Model Redlich-Kister equation

An accurate representation of the phase equilibrium behavior is required to design or simulate any separation process. Equilibrium data for salt-free systems are usually correlated by one of a number of possible equations, such as those of Wilson, Van Laar, Margules, Redlich-Kister, etc. These correlations can then be used in the appropriate process model. It has become common to utilize parameters from such correlations to obtain insight into the fundamentals underlying the behavior of solutions and to predict the behavior of other solutions. This has been particularly true of the Wilson equation, which is shown below for a binary system. 

For mixtures that do not obey the Porter or Margules equations, additional high-order terms must be kept in the Redlich-Kister expansion hence, more parameters must be evaluated from experimental data. Alternatively, if we want to keep only two parameters, then we must abandon the Redlich-Kister expansion for some more complicated representation of g. Many functional forms have been proposed [1, 2], but here we restrict our attention to a useful expression proposed by Wilson in 1964 [14] and now identified as one of the class of "local-composition" models [2], For binary mixtures Wilson s equation takes the form... 

Many approximations used in modeling thermodynamic properties are based on the Taylor series. Examples are the virial expansions for the equation of state and the Redlich-Kister expansion of the excess Gibbs energy. Let/(x) and all its derivatives be continuous and single-valued on [a, b]. Then the Taylor series provides an approximation to f b) if we know/at a nearby point x = a and if we can evaluate derivatives of/ atx = a,... 

The Margules equation models the excess Gibbs free energy by a two-parameter Redlich-Kister polynomial. The excess Gibbs energy and the activity coefficients are given by the following equations ... 

You may have noticed that the Margules model is equivalent to the two-parameter Redlich-Kister polynomial used in Examples 12.6 and 12.7. This maybe confirmed by setting A12 = Oo andA = Oo + Oi to the above equations (see Example 12.8. below). In the form given here, the expressions for the activity coefficients are symmetric in the two components such that each expression is obtained from each other by switching the subscripts 1 and 2. 

Obtaining the derivatives on the right-hand side requires a fitting equation for the excess mixing quantities. The Wilson, Redlich-Kister, and nonrandom two liquid (NRTL) model equations are some of the most commonly used (Poling, Praunitz, and O Connell 2000). Some additional practical considerations are also provided in Section 1.3.9 and Section 4.2 in Chapter 4. 

Many equations, either empirical or derived from models, are available to represent excess thermodynamic properties as a function of the liquid mole fraction and of a number of adjustable parameters for use in Equations 4.3 and 4.4 (Prausnitz, Lichtenthaler, and Gomes de Azevedo 1999). In the treatment of our experimental data, we have examined the capability of many of these equations to fit data with the smallest number of parameters and with the least standard deviation of the O.F. (Lepori et al. 1998). For the majority of systems, either binary or ternary, examined by us in the course of about two decades, the rational form (Myers and Scott 1963) of the Redlich-Kister (RK) expression (Redlich and Kister 1948), and the Wilson (Wilson 1964) equation in the extended form (Novtik, MatouS, and Pick 1987), have resulted in the most appropriate representations of and For ternary systems, the excess functions can be expressed as the sum of a contribution (subscript B), which depends only on the parameters of the three binary systems, and a ternary contribution (subscript T), which involves additional parameters. 

Smith and Brown have recently completed an extensive study of alcohol -I-alkane mixtures in which Barker s method is compared directly with chemical association models. They conclude that the equations of Barker s theory are more satisfactory than those of Kretschmer and Wiebe with concentration equilibrium constants and those of Redlich and Kister with mole fraction equilibrium constants. However, even Barker s equations are inadequate for predicting accurate excess Gibbs energies and enthalpies. Sosnkowska-Kehiaian, Hryniewicz, and Kehiaian accounted for their measurements of enthalpies of mixing of alkanes and other hydrocarbons with n-alkyl ethers using a zeroth approximation formula with one temperature-independent energy parameter. 

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