Library / Heat transfer and thermal systems / HTS-11
Heat transfer and thermal systems
Tip temperature of an insulated-tip fin
Temperature at the tip of a straight uniform fin with an insulated tip, from the base temperature Tb, the ambient temperature T∞ and the product mL, where m = √(hP/(kAc)) in 1/m.
- Ref
- HTS-11
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- Ttip [K]
A straight fin with an insulated tip has the excess temperature θ(x)/θb = cosh[m(L − x)]/cosh(mL), so at the tip θ/θb = 1/cosh(mL). The tip approaches ambient as mL grows. At mL of about 5 the tip excess is under 1 % of the base excess, so extra length adds almost nothing.
Temperature at the tip of a straight uniform fin with an insulated tip, from the base temperature Tb, the ambient temperature T∞ and the product mL, where m = √(hP/(kAc)) in 1/m.
Calculator
- Fin parameter product, mL1
- Tip excess as a fraction of the base excess, 1/cosh(mL)0.6481
- Length at which the tip excess is 1 % of the base excess, 5.298/m0.5298 m
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 = 1 the tip is at 338.9 K, only 64.8 % of the base excess above ambient.
- Beyond a length of 0.53 m (mL ≈ 5.3) the fin adds under 1 % more tip excess, so a longer fin gains almost nothing.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (K) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Tb Base temperature | 250 to 1,000 K | 267.6 | 753.6 | 486 | +10% | +6.88 % |
| T_∞ Ambient temperature | 250 to 1,000 K | 321.3 | 585.2 | 264 | +10% | +3.12 % |
| m Fin parameter √(hP/(kAc)) | 0.1 to 100 1/m | 300 | 360 | 59.99 | +10% | −0.863 % |
| L Fin length | 0.001 to 1 m | 300 | 360 | 59.99 | +10% | −0.863 % |
Sweep of Fin length
| Fin length (m) | Tip temperature (K) |
|---|---|
| 0.001 | 360 |
| 0.002371 | 360 |
| 0.005623 | 359.9 |
| 0.01334 | 359.5 |
| 0.03162 | 357.1 |
| 0.07499 | 346.3 |
| 0.1778 | 319.7 |
| 0.4217 | 301.8 |
| 1 | 300 |
Sweep one variable
Variables and how each one contributes
Tip temperature
What the equation gives you. Everything else feeds it.
Ambient temperature
The ambient temperature T∞, in K. Raises the tip temperature by the same amount. Raising it by 10 % (to 330) moves the result by +3.12 %, from 338.9 to 349.4 K.
Base temperature
The base temperature, in K. Raises the tip temperature in proportion to the excess above ambient. Raising it by 10 % (to 396) moves the result by +6.88 %, from 338.9 to 362.2 K.
Fin parameter √(hP/(kAc))
The fin parameter √(hP/(kAc)), in 1/m. A larger m means faster loss along the fin, so the tip is cooler. Raising it by 10 % (to 11) moves the result by −0.863 %, from 338.9 to 336 K.
Fin length
The fin length, in m. A longer fin has a cooler tip. Raising it by 10 % (to 0.11) moves the result by −0.863 %, from 338.9 to 336 K.
Worked example with the values above
- Fin number: mL = 10 × 0.1 = 1.
- Tip fraction: 1/cosh(1) = 0.64805.
- Ttip = 300 + (360 − 300) × 0.64805 = 338.88 K.
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