Uniform internal heat generation at $700\frac{W}{m^3}$ is occurring in a cylindrical nuclear reactor fuel rod of $70-Fin{m}^{-3}$ diameter, and under steady$-$state conditions the temperature distribution is of the form $T(r)=a+b{r}^3,$ where: $T$ is in degrees Celsius, $r$ is in meters and $a=300C$ and $b=-300C^3.$ The fuel rod properties are $k=70\frac{W}{m}.K,cp=700\frac{J}{k}g.K$ and density is $1000k\frac{g}{m^3}.$ Please answer the following: $($a$)$ What is the rate of heat transfer per unit length of the rod at centerline and at the surface? $($b$)$ If the reactor power level is suddenly increased to $30,000\frac{W}{m^3},$ what is the rate of temperature change with respect to time at centerline and at the surface?

Question $2$:

The thermal conductivity of a sheet $(WL=2m2m)$ of insulation is reported to be $k=0\mathrm{.}2\frac{W}{m}K.$ The measured temperature difference across the $22-$mm$-$thick sheet is $T_1-T_2=22K.$

a$)$ What is the heat flux through the insulation?

b$)$ What is the rate of heat transfer per unit length through the sheet of insulation?