MQH correlation

The MQH correlation is a simple formula for estimating the average temperature of the hot smoke layer in a room fire with a door or window open to the outside. It is named after McCaffrey, Quintiere and Harkleroad, who developed it in 1981 by fitting a simple energy balance to data from about 100 experimental fires. The formula gives the rise of the upper layer temperature above ambient from the heat release rate of the fire, the size of the opening and the heat lost to the walls, ceiling and floor. It belongs to the methods for compartment fire temperatures that are most widely accepted in fire protection engineering.

The correlation is one of the standard hand calculations of fire safety engineering. It is used to estimate the gas temperature and, in a rearranged form, the heat release rate needed for flashover. It is valid for upper layer temperatures up to about 600 °C and predicts room averages, not local temperatures near the fire. In a validation study against full-scale experiments, it overpredicted the rise of the layer temperature above ambient by about 17 % on average.

Origin and data

McCaffrey, Quintiere and Harkleroad published the correlation in 1981, and Walton and Thomas later reported it in the SFPE Handbook of Fire Protection Engineering. The constants were fitted to data from over 100 experimental fires, both steady and transient. The fuels included cellulosic materials, synthetic polymers and gaseous hydrocarbons. The test compartments were 0.3 to 2.7 m high, with floor areas of 0.14 to 12.0 m² and a variety of window and door sizes.

NUREG-1805, a set of fire hazard calculation methods of the United States Nuclear Regulatory Commission (NRC), describes the data as 100 fires from eight test series. The rooms were of conventional size, and the temperature rises ranged from 20 to 600 °C. In these tests the fire was placed away from the walls, and most of the fuel was near the centre of the room. NUREG-1805 summarises the main trend: the larger the heat release rate and the smaller the opening, the higher the upper layer temperature.

Energy balance

The correlation rests on the two-layer picture of a room fire. The fire plume carries hot gases to the ceiling, where they form a hot gas layer that deepens with time. A fairly sharp interface separates this layer from the cooler air below. Once the layer reaches the top of a door or window, hot gas flows out through the upper part of the opening and outside air flows in below. Walton and Thomas call this description a two-layer or zone model.

Van Rossum describes the same idea as an energy balance on the smoke layer. The convective part of the heat release rate enters the layer, and energy leaves with the outflowing smoke and by heat transfer to the walls and ceiling. Whatever remains changes the temperature of the layer. For a quasi-steady fire, McCaffrey, Quintiere and Harkleroad state that the heat released by the fire equals the heat carried out with the hot gas plus the heat lost to the boundaries: $$\dot{Q}={\dot{m}}_g{c}_p(T_g-T_{\mathrm{\infty }})+{\dot{q}}_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}},{\dot{q}}_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}=h_k{A}_T(T_g-T_{\mathrm{\infty }})$$

The heat loss is written as an effective heat transfer coefficient multiplied by the total area of the enclosing surfaces and the temperature rise. Solving for the temperature rise gives two dimensionless groups. The first compares the heat released with the heat that the outflowing gas can carry away. The second compares the heat lost to the boundaries with the heat carried away by the gas. The outflow depends on the height of the neutral plane in the opening, which in turn depends mainly on the gas temperature, the heat release rate and the geometry. The outflow is therefore replaced by the term $\sqrt{g}{\rho }_{\mathrm{\infty }}{A}_0\sqrt{H_0}$, and the remaining effects are absorbed into the fitted constants.

Measurements show how the heat divides between these terms. In experimental wood crib fires in a compartment with a floor area of 29 m², 47 to 65 % of the heat left with the hot gases through the windows. Another 15 to 30 % went into the walls, ceiling and floor, and the rest returned to the fuel or radiated out through the windows. With smaller windows, a larger share of the heat went into the enclosing surfaces.

The correlation

Fitting the two groups to the test data gave a power law with a constant of 480 and exponents of 2/3 and −1/3: $$\mathrm{\Delta }{T}_g=480{\left(\frac{\dot{Q}}{\sqrt{g}{c}_p{\rho }_{\mathrm{\infty }}{T}_{\mathrm{\infty }}{A}_0\sqrt{H_0}}\right)}^{2/3}{\left(\frac{h_k{A}_T}{\sqrt{g}{c}_p{\rho }_{\mathrm{\infty }}{A}_0\sqrt{H_0}}\right)}^{-1/3}$$

With standard ambient values of $g$ = 9.8 m/s², $c_p$ = 1.05 kJ/(kg·K), ${\rho }_{\mathrm{\infty }}$ = 1.2 kg/m³ and $T_{\mathrm{\infty }}$ = 295 K, the formula reduces to the simpler form that NUREG-1805 also uses: $$\mathrm{\Delta }{T}_g=6.85{\left(\frac{{\dot{Q}}^2}{A_0\sqrt{H_0}{h}_k{A}_T}\right)}^{1/3}$$

