and the evaluaton of the dervatves dx /dt, dx /dx and dx /dt for all components of ths reactve mxture. It s observed that the reactve flash problem shows multple solutons for the vapor mole fracton of n-butane at 5 atm, whle sobutene shows multplcty at 20 atm. As stated, the mole fractons of these components are concave or convex functons wth respect to T when multple solutons are present for reactve flash separaton. It s nterestng to note that Eq. (17) s satsfed f only f: dx /dt /= 0 and dx /dx =0ordX /dt =0 and dx /dx = 0 (.e., Condtons II and III gven below). Smlar results have been obtaned for the remanng operatng condtons where nput multplcty occurs. In fact, f dx /dt =0 and dx /dx /= 0, ths condton ndcates that x does not have multplcty of solutons n tested phase. These results are llustrated n Fgs. 7 and 8 for ths reactve system at Fg.7 X versus T and x versus X for the vapor phase of MTBE producton n a reactve flash separaton at 5 atm.

10 1864 chemcal engneerng research and desgn 9 0 ( ) Fg.8 x versus X for the vapor phase of MTBE producton n a reactve flash separaton at 5 atm. 5 atm, where the dependence of X wth T and the dependence of x wth X are shown. For example, the dervatves of the sobutene (x 1 )aredx 1 /dx /= 0 (for = 1, 2 and 4), whle dx 1 /dt =0,dX 2 /dt /= 0 and dx 4 /dt = 0; therefore, based on prevous analyss, we can conclude that ths component does not show multple solutons n the vapor phase at tested condtons. Smlar analyss has been performed for determnng the presence of multple states n the remanng components of ths mxture. In ths context, t s convenent to recall that Tscareño et al. (1998) proposed a set of condtons to predct nput multplcty n non-reactve flash systems. These condtons are based n the values of phase equlbrum constants K. In partcular, these authors have suggested that the vapor mole fracton of component can show a statonary pont only f K >1, whch s consdered a necessary but not suffcent condton for the presence of multple solutons n non-reactve flash separaton. By analyzng the values of K for both sobutene and n-butane, our results ndcate that ths condton s satsfed for all operatng condtons where the multplcty s present. Wth llustratve purposes, Fg. 9 shows the temperature dependence of K n the range of T bub and T dew for ths reactve system at 5 and 20 atm. Both K 1 and K 4 are hgher than 1 wthn the tested temperature nterval. On the other hand, the less volatle component cannot have multplcty n the vapor phase (Tscareño et al., 1998). In ths reactve system, MTBE s the less volatle component and also meets ths condton. In general, ths agreement may suggest that the necessary K-based condtons for multple solutons proposed for non-reactve systems could be useful to explan the presence of nput multplctes n a reactve flash separaton Reactve system for TAME synthess wth and wthout nert TAME s commonly produced by lqud-phase etherfcaton between methanol and so-amylenes, n the presence of an acdc catalyst. Among the three so-amylenes, only 2-methyl- 1-butene (2M1B) and 2-methyl-2-butene (2M2B) are reactve, whereas 3-methyl-1-butene (3M1B) s non-reactve. In ths study, we have consdered the lumped sngle reacton wth and wthout n-pentane (x 5 ) as nert, whch can be wrtten as: 2M1B(x 1 ) + 2M2B(x 2 ) + 2 Methanol (x 3 ) 2TAME(x 4 ) (23) Wlson and deal gas models have been also used to calculate thermodynamc propertes of ths mxture. Model parameters are taen from Chen et al. (2002) and Bonlla-Petrcolet et al. (2008a,b). Ths reactve system has been also analyzed by Ruz et al. (2006) usng Hopf bfurcatons and shows multple solutons for the sothermal reactve flash process. Reacton-nvarant mole fractons, consderng TAME as reference component (x 4 ) and wthout nert, are defned as X 1 = x x x 4 (24)

