Ind. Eng. Chem. Res. 1992,31, 1830-1833
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Comparison of Generalized Methods To Predict Gas-Phase Heat Capacity A comparison was made of seven generalized equations of state on the ability to predict Cpvalues. They were tested using published critically evaluated data. The results showed that all equations give relatively large errors. Introduction The basic thermodynamic properties of most gases at zero pressure (ideal gases) are either available or can be calculated from literature data (Rossini et al., 1952; Thinh et al., 1971; TRC Thermodynamic Tables, 1985; Technical Data Book, 1983). These properties at higher pressures deviate sometimes substantially from the ones of ideal gas at the same conditions. In many chemical engineering calculations, it is imperative to have accurate values of real gas thermodynamic properties. Since these are not always available or the available data do not cover a given condition, estimating methods must be developed. One usual method is to obtain these properties from generalized equations of state by using rigorous thermodynamic relationships. There are many generalized equations of state in the literature to be used for this purpose, and a review of this field is given by Anderko (1990). Comparison studies have been made in the past for calculating enthalpy (Tarakad and Danner, 1976; Toledo and Reich, 19881, density and fugacity (Tarakad et al., 1979), and entropy (Ormanoudisand Stamatoudis, 1988). Garipis and Stamatoudis (1992) compared the ability of eight well-known generalized equations of state (Barner and Adler, 1970; Sugie and Lu, 1970, 1971; Soave, 1972; Lee et al., 1973; Yamada, 1973; Lee and Kesler, 1975; Peng and Robinson, 1976) to predict heat capacity values. All examined equations gave relatively large errors. Since heat capacity is an important property and there is a scarcity of experimental data, further search for an accurate estimating method for Cpis necessary. The heat capacity, CR is given below as the s u m of ideal gas value, Cpo, and the residual heat capacity, AC,, C p = C p o + ACp The residual heat capacity can be calculated by substituting the experimental pressure-volume-temperature data (or the corresponding equation of state) in the following rigorous relationship
constant T (1) Purpose of this work is to extend the previous work of Garipis and Stamatoudis (1992) on Cp predictions by Table I. Equations Tested equations of state Starling-Han (Starling and Han, 1972) Nishiumi-Saito modification of SH (Nishiumi and Saito, 1975) Adachi-Lu-Sugie (Adachi et al., 1983) Chao-Zhong modified Lee-Kesler (Chao and Zhong, 1986) Orbey-Vera (Orbey and Vera, 1989) Platzer-Maurer generalization of Bender (Platzer and Maurer, 1989) Valderrama generalization of Patel-Teja (Valderrama, 1990)
abbreviation SH NS
input data required T,,V,, w Tc, V,, w
ALS CZ
T,,Pc,w T,,P,,w
OV PM
T,,V,, w T,,V,, w
VAL
T,,Pc, w
Table 11. Components Used for Heat Capacity Data Base pressure temperature no. of component range (bar) range (K) values (1) . . Light - Hvdrocarbons (C < 5) 1-loo0 140-650 339 methane 1-1000 250-550 323 ethane 270-1200 1-700 362 propane 1-400 280-1200 276 n-butane 1-400 270-1200 273 isobutane 1-lo00 320-1000 219 1-butene 1-1000 320-1000 266 isobutene 222 1-50 280-1000 ProPYne 1-120 propylene 80 298-473 (2) Intermediate Hydrocarbons (Cs-Cs) n-hexane 1-30 400-1000 n-heptane 1-1000 440-1000 n-octane 1-1000 480-1000 isooctane 1-1000 440-1000 cyclohexane 1-1000 420-1000 nonane decane
