www.elsevier.nlrlocaterfluid
High-pressure phase equilibria of binary and ternary mixtures
containing the methyl-substituted butanols
Hyun-Song Lee, Sung Yong Mun, Huen Lee
)Department of Chemical Engineering, Korea AdÕanced Institute of Science and Technology, 373-1, Kusong-dong, Yusong-gu, Taejon, 305-701, South Korea
Received 5 February 1999; accepted 12 October 1999
Abstract
High-pressure VLE and VLLE for both the binary mixtures of the carbon dioxide–3-methyl-1-butanol and carbon dioxide–3-methyl-2-butanol and the ternary mixtures of the carbon dioxide–3-methyl-1-butanol–water, carbon dioxide–3-methyl-2-butanol–water and carbon dioxide–2-methyl-2-butanol–methanol were measured at 313.2 K. The phase equilibrium apparatus used in this work is of the circulation type in which the coexisting phases are recirculated, on-line sampled and analyzed. The critical pressures and corresponding mole fractions at 313.2 K were also carefully determined for two binary mixtures. Two water-containing ternary mixtures showed the liquid–liquid–vapor phase behavior over the range of pressure up to their critical point, while for a methanol-containing ternary mixture only two phases coexist at equilibrium. The binary equilibrium data were all reasonably well correlated with the Redlich–Kwong, Soave–Redlich–Kwong, Peng–Robinson, and Patel– Teja equations of state incorporated with eight different mixing rules; the van der Waals, Panagiotopoulos–Reid, and six modified Huron–Vidal mixing rules with UNIQUAC parameters. q 2000 Elsevier Science B.V. All rights reserved.
Keywords: High-pressure; Vapor–liquid equilibria; Mixing rule; Equation of state; Carbon dioxide; Alcohol
1. Introduction
Alcohol is typically synthesized in aqueous solution and then separated from water by distillation or evaporation. Water–alcohol separation is one of the most energy intensive processes in chemical industry. Preliminary evaluation reveals that the supercritical fluid extraction can be adopted as one of the potential separation technologies that satisfy the lower energy requirement and some other process advantages over traditional separation processes. This process can also avoid the current
environmen-)
Corresponding author.
0378-3812r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.
Ž .
Table 1
Physical properties of chemicals used in this work
Ž . Ž . Ž . Ž . Ž .
Name M.W. grmol M.P. K B.P. K Tc K Pc MPa v
Carbon dioxide 44.01 194.7 216.6 304.1 7.38 0.239 Water 18.02 273.2 373.2 647.3 22.12 0.344 Methanol 32.04 175.5 337.7 512.6 8.09 0.556 a 3-Methyl-1-butanol 88.15 156.0 405.2 579.4 4.05 0.596 a 3-Methyl-2-butanol 88.15 203.0 386.1 554.0 4.16 0.589 a 2-Methyl-2-butanol 88.15 264.4 375.5 545.0 3.95 0.506 a w x
Estimated from the Ref. 14 .
tal and health concerns associated with many organic solvents. In connection with these concerns, many researchers have investigated the phase equilibria of a variety of carbon dioxide-containing systems at high-pressure conditions. The related researches for the mixtures containing both carbon dioxide and alcohol are abundant up to the 4-carbon alcohols. The phase equilibrium data for the higher-carbon alcohol systems are particularly limited to the normal alcohols. There are several review articles concerning the experimental techniques and high-pressure phase equilibrium data;
w x w x w x w x w x w x
Tsiklis 1 , Schneider 2 , Eubank et al. 3 , Deiters and Schneider 4 , and Fornari et al. 5 . Hicks 6
w x
and Knapp et al. 7 published the review papers covering the period from 1900 to 1980, Fornari et al.
w x5 from 1978 to 1987, and Dohrn and Brunner 8 from 1988 to 1993.w x
The several high-pressure phase equilibria of the systems containing alcohols were measured in the
w x
previous studies 9–13 . As a continuing research, the high-pressure phase equilibria of the binary carbon dioxide–3-methyl-1-butanol, carbon dioxide–3-methyl-2-butanol and ternary carbon dioxide– 3-methyl-1-butanol–water and carbon dioxide–3-methyl-2-butanol–water systems were measured at 313.2 K and pressures up to the critical point in this study. The experimental equilibrium data were correlated with the four cubic equations of state incorporated with several different types of mixing rules.
