Grey-Surface Radiation Exchange

Calculate net radiation between two large, facing, parallel grey surfaces.

How this calculation works

Radiation between two facing grey surfaces depends on both emissivities and the fourth powers of their absolute temperatures. Enter each surface temperature in Celsius, each emissivity as a fraction above zero and no greater than one, and the facing area. The calculator converts temperatures to kelvin and combines the two emissivities into an effective exchange emissivity. Net flux is positive from surface 1 to surface 2 when the first surface is hotter; reversing their temperatures reverses the sign. Multiplying flux by area gives the net radiation duty. The geometry assumes effectively infinite, parallel, isothermal planes with a view factor of one and a nonparticipating medium. Arbitrary enclosures, small objects, solar input, conduction and convection require separate treatment.

Inputs and units

Surface 1 temperature (°C): Must be above absolute zero
Surface 2 temperature (°C): Must be above absolute zero
Surface 1 emissivity (1)
Surface 2 emissivity (1)
Facing area (m²)

Method and formula

εeff = 1/(1/ε1 + 1/ε2 − 1); q = σ εeff (T1⁴ − T2⁴); σ = 5.670374419×10⁻⁸ W/(m²·K⁴); temperatures use kelvin; Q = qA.

Worked example

Example inputs

Surface 1 temperature: 100 °C
Surface 2 temperature: 20 °C
Surface 1 emissivity: 0.8 1
Surface 2 emissivity: 0.8 1
Facing area: 10 m²

Calculation steps

Convert temperatures to kelvin: T1 = 100 + 273.15 = 373.15 K; T2 = 20 + 273.15 = 293.15 K.
Combine emissivities: εeffective = 1/(1/0.8 + 1/0.8 − 1) = 0.666667, dimensionless.
Evaluate the Stefan–Boltzmann exchange: q = 5.670374419 × 10⁻⁸ × (2/3) × (373.15⁴ − 293.15⁴) = 453.738819 W/m².
Multiply by facing area: Q = 453.738819 × 10/1000 = 4.537388 kW, from surface 1 to surface 2.

Example results

Effective emissivity: 0.6666666667 1
Signed net flux, surface 1 to 2: 453.738819 W/m²
Signed net radiation duty: 4.53738819 kW

Assumptions

Opaque diffuse-grey, isothermal, effectively infinite parallel planes; view factor equals one
The intervening medium does not absorb or emit radiation

Limitations

Unsuitable for arbitrary shapes, small objects in large rooms, spectral surfaces or participating gases
Does not include conduction, convection or solar input

Sources

MIT 2.810: Thermoforming: Slide 6, radiation exchange between parallel plates
NIST CODATA: Stefan-Boltzmann constant: NIST SP 961 (May 2024), SI Stefan-Boltzmann constant