Skip to content
Upcoming courses

T-squared fire

Tools
From Fire Safety Academy Wiki
Revision as of 18:12, 2 November 2026 by Lisa (talk | contribs) (Add Intermediate SVG figures with Fire Safety Academy watermark at planned locations; preserve article text)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Contents
  1. 1Definition and growth classes
  2. 2Calculating the heat release rate
  3. 3Physical basis
  4. 4Choosing a growth rate
  5. 5Use in design
  6. 6Limits and common misconceptions
  7. 7See also
  8. 8Notes
  9. 9References

A t-squared fire is an idealised growing fire whose heat release rate increases with the square of the time since ignition.[1][2] It is the most widely used way to describe the growth phase of a design fire.[3] One constant, the fire growth coefficient α, sets how fast the fire grows. The same information is often given as a growth time: the time the fire needs to reach a reference heat release rate of about 1 MW.[4][1]

Four standard classes are used: slow, medium, fast and ultrafast, with growth times of 600, 300, 150 and 75 seconds.[1][5] The concept was developed in the 1970s to predict when fire detectors respond, and it became popular after it was included in the American fire alarm code NFPA 72.[3] Today it is used for detector and sprinkler design, smoke control, escape analysis and as input for fire models.[6][7] The t-squared curve is a simplification. It fits the growth of many real fires reasonably well, but it ignores the slow early phase after ignition. It is also unsuitable for fires that spread upwards as well, such as in high storage racks.[8][2]

Definition and growth classes

In a t-squared fire the heat release rate is proportional to time squared. The heat release rate is expressed in kilowatts, the time in seconds and the growth coefficient α in kW/s2.[1][4] The growth coefficient is also called the fire intensity coefficient or the proportionality constant.[1][4] Fires are grouped into classes by the speed of growth, defined by the time needed to reach the reference heat release rate.[1]

Line chart of four parabolic heat release rate curves, ultrafast to slow, crossing a horizontal line at 1,055 kW at 75, 150, 300 and 600 s.
The four standard t-squared growth curves; each class is defined by the time to reach 1,055 kW. Based on Iqbal and Salley (2004), p. B-3.
Growth class Growth time tg (s)[1] Growth coefficient α (kW/s2)[1] Growth time range in fire statistics (s)[9]
Very slow 1,600 0.000412 more than 1,600
Slow 600 0.00293 400 to 1,600
Medium 300 0.01172 200 to 400
Fast 150 0.0469 100 to 200
Ultrafast 75 0.1876 less than 100

The reference heat release rate differs slightly between sources. The International Fire Engineering Guidelines, the Australian Data Sheets and the SFPE Handbook define the growth time as the time to reach 1,055 kW, which equals 1,000 Btu/s.[8][9][10] The Fire Dynamics Tools of the US Nuclear Regulatory Commission and Introduction to Fire Safety Engineering use 1,000 kW, or 1 MW.[1][5] The tabulated coefficients follow from 1,055 kW. For example, 0.0469 × 1502 gives 1,055 kW.

The book writes the relation directly in megawatts as the square of the time divided by the growth time. That form gives values about 5% lower than the tabulated coefficients.[2] For a fast fire after 300 s, for example, it gives 4.0 MW, whereas 0.0469 × 3002 gives 4.2 MW.[2][1] The difference is small compared with the uncertainty in choosing a growth class.

Some documents add classes or alternative forms. The Australian Data Sheets add a very slow class with a growth time of 1,600 s, used in a statistical study of fire incidents.[9] Some Dutch models describe growth by the time needed to reach a burning area of 2 m2 instead of 1 MW, for example NEN 6093 on smoke and heat exhaust systems.[5] For storage higher than 5.0 m, C/VM2 (second edition, 2025) uses a growth with the cube of time, multiplied by the storage height.[7]

Calculating the heat release rate

The heat release rate at any time during growth follows from the growth coefficient and the time since ignition. In words, doubling the time multiplies the heat release rate by four.[1] The relation used in the American, Australian and New Zealand sources is:[1][4]

Q˙=αt2

Symbol Description Unit
Q˙ heat release rate of the fire kW
α fire growth coefficient, equal to 1,055 kW divided by the square of the growth time kW/s2
t time from the effective ignition time s

