Critical condition relevant to the extinction of a carbon particle in the course of combustion

In order to examine extinction of carbon particle in the course of combustion, an attempt has been made to obtain its critical condition. Main concern has been put on the particle extinction in the lower limit that can occur at the end of the particle combustion although not only the lower limit but also the upper limit of the critical conditions has been obtained. By conducting asymptotics, with focusing on the temporal variation of the particle temperature, an analytical expression has been obtained for the limit of the particle diameter, as functions of the pressure ratio, oxygen mass-fraction, ambient temperature, and/or radiative heat flux. An approximate expression is also obtained from the analytical solution. It is found that the approximate expression can fairly represent the limits. In addition, use has been made of the Arrhenius plot of a comprehensive parameter, consisting of the particle diameter and pressure ratio, in order to identify regions for the particle combustion sustained, which are next to those for the particle extinction. Comparisons have also been conducted by use of experimental data in the literature, with presenting a fair degree of agreement, as far as the trend and approximate magnitude are concerned. It has been confirmed that the formulation has captured the essential feature, that the reduction in particle size does not necessarily favor the particle combustion, and that the particle extinction can occur when the particle diameter is reduced to the critical value.