Two Airways in Series

Air that must pass through one opening and then the next carries the same quantity through both. The pressure drops therefore add, and since each is $R_iQ^{2}$ with the same $Q$, the resistances add too: $R_{eq} = R_1 + R_2$. It is the simplest relation in ventilation network analysis, and it is worth stating because the parallel case is so different.

What the sum is FOR is finding out where to spend money. In a series run the pressure drops divide in proportion to the resistances, so if one branch carries 80 % of the resistance it carries 80 % of the loss, and halving the resistance of the other one buys you almost nothing. Enlarging the easy airway in a series pair is close to wasted work. The whole art of improving an existing circuit is finding the worst branch first, and this arithmetic is how you find it.

Series and parallel are not alike and the difference always runs the same way. Two airways of 0.08 and 0.15 N·s²/m⁸ in series give 0.23. The same two side by side in parallel give about 0.033 — roughly a seventh. Confusing the two in a hand calculation is a common error and it always errs in the direction of a circuit that looks easier than it is, which means a fan that turns out too small.

The honest limitation of the simple sum is that it is FRICTION only. The bend, the junction or the step change in section between the two airways has a shock loss of its own that belongs in neither branch. In a long straight drift those losses are negligible; in a short crooked circuit with several right-angle junctions they can rival the friction entirely. The trade handles this by adding an equivalent length to one airway — a right-angle bend might be worth twenty or thirty metres of straight drift — which puts the shock loss back into the sum honestly rather than ignoring it.

One more thing a series sum quietly assumes: that the same quantity really does pass through both. In a real mine it usually does not, because leakage through stoppings, doors and old workings bleeds air out along the way. A circuit modelled as two airways in series and measured as carrying 40 m³/s at the start and 28 m³/s at the end is not a series pair at all — it is a series pair with a large parallel leakage path, and the honest model has three branches in it.