Therefore, the fin effectiveness can be determined easily when the fin efficiency is known, or vice versa.

The rate of heat transfer from a sufficiently long fin or uniform cross section under steady conditions is given by Equation 3.34. Substituting this relation into Equation 3.40, the effectiveness of such a long fin is determined to be

Since *A*_{c}*=A*_{b}in this case. We can draw several important conclusions from the fin effectiveness relation above for consideration in the design and selection of the fins

• The *thermal conductivity **k* of the fin material should be as high as possible. Thus it is no coincidence that fins are made from metals, with copper, aluminium, and iron being the most common ones. Perhaps the most widely used fins are made of aluminum because of its low cost and weight and its resistance to corrosion.

• The ratio of the *perimeter *to the *cross sectional area *of the fin *P/A*_{c} should be as high as possible. This criterion is satisfied by *thin plate *fins or *slender pin fins *

• The use of fins is *most effective *in applications involving low convection heat transfer coefficient*. *Thus, the use of fins is more easily justified when the medium is a gas instead of a liquid and the heat transfer is by *natural convection *instead of by forced convection. Therefore, it is no coincidence that in liquid-to-gas heat exchangers such as the car radiator, fins are placed on the gas side.

When determining the rate of heat transfer from a finned surface, we must consider the *unfinned portion *of the surface as well as the *fins *. Therefore, the rate of heat transfer for a surface containing *n *fins can be expressed as

*q*_{total,fin }= q_{unfin}+ q_{fin}= hA_{unfin}(*T*_{b} -)* + h*_{fin} hA_{fin} (*T*_{b} -)

Where *A*_{nofin} is the area of the surface when there are no fins, *A*_{unfin}is the total surface area of all the fins on the surface, and *A *unfin is the area of the unfinned portion of the surface (Figure. 3.13). Note that the overall fin effectiveness depends on the fin density (number of fins per unit length) as well as the effectiveness of the individual fins. The overall effectiveness is a better measure of the performance of a finned surface than the effectiveness of the individual fins.

Figure 3.13 Various Surface Areas Associated With A Rectangular Surface With Three Fins