Combustion waves in a model with chain branching reaction and their stability

In this paper the travelling wave solutions in the adiabatic model with two-step chain branching reaction mechanism are investigated both numerically and analytically in the limit of equal diffusivity of reactant, radicals and heat. The properties of these solutions and their stability are investigated in detail. The behaviour of combustion waves are demonstrated to have similarities with the properties of nonadiabatic one-step combustion waves in that there is a residual amount of fuel left behind the travelling waves and the solutions can exhibit extinction. The difference between the nonadiabatic one-step and adiabatic two-step models is found in the behaviour of the combustion waves near the extinction condition. It is shown that the flame velocity drops down to zero and a standing combustion wave is formed as the extinction condition is reached. Prospects of further work are also discussed.Comment: pages 32, figures 2

Linear stability of planar premixed flames: reactive Navier-Stokes equations with finite activation energy and arbitrary Lewis number

A numerical shooting method for performing linear stability analyses of travelling waves is described and applied to the problem of freely propagating planar premixed flames. Previous linear stability analyses of premixed flames either employ high activation temperature asymptotics or have been performed numerically with finite activation temperature, but either for unit Lewis numbers (which ignores thermal-diffusive effects) or in the limit of small heat release (which ignores hydrodynamic effects). In this paper the full reactive Navier-Stokes equations are used with arbitrary values of the parameters (activation temperature, Lewis number, heat of reaction, Prandtl number), for which both thermal-diffusive and hydrodynamic effects on the instability, and their interactions, are taken into account. Comparisons are made with previous asymptotic and numerical results. For Lewis numbers very close to or above unity, for which hydrodynamic effects caused by thermal expansion are the dominant destablizing mechanism, it is shown that slowly varying flame analyses give qualitatively good but quantitatively poor predictions, and also that the stability is insensitive to the activation temperature. However, for Lewis numbers sufficiently below unity for which thermal-diffusive effects play a major role, the stability of the flame becomes very sensitive to the activation temperature. Indeed, unphysically high activation temperatures are required for the high activation temperature analysis to give quantitatively good predictions at such low Lewis numbers. It is also shown that state-insensitive viscosity has a small destabilizing effect on the cellular instability at low Lewis numbers

Properties of reaction fronts in a non-adiabatic two stage exothermic-endothermic competitive reaction scheme

