Ind. Eng. Chem. Res. 2002, 41, 4899-4905
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Representation of the Solubility of Solids in Supercritical Fluids Using the SAFT Equation of State Chongli Zhong* and Hongyu Yang Department of Chemical Engineering, P.O. Box 100, Beijing University of Chemical Technology, Beijing 100029, China
A SAFT equation of state (EOS) combined with eight mixing rules was used to evaluate the capability of the SAFT approach for modeling the solubility of solids in supercritical fluids (SCFs). The results show that the SAFT approach gives good correlative accuracy in general, and the results are satisfactory when the three-parameter mixing rules are used, where the average absolute deviation of solid solubility in the SCF phase is normally smaller than 10%. The present work shows that the SAFT approach can be used to model solid-SCF equilibria, which gives slightly better correlative accuracy than the cubic EOSs. Introduction Supercritical fluid extraction (SCFE) is a relatively new and promising separation technology that has a great application potential in many separation and purification processes, such as in food, pharmaceutical, polymer processing, and biochemical industries and so forth. Since the solubility of solids can be easily tuned with solvent density near the solvent critical point, it makes supercritical fluids (SCFs) attractive solvent candidates for separating heavy compounds, which becomes one of the main applications of SCFE technology. For the process design, development, and optimization, an accurate model for the calculation of the solubility of solids in SCFs is needed, where an equation of state (EOS) is the most suitable model because such processes deal with high-pressure phase equilibria. Both cubic and noncubic EOSs have been used to model the solubility of solids in SCFs, and a summary of the models and methods used was given by Ashour et al.1 They also carried out a comprehensive study on the capability of the cubic EOSs to model the solubility of solids in SCFs, and they concluded that no single cubic EOS currently exists that is equally suitable for the quantitative prediction of all SCF mixtures. The statistical associating fluid theory (SAFT) proposed by Chapman et al.2,3 has received great attention since its publication, it has been applied to model the phase equilibria of many kinds of systems, and a recent review shows that more than 200 papers dealing directly with the SAFT approach and its applications have been published in the past 10 years.4 The applications illustrate that the approach is very successful in modeling the phase equilibria of complex systems; however, research dealing with its application to the modeling of the solubility of solids in SCFs is very scarce.4 Economou et al.5 used the SAFT EOS to correlate the solubility of * Corresponding author. E-mail:
[email protected]. Fax: +86-10-64436781. Telephone: +86-10-64419862.
polynuclear aromatics in supercritical ethylene and ethane, where a total of six systems were adopted and the correlated results show that good agreement with experimental data can be obtained with the SAFT EOS. Since a comprehensive investigation of the capability of the SAFT approach for modeling the solubility of solids in SCFs has not been carried out, we adopted an extended SAFT EOS combined with eight mixing rules to evaluate its capability for modeling this kind of process in this work. Thermodynamic Model The SAFT EOS. Since the publication of the first papers2,3 on SAFT, many modified versions and related approaches have appeared.4 Among the various investigations, Pfohl and Brunner pursued the particular improvement of SAFT for gas-extraction processes, and an extended SAFT EOS was proposed.6 They found the original SAFT7,8 cannot simultaneously reproduce critical temperatures, critical pressures, and saturation properties of selected pure nonpolar components with physically meaningful parameters, while the BACK EOS by Chen and Kreglewski9 performed excellently under these constraints. Therefore, they used the BACK EOS to modify the SAFT EOS, and they proposed an extended SAFT EOS. In their work, the supercritical solvent is modeled as a convex body, allowing an accurate description in the near critical region, while all other compounds are modeled as sphere chains, utilizing the benefits of the original SAFT. Details of the EOS are giving in the following paragraphs. The residual molar Helmholtz free energy in the extended SAFT EOS is a sum of the following contributions:
ares ) ahcb + achain + aassoc + adis
(1)
where the superscripts hcb, chain, assoc, and dis represent the contributions of hard convex bodies, chain, association, and dispersion, respectively.
