Correction of kinetic data in non-isothermal reactions with non-uniform temperatures: analytical treatments for spherical reactant masses

Abstract

Exothermic reactions normally proceed non-isothermally, and to treat them quantitatively allowance must be made for variations in internal temperature. Except for the special cases of the infinite slab and infinite cylinder, this has so far had to be done solely by numerical computation. The sphere and other finite, three-dimensional geometries have eluded analytical solution, even in the stationary state. The present paper presents two useful treatments for spherical geometry. They provide families of compact analytical expressions of very satisfactory precision for all the important aspects, errors rarely being greater than 2% right up to criticality. The aspects studied include temperature-position profiles, central temperature-excesses and surface temperature-gradients. (Both stable and unstable subcritical conditions as well as critical cases can be treated.) From these there follow values for rate-constant correction-factors f (and their reciprocals, the effectiveness factors η) and, in turn, the means of correcting errors in Arrhenius activation energies and in reaction orders - again in expressive, simple forms. In addition, explicit equations have been set out for the first time to cover the whole range of boundary conditions (arbitrary Biot number, β ) from Frank-Kamenetskii ( β → ∞) to Semenov ( β ═ 0) extremes. This is a major development, since mere tabulation of numerical solutions has hitherto been a formidable task. Calculating procedures are detailed in an Appendix. Endothermic reactions show self-cooling and self-repression instead of self-stimulation. These aspects are all expressively encapsulated in the change of sign of single coefficients in tidy equations. The two routes - quintic approximation (q. a.) and second-order reversion (r2) - can both be applied most conveniently in parametric form (this is necessary for r2 but not for q. a). The parameters l and x that appear naturally in the two treatments bear a close relation to corresponding integration constants encountered in one- and two-dimensional geometry. Each has striking physical significance, l being most directly related to the effectiveness factor and x to the central temperature excess.