Rigorous Bounds and Relations among Spatial and Temporal Approximations in the Theory of Combustion

Abstract

Rigorous bounds on temperature and concentration are obtained for a model reacting system incorporating spatial variation, reactant consumption, and Arrhenius kinetics. Similar bounds are obtained for the Semenov (spatially uniform) approximation. Temperatures predicted by the Frank-Kamenetskii (stationary) approximation are shown to be upper bounds on the temperature of the complete time-dependent system. The latter supports numerical and approximate analytic arguments that the critical value of the Frank-Kamenetskii heat generation parameter in the stationary approximation is a lower bound on analogously defined critical parameter values in the complete system. The fundamental mathematical tool is a comparison result for systems of parabolic partial differential equations