10.1021/ie020374w CCC: $22.00 © 2002 American Chemical Society Published on Web 08/20/2002
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Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 Table 1. Pure Component Parametersa
Hard Convex Body Term
{
(ξ2)3 + 3ξ1ξ2ξ3 - 3ξ1ξ2(ξ3)2 ahcb 6 ) RT πNAVF ξ (1 - ξ )2
[
3
3
ξ0 -
ξk ) (πNAV/6)F ξk ) (πNAV/6)F
(ξ2)
]
3
(ξ3)2
∑i ximi(dii)k,
k ) 0 and 3
∑i ximi(Ri1/3dii)k,
[
(3)
(5)
πNAV 3 σ 6τ ii
(6)
where τ ) 0.740 48 and c ) 0.12. xi and mi are the mole fraction and the number of segments per molecule of component i, respectively. σii and dii are the temperature-independent segment diameter and the effective temperature-dependent segment diameter, respectively. F is the molar density, and Ri is the shape parameter for the convex body, which equals unity in SAFT. v°° ii and u° ii/k are two pure component parameters. Chain Term
achain
)
RT
∑i xi(1 - mi) ln[gii(dii)hcb]
(7)
and
gij(dij)hcb )
3diidjj ξ2 1 + +2 1 - ξ3 dii + djj (1 - ξ )2 3
(
)
diidjj 2 ξ22 (8) dii + djj (1 - ξ )3 3
gij(dij)hcb is the pair correlation function for a mixture of hard convex bodies. Association Term
aassoc RT
)
[(
∑i ∑ A xi
ln XAi -
) ]
XAi
i
v°°
K
mL/mol
carbon dioxide ethane ethene naphthalene anthracene phenanthrene pyrene biphenyl octacosane benzoic acid phenol
274.37 301.38 279.85 304.80 352.65 352.00 369.38 280.54 209.96 272.66 290.08
19.801 31.071 27.872 13.704 16.297 16.518 18.212 12.068 12.0 12.0 12.0
a
( )]
v°° ii )
(2)
k ) 1 and 2 (4)
-3u°ii kT
dii ) σii 1 - c exp
}
ln(1 - ξ3)
2
+
Mi 2
X ) (1 + NAVF
xjX ∑j ∑ B
Bj
AiBj -1
∆
)
(10)
adis
where σij ) (σii + σjj)/2, AiBj/k is the association energy of the interaction between site A on molecule i and site B on molecule j, and κAiBj is the volume of interaction between site A on molecule i and site B on molecule j. In this work, phenol and benzoic acid were considered as self-associating fluids, as treated by Huang and
K
K
102κAB
1.0560 1 52 1.0402 1 16 1.0384 1 13 1 4.671 10 1 5.344 10 1 5.327 10 1 5.615 10 1 6.136 10 1 19.287 10 1 4.608 10 5930 0.3149 1 4.103 10 1894 4.315
RT
[ ][ ] u
i
∑i ∑j Dij kT
)m
j
η τ
(12)
where Dij are universal constants, and for a pure component
(
η)
e kT
)
(13)
πNAV 3 Fd m 6
(14)
u ) u° 1 +
where e/k is a pure component parameter. Equation 12 requires two mixing rules for the mixture properties of m and u/kT, which will be given in the following section. The pure component parameters for the substances used in this work are shown in Table 1. Mixing Rules. It is commonly known that mixing rules play an important role in the correlation performance of cubic EOSs in phase equilibrium calculations. Therefore, a variety of mixing rules, including some existing mixing rules and several new ones constructed in this work, were used to evaluate the capability of the SAFT approach for modeling the solubility of solids in SCFs. Details of the mixing rules adopted in this work are given in the following paragraphs. One-Parameter Mixing Rules. I-1 Mixing Rule. Huang and Radosz adopted the following one-parameter mixing rules in their work:8
m)
u
)
∑i ∑j xixjmij
∑i ∑j
mij ) v°ij )
(15)
[]
xixjmimj
∑i ∑j
kT
and the association strength ∆AiBj
∆AiBj ) gij(dij)hcb[exp(AiBj/kT) - 1](σij)3κAiBj (11)
m
Radosz,8 while all the other compounds were considered as nonassociating fluids. Dispersion Term
(9)
j
R
The pure component parameters are taken from refs 6 and 7.
