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Transient diffusion from a well-stirred reservoir to a body of arbitrary shape



A theoretical study was made of transient diffusion to a body immersed in a finite volume of well-stirred fluid. The major contribution of this work was the development of a technique for solving the problem for a three-dimensional body of arbitrary shape. The solutions are in a form that is useful for determining diffusion coefficients insolids by means of the constant-volume experimental technique.

The partial differential equation coupled with the ordinary differential equation describing the diffusion process is transformed into a single integral equation in terms of the solute concentration in the reservoir. A numerical technique is then presented for solving the integral equation. Numerical solutions were computed for the three geometries that possess analytical solutions: the infinite slab, the infinite cylinder, and the sphere. By properly choosing the step-size numerical results were easily obtained that agreed with the exact solution to four decimal places.

New solutions were computed for two three-dimensional geometries: the finite cylinder and the rectangular prism. A range of shape factors and ratios of the volume of the reservoir to that of the solid body were employed for each geometry. It was shown that by selecting the ratio of volume to external surface area as the characteristic length of each shape object, the solutions for all shapes were brought close together and were identical during the initial part of the transient.

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