Ind. Eng. Chem. Res. 1993,32, 2841-2848
2841
Solubility of Cholesterol in Supercritical Ethane and Binary Gas Mixtures Containing Ethane Harcharan Singh, S. L. Jimmy Yun, Stuart J. Macnaughton, David L. Tomasko,t and Neil R. Foster' School of Chemical Engineering and Industrial Chemistry, University of New South Wales, P.O. Box 1, Kensington, NSW 2033, Australia
The solubility of cholesterol in supercritical ethane a t 313.1,323.1,and 333.1 K and 70-190 bar is presented as well as the solubility in ethane with 3.5 and 14 mol % propane and ethane with 3.5, 14,50,and 96.5 mol % carbon dioxide a t temperatures from 308.1 to 338.1 K and 85-220 bar. The results are compared with previous measurements in pure carbon dioxide and are discussed in terms of the effect of mixture density and properties of the solute. Cholesterol appears to behave as a predominantly hydrocarbon molecule despite the presence of an OH group and a dipole moment of 1.9D. The cosolvent effects observed with propane and with COZcan be explained by the density behavior of the solvent mixtures. The data are correlated with the Peng-Robinson and Patel-Teja equations of state. It is shown that estimated critical properties for cholesterol are unsatisfactory for phase equilibrium calculations, and a modified Peng-Robinson equation is introduced in which the solute parameters a2 and bz are made adjustable. The resulting equation of state has three adjustable parameters for an entire binary solidaupercritical fluid data set instead of one parameter per isotherm. With the modified Peng-Robinson equation, the solubility of cholesterol in the ethane-C02 mixtures was predicted with an average absolute relative deviation of 21 % using only parameters from the binary systems.
Introduction The solubility of cholesterol at supercritical conditions has been investigated by several researchers (Chrastil, 1982;Wong and Johnston, 1986;Yeh et al., 1991;Yun et al., 1991;Kosalet al., 1992). The incentivefor such studies is that considerable market advantage is provided by food products with reduced cholesterol contents. Application studies have been performed in which cholesterol has been extracted from egg yolk (Levi and Sim, 1988;Froning et al., 1990;Bulley and Labay, 1991;Pasin et al., 19911,milk (Shishikuraet al., 1986;Lim et al., 1991;Watts, 1989),and fish muscle (Hardardottir and Kinsella, 1988). The challenge for the food industry is to reduce or remove the cholesterol from food products without altering the taste or appearance. Supercritical fluid (SCF) extraction has several advantages over conventional liquid extraction including the ability to change dramatically the solubility and/or selectivity through small changes in pressure or temperature. In particular, for application to biomolecule and pharmaceutical extractions the most important features are the ability to leave the substrate free of any solvent residue and the ability to achieve separations at much lower operating temperatures than required by distillation, usually only a few degrees above ambient. In most studies to date, COZhas been used as the primary solvent. While the technical feasibility of extracting cholesterol with pure COZhas been established, the use of alternative supercriticalfluids has not been investigated. The addition of a liquid organic cosolvent has been shown to increase solubilities and selectivities in SCF processes (Schmitt and Reid, 1986; Dobbs and Johnston, 1987). However, this technique has the potential problem of leaving residual cosolvent in the extract, thus negating some of the advantages of using SCF's. It is therefore desirable to investigate other primary solvents and gas mixtures in an attempt to eliminate this problem.
* Author to whom correspondence should be addressed.
+ Present address: Department of Chemical Engineering,The Ohio State University, Columbus, Ohio 43210-1180.
Consequently, the solubility of cholesterol in supercritical ethane and gas mixtures containing ethane was determined. Ethane has a lower critical pressure than COz, thus allowing lower operating pressures, and it is an acceptable solvent for food processing. Data are reported for the solubility of cholesterolin supercritical (SC) ethane at 313.1,323.1,and 333.1 K and in SC ethane containing propane (3.5 and 14.0 mol %) and C02 (3.5,14,50, and 96.5 mol % ) as cosolvents at temperatures from 308.1to 338.1 K. The operating conditionswere selected to ensure operation in the critical region of each binary mixture of defined composition. The results are compared with those obtained in supercritical COZby Yun et al. (1991). The solubility data were correlated using the PengRobinson (PR) and Patel-Teja (PT) equations of state with one adjustable parameter for the cosolventaolute interactions. However, the results were better correlated with a modified version of the PR equation of state (EOS) where the optimized pure component parameters (a and b) were obtained from the binary ethane-cholesterol solubility data.
