Optimization of High-Pressure Vapor-Liquid Equilibrium Modelling of Binary Mixtures (Supercritical Fluid + Ionic Liquid) by Particle Swarm Algorithm

Transcription

1 MATCH Communcatons n Mathematcal and n Computer Chemstry MATCH Commun. Math. Comput. Chem. 73 (215) ISSN Optmzaton of Hgh-Pressure Vapor-Lqud Equlbrum Modellng of Bnary Mxtures (Supercrtcal Flud + Ionc Lqud) by Partcle Swarm Algorthm Juan A. Lazzús* Departamento de Físca, Unversdad de La Serena, Caslla 554, La Serena, Chle jlazzus@dfuls.cl (Receved December 13, 214) Abstract Hgh-pressure vapor lqud equlbrum data of bnary mxtures contanng supercrtcal fluds and mdazolum onc lquds were correlated usng a thermodynamc model optmzed wth a partcle swarm algorthm. The Peng Robnson (PR) equaton of state and the Wong Sandler (WS) mxng rules ncludng the van Laar (VL) model for the excess Gbbs free energy, were used as thermodynamc model. Forty sx bnary systems taken from lterature were selected for ths study, and the optmzaton algorthm was used to determne the bnary nteracton parameters of each system. The algorthm was development to mnmze the dfference between calculated and expermental bubble pressure. The results gven by the model show that the proposed algorthm s a good tool to correlate and descrbe the vapor lqud equlbrum of ths type of systems. 1. Introducton In the recent years, room temperature onc lquds (RTILs) or just onc lquds (ILs) came nto focus because of ther potental as alternatves for several engneerng applcatons [1]. ILs are typcally composed of a large organc caton and an norganc polyatomc anon.

2 -664- There s vrtually no lmt n the number of possble onc lquds snce there s a large number of catons and anons that can be combned [2]. At ambent room temperature, they exst as lquds and have a wde varety of unque propertes (for nstance, neglgble vapor pressure, favorable chemcal behavor, low vscosty, and hgh reactvty and selectvty) [3]. The most commonly used caton n room-temperature onc lquds (RILs) s dalkylmdazolum. And n recent years, 1-alkyl-3-methylmdazolum ([Cnmm] + ) ILs have been ntensvely studed [1]. The ncreasng utlzaton of ILs n chemcal and ndustral processes requres relable and systematc thermophyscal propertes such as actvty coeffcents, heats of mxng, denstes, solubltes, vapor lqud equlbra (VLE), and lqud lqud equlbra (LLE). In addton, the transport propertes are also needed (vscosty, electrc conductvty, mutual dffuson coeffcents, etc.) [4]. For a better understandng of ther thermodynamc behavor and for the development of thermodynamc models relable expermental phase equlbrum data s requred [5]. Phase equlbrum data of mxtures contanng onc lquds are necessary for further development of some separaton processes nvolvng supercrtcal fluds. Blanchard et al. [6] descrbed several potental applcatons of supercrtcal fluds wth ILs. They demonstrated the possblty of usng supercrtcal carbon doxde (CO2) to remove a solute from an IL, wthout contamnaton of the extracted solute, solvng one of the shortcomngs of the use of onc lquds n solvent extracton processes: the recovery of the compounds from the onc lqud meda [7]. Scurto et al. [8] proposed an nnovatve process of separatng ILs from organc solvents usng supercrtcal CO2 that nduces a phase separaton, due to the organc lqud phase expanson and the delectrc constant decrease, forcng the IL to separate nto a second lqud phase [7]. Later, Scurto et al. [9] also demonstrated that separaton of aqueous solutons of both hydrophobc and hydrophlc ILs can be performed usng supercrtcal CO2 [7]. The solublty of carbon doxde n a varety of ILs has been determned at low pressures and hgh pressures [1]. The gas solubltes data provdes mportant nformaton for the characterzaton of solute-solvent nteractons and so contrbute to understand the mechansms of dssoluton. From a practcal pont of vew, gas solublty can be useful n the calculaton of vapor-lqud equlbrum (VLE) [11]. On ths lne, VLE data for bnary systems ncludng onc lquds, although essental for the desgn and operaton of separaton processes, are stll scarce. Recently, some works have presented bnary VLE data nvolvng several onc lquds and such organc compounds as alkanes, alkenes and aromatcs, as well as supercrtcal fluds [12]. Varous models have been used to correlate expermental data of phase equlbra of these systems [13]. One of the

