R = gasconstant T = temperature, K W ( r ) = spherically symmetric cell potential, ergs
Xw Y,
= mole fraction of liquid water = mole fraction of component j in vapor phase
Greek Letters = binary interaction constant Hnli = fraction of cavities type m occupied by component j p = chemical potential '!Jivr' = potentia' difference Of water between liquid phase and empty hydrate &wH = chemical potential difference of water between hydrate phase and empty hydrate per water molecule in hy'nl = number Of cavities type drate N
Clausen, W. F., J. Cbem. Pbys., 19, 1425 (1951). Deaton, W. N., Frost, E. M., pp 122-128, Proceedings of American Gas Association Houston, Texas, May 1940. Deaton, W. N.. Frost, E. M., U.S. Bureau of Mines, Monograph 8 (1949). Hammerschmidt. E. G., Am. Gas Association Monthly, 18, 273 (1936). 851 (1934). Hammerschmidt, E. G., lnd. Eng. Cbem.. 26 (8), Jhaveri. J., Robinson, D. B., Can. J, Cbem. Eng., 43, 75 (1965). Katz, D. L., J. Pet. Technol., 23, 419 (1971). McKoy. V.. Sinanoglu, O., J. Cbem. Pbys., 38, 2946 (1963). Otto, F . D., Robinson, D. B., A.I.Ch.E. J., 6, 602 (1960). Parrish, W. R., Prausnitz. J. M.. lnd. Eng. Cbem., Process Des. Dev., 11, 26 (1972). Peng, D.-Y.,Robinson, D. B., ind. Eng. Chem., Fundam., 15, 59 (1976). Robinson, D. B., Mehta, B. R., J. Can. Pet. Techno/., 49 (5), 642 (1971). Stackelberg, M. Von, Muller, H. R.. Z.Elektrocbem., 58, 25 (1954). Stackelberg. M. Von, Muller, H. R., J. Chem. Pbys., 19, 1319 (1951). van Der Waals, J. H., Platteeuw, J. C., Adv. Cbem. Pbys., 2, 1 (1959). Wilcox, W. I., Carson, D. E., Katz, D. L., Ind. Eng. Cbem., 33, 662 (1941). Wu. B.-J.. Robinson. D. 13.. No. H.-J.. J. Chem. Thermodyn., 8. 461 (1976).
Literature Cited Received f o r review December 22, 1975 Accepted June 21,1976
Carson, D. 8.. Katz, D. L., Trans. A.I.M.E., 146, 150 (1942). Chersky, N , Makagon. Ya. F., OilGas lnt., IO, (8),82 (1970)
Vapor-Liquid Equilibria by the Pseudocritical Method Joseph Joffe Department of Chemical Engineering and Chemistry, New Jersey institute of Technology, Newark, New Jersey 07102
The pseudocritical method of Lee and Kesler has been extended to the calculation of component fugacity coefficients and of vapor-liquid equilibria. The Lee-Kesler mixing rules have been modified by the introduction of binary interaction constants, which are evaluated from experimental pseudocritical constants. The critical compressibility factor is introduced as a correlating fourth parameter. The resulting component fugacity coefficients agree with experimental values and with experimentally determined vapor-liquid equilibrium ratios. The proposed method may be applied to systems with any number of components. It is well adapted to computer programming; being based on the modified BWR equation of Lee and Kesler.
Introduction
Proposed Pseudocritical Method
The use of the pseudocritical method for the calculation of component fugacities in liquid as well as in gaseous mixtures and the application to vapor-liquid equilibria have been proposed by a number of investigators, among them Pitzer and Hultgren (1958), Leland et al. (1962), Leach et al. (1968), and Leyendecker and Gunn (1972). Since in practice vaporliquid equilibrium calculations are carried out on a computer, it is necessary to have a precise equation of state describing the behavior of the reference substance or substances, or else a computer capable of storing tables of compressibility factors, fugacity coefficients, and enthalpy deviations of the reference substances and a program for precise interpolations. Thus, Leyendecker and Gunn (1972) utilized the well-known Pitzer-Curl tables (Pitzer et al., 1955,1957;Curl and Pitzer, 1958) in their computer program. Lee and Kesler (1976) have recently published an extended and improved version of the Pitzer-Curl tables, based on a modified Benedict-Webb-Rubin (BWR) equation of state, in a convenient form for computer use. In applying their correlation to mixtures, Lee and Kesler have used the pseudocritical method and have proposed a set of mixing rules for calculating the pseudocritical constants of a mixture.