Background Typical primary supercriticalfluidssuch as ethane, COZ, NzO, and propane have also been used in small quantities as cosolvents (Joshiand Prausnitz, 1984,Schmitt and Reid, 1986;Hollar and Ehrlich, 1990;Smith and Wormald, 1990; Kosal et al., 1992). One potential advantage of using a mixed-gas solvent is the lowering of the mixture critical temperature. For example, the ethane-COa mixture critical locus displays a minimum in temperature at 291 K and 57 mol % C02, due to the formation of an azeotrope (Kuenen, 1897;Ohgakiand Katayama, 1977). This enables operation at lower temperatures than the critical temperature of either pure component, which is attractive for extraction of thermally labile biomolecules and pharmaceutical products. Such a decrease in operating temperature would of course be contingent upon maintaining a high solubility.
0888-5885/93/2632-2841$04.00/00 1993 American Chemical Society
2842 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 TBERMOCOUPLE
scF SUPPLY
PRESSURE
4
PRJUDMTJNG
con,
BATH
SAMPLE BOMB
Tm I
\
.
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I
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.............
Is WET GAS
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i TO VENT
____________________________________
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WATER BATH Figure 1. Continuous flow apparatus for measuring solid solubilities in supercritical fluids. Table I. Source, Purity, and Physical Properties of Solute and Solvents compound cholesterol ethane propane
coz
source Sigma Matheson
CIG Liquid Air
purity(%) 99+ 99.5 99.0 99.8
MW 386.7
T, (K) 421.6
30.1 44.1 44.0
Carbon dioxide has been the most frequently used supercritical fluid because it is nontoxic, nonflammable, and easily obtained. However, depending on the nature of the solute, hydrocarbon SCFs can be used to achieve higher solubilities. Schmitt and Reid (1986) observed higher solubilities in ethane than in C02 for nonpolar aromatic hydrocarbons while the opposite was observed for polar compounds. In addition to the practical significance of cholesterol extraction, this study provides the basis for determining the nature of cholesterol as a solute. Cholesterol is a bulky, predominantly hydrocarbon compound (C27H4~0)with one OH polar functional group which gives it a dipole moment of approximately 1.9 D (McClellan, 1963). The relative solubility in COZ and ethane illustrates which characteristic dominates the solubility behavior. Experimental Section Cholesterol solubilities in supercritical ethane were determined at temperatures of 313.1,323.1, and 333.1 K and at pressures ranging from 70 to 190 bar. Measurements were obtained in a continuous flow type apparatus, a schematic of which is shown in Figure 1. The experimental procedure has been described by Yun et al. (1991). Materials. The purity, sources, and some physical properties of the solute and solvents are listed in Table I. These materials were used without further purification. Preparation of Solvent43aseous Cosolvent Mixture. The preparation of binarygas mixtures was achieved with two separate high-pressure vessels. The barrels of two syringe pumps were used as the mixing vessels: an Isco Model 260D with an internal volume of 266 mL and an Isco Model 5000LC with an internal volume of 500 mL. To prepare an ethane402 mixture, the syringe of the 266-mL pump was fully retracted and the barrel was completely filled with ethane from an inverted ethane cylinder. The cooling water jacket on the pump was
Table 11. Solubility of Cholesterol in Supercritical Ethane press. (bar) 70 85 100 115 130 145 160 190
solubility (mole fraction X 106) 313.1 K 323.1 K 333.1 K 3.75 1.30 5.90 4.50 2.61 8.1 7.92 6.10 10.7 11.4 10.8 13.4 15.0 16.2 15.7 18.7 21.1 18.4 23.6 26.5 23.2 30.5 38.0
maintained at 274 K, and the barrel was allowed to equilibrate to the cylinder pressure (44 bar) for 15 min to ensure that it was completelyfilled with liquid. The ethane was then transferred into the barrel of the 500-mL pump. The desired amount of liquid COZwas then drawn into the 266-mL pump and delivered into the 500-mL pump containing the ethane. The liquids were mixed in the 500-mL barrel by alternately heating and cooling the barrel between approximately 274 and 324 K. The temperature cycling was continued for approximately 30 min, after which the mixture was allowed to equilibrate for several hours before being transferred into the barrel of the 266-mL syringe pump. The ethane-propane mixtures were similarly prepared. The compositions of the mixtures were checked using gas chromatography. The relative precision of the mixture composition was f 3 5%. Procedure. For allthe ternary systems studied, at least 500 