Roslyn Hickson

Finite difference schemes for multilayer diffusion

Although numerical methods have been developed for diffusion through single layer materials, few have been developed for multiple layers. Diffusion processes through a multilayered material are of interest for a wide range of applications, including industrial, biological, electrical, and environmental areas. We present finite difference schemes for multilayered materials with a range of matching conditions between the layers, in particular for a jump matching condition. We show the finite difference methods are flexible, simple to implement, and help illustrate interesting behaviour in multilayered diffusion.

Critical times in one- and two-layered diffusion

Summary The study of diffusion is commonplace in engineering mathematics courses as a classic example of partial differential equations and separation of variables. However, many textbooks stop at a derived solution without going further to explore what the solution means. The analysis of the critical diffusion time is used here to demonstrate how the solutions obtained can be used to explore additional useful results for diffusion through a single layer of material. We also show how consideration of diffusion through two layers gives rise to some surprising new results. This problem was motivated by analysing the annealing of steel coils, where knowledge of the time to heat a system of air and steel layers is critical in the manufacturing process.

A COMPARISON OF CRITICAL TIME DEFINITIONS IN MULTILAYER DIFFUSION

There are many ways to define how long diffusive processes take, and an appropriate “critical time” is highly dependent on the specific application. In particular, we are interested in diffusive processes through multilayered materials, which have applications to a wide range of areas. Here we perform a comprehensive comparison of six critical time definitions, outlining their strengths, weaknesses, and potential applications. A further four definitions are also briefly considered. Equivalences between appropriate definitions are determined in the asymptotic limit as the number of layers becomes large. Relatively simple approximations are obtained for the critical time definitions. The approximations are more accessible than inverting the analytical solution for time, and surprisingly accurate. The key definitions, their behaviour and approximations are summarized in tables.