# Heat transfer and diffusion of particles

# A lake freezes

Take 15 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.

The temperature of the air is and the temperature of the liquid water is .

We give the massic latent heat of fusion of ice , its density and its thermal conductivity .

The thermal capacity of ice is supposed to be negligible.

Let be the height of ice at time . At , .

## Question

Establish the differential equations which have and as solutions.

Solve them and determine .

### Hint

### Solution

The heat equation in the ice :

So if we suppose that is almost :

So :

We make the approximation of the quasi-stationary mode.

The heat flow in the ice is due to its solidification, that is to say, between and :

So :

That is to say :

And :

After integration :