In the current study the Lee-Kesler correlation has been applied to vapor-liquid equilibrium calculations. The LeeKesler mixing rules for mixtures have been modified by introducing two binary interaction constants for each pair of components and by treating the pure component critical compressibility factor as an independent fourth parameter. The proposed modified mixing rules are as follows:
298
lnd. Eng. Chem.,
Fundam., Vol. 15, No. 4, 1976
2,'
=
2 XJCl 1
P,' = zc'RTc'/Vc'
Lee and Kesler have proposed the relationship, z,/ = 0.2905
Table I
Ta, O
Component Methane Propane n -Butane Carbon dioxide
R
P,,, lb/in.2
V,,, cm3
C1
ai
667.2 615.8 551.1 1069.9
99.0 203.0 255.0 94.0
0.2874 0.2803 0.2741 0.2742
0.007 0.145 0.193 0.225
343.1 665.6 765.4 547.6
Table 11. n-Butane-Carbon Dioxide System Data
Mole fraction n-butane
Calculated values
P,‘, 1 b h 2
V:, cm3
Vc12, cm3
V:, cm3
Tc12,OR
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
0.1000 0.3000 0.5000 0.7000 0.9000
558.4 591.3 635.6 685.3 737.7
937.0 771.0 671.5 609.8 565.7
109.47 140.86 173.84 206.38 239.47
171.01 171.07 173.18 173.74 177.67 173.33
109.89 141.81 173.92 206.21 238.69
567.8 564.1 564.0 563.8 560.0 563.9
557.3 591.2 635.5 685.3 738.2
931.6 765.7 671.1 610.3 567.9
T:,
OR
Av values:
- O.O85w~,instead of eq 4. Such a Drocedure introduces a discontinuity into the relation between pseudocritical pressure and composition a t either end of the composition range and is therefore avoided in the present study. In eq 2 and 3, when i = j , V‘,, and Tcr,are the critical properties of the pure i t h component. When i # j , VCiiand Tcrlare binary constants to be established from binary mixture data. Equations 2 and 3 are analogous to the combining rules for van der Waals’ constants b and a , respectively, since we can write b, = (%) V,, and ai = (9/8)RTc1Vc1. Lee and Kesler have utilized the so-called Lorentz rule for vci,:
and the simple geometric mean relation for T C L J :
Tcq =
(8)
As pointed out by Reid and Leland (1965), a better combining rule than eq 8 is: but even this rule is only valid for mixtures of paraffin hydrocarbons (Prausnitz and Chueh, 1968). In the present study eq 7,8, and 9 have not been used and the binary constants V,, and Tcr,have been determined from binary mixture data. Values of pure component properties used in the current work are shown in Table I. Following Pitzer and Hultgren (1958), binary constants have been established in the present study from experimental values of the pseudocritical constants of binary mixtures of known composition. A relatively simple procedure for obtaining the pseudocritical constants of a gaseous mixture is based on the existence of minima when experimental isothermal curves of compressibility factor are plotted against pressure. These minima may then be correlated with the minima on a generalized compressibility factor chart. This procedure was first employed by the author (Joffe, 1949) and was later elaborated and extended to the three-parameter Law of Corresponding States by Prausnitz and Gunn (1958) and by Case and Weber (1960). Bosler (1967) has analyzed a number of systems by the methods of Prausnitz and Gunn and of Case and Weber, and has provided sets of pseudocriticals a t various compositions. These data have been utilized in the
T:,
O R
P:, lbhn.2
current study to evaluate binary constants for selected systems. The procedure is illustrated with the help of Table I1 for the n-butane-carbon dioxide system. In columns 2 and 3 are entered the pseudocritical temperatures and pressures determined by Bosler (1967) using the method of Prausnitz and Gunn (1958). Equations 4 and 5 are used along with the data of columns 2 and 3 to calculate the pseudocritical volume a t each composition. The values are entered in column 4.The pseudocritical volume is entered into eq 2 and the binary parameter Vc12is calculated a t each composition. The average value of this parameter, Vc12 = 173.33, is taken as the value of this binary constant. Using the value Vc12= 173.33 and eq 2, the pseudocritical volumes are recalculated. The results are entered in column 6. The pseudocritical temperatures in column 2 along with the calculated pseudocritical volumes in column 6 are substituted into eq 3 and the parameter Tc12is evaluated a t each composition. I t is shown in column 7. The average value of this parameter, Tc12 = 563.9, is taken as the second binary constant for this system. The two binary constants are substituted into eq 3 to obtain