The slowly reacting mode of combustion of gaseous mixtures in spherical vessels. Part 1: Transient analysis and explosion limits

Frank-Kamenetskii's analysis of thermal explosions is revisited, using also a single-reaction model with an Arrhenius rate having a large activation energy, to describe the transient combustion of initially cold gaseous mixtures enclosed in a spherical vessel with a constant wall temperature. The analysis shows two modes of combustion. There is a flameless slowly reacting mode for low wall temperatures or small vessel sizes, when the temperature rise resulting from the heat released by the reaction is kept small by the heat-conduction losses to the wall, so as not to change significantly the order of magnitude of the reaction rate. In the other mode, the slow reaction rates occur only in an initial ignition stage, which ends abruptly when very large reaction rates cause a temperature runaway, or thermal explosion, at a well-defined ignition time and location, thereby triggering a flame that propagates across the vessel to consume the reactant rapidly. Explosion limits are defined, in agreement with Frank-Kamenetskii's analysis, by the limiting conditions for existence of the slowly reacting mode of combustion. In this mode, a quasi-steady temperature distribution is established after a transient reaction stage with small reactant consumption. Most of the reactant is burnt, with nearly uniform mass fraction, in a subsequent long stage during which the temperature follows a quasi-steady balance between the rates of heat conduction to the wall and of chemical heat release. The changes in the explosion limits caused by the enhanced heat-transfer rates associated with buoyant motion are described in an accompanying paper.