Isothermal Gaseous Detonation Model

Abstract

Introduction. Approximate models, as compared to exact ones, usually use additional physical assumptions that permit "screening" unimportant details of the investigated phenomenon and simplifying the system of equations to be solved. Therefore, approximate models enhance the graphicness of obtained results, facilitate the theoretical analysis of the phenomenon, and shorten the time of computer calculations. If, moreover, the above models feature also a high accuracy, then their practical importance increases, since it enables them to describe the phenomenon not only qualitatively but quantitatively as well. In modeling the propagation of the detonation wave (DW) in a chemically active gas, its front is often considered as a compression shock with instantaneous heat release, at which the mass, momentum, and energy conservation laws should be observed [1–3]. Such a situation takes place when the thickness of the zone of major energy release due to the chemical reaction (for a gaseous detonation of the order of the size of the detonation cell [4]) is small compared to the characteristic linear scale of the whole fl ow, for example, the diameter of the tube in which the detonation propagates. Such a tube can be considered to be "wide" [5], and the difference of the gas-dynamic parameters of the DW front from the ideal detonation (without friction and heat transfer losses) is negligibly small. There exist well-tested methods (for example, the method described in [6] for calculating the parameters of the steady ideal detonation, often called the Chapman–Jouguet detonation) [3, 7 that are based on the assumption of chemical equilibrium of the composition of gaseous detonation products. The condition of chemical equilibrium of detonation products means that the chemical reaction rate is much higher than the rate of change in the thermodymamic parameters of the substance, i.e., the chemical reaction is instantaneous. Estimates of the characteristic times of the processes in which the chemical equilibrium is reestablished are given in [8]. Data of equilibrium calculations agree fairly well with experimental data; in the fi rst place it concerns the detonation front velocity DCJ measured with a high accuracy in experiments [9]. In spite of its apparent simplicity, the equilibrium detonation model is rather complicated for theoretical analysis, since in the computational iterative process one has to solve an expanded system of equations containing, apart from the gas dynamics equations, chemical equilibrium formulas, thermochemical relations, and atomic balance equations for all possible components of the detonation reaction. Therefore, other approaches are needed for modeling the detonation in gases. For instance, from the viewpoint of the classical one-dimensional theory for the analytical description of the gas dynamics of the detonation process [1–3], one usually uses an approximate model with highly idealized notions on detonation products, which is justifi ed only by the possibility of simple algebraic transformations with the use of conservation laws. In particular, it is assumed that detonation products represent an ideal inert gas with a constant adiabatic index �� and the chemical energy is released in the form of heat Q = const instantaneously at the DW front. The reasoning behind what values of �� and Q should be used in the given model is usually omitted thereby. The accuracy of the description of the thermodynamic part of internal