948
Ind. Eng. Chem. Res. 1996,34, 948-957
Prediction of Gas Solubilities Using the LCVM Equation of State/ Excess Gibbs Free Energy Model Dimitrios A. Apostolou, Nikolaos S. Kalospiros, and Dimitrios P. Tassios" Laboratory of Thermodynamics and Transport Phenomena, Department of Chemical Engineering, National Technical University of Athens, Zographou Campus, 9,Heroon Polytechniou Str., Zographos GR-15780,Greece
A recently developed EoS/GE model, the LCVM one, is applied to the prediction of Henry constants of nine gases ( 0 2 , NO,COZ,CO, HzS,CH4, CzHs, C3H8, C 4 H d in a wide range of pure and mixed solvents, including heavy hydrocarbons, polar solvents, and water. When not available, LCVM interaction parameters are estimated by correlating vapor-liquid equilibrium data. LCVM yields very good predictions for Henry constants in pure solvents, with typical errors less than 8%;it also performs very satisfactorily in the prediction of high-pressure vapor-liquid equilibria. On the other hand, widely-used E o S P models, such as PSRK and MHV2, are shown to result in progressively poorer behavior with increasing system asymmetry. Finally, LCVM predictions for Henry constants in mixed solvents are also satisfactory, especially when combined with the method of Catte et al. The results presented here, combined with those from previous ones, render LCVM a valuable tool for predicting vapor-liquid equilibria.
Introduction Solubilities of gases in pure and mixed solvents, characterized by Henry constants, are fundamental properties for many industrial processes from the efficient design of gas absorption and stripping columns t o the composition of artificial atmospheres. The development of methods for the satisfactory prediction of gas solubilities in liquids is limited by the finite experimental reliability (Fog and Gerrard, 1991)coupled with the lack of data, especially at temperatures remote from 25 "C. Several such methods have been developed recently, most of them based on the group contribution concept. Some of them are based on original or modified UNIFAC (Antunes and Tassios, 1983; Gani et al., 1989) or are group contribution methods specifically developed for Henry constant predictions (Catte et al., 19931, and others are based on group contribution equations of state (GC-EoS model, Skjold-Jorgensen, 1984, 1988; Wolff et al., 1992). The most successful has been the one of Catte et al. (1993) with typical errors less than 5% in the predicted Henry constants in the temperature range between 273 and 323 K. None of these methods that can simultaneously provide good predictions for both high-pressure vapor-liquid equilibrium (VLE) and gas solubilities has been developed so far. Furthermore, some of them, such as the ones of Catte et al. and Antunes et al., are only applicable at low pressures and near-ambient temperatures. On the other hand, recent developments have led to predictive and computationally simple EoS/@ models incorporating an equation of state (EoS) with the UNIFAC GE expression, capable of describing binary and multicomponent high-pressure vapor-liquid equilibria in complex systems. The most successful of these models are the PSRK (Holderbaum and Gmehling, 19911, MHVB (Dahl and Michelsen, 1990; Dahl et al., 19911, and LCVM (Boukouvalaset al., 1994)ones. None of them, however, has been extensively applied to the prediction of Henry constants of supercritical gases in pure and mixed solvents. The main feature in the development of such models is to obtain the mixing rule for the mixture attractive-
* Author t o whom correspondence should be addressed. oaaa-5a~519512634-0948$09.0010
term parameter, a , of the EoS by setting the expression for P obtained from the EoS equal to that of an existing @ model; e.g., UNIFAC, ASOG, NRTL, etc. This matching was initially postulated at infinite pressure (Vidal, 1978). In the case of MHV2 and PSRK, however, these models are based on matching GE's at zero pressure, &r a suggestion made by Michelsen (1991a,b). Finally, LCVM is a linear combination of the Vidal (1978) and MHVl (Michelsen, 1991a,b) mixing rules, developed so as to give satisfactory bubble point pressure predictions for a variety of systems, and has no specified reference pressure. Although MHVB and PSRK have been quite successful in the prediction of vapor-liquid equilibrium (VLE) in systems with components of similar size, both models perform poorly when applied to systems with components that differ appreciably in size, such as those containing gases with large n-alkanes (Boukouvalas et al., 1994;Voutsas et al., 1994). This behavior is related to the inability of these models to reproduce the P model they are combined with, for systems characterized by high degree of asymmetry (Kalospiros et al., 1994). LCVM, on the other hand, gives satisfactory VLE results even for highly asymmetric systems. Considering the success of LCVM in the prediction of VLE behavior in asymmetric systems containing supercritical gases, the objective of this work is to examine the applicability of the same model in the prediction of Henry constants of such gases in pure and mixed solvents. The obtained results are compared with those from the M W 2 and PSRK models. All three models are used as suggested in the original publications; i.e., the same EoS and P model. For further details on the LCVM, PSRK, and MHV2 models, see the original publications. Available interaction parameters for pairs of groups involving supercritical gases are obtained from Holderbaum and Gmehling (1991) and Holderbaum (1993) for PSRK, Dahl et al. (1991) for M W 2 , and Boukouvalas et al. (19941, Spiliotis et al. (19941, and Vlachos et al. (1994) for LCVM. In the context of the present study, LCVM's interaction parameter table is expanded to cover new gas/ UNIFAC groups by correlating solubility and, when available, also VLE data. Although emphasis of this study is put on the prediction of Henry constants, the
0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34,No. 3, 1995 949 performance of LCVM in the prediction of the available
VLE data used to obtain the new interaction parameters is also reported. The remainder of the paper is organized as follows: First, we present the expressions for the Henry constants with the EoSlGE models considered followed by the data base used. The results obtained are then presented, followed by a discussion of the performance of the LCVM and the other two EoS/GE models. We close with our conclusions.