3 -665- common approaches used n the lterature to correlate and predct phase equlbrum requres an equaton of state that well relates the varables temperature, pressure and volume and approprate mxng rules to express the dependence of the equaton of state parameters on the concentraton [14]. On equatons of state, the Peng Robnson equaton has been used to descrbe the solublty of ILs n supercrtcal fluds [15]. The exstng methods to solve phase equlbrum systems gve only local solutons. It has been demonstrated that for cases of systems contanng supercrtcal CO2, the optmum values of the nteracton parameters depend on the searchng nterval and on the ntal value of used nteracton parameters [16]. Parameter estmaton procedures are very mportant n engneerng, ndustral, and chemcal process for development of mathematcal models, snce desgn, optmzaton and advanced control of boprocesses depend on model parameter values obtaned from expermental data. The am of optmzaton s to determne the best-suted soluton to a problem under a gven set of constrants. Mathematcally, an optmzaton problem nvolves a ftness functon descrbng the problem, under a set of constrants representng the soluton space for the problem. The optmzaton problem, now-a-days, s represented as an ntellgent search problem, where one or more agents are employed to determne the optmum on a search landscape [17]. Modern optmzaton technques have aroused great nterest among the scentfc and techncal communty n a wde varety of felds recently, because of ther ablty to solve problems wth non-lnear and non-convex dependence of desgn parameters [18]. Thus, the use of heurstc optmzaton methods, such partcle swarm optmzaton (PSO) [19], for the parameter estmaton s very promsng [2]. Ths bologcally-derver method represents an excellent alternatve to fnd a global optmum for phase equlbrum calculatons [14-18]. In ths work, forty-sx bnary vapor-lqud phase systems contanng supercrtcal fluds (CO2 or CHF3) + 1-alkyl-3-methylmdazolum ILs were correlated usng a thermodynamc model optmzed wth a PSO algorthm. The Peng Robnson (PR) equaton of state and the Wong Sandler (WS) mxng rules ncludng the van Laar (VL) model for the excess Gbbs free energy were used as thermodynamc model. The algorthm was development to calculate the bnary nteracton parameters, and used for mnmze the dfference between calculated and expermental bubble pressure.

4 Partcle swarm optmzaton Adjustable parameters are a common feature of most thermodynamc models for phase equlbrum calculatons. Most of the exstng methods for solvng phase equlbrum and stablty problems are local n nature and at best yeld only local solutons. Use of global technques n these problems s relatvely unexplored and deserves greater nvestgaton [21]. Because of the dffcultes n evaluatng the frst dervatves, to locate the optma for many rough and dscontnuous optmzaton surfaces, n recent tmes, several dervatve algorthms have emerged [2]. Partcle swarm optmzaton (PSO) s a relatvely recently devsed populaton-based stochastc global optmzaton algorthm. As descrbed by Eberhart and Kennedy, the PSO algorthm s an adaptve algorthm based on a socal-psychologcal metaphor; a populaton of ndvduals (referred to as partcles) adapts by returnng stochastcally toward prevously successful regons [19]. The PSO algorthm s ntalzed wth a populaton of random partcles and the algorthm searches for optma by updatng generatons [22]. In a PSO system, each partcle s flown through the multdmensonal search space, adjustng ts poston n search space accordng to ts own experence and that of neghborng partcles. The partcle therefore makes use of the best poston encountered by tself and that of ts neghbors to poston tself toward an optmal soluton [23]. The performance of each partcle s evaluated usng a predefned ftness functon, whch encapsulates the characterstcs of the optmzaton problem [24]. Each partcle s assocated wth velocty that ndcates where the partcle s travelng. Let k be a tme nstant. The new partcle poston s computed by addng the velocty vector to the current poston s = s + v (1) k + 1 k k + 1 when s and v denote a partcle poston and ts correspondng velocty n a search space, respectvely. Beng s k partcle poston, = 1,,ρ, at tme nstant k, v k + 1 new velocty (at tme k+1) and ρ s populaton sze. The velocty update equaton s gven by: ( ) ( ) g v = k 1 wk v + k c1r 1 p k s + + k c2r2 p k sk (2) where k s the current step number, w s the nerta weght, c1 and c2 are the acceleraton constants, and r1, r2 are element from two random sequences n the range [, 1]. The current poston of the partcle s determned by reached, s k ; pk s the best one of the solutons ths partcle has g pk s the best one of the solutons all the partcles have reached [22].

5 -667- The varable w [25] s responsble for dynamcally adjustng the velocty of the partcles, so t s responsble for balancng between local and global search, hence requrng fewer teratons for the algorthm to converge. A low value of nerta weght mples a local search, whle a hgh value leads to a global search. Applyng a large nerta weght at the start of the algorthm and makng t decay to a small value through the PSO executon makes the algorthm search globally at the begnnng of the search, and search locally at the end of the executon [23]. The followng weghtng functon w s used n Eq. (2): wmax wmn w= wmax k k Generally, the value of each component n v can be clamped to the range [ vmax,vmax] control excessve roamng of partcles outsde the search space [17]. After calculatng the velocty, the PSO algorthm performs repeated applcatons of the update equatons above untl a specfed number of teraton has been exceeded, or untl the velocty updates are close to zero [23]. The PSO algorthm s presented n detal n Table 1. Fgure 1 shows the update systems of the PSO algorthm. Fgure 2 shows the flow dagram of the PSO algorthm used. In PSO, the nertal weght w, the constant c1 and c2, the number of partcles Npart and the maxmum speed of partcle summarze the parameters to synchronze for ther applcaton n a gven problem. An exhaustve tral and error procedure was appled for tunng the PSO para meters. Frstly, the effect of w was analyzed for values from.1 to.9. Fgure 3a shows the values of w that favored the search of the partcles and accelerated the convergence. Next, the effect of Npart was analyzed for values from 1 to 1 partcles n the swarm. Fgure 3b max (3) Table 1. Scheme of the PSO algorthm development n ths study. Step Descrpton Intalze algorthm: populaton sze and number of weghts and bases. 1 Set constants: kmax, vmax, w, c1, c2 n 2 Randomly ntalze the swarm postons s R for = 1,, ρ 3 Randomly ntalze partcle veloctes v for = 1,, ρ 4 Set k = 1 Evaluate functon value F k usng desgn space coordnates s k : 5 If F F then F = F, p = s k best best g If Fk Fbest then g g Fbest = Fk, pk = sk 6 If stoppng condton s satsfed then stop algorthm 7 Update all partcle veloctes v k for = 1,, ρ 8 Update all partcle postons s k for = 1,, ρ 9 Otherwse set k = k + 1goes to step 5 k k k

