Theory of Detonations. III. Ignition Temperature Approximation

Abstract

This paper is one of a series in which we discuss the effect of viscosity, diffusion, and heat conductivity on the structure of steady-state plane detonation waves. It is shown that the consideration of these phenomena may lead to considerable overlapping of the reaction and shock zones and also to the existence of detonation limits. In order to discuss in more detail the topological nature of the solutions and the nature of the detonation limits, the equations are simplified by idealizing the chemical kinetics in the following manner: At temperatures below an ignition temperature, the reaction rates are taken to be zero; at temperatures above this ignition temperature, the reaction rates are taken to be functions of the chemical composition but independent of the temperature. In order to further simplify the equations, the Prandtl number is taken to be ¾ and the Lewis number to be unity. These idealizations lead to an uncoupling of the detonation equations, although the importance of the transport properties remains unchanged. In the high-temperature range, the fractions of the mass rate of flow, Gi, are functions of the chemical composition only. In the low-temperature range, the Gi are constant. With these simplifications, we consider in detail both flames and detonations involving the unimolecular A→B or the equivalent bimolecular reaction A+X→B+X where X represents either a molecule of A or B. Calculations are made for values of the ignition temperature greater than, equal to, and less than the von Neumann spike temperature. Comparison of these calculations with numerical calculations previously performed, using an Arrhenius type variation of the reaction rate with temperature, gives the best agreement when the ignition temperature is taken to be greater than the von Neumann spike temperature. If the ignition temperature is greater than the von Neumann spike temperature, it is found that the thickness of the detonation wave is of the order of one mean free path. This suggests the conclusion that under the extreme conditions of a detonation the usual concept of chemical equilibrium between activated state and reactant molecules cannot apply. Instead, the molecules gain their energies of activation in successive stages. With this more complicated reaction mechanism, the detonation wave would be much thicker. These conclusions, regardless of the functional form of the rate law, depend upon the existence of a lower limit for the value of the rate constant which can support a steady-state detonation. Such a lower limit was suggested by our previous calculations with the Arrhenius rate law. However, more detailed studies are required to determine under what conditions such a lower limit may exist.