Expressions for the Henry Constants with EoS/GE Models The fugacity of a supercritical compound of a mixture is usually expressed through the asymmetric convention:
T is the temperature of the system, V t , and ~ bi are the liquid volume at T and co-volume parameter of pure component i, respectively, and t2 is the translation factor of the solvent [a parameter used in the t-m PR EoS to improve the prediction of pure component saturated liquid volumes (Magoulas and Tassios, 1990) that has a negligible effect on the predicted Henry constants-less than 0.5%1. The partial molar value 6i of the mixture reduced attractive term parameter, a = aIbRT, is defined as
4=
(””)
where HI is the Henry constant defined as
In eqs 1and 2, x1 denotes the composition of the gas in the liquid phase, f1 the fugacity of the same component in the mixture and 71” is the activity coefficient. Henry constants in this work are calculated using the following analytical expression, derived from the definition
where $1 is the fugacity coefficient of the gas in the liquid phase of the mixture and P is the pressure of the system. Henry Constants in Pure Solvents. For a twocomponent system, the pressure of the system as XI 0 is the vapor pressure, PzSat,of the pure solvent at the same temperature. Thus, we have
-
H, = $;PTt
b P aat(V2,L - t2) b2 2 RT -1)-
while for SRK used in the PSRK and MHV2 models
T,P,nCzj
with n denoting the number of moles. This parameter is calculated by applying the above definition to the mixing rules for a of the LCVM, MHV2, and PSRK models. Henry Constants in Mixed Solvents. Henry constant calculations of gases in binary mixed solvents were carried out using two methods. Method a: A procedure similar to the one above for Henry constants in pure solvents was followed. The only difference is the pressure of the system which now is the bubble pressure of the mixed solvent. This pressure is evaluated using the corresponding EoSIGE model for the binary system of the two solvents. Method b: This is a modification of the simplified Krischevsky equation proposed by Catte et al. (1993), which assumes that the energetic interactions between solvent molecules do not affect the solubility of the gas. The value of Henry constant in the mixed solvent is given by In H =
cQiHi In
i
where is the volume fraction of component i, calculated using the expression proposed by Kikic et al. (1980):
(4)
The value of PzEat is calculated with the corresponding EoS, while 41- is the infinite dilution fugacity coefficient of the gas in the liquid phase, obtained from the corresponding model. For the t-m PR EoS used in the LCVM model 1nm;=-i
(7)
ani
f1.L = “1Yl”HI
a,=-
xiri2/3 (9)
c3Ciriy3 i
with ri denoting the volume parameter of molecule i as defined in the UNIFAC model (Fredenslund et al., 1975). Hi is the corresponding Henry constant value of the gas in pure solvent i, calculated here using eqs 4-6.
Data Base and Parameter Evaluation The data base used consisted of the following systems: (a) Henry constants and high-pressure VLE data in binary systems containing pure solvents with CH4, C2H6, C3Hs, C4H10, COS,CO, 0 2 , N2, and H2S; (b) Henry constants in systems of binary mixed solvents mostly with N2,02, CH4, and C2H6. References for all the experimental data used are given in the supplementary material of this paper (see paragraph at the end of the paper regarding the
960 Ind. Eng. Chem. Res., Vol. 34, No. 3, 1995
Table 1. Gas Parameters Used in the LCVM Model gas N2 0 2
COz
co
CHI CzHs H2S
T,(K) 126.10 154.58 304.19 132.92 190.55 305.42 373.55
P,(bar) 33.94 50.43 73.82 34.99 45.99 48.80 90.07
w
R
Q
0.0403 0.0218 0.2276 0.0663 0.0110 0.0990 0.0814
0.9340 0.8570 1.2960 1.0679 1.1290 1.8022 1.1665
0.985 0.940 1.261 1.112 1.124 1.696 1.163
Table 2. Pure-ComponentParameters Used in the LCVM (t-m PR) Model component methanol ethanol acetone benzene toluene p-xylene l-methylnaphthalene chloroform water methyl acetate ethyl acetate tert-butanol 2-propanol 1-butanol 1-propanol 2-propanol isobutanol 1-pentanol 1-octanol 1-decanol 2-butanone 2-pentanone diethyl ether 1,4-dimethylbenzene diethyl ether
T,(K) P,(bar)
c1
c2
c3
1.224 003 1.206 189 0.818 753 0.712 993 0.769 212 0.984 128 0.794 445
-0.273 495 0.789 831 0.036 533 -0.396 127 -0.181 845 -1.280 922 0.702 066
-0.398 234 -2.308 417 -0.168 438 1.369 983 0.705 585 2.776 570 -1.197 180
