Extension to multicomponent systems

The above is not intended to be a definitive list but rather to indicate some of the more commonly used models at the present time. Other, more historical, models have been used extensively, for example the polymerisation models of Toop and Samis (1962) and Masson (1965), the models of Flood (1954), Richardson (1956) and Yokakawa and Niwa (1969). More recently the central atom model by Satsri and Lahiri (1985, 1986) and the complex model of Hoch and Arpshofen (1984) have been proposed. Each has been used with some success in lower-order systems, but the extension to multicomponent systems is not always straightforward. 

The Redlich/Kister expansion, the Margules equations, and the van Laar equations are all special cases of a general treatment based on rational functions, i.e., on equations for G /x X2RT given by ratios of polynomials. They provide great flexibility in the fitting of VLE data for binary systems. However, they have scant theoretical foundation, and therefore fail to admit a rational basis for extension to multicomponent systems. Moreover, they do not incorporate an explicit temperature dependence for the parameters, though this can be supplied on an ad hoc basis. 

Summing the resistance in both phases in order to obtain a single expression for computing the fluxes without knowledge of the interface composition is widely discussed in the literature on binary mass transfer. Here we consider the extension to multicomponent systems (Toor, 1964a Krishna and Standart, 1976b). 

This extension to multicomponent systems of Lewis s tray efficiency model is due to Toor... 

For simplicity, we consider here only a binary mixtures (A, B) and do not discuss the complications posed by extensions to multicomponent systems. Figure 1 shows a schematic phase diagram in the plane of variables temperature T and concentration c of species B. Kinetics of phase separation in bulk fluid mixtures is triggered by a rapid quench (at time t = 0) from the one-phase region into the miscibility gap. The initial equilibrium state (f < 0) is spatially homogeneous, apart from small-scale concentration inhomogeneities. The final equilibrium state towards which the system ultimately evolves (t oo) consists of... 

The extension of the simple statistical model to adsorption of a binary mixture is given by Eq. (3.102) and further extension to multicomponent systems follows naturally. " The parameters of the model (the Henry con-stant and effective molecular volume for each component) are derived from the single-component isotherms so that an a priori prediction of the mixture... 

For all isotherms mentioned in Table 7.3, extensions to multicomponent systems exist and have been mentioned in Sects. 2-4. However, it must be emphasized that in mixture adsorption it is often very difficult to predict or calculate the amounts of the lesser (or weakly) adsorbed components. Hence reliable measurements of coadsorption equilibria are recommended again. 

For simplicity, we consider here only a binary mixtures (A,B), and do not discuss the complications posed by extensions to multicomponent systems. Figure 1 shows a schematic phase diagram in the plane of variables tem-... 

Hildebrand and Hansen parameters can be calculated using other thermodynamic quantities. This section contains some of the relationships for binary systems. Extensions to multicomponent systems are described by Flory (37) and Olabisi et al. (89). 

Cullinan This is an extension of Vignes equation to multicomponent systems ... 

The extension of the cell model to multicomponent systems of spherical molecules of similar size, carried out initially by Prigogine and Garikian1 in 1950 and subsequently continued by several authors,2-5 was an important step in the development of the statistical theory of mixtures. Not only could the excess free energy be calculated from this model in terms of molecular interactions, but also all other excess properties such as enthalpy, entropy, and volume could be calculated, a goal which had not been reached before by the theories of regular solutions developed by Hildebrand and Scott8 and Guggenheim.7... 

The extension of vector-algebraic techniques to multicomponent systems of higher dimensionality (degrees of freedom /> 2) can be carried out straightforwardly, even though one loses the convenience of mutually complementary pairs (X, X ) and orthogonal complementary conjugates (X, x ) that are a special feature of /= 2. In a space of / dimensions, a... 

The extension of the isotherm equation to multicomponent systems is straightforward. The configuration integral for a cavity containing i molecules of species A and j molecules of species B is approximated by the expression... 

Chan and Fair (145) extended their correlation to multicomponent systems. Unfortunately, the extension was tested only against few data points, all derived from laboratory-scale columns. However, this extension represents a large improvement over most alternative theoretical correlations. 

Cullinan /Can. J. Chem. Eng. 45, 377 (1967)] This is an extension of Vignes equation to multicomponent systems ... 

There are several models to describe intracrystalline diffusion (step 3) in microporous media. Diffusion in zeolites is extensively described in Ref. 30. For the modeling of permeation through zeolitic membranes, such a model should take the concentration dependence of zeolitic diffusion into account. Moreover, it should be easy applicable to multicomponent systems. In Section III.C, several models will be discussed. 

Let us now try to extend the model of binary distillation developed in Section 12.1.1 to multicomponent systems. The extension is based on the work of Toor (1964b) and the starting point is the material balance Eqs. 12.1.4, which must now be combined in n — 1 dimensional matrix form as... 

The extension of the potential theory was studied by Bering et al (1963), Doong and Yang (1988) and Mehta and Dannes (1985) to multicomponent systems. We shall present below a brief account of a potential theory put forward by Doong and Yang (1988). The approach is simple in concept, and it results in analytical solution for the multicomponent adsorption isotherm. The basic assumption of their model is that there is no lateral interaction between molecules of different types and pure component isotherm data are described by the DA equation. With this assumption, the parameters of the DA equation (Wq, Eq, n) of each species are not affected by the presence of the other species, but the volume available for each species is reduced. This means that the volume available for the species i is ... 

---