Preferred analytical correlations are less empirical in nature and most often are theoretically based on one of two exact thermodynamic formulations, as derived in Sec. 4. When a single pressure-volume-temperature (PVT) equation of state is applicable to both vapor and liquid phases, the formulation used is
where the mixture fugacity coefficients <t>L for the liquid and <t>,V for the vapor are derived by classical thermodynamics from the PVT expression. Consistent equations for enthalpy can similarly be derived.
Until recently, equations of state that have been successfully applied to Eq. (13-3) have been restricted to mixtures of nonpolar compounds, namely, hydrocarbons and light gases. These equations include those of Bene dict-Webb-Rubin (BWR), Soave (SRK) [Chem. Eng. Sci., 27, 1197 (1972)], who extended the remarkable Redlich-Kwong equation, and Peng-Robinson (PR) [Ind. Eng. Chem. Fun-dam., 15,59 (1976)]. The SRK and PR equations belong to a family of so-called cubic equations of state. The Starling extension of the BWR equation (Fluid Thermodynamic Properties for Light Petroleum Systems, Gulf, Houston, 1973) predicts K values and enthalpies of the normal paraffins up through n-octane, as well as isobutane, isopen-tane, ethylene, propylene, nitrogen, carbon dioxide, and hydrogen sul-
fide, including the cryogenic region. Computer programs for K values derived from the SRK, PR and other equations of state are widely available in all computer-aided process design and simulation programs. The ability of the SRK correlation to predict K values even when the pressure approaches the convergence pressure is shown for a multicomponent system in Fig. 13-18. Similar results are achieved with the PR correlation. The Wong-Sandler mixing rules for cubic equations of state now permit such equations to be extended to mixtures of organic chemicals, as shown in a reformulated version by Orbey and Sandler [AlChE J., 41, 683 (1995)].
An alternative K-value formulation that has received wide application to mixtures containing polar and/or nonpolar compounds is
where different equations of state may be used to predict the pure-component liquid fugacity coefficient Of and the vapor-mixture fugacity coefficient, and any one of a number of mixture free-energy models may be used to obtain the liquid activity coefficient y,L. At low to moderate pressures, accurate prediction of the latter is crucial to the application of Eq. (13-4).
When either Eq. (13-3) or Eq. (13-4) can be applied, the former is generally preferred because it involves only a single equation of state
applicable to both phases and thus would seem to offer greater consistency. In addition, the quantity OL in Eq. (13-4) is hypothetical for any components that are supercritical. In that case, a modification of Eq. (13-4) that uses Henry's law is sometimes applied.
For mixtures of hydrocarbons and light gases, Chao and Seader (CS) [AIChE, 7,598 (1961)] applied Eq. (13-4) by using an empirical expression for OL based on the generalized corresponding-states PVT correlation of Pitzer et al., the Redlich-Kwong equation of state for OV, and the regular solution theory of Scatchard and Hildebrand for The predictive ability of the last-named theory is exhibited in Fig. 13-19 for the heptane-toluene system at 101.3 kPa (1 atm). Five pure-component constants for each species (Tc, Pc, ffl, 8, and vL) are required to use the CS method, which when applied within the restrictions discussed by Lenoir and Koppany [Hydrocarbon Process., 46(11), 249 (1967)] gives good results. Revised coefficients of Grayson and Streed (Gs) (Pap. 20-P07, Sixth World Pet. Conf., Frankfurt, June, 1963) for the OL expression permit application of the CS correlation to higher temperatures and pressures and give improved predictions for hydrogen. Jin, Greenkorn, and Chao [AlChE J, 41,1602 (1995)] present a revised correlation for the standard-state liquid fugacity of hydrogen, applicable from 200 to 730 K.
For mixtures containing polar substances, more complex predictive equations for ytL that involve binary-interaction parameters for each pair of components in the mixture are required for use in Eq. (13-4), as discussed in Sec. 4. Six popular expressions are the Mar-gules, van Laar, Wilson, NRTL, UNIFAC, and UNIQUAC equations. Extensive listings of binary-interaction parameters for use in all but the UNIFAC equation are given by Gmehling and Onken (op. cit.). They obtained the parameters for binary systems at 101.3 kPa (1 atm) from best fits of the experimental T-y-x equilibrium data by setting OV and OL to their ideal-gas, ideal-solution limits of 1.0 and Psat/P respectively, with the vapor pressure Psat given by a three-constant Antoine equation, whose values they tabulate. Table 13-2 lists their parameters for some of the binary systems included in
Table 13-1, based on the binary-system activity-coefficient-equation forms given in Table 13-3. Consistent Antoine vapor-pressure constants and liquid molar volumes are listed in Table 13-4. The Wilson equation is particularly useful for systems that are highly nonideal but do not undergo phase splitting, as exemplified by the ethanol-hexane system, whose activity coefficients are shown in Fig. 13-20. For systems such as this, in which activity coefficients in dilute regions may
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