## Prediction of Vapour Liquid Equilibria

Although vapour-liquid equilibria have been investigated for more than 10 000 systems, values resulting from various combinations are still unknown. It would be impractical to determine experimentally all the systems needed individually.

In principle, experimental values of some thermo-dynamic properties can be used to estimate other properties. For examples, binary vapour-liquid equilibrium can be estimated from the liquid activity coefficients calculated from mutual solubility data for the same mixture, and the infinite-dilution activity coefficients measured from gas-liquid chromatogra-phy can be used to predict the vapour-liquid equilibria over the complete concentration range. Some prediction methods are briefly described below with emphasis placed on binary mixtures. Extending the

Table 7 Some mixing and combining rules for cubic equations of state van der Waals/Berthelot a,= {a,a,)1/2

Modified van der Waals/Berthelot a = EE yya, b = EE yyA

aij=(aia,)1/2(1 - kj) bj = Hbi + bj)(1 -cj) Wong-Sandler b-~RT=EE (bj-~RT, b # b, b, = ^-j(1 - kj)

b = E b CRT where: GE is a selected excess Gibbs energy model C is characteristic of the equation of state

(A) Mole fraction ethanol (B) Temperature (K)

Figure 3 (A) Correlating the phase behaviour of the (ethanol # benzene) system at 323.15 Kby means of the gamma-phi approach. The lines represent the values calculated by using the Margules equations and the points represent the experimental values reported by ND. Litvinov (1952) Isothermal equilibrium of vapor and liquid in systems of three fully miscible liquids. ZhurnalFizicheskoi Khimii 26:1405-1412. (B) Predicting the phase behaviour of the (0.2654 mole fraction ethane # 0.7346 mole fraction n-heptane) mixture by means of the equation-of-state approach. The smooth curve represents the values calculated by using the Peng-Robinson equation, and the points represent the experimental values reported by WB Kay (1938) Liquid-vapor phase equilibrium relations in the ethane-n-heptane system. IndustrialandEngineering Chemistry 30: 459-464.

(A) Mole fraction ethanol (B) Temperature (K)

Figure 3 (A) Correlating the phase behaviour of the (ethanol # benzene) system at 323.15 Kby means of the gamma-phi approach. The lines represent the values calculated by using the Margules equations and the points represent the experimental values reported by ND. Litvinov (1952) Isothermal equilibrium of vapor and liquid in systems of three fully miscible liquids. ZhurnalFizicheskoi Khimii 26:1405-1412. (B) Predicting the phase behaviour of the (0.2654 mole fraction ethane # 0.7346 mole fraction n-heptane) mixture by means of the equation-of-state approach. The smooth curve represents the values calculated by using the Peng-Robinson equation, and the points represent the experimental values reported by WB Kay (1938) Liquid-vapor phase equilibrium relations in the ethane-n-heptane system. IndustrialandEngineering Chemistry 30: 459-464.

application to multicomponent mixtures is feasible once good correlation of the vapour-liquid equilibria of its constituent binary systems becomes available.

### Prediction from Pure Component Properties

Application of the regular solution theory For mixtures containing nonpolar components that are not much different in size and shape, the regular solution theory of Hildebrand leads to a semi-quantitative prediction of yk values of all components in a mixture. In terms of the solubility parameter, the activity coefficients of the components in a regular solution can be calculated from the equation:

where the volume average solubility parameter is given by:

and the volume fraction is defined by:

The solubility parameter for substance k, àk, is defined by:

where AUV and AHV are, respectively, the molar energy and enthalpy of vaporization of pure liquid k at temperature T. The assumption involved here is that T is well below the critical temperature in order to make the approximation valid. The calculated liquid activity coefficients can then be used to obtain the desired vapour-liquid equilibrium values. For a binary mixture:

Should a binary interaction parameter be required to improve the data representation, an extension of the approach to the prediction of multicomponent vapour-liquid equilibrium may not be practical; attempts made to correlate the binary parameters have not been successful.

Liquid activity coefficients at infinite dilution y"

Values of y" are particularly useful for obtaining the parameters of any of the two-constant equations for the excess Gibbs energy; the y" values for a binary system are the parameter values. For example, y" = A and y" = B for the van Laar and Margules equations presented in Table 3. If a three-parameter equation is used, the third parameter must be determined by an independent approach.

The modified separation of cohesive energy density (MOSCED) method proposed by Thomas and Eckert in 1984 may be used to predict y00 values from pure component parameters. This method is based on a modified regular solution theory and the assumption that the forces contributing to the cohesive energy are additive. It has been reported that the average error of 3357 y00 values predicted by this method was 9.1%.

In general, calculated equilibrium vapour compositions are relatively insensitive to moderate errors in the yo used in the calculation.

