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Ideal gas law

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Contents
  1. 1Statement of the law
  2. 2Density and temperature
  3. 3Expansion and pressure in a fire compartment
  4. 4Buoyancy, stack effect and the neutral plane
  5. 5Use in fire models
  6. 6Limits and common misconceptions
  7. 7See also
  8. 8Notes
  9. 9References

The ideal gas law is the relation between the pressure, the temperature, the volume and the amount of a gas. It states that, for a given gas, the pressure divided by the density is proportional to the absolute temperature.[1][2] A gas that follows this law exactly is called a perfect gas or ideal gas. Real gases follow it closely at temperatures and pressures well away from liquefaction or dissociation.[1] For air at atmospheric pressure the error is no more than a fraction of a percent.[2]

In fire safety engineering the ideal gas law explains why hot smoke rises and spreads. When air and combustion gases are heated at constant pressure they expand, so the same mass occupies more space and the density falls.[2] The lighter smoke is pushed upwards by the colder, heavier air around it. This buoyancy drives fire plumes, the stack effect and natural smoke venting.[2] If a heated gas cannot expand, its pressure rises instead.[2]

At atmospheric pressure the law reduces to a simple rule: the density of air or smoke in kg/m³ is about 353 divided by the absolute temperature in kelvin.[2] Air at 20 °C therefore has a density of about 1.2 kg/m³, and smoke at 200 °C about 0.75 kg/m³.[2] Zone models such as CFAST and computational fluid dynamics models such as the Fire Dynamics Simulator use the ideal gas law as the equation of state that links temperature, density and pressure.[3][4]

Statement of the law

For a gas of uniform properties and fixed composition, experiments show that the density depends only on the pressure and the temperature. If two of these three quantities are known, the third follows.[1] The ideal gas law expresses this dependence in its simplest form. Introduction to Fire Safety Engineering describes it as the relation between temperature, volume, pressure and the amount of gas.[2] In terms of density, the law reads:[1][5]

p=ρRT

Symbol Description Unit
p absolute pressure of the gas Pa
ρ density of the gas kg/m³
R specific gas constant: the universal gas constant divided by the molar mass of the gas; for air 287 J/(kg·K)
T absolute temperature K

The specific gas constant depends on the gas. It equals the universal gas constant divided by the molar mass of the gas or gas mixture.[3] NUREG-1805 gives a value of 287.045 J/(kg·K) for air and 461.52 J/(kg·K) for water vapour.[6] The zone model CFAST uses a constant value of about 289 J/(kg·K), because it assumes that the gas mixture in a fire has the molar mass of air.[3] Air is in reality a mixture of gases, but the law is accepted for air flows with an acceptable degree of accuracy. It is also assumed to hold for a gas in motion as well as at rest.[1]

Because the gas constant contains the molar mass, gases at the same pressure and temperature have densities in proportion to their molar masses. Kandola lists molar masses of 2.0 for hydrogen, 16.0 for methane, 28.0 for carbon monoxide, 44.0 for propane and 28.9 for air.[1] Propane gas is therefore about 1.5 times as dense as air at the same temperature, while methane has little more than half the density of air. For a perfect gas, the coefficient of thermal expansion at constant pressure equals one divided by the absolute temperature.[1] At 20 °C, a gas thus expands by about 1/293 of its volume for every kelvin of heating. CFAST also uses the link between the gas constant and the specific heat capacity of the gas: the gas constant equals the difference between the specific heats at constant pressure and at constant volume.[3]

The temperature in the law is always the absolute temperature in kelvin. Introduction to Fire Safety Engineering gives the conversion as the temperature in degrees Celsius plus 273.15.[2] For a temperature difference it does not matter whether degrees Celsius or kelvin are used, because a rise of 100 °C equals a rise of 100 K. For the density of smoke at 200 °C, however, 473 K must be entered, not 200.[2]

Density and temperature

The most common use of the law in fire safety engineering is to find the density of hot air or smoke. At a pressure of about one atmosphere, as at sea level, the law gives the density of air and smoke as approximately 353 divided by the absolute temperature.[2] The book derives this rule from the ideal gas law and uses it throughout, for instance for smoke and heat exhaust and in zone models.[7][8]

A worked example shows the size of the effect. Air at 20 °C, or 293 K, has a density of 353/293, which is about 1.20 kg/m³.[2] In a hot gas layer at 600 °C, or 873 K, the density falls to 353/873, about 0.40 kg/m³. One kilogram of gas then fills about 2.47 m³ instead of 0.83 m³, so it has expanded by a factor of almost three. The volume of a gas at constant pressure is proportional to its absolute temperature.[2] In a smoke and heat exhaust example in the book, smoke at 463 °C has a density of about 0.48 kg/m³.[7]

Falling curve of gas density against temperature from 0 to 1000 °C: 1.20 kg/m³ at 20 °C, 0.75 at 200 °C, 0.40 at 600 °C.
Density of air and smoke at atmospheric pressure, ρ ≈ 353/T with T in kelvin. Based on Van Rossum (2027), §3.2.

