I Process Design D a t a .
ALVIN 61. WEISSl and JOSEPH JOFFE Newark College of Engineering, Newark, N. 1. e e
eneralize obaric He
ti
f
IN
VIEW of the wide use of high pressures in the chemical industry, thermodynamic data a t high pressures are an invaluable aid to chemical process design. Experimental data. though preferred, are not available for a number of substances encountered in industrial practice. This is particularly true of heat capacities of gases at high pressure, for which only scattered experimental data are found in the literature, even in the case of common gases. As a practical solution to this difficulty, generalized correlations of p-V-T and thermodynamic properties of gases have been developed, based on the law of corresponding states. The isobaric heat capacity, C,. is one of the properties for which generalized correlations have been published. T h e most desirable correlation is in terms of the difference between the values of C, and the isobaric heat capacity in the ideal gas state, C ,, as a function of reduced temperature reduced pressure.
Early Generalized Correlations for C,
- C;
The original presentation of this nature was by Watson and Smith (ZZ), who based their work on a generalized compressibility factor plot, constructed from experimental determinations of compressibility factors. These data were graphically integrated to give a generalized activity coefficient plot, and then two successive graphical differentiations resulted in the data required for a generalized plot of C, - Cp” us. reduced pressure. Their curves were later modified by Hougen and Watson ( 9 ) . Edmister (5, 6 ) has presented a generalized correlation for C, - Cp* in tabular and graphical forms, based on the difference between ideal and actual gas volumes, rather than their ratio, as in the case of the Watson and Smith correlation. This difference, defined as the Lewis and Randall volume residual quantity (729, is represented by the equation a = R T / P - V and has been discussed by Deming and Shupe ( 3 ) . I t is considered by them to be much better adapted to graphical treatment than the compressibility factor. Present address, Houdry Process Corp., Marcus Hook, Pa.
120
INDUSTRIAL AND ENGINEERING CHEMISTRY
Table
-
I.
~ e n ~ ~~ ~ l i ~~fore C,~ ~ C;
~
e
(Parameters of reduced pressure and temperature)
T, Pr
0.2 0.3 0.4 0.6 0.8 1.0
I .O
1.1
1.2
1 I65 2-09 3.00
0 758 1,315 1.83
0.550 0.882 1.25
3.21 5.35 8.4
e
5.90 14.0
23.7
1.5 2.0 3.0
I______.
1,s
1.4
1.5
0,419 0.640 0 905
0.322 0.490 0.705
0.260 0.392 0,555
2.06 3.23 4.58
1.46 2,21 2.90
1.10
1.55 2.09
0.855 1.15 1.59
8.90
5.27 8.50 10.8
3.44 5.00 7.60
2.50 3.59 5.53
10.8
8.40
6.52 6.47 6.03
4.0 6.0 8.0 10.0 12.0 15.0
4.85 Tr
__
-.___
1.6
1.75
2.0
2.25
2.5
2.75
0.2 0.3 0.4
0.216 0.321 0.442
0.169 0.247 0.331
0.1165 0.I675 0.227
0.0818 0.119 0.1615
0.0609 0.0873 0.1205
0.0476 0.0667 0.0940
0.6 0.8 1.0
0.688 0.907 1.225
0.513 0.687 0.875
0.340 0.461 0.571
0.243 0.323 0.401
0.1805 0.237 0.294
0.138 0.1805 0.227
1.5 2.0 3.0
1.91 2.71 4.17
1.355 1.89 2.87
0.864 1.185 1.75
0.608 0.810 1.20
0.449 0.590 0.877
0.344 0.455 0.670
4.0 6.0 8.0
5.04 5.45 5.35
3.58 4.23 4.48
2.24 2.90 3.31
1.555 2.07 2.48
1.135 1.547 1.89
0.862 1.195 1.465
10.0 12.0 15.0
4.61 4.25
4.19 3.99 3.64
3.35 3.32 3.22
2.60 2.69 2.78
2.05 2.19 2.31
1.635 1.79 1.93
3.0
3.5
4.0
4.5
5.0
5.5
0.2 0.3 0.4
0.0540 0.0747
0.0490
0.0354
0.6 0.8 1.0
0.108 0.1445 0.180
0.0737 0.0987 0.1235
0.0542 0.0713 0.0895
0.0419 0.0547 0.0685
0.0333 0.0439 0.0544
0.0270 0.0361 0.0442
1.5 2.0 3.0
0.273 0.359 0.529
0.183 0.242 0.352
0.132 0.174 0.254
0.101 0.132 0.193
0.0793 0.1025 0.1515
0.0640 0.0834 0.122
4.0 6.0 8.0
0.673 0.845 1.165
0.456 0.635 0.797
0.329 0.462 0.582
0.249 0.352 0.444
0.195 0.277 0.349
0.157 0.224 0.283
10.0 12.0 15.0
1.325 1.48 1.60
0.935 1.05 1.17
0.690 0.788 0.890
0.525 0.603 0.698
0.412 0.478
0.338 0.389 0.462
__
17
0.555
~
100
/
/ IO
h
a
5 0
f3
la 1.0
* P
u I
0"
0.1
0.01 !
