- Email: [email protected]
Correlations for binary phase equilibria in high-pressure carbon dioxide
Fluid Phase Equilibria 238 (2005) 174–179
Correlations for binary phase equilibria in high-pressure carbon dioxide Sujit Kumar Jha, Giridhar Madras ∗ Department of Chemical Engineering, Indian Institute of Science, Bangalore-560 012, India Received 17 December 2004; received in revised form 29 September 2005; accepted 30 September 2005 Available online 4 November 2005
Abstract The high-pressure phase equilibrium in binary mixtures of carbon dioxide and various liquids such as benzene, toluene, m-xylene, chlorobenzene, 1,2-dichlorobenzene, low molecular weight alcohols, amides and a solid, hexachlorobenzene, was modeled using the Peng–Robinson equation of state with quadratic mixing rules. Though the technique of modeling is not new, the key conclusion of the study is that the two adjustable parameters, kij and lij , vary linearly with the addition of functional groups. In addition, the adjustable parameters are found to be nearly independent of temperature and the same values of kij and lij can be used for a range of temperatures for the systems considered in the present study. © 2005 Elsevier B.V. All rights reserved. Keywords: Phase equilibria; Supercritical fluids; Solubility; Model; Equation of state
1. Introduction The increasing use of supercritical fluids (ScFs) [1] in the food [2], pharmaceutical [3], and petroleum [4] industries has induced a lot of theoretical and experimental work on highpressure fluid phase equilibria (HPE). Thus, the thermodynamics of phase equilibria encountered in design of chemical products and processes is of utmost importance. Dohrn and Brunner [5] and Christov and Dohrn [6] have reviewed the experimental methods and systems investigated in HPE. Since the experimental determination of HPE at various temperatures is time consuming and expensive, modeling and prediction is of paramount importance for the successful design of high-pressure systems. The equation of state approach is used [7] for phase equilibrium calculations. Various cubic equations of state and mixing rules used commonly have been reviewed by Valderrama [8]. The Peng–Robinson equations of state (PR-EOS) with van der Waal’s quadratic mixing rules has been reported to correlate the HPE reasonably well [9–12]. The carbon dioxide–toluene binary system has been previously modeled using the PR-EOS with a single energy interaction parameter [13,14]. However, Fink and Hershey [9] have shown that both the energy and volume parameters are needed to reproduce the high-pressure experimental data well. ∗
Corresponding author. Tel.: +91 80 2293 2321; fax: +91 80 2360 0683. E-mail address: [email protected] (G. Madras).
0378-3812/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2005.09.029
In this study, PR-EOS with two parameter quadratic mixing rules (PR2) has been used to estimate the HPE in binary mixtures of carbon dioxide and benzene, toluene, m-xylene and chlorobenzenes at various temperatures and pressures. Though more accurate combinations of EOS and mixing rules, e.g. Patel–Teja EOS with Wong–Sandler mixing rules provide better correlations [15], PR2 has been used in this work. The key conclusion of the study is that the adjustable binary interaction parameters in the model vary linearly with the addition of functional groups. It should be noted that the linearity extends from the phase equilibria for both liquids (chlorobenzenes) and solid (hexachlorobenzene). Thus, by using the correlation, this model can predict the binary interaction parameters and hence, the phase equilibria of other species in the homologous series. 2. Theoretical background The phase equilibria in high-pressure systems have been modeled for both solids and liquids. For vapor–liquid equilibria (VLE): yi =
fˆ il g pφˆ i
(1)
where fˆ il = xi pφˆ il is the fugacity of component i in liquid phase, p, the pressure, xi and yi , respectively, the mole fractions of g component i in liquid and vapor phases, and φˆ il and φˆ i are, respectively, the fugacity coefficients of i in liquid and vapor
S.K. Jha, G. Madras / Fluid Phase Equilibria 238 (2005) 174–179
175
phases. Thus, yi and p can be calculated for given values of xi and temperature, T. For solid-ScF phase equilibria [16]: (p−psat i )vis exp psat i RT yiScF = (2) φˆ iScF where yiScF is the solubility of the solute (in mole fraction) in the supercritical fluid phase, psat i , the sublimation pressure of the pure solid at the system temperature, vis , the molar volume of the pure solid, R, the universal gas constant, and φˆ iScF is the fugacity coefficient of the solute in the supercritical fluid phase. The fugacity coefficient of a component in a homogeneous mixture is calculated using an equation of state with appropriate mixing rule. The PR-EOS [17] is p=
RT a − 2 V − b V + 2Vb − b2 a=
0.457235R2 Tc2 γ ; pc
γ = (1 + (0.37464 + 1.54226ω − 0.26992ω2 )(1 −
√
(3a)
Fig. 1. Vapor–liquid equilibria for the system carbon dioxide + benzene at 313.2 K. (䊉) Experimental data [20]; (—) PR2 prediction with kij = 0.0644 and lij = −0.1258.
