How to use this calculator

Choose a pin (round) or rectangular fin and enter its size and length. Pick the material to fill in the thermal conductivity, then enter the convection coefficient and the base and fluid temperatures. The tip condition sets the boundary at the free end: insulated is the usual textbook choice, convection from the tip is the most accurate, and very long applies when mL is above about 2.65.

Formulas

Fin equation and parameterd²θdx² − m²θ = 0 m = √(hPkA_c) M = √(hPkA_c) θ_b

Heat rate by tip conditioninsulated: q = M tanh mL very long: q = M convective: q = M sinh mL + (h/mk) cosh mLcosh mL + (h/mk) sinh mL

Performanceη_f = qh A_f θ_b ε_f = qh A_c θ_b θ = T − T_∞

For a pin of diameter D, A_c = πD²/4 and P = πD. For a rectangular fin of thickness t and width w, A_c = wt and P = 2(w + t). A_f is the fin’s convecting surface, PL, plus the tip area A_c when the tip convects.

Worked example

A 5 mm aluminum pin, 50 mm long, in air with h = 25 W/m²·K (the default)

A_c = 1.963 × 10⁻⁵ m², P = 0.01571 m, and m = √(25 × 0.01571 / (167 × 1.963 × 10⁻⁵)) = 10.94 m⁻¹.
θ_b = 100 − 25 = 75 °C, so M = √(25 × 0.01571 × 167 × 1.963 × 10⁻⁵) × 75 = 2.691 W.
mL = 0.547, so q = 2.691 × tanh 0.547 = 1.34 W.
Efficiency 91.1%, effectiveness 36.4, and the tip runs at 90.0 °C.

Designing good fins

Effectiveness tells you whether a fin is worth adding: below about 2 it hardly helps. Fins work best where h is small (air, natural convection) and the material conducts well, which is why heat sinks are aluminum or copper and car radiators put fins on the air side, not the coolant side. Length has diminishing returns: tanh(mL) is already 0.76 at mL = 1 and 0.99 at mL = 2.6, so beyond that extra length only adds weight. Thin, closely spaced fins give more surface, until the gaps get so narrow that air can’t flow through them.

Assumptions and limits

Steady, one-dimensional conduction along the fin: the cross-section is thin enough that its temperature is uniform across it (Biot number h t/k well below 0.1).
Constant thermal conductivity, a uniform convection coefficient over the whole fin and no radiation.
The base is held at T_b. In a real heat sink, adding fins also cools the base a little.

Frequently asked questions

What is the corrected-length trick?

Use the insulated-tip formula with L_c = L + D/4 for a pin, or L + t/2 for a rectangular fin. It is a close shortcut for the convective-tip result.

Why can efficiency be high while effectiveness is low?

Efficiency compares the fin with an ideal fin at the base temperature; effectiveness compares it with having no fin. A short, thick fin is nearly isothermal (efficient) but adds little surface (not effective).

Where do I get h?

Roughly 5 to 25 W/m²·K for natural convection in air, 25 to 250 for forced air, and hundreds to thousands for liquids. The convection calculator uses h directly.