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Ind. Eng. Chem. Res. 1998, 37, 3786-3792
CORRELATIONS A Density-Dependent Solute Solubility Parameter for Correlating Solubilities in Supercritical Fluids Selma E. Guigard and Warren H. Stiver* School of Engineering, University of Guelph, Guelph, Ontario, Canada N1G 2W1
A density-dependent solute solubility parameter has successfully correlated solubilities in supercritical fluids (SCFs). Fifteen solutes, including polar and nonpolar compounds, have been studied in up to four SCFs. A two-parameter, linear fit resulted in a median average absolute relative deviation (AARD) of 20%, ranging from 6.1% to 42%, for the 34 systems studied. A three-parameter power fit provided a median AARD of 12%, ranging from 2.3% to 38%. Introduction Supercritical fluids (SCFs) are used on a commercial scale for the decaffeination of coffee and tea and for the extraction of hops and natural fragrances (McHugh and Krukonis, 1994). Analytical-scale supercritical fluid extraction systems are commercially available, and numerous methodologies have been developed for the recovery of many analytes from different matrixes. A review of the literature immediately demonstrates that a wide range of additional applications are at various stages of research and development ranging from the extraction of heavy metals from soil (Yazdi and Beckman, 1996) to the extraction of various compounds from coal (Kershaw, 1989; Sunol and Beyer, 1990). All of these developments require accurate and reliable solubilities of the solute in the chosen supercritical fluid. Design and optimization of supercritical fluid extraction (SFE) processes depend on solubility data. Although solubility is often not attained in extraction procedures, it can provide insight into the efficiency of the extraction process. Solubility data have been tabulated in the literature for a number of different compounds (Bartle et al., 1991). However, an extensive amount of data is required due to the strong dependence of solubility on density, temperature, and pressure. For this reason, correlations and predictive techniques are currently being developed to generate solubility data without investing the time and money required to generate the data experimentally. These models generally fall into one of three approaches: an equation of state (EOS) approach where the SCF is treated as a high-pressure gas, a density-based approach, or a solubility parameter approach where the SCF is treated as a liquid. These approaches as well as others are discussed in Brennecke and Eckert (1989). Equation of state (EOS) modeling is a first approach. This type of modeling is the most commonly used technique to correlate solubilities in SCFs. There exist a large number of EOS models with Peng-Robinson * To whom correspondence should be addressed. Phone: (519) 824-4120, ext. 4862. Fax: (519) 836-0227. E-mail:
[email protected].
(PR) and the Soave-Redlich-Kwong equations in common usage (McHugh and Krukonis, 1994). Brennecke and Eckert (1989) and Johnston et al. (1989) provide a review of some of the work that has been done in EOS modeling for SCFs. More recent work includes that of Mukhopadhyay and Rao (1993), Macnaughton and Foster (1994), and Cross et al. (1996). Liu and Nagahama (1996) describe EOS modeling for binary and ternary systems involving naphthalene and phenanthrene in supercritical carbon dioxide (SC CO2) and supercritical fluoroform. When dealing with the EOS approach, it is necessary to select the correct equation, choose the mixing rules to be applied, and evaluate the pure component parameters. In some cases the parameters can be determined a priori, while in other cases either the required data are not available or the resulting correlations are poor. These latter cases lead to a fitting exercise that introduces empiricism into an otherwise theoretically sound structure. In reference to selecting the appropriate equation, Johnston et al. (1989, p 1122) state that “no single model will work for all situations”. Density-based correlations are a second type of