Ind. Eng. p = pL
density
= density of liquid, lb/ft3
= air or vapor density, lb/ft3 = surface tension, dyn/cm 4 = froth density factor +,.J = as given in eq 15 $h = as given in eq 14
pg,pv 0
& = entrainment factor eq 28 Literature Cited
-
A.1.Ch.E. Research Studies, Final ReDorts. Delaware. Michiaan. North Carolina. Dee 1958. Bain, J. L., Van Winkle, M., AIChE J . , 7 (3),363 (1961). Calderbank, P. H., Trans. Inst. Chem. Eng., 34, 79 (1956). Chase, J. D., Chem. Eng., 105 (July 1967);139 (Aug 1967). Eduljee, H. E., Chem. Age (India), 17, (7),560 (1966). Fair. J. R.. Matthewa. R. L.. Pet. Refiner. 37 14). 153 11958). Foss, A. S., Gerster, J. A., Chem. Eng. Prog:, 52, 28'(1956). Hutchinson, M. H., Buran, A. G , O'Connell, H. E., Chem. Eng. Prog.. 53, 127
Chem. Process Des. Dev., Vol. 17, No. 4, 1978
443
Hunt, C. d'A., Hanson, D. N., Wilke, C. R., AIChE J., 1, 441 (1955). Liebson, J., Kelley, R. E., Bullington, L. A,, Pet. Refiner, 36, 127 (1957). McAllister, R. A., Mcginnis, P. H., Plank, C. A,, Chem. Eng. Sci., 9, 25 (1958). Mayfield,F. D., Chwch, W. L., Green, A. C., Rasmussen, R. W., Id.Eng. Chem., 44 (9),2238 (1952). Ogboja, O., Ph.D Thesis, University of Surrey, 1974. Prince, R. G. H., Int. Symp. Dist., Inst. Chem. Eng., 177 (1960). Sargent, R. W. H., Bernard, J. D. T., Trans. Inst. Chem. Eng., 44, 314 (1966). Smith, B. D., Fair, J. R., "Design of Equilibrium Stage Processes", (Chapter on Perforated Trays by Fair), McGraw-Hill, New York, N.Y., 1963. Souders, M., Brown, G. G., Ind. Eng. Chem., 26,98 (1934). Thomas, W. J., Final Report ABCMIBCPMA Distillation Symposium, London, p 181, 1964. Thomas, W. J., Campbell, M. M., Trans. Inst. Chem. Eng., 45, 53 (1967). Thomas, W. J., Haq, A. M., I d . Eng. Chem., Process Des. Dev.. 15, 509 (1976). Van Winkle, M., "Distillation", p 536, McGraw-Hill, New York, N.Y., 1967. West, F. B., Chem. Eng. Sci., 266 (1952). Zanelli, S.,Del Blanco, R., Chem. Eng. J., 6, 181 (1973). Zenz, F. A., Pet. Refiner, 33 (2),99 (1954):36 (3),179 (1957).
Received for review August 1, 1977 Accepted April 25, 1978
(1952). Hughmark, G. A., O'Connell, H. E., Chem. Eng. Prog., 53, 127 (1957)
A Modified Soave Equation of State for Phase Equilibrium Calculations. 1 Hydrocarbon Systems Michael S. Graboski and Thomas E. Daubert' Department of Chemical Engineering, The Pennsylvania State University, 165 Fenske Laboratory, University Park, Pennsylvania 16802
The Soave modification of the Redlich-Kwong equation of state is shown to work well for the estimation of the vapor-liquid equilibrium behavior of a wide variety of technically important hydrocarbon mixtures. The American Petroleum Institute's Technical Data Book-Petroleum Refining recently adopted a modified Soave procedure for vapor-liquid equilibria calculations. It is the purpose here to present the finished form of the correlation for treating mixtures of hydrocarbons of importance to the natural gas and petroleum refining industries. The complete correlation is fully generalized, requiring only the readily available characterization parameters to make equilibrium calculations. The utility of the correlation is demonstrated by equilibria predictions for binary and multicomponent defined and
undefined hydrocarbon systems.
