Isobaric Heat Capacity of Methane EDWARD W. SLEDJESBI' Newark College of Engineerirtg, Newark, N. J. Development of industrial processes using higher temperatures and pressures requires better knowledge of heat capacities of fluids used. Experimental values are limited for practical reasons to narrow ranges of variables and recourse must be made to values computed from equations of state. Isobaric heat capacities for methane relative to that of an ideal gas, (Cp-C*,), were computed for pressures up to 10,000 pounds per square inch absolute and temperatures up to 1300" F., using the Benedict-Webb-Rubin equation of state. These values were found to be more accurate than data derived from simpler equations. Heat capacities derived from experimental work by Budenholtzer, Sage, and Lacey over a relatively narrow range of variables were . used as a basis for comparison. Methane isobaric heat capacity data presented are believed to be the best available over-the extreme ranges covered.
R
ELIABLE heat capacity data for methane, covering the widc range of values required by the trend of modern technology toward high pressure processes, are limited to several recent compilations covering a relatively narrow range and another study extending over a wide range up to and including 10,000 pounds per square inch absolute. The values of isobaric heat capacity of methane reported by Edmister ( 5 ) were computed from work based on the pressure-volume-temperature data of Kvalnes and Gaddy (10) and of Keyes and Burks (Q),and the data of Eucken and Lude (7) on atmospheric specific heats of methane. The range covered included temperatures of -70" to 200" C. and pressures of 1 to 120 atmospheres. -4s the isothermal change in the isobaric heat capacity with change in pressure is sensitive to minor uncertainties in pressurevolume-temperature data when based upon them, Edmister's results are not in satisfactory agreement with those obtained from Joule-Thomson data. Specific heats calculated from JouleThomson coefficients are less subject to error than those calculated from pressure-volume-temperature data. Budenholtzer, Sage, and Lacey ( 4 ) evaluated isobaric heat capacities for methane from experimental Joule-Thomson coefficients for a range of 70" to 220' F. and 0 to 1500 pounds per square inch absolute. They report an uncertainty of not more than 1% in their computed values of heat capacities, except where the specific heats are subject to possible later modifications of the spectroscopic value of the heat capacity at infinite dilution. Ellenwood, Kulik, and Gay ( 6 ) applied the Beattie-Bridgeman equation of state ( 8 )for specific heats of methane throughout the range of 400' to 4500' R. and 0 to 10,000pounds per square inch absolute. Their results are subject to the uncertainties attending extensive extrapolations of data by means of an equation of state. Because of practical difficulties, methane heat capacities based on direct measurements of thermodynamic properties are not available above the relatively moderate pressures and temperatures readily controlled and measured in the laboratory. However, values of heat capacity may be calculated with a varying degree of accuracy from equations of state for regions not covered by experimental work. The equation of state of Benedict, Webb, and Rubin ( 8 ) may be regarded as a modification of the Beattie-Bridgeman equation 1 Present
to obtain a more accurate representation of the properties of fluids a t high densities. The pressure-volume-temperature properties of gases above their critical densities are not reflected accurately by the Beattie-Bridgeman equation, whereas the equation of Benedict, Webb, and Rubin fib data well for all four hydrocarbons-methane, ethane, propane, and butane-up t o about 1.8 times the critical density; but above this density, calculated pressures are high. Benedict, Webb, and Rubin note that their equation reproduces observed properties of the gas phase in the critical region, a property the Beattie-Bridgeman equation does not possess. The Benedict equation, therefore, should reproduce thermodynamic properties, such as isobaric specific heat, of a reali gas more accurately than the simpler Beattie-Bridgeman equation. The author has accordingly selected and used the BenedictWebb-Rubin equation to calculate the isobaric specific heat of methane over a range of 0 to 10,OOO pounds per square inch absolute and 100' to 1350"F. METHOD OF CALCULATION
Reference to any text on thermodynamics such as Hougen and Watson (8)will establish that
Cp-C%=Cp-CP
- R
(1)
By adding and subtracting C , on one side and rearranging,
C p - C,+ = (Cp
- Co)
+ (Cv - C:)
-R
(2)
The Benedict-Webb-Rubin equation of state may be written follows:
a8
Relationships for C p - C, and (C, - C 7 ) in terms of the Benedict-Webb-Rubin equation of state may be developed from the following thermodynamic relationships:
(4) (5)
v
3 (bRT -
vz
T4V2
a)
+ 6aa ~6 +
(3
+ % - g)](8)
Substituting in Equations 4 and 5 and performing the indicated mathematical manipulation, there is obtained (9)
address, Arthur G. MoKee Co., Union, N. J.
