Ind. Eng. Chem. Res. 1994,33, 1955-1961
1955
GENERAL RESEARCH High-pressure Vapor-Liquid Equilibria for Mixtures Containing a Supercritical Fluid Wen-Lu Weng Department of Chemical Engineering, Ming-Hsin Engineering College, Hsinchu, 304, Taiwan
Jui-Tang Chen and Ming-Jer Lee' Department of Chemical Engineering, National Taiwan Institute of Technology, Taipei, 106, Taiwan
High pressure vapor-liquid phase equilibrium compositions were measured for the binary systems of carbon dioxide 2-methyl-1-pentanol, carbon dioxide + 1-octanol, and carbon dioxide + 1-decanol over a temperature range between 348.15 and 453.15 K. In addition to the new data, a variety of supercritical fluid systems was used to test the validity of the Peng-Robinson and Patel-Teja equations of state accompanied by several types of mixing rules. In general, the Peng-Robinson equation incorporated with the cubic mixing rule yielded the best representation.
+
Table 1. Phase Equilibrium Composition of COz (1) 2-Methyl-1-Pentanol(2)
Introduction Supercritical fluid extraction has been proven as an efficient separation method for some specific industrial applications. The knowledge of the phase behavior of supercitical systems plays an important role in the process design. Unfortunately, most conventional models are inadequate to correlate those phase equilibrium data due to highly asymmetric interactions between solvent and solute molecules (Brennecke and Eckert, 1989). The accuracy of calculations become worse if the systems contain a polar compound, especially near the critical region. It should be of worthwhile to measure new vaporliquid equilibrium (VLE) data and improve the phase equilibrium calculation for supercritical fluid systems. The first objective of this work is the observation of the phase equilibrium behavior for carbon dioxide + alcohol systems at elevated pressures. In addition to the VLE data of carbon dioxide + benzyl alcohol (313-393 K and 60-160 bar) reported recently by Walther and Maurer (1993),the related literature has been summarized by Lee and Chen (1994). It was found that the mixtures composed of carbon dioxide and a heavy alcoholwere not sufficiently investigated. In the previous works, Wilcock et al. (978) measured the carbon dioxide solubilities in 1-octanoland in 1-decanol at 283-313 K under atmospheric pressure. Weng and Lee (1992b) conducted VLE measurements for carbon dioxide + 1-octanol at temperatures from 313 to 348 K and pressures up to 190 bar. Recently, Lee and Chen (1994) reported the VLE data for carbon dioxide + 2-methyl-1-pentanol, carbon dioxide + 1-octanol, and carbon dioxide 1-decanolat higher temperatures (348453 K) and at subcritical pressures of carbon dioxide. The phase equilibrium compositions of these three binary systems were measured experimentally from 60 bar up to 190 bar in the present study. Examining the validity of simple equations of state and mixing rules for supercritical fluid systems is the second objective of this work. Since supercritical extraction is generally operated at high pressures, an equation of state,
+
~~
~
* To whom correspondence should be addressed.
TW)
POW
YZ
348.15
65 75 90 105 120 65 80 100 120 140 155 65 80 100 120 140 160 180
0.0033 0.0037 0.0061 0.0086 0.0166 0.0195 0.0197 0.0243 0.0302 0.0384 0.0482 0.0802 0.0773 0.0788 0.0840 0.0911 0.111 0.167
403.15
453.15
x1 0.341 0.400 0.520 0.629 0.737 0.246 0.302 0.376 0.446 0.520 0.581 0.212 0.272 0.339 0.404 0.473 0.561 0.712
Ki 2.92 2.49 1.91 1.58 1.34 3.99 3.25 2.60 2.17 1.85 1.64 4.34 3.40 2.72 2.27 1.92 1.59 1.17
+ Kz 0.0050 0.0062 0.0127 0.0232 0.0630 0.0259 0.0282 0.0389 0.0545 0.0801 0.115 0.102 0.106 0.119 0.141 0.173 0.252 0.578
