Library / Heat transfer and thermal systems / HTS-14
Heat transfer and thermal systems
Base heat rate with prescribed fin-tip temperature
Heat entering a uniform fin when both the base and tip temperatures are imposed, from their excess temperatures θb and θL above the common ambient.
- Ref
- HTS-14
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- Qb [W]
If both ends of a fin are held at fixed temperatures, with excess temperatures θb at the base and θL at the tip, the heat entering at the base is Qb = kmAc[θb cosh(mL) − θL]/sinh(mL). It reduces to the insulated-tip result when θL = θb/cosh(mL), and the sign reverses if the tip is hot enough to feed heat back towards the base.
Heat entering a uniform fin when both the base and tip temperatures are imposed, from their excess temperatures θb and θL above the common ambient. The rate can change sign if the tip is hotter than the fin would be.
Calculator
- Heat rate if the tip were at ambient, kmAcθb/tanh(mL)6.565 W
- Heat flowing out at the tip end1.629 W
- Heat lost from the fin sides, Qb − Q at the tip3.235 W
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 the tip held 20 K above ambient, the fin takes 4.863 W at the base, against 6.565 W if the tip were at ambient.
- Raising the tip temperature reduces the heat drawn from the base, and above a critical value the fin would carry heat back to it.
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 | 0.04863 | 486.3 | 486.3 | +10% | +10 % |
| L Fin length | 0.001 to 1 m | 4.583 | 300 | 295.4 | +10% | −2.36 % |
| θb Base excess temperature | 0 to 500 K | −1.702 | 63.95 | 65.65 | +10% | +13.5 % |
| m Fin parameter √(hP/(kAc)) | 0.1 to 100 1/m | 3 | 50 | 47 | +10% | +7.41 % |
| θL Tip excess temperature | 0 to 500 K | −35.98 | 6.565 | 42.55 | +10% | −3.5 % |
| k Fin conductivity | 1 to 500 W/(m K) | 0.04863 | 24.32 | 24.27 | +10% | +10 % |
Sweep of Tip excess temperature
| Tip excess temperature (K) | Base heat rate (W) |
|---|---|
| 0 | 6.565 |
| 62.5 | 1.247 |
| 125 | −4.071 |
| 187.5 | −9.39 |
| 250 | −14.71 |
| 312.5 | −20.03 |
| 375 | −25.34 |
| 437.5 | −30.66 |
| 500 | −35.98 |
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 at fixed m. Raising it by 10 % (to 110) moves the result by +10 %, from 4.863 to 5.35 W.
Fin parameter √(hP/(kAc))
The fin parameter √(hP/(kAc)), in 1/m. Raises the heat rate, with the hyperbolic terms saturating at large mL. Raising it by 10 % (to 11) moves the result by +7.41 %, from 4.863 to 5.224 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 4.863 to 5.35 W.
Base excess temperature
The base excess temperature, in K. Raises the heat rate in direct proportion. Raising it by 10 % (to 55) moves the result by +13.5 %, from 4.863 to 5.52 W.
Tip excess temperature
The tip excess temperature, in K. A hotter tip lowers the heat taken at the base. Raising it by 10 % (to 22) moves the result by −3.5 %, from 4.863 to 4.693 W.
Fin length
The fin length, in m. A longer fin carries more heat from the base, saturating at large mL. Raising it by 10 % (to 0.11) moves the result by −2.36 %, from 4.863 to 4.749 W.
Worked example with the values above
- Hyperbolic terms: cosh(mL) = 1.5431, sinh(mL) = 1.1752.
- Numerator: θb cosh(mL) − θL = 50 × 1.5431 − 20 = 57.154 K.
- Qb = 0.1 × 57.154 / 1.1752 = 4.8633 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