Uniform internal heat generation at \(\dot{q}=6 \times 10^{7} \mathrm{~W} / \mathrm{m}^{3}\) is occurring in a cylindrical nuclear reactor fuel rod of \(60-\mathrm{mm}\) 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 \(a=900^{\circ} \mathrm{C}\) and \(b=-5.26 \times 10^{5}{ }^{\circ} \mathrm{C} / \mathrm{m}^{2}\). The fuel rod properties are \(k=30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, ho=1100 \mathrm{~kg} / \mathrm{m}^{3}\), and \(c_{p}=800 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\).

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

(b) If the reactor power level is suddenly increased to \(\dot{q}_{2}=10^{8} \mathrm{~W} / \mathrm{m}^{3}\), what is the initial time rate of temperature change at \(r=0\) and \(r=30 \mathrm{~mm}\) ?