Symbol	Description	Unit
$\mathrm{\Delta }{T}_g$	rise of the upper layer temperature above ambient, $T_g-T_{\mathrm{\infty }}$	K
$\dot{Q}$	heat release rate of the fire	kW
$A_0$	area of the opening	m²
$H_0$	height of the opening	m
$A_T$	total area of the enclosing surfaces, without the openings	m²
$h_k$	effective heat transfer coefficient of the boundaries	kW/(m²·K)
$g$	acceleration due to gravity	m/s²
$c_p$	specific heat of the gas	kJ/(kg·K)
${\rho }_{\mathrm{\infty }}$	ambient air density	kg/m³
$T_{\mathrm{\infty }}$	ambient temperature	K
$k$	thermal conductivity of the lining	kW/(m·K)
$\rho$, $c$	density and specific heat of the lining	kg/m³, kJ/(kg·K)
$\delta$	thickness of the lining	m
$t$, $t_p$	time since ignition, thermal penetration time	s

The term $A_0\sqrt{H_0}$ is the ventilation factor of the opening. For several openings, the values of $A_0\sqrt{H_0}$ are added together. The area $A_T$ covers the walls, ceiling and floor, with the area of the openings subtracted. Because the temperature rise grows with the heat release rate to the power 2/3, doubling the heat release rate raises it by a factor of about 1.6. Halving the ventilation factor raises it by a factor of about 1.26, the cube root of 2.

Heat losses to the boundaries

The effective heat transfer coefficient depends on the lining and on how long the fire has burnt. Heat first soaks into the lining, and only later flows through it to the back face. The time this takes is the thermal penetration time: $$t_p=\frac{\rho c}{k}{\left(\frac{\delta }{2}\right)}^2$$

After this time, heat conduction through the lining is close to steady, and the coefficient is the conductivity divided by the thickness. Before it, the lining behaves like a thick solid that keeps absorbing heat, and the coefficient falls with time: $$h_k=\frac{k}{\delta }\text{for }t>t_p,h_k={\left(\frac{k\rho c}{t}\right)}^{1/2}\text{for }t\le t_p$$

The thermal inertia of the lining is the product $k\rho c$. NUREG-1805 lists typical values, for example 0.18 (kW/(m²·K))²·s for gypsum board and 2.9 (kW/(m²·K))²·s for concrete. A lining with a low thermal inertia gives a low coefficient, and the same fire then produces a hotter layer. If walls and ceiling are made of different materials, an area-weighted average of $h_k$ is used.

Worked example

Walton and Thomas apply the method to a room of 3 m by 3 m with a height of 2.4 m. The room has a door 1.8 m high and 0.6 m wide, and a steady fire of 750 kW. The lining is 16 mm of gypsum plaster on metal lath with a density of 1,440 kg/m³. Its conductivity is 0.48 × 10⁻³ kW/(m·K) and its specific heat 0.84 kJ/(kg·K). The ambient air is at 27 °C (300 K), with a density of 1.18 kg/m³ and a specific heat of 1.0 kJ/(kg·K).

The door has an area of 1.08 m², which gives a ventilation factor of 1.08 × √1.8 ≈ 1.45 m^5/2. The enclosing surfaces add up to 28.8 m² of wall, 9 m² of floor and 9 m² of ceiling, minus the door, which gives 45.72 m². The thermal penetration time is (1,440 × 0.84 / 0.00048) × 0.008², or about 161 s. At 10 s and 60 s the lining is still absorbing heat, so $h_k$ is 0.24 and 0.098 kW/(m²·K). At 600 s the steady value applies: 0.00048 / 0.016 = 0.030 kW/(m²·K).

The full correlation then gives temperature rises of 227 K after 10 s, 306 K after 60 s and 455 K after 600 s. The handbook reports 227 K, 307 K and 453 K, because it rounds the intermediate terms. After ten minutes the layer is at about 480 °C. This is below the 500 °C often used as a flashover criterion and within the 600 °C range of the data. The 750 kW fire is also well below the stoichiometric heat release rate for this door, 1,500 × $A_0\sqrt{H_0}$ ≈ 2,170 kW, which is the most that the incoming air can burn completely.

Heat release rate for flashover

Rearranging the correlation gives the heat release rate needed to reach a chosen layer temperature. Walton and Thomas take a temperature rise of 500 °C above an ambient of 295 K as the condition for flashover, with standard air properties. This gives: $${\dot{Q}}_{\mathrm{f}\mathrm{o}}=610{\left(h_k{A}_T{A}_0\sqrt{H_0}\right)}^{1/2}$$ where ${\dot{Q}}_{\mathrm{f}\mathrm{o}}$ is the heat release rate for flashover in kilowatts. Starting from the simplified form gives a coefficient of 620 instead, and either value is acceptable within the accuracy of the expression. For the example room with $h_k$ = 0.03 kW/(m²·K), the result is about 860 kW.

