Fluid Mechanics

Darcy-Weisbach Equation Explained

The Darcy-Weisbach equation in practical terms — the friction factor, the length-to-diameter ratio, the velocity head term, and the Darcy-vs-Fanning trap. Learn when Darcy-Weisbach is the right tool for pipe friction loss.

TypeEngineering guide — concept explainer

Definition

The Darcy-Weisbach equation gives the friction head loss in a length of straight pipe as h_f = f · (L/D) · v²/(2g), or equivalently the pressure drop ΔP = f · (L/D) · ρv²/2. It is the physically general, dimensionally consistent way to calculate pipe friction for any Newtonian fluid in any unit system. The single empirical quantity it needs is the Darcy friction factor f, which captures the effect of the flow regime and the pipe roughness.

Why it matters

Darcy-Weisbach is the workhorse equation for pipe friction in process and mechanical engineering. Unlike the older Hazen-Williams formula (which is tuned to water near room temperature), Darcy-Weisbach works for any liquid or gas, any temperature, and any roughness as long as you supply the right friction factor. Understanding its three factors — the friction factor, the length-to-diameter ratio, and the velocity head — tells you immediately how a design change moves the pressure drop.

Formula

Head loss form

h_f = f × (L/D) × v²/(2g)

Pressure drop form

ΔP = f × (L/D) × ρv²/2

Velocity head

v²/(2g)

Darcy ↔ Fanning

f_Darcy = 4 × f_Fanning

Units involved

•h_f — friction head in m or ft of fluid
•ΔP — pressure drop in Pa, kPa, bar, or psi
•f — Darcy friction factor, dimensionless
•L — pipe length, m or ft
•D — internal diameter, m (consistent with L)
•v — average velocity, m/s
•ρ — density, kg/m³
•g — 9.80665 m/s²

Concept diagram

Worked example

Water (ρ = 998 kg/m³) flows at 2.5 m/s through a 150 mm pipe, 100 m long, with a Darcy friction factor of 0.020. g = 9.80665 m/s².

01L/D = 100 / 0.15 = 666.7
02Velocity head = v²/(2g) = 2.5² / (2 × 9.80665) = 0.3187 m
03h_f = f × (L/D) × v²/(2g) = 0.020 × 666.7 × 0.3187 = 4.25 m
04ΔP = ρ·g·h_f = 998 × 9.80665 × 4.25 ≈ 41,600 Pa ≈ 41.6 kPa

Result

Friction head ≈ 4.25 m; pressure drop ≈ 41.6 kPa for the 100 m run.

Common mistakes

•Mixing up the Darcy and Fanning friction factors — they differ by a factor of 4. The Darcy-Weisbach equation above uses the Darcy factor; if your chart or correlation gives Fanning, multiply by 4 first.
•Forgetting that f itself depends on velocity (through Reynolds number) — you cannot treat f as a fixed constant across the whole operating range.
•Using inconsistent length and diameter units in L/D — both must be in the same unit so the ratio is dimensionless.
•Applying Darcy-Weisbach to fittings — it covers straight-pipe friction only; fittings are added separately as minor losses.
•Assuming Darcy-Weisbach handles compressible gas with large density change — for that you need a compressible-flow treatment, not a single average density.

When to use the calculator

Use the Darcy-Weisbach Pressure Drop calculator when you already have a friction factor and want the friction loss for a straight run. Use the Pipe Pressure Drop calculator when you want the friction factor computed for you from flow and fluid properties, and the Friction Factor calculator when you need f on its own.

Darcy-Weisbach Pressure Drop Calculator →Pipe Pressure Drop Calculator →Friction Factor Calculator →Pipe Velocity Calculator →Pipe Head Loss Calculator →

FAQ

What is the difference between Darcy-Weisbach and Hazen-Williams?

Darcy-Weisbach is physically general and works for any fluid, temperature, and roughness through the friction factor. Hazen-Williams is an empirical water-only formula using a C-coefficient, valid only for water near ambient temperature in turbulent flow. For process work with varied fluids, Darcy-Weisbach is the correct default.

Why is the velocity-head term v²/(2g) important?

It is the kinetic energy of the flow per unit weight. Friction loss is a multiple of the velocity head set by f·(L/D). Because the velocity head goes with v², doubling the velocity quadruples the friction loss — the single biggest lever on pressure drop.

How do length and diameter enter the equation?

Through the dimensionless ratio L/D. Loss is linear in length but, once you account for velocity rising as 1/D² at fixed flow, very strongly inverse in diameter. That is why increasing the pipe size is usually the most effective way to cut pressure drop.

Where does the friction factor come from?

From the flow regime and relative roughness: f = 64/Re in laminar flow, and from the Moody chart or a correlation such as Colebrook-White or the explicit Swamee-Jain approximation in turbulent flow. See the Friction Factor Explained guide.