$*$OMIT PART B$*$ Water flows through a pipe at an average temperature of T$\backslash$infty $=70\backslash$deg C$.$ The inner and outer radii of the pipe are r$1=6$ cm and r$2=6\mathrm{.}5$ cm$,$ respectively. The outer surface of the pipe is wrapped with a thin electric heater that consumes $300$ W per m length of the pipe. The exposed surface of the heater is heavily insulated so that all heat generated in the heater is transferred to the pipe. Heat is transferred from the inner surface of the pipe to the water by convection with a heat transfer coefficient of h $=85$ W$/$m$2$ K$.$ Assuming constant thermal conductivity and one$-$dimensional heat transfer, express the mathematical formulation $($the differential equation and the boundary conditions$)$ of the heat conduction in the pipe during steady operation. Solve the problem and give the particular solution. $($a$)$ Specified temperature of $50$oC: $()150$o T r $=$ C $($c$)$ Convection to a medium at T$$ with a heat transfer coefficient of h$=85$ W$/$m$2/$K$.$ The thermal conductivity of the pipe is $15$ W$/$K$*$m Water flows through a pipe at an average temperature of $T_{}=70C.$ The inner and outer radii

of the pipe are $r_1=6cm$ and $r_2=6\mathrm{.}5cm,$ respectively. The outer surface of the pipe is

wrapped with a thin electric heater that consumes $300$ W per $m$ length of the pipe. The

exposed surface of the heater is heavily insulated so that all heat generated in the heater is

transferred to the pipe. Heat is transferred from the inner surface of the pipe to the water by

convection with a heat transfer coefficient of $h=85\frac{W}{m^2}$ K$.$ Assuming constant thermal

conductivity and one$-$dimensional heat transfer, express the mathematical formulation

$($the differential equation and the boundary conditions$)$ of the heat conduction in the pipe

during steady operation. Solve the problem and give the particular solution.

$($a$)$ Specified temperature of $50C$:$T(r_1)=50C$

$($b$)$ Specified heat flux of $45\frac{W}{m^2}$ towards the center: $-k\frac{dT}{dr}{|}_{\text{t}\text{i}\text{l}\text{d}\text{e}(r)}=-45\frac{W}{m^2}$

$($c$)$ Convection to a medium at $T_{}$ with a heat transfer coefficient of $h=85\frac{W}{m^2}\frac{?}{K}$