When the ceiling surface temperature in the PSU Ice Pavilion drops below the dew point of the air in the ice rink, water condensing on the ceiling will drip onto the ice surface, causing bumps to form. Your assignment is to examine the effect of the emissivity of the ice rink ceiling on the propensity for condensation; specifically, is it best to make the ceiling have a high or low emissivity? The ice rink can be modeled as a large rectangular building with a $\backslash (60\backslash$mathrm$\{$~m$\}\backslash$times $20\backslash$mathrm$\{$~m$\}\backslash )$ floor space and a $10\mathrm{.}0$ m high ceiling. You may consider the ceiling to be a diffuse gray surface, the walls $($all $4$ of them$)$ to be blackbodies at $\backslash (10^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ),$ and the ice to be a blackbody maintained at $\backslash (-7^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The air in the rink is maintained at $\backslash (15^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ with a relative humidity of $\backslash (70\backslash \%\backslash ).$ The convection heat transfer coefficient between the air and the ceiling, ice surface, and walls is $\backslash (5\mathrm{.}0\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash$mathrm$\{$~K$\}\backslash ).$ The ceiling is separated from $\backslash (-5^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ external air by a $0\mathrm{.}30$ m $-$thick layer of insulation $(\backslash ($ k$\_\{$i$\}=0\mathrm{.}045\backslash$mathrm$\{$Wm$\}^\{-1\}\backslash$mathrm$\{$~K$\}^\{-1\}\backslash )),$ and the external convective resistance is negligible.

a$.$ Draw a resistance network for a $3-$surface enclosure that represents the radiation heat transfer between the ceiling, ice, and side walls. Label the nodes and resistances and calculate the value of each resistance. The ice$-$ceiling view factor may be calculated using the result for aligned parallel rectangles shown below. Other view factors can be determined using the summation and reciprocity rules.

b$.$ Consider a flat ceiling with highly reflective panels $\backslash (\backslash$left$(\backslash$varepsilon$\_\{$r$\}=0\mathrm{.}05\backslash$right$)\backslash )$ or painted panels $\backslash (\backslash$left$(\backslash$varepsilon$\_\{$p$\}=0\mathrm{.}95\backslash$right$)\backslash ).$ Calculate the ceiling temperature in each case to determine if condensation would occur on the ceiling. The dewpoint for the specified air temperature and relative humidity can be estimated using the Antoine equation.