6 -668- g p k v k s k wv k c r p 1 1( s k k ) v k + 1 p k s k + 1 c r p s g 2 2 ( k k ) Fgure 1. PSO poston and velocty update. shows that the best populaton to solve the problem s of 25 partcles. Table 2 shows the selected parameters for the PSO algorthm. 3. Equatons of vapor lqud equlbrum As known, the phase equlbrum problem to be solved conssts of the calculaton of some varables of the set T P x y (temperature, pressure, lqud phase concentraton and vapor-phase concentraton, respectvely), when some of them are known. For a vapor lqud mxture n thermodynamc equlbrum, the temperature and the pressure are the same n both phases, and the remanng varables are defned by the materal balance and the fundamental equaton of phase equlbrum [26]. The applcaton of ths fundamental equaton requres the use of thermodynamc models whch normally nclude bnary nteracton parameters. These bnary parameters must be determned usng expermental data for bnary systems. Theoretcally, once these bnary parameters are known one could predct the behavor of multcomponent mxtures usng standard thermodynamc relatons and thermodynamcs models [27]. The fundamental equaton of vapor lqud equlbrum can be expressed as the equalty of fugactes of each component n the mxture n both phases [26]: f L = f (4) where the superscrpts L and V represent lqud and vapor, respectvely. V

7 -669- Start Intalze algorthm constants: k max, v max, w, c 1, c 2 Set k = 1, = 1 Intalze partcles wth random poston s k and velocty vectors v k. No Set = 1 k = k + 1 Yes > total number of partcles? Evaluate objectve functon f(s) for partcle. Update partcle and swarm best values F, F. Update velocty best g best vk and poston s k. = + 1 No Stoppng crteron satsfed? Yes The best partcles are found. Stop PSO Fgure 2. Flow dagram of the PSO algorthm used n ths study.

8 (a) 1 F Number of teratons 2 15 (b) 1 F Number of teratons Fgure 3. Convergence graphcs. (a) Determnaton of the best values for w as:.3( ),.5(- - -),.7( ),.9 ( ). (b) Effect of Npart for: 25(- - -), 5( ), 1( ).

9 -671- Table 2. Parameters used n the PSO algorthm. PSO Parameter Value Number of partcles n swarm (Npart) 25 Number of teratons (kmax) 1 Cogntve component (c1) Socal component (c2) Maxmum velocty (vmax) 12 Mnmum nerta weght (wmn).5 Maxmum nerta weght (wmax).7 The fugacty of a component n the vapor phase s usually expressed through the fugacty coeffcent φ : V f = y φ P (5) V V And the fugacty of a component n the lqud phase s expressed through ether the L fugacty coeffcent φ or the actvty coeffcentγ : f = xφ P (6) L L f = x γ f (7) L In these equatons, y s the mole fracton of component n the vapor phase, x s the mole fracton of component n the lqud phase, and P s the pressure. The fugacty s related to the temperature, the pressure, the volume and the concentraton though a standard thermodynamc relaton [28]. If the fugacty coeffcent s used n both phases, the method of soluton of the phase equlbrum problem s known as the equaton of state method. Then, equaton of state (EoS) and a set of mxng rules are needed to express the fugacty coeffcent as functon of the temperature, the pressure and the concentraton [26]. Modern EoS methods nclude an excess Gbbs free energy model (G E ) n the mxng rules of the EoS, gvng orgn to the so called equaton of state + G E model [27]. Ths means that an actvty coeffcent model (γ) s used to descrbe the complex lqud phase, and the fugacty coeffcent (φ) s calculated usng a smple equaton of state. If the fugacty coeffcent s used for the vapor phase and the actvty coeffcent s used for the lqud phase the equlbrum problem s known as the gamma ph method (γ φ) [26].

