The Effect of Square Corners on the Ignition of Solids

Abstract

The influence of square corners on the ignition of a solid exposed to a step in surface temperature is analyzed by means of large activation energy asymptotics. This study begins by considering the case of a semi-infinite square corner, applicable to the ignition of finite bodies with square corners when the reaction is very fast. Two spatial zones (reactive and inert) and two time stages (initial and transition) are identified. During the initial stage, the structure of the reaction zone is determined by a quasi-stationary problem of the Frank-Kamenetskii type, where the time variable plays the role of the Damköhler number. There is no solution to this problem if the time $\tau $ is larger than a certain critical value $\tau_0 $, which is a first approximation for the ignition time. In a transition stage, for $0 < \tau_0 - \tau \ll 1$, the nonstationary effects cannot be neglected; when these are taken into account, a first correction to the ignition time is obtained. The ignition of a two-dimensional rectangular solid is also described, for which the previous analysis is partially applicable if the Damköhler number $D_a$ is large enough. For $D_a$ of order unity, an asymptotic analysis is given, in which the process is described in terms of a first inert heating stage and a second reacting stage ending in a thermal runaway. A numerical description is given for the second stage to determine the ignition time in terms of the Damköhler number.