Any Number of Components

Let s turn now to the mixtures with any number of components, and let s discuss general conditions of existence of sections trajectory bundles and their structure. 

It follows from Eqs. (5.15) and (5.16) that distillation trajectory tear-off at finite reflux from A -component product boundary element inside concentration simplex is feasible in that case, if in tear-off point x conditions of tear-off into all the (k + l)-component boundary elements, adjacent with the product boundary element are valid. 

To check conditions that possible product point at some k-component boundary element(Cj j, ) should meet, it is necessary (1) for the product point x or xy under exanunation to construct reversible distillation trajectory inside the product boundary element and (2) to define all the first and second (if they are) reversible trajectory tear-off points x[] and x from the product boundary element into all the adjacent (k + l)-component boundary elements. If there is only one reversible distillation trajectory tear-off point into each adjacent boundary element, the point under examination is possible product point and part of reversible distillation trajectory from the most remote from it tear-off point 

If there are two points of reversible distillation trajectory tear-off into, at least, one of the adjacent boundary elements and if there is segment of reversible distillation trajectory, limited by the most distant from the product point under 

At marked in such a way trajectory tear-off segments RegP, Eqs. (5.13) (5.16) are valid. 

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This chapter presents a general method for estimating nonidealities in a vapor mixture containing any number of components this method is based on the virial equation of state for ordinary substances and on the chemical theory for strongly associating species such as carboxylic acids. The method is limited to moderate pressures, as commonly encountered in typical chemical engineering equipment, and should only be used for conditions remote from the critical of the mixture. 

Equations (4) and (5) are not limited to binary systems they are applicable to systems containing any number of components. ... 

Individual components. Components that typically make up most industrial and municipal solid wastes and their relative distribution are reported in Table 25-50. Although any number of components could be selected, those listed in the table have been chosen because they are readily identifiable, are consistent with component categories reported in the literature, and are adequate for the characterization of solid wastes for most applications. 

The unsymmetric convention of normalization is readily applicable to multicomponent solutions, but care must be taken to specify exactly the conditions that give y - 1. Whereas Eq. (35) is immediately applicable to solutions containing any number of components, Eq. (36) is not complete for a solution containing components in addition to 1 and 2. For a solute 2 dissolved in a mixture of solvents 1 and 3, the normalization conditions are completely specified if we write, for a fixed ratio xl/(x1 + x3),... 

By adopting mixing rules similar to those given in Section II, Chueh showed that Eq. (55) can be used for calculating partial molar volumes in saturated liquid mixtures containing any number of components. Some results for binary systems are given in Figs. 7 and 8, which compare calculated partial molar volumes with those obtained from experimental data. 

This result is generalized for any number of components. For this special case Equation 11-9 reduces to... 

We assume that no precipitation occurs, although this assumption will later be proved to be inadequate. Using a non-linear system is a complicated way of solving this particular problem, but this example is quite illustrative and can be extended to any number of components. Although ionized atoms like H+ and Ca2 + are natural components of the dilute solution under consideration, carbon and oxygen do not appear as such in natural systems. Since the group C032- is not destroyed in any reaction it will therefore be taken as the carbon host. The component matrix B is shown in Table 6.1. As explained above, the H20 row is subtracted from the OH-row, which is left with — 1 in the H + column, which produces the new component matrix of Table 6.2. 

It is straightforward to generalise equations (3.13) and (3.14) to allow for the formation of species of any complexity from any number of components. Discussion here is limited to species formed by only three components X, Y, and Z. The generalisation to 4, 5 or any number of components is self-evident. 

And maybe a last reminder, the above package of short routines is actually a fairly complete pH titration fitting package. Theoretically, it deals with any number of components, species, equilibria, concentrations and whatever the chemist wants to fit All that lacks is a user friendly interface. 

The creation of such a superstructure can be generalized for any number of components. The number of columns is determined from the number of components - one column for each separation breakpoint. This superstructure consists of ... 

Equations (5.1) and (5.2) give the basic criteria of equilibrium, but it is advantageous to obtain more-useful conditions from them. We first consider an isolated heterogenous system composed of any number of phases and containing any number of components at equilibrium. These phases are in thermal contact with each other, and there are no walls separating them. 

First, for any homogenous system containing any number of components, both (d2E/dS2)v and (d2A/dV2)Tn must be positive. The first derivative requires that... 

