Thermal explosion in the sphere and the hollow spherical shell heated at the inner face

Abstract

The steady state heat conduction equation for reactants of spherical symmetry decomposing by a zero-order exothermic process has been converted by means of the fifth-power (quintic) approximation ▪ (where α and β are constants) into a form which is soluble analytically. The method of solution depends upon the physical model under examination: two models have been discussed, that of self ignition of a hollow spherical shell of reactant heated at the inner face and cooling to the surroundings from the outer face and the solid sphere immersed in a constant temperature bath. Approximate Frank-Kamenetsky criticality criteria for hollow spheres of varying thickness and hot face temperatures were derived analytically by a method involving the use of incomplete elliptic integrals of the first kind. The results were compared with numerical solutions of the equation wherein the exponential approxiamation had been made and found to be in good agreement. It has been shown how these results fit into the pattern established by those for the asymmetrically heated slab and the infinite hollow cylindrical shell heated at the inner face. For the first time an analytical criticality criterion (δ crit. = 3/4α 4β), critical centre temperature rise [θ mcrit. = (□2−1)(α/β)] and temperature distribution, given by ( α + β θ ) 2 = 1 ⋅ 5 α 2 [ ( 1 + z 2 ) − ( 1 − z 2 ) { 1 − ( 4 α 4 δ β / 3 ) } 1 2 ] [ δ β ( 1 − z 2 ) 2 α 4 + 3 z 2 ] have been determined for the sphere by a method in which the phenomenon of criticality emerges naturally from the analysis.