The slowly reacting mode of combustion of gaseous mixtures in spherical vessels. Part 2: buoyancy-induced motion and its effect on the explosion limits

Abstract

This paper investigates the effect of buoyancy-driven motion on the quasi-steady “slowly reacting” mode of combustion and on its thermal-explosion limits, for gaseous mixtures enclosed in a spherical vessel with a constant wall temperature. Following Frank-Kamenetskii’s seminal analysis of this problem, the strong temperature dependence of the effective overall reaction rate is taken into account by using a single-reaction model with an Arrhenius rate having a large activation energy, resulting in a critical value of the vessel radius above which the slowly reacting mode of combustion no longer exists. In his contant-density, convection-free analysis, the critical conditions were found to depend on the value of a Damk¨ohler number, defined as the ratio of the time for the heat released by the reaction to be conducted to the wall, to the homogeneous explosion time evaluated at the wall temperature. For gaseous mixtures under normal gravity, the critical Damk¨ohler number increases through the effect of buoyancy-induced motion on the rate of heat conduction to the wall, measured by an appropriate Rayleigh number Ra. In the present analysis, for small values of Ra, the temperature is given in the first approximation by the spherically symmetric Frank-Kamenetskii solution, used to calculate the accompanying gas motion, an axisymmetric annular vortex determined at leading order by the balance between viscous and buoyancy forces, which we call the FrankKamenetskii vortex. This flow is used in the equation for conservation of energy to evaluate the influence of convection on explosion limits for small Ra, resulting in predicted critical Damk¨ohler numbers that are accurate up to values of Ra on the order of a few hundred.