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Alpert correlations

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Contents
  1. 1Ceiling jets and the purpose of the correlations
  2. 2The equations
  3. 3Walls, corners and the method of reflection
  4. 4Worked examples
  5. 5Use in engineering practice
  6. 6Related correlations
  7. 7Limits and uncertainty
  8. 8See also
  9. 9Notes
  10. 10References

Alpert correlations are four empirical equations for the maximum gas temperature and the maximum gas velocity in a ceiling jet.[1] A ceiling jet is the thin layer of hot gas that flows beneath a ceiling above a fire.[2] They need only three inputs: the heat release rate of the fire, the height of the ceiling above the fuel and the horizontal distance from the axis of the fire.[3] Ronald L. Alpert published the equations in 1972 in the journal Fire Technology.[4] They rest on large-scale fire tests with heat release rates from 668 kW to 98 MW under ceilings 4.6 to 15.5 m high.[1]

Engineers mainly use the correlations to predict when heat detectors and sprinklers below a ceiling will respond to a fire.[5] They appear in the fire dynamics tools of the United States Nuclear Regulatory Commission and in the International Fire Engineering Guidelines.[3][6] The equations hold for a steady fire under a smooth, flat and horizontal ceiling, away from walls, before a layer of hot smoke has formed under the ceiling.[7] When such a layer is present, applying them can under-predict the measured ceiling jet temperature by about 70%.[8]

Ceiling jets and the purpose of the correlations

When a fire burns below a ceiling, the hot gases of the fire plume rise, strike the ceiling and turn to flow horizontally away from the fire.[2] The plume rises because the hot gases are less dense than the surrounding air, and it mixes in cold room air as it rises.[9] The horizontal flow under the ceiling is the ceiling jet, a relatively fast and shallow gas layer driven by the buoyancy of the hot combustion products.[2] As the jet spreads, it entrains room air at its lower boundary, which cools the gases and slows them down.[2] Heat transfer to the ceiling also cools the part of the jet that touches the ceiling surface.[2]

Much of the equipment for fire detection and suppression, such as smoke detectors, heat detectors and sprinklers, is mounted close to the ceiling.[2] These devices sit in the hot flow, and their response is the basis of automatic fire protection in buildings.[2] The ceiling jet is about 5 to 12% of the fire-to-ceiling height deep, and its maximum temperature and velocity occur at about 1% of that height below the ceiling.[10] Alpert developed his equations as easy-to-use expressions for these maximum values, and they are widely used in fire hazard analysis.[1] The title of his 1972 paper, "Calculation of response time of ceiling-mounted fire detectors", states the purpose of the work.[11][12]

Ceiling flows have been studied since the 1950s.[2] Research took place at the Fire Research Station in Great Britain, at Factory Mutual Research Corporation, at the National Institute of Standards and Technology and elsewhere.[2] Alpert's correlations are based on fire tests with wood and plastic pallets, empty cardboard boxes, plastic goods in cardboard boxes and liquid fuels.[1] The heat release rates in these tests ranged from 668 kW to 98 MW, and the total ceiling heights from 4.6 to 15.5 m.[1] Alpert notes that analyses for unconfined ceilings may be sufficient for large industrial or commercial storage buildings, whereas in smaller rooms a heated gas layer collects under the ceiling.[13]

The equations

The correlations divide the area under the ceiling into two regions. Near the fire axis lies the turning region, where the rising plume bends to flow outwards under the ceiling.[1] In this region the temperature and velocity do not depend on the radial distance. The equations there actually give the temperature and velocity on the plume axis at ceiling height.[1] Outside the turning region, the gas cools and slows as the jet spreads, so the equations contain the ratio of the radial distance to the ceiling height.[1] The boundary between the regions lies at a ratio of 0.18 for temperature and 0.15 for velocity.[1]

The first pair of equations gives the excess temperature, which is the rise of the gas temperature above the ambient temperature. It grows with the heat release rate to the power 2/3 and falls with the ceiling height to the power 5/3. The second pair gives the gas velocity, which grows with the ratio of heat release rate to ceiling height to the power 1/3. In SI units, the four equations read:[1] T−T∞=16.9Q˙2/3H5/3for r/H≤0.18 T−T∞=5.38Q˙2/3/H5/3(r/H)2/3for r/H>0.18 U=0.96(Q˙H)1/3for r/H≤0.15 U=0.195(Q˙/H)1/3(r/H)5/6for r/H>0.15

Symbol Description Unit
T maximum gas temperature in the ceiling jet °C
T∞ ambient temperature °C
U maximum gas velocity in the ceiling jet m/s
Q˙ total heat release rate of the fire kW
H height of the ceiling above the top of the fuel m
r horizontal distance from the plume axis m

Because of these powers, doubling the heat release rate raises the excess temperature by a factor of about 1.59, but the velocity only by a factor of about 1.26.[1] Doubling the ceiling height lowers the excess temperature directly above the fire to about 31% of its former value.[1] The two temperature equations give practically the same value at the boundary, since 5.38 divided by 0.18 to the power 2/3 is about 16.9.[1] The predicted temperature therefore does not jump when a position moves from one region into the other.

