Thermal explosion and times to ignition in systems with distributed temperatures. III. The influence of a steadily increasing (ramped) ambient temperature

Abstract

In classical treatments of thermal explosion, reactant consumption is ignored and ambient temperatures are assumed constant. We turn here to the very important case where the temperature of the surroundings is varying at a steady rate. The influence of this ramping and the importance of reactant consumption are considered for the whole range of Biot numbers from the Semenov extreme (β = 0) to the Frank-Kamenetskii extreme (β→∞) for arbitrary geometry and for any concentration- and temperature-dependence of reaction rate. External heating modifies the critical value for the Frank-Kamenetskii parameter δ ; δ cr /δ 0 = 1 + P[Qg w - α )/B] ⅔ . Here δ 0 is the critical value without reactant consumption, α is the dimensionless rate of change of ambient temperature (and is negative for external cooling), B is the dimensionless adiabatic temperature rise, and g w is the effective order of reaction. The numbers P and Q are of order unity and are determined here for all important circumstances. Illustrative values are presented for the sphere, the infinite cylinder and the infinite slab. Times to ignition are also evaluated and, for limiting cases, simple asymptotic formulae are given which are good approximations over wide ranges. Many of the new results parallel those for constant ambient temperature but a new branch of solutions is found if α > Qg w where thermal runaway becomes inevitable and criticality is lost.