Ind. Eng. Chem. Res. 1989, 28, 1073-1081 Lee, J. I.; Otto, F. D.; Mather, A. E. Equilibrium in Hydrogen Sulfide-Monoethanolamine-Water System. J . Chem. Eng. Data 1976b,21, 2, 207. Mahajani, V. V.; Dankwerts, P. V. Carbamate-bicarbonate Equilibrium for Several Amines at 100 OC in 30% Potash. Chem. Eng. Sci. 1982, 37(6), 943. Matin, N. B.; Danov, S. M.; Efremov, R. V. Vapor Pressures in the Systems Diethanolamine-Water and Triethanolamine-Water. Tr. Khim. Khim. Tekhnol. 1969,2, 7. Maurer, G. On the Solubility of Volatile Weak Electrolytes in Aqueous Solutions. In Thermodynamics of Aqueous Systems with Industrial Applications; Newman, S . A,, Ed.; ACS Symposium Series 133; American Chemical Society: Washington, DC, 1980; pp 139-186. Meissner, H. P.; Tester, J. W. Activity Coefficients of Strong Electrolytes in Aqueous Solutions. Ind. Eng. Chem. Process Des. Deu. 1972, 11(1), 128. Meissner, H. P.; Kusik, C. L.; Tester, J. W. Activity Coefficients of Strong Electrolytes in Aqueous Solution-Effect of Temperature. AIChE J . 1972, 18(3), 661. Meyer, B.; Ward, K.; Koshlap, K.; Peter, L. Second Dissociation Constant of Hydrogen Sulfide. Inorg. Chem. 1983,22, 2345. Mock, B.; Evans, L. B.; Chen, C. C. Thermodynamic Representation of Phase Equilibria of Mixed-Solvent Electrolyte Systems. AIChE J . 1986, 32(10), 1655. Muhlbauer, H. G.; Monaghan, P. R. Sweetening Natural Gas With Ethanolamine Solutions. Oil Gas J . 1957, 55(17), 139. Nath, A.; Bender, E. Isothermal Vapor-Liquid Equilibria of Binary and Ternary Mixtures Containing Alcohol, Alkanolamine, and Water with a New Static Device. J . Chem. Eng. Data 1983, 28, 370. Pitzer, K. S. Thermodynamics of Electrolytes. I. Theoretical Basis and General Equations. J . Phys. Chem. 1973, 77(2), 268.
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Pitzer, K. S. Electrolytes. From Dilute Solutions to Fused Salts. J. Am. Chem. SOC.1980, 102(9), 2902. Prausnitz, J. M.; Anderson, T. F.; Grens, E. A.; Eckert, C. A.; Hsieh, R.; O'Connell, J. P. Computer Calculations for Multicomponent Vapor-Liquid Equilibria and Liquid-Liquid Equilibria; Prentice Hall: Englewood Cliffs, NJ, 1980. Redlich, 0.;Kwong, J. N. S. On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions. Chem. Reo. 1949, 44, 233. Renon, H.; Prausnitz, J. M. Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures. AIChE J. 1968,14(1), 135. Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworth and Co.: London, 1970. Sander, B.; Fredenslund, A.; Rasmussen, P. Calculation of VapourLiquid Equilibria in Mixed Solvent/Salt Systems Using An Extended Uniquac Equation. Chem. Eng. Sci. 1986, 41(5), 1171. Scauflaire, P.; Richards, D.; Chen, C. C. Ionic Activity Coefficients of Mixed-Solvent Electrolyte Systems. AIChE J. 1989, submitted. Sivasubramanian, M. S.; Sardar, H.; Weiland, R. H. Simulation of Fully-Integrated Amine Units For Acid Gas Removal. Presented at the AIChE National Meeting, Houston, TX, 1985; paper 87e. Soave, G. Equilibrium Constants from a Modified Redlich-Kwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197. Touhara, H.; Okazaki, S.; Okino, F.; Tanaka, H.; Ikari, K.; Nakazishi, K. Thermodynamic Properties of Aqueous Mixtures of Hydrophilic Compounds 2. Aminoethanol and its Methyl Derivatives. J . Chem. Thermodyn. 1982, 14, 145. Van Krevelen, D. W.; Hoftijzer, P. J.; Huntjens, F. J. Composition and Vapour Pressures of Aqueous Solutions of Ammonia, Carbon Dioxide and Hydrogen Sulfide. Recueil 1949, 68, 191.
Received for review July 12, 1988 Accepted March 20, 1989
Measurement and Model Prediction of Vapor-Liquid Equilibria of Mixtures of Rapeseed Oil and Supercritical Carbon Dioxide Thomas Kleint and Siegfried Schulz* Lehrstuhl f u r Thermodynamik, Abteilung Chemietechnik, Unioersitat Dortmund, 0-4600 Dortmund 50, Federal Republic of Germany
Isothermal vapor-liquid equilibrium data of the rapeseed oil-carbon dioxide system were measured with a vapor-recirculation apparatus a t 313.15, 333.15, 353.15, and 373.15 K a t pressures between 10 and 85 MPa. T h e solubilities of rapeseed oil in carbon dioxide were compared with solubility data of other vegetable oils from the literature. A lattice model equation of state (EOS) recently developed by Kumar e t al. was used to describe the rapeseed oil-carbon dioxide system. T h e lattice EOS was improved by introducing a more accurate EOS for carbon dioxide (Bender EOS) into the model. With the Kumar-Bender model and a three-binary-parameter, composition- and densitydependent combining rule proposed by Wilczek-Vera and Vera, a quantitative description of phase equilibrium was obtained. T h e temperature dependence of the binary parameters was evaluated for interpolation or extrapolation purposes. The considerable interest in the development of industrial processes using supercritical fluids is shown by many publications (e.g., Brunner (1987), Stahl et al. (1987), and McHugh and Krukonis (1986)). Recent research and development activities on supercritical fluid processing in the fat and oil industry are reviewed by Brunner (1986). Examples are the extraction (Eggers et al., 1985; Friedrich and Pryde, 1984) and refining (Coenen and Kriegel, 1983) of vegetable oils, the separation of mono- and diglycerides (Riha, 1976), fatty acids (Tiegs and Peter, 1985) or fatty acid methyl esters (Wu et al., 1987; Ikushima et al., 1988), and the separation of lecithin and soybean oil (Peter et al., 1987). Carbon dioxide is a particularly desirable su'Present address: Department of Chemical Engineering, University of California, Berkeley, CA 94720.
