Chapter 3, Problem 3.30P

The performance of gas turbine engines may beimproved by increasing the tolerance of the turbineblades to hot gases emerging from the combustor. Oneapproach to achieving high operating temperaturesinvolves application of a thermal barrier coating (TBC) to the exterior surface of a blade, while passingcooling air through the blade. Typically, the blade ismade from a high-temperature superalloy, such asInconel $\left(k\approx 25\text{W/m}\cdot \text{K}\right),$ while a ceramic, such aszirconia $\left(k\approx 1.3\text{W/m}\cdot \text{K}\right),$ is used as a TBC.

Consider conditions for which hot gases at $T_{\infty ,o}=1700\text{K}$ and cooling air at $T_{\infty ,i}=400\text{K}$ provide outerand inner surface convectioncoefficients of $h_o=1000{\text{W/m}}^2\cdot \text{K}$ and $h_i=500{\text{W/m}}^2\cdot \text{K,}$ respectively. If a0.5-mm-thick zirconia TBC is attached to a 5-mm-thickInconel blade wall by means of a metallic bondingagent, which provides an interfacial thermal resistanceof $R_{t,c}^{"}={10}^{-4}{\text{m}}^2\cdot \text{K/W,}$ can the Inconel be maintainedat a temperature that is below its maximumallowable value of 1250 K? Radiationeffects may beneglected, and the turbine blade may be approximatedas a plane wall. Plot the temperature distribution withandwithout the TBC. Are there any limits to the thicknessof the TBC?