SUBSCRIPTS R = outer surface of alumina pellet 1
= inner surface of alumina pellet = surroundings of pellet
o
literature Cited
s., Natl. Ad~~isory Comm. for Aeronaut., RM E52C05 (1952). (2) Eucken, A., Forsch. Gebiete znEenieu7w. 133, Forschungsh. 353, D. 16. (3j Harper, J. C., Prof. Rept. 3, Quartermaster Corps, Contract DA 19-129-QM-1349, File 306, Oct. 14, 1959. (1) Deissler. R. G., Eian, c .
(4) Henry, J. P., Chennakesavan, B., Smith, J. M., A.I.Ch.E. Journal 7, 10 (1961). (5) Knudsen, M., Ann. Physik 34, 593 (1911). (6) Kunii, D., Smith, J. M., A.Z.Ch.E, Journal 6, 71 (1960). (7) Mischke, Roland, Ph.D. thesis, Northwestern University, Evanston, Ill., August 1961. (8) Petersen, E. E., A.Z.CI1.E. Journal 4, 343 (1958). (9) Robertson, J. L., Smith, J. M., Zbid., to be published. (10) Schotte, W., Zbid., 6, 63 (1960). (11) Scott, D. c.,COX,K. E., J . Chem. Phys. 57, 1010 (1960). (12) Sehr, R. A , , Chem. Eng. Sci. 9, 145 (1958). (13) Villet, R.H., \$'ilhelm, R. H., IND. ENG.CHEM. 53,837 (1961). RECEIVED for review March 19, 1962 ACCEPTED August 30, 1962
DERIVATIVE COMPRESSIBILITY FACTORS ROBERT C. R E I D AND JON R. VALBERT Department of Chemical Engineering, Massachusetts Znstitute of Technology, Cambridge, Mass.
Two new generalized compressibility functions are defined and evaluated in terms of Pitzer's acentric factors for various reduced conditions. These new factors allow rapid estimation of those thermodynamic properties which involve the partial derivatives of volume with either temperature or pressure.
HE COMPRESSIBILITY FACTOR has proved extremely useful Tin expressing quantitatively P-V- T relations for nonideal gases. A logical extension to this concept involves the expression of common thermodynamic derivative functions in terms of a similar type of factor.
Theoretical
and is correlated in terms of the reduced parameters T, and P R and often with a n additional parameter which is a characteristic property of the material (7-4). From such correlations it is possible to estimate, with sufficient engineering accuracy, a value of Z for most gases under a given set of conditions. Often, however, it is also desirable to determine isothermal variations in the common energy functions, and the
The compressibility factor is defined as: Z = Pv/RT
1
(1)
-
-.-
0
0
-N n
N
0. I
0
IO
20
40
3 0
0
Figure 1. 292
Effect of
20
IO
30
PR
PR
PR
l&EC FUNDAMENTALS
and
TR
on
ZP'"
Figure 2.
Effect of
PE and
T R on
ZT"'
41
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I I I I I
I&EC FUNDAMENTALS
I I I I
I I I I I I I I
oon----*r----*--
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I I I I I
n
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IbEC FUNDAMENTALS
derivatives (bv/bT)p and (av/bP) T are then required. defining two new factors, ZPand Z T :
ZP = i?
- PR(~Z/~PR)T,
ZT = 2 f T R ( ~ Z / ~ ~ R ) P ,
By (2) (3)
it is easy to show that:
d (ZRT/P)p = RZT (du/dT)p = p dT
(4)
and 13
( d v / d P ) ~= ~ ( Z R T / P ) =T
TZp - RP=
(5)
Equations 4 and 5 mary be used to calculate derivative functions after Zp and Z T values are obtained. I n determining the relation of Zp and 2: T to reduced properties, two separate compressibility correlations are available. Lydersen, Greenkorn, and Hougen (2) presented Z as a function of T R , PR, and Z,. Pitzer (3) correlated Z also with T Rand PR but used the acentric factor o as a third parameter. Pitzer's tables were used in this work to calculate Zp and Z T . T h e acentric factor is related to the slope of the vapor pressure-temperature curve by Equation 6. W
= --log P R O , - '1.000
(6)
where P R ~ ,is, the reduced vapor pressure a t a reduced temperature of 0.7. The acentric factor o varies for different compounds and approaches zero for nonassociating, unimolecular gases such as argon. I t may be estimated from Z, values as: w ~ ( 0 . 2 9 1- Zc)/ 0.080. Pitzer has shown (3) that for molecules that are not strongly polar or very elongated Z is linearly related to w for each T Rand PR:
z
Z(0) + ,Z(C
(7)
I n this expression Z(0)refers to the value of Zfor nonassociating, unimolecular gases and Z(1) = (dZ/dw) T ~ p, R . Both Z(0) and Z(I) have been published as tabulated functions of T R and P R (7,3). I n terms of the new parameters, Z, and Zp:
ZT
:=
ZT(OI f WZT(l)
(8)
where
ZT(') =
f T~(dz(~,/bT~)p,
7 ~ (= l Z(l)
f T~(dz(~)/dT~)p,
and zp -=
Zp(0)
+ wZp(')
(9)
where
ZP(O)= Z(0) - p R ( ~ Z ( O ) /)TR ~PR Zp(l) = Z(I) - P R ( ~ Z ( ' ) / ~ P R ) T , Calculations
To determine values of Z , and ZP from Equations 8 and 3 0 ) and Z(I) tables were used and the derivatives obtained by arithmetic: and graphical techniques. I n the arithmetic technique, five evenly spaced Z(0)- or Z(l)-reduced coordinate points were fitted to a second-order Gram polynomial. The slopes were calculated a t the center point of the five points. A standard error was also calculated. The method of calculation follows closely that described by Whitaker and Pigford ( 8 ) . Since the grids had to contain evenly spaced data, numerous points were common to more than one grid (Le., the slope for 1.0 could be obtained from the data points 0.8, 0.9, 1.0: 1.1, and 1.2 or 0.6, 0.8, 1.0, 1.2, and 1.4).
