The temperature distribution in fins and pins

As already shown in section 1.2.3, heat transfer between two fluids can be improved if the surface area available for heat transfer, on the side with the fluid which has the lower heat transfer coefficient, is increased by the addition of fins or pins. However this enlargement of the area is only partly effective, due to the existence of a temperature gradient in the fins without which heat could not be conducted from the hn base. Therefore the average overtemperature decisive for the heat transfer to the fluid is smaller than the overtemperature at the hn base. In order to describe this effect quantitatively, the fin efficiency was introduced in section 1.2.3. Its calculation is only possible if the temperature distribution in the hn is known, which we will cover in the following. Results for the hn efficiencies for different hn and pin shapes are given in the next section. 

In order to calculate the temperature distribution some limiting assumptions have to be made  

The hn (or pin) is so thin that the temperature only changes in the direction from hn base to hn tip. 

The hn material is homogeneous with constant thermal conductivity Af. 

The heat transfer at the hn surface will be described by a constant heat transfer coefficient af. 

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