2. Experimental
2.1. Chemicals
The carbon dioxide obtained from Express Gas Co. in South Korea was used and its purity was checked by gas chromatography and found at least 99.9 mol%. The 2-methyl-2-butanol,
3-methyl-1-Table 2
Ž . Ž .
Equilibrium compositions and critical point of the carbon dioxide 1 –3-methyl-1-butanol 2 system at 313.2 K
Ž . Ž . P MPa x1 y1 P MPa x1 y1 2.00 0.113 0.989 8.00 0.812 0.981 a 4.00 0.264 0.995 8.35 0.967 0.967 6.00 0.435 0.996 a
Table 3
Ž . Ž .
Equilibrium compositions and critical point of the carbon dioxide 1 –3-methyl-2-butanol 2 system at 313.2 K
Ž . Ž . P MPa x1 y1 P MPa x1 y1 2.00 0.117 0.992 8.00 0.940 0.990 a 4.00 0.290 0.995 8.22 0.974 0.974 6.00 0.512 0.994 a
Measured critical point.
butanol and 3-methyl-2-butanol supplied by Aldrich had a purity better than 99.0 mol%. The HPLC-grade distilled water was supplied by Merck. The methanol supplied by Merck had a minimum
˚
Ž .
purity of 99.8 mol%. The molecular sieves 4 A, beads, 8–12 mesh supplied from Aldrich were used to remove water from the 2-methyl-2-butanol, 3-methyl-1-butanol and 3-methyl-2-butanol. The carbon dioxide, methanol and water were used without any further purification.
2.2. Apparatus and procedure
The apparatus and experimental procedures are almost the same as those used in previous works
w13 . First, the equilibrium cell was charged with a mixture of liquid and then slightly pressurized byx
Fig. 1. Vapor–liquid equilibria and critical point of the carbon dioxide–3-methyl-1-butanol system at 313.2 K: ` experimental data; v measured critical point.
Ž . Fig. 2. Vapour–liquid equilibria and critical point of the carbon dioxide–3-methyl-2-butanol system at 313.2 K: `
Ž .
experimental data; v measured critical point.
carbon dioxide. The cell was heated to the experimental temperature. When the desired temperature reached a steady state, the cell was pressurized to the experimental pressure with carbon dioxide using
Ž .
a simplex mini-pump Milton Roy, 396-31 . To supply carbon dioxide to the cell in the liquid state, a
Table 4
Ž . Ž . Ž .
Equilibrium compositions of the carbon dioxide 1 –3-methyl-1-butanol 2 –water 3 system at 313.2 K
Ž . Ž .
P MPa Vapor Liquid1 Liquid2 Middle
y1 y2 y3 x1 x2 x3 x1 x2 x3 2.00 0.991 0.009 0.000 0.002 0.009 0.989 0.070 0.567 0.364 4.00 0.996 0.004 0.000 0.009 0.009 0.982 0.176 0.511 0.313 a 4.00 0.996 0.004 0.000 0.206 0.579 0.215 6.00 0.997 0.003 0.000 0.015 0.009 0.976 0.309 0.433 0.257 8.00 0.929 0.046 0.025 0.021 0.007 0.972 0.578 0.253 0.168 b 8.35 0.960 0.028 0.013 0.020 0.006 0.974 a
Two-phase tie line. b
Fig. 3. Liquid–liquid–vapor equilibria of the carbon dioxide–3-methyl-1-butanol–water system at 313.2 K and pressures of Ž .
2.00, 4.00, 6.00, 8.00, and 8.23 MPa: ` measured binary data.
bomb with a deep-tube siphon was used. Fine control of the system pressure could be obtained by
Ž . Ž .
using a pressure generator HIP, 62-6-10 . A duplex recirculating pump Milton Roy, 2396-31 was used to rapidly attain equilibrium, and each phase was recirculated through each sampling valve under equilibrium conditions. The three-way valve was installed between the middle and bottom liquid phases in order to switch two phases. A liquid sampling valve was used to collect the liquid-phase samples. The accuracies of measured temperatures and pressures are "0.1 K and "0.01 MPa, respectively. The equilibrium compositions of each phase were determined by injecting the high-pres-sure sample into the gas chromatograph for the on-line composition analysis. Each sample was analyzed at least ten times, and the vapor- and liquid-phase compositions were found reproducible within a mole fraction of "0.002 and "0.003, respectively.