rather than an activity coefficient model, is preferable for correlating the VLE data. However, a simple cubic equation of state with a one-fluid, one-parameter van der Waals mixing rule is rarely accurate enough for most supercritical systems at elevated pressures. Using some other sophisticated mixing rules should be necessary to improve the phase equilibriumcalculation. A cubic mixing rule (Le., composition-dependent combining rule) was developed by Panagiotopoulos and Reid (1986a) for aqueous supercritical fluid systems. This mixing rule, however, does not meet the quadratic form of the second virial coefficient of mixtures at the low-density limit. Panagiotopoulos and Reid (1986b) further introduced a density-dependent,asymmetricmixing rule into the PengRobinson (PR) equation of state (Peng and Robinson, 1976) for quantitatively reproducing the phase diagram of the carbon dioxide + n-butanol + water system. The discrepancy of the low-density limit is removed from this mixing rule because the cubic compositionterm disappears as density approches zero. Another density-dependent mixing rule with two temperature-specificbinary interaction parameters was used by Mohamed and Holder (1988). Some degree of improvement was obtained when the
0888-588519412633-1955$04.50/0 0 1994 American Chemical Society
1956 Ind. Eng. Chem. Res., Vol. 33, No. 8, 1994 Table 2. Phase Equilibrium Composition of COz (1) t 1-Octanol (2) ~~
T(K) 403.15
453.15
P(bar) 65 80 95 115 130 145 160 170 185 65 80 100 120 145 170 190
Y2
Zl
Ki
K2
0.0055 0.0055 0.0065 0.0083 0.0107 0.0132 0.0170 0.0193 0.0269 0.0216 0.0204 0.0210 0.0244 0.0310 0.0426 0.0527
0.255 0.306 0.357 0.422 0.471 0.514 0.557 0.590 0.634 0.231 0.281 0.346 0.397 0.467 0.532 0.595
3.91 3.25 2.79 2.35 2.10 1.92 1.77 1.66 1.54 4.23 3.49 2.83 2.46 2.08 1.80 1.59
0.0074 0.0079 0.0101 0.0144 0.0202 0.0272 0.0384 0.0470 0.0735 0.0281 0.0284 0.0321 0.0405 0.0581 0.0910 0.130
0
Table 3. Phase Equilibrium Composition of C02 (1) + 1-Decanol (2)
T(K)
P(bar)
Ya
Z1
Ki
K2
348.15
70 85 100 115 130 145 170 190 60 75 90 105 125 140 165 190 65 80 95 110 130 150 170 190
0.00025 o.Ooo40 0.00089 0.0017 0.0033 0.0062 0.0154 0.0298 0.0018 0.0023 0.0026 0.0032 0.0042 0.0052 0.0080 0.0135 0.0087 0.0090 0.0096 0.0108 0.0124 0.0150 0.0176 0.0231
0.367 0.437 0.501 0.572 0.640 0.690 0.736 0.795 0.255 0.320 0.393 0.456 0.543 0.594 0.655 0.745 0.239 0.285 0.338 0.385 0.452 0.530 0.594 0.661
2.73 2.29 2.00 1.74 1.56 1.44 1.34 1.22 3.91 3.12 2.54 2.19 1.84 1.68 1.51 1.33 4.15 3.47 2.93 2.57 2.18 1.86 1.65 1.48
0.00039 0.00071 0.0018 0.0040 0.0092 0.0168 0.0582 0.145 0.0024 0.0034 0.0043 0.0059 0.0092 0.0128 0.0232 0.0529 0.0114 0.0126 0.0145 0.0176 0.0226 0.0319 0.0434 0.0680
403.15
453.15
mixing rule was incorporated into the PR equation. On the basis of statistical mechanics, Park et al. (1987) developed a conformal solution van der Waals mixing rule. With the aid of this new mixing rule, the PR equation calculated the solubilities of heavy liquid in supercritical gases more accurately than those from the van der Waals and the Redlich-Kwong equations of state. Numerous types of mixing rules were tested by using the isothermal phase equilibrium and density data of nitrogen + n-butane (Shibata and Sandler, 1989). Those experimental data included several points around the critical region. The comparison showed that the one-fluid, two-parameter van der Waals mixing rule with fluid-specific equation constants (denoted as VDW2FS) was the best one for the binary system. A few predictive models have also been developed. Trebble and Sigmund (1990) employed the PR equation together with a generalized k,, correlation to predict the VLE behavior of nonpolar supercritical systems within reasonable accuracy. Sheng et al. (1992) proposed a set of modified UNIFAC group parameters that enabled the Patel-Teja (PT)equation of state (Pate1 and Teja, 1982) to calculate the solubilities of aromatic compounds in supercritical carbon dioxide satisfactorily. Recently, the k9 of the Soave equation (Soave, 1972) was generalized with alcohol's critical volume or normal boiling point by
1 A A A A A 453.15K 0 0 ~ 0 0 403.15K
00000 A
0.1
348.1 5K Eq.(l)
;oL-A A/
38
Figure 2. Correlation between saturated vapor composition of 2-methyl-1-pentanol and density of C02.
