Library / Heat Transfer / HTR-13
Heat Transfer
Fin parameter of a straight fin
Combined measure of how quickly heat leaves a straight fin by convection compared with how easily it is conducted along it.
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- HTR-13
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- v1
- Field
- Heat Transfer
- Result
- mf [1/m]
The fin parameter m = √(hP/(kAc)) sets how quickly the temperature drops along a fin: the excess temperature falls off roughly as e−mx. A large m (strong convection, thin fin, poor conductor) means the fin is cold after a short distance, so a long fin is wasted. 1/m is the characteristic length. For a round pin, m = √(4h/(kd)).
Combined measure of how quickly heat leaves a straight fin by convection compared with how easily it is conducted along it. A large value means the temperature falls off quickly along the fin.
Calculator
- Characteristic length, 1/m141.4 mm
- mL for a 100 mm fin0.7071
- Length beyond which the fin behaves as infinitely long (about 2/m)282.8 mm
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.
- For a round pin 9.99 mm in diameter (perimeter 31.4 mm), m = 7.07 m⁻¹, so the temperature excess falls to 37 % in 141 mm. A pin about 283 mm long (2/m) already behaves as if it were infinitely long.
- Using a better conductor (higher k) or a thicker fin lowers m and lets the fin stay warm for longer; stronger cooling (higher h) raises m and shortens the useful length.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (1/m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| k Thermal conductivity of the fin | 1 to 400 W/(m K) | 5 | 100 | 95 | +10% | −4.65 % |
| h Convection coefficient | 5 to 500 W/(m² K) | 3.162 | 31.62 | 28.46 | +10% | +4.88 % |
| Ac Cross-sectional area of the fin | 5 to 5,000 mm² | 0.886 | 28.02 | 27.13 | +10% | −4.65 % |
| P Exposed perimeter of the fin | 5 to 500 mm | 2.822 | 28.22 | 25.39 | +10% | +4.88 % |
Sweep of Convection coefficient
| Convection coefficient (W/(m² K)) | Fin parameter (1/m) |
|---|---|
| 5 | 3.162 |
| 66.88 | 11.57 |
| 128.8 | 16.05 |
| 190.6 | 19.53 |
| 252.5 | 22.47 |
| 314.4 | 25.07 |
| 376.3 | 27.43 |
| 438.1 | 29.6 |
| 500 | 31.62 |
Sweep one variable
Variables and how each one contributes
Fin parameter
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). Raises m with the square root of h: more convection removes heat faster so the temperature drops more quickly. Raising it by 5 (to 30) moves the result by +9.54 %, from 7.071 to 7.746 1/m.
Exposed perimeter of the fin
The exposed perimeter of the fin cross-section, in mm (for a round pin, πd). Raises m with the square root of the perimeter: more surface per unit length loses more heat. Raising it by 5 mm (to 36.4 mm) moves the result by +7.67 %, from 7.071 to 7.613 1/m.
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). Lowers m with the inverse square root of the conductivity: a better conductor carries heat further. Raising it by 20 (to 220) moves the result by −4.65 %, from 7.071 to 6.742 1/m.
Cross-sectional area of the fin
The cross-sectional area of the fin, in mm² (for a round pin, πd²/4). Lowers m with the inverse square root of the area: a thicker fin carries more heat along its length. Raising it by 10 mm² (to 88.5 mm²) moves the result by −5.82 %, from 7.071 to 6.66 1/m.
Worked example with the values above
- Convert units: P = 0.0314 m and Ac = 7.85×10−5 m². Numerator hP = 25 × 0.0314 = 0.785 W/(m K).
- Denominator k Ac = 200 × 7.85×10−5 = 0.0157 W m/K; ratio = 50 m⁻².
- m = √50 = 7.071 m⁻¹.
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
- Mixing mm and m for P and Ac.
- Using the perimeter of the base block rather than the fin cross-section.
Statistics and uncertainty
The convection coefficient h is the least certain input, often only known to ±25 %. Because m depends on √h, that becomes about ±12 % in m, and the useful fin length changes by the same fraction.
Where it is used
- Judging how long a fin on a heat sink, cylinder barrel or electronics housing is useful.
- Input to the heat transfer and efficiency of a fin.
Maths you need
- Convection coefficient: heat per area per temperature difference
- Conduction: heat flow through a solid by temperature gradient