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Prediction of the solubility and gas-liquid equilibria for gas-water and light hydrocarbonwater systems at high temperatures and pressures with a group contribution equation of state
B
ELSEVIER
Fluid Phase Equilibria 131 (1997) 107-118
Prediction of the solubility and gas-liquid equilibria for gas-water and light hydrocarbon-water systems at high temperatures and pressures with a group contribution equation of state Jiding Li ", Isabelle Vanderbeken b, Suyu Ye b, Herve Carrier b, Pierre Xans h,, a Department of Chemical Engineering, Tsinghua University, Beijing 100084, China b Laboratoire Haute Pression, Universit£ de Pau et des Pays de l'Adour, 64000 Pau, France
Received 7 February 1996; accepted 2 December 1996
Abstract A group contribution equation of state has been proposed. The model is based on the Weidlich-Gmehling modified UNIFAC and Soave-Redlich-Kwong equation of state, in which the modified Huron-Vidal excess Gibbs free energy mixing rule is used, the residual term of the modified UNIFAC is changed a little and the combinatorial term is improved. Nine interaction parameters between gas groups CO 2, CO, N 2, H 2, HES, CH4, C2H6, C3H 8, C4nlo and water are fitted. The model parameters have been used to predict solubility and gas-liquid equilibria for seven gas-water, and thirty-nine light hydrocarbon-water data sets in large temperature and pressure ranges (278-637 K, 1-1972 bar). The predicted results are in good agreement with the experiments, which makes it possible that the real phase behavior at high temperatures and pressures for gas-water and light hydrocarbon-water systems which are of interest to petroleum and natural gas exploitation industries, can be predicted reliably. © 1997 Elsevier Science B.V. Keywords: Theory; Vapour-liquid equilibria; Equation of state; Excess function; Method of calculation; Activity coefficient
1. Introduction
Gas (CO 2, CO, N2, H 2, H2S)-water and light hydrocarbon (CH4, C2H6, C3Hs, nC4Hi0)-water systems are quite common in many industrial processes, especially in petroleum and natural gas exploitation, petroleum refining and coal gasification processes. For the design and optimization of these processes information on the solubility and phase behavior of these systems at high temperatures and pressures has to be known since the size of equipment and process condition (T, P) are mainly * Corresponding author. 0378-3812/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. Pll S0378-3812(96)03234-7
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J. Li et a l . / Fluid Phase Equilibria 131 (1997) 1 0 7 - 1 1 8
determined by this knowledge. Measuring all data needed for the design is very difficult or impossible in some cases at high temperatures and pressures. It thus is necessary to use a general thermodynamic model so that the solubility and phase equilibria for these systems can be confidently predicted where data do not exist or where they are of poor quality. The use of group contribution models is an effective method to predict phase equilibria with the help of limited experimental data or when reliable experimental information can not be found. Since Huron and Vidal [1] proposed an approach that incorporates an excess free energy (gE) model into the mixing rules for the Redlich-Kwong equation of state, group contribution models like UNIFAC [2], ASOG [3], modified UNIFAC [4-6] coupled with equations of state, have been used in high pressure systems [7,8], in polymer solution systems [9,10], and in multi-phase equilibria systems [11] and so on. However, up to now most of the research on group contribution equations of state is focused on the modification of the Huron-Vidal excess Gibbs free energy mixing rule [12,13,7,14-16]. The research on the improvement or the modification of the group contribution models in order to obtain a good g e expression to be used with cubic equations of state has received only minor attention. In addition, the results predicted by current group contribution equations of state at high temperatures and pressures for some systems, especially for light hydrocarbon-water systems [17] cannot meet the needs of the design and optimization of the industrial processes because of the quite large deviations. The purpose of this work is (1) to modify group contribution models in order to develop a group contribution equation of state, i.e., an equation of state-group contribution model applicable for gas-water and light hydrocarbon-water systems at high temperatures and pressures, (2) to fit the model parameters between nine gas groups (CO 2, CO, N 2, H 2, H2S, Ch 4, C z H 6, C3H 8, nC4H~0 ) and water group, (3) to use these parameters to predict solubility and gas-liquid equilibria for seven gas (CO 2, Co, N 2, H 2, HzS)-water and thirty-nine light hydrocarbon (CH 4, C z H 6, C3H 8, nC4H~0)-water data sets in large temperature and pressure ranges (278-637 K, 1-1972 bar). The results for these systems have not been published in many papers on group contribution equations of state. Only a few results can be found in the papers on the MHV2 model [18,8] in a small pressure range (7-200 bar).
2. A group contribution equation of state As well known, the Soave-Redlich-Kwong equation of state [19] describes the phase behavior of most systems as effectively as the Peng-Robinson equation of state [20]. Moreover, the expression of Soave-Redlich-Kwong is a little simpler than that of Peng-Robinson. In this paper the Soave-Redlich-Kwong equation of state is used. The Soave-Redlich-Kwong equation of state may be written in compressibility factor form: Z 3 -
Z 2 -I-
(A - B - B2)Z-AB
= 0
where Pv
aP A = - - B RT (RT) 2
Z~-----
bP RT
where the mixture b-parameter is calculated by the conventional linear mixing rule.
(1)
J. Li et al. / Fluid Phase Equilibria 131 (1997) 107-118
109
The mixture a-parameter in Eq. (1) is given by the modified Huron-Vidal excess Gibbs free energy mixing rule [13,21]:
ql(ol__~iZiO~i)+q2(ol
2
~iZiOt?) = - R T +
b
in Eq. (2), q~ and q2 are the coefficients of the equation ql = - 0 . 4 7 8 used in this w o r k . a and a i are defined as
a
bRT
and q2 = - 0 . 0 0 4 7
that are
ai b i RT
where a~ is the parameter of pure component i and is obtained from a~=0.42748
R2T2 Pci [f(Tri)]2
(3)
where Tr = T / T c, f(Tr) is given by Mathias and Copeman [22] f(Tr) :
1 q-- C l ( 1 -
/(Tr)
1 + Cl(1 - ~-~) (Tr > 1)
=
~ r r ) "[- C 2 ( 1 -
~ r r ) 2"t- C 3 ( 1 -
}/Trr) 3 (Tr ~ 1)
(4a)
(48)
in Eqs. (4a) and (4b) constants C 1, C 2 and C 3 are estimated using the vapor pressure of the pure components [23]. For hydrogen and carbon monoxide, C~ is calculated directly from the acentric factor to [19]: C l = 0.48 + 1.574o9 - 0.176to 2
(5)
The values of the constants C i and the critical point constants of the pure components used in this paper are listed in Table 1.
