The Flux Equation

Let us now turn attention to situations in which the flux equations can be replaced by simpler limiting forms. Consider first the limiting case of dilute solutions where one species, present in considerable excess, is regarded as a solvent and the remaining species as solutes. This is the simplest Limiting case, since it does not involve any examination of the relative behavior of the permeability and the bulk and Knudsen diffusion coefficients. 

The limiting form of the flux equations for large pore diameters or high pressure is best approached starting from equations (5.7) and (5.8). 

There is a further simplification which is often justifiable, but not by consideration of the flux equations above. The nature of many problems is such that, when the permeability becomes large, pressure gradients become very small ialuci uidiii iiux.es oecoming very large. in catalyst pellets, tor example, reaction rates limit Che attainable values of the fluxes, and it then follows from equation (5,19) that grad p - 0 as . But then the... 

The above estimates of pressure variations suggest that their magni-tude as a percentage of the absolute pressure may not be very large except near the limit of Knudsen diffusion. But in porous catalysts, as we have seen, the diffusion processes to be modeled often lie in the Intermediate range between Knudsen streaming and bulk diffusion control. It is therefore tempting to try to simplify the flux equations in such a way as to... 

Hite s treatment is based on equations (5.18) and (5.19) which describe the dusty gas model at the limit of bulk diffusion control and high permeability. Since temperature Is assumed constant, partial pressures are proportional to concentrations, and it is convenient to replace p by cRT, when the flux equations become... 

The flux equation assumes constant temperature. As T rises, H rises slowly, but around 25°C the viscosity of water drops enough to produce about a 3 percent rise in flux per °C. 

The analysis of oxidation processes to which diffusion control and interfacial equilibrium applied has been analysed by Wagner (1933) who used the Einstein mobility equation as a starting point. To describe the oxidation for example of nickel to the monoxide NiO, consideration must be given to tire respective fluxes of cations, anions and positive holes. These fluxes must be balanced to preserve local electroneutrality tliroughout the growing oxide. The flux equation for each species includes a term due to a chemical potential gradient plus a term due to the elecuic potential gradient... 

The term hID is often called the diffusional resistance, denoted by R. The flux equation, therefore, can be written as... 

Figure 8. Variation of the hydrodynamic boundary layer thickness (So, equation (26), continuous line), the diffusion layer thickness (<5,-, equation (34), dotted line) and the ensuing local flux (/, equation (32), dashed line) with respect to the distance from the leading edge (y) in the case of laminar flow parallel to an active plane (the surface is a sink for species i). Parameters /), = 10-9nrs, v= 10 3ms, c = lmolm-3, and v — 10-6 m2 s 1. Notice that c5, o (as required for the derivation of the flux equation (32)), and that the flux decreases when <5, increases...

Pick s first law relates the diffusive flux to the concentration gradient but does not provide an equation to solve for the evolution of concentration. In general, diffusion treats problems in which the concentration of a component or species may change with both spatial position and time, i.e., C = C(x, t), where x describes the position along one direction. Therefore, a differential equation for C(x, t) must include differentials with respect to both t and x. That is, a partial differential equation is necessary to describe how C would vary with x and t. It can be shown that under simple conditions, the flux equation and mass conservation can be transformed to the following equation ... 

The diffusion equation in an anisotropic medium is complicated. Based on the definition of the diffusivity tensor, the diffusive flux along a given direction (except along a principal axis) depends not only on the concentration gradient along this direction, but also along other directions. The flux equation is written as F = —D VC (similar to Fick s law F= -DVC but the scalar D is replaced by the tensor D), i.e.. 

Writing the diffusive flux of a component i in terms of activity gradients of all independent components in an N-component system, the flux equation is... 

Now we insert Eq. 19-38 into one of the flux equations, for instance, into Eq. 19-36. After some rearrangement, we get ... 

Let us discuss an L matrix transformation for isothermal and isobaric atomic fluxes when there is one additional electronic species present. We start with the flux equations in which the index j denotes the atomic species and e denotes the electric charge carriers (eg., electrons). 

It is a straightforward but rather lengthy exercise to write down and evaluate the flux equations jA, jA, jB, jB under the assumption of local (vacancy) equilibrium (A v = 0). We find that five independent L-ti are needed to fully describe the transport in such a system. However, only four experimental parameters DA, DB, DA, and Db are available from flux measurements. Since DA = DB, jA jB in the solid solution crystal. Lattice site conservation requires that the sum of the fluxes /a + 7b + 7v = 0, that is,, /v = 0, despite X = 0. The external observer of the A-B interdiffusion process therefore sees the fluxes... 

Equation (5.118) is the condition of mechanical equilibrium. Only two fluxes are independent in the isothermal ternary system. If we choose them to be jA and jB, and the independent forces to be XA and X0, we obtain by inserting Eq. (5.118) into the flux equations for A and B... 

The kinetic decomposition process is illustrated in Figure 8-4. In order to define the transport coefficients, we assume that the spinel is a semiconducting oxide with immobile oxygen ions. As before, the flux equations will then have the following forms... 

Solutions for this type of kinetics can only be achieved numerically. Model calculations with constant kinetic parameters have been made [H. Wiedersich, et al. (1979)], however, the modeling of realistic transport (diffusion) coefficients which enter into the flux equations is most difficult since the jump rate vA vB. Also, the individual point defects have limited lifetimes which determine the magnitude of correlation factors (see Section 5.2.2). Explicit modeling for dilute or non-dilute alloys can be found in [A.R. Allnatt, A.B. Lidiard (1993)]. 

Although the flux equations for grain boundary and volume transport are of the same type, the creep kinetics are different because the boundary conditions of the transport differ for the two models (Fig. 14-3). Finally, we observe that creep in compound crystals requires the simultaneous motion of all components [R.L. Coble (1963)] so that the slow ones necessarily determine the creep rate. 

An applied stress, as in Fig. 16.1, can reverse the situation by modifying the diffusion potential on interfaces if their inclinations are not perpendicular to the loading direction. With applied stress and capillary forces, the flux equations for crystal diffusion and surface diffusion are given by Eqs. 13.3 and 14.2. For grain-boundary... 

This surface-diffusion problem can be mapped to a one-dimensional problem by approximating the neck region as a cylinder of radius x as shown in Fig. 16.36. The fluxes along the surface in the actual specimen (indicated by the arrows in Fig. 16.3a) are mapped to a corresponding cylindrical surface (indicated by the arrows in Fig. 16.36). The zone extends between z = 2np/Z. The flux equation has the same form as Eq. 14.4, so that8... 

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