Equivalence ratio

The equivalence ratio is the actual fuel-to-air ratio of a burning mixture or a fire, divided by the stoichiometric fuel-to-air ratio. At the stoichiometric ratio, the oxygen is exactly sufficient to burn all of the fuel. It is usually written with the Greek letter phi (φ). A value below 1 describes a fuel-lean or over-ventilated fire with air to spare, and a value above 1 a fuel-rich or under-ventilated fire. In fire safety engineering, the equivalence ratio of a fire compartment or a fire plume, often called the global equivalence ratio, is the main parameter for estimating the production of toxic gases.

Carbon monoxide is the most common fire toxicant, and over half of all fire deaths have been attributed to its inhalation. Well-ventilated fires produce little of it. Once the equivalence ratio rises above about 0.6, however, the carbon monoxide yield increases steeply, and in fuel-rich fires it reaches about 0.2 g per gram of fuel burned. Engineers use the equivalence ratio to classify fires as fuel- or ventilation-controlled, to choose species yields for design fires and to check whether a fire model has been validated for the conditions of a design.

Definition

The equivalence ratio builds on stoichiometry, which describes the proportions in which substances are consumed and formed in a chemical reaction. The stoichiometric ratio is the ideal mixture in which all of the fuel and all of the oxygen take part in the combustion. For a fire, the equivalence ratio compares the rate at which fuel is supplied with the rate at which air is supplied, and normalises this ratio by the stoichiometric value:

$$\varphi =\frac{{\dot{m}}_f/{\dot{m}}_a}{r}$$

The stoichiometric constant expresses the same chemistry as a mass of oxygen per mass of fuel. It is about 1.25 for cellulosic fuels such as wood and about 3.46 for hydrocarbon fuels such as petrol and diesel. Gottuk and Lattimer also write the equivalence ratio with this oxygen-to-fuel ratio and the mass fraction of oxygen in air, which is 0.23. Multiplying numerator and denominator by the heat of combustion gives a third form, which needs no knowledge of the fuel chemistry:

$$\varphi =\frac{{\dot{m}}_f{r}_o}{{\dot{m}}_a{Y}_{{\mathrm{O}}_{\mathrm{2}},\mathrm{a}\mathrm{i}\mathrm{r}}}=\frac{\dot{Q}}{{\dot{m}}_{a}E{Y}_{{\mathrm{O}}_{\mathrm{2}},\mathrm{a}\mathrm{i}\mathrm{r}}}$$

Here the heat released per kilogram of oxygen consumed is close to 13,100 kJ/kg for most fuels. In this form the heat release rate is the ideal value, the mass loss rate of the fuel times its heat of combustion, and is not limited by the air supply. The verification and validation study NUREG-1824 uses the same idea with the mass flow of oxygen instead of air, $\varphi =\dot{Q}/(\mathrm{\Delta }{H}_{{\mathrm{O}}_{\mathrm{2}}}{\dot{m}}_{{\mathrm{O}}_{\mathrm{2}}})$.

Symbols used in the definitions

Symbol	Description	Unit
$\varphi$	equivalence ratio	–
${\dot{m}}_f$	mass burning rate of the fuel	kg/s
${\dot{m}}_a$	mass flow rate of air reaching the fire or the control volume	kg/s
$r$	stoichiometric fuel-to-air mass ratio	–
$r_o$	stoichiometric oxygen-to-fuel mass ratio	–
$Y_{{\mathrm{O}}_{\mathrm{2}},\mathrm{a}\mathrm{i}\mathrm{r}}$	mass fraction of oxygen in air (0.23)	–
$\dot{Q}$	ideal heat release rate of the fire	kW
$E$, $\mathrm{\Delta }{H}_{{\mathrm{O}}_{\mathrm{2}}}$	heat released per mass of oxygen consumed (about 13,100)	kJ/kg
${\dot{m}}_{{\mathrm{O}}_{\mathrm{2}}}$	mass flow rate of oxygen	kg/s

Typical stoichiometric air-to-fuel ratios, the inverse of $r$, show how much air different fuels need. Burning 1 kg of methane completely takes 17.2 kg of air, and 1 kg of propane 15.6 kg. Polyurethane foam needs 8.78 kg and the volatiles of spruce wood only 3.87 kg. For the foam, with the formula CH_1.74O_0.323N_0.07, Gottuk and Lattimer derive a stoichiometric fuel-to-air ratio of 0.1139.

