Pipe Friction and the Moody Diagram
Darcy–Weisbach, the friction factor, roughness, the Colebrook and Swamee–Jain equations, Hazen–Williams and a fully worked pressure drop.
The Darcy–Weisbach equation
Fluid flowing in a pipe rubs on the wall, and the pressure falls along the pipe to pay for it. The pressure drop for any fluid, any pipe size and any regime comes from one equation.
ΔP = f × (L ÷ D) × (ρ v² ÷ 2) hf = f (L ÷ D) v² ÷ (2 g)
The loss grows with length, with the square of the velocity (so about with the square of the flow) and falls with a bigger diameter: because v = Q ÷ (π D² ÷ 4), the loss at a fixed flow goes roughly as 1 ÷ D⁵. Doubling the pipe diameter cuts the friction loss by a factor of about 30.
The friction factor
The friction factor f depends on the Reynolds number Re and the relative roughness ε ÷ D (the height of the wall roughness over the diameter).
- Laminar flow (Re < 2 300): f = 64 ÷ Re. Roughness does not matter and the loss is proportional to the velocity, not its square.
- Turbulent flow: the Colebrook–White equation, 1 ÷ √f = −2 log₁₀( (ε ÷ D) ÷ 3.7 + 2.51 ÷ (Re √f) ), which has f on both sides and is solved by iteration (a calculator or a few passes in a spreadsheet).
- Swamee–Jain, an explicit fit good to about 1 to 2 percent over the working range: f = 0.25 ÷ [log₁₀( (ε ÷ D) ÷ 3.7 + 5.74 ÷ Re⁰·⁹ )]².
- Fully rough flow (high Re): f depends only on ε ÷ D, f ≈ [−2 log₁₀( (ε ÷ D) ÷ 3.7 )]⁻², and the loss goes exactly as v².
Do not confuse the Darcy factor with the Fanning friction factor used in some chemical-engineering texts: fDarcy = 4 fFanning. Every equation on this site uses the Darcy factor.
Roughness
| Pipe material | Absolute roughness ε (mm) |
|---|---|
| Drawn tubing, copper, brass, glass, PVC, PE | 0.0015 |
| Commercial steel, welded or seamless (new) | 0.045 |
| Galvanized iron | 0.15 |
| Cast iron (new) | 0.26 |
| Ductile iron, cement lined | 0.12 |
| Concrete | 0.3 to 3 |
| Riveted steel | 0.9 to 9 |
| Old, corroded or scaled steel | 0.5 to 3 (grows with age) |
Roughness is the biggest uncertainty in a friction calculation. Pipes get rougher with age: for the worked example below, scale that raises ε from 0.045 mm to 0.5 mm raises the pressure drop from 55.1 to 93.8 kPa, a 70 percent increase.
The Moody diagram
The Moody diagram plots the friction factor against Reynolds number for lines of constant relative roughness. It is the Colebrook equation drawn, and gives a quick check on any calculation.
Reading it: on the left is the straight laminar line; the shaded band is the unstable critical zone, which should be avoided in design; to the right each roughness curve falls with rising Re and then flattens into the fully rough zone. Smooth pipes follow the lowest curve.
A complete worked example
Worked example: pressure drop in 100 m of DN80 steel pipe
- Given: water at 20 °C (ρ = 998.2 kg/m³, μ = 1.002 mPa·s), Q = 10 L/s, DN80 Schedule 40 (inside diameter 77.92 mm), commercial steel ε = 0.045 mm, L = 100 m.
- 1. Velocity: v = Q ÷ A = 0.010 ÷ 0.00477 = 2.097 m/s.
- 2. Reynolds number: Re = ρ v D ÷ μ = 998.2 × 2.097 × 0.07792 ÷ 0.001002 = 162 784: turbulent.
- 3. Relative roughness: ε ÷ D = 0.045 ÷ 77.92 = 0.00058.
- 4. Friction factor (Colebrook, iterated): f = 0.0196. The explicit Swamee–Jain gives 0.0197, a difference of only 0.62 percent.
