Froude scaling

Froude scaling, also called Froude modelling, is a method for studying fire-driven flows in a physical model that is smaller than the real building. The model is designed so that the Froude number, the ratio of the inertia force to the gravity force in a flow, has the same value as at full scale. This number is relevant to the buoyant flows produced by fires, in which hot gases rise because they are lighter than the air around them. When the Froude number is preserved, temperatures in the model equal those at full scale. Lengths, velocities, times, heat release rates and air flows then change by fixed powers of the ratio between model size and full size.

A model at one eighth of full size, for example, needs a fire of less than 1 % of the real heat release rate. Engineers use the method to study smoke movement in atria and covered malls with complex shapes, where simple calculation models are of limited value. Froude scaling does not reproduce all of the physics. The Reynolds number and the groups that govern heat transfer are not kept equal, and the method does not apply close to the flames. It therefore gives useful information on smoke transport away from the fire, but not on the combustion itself.

Similarity and the Froude number

A scale model is a physical representation of a space at a reduced scale. Physical modelling rests on the hypothesis that a full-scale phenomenon can be reproduced in a small-scale experiment, provided that certain non-dimensional parameters are preserved. Geometric similarity alone is not enough. Kinematic similarity requires that the paths of the fluid particles are geometrically similar. Dynamic similarity exists between two such systems when, in addition, the ratios of all the forces involved are equal.

Each of these force ratios is expressed by a dimensionless group. The Froude number is the ratio of the inertia force to the gravity force, and it is the group that matters for the buoyant flows associated with fires. It is defined as follows:

$$\mathrm{F}\mathrm{r}=\frac{u^2}{lg}$$

Symbol	Description	Unit
$\mathrm{F}\mathrm{r}$	Froude number	none
$u$	characteristic flow velocity	m/s
$l$	characteristic length	m
$g$	acceleration due to gravity	m/s²
$\mathrm{R}\mathrm{e}$	Reynolds number	none
$\rho$	density of the fluid	kg/m³
$\mu$	dynamic viscosity of the fluid	kg/(m·s)

Other groups describe other force ratios. The Reynolds number, $\mathrm{R}\mathrm{e}=\rho ul/\mu$, is the ratio of the inertia force to the friction force. The Grashof number compares buoyancy with viscous forces, as in fire plumes. Because gravity is the same in a laboratory and in a real building, a constant Froude number means that velocities must change with the square root of the length. A model with all lengths reduced by a factor of four must therefore have all velocities reduced by a factor of two.

Scaling relations

Milke summarises the relations that follow from preserving the Froude number, based on a review by Quintiere. In the table, the subscript m refers to the model and F to the full-scale facility. The ratio $l_m/l_F$ is a model length divided by the corresponding full-scale length.

Quantity	Scaling relation	Unit of the quantity
Temperature	$T_m=T_F$	K
Geometric position	$x_m=x_F(l_m/l_F)$	m
Pressure difference	$\mathrm{\Delta }{p}_m=\mathrm{\Delta }{p}_F(l_m/l_F)$	Pa
Velocity	$v_m=v_F(l_m/l_F{)}^{1/2}$	m/s
Time	$t_m=t_F(l_m/l_F{)}^{1/2}$	s
Convective heat release rate	${\dot{Q}}_{c,m}={\dot{Q}}_{c,F}(l_m/l_F{)}^{5/2}$	kW
Volumetric flow rate of a fan	${\dot{V}}_m={\dot{V}}_F(l_m/l_F{)}^{5/2}$	m³/s

In words, temperatures stay the same, positions and pressure differences shrink in proportion to the size, and velocities and times shrink with the square root of the size. Heat release rates and fan flow rates shrink with the size to the power 5/2. The 5/2 power follows from the velocity rule. A volume flow is a velocity times an area, so it changes with the square root of the length times the length squared. With the gas temperatures unchanged, the heat carried by such a flow changes in the same way. Experiments based on Froude modelling can be carried out with air at atmospheric pressure.

Factor by which a full-scale value is multiplied in a Froude-scaled model, as a function of the length ratio. Based on Milke (2002b), p. 4-294.

The walls and ceilings of a space also exchange heat with the smoke. Some of these surface effects can be preserved by scaling the thermal properties of the construction materials. The product of thermal conductivity, density and specific heat is then reduced by the length ratio to the power 0.9. Because this scaling has only a secondary effect on the flow, practical needs such as convenient construction and flow visualisation may lead to model materials that do not follow the rule.

