Díaz Díaz, Jesús Ildefonso and Stakgold, Ivar
(1995)
*Mathematical aspects of the combustion of a solid by a distributed isothermal gas reaction.*
Siam Journal on Mathematical Analysis , 26
(2).
pp. 305-328.
ISSN 0036-1410

PDF
Restricted to Repository staff only until 31 December 2020. 903kB |

Official URL: http://epubs.siam.org/simax/resource/1/sjmaah/v26/i2/p305_s1?isAuthorized=no

## Abstract

When a diffusing gas reacts isothermally with an immobile solid phase, the resulting equations form a semilinear system consisting of a parabolic partial differential equation for the gas concentration coupled with an ordinary differential equation for the solid concentration. Existence and uniqueness proofs are given which include the important case of nonlipschitzian reaction rates such as those of fractional-power type. Various qualitative features of the solution are studied: approach to the steady state; monotonicity in time; and dependence on initial conditions, on the porosity, and on the geometry. The relationship between the original problem and the pseudo-steady-state approximation of zero porosity is investigated. When the solid reaction rate is nonlipschitizian, there is a conversion front separating a fully converted region adjacent to the boundary and a partially converted interior core. Estimates are given for the time to full conversion. If the gas reaction rate is nonlipschitzian the gas may not at first fully penetrate the solid. Estimates are given for the time at which full penetration occurs.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | diffusion-systems; boundary; gas-solid reactions; reaction-diffusion; combustion; pseudo-steady state |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 15947 |

References: | R. ARIS, The Mathematieal Theory of Diffusion and Reaetion in Permeable Catalysts, Clarendon Press, Oxford, 1975. C. BANOLE, Isoperimetrie Inequalities and Their Applications, Pitman, London, 1980. C. BANDLE, R. SPERB, AND I. STAKGOLD, Diffusion-reaciion with monotone kinetics, Nonlinear Anal., 8 (1984), pp. 321-333. C. BANDLE AND I. STAKGOLD, The formation of the dead eore in pambolie reaction-diffusion equations, Trans. Amer. Math. Soc., 286 (1984), pp. 275-293. M. BERTSCH, R. KERSNER, ANO L. A. PELETIER, Positivity versus localization in degenerate diffusion problems, Nonlinear Anal., 9 (1985), pp. 831-847. D. BLANCHARD, A. DAMLAMIAN, ANO H. GHIDOUCHE, A nonlinear system for phase change with dissipation, Differential and Integral Equations, 2 (1989), pp. 344-362. J. I. DÍAZ, Nonlinear Partial Differential Equations and Free Boundaries, Vol.1. Elliptic Equations, Pitman, London, 1985. J. I. DÍAZ AND J. HERNÁNDEZ, On the existence of a free boundary for a class of reaction diffusion systems, SIAM J. Math. Anal., 15 (1984), pp 670-685. J. I. DÍAZ AND J. HERNÁNDEZ, Some results on the existenee of free boundaries for parabolic reaction-diffusion systems, In Trends in Theory and Praetiee of Nonlinear Differential Equations, V. Lakshmikantham, ed. Marcel Dekker, 1984, pp. 149-156. J. I. DÍAZ AND I. STAKGOLO, Mathematical analysis of the conversion of a porous solid by a distributed gas reaetion, in Proc. of the XI CEDYA, Univ. of Malaga, Spain 1989, pp. 217-223. J. I. DÍAZ AND I. I. VRABIE, Existence for reaetion diffusion systems, A compaetness method approach,submitted. A. DI LIDDO AND L. MADDALENA, Mathematieal Analisis of a Chemical Reaetion with a Lumped Temperature and Strong Absorption, J. Math.Anal. Appl., 163 (1992), pp. 86-102. A. DI LIDDO, L. MADDALENA, AND I. STAKGOLD,Travelng waves for distributed gas-solid reactions, J. Differential Equations, to appear. A. DI LIDDO AND I. STAKGOLD, Isothermal combustion with two moving fronts, J. Math. Anal. Appl., 152 (1990), pp. 584-599. A. FRIEDMAN AND A. E. TZAVARAS, A quasilinear pambolie system arising in modelling of catalytie reactors, J. Differential Equations, 70 (1987), pp. 167-196. G. F. FROMENT AND K.B. BISCHOFF,Chemical Reactor Analysis and Design, Wiley, New York 1979. J. GRAHAM-EAGLE AND I. STAKGOLD, A steady-state diffusion prob/em with fraetional power absorption rate, IMA J. Appl. Math., 39 (1987), pp. 67-73. G. S. LADDE, V. LAKSHMIKANTHAM, AND A. S. VATSALA, Monotbne Iterative Teehniques for Nonlinear Differential Equations, Pitman, London, 1985. A. McNABB, Asymptotie behaviour of solutions of diffusion equations, J. Math. Anal. Appl., 51 (1975), pp. 219-222. A. McNABB AND G. KEADY, Sorne explicit solutions of -w= 1 with zero boundary data, Report 42, Dept. of Mathematics, The University of Western Australia, Nedlands, Australia, 1988. C. V. PAO, Asymptotic behavior and nonexistenee of global solutions for a class of nonlinear boundary value problems of parabolie type, J. Math. Anal. Appl., 65 (1978), pp. 616-637. ---, On nonlinear reaetion-diffusion systems, J. Math.Anal. Appl., 87 (1982), pp. 165-198. D. H. SATTINGER, Monotone methods in nonlinear elliptie and parabolie boundary value problems, Indiana Univ. Math. J., 21 (1972), pp. 979-1000. I. STAKGOLD, Gas-solid reaetions, in Dynamieal Systems II, A. R. Bednarek and L.Cesari, eds., Academic Press, New York, 1982, pp. 403-417. ---, Partial extinction in reacition-diffusion, Conferenze del Seminario di Matematica, Universitá di Bari, Bari, ItaIy, 224, 1987. --- Localization and extinction in reaction-diffusion, in Free Boundary Problems: Theory and Applications, Vol.1, K. H. Hoffmann et al., eds., Longman, London, 1988, pp. 208-221. I. STAKGOLD, K. B. BISCHOFF, AND V. GOKHALE, Validity of the pseudo-steady-state approximation, Intenat. J. Eng. Sci., 21 (1983), pp. 537-542. I. STAKGOLD AND A. McNABB, Conversion estimates lor gas-solid reactions, Math. Modelling,5 (1984), pp. 325-330. J. SZEKELY, J. W. EVANS, AND H. Y. SOHN, Gas-solid reactions, Academic Press, New York, 1976. J. L. VÁZQUEZ, A strong maximum principie for some quasilinear elliptic equations, Appl.Math. Optim., 12 (1984), pp. 191-202. I. I. VRABIE, Compactness Methods for Nonlinear Evolutions, Longman, London, 1987. C. BANDLE AND I. STAKGOLD, Reaction-Diffusion and Dead Cores, in Free Boundary Problems:Applications and Theory, Vol. IV, Pitman, London, 1985. R. MARTIN, Mathematical models in gas-liquid reactions, Nonlinear Anal., Theory and Appl.,4 (1980), pp. 509-527. |

Deposited On: | 13 Jul 2012 07:27 |

Last Modified: | 06 Feb 2014 10:35 |

Repository Staff Only: item control page