The relation applies only to the growth phase. Growth stops when flashover occurs, when no more fuel is available, when the oxygen runs out or when the fire is suppressed.[11] In design, the curve is therefore capped at a peak heat release rate or at the ventilation limit. The time is counted from an effective ignition time after any slow early phase.[8][7]

As a worked example, consider the fast fire that C/VM2 (second edition, 2025) prescribes for most buildings, with α = 0.0469 kW/s2.[7] After 100 s the fire releases 0.0469 × 1002 = 469 kW. It reaches 1 MW after about 146 s and 5 MW after about 327 s. Without flashover or ventilation limits it would reach the C/VM2 peak of 20 MW after about 653 s, or 10.9 minutes.[7] A slow fire with α = 0.00293 kW/s2 needs about 584 s to reach 1 MW, four times as long as the fast fire.[1]

Real fires rarely start growing at the moment of ignition. Alpert describes a burning foam sofa whose heat release rate followed 0.1736 times the square of the time after 80 s.[12] This sofa reaches 1 MW about 156 s after ignition, after an incubation period of 80 s. From the end of that period, it grows slightly more slowly than an ultrafast fire.[12][1]

Line chart: a sofa fire curve stays at zero for 80 s, then rises steeply between the standard fast and ultrafast curves.
Heat release rate of a burning foam sofa fitted by 0.1736 (t − 80)², compared with the standard fast and ultrafast t-squared fires. Based on Alpert (2002), p. 2-23 and Iqbal and Salley (2004), p. B-3.

Physical basis

Fire growth is driven mainly by flames spreading across a fuel item or from one item to the next.[11] Once the fire is established, its growth rate depends on the ignition process, on flame spread, which defines the perimeter of the fire, and on the mass burning flux over the area involved.[11] The 2013 commentary on C/VM2 adds that the growth phase begins when radiation from the flame governs the burning rate. In a vented space the growth rate is then governed mainly by the fuel properties and orientation, rather than by the room.[3]

A simple picture explains the square of time. Suppose a fire burns with a constant heat release rate per unit area and spreads outwards as a circle at a steady flame speed. The radius then grows in proportion to time, the burning area with the square of the radius, and the heat release rate with the square of time.[6][13] Fuels without such a regular arrangement may or may not follow a t-squared curve. For design purposes the approximation is usually considered close enough.[6]

Left: plan view of a fire spreading as circles of radius 1, 2 and 3 units. Right: matching bars of burning area and heat release rate of 1, 4 and 9 units.
A fire that spreads as a circle at a steady flame speed and burns at a constant rate per unit area: the radius grows with time, the area and heat release rate with time squared. Based on Iqbal and Salley (2004), p. B-4.

Tests support the shape. According to the Fire Dynamics Tools, the heat release rate of many growing fires can be described by simple polynomial or exponential functions of time. After an incubation period, it grows continuously with the square of time.[14] Schifiliti and co-workers state that this power-law model with an exponent of 2 can represent a wide range of fuels.[10] The International Fire Engineering Guidelines likewise state that many natural fires follow this law in the initial growth phase.[8]

The slow phase before steady growth is called the incubation, induction or establishment phase. Its length cannot be determined reliably, and most design fires ignore it, which is usually a conservative assumption.[8] Experiments often start fires with an ignition burner of 10 to 40 kW, which effectively removes this phase. The 2013 commentary therefore expects C/VM2 to overpredict the early growth slightly.[13]

Choosing a growth rate

The International Fire Engineering Guidelines rank the ways to determine fire growth in order of preference. Carefully designed full-scale experiments come first, followed by furniture calorimeter data, statistical data from fire incidents and finally t-squared fires.[15] They note an absence of good data on growth rates, particularly outside residential buildings. Where the actual fuel items are known, test data can be used to select a suitable t-squared curve.[15][16]

