Sc = Schmidt number Sh, Sh’ = Sherm-ood number, as for Nu, Nu’ t’ = residence time, contact time, t = velocity of particle relative to continuous phase, V L/t V = volume, L 3 = distance from interface into continuous phase, L X = diameter of particle normal to motion, L Y 2 = maximum dimension of particle in direction of motion, L P = mass concentration, M / L 3
SUBSCRIPTS A
= diffusing component
HT ;ZIT
heat transfer = mass transfer = initial = a t interface
-4
=
0
s
literature Cited
Batchelor, G. K., “An Introduction to Fluid hlechanics,” Cambridge University Press, p 475, 1967. Bowman, C. W7.,Johnson, A. I., Can. J . Chem. Eng. 40, 139 ( l a w ,.) Calderbank, P. H., Chem. Eng. (London)Rev. Ser. No. 3, CE 209 Figure 9A, (1967). ,A
-
Cheh, H. Y., Tobias, C. W., IND. ENG.CHEM.,FUXDAM. 7, 48 (1968). \ - - - - I
Datta, R. L., Napier, D. H., Newitt, D. bf., Trans. Inst. Chem. Eng. 28,14, (1950). Elmore, W. C., and Heald, X.A., “Physics of Waves,” p 204, McGraw-Hill, New York, N. Y., 1969. Grassmann, P., “Ph sikalische Grundlagen der Chemie-Ingenieur Technik,” Czapter 10, Sauerlander and Co., Aarau and Frankfurt-am-Main, 1961. Heertjes, P. M., Holve, W. A., Talsma, H., Chem. Eng. Sci. 3, 122 119541. Krishna, P. ’M., Venkateswarlu, D., Narasimhamurty, G. S. R., J . Chem. Eng. Datu 4, 336 (1959). Mendelson, H. D., A.I.Ch.E. J . 13, 2, 260 (1967). Moore, D. W., J . Fluid Mech. 6 , 113 (1959). Perry, R. H., Chilton, C. H., Kirkpatrick, S. D., Ed., “Perry’s Chemical Engineers’ Handbook,” 4th ed. NcGraw-Hill, New York, N. Y., 1963.
KEVITU’ GEHAK
ISAAC H. LEHRER’
Monash Cniversity Clayton, Victoria, Australia 5168
‘To whom correspondence should be sent. RECEIVED for review December 15, 1969 ACCEPTED March 29, 1971
Combining Rules for the Third Parameter in the Pseudocritical Method for Mixtures Combining rules for the third parameter are derived from a single assumption, that the residual property of a mixture i s the molal average of the residual properties of its components, all evaluated at the same reduced temperature and pressure.
A c c o r d i n g to the two-parameter Law of Corresponding States, all fluids should hare the same values of their residual properties when compared at the same values of the two parameters, reduced temperature and reduced pressure. B y residual properties are meant here thermodynamic properties such as the compressibility factor, z , the enthalpy departure, (H* - H ) / T , , and the entropy departure, (S* S), which represent deviations from ideal gas behavior. The pseudocritical hypothesis assumes that values may be assigned to the critical constants of a mixture (the so-called pseudocritical constants) such that a t a given pseudo-reduced temperature and pseudo-reduced pressure the residual properties of the mixture are the same as those of a pure reference substance a t the specified reduced conditions. If the residual properties of the mixture are to correspond closely to those of the reference substance a t the same reduced conditions, then the latter should be so chosen as t o have as much molecular similarity as possible with the components of the mixture (Leland, et al., 1962). It has been proposed that the components of a mixture may themselves be used as reference substances and that the residual property of the mixture be obtained by averaging in some manner the residual properties of the pure components a t the same reduced conditions (Yesavage and Powers, 1969). I n the absence of any specific data on the mixture the following assumption appears reasonable: a residual property of 532 Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
a mixture is best represented by the molal average of the values of this property for each of the components, determined a t the same reduced conditions as those of the mixture. Thus, for the compressibility factor, z , of a mixture of n components