Table 1 The Ci-constants (Eqs. (4a) and (4b)) and critical point constants of pure components Component
C1
C2
C3
Tc (K)
Pc (bar)
Hydrogen Nitrogen Carbon monoxide Carbon dioxide Methane Ethane Propane n-Butane Hydrogen sulphide Water
0.1332 0.5427 0.5836 0.8653 0.5472 0.6853 0.7726 0.8487 0.5507 1.0873
0.0000 - 0.0524 0.0000 - 0.4386 - 0.3992 - 0.4284 - 0.5090 - 0.5520 0.4534 - 0.6377
0.0000 - 0.3381 0.0000 1.3447 0.5751 0.7382 1.0306 1.0774 - 0.5841 0.6345
33.2 126.2 132.9 304.2 190.6 305.4 369.8 425.2 373.2 647.3
12.97 33.94 34.96 73.76 46.00 48.48 42.46 38.00 89.37 220.48
110
J. Li et al./Fluid Phase Equilibria 131 (1997) 107-118
In Eq. (2), g E(0) is the mole excess Gibbs free energy at the zero pressure. It is assumed that: E(O)
g RT
(6)
= E z/In'y/ i
where Yi is the activity coefficient of the component i, and Yi is obtained from the modified UNIFAC model. In the modified UNIFAC model, the activity coefficient is expressed in the following form:
lny i = lny/c + lny/R
(7)
The first term on the right-hand side of Eq. (7) represents the combinatorial part of the activity coefficient and the second term refers to the residual part. In the modified UNIFAC-Lyngby [5], the combinatorial part is described as lny c = 1 -
(8)
~ i qt- I n
where
/,.}/3 ~i =
ri = Y'~ v(~i)Rk
E J
In the modified UNIFAC-Dortmund [4], it is given by Vi + in V/t lny/c = 1 - V / + INV.'- 5qi 1 - ~ii Fi ]
(9)
where r}/4 Vi' =
Exjr:3./4
ri gi -
J
q~
- -
Fi = -
Y'.xjrj J
-
Y'.xjqj
qi = E
k
v(i)Qk
J
In Eqs. (8) and (9), R k and Qk are van der Waals volume and surface area of subgroup k respectively. They can be calculated directly [2] and don't need to be fitted. In this work, it is found that modification of the combinatorial term may improve the final results even though the number of the parameters which has to be fitted to experimental data has not been increased. In this paper the combinatorial term is modified as lny/c
---- I - V,.'+ InV/-
5qi
|
-
-
-~i -'l-
In Eq. (10) all symbols have the same meaning as in Eq. (9). It should be noted that Eq. (10) is slightly different from Eq. (9) in the last term, and different from Eq. (8). The group assignment and the values of R and Q used in this paper are listed in Table 2.
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J. Li et al. / Fluid Phase Equilibria 131 (1997) 107-118
Table 2 The group assignment and R and Q values Main Group
Subgroup
R
Q
H2 N2 CO CO 2 CH 4 C2H 6 C3H 8 C4HIo
H2 N2 CO CO 2 CH 4 C2H 6 C3H 8 nC4H lO iCaH ,o H2S H 20
0.8320 1.8680 2.0940 2.5920 2.2440 3.6044 4.9532 6.3020 6.3020 2.3330 0.9200
1.1410 1.9700 2.1200 2.5220 2.3120 3.3920 4.4720 5.5520 5.5520 2.3260 1.4000
HzS H 20
The following relations are used to represent the residual part in this paper: lny R =
E , i'(lnS-lnF
(11)
i')
k
O.,~k m~Omll3mk) -- ~m E On~I~nm ,.
lnFk= Qkl-ln(
(12)
n
where U(j) g....a m "~j J
Q,. Xm Ore=
E Qnx°
X,.=
n
~m. = exp
E E ,?xj j
n
(amn+bmnT)
(13)
T
where a., n and bran are UNIFAC interaction parameters between groups m and n, and estimated from experimental data. Eq. (13) is close to the equations which are used by Weidlich and Gmehling for their modified UNIFAC, in which
~.,n =
( amn WbmnTW CmnT2 ) exp -
in which in many cases
~m,,=exp -amn-~
T
Cm. =
0 and differs Larsen's modified UNIFAC, where
- b., ~
~
cm~
ln-~-+
112
J. Li et al. / Fluid Phase Equilibria 131 (1997) 107-118
3. Estimation of parameters The interaction parameters for the group contribution model proposed in this work were determined by minimization of the following objective function using the Simplex-Nelder-Mead method [24]:
[
F ( a .... a,,m,bm,,,b,,m) = Y'~ Y'~ gl(Yexp--Yca'c) 2 + g2 np
nc
txpcc)21 Pexp
(14)