Lee and Chen (1994) for carbon dioxide + alcohol systems. The generalizationsled to the increase of the grand average absolute deviation (AAD) of equilibrium vaporizationratio (Kvalue)calculationsby about only 1% . A neural network was also developed by Lee and Chen (1994) for correlating the K values of carbon dioxide + alcohol systems. The trained network predicted the VLE isotherms of carbon dioxide + 1-propanol reasonably well. As described above, most previous studies focused on the mixing rules for carbon dioxide + nonpolar solute systems. The applicability of those mixing rules for polar supercritical systems is still questionable. In the present study, the performances of the PR and PT equations of state incorporated with one of four representative mixing rules were tested by a diversity of supercritical systems, respectively. Generally, the PR equation with the cubic mixing rule of Panagiotopoulos and Reid (1986a) yields the best results. Experimental Work A semiflow type apparatus was employed to the phase equilibrium composition measurements. Detailed illustration of the apparatus and operation is given elsewhere (Lee and Chao, 1988;Weng and Lee, 1992a,Lee and Chen, 1994). Carbon dioxide (99.8%) was purchased from SanFu Chemical Compnay (Taiwan), 2-methyl-1-pentanol (99%) and 1-octanol(99.55% ) were purchased from Aldrich Chemicals (USA), and 1-decanol (99% ) was purchased
Ind. Eng. Chem. Res., Vol. 33, No. 8,1994 1957 Table 4. Results of Empirical Correlation for Saturated Vapor Composition mixture" T(K) nb A B C(106) A A D C ( 1 0 - 2 ) I 348.15 10 -5.1170 -398.5 0.7698 0.084 403.15 453.15 403.15 453.15 348.15 403.15 453.15
I1
I11
11 12 14 12 13 13 13
-680.7 -787.2 -319.4 -777.7 235.1 -193.3 -625.8
-2.8481 -1.4786 -4.6618 -2.7142 -8.7697 -5.8659 -3.7844
1.0924 1.3075 0.6172 1.2753 0.1138 0.5109 1.0862
0.494 1.557 0.145 0.532 0.047 0.056 0.220
from Janssen Chimica (Belgium). The chemicals were used without further purification. The accuracy of measurements is better than f0.1% for pressure and f0.02 K for temperature. In general, the phase composition was reproduced within f 2 % . The equilibrium phase compositions as well as the equilibrium vaporization ratios (Ki = yi/xi) for carbon dioxide 2-methyl-1-pentanol,carbon dioxide+ 1-octanol, and carbon dioxide + 1-decanol are listed in Tables 1, 2, and 3, respectively. Figure 1 illustrates the variation of K valueswith pressure for 2-methyl-1-pentanol. Minimum K2 was observed at pressures around 40-50 bar for each isotherm, and a crossover of KZisotherms may occur at pressures higher than 120 bar. Similar phase behavior was also found in the other two systems. Each saturated vapor composition isotherm (yz) was correlated empirically with the density of carbon dioxide ( p ) by the following equation:
+
+ Bp + Cp2
C16 C17 C18 c19 c20
n T(K) P(bar) datasource Solvent: Carbon Dioxide 1-pentene 37 303-329 14-84 Wu et al. (1988) n-octane 20 313-348 5-113.5 Wengandhe (1992a) 13 313-383 27-140 King et al. (1983) benzene 28 313-393 5-63 Kimetal. (1986) toluene 34 311-477 3.3-153 Ng and Robinson (1978) 22 353-393 5.2-65 Kim et al. (1986) m-xylene 21 313-366 13-154 Mohamedand Holder (1987) p-xylene 20 313-366 12-142 Mohamedand Holder (1987) 20 353-393 4.6-62 Kim et al. (1986) o-xylene 64 312-370 24-142 Mohamedand Holder (1987) 22 312-366 14-149 Mohamedand Holder (1987) .~ ethylbenzene 19 313-366 24-143 Mohamedand Holder (1987) l-methvl5 373 37-207 Kim et al. (1989) naphthalene 32 308-328 20-240 Lee and Chao (1988) tetralin 11 344-373 32-221 Kim et al. (1989) 1 2 298 2.2-61 Katayamaetal. methanol (1975) 2-propanol 21 317-395 14-124 Radosz (1986) 33 348-453 10-180 Lee and Chen (1994) 2-methyl-lpentanol and this work 31 348-453 10-190 Lee and Chen (1994) 1-octanol and this work 20 313-348 40-190 Weng and Lee (1992b) 1-decanol 39 348-453 10-190 Lee and Chen (1994) and this work m-cresol 27 308-328 20-165 Lee and Chao (1988) acetone 24 298-313 4-74 Katayama et al. (1975) acetophenone 20 313-348 20-165 Weng and Lee (1992d) benzaldehyde 10 343-373 28-183 Kim et al. (1989) methyl acetate 22 298-313 6.5-80 o 0 akiand ~ ~(1975) ~ a
c21
diethyl ether
c22
dibutyl ether
C23
methyl butyl ether anisole methyl benzoate
C3 C4
+
ln(y,) = A
mixture ID C1 C2
0 I, C02 + 2-methyl-1-pentanol; 11, CO2 1-octanol; 111, C02 + 1-decanol. Includes low-pressure data reported by Lee and Chen (1994). e AAD (IOO/n)C;.,([y,* - y2'xPl)k.
*
Table 5. Mixture Identifications, Components, Experimental VLE Data Ranges, and Sources
(1)
where p was calculated from the equation proposed by Huang et al. (1985) a t the same T and P as the mixtures. The determined coefficients are given in Table 4 for each isotherm. Figure 2 shows that the empirical equation interpolates the saturated vapor composition to within the experimental accuracy. Those empirical correlations could be useful for application purposes.