calculated values of the pseudocritical temperature a t each composition. These values are shown in column 8. The values of V,’ and T,‘ in columns 6 and 8 are substituted into eq 5 and the pseudocritical pressure is calculated a t each composition with the help of eq 4. These values are entered in column 9. A comparison of the pseudocritical parameters in columns 2 and 3 with the calculated values in columns 8 and 9 gives some indication of the validity of the mixing rules, eq 2, 3, and 4, for this system. I t should be noted that values for the binary constants obtained with eq 7 and 8 are Vc12 = 161.35 and Tc12 = 647.4, differing considerably from the ones which fit Bosler’s data. The expression for Tc12 recommended by Prausnitz and Chueh (1968)
with k = 0.16, appears to over-correct the geometric mean, yielding a value Tc12 = 543.9. Equations 7 and 9 yield a value Tc12 = 621.2 which is too high. Since the equations for component fugacities, to be considered below, involve the derivatives of the pseudocritical temperature and pressure with respect to the composition Ind. Eng. Chern., Fundarn., Vol. 15,No. 4 , 1976
299
variable, it is essential that the mixing rules reproduce faithfully the relationships between the pseudocritical parameters and composition.
Application to Vapor-Liquid Equilibria The basic thermodynamic requirement for vapor-liquid equilibrium is that the fugacity of each component is the same in both phases: fiG
= fiL
Since the pseudocritical pressure, P l , does not appear directly in the mixing rules, eq 2,3, and 4, it is convenient to eliminate its derivative from eq 21. This may be done by utilizing the relation among the pseudocritical constants, eq 5 , as pointed out by Leach et al. (1968). There follows from eq 5
(11)
The fugacity coefficient of a component in a mixture, &, is defined as the ratio of the fugacity to its partial pressure. Replacing fugacities with fugacity coefficients, eq 11 becomes, 4 i ~ y i P= q+~xiP,or better $JiL/@iG
(21)
= Yi/xi = Ki
(12)
where Ki = yJxi is by definition the vapor-liquid equilibrium ratio, the so-called K constant. The problem of predicting vapor-liquid equilibrium resolves itself into finding the effect of system temperature, pressure, and composition on the fugacity coefficients of the components present in the two phases. Following Van Ness (1964), the fugacity coefficient of a component in either phase can be related to that of the mixture by the equation
Substituting eq 22 into eq 21
dTc' z - 1av,' d In 4M - - (H* - H ) + ( z - 1 ) R T-+--
ax k
RTT,'
dxk
vc'
axk
Equation 23 may also be written:
-dln4M ax k
U*-UaTc' + t - l d v , ' RTT,' d X k v,' axk
. ~ - l a t ~ ~ t c f dxk
+ wk (In @M)(')
The partial derivatives are evaluated from the mixing rules, eq 2, 3, and 4: (25)
In the above expression the partial derivatives are taken a t constant temperature and pressure and with all other mole fractions held constant. As a consequence of the three-parameter theorem of corresponding states we may write: aln4M -=ax k
a l n h .-+aT, aT, axk
.-
a l n h aP, ap, axk
+-.-a l n 4 M &OM
awM
dxk
(14)
All partial derivatives in eq 14 are taken a t constant system temperature and pressure. As previously demonstrated by Gamson and Watson (1944) and by the author (Joffe, 1948), the partial derivatives in eq 14 may be replaced as follows:
(24)
(26)
Substituting eq 24,25,26,and 27 into eq 13, and in view of eq 2,3, and 4, the following expression is obtained for the fugacity coefficient of component j in a multicomponent mixture:
2(2 - 1) 2-1 + ~ v( c~ x l v c l , - v c ~ ) z- C'-
(ZCl
- zc')
+ (w; - wM)(ln 4 ~ ) ( ' ) (28) The expressions for the fugacity coefficients of the components of a binary mixture follow from eq 28:
a In 4 M
=-2
- 1
(19) a_ w M-
(20) - Wk ax k Equation 17 was introduced by Pitzer and Hultgren (1958). Equation 20 follows from eq 6. I t is assumed in the current study that the pseudocritical parameters, T,' and P i , and the acentric factor of the mixture, W M , are functions of composition only, and that they are independent of such other system variables as temperature, pressure, or density. Substituting eq 15 through 20 into eq 14, there follows: 300
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
--2 - 1 c'
(Z,Z
- z,')
+
(WZ
- wM)(ln @MI(')
(30)
The above equations are equally valid for liquid and vapor phases. Conforming to the usual notation for vapor-phase mole fractions, yi would replace xi in the above expressions in vapor-phase calculations.