536.40 54.72 0.756 839 -0.561 065 647.13 220.55 0.923 656 -0.379 370 506.80 49.91 0.855 649 -0.152 987 523.25 38.30 0.866 220 0.189 328 506.20 39.72 1.019 044 1.467 049 508.30 47.62 1.105 253 1.837 749 562.93 44.13 0.953 781 2.007 034 536.71 51.70 1.085 974 1.384 980 508.31 47.64 1.174 225 1.082 550 547.73 42.95 1.021 551 1.102 856 586.15 38.80 1.124 403 0.436 450 652.50 28.60 1.251 135 -0.754 595 690.00 23.70 1.142 313 0.299 890 535.50 41.54 0.866 759 -0.176 550 564.00 38.40 0.787 561 0.112 740 466.70 36.38 0.867 092 -0.778 236 616.26 35.11 0.984 128 -1.280 972
1.370 657 0.442 429 0.600 531 -0.018 160 -0.838 356 -3.853 776 -2.924 842 -2.356 550 -1.922 220 -0.009 568 0.534 816 3.621 880 1.682 867 0.231 901 0.519 750 1.986 702 2.776 500
512.64 513.92 508.20 562.16 591.80 616.26 772.04
466.70
80.97 61.48 47.01 48.98 41.06 35.11 36.60
36.38 0.867 092 -0.778 236
calculations were performed except, of course, for the case of n-alkanes. The T,, P,, and w values of Magoulas and Tassios (1990) were used in this case with the SRK
EoS. The gases in Table 1were considered to be separate groups and, when not available, temperature-dependent UNIFAC interaction parameters were determined through regression for LCVM only. The UNIFAC expression Y v= ~ exp(-C,//T) is replaced by
[
Y, = exp -
A,
+ B,(T
- 298.15) T
with Tin kelvin. For MHV2 all organic gases, including propane and butane, are considered as separate groups, while for PSRK only methane is. For MHV2 the same type of expression for Y, is used, while for PSRK Yo = exp
[a, + b,:
+c u p
I
(11)
New parameters were estimated in this study from binary VLE and Henry constant data using the LCVM model, by minimizing the following objective function:
NSYS NDAT
1.986 702
availability of supplementary material). Published Henry constants often display wide deviation and discrepancies. Thus, all data were carefully checked, and any which appeared unreliable were excluded from the calculations. The experimental data in this study are at temperatures larger than 273 K, reaching up to 623 K for some systems. The critical properties T,and P,, acentric factor w , and volume and area parameters R and Q used in the LCVM model for the gases considered in this study are presented in Table 1. For 02,CH4, and CO the method of Bondi (1968) was used for the estimation of the R and Q parameters. The values for the remaining gases are the ones used by Spiliotis et al. (1994). The R and Q values used with the PSRK and MHV2 models are the ones suggested in the pertinent publications (Holderbaum and Gmehling, 1991; Dahl et al., 1991). Pure-component parameters for the t-m PR EoS of LCVM, for all solvents encountered in this study but alkanes, are shown in Table 2. The parameters c1, c2, and cg are obtained by fitting available vapor pressure data (Daubert and Danner, 1989) with the MathiasCopeman (1983) expression for the temperature correction of the attractive term of the EoS. For n-alkanes, the T,, P,,and w values suggested by Magoulas and Tassios (1990) were used with the t-m PR EoS. The Mathias-Copeman parameters c1, c2, and cg, for the SRK EoS as well as the T, and P,values used in MHV2 and PSRK, are the same as the ones suggested by Dahl et al. (1991) and Holderbaum and Gmehling (1991), respectively. When no c1, C Z , and c3 values were given in the aforementioned publications, no Henry constant
where NSYS denotes the number of systems and NDAT the number of data points per system. The first two terms of the objective function may be omitted depending on the availability of experimental VLE data. In cases where all terms are used in the regression, the second term is multiplied by a factor equal to 1000 so that its magnitude becomes comparable to the magnitudes of the other two terms, minimizing the percent errors in bubble point pressure and Henry constants, respectively. The values of the LCVM group interaction parameters so obtained are presented in Tables 3 and 4 in addition to the ones already available. The boxes marked with the symbol # in Tables 3 and 4 correspond to parameters already available for the LCVM model (Boukouvalas et al., 1993; Spiliotis et al., 1994;Vlachos et al., 1994) and were determined by fitting only highpressure VLE data. Consequently, Henry constants for systems containing the respective groups correspond to pure predictions of the LCVM model. The type of data used to obtain the new LCVM interaction parameters presented in this study (boxes not marked with the symbol # in Tables 3 and 4) is shown in Table 5. No new parameters were determined for the two other models. Thus, comparison of the three models is based on systems for which parameters are available for all of them. Our intention was to include in the data base both VLE and solubility data. However, for a few systems such as those containing oxygen in n-alkanes, only solubility data were available. Systems for which only high-pressure VLE data were available were not in-