C5 C6 C7
~
C8 C9 c10 c11 c12 C13
C14 C15
C24 C25 El
Equations of State Two cubic equations of state (PR and PT) with four different variations of mixing rules were investigated here. The mixture constants 8, including a, b,, and c, were calculated from
e, = 7,+ppij &
I
The combining rule of cij (for the PT equation only) was given by cij = (Ci
+ Cj)/2
(3)
In mixing rule A (one-fluid, one-parameter van der Waals mixing rule), the combining rules of aijand bijwere defined respectively as ~ i =j (1
- kaj)(ataj)0.6
(4)
and
where k,, is a binary interaction constant. If an additional
solute
E2 E3 E4 E5 E6 E7 E8 E9 N1 N2 N3 N4 N5 N6
I
18 298-313 7.0-72
atayama (1975) 22 313-366 12-142 Mohamed and Holder (1987) 34 310-329 5.4-91 Wu et al. (1988)
10 343-372 25-168 Kim et al. (1989) 19 313-348 30-145 WengandLee (1992~) Solvent: Ethane 18 313-348 25-68 WengandLee n-octane (1992a) isooctane 8 346 9.0-67 King et al. (1983) l-methyl5 373 32-122 Kim et al. (1989) naphthalene 29 308-328 15-145 Lee and Chao (1988) 18 313-348 20-110 Weng and Lee (1992b) 1-odanol 30 308-328 15-240 Lee and Chao (1988) m-cresol acetophenone 20 313-348 25-185 Weng and Lee (1992d) benzaldehyde 5 373 24-94 Kim et al. (1989) anisole 6 373 24-97 Kim et al. (1989) methyl 21 313-348 20-130 Weng and Lee (1992~) benzoate Solvent: Ethylene 20 313-338 15-95 Weng and Lee (1992a) n-octane 3 348 32-91 King et al. (1983) benzene toluene 10 347 7.6-88 King et al. (1983) 1-octanol 19 313-338 30-180 Weng and Lee (1992b) acetophenone 21 313-338 30-190 Weng and Lee (1992d) methyl benzoate! 19 313-338 25-150 Weng and Lee (1992~)
cross parameter kbij was introduced into the combining rule of bij, bij = (1 - kbJ(bi
+ bj)/2
(6)
the mixing rule was designated as mixing rule B (onefluid, two-parameter van der Waals mixing rule). Mixing rule C included eq 5 for bij and the linear density-dependent combining rule for aij (Mohamed and Holder, 1987):
n
~
1958 Ind. Eng. Chem. Res., Vol. 33, No. 8, 1994 Table 6. Fluid-Specific Parameters of the Peng-Robinson and Patel-Teja Equations of State for the Pure Components Peng-Robinson Patel-Teja compound K 50 F 0.965347' 1.1708 0.274" methanol 0.338 1.4830 2-propanol 1.3108 0.338 1.2954 2-methyl-1-pentanol 1.1458 1-octanol 1.2207 0.308' 1.270267' 0.336 1.4805 1-decanol 1.3609 0.8231 0.283' 0.701112' acetone 0.309 0.9792 acetophenone 0.9757 0.8711 0.8638 0.312 benzaldehyde 0.330 0.9528 methyl acetate 0.8494 0.308" 0.787322' diethyl ether 0.8020 0.334 1.2062 dibutyl ether 1.0682 butyl methyl ether 0.8421 0.326 0.9226 1.0287 0.8857 0.338 anisole methyl benzoate 1.0182 0.338 1.1536 carbon dioxide b 0.3090 0.707727' b 0.317' 0.561567' ethane b 0.313' 0.554364 ethylene TakenfromGeorgetonet al. (1986). Estimated fromgeneralized equation.
(7)
In mixing rule D, bij was also evaluated from eq 5 and aij was computed from the composition-dependent combining rule (Panagiotopoulos and Reid, 1986a):
a modified Levenberg-Marquardt algorithm:
The experimental data being correlated in this work are summarized in Table 5. Among the 40 binary systems, there are 25 systems containing carbon dioxide(Cl-C25), 9 systems containing ethane(E1-E9) and 6 systems containing ethylene (Nl-N6). The heavy components (solutes) include nonpolar substances (e.g., alkanes and aromatics) and highly polar substances (e.g., alcohols, ketones, ethers, and esters). In the following VLE calculations,the pure-component parameters ( K in the PR equation and tC,F in the PT equation) were estimated from the original generalized expressions for the nonpolar solutes, wheras those were determined from vapor pressure data for the polar solutes. The fluid-specific parameters employed to the present calculation are listed in Table 6. Among those tabulated values, some of the tCand Fvalues were taken from the literature (Georgeton et al., 1986). Also, the critical properties and acentric factor listed in the databank of Reid et al. (1987) were used in the present study for each pure component. Tables 7 and 8 present the VLE calculated results from the PR and PT equations for supercritical carbon dioxide systems, respectively. The entries of APIP and Ay2 are defined as