where n p and n c a r e the number of data points and components respectively, g refers to a weighting factor. The subscript "exp" means experimental and "calc" calculated. The Yca~c and ecalc are determined by the iterative method. Experimental (T, P, x, y) vapour-liquid equilibrium data were used to estimate the interaction parameters between ten main groups. Only propane-water interaction parameters were determined using the incomplete (solubility) data (T, P, x). The data sets selected to fit the parameters are given in Table 3. It should be noted that for the group interaction C H a - H 2 0 , C O - H 2 0 , CO2-H20, H 2 S - H 2 0 , N 2 - H 2 0 and H2-H~O the experimental data used to estimate the interaction parameters are suitable but for the group CzH6-H20, C3H8-H20 and nC4H1o-H20 new data bases are desirable. The fitted interaction parameters between ten groups are given in Table 4. It is known that using parameters fitted from experimental data (Table 3), may lead to erroneous results when the model is used outside temperature and pressure range of the data base. In this paper the reliability of the parameters used to predict vapor-liquid equilibria at the extrapolated temperature and pressure has been checked and judged from the pure component properties and by means of the typical
Table 3 The data sets used to fit the interaction parameters System T (K) H 2-H20
N2 - H
2°
CO-H 20 CO2-H 2°
CH 4-H 20
C2H6-H20
C3H8-H20
nC4Hlo-H20 H2S-H20
310-588 310-588 310-588 288-533 383-623 297-518 323-588 423-633 298-338 473-573 288-410 285-422 310-410 628-637 310-588
P (bar)
Reference
3-137 3-137 3-137 6-202 100-1500 13-64 13-169 98-1078 25-125 27-224 1-34 5-192 3-40 255-1102 3-208
[25] [25] [25] [26] [27] [28] [26] [29] [30] [31] [32] [32] [33] [31 ] [25]
J. Li et al. / Fluid Phase Equilibria 131 (1997) 107-118
113
Table 4 Modified UNIFAC (proposed in this paper) group interaction parameters
i
j
aij
aji
H2 N2 CO CO 2 CH 4 C2H 6 C 3H 8
H 20 H 20 H 20 H 20 H 20 H20 H 20
C4H 1o
H20
H2S
H20
411.4 2280 2324 624.0 2435 1478 2699 2965 1019
1044 403.8 452.0 257.3 477.3 324.9 326.6 420.4 349.5
big -
3.891 3.607 2.878 0.320 3.057 1.509 3.559 3.773 0.8687
bji
T (K)
P (bar)
- 0.3553 0.5907 0.05167 0.01808 - 0.03417 0.2567 0.1518 - 0.1765 - 0.3832
275-570 275-570 275-570 275-620 275-600 275-550 275-420 275-580 275-580
1-500 1-600 1-500 1- 1500 1- 1900 1-600 1-400 1-1000 1-500
binary-system properties. So most of the parameters can be used with confidence in the given temperature and pressure range shown in the last two columns in Table 4. 4. Results and discussion The overall results of 46 data sets predicted by the model proposed in this work are summarized in Table 5, where AP IPe~p- P¢.lcl
P ( % ) = 100×
P~xp
' Ay=lYexP-Ycalc[
Table 5 shows that in the temperature range 278-637 K, and pressure range 1-1972 bar the total Table 5 Deviations between experimental and predicted gas-liquid equilibria
System
P (bar)
Data points
A p/p
I. Data type one: T, P, x, y H 2- H 2 0 310-588 Nz-H20 310-588 CO-H20 310-588 CO 2 - H 2O 288-623 CH 4-H 2° 279-633 C 2H6 - H 2° 473-573 nC 4H Io - H 20 310-637 H 2S - H 2 ° 310-588 Mean deviation
3-138 3-138 3-138 6-1500 13-1079 13-241 3-1103 3-206
17 16 18 170 93 24 12 28 378
2.44 4.96 4.26 6.66 5.81 26.73 18.82 4.27 7.56
II. Data type two: T, P, x N 2 - H 20 324-398 CH 4 - H 2O 298-627 C2H6-H 20 310-444 C3Hs-H20 278-422 nCaHt0-HzO 310-410 Mean deviation
91-515 3-1972 4-685 1-192 1-34
18 243 80 186 49 576
5.94 11.35 9.68 18.16 11.92 13.20
T (K)
(%)
Ay 0.0043 0.0038 0.0034 0.0245 0.0381 0.0573 0.0609 0.0077 0.0271
114
J. Li et al. / Fluid Phase Equilibria 131 (1997) 107-118
160( 120 I
1200 "Z
eo
°8oQ
n 4(
400
0
I
0
0.50 x I , Yl
0
0.40 Xltyl
0.80
D
c i 120
8o ~-
,
I00~
4o~
I 1
,
0 0
0.50 x.yf
~
o! p
I
i
I
0,40
0
0.80
x I , Yl
Fig. 1. Experimental and predicted gas-liquid equilibria. (A) Carbon monoxide(1)-water(2) at [] 346.48 K; (B) carbon dioxide(l)-water(2) at [] 383.15 K, zx 523.15 K, +543.15 K, * 573.15 K, O 623.15 K; (C) hydrogen(l)-water(2) at [] 366.48 K, zx 477.59 K; (D) hydrogen sulphide(1)-water(2) at [] 310.93 K. ( - - ) model proposed in this work, (•, zx, + , *, O) experimental values from (A) Gillespie and Wilson [25], (B) Takenouchi and Kennedy [27], (C) Gillespie and Wilson [25], (D) Gillespie and Wilson [25].
20OO 160C
"~ 1200 .o
~" e00 400
y z
o
i
0.04
8
80( 60(
~4~
i
0.002 x,I
0,08
x1
0.OO4
D n
60o
~,00 I1 100
0
0
o_ 2 0 0
0.0004 xq
0.0008
0
i
I 0.002 11
I 0.004
Fig. 2. Experimental and predicted solubility of gases in water. (A) Methane(l)-water(2) at [] 479.15 K, zx 553.15 K, (D 589.15 K; (B) ethane(1)-water(2) at [] 310.93 K, Lx 377.59 K, C) 444.26 K; (C) propane(1)-water(2) at [] 360.93 K, zx 399.82 K, O 422.04 K; (D) nitrogen(l)-water(2) at [] 375.65 K. ( - - ) model proposed in this work, (•, LX, O) experimental values from (A) Price [34], (B) Culberson and McKetta [35], (C) Kobayashi and Katz [36], (D) O'Sullivan and Smith [37].
