Criticality in reactors under domain or external temperature perturbations

Abstract

The conditions for the onset of thermal runaway in reactors with small non-uniformities is investigated. The reaction is modelled by an Arrhenius heat generation term with a finite activation energy and the dimensionless temperature, u 0 , is taken to satisfy a nonlinear equation of the form Δ u 0 + λ 0 F ( u 0 ) = 0, x ∈ D ; ∂ v u 0 + bu 0 = 0. x ϵ∂ D . We investigate three classes of perturbations of this problem. First, we treat a small temperature variation maintained on the boundary of the domain. Secondly, we consider a small distortion of the boundary of a circular cylindrical domain, and thirdly, we analyse the effect of a small hole in the domain. In each case we derive asymptotic expansions for the critical Frank-Kamenetskii parameter, λ c ( ϵ ), where ϵ is a measure of the size of the perturbation. A numerical scheme is then used to determine numerical values for the coefficients in the asymptotic expansion of λ c . Finally, some of the asymptotic results are compared with corresponding numerical results obtained from a full numerical solution of the perturbed problem.