10 Equaton of state method From the relaton between the fugacty, the Gbbs free energy, and an EoS, the fugacty n a vapor can be calculated as: ( T, P, y ) V f ln = lnφ yp 1 V RT P V lnφ = dv ln Z RT (9) V = V N T, V, N j where V s the total volume, and Z PV ( RT ) 1 (8) = s the compressblty factor calculated from as EoS, and V s the molar volume of the mxture [27]. The most common EoS used for the correlaton of phase equlbra n mxtures at hgh and low pressure are the cubc equatons derved from van der Waals EoS [29]; among these, the Peng Robnson equaton has proven to combne the smplcty and accuracy requred for the predcton and correlaton of volumetrc and thermodynamc propertes of fluds [3]. The Peng Robnson EoS was expressed as follows [3]: wth RT a P = + V b V V +b +b V b ( ) ( ) 2 2 R Tc a = α Tr P RTc b = P c c ( ) ( ) 1 ( 1 ) 2 α T r = +κ Tr (1) (11) (12) (13) 2 κ = ω.26992ω (14) where Tr = T/Tc s the reduced temperature. In ths form, the Peng Robnson EoS s completely predctve once the constants (crtcal temperature Tc, crtcal pressure Pc, and acentrc factor ω) are gven. Consequently, ths equaton s a two-parameter EoS (a and b) that depends upon the three constants (Tc, Pc, and ω) [27]. For mxtures, the parameters a and b are expressed as functons of the concentraton of the dfferent components n the mxture, through the so-called mxng rules [26]. Untl recent years, most of the applcatons of EoS to mxtures consdered the use of the classcal van der

11 -673- Waals mxng rules, wth the ncluson of an nteracton parameter for the force constant a and volume constant b. The Peng Robnson EoS for a mxture s: RT am P = + V b V V +b +b V b ( ) ( ) m m m m The classcal van der Waals mxng rules are [27]: a = x x a m j j j b = x x b m j j j (15) (16) (17) and the combnng rules for aj and bj, wth nteracton parameters for the force and volume constants, are: ( 1 ) a = a a k (18) j j j b + bj b j = 2 ( 1 lj ) (19) The parameters kj and lj n the above combnng rules for the equaton of state are usually calculated by regresson analyss of expermental phase equlbrum data. The modern equaton of state ncludes an excess Gbbs free energy model n the mxng rules of the EoS. Thus, the connecton between equatons of state + excess Gbbs free energy, seem to be the most approprate for modelng complex mxtures [29]. The Wong Sandler mxng rule s an example of these types of mxng rules, and can be summarzed as follows [31]: b = m N 1 N j j j N E x A a ( x) a x x b RT b RT ΩRT a 1 a a b = b +b k RT 2 RT j j ( j ) ( 1 j ) (2) (21) N a m = b m + b ( x) E x a A (22) Ω In these equatons am and bm are the equaton of state constants wth kj as adjustable E parameter, Ω= for the PR EoS, and A ( x) E E E that A ( x) A ( x) ( ) G x. s calculated assumng

12 -674- For a bnary mxture: a a a x b + 2x x b + x b b = m RT 1 RT 12 RT 2 E x G 1a1 x2a2 ( x) 1 b RT b RT ΩRT a 1 a a b = b +b k RT 2 RT 1 2 ( ) ( 1 ) E x a x a G b RT b RT Ω 1 1 a m = b m ( x) (23) (24) (25) E The excess Gbbs free energy G ( ) E approprate lqud phase model. In ths work, G ( ) x n the mxng rules s calculated usng an x has been calculated usng the van Laar model that has been shown to perform well n hgh pressure phase equlbrum calculatons [15]. E The van Laar model for ( ) G x s descrbed by the followng equaton [26]: N N x j Aj x x j Aj j j 1 1 x j x x j A j + ( 1 x ) x x j Aj j j E N G = x N N RT 2 (26) For a bnary mxture, the model reduces to: E G A12 x1x2 = RT A x + x A21 Thus, the problem s reduced here to determne the nteracton parameters A12, A21, and the k12 parameter ncluded n the combnng thermodynamc model (PR-WS-VL), usng avalable hgh pressure T P x data of vapor lqud phase equlbrum of complex mxtures. These optmal nteracton parameters were determned by mnmzng the followng objectve functon n data regresson, usng a hybrd algorthm based on partcle swarm optmzaton and ant colony optmzaton: D = 1 (27) ND calc exp 1 P P mn F = N P exp (28)