Ind. Eng. Chem. Res., Vol. 34, No. 3, 1995 951 Table 3. W A C Group (2)-Gas (1) Interaction Parameters for the LCVM Model (First Row Gives A21 and Second Row Gives B21)
02 CH2
153.1 1.3294 167.1 -0.0637 102.5 -1.815 272.2 - 1.443 104.1 -0.8877 278.0 0.6419 458 4 1037.8 1.400 146 0.27
CCHV cCHCH3 ACH ACHCHz CH30H OH HzO CHzCO
c02 116.7# -0.911 197.8 -0.8975 128.4 -0.543 187.0# 1.098 87.4# 0.3087 33.84# 0.0242 471.W 2.5883 271.8 2.75 -79.5 -0.1441 358.M -0.3666 102.8# -0.4999
N2 179.9# 2.3079 414 -0.1466 127.2 -0.888 249.2# 0.877 152.4# 2.5958 145.0 0.2394 820 0.0 969.2 3.5853 290.8 -1.2352
COOH
ccoo
318.2 -0.3796 231.2 -0.0887
CH2O
co
HzS 112.w -0.5823
45 0
62 0.84 286.9 0 637.4 5.5539
CHI 88.0#
0.300 188.9 0.6564 105.6# 0.257 91.M 0.1498 -5.499 -0.0984 334 -0.7 950.5 2.5081 1530
339.5 1.7145
0
C2H6 -87.0# -1.39 25.0 0
58.M 0.769 18.7# -0.936 164.1 0.5068 542.8 4.0315 926.3 0.3445 55 0
Table 4. Gas (l)-UMFAC Group (2) Interaction Parameters for the LCVM Model (First Row Gives A12 and Second Row Gives B12) CHz 34.8 -2.173 N2 19.81# -1.758 COz 110.6# 0.5003 CO 41 0 H2S 188.5# 0.4882 CH4 -25.0# -0.300 CzH6 120.# 2.35
0 2
cCH~ 139.5 -0.104 -92.8 -0.114 129.7 -0.610
cCHCH3 ACH -55.9 13.2 -1.213 0.7387 232.9 114.8# 0.618 -1.489 -38.2 -26.8# -0.17 -1.235
-67.7 -0.668 6.6 0
-13.6# -0.453 -4.1# -0.919
ACHCHz 133.1 0.191 143.6# -1.987 175.7# -2.958
CH30H OH 131.4 1540 0.38 -4 969.6 1823 0.215 0 283.8# 87.1# 0.0315 3.9273 407 2000 -1.4 0
-1.39# 892.6 1900 -1.589 -0.7099 75.16# 179.1 -0.793 0.8259
HzO CHzCO COOH CCOO CHzO 14.4 2509 245.2 200.2 6.8 2.6 -0.3734 -0.8054 238.5 432.1 9.748 -0.5989 601.1 100.7 218.6# -126.9# -2.91 -0.176 -0.7217 -1.8187 497.7 -3.4784 768.3 -3.1983 41 -62
Table 5. Specification of Experimental Information Used in the Estimation of New Parametersa 0 2
C02
CO A2
H2S
B,C1 A, C 1 C8 A, C 6 A , C 1 B , C 1 A,C A,C1 A,C1 C3 C1 A,C1
CH2O
C1
ccoo
C5 C2
N2
CH2 cCH~ cCHCH~ ACH ACHCHz CH30H OH H2O CHzCO
C1 C1
C2 C1 C6
c1
B,C 1 A,C3 A , C 1 A,C 1
CH4
C2He
A,C1
A,C1
A,C1 A , C 10 A,B,Cl A,C1
A,C1 A,C7 B,C1 A,C1
=A: P , T,2,y data. B: P , T,x data. C: solubilities (P = 1 atm, T,x)/Henry constant. The number of different mixtures used in the regression is also shown.
cluded in the data base for the estimation of the corresponding interaction parameters, because the emphasis of the present study was put mainly on lowpressure solubilities and Henry constants. Finally, in order to account for the variation in the solubility of a gas in linear and cyclic alkanes the groups CH2 and cyclic CH2 are distinguished, both having the same UNIFAC group parameters, R and Q. Also cyclic CHCH3, a new UNIFAC group, is introduced for the distinction of substituted cyclic compounds such as methylcyclohexane. The group parameters for this group were estimated additively from the two groups CH and CH3 (R = 1.348, Q = 1.076).
1
NDP
avAH% = -
NDPZ
I
H;d-H?P HFP
x 100 (13)
952 Ind. Eng. Chem. Res., Vol. 34, No. 3, 1995 Table 6. Average Absolute Percent Deviations (avA€Z%) between Experimental and Predicted Henry Constants in Pure Solventsa
cCH~ cCHCH~ ACH ACHCHz CH30H OH HzO CHzCO COOH
ccoo CHzO
coz
Nz
02 PSRK
co
MHV2 LCVM PSRK MHV2 LCVM PSRK MHV2 LCVM PSRK MHV2 LCVM PSRK MHV2 LCVM PSRK MHV2 LCVM PSRK MHV2 LCVM PSRK MHV2 LCVM PSRK MHV2 LCVM PSRK MHV2 LCVM PSRK MHV2 LCVM PSRK MHV2 LCVM
HzS 27 40 7e
>
17 15 2'2' 28 41 7'2) 9 13
>
6d
CH4 26 25 5f 19 42
CZH6 41
0'56)
1'56)
20 17 22'56) 11 28
40 1 8'56) 44 32 2'56) 42 15 2'4) 43 27 7'4,
>
78 8 35
C3He 29 35 6h
C4H10 36
23 14 22'33)
19 20 19(33)
>
5'
0'2)
*
47
0'2)
*
10 4'2)
0'56)
*
17 6
30 1'2'
9 9 14(4) 3
$56)
*
18
15 4'2'
>
*
*
>
>
12 7'57)
21 4'57'
4'2)
*
5'4' 13 27
>
>
>
16
31
14
17 2'57) 42 1
$561
1(56)
1'57)
*
31 3'21
> (57)
>'57)
* * 1'2) * 23 1'2'
0 Numbers in parentheses denote the reference where the data were obtained from. A complete list of these references is given in the supplementary material of this paper. (29,30). (6,8,16,21,23,26,27,51, 52,531. (2,6,21,51). e (3, 13, 14,51,60). f(S,9, 11,22,25, 26, 27, 43, 53). 8 (9, 26, 43, 45, 53). (6, 23, 43, 45, 56, 61). (43, 61).
