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Ind. Eng. Chem. Fundam.
phenyloctane, coz + 1-hexadecene, cop + n-propyl-k water + licyc1ohexane, and 2“‘ quids and their mixtures with hydrogen at temperatures to 710 K and pressures to 25 Mpa have also been investiaated. The new experimental results will be presented shortly. Acknowledgment Funds of this research were provided by the Electric Power Research Institute through Research Project RP367. Literature Cited Connolly, J. F. J . Chem. Phys. 1962, 3 6 , 2897
1985,2 4 , 262-265 Grayson, H. G.; Streed, C. W. Sixth World Petroleum Congress, Frankfurt/ Main, Section V I I , paper 20, June 19-26, 1963. Laugier. S.; Richon, D.; Renon, H. J . Chem. Eng. Data 1980,25, 274. Nasir, P.: Martin, R. J.; Kobayashl, R. N U ~ I Phase Equilib. 1980, 5 , 279. Robinson, D. 6 . ; Ng, H. “The Equilibrium Phase Compositions of Selected Aromatic and Naphthenic Binary Systems”; GPA RR-29;Gas Processors Association: Tulsa. OK. 1978:J . Chem. Data 1978. 23. 325. Sebastian, H. M.; Simnick, J. J.; Lin, H. M.;Chao. K. C. J . &em. Eng. Data 1980,25, 246. Simnick, J. J.; Lawson, C. C.; Lin, H. M.; Chao, K. C. AIChE J . 1977, 23, A69 .--.
Sung, S.PhD Thesis, University of Pittsburgh, 1981. Wilson, G. M.: Johnston, R. H.: Hwang, S. C.; Tsonopouios, C. Ind. Eng. Chem. Process Des. Dev. 1981,2 0 , 94.
Received for reuiew April 26, 1984 Accepted January 16, 1985
COMMUNICATIONS Second Virial Coefficient as a Corresponding States Temperature Variable for Pure Fluids: A Universal Saturation Curve A recently proposed corresponding states theory for pure fluids is extended to saturation conditions and comparisons are made to experimental data. The new theory is similar to simple corresponding states theory with the reduced temperature being replaced by a new corresponding states temperature variable which is an approximation to B,(T)IB,(T,) where B,(T) is the second viriai coefficient and T , is the critical temperature. The theory predicts that the saturation properties of pure fluids may be represented by a universal surface, called UNISAT, upon which the reduced vapor pressure and reduced saturated gaseous and liquid densities are universal functions of the new corresponding states temperature variable. When compared to data on the saturation properties of 12 pure fluids, UNISAT is found to do at least as well as, and usually better than, the perturbed-hard-chain equation and the Leland-Leach shape factors calculation using McCarty’s 33-constant modified Benedict-Webb-Rubin equation for methane as the reference equation. UNISAT is also compared to the Lee-Kesler form of the Rackett equation for vapor pressures, with a modified form of the Rackett equation by Yamada and Gunn for saturated liquid densities, and with the Edwards-Thodos equation for saturated vapor densities; UNISAT did almost as well as these three correlations.
Introduction A quantitative knowledge of the thermodynamic and phase equilibria properties of fluids and fluid mixtures is necessary for the design of many processes in chemical engineering. Although such knowledge can be obtained experimentally by gathering data for the particular process at the conditions of interest, a simpler approach is to make use of one of the many correlations which have been developed to estimate thermophysical properties. The principle of corresponding states and its extensions are often used for this purpose. Although simple corresponding states theory, as originally introduced by van der Waals (1878,1881)is applicable only to spherical molecules such as Ar, Xe, and Kr, various modifications of the theory have been developed which allow the application of corresponding states theories to more complex molecules. Examples of such improvements include the so-called shape factors approach (Leland and Chappelear, 1968; Rowlinson and Watson, 1969; Mentzer et al., 1980; Mollerup, 1980) in which the critical properties are modified by temperature- and density-dependent functions, or the three-parameter corresponding states theories (Pitzer et al., 1955; Lydersen et al., 1955; Leland et al., 1962) in which a third parameter such as the acentric factor is used to characterize different classes of molecules.
