922
Ind. Eng. C h e m . Res. 1993,32, 922-930
Thermodynamic Modeling for Supercritical Fluid Process Design M. Mukhopadhyay' and G. V. Raghuram Rao Chemical Engineering Department, Indian Institute of Technology, Bombay, India
The present paper describes a predictive model for ascertaining the process parameters toward establishing an economically viable supercritical fluid technology. The model utilizes the PengRobinson equation of state with a modified covolume-dependent (CVD)mixing rule to be able to predict the solubility of mixed solids in pure and mixed supercritical carbon dioxide from minimum information, such as the pure component solubilities. The model can predict the solubility data at various temperatures by virture of its temperature-insensitive CVD interaction parameter. In order to make the model completely predictive, the CVD interaction parameter has been correlated in terms of the easily available pure component properties, such as solid molar volume, van der Waals volumes, and dipole moments. For mixed solvent, however, a more rigorous correlation is needed to account for stronger solute-cosolvent interactions. The validity of the model has also been tested for the solubility predictions of liquid solutes in the supercritical solvent. Slightly higher deviations of the liquid-phase compositions indicate that the mixing rule is rather well suited for the dilute supercritical solution. The utility of the model has been demonstrated for separation of palmitic acid from tripalmitin. The process parameters have been established under the equilibrium condition utilizing the well-predicted crossover pressures and solubility behavior with temperature.
Introduction The commercial success of supercritical fluid technology greatly depends on the reliable design, simulation, and scale-up operation of the process plant. As a first step toward achieving this goal, it is necessary to screen the process design parameters based on theoretical predictions, prior to extensive experimentation. Accurate thermodynamic modeling of the multicomponent supercritical fluid phase behavior is useful for developing such a predictive model. The model should be able to give accurate predictions from minimum input information, preferably from the knowledge of the pure component properties. The motivation of the present work is to develop such a model which meets the above requirement. The development of a model for supercritical solutions is a challenging task in view of the highly compressible and asymmetric nature of the systems. The compressed gas model has been the most widely used method for representing solid-fluid phase equilibria. Out of the several equations of states (EOS)which have been lately used for the purpose, it has been realized that the cubic EOS must be the choice for process design of any highly complex system, because the molecular interactions are too involved to justify the use of more fundamental equations (Brennecke and Eckert, 1989). In spite of certain limitations the Peng-Robinson (P-R) EOS (1976) performed as satisfactorily as the more complicated perturbed hardsphere equations. Accordingly, in the present work, the P-R EOS has been utilized to calculate the fugacity coefficients. A general review of the various mixing rules reveals that any improvement over the van der Waals (VDW) mixing rules is achieved at the cost of the inclusion of a greater number of adjustable parameters. However, this added complexity is not fully justified in that the uncertainty of one parameter is forced upon the others during regression and the calculated compositions are still extremely sensitive to the regressed binary interaction parameters. Hence with a view to satisfying the requirements of restricting the number of adjustable parameters to one and reducing their sensitivity as well, Rao and Mukho-
* To whom correspondence should be addressed. 0S8S-5885/93/2632-0922$04.0QIQ
padhyay (1988) developed a covolume-dependent(CVD) mixing rule. This emerged from the philosophy that, in a dilute supercritical mixture comprising molecules having large size differences, the probability of a molecule interacting with another in its vicinity depends on what fraction of the surface it can "see" of the other molecule, rather than its relative number or mole fraction. Furthermore, the repulsive forces become increasingly important in quantifying the energy of interaction for largesized molecules. Hence in the CVD mixing rule, an inverse covolume (bij) dependency has been introduced for the attraction energy parameter aij in order to account for the asymmetry and nonrandomness. The success of the original CVD mixing rule was amply demonstrated for pure solid-fluid binaries (Rao and Mukhopadhyay, 1989). However, in the present work its range of applicability is extended to multicomponent systems with varied functional groups for both solid-fluid and liquid-fluid equilibria, upon suitably modifying it.
