Ind. Eng. Chem. Res. 1987, 26, 1686-1691
1686
Y = mole fraction in the gas phase mole fraction
z =
Greek Symbols = parameter describing nonideality in the adsorbed phase induced by interaction with surface (vacancy) y = activity coefficient 7 = lateral interaction parameter 0 = fractional amount adsorbed based on a monolayer coverage T = spreading pressure 4 = fugacity coefficient a!
Subscripts
i = species or component m = mixture t = total v = vacancy e = experimental quantity p = predicted quantity Superscripts s = surface phase m = limiting or maximum value Registry No. H2, 1333-74-0; CO, 630-08-0; CH,, 74-82-8; COP, 124-38-9; HzS, 7783-06-4; C, 7440-44-0.
Literature Cited Cochran, T. W.; Kabel, R. L.; Danner, R. P. AZCHE J . 1985,31,268. Costa, E.; Sotela, J. L.; Calleja, G.; Marron, C. AZCHE J . 1981,27, 5.
Friederich, R. 0.;Mullins, J. C. Znd. Eng. Chem. Fundam. 1972, 11, 439. Grant, R. J.; Manes, M. Ind. Eng. Chem. Fundam. 1966, 5, 490. Hyun, S. H.; Danner, R. P. J . Chem. Eng. Data. 1982,27, 196. Lee, A. K. K. Can. J . Chem. Eng. 1973, 51, 688. Lee, C. S.; O’Connell, J. P. AIChE J . 1986, 32, 96. Lewis, W. K.; Gilliland, E. R.; Chertow, B.; Cadogan, W. P. Znd. Eng. Chem. 1950, 42, 1319. Markham, E. D.; Benton, A. F. J. Am. Chem. Soc. 1931, 53, 497. Miller, G. W.; Knaebel, K. S.; Ikels, K. G. AIChE J . 1987, 33, 194. Myers, A. L. Ind. Eng. Chem. 1968, 60, 45. Myers, A. L.; Prausnitz, J. M. AIChE J . 1965, 11, 121. Peng, D.-Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976,15,59. Reich, R.; Zeigler, W. T.; Rogers, K. A. Ind. Eng. Chem. Process Des. Deu. 1980, 19, 336. Ritter, J. A. M. S. Thesis, State University of New York, Buffalo, 1986. Ruthven, D. M.; Loughlin, K. F.; Holborow, K. A. Chem. Eng. Sci. 1973, 28, 701.
Saunders, J. T. M. S. Thesis, State University of New York, Buffalo, 1983. Schay, G. J. Chem. Phys. Hungary, 1956,53, 691. Talu, 0.; Zwiebel, I. AZChE J. 1986, 32, 1263. Van Ness, H. C. Ind. Eng. Chem. Fundam. 1969,8, 464. Wilson, R. J.; Danner, R. P. J . Chem. Eng. Data 1983, 28, 14. Yang, R. T. Gas Separation by Adsorption Processes; Butterworth Boston, 1987; Chapter 3. Yon, C. M.; Turnock, P. H. AZChE Symp Ser. 1971, 67(117), 3. Received for review October 28, 1986 Revised manuscript received April 24, 1987 Accepted May 1, 1987
Evaluation of an Equation of State Method for Calculating the Critical Properties of Mixtures J. Richard Elliott, Jr. Department of Chemical Engineering, The University of Akron, Akron, Ohio 44325
Thomas E. D a u b e r t * Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
The accuracy of an equation of state method for predicting critical properties of mixtures is evaluated and compared to several empirical methods. The Soave equation of state is used with binary interaction coefficients predicted from vapor-liquid equilibrium data. An extensive data base of about 1500 points each for critical temperature and critical pressure is used in the evaluation. These data include hydrocarbon mixtures and hydrocarbon-non-hydrocarbon mixtures with H2, N2,CO, C 0 2 ,and H2S. The equation of state method is determined to be more accurate than the empirical methods for critical temperature and critical pressure and slightly less accurate for critical volume. Furthermore, the equation of state method predicts anomalous trends in critical loci which were previously noncalculable by empirical methods. When phase equilibria at high temperatures and pressures are considered, knowledge of the critical properties of mixtures is essential. Many processes involving high pressure are designed specifically to take advantage of the unique phase behavior in the critical region. Enhanced oil recovery with carbon dioxide and supercritical extraction provide two examples of such processes. Accurate knowledge of the critical properties of the mixtures is especially important for these types of processes. Many correlations have been proposed for predicting the critical properties of mixtures. Most of these correlations have been empirical, and they have been limited in the types of systems which they could represent. Empirical correlations were evaluated by Spencer et al. (1973), and the most accurate methods were recommended. One 0888-5885/87/2626-1686$01.50 f 0
