Point source model

A point source model represents the radiation from a fire as though its radiative power came from one location. The power spreads outward, and the predicted flux decreases with the square of distance from that point. For a pool fire, a common representation places the point on the source centreline at half the mean flame height.

The method provides a simple estimate of flame radiation reaching a target outside the flames. It requires the radiative power, source position, target position and receiving surface orientation. It does not retain the full size or shape of the emitting flame. This simplification becomes more appropriate at large distances relative to the source dimensions. Close to the fire, the neglected geometry can produce substantial errors, including underestimated exposure in the pool fire comparisons reviewed by Beyler.

What the equivalent point represents

The equivalent source is a mathematical representation of the fire's radiation field. It does not mean that combustion physically occurs at one point. A real flame occupies a volume and has varying temperatures, soot concentrations and visible boundaries. The point model replaces those details with an overall radiative power and a selected position.

For an ideal isotropic point, radiation is distributed equally in all directions. Its power crosses a spherical area that grows with the square of radius. This geometry gives the inverse square distance relationship. A real extended flame approaches that representation only when its dimensions have sufficiently little influence on the radiation arriving at the target.

The prediction concerns incident radiant heat flux at a particular receiving surface. It is not the temperature of that surface, the energy it ultimately absorbs or its time to ignition. Those outcomes require additional information about material response and exposure duration. The US nuclear inspection guidance explicitly distinguishes a flame temperature from the temperature reached by an exposed object.

Source location also depends on the represented scenario. A floor spill and a fire in elevated equipment do not necessarily have the same reference height. NUREG-1805 discusses both fixed and transient combustibles, including equipment and cable arrangements. The relevant distance must therefore connect the chosen emitting point to the actual target, rather than simply reuse a floor plan separation.

Distance and receiving orientation

The general expression divides radiative power by the surface area of a sphere and includes a target orientation factor. The cosine accounts for the angle between the target's outward facing normal and its line of sight to the source. A surface facing the point directly receives the largest flux at that location under this representation.

Distance and incidence angle are measured from the equivalent source point. Based on Beyler (2002), chapter 3-11, p. 3-274.

q̇″ = Q̇_r cos θ / (4πR²)

For the illustrated pool fire method, the point is halfway up the mean flame height. The straight line distance to the target follows from the horizontal separation and their vertical difference. Horizontal distance alone equals the required distance only when the target and equivalent point are at the same elevation.

R = √(L² + H_T²)

Symbol	Description	Unit
q̇″	Incident radiant heat flux on the target	kW/m²
Q̇r	Total radiative power represented by the source	kW
R	Straight line distance from source point to target	m
θ	Angle between receiving surface normal and line of sight	radians
L	Horizontal distance from source centreline to target	m
HT	Vertical difference between target and source point	m

A target's orientation and position are separate inputs. Moving a target changes its distance and line of sight, while rotating it changes the angle relative to its surface normal. A vertical wall at ground level therefore need not receive the same flux as a surface tilted to face the equivalent point. Both can occupy the same target location.

Care is also needed with the meaning of separation distance. Distance from the edge of a pool differs from distance from its centreline. The pool method uses centreline distance when locating the equivalent source. Substituting a shorter edge clearance into the same equation would represent a different geometry, particularly for a large pool.

Obtaining a consistent source power

Radiative power is obtained by multiplying total heat release by the radiative fraction. The fraction describes the part of actual combustion power represented as outgoing radiation. It is not a geometric factor and does not account for distance to the target. Geometry enters separately through distance and orientation.

Q̇_r = χ_r Q̇

Beyler's particular pool fire procedure estimates the radiative fraction from pool diameter. The relationship was fitted to experimental results interpreted through the point source method. It must therefore remain associated with that method and its source assumptions. It is not a universal material property for all fires involving the same fuel.

