Steam at $$$1=320$C flows in a cast iron pipe $($k $=80$ W$/$m $$C$)$ whose inner and outer diameters are $1=5$ cm and $2=5\mathrm{.}5$ cm$,$ respectively.

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The pipe is covered with $17-$cm$-$thick glass wool insulation with k $=0\mathrm{.}05$ W$/$m $$C$.$

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Heat is lost to the surroundings at $$$2=5$C by natural convection and radiation, with a combined heat transfer coefficient of $2=18$ W$/$m$^2$C$.$

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Taking the heat transfer coefficient inside the pipe to be $1=60$ W$/$m$^2$C$.$

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Investigate the temperature and the rate of heat loss from the steam per unit length of the pipe.