- 5. Pressure drop: ΔP = f (L ÷ D) ρ v² ÷ 2 = 0.0196 × (100 ÷ 0.07792) × (998.2 × 4.398 ÷ 2) = 55.1 kPa, which is 5.63 m of water (551 Pa for every meter of pipe).
| Flow | Velocity | Re | f | ΔP over 100 m |
|---|---|---|---|---|
| 5 L/s | 1.05 m/s | 81 392 | 0.0212 | 14.9 kPa |
| 10 L/s | 2.10 m/s | 162 784 | 0.0196 | 55.1 kPa |
| 15 L/s | 3.15 m/s | 244 176 | 0.0189 | 119.8 kPa |
| 20 L/s | 4.19 m/s | 325 567 | 0.0185 | 208.9 kPa |
Quadrupling the flow raises the pressure drop by a factor of 14.0, so ΔP ∝ Q1.90: slightly less than the square, because f falls a little as Re rises. A smaller pipe makes a big difference: the same 10 L/s in DN65 (inside diameter 62.68 mm) runs at 3.24 m/s with ΔP = 166 kPa, 3.0 times as much. The Pipe Flow and Pressure Drop Calculator does these steps for any fluid and pipe, with fittings and a pressure-against-flow chart.
Hazen–Williams for water
Water-supply and fire-protection engineers often use the empirical Hazen–Williams equation, which needs no viscosity or Reynolds number but only works for water at ordinary temperatures in turbulent flow:
hf = 10.67 L Q1.852 ÷ (C1.852 D4.87)
For the worked example with C = 130: hf = 6.41 m, ΔP = 62.7 kPa, against 55.1 kPa from Darcy–Weisbach: within 14 percent, which is typical. Use Darcy–Weisbach for anything but plain cold water.
Laminar flow: oils and small bores
Worked example: viscous oil in the same pipe
- ISO VG 46 oil (ν = 46 mm²/s, ρ = 870 kg/m³) at 0.5 m/s in the DN80 pipe: Re = 0.5 × 0.07792 ÷ 46 × 10⁻⁶ = 847: laminar.
- f = 64 ÷ Re = 0.0756, so ΔP = 0.0756 × (100 ÷ 0.07792) × (870 × 0.25 ÷ 2) = 10.5 kPa.
- In laminar flow ΔP is proportional to v (Hagen–Poiseuille: ΔP = 32 μ L v ÷ D²), so doubling the flow only doubles the loss. Cooling the oil raises ν steeply and the loss with it.
Related
- Previous: Continuity, Bernoulli and the Energy Equation. Next: Fittings, Valves and Minor Losses for valves and bends.
- Pipe Flow and Pressure Drop Calculator.
- Tolerances by Process and Surface Finish: the roughness side of surface finish.
- Fluid Mechanics and Pipe Flow: the whole guide.
FAQ
Why do I need an iterative equation for f?
Because the friction factor appears on both sides of the Colebrook equation. Swamee–Jain or Haaland give explicit answers within 1 to 2 percent, which is enough for engineering; the calculator on this site iterates exactly.
How rough is a real pipe?
Use the table for new pipe and add margin for age: scale, corrosion and deposits can increase roughness tenfold in water systems. For critical designs, calculate with both new and aged roughness.
Is the pressure drop the same for any liquid?
The head loss hf does depend on the liquid only through f (which depends on Re), but the pressure drop ΔP is proportional to the density, so a denser liquid loses more pressure for the same head.
Does elevation matter?
Yes, separately: add (or subtract) ρ g Δz for rise (or fall) of the pipe. Friction depends on the length of the pipe, not its height.
References
- Colebrook, C. F., “Turbulent flow in pipes”, Journal of the Institution of Civil Engineers, 1939; Moody, L. F., “Friction factors for pipe flow”, Trans. ASME, 1944.
- Swamee, P. K. and Jain, A. K., “Explicit equations for pipe-flow problems”, Journal of the Hydraulics Division, ASCE, 1976.
- White, F. M., Fluid Mechanics, McGraw-Hill; Crane Technical Paper 410.
- Williams, G. S. and Hazen, A., Hydraulic Tables (1920) for the Hazen–Williams coefficients.
Last reviewed: October 2026. Roughness values are typical.