In the equations, the heat release rate is the total rate, because Alpert correlated the test data with the total rate of heat release in the fire.[1] The buoyancy that drives the flow, however, comes only from the convective part of that heat.[1] For the alcohol pool fires that formed the main basis of the correlations, the two are roughly equal. For solid goods and pallets, however, the convective part is only about 60% of the total.[1] A common rule of thumb states that a fire releases about 70% of its energy by convection and 30% by radiation, although the real split differs from fire to fire.[14]

Walls, corners and the method of reflection

The equations describe a plume that draws in air from all sides. They were developed for fires at least 1.8 times the ceiling height away from the walls of the space.[15] A wall close to the fire reduces the amount of air entrained into the plume, which lengthens the flames and keeps the plume hotter at a given height.[7] The method of reflection accounts for this by symmetry. For a fire against a flat wall the analyst enters twice the heat release rate, and for a fire in a 90-degree corner four times.[15] The wall acts as a mirror, so that a half plume against the wall behaves like a full plume with twice the heat release rate.[9]

The rule only holds when the fuel really touches the wall along its edge. Zukoski and co-workers found that a circular burner touching a wall at a single point behaved almost like a fire far from the wall, with only 3% less entrainment.[15] Such a fire is better represented by 1.05 times the heat release rate than by twice it. A factor of 2 suits a semicircular burner with its whole flat side against the wall.[15] In fire risk assessment for nuclear power plants, a location factor of 2 is applied in the same way to fires that burn along a wall.[16]

Worked examples

Alpert's chapter in the SFPE Handbook demonstrates the correlations with three short calculations, which are repeated here.[15] In the first, a steady fire of 1.0 MW burns 10 m below a ceiling, and the task is to find the maximum excess temperature directly above the fire. Directly above the fire the radial distance is zero, so the turning-region equation applies: T−T∞=16.9×10002/3105/3=16.9×10046.42=36.4 ∘C With the third equation, the maximum velocity at the same point is 4.45 m/s.[1] At a sprinkler 5 m from the fire axis, the ratio r/H is 0.5. There the outer equations give an excess temperature of 18.4 °C and a velocity of 1.61 m/s.[1] Both values are much lower than directly above the fire, because the jet cools and slows as it entrains room air.[2]

In a room at 20 °C, the gas at that sprinkler therefore reaches only about 38 °C. A sprinkler with the generic ordinary temperature rating of 74 °C would never operate in this steady fire.[17] Solved for the heat release rate, the outer temperature equation shows that a steady fire of about 5.03 MW is needed before the gas there even reaches 74 °C.[1] The sensing element would respond later still, because its thermal inertia makes it lag behind the temperature of the surrounding gas.[18]

The second example concerns a fire in a corner formed by two non-combustible walls, 12 m below the ceiling.[15] The question is how large the fire must be to raise the gas temperature under the ceiling by 50 °C at 5 m from the corner. Here r/H equals 5/12, or 0.42, so the outer temperature equation applies with four times the heat release rate for the corner: 50=5.38(4Q˙)2/3/125/3(5/12)2/3⇒Q˙=54(50×125.38)3/2=1472 kW The required fire is therefore about 1.48 MW.[15] With the same factor of four, the outer velocity equation gives a maximum gas velocity of about 3.2 m/s at that position.[15] A fire away from the walls would need four times this heat release rate, about 5.89 MW, to give the same temperature rise.[1]

Use in engineering practice

Calculations like these are typical hand calculations: a few formulas that can be worked out by hand, in a spreadsheet or with a modest computer application.[19] They take less than a minute, whereas a computational fluid dynamics (CFD) model may need hours or days.[20] Their main use is to estimate the detector activation time of heat detectors and sprinklers.[5] The gas temperature and velocity from the correlations are entered in a heat balance for the heat-sensitive element of the device.[6] The sensitivity of that element is expressed by its response time index, which depends on its mass, specific heat and surface area and on the heat transfer coefficient.[21]

