Diffusion Calculations: Interrelation Between Two Solutions of the Fourier Equation

Abstract

An examination of the interrelation between the sine series solution (for a slab with sealed periphery) and the error function solution (for a semi‐infinite body) of the fundamental Fourier equation has yielded the following results: (1) Distribution functions according to the two solutions begin to diverge soon after diffusion has begun. (2) The error function yields an excellent simple approximation, valid until half the material initially present has been lost, for the loss of material from a slab with sealed periphery. (3) A simple, but extremely good, approximation is given for an infinite exponential series in the region of slow convergence. (4) Two simplifications in the calculation of diffusion constants have been given. (5) While the discussion has been restricted to diffusion it is applicable, with obvious modifications, to other cases of flow governed by the fundamental Fourier equation.