Optical density

Optical density describes the attenuation of light along a path through smoke. In fire engineering, the term commonly means the base-ten optical density per unit distance, expressed in inverse metres. It connects a measured or calculated aerosol concentration with light transmission and, through an additional empirical relationship, with visibility in smoke.

The logarithm convention matters. Optical density per metre and the natural-logarithm extinction coefficient describe the same physical attenuation, but their numerical values differ. Neither quantity is a mass concentration, a percentage obscuration, or a direct measure of toxicity. Confusing these quantities can substantially change the interpretation of a fire calculation.

A defensible application therefore identifies the optical quantity, path length, aerosol properties and intended use. Laboratory smoke measurements, predicted visibility and detector response require related information, but they do not use that information in identical ways. Particle size, lighting and the exposure of the observer remain relevant after an optical density has been calculated.

Definitions and logarithm conventions

For a homogeneous smoke path, the transmitted fraction of monochromatic light follows the attenuation relationship given by Mulholland. The same relationship can be written with either a natural exponential or a base-ten exponential:

The two logarithm conventions describe the same transmission but use different coefficients. Based on Mulholland (2002), p. 2-263.

I/I₀ = exp(−KL) = 10^−DL

D = log₁₀(I₀/I)/L = K/ln(10)

Here the path-integrated logarithmic attenuation is DL. Dividing by the path length produces the optical density per unit distance used throughout this article. A reported optical density is consequently incomplete unless its normalisation is clear. A dimensionless path measurement cannot be inserted directly into an equation requiring inverse metres.

Symbol	Meaning	Unit
I0, I	Incident and transmitted light intensity, measured consistently	Same units
L	Optical path length	m
D	Base-ten optical density per unit distance	m−1
K	Natural-logarithm extinction coefficient	m−1
Km	Mass-specific extinction coefficient, based on aerosol mass	m2/kg
C	Aerosol mass concentration	kg/m3
S	Visibility distance for a specified target and correlation	m

The conversion factor is ln(10), approximately 2.303. Thus D and K are not alternative symbols for an identical number. The unit alone cannot identify the convention because both quantities have inverse-length dimensions. Model documentation and measurement reports must identify the defining equation as well as the symbol.

For example, D = 1 m⁻¹ means that a one-metre path transmits one tenth of the incident light. It does not mean complete obscuration. The exponential relationship also means that doubling the path length multiplies two transmission fractions; it does not simply double a percentage light loss.

From aerosol mass to optical attenuation

The extinction coefficient can be related to the mass concentration of suspended particles through a mass-specific coefficient. With the definitions above, the corresponding optical density is:

K = K_mC

D = K_mC/ln(10)

This formulation separates the amount of aerosol from its optical effectiveness. The soot yield contributes to the amount released by the fire. The mass-specific extinction coefficient describes attenuation per unit suspended mass. Increasing either input raises the calculated optical density when the other input remains fixed.

The coefficient is not determined by particle mass alone. Mulholland relates extinction to particle size distribution, refractive properties and wavelength. Different aerosols can therefore produce different attenuation at the same mass concentration. The distinction is particularly relevant when comparing dark particles from flaming combustion with condensed droplets from pyrolysis or smouldering.

The supplied CFAST technical reference uses a default mass-specific extinction coefficient of 8700 m²/kg. It calculates layer optical density from predicted soot concentration using the logarithm conversion above. That value is a documented model parameter; it is not evidence that every fuel and combustion condition produces an identical aerosol.

Concentration also changes after production. Dilution and transport alter the amount of aerosol within a particular volume. Coagulation changes particle size, while deposition removes particles from the suspended phase. Condensation and evaporation can change the aerosol mass and properties. A source yield therefore does not, by itself, specify the optical conditions at a remote location.

This distinction prevents a common interpretation error. A high mass of smoke produced by a fire does not specify a unique room optical density. The receiving volume, distribution, losses and optical properties must also be known. A well-mixed estimate may provide a useful calculation, but its uniform concentration is an explicit assumption.

Laboratory quantities and measurement practice

Several laboratory smoke quantities contain the words optical density. They normalise the measurement in different ways. Mulholland defines specific optical density, D_s, using the chamber volume V_c and exposed specimen area A:

D_s = DV_c/A

This quantity is dimensionless. It describes a test result normalised to specimen area, rather than the optical density per metre within an occupied space. Mulholland notes that specimen thickness affects the result. Comparing specimens therefore requires attention to the test configuration as well as the reported number.

