The Effect of Square Corners on the Ignition of Solids

Abstract

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 Damkohler number. There is no solution to this problem if the time r is larger than a certain critical value To, which is a first approximation for the ignition time. In a transition stage, for 0 < TO -r <C 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 twodimensional rectangular solid is also described, for which the previous analysis is partially applicable if the Damkohler 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 Damkohler number.Key words, combustion, ignition, thermal runaway, hot spot, activation energy asymptotics, Arrhenius kinetics AMS subject classifications. 35C20, 35K57, 80A251. Introduction. When the chemical reactions occurring in combustion processes are modeled by an overall reaction, their activation energy E is large compared with the thermal energy RT. Then the reaction rate is very sensitively dependent on the temperature, so that only a small temperature increment (of the order of the Frank-Kamenetskii temperature RT 2 /E) is required to increase the rate by a factor e (see [1] and [2]). The ratio of this temperature to the typical spatial temperature difference is considered as a small parameter e, whose inverse is called the Zeldovich number in large activation energy asymptotics.For small values of e, we find ignition events characterized by a thermal runaway or abrupt rise in temperature occurring at a hot spot at a well-defined ignition time. During the ignition stage the reaction heat release can be neglected outside the region where the temperature differs from the maximum by an amount of the order of the Frank-Kamenetskii temperature.It is well known that when the surface of a reacting solid is maintained at a constant temperature, equal to the initial temperature To, and the Damkohler number D a , defined as the ratio of the heat conduction time t c to the homogeneous or adiabatic ignition time t qi is lower than a critical value, of order unity, no ignition event in the form of a thermal runaway takes place. In this case, only a small increment in temperature is required for the conduction ...

Ignition of a Rectangular Solid by an External Heat Flux

“…We found regimes similar to those described in this paper in our analysis of the ignition of a rectangular solid, with Dirichlet type conditions resulting from a step in surface temperature [11]. In that case we found a critical value of the Damkohler number, such that no ignition occurred for Damkohler numbers less than the critical one.…”

“…a (0,9, t) = y/nt/a for any real value of a. This problem, to be solved numerically, is very similar to that considered in [11]. We can expect, as in similar reactive-diffusive systems, two solutions to exist for values of 6 smaller than a certain critical value, i.e., the solution branch [ll^ll?…”

“…We have recently used [11] these techniques to analyze the ignition of rectangular bodies under a step rise in surface temperature. We will present in this paper a large activation energy analysis of the ignition of wedges by an external heat flux, the problem treated in [8]- [10].…”

The virial theorem for nonlinear problems

“…In Sec. 3 we develop the virial theorem, apply it to conservative nonlinear oscillators, and compare its results with those obtained by Beléndez et al [1]. In Sec.…”

“…that appears in simple models for the study spontaneous explosion due to internal heating in combustible materials [3,4]. It is also interesting for another reason: it is a simple strongly nonlinear problem that can be exactly solved.…”