Point Source Radiant Heat Flux

The crudest radiation model in fire protection, and usually the first one anyone reaches for: pretend the fire is a point, pretend it radiates a fixed fraction of its heat release rate equally in all directions, and the flux at a target falls off as the inverse square of the distance. There is no fitted constant — the $4\pi R^{2}$ is the surface area of a sphere and nothing else. The equation is dimensionally honest. The model is not.

A fire is not a point. The point-source approximation is generally accepted beyond about twice the flame height or twice the fire diameter, whichever is larger. Closer than that it UNDERSTATES the flux, and badly, because a target near a tall flame sees a large hot surface filling much of its field of view rather than a distant speck. That error runs in the dangerous direction: the model is optimistic exactly where being optimistic matters. Close-in work needs a configuration-factor or solid-flame model, where the flame is treated as a cylinder of known emissive power and the view factor from target to flame is computed properly.

The radiative fraction is the weak input. $\chi_r$ runs from about 0.15 for clean-burning fuels like methanol to 0.6 for heavily sooting ones, and 0.3 is the usual default when nothing better is known. Soot is what radiates: a luminous yellow flame is losing heat by radiation and a clean blue one is not. It also FALLS for large fires, because the smoke shrouding a big flame absorbs radiation before it escapes — so a $\chi_r$ measured on a small pool overstates the flux from a large one. If you fit $\chi_r$ from a measurement and get a number above 0.6, the likeliest explanation is not an extraordinary fuel; it is that the radiometer was too close for a point-source model.

Two further omissions. The model neglects atmospheric attenuation, which is real over tens of metres in humid air and which makes the answer conservative. And it gives the flux incident on a surface FACING the fire squarely; a surface at an angle sees less, by the cosine of the angle between its normal and the line to the fire.

The benchmark flux figures are worth knowing and worth being careful with. Around 1 kW/m² is roughly summer sunshine. About 2.5 kW/m² is the commonly cited limit for prolonged firefighter exposure in full protective clothing. About 4 kW/m² causes pain on bare skin within a few seconds and blistering shortly after, and is often used as a tenability limit on an escape route. Around 12.5 kW/m² will melt plastics and pilot-ignite many materials given time. Around 20 kW/m² is the piloted ignition threshold commonly quoted for wood, and it is roughly the flux a flashover-level layer puts on the floor. None of these is a code requirement. They come from a range of sources with a range of exposure durations behind them, and what an authority having jurisdiction actually requires is the thing to check.

Separation distances between buildings, around tank farms, and for apparatus positioning are governed by codes and standards that have already settled the question. This equation is a way of understanding why those requirements look the way they do. It is not a way of replacing them.