percritical solvent because it is nonreactive, nontoxic, and available at high purity and at low cost. Alternative supercritical solvents which are considered in the fat and oil industry are propane, mixtures of propane and carbon dioxide, or mixtures with entrainers. For the development of supercritical fluid extraction processes, the knowledge of the relevant phase equilibria over a wide range of temperatures and pressures is of fundamental importance. The considerable expense of phase equilibrium measurements can be reduced, when reliable thermodynamic models are available for the calculation of the phase equilibria. Furthermore, for economic considerations, an analysis of the energy requirements of the process is needed. If the concentration of the solutes in the supercritical solvent is low, the extraction process may be analyzed
0888-5885/89/2628-1073$01.50/0 0 1989 American Chemical Society
1074 Ind. Eng. Chem. Res., Vol. 28, No. 7 , 1989
1 = 343.15 K
0
0.6
without taking the solutes into account. The thermodynamic properties of the pure solvents commonly encountered in supercritical fluid extraction are well-known and may be calculated by accurate semiempirical or empirical multiconstant equations of state (EOS’s). For higher solute concentrations, however, the calculations must be carried out for the mixtures. The objective of this work is to develop a method that will allow accurate EOS’s for supercritical fluids to be connected with other EOS’s suitable for the description of the thermodynamic properties of the solutes. We use the Bender EOS (Bender, 1970) for the supercritical solvent and a recently published statistical mechanics-based lattice model EOS (Kumar et al., 1986, 1987a,b) for the solutes. The capability of this method in the modeling of vapor-liquid equilibria of asymmetric mixtures is examined by applying it to the rapeseed oil-carbon dioxide system. The phase equilibria of mixtures of rapeseed oil and carbon dioxide were measured in a vapor-recirculation apparatus.
Experimental Section Apparatus and Operations. In the past few years, mainly solubilities of vegetable oils in carbon dioxide were published (Stahl et al., 1983; Friedrich, 1984; Lack, 1985). Only in a few cases is the composition of the coexisting liquid phases given (Brunner, 1978; Zobel, 1985; Kalra et al., 1987). Therefore, we completed the experimental data by measuring vapor-liquid equilibria in the rapeseed oil-carbon dioxide system. The vapor-recirculation apparatus for measurement of vapor-liquid equilibria a t temperatures from ambient to 423 K and pressures to 85 MPa built for this investigation is shown in Figure 1. The vapor phase is continuously recirculated by a magnetic pump to achieve thorough mixing of the equilibrium phases. To avoid entrainment of droplets, a droplet separator made of borosilicate glass fiber tissue is installed in the upper half of the cell. Separating a part of the recirculation line from the equilibrium cell, a large vapor sample can be removed without disturbing the equilibrium pressure in the cell. Samples of both equilibrium phases can also be withdrawn directly from the cell through two sampling capillaries. If the density differences between the coexisting phases become too small for complete separation of the phases in the droplet separator, these capillaries are used for both liquid and vapor sampling. On-line sampling with a gas chromatograph might be very difficult because of the low oil solubilities and the
0.7
0.8
-
0.9
1.0
JICO, / Figure 2. Vapor-liquid equilibrium of the squalane-carbon dioxide This work; (A)Lohrl-Thiel (1978); (*) interpolated from system. (0) Liphard and Schneider (1975); ( X ) Brunner (1978); (-) calculated with the mean-field lattice-gas model (Kleintjens and Koningsveld, 1980; Marouschek, 1985).
extremely different volatilities of carbon dioxide and the vegetable oil. Therefore, the withdrawn samples are expanded into the separation and sampling section and separated into a gaseous carbon dioxide and a liquid vegetable oil fraction in cooling traps. The vegetable oil remaining in the sampling capillaries can be flushed into the cooling traps with a volatile solvent. Subsequently, the solvent is evaporated in a vacuum shelf dryer at 323 K. The difference between the boiling points of the solvent (n-pentane) and the rapeseed oil is sufficient to make sure that the rapeseed oil remains completely in the cooling traps. The collected liquid is then weighed on an analytical balance. The gaseous