Unsteady combustion modelling of energetic solids, revisited

Abstract

The mathematical problem of unsteady combustion of a homogeneous solid propellant with zero-order, high activation-energy, single-step decomposition and zero-activation-energy, quasi-steady gas-phase reaction is considered. Two approximate decomposition models, simple pyrolysis and leading-order asymptotic, are compared with each other and with the full, distributed reaction solution. It is shown that the leading-order asymptotic model is a reasonably good approximation of the exact solution for steady, linear-oscillatory and, with some additional limitations noted herein, nonlinear transient combustion conditions. Further, it is shown that the pyrolysis model can be made equivalent to the leading-order asymptotic model locally, i.e. for given pressure and temperature conditions, at the sensitivity parameter or linear behaviour level, but not beyond. Several misconceptions and unresolved issues associated with these models are resolved. It is shown that when equivalence is enforced, which necessarily involves a non-zero—usually negative—Jacobian parameter, n_s, the previously held approximate relation between the pyrolysis and bulk decomposition activation energies, E_s ≈ E_c/2, is not valid. Rather, the value of E_s may exceed that of E_c. Other clarifications include the demonstration of an asymptotic inconsistency among reported leading-order model sensitivity parameters that gives rise to an error of the order of 10% in certain parameters (r, δ, n_s). A blow-off behaviour is also shown to exist in the zero-activation-energy, gas-phase sub-model that can appear under nonlinear dynamic burning conditions. It is also demonstrated that previously reported nonlinear, high-frequency oscillatory behaviour associated with the leading-order asymptotic model is an artefact of the approximate asymptotic solution and not physical.