The Airway Square Law

Everything expensive about mine ventilation follows from one exponent. Pressure drop rises with the SQUARE of the airflow, $p = RQ^{2}$, and air power is pressure times flow, so power rises with the CUBE. Double the air and you pay four times the pressure and eight times the power. Halve the air and you keep an eighth of the bill.

That cube is the entire business case for ventilation on demand — the practice of routing air to the places people and equipment actually are, rather than ventilating the whole mine at the peak rate all the time. A system that cuts average airflow by 30 % cuts fan power by 66 %, and on a mine where ventilation is a quarter to a half of the total electrical load, that is not a rounding error. It is also why nobody with any experience answers a ventilation shortage by turning the fan up: the square root running the other way means a 21 % increase in pressure buys 10 % more air, and doubling the pressure buys only 41 %.

Plot $p$ against $Q$ and you have drawn the mine characteristic. It is a parabola through the origin whose steepness is $R$. A fan characteristic falls as flow rises. The two curves cross at exactly one point, and that crossing is the operating point of the mine — not what the fan is rated at, not what the plan says, but where those two curves meet. Every action a ventilation engineer takes moves one of them. Driving a new raise or opening a door lowers $R$, flattens the parabola, and slides the crossing to the right: more air, at less pressure. Setting a regulator raises $R$ and does the opposite. Changing the fan speed or the blade pitch moves the other curve.

The square law also explains why leakage is so stubborn. A door, a stopping or a bulkhead with a hole in it is an airway with a resistance of its own, in parallel with the circuit it is meant to seal, and its leakage follows the same square root of pressure. That means leakage is remarkably insensitive to fan pressure and extremely sensitive to the size of the hole. Sealing well is worth far more than pushing harder, and in an old mine with many stoppings the accumulated leakage can exceed the air reaching the faces.

Finally, a word on where the square law comes from and where it stops. It is the fully-turbulent branch of the friction relation, in which the friction factor no longer depends on Reynolds number — which is true of essentially every mine airway, at every velocity anyone works at. In the laminar regime the exponent would be 1 rather than 2. You will never meet that in an airway. You will meet it in the pore space of a coal pillar or a gob, and that is exactly where mine ventilation stops using this equation and starts using Darcy's law instead.