How a fire heats fuel it hasn’t touched yet

A flame front is a hot, glowing surface. All the fuel in front of it receives radiant heat. The amount depends on two things: the flame temperature to the fourth power, and the angles between the flame and the fuel.

Drag to turn the view. Scroll or pinch to zoom. Click the ground to move the leaf. The ground colour shows the radiant flux on a surface that faces the same way as the leaf.

Stefan-Boltzmann law: why temperature has the largest effect

Every surface radiates heat. A perfect emitter at absolute temperature T gives off σT⁴ watts per square metre. A real flame emits less than this, so the equation includes an emissivity ε between 0 and 1. The fourth power is the important part. A flame at 1,200 K is only 10% hotter than one at 1,090 K, but it radiates about 47% more.

The page calculates the flame colour from Planck’s law for the temperature you set. So the colour is the one that this temperature really produces. Most of the energy is invisible. A 1,090 K flame emits most strongly near 2.7 µm, in the infrared.

The view factor: angles and distance

Only part of the flame’s radiation lands on one leaf. This fraction is the view factor F. It adds up each small piece of the flame, multiplied by two cosines and divided by πr². θ₁ is the angle between the ray and the normal to the flame. θ₂ is the angle between the ray and the normal to the leaf.

Lean the flame forward and watch the ground below it get brighter. Wind does this to real fires. The flame tilts over the unburnt fuel, so both cosines increase and the distances decrease. The fuel ahead then receives much more heat. Turn the leaf from the sky to the fire. Its flux changes, but nothing else has moved.

The model treats each flame as a flat rectangular panel that radiates. AS 3959, the Australian standard for buildings in bushfire areas, uses the same simplification. That standard assumes a 1,090 K flame. Its bushfire attack levels are bands of radiant flux at 12.5, 19, 29 and 40 kW/m². The model ignores absorption by the air, so fluxes far from the flame are slightly high.

Radiation, convection, conduction

Heat gets to unburnt fuel in three ways. Each one acts over a very different distance. Which one matters depends on how far ahead of the flame you look.

Heating available to cold fuel, by distance ahead of the fire lineSame flame as section 1. The fuel is at ground level and faces the same way as the leaf. Both axes are logarithmic.

Radiation against convection as a surface heats upHeat that a surface at T loses to 30 °C surroundings: radiation εσ(T⁴ − Tₐ⁴) compared with convection h(T − Tₐ).

Radiation travels at the speed of light and needs no contact. It is the only path that reaches metres ahead. It preheats and dries the fuel long before the flame arrives.

Convection is heat that moving hot gas carries. It needs contact. An upright flame sends its hot gas up and away from the fuel ahead. Wind leans the flame over the fuel, and then the hot gas flows directly over it. The second chart shows why radiation is larger at flame temperatures. It grows as T⁴, but convection grows only as T.

Conduction through a loose bed of litter is real but slow. Ahead of a moving front, its warm zone is less than a millimetre long. So it has almost no effect.

There is a complication. Fine particles such as leaves and grass lose heat to the air very quickly. So radiation alone often cannot ignite them. Laboratory work by Finney et al. (2015, PNAS) found that fine fuels ignite from direct flame contact. Radiation only preheats them. You can test this in section 3.

Heating, drying, pyrolysis, ignition

Solid fuel does not burn. Heat breaks it down into flammable gas, and the gas burns. Follow one leaf as it heats. It gets warmer, then stays at 100 °C while its water boils off. Next it decomposes. It catches fire when gas leaves its surface fast enough to keep a flame alight.

Leaf temperature

Remaining dry mass and flux of flammable gasThe leaf ignites when the gas flux reaches the critical value (dashed line).

An energy balance for one leaf

A leaf 0.3 mm thick is thin enough to have the same temperature all the way through. So its temperature comes from simple accounting. Absorbed radiation and any heat from hot gas come in. Radiation from the leaf and cooling by the air go out. The remainder raises the temperature, or boils water while some is left.

Pyrolysis follows the Arrhenius law. The rate increases very quickly with temperature. Near 600 K, a 50 K rise multiplies it by about 40. This is why ignition looks like a threshold.

Rate of mass loss of cellulose in a 10 K/min heating testThe kinetics behind the leaf model

From radiant heat to a rate of spread

Now put the parts together. A fire front that moves at speed R must heat each kilogram of fuel ahead of it to ignition. Here, the flame’s radiation is the only heat source. Conservation of energy then sets R.

Side view along the axis of the fire. The colour of the fuel bed ahead shows its steady preheating temperature. The rays show the angles at which radiation from the flame hits it.

