Solid phase, energy transport

This chapter describes some of the properties of solids that affect transport across phases and membranes, with an emphasis on biological membranes. Four aspects are addressed. They include a comparison of crystalline and amorphous forms of the drug, transitions between phases, polymorphism, and hydration. With respect to transport, the major effect of each of these properties is on the apparent solubility, which then affects dissolution and consequently transport. There is often an opposite effect on the stability of the material. Generally, highly crystalline substances are more stable but have lower free energy, solubility, and dissolution characteristics than less crystalline substances. In some situations, this lower solubility and consequent dissolution rate will result in reduced bioavailability. 

Axial transport of energy in the solid phase by conduction. 

In equation 13, C1 and Cs are the total concentrations in the liquid and solid phases, respectively. This statement of the problem assumes that the convective flux due to the moving boundary (growing surface) is small, the diffusion coefficients are mutual and independent of concentration, the area of the substrate is equal to the area of the solution, the liquid density is constant, and no transport occurs in the solid phase. Further, the conservation equations are uncoupled from the equations for the conservation of energy and momentum. Mass flows resulting from other forces (e.g., thermal diffusion and Marangoni or slider-motion-induced convective flow) are neglected. 

The only dynamic phenomena of interest within the solid materials of an SOFC (cell, interconnects, end plates, etc.) is thermal energy transport. Here, the same formulation used for the gas-phase, Equation (9.10), can be used, except that here the velocity, V, is zero. While the solid body components do not show chemical reactions, Ri, there is internal heat generation through electrical ohmic losses. Hence, the model equation is ... 

When the gas-solid flow in a multiphase system is dominated by the interparticle collisions, the stresses and other dynamic properties of the solid phase can be postulated to be analogous to those of gas molecules. Thus, the kinetic theory of gases is adopted in the modeling of dense gas-solid flows. In this model, it is assumed that collision among particles is the only mechanism for the transport of mass, momentum, and energy of the particles. The energy dissipation due to inelastic collisions is included in the model despite the elastic collision condition dictated by the theory. 

The fluxes of mass, momentum, and energy of phase k transported in a laminar or turbulent multiphase flow can be expressed in terms of the local gradients and the transport coefficients. In a gas-solid multiphase flow, the transport coefficients of the gas phase may be reasonably represented by those in a single-phase flow although certain modifications... 

With regard to energy and power density, Li as the element to be stored is the element of choice because, according to its position in the periodic table, Li is light, small, and very electropositive. These are excellent prerequisites for low weight, high voltages, fast transport, and easy accommodation in solid phases. 

A major limitation of the present work is that it deals only with well-defined (and mostly unidirectional) flow fields and simple homogeneous and catalytic reactor models. In addition, it ignores the coupling between the flow field and the species and energy balances which may be due to physical property variations or dependence of transport coefficients on state variables. Thus, a major and useful extension of the present work is to consider two- or three-dimensional flow fields (through simplified Navier-Stokes or Reynolds averaged equations), include physical property variations and derive lowdimensional models for various types of multi-phase reactors such as gas-liquid, fluid-solid (with diffusion and reaction in the solid phase) and gas-liquid-solid reactors. 

Heterogeneous Models. The two-phase character of a packed-bed is preserved in a heterogeneous model. Thus mass and energy conservation equations are written separately for the fluid and solid phases. These equations are linked together by mass and heat transport between the phases. 

Mixing in crystallization involves all elements of transport phenomena momentum transport, energy transport, and material transport in both the solution phase and the solid phase. In many cases, the interactions of these elements can affect every aspect of a crystallization operation including nucleation, growth, and maintenance of a crystal slurry. To further complicate the problem, mixing optimization for one aspect of an operation may require different parameters than for another aspect even though both requirements must be satisfied simultaneously. In addition, these operations are intrinsically scale dependent. 

It is convenient to classify an overall soil reaction as a slow or rapid reaction. A soil reaction is slow when the kinetics are associated with an energy of activation. Slow reactions are those in which processes taking place at the solid phase are rate determining whether transport processes (such as surface diffusion, diffusion in micropores, penetration into the bulk, etc.), or chemical interactions. Mechanisms for slow soil reactions are discussed in detail in the following sections. 

The above set of equations must be augmented by an energy balance for the solution and/or the solid phase if temperature effects are important. An example is high rate etching or deposition effected by a laser beam [265]. Also, potential depended transport of charge carries (electrons and holes) in the semiconductor must be accounted for in photochemical and photoelectrochemical etching [266, 267]. 

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