Ind. Eng. Chem. Res. 2000, 39, 1115-1117
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A Proposal for Parametrizing the Sanchez-Lacombe Equation of State Kerstin Gauter and Robert A. Heidemann* Department of Chemical and Petroleum Engineering, University of Calgary, 2500 University Drive, Calgary, Alberta, Canada T2N 1N4
It is shown that the three pure-component parameters in the Sanchez-Lacombe equation of state can be determined from the critical temperature, critical pressure, and acentric factor. Vapor pressures from the equation have accuracy comparable to that obtained with the original Soave-Redlich-Kwong equation and Peng-Robinson equation, even without introducing any temperature dependence in the parameters. If temperature dependence is added to the SanchezLacombe energy parameter, the accuracy in the vapor pressures can be improved. Introduction The Sanchez-Lacombe equation is one of the “latticegas” models that have been proposed to describe polymer/ solvent phase behavior. The model contains three parameters for fitting pure substance behavior. Typically,1-3 the parameters have been obtained by a leastsquares fitting of density and vapor pressure data. As was shown by Koak and Heidemann,4 the parameter sets reported in the literature may produce significant errors in the vapor pressures and can overpredict the critical temperature by a wide margin. The proposal in this note is to match exactly the critical temperature and critical pressure of the pure substances and to use vapor pressure data to provide a third equation for the three Sanchez-Lacombe parameters. Alternative Parametrization In the notation proposed by Koak and Heidemann,4 the Sanchez-Lacombe pressure for a pure component can be written
P)-
ν*d2 1 dν* d-1 - RT - RT (1) ln 1 v ν* v v2
(
)
(
)
The pure component parameters are d, ν*, and which denote the number of lattice sites per molecule, the volume of a mole of lattice sites for the molecule type, and an energy parameter, respectively. The parameters are related to the critical temperature and pressure through
) and
ν* )
RTc (1 + xd)2 2 d
[( )
(2)
]
xd - 1/2 RTc 1 + xd ln Pc d xd
(3)
The critical compressibility factor is * To whom correspondence should be addressed. E-mail:
[email protected]. Phone: (403) 220-8755. Fax: (403) 284-4852.
[(
Zc ) d(1 + xd) ln
)
]
xd - 1 1 + xd d xd
(4)
Sanchez and Lacombe presented equivalent critical point equations but did not emphasize that the parameters could be evaluated from the pure substance critical temperature and pressure as is done routinely with cubic equations of state. Corresponding States Theory From the previous equations, the Sanchez-Lacombe equation implies a three-parameter corresponding states theory in that all substances with the same d parameter have the same equation of state in reduced temperature, reduced pressure, and reduced volume. Furthermore, substances with the same d parameter will have the same reduced vapor pressure curve and, therefore, the same acentric factor ω since
ω ≡ -log10(Pvap/Pc)|Tr)0.7 - 1
(5)
The acentric factors corresponding to given values of d were found through vapor pressure calculations at a reduced temperature of 0.7. At each d, the acentric factor was calculated from the vapor pressure via eq 5. The resulting ω is independent of the critical temperature and critical pressure used in the calculation. The relationship between ω and d is well fit by the following cubic polynomial:
d ) 5.1178 + 13.5698ω + 5.9404ω2 - 1.2952ω3 (6) This expression reproduces the equation of state relationship with a maximum error in d of 0.004 over the range of acentric factors from zero to 1.00. Given the critical temperature, critical pressure, and acentric factor for a pure substance, the three SanchezLacombe parameters can be evaluated. The acentric factor is used to calculate d from eq 6. Then, and ν* are evaluated from the substance critical temperature, critical pressure, and d, using eqs 2 and 3. This procedure ensures a match of vapor pressure data at the critical point and at a reduced temperature of 0.7. Vapor Pressures Vapor pressures have been calculated using this parametrization for seven substances with a range of acentric factors from near zero (methane) to 0.64 (etha-
10.1021/ie990800m CCC: $19.00 © 2000 American Chemical Society Published on Web 02/26/2000