phase is trapped in glass cylinders of known volume. When the pressure and temperature of the gaseous phase are measured, the amount of carbon dioxide is calculated from its p-V,-T properties. The system pressure is measured within an accuracy of 3~0.03MPa by means of a pressure transducer, which was previously calibrated with a precision dead-weight gage. The temperature is measured in the oil bath by means of a Pt25 resistance thermometer. The oil bath temperature is controlled within fO.O1 K by a temperature controller. The recirculation line for the vapor phase is maintained at the equilibrium cell temperature by heating tapes. Details of the apparatus and its operation are described elsewhere (Klein, 1988). Test Measurements. To verify the attainment of equilibrium, the apparatus has been tested with mixtures of carbon dioxide and squalane. The carbon dioxide used in this work was obtained from Messer-Griesheim with a purity of 99.995% and squalane from Riedel-de Haen with a 99% purity. As shown in Figure 2, the new data of the vapor phase agree well with the results obtained by Lohrl-Thiel (1978) from a static apparatus and the values of Liphard and Schneider (1975), which were obtained from a synthetic optical method. In the liquid phase, our results are located between those of Lohrl-Thiel and Liphard and Schneider but agree with the data of Brunner (1978) from a synthetic method. Brunner obtained his results by recording the p V curves, which show a sudden change in compressibility at the phase boundary. The accuracy of our test measurements was better than fO.OO1 mole fraction in the vapor phase and f0.004 mole fraction in the liquid phase. Rapeseed Oil-Carbon Dioxide System. For this investigation, refined rapeseed oil with a low content of erucic acid was used. The vapor-liquid equilibria of the
Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989 1075 Table I. Vapor-Liquid Equilibria of the Rapeseed Oil (1)-Carbon Dioxide (2) System, Liquid P h a s e P, MPa wcoz Awcoz T = 313.15 K 0.2771 -3.92 x 10-3 9.92 14.18 0.3061 5.81 x 10-3 0.3073 -1.09 x 10-3 16.11 0.3224 7.82 x 10-4 19.49 0.3346 -3.13 x 10-4 23.22 0.3557 2.50 x 10-3 29.07 0.3600 -4.80 x 10-3 33.35 0.3909 2.02 x 104 45.28 0.4015 6.40 X lo4 51.20 0.4068 4.33 x 10-4 54.83 0.4063 -5.68 x 10-4 55.21 0.4142 -1.23 x 10-3 61.49 0.4150 -2.11 x 10-3 62.75 0.4225 1.67 x 10-3 65.77 0.4260 3.98 x 10-3 66.67 0.4256 -9.25 x 10-4 70.35 0.4386 1.96 x 10-3 78.55 0.4350 -2.75 x 10-3 79.39 0.4365 -1.29 x 10-3 79.45 1.03 x 10-3 0.4430 82.62 0.2318 0.2587 0.2857 0.3156 0.3394 0.3584 0.4028 0.3841 0.4366 0.4568 0.4621 0.4727 0.4617 0.4843 0.4925 0.5102 0.5174
3.02 X 6.27 x -1.81 x 2.36 x 6.62 x -8.68 X 8.73 x -1.15 X -3.76 x 1.05 X -1.80 x 3.64 x -8.32 x -3.65 X 2.58 x -1.13 x 7.04 x
lo4
2.18 x -1.87 x 3.42 x -1.80 x 3.68 x -5.15 x -2.35 x 4.43 x -3.23 x 1.97 x -2.43 x -3.09 x 4.89 x -1.61 x
10-4 10-3 10-3 10-3 10-3 10-3 10-3 10-3 10-4 10-3 10-3 10-3 10-3 10-3
10-4 10-3 10-3 10-4
0.1628 0.2094 0.2560 0.2830 0.3242 0.3405 0.3719 0.4061 0.4501 0.4798 0.5025 0.5561 0.5803 0.5904
10-3 lo-' 10-3 lo-' 10-3 10-3 10-3 lo4 10-3 10-3 10-4
T = 373.15 K 9.89 14.99 18.67 24.22 28.52 32.89 39.85 44.33 49.00 54.01 58.68 63.73 70.40 71.36 74.01 78.23
0.1484 0.2067 0.2512 0.2858 0.3193 0.3470 0.3940 0.4204 0.4369 0.4720 0.5058 0.5494 0.6233 0.6239 0.6750 0.7524
f -
0.97
0.98
0.99 w
cot
1.00
/ - -
Figure 3. Solubility of vegetable oils in carbon dioxide at 313.15 K. (-) Rapeseed oil (this work); ( X I rapeseed oil (Lack, 1985); (Y) rapeseed oil (Eggers and Stein, 1984); (0) soybean oil (Stahl et al., 1983); (*) soybean oil (Friedrich, 1984). m
n
f
0
P E \
-
a m
Y
N
lo4
T = 353.15 K 9.89 14.61 19.22 23.27 28.29 32.15 36.89 41.74 50.93 56.35 61.93 74.33 78.83 84.67
GI
"
T = 333.15 K 10.64 14.28 18.78 23.13 27.90 32.12 40.26 40.63 53.47 55.37 61.45 63.30 63.68 69.25 71.34 80.25 82.63
0
-9.22 x 10-4 -1.11 x 10-3 7.17 x 10-3 -4.82 x 10-3 -2.22 x 10-3 -2.48 x 10-3 4.24 x 10-3 5.66 x 10-3 -4.82 x 10-3 -1.17 x 10-3 -6.07 x 10-4 8.34 x 10-4 5.19 x 10-3 -6.87 x 10-3 4.61 x 10-3 2.46 x 10-4
rapeseed oil-carbon dioxide system were measured at 313.15, 333.15, 353.15, and 373.15 K at pressures between
0
0.92
0.94 w
0.96 / - -
0.98
1.00
coz
Figure 4. Solubility of vegetable oils in carbon dioxide at 353.15 K. (-) Rapeseed oil (this work); (Y) rapeseed oil (Eggers and Stein, 1984); (0)soybean oil (Stahl et al., 1983); (*) soybean oil (Friedrich, 1984); (A) palm oil at 348.15 K (Zobel, 1985); ( X ) palm oil at 348.15 K (Brunner, 1978).
