In this paper, we explore the effect of oxygen supply on the conditions necessary to sustain a self-propagating front from a spherical source of heat embedded in a much larger volume of solid. The ignition characteristics for a spherical hot-spot are investigated, where the reaction is limited by oxygen, that is, reactant + oxygen ? product. It is found that over a wide range of realistic oxygen supply levels, constant heating of the solid by the hot-spot results in a self-propagating combustion front above a certain critical hot-spot power; this is clearly an important issue for industries in which hazard prevention is important. The ignition event leading to the formation of this combustion wave involves an extremely sensitive balance between the heat generated by the chemical reaction and the depletion of the reactant. As a result, for small hot-spot radii and infinite oxygen supply, not only is there a critical power above which a self-sustained combustion front is initiated there also exists a power beyond which no front is formed, before a second higher critical power is found. The plot of critical power against hot-spot radius thus takes on a Z-shape appearance. The corresponding shape for the oxygen-limited reaction is qualitatively the same when the ratio of solid thermal diffusion to oxygen mass diffusion (N) is small and we establish critical conditions for the initiation of a self-sustained combustion front in that case. As N gets larger, while still below unity, we show that the Z-shape flattens out. At still larger values of N, the supercritical behaviour becomes increasingly difficult to define and is supplanted by burning that depends more uniformly on power. In other words, the transition from slow burning to complete combustion seen at small values of N for some critical power disappears. Even higher values of N lead to less solid burning at fixed values of power
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