Influence of pressure-driven gas permeation on the quasi-steady burning of porous energetic materials

A theoretical two-phase-flow analysis is developed to describe the quasi-steady propagation, across a pressure jump, of a multi-phase deflagration in confined porous energetic materials. The difference, or overpressure, between the upstream (unburned) and downstream (burned) gas pressure leads to a more complex structure than that which is obtained for an unconfined deflagration in which the pressure across the multi-phase flame region is approximately constant. In particular, the structure of such a wave is shown by asymptotic methods to consist of a thin boundary layer characterized by gas permeation into the unburned solid, followed by a liquid-gas flame region, common to both types of problem, in which the melted material is preheated further and ultimately converted to gaseous products. The effect of gas flow relative to the condensed material is shown to be significant, both in the porous unburned solid as well as in the exothermic liquid-gas melt layer, and is, in turn, strongly affected by the overpressure. Indeed, all quantities of interest, including the burn temperature, gas velocity and the propagation speed, depend on this pressure difference, leading to a significant enhancement of the burning rate with increasing overpressure. In the limit that the overpressure becomes small, the pressure gradient is insufficient to drive gas produced in the reaction zone in the upstream direction, and all gas flow relative to the condensed material is directed in the downstream direction, as in the case of an unconfined deflagration. The present analysis is particularly applicable to those types of porous energetic solid, such as degraded nitramine propellants that can experience significant gas flow in the solid preheat region and which are characterized by the presence of exothermic reactions in a bubbling melt layer at their surfaces.