Head loss in pipes: Reynolds number and Darcy–Weisbach

Water flowing through a pipe loses energy to friction against the wall. Estimating that head loss sizes every pump and water main.

Reynolds number

Re = ρ V Dμ = V Dν

Re	Flow
below 2000	Laminar
2000 to 4000	Transition
above 4000	Turbulent

Water at 20 °C has ν ≈ 1.0 × 10⁻⁶ m²/s, so almost every water supply pipe runs turbulent.

Darcy–Weisbach equation

h_f = f LD V²2 g

f is the Darcy friction factor (dimensionless). Some books use f' = f/4 with 4f'.

Getting f

• Laminar: f = 64 / Re.
• Turbulent, smooth pipe (Re up to 10⁵): Blasius, f = 0.316 / Re^1/4.
• Turbulent, rough pipe: Colebrook–White, read from the Moody chart, or the explicit Swamee–Jain formula:

f = 0.25[ log₁₀ ( ε/3.7D + 5.74/Re^0.9 ) ]²

where ε is the wall roughness (about 0.26 mm for cast iron, 0.0015 mm for PVC).

Worked example

A 300 mm cast iron main, 1.5 km long, carries 0.1 m³/s of water. Take f = 0.02.

V = 0.1π × 0.3² / 4 = 1.41 m/s

h_f = 0.02 × 15000.3 × 1.41²2 × 9.81 = 10.2 m

Minor losses

Bends, valves and entrances add losses of K × V²/2g. A sudden enlargement loses (V₁ − V₂)²/2g. In long mains these are small; in pump houses full of fittings they can be large.

Work through these steps with a Moody chart in our pipe flow tool.