Lean, stoichiometric and rich burning

The equivalence ratio separates two burning regimes. Values below 1 mean over-ventilated or fuel-lean conditions, and values above 1 mean under-ventilated or fuel-rich conditions. At a value of exactly 1, ideal burning converts all of the fuel to carbon dioxide and water with no oxygen left over. Under fuel-rich conditions there is not enough oxygen to burn all of the fuel. The products then also contain unburnt hydrocarbons, carbon monoxide and hydrogen, which are products of incomplete combustion.

A simple model of the most complete combustion possible expresses this in terms of normalised yields, the yield of a product divided by its theoretical maximum. Below an equivalence ratio of 1, the normalised yields of carbon dioxide and water and the use of oxygen are all 1. Above 1, they fall to $1/\varphi$, and a fraction $1-1/\varphi$ of the fuel leaves unburnt. Real fires depart from this ideal. Experiments show that carbon monoxide forms mainly at the expense of carbon dioxide, while the use of oxygen stays close to the ideal.

Fuel-rich gases that leave a fire compartment can mix with fresh air and burn outside it. This external burning, and the ignition of a fuel-rich layer at its lower boundary, occur only under under-ventilated conditions. Unburnt pyrolysis gases mixed with oxygen can form a mixture within the flammability limits that only needs an ignition source to burn. The length of the external flames depends among other things on the equivalence ratio. External burning can even reduce the hazard further away, because it oxidises carbon monoxide and smoke that leave the compartment.

Global equivalence ratio in compartment fires

In a compartment fire, the global equivalence ratio expresses the overall ventilation of a chosen control volume. Because of the complex interaction between the plume and the two gas layers, and the possible spread of the fire beyond the room, the term has no unique definition. It must always be linked to a defined control volume. The plume equivalence ratio relates the fuel burning rate to the air entrained into the fire plume below the smoke layer. The upper layer equivalence ratio describes the accumulated composition of the smoke layer, and the compartment equivalence ratio relates all fuel to all air entering the room.

These values are not always the same. The upper layer reflects the history of the fire, so it lags behind the plume when the fire grows quickly. Gottuk and Lattimer give the example of a plume equivalence ratio that rises from 0.5 to 1.5 within about 5 s, while the layer gases need 20 s to be replaced. During that time the layer value lies somewhere between 0.5 and 1.5. For a fire within a single compartment, the equivalence ratio is in practice calculated from the instantaneous burning rate and the air inflow.

The equivalence ratio links directly to the distinction between fuel-controlled and ventilation-controlled fires. A fire in its development phase is always fuel-controlled, because the fire is small compared with the room and there is enough oxygen. Before flashover a fire is local and fuel-controlled, while afterwards it is usually ventilation-controlled. In a ventilation-controlled fire, the available oxygen rather than the fuel determines the heat release rate. As a compartment fire grows, conditions therefore move from over-ventilated to under-ventilated, and it is mainly in the under-ventilated phase that high levels of carbon monoxide form.

Species yields and fire toxicity

Carbon monoxide dominates the fire toxicity of most compartment fires. Concentrations as low as 4,000 ppm (0.4 % by volume) can be fatal in less than an hour, and levels of several percent have been measured in full-scale compartment fires. Nearly 75 % of the deaths due to smoke inhalation occur outside the room where the fire started, so the gases transported away from the fire room matter most.

In hood experiments with a range of fuels, the yields of all major species correlated well with the plume equivalence ratio. Below an equivalence ratio of 0.6, little carbon monoxide formed. Above 0.6 the yield increased with the equivalence ratio and, for most fuels, levelled off above 1.2. Toluene, which produces large amounts of soot, was an exception, with a carbon monoxide yield of about 0.09 g/g under all conditions. Typical peak concentrations in these and similar compartment tests were 2 to 4 % carbon monoxide.

