Dimensionless heat release rate

The dimensionless heat release rate, usually written ${\dot{Q}}^{*}$, is a number that compares the heat release rate of a fire with the size of the area from which it burns. It divides the heat release rate by a reference quantity built from the fire diameter, the acceleration due to gravity and the density, specific heat and temperature of the surrounding air. Because all units cancel, a small laboratory flame and a large accidental fire can be placed on the same scale. The group is also called the fire Froude number, and it expresses the ratio of inertial and buoyancy-induced velocities in a fire.

Low values belong to broad fires with short flames, such as a brush fire, and high values to narrow fires with long flames, such as an oil well blowout. Most accidental fires have a value of the order of 1. Engineers use the number to estimate flame height, to describe the fire plume above a fire and to check whether a fire model has been validated for a given scenario. A related length, the characteristic fire diameter, guides the choice of grid cell size in computer fire models.

Definition

The dimensionless heat release rate divides the total heat release rate of a fire by a product of five factors. These are the density of the ambient air, the specific heat of air, the ambient temperature, the square root of gravity times the fire diameter, and the square of that diameter. Heskestad gives the definition in the following form:

The same heat release rate of 7.4 MW over a wide base (left) and a narrow base (right). The narrow fire has the higher dimensionless heat release rate and the taller flame. Based on Heskestad (2002), pp. 2-2–2-3.

$${\dot{Q}}^{*}=\frac{\dot{Q}}{{\rho }_{\mathrm{\infty }}{c}_p{T}_{\mathrm{\infty }}\sqrt{gD}{D}^2}$$

Symbol	Description	Unit
${\dot{Q}}^{*}$	dimensionless heat release rate (fire Froude number)	none
$\dot{Q}$	total heat release rate of the fire	kW
${\rho }_{\mathrm{\infty }}$	density of the ambient air	kg/m³
$c_p$	specific heat of air at constant pressure	kJ/(kg·K)
$T_{\mathrm{\infty }}$	ambient temperature	K
$g$	acceleration due to gravity	m/s²
$D$	diameter of the fire source	m

The units in the table follow the definition used in the NUREG-1824 validation study. The total heat release rate is the mass burning rate of the fuel multiplied by its heat of combustion. Many fires are not round, so the diameter is then taken as that of a circle with the same area as the fire source. With the units in the table, the numerator and the denominator are both in kilowatts, so the result is a pure number. The diameter appears with the power 5/2, so for the same heat release rate a doubling of the diameter lowers the number by a factor of about 5.7.

For ordinary room air the density, specific heat, temperature and gravity can be combined into one constant. Cadorin and co-workers insert standard values for air and obtain the more convenient form ${\dot{Q}}^{*}=\dot{Q}/(1.11\times 10^6{D}^{5/2})$, with the heat release rate in watts. The same constant appears as 1,110 in the New Zealand commentary to C/VM2, where the heat release rate is given in kilowatts. With the values of 1.2 kg/m³, 1 kJ/(kg·K), 293 K and 9.81 m/s² used in a NUREG-1824 example, the combined factor comes to about 1,100. The small difference between 1,100 and 1,110 reflects slightly different assumed values for the air properties.

What the number expresses

A fire plume rises because its hot combustion gases are less dense than the cooler air around them, and this upward drive is called buoyancy. Some fires also push gas upward with the speed of the fuel release itself, for example when a flammable gas escapes from a pressurised source. The fire Froude number compares these two effects as a ratio of inertial and buoyancy-induced velocities. Buoyancy-driven fire plumes have relatively low values, while momentum-driven plumes such as jet flares have relatively high values.

The NUREG-1824 validation reports illustrate the scale with two familiar examples. An oil well blowout releases a large amount of energy compared with its physical diameter and therefore has a large value. A brush fire releases little energy compared with its diameter and has a low value. The number thus describes how concentrated the heat release is over the base of the fire, not how large the fire is in absolute terms. Most conventional accidental fires have values of the order of 1, and the main use of the number in those reports is the assessment of flame height.

