Consider the surface of the plane wall at temperature T_{b} exposed to a medium at temperature . Heat is lost from the surface to the surrounding medium by convection with a heat transfer coefficient of h. Disregarding radiation, heat transfer from a surface area A is expressed as . Now let us consider a fin of constant cross sectional area A_{c}= A_{b} and length L that is attached to the surface with a perfect contact (Figure 3.8).
Figure 3.8 Fins Enhance Heat Transfer From A Surface By Enhancing Surface
This time heat will flow from the surface to the fin by conduction and from the fin to the surrounding medium by convection with the same heat transfer coefficient h. The temperature of the fin will be T_{b} at the fin base and gradually decrease toward the fin tip. Convection from the fin surface causes the temperature at any cross section to drop somewhat from the midsection toward the outer surfaces. However, the cross sectional area of the fins is usually very small, and thus the temperature at any cross section can be considered to be uniform. Also, the fin tip can be assumed for convenience and simplicity to be insulated by using the corrected length for the fin instead of the actual length.
In the limiting case of zero thermal resistance or infinite thermal conductivity ( ), the temperature of the fin will be uniform at the base value of T_{b}. The heat transfer from the fin will be maximum in this case and can be expressed as

(3.35) 
In reality, however, the temperature of the fin will drop along the fin, and thus the heat transfer from the fin will be less because of the decreasing temperature difference T(x)  toward the fin tip, as shown in Figure. 3.9.
To account for the effect of this decrease in temperature on heat transfer, we define a fin efficiency as

(3.36) 
where A_{fin} is the total surface area of the fin. This relation enables us to determine the heat transfer from a fin when its efficiency is known. For the cases of constant cross section of very long fins and fins with insulated tips, the fin efficiency can be expressed as

(3.37) 
and
SinceA_{fin}=PL for fins with constant cross section. Equation 3.38 can also be used for fins subjected to convection provided that the fin length L is replaced by the corrected length L_{c}.
Fin efficiency relations are developed for fins of various profiles and are plotted in Figure. 3.10 for fins on a plain surface and in Figure 3.11 for circular fins of constant thickness. The fin surface area associated with each profile is also given on each figure. For most fins of constant thickness encountered in practice, the fin thickness t is too small relative to the fin length L , and thus the fin tip area is negligible. Note that fins with triangular and parabolic profiles contain less material and are more efficient than the ones with rectangular profiles, and thus are more suitable for applications requiring minimum weight such as space applications.
Figure 3.9 Ideal And Actual Temperature Distribution In A Fin
An important consideration in the design of finned surfaces is the selection of the proper fin length L. Normally the longer the fin, the larger the heat transfer area and thus the higher the rate of heat transfer from the fin. But also the larger the fin, the bigger the mass, the higher the price, and the larger the fluid friction. Therefore, increasing the length of the fin beyond a certain value cannot be justified unless the added benefits outweigh the added cost. Also, the fin efficiency decreases with increasing fin length because of the decrease in fin temperature with length. Fin lengths that cause the fin efficiency to drop below 60 percent usually cannot be justified economically and should be avoided. The efficiency of most fins used in practice is above 90 percent.
Figure 3.10 Efficiency Of Circular, Rectangular And Triangular Fins On A Plain Surface Of Width W
Figure 3.11 Efficiency Of Circular Fins Of Length L And Constant Thickness T