We recently proposed a new corresponding states theory for pure fluids containing asymmetric and/or chain molecules (Hall and Hacker, 1983). This theory is similar to simple corresponding states theory with the reduced temperature being replaced by a new corresponding states temperature variable which is an approximation to B,(7‘j/B,(Tc)where B , ( n is the second virial coefficient and T,is the critical temperature. The use of the second virial coefficient as a corresponding states temperature variable was motivated by arguments based on the recently developed polymer scaling theories and on the hypothesis that the second virial coefficient is a good measure of the coupling between the temperature and the intermolecular potential in the partition function. In order to &estthis hypothesis, the new corresponding states theory was then used to correlate equation of state “data” generated by the highly accurate perturbed-hard-chain theory developed by Donohue and Prausnitz (1977, 1978). The correlations worked quite well in the supercritical region for reduced volumes between 0.5 and 6.0. In this paper we consider the extension of this theory to the correlation of the saturation properties of pure fluids. The theory predicts that the saturation properties of pure fluids may be represented by a universal vapor pressure curve in which the reduced vapor pressure is a 1985 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
universal function of the new corresponding states temperature variable and by a universal coexistence curve in which the reduced gaseous and liquid densities are universal functions of the new corresponding states temperature variable. The universal vapor pressure and coexistence curves are referred to together as UNISAT. Theoretical Argument Concerning a Universal Saturation Curve Here we present theoretical arguments leading to the development of a universal saturation curve, which we have called UNISAT. In the next section we will describe the results obtained when UNISAT is used to correlate real saturation data. We begin by assuming that the configurational partition function has the following functional form
where h and f are scale factors which may depend on both temperature and density. This equation is similar to the one used in the development of the shape factors approach to pure fluids. The configurational Helmholtz free energy, A, is given by (2)
and a universal vapor pressure curve (7)
In the new corresponding states theory described previously (Hall and Hacker, 1983), h and f are given by h
= Vc
(8)
f = T/X(TI)
(9)
X(TJ = 0 / T I - T c / T ~ ) / (-l T,/TB)
(10)
where Since these are both volume independent, the arguments described in this section may be used to obtain a universal coexistence curve
and a universal vapor pressure curve
where F1,F2, and F3 are universal functions. Since X(TI = 1) 1, it can be shown from eq 12 that PcVc/kTcis the same for all substances. Thus eq 1 can be written as
where k is Boltzmann's constant, so that
(13)
Eh ( % av) T
(3)
Equation 3 can be written as
kT
-(;, ;)
=z - -
or
=( );
ph = z ,;
-
kT if h and f are constants or if the proper cancellation occurs in those terms which involve derivatives of h and f. Such a cancellation of terms is implicitly assumed in the shape factors approach (Rowlinson and Watson, 1969) which has as its basis eq 2 and 4 with f and h taken to be temperature and density dependent. For our purposes we will assume that both h and f are independent of volume since it is only under these conditions that a universal saturation curve will result. Thus we obtain the following conditions for equilibrium between gaseous (g) and liquid (1) phases =
(
V(1) T(1)
z -,-
263
)=i(y,y)
(Sa)
For a given T/f, eq 5a and 5b are two simultaneous equations in two unknowns, V(l)/h and V(g)/h. This gives a universal coexistence curve governed by the equations
