Library / Heat Transfer / HTR-15
Heat Transfer
Efficiency of a straight fin with an insulated tip
Fraction of the heat a fin would lose if all of it were at the base temperature that it actually loses.
- Ref
- HTR-15
- Rev
- v1
- Field
- Heat Transfer
- Result
- ηf
Fin efficiency ηf = tanh(mL)/(mL) compares the actual heat flow of a fin with the ideal flow if the whole fin were at the base temperature. It is 1 for a very short fin (mL → 0) and falls as the fin gets longer, because the tip becomes cooler. It is used to size fins and to decide how much extra surface area is worth adding.
Fraction of the heat a fin would lose if all of it were at the base temperature that it actually loses. It falls as the fin gets longer, because the far end gets cooler.
Calculator
- mL0.7071
- tanh(mL)0.6089
- Efficiency of a fin twice as long0.6282
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With mL = 0.707 the fin is 86.1 % efficient: it transfers 86.1 % of what it would if it were all at the base temperature.
- Doubling the length gives mL = 1.41 and an efficiency of 62.8 %: the extra length adds surface but each added centimetre is less effective.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| k Thermal conductivity of the fin | 1 to 400 W/(m K) | 0.1 | 0.9242 | 0.8242 | +10% | +1.24 % |
| L Fin length | 5 to 500 mm | 0.2824 | 0.9996 | 0.7172 | +10% | −2.73 % |
| h Convection coefficient | 5 to 500 W/(m² K) | 0.3151 | 0.9679 | 0.6529 | +10% | −1.32 % |
| Ac Cross-sectional area of the fin | 5 to 5,000 mm² | 0.3543 | 0.9974 | 0.6431 | +10% | +1.24 % |
| P Exposed perimeter of the fin | 5 to 500 mm | 0.3519 | 0.9743 | 0.6224 | +10% | −1.32 % |
Sweep of Fin length
| Fin length (mm) | Fin efficiency |
|---|---|
| 5 | 0.9996 |
| 66.88 | 0.9316 |
| 128.8 | 0.7923 |
| 190.6 | 0.6481 |
| 252.5 | 0.5294 |
| 314.4 | 0.4394 |
| 376.3 | 0.3722 |
| 438.1 | 0.3215 |
| 500 | 0.2824 |
Sweep one variable
Variables and how each one contributes
Fin efficiency
What the equation gives you. Everything else feeds it.
Convection coefficient
The convection coefficient between the fin and the surrounding fluid, in W/(m² K) (about 10 to 25 for natural convection in air, higher for forced). Lowers the efficiency: stronger cooling makes the fin cooler along its length. Raising it by 5 (to 30) moves the result by −2.61 %, from 0.8611 to 0.8386.
Exposed perimeter of the fin
The exposed perimeter of the fin cross-section, in mm (for a round pin, πd). Lowers the efficiency: more perimeter loses more heat per unit length and cools the fin faster. Raising it by 5 mm (to 36.4 mm) moves the result by −2.09 %, from 0.8611 to 0.8431.
Thermal conductivity of the fin
The thermal conductivity of the fin material, in W/(m K) (about 200 for aluminium, 400 for copper, 50 for steel). Raises the efficiency: a better conductor keeps the fin closer to the base temperature. Raising it by 20 (to 220) moves the result by +1.24 %, from 0.8611 to 0.8718.
Cross-sectional area of the fin
The cross-sectional area of the fin, in mm² (for a round pin, πd²/4). Raises the efficiency: a thicker fin carries heat further along its length. Raising it by 10 mm² (to 88.5 mm²) moves the result by +1.55 %, from 0.8611 to 0.8744.
Fin length
The length of the fin from base to tip, in mm. Lowers the efficiency: a longer fin has a cooler tip. Raising it by 10 mm (to 110 mm) moves the result by −2.73 %, from 0.8611 to 0.8375.
Worked example with the values above
- Fin parameter times length: mL = 0.7071.
- Hyperbolic tangent: tanh(0.7071) = 0.6089.
- ηf = 0.6089 / 0.7071 = 0.8611.
Assumptions and limits
- A straight fin of uniform cross-section with one-dimensional conduction along its length.
- A constant convection coefficient and a constant fin conductivity, with an insulated (adiabatic) tip.
Common mistakes
- Confusing the efficiency of a single fin with the overall efficiency of a finned surface.
- Using it for a fin that is tapered or has a convecting tip without correction.
Statistics and uncertainty
Efficiency is a smooth function of mL, with the relative error in mL being half the relative error in h. A 25 % error in h therefore shifts mL by about 12 % and the efficiency by a few per cent for typical fins.
Where it is used
- Deciding whether a longer fin is worthwhile.
- Combining fin efficiency with the surface area to find the finned heat loss.
Maths you need
- Convection coefficient: heat per area per temperature difference
- Conduction: heat flow through a solid by temperature gradient