Critical water application rate

Critical water application rate is the lowest rate at which water must be applied to each square metre of a burning surface for the flames on that surface to go out. Below this rate a fire is not extinguished, however long the water is applied. The rate is usually expressed in grams of water per square metre of fuel surface per second, written g/(m²·s). For common plastics burning in normal air without outside heating, calculated values lie between about 2 and 5 g/(m²·s).

Real fires need much more water than this minimum. Strong flames heat the fuel surface, and every extra unit of heat must be removed by extra water. For burning solids arranged in three dimensions, such as stacked boxes, Tewarson estimates rates of 19 to 34 g/(m²·s), about ten times the critical values. Sprinkler systems, water mist systems and fire service attacks are accordingly specified by a minimum flow of water, not only by a total quantity.

Definition and units

In the SFPE Handbook of Fire Protection Engineering, Tewarson defines the critical water application rate for one specific case. The burning material receives no heat from outside its own flame, and the applied water evaporates on the surface without forming a puddle. Under these conditions the rate depends only on properties of the material, which is why Tewarson relates it to the fundamental fire properties of that material. Larger or more complex fires need higher rates, because their flames send more heat to the fuel surface.

Foam engineers use the term in a wider sense. Scheffey describes the critical application rate as the rate below which fires will not be extinguished, independent of the duration of application. He separates it from the application density, which is the total amount of agent per unit area needed to control or extinguish a fire. A rate describes how quickly the agent arrives, while a density describes how much arrives in total. A large quantity of agent delivered below the critical rate therefore never puts the fire out.

Tewarson expresses water application as a mass flux in g/(m²·s). Foam and sprinkler sources use a volume flux, in litres per minute per square metre or in US gallons per minute per square foot (gpm/ft²). One gram per square metre per second equals 0.06 L/(m²·min), which is also a water depth of 0.06 mm per minute. Tewarson quotes 407 g/(m²·s) as equivalent to 0.6 gpm/ft², which agrees with this conversion.

How water puts out a burning surface

Tewarson explains extinction by water with an energy balance at the fuel surface. The flame and any external source heat the surface, while the hot surface loses heat by re-radiation. The remaining heat turns solid fuel into vapour, and each gram of vapour needs a fixed amount of energy called the heat of gasification. The flame goes out when the release of vapour falls to a critical mass loss rate. This rate is similar to the lowest burning rate that can establish a sustained flame at ignition.

Water applied to the surface removes heat by vaporisation. The heat flux it removes equals the water application rate times the heat of gasification of water, 2.58 kJ/g, times a water application efficiency. Extinction occurs once this extra loss pushes the release of fuel vapour down to the critical rate. Small-scale tests on wetted filter paper confirmed the balance, because the heat removed from the surface matched the heat needed to gasify the applied water.

On a horizontal surface, part of the water may collect as a puddle instead of evaporating at once. The puddle then also blocks heat from the flame and hinders the escape of fuel vapour. On vertical slabs of polymethylmethacrylate (PMMA) and polyoxymethylene, each extra gram of water offset 2.7 and 2.3 kJ of external heat, close to the 2.6 kJ/g heat of gasification of water. Puddles therefore played no part on vertical surfaces. On horizontal slabs each extra gram offset 4.1 to 4.6 kJ, almost twice as much, so blocking mattered as much as evaporation.

Energy balance at a burning surface under water spray. Left: evaporating water removes heat. Right: on a horizontal surface a puddle also blocks flame heat and fuel vapour. Based on Tewarson (2002), pp. 3-147 to 3-148.

Water can also act in the flame itself, which brings in further extinguishing mechanisms. Mawhinney and Back list gas-phase cooling, oxygen depletion with dilution of flammable vapour, and wetting and cooling of the fuel surface as the main mechanisms of water mist. Radiation attenuation and kinetic effects play a secondary role. A diffusion flame goes out when it is cooled below a limiting adiabatic flame temperature of about 1600 K (1326 °C). The critical water application rate as defined by Tewarson covers only the cooling of the fuel surface.

Calculating the critical rate

Tewarson combines the surface energy balance with the heat removed by water into one equation for the water application rate at flame extinction. Without external heating and without a puddle, the equation reduces to the critical water application rate. The numerator is the surplus heat flux that the flame still delivers to the surface when burning has dropped to the critical rate. It equals the convective heat from the flame, minus the heat used to gasify the fuel, minus the re-radiation loss. The denominator is the heat that each gram of applied water removes.

$${\dot{m}}^{\prime }{\text{'}}_{w,\mathrm{c}\mathrm{r}}=\frac{{\dot{m}}^{\prime }{\text{'}}_{\mathrm{c}\mathrm{r}}(\varphi \mathrm{\Delta }{H}_T-\mathrm{\Delta }{H}_g)-{\dot{q}}^{\prime }{\text{'}}_{rr}}{{\epsilon }_w\mathrm{\Delta }{H}_w}$$

