Darcy-Weisbach Head Loss

Darcy–Weisbach head loss: h_f = f·(L/D)·(v²/(2g))

Darcy–Weisbach tells you the friction head loss in a pipe, expressed in meters of water column (mwc): h_f = f·(L/D)·(v²/(2g)). Here f is the Darcy friction factor, L the pipe length (m), D the internal diameter (m), v the mean velocity (m/s) and g = 9.81 m/s². What makes f tricky is that it varies with the Reynolds number Re and the relative roughness ε/D, which is exactly what the Moody diagram captures. In laminar flow (Re < 2,300) it's simply f = 64/Re. Once the flow turns turbulent you reach for the implicit Colebrook equation, or the explicit Swamee–Jain and Haaland approximations if you'd rather skip the iteration. As a feel for the numbers, Re = 10⁵ in smooth PVC (ε/D ≈ 10⁻⁶) gives f ≈ 0.018. Local losses at elbows, valves and reductions get added on through K coefficients. When the fluid is water near room temperature, the empirical Hazen–Williams formula is another route.

Applications

It comes up when sizing hydraulic pipework under NBR 5626 (cold and hot water for buildings), picking the right pump head, designing sanitary drainage, and laying out oil pipelines, gas pipelines or irrigation networks.

FAQ

Darcy factor vs Fanning factor? There's a factor of 4 between them: f_Darcy = 4·f_Fanning. Civil and hydraulic work tends to use Darcy, while chemical engineering often goes with Fanning, so confirm which one your formula expects before you plug a value in.

When does relative roughness ε/D matter? Deep in the turbulent regime, at high Re, f rides on ε/D alone and viscosity drops out of the picture. In transitional flow both Re and ε/D have a say, which is where Colebrook comes in. And in laminar flow ε/D plays no role whatsoever.

Why use Darcy instead of Hazen–Williams? Darcy–Weisbach rests on physics, so it holds for any fluid and any flow regime. Hazen–Williams is empirical and was calibrated only for water around 15 °C in turbulent flow. It's easier to apply, but step outside that window and the accuracy falls apart.