Connection with fire correlations

Several fire correlations are written in terms of Froude-type groups, which makes them consistent with Froude scaling. The dimensionless heat release rate is also called the fire Froude number and expresses the ratio of inertial and buoyancy-induced velocities in a fire. According to Heskestad, Zukoski first derived this group from an analysis of the turbulent fire plume above a fire. The group divides the heat release rate by the fire diameter to the power 5/2, together with properties of the ambient air. A model fire whose heat release rate follows the 5/2 rule and whose diameter follows the length rule therefore has the same value as the full-scale fire.

The same reasoning applies to flame height. Heskestad's correlation gives the mean flame height divided by the fire diameter as a function of the dimensionless heat release rate alone. A correctly scaled model fire thus produces flames with the same ratio of height to diameter, as long as the correlation applies. This holds in the buoyancy-dominated regime, not for high-momentum jet flames.

A third example is the critical velocity needed to stop smoke from flowing upstream in a corridor. The fire may be in the corridor itself, or the smoke may enter through an open door or another opening. Thomas developed the following empirical relation for this velocity:

$$V_k=K{\left(\frac{g\dot{Q}}{W\rho cT}\right)}^{1/3}$$

Symbol	Description	Unit
$V_k$	critical air velocity to prevent smoke backflow	m/s
$K$	constant of the order of 1	none
$\dot{Q}$	heat release rate into the corridor	kW
$W$	corridor width	m
$\rho$	density of the upstream air	kg/m³
$c$	specific heat of the downstream gases	kJ/(kg·K)
$T$	absolute temperature of the downstream mixture of air and smoke	K

If the heat release rate shrinks with the length to the power 5/2 and the width with the length, the term in brackets shrinks with the length to the power 3/2. Its cube root then shrinks with the square root of the length, which is exactly the Froude rule for velocities. With standard air properties and K equal to 1, the relation gives a critical velocity of 1.45 m/s for a wastebasket fire of 150 kW in a corridor 1.22 m wide. In a model at a scale of 1/4, the same fire becomes 4.7 kW in a corridor 0.31 m wide, and the relation gives 0.73 m/s. That is half the full-scale value, as the Froude rule requires for a length ratio of 1/4.

Critical velocity to stop smoke backflow in a corridor at full scale and in a 1/4 scale model, from Thomas's relation. Based on Klote (2002), pp. 4-279–4-280.

Applications

Smoke management in atria is the classic application. Milke lists four groups of design tools for atria: zone models based on algebraic equations, computer-based zone models, field models and scale models. Scale models are especially useful for atria with irregular shapes or numerous projections, and Milke and Klote have reviewed their use as a design aid for atrium smoke control. In an atrium whose horizontal cross-section changes with height, algebraic and computer-based zone models are of limited value, and a scale model can show the smoke layer position directly.

The International Fire Engineering Guidelines of 2005 list experiments with physical scale models as one of the methods of analysis in a fire engineering design. The other methods include hand calculations, statistical studies, full-scale fire testing and computer simulation of fire development, smoke spread and people movement. Whatever method is chosen should be well documented and validated, suitable for the task, and able to produce outputs that can be compared with the acceptance criteria. Validation studies use dimensionless groups in the same spirit to describe tests of very different sizes. Supplement 1 to the NUREG-1824 validation study, for example, summarises its experiments with normalised parameters such as the fire Froude number, which ranged from 0.2 to 9.1.

Scale models also have a place in the wider field of fire modelling. The NUREG-1805 report on fire dynamics tools quotes the ASTM E176 definition of a fire model as a physical or mathematical representation of burning or of other processes associated with fire. Physical models reproduce fire phenomena in a simplified physical situation. The same report notes that scale models are a very widespread form of modelling, because full-scale experiments are expensive, difficult and sometimes wholly infeasible.

Worked example

Milke describes a scale model to find the equilibrium smoke layer position in an atrium with a cross-section that varies with height. The full-scale atrium is 30.5 m high, the design fire is a steady fire of 5 MW, and the proposed exhaust fan capacity is 142 m³/s. The resulting model is 3.8 m tall at a scale of 1/8, with a fire of 28 kW and a fan capacity of 0.78 m³/s. All numbers below were recalculated for this article with the scaling relations.

