Problem #$2(40$ pts$)$

Steam is flowing through a stainless steel AISI $304$ pipe with thermal conductivity of $19\frac{W}{m}.K.$ The steam temperature is $200C$ and the convective heat transfer coefficient between the steam and the inner surface of the pipe is $h_s=65\frac{W}{m^2}.K.$ The inner radius of the steel pipe is $r_l=0\mathrm{.}08m,$ the outer radius of the pipe is $r_2=0\mathrm{.}085m,$ and the length of the pipe is $L=10m.$ An insulation made of glass fiber, coated, duct liner is applied on the outer surface of the pipe to reduce the heat loss. The outside radius of insulation is $r_3=0\mathrm{.}1m.$ The temperature of the air surrounding the pipe is $T_{}=25C$ and the convective heat transfer coefficient is $h_{}=15\frac{W}{m^2}.K.$

$($i$)$ Draw a schematic of the problem.

$($ii$)$ What is the total heat loss from the steam to the surrounding air?

$($iii$)$ What is the temperature at the interface between the steel pipe and the insulation?

$($iv$)$ Will the given thickness of insulation reduce the heat loss?