Mixing Rule In the original CVD mixing rule, the degree (or the exponent) of covolume dependency was uniform for both like and unlike interactions. However, considering the fact that solute-solvent interactions are significantly different from solvent-solvent or solute-solute interactions, in dilute supercritical mixtures, it is proposed to use the degree of covolume dependencyfor unlike interactions as the adjustable parameter, while fixing ita value to be unity for the like interactions. Thus,
where mi; = mjj = 1.0. This choice of mii or mjj reduces the sensitivity of the adjustable parameter mi, while restricting the number of the adjustable parameters to only one. Vidal(1978) had used a similar dependency of aii on bii for the like interactions while developing an empirical mixing rule for the parameters alb, which however required an activity coefficient model with three adjustable parameters and hence was considered less useful 0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 5, 1993 923 for predictive purposes. The other quantities in eq 1are given as b = xyibii
(2)
aij = (a..a.,)'/' 11 I1
(3)
Table I. Critical Properties of the Solids solid acridine 2-aminobenzoicacid anthracene
bij = (b..b.,)'/' 15 I1 (4) With this mixing rule for the P-R EOS the fugacity coefficient for the kth component is expressed as
benzoic acid biphenyl
ln4k = (Z-l)B,-ln(Z-B)-
carbazole 1,lO-decanediol 2,3-DMN 2,6-DMN
where
fluorene hexamethylbenzene 5-methoxyindole naphthalene 2-naphthol palmitic acid phenanthrene
and A, = -[B, 1 -ll~,~y~pijmij[b/bijlmi~ a The like parameters for the solute, a22 and b22, are calculated by either (i) the conventional corresponding states (CS)method of Peng-Robinson (1976) using critical constants or (ii) a group contribution (GC)method using heats of sublimation and molar volumes etc. as b,, = 1.71bv,,
+ 1.28
(6)
where bvDw is the van der Waals volume and AHH~~ is the heat of sublimation of the solid solute calculated by Bondi's group contribution method (Bondi, 1968). The empirical expressions (6) and (7) have been developed in the present work based on their behavioral trends with size and cohesive energy parameters. The critical constants for several solids are either not available in the literature, or there are discrepancies in the reported values. Often they cannot be measured correctly if they decompose at high temperatures. For 21 solids considered in this study, the values of a22 and b22 calculated by the two methods differ by an average deviation of 9.6% and 8.7 % ,respectively. For solid-fluid equilibria, the fugacity of the solute in pure solid phase is as usual calculated from ita vapor pressure with the Poynting correction using the vapor pressure data given in the same literature from where solubility data are taken. The adjustable parameter m12 for the solvent-solute interaction is evaluated by regressing the solubility of pure solid (2) in solvent (1)without and with a cosolvent (3) separately, utilizing the Hooke and Jeeves optimization technique (Beveridge and Schechter, 1970) with 9% absolute average relative deviation (AAFtD), as defined below, as the objective function.
where NDP is the number of data pointa and y2,d and are the calculated and experimental solubility data, respectively. For liquid-fluid equilibria, the fugacities of all components are equated in the two phases and m12 is
phenol skatole stearic acid
Tc (K) 890.10 891.15 800.15 869.00 883.15 869.30 752.80 789.00 769.10 899.10 720.40 771.00 769.20 771.90 779.30 826.40 758.00 777.15 745.00 751.70 815.50 825.15 709.85 791.00 882.60 877.15 692.20 789.00 810.00
Pc(bar) 29.72 32.10 41.00 30.80 32.60 21.90 45.83 38.50 33.90 32.65 23.70 28.90 29.06 30.05 29.06 29.91 24.40 24.10 35.50 40.60 35.26 42.90 15.10 19.00 31.30 31.60 61.30 37.20 16.50
ref
w
0.428 0.450 0.600 0.353 0.460 0.370 0.620 0.372 0.416 0.496 1.325 0.450 0.417 0.455 0.417 0.406 0.515 0.380 0.518 0.302 0.558 0.460 1.064 1.047 0.330 0.440 0.450 0.431 1.085
a b b c b a c c
a a d c a
c a a c b c c
c b e
a c b c C
a
a Bartle et al., 1991. Dobbs, 1986. Rao, 1990. Chimowitz and Pennissi, 1986. e Oghaki et al., 1989.
Table 11. Physical Properties of the Solids VDW AHZS" solid volume" (J/mol) p b (D) 96936.0 1.95 acridine 101.0 115 500.0 2-aminobenzoic acid 66.7 1.50 anthracene 100.7 100930.0 0.00 benzoic acid 65.6 91 510.0 1.70 75850.0 0.00 biphenyl 91.7 102900.0 2.11 carbazole 97.1 148 260.0 2.54 lJ0-decanediol 118.9 2,3-DMN 94.4 95 OOO.0 0.69 85060.0 0.14 2,6-DMN 95.8 fluorene 96.8 82660.0 0.53 hexamethylbenzene 104.3 80 580.0 0.00 5-methoxyindole 86.6 108800.0 3.64 naphthalene 75.9 72 500.0 0.00 2-naphthol 79.7 93 300.0 1.5 154629.0 1.77 palmitic acid 176.7 phenanthrene 100.2 90900.0 0.00 phenol 54.1 67 838.0 1.55 phthalic anhydride 75.0 88 765.0 5.20 93 977.0 0.00 pyrene 112.0 71 910.0 2.1 skatole 80.9 185.1 166645.0 1.70 stearic acid
up (cm3/mol) 178.0 106.0 142.6 96.5 155.4 151.5 158.4 150.0 155.0 138.2 160.0 127.0 112.0 118.0 300.7 151.6 89.0 112.0 159.1 134.0 302.4
a Bondi, 1963. McClellan, 1974. Fromrespective solubilitydata references.
regressed by utilizing a bubble-point calculation algorithm, minimizing the % AARD of the mole fractions in both phases.
