Byram's Flame Length from Fireline Intensity

Fireline intensity is the heat a fire releases per metre of its front, in kilowatts per metre. Flame length is how long the flames are, and this relation turns the first into the second: $L = 0.0775\,I^{0.46}$, with $L$ in metres and $I$ in kW/m. The exponent is below one, so the flames lengthen more slowly than the intensity grows: doubling the intensity lengthens the flames by 38 percent. Run in reverse, the exponent becomes $1/0.46$, so a small misjudgement of the flame length turns into a larger error in the intensity.

The length meant is the one measured along the flame, not its height, and the equation is for the flaming front of a surface fire in the open. In a crown fire or in strong wind the flames tilt and the length is longer than this gives. Where the intensity itself comes from, the fuel burned, its heat yield and the rate of spread, is the job of the library's fireline-intensity page; this page starts from the intensity.

The equation is Byram's. The United States Forest Service's RMRS-GTR-371 (Andrews, 2018) prints it in feet and Btu per foot per second as $F = 0.45\,I^{0.46}$, citing Byram's equation as given by Brown and Davis (1973), and in its metric table A.3 as $0.0775\,I^{0.46}$ after Wilson (1980). The two forms agree, which is a check on both. At 100 Btu per foot per second (346 kW/m) the foot form gives 3.74 ft, and the metric form gives 1.14 m, which is the same 3.74 ft. At 1 000 Btu per foot per second the foot form gives 10.795 ft against the page's 10.798.

The page's example is a fire of 9 000 kW/m, which has flames 5.1 m long; the page gives 5.11 m, and entered backwards, 5.10802785 m of flame returns 9 000 kW/m.