Hostname: page-component-cc8bf7c57-xrnlw Total loading time: 0 Render date: 2024-12-12T01:54:26.193Z Has data issue: false hasContentIssue false
  • English
  • Français

A combustion wave of permanent form in a compressible gas

Published online by Cambridge University Press:  17 February 2009

Lawrence K. Forbes  and
William Derrick
Lawrence K. Forbes
Affiliation:
School of Mathematics and Physics, The University of Tasmania, Hobart TAS 7001, Australia.
William Derrick
Affiliation:
Department of Mathematics, University of Montana, Missoula, Montana 59812, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A model is presented for describing the propagation of a one-dimensional wave of permanent form in a compressible gas in a pipe. Energy is lost to the system through the walls of the pipe, but the combustion wave produces heat through an exothermic chemical reaction. The full set of equations for the model is reduced to a phase-plane system, and it is shown that, for small amplitude waves, a weakly non-linear analysis leads to a temperature profile that is a classical solitary wave. A novel shooting method is developed for the full non-linear problem, and this confirms and extends the solitary-wave solution, up to a value of the temperature amplitude at which the wave begins to develop a shock. The jump conditions across the shock are presented, and numerical integration is used to continue the solution for temperature amplitudes at which a shock is present. An example is given of the extreme situation in which the shock is so strong that all the fuel behind the shock is exhausted.

Type
Research Article
Information
The ANZIAM Journal , Volume 43 , Issue 1 , July 2001 , pp. 35 - 58
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Bi, H. and Agrawal, A. K., “Study of autoignition of natural gas in diesel environments using computational fluid dynamics with detailed chemical kinetics”, Combust. Flame 113 (1998) 289–302.CrossRefGoogle Scholar
[2]Buckmaster, J. D., The mathematics of combustion, Frontiers in Appl. Math. (SIAM, Philadelphia, 1985).CrossRefGoogle Scholar
[3]Clarke, J. F., “On changes in the structure of steady plane flames as their speed increases”, Combust. Flame 50 (1983) 125–138.CrossRefGoogle Scholar
[4]Coppersthwaite, D. P., Griffiths, J. F. and Gray, B. F., “Oscillations in the H2 + Cl2 reaction: Experimental measurements and numerical simulation”, J. Phys. Chem. 95 (1991) 6961–6967.CrossRefGoogle Scholar
[5]Forbes, L. K., “Limit-cycle behaviour in a model chemical reaction: the Sal'nikov thermokinetic oscillator”, Proc. Roy. Soc. London Sen A 430 (1990) 641–651.Google Scholar
[6]Forbes, L. K., “One-dimensional pattern formation in a model of burning”, J. Austral. Math. Soc. Ser. B 35 (1993) 145–173.CrossRefGoogle Scholar
[7]Forbes, L. K., “Stationary circular target patterns in a surface burning reaction”, J. Engin. Math. 30 (1996) 471–486.CrossRefGoogle Scholar
[8]Forbes, L. K., “A two-dimensional model for large-scale bushfire spread”, J. Austral. Math. Soc. Ser. B 39 (1997) 171–194.CrossRefGoogle Scholar
[9]Forbes, L. K., Myerscough, M. R. and Gray, B. F., “On the presence of limit-cycles in a model exothermic chemical reaction: Sal'nikov's oscillator with two temperature-dependent reaction rates”, Proc. Roy. Soc. London Ser. A 435 (1991) 591–604.Google Scholar
[10]Gray, B. F. and Forbes, L. K., “Analysis of chemical kinetic systems over the entire parameter space IV. The Sal'nikov oscillator with two temperature-dependent reaction rates”, Proc. Roy. Soc. London Ser. A 443 (1994) 621–642.Google Scholar
[11]Gray, B. F. and Roberts, M. J., “An asymptotic analysis of the Sal'nikov thermokinetic oscillator”, Proc. Roy. Soc. London Ser. A 416 (1988) 425–441.Google Scholar
[12]Gray, P. and Griffiths, J. F., “Thermokinetic combustion oscillations as an alternative to thermal explosion”, Combust. Flame 78 (1989) 87–98.CrossRefGoogle Scholar
[13]Gray, P. and Kordylewski, W., “Travelling waves in exothermic systems”, Proc. Roy. Soc. London Ser. A 416 (1988) 103–113.Google Scholar
[14]Gray, P. and Scott, S. K., Chemical oscillations and instabilities. Non-linear chemical kinetics (Clarendon Press, Oxford, 1990).CrossRefGoogle Scholar
[15]Griffiths, J. F., “The fundamentals of spontaneous ignition of gaseous hydrocarbons and related organic compounds”, Adv. Chem. Phys. 64 (1986) 203–304.Google Scholar
[16]Kay, S. R. and Scott, S. K., “Oscillations of simple exothermic reactions in a closed system II. Exact Arrhenius kinetics”, Proc. Roy. Soc. London Ser. A 416 (1988) 343–359.Google Scholar
[17]Liepmann, H. W. and Roshko, A., Elements of gasdynamics (John Wiley and Sons, New York, 1957).CrossRefGoogle Scholar
[18]Murray, J. D., Mathematical biology, Biomathematics Texts 19 (Springer, Berlin, 1989).CrossRefGoogle Scholar
[19]Norbury, J. and Stuart, A.M., “Travelling combustion waves in a porous medium. Part I—Existence”, SIAM J. Appl. Math. 48 (1988) 155–169.CrossRefGoogle Scholar
[20]Sal'nikov, I. Ye., “Contribution to the theory of the periodic homogeneous chemical reactions”, Zh. Fiz. Khim. 23 (1949) 258–272.Google Scholar
[21]Tam, K. K., “Traveling wave solutions for a combustion problem”, Stud. Appl. Math. 81 (1989) 117–124.CrossRefGoogle Scholar
[22]Weber, R. O., Mercer, G. N., Sidhu, H. S. and Gray, B. F., “Combustion waves for gases (Le = 1) and solids (Le → ∞)”, Proc. Roy. Soc. London Ser. A 453 (1997) 1105–1118.CrossRefGoogle Scholar
[23]Whitham, G. B., Linear and nonlinear waves (Wiley, New York, 1974).Google Scholar
[24]Williams, F. A., Combustion theory (Addison-Wesley, Massachusetts, 1965).Google Scholar