Ind. Eng. Chem. Res. 1988,27, 364-366
364
A Comparison of Eight Generalized Equations of State To Predict Gas-Phase Entropy Eight generalized equations of state are evaluated for their ability to predict gas-state entropy. The equations studied are the Lee-Kesler, Lee-Erbar-Edmister, Sugie-Lu, Redlich-Kwong-Soave, Barner-Adler, the two Yamada's modifications of the Benedict-Webb-Rubin (BWR) equation, and the Peng-Robinson equation of state. Studies include organic, inorganic, and polar substances. Pure components and mixtures are considered. Reliable methods for estimating thermodynamic properties of substances are of great importance in many engineering calculations and design situations encountered in industry. In the past, comparisons have been made between several equations of state for predicting enthalpy (Tarakad and Danner, 1976) and density and fugacity (Tarakad et al., 1979). There is no similar work available for predicting entropy. A t zero pressures, all gases behave ideally and the entropy can be easily calculated from ideal gas heat capacities, available in the literature (Kobe et al., 1949-1958; Rossini et al., 1952; Selected Values of Properties of Hydrocarbons and Related Compounds, 1983; Technical Data Book, 1983). At higher pressures, the entropy of a fluid relative to ita entropy as an ideal gas can be calculated by substituting the experimental pressure-volume-temperature data (or the corresponding equation of state) in the following rigorous relationship:
In case there are no P-V-T data for a certain compound or the available data do not cover a given condition, generalized correlations are of great help. The purpose of this paper is to compare eight generalized equations and evaluate their ability to predict entropy departures of the real gas from the corresponding ideal gas values.
Equations Studied There are many equations of state available in the literature. The eight generalized equations evaluated in this study and their input data required are listed in Table I. Several other less important literature equations are not included, in order to keep this study in reasonable bounds.
Sources of Entropy Data Base The eight generalized equations were tested against a data base set consisting of 2021 calculated entropy values, which comprised 20 single components and 6 binary mixtures. Table I1 gives a listing of them, the source, and the data range. The data used were entropy departures from the ideal gas values and were calculated for the purpose of the present work by using the respective Benedict-WebbRubin (BWR) equations of state found in the above-mentioned sources. The respective authors obtained the BWR constants by fitting to experimentalP-V-T data. Entropy data base values were obtained only for the region where P-V-T data exist. The error in the calculated entropy data base is estimated to be less than 1%.
Results and Discussion Tables I11 and IV summarize the results of the evaluations for the eight generalized equations when tested with pure components and mixtures. The results are grouped into three major categories, depending on whether the substances tested are polar (NO and NH3) or nonpolar (including the mildly polar COz and SOz). The nonpolar substances are further grouped into organic and inorganic ones. Tests were made separately for the regions of reduced pressures, P, < 1and P, > 1. The investigation was accomplished by making a statistical analysis of the entropy departure predictions, as compared to the calculated 2021 base entropy departure values. This is done because there are no experimental entropy data available. The average, biased, standard, and standard fractional deviations are calculated for each equation and the results compared. The eight generalized equations give much more reliable results when they are used for the predictions of entropy departures of nonpolar organic compounds (mainly hydrocarbons). This is expected, since the state equations
Table I. Equations Evaluated in This Study
no.
eauation of state Lee-Kesler (1975) 3-parameter corresponding states correlation Lee-Erbar-Edmister (1973) eq of state Sugie-Lu (1970, 1971) eq of state Soave (1972) modification of the Redlich-Kwong eq of state Barner-Adler (1970) modification of the Joffe eq of state Yamada (1973) generalization of the BWR eq of state using 8 parameters Yamada (1973) generalization of the BWR eq of state using 16 parameters Peng-Robinson (1976) eq of state
abbreviation
input data required mixtures (additional Dure comDonents data)
"The parameters m, and m2 are given by Lee et al. (1973); if they are not available, they can be set equal to zero without an appreciable error. *The interaction parameter kij is given by Chueh and Prausnitz (1967); if it is not available, it can be set equal to zero without an appreciable error. cThe interaction parameter K , is given by Barner and Quinlan (1969); if it is not available, it can be set equal to one without an appreciable error. dIf the interaction coefficient b,, is not available, it can be set equal to zero.
