Engineering & Science
Cooling Fin Efficiency (eta_fin) & Fin Effectiveness (epsilon_fin) Heat-Sink Formula
Calculate the extended-surface fin parameter m*L = L * SQRT((h * P) / (k * Ac)), adiabatic-tip fin efficiency eta_fin = tanh(m*L) / (m*L) (%) via exact EXP identity, and fin effectiveness epsilon_fin.
Design CPU heat sinks, EV inverter cold plates, and air-cooled cylinder fins!
Adding aluminum or copper fins to a power-electronics baseplate multiplies convective surface area—but as heat conducts along fin length L from root to tip, the fin cools down, reducing temperature difference toward the tip! Thermal engineers quantify that conductive drop with dimensionless fin parameter m * L = L * SQRT((h * P) / (k * Ac)), where h is convection coefficient (W/(m^2*K)), P is fin perimeter (m), k is fin thermal conductivity (W/(m*K)), and Ac is cross-sectional area (m^2)!
For an adiabatic-tip straight or pin fin, Fin Efficiency is eta_fin = tanh(m * L) / (m * L) (actual heat transfer divided by ideal isothermal-fin heat transfer), and Fin Effectiveness is epsilon_fin = tanh(m * L) * SQRT((k * P) / (h * Ac)) (heat transfer WITH the fin divided by bare-base heat transfer WITHOUT the fin—must exceed 2.0 to justify adding fins!). Writing hyperbolic tangent via its exact exponential definition tanh(x) = (EXP(2*x) - 1) / (EXP(2*x) + 1) runs identically in every spreadsheet engine: an aluminum fin (k = 205 W/(m*K), P = 0.204 m, Ac = 0.0002 m^2, L = 0.06 m) in forced air (h = 50 W/(m^2*K)) achieves m*L = 0.9464, eta_fin = 78.00%, and whopping effectiveness epsilon_fin = 47.73!
| Convection Regime (60 mm Al Fin) | Dimensionless m * L | Fin Efficiency eta_fin (%) | Fin Effectiveness epsilon_fin |
|---|---|---|---|
| Moderate Forced Air (h = 50 W/m^2*K) | 0.9464 | 78.00% | 47.73 (Highly justified > 2!) |
| High-Velocity Air / Spray (h = 150 W/m^2*K) | 1.6392 | 56.58% (Tip cools more) | 34.62 |
| Invalid Fin Geometry (L <= 0 or k <= 0) | Domain Guard | Guarded Input Error | "Invalid Fin Parameters" |
Adiabatic-Tip Cooling Fin Efficiency eta_fin (%) via Exact Exponential tanh(m*L)
Evaluates 100 * ((EXP(2*mL) - 1) / (EXP(2*mL) + 1)) / mL where mL = L * SQRT((h * P) / (k * Ac)).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0), "Invalid Fin Parameters", ROUND(100 * (((EXP(2 * B6 * SQRT((B2 * B3) / (B4 * B5))) - 1) / (EXP(2 * B6 * SQRT((B2 * B3) / (B4 * B5))) + 1)) / (B6 * SQRT((B2 * B3) / (B4 * B5)))), 2))Fin Effectiveness Ratio epsilon_fin = Q_fin / Q_bare_base (Target > 2.0)
Computes tanh(m*L) * SQRT((k * P) / (h * Ac)) to verify that the fin multiplies bare-footprint heat rejection.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0), "Invalid Fin Parameters", ROUND(((EXP(2 * B6 * SQRT((B2 * B3) / (B4 * B5))) - 1) / (EXP(2 * B6 * SQRT((B2 * B3) / (B4 * B5))) + 1)) * SQRT((B4 * B3) / (B2 * B5)), 2))Try the example
Use the original values or change them to test your understanding.