Results and Discussion Regression of m12 and m23 from Solubilities of Pure Solids. For testing the capability of the new mixing rule, the solubilities of 21 solids with varying polarities in supercritical carbon dioxide have been considered for which experimental data are available in the literature. The list of solids and their physical properties are given in Tables I and 11. The regressed values of m12 along with the % AARD in correlating the solubilitiesfor 52 isotherms
924 Ind. Eng. Chem. Res., Vol. 32, No. 5, 1993 Table 111. Solubility Calculations with Modified CVD Mixing Rule for CO, (1) + Solid (2) Binary Systems by GC method by CS method solute acridine
ref
T (K)
a
308.3 318.3 328.3 343.3 308.1 308.1 313.1 318.1 343.1 308.1 318.1 328.1 343.1 309.0 318.5 322.6 328.4 313.1 318.0 323.0 328.0 308.0 318.0 328.0 308.0 318.0 328.0 308.1 313.1 323.1 343.1 303.0 308.0 323.0 343.0 308.1 308.1 328.1 333.5 308.3 318.3 328.1 343.1 318.0 328.0 308.0 313.1 318.1 328.0 309.1 308.1 308.1 323.1 343.1 308.1 318.0 328.0 313.1
a
a a
2-aminobenzoicacid anthracene
b C
d C
e
benzoic acid
a a
a a
biphenyl
f
f carbazole 1,lO-decanediol 2,3-DMN 2,6-DMN
fluorene
f f
d
g g g
h h h h h h e e e e
hexamethylbenzene
e
i e e
5-methoxyindole naphthalene 2-naphthol
;
f f
a
a a a
palmitic acid
k k
phenanthrene
2
phenol phthalic anhydride pyrene
d h h 1 b e e
e
skatole stearic acid
i
m m
tripalmitin
n
m,, 0.87 0.86 0.85 0.83 0.77 0.85 0.88 0.84 0.82 0.61 0.58 0.55 0.46 0.83 0.79 0.75 0.80 1.05 0.92 0.91 0.91 0.90 0.91 0.90 0.96 0.94 0.93 0.84 0.90 0.88 0.85 1.03 1.03 1.02 0.99 0.88 0.89 0.81 0.80 0.86 0.85 0.85 0.84 0.77 0.79 0.85 0.85 0.82 0.80 0.92 0.69 0.92 0.89 0.86 1.02 0.82 0.79
?6 AARD 8.2 6.5 6.2 9.8 13.9 5.5 16.8 11.5 11.8 11.9 17.4 19.5 9.7 5.5 5.7 12.0 29.9 16.8 2.0 3.7 8.2 2.9 3.2 2.7 5.5 4.9 6.1 7.3 12.3 10.4 9.1 14.7 13.1 9.8 12.1 2.0 2.8 3.4 31.0 24.5 10.8 17.3 19.1 9.1 16.9 6.1 19.1 8.6 9.2 14.7 15.9 12.8 18.7 9.2 7.9 29.0 29.4
mi,
0.96 0.94 9.94 0.93 0.81 0.89 0.92 0.89 0.87 0.58 0.57 0.54 0.48 0.87 0.84 0.81 0.86 0.95 0.76 0.75 0.75 0.89 0.91 0.91 0.92 0.91 0.91 0.83 0.88 0.87 0.85
0.92 0.92 0.91 0.89 1.20 0.93 0.88 0.88 0.78 0.77 0.77 0.78 0.84 0.87 0.89 0.92 0.88 0.87 0.89 0.80 1.01 0.99 0.97 0.89 0.93 0.91 1.28
% AARD 15.1 12.7 17.7 15.6 18.0 6.1 18.8 11.7 14.7 8.9 13.6 14.8 7.3 5.1 9.8 16.2 32.1 19.5 3.0 3.7 5.9 1.9 2.5 6.0 3.6 2.5 6.4 3.5 15.7 7.9 6.0 2.6 3.7 7.0 4.3 3.4 2.9 5.9 32.9 17.9 18.7 23.3 24.8 21.4 27.5 8.7 18.8 11.7 14.7 4.5 6.9 3.6 6.2 11.0 6.7 29.8 21.5 17.0
11.3 overall % AARD 11.8 Schmitt and Reid, 1986. * Dobbs et al., 1987. Kosal and Holder, 1987. Kwiatkowsky et al., 1984. e Johnston et al., 1982. f McHugh and Paulaitis, 1980. 8 Chimowitz and Pennissi, 1986. Kurnick et al., 1981. Dobbs et al., 1986. Sako et al., 1989. Krammer and Thodos, 1988. Vanleer and Paulaitis, 1980. Krammer and Thodos, 1989. Bamberger et al., 1988.