H [bar]
H [bar]
3,500 1,250
* LCVM
* MHV2
3,000
1,150
1,050
950
850
- - - - - - -m- - - - m- _- - m
-ACETONE '
1,500
"2-BUTANONE
-' 2-PENTANONE
750
2 3
T
1,000 L 201
333
313
293
the LCVM model.
VLE. Results for some of the above gases with various pure solvents are presented in Table 9. In each case, the results obtained with the LCVM model are compared with those from MHV2 and PSRK models. In this table, the following notation is used: AP% is the average absolute percent error in bubble point pressure: 1
NDP p d - p x p PXP
501
401
I
801
1001
Figure 2. Prediction of Henry constants of nitrogen in n-alcohols
[kelvin]
Figure 1. Prediction of Henry constants of oxygen in ketones with
AP%=-ZI NDP i=l
301
1
x 100
(14)
AY is the average absolute deviation in vapor phase
at 298.15 K with LCVM and MHV2.
mole fraction: 4
AY=NDP
NDP
lYF'
- YiexPl x 1000
(15)
i=l
The acronym NDP and the symbols same meaning as in Tables 6-8.
* and
>
have the
Discussion of the Results Henry Constants in Pure Solvents. Close examination of the results presented in the preceding section
Ind. Eng. Chem. Res., Vol. 34,No. 3, 1995 963 H [bar]
950
%
EXP
* LCVM
* MHV2
A&
)C\
* LCVM
* PSRK
0
% EXP
0
0
c20
Cl6
C28
C36
Figure 6. Prediction of Henry constants of carbon monoxide in n-alkanes at T = 473 K with the LCVM, MHV2, and PSRK models. H [bar] 1,000,000
* LCVM
* MHV2
80
1
C13
c10
too
Figure 4. Prediction of Henry constants of hydrogen sulfde in n-alkanes at T = 473 K with the LCVM, MHV2, and PSRK models.
C3
C2
C1
02
N2
CO
C02
H2S
Figure 7. Prediction of Henry constants of a series of gases in water at 303.15 K with the LCVM, MHV2, and PSRK models. H [bar]
A.A.E. i n H e n r y C o n s t a n t
100
nC4
C16
200 r
* LCVM
t
-i. PSRK
f,
0,
80 +MHV2
, ,
.
,
,I
,
I
I
'\
I
60
40
upper limit for LCVM's 20
:
I
- LCVM
321
0 (
2
5
Figure 6. Average percent error for Henry constants of methane in n-alkanes with the LCVM, MHV2, and PSRK models.
yields the following comments on the performance of LCVM and its comparison with MHV2 and PSRK 1. Very satisfactory predictions are obtained using the LCVM model for Henry constants of all gases considered in a variety of pure solvents, with typical errors less than 8% (see Figures 1-7 and Table 6). 2. Notice that this applies to both the Henry constants obtained from the parameters developed here by the simultaneous fit of Henry constant data and, when available, of VLE ones (Table 5), and those obtained by using existing parameters obtained from VLE data
361
401
441
481
521
561
T [kelvin]
Figure 8. Prediction of Henry constants of carbon dioxide in n-eicosane with the LCVM model. The vertical line denotes the higher temperature at which data were used to estimate the interaction parameters of the LCVM model.
(Boukouvalas et al., 1994;Spiliotis et al.,1994;Vlachos et al., 1994). 3. Results with the first approach are presented in Table 6 and typical ones in Figures 1-3. Results with the second approach are presented in Table 6 for the following cases: N2 with n-alkanes and aromatics; CO2 with the same type of solvents plus acids; H2S with n-alkanes; CH4 with n-alkanes and aromatics; CzHs with the same type of solvents. Typical results are shown graphically in Figures 4 and 5.
964 Ind. Eng. Chem. Res., Vol. 34,No. 3, 1995 Table 8. Average Absolute Percent Deviations (avAH%) between Experimental and Predicted Henry Constants in Binam Solvents Containing Water
H (bar]
avAH% rep T ( K ) PSRK MHV2 LCVM
system
1
l o w e r limit for LCVM's
20,
0
213.15
I.PS
233.15
253.15
LCVM+ Catte
- LCVM
273.15
293.15
T [kelvin]
Figure 9. Prediction of Henry constants of nitrogen in n-hexane with the LCVM model. The vertical line denotes the lower temperature at which data were used to estimate the interaction parameters of the LCVM model.
Table 7. Average Absolute Percent Deviations (avAZZ%) between Experimental and Predicted Henry Constants in Anhydrous Binary Solvents avAH%
Ndcyclohexane/2,2,4trimethylpentane Nz/benzene/cyclohexane NdethanoVbenzene NdethanoY2,2,4trimethypentane Ndl-propanoY2,2,4trimethwentane Ndethanofiacetone CzH$benzene/ cyclohexane C*H&enzene/2,2,4trimethylpentane
rep T(K) PSRK MHV2 LCVM 30 298.15 7.7 27.8 15.0 38 302.75 22.0 30 298.15 * 30 298.15 * 30 298.15 30 273.15
* *
44.6 43.4
* *
*
4.7 7.4
LCVM+ Catte 10.0
1.6
2.3 3.6 8.4
7.8
5.6
0.4 7.2 9.6 5.5 2.8
)1(
EXP
- LCVMcCATTE '.MHV2 LCVM 6o
7o
'
'
298.15 323.15 1 302.75 29.6
18.2
3.9 1.6 2.3 2.3
1 302.75 21.5
4.6
11.1
"The numbers denote the reference where the data were obtained from. A complete list of these references is given in the supplementary material of this paper.