Table 111. n-Butane-Carbon Dioxide System at 160 OF. Comparison of Calculated Fugacity Coefficients with Experimental Valuesa % Deviation in
n-Butane Pressure, lb/in.* 200 300 400 500 600 700 800 900 1000 1100 1184 (crit. point)
Mole fraction n butane Liquid Gas
Sat. liq.
-
0.955 0.897 0.838 0.778 0.717 0.655 0.591 0.526 0.457 0.382 0.287
0.645 0.464 0.365 0.306 0.268 0.246 0.230 0.220 0.216 0.222 0.287
Av abs. % deviation
Calculated Values Carbon dioxide
Sat. gas
Sat. liq.
A
B
A
B
A
-3.7 0.8
1.2 0.6 0.8 2.5 -3.7 -4.7 -5.5 -4.9 -3.6 0.7 5.5
-0.8 0.6
-0.3 0.6 -0.6 -2.7 3.5 1.3 3.1
-14.8 -8.3
3.1
2.9
0.0
-1.3 -1.4 -1.6 -2.1
-2.5 -3.4 -4.7 -4.9 2.4
0.0
-1.1 -2.0 -2.3 -2.8 -4.8 -5.9 -6.5 -4.9
1.8
0.3 2.1
5.5 2.0
Sat. gas
-
B
A
B
-16.1 -11.3
-0.1 -1.1
-0.6 0.2 0.3 -0.6
- 1.0
n m -1.1
-0.3
-7.4
-5.7 -1.2 0.5 2.5 0.9 0.7 -2.2 -3.0
-0.2 1.9 0.5 0.4
4.7
0.7
C
r
-5.3
-3.8 -2.4 -1.5 -0.9 -0.8 -1.7 4.9
0.5
0.2 -0.9 -1.7
-0.5
0.6 0.2 0.4 0.6 -1.2 -3.0
0.7
Method A, this paper; method B, Leyendecker and Gunn (1972) Comparison of Results Following Leyendecker and Gunn (1972), the n-butanecarbon dioxide system was selected for detailed study, because mixtures of hydrocarbons with carbon dioxide are strongly nonideal (Joffe and Zudkevitch, 1966), and because component fugacities of the equilibrium phases have been determined experimentally for this system (Olds et al., 1949). Since Leyendecker and Gunn used this system to test their procedure, a two-way comparison was carried out. The calculated results of Leyendecker and Gunn and those of the author are compared with experimentally determined fugacities in Table 111. Method A illustrated in the table is the procedure developed in the current study. The binary constants, Vc12 and Tc12,were determined for the n-butane-carbon dioxide system from the pseudocritical data of Bosler (19671, as discussed earlier. At each system temperature and pressure the experimental compositions were substituted into eq 2,3, and 4 along with the critical constants of the pure components and the binary constants. The pseudocritical constants of the mixture were found from eq 3 and 5, the reduced temperature, pressure, and the acentric factor of the mixture (eq 6) were then calculated. The three-parameter tables of Lee and Kesler (1975) were used to establish values of the compressibility factor, enthalpy deviation, and fugacity coefficient of the mixture. These values and the experimental compositions were then substituted into eq 29 and 30 to obtain fugacity coefficients. The calculations were carried out separately for each phase. Method B is the pseudocritical method of Leyendecker and Gunn (1972). Their calculations are also based on experimental compositions of the two phases. A comparison of the deviations of the calculated results from the experimental fugacities indicates the same order of accuracy for the two methods. Method B is not significantly better for this system a t low pressure, even though it is based on the second virial coefficient and should do well a t low-pressure conditions, a t least in the vapor phase (Gunn, 1972). Another system subjected to a test in the current study is the methane-carbon dioxide system, for which the vaporliquid equilibrium data of Donnely and Katz (1954) are available. The binary constants of this system, VcI2 = 92.36 and Tc12 = 430.33, were calculated from Bosler's pseudocritical data in the same manner as for the n-butane-carbon
