In multicomponent systems

The calculation of single-stage equilibrium separations in multicomponent systems is implemented by a series of FORTRAN IV subroutines described in Chapter 7. These treat bubble and dewpoint calculations, isothermal and adiabatic equilibrium flash vaporizations, and liquid-liquid equilibrium "flash" separations. The treatment of multistage separation operations, which involves many additional considerations, is not considered in this monograph. 

To predict vapor-liquid or liquid-liquid equilibria in multicomponent systems, we require a method for calculating the fugacity of a component i in a liquid mixture. At system temperature T and system pressure P, this fugacity is written as a product of three terms... 

The exponents apply not only to solid systems (e.g. order-disorder phenomena and simple magnetic systems), but also to fluid systems, regardless of the number of components. (As we have seen in section A2.5.6.4 it is necessary in multicomponent systems to choose carefully the variable to which the exponent is appropriate.)... 

Griffiths R B and Wheeler J C 1970 Critical points in multicomponent systems Phys. Rev. A 2 1047-64... 

We see, then, that pressure gradients must necessarily exist in catalyst pellets to free the fluxes from the constraints Imposed by Graham s relation (11,42), or Its generalization = 0 in multicomponent systems. Without this freedom the fluxes are unable to adjust to the demands... 

Multicomponent Diffusion. In multicomponent systems, the binary diffusion coefficient has to be replaced by an effective or mean diffusivity Although its rigorous computation from the binary coefficients is difficult, it may be estimated by one of several methods (27—29). Any degree of counterdiffusion, including the two special cases "equimolar counterdiffusion" and "no counterdiffusion" treated above, may arise in multicomponent gas absorption. The influence of bulk flow of material through the films is corrected for by the film factor concept (28). It is based on a slightly different form of equation 13 ... 

Problem Solving Methods Most, if not aU, problems or applications that involve mass transfer can be approached by a systematic-course of action. In the simplest cases, the unknown quantities are obvious. In more complex (e.g., iTmlticomponent, multiphase, multidimensional, nonisothermal, and/or transient) systems, it is more subtle to resolve the known and unknown quantities. For example, in multicomponent systems, one must know the fluxes of the components before predicting their effective diffusivities and vice versa. More will be said about that dilemma later. Once the known and unknown quantities are resolved, however, a combination of conservation equations, definitions, empirical relations, and properties are apphed to arrive at an answer. Figure 5-24 is a flowchart that illustrates the primary types of information and their relationships, and it apphes to many mass-transfer problems. 

We first encountered in Chapter 3 on mixing in multicomponent systems the problem of... 

Gibb s Phase Rule. The phase rule derived by W. J. Gibbs applies to multiphase equilibria in multicomponent systems, in the absence of chemical reactions. It is written as... 

The situation becomes most complicated in multicomponent systems, for example, if we speak about filling of plasticized polymers and solutions. The viscosity of a dispersion medium may vary here due to different reasons, namely a change in the nature of the solvent, concentration of the solution, molecular weight of the polymer. Naturally, here the interaction between the liquid and the filler changes, for one, a distinct adsorption layer, which modifies the surface and hence the activity (net-formation ability) of the filler, arises. Therefore in such multicomponent systems in the general case we can hardly expect universal values of yield stress, depending only on the concentration of the filler. Experimental data also confirm this conclusion [13],... 

In multicomponent systems such as solutions, diffusion will arise when at least one of the components is nonuniformly distributed, and its direction will be such as to level the concentration gradients. The diffusion flux (in the direction of decreasing concentrations) is proportional to the concentration gradient of the diffusing substance ... 

In activity studies in multicomponent systems, G. N. Lewis and M. Randall found in 1923 that in the case of dilute solutions, when a foreign electrolyte is added, the activity change of the substance studied depends only on the concentration and valence type of the substance added, not on its identity. For a quantitative characterization of solutions, they introduced the concept of ionic strength / of a solution (units mol/L),... 

Diffusion of ions can be observed in multicomponent systems where concentration gradients can arise. In individnal melts, self-diffnsion of ions can be studied with the aid of radiotracers. Whereas the mobilities of ions are lower in melts, the diffusion coefficients are of the same order of magnitude as in aqueous solutions (i.e., about 10 cmVs). Thus, for melts the Nemst relation (4.6) is not applicable. This can be explained in terms of an appreciable contribntion of ion pairs to diffusional transport since these pairs are nncharged, they do not carry cnrrent, so that values of ionic mobility calculated from diffusion coefficients will be high. 

Conductometric analysis is performed in both concentrated and dilute solutions. The accuracy depends on the system in binary solutions it is as high as 0.1%, but in multicomponent systems it is much lower. 

In multicomponent systems, large solutes with lower diffusivity polarize more and exclude smaller solutes from the membrane surface, decreasing their passage. Operation at the knee of the flex curve reduces this effect. 

Thermo- and photospillover of hydrogen atoms in multicomponent systems... 

In multicomponent systems, the number of signals is (at least) equal in size to the number of components ... 

