Chapter 4

Heat transfer and diffusion of particles

Temperature wave

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 basement is considered to be a half-infinite and homogeneous medium, with its thermal conductivity, its density, its mass thermal capacity and located in the half-space .

We suppose that the temperature of the ground ( ) is subjected to sinusoidal variations :

Question

Determine the temperature at the depth (use the complex notations ) in stationary mode.

Hint

  • Solve the heat equation using the complex numbers method.

  • This exercise deals with the equivalent of the skin effect in the domain of electromagnetism.

Solution

The heat equation is :

We use the complex numbers method and we have :

The heat equation leads to :

That is to say :

Therefore :

We note :(skin thickness)

We go back to real numbers, and we keep only the solution that does not diverge in the infinite :

Question

Calculate the velocity of the thermal wave that has been obtained.

Hint

Solution

The velocity of the wave is :

Question

We consider daily temperature changes, the one on the floor varying between at night and during the day.

From which depth are the temperature changes less than ?

Calculate .

We give :

Let us consider annual temperature changes, from to . Answer the same questions.

Hint

Solution

First case : and .

second case : and  :

The temperature in an inter cave is fresh in summer and mild in winter.

Indeed, at a depth, the evolution of the temperature is the same than outside days late.

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Cooling fin
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Probabilistic approach of scattering