modification has been made to the Chueh and Prausnitz (1967) method in the API book (1986) as described later. Since that time, the calculation of the critical properties via an equation of state by applying the rigorous thermodynamic criteria at the critical point has become more practical for common application. Peng and Robinson (1977) evaluated their equation of state for 30 mixture critical points by this method. Heidemann and Khalil (1980) published an improved algorithm which was considerably more rapid and more robust than its predecessors. Michelsen and Heidemann (1981) have improved the computation speed of this latter algorithm. Considering these developments, a thorough evaluation of the merits of a rigorous method relative to the empirical methods was undertaken. 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1687
Methods Compared The empirical methods used to compare to the rigorous method were those determined to be most accurate by Spencer et al. (1973). These were the Li (1971) method for critical temperature, the Kreglewski and Kay (1969) method for critical pressure, and the Chueh and Prausnitz (1967) method for all three critical properties. Li (1971) Method for Critical Temperature: n
Tcm = C ~ L T C ,
(1)
i=l
CxiVci
i=l
Kreglewski and Kay (1969) Method for Critical Pressure:
r pcm=
1
n
1
+ [5.808 + 4.93(C~i0i)] X i=l
[
]2XiPCi
Tc?;]
i=1
ponent mixtures. To evaluate the binary parameters, binary critical data must be available for each component pair in the mixture. Because of these limitations, the method was not evaluated in this investigation. As representative of the equation of state approach, the Soave (1972) equation of state was investigated. The Soave (1972) equation was implemented as described in Chapter 8 of the API book (1986). The binary interaction coefficients and correlations of binary interaction coefficients recommended by that source were used directly with no optimization of the parameters to fit the critical data. An adaptation of the procedure of Moysan et al. (1983) for binary interaction coefficients of hydrogen systems was also implemented as described in the API book (1986). Thus, the implementation of the equation of state method for critical properties could fairly be termed predictive. The implementation of the equation of state approach requires trial and error solution for the two variables V,, and T,,. The two equations which must be satisfied are det [&I = 0
(15)
C=O
(16)
a2A nTQij= nT anidnj
(3)
Chueh and Prausnitz (1967) Method for All Critical Properties: It
T,, = i=l CBiT,, n
V,, = C8iVC,+ i=l
P,, =
~
n
n
+ i=l jCBiBj~ij =1
(4)
C0i0j~ij
i=lj=1
U
V,,
+ b)
+ Raj)RTcij1.5(Vci + Vcj) 0.291 - 0.04(~i+ ~ j )
(5) (6)
1/4(aai aij
=
+
(11)
= 0.0867 - 0.0125~i 0.011W: (14) Correlations for the binary parameters, T ~ and , vi,, are available from Chapter 4 of the API book (1986). These correlations were based on the data base used at the time of the evaluation by Spencer et al. (1973). One empirical method which has been proposed for calculating critical properties of defined mixtures since the previous investigations is that of Teja et al. (1983). The method is limited, in that binary critical data cannot be predicted. The method requires evaluation of binary interaction parameters to predict properties of multicomRbi
[
6 i # j # k , i # k
n n
RT Vcm - b T,,'/2V,,(
1 i=j=k
hi,,= 3 i = j # k , i = k # j , j = k # i
where A = total Helmholtz free energy of the mixture. Formulas for the derivatives of the Helmholtz energy with respect to mole number are given by Heidemann and Khalil (1980). Once V,, and T,, have been determined, these values are substituted into the Soave (1972) equation of state to obtain a value of P,,. The strategy of the Heidemann and Khalil algorithm is to iterate on Tcmand V,, in a nested manner instead of performing iterations on both variables simultaneously as in a typical solution of multiple nonlinear equations. Based on an initial guess of V,,, T,, is iterated until eq 15 is satisfied. Closure of eq 16 is checked, a new estimate for V,, is generated, and the iteration on Tcmis carried out again. The advantage of this strategy is that it leads to convergence on the desired root much more often than by use of simultaneous iteration. Iteration on T , and V,, by the secant method was found to be satisfactory for all calculations performed in this investigation.
