Fin Efficiency (Straight Fin)

A fin can never be as good as its base, because heat has to travel out along the metal and the fin cools as it goes. Fin efficiency is the honest accounting: real duty divided by the duty the same surface would deliver if the whole fin sat at base temperature. For a straight fin with an insulated tip the answer is the compact and slightly magical $\eta_f = \tanh(mL)/mL$. At mL = 0.5, $\eta_f = \tanh(0.5)/0.5 = 0.924$ — the fin is working at 92% of ideal. At mL = 1.5 it has slumped to 0.603, and at mL = 3 to 0.332, where two-thirds of the aluminium you paid for is decoration.

The tanh form is exact for a straight rectangular fin of constant cross-section with a uniform h and an adiabatic tip, and it is a good approximation for a real tip if you use the corrected length L + t/2. Triangular, annular and pin fins have their own expressions — annular fins bring in Bessel functions, which is why the charts in Incropera exist. The trap that catches people is confusing efficiency with effectiveness: efficiency compares the fin with an ideal fin and is always below 1, while fin effectiveness compares the finned surface with the bare surface and had better be well above 1 or there was no point adding fins at all. A fin with $\eta_f$ = 0.6 can still multiply the surface duty fivefold, and that is what pays for it. This page solves for $\eta_f$ only — recovering mL from an efficiency means inverting tanh(x)/x, which has no closed form.