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Ind. Eng. Chem. Res., Vol. 39, No. 4, 2000
Table 1. Properties of Substances substance
Tc, K
Pc, bar
acentric factor (est. from data)
data source
methane ethylene benzene cyclohexane n-hexane methanol ethanol
190.56 282.34 562.60 553.05 507.87 512.64 513.92
45.95 50.40 49.24 40.30 30.32 80.97 61.48
0.010 44 0.086 78 0.207 17 0.208 53 0.298 41 0.563 78 0.644 47
7 8 9 9 10 11, 12 11, 12
nol). Table 1 lists the data sources for the vapor pressure data employed, the critical properties reported with the data, and the acentric factors that we estimated from the data. Table 2 shows the maximum error in vapor pressures calculated from the Sanchez-Lacombe equation and compares it with the results given by the SoaveRedlich-Kwong5 equation and the Peng-Robinson6 equation. In the interval 0.7 < Tr < 1.0, the maximum error in vapor pressure for most of the materials examined is less than 4%. As an exception, the maximum error for methanol is 5.54%. Extrapolation to low reduced temperatures, in general, results in larger errors. For most of the substances, the maximum error at reduced temperatures, Tr < 0.7, occurs at the lowest temperature examined. The performance of the Sanchez-Lacombe equation, as shown in Table 2, is somewhat inferior to the results from the familiar cubic equations of state but is comparable. It should be noted, however, that no temperature dependence has been introduced into the three Sanchez-Lacombe parameters. A possibility for improving the results is to add temperature dependence to the parameter. The form
) c[1 + κ(1 - Tr/0.7)(1 - Tr)]
(7)
would preserve the vapor pressure at reduced temperature 0.7 and at the critical point. The constant κ may be chosen for a given substance to fit the vapor pressure at a temperature of particular interest. Equation 7 proves to be particularly effective for methane and methanol. With κ ) -0.021 for methane and κ ) -0.173 for methanol, the maximum vapor pressure errors over the whole temperature range were found to be 1.6% and 0.8%, respectively. Note the very remarkable maximum error reduction for methanol, from about 22% to less than 1%. The improvement possible for the ethylene vapor pressure is not so dramatic but is still significant. With κ ) +0.034, the vapor pressure error at the lowest temperature datum
Figure 1. Ethylene: calculated error in vapor pressure [%] versus reduced temperature Tr; experimental data from ref 8; calculated from eq 7.
Figure 2. Methanol: calculated error in vapor pressure [%] versus reduced temperature Tr; experimental data from refs 11 and 12; calculated from eq 7.
is reduced to 8.33%. The error elsewhere is somewhat increased but remains less than 3.85% over the temperature interval from 0.39 < Tr < 1.0. Figures 1 and 2 present the errors in vapor pressure for ethylene and methanol, respectively, plotted against reduced temperature. Volumetric Behavior As ω varies over the range from zero to 1.5 (the interval where virtually all the experimental data fall), the d parameter varies from near 5 to near 25. The
Table 2. Maximum Percent Error in Vapor Pressure Tr > 0.7 methane (κ ) 0) (κ ) -0.021) ethylene (κ ) 0) (κ ) +0.034) benzene cyclohexane n-hexane methanol (κ ) 0) (κ ) -0.173) ethanol
Tr < 0.7
S-L
PR
SRK
S-L
1.64 1.31
0.89
1.54
-2.45 1.44
1.48 2.07 1.99 2.75 3.83
0.47
1.20
28.7
-0.79 0.96 1.96
1.36 2.09 3.01
28.3 8.33 3.06 4.57 -2.87
3.31
3.72
-11.4
2.83
0.63
-22.0 0.29 -13.1
5.54 0.81 2.67
PR 1.69
9.16 10.39 3.41
4.06
SRK
lowest Tr in data set
-7.68
0.476
3.08
0.368
1.74 3.56 -2.10
0.498 0.512 0.538
-19.2 -9.05
0.562 0.570
Ind. Eng. Chem. Res., Vol. 39, No. 4, 2000 1117
critical compressibility factor is 0.363 when d ) 5 and is 0.348 when d ) 25. (Sanchez and Lacombe point out that the limiting critical compressibility for infinite d is 1/3.) These values are much too high when compared with experimental data. Consequently, the saturated liquid volumes calculated from the Sanchez-Lacombe equation with parameters fitted to the critical pressure and temperature will tend to be too large, especially at high reduced temperature. This deficiency can be compensated to some degree by employing a “volume shift” as suggested by Pe´neloux et al.13 for the cubic equations of state. The volume of a mixture can be calculated from the equation of state volume through nc