10 and 85 MPa. The mass fractions obtained are listed in Tables I and 11. Additionally, polynomials were fitted to the reported data. The accuracy of the data, estimated as the difference between the experimental value and the mass fraction calculated by the appropriate polynomial, is also listed in the tables. Phase inversion occurs at 313.15 K at about 40 MPa, at 333.15 K at about 50 MPa, and at 353.15 K a t about 70 MPa. Because of the change in density of pure carbon dioxide with temperature and pressure, the barotropic point shifts to higher pressures with rising temperature. A t 373.15 K, the mixture's critical point is reached at approximately 80 MPa. In the liquid phase, the deviations between the mass fractions measured and the mass fractions calculated by the polynomials are less than f0.005 except at 333.15 K, in the vicinity of the barotropic point. There, the deviations increase to 0.012. Because of the low solubility of rapeseed oil in carbon dioxide, the deviations in the vapor phase are lower than in the liquid phase and do not exceed h0.003 mass fraction generally. At 373.15 K, the deviations increase up to =k0.008 mass fraction in the near-critical region. The larger inaccuracies are due to incomplete phase separation in the near-critical and near-barotropic region. Figures 3 and 4 compare our results with solubility data of other investigators. The solubilities of rapeseed oil in carbon dioxide measured by Lack (1985) confirm our values. Because of its lower mean molecular weight, the solubilities of palm oil (Brunner, 1978: Zobel. 1985) are higher than thbse of rapeseed oil.' Stahi et al. (1983) and Friedrich (1984) applied dynamic methods to determine
1076
Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989
Table 11. Vapor-Liquid Equilibria of the Rapeseed Oil (1)-Carbon Dioxide (2) System, Vapor Phase
p, MPa
uoii
10.11 14.22 18.56 23.58 29.14 33.57 42.74 43.45 58.41
2.44 x 10-4 5.60 x 10-4 2.51 x 10-3 4.81 x 10-3 6.70 x 10-3 9.32 x 10-3 0.0127 0.0139 0.0188
p, MPa T = 313.15 K 4.15 X 62.21 -2.64 X 67.54 8.34 X 69.32 1.34 X 70.82 -5.11 X 71.41 1.11 X 76.88 -4.47 X 78.51 4.31 X 80.34 5.18 x 10-5
10.70 14.48 15.52 18.27 23.11 27.93 32.16 35.80 38.43 40.59 43.76 45.48
5.40 x 10-4 4.92 X lo4 2.30 x 10-4 1.70 x 10-3 3.68 x 10-3 6.04 x 10-3 8.56 x 10-3 0.0130 0.0145 0.0160 0.0179 0.0212
T = 333.15 K 1.04 X 10'' 49.90 -1.44 X 58.11 -4.38 X 58.99 3.42 X 66.21 2.90 X 67.02 -2.09 X 69.28 -7.41 X low4 72.93 8.14 X 73.20 1.70 X 73.27 -1.70 X 79.10 -1.06 X lo" 80.11 6.61 x 10-4
9.90 14.62 19.19 23.29 28.26 32.18 36.86 41.77 47.03
6.93 X 6.29 x 5.94 x 1.16 x 3.55 x 5.92 x 9.08 x 0.0144 0.0176
lo4
T = 353.15 K 1.25 X 49.19 -7.76 X 54.50 61.99 -2.45 X -2.68 X 63.54 4.00 X 72.17 7.07 X 73.71 76.46 5.27 X 1.20 x 10-3 77.77 -1.96 X 81.65
9.91 15.28 19.71 24.24 28.54 32.91 39.86 44.39 48.97
6.35 x 5.45 x 5.22 x 7.88 x 2.52 x 4.87 x 0.0125 0.0211 0.0309
10-4 7.61 X 10-4 -6.04 X 10-4 -1.21 X 10-4 2.36 X 10-3 3.96 X 10-3 -4.01 X
AWOIl
10-4 10-4 10-3 10-3 10-3 10-3
Woii
AWoii
0.0209 0.0205 0.0211 0.0216 0.0206 0.0216 0.0220 0.0226
1.14 x -4.36 x -1.38 x 1.90 x -9.40 x -3.26 x 4.75 x 6.25 x
10-3 10-4 10-4 10-4 10-4 10-4 10-7 10-4
0.0252 0.0323 0.0309 0.0390 0.0387 0.0398 0.0434 0.0451 0.0423 0.0489 0.0470
7.40 x 6.61 x -1.47 x 7.26 x -1.47 x -8.15 x 1.45 x 1.61 x -1.28 x 1.31 x -1.23 x
10-4 10-4 10-3 10-4 10-4 10-4 10-4 10-3 10-3 10-3 10-3
0.0222 0.0294 0.0448 0.0530 0.0756 0.0784 0.0879 0.0898 0.1052
-3.48 x -1.89 x -1.70 x 2.83 x 2.10 x 1.69 x 7.63 x -1.79 x -4.98 x
10-4 10-3 10-3 10-3 10-3 10-4 10-4 10-3 10-4
0.0409 0.0552 0.0756 0.0775 0.1050 0.1215 0.1387 0.1462 0.1789
-1.40 x 10-3 4.97 x 10-4 -1.81 x 10-3 -1.96 x 10-3 1.59 x 10-3 6.84 x 10-3 -6.95 x 10-3 -1.65 x 10-3 3.23 x 10-3
Modeling Modeling of mixtures of molecules of dissimilar size has in the past been performed by the use of either perturbation theory (e.g., Donohu and Vimalchand (1988)) or lattice models (e.g., Kleintjens and Koningsveld (19801, Panayiotou and Vera (1982), Kumar et al. (l986,1987a,b)). Recently, Bamberger et al. (1988) were successful in applying the lattice model of Kumar et al. to mixtures of fatty acids or triglycerides and carbon dioxide a t 313.15 K. In this work, we also use their lattice EOS to model the rapeseed oil-carbon dioxide system. Taking into account the need for an accurate description of the thermodynamic properties of the supercritical solvent, we modify the lattice EOS to allow the use of a more appropriate EOS for the supercritical fluid. For carbon dioxide, we have chosen the Bender EOS (Bender, 1970) instead of the IUPAC EOS (Angus et al., 1976) or the EOS of Huang et al. (1985), because pure-component parameters of the Bender EOS are also available for other potential supercritical solvents, e.g., propane (Buhner et al., 1981). The Bender EOS is able to reproduce the thermodynamic properties of carbon dioxide up to pressures of 100 MPa with good accuracy as shown by Sievers (1986). A method of combining different EOS's was described by Peneloux et al. (1986) and Rauzy and Peneloux (1986), who used the IUPAC EOS for carbon dioxide and the Peng-Robinson EOS for alkanes to calculate phase equilibria in mixtures of carbon dioxide and alkanes. In accordance with Rauzy and Peneloux (1986), the residual Helmholtz energy of a mixture is
T = 373.15 K
-8.06
X
5.49 x 10-4 1.13 X
54.02 58.16 63.77 64.22 68.59 70.27 74.10 74.33 77.03
the solubilities of soybean oil in carbon dioxide. In most cases, the solubilities obtained by Stahl et al. (1983) and Friedrich and Pryde (1984) are higher than those of rapeseed oil or palm oil, though the mean molecular weights of soybean oil and the rapeseed oil used in this work are similar. We suppose that oil droplets were entrained by the supercritical phase because of small density differences. A t 353.15 K, the values of Stahl et al. (1983) approach our results asymptotically with increasing pressure. A t 70 MPa, where both values agree, the carbon dioxide, which is needed at the beginning of the experiment to generate this pressure in their equilibrium cell, dissolved nearly the maximum amount of soybean oil the carrier material in the cell could be coated with. There was only a small amount of soybean oil remaining on the carrier material so that entrainment became negligible. To our knowledge, there are no liquid-phase solubilities published for the soybean oil-carbon dioxide system. Therefore, proof of the entrainment hypothesis by comparing liquid-phase data is not possible. Additionally, Figures 3 and 4 show solubilities of rapeseed oil in carbon dioxide measured by Eggers and Stein (1984) during extraction of rapeseed oil from seed. A t 313.15 K, their values are higher than our mass fractions but confirm our results in principle. However, at 353.15 K, the mass fraction obtained by Eggers and Stein (1984) is only half of the equilibrium solubility.
