Thermal explosion and times-to-ignition in systems with distributed temperatures I. Reactant consumption ignored

Abstract

Hitherto the time evolution of temperature profiles within systems capable of reacting exothermically and perhaps of exploding has only been known for the limiting case in which the temperature remains essentially uniform throughout. In this paper we consider the more difficult and general problem in which the temperature profile is not uniform. We concentrate on numerical results for the three simplest geometries (infinite slab, infinite cylinder, sphere) but our solutions are valid for arbitrary geometry. Similarly we concentrate on Arrhenius temperature-dependence; the solutions can again cope readily with general forms. The principal result of the paper is a description of the evolution of the central temperature when circumstances are marginally supercritical. When the value of Frank-Kamenetskii’s reduced reaction rate parameter ( δ ) slightly exceeds its critical value ( δ cr ), the temperature at first increases rapidly and then moves very slowly for a long while before a final, rapid acceleration leads to ignition. Times to ignition ( t ) all have the form t / t ad = M /( δ / δ cr - 1) ½ where t ad is a known time closely corresponding to the time to ignition under adiabatic conditions. The value of the constant M is readily calculated from certain simple integrals. Its variation with geometry and Biot number β is presented and discussed. The limiting form of our expressions for β → 0 is precisely the well known solution of the problem for Semenov boundary conditions. We also compare our results with an exact (numerical) solution of the time-dependent problem for a sphere with fixed surface temperature. Not only is the agreement close to criticality very good but also it yields a fair approximation up to δ ≈ 2 δ cr .