Symbol	Description	Unit
${\dot{m}}^{\prime }{\text{'}}_{w,\mathrm{c}\mathrm{r}}$	critical water application rate	g/(m²·s)
${\dot{m}}^{\prime }{\text{'}}_{\mathrm{c}\mathrm{r}}$	critical mass loss rate of the fuel at flame extinction	g/(m²·s)
$\varphi$	kinetic parameter: convective energy from the flame per gram of fuel gasified, divided by $\mathrm{\Delta }{H}_T$	none
$\mathrm{\Delta }{H}_T$	net heat of complete combustion of the fuel	kJ/g
$\mathrm{\Delta }{H}_g$	heat of gasification of the fuel	kJ/g
${\dot{q}}^{\prime }{\text{'}}_{rr}$	surface re-radiation loss	kW/m²
${\epsilon }_w$	water application efficiency (taken as 1 in the calculated values)	none
$\mathrm{\Delta }{H}_w$	heat of gasification of water, 2.58 kJ/g	kJ/g

The equation applies to a burning solid in normal air, without external heat flux and without a puddle. The kinetic parameter compares the convective energy that the flame returns to the fuel with the heat of complete combustion. All inputs can be measured in small-scale apparatus, such as the oxygen bomb calorimeter, the ASTM E2058 fire propagation apparatus and the cone calorimeter. Materials with highly reactive vapours, such as polyethylene, need large amounts of agent to reduce their heat release rate to the critical value.

Material	Critical water application rate (g/(m²·s))
Polyoxymethylene	2.3
Polymethylmethacrylate (PMMA)	2.5
Polypropylene	3.0
Polyethylene	3.8
Polystyrene	5.1
Polyethylene with 25 % chlorine	2.1
Polyethylene with 36 % or 48 % chlorine	0

The table gives calculated values from the SFPE Handbook, with a water application efficiency of one. Polystyrene needs the most water of the listed plastics, while polyoxymethylene, PMMA and polyethylene with 25 percent chlorine need the least of the materials that burn in normal air. Polyethylene with 36 or 48 percent chlorine does not burn in normal air without external heating, so its critical rate is zero.

Worked example

For polystyrene, the SFPE Handbook gives a critical mass loss rate of 4.0 g/(m²·s), a kinetic parameter of 0.21 and a heat of complete combustion of 39.2 kJ/g. The heat of gasification is 1.7 kJ/g and the re-radiation loss is 13 kW/m². The flame then returns 0.21 × 39.2 × 4.0 = 32.9 kW/m² to the surface by convection. Gasifying the fuel uses 1.7 × 4.0 = 6.8 kW/m², and re-radiation removes 13 kW/m², which leaves a surplus of 13.1 kW/m². With an efficiency of one and 2.58 kJ/g for water, the critical rate is 13.1 / 2.58 = 5.1 g/(m²·s), the tabulated value. This is only about 0.31 L/(m²·min), a film of water about 0.3 mm deep every minute.

Effect of fire size, orientation and fuel arrangement

The general equation for extinction contains an additional term for heat that reaches the surface from outside the flame. Each extra kilowatt per square metre must be removed by extra water, so the required rate rises as fires become larger and more intense. As the burning area grows, the flame heat flux to the surface rises towards an asymptotic limit, which is reached in large fires. At this limit, Tewarson replaces the external heat flux by the flame heat flux to the surface. This flame term then dominates, and the critical rate becomes a small part of the total.

Material	Flame heat flux to the surface (kW/m²)	Water application rate at the asymptotic limit (g/(m²·s))
Polyoxymethylene	50	22
Polymethylmethacrylate (PMMA)	60	26
Polyethylene	61	27
Polypropylene	67	29
Polystyrene	75	34
Polyvinyl chloride	50	19
n-Heptane (pool fire)	37	14

For polystyrene, a flame heat flux of 75 kW/m² divided by 2.58 kJ/g adds about 29 g/(m²·s) to the critical 5.1 g/(m²·s). The result is the tabulated 34 g/(m²·s), almost seven times the critical rate. For liquid pool fires the values are less reliable, because water does not stay on the liquid surface. Tewarson therefore expects actual rates for pool fires to be higher than calculated.

Critical water application rate and calculated rate at the asymptotic limit of a large fire for five plastics, with a water application efficiency of one. Based on Tewarson (2002), pp. 3-148 to 3-149.