I
I
I
/
I
0.I
I I I l l l
I
I
I 1
L
1.0
P, I
Figure 1 . The effects of pressure on the isobaric heat capacity of gases. Parameters of reduced temperature
The fact that both Watson and Smith and Edmister had to resort to graphical integration and differentiations must of necessity result in a certain amount of arithmetical error, aside from any question of accuracy of the original data.
Dodge (dl has Presented an work in which is plotted as a function of p , with parameters of T,. Comings and Nathan (2) have shown that this method of correlation is not valid. Most recently, Lydersen, Greenkorn,
cp/cz
and Hougen (73) have submitted a correlation for C, - Cp* based upon a generalized correlation of compressibility factors. They have included a third parameter, critical compressibility factor, zo, in addition to T, and p , , for the purpose of eliminating some of the inadequacies of the law of corresponding states. Lydersen, Greenkorn, and Hougen, calculated C, - C,* only for z, = 0.27, and have indicated that this refinement does not compensate for the inherent inaccuracy of a double graphical differentiation. Consequently, they compiled no tables for heat capacity corrections (73), but because their publication is the most recent on the subject, it has been included in this work for comparison.
C, - Data Based on Benedicb Webb-Rubin ~~~~~i~~ of state Considering the inherent inaccuracies of graphical differentiation processes, there still remains a need for a more VOL. 49, NO. 1
JANUARY I957
121
0.01
I.o
0.I
Figure 2.
A comparison of
-- -- ----
C,
Weiss and Joffe Hougen and Watson Edmister Lydersen et a / .
accurate generalized heat capacity correlation. Because of the difficulties and inaccuracies involved in experimental C, measurements, it is expedient to calculate C, - C,* from a n equation of state based on accurately measured
1 22
- C; correlations p-V- T values. Of the available equations of state, the Benedict-Webb-Rubin equation ( 7) seems the most applicable for calculating C, - C,* values. By application of well known thermodynamic and mathe-
INDUSTRIAL A N D ENGINEERING CHEMISTRY
matical principles, Sledjeski (78, 79) has derived from the Benedict-WebbRubin equation a relationship for C, C,* which makes it possible to calculate C, - C,* without resorting to graphical differentiation processes. However, because of its cumbersome nature, the equation itself is of limited utility to design engineers. Furthermore, it can be applied only to gases for which the Benedict-Webb-Rubin constants are knomm. A program has been undertaken at the Newark College of Engineering to solve the relation for C, - C,* for specific gases over a wide range of temperatures and pressures. Six hydrocarbon gases have been studied to date, and their specific heat properties have been calculated and reported (7, 74, 75, 77, 79-27). Sledjeski (78) and Seifarth and Joffe (76) have presented cii - cp data for methane and propane. The five hydrocarbons whose c, - c,* values were used in the present work are methane, ethylene, ethane, propane, and n-butane. Assuming that the law of corresponding states applies to these five gases, a generalized correlation of c p - c,* has been constructed in the present study from the E , - c,* values for the individual gases. Method of Calculation
To carry out the generalized correlation, the c, - c,* values and conditions
Table II. Hougen and Watson Pr
T?
