Opposed-flow flame spread

Opposed-flow flame spread is the spread of a flame over the surface of a solid or liquid fuel in the direction opposite to the surrounding gas flow. The flow can be an external wind or the air that the fire itself draws in, as in downward spread on a wall or sideways spread across a vertical panel. The flame heats the fuel ahead of it mainly by conduction through the gas, over a short strip of about 1 to 3 mm. This form of flame spread is therefore slow: lateral or downward spread on thick solids has a typical rate of about 0.1 cm/s, against 1 to 100 cm/s for upward spread.

The slow rate still matters, because even spread at about 1 mm/s enlarges a small burning area quickly and increases the heat transfer that drives further spread. Simple energy balances give the spread rate for thermally thin fuels, such as paper, and for thermally thick fuels, such as boards and wall linings. The lateral ignition and flame spread test (LIFT) provides the material data that these formulas need. Attempts to include configuration effects, such as spread in corners and on ceilings, have not yet led to fully established methods but show promise in replacing older flammability indices. Flame spread equations combined with cone calorimeter data have also been used to predict room-corner fire tests.

Definition and terminology

Quintiere describes surface flame spread as the movement of a flame close to the condensed fuel that feeds it. Two fronts advance together: the flame in the gas phase and the edge of the vaporising or pyrolysing region on the fuel surface. The flame spread velocity is defined as the speed of this pyrolysis front.

The direction of the gas flow relative to the spread defines two classes. When the flame spreads upwind, against the flow, the process is called opposed-flow spread; when it spreads downwind, it is called wind-aided spread. The flow can come from an external wind or from the natural convection that the spreading flame, or another fire, sets up. Downward spread on a wall is a common case: the flame moves down while the air that feeds it rises, driven by buoyancy, at roughly 30 cm/s. Lateral spread across a vertical surface behaves in a similar way, and so does the outward spread from a small fire on a horizontal surface.

Quintiere summarises the process with the fundamental equation of flame spread. It considers the net heat that the flame transfers, per unit area and time, to the fuel ahead of it. For steady spread, this heat equals the heat needed to raise the fuel from its initial temperature to its ignition temperature:

$$\rho V\mathrm{\Delta }h={\dot{q}}^{″},\mathrm{\Delta }h=c(T_{\mathrm{i}\mathrm{g}}-T_{\mathrm{s}})$$

The ignition temperature in this balance is the surface temperature at which the fuel releases enough vapour to sustain piloted ignition. For liquids it is the fire point, and for solids it depends on the kinetics of thermal decomposition. The balance shows that the spread rate rises as the fuel density falls. It also grows without bound as the surface temperature ahead of the flame approaches the ignition temperature.

Symbol	Description	Unit
$V$	flame spread rate (speed of the pyrolysis front)	m/s
$\rho$	density of the fuel	kg/m³
$c$	specific heat capacity of the fuel	J/(kg·K)
$\mathrm{\Delta }h$	heat needed per unit mass to raise the fuel from $T_{\mathrm{s}}$ to $T_{\mathrm{i}\mathrm{g}}$	J/kg
${\dot{q}}^{″}$	net heat flux from the flame to the fuel ahead of it	W/m²
$T_{\mathrm{i}\mathrm{g}}$	ignition temperature of the fuel surface	K or °C
$T_{\mathrm{s}}$	surface temperature of the fuel ahead of the flame	K or °C
$T_{\mathrm{f}}$	flame temperature	K
$k_{\mathrm{g}}$	thermal conductivity of the gas	W/(m·K)
$\delta$	thickness of a thermally thin fuel	m
$V_{\mathrm{g}}$	speed of the opposed gas flow	m/s
$(k\rho c{)}_{\mathrm{g}}$	product of conductivity, density and specific heat of the gas	W²·s/(m⁴·K²)
$k\rho c$	thermal inertia of the fuel	kW²·s/(m⁴·K²)
$\mathrm{\Phi }$	flame heating parameter of the LIFT method	kW²/m³
$T_{\mathrm{s},\mathrm{m}\mathrm{i}\mathrm{n}}$	minimum surface temperature for continued spread	°C

Heat transfer ahead of the flame

In opposed-flow spread, the flame heats the fuel ahead of it mainly by conduction through the gas. The heated length is short, about 1 to 3 mm. In wind-aided spread it is of the order of the flame length, 0.1 to 10 m. Measurements on polymethyl methacrylate (PMMA) by Ito and Kashiwagi show a maximum heat flux of about 70 kW/m² just ahead of the front in downward spread. Upward spread gives a nearly constant flux of about 25 kW/m², spread over a much longer distance.

The flame temperature that drives this heat transfer depends mainly on the oxygen concentration of the surrounding air. The reason is that the heat released per unit mass of oxygen is nearly constant for most hydrocarbons, at about 13 kJ/g. Spread under a ceiling, where the oxygen concentration has fallen through oxygen depletion, is therefore expected to slow down. The heat of combustion, the heat of gasification and the specific heat of the gas also enter the full expression for the flame temperature.

