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_o, is taken to satisfy a nonlinear equation of the form Δu_0+λ_0F(u_0) = 0, xєD; ∂_vu_0+bu_0 = 0, xє∂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(є), where є 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.

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