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A comparison of two- and three-variable models for combustion in sealed containers

Published online by Cambridge University Press:  17 February 2009

Claire E. Trenham  and
Larry K. Forbes
Claire E. Trenham
Affiliation:
School of Mathematics and Physics, University of Tasmania, G.P.O. Box 252–37, Hobart 7001 TAS, Australia; e-mail: [email protected].
Larry K. Forbes
Affiliation:
School of Mathematics and Physics, University of Tasmania, G.P.O. Box 252–37, Hobart 7001 TAS, Australia; e-mail: [email protected].
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Abstract

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This paper analyses a model for combustion of a self-heating chemical (such as pool chlorine), stored in drums within a shipping container. The system is described by three coupled nonlinear differential equations for the concentration of the chemical, its temperature and the temperature within the shipping container. Self-sustained oscillations are found to occur, as a result of Hopf bifurcation. Temperature and concentration profiles are presented and compared with the predictions of a simpler two-variable approximation for the system. We study the period of oscillation and its variation with respect to the ambient temperature and the reaction parameter. Nonlinear resonances are found to exist, as the solution jumps between branches having different periods.

Keywords

combustion modelslimit cyclesnonlinear resonancesrelaxation oscillations.
Type
Research Article
Information
The ANZIAM Journal , Volume 47 , Issue 4 , April 2006 , pp. 527 - 540
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Abramowitz, M. and Stegun, I. A. (eds.), Handbook of Mathematical Functions (Dover, New York, 1972).Google Scholar
[2]Arnold, L., Namachchivaya, N. S. and Schenk-Hoppe, K. R., “Toward an understanding of stochastic Hopf bifurcation: A case study”, Internat. J. Bifur. Chaos Appl. Sci. Engng 6 (1996) 1947–1975.CrossRefGoogle Scholar
[3]Atkins, P. W., The Elements of Physical Chemistry, 3rd ed. (Oxford University Press, Oxford, 2000).Google Scholar
[4]Bowes, P. C., Self heating: evaluating and controlling the hazards (H. M. Stationery Office, 1984).Google Scholar
[5]Ebert, K. H., Deuflhard, P. and Jäger, W. (eds.), Modelling of Chemical Reaction Systems, Springer Series in Chemical Physics 18, Ch. 12 and others, (Springer, Berlin, 1981).CrossRefGoogle Scholar
[6]Forbes, L. K., “Limit-cycle behaviour in a model chemical reaction: the Sal'nikov thermokinetic oscillator”, Proc. Roy. Soc. Lond. Ser A Math. Phys. Eng. Sci. 430 (1990) 641–651.Google Scholar
[7]Forbes, L. K., “One-dimensional pattern formation in a model of burning”, J. Austral. Math. Soc. Ser. B 35 (1993) 145–173.CrossRefGoogle Scholar
[8]Forbes, L. K., “Stationary circular target patterns in a surface burning reaction”, J. Engng Math. 30 (1996) 471–486.CrossRefGoogle Scholar
[9]Forbes, L. K. and Gray, B. F., “Forced oscillations in an exothermic chemical reaction”, Dyn. Stab. Syst. 9 (1994) 253–269.Google Scholar
[10]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. Lond. Ser. A Math. Phys. Eng. Sci. 435 (1991) 591–604.Google Scholar
[11]Glasstone, S., Laidler, K. J. and Eyring, H., The Theory of Rate Processes, 1st ed. (McGraw-Hill, New York, 1941).Google Scholar
[12]Gray, B. F., “On the critical conditions for an assembly of interacting thermons”, ANZIAM J. 43 (2001) 1–11.CrossRefGoogle Scholar
[13]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. Lond. Ser A Math. Phys. Eng. Sci. 444(1994) 621–642.Google Scholar
[14]Gray, B. F. and Roberts, M. J., “Analysis of chemical kinetic systems over the entire parameter space. I. The Sal'nikov thermokinetic oscillator”, Proc. Roy. Soc. Lond. Ser A Math. Phys. Eng. Sci. 416 (1988) 391–402.Google Scholar
[15]Gray, B. F. and Roberts, M. J., “An asymptotic analysis of the Sal'nikov thermokinetic oscillator”, Proc. Roy. Soc. Lond. Ser A Math. Phys. Eng. Sci. 416 (1988) 425–441.Google Scholar
[16]Gray, P., Kay, S. R. and Scott, S. K., “Oscillations of an exothermic reaction in a closed system. I. Approximate (exponential) representation of Arrhenius temperature-dependence”, Proc. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 416 (1988) 321–341.Google Scholar
[17]Gray, P. and Scott, S. K., Chemical oscillations and instabilities: non-linear chemical kinetics (Oxford University Press, Oxford, 1990).CrossRefGoogle Scholar
[18]Guckenheimer, J. and Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer Applied Mathematical Sciences 42 (Springer, New York, 2002).Google Scholar
[19]Hopf, E., “Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems”, Ber Verh. Sachs. Akad. Wiss. Leipzig Math.-Nat. 94 (1942) 3–22.Google Scholar
[20]Kay, S. R. and Scott, S. K., “Oscillations of simple exothermic reactions in a closed system. II. Exact Arrhenius kinetics”, Proc. Roy. Soc. Lond. Ser A Math. Phys. Eng. Sci. 416 (1988) 343–359.Google Scholar
[21]Merkin, J. H., Needham, D. J. and Scott, S. K., “Oscillatory chemical reactions in closed vessels”, Proc. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 406 (1986) 299–323.Google Scholar
[22]Nelson, M., “Mechanisms for hydrocarbon oxidation—The Sal'nikov thermokinetic oscillator”, Short articles on combustion, Dec 21 2002, available at: http://www.uow.edu.au/~mnelson/review.dir/salnikov.html.Google Scholar
[23]Perko, L., Differential Equations and Dynamical Systems (Springer, New York, 1991).CrossRefGoogle Scholar
[24]Sal'nikov, I. Y., “A thermokinetic model of homogeneous periodic reactions”, Dokl. Akad. Nauk SSSR 60 (1948) 405–408, Russian.Google Scholar
[25]Sal'nikov, I. Y., “Contribution to the theory of the periodic homogenous chemical reactions: II A thermokinetic self-excited oscillating model”, Zh. Fiz. Khim. 23 (1949) 258–272, Russian.Google Scholar
[26]Scott, S. K., Chemical Chaos (Oxford Science Publications, Clarendon Press, Oxford, 1991).CrossRefGoogle Scholar
[27]Sexton, M. J. and Forbes, L. K., “A note on oscillation in a simple model of a chemical reaction”, J. Austral. Math. Soc. Ser. B 37 (1996) 451–457.CrossRefGoogle Scholar
[28]Sidhu, H. S., Sexton, M. J., Nelson, M. I., Mercer, G. N. and Weber, R. O., “A simple combustion process in a semibatch reactor”, in EMAC 2000 Proceedings (eds. May, , Fitz-Gerald, and Grundy, ), (Institution of Engineers, Australia, 2000) 251–254.Google Scholar