are presented in Table 111. The overall deviations are 11.9% and 11.7%bytheCSandGC methods,respectively. For most isotherms, the present mixing rule works better than the previous ones in that the adjustable parameter is less sensitive even though in no case are the deviations worse. This is attributed to the proper composition dependency of the covolume-dependent attraction energy parameter; as in the fugacity coefficient calculations, a is corrected for the asymmetry in the size and interaction energy parameters. The fact that for certain complex
solids, such as hexamethylbenzene, benzoic acid, and pyrene, the GC method seems to be superior to the CS method suggests preference to the former method. For the 17 isotherms of 7 solids that were common to both the original and modified CVD mixing rule, the 7% AARD in the present work of 13.3% justifies the improvement, as 15.3 % AARD was obtained with the original CVD mixing rule (Rao, 1990). The improvement over the conventional VDW mixing rule with one adjustable parameter, kij for aij, is demonstrated in Figures 1-3 by comparing the
Ind. Eng. Chem. Res., Vol. 32, No. 5,1993 925
I -=
V O W M I X I N G RULE C V D M I X I N G RULE
80
100
120 1 4 0 160 780 PRESSURE ( b a r )
200
220 PRESSURE ( b a r )
Figure 1. Comparison of solubility predictions for the COznaphthalene system at 328.15 K (experimental data of McHugh and Paulaitis (1980)).
____
L
VOW
-CVD
MIXING RULE
MIXING RULE
z
0
+ V
I
/
100
150
200
250
PRESSURE (bar 1
Pigure 2. Comparison of solubility predictions for the COz-phenol system at 309.15 K.
computed solubilitiesof three solideof different functional groups. In another similar work Wong et al. (1985) correlated solubilities of 8 solids in supercritical carbon dioxide with 8.6 % AARD for 30 isotherms, for which, in the present work, % AAFtDs by the CS and GC methods are within 9.1% and 8.1%, respectively. For a particular system, m12 has been found to decrease with increasing temperature, though the temperature dependency is not significant for the nonpolar hydrocarbons, as can be seen from Table 111. A constant value would suffice for the entire range of temperature without much affecting the accuracy. However, for polar solutes, the calculated solubilitiesdeviate more near the solvent’s critical pressure owing to probable association. For correlating solubilities of pure solids (2) in the supercritical solvent (1)mixed with a cosolvent (31, there appear two additional binary parameters, namely m13 for the solvent (l)-cosolvent (3) interaction and m23 for the solute (2)-cosolvent (3) interaction, apart from m12 for the solute (2)-eolvent (1)interactions, the latter being taken from the binary regressions. Considering the fact
Figure 3. Comparisonof solubilitypredictions for the COZ-palmitic acidsystemat313.15K (experimentaldataofBambergeretal. (1988)).
the solute-cosolvent interactions are much stronger than the solvent-cosolvent interaction, only m23 is adjusted, keeping mi3 constant at 1.0, as in the case of like interactions. However, m13, when evaluated from vaporliquid equilibrium (WE) data as a function of temperature, justifies this value, as it is quite close to 1.0. For the sake of convenience,mi3 can be taken as 1.0instead of regressing from VLE data, in case such data are not available. Both the CS and GC methods are employed for the regression of r n 2 3 for 14 systems comprising 8 solids and 3 cosolvents. Compared to 6.3% AAFtD as obtained by Dobbs (1986)using the more rigorous hard-sphere van der Waals (HS-VDW)EOS,in the present work the overall % AARD is found to be 10.3 and 9.1 by the two methods, respectively. Slightlyhigher deviationsin the present work are solely the result of relatively poorer representations of the systems involving benzoic acid and 2-aminobenzoic acid in polar cosolvents. This is attributed to the acidbase interactions and hydrogen bonding between the solid molecules themselves and with the polar cosolvent. The complexityof the interactionsis manifested by the negative values of m23 obtained for these systems, indicating complexation and consequently a need for correction for the local compositions. The temperature dependency of m23 could not be studied due to lack of experimental data. Figures 4 and 5 describe the ability of the model to represent a ternary (solute-eolvent-cosolvent)system. Solubility Predictions of Mixed Solids. For the design of a separation process utilizing a supercritical fluid solvent, it is often necessary to investigate the optimum operating conditions by predicting the ratio of the component solubilitiesof the mixture or the separation factor. In modeling the solubility data of the mixed solide, it is assumed that the solids are completely immiscible and each solid component exists as a pure solid phase in equilibrium with the fluid phase. However, the solutesoluteinteractionsin the supercritical phase are considered using the degree of covolume dependency, m22, as 1.0,as in the case of the like interactions. The solute-eolvent adjustable parameters, m12 and m12!,are taken from the correspondingpure component solubility data regressions. The accuracy of the predicted solubilities,thus obtained for solid mixtures by both the CS and GC methods, is displayed in Table IV corresponding to 14 isotherms. It is interesting to note that the predictions are reasonably
926 Ind. Eng. Chem. Res., Vol. 32, No. 5, 1993
1 7 1
Table IV. Solubility Predictions with Modified CVD Mixing Rule for COz (1) + Solid (2) + Solid (2‘) Systems
OCTANE ( 3 . 5’/0)
I AC E TONE (3.5 ’/* )
svstem
ref
anthracene (2) carbazole (2’) anthracene (2) fluorene (2’) anthracene (2) phenanthrene (2’) benzoic acid (2) l,lO-decanediol(2’)
1
c3
80
130
180
230
280
330
380
PRESSURE ( b a r )
Figure 4. Solubility predictionsfor hexamethylbenzene in COZdoped with cosolvents at 308.15 K (experimental data of Dobbs (1986)).