Data Base Attention was restricted for this evaluation to hydrocarbon mixtures and mixtures containing hydrogen, nitrogen, carbon monoxide, carbon dioxide, and hydrogen sulfide. An extensive compilation of critical properties which included these types of mixtures was published by Hicks and Young (1975). A second source which provided data for a few systems not listed in Hicks and Young (1975) was due to Kay (1972). Data from references after 1975 were also included. Points obviously inconsistent with other similar data were eliminated so as not to bias the evaluation. The complete data set was recorded on magnetic tape. A copy of the data set as well as the Fortran subroutine for implementation of the Heidemann and Khalil (1980) algorithm is available from the authors. Table I summa-
1688 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 Table I. Summary of Systems Studied for Critical Points of Defined Mixtures no, of
no. of points
binary systems methane-hydrocarbon hydrocarboh-hydrocarbon
systems T, P, V , 15 139 137 85 138 1108 878 182 33 199 186 57 hydrocarbon-non-hydrocarbon non-hydrocarbon-non-hydrocarbon 5 21 17 9 30 194 169 10 multicomponent
rizes the contents of the data set.
Results The results of the investigation are summarized in Tables 11-IV for binary mixtures and in Table V for multicomponent mixtures. For critical temperatures of binary mixtures, Table I1 shows that the accuracy of all the methods was roughly equivalent. The Soave equation is slightly more accurate for the non-hydrocarbon mixtures. The Chueh-Prausnitz correlation failed for the n-hexane-acetylene system, as negative absolute temperatures were calculated. The reason for the failure is that the value of the correlation
parameter was outside the range of the polynomial correlation used to adapt the graphs of Chueh and Prausnitz, showing that extrapolation of the method may be unreliable. Table I11 shows that the critical pressure calculations for binary mixtures are much more accurate when the equation of state approach is used. This observation is especially true for methane systems and for hydrocarbon-non-hydrocarbon systems. Calculations of critical volumes of binary mixtures are summarized in Table N.The Soave equation is inherently wrong for the prediction of critical volumes because the critical compressibility factor for pure compounds is fixed at 2, = l j 3 . Two corrections of this shortcoming were tested. Peneloux et al. (1982) have suggested a correction scheme based on the Rackett equation being applied to the liquid at a reduced temperature of 0.7. Unfortunately, it was determined that this correction is not accurate for the critical region. The second scheme involved calculating correction factors for the pure compounds by using the critical volumes of pure compounds from the data base and then applying the molar average correction factor to the mixture critical volume calculated from the Soave equa-
Table XI. Results of Evaluations for Critical Temperatures of Defined Binary Mixtures NPTSO NCANT* % AAD' % BIASd
AAD,' K
BIAS! K
A. Hydrocarbon-Hydrocarbon Systems 1. methane system Soave Li Chueh-Prausnitz Chueh-Prausnitz (revised) 2. non-methane systems Soave Li Chueh-Prausnitzg Chueh-Prausnitz (revised)