Taken together, eq 11 and 13 represent a universal saturation surface, which shall henceforth be referred to as UNISAT. In the following section we will use the saturation data for 12 nonpolar compounds to test the correlating abilities of UNISAT. Test of UNISAT Using Experimental Data The UNISAT correlation was tested by comparing its predictions with saturated property data on the 1 2 compounds: methane, ethane, propane, butane, pentane, octane, benzene, ethylene, carbon dioxide, carbon monoxide, nitrogen, and sulfur dioxide. These 12 compounds were chosen because reliable data exist for vapor pressures and for saturated gaseous and liquid densities. The parameters used in the comparison between real data and UNISAT were determined in the following manner. Real critical constants, as given by the data source for each compound, were used. The Boyle temperatures were determined by first choosing the Boyle temperature of methane to be T B F e h e = 505 K (Goodwin, 1974) and then least-squares fitting the Boyle temperatures of the other compounds so that the vapor pressures of all the compounds are accurately represented by the equation In (Psat,r/TI) = a1 + a&(Tr)
+ A&(T1P
(14)
where al, a2, and u3 were determined by the data for methane. Note that the curve is not constrained to go through the critical point and that the parameter T B can no longer be interpreted as the Boyle temperature. These new fitted values of TB are listed in Table I. Alternatively, one can solve eq 14 for TBat the normal boiling point ( P = 1 atm at T = boiling point) thus, needing only one readily available data point. There is only a slight compromise in accuracy if the Boyle temperatures are determined from the normal boiling point. In Figure 1 the vapor pressure data are plotted as In (Psat,r/TI) vs. X(TI) and vs. l / T I using the new fit TB's. The new variable X( TI) does an excellent job of correlating the reduced vapor pressures, as expected since the values of TBwere determined for this purpose. The reduced
264
Ind. Eng. Chem. Fundam.. Vol. 24, No. 2, 1985
Table I. Data Sources and Fitted Values of T n compound methane ethane propane butane pentane octane benzene carbon dioxide carbon monoxide ethylene nitrogen sulfur dioxide
data source
TB,K
Goodwin (1974) Goodwin et al. (1976) Das and Eubank (1973) Das et al. (1973) Das et al. (1977) Connolly and Kandalic (1962) Connolly and Kandalic (1962) Newitt et al. (1956) Hust and Stewart (1963) Douslin and Harrison (1976) Strobridge (1962) Rynning and Hurd (1945)
505.0 698.8 794.7 871.4 916.1 1013.8 1153.1 620.9 329.0 665.1 316.5 832.3
saturated density data are shown in Figure 2 as l / X ( T r ) vs. pr and T, vs. p, where pr = l / V , . It can be seen that an essentially universal coexistence curve for the real data is obtained when the new variable X(T,) is used. In order to increase the usefulness of the UNISAT correlation, the universal functions Fl and F2which occur in eq 12 have been determined by fitting the data appearing in Figure 2 to single curves. (The function F3 has already been determined). The final forms of the UNISAT equations are the following vapor pressure
where a l = 2.8225, u2 = -2.8331, and a 3 = -5.3727 saturated liquid density
X
lo4
-_ 1=
where bl = 1.5914, b2 = 0.26942, b3 = 0.53463, and b4 = -0.013916, and saturated vapor density
where c1 = 1.4794, c2 = 4.7394, c3 = -6.4486, and c4 = 17.652.
The saturated property predictions of UNISAT, eq 15-17, were compared with the predictions of the per-
turbed-hard-chain equation of state and with the predictions of the corresponding states theory using shape fadors of Leach et al. (1966) for the 12 compounds. A 33-constant modified Benedict-Webb-Rubin equation of state by McCarty (1974) for methane was used as the reference system equation for the shape factor calculations. The
Table 11. Predictions of Saturated Properties, Average Absolute Percentage Deviation"
methane (T= 100 to 185 K)
ethane (T= 180 to 298.15 K)
propane (T= 240 t o 365 K) butane (T= 280 to 420 K) pentane (T= 310 to 460 K) octane (T= 463 to 553 K)
benzene (T= 463 to 553 K) ethylene (T= 235.15 to 280.15 K) carbon dioxide (T= 253 to 298 K) carbon monoxide (T= 81.6 to 129.8 K) nitrogen (T= 65 to 125 K) sulfur dioxide (T= 277.4 to 421.9 K)
UNISAT
P.H.C.
Leach et al.