mL of the gas mixture was flushed through the experimental apparatus before solubilities were measured. Results The solubility of cholesterol in supercritical ethane was determined at 313.1, 323.1, and 333.1 K from 70 to 190 bar. The solubility of cholesterol in supercritical ethane containing propane (3.5 and 14mol % 1 and CO2 (3.6,14.0, 50, and 96.5 mol %) as cosolvents was determined a t various temperatures from 308.1 to 338.1 K and pressures from 85 to 220 bar. The experimental results are presented in Tables I1 and 111. The reported values are the average of at least two replicate measurements with a deviation of less than 4%. Reproduction of the data points was more difficult at higher pressures (>145 bar), due to
Ind. Eng. Chem. Res., Vol. 32, No. 11,1993 2843 Table 111. Solubility of Cholesterol in Supercritical Ethane-Propane and Ethandarbon Dioxide Mixtures
Table IV. Mixture Densities and Cosolvent Effects in Ethane-Propane and E t h a n 4 0 2 Systems at 328.1 K
(a) Supercritical EthanePropane Mixtures
press. (bar) 70 85 100 120 140 160 190
solubility (mole fraction x 104) 318.1 K 328.1 K 3.5 mol %" 14.0 mol %" 3.5 mol %" 0.121 0.497 1.43 0.270 0.880 1.91 0.754 1.48 2.80 1.58 2.19 3.59 2.47 2.88 4.50 3.48 3.99 5.74 5.17
(b) Supercritical EthaueCarbon Dioxide Mixtures solubility (mole fraction X 10') at 328.1 K press. (bar) 3.5 mol % 14.0 mol % 50.0 mol % 96.5 mol % 85 0.239 100 0.548 0.319 120 1.07 0.701 0.186 0.0741 140 1.67 1.30 0.449 0.219 160 2.30 1.90 0.861 0.463 190 3.34 2.95 1.61 0.801 220 3.98 2.41 1.18
press. (bar) 85 100 120 130 140 160 190 220
solubility (mole fraction X 104) 318.1 K 308.1 K 338.1 K 50.0 mol % 50.0 mol % 50.0 mol % b 0.125 0.0804 0.382 0.111 0.200 0.979 0.300 0.404 0.437 0.637 1.73 0.571 0.890 2.66 0.901 1.28 4.03 1.45 1.98 1.64
a Solute-freeconcentration of propane. Solute-freeconcentration of carbon dioxide.
frequent blockages of the metering valve resulting from the high solubilities encountered. A preliminary study to determine the solubility of cholesterol in propane was also undertaken. Since propane has a high critical temperature (T,= 369.8K))the solubility apparatus was placed in an air bath. At 400 K and 23 bar and 374 K and 64 bar a liquid phase was observed in the Jerguson sight gauge. This corresponds to a decrease in the melting temperature of cholesterol of 22 and 48 K, respectively, and is consistent with previous studies of the pressure dependence of the solid-liquid-gas (SLG) curve. These data are only approximate, as an accurate determination of the melting point depression was not the objective of this study. Such behavior does however indicate the potential of hydrocarbons for producing large solubilities when added as cosolvents. The solubilities obtained for the ethane-cosolvent mixtures are described with respect to mixture densities and cosolvent effect. The mixture densities were calculated by EOS
PTERNARY PMIXTURE= EOS PPUREETHANE PPUREETHANE
(1)
where, for each operating pressure and temperature) EOS PPUREETHANE is the density of pure ethane (g/cm3) calculated using the Peng-Robinson EOS and is the density of the ternary cholesterol-supercritical fluid cosolvent mixture. The ternary densities were obtained from the compressibility factors resulting from the correlation of ternary solubility data. The equation of state
press. 3.5 mol % propane (bar) p(g/L) 85 278 1.38 309 100 1.21 120 332 1.17 140 348 1.18 360 160 1.14 375 190 1.16
+
press. (bar) 100 120 140 160 190 220
14 mol % COZ P (g/L) 298 0.44 336 0.56 360 0.70 378 0.75 399 0.85 415 0.93
+
14 mol % propane P (g/L) 325 3.97 347 2.63 365 2.22 317 1.98 387 1.78 400 1.66
+
50 mol % C02 P (g/L)
+
354 401 435 471 498
0.15 0.24 0.34 0.47 0.57
3.5 mol % COz P (g/L) 263 0.66 301 0.76 330 0.85 349 0.90 364 0.91 380 0.97
+
96.5 mol % COz P (g/L)
+
455 545
602 660 701
0.06 0.12 0.18 0.23 0.28
used was a modified version of the Peng-Robinson EOS described below. This method assumes that the errors in calculated compressibility factors are similar for the pure ethane and the ternary systems. Pure ethane densities were obtained from an accurate equation of state for pure ethane (Younglove and Ely, 1987). The cosolvent effect is defined as
rc/=
YTERNARY YBINARY
In order to calculate cosolvent effects a t the temperatures of the ternary data, the binary solubility data had to be interpolated with respect to temperature. Rather than rely on any model that may or may not be accurate, it was considered best to interpolate the data directly as a function of temperature at constant pressure. The pure ethane data at 55 "C were calculated in this way, and the cosolvent effects at this temperature are tabulated along with the calculated mixture densities in Table IV.
