We consider travelling front solutions of a one-dimensional reaction-diffusion system corresponding to two-stage competitively exothermic reactions. We suppose all reactions occurring during the combustion may be lumped together as two different paths. Both exothermic reactions compete for the same reactant. Properties of travelling wave fronts, particularly flame speed, are determined numerically by solving the governing partial differential equations. The flame speed is analysed for different values of the heat loss parameter. It is demonstrated that, as the heat loss coefficient increases, the flame speed decays gradually until the front ceases to exist due to insufficient energy being available to sustain the flame front. Earlier studies for the adiabatic case showed the existence of bi-stability (fast and slow waves co-exist for the same parameter values). We study how heat loss affects the size of the bi-stable region. Furthermore, we investigate how the extinction limit depends on the heat loss parameter as well as the parameter representing the ratio of the activation energy to the heat release of the second reaction. Numerical solutions show that there is no travelling front when these parameters are above threshold values. The dependence of flame speed on the temperature profile is also investigated. The bi-stability phenomenon is demonstrated by perturbing the temperature profile. References W. Choi, S. Hong, J. T. Abrahamson, J.-H. Han, C. Song, N. Nair, S. Baik and M. S. Strano. Chemically driven carbon-nanotube-guided thermopower waves. Nat. Mater. 9:423–429, 2010. doi:10.1038/nmat2714 A. G. Merzhanov. Combustion and explosion processes in physical chemistry and technology of inorganic materials. Russ. Chem. Rev. 72:289–310, 2003. doi:10.1070/rc2003v072n04abeh000766 J. K. Bechtold and C. K. Law. The structure of premixed methane-air flames with large activation energy. Combust. Flame 97:317–338, 1994. doi:10.1016/0010-2180(94)90024-8 K. Seshadri, N. Peters and F. A. Williams. Asymptotic analysis of stoichiometric and lean hydrogen-air flames. Combust. Flame 96:407–427, 1994. doi:10.1016/0010-2180(94)90108-2 A. L. Sanchez, A. Lepinette, M. Bollig, A. Linan and B. Lazaro. The reduced kinetic description of lean premixed combustion. Combust. Flame 123:436–464, 2000. doi:10.1016/S0010-2180(00)00177-2 A. L. Sanchez, G. Balakrishnan, A. Linan and F. A. Williams. Relationships between bifurcation and numerical analyses for ignition of hydrogen-air diffusion flames. Combust. Flame 105:569–590, 1996. doi:10.1016/0010-2180(95)00241-3 V. V. Gubernov, A. V. Kolobov, A. A. Polezhaev and H. S. Sidhu. Analysing the stability of premixed rich hydrogen-air flame with the use of two-step models. Combust. Flame 160:1060–1069, 2013. doi:10.1016/j.combustflame.2013.01.021 R. Ball, A. McIntosh and J. Brindley. Thermokinetic models for simultaneous reactions: a comparative study. Combust. Theor. Model. 3:447–468, 1999. doi:10.1088/1364-7830/3/3/302 N. A. Martirosyan, S. K. Dolukhanyan and A. G. Merzhanov. Nonuniqueness of stationary states in combustion of mixtures of zirconium and soot powders in hydrogen. Combust. Explo. Shock 19:569–571, 1983. doi:10.1007/BF00750423 H. S. Sidhu, I. N. Towers, V. V. Gubernov, A. V. Kolobov and A. V. Polezhaev. 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 I. N. Towers, V. V. Gubernov, A. V. Kolobov, A. V. Polezhaev and H. S. Sidhu. Bistability of flame propagation in a model with competing exothermic reactions. Proc. R. Soc. Lond. A 469:20130315, 2013. doi:10.1098/rspa.2013.0315 L. K. Forbes and W. Derrick. A combustion wave of permanent form in a compressible gas. Anziam J. 43:35–58, 2001. doi:10.1017/S144618110001141X Z. Huang, H. S. Sidhu, I. N. Towers, Z. Jovanoski and V. V. Gubernov. Investigating flame fronts in competitive exothermic reactions. MODSIM2015 , 21st International congress on Modelling and simulation, Gold coast, Australia, 29 November–4 December, 2015. http://www.mssanz.org.au/modsim2015/A1/huang.pdf F. A. Williams. Combustion theory . Westview Press, 1985. https://westviewpress.com/?s=Combustion+theory FlexPDE\(^TM\), PDE Solutions Inc. http://www.pdesolutions.com W. E. Schiesser. The numerical method of lines: integration of partial differential equations . Academic Press, San Diego, 1991. http://store.elsevier.com/The-Numerical-Method-of-Lines/isbn-9780128015513/ A. C. McIntosh, R. O. Weber and G. N. Mercer. Non-adiabatic combustion waves for general Lewis numbers: wave speed and extinction conditions. Anziam J. 46:1–16, 2004. doi:10.1017/S1446181100013638 A. R. Hall and H. G. Wolfhard. Multiple reaction zones in low pressure flames with ethyl and methyl nitrate, methyl nitrite and nitromethane. Proc. Combust. Inst , 6:190–199, 1957. doi:10.1016/S0082-0784(57)80029-0
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