A.R. Shouman

The Frank-Kamenetskii problem revisited. Part I. Boundary conditions of first kind

Abstract The classical steady-state thermal ignition problem in the three one-dimensional geometries is considered. The well-known method of expanding the exponent first suggested by Frank-Kamenetskii is modified by performing the expansion about the unknown midplane temperature. Analytical expressions were obtained for the critical conditions that will allow the handling of more complex boundary conditions. The results are compared with the existing exact and numerical solutions.

Accounting for reactant consumption in the thermal explosion problem. III. Criticality conditions for the Arrhenius problem

Abstract The rigorously defined criticality condition established by the authors is applied to the Arrhenius model. The dependence of the critical temperature θ∗ on both the ambient temperature θ a and the order of reaction n is demonstrated. All solutions for θ∗ as a function of θ a for any value of n pass through the point of θ a = 0.25, θ∗ = 0.5 (the transition point for n = 0). It is shown that a transition temperature exists for all degrees of reaction except for 0 n ≤ 1.0. For a first order reaction, θ∗ monotonically increases with the ambient temperature and reaches ∞ at θ a = 0.5. For 0 n ≤ 1.0, no solution exists for θ∗ > 1/2(1 − n ), no transition temperature exists, and the solution for θ∗ as a function of θ a passes through an inflection point at θ a = 0.25, θ∗ = 0.5. It is also shown that there is significant difference between the results obtained from the Arrhenius model and those from the Frank-Kamenetskii approximated model. The critical state in the θ-τ plane is found to be subcritical in both the θ- Z and the Ψ-θ∗ planes, which produce identical results. Proof is provided that criticality in the θ-τ plane coincides with that in the θ- Z and Ψ-θ∗ planes only when n = 0 or B = ∞. The criticality limits with different ambient temperature θ a for zero order reaction ( n = 0) are established. The solution also treats various initial conditions for a zero order reaction.

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

Abstract 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.

Accounting for reactant consumption in the thermal explosion problem part II: A direct solution with application to the Frank-Kamenetskii problem

The thermal explosion problem with reactant consumption is investigated. A direct solution for the problem using a marching Taylor series expansion was obtained. The derivatives in the Taylor series are obtained from the original equations. The states defining criticality are accurately determined for various degrees of reaction. The results obtained are compared with those in the literature by other authors. An approximate method for determining the flammability limits as well as the critical states is presented.

Accounting for reactant consumption in the thermal explosion problem. part IV. numerical solution of the arrhenius problem

Abstract The Marching Taylor expansion method is used to determine the critical conditions for the thermal explosion problem with the reaction governed by the Arrhenius reaction kinetic equation. The contrast between the behavior with high θ ad and high θ a , low θ ad and low θ a as well as high θ ad and low θ a is demonstrated. The effect of the degree of reaction is examined. The criticality boundaries with different degrees of reaction are established as a function of the ambient temperature. Comparison with results existing in the literature is shown. Finally an approximate method for determining the critical state is presented. This reduces the computation time needed for establishing accurately the critical states.

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.

The stationary problem of thermal ignition in a reactive slab with unsymmetric boundary temperatures

Abstract An exact solution is presented to the stationary problem of thermal ignition in a reactive slab with unsymmetric boundary temperatures and the solution values thereof are tabulated. The results are examined for the condition of incipient thermal runaway as a function of boundary temperatures and slab thickness.

Solution to the dusty gas explosion problem with reactant consumption part I: the adiabatic case

Abstract The solution to the adiabatic thermal explosion problem of a dusty gas mixture is presented. The same definition of the critical state used for the homogeneous solid is used for the gas mixture. The critical trajectory is defined as the trajectory containing only one inflection point before the maximum temperature is reached in the temperature–concentration domain. In the case of the solid, the inflection point occurred before the maximum temperature. In the case of the dusty gas mixture it was found that the inflection point could occur as well at the maximum temperature. The presence of the gas produces a second critical temperature, and the limiting cases of no gas present or infinite amount of gas present lead to the classical adiabatic solid case or the case of the solid losing heat to the gas as its environment. The solution led to the discovery of three different singular solutions that help in understanding the various trajectories in the subcritical, critical, or supercritical state.

The Frank-Kamenetskii problem revisited, part III: The moving flame front

Abstract The basic flame propagation theory proposed in part by Frank-Kamenetskii was modified to show that flame behavior can be predicted in a two-zone flame structure without eliminating the ignition temperature. The ignition temperature can be retained when the inert zone maintains the reaction zone in a critical condition. By retaining the ignition temperature and by maintaining the integrity of the zone matching equations, the two step flame model we have developed predicts flame behavior in either zone.

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

Abstract 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].