modeling used for supercritical-phase solubilities. This type of modeling attempts to explain the common observation that the logarithm of the solubility is linearly dependent on the density or the log of the density of the supercritical fluid (Kumar and Johnston, 1988). This method has been used by many including Kumar and Johnston (1988), Chrastil (1982), Mitra and Wilson (1991), and Liu and Nagahama (1996). For the studied solubility data, Liu and Nagahama (1996) found that the densitybased modeling yielded better results than the EOS modeling; however, a quadratic equation with three fitted parameters was required. The use of solubility parameters to predict solubilities in SCFs is a third approach. This type of modeling is based on regular solution theory, and the activity coefficient of the solute in the supercritical phase is calculated using the Scatchard-Hildebrand equation. The solubility parameter approach has been investigated by Lagalante et al. (1995) and Wai et al. (1996). Lagalante et al. (1995) concluded that the solubility parameter approach was limited in predicting the
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Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 3787
solubility of chromium(III) β-diketonates in SC CO2. Wai et al. (1996) suggest that the solubility parameter approach, along with a group contribution method to evaluate solubility parameters, could be used to qualitatively evaluate solubilities of metal-dithiocarbamate complexes in SCFs. Ziger and Eckert (1983) developed an equation that uses the solubility parameter approach combined with the van der Waals equation of state. This equation was used by Ziger and Eckert (1983), Gurdial and Foster (1991), and Li et al. (1991) to correlate solubilities of solids in SCFs. Ziger and Eckert (1983) found average standard deviations ranging from 10% to 60% for the 24 systems studied. Gurdial and Foster (1991) found average absolute relative deviations varying from 9.6% to 15.5% for o-hydroxybenzoic acid solubility in SC CO2. Li et al. (1991) found that this method was limited for correlating solubilities of caffeine and theobromine solubility in SC CO2. This paper provides a modification of the solubility parameter approach that leads to quantitative success in correlating solubilities in SCFs. Theory Regular solutions are defined as solutions for which, upon mixing of the components, there is no excess entropy or excess volume of mixing (Prausnitz et al., 1986). For these solutions, the Scatchard-Hildebrand equation can be written
RT ln γ2 ) vL2 φ12(δ2 - δ1)2
(1)
with φ1, the volume fraction of the solvent, defined as
φ1 )
x1vL1
(2)
x1vL1 + x2vL2
f s2 ) f2(solute in solution)
(3)
) γ2 y2 f L2
() f L2
vL2 φ12 (δ2 - δ1)2 - ln y2 ) RT
(4)
According to Prausnitz et al. (1986),
ln
() f s2
f
L 2
)
( ) ( )
-∆Hm ∆cp Tm fus Tm -1 + -1 RTm T R T
Prausnitz et al. (1986) state that, to a fair approximation, the heat capacity terms can be neglected. Equations 4 and 5 then combine to yield an expression for the solute solubility:
(
CO2
naphthalene phenanthrene anthracene benzoic acid acridine naphthol biphenyl 2,3-dimethylnaphthalene 2,6-dimethylnaphthalene caffeine theobromine phenol 2-hydroxybenzoic acid Cu(acac)2 Cr(acac)3
a,b,c,d,e,f,g f,d,l,n l,o f,m,v m m a,m f f p,q,r p,q s w n,t,u t
ethylene ethane g,h,i,j,k f,j j f
)
]
vL2 φ12 -∆Hm fus Tm -1 (δ - δ1)2 y2 ) exp RTm T RT 2
(6)
Equation 6 represents the functional form of regular
l,m l l m m m m
fluoroform m m m m m m
f f
a McHugh and Paulaitis, 1980. b Lamb et al., 1986. c Chang and Morrell, 1985. d Dobbs et al., 1986. e Mitra et al., 1988. f Kurnik et al., 1981. g Tsekhanskaya et al., 1964. h Diepen and Scheffer, 1948. i Diepen and Scheffer, 1953. j Johnston and Eckert, 1981. k Masuoka and Yorizane, 1982. l Johnston et al., 1982. m Schmitt and Reid, 1986. n Cross et al., 1996. o Hampson, 1996. p Johannsen and Brunner, 1994. q Li et al., 1991. r Ebeling and Franck, 1984. s van Leer and Paulaitis, 1980. t Lagalante et al., 1995. u M’Hamdi et al., 1991. v Dobbs et al., 1987. w Gurdial and Foster, 1991.