The energy balance

Each square metre of ground holds w kg of fine fuel. Each kilogram needs Q_ig joules to dry it and bring it to ignition. A front that moves at R uses R·w kg of fuel per second for each metre of fire line. So it needs R·w·Q_ig watts. The flame puts radiant power into the bed ahead, and the warm bed loses some of it again by radiation and to the air. What remains must supply exactly R·w·Q_ig.

Turn on Flame sized by the fire and the flame length is no longer an input. A faster fire releases more energy and so makes a longer flame (Byram’s correlation). The longer flame radiates more and pushes the fire faster. The model repeats this loop until it reaches a fixed point.

This rate changes only a little with wind. Wind leans the flame and the radiation increases. But real fires in strong wind speed up much more. The model leaves out flame contact, gusts that push flame into the fuel, and embers. Section 6 measures this gap against a real fire.

Spread across the Brindabellas

Operational models do not calculate the physics. They use relations fitted to observed fires. This section runs McArthur’s Mk5 forest model over real terrain west of Canberra. Click the map to light a fire. Then set the weather and press Play.

0 h 00 min

Ignite at x y

How the map model works

The fire reaches each cell along the quickest path from an ignition. Each step takes its distance divided by the local rate of spread. That rate starts from McArthur’s Mk5 rate for the head fire, R = 0.0012 · FFDI · w. A slope correction doubles it for each 10° uphill. An ellipse around the wind direction sets the rate in the other directions.

The ratio of length to breadth of the ellipse comes from Equation 79 of the Canadian Fire Behavior Prediction System. That equation is fitted to Alexander’s (1985) observations of the shapes of forest fires. CSIRO’s Spark framework uses the same source. At 46 km/h it gives 5.7. Maps of the Kilmore East fire show about 5 in that wind.

Ratio of length to breadth against wind speedPublished relation and a simple linear rule for comparison

Checked against a real fire

CSIRO researchers rebuilt the Black Saturday Kilmore East fire of 7 February 2009 hour by hour (Cruz et al. 2012). They published the weather, fuel loads and measured spread rates. This makes the fire a hard test for any model.

Forward rate of spread against 10 m wind, dry sclerophyll forestKilmore East weather and fuel. Points show the observed burning periods.

What this model leaves out

No ember spotting, which carried the Kilmore East fire 30 km or more ahead of its front.
No coupling between the fire and the atmosphere. No winds driven by the smoke plume, and no fire channelling in rugged terrain.
The weather stays constant for the whole run. The wind is uniform, and hills do not change it.
Fuel is uniform and continuous. There are no roads, rivers, clearings or different vegetation types.
Mk5 predictions of summer wildfire spread are often 2 to 3 times too low (Burrows 1999; McCaw et al. 2008). CSIRO now recommends Vesta Mk 2 for dry eucalypt forest.
The physics sections treat the flame as a flat panel at one temperature. Real flames flicker, vary in temperature and absorb their own radiation.
The terrain is a 62 m grid from AWS Terrain Tiles, based on SRTM data. It is not Geoscience Australia’s ELVIS data.

This is a teaching tool. It is not validated for operational use. Do not use it for decisions about evacuation, planning or firefighting.

Sources

Cruz M.G. et al. (2012) Anatomy of a catastrophic wildfire: the Black Saturday Kilmore East fire. For. Ecol. Manage. 284, 269 to 285.
Noble I.R., Bary G.A.V., Gill A.M. (1980) McArthur’s fire-danger meters expressed as equations. Aust. J. Ecol. 5, 201 to 203.
Forestry Canada Fire Danger Group (1992) Development and structure of the Canadian Forest Fire Behavior Prediction System, ST-X-3 (Eq. 79); Alexander M.E. (1985) Estimating the length-to-breadth ratio of elliptical forest fire patterns.
Cruz M.G., Alexander M.E. (2019) The 10% wind speed rule of thumb for estimating a wildfire’s forward rate of spread. Ann. For. Sci. 76, 44.
Finney M.A. et al. (2015) Role of buoyant flame dynamics in wildfire spread. PNAS 112, 9833 to 9838.
Várhegyi G., Jakab E., Antal M.J. (1994) Is the Broido-Shafizadeh model for cellulose pyrolysis true? Energy & Fuels 8, 1345 to 1352.
Byram G.M. (1959) Combustion of forest fuels. In Forest Fire: Control and Use.
Frandsen W.H. (1971) Fire spread through porous fuels from the conservation of energy. Combust. Flame 16, 9 to 16.
Wyman C., Sloan P.-P., Shirley P. (2013) Simple analytic approximations to the CIE XYZ colour matching functions. JCGT 2(2).