2
dioxide–hexachlorobenzene, these parameters were obtained from VLE isotherms by minimizing the objective function: N pi,obs − pi,model 2 AARD = (6) pi,obs
T/Tc ) (3b)
i=1
b=
0.077796RTc pc
(3c)
where V is the molar volume of the fluid phase, Tc , the critical temperature, pc , the critical pressure, and ω is the accentric factor. The combining rules [18], used with van der Waal’s quadratic mixing rules, are √ √ aij = (1 − kij ) aii ajj ; aij = (1 − kij ) aii ajj (4a) b=
i
1/3 3
1/3
j
yi yj bij ;
bij = (1 − lij )
(bii + bjj ) 8
(4b)
where indices i and j refer to different components present in the fluid phase and kij and lij are two adjustable binary interaction parameters. The fugacity coefficient of ith component in a homogeneous mixture, is given by pV p(V − b) 1 ∂nb a m ˆ lnφi = −ln + −1 + √ RT b ∂ni RT 2 2bRT √ 1 1 ∂n2 a 1 ∂nb V + b(1 − 2) √ × − ln a n ∂ni b ∂ni V + b(1 + 2) (5) where n is the total number of moles. 3. Results and discussion The binary interaction parameters, kij and lij , are essential for phase equilibrium calculation using the equation of state approach. For all the binary systems except carbon
where N is the number of data points. For each value of x and y, the value of pi,model is computed from the model and compared against the value of pi,obs (observed experimentally) based on AARD and this is minimized. This ensures that the deviations in both x and y are considered. However, for the system of hexachlorobenzene in carbon dioxide (which is a solid-ScF phase equilibria), Eq. (2) was used to determine the solubility. The binary interaction parameters, kij and lij , were then obtained by minimizing the sum of the square of relative errors in the correlation of the solubility of hexachlorobenzene in ScF at a fixed temperature and various pressures. The Gauss–Newton algorithm with Levenberg–Marquardt modifications for global convergence [19] was used. The optimum values of kij and lij for different systems along with the relative deviation (AARD) for the correlation have been listed in Table 1. Fig. 1 shows the VLE correlation of carbon dioxide–benzene mixture at 313.2 K using the optimum values of kij and lij . Fig. 2a shows the VLE correlation of carbon dioxide–1,2dichlorobenzene mixture at 313.2 K. The sensitivity of the parameters was investigated by increasing kij by 20% and keeping lij at the optimum value and the AARD was 15%, while increasing lij by 20% and keeping kij at the optimum value results in AARD of 45%. Fig. 2b shows the dependence of AARD on the binary interaction parameters, kij and lij for the system of toluene in high-pressure carbon dioxide at 311.11 K. Though over a limited range of kij and lij of 0.075–0.08 and −0.13 to −0.16, respectively, the AARD is less than 10%, the values of the binary interaction parameters listed in Table 1 yield the lowest AARD. This shows that the modeling of phase equilibria is sensitive to binary interaction parameters in addition to the values of the critical constants and the acentric factor. Further, reducing the
S.K. Jha, G. Madras / Fluid Phase Equilibria 238 (2005) 174–179
176
Table 1 Binary interaction parameters, kij and lij , for carbon dioxide + X X
T (K)
kij
lij
AARD (%)
AARDb (%)
Benzene [20] Toluene [13] m-Xylene [21] Chlorobenzene [22] 1,2-Dichlorobenzene [22] Hexachlorobenzene [23] Methanol [24] Ethanol [24] 1-Propanol [24] 1-Butanol [25] 1-Octanol [26] 1-Decanol [26] N,N-dimethylformamide [27] N,N-diethylformamide [27] N,N-dibutylformamide [27]
313.20 311.11 310.90 313.20 313.20 318.00 313.40 313.40 313.40 313.20 318.19 318.14 318.2 318.2 318.2
0.0644 0.0763 0.0872 0.0694b 0.0735 0.1042 0.0777 0.0790 0.0810 0.0845c 0.0950 0.1003 0.045 0.04 0.03
−0.1258 −0.1524 −0.1746 −0.1720b −0.2157 −0.4318 −0.0049 −0.0520 −0.0900 −0.1231b −0.2702 −0.3484 −0.18 −0.22 −0.29
1.0 1.7 2.3 4.0 0.2 13.4a 2.2 6.9 6.6 1.0 1.3 0.8 17.1 16.2 11.2
1.1 1.7 3.2 4.0 4.5 14.1a 2.4 6.9 6.8 1.1 1.4 0.9 17.1 16.2 11.5
a b
Based on minimization of solid solubilities. Predicted from the linear regression.
model to a single parameter model by equating kij or lij to zero resulted in very large AARD (>50%) indicating that such models were unsuitable for the correlation of VLE of the systems investigated here. In general, the parameter kij is taken to be dependent on temperature and, therefore, the effect of temperature was investigated. Fig. 3 shows the VLE correlation of carbon dioxide–toluene mixture at 311.11 and 393.55 K using the optimum values of kij and lij at 311.11 K. The AARD for the correlations at 393.55 K is found to be 5.48%. Similar results (not shown) were obtained for the other systems considered here and it is concluded that kij and lij are nearly independent of temperature for these systems. Thus, the same values listed in Table 1 can be used over a considerably wide range of temperatures for the systems investigated in this study. The variation of the binary interaction parameters, kij and lij , with number of chlorine (−Cl) substituents on benzene ring for the binary mixtures of carbon dioxide and benzene,
Fig. 2. (a) Vapor–liquid equilibria for the system carbon dioxide + 1,2dichlorobenzene at 313.2 K. (䊉) Experimental data [22], PR2 prediction; (—) kij = 0.0735 and lij = −0.2157; . . . kij = 0.0735 and lij = −0.2588; — kij = 0.0882 and lij = −0.2157. (b) Dependence of AARD on the binary interaction parameters, kij and lij for the system carbon dioxide + toluene at 311.11 K.
Fig. 3. Vapor–liquid equilibria for the system carbon dioxide + toluene. (䊉) Experimental data at 311.11 K [13];() experimental data at 393.55 K [13]; (—) PR2 prediction with kij = 0.0763 and lij = −0.1524.
