Wilson equation, vapor-liquid equilibrium

Example 4.5 2-Propanol (isopropanol) and water form an azeotropic mixture at a particular liquid composition that results in the vapor and liquid compositions being equal. Vapor-liquid equilibrium for 2-propanol-water mixtures can be predicted by the Wilson equation. Vapor pressure coefficients in bar with temperature in Kelvin for the Antoine equation are given in Table 4.113. Data for the Wilson equation are given in Table 4.126. Assume the gas constant R = 8.3145 kJ-kmol 1-K 1. Determine the azeotropic composition at 1 atm. 

A model is needed to calculate liquid-liquid equilibrium for the activity coefficient from Equation 4.67. Both the NRTL and UNIQUAC equations can be used to predict liquid-liquid equilibrium. Note that the Wilson equation is not applicable to liquid-liquid equilibrium and, therefore, also not applicable to vapor-liquid-liquid equilibrium. Parameters from the NRTL and UNIQUAC equations can be correlated from vapor-liquid equilibrium data6 or liquid-liquid equilibrium data9,10. The UNIFAC method can be used to predict liquid-liquid equilibrium from the molecular structures of the components in the mixture3. 

The vapor-liquid x-y diagram in Figures 4.6c and d can be calculated by setting a liquid composition and calculating the corresponding vapor composition in a bubble point calculation. Alternatively, vapor composition can be set and the liquid composition determined by a dew point calculation. If the mixture forms two-liquid phases, the vapor-liquid equilibrium calculation predicts a maximum in the x-y diagram, as shown in Figures 4.6c and d. Note that such a maximum cannot appear with the Wilson equation. 

Measurements of binary vapor-liquid equilibria can be expressed in terms of activity coefficients, and then correlated by the Wilson or other suitable equation. Data on all possible pairs of components can be combined to represent the vapor-liquid behavior of the complete mixture. For exploratory purposes, several rapid experimental techniques are applicable. For example, differential ebulliometry can obtain data for several systems in one laboratory day, from which infinite dilution activity coefficients can be calculated and then used to evaluate the parameters of correlating equations. Chromatography also is a well-developed rapid technique for vapor-liquid equilibrium measurement of extractive distillation systems. The low-boiling solvent is deposited on an inert carrier to serve as the adsorbent. The mathematics is known from which the relative volatility of a pair of substances can be calculated from the effluent trace of the elutriated stream. Some of the literature of these two techniques is cited by Walas (1985, pp. 216-217). 

Related Calculations. These calculations show how to use vapor-liquid equilibrium data to obtain parameters for activity-coefficient correlations such as those of Van Laar and Wilson. (Use of liquid-liquid equilibrium data for the same purpose is shown in Example 1.20.) If the system forms an azeotrope, the parameters can be obtained from a single measurement of the azeotropic pressure and the composition of the constant boiling mixture. If the activity coefficients at infinite dilution are available, the two parameters for the Van Laar equation are given directly, and the two in the case of the Wilson equation can be solved for as shown in the example. 

In principle, the parameters can be evaluated from minimal experimental data. If vapor-liquid equilibrium data at a series of compositions are available, the parameters in a given excess-free-energy model can be found by numerical regression techniques. The goodness of fit in each case depends on the suitability of the form of the equation. If a plot of GE/X X2RT versus X is nearly linear, use the Margules equation (see Section 3). If a plot of Xi X2RT/GE is linear, then use the Van Laar equation. If neither plot approaches linearity, apply the Wilson equation or some other model with more than two parameters. 

Compute the Gfj parameters for the Wilson equation. General engineering practice is to establish liquid-phase nonideality through experimental measurement of vapor-liquid equilibrium. Models with adjustable parameters exist for adequately representing most nonideal-solution behavior. Because of these models, the amount of experimental information needed is not excessive (see Example 3.9, which shows procedures for calculating such parameters from experimental data). 

So that an azeotrope with acetone does not form, the alcohol used must have a high enough boiling point. This requirement is reliably established only if vapor-liquid equilibrium data for at least two, preferably three, of the members of the series with acetone are known. The Pierotti-Deal-Derr method (4) (discussed later) or the Tassios-Van Winkle method (5) can be used in this case. In the latter method a log-log plot of y°i vs. P°i should yield a straight line. Figure 1 presents results for n-alco-hols and benzene from the isobaric (760 mm Hg) data of Wehe and Coates (6). Reliable infinite dilution activity coefficients are established for the other n-alcohols from data for at least two, and preferably three, of them. These y° values are used with equations like those of Van Laar or Wilson (7) to generate activity coefficients at intermediate compositions and to check for an existing azeotrope or a difficult separation (x-y curve close to the 45° line). 

