Solid Mixture Models

An ideal binary crystalline mixture obeys the equations 

By application of Stirling s approximation for large numbers (note that Nq = 6.022 X 10 3)  

Equations 3.121 and 3.122 distinguish the bulk Gibbs free energy of the mixture ( mixture) from the Gibbs free energy term involved in the mixing procedure 

Let us now consider in detail the mixing process of two generic components AN and BN, where A and B are cations and N represents common anionic radicals (for instance, the anionic group SiO )  

If mixture (A,B)N is ideal, mixing will take place without any heat loss or heat production. Moreover, the two cations will be fully interchangeable in other words, if they occur in the same amounts in the mixture, we will have an equal opportunity of finding A or B over the same structural position. The Gibbs free energy term involved in the mixing process is 

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Tsuji et al. (1990) have modeled the flow of plastic pellets in the plug mode with discrete dynamics following the behavior of each particle. The use of a dash pot/spring arrangement to account for the friction was employed. Their results show remarkable agreement with the actual behavior of real systems. Figure 28 shows these flow patterns. Using models to account for turbulent gas-solid mixtures, Sinclair (1994) has developed a technique that could have promise for the dense phase transport. 

Sohn et al.27 used GAs to try to determine the composition of a high luminescence phosphor at 400 nm. Rather than relying on a theoretical model to assess the fitness of solid mixtures proposed by the GA, they synthesized each mixture and measured the emission efficiency experimentally. This is in... 

It is important to emphasize here that, theoretically, if a solid mixture is ideal, intracrystalline distribution is completely random (cf section 3.8.1) and, in these conditions, the intracrystalline distribution constant is always 1 and coincides with the equilibrium constant. If the mixture is nonideal, we may observe some ordering on sites, but intracrystalline distribution may still be described without site interaction parameters. We have seen in section 5.5.4, for instance, that the distribution of Fe and Mg on Ml and M3 sites of riebeckite-glaucophane amphiboles may be approached by an ideal site mixing model—i.e.. 

Determination of T y. In the formulation of the phase equilibrium problem presented earlier, component chemical potentials were separated into three terms (1) 0, which expresses the primary temperature dependence, (2) solution mole fractions, which represent the primary composition dependence (ideal entropic contribution), and (3) 1, which accounts for relative mixture nonidealities. Because little data about the experimental properties of solutions exist, Tg is usually evaluated by imposing a model to describe the behavior of the liquid and solid mixtures and estimating model parameters by semiempirical methods or fitting limited segments of the phase diagram. Various solution models used to describe the liquid and solid mixtures are discussed in the following sections, and the behavior of T % is presented. 

Parameter Estimation from Phase Diagram Assessment. The second approach used to determine solid-solution behavior is to estimate solution model parameters from a fit to the measured phase diagram. Many of the solution models used to describe the liquid solution have been used to model the solid mixture. The simple-solution expression and its special cases have been used most extensively. 

This chapter describes the fundamental principles of heat and mass transfer in gas-solid flows. For most gas-solid flow situations, the temperature inside the solid particle can be approximated to be uniform. The theoretical basis and relevant restrictions of this approximation are briefly presented. The conductive heat transfer due to an elastic collision is introduced. A simple convective heat transfer model, based on the pseudocontinuum assumption for the gas-solid mixture, as well as the limitations of the model applications are discussed. The chapter also describes heat transfer due to radiation of the particulate phase. Specifically, thermal radiation from a single particle, radiation from a particle cloud with multiple scattering effects, and the basic governing equation for general multiparticle radiations are discussed. The discussion of gas phase radiation is, however, excluded because of its complexity, as it is affected by the type of gas components, concentrations, and gas temperatures. Interested readers may refer to Ozisik (1973) for the absorption (or emission) of radiation by gases. The last part of this chapter presents the fundamental principles of mass transfer in gas-solid flows. 

One-dimensional flow models are adopted in the early stages of model development for predicting the solids holdup and pressure drop in the riser. These models consider the steady flow of a uniform suspension. Four differential equations, including the gas continuity equation, solids phase continuity equation, gas-solid mixture momentum equation, and solids phase momentum equation, are used to describe the flow dynamics. The formulation of the solids phase momentum equation varies with the models employed [e.g., Arastoopour and Gidaspow, 1979 Gidaspow, 1994], The one-dimensional model does not simulate the prevailing characteristics of radial nonhomogeneity in the riser. Thus, two- or three-dimensional models are required. 

Assuming the riser flow to be characterized by one-dimensional steady flow and applying the two-fluid model to the flow, one can obtain the momentum equation for the gas-solid mixture as... 

Obata, M., Banno, S. Mori, T. (1974) The iron-magnesium partitioning between naturally occurring coexisting olivine and Ca-rich clinopyroxene an application of the simple mixture model to olivine solid solution. Bull. Soc. Franc. Mineral., 97,101-7. 

Figure 10. Vapor-liquid equilibria for an argon-krypton mixture (modeled as a Lennard-Jones mixture) for the bulk fluid (R = >) and for a cylindrical pore of radius R = / /Oaa = 2.5. The dotted and dashed lines are from a crude form of density functional theory (the local density approximation, LDA). The points and solid lines are molecular dynamics results for the pore. Reprinted with permission from W. L. Jorgensen and J. Tirado-Rives, J. Am. Chem. Soc.

Mixture models (such as those of Scheffe) are still useful, especially when there are three or more such excipients with fairly large ranges of variation. In solid formulations, this is often the case for diluents (or fillers) and also for the polymers or waxes incorporated into controlled-release tablets to form a matrix through which the drug diffuses slowly out when immersed in aqueous fluid, i.e., in the gastrointestinal tract. 

Competent design of chemical processes requires accurate knowledge of such process variables as the temperature, pressure, composition and phase of the process contents. Current predictive models for phase equilibria Involving supercritical fluids are limited due to the scarcity of data against which to test them. Phase equilibria data for solids In equilibrium with supercritical solvents are particularly sparse. The purpose of this work Is to expand the data base to facilitate the development of such models with emphasis on the melting point depressions encountered when solid mixtures are contacted with supercritical fluids. 

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