i=l
i=l
ALi is the residual Helmholtz energy of pure com onent i at the packing fraction 3 of the mixture and A,Pis the excess Helmholtz energy at constant packing fraction. Constant packing fraction means that rMVH Vm
17=-=--
riVH - constant Vmi
We assign a value of 9.75 X lo4 m3 mol-' (Kumar et al., 1987b) to the lattice unit cell size VH. ri is the chain length of chain molecule i. Each molecule i is assumed to occupy ri lattice cells, so that riVH is the molar hard-core volume of molecule i. riVH is one of the two pure-component parameters of the lattice EOS. r M is calculated by (3) where Jii is the mole fraction of component i. V, and Vmi are the molar volumes of the mixture and the pure component i, respectively. As Kumar et al. (1987b) do not give expressions for the residual Helmholtz energy of a mixture obeying their lattice model, these equations are presented below. The residual Helmholtz energy of a pure component, Ahi, and the excess Helmholtz energy at constant packing fraction, A,: are derived from the canonical partition functions, Q,, of pure components and mixtures described by Kumar et al. (198713) by applying eq 4 and eq 4 and 1, respectively.
A&(T,V,,+) = -R,T
In Q, - A$(T,V,,+)
(4)
In eq 4 superscript ig denotes the ideal gas state. We then obtain
Ind. Eng. Chem. Res., Vol. 28, No. 7 , 1989 1077 with
The residual Helmholtz energy of pure carbon dioxide may now be given by either the lattice EOS (eq 5) or the Bender EOS (eq 16). With eq 5, the original lattice model remains unchanged, while we obtain the Kumar-Bender model when eq 16 is used for carbon dioxide.
where the lattice coordination number is set to z = 10. zqi is a measure of the effective number of external contacts per molecule i and is related to the molecule’s chain length, riy by zqi = ( z - 2)ri 2 (7)
+
The lattice consists of lattice sits occupied by the molecules and empty sites called holes. GOi (eq 8) and 8 (eq 9,
(9)
the effective surface area fractions of all molecules in the pure-component lattice and the mixture lattice, respectively, represent the ratio of occupied t~ empty lattice sites on a “surface area” basis, while the packing fraction, 1, defines this ratio on a “volume” basis. The surface area fractions on a “hole-free” basis, 8i, given by eq 11 are used to describe the composition of mixtures. ai = +iqi/qM (11) The square-well potential, tii, is the second pure-component parameter of the lattice EOS. The interaction energies between segments of unlike molecules, ti,, are given by the combining rule (12) t i j = 0 . 5 ( ~+ i tjj)(l - k ~ j ) The binary parameter kij is used to describe the deviation of tij from the arithmetic mean of tii and e,? As the lattice EOS allows molecular segments to mix nonrandomly by following the quasi-chemical treatment of Guggenheim (1952), k , is also a measure of the local composition about a molecular segment. When all ki:s are zero, the mixture is random and the local surface area fractions on a hole-free basis are identical with the bulk surface area fractions, With increasing ki;s, the degree of nonrandomness is rising. Instead of solving the quasi-chemical equations, which are used to calculate the local surface area fractions directly, the nonrandomness corrections, rij, are determined from the general first-order solution (eq 13) of the quasi-chemical equations published by Kumar et al. (1987b).
ai.
n
n
B = alT - a 2 - a3/T - a 4 / T 2 - a 5 / T 3 c = a6T + a7 + a,/T
(16b)
D = a9T + a,, E = a l l T + a12 F = a13
(16d) (16e)
G = a14/T2 + a15/T3 + a16/T4
(16g)
H = a17/T2 + aI8/T3 + a19/T4
(16h)
(164
( 16f)
In eq 16, the molar density is expressed as a function of the packing fraction, 71. The hard-core volume introduced into the Bender EOS in this way may be chosen as desired. Here it is obvious to use the hard-core volume of the lattice
EOS. The expressions for the pressure and fugacity coefficients of the mixture components are derived from the common thermodynamic equations, assuming that the binary parameter is independent of density or composition. Pure-ComponentParameters. Since we use the same values for the lattice unit cell size, VH,and the coordination number, 2, as did Kumar et al. (198713) and Bamberger et al. (1988), we can adopt the pure-component parameters of carbon dioxide from Bamberger et al. Pure-component parameters of saturated triglycerides were determined as a function of the triglyceride carbon number, which is the number of carbon atoms in the fatty acid rests. For this, we used the vapor pressure equations of Mathias et al. (1986) and liquid densities published by Formo (1979). Because experimental data about pure triglycerides are scarce, we were not able to distinguish between triglycerides with different degrees of unsaturation or even between isomers. Therefore, the rapeseed oil is treated as a triglyceride with an effective carbon number equal to the mean carbon number of the oil. This carbon number was evaluated by determination of the triglyceride molecular weight distribution with gas chromatography. In the analysis, the triglycerides were separated with regard to their carbon number only. The value of the effective carbon number of the rapeseed oil, 53.4, is also representative of other vegetable oils. For example, we calculated effective carbon numbers of 53.3 for corn oil, 53.5 for soybean and sesame oil, and 53.6 for sunflower and safflower oil based on triglyceride compositions given by D’Alonzo et al. (1982). The rapeseed oil parameters evaluated as well as the carbon dioxide parameters of the lattice EOS and the Bender EOS are listed in Table 111.