11 chemcal engneerng research and desgn 9 0 ( ) Fg. 10 Lqud phase composton behavor of TAME as a functon of temperature and vaporzaton fracton for TAME producton wthout nert. Fg. 9 Temperature dependence of K for the MTBE producton n a reactve flash separaton at (a) 5 and (b) 20 atm. X 2 = x x x 4 (25) X 3 = x 3 + x x 4 = 1 X 1 X 2 (26) For ths reactve system wthout nert, we have selected a feed transformed composton: Z = (0.1382, , ) and multplcty analyss has been performed at dfferent pressures. Fg. 10 shows the response surfaces of vapor lqud equlbrum for that component wth multple solutons, whch corresponds to TAME (x 4 ). In partcular, we have analyzed the reactve flash separaton of ths system at 2 atm. At these operatng condtons, dervatves for ths component present change of sgn only n the lqud phase. None of all components presents a change of the dervatve sgns between the bubble and dew ponts n the vapor phase and, as a consequence, the multplcty does not occur n ths phase. The multplcty analyss for the lqud phase s gven n Fg. 11 where the dervatves of all components are reported. Agan, the Condtons II and III are satsfed f multple solutons are present for the reactve flash separaton (see Fgs. 12 and 13). In ths reactve, the lqud mole fracton of TAME satsfes the necessary and suffcent condtons for detectng the presence of nput multplcty n the flash separaton. Note that the evaluaton of Eq. (17) s an easy and straghtforward approach for predctng the presence of multple solutons n a reactve flash separaton. Fg. 14 shows the temperature dependence of K for each component that partcpates n ths reactve system at 2 atm. As stated, TAME showed nput multplcty n lqud phase and ts K < 1 for the nterval T bub T T dew where the multplcty s Fg. 11 Input multplcty (IM) analyss of the lqud phase for the TAME producton wthout nert n a flash separaton at 2 atm.

12 1866 chemcal engneerng research and desgn 9 0 ( ) Fg. 12 X versus T and x versus X for the lqud phase of TAME producton wthout nert n a reactve flash separaton at 2 atm. Fg. 13 x versus X for the lqud phase of TAME producton wthout nert n a reactve flash separaton at 2 atm.

13 chemcal engneerng research and desgn 9 0 ( ) thermodynamc models. Transformed mole fractons of ths qunary system are gven by Eqs. (24) (26) and X 5 = x x 4 = 1 X 1 X 2 X 3 (27) Fg. 14 Temperature dependence of K for the TAME producton n a reactve flash separaton at 2 atm (a) wthout and (b) wth the presence of an nert. present. These results also agree wth the necessary condton proposed by Tscareño et al. (1998) for nput multplcty n the lqud phase, whch ndcates that the lqud mole fracton of a component n a flash separaton can show multplcty f K <1. Ths condton also mples that the more volatle component cannot present nput multplcty n the lqud phase. In ths reactve system, 2M1B s the lghtest component and does not show multple solutons n lqud phase. Fnally, we have studed ths reactve system but consderng the presence of n-pentane (x 5 ) as nert usng the same We have analyzed a feed composton Z = (0.0907, , 0.190, ) and results of reactve flash calculatons from 1 to 4 atm are reported n Fg. 15. In partcular, our results ndcate that methanol (x 3 ) shows nput multplcty n both the lqud and the vapor phases at 2 atm, see Fgs. 16 and 17. As expected, there are convex functons (.e., x 3 versus T) for the composton of ths component n both phases. Therefore, ts dervatves show a change of slope sgn at bubble and dew ponts and satsfy Eq. (17) f Condtons II and III are met, ndcatng the presence of multple solutons n a reactve flash separaton problem (see Fg. 16). Partcularly, ths component shows K 3 > 1 n all operatng condtons where the multplcty s present. For example, we report the K values for ths reactve system at 2 atm n Fg. 14b. Note that both the lghtest and the heavest components (.e., 2M1B and TAME) do not have nput multplcty for the reactve flash separaton n the lqud phase and n the vapor phase, respectvely. In fact, these components meet the necessary condtons establshed by Tscareño et al. (1998) for nput multplcty n non-reactve systems. In summary, our results ndcate that the necessary condtons proposed by Tscareño et al. (1998) to determne the nput multplcty n non-reactve systems, whch are based n the analyss of the equlbrum constants values (K ), may be applcable for studyng multple solutons n reactve systems. However, t s convenent to remar that the nteracton of smultaneous chemcal and physcal equlbrum may cause a complex phase behavor n systems subject to chemcal reactons. Therefore, we could expect that, for some reactve mxtures wth complex phase behavor, the based-k condtons may not apply. Fnally, based on our numercal calculatons and analyss, we conclude that the necessary condtons for a statonary pont of x (.e., multple solutons) to occur n the feasble doman of n a reactve flash separaton are: (1) The mole fracton of component n phase j shows a statonary pont wth respect to n a reactve flash separaton f dx j /d /= 0 and dxj /dxj = 0 for at least one Xj where =1,...,c r. (2) The mole fracton of component n phase j shows a statonary pont wth respect to n a reactve flash Fg. 15 (a) Vapor and (b) Lqud phase composton behavor of TAME as a functon of temperature and vaporzaton fracton for TAME producton wth n-pentane as nert.

