$11$ A thin electrical heater is wrapped around the outer

surface of a long cylindrical tube whose inner surface

is maintained at a temperature of $6C.$ The tube wall

has inner and outer radii of $24$ and $78$ mm $,$ respec$-$

tively, and a thermal conductivity of $10\frac{W}{m}*K.$ The

thermal contact resistance between the heater and the

outer surface of the tube $($per unit length of the tube$)$

is $R_{t,c}^{\text{'}}=0\mathrm{.}01m*\frac{K}{W}.$ The outer surface of the heater

is exposed to a fluid with $T_{}=-10C$ and a convection

coefficient of $h=100\frac{W}{m^2}*K.$ Determine the heater

power per unit length of tube required to maintain the

heater at $T_n=25C.$