Heat transfer, via conduction, plays an important role in daily life. This is

especially true for the Inuits, who find shelter in their snow$-$built Igloos that protect them

from the low outside temperature.

In this problem, we will consider a half spherical igloo of radius $R=2m.$ The wall of the

igloo consists of $30cm$ of thickly compressed snow blocks. The heat conductivity of the

compressed snow is $0\mathrm{.}3\frac{W}{m*(K)}.$

If the outside temperature is $-20C$ and the inside needs to be at $+20C$ : how many

humans need to be inside the igloo to achieve this with only the heat produced by the

humans?

Use the following data and assumptions:

the situation is steady state

heat is only lost through the wall of the igloo $($so we ignore the floor$)$

the only resistance against the heat flow is in the walls of the igloo

a human produces about $150$ Watt

a$.$ In a steady state, what is the relation between the heat flow through the wall and

the temperature difference across the wall?

b$.$ Set up a heat balance for the igloo.

c$.$ Compute how many people are roughly needed for the situation sketched above.