Most architects know that the ceiling of an

Chapter 13, Problem 13.97

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Most architects know that the ceiling of an ice-skatingrink must have a high reflectivity. Otherwise, conden-sation may occur on the ceiling, and water may driponto the ice, causing bumps on the skating surface.Condensation will occur on the ceiling when its sur-face temperature drops below the dew point of therink air. Your assignment is to perform an analysis todetermine the effect of the ceiling emissivity on theceiling temperature, and hence the propensity for condensation. The rink has a diameter of D?50 m and a height ofL?10 m, and the temperatures of the ice and wallsare?5?C and 15?C, respectively. The rink air tempera-ture is 15?C, and a convection coefficient of 5 W/m2?Kcharacterizes conditions on the ceiling surface. Thethickness and thermal conductivity of the ceiling insu-lation are 0.3 m and 0.035 W/m?K, respectively, andthe temperature of the outdoor air is?5?C. Assumethat the ceiling is a diffuse-gray surface and that thewalls and ice may be approximated as blackbodies.(a) Consider a flat ceiling having an emissivity of0.05 (highly reflective panels) or 0.94 (paintedpanels). Perform an energy balance on the ceilingto calculate the corresponding values of the ceil-ing temperature. If the relative humidity of therink air is 70%, will condensation occur for eitheror both of the emissivities?(b) For each of the emissivities, calculate and plot theceiling temperature as a function of the insulationthickness for 0.1t1 m. Identify conditionsfor which condensation will occur on the ceiling.