solution theory, using solubility parameters to calculate solubilities. The use of this equation relies on the availability of reliable values for the solute properties m ), melting point (Tm), and molar of heat of fusion (∆H fus L volume (v2 ). The model becomes empirical through the values of the solubility parameters, δ1 and δ2. The solubility parameter is defined by Hildebrand and Scott (1950) as
( ) -E vL
1/2
(7)
Through a combination of the van der Waals equation of state and existing solubility data, Hildebrand and Scott (1950) proposed the following relationship to relate the solubility parameter to the critical pressure, Pc:
(8)
Giddings (1968) further extended this equation by assuming the equivalence of gases and liquids at a common density. This assumption led to the following equation that describes the solubility parameter of a solvent:
( )
δ1 ) 1.25Pc1/2
∆cp Tm ln (5) R T
[
solute
δ ) 1.25Pc1/2
Combining eqs 1 and 3 yields
ln
solvent
δ)
Fugacities are used to determine the activity coefficient
f s2
Table 1. Summary of Systems Studied and Solubility Data Sources
Fr Fr(liq)
(9)
where Fr(liq), the reduced density of the solvent at its normal boiling point, is usually taken to be equal to 2.66. Equation 9 has since been used to calculate the solubility parameter for solvents in the supercritical state. In calculating δ2, the conventional solubility parameter approach assumes that the δ2 is a constant, independent of the solvent in which the solute is dissolved. Solute solubility parameter values are widely available in various references (Barton, 1991) or can be calculated from group contribution techniques such as Fedors (1974a,b). This conventional approach leads to the solubility parameter approach being limited to
3788 Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 Table 2. Solute Properties ∆Hfus (J/mol) ref
solute naphthalene phenanthrene anthracene benzoic acid acridine naphthol biphenyl 2,3-dimethylnaphthalene 2,6-dimethylnaphthalene caffeine theobromine phenol 2-hydroxybenzoic acid Cu(acac)2 Cr(acac)3
19 123 16 465 28 829 17 317 19 700 17 511 18 601 25 101 25 055 21 118 41 110 11 289 19 585 37 121 28 400
a a a a a a a a a b b a c d e
Tm (K) 353.2 372.2 492.5 395.4 384.0 396.0 342.0 377.8 383.2 511.0 610.0 313.9 432 557.0 489.0
vL2 ref (cm3/mol) ref a a a a a a a a a f a a g h e
125.0 182.0 138.9 112.4 178.0 118.0 132.0 156.0 199.0 157.9 125.5 87.79 92 874.3 257.7
i i i c k k k i i j j j g i l
a
Lide, 1993. b Johannsen and Brunner, 1994 c Daubert and Danner, 1989. d Estimate from Koshimura, 1976. e Lagalante et al., 1995. f Li et al., 1991. g Gurdial and Foster, 1991. h Aldrich Chemical Co., 1994. i Mackay et al., 1992. j Calculated from density and molecular weight. k Schmitt and Reid, 1986. l Cross et al., 1996.
Figure 1. Solubility parameters of naphthalene and SC CO2 as a function of SCF density.
Table 3. Supercritical Fluid Properties SCF
Tc (K)
Pc (MPa)
Fc (g/cm3)
ref
P-v-T data
CO2 ethylene ethane fluoroform
304.2 282.4 305.4 299.1
7.38 5.04 4.88 4.95
0.468 0.217 0.233 0.525
a a a b
c d, e f
a Prausnitz et al., 1986. b Schmitt and Reid, 1986. c IUPAC, 1976. d IUPAC, 1988. e IUPAC, 1972. f Sychev et al., 1987.
qualitative or semiquantitative success in SCFs. In this paper, a break from this convention is attempted to explore whether quantitative success can be achieved in correlating and predicting solubility behavior in SCFs. Two proposed relations between δ2 and the density of the SCF are tested. The first is a linear fit and the second is a power fit. The linear fit is a limiting case of the power fit in which the power is forced to be 1.0. The forms of the relations are given below:
δ2 ) A + BFSCF
(10)
δ2 ) a + bFcSCF
(11)
Figure 2. Naphthalene solubility in SC CO2 at 308 K.
