Adiabatic wall temperature or recovery temperature is the temperature acquired by a perfect insulation near compressible flow. - If the velocity of the flow is very high viscous dissipation effects cannot be ignored. - An adiabatic wall temperature will be higher than free stream temperature since the flow has to be brought to rest near the wall. The flow is not brought to rest isentropically since viscous action is irreversible. To take this into account we use a *recovery factor* $r=\frac{T_{aw}-T_{\infty}}{T_0-T_{\infty}}$ where, $T_{\infty}$ is the free stream temperature (static temperature) $T_{aw}$ - is adiabatic wall temperature $T_{0}$ is stagnation temperature or chamber temperature ## 1. Walls with heat transfer Consider a wall at a temperature $T_w$ in high speed flow. The heat transfer from this wall can be calculated with similar relations used for incompressible flow $q=hA(T_w-T_{aw})$ ## 2. Recovery factor For laminar flow $r=Pr^{1/2}$ For turbulent flow $r=Pr^{1/3}$ ## 3. Heat transfer coefficient To calculate heat transfer coefficient using empirical correlations use properties calculated at $T^*$ $T^*=T_{\infty} + 0.5(T_w-T_{\infty})+0.22(T_{aw}-T_{\infty})$ ### 3.1. Local Nusselt's number For laminar flow $Nu^*_x=0.332{Re^*_x}^{1/2}{Pr^*}^{1/3}$ For turbulent flow $5\times 10^5 <Re_x<10^7$ (almost close to [[Dittus-Boelter]]) $Nu^*_x=0.0296{Re^*_x}^{4/5}{Pr^*}^{1/3}$ For turbulent flow $10^7 <Re_x<10^9$ $Nu^*_x=0.185Re^*_x(log{Re^*_x})^{-2.584}{Pr^*}^{1/3}$ To obtain average Nusselts number the equations above need to be integrated over length. Note that recovery factor is different for laminar and turbulent portions of flow so it may lead to a different T* at both locations Reference: - Holman, Heat Transfer