Problem $1.(40$ pts$)$ A $1-$m long cylindrical fuel rod is sheathed in a cylindrical metal sleeve. The

metal sleeve has a thermal conductivity, $k.$ The outer sleeve surface has an emissivity, $,$ and

outer surface temperature, $T_2$ at $r_2.$ Convection and radiation occur at the outer sleeve surface.

Contact resistance is negligible. Determine

$($a$)$ The volumetric heat generation of the fuel in $\{$:$\frac{W}{m^3}).$

$($b$)$ The inner surface temperature of the metal sleeve, $T_1$ at $r_1.$

$($Note: the Stefan$-$Boltzmann constant is $=5\mathrm{.}67{10}^{-8}\frac{W}{m^2}-K^4)$

Surrounding: $T_{\text{s}\text{u}\text{r}\text{r}}$

Known variables:

Ambient: $T_{},h$

$r_1=5cm$

$r_2=10cm$

$k=12\frac{W}{m}*K$

$T_2=200C$

$T_{}=T_{\text{s}\text{u}\text{r}\text{r}}=20C$

$h=15\frac{W}{m^2}-K$

$=0\mathrm{.}8$