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INDUSTRIAL AND ENGINEERING CHEMISTRY
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Vol. 43, No. 12
Figure 1. Constant Pressure Specific Heat of Methane Relative to an Ideai Gas
DISCUSSION OF RESULTS
divided by
B,RT - A ,
V
- &) + $ (bRT
- u)
+
Since Equations 3, 9, and 10 are impficit in temperature, T , and specific volume, V , they are solved most readily by assuming a series of values of volume, V , for a particular temperature, 7'. The solutions of Equations 9 and 10, (C, - C Y ) and (C, - Cv),respectively, are then substituted in Equation 2, and (C, - 12:)is calculated. A plot of (C, - C,*) versus pressures for a specific temperature was made and interpolated values of (C, - C*,) for pressures of 500, 1000, 1500, and 2000 to 10,000 pounds per square inch absolute in steps of 1000 picked off. Temperatures selected for purposes df calculation were 300 350 ", 400 500 600 O, 800 ', and 1000° K. Five-place logarithms were used throughout the calculations, A final plot of (C, - C:) versus temperature with pressure as parameters was then made and is shown in Figure 1. Spectroscopic values of heat capacity at infinite dilution are subject to constant modification. Therefore, values of (C, C ; ) , heat capacity of methane relative to that of an ideal gas, are presented. For purposes of application to a specific problem involving heat capacity of methane, values at infinite dilution, C,*, at the required temperatures should be added to values of (C, 12:)presented in this paper. The American Petroleum Institute Tables ( 1 ) may be consulted for values of C,*. O,
O,
O,
The value of C*,,heat capacity of methane a t infinite dilution, is subject to constant revision. Accordingly, a discussion of the merits of various investigations of high pressure methane heat capacity data should be limited to the values of (C, - C:), heat capacity of methane relative t o that of a gas exhibiting ideal behavior. Values of (C, - C:) were calculated by subtracting C, a t 0 pounds per square inch absolute from C, at the various pressures covered by the investigators. A plot of (C, - C,*) versus temperature with pressure parameters of 500, 1000, and 1500 pounds per square inch absolute was developed from three literature sources and from the work presented in this paper for comparative purposes. This plot is shown in Figure 2. Since data derived from Joule-Thomson coefficients are believed to be more accurate than those based on pressure-volumetemperature work, the curves of (C, - C:) obtained from Budenholtzer's tabulation (4)were used as a basis for evaluating the accuracy of the heat capacity data calculated in this paper. Inspection of the three plots for 500, 1000, and 1500 pounds per square inch absolute reveals quite clearly that Edmister's data, as stated in the opening discussion, deviate substantially from Budenholtaer's, indicating errors in Edmister's values of (C, C,*)as well as discrepancies in atmospheric heat capacity values. A comparison of the results derived from application of the Beattie-Bridgeman equation by Ellenxvood et al. and the BenedictWebb-Rubin equation by the author, indicate that for temperatures above 150" F., the Benedict equation reproduces values of (C, - C:) more accurately than the Bridgeman equation does. Below 150" F. both equations show a deviation from Budenholtzer's figures that obviously places low temperature data de-
December 1951
INDUSTRIAL AND ENGINEERING CHEMISTRY
Figure 2.
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Constant Pressure Specific Heat of Methane Relative to an Ideal Gas Comparison of results of different investigations
rived from either equation of state in the questionable category. At 500 pounds per square inch absolute, the author’s values are virtually identical with those of Budenholtzer, both curves appearing to be superimposed, whereas the values calculated by Ellenwood are approximately 0% low. I n all cases, the author’s data appear to be a few per cent higher than those of Ellenwood et al., a trend apparently followed by Budenholtzer’s values. I n general, the upper limits of Budenholtaer’s data are more nearly approximated by the results of the application of the BenedictWebb-Rubin equation of state. It may be concluded, t’herefore, that the calculation of isobaric heat capacities of methane by means of the Benedict-Webb-Rubin equation of state gives more nearly accurate results than use of the Beattie-Bridgeman equation, as was expected. The author’s values, based on the Benedict-Webb-Rubin equation of state, are recommended for application to specific problems involving heat capacity of methane above 220’ F. and above 1500 pounds per square inch absolute. Below this range, the most accurate work is believed to be that of Budenholtzer, Sage, and Lacey. NOMENCLATURE
C , = constant-pressure heat capacities, molal basis C, = constant-volume heat capacities, molal basis C t = constant-volume specific heat a t zero pressure or infinite volume T = temperature, K. V = molal volume, liters per gram-mole d = molal density, gram moles per liter P = pressure, atmospheres
R = ga:
constant, 0.08207 liter-atmospheres per gram-mole,
K.
Bo, A,, C,, b, a, c
C,
- C:
Y, CY = Benedict-Webb-Rubin equation of state constants = heat capacity of a real gas relative to heat capacity of gas exhibiting ideal behavior
ACKNOWLEDGMENT
This work was carried out under the guidance of Joseph Joffe, Department of Chemical Engineering, Newark College of Engineering. LITERATURE CITED
(1) American Petroleum Institute Research Project 44, “Selected Values of the Properties of Hydrocarbons,” Washington, D. C., Natl. Bur. Standards (1944). (2) Beattie, J. A., and Bridgeman, 0. C., J . Am. Chem. SOC.,49,1665 (1927);50,3133(1928). (3) Benedict, M.,Webb, G. W., and Rubin, L. C., J , Chem. Phys., 8, 334-5 (1940). (4) Budenholtzer, R. A., Sage, B. H., and Lacey, W. N., IND.ENG. CHEM.,31,369 (1939). (5) Edmister, W. C.,Ibid., 28,1112 (1936). (6) Ellenwood, F.O., Kulik, N., and Gay, N. R., “The Specific Heat of Certain Gases over Wide Range of Pressure and Temperature. Air, CO, COZ,CHI, CZH4, Hz, Nz, and 0 2 , ” Cornell Univ. Eng. Erpt. Sta. Bull. No.30 (1942). (7) Eucken and Lude, 2.physik. Chem., B5,413(1929). (8) Hougen, 0.A., and Watson, K. M., “Chemical PrQcessPrinciplea-Part Two,” New York, John Wiley & Sons, 1947. (9) Keyes and Burks, J. Am. Chem. SOC.,49,1403 (1927). (10)Kvalnes and Gaddy, Ibid., 53,397 (1931). REOEIVED March 30, 1951. This paper is part of a thesis submitted in partial fulfillment of the requirements for the degree of M.S. in Chemical Engineering, Newark College of Engineering, Newark, New Jersey.
