140 FUNDAMENTALS OF HEAT AND MASS TRANSFER Higher values of effectiveness can result from higher values of convection resistance compared to the fin resistance. In effect it can be said that fins reduce to resistance to heat flow at the surface. 4.4.2. Fin efficiency, η f : This quantity is more often used to determine the heat flow when variable area fins are used. Fin efficiency is defined as the ratio of heat transfer by the fin to the heat transfer that will take place if the whole surface area of the fin is at the base temperature. η f = q hA T T f s b ( − ∞ ) ...(4.20) where A s is the surface area of the fin. For constant area fin with adiabatic tip, η f = hPkA T T mL hPL T T mL mL b b ( ) . tanh ( ) ( ) tanh ( ) − − = ∞ ∞ ...(4.21) The surface area A s = PL . This equation can be used in general without significant error by increasing the fin length with surface area equal to the area at the tip. In the case of plate fins new length L c = L + t 2 where t is the thickness. In the case of circular fins L c = L + D /4, where D is the diameter of the fin. Error associated with this assumption is negligible if ht k hD k or 2 ≤ 0.0625.
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