Prediction of Binary y Values Using Azeotropic or Mutual Solubility Data

Prediction from azeotropic data Many binary systems exhibit azeotropic behaviour. At an azeotropic condition, the compositions of the liquid and vapour phases are identical. At low pressures, the liquid activity coefficients can be simply calculated by:

The parameters of any two-parameter expression of the excess Gibbs energy can then be obtained and used for extrapolating vapour-liquid equilibrium over the complete concentration range.

### Prediction of y values from mutual solubility data

The thermodynamic consideration applicable to a binary system at vapour-liquid equilibrium is also applicable to a partially miscible binary liquid mixture at equilibrium. Hence, the activity coefficients of the two liquids at the temperature at which the solubilities were experimentally determined can be expressed by:

where the two superscripts refer to the two liquid phases. Applying these relationships to any two-parameter expression of the excess Gibbs energy leads to the determination of the parameter values, which permit vapour-liquid equilibrium estimation of the mixture.

### Prediction of y from Group Contribution Methods

In group contribution methods, the calculation of thermodynamic properties of pure fluids is based on the assumption that each molecule is an aggregate of functional groups. Langmuir in 1925 extended the concept to mixtures. Redlich, Derr, Pierotti and Papadopoulos developed a group interaction model for heats of solution in 1959. Adopting the concepts presented by these authors, Wilson and Deal suggested in 1962 a solution of the groups approach by which liquid activity coefficients can be estimated on the basis of group contributions. In this approach, the logarithm of the activity coefficient of a component is assumed to be the sum of two contributions: the configurational contribution, which accounts for the differences in molecular size, and the group interaction contribution, which accounts for the intermolecular forces originating from the different functional groups.

The group contribution approach to calculating y is attractive because through this approach it is possible to estimate vapour-liquid equilibria of nonideal mixtures without experimentation. Although a large number of mixtures can result from pure compounds, the functional groups, such as CH2, OH, CO, COO and COOH, that constitute these compounds are limited. If the activity coefficients of the mixture components could be obtained from a knowledge of the interactions of these groups, and with the assumption that the contribution to y by one group within a molecule is independent of that made by any other group in that molecule, a relatively small number of parameters would suffice for the prediction of the activity coefficients for mixtures containing the same groups. This assumption implies that the contribution of the group interaction is independent of the nature of the molecule.

Two of these approaches are mentioned here.

Analytical solutions of groups (ASOG) method

Following the concept of the solutions of groups of Wilson and Deal, the analytical solutions of groups (ASOG) method was first presented by Derr and Deal in 1969. Basically, the practical application of the solutions of groups concept involves the reduction of liquid activity coefficients obtained from experimental data for vapour-liquid equilibria into a number of binary group interaction parameters. The working equations of the ASOG method are presented in Table 8.

Kojima and Tochigi in 1979 calculated the group interaction parameters for 31 groups and used the method to predict the vapour-liquid equilibria for 936 binary systems, 103 ternary systems, five quaternary systems, and two quinary systems at low pressures. They reported that the average absolute deviation of the predicted vapour compositions was 1.2%.

Universal functional group activity coefficient (UNIFAC) method The universal functional group activity coefficient (UNIFAC) method proposed by Fredenslund, Gmehling and Rasmussen and the ASOG method are based on the same principle of group contributions. The main difference between these two methods is in the equations used for

Table 8 The analytical solution of groups (ASOG) method ln y, = ln yS + ln yG ln yS = 1 -r-lnr, where:

r"

Xj= mole fraction of molecule j v = number of atoms other than hydrogen in molecule j ln yG = X v(ln rk- ln rip)

where: vki"number of atoms other than hydrogen in group k in molecule i ln rk= 1 — Ck — ln Dk ln rk = 1 — Ck — ln Dk

Dk = yxXjAkj, Df-yxXfAj

S = Y x,Y Vki i k representing the excess Gibbs energy. The Wilson equation is used in the ASOG method, whereas the two-parameter universal quasi-chemical (UNIQUAC) equation of Abrams and Prausnitz is used in the UNIFAC method. The working equations of the UNIFAC method are presented in Table 9. There are 50 main groups together with their subgroups for the determination of the parameters involved in the calculation. For y calculations for multicomponent systems, the adjustable binary parameters are evaluated from binary vapour-liquid equilibrium data.

### Prediction using Equations of State

The equations of state successfully used for correlation of binary vapour-liquid equilibria can be used for the purposes of predicting multicomponent vapour-liquid equilibria, provided that the binary interaction parameters of all the constituent binary systems are available. All the parameters should be obtained by regression of the binary data using the same mixing and combining rules. Interpolated and estimated values of these parameters are available for some systems, but their values are subject to frequent revision.

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