Temperature changes the picture. Between about 800 and 900 K, the oxidation of carbon monoxide to carbon dioxide changes from a very slow to a fast reaction. In compartment fires with hexane and plastic sheets above 920 K, typical of fires after flashover, hardly any carbon monoxide formed until the fire became under-ventilated. Production of carbon monoxide therefore depends mainly on the equivalence ratio and on the temperature of the upper layer.

Additional fuel in the hot layer raises the levels much further. With wood panelling burning in the upper layer, average concentrations of 10 % carbon monoxide were measured, against 3.2 % without the wood. Wood contains oxygen in its own structure and can therefore form carbon monoxide even in a vitiated layer. The equivalence ratio correlations still held in these tests, up to equivalence ratios of about 5.5.

Under-ventilation also changes the heat release and the smoke. Tewarson found that the chemical heat of combustion decreases as the equivalence ratio increases. For red oak, it falls from 12.4 kJ/g in a well-ventilated fire to 8.3 kJ/g at an equivalence ratio of 2 and 6.2 kJ/g at 3. The soot yield of polystyrene rises from 0.165 kg/kg to 0.416 kg/kg over the same range. In flaming wood crib fires in enclosures, the carbon monoxide generation efficiency peaks at equivalence ratios of about 2.5 to 4.0.

Correlations and worked example

Gottuk and Lattimer present several correlations for the carbon monoxide yield. The simplest assumes no carbon monoxide below an equivalence ratio of 1 and a yield of 0.2 g per gram of fuel above it, and suits fires with upper layers hotter than 900 K. Two further correlations include the temperature effect. The first applies to layers below 800 K and the second to layers above 900 K:

$$Y_{\mathrm{C}\mathrm{O}}=\frac{0.19}{180}{\tan }^{-1}[10(\varphi -0.8)]+0.095(T<800\mathrm{K})$$

$$Y_{\mathrm{C}\mathrm{O}}=\frac{0.22}{180}{\tan }^{-1}[10(\varphi -1.25)]+0.11(T>900\mathrm{K})$$

In both equations, the inverse tangent is taken in degrees. $Y_{\mathrm{C}\mathrm{O}}$ is the carbon monoxide yield in grams per gram of fuel burned and $T$ the average temperature of the upper layer. The yield therefore rises from close to zero for lean fires to at most 0.19 g/g for cool layers and 0.22 g/g for hot layers. The correlations represent a lower bound, the minimum carbon monoxide production to be expected in a hazard analysis.

NUREG-1824 contains a worked example for a pump room of 8 m by 6 m and 7 m high. An oil spill of 5.3 m² burns there with a heat release rate of 7,416 kW. The maximum oxygen supply through an open door follows from the area $A_o$ and height $h_o$ of the opening:

$${\dot{m}}_{{\mathrm{O}}_{\mathrm{2}}}=0.23\times \frac{1}{2}{A}_o\sqrt{h_o}$$

For a door of 2 m by 2 m, this gives 0.23 × 0.5 × 4 × √2 = 0.650 kg/s of oxygen. The equivalence ratio becomes 7,416/(0.650 × 13,100) = 0.87, so the fire is still over-ventilated. If the door is closed and the room is ventilated mechanically with 8 air changes per hour, only 0.2026 kg/s of oxygen arrives. The equivalence ratio then rises to 2.8, and the fire is strongly under-ventilated.

Carbon monoxide yield (g/g) for the two ventilation cases, calculated with the correlations above

Case	Equivalence ratio	Upper layer below 800 K	Upper layer above 900 K
Open door, 2 m by 2 m	0.87	0.13	0.018
Mechanical ventilation, 8 air changes per hour	2.8	0.19	0.22

The table shows how strongly the temperature matters near stoichiometric conditions. In the open-door case, a cool layer gives about seven times more carbon monoxide than a hot layer. In the strongly under-ventilated case, both correlations approach their upper limits of about 0.2 g/g.