benzoic acid (2) hexamethylbenzene (2’) naphthalene (2) biphenyl (2’) naphthalene (2) 2.6-DMN (2’) naphthalene.(2) phenanthrene (2’) 2-naphthol (2) anthracene (2’) 2-naphthol (2) phenanthrene (2’)
IdZ 0 I-
a
T(K) 313.1
a
313.1
b
308.0
b
318.0
C
308.0
C
318.0
d
308.0
e
308.0
f
308.0
e
308.0
d
308.0
f
308.1
f
318.1
f
328.1
% AARD by CS by GC method method 36.0 37.5 18.0 19.6 23.9 26.2 14.4 17.4 17.2 19.0 5.6 10.2 10.8 9.6 2.0 4.6 14.6 14.2 28.9 37.2 16.4 16.3 25.1 24.9 24.7 23.4 15.4 7.5 19.3 24.3 24.6 24.4 48.4 32.9 9.6 8.5 9.5 6.9 23.3 39.0 13.9 17.0 12.2 11.6 10.8 12.7 16.6 16.2 15.6 13.2 7.3 4.8 11.9 12.1 10.3 14.9
overall % AARD
V
a
17.4
18.1
*
Kwiatkowskyetal., 1984. KosalandHolder, 1987. Chimowitz and Pennissi, 1986. Dobbs and Johnston, 1987. e Gopal et al., 1985. f Lemert and Johnston, 1990.
n
U
w J
0
I
t/ L I BO
1
I
1
I
I
1
130
180
230
280
330
380
PRESSURE ( b a r )
Figure 5. Solubility predictions with modified CVD mixing rule for benzoic acid in COz mixed with cosolventsat 308.15 K (experimental data of Dobbs (1986)).
accurate with overall % AAFtD less than 19% by either method. It is considered as an appreciable achievement to be able to predict the mixture solubility data without using any adjustable parameter for solute-solute interactions. Earlier efforts with the VDW and the original CVD mixing rules yielded solubility predictions with overall deviations aa high as 50%, even after inclusion of an adjustable solute-solute interaction parameter in each method. For the process synthesis utilizing the “crossover region” in the solubility isotherms, it is crucial to be able to predict crossover pressures accurately. The success of the present model in this regard is clearly evident from Figures 6 and 7. The predicted crossover pressures of the individual components differ from the corresponding experimental values by less than 4% U R D , irrespective of relatively larger deviations in the solubility predictions. This agreement is attributed to the built-in temperature insensitivity of the binary interaction parameters, as in the highly compressible region of the crossover pressures a temperature variation has a pronounced effect.
I-
::
/
(r
LL W
0 -I
lo4
I
318.15 K
I
-5
10
80
130
180
230
PRESSURE ( b a r )
Figure 6. Solubility predictions with modified CVD mixing rule for the anthracene + phenanthrene mixture (experimentaldata of Kosal and Holder (1987)).
