A hemispherical igloo in the Arctic has an inner radius $r_i=4m$ and is composed of a composite wall of hardwood tiles of thickness $t_{\text{t}\text{i}\text{l}\text{e}}=1cm$ on the inside, and ice blocks of thickness $t_{\text{i}\text{c}\text{e}}=10cm$ on the outside.

no

A A thin water barrier is placed between the wood and the ice. The contact between the barrier and ice, and thick barrier and wood, results in a combined thermal contact resistance, $R_c$ of $5{10}^{-2}{m}^2-\frac{K}{W}.$ The air in interior of the igloo needs to be maintained at a temperature $T_{\text{i}\text{n}\text{f}\text{.}}=298K$ by a circulating heater fan that creates a convective heat transfer coefficient $h_i=20\frac{W}{m^2}-K.$ It is the winter season in the Arctic, and the external convective heat transfer coefficient, $h_o=45\frac{W}{m^2}-K$ at an air temperature of $T_{\text{i}\text{n}\text{f}\text{,}o}=265K.$ The floor of the igloo is also made up of hardwood tiles of the same thickness of $1$ cm and the igloo is built over a very thick layer of ice $(t_{\text{f}\text{l}\text{o}\text{o}\text{r}\text{,}\text{i}\text{c}\text{e}}=5m).$ Similar to the shell, a water barrier with the same combined thermal contact resistance separates the wooden floor tiles from the ice. The temperature of the soil underneath the thick layer of ice is $T_{\text{s}\text{o}\text{i}\text{l}}=270K.$ The emissivity of ice to long wavelength radiation is ${0\mathrm{.}96}_{?}.$

Under steady state conditions, determine the heat rate that needs to be provided to the heater fan to keep the interior at the desired temperature $T_{\text{i}\text{n}\text{f}\text{,}i}.$

What is the heat loss $($in W $)$ to radiation exchange with the sky$,$ convection to the surroundings, and conduction to the soil beneath? Suggest a method by which the heater fan power can be reduced. There is no need to show calculations.

The thermal conductivity of the wooden tiles is $k_{wd}=0\mathrm{.}5\frac{W}{m}-K,$ and that of ice is $k_{\text{i}\text{c}\text{e}}=2\mathrm{.}22\frac{W}{m}-K.$

Neglect radiation interactions within the igloo and only consider convection within. Assume that $h_i$ is the same for air interaction with the igloo shell and the floor.