dioxide system. The fugacity coefficients of each component in each phase were calculated a t the experimental compositions, as discussed above for the n-butane-carbon dioxide system. There are no experimental fugacities with which to compare the calculated values for methane and carbon dioxide. However, the ratio of the fugacity coefficients of each component may be compared in each case with the experimental y / x ratio, since in accordance with eq 12 the two ratios should be equal. The comparison is made in Table IV. Inspection of the results indicates that the agreement between calculated and observed equilibrium ratios is quite satisfactory. Since for purposes of analysis of the proposed method it is desirable to obtain some indication of the accuracy of calculated fugacity coefficients in each phase, a method has been devised to estimate these coefficients from experimental x-y data and without the use of eq 24, which is basic to the proposed method. The alternative procedure may be applied to a binary system. I t rests on the thermodynamically rigorous relationship between the fugacity coefficients of the components and that of the mixture (Gamson and Watson, 1944; Joffe, 1948): 1n ~
I =LX I In @ I L
+ x2 In ~
Z L
(31)
for the liquid phase. Similarly, for the vapor phase: In 4 M G = 3'1 In 4 l C
f
Y2
42G
(32)
The fugacity coefficients of the mixture are determined by the pseudocritical method based on the experimental compositions of the phases. The vapor-phase component fugacity coefficients may be eliminated from eq 32 in terms of the liquid-phase fugacity coefficients with the help of the experimental compositions and eq 12 applied to each component. Equations 31 and 32 are then solved simultaneously to obtain the liquid-phase component fugacity coefficients. The vapor-phase coefficients follow from eq 12. These calculations have been applied to the methane-carbon dioxide system and the results have been compared with the values obtained by the proposed method. Table V shows the percent deviation of fugacity coefficient values obtained by the proposed method from those obtained by the alternate method which by-passes eq 24. The close agreement of the two methods to some extent validates eq 24, and eq 29 and 30 which have been derived from it. Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
301
Table IV. Methane-Carbon Dioxide System Methane mole fraction Temp, OF
Calculated ratio of fugacity coeff.
Obsd y l x ratio
Absolute % dev
Press., 1 b h 2
Liquid
Vapor
CHI
COn
CH4
COn
CH4
COz
870 1052 733 988 591 992 498 899 659 900 755 776
0.103 0.165 0.1095 0.224 0.137 0.322 0.1465 0.465 0.261 0.562 0.611 0.642
0.329 0.387 0.425 0.505 0.605 0.686 0.751 0.805 0.813 0.833 0.878 0.882
3.166 2.360 3.924 2.402 4.589 2.131 5.114 1.716 3.103 1.443 1.420 1.346
0.743 0.730 0.637 0.620 0.457 0.466 0.299 0.377 0.252 0.409 0.339 0.371
3.194 2.345 3.881 2.254 4.416 2.130 5.126 1.731 3.115 1.482 1.437 1.374
0.748 0.734 0.646 0.638 0.458 0.463 0.292 0.364 0.253 0.381 0.314 0.330
0.9 0.6
0.7 0.5 1.4 2.8
29 29 8 8
-25 -25 -57 -57 -65 -65 -83 -83
Av abs % deviation:
1.1
6.6 3.9
0.1
2.0
0.6 2.5 3.5 0.3 7.4 8.0 12.6
1.7
3.4
0.0
0.2 0.9 0.4 2.6 1.2
Table V. Methane-Carbon Dioxide System. Comparison of Fugacity Coefficients Calculated by Two Methods Temp O F
29 29 8
8 -25 -25 -57 -57 -65 -65 -83 -83
Press 1bh2 870 1052 733 988 591 992 498 899 659 900 755 776
Methane mole fraction Liquid Vapor 0.103 0.165 0.1095 0.224 0.137 0.322 0.1465 0.465 0.261 0.562 0.611 0.642
% Dev fug. coeff. CH4 Liquid Vapor
0.329 0.387 0.425 0.505 0.605 0.686 0.751 0.805 0.813 0.833 0.878 0.882
-3.1 -0.3 -0.9 4.3 0.4 0.6 0.1 -0.5 -1.2 -0.1 -0.6