similar equations holding for each of the other residual properties. Since the two-parameter Law of Corresponding States is known to be approximate even for pure substances of similar molecular structure, the law has been modified by introducing a third parameter. The best known three-parameter correlations are those of Pitzer and coworkers (1955) and of Lydersen and coworkers (1955). Pitzer and coworkers use as the third parameter the acentric factor, w , which is related to the vapor pressure. They have found that each residual property may be expressed as a linear function of the acentric factor. Thus, in the case of the compressibility factor they have proposed Z( =
z(0)
+
WiZ(’)
(2)
where z ( 0 ) and z(1) are tabular functions of the two main parameters, reduced temperature and reduced pressure. I n
accordance with the pseudocritical hypothesis a similar equation may be written for the mixture 2.w = z ( 0 )
+
WMZ(1)
(3)
A t t h e same reduced conditions z(O) and likewise d1) are identically the same for all pure components and also for the mixture. Hence, no subscripts are used on do)and on ~ ( 1 ) .Substituting eq 2 and 3 in eq 1, we have
combiiiing rule for the third parameter, z c , as suggested b y Leland and Mueller (1959) and by Hougeii, et al. (1959). To summarize, the two combining rules for the third parameter, eq 5 and 9, have been derived from a single basic assumption, eq 1, and are thus seen to be related. Nomenclature
D H T z
= = = = =
z
=
n
S
It follows t h a t n W M
=
(5)
ZiW(
i
Thus, we have derived a combining rule for the third parameter, W . This combining rule was suggested b y Pitzer a n d Hultgren (1958) on the basis of empirical evidence on compressibilities of binary mixtures. If the three-parameter correlation of Lydersen and coworkers (1955) is used, a combining rule similar t o eq 5 may be deduced for the third parameter, z,, the critical compressibility factor. Hougen, et al. (1959), have shown t h a t residual properties may in general be expressed as linear functions of z c . Thus, for the compressibility factor of the i t h component we may write ~t =
z
+ D ( Z , -~ 0.27)
(6)
where z and D are functions of only the reduced temperature and reduced pressure. The corresponding equation for the compressibility factor of the mixture is ZM =
z
+ D(z,M - 0.27)
(7)
+ D ( z , M - 0.27)
+ D ( z C z- 0 . 2 7 ) ]
=
Z ~ [ Z
i
function of reduced temperature and reduced pressure enthalpy number of components entropy absolut,e temperature mole fract,ioii compressibility fact’or
GREEKLETTER = Pitzer’s acentric factor
w
SUPERSCRIPT * = pertaining to ideal gas state SUBSCRIPTS c = critical property i = pertaining to i t h component ill = pertaining to mixture literature Cited
Hougen, 0. A4.JWatson, K. AI., Ragatz, R. A,, “Chemical Process Principles,” Part 11, 2nd ed, pp 574-576, 859, 860, Wiley, New York, N. Y., 1959. Leland, T. W., Chappelear, P. S., Gamaon, B. W., A.I.Ch.E. J . 8 , 482 (1962). Leland, T. W., Nueller, W. H., I n d . Eng. Chem. 51, 597 (1959). Lydersen, A. L., Greenkorri, R. A., Hougen, O., “Generalihed Thermodynamic Properties of Pure Fluids,” University of Wisconsin, Engineering Experiment Station Report No. 4, Oct 1955. Pitzer, K. S., Hultgren, G. O., J . Amer. Chem. SOC.80, 4793 (19.58) \----,
Substituting eq 6 and 7 into eq 1, there follows z
=
(8)
Pitzer, K. S., Lippmann, I). Z. Curl, R. F., Huggins, C. AI., Petersen, D. E., J . Amer. Cheni. SOC.77, 3427, 3433 (1955~. Yesavage, V. F., Powers, J. E., “Equations of State and the Corresponding States Principle,” 65th National hleeting, American Institute of Chemical Engineers, Cleveland, May 4-7, 1969.
and therefore
JOSEPH JOFFE
n ZeM
=
ZiZCi
(9)
i
Equation 9 has been used b y many workers as an empirical
Newark College of Engineering Newark, N . J . 07108 RECEIVED for review December 24, 1970 ACCEPTED April 27, 1971
Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
533
