Thermal theory of spontaneous ignition: criticality in bodies of arbitrary shape

Abstract

For an exothermic reaction to lead to explosion, critical criteria involving reactant geometry, reaction kinetics, heat transfer and temperature have to be satisfied. In favourable cases, the critical conditionstreatments are either confined to idealized geometries, namely, the sphere, infinite cylinder or infinite slab, or require the simplest representations of heat transfer. may be summarized in a single parameter, Frank-Kamenetskii’s δ being the best known, but analytical. In the present paper a general steady-state description is given of the critical conditions for explosion of an exothermic reactant mass of virtually unrestricted geometry in which heat flow is resisted both internally (conductive flow) and at the surface (Newtonian cooling). The description is founded upon the behaviour of stationary-state systems under two extremes of Biot number—that corresponding to Semenov’s case ( B i → 0 ) and that corresponding to Frank-Kamenetskii’s case ( B i → ∞ ) it covers these and intermediate cases. For Semenov’s conditions, the solution is already known, but a fresh interpretation is given in terms of a characteristic dimension—the mean radius R s . A variety of results for criticality is tabulated. For Frank-Kamenetskii’s conditions, the central result is an approximate general solution for the stationary temperature distribution within any body having a centre. Critical conditions follow naturally. They have the simple form: q σ A exp ⁡ ( − E / R T a ) k R T a 2 / E R 0 2 er ≡ δ er ( R 0 ) = 3 F ( j ) , where F ( j ) is close to unity, being a feeble function of shape through a universally defined shape parameter j , and δ cr ( R 0 ) is Frank-Kamenetskii’s δ evaluated in terms of a universally defined characteristic dimension R 0 —a harmonic square mean radius weighted in proportion to solid angle: 1 R 0 2 = 1 4 π ∫ ∫ d ω a 2 . Expressions for the mean radius R 0 have been evaluated and are tabulated for a broad range of geometries. The critical values generated for δ are only about 1 % in error for a great diversity of shapes. No adjustable parameters appear in the solution and there is no requirement of an ad hoc treatment of any particular geometric feature, all bodies being treated identically. Critical sizes are evaluated for many different shapes. For arbitrary shape and arbitrary Biot number (0 < < ∞) an empirical criterion is proposed which predicts critical sizes for a great diversity of cases to within a few parts per cent. Rigorous, closely adjacent upper and lower bounds on critical sizes are derived and compared with our results and with previous investigations, and the status of previous approaches is assessed explicitly. For the most part they lack the generality, precision and ease of application of the present approach.