S.K. Jha, G. Madras / Fluid Phase Equilibria 238 (2005) 174–179
1,2-dichlorobenzene and hexachlorobenzene is linear with kij = 0.0068NCl + 0.0626; lij = −0.0514NCl − 0.1206, where NCl is the number of chlorine substituents on benzene ring. The variation of kij and lij with the number of methyl (−CH3 ) substituents on benzene ring for binary mixtures of carbon dioxide and benzene, toluene and m-xylene is linear with kij = 0.0114Nm + 0.0646; lij = −0.0244Nm − 0.1265, where Nm is the number of methyl substituents on benzene ring. The last column of Table 1 shows the AARD for all cases based on calculating the binary interaction parameters using the linear relationship for the homologous series. The AARD does not change significantly indicating that the degree of accuracy that is sacrificed for generality by the correlation is minimal. To demonstrate the utility of the model, we attempt to predict the VLE of carbon dioxide–chlorobenzene. The parameters, kij and lij , determined from the linear relationship are 0.0694 and −0.172, respectively. These values are then used to model VLE of carbon dioxide–chlorobenzene at 313.2 and 393.1 K (Fig. 4) and the AARD for the prediction are 3.98 and 8.17%, respectively. This shows that the model can be used to successfully correlate the phase equilibria of binary mixtures of carbon dioxide and chlorobenzenes. However, one of the limitations of the theory is that the same binary interactions parameters apply for all isomers of the organics. Further, the model is based on the mixing and combination rules listed in Eqs. (4a) and (4b). The usage of a linear combination rule for bij , for example, leads to values of lij that do not follow the linear relationship discussed here. The correlation of high-pressure fluid phase equilibria becomes challenging because of significant deviation of vapor phase from ideal gas behavior. This can be observed by comparing the ideal gas vapor phase density with the experimentally observed [26] vapor phase density of carbon dioxide–1-octanol mixture at 318.19 K (Fig. 5). The PR-EOS without the interaction parameters showed good agreement with the experimentally observed values of vapor phase density but did not predict the liquid phase density. However, the PR-EOS with the interaction
Fig. 4. Vapor–liquid equilibria for the system carbon dioxide + chlorobenzene. (䊉) Experimental data at 311.2 K [22]; () experimental data at 393.1 K [22]; (—) prediction by the proposed model.
177
Fig. 5. Liquid and vapor phase density of carbon dioxide–1-octanol mixture at 318.19 K. () Liquid phase density [26]; () vapor phase density [26]; (—) ideal gas law prediction of vapor phase density; (. . .) PR-EOS prediction with kij = lij = 0; (—) PR2 prediction with kij = 0.095 and lij = −0.270.
parameters (PR2) showed good predictions of both the vapor phase and liquid phase densities. The same values of interaction parameters used for the correlation of the densities were used for the correlation of VLE of the system. To extend the idea of linearity of binary interaction parameters, the HPE of binary mixtures of various alcohols, from methanol to decanol, in carbon dioxide were modeled. Fig. 6 shows the VLE prediction of methanol–carbon dioxide at 313.14 K. Table 1 includes the values of kij and lij for various alcohol–carbon dioxide binaries. The interaction parameters, kij = 0.0741 + 0.0026Nalc ; lij = 0.0265 − 0.0374Nalc , were found to vary linearly with the number of carbon atoms in the main chain of alcohol, Nalc . If more experimental data is available, one can investigate how far the linearity extends. Similar to the previous case, the utility of the linearity is demonstrated by determining the values of kij and lij for the carbon dioxide–1butanol system using the linear relations and the VLE prediction is shown against the experimental [25] data (Fig. 7). Thus, the
Fig. 6. Vapor–liquid equilibria for the system carbon dioxide + methanol at 313.14 K. (䊉) Experimental data [24]; (—) PR2 prediction with kij = 0.0777 and lij = −0.0049.
178
S.K. Jha, G. Madras / Fluid Phase Equilibria 238 (2005) 174–179
adjustable binary interaction parameters, kij and lij , vary linearly with the addition of functional groups and, therefore, can be used to predict the HPE of binary mixtures in the homologous series. The parameters, kij and lij , for carbon dioxide–chlorobenzene and carbon dioxide–1-butanol obtained from the linear relationships successfully predict the VLE data. References
Fig. 7. Vapor–liquid equilibria for the system carbon dioxide + 1-butanol at 313.14 K. (䊉) Experimental data [25]; (—) prediction by the proposed model.
Fig. 8. Vapor–liquid equilibria for the systems: carbon dioxide + N,Ndimethylformamide (DMFA), carbon dioxide + N,N-diethylformamide (DEFA), carbon dioxide + N,N-dibutylformamide (DBFA) at 318.2 K. (䊉) Experimental data [27]; (—) PR2 prediction.