Discussion
Pure Ethane. A plot of the solubility of cholesterol in ethane as a function of density is shown in Figure 2. For comparison)the data of Yun et al. (1991)in SC COZhave also been included. For both systems the isotherms are almost parallel and exhibit a linear relationship between log solubility and density which is consistent with the trends observed in previous studies (Gurdial and Foster, 1991; Yun et al., 1991). However, the solubility of cholesterol in supercritical ethane is significantly higher than in COZ (approximately 2 orders of magnitude at a solvent density of about 400 g/L) even though ethane generally has a much lower density. The slopes of the isotherms are also larger for ethane than COz. As a consequence of this difference in slope, the difference in solubility varies with density. The difference is greatest at low pressures and high temperatures; for example at 333.1 K and 130 bar the ratio of solubility in ethane to solubility in COz is approximately 15.3 compared with 3.4 at 313.1 K and 190 bar. The extent of solubility enhancement due to solutesolvent interactions in the two systems can be determined from a direct comparison of solubility versus reduced density plots. Treatment of solubility data in this manner removes the effect of proximity to the critical point. The data for the pure COn and ethane systems are presented in this manner in Figure 3. This figure shows that the data for the two systems were obtained at a similar
~Fgi~~~
2844 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
&I -
UrWn Dbrld. D.L.
3
d 5E-6 v)
2E-6 1E-6
Irn
1E-3
., 0
-
400 600 DENSITY (glL)
200
800
r'
Z5E-4
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1.75 mol% Hx
i2E-4 E
-
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3.5 mol% Hx
=!
2 1E-4 -
I
3E-5 200
1,000
Figure 2. Solubilityof cholesterol in supercriticalethane and carbon dioxide as a function of solvent density.
-
I
x
I
250
I
I
300 350 DENSITY (g/L)
400
450
Figure 4. Solubility of cholesterol in ethane, ethanepropane, and ethane-hexane mixtures at 328 K as a function of solvent mixture density.
5E-4
2 2E-4
-
s5 1E-4 E ,-,E.,
-
8
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1.4 1.6 1.8 REDUCED DENSITY
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Figure 3. Solubilityof cholesterol in supercriticalethane and carbon dioxide as a function of reduced solvent density.
1
I
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4
5E-6
200
I
I
300
400
I
I
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5w
600
700
800
WO
DENSITY (g/L)
Figure 5. Solubility of cholesterol in ethane-carbondioxidemixtures at 328 K.
displacement from the respective pure solvent critical densities. Any solubility difference can now be explained in terms of solute-solvent interactions. Unlike the data in Figure 2, the isotherms for both systems have similar slopes. Since the cholesterol solubilities in ethane are larger than those obtained in C02 at a similar reduced density, it can be inferred that the degree of solutesolvent interactions in ethane is greater. This observation indicates that cholesterol behaves primarily as a hydrocarbon molecule, an interpretation consistent with previous studies comparing ethane and COZ. Schmitt and Reid (1986) and Ekart (1992) observed that ethane was a better solvent for nonpolar aromatic hydrocarbons and that C02 was better for polar compounds. Ethane-Propane Mixture. When cosolvents are added to supercritical fluids, a concomitant increase in solvent density is observed. In the absence of specific interactions, this increase in density can be the dominant cause of solubilityenhancement. For the ethane-propane mixture this appears to be the case. In Figure 4 the solubilities of cholesterol in SC ethane, ethane-propane mixtures, and ethane-hexane mixtures (Singh, 1993) are plotted as a function of mixture density. In this format, and allowing for experimental error, all the isotherms collapse onto a single line. The interactions between n-alkanes and cholesterol appear to be strictly density dependent (e.g., dispersion forces). This result is in agreement with that of Ekart (1992) who found that the cosolvent effect for anthracene in an ethane-ethanol mixture (at 323 K)was solelydue to the increase in mixture bulk density. Conversely, this type of behavior has not been reported for COzsolvent systems. Ethandarbon Dioxide Mixtures. The solubility of cholesterol in ethane402 mixtures is intermediate to the pure SCF solubilitiesat all densities as shown in Figure 5. This behavior is not unexpected, but since data were taken over the entire range of solvent composition, the
0
120
140 0
160
im 0
220
w 0.2
0
0.2
0.4
0.6
0.8
1
MOLE FRACTION CARBON DIOXIDE Figure 6. Normalized relative solubility at 328 K as a function of solvent composition showing effect of pressure.