While the main driving force in [43, 44] was to avoid direct particle transfers, Escobedo and de Pablo [38] designed a pseudo-NPT method to avoid direct volume fluctuations which may be inefficient for polymeric systems, especially on lattices. Escobedo [45] extended the concept for bubble-point and dew-point calculations in a pseudo-Gibbs method and proposed extensions of the Gibbs-Duhem integration techniques for tracing coexistence lines in multicomponent systems [46]. 

Fortunately, few of these variables are truly independent. Geochemists have developed a variety of numerical schemes to solve for equilibrium in multicomponent systems, each of which features a reduction in the number of independent variables carried through the calculation. The schemes are alike in that each solves sets of mass action and mass balance equations. They vary, however, in their choices of thermodynamic components and independent variables, and how effectively the number of independent variables has been reduced. 

This choice of basis follows naturally from the steps normally taken to study a geochemical reaction by hand. An aqueous geochemist balances a reaction between two species or minerals in terms of water, the minerals that would be formed or consumed during the reaction, any gases such as O2 or CO2 that remain at known fugacity as the reaction proceeds, and, as necessary, the predominant aqueous species in solution. We will show later that formalizing our basis choice in this way provides for a simple mathematical description of equilibrium in multicomponent systems and yields equations that can be evaluated rapidly. 

Brown, T. H. and B. J. Skinner, 1974, Theoretical prediction of equilibrium phase assemblages in multicomponent systems. American Journal of Science 274, 961-986. 

Covalently bonded substructures having compositions distinguishable from their surroundings are formed in multicomponent systems they are called chemical clusters. The adjective chemical defines covalency of bonds between units in the cluster. To be a part of a cluster, the units must have a common property. For example, hard clusters are composed of units yielding Tg domains. Hard chemical clusters are formed in three-component polyurethane systems composed of a macromolecular diol (soft component), a low-molecular-weight triol (hard component) and diisocyanate (hard component). Hard clusters consist of two hard... 

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... 

Our results lead to good predictions of activity coefficients in multicomponent systems from data measured 1n simple solutions. Also, they yield values similar to those of Pitzer and Kim (8 ) as is shown in Table II. 

In multicomponent systems A"0 can be written as a sum of the individual absorption coefficients A ot = 2TA , where each AT,(A ) depends in a different way on the wavelength. If one or more of the components are fluorescent, their excitation spectra are mutually attenuated by absorption filters of the other compounds. This effect is included in Eqs. (8.27) and (8.28) so that examples like that of Figure 8.4 can be quantified. The two fluorescent components are monomeric an aggregated pyrene, Mi and Mn. The fluorescence spectra of these species are clearly different from each other but the absorption spectra overlap strongly. Thus the excitation spectrum of the minority component M is totally distorted by the Mi filter (absorption maxima of Mi appear as a minima in the excitation spectrum ofM see Figure 8.4, top). In transparent samples this effect can be reduced by dilution. However, this method is not very efficient in scattering media as can be seen by solving Eqs. (8.27 and 8.28) for bSd — 0. Only the limit d 0 will produce the desired relation where fluorescence intensity and absorption coefficient of the fluorophore are linearly proportional to each other in a multicomponent system. 

The Langmuir model for competitive adsorption can be used as a common model for predicting adsorption equilibria in multicomponent systems. This was first developed by Butler and Ockrent [77] and is based on the same assumptions as the Langmuir model for single adsorbates. It assumes, as in the case of the Langmuir model, that the rate of adsorption of a species at equilibrium is equal to its desorption rate. This is expressed by Eq. (18) ... 

If bottoms composition is to be controlled by vapor boilup, the control tray should be located as dose to the base of the column as possible in a binary system. In multicomponent systems with heavy components in the feed which have their highest concentration in the base of the column, the optimum control tray moves up in the column. 

Figure 8.11 shows how the equilibrium curve shrinks in the presence of inefficiencies. In multicomponent systems where there is mutual interference in extraction by several components, the efficency shrinkage comes on top of the other reductions in the equilibrium curve, and for this reason there is stress in such systems on achieving high efficiency. 

The basic problem in determining phase equilibria in multicomponent systems is the existence of a large number of variables, necessitating extensive experimental work. If ten measurements are considered satisfactory for acceptable characterization of the solubility in a two-component system in a particular temperature range, then the attainment of the same reliability with a three-component system requires as many as one hundred measurements. Therefore, a reliable correlation method permitting a decrease in the number of measurements would be extremely useful. Two different methods - the first of them based on geometrical considerations, and the second on thermodynamic condition of phase equilibria - are presented and their use is demonstrated on worked examples. 

The adjustable interaction constants Q can be evaluated from the experimental data for three-component systems these constants can then be employed for concentration of temperature interpolations and also for calculation of phase equilibria in multicomponent systems. Moreover, the constants Q usually depend very little on temperature, as the relative molalities, related to the solubility of the substance in the pure solvent, are employed hence calculations of other isotherms can be carried out easily. 

Uphill diffusion in a binary system is rare and occurs only when the phase undergoes spinodal decomposition. In multicomponent systems, uphill diffusion occurs often, even when the phase is stable. The cause for uphill diffusion in multicomponent systems is different from that in binary systems and will be discussed later. 

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