where XAi is the mole fraction of molecules i not bonded at site A, which is given by Ai
e/k AB/k
u°/k substance
uij
kT
v°ij (16)
xixjmimj v°ij
mi + mj 2
[21[(v°)
1/3
i
+ (v°j)1/3]
uij ) (uiiujj)1/2(1 - kij)
(17) 3
]
(18) (19)
where v°i is the temperature-dependent segment vol-
Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4901
components in size is large. Therefore, an adjustable parameter is introduced into eq 17:
Table 2. Solid Molar Volumes and Sublimation Pressures VS substance
dm3/mol
Psat (mmHg)
ref
naphthalene anthracene phenanthrene pyrene biphenyl octacosane benzoic acid phenol
0.111 0.143 0.153 0.1585 0.132 0.416 0.0965 0.0890
10[9.5810-2619.9/(T-52.5)]a 10(12.630-5313.7/T) 10(11.420-4567.7/T) 10(11.2697-4904.0/T) 10(12.679-4367.4/T) exp(74.45723 - 29552.37/T) 10(12.283-4618.1/T) 0.9068 (309.15 K)
12 12 12 13 14 13 14 15
a
mij )
mi + mj (1 - lij) 2
(20)
To construct a one-parameter mixing rule and compare it with the I-1 mixing rule, the adjustable parameter in eq 19 is deleted, which gives
uij ) (uiiujj)1/2
T is in kelvin.
Table 3. Temperature and Pressure Ranges for the Binary Systems temp range
press range
system
K
MPa
CO2-naphthalene CO2-phenanthrene CO2-anthracene CO2-pyrene CO2-biphenyl CO2-octacosane CO2-benzoic acid CO2-phenol C2H4-naphthalene C2H4-phenanthrene C2H4-anthracene C2H4-pyrene C2H4-benzoic acid C2H6-naphthalene C2H6-phenanthrene C2H6-anthracene C2H6-pyrene C2H6-biphenyl C2H6-octacosane C2H6-benzoic acid
308.15-328.15 318.15-338.15 303.15-343.15 308.15-343.15 308.95-322.65 307.85-323.35 318-338 309.15 298.15-323.15 318-338 323.15-358.15 318.15-348.15 318-338 293.15-318.15 313.15-333.15 308.15-343.15 333 308.15-318.15 308.2-318.2 308.15-343.25
7.95-33.4 12.0-28.0 9.06-41.45 8.36-48.34 10.6-47.52 11.91-28.36 12.00-28.00 7.93-24.94 5.61-17.34 12.00-28.00 10.44-48.36 8.37-48.36 12.00-28.00 3.54-27.67 6.99-41.45 10.43-48.35 10.02-31.45 7.05-28.1 6.90-20.72 5.48-36.36
no. of data points ref 40 15 23 22 24 29 15 25 19 15 26 15 15 28 11 19 15 10 10 30
16 17 18 18 19 20 17 21 22 17 22 18 17 18 18 18 23 24 25 24
ume of component i. This mixing rule is denoted as the I-1 mixing rule in this work. I-2 Mixing Rule. The systems considered in this work are highly asymmetric, and the difference of the
(21)
Therefore, eqs 15, 16, 18, 20, and 21 constitute the I-2 mixing rule proposed in this work. I-3 Mixing Rule. Huang and Radosz proposed a mixing rule for u/kT in their work as follows:8
u
)
kT
[] uij
∑i ∑j fifj kT
(22)
where fi is the volume fraction of component i, defined as
fi )
ximiv°i
∑j xjmj v°j
(23)
Equations 15, 17, 19, 22, and 23 constitute the second mixing rule proposed by Huang and Radosz,8 which is denoted as the I-3 mixing rule in this work. Two-Parameter Mixing Rules. II-1 Mixing Rule. Pfohl and Brunner adopted the following mixing rule for u/kT in their work:6
Table 4. Results of One-Parameter Mixing Rules AADya (%)
binary interaction parameters system CO2-naphthaleneb
CO2-phenanthrene CO2-anthracene CO2-pyrene CO2-biphenylc CO2-octacosane CO2-benzoic acid CO2-phenol C2H4-naphthalened C2H4-phenanthrene C2H4-anthracene C2H4-pyrene C2H4-benzoic acid C2H6-naphthalene C2H6-phenanthrene C2H6-anthracene C2H6-pyrene C2H6-biphenyl C2H6-octacosane C2H6-benzoic acid average error d
(1/n)∑|(yexp 2
exp ycal 2 )/y2 |
I-1 (kij)
I-2 (lij)
I-3 (kij)
I-1
I-2
I-3
0.1135 0.1012 0.1133 0.1282 0.1390 0.1731 0.0734 0.1031 0.0724 0.0391 0.0681 0.0570 0.0699 0.1144 0.0831 0.0916 0.0903 0.1500 0.1809 0.1257
0.1376 0.1566 0.1791 0.2203 0.1526 0.1998 0.0835 0.1144 0.0770 0.0528 0.0932 0.0839 0.0711 0.1168 0.1019 0.1149 0.1228 0.1420 0.1636 0.1176
0.1286 0.1040 0.1180 0.1296 0.1768 0.2911 0.1122 0.1363 0.0784 0.0186 0.0562 0.0404 0.1173 0.1633 0.0867 0.1011 0.0903 0.2443 0.3835 0.2469
30.07 23.42 20.49 15.69 18.18 36.12 30.16 23.07 26.18 19.56 20.21 18.03 35.20 19.64 23.67 12.02 14.62 18.65 58.71 39.97
15.09 18.41 18.56 13.84 8.46 27.70 26.77 18.77 20.62 18.66 18.94 16.03 33.38 16.41 20.51 10.63 12.32 8.73 50.59 35.97
25.29 23.25 20.31 15.72 9.63 29.98 29.34 20.41 21.33 19.44 20.09 17.79 33.92 18.24 22.33 12.00 14.45 11.22 33.72 37.23
25.18
20.52
21.78
a AADy ) × 100. b Calculations for four points are not possible. c Calculations for four points are not possible. Calculation for one point is not possible.
