Extinction coefficient

The extinction coefficient K [1/m] measures how strongly smoke attenuates light per metre of path. It is the quantity from which visibility, walking speed in smoke and most smoke tenability limits are computed.

Symbols follow the notation table; here L is the length of a light path, not the route length \(L_k\) of the table.

The equations as published

Light of intensity \(I_0\) that crosses a path of length L [m] through smoke leaves with intensity I given by the Beer–Lambert law, written in the Fire Dynamics Simulator (FDS) User Guide (McGrattan et al., FDS 6.10.1, §22.10.5) as

$$ \frac{I}{I_0} = e^{-KL} \qquad \text{(FDS UG Eq. 22.21)} $$

and K is the product of a mass-specific extinction coefficient \(K_m\) [m²/kg] and the mass concentration of smoke particulate \(\rho Y_S\) [kg/m³]:

$$ K = K_m\, \rho Y_S \qquad \text{(Eq. 22.22)} $$

Along a path on which K varies, the exponent becomes the integral of K along the path, which FDS evaluates as a sum over cells for its beam detector (Eq. 18.5). That dimensionless integral is the optical depth. The optical density per metre D [1/m] uses base-10 logarithms instead of natural ones:

$$ D \equiv -\frac{1}{L}\log_{10}\frac{I}{I_0} = K \log_{10} e \approx K/2.3 \qquad \text{(Eq. 22.24)} $$

The distinction matters when reading the literature. Tenability limits are often quoted as OD/m (optical density per metre, D), and Purser and McAllister (2016), in the Society of Fire Protection Engineers (SFPE) Handbook, give both forms: for example OD/m = 0.2 corresponds to an extinction coefficient of about 0.5 1/m (Ch. 63, Table 63.5 and the text beside it).

The data behind \(K_m\)

FDS uses \(K_m\) = 8700 m²/kg by default (MASS_EXTINCTION_COEFFICIENT). The User Guide describes it as a value suggested for flaming combustion of wood and plastics, and gives 8700 ± 1100 m²/kg at a wavelength of 633 nm for most flaming fuels (FDS User Guide §22.10.5, footnote 7), citing Mulholland and Croarkin (2000).

Known limits

K depends on wavelength, and the 8700 m²/kg figure is for red light at 633 nm. It is a property of flame-generated soot, not of every fire effluent. In a fire model K also inherits every uncertainty of the predicted soot yield and soot transport. FDS writes the local K as the EXTINCTION COEFFICIENT output quantity (§22.10.5); the unrelated quantity EXTINCTION is a combustion-suppression flag (§22.10.29).

Sources

• McGrattan, K., Hostikka, S., Floyd, J., McDermott, R., Vanella, M., Mueller, E., & Paul, C. (2025). Fire Dynamics Simulator User’s Guide. NIST Special Publication 1019, 6th ed., revision FDS-6.10.1-0-g12efa16, §18.3.6 and §22.10.5. github.com/firemodels/fds/releases/tag/FDS-6.10.1
• Mulholland, G. W., & Croarkin, C. (2000). Specific extinction coefficient of flame generated smoke. Fire and Materials, 24(5), 227–230. doi:10.1002/1099-1018(200009/10)24:5<227::AID-FAM742>3.0.CO;2-9 (value quoted from the FDS User Guide).
• Purser, D. A., & McAllister, J. L. (2016). Assessment of hazards to occupants from smoke, toxic gases, and heat. SFPE Handbook of Fire Protection Engineering, 5th ed., Ch. 63, 2308–2428. doi:10.1007/978-1-4939-2565-0_63

How pyFDS-Evac uses this: see the smoke-speed model and route rerouting.

How it is verified: ISO 20414 Test 18 and the S2 corridor test.