The properties of the three product oils are shown in Table 111. The high pour point of the Utah oil product is caused by a larger paraffin content. A comparison of the reformer stock and middle distillate fractions from Utah and Pittsburgh-seam oils shows this difference. The two reformer stocks shown in Table IV are satisfactory reformer feeds after conventional pretreat steps to remove the residual oxygen and nitrogen. The high concentration of ring structures indicates that only a very mild reforming severity is needed to produce a high octane naphtha. The middle distillates can be used as heating oils, or after oxygen and nitrogen removal b y conventional pretreating, they can be charged to a hydrocracker for additional gasoline production. The high cycloparaffin content of the hydrogenated COED oils would classify the syncrudes as naphthenic-type, Although the "API gravity of these oils may be lower than most crude oils, they contain much less residuum than typical crudes. These hydrogenated COED coal oil syncrudes can be processed in typical petroleum refinery units.
Conclusion
I n summary, the removal of heteroatoms and the gross consumption of hydrogen occurring during the hydrogenation of COED coal oils can be correlated by simple, first-order kinetics. The oil from Pittsburgh-seam coal is more difficult to hydrogenate than the oils from Utah A-seam and Illinois No. 6-seam coals. The Utah oil requires less hydrogen than the other two to produce a superior oil. The product oils are naphthenic, containing high cycloparaffin concentrations. They contain little residuum and can be processed by conventional petroleum refining methods. literature Cited Qader, S. A., Wiser, W. H., Hill, G. R., Ind. Eng. Chem. Process Des. Develop., 7, 390-7 (1968). White, P. J., Jones, J. F., Eddinger, R. T., Hydrocarbon Process., 47 (12), 97-102 (1968).
RECEIVED for review January 11, 1971 ACCEPTEDApril 26, 1971 Presented at symposium on Hydrogen Processing of Solid and Liquid Fuels, Division of Fuel Chemistry, 160th Meeting, ACS, Chicago, Ill., September 1970.
One-Parameter Equation for Excess Gibbs Energy of Strongly Nonideal liquid Mixtures Solke Bruin' Western Utilization Research and Development Division, U . S. Department of Agriculture, Albany, Calif. 54710
John M. Prausnitz2 Department of Chemical Engineering, University of California, Berkeley, Calif. 54720
The local composition (nonrandom, two-liquid) equation proposed b y Renon was modified by substituting local volume fractions for local mole fractions. When a physically reasonable approximation i s introduced, the number of adjustable parameters can be reduced from two to one. Correlations linking the value of this parameter to molecular structure were established for several classes of binary systems. Approximately l 30 sets of binary data were considered; special attention was given to 50 sets of data for aqueous systems. Brief consideration i s given to prediction of phase separation in binary systems and to generalization of the new equation to multicomponent systems. Illustrations for some aqueous mixtures are presented.
F o r design work in chemical engineering and in food engineering, it is frequently necessary t o estimate the thermodynamic properties of a liquid mixture from severely limited experimental data. Because of the complexity of liquid mixtures and the well-known difficulty in obtaining reliable experimental equilibrium data, simple physical models have proved useful for representing phase equilibria in a thermodynamically consistent way. Not only do such models provide efficient tools for data reduction, but they also provide sound guidelines for estimation of other equilibrium mixture properties such as liquid-liquid equilibria from vapor-liquid equilibria and vice versa or multicomponent vapor-liquid equilibria from equilibrium data for binary systems. Present address, Koninklijke/Shell-Laboratorium, Shell Research N.V., Amsterdam, The Netherlands. To whom correspondence should be addressed. 562
Ind. Eng. Chem. Process Des. Develop., Vol. 10, NO. 4, 1971
All simple models of liquid mixtures-Van Laar, ScatchardHildebrand, lattice theory, chemical theories, and two-liquid theory -aim at expressing the dependence of the molar excess Gibbs energy g E on temperature and composition (Prausnitz, 1969). Differelltiation of gE with respect to composition yields the activity coefficients. I n every theory parameters have to be introduced: first, because we do not possess an accurate statistical mechanical description of liquid mixtures and, second, because of the intrinsic flexibility one can achieve by following such a procedure. From both a practical and a theoretical standpoint, it is desirable to minimize the number of parameters needed to describe as wide a variety of systems as possible. A t the same time the tremendous complexity and variety of liquid mixtures prevent a n accurate representation of many mixture properties with one mathematical formula containing a small number of parameters.
