A heat flux of 1.0 MW/m² passes uniformly through the thickness of the top coat of a thermal barrier coating (thickness 250 µm, thermal conductivity 1.2 W/(m·K)). Treating it as steady one-dimensional heat conduction, which is closest to the temperature difference between the two faces of the top coat?

Answer: C. About 208 K

From Fourier's law q = kΔT/L, ΔT = qL/k = (1.0×10⁶ × 250×10⁻⁶)/1.2 ≈ 208 K. That a coating only about 0.25 mm thick can lower the metal temperature by about 200 K is the benefit of a thermal barrier coating. The creep strength of nickel-based superalloys is sensitive to temperature, so this temperature difference strongly affects the life and the allowable gas temperature. About 300 K multiplies by the thermal conductivity instead of dividing, and about 21 K converts the thickness as 25 µm, an order of magnitude too small. The lower the thermal conductivity and the thicker the coating, the larger the temperature difference, but thicker coatings spall more easily under thermal stress, so the thickness is limited.