Computational Studies On Certain Problems Of Combustion Instability In Solid Propellants
This thesis presents the results and analyses of computational studies on certain problems of combustion instability in solid propellants. Specifically, effects of relaxing certain assumptions made in previous models of unsteady burning of solid propellants are investigated. Knowledge of unsteady burning of solid propellants is essential in studying the phenomenon of combustion instability in solid propellant rocket motors. In Chapter 1, an introduction to different types of unsteady combustion investigated in this thesis, such as 1) intrinsic instability, 2) pressure-driven dynamic burning, 3) extinction by depressurization, and 4) L* -instability, is given. Also, a review of previous experimental and theoretical studies of these phenomena is presented. From this review it is concluded that all the previous studies, which investigated the unsteady combustion of solid propellants, made one or more of the following assumptions: 1) quasi-steady gas-phase (QSG), 2) quasi-steady condensed phase reaction zone (QSC), 3) small perturbations, and 4) unity Lewis number. These assumptions limit the validity of the results obtained with such models to: 1) relatively low frequencies (< 1 kHz) of pressure oscillations and 2) small deviations in pressure from its steady state or mean values. The objectives of the present thesis are formulated based on the above conclusions. These are: 1) to develop a nonlinear numerical model of unsteady solid propellant combustion, 2) to relax the assumptions of QSG and QSC, 3) to study the consequent effects on the intrinsic instability and pressure-driven dynamic burning, and 4) to investigate the L* -instability in solid propellant rocket motors. In Chapter 2, a nonlinear numerical model, which relaxes the QSG and QSC assumptions, is set up. The transformation and nondimensionalization of the governing equations are presented. The numerical technique based on the method of operator-splitting, used to solve the governing equations is described. In Chapter 3, the effect of relaxing the QSG assumption on the intrinsic instability is investigated. The stable and unstable solutions are obtained for parameters corresponding to a typical composite propellant. The stability boundary, in terms of the nondimensional parameters identified by Denison and Baum (1961), is predicted using the present model. This is compared with the stability boundary obtained by previous linear stability theories, based on activation energy asymptotics in the gas-phase, which employed QSC and/or QSG assumptions. It is found that in the limit of large activation energy and low frequencies, present result approaches the previous theoretical results. This serves as a validation of the present method of solution. It is confirmed that relaxing the QSG assumption widens the stable region. However, it is found that a distributed reaction in the gas-phase destabilizes the burning. The effect of non-unity Lewis number on the stability boundary is also investigated. It is found that at parametric values corresponding to low pressures and large flame stand-off distances, small amplitude, high frequency (at frequencies near the characteristic frequency of the gas-phase) oscillations in burning rate appear when the Lewis number is greater than one. In Chapter 4, the effect of relaxing the QSG assumption is further investigated with respect to the pressure-driven dynamic burning. Comparison of the pressure-driven frequency response function, Rp, obtained with the present model, both in the QSG and non-QSG framework, with those obtained with previous linear stability theories invoking QSG and QSC assumptions are made. As the frequency of pressure oscillations approaches zero, |RP| predicted using present models approached the value obtained by previous theoretical studies. Also, it is confirmed that the effect of relaxing QSG is to decrease the |Rp| at frequencies near the first resonant frequency. Moreover, relaxing QSG assumption produces a second resonant peak in |Rp| at a frequency near the characteristic frequency of the gas-phase. Further, Rp calculated using the present model is compared with that obtained by a previous linear theory which relaxed the QSG assumption. The two models predicted the same resonant frequencies in the limit of small amplitudes of pressure oscillations. Finally, it is found that the effect of large amplitude of pressure oscillations is to introduce higher harmonics in the burning rate and to reduce the mean burning rate. In Chapter 5, first the present non-QSC model is validated by comparing its results with that of a previous non-QSC model for radiation-driven burning. The model is further validated for steady burning results by comparing with experimental data for a double base propellant (DBP). Then, the effect of relaxing the QSC assumption on steady state solution is investigated. It is found that, even in the presence of a strong gas-phase heat feedback, QSC assumption is valid for moderately large values of condensed phase Zel'dovich number, as far as steady state solution is concerned. However, for pressure-driven dynamic burning, relaxing the QSC assumption is found to increase |RP| at all frequencies. The error due to QSC assumption is found to become significant, either when |Rp| is large or as the driving frequency approaches the characteristic frequency of the condensed phase reaction zone. The predicted real part of the response function is quantitatively compared with experimental data for DBP. The comparison seems to be better with a value of condensed phase activation energy higher than that suggested by Zenin (1992). In Chapter 6, burning rate transients for a DBP during exponential depressurization are computed using non-QSG and non-QSC models. Salient features of extinction and combustion recovery are predicted. The predicted critical initial depressurization rate, (dp/dt)i, is found to decrease markedly when the QSC assumption is relaxed. The effect of initial pressure level on critical (dp/dt)i is studied. It is found that the critical (dp/dt)i decreases with the initial pressure. Also, the overshoot of burning rate during combustion recovery is found to be relatively large with low initial pressures. However as the initial pressure approached the final pressure, the reduction in initial pressure causes a large increase in the critical (dp/dt)i. No extinction is found to occur when the initial pressure is very close to the final pressure. In Chapter 7, a numerical model is developed to simulate the L* -instability in solid propellant motors. This model includes a) the propellant burning model that takes into account nonlinear pressure oscillations and that takes into account an unsteady gas- and condensed phase, and b) a combustor model that allows pressure and temperature oscillations of finite amplitude. Various regimes of L* -burning of a motor, with a typical composite propellant, namely 1) steady burning, 2) oscillatory burning leading to steady state, 3) oscillatory burning leading to extinction, 4) reignition and 5) chuffing are predicted. The predicted dependence of frequency of L* -oscillations on mean pressure is compared with one set of available experimental data. It is found that proper modeling of the radiation heat flux from the chamber walls to the burning surface may be important to predict the re-ignition. In Chapter 8, the main conclusions of the present study are summarized. Certain suggestions for possible future studies to enhance the understanding of dynamic combustion of solid propellants are also given.