Criticality in reactors under domain or external temperature perturbations
Eric F. Van de Velde, Michael J. Ward · Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences · 1991
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, u0, is taken to satisfy a nonlinear equation of the form Δu0 + λ0F(u0) = 0, x∈D; ∂vu0 + bu0 = 0. xϵ∂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, λc(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 λc. Finally, some of the asymptotic results are compared with corresponding numerical results obtained from a full numerical solution of the perturbed problem.