9, Pitzer's
VOL. 1
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297
Wherever the standard error was small and the slopes for points common to multiple grids were in reasonable agreement, the computed value was accepted. All calculations were done on the IBM 709 computer. The remaining points were obtained graphically by plotting the points, drawing a smooth curve, and measuring the tangent angle a t the desired point. Most of the Z P values were calculated arithmetically, but the Z T values had to be done entirely by the graphical method, since it was impossible to express analytically the Z(0)-or Z(')-reduced temperature relationship with any accuracy. Results. Complete tables of ZT(o),Z T ( l ) ,Zp(0),and Z p ( l ) values for different reduced temperatures and pressures are given in Tables I through I V . Plots of Z ~ ( 0 )and Zp(0)us. P, for a few values of T Rare shown in Figures 1 and 2.
Example 2. Repeat Example 1 for ethyl alcohol, a polar molecule, a t 422' F. (TR= 0.95) and 371 p.s.i.a. (PR = 0.40). The vapor pressure a t T R = 0.7 (190.6' F.) is 21.3 p.s.i.a. and P, = 928 p.s.i.a. w =
Table V expresses a portion of the Bridgman tables (6) in terms of Z T and ZP. T o use the table, for example, suppose that (bH/bP)T is desired. Since (dH) T = ( R T / P ) ( Z , Z ) and (dP) T = -1, then: ( d H / d p ) ~= ( R T / P ) ( Z - Z T )
Thus, a t any given T and P, (bH/bP) T may be calculated from reduced state correlations of Z and Z r . The former is available in the references cited previously and in most textbooks on engineering thermodynamics, while the latter are reported here. Example 1. Estimate u, (du/dT),, and ( d u / d P ) , for propane a t reduced values of TR = 0.9 and P R = 2.0. (These reduced conditions indicate T = 140" F. and P = 1235 0.152 ( 3 ) . From Pitzer's tables, p.s.i.a.) For propane, w Z(0) = 0.316 and Z(1) = -0.14; so from Equation 7, Z = 0.316 - (0.152)(0.14) = 0.298. From Tables I through IV, Zp = 0.028 (0.152)(0.04) = 0.022 Z T = 0.351 - (0.152)(0.010) = 0.336
-
Thus :
1.000 = 0.649
The calculated and experimental (5)values are shown below: Error,
cu. ft./lb. - - ( d v / d P ) ~ ,cu. ft./lb.-p.s.i.a. X lo3 (du/dT)p,cu. ft./lb.-" F. X 103 u,
Calcd. 0.437 1 .58 1.19
Exptl. 0.447 1 .61 1.295
% '
4-2 +2 +8
Nomenclature
A
CP Applications
- log 21.3/928 -
= specific Helmholtz free energy, = E = heat capacity a t constant pressure = specific internal energy
E F = specific Gibbs free energy, = E H = specific enthalpy, = E Pu P = absolute pressure Pao,, = reduced pressure a t T R = 0.7 R = gas constant
+
S T
Z Z(0) Z(1) Zp
= = = = = = =
Zp(0)
=
u
Zp(1) =
ZT = ZT(O)= Z+) = =
- TS
+ Pv - TS
specific entropy temperature specific volume compressibility factor, = Pu/RT compressibility factor for nonassociating gases (bZ/bw)r,,,, Z - PR(~Z/~PR)T, Z(0) - P R ( ~ Z ( O ) / ~ P R ) T , Z(1) - P R ( ~ Z ( ' ) / ~ P R ) T ~ Z TR(~Z/~TR)P, Z(O) T R ( ~ Z ( O ) / ~ T R ) P , Z(1) T ~ ( b z ( l ) / b T ~ ) p , - log P R O , , - 1.000
++ +
Subscripts R = reduced C = critical Acknowledgment
This work was done in part a t the Computation Center a t the Massachusetts Institute of Technology, Cambridge, Mass. literature Cited
(10.73)(0.336) ~. (44,07)(1235) = 6.62 X 1 0 - 5 ~ ft./lb.-'F. T h e experimental values for these three quantities are (7): 0.0344, -2.104, and 6.156, and the percentage errors are - 3 , 0 , and -8%, respectively.
298
I&EC FUNDAMENTALS
(1) Lewis, G. N, Randall, M., "Thermodynamics," 2nd ed., rev. by K. S. Pitzer and L. Brewer, Appendix 1, McGraw-Hill, New York, 1961. ( 2 ) Lyderson, A. L., Greenkorn, R. A., Hougen, 0. A., Wis. Univ., Eng. Expt. Sta., Rept. No. 3, 1955. (3) Pitzer. K. S.. J . Am. Chem. Soc. 77. 3427 (1955). (4) Pitzer; K. S:, Lippman, D. Z., Curl, R: F., Huggins, C. M., Peterson, D. E., Zbid.,77, 3433 (1955). (5) Reid, R. C., Smith, J. M., Chem. Eng. Progr. 47, 415 (1951). (6) ~, Sage, B. H., Lacey, W. N., "Thermodynamics of One Compon&t Systems," Academic Press, New York, 1957. (7) Sage, B. H., Schaafsma, J. G., Lacey, 1%'.N., IND.ENG.CHEM. 26, 1218 (1934). (8) Whitaker, S., Pigford, R. L., Zbid., 5 2 , 185 (1960). RECEIVED for review March 26, 1962 ACCEPTEDSeptember 4, 1962
