Correlation of Ideal Gas Enthalpy, Heat Capacity, and Entropy Charles A. Passut and Ronald P. Dannerl Department of Chemical Engineering, The Pennsylvania State University, University Park, Pa, 16802
The accepted values of the ideal gas enthalpy, heat capacity, and entropy of 89 compounds, mostly hydrocarbons, have been represented by a set of three thermodynamically consistent polynomials. The constants for these equations, derived by minimizing the deviations in all three properties simultaneously, are tabulated together with the average and maximum percent deviation for each compound. The overall error analysis indicates that thermodynamic consistency can be maintained while yielding excellent predictions for the three properties.
For the heat. capacit'y, (Cp* = dH*/dT): Ideal gas properties are frequently required in correlations of enthalpy, heat capacity, and entropy. In general, these properties are correlated in terms of their deviations from ideality and, thus to obtain reliable absolute values, accurate predictions of the ideal gas values are required. The American Petroleum Instit'ute, Research Project 44 publishes ideal gas enthalpy, heat capacity, and entropy values in tabular form. These values were corrected for the molecular weight change to the carbon-12 basis. Frequently it is more convenient t'o have these values represented by analytical equations for comput,er applications. Several methods have been used to obtain analytical expressions. I n the M I "Technical Dat'a Book-Petroleum Refining" (1971), a set of three independent polynomial equations for ideal gas enthalpy, heat capacity, and entropy are given. Xn independent set of constants is provided for each polynomial and thus one cannot go from one equation to the other by the standard t'hermodynamic formula-Le., AH* = J C p * d T and AS* = J ( C p * / T ) d T . This method provides greater accuracy in fitting of the individual properties at the expense of being thermodynamically inconsistent. I n a recent publication, Thinh et' al. (19i1) reviewed the published equations for predicting ideal gas heat capacity and presented a listing of constants for the ideal gas heat capacity of numerous hydrocarbons and relat'ed compounds in t'he form: Cp*
=
a
+ bT + cT2 + dT3
(1)
Values for the entropy and ent'halpy can be predicted from this equat.ion by the appropriate thermodynamic relations. However, some loss in accuracy for enthalpy and entropy values is inevitable by t'his procedure, since t'he actual heat capacit,y behavior is not theoretically constrained to follow any particular polynomial. The method described below yields thermodynamically consistent equations with good accuracy for all three properties. The forms of the equations used were as follows. For the enthalpy: H* = A
+ BT + CT2 + D T 3 -+ ET4 + FT5
(2)
where 9, B, C, D!E , and F are derived coefficient's, with the enthalpy in Btu/lb and the temperature in "R. T o whom correspondence should be addressed.
Cp*
B
=
+ 2 CT + 3 CT2 + 4 E T 3 + 5 F T 4
and for the entropy,
S*
=
B In
T
-
TB "3
(3)
[S*= J ( C p * / T ) d T ] :
+ 2 C(T - TB) + E ( T 3 - T B ~+)
'/4
"2
+
D ( T 2- TB2) F(T'
- TB') + G' (4)
where T B is t'he base temperat'ure for the ent'ropy equation (l"R), and G' is the constant of int'egration. Equation 4 may be further simplified by combining the constant base temperature terms with the consbant G' t'o give Equation 5 :
S*
=
B 111 T
+ 2 CT + ' 1 2 DT2 + 4/3ET3
+ 5/4FT4+ G
(5)
where t'he constant G now contains the base correction. The data bases used were 0 Utu/lb a t 0"R for the enthalpy and 0 I3tu/lb " R a t 0"R and 1 a t m pressure for the entropythe same as those used for tmheXPI Research Project 44 tables. (It should be noted t h a t these bases cannot be used for a system in which a chemical react,ion has occurred.) The enthalpy, heat capacity, and entropy data used in these correlations were taken from API Research Project 44 rvith the exception of SO2, H2S, and XI-1; which were obtained from AIcBride et al. (1963). The entire temperature range available from t'he API-RP-44 data was included. The degree of the polynomial in Equation 2 was chosen by balancing t'he improved accuracy obtained with additional terms against the increased complexity of the equations. Correlational Method
Since the constants in the equation are linear, a linear method of solution xas indicated. The three different data t'ypes (enthalpy, heat capacity, and entropy) were arranged in one matrix [VI. The seven coefficients t'o the three equat'ions rvere arranged in the coefficient matrix [A]. A temperature matrix was prepared to sat.isfy Equation 6 : [ T I n x 7 [ ~ 1 5 X l= [VInYJ
(6)
The subscripts denote rows and columns, respectively. T o solve for the values in the [ A ]matrix, the [TImatrix is first transformed into a square matrix by multiplying Equation 6 by the t'ranspose of the [TI matrix, Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 4, 1972
543
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545
(7)
Table II. Comparison of Results with Previous Work
The result of this operation is:
Av Heat capacity