National documents assign classes to building uses. Introduction to Fire Safety Engineering lists the classes of the Dutch national annex to EN 1991-1-2. Dwellings, classrooms, hotels and sleeping rooms in health care grow at a medium rate, and theatres, cinemas, libraries and retail at a fast rate. Museums with paintings grow slowly, and chemical plants and storage of alcohol or upholstered furniture ultrafast.[2] The book adds that these values are meant to give a safe estimate, since many real fires grow more slowly.[2]

In New Zealand, C/VM2 (second edition, 2025) uses a fast fire with α = 0.0469 kW/s2 for most buildings and a medium fire with 0.0117 kW/s2 for car parks without stacking. Storage of 3.0 to 5.0 m high grows ultrafast, with 0.188 kW/s2.[7] The 2013 commentary explains that these values came from a sensitivity study on case study buildings. Car parks were treated as an exception because the fuel can be bounded and its layout is controlled.[13]

Fire statistics give a different view. An Australian analysis of fire incidents by Holborn and co-workers derived the share of each growth class per occupancy. For offices, 35% of fires grew very slowly, 53% slowly, 9% at a medium rate and 3% fast. For retail, the shares were 10%, 47%, 25%, 13% and 5%, including ultrafast fires.[17] The growth rates in the Engineers Australia practice note were equal to or higher than those found in this study for at least 95% of the fires.[18]

Stacked bars for residential, office, retail and factory fires: mostly very slow and slow growth, with more medium to ultrafast fires in retail and factories.
Share of fire incidents per t-squared growth class for four occupancies, from the analysis by Holborn and co-workers. Based on Australian Building Codes Board (2020a), Data Sheet B2, pp. 4–5.

Tests on fuel arrays point the other way. Free-burn tests on standard commodities and upholstered furniture showed that many common fuel arrays grow faster than the medium curve.[19] The Australian Data Sheets therefore recommend checking a range of growth rates, so that a design is not overly sensitive to the choice of one class.[20]

Use in design

The t-squared curve describes the growth stage of the fire development stages. In the 2013 commentary on C/VM2, this stage follows the incipient phase, which is considered too unpredictable to include in design.[3] In C/VM2 (second edition, 2025) the fire grows along the t-squared curve until flashover, until it reaches the prescribed peak heat release rate or until it becomes ventilation limited. In a sprinklered building it grows until sprinklers activate, after which the heat release rate is held constant.[7]

Detection was the first application. NFPA 72 uses t-squared fires of different growth rates to space heat detectors.[21] Correlations by Heskestad and Delichatsios give the gas temperature and velocity in the ceiling jet of a t-squared fire. From these, the detector activation time of a heat detector can be calculated for a given growth coefficient and ceiling height.[10][22] Detector response depends on the fire size and growth rate, the ceiling height and the characteristics of the detector.[23]

The same curves are used for smoke control and escape analysis. The Fire Dynamics Tools note their adoption in NFPA 92B for smoke management in malls, atria and large spaces.[6] In fire models such as zone and CFD models, a heat release rate curve is prescribed as input, and the t-squared curve is a common choice.[2] Growing fires are often described as one t-squared fire or as a series of them, to cover the range of credible design fires in an enclosure.[4]

The shape is also used in fire experiments. In the American validation study of fire models for nuclear power plants, burners in several test series followed a t-squared growth. In one series the fires grew over 4 minutes to peaks of 500 to 2,000 kW.[24] In NIST experiments on residential sprinklers, analysed by Vettori, a methane burner mimicked slow, medium and fast growth.[25] In tests in a manufactured home, the heat release rate of burning couches and mattresses could not be measured reliably. A t-squared ramp with a time offset was therefore fitted to the measured ceiling temperatures.[26]

Limits and common misconceptions

The t-squared curve assumes that the fire spreads horizontally in all directions. Introduction to Fire Safety Engineering therefore considers it unsuitable for fires that also spread vertically, such as in high racks.[2] For such storage, C/VM2 (second edition, 2025) uses a cubic growth with time, multiplied by the storage height, based on research by Ingason.[7][13]

A common misconception is that a t-squared fire keeps growing until it becomes very large. The 2013 commentary notes that the curve has been applied well beyond its original intent, in some cases to fires of 30 MW, and that this has been questioned. It led to separate rules for post-flashover and under-ventilated fires in C/VM2.[13] In a room, growth is limited by flashover, fuel or oxygen, and these limits must be applied separately.[11]