13 -675- where ND s the number of ponts n the expermental data set and P s the pressure of the onc lqud n the vapor phase, the superscrpt denotes the expermental (exp) data pont and calculated (calc) values. Fgure 4 shows the flow dagram of the total algorthm development for the vapor lqud equlbrum modelng. Forty-sx bnary vapor lqud phase systems contanng supercrtcal carbon doxde and mdazolum-based onc lquds were consdered n ths study. The anons: bs(trfluoromethylsulfonyl)mde ([Tf2N]), hexafluorophosphate ([PF6]), tetrafluoroborate ([BF4]), ethyl sulfate ([EtSO4]), dcyanamde ([DCA]), ntrate ([NO3]), trfuoromethanesulfonate ([TfO]), and trs(trfluoromethylsulfonyl)methde ([methde]), are the ones presentng the hghest supercrtcal carbon doxde solublty. Although both anon and caton nfluence the carbon doxde solublty, the anon has the strongest nfluence [32]. And the most common 1-alkyl-3-methylmdazolum catons were used: 1-ethyl-3- methylmdazolum ([C2mm]), 1-butyl-3-methylmdazolum ([C4mm]), 1-pentyl-3- methylmdazolum ([C5mm]), 1-hexyl-3-methylmdazolum ([C6mm]), and 1-octyl-3- methylmdazolum ([C8mm]). Table 3 shows the thermodynamc propertes of the substances nvolved n the study. In ths Table, Tc s the crtcal temperature, Pc s the crtcal pressure, and ω s the acentrc factor. The data for the onc lquds were taken from the lterature [33]. The data for supercrtcal fluds were taken from Daubert et al [34]. The detals of the expermental vapor lqud equlbrum data taken from references [35-4] are presented n Table 4. As seen n the Table, the temperature and pressure ranges are narrow and go from 313K to 333K and from to 43 MPa, respectvely. 5. Results and dscusson The PR-WS-VL model and the PSO algorthm were used to calculate k12, A12 and A21, and P by mnmzng the Eq. (28), and consderng the absolute devatons between expermental and calculated values of bubble pont n the vapor lqud phase of the onc lquds on the supercrtcal carbon doxde. In order to provde a substantal margn of safety, the range for the nteracton parameters (A12 and A21 for VL model for the excess Gbbs free energy) was defned as [ 5, 5]. Ths wde range was based on physcal consderatons [27], and s extremely lkely that t wll contan the optmal parameter values. In addton, the range for the WS parameter k12 wth theoretcal bases [31] was defned as [ 1, 1]. Fgure 5 shows

14 -676- the nteracton parameters determned wth the proposed algorthm and based on the mnmzaton Start Specfy the parameter for PSO. Specfy mole fractons x and T. Generate ntal populaton. Possble values of the actvty coeffcent model. Guess bubble-pont pressure P. Guess set of K = y /x for all components. k = 1 y = K x Tme-doman smulaton. Fnd the ftness of each partcle n the current populaton. Calculate f L for all components. Calculate f V for all components. k = k + 1 k > k max? Yes The best partcles are found. No Update PSO operators: poston and velocty. Stop PSO Optmum parameters for PR-WS-VL model are obtaned

15 -677- Fgure 4. Flow dagram of the total algorthm used for the vapor lqud equlbrum modelng. Table 3. Thermodynamc propertes of the substances nvolved n ths study. Substance Tc (K) Pc (MPa) ω [C2mm][Tf2N] [C4mm][Tf2N] [C5mm][Tf2N] [C6mm][Tf2N] [C8mm][Tf2N] [C4mm][PF6] [C8mm][PF6] [C4mm][BF4] [C8mm][BF4] [C2mm][EtSO4] [C4mm][DCA] [C4mm][NO3] [C4mm][TfO] [C4mm][methde] CO CHF of bubble pressure. These results show that the pressures of the onc lquds n the vapor phase were correlated wth low devatons between expermental and calculated values (devatons are below 4%). Results of the modelng are presented n Tables 5 to 7. Table 5 shows the optmum values and devatons calculated for the bnary nteracton parameters k 12, A 12 and A 21 at 313K (19 systems). Table 6 shows the optmum values and devatons calculated for the bnary nteracton parameters k 12, A 12 and A 21 at 323K (8 systems). Table 7 shows the optmum values and devatons calculated for the bnary nteracton parameters k 12, A 12 and A 21 at 333K (19 systems). From the results contaned n these Tables, s possble to determne the capablty of the algorthm to correlate the expermental data accordng to the anon type: [Tf2N] (1.5%) ~ [methde] (1.5%) < [PF6] (2.%) < [EtSO4] (2.2%) < [NO3] (2.4%) < [DCA] (2.8%) < [BF4] (2.9%) < [TfO] (3.7%). And for the case of caton type: [C5mm] (1.5%) < [C2mm] (1.6%) ~ [C6mm] (1.6%) < [C8mm] (2.%) < [C4mm] (2.4%). One reason for the better results s the electon of the thermodynamc model selected. In partcular the parameters of the van Laar model ncluded n the Wong Sandler mxng rules. Among the many cubc EoS of van der Waals type nowadays avalable, the one proposed by Peng Robnson EoS s wdely used because of ts smplcty and flexblty [27]. Ths equaton has proven to combne the smplcty and accuracy requred for the predcton and correlaton of flud

16 -678- propertes, n partcular of phase equlbra [3,31]. The effect of the uncertanty of the crtcal propertes n the phase equlbra calculatons usng PR-EoS has been nvestgated for several Table 4. Detals on the phase equlbrum data of all systems used n ths study. Component (1) Component (2) T (K) xscf P (MPa) Ref CHF3 [C2mm][PF6] [35] [C4mm][PF6] [36] CO2 [C2mm][Tf2N] [37] [C4mm][Tf2N] [38] [C5mm][Tf2N] [37] [C6mm][Tf2N] [39] [38] [C8mm][Tf2N] [38] [C4mm][NO3] [39] [38] [C4mm][PF6] [39] [38] [4] [C8mm][PF6] [39] [C4mm][BF4] [38] [C8mm][BF4] [39] [C4mm][DCA] [38] [C4mm][TfO] [38] [C2mm][EtSO4] [39] [C4mm][methde] [38] systems, but the general trend and curvature of the phase equlbrum curve s not altered [41]. The nteracton parameters represent the functonalty of the constants of the equaton wth the concentraton. It has been recognzed that van der Waals mxng rules wth one or two

