Prediction of phase equilibria in waterralcoholralkane systems

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1 Fluid Phase Equilibria Prediction of phase equilibria in waterralcoholralkane systems Epaminondas C. Voutsas ), Iakovos V. Yakoumis, Dimitrios P. Tassios Laboratory of Thermodynamics and Transport Phenomena, Department of Chemical Engineering, National Technical UniÕersity of Athens, 9, Heroon Polytechniou Str., Zographou Campus, 15780, Athens, Greece Received 22 March 1998; accepted 9 October 1998 Abstract The Cubic Plus Association Equation of State CPA EoS. is applied to the prediction of Vapor Liquid Equilibrium VLE. and Liquid Liquid Equilibrium LLE. in ternary associating mixtures containing water, alcohols and alkanes. The appropriate set of combining rules for the cross-association energy and volume parameters of the EoS is identified by evaluating the performance of four such sets in the correlation of VLE and LLE for waterralcohol mixtures. Using only one interaction parameter per binary mixture, in the physical part of the EoS, CPA gives very satisfactory predictions of ternary VLE and LLE. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Associating fluids; Equation of state; Vapor liquid equilibrium; Liquid liquid equilibrium 1. Introduction Knowledge of phase equilibria VLE and LLE. in waterralcoholralkane mixtures is very important for many practical applications such as the extraction of alcohols from aqueous solutions using near critical light hydrocarbon as solvents w1,2 x; or in the petroleum industry, since alcohols are widely used as fuel additives or to inhibit hydrate formation. Mixtures containing associating compounds are also of great interest from the theoretical point of view for testing molecular as well as statistical models. Species that form hydrogen bonds often exhibit unusual thermodynamic behavior. In pure fluids, strong attractive interactions between like molecules result in the formation of molecular clusters that considerably affect their thermodynamic properties. In mixtures, hydrogen bonding interactions can occur between molecules of the same species self-association or between molecules of different species cross-association or solvation. ) Corresponding author. Tel.: q ; fax: q ; evoutsas@orfeas.chemeng.ntua.gr r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S

2 152 ( ) E.C. Voutsas et al.rfluid Phase Equilibria Classical engineering models, such as UNIFAC or cubic equations of state, cannot accurately represent phase equilibria in mixtures containing associating compounds w3,4x because they do not take explicitly into account the special interactions encountered in them. In the last few years, many equations of state that take explicitly into account hydrogen bonding have been developed w2,5 7 x. One of these is the Cubic Plus Association CPA., which combines the SRK EoS with the perturbation theory of Wertheim which is also used in SAFT wx 6. In a series of papers, CPA has been successfully applied to pure associating fluids as well as mixtures of associating fluids with inert compounds w3,8 10 x. In this study, we extend CPA to systems involving cross-association and proceed to investigate its predictive capabilities in ternary mixtures containing water, alcohols and alkanes. Only binary interaction parameters are used, which are determined by fitting to binary VLE and LLE data. 2. The CPA EoS The pressure equation of the CPA EoS is given by the following expression: RT a RT 1 1 E X A j Ps y q ÝxiÝrjÝ y, 1 A j. Vyb V Vqb. V i j A j X 2 Eri where the physical term is that of the SRK EoS and the association term is taken from SAFT wx 6. The A summation is over all association sites and the mole fraction of molecules not bonded at site A X i. is defined by: ž / y1 Ai Bj AiBj X s 1qrÝÝxX j D. 2 j B j In the above equation, D A i B j where g AB i j AB i j AB i j ž / RT is the association strength given by: D sg exp y1 b b, 3. is the radial distribution function defined by: b 2y 4V gs b 21y 4V ž / Finally, the energy parameter of the EoS a is defined using a Soave-type temperature dependency: ( r ž ž // asa 1qc 1y T, 5 and the co-volume parameter b. is assumed to be temperature-independent.

