A $0\mathrm{.}25-$m$-$diameter, thin$-$walled steel pipe is used to transport saturated steam at a pressure of $20$ bars in a room for which the air temperature is $25C$ and the convection heat transfer coefficient at the outer surface of the pipe is $20\frac{W}{m^2}*K.($a$)$ What is the heat loss per unit length from the bare pipe $($no insulation$)?($b$)$ Estimate the heat loss per unit length if a $50-$mm$-$thick layer of insulation $($magnesia$,85\%,k=0\mathrm{.}058\frac{W}{m}*K)$ is added. The steel and magnesia may each be assumed to have an emissivity of $0\mathrm{.}8,$ and the steam$-$side convection resistance may be neglected. $($c$)$ The costs associated with generating the steam and installing the insulation are known to be $$\frac{4}{{10}^9}J$ and $$\frac{100}{m}$ of pipe length, respectively. If the steam line is to operate $7500\frac{h}{y}r,$ how many months are needed to pay back the initial investment in insulation?

$($a$)q_1^{\text{'}}=$

$\frac{W}{m}$

$($b$)q_2^{\text{'}}=$

$\frac{W}{m}$

$($c$)$ Pay Back Period$=$ mo