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Pipe friction losses and the Moody chart, explained

In Hebrew: הפסדי עומד בצינורות

Friction at a pipe wall costs pressure. The Darcy–Weisbach equation gives that loss from the pipe’s length, diameter and flow speed, and the Moody chart gives the friction factor in it, from the Reynolds number and the wall roughness.

Darcy–Weisbach equation calculator

Pre-filled with the worked example below. Change any value.

Reynolds number
100,000
Darcy friction factor
0.02161
Head loss
8.813m
Pressure drop
86.28kPa

Turbulent: Haaland.

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Darcy–Weisbach: where the loss goes

Fluid moving through a pipe loses mechanical energy to friction at the wall. Darcy–Weisbach writes that loss as a head h_f, a height of the fluid: h_f = f (L/D) V²/2g. Multiply by ρg and it becomes a pressure drop.

The loss grows in proportion to length and with the square of speed. Push twice the flow through the same pipe and the loss roughly quadruples. Halve the diameter at the same flow rate and the speed goes up four times, so the loss climbs far faster than intuition expects. That is why pipe sizing is mostly a decision about pumping cost.

The friction factor and the Moody chart

The friction factor f is dimensionless and depends on two numbers: the Reynolds number Re = VD/ν and the relative roughness ε/D. In laminar flow, below about Re = 2,300, it is exactly 64/Re and roughness does not matter.

In turbulent flow it comes from the Colebrook equation, which is implicit in f and has to be solved by iteration. The Moody chart is that equation drawn as a family of curves, one per relative roughness. The Haaland equation is an explicit fit to Colebrook, close enough for design, and it is what the calculator on this page uses.

At very high Reynolds numbers in a rough pipe the curves go flat: f stops depending on Re and is set by roughness alone. This is called fully rough flow.

Minor losses

Bends, valves, tees, entrances and exits each cost a loss of their own, written as h_m = K V²/2g with a loss coefficient K from tables. In a long straight line they are small. In a short system full of fittings they can be larger than the pipe friction itself, which is why “minor” is a misleading name.

A worked number

Water with ν ≈ 1.0 × 10⁻⁶ m²/s flows at 2 m/s through 100 m of 50 mm commercial steel pipe, roughness ε ≈ 0.045 mm. That is about 3.9 litres a second. Re = 2 × 0.05 / 10⁻⁶ = 100,000, so the flow is turbulent, and ε/D = 0.0009.

Haaland gives f ≈ 0.0216. The head loss is 0.0216 × (100 / 0.05) × 2² / (2 × 9.81) ≈ 8.8 m of water, a pressure drop of about 86 kPa. The pump has to supply that before it lifts the water a single metre.

Common mistakes

  • Mixing units: the diameter in millimetres and the length in metres, or the radius where the formula wants the diameter.
  • Using the Fanning friction factor, which is a quarter of the Darcy factor, in the Darcy–Weisbach equation.
  • Using 64/Re in turbulent flow, or reading the absolute roughness ε off a table and entering it as ε/D.

Mechanical engineering, explained