The Darcy–Weisbach equation gives the pressure lost to friction when fluid flows through a straight pipe: Δp = f·(L/D)·ρV²/2. Everything about the flow regime and the pipe wall is packed into the Darcy friction factor f, which is 64/Re for laminar flow and comes from the Colebrook equation or the Moody chart for turbulent flow.
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The equation
Friction loss grows with pipe length L, shrinks with diameter D, and scales with the dynamic pressure ρV²/2 of the flow. The Darcy friction factor f is the constant of proportionality: Δp = f·(L/D)·ρV²/2.
Dividing by ρg gives the head loss in metres of the flowing fluid, h_f = f·(L/D)·V²/(2g). Head is convenient for pumps, which are rated in head, and it does not depend on the fluid’s density.
Find the regime first with the Reynolds number. It decides which friction factor formula applies.
The friction factor
For laminar flow, f = 64/Re exactly, and the result is the Hagen–Poiseuille law. Wall roughness does not matter.
For turbulent flow, f depends on Re and the relative roughness ε/D. The Colebrook equation, 1/√f = −2·log10(ε/(3.7D) + 2.51/(Re·√f)), is implicit in f, so it is solved by iteration. The Moody chart is a plot of the same equation. Explicit fits such as Swamee–Jain give a close answer in one step and are a good starting guess.
The Darcy factor is four times the Fanning friction factor used in some chemical engineering texts. Check which one a chart or formula uses before you mix sources.
What the equation leaves out
Darcy–Weisbach covers straight, full pipe only. Valves, elbows, entrances and exits add minor losses, usually written as K·ρV²/2 per fitting, and elevation change adds ρgΔz. A pump must supply the sum. For compressible gases with a large pressure drop, density changes along the pipe and the line has to be split into short segments.
Key formulas
Formulas used on this page, with the textbook each is cited from
Formula
Expression
Where
Cited from
Darcy–Weisbach equation
Δp = f·(L/D)·ρ·V²/2
f = Darcy friction factor
White, Fluid Mechanics
Head loss
h_f = Δp / (ρ·g) = f·(L/D)·V²/(2g)
g = 9.81 m/s²
Munson et al., Fundamentals of Fluid Mechanics
Laminar friction factor
f = 64 / Re
Re < about 2300
White, Fluid Mechanics
Colebrook equation
1/√f = −2·log10(ε/(3.7·D) + 2.51/(Re·√f))
turbulent flow; solve by iteration
Colebrook (1939), Journal of the Institution of Civil Engineers
Swamee–Jain approximation
f = 0.25 / [log10(ε/(3.7·D) + 5.74/Re^0.9)]²
explicit fit to Colebrook
Swamee & Jain (1976), Journal of the Hydraulics Division, ASCE
Worked examples
Example: Laminar pressure drop in a hydraulic line
Given
Oil flows at 6 L/min through 5 m of line with a 12 mm bore.
Take ρ = 870 kg/m³ and μ = 0.040 Pa·s, typical values for a mid-grade hydraulic oil at 40 °C.
Find
The Reynolds number and friction factor.
The pressure drop and head loss.
Solution
Velocity, from Q = 1.0 × 10⁻⁴ m³/s and A = π × 0.012²/4.V = 1.0 × 10⁻⁴ / 1.131 × 10⁻⁴ = 0.884 m/s
Reynolds number. It is far below 2300, so the flow is laminar.Re = 870 × 0.884 × 0.012 / 0.040 = 231
Check with Hagen–Poiseuille, which should agree exactly for laminar flow.Δp = 32·μ·L·V / D² = 32 × 0.040 × 5 × 0.884 / 0.012² = 39.3 kPa
Head loss in metres of oil.h_f = 39 300 / (870 × 9.81) = 4.61 m
Answer: Re ≈ 231 (laminar), f ≈ 0.277, Δp ≈ 39.3 kPa and h_f ≈ 4.6 m of oil over the 5 m line.
Example: Turbulent head loss in a cooling-water line
Given
Water flows at 8 L/s through 120 m of steel pipe with an 80 mm bore.
Take ρ = 998 kg/m³ and μ = 1.0 × 10⁻³ Pa·s (typical for water near 20 °C) and ε = 0.045 mm, a value commonly used for new commercial steel.
Find
The friction factor.
The pressure drop and head loss.
How much the loss rises if the flow doubles.
Solution
Velocity and Reynolds number. Re is far above 4000, so the flow is turbulent.V = 0.008 / (π × 0.08²/4) = 1.59 m/s, Re = 998 × 1.59 × 0.08 / 0.001 = 127 000
Relative roughness, and a first estimate from Swamee–Jain.ε/D = 0.045 / 80 = 0.000563, f₀ = 0.25 / [log10(0.000563/3.7 + 5.74/127 000^0.9)]² = 0.0201
Iterate Colebrook from f₀. It settles in two or three passes.1/√f = −2·log10(0.000563/3.7 + 2.51/(127 000·√f)) → f = 0.0200
At 16 L/s, V doubles and Re = 254 000. Colebrook gives f = 0.0188, so the loss rises by less than four times.Δp = 0.0188 × 1500 × 998 × 3.18² / 2 = 142 kPa, a ratio of 3.75
Answer: f ≈ 0.0200, Δp ≈ 37.9 kPa and h_f ≈ 3.9 m. Doubling the flow raises the loss about 3.75 times, slightly less than V² because f falls as Re rises.
Common mistakes
Mixing Darcy and Fanning friction factors. Darcy f = 4 × Fanning f. Using a Fanning value in Darcy–Weisbach underestimates the loss by a factor of four.
Using roughness with the wrong units. ε/D must be dimensionless. Put ε and D in the same unit, usually millimetres.
Applying 64/Re to turbulent flow. Check Re first. 64/Re only holds below about 2300.
Forgetting minor losses. In short plant runs, fittings and valves can lose as much as the straight pipe. Add them separately.
Confusing pressure drop with head. h_f is in metres of the flowing fluid. A metre of oil is a smaller pressure than a metre of water.
In industry
Frequently asked questions
What is the difference between Darcy–Weisbach and Hazen–Williams?
Hazen–Williams is an empirical formula for water at ordinary temperatures. Darcy–Weisbach is dimensionally consistent and works for any Newtonian fluid, regime and pipe, which is why most fluid mechanics texts prefer it.
Why does Colebrook need iteration?
f appears on both sides of the equation, inside a logarithm, so there is no closed-form solution. A few rounds of substitution converge quickly, and explicit fits like Swamee–Jain give a close first guess.
Does roughness matter in laminar flow?
No. In laminar flow f = 64/Re regardless of the wall. Roughness starts to matter as the flow becomes turbulent and, at high Re, dominates.
Is there a calculator for this?
Yes. Coplain’s pipe pressure drop calculator, linked below, solves Colebrook and shows each step in SI or US units.
B. R. Munson, T. H. Okiishi, W. W. Huebsch, A. P. Rothmayer, Fundamentals of Fluid Mechanics, Wiley
C. F. Colebrook, Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws (1939), Journal of the Institution of Civil Engineers
P. K. Swamee, A. K. Jain, Explicit equations for pipe-flow problems (1976), Journal of the Hydraulics Division, ASCE
L. F. Moody, Friction factors for pipe flow (1944), Transactions of the ASME
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