AN ASYMPTOTIC THEORY OF CONDENSED TWO-PHASE FLAME PROPAGATION

A model is presented for flame propagation though a condensed combustible mixture in which the limiting component of the mixture melts during the reaction process. An asymptotic analysis, valid for large activation energies, is employed to derive a two-term expansion for the steady, planar adiabatic flame speed. A linear stability analysis is then used to show that for sufficiently large values of the activation energy and/or a special group of melting parameters, the steady, planar solution loses stability to various types of planar and nonplanar pulsating modes. The effect of melting is found to be destabilizing in the sense that these pulsating modes occur for lower values of the activation energy than would be the case for strictly solid fuel combustion. 1. Introduction. The propagation of reaction fronts in condensed phase combus- tion occurs in various metallurgical and pyrotechnical applications. In exothermic metal alloying processes, for example, two metals are ground into a fine powdered mixture. When ignited, a reaction front is established which propagates through the mixture, converting the metal reactants into an alloy product. Although this phenomenon is often referred to as solid fuel combustion, melting of at least one of the reactants is thought to be necessary in order to sustain the combustion process (cf. Hardt and Phung (1973)). An obvious explanation for this is that microscopic species diffusion, which determines the local reaction rate, is small when both reactants are solid due to the limited amount of surface-to-surface contact between the solid particles. When one of the reactants melts, the melted reactant can coat the unmelted particles, thereby significantly increasing the local reaction rate. One fascinating feature of condensed phase combustion is the confirmed existence of nonsteady, nonplanar modes of propagation of the reaction front. Experimental studies due to Merzhanov et al. (1973) and Maksimov et al. (1979) show that in addition to the usual steady, planar mode of propagation, there exist parameter regimes for which "auto-oscillatory" and spinning combustion fronts are observed. In the former case, the reaction front propagates in a pulsating fashion and the burned samples have a layered appearance. In the latter situation, the front has a nonuniform structure in which one or more hot spots are observed to rotate about the surface of the cylindrical fuel sample as the front propagates. Numerical predictions of these phenomena in condensed single-phase systems (i.e., no melting) have been reported by Shkadinsky et al. (1971) and Ivleva et al. (1978). The former paper showed that the activation energy is a critical parameter in determining the transition between the steadily propagating and pulsating solutions. In addition, due to what appears to be secondary and higher order bifurcations, the pulsating modes can be multiply periodic. That is, a complete cycle can consist of several pulsations. A similar phenomenon in gaseous combustion was predicted by Margolis (1980). The nature and transition characteristics of these various modes of propagation have been greatly clarified by several recent analytical studies (Matkowsky and Sivashinsky (1978), Sivashinsky (1981), Kaper, Leaf and Matkowsky (1982)). Using a single-phase model suggested by the asymptotic (large activation energy) models derived for gaseous flames (Sivashinsky (1977), Matkowsky and Sivashinsky (1979)),