17 -679- parameters do not gve good results for systems complex [42]. The Wong Sandler mxng rules have shown to be successful n these applcatons. In other works to mprove the F A A Fgure 5. Devatons of the bnary nteracton parameters estmated by mnmzaton of the objectve functon wth PSO algorthm. predctons n mxtures, a thrd nteracton parameter has been ntroduced and has been shown that these mxng rules allow an accurate representaton that when the van der Waals mxng rules are used [29]. Fgure 6 shows the varaton of the bnary nteracton parameters as a functon of the absolute temperature. It can be observed the behavor of the parameters ncluded n the PR-WS-VL model. The parameter k12 decreases wth the temperature n most of the cases studed. For the mxng rules, parameter A12 shows a smooth behavor, and A21 shows a dynamcal behavor. Ths s not unusual n complex systems and n partcular n mxtures contanng onc lquds. Fgure 7 shows the nner behavor among the parameters. The dfferent nfluence of the parameters and ther range of varaton provde the PR-WS-VL

18 -68- model greater flexblty n the sense that the model can better capture the dfferent behavor of the mxtures studed. Table 5. Optmum values and devatons calculated for the nteracton parameters at 313K. No. Comp. (1) Comp. (2) N D k 12 A 12 A 21 F 1 CO 2 [C 2mm][Tf 2N] CO 2 [C 4mm][Tf 2N] CO 2 [C 5mm][Tf 2N] CO 2 [C 6mm][Tf 2N] CO 2 [C 6mm][Tf 2N] CO 2 [C 8mm][Tf 2N] CO 2 [C 4mm][NO 3] CO 2 [C 4mm][NO 3] CO 2 [C 4mm][PF 6] CO 2 [C 4mm][PF 6] CO 2 [C 4mm][PF 6] CO 2 [C 8mm][PF 6] CO 2 [C 4mm][BF 4] CO 2 [C 8mm][BF 4] CO 2 [C 4mm][DCA] CO 2 [C 4mm][TfO] CO 2 [C 2mm][EtSO 4] CO 2 [C 4mm][methde] CHF 3 [C 2mm][PF 6] Table 6. Optmum values and devatons calculated for the nteracton parameters at 323K. No. Comp. (1) Comp. (2) N D k 12 A 12 A 21 F 1 CO 2 [C 2mm][Tf 2N] CO 2 [C 5mm][Tf 2N] CO 2 [C 4mm][NO 3] CO 2 [C 4mm][PF 6] CO 2 [C 8mm][PF 6] CO 2 [C 8mm][BF 4] CO 2 [C 2mm][EtSO 4] CHF 3 [C 4mm][PF 6]

19 -681- Table 7. Optmum values and devatons calculated for the nteracton parameters at 333K. No. Comp. (1) Comp. (2) N D k 12 A 12 A 21 F 1 CO 2 [C 2mm][Tf 2N] CO 2 [C 4mm][Tf 2N] CO 2 [C 4mm][Tf 2N] CO 2 [C 5mm][Tf 2N] CO 2 [C 6mm][Tf 2N] CO 2 [C 6mm][Tf 2N] CO 2 [C 8mm][Tf 2N] CO 2 [C 4mm][NO 3] CO 2 [C 4mm][NO 3] CO 2 [C 4mm][PF 6] CO 2 [C 4mm][PF 6] CO 2 [C 4mm][PF 6] CO 2 [C 8mm][PF 6] CO 2 [C 4mm][BF 4] CO 2 [C 8mm][BF 4] CO 2 [C 4mm][DCA] CO 2 [C 4mm][TfO] CO 2 [C 2mm][EtSO 4] CO 2 [C 4mm][methde] A comparson was made between of the results obtaned wth the PSO algorthm and results obtaned wth Levenberg Marquart algorthm (LM). Note that, LM [43] s commonly used n these problems. Fgure 8 shows the average pressure devatons found wth PSO and LM for all onc lquds consdered n ths study. As s observed n the fgures, the best method to estmate the vapor lqud equlbrum of the systems used s the PSO algorthm. Thus, the results show that the applcaton of PSO algorthm on thermodynamc model (PR-WS-VL), was crucal, and that the proposed algorthm s a good tool to optmze the nteracton parameters to descrbe the vapor lqud equlbrum of several systems contanng supercrtcal fluds and onc lquds at hgh-pressures.

20 (a) (b) (c) k 12 A 12 A T T T Fgure 6. Varaton of the bnary nteracton parameters as a functon of the absolute temperature. (a) Wong-Sandler parameter k12, (b) van Laar parameter A12, (c) van Laar parameter A21.