(3) Heavy Hydrocarbons (Cg-C,,) 1-20 500-1000 1-20 520-1000
116 102
(4) Aromatic Hydirocarbons 520-1000 1-1000 540-1000 1-lo00 1-1000 580-1000 560-1000 1-1000 490-1000 1-1000 580-1000 1-1000 1-1000 430-1000 1-1000 600-1000 600-1000 1-1000 490-1000 1-1000 490-1000 1-1000 490-1000 1-1000
146 133 134 147 136 139 189 134 129 173 155 173
cumene p-cymene cis-decalin trans-decalin naphthalene tetralin toluene quinoline isoquinoline rn-xylene o-xylene p-xylene methanol ethanol 1-propanol 2-propanol
(5) Alcohols 1-1000 1-1000 1-1000 1-1000
400-1000 400-1000 420-1000 420-1000
247 224 215 224
ethanethiol methanethiol sulfur dioxide thiophene
(6) Sulfur Components 1-1000 380-1000 1-1000 340-1000 1-1000 310-1000 1-1000 430-1000
190 224 223 172
Freon-11 Freon-12 Freon-13 Freon-21 Freon-22 Freon-23 Freon-113 Freon-114 Freon-ll4a Freon-152a diethyl ether methanamine
(7) Refrigerants 1-40 360-1000 0.1-40 220-520 1-30 240-1000 1-50 340-1000 1-40 240-1000 1-40 220-1000 1-30 340-1000 1-30 280-1000 1-30 300-1000 0.1-250 220-520 1-1000 380-1000 1-50 300-1000
155 159 161 168 167 175 133 135 168 182 199 174
(8) Unclassified Components 1-1000 400-1000 1-50 110-600
186 150
acetone oxygen
DDatafor all components, with the exception of those for propylene (Bier et al., 1974), are taken from TRC Thermodynamic Tables (1985).
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128 193 124 188 226
Ind. Eng. Chem. Res., Vol. 31, No. 7, 1992 1831
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1832 Ind. Eng. Chem. Res., Vol. 31, No. 7,1992 Table IV. Number of ur Values Falling in Each Interval of up intervals S H a n d N S CZ ALS VAL ov of region A region A region A region B region A region B region A redon B 254 total no. of UF values 102 158 153 154 174 107 124 0-10 12 2 2 0 2 0 3 0 light hydrocarbons (C1-C4) 26 10-20 9 7 0 5 0 5 3 2 15 33 20 29 12 22 6 >20 0 0 0 0 0 0 2 0 0-10 medium hydrocarbons (Cs-C8) 5 0 7 10-20 0 8 0 1 0 20 >20 10 5 18 4 24 3 14 0 0-10 heavy hydrocarbons (Cs-Clo) 0 0 0 0 0 0 0 0 10-20 0 0 0 0 0 0 0 8 4 4 4 4 4 0 4 >20 0 aromatic hydrocarbons (1 or 2 rings) 0-10 0 0 0 0 0 0 0 8 10-20 0 15 0 15 0 10 0 44 >20 20 29 26 26 38 19 27 2 0-10 refrigerants 0 3 0 3 0 1 0 37 10-20 4 13 0 16 0 19 2 >20 40 32 36 32 25 33 11 23 0-10 3 sulfur components 0 0 0 0 0 1 0 10-20 10 3 4 0 3 0 4 0 4 >20 4 3 18 4 18 3 12 0 0-10 0 0 0 0 0 0 0 alcohols 0 10-20 0 0 0 0 0 0 0 >20 18 6 10 14 14 20 8 11 0-10 5 oxygen 1 0 0 0 0 0 0 0 10-20 2 0 0 0 0 2 1 0 0 2 3 2 3 1 1 >20 0 0-10 0 0 0 0 0 0 0 acetone 5 10-20 0 2 0 1 0 0 0 >20 1 3 1 5 2 5 2 4 ~~~
comparing seven more equations of state, most of them proposed in the past decade.
Equations Tested The equations teated are listed in Table I. Two of them are of cubic type (Adachi et al., 1983;Valderrama, 1990), four are of virial type (Starling and Han, 1972;Nishiumi and Saito, 1975; Chao and Zhong, 1986; Platzer and Maurer, 1989),and one is of augmented hard core type (Orbey and Vera, 1989). It should be noted that the critical temperature, critical pressure, and the acentric factor were used as inputs to them. Critical property values and acentric factors are taken from the literature (TRC Thermodynamic Tables, 1985;Reid et al., 1977;Daubert and Danner, 1985;Cholinski et al., 1986).