1078 Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989 Table 111. Pure-Component Parameters of Rapeseed Oil a n d Carbon Dioxide T , K c;,/R,, K riV", m3 mol" substance rapeseed oil (this work) 313.15 102.881 878.606 X 10+ 333.15 104.440 878.606 X lob 353.15 105.968 878.606 X lob 373.15 107.456 878.606 X lo4 COZ (Bamberger et al., 310-375 82.026 36.0822 X lob 1988) Carbon Dioxide (Severs, 1984) J cm3 g'z K-' a1 = 0.224 885 58 a2 = 137.17965 J cm3 g-2 a3 = 14430.214 J K cm3 g-2 a4 = 2963049.1 J K2 cm3 g-2 ab = 206060390 J K3 cm3 g-2 J cm6 g-3 K" a6 = 0.045 554 393 a7 = 77.042840 J cm6 g-3 = 40 602.371 J K cm6 g-3 J cm9 g-4 K-' 0 9 = 0.400 295 09 alo = -394.36077 J cm9 g-4 all = 0.121 152 86 J cm12 g+ K-' a12 = 107.83386 J cmlZg-' aI3 = 43.962336 J cm15 gb J K2 cm6 g-3 a14 = 36 505 545 J K3 cm6 g-3 a15 = 0.194905 11 X 10" a16 = -0.291 867 18 x l O I 3 J K' cm6 g-3 J K2 cm12 g-5 a17 = 24 358 627 J K3 cml2 g-5 a18 = -0.37546530 X 10" aI9 = 0.11898141 X 1014 J K4 cm12g'5 a20 = 5.0 cm6 g-*
0.2
0.4 w
0.6
0.8
~
0
,
,
,
0.2
,
,
,
,
0.4 w
,
,
,
3.6
,
,
,
,
0.8
,
,
,
,
:.0
/ - - + CD?
Figure 6. Vapor-liquid equilibrium of the rapeseed oil-carbon dioxide system a t 373.15 K. (0)Smoothed experimental data; (-1 Kumar-Bender model, k12 = 0.101 484; (- - -) lattice model, k12 = 0.107 690.
i
0
__I/
4 ' . O
0
a
Q
1.0
/ - co2
Figure 5. Vapor-liquid equilibrium of the rapeseed oil-carbon dioxide system a t 313.15 K. (0)Smoothed experimental data; (-) Kumar-Bender model, klZ = 0.109 711; (- - -) lattice model, k,, = 0.112 789.
Results and Discussion. The expressions derived for the residual Helmholtz energy of a mixture were used to model the phase equilibria of the rapeseed oil-carbon dioxide system. Model fitting is carried out by minimizing the sum of squares of relative deviations between experimental data and calculated mole fractions. Because of the solubility of rapeseed oil in carbon dioxide is low, the absolute deviations between measured and evaluated mole fractions are likewise low. Consequently, minimizing the sum of squares of absolute deviations leads to a good representation of liquid-phase compositions at the cost of the accuracy of calculated vapor-phase mole fractions. However, with regard to supercritical fluid extraction, we are more interested in an accurate description of the vapor-phase composition. First we compared both EOS's, the lattice model and the Kumar-Bender model, using binary parameters which were separately fitted to isothermal data. Our results are shown in Figures 5 and 6. We present only the coexistence curves at 313.15 and 373.15 K, because the computed model behavior a t 333.15 and 353.15 K is similar to that at 313.15 K. In all cases, agreement is only poor. With
either EOS, a two-phase region open to high pressures is predicted at 313.15, 333.15, and 353.15 K. This seems to be correct at 313.15 and 333.15 K, while mixture critical points exist at 353.15 and 373.15 K. At 373.15 K, a mixture critical point is calculated only with the Kumar-Bender model. Otherwise, the two-phase region remains open to high pressures. The deviations between the coexisting curves calculated with the two models increase with rising pressure. This is because the density of the vapor phase does not increase fast enough, when the lattice EOS is applied. As the Kumar-Bender model gives slightly better results, this EOS is used in our further examinations. The method of Rauzy and Peneloux (1986) cannot be responsible for the lack of fit because this method does not affect the EOS or the fugacity expression, when the lattice model is used for all mixture components. In this case the original lattice model remains unchanged. In calculations published elsewhere (Klein, 1988), we considered the rapeseed oil to be a mixture of 10 triglycerides. T o reduce the number of binary parameters, which must be fitted to the experimental data, we evaluated the binary triglyceride-carbon dioxide parameters assuming the interaction energy between different triglycerides to obey the arithmetic mean. As the model behavior was only slightly improved, we supposed that the simple combining rule (eq 12) is the source of the poor modeling results. Therefore, we made K,, a linear function of pressure, molar density, or composition of the coexisting phases as suggested by Mohamed et al. (1987), Stryjek and Vera (1986), or Panagiotopoulos and Reid (1986). Though model behavior is improved by these mixing rules, the agreement with the experimental data remains poor. Recently, Wilczek-Vera and Vera (1987) published a three-binary-parameter, density- and composition-dependent combining rule. Stryjek and Vera (1986) and Panagiotopoulos and Reid (1986) use mole fractions to describe the composition dependence. Though the phase diagrams shown here suggest that the compositions of the coexisting phases cover a wide range of concentrations, the carbon dioxide mole fractions lie only between 0.87 and 1. Carbon dioxide mass fractions of 0.5 and 0.8 correspond to mole fractions of 0.9525 and 0.9877, respectively. Therefore, we substituted surface fractions, a,, for mole fractions in the combining rule of Wilczek-Vera and Vera. In this work, we have used the so-called van Laar version (eq 17), because it performs slightly better than the
~
Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989 1079 0
0
N
D
.