model can be used to predict the HPE of binary mixture of carbon dioxide and other alcohols in the homologous series. Finally, the experimental results [27] for the high-pressure vapor liquid equilibria of binary systems of carbon dioxide with N,N-dimethylformamide (DMFA), N,N-diethylformamide (DEFA) and N,N-dibutylformamide (DBFA) is modeled (Fig. 8). The binary interaction parameters kij and lij are shown in Table 1 and these interaction parameters, kij = 0.05 + 0.0025Nam ; lij = −0.145 − 0.0182Nam , were found to vary linearly with the number of carbon atoms in the main chain of the amide, Nam . 4. Conclusions The high-pressure phase equilibrium in binary mixtures of carbon dioxide and benzene, toluene, m-xylene, chlorobenzenes, amides and various low molecular weight alcohols has been modeled. Peng–Robinson equation of state with two parameter quadratic mixing rules has been used to estimate the mixture phase equilibria. The key conclusion of the study is that the
[1] A.S. Teja, C.A. Eckert, Commentary on supercritical fluids: research and applications, Ind. Eng. Chem. Res. 39 (2000) 4442. [2] M. Sihvonen, E. Jarvenpaa, V. Hietaniemi, R. Huopalahti, Advances in supercritical carbon dioxide technologies, Trends Food Sci. Technol. 10 (1999) 217. [3] B. Subramaniam, R.A. Rajewski, K. Snavely, Pharmaceutical processing with supercritical carbon dioxide, J. Pharm. Sci. 86 (1997) 885. [4] D.E. Knowles, W.D. Felix, N.L. Porter, B.A. Jones, D.L. Knowles, Supercritical fluid chromatography applications for the petroleum industry, in: Proceedings of the International Symposium on Fuels and Lubricants, New Delhi, 2000, p. 367. [5] R. Dohrn, G. Brunner, High-pressure fluid-phase equilibria: experimental methods and systems investigated, Fluid Phase Equilib. 106 (1995) 213. [6] M. Christov, R. Dohrn, High-pressure fluid phase equilibria experimental methods and systems investigated (1994–1999), Fluid Phase Equilib. 202 (2002) 153. [7] S.M. Walas, Phase Equilibria in Chemical Engineering, Butterworths, MA, 1985. [8] J.O. Valderrama, The state of the cubic equations of state, Ind. Eng. Chem. Res. 42 (2003) 1603. [9] S.D. Fink, H.C. Hershey, Modeling the vapor–liquid equilibria of 1,1,1trichloroethane + carbon dioxide and toluene + carbon dioxide at 308, 323 and 353 K, Ind. Eng. Chem. Res. 29 (1990) 295. [10] J.M. Prausnitz, R.N. Lichtenthaler, E.G. de Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria, Prentice Hall PTR, New Jersey, 1999, p. 718. [11] G. Madras, C. Kulkarni, J. Modak, Modeling the solubilities of fatty acids in supercritical carbon dioxide, Fluid Phase Equilib. 209 (2003) 207. [12] S.K. Jha, G. Madras, Modeling of adsorption equilibria in supercritical fluids, J. Supercrit. Fluid 32 (2004) 161. [13] H.-J. Ng, D.B. Robinson, Equilibrium phase properties of the toluene–carbon dioxide system, J. Chem. Eng. Data 23 (1978) 325. [14] H.M. Lin, Peng–Robinson equation of state for vapor–liquid equilibrium calculations for carbon dioxide + hydrocarbon mixtures, Fluid Phase Equilib. 16 (1984) 151. [15] S. Guha, G. Madras, Modeling of ternary solubilities of organics in supercritical carbon dioxide, Fluid Phase Equilib. 187 (2001) 255. [16] M.A. McHugh, V.J. Krukonis, Supercritical Fluid Extraction Principles and Practice, Butterworth-Heinemann, MA, 1994, p. 127. [17] D.Y. Peng, D.B. Robinson, A new two-constant equation of state, Ind. Eng. Chem. Fund. 15 (1976) 59. [18] T.Y. Kwak, G.A. Mansoori, Van der Waal’s mixing rules for cubic equations of state, applications for supercritical fluid extraction modeling, Chem. Eng. Sci. 41 (1986) 1303. [19] W.H. Press, S.A. Teukolsky, W.T. Vellerling, B.P. Flannery, Numerical Recipes in C, Cambridge University Press, 1992, p. 683. [20] M.K. Gupta, Y.-H. Ll, B.J. Hulsey, R.L. Robinson Jr., Phase equilibrium for carbon dioxide–benzene at 313.2, 353.2, and 393.2 K, J. Chem. Eng. Data 27 (1982) 55. [21] H.-J. Ng, Sam S.-S. Huang, D.B. Robinson, Equilibrium phase properties of selected m-xylene binary systems. m-xylene–methane and mxylene–carbon dioxide, J. Chem. Eng. Data 27 (1982) 119. [22] D. Walther, G. Maurer, High pressure vapour–liquid equilibria in binary mixtures of carbon dioxide and benzaldehyde, bromobenzene, chlorobenzene 1,2-dichlorobenzene and 2-chloro-1-methyl benzene at
S.K. Jha, G. Madras / Fluid Phase Equilibria 238 (2005) 174–179 temperatures between 313 and 393 K and pressures up to 22 MPa, Ber. Bunsen. Phys. Chem. 96 (1992) 981. [23] W.M. Cross Jr., A. Akgerman, Single component and mixture solubilities of hexachlorobenzene and pentachlorophenol in supercritical carbon dioxide, Ind. Eng. Chem. Res. 37 (1998) 1510. [24] 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: carbon dioxide–methanol, carbon dioxide–ethanol, carbon dioxide–1-propanol, methane–ethanol, methane–1-propanol, ethane–ethanol, and ethane–1-propanol systems, J. Chem. Eng. Data 35 (1990) 63.
179
[25] T. Hiaki, H. Miyagi, T. Tsuji, M. Hongo, Vapor–liquid equilibria of supercritical carbon dioxide + butanol systems at 313.2 K, J. Supercrit. Fluid 13 (1998) 23. [26] C. Chiehming, J.C. Kou-Lunga, D. Chang-Yihb, A new apparatus for the determination of P–x–y diagrams and Henry’s constants in high pressure alcohols with critical carbon dioxide, J. Supercrit. Fluid 12 (1998) 223. [27] H.-S. Byun, N.-H. Kim, C. Kwak, Measurements and modeling of highpressure phase behavior of binary CO2 –amides systems, Fluid Phase Equilib. 208 (2003) 53.