Addition of a cosolvent is viewed as an important consideration, as any effect that the cosolvent would have on improving the selectivity of separation, apart from enhancing the solubilities, could be of vital significance in improving the process economics. The predictability of the present model has been tested for solubilities of three solid mixtures in supercritical carbon dioxide doped with methanol (Dobbs and Johnston 1987). Each solid mixture consists of a polar and a nonpolar solid component. The prediction requires no additional parameters other than
Ind. Eng. Chem. Res., Vol. 32, No. 5, 1993 927
c
318.15 K a/
0,001 I-
V
a a LL L
I
W
0
I
-1
a
’ ‘ A #
a/,
L
-L
10
100
-
_ _ _- --308.15K --
/-
I
I
150
200
300
Table V. Solubility Predictions with Modified CVD Mixing Rule for COz (1) + Solid (2) + Solid (29 + Cosolvent (3) Systems (Cosolvent: Methanol (3.5 mol %)) % AARD by CS by GC system ref T(K) method method anthracene (2) a 308.1 6.2 3.5 2-aminobenzoicacid (2’) 15.8 21.6 anthracene (2) a 308.1 8.6 8.4 2-naphthol (2’) 4.8 9.7 a 308.1 9.7 4.1 benzoic acid (2) hexamethylbenzene (2’) 32.4 20.4 12.9
1~.-
f‘-
~
200
150
350
Figure 7. Solubility predictions with modified CVD mixing rule for the benzoic acid + 1,lO-decanediol mixture (experimental data of Chimowitz and Pennissi (1986)).
overall 7% AAFZD
0.0001
t
100
I
I
250
I
PRESSURE ( b a r )
a
W 2
11.4
Dobbs and Johnston, 1987.
the pure solute-solvent and pure solute-cosolvent adjustable parameters regressed from the pure component solubility data. The overall % AARD is found to be less than16% byeither method as reported in Table V, whereas the HS-VDW EOS (Dobbs and Johnston, 1987)could only qualitatively represent the solubility behavior of the quaternary systems. The quantitative agreement, as demonstrated by Figure 8, speaks for the ability of the model presented. Solubility Predictions of Solids from Pure Component Properties. In the prediction model discussed above, though the usage of the solvent-cosolvent and solute-solute binary adjustable parameters, m13 and m22, has been obviated, the solubility data of the pure solid in pure and mixed solvents are, however, needed for regression of the other two adjustable parameters, m12 and m23, respectively. In order to make the model completely predictive, i.e., to be able to predict the mixture solubility data even without having any pure component experimental solubility data, it is necessary to develop a correlation function for the binary adjustable parameters in terms of the easily available pure component properties. Similar attempts (Johnston, 1982; Wong, 1985; Dobbs, 1986) toward achieving this objective gained limited success due to inadequacy of the mixing rule to reduce the sensitivity of the adjustable parameter or to represent its temperature dependency. Ftao and Mukhopadhyay (1989) utilized the original CVD mixing rule in correlating either
- -,- - - 2 - - -X 250
3 01 0
300 1
4
PRESSURE
Figure 8. Solubility predictions with modified C M mixing rule for the 2-naphthol + anthracene mixture in the presence of methanol (experimental data of Dobbs and Johnston (1987)).
the adjustable parameters in three different approaches in terms of the critical properties of the pure component or the binary interaction energy parameter itself in terms of the molecular weights and VDW volumes (Rao, 1990). However, the latter model did not account for the temperature dependency and consequently could not predict the temperature effects of the solubility data. It may be noted here that in the binary interaction energy term of the present mixing rule, aij(b/bjj)”v,the term U i j / b p u actually represents the unlike interaction characteristics of the solute-solvent system, as b ‘Y bl. Correlating this CVD interaction parameter, rather than mij, is preferable for two reasons, namely, &j/bijmCJ is less ternperature dependent and less sensitive to the variation of mjj itself. A temperature-independent constant value of u12/b12m12 for a representative binary system marginally increases the AAFtD from 11.9% to 18.0% by the CS method and from 11.7% to 15.9% by the GC method. Thus u12/b12m12has been correlated as a function of VDW volume, molar volume, and the molecular weights of the solids and carbon dioxide for 21 solids, independent of temperature, as u12/b12m1* =
[a+ B(M,/M,)((b,b,,,,)”2/v,B) + y(p1P2)1’21 x
lo6 (9)
where p1 and pp are the dipole moments and M Iand M Z are the molecular weights. The van der Waals volume bp’ww and the solid molar volume u28 are listed in Table 11. The constants a,8, and y for the oxygen-free (I) and oxygen-containing (11) solids are type I type I1
Ly
i3
Y
-0.282 0.886
0.743 -0.409
-0.045 0.680
Two separate correlations are needed for nonoxygenated and oxygenated solutes, as the values of the CVD interaction parameter fall in a close range which is different for these two classes respectively. Their marginal variation within the range is found to depend only on properties like molecular weight, van der Waals volume, and dipole moment.
