Thermal explosion problem with a stochastic boundary: quasi-stationary approximation and direct numerical modelling

Abstract

This paper considers a stochastic modification of the Frank-Kamenetskiy problem of exothermic reaction dynamics in a plane-parallel layer with random temperature fluctuations at the outer boundary as a means of modeling the behavior of chemical reactors when operating under uncontrolled environment impacts. Unlike deterministic formulations, such approaches take into account the possibility of a thermal explosion whose probability depends on the noise intensity. Based on random process theory, the conditions for achieving ignition in the quasi-stationary approximation (i.e., when the thermal relaxation rate is much higher than the rate of temperature change) are estimated. The possibility of using such a formulation to obtain an approximate relationship between the parameters of the noise and the dynamic characteristics of ignition (expected thermal explosion time) is demonstrated. The equation of non-stationary heat transfer in the reacting medium is solved numerically for a large number of random temperature trajectories at the boundary of the region of interest using a scheme combining explicit approximation of the nonlinear source with implicit approximation of the temperature field. By comparing the two approaches, the main regularities of non-stationary development of a thermal explosion in a stochastic environment can be approximated with good accuracy. Such a comparison relies on dependencies obtained when solving the quasi-stationary problem, taking into account a small correction for the critical temperature (marking the stability boundary for the stationary problem). Distributions of ignition characteristics (ignition temperature, maximum ambient temperature, and ignition time) and their dependence on input parameters (reactivity and noise intensity) are discussed.