Investigation of unsteady behaviors of forward and opposed flow combustion of solid fuel

We theoretically examine the characteristics of a combustion wave of solid fuel in a porous medium. We consider two problems (i) a sample is ignited at one of the either ends and (ii) a sample is ignited around the middle. Depending on the ignition process, forward and opposed modes of a combustion wave are found in the first problem while they simultaneously occurred in the second problem. Firstly, we make a comparative study between forward and opposed modes and secondly, we investigate the combined effects of these modes. Analytical expressions of combustion characters, namely, the moving speed of the reaction front and the spatial temperature and species profiles of the solid combustible, are obtained employing large activation energy asymptotics. It is found that the reaction fronts of both forward and opposed flows demonstrate unsteady behaviors for a higher thermal conductivity porous medium, a lower initial mass fraction of oxidizer and a lower gas flow velocity. The response of the moving speed of the reaction front of the forward mode to an increasing heat transfer coefficient is completely opposite to that of the opposed mode. In particular, the reaction front of the forward (opposed) flow becomes unsteady for lower (higher) heat transfer coefficient. However, for the middle ignition of a sample, the moving speed of the reaction front of the forward mode is found to monotonically increase with an increase of heat transfer coefficient, while in opposed mode it first increases and then decreases. The results also provide a remarkable finding that an “unsteady solution region” occurs in between the critical heat transfer coefficients of forward and opposed modes where the reaction fronts of both modes become unsteady. Moreover, the present solutions have been compared with previous numerical and experimental observations that show a qualitative agreement.