An important step in the design of a fin is the determination of the appropriate length of the fin once the fin material and the fin cross section are specified. You may be tempted to think that the longer the fin, the larger the surface area and thus the higher the rate of heat transfer. Therefore, for maximum heat transfer, the fin should be infinitely long. However, the temperature drops along the fin exponentially and reaches the environment temperature at some length. The part of the fin beyond this length does not contribute to heat transfer since it is at the temperature of the environment, as shown in Figure 3.14. Therefore, designing such an “extra long” fin is out of question since it results in material waste, excessive weight, and increased size and thus increased cost with no benefit in return (in fact, such a long fin will hurt performance since it will suppress fluid motion and thus reduce the convection heat transfer coefficient). Fins that are so long that the temperature approaches the environment temperature cannot be recommended either since the little increase in heat transfer at the tip region cannot justify the large increase in the weight and cost.
To get a sense of proper length of a fin, we compare heat transfer from a fin of finite length to heat transfer from an infinitely long fins under the same conditions. The ratio of these two heat transfers is heat transfer ratio
 |
(3.45) |
The values of TanhmL are evaluated for some values of mL and the results are given in Table 3.2.
Table 3.2 The variation of heat transfer from a fin relative to that from an infinitely long fin
mL |
TanhmL |
0.1 |
0.1 |
0.2 |
0.197 |
0.5 |
0.462 |
1.0 |
0.762 |
1.5 |
0.905 |
2.0 |
0.964 |
2.5 |
0.987 |
3.0 |
0.995 |
4.0 |
0.999 |
5.0 |
1.000 |
We observe from the table that heat transfer from a fin increases with mL almost linearly at first, but the curve reaches a plateau later and reaches a value for the infinitely long fin at about mL=5.Therefore, a fin whose length is L=m/5 can be considered to be an infinitely long fin. We also observe that reducing the fin length by half in that case (from mL=5 to mL=2.5 ) causes a drop of just 1 percent in heat transfer. We certainly would not hesitate sacrificing 1 percent in heat transfer performance in return for 50 percent reduction in the size and possibly the cost of the fin. In practice, a fin length that corresponds to about mL=1 will transfer 76.2 percent of the heat that can be transferred by an infinitely long fin, and thus it should offer a good compromise between heat transfer performance and the fin size.