Lee-Kesler
0.18 0.18 3.66
0.28 1.72 2.94
0.63 0.10 1.80
0.26 2.56 5.24
0.82 0.34 1.88
0.55
0.62 0.42 1.02
0.79 1.68 6.56
1.09 0.91 3.40
0.39
0.78 0.15 1.77
0.75 1.81 7.83
1.08 1.06 3.74
0.30
0.84 1.92 4.14
1.41 1.10 6.72
1.50 0.54 4.81
0.60
0.15 1.12 5.59
0.48 4.06 7.73
3.09 0.82 7.87
0.38
0.16 0.63 1.47
0.23 1.90 9.14
2.02 0.87 5.73
0.12
0.23 0.71 1.49
0.27 8.70 11.88
0.82 1.14 2.56
0.05
0.20 0.24 1.52
0.17 1.32 8.44
4.35 3.70 8.20
(1.96)b
0.65 0.63 4.92
0.63 2.43 3.77
1.16 2.86 0.97
0.55
0.72 0.59 4.77
0.28 1.42 3.40
1.53 2.02 1.65
0.35
0.88 1.74 1.56
1.81 2.21 5.75
1.25 1.88 3.62
1.84
YamadaGunn
EdwardsThodos
0.39 0.23 0.92 0.19 0.89 0.39 1.00 0.19 1.51 0.43 1.24 0.55 2.35 0.41 1.54 0.83 1.74 0.50 2.05 0.56 0.58 0.25 1.39 0.59 1.17
"Average absolute percentage deviation, aapd = ~ ( 1 0 0 % ( a c t u a lcalcd)/actual)/N. bSince C02 does not have a normal boiling point, the standard value for the acentric factor was used.
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
-1.0 OI 0-
Y.
'..
i -1
-501
IO
'
"
"
15
'
"
X (T,)
I
20 [a]
'
"
and I / T r
'
[
25
'
"
'
1
30
A]
F i g u r e 1. Experimentally determined vapor pressures for 12 compounds. The triangles are for In (Pmfr/Tr) vs. l/Tr and the circles vs. X(T,). are for In (Pmt,JTr) IO
1
0.0
0.6
... -I
0.4!-
265
equations by Lee and Kesler (1975) for vapor pressure, by Yamada and Gunn (1973) for saturated liquid density, and by Edwards and Thodos (1974) for saturated vapor density. It should be pointed out that the Yamada-Gunn correlation requires one empirical parameter, or at least one data point for the saturated liquid density for each compound. The UNISAT equation does not require any saturated liquid density data or any parameters unique to the liquid density equation. Also, the Edwards-Thodos correlation requires the vapor pressure at the temperature of interest which must be either known from experiment or estimated from a correlation such as the Lee-Kesler equation. For the UNISAT equations for saturated liquid density and saturated vapor density, once the value of TB has been determined no additional parameters, besides the critical constants, are needed. Acknowledgment The support of the Gas Research Institute (Grant No. 5082-260-0724) and of the Petroleum Research Fund (Grant No. 14581-AC5) is gratefully acknowledged. The authors also wish to thank Professor M. Donohue for providing copies of his programs and for helpful discussions. Literature Cited Connolly, J. F.; Kandallc, G. A. J. Chem. Eng. Data 1962, 7, 137. Das, T. R.; Eubank, P. T. A&. Ctyog. Eng. 1973, 78, 208. Das, T. R.; Reed, C. O.,Jr.; Eubank, P. T. J. Chem. Eng. Data 1973, 78, 244. Das, T. R.; Reed, C. O., Jr.; Eubank. P. T. J. Chem. Eng. Data 1977, 2 2 , 3. Donohue, M. 0.; Prausnitz, J. M. Gas Processors Association Research Report Number RR-26, Tulsa, OK, 1977. Donohue. M. 0.; Prausnltz, J. M. AIChE J . 1978, 2 4 , 849. Douslin, D. R.; Harrison, R. H. J. Chem. Thermodyn. 1978, 8 , 301. Edwards, M. N. 8.; Thodos, G. J. Chem. Eng. Data. 1974. 79, 14. Goodwin, R. D. NBS Technical Note 653, 1974. Hall, C. K.; Hacker, B. A. I n "Chemical Engineering at Supercrltlcai Condltlons"; Paulattls. J. M. L.; Penninger, M.; Gray, R.; Davidson, P..Ed.; Ann Arbor Science Publishers: Ann Arbor, 1983; Chapter 16. Hust, J. 0.; Stewart, R. B. NBS Technical Note 202 1981. Leach, J. W.; Chappelear, P. S.; Leland, T. W., Jr. Roc. Am. Pet. Inst., Dlv. Refining 1968, 46, 223. Lee, B. I.; Kesier, M. 0. AIChE J . 