Mass optical density instead uses the specimen mass loss, ΔM, as the denominator:

Aerosol-based extinction and fuel-based mass optical density use different denominators. Based on Mulholland (2002), pp. 2-263–2-266.

D_m = DV_c/ΔM

When mass is expressed in kilograms, D_m has units of square metres per kilogram of fuel lost. This denominator differs from the aerosol mass used for K_m. Replacing one coefficient with the other would combine a different mass basis with a different logarithm convention. The identical dimensional form does not make the coefficients interchangeable.

A direct mass optical density measurement incorporates the optical consequence of the aerosol produced during that test. A calculation using yield and mass-specific extinction instead combines two separately selected inputs. Either approach requires representative combustion conditions. Agreement between small and large tests can deteriorate when the fire becomes more complex.

Measurement geometry matters as well. The attenuation law is formulated for monochromatic light. Mulholland describes an extinction instrument that uses a monochromatic source and excludes forward-scattered light from the receiver. A broad-spectrum source or a receiver collecting scattered light can change the measured relationship between transmitted intensity and aerosol concentration.

An engineering record should therefore preserve the original test quantity, specimen condition and mass basis. Converting every reported number into a generic smoke coefficient can conceal information needed to judge applicability. Heat flux and ventilation during the test are especially relevant because both can change smoke production.

Worked calculation for a uniformly mixed room

Mulholland illustrates the calculation using a 200 g polyurethane foam pillow burning in a room measuring 6 m by 6 m by 2.5 m. The example selects an aerosol conversion factor of 0.03 and assumes all produced aerosol remains uniformly distributed. Its mass-specific extinction coefficient is 7.6 m²/g.

Using kilograms consistently gives a fuel mass of 0.200 kg, an aerosol mass of 0.00600 kg and a room volume of 90.0 m³. The resulting concentration is 6.67 × 10⁻⁵ kg/m³. Converting the coefficient to 7600 m²/kg gives:

K = 7600 × (0.00600/90.0) = 0.507 m⁻¹

D = 0.5067/ln(10) = 0.220 m⁻¹

A calculated extension of the example gives a one-metre transmission of approximately 60.3%. Over five metres, transmission falls to approximately 7.94%. These values follow from the same concentration and attenuation coefficient. They illustrate why the length of the viewing path must accompany a statement about light transmission.

Transmission decreases exponentially with path length in the uniformly mixed example. Based on Mulholland (2002), pp. 2-263, 2-265–2-266.

The source also presents an alternative estimate using a fuel-based mass optical density of 0.22 m²/g. Applying that value directly to the 200 g mass loss gives D = 0.489 m⁻¹ and K = 1.13 m⁻¹. The two routes therefore produce materially different predictions.

This difference is informative. The alternative calculation uses different experimental input information; it is not a correction obtained by changing the logarithm convention. Averaging the two answers would not resolve their differing assumptions. The source discusses the more direct character of the mass optical density approach while retaining the need for relevant test data.

The example has substantial physical limitations. Real smoke may concentrate near the ceiling, and the selected yield came from small-scale generic foam data. The pillow could smoulder before flaming, producing a different aerosol yield. The extinction coefficient also came from limited tests. The calculation is consequently an illustration of the method, not a product-specific room prediction.

Visibility and the observer

Optical attenuation becomes a visibility estimate only after a target and a visibility relationship are selected. Mulholland presents empirical correlations of KS = 3 for a light-reflecting sign and KS = 8 for a light-emitting sign. These relationships use K, not the base-ten optical density D.

For the first room calculation, the reflecting-sign relationship gives a visibility distance of approximately 5.9 m. The alternative mass optical density input gives approximately 2.7 m. Both values are calculations within the source's assumptions. Neither establishes that an occupant can necessarily identify or reach an exit at that distance.

Visibility depends on contrast between an object and its background. Illumination, the optical properties of the aerosol and the observer's visual acuity affect that contrast. Adaptation to dark or bright conditions also matters. An attenuation coefficient alone therefore cannot represent every aspect of visual performance during evacuation.

