Natural convection effects on thermal ignition in a porous medium. I. Semenov model

A one-dimensional diffusion-convection-reaction model is formulated to account for natural convection effects on thermal ignition in an open system consisting of a porous medium. Various limiting cases of the model are considered. A detailed analysis of the Semenov (lumped) model is presented. Explicit relations are derived for the dependence of the critical Semenov number (ψc) on the Rayleigh number (R*). It is shown that for R* → 0, ψc approaches the classical (conduction) limit e-1, while for R* ≫ 1, the ignition locus is given by the convection asymptote ψc/R* = 4 e-2. Inclusion of reactant consumption shows that the conduction asymptote disappears at B = 4 while the convection asymptote ceases to exist for B Ls < 3 + 2√2, where Ls is a modified Lewis number and B is the heat of reaction parameter. It is shown that the Semenov model has five different types of bifurcation diagrams of temperature against Rayleigh number (particle size), (single-valued, inverse S, isola, inverse S + isola and mushroom). This behaviour is found to be qualitatively identical to that of the forced convection problem investigated by Zeldovich & Zysin.