The pressure lost along an air pipe rises with the flow to the power 1.75 and falls with the inside diameter to the power 4.75, so a small increase in bore cuts the loss sharply. At 235 L/min of free air over 20 m, a smooth 8 mm bore loses about 0.253 bar, a 12 mm bore about 0.037 bar and a 20 mm bore about 0.0033 bar.
Part of Pneumatic Systems
Choosing the bore of an air line by computing the pressure it loses at the flow it carries, instead of by a rule of thumb.
The Method: Darcy-Weisbach With a Smooth-Pipe Friction Factor
Pressure drop along a pipe is
Δp = f x (L / D) x (ρ x v² / 2)
with f the friction factor, L the length, D the inside diameter, ρ the air density at line pressure and v the velocity. For smooth pipe and Reynolds numbers below about 10⁵, the Blasius formula gives f = 0.316 x Re^-0.25, where Re = ρ x v x D / μ. These formulas, and air’s viscosity of 1.8 x 10⁻⁵ Pa·s, are recalled from fluid-mechanics texts and not from this corpus; the same Darcy-Weisbach structure appears in the corpus’s water pipe pages (Friction Losses).
Worked Example: 235 L/min Through 20 m of 12 mm Pipe
Supply at 7 bar gauge (8 bar absolute) and 20 °C. Free-air density 1.204 kg/m³.
- Mass flow = 1.204 x (0.235 / 60) = 4.716 x 10⁻³ kg/s
- Air density at 8 bar absolute = 8 x 101,325 / (287.05 x 293.15) = 9.63 kg/m³
- Pipe area = π / 4 x 0.012² = 1.131 x 10⁻⁴ m²; velocity = 4.716 x 10⁻³ / (9.63 x 1.131 x 10⁻⁴) = 4.33 m/s
- Re = 9.63 x 4.33 x 0.012 / 1.8 x 10⁻⁵ = 27,800
- f = 0.316 x 27,800^-0.25 = 0.0245
- Δp = 0.0245 x (20 / 0.012) x (9.63 x 4.33² / 2) = 0.0245 x 1,667 x 90.3 = 3,690 Pa = 0.037 bar
Results for Three Bores and Three Flows (20 m, Smooth Pipe)
| Free air | 8 mm inside diameter | 12 mm | 20 mm |
|---|---|---|---|
| 150 L/min | 0.115 bar (6.2 m/s) | 0.017 bar | 0.0015 bar |
| 235 L/min | 0.253 bar (9.7 m/s) | 0.037 bar (4.3 m/s) | 0.0033 bar (1.6 m/s) |
| 500 L/min | 0.947 bar (20.7 m/s) | 0.138 bar | 0.012 bar |
The Scaling Law Replaces the Table
Velocity goes as flow over diameter squared, f as (flow / diameter)^-0.25, and the loss as f x velocity² / diameter, so
Δp is proportional to Q^1.75 x D^-4.75 (density held constant)
Checks against the table: going from 8 mm to 12 mm should cut the loss by (12 / 8)^4.75 = 6.9, and 0.253 / 0.037 = 6.9. Going from 12 mm to 20 mm: (20 / 12)^4.75 = 11.3, and 0.037 / 0.0033 = 11. Raising flow from 235 to 500 L/min multiplies the loss by (500 / 235)^1.75 = 3.75, and 0.947 / 0.253 = 3.75. The carrier’s rule, to oversize pipe because lost pressure is expensive, follows directly: a half-again bore cuts the loss seven-fold.
The carrier’s own figures do not reproduce. The carrier lists drops of 0.3 bar for 12 mm, 1.2 bar for 8 mm and 0.05 bar for 20 mm at “15 L/min” over 20 m. At 15 L/min the calculation gives about 0.002 bar for 8 mm, and the three figures are not in the 6.9 and 11 ratios the scaling law requires. Use the table above.
Limits of This Calculation
- The pipe is assumed smooth. Corroded or scaled steel loses more.
- Fittings, elbows, valves and couplers add losses, not included here.
- Density is taken at the supply pressure, a good approximation while the drop is small against the line pressure; at 0.947 bar it is already a rough guide.
- The Blasius formula holds below Re of about 10⁵, which covers all the cases above.
Common Mistakes
- Using nominal pipe sizes as inside diameters. An 8 mm bore is the inside diameter, not the label on a pipe.
- Sizing for average flow. Size for the largest flow that runs at once.
- Adding a tool without rechecking the main. The loss goes as the flow to the 1.75 power.
Safety
Starved tools tempt people to raise the pressure
When a long, narrow line leaves a tool short of pressure, the temptation is to raise the regulator or the relief setting. Do not: the pipe, hose and fittings were chosen against the relief setting, and the weakest part is the limit (Piping and Fittings). Fix the bore instead. A calculated pressure drop is not a pressure rating; each part still needs its own.
What’s Next
- Learn next: Quick-Connect Couplers — the fittings that also add loss
- Combine with: Pipe Sizing for Water — the same idea with a liquid