Generalized binary interaction parameters in the Wong Sandler mixing rules for mixtures containing n-alkanols and carbon dioxide

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Fluid Phase Equilibria 234 (2005) 136 143 Generalized binary interaction parameters in the Wong Sandler mixing rules for mixtures containing n-alkanols and carbon dioxide José O.

724, La Serena, Chile c Faculty of Engineering, Chemical Engineering Department, National Engineering University, Lima, Perú Received 28 August 2004; received in revised form 23 May 2005; accepted 26

1 Fluid Phase Equilibria 234 (2005) Generalized binary interaction parameters in the Wong Sandler mixing rules for mixtures containing n-alkanols and carbon dioxide José O. Valderrama a,b,, Jack Zavaleta b,c a Faculty of Engineering, Mechanical Engineering Department, University of La Serena, Casilla 554, La Serena, Chile b Centro de Información Tecnológica, Casilla 724, La Serena, Chile c Faculty of Engineering, Chemical Engineering Department, National Engineering University, Lima, Perú Received 28 August 2004; received in revised form 23 May 2005; accepted 26 May 2005 Abstract Generalized correlations for mixing rule interaction parameters as functions of the reduced temperature and a polar parameter in n- alkanol + carbon dioxide mixtures, were obtained. Ten mixtures containing sub and supercritical carbon dioxide and one n-alkanol (from methanol to decanol) were considered in the study. The Peng Robinson equation of state has been used and the Wong Sandler mixing rules, that include a model for the excess Gibbs free energy g E, have been incorporated into the equation of state constants. In the Wong Sandler mixing rules the van Laar model for g E has been used. The experimental data were obtained from literature sources and the adjustable parameters were found by minimizing the errors between predicted and experimental data of pressure and solute concentration in the gas phase. It is shown that the proposed generalized correlations have predictive capabilities with accuracy similar to the best correlating tools available in the literature Elsevier B.V. All rights reserved. Keywords: Equations of state; Peng Robinson; Supercritical fluids; Wong Sandler mixing rules 1. Introduction Supercritical extraction processes of natural products usually require a cosolvent, a substance that serves to release the product of interest from the matrix of source materials such as limonene from lemon peels, astaxantine from micro algae, or caffeine from coffee grains, just to name some. Although water is the preferred cosolvent for the extraction of natural products, sometimes n-alkanols of low and high molecular weight are needed [1 3]. For example, the study of these mixtures is of special interest in the extraction of biomolecules with supercritical carbon dioxide [4,5], in the extraction of n-alkanols from aqueous solutions with carbon dioxide [6], and in the production of n-alkanols from syngas [7]. Corresponding author. address: citchile@entelchile.net (J.O. Valderrama). To analyze the feasibility of supercritical extraction processes, to design new processes or to analyze and simulate existing processes using carbon dioxide as the supercritical solvent, phase equilibrium properties in carbon dioxide + n-alkanols at high pressures are required. These properties must be either experimentally obtained or semi empirically correlated. Several sets of data have been presented in the literature for this type of mixtures and some correlating methods have been used [7 9]. One of the most common methods used for the correlation and prediction of phase equilibrium in mixtures is the use of equations of state (EoS). Common and industrially important EoS are the cubic equations derived from van der Waals equation of state (VdW). Among the many cubic EoS of VdW type nowadays available, the model proposed by Peng and Robinson [10] is widely used because of its simplicity and flexibility [11]. The Peng Robinson EoS can be /$ see front matter 2005 Elsevier B.V. All rights reserved. doi: /j.fluid

