Criticality in reactors under domain or external temperature perturbations

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 $\Delta $u$_{0}$ + $\lambda _{0}$F(u$_{0}$) = 0, x$\in $D; $\partial _{\nu}$u$_{0}$ + bu$_{0}$ = 0, x $\in \partial $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, $\lambda _{\text{c}}(\epsilon)$, where $\epsilon $ 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 $\lambda _{\text{c}}$. Finally, some of the asymptotic results are compared with corresponding numerical results obtained from a full numerical solution of the perturbed problem.