J. Li et al. / Fluid Phase Equilibria 131 (1997) 107-118
115
mean relative deviation between experimental and predicted pressure in this work is not more than 10.97% and the total mean absolute deviation of mole fractions in the vapor phase is 0.0271. These results show that the model proposed in this paper can be used reliably to predict gas-liquid equilibria for gas (CO z, CO, N 2, H 2, H2S)-water and light hydrocarbon (CH4, C2H 6, C3H 8, nCnHlo)-water systems in large temperature and pressure ranges. It should be pointed out that the above results include all data sets of C2H6-H20, C3H8-H20 and nCaHI0-H20 systems in which the experimental results for some points are not reasonable. A few typical prediction results of gas-liquid equilibria are presented in Fig. 1, where Fig. I(A) and Fig. I(D) show isotherms at 346.48 K and 310.93 K for the carbon monoxide-water and hydrogen sulphide-water systems respectively, Fig. I(B) describes in detail gas-liquid equilibrium behavior for the carbon dioxide-water system (temperature from 383 to 623 K, pressure from 100 to 1500 bar) and Fig. I(C) shows isotherms at 366.48 K and 477.59 K for the hydrogen-water system. In order to test the ability of the model to predict gas solubility we have studied 26 data sets. The predicted results of solubility of gases (methane, ethane, propane and nitrogen) in water are shown in Fig. 2. It can be seen that the predicted results are in good agreement with the experimental ones, not only for the high temperature and pressure cases (Fig. 2(A), methane(1)-water(2) system, temperature up to 590 K, pressure up to 1950 bar), but also at other cases (Fig. 2(B), ethane(1)-water(2) system, 310-444 K, 10-700 bar; Fig. 2(C), propane(1)-water(2) system, 360-420 K, 5-200 bar; Fig. 2(D), nitrogen(l)-water(2) system, 375 K, 100-700 bar).
5. Conclusion A group contribution equation of state has been proposed. This model is based on the modified Weidlich-Gmehling UNIFAC and Soave-Redlich-Kwong equation of state, in which the modified Huron-Vidal excess Gibbs free energy mixing rule is used, the residual term of the modified UNIFAC is changed a little: all the interaction parameters cij are set equal to zero in all cases, and the combinatorial term is improved. With the help of the experimental data the interaction parameters between nine gas groups CO 2, CO, N2, H2, H2S, CH4, C2H6, C3H8, C4HIo and water group are fitted. The model parameters have been used to predict solubility and gas-liquid equilibria for seven gas-water, and thirty-nine light hydrocarbon-water data sets in large temperature and pressure ranges (temperature from 278 to 637 K, pressure from 1 to 1972 bar). The total mean relative deviation between experimental and predicted pressure is no more than 10.97% and total mean absolute deviation of mole fractions in the vapor phase is 0.0271. These results show that phase equilibria at high temperatures and pressures for gas (CO 2, CO, N 2, H2, H2S)-water and light hydrocarbon (CH4, C2H6, C3H 8, nC4Hlo)-Water systems, which are of interest to petroleum and natural gas exploitation, petroleum refining and coal gasification industries, can be predicted by this model with the desired accuracy.
6. List of symbols a
ai
parameter of mixture in equation of state parameter of pure component i in equation of state
116
a,,n b b~ bin,
Cm, F Fi g P q q~ Qk r~ Rk T v V~ Vi' xi Xm y~ z~ Z
J. Li et a L / Fluid Phase Equilibria 131 (1997) 107- 118
modified UNIFAC interaction parameter between groups m and n parameter of mixture in equation of state parameter of pure component i in equation of state modified UNIFAC interaction parameter between groups m and n modified UNIFAC interaction parameter between groups m and n objective function auxiliary property for component i molar Gibbs free energy pressure coefficient of equation in the modified Huron-Vidal mixing rule van der Waals surface area of component i van der Waals surface area of subgroup k van der Waals volume of component i van der Waals volume of subgroup k absolute temperature molar volume of mixture auxiliary property for component i empirically modified V/-value mole fraction of component i in the liquid phase group mole fraction of group m in the liquid phase mole fraction of component i in the vapor phase mole fraction of component i in the phase compressibility factor
6.1. Greek symbols a ai yi q~v qbgL Fk F~ ;) L,~~) ~9,. ~mn
auxiliary property for mixture auxiliary property for component i activity coefficient of component i fugacity coefficient for component i in the vapor phase fugacity coefficient for component i in the liquid phase group activity coefficient of group k in the mixture group activity coefficient of group k in the pure component i number of structural groups of type k in molecule i surface fraction of group m in the phase modified UNIFAC group interaction parameter between group m and n
6.2. Superscripts C E i R
combinatorial part excess quantity component i residual part
J. Li et al. / Fluid Phase Equilibria 131 (1997) 107-118
117
6.3. S u b s c r i p t s
calc exp i k m n
calculated quantity e x p e r i m e n t a l quantity component i group k group m group n
Acknowledgements T h e authors thank E l f A q u i t a i n e P r o d u c t i o n Society and Services o f Cultural and Scientific C o o p e r a t i o n o f the F r a n c e E m b a s s y in C h i n a for the financial support.
