For multicomponent systems

The preceding material of this section has focused on the most important phenomenological equation that thermodynamics gives us for multicomponent systems—the Gibbs equation. Many other, formal thermodynamic relationships have been developed, of course. Many of these are summarized in Ref. 107. The topic is treated further in Section XVII-13, but is worthwhile to give here a few additional relationships especially applicable to solutions. 

In order to specify fhe size of fhe sysfem, af leasf one of fhese variables ought to be extensive (one that is proportional to the size of the system, like n or the total volume V). In the special case of several phases in equilibrium several extensive properties, e.g. n and Vfor two phases, may be required to detennine the relative amounts of the two phases. The rest of the variables can be intensive (independent of the size of the system) like T,p, the molar volume V = V/n,or the density p. For multicomponent systems, additional variables, e.g. several ns, are needed to specify composifion. 

The analogue of the Clapeyron equation for multicomponent systems can be derived by a complex procedure of systematically eliminating the various chemical potentials, but an alternative derivation uses the Maxwell relation (A2.1.41)... 

Discussions of shortcut methods have appeared many times in the Hterature (16,36), accompanied by the usual admonition to use such methods only for approximate designs or analyses. For multicomponent systems having significant nonideaUties, the shortcut methods can be grossly in error. 

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 order to determine the packed height it is necessary to obtain a value of the overall number of transfer units methods for doing this are available for binary systems in any standard text covering distillation (73) and, in a more complex way, for multicomponent systems (81). However, it is simpler to calculate the number of required theoretical stages and make the conversion ... 

The local-composition models have hmited flexibility in the fitting of data, but they are adequate for most engineering purposes. Moreover, they are implicitly generalizable to multicomponent systems without the introduction of any parameters beyond those required to describe the constituent binaiy systems. For example, the Wilson equation for multicomponent systems is written ... 

FIG. 14-9 Graphical design method for multicomponent systems absorption of hiitane and heavier components in a soliite-free lean oil. 

It is basically a fractionation process that depends not only on molecular size, but also on chemical composition, stereo-configuration, branching, and crosslinking. For multicomponent systems, fractionation with different ion polymolecularity, chemical heterogeneity and sequence length distribution, solubility or elution fractionation is of primary importance. Therefore, gel permeation chromatography or size exclusion chromatography is used as an important tool for the characterization of PBAs. 

For multicomponent systems, the relation of the system can be expressed using the relative volatility ... 

Martinez-Ortiz, J. A., and D. B. Manley, Direct Solution of the Isothermal Gibbs-Duhem Equation for Multicomponent Systems, Ind. Eng. Chem. Process Des. Dev., 17, 3, (1978) p. 346. 

Tersoff, J., Modeling Solid-State Chemistry Interatomic Potentials for Multicomponent Systems," Phys. Rev. B, Vol. 39, No. 8,1989, pp. 5566-5568. 

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 vapour-liquid data collection, DECHEMA (1977), and by Hirata (1975). Hirata gives methods for calculating the Wilson coefficients from vapour liquid equilibrium experimental data. 

The distillation of binary mixtures is covered thoroughly in Volume 2, Chapter 11, and the discussion in this section is limited to a brief review of the most useful design methods. Though binary systems are usually considered separately, the design methods developed for multicomponent systems (Section 11.6) can obviously also be used for binary systems. With binary mixtures fixing the composition of one component fixes the composition of the other, and iterative procedures are not usually needed to determine the stage and reflux requirements simple graphical methods are normally used. 

The prediction of efficiencies for multicomponent systems is also discussed by Chan and Fair (1984b). For mixtures of dissimilar compounds the efficiency can be very different... 

For multicomponent systems, mixing rules are needed to determine the values of a and b3 5 ... 

For multicomponent systems, Equation 10.4 can be written for the limiting component, that is, the component with the highest Kt. Having determined the number of stages, the concentrations of the other components can be determined from Equation 10.6. 

Equations 10.8 to 10.10 can be written in terms of a stripping factor (S), where S = 1/A. For multicomponent systems, Equations 10.8 and 10.9 can be used for the limiting component (i.e. the component with the lowest K,) and then Equation 10.10 can be used to determine the distribution of the other components. 