0888-5885/88/2627-0364$01.50/0
0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 365 Table 11. Temperature and Pressure Ranges and Literature Sources for Entropy Data Base system methane
temp range, K 250-470
pressure range, atm 25-250
ethane
300-540
11-110
propane
400-540
25-90
n-butane
450-570
15-70
isobutane
380-510
13-65
n-pentane
470-550
12-65
isopentane
440-550
12-65
n-hexane
520-548
26-50
n-heptane
550-620
27-48
ethylene
290-470
20-75
propylene
330-570
15-75
isobutylene
430-540
40-90
benzene
530-625
25-65
pentafluoromonochloroethane argon
320-450
12-69
170-600
1-240
sulfur dioxide
350-520
10-250
nitrogen
110-370
5-150
carbon dioxide ammonia
330-390 320-580
10-220 1-500
nitrogen oxide
280-370
1-170
methane-nitrogen (90%-10%) methane-nitrogen (70%-30% ) e thylene-carbon dioxide (80%-20% ) ethylene-carbon dioxide (60%-40%) ethylene-carbon dioxide (40%-60%) ethylene-carbon dioxide (20%-80%)
200-350
1-100
200-350
1-100
330-390
20-200
Zudkevitch and Kaufmann, 1966 Kang and McKetta, 1961 Crain and Sonntag, 1967 Sass et al., 1967 Ramalho and Frizelle, 1965 Seshadri et al., 1967 Bloomer et al., 1955 Bloomer et al., 1955 Sass et al., 1967
330-390
20-220
Sass et al., 1967
330-390
20-250
Sass et al., 1967
a
330-390
20-250
Sass et al., 1967
c
lit. source Benedict et al., 1951 Benedict et al.. 1951 Benedict et al., 1951 Benedict et al., 1951 Benedict et al., 1951 Benedict et al., 1951 Benedict et al., 1951 Benedict et al., 1951 Benedict et al.. 1951 Benedict et al., 1951 Benedict et al., 1951 Benedict et al., 1951 Organick and Studhalter, 1948 Mears et al., 1966
were developed using mainly hydrocarbon data. The equations, with the exception of BWR44, give better predictions in the region of pressures P, > 1. For these substances, the best results for engineering calculations are obtained by using the BWR24, JBA, LK, LEE, and BWR44 equations. Greater errors are observed when the equations are applied to inorganic substances. For these substances, the relatively best equation is the JBA one. In the case of polar compounds, all eight equations fail, giving results which deviate substantially from the data base. Finally, a test was made to investigate the ability of the equations to predict entropy departures for gas mixtures. Extensive study was not possible, due to limited mixtures available for comparisons. The mixing rules suggested by the original authors were used with required input data
I
a t
I
E
c 0
E
E
2
9
366 Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988
referred to in Table I. The best predictions are obtained by using the BWR24, JBA, and BWR44 equations.
Conclusions and Recommendations The following conclusions can be drawn and recommendations given, after evaluating and comparing the eight generalized equations for their ability of predicting gasphase entropy departures: 1. Organic Substances (Mainly Hydrocarbons). Equations BWR24, JBA, LK, LEE, and BWR44 are good for engineering calculations. 2. Inorganic Substances. The JBA equation should be used. 3. Polar Compounds. None of the equations give satisfactory results. 4. Gas Mixtures. The recommended equations are the BWR24, JBA, and BWR44 ones.
