Numerical calculation of critical points for a slab with partial insulation

Abstract

Numerical solutions of a nonlinear partial differential equation arising in thermal explosion theory are obtained for a finite rectangular strip domain with mixed boundary conditions. Critical points in the solution (marking loss of stability and loss of criticality) are computed directly for the case of a self-heating slab with its upper and lower surfaces partially insulated, retaining the full Arrhenius rate term. For the special case of the Frank-Kamenetskii approximation it is shown numerically that the critical value of the rate parameter satisfies λ c (ϵ)=λ c (0)(1+O(ϵ 2)), where λ c(0) ≅ 0.878 and ϵ is the ratio of the insulation width to slab thickness.