3. Result and discussion
3.1. Binary VLE
The physical properties of pure chemicals were presented in Table 1 in which some values were
w x
estimated from the Ref. 14 . The equilibrium compositions and critical points of the binary carbon dioxide–3-methyl-1-butanol and carbon dioxide–3-methyl-2-butanol systems were measured at 313.2 K and listed in Tables 2 and 3, respectively. The corresponding isothermal pressure–composition diagrams are shown in Figs. 1 and 2. There was little change of carbon dioxide composition in the
Table 5
Ž . Ž . Ž .
Equilibrium compositions of the carbon dioxide 1 –3-methyl-2-butanol 2 –water 3 system at 313.2 K
Ž . Ž .
P MPa Vapor Liquid1 Liquid Middle2
y1 y2 y3 x1 x2 x3 x1 x2 x3 2.00 0.993 0.007 0.000 0.002 0.013 0.986 0.066 0.521 0.413 4.00 0.996 0.004 0.000 0.009 0.011 0.979 0.175 0.465 0.360 6.00 0.995 0.005 0.000 0.015 0.010 0.975 0.349 0.389 0.262 8.00 0.971 0.015 0.014 0.023 0.007 0.971 0.824 0.061 0.116 a 8.23 0.028 0.006 0.966 0.966 0.021 0.012 a Two-phase region.
vapor phase while the liquid phase composition increased rapidly with pressure. The critical pressure was determined by visual observation, and the overall range of the critical opalescence was less than 0.01 MPa. Furthermore, the critical mole fractions repeatedly measured through the liquid sampling valve were reproducible within "0.002. The measured critical pressure and composition at 313.2 K were 8.35 MPa and 0.967 mole fraction of carbon dioxide, respectively.
3.2. Ternary VLE and VLLE
The equilibrium compositions and pressures of the ternary carbon dioxide–3-methyl-1-butanol– water system were measured at 313.2 K and several different pressures and presented in Table 4 and
Fig. 4. Liquid–liquid-vapor equilibria of the carbon dioxide–3-methyl-2-butanol water system at 313.2 K and pressures of Ž .
Fig. 3. This ternary system showed the liquid–liquid–vapor three-phase behavior over the range of pressure up to the critical pressure of 8.35 MPa. Since all binary mixture combinations of three components, carbon dioxide–3-methyl-1-butanol, carbon dioxide–water, and 3-methyl-1-butanol– water become immiscible below the critical pressure of the carbon dioxide–3-methyl-1-butanol binary system, the LLV phases are expected to exist in the ternary mixture under this condition. The equilibrium compositions of the water-rich liquid and carbon dioxide-rich vapor phases changed slightly with pressure. The 3-methyl-1-butanol compositions in the middle phase changed rapidly with pressure, particularly near the critical point of the binary carbon dioxide–3-methyl-1-butanol system. This middle phase gradually merged into the vapor phase and finally disappeared at 8.35 MPa that was identical with the critical pressure of the corresponding binary system. The color of the middle phase was dark brown just below 8.35 MPa while it became clear just above 8.35 MPa. The equilibrium compositions and pressures of the ternary carbon dioxide–3-methyl-2-butanol–water system were also measured at 313.2 K and several different pressures and presented in Table 5 and Fig. 4. The phase behavior of this ternary system was found to be very similar to that of the carbon dioxide–3-methyl-1-butanol–water system. This ternary system also showed the three-phase LLV behavior over the range of pressure up to the critical pressure of 8.23 MPa. Similarly to the 3-methyl-1-butanol rich liquid phase, the 3-methyl-2-butanol rich liquid in the middle phase merged into the vapor phase and finally disappeared at 8.23 MPa that was slightly higher than the critical pressure of the binary system. The ternary mixture containing methanol was attempted to examine the phase behavior difference from the previous two ternary mixtures. The equilibrium data for the carbon dioxide–methanol-2-methyl-2-butanol system are listed in Table 6 and presented in Figs. 5–7. This
Table 6
Ž . Ž . Ž .
Vapor–liquid equilibrium compositions of the carbon dioxide 1 qmethanol 2 q2-methyl-2-butanol 2 system at 313.2 K Ž .