The simple theory makes the spread rate independent of the speed of the opposed flow. That holds only while the chemical reactions in the flame are fast compared with the time the gas takes to pass through it. The ratio of these two times is the Damköhler number, which varies with the inverse square of the flow speed. When the flow is too fast, the reactions are incomplete and the flame weakens. Below a critical value of the Damköhler number no further spread is possible, and the flame is extinguished.

Thermally thin fuels

A fuel is thermally thin when it is so thin that its temperature is uniform across its thickness. For this case Quintiere writes an energy balance over the heated strip and expresses the flame heat flux as conduction through the gas. The result is nearly identical to the formula that de Ris derived more formally:

$$V=\frac{\sqrt{2}{k}_{\mathrm{g}}(T_{\mathrm{f}}-T_{\mathrm{s}})}{\rho c\delta (T_{\mathrm{i}\mathrm{g}}-T_{\mathrm{s}})}$$

The spread rate is inversely proportional to the thickness, density and specific heat of the fuel, so thinner and lighter sheets spread faster. The flame temperature should ideally be the adiabatic stoichiometric value, but heat losses and chemical kinetic effects lower it in practice. Measured rates on thin paper fall as the opposed flow speed increases, in line with the effect of the Damköhler number.

For common combustible solids, it has been reasoned that the thin analysis holds up to a thickness of about 1 mm. For downward spread on PMMA, Fernandez-Pello and Hirano found thin behaviour below 2 mm and thick behaviour above 2 cm, with a non-uniform temperature in between. Apart from items such as paper, garments and draperies, most solids behave as thermally thick under flame spread conditions.

Thermally thick fuels

In a thick fuel, only a surface layer is heated while the flame passes. Its depth follows from heat conduction theory as the square root of the thermal diffusivity multiplied by the heating time. The heated length in the gas follows from a balance between conduction and convection. Combining these steps gives the thick-fuel result, again identical to that of de Ris:

$$V=\frac{V_{\mathrm{g}}(k\rho c{)}_{\mathrm{g}}{(T_{\mathrm{f}}-T_{\mathrm{i}\mathrm{g}})}^2}{k\rho c{(T_{\mathrm{i}\mathrm{g}}-T_{\mathrm{s}})}^2}$$

The spread rate now depends on the thermal inertia $k\rho c$ of the fuel and falls with the square of the gap between the ignition temperature and the surface temperature. Unlike in the thin case, the ideal rate grows with the speed of the opposed flow. Chemical kinetics act against this, so the actual rate may rise or fall with flow speed, depending on the oxygen concentration.

For PMMA in normal air, with an oxygen mass fraction of 0.233, the rate is insensitive to flow speeds up to about 30 cm/s and falls at higher speeds. The buoyancy-induced flow in downward spread on a wall is roughly 30 cm/s. Under normal fire conditions the spread rate therefore depends mainly on the oxygen concentration, at least for PMMA. In turbulent opposed flow, the rate on thin paper fell as the turbulence intensity rose, whereas thick PMMA showed a maximum at a flow speed of about 0.5 m/s. Quintiere suggests that, for engineering purposes, solids thicker than about 1 mm may be regarded as thermally thick. Up to a thickness of 1 to 2 cm, the spread rate can still depend on the thickness and on the material behind the fuel.

Measurement and material data

The cone calorimeter and the lateral ignition and flame spread test have provided the basis for the material data that flame spread calculations need. Quintiere and Harkleroad developed a practical test procedure that yields the parameters for opposed-flow spread on thick materials burning in air under natural convection. The thick-fuel result is rewritten with a single flame heating parameter $\mathrm{\Phi }$, which depends on the flow speed and the oxygen concentration:

$$V=\frac{\mathrm{\Phi }}{k\rho c{(T_{\mathrm{i}\mathrm{g}}-T_{\mathrm{s}})}^2}$$

In the test, the flame spreads sideways along a vertical sample that is exposed to an external radiant heat flux. The parameters $\mathrm{\Phi }$, $k\rho c$ and $T_{\mathrm{i}\mathrm{g}}$ are deduced from the measurements, and the procedure is standardised as the LIFT test. A wide range of tests gave ignition temperatures from 280 °C for PMMA to 570 °C for paper on gypsum wallboard. Values of $\mathrm{\Phi }$ ranged from about 1 to 15 kW²/m³.

Each material also has a minimum surface temperature below which continued spread is not possible. The minimum irradiance that allows piloted ignition marks a critical condition for spread. Results from the lateral test were similar to those for downward spread, except for materials that melt and drip heavily. The table lists selected values, which are significant to two places only.

