We investigate travelling premixed reaction waves in a diffusional-thermal model with a two-step competitive reaction mechanism where both reactions are exothermic. Travelling waves are assumed to propagate at constant speed. An approximation of the Arrhenius reaction rate is adopted to simplify the combustion model. Based on this assumption, an asymptotic theory is presented for solid fuels under adiabatic conditions. This approach provides a convenient way to analyse the system in the phase plane. The asymptotic speeds for the flame fronts are compared with numerical solutions by solving the governing partial differential equations. In addition, piecewise approximate solutions for the temperature and fuel mass fraction profiles are presented and compared with those obtained numerically. Our results can be applied to combustion synthesis in the production of advanced materials. References Martirosyan, N. A., Dolukhanyan, S. K., and Merzhanov, A. G., Experimental observation of the nonuniqueness of stationary combustion in systems with parallel reactions. Combust. Explos. Shock Waves 19(6):711–712, 1983. doi:10.1007/BF00750777 Martirosyan, N. A., Dolukhanyan, S. K., and Merzhanov, A. G., Nonuniqueness of stationary states in combustion of mixtures of zirconium and soot powders in hydrogen. Combust. Explos. Shock Waves 19(5):569–571, 1983. doi:10.1007/BF00750423 Sidhu, H. S., Towers, I. N., Gubernov, V. V., Kolobov, A. V., and Polezhaev, A. A., Investigation of flame propagation in a model with competing exothermic reactions. Chemeca 2013 , Brisbane, Australia, 29 September–2 October, 2013. http://www.conference.net.au/chemeca2013/papers/29350.pdf Towers, I. N., Gubernov, V. V., Kolobov, A. V., Polezhaev, A. A., and Sidhu, H. S., Bistability of flame propagation in a model with competing exothermic reactions. P. R. Soc. Lond. A Mat. 469:2158, 2013. doi:10.1098/rspa.2013.0315 Hmaidi, A., McIntosh, A. C., and Brindley, J., A mathematical model of hotspot condensed phase ignition in the presence of a competitive endothermic reaction. Combust. Theory Model. 14(6):893–920, 2010. doi:10.1080/13647830.2010.519050 Matkowsky, B. J., and Sivashinsky, G. I., Propagation of a pulsating reaction front in solid fuel combustion. SIAM J. Appl. Math. 35(3):465–478, 1978. doi:10.1137/0135038 Forbes, L. K., A note on travelling waves in competitive reaction systems. ANZIAM J. 55(1):1–13, 2013. doi:10.1017/S1446181113000333 Barenblatt, G., The Mathematical theory of combustion and explosions . Springer, 1985. http://www.springer.com/us/book/9781461294399 Jovanoski, Z., and Robinson, G., Piecewise linear approximation to Fisher's equation. ANZIAM J. 53:C465–C477, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/5129 Schiesser, W. E., The numerical method of lines: integration of partial differential equations . Academic Press, 1991.
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