where [SIand [C] are the resultant matrices. The [ A ] matrix may be evaluated by inverting the [SI matrix The matrix [SIis equivalent to a simultaneous leastsquares analysis for the three sets of ideal equations. A proof of this procedure is given by Zeleznik and Gordon (1961). The [SImatrix in Equation 8 can be inverted by standard reduction techniques to find the values in the coefficient matrix [A]. However, this matrix possesses certain special properties which can simplify the inversion. The [SImatrix is positive, symmetric, and nonsingular. Therefore, the method of Cholesky can be used to invert the matrix (Lapidus, 1962). This method minimizes round-off errors in the inversion. The Cholesky method will not be described here, but v a s built into the computer program developed to solve for the ideal gas property coefficients. Initial work with the equations showed that forcing the enthalpy equation through 0 a t 0”R was an unnecessary and undesirable restriction even though the base for the values is 0 a t 0”R. Since none of the data are in the range of 0”R and since no empirical equation for the enthalpy can be extrapolated outside the range of the available data, there is no apparent loss from the addition of a constant, temperature-independent term, to the enthalpy equation. I n the same way the heat capacity equation has a constant term when the enthalpy equation is differentiated. Again the heat capacity is not predicted to be zero a t absolute zero but the equation for Cp* is not applicable in this range. The constant term in Equation 4 (G’) is the predicted entropy of the ideal gas a t the chosen base temperature of 1”R and unit pressure. This value comes out of the regression analysis, and once again, since there is no data in this region, the predicted value has no real significance. The temperature limitations of the data are shown in Table I.
Complete temperature range, 89 compounds This work 0 XPITechnicalDataBook 0 298-1500°K, 7 5 compounds This work 0 0 Thinh et al. (1971) a ( l O O / S ) L: [(predicted value value]. These values are based from 298°K.
436 877
% errora
Enthalpy
Entropy
0 064 0 388
0 040 0 036
249 0 14gb 0 721b 194 0 2746 0 721b - tabulated value)/tabulated on predicted property change
from 298’K since Thinh’s equations cannot be used to obtain absolute values. The results given in Table I1 show marked improvement over the independent set of three equations reported in the A P I Technical Data Book. I n comparison to the correlations of Thinh et al. for the 75 common hydrocarbons, there is a loss in accuracy in the prediction of heat capacity, but a decided improvement was obtained in the enthalpy predictions. Since the enthalpy differences are, in general, the most useful quantities, this improvement is significant. Furthermore the equations presented in this work have the advantage of extending the temperature range. On the other hand, the newly developed equations have the disadvantage of being one degree higher in their powers of T compared to the equations of Thinh et al., but this added complexity is a necessity when the equations are applied over the larger temperature range available for many compounds. In conclusion, Equations 2, 3, and 5, together with the coefficients in Table I, are recommended for predicting the ideal gas properties of the compound listed. They represent a thermodynamically consistent set of equations with minimum overall predictive errors.
Results
literature Cited
The constants obtained for the enthalpy, heat capacity, and entropy equations of the 89 compounds are given in Table I. The overall average percent error and maximum error obtained in predicting the absolute values of the enthalpy, heat capacity, and entropy are also given. A comparison of the results of this work, those of the API Technical Data Book (1971) and Thinh et al. (1971) are shown in Table 11. The equations of Thinh have been compared only for the 7 5 hydrocarbons common to both studies. Furthermore since Thinh’s equations have been developed only for the temperature range of 298-1500°K for the comparison given in Table 11, the equations of this work were evaluated in this same temperature range. The values of enthalpy and entropy for comparison with the correlations of Thinh et al. had to be calculated as the property change
American Petroleum Institute, “Technical Data Book-Petroleuin Refining,’’ 2nd ed., Chap. 7, Washington, D.C., 1971. American Petroleum Institute Research Project 44, “Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center, Texas -4&M University, College Station, Tex. (looseleaf data sheets extant 1972). Lapidus, L., “Digital Computation for Chemical Engineers,” pp 240-57, McGraw-Hill, Kew York, N.Y., 1962. NcBride, B. J., Heimel, S., Ehlers, J. G., Gordon, S., SASA Report hTo. SP-3001, 1963. Thinh, T. P., Duran, J. L., Ramalho, R. S.,Kaliaguine, S., Hydrocarbon Proems., 50 ( l ) ,98-104 (1971). Zeleznik. F. J.. Gordon. S..N A S A Tech. Kote D-767, Washington, D’.C , 1961. RECEIVED for review January 27, 1972 ACCEPTEDJune 1, 1972
546 Ind. Eng. Chem. Process Des. Develop., Vol. 11, No. 4, 1972
This work was sponsored by the American Petroleum Institute, Division of Refining.
