COMMUNlCATlONS Generalized Enthalpy and Entropy Deviation Functions of Normal Liquids at Low Temperatures
T h e generalized enthalpy and entropy deviation functions of normal liquids at low temperatures (0.8 > T r 2 0.35) previously prepared by Chao and Greenkorn to p r = 3.0 are improved and extended to Pr = 9.0.
The enthalpy and entropy of low-temperature liquids are needed in engineering calculations for cryogenic systems, for instance, the liquefaction of natural gases. Chao and Greenkom (1971) presented enthalpy and entropy deviation functions in the form of three-parameter (Tr, P r , w ) generalized correlations for normal (nonpolar and slightly polar) fluids. The enthalpy deviation function was obtained by combining data of two types: (1) temperature derivatives of generalized fugacity coefficients prepared by Chao, et al. (1971), and (2) calorimetric data measured at the University of Michigan, (Jones, et al., 1963; Yesavage, 1968; Yesavage, et al., 1970). The entropy deviation functions were obtained by combining the enthalpy deviation functions with the generalized fugacity coefficients. The range of conditions covered in the work of Chao and Greenkom is 0.80 > Tr I 0.35 and up to P r = 3.0. This upper pressure is lower than desirable. In this work the pressure is extended to P r = 9.0. Systematic calorimetric data are not available a t this level of pr and T,. The extension is made with thermodynamic derivations based on established volumetric properties of liquids. Spencer and Danner (1972) found the density of saturated liquids to be described well by the Rackett equation (Rackett, 1970)
The density dependence of the isothermal compressibility
p
I
-’(”) T
of liquids a t a constant temperature has been found by Chueh and Prausnitz (1969) to be accurately described by the Wada equation
where the subscript S refers to saturated liquids. Chueh and Prausnitz (1969) also give an equation for d, RPSTC - (1.0 - 0 . 8 9 ~ ” ~ )exp(6.9547 brc 76.2853Tr + 191.3060T: 203. 5472TT3 + 82.7631T:)
By combining eq 1, 2, and 3 and integrating, we obtain an expression for liquid molal volume as a function of T , , p r , and w . The pressure dependence of enthalpy is then derived by integrating the thermodynamic identity
(g)
= lr- T($)
T
With Z R treated as an adjustable parameter having a constant value for each substance, Spencer and Danner found the equation has an average deviation of 0.38% from data for 36 hydrocarbons over wide ranges of temperatures.
(3)
P
The integrated equation is lengthy and may be found elsewhere (White, 1973). Figure 1 shows the enthalpy deviation function for simple fluids a t several temperatures. The curves were ob-
Table I. Values of [ (fi*- f i ) / R T c .~
T,
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
5.81 5.63 5.48 5.31 5.18 5.05 4.91 4.81 4.69
5.80 5.62 5.47 5.30 5.17 5.04 4.90 4.80 4.68
5.79 5.61 5.45 5.29 5.15 5.03 4.90 4.78 4.67
5.77 5.60 5.42 5.27 5.12 5.01 4.88 4.76 4.66
5.75 5.58 5.40 5.25 5.10 4.98 4.86 4.75 4.64
5.71 5.56 5.37 5.23 5.08 4.96 4.84 4.73 4.62
5.69 5.54 5.35 5.21 5.06 4.94 4.82 4.71 4.61
5.67 5.52 5.33 5.19 5.04 4.92 4 .BO 4.69 4.60
5.65 5.50 5.31 5.17 5.02 4.90 4.78 4.68 4.58
5.63 5.48 5.30 5.16 5 .OO 4.88 4.76 4.66 4.56
5.61 5.46 5.28 5.14 4.98 4.86 4.75 4.65 4.55
5.59 5.44 5.26 5.12 4.97 4.83 4.73 4.63 4.53
5.57 5.42 5.25 5.10 4.95 4.82 4.72 4.62 4.52
T,
2.8
3.0
3.2
3.4
3.6
3.8
4.0
4.5
5.0
6.0
7.0
8.0
9.0
0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
5.55 5.40 5.23 5.09 4.93 4.81 4.70 4.60 4.51
5.53 5.38 5.21 5.07 4.91 4.79 4.68 4.58 4.49
5.51 5.37 5.20 5.06 4.90 4.77 4.67 4.57 4.48
5.49 5.35 5.18 5.03 4.88 4.76 4.65 4.55 4.47
5.47 5.34 5.17 5.02 4.87 4.74 4.64 4.54 4.45
5.46 5.32 5.15 5 .OO 4.85 4.72 4.62 4.52 4.43
5.44 5.30 5.13 4.98 4.83 4.71 4.60 4.51 4.42
5.40 5.26 5.10 4.95 4.80 4.67 4.57 4.47 4.39
5.36 5.22 5.05 4.91 4.76 4.63 4.53 4.43 4.35
5.29 5.15 4.98 4.82 4.68 4.55 4.46 4.37 4.27
5.21 5.07 4.90 4.74 4.60 4.47 4.38 4.29 4.20
5.13 5.00 4.82 4.65 4.51 4.38 4.30 4.21 4.12
5.05 4.92 4.73 4.55 4.42 4.29 4.21 4.12 4.03
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 4,1974
453
- I?)/RTc](~!