McCaffrey compiled flame height measurements from many researchers and plotted them against this Froude number. Pool fires, whose flames are about as tall as the base is wide, lie at the low end of the plot. An intermediate regime follows in which all flames behave in a similar way. At the high end lie high-momentum jet flames, whose flame height stops changing with the fuel flow rate and reaches several hundred times the source diameter. According to Heskestad, the group was first derived by Zukoski from an analysis of the turbulent plume above a fire, without taking the combustion into account.

The name places the group in the family of Froude numbers used in fluid mechanics. The classic Froude number is the ratio of the inertia force to the gravity force, and it is relevant to the buoyant flows associated with fires. Small-scale physical models of fire flows rely on keeping such non-dimensional ratios equal between the model and the full-scale case, which is the idea behind Froude scaling.

Characteristic fire diameter

The physical diameter of a fire is not always well defined. A burning liquid in a round pan has an obvious diameter, but a fire that involves the contents of a room does not. For this reason, fire model validation studies also use a characteristic fire diameter, written $D^{*}$, which depends only on the heat release rate and the ambient conditions:

$$D^{*}={\left(\frac{\dot{Q}}{{\rho }_{\mathrm{\infty }}{c}_p{T}_{\mathrm{\infty }}\sqrt{g}}\right)}^{2/5}$$

The symbols have the same meaning and units as in the definition of ${\dot{Q}}^{*}$. In many cases the characteristic diameter is comparable to the physical diameter of the fire, and the dimensionless heat release rate is then of the order of 1. There is a direct link between the two quantities. Dividing the definitions shows that ${\dot{Q}}^{*}$ equals the ratio $D^{*}/D$ raised to the power 5/2, so a fire with a characteristic diameter larger than its real diameter has a value above 1.

The characteristic diameter gives an everyday feel for fire size. An average living-room sofa fire releases about 2.5 MW. With the property values used above, its characteristic diameter is about 1.4 m. A second useful ratio divides the ceiling or target height by the characteristic diameter. This ratio, $H/D^{*}$, shows how important the plume is compared with other parts of the fire-driven flow, such as the ceiling jet or a doorway flow. Much of the mixing of fresh air into the smoke happens in the plume, and this dilution determines the temperature of the hot gas layer.

Flame height

The best-known use of the dimensionless heat release rate is the prediction of the mean flame height, the height at which flames are present half of the time. Heskestad gives the following correlation in the form presented by McCaffrey. It covers the whole range of the number except the momentum regime of jet flames.

Mean flame height divided by fire diameter as a function of the dimensionless heat release rate (Heskestad correlation). Based on Heskestad (2002), p. 2-3, and Joglar (2007), p. 3-3.

$$\frac{L}{D}=3.7{\dot{Q}}^{*2/5}-1.02$$

Here $L$ is the mean flame height in metres and $D$ the fire diameter in metres. Under normal atmospheric conditions of 293 K and 760 mmHg, the same relation can be written with the heat release rate in kilowatts as $L=0.235{\dot{Q}}^{2/5}-1.02D$. The coefficient 0.235 is a typical value within a range of 0.226 to 0.240. That range applies to fuels that release 2,900 to 3,200 kJ for each kilogram of air consumed. Fuels outside this range include hydrogen, with a coefficient of 0.211, and gasoline, with 0.200.

The correlation has clear limits. For very small fires it gives negative flame heights, which have no physical meaning. The NUREG-1824 documentation of the FIVE-Rev1 model states that it agrees well with data for flame heights of up to 20 fire diameters and dimensionless heat release rates of up to 100. It is not suitable for jet flames or for atmospheric conditions that differ strongly from normal. The same documentation notes that representing an irregular fuel source by a circle of equal area may overestimate the flame height.

