The Confined Nondiffusive Thermal Explosion with Spatially Homogeneous Pressure Variation

A solution is developed for a nondiffusive thermal explosion in a reactive gas confined to a bounded container Ω with a characteristic length l'. The process evolves with a spatially homogeneous time-dependent pressure field because the characteristic reaction time t'_R is large compared to the acoustic time l'/C'₀ where C'₀ is the initial sound speed. Exact solutions, in terms of a numerical quadrature are obtained for the induction period temperature, density and pressure perturbations as well as for the induced velocity field. Traditional single-point thermal runaway singularities are found for temperature and density when the initial temperature disturbance has a single point maximum. In contrast, if the initial maximum is spread over a finite subdomain of Ω, then the thermal runaway occurs everywhere. Asymptotic expansions of the exact solutions are used to provide a complete understanding of the singularities. The perturbation temperature and density singularities have the familiar logarithmic form — ln (t'_e — t') as the explosion time l_e is approached. The spatially homogeneous pressure is bounded for single-point explosions but is logarithmically singular when global runaway occurs. Compression heating associated with the unbounded perturbation pressure rise is the physical source of the global thermal runaway.