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

state heat conduction through a cylindrical wall with inner radius $r_1=0\mathrm{.}1m$ and outer radius

$r_2=0\mathrm{.}2m,$ 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 generation $($if there is any$)q^{}($in $\frac{W}{m^3})$ in the middle of the tube wall, $r$

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

$($b$)$ Which case$($s$)$ of $T(r)$ have uniform $q^{}$ in the entire wall.

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