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Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002
Table 5. Results of Two-Parameter Mixing Rules binary interaction parameters II-1
II-2
II-3
AADy (%)
system
kij
λij
kij
lij
kij
lij
II-1
II-2
II-3
CO2-naphthalene CO2-phenanthrene CO2-anthracene CO2-pyrene CO2-biphenyl CO2-octacosane CO2-benzoic acid CO2-phenol C2H4-naphthalene C2H4-phenanthrene C2H4-anthracene C2H4-pyrene C2H4-benzoic acid C2H6-naphthalene C2H6-phenanthrene C2H6-anthracene C2H6-pyrene C2H6-biphenyl C2H6-octacosane C2H6-benzoic acid
0.1396 1.0668 2.7715 0.6314 0.1519 0.4590 0.6118 0.3463 0.1043 0.2460 3.0591 0.4469 1.1362 0.1343 0.4504 0.8085 1.5432 0.1562 0.2240 0.7275
0.0790 2.0234 5.3319 1.0142 0.0553 0.5938 1.1653 0.6426 0.1036 0.4515 6.0333 0.8038 2.2578 0.0624 0.8108 1.4437 2.9607 0.0391 0.1897 1.2620
-0.0087 -0.3055 -0.0293 0.0004 0.0289 -0.2258 -0.3949 -0.3525 -0.1524 -0.2492 -0.5120 -0.2784 -0.6981 -0.0960 -0.4189 -0.3852 -0.3766 -0.0053 -0.2271 -0.4868
0.1476 0.6272 0.2256 0.2196 0.1208 0.4604 0.5274 0.4846 0.2396 0.3855 0.7932 0.4924 0.7664 0.2132 0.6066 0.5977 0.6333 0.1470 0.3543 0.5691
-0.0246 -0.3465 -0.0381 -0.0035 0.0904 -0.2240 -0.5479 -0.5110 -0.3679 -0.4125 -0.7163 -0.4025 -1.2280 -0.3364 -0.6823 -0.5988 -0.5496 -0.2153 -0.5220 -0.9076
0.1648 0.6408 0.2254 0.2212 0.0818 0.4570 0.5799 0.5519 0.3486 0.4473 0.8059 0.5281 0.8642 0.3478 0.6802 0.6283 0.6629 0.2924 0.5201 0.6471
19.03 16.11 18.22 16.71 8.51 19.42 20.34 6.42 20.71 18.75 13.32 18.07 17.27 18.03 11.98 11.74 6.94 8.68 29.88 30.04
15.78 10.62 18.22 13.84 7.90 18.25 13.88 3.27 18.11 16.20 10.93 10.82 10.03 17.00 11.20 7.11 4.17 8.60 28.26 21.44
16.13 10.44 18.08 13.86 8.88 18.83 13.20 2.93 15.95 15.69 11.39 9.78 8.63 15.37 10.36 6.67 4.00 7.10 27.41 19.90
16.51
13.28
12.73
average error
rule, which is denoted as the II-2 mixing rule. Apparently, the II-2 mixing rule is similar to the twoparameter van der Waals mixing rule, where two binary interaction parameters are introduced into coenergy and cosegment number parameters, respectively. II-3 Mixing Rule. Furthermore, we combined eqs 15, 19, 20, 22, and 23 to construct the third two-parameter mixing rule adopted in this work. The difference between the II-2 and II-3 mixing rules is the expressions used for u/kT. Three-Parameter Mixing Rules. Two three-parameter mixing rules were further constructed to evaluate the capability of the SAFT approach for modeling the solubility of solids in SCFs in this work. III-1 Mixing Rule. Brenneke and Eckert10 showed that solute-solute interactions in the supercritical phase are not neglectable even at low concentrations, and they recommended using solute-solute interactions in calculating solid-SCF equilibria. Therefore, we proposed the following mixing rule for u/kT:
[]
u
uij
u kT
)
∑i ∑j xixjmimj kT v°ij(1 - kij)
kT +
∑i ∑j xixjmimj v°ij
[ ([ ] ) ]
∑i ximi ∑j xjmj (
uji
kT
1/3 3
v°ijλji
∑i ximi) ∑i ∑j xixjmimj v°ij
(24)
2
where λij ) -λji. Equations 15, 17, 18, 21, and 24 constitute the mixing rule adopted by Pfohl and Brunner,6 which is a two-parameter mixing rule and is denoted as the II-1 mixing rule in this work. II-2 Mixing Rule. To compare with the II-1 mixing rule adopted by Pfohl and Brunner,6 eqs 15, 16, and 1820 were combined to constitute a two-parameter mixing
[] uij
Figure 1. Experimental and calculated solubility of phenanthrene in supercritical CO2 at 318.15 K using the three mixing rules.