P MPa Vapor Liquid
y1 y2 y3 x1 x2 x3 2.00 0.990 0.000 0.010 0.129 0.069 0.802 2.00 0.983 0.009 0.008 0.132 0.261 0.606 2.00 0.980 0.012 0.007 0.142 0.424 0.435 2.00 0.985 0.015 0.000 0.158 0.618 0.224 4.00 0.993 0.000 0.007 0.314 0.056 0.630 4.00 0.988 0.008 0.005 0.315 0.372 0.314 4.00 0.986 0.011 0.003 0.344 0.551 0.105 6.00 0.991 0.005 0.004 0.514 0.172 0.315 6.00 0.990 0.007 0.003 0.512 0.318 0.170 6.00 0.990 0.008 0.002 0.521 0.404 0.075 8.00 0.985 0.009 0.006 0.932 0.032 0.035 8.00 0.985 0.014 0.001 0.905 0.085 0.010 8.07 0.985 0.004 0.011 0.952 0.014 0.034 8.09 0.983 0.012 0.005 0.947 0.031 0.022 8.11 0.980 0.018 0.002 0.945 0.042 0.013 8.16 0.972 0.020 0.009 0.956 0.026 0.018 8.16 0.976 0.006 0.018 0.959 0.014 0.027
Fig. 5. Vapor–liquid equilibria of the carbon dioxideqmethanolq2-methyl-2-butanol system at 313.2 K and pressures of 2.00 and 4.00 MPa.
ternary mixture exhibits only two phases of vapor and liquid at pressures of 2.00, 4.00, 6.00, and 8.00 MPa. Only two phases could be expected to appear for this ternary mixture by two facts that the binary combinations of the carbon dioxide–2-methyl-2-butanol and carbon dioxide–methanol mix-tures show the immiscible mixmix-tures below the critical pressure of each binary mixture and the methanol–2-methyl-2-butanol mixture is completely miscible in this pressure range. By increasing pressure the liquid phase moves toward the carbon dioxide-rich vapor phase and finally two phases become one phase above 8.16 MPa. This behavior can be expected since a mixture of carbon dioxide and alcohol showed one phase above its critical pressure.
3.3. Equations of state and mixing rules
The VLE data were correlated with the four conventional cubic equations of state, Redlich–Kwong
ŽRK. w15 , Soave–Redlich–Kwong SRKx Ž . w16 , Peng–Robinson PRx Ž . w17 , and Patel–Teja PTx Ž . w18 .x
The following several mixing rules were incorporated with the specific equation of state; van der
Ž . w x w x
Waals, Panagiotopoulos–Reid P and R 19 and six modified Huron–Vidal mixing rules 20–26 .
Fig. 6. Vapor–liquid equilibria of the carbon dioxideqmethanolq2-methyl-2-butanol system at 313.2 K and pressures of 6.00 and 8.00 MPa.
two binary interaction parameters for the Panagiotopoulos–Reid mixing rule. The fugacity coefficient for SRK-EOS with the six modified Huron–Vidal mixing rules is given by:
RT 1 a Õ q b E na
Ž
.
lnf s ln q y b y ln
Ž .
1i
ž
/
iž
/
P Õ y b
Ž
.
Õ y b Õ q b Õ Eni T , nj/ iwhere the composition derivative of na can be calculated from each mixing rule.
Ž .1 PSRK Predictive Soave–Redlich–Kwong mixing rule 20,21 :Ž . w x
E na
Ž
.
1 b bs lng q ln q y1 q a
Ž .
2i i
PSRK
ž
/
Eni T , nj/ i C bi bi
Ž .2 MHV1 Modified Huron–Vidal 1st order mixing rule 22 :Ž . w x
E na
Ž
.
1 b bis lng q ln q y1 q a
Ž .
3i i
MHV 1
ž
/
Eni T , nj/ i C bi b
Ž .3 HVOS Huron–Vidal Orbey–Sandler mixing rule 23 :Ž . w x
E na
Ž
.
1 b bis lng q ln q y1 q a
Ž .
4i i
U
ž
/
Fig. 7. Vapor–liquid equilibria of the carbon dioxideqmethanolq2-methyl-2-butanol system at 313.2 K and pressures of 8.07, 8.09, 8.11 and 8.16 MPa.
Ž .4 MHV2 Modified Huron–Vidal 2nd order mixing rule 24 :Ž . w x
E na
Ž
.
