A glowing furnace, the warmth of the sun, a red-hot element — all reach you through thermal radiation, no contact or air required. Unlike conduction and convection, radiation needs no medium and grows explosively with temperature: it scales with the fourth power of absolute temperature. The Stefan-Boltzmann law captures this, and this free Stefan-Boltzmann Radiation Calculator computes the radiative heat transfer from a surface to its surroundings (or between two surfaces), plus blackbody emissive power, from temperature, emissivity and area.
Table of Contents
The Stefan-Boltzmann Radiation Calculator
Choose the case, enter temperatures (°C), emissivity and area, and get the net radiative heat plus blackbody emissive power. Temperatures are converted to kelvin internally.
☀ Stefan-Boltzmann Radiation Calculator
The Stefan-Boltzmann Law
Every surface above absolute zero radiates. The Stefan-Boltzmann law gives the emitted power per unit area of a real surface:
and the net radiative heat exchanged with large surroundings:
where σ = 5.67×10−8 W/m²K⁴ is the Stefan-Boltzmann constant, ε emissivity, A area, and temperatures are absolute (kelvin). Radiation is the third heat-transfer mode alongside conduction and the convection captured by our Nusselt number calculator.
Emissivity Explained
Emissivity (ε) rates how well a real surface radiates versus an ideal blackbody (ε = 1):
| Surface | Emissivity ε |
|---|---|
| Polished aluminium / silver | 0.02 – 0.1 |
| Oxidised metal | 0.6 – 0.85 |
| Painted / anodised surface | 0.8 – 0.95 |
| Human skin, water, wood | 0.9 – 0.98 |
| Perfect blackbody (ideal) | 1.0 |
The Fourth-Power Effect
The T⁴ dependence is what makes radiation special:
This is why radiation is negligible for a warm wall but dominant in a furnace or flame. It also means you must use absolute temperature — a common and serious error is plugging in Celsius.
Surface-to-Surface Exchange
Between two large parallel gray surfaces, emissivities combine:
For more complex geometries you also need a view factor (how much of one surface "sees" the other). Real furnace and enclosure radiation is a major use of CFD radiation models (DO, P-1, S2S).
Radiation vs Convection
- High temperature (furnaces, combustion, glowing metal) → radiation often dominates.
- Vacuum / space → radiation is the only mode (no fluid for convection).
- Near room temperature → convection usually larger, but radiation is never zero.
In building energy, radiant exchange works alongside the conduction in the U-value / R-value calculation to set total heat flow.
Worked Example
A 1 m² oxidised surface at 500 °C (773 K) radiating to a room at 25 °C (298 K), ε = 0.9:
- Q = 0.9 × 5.67e-8 × 1 × (773⁴ − 298⁴)
- 773⁴ = 3.57×1011; 298⁴ = 7.89×109
- Q = 0.9 × 5.67e-8 × (3.57e11 − 0.079e11) ≈ 17,831 W
Nearly 18 kW from one square metre — the fourth-power law in action. The cool surroundings contribute little because 298⁴ « 773⁴.
Common Mistakes
- Using Celsius instead of kelvin. The #1 radiation error — T must be absolute.
- Forgetting the surroundings term. Net heat uses (Ts⁴ − Tsur⁴), not just Ts⁴.
- Assuming ε = 1. Real surfaces radiate less; use the actual emissivity.
- Ignoring view factors for surfaces that don't fully face each other.
- Neglecting radiation at high T. Above a few hundred °C it often beats convection.
Frequently Asked Questions
What is the Stefan-Boltzmann law?
Radiated power per unit area is proportional to absolute temperature to the fourth power: E = εσT⁴. It makes hot surfaces radiate far more than warm ones.
What is the Stefan-Boltzmann constant?
σ ≈ 5.67×10−8 W/m²K⁴ — the constant linking absolute temperature to radiated power.
What is emissivity?
A 0–1 property of how well a surface radiates vs an ideal blackbody. Polished metals < 0.1; painted/oxidised surfaces > 0.8; blackbody = 1.
Why does radiation depend on temperature to the fourth power?
It comes from integrating Planck's law over all wavelengths. Doubling absolute temperature gives 16× the radiated power.
When is radiation important compared to convection?
At high temperatures (furnaces, combustion) and in vacuum/space it dominates. Near room temperature convection is usually larger, but radiation is never zero.
Conclusion
Thermal radiation is the heat-transfer mode that needs no medium and explodes with temperature: Q = εσA(Tₛ⁴ − Tₛ₰₨⁴). Remember to work in kelvin, use the real emissivity, include the surroundings, and respect the fourth-power law — and you can predict radiant heat from a hot surface or between two surfaces in seconds with the calculator above.
For more heat transfer, thermal-radiation and CFD tutorials plus free engineering calculators, explore Free CFD Tutorial. If this tool helped you, please share it with your fellow engineers and students.
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