Sources of Heat Capacity Data Base The published critically evaluated C p data (TRC Thermodynamic Tables, 1986)of 49 components were used as the data base upon which the comparison was made. Data for propylene was taken from Bier et al. (1974).The components and the range of the data base are given in Table II. They are the same as the ones used in an earlier report (Garipis and Stamatoudis, 1992). The data base consisted of 9260 base C pdeparture values of 50 substances (2inorganics, 29 hydrocarbons, and 19 non-hydrocarbon organics). These values were classified into 345 groups (regions) to expedite the calculations. Similarly, the corresponding 9260 calculated C p departure values are obtained for each tested equation and classified to the previously corresponding 345 groups (regions). A comparison then is made of each of the 345 groups of calculated C p departure with ita corresponding base C p departure group. The root mean square error (group standard fractional deviation), uF, and the group maximum fractional deviation are calculated using the following equations: 1 112 uF = lm( ~ ~ ( ( A c P , c a-l AcP,bam)/AcP,bu.)I) c (2)
Results and Conclusions A comparison between the seven equations is made on the ability to predict C p values using the 345 group values for uF, the maximum fractional deviation, and the 85% confidence intervals for the uF values. The results of comparison are presented in Tables I11 and IV. Table IIIgives the values for 3, the maximum fractional deviation, and the 85% confidence interval containing the 85% smaller UF values for each of the equations studied. It should be noted that the results here made it necessary to classify the compounds in eight categories (as presented in Table 11): small, intermediate, heavy and aromatic hydrocarbons, alcohols, compounds containing sulfur, refrigerants, and the category of unclassified compounds of oxygen and acetone. Each of these categories was subdivided usually into three regions A, B, and C in order of decreasing Cp departure prediction ability. Category C has the worst predictions and is not included in this work. Regions A and B are defined in Table 111. Table IV presents the distribution of UF values. It gives the number of UF values falling in each oF interval. The comparison between the equations show that they give various results depending on the temperature and pressure. None of the equations studied in this work gives accurate results. The following are observed. PM Equation. It gives C p predictions almost always larger than the base values, and the U F values are almost always higher than 300. Thus, it is not discussed further in this work. SH and NS Equations. In the ranges studied, both equations give exactly the same results. This is expected because equation NS is a modified form of equation SH. In region A, the uF values range between 4 and 45. At greater pressures, oF becomes very large. CZ Equation. In region A, the U F values range from 9 to 55. At greater pressures, UF values exceed 100.
Ind. Eng. Chem. Res., Vol. 31,No. 7, 1992 1833
ALS and VAL Equations. These equations give simvalues range
Garipis, D.; Stamatoudis, M. Comparison of Generalized Equations of State to Predict Gas Phase Heat Capacity. AIChE J . 1992,38,
OV Equation. The estimated Cpvalues by this equation are always smaller than the base ones. In region A, the uF values range from 18 to 31. At pressures PI > 6 predictions become unacceptable. The relatively best equations for each group of compounds in region A are given below (the 85% confidence intervals are given in parentheses). Small Hydrocarbons. Equations SH and NS (13-17 and 13-21, respectively). Intermediate Hydrocarbons. Equations ALS and VAL (18-22). Heavy Hydrocarbons. Based on only two components, all equations give large errors. Aromatic Hydrocarbons. Equation OV (24-36). Refrigerants. Equations SH and NS (21-25). Components Containing Sulfur. Equations SH and NS (12-17). Alcohols. Equation OV (27-32). Unclassified. Equations SH and NS (2-6)for oxygen and acetone (13-21).
Lee, B.-I.; Kesler, M. G. A Generalized Thermodynamic Correlation on Three-Parameter Corresponding States. AZChE J . 1975,21,
ilar results in both regions. In region A, the from 19 to 48.
