The progressive variation of density through a reacting particle, due to internal reaction in the particle pores, is shown to modify the reaction rate expressions by inclusion of a time-dependent parameter, b(t), which is the ratio of the surface density (r s ) to the mean particle density
The values of b lie in the ( r). range 0 Յ b Յ 1. This parameter modifies, in particular, the second effectiveness factor, e ס R s /R e ס (1 ם ␣/3)/b, which is a measure of the penetration depth of the reaction, and where ␣ is proportional to the gradient of the (mean) density with particle radius. This gradient can be negative as well as d r/da, positive, in agreement with known experimental results where, at high temperature, the variation of density with burn-off (B) is an inverted U shape, going through a minimum and then rising. This new analytical result is in contrast to earlier theoretical work that could not account for the rise, since the negative values of in that earlier work were prohibited. At the upper limit of b Ϸ 1, this parameter has marginal d r/da influence on predicted behavior. At the lower limit, as b → 0, however, the corresponding potential error can be major since the predicted penetration depth, with corresponding transition from zone III to II to I, can alter the effective or apparent activation energy by as much as a factor of 2 on account of exclusion or inclusion of the Thiele behavior. In other results, obtained by comparison of the theoretical extensions with earlier experimental work, and notably the variation of the unburned (U) with radius (a) in addition to the variation of mean particle density with burn-off (B), we show the improved interpretation of mechanistic behavior.