Fin Efficiency⚠ unverified
Physics / Heat Transfer · Compute the efficiency of a long fin with tip convection
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| m | m | 1/m | 1.0 | F |
| L | L | m | 1.0 | F |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | eff | — | Fin efficiency (dimensionless). Returns 1.0 when the product ``m * L`` is zero |
The science & history
Understanding the Parameters
- $mL$ — dimensionless fin size; large $mL$ → low efficiency (tip much cooler than base).
- $\eta_f$ — actual heat transfer over heat transfer if the entire fin were at base temperature: $Q = \eta_f h A_f (T_b - T_{\infty})$.
Derivation (Approaching a Proof)
Solve the 1-D fin ODE $d^{2}\theta/dx^{2} = m^{2}\theta$ with $\theta = T - T_{\infty}$, base $\theta(0)=\theta_b$, tip $d\theta/dx=0$. Heat in at the base over ideal $h A_f \theta_b$ yields $\eta_f = \tanh(mL)/(mL)$.
History
Fin efficiency charts (Harper–Brown, later textbooks) are standard electronics and heat-exchanger design tools.
Related Concepts: Convection Heat Transfer, Biot Number, Nusselt Number
Notes: Registry calculator fin-efficiency (unverified). Insulated-tip straight fin; $m$ given.