effect of mixed gas solvents can be studied in more detail. An unusual feature of this system is that the change in solubility is not proportional to the amount of cosolvent added. This can be explained more clearly by referring to Figure 6, in which the normalized solubility is plotted as a function of solvent composition. The normalized solubility defined by eq 3 enables the data at different pressures to be presented on the same scale and is the ratio of the difference between the mixture solubilityand the pure COZsolubility and the difference between the pure ethane and pure C02 solubilities. normalized solubility =
YMIXTURE - Yco, YETItANE - Yco,
(3)
If the solvent mixture behaved ideally, the solubility would follow the straight line drawn between the two pure solvent solubilities. However, the data show a distinct negative deviation from this ideal behavior at all pressures.
Ind. Eng. Chem. Res., Vol. 32,No. 11,1993 2845 Table V. Normalized Relative Solubility at 160 bar as a Function of Solvent Composition Showing Effect of Temperature temp (K) 308.1 318.1 328.1
normalized re1 soly 0.336 0.260 0.206
mole fraction COS 0.60 0.60 0.60
This result indicates that the addition of a small amount of C02 to ethane has a much stronger effect on cholesterol solubility than the addition of a similar amount of ethane to COZ. This is actually contrary to what one expects with a cosolvent in a SCF. With liquid cosolvents,for example, a positivedeviation from linear behavior would be expected because of local density and composition enhancements. A qualitative explanation for this behavior can be found in the volumetric properties of the ethane402 mixture. At 35 "C the binaryethaneCO2 system exhibits a positive excess volume of mixing for all compositions at pressures above 75bar (Wormaldand Eyears, 1988). The magnitude of the excess volume decreaseswith both temperature and pressure. Therefore, the density of the mixture is less than the density linearly interpolated between the pure components (i.e., assuming ideal mixing) and the difference between the actual and ideal densities decreases with increasing pressure. This density effect is manifested in the solubility results as the deviation from ideal behavior becomes smaller with increasing pressure. The temperature effect on the solubility is also consistent with the excess volume data. Although the temperature dependence of the excess volume varies with pressure, at pressures above 100bar, the excess volume increases with temperature. The normalized relative solubilityis shown as a function of solvent composition at 160 bar and three temperatures in Table V. The solubility in the mixture becomes more nonideal as the temperature is increased. It is apparent that the solubility of cholesterol mirrors the density behavior of the solvent mixture and is thus influenced by the density dependent dispersion interactions with the solvent. The observed solubility behavior in ethane402 mixtures may also yield insight into the relative strength of the different molecular pair interactions. For instance, since cholesterol is less soluble in CO2 than in ethane, it is reasonable to say that the ethane-cholesterolinteraction is stronger than the CO2-cholesterol interaction. The strength of the ethane402 interaction then determines whether the addition of a second solvent component will increase or decrease the potential number of interactions with cholesterol. In this case, it appears that the ethaneC02 interaction is similar in magnitude to the ethanecholesterol interaction. On the basis of this, the addition of a small amount of C02 to ethane would merely tie up ethane molecules and decrease the number of solventsolute interactions available to cholesterol. As a result, the CO2 interacts preferentially with the ethane (which is present in excess anyway), and since the cholesterol must now compete for ethane interactions, its solubility is decreased. On the other hand, ethane added to COz would interact equally with cholesteroland CO2, resulting in little or no change to the solubility. In a different study usinggas mixtures, Kosal et al.(1992) observed no change in cholesterol solubility when 10 vol % N2O was added to CO2. Nitrous oxide differs from COZ only in terms of polarity (1.1 = 0.116 D)and apparently causes little change in the density. These observations
-1 4 .L
.
t
sa
I
100
150
200
aa
(W
Figure 7. Cosolvent effect as a function of pressure from propane and carbon dioxide cosolventa.
