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This paper investigates effects of buoyancy-driven motion on the "slowly reacting" mode of combustion, and its thermal-explosion limits, of an initially cold gaseous mixture enclosed in a spherical vessel with a constant wall temperature. As in Frank-Kamenetskii's seminal analysis, the strong temperature dependence of the effective overall reaction is modeled with a single irreversible reaction with an Arrhenius rate having a large activation energy. Besides the classical Damkohler number Da, measuring the ratio of the heat-release rate by chemical reaction evaluated at the wall temperature to the rate of heat removal by heat conduction to the wall, the solution is seen to depend on the Rayleigh number Ra, measuring the effect of buoyancy-induced motion on the heat-transport rate. For values of Da below a critical value Da, the system evolves in a slowly reacting mode where the heat losses to the wall limit the temperature increase associated with the chemical reaction, whereas for Da > Da(c) the initial stage of slow reaction ends abruptly at a well-defined ignition time, at which a thermal runaway occurs. Transient numerical integrations of the initial stage of slow reaction, formulated in the distinguished limit Da 1 and Ra 1 with account taken of the effects of the temporal pressure variation, are used to investigate influences of natural convection on thermal-explosion development, including changes in ignition times for Da > Da(c) and modified explosion curves. Our analysis reveals that Frank-Kamenetskii's criterion for the determination of critical explosion conditions, based on the investigation of existence of steady solutions, provides values of Da(c)(Ra) that are identical to those extracted from the transient computations.
thermal explosions; laminar reacting flows; natural convection; boundary-layer theory