Homogeneity, fractional diffusion equations

Exemplifying the convenience of the fractional approach, we address the imposition of boundary value problems on the fractional diffusion equation which was demonstrated in Ref. 62. In this force-fiee case for which the kernel, Eq. (27), takes on the homogeneous form K(x,x u) = uw(u) (2( c — x ) — <5(x))/(l - w(u)), one can apply the method of images in order to construct the solution [12]. 

In addition to overall mass conservation, we are concerned with the conservation laws for individual chemical species. Beginning in a way analogous to the approach for the overall mass-conservation equation, we seek an equation for the rate of change of the mass of species k, mk. Here the extensive variable is N = mu and the intensive variable is the mass fraction, T = mk/m. Homogeneous chemical reaction can produce species within the system, and species can be transported into the system by molecular diffusion. There is convective transport as well, but it represented on the left-hand side through the substantial derivative. Thus, in the Eulerian framework, using the relationship between the system and the control volume yields... 

Both Knudsen and molecular diffusion can be described adequately for homogeneous media. However, a porous mass of solid usually contains pores of non-uniform cross-section which pursue a very tortuous path through the particle and which may intersect with many other pores. Thus the flux predicted by an equation for normal bulk diffusion (or for Knudsen diffusion) should be multiplied by a geometric factor which takes into account the tortuosity and the fact that the flow will be impeded by that fraction of the total pellet volume which is solid. It is therefore expedient to define an effective diffusivity De in such a way that the flux of material may be thought of as flowing through an equivalent homogeneous medium. We may then write ... 

The cumulative effect of the instantaneous fractionations given by Equations (6)-(9) is easily calculated if it is further assumed that mass transport processes (e.g., chemical diffusion) are sufficiently fast to maintain chemical and isotopic homogeneity in both the gas and in the condensed phase. There are cases where diffusion in the residue or gas limits mass transport and these effects on isotopic and chemical fractionation have been explored by Richter et al. (2002). Let us consider first the isotopic fractionations associated with condensation in a supersaturated closed system. The change in the moles of isotope 1 of element k in the gas can be written as... 

A convenient way to formulate a dynamical equation for a Levy flight in an external potential is the space-fractional Fokker-Planck equation. Let us quickly review how this is established from the continuous time random walk. We will see below, how that equation also emerges from the alternative Langevin picture with Levy stable noise. Consider a homogeneous diffusion process, obeying relation (16). In the limit k — 0 and u > 0, we have X(k) 1 — CTa fe and /(w) 1 — uz, whence [52-55]... 

Some rapid homogeneous reactions are controlled by the rate of diffusion of reagents towards each other. In some cases, as pressure increases in an SCF, a reaction may pass from activation control to diffusion control [10]. In the simplest analysis of diffusion control, the second-order rate coefficient in terms of mole fractions, kx, is given by the Smoluschowski equation. 

However, the void area fraction is equivalent to the void volume fraction, based on equation (21-76) and the definition of intrapellet porosity Sp at the bottom of p. 555. Effectiveness factor calculations in catalytic pellets require an analysis of one-dimensional pseudo-homogeneous diffusion and chemical reaction in a coordinate system that exploits the symmetry of the macroscopic boundary of a single pellet. For catalysts with rectangular symmetry as described above, one needs an expression for the average diffusional flux of reactants in the thinnest dimension, which corresponds to the x direction. Hence, the quantity of interest at the local level of description is which represents the local... 

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