v
shift
)v+
xici ) v + c ∑ i)1
(8)
So long as the shift parameter is linear in the mole fractions, there is no effect on the phase equilibrium. The consequences of the volume shift process can be shown by starting with the Helmholtz free energy. The “shifted” Helmholtz free energy is taken as
Ashift ) A(T,V + nc, n1, ..., nc)
(9)
The expressions for the pressure and chemical potential in the shifted model are found by differentiation as usual. For the pressure,
Pshift ) P(T,V + nc)
(10)
The shifted chemical potential is
) µi + ciP µshift i
(11)
As a result of this last equation, shifting the volume does not change the compositions in the equilibrium phases at a fixed temperature and pressure. The equations also hold even if the ci volume shift parameters are temperature dependent (although there will be an effect on the temperature derivatives). Summary The Sanchez-Lacombe parameters for a pure substance may be calculated from the critical temperature and pressure and the acentric factor through eqs 6, 2, and 3. Without any temperature dependence being introduced into the parameters, calculated vapor pressures agree with experimental values over very wide temperature ranges with an accuracy comparable to that achieved by the original forms of the SoaveRedlich-Kwong and Peng-Robinson equations. The vapor pressure prediction can be improved by introducing temperature dependence into the energy parameter, such as eq 7. Adding substance-dependent temperature dependence to the a parameter in the cubic
equations of state has proved to be an effective tool for matching vapor pressure data precisely.14-16 Volumetric predictions from the equation can be improved through a volume shift, as in eq 8. Acknowledgment This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. Literature Cited (1) Sanchez, I. C.; Lacombe, R. H. An Elementary Molecular Theory of Classical Fluids. Pure Fluids. J. Phys. Chem. 1976, 80, 2352. (2) Lacombe, R. H.; Sanchez, I. C. Statistical Thermodynamics for Fluid Mixtures. J. Phys. Chem. 1976, 80, 2568. (3) Sanchez, I. C.; Lacombe, R. H. Statistical Thermodynamics of Polymer Solutions. Macromolecules 1978, 11, 1145. (4) Koak, N.; Heidemann, R. A. Polymer-Solvent Phase Behavior near the Solvent Vapor Pressure. Ind. Eng. Chem. Res. 1996, 35, 4301. (5) Soave, G. Equilibrium Constants from a Modified RedlichKwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197. (6) Peng, D.-Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59. (7) Angus, S.; Armstrong, B.; de Reuck, K. M. International Thermodynamic Tables of the Fluid State, Methane; Pergamon Press: New York, 1976. (8) Younglove, B. A. Thermophysiscal Properties of Fluids. I. Argon, Ethylene, Parahydrogen, Nitrogen, Nitrogen Trifluoride, and Oxygen. J. Phys. Chem. Ref. Data 1982, 11 (Supplement No. 1). (9) Vargaftik, N. B. Handbook of Physical Properties of Liquids and Gases. Pure Substances and Mixtures, 2nd ed.; Hemisphere Publishing Corporation: Washington, 1983. (10) Canjar, L. N.; Manning, F. S. Thermodynamic Properties and Reduced Correlations for Gases; Gulf Publishing Co.: Houston, TX, 1967. (11) Ambrose, D.; Sprake, C. H. S. Thermodynamic Properties of Organic Oxygen Compounds. XXV. Vapor Pressures and Normal Boiling Temperatures of Aliphatic Alcohols. J. Chem. Thrermodyn. 1970, 2, 631. (12) Ambrose, D.; Sprake, C. H. S.; Townsend, R. Thermodynamic Properties of Organic Oxygen Compounds. XXXVII. Vapor Pressures of Methanol, Ethanol, Pentan-1-ol, and Octan-1-ol from the Normal Boiling Temperature to the Critical Temperature. J. Chem. Thermodyn. 1975, 7, 185. (13) Pe´neloux, A.; Rauzy, E.; Fre´ze, R. A Consistent Correction for Redlich-Kwong-Soave Volumes. Fluid Phase Equilibr. 1982, 8, 7. (14) Mathias, P. M. A Versatile Phase Equilibrium Equation of State. Ind. Eng. Chem. Process Des. Dev. 1983, 22, 385. (15) Stryjek, R.; Vera, J. H. An Improved Cubic Equation of State. In ACS Symposium Series; Chao, K. C., Robinson, R. L., Jr., Eds.; American Chemical Society: Washington, DC, 1985; Vol. 300, p 560. (16) Holderbaum, T.; Gmehling, J. PSRK: A Group Contribution Equation of State Based on UNIFAC. Fluid Phase Equilibr. 1991, 70, 251.
Received for review November 8, 1999 Revised manuscript received January 27, 2000 Accepted February 1, 2000 IE990800M
