Library / Heat transfer and thermal systems / HTS-13
Heat transfer and thermal systems
Base heat rate of a fin with a convecting tip
Heat entering a uniform fin at the base when the tip also convects, with B = ht/(km) for a tip area equal to the cross-section and the same ambient.
- Ref
- HTS-13
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- Qb [W]
When the tip also convects to the same ambient, a uniform fin delivers Qb = kmAcθb[tanh(mL) + B]/[1 + B tanh(mL)], with B = ht/(km). B = 0 gives the insulated-tip result kmAcθb tanh(mL), and a large B gives the result for a tip held at the ambient temperature. A short fin gains most from tip convection.
Heat entering a uniform fin at the base when the tip also convects, with B = ht/(km) for a tip area equal to the cross-section and the same ambient. Setting B = 0 recovers the insulated-tip result.
Calculator
- Insulated-tip heat rate, kmAcθb tanh(mL)3.808 W
- Gain from tip convection5.124 %
- Fin parameter product, mL1
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.
- The fin delivers 4.003 W at the base, 5.12 % more than with an insulated tip.
- For long fins (mL above about 3) the tip carries almost no heat and the two results converge.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (W) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Ac Cross-sectional area | 1e-6 to 0.01 m² | 0.04003 | 400.3 | 400.3 | +10% | +10 % |
| m Fin parameter √(hP/(kAc)) | 0.1 to 100 1/m | 0.005494 | 50 | 49.99 | +10% | +14.6 % |
| θb Base excess temperature Tb−T∞ | 1 to 500 K | 0.08006 | 40.03 | 39.95 | +10% | +10 % |
| k Fin conductivity | 1 to 500 W/(m K) | 0.04003 | 20.02 | 19.98 | +10% | +10 % |
| L Fin length | 0.001 to 1 m | 0.5494 | 5 | 4.451 | +10% | +4.14 % |
| B Tip Biot ratio ht/(km) | 0 to 5 | 3.808 | 5.992 | 2.184 | +10% | +0.45 % |
Sweep of Tip Biot ratio ht/(km)
| Tip Biot ratio ht/(km) | Base heat rate (W) |
|---|---|
| 0 | 3.808 |
| 0.625 | 4.697 |
| 1.25 | 5.153 |
| 1.875 | 5.43 |
| 2.5 | 5.616 |
| 3.125 | 5.749 |
| 3.75 | 5.85 |
| 4.375 | 5.929 |
| 5 | 5.992 |
Sweep one variable
Variables and how each one contributes
Base heat rate
What the equation gives you. Everything else feeds it.
Fin conductivity
The fin conductivity, in W/(m K). Raises the heat rate in proportion to k at fixed m. Raising it by 10 % (to 110) moves the result by +10 %, from 4.003 to 4.403 W.
Fin parameter √(hP/(kAc))
The fin parameter √(hP/(kAc)), in 1/m. Raises the heat rate, with the tanh term saturating once mL is large. Raising it by 10 % (to 11) moves the result by +14.6 %, from 4.003 to 4.586 W.
Cross-sectional area
The cross-sectional area, in m². Raises the heat rate in direct proportion. Raising it by 10 % (to 1.1×10−4) moves the result by +10 %, from 4.003 to 4.403 W.
Base excess temperature Tb−T∞
The base excess temperature Tb − T∞, in K. Raises the heat rate in direct proportion. Raising it by 10 % (to 55) moves the result by +10 %, from 4.003 to 4.403 W.
Fin length
The fin length, in m. A longer fin carries more heat, with falling returns once mL exceeds about 2. Raising it by 10 % (to 0.11) moves the result by +4.14 %, from 4.003 to 4.169 W.
Tip Biot ratio ht/(km)
The tip Biot ratio ht/(km), dimensionless. More tip convection raises the heat rate, most for short fins. Raising it by 10 % (to 0.11) moves the result by +0.45 %, from 4.003 to 4.021 W.
Worked example with the values above
- Hyperbolic term: tanh(mL) = tanh(1) = 0.76159; tip factor (t + B)/(1 + B t) = 0.80062.
- Prefactor: kmAcθb = 100 × 10 × 1×10−4 × 50 = 5 W.
- Qb = 5 × 0.80062 = 4.0031 W.
Assumptions and limits
- Uniform cross-section, constant k and h, and one-dimensional conduction along the fin.
- Negligible radiation, with the base held at a uniform temperature.
Common mistakes
- Using the cross-section perimeter wrongly when finding m.
- Mixing absolute temperatures and excess temperatures above ambient.
- Using an insulated-tip formula when the tip convects, which matters for short fins.
Statistics and uncertainty
Fin results depend strongly on the heat-transfer coefficient h, which is usually known to only 20 % to 30 %. Because m varies as √h, the temperature and heat-rate errors are somewhat smaller, but still significant.
Where it is used
- Pin, rib and rod fins on heat sinks, engines and pipes.
- Rating finned tubes and checking tip temperatures.
Maths you need
- Fin parameter: m = √(hP/(kAc))
- Excess temperature: θ = T − T∞
- Fin efficiency: actual heat rate divided by the rate if the whole fin were at the base temperature