Fins are used in a large number of applications to increase the heat transfer from surfaces. Typically, the fin material has a high thermal conductivity. The fin is exposed to a flowing fluid, which cools or heats it, with the high thermal conductivity allowing increased heat being conducted from the wall through the fin. The design of cooling fins is encountered in many situations and we thus examine heat transfer in a fin as a way of defining some criteria for design. A model configuration is shown in Figure 5.1. The fin is of length L. The other parameters of the problem are indicated. The fluid has velocity c∞ and temperature T∞.
We assume (using the Reynolds analogy or other approach) that the heat transfer coefficient for the fin is known and has the value h. The end of the fin can have a different heat transfer coefficient, which we can call hL. The approach taken will be quasi-one-dimensional, in that the temperature in the fin will be assumed to be a function of x only. This may seem a drastic simplification, and it needs some explanation.
With a fin cross-section equal to A and a perimeter P, the characteristic dimension in the transverse direction is A / P (For a circular fin, for example, A / P = r / 2). The regime of interest will be taken to be that for which the Biot number is much less than unity, ( ) 1 / = << k h A P Bi , which is a realistic approximation in practice.
The physical content of this approximation can be seen from the following. Heat transfer per unit area out of the fin to the fluid is roughly of magnitude ~ h(Tw – T∞ ) per unit area . The heat transfer per unit area within the fin in the transverse direction is (again in the same approximate terms)
capability for heat transfer per unit area across the fin than there is between the fin and the fluid, and thus little variation in temperature inside the fin in the transverse direction. To emphasize the point, consider the limiting case of zero heat transfer to the fluid i.e., an insulated fin. Under these conditions, the temperature within the fin would be uniform and equal to the wall temperature.