Recent Advances in Gas-Particle Nozzle Flows

T USE of metallic fuel constituents in modern rocket engines has brought attention to nonequilibrium aspects of two-phase nozzle expansion processes. Since condensed metal oxide combustion products (which comprise 30 to 40% by weight of the total products of contemporary solid rockets) can do no expansion work, their presence in the rocket nozzle can only be deleterious to the effectiveness of the nozzle expansion process in converting thermal to kinetic energy. The condensed particles are accelerated in a nozzle almost exclusively by drag forces associated with lag or slippage of the particles relative to the expanding gas. Some performance loss relative to the calculated ideal no-slip expansion process must always be associated with macroscopic size particles, and experience has shown that the magnitude of the loss increases with the weight fraction of particles. Significant velocity and thermal lags thus have been suspected as a prime cause of rocket performance losses, and a number of studies have been directed toward delineation of the mechanism and magnitude of these lag effects. Early studies were summarized and extended in the review of Altman and Carter (l). Primarily, these early studies served to place bounds on the performance losses by examining the limiting cases of no-lag and complete lag. They demonstrated that thermal lag ordinarily has a lesser effect on specific impulse than does velocit}^ lag. Gilbert, Davis, and Altman (2) were the first to relate the losses to particle size. They solved the linear equation that results from assuming the drag force to be proportional to the velocity difference (Stokes' law) for the case of linearly accelerated nozzle gas. They demonstrated that, typically, a 1-ju diam particle follows the gas velocity closely, whereas a IQ-fj, diam particle has a significant lag. All of these early studies treated the nozzle expansion processes as though they are uncoupled, i.e., the thermal lag