Problem$1$

A copper #$12$ AWG wire with PTFE insulation.

The bare wire has a $r_i=0\mathrm{.}001026m$ and the insulation is $0\mathrm{.}000381m$ thick. The thermal conductivity of the insulation is $k=0\mathrm{.}25\frac{W}{m}C.$ The temperature $T_1$ at the interface between the wire and the insulation is kept at $100C,$ and the surrounding air is at $20C.$ The convective coefficient $h$ at the outer surface of the insulation is $10\frac{W}{m}2C.$

$($a$)$ What is the heat transfer rate from the wire per meter length before additional insulation is added?

$($b$)$ If more insulation is added to the wire, will the heat transfer rate increase or decrease?

$($c$)$ If the answer to Part $($b$)$ is that the heat transfer will increase, then how much insulation has to be added for the heat transfer to reach a maximum? Also, what is the value of this maximum heat transfer rate?