Introduction to Fire Safety Engineering uses the flame height to choose between plume models, following the Dutch standard NEN 6093. When the mean flame height is smaller than the fire diameter, the Thomas plume model is the more reliable choice. When it is equal to or larger than the diameter, the Heskestad plume model is the better choice. Setting the flame height equal to the diameter in the correlation above gives a dimensionless heat release rate of about 0.2. In this way the number marks the boundary between fires that are wider than their flames are tall and fires with taller flames.

Use in fire model validation and grid design

Validation studies use the dimensionless heat release rate to describe the range of fires covered by the tests of a model. In the original NUREG-1824 study of 2007, the experiments covered values from 0.4 to 2.4. Supplement 1 of 2016 added many experiments, raising the total from 26 to almost 500, and widened the range to 0.2 to 9.1. Such ranges are used to judge whether the validation results apply to a new scenario. Supplement 1 also uses a flame length ratio, which expresses the size of the fire relative to the height of the compartment. A value of 1 means that the flames reach the ceiling.

Range of the dimensionless heat release rate covered by the NUREG-1824 validation experiments in 2007 and in Supplement 1 of 2016. Based on Stroup et al. (2016), p. 7-3.

The characteristic diameter is also the basis of a rule of thumb for mesh resolution in computational fluid dynamics. In the Fire Dynamics Simulator, the success of the large eddy simulation technique depends largely on the ratio of the characteristic diameter to the size of a grid cell, $D^{*}/\delta x$. The larger this ratio, the more of the fire dynamics is resolved directly and the more accurate the simulation. A ratio of 5 to 10 usually gives favourable results at a moderate computational cost. In the investigation of the World Trade Center fires, NIST chose a uniform grid of 10 cm cubes on this basis for its simulations of a series of compartment experiments. Citing earlier experience, that investigation reported that a ratio of 10 gives favourable results at a moderate cost.

Characteristic fire diameter D* and the grid cell sizes that give D*/δx = 5 and 10, for air at 293 K. Based on McGrattan (2007), p. 6-5.

The rule of thumb does not replace a grid resolution study, in which the grid is refined until the results stop changing. The importance of the ratio also depends on the geometry. In the NBS multi-room experiments, a 100 kW burner with a characteristic diameter of 0.4 m was spanned by only four grid cells of 0.10 m. Because the ceiling was low and the plume less important, the predicted temperatures in all three rooms were still within the experimental uncertainty.

Other reference lengths

The same group can be formed with a length other than the fire diameter. In the localised fire model of Hasemi, as implemented in the OZone design tool, the form based on the fire diameter fixes the vertical position of a virtual point source relative to the fire surface. For a value up to 1, this position divided by the diameter equals $2.4({\dot{Q}}^{*2/5}-{\dot{Q}}^{*2/3})$. For larger values, it equals $2.4(1-{\dot{Q}}^{*2/5})$. When the flame reaches the ceiling, the model uses the distance from the fire to the ceiling instead of the diameter. The flame then spreads horizontally under the ceiling, and its total length from the burner divided by that distance equals 2.90 times this ceiling-based number to the power 1/3.

In New Zealand, the 2013 commentary to the verification method C/VM2 uses the compartment height as the length scale. It defines the number as the maximum heat release rate in kilowatts divided by 1,110 times the compartment height to the power 5/2. A single-room two-zone model is considered satisfactory when this value is at most 0.15 and the floor area divided by the square of the height lies between 0.4 and 70.

Rooms that are very low compared with their floor area, with a ratio above 70, can be modelled as several virtual rooms. Tall, narrow spaces with a ratio below 0.4 are preferably treated as shafts, because the plume is likely to reach the walls before the ceiling. The commentary stresses that these criteria are guidance and not absolute limits. In its worked table, a 50 MW fire gives a value of 0.09 in spaces 12 m high, while spaces of 6 m and 9 m fall outside the range.