2 J.O. Valderrama, J. Zavaleta / Fluid Phase Equilibria 234 (2005) written in a general form as follows: P = RT V b a c α( ) (1) V (V + b) + b(v b) In this equation, a c and b are parameters specific for each substance. These parameters a c and b for pure substances are determined using the critical properties, T c and P c. Also, ( ) is a function of the reduced temperature = T/T c and of the acentric factor ω, as follows: a c = (R 2 Tc 2/P c) b = (RT c /P c ) α( ) = [1 + F(1 TR 0.5 (2) )]2 F = ω ω 2 For mixtures, the parameters a and b are also concentration dependent, dependency expressed through defined mixing rules, as described later here. During the period , most applications of EoS to mixtures considered the use of the classical van der Waals mixing rules. An interaction parameter has been introduced into the force parameter a to improve predictions of mixture properties. It has been recognized, however, that even with the use of an interaction parameter the classical VdW mixing rules give accurate results for simple fluid mixtures only. During the last 25 years, efforts have been put on extending the applicability of cubic equations of state to obtain accurate representation of phase equilibria in many industrially important mixtures. The different approaches presented in the literature, include the use of multiple interaction parameters in the mixing rules, the introduction of the local-composition concept, and the use of non-quadratic mixing rules. A recent review on EoS gives more details on these different approaches [11]. Another attractive way, which has been proposed to develop more accurate mixing rules, has been the combination of an EoS with a model for the excess Gibbs free energy (or activity coefficient model). Two main approaches have been used for applying these models. In the first approach, the link between the EoS and the excess Gibbs free energy model is done at infinite pressure [12,13]. In the second approach, the link between the EoS and the excess Gibbs free energy model is done at low or zero pressure [14]. In this work, the Peng Robinson EoS in combination with the Wong and Sandler mixing rules including different models for the excess Gibbs free energy are applied to CO 2 /n-alkanols mixtures. The study considers data for ten binary mixtures, including CO 2 with methanol, ethanol, 1-propanol, 1-butanol, 1- pentanol, 1-hexanol, 1-heptanol, 1-octanol, 1-nonanol and 1-decanol. 2. Equations of state and mixing rules used The Peng Robinson equation of state with the Wong Sandler mixing rules has been used to correlate Table 1 Models for the Gibbs free energy and the activity coefficients for the van Laar model used with the Wong Sandler mixing rule Van Laar N N N ]2 g E RT = j x x ja ij x i j i 1 x i [1 x ja ij N N x i i j x ja ij +(1 x i )x i j x ja ij For a binary mixture g E RT = (A 12/RT )x 1 x 2 x 1 (A 12 /A 21 )+x 2 VLE data for CO 2 /n-alkanol mixtures. The Wong Sandler mixing rules can be summarized as follows [14]: Ni Ni x i x j (b a ) RT ij b m = Ni x i x i 1 AE (x) b i RT i ΩRT ( b a ) RT = 1 ai ij 2 [b a j i + b j ] RT (1 k (3) ij) [ Ni ] x i a i a m = b m + AE (x) b i Ω In these equations, Ω = for the PR equation, and A E (x) is calculated using an appropriate model and assuming that A E (x) AE 0 (x) GE 0 (x) = being GE 0 (x) = ge, the excess Gibbs free energy at low pressure [15]. The combining rule for (b a/rt) ij includes one adjustable parameter k ij. The van Laar model for the excess Gibbs free energy g E included in the Wong Sandler was used. This model contains two empirical parameters for a binary mixture. Therefore, for a binary mixture the Wong Sandler mixing rule includes one adjustable binary interaction parameter k 12 for (b a/rt) ij, besides the two parameters included in the g E model. These three adjustable parameters for each of the models have been determined using experimental phase equilibrium data at constant temperature, available in the literature for the mixtures studied. The van Laar model has been presented in the literature using different expressions for the parameters of the models. The expressions used in this work for multicomponent mixtures and those for binary mixtures are presented in Table 1. In summary, the thermodynamic model include the Peng Robinson equation of state, the Wong Sandler mixing rule, and the van Laar model for g E in the mixing rules, model designated as PR/WS/VL in the rest of the paper. 3. Applications Ten gas + liquid binary systems containing sub and supercritical carbon dioxide with an n-alkanol, were selected for study (CO 2 +: methanol, ethanol, 1-propanol, 1-butanol, 1- pentanol, 1-hexanol, 1-heptanol, 1-octanol, 1-nonanol and 1-decanol). The experimental data were taken from the literature. Table 2 shows the basic properties of the fluid substances