m
o
\
,
,
30 3
,
/
,
,
323
,
\
,
343
,
,
\
363
,
,
o
,
~
0
383
,
,
,
0.2
,
,
,
0.4
,
,
,
0.8
,
,
,
1.0
/ - -
w
T / K -
,
0.6
cog
Figure 7. Binary parameters lZ1, 112, and mI2as a function of temperature. (- - -) Linear temperature dependence obtained from fit to isothermal data; (-) eq 18.
Figure 9. Vapor-liquid equilibrium of the rapeseed oil-carbon dioxide system at 333.15 and 373.15 K. (0)Smoothed experimental data; (-) calculated with eq 18.
0
P
0
H
\
c
E]
A
o
J
0
,
0
313.15 K 353.15 K
,
0.2
,
t
,
,
0.4 w
,
,
,
0.6 / - - +
,
,
0.8
I
,
,
1.0
,
,
0 3
,
I
O O
25
50 p /
co2
nPa
-
I
75
100
Figure 8. Vapor-liquid equilibrium of the rapeseed oil-carbon dioxide system a t 313.15 and 353.15 K. (0)Smoothed experimental data; (-) calculated with eq 18.
Figure 10. Change of binary parameters k,, in the coexisting phases Optimal values in the liquid phase; with pressure at 353.15 K. (0) (A) optimal values in the vapor phase; (-) calculated with eq 18.
equivalent Margules version. In eq 17, pm' is the molar density of the liquid phase and pm is the molar density in the phase we are looking at. All three binary parameters, 121, 112, and m12,were first fitted separately to the isothermal data. Figure 7 shows that 112 and m12increase linearly with rising temperature (dashed lines), while a linear function is not sufficient for the temperature dependence of 121. Employing linear functions for 112 and m12and an exponential temperature dependence for 121, we fitted the function parameters to the complete experimental data. The resulting temperature dependences, shown in Figure 7, are given in eq 18.
obtained from fitting k12 in the liquid and vapor phases separately to the experimental data of each pressure. In the liquid phase, k12 decreases with increasing pressure. In the vapor phase, k12rises steeply at first, passes through a maximum, and then decreases a t higher pressures. The behavior of the rapeseed oil-carbon dioxide system shown in Figure 10 can be explained, when we consider the meaning of k, in the quasi-chemical theory used in the lattice model. There, k, is a measure of local ordering. Therefore, the decrease of kjj in the liquid phase indicates that the molecules mix more randomly at higher pressures. This is expected from computer simulation data (e.g., Lee and Chao (1986)). Computer simulations have shown that local ordering decreases with increasing density. In the liquid rapeseed oil-carbon dioxide mixture, densities are higher the more carbon dioxide is dissolved in the rapeseed oil. In the combining rule (eq 17), the change of kij in the liquid phase is described by a composition dependence. This is reasonable, because the liquid rapeseed oil-carbon dioxide mixture is very incompressible and because changes in composition influence the density more than pressure changes. For the supercritical phase, the decrease of kij at high pressures may also be explained by the rising density. In this case, the changes in density are caused by pressure instead of composition because of the higher compressibility of the supercritical fluid. Furthermore, the behavior of the supercritical rapeseed oil-carbon dioxide mixture is also in accordance with recent experimental results. Kim and Johnston (1987a,b) investigated clustering in supercritical fluid mixtures using spectroscopic measurements. They observed that the supercritical solvent clusters about the solute such that the local density exceeds
121 = 0.173235 - 0.004327 exp[0.741073 X 10-4t2.24516 1 (184
+ 0.280946 X 10-3t m12= 0.032604 + 0.366640 X 10-3t 112 = 0.063016
(18b) (18c)
In Figures 8 and 9, the model behavior is compared to the experimental data at all temperatures. The model is now in good agreement with the measured data over the whole temperature range and pressure range, though the curvature of the coexisting curves seems to be too large at 313.15 K. Phase inversion observed during the experiments is predicted by the model at 313.15 K and 37.32 MPa, a t 333.15 K and 52.97 MPa, and at 353.15 K and 69.12 MPa. Consequently, the barotropic points calculated are in good agreement with the experimental results. For example, in Figure 10, the binary parameter k12in the coexisting phases is shown as a function of pressure at 353.15 K. While the lines were calculated from eq 18, the binary parameters represented by the symbols were
,
,
,
,
1080 Ind. Eng. Chem. Res., Vol. 28, No. 7 , 1989
the bulk density. They also found that the ratio of the local density to the bulk density increases with rising compressibility. Randolph et al. (1988) studied aggregate formation in mixtures of cholesterol and carbon dioxide by means of EPR spectroscopy. They observed maximum local concentrations of solute molecules a t 308 K and pressures between 8.4 and 8.8 MPa, where carbon dioxide compressibilities are very large. Furthermore, they reported that the solute is present in monomeric form in the supercritical carbon dioxide a t lower pressures. This is equivalent to a low degree of nonrandomness in the mixture. Looking a t the supercritical rapeseed oil-carbon dioxide phase, this means that k, values are low a t low pressures and rise steeply with pressure as compressibility and local ordering in the supercritical phase increase. Nonrandomness as well as compressibility passes through a maximum in accordance with the observations of Randolph et al. and decreases with further increasing pressures, leading to decreasing k , values. Conclusions For the design of processes