Chapter 2, Problem 2.28P

Uniform internal heat generation at $q=5\times {10}^7{\text{W/m}}^{\text{3}}$ is occurring in a cylindrical nuclear reactor fuel rod of 50-mm diameter, and under steady-state conditions the temperature distribution is of the form $T\left(r\right)=a+b{r}^2,$ where T is in degrees Celsius and r is in meters, while $a=800°\text{C}$ and $b=-4.167\times {10}^5°{\text{C/m}}^{\text{2}}.$ The fuel rod properties are $k=30\text{W/m}\cdot \text{K,}\rho =1100{\text{kg/m}}^{\text{3}},$ and $c_{\rho }=800\text{J/kg}\cdot \text{K}\text{.}$

1. What is the rate of heat transfer per unit length of the rod at $r=0$ (the centerline) and at $r=25$ mm (the surface)?
2. If the reactor power level is suddenly increased to $q_2={10}^8{\text{W/m}}^{\text{3}},$ what is the initial time rate of temperature change at $r=0$ and $r=25\text{mm}?$