A simpler method by Babrauskas gives about 1,090 kW for the same room, using 750 × $A_0\sqrt{H_0}$. Babrauskas compared the methods and found similar results over the range of room sizes of most interest. The MQH method departs from the others for small rooms, and all the methods represent the test data conservatively. NUREG-1805 adds that the minimum heat release rate for flashover grows with the size of the compartment. With too little ventilation flashover cannot occur, while too much ventilation dilutes and cools the smoke.

The result depends on the chosen flashover criteria. NUREG-1805 uses a temperature rise of 500 °C, sometimes 600 °C, or a radiant heat flux of 20 kW/m² at the floor. Van Rossum links a layer temperature of 500 °C to a radiation of about 20 kW/m² at the floor, at which many materials pyrolyse rapidly. In Australia, the data sheets of the Fire Safety Verification Method cite the SFPE engineering standard S.01. This standard bases its approach on the MQH and Thomas methods and uses 600 °C or 20 kW/m² as flashover criteria in fire models.

Accuracy and validation

The NRC and the Electric Power Research Institute (EPRI) compared the correlation with full-scale experiments in naturally ventilated rooms. All results were within or above the experimental uncertainty, with no sign of systematic underprediction. For the rise of the layer temperature, the bias factor, which shows the average overprediction or underprediction, was 1.17 with a relative standard deviation of 0.15. For comparison, the zone model CFAST scored 1.21 and 0.38, and the Fire Dynamics Simulator (FDS) scored 1.02 and 0.07.

The power 2/3 also governs how errors in the input carry through. A relative error in the heat release rate gives two thirds of that relative error in the temperature rise, so 7.5 % uncertainty in the heat release rate gives about 5 % in temperature. In the investigation of the World Trade Center fires, the National Institute of Standards and Technology (NIST) used the same reasoning. A 5 % uncertainty in the measured heat release rate corresponded to 3.3 % in the temperature rise, or roughly ±20 °C at 600 °C. The observed differences between simulations and measurements were 5 to 20 °C.

NIST also noted that the correlation rests on test data below 600 °C. In a test with a layer temperature of about 750 °C, a 10 % higher heat release rate raised the simulated temperature by 9 %. The correlation implies 6.7 %, and NIST attributed the difference to its simplifications and its limited range.

Limitations

Walton and Thomas list the limits given by McCaffrey and co-workers. The correlation holds up to an upper layer temperature of about 600 °C. It applies to steady and growing fires, as long as heat conduction into the walls is the main transient process. It does not apply to rapidly growing fires in large enclosures, where much of the growth happens before smoke leaves the room. The heat release rate and the thermal properties of the lining must be known, and the data contain few ventilation-controlled fires and no combustible linings.

NUREG-1805 adds several conditions. The correlation predicts average temperatures and should not be used for the activation of detectors or sprinklers, or for heating by direct flame contact. It treats heat flow into the boundaries as one-dimensional and uses one heat transfer coefficient for the whole room. It is not valid with the coefficient 6.85 for a fire against a wall or in a corner, and it does not allow an opening in the ceiling. The validation study also notes that the correlation assumes a single material for ceiling, walls and floor.

The size of the room is less critical than it may seem. NUREG-1805 advises caution for large compartments, but it also reports Quintiere's view that the correlation works for any room size because it is dimensionless. According to the same report, Quintiere also gave an upper limit for the heat release rate, based on about 3,000 kJ per kilogram and the air inflow term 0.5 $A_0\sqrt{H_0}$. Walton and Thomas use the same two numbers for the heat released per kilogram of air and for the air inflow of a room near flashover. Rooms with several connected spaces, ceiling vents or forced ventilation need other correlations or a computer fire model.

Related correlations

Foote, Pagni and Alvares (FPA) applied the same energy balance to rooms with mechanical ventilation. Their correlation uses the measured ventilation flow instead of the flow through an opening, with a coefficient of 0.63 and exponents of 0.72 and −0.36. It was fitted to well-ventilated tests in a compartment of 6 m by 4 m and 4.5 m high, with ventilation rates of 110 to 325 g/s. For closed rooms, NUREG-1805 gives the method of Beyler, which assumes that the room leaks enough to prevent a pressure build-up. In the validation study of the NRC and EPRI, the FPA correlation had a bias factor of 1.29 and the Beyler method 1.04.

See also

• Compartment fire
• Ventilation-controlled fire
• Smoke filling
• Vent flow
• Fire-induced pressure

References

• 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.
• McGrattan, K. B., Bouldin, C., & Forney, G. P. (2005). Computer simulation of the fires in the World Trade Center towers (NIST NCSTAR 1-5F). National Institute of Standards and Technology.
• 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.
• Walton, W. D., & Thomas, P. H. (2002). Estimating temperatures in compartment 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. 3-171–3-188). National Fire Protection Association; Society of Fire Protection Engineers.