Airway Resistance from Geometry

Atkinson's equation splits cleanly into two halves. One half depends only on the opening — its lining, its rubbing surface, its length and its area — and the other depends only on the airflow. $R = kOL/A^{3}$ is the first half, and it is what a ventilation network model stores for every branch in the mine. Feed it a quantity and the square law gives the pressure; that is all a network solver is doing, thousands of times, until the flows and pressures around every loop agree.

The unit is the awkward part. Resistance has dimensions of pressure divided by flow squared, which comes out as kg/m⁷ and is written in mine ventilation as N·s²/m⁸, sometimes called the atkinson. This site has no unit type for it, so $R$ is carried as a bare number and it is a bare number in SI. If you are reading a North American ventilation report that quotes resistance in inches water gauge per thousand cfm squared — the "practical unit" — that figure is about 5.98 times the SI one, and putting it into an SI field unconverted will make every airway look six times worse than it is.

The cube on the area is the design lesson, and it deserves to be felt rather than read. Take an airway and shrink its section by 10 % to save development cost. Resistance rises by $1/0.9^{3} = 1.37$, so 37 % more. Shrink it by 20 % and resistance nearly doubles. Because fan power goes as $RQ^{3}$, that saving in excavation is paid back in electricity every hour for the life of the opening, and no fan upgrade ever gets it back — the opening is in the ground. The reverse is the single best move available in ventilation planning, and it is available only once, at the design stage.

Resistance is also the number that makes the two combination laws intelligible. Two airways in series carry the same quantity, so their resistances add. Two in parallel share the same pressure, and the square law then forces a reciprocal-square-root combination that is nothing like the electrical formula. Keeping resistance as an explicit quantity, rather than always working in pressures and flows, is what makes a ventilation network tractable by hand at all.

One practical note on measurement. A resistance measured in the field always comes out higher than one computed from a design section, and the difference is real: the muck on the floor, the pipe run along the rib, the cables, the vent bag, the conveyor and the partly closed regulator are all in the airway and none of them is in the drawing. Survey annually. Airways fill up with services, and resistance only ever goes one way.