Simple solutions of the partial differential equation for diffusion (or heat conduction)

Abstract
It is shown that simple approximate solutions of the partial differential equation for diffusion (or heat conduction) in finite solids of various shapes and under various conditions can be derived from the simple solutions which are rigorously applicable to linear diffusion in a semi-infinite slab. The case in which the initial volume concentration is constant and the surface concentration is zero is considered in detail. For linear diffusion in a finite slab, the solutions show that each end of the slab can be regarded as functioning as the end of a semi-infinite slab for a time during which the central and the average fractional concentrations fall to 0$\cdot $6 and 0$\cdot $3, respectively. For a small region near the centre, this is true for a much longer time range, i.e. till the central and the average fractional concentrations fall to 0$\cdot $2 and 0$\cdot $1, respectively. Hence, very simple expressions for the concentration distribution or for average concentration in solids of various shapes are obtained without using any special mathematical method. The condition under which a solid of any shape or dimensions behaves as a linear semi-infinite slab is formulated. Some empirical and experimental findings of other workers are examined and found to be consistent with the theoretical conclusions. To illustrate the general applicability of the method, linear diffusion in a finite slab when the material is generated inside it at a constant rate or when the surface concentration increases linearly with time is briefly discussed and explicit results given. All expressions are obtained in terms of a dimensionless parameter, and it is shown that the concentration distribution in solids of any material and of various shapes can be derived from one single universal curve. Tables and graphs are given showing the relation between the numerical values calculated from the present simple solutions and those obtained by other much more laborious methods.

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