124 139 139 139
15 0 0 0
4.94 5.73 5.59 5.72
3.80 2.73 3.49 4.27
15.90 17.24 16.53 16.73
13.21 6.58 9.19 11.78
1108 1108 1104 1108
0 0 4 0
0.81 0.61 1.01 1.37
0.45 0.21 -0.88 -0.64
3.86 2.86 4.72 6.80
1.78 0.67 -4.10 -3.80
Soave Li Chueh-Prausnitz Chueh-Prausnitz (revised)
B. Hydrocarbon-Non-Hydrocarbon Systems 188 11 1.81 1.01 0 5.02 -1.18 199 0 2.55 -0.09 199 0 2.24 -0.83 199
6.92 19.92 10.24 8.70
3.88 -5.93 -0.56 -3.84
Soave Li Chueh-Prausnitz Chueh-Prausnitz (revised)
C. Non-Hydrocarbon-Non-HydrocarbonSystems 16 5 1.33 0.90 21 0 3.20 0.53 21 0 8.92 6.35 21 0 6.74 5.37
2.30 6.55 9.51 6.53
1.74 0.71 2.48 4.02
NPTS = total number of points. NCANT = number of points for which the method failed to provide a reasonable answer. For the Soave equation, these failures could be overcome by a different initial guess which will vary according to the mixture being considered. % AAD = ( l / N P T S ) C F [ l c a l c d a t l l / e x p t 1 ] 1 0 0 . % BIAS = ( l / N I " S ) C F [ ( c a l c d - exptl)/expt1]100. = (l/NPTS)zFlcalcd - exptll. BIAS = (l/NPTS)C, (calcd - exptl). #Failed for n-hexane-acetylene system.
Table 111. Results of Evaluations for Critical Pressures of Defined Binary Mixtures NPTS NCANT % AAD % BIAS
h, bar
BIAS, bar
A. Hydrocarbon-Hydrocarbon Systems 1. methane systems Soave Kreglewski-Kay Chueh-Prausnitz 2. non-methane systems Soave Kreglewski-Kay Chueh-Prausnitz
122 137 137
15 0 0
6.57 23.79 7.79
-3.41 -23.15 -1.73
9.22 48.91 14.83
-5.33 -48.47 -2.83
878 878 878
0 0 0
2.35 3.82 7.67
-0.86 1.58 1.79
1.58 2.06 3.55
0.26 0.03 1.38
1.77 -5.73 -0.17
10.46 52.53 36.89
2.28 -37.21 -21.42
C. Non-Hydrocarbon-Non-HydrocarbonSystems 12 5 4.40 3.45 0 37.62 -36.59 17 0 30.45 -25.81 17
4.35 82.78 69.18
3.17 -82.00 -65.47
B. Hydrocarbon-Non-Hydrocarbon Systems Soave Kreglewski-Kay Chueh-Prausnitz Soave Kreglewski-Kay Chueh-Prausnitz
176 186 186
10 0
0
9.41 21.91 17.36
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1689 Table IV. Results of Evaluations for Critical Volumes of Defined Binary Mixtures NPTS NCANT %AAD % BIAS A. Hydrocarbon-Hydrocarbon Systems 1. methane systems Soave 85 0 19.56 19.02 Soave (corrected) Chueh-Prausnitz Chueh-Prausnitz (revised) Kay’s rule 2. non-methane systems Soave S6ave (corrected) Chueh-Prausnitz Chueh-Prausnitz (revised) Kay’s rule
85 85 85 85
0 0 0 0
13.96 9.02 11.24 44.92
-12.54 2.60 7.78 44.60
24.11 16.23 9.82 11.12 44.78
23.38 -14.78 0.93 5.97 44.22
182 182 182 182 182
0 0 0 0 0
21.80 7.58 8.02 8.06 12.70
18.60 -4.96 -4.61 -3.74 9.65
52.86 16.37 18.88 18.10 25.01
44.11 -9.86 -11.62 -10.45 17.75
Soave Soave (corrected) Chueh-Prausnitza Chueh-Prausnitz (revised) Kay’s rble
55 55 47 57 57
B. Hydrocarbon-Non-Hydrocarbon Systems 2 22.34 22.34 2 8.12 -1.55 10 5.51 1.92 0 9.02 3.36 0 13.60 13.37
32.43 9.93 7.74 10.79 16.03
32.43 -2.26 2.58 2.83 15.67
C. Non-Hydrocarbon-Non-Hydrocarbon Systems 9 0 15.01 15.01 9 0 3.39 -3.35 9 0 7.64 -6.34 8.84 8.84 9 0 9 0 4.31 4.31
13.93 2.90 7.03 7.81 3.68
13.93 -2.86 -6.02 7.81 3.68
Soave Soave (corrected) Chueh-Prausnitz Chueh-Prausnitz (revised) Kay’s rule (I
AAD, cm3/amol BIAS,cm3/mol
Failed for hydrogen-n-decane and carbon dioxide-n-decane.
