Analytical Model of Transpired-Coolant Concentration at Downstream Wall in High-Speed Laminar Flow

An analytical model based on one-dimensional diffusion is proposed to predict the mixing between the coolant gas and the boundary-layer gas at the wall downstream of a transpiring injector in a laminar flow. The model is validated against coolant concentration data experimentally obtained at Mach 7 over a flat plate at a zero pressure gradient. It successfully predicts the mixing at the wall downstream within 17% of the experimental data. It is shown that this mixing mechanism at the wall in laminar flows is fully described by the process of diffusion. The coolant coverage at a given downstream location is promoted when the streamwise velocity decreases, the blowing ratio increases, or the diffusion coefficient drops. Finally, a mass budget calculation is performed for a transpiration-cooled hypersonic vehicle employing the analytical model. The model predicts a mass requirement of 3.6 times less coolant when helium is used as the coolant gas as opposed to nitrogen for the chosen trajectory. However, helium requires twice the storage volume as compared to nitrogen.