A second misconception is that the growth class is a fixed property of a building use. Growth depends on the ignition source, the orientation of the item, the nearness of other items, walls and openings, and partly on chance.[3] The commentary stresses that the reasonable worst case fire, not the most likely fire, must be used for design.[3] Statistical data show that most fires grow more slowly than the design classes, which is intended.[17][2]

Finally, the growth curve says nothing about how long a fire lasts. The duration depends on the amount of fuel and on ventilation, which are described by other parameters of the design fire.[3][27] When fire resistance is determined with the time equivalence formula, the growth phase makes little difference.[3]

See also

Notes

  1. ↑ 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 Iqbal and Salley (2004), p. B-3.
  2. ↑ 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Van Rossum (2027), §8.6.
  3. ↑ 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Ministry of Business, Innovation and Employment (2013), p. 53.
  4. ↑ 4.0 4.1 4.2 4.3 4.4 Australian Building Codes Board (2020a), Data Sheet B1, p. 10.
  5. ↑ 5.0 5.1 5.2 Van Rossum (2027), §4.5.
  6. ↑ 6.0 6.1 6.2 6.3 Iqbal and Salley (2004), p. B-4.
  7. ↑ 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Ministry of Business, Innovation and Employment (2025b), p. 20.
  8. ↑ 8.0 8.1 8.2 8.3 8.4 National Research Council of Canada et al. (2005), p. 2.4-11.
  9. ↑ 9.0 9.1 9.2 Australian Building Codes Board (2020a), Data Sheet B2, p. 3.
  10. ↑ 10.0 10.1 10.2 Schifiliti et al. (2002), p. 4-10.
  11. ↑ 11.0 11.1 11.2 11.3 Iqbal and Salley (2004), p. B-1.
  12. ↑ 12.0 12.1 Alpert (2002), p. 2-23.
  13. ↑ 13.0 13.1 13.2 13.3 13.4 Ministry of Business, Innovation and Employment (2013), p. 55.
  14. ↑ Iqbal and Salley (2004), p. B-2.
  15. ↑ 15.0 15.1 National Research Council of Canada et al. (2005), p. 2.4-10.
  16. ↑ National Research Council of Canada et al. (2005), p. 2.4-12.
  17. ↑ 17.0 17.1 Australian Building Codes Board (2020a), Data Sheet B2, p. 4.
  18. ↑ Australian Building Codes Board (2020a), Data Sheet B2, p. 5.
  19. ↑ Iqbal and Salley (2004), p. B-5.
  20. ↑ Australian Building Codes Board (2020a), Data Sheet B1, p. 11.
  21. ↑ Alpert (2002), p. 2-24.
  22. ↑ Schifiliti et al. (2002), p. 4-11.
  23. ↑ Schifiliti et al. (2002), p. 4-4.
  24. ↑ Stroup et al. (2016), p. 3-4.
  25. ↑ Stroup et al. (2016), p. 3-23.
  26. ↑ Stroup et al. (2016), p. 3-14.
  27. ↑ Ministry of Business, Innovation and Employment (2025b), p. 21.

References

  • Alpert, R. L. (2002). Ceiling jet flows. 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. 2-18–2-31). National Fire Protection Association; Society of Fire Protection Engineers.
  • Australian Building Codes Board. (2020a). Data sheets: Fire Safety Verification Method (Handbook annex, Print version 1.1).
  • 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, Office of Nuclear Reactor Regulation.
  • Ministry of Business, Innovation and Employment. (2013). Commentary for Building Code clauses C1–C6 and Verification Method C/VM2.
  • Ministry of Business, Innovation and Employment. (2025b). Verification Method C/VM2: Framework for fire safety design (2nd ed.).
  • 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.
  • Schifiliti, R. P., Meacham, B. J., & Custer, R. L. P. (2002). Design of detection 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. 4-1–4-43). National Fire Protection Association; Society of Fire Protection Engineers.
  • 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.
  • Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.
Retrieved from "https://fire-safety.academy/w/index.php?title=T-squared_fire&oldid=2571"