### Reaction waves in solid fuels for adiabatic competitive exothermic reactions

#### Abstract

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.

#### Keywords

competitive exothermic reactions, reactive waves, phase plane, piecewise linear approximation.

DOI: http://dx.doi.org/10.21914/anziamj.v56i0.9343

**Remember,**for most actions you have to record/upload into this online system

**and then**inform the editor/author via clicking on an email icon or Completion button.

ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.