Propagation and diffusion-limited extinction of nonadiabatic heterogeneous flame in the SHS process

Nonadiabatic heterogeneous flame propagation and extinction in self-propagating high-temperature synthesis (SHS) are analyzed based on a premixed mode of propagation for the bulk flame supported by the nonpremixed reaction of dispersed nonmetals in the liquid metal. The formulation allows for volumetric heat loss throughout the bulk flame, finite-rate Arrhenius reaction at the particle surface, and temperaturesensitive Arrhenius mass diffusion in the liquid. Results show that, subsequent to melting of the metal, the flame structure consists of a relatively thin diffusion-consumption/convection zone followed by a relatively thick convection-loss zone, that the flame propagation rate decreases with increasing heat loss, that at a critical heat-loss rate the flame extinguishes as indicated by the characteristic turning-point behavior, that the surface reaction is diffusion limited such that the nonlinear, temperature-sensitive nature of the system is actually a consequence of the Arrhenius mass diffusion, and that extinction is sensitively affected by the mixture ratio, the degree of dilution, the initial temperature of the compact, and the size of the nonmetal particles. An explicit expression is derived for the normalized mass burning rate, which exhibits the characteristic turning point and shows that extinction occurs when this value is reduced to e^-1/2, which is the same as that for the nonadiabatic gaseous premixed flame. It is further shown that the theoretical results agree well with available experimental data, indicating that the present formulation captures the essential features of the nonadiabatic heterogeneous SHS processes and its potential for extension to describe other SHS phenomena.