Chapter 12, Problem 12.96P

One scheme for extending the operation of gas turbine blades to higher temperatures involves applying a ceramic coating to the surfaces of blades fabricated from a superalloy such as inconel. To assess the reliability of such coatings, an apparatus has been developed for testing samples under laboratory conditions. The sample is placed at the bottom of a large vacuum chamber whose walls are cryogenically cooled and which is equipped with a radiation detector at the top surface. The detector has a surface area of $A_d={10}^{-5}{m}^2$ ,Is locate data distance of $L_{s-d}=1m$ from the sample, and views radiation originating from a portion of the ceramic surface having an area of $\Delta A_c={10}^{-4}{m}^2$ . An electric heater attached to the bot- tom of the sample dissipates a uniform heat flux, $q_h^{"}$ , which is transferred upward through the sample. The bottom of the heater and sides of the sample are well insulated.

Consider conditions for which a ceramic coating of thickness $L_c=0.5mm$ and thermal conductivity $K_c=6W/m\cdot k$ has been sprayed on a metal substrate of thickness $L_s=8mm$ and thermal conductivity $K_s=25W/m\cdot k$ . The opaque surface of the ceramic may be approximated as diffuse and gray, with a total, hemi- spherical emissivity of ${\epsilon }_c=0.8$ .

(a) Consider steady-state conditions for which the bottom surface of the substrate is maintained at $T_1=1500K$ , while the chamber walls (including the surface of the radiation detector) are maintained at $T_w=90K$ . Assuming negligible thermal contact resistance at the ceramic−substrate inter- face, determine the ceramic top surface temperature $T_2$ and the heat flux $q_h^{"}$ .

(b) For the prescribed conditions, what is the rate at which radiation emitted by the ceramic is intercepted by the detector?

(c) After repeated experiments, numerous cracks develop at the ceramic−substrate interface, creating an interfacial thermal contact resistance. If $T_w$ and $q_h^{"}$ are maintained at the conditions associated with part (a), will $T_1$ increase, decrease, or remain the same? Similarly, will $T_2$ increase, decrease, or remain the same? In each case, justify your answer.