Eni T , nj/ i 1 b bi MHV 1 MHV 2 2 2 s C a q CŽ
a q a.
qlng q ln q y1 q aŽ .
5 1 i 1 i i i MHV 2 MHV 2ž
/
C q2 a C b bŽ
1 2.
iŽ .5 LCVM Linear Combination of the Vidal and Michelsen mixing rule 25 :Ž . w x
E na
Ž
.
l 1 y l 1 y l b bi s q lng q ln q y1 q aŽ .
6 i i U MHV 1 MHV 1ž
/
ž
/
Eni T , nj/ i C C C bi bŽ .6 CHV Corrected Huron–Vidal mixing rule 26 :Ž . w x
E na
Ž
.
1 1 y l b bis lng q ln q y1 q a
Ž .
7i i
U U
ž
/
E ni T , nj/ i C C bi b
where a s arbRT and a s a rb RT. The activity coefficient of component i, g , is calculated fromi i i i
w x w x
constants was well reviewed by Orbey and Sandler 26 . In this study, the UNIQUAC equation 27 was chosen as an appropriate excess Gibbs energy model. The binary interaction parameters were determined by a fitting procedure to minimize the following objective function:
2 2
V L V L
f k , k
Ž
.
sÝ
Ž
y f P y x f P.
qŽ
y f P y x f P.
Ž .
812 21 1 1 1 1 2 2 2 2
nbin
where Snbin means the summation over all the binary data points. The superscripts V and L refer to
vapor and liquid phase. The calculated equilibrium compositions of the binary carbon dioxide–3-methyl-1-butanol system are shown in Fig. 1 along with the experimental data. The resulting binary interaction parameters and average absolute deviations between the experimental data and calculated values are listed in Table 7. As shown in Fig. 1, both are in good agreement over the entire pressure
w x
range considered. Takishima et al. 28 used the Patel–Teja equation of state with the Panagiotopou-los–Reid mixing rule in order to predict the VLE of the carbon dioxide–ethanol system better than the Peng–Robinson equation of state. In the present study, the Redlich–Kwong equation of state poorly predicted the equilibrium behavior of both liquid and vapor phases compared with other three equations of state. The equilibrium values calculated from the Soave–Redlich–Kwong, Peng–Robin-son, and Patel–Teja equations of state incorporated with any specified mixing rule were almost the same as could be seen in Fig. 1. It should be also noted that the van der Waals one-fluid mixing rule underestimates to a great extent the liquid carbon dioxide concentrations above 5.00 MPa, while the Panagiotopoulos and Reid mixing rule predicts reasonably well. However, both mixing rules completely fail to predict the equilibrium behavior near the critical point, while the six modified Huron–Vidal mixing rules slightly underestimate the equilibrium composition of carbon dioxide in the liquid phase near the critical point. The predictive results obtained by using these six mixing rules showed no great differences among them. The vapor–liquid equilibrium data of the binary carbon dioxide–3-methyl-2-butanol system were also correlated with the same equations of state and mixing
Table 7
Binary interaction parameters and average absolute deviations of the carbon dioxide–3-methyl-1-butanol system at 313.2 K
a a
Model k12 k21 AADx AADy
RKqvan der Waals 0.2128 0.2128 19.98 2.57
SRKqvan der Waals 0.1206 0.1206 21.54 1.60
PRqvan der Waals 0.1212 0.1212 21.88 1.11
PTqvan der Waals 0.1021 0.1021 21.59 0.93
RKqP and R 0.2515 0.1691 5.70 8.18 SRKqP and R 0.1668 0.0709 8.47 3.36 PRqP and R 0.1665 0.0721 8.48 3.36 PTqP and R 0.1482 0.0558 11.54 0.97 a a Ž . Ž .