)
∑i ∑j xixjmimj kT v°ij(1 - kij) ∑i ∑j
(25)
xixjmimjv°ij
where k12 ) k21, k11 ) 0.0 (solvent), and k22 * 0.0 (solute). Equations 15, 18, 20, 21, and 25 constitute the first three-parameter mixing rule proposed in this work, which is denoted as the III-1 mixing rule. III-2 Mixing Rule. Panagiotopoulos and Reid11 proposed a mixing rule for systems where there is a large discrepancy in molecular size and interactions between the components, which is introduced to the SAFT EOS as follows:
[
uij ) xuiuj 1 - kij + (kij - kji)
ximi
∑j
xjmj
]
(26)
where kij * kji. Equations 15, 16, 18, 20, and 26
Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4903 Table 6. Results of Three-Parameter Mixing Rules binary interaction parameters III-1
III-2
AADy (%)
system
lij
k12
k22
lij
k12
k21
III-1
III-2
CO2-naphthalene CO2-phenanthrene CO2-anthracene CO2-pyrene CO2-biphenyl CO2-octacosane CO2-benzoic acid CO2-phenol C2H4-naphthalene C2H4-phenanthrene C2H4-anthracene C2H4-pyrene C2H4-benzoic acid C2H6-naphthalene C2H6-phenanthrene C2H6-anthracene C2H6-pyrene C2H6-biphenyl C2H6-octacosane C2H6-benzoic acid
0.5211 0.9568 0.1637 0.2705 0.6009 0.3426 0.8198 0.7828 0.8310 1.3407 0.6770 0.8375 1.2064 0.6027 1.0354 0.7343 1.0881 0.3604 0.3175 0.8254
-0.2768 -0.4941 0.0064 -0.0270 -0.3579 -0.1302 -0.6279 -0.5729 -0.6354 -0.9075 -0.4326 -0.4996 -1.1036 -0.4527 -0.7350 -0.4909 -0.6926 -0.2163 -0.1898 -0.7478
-0.9085 -2.5834 4.8127 -1.1518 -1.0771 1.3626 -2.4476 -1.7883 -2.3517 -5.5586 3.1896 -2.5386 -5.7476 -1.1693 -3.2120 -2.7978 -5.7960 -0.3444 0.1067 -3.0462
0.5740 0.9669 0.1633 0.2710 0.4349 0.3424 0.8302 0.7511 0.7962 1.4145 0.6766 0.8454 1.2222 0.6752 1.0610 0.7345 1.0926 0.2845 0.2812 0.8449
-0.9836 -2.1860 2.9766 -0.8050 -0.5794 0.4383 -1.8345 -1.3544 -1.7429 -4.5038 1.2851 -2.0499 -3.5147 -1.3133 -2.5097 -1.8465 -3.8054 -0.2376 -0.0751 -1.9641
-0.3087 -0.4995 0.0066 -0.0272 -0.2146 -0.1301 -0.6355 -0.5471 -0.5975 -0.9535 -0.4324 -0.5045 -1.1176 -0.5024 -0.7524 -0.4910 -0.6956 -0.1376 -0.1539 -0.7674
7.66 5.78 16.15 12.22 4.57 10.62 7.95 1.76 7.35 5.94 10.53 5.09 3.65 12.01 8.19 5.51 0.84 4.42 28.51 16.12
6.74 5.74 16.14 12.22 6.45 10.57 8.48 2.36 9.58 6.95 10.53 5.02 3.53 10.26 8.15 5.51 0.84 8.31 28.65 15.83
8.74
9.09
average error
constitute the second three-parameter mixing rule proposed in this work, denoted as the III-2 mixing rule. Solubility of Solids in SCFs. The solubility of a solid (2) in a SCF (1) can be calculated using the following equation:
y2 )
Psat 2 P
φsat 2
[
]
VS2 (P - Psat 2 ) exp RT SCF φ2
(27)
where y2 is the mole fraction of the solid in the SCF is the sublimation pressure of the pure phase, P sat 2 solid at the system temperature, T, φsat 2 is the fugacity coefficient of the pure solid at temperature T, which is set to unity, V S2 is the molar volume of the pure solid, is the fugacity coefficient of the solid in the and φSCF 2 SCF at the system temperature and pressure. The physical properties for the pure solids used in this work are shown in Table 2. Results and Discussion A total of 20 systems were collected from the literature, as shown in Table 3, and were used to evaluate the SAFT EOS. One-Parameter Mixing Rules. The calculated results are shown in Table 4, where the binary interaction parameters are also