Finite transient diffusion planar

Transient Finite (Symmetric) Spherical Diffusion So far, we have only examined ID (Cartesian) examples of Fick s second law. Solving Fick s second law in alternative coordinate systems (e.g., for radial, spherical, 2D, or 3D problems) is not really any different. As an example, we examine here the case of transient finite spherical diffusion, which is essentially analogous to the transient finite planar diffusion problem that we just finished discussing. 

Pick s second law is a second-order partial differential equation. Solving it in order to predict transient diffusion processes can be fairly straightforward or quite complex, depending on the specific situation. In this chapter, analytical solutions were discussed for a number of cases, including ID transient infinite and semi-infinite diffusion, ID transient finite planar diffusion, and transient spherical finite diffusion as summarized in Table 4.4. In all cases, solution of Pick s second law requires the specification of a number of boundary conditions and initial conditions. 

Transport by combined migration—diffusion in a finite planar geometry can achieve a true steady state when only two ions are present, as we saw in Sect. 4.2. The same holds true when there are three or more ions present. Under simplifying conditions [see eqn. (89) below], it is possible to predict the steady-state behaviour with arbitrary concentrations of many ions. However, the corresponding transient problem is much more difficult and we shall not attempt to derive the general transient relationship, as we were able to do in deriving eqn. (82) in the two-ion case. 

Transient Finite (Symmetric) Planar Diffusion In this section, we progress from transient infinite diffusion problems to transient finite diffusion problems. In many cases, the approaches and solutions to finite problems are quite similar to those just discussed in the context of infinite diffusion problems. Transient finite diffusion problems can often be solved using the separation-of-variables technique, which... 

Solution This is a transient ID finite (symmetric) planar diffusion problem. The initial and boundary conditions for this problem are ... 

Problem 4.3. Equation 4.43 provides the solution for transient finite (symmetric) planar diffusion in a plate of thickness L starting from a uniform initial concentration of c° when the concentrations at the edges of the plate are set to c at time t = 0 ... 

Bieniasz LK (2012) Automatic simulation of electrochemical transients assuming finite diffusion space at planar interfaces, by the adaptive Huber method for Volterra integral equations. J Electroanal Chem 684 20-31... 

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