(8) As a consequence, mixing rule A has a single cross parameter (kaJ for each specific binary system and the rest of mixing rules contain two binary interaction constants. The optimal cross parameters (e.g., kaU,kbU, kl.. kzU,and k ,) were determined from bubble-pressure cahlations by minimizing the objective function a with
AYZ =
loo " ---(lYZdc
- YFPl)k
The PR equation with the cubic mixing rule (mixing rule D) and the PT equation with the one-fluid, two-
Table 7. Bubble-Point Calculations from the Peng-Robinson Equation of State for COrContaining Systems mixing rule A' mixing rule Bb mixing rule Cc mixing rule Dd mixture APIP Ay2 APIP 4 y 2 M l P Ay2 APJP ID k, (%) (le2) k, kbg (%) (le2) ki, kz, (%) (le2) k, k, (%)
c1
c2 c3 c4 c5 C6 c7 C8 c9 c10
c11 c12 C13 C14 C15 C16 C17 C18 c19 c20 c21 c22 C23 C24 C25 grand AAD
0.0675 0.0536 0.1021 0.1166 0.0848 0.0858 0.0652 0.0841 0.1263 0.0954 0.0399 0.0910 0.1098 0.0976 0.0883 0.1045 0.0045 0.0404 0.0330 -0.0408 0.0461 0.0572 -0.0198 0.0524 0.0411
1.96 7.76 3.91 7.21 6.13 6.16 16.35 5.63 10.58 7.17 4.61 5.57 11.39 10.33 12.41 10.01 1.37 8.79 3.29 5.82 2.55 6.33 4.19 4.90 5.70 8.06
0.0700 0.0031 1.95 0.43 0.36 0.1018 -0.0211 6.51 1.87 0.0847 -0.0159 2.70 1.25 0.1036 -0.0077 5.30 0.41 0.0612 -0.0345 1.74 1.22 0.0790 -0.0098 4.74 0.39 0.0420 -0.0413 13.01 0.51 0.0634 -0.0379 1.75 0.68 0.1244 -0.0028 9.88 0.34 0.0914 -0.0179 4.53 0.23 0.0513 0.0084 3.67 0.24 0.0725 -0.0313 3.37 0.73 0.0499 -0.0616 2.97 0.32 0.0882 -0.0261 6.59 0.15 0.0760 -0.0263 8.54 0.30 0.0847 -0,0230 3.96 0.21 0.0046 -0.0051 1.27 0.33 0.0272 -0.0207 3.27 0.24 0.0265 -0.0118 2.20 0.38 -0.0418 -0.0015 5.77 0.50 0.0455 0.0010 2.54 0.43 0.0517 -0.0101 5.67 0.32 -0.0064 0.0382 2.93 0.0406 -0.0144 2.63 0.52 0.12 0.0326 -0.0170 2.72 0.55 5.53
0.41 0.44 2.02 1.05 0.85 1.22
(11)
0.0674 0.039 1.95 0.1074 -0.738 7.72 0.1514 -8.527 2.19 0.1066 0.100 5.55 0.1567 -11.564 3.52 0.1373 -9.130 4.39 0.55 0.1582 -14.059 14.48 0.64 0.1508 -10.375 3.48 0.65 0.1300 -0.468 8.65 0.48 0.1506 -7.241 4.48 0.22 0.0391 0.115 4.51 0.24 0.0909 0.015 5.57 0.34 0.1370 -4.810 9.72 0.31 0.1034 -0.819 9.74 0.23 0.1126 -3.916 11.21 0.08 0.1045 -0.099 9.92 0.21 -0.OOO9 1.435 0.88 0.48 0.0408 -0.209 8.48 0.28 0.0644 -4.156 2.00 0.38 0.0413 -13.135 3.74 0.49 0.0400 1.078 2.48 0.42 0.0578 0.151 6.29 0.21 -0.0293 -0.016 4.12 0.30 0.0551 -0.584 4.64 0.15 0.0832 -5.740 2.41 0.55 6.89
0.42 0.0642 0.0692 1.96 0.38 0.1466 0.1086 6.58 1.80 0.1160 0.0765 1.87 0.99 0.1271 0.0765 3.02 0.18 0.1319 0.0738 2.11 1.49 0.0858 0.0858 6.16 0.23 0.1301 0.0539 13.19 0.15 0.1441 0.0743 2.94 0.32 0.1731 0.1175 4.79 0.32 0.0954 0.0954 7.17 0.22 0.0374 0.0564 2.79 0.24 0.1230 0.0811 3.75 0.74 0.1809 0.0705 3.69 0.31 0.1510 0.0982 6.95 0.17 0.1530 0.0862 8.84 0.32 0.1387 0.0938 4.26 0.19 0.0050 0.0118 0.80 0.31 0.0766 0.0348 3.34 0.41 0.0506 0.0311 2.16 0.48 -0.0384 -0.0423 5.57 0.49 0.0391 0.0492 2.51 0.46 0.0791 0.0554 5.67 0.26 -0.0736 -0.0216 2.67 0.44 0.0750 0.0432 2.76 0.25 0.0743 0.0388 2.79 0.49 5.30
Ay2
(le2) 0.42 0.40 2.03 0.93 0.39 1.22 0.36 1.14 0.38 0.34 0.21 0.20 0.41 0.32 0.22 0.07 0.21 0.48 0.27 0.38 0.50 0.42 0.19 0.25 0.15 0.50
One-fluid, one-parameter van der Waals mixing rule. One-fluid, two-parameter van der Waals mixing rule. Linear density-dependent mixing rule. d Cubic mixing rule.
