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It can be seen also that the criticaL isotherm is best reproduced with a c value corresponding to a critical compressibility equal to, or slightly lower than L/3, i.e. with c values very similar to those determined from low-temperature liquid densities. That justifies the common assumption of a constant c va1ue, but prevents one from expecting miracles in the critical area, which is the weak point of aII cubic equations. c) Parameter mixinq rules From theoretical considerations, a linear mixing rule for b and a quadratic one for ilart can be derived. A linear mixing rule involves no binary parameter, but for the attractive parameter, in a few cases only (e.9., mixtures of hydrocarbons of the same type), the generalized expression can be applied: a- (Ex ifr)' resulting from the more general quadratic rule: a=EEx.x .a.LJ I J rJ when letting- a. .=ú.-IJ I f (12) (13) (14) 354 More generaIJ.y, even for by hydrocarbons, dr empirical mean must be applied: regular mixtures such as those formed correction term over the geometric cl- .=l1l l-k..)r'a.a.Ll' 1 l (1s) which, in principle, has to be determined by correlation of experimental VLE data. Fortunately, Urj values can be considered as independent of the temperature and are almost constant for each class of compounds (e.9., all COr-alkane pairs), so that their values can be estimated from similar systems, when experimental data are lacking. The above mixing rules are quite adequate with mixtures of hydrocarbons and sIÍghtly polar compounds, such as carbon dioxide and hydrogen sulfide, but with systems containing polar components they give only a first-order approximation and must be replaced by other, more sophisticated rules, involving at Ieast two empirical parameters for each binary pair. Theoretical considerations prevent assuming other than quadratic mixÍng rules for a and b, in order to keep a quadratic mixing rule of the second virial coefficients (B=b-a/RT); so one proposal bras to use a quadratic rule for b also, thus introducing a second binary constant. UnfortunateJ.y the two binary constants in a and b are correlated and for strongly non-ideal systems they tend to assume absurd val-ues. Furthermore, it can be seen that, in order to have a correct mixing rule for higher virial coefficients also, a linear mixing rule of b would be required. Another solution satisfying a quadratic mixing rule of the second virÍal coeffieient is given by the so-calIed density-dependent mixing rules ( Lr-idecke, Prausnitz, 1985) ,(MolIerup,1985), where a quadratic rule is applied for the attractive parameter when the density tends to zero, and a non-guadratic one at non-zero densÍties. A major drawback of this solution is that, at least with mixtures, the resulting equation of state is no longer cubic. Most commonly, authors applied a Iinear rule for b, focusing on the attractive parameter to give the required flexibility (and forgetting the quadratic rule). One proposal (Kabadi,Danner,L985), arising from the attempt to reproduce the mutual solubility of water and hydrocarbons, was to 355 use Iinearly composition-dependent Orj values, which results in different k. - values in the two liquid phases and a cubic mixing1J ruLe f or ttatt . Another, very attractive proposal, which opened a way to the rich fields of the liquid state theories, was that by Huron and Vidal (Huron,Vidal ,L979). By the simple, reasonable assumption that, at infinite pressure, the excess Gibbs energy tends to a finite limit, a linear mixing rule of b was confirmed and the following rule was derived: a/b=Ex . a, /b, -g:/ 1n/=Ex, ( a, /b, -IrV:/ 1n2 ) (l_6) Eq.(16) in fact is useLess in determining the interaction parameters, but it allows to defÍne the analytical form of the mixing rules, by simply assuming tor gl one of the models available -'E from Iiquid-state theory. Besides, connecting the nixing rules to an excess Gibbs energy makes it possible even to predict nI O" applying the principle of group contributions (just like in the well known UNIFAC method). In the Iast years, two predictÍve models (Tochigi et aI.,1990), (Fredenslund et a1.,1991) have been presented, based on the SRK EOS and a UNIFAC-type f@ model. Referring to the infinite-pressure excess properties avoids any reference to a physical state and allows the treatment of supercritical components aIso, which was impossible with all the methods based on liquid-phase activity coefficients. If a table of infinite-pressure group contributions is developed, it can be thus extended to substances like hydrogen, nitrogen, methane, carbon dioxide, etc. , what r^ras impossibi le with UNIFAC and simi Iar methods. A similar, but not identical approach has been followed with the Peng-Robinson EOS by Abdoul et al. (1991-), and with the R-K and P-R EOS by Lermite and Vidal ( 1-992 ) . Another attractive solution was proposed by Michelsen (Michelsen, 1990ab), (Michelsen,Dahl, 1990), to predict high-pressure V-L equilibria, when no experimental data are available. It uses T values generated by a group- contributÍon method like UNIFAC, to defÍne the mixing rules of the SRK equation of state and so predict directly VLE K values. The method appears simpler and more efficient than previous versions (MoIIerup, L986), (Soave,1986), 356 (Gupte et a1.,1986); although restricted theoretically to mixtures of subcritical compounds (or better, to components with reduced temperatures lower than about 0.9) it can be easily extended to the light gases. Basing on Michelsen's work, Gmehling and coworkers (HoIderbaum, Gmeh1ing, 1991) are no\^r extending the UNIFAC group contribution table to Iight compounds such as methane, carbon dioxide and even hydrogen. Fig.6 shows the phase envelope of the propane-benzene system at 444.3 K (beyond propane's critical temperature), ds calculated by Michelsenrs method, with UNIFAC f values: experimental VLE data hrere correctly reproduced up to the critical point, over 60 bars. o 05 1 \./ I ^ , ^ ^ ^ ^ ^ \ ^,. ljf uudf ìe / Fig.6. VLE of propane-benzene system Crosses:experimental- SoIid lines:calc., Michelsen method FUTURE DEVELOPMENTS In spite of the development of high-speed computers, enabling the use of more complex (and more accurate) equations of state, cubic equations, and in particular the R-K equation in its many versions, seem still far from having concluded their rday. Not particularly accurate, but easy to solve and program, moderate in computer requirements, easy to adjust, they remain the work-horse of students, researchers and designers. Further developments are stilI needed anyway, in particular to improve their predictive ability; some of them are outlined here: a) improve the vapor pressure prediction in the low temperature i'essrre t ar ItÈ + b) 357 range: expressions used normally for o(T*) are usually accurate in a pressure range from, say, I/IOO bar upwards, but the heaviest components of a mixture can have much Lower saturation pressures and any error in their evaluation can affect seriously the calculated dew-point conditions. It is necessary to develop new ot functions which can be extrapolated with confidence to much Lower reduced temperatures than presently. improve the treatment of heavy hydrocarbons: first, by making their parameters in the a function predictable from available properties. On such matter, a good work is being carried on at the University of MarseiIIe for the Peng-Robinson EOS (Carrier et aI.,1988; Rogalski et al., l-990). Second, methods for the estimatÍon of pseudocomponent binary interaction constants should keep into account their higher or lower aromatic nature, by the use of parameters bound to it (Iike, e.9., the blatson K factor). improve high-temperature, high-pressure behaviour: current ot(T*) functions are