The difference between mass flow and volume flow follows directly from this. The mass flow through an opening equals the density times the volume flow.[9] Air flowing at 2 m/s through an opening of 10 m² gives a volume flow of 20 m³/s. At a density of 0.7 kg/m³ this is a mass flow of 14 kg/s.[9] Because hot smoke has a lower density, the same mass of smoke takes up a larger volume. Supply and exhaust flows must therefore be compared with their temperatures and densities taken into account.[9]

In a two-zone model, the hot gas layer is described by its mass and its temperature. The mass follows from the plume model, and the density follows from the temperature with the same rule. The volume of the layer, and with it the layer thickness, is then the mass divided by the density.[8]

Expansion and pressure in a fire compartment

What happens to a heated gas depends on whether it can expand. Introduction to Fire Safety Engineering distinguishes two cases. If the pressure stays the same, the volume increases and the density falls. If the amount of gas and the volume stay the same, the density stays the same, but the pressure rises.[2] A real fire compartment lies between these two cases, depending on how leaky it is.

Left: a cylinder with a free piston that rises when the gas is heated. Right: a closed box whose pressure gauge rises when the gas is heated.
Heating a gas. Left: at constant pressure the volume grows and the density falls. Right: at constant volume the density stays the same and the pressure rises. Based on Van Rossum (2027), §3.2.

In a compartment with only one opening, cool air flows in and hot smoke flows out. Klote shows that, neglecting the mass of the fuel, the ratio of the volume flows out and in equals the ratio of the absolute temperatures.[10] For smoke at 700 °C the ratio is 3.32. If 1.5 m³/s of air flows into the compartment, about 4.98 m³/s of smoke flows out.[10] With open doors or windows, the pressure difference caused by this expansion is negligible. In a tightly sealed compartment it may be important.[10]

In a closed volume, the ideal gas law requires the pressure to rise with the temperature. In building fires, gas leaks through cracks around windows and doors usually keep this fire-induced pressure and its rate of rise very small. Some enclosures, such as certain compartments on ships, can be well sealed.[11] Introduction to Fire Safety Engineering notes that expansion is strongest around the moment of flashover, when smoke is forced through seams, cracks and holes.[12] For a confined gas explosion, NUREG-1805 estimates the maximum pressure as the initial pressure multiplied by the ratio of the adiabatic flame temperature to the initial temperature. Both temperatures must be absolute.[13]

Buoyancy, stack effect and the neutral plane

Because hot gas is less dense than cold air, a column of hot smoke weighs less than a column of cold air of the same height. The colder air pushes the smoke upwards.[2] In a gas at rest, the pressure difference over a height equals approximately the density times the acceleration of gravity times the height. Between a column of cold air and a column of hot smoke, the pressure difference is therefore the acceleration of gravity times the height times the difference in density.[2] The larger the density difference and the height, the larger the pressure difference.

The same principle drives the stack effect in buildings. When it is cold outside, the warmer, less dense air inside rises in stairwells, lift shafts and other shafts.[14] Klote gives an example of a building 60 m tall with the neutral plane at mid-height, an outside temperature of −18 °C and an inside temperature of 21 °C. The maximum pressure difference due to stack effect is then 55 Pa.[15] This normal stack effect is stronger for low outside temperatures and tall shafts. When the outside air is warmer than the inside air, the flow in shafts reverses.[14]

For a fire compartment, the same type of equation gives the buoyancy pressure of the hot smoke. At a compartment temperature of 800 °C, the pressure difference 1.52 m above the neutral plane is about 13 Pa.[16] In full-scale tests by Fang, the largest pressure difference across the wall of the burn room was 16 Pa at the ceiling. For a compartment at 700 °C and a height of 10.7 m above the neutral plane, the difference reaches 88 Pa.[16] The neutral plane is the level where the pressures inside and outside are equal. Below it, air flows into the compartment, and above it, smoke flows out.[2]

Fire room with a door: smoke out above the neutral plane, air in below; a graph shows the pressure difference growing with height.
Buoyancy in a fire room with one opening: the pressure difference between inside and outside grows with height above the neutral plane, where it is zero. Based on Van Rossum (2027), §3.2, and Klote (2002), p. 4-276.

These pressure differences are small, a few tens of pascals, yet they drive most smoke movement in buildings. Klote notes that in hot smoke the buoyancy forces can be so great that smoke flows upwards even during reverse stack effect.[16] Together with expansion, wind and ventilation systems, they are the driving forces behind smoke spread.[12][14]

Use in fire models

Zone models are built on conservation of mass and energy together with the ideal gas law. CFAST derives its equations from these principles, using the ideal gas law as the equation of state that relates temperature and density.[3] The model applies the law to each gas layer, linking the compartment pressure, the layer volume, the layer mass and the layer temperature.[5] The zone model MAGIC also calculates the gas densities with the ideal gas law.[17]

The Fire Dynamics Simulator solves the conservation equations for an expandable mixture of ideal gases in the low Mach number limit.[18] In this approach, the pressure is split into a background pressure and a small perturbation. Only the background pressure is kept in the equation of state.[4] The temperature is then found from the density through the equation of state.[18] Both types of model therefore rest on the same simple relation between pressure, density and temperature.

