Fins and Extended Surfaces
The fin equation, fin efficiency, heat sinks and finned tubes, with a worked aluminum fin.
Why add fins?
The heat flow from a surface is Q̇ = h A ΔT. When h is small (any gas, natural convection) the easiest way to move more heat is to increase the area A. Fins are thin projections that add area cheaply: the heat sink on a processor, the fins on a motor, a radiator, an air-cooled engine, a finned tube in an air heater.
The catch is that a fin is not at the base temperature all along its length: heat has to conduct along the fin, which costs a temperature drop, so the outer part of the fin is cooler and transfers less heat. The fin equations quantify this.
The fin equation
For a straight fin of constant cross-section (area Ac, perimeter P), conduction along the fin balances convection from its sides. The temperature excess θ = T − T∞ then decays along the fin:
θ(x) = θb cosh(m (L − x)) ÷ cosh(m L) m = √(h P ÷ (k Ac))
Q̇fin = √(h P k Ac) θb tanh(m L)
A larger m means a faster temperature fall along the fin: a good fin has a high conductivity k (copper, aluminum) and a thick cross-section, and works in a low h (air).
Fin efficiency
The fin efficiency is the heat actually transferred divided by the heat if the whole fin were at the base temperature:
ηf = tanh(m L) ÷ (m L) Q̇ = ηf h Afin θb
A worked aluminum fin and heat sink
Worked example: one fin, then a heat sink
- An aluminum fin (k = 205 W/m·K), 2 mm thick, 100 mm wide and 30 mm high, in air with h = 25 W/m²·K, base 60 K above ambient.
- Perimeter P = 2 (0.002 + 0.100) = 0.204 m; Ac = 200 mm². m = √(25 × 0.204 ÷ (205 × 0.000200)) = 11.15 m⁻¹, so m L = 0.335.
- Efficiency ηf = tanh(0.335) ÷ 0.335 = 0.964. Heat per fin: Q̇ = √(h P k Ac) θb tanh(m L) = 8.9 W. The tip temperature excess is θb ÷ cosh(m L) = 95 percent of the base.
- The fin has an effectiveness of Q̇ ÷ (h Ac θb) = 30: it moves 30 times the heat that the bare base area it sits on would. A fin is worth adding when the effectiveness is above about 2.
- Heat sink: a 100 mm × 100 mm base with 15 of these fins (6.4 mm pitch). Fin area 918 cm² plus 70 cm² of exposed base. Q̇ = h (Abase + ηf Afins) θb = 143 W, against 15 W for the plain plate: 9.6 times more. The heat sink’s thermal resistance is θSA = 60 ÷ 143 = 0.42 K/W (overall surface efficiency 97 percent). The Heat Sink and Thermal Resistance Calculator applies the same idea.
Fin design: material, length, thickness, spacing
| Fin material (same 30 mm fin) | k (W/m·K) | m L | Efficiency | Heat per fin |
|---|---|---|---|---|
| Copper | 390 | 0.24 | 98 % | 9.0 W |
| Aluminum 6063 | 205 | 0.33 | 96 % | 8.9 W |
| Aluminum 1050 | 230 | 0.32 | 97 % | 8.9 W |
| Carbon steel | 50 | 0.68 | 87 % | 8.0 W |
| Stainless steel | 15 | 1.24 | 68 % | 6.3 W |
Copper gives the best fins but at about three times the mass and cost; steel and stainless are poor fins (efficiency under 70 percent for stainless). Now the length, with aluminum:
| Fin height | m L | Efficiency | Heat per fin | Heat relative to an ideal fin |
|---|---|---|---|---|
| 15 mm | 0.17 | 99 % | 4.5 W | 99 % |
| 30 mm | 0.33 | 96 % | 8.9 W | 96 % |
| 60 mm | 0.67 | 87 % | 16.0 W | 87 % |
| 100 mm | 1.12 | 72 % | 22.1 W | 72 % |
A fin four times as high does not carry four times the heat: going from 30 to 100 mm only lifts the heat per fin from 8.9 to 22.1 W, because the outer part is nearly at the air temperature. Long thin fins waste material: use more, shorter fins. A thicker fin lowers m, so the temperature falls more slowly along it: at 4 mm thickness the same 30 mm fin has ηf = 0.981 and carries 9.2 W (1.04 times) for twice the metal.
- Fin spacing: too close and the boundary layers merge and choke the flow (natural convection wants 6 to 12 mm between fins for fins 30 to 60 mm tall; forced air allows 2 to 4 mm); too wide and you lose area.
- Orientation: in natural convection, vertical fins let the warm air rise through them (the chimney effect); horizontal fins facing up trap the air.
- Fins and radiation: in still air, black anodizing raises the heat sink’s output by about 10 to 20 percent.
- The base: a thick base spreads the heat from a small source to all the fins, and its spreading resistance matters (see Cooling Electronics and Thermal Design).
Finned tubes and other extended surfaces
Round pins are used where the flow can come from any direction: a 3 mm aluminum pin 20 mm long in the same conditions has m L = 0.26 and ηf = 0.979. Finned tubes (a spiral or plate fin pressed on a tube) are the standard for air coolers, radiators and economisers: the tube carries a liquid with a high h inside, and the fins multiply the area on the air side, where the resistance is. The overall coefficient is set by the air side, so extend that surface. The pressure drop of the air through the fin bank is the price (see Fluid Mechanics and Pipe Flow).
Related
- Previous: Thermal Radiation. Next: Heat Exchangers: LMTD and NTU.
- Heat Sink and Thermal Resistance Calculator.
- Cooling Electronics and Thermal Design for heat sinks on power devices.
- Heat Transfer: the whole guide.
FAQ
What does fin effectiveness mean?
It is the heat the fin transfers divided by the heat the bare base area under it would transfer. A value of 2 or more justifies a fin; typical metal fins in air have effectiveness of 10 to 50.
Why is fin efficiency less than one?
Because heat conducting along the fin makes the outer part cooler than the base, so it transfers less per unit area than the base. Only a perfectly conducting fin has an efficiency of one.
When do fins not help?
When h is already high (liquids, boiling, a fan-forced fast stream) so that the fin resistance dominates, or when fins are so close that they choke the flow. Fins pay off on the low-h (gas) side.
Do I correct for the tip?
Yes, for accuracy, use the corrected length L + t ÷ 2 for a thin fin with a convecting tip. The examples here use an insulated tip, which is within a few percent for a thin fin.
References
- Incropera, F. P., et al., Fundamentals of Heat and Mass Transfer, Wiley (extended surfaces).
- Kraus, A. D., Aziz, A. and Welty, J., Extended Surface Heat Transfer, Wiley.
- Çengel, Y. A. and Ghajar, A. J., Heat and Mass Transfer, McGraw-Hill; Lienhard, J. H., A Heat Transfer Textbook (free online).
- Heat sink manufacturers’ catalogs for θSA against airflow.
Last reviewed: October 2026. Heat transfer coefficients are typical for air.