Methodology Table 1 outlines the 34 systems studied in this paper and the literature sources used for the solubility data. It is acknowledged that the systems studied do not include all of the data available but should be sufficient to provide a thorough testing of the potential of eqs 10 and 11. It should be noted that only data that was available in numerical form has been used and no interpolation from graphically presented data has been relied on. The thermodynamic properties of the different solutes and supercritical fluids are summarized in Tables 2 and 3. For each data point available, eq 6 was used to solve for the solubility parameter of the solute (δ2). For each m , Tm, and vL2 from Table 2 and case, the values of ∆H fus the solubility parameter for the SCF (δ1), calculated from eq 9, were used to determine δ2. In instances where the solubility data from the literature did not provide the SCF density, the density was calculated by linear interpolation of P-v-T data referenced in Table
Figure 3. Naphthalene solubility in SC CO2 at 328 K.
3. The final result is a value of δ2 for each experimental data point. The solute solubility parameters were then regressed according to eqs 10 and 11 for the linear and power fit, respectively. The best regression was based on minimizing the errors between the regressed and observed solubilities (y2) rather than based on minimizing the error between the regressed and observed values of δ2. The error was calculated as the “average absolute relative deviation” (AARD) according to eq 12:
AARD(%) )
|
1 n
∑
|
y2(exp) - y2(regr) y2(exp)
× 100
(12)
Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 3789 Table 4. Results of the Linear and Power Fits of δ2 As a Function of Solvent Density, with Minimization of the Error between Experimental and Calculated Solubilities linear fit SCF
no. of data points
A
naphthalene
CO2 ethylene ethane fluoroform
175 192 54 18
phenanthrene
CO2 ethylene ethane fluoroform
anthracene
power fit
B
% AARD
a
b
c
% AARD
12.675 13.088 13.239 11.828
12.203 16.936 18.532 8.089
25.7 25.3 25.4 25.8
15.543 14.639 15.547 13.876
10.471 25.835 39.018 9.472
1.929 1.700 2.201 2.419
12.6 17.8 13.7 6.3
40 45 23 8
13.058 13.163 13.156 12.359
12.539 18.685 20.695 8.118
30.0 16.3 13.7 17.1
16.288 14.033 13.823 14.162
9.950 21.821 21.367 8.128
1.776 1.286 1.129 2.015
15.5 13.8 13.0 3.4
CO2 ethylene ethane fluoroform
42 29 21 6
14.506 14.505 13.683 14.640
12.487 18.733 22.307 6.722
23.1 8.6 7.0 20.2
15.543 14.562 18.009 17.051
11.557 18.766 30.470 10.657
1.184 1.011 2.077 3.670
20.8 8.5 5.9 9.7
benzoic acid
CO2 ethylene ethane fluoroform
60 15 30 10
15.794 16.004 16.677 15.566
10.893 17.114 18.120 5.961
18.7 10.7 27.4 28.0
17.916 17.698 16.461 16.832
9.194 20.478 17.786 8.694
1.541 1.494 0.946 2.690
14.0 10.3 27.4 9.6
acridine
CO2 ethane fluoroform
28 32 12
13.281 13.585 12.856
12.463 21.716 9.266
16.7 25.2 20.4
15.072 14.059 14.519
10.932 22.656 7.638
1.316 1.103 1.386
11.5 24.7 11.4
naphthol
CO2 ethane fluoroform
26 28 10
15.709 16.287 15.911
12.628 23.080 8.781
20.7 31.2 21.5
17.467 16.900 17.971
11.123 23.983 6.803
1.352 1.118 1.710
15.2 30.7 8.0
biphenyl
CO2 ethane
43 10
11.032 11.791
14.745 23.144
8.5 6.1
14.620 14.075
11.405 25.432
1.527 1.412
4.5 5.7
2,3 dimethylnaphthalene
CO2 ethylene
15 18
12.614 13.255
11.873 14.802
15.7 12.2
15.088 13.002
9.941 14.325
1.657 0.925
10.3 12.3
2,6 dimethylnaphthalene
CO2 ethylene
15 18
10.722 11.666
13.114 17.267
11.6 9.9
12.711 12.130
11.479 18.606
1.364 1.149
4.0 9.2
caffeine
CO2
98
15.751
9.668
25.4
16.255
9.349
1.162
22.5
theobromine
CO2
37
18.957
8.783
38.4
16.197
11.366
0.667
36.3
phenol
CO2
41
13.257
15.035
8.3
16.079
12.470
1.373
5.1
2-hydroxybenzoic acid
CO2
49
18.060
11.165
20.1
20.675
9.423
1.732
12.3
Cu(acac)2
CO2
29
7.475
13.218
42.2
8.892
12.123
1.301
37.7
Cr(acac)3
CO2
8
12.145
11.825
8.4
17.176
7.112
2.273
2.3
solute
median
Results and Discussion Figure 1 illustrates the resulting solute solubility parameters from fitting the 175 data points for naphthalene in SC CO2 using eq 6. The solubility parameter for each data point is plotted against the supercritical fluid density. Included on the figure are the solubility parameters for the SCF (δ1) on which the solute values depend. Although a given density may be arrived at by a number of combinations of temperature and pressure, it is visually obvious that naphthalene’s solubility parameter in CO2 is dependent on the SCF’s density. The same observation is true for all of the other solute and solvent combinations. Figures 2 and 3 show experimental solubility and the linear and power fits for naphthalene in SC CO2 as a function of SCF density, at two temperatures. From these figures, it appears that the linear fit does not have the correct functional form. The linear fit predicts solubilities to continually increase with increasing density while the data demonstrates a leveling off or a small reduction. The power fit more adequately captures the observed form. Table 4 provides the quantitative results of the regression for both the linear and power fits. The AARDs are included for each case and the generally low