21 k A A Fgure 7. Inner behavor among the parameters of the PR-WS-VL model. 6. Conclusons In ths study, hgh-pressure vapor lqud equlbrum data of bnary mxtures contanng supercrtcal fluds and mdazolum onc lquds were correlated usng a thermodynamc model optmzed wth a partcle swarm algorthm. The Peng Robnson (PR) equaton of state the Wong Sandler (WS) mxng rules ncludng the van Laar (VL) model for the excess Gbbs free energy, were used as thermodynamc model. Forty-sx bnary systems taken from lterature were selected for ths study, and the optmzaton algorthm was used to determne the bnary nteracton parameters of each system. The algorthm was development to mnmze the dfference between calculated and expermental bubble pressure. Based on the results and dscusson presented n ths study, the followng man conclusons were derved: () PSO algorthm s approprate to descrbe the vapor-lqud equlbrum of bnary systems contanng supercrtcal fluds and onc lquds; () the low

22 -684- F 1 8 (a) System (Table 5) F (b) System (Table 6) F (c) System (Table 7) Fgure 8. Comparson between PSO ( ) and LM ( ) optmzatons used n the VLE modelng. (a) Systems at 313K, (b) systems at 323K, and (c) systems at 333K. In these fgures, the systems are lsted as n Table 5 to 7.

23 -685- devatons obtaned wth the proposed PSO algorthm ndcate that t can estmate the bnary nteracton parameters wth better accuracy than other algorthms avalable n the lterature; () the values calculated wth the PSO algorthm are beleved to be accurate enough for engneerng calculatons, among other uses. Acknowledgments: The author thank the Drecton of Research of the Unversty of La Serena (DIULS), and the Department of Physcs of the Unversty of La Serena (DFULS) for the specal support that made possble the preparaton of ths paper. References [1] J. A. Lazzús, Estmaton of densty as a functon of temperature and pressure for mdazolum based onc lquds usng a multlayer net wth partcle swarm optmzaton, Int. J. Thermophys. 3 (29) [2] J. A. Lazzús, Hybrd partcle swarm ant colony algorthm to descrbe the phase equlbrum of systems contanng supercrtcal fluds wth onc lquds, Commun. Comput. Phys. 14 (213) [3] A. Hentz, Recent developments n thermodynamcs and thermophyscs of non aqueous mxtures contanng onc lquds: a revew, J. Chem. Thermodyn. 35 (25) [4] T. Wang, C. Peng, H. Lu, Y. Hu, J. Jang, Equaton of state for the vapor lqud equlbra of bnary systems contanng mdazolum based onc lquds, Ind. Eng. Chem. Res. 46 (27) [5] R. Kato, J. Gmehlng, Measurement and correlaton of vapor lqud equlbra of bnary systems contanng the onc lquds [EMIM][(CF3SO2)2N], [BMIM][(CF3SO2)2N], [MMIM][(CH3)2PO4] and oxygenated organc compounds respectvely water, Flud Phase Equlb. 231 (25) [6] L. A. Blanchard, D. Hancu, E. J. Beckman, J. F. Brennecke, Green processng usng onc lquds and CO2, Nature 399 (1999) [7] J. A. Lazzús, J. Marín, Actvty coeffcent model to descrbe sothermal vapor-lqud equlbrum of bnary systems contanng onc lquds, J. Eng. Thermophys. 19 (21) [8] A. Scurto, S. Ak, J. Brennecke, CO2 as a separaton swtch for onc lqud/organc mxtures, J. Am. Chem. Soc. 124 (22) [9] A. M. Scurto, S. N. V. K. Ak, J. F. Brennecke, Carbon doxde nduced separaton of onc lquds and water, Chem. Commun. 5 (23)

24 -686- [1] V. H. Álvarez, M. Aznar, Applcaton of a thermodynamc consstency test to bnary mxtures contanng an onc lqud, Open Thermodyn. J. 2 (28) [11] G. Hong, J. Jacquemn, M. Deetlefs, C. Hardacre, P. Husson, M. F. Costa Gomes, Solublty of carbon doxde and ethane n three onc lquds based on the bs{(trfluoromethyl)sulfonyl}mde anon, Flud Phase Equlb. 257 (27) [12] R. Bogel-Łukask, D. Matkowska, E. Bogel Łukask, T. Hofman, Isothermal vaporlqud equlbra n the bnary and ternary systems consstng of an onc lquds, 1- propanol and CO2, Flud Phase Equlb. 293 (21) [13] J. Yang, C. Peng, H. Lu, Y. Hu, Calculaton of vapor lqud and lqud lqud phase equlbra for systems contanng onc lquds usng a lattce model, Ind. Eng. Chem. Res. 45 (26) [14] J. A. Lazzús, Thermodynamc modellng based on partcle swarm optmzaton to predct phase equlbrum of bnary systems contanng onc lquds, J. Mol. Lq. 186 (213) [15] J. A. Lazzús, A. A. Pérez Ponce, L. Palma Chlla, Applcaton of partcle swarm optmzaton to model phase equlbrum of complex mxtures, Flud Phase Equlb. 317 (212) [16] J. A. Lazzús, M. Rvera, Gas sold phase equlbrum of bosubstances by two bologcal algorthms, Rev. Mex. Fís. 59 (213) [17] J. A. Lazzús, Optmzaton of actvty coeffcent models to descrbe vapor lqud equlbrum of (alcohol + water) mxtures usng a partcle swarm algorthm, Comput. Math. Appl. 6 (21) [18] J. A. Lazzús, Optmzaton of a cubc equaton of state and van der Waals mxng rules for modelng the phase behavor of complex mxtures, Rev. Mex. Fís. 58 (212) [19] J. Kennedy, R. C. Eberhart, Y. Sh, Swarm Intellgence, Academc Press, San Dego, 21. [2] M. Schwaab, E. Chalbaud, J. L. Montero, J. C. Pnto, Nonlnear parameter estmaton through partcle swarm optmzaton, Chem. Eng. Sc. 63 (28) [21] V. H. Alvarez, R. Larco, Y. Ianos, M. Aznar, Parameter estmaton for VLE calculaton by global mnmzaton: the genetc algorthm, Braz. J. Chem. Eng. 25 (28) [22] R. C. Eberhart, J. Kennedy, A new optmzer usng partcle swarm theory, n: T. Nsho, K. Ahe, N. Kato, H. Ota (Eds.), Proceedngs of the Sxth Internatonal Symposum on Mcro Machne and Human Scence, IEEE, Pscataway, 1995, pp [23] Y. Jang, T. Hu, C. C. Huang, X. Wu, An mproved partcle swarm optmzaton algorthm, Appl. Math. Comput. 193 (27)