Av abs % deviation:
1.4
5.1
-2.2 -0.9 -2.0 -1.3 0.3 0.3 0.9 0.8 -0.1 1.5
% Dev fug. coeff. C02 Liqui Vapor d
0.4 0.1 0.1 -1.4 -0.6 -0.2 -0.1
1.1
1.5
0.1 -0.1 0.2 1.0
0.6 1.5 1.4 -0.5 -0.8 -2.6 3.4 0.5 -6.9 -7.2 -10.2
1.1
0.4
3.1
1.1
0.0
Table VI. Methane-Propane System Temp, O F
Press, lb.iin.2
160 160
1000
100
500
100 100 40 40 40 0 0
1000 1250
0
-50 -50
500
500
1000 1250 600 1000 1300 600
1000
Methane mole fraction Liquid Vapor 0.0433 0.2800 0.1235 0.3271 0.4511 0.1923 0.4226 0.5492 0.311 0.522 0.708 0.438 0.736
0.1550 0.3558 0.5209 0.6635 0.6766 0.7819 0.8208 0.8222 0.891 0.895 0.845 0.9585 0.9458
Calc ratio of fugacity coeff. CH4 C3H8
Obsd ratio CH4 C3H8
3.691 1.296 4.380 2.113 1.522 4.085 1.994 1.542 2.956 1.740 1.225 2.201 1.296
3.580 1.271 4.218 2.028 1.500 4.066 1.942 1.497 2.865 1.715 1.194 2.188 1.286
0.878 0.885 0.529 0.484 0.593 0.262 0.288 0.385 0.148 0.200 0.484 0.0702 0.194
ylx
0.883 0.895 0.547 0.500 0.589 0.270 0.310 0.394 0.158 0.220 0.531 0.0738 0.205
Av abs % deviation:
To explore the application of the proposed method to paraffin hydrocarbon systems, the methane-propane binary has been put to the test. The vapor-liquid equilibrium data of Wiese et al. (1970) a t 160,100, and 40 O F and the data of Price and Kobayashi (1959) at 0 and -50 O F have been utilized. The binary constants of this system, Vc12 = 144.26 and Tc12 = 476.02, were determined from Bosler’s pseudocritical data as in the two previously described cases. It should be pointed out 302
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
Abs % dev CH4 CiH8 3.1 2.0 3.8 4.2 1.5 0.5 2.6 3.0 3.2
0.6 1.1
2.6 0.6 0.8
3.1 3.2 0.6 2.9 7.3 2.5 6.5 8.9 8.9 5.0 5.3
2.3
4.3
1.5
that for this binary the Lee-Kesler mixing rules, eq 7 and 8, yield the values, Vc12 = 144.83 and Tc12 = 477.88, close t o those used in the present investigation. The fugacity coefficients of each component and their ratios were calculated as previously described for the methane-C02 system, and the ratios were compared in each case with experimental y / x ratios. The results are shown in Table VI. The agreement between calculated and observed values is quite satisfactory.
Table VII. Methane-Propane System. Comparison of Fugacity Coefficients Calculated by Two Methods Temp, OF 160 160 100 100 100 40 40 40 0 0 0
-50 -50
Press., 1bh2 500 1000 500 1000 1250 500 1000 1250 600 1000 1300 600 1000
Methane mole fraction Liquid Vapor 0.0433 0.1550 0.2800 0.3558 0.5209 0.1235 0.6635 0.3271 0.4511 0.6766 0.1923 0.7819 0.4226 0.8208 0.5492 0.8222 0.891 0.311 0.522 0.895 0.708 0.845 0.9585 0.438 0.736 0.9458 Av abs % deviation:
The alternative procedure of calculating the individual component fugacity coefficients from experimental y-x data and simultaneous solution of eq 31 and 32 was also applied, as for the methane-carbon dioxide system, and the results were compared with fugacity coefficient values obtained by the proposed method. The comparative results are exhibited in Table VII. Again there is satisfactory agreement. In discussing the limitations of their pseudocritical method, Leyendecker and Gunn (1972) pointed out that their procedure becomes increasingly inaccurate whenever the ratio Vc,Tc,/Vc,Tc,falls outside the limits 0.25 to 4.0 for a binary pair. Trial calculations with systems, such as the methanen-pentane binary, for which this ratio is 7.6, indicate that the same limitation applies to the method proposed in this paper. In conclusion, it may be stated that the proposed method for calculating component fugacity coefficients should