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118
[t5] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
[26]
[27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
J. Li et al. / Fluid Phase Equilibria 131 (1997) 107-118
H. Huang and S.I. Sandier, Ind. Eng. Chem. Res., 32 (1993) 1498. K. Tochigi, Fluid Phase Equil., 104 (1995) 253. K. Fischer and J. Gmehling, Fluid Phase Equil., 112 (1995) 1-22. S. Dahl, Phase Equilibria for Mixtures Containing Gases and Electrolytes, Ph.D. Dissertation, Technical University of Denmark, 1991. G. Soave, Equilibrium constants from a modified Redlich-Kwong equation of state, Chem. Eng. Sci., 27 (1972) 1197. D.Y. Peng and D.B. Robinson, A new two constant equation of state, Ind. Eng. Chem. Fundamen., 15 (1976) 59. M.L. Michelsen, A modified Huron-Vidal mixing rule for cubic equations of state, Fluid Phase Equil., 60 (1990) 213-219. P.M. Mathias and T.W. Copeman, Extension of the Peng-Robinson equation of state to complex mixtures: Evaluation of the various forms of the local composition concept, Fluid Phase Equil., 13 (1983) 91-108. T.E. Daubert and R.P. Danner, DIPPR data compilation, AIChE, New York, 1986. J.A. Nelder and R.A. Mead, A simplex method for function minimization, Comput. J., 7 (1965) 308. P.C. Gillespie and G.M. Wilson, Vapor-liquid equilibrium data on water-substitute gas components: N2-H20, H 2- H20, C O - H20, H 2 - C 0 - H20 and H2S- H20, Gas Processors Association Research Report No.41, Wiltec Research Co., Provo, LIT, April, 1980. P.C. Gillespie and G.M. Wilson, Vapor-liquid equilibria, water-methane, water-carbon dioxide, water-hydrogen sulphide, water-pentane, Gas Processors Association Research Report No.48, Wiltec Research Co., Provo, LIT, April, 1982. S. Takenouchi and G.C. Kennedy, The binary system H 2 0 - C O 2 at high temperatures and pressures, Am. J. Sci., 262 (1964) 1055-1074. R. Crovetto, R. Fernandez-Prini and M.L. Japas, J. Chem. Phys., 76 (1982) 1077-1086. R.G. Sultanov, V.G. Skripka and A.Y. Namoit, Zh. Fiz. Khim., 46 (1972) 2160. N.L Yarym-Agaev, R.P. Sinyavskaya, I.L. Koliushko and L.Y. Levinton, Zh. Prikl. Khim., 58(1) (1985) 165-168. A. Danneil, K. Todheide and E.U. Franck, Chem. Ing. Tech., 13 (1967) 816-821. A.S Kertes, IUPAC Solubility Data Series, Pergamon Press, New York, 1989. H.H. Reamer, B.H. Sage and W.N. Lacey, Ind. Eng. Chem., 44 (1952) 609-615. L.C. Price, Am. Ass. Pet. Geol. Bull., 63 (1979) 1. O.L. Culberson and J.J. McKetta, J. Pet, Tech. Trans. Aime Pet. Div., 189 (1950) 319-322. R. Kobayashi and D.L. Katz, Ind. Eng. Chem., 45 (1953) 440-446. T.D. O'Sullivan and N.O. Smith, The solubility and partial molar volume of nitrogen and methane in water and in aqueous sodium chloride from 50 to 125° and 100 to 600 atm., J. Phys. Chem., 74 (1970) 1460-1466.
ELSEVIER
Fluid Phase Equilibria 131 (1997) 107-118
Prediction of the solubility and gas-liquid equilibria for gas-water and light hydrocarbon-water systems at high temperatures and pressures with a group contribution equation of state Jiding Li ", Isabelle Vanderbeken b, Suyu Ye b, Herve Carrier b, Pierre Xans h,, a Department of Chemical Engineering, Tsinghua University, Beijing 100084, China b Laboratoire Haute Pression, Universit£ de Pau et des Pays de l'Adour, 64000 Pau, France
Received 7 February 1996; accepted 2 December 1996
Abstract A group contribution equation of state has been proposed. The model is based on the Weidlich-Gmehling modified UNIFAC and Soave-Redlich-Kwong equation of state, in which the modified Huron-Vidal excess Gibbs free energy mixing rule is used, the residual term of the modified UNIFAC is changed a little and the combinatorial term is improved. Nine interaction parameters between gas groups CO 2, CO, N 2, H 2, HES, CH4, C2H6, C3H 8, C4nlo and water are fitted. The model parameters have been used to predict solubility and gas-liquid equilibria for seven gas-water, and thirty-nine light hydrocarbon-water data sets in large temperature and pressure ranges (278-637 K, 1-1972 bar). The predicted results are in good agreement with the experiments, which makes it possible that the real phase behavior at high temperatures and pressures for gas-water and light hydrocarbon-water systems which are of interest to petroleum and natural gas exploitation industries, can be predicted reliably. © 1997 Elsevier Science B.V. Keywords: Theory; Vapour-liquid equilibria; Equation of state; Excess function; Method of calculation; Activity coefficient
1. Introduction
Gas (CO 2, CO, N2, H 2, H2S)-water and light hydrocarbon (CH4, C2H6, C3Hs, nC4Hi0)-water systems are quite common in many industrial processes, especially in petroleum and natural gas exploitation, petroleum refining and coal gasification processes. For the design and optimization of these processes information on the solubility and phase behavior of these systems at high temperatures and pressures has to be known since the size of equipment and process condition (T, P) are mainly * Corresponding author. 0378-3812/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. Pll S0378-3812(96)03234-7
108
J. Li et a l . / Fluid Phase Equilibria 131 (1997) 1 0 7 - 1 1 8
determined by this knowledge. Measuring all data needed for the design is very difficult or impossible in some cases at high temperatures and pressures. It thus is necessary to use a general thermodynamic model so that the solubility and phase equilibria for these systems can be confidently predicted where data do not exist or where they are of poor quality. The use of group contribution models is an effective method to predict phase equilibria with the help of limited experimental data or when reliable experimental information can not be found. Since Huron and Vidal [1] proposed an approach that incorporates an excess free energy (gE) model into the mixing rules for the Redlich-Kwong equation of state, group contribution models like UNIFAC [2], ASOG [3], modified UNIFAC [4-6] coupled with equations of state, have been used in high pressure systems [7,8], in polymer solution systems [9,10], and in multi-phase equilibria systems [11] and so on. However, up to now most of the research on group contribution equations of state is focused on the modification of the Huron-Vidal excess Gibbs free energy mixing rule [12,13,7,14-16]. The research on the improvement or the modification of the group contribution models in order to obtain a good g e expression to be used with cubic equations of state has received only minor attention. In addition, the results predicted by current group contribution equations of state at high temperatures and pressures for some systems, especially for light hydrocarbon-water systems [17] cannot meet the needs of the design and optimization of the industrial processes because of the quite large deviations. The purpose of this work is (1) to modify group contribution models in order to develop a group contribution equation of state, i.e., an equation of state-group contribution model applicable for gas-water and light hydrocarbon-water systems at high temperatures and pressures, (2) to fit the model parameters between nine gas groups (CO 2, CO, N 2, H 2, H2S, Ch 4, C z H 6, C3H 8, nC4H~0 ) and water group, (3) to use these parameters to predict solubility and gas-liquid equilibria for seven gas (CO 2, Co, N 2, H 2, HzS)-water and thirty-nine light hydrocarbon (CH 4, C z H 6, C3H 8, nC4H~0)-water data sets in large temperature and pressure ranges (278-637 K, 1-1972 bar). The results for these systems have not been published in many papers on group contribution equations of state. Only a few results can be found in the papers on the MHV2 model [18,8] in a small pressure range (7-200 bar).