25 -687- [24] Y. Da, G. Xurun, An mproved PSO-based ANN wth smulated annealng technque, Neurocomputng 63 (25) [25] Y. Sh, R. C. Eberhart, A modfed partcle swarm optmzer, n: P. K. Smpson, D. B. Fogel (Eds.), The 1998 IEEE Internatonal Conference on Evolutonary Computaton Proceedngs, IEEE Press, Pscataway, 1998, pp [26] S. M. Walas, Phase Equlbra n Chemcal Engneerng, Butterworth, Storeham, [27] H. Orbey, S. I. Sandler, Modelng Vapor Lqud Equlbra. Cubc Equatons of State and Ther Mxng Rules, Cambrdge Unv. Press, Cambrdge, [28] J. M. Prausntz, R. N. Lchtenthaler, E. Gomes de Azevedo, Molecular Thermodynamcs of Flud Phase Equlbra, Prentce Hall, New Jersey, [29] F. M. Maa, I. Tsvntzels, O. Rodrguez, E. A. Macedo, G. M. Kontogeorgs, Equaton of state modellng of system wth onc lquds: lterature revew and applcaton wth the cubc plus assocaton (CPA) model, Flud Phase Equlb. 332 (212) [3] D. Y. Peng, D. B. Robnson, A new two-constant equaton of state, Ind. Eng. Chem. Fundam. 15 (1976) [31] D. S. Wong, S. I. Sandler, A theoretcally correct mxng rule for cubc equatons of state, AIChE J. 38 (1992) [32] P. Wassersched, T. Welton, Ionc Lquds n Synthess, Wley VCH, Wenhem, 28. [33] J. O. Valderrama, W. W. Sanga, J. A. Lazzús, Crtcal propertes, normal bolng temperature, and acentrc factor of another 2 onc lquds, Ind. Eng. Chem. Res. 47 (28) [34] T. E. Daubert, R. P. Danner, H. M. Sbul, C. C. Stebbns, Physcal and Thermodynamc Propertes of Pure Chemcals. Data Complaton, Taylor & Francs, London, [35] A. Sharat, C. J. Peters, Hgh pressure phase equlbra of systems wth onc lquds, J. Supercrt. Fluds 25 (23) [36] A. Sharat, K. Gutkowsk, C. J. Peters, Comparson of the phase behavor of some selected bnary systems wth onc lquds, AIChE J. 51 (25) [37] P. J. Carvalho, V. H. Alvarez, J. J. B. Machado, J. Pauly, J. L. Dardon, I. M. Marrucho, M. Aznar, J. A. P. Coutnho, Hgh pressure phase behavor of carbon doxde n 1-alkyl-3-methylmdazolum bs(trfluoromethylsulfonyl)mde onc lquds, J. Supercrt. Fluds 48 (29) [38] S. N. V. K. Ak, B. R. Mellen, E. M. Saurer, J. F. Brennecke, Hgh pressure phase behavor of carbon doxde wth mdazolum-based onc lquds, J. Phys. Chem. 18B (24) [39] L. A. Blanchard, Z. Gu, J. F. Brennecke, Hgh pressure phase behavor of onc lqud/co2 systems, J. Phys. Chem. 15B (21)

26 -688- [4] A. Pérez Salado Kamps, D. Tuma, J. Xa, G. Maurer, Solublty of CO2 n the onc lqud [bmm][pf6], J. Chem. Eng. Data 48 (23) [41] P. Coutskos, K. Magoulas, G. M. Kontogeorgs, Predcton of sold gas equlbra wth the Peng Robnson equaton of state, J. Supercrt. Fluds 25 (23) [42] M. A. Trebble, P. R. Bshno, Extenson of the Trebble Bshno equaton of state to flud mxtures, Flud Phase Equlb. 4 (1988) [43] M. Relly, Computer Programs for Chemcal Engneerng Educaton, Sterlng Swft, Manchaca, 1972.