What the calculation gives you
For a given flow through a pipe of known length, diameter and roughness, the Darcy–Weisbach equation gives the pressure lost to friction. You use it to size a pump, to check whether a line delivers the flow you need, or to find the diameter that keeps the loss within a budget. The same calculation, turned around, tells you how much flow a given pressure can push.
The equation
Δp = f · (L / D) · (ρ v2 / 2)
Δp pressure drop · f Darcy friction factor (dimensionless) · L pipe length · D inside diameter · ρ fluid density · v mean velocity
Expressed as a head of fluid instead of a pressure, which is how pump curves are drawn, the same loss is hf = f (L/D) v2/(2g). The two are linked by Δp = ρ g hf.
The velocity comes from the volume flow and the pipe area:
v = Q / A, with A = π D2 / 4
The friction factor
Everything difficult is in f. It depends on the Reynolds number and, in turbulent flow, on the relative roughness of the pipe wall:
Re = ρ v D / μ
μ dynamic viscosity. With the kinematic viscosity ν = μ/ρ this is Re = v D / ν.
- Laminar flow, Re below about 2300: f = 64 / Re. Roughness does not matter.
- Turbulent flow, Re above about 4000: the Colebrook equation, which is implicit in f:
1 / √f = −2 log10 ( ε / (3.7 D) + 2.51 / (Re √f) )
ε absolute roughness of the wall (about 0.045 mm for commercial steel, 0.0015 mm for drawn copper or plastic)
Because f appears on both sides, Colebrook is solved by iteration: guess f, compute the right-hand side, repeat until it stops changing. This is what the worksheet below does with a Newton–Raphson solver, starting from a guess of 0.01. If you only want a close answer by hand, the Swamee–Jain approximation is explicit and within about one percent:
f ≈ 0.25 / [ log10 ( ε / (3.7 D) + 5.74 / Re0.9 ) ]2
Between Re 2300 and 4000 the flow is in transition and no formula is reliable; design out of that range if you can.
Fittings and valves
Elbows, tees, valves and entries add losses on top of the straight pipe. Each is given a loss coefficient K, and its loss is K ρ v2/2. Add the K values of all fittings in a run to the straight-pipe term f L/D. Typical values: a standard 90° elbow about 0.9, a fully open gate valve about 0.2, a globe valve about 10, a sharp pipe entrance 0.5. In a short run with many fittings these losses dominate the straight pipe.
Worked example
The worksheet below takes 500 gal/min of water at about 100 °F through 30 ft of pipe, with a design pressure drop of 20 psi. It defines the friction factor as a function of the diameter and a guess, solves the Colebrook form for the friction factor with the built-in numeric solver, then evaluates the pressure drop for a 2 in pipe and plots the drop against diameter from 1 to 5 in. The units are mixed on purpose, gallons per minute, slugs per cubic foot, metres per second squared, inches: the sheet carries them through and the result comes out in psi.
Common mistakes
- Nominal size instead of inside diameter. A "2 inch" schedule 40 steel pipe has an inside diameter of 2.067 in; schedule 80 is 1.939 in. The loss goes with the fifth power of D, so a 6% error in diameter is a 30% error in Δp.
- Roughness in the wrong unit. ε and D must be in the same unit before dividing. 0.045 mm against a diameter in inches is wrong by a factor of 25.
- Laminar formula in turbulent flow. Check Re first; most liquid lines above a few millimetres in diameter are turbulent.
- Viscosity at the wrong temperature. Water at 100 °F is about half as viscous as at 50 °F, which moves Re and f.
- Forgetting the fittings. In plant piping the fittings often lose more than the pipe.
- Head and pressure mixed up. Pump curves are in head; losses computed as pressure must be divided by ρg before comparing.
Questions
- Darcy–Weisbach or Hazen–Williams?
- Hazen–Williams is an empirical formula for water near room temperature in turbulent flow, convenient for municipal water networks. Darcy–Weisbach works for any Newtonian fluid, temperature and flow regime, which is why it is the general engineering choice.
- Does the equation work for gases?
- Yes when the pressure drop is small compared with the absolute pressure (under about 10%), so density can be taken as constant. For larger drops the density changes along the pipe and a compressible-flow calculation is needed.
- What velocity should I design for?
- Common targets are 1 to 3 m/s for water in steel pipe; higher velocities raise losses with the square of v and bring noise and erosion. Suction lines of pumps are kept slower.
