Question

A turbine blade made of a metal alloy (k = 17 W/m·K) has a length of 5.3 cm, a perimeter of 11 cm, and a cross-sectional area of 5.13 cm2. The turbine blade is exposed to hot gas from the combustion chamber at 1053°C with a convection heat transfer coefficient of 538 W/m2·K. The base of the turbine blade maintains a constant temperature of 450°C and the tip is adiabatic.

Answer 1

The base of the turbine blade maintains a constant temperature of 450°C and the tip is adiabatic is 1076.67° C.

Explanation:

The rate of heat transfer for the turbine blade is expressed.

$Q=\sqrt{h P k A}\left(T_{b}-T_{\infty}\right) \tanh (m L)$

The parameters are calculated.

$m=\sqrt{\frac{h P}{k A}}$

$=\sqrt{\frac{\left(538 \mathrm{W} / \mathrm{m}^{2} \mathrm{K}\right)(0.11 \mathrm{m})}{(17 \mathrm{W} / \mathrm{m} \cdot \mathrm{K})\left(0.000513 \mathrm{m}^{2}\right)}}$

m = 82.38

Hence the rate of heat transfer for the turbine blade is

Q = $\sqrt{(538)(0.11)(17)(0.000513)(450-1093)tanh(82.38)(0.053)}$

Q = -461.758 W

The temperature at the blade tip is calculated.

$\frac{T_{x}-T_{\infty}}{T_{b}-T_{\infty}}$ = $\frac{1}{\cosh (m L)}$

$\frac{T_{x}-1093^{\circ} \mathrm{C}}{450^{\circ} \mathrm{C}-1093^{\circ} \mathrm{C}}$ = $\frac{1}{\cosh (82.38 \times 0.053)}$

$\frac{T_{x}-1093^{\circ} \mathrm{C}}{450^{\circ} \mathrm{C}-1093^{\circ} \mathrm{C}}$ = 0.0254

$T_{x}$ = 1076.67° C