An aluminum plate 4 mm thick is mounted in a horizontal position and its bottom surface is well insulated. A special, thin coating is applied to the top surface such that it absorbs 80% of any incident solar radiation, while having an emissivity of 0.25. The density p and specific heat c of aluminum are known to be 2700 kg/m3 and 900 J/kg ∙ K, respectively.

(a) Consider conditions for which the plate is at a temperature of 25°C and its top surface is suddenly exposed to ambient air at T = 20°C and to solar radiation that provides an incident flux of 900 W/m2. The convection heat transfer coefficient between the surface and the air is h = 20 W/m2. K. What is the initial rate of change of the plate temperature?

(b) What will be the equilibrium temperature of the plate when steady-state conditions are reached?

(c) The surface radiative properties depend on the specific nature of the applied coating. Compute and plot the steady-state temperature as a function of the emissivity for 0.05 < ε < 1, with all other conditions remaining as prescribed. Repeat your calculations for values of as = 0.5 and 1.0, and plot the results with those obtained for as= 0.8. If the intent is to maximize the plate temperature, what is the most desirable combination of the plate emissivity and its absorptivity to solar radiation?