14 1868 chemcal engneerng research and desgn 9 0 ( ) Fg. 16 Input multplcty (IM) analyss of the lqud phase for the TAME producton wth nert n a flash separaton at 2 atm. Fg. 17 Input multplcty (IM) analyss of the vapor phase for the TAME producton wth nert n a flash separaton at 2 atm.

15 chemcal engneerng research and desgn 9 0 ( ) separaton f dx j /d = 0 and dxj /dxj = 0 for at least one X j where =1,...,c r. Our numercal experence ndcates that f the dervatves dx j /d = 0 and dxj /dxj /= 0, ths result s a relable ndcator of the absence of multple solutons of component n phase j for a reactve flash separaton at tested operatng condtons. Fnally, t s mportant to remar that any thermodynamc model and reactve system, assumng that all reactons are reversble and n thermodynamc equlbrum, can be used n our approach because these parameters do not affect the structure and assumptons of the proposed algorthm for predctng nput multplcty. In general, our strategy s based on straghtforward phase equlbrum calculatons and the evaluaton of smple mathematcal condtons for predctng the presence of multple states n reactve flash separaton and, as a consequence, t appears to be compettve wth respect to other methods reported n the lterature. As stated, several methods for predctng nput multplctes manly reles on the applcaton of homotopy contnuaton methods due to ther capabltes for fndng multple solutons. In general, these methods mply the resoluton of non-lnear equaton systems usng an ntal value problem to defne the soluton s path. Even though these numercal methods are generally robust, some authors have recognzed that the homotopy methods may be tmeconsumng and requre consderable computatonal effort (Malnen and Tansanen, 2010). Based on ths fact, we consder that our approach s an alternatve strategy, relatvely flexble an easy to use, for predctng nput multplcty n flash separaton of systems subject to chemcal reactons. 4. Conclusons Ths study ntroduces a new approach and condtons to dentfy nput multplctes n reactve flash separaton process, whch are based on the applcaton of reacton-nvarant composton varables. Ths approach has been tested and appled for predctng and analyzng nput multplcty n two reactve systems of ndustral relevance. Numercal results support and valdate the proposed condtons and they can be regarded as necessary condtons to predct the presence or absence of nput multplcty n reactve flash separaton. Our calculatons ndcate that the proposed approach s easy to use and effectve for determnng the presence of multple solutons n reactve flash calculatons. In fact, our strategy can be appled wth any reactve system and model for determnng thermodynamc propertes, assumng that all reactons are reversble and n thermodynamc equlbrum and the operatng condtons are away from the retrograde regon. In fact, our method seems sutable and robust to perform the multplcty analyss of mult-reactve and mult-component systems. Fnally, t appears that the necessary condtons proposed to determne the nput multplcty n non-reactve systems, whch are based n the analyss of the equlbrum constants values (K ), may be applcable for studyng multple solutons n reactve systems. Further wor wll be focused on the study of multple solutons n reactve flash separatons for netcally controlled reactve systems. Acnowledgements We acnowledge the fnancal support provded by Insttuto Tecnológco de Aguascalentes, Unversdad de Guanajuato, CONACyT and DGEST (Mexco). References Baur, R., Taylor, R., Krshna, R., Bfurcaton analyss for TAME synthess n a reactve dstllaton column: comparson of pseudo-homogeneous and heterogeneous reacton netcs models. Chem. Eng. Process. 42, Bonlla-Petrcolet, A., Acosta-Martínez, A., Bravo-Sánchez, U.I., Segova-Hernández, J.G., 2006a. Bubble and dew pont calculatons n multcomponent and multreactve systems. Chem. Bochem. Eng. Q. 20, Bonlla-Petrcolet, A., Acosta-Martínez, A., Tapa-Pcazo, J.C., Segova-Hernández, J.G., 2008a. A method for flash calculatons n reactve mxtures. Afndad 65, Bonlla-Petrcolet, A., Bravo-Sanchez, U.I., Castllo-Borja, F., Frausto-Hernandez, S., Segova-Hernandez, J.G., 2008b. Gbbs energy mnmzaton usng smulated annealng for two-phase equlbrum calculatons n reactve systems. Chem. Bochem. Eng. Q. 22, Bonlla-Petrcolet, A., Vázquez-Román, R., Iglesas-Slva, G.A., Hall, K.R., 2006b. Performance of stochastc optmzaton methods n the calculaton of phase stablty analyses for nonreactve and reactve mxtures. Ind. Eng. Chem. Res. 45, Carrera-Rodrguez, M., Segova-Hernandez, J.G., Bonlla-Petrcolet, A., 2011a. A short method to calculate resdue curve maps n multreactve and multcomponent systems. Ind. Eng. Chem. Res. 50, Carrera-Rodrguez, M., Segova-Hernandez, J.G., Bonlla-Petrcolet, A., 2011b. A short method for the desgn of reactve dstllaton columns. Ind. Eng. Chem. Res. 50, Chen, F., Huss, R.S., Doherty, M.F., Malone, M.F., Multple steady states n reactve dstllaton: netc effects. Comput. Chem. Eng. 26, Eldars, H.S., Douglas, P.L., Methyl-tert-butyl-ether catalytc dstllaton column. Part I: multple steady states. Chem. Eng. Res. Des. 76, Güttnger, T.E., Morar, M., 1999a. Predctng multple steady states n equlbrum reactve dstllaton. 1. Analyss of nonhybrd systems. Ind. Eng. Chem. Res. 38, Güttnger, T.E., Morar, M., 1999b. Predctng multple steady states n equlbrum reactve dstllaton. 2. Analyss of hybrd systems. Ind. Eng. Chem. Res. 38, Hauan, S., Hertzberg, T., Len, K.M., Multplcty n reactve dstllaton of MTBE. Comput. Chem. Eng. 21, Kumar, M.V.P., Kastha, N., Role of multplcty n reactve dstllaton control system desgn. J. Process Control 18, Laerveld, R., Bldea, C.S., Almeda-Rvera, C.P., Exothermc somerzaton reacton n a reactve flash: steady state behavor. Ind. Eng. Chem. Res. 44, Luca, A., Unqueness of solutons to sngle-staged sobarc flash process nvolvng homogeneous mxtures. AIChE J. 32, Maer, R.W., Brennece, J.F., Stadtherr, M.A., Relable computaton of reactve azeotropes. Comput. Chem. Eng. 24, Malnen, I., Tansanen, J., Homotopy parameter boundng n ncreasng the robustness of homotopy contnuaton methods n multplcty studes. Comput. Chem. Eng. 34, Mchelsen, M.L., The sothermal flash problem. Part I. Stablty. Flud Phase Equlb. 9, Mohl, K.D., Kenle, A., Glles, E.D., Rapmund, P., Sundmacher, K., Hoffmann, U., Steady-state multplctes n reactve dstllaton columns for the producton of fuel ethers MTBE

