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New flame extinction conditions for the critical mass pyrolysis rate are developed when extinction occurs by interaction of flames with the pyrolyzing surface of a condensed mated The extinction conditions provide the critical mass pyrolysis rate and the corresponding convective heat fhur to the surface. A novel formulation shows that the sum of fuel mass fraction near the surface and the ambient oxygen mass fraction corrected for stoichiometry and incompleteness of combustion is constant. The extinction conditions are derived from simple analysis of combustion and heat transfer, and they are shown to be applicable for various experimental conditions such as fuel dilution by inert gas, oxygen dilution by inert gas, effects of external heat flux, material preheating, transient (charring) pyrolysis, including geometric effects which influence the critical mass pyrolysis rate through an effective heat transfer coefficient. Additional validation of the proposed extinction conditions is provided by numerical simulation reported in the literature in the regime of low straining rates for a stagnation flow on a cylinder. The present approach can be used to obtain the critical extinction conditions from measurements in a standard flammability apparatus.