A Singular-Perturbation Analysis of the Burning-Rate Eigenvalue for a Two-Temperature Model of Deflagrations in Confined Porous Energetic Materials

Deflagrations in porous energetic materials are characterized by regions of two-phase flow, where, for sufEciently large flow velocities, temperature-nonequilibration effects can significantly affect the overaI1 burning rate. In the present work, we analyze a two-temperature model of defiagrations in confined porous propellants that exhibit a bubbling melt layer at their surfaces. For appropriately scaled rates of interphase heat transfer, the problem reduces to a nontrivial eigenvalue calculation in the thin reaction region where final conversion of the liquid to gaseous products occurs. For realistically small values of the ratio of gas-to-liquid thermal conductivity, solutions in the reaction zone take on a singular-perturbation character that can be exploited to derive an asymptotic expansion of the burning-rate eigenvalue. The resulting problem requires a , rather sophisticated application of techniques in matched asymptotic expansions stemming from the appearance of an infinite number of logarithmic terms in the asymptotic development that must . be summed to arrive at the desired level of approximation. The physical effects of temperature nonequilibrium, which decreases the rate of heat transfer from the reacting liquid phase to the gas-phase products and thus allows a greater amount of thermal energy to remain in the reacting phase, is to increase the burning rate relative to the singl~temperature limit and to sharpen the transition from “conductive” to “convective” burning.