3 138 J.O. Valderrama, J. Zavaleta / Fluid Phase Equilibria 234 (2005) Table 2 Properties of the pure substances included in the mixtures studied Substance T c (K) P c (bar) v c (L/mol) ω z c µ (Debye) ψ CO Methanol Ethanol Propanol Butanol Pentanol Hexanol Heptanol Octanol Nonanol Decanol The data are from the DIPPR database [17]. involved in the study. In Table 2, T c is the critical temperature, P c is the critical pressure, v c is the critical volume, ω is the acentric factor, z c is the critical compressibility factor, µ is the dipole moment and ψ is a polar parameter defined by Nishiumi [16], a function of the dipole moment, the critical temperature and the critical volume. The data for carbon dioxide and all the n-alkanols were obtained from the DIPPR database [17]. Table 3 gives details on the experimental vapor liquid equilibrium data for the ten mixtures studied including the literature source for each data set. As seen in Table 3, data for 41 isotherms with a total of 416 data points were considered. The temperature ranges from 291 to 453 K and the pressure from 0.5 to 19.8 MPa. Bubble pressure calculations for binary mixtures were performed using the PR/WS/VL model. The adjustable parameters (A 12, A 21 and k 12 ) the model are determined by optimization of the objective function given by Eq. (4). The program designed considers the use of the Levenberg Marquardt algorithm [18] as the optimization method. The objective function was defined as the relative error between calculated and experimental values of the pressure: F = N i=1 P exp i P calc i P exp i Once the interaction parameter k ij included in the mixing rules for the force constant a and the parameters of the van Laar model (A 12 and A 21 ) were calculated from vapor liquid equilibrium data, they were correlated to a good approximation in terms of the reduced temperature ( = T/T c ) and the reduced polar parameter ψ of the n-alkanol, as follows: p ij = p 0 + p 1 ψ + p 2 ψ 2 (5) (4) Here, p ij represents any of the parameters involved in the mixing rules and in the excess Gibbs free energy model (A 12, A 21 and k 12 for PR/WS/VL). The terms designated as p 0 and p 1 were found to have linear dependency of the reduced temperature of the n-alkanol, as follows: p 0 = m 0 + n 0 p 1 = m 1 + n 1 p 2 = m 2 + n 2 The polar parameter ψ was defined by Nishiumi [16], in terms of the dipole moment µ in Debye, the critical temperature T c in Kelvin and the critical volume v c in L/mol, as follows: ψ = 2.83 µ2 (7) T c v c The parameters m i and n i are constant for a CO 2 /n-alkanol and are presented in Table 4. It should be mentioned that other correlations were also attempted, following information presented in the literature. The acentric factor, for instance, has also been considered, but we have found that ψ is a more relevant term for correlating the model parameters for these mixtures containing polar fluids. Nishiumi [16] has discussed the use of different parameters to generalize different properties of polar fluids and obtained good results using the defined parameter ψ. Therefore, the perturbation-type expression (5), with ψ as the perturbation parameter, has a reasonable basis to correlate the model parameters for carbon dioxide/n-alkanol mixtures. 4. Results and discussion Results of the application of the different mixing rules are presented in Table 5. All these calculations were done using the generalized parameters described by Eq. (6) with the constants defined in Table 4. The results are shown as the absolute deviations given by the different models for predicting the pressure ( P) (%) and the solute concentration in the gas phase ( y 2 ) (%). The deviations in the gas phase concentration of the solute are lower than those reported in the literature for similar type of mixtures, as presented in Table 6. The calculations presented in the literature were done using other similar models, (6)