The parameter An was calculated from vapor—liquid equilibrium data for binary solvents using Eq. (29). The activity coefficients of the components in the binary solvents were expressed via the Wilson equation (Wilson, 1964) and the Wilson parameters Ln and Ln were taken from GmehUng s vapor—liquid equilibrium data compilation (Gmehling et al., 1977—2003). 

FIGURE 2. Solubilities of naphthalene (S is the mole fraction of naphthalene) in the mixtures a) methanol + water and b) ethanol + water. The experimental data (0) were taken from Ref. (2). The solid lines represent the solubilities of naphthalene predicted using equation M4. The Wilson constants were taken from Gmeling s vapor-liquid equilibrium compilation (2 ). Thus, the only solubilities in pure water and cosolvents were used for prediction. 

A significant advantage of the Wilson equation is that it can be used to calculate the equilibrium compositions for multicomponent systems using only the Wilson coefficients obtained for the binary pairs that comprise the multicomponent mixture. The Wilson coefficients for several hundred binary systems are given in the DECHEMA vapor-liquid data collection (DECHEMA, 1977) and by Hirata (1975). Hirata gives methods for calculating the Wilson coefficients from vapor-liquid equilibrium experimental data. 

Fig. 2 Isobaric vapor-liquid equilibrium of acetone and chlorobenzene at 1.01325 Bars. Solid lines are calculated from Wilson s equation. Dashed lines are from Raoult s law.

The activity coefficients can be calculated from Wilson s equations or from UNIFAC if the parameters of the models are known. There are some parameters for UNIFAC in Magnussen, Rasmussen, and Fredenslund, Gupte and Danner,and Hooper, Michel, and Prausnitz. These parameters are not as accurate as those for vapor-liquid equilibrium. 

The Wilson equation is widely used for many nonpolar, polar, and associated solutions in vapor-liquid equilibrium systems. It is often best for hydrogen-bonded substances. For multicomponent solutions, it makes effective use of binary-solution parameters to give good results, but it cannot predict the liquid immiscibihty phenomena. 

Comparison with experimental data shows that the complete local-composition equation preserves the quality of Wilson s equation in describing vapor-liquid equilibrium of completely miscible systems. There are no more than slight differences between the complete equation and Wilson s equation in the fitting of data. But the complete local-composition (CLC) equation extends Wilson s local-composition equation to partially miscible solutions. Good predictions of the coexistent liquid compositions of ternary mixtures based on the binary parameters have been found for water + ethyl acetate + ethanol, for water + methyl acetate + acetone, and for water + acrylonitrile + acetonitrile. 

Non ideal Compute 7-Wilson equation Vapor-liquid equilibrium data Muir and Howal3... 

Binary and ternary forms of the NRTL equation were evaluated and compared to other equations for vapor-liquid equilibrium applications by Renon and Prausnitz, Larson and Tassios, Mertl, Marina and Tassios, and Tsu-boka and Katayama. In general, the accuracy of the NRTL equation is comparable to that of the Wilson equation. Although is an adjustable constant, there is little loss in accuracy over setting its value according to the rules described above. Methods for determining best values of NRTL binary parameters are considered in detail in the above references. Mertl tabulated NRTL parameters obtained from 144 sets of data covering 102 different binary systems. Other listings of NRTL parameters are also available. 

Vapor-Liquid Equilibrium Data Collection (Gmehling et al., 1980). In this DECHEMA data bank, which is available both in more than 20 volumes and electronically, the data from a large fraction of the articles can be found easily. In addition, each set of data has been regressed to determine interaction coefficients for the binary pairs to be used to estimate liquid-phase activity coefficients for the NRTL, UNIQUAC, Wilson, etc., equations. This database is also accessible by process simulators. For example, with an appropriate license agreement, data for use in ASPEN PLUS can be retrieved from the DECHEMA database over the Internet. For nonideal mixtures, the extensive compilation of Gmehling (1994) of azeotropic data is very useful. 

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