20
12
values are indicative of a good fitting form. The linear fit achieves a median AARD of 20% with a range of 6.1% to 42% for the various solutes and solvents. The additional parameter in the power fit leads to a median AARD of 12% with a range of 2.3-38%. The improved AARDs of the power fit over the linear fit is likely due to its better representation of the form of the solubility curve. It is believed that the power fit of the solute solubility parameter gives a better correlation of the experimental solubility data. The values of the fitting parameters, A and B for the linear case and a, b, and c for the power case, are dependent on the thermodynamic parameters provided in Table 3 and on the validity of eq 9 for the value of δ1. To test whether the success of the fitting was due to potentially incorrect values in Table 3, a sensitivity analysis was completed for naphthalene in SC CO2. The analysis was completed by varying the values of molar volume and heat of fusion by (10%. The numerical values for the fitting parameters changed, yet for both the linear and power fits, the performance remained unchanged. The AARD was within 1% in all cases. The correlation success depends on the experimental accuracy of each data point. Tsekhanskaya et al. (1964), McHugh and Paulaitis (1980), and Dobbs et al. (1986)
3790 Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998
report experimental accuracies of the range of 1-4%. These uncertainties will influence the correlation and become buried in the error in the correlation or represent a limiting value for a sound correlation structure. For some compounds, the experimental data may be unknowingly of much lower quality and could lead to a weaker correlation performance even with a strong correlation structure. For example, Li et al. (1991) found in their correlation that the values for theobromine were very different from those for caffeine. Li et al. (1991) attributed these differences to possible errors in the experimental data used in the correlation. They suggest that errors may have occurred when measuring solubilities as low as 10-6 (mole fraction). In reality, all correlations are limited by the accuracy of the original data. A second factor-limiting correlation success is the variability that exists between different experimental data sets. Schmitt and Reid (1986) suggest that this variability is in the range of 5-10%. Seven different data sets have been included for naphthalene in the carbon dioxide system. Treating each of these data sets independently, the power fit correlation achieves AARDs of 0.87-19% compared with 12.6% for the combined data set. For five of the seven data sets, using solubility data from a single experimental source resulted in a noticeable improvement in performance. The PR-EOS is a commonly used approach for correlating solubility in supercritical fluids. Schmitt and Reid (1986) used the data of Tsekhanskaya et al. (1964) to arrive at best fit parameters for naphthalene in carbon dioxide, and on the basis of these parameters, the calculated AARD is 37%. For this single data set, the power fit correlation achieved an AARD of 19%. It is recognized that this is a single direct comparison and that other EOSs may perform better while some worse. The message from this comparison is that this new approach provides comparable quantitative capabilities with one common EOS approach. Four alternative correlation approaches are compared with the linear and power fit for 2-hydroxybenzoic acid (Table 5). Gurdial and Foster (1991) measured the solubilities for 2-hydroxybenzoic acid at different temperatures and tested the correlations of Kumar and Johnston (1988), Schmitt and Reid (1985), and Ziger and Eckert (1983). The correlations of Kumar and Johnston (1988) and Schmitt and Reid (1985) provide the best performance at an AARD of approximately 7%; however, these correlations require eight and four fitting parameters, respectively. The correlations of Chrastil (1982), Ziger and Eckert (1983), and the power fit model of this work all provide performance at around an AARD of 11-12%. Ziger and Eckert’s model has only two fitting parameters but requires the estimation of a number of pure component solute and solvent properties (Gurdial and Foster, 1991). Chrastil’s (1982) model and the power fit model both offer good performance, within a simple structure and with only three fitting parameters.
