Library / Heat transfer and thermal systems / HTS-39
Heat transfer and thermal systems
Radiative heat-transfer coefficient to large surroundings
Secant radiative coefficient for a small diffuse grey surface at Ts facing large black surroundings at Tr, so that q″ = hr(Ts − Tr) exactly for the pair of temperatures.
- Ref
- HTS-39
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- hr [W/(m² K)]
The radiative heat-transfer coefficient lets radiation be treated like convection. For a small grey surface at Ts facing large black surroundings at Tr, hr = εσ(Ts + Tr)(Ts² + Tr²), which gives q″ = hr(Ts − Tr) exactly, because Ts⁴ − Tr⁴ factors that way. It stays finite and sensible when the two temperatures are equal, unlike a ratio of differences.
Secant radiative coefficient for a small diffuse grey surface at Ts facing large black surroundings at Tr, so that q″ = hr(Ts − Tr) exactly for the pair of temperatures. It stays finite when the two are equal.
Calculator
- Net radiative flux, hr(Ts − Tr)793.9 W/m²
- Coefficient for a black surface (ε = 1)9.923 W/(m² K)
- Mean-temperature estimate, 4εσTm³7.78 W/(m² K)
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 radiation acts like a convection coefficient of 7.939 W/(m² K), and moves 793.9 W/m² across the 100 K difference.
- Natural convection in air gives about 5 to 10 W/(m² K), so near room temperature radiation is comparable to it, and it dominates when surfaces are hot.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (W/(m² K)) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Tr Surroundings temperature | 100 to 2,000 K | 3.856 | 452.9 | 449 | +10% | +12.2 % |
| Ts Surface temperature | 100 to 2,000 K | 1.815 | 426.7 | 424.9 | +10% | +19.9 % |
| ε Emissivity | 0 to 1 | 0 | 9.923 | 9.923 | +10% | +10 % |
Sweep of Surface temperature
| Surface temperature (K) | Radiative coefficient (W/(m² K)) |
|---|---|
| 100 | 1.815 |
| 337.5 | 5.897 |
| 575 | 16.7 |
| 812.5 | 37.86 |
| 1,050 | 73.03 |
| 1,288 | 125.9 |
| 1,525 | 200 |
| 1,763 | 299.1 |
| 2,000 | 426.7 |
Sweep one variable
Variables and how each one contributes
Radiative coefficient
What the equation gives you. Everything else feeds it.
Emissivity
The emissivity of the surface, between 0 and 1. Raises the coefficient in direct proportion. Raising it by 10 % (to 0.88) moves the result by +10 %, from 7.939 to 8.732 W/(m² K).
Surface temperature
The surface temperature, in K. Raises the coefficient, roughly as the cube of the temperature. Raising it by 10 % (to 440) moves the result by +19.9 %, from 7.939 to 9.52 W/(m² K).
Surroundings temperature
The surroundings temperature, in K. Raises the coefficient, roughly as the cube of the temperature. Raising it by 10 % (to 330) moves the result by +12.2 %, from 7.939 to 8.905 W/(m² K).
Worked example with the values above
- Temperature terms: Ts + Tr = 700 K; Ts² + Tr² = 250,000 K².
- Prefactor: εσ = 0.8 × 5.67037×10−8 = 4.5363×10−8.
- hr = 4.5363×10−8 × 700 × 250,000 = 7.9385 W/(m² K).
Assumptions and limits
- Blackbody or diffuse grey behaviour as stated, in a vacuum or non-participating medium.
- Absolute temperatures and the exact values of σ, h, c and kB given on the page.
Common mistakes
- Using °C in Planck’s law or Wien’s law.
- Mixing per-metre and per-micrometre spectral units.
- Treating a grey surface as if it followed the blackbody spectrum without considering emissivity.
Statistics and uncertainty
The radiation laws for blackbodies are exact, but real-surface emissivity and its variation with wavelength add uncertainty of 10 % or more, and measured view factors depend on exact geometry.
Where it is used
- Infrared sensing, lamps and radiant heaters.
- Setting up radiosity networks and linearised radiation coefficients.
Maths you need
- Planck’s law: spectral emissive power of a blackbody
- Wien’s law: the peak wavelength is inversely proportional to T
- Radiosity: the total radiation leaving a surface per unit area