Criteria for thermal stability of parabolically oxidable heterogeneous systems

Abstract

A system of equations that interaction of an isolated particle or a dense ensemble of particles with an oxidizing agent in the case of parabolic growth of the product layer is analytically treated in the one-temperature approximation. It is shown that a jumplike transition from the lower reacting mode (regime with a temperature maximum) to the upper reacting mode (regime with a monotonically increasing temperature) occurs with a continuously changing ignition parameter. The critical ignition parameter obtained differs from its numerically predicted value by 7%.Key words: parabolic law, phase trajectory, critical conditions.In the combustion and thermal-explosion theory, in problems where heterogeneous chemical interaction of components has to be analyzed, one has often to deal with a parabolic law of product-layer growth, which describes the mechanism of diffusive self-retardation of the self-heating rate. In dimensionless variables, the system of equations has the form [1, 2](E is the activation energy, R is the universal gas constant, and T 0 is the temperature of the ambient medium to which the heat is removed), z is the dimensionless thickness of the oxide film, δ is the thermal-explosion criterion, and z 0 is the dimensionless initial thickness of the oxide film. Expressions for the dimensionless quantities δ and τ can be found in [1,2]. System (1) is an autonomous dynamic system of the first order, which has no stationary solutions; this system describes a class of problems that arise in the heterogeneous-ignition theory, which includes ignition of an isolated metal particle during formation of the product oxide film [1], ignition of a dense aggregation of particles in the onetemperature approximation [2], and ignition of a particle during formation of an intermetallide layer with a narrow zone of intermetallic-compound homogeneity in the equilibrium phase diagram of components [3]. Specific features of individual problems are determined by a particular form of the dimensionless parameter δ, which does not need to be refined in statement (1). System (1) was numerically analyzed for the first time by Khaikin et al. [1], who determined the critical value of δ from the breakdown of the curve Θ(τ ) and from the transition from the regime with a maximum to the regime with a monotonically increasing temperature. The physical mechanism of the temperaturecurve breakdown, however, remained unclear, allowing no mathematical formulation of the ignition criterion.Rozenband and Chernenko [2] gave an approximate analytical solution to problem (1) based on examining the dependence z(τ ) from the viewpoint of the lower ignition limit. The analysis was based on a criterion responsible for merging of inflection points in the curve z(τ ). Results obtained by this method agree well with simulated data, though this criterion is not a true ignition criterion. The true criterion, obviously, must be capable of defining the qualitative change in the behavior of Θ(τ ) and the transition from the regime with a temperature maximum to ...