Chapter 1, Problem 1.45P

An aluminum plate 4 mm thick is mounted in a horizontal position, and its bottom surface ¡s 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 $\rho$ and specific heat c of aluminum are known to be $2700\text{Kg}/{\text{m}}^3$ and $900\text{J/Kg}\cdot \text{K}$ , respectively.
(a) Consider conditions for which the plate ¡s at a temperature of 25°C and its top surface is suddenly exposed to ambient air at $T_{\infty }=20°C$ and to solar radiation that provides an incident flux of $900\text{W}/{\text{m}}^2$ . The convection heat transfer coefficient between the surface and the air is $h=20\text{W}/{\text{m}}^{\text{2}}\cdot \text{K}$ . What ¡s 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 $0.05\le \in \le 1$ , with all other conditions remaining as prescribed. Repeat your calculations for values of ${\alpha }_s=0.5$ and 1.0, and plot the results with those obtained for ${\alpha }_s=0.8$ . 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?