Chapter 3, Problem 3.126P

Turbine blades mounted to a rotating disc in a turbine engine are exposed to a gas stream that is at $T_{\infty }=1200°\text{C}$ and maintains a convection coefficient of $h=250{\text{W/m}}^{\text{2}}\cdot \text{K}$ over the blade.

The blades, which are fabricated from Inconel, $k\approx 20\text{W/m}\cdot \text{K,}$ have a length of $L=50\text{mm}\text{.}$ The blade profile has a uniform cross-sectional area of $A_c=6\times {10}^{-4}{\text{m}}^{\text{2}}$ and a perimeter of $P=110\text{mm}\text{.}$ 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 $T_b=300°\text{C}\text{.}$

1. If the maximum allowable blade temperature is 1050°C and the blade tip may be assumed to be adiabatic is the proposed cooling scheme satisfactory?
2. For the proposed cooling scheme, what is the rate at which heat is transferred from each blade to the coolant?