Ind. Eng. Chem. Res., Vol. 33, No. 8, 1994 1969 Table 8. Bubble-Point Calculations from the Patel-Teja Equation of State for COtContaining Systems
c1 c2
c3 c4 c5 C6 c7 C8 c9 ClO c11 c12 C13 C14 C15 C16 C17 C18 c19 c20 c21 c22 C23 C24 C25 grand AAD
0.0650 0.0967 0.1051
2.04 7.75 3.99 0.0800 9.01 0.0815 5.99 0.0809 5.95 0.0604 16.29 0.0809
5.55
0.1228 0.0926 0.0347 0.1165 0.1471 0.0932 0.1294 0.1000 0.0019 0.0428 0.0418 0.0057 0.0438
10.34 6.99 6.41 5.19 12.48 10.41 14.01 9.79 2.78
8.55
3.04 11.84 2.48 0.0900 6.70 0.0010 4.20 0.0840 5.34 0.0708 6.16 8.56
0.40 0.38 1.86 0.86 0.42 1.24 0.41 0.51 0.67 0.36 0.39 0.33 0.99 0.31 0.08 0.23 0.12 0.34 0.27 0.35 0.41 0.29 0.30 0.50 0.15 0.52
0.0743 0.0922 0.0848 0.0660 0.0573 0.0810 0.0361 0.0622 0.1214 0.0892 0.0356 0.0920 0.0964 0.0817 0.1138 0.0805 0.0026 0.0300 0.0386 0.0057 0.0438 0.0830 0.0141 0.0707 0.0590
0.0119 -0.2200 -0.0202 -0.0380 -0.0381 -0.0061 -0.0485 -0.0386 -0.0022 -0.0155 -0.0033 -0.0381 -0.0555 -0.0335 -0.0338 -0.0262 0.0160 -0.0233 -0.0063 0.0316 -0.0003 -0.0164 0.0446 -0.0191 -0.0244
2.01 6.64 2.59 2.81 1.82 5.28 13.04 1.81 9.85 4.59 6.05 2.42 3.30 6.41 5.20 3.88 0.88 3.12 2.28 8.21 2.48 5.60 2.74 2.64 2.49 5.25
0.36 0.40 1.97 0.89 0.85 1.27 0.64 0.66 0.65 0.46 0.35 0.21 0.56 0.27 0.14 0.13 0.13 0.48 0.23 0.43 0.41 0.29 0.23 0.31 0.21 0.55
0.0670 0.0972 0.1091 0.1013 0.0834 0.0809 0.0590 0.0925 0.1230 0.1147 0.0409 0.1126 0.1496 0.0935 0.1287 0.0989 0.0016 0.0412 0.0187 0.0008 0.0491 0.0901 -0.0019 0.0829 0.0709
0.011 0.123 1.058 -0.268 0.395 0.005 0.034 1.838 -0.025 2.132 -0,041 0.003 -0.007 0.000 0.000 0.000 -0.013 -0.073 -5.575 -2.098 0.845 0.001 -0.088 0.011 0.00
1.96 7.67 3.90 6.03 5.48 5.94 14.05 2.99 10.02 6.21 5.62 4.42 11.91 9.92 13.98 9.63 2.72 7.82 2.63 8.69 2.47 6.51 4.09 4.89
5.84 7.70
0.40 0.36 1.85 1.13 0.32 1.24 0.71 0.20 0.53 0.38 0.32 0.34 1.03 0.58 0.08 0.27 0.12 0.53 0.53 0.40 0.38 0.31 0.28 0.50 0.15 0.59
0.0676 0.0686 1.95 0.1357 0.0987 6.71 0.1157 0.0913 2.57 0.1709 0.0359 8.09 0.1201 0.0691 2.36 0.0809 0.0809 5.95 0.1223 0.0487 13.24 0.1408 0.0708 2.14 0.1677 0.1137 5.10 0.1232 0.0928 4.72 0.0385 0.0380 5.79 0.1506 0.1055 2.90 0.2273 0.1088 3.03 0.1287 0.0912 7.34 0.2153 0.1259 5.52 4.33 0.1330 -0.0889 0.0026 0.0002 2.65 0.0772 0.0378 3.18 0.0572 0.0382 2.21 0.0091 0.0025 11.57 0.0413 0.0490 2.38 0.1208 0.0879 5.53 -0.0004 0.0002 4.13 0.1035 0.0762 2.74 0.1171 0.0679 2.59 5.82
0.39 0.39 1.95 1.48 0.43 1.24 0.36 0.43 0.40 0.43 0.36 0.21 0.50 0.31 0.14 0.12 0.12 0.46 0.25 0.34 0.42 0.29 0.29 0.33 0.21 0.53
Table 9. Bubble-Point Calculations from the Peng-Robinson and Patel-Teja Equations of State for CzH&ontaining Systems mixing rule A mixing rule B mixing rule C mixing rule D mixture M l P AYZ M I P AY2 M I P AYZ MlP AYZ ID k,, (%) (10-2) k,, kbg (%) (1t2) kl, kz, (%) (IO-') ka, ka# (%) (IO-') Peng-Robinson Equation of State 0.0105 1.40 0.45 0.00394 -0.00911 0.83 0.45 0.0308 -4.795 0.0225 4.74 2.19 0.04803 0.03237 3.93 1.96 0.0256 0.808 0.010 0.0535 17.44 1.82 0.04378 -0.02716 9.60 1.01 0.0580 0.0526 6.15 0.55 0.04918 -0.01460 1.65 0.59 0.0772 -3.810 0.000 0.0679 6.77 0.62 0.06728 -0.00059 6.67 0.63 0.0679 0.0722 6.36 0.63 0.07171 -0.00056 6.25 0.63 0.0902 -3.236 0.03860 -0.07588 2.98 0.14 0.1730 -8.204 0.1348 11.54 0.26 0.0693 15.13 0.27 -0.00563 -0.08568 4.76 0.25 0.1358 -13.451 