are consistentwith the hypothesis that cholesterolbehaves primarily as a hydrocarbon. To compare between the different cosolvents used, the cosolvent effect as a function of pressure at 328.1 K is shown in Figure 7 for 3.5% and 14% propane and C02. Propane exhibits a large positive cosolvent effect at low pressures (up to 4)which generallydecreaseswith pressure whereas the cosolvent effect for C02 increases with pressure. The figure shows that as the pressure is increased, the cosolvent effect approaches unity regardless of the value at low pressure. This is a result of the solvent becoming more uniform with regard to density and composition as the conditions move away from the critical point and the isothermal compressibility begins to decrease. The different trends exhibited are similar to those of the mixture densities in these systems and demonstrate the pronounced role of mixture density on cosolvent effect in mixed-gas systems. Modeling The cholesterol solubility data were correlated using the Peng-Robinson (original and a modified version) and the Patel-Teja equations of state. The former has two pure component parameters (a, b) while the latter has an additional parameter, c. The conventional van der Waals (vdW) mixing rules, with one adjustable parameter, were used with both of the EOS methods. Since all the critical properties for cholesterol that were required for the EOS parameters had to be estimated, a modification to the Peng-Robinson equation was made so that the solute pure component parameters, a2 and b2, were made adjustable. The attractive parameter, 02, was allowed to be temperature dependent which resulted in three adjustable parameters over all the isotherms compared to the one adjustable parameter for each isotherm for the original PR EOS. Peng-Robinson EOS. Most common cubic equations of state use the critical point as a corresponding state, and as a result, the critical properties of all componenta must be known in order to do phase equilibrium calculations. The physical and critical properties required to evaluate the pure component parameters in the Peng-Robinson (PR) equation of state (Peng and Robinson, 1976)are listad in Table VI. The critical properties of ethane were obtained from Reid et al. (19871,but the properties of cholesterol were not available and were estimated using Lydersen's method (Lymanet al., 1982). These estimation techniques were not intended for large molecules such as cholesterol, and indeed, cholesterol decomposes at high temperature, precluding the existence of a critical point.
2846 Ind. Eng. Chem. Res., Vol. 32,No. 11,1993 Table VI. Physical and Critical Properties of Ethane and Cholesterol property ethane cholesterol 305.4 778.7 Tc (K) 48.8 12.2 Pc (bar) 0.099 1.011” w 362.4 molar vol (cm3/mol) Pt (10-9 bar) 2.6 313.1 K 9.4 323.1 K 18 328.1 K 32 333.1 K 0
Edminster’s method (Lymanet al., 1982).
Table VII. Equation of State Correlation of Ethandholesterol Binary System Peng-Robinson Patel-Teja temp(K) 313.1 323.1 333.1 av
klz
0.111 0.113 0.110
URD(%) 20.6 20.6 23.2 21.5
kiz 0.0862 0.0881 0.0879
AARD(%) 10.5 13.9 17.9 13.9
Therefore, these parameters must only be considered mathematically convenient for calculating solubilities. The cholesterol molar volume and interpolated saturation vapor pressures were obtained from Wong and Johnston (1986). The solubility data in pure supercritical ethane were calculated using the PR EOS with optimization of the binary interaction parameter, k12. The optimized k12 values and the average absolute relative deviation (AARD) obtained are listed in Table VII. The k12 values were observed to be temperature independent, and the overall AARD was about 21% for all the isotherms studied. However, if the cholesterol P, was increased by 5 bar, a better fit was observed. For example, at 323.1 K the AARD reduced from 20.6% to about 10%. This suggests that the critical properties themselves may as well be made adjustable as the EOS appears to be sensitive to their values. Patel-Teja EOS. A three-parameter (a,b, and c) cubic EOS was developed by Patel and Teja (1982)which is superior in determining mixture critical compressibilities, ultimately improving solubility estimation. For vaporliquid equilibrium (VLE)calculations, the Patel-Teja (PT) EOS is as good as the Soave-Redlich-Kwong and PengRobinson EOS for mixtures of light hydrocarbons and is slightly better for systems containing heavy hydrocarbons and polar substances (Patel and Teja, 1982). The PT EOS uses pure component parameters calculated from critical properties and two empirical constants (Fand fc) which have been correlated with acentric factor. The mixing rules and binary interaction parameter are similar to those of the PR EOS. The F and lc for compounds other than cholesterol were obtained from the literature (Patel and Teja, 1982;Georgeton et al., 1986), and those for cholesterol were determined from the estimated acentric factor. The kl2 is again slightly temperature dependent as also shown in Table VII. Although both EOS use only one adjustable parameter, 1212, the PT EOS is slightly superior to the PR EOS in correlating solubilities. This is due to the additional pure component parameter, c, which provides flexibility to the EOS and gives an improved estimation of the mixture compressibilities, especially at higher pressures. Modified Peng-Robinson EOS. The solute pure component parameters, a2 and bz, calculated for the original PR EOS are not well represented because they
Table VIII. Modified Peng-Robinson Correlation of Ethane Cholesterol Binary System KI (bar temp (~rn~/rnol)~ KZ(bar bz AARD solvent range (K) K-l) (~rn~/rnol)~) (cms/mol) ( % ) 5.9 ethane 313.1-333.1 -469 557 4.1052 X 1oB 329.9 5.7 313.1-333.1 -532 414 3.6299X 108 269.3 carbon dioxide0 a
Regressed from data of Yun et al. (1991).