Correlations for binary phase equilibria in high-pressure carbon dioxide Sujit Kumar Jha, Giridhar Madras ∗ Department of Chemical Engineering, Indian Institute of Science, Bangalore-560 012, India Received 17 December 2004; received in revised form 29 September 2005; accepted 30 September 2005 Available online 4 November 2005
Abstract The high-pressure phase equilibrium in binary mixtures of carbon dioxide and various liquids such as benzene, toluene, m-xylene, chlorobenzene, 1,2-dichlorobenzene, low molecular weight alcohols, amides and a solid, hexachlorobenzene, was modeled using the Peng–Robinson equation of state with quadratic mixing rules. Though the technique of modeling is not new, the key conclusion of the study is that the two adjustable parameters, kij and lij , vary linearly with the addition of functional groups. In addition, the adjustable parameters are found to be nearly independent of temperature and the same values of kij and lij can be used for a range of temperatures for the systems considered in the present study. © 2005 Elsevier B.V. All rights reserved. Keywords: Phase equilibria; Supercritical fluids; Solubility; Model; Equation of state
1. Introduction The increasing use of supercritical fluids (ScFs) [1] in the food [2], pharmaceutical [3], and petroleum [4] industries has induced a lot of theoretical and experimental work on highpressure fluid phase equilibria (HPE). Thus, the thermodynamics of phase equilibria encountered in design of chemical products and processes is of utmost importance. Dohrn and Brunner [5] and Christov and Dohrn [6] have reviewed the experimental methods and systems investigated in HPE. Since the experimental determination of HPE at various temperatures is time consuming and expensive, modeling and prediction is of paramount importance for the successful design of high-pressure systems. The equation of state approach is used [7] for phase equilibrium calculations. Various cubic equations of state and mixing rules used commonly have been reviewed by Valderrama [8]. The Peng–Robinson equations of state (PR-EOS) with van der Waal’s quadratic mixing rules has been reported to correlate the HPE reasonably well [9–12]. The carbon dioxide–toluene binary system has been previously modeled using the PR-EOS with a single energy interaction parameter [13,14]. However, Fink and Hershey [9] have shown that both the energy and volume parameters are needed to reproduce the high-pressure experimental data well. ∗
Corresponding author. Tel.: +91 80 2293 2321; fax: +91 80 2360 0683. E-mail address: [email protected] (G. Madras).
0378-3812/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2005.09.029
In this study, PR-EOS with two parameter quadratic mixing rules (PR2) has been used to estimate the HPE in binary mixtures of carbon dioxide and benzene, toluene, m-xylene and chlorobenzenes at various temperatures and pressures. Though more accurate combinations of EOS and mixing rules, e.g. Patel–Teja EOS with Wong–Sandler mixing rules provide better correlations [15], PR2 has been used in this work. The key conclusion of the study is that the adjustable binary interaction parameters in the model vary linearly with the addition of functional groups. It should be noted that the linearity extends from the phase equilibria for both liquids (chlorobenzenes) and solid (hexachlorobenzene). Thus, by using the correlation, this model can predict the binary interaction parameters and hence, the phase equilibria of other species in the homologous series. 2. Theoretical background The phase equilibria in high-pressure systems have been modeled for both solids and liquids. For vapor–liquid equilibria (VLE): yi =
fˆ il g pφˆ i
(1)
where fˆ il = xi pφˆ il is the fugacity of component i in liquid phase, p, the pressure, xi and yi , respectively, the mole fractions of g component i in liquid and vapor phases, and φˆ il and φˆ i are, respectively, the fugacity coefficients of i in liquid and vapor
S.K. Jha, G. Madras / Fluid Phase Equilibria 238 (2005) 174–179
175
phases. Thus, yi and p can be calculated for given values of xi and temperature, T. For solid-ScF phase equilibria [16]: (p−psat i )vis exp psat i RT yiScF = (2) φˆ iScF where yiScF is the solubility of the solute (in mole fraction) in the supercritical fluid phase, psat i , the sublimation pressure of the pure solid at the system temperature, vis , the molar volume of the pure solid, R, the universal gas constant, and φˆ iScF is the fugacity coefficient of the solute in the supercritical fluid phase. The fugacity coefficient of a component in a homogeneous mixture is calculated using an equation of state with appropriate mixing rule. The PR-EOS [17] is p=
RT a − 2 V − b V + 2Vb − b2 a=
0.457235R2 Tc2 γ ; pc
γ = (1 + (0.37464 + 1.54226ω − 0.26992ω2 )(1 −
√
(3a)
Fig. 1. Vapor–liquid equilibria for the system carbon dioxide + benzene at 313.2 K. (䊉) Experimental data [20]; (—) PR2 prediction with kij = 0.0644 and lij = −0.1258.
