Isentropic Compression of Nonideal Gases J. M. PRAUSNITZ Department of Chemical Engineering, Princeton University, Princeton, N . J .
T
HE thermodynamic treatment of isentropic compression of ideal gases is discussed in all standard texts on engineering
thermodynamics. These texts, however, do not give any general treatment for the case of nonideal gases. In some cases references are made to the use of Mollier diagrams, to empirical methods such as that of Joffe (4),or t o trial-and-error methods based on the law of corresponding states such as those of York (7) and Edmister and McGarry (8). The only theoretical treatment appears to be that of Iberall (S), which, however, is useful only in the case of very small volume changes. This brief paper presents a perfectly general analytical treatment utilizing P-V-T data in the form of an equation of state. An illustration of the method follows its derivation. Although the techniques employed are those of standard thermodynamic procedure, the author believes that their application to the isentropic compression of nonideal gases has not heretofore appeared in print. The differential equation for isentropic compression is
-
Ai3111
spz [(E),
- R / p ] dP
P=O
evaluated a t T = T Z (7) Substituting Equations 5, 6 and 7 in 4 we obtain
Jpl
(g)p+ dP
JT:
$ - Jp2
CPo
L -
(g)
dP = 0
L -
a t Tz
at TI
Equation 8 can be considered an integral transform of the differential Equation 1. Equation 8 is perfectly general and can readily be solved to give the final temperature, provided a volumeexplicit equation of state is available. Once the final temperature is known, the work of compression can readily be calculated. It can be shown that W , the work of compression, is given by
W = AH Equation 1 can easily be integrated to give the final temperature for the case of an ideal gas. Upon substituting V = R T / P , integration readily gives the familiar result:
(9)
where AH is the change in enthalpy of the gas between TIP1 and TsP2. As enthalpy is also a state function independent of the path, we can, in a manner perfectly analogous to that used in the derivation of equation 8, show that AH =
Lpl
[T(g),
It would appear to be a simple matter t o extend this treatment to a nonideal gas by replacing the ideal gas law by a suitable equation of state of the form
- V ] dP
+
I. Formulate the entropy change for the transition from nonideal gas a t T I and PI to ideal gas a t T I and PI. 11. Formulate the entropy .. change for the ideal gas from Ti and Pi to TZand Pz. 111. Formulate the entropy change for the transition from ideal gas a t T Zand PZto nonideal gas a t T Zand Pz.
= ASI
spi [(g), $1
ASII
+ AS111 = 0
(4)
By standard thermodynamic procedure it can be shown that
asI =
-
P=O
dP evaluated a t T = Ti
GPOdT
~''[T(%)
-
- V ] dP
(10)
7--
at Tz Equation 10 is perfectly general and can easily be evaluated, provided a volume-explicit equation of state is available. To illustrate the use of Equation 8, calculations have been performed using the volume-explicit equation of state proposed by Beattie and Bridgeman (1).
V
= (r
+ B)(l -
e)
- A/RT
(11)
where B = Bo(1 - b / x ) A = Ao(1 - U / T ) 6 = C/T3r r = RT/P
The fundamental equation for isentropic compression then becomes ASoompression
sT:
a t T1
L -
Substitution of Equation 3 in 1, however, results in a nonintegrable differential equation except for the trivial case where f = RT/P. This difficulty can be overcome by using the familiar technique of choosing a more convenient path for integration. Because entropy is a state function which is independent of the path, any convenient path may be used. A well known and convenient path is the following:
(8)
Substituting I1 in 8 and performing all the indicated differentiations and integrations, we have
(5)
where C,, is a t aero pressure.
1032
INDUSTRIAL AND ENGINEERING CHEMISTRY
May 1955
Equation 12 is, in fact, the integrated form of the differential Equation 1 when integrated between the limits PITI and PZT,. If the specific heat at low pressure is known as a function of temperature and if the constants in the Beattie-Bridgeman equation are available, Equation 12 contains the final temperature as an unknown. Equation 12, however, is inconvenient because Tz,the unknown, appears as a logarithm and as a polynomial of the sixth degree. It is possible, however, to rewrite Equation 12 in an approximate form with which a close first approximation of T,can be calculated easily. The approximate form of Equation 12 can be derived as follows: First, neglect all terms where 2’ appears to a power of 3 or larger.
Table I.
