The Darcy-Weisbach equation: pressure drop and head loss

The Darcy-Weisbach equation is how engineers put a number on the energy a fluid loses to friction as it travels down a pipe. That number decides how powerful a pump you need, what pipe diameter to use, and whether gravity alone can push the fluid to its destination. This guide gives you the formula in both SI and imperial units, defines every symbol, and works it end to end on real numbers.

Why pressure drop matters

Consider a water distribution system for a building where the pump delivers 5 bar at the inlet. Will that pressure be enough at the top floor, 30 meters up, after flowing through 200 meters of pipe with bends and valves? Only a pressure drop calculation tells you.

Get it right and your system delivers the required flow at every outlet. Get it wrong and you either undersize the pump (insufficient flow) or oversize it (wasted energy cost for the life of the system).

The Darcy-Weisbach equation

The industry standard for pipe pressure drop is the Darcy-Weisbach equation:

• ΔP - pressure drop (Pa)
• h_f - head loss (m of fluid)
• f - Darcy friction factor (from the Moody Chart)
• L - pipe length (m)
• D - internal diameter (m)
• ρ - fluid density (kg/m³)
• V - average flow velocity (m/s)
• g - 9.81 m/s²

The two forms carry the same physics. The first gives you a pressure in pascals, which is what you want when you are checking a rating or a gauge reading. The second gives you head in metres of the fluid itself, which is what you want when you are reading a pump curve. Convert between them with ΔP = ρ × g × h_f.

The equation in imperial units

The equation is dimensionally consistent, so it works in any coherent unit set. The catch in US customary units is that pressure is usually wanted in psi while length, density and velocity come in feet, lb/ft³ and ft/s. The version that handles the bookkeeping for you is:

SG is specific gravity (1.0 for water at ambient temperature). Keep L and D in the same unit — feet with feet, or inches with inches. Mixing a length in feet with a diameter in inches is the most common arithmetic error in imperial pressure drop work. It inflates the answer twelvefold.

Where the equation comes from

Julius Weisbach published the form above in 1845. It was dimensionally sound but of limited practical use, because nobody could yet predict f. Henry Darcy supplied the missing piece in 1857. His experiments on cast iron and wrought iron pipe in Dijon showed that friction depended on the condition of the pipe wall, not on the fluid alone. Weisbach's form plus Darcy's roughness insight is why the equation carries both names, and why the factor inside it is the Darcy friction factor.

The last gap closed in 1944, when Lewis Ferry Moody plotted the Colebrook-White solution across the whole turbulent range. That plot is the Moody chart, and it is still the fastest way to get the f that this equation needs.

Darcy friction factor vs Fanning friction factor

Two friction factors are in circulation and they differ by a factor of four: f_Darcy = 4 × f_Fanning. The equation above expects the Darcy factor. Chemical engineering texts often tabulate the Fanning factor instead. Drop a Fanning value into a Darcy equation and you understate pressure drop by 75%.

The quick sanity check: in laminar flow the Darcy factor is 64/Re and the Fanning factor is 16/Re. At Re = 2000, that is 0.032 versus 0.008. If a published chart bottoms out near 0.008 at low Reynolds number, you are holding a Fanning chart. Everything on this site uses the Darcy convention.

Step-by-step calculation

Follow this workflow for any pipe pressure drop problem:

Step 1: find flow velocity

If you know flow rate Q (m³/s) and pipe diameter D (m): V = Q / (π D²/4).

Use the Pipe Flow Velocity Calculator if needed.

Step 2: calculate Reynolds number

Re = V × D / ν. This tells you whether flow is laminar or turbulent.

Use the Reynolds Number Calculator.

Step 3: get friction factor (f)

Find relative roughness ε/D using the Pipe Roughness Calculator, then enter Re and ε/D into the Moody Chart Calculator to get friction factor f.

Step 4: calculate pressure drop

Enter f, L, D, ρ, V into the Pressure Drop Calculator.

Worked example: water in a 100 mm steel pipe

Problem: Water flows at 1.5 m/s through 100 m of 100 mm commercial steel pipe. Fluid density ρ = 1000 kg/m³. Friction factor from Moody Chart: f = 0.019.

This means the pump must supply an extra 2.18 m of head (or 21.4 kPa) just to push water through this 100 m pipe section.

Major losses vs minor losses

The Darcy-Weisbach equation calculates major losses: friction along straight pipe runs. Real systems also have minor losses from fittings:

• Elbows: K ≈ 0.3 to 1.5 depending on bend radius and angle
• Gate valve (fully open): K ≈ 0.1 to 0.2
• Ball valve (fully open): K ≈ 0.05
• Globe valve: K ≈ 6 to 10
• Tee (flow-through): K ≈ 0.4 to 0.9
• Sudden expansion: K ≈ (1 - A₁/A₂)²

Minor losses: h_minor = K × V²/(2g). Total head loss = major losses + Σ(minor losses). For long pipe runs, major losses dominate. For short systems with many fittings, minor losses can match or exceed major losses.

Pressure drop sensitivity to pipe diameter

Head loss is proportional to 1/D⁵ (substituting Q = V × πD²/4 into the Darcy-Weisbach equation). This means:

This is why pipe sizing is one of the most high-impact decisions in hydraulic system design.

Working a minor loss, using the same pipe

K values are only useful once you put them through a system. Take the 100 m steel line from the worked example, still at 1.5 m/s, and give it realistic fittings: four elbows, two gate valves, and one globe valve for throttling.

The fittings add 37% on top of the pipe friction, which is already enough to matter for pump selection. Look at where it comes from, though: the single globe valve contributes 0.92 m on its own. That one component costs more head than 40 m of the pipe it sits in.

Swapping it for a ball valve at K ≈ 0.05 would cut the minor losses to 0.38 m and the total to 2.56 m. When a system comes in over its head budget, the fittings schedule is usually a cheaper place to look than the pipe diameter.

Where Darcy-Weisbach stops being valid

The equation assumes the fluid is incompressible, so density stays put from inlet to outlet. Liquids satisfy this comfortably at any pressure you are likely to see in a pipe.

Gases do not. Compressed air, steam, and natural gas expand as pressure falls along the run, so velocity rises while density drops, and the single V in the formula no longer describes the whole pipe. The usual working rule is that Darcy-Weisbach stays acceptable while total pressure drop stays under roughly 10% of inlet absolute pressure. Past that, use a compressible method such as the isothermal flow equation, or split the line into short segments and recalculate density in each.

Two other cases fall outside it entirely. Slurries and polymer solutions are non-Newtonian, so there is no single friction factor to look up. Two-phase flow, where liquid and gas travel together, follows different correlations again. In both cases the calculator here will still return a number, and that number will not describe your pipe.

Frequently asked questions