2
dioxide–hexachlorobenzene, these parameters were obtained from VLE isotherms by minimizing the objective function: N pi,obs − pi,model 2 AARD = (6) pi,obs
T/Tc ) (3b)
i=1
b=
0.077796RTc pc
(3c)
where V is the molar volume of the fluid phase, Tc , the critical temperature, pc , the critical pressure, and ω is the accentric factor. The combining rules [18], used with van der Waal’s quadratic mixing rules, are √ √ aij = (1 − kij ) aii ajj ; aij = (1 − kij ) aii ajj (4a) b=
i
1/3 3
1/3
j
yi yj bij ;
bij = (1 − lij )
(bii + bjj ) 8
(4b)
where indices i and j refer to different components present in the fluid phase and kij and lij are two adjustable binary interaction parameters. The fugacity coefficient of ith component in a homogeneous mixture, is given by pV p(V − b) 1 ∂nb a m ˆ lnφi = −ln + −1 + √ RT b ∂ni RT 2 2bRT √ 1 1 ∂n2 a 1 ∂nb V + b(1 − 2) √ × − ln a n ∂ni b ∂ni V + b(1 + 2) (5) where n is the total number of moles. 3. Results and discussion The binary interaction parameters, kij and lij , are essential for phase equilibrium calculation using the equation of state approach. For all the binary systems except carbon
where N is the number of data points. For each value of x and y, the value of pi,model is computed from the model and compared against the value of pi,obs (observed experimentally) based on AARD and this is minimized. This ensures that the deviations in both x and y are considered. However, for the system of hexachlorobenzene in carbon dioxide (which is a solid-ScF phase equilibria), Eq. (2) was used to determine the solubility. The binary interaction parameters, kij and lij , were then obtained by minimizing the sum of the square of relative errors in the correlation of the solubility of hexachlorobenzene in ScF at a fixed temperature and various pressures. The Gauss–Newton algorithm with Levenberg–Marquardt modifications for global convergence [19] was used. The optimum values of kij and lij for different systems along with the relative deviation (AARD) for the correlation have been listed in Table 1. Fig. 1 shows the VLE correlation of carbon dioxide–benzene mixture at 313.2 K using the optimum values of kij and lij . Fig. 2a shows the VLE correlation of carbon dioxide–1,2dichlorobenzene mixture at 313.2 K. The sensitivity of the parameters was investigated by increasing kij by 20% and keeping lij at the optimum value and the AARD was 15%, while increasing lij by 20% and keeping kij at the optimum value results in AARD of 45%. Fig. 2b shows the dependence of AARD on the binary interaction parameters, kij and lij for the system of toluene in high-pressure carbon dioxide at 311.11 K. Though over a limited range of kij and lij of 0.075–0.08 and −0.13 to −0.16, respectively, the AARD is less than 10%, the values of the binary interaction parameters listed in Table 1 yield the lowest AARD. This shows that the modeling of phase equilibria is sensitive to binary interaction parameters in addition to the values of the critical constants and the acentric factor. Further, reducing the
S.K. Jha, G. Madras / Fluid Phase Equilibria 238 (2005) 174–179
176
Table 1 Binary interaction parameters, kij and lij , for carbon dioxide + X X
T (K)
kij
lij
AARD (%)
AARDb (%)
Benzene [20] Toluene [13] m-Xylene [21] Chlorobenzene [22] 1,2-Dichlorobenzene [22] Hexachlorobenzene [23] Methanol [24] Ethanol [24] 1-Propanol [24] 1-Butanol [25] 1-Octanol [26] 1-Decanol [26] N,N-dimethylformamide [27] N,N-diethylformamide [27] N,N-dibutylformamide [27]
313.20 311.11 310.90 313.20 313.20 318.00 313.40 313.40 313.40 313.20 318.19 318.14 318.2 318.2 318.2
0.0644 0.0763 0.0872 0.0694b 0.0735 0.1042 0.0777 0.0790 0.0810 0.0845c 0.0950 0.1003 0.045 0.04 0.03
−0.1258 −0.1524 −0.1746 −0.1720b −0.2157 −0.4318 −0.0049 −0.0520 −0.0900 −0.1231b −0.2702 −0.3484 −0.18 −0.22 −0.29
1.0 1.7 2.3 4.0 0.2 13.4a 2.2 6.9 6.6 1.0 1.3 0.8 17.1 16.2 11.2
1.1 1.7 3.2 4.0 4.5 14.1a 2.4 6.9 6.8 1.1 1.4 0.9 17.1 16.2 11.5
a b
Based on minimization of solid solubilities. Predicted from the linear regression.
model to a single parameter model by equating kij or lij to zero resulted in very large AARD (>50%) indicating that such models were unsuitable for the correlation of VLE of the systems investigated here. In general, the parameter kij is taken to be dependent on temperature and, therefore, the effect of temperature was investigated. Fig. 3 shows the VLE correlation of carbon dioxide–toluene mixture at 311.11 and 393.55 K using the optimum values of kij and lij at 311.11 K. The AARD for the correlations at 393.55 K is found to be 5.48%. Similar results (not shown) were obtained for the other systems considered here and it is concluded that kij and lij are nearly independent of temperature for these systems. Thus, the same values listed in Table 1 can be used over a considerably wide range of temperatures for the systems investigated in this study. The variation of the binary interaction parameters, kij and lij , with number of chlorine (−Cl) substituents on benzene ring for the binary mixtures of carbon dioxide and benzene,
Fig. 2. (a) Vapor–liquid equilibria for the system carbon dioxide + 1,2dichlorobenzene at 313.2 K. (䊉) Experimental data [22], PR2 prediction; (—) kij = 0.0735 and lij = −0.2157; . . . kij = 0.0735 and lij = −0.2588; — kij = 0.0882 and lij = −0.2157. (b) Dependence of AARD on the binary interaction parameters, kij and lij for the system carbon dioxide + toluene at 311.11 K.
Fig. 3. Vapor–liquid equilibria for the system carbon dioxide + toluene. (䊉) Experimental data at 311.11 K [13];() experimental data at 393.55 K [13]; (—) PR2 prediction with kij = 0.0763 and lij = −0.1524.