rely on estimated critical properties. An alternative approach to modeling solubility data with a cubic EOS without relying on estimated critical properties is to treat the a2 and b2 as adjustable parameters as suggested by Schmitt and Reid (1986). They proposed optimizing each isotherm individually with k12 equal to 0. However, this approach causes both a2 and b2 to decrease slightly with temperature and essentially results in two adjustable parameters per isotherm. In this work, we have modified their approach by assuming that u2, over the small temperature range considered, can be approximated by a linear function of temperature (eq 4)and bz is temperature-
a, = K I T + K 2
(4)
independent. This results in three adjustable parameters (K1, K z , and b2) regardless of the number of isotherms. Although the temperature dependence is arbitrary, it is in fact close to that used in the original form of the EOS. The optimization is then carried out on all isotherms simultaneously by minimizing the average absolute relative deviation defined as
Where Nt is the number of isotherms, Nj is the number of points in isotherm j , and NTOTis the total number of data points. The b2 values obtained using the modified PengRobinson (MPR) EOS are listed in Table VIII. An overall AARD of about 6% is obtained, which is significantly superior to that obtained with the original Peng-Robinson EOS, even though for three isotherms the number of adjustable parameters is the same. This demonstrates that estimated critical properties are probably not the best reducing properties for large biomolecules. Ternary Systems. For correlating ternary systems, the cosolventaolute interaction parameter, k23, was optimized. Since the ternary data were obtained at temperatures different from those for the binaries, the k12 required in the PR and PT equations were linearly interpolated from the binary k12 values. The cholesterol pure component parameter a2 in the MPR equation was determined from eq 4. The solvent-cosolvent interaction parameter, k13, for the ethane-propane system was regressed from VLE data (Matschke and Thodos, 1962)using a flash program. A single value of k13 (0.001for PR and 0.011 for PT) was used for all temperatures. The optimized k 2 3 values obtained for the PR, PT, and MPR a t each temperature are presented in Table IX. It can be observed that the ternary solubility data were best correlated with the MPR EOS. The k23 values are very small for the PR and PT EOS and are negative for the MPR EOS. Negative values indicate that the attractive interaction between solute and cosolvent is stronger than that approximated by the geometric mean. The significantly improved correlation of the modified PR EOS over the others is attributed to the use of the optimized a2 and
Ind. Eng. Chem. Res., Vol. 32, No. 11,1993 2847 Table IX. Equation of State Correlation of Ethane-Propandholesterol Ternary System
~
EOS Peng-Robinson Patel-Teja modified PR
328 K AARD 0.037 21.5 0.017 12.7 -0.065 6.0 kzs
338 K km AARD 0.059 25.8 0.015 21.0 -0.068 2.1
b2. While the PT EOS gives slightly better results than the PR EOS because of its three-constant form, the gross estimationsrequired to determine soluteparameters offset any advantage. The ethane402 solubility data provides a unique opportunity for modeling since the entire range of solvent composition has been studied. In this case it is improper to simply model the system as solvent-cosolvent-solute, since the roles of solvent and cosolvent would be confused. Instead, an attempt has been made to predict the ternary solubility data using parameters from the individual binaries. The MPR EOS was used to obtain solute parameters in each pure solvent; see Table VIII. Excess volume data for the ethane402 system at 323.1 K (Wormald and Eyears, 1988) was regressed to obtain the k13 interaction parameter (0.13). The solute parameters for the mixed solvent were linearly weighted according to the solvent composition, and the solvent-solvent interaction was accounted for in the quadratic mixing rule for the attractive parameter. All other interaction parameters were set to zero. This method predicted the mixture data with an overall AARD of 21 % as shown in Figure 8. The results are very good and suggest that nonspecific interactions (e.g., dispersion forces) dominate the phase behavior in this system.