Procedure Soave (1972) originally proposed a modification of the Redlich-Kwong equation of state which introduced a third parameter, the acentric factor, and a temperature dependency into the cohesive energy term to account for the effect of nonsphericity on fluid P-V-T properties. The modified equation of state is given by aa p = - -R T u - b (u)(u + b ) where a = 0.42747R2T,:!/P,, b = Q.08664RTc/P,, and a = f (TIT,, w ) . a is an empirical function developed from pure hydrocarbon vapor pressure data. The equation of state was extended to mixtures by Soave using the van der Waals one-fluid (Reid and Leland (1965)) combining rules given as n
effect of deviation from geometric mean combining rule for a6 and must be obtained from binary mixture data. To a good approximation C , is independent of temperature, pressure, and composition. For hydrocarbons, Soave suggested that Cij was zero. Vapor-liquid equilibria calculations for hydrocarbon mixtures presented in this paper indicate that this is a reasonable approximation. The fugacity coefficient of a component in a phase is needed to predict phase equilibrium. Based on the Soave equation of state, the fugacity coefficient is given by eq 4 for component i in an n-component system, where $i is In
$i
bi = -(Z b
-
1) - In (2 - B ) -
n
n
6= C j=1
xjb,
(3)
where cyy a,, = (1- CL,)(a,aLalal)1/2, The constant C,, is the interaction coefficient as described by Chueh and Prausnitz (1967). It corrects for the 0019-7882/78/1117-0443$01.00/0
a fugacity coefficient of component i, bi is the covolume constant of component i, aijaij is given by eq 2, A = ~ Mf R2F, P B = bPf RT, and Z is the compressibility factor. Correlation Development for Pure Hydrocarbons The original Soave development was based on an accurate correlation of hydrocarbon vapor pressures in terms of reduced temperature, reduced pressure, and the acentric 0 1978 American Chemical Society
444
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 4, 1978
Table I. Summary of Vapor Pressure Errors of Hydrocarbons Based on the Soave Procedure ~~
~~
Table 11. Results for Vapor Pressures of Nonhydrocarbons by Penn State Soave ~
average deviation no. of data points psia %
average deviations
no. of data points psia
compound class paraffins (Cl-C20) olefins (C2-Clo) naphthenes (C,-C, ,) aromatics (C6-C,o)
1946 743 597 1157
0.83 1.16 0.46
1.10
%
compound
1.74 2.99 4.24 4.00
ammonia carbon dioxide hydrogen sulfide nitrogen sulfur dioxide water
factor. By accurately correlating vapor pressure, the equation of state is capable of accurately predicting the fugacity of fluids and therefore the phase equilibrium behavior. Specifically, Soave used the critical point and the vapor pressure a t a reduced temperature of 0.7 (i.e., the acentric factor) for a number of compounds to determine the function required to correlate vapor pressures. A limited set of critical pressures, critical temperatures, and acentric factors which are not totally consistent with the recommended American Petroleum Institute Data Book (1977) parameters were used in the original development. Thus, the Soave “a” equation was refit with the expanded property set. Instead of anchoring the equation a t T , = 0.7, the regression was based on a detailed set of hydrocarbon vapor pressure data compiled by the Penn State API research staff. According to Soave, the term “a” is defined as a = (1
+ s (1 - f i ) ) 2
(5)
where S = a + bw + cu2,T, = reduced temperature TIT,, w is the acentric factor, and a, b, and c are constants. The “a” function was determined as a function of reduced temperature and acentric factor for the API vapor pressure data set. It was found that eq 5 correlated the “a” data well. The regression equation for “S”based on the a data was derived to be S = 0.48508
+ 1.55171~ 0 . 1 5 6 1 3 ~ ~ -
(6)
Table I summarizes the results of the vapor pressure correlation effort. For 4443 vapor pressure points Soave predicted the pressure with an average error of 2.87% and 0.91 psi, respectively. Errors are smallest for paraffins. For olefins, napthenes, and aromatics the errors are larger. The analysis for many of the compounds in the later data set were based on critical constants which were estimated by the recommended correlations given in the API Data Book (1977). This could explain the slightly poorer correlation. For paraffins, the vapor pressure analysis extended to Clo with an acentric factor of about 0.91 and a normal boiling point of 650 O F . The heaviest olefin, naphthene, and aromatic compounds were Cl
