Radiation from a participating gas
A flame is not a gray surface. CO₂ and H₂O absorb and emit in narrow bands, so the gas is transparent at some wavelengths and nearly opaque at others — which is why a furnace calculation cannot use a single emissivity. The usual engineering treatment is the weighted sum of gray gases: a handful of fictitious gray gases fitted to the band data.
| Quantity | Value | Note |
|---|---|---|
| Gas temperature | 1200 °C | 1473 K |
| Wall temperature | 400 °C | 673 K |
| Partial pressure of CO₂ + H₂O | 0.250 atm | typically 0.2–0.3 for flue gas |
| Mean beam length | 2.00 m | 3.6 V/A for a box |
| pL | 0.500 | the scaling group |
| Gas emissivity | 0.461 | WSGG at the gas temperature |
| Gas absorptivity | 0.808 | at the wall temperature |
| Wall emissivity | 0.01 | refractory or tube |
| Effective emissivity | 0.010 | 1/(1/ε_g + 1/ε_w − 1) |
| Heat flux | 1.14 kW/m² | band model |
| Gray approximation | 2.52 kW/m² | single emissivity |
| Difference | -55% | band model lower |
At pL = 0.50 atm·m the gas emits with an emissivity of 0.46, and the band model gives 1.1 kW/m² against 2.5 kW/m² from a single-gray-gas approximation — -55% different. The discrepancy is why furnace models use band data rather than one emissivity.
ε_g = Σ a_i(T)(1 − e^(−b_i pL)) · Three gray gases.
q = ε_w σ(ε_g T_g⁴ − α_g T_w⁴) · Hottel form.
α_g = ε_g(T_w)(T_g/T_w)^0.5 · Temperature correction.
L = 3.6 V/A · Rectangular box.
A clear path from inputs to results.
A flame is not a gray surface. CO₂ and H₂O absorb and emit in narrow bands, so the gas is transparent at some wavelengths and nearly opaque at others — which is why a furnace calculation cannot use a single emissivity.
How to use
- Enter the gas and wall temperatures.
- Set the partial pressure of CO₂ and H₂O.
- Add the mean beam length and the wall emissivity.
- Read the gas emissivity, the absorptivity and the flux.
The formula
Reference: Three-term weighted-sum-of-gray-gases fit; flux in the Hottel form.
Worked example
Flue gas at 1,200 °C against a 400 °C wall
- Gas 1,200 °C, wall 400 °C.
- 25% CO₂ and H₂O by volume, 2 m beam length.
- Wall emissivity 0.8.
- 1
The pressure-path-length product is 0.25 × 2 = 0.5 atm·m.
- 2
At 1,473 K the WSGG weights give a gas emissivity of 0.46.
- 3
At the wall temperature the absorptivity is higher, 0.81, because the bands narrow as the gas cools.
- 4
The flux is 0.8 × σ × (0.46 × 1473⁴ − 0.81 × 673⁴) = 91 kW/m².
- 5
A single-gray-gas approximation gives a different answer, which is why band models exist.
ε 0.46 · 91 kW/m² · absorptivity 0.81.
Interpreting the result
The emissivity falls with temperature and rises with the path length, so a large furnace is more radiative per unit area than a small one at the same temperature. That is the opposite of a gray surface, whose emissivity does not depend on size at all.
Assumptions & limitations
- A three-term WSGG fit for a 1 atm flue-gas mixture.
- Gray wall with a stated emissivity.
- No soot, ash or luminous flame contribution.
- Uniform gas temperature and a mean beam length.
Frequently asked questions
Why not use a single emissivity?
Because the gas is not gray. The error from a single-gray-gas approximation is typically 10–30%, and it goes the wrong way at high temperature.
Why is absorptivity different from emissivity?
Because the gas absorbs at the wall temperature, where the bands are narrower. The correction factor (T_g/T_w)^0.5 accounts for it.
Does this work for a boiler furnace?
Yes, as the radiation term. A full furnace calculation couples it with the heat balance and the gas flow, which is what a zone model or CFD does.
What is the mean beam length?
The effective path length for radiation, 3.6 V/A for a rectangular box. A large furnace has a long one, and that raises the emissivity.
What about a luminous flame?
Soot particles radiate continuously and add a large gray component. A WSGG model alone underestimates a sooty flame, sometimes badly.
Methodology last reviewed 2026-10-22. Results are informational and not financial or engineering advice — see our disclaimer.