Chapter 4

# Cooling fin

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.

Let's consider a solid body (for instance, a power transistor housing) with its temperature, which is higher than , temperature of the room air.

In order to cool the body down, we put into place a cooling fin, made of a cylinder of length and with its section.

We will study this fin in stationary mode.

We will also suppose that the temperature of the bar only depends on , in the direction of the length of that bar, that is to say .

The fin is not insulated and loses heat through its side walls, those losses being defined by Newton's law :

represents the loss energy per unit of surface, on the side wall of the fin, at the abscissa .

is the thermal conductivity of the fin.

Let's suppose that the length of the fin is infinite.

Cooling fin

A video about "Cooling fin" (Ecole Centrale de Paris)

Conduction stationnaire linéaire : bilan d'une ailette

## Question

Determine , the temperature in the fin.

### Hint

Apply the first law of thermodynamics to a element of the fin of length, and take into account the conduction but also the convection on the side surface.

### Solution

We apply the first law of thermodynamics to an element of the fin of length :

That is to say :

Using Fourier's law :

The solution to this differential equation is :

The length of the fin is infinite, so :

The boundary condition at allows us to calculate :

We notice that the temperature of the fin tends to be the same as the room air when the distance between and the origin is than the characteristic distance .

## Question

Calculate in two ways the power provided by the housing to the bar.

### Solution

The objective of the cooling fin is that we can finally wonder about the value of the thermal flow eliminiated by the fin into the atmosphere.

• We can determine this flow with Fourier's law at .

Indeed, in stationary mode, those two thermal fluxs are identical since the fin transfers to the room air everything that comes in it.

So this flow is :

• We could have obtained the same result by integrating the conducto-convective flow along the whole side surface of the bar :

So :

We finally have the expression for the power.

Without the fin, the flow would have been :

The ratio between those flows is :

With usual numeric values, the ratio is about ; we can see here the interest of having a cooling fin.

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Thermal convection
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Temperature wave