Ignition Of An Absorbing/Emitting Medium by an Imposed Radiant Heat Flux

Abstract

We consider the unsteady ignition of a reactive slab subjected to external heating, in conditions where all the heat transfers are of radiative origin and, consequently, introduce a reference length L (Planck's). Radiant exchanges in the bulk are modelled by the Eddington equation, and an Arrhenius law with a large activation temperature is postulated for the rate of heat release. The problem is thus solved by activation energy asymptotics. Similarly to ignitions by conductive heating, one can distinguish two main stages: an inert stage, followed by an ignition stage. As is first shown upon use of a simplified source function, different situations must be distinguished depending on the ignition time t_ig : 1) for early ignition, the thermal runaway occurs in an optically-thin surface layer where the excess-emission due to heat release may be neglected. 2) moderately late ignitions still take place in optically-thin layers but excess emission must be retained. 3) late ignitions occur in reaction layers of O(1) optical thicknesses, implying fully nonlocal exchanges. 4) very late ignitions are governed by the optically-thick approximation and the problem assumes a structure encountered in conductive ignitions. In any case asymptotic estimates of l. are provided analytically. In the last parts of the paper we show that the aforementioned reaction layers have rather universal structures, in the sense that they are also encountered when nonlinearized radiative transfer and/or, in some instances, when reactive bodies of more complicated shapes are considered. In the case of finite reactive bodies, ignition may not occur and generalized Semenov-Frank-Kamenetskii problems define the conditions for the existence of a thermal runaway.