$2\mathrm{.}1$ Steam at $150$ C is flowing through a $100$ mm nominal size standard metric pipe. The pipe is carbon steel $)$ and is located in a factory where the air temperature is $20$ C $.$ The convective coefficient between the pipe and the room air is $4\frac{W}{m}2C.$ The pipe is $50$ m long and is uninsulated.

$($a$)$ What is the rate of heat transfer from the bare pipe to the room air?

$($b$)$ Insulation is added to the pipe to reduce the heat transfer. The insulation is $51$ mm thick and has a conductivity of $0\mathrm{.}05\frac{W}{m}C.$ What is the new rate of heat transfer?