$9\mathrm{.}43$ Saturated steam at $4$ bars absolute pressure with a mean velocity of $\backslash (3\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ flows through a horizontal pipe whose inner and outer diameters are $55$ and $65$ mm $,$ respectively. The heat transfer coefficient for the steam flow is known to be $\backslash (11,000\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash$cdot $\backslash$mathrm$\{$~K$\}\backslash ).$

$($a$)$ If the pipe is covered with a $25-$mm$-$thick layer of $\backslash (85\backslash \%\backslash )$ magnesia insulation and is exposed to atmospheric air at $\backslash (25^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ),$ determine the rate of heat transfer by free convection to the room per unit length of the pipe. If the steam is saturated at the inlet of the pipe, estimate its quality at the outlet of a pipe $30$ m long.

$($b$)$ Net radiation to the surroundings also contributes to heat loss from the pipe. If the insulation has a

surface emissivity of $\backslash (\backslash$varepsilon$=0\mathrm{.}8\backslash )$ and the surroundings are at $\backslash ($ T$\_\{\backslash$text $\{$sur $\}\}=$T$\_\{\backslash$infty$\}=25^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ),$ what is the rate of heat transfer to the room per unit length of pipe? What is the quality of the outlet flow?

$($c$)$ The heat loss may be reduced by increasing the insulation thickness and$/$or reducing its emissivity. What is the effect of increasing the insulation thickness to $50$ mm if $\backslash (\backslash$varepsilon$=0\mathrm{.}8\backslash )?$ Of decreasing the emissivity to $0\mathrm{.}2$ if the insulation thickness is $25$ mm $?$ Of reducing the emissivity to $0\mathrm{.}2$ and increasing the insulation thickness to $50$ mm $?$