2. A group contribution equation of state As well known, the Soave-Redlich-Kwong equation of state [19] describes the phase behavior of most systems as effectively as the Peng-Robinson equation of state [20]. Moreover, the expression of Soave-Redlich-Kwong is a little simpler than that of Peng-Robinson. In this paper the Soave-Redlich-Kwong equation of state is used. The Soave-Redlich-Kwong equation of state may be written in compressibility factor form: Z 3 -
Z 2 -I-
(A - B - B2)Z-AB
= 0
where Pv
aP A = - - B RT (RT) 2
Z~-----
bP RT
where the mixture b-parameter is calculated by the conventional linear mixing rule.
(1)
J. Li et al. / Fluid Phase Equilibria 131 (1997) 107-118
109
The mixture a-parameter in Eq. (1) is given by the modified Huron-Vidal excess Gibbs free energy mixing rule [13,21]:
ql(ol__~iZiO~i)+q2(ol
2
~iZiOt?) = - R T +
b
in Eq. (2), q~ and q2 are the coefficients of the equation ql = - 0 . 4 7 8 used in this w o r k . a and a i are defined as
a
bRT
and q2 = - 0 . 0 0 4 7
that are
ai b i RT
where a~ is the parameter of pure component i and is obtained from a~=0.42748
R2T2 Pci [f(Tri)]2
(3)
where Tr = T / T c, f(Tr) is given by Mathias and Copeman [22] f(Tr) :
1 q-- C l ( 1 -
/(Tr)
1 + Cl(1 - ~-~) (Tr > 1)
=
~ r r ) "[- C 2 ( 1 -
~ r r ) 2"t- C 3 ( 1 -
}/Trr) 3 (Tr ~ 1)
(4a)
(48)
in Eqs. (4a) and (4b) constants C 1, C 2 and C 3 are estimated using the vapor pressure of the pure components [23]. For hydrogen and carbon monoxide, C~ is calculated directly from the acentric factor to [19]: C l = 0.48 + 1.574o9 - 0.176to 2
(5)
The values of the constants C i and the critical point constants of the pure components used in this paper are listed in Table 1.
Table 1 The Ci-constants (Eqs. (4a) and (4b)) and critical point constants of pure components Component
C1
C2
C3
Tc (K)
Pc (bar)
Hydrogen Nitrogen Carbon monoxide Carbon dioxide Methane Ethane Propane n-Butane Hydrogen sulphide Water
0.1332 0.5427 0.5836 0.8653 0.5472 0.6853 0.7726 0.8487 0.5507 1.0873
0.0000 - 0.0524 0.0000 - 0.4386 - 0.3992 - 0.4284 - 0.5090 - 0.5520 0.4534 - 0.6377
0.0000 - 0.3381 0.0000 1.3447 0.5751 0.7382 1.0306 1.0774 - 0.5841 0.6345
33.2 126.2 132.9 304.2 190.6 305.4 369.8 425.2 373.2 647.3
12.97 33.94 34.96 73.76 46.00 48.48 42.46 38.00 89.37 220.48
110
J. Li et al./Fluid Phase Equilibria 131 (1997) 107-118
In Eq. (2), g E(0) is the mole excess Gibbs free energy at the zero pressure. It is assumed that: E(O)
g RT
(6)
= E z/In'y/ i
where Yi is the activity coefficient of the component i, and Yi is obtained from the modified UNIFAC model. In the modified UNIFAC model, the activity coefficient is expressed in the following form:
lny i = lny/c + lny/R
(7)
The first term on the right-hand side of Eq. (7) represents the combinatorial part of the activity coefficient and the second term refers to the residual part. In the modified UNIFAC-Lyngby [5], the combinatorial part is described as lny c = 1 -
(8)
~ i qt- I n
where
/,.}/3 ~i =
ri = Y'~ v(~i)Rk
E J
In the modified UNIFAC-Dortmund [4], it is given by Vi + in V/t lny/c = 1 - V / + INV.'- 5qi 1 - ~ii Fi ]
(9)
where r}/4 Vi' =
Exjr:3./4
ri gi -
J
q~
- -
Fi = -
Y'.xjrj J
-
Y'.xjqj
qi = E
k
v(i)Qk
J
In Eqs. (8) and (9), R k and Qk are van der Waals volume and surface area of subgroup k respectively. They can be calculated directly [2] and don't need to be fitted. In this work, it is found that modification of the combinatorial term may improve the final results even though the number of the parameters which has to be fitted to experimental data has not been increased. In this paper the combinatorial term is modified as lny/c
---- I - V,.'+ InV/-
5qi
|
-
-
-~i -'l-
In Eq. (10) all symbols have the same meaning as in Eq. (9). It should be noted that Eq. (10) is slightly different from Eq. (9) in the last term, and different from Eq. (8). The group assignment and the values of R and Q used in this paper are listed in Table 2.
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J. Li et al. / Fluid Phase Equilibria 131 (1997) 107-118
Table 2 The group assignment and R and Q values Main Group
Subgroup
R
Q
H2 N2 CO CO 2 CH 4 C2H 6 C3H 8 C4HIo
H2 N2 CO CO 2 CH 4 C2H 6 C3H 8 nC4H lO iCaH ,o H2S H 20
0.8320 1.8680 2.0940 2.5920 2.2440 3.6044 4.9532 6.3020 6.3020 2.3330 0.9200
1.1410 1.9700 2.1200 2.5220 2.3120 3.3920 4.4720 5.5520 5.5520 2.3260 1.4000
HzS H 20
The following relations are used to represent the residual part in this paper: lny R =
E , i'(lnS-lnF
(11)
i')
k
O.,~k m~Omll3mk) -- ~m E On~I~nm ,.