3 ( ) E.C. Voutsas et al.rfluid Phase Equilibria Table 1 CPA parameters for the components examined in this work Component b lrmol. a bar l rmol. c K. b 2 AB AB 0 1 water methanol ethanol i-propanol n-butanol butanol propane i-butane n-hexane The CPA EoS uses three pure-compound parameters for nonassociating inert. compounds, a 0, c1 and b, while for associating compounds, two additional parameters are needed: the association energy AB. AB of interaction between sites A and B on a molecule and the parameter b, which is an association volume parameter. Pure-compound parameters are determined by fitting experimental saturated vapor pressure and liquid volume data. For inert compounds, these parameters can be also estimated using their critical properties and acentric factor as in the original SRK EoS. The latter represents an advantage of CPA over SAFT in cases where supercritical compounds are involved. The pure-compound parameters used in this study are presented in Table Phase equilibria calculations in binary systems 3.1. Systems containing only one self-associating compound and one inert compound Such systems involved in this study are the alcoholralkane and waterralkane binary systems. The alcohol molecules are treated as two-site molecules, while water as a four-site one. The extension of the CPA EoS to such mixtures requires explicit mixing rules only for the parameters a and b of the physical part, while the extension of the association term is straightforward. The classical van der Waals one-fluid mixing rules are used for the parameters a and b with a single binary interaction coefficient k. in the a parameter. A detailed study of the performance of CPA in the correlation and prediction of phase equilibria in alcoholralkane and waterralkane binary mixtures has been presented elsewhere w3,9,10 x. Here we consider only typical systems pertinent to the objective of this study. Thus, Table 2 and Figs. 1 and 2 present results for mixtures containing alcohols and light hydrocarbons at temperatures near the critical ones of the hydrocarbons, where the latter have been proved to be very promising solvents for the dehydration and recovery of alcohols wx 1. As it can be seen, CPA gives excellent correlation of the VLE data and it also accurately predicts the azeotropic behavior of the mixture ethanolri-butane Fig. 2., which hinders the application of i-butane as solvent for the supercritical extraction of ethanol from aqueous solutions. Results, finally, for a waterrlight hydrocarbon system is shown in Fig. 3 for the waterrpropane system.

4 154 ( ) E.C. Voutsas et al.rfluid Phase Equilibria Table 2 Correlation performance of the CPA EoS in alcoholrlight hydrocarbon binaries a Alcohol Light hydrocarbon Ref. T K k D P % D y 100 Ethanol propane w13x y y y i-butane w13x i-propanol propane w14x i-butane wx Butanol propane w15x y a D P %. is average absolute percentage error in bubble point pressure. b D y is the average absolute deviation in vapor phase mole fraction. U b 3.2. Systems containing two self-associating compounds In such systems, like the waterralcohol systems, cross-association occurs. Since the type of association we used for the four-sitertwo-site waterralcohol systems is discussed in details in Refs. Fig. 1. Correlation of VLE of the binary ethanolrpropane system. Experimental data from Ref. w15 x.

5 ( ) E.C. Voutsas et al.rfluid Phase Equilibria Fig. 2. Correlation of VLE of the ethanolr i-butane system. Experimental data from Ref. w15 x. w6,11 x, we will concentrate here to the combining rules for the cross-association energy and volume A i B j A i B parameters and b j Ai and Bj denote two different sites on two different molecules, e.g., one on the oxygen atom of the water molecule and one on the hydrogen atom on the alcohol molecule.. Different sets of combining rules have been used by different authors working with equations of state based on Wertheim s theory w7,11,12 x. In this study, four different sets of Fig. 3. VLLE calculations for the system waterrpropane. Experimental data from Ref. w25 x.