Small-scale tests measured the water rate needed for extinction of plastic slabs exposed to external heat fluxes of 0 to 17 kW/m². For a vertical slab of PMMA, the results fitted the line ${\dot{m}}^{\prime }{\text{'}}_w=0.37{\dot{q}}^{\prime }{\text{'}}_e+1.67$, with the water rate in g/(m²·s) and the external heat flux in kW/m². At an external heat flux of 10 kW/m² this gives 5.4 g/(m²·s), more than three times the value without external heating. Under a constant 5.2 g/(m²·s), the time to extinction of a vertical PMMA slab grew with the external heat flux. Close to its critical heat flux of 11 kW/m², the flame was no longer extinguished at all.

Water application rate for flame extinction of plastic slabs against external heat flux, from fitted test correlations. Based on Tewarson (2002), pp. 3-152 to 3-153.

Stored goods are usually heterogeneous and three-dimensional, and sprinklers wet them from the ceiling or from just above the stack. In large-scale tests at Factory Mutual, water was spread evenly over the top of stacks 2.3 m by 2.3 m in plan and 2.9 m high. The rates ranged from 0 to 407 g/(m²·s), about ten times the range predicted for three-dimensional arrangements. Tewarson concludes that blocking of flame heat and fuel vapour matters as much as evaporation in such stacks. Goods containing more polystyrene, such as Class IV commodities and Group A plastics, need higher rates because their flames send more heat to the surface.

Critical rate, optimum rate and application density

Fire control time depends strongly on the application rate. In aviation fire tests with foam on jet fuel fires, control times increased dramatically as the rate fell towards a critical value. No fire control or extinguishment was achieved at 0.03 gpm/ft² with aqueous film-forming foam on fires of 743 m². At the other end, control times hardly improved above about 4.1 L/(m²·min) for aqueous film-forming foam and 8.2 L/(m²·min) for protein foam. Between these limits, control time falls as the rate rises.

Schematic relation between application rate and fire control time, as found in aviation foam tests: no control below the critical rate, little gain above about 4.1 L/(m²·min) for AFFF. Based on Scheffey (2002), pp. 4-100 to 4-101.

These data concern foam solution on burning fuel, not plain water, but they show why a rate and a total quantity are separate requirements. Standard fire tests use lower rates than practice, partly because differences between agents show up more clearly near the critical rates. Rates used in practice also include a safety factor. Water applied early, while a fire is still small, faces a lower heat flux than water applied to a large fire. An active suppression system generally intervenes during the growth stage, which prevents the fire from developing fully.

Use in sprinkler design, water mist and firefighting

Fleming notes that only very rudimentary methods exist to calculate how well a water spray suppresses a fire, so the suppression side of sprinkler design is empirical. In the United States, sprinkler design follows NFPA 13, a standard of the National Fire Protection Association. It links each occupancy hazard class to a minimum water spray density and a minimum design area, the inputs for the hydraulic design. For light hazard occupancies such as offices, the 2002 edition requires a sprinkler design density of 0.1 gpm/ft² over 1,500 ft², about 4.1 L/(m²·min) over 139 m². Converted to a mass flux, 0.1 gpm/ft² is about 68 g/(m²·s).

That value is roughly twice the 34 g/(m²·s) calculated for polystyrene at its asymptotic limit, but the two numbers measure different things. The design density describes the water discharged per unit of area covered by the sprinklers, while the critical rate refers to water that reaches and evaporates on a burning surface. In full-scale compartment fires, the efficiency of water application is very much less than 100 percent. Conventional sprinkler systems aim to control a fire rather than extinguish it, leaving full extinguishment to the fire service where needed. Conventional sprinklers achieve control partly by pre-wetting the surrounding fuel, which limits fire spread.

In a water mist system, fine droplets add gas-phase cooling and the dilution of oxygen and fuel vapour to the cooling of the fuel surface. Wighus defined the spray heat absorption ratio as the rate at which evaporating water absorbs heat, divided by the rate at which the fire releases heat. For an unconfined propane flame this ratio could be as low as 0.3 under optimum conditions, and about 0.6 in realistic machinery spaces. Andersson et al. extinguished propane flames with water applied at 1.2 to 2.2 times the mass burning rate of the propane. Both results describe ideal mixing of mist and flame, so real systems need higher flows. Measured critical concentrations of mist around a diffusion flame range from 100 to 200 g/m³.

Manual firefighting raises the same question of rate. Van Rossum describes how the fire service uses the heat release rate of a burning sofa to judge whether a fire can still be fought with a given flow of water. Low-, medium- and high-pressure hose systems each deliver a different flow, and the crew matches a cooling capacity to the expected fire size. An average living-room sofa releases about 2.5 MW, and a burning car peaks at about 8.5 MW. In the International Fire Engineering Guidelines, Chart 15 of the fire brigade intervention model calculates the amount and rate of water needed for fires of varying heat release rate. With too little water or too few firefighters, the strategy may change from fighting the fire to stopping its spread to adjoining buildings.

