Heat transfer and diffusion of particles
Cooling fin
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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.
A video about "Cooling fin" (Ecole Centrale de Paris)
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.
Hint
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 conductoconvective 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.