Answer:
The maximum temperature at the center of the rod is found to be 517.24 °C
Explanation:
Assumptions:
1- Heat transfer is steady.
2- Heat transfer is in one dimension, due to axial symmetry.
3- Heat generation is uniform.
Now, we consider an inner imaginary cylinder of radius R inside the actual uranium rod of radius Ro. So, from steady state conditions, we know that, heat generated within the rod will be equal to the heat conducted at any point of the rod. So, from Fourier's Law, we write:
Heat Conduction Through Rod = Heat Generation
-kAdT/dr = qV
where,
k = thermal conductivity = 29.5 W/m.K
q = heat generation per unit volume = 7.5 x 10^7 W/m³
V = volume of rod = π r² l
A = area of rod = 2π r l
using these values, we get:
dT = - (q/2k)(r dr)
integrating from r = 0, where T(0) = To = Maximum center temperature, to r = Ro, where, T(Ro) = Ts = surface temperature = 120°C.
To -Ts = qr²/4k
To = Ts + qr²/4k
To = 120°C + (7.5 x 10^7 W/m³)(0.025 m)²/(4)(29.5 W/m.°C)
To = 120° C + 397.24° C
To = 517.24° C