Model a12 K a21 K AADx AADy
w x SRKqPSRK 20,21 y63.4 276.1 4.90 0.61 w x SRKqMHV1 22 y75.0 271.9 6.31 0.48 w x SRKqHVOS 23 y56.1 284.4 6.39 0.53 w x SRKqMHV2 24 y95.2 492.4 5.50 0.48 w x SRKqLCVM 25 y35.7 287.5 6.36 0.50 w x SRKqCHV 26 y22.7 294.0 5.75 0.44 a
Ž . NP <Ž cal exp. exp<
rule combinations used in the binary carbon dioxide–3-methyl-1-butanol system. The calculated equilibrium compositions of the binary carbon dioxide–3-methyl-2-butanol system are shown in Fig. 2 along with the experimental data. The resulting binary interaction parameters and average absolute deviations between the experimental and calculated values are listed in Table 8. Only the Redlich– Kwong equation of state incorporated with the simple van der Waals mixing rules poorly predicts the equilibrium behavior of both liquid and vapor phases near the critical point compared with other three equations of state. The equilibrium values calculated from the Soave–Redlich–Kwong, Peng–Robin-son, and Patel–Teja equations of state incorporated with any specified mixing rule were almost the same as could be seen in Fig. 2. It should be also noted that the van der Waals one-fluid mixing rule predicts the equilibrium phase behavior near the critical point while the Panagiotopoulos and Reid mixing rule fails. In the carbon dioxide–3-methyl-1-butanol system, both mixing rules completely fail to predict the equilibrium behavior near the critical point. The six modified Huron–Vidal mixing rules calculate the equilibrium composition reasonably well near the critical point. The predictive results obtained by using these six mixing rules showed no great differences among them.
The UNIQUAC binary interaction parameters for the 3-methyl-1-butanol–water, 3-methyl-2-butanol–water, and 2-methyl-2-butanol–methanol systems were not available in the literature. Only liquid–liquid equilibria of 3-methyl-1-butanol–water and 3-methyl-2-butanol–water at 293.2, 298.2,
w x w x
and 303.2 K were only available in the literature 29 . The UNIFAC group contribution method 30 was attempted to correlate these ternary systems, but failed over the entire pressure range. For the better prediction of these ternary systems, the binary vapor–liquid equilibria of the 3-methyl-1-butanol–water, 3-methyl-2-3-methyl-1-butanol–water, and 2-methyl-2-butanol–methanol systems should be first measured. In addition, the new excess Gibbs energy model describing the alcohol-containing system,
w x
e.g., the DISQUAC 31,32 based on association theory, should be also closely examined.
Table 8
Binary interaction parameters and average absolute deviations of the carbon dioxide-3-methyl-2-butanol system at 313.2 K
a a
Model k12 k21 AADx AADy
RKqvan der Waals 0.2080 0.2080 12.39 8.05
SRKqvan der Waals 0.1145 0.1145 20.04 0.99
PRqvan der Waals 0.1153 0.1153 20.40 0.99
PTqvan der Waals 0.0976 0.0976 15.78 1.60
RKqP and R 0.2582 0.1518 8.03 8.09 SRKqP and R 0.1742 0.0579 4.06 4.12 PRqP and R 0.1727 0.0600 4.31 4.31 PTqP and R 0.1552 0.0448 4.20 4.24 a a Ž . Ž .
Model a12 K a21 K AADx AADy
w x SRKqPSRK 20,21 y107.2 361.1 3.27 0.40 w x SRKqMHV1 22 y113.7 349.1 3.28 0.40 w x SRKqHVOS 23 y100.8 370.0 2.87 0.25 w x SRKqMHV2 24 y117.5 510.2 2.51 0.25 w x SRKqLCVM 25 y80.6 354.6 3.32 0.40 w x SRKqCHV 26 y68.8 370.4 3.27 0.40 a
Ž . NP <Ž cal exp. exp<
List of symbols
AAD Average absolute deviation
Ž .
a UNIQUAC binary interaction parameter K
CHV Corrected Huron–Vidal
HVOS Huron–Vidal Orbey–Sandler
k Binary interaction parameter
LCVM Linear combination of the Vidal and Michelsen
M.W. Molecular weight
MHV1 Modified Huron–Vidal 1st order
MHV2 Modified Huron–Vidal 2nd order
Ž .
P Pressure MPa
Ž .
Pc Critical pressure MPa
PSRK Predictive Soave–Redlich–Kwong Ž . T Temperature K Ž . Tb Boiling point K Ž . Tc Critical temperature K Ž . Tm Melting point K
x Liquid-phase mole fraction
y Vapor-phase mole fraction
Greek letters f Fugacity coefficient v Acentric factor Subscripts i Component i j Component j Superscripts L Liquid phase V Vapor phase Acknowledgements
This work was supported by non-directed research fund of the Korea Research Foundation.
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