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The dynamics of chemical reactors in porous media. (English) Zbl 1254.80004
Summary: Is ignition or extinction the fate of an exothermic chemical reaction occurring in a bounded region within a heat conductive solid consisting of a porous medium? In the spherical case, the reactor is modeled by a system of reaction-diffusion equations that reduces to a linear heat equation in a shell, coupled at the internal boundary to a nonlinear ODE modeling the reaction region. This ODE can be regarded as a boundary condition. This model allows the complete analysis of the time evolution of the system: there is always a global attractor. We show that, depending on physical parameters, the attractor contains one or three equilibria. The latter case has special physical interest: the two equilibria represent attractors (“extinction” or “ignition”) and the third equilibrium is a saddle. The whole system is well approximated by a single ODE, a “reduced” model, justifying the “heat transfer coefficient” approach of Chemical Engineering.
MSC:
80A32 Chemically reacting flows
76S05 Flows in porous media; filtration; seepage
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
35Q79 PDEs in connection with classical thermodynamics and heat transfer
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35K57 Reaction-diffusion equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
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Full Text: Euclid