Library / Heat transfer and thermal systems / HTS-36
Heat transfer and thermal systems
Blackbody spectral emissive power
Hemispherical emissive power per unit wavelength of a black surface at temperature T, in W/m² per metre of wavelength (multiply by 10⁻⁶ for per micrometre).
- Ref
- HTS-36
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- E_λ [W/m³]
Planck’s law gives the hemispherical spectral emissive power of a blackbody, Eλ = 2πhPc²/{λ⁵[exp(hPc/(λkBT)) − 1]}, in W per square metre per metre of wavelength. Multiply by 10⁻⁶ for per micrometre. The exponential becomes huge at short wavelengths and low temperatures, so the calculation uses the form exp(−y)/(1 − exp(−y)) to avoid overflow.
Hemispherical emissive power per unit wavelength of a black surface at temperature T, in W/m² per metre of wavelength (multiply by 10⁻⁶ for per micrometre). The calculation avoids overflow at short wavelengths.
Calculator
- Emissive power per micrometre, 10⁻⁶ Eλ31.18 W/(m² µm)
- Wavelength of the peak, 2.898×10⁻³/T9.659×10−6 m
- Total emissive power of the blackbody, σT⁴459.3 W/m²
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.
- At 10 µm and 300 K a blackbody emits 3.118×107 W per m² per metre, which is 31.18 W/(m² µm).
- The peak for this temperature is at 9.659 µm; the spectrum is far from the peak when the wavelength is a lot shorter or longer than that.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (W/m³) | Local step | Change in result |
|---|---|---|---|---|---|---|
| T Absolute temperature | 100 to 10,000 K | 2,111 | 2.418e10 | 2.418e10 | +10% | +55.4 % |
| λ Wavelength | 1e-7 to 0.001 m | 1.944e-189 | 3.118e7 | 3.118e7 | +10% | −3.54 % |
Sweep of Absolute temperature
| Absolute temperature (K) | Spectral emissive power (W/m³) |
|---|---|
| 100 | 2,111 |
| 1,338 | 1.937e9 |
| 2,575 | 4.999e9 |
| 3,813 | 8.162e9 |
| 5,050 | 1.135e10 |
| 6,288 | 1.455e10 |
| 7,525 | 1.776e10 |
| 8,763 | 2.097e10 |
| 10,000 | 2.418e10 |
Sweep one variable
Variables and how each one contributes
Spectral emissive power
What the equation gives you. Everything else feeds it.
Wavelength
The wavelength, in m. Moves along the spectrum; the value peaks at λmax = b/T and falls on either side. Raising it by 10 % (to 1.1×10−5) moves the result by −3.54 %, from 3.118×107 to 3.008×107 W/m³.
Absolute temperature
The absolute temperature, in K. Raises the emissive power at every wavelength, most steeply at short wavelengths. Raising it by 10 % (to 330) moves the result by +55.4 %, from 3.118×107 to 4.844×107 W/m³.
Worked example with the values above
- Exponent: hPc/(λkBT) = 4.7959.
- Prefactor: 2πhPc²/λ⁵ = 3.7418×109; exponential term 1/(e^y − 1) = 0.0083322.
- Eλ = 3.1177×107 W/(m² m).
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 instead of kelvin.
- Mixing per-metre and per-micrometre units.
- Applying a blackbody spectrum to a surface of low or wavelength-dependent emissivity without correcting for it.
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