χ_r = 0.21 − 0.0034 D

For diameters above 50 m, the source specifies retaining the fraction calculated at 50 m. Continuing the straight line indefinitely would depart from the stated treatment. The source also uses a representative mean flame height correlation to locate the equivalent point. This relationship requires total heat release in kilowatts and source diameter in metres.

H = 0.235 Q̇^2/5 − 1.02 D

Symbol	Description	Unit
χr	Radiative fraction in the specified pool method	dimensionless
Q̇	Actual total heat release rate	kW
D	Pool diameter	m
H	Mean flame height	m

For a nearly circular noncircular pool, an equivalent area diameter can be used within the method's geometric assumptions. A strongly elongated fire requires further consideration because equal area does not preserve its full radiating geometry. Fuel burning rate, flame dimensions and radiative properties therefore remain linked parts of the source description.

Beyler cautions that empirical components from different pool fire methods cannot be combined indiscriminately. A fraction fitted through a point representation and an emissive power fitted through a cylinder represent different models. Selecting apparently conservative pieces from each does not establish a consistent prediction. The complete method, including its calibration and limitations, is the relevant unit of comparison.

The radiative fraction fit was obtained using the most remote measurement location in each pool experiment. Radiation measured at that location was interpreted through the equivalent point representation. This procedure estimates a source strength consistent with the assumed geometry. It is different from independently measuring the total radiant output in every direction around a fire.

The geometry is likewise based on a mean visible flame envelope. Real pool flame tips fluctuate, so an instantaneous photograph can show a different height from the mean used in the calculation. Beyler describes several criteria used to characterise flame height and the variation associated with those definitions. A consistent estimate therefore identifies what the quoted dimension represents, rather than treating every visible tip as a fixed source boundary.

These two forms of averaging operate together. An average radiative power placed at an average height describes a mean exposure field under the selected assumptions. It does not reproduce short local peaks caused by flame motion. That distinction is relevant when comparing the result with a measurement taken over a different time interval or at a rapidly changing position.

A toluene pool calculation

Beyler's example considers a toluene spill forming a 12 m diameter pool, with a ground level target 30 m from its centre. The stated mass burning rate is 0.1126 kg/(m²·s), and the heat of combustion is 40,550 kJ/kg. Multiplying these by the circular burning area gives approximately 516 MW of total power. These inputs describe the large open pool example.

The diameter dependent radiative fraction is 0.1692, giving approximately 87.4 MW of radiative power. Applying the main flame height equation with its coefficient of 0.235 gives 33.08 m. The equivalent point therefore lies about 16.54 m above the pool. The straight line distance to the ground level target becomes approximately 34.26 m.

For a vertical receiving surface facing the pool, the cosine of the incidence angle is about 0.876. Substitution into the point source equation gives approximately 5.19 kW/m². This recalculation consistently uses the main equation's coefficient of 0.235 and retains intermediate precision. The source's worked arithmetic uses 0.23 and reports approximately 5.3 kW/m².

The distinction between a vertical target and one facing the source point directly is visible in the calculation. Omitting the cosine at the same location gives approximately 5.92 kW/m². That value represents a different receiving orientation. It is not an alternative rounding of the vertical target's result.

Moving the vertical target to a horizontal distance of 60 m gives approximately 1.73 kW/m² under the same source assumptions. This is a calculated extension of the example, rather than another reported experiment. Doubling horizontal distance does not exactly quarter the result because the vertical separation and incidence angle also influence the geometry.

The calculated exposure depends on both separation and receiving orientation. Based on Beyler (2002), chapter 3-11, pp. 3-274–3-275 and 3-286–3-287.

The result at 30 m also exposes a limitation of the screening method. It lies slightly above the approximately 5 kW/m² range recommended in Beyler's evaluation of point source predictions. The arithmetic alone therefore does not demonstrate suitability for an ignition assessment at that exposure. The chapter directs attention to more detailed representations where the point approximation performs poorly.

