# Heat transfer and diffusion of particles

# 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 .

In its core, the nuclear reaction releases a volumetric thermal power .

The thermal conductivity of uranium is .

## Question

Determine, in stationary mode, the temperature distribution in the bar.

At the lateral walls of the bar, the temperature is . Calculate ?

### Hint

Define properly the thermal system for which we will apply the first law of thermodynamics : the cylinder of height and radius .

### Solution

The thermal system is the cylinder of height and radius.

The first law of thermodynamics gives in stationary mode :

So :

After an integration :

The maximum temperature is at and is :