Use in design and regulation

Gottuk and Lattimer propose a stepwise method for estimating the species carried to other rooms. The first step is to determine the compartment equivalence ratio. Below 1, the yield correlations apply directly. Above 1, the next question is whether external burning will occur, which can be assumed at an equivalence ratio of 1.6. If it does, the depth of the smoke layer decides whether the external flames can oxidise the carbon monoxide before the gases spread. The authors note that the method may not give the highest possible levels of products of incomplete combustion.

In New Zealand, Verification Method C/VM2 (second edition, 2025) prescribes fixed yields for design fires. Before flashover, the design fire has a soot yield of 0.07 kg/kg and a carbon monoxide yield of 0.04 kg/kg. Once flashover occurs or the fire becomes ventilation-limited, C/VM2 requires a soot yield of 0.14 kg/kg and a carbon monoxide yield of 0.40 kg/kg of fuel. The ventilation limit is the heat release rate at which the modelled energy release first falls below the design fire through lack of oxygen. In ventilation-limited cases, the fire is then ramped up to 1.5 times that limit within 15 s.

In Australia, the Fire Safety Verification Method data sheets of the Australian Building Codes Board (2020) offer non-mandatory guidance. They use Tewarson's correlations to convert soot yields and heats of combustion from well-ventilated to ventilation-controlled conditions, at equivalence ratios of 1, 2 and 3. The equivalence ratio is also central to the validation of fire models. The experiments used to validate fire models for nuclear power plants under NUREG-1824 covered equivalence ratios of at most 0.6, so all of them were well ventilated. In the pump room example, both ventilation cases exceed this range.

Computer models use the concept as well. The Fire Dynamics Simulator applies a modified equivalence ratio in one of its extinction models to decide whether the gases in a grid cell can still burn. In that model, excess fuel acts as a diluent, so a cell that is mostly fuel with little air will not burn.

Limitations and misconceptions

The most common pitfall is the direction of the ratio. Gottuk and Lattimer warn that the stoichiometric ratio is sometimes defined as an air-to-fuel ratio instead of a fuel-to-air ratio, so tabulated values need checking. Some authors also define the equivalence ratio itself as air over fuel. Drysdale, for example, compares the ratio of the air flow to the fuel burning rate with the stoichiometric ratio. Tewarson correlated wood crib data with the air-to-fuel stoichiometric fraction, the reciprocal of the usual equivalence ratio. A value of 2 can therefore mean a rich fire in one source and a lean fire in another.

The correlations come from steady or quasi-steady tests, mostly in hoods and reduced-scale enclosures. Real fires are usually still growing and therefore more transient. In some test series, the air inflow was estimated from the size of the opening rather than measured, which makes the reported equivalence ratios uncertain. Layers are not always uniform either. In one compartment, carbon monoxide concentrations near the door were 30 to 60 % higher than at the back, so a single global value cannot describe every location.

The yields also depend on factors that the equivalence ratio does not capture. The temperature of the upper layer can change the carbon monoxide yield of a near-stoichiometric fire several times over. Oxygen-containing fuels in the hot layer raise the levels further, and it is not yet clear whether the correlations hold for other fuels in the layer. The form of the equivalence ratio based on the heat release rate is convenient, but at the time of writing it had not been established in the literature.

See also

• Soot yield
• Adiabatic flame temperature
• Oxygen consumption principle
• Effective heat of combustion

References

• Australian Building Codes Board. (2020a). Data sheets: Fire Safety Verification Method (Handbook annex, Print version 1.1).
• Drysdale, D. D. (2002). Thermochemistry. 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. 1-90–1-98). National Fire Protection Association; Society of Fire Protection Engineers.
• Gottuk, D. T., & Lattimer, B. Y. (2002). Effect of combustion conditions on species production. 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. 2-54–2-82). 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
• Ministry of Business, Innovation and Employment. (2025b). Verification Method C/VM2: Framework for fire safety design (2nd ed.).
• Najafi, B., Joglar, F., & Dreisbach, J. (2007). Verification and validation of selected fire models for nuclear power plant applications: Vol. 1. Main report (NUREG-1824; EPRI 1011999). U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research; Electric Power Research Institute.
• 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.
• 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.