928 Ind. Eng. Chem. Res., Vol. 32, No. 5,1993 Table VII. Correlation of High-pressure Vapor-Liquid Equilibria with Modified CVD Mixing Rule % AARD alz/blzml+eg 7% a 1 2 / b 1 ~ ~%~ ~ ~ ~ ~ ~ solid T(K) X10-6 AARD X10-6 AARD system T(K) mi, inyz inxz ref Oxygen-Free Solids COz + methanol 310.1 1.09 0.3 15.9 a 0.501 18.9 15.1 0.470 308.3 acridine 313.4 1.10 0.1 15.7 b 12.7 12.7 0.499 318.3 1.15 0.1 15.9 b COz+ ethanol 313.4 17.6 17.7 0.505 328.3 1.07 0.5 14.9 b 333.4 15.6 15.6 0.506 343.3 0.99 0.4 19.9 a COz + hexane 298.15 0.571 7.8 6.1 0.573 anthracene 308.1 1.04 0.1 15.3 a CO2 + benzene 298.15 29.7 18.8 0.502 313.1 11.7 13.8 Oghaki et al., 1989. * Suzuki et al., 1990. 0.571 318.1 14.7 13.3 0.598 343.1 13.2 0.594 5.1 0.555 biphenyl 309.0 10.8 9.8 0.620 318.5 23.0 16.2 0.688 322.6 34.3 32.1 0.552 328.4 41.1 0.496 19.5 0.418 313.1 carbazole 0.475 11.0 1.9 0.502 2,3-DMN 308.0 6.5 2.5 30 0.470 318.0 7.4 6.0 0.461 328.0 0.479 19.9 3.6 0.435 2,6-DMN 308.0 8.2 2.5 0.462 318.0 9.7 6.4 0.451 328.0 0.501 56.5 3.5 0.702 fluorene 308.1 28.8 15.7 0.543 313.1 26.3 7.9 0.566 323.1 26.9 6.0 0.600 343.1 0.437 3.3 2.6 0.431 hexamethyl303.0 3.7 3.7 0.437 benzene 308.0 6.3 7.0 0.449 323.0 8.1 4.3 0.464 343.0 I TRI P A L M I l t N 0.543 47.3 2.9 0.399 naphthalene 308.1 I 10.2 I 5.9 0.466 328.1 I 29.1 32.9 0.475 333.5 i 5.7 0.569 8.7 0.572 phenanthrene 308.0 10I F I I I I 23.3 18.8 0.502 313.1 ‘0 100 150 200 250 50 33.4 23.3 0.571 318.1 21.5 24.8 0.598 328.0 PRESSURE ( b a r ) 0.440 11.0 6.7 0.460 skatole 308.1 Figure9. Solubilitiesof pure palmitic acid and tripalmitin (predicted Oxygenated Solids with modified CVD mixing rule). 0.812 18.8 0.788 18.0 2-aminobenzoic 308.1 acid point calculations of four liquids in supercritical carbon 0.816 78.9 1.710 8.9 benzoic acid 308.1 dioxide are presented in Table VII. It can be observed 76.4 13.6 1.760 318.1 that the fluid-phase compositions, as expected, can be very 14.8 74.7 1.960 328.1 67.4 2.450 7.3 343.1 accurately predicted as the mixing rule is designed to 0.881 2.3 3.0 1,lO-decanediol 318.0 0.888 represent the molecular interactions in highly dilute 3.7 3.7 0.887 323.0 supercritical mixtures. At lower pressures, the deviations 4.7 5.9 0.881 328.0 of the liquid-phase compoeitions are relatively large. Thie 0.825 19.2 17.9 0.811 2-naphthol 308.3 could be a consequence of adjusting the degree of covolume 18.4 18.7 318.3 0.838 dependency for the unlike interactions at the cost of the 23.3 23.2 328.1 0.827 25.0 24.8 0.774 343.1 like interactions. A uniform adjustable parameter may 0.662 21.4 51.8 0.788 palmitic acid 318.0 be a better choice for representing a liquid-fluid system 27.8 27.5 0.662 328.0 with relatively less asymmetry. Further, as the model 0.676 35.3 29.8 318.0 0.615 stearic acid works well for near-critical pressures, it is expected to 21.4 21.5 0.676 328.0 perform better for liquid-fluid systems with high asymoverall % U R D 12.8 23.6 metry.