E8 0.000 E9 0.0533 9.65 1.41 0.04405 -0.01965 3.94 1.21 0.0533 5.34 0.80 9.05 1.01 grand AAD Patel-Teia Eauation of State El -0.0084 1.10 0.45 -0.0147 -0.00786 0.82 - 0.45 -0.0084 -0.0013 0.00780 4.18 1.64 0.0040 0.1580 E2 0.0031 4.63 2.22 0.0100 0.0269 E3 0.0421 15.94 1.72 0.0336 -0.02597 9.49 1.07 0.0416 0.0110 E4 0.0389 5.99 0.39 0.0362 -0.01560 1.57 0.41 0.0391 E5 0.0589 4.85 0.72 0.0532 -0.00614 4.33 0.85 0.0590 0.0011 0.0662 0.0049 E6 0.0662 6.10 0.60 0.0605 -0.00780 4.95 0.47 0.10940 E7 0.1335 11.23 0.23 0.0757 -0.05033 5.77 0.13 0.1378 0.0885 -0.0102 E8 0.0851 15.22 0.28 0.0115 -0.08639 4.74 0.30 0.0003 E9 0.0664 9.21 1.57 0.0588 -0.01961 3.26 1.29 0.0663 grand AAD 8.22 1.00 4.71 0.81 El E2 E3 E4 E5 E6 E7
parameter van der Waals mixing rule (mixing rule B) appear to yield better results for a majority of mixtures. The linear density-dependent mixing rule (mixing rule C) is not better than the mixing rule B, although the former is more complicated than the latter. Figure 3 is an illustration for comparing the calculated results from the PT equation with four different types of mixing rules. It is shown that the mixing rule A is applicable provided that the mixtures are well below their critical points. This mixing rule, however, always overestimates the bubble pressures around the critical region. Similar results were obtained from the linear density-dependent mixing rule (mixing rule C). The figure also illustrates that either the one-fluid, two-parameter van der Waals mixing rule (mixing rule B) or the cubic mixing rule (mixing rule D)
1.12 4.65 13.49 2.01 6.74 5.44 10.23 11.65 9.65 7.42
0.46 1.55 0.67 0.35 0.62 0.60 0.31 0.47 1.41 0.71
0.0078 0.0332 0.0540 0.0546 0.0583 0.0654 0.0562 0.0183 0.0513
0.85 4.27 9.29 1.59 5.12 4.36 3.07 4.99 4.05 4.78
0.45 2.15 1.14 0.61 0.94 0.34 0.13 0.22 1.21 0.86
1.10 4.53 14.38 5.67 4.68 6.08 11.11 13.86 9.04 7.73
0.45 -0.0021 -0.0122 2.23 0.0031 0.0031 0.92 0.0817 0.0421 0.31 0.0641 0.0407 0.72 0.0687 0.0518 0.58 0.0847 0.0599 0.24 0.1841 0.0533 0.32 0.1652 0.0328 1.58 0.1011 0.0655 0.83
0.83 4.63 9.24 1.64 3.97 4.61 3.06 4.95 3.33 4.51
0.45 2.22 1.16 0.42 0.94 0.32 0.13 0.34 1.28 0.86
0.0209 0.0046 0.0986 0.0816 0.0820 0.0914 0.1851 0.1464 0.0852
is able to make a substantial improvement as the system approaches the critical points. The results of ethane- and ethylene-containing systems are given in Tables 9 and 10,respectively. It is found that the cubic mixing rule (mixing rule D) is generally superior to the others. With this mixing rule, the PT equation is slightly better than the PR equations, whereas the PT equation is comparable to the PR equation for the ethylene supercritical systems. Conclusion
High-pressure VLE data for three binary systems composed of supercritical carbon dioxide plus 2-methyl-
1960 Ind. Eng. Chem. Res., Vol. 33, No. 8, 1994 Table 10. Bubble-Point Calculations from the Peng-Robinson and Patel-Teja Equations of State for CIIi&ontaining Systems mixing rule A mixing rule B mixing rule C mixing rule D mixture M I P AYZ apIP A.YZ MIP b 2 APlP ID k,, (%) (10-2) k, kb, (%) (lo-') kl, kg (%I (10-') k, kq (%) Pew-Robinson Equation of State N1 0.0069 2.69 0.10 0.00737 -0.01186 1.06 0.11 0.0149 -0.042 1.37 0.17 0.0272 0.0130 0.99 N2 0.0070 2.18 0.80 0.00820 0.00518 2.21 0.70 0.0288 -4.498 1.55 0.27 