You may input any of the components selected from Table 1 into Table 2 simply by inserting into Table 2 the ID number of the component from Table 1. Any number of components up to 60 may be installed. Those of you with a copy of VB may change a few program code lines to any number of components desired. 

The wave and pulse patterns of nonreactive separation processes, as well as the integrated reaction separation processes illustrated above, can be easily predicted with some simple graphical procedures derived from Eqs. (4) and (5). The behavior crucially depends on the equilibrium function y(x) in the nonreactive case, and on the transformed equilibrium function Y(X) in the reactive case. In addition to phase equilibrium, the latter also includes chemical equilibrium. An explicit calculation of the transformed equilibrium function and its derivatives is only possible in special cases. However, in Ref. [13] a numerical calculation procedure is given, which applies to any number of components, any number of reactions, and any type of phase and reaction equilibrium. 

Equation (7) is the general form of Gibbs s relation between surface tension, temperature, surface excesses, and chemical potentials for a system of any number of components, and if the surfaces are not very highly curved it holds good whatever convention is adopted for defining I, with any arbitrary position of the surface XY in the idealized system. 

Guggenheim shows that the change in surface tension with temperature, for a solution of any number of components, in a change in which the composition is kept constant, is /Q x... 

We return to the problem of stability with respect to diffusion studied in 12, but now we consider the case of a system containing any number of components. To achieve the greatest symmetry we introduce as many reaction variables. .. c there are components. Each extent of change corresponds to passage of component i from the volume element a to 6. 

Presentation of phase phenomena for mixtures that are completely miscible in the liquid state involves some rather complex reductions of three-, four-, and higher-dimensional diagrams into two dimensions. We shall restrict ourselves to two-component systems because, although the ideas discussed here are applicable to any number of components, the graphical presentation of more complex systems in an elementary text such as this is probably more confusing than helpful. 

The /z-transf orm method was also applied to the calculation of the breakthrough curve for the separation of a binary mixture on a charcoal bed by Tien et al. [36]. By using the li-transform, the course of the separation can be calculated by transforming the concentration variable via the /i-transform into a coordinate system in which algebraic equations describe the process for any number of components. 

What we have said so far about ideal solutions applies to any number of components. Let s now restrict our attention to systems with only two components, which we will call A and B. 

These results, obtained for a dilute ternary vapor mixture may be generalized with the help of Eq. 4.2.2 to a mixture of any number of components where one component is present in a large excess x 1, x 0, i = 1, 2,- (see discussion below Eq. 3.2.10)... 

Equations 8.5.1.3 are quite general they involve no assumptions regarding the constancy of particular matrices they apply to mixtures with any number of components and for any relationship between the fluxes. It is at this point where any assumptions necessary to solve Eqs. (8.5.1-8.5.3) must be made. In the three methods to be discussed below we proceed in exactly the same way as we did when deriving the exact solution and the solution to the linearized equations first obtain the composition profiles, then differentiate to obtain the gradients at the film boundary, and combine the result with Eq. 8.5.3 to obtain the working flux equations. 

The material balance relations presented above are valid for any number of components. We shall discuss solutions to this system of equations for binary mixtures in the remainder of this section of Chapter 12 before moving on to obtain generalized results for multicomponent systems in Section 12.2. In the analyses that follow we shall ignore the effects of heat transfer between the vapor and liquid phases. 

The attention of many workers has been given to the equilibrium-limited case (proportionate-pattern) of multiple adsorption (D2, Wl, W3, W7, among others). In the constant-pattern case, Fujita s work has already been discussed (Section III, B, 2b). Also, using the theoretical-plate approximation to a packed column, plate-by-plate calculations can be carried out in the constant-pattern case exactly as for continuous (countercurrent) distillation this treatment is suggested from work on chromatographic and displacement problems by Mair (M2), Spedding (S6), and Glueckauf (G3). Moreover the linear-equilibrium result can be extended, in a nearly trivial fashion, to any number of components. 

We have defined the solvation process as the process of transfer from a fixed position in an ideal gas phase to a fixed position in a liquid phase. We have seen that if we can neglect the effect of the solvent on the internal partition function of the solvaton s, the Gibbs or the Helmholtz energy of solvation is equal to the coupling work of the solvaton to the solvent (the latter may be a mixture of any number of component, including any concentration of the solute s). In actual calculations, or in some theoretical considerations, it is often convenient to carry out the coupling work in steps. The specific steps chosen to carry out the coupling work depend on the way we choose to write the solute-solvent interaction. 

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