prove useful in vapor-liquid equilibrium calculations. The method is well adapted to computer programming, being based on the modified BWR equation of Lee and Kesler (1975). I t employs mixing rules which describe well the composition dependence of the pseudocritical parameters. Following the suggestion of Pitzer and Hultgren (1958), two binary interaction constants are used to describe the behavior of a binary system. This gives the method an inherently greater flexibility than is possible with the more common approach which relies on a single interaction constant for each binary pair (Prausnitz and Chueh, 1968). Nomenclature
a , b = constants in van der Waals' equation f = fugacity H = enthalpy per mole k = binary interaction constant K = vapor-liquid equilibrium ratio n = total number of components P = absolute pressure R = gasconstant T = absolute temperature b' = internal energy = H - PV V = molar volume x = mole fraction; liquid-phase mole fraction y = vapor-phase mole fraction z = compressibility factor = PVIRT
% Dev fug. coeff. CH4 Liquid Vapor
-0.29 -0.25 0.99 3.30 2.92 -0.38 1.13 3.35 2.47 0.59 1.61 0.35 0.60 1.40
% Dev fug. coeff. C3H8 Liquid ' Vapor
-3.29 -2.18 -2.74 -0.84 1.43 -0.85 -1.47 0.34 -0.69 -0.88 -0.97 -0.21 -0.21 1.24
0.01 0.11 -0.16 -1.55 -2.34 0.07 -0.82 -3.99 -1.14 -0.62 -3.81 -0.26 -1.76 1.28
0.62 1.22 3.07 1.67 -2.91 3.10 7.02 -1.55 5.78 9.05 5.54 4.96 3.73 3.86
Greek Letters 4 = fugacity coefficient o = acentric factor Subscripts c = critical G = gasor vapor phase i,..j , k = component i, j , or k , respectively LJ = pertaining to components i and j M = mixture property r = reduced 1 , 2 = component 1,component 2 12 = pertaining to components 1 and 2 Superscripts * = idealgas state ' = pseudo property of a mixture (1) = deviation function of Pitzer-Curl or Lee-Kesler correlation Literature Cited Bosler, W. H., M.S. Thesis, Pennsylvania State University, University Park, Pa., 1967. Case, L. C., Weber, H. C., AlChEJ., 6, 171 (1960). Curl, R. F., Pitzer, K. S., lnd. Eng. Chem., 50, 265 (1958). Donnely, H. G., Katz, D. L.. lnd. Eng. Chem.. 46, 51 1 (1954). Gamson, B. W., Watson, K. M., Nat. Petrol. News, 36,R623 (1944). Gunn, R. D., AlChEJ., 18, 183(1972). Joffe, J., lnd. Eng. Chem., 40, 1738 (1948). Joffe, J., Chem. Eng. Prog., 45, 160 (1949). Joffe, J., Zudkevitch, D., Ind. Eng. Chem., Fundam., 5, 455 (1966). Leach, J. W.. Chappelear, P. S., Leland, T.W.. AlChE J., 14, 568 (1968). Lee, B. I., Kesler, M. G., AlChE J., 21, 510 (1975). Leland, T.W.. Chappelear, P. S.,Gamson, B. W., AlChEJ., 8, 482 (1962). Leyendecker, W. R., Gunn, R. D., AlChEJ., 18, 188 (1972). Olds, R. H., Reamer, H. H., Sage, B. H., Lacey, W. N.. h d . Eng. Chem., 41,475 (1949). Pitzer. K. S., Lippmann, D. Z., Curl, R . F., Huggins. C. M., Petersen, D. E., J. Am. Chem. SOC.,77, 3427, 3433 (1955); 79, 2369 (1957). Pitzer, K. S., Hultgren, G. O., J. Am. Chem. SOC., 80, 4793 (1958). Prausnitz, J. M., Gunn, R. D.. AIChEJ., 4, 494 (1958). Prausnitz, J. M., Chueh, P. L., "Computer Calculations for High-pressure Vapor-Liquid Equilibria," pp 58-59, Prentice-Hall, Englewood Cliffs, N.J., 1968. Price, A. R., Kobayashi, R., J. Chem. Eng. Data, 4, 40 (1959). Reid, R . C., Leland, T.W., AIChE J., 11, 228 (1965). Van Ness, H. C., "Classical Thermodynamics of Non-Electrolyte Solutions," p 91, Macmillan. New York, N.Y., 1964. Wiese, H. C., Jacobs, J., Sage, B. H., J. Chem. Eng. Data, 15, 82 (1970).
Receiued for reuieu: December 24, 1975 A c c e p t e d May 27, 1976
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