Solutions of the diffusion equation parallel flux

Different techniques are commonly used to solve the diffusion equation (Carslaw and Jaeger, 1959). Analytic solutions can be found by variable separation, Fourier transforms or more conveniently Laplace transforms and other special techniques such as point sources or Green functions. Numerical solutions are calculated for the cases which have no simple analytic solution by finite differences (Mitchell, 1969 Fletcher, 1991), which is the simplest technique to implement, but also finite elements, particularly useful for complicated geometry (Zienkiewicz, 1977), and collocation methods (Finlayson, 1972). 

A series of cases relevant to geochemical problems in parallel and radial fluxes will now be described as examples of a few methods. 

---

Generally, a set of coupled diffusion equations arises for multiple-component diffusion when N > 3. The least complicated case is for ternary (N = 3) systems that have two independent concentrations (or fluxes) and a 2 x 2 matrix of interdiffusivities. A matrix and vector notation simplifies the general case. Below, the equations are developed for the ternary case along with a parallel development using compact notation for the more extended general case. Many characteristic features of general multicomponent diffusion can be illustrated through specific solutions of the ternary case. 

For some biological systems, the species that eventually crosses the cell membrane has travelled through different media, each one with its own mass transfer characteristics. As an example, we deal with the case where the two media are the bulk solution and the cell wall (with the separation surface parallel to the cell membrane) with diffusion as the only relevant mass transfer phenomenon in each medium. Apart from having different parameters in the differential equations in each medium (due to the unequal diffusion coefficients), we need to impose two new boundary conditions at the separating plane which we denote as a. The first boundary condition follows from the continuity of the material flux ... 

Boundary layer formulation. Many membrane processes are operated in cross-flow mode, in which the pressurised process feed is circulated at high velocity parallel to the surface of the membrane, thus limiting the accumulation of solutes (or particles) on the membrane surface to a layer which is thin compared to the height of the filtration module [2]. The decline in permeate flux due to the hydraulic resistance of this concentrated layer can thus be limited. A boundary layer formulation of the convective diffusion equation can give predictions for concentration polarisation in cross-flow filtration and, therefore, predict the flux for different operating conditions. Interparticle force calculations are used in two ways in this approach. Firstly, they allow the direct calculation of the osmotic pressure at the membrane. This removes the need for difficult and extensive experi-... 

In this chapter we formulate the thermodynamic and stochastic theory of the simple transport phenomena diffusion, thermal conduction and viscous ffow (1) to present results parallel to those listed in points 1-7, Sect. 8.1, for chemical kinetics. We still assume local equilibrium with respect to translational and internal degrees of freedom. We do not assume conditions close to chemical or hydrodynamic equilibrium. For chemical reactions and diffusion the macroscopic equations for a given reaction mechanism provide sufficient detail, the fluxes in the forward and reverse direction, to write a birth-death master equation with a stationary solution given in terms of For thermal conduction and viscous flow we derive the excess work and then find Fokker-Planck equations with stationary solutions given in terms of that excess work. 

In some cases, however, our dilute-solution analyses do not successfully correlate our experimental observations. Consequently, we must use more elaborate equations. This elaboration is best initiated with the physically based examples given in Section 3.1. This is followed by a catalogue of flux equations in Section 3.2. These flux equations form the basis for the simple analyses of diffusion and convection in Section 3.3 that parallel those in the previous chapter. 

---