Approximate solutions in the theory of thermal explosions for semi-infinite explosives

Abstract

If the solution, of the heat conduction equation θ τ ( 0 ) = θ ξ ξ ( 0 ) , ξ > 0 , τ > 0 of a chemically ‘inert’ material is known, then an approximate formula for the explosion time, ד expl. , of an explosive satisfying the heat conduction equation with zero order reaction, θ ד = θ ξξ +exp(-1/θ), ξ > 0, ד 0, and the same initial and boundary conditions as the ‘inert’, is given by the root of the equation, − ∂ θ ( 0 ) ( ξ , τ expt . ) / ∂ ξ | ξ − 0 = ∫ 0 ∞ exp ⁡ [ − 1 / θ ( 0 ) ( ξ , τ expl . ) ] d ξ provided 1/θ (0) (ξ, ד) is suitably expanded about the surface ξ = 0 such that the integrand vanishes as ξ→∞. Similar results hold for one-dimensional cylindrically and spherically symmetric problems. The derivation of the explosion criterion is based on observation of existing numerical solutions where it is seen that (i) almost to the onset of explosion, the solution θ(ξ, ד )does not differ appreciably from θ (0) (ξ, ד ) (ii) the onset of explosion is indicated by the appearance of a temperature maximum at the surface. Simple formulas for ד expl. readily obtainable for a wide variety of boundary conditions, are given for seven sample problems. Among these are included a semi-infinite explosive with constant surface flux, convective surface heat transfer, and constant surface temperature with and without subsurface melting. The derived values of ד expl. are in satisfactory agreement with those obtained from finite-difference solutions for the problems that can be compared.