Generic values of this index in NUREG-1805 range from 34 (m·s)1/2 for a quick-response link to 235 (m·s)1/2 for a standard-response bulb.[21] For a steady fire, the activation time follows from a single equation that combines this index with the ceiling jet temperature and velocity.[10] A fire with a changing heat release rate can be treated as a series of steady fires that follow one another.[10] Alpert calls this the quasi-steady assumption and considers it reasonable for a slowly growing fire in a room-sized space.[22] It may not be appropriate when the heat release rate changes significantly within the time that the gas needs to travel from the fire to the detector.[22]

In New Zealand, C/VM2 (second edition, 2025) requires fire modelling to determine the detection time of an automatic warning system.[23] The model must be either an algorithm that includes a ceiling jet correlation or a CFD model.[23] The International Fire Engineering Guidelines (2005) present Alpert's model for typical room enclosures.[6] They round two of the constants slightly, to 0.95 and 0.20 instead of 0.96 and 0.195.[24]

In fire probabilistic risk assessment for nuclear power plants in the United States (NUREG/CR-6850, 2005), Alpert's ceiling jet temperature correlation sets a critical horizontal distance from the plume.[25] A cable or other target in the ceiling jet within this distance is assumed to meet gases at or above its damage temperature.[25] Zone models, which divide a room into layers with uniform properties, contain ceiling jet correlations as well.[26][8] The zone model CFAST, for example, calculates the ceiling jet temperature with a dimensionless form from the work of Alpert and Heskestad.[27] CFAST scales this form so that it matches the calculated plume temperature near the axis.[27]

Related correlations

Heskestad developed dimensionless correlations for the same two quantities from alcohol pool fire tests carried out at the Fire Research Station in the 1950s.[15] These predict slightly higher excess temperatures and substantially greater gas velocities than Alpert's equations.[15] For steady ceiling jets beneath unobstructed ceilings, Alpert's handbook chapter therefore recommends Heskestad's dimensionless temperature together with the velocity from Alpert's later theoretical work.[15] For realistic fires, the chapter recommends the convective heat release rate in these dimensionless expressions.[28]

For fires that grow with the square of time, Heskestad and Delichatsios developed correlations that also account for the time the gas needs to reach a detector.[29] These correlations are the basis of the detector spacing values in NFPA 72, which work with slow, medium, fast and ultrafast fire growth.[29] In an example in the same chapter, a foam sofa fire grows to 2.5 MW under a 5 m ceiling.[30] The transient correlations then give 167 °C at a detector 4 m from the fire axis.[30] The quasi-steady limit of the same correlations gives 197 °C at that moment.[30]

Separate correlations exist for other ceiling geometries. Kung and co-workers studied the flow under sloped ceilings.[22] Delichatsios developed correlations for flows confined between deep beams or along corridors.[13] Motevalli and Marks measured the depth of the jet, which stays at about 10 to 12% of the ceiling height beyond a radial distance of about one ceiling height.[1]

Limits and uncertainty

NUREG-1805 lists the main assumptions behind the method. The ceiling must be unconfined, unobstructed, smooth, flat and horizontal, and the fire must be away from walls and corners.[7] The fire must be steady, and no smoke layer may form below the ceiling jet during the time of interest.[7] Device calculations based on the correlations consider only convective heating of the sensing element and ignore direct radiation from the flames.[7]

The correlations give the maximum values in the jet, which occur close to the ceiling. Detectors or sprinklers mounted further below the ceiling meet cooler and slower gas than the equations suggest.[1] Under a low ceiling, a device placed more than 12% of the fire-to-ceiling height below the ceiling may lie outside the jet altogether. Its response time can then increase drastically.[1] Under very high ceilings, a device may instead sit within the thin thermal and viscous boundary layers next to the ceiling surface.[1]

The most serious limitation concerns the hot gas layer. In a room, a quiescent layer of hot gas collects under the ceiling, and the ceiling jet then mixes with this warm gas instead of cool air.[13] The 2016 supplement to the validation study NUREG-1824 examined this situation. Using the correlation in a compartment with a hot gas layer can under-predict the measured ceiling jet temperature by about 70%.[8] Zone models such as CFAST and MAGIC therefore use a modified plume algorithm.[31] Adding the predicted layer temperature rise to the result of the correlation has been suggested as a simpler engineering approximation.[31]