0.0076 -0,0043 2.06 0.0011 4.91 0.47 -0.01588 -0.04174 3.64 0.41 0.1123 -19.519 1.68 0.61 0.0361 -0,0062 1.48 N3 N4 0.0433 1.86 0.20 0.04088 -0.00464 1.29 0.20 0.0544 -1.744 1.38 0.18 0.0511 0.0424 1.29 0.0230 3.51 0.68 0.01779 -0.00864 1.71 0.66 0.0437 -3.197 2.05 0.76 0.0376 0.0209 1.74 N5 0.0138 6.12 0.90 0.00534 -0.01588 2.34 0.92 0.0574 -7.128 2.30 1.28 0.0424 0.0101 2.34 N6 grand AAD 3.64 0.48 1.84 0.47 1.76 0.59 1.59 Patel-Teja Equation of State N1 -0.00756 1.13 0.10 -0.00956 -0.00493 1.00 0.11 -0.0060 -0.OOO1 1.02 0.12 O.ooo8 -0.0083 0.99 N2 -0.00860 1.98 0.82 -0.00618 0.00271 2.04 0.55 -0.0061 0.2966 1.79 0.87 -0.0086 0.0086 1.97 N3 -0.OlOOO 4.53 0.04 -0.02592 -0.02948 1.51 0.39 0.0107 2.1500 2.91 0.41 0.0217 -0,0196 1.49 N4 0.03054 1.93 0.17 0.02896 -0.00420 1.47 0.18 0.0307 0.1197 1.82 0.19 0.0385 0.0301 1.48 N5 0.01788 3.17 0.68 0.01382 -0.00767 1.97 0.67 0.0179 -0.0098 3.16 0.69 0.0301 0.0159 2.00 N6 0.02582 7.10 1.07 0.02279 -0.01665 2.31 1.02 0.0248 0.0010 6.36 0.74 0.0630 0.0283 2.29 grand AAD 3.39 0.46 1.68 0.48 3.01 0.45 1.68
4\\I
4v2 (IO-*) 0.11 0.86 0.39 0.19 0.66 0.96 0.48 0.11 0.82 0.39 0.18 0.68 1.03 0.50
kaU, ka.., kl", k = binary interaction parameters in the combining ruyes of aij kb.. = binary interaction parameter in the combining rule of l b i j for the mixing rule B K = equilibrium ratio n = number of data points P = pressure (bar) T = temperature (K) x = mole fraction of liquid phase y = mole fraction of vapor phase Greek Symbols K
x19
Y1
= pure-component parameter in the Peng-Robinson equa-
tion SC = pure-component parameter in the Patel-Teja equation A = objection function p = density (mol cm")
Figure 3. Comparison of phase equilibrium calculations from the Patel-Teja equation of state with different rules for the CO2 (1) + 2-methyl-1-pentanol(2)system.
Superscripts
1-pentanol, 1-octanol, and 1-decanol were reported over a temperature range from 345.15 to 453.15 K. The saturated vapor compositions could be correlated accurately by an empirical equation in terms of the density of carbon dioxide. It was also found that the PengRobinson equation with the cubic mixing rule gave the best results for a majority of supercritical systems. Using the one-fluid, one-parameter van der Waals mixing rule could produce erroneous phase envelopes around the critical regions. The linear density-dependent mixing rule also failed as the mixtures neared their critical points. However, either the one-fluid, two-parameter van der Waals mixing rule or the cubic mixing rule could improve substantially the VLE calculations near the critical region.
Subscripts
Acknowledgment Financial support from the National Science Council, ROC, through Grant No. NSC82-0402-EOll-066 is gratefully acknowledged. Nomenclature a , b, c = constants in the equations of state A, B, C = coefficienta in eq 1 F = pure-component parameter in the Patel-Teja equation
calc = calculated value exp = experimental value i = component i ij = i-j pair interaction j = component j j i = j-i pair interaction m = mixture 1 = component 1 (light component) 2 = component 2 (heavy component)
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Received for review November 3, 1993 Revised manuscript received May 10,1994 Accepted May 24, 1994. 0
Abstract published in Advance ACS Abstracts, July 1,1994.