Limits of Generalized Correlations Lydersen, Edmister Others
0.01 -7.0 0.625-3.0
of each of the five hydrocarbons were first converted to molal quantities and reduced temperatures and pressures. For each individual gas, plots were made of C, - C; us. reduced pressure with parameters of reduced temperature. Parameters of reduced pressure were then arbitrarily chosen, and cross plots were made of C, C,* us. reduced temperature for each chosen parameter of reduced pressure. T h e data for all five gases were included on these graphs, and the resulting averaged lines provided a generalized correlation of C, C; us. reduced temperature with parameters of reduced pressure. Another cross plotting provided a correlation of C, - C; us. reduced pressure with parameters of reduced temperature. The values obtained are shown in Table I. Figure 1 is the generalized correlation.
-
Table 111.
Comparison of
Discussion of Results
Generalized Correlation. T h e range of variables included in this correlation is compared in Table I1 to those included by Hougen and Watson, Edmister, and Lydersen. Outside of the limits of this correlation, it is recommended that the plot of Edmister be used-Le., from p r 2 0.01 to pr < 0.2 and T, 2 0.6 to T, < 1.0. Where neither the data of Edmister nor Figure 1 apply, the plots of Hougen and Watson and of Lydersen may be used, although they may be subject to severe errors.
T T
1.0
Edmister Hougen and Watson Lydersen, others
11 to 37% 59 to 62Y0 - 9 to 35%
T,
Pr
c, - c,*
5.97 14.91 14.91 14.91 14.91
1.47 2.30 2.00 1.46 1.12
c, - cg
0.199 0.597 0.996 1.392 1.792 2.39 2.58
0.1116 0.363 0.614 0.875 1.141 1.41 1.68
0.295 0.895 1.478 2.06 2.66 3.24 3.84
0.0671 0.223 0.382 0.540 0.686 0.826 0.962
- 1 to -11% 26 to
% Deviations from Experimental Values Weiss and Hougen and Joff e Edmister Watson 21 9 13 10 13
-1
77
C, - C,*Correlations with ExperimentalData of
Workman
% Deviations from Experimental Values Pr
been compared with those of Edmister, Hougen and Watson, and Lydersen for the parameters of T,= 1.0, 1.2, 1.5, and 2.5. Figure 2 graphically depicts the differences among the four correlations. Using the curves calculated in this work as a reference, the range of deviations of the other three may be tabulated as follows: 1.2
for Nitrogen and Oxygen Exptl.
Comparison of Generalized Correlations. This generalized correlation has
Correlations with Experimental Data of Krase and Mackey for Nitrogen
2.37 2.40 2.56 2.96 3.36
Comparison of
Weiss and Joffe 0.2-15 1.0- 5.5
C, -C;
Exptl.
Table IV.
0.1-4.0 0.8-3.0
0.01-6.0 0.6 4 . 0
T h e calculated values of C, - C; (on which this correlation has been based) were representatively sampled and compared to the values read from Figure 1. T h e results indicate that the deviations of the generalized correlation from the original data range from 0 to 16%. T h e higher deviations occur in the regions of either extreme curvature or extremely high values of C, - C;, where the differences between individual gases may be expected to increase. T h e average deviation is estimated to be *4.2%.
