Library / Heat Transfer / HTR-14
Heat Transfer
Heat transfer from a straight fin with an insulated tip
Heat carried away by a uniform straight fin whose base is held at a fixed temperature and whose tip loses negligible heat.
- Ref
- HTR-14
- Rev
- v1
- Field
- Heat Transfer
- Result
- Q̇f [W]
For a straight fin of uniform cross-section with an insulated tip, the heat flow at its base is Q = √(hPkAc)(Tb − T∞) tanh(mL), with m = √(hP/(kAc)). The tanh factor tends to 1 for a long fin, so the maximum heat flow is √(hPkAc) times the temperature difference. For a fin whose tip is also cooled, use a corrected length L + Ac/P.
Heat carried away by a uniform straight fin whose base is held at a fixed temperature and whose tip loses negligible heat. The tanh term shows that a longer fin helps less and less.
Calculator
- Fin parameter m7.071 m⁻¹
- Heat flow if the whole fin were at the base temperature, hPL(Tb − T∞)6.28 W
- Heat flow with no fin, hAc(Tb − T∞)0.157 W
- Fin effectiveness, with-fin over without-fin heat flow34.44
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.
- A 100 mm round pin (9.99 mm diameter) at 100 °C in 20 °C air sheds 5.407 W. The bare stub of the same cross-section would lose only 0.157 W, so the fin multiplies the heat flow by 34.4.
- Doubling the fin length does not double the heat: here it would raise it by only about 45.9 %, since the far end is cooler.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (W) | Local step | Change in result |
|---|---|---|---|---|---|---|
| h Convection coefficient | 5 to 500 W/(m² K) | 1.216 | 39.58 | 38.36 | +10% | +8.54 % |
| P Exposed perimeter of the fin | 5 to 500 mm | 0.9743 | 35.19 | 34.22 | +10% | +8.54 % |
| Tb Base temperature | 20 to 500 °C | 0 | 32.44 | 32.44 | +10% | +12.5 % |
| L Fin length | 5 to 500 mm | 0.3139 | 8.866 | 8.552 | +10% | +7 % |
| T_∞ Surrounding fluid temperature | −20 to 100 °C | 0 | 8.111 | 8.111 | +2.4 °C | −3 % |
| k Thermal conductivity of the fin | 1 to 400 W/(m K) | 0.628 | 5.804 | 5.176 | +10% | +1.24 % |
| Ac Cross-sectional area of the fin | 5 to 5,000 mm² | 2.225 | 6.264 | 4.039 | +10% | +1.24 % |
Sweep of Fin length
| Fin length (mm) | Fin heat transfer (W) |
|---|---|
| 5 | 0.3139 |
| 66.88 | 3.912 |
| 128.8 | 6.406 |
| 190.6 | 7.758 |
| 252.5 | 8.395 |
| 314.4 | 8.675 |
| 376.3 | 8.795 |
| 438.1 | 8.845 |
| 500 | 8.866 |
Sweep one variable
Variables and how each one contributes
Fin heat transfer
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 the heat flow: more convection takes more heat, but with diminishing returns because the fin cools along its length. Raising it by 5 (to 30) moves the result by +16.9 %, from 5.407 to 6.32 W.
Exposed perimeter of the fin
The exposed perimeter of the fin cross-section, in mm (for a round pin, πd). Raises the heat flow: more exposed perimeter means more surface to lose heat. Raising it by 5 mm (to 36.4 mm) moves the result by +13.5 %, from 5.407 to 6.138 W.
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 heat flow: a better conductor keeps more of the fin warm, though the benefit levels off. Raising it by 20 (to 220) moves the result by +1.24 %, from 5.407 to 5.475 W.
Cross-sectional area of the fin
The cross-sectional area of the fin, in mm² (for a round pin, πd²/4). Raises the heat flow: a thicker fin carries more heat from the base. Raising it by 10 mm² (to 88.5 mm²) moves the result by +1.55 %, from 5.407 to 5.491 W.
Fin length
The length of the fin from base to tip, in mm. Raises the heat flow, with diminishing returns as the tip region is already close to the surrounding temperature. Raising it by 10 mm (to 110 mm) moves the result by +7 %, from 5.407 to 5.786 W.
Base temperature
The temperature at the base of the fin, in °C. Raises the heat flow in direct proportion to the temperature difference. Raising it by 5 °C (to 105 °C) moves the result by +6.25 %, from 5.407 to 5.745 W.
Surrounding fluid temperature
The temperature of the surrounding fluid, in °C. Lowers the heat flow in direct proportion to the temperature difference. Raising it by 5 °C (to 25 °C) moves the result by −6.25 %, from 5.407 to 5.069 W.
Worked example with the values above
- Fin parameter: m = √(hP/(kAc)) = 7.071 m⁻¹, so mL = 0.7071 and tanh(mL) = 0.6089.
- Prefactor: √(hPkAc) = 0.111 W/K; temperature excess 80 K.
- Q = 0.111 × 80 × 0.6089 = 5.407 W.
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
- Using the formula for a fin with a convecting tip without adding the correction length.
- Mixing mm and m for the dimensions.
- Taking Tb as the wall temperature when there is significant contact resistance.
Statistics and uncertainty
The heat flow depends on h through √h in the prefactor and through m, so a 25 % error in h gives roughly a 10 to 15 % error in Q for a fin of moderate length. The base temperature is not always known: it can be lowered by the contact resistance between the fin and the wall.
Where it is used
- Sizing pins and plates for cooling hot components, such as electronics, gun barrels and engine cylinders.
- Deciding how many fins and what length give the heat flow required.
Maths you need
- Convection coefficient: heat per area per temperature difference
- Conduction: heat flow through a solid by temperature gradient