Nomenclature av = average deviation defined in the footnote of Table 111, J/(mol.K) bias = bias deviation defined in the footnote of Table 111, J/(mol.K) Itij, Kij = interaction parameters ml, m2 = mixture parameters N = number of points P = pressure, atm P, = critical pressure, atm P, = reduced pressure, dimensionless R = ideal gas constant, J/(mol.K) A S = entropy departure, J/(mol.K) ASbase= entropy departure of data base, J/(mol.K) AScdcd = calculated entropy departure, J/ (mo1.K) T = temperature, K T,= critical temperature, K V = volume, m3/mol V , = critical volume, m3/mol xi = mole fraction, dimensionless
Literature Cited Barner, H. E.; Adler, S. B. Ind. Eng. Chem. Fundam. 1970, 9, 521. Barner, H. E.; Quinlan, C. W. Ind. Eng. Chem. Process Des. Deu. 1969, 8, 407. Benedict, M.; Webb, G. B.; Rubin, L. C. Chem. Eng. Prog. 1951,47, 419. Bloomer, 0. T.; Eakin, B. E.; Ellington, R. T.; Gami, D. C. "Thermodynamic Properties of Methane-Nitrogen Mixtures". Research Bulletin No. 21, Feb 1955; Institute of Gas Technology, Chicago, IL. Chueh, P. L.; Prausnitz, J. M. Ind. Eng. Chem. Fundam. 1967, 6, 492. Crain, R. W.; Sonntag, R. E. J. Chem. Eng. Data 1967, 12, 73. Kang, T. L.; McKetta, J. J. J . Chem. Eng. Data 1961, 6, 227. Kobe, K. A., et al. Thermochemistry of Petrochemicals;Petroleum Refiner, Gulf Publishing: Houston, Jan 1949-July 1958; reprint. Lee, B. I.; Erbar, J. H.; Edmister, W. C. AIChE J . 1973, 19, 349. Lee, B. I.; Kesler, M. G. AIChE J . 1975,21, 510. M e m , W. H.; Rosental, E.; Sinka, J. V. J . Chem. Eng. Data 1966, 11, 338. Organick, E. I.; Studhalter, W. R. Chem. Eng. Prog. 1948,44, 847. Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976,15,59. Ramalho, R. S.; Frizelle, W. G. J. Chem. Eng. Data 1965, 10, 366. Rossini, F. D., et al. Selected Values of Chemical Thermodynamic Properties;National Bureau of Standards Circular 500; Government Printing Office: Washington, D.C., 1952. Sass, A.; Dodge, B. F.; Bretton, R. H. J . Chem. Eng. Data 1967,12, 168. Selected Values of Properties of Hydrocarbons and Related Compounds; Thermodynamics Research Center, Texas A&M University: College Station, 1983. Seshadri, D. N.; Viswanath, D. S.;Kuloor, N. R. J. Chem. Eng. Data 1967, 12, 70. Soave, G. Chem. Eng. Sci. 1972,27, 1197. Sugie, H.; Lu, B. C.-Y. Ind. Eng. Chem. Fundam. 1970, 9, 428. Sugie, H.; Lu, B. C.-Y. AIChE J . 1971, 17, 1068. Tarakad, R. R.; Danner, R. P. AIChE J . 1976,22, 409. Tarakad, R. R.; Spencer, C. F.; Adler, S. B. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 726. Technical Data Book; American Petroleum Institute: Washington, D.C., 1983. Yamada, T. AIChE J. 1973,19, 286. Zudkevitch, D.; Kaufmann, T. G. AIChE J. 1966, 12, 577.
Harisios Ormanoudis, Michael Stamatoudis* Greek Symbols
fii, = interaction coefficient, dimensionless cA = standard deviation defined in the footnote of Table IV, J/ (mo1.K) gF = standard fractional deviation defined in the footnote of Table IV, dimensionless w = acentric factor
Department of Chemical Engineering Aristotle University of Thessaloniki GR-54006 Thessaloniki, Greece Received for review January 21, 1987 Revised manuscript received July 13, 1987 Accepted July 27, 1987
