A Simplified Model of Unstable Burning in Solid Propellants

Abstract

An analysis of the surface temperature, and hence mass flux, response of a solid propellant to a disturbance in gas pressure has been developed. Time lags in the gas phase are neglected while transient heat conduction in the solid is considered. The results are obtained by perturbing the conservation equations in both the gas phase and the solid phase. Stability conditions are obtained in terms of a few dimensionless parameters which depend upon the steady state conditions. T HE COUPLING of pressure disturbances with propellant combustion has been the subject of several recent investigations (1 through 5). 4 It is now generally accepted that this feature plays an important role in the mechanism of instability (6). The most elaborate of the recent analyses (2) obtains results in terms of eleven sets of parameters which can be varied independently. Many simplifications were required in order to obtain even this complex solution. The value of such an analysis is not in obtaining explicit mathematical relationships suitable for design calculations, but rather in obtaining an understanding of what physical conditions may be expected to have a large influence on the unstable burning phenomenon. Therefore, an analysis of comparable sophistication with results in a simpler form is desirable.It was indicated by Hart and McClure (2) that a characteristic time for heat conduction in the solid is at least an order of magnitude greater than the characteristic times for the gas phase transport processes and the chemical reactions. This suggests that the gas may adjust very quickly to changes in conditions when compared to the response of the. solid. In the case of periodic pressure disturbances, time lags in the solid phase may dominate, provided the period of the disturbance is large compared to the characteristic time for gas phase adjustment. Thus, there is an upper limit in the frequency of the pressure disturbance above which an analysis neglecting gas phase time lags does not apply. This limit, which would vary depending on the physical properties of the combustion gases, is on the order of 10 4 -10 5 cps for many propellants.The physical processes presumed to take place are as follows:Consider first a step function increase in pressure. The response may be thought of as occurring in two phases. After the rapid application of the disturbance, the combustion zone moves quickly to a new position nearer the surface. Later it moves slowly as the surface temperature of the solid responds to the changed condition.The mass flux leaving the surface is less than would normally occur at the new pressure because, during the first period, the surface temperature does not have time to adjust to its steady state value. When relative motion between the combustion zone and the surface becomes small, the mass Received Nov. 25, 1960.

A review of calculations for unsteady burning of a solid propellant.

“…30 They use the quasi-static assumption for the gas phase, and, although diffusion is ostensibly included, its influence does not appear in the final results because the Lewis number is taken to be unity. However, they unjustifiably use a result found in adiabatic laminar flame theory for premixed gases.…”

Influences of combustion dynamics on linear and nonlinear unsteady motions in solid propellant rockets

“…In terms of the A-B formula (references [9] and [19]), Novozhilov finds that the parameter A is unchanged but B is replaced by BeiwTd : Figure 4.1 shows some results computed with (4.2). Although there is a pleasing, though small, effect in the vicinity of the peak of the real part, if Td is given an appropriate value, the behavior at higher frequencies is unacceptable.…”

“…The analysis of the effect of the shape of the response function on the combustion dynamics is based on parametric variations of a function expressed in the classical Denison and Baum form (reference [9]): Other geometrical characteristics of the chamber and physical properties of the propellant used for computations can be found in [2] and are not repeated here. The frequency of the fundamental mode (first longitudinal mode) is 900 Hz.…”

The Effect of Oxidizer Particle Size Distribution on the Steady and Nonsteady Combustion of Composite Propellants.

“…Several of the more pertinent models which have appeared in the literature are: the Denison and Baum (DB) model (49), the Cohen (C) model (51), the Zel'dovlch Novozhilov (Z N) model (50) and the veloc i ty coupling model of Lengellé (43). A discuss ion of these models follows .…”