Atkinson's Equation for Airway Pressure Drop

John Atkinson was a mining engineer in the north of England, and in 1854 he published what remains the working equation of mine ventilation: $p = kOLQ^{2}/A^{3}$. It is Darcy-Weisbach in disguise. A mine airway is not a pipe and has no diameter worth the name — it is an irregular hole in rock, described by the area you can get air through and the perimeter it rubs against — so Atkinson wrote the friction term in those two quantities instead, and buried the friction factor, the air density and a fistful of constants into a single coefficient $k$.

The two exponents are the whole economics of the subject. Airflow is squared, so twice the air costs four times the pressure, and since power is pressure times flow, eight times the fan power. Area is cubed, so an airway driven 10 % smaller in section carries 37 % more resistance for as long as it exists. Those two facts, taken together, say something that mine planners re-learn every generation: bigger openings are almost always cheaper than bigger fans. A shaft or a decline is excavated once and ventilated for thirty years, and the electricity saved by a generous section is compounded over every hour of that.

The perimeter term is easy to skip past and worth pausing on. It is the rubbing surface per metre of airway, and it is why SHAPE matters independently of size. For a fixed area, a circle has the least perimeter of any shape, an arch or horseshoe section is close behind, and a wide flat entry has appreciably more. A bored raise is a low-resistance airway for two separate reasons — a good shape and a smooth wall — and that is why the friction factor and the perimeter tend to move together in practice.

Now the honest part, and it is $k$. Atkinson's friction factor is tabulated by airway LINING and CONDITION, and the published values span roughly 0.0037 kg/m³ for smooth concrete up to 0.02 kg/m³ and beyond for rough-blasted rock, timbered sets, or an airway strung with pipes, cables, vent bag and a conveyor. That is a factor of five or more, and pressure drop is directly proportional to it. A friction factor borrowed from a different lining is the single commonest way to get a mine ventilation answer that is wrong by a factor of several and looks entirely reasonable. This site takes $k$ as an input and always will, because the number belongs to the airway in front of you and not to a category called "rock".

Note also that $k$ is genuinely DIMENSIONAL — kilograms per cubic metre — because Atkinson's equation is a real physical relation rather than a fitted correlation. It carries the air density inside it, normalised to about 1.2 kg/m³, which is why deep, hot or high-altitude workings need a density correction that the equation does not give you.

Finally, what the equation does not contain. This is FRICTIONAL loss only. Shock losses at bends, splits, junctions, doors, regulators and sudden changes of section are additional, and in a short crooked circuit they can exceed the friction entirely. The trade handles them by adding an "equivalent length" to the airway, which puts them honestly back into $L$. And the best value of $k$ is never a table value: it is one you measured yourself, in your own mine, with a gauge and an anemometer over a straight length of airway, because that number knows about the muck pile and the pipe run and no table does.