16 1870 chemcal engneerng research and desgn 9 0 ( ) and TAME: theoretcal analyss and expermental verfcaton. Chem. Eng. Sc. 54, Monroy-Loperena, R., A fast method to chec steady-state nput multplctes n vapor lqud flash separaton. Ind. Eng. Chem. Res. 40, Ramzan, N., Faheem, M., Gan, R., Wtt, W., Multple steady states detecton n a paced-bed reactve dstllaton column usng bfurcaton analyss. Comput. Chem. Eng. 34, Rodríguez, I.E., Zheng, A., Malone, M.F., The stablty of a reactve flash. Chem. Eng. Sc. 56, Rodríguez, I.E., Zheng, A., Malone, M.F., Parametrc dependence of soluton multplcty n reactve flashes. Chem. Eng. Sc. 59, Ruz, G., Srdhar, L.N., Rengaswamy, R., Isothermal sobarc reactve flash problem. Ind. Eng. Chem. Res. 45, Sngh, B.P., Sngh, R., Kumar, M.V.P., Kastha, N., 2005a. Steady state analyss of reactve dstllaton usng homotopy contnuaton. Chem. Eng. Res. Des. 83, Sngh, B.P., Sngh, R., Kumar, M.V.P., Kastha, N., 2005b. Steady state analyses for reactve dstllaton control: an MTBE case study. J. Loss Prev. Process Ind. 18, Svandová, Z., Labovsý, J., Maros, J., Jelemensý, L., Impact of mathematcal model selecton on predcton of steady state and dynamc behavour of a reactve dstllaton column. Comput. Chem. Eng. 33, Taylor, R., Krshna, R., Modellng reactve dstllaton. Chem. Eng. Sc. 55, Tscareño, F., Gómez, A., Jménez, A., Chávez, R., Multplcty of the solutons of the flash equatons. Chem. Eng. Sc. 53, Ung, S., Doherty, M.F., 1995a. Vapor lqud phase equlbrum n systems wth multple chemcal reactons. Chem. Eng. Sc. 50, Ung, S., Doherty, M.F., 1995b. Theory of phase equlbra n multreacton systems. Chem. Eng. Sc. 50, Vaca, M., Jmenez, A., Monroy-Loperena, R., On the multple solutons of the flash equatons. Chem. Eng. Sc. 61, Yang, B., Wu, J., Zhao, G., Wang, H., Lu, S., Multplcty analyss n reactve dstllaton column usng ASPEN PLUS. Chnese J. Chem. Eng. 14,