S.K. Jha, G. Madras / Fluid Phase Equilibria 238 (2005) 174–179
1,2-dichlorobenzene and hexachlorobenzene is linear with kij = 0.0068NCl + 0.0626; lij = −0.0514NCl − 0.1206, where NCl is the number of chlorine substituents on benzene ring. The variation of kij and lij with the number of methyl (−CH3 ) substituents on benzene ring for binary mixtures of carbon dioxide and benzene, toluene and m-xylene is linear with kij = 0.0114Nm + 0.0646; lij = −0.0244Nm − 0.1265, where Nm is the number of methyl substituents on benzene ring. The last column of Table 1 shows the AARD for all cases based on calculating the binary interaction parameters using the linear relationship for the homologous series. The AARD does not change significantly indicating that the degree of accuracy that is sacrificed for generality by the correlation is minimal. To demonstrate the utility of the model, we attempt to predict the VLE of carbon dioxide–chlorobenzene. The parameters, kij and lij , determined from the linear relationship are 0.0694 and −0.172, respectively. These values are then used to model VLE of carbon dioxide–chlorobenzene at 313.2 and 393.1 K (Fig. 4) and the AARD for the prediction are 3.98 and 8.17%, respectively. This shows that the model can be used to successfully correlate the phase equilibria of binary mixtures of carbon dioxide and chlorobenzenes. However, one of the limitations of the theory is that the same binary interactions parameters apply for all isomers of the organics. Further, the model is based on the mixing and combination rules listed in Eqs. (4a) and (4b). The usage of a linear combination rule for bij , for example, leads to values of lij that do not follow the linear relationship discussed here. The correlation of high-pressure fluid phase equilibria becomes challenging because of significant deviation of vapor phase from ideal gas behavior. This can be observed by comparing the ideal gas vapor phase density with the experimentally observed [26] vapor phase density of carbon dioxide–1-octanol mixture at 318.19 K (Fig. 5). The PR-EOS without the interaction parameters showed good agreement with the experimentally observed values of vapor phase density but did not predict the liquid phase density. However, the PR-EOS with the interaction
Fig. 4. Vapor–liquid equilibria for the system carbon dioxide + chlorobenzene. (䊉) Experimental data at 311.2 K [22]; () experimental data at 393.1 K [22]; (—) prediction by the proposed model.
177
Fig. 5. Liquid and vapor phase density of carbon dioxide–1-octanol mixture at 318.19 K. () Liquid phase density [26]; () vapor phase density [26]; (—) ideal gas law prediction of vapor phase density; (. . .) PR-EOS prediction with kij = lij = 0; (—) PR2 prediction with kij = 0.095 and lij = −0.270.
parameters (PR2) showed good predictions of both the vapor phase and liquid phase densities. The same values of interaction parameters used for the correlation of the densities were used for the correlation of VLE of the system. To extend the idea of linearity of binary interaction parameters, the HPE of binary mixtures of various alcohols, from methanol to decanol, in carbon dioxide were modeled. Fig. 6 shows the VLE prediction of methanol–carbon dioxide at 313.14 K. Table 1 includes the values of kij and lij for various alcohol–carbon dioxide binaries. The interaction parameters, kij = 0.0741 + 0.0026Nalc ; lij = 0.0265 − 0.0374Nalc , were found to vary linearly with the number of carbon atoms in the main chain of alcohol, Nalc . If more experimental data is available, one can investigate how far the linearity extends. Similar to the previous case, the utility of the linearity is demonstrated by determining the values of kij and lij for the carbon dioxide–1butanol system using the linear relations and the VLE prediction is shown against the experimental [25] data (Fig. 7). Thus, the
Fig. 6. Vapor–liquid equilibria for the system carbon dioxide + methanol at 313.14 K. (䊉) Experimental data [24]; (—) PR2 prediction with kij = 0.0777 and lij = −0.0049.
178
S.K. Jha, G. Madras / Fluid Phase Equilibria 238 (2005) 174–179
adjustable binary interaction parameters, kij and lij , vary linearly with the addition of functional groups and, therefore, can be used to predict the HPE of binary mixtures in the homologous series. The parameters, kij and lij , for carbon dioxide–chlorobenzene and carbon dioxide–1-butanol obtained from the linear relationships successfully predict the VLE data. References
Fig. 7. Vapor–liquid equilibria for the system carbon dioxide + 1-butanol at 313.14 K. (䊉) Experimental data [25]; (—) prediction by the proposed model.
Fig. 8. Vapor–liquid equilibria for the systems: carbon dioxide + N,Ndimethylformamide (DMFA), carbon dioxide + N,N-diethylformamide (DEFA), carbon dioxide + N,N-dibutylformamide (DBFA) at 318.2 K. (䊉) Experimental data [27]; (—) PR2 prediction.
