Problem $3.(30$ pts$)$ Consider the following temperature profiles for one$-$dimensional, steady$-$

siate heat conduction through a glindrical wall with inner redius $n=0\mathrm{.}1m$ and outer ratio

$n=0\mathrm{.}2m,$ and constant thermal conductivity $k=10\frac{W}{m}*K.$

Case $1$: $T(r)=5ln(\frac{r}{r_2})+10$

Case $2$: $T(r)=-5r^2+10r+50$

Case $3$: $T(r)=\frac{1}{9}{r}^3-\frac{2}{5}{r}^2$

Determine

$($a$)$ The volumetric heat gencration $($if there is any$)($in $\frac{W^{\text{'}}}{m^{\text{'}}})$ in the middle of the wbe wall, $r$

$=0\mathrm{.}15m,$ for each case.

$($b$)$ Which caso$(s)$ of $T(r)$ have uniform $$ in the entire wall.

$($Note: The units for $T$ and $r$ are $C$ and $m,$ respectively.$)$