lnFk= Qkl-ln(
(12)
n
where U(j) g....a m "~j J
Q,. Xm Ore=
E Qnx°
X,.=
n
~m. = exp
E E ,?xj j
n
(amn+bmnT)
(13)
T
where a., n and bran are UNIFAC interaction parameters between groups m and n, and estimated from experimental data. Eq. (13) is close to the equations which are used by Weidlich and Gmehling for their modified UNIFAC, in which
~.,n =
( amn WbmnTW CmnT2 ) exp -
in which in many cases
~m,,=exp -amn-~
T
Cm. =
0 and differs Larsen's modified UNIFAC, where
- b., ~
~
cm~
ln-~-+
112
J. Li et al. / Fluid Phase Equilibria 131 (1997) 107-118
3. Estimation of parameters The interaction parameters for the group contribution model proposed in this work were determined by minimization of the following objective function using the Simplex-Nelder-Mead method [24]:
[
F ( a .... a,,m,bm,,,b,,m) = Y'~ Y'~ gl(Yexp--Yca'c) 2 + g2 np
nc
txpcc)21 Pexp
(14)
where n p and n c a r e the number of data points and components respectively, g refers to a weighting factor. The subscript "exp" means experimental and "calc" calculated. The Yca~c and ecalc are determined by the iterative method. Experimental (T, P, x, y) vapour-liquid equilibrium data were used to estimate the interaction parameters between ten main groups. Only propane-water interaction parameters were determined using the incomplete (solubility) data (T, P, x). The data sets selected to fit the parameters are given in Table 3. It should be noted that for the group interaction C H a - H 2 0 , C O - H 2 0 , CO2-H20, H 2 S - H 2 0 , N 2 - H 2 0 and H2-H~O the experimental data used to estimate the interaction parameters are suitable but for the group CzH6-H20, C3H8-H20 and nC4H1o-H20 new data bases are desirable. The fitted interaction parameters between ten groups are given in Table 4. It is known that using parameters fitted from experimental data (Table 3), may lead to erroneous results when the model is used outside temperature and pressure range of the data base. In this paper the reliability of the parameters used to predict vapor-liquid equilibria at the extrapolated temperature and pressure has been checked and judged from the pure component properties and by means of the typical
Table 3 The data sets used to fit the interaction parameters System T (K) H 2-H20
N2 - H
2°
CO-H 20 CO2-H 2°
CH 4-H 20
C2H6-H20
C3H8-H20
nC4Hlo-H20 H2S-H20
310-588 310-588 310-588 288-533 383-623 297-518 323-588 423-633 298-338 473-573 288-410 285-422 310-410 628-637 310-588
P (bar)
Reference
3-137 3-137 3-137 6-202 100-1500 13-64 13-169 98-1078 25-125 27-224 1-34 5-192 3-40 255-1102 3-208
[25] [25] [25] [26] [27] [28] [26] [29] [30] [31] [32] [32] [33] [31 ] [25]
J. Li et al. / Fluid Phase Equilibria 131 (1997) 107-118
113
Table 4 Modified UNIFAC (proposed in this paper) group interaction parameters
i
j
aij
aji
H2 N2 CO CO 2 CH 4 C2H 6 C 3H 8
H 20 H 20 H 20 H 20 H 20 H20 H 20
C4H 1o
H20
H2S
H20
411.4 2280 2324 624.0 2435 1478 2699 2965 1019
1044 403.8 452.0 257.3 477.3 324.9 326.6 420.4 349.5
big -
3.891 3.607 2.878 0.320 3.057 1.509 3.559 3.773 0.8687
bji
T (K)
P (bar)
- 0.3553 0.5907 0.05167 0.01808 - 0.03417 0.2567 0.1518 - 0.1765 - 0.3832
275-570 275-570 275-570 275-620 275-600 275-550 275-420 275-580 275-580
1-500 1-600 1-500 1- 1500 1- 1900 1-600 1-400 1-1000 1-500
binary-system properties. So most of the parameters can be used with confidence in the given temperature and pressure range shown in the last two columns in Table 4. 4. Results and discussion The overall results of 46 data sets predicted by the model proposed in this work are summarized in Table 5, where AP IPe~p- P¢.lcl
P ( % ) = 100×
P~xp
' Ay=lYexP-Ycalc[
Table 5 shows that in the temperature range 278-637 K, and pressure range 1-1972 bar the total Table 5 Deviations between experimental and predicted gas-liquid equilibria
System
P (bar)
Data points
A p/p
I. Data type one: T, P, x, y H 2- H 2 0 310-588 Nz-H20 310-588 CO-H20 310-588 CO 2 - H 2O 288-623 CH 4-H 2° 279-633 C 2H6 - H 2° 473-573 nC 4H Io - H 20 310-637 H 2S - H 2 ° 310-588 Mean deviation
3-138 3-138 3-138 6-1500 13-1079 13-241 3-1103 3-206
17 16 18 170 93 24 12 28 378
2.44 4.96 4.26 6.66 5.81 26.73 18.82 4.27 7.56
II. Data type two: T, P, x N 2 - H 20 324-398 CH 4 - H 2O 298-627 C2H6-H 20 310-444 C3Hs-H20 278-422 nCaHt0-HzO 310-410 Mean deviation
91-515 3-1972 4-685 1-192 1-34
18 243 80 186 49 576
5.94 11.35 9.68 18.16 11.92 13.20
T (K)
(%)
Ay 0.0043 0.0038 0.0034 0.0245 0.0381 0.0573 0.0609 0.0077 0.0271
114
J. Li et al. / Fluid Phase Equilibria 131 (1997) 107-118
160( 120 I
1200 "Z
eo
°8oQ
n 4(
400
0
I
0
0.50 x I , Yl
0
0.40 Xltyl
0.80
D
c i 120
8o ~-
,
I00~
4o~
I 1
,
0 0
0.50 x.yf
~
o! p
I
i
I
0,40
0
0.80
x I , Yl
Fig. 1. Experimental and predicted gas-liquid equilibria. (A) Carbon monoxide(1)-water(2) at [] 346.48 K; (B) carbon dioxide(l)-water(2) at [] 383.15 K, zx 523.15 K, +543.15 K, * 573.15 K, O 623.15 K; (C) hydrogen(l)-water(2) at [] 366.48 K, zx 477.59 K; (D) hydrogen sulphide(1)-water(2) at [] 310.93 K. ( - - ) model proposed in this work, (•, zx, + , *, O) experimental values from (A) Gillespie and Wilson [25], (B) Takenouchi and Kennedy [27], (C) Gillespie and Wilson [25], (D) Gillespie and Wilson [25].
