Thermal Radiation Calculator (Stefan-Boltzmann)

Understanding the thermal radiation calculator — concept and principle

Every surface above absolute zero emits electromagnetic radiation. This calculator uses the Stefan–Boltzmann law to find the net heat a surface exchanges by radiation with its surroundings, and Wien's displacement law to find the wavelength at which its emission peaks. It is used for heaters, furnaces, pipes, electronics, building surfaces and thermal imaging.

How it is calculated

Q = ε · σ · A · (T_s⁴ − T_sur⁴), with σ = 5.670374419 × 10⁻⁸ W/m²K⁴.

• ε: emissivity, from 0 (perfect reflector) to 1 (black body).
• A: radiating area (m²).
• T_s, T_sur: surface and surroundings temperatures, entered in °C and converted to kelvin by adding 273.15.

Peak wavelength: λ_max = 2897.77 µm·K / T_s.

The principle

A black body emits σT⁴ watts per square metre, where T is absolute temperature; a real surface emits ε times that. The surroundings radiate back, and for a small grey surface inside a large enclosure the net exchange is εσA(T_s⁴ − T_sur⁴). The fourth power is why radiation dominates at high temperature: doubling the absolute temperature multiplies emission by 16. Wien's law follows from Planck's spectrum and explains why warm objects glow in the infrared while very hot ones become visibly red, then white.

Worked example

1. Inputs: ε = 0.9, A = 1 m², T_s = 100 °C, T_sur = 20 °C.
2. Kelvin: 373.15 K and 293.15 K.
3. Fourth powers: 373.15⁴ ≈ 1.93880 × 10¹⁰ and 293.15⁴ ≈ 7.38515 × 10⁹; difference ≈ 1.20029 × 10¹⁰ K⁴.
4. Q = 0.9 × 5.670374419 × 10⁻⁸ × 1 × 1.20029 × 10¹⁰ ≈ 612.55 W.
5. λ_max = 2897.77 / 373.15 ≈ 7.77 µm (thermal infrared).

How to read the result

The surface emits about 989 W and absorbs about 377 W from the room, leaving 612.5 W net. Positive values mean net heat loss. Emissivity has a large effect: polished metals can be below 0.1, while paint, brick, skin and water are typically above 0.9. At room-temperature levels, radiation from a surface is often comparable to natural convection.

Common mistakes

• Using °C directly in the T⁴ terms instead of kelvin.
• Assuming shiny metal has ε near 1 (it is low, which is why foil is used as a radiant barrier).
• Applying the formula to two surfaces of similar size facing each other, which needs view factors.
• Ignoring that emissivity varies with wavelength and temperature.

Frequently asked questions

Why does a thermal camera need the emissivity setting?

The camera measures emitted radiation; a low-emissivity surface emits less and would otherwise read colder than it is.

Does radiation need air to travel?

No. It travels through vacuum, which is how the Sun heats the Earth.

When can I linearise radiation?

For small temperature differences, Q ≈ h_r A ΔT with h_r ≈ 4εσT_m³, where T_m is the mean absolute temperature.

Sources: CODATA 2018 recommended values of the fundamental physical constants (NIST) · Incropera, DeWitt, Bergman and Lavine, Fundamentals of Heat and Mass Transfer, Wiley