A $0\mathrm{.}20-$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 $250$C and the convection

heat transfer coefficient at the outer surface of the pipe is $20$ W$/$m$2$

$.$K$.$

$($a$)$ What is the heat loss per unit length from the bare pipe $($no insulation$)?[6$ marks$]$

$($b$)$ Estimate the heat loss per unit length if a $50-$mm$-$thick layer of insulation

$($magnesia$,85\%)$ 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.

$[9$ marks$]$

$($c$)$ The costs associated with generating the steam and installing the insulation are

known to be $$4/109$ J and $$100/$m of pipe length, respectively. If the steam line is to

operate $7500$ h$/$yr$,$ how many years are needed to pay back the initial investment in

insulation? $[10$ marks$]$

Given: Saturated water $($p$=20$ bar$),$ Tsat $=$ Ts $=486$ K$.$

Properties of Magnesia $(85\%)($T $=302$ K$),$ Thermal conductivity, k $=0\mathrm{.}058$ W$/$m$.$K

$[$Total: $25$ marks$]$