Modelling suppression

The fire models CFAST (Consolidated Fire and Smoke Transport) and FDS (Fire Dynamics Simulator) describe suppression with empirical rules rather than with the critical rate. In CFAST, the heat release rate falls by the same fraction in every equal time step after the sprinkler activates. The time constant of this decay becomes shorter as the water spray density increases.

$$\dot{Q}(t)=\dot{Q}(t_{\mathrm{a}\mathrm{c}\mathrm{t}}){\mathrm{e}}^{-(t-t_{\mathrm{a}\mathrm{c}\mathrm{t}})/\tau },\tau =3u_w^{-1.8}$$

Symbol	Description	Unit
$\dot{Q}(t)$	heat release rate at time $t$	kW
$t_{\mathrm{a}\mathrm{c}\mathrm{t}}$	time of sprinkler activation	s
$\tau$	time constant of the decay	s
$u_w$	water spray density	mm/s

Fleming describes the model of the National Institute of Standards and Technology (NIST) behind this equation, which was derived from the suppression of wood crib fires by pendent spray sprinklers. In the SFPE Handbook the exponent is 1.85 instead of 1.8. The model is stated to apply only where the fuel is not shielded from the spray and the density is at least 0.07 mm/s, which is 0.1 gpm/ft². Because it ignores further fire growth after activation, Fleming restricts it to light hazard applications.

For a light hazard room fire of 500 kW at activation, the minimum density reduces the fire to about 465 kW after 30 s and 432 kW after 60 s. A density of 0.205 mm/s, or 0.3 gpm/ft², reduces the same fire to about 293 kW and 172 kW. The CFAST documentation adds that the model assumes enough water is applied, so it cannot handle a fire that overwhelms the sprinkler.

Heat release rate after sprinkler activation for a 500 kW light hazard fire in the NIST suppression model, for two spray densities. Based on Fleming (2002), p. 4-87.

The Fire Dynamics Simulator reduces the local burning rate exponentially with the mass of water per unit area on the fuel surface. The rate of decay contains an empirical constant that depends on the fuel and its geometry. This approach follows work at Factory Mutual by Yu et al., who described suppression of rack storage fires with a few global parameters. Such rules are needed because most real fuels have complex shapes at scales that the computational grid cannot resolve.

Limits and common misconceptions

Calculated critical rates rest on simplifying assumptions. The values in the SFPE Handbook take the water application efficiency as one and assume no puddle and no external heating. Their inputs come from small samples in laboratory apparatus, while real fires involve mixed materials in complex arrangements. The critical rate also describes surface cooling only, so it does not cover extinction in the gas phase by fine mist.

A common misconception is that any amount of water will put out a fire if it is applied long enough. Below the critical rate this does not happen, whatever the duration of application. A second misconception is that a sprinkler design density equals the critical rate of the stored goods. Design densities are set per hazard class to control a fire over a design area, and only part of the discharged water reaches burning surfaces. A third misconception is that more water always gives faster control. Above a certain rate, extra agent hardly shortens the control time.

See also

• Hydraulic calculation
• Sprinkler hazard classification
• ESFR sprinkler
• Fire water supply
• Firefighting foam

References

• Fleming, R. P. (2002). Automatic sprinkler system calculations. 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. 4-72–4-87). National Fire Protection Association; Society of Fire Protection Engineers.
• Hopkins, M., Schoenrock, J., & Budnick, E. (2005). Fire suppression systems (NIST NCSTAR 1-4B). National Institute of Standards and Technology.
• Mawhinney, J. R., & Back, G. G., III. (2002). Water mist fire suppression systems. 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. 4-311–4-337). National Fire Protection Association; Society of Fire Protection Engineers.
• McGrattan, K., Hostikka, S., Floyd, J., McDermott, R., Vanella, M., Mueller, E., & Paul, C. (2026). Fire Dynamics Simulator technical reference guide: Vol. 1. Mathematical model (6th ed., NIST Special Publication 1018-1). National Institute of Standards and Technology. https://doi.org/10.6028/NIST.SP.1018
• National Research Council of Canada, International Code Council, Department of Building and Housing, & Australian Building Codes Board. (2005). International fire engineering guidelines (2005 ed.). Australian Building Codes Board.
• Peacock, R. D., McGrattan, K. B., Forney, G. P., & Reneke, P. A. (2026). CFAST – Consolidated Fire and Smoke Transport: Vol. 1. Technical reference guide (Version 7, NIST Technical Note 1889v1). National Institute of Standards and Technology. https://doi.org/10.6028/NIST.TN.1889v1
• Scheffey, J. L. (2002). Foam agents and AFFF system design considerations. 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. 4-88–4-122). National Fire Protection Association; Society of Fire Protection Engineers.
• Tewarson, A. (2002). Generation of heat and chemical compounds in fires. 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. 3-82–3-161). National Fire Protection Association; Society of Fire Protection Engineers.
• Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.