Validity and near field errors

At sufficiently large distance, an extended flame can behave approximately like a point radiator. Closer to the source, its finite height, width and orientation matter. Different parts of the flame then occupy different directions and distances from the target. The single equivalent point cannot retain that spatial distribution.

Beyler cites a theoretical analysis giving agreement within 5% beyond a horizontal distance to diameter ratio of 2.5. The chapter explicitly states that this result concerned a particular modelled flame and was not derived from experimental validation. It therefore does not establish a universal distance threshold for every fuel, flame shape or target arrangement.

The reviewed pool fire comparisons show systematically nonconservative predictions at fluxes above about 5 kW/m². In that context, nonconservative means that predicted exposure is lower than measured exposure. The chapter identifies poor near field performance and limits the recommended point source application. A simple inverse square calculation must not be assumed to provide an upper bound merely because it is simple.

The supplied US nuclear inspection report describes near field error differently, stating that the point method can overestimate exposure close to a fire. That statement differs from Beyler's reported underprediction in the evaluated pool data. These sources do not support a universal direction of error across all implementations. The particular formulation, geometry and comparison evidence remain necessary when interpreting a result.

Beyler recommends a factor of two for design use of the evaluated point method below 5 kW/m². The chapter distinguishes that design treatment from a calculation intended to estimate realistic exposure without an added factor. The recommendation is part of the documented methodology, rather than a universal regulatory requirement. It also does not extend the model's validity to every higher flux.

Relationship to finite flame models

A solid flame model represents the visible flame envelope as an emitting surface with finite dimensions. Its source may be approximated by a cylinder or another defined shape. Radiation arriving at a target is then related to the emitting surface through a view factor. This retains geometric information that the point model discards.

A finite flame representation retains geometry discarded by an equivalent point. Based on Beyler (2002), chapter 3-11, pp. 3-273–3-276.

The additional geometry does not remove uncertainty in the radiative source. Average emissive power must still be estimated, and flame dimensions themselves depend on burning behaviour and wind. Beyler's assessment identifies radiative properties as a substantial uncertainty in detailed methods. A more elaborate geometric expression therefore needs compatible source data as well as careful implementation.

Fire spread between buildings assessments can involve extended openings rather than a compact open pool. Van Rossum demonstrates finite rectangular source calculations for such arrangements. The positions of separate emitting openings influence the exposure at the receiving facade. Replacing those areas by one point would be an additional approximation requiring its own justification.

Atmospheric absorption can also affect transmission over long paths. Beyler identifies dependence on path length, flame characteristics and atmospheric moisture. That process is separate from geometric dilution with distance. A model that neglects it and a model that includes it are not identical merely because both begin with the same total radiative power.

Interpreting the result

An incident flux is one part of an exposure assessment. The duration of heating and the properties of the receiving material determine how its temperature evolves. A brief exposure and a sustained exposure at the same flux need not produce the same outcome. The point model itself contains no thermal response calculation to resolve that difference.

The fire description also needs a credible burning rate and geometry. A prescribed total power does not automatically establish flame height, radiative fraction or source elevation without additional assumptions. The method makes those assumptions visible through separate inputs. Their consistency matters as much as evaluating the final inverse square expression.

A calculation record can therefore identify the exact source formulation, reference levels, target orientation and treatment of uncertainty. These details allow another assessment to reproduce the same physical representation. They also explain why apparently similar point source calculations can return different answers. Differences in source location or the definition of distance can be material even when total power is identical.

References

• Beyler, C. L. (2002). Fire hazard calculations for large, open hydrocarbon fires. In P. J. DiNenno (Ed.), SFPE handbook of fire protection engineering (3rd ed., pp. 3-268–3-314). National Fire Protection Association.
• Iqbal, N., & Salley, M. H. (2004). Fire dynamics tools (FDTs): Quantitative fire hazard analysis methods for the U.S. Nuclear Regulatory Commission fire protection inspection program (NUREG-1805). U.S. Nuclear Regulatory Commission.
• Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.