Interaction of acoustic waves with burning surfaces of condensed systems

Abstract

The principal topic of the linear theory of the acoustic burning stability of solid propellant systems is the possibility of amplifying acoustic pressure waves reflected from a burning surface. The investigation of reflected acoustic waves reduces to the study of the reorganization of the physico-chemical processes in the combustion zone, caused by harmonic pressure disturbances, and to the calculation of the acoustic perturbation of the rate at which the gas flows off the burning surface. The reflectivity of the combustion zone is characterized by the value of the acoustic admittance.

It is not possible to obtain an exact solution of the problem of the acoustic admittance of a burning surface, because the complex processes taking place in the combustion zone of a solid propellant are not amenable to rigorous analytical description, even in the stationary case. Analysis is based on simplified models of the combustion zone which take into account only the most significant peculiarities of the combustion mechanism of solid propellants [1–7].

The problem of acoustic admittance of a burning surface has been investigated in detail in [1,2, 6]. In [1,2], the assumptions concerning the combustion zone include the hypothesis of constant combustion temperature in steady-state conditions. It may be shown that this is possible only when the value of the total heat release in the combustion zone does not remain constant in steady-state conditions and, at certain moments of time, exceeds the initial enthalpy of the solid propellant, i.e., the magnitude of the thermal effect of the combustion reaction. Such an assumption can be hardly justified from the physical point of view.

In [5], the acoustic admittance of a burning surface was calculated on the basis of Zel'dovich's theory of solid-propellant combustion under variable pressure [8].

This paper did not take into account the exothermic chemical reaction in the condensed phase (k-phase) and its effect on the burning rate, nor did it allow for the change in surface temperature in the k-phase in unsteady conditions.

The aim of the present paper is to obtain formulas for the acoustic admittance of a burning surface with allowance for these factors. As distinct from [1, 2], we assume as in [5] that the total heat release rather than the combustion temperature is constant in unsteady conditions. This, and some other differences in the steady-state combustion model, leads to results that differ from the conclusions of [1,2].

The problem is solved on the assumption that the period of the acoustic oscillations is large compared to the characteristic time of the processes that take place in the gas and in the chemical reaction zone of the k-phase (quasi-stationary approximation). Such an approach is justified for frequencies not higher than the order of 10⁴ cps. This frequency range includes essentially the frequencies characteristic of the vibrational modes of powder burning.

In order to investigate the reflection of acoustic waves of a frequency higher than 10 cps from a burning surface it is necessary to take into account the inertial properties of the processes occurring in the gas and the reaction zones.

In the frequency range studied, the dimensions of the combustion zone in the gas are much smaller than the acoustic wavelength, and therefore it is safe to assume that the pressure fluctuations within the limits of the combustion zone are independent of the space coordinate.