4. The main advantage of the LCVM model demonstrated by Boukouvalas et al. (1994);i.e., its satisfactory performance with asymmetric systems, is observed here as well, as demonstrated with the typical cases of H2S in n-alkanes (Figure 4) and CHI in n-alkanes (Figure 5). 5. The satisfactory performance of the LCVM model is also demonstrated with the case of CO with n-alkanes shown in Figure 6 , where the interaction parameters for all three models were obtained using only highpressure VLE data for CO with n-alkanes up to n-octane for the PSRK model and n-decane for the LCVM and MHV2 ones. 6. Propane and n-butane are not considered as separate UNIFAC groups with the LCVM model, and the same applies to the PSRK one. Prediction results are presented in Table 6 . Results in water are contained in Figure 7. All results are very satisfactory except for the case of water. Poor prediction of the same gases in water are also obtained with PSRK. On the other hand, the good performance of MHV2 with water is explained by the fact that propane and n-butane are considered as separate groups in its development and thus, specific interaction parameters with water are available. This also explains the good results for the other gases in water, shown in Figure 7, for the PSRK and LCVM model (ethane is not a group for PSRK).
+-PSRK
50
b
40
30"
20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.8
0.7
0.9
1
benzene m o l e f r a c t i o n
Figure 10. Prediction of Henry constants of ethane in benzene/ cyclohexane at 302.75 K with the LCVM, LCVMfCatte, MHV2, and PSRK models. H [bar] Thousands
-LCVM+CATTE ' '-
LCVM
-
MHV2
'
k\
60 40
'
'
.. 0o
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
i
alcohol mole fraction
Figure 11. Prediction of Henry constants of oxygen in ethanoY water at 313 K with the LCVM, LCVM+Catte, and MHV2 models.
7. For high molecular weight solvents, such as large alkanes, the temperatures of the experimental Henry constants were often much higher than those of the data correlated in order to obtain the LCVM interaction parameters. However, such extrapolation yields satis-
Ind. Eng. Chem. Res., Vol. 34, No. 3, 1995 955 Table 9. VLE Results for the Systems Considered in this Study
PSRK NDP
T (K)
47
273-533
13-200
*
19 20 21 53
373-433 376-543 273-343 298-513
47-101 19-882 49-284 25-1013
21 14 4
23 23 39 17
473-533 311-477 304-533 298-313
34 64 48 18
P(bar)
AP%
MHV2
AY x 1000
-
LCVM
AP%
AY x 1000
AP%
AY x 1000
21
-
24
-
-
3 8 3 13
-
-
>
-
36 16 17 18
20-101 3-136 6-200 4-56
2 2 26
27 9 21
13 7 8 3
41 15 9 3
5 8 8 2
17 15 15 2
198-339 298-498 298-323 310-588
12-317 22-1002 10-40 3-137
11 9 26
19 22 -
16 15 15 4
27 45
3
4 5 12 4
18 20 9
22
310-588
3-206
13
20
4
9
6
9
54 47 20 29 22
294-378 273-373 313-333 323-588 298-323
14-207 14-368 14-104 14-600 10-118
16 5 6 13
5 9 2
25 8 8 9 4
10 2 1 12 3
2 9 6 9 3
3 3 1 14 5
54 19 17 26 6
283-466 298-373 313-333 310-444 298
6-82 10-60 13-67 4-137 5-36
3 28 36
15 6 8 12 3
35 32 4 2
4 5 8 12 3
14 34 4 2
*
*
*
>
32
>
13 15 7 14
-
-
-
a Numbers in parentheses denote the reference where the data were obtained from. A complete list of these references is given in the supplementary material of this paper.
factory results. For example as shown in Figure 8 for the system COdn-eicosane, although the maximum temperature of the data used t o obtain the parameters in Tables 3 and 4 is 423 K, good results are obtained even a t T = 573 K. On the other hand, extrapolation to temperatures below the low end of the temperature range of the data base may lead to poor results and thus use of LCVM a t such temperatures should be avoided. An example is shown in Figure 9 for the system Ndnhexane (low-temperature end of the data base used is 273.15 K). However, the poor performance of LCVM at such low temperatures does not constitute a major problem, since the value of the lower temperature limit of the data base used is typically 273.15 K and Henry constant values at lower temperatures are rarely needed. Henry Constants in Mixed Solvents. The following observations can be made considering the results for Henry constants in binary mixed solvents: 1. LCVM yields very satisfactory predictions of Henry constants in anhydrous mixed solvents with both methods considered, with typical errors less than 10% as shown in Table 7 and in Figure 10. For the same type of mixed solvents, MHV2 and PSRK with method a yield results generally poorer than LCVM (with the same method), most probably due to poorer predictions of Henry constants in the two pure solvents. 2. On the other hand, LCVM predictions with method a in aqueous solutions of alcohols are rather unsatisfactory as shown in Table 8 and Figure 11. In this case, as shown in the same table and figure, coupling of LCVM with the method of Catte et al. (1993; method b) yields much better results with typical errors less than 15%. Notice, however, that the experimental local maximum of Henry constant observed in water-rich mixed solvents is not captured by this method (see
Figure 11). This maximum is typical with the systems shown in Table 8 and is not captured using any of the models considered in this study coupled with either method a orb. Furthermore, this phenomenon becomes stronger with decreasing temperature. Thus, due to the inability of the models to capture it, the results in Table 8 for the systems Oz/alcohoYwater at 273 K are characterized by larger errors (25-35% for LCVM coupled with method b) than the ones at higher temperatures. Finally, since the method of Catte et al. assumes no energetic interactions between the solvents and, therefore, requires only Henry constants in the pure solvents, coupling with MHV2 and PSRK will not yield as good predictions because both models perform poorer in the prediction of Henry constants in pure solvents.
