An overhead $25-m-$long, uninsulated industrial steam pipe of $162-mm$ diameter is routed through a building whose walls and air are at $25C.$ Pressurized steam maintains a pipe surface temperature of $188C,$ and the coefficient associated with natural convection is $h=10\frac{W}{m^2}*K.$ The surface emissivity is $=0\mathrm{.}8.$

$($a$)$ What is the rate of heat loss from the steam line?

$($b$)$ If the steam is generated in a gas$-$fired boiler operating at an efficiency of ${}_f=0\mathrm{.}90$ and natural gas is priced at $C_g=$$$0\mathrm{.}02$ per MJ$,$ what is the annual cost of heat loss from the line?

Physical Properties Mathematical Functions