Problem $4$: $(20$ pts$)$

The wind chill, which is experienced on a cold, windy day, is related to increased heat transfer from exposed human skin to the surrounding atmosphere.

Consider a layer of fatty tissue that is $3$ mm thick and whose interior surface is maintained at a temperature of $36C.$ On a calm day the convection heat transfer

coefficient at the outer surface is $25\frac{W}{m^2}*K,$ but with $30k\frac{m}{h}$ winds it reaches $65\frac{W}{m^2}*K.$ In both cases the ambient air temperature is $-15C.$

$($a$)$ What is the ratio of the rate of heat loss per unit area from the skin for the calm day to that for the windy day?

$($b$)$ What will be the skin outer surface temperature for the calm day? For the windy day?

$($c$)$ What temperature would the air have to assume on the calm day to produce the same heat rate occurring with the air temperature at $-15C$ on the windy day?

$[$please use $k=0\mathrm{.}2\frac{W}{m}-k$ for human skin$]$