Problem $3$

Uniform internal heat generation at $q^{}=6{10}^7\frac{W}{m^3}$ is occurring in a cylindrical nuclear reactor fuel rod of $60-mm$ diameter, and under steady$-$state conditions the temperature distribution is of the form $T(r)=a+b{r}^2,$ where T is in degrees Celsius and $r$ is in meters, while $a=900C$ and $b=-5\mathrm{.}26{10}^5\frac{C}{m^2}.$ The fuel rod properties are $k=30\frac{W}{mK},=1100k\frac{g}{m^3},$ and $c_p=800\frac{J}{kgK}.$

$($a$)$ What is the rate of heat transfer per unit length of the rod at $r=0($the centerline$)$ and at $r=30mm($the surface$)?$

$($b$)$If the reactor power level is suddenly increased to $q^{}={10}^8\frac{W}{m^3},$ what is the initial time rate of temperature change at $r=0$ and $r=30mm?$