The histogram reweighting methodology for multicomponent systems [52-54] closely follows the one-component version described above. The probability distribution function for observing Ni particles of component 1 and No particles of component 2 with configurational energy in the vicinity of E for a GCMC simulation at imposed chemical potentials /. i and //,2, respectively, at inverse temperature ft in a box of volume V is... 

This model was shown to be applicable for describing moisture uptake kinetics (in vacuum) above RH0 for single-component systems of alkali halides, sugars, and choline salts [31]. The model later was extended to consider the moisture uptake kinetics above RH0 for multicomponent systems of these substances [33]. 

Choices of precursor(s) may be dictated by solubility, reactivity, or other property. For multicomponent systems, mutual solubility is another factor that must be considered. For such solutions, the solvent selected must facilitate dissolution of all precursors. 

Cederberg, G. A., R. L. Street and J. O. Leckie, 1985, A groundwater mass transport and equilibrium chemistry model for multicomponent systems. Water Resources Research 21, 1095-1104. 

For multicomponent systems, components A and B refer to the light and heavy keys respectively. In this problem, o-cresol is the light key and m-cresol is the heavy key. A mass balance may be carried out in order to determine the bottom composition. Taking as a basis, 100 kmol of feed, then ... 

For multicomponent systems with boiling range greater than 80°C, a single adiabatic flash calculation to 80 to 90 percent of the inlet pressure P0 yields the two-phase specific volume oI at pressure P1 and co is calculated from (Nazario and Leung, Sizing Pressure Relief Valves in Flashing and Two-Phase Service An Alternative Procedure, J. Loss Prev. Process lnd. 5(5), pp. 263-269, 1992)... 

Assuming the liquid phases remain immiscible, the modelling approach for multicomponent systems remains the same, except that it is now necessary to write additional component balance equations for each of the solutes present, as for the multistage extraction cascade with backmixing in Section 3.2.2. Thus for component j, the component balance equations become... 

Mass transport models for multicomponent systems have been developed where the equilibrium interaction chemistry is solved independently of the mass transport equations which leads to a set of algebraic equations for the chemistry coupled to a set of differential equations for the mass transport. (Cederberg et al., 1985). 

In developing the thermodynamic framework for ECES, we attempted to synthesize computer software that would correctly predict the vapor-liquid-solid equilibria over a wide range of conditions for multicomponent systems. To do this we needed a good basis which would make evident to the user the chemical and ionic equilibria present in aqueous systems. We chose as our cornerstone the law of mass action which simply stated says "The product of the activities of the reaction products, each raised to the power indicated by its numerical coefficient, divided by the product of the activities of the reactants, each raised to a corresponding power, is a constant at a given temperature. ... 

Structure parameters. For a single compound, the structure parameters include the proportion of atoms and their connectivity, the geometric and energetic parameters of bonds, angles, and conformation, and the electronic parameters of electron distribution and polarization. For multicomponent systems of solutions, microstmctural material, and composite material, the additional structure parameters include the proportion of the various components, and the relations of their phases as solutions, colloids, or composite solids. 

The general method of stage-by-stage calculation for multicomponent systems was first shown by Lewis and Matheson (LI) and by Underwood (Ul) in 1932. The method of Lewis and Matheson was further improved by Robinson and Gilliland (Rl), but substantially unchanged. In its most basic form, the concept of the method is simple. Consider the example cited above. If the amounts of each component in both of the products could be exactly calculated, it would only be necessary to start at one end and calculate until a stage was reached at which the composition matched that of the other product. 

Pick s law is an empirical diffusion law for binary systems. For multicomponent systems. Pick s law must be generalized. There are several ways to generalize Pick s law to multicomponent systems. One simple treatment is called the effective binary treatment, in which Pick s law is generalized to a multicomponent system in the simplest way ... 

J. W. Gorman and J.E. Hinman, Simplex lattice designs for multicomponent systems, Technometrics, 4 (1962) 463-487. 

Sedimentation velocity. Tire relative molecular mass Mr can also be measured from observation of the velocity of movement of the boundary (or boundaries for multicomponent systems) between solution and solvent from which the macromolecules have sedi-... 

Since linear variation of hardness is not always the case, equation (5.7) is approximate. But Glazov and Vigdorovich consider that the production of many very complex solid solution systems does require some method if only rough, for hardness analysis of such systems. They formulate the additivity principle for multicomponent systems as follows the numerical increase in hardness of multi-component solid solutions equals the sum of hardness increments in bi-component solutions... 

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