Table 11. Values of [ (I?*
p,
T,
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80
10.40 9.93 9.48 8.93 8.46 7.88 7.31 6.79 6.21 -
10.46 9.98 9.51 8.97 8.50 7.91 7.34 6.81 6.24 5.60
10.55 10.05 9.55 9 .oo 8.52 7.93 7.38 6.83 6.26 5.62
10.61 10.11 9.59 9.04 8.55 7.95 7.42 6.87 6.28 5.65
10.68 10.17 9.63 9 .OB 8.57 7.97 7.45 6.90 6.30 5.68
10.74 10.22 9.67 9.10 8.58 7.99 7.48 6.94 6.32 5.70
10.80 10.26 9.71 9.13 8.60 8.01 7.51 6.97 6.35 5.72
10.85 10.30 9.74 9.15 8.61 8.02 7.54 6.99 6.37 5.75
10.90 10.34 9.77 9.18 8.63 8.04 7.56 7.03 6.39 5.77
T,
2.4
2.6
0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80
11.02 10.44 9.85 9.24 8.67 8.08 7.63 7.10 6.45 5.83
11.06 10.47 9.87 9.26 8.68 8.10 7.65 7.11 6.47 5.86
2.8 11.09 10.50 9.89 9.27 8.70 8.11 7.68 7.13 6.49 5.87
2.0
2.2
10.94 10.37 9.80 9.20 8.65 8.05 7.59 7.06 6.41 5.80
3.0
4.0
5.0
6.0
7.0
8.0
9.0
11.12 10.52 9.91 9.29 8.71 8.13 7.70 7.15 6.52 5.90
11.27 10.63 10.00 9.36 8.77 8.20 7.78 7.25 6.65 6 .OO
11.38 10.72 10.07 9.42 8.82 8.27 7.86 7.33 6.75 6.13
11.47 10 .BO 10.13 9.48 8.88 8.34 7.92 7.41 6.85 6.25
11.55 10.86 10.18 9.52 8.93 8.41 7.98 7.48 6.94 6.36
11.61 10.92 10.23 9.57 8.98 8.47 8.04 7.54 7.02 6.46
11.67 10.97 10.27 9.61 9.02 8.52 8.09 7.60 7.10 6.56
10.98 10.41 9.82 9.22 8.66 8.07 7.61 7.08 6.43 5.82
Table 111. Values of [ (S* - s')/R](o!