1975, 2 7 , 510. Leland, T. W., Jr.; Chappelear, P. S.; Gamson, G. W. A I C M J. 1982, 8 , 482. Leland. T. W., Jr.; Chappelear, P. S. Ind. Eng. Chem. 1988, 60(7). 15. Lyderson, R. A.; Greenkorn, R. A.; Hougen. 0. A. "A Generailzed Correlation of Thermodynarnlc Properties of Pure Fluids"; Engineering Experimental Statlon Report No. 4, University of Wisconsin, Madison, WI, 1955. McCarty, R. D. Ctyogenlcs 1974, 74, 278. Mentzer, R. A.; Greenkorn, R. A.; Chao. K. C. Sep. Sci. Techno/. 1980, 75, 1613. Mollerup, J. Fluid Phase Equlllb. 1080, 4 , 1 1 . Newitt, D. M.;Pal, M. U.; Kuloor, N. R.; Huggiil, J. A. W. In "Thermodynamic Functions of Gases"; Din, F., Ed.; Butterworths Scientific Publications: London, 1958; Vol. 1, p 102. Pitzer, K. S.; Lippman, D. 2.; Curl, R. F., Jr.; Huggins, C. M.; Petersen, D. E., J . Am. Chem. SOC.1955, 77, 3433. Rowiinson, J. S.; Watson, I.D. Chem. Eng. Sci., 1969, 2 4 , 1565. Rynning, D. F.; Hurd, C. 0. AIChE Transectlons, 1945, 47, 265. Strobridge, T. R. NBS Tech. Note 129, 1962. Waals. J. D. van der Amsterdam Verslagen 1870, 10, 321. Waals, J. D. van der "Die Continuitit des garf&rnigen und Fiussigen Zustandes"; Roth, F., Trans.; Barth: Leipzig, E. Germany, 1881. Yarnada, T.; Gunn, R. D.J. Chem. Eng. D8ta 1973, 78, 234.
k
0
20
10
30
Pr
F i g u r e 2. Experimentally determined gaseous and liquid densities for 12 compounds. The triangles are for l/X(Tr) vs. pr and the circles are for TI VB. pr.
results of these three calculations are listed in the first three columns of Table I1 as the average absolute percentage deviation for each of the 12 compounds considered. Also listed in Table 11is the temperature range of the data; for each compound ten data points were used. It is clear from Table I1 that the UNISAT equations usually do at least aa well aa the perturb I-hard-chain equation of state and the shape factors method of Leach et al. and usually better. The UNISAT equations offer a direct means of calculating the saturated properties. This is in contrast to the two equation of state approaches in which the saturated properties are calculated by iterating until the temperature, pressure, and chemical potential of the two phases are equal. It is also of interest to compare UNISAT to other correlations of saturation properties, particularly those which are given in terms of universal parameters. Accordingly, UNISAT was compared with the Lee-Kesler (1975) form of the Pitzer equation for vapor pressures, with a modified form of the Rackett equation by Yamada and Gunn (1973) for saturated liquid densities, and with an equation by Edwards and Thodos (1974) for saturated vapor densities. Columns 4, 5, and 6 list the average absolute percentage deviation of these three equations for the data on the 12 compounds. Comparison with column 1 shows that the UNISAT equation did almost as well as the specific
-
Department of Chemical Engineering Princeton University Princeton, New Jersey 08544
Barbara A. H a c k e r Carol K.Hall*
Received for review October 19, 1983 Revised manuscript received June 25, 1984 Accepted November 20,1984