Prediction of High-pressureVLE Behavior for the SupercriticalGas/Pure Solvent Systems Considered in This Work. The following comments summarize our observations on the performance of LCVM in the description of the VLE data, used for the extension of the corresponding interaction parameters (Tables 3 and 4). 1. As shown in Table 9, LCVM performs very satisfactorily even at very high temperatures and pressures (see for example the water-containing systems and CO with methanol). This observation is in agreement with previously reported results (Boukouvalas et al., 1994; Spiliotis et al., 1994). 2. PSRK and MHV2 yield results of comparable quality with the LCVM model, unlike the case of Henry constants discussed above. This is due to the fact that most of the VLE data considered here refer to relatively symmetric systems, and thus satisfactory results are expected with all models (Kalospiros et al., 1994). This is not the case with asymmetric systems, as shown by
956 Ind. Eng. Chem. Res., Vol. 34,No. 3, 1995
Boukouvalas et al. (1994), Voutsas et al. (1994), and Spiliotis et al. (1994). Finally, as under Henry Constants in Pure Solvents above, the poorer behavior of PSRK and MHV2 compared to LCVM, occasionally observed for systems containing cyclic hydrocarbons (see relevant results in Table 61,is due to the fact that cCHz and cCHCH3 are considered as separate UNIFAC groups only in LCVM.
Conclusions The applicability of the LCVM EoS/@ model to the prediction of Henry constants of several gases in a range of pure and mixed solvents is illustrated in this work. Very satisfactory predictions are obtained for Henry constants in pure solvents with errors generally less than 8%. This model also performs much better than the other two commonly used EoS/GE ones: MHV2 and PSRK. The latter yield progressively poorer results, as the size difference between the gas and the solvent increases. For mixed solvents, experimental data are predicted to within 10% by LCVM, except for watercontaining systems where poor results are obtained. However, if LCVM is combined with the method for Henry constant in mixed solvents of Catte et al. (1993)) satisfactory predictions are obtained. MHV2 and PSRK give generally poorer results. Finally VLE results are also very satisfactory, similar to those obtained with PSRK and MHV2 except at asymmetric systems, where LCVM performs better. The success of LCVM in the prediction of Henry constants and VLE demonstrated in this study, in combination with its previously reported excellent performance in predicting high-pressure VLE (Boukouvalas et al., 1994; Spiliotis et al., 19941, for both symmetric and asymmetric systems, renders this model a valuable tool for phase equilibrium predictions. To this purpose, an extensive list of gasAJNIFAC group interaction parameters is available (Tables 3 and 4).
Nomenclature a = EoS attractive term (cohesion) parameter (barm6/ kmo12) b = EoS co-volume parameter (m3/kmol) c1, c2, c3: pure compound parameters of the MathiasCopeman expression F = objective function for minimization GE = excess Gibbs free energy ( J h o l ) H = Henry constant (bar) NDAT, NDP = number of data points per system NSYS = number of systems P = pressure (bar) Q = structural group or molecular surface area R = structural group or molecular volume t = volume translation ( m 3 h 0 l ) T = temperature (K) V = molar volume ( m 3 h 0 l ) x = mole fraction in liquid phase Greek Symbols
a = reduced attractive term EoS parameter, a = a/bRT ai = partial molar value of a for component i yi* = activity Coefficient \vi = interaction energy parameter in UNIFAC CP = volume fraction of component 4 = fugacity coefficient 41- = infinite dilution fugacity coefficient of component i o = acentric factor
Subscripts
1 = gas 2 = liquid solvent c = critical value i = component i
Table and Figure Symbols * = no parameters available # = LCVM interaction parameters available from previous publications > = avAH% greater than 50, AP% greater than 50 avAH%, A.A.E. = average absolute error % in Henry constants I.P. = interaction parameters AP% = average absolute error % in bubble pressure AY = average absolute deviation of vapor phase composition
