Modeling the combustion response function with surface and gas phase dynamics

Considerable data exists suggesting that the response functions for many solid propellants tend to have higher values, in some ranges of frequencies, than predicted by the conventional QSHOD theory. It is a familiar idea that such behavior is associated with dynamical processes possessing characteristic times shorter than that of the thermal wave in the condensed phase. The QSHOD theory, and most of its variants, contains only the dynamics of that process, which normally has a characteristic frequency in the range of a few hundred hertz. Two previous works seeking to correct this deficiency (T’ien, 1972; Lazima and Clavin, 1992) have focused their attention on including the dynamics of the thermal wave in the gas phase. Both include effects of diffusion that complicate the analysis although the second achieves some simplification by applying the ideas of ‘activation energy asymptotics’. While their results differ in detail, both works show influences at frequencies higher than those near the broad peak of the response due to the thermal wave. The work reported in this paper has the primary purpose of constructing a simple model of the problem so that the possible dynamics of the thin region adjacent to the interface of the condensed phase may be incorporated and investigated in heuristic fashion, with and without approximations in the gas phase. It is well known from many observations, both with high-speed films and from pictures taken with scanning electron microscopes, that the surface of a burning solid propellant is certainly not smooth and in general contains both liquid and solid particles. For metallized propellants the agglomeration of aluminum drops is an important process affected, for example, by small amounts of impurities or additives. The dynamics of this region may be significant to the response of a burning propellant to external disturbances, but this phenomenon has not been previously been studied. In OUT analysis we include both phenomenological modeling of that surface layer as well as the thermal waves in both the gas and solid phase. Particular attention is given to the selection of the boundary conditions and their effect on the solution of the problem. Assumptions made in the analytical approach are tested against direct numerical integration of the relevant equations. Response functions are shown for realistic ranges of the chief parameters characterizing the dynamics of solid phase and surface layer. The results are also incorporated in the dynamical analysis of a small rocket motor to illustrate the consequences of the combustion dynamics for the stability and nonlinear behavior of unsteady motions in a motor. That is part of the primary objective of the Caltech MURl program, to understand the influences of propellant composition and chemistry on the global dynamical behavior of a solid rocket combustor by connecting the microscopic and macroscopic through the response function.