model can be used to predict the HPE of binary mixture of carbon dioxide and other alcohols in the homologous series. Finally, the experimental results [27] for the high-pressure vapor liquid equilibria of binary systems of carbon dioxide with N,N-dimethylformamide (DMFA), N,N-diethylformamide (DEFA) and N,N-dibutylformamide (DBFA) is modeled (Fig. 8). The binary interaction parameters kij and lij are shown in Table 1 and these interaction parameters, kij = 0.05 + 0.0025Nam ; lij = −0.145 − 0.0182Nam , were found to vary linearly with the number of carbon atoms in the main chain of the amide, Nam . 4. Conclusions The high-pressure phase equilibrium in binary mixtures of carbon dioxide and benzene, toluene, m-xylene, chlorobenzenes, amides and various low molecular weight alcohols has been modeled. Peng–Robinson equation of state with two parameter quadratic mixing rules has been used to estimate the mixture phase equilibria. The key conclusion of the study is that the
[1] A.S. Teja, C.A. Eckert, Commentary on supercritical fluids: research and applications, Ind. Eng. Chem. Res. 39 (2000) 4442. [2] M. Sihvonen, E. Jarvenpaa, V. Hietaniemi, R. Huopalahti, Advances in supercritical carbon dioxide technologies, Trends Food Sci. Technol. 10 (1999) 217. [3] B. Subramaniam, R.A. Rajewski, K. Snavely, Pharmaceutical processing with supercritical carbon dioxide, J. Pharm. Sci. 86 (1997) 885. [4] D.E. Knowles, W.D. Felix, N.L. Porter, B.A. Jones, D.L. Knowles, Supercritical fluid chromatography applications for the petroleum industry, in: Proceedings of the International Symposium on Fuels and Lubricants, New Delhi, 2000, p. 367. [5] R. Dohrn, G. Brunner, High-pressure fluid-phase equilibria: experimental methods and systems investigated, Fluid Phase Equilib. 106 (1995) 213. [6] M. Christov, R. Dohrn, High-pressure fluid phase equilibria experimental methods and systems investigated (1994–1999), Fluid Phase Equilib. 202 (2002) 153. [7] S.M. Walas, Phase Equilibria in Chemical Engineering, Butterworths, MA, 1985. [8] J.O. Valderrama, The state of the cubic equations of state, Ind. Eng. Chem. Res. 42 (2003) 1603. [9] S.D. Fink, H.C. Hershey, Modeling the vapor–liquid equilibria of 1,1,1trichloroethane + carbon dioxide and toluene + carbon dioxide at 308, 323 and 353 K, Ind. Eng. Chem. Res. 29 (1990) 295. [10] J.M. Prausnitz, R.N. Lichtenthaler, E.G. de Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria, Prentice Hall PTR, New Jersey, 1999, p. 718. [11] G. Madras, C. Kulkarni, J. Modak, Modeling the solubilities of fatty acids in supercritical carbon dioxide, Fluid Phase Equilib. 209 (2003) 207. [12] S.K. Jha, G. Madras, Modeling of adsorption equilibria in supercritical fluids, J. Supercrit. Fluid 32 (2004) 161. [13] H.-J. Ng, D.B. Robinson, Equilibrium phase properties of the toluene–carbon dioxide system, J. Chem. Eng. Data 23 (1978) 325. [14] H.M. Lin, Peng–Robinson equation of state for vapor–liquid equilibrium calculations for carbon dioxide + hydrocarbon mixtures, Fluid Phase Equilib. 16 (1984) 151. [15] S. Guha, G. Madras, Modeling of ternary solubilities of organics in supercritical carbon dioxide, Fluid Phase Equilib. 187 (2001) 255. [16] M.A. McHugh, V.J. Krukonis, Supercritical Fluid Extraction Principles and Practice, Butterworth-Heinemann, MA, 1994, p. 127. [17] D.Y. Peng, D.B. Robinson, A new two-constant equation of state, Ind. Eng. Chem. Fund. 15 (1976) 59. [18] T.Y. Kwak, G.A. Mansoori, Van der Waal’s mixing rules for cubic equations of state, applications for supercritical fluid extraction modeling, Chem. Eng. Sci. 41 (1986) 1303. [19] W.H. Press, S.A. Teukolsky, W.T. Vellerling, B.P. Flannery, Numerical Recipes in C, Cambridge University Press, 1992, p. 683. [20] M.K. Gupta, Y.-H. Ll, B.J. Hulsey, R.L. Robinson Jr., Phase equilibrium for carbon dioxide–benzene at 313.2, 353.2, and 393.2 K, J. Chem. Eng. Data 27 (1982) 55. [21] H.-J. Ng, Sam S.-S. Huang, D.B. Robinson, Equilibrium phase properties of selected m-xylene binary systems. m-xylene–methane and mxylene–carbon dioxide, J. Chem. Eng. Data 27 (1982) 119. [22] D. Walther, G. Maurer, High pressure vapour–liquid equilibria in binary mixtures of carbon dioxide and benzaldehyde, bromobenzene, chlorobenzene 1,2-dichlorobenzene and 2-chloro-1-methyl benzene at
S.K. Jha, G. Madras / Fluid Phase Equilibria 238 (2005) 174–179 temperatures between 313 and 393 K and pressures up to 22 MPa, Ber. Bunsen. Phys. Chem. 96 (1992) 981. [23] W.M. Cross Jr., A. Akgerman, Single component and mixture solubilities of hexachlorobenzene and pentachlorophenol in supercritical carbon dioxide, Ind. Eng. Chem. Res. 37 (1998) 1510. [24] 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: carbon dioxide–methanol, carbon dioxide–ethanol, carbon dioxide–1-propanol, methane–ethanol, methane–1-propanol, ethane–ethanol, and ethane–1-propanol systems, J. Chem. Eng. Data 35 (1990) 63.
179
[25] T. Hiaki, H. Miyagi, T. Tsuji, M. Hongo, Vapor–liquid equilibria of supercritical carbon dioxide + butanol systems at 313.2 K, J. Supercrit. Fluid 13 (1998) 23. [26] C. Chiehming, J.C. Kou-Lunga, D. Chang-Yihb, A new apparatus for the determination of P–x–y diagrams and Henry’s constants in high pressure alcohols with critical carbon dioxide, J. Supercrit. Fluid 12 (1998) 223. [27] H.-S. Byun, N.-H. Kim, C. Kwak, Measurements and modeling of highpressure phase behavior of binary CO2 –amides systems, Fluid Phase Equilib. 208 (2003) 53.