Thermal explosions, criticality and the disappearance of criticality in systems with distributed temperatures. ।. Arbitrary Biot number and general reaction-rate laws

Abstract

The conductive theory of thermal explosion in its original form (Frank-Kamenetskii, Acta phys. -chim . URSS (1939)) expresses the balance between heat generation and heat conduction in terms of the dimensionless parameter δ = [QEA a 2 0 c n 0 exp ( - E/RT a )]/ K RT 2 a Stability is lost when δ exceeds a critical value which, in this approximation, depends only on the geometry of the system. Matters are usually more complicated than this. First, heat transfer is often impeded both in the interior (by conduction) and at the surface; the relative importance of these impedances is expressed by the Biot number B i = X a o / K - Second, the temperature dependence of reaction rate may not be well enough represented by the 'exponential approximation’ (which simply implies a doubling of rate every so many degrees). The natural and convenient dimensionless measure here is the parameter ϵ = RT a /E. In the present paper, critical values for the parameter δ and for the dimensionless central-temperature excess Ɵ o have been evaluated for the whole range of Biot number from the uniform case (Semenov extreme, B i → 0) to the Frank-Kamenetskii extreme (B i →∞). The procedures can handle any temperature-dependence of rate and are illustrated here for the Arrhenius and 'bimolecular’ forms for which, k∝ exp ( — E /RT ) and k ∝T ½ exp ( - E /R T ) respectively. When E /RT a is not large, criticality is lost at ϵ = ϵ tr ≤ ¼. Such transitional values for the reduced ambient temperature ϵ, for the critical value of δ, and for the dimensionless central temperature excess Ɵ 0 have also been obtained. They are represented both graphically and numerically. The present results are also compared with earlier work.