T,
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
8.15 7.58 7.2'7 6.93 6.69 6.46 6.24 6.10 5.92
7.45 6.91 6.59 6.26 6.02 5.80 5.57 5.42 5.25
7.05 6.53 6.19 5.87 5.61 5.40 5.20 5.03 4.86
6.76 6.29 5.88 5.59 5.31 5.12 4.92 4.74 4.60
6.54 6.05 5.66 5.36 5.09 4.88 4.70 4.54 4.38
6.35 5.86 5.45 5.19 4.91 4.70 4.52 4.36 4.20
6.20 5.71 5.30 5.03 4.76 4.55 4.37 4.21 4.07
6.06 5.57 5.16 4.92 4.65 4.42 4.24 4.09 3.95
5.93 5.46 5.05 4.79 4.51 4.31 4.13 3.99 3.84
5.83 5.35 4.97 4.70 4.41 4.20 4.03 3.89 3.74
5.73 5.25 4.87 4.61 4.32 4.11 3.95 3.81 3.67
5.62 5.17 4.80 4.53 4.25 4.04 3.87 3.73 3.58
5.53 5.09 4.73 4.46 4.17 3.97 3.80 3.67 3.52
T,
2.8
3.0
3.2
3.4
3.6
3.8
4.0
4.5
5.0
6.0
7.0
8.0
9 .o
0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
5.45 5 .OO 4.66 4.40 4.10 3.90 3.74 3.60 3.46
5.38 4.94 4.60 4.32 4.03 3.83 3.67 3.53 3.40
5.31 4.89 4.55 4.28 3.99 3.78 3.62 3.49 3.35
5.25 4.83 4.50 4.23 3.93 3.74 3.58 3.44 3.30
5.20 4.79 4.45 4.18 3.89 3.69 3.53 3.40 3.26
5.15 4.75 4.40 4.13 3.85 3.64 3.49 3.36 3.22
5.10 4.70 4.36 4.09 3.80 3.60 3.45 3.32 3.18
5.01 4.61 4.26 3.98 3.71 3.50 3.35 3.22 3.09
4.92 4.52 4.17 3.90 3.64 3.41 3.27 3.13 3.02
4.81 4.40 4.04 3.75 3.49 3.27 3.14 3.01 2.89
4.70 4.29 3.94 3.63 3.38 3.16 3.04 2.91 2.79
4.64 4.22 3.84 3.52 3.27 3.06 2.94 2.81 2.68
4.53 4.1s 3.75 3.42 3.18 2.96 2.85 2.73 2.60
I 0
-
4
I 05
FROM CHAO AND GREENKORN (1971) FROM I N T E G R A T I N G EO. I41
I 20
I 10
I 30
I
50
\ J 10
pr
Figure 1. Generalized enthalpy deviation function for simple fluids. 454
Ind. Eng. Chem., Process Des. Develop., Val. 13,No. 4 , 1974
tained from the integrated form of eq 4. At each temperature a constant of integration is selected so as to give the best fit of the tubular values of Chao and Greenkorn in the range of p , = 2.0 to 3.0. As a check on the integrated results we extended the calculations back to low pressures. Figure 1 shows the extension to be in good agreement with Chao and Greenkorn's table values at low pressures. Chueh and Prausnitz recommend eq 3 for the range 0.98 2 T , 1 0.40. The use of eq 3 a t T , = 0.35 is an extrapolation beyond the recommended range. However, the results shown in Figure 1 are reasonable and agree with the general behavior a t other temperatures. We include these results here but with a warning that the extension a t T , = 0.35 may not be as reliable as the other results. For the extension of the enthalpy deviation function
Table 1V. Values of [ (3. -
s)/ R I(')
T,
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
14.40 13.26 12.08 11.06 10.26 9.27 8.31 7.53 6.70
14.59 13.42 12.13 11.13 10.30 9.28 8.35 7.55 6.72
14.77 13.55 12.20 11.19 11.34 9.30 8.39 7.58 6.74
14.93 13.67 12.27 11.24 11.36 9.32 8.42 7.60 6.75
15.09 13.77 12.33 11.28 10.39 9.34 8.46 7.63 6.77
15.24 13.87 12.39 11.32 11.40 9.35 8.50 7.66 6.79
15.37 13.95 12.45 11.35 11.41 9.36 8.53 7.68 6.81
15.49 14.02 12.50 11.38 11.42 9.38 8.56 7.71 6.83
15.60 14.10 12.55 11.40 11.43 9.39 8.58 7.73 6.84
15.70 14.16 12.60 11.42 10.44 9.41 8.61 7.75 6.85
15.80 14.22 12.63 11.45 10.45 9.42 8.63 7.77 6.87
15.88 14.27 12.67 11.47 10.46 9.43 8.65 7.80 6.89
15.96 14.33 12.70 11.48 10.47 9.44 8.67 7.82 6.91
3.6
3.8
4.0
4.5
5.0
6.0
7.0
8.0
9.0
12.28 14.51 12.81 11.55, 10.50 9.50 8.75 7.90 7.01
16.33 14.55 12.83 11.56 10.50 9.51 8.76 7.91 7.03
16.59 14.70 12.91 11.60 10.52 9.56 8.83 8.00 7.16
16.72 14.77 12.94 11.62 10.55 9.61 8.87 8.06 7.25
16.84 14.84 12.97 11.64 10.58 9.66 8.90 8.12 7.32