4 J.O. Valderrama, J. Zavaleta / Fluid Phase Equilibria 234 (2005) Table 3 Experimental data for the systems considered in this study System CO 2 + N T (K) Range P (MPa) Range (x ) Range (y ) Reference Methanol [9] [21] Ethanol [7] [1] [21] 1-Propanol [22] Butanol [1] Pentanol [23] [1] [23] Hexanol [3] Heptanol [3] Octanol [9] Nonanol [8] Decanol [9,24] In the table, the temperature and pressure values have been rounded to the closest integer. In the table, x 2 and y 2 refer to the concentration of the solute amplified by Table 4 Coefficients for the generalized correlations for the parameters p ij in Eqs. (6) and (7) p ij m o n o m 1 n 1 m 2 n 2 k A A p ij = ( m 0 + n 0 ) + ( m1 + n 1 ) Ψ + ( m2 + n 2 ) Ψ 2. but comparison are still valid because the thermodynamic treatment is the same, so the improvements found in this study can be well observed. Elizalde-Solís [3] did similar type of calculations for the system CO 2 /1-hexanol and found average absolute deviations of 4.3% for the pressure and of 108% for the solute concentration in the gas phase. Chiehming et al. [8] modeled the system CO 2 /1-nonanol using the equation of state of Patel and Teja [19], and found average absolute deviations of 1.1% for the pressure and of 127% for the solute concentration in the gas phase. For all this cases and for the systems considered in this study, deviations in the solvent concentration in the gas phase is below 1%. However, con-

5 140 J.O. Valderrama, J. Zavaleta / Fluid Phase Equilibria 234 (2005) Table 5 Deviations in the correlated values for the pressure and the solute concentration in the gas phase ψ T (K) System CO 2 + P (%) y 2 (%) Methanol Ethanol Propanol Butanol Pentanol Hexanol Heptanol Octanol Nonanol Decanol Average Deviations for the solvent concentration in the gas phase (CO 2 in the mixtures studied) are below 2% for all cases, so details are not shown. Table 6 Deviations in the correlated values for the pressure and the solute concentration in the gas phase presented in the literature for some cases studied in this work System CO 2 + T (K) This work PR/WS/VL Literature P (%) y 2 (%) P (%) y 2 (%) Model Reference Methanol PR/VDW [8] 1-Pentanol PT/VDW [23] PT/VDW 1-Hexanol PR/WS/NRTL [3] 1-Octanol PT/VDW [9] 1-Nonanol PR/VDW [8]

6 J.O. Valderrama, J. Zavaleta / Fluid Phase Equilibria 234 (2005) Fig. 1. Predicted phase equilibrium diagram for selected CO 2 /n-alkanol mixtures using the generalized correlations for the mixing rules parameters for the model PR/WS/VL. sidering that in supercritical fluid extraction one is usually interested in the solute being extracted, a severe test of the accuracy of a model must be the correct correlation of the solute concentration in the gas phase. As shown in the literature, using the solvent concentration in the gas phase is at least a misleading way of analyzing the capabilities of a given model [20]. The deviations for the pressure found in this work are a little higher than those presented in the literature. This observation agrees with information already reported in the literature indicating that, for complex systems such as those studied in this work, deviations for the pressure increase as deviations for the solute concentration in the gas phase decrease [11,20]. One should notice that the highest deviations are found for those cases in which the solute concentration in the gas phase is very low. Table 3 shows that while for most systems de solute mole fraction in the gas phase is of the order of 10 2, this concentration is of the order of 10 4 to 10 3 for the higher n-alkanols (heptanol to decanol). To better show the capabilities of the proposed correlations for the parameters of the PR/WS/VL model, graphical results for four systems but for different temperatures are shown in Fig. 1. As seen in the figure, good agreement between calculated values and experimental data is found. Pressure is a little overestimated for the system CO 2 /1-octanol at K in the higher pressure range (over 100 bar). The same observation can be done for CO 2 /methanol for pressures over 60 bar. All these results show that the generalized correlations for the model parameters (A 12, A 21 and k 12 for van Laar), all represented by p ij in Eq. (5) have good capabilities for predicting VLE properties in CO 2 /n-alkanol mixtures of interest in several engineering processes. 5. Conclusions 1. The concentration of the solvent (CO 2 ) in the gas phase can be obtained with good accuracy with the model studied. 2. Solute (n-alkanol) concentration in the gas phase predicted using the proposed generalized parameters give deviations similar and lower than other correlating models presented in the literature.