Table VI. Comparison of Solubility Predictions with Modified CVD Mixing Rule (GCMethod)
(2
3k
The overall 7’ 5 AARD of the predicted solubilities utilizing the temperature-independent CVD interaction parameters from the above correlation for the same 52 isotherms is 23.6%,as can be seen from Table VI. The slightly higher overall 5% AARD, as compared to that for the regression, is due to relatively higher deviations for the polar solids. A similar correlation for the CVD interaction parameter, a23/ b23m23 for the solute-cosolvent interaction, also yields large deviations in the solubility predictions, hinting at the possibility of chemical complexation or similar strong interactions, which calls for a different treatment. This will be discussed in a subsequent publication. Solubilities of Liquid Solutes. The utility of the modified CVD mixing d e has been tested for high pressure fluid-liquid equilibria. m12 values regressed from bubble-
Process Parameters for uCrossover” Separation. To demonstrate the utility of the predictive model developed in the present work, the process parameters for the separation of palmitic acid from tripalmitin have been established. Solubility predictions for the palmitic acidtripalmitin mixture have been carried out at temperatures from 35 to 50 OC using the temperature-independent CVD interaction parameter, Uij/bijmlJ. The experimental data are reported by Bamberger et al. (1988) at only 4OOC. The crossover pressures for pure palmitic acid and pure tripalmitin (Figure 9) are approximately 120 and 140 bar, respectively, whereas in the mixture (Figure lo), they are 120 and 136 bar, respectively. The prediction model indicates that the solubility of pure palmitic acid is almost same as that in the mixture. This is probably due to the weak solute-solute interactions between the two compo-
PALMITIC ACID
323.5 K
c
,
I
/
308.5 K /
E 1 90
/
, /'
TRI P A L M I T I N
/
/
I
I
110
130
I I 1s 0
PRESSURE ( b a r )
Figure 10. Solubility isotherms of the palmitic acid + tripalmitin mixture (predicted with modified CVD mixing rule).
nents with similar polarity. Also, with the solubility of tripalmitin being very low, its interactions with palmitic acid would not be of significance. This prediction result was in conformity with the experimental observation (Ftao, 1992). In the proposed scheme of separation, the extraction is carried out at 130 bar and 50 "C, and then palmitic acid is separated out of the extract phase by isobarically changing the temperature to 35 "C. Carbon dioxide released from the separator is isobarically heated to 50 "C prior to recycling. Calculations based on the predictive model indicate that 2.1 m3 COdh flow rate per pass would yield 1kg palmitic acid/h. These prediction results compare satisfactorily with the experimental observation under the equilibrium condition.
Conclusion The compressed gas model utilizing the P-R EOS has been employed for representing the supercritical fluid base behavior of solid-fluid and liquid-fluid systems. The covolume-dependentmixing rule has been suitably modified to be able to predict the solubilities of mixed solids in pure supercritical carbon dioxide without and with a cosolvent from the knowledge of the corresponding pure component solubility data. The group contribution (GC) method developed in the present work seems to be preferable to the correspondingstates (CS)method. The former predicts better and obviates the need for the critical properties of the solids which are often not available in the literature. The overall deviations indicate that the modified CVD mixing rule used in the present work gives better predictions for the solubilities of mixed solids in comparison to any other mixing rule without an unlike solute-solute adjustable parameter. The ability of the model to accurately predict the crossover pressures is considered to be of vital significance. Further, it is observed that the CVD interaction parameter, ajj/bjjmlJ, is insensitive to temperature or mij itself. A temperature independent value of a1J b 1 p 2 , calculated from a correlation function developed in terms of pure component properties, is able to generate solubility data at various temperatures. The pure component properties needed are VDW molecular volume, dipole moment, molecular
Ind. Eng. Chem. Res., Vol. 32,No. 5,1993 929 weight, and molar volume, in order that they are easily available in the literature. However, such simple properties are not adequate in representing the solutecosolvent CVD interaction parameter, a23/bzp3, and hence a different approach is called for in the case of solubility predictions in mixed solvent from only the pure component properties. The validity of the present mixing rule has been tested for supercritical fluid-liquid equilibria. It is observed that the fluid-phase compositionsare well predicted although the deviations are more for the liquid-phase compositions. This leads to the conclusion that the present mixing rule is well suited for dilute supercritical solutions. Utilizing the predicted solubility data and crossover pressures, a process scheme has been evolved for separation of palmitic acid from a tripalmitin and palmitic acid mixture and verified experimentally. This study shows how a predictive model can be used for preliminary screening of process design parameters.
Nomenclature AI, Az = terms in the fugacity coefficient expression a = energy parameter of EOS B, B1 = terms in the fugacity coefficient expression b = covolume parameter of EOS AH = heat of sublimation, J/mol k = binary interaction of a M = molecular weight m = adjustable parameter in CVD mixing rule NDP = number of experimental data points P = pressure, bar R = universal gas constant, (cm3)(bar)/(rnol)(K) T = temperature, K u = molar volume, cm3/mol 1c = mole fraction, solid or liquid phase y = mole fraction, vapor or SCF phase 2 = compressibility factor
Greek Letters a = empirical constant /3 = empirical constant y = activity coefficient p = dipole moment, D (b = fugacity coefficient Superscript s = solid phase or saturation Subscripts 1 = solvent 2, 2' = solute 3 = cosolvent c = critical cal = calculated exp = experimental i, j = components i and j i j = binary parameter for constituenta i and j vdW = van der Waals volume
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Received for review August 3, 1992 Revised manuscript received December 7, 1992 Accepted January 28, 1993