Weiss and Joffe
Edmister
Hou,gen and Watson
Lydersen, others
Oxygen (T,= 1.948, ze = 0.290) - 1
- 1 1 - 1
- 2 9 2
-
4 2 4 4 3
- 6
59 46 45 39 38 49 34
-. 2 9 40 31 24 7
Nitrogen (T,= 2.64,zo = 0.291)
-
13 2 1 4 2 2 1
- 15 - 23 - 26 - 30 - 30 - 30 - 28
49 75 68 54 53 46 40
- 48 - 41
- 37 - 37 - 39
5 to
55% 29%
1.5 4 t o - 89’0 33 to 64v0 -61 to 20%
-
2.5 -22 to - 23% 43 to 63y0 -30 to -108%
Comparison of Generalized Correlation with Experimental Data for Nitrogen and Oxygen. I n order to test the applicability of this correlation to gases other than hydrocarbons, experimental data of Krase and Mackey (70, 71) for nitrogen were utilized. Table I11 lists their results, and includes the deviations of the corresponding values of C, - C; read from Figure 1. The agreement is sufficiently good to indicate that Figure 1 is suitable for use as a generalized correlation for nonpolar gases. I n addition, Krase and Mackey’s data for nitrogen have been compared to the values read from the curves presented by Hougen and Watson and by Edmister. Unfortunately, both of these correlations are applicable to only one of Krase and Mackey’s points. This point corresponds to the point of maximum disagreement of Krase and Mackey’s experimental data from Figure 1 and of exceedingly high disagreement for Hougen and Watson, while the value presented by Edmister is only 1% lower than the experimental value. However, one point is not a sufficient basis for any significant conclusions. Table IV lists some of the experimental data (arbitrarily chosen) obtained by Workman (23) for nitrogen and oxygen, compared to Figure 1, Edmister’s, Lydersen’s, and Hougen and Watson’s correlations. A study of Table IV will show that there is essentially no disagreement between the experimental values for oxygen and the values read from Figure VOL. 49, NO. 1
JANUARY 1957
123
1 and Edmister’s curves. T h e data of Table IV for nitrogen correspond to T , = 2.64, where there is significant disagreement between Figure 1 and Edmister. In this instance, Table IV indicates that Figure 1 is significantly closer to the experimental values (1 to 1370 different). This test would indicate that Figure 1 is not only applicable to gases other than hydrocarbons, but may be preferable to the other published correlations. However, this must be a strictly qualified conclusion, in that only one set of experimental data (nitrogen) indicates that Figure 1 is more accurate than Edmister’s work. T h e data of Krase and Mackey and of Workman have provided checks of C, - Cp* only in the range of T , = 1.9 to 3.4. No such checks in a lower reduced temperature range have been made because of lack of experimental data. Effect of Deviations among GeneralAlized Correlations on Values of C., though the Cp Cp” corrections presented by the different correlations differ considerably on a percentage basis, the actual effect of these differences on C, is often negligible in terms of absolute values. As a n illustration, Table V represents data for nitrogen at p , = 0.5. C, values have been calculated a t T , = 1.0 and 2.5, representing, respectively, cases of high and low absolute C, - C; corrections. These data illustrate that only where C, - Cp*itself is of a significant magnitude will the differences in correlations have a n effect on the resulting value of C,. If the C, - Cp* correction is of sufficient magnitude to justify its use, differences in correlations which are of the order of one half the magnitude of the correction itself cannot be dismissed.
-
considered to be the most accurate source of C, - C,* data now available for these hydrocarbons. This correlation (Figure 1) differs from previously published correlations in both the source of data and the method of calculation. Earlier correlations were based on averaged compressibility data, which have been found less accurate than the Benedict-WebbRubin relationships used for this work. I n addition, to obtain the C, - C; correlation, it was necessary for earlier authors to differentiate graphically a generalized compressibility factor (or a) correlation. Figure 1 was based upon the drastically different method of averaging specific C, - C,* data for five gases, the data having been obtained by analytical differentiation processes. These considerations should all tend to strengthen confidence in the accuracy of Figure 1 in comparison to the earlier C, - Cp* correlations. Comparison of Figure 1 with experimental data for oxygen and nitrogen has indicated that Figure 1 is suitable for gases other than hydrocarbons, and there are indications that Figure 1 approaches the experimental C, - Cp* values more closely than the most accurate earlier work (Edmister). Considering these factors, it is recommended that the generalized correlation for C, - C; presented in this work be used in preference to the earlier generalized correlations that have been published. However, this must be qualified by the understanding that a generalized correlation is inherently subject to error (Figure 1 differs as much as 16y0 from the data for specific gases on which it is based). Nomenclature
C,
Conel usions
This work has provided a generalized correlation for C, - Cp* which is believed to be more accurate than correlations previously published. This has been effected by averaging C, - C: data for five hydrocarbon gases. T h e data for the individual hydrocarbons were obtained by previous authors by mathematical differentiation of the BenedictWebb-Rubin equation of state and are
Table V.