Conclusion Solubilitydata presented for cholesterol in supercritical ethane show that cholesterol is 3-15 times more soluble in supercritical ethane than in supercritical C02 over a similar range of conditions. The solubility difference is more pronounced at higher temperatures and lower pressures. Although cholesterolliquefied in the presence of supercritical propane, the addition of 3.5 and 14 mol % propane to supercriticalethane resulted in cosolventeffects up to 4 at low pressures. It was shown that, for n-alkane cosolvents in ethane, the cosolvent effects on solubility were solely a result of the increased solvent density. Due to the solubility behavior exhibited by cholesterol in hydrocarbon supercritical fluids compared with carbon dioxide, it appears that the bulky, predominantly hydrocarbon nature of cholesterol overshadows the presence of the OH group. The higher solubility obtained in ethane is consistent with literature findings that generally nonpolar hydrocarbons are more soluble in ethane than in c02. The solubility of cholesterol in ethane402 mixtures is less than that in pure ethane, and successive additions of C02 moved the solubility steadily toward that obtained in pure CO2. This is expected since the binary solubilities showed that ethane was a better solvent than COz. The change in solubility,however, is not linearly related to the change in solvent composition. Negative deviations from ideal solubility behavior were observed, and these are attributed to the positive excess volume exhibited by the ethane402 system at these conditions, resulting in a lower mixture density than the mole fraction average of pure densities. The experimental results were correlated using the Peng-Robinson and the Patel-Teja equations of state. A
1E 4
5E4
2E4
11
PURE
I / & I
E
I
- PREDlCTEOFROU lllw I
1E 4
60
BO
100
120
,
,
110 160 180 PRESSURE(BAR)
PARAMETERS ki3plS
2w
220
210
Figure8. Prediction of solubility in ethane-carbon dioxide mixtures using modified Peng-Robinson equation of state.
modified Peng-Robinson equation was introduced in which the pure component solute parameters were made adjustable to alleviate the problem of estimating critical properties. The attractive parameter, a2, was allowed to be temperature dependent and the size parameter, bz, was assumed constant. This resulted in three adjustable parameters over all the isotherms compared to the one adjustable parameter for each isotherm for the original PRand PT EOS. The PT EOS was superior to the original PR EOS,especially at lower temperatures; however, both equations were found to be extremely sensitive to the estimated criticalproperties for cholesterol. The modified PR EOS significantly improved the correlation of the binary solubility data to an overall AARD of less than 7 9%. The ethane-propane-cholesterol ternary data were also correlated using the PR, PT, and MPR equations with results similar to those obtained for the binary systems. The MPR equation was used to predict solubilities in ethane402 mixtures using parameters obtained from binary data. The data were predicted to within 21 % . The results show that estimated critical properties are not at all sufficient for calculatingphase equilibriaof large, bulky molecules, many of which do not even have critical points. For systems containing thermally labile biomolecules, other properties at lower temperatures are needed in order to develop corresponding states theory. Acknowledgment This work was funded in part by the Australian Research Council through Grant No. A89030392. D.L.T. gratefully acknowledgesfinancialsupport from the National Science Foundation through Grant No. INT-9203312. Literature Cited Bulley, N. R.; Labay, L. Extraction/Fractionationof Egg Yolk using SC C02 and Alcohol Entrainers. Proceedings of the Second International Symposium on Supercritical Fluids, Boston, MA; 1991;p 10. Chrastil, J. Solubility of Solids and Liquids in Supercritical Gases. J. Phys. Chem. 1982,86,3016. Dobbs, J. M.; Johnston, K. P. Selectivities in Pure and Mixed Supercritical Mixed Solvents. Ind. Eng. Chem. Re8. 1987,26, 1476. Ekart, M. P. Specific Intermolecular Interactions for Tailoring Supercritical Fluid Solutions. Ph.D. Thesis, University of Illinois at Urbana-Champaign, Urbana, IL, 1992. Froning, G. W.; Wehling, R. L.; Cuppett, R. L.; Pierce, M. M.; Niemann, L.; Siekman, D. K. Extraction of Cholesterol and Other Lipids from Dried Egg Yolk using Supercritical Carbon Dioxide. J. Food Sci. 1990,55 (l), 95.
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Received for review March 2, 1993 Revised manuscript received June 22, 1993. e Abstract published in Advance ACS Abstracts, August 15, 1993.