The experimental conditions behind the correlations provide another boundary. Subjects viewed smoke through glass, which excluded the direct effect of irritants on their eyes. Irritating smoke can reduce visual acuity beyond the effect of light attenuation. Applying the optical correlation does not separately account for this physiological response.

These limitations matter when selecting tenability criteria. A visibility threshold is a criterion for a specified assessment, rather than a conversion of optical density into a universal safe exposure. The visual target, lighting assumptions and required occupant task should remain identifiable in the engineering interpretation.

Percentage obscuration and detector response

Percentage obscuration describes the fraction of light lost over a stated path. For a one-metre reference path, the conversion from D is:

O_1m = 100[1 − 10^{−D × 1 m}]

The one-metre reference is part of the definition. A percentage measured over another path cannot be converted by simply dividing that percentage by the number of metres. The transmission fractions must first be related through the exponential attenuation law. This distinction becomes increasingly significant as attenuation rises.

For the first room example, the one-metre obscuration is approximately 39.7%. This percentage and D = 0.220 m⁻¹ describe the same optical state. The numerical percentage is larger because it is a different expression of transmission, not because it represents a greater aerosol concentration.

A smoke detector introduces another relationship. Light-scattering and ionisation detectors respond differently to particle size distributions. Mulholland shows that a detector's output depends on its response function and the distribution of particle sizes. Matching optical density alone does not guarantee an equal response from different detector principles or different aerosols.

CFAST represents smoke detector activation by comparing layer obscuration with a specified threshold. Its technical reference describes this as a limited approach that omits transport delay and smoke penetration into the detector. A calculated threshold crossing should therefore be interpreted within that model, rather than as a complete physical simulation of detector operation.

Engineering interpretation and uncertainty

A useful calculation keeps the full chain visible: fuel loss, aerosol yield, transport, local concentration, extinction coefficient, optical density and the final visibility or detection relationship. Uncertainty can enter at every stage. Reporting only the final distance or activation time makes it harder to identify which assumption controls the result.

Sensitivity to the optical coefficient can be separated from sensitivity to aerosol production. Holding the room-example concentration fixed while using 8700 m²/kg gives K = 0.580 m⁻¹. The reflecting-sign estimate then becomes approximately 5.17 m. This is a parameter comparison, not evidence that one room calculation has been validated.

The CFAST formulation also illustrates a modelling boundary. Soot is transported with the gas species, and each layer has a representative concentration. That treatment provides an optical density for the modelled layer. It does not resolve every individual viewing path or the detailed particle processes that can change an aerosol during transport.

A laboratory comparison needs a similarly careful boundary. An area-normalised chamber result describes the smoke produced by the exposed specimen configuration. A fuel-mass-normalised result describes optical attenuation relative to measured specimen loss. Selecting between them depends on the information required for the subsequent calculation, rather than which number appears larger.

Direct mass-loss measurements have their own requirements. The sample mass change must be measured accurately alongside attenuation. An apparent improvement in the optical measurement cannot compensate for an uncertain denominator. Conversely, a reliable fuel mass loss does not establish that the smoke has the same optical properties under another ventilation condition.

These distinctions explain why a conversion between units can be exact while the resulting engineering prediction remains uncertain. The logarithmic relationship follows directly from the attenuation definitions. The relationship between a laboratory specimen and a developing room fire depends on physical representativeness. Keeping those two issues separate makes a sensitivity calculation easier to interpret.

The most useful checks are consequently definitional and physical. The logarithm, units, mass basis and optical path must agree. The selected aerosol data must suit the combustion conditions, and the assumed distribution must suit the location being assessed. These checks establish what the optical result represents before it is used in a broader fire safety decision.

See also

• Soot

References

• Mulholland, G. W. (2002). Smoke production and properties. In P. J. DiNenno (Ed.), SFPE handbook of fire protection engineering (3rd ed., pp. 2-258–2-268). National Fire Protection Association.
• Peacock, R. D., McGrattan, K. B., Forney, G. P., & Reneke, P. A. (2026). CFAST: Consolidated model of fire growth and smoke transport, version 7. Volume 1: Technical reference guide (NIST Technical Note 1889v1, revision 7.7.7). National Institute of Standards and Technology.