The influence of initial temperature-excess on critical conditions for thermal explosion

In the conductive theory of thermal explosion, an exothermic system is considered to be an ignition hazard only if the numerical value of the dimensionless group δ = Qσa₀ ²A exp (- E/RT_a κ(RT_a ²/E) exceeds a critical value. [Here Q is the exothermicity, σ the density, a₀ the half-width, A exp(- E RT_a) the rate constant at ambient temperature, and κ the thermal conductivity.] This classical approach implicitly assumes that the material is initially assembled at or near to ambient temperature. Such initial conditions represent only a subclass of the whole problem. In some situations of technical importance the reactant may be initially considerably above T_a. The present paper considers the temperature evolution in bodies subject to Frank-Kamenetskii boundary conditions but which are assembled with a positive, uniform temperature-excess. It is shown that in the usual exponential approximation, any system with δ ≤ δ_cr has a critical initial temperature, above which thermal runaway occurs. Exact numerical results for the critical conditions are presented for the three class A geometries (infinite slab, infinite cylinder, and sphere). Very accurate analytical approximations are also provided. For an Arrhenius rate-law, ignition cannot occur in this way for very low δ < δ_ex, where δ_ex ∼ O[exp(-ε{lunate}^{- 1 2})] and ε{lunate} = RT_a E. A comparison is made between the predictions of this model and the critical conditions observed in an actual, practical example; the spontaneous ignition of piles of bagasse (extracted sugar cane) for which δ ≈ 0.02 ≪ δ_cr.