Two Airways in Parallel

When air can choose between two openings it takes both, dividing itself so that the pressure drop across each branch comes out identical. That single condition — equal pressure, shared quantity — plus the square law is enough to derive everything on this page.

Set $p = R_1Q_1^{2} = R_2Q_2^{2} = R_{eq}(Q_1+Q_2)^{2}$. Solve each branch for its quantity, $Q_i = \sqrt{p/R_i}$, add them, and the pressure cancels: $1/\sqrt{R_{eq}} = 1/\sqrt{R_1} + 1/\sqrt{R_2}$. The reciprocals are of the SQUARE ROOTS, and this is the trap. Resistors in parallel combine as $1/R_{eq} = 1/R_1 + 1/R_2$ because Ohm's law is linear. Airways are quadratic. Reaching for the electrical formula out of habit gives an equivalent resistance far too small and a fan far too small with it, and the mistake survives review because the formula looks right.

The headline consequence: two identical airways in parallel have one quarter the resistance of either, not one half. Put $R_1 = R_2 = R$ into the relation and $R_{eq} = R/4$. This is why a second return, a second decline, or a second egress raise is so much more valuable to ventilation than it appears on a plan. The second opening does not merely share the load — at constant total flow it drops the pressure the fan must raise by a factor of four, and the fan power with it.

The other consequence is that you do not choose where the air goes. The split is fixed by the resistances alone: branch 1 takes a fraction $(1/\sqrt{R_1})/(1/\sqrt{R_1}+1/\sqrt{R_2})$ of the total, and no amount of wanting changes it. This is what mine ventilation calls "natural splitting". If the division is wrong — too much air down the easy drift and not enough up the working stope — the only remedies are to change a resistance, by setting a regulator in the easy branch or by opening the difficult one up, or to add pressure to one branch with a booster fan. Controlled splitting is regulators; natural splitting is what you get if you do nothing.

A word of caution on boosters. Putting a fan inside one branch of a parallel pair changes the pressure balance, and if it is pushed too hard it can reverse the flow in the neighbouring branch — which in a mine means pulling contaminated return air into an intake. Booster fans are regulated for this reason in most jurisdictions, and a network model rather than a hand calculation is the right tool once one is in the circuit.