Except for some simple systems (symmetric mixtures), a t least two parameters are commonly introduced in isothermal relationships for the excess Gibbs energy-e.g., Van Laar, Margules, Wohl, Wilson, Enthalpic Wilson, Orye, Heil, and Renon nonrandom, two-liquid equations (hereafter referred to as the N R T L equations), The Enthalpic Wilson equations were discussed by Bruin (1970a). The Enthalpic Wilson and N R T L equations have proved to be most useful for strongly nonideal systems; Renon’s N R T L equations are especially flexible (Bruin, 1970a; Renon, 1966; Renon and Prausiiitz, 1968,1969a,b). I n this work we have modified the Renon model to yield oneparameter equations for the excess Gibbs energy and the activity coefficients. While our modification has a close resemblance to Renon’s equations, we have used local volume fractions rather than local mole fractions and have achieved simplification through our interpretation of the physical significance of the interaction energies. It appears that our single binary parameter can be correlated a t least approximately with molecular structure. This correlation makes it possible to interpolate and, in some cases, to extrapolate from limited experimental data in a manner simpler than that used previously (Pierotti et al., 1959). First, we give a general discussion of possible ways to derive one-parameter equations for f, using the concept of the probability of local interactions. The two-liquid model (Prigogine and Bellemans, 1953; Scott, 1956) of a binary system is used although other models may lead to the same results. Our final equations are used to reduce vapor-liquid equilibrium data of 105 different binary systems, including 39 aqueous solutions of organic nonelectrolytes. For 82 systems, data over the whole composition range are used (a total of 136 different data sets, Tables I1 and 111). For 23 additional systems, data for infinite-dilution activity coefficients are used to predict parameter values. Our calculated results are compared with those obtained by the original two-parameter NRTL equations.
1-CELL
= 0=
MOLECULES 1 : q , 2 = 1 MOLECULES 2 :
Two dimensional representation, z = 6
tionship for the excess Gibbs energy. For a binary mixture we consider two types of cells: first, a cell with a molecule 1 a t its center (1-cell) and, second, a cell with a molecule 2 a t its center (2-cell). I n Figure 1, the two kinds of cells are indicated. Molecule 2 is assumed to be considerably larger than molecule 1. The residual Gibbs energy for each cell is equal to the sum of the residual Gibbs energies of all the contacts experienced by the central molecule of the cell with its immediate neighbors (Renon and Prausnitz, 1968). I n the 1-cell are a total of 2412 contacts available; 912 is a measure of the number of sites a molecule of type 1 occupies in a pseudolattice structure with coordination number z. If @ll is the local volume fraction of molecules 1 around ~ ~ ) contributing another molecule 1, there are ( Z Q ~ ~ @contacts 1-1 interaction energies. The same reasoning for molecules 2 around the central molecule 1 shows zq12@?lcontacts with 1-2 interaction energies. The residual Gibbs energy g(1) for the 1-cell thus becomes g(1)
=
2q12911g11p
g(2) =
zq21@1zg12P
2q12921g12p
(3)
+
zy21@22g21Q
(4)
For a pure liquid 1, 911= 1 and @21 = 0, giving for the residual Gibbs energy of the pure liquid 1 : lini 2 1 --*
From these definitions, g1Zp in Equation 2 equals g12pin Equation l . Equations l and 2 have a close relationship to the quasichemical approximation equations for mixtures of molecules of different size (Guggenheim, 1944, 1952). Renon (1966) combined the local composition concept and the two-liquid theory of a binary mixture (Scott, 1956) to derive a relationship for gE. I n his derivation he implicitly assumed that molecules 1 and 2 are of the same size. We now remove this assumption to arrive a t a slightly modified rela-
+
Similar reasoning for a 2-cell gives for g(2)
gl1(l) =
I n Equation 1, g l Z p is proportional to the interaction energy between one molecule of type 1 and one molecule of type 2 and represents a pair-interaction energy. Similarly, glIp is proportional to the interaction energy between two molecules of type 1. A relationship similar to Equation 1 holds for the local volume fraction of a molecule 1 around a central molecule 2:
q2,= (V,/V,)”
Figure 1 . Two types of cells for binary mixture containing molecules of different size
Excess Gibbs Energy Equations
Local Volume Fractions. Wilson (1964) defined the local volume fraction a21 of a molecule of type 2 with molar volume v2 around a molecule of type 1 with molar volume VI by the relationship
2-CELL
1
g(l) =
zyl2gllp
(5)
For component 1 infinitely dilute in component 2, Equation 3 predicts (al1