16.93 14.88 12.98 11.65 10.59 9.69 8.94 8.16 7.39
16.97 14.92 12.99 11.66 10.60 9.73 8.97 8.20 7.46
T,
2.8
3.0
3.2
3.4
0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
16.04 14.37 12.73 11.50 10.47 9.45 8.68 7.83 6.93
16.09 14.41 12.75 11.51 10.48 9.46 8.71 7.85 6.94
16.17 14.45 12.77 11.52 10.48 9.47 8.72 7.87 6.97
16.23 14.49 12.79 11.53 10.49 9.48 8.73 7.88 7 .OO
[(A* - A ) / R T c ] [ 1additional ) calculations are made ip the same way for the enthalpy deviation function [(H* @ / R T r ] of fluids a t w = 0.225. By combining the latter results with the simple fluid calculations a t w = 0 the (1) function is determined. The choice of w = 0.225 for this purpose is arbitrary, but is justified on the basis that the w of normal fluids of interest are generally in the range 00.45 with 0.225 being the average value. Tables I and I1 present the (0) and (1) enthalpy deviation functions in the entire range of pressure up to P r = 9.0. The region beyond P r = 3.0 is new. The region below pr = 3.0 now contains more entries than in Chao and Greenkorn's previous tables. There are also slight changes from the earlier table due to smoothing. Generalized entropy deviation functions are obtained by combining the enthalpy deviation functions with fugacity coefficient functions according to
The fugacity coefficient functions that are used here are presented by White (1973) and represent an improvement over the values of Chao, et al. (1971). The improvement is mainly due to the new vapor pressure data that are available from Carruth and Kobayashi (1972). Tables 111 and IV present the new entropy values.
Nomenclature H = enthalpy p = pressure R = universal gas constant S = entropy T = absolute temperature z = compressibility factor
16.38 16.49 14.57 14.63 12.84 12.87 11.57 11.58 10.50 10.51 9.52 9.54 8.80 8.77 7.93 7.97 7.10 7.05
Greek Symbols p = isothermal compressibility coefficient w = acentric factor Subscripts C = critical property r = reduced quantity formed by division with the corresponding critical property S = saturated state Superscripts = molal quantity (0)= simple fluid property function (1) = correction function that expresses departure from simple fluid behavior * = ideal gas
-
Literature Cited Carruth, G. F., Kobayashi, R . , lnd. Eng. Chem., Fundam. 11, 509 (1972). Chao, K. C., Greenkorn, R. A., Proceedings of National Gas Processors Association, 50th Annual Meeting, p 42-6, Mar 1971; also in Research Report RR-3, Natural Gas Processors Association, Tulsa, Okla., Apr 1971. Chao, K. C., Greenkorn. R. A., Olabisi. O., Hensel, B. H., AlChE J., 17, 353 (1971). Chueh. P. L., Prausnitz, J. M., AlChE J., 15, 471 (1969). Jones, M. L., Mage, D. T., Faulkner, R. C., Jr., Katz. D. L., Chem. Eng. Progr. Symp. Ser., 59 (44), 52 (1963). Rackett, H. G.. J. Chem. Eng. Data, 15, 514 (1970) Spencer, C. F., Danner, R. P., J. Chem. Eng. Data, 17, 236 (1972). White, M. G. M.S. Thesis, Purdue University, 1973. Yesavage, V. F., Ph.D. Thesis, University of Michigan, 1968. Yesavage, V. F., Katz. D. L., Powers, J. E.. AlChE J., 16, 867 (1970).
School of Chemical Engineering h r d u e University West Lafayette, Indiana 47907
Mark G . White Robert A. Greenkorn Kwang-Chu Chao*
Received for Review M a r c h 28,1974 Accepted M a y 17,1974 This work was supported by N a t i o n a l Science F o u n d a t i o n G r a n t GK-42051.
Ind. Eng. Chem., Process Des. Develop., Vol. 13,No. 4, 1974
455
