Problem 3
Uniform internal heat generation at $\dot{q} = 6 \times 10^7 W/m^3$
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 $r_o = 0.030 m$
$T(r) = a + br^2$, where T is in degrees Celsius and r is in
meters, while $a = 900^\circ C$ and $b = -5.26 \times 10^5 ^\circ C/m^2$.
The fuel rod properties are $k = 30 W/(m K)$,
$\rho = 1100 kg/m^3$, and $c_p = 800 J/(kg K)$.
$T(r) = 900 - 5.26 \times 10^5 r^2$
$\dot{q} - \dot{q}_i = 6 \times 10^7 W/m^3$
$k = 30 W/(m \cdot K)$
$\rho = 1100 kg/m^3$, $c_p = 800 J/(kg \cdot K)$
(a)What is the rate of heat transfer per unit length of the rod at $r = 0$ (the centerline) and at $r = 30 mm$
(the surface)?
(b) If the reactor power level is suddenly increased to $\dot{q} = 10^8 W/m^3$, what is the initial time rate of temper-
ature change at $r = 0$ and $r = 30 mm$?