Under unconfined ceilings, the same study found a bias factor of 0.86 and a relative standard deviation of 0.11 for the ceiling jet temperature rise predicted by Alpert's correlation.[32] A bias factor below 1 means that a model under-predicts the measurements on average, in this case by about 14%.[33] The uncertainty of the input matters as well. The temperature rise scales with the heat release rate to the power 2/3. An uncertainty of 7.5% in the heat release rate therefore gives an uncertainty of about 5% in the temperature rise.[34]

The correlations describe gas temperatures, not the temperatures of solid objects. Cadorin and co-workers note that Alpert's work aimed at the response of detectors, while the temperature of a structural member also depends on radiation from the fire itself.[12] For steel members near a localised fire, they therefore use a model that gives the heat flux directly.[35] In the nuclear fire risk method, the authors state that Alpert's correlation had not been verified and validated for their control panel model beyond internal checks.[36]

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 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 Alpert (2002), p. 2-19.
  2. ↑ 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 Alpert (2002), p. 2-18.
  3. ↑ 3.0 3.1 Iqbal and Salley (2004), p. 10-10.
  4. ↑ Alpert (2002), p. 2-30.
  5. ↑ 5.0 5.1 National Research Council of Canada et al. (2005), p. 2.7-2.
  6. ↑ 6.0 6.1 6.2 National Research Council of Canada et al. (2005), p. 2.7-3.
  7. ↑ 7.0 7.1 7.2 7.3 7.4 Iqbal and Salley (2004), p. 10-12.
  8. ↑ 8.0 8.1 8.2 Stroup et al. (2016), p. 6-17.
  9. ↑ 9.0 9.1 Van Rossum (2027), §6.7.
  10. ↑ 10.0 10.1 10.2 Iqbal and Salley (2004), p. 10-9.
  11. ↑ Iqbal and Salley (2004), p. 10-14.
  12. ↑ 12.0 12.1 Cadorin et al. (2003), p. 441.
  13. ↑ 13.0 13.1 13.2 Alpert (2002), p. 2-26.
  14. ↑ Van Rossum (2027), §4.3.
  15. ↑ 15.00 15.01 15.02 15.03 15.04 15.05 15.06 15.07 15.08 15.09 15.10 Alpert (2002), p. 2-20.
  16. ↑ Nowlen et al. (2005b), p. L-5.
  17. ↑ Iqbal and Salley (2004), p. 10-7.
  18. ↑ Iqbal and Salley (2004), p. 12-3.
  19. ↑ Van Rossum (2027), §8.2.
  20. ↑ Stroup et al. (2016), p. 6-56.
  21. ↑ 21.0 21.1 Iqbal and Salley (2004), p. 10-8.
  22. ↑ 22.0 22.1 22.2 Alpert (2002), p. 2-23.
  23. ↑ 23.0 23.1 Ministry of Business, Innovation and Employment (2025b), p. 27.
  24. ↑ National Research Council of Canada et al. (2005), p. 2.7-4.
  25. ↑ 25.0 25.1 Nowlen et al. (2005b), p. F-5.
  26. ↑ Van Rossum (2027), §9.1.
  27. ↑ 27.0 27.1 Peacock et al. (2026), p. 16.
  28. ↑ Alpert (2002), p. 2-21.
  29. ↑ 29.0 29.1 Alpert (2002), p. 2-24.
  30. ↑ 30.0 30.1 30.2 Alpert (2002), p. 2-25.
  31. ↑ 31.0 31.1 Stroup et al. (2016), p. 6-18.
  32. ↑ Stroup et al. (2016), p. 6-57.
  33. ↑ Stroup et al. (2016), p. 6-1.
  34. ↑ Stroup et al. (2016), p. 3-37.
  35. ↑ Cadorin et al. (2003), p. 442.
  36. ↑ Nowlen et al. (2005b), p. L-1.

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.
  • Cadorin, J.-F., Pintea, D., Dotreppe, J.-C., & Franssen, J.-M. (2003). A tool to design steel elements submitted to compartment fires–OZone V2. Part 2: Methodology and application. Fire Safety Journal, 38, 429–451. https://doi.org/10.1016/S0379-7112(03)00015-8
  • 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. (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.
  • Nowlen, S. P., Najafi, B., Wyant, F. J., Forester, J., Kazarians, M., Kolaczkowski, A., Joglar, F., Funk, D., Hannaman, G., & Anoba, R. (2005b). EPRI/NRC-RES fire PRA methodology for nuclear power facilities: Vol. 2. Detailed methodology (NUREG/CR-6850; EPRI 1011989). Electric Power Research Institute; U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research.
  • 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
  • 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.
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