Chapter 1, Problem 1.45P

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 $\rho$ and specific heat c of aluminum are known to be $2700{\text{kg/m}}^{\text{3}}$ and $900\text{J/kg}\cdot \text{K,}$ respectively.

1. Consider conditions for which the plate is at a temperature of $25°\text{C}$ and its top surface is suddenly exposed to ambient air at $T_{\infty }=20°\text{C}$ and to solar radiation that provides an incident flux of $900{\text{W/m}}^{\text{2}}.$ The convection heat transfer coefficient between the surface and the air is $h=20{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ What is the initial rate of change of the plate temperature?
2. What will be the equilibrium temperature of the plate when steady-state conditions are reached?
3. 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 $0.05\le \epsilon \le 1,$ with all other conditions remaining as prescribed. Repeat your calculations for values of as ${\alpha }_S=0.5$ and 1.0, and plot the results with those obtained for ${\alpha }_S=\mathrm{0.8.}$ 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?