The law is also used in fire measurements. In the experiments for the World Trade Center investigation, the molar flow of smoke-laden gas drawn through a sampling system was calculated from the measured pressure, temperature and volume flow with the ideal gas equation.[19] The uncertainties in these measured values were 0.3 kPa, 1 °C and 1.1 × 10−3 L/s, respectively.[19] Hand calculations of smoke and heat exhaust, smoke layer filling and flows through openings use the ideal gas law in the same way, through the density of the hot gas.[2][9]

Limits and common misconceptions

The ideal gas law is a theoretical assumption. It is a good approximation for many common gases, although some limitations exist.[2] Ideal gases do not exist, but the law is useful at sufficiently low densities.[2] The simple rule of 353 divided by the temperature also assumes a pressure of about one atmosphere.[2] The book's pressure formulas are simplifications as well. In practice the shape and position of openings, flow resistance, wind and mechanical ventilation also influence pressure differences and flows.[2]

A frequent error is to enter the temperature in degrees Celsius. Using 200 instead of 473 for smoke at 200 °C would give a density of about 1.8 kg/m³, higher than that of cold air, which is clearly wrong.[2] Absolute temperatures are also required in the expansion ratio and in the pressure rise of a confined explosion.[10][13]

A second misconception is that hot smoke rises because it is at a higher pressure. In an open compartment the pressure hardly changes; the gas expands and its density falls.[2][10] The flow is driven by the small pressure differences that arise from the density difference over a height. A third misconception is that a fire always produces a large overpressure. In ordinary buildings, leakage keeps the pressure rise from expansion small. Only in well-sealed enclosures does the full pressure rise predicted by the ideal gas law become relevant.[11]

See also

Notes

  1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Kandola (2002), p. 1-2.
  2. ↑ 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 Van Rossum (2027), §3.2.
  3. ↑ 3.0 3.1 3.2 3.3 3.4 Peacock et al. (2026), p. 7.
  4. ↑ 4.0 4.1 McGrattan et al. (2026), p. 10.
  5. ↑ 5.0 5.1 Peacock et al. (2026), p. 8.
  6. ↑ Iqbal and Salley (2004), p. I-17.
  7. ↑ 7.0 7.1 Van Rossum (2027), §5.3.
  8. ↑ 8.0 8.1 Van Rossum (2027), §9.3 (How a zone model works).
  9. ↑ 9.0 9.1 9.2 9.3 Van Rossum (2027), §3.6.
  10. ↑ 10.0 10.1 10.2 10.3 10.4 Klote (2002), p. 4-277.
  11. ↑ 11.0 11.1 Iqbal and Salley (2004), p. 14-3.
  12. ↑ 12.0 12.1 Van Rossum (2027), §6.6.
  13. ↑ 13.0 13.1 Iqbal and Salley (2004), p. 15-14.
  14. ↑ 14.0 14.1 14.2 Klote (2002), p. 4-274.
  15. ↑ Klote (2002), p. 4-275.
  16. ↑ 16.0 16.1 16.2 Klote (2002), p. 4-276.
  17. ↑ Joglar et al. (2007), p. 3-1.
  18. ↑ 18.0 18.1 McGrattan (2007), p. 3-2.
  19. ↑ 19.0 19.1 Hamins, Maranghides, McGrattan, Johnsson, et al. (2005), p. 72.

References

  • Hamins, A., Maranghides, A., McGrattan, K. B., Johnsson, E. L., Ohlemiller, T. J., Donnelly, M. K., Yang, J. C., Mulholland, G. W., Prasad, K. R., Kukuck, S. R., Anleitner, R. L., & McAllister, T. P. (2005). Experiments and modeling of structural steel elements exposed to fire (NIST NCSTAR 1-5B). National Institute of Standards and Technology.
  • 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.
  • Joglar, F., Gautier, B., Gay, L., & Texeraud, J. (2007). Verification and validation of selected fire models for nuclear power plant applications: Vol. 6. MAGIC (NUREG-1824; EPRI 1011999). U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research; Electric Power Research Institute.
  • Kandola, B. S. (2002). Introduction to mechanics of fluids. 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. 1-1–1-26). National Fire Protection Association; Society of Fire Protection Engineers.
  • Klote, J. H. (2002). Smoke control. 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. 4-274–4-291). National Fire Protection Association; Society of Fire Protection Engineers.
  • McGrattan, K. (2007). Verification and validation of selected fire models for nuclear power plant applications: Vol. 7. Fire Dynamics Simulator (NUREG-1824; EPRI 1011999). U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research; Electric Power Research Institute.
  • 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
  • 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
  • Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.
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