Turbine blades mounted to a rotating disc in a gas turbine engine are exposed to a gas stream that is at $T_{}=1200C$ and maintains a convection coefficient of $h=200\frac{W}{m^2}*K$ over the blade. The blades, which are fabricated from Inconel, $k$~~$20\frac{W}{m}*K,$ have a length of $L=50mm.$ The blade profile has a uniform cross$-$sectional area of $A_c=6{10}^{-4}{m}^2$ and a perimeter of $P=110mm.$ A proposed blade$-$cooling scheme, which involves routing air through the supporting disc, is able to maintain the base of each blade at a temperature of $T_b=300C.$

$($a$)$ If the maximum allowable blade temperature is $1050C$ and the blade tip may be assumed to be adiabatic, is the proposed cooling scheme satisfactory?

$($b$)$ For the proposed cooling scheme, what is the rate at which heat is transferred from each blade to the coolant, W$?$

$q_b=$

W

Physical Properties Mathematical Functions