listed. Generally, the I-2 and I-3 mixing rules give similar correlative accuracy, with the average absolute deviations (AADs) being 20.52% and 21.78%, respectively. The I-1 mixing rule shows the worst correlative accuracy, with the AAD of 25.18%. It seems that the correction to the size difference is more effective for solid-SCF equilibria, where the constituents show large discrepancy in molecular size. Comparing the I-1 and I-3 mixing rules, it is evident that the main difference is that volume fractions are used in the latter for u/kT. The evidence that the I-3 mixing rule shows better correlative accuracy than the I-1 mixing rule again illustrates that the size difference between components plays an important role in the phase equilibria calculation for solid-SCF systems.
Ashour et al.1 used cubic EOSs to correlate the solubility of solids in SCFs; the results of this work show that the SAFT EOS gives comparable correlated accuracy to that of cubic EOSs with one-parameter mixing rules. Two-Parameter Mixing Rules. The correlated results and the binary interaction parameters are shown in Table 5. Obviously, the two-parameter mixing rules give improved correlative accuracy than that of the oneparameter mixing rules. In the II-1 mixing rule two binary interaction parameters are introduced into u/kT, while in the II-2 and II-3 mixing rules two binary interaction parameters are introduced into uij and mij, respectively. From Table 5 it is evident that the II-2 and II-3 mixing rules show much better correlative accuracy than the II-1 mixing rule, which illustrates again that, for solid-SCF systems, the feature of high asymmetry in molecular size is important and that the introduction of two adjustable parameters into the attractive and size terms, respectively, is effective. Compared with the cases of the II-2 and II-3 mixing rules, for which the binary interaction parameters are larger than unity for some systems, for the II-1 mixing rule they are physically meaningless for thesee systems. Therefore, the II-2 and II-3 mixing rules are more suitable for modeling the solid-SCF equilibria and can give reasonable correlative accuracy. A comparison with the work of Ashour et al.1 shows that the SAFT EOS gives similar correlative accuracy to that of the cubic EOSs when twoparameter mixing rules are used. Figure 1 shows the experimental and the calculated solubility of phenanthrene in supercritical CO2 at 318.15 K using the three mixing rules. As can be seen, the II-2 and II-3 mixing rules give comparable results, which are better than those for the II-1 mixing rule. Three-Parameter Mixing Rules. The correlated results and the binary interaction parameters are shown in Table 6. Obviously, both the mixing rules show very good correlative accuracy, with the AAD smaller than 10%. However, for some systems the adjustable parameters have large values (normally negative values), which are physically meaningless. Particularly, for the III-1 mixing rule the parameter k22 accounting for the
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Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002
Nomenclature
Figure 2. Experimental and calculated solubility of phenol in supercritical CO2 at 309.15 K using the I-2, II-2, and III-1 mixing rules.
