Analysis of shape of porous cooled medium for an imposed surface heat flux and temperature

The surface of a porous cooled medium is to be maintained at a specified design temperature while being subjected to uniform heating by an external source. An analytical method is given for determining the shape of the medium surface that will satisfy these boundary conditions. The analysis accounts for temperature dependent variations of fluid density and viscosity and for temperature dependent matrix thermal conductivity. The energy equation is combined with Darcy's law in such a way that a potential can be defined that satisfies Laplace's equation. All of the heat-transfer and flow quantities are expressed in terms of this potential. The determination of the shape of the porous cooled region is thereby reduced to a free-boundary problem such as in inviscid free jet theory. Two illustrative examples are carried out: a porous leading edge with coolant supplied through a slot and a porous cooled duct with a rectangular outer boundary.