Nomenclature Cp = heat capacity of real gas, J/(mol.K) Cpo = heat capacity of ideal gas, J/(mol.K) N = number of points P = pressure, bar Pc = critical pressure, bar PI = reduced pressure, dimensionless R = ideal gas constant, J/(mol.K) ACp = heat capacity departure (real-ideal), J/ (mo1-K) ACP,bm = heat capacity departure of data base, J/(mol-K) ACp,calc= calculated heat capacity departure, J/ (mol-K) T = temperature, K Tc = critical temperature, K TI = reduced temperature, dimensionless V = volume, m3/mol Greek Symbols uF = root mean square error (group standard fractional deviation), dimensionless w = acentric factor, dimensionless
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Lee, B.-I.; Erbar, J. H.; Edmister, W. C. Prediction of Thermodynamic Properties for Low Temperature Hydrocarbon Process Calculations. AIChE J. 1973,19,349. Nishiumi, H.; Saito,S. An Improved Generalized BWR Equation of State Applicable to Low Reduced Temperatures. J . Chem. Eng. Jpn. 1975,8, 356. Orbey, H.; Vera, J. H. An Augmented Hard Core Equation of State Generalized in Terms of T,, P,, and w. Pure Appl. Chem. 1989, 61, 1413.
Ormanoudis, H.; Stamatoudis, M. A Comparison of Eight Generalized Equations of State to Predict Gas-Phase Entropy. Ind. Eng. Chem. Res. 1988,27, 364. Peng, D.-Y.; Robinson, D. B. A New Two-Constant Equation of State. Znd. Eng. Chem. Fundam. 1976,15,59. Platzer, P.; Maurer, G. A Generalized Equation of State for Polar and Nonpolar Fluids. Fluid Phase Equilib. 1989,51, 223. Reid, R. C.; Prausnitz, J. M.; Sherwocd, T. K. The Properties of Gases and Liquids; McGraw-Hill: New York, 1977. Rossini, F. D.; Wagman D. W.; Evans, W. H.; Levine, S.; Jaffe, I. Selected Values of Chemical Thermodynamic Properties; National Bureau of Standards Circular 500; Government Printing Office: Washington, DC, 1952. Soave, G. Equilibrium Constants From a Modified Redlich-Kwong Equation of State. Chem. Eng. Sci. 1972,27, 1197. Starling,K. E.; Han, M. S. Thermo Data Refined for LPG. Part 1 4 Mixtures. Hydrocarbon Process. 1972,51, 129. Sugie, H.; Lu, B. C.-Yu. Generalized Equation of State for Gases. Znd. Eng. Chem. Fundam. 1970,9,428. Sugie, H.; Lu, B. C.-Yu. Generalized Equation of State for Vapors and Liquids. AIChE J . 1971,17, 1068. Tarakad, R. R.; Danner, R. P. A Comparison of Enthalpy Prediction Methods. AIChE J . 1976,22,409. Tarakad, R. R.; Spencer, C. F.; Adler, S B. A Comparison of Eight Equations of State to Predict Gas-Phase Density and Fugacity. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 726. Technical Data Book; American Petroleum Institute: Washington, DC, 1983. Thinh, T. P.; Duran, J. L.; Ramalho, R. S.; Kaliaguine, S. Equations Improve Cp Predictions. Hydrocarbon Process. 1971,50,98. Toledo, P. G.; Reich, R. A Comparison of Enthalpy Prediction Methods for Nonpolar and Polar Fluids and Their Mixtures. Znd. Eng. Chem. Res. 1988,27,1004. TRC Thermodynamic Tables (formerly API Research Project 44); Thermodynamics Research Center, The Texas A&M University System: College Station, TX, 1985. Valderrama, J. 0. A Generalized Patel-Teja Equation of State for Polar and Nonpolar Fluids and Their Mixtures. J . Chem. Eng. Jpn. 1990,23, 87. Yamada, T. An Improved Generalized Equation of State. AIChE J . 1973, 19, 286.
* To whom correspondence should be addressed. Antonio6 Karkaris, Theodore Kalfopoulos Michael Stamatoudis* Department of Chemical Engineering Aristotle University of Thessaloniki Thessaloniki, Greece Received for review December 17, 1991 Accepted April 17, 1992