20OO 160C
"~ 1200 .o
~" e00 400
y z
o
i
0.04
8
80( 60(
~4~
i
0.002 x,I
0,08
x1
0.OO4
D n
60o
~,00 I1 100
0
0
o_ 2 0 0
0.0004 xq
0.0008
0
i
I 0.002 11
I 0.004
Fig. 2. Experimental and predicted solubility of gases in water. (A) Methane(l)-water(2) at [] 479.15 K, zx 553.15 K, (D 589.15 K; (B) ethane(1)-water(2) at [] 310.93 K, Lx 377.59 K, C) 444.26 K; (C) propane(1)-water(2) at [] 360.93 K, zx 399.82 K, O 422.04 K; (D) nitrogen(l)-water(2) at [] 375.65 K. ( - - ) model proposed in this work, (•, LX, O) experimental values from (A) Price [34], (B) Culberson and McKetta [35], (C) Kobayashi and Katz [36], (D) O'Sullivan and Smith [37].
J. Li et al. / Fluid Phase Equilibria 131 (1997) 107-118
115
mean relative deviation between experimental and predicted pressure in this work is not more than 10.97% and the total mean absolute deviation of mole fractions in the vapor phase is 0.0271. These results show that the model proposed in this paper can be used reliably to predict gas-liquid equilibria for gas (CO z, CO, N 2, H 2, H2S)-water and light hydrocarbon (CH4, C2H 6, C3H 8, nCnHlo)-water systems in large temperature and pressure ranges. It should be pointed out that the above results include all data sets of C2H6-H20, C3H8-H20 and nCaHI0-H20 systems in which the experimental results for some points are not reasonable. A few typical prediction results of gas-liquid equilibria are presented in Fig. 1, where Fig. I(A) and Fig. I(D) show isotherms at 346.48 K and 310.93 K for the carbon monoxide-water and hydrogen sulphide-water systems respectively, Fig. I(B) describes in detail gas-liquid equilibrium behavior for the carbon dioxide-water system (temperature from 383 to 623 K, pressure from 100 to 1500 bar) and Fig. I(C) shows isotherms at 366.48 K and 477.59 K for the hydrogen-water system. In order to test the ability of the model to predict gas solubility we have studied 26 data sets. The predicted results of solubility of gases (methane, ethane, propane and nitrogen) in water are shown in Fig. 2. It can be seen that the predicted results are in good agreement with the experimental ones, not only for the high temperature and pressure cases (Fig. 2(A), methane(1)-water(2) system, temperature up to 590 K, pressure up to 1950 bar), but also at other cases (Fig. 2(B), ethane(1)-water(2) system, 310-444 K, 10-700 bar; Fig. 2(C), propane(1)-water(2) system, 360-420 K, 5-200 bar; Fig. 2(D), nitrogen(l)-water(2) system, 375 K, 100-700 bar).
5. Conclusion A group contribution equation of state has been proposed. This model is based on the modified Weidlich-Gmehling UNIFAC and Soave-Redlich-Kwong equation of state, in which the modified Huron-Vidal excess Gibbs free energy mixing rule is used, the residual term of the modified UNIFAC is changed a little: all the interaction parameters cij are set equal to zero in all cases, and the combinatorial term is improved. With the help of the experimental data the interaction parameters between nine gas groups CO 2, CO, N2, H2, H2S, CH4, C2H6, C3H8, C4HIo and water group are fitted. The model parameters have been used to predict solubility and gas-liquid equilibria for seven gas-water, and thirty-nine light hydrocarbon-water data sets in large temperature and pressure ranges (temperature from 278 to 637 K, pressure from 1 to 1972 bar). The total mean relative deviation between experimental and predicted pressure is no more than 10.97% and total mean absolute deviation of mole fractions in the vapor phase is 0.0271. These results show that phase equilibria at high temperatures and pressures for gas (CO 2, CO, N 2, H2, H2S)-water and light hydrocarbon (CH4, C2H6, C3H 8, nC4Hlo)-Water systems, which are of interest to petroleum and natural gas exploitation, petroleum refining and coal gasification industries, can be predicted by this model with the desired accuracy.
6. List of symbols a
ai
parameter of mixture in equation of state parameter of pure component i in equation of state
116
a,,n b b~ bin,
Cm, F Fi g P q q~ Qk r~ Rk T v V~ Vi' xi Xm y~ z~ Z
J. Li et a L / Fluid Phase Equilibria 131 (1997) 107- 118
modified UNIFAC interaction parameter between groups m and n parameter of mixture in equation of state parameter of pure component i in equation of state modified UNIFAC interaction parameter between groups m and n modified UNIFAC interaction parameter between groups m and n objective function auxiliary property for component i molar Gibbs free energy pressure coefficient of equation in the modified Huron-Vidal mixing rule van der Waals surface area of component i van der Waals surface area of subgroup k van der Waals volume of component i van der Waals volume of subgroup k absolute temperature molar volume of mixture auxiliary property for component i empirically modified V/-value mole fraction of component i in the liquid phase group mole fraction of group m in the liquid phase mole fraction of component i in the vapor phase mole fraction of component i in the phase compressibility factor
6.1. Greek symbols a ai yi q~v qbgL Fk F~ ;) L,~~) ~9,. ~mn
auxiliary property for mixture auxiliary property for component i activity coefficient of component i fugacity coefficient for component i in the vapor phase fugacity coefficient for component i in the liquid phase group activity coefficient of group k in the mixture group activity coefficient of group k in the pure component i number of structural groups of type k in molecule i surface fraction of group m in the phase modified UNIFAC group interaction parameter between group m and n
6.2. Superscripts C E i R
combinatorial part excess quantity component i residual part
J. Li et al. / Fluid Phase Equilibria 131 (1997) 107-118
117
6.3. S u b s c r i p t s
calc exp i k m n
calculated quantity e x p e r i m e n t a l quantity component i group k group m group n
Acknowledgements T h e authors thank E l f A q u i t a i n e P r o d u c t i o n Society and Services o f Cultural and Scientific C o o p e r a t i o n o f the F r a n c e E m b a s s y in C h i n a for the financial support.
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