Chapter 13, Problem 13.97P

Most architects know that the ailing of an ice-skating rink must have a high reflectivity. Otherwise, condensation may occur on the ceiling, and water may drip onto the ice, causing humps on the skating surface. Condensation will occur on the ceiling when its surface temperature drops below the dew point of the rink air. Your assignment is to perform an analysis to determine the effect of the ceiling emissivity on the ceiling temperature, and hence the propensity for condensation.

The rink has a diameter of $D=50\text{m}$ in and a height of $L=10\text{m}$ , and the temperatures of the ice and walls are −5°C and 15°C, respectively. The rink air temperature is 15°C, and a convection coefficient of $5{\text{W/m}}^{\text{2}}\cdot \text{K}$ characterizes conditions on the ceiling surface. The thickness and thermal conductivity of the ceiling insulation are 0.3 m and $0.035\text{W/m}\cdot \text{K}$ , respectively, and the temperature of the outdoor air is −5°C. Assume that the ceiling is a diffuse-gray surface and that the walls and ice may be approximated as blackbodies.
(a) Consider a flat ceiling having au emissivity of 0.05 (highly reflective panels) or 0.94 (painted panels). Perform an energy balance on the ceiling to calculate the corresponding values of the ceiling temperature. If the relative humidity of the rink air is 70%, will condensation occur for either or both of the emissivities?
(b) For each of the emissivities, calculate and plot the ceiling temperature as a function of the insulation thickness for $0.1\le t\le 1\text{m}$ . Identify conditions for which condensation will occur on the ceiling.