Material	$T_{\mathrm{i}\mathrm{g}}$ (°C)	$k\rho c$ (kW²·s/(m⁴·K²))	$\mathrm{\Phi }$ (kW²/m³)	$T_{\mathrm{s},\mathrm{m}\mathrm{i}\mathrm{n}}$ (°C)
PMMA Polycast (1.59 mm)	278	0.73	5.4	120
PMMA type g (1.27 cm)	378	1.02	14.4	90
Douglas fir particleboard (1.27 cm)	382	0.94	12.7	210
Plywood, plain (1.27 cm)	390	0.54	12.9	120
Flexible foam (2.54 cm)	390	0.32	11.7	120
Gypsum board, common	565	0.45	14.4	425
Polystyrene (5.08 cm)	630	0.38	not measurable	630

A second test arrangement, the fire propagation apparatus of standard ASTM E2058 of the American Society for Testing and Materials, measures downward propagation against an imposed airflow. For a vertical PMMA slab 25 mm thick in oxygen-enriched air, with an oxygen mass fraction of 0.334 and an airflow of 0.09 m/s, the pyrolysis front advanced linearly with time. The heat release rate during downward propagation also grew linearly with time. Upward propagation in the same apparatus, in a test in normal air with an oxygen mass fraction of 0.233, was much faster and accelerated with time.

Worked example

A vertical lining of Douglas fir particleboard, 1.27 cm thick, is heated by radiation from a fire elsewhere in the room. The LIFT data give an ignition temperature of 382 °C, a thermal inertia of 0.94 kW²·s/(m⁴·K²), $\mathrm{\Phi }=12.7$ kW²/m³ and a minimum spread temperature of 210 °C. The example estimates how fast a flame spreads downward or sideways over the board for several surface temperatures ahead of the flame. At the minimum spread temperature of 210 °C, the spread rate is:

$$V=\frac{12.7}{0.94\times (382-210{)}^2}\approx 4.6\times 10^{-4}\mathrm{m}/\mathrm{s}=0.46\mathrm{m}\mathrm{m}/\mathrm{s}$$

At surface temperatures of 250 °C, 300 °C and 350 °C, the same formula gives 0.78 mm/s, 2.0 mm/s and 13 mm/s. Crossing 0.1 m of board then takes about 210 s at 210 °C, but only about 7.5 s at 350 °C, with both times rounded down to the safe side. Raising the surface temperature from 210 °C to 300 °C multiplies the spread rate by about 4.4.

Below 210 °C the flame cannot continue along this board, so an unheated board at room temperature does not support this mode of spread. Piloted ignition of the same material requires a heat flux of at least 16 kW/m². The steep rise in rate with surface temperature explains why far-field radiation and convection from the fire make the spread accelerate.

Role in fire growth

Quintiere's table of typical spread rates puts lateral or downward spread on thick solids at about 0.1 cm/s. Upward spread on thick solids reaches 1 to 100 cm/s, and smouldering only 0.001 to 0.01 cm/s. Opposed-flow spread is therefore much slower than upward flame spread, in which the flame lies over the fuel ahead of it and preheats it over a long distance. The fire dynamics tools of the U.S. Nuclear Regulatory Commission illustrate this with a window curtain. A pilot flame at the bottom causes rapid upward spread, while the same flame at the top gives much slower fire growth through slow downward spread.

Even slow spread enlarges a fire quickly. In Quintiere's example, an ignited region 6 cm across on an upholstered chair spreads at about 1 mm/s and reaches 18 cm across within a minute. The burning area then grows nearly tenfold, which increases the heat release and the heat transfer to the fuel. Fire growth accelerates when the time for spread is short compared with the burning time. It can slow down and stop when the spread time is long compared with the burning time.

In an enclosure, heat from the fire and its hot combustion gases accumulates and reaches other combustible objects by radiation. The surface temperature ahead of a spreading flame then rises, and the spread accelerates. Orientation and geometry can also change the mode of spread. In the London King's Cross fire, flames spread up a wooden escalator, and the sidewalls of the escalator caused the flames to hug the steps. An experimental study of that fire, listed by Quintiere, refers to it as the trench effect.

Limits and uncertainty

The formulas are heuristic. They come from energy balances with simple assumptions, and their results are consistent with more formal analyses but remain approximate. The ignition temperature is assumed constant for each material. Kanury stresses that the surface temperature at ignition is an effect rather than a cause, and that it depends on the conditions of the experiment.

Computing the flame heat flux from first principles is a complex problem that has limited the ability to predict flame spread. For materials that melt and drip heavily, results from the lateral test may not represent downward spread. Some materials, such as fire-retardant plywood and polystyrene, gave no measurable spread at all.

Where conditions are simple, the theory performs well. For filter paper burning in microgravity at 50 % oxygen in still gas, a measured spread rate of 0.44 cm/s compared with a predicted 0.42 cm/s, once gas-phase radiation was included. A complete numerical simulation by Di Blasi and co-workers, including solid- and gas-phase kinetics, reached qualitative success.

Quintiere concludes that flame spread depends on the heat transfer at the flame front, which in turn depends on the fuel, its configuration and orientation, and the surrounding conditions. Flame spread in enclosures, mines, ducts and buildings adds a further complex array of conditions. Flame spread on materials therefore has to be evaluated in the context of their use, with data that suit the assessment.

See also

• Critical heat flux for ignition
• Time to ignition
• Fire spread
• Ignition

References

• 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.
• Kanury, A. M. (2002a). Flaming ignition of solid fuels. 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. 2-229–2-245). National Fire Protection Association; Society of Fire Protection Engineers.
• Quintiere, J. G. (2002b). Surface flame spread. 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. 2-246–2-257). National Fire Protection Association; Society of Fire Protection Engineers.
• 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.