Models for Gasless Combustion in Layered Materials and Random Media

Abstract

It is generally assumed in this work, as elsewhere, that the mass diffusion coefficient has an Arrhenius temperature dependence. This allows a combustion wave propagated by thermal conduction to develop similar to more conventional combustion systems. It is found that for realistic systems the problem must involve a three length scale analysis. Similar to the classic theory on premixed, gas-phase flames, the largest scale is identified with thermal conduction and a “reaction zone” scale proportional to the inverse of the mass diffusion activation energy is associated with a small length over which significant mass diffusion is possible. Unlike classic flame theory a still smaller scale is identified with the length over which mass diffusion takes place (i.e., the size of a typical domain of alloyable constituent). Flame speeds are derived for three different geometric configurations of a binary system using a singular perturbation analysis. First a system where the constituents are arrayed in alternating lamina stacked perpendicular to the flame propagation direction is considered. Next, the same system is undertaken except the laminae are stacked parallel to the flame direction. Finally, the elements of these previous problems are combined to investigate the more realistic system of a random media. It is found that the physics for all three of these cases is essentially the same and their resulting flame speeds differ only by a numerical factor (if the correlation length statistic for the random media is considered to be the same as a quarter of the lamina thickness). Thus the more simplified nonrandom models are given more credence at predicting the behavior of systems in practicality.