7 142 J.O. Valderrama, J. Zavaleta / Fluid Phase Equilibria 234 (2005) The proposed generalized correlation given by Eq. (5) has a predictive character, since one needs only pure component properties for the n-alkanol to predict the pressure and the gas solute concentration in binary carbon dioxide/nalkanol mixtures. 4. The generalized correlations for the model parameters have shown good capabilities for predicting VLE properties in CO 2 /n-alkanol mixtures of interest in several engineering processes. List of symbols A E, AE 0 Helmholtz free energy at infinite and zero pressure A ij, A ji, A 12, A 21 parameters in the van Laar model a i, b i EoS parameters for pure components a ij, b ij EoS interaction coefficients between components i and j in a mixture a m, b m EoS parameters for mixtures F parameter in the function of the PR EoS G E o Gibbs free energy at zero pressure kij, k 12 binary interaction parameter in an EoS m, m 0, m 1, m 2 parameter in the correlation for the interaction parameter p ij N number of points in a data set n, n 0, n 1, n 2 parameter in the correlation for the interaction parameter p ij p ij interaction parameter p 0, p 1 adjustable parameters in p ij (Eq. (3)) P pressure P c critical pressure R ideal gas constant T temperature T c critical temperature reduced temperature ( = T/T c ) V volume v c critical volume x i, x j mole fraction of components i and j in the liquid phase y i, y j mole fraction of components i and j in the vapor phase Abbreviations EoS equation of state PR Peng Robinson EoS PT Patel Teja EoS VDW van der Waals VL van Laar WS Wong Sandler mixing rule Greek letters deviation Ω constant in the Wong Sandler mixing rule α temperature function in an EoS µ dipole moment ω acentric factor ψ Nishiumi s polar parameter (Eq. (7)) Acknowledgments The authors thank the support of the National Commission for Scientific and Technological Research (CONICYT- Chile), through the research grants FONDECYT and , the Direction of Research of the University of La Serena, Chile for permanent support through several research grants and of the Center for Technological Information (CIT, La Serena, Chile), for computer and library support. References [1] D.W. Jennings, R.J. Lee, A.S. Teja, Vapor liquid equilibria in the CO 2 -ethanol and CO 2-1-butanol systems, J. Chem. Eng. Data 36 (3) (1991) [2] M. Artal, V. Pauchon, J. Muñoz, J. Jose, Solubilities of 1-nonanol, 1-undecanol, 1-tridecanol, and 1-pentadecanol in supercritical carbon dioxide at T = K, J. Chem. Eng. Data 43 (1998) [3] O. Elizalde-Solis, L.A. Galicia-Luna, S.I. Sandler, J.G. Sampayo- Hernandez, Vapor liquid equilibria and critical points for the carbon dioxide + 1-hexanol and carbon dioxide + 1-heptanol systems, Fluid Phase Equil. 200 (2002) [4] S.T. Schaeffer, L. Zalkow, A.S. Teja, Solubility of monocrotaline in supercritical carbon dioxide and carbon dioxide-ethanol mixtures, Fluid Phase Equil. 43 (1988) [5] J.M. Wong, K.P. Johnston, Solubilization of biomolecules in carbon dioxide based supercritical fluids, Biotechnol. Prog. 2 (1986) [6] M. de la Ossa, E.V. Brandani, G. Del Re, G. Di Giacomo, E. Ferri, Binary and ternary phase behavior of the system water-ethanolcarbon dioxide, Fluid Phase Equil. 56 (1990) [7] K. Suzuki, H. Sue, M. Itou, R.L. Smith, H. Inomata, K. Arai, S. Saito, Isothermal vapor liquid equilibrium data for binary systems at high pressures: CO 2 -methanol, CO 2 -ethanol, CO 2-1-propanol, methane-ethanol, methane-1-propanol, ethane-ethanol, and ethane-1- propanol systems, J. Chem. Data 35 (1990) [8] J.C. Chiehming, C. Kou-Lung, D. Chang-Yih, A new apparatus for the determination of P-x-y and Henry s constants in high pressure alkanols with critical carbon dioxide, J. Supercrit. Fluids 12 (1998) [9] W.L. Weng, J.T. Chen, M.J. Lee, High-pressure vapor liquid equilibria for mixtures containing a supercritical fluid, Ind. Eng. Chem. Res. 33 (8) (1994) [10] D.Y. Peng, D.B. Robinson, A new two-constant equation of state, Ind. Eng. Chem. Fundam. 15 (1) (1976) [11] J.O. Valderrama, The state of the cubic equations of state, Ind. Eng. Chem. Res. 42 (7) (2003) [12] M.J. Huron, J. Vidal, New mixing rules in simple equations of state for representing vapor liquid equilibria of strongly non-ideal solutions, Fluid Phase Equil. 3 (1979) [13] D.S. Wong, S.I. Sandler, A theoretically correct mixing rule for cubic equations of state, AIChE J. 38 (1992) [14] S. Dahl, M.L. Michelsen, High-pressure vapor liquid equilibrium with a UNIFAC-based equation of state, AIChE J. 36 (1990) [15] H. Orbey, S.I. Sandler, Modeling vapor liquid equilibria, in: Cubic Equations of State and Their Mixing Rules, Cambridge University Press, USA, [16] H. Nishiumi, An improved generalized BWR equation of state with three polar parameters applicable to polar substances, J. Chem. Eng. Jpn. 13 (1980)