To further compare Chrastil’s model to the power fit model, two additional systems were investigated: naphthalene in CO2 and naphthalene in ethylene. For naphthalene in CO2, Chrastil’s model resulted in an AARD of 16.5% compared with 12.6% for the power fit model. While for naphthalene in ethylene, the AARD was 20.5% and 17.8% for the Chrastil and the power fit models, respectively. The power fit model appears to perform on par and in some cases better than Chrastil’s model. Upon inspection of the parameters in Table 4, it seems that the parameters A and a are solute-dependent, the parameters B and b are solvent-dependent, and the parameter c depends on both. These trends seem to indicate that a predictive capability may exist and is currently being explored. Conclusions A density-dependent solute solubility parameter successfully correlated solubilities in supercritical fluids. A linear (two-parameter) and power (three-parameter) fit between the solute solubility parameter and the solvent density resulted in an average performance of an AARD of 20% and 12%, respectively. This performance is similar and sometimes superior to current methods in use. The advantages of the correlation include the following: one, it is simple to use; two, it requires a maximum of three fitting parameters to cover all temperature, pressure, and densities in the supercritical region; three, it is successful for all systems studied including both polar and nonpolar species. Acknowledgment Funding for this research has been provided by the Natural Sciences and Engineering Research Council of Canada and an Ontario Graduate Scholarship. Nomenclature m ∆Hfus ) enthalpy of fusion, J/mol ∆cp ) heat capacity, J/(mol‚K) A,B ) parameters for the linear fit (eq 10) a,b,c ) parameters for the power fit (eq 11) -E ) energy of vaporization to the gas at zero pressure, J/mol f ) fugacity, Pa n ) number of data points P ) pressure, Pa (except eq 9 where P is in atm) R ) universal gas constant, 8.314 J/(mol‚K) T ) temperature, K v ) molar volume of the pure component, cm3/mol x ) mol fraction y ) solubility, mol fraction
Greek Symbols δ ) solubility parameter, MPa1/2 (except eq 9 where δ is in (cal/cm3)1/2)
Table 5. Comparison of Correlation Results for 2-Hydroxybenzoic Acid AARD (%) T (°C)
Kumar and Johnston (1988) (two × four parameters)
Chrastil (1982) (three parameters)
Schmitt and Reid (1985) (four parameters)
Ziger and Eckert (1983) (two parameters)
35 40 45 55
5.8 7.2 8.1 6.2
8.7 11.8 13.9 8.3
4.5 8.5 13.4 4.0
10.6 9.6 11.4 15.5
this work linear fit power fit (two parameters) (three parameters) 14.6 18.4 23.2 24.7
5.7 12.9 11.0 19.3
Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 3791 γ ) activity coefficient φ ) ideal volume fraction F ) density, g/cm3 Subscripts 1 ) solvent 2 ) solute c ) critical exp ) experimental liq ) liquid m ) melting point r ) reduced regr ) regressed SCF ) supercritical fluid Superscripts L ) liquid s ) solid
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Resubmitted for review May 14, 1998 Accepted June 30, 1998 IE9702946