6 156 ( ) E.C. Voutsas et al.rfluid Phase Equilibria combining rules are compared with respect to their performance in the correlation of VLE and LLE in alcoholrwater mixtures: Ai q B j b A i qb B j. AB i j AB i s, b i j s, which will be referred as CR-1 set. 2 2 A i B q j AB i j AB i j A i B ii. s, b s( b b j, which will be referred as CR-2 set. 2 AB i j ' Ai Bj AB i j Ai B ( j iii s, b s b b, which will be referred as CR-3 set. AB ' i j A i B iv. D s D D j. In this approach, which has been proposed by Suresh and Elliott wx 7, no combining rules are needed for the Ai B j and b A i B j parameters. Furthermore, unlike the other three sets, it gives an analytical solution for the mole fraction of molecules not bonded at site A of the A molecule i X i.. This combining rule will be referred as CR VLE in waterr alcohol binary systems Table 3 presents VLE correlation results using the four sets of combining rules for some waterralcohol systems using a single binary interaction parameter in the physical part of the EoS. It is A i B apparent that: a the combining rule for the b j parameter is the crucial one, since the arithmetic mean CR-1. gives poor results, while the geometric one CR-2 and CR-3. gives very good ones; b. the arithmetic mean and the geometric mean combining rules for the Ai B j parameter are equivalent A i B when coupled with the geometric mean for the b j parameter CR-2 and CR-3.; and c. the CR-4 combining rule gives very good results similar to the ones obtained by the CR-2 and CR-3 sets LLE in waterr alcohol binary systems Fig. 4 presents LLE correlation results with the CPA and the CR-2, CR-3 and CR-4 combining rules for the waterrn-butanol mixture. The CR-2 and CR-3 sets give similar satisfactory results, eventhough they tend to overestimate the upper critical solution temperature probably due to the use of a temperature independent k. On the other hand, the Elliott s combining rule CR-4. strongly overestimates the n-butanol solubilities in the water-rich phase A theoretical explanation of the CR-2 set of combining rules As suggested by Prausnitz w16 x, the standard Gibbs energy of forming cross-associates can be approximated by: 1 o. o. o. Dg s ž Dgii q Dg jj / It follows from Eq. 6 that the cross-association constant K can be approximated by: ( K s K K. 7. ii jj On the other hand: DH DS ln K sy q, 8. RT R where DH and DS are the enthalpy and entropy of forming cross-associates, respectively.

7 Table 3 VLE correlation results for waterralcohol systems with the CPA EoS System Ref. NP T K CR-1 CR-2 CR-3 CR-4 k D P %. D y 100 k D P %. D y 100 k D P %. D y 100 k D P %. D y 100 U U U U Waterr w18x y y y methanol w19x y y y w19x y y y w19x y y y Waterr w18x y y y ethanol w20x y y y w20x y y y w18x y y y w21x y y y w21x y y y Waterr w22x y y y i-propanol w20x y y y w21x y y y w21x y y y OÕerall errors E.C. Voutsas et al.rfluid Phase Equilibria ( 1999 )

8 158 ( ) E.C. Voutsas et al.rfluid Phase Equilibria Fig. 4. LLE correlation of the waterr n-butanol mixture. Experimental data from Ref. w26 x. Combination of Eqs. 7. and 8. leads to the following set of combining rules for DH and DS : DHiiqDHjj DSiiqDSjj DHs and DSs Economou and Donohue w17x and Kontogeorgis et al. w8x have shown that the chemical and perturbation theories yield essentially the same expressions for the compressibility factor and the mole fraction of monomers. They have concluded that KRTAD, which after some easy algebra, leads to the following relationships: AB DS i j R AB i j DH A and e Ab. 10. RT RT Eq. 10. suggests that the arithmetic mean combining rule for the cross-association enthalpy DH. A i B is equivalent to the arithmetic mean one for the cross-association energy parameter j. and that the arithmetic mean combining rule for the cross-association entropy DS. is equivalent to the A i B geometric mean one for the cross-association volume parameter b j.. Note that the similar results Table 4 VLE prediction results in ternary mixtures with the CPA EoS U U System Ref. NP T K. D P %. D y 100 D y Waterr ethanolr propane w23x w23x w23x Waterr i-propanolr i-butane wx wx wx Waterr i-propanolr propane w24x

