Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-03T02:36:40.031Z Has data issue: false hasContentIssue false
  • English
  • Français

Slowly varying filtration combustion waves†

Published online by Cambridge University Press:  26 September 2008

M. R. Booty  and
B. J. Matkowsky
M. R. Booty
Affiliation:
Department of Mathematics, Southern Methodist University, Dallas, TX 75275, USA
B. J. Matkowsky
Affiliation:
Department of Engineering Science and Applied Mathematics, Northwestern University, Evanston, IL 60208, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We describe the slow evolution of the wave speed and reaction temperature in a model of filtration combustion. In the counterflow configuration of the process, a porous solid matrix is converted to a porous solid product by injecting an oxidizing gas at high pressure into one end of a fresh sample of the solid while igniting it at the other end. The solid and gas react exothermically at high activation energy and, under favourable conditions, a self-sustaining combustion wave travels along the sample, converting reactants to product. Since the reaction rate depends on the gas pressure p in the pores, small gradients in p cause variations in the conditions of combustion, which, in turn, cause inhomogeneities in the physical properties of the product. We determine the slow evolution of the wave speed, the reaction temperature, and the mass flux of the gas downstream of the reaction zone. In the absence of a pressure gradient, there is a branch of steadily propagating solutions which has a fold. For planar disturbances on the slow time scale, we show that the middle part of the branch is unstable, with the change of stability occurring at the turning points of the branch. When the pressure gradient is nonzero, there are no steadily propagating solutions and the wave continually evolves. Conditions on the state of the gas at the inlet are described such that the variation in the wave speed and reaction temperature throughout the process can be minimized.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 4 , Issue 2 , June 1993 , pp. 205 - 224
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aldushin, A. P., Martemyanova, T. M., Merzhanov, A. G., Khaikin, B. I. & Shkadinsky, K. G. 1972 Propagation of the front of an exothermic reaction in condensed mixtures with the interaction of the components through a layer of high-melting product. Combust., Explosion and Shock Waves 8(2), 159–167.Google Scholar
Aldushin, A. P., Merzhanov, A. G. & Khaikin, B. I. 1974 Conditions for the layer filtration combustion of porous metals. Dokl. Phys. Chem. 215(3), 295–298.Google Scholar
Aldushin, A. P., Seplyarskii, B. S. & Shkadinskii, K. G. 1980 Theory of filtrational combustion. Combust., Explosion and Shock Waves 16(1), 33–40.Google Scholar
Booty, M. R. & Matkowsky, B. J. 1991 Modes of burning in filtration combustion. Euro. J. Appl. Math. 2(1), 17–41.Google Scholar
Booty, M. R. & Matkowsky, B. J. 1991 On the stability of counterflow filtration combustion. Combust. Sci. and Tech. 80, 231–264.CrossRefGoogle Scholar
Buckmaster, J. D. & Ludford, G. S. S. 1982 Theory of Laminar Flames. Cambridge University Press.CrossRefGoogle Scholar
Hardt, A. P. & Phung, P. V. 1973 Propagation of gasless reactions in solids 1. Analytic study of exothermic intermetallic reaction rates. Combust, and Flame 21(1), 77–97.CrossRefGoogle Scholar
Merzhanov, A. G. 1990 Self-propagating high temperature synthesis: Twenty years of search and findings. In: Holt, J. B. and Munir, Z., eds., Proc. Int. Symp. Comb, and Plasma Synth. High-Temperature Materials. Verlag Chemie.Google Scholar
Munir, Z. A. & Anselmi-Tamburini, U. 1989 Self-propagating exothermic reactions: The synthesis of high-temperature materials by combustion. Mater. Sci. Rep. 3(7), 277–365.CrossRefGoogle Scholar
Margolis, S. B. 1983 An asymptotic theory of condensed two-phase flame propagation. SIAM J. Appl. Math. 43(2), 351–369.CrossRefGoogle Scholar
Margolis, S. B. 1992 The Asymptotic Theory of Gasless Combustion Synthesis. Metall. Trans. 23, 15–22.CrossRefGoogle Scholar
Shkadinsky, K. G., Shkadinskaya, G. V., Matkowsky, B. J. & Volpert, V. A. 1992(a) Combustion synthesis of a porous layer (to appear in Combust. Sci. and Tech.).Google Scholar
Shkadinsky, K. G., Shkadinskaya, G. V., Matkowsky, B. J. & Volpert, V. A. 1992(b) Self compaction or expansion in combustion synthesis of porous metals (to appear in Combust. Sci. and Tech.).Google Scholar
Sivashinsky, G. 1981 On spinning propagation of combustion waves. SIAM J. Appl. Math 40, (3), 432–438.Google Scholar