Supplementary Material Available: A list of the data references used to obtain the results in Tables 6-9 (7pages). Ordering information is given on any current masthead page. Literature Cited Antunes, C.; Tassios, D. Modified UNIFAC Model for the Prediction of Henry's Constants, Znd. Eng. Chem. Process Des. Dev. 1983,22,457-462. Bondi, A. Physical Properties of Molecular Crystals, Liquids and Glasses; John Wiley and Sons, Inc.: New York, London, Sydney, 1968. Boukouvalas, C.; Spiliotis, N.; Coutsikos, P.; Tzouvaras, N.; Tassios, D. Prediction of Vapor-Liquid Equilibrium with the LCVM model: A Linear Combination of the Vidal and Michelsen Rules coupled with the Original UNIFAC and the t-mPR EoS. Fluid Phase Equilib. 1994, 92, 75-106. Catte, M.; Achard, C.; Dussap, C.-G.; Gros, J.-B.Prediction of Gas Solubilities in Pure and Mixed Solvents Using a Group Contribution Method. Znd. Eng. Chem. Res. 1993,32,2193-2198. Dahl, S.; Michelsen, M. L. High-pressure Vapor-Liquid Equilibrium with a UNIFAC-Based Equation of State. AIChE J . 1990, 36,1829-1836. Dahl, S.; Fredenslund, A.; Rasmunssen, P. The MHV2 Model: A UNIFAC-Based Equation of State Model for Prediction of Gas Solubility and Vapor-Liquid Equilibria at Low and High Pressures. Znd. Eng. Chem. Res. 1991,30,1936-1945. Daubert, T. E.; Danner, R. P. DZIPR Data Compilation; AIChE: New York, 1989. Fogg, P. G. T.; Gerrard, W. Solubility of Gases in Liquids. A Critical Evaluation of Gas ILiquid Systems in Theory and Practice; Wiley: New York, 1991. Fredenslund, A,; Jones, R. L.; Prausnitz, J. M. Group-Contribution Estimation of Activity Coefficients in Non-Ideal Liquid Mixtures. AIChE J. 1975,21, 1086-1099. Gani, R.; Tzouvaras, N.; Rasmussen, P.; Fredenslund, A. Prediction of Gas Solubility and Vapor-Liquid Equilibria by Group Contribution. Fluid Phase Equilib. 1989, 47, 133-152. Holderbaum, T. Die Vorausberechnung von Dampf-Flussig-Gleichgewichten mit einer Gruppenbeitragszustandsgleichung. Cortschrittsbericht VDI-Verlag: Dusseldorf, 1991; No. 243. Holderbaum, T.; Gmehling, J. PSRK. A Group Contribution Equation of State Based on UNIFAC. Fluid Phase Equilib. 1992, 70, 251-265. KalosDiros. N. S.: Tzouvaras. N.: Coutsikos. Ph.: Tassios. D. Analysis of Zero-Reference-Pressure EoS/@' Models. MChk J . 1994, in press. Kikic, I.; Alessi, P.; Rasmussen, P.; Frendeslund, A. On the Combinatorial Part of the UNIFAC and UNIQUAC Models. Can. J. Chem. Eng. 1980,58, 253-258. Magoulas, K.; Tassios, D. "hermophysical Properties of n-Alkanes from C1 to C ~ and O "heir Prediction for Higher Ones. Fluid Phase Equilib. 1990, 56, 119. Mathias, P. M.; Copeman, T. W. Extension of the Peng-Robinson Equation of State to Complex Mixtures: Evaluation of the Various Forms of the Local Composition Concept. Fluid Phase Equilib. 1983, 13, 91.
Ind. Eng. Chem. Res., Vol. 34, No. 3, 1995 957 Michelsen, M. L. A Method for Incorporating Excess Gibbs Energy Models in Equations of State. Fluid Phase Equilib. 1990a, 60, 42-58. Michelsen, M. L. A Modified Huron-Vidal Mixing Rule for Cubic Equations of State. Fluid Phase Equilib. 1990b, 60, 213-219. Skjold-Jorgensen, S. Gas Solubility Calculations. 11. Application of a new Group Contribution Equation of State. Fluid Phase Equilib. 1984, 16, 317-351. Skjold-Jorgensen, S. Group Contribution Equation of State (GC EoS): a Predictive Method for Phase Equilibrium Computations over wide Ranges of Temperature and Pressure up to 30MPa. Znd. Eng. Chem. Res. 1988,27, 110-118. Spiliotis, N.; Boukouvalas, C.; Tzouvaras, N.; Tassios, D. Application of the LCVM Model to MulticomponentSystems: Extension of the UNIFAC Interaction Parameter Table and Prediction of the Phase Behavior of Synthetic Gas Condensate and Oil Systems. Fluid Phase Equilib. 1994,101, 187-210. Stelmachowski, M.; Ledakowicz, S. Prediction of Henry's Constants by the UNIFAC-FV Model for Hydrocarbon Gases and Vapors in High-boiling Hydrogarbon Solvents. Fluid Phase Equilib. 1993, 90, 205-217.
Vidal, J. Mixing Rules and Excess Properties in Cubic Equations of State. Chem. Eng. Sci. 1978,33,787-791. Vlachos, K.; Kalospiros, N. S.;Kolisis, F.; Tassios, D. Solubilities of Alcohols, Acid, Esters in Supercritical Carbon Dioxide: Correlation and Prediction. Fluid Phase Equilib., submitted. Voutsas, E. C.; Spiliotis, N.; Kalospiros, N. S.; Tassios, D. Prediction of Vapor-Liquid Equilibrium a t Low and High Pressures Using UNIFAC-based Models. Znd. Chem. Eng. Res. 1995, in press. WOW,S. R.; Danner, R. P.; Frendeslund, A. Extension of the Group Contribution Equation of State for the Calculation of Gas Solubilities. Fluid Phase Equilib. 1992, 81, 109-127.
Received for review July 5, 1994 Accepted November 21, 1994 @
IE940410U Abstract published in Advance ACS Abstracts, February 15, 1995. @