We numerically derive the properties of reaction fronts arising in a pre-mixed one dimensional two staged non-adiabatic competitive exothermic-endothermic reaction scheme where both reaction pathways compete for the same fuel. We utilise FlexPDE and the method of lines to obtain numerical solutions for properties such as the front speed and stability over a range of parameter values such as the Lewis number and the ratios of enthalpies and activation energies. Steady and pulsating speeds are demonstrated for specific regions of the parameter space. We also show that in some circumstances there exists a chaotic regime of combustion wave propagation. References M. J. Antal and G. Varhegyi, Cellulose Pyrolysis Kinetics - The Current State of Knowledge, Industrial and Engineering Chemistry Research, 34(3):703–717, 1995 doi:10.1021/ie00042a001 R. Ball, A. C. McIntosh and J. Brindley, Thermokinetic Models for Simultaneous Reactions: a Comparative Study, Combustion Theory and Modelling, 3(3):447–468, 1999. doi:10.1088/1364-7830/3/3/302 S. K. Chan and R. Turcotte, Onset Temperatures in Hot Wire Ignition of AN-Based Emulsions, Propellants Explosives Pyrotechnics, 34(1):41–49, 2009. doi:10.1002/prep.200700288 FlexPDE$^{TM}$, PDE Solutions Inc, http://www.pdesolutions.com. V. V. Gubernov, A. Kobolov, A. Polezhaev and H. Sidhu, Period Doubling and Chaotic Transient in a Model of Chain-Branching Combustion Wave Propagation, Proceedings of the Royal Society A, 2011. doi:10.1098/rspa.2009.0668 V. V. Gubernov, J. J. Sharples, H. S. Sidhu, A. C. McIntosh and J. Brindley, Properties of Combustion Waves in the Model with Competitive Exo- and Endothermic Reactions, Journal of Mathematical Chemistry, 50(8):2130–2140, 2012. doi:10.1007/s10910-012-0021-y V. V. Gubernov, H. S. Sidhu, G. N. Mercer, The Effect of Ambient Temperature on the Propagation of Nonadiabatic Combustion Waves, Journal of Mathematical of Mathematical Chemistry, 37(2):149–162, 2005. doi:10.1007/s10910-004-1447-7 A. Hmaidi, A. C. McIntosh and J. Brindley, A Mathematical Model of Hotspot Condensed Phase Ignition in the Presence of a Competitive Endothermic Reaction, Combustion Theory and Modelling, 14(6):893–920, 2010. doi:10.1080/13647830.2010.519050 D. A. Kessler, V. N. Gamezo and E. S. Oran, Simulations of Flame Acceleration and Deflagration-to-Detonation Transitions in Methane-Air Systems, Combustion and Flame, 157(11):2063–2077, 2010. doi:10.1016/j.combustflame.2010.04.011 F. Liu, D. L. S. McElwain and C. P. Please, Simulation of Combustion Waves for Two-Stage Reactions, Proceedings of the 8th Biennial Computational Techniques and Applications Conference (CTAC97), pages 385–392, 1998. A. Makino, Fundamental Aspects of the Heterogeneous Flame in the Self-Propagating High-Temperature Synthesis (SHS) Process, Progress in Energy and Combustion Sciences, 27(1):1–74, 2001. doi:10.1016/S0360-1285(00)00004-6 A. G. Merzhanov and E. N. Rumanov, Physics of Reaction Waves, Reviews of Modern Physics, 71(4):1173–1211, 1999. doi:10.1103/RevModPhys.71.1173 C. P. Please, F. Liu and D. L. S. McElwain, Condensed Phase Combustion Travelling Waves with Sequential Exothermic or Endothermic Reactions, Combustion Theory and Modelling, 7(1):129–143, 2003. doi:10.1088/1364-7830/7/1/307 W. E. Schiesser, The numerical method of lines: Integration of Partial Differential Equations, Academic Press, Inc, 1991. J. J. Sharples, H. S. Sidhu, A. C. Mcintosh, J. Brindley and V. V. Gubernov, Analysis of Combustion Waves Arising in the Presence of a Competitive Endothermic Reaction, IMA Journal of Applied Mathematics, 77(1):18–31, 2012. doi:10.1093/imamat/hxr072 V. P. Sinditskii, V. Y. Egorshev, A. I. Levshenkov and V. V. Serushkin, Ammonium nitrate: Combustion Mechanism and the Role of Additives, Propellants Explosives Pyrotechnics, 30(4):269–280, 2005. doi:10.1002/prep.200500017 J. Subrahmanyam and M. Vijayakumar, Self-Propagating High-Temperature Synthesis, Journal of Materials Science, 27(23):6249–6273, 1992. doi:10.1007/BF00576271 R. Turcotte, S. Goldthorp, C. M. Badeen and S. K. Chan, Hot-Wire Ignition of AN-Based Emulsions, Propellants Explosives Pyrotechnics, 33(6):472–481, 2008. doi:10.1002/prep.200700276 S. Walia, R. O. Weber, K. Latham, P. Petersen, J. T. Abrahamson, M. S. Strano, and K. Kalantar-zadeh, Oscillatory Thermopower Waves Based on Bi$_2$Te$_3$ Films, Advanced Functional Materials, 21(11):2072–2079, 2011. doi:10.1002/adfm.201001979 R. O. Weber, G. N. Mercer, H. S. Sidhu and B. F. Gray, Combustion Waves for Gases ($Le=1$) and Solids ($Le\rightarrow \infty$), Proceedings of the Royal Society of London A, 453(1960):1105–1118, 1997. doi:10.1098/rspa.1997.0062 W. Y. S. Wee, J. J. Sharples, H. S. Sidhu and V. V. Gubernov, Analysis of a Two-Stage Competitive Endothermic-Exothermic Reaction Scheme, Proceedings of the 40th Australian Chemical Engineering Conference (CHEMECA 2012), submitted June 2012

A wildland fire model with data assimilation

A wildfire model is formulated based on balance equations for energy and fuel, where the fuel loss due to combustion corresponds to the fuel reaction rate. The resulting coupled partial differential equations have coefficients that can be approximated from prior measurements of wildfires. An ensemble Kalman filter technique with regularization is then used to assimilate temperatures measured at selected points into running wildfire simulations. The assimilation technique is able to modify the simulations to track the measurements correctly even if the simulations were started with an erroneous ignition location that is quite far away from the correct one.Comment: 35 pages, 12 figures; minor revision January 2008. Original version available from http://www-math.cudenver.edu/ccm/report

Numerical methods for the travelling wave solutions in reaction-diffusion equations

In this work we consider how shooting and relaxation methods can be used to investigate propagating waves solutions of PDEs. Particular attention is paid to overcoming some of the numerical difficulties. The linear stability of these solutions are analyzed by using the Evans function approach. As an illustration, we shall apply the above methods to an autocatalytic reaction involving two diffusing chemicals in one spatial dimension. Prospects of further work are also discussed

