Heat transfers in a cylinder
Take 10 minutes to prepare this exercise.
Then, if you lack ideas to begin, look at the given clue and start searching for the solution.
A detailed solution is then proposed to you.
If you have more questions, feel free to ask them on the forum.
A cylindrical uranium bar has a radius \(R = 30\) \(mm\).
In its core, the nuclear reaction releases a volumetric thermal power \(p = 700\;MW.m^{-3}\).
The thermal conductivity of uranium is \(\lambda = 27 \;W.m^{-1} .K^{-1}\).
Question
Determine, in stationary mode, the temperature distribution in the bar.
At the lateral walls of the bar, the temperature is \(T_0=200°C\). Calculate \(T_{max}\) ?
Indice
Define properly the thermal system for which we will apply the first law of thermodynamics : the cylinder of height \(h\) and radius \(r<R\).
Solution
The thermal system is the cylinder of \(h\) height and \( r<R\) radius.
The first law of thermodynamics gives in stationary mode :
\(dU = 0 = \pi {r^2}hpdt - 2\pi rh{j_{th}}(r)dt\)
So :
\({j_{th}}(r) = \frac{p}{2}r = - \lambda \;\frac{{dT(r)}}{{dr}}\)
After an integration :
\(T(r) = {T_0} + \frac{p}{{4\lambda }}\left( {{R^2} - {r^2}} \right)\)
The maximum temperature is at \(r=0\) and is :
\({T_{\max }} = {T_0} + \frac{p}{{4\lambda }}{R^2}\)