solute-solute interaction is a large negative value for most systems, which may mean, to some extent, the solute-solute interactions in the supercritical phase are evident. The work of Ashour et al.1 shows that three-parameter mixing rules do not improve the correlative accuracy of cubic EOSs much compared to the case of the two-parameter mixing rules. However, this work shows that for the SAFT EOS the improvement over the twoparameter mixing rules is large, and the correlative accuracy of the two three-parameter mixing rules is satisfactory. Figure 2 shows the experimental and the calculated solubility of phenol in supercritical CO2 at 309.15 K using three mixing rules, say the I-2, II-2, and III-1 mixing rules. Obviously, the improvement of the correlative accuracy by increasing the number of the adjustable parameters is large, and the III-1 mixing rule shows very good correlative accuracy. Conclusion An extended SAFT EOS was adopted to evaluate the capability of the SAFT approach for modeling the solubility of solids in SCFs, where eight mixing rules are used. The correlated results show that the SAFT EOS shows good correlative accuracy, which is comparable to that for cubic EOSs when the one- and twoparameter mixing rules are used. When three-parameter mixing rules are adopted, the SAFT EOS shows very good correlative accuracy, which is better than that for the cubic EOSs with the three-parameter mixing rules. This illustrates, to some extent, that the SAFT EOS, with a much better theoretical basis, has advantages over the semiempirical cubic EOSs for the modeling of the solubility of solids in SCFs. Currently, the representation of the solubility of solids in SCFs with cosolvents using the SAFT approach is under investigation, and the results will be reported in the coming paper. Acknowledgment The financial support of the Natural Science Foundation of China (Contract 20106001) and the State Key Fundamental Research Plan (No. G2000048) is greatly appreciated.
a ) molar Helmholtz energy, J/mol c ) constant in eq 5 d ) temperature-dependent segment diameter, m3 Dij ) 24 universal constants e/k ) pure component parameter f ) volume fraction, defined by eq 23 gij(dij)hcb ) segment distribution function, defined by eq 8 k ) Boltzman’s constant ≈ 1.380 66 × 10 - 23, J/K kij ) binary adjustable parameter lij ) binary adjustable parameter m ) effective number of segments within the molecule (segment number) Mi ) number of association sites on molecule i n ) number of data points NAV ) Avogadro’s number ≈ 6.022 14 × 1023 P ) pressure, Pa R ) universal gas constant, J/mol‚K T ) temperature, K u/k ) temperature-dependent dispersion energy of interaction between segments, K u°/k ) temperature-independent dispersion energy of interaction between segments, K VS ) solid molar volume, m3/mol v°° ) temperature-independent segment volume, m3/mol v° ) temperature-dependent segment volume, m3/mol x ) mole fraction XAi ) monomer mole fraction of component i not bonded at site A y ) solubility, mole fraction Greek Letters R ) shape parameter for convex body AiBj/k ) association energy of interaction between site A on molecule i and site B on molecule j, K φ ) fugacity coefficient η ) reduced density κAiBj ) volume of interaction between site A on molecule i and site B on molecule j λij ) binary adjustable parameter F ) molar density, mol/m3 σ ) temperature-independent segment diameter, m ∆AiBj ) “strength of interaction” between site A on molecule i and site B on molecule j, m3 τ ) constant, 0.740 48 ξk ) function of the molar density, defined by eqs 3 and 4 Subscripts 2 ) solute i ) properties of component i ij ) cross properties for components i and j Superscripts A, B ) association sites assoc ) associating or due to association cal ) calculated value dis ) dispersion exp ) experimental value hcb ) hard convex body res ) residual properties S ) solid sat ) saturated value SCF ) supercritical fluid
Literature Cited (1) Ashour, I.; Almehaideb, R.; Fateen, S.-E.; Aly, G. Representation of Solid-Supercritical Fluid Phase Equilibria Using Cubic Equations of State. Fluid Phase Equilib. 2000, 167, 41.
Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4905 (2) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. SAFT: Equation-of-State Solution Model for Associating Fluids. Fluid Phase Equilib. 1989, 52, 31. (3) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New Reference Equation of State for Associating Liquids. Ind. Eng. Chem. Res. 1990, 29, 1709. (4) Muller, E. A.; Gubbins, K. E. Molecular-Based Equations of State for Associating Fluids: A Review of SAFT and Related Approaches. Ind. Eng. Chem. Res. 2001, 40, 2193. (5) Economou, I. G.; Gregg, C. J.; Radosz, M. Solubilities of Solid Polynuclear Aromatics (PNA’s) in Supercritical Ethylene and Ethane from Statistical Associating Fluid Theory (SAFT): Toward Separating PNA’s by Size and Structure. Ind. Eng. Chem. Res. 1992, 31, 2620. (6) Pfohl, O.; Brunner, G. 2. Use of BACK to Modify SAFT in Order to Enable Density and Phase Equilibrium Calculations Connected to Gas-Extraction Processes. Ind. Eng. Chem. Res. 1998, 37, 2966. (7) Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse, and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284. (8) Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse, and Associating Molecules: Extension to Fluid Mixtures. Ind. Eng. Chem. Res. 1991, 30, 1994. (9) Chen, S. S.; Kreglewski, A. Applications of the Augmented van der Waals Theory of Fluids. I. Pure Fluids. Ber. Bunsen-Ges. 1977, 81, 1048. (10) Brennecke, J. F.; Eckert, C. A. ACS Symp. Ser. 1989, 406, 14. (11) Panagiotopoulos, A. Z.; Reid, R. C. New Mixing Rule for Cubic Equations of State for Highly Polar, Asymmetric Systems. In Equations of State Theories and Applications; Chao, K. C., Robinson, R. L., Jr., Eds.; ACS Symposium Series 300; American Chemical Society: Washington, DC, 1986; pp 571-582. (12) Zhong, C.; Masuoka, H. An EOS/GE Type Mixing Rule for Perturbed Hard-Sphere Equation of State and Its Application to the Calculation of Solid Solubility in Supercritical Carbon Dioxide. Fluid Phase Equilib. 1997, 141, 13. (13) Barna, L. R.; Rauzy, E.; Berro, C.; Blanchard, J. M. An “Excess Function-Equation of State” Model for Solubility of Hydrocarbon Solids in Supercritical Carbon Dioxide. Fluid Phase Equilib. 1994, 100, 191. (14) Schmitt, W. J.; Reid, R. C. Solubility of Monofunctional Organic Solids in Chemically Diverse Supercritical Fluids. J. Chem. Eng. Data 1986, 31, 204.
(15) Iwai, Y.; Uchida, H.; Koga, Y.; Arai, Y. Monte Carlo Simulation of Solubilities of Aromatic Compounds in Supercritical Carbon Dioxide by A Group Contribution Site Model. Ind. Eng. Chem. Res. 1996, 35, 3782. (16) Tsekhanskaya, Y. V.; Iomtev, M. B.; Mushkina, E. V. Solubility of Naphthalene in Ethylene and Carbon Dioxide under Pressure. Russ. J. Phys. Chem. 1964, 38, 1173. (17) Kurnik, R. T.; Holla, S. J.; Reid, R. C. Solubility of Solids in Supercritical Carbon Dioxide and Ethylene. J. Chem. Eng. Data 1981, 26, 47. (18) Johnston, K. P.; Ziger, D. H.; Eckert, C. A. Solubilities of Hydrocarbon Solids in Supercritical Fluids. The Augmented van der Waals Treatment. Ind. Eng. Chem. Fundam. 1982, 21, 191. (19) McHugh, M. A.; Paulaitis, M. E. Solid Solubilities of Naphthalene and Biphenyl in Supercritical CO2. J. Chem. Eng. Data 1980, 25, 326. (20) Mchugh, M. A.; Seckner, A. J.; Yogan, T. J. High-Pressure Phase Behavior of Binary Mixtures of Octacosane and Carbon Dioxide. Ind. Eng. Chem. Fundam. 1984, 23, 493. (21) Van Leer, R. A.; Paulaitis, M. E. Solubilities of Phenol and Chlorinated Phenols in Supercritical Carbon Dioxide. J. Chem. Eng. Data 1980, 25, 257. (22) Johnston, K. P.; Eckert, C. A. An Analytical CarnahanStarling-van der Waals Model for Solubility of Hydrocarbon Solids in Supercritical Fluids. AIChE J. 1981, 27, 773. (23) Anitescu, G.; Tavlarides, L. L. Solubilities of Solids in Supercritical Fluids?I. New Quasistatic Experimental Method for Polycyclic Aromatic Hydrocarbons (PAHs) + Pure Fluids. J. Supercrit. Fluids 1997, 10, 175. (24) Schmitt, W. J.; Reid, R. C. Solubility of Monofunctional Organic Solids in Chemically Diverse Supercritical Fluids. J. Chem. Eng. Data 1986, 31, 204. (25) Kalaga, A.; Trebble, M. Density Changes in Supercritical Solvent + Hydrocarbon Solute Binary Mixtures. J. Chem. Eng. Data 1999, 44, 1063.
Received for review May 20, 2002 Revised manuscript received July 24, 2002 Accepted July 26, 2002 IE020374W