Front propagation in a phase field model with phase-dependent heat absorption

Copyright © 2006 Elsevier. NOTICE: This is the author’s version of a work accepted for publication by Elsevier. Changes resulting from the publishing process, including peer review, editing, corrections, structural formatting and other quality control mechanisms, may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Physica D, Vol 215, Issue 2, 2006, DOI: 10.1016/j.physd.2006.01.024We present a model for the spatio-temporal behaviour of films exposed to radiative heating, where the film can change reversibly between amorphous (glassy) and crystalline states. Such phase change materials are used extensively in read-write optical disk technology. In cases where the heat absorption of the crystal phase is less than that in the amorphous state we find that there is a bi-stability of the phases. We investigate the spatial behaviours that are a consequence of this property and use a phase field model for the spatio-temporal dynamics in which the phase variable is coupled to a suitable temperature field. It is shown that travelling wave solutions of the system are possible and, depending on the precise system parameters, these waves can take a range of forms and velocities. Some examples of possible dynamical behaviours are discussed and we show, in particular, that the waves may collide and annihilate. The longitudinal and transverse stability of the travelling waves are examined using an Evans function method which suggests that the fronts are stable structures

Analysing instability of combustion waves using the Evans function

We consider travelling wave solutions of a reaction-diffusion system corresponding to a single step, homogeneous, premixed combustion scheme with Newtonian heat loss and general Lewis number. Particular attention is paid to unstable combustion wave regimes, especially those associated with oscillatory behaviour. The instability analysis is conducted with the use of Evans function techniques, which we use to derive eigenvalues of the linear stability problem via the argument principle and Nyquist plots. These techniques permit the study of transitions to different modes of unstable behaviour in great detail. Threshold values of the parameters corresponding to Hopf and Bogdanov--Takens bifurcation are established and it is shown that for certain parameter values the system exhibits a period doubling route to chaotic behaviour. References Afendikov, A. L. and Bridges, T. J. Instability of the Hocking--Stewartson pulse and its implications for three-dimensional Poiseuille flow. P. R. Soc. A, 457:1--16, 2001. doi:10.1098/rspa.2000.0665 Alexander, J., Gardner, R. and Jones, C. A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math., 410:167--212, 1990. Dold, J. W. Premixed flames modelled with thermally sensitive intermediate branching kinetics. Combust. Theor. Model., 11:909--948, 2007. doi:10.1080/13647830701294599 Dold, J. W., and Zinoviev, A. Fire eruption through intensity and spread rate interaction mediated by flow attachment. Combust. Theor. Model., 13:763--793, 2009. doi:10.1080/13647830902977570 FlexPDE. http://www.pdesolutions.com Gubernov, V. V., Mercer, G. N., Sidhu, H. S. and Weber, R. O. Evans function stability of combustion waves. SIAM J. Appl. Math., 63:1259--1275, 2003. http://www.siam.org/journals/siap/63-4/40024.html Gubernov, V. V., Mercer, G. N., Sidhu, H. S. and Weber, R. O. Evans function stability of non-adiabatic combustion waves. P. R. Soc. A, 460:2415--2435, 2004. doi:10.1098/rspa.2004.1285 Gubernov, V. V., Sidhu, H. S. and Mercer, G. N.. Generalized compound matrix method. Appl. Math. Lett., 19:458--463, 2006. doi:10.1016/j.aml.2005.07.002 Gubernov, V. V, Kolobov, A. V., Polezhaev, A. A., Sidhu, H. S. and Mercer, G. N. Period doubling and chaotic transient in a model of chain-branching combustion wave propagation. Proc. Roy. Soc. Lond. A, 466:2747--2769, 2010. doi:10.1098/rspa.2009.0668 Makino, A. Fundamental aspects of the heterogeneous flame in the self-propagating high-temperature synthesis (SHS) process. Prog. Energ. Combust., 27:1--74, 2001. Merzhanov, A. G., and Rumanov, E. N. Physics of reaction waves. Rev. Mod. Phys., 71:1173--1211, 1999. Pego, R. L., Smereka, P. and Weinstein, M. I. Oscillatory instability of travelling waves for a KdV-Burgers equation. Physica D, 67:45--65, 1993. http://www.sciencedirect.com/science/journal/01672789 Sandstede, B. Stability of travelling waves. In B. Fiedler, editor, Handbook of Dynamical Systems II, pp. 983--1055. Elsevier, 2002. Sharples, J. J., Sidhu, H. S. and Gubernov, V. V. Properties of nonadiabatic premixed combustion fronts arising in single-step reaction schemes. In Proceedings of Chemeca 2010, Paper No. 357, 26--29 September 2010, Adelaide. ISBN 978 085 825 9713. Thomas, P. H., Bullen, M. L., Quintiere, J. G. and McCaffrey, B. J. Flashover and instabilities in fire behaviour. Combust. Flame, 38:159--171, 1980. http://www.sciencedirect.com/science/article/pii/0010218080900486 Weber, R. O., G. N. Mercer, H. S. Sidhu, and Gray, B. F. Combustion waves for gases (Le$=1$) and solids (Le$=\infty$). Proc. Roy. Soc. Lond. A, 453:1105--1118, 1997. doi:10.1098/rspa.1997.006