8 J.O. Valderrama, J. Zavaleta / Fluid Phase Equilibria 234 (2005) [17] T.E. Daubert, R.D. Danner, H.M. Sibul, C.C. Stebbins, Physical and thermodynamic properties of pure chemical, in: Data Compilations, Taylor and Francis, Washington, DC, USA, [18] M. Reilly, Computer Programs for Chemical Engineering Education, vol. 2, Sterling Swift, TX, USA, 1972, pp [19] N.C. Patel, A.S. Teja, A new cubic equation of state for fluids and fluid mixtures, Chem. Eng. Sci. 37 (1982) [20] J.O. Valderrama, V.H. Alvarez, Vapor liquid equilibrium in mixtures containing supercritical CO 2 using a new modified Kwak-Mansoori mixing rule, AIChE J. 50 (2) (2004) [21] S.N. Joung, C.W. Yoo, H.Y. Shin, S.Y. Kim, K.P. Yoo, C.S. Lee, Measurements and correlation of high-pressure VLE of binary CO 2 -alcohol systems (methanol, ethanol, 2-methoxyethanol and 2-ethoxyethanol, Fluid Phase Equil. 185 (1 2) (2001) [22] V. Vandana, A.S. Teja, Vapor liquid equilibria in the carbon dioxide + 1-propanol system, J. Chem. Eng. Data 40 (1995) [23] G. Silva-Oliver, L.A. Galicia-Luna, S.I. Sandler, Vapor liquid equilibria and critical points for the carbon dioxide + 1-pentanol and carbon dioxide + 2-pentanol systems at temperatures from 332 to 432 K, Fluid Phase Equil. 200 (2002) [24] M.-J. Lee, J.T. Chen, Vapor liquid equilibrium for CO 2 /alkanol systems, Fluid Phase Equil. 92 (1994)

THERMODYNAMIC CONSISTENCY TESTS FOR PHASE EQUILIBRIUM IN LIQUID SOLUTE+SUPERCRITICAL SOLVENT MIXTURES

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