Worked example

The NUREG-1824 main report uses an oil spill in a pump room to show how the number is used in practice. In the scenario, 3.8 L of oil spreads over 5.3 m² of floor in a room 7 m high, and a cable crosses the room near the ceiling. The oil has a heat of combustion of 46,400 kJ/kg and burns at 0.036 kg/(m²·s). The report finds a heat release rate of 7,416 kW for this spill. All the following steps were recalculated for this article.

The first step converts the spill area into an equivalent diameter. A circle of 5.3 m² has a diameter of 2.6 m, as in the report. Inserting the heat release rate, this diameter and the air properties of 1.2 kg/m³, 1 kJ/(kg·K), 293 K and 9.81 m/s² into the definition gives:

$${\dot{Q}}^{*}=\frac{7416}{1.2\times 1.0\times 293\times \sqrt{9.81\times 2.6}\times 2.6^2}=0.62$$

This value lies within the range of 0.4 to 2.4 covered by the 2007 validation study, so the fire size is within the validated range. The characteristic diameter of the same fire is 2.14 m. With the cable 7 m above the fire, the ratio $H/D^{*}$ is 3.3. That is below the tested range of 3.6 to 16, so a target inside the plume at this height falls outside the validated conditions.

The flame height correlation gives a ratio $L/D$ of 2.03 and thus a mean flame height of about 5.3 m, roughly three quarters of the room height. The dimensional form with the coefficient 0.235 gives 5.7 m, about 7 % more. With the air properties used here, the factor 3.7 corresponds to a dimensional coefficient of about 0.225, slightly below the typical value. For a computer simulation of this fire, a ratio $D^{*}/\delta x$ of 5 to 10 points to grid cells between about 0.42 m and 0.21 m. The finer cells would resolve the plume, which carries the heat to the cable, in more detail.

Limitations

The dimensionless heat release rate describes the fire source with a single number, and this brings several limits. Heskestad tested flame heights at widely varying ambient temperatures and found that this number did not capture the observed increase of flame height with ambient temperature. A related parameter that also includes the heat of combustion per unit mass of air did capture it. Heskestad therefore considers that parameter more appropriate for flame heights. For ordinary use, the same chapter regards the standard coefficient of 0.235 as adequate unless the fuel properties or the atmospheric conditions differ significantly from normal.

The choice of diameter is a second source of uncertainty, because a fire that involves the contents of a room has no well-defined diameter. The characteristic diameter avoids this problem, but it depends only on the heat release rate and therefore says nothing about the shape or spread of the fire. A third limit concerns the purpose of the number. It describes the fire, not the room, and judgements about a whole compartment need further parameters, such as the ratio of room height to characteristic diameter or the room shape factor used in New Zealand.

A common misconception is that a high dimensionless heat release rate means a large fire. The number compares the heat release rate with the size of the fire base, so a wide fire with a large total output can still have a low value. Another misconception is that the grid rule $D^{*}/\delta x$ guarantees an accurate simulation. The validation reports present it only as a starting point that must be confirmed by a grid resolution study.

See also

• Heskestad plume
• Virtual origin
• Ceiling jet
• Point source model

References

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• Heskestad, G. (2002). Fire plumes, flame height, and air entrainment. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 2-1–2-17). National Fire Protection Association; Society of Fire Protection Engineers.
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• McGrattan, K. B., Bouldin, C., & Forney, G. P. (2005). Computer simulation of the fires in the World Trade Center towers (NIST NCSTAR 1-5F). National Institute of Standards and Technology.
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• Najafi, B., Joglar, F., & Dreisbach, J. (2007). Verification and validation of selected fire models for nuclear power plant applications: Vol. 1. Main report (NUREG-1824; EPRI 1011999). U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research; Electric Power Research Institute.
• Stroup, D., Lindeman, A., McGrattan, K., Peacock, R., Overholt, K., Joglar, F., LeStrange, S., & Montanez, S. (2016). Verification and validation of selected fire models for nuclear power plant applications: Supplement 1 (NUREG-1824, Supplement 1; EPRI 3002002182). U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research; Electric Power Research Institute.
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