Electric Discharge Sintering: A Mathematical Model

In this paper we mathematically model the densification of metallic powders and the sintering of ceramic powders by electric discharge. The ordinary and partial differential equations governing these processes are the same with the exception of the effective electrical conductivity. This function is a monotonically decreasing (increasing) function of temperature for the metallic (ceramic) powders. We employ asymptotic methods to approximate the solution to these equations in the limit as ǫ → 0, where ǫ is the ratio of the discharge to diffusion time scales. We find on the shortest time scale that the temperature, voltage, and density satisfy a system of nonlinear, coupled ordinary equation. We solve these and find the relationship between the temperature and density, as functions of the input energy. The results on the short or discharge time scale do not take into account diffusion and heat loss into the surrounding medium. These occur on a much longer time scale which we identify and exploit to deduce a new approximation. On this time scale the capacitor has no more energy to deposit into the powder. The temperature relaxes to that of its surroundings and the density increases to its final value. Our results show the functional relationship between the final density and the initial energy stored in the capacitor, as well as the initial density of the powder.