Diffusion equation finite slab

Occasionally (e.g., thin-layer electrochemistry, porous-bed electrodes, metal atoms dissolved in a mercury film), diffusion may be further confined by a second barrier. Figure 2.7 illustrates the case of restricted diffusion when the solution is confined between two parallel barrier plates. Once again, the folding technique quickly enables a prediction of the actual result. In this case, complete relaxation of the profile results in a uniform finite concentration across the slab of solution, in distinct contrast to the semi-infinite case. When the slab thickness t is given, the time for the average molecule to diffuse across the slab is calculable from the Einstein equation such that... 

In my experience, most novices measuring diffusion do not concentrate on this experimental measurement but rather on improving the mathematics behind Equation 5.6-7. These novices assume a finite slab and solve the diffusion equations for that case, getting results like those in Section 3.5. They include the variation of solution concentration with time, performing an analysis like that in Example 3.5-3. These novices are then dismayed that their results are poorly reproduceable, and they conclude that their mathematics is incorrect. It often isn t it is unnecessary. The novices need instead to focus on their measurement of concentration. 

Problems involving unsteady diffusion in more then one coordinete direction, such as a cylinder or finite length or a long slab of comparable width and depth dimensions, cun usually be solvnd by separation of variables. For example, for a cylinder of radius R and length L the governing equation and boundary conditions for an ideal gas mixture and constant surface conditions would be... 

As an alternative to the previous example, we can also solve the problems with inhomogeneous boundary conditions by direct application of the finite integral transform, without the necessity of homogenizing the boundary conditions. To demonstrate this, we consider the following transient diffusion and reaction problem for a catalyst particle of either slab, cylindrical, or spherical shape. The dimensionless mass balance equations in a catalyst particle with a first order... 

This equation describes many transient heat and mass transfer processes, such as the diffusion of a solute through a slab membrane with constant physical properties. The exact solution, obtained by either the Laplace transform, separation of variables (Chapter 10) or the finite integral transform (Chapter 11), is given as... 

An approach was reported by Duque et al. for calculating the diffusion coefficient perpendicular to the interface. It takes into account the finite residence time (by employing a solution to a stochastic differential equation), but avoids using slabs and considers the molecule positions relative to the intrinsic surface. The main conclusion of their study is that, even if a diffusion coefficient can still be computed, the turnover processes by which molecules enter and leave the intrinsic surface are as important as diffusion itself. 

This function pdeivbv() provides a versatile multi-time step function for solving sets of PDLs of the IV, BV type. As a simple example let s consider a linear diffusion problem into a slab of finite thickness defined by the following equation set ... 

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