A 30.5 m atrium and its 1/8 scale model. Heat release rate and fan flow shrink with the length ratio to the power 5/2, time with its square root; temperatures stay the same. Based on Milke (2002b), pp. 4-294–4-295.

The length ratio is 1/8, so the model height is 30.5 m divided by 8, or 3.81 m. This matches the model height of 3.8 m given by Milke. The heat release rate scales with the length ratio to the power 5/2:

$${\dot{Q}}_{c,m}=5000\times {\left(\frac{1}{8}\right)}^{5/2}=5000\times 0.00552=27.6\mathrm{k}\mathrm{W}$$

The fan flow scales in the same way, from 142 m³/s to 0.784 m³/s. Velocities and times scale with the square root of 1/8, which is 0.354. A smoke velocity of 1 m/s in the real atrium corresponds to 0.35 m/s in the model, and one hour of full-scale time corresponds to about 21 minutes in the model. A pressure difference of 10 Pa becomes 1.25 Pa, while the smoke temperatures stay the same. The model therefore runs faster and with smaller pressure differences, but with the same temperatures as the real building.

The model must also be large enough for its flow to stay turbulent. Milke states that a critical dimension of at least 0.3 m in the areas of interest is often sufficient. At a scale of 1/8, this corresponds to 2.4 m at full scale. A balcony with a floor-to-ceiling height below 2.4 m would therefore fall below this guidance in the model, and a larger model scale would be needed if that balcony were important.

Limitations

Froude modelling does not preserve the Reynolds number, the ratio of inertia to friction forces. In a small model the Reynolds number is lower, so friction plays a relatively larger role than in the real building. This effect is limited by choosing a model large enough for the flow to be fully developed and turbulent in the areas of interest. In shopping centres and atria, the critical dimension is usually the floor-to-ceiling height of one of the balconies.

The method also does not preserve the dimensionless groups that describe heat transfer. Generally this has little effect, because the temperatures are the same in the model and at full scale. Froude modelling does not apply to locations with high temperatures and low Reynolds numbers, such as the region near the flame. It does, however, provide useful information about smoke transport away from the fire.

Combustion chemistry does not follow simple scaling rules. In a NIST study, fires of 0.5 to 3.4 MW were burned in a standard ISO 9705 room and compared with tests in a geometrically similar reduced-scale enclosure. The burner was scaled to give the same gas exit velocities as in the reduced-scale tests. The predicted mass flows agreed well, but carbon monoxide concentrations in the full-scale room were up to two times higher, together with upper layer temperatures of 1,400 to 1,500 K. The researchers concluded that the generation of combustion products is not entirely controlled by the ventilation of the compartment.

A common misconception is that a correctly scaled model reproduces every aspect of a real fire. The scaling relations apply to the convective heat release rate and to the transport of smoke, not to flame chemistry, toxic gas yields or the response of hot surfaces close to the fire. Results from a scale model therefore need the same clear statement of limitations and assumptions that the 2005 guidelines expect for every method of analysis.

Froude numbers in other fire calculations

Froude numbers also appear inside fire calculation methods that have nothing to do with physical models. Milke gives a correlation for the minimum average air velocity needed to stop smoke from a communicating space from entering an atrium through an opening. That velocity equals 0.64 times the square root of the product of gravity, a height and the relative temperature difference between smoke and ambient air. The square root of gravity times a height is the velocity scale of the Froude number, so this correlation also follows the Froude rule. Emmons notes that accurately measured buoyant flow coefficients for vents would depend on the Reynolds number, a Froude number and a depth parameter. In practice, the best available option is a flow coefficient of 0.68, with errors of about 10 % in the calculated flows.

The CFAST zone model uses a Froude number to describe plug-holing. When the hot layer under a ceiling vent is not deep enough, air from the lower layer is also drawn through the vent, which reduces the amount of smoke removed. Following Cooper, the degree of plug-holing depends on a Froude number formed from the vent flow, the depth of the hot layer and its temperature rise. A correlation fitted to calculations with the Fire Dynamics Simulator then gives the fraction of the vent flow that is taken from the hot upper layer. In each of these cases the Froude number compares the inertia of a flow with the effect of gravity on hot, light gases, in line with its general definition.

See also

• Verification and validation
• Computational fluid dynamics
• Hand calculations
• Large eddy simulation

References

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• Milke, J. A. (2002b). Smoke management in covered malls and atria. 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. 4-292–4-310). National Fire Protection Association; Society of Fire Protection Engineers.
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