Table V. Results of Evaluations for Critical Properties of Defined Multicomponent Mixtures critical temDerature Soave Li Chueh-Prausnitz critical pressure Soave Kreglewski-Kay critical volume Soave (corrected) Chueh-Prausnitz Kay’s rule
NPTS 193 194 194 NPTS 168 169 NPTS 10 10 10
NCANT 1 0 0 NCANT 1 0 NCANT 0 0 0
tion. This last approach is denoted as the Soave (corrected) method and proves to be reasonably accurate. Of the empirical methods, the Chueh-Prausnitz is accurate but failed for two mixtures, n-decane-hydrogen and ndecane-carbon dioxide, for the same reason as for failure with critical temperatures. Table V summarizes the results for multicomponent systems. The Soave method yields reasonable results for multicomponent systems and compares favorably with the best empirical method ascertained in previous work.
% AAD
AAD, K BIAS, K 6.70 5.69 4.62 -2.64 5.97 -4.43 AAD, bar BIAS, bar 3.06 0.21 5.10 -4.63 AAD, cm3/gmol BIAS, cm3/mol 23.93 -23.93 18.21 -18.21 58.27 58.27
% BIAS
1.70 1.17 1.60 % AAD 3.33 4.62 % AAD 15.94 11.82 38.79
1.44 -0.61 -1.15 % BIAS 0.01 -3.86 % BIAS -15.94 -11.82 38.79
PURE ACETYLENE 890 880 870 880 850
840 850 820
.-
e10
% 800
700 700
Discussion The use of the Soave equation of state to predict the critical points of defined mixtures is considerably more difficult than the empirical methods in that the method is only practical on a computer. This method offers two important rewards as discussed below. First, the equation of state method offers versatility in representing anomalous critical behavior. Figure 1is an example illustrating that the critical behavior of the etheneethyne system can be accurately represented by the Soave equation if a nonzero interaction coefficient is used. Figures 2 and 3 show excellent reproduction of the qualitative features of the experimental data smoothed and plotted in Figures 4 and 5, respectively. Figure 6 shows that qualitative agreement is also reasonable for a more complicated system. Furthermore, the previous methods cannot predict opposite signs of excess critical temperature
770 780 750
PURE ETHYLENE
720 710 ..
I
J
I
35
40
45
I
50
I
55
I
I
I
I
BO
85
70
75
I
80
I
85
I
,
80
95
Tc, F
Figure 1. Critical locus for the ethene-ethyne system.
and excess critical pressure (e.g., the benzene-n-decane mixture) nor can they predict the changing sign of the excess critical pressure (e.g., the benzene-n-tridecane mixture). Thus, the equation of state method appears to be more reliable than the empirical method. Second, the use of the equation of state method for critical points and vapor-liquid equilibrium permits ex-
1690 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987
U W
U I3
U U
a W
5 _1
