A typical snow igloo (thermal conductivity about 1/10 of ice) is shaped like a h

Question

A typical snow igloo (thermal conductivity about 1/10 of ice) is shaped like a hemisphere of inner radius 1.5 m with 0.36-m thick wall.

What is the conductive heating rate through the walls if the inside temperature is 12 C and the outside temperature is -5 C ?
Hint: Use the conductive heating equation
HConduction=KA(T2T1)/l, where A is surface area, l is the distance that the thermal energy travels, and K is thermal conductivity. To calculate the surface area use the mean radius of the wall that is an average of inner and outer radii.

Explanation / Answer

Heating rate through a spherical shell = 4kr1r2 (T1 - T2) / (r1 - r2)

For a hemishphere the rate will be half that of a sphere

Hence heating rate = 2kr1r2 (T1 - T2) / (r1 - r2)

k for ice = 2.23 W/mK

k for snow gloo = 0.223 W/mK

Hence heating rate = 2 *0.223*1.5*0.36*(12 - (-5)) / (1.5 - 0.36)

heating rate = 2 *0.223*1.5*0.36*(17) / (1.14) = 11.283 W

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