Fin Parameter mL (Straight Fin)

Solve the fin equation for a straight rectangular fin and every geometric and material detail collapses into one group: m = √(2h/kt), with units of 1/m, multiplied by the fin length. mL compares how fast heat leaks off the sides with how fast the metal can carry it to the tip. Small mL means a short, thick, highly conductive fin that is nearly isothermal — very effective, but you have spent a lot of metal. Large mL means a long, thin or poorly conducting fin whose tip has already cooled to the air temperature and is doing nothing at all.

Worked example: an aluminium fin, k = 180 W/(m·K), 2 mm thick, 30 mm long, in air at h = 50 W/(m²·K). 2h/(kt) = 100/0.36 = 278 per m², m = 16.7 /m, and mL = 0.5 — a well-proportioned fin. Swap the aluminium for stainless at k = 15 and mL jumps to 1.7, which is why nobody makes stainless heat sinks. Designers aim for mL in the 0.5–1.5 band; below 0.5 the fin is over-built and below 0.3 you would be better off adding more fins. Trap: L should be the corrected length L + t/2 for an adiabatic-tip analysis of a fin that actually loses heat off its end, and t is the full thickness, not the half-thickness that appears in some derivations.