QUESTION $1$

$1\mathrm{.}1$ A steam pipe is lagged with $75$ mm thick insulation material having a thermal conductivity of $\backslash (0,069\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}.\backslash$mathrm$\{$K$\}\backslash )$ and covered with a thin layer of material which has a transfer coefficient of $\backslash (6,8\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash$mathrm$\{$~K$\}\backslash ).$ The pipe thickness is thin compared to its radius and has a temperature of $\backslash (180^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ throughout its wall thickness. The diameter of the pipe is $100$ mm and the ambient temperature is $\backslash (19^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$

Calculate each of the following:

$1\mathrm{.}1\mathrm{.}1$ Temperature on the outside surface

$1\mathrm{.}1\mathrm{.}2$ Rate of surface heat transfer per metre length of pipe

$1\mathrm{.}1\mathrm{.}3$ Condition of steam at the delivery end if the pipe is $120$ m long, $\backslash (500\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$h$\}\backslash )$ steam is transmitted, and the initial dryness fraction of the steam is $0,9.$ Ignore the pressure drop due to friction.