During transient operation the steel nozzle of a rocket engine must not exceed a maximum allowable operating temperature of 1500 K when exposed to combustion gases characterized by a temperature of 2300 K and a convection coefficient of 5000 W/m² ∙ K. To extend the duration of engine operation, it is proposed that a ceramic thermal barrier coating (k = 10 W/m ∙ K, a = 6 X 10^-6 m²/s) be applied to the interior surface of the nozzle.

(a) If the ceramic coating is 10 mm thick and at an initial temperature of 300 K. obtain a conservative estimate of the maximum allowable duration of engine operation. The nozzle radius is much larger than the combined wall and coating thickness.

(b) Compute and plot the inner and outer surface temperatures of the coating as a function of time for o < t < 150 s. Repeat the calculations for a coating thickness of 40 mm.