Multicomponent systems theory

Most distillation systems ia commercial columns have Murphree plate efficiencies of 70% or higher. Lower efficiencies are found under system conditions of a high slope of the equiHbrium curve (Fig. lb), of high Hquid viscosity, and of large molecules having characteristically low diffusion coefficients. FiaaHy, most experimental efficiencies have been for biaary systems where by definition the efficiency of one component is equal to that of the other component. For multicomponent systems it is possible for each component to have a different efficiency. Practice has been to use a pseudo-biaary approach involving the two key components. However, a theory for multicomponent efficiency prediction has been developed (66,67) and is amenable to computational analysis. 

In contrast to quantitative analyses, the results of qualitative tests and of identifications cannot be evaluated by means of mathematical statistics. Instead, information theory is a helpful tool to characterize qualitative analyses, in particular in case of 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... 

As will be seen later (Section V.l), meaningful molecular weights in multicomponent systems can be determined, if the specific refractive index increment appertains to conditions of constant chemical potential of low molecular weight solvents (instead of at constant composition). Practically, this can be realised by dialysing the solution against the mixed solvent and then measuring the specific refractive index increment of the dialysed solution. The theory and practice have been reviewed4-14-1S> 72>. 

It is fortunate that theory has been extended to take into account selective interactions in multicomponent systems, and it is seen from Eq. (91) (which is the expression used for the plots in Fig. 42 b) that the intercept at infinite dilution of protein or other solute does give the reciprocal of its correct molecular weight M2. This procedure is a straightforward one whereby one specifies within the constant K [Eq. (24)] a specific refractive index increment (9n7dc2)TiM. The subscript (i (a shorter way of writing subscripts jUj and ju3) signifies that the increments are to be taken at constant chemical potential of all diffusible solutes, that is, the components other than the polymer. This constitutes the osmotic pressure condition whereby only the macromolecule (component-2) is non-diffusible through a semi-permeable membrane. The quantity... 

In this chapter consideration is given to the theory of the process, methods of distillation and calculation of the number of stages required for both binary and multicomponent systems, and discussion on design methods is included for plate and packed columns incorporating a variety of column internals. 

Two different methods have been presented in this contribution for correlation and/or prediction of phase equilibria in ternary or mul> ticomponent systems. The first method, the clinogonial projection, has one disadvantage it is not based on concrete concepts of the system but assumes, to a certain extent, additivity of the properties of individiial components and attempts to express deviations from additivity of the properties of individual components and attempts to express deviations from additivity by using geometrical constructions. Hence this method, although simple and quick, needs not necessarily yield correct results in all the cases. For this reason, the other method based on the thermodynamic description of phase equilibria, reliably describes the behaviour of the system. Of cource, the theory of concentrated ionic solutions does not permit a priori calculation of the behaviour of the system from the thermodynamic properties of pure components however, if a satisfactory equation is obtained from the theory and is modified to express concrete systems by using few adjustable parameters, the results thus obtained are still substantially more reliable than results correlated merely on the basis of geometric similarity. Both of the methods shown here can be easily adapted for the description of multicomponent systems. 

There is no well developed theory which describes the sputtering of multicomponent systems (e.g., oxides), but some authors (sec, for example, Ref. 88) have assumed that the results of Sigmund s theory are valid and have applied reasonable values for the stopping power and binding energy to obtain results for various multi-component systems. 

In the preceding section it is seen how the theory of multicomponent systems gives the correct starting formulas for the analysis of ordinary and thermal diffusion in binary systems. Whereas these latter topics have been the subject of considerable investigation, there are a number of types of more complex diffusion problems of engineering interest for which little has been done. Several of these topics are discussed here, and an attempt is made to indicate to what extent they can be interpreted in terms of the theoretical development in the preceding sections. 

Expressions for the transport coefficients suitable for use in computational simulations of chemically reacting flows are usually based on the Chapman-Enskog theory. The theory has been extended to address in detail transport properties in multicomponent systems [103,178]. 

The first mean-field theories, the lattice models, are typified by the Flory-Huggins model. Numerous reviews (see, e.g., de Gennes, 1979 Billmeyer, 1982 Forsman, 1986) describe the assumptions and predictions of the theory extensions to polydisperse and multicomponent systems are summarized in Kurata s monograph (1982). The key results are reiterated here. 

The investigation of the copolymerization dynamics for multicomponent systems in contrast to binary ones becomes a rather complicated problem since the set of the kinetic equations describing the drift of the monomer feed composition with conversion in the latter case has no analytical solution. As for the numerical solutions in the case of the copolymerization of more than three monomers one can speak only about a few particular results [7,8] based on the simplified equations. A simple constructive algorithm [9] was proposed based on the methods of the theory of graphs, free of the above mentioned shortcomings. 

A generalization of the theory of the binary copolymerization for multicomponent systems in the case of the terminal model (2.8) is not difficult since the copolymer microstructure is still described by the Markov chain with states S corresponding to the monomer units Mj. The number m of their types determines the order of... 

Examples of calculations of such characteristics for the concrete terpolymer are presented, for instance, in the monograph listed as Ref. [6]. Similar computer calculations of the trajectories i(p) and X(p) which determine the values of the statistical characteristics (5.3) and (5.7) can be in principle carried out for other processes of multicomponent copolymerization. However, the original approach [6, 201, 13-15, 18, 202, 203] allows one to use traditional methods of dynamic systems theory [204] to reveal the main qualitative peculiarities of the behavior of the trajectories X(p) and X(p) only via the elementary arithmetical operations by means of a pocket microcalculator. 

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