Bonus problem: Uniform internal heat generation is occurring in a cylindrical nuclear reactor

fuel rod of $50$ mm diameter. Under steady$-$state conditions the temperature distribution is of the

form $T(r)=a+b{r}^2,$ where $T$ is $C$ and $r$ is in meters, while $a=800C$ and $b=-4\mathrm{.}167$

$105\frac{C}{m^2}.$ The fuel rod properties are $k=30\frac{W}{m}*K,=1100k\frac{g}{m^3},Cp=800\frac{J}{k}g*K$

a$.$ What is the rate of heat transfer per unit length of the rod at $r=0($centerline$)$ and $r=25mm$

$($surface$)?$

b$.$ Determine the volumetric heat generation rate $(\frac{W}{m^3})$ in the rod.

b$.$ If the reactor power level is suddenly increased to ${10}^8\frac{W}{m^3},$ what is the initial time rate of

temperature change at $r=0$ and $r=25mm?$