Problem Of Thermal Explosion Research Articles

Duck-trajectories in a thermal explosion problem

The critical values of a parameter in a thermal explosion problem in the autocatalytic case are examined by means of the method of integral manifolds and French-ducks techniques.

Convective stability of the horizontal reacting liquid layer in the presence of various complicating factors

The convective stability of a horizontal chemically active liquid layer is studied within which an exothermal zero-order reaction takes place in the presence of a number of complicating factors: warming of the layer, different thermal conductivities of the boundaries and transverse motion of the reactant. Using the Runge-Kutta method, numerical investigation of the linear spectral stability problem of the steady-state regimes of heat transfer determined by solving the generalized non-linear problem of thermal explosion is carried out. The results obtained demonstrate a substantial influence of the above-listed complicating factors on the critical conditions for the thermal explosion and on the limit of the incipience of the reactant convective motion.

Accounting for reactant consumption in the thermal explosion problem. Part I: Mathematical foundation

New definitions for the criticality conditions of the thermal explosion problem are founded on the mathematical behavior of the governing equations. The paper deals with uniform temperature and concentration (Semenov problem). It is well known that the results can be applied to the distributed temperature and concentration case by the use of correction factors. It is shown that criticality can be defined in the temperature-time plane as accepted by most authors. However, using our definitions of criticality in the temperature-concentration plane confirms the previous findings of Adler and Enig. They showed that the classically defined critical state in the temperature-time plane is always a subcritical state in the temperature-concentration plane. At the same time, the critical state in the temperature-concentration plane is always a supercritical state in the temperature-time plane. However, the critical state in the temperature-concentration plane is in agreement with that in the Semenov number (Ψ)-temperature plane. It is shown that the critical states in all planes coincide only when n = 0 or B = ∞ and agree with the well known results neglecting reactant consumption. The difference between the critical and ignition temperatures is discussed. It is shown that as B approaches infinity, the solution for Φ as a function of τ for any value of n approaches the solution for n = 0. Hence for B = ∞, substituting for n = 0 in Alder and Enig results produces the classical Semenov result. This resolves the objections expressed against these results before. At the same time the locus of the critical states for n = 0, with finite B is determined. The effect of the degree of reaction on the induction time for adiabatic systems is demonstrated. The conditions required for ignition of subcritical systems is demonstrated. The conditions required for ignition of subcritical systems are discussed as well as the effect of initial conditions on criticality.

Thermal explosion theory with Arrhenius kinetics: homogeneous and inhomogeneous media

The classical thermal explosion problem for a reactive slab with Arrhenius kinetics and small activation energy parameter is considered. Perturbation series expansions are obtained that give an expression for the critical Frank-Kamenetskii parameter. We verify that the often used ‘Frank-Kamenetskii approximation’ is the first-order problem in such an expansion procedure. The analysis is extended to a reactive slab with a random distribution of reactant. The statistical properties are incorporated in an effective thermal diffusivity that may be expressed as a function of its mean and the variance about the mean. Some other expressions are also considered. It is shown that a random distribution decreases the thermal stability when compared with criticality with mean properties.

Newton-variational solution of the Frank–Kamenetskii thermal explosion problem

The Newton method was shown by Vatsya to be suitable for solving the generalised elliptic problem, and we show that this approach can be used to treat the Frank–Kamenetskii model of a thermal explosion, by using a variational solution of the sequence of linear equations that are encountered in the Newton method. Convergence to the desired solution is rapid in the case of spherical geometry. The method produces converging upper and lower bounds to the critical value of the dimensionless heat production rate, δc, and lower bounds to the dimensionless temperature excess distribution function θ and its critical form θc. Keywords: thermal explosions, Frank–Kamenetskii model.

Critical local disturbance of temperature field necessary for ignition, and the effects of reactant consumption

A minimal disturbance is necessary to bring about thermal runaway in a system on the verge of criticality. The nature and size of this disturbance is governed by the unstable standing solitary wave whose existence was proved in a previous paper (Gray & Kordylewski, Proc. R. Soc. Lond . A 398, 281—288 (1985)). The effect of reactant consumption on the minimal disturbance necessary is important. A proportionally greater increase is necessary here than in the ‘classical’ problem of thermal explosion with reactant consumption.

The Frank-Kamenetskii problem revisited, part II: Gradient boundary conditions

The thermal explosion problem which was posed by Frank-Kamenetskii to explain ignition has been examined in Part I. In the present work, we extend the analysis to include the gradient boundary conditions. Cooling boundary conditions are shown to cause ignition when the steady state solutions occur within the unstable region found in the previous work of Part I. A three dimensional plot of characteristic parameters for the heating boundary conditions is found to form an unstable region that defines the critical conditions. The symmetrical temperature boundary condition studied in Part I is shown to be only a special case of the heating problem. A summary of the results obtained is presented here while the details of the study are published in Reference [1].

Prediction of critical conditions for thermal explosion problems by a series method

Owing to the high degree of nonlinearity of thermal explosion problems, a variety of procedures has evolved for the prediction of the critical conditions for ignition. In this communication, a series solution method is applied to the general one-dimensional, steady-state thermal explosion problem. The method has as advantages: (1) conceptual simplicity; (2) application to Cartesian, cylindrical and spherical geometry; (3) simple expression when flexibility, but not accuracy is required; and (4) direct extension to higher order series when accuracy is required.

Approximate solution of the nonstationary problem of thermal explosion in the presence of a warmup stage

Approximate solution of the nonstationary problem of thermal explosion in the presence of a warmup stage