An Exact Solution Based on a Three-Energy Equation Model for Gaseous Transpiration Cooling Through a Bi-Disperse Porous Medium

A local thermal nonequilibrium analysis was made to assess the potential use of bi-disperse porous walls for a transpiration cooling system. A three-energy equation model successfully used for the transient thermal analysis of bi-disperse packed bed thermocline storage systems was introduced to investigate various heat transfer aspects of transpiration cooling through a bi-disperse porous wall made of combination of large and small particles. Three independent energy balance equations, namely, the energy equation of the coolant gas phase, that of the solid phase of large particles, and that of small particles are coupled with one another to obtain a set of exact expressions for all three individual temperature distributions across the porous wall for given thermal boundary conditions of the third kind. It has been revealed that the solid wall temperature of the bi-disperse porous wall stays lower than that of the monodisperse porous wall in the high Peclet number range, resulting in a higher overall cooling efficiency for a given blowing flowrate. Furthermore, the analysis provides a suitable range of the Peclet number, under which the transpiration cooling should be operated to suppress excessive heat loss to the coolant reservoir at the same time to ensure a high overall cooling efficiency.