that take advantage of the favorable qualities of supercritical fluids, there is a demand for models that allow for a reliable description not only of phase equilibria but also of other thermodynamic properties. Therefore, we have displayed a method of combining accurate EOS's for the supercritical solvent with EOS's suitable for the description of the solute properties. To take into account the extreme asymmetry of mixtures often occurring in supercritical fluid processing (for instance, in the oil and fat industry), a statisitical mechanics-based lattice model EOS recently published by Kumar et al. was used for the solute. The Bender EOS was chosen to represent carbon dioxide more accurately. The Kumar-Bender model was examined by fitting it to vaporliquid equilibrium data of the rapeseed oil-carbon dioxide system measured within this work. Though the Kumar-Bender model improves the description of the phase equilibria compared to the original lattice model, calculations are not accurate enough with only one binary interaction parameter. Only when a three-binary-parameter, density- and composition-dependent combining rule (Wilczek-Vera and Vera) is applied, calculations are significantly improved. Then, agreement between calculated and experimentally obtained barotropic points indicates that the p-V,-T properties of the coexisting phases are also well described by the model. The good representation of phase equilibria in the extremely asymmetric rapeseed oil-carbon dioxide system over a wide range of pressures proves that EOS's like the Kumar-Bender model may be useful tools for the development of supercritical fluid extraction processes. Furthermore, the agreement between the experimental observations of Kim and Johnston (1987a,b) and Randolph et al. (1988) and the behavior of the rapeseed oil-carbon dioxide mixture indicated by the model is striking evidence for the physical significance of the modeling results and shows that local compositions and their density and composition dependences must be taken into account, when supercritical fluid mixtures are modeled with EOS's. Nomenclature Ql-Uzo = parameters in the Bender EOS A , = molar Helmholtz energy, J mol-' B-H = temperature-dependent parameters in the Bender EOS k,, l,J, m, = binary interaction parameters between segments of i and j molecules n = number of mixture components p = pressure, MPa
q =
effective chain length of a chain molecule
r = chain length of a chain molecule R , = universal gas constant, 8.3143 J mol-' K-l t = temperature, "C T = temperature, K VH = lattice unit cell size, m3 mol-'
V , = molar volume, m3 mol-' w = mass fraction z = lattice coordination number Greek Symbols r = nonrandomness correction 6 = interaction energy between molecular segments, J mo1-l 17 = packing fraction
9 = surface area fraction
8 = surface area fraction on a hole-free basis ( = term in the quasi-chemical equations p, = molar density, mol m-3 = mole fraction Qc = canonical partition function
Subscripts
0 = pure component
i = molecule i ij = quantity related to the i-j molecule segment pair M = mixture property Superscripts E = excess property at constant packing fraction ig = ideal-gas property r = residual property at constant volume ' = liquid-phase property Registry No. COz, 124-38-9.
L i t e r a t u r e Cited Angus, S.; Armstrong, B.; deReuck, K. M.; Altunin, V. V.; Gadetskii, 0. G.; Chapela, G. A.; Rowlinson, J. S. Carbon Dioxide In International Thermodynamic Tables of t h e Fluid State; Pergamon Press: Oxford, 1976. Bamberger, T.; Erickson, J. C.; Cooney, C. L.; Kumar, S. K. Measurement and Model Prediction of Solubilities of Pure Fatty Acids, Pure Triglycerides and Mixtures of Triglycerides in Supercritical Carbon Dioxide. J . Chem. Eng. Data 1988, 33(3), 327-333. Bender, E. Equations of State exactly representing the Phase Behavior of pure Substances. Proceedings of the 5th Symposium on Thermophysical Properties, ASME, New York, 1970; pp 227-235. Brunner, G. Phasengleichgewichte in Anwesenheit komprimierter Gase and ihre Bedeutung bei der Trennung schwerfluchtiger Stoffe. Habilitation Thesis. University of Erlangen-Nurnberg, FRG, 1978. Brunner, G. Anwendungsmoglichkeiten der Gasextraktion im Bereich der Fette und Ole. Fette, Seifen, Anstrichm. 1986, 88(12), 464-474. Brunner, G. Stofftrennung mit uberkritischen Gasen (Gasextrak1987, 59(1), 12-22. tion). Chem.-Ing.-Tech. Buhner, K.; Maurer, G.; Bender, E. Pressure-enthalpy diagrams for methane, ethane, propane, ethylene and propylene. Cryogenics 1981, 21, 157-164. Coenen, H.; Kriegel, E. Anwendungen der Extraktion mit uberkritischen Gasen in der Nahrungsmittel-Industrie.Chem.Ing.-Tech. 1983, MS 1162. Donohue, M. D.; Vimalchand, P. The Perturbed-Hard-Chain Theory. Extensions and Applications. Fluid Phase Equilib. 1988,40, 185-21 1. D'Alonzo, R. P.; Kozarek, W. J.; Wade, R. L. Glyceride Composition of Processed Fats and Oils as Determined by Glass Capillary Gas Chromatography. J . Am. Oil Chem. SOC.1982, 59(7), 292-295. Eggers, R.; Stein, W. Moglichkeiten der Bearbeitung von Olsaaten mit uberkritischen Gasen. Tech. Mitt. K r u p p , Werksber. 1984, 42(1), 7-16. Eggers, R.; Sievers, U.; Stein, W. High Pressure Extraction of Oil Seed. J . Am. Oil Chem. SOC.1985,62(8), 1222-1230. Formo, M. W. Physical Properties of Fats and Fatty Acids. In Bailey's Industrial Oil and Fat Products; Swern, D., Ed.; Wiley: New York, 1979; Vol. 1, pp 177-232.
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