Theory of thermal explosions with simultaneous parallel reactions. II. The two- and three-dimensional cases and the variational method

Abstract

Many real, exothermic systems involve more than one simultaneous reaction. In part I of this series (Boddington, T., Gray, P. & Wake, G. C. Proc . R . Soc . Lond . A 393, 85 (1984)) a simple, unifying approach to the behaviour of such multi-reaction systems was proposed. It built on a communal activation energy as a basis for dimensionless quantities (such as the θ, δ and ϵ of Frank-Kamenetskii’s treatment of single reactions). The procedures were illustrated for the infinite slab where solution by quadratures was possible. The present paper extends the investigation to the other two simple geometries of the infinite circular cylinder and the sphere. As the method of solution by quadratures does not generalize to these cases, a variational method (which is applicable to many other problems of this type) is proposed and used. The results for the critical values of the Frank-Kamenetskii parameter δ cr (beyond which no steady solution exists) and corresponding dimensionless temperature rise θ m,cr are slightly less accurate when a variational method is used. However, errors verified as no greater than 0.2 % arise for standard cases of one reaction where δ cr and θ m,cr are known to high accuracy (for example, δ cr = 2, θ m,cr = 2 In 2 in an infinite cylinder with one reaction). As with the infinite slab the same qualitative effects are found: as the heat release rate for the second reaction increases, the stable régime is reduced, but this reduction is small for ratios of activation energies in the region of 0.2–5. The results are presented numerically in tabular form only (the general shape of the graphs are exactly as in part I).