9 ( ) E.C. Voutsas et al.rfluid Phase Equilibria Fig. 5. Waterrpropane relative volatilities in mixtures with ethanol. Experimental data from Ref. w23 x. A i B j between arithmetic and geometric mean for CR-2 and CR-3 are due to the fact that the corresponding values differ by less than 1%. 4. Prediction of ternary phase equilibrium Phase equilibrium predictions with the CPA have been performed for ternary waterralcoholralkane mixtures using the CR-2 set of combining rules for the waterralcohol cross-associating wx Fig. 6. i-propanol distribution coefficients between water and i-butane. Experimental data from Ref. 1.

10 160 ( ) E.C. Voutsas et al.rfluid Phase Equilibria Fig. 7. Binodal curve predictions for the mixture waterrmethanolrpropane at 293 K. Experimental from Ref. w27 x. mixtures. All k values are obtained from the corresponding binary data except for waterri-butane where, in the absence of data, the value for the waterrbutane was used. Table 4 presents VLE prediction results with the CPA for the waterrethanolrpropane, waterripropanolri-butane and the waterri-propanolrpropane mixtures and typical ones in Figs. 5 and 6. Very satisfactory predictions are obtained for the three systems. Thus, Figs. 5 and 6 indicate that CPA correctly predicts the ability of the two light hydrocarbons to act as dehydrating agents: the larger than Fig. 8. Prediction of methanol distribution between water and propane. Experimental data from Ref. w27 x.

11 ( ) E.C. Voutsas et al.rfluid Phase Equilibria Fig. 9. Prediction of methanol distribution coefficients between water and n-hexane at 293 K. Experimental data from Ref. w28 x. 1 relative volatilities of water for high propane concentrations in Fig. 5; and the low distribution coefficients of i-propanol in Fig. 6 combined with the correctly predicted high values for water. Very satisfactory, finally, LLE predictions are obtained as suggested by the results for the ternary waterrmethanolrpropane, waterrmethanolrn-hexane and waterrethanolrn-hexane mixtures presented in Figs Fig. 10. Prediction of ethanol distribution coefficients between water and n-hexane at 298 K. Experimental data from Ref. w29 x.

12 162 ( ) E.C. Voutsas et al.rfluid Phase Equilibria Conclusions The performance of the CPA EoS in the prediction of VLE and LLE in mixtures containing water, alcohols and alkanes is investigated. Four different sets of combining rules for the cross-association energy and volume parameters of the EoS were evaluated on the basis of their performance in the correlation of VLE and LLE in waterralcohol mixtures. It is concluded that the combining rule for the cross-association volume parameter is the crucial rule. The geometric mean gives good results, while the arithmetic mean gives poor ones. For the cross-association energy parameter, the arithmetic mean represents the appropriate choice. Finally, the Elliott s combining rule gives good results in the correlation of VLE, but poor ones in the correlation of LLE in waterralcohol mixtures. Very satisfactory VLE and LLE prediction results are obtained for waterralcoholralkane mixtures suggesting that CPA is a reliable tool for modelling both high-pressure separation methods such as the supercritical extraction as well as low-pressure ones such as liquid liquid extraction. 6. List of symbols a a0 b c1 g Dg DH k K P R DS T Tr V X A energy parameter attractive-term parameter co-volume parameter parameter in the attractive term radial distribution function standard Gibbs energy of forming cross associates enthalpy of forming cross associates binary interaction coefficient cross-association constant pressure gas constant entropy of forming cross associates temperature reduced temperature molar volume mole fraction of the component not bonded at site A Greek letters b AB parameter in the association term of the CPA EoS D AB strength of interaction between sites A and B AB association energy of interaction between sites A and B r molar density Subscripts A, B for site A, B, on the molecule i component i j component j

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