cp
- c;
CP
c, - c; CP
124
INDUSTRIAL
4.2
10.7 0.15 6.89
p
R T V
z a:
in isobaric heat capacities of a real gas and a gas exhibiting ideal behavior - cg = difference in isobaric specific heats of a real gas and a gas exhibiting ideal behavior = absolute pressure = universal gas constant = absolute temperature = molal volume = compressibility factor = Lewis and Randall volume residual quantity
Effect of Cp-C*, Deviations on C, Values
1 Weiss and
Joffe
cp
- C,* = difference
% Deviation
2
Hougen and Watson 6.1 12.6 0.25 6.99
AND ENGINEERING CHEMISTRY
of 2 from 1
T, =
45 18
large correction
1.0
67 1.5
Tr = 2 . 5 small correction
SUBSCRIPTS = at critical conditions r = reduced conditions c
literature Cited
Benedict, M., Webb, G. W., Rubin, L. C., J . Chem. Phys. 8, 334 (1940). Comings, W., Nathan, M. F , , IND. ENG.CHEM.39. 964 (1947). Deming, W. E.,‘and Shupk, L. S., Phys. Rev. 37, 638 (1931). Dodge, B. F., IKD.ENG. CHEM.24, 1353 (1932). Edmister, W. C., lbid.,30, 352 (1938). Edmister, W. C . , Petroleum Rejner 27, 314 (1948). Glueck, R. M., “Isobaric Heat Capacity of n-Butane over a Wide Range of Temperature and Pressure,” thesis in chemical engineering, Newark College of Engineering, Yewark, N. J., 1952. Hougen, 0. A., private communication. Hougen, 0. A , , Watson, K. M., “Chemical Process Principles Charts,’’ Wiley, New York, 1946. Krase, N. W., Mackey, B. H., J . Am. Chem. SOC.52, 108-15 (1930). Zbid.,pp. 5111-4. Lewis, G. N., and Randall, Lf., “Thermodynamics,” McGraw-Hill New York, 1923. Lydersen, A. L., Greenkorn, R. A,, Hougen, 0. A., “Generalized Thermodynamic Properties of Pure Fluids,” Engineering Experiment Station, University of Wisconsin, Rept. 4 (October 1955). Reiter, T. A., “Heat Capacities of Ethane over a Wide Range of Temperature and Pressure,” thesis in chemical encrineerincr. Newark College of Eng‘ineering’ Newark, N. J., 1954. Seifarth, J. H.. “Isobaric Heat Cauacity of Propane over a Wide Range of Temperature and Pressure,” thesis in chemical engineering, Newark College of Engineering, Newark, N. J., 1951. Seifarth, J. H., -Joffe, J., IND.EKC. CHEDI.44, 2894 (1952). Sibilia, R. J., “Isobaric Heat Capacity of Ethylene over a Wide Range of Temperature and Pressure,” thesis in chemical engineering, Newark College of Engineering, Newark, N. J., 1952. Sledjeski, E. W., IND.ENG.CNEY.43, 2913 (1951). Sledjeski, E. \V., “Isobaric Heat Capacity of Methane over a Wide Range of Temperature and Pressure,” thesis in chemical engineering, Newark College of Engineering, Newark: N. J., 1950. Sledjeski, E. W., private comunication. (21) Tassoney, J., ‘(Isobaric Heat Capacity of Propylene over a Wide Range of Temperature and Pressure,” thesis in chemical engineering, Newark College of Engineering, Newark, N. J., 1955. (22) Watson, K. M., Smith, R. L., Nutl. Petroleum brews 28,29 (July 1, 1936). (23) Workman, E. J., Phys. Rev. 37, 1345 (1931). RECEIVED for review November 16, 1955 ACCEPTED July 1 6 , 1956 Based on a thesis in chemical engineering, presented by A. H. Weiss to the Newark College of Engineering, 1955.