Temperatures
FL
Initial
~ i ~ Final ~ Temperature, l Pressure, Pressure Calcd. Temp., lb./sq. Lb./Sq.’ ideal Eq. Eq. Gas O F. inoh Inch gas 20 12 Obsd.5 ”I 150 93 300 343 332 330 330 CZH4 0 147 882 238 213 210 210 coz 8 300 1150 191 182 181 180 220 270 214 219 220 CCIzFz g8 46 260 287 269b 271b 272 CHsClb 148 100 a From data quoted by Edmister ( 8 ) . b Beattie-Bridgeman constants calculated by Su’s method.
Table 11. Constants in Equation 20 for Ethylene (Units in atmospheres, liters per gram-mole,
Let -R In P2/Pl
1033
a
K.)
PI = 10 Pz = 60 T I = 256 = 0.0265 A,/R = 74.9 @ . ’ = -0.136 N = 4585 C ,, [averaged between T I and T Z (ideal)] = 0.454
b Pa A, P + B2 +2 =M 2R T: R T:
B,b 2R
2
and let
Also, let (15)
A summary of calculations for ethylene for, Equation 20 is shown in Table 11.
Then
NOMENCLATURE
Equation 16 can be rewritten
In almost all practical cases the ratio of TI to T Zis such that m
0.7
< T2 11 < 1.0
The logarithmic term is therefore closely approximated by the first two terms of the expansion
A , A , , a = Beattie-Bridgeman constants B, Bo,b = Beattie-Bridgeman constants C = Beattie-Bridgeman constant = specific heat a t constant pressure CP = specific heat a t constant volume cv H = enthalpy per mole k = CP/CV M = defined by Equation 13 N = defined by Equation 14 P = pressure R = gas constant s = entropy per mole T = absolute temperature V = volume per mole W = work of compression E = Beattie-Bridgeman constant r
Substituting 18 in 17 \ye have
[z-CP, + 3/21
Ti M
Z-] CPO
- 2T1T2 + [ a T : - N
and, by the quadratic formula,
T~
Tz =
+ .\IT:
-
[E + [i 3/21
;1;
+ 3/21
T:
= 0
(19)
-1 0
= RT/P
LITERATURE CITED
- N
CP
.
Subscripts 1 = initial state 2 = final state 0 = zero pressure
(20)
Equation 20 gives a good first approximation for the final temperature. If a more accurate answer is needed, the value found by equation 20 can then be used as a suitable starting point for the graphical solution of equation 12. Equations 12 and 20 require only specific heat data a t low pressure and the constants for the Beattie-Bridgeman equation. These constants have been determined for numerous gases ( 6 ) , and a generalized method of Su and Chang (6) may be used to estimate them for gases for which they are not available. Equations 12 and 20 have been used to calculate the final temperatures of compression for a few representative cases which have been discussed by Edmister and McGarry ( 8 ) . Table I compares the final temperature as calculated by the ideal gas law to those calculated by Equations 12 and 20. As expected, the ideal gas law gives unreliable results; only in fortuitous cases does it predict the final temperature reasonably well. The Beattie-Bridgeman equation, however, appears t o give consistently reliable results,
(1) Beattie, J. A., Proc. Natl. Acad. Sci., 16, 14 (1930). (2) Edmister, W. C., and McGarry, R. J., Chem. Eng. Progr., 45, 421 (1949). (3) Iberall, A. S., J . A p p l . Phys., 19, 997 (1948). (4) Joffe, J., Chem. Eng. Progr., 47, 80 (1951). (5) Su, G. J., and Chang, C. H., J. Am. Chem. Soc., 60, 1000 (1946). (6) Taylor, H. S., and Glasstone, S., “Treatise on Physical Chemistry,’’ Vol. 11, 3rd ed., p. 206, Van Nostrand, New York, 1951. (7) York, R., IND.E m . CHEM.,34, 535 (1942). RECEIVED for review March 27, 1954.
ACCEPTED December
21, 1984.
Correction In the article, “Solvent Extraction with Liquid Carbon Dioxide” [Alfred W. Francis, IND. ENG.CHEM.,47, 230-3 (1955)], the references to patents should be as follows: Page 231, Page 232, Page 232, Page 233, Page 233, Page 233,
col. 2, par. col. 1, par. col. 1, par. col. 1, par. col. 1, par. col. 2, par.
4, line 4 2, line 2 4, line 3 1, line 5 2, line 3 1, line 3
(10)
(14) (11) (8, 13) (9) (12 1
