The interaction of shock waves and heat addition in the design of supersonic combustors

Analytical models are needed that will permit the prediction of thrust efficiency of a supersonic combustor as a function of combustor geometry and heat release. An earlier, more general procedure for the one-dimensional analysis of supersonic combustors was developed on the basis of exponential pressure-area dependence. This paper extends that analysis by establishing three regimes for specific pseudo-one-dimensional solutions for thrust efficiency: (1) a low-heat-release regime in which the flow can be considered essentially shock-free; (2) an intermediate regime in which an oblique shock wave is sustained with a pressure ratio equal to or greater than that required for turbulent boundary-layer separation; and (3) a high-heat-release regime in which the combustion is preceded by a normal shock. The key to solution is the formulation for the wall-pressure force needed for the momentum equation. With it, a set of integral equations can be solved simultaneously for relationships between pressure ratio, total temperature ratio and area ratio across the combustor in the three regimes. It is argued that the most probable solutions are those that yield the lowest back pressure without violating the entropy limit (second law of thermodynamics). It is shown that such limiting processes, including shock waves, can yield higher over-all efficiencies than shock-free solutions, because the total pressure loss across the shock is more than compensated by the reduction in total pressure loss due to heat addition, which occurs at a lower Mach number. Results of a typical substantiating experiment falling in this class are presented, for a combustor-inlet Mach number of 3.2 and an alkyl borane fuel.