Pipe friction losses

head lossDarcy-WeisbachHazen-Williamsfriction factorminor lossesequivalent length

Darcy–Weisbach, Hazen–Williams, friction factors, minor losses and equivalent length — every way the trade computes head loss in pipe.

Darcy–Weisbach Head Loss

h_f = f \, \frac{L}{D} \, \frac{v^{2}}{2g}

The rigorous pipe friction equation: head loss from friction factor, length-to-diameter ratio and velocity head, with g = 9.80665 m/s².

Laminar Friction Factor (f = 64/Re)

f = \frac{64}{Re}

In laminar pipe flow the Darcy friction factor depends only on Reynolds number — roughness plays no part below about Re = 2300.

Swamee–Jain Friction Factor

f = \frac{0.25}{\left[\log_{10}\!\left(\frac{\varepsilon}{3.7D} + \frac{5.74}{Re^{0.9}}\right)\right]^{2}}

An explicit turbulent friction factor within about 1% of the implicit Colebrook–White equation, valid for Re from 5000 to 10⁸.

Hazen–Williams Head Loss

h_f = \frac{10.67 \, L \, Q^{1.852}}{C^{1.852} D^{4.8704}}

The waterworks head-loss equation in SI form, with Q in m³/s and D in m; the 10.67 constant is 4.727 when working in feet and cubic feet per second.

Hazen–Williams Velocity

v = 0.849 \, C \, R^{0.63} S^{0.54}

Mean water velocity from hydraulic radius and hydraulic gradient; the 0.849 SI constant becomes 1.318 when R is in feet and v in feet per second.

Minor Loss from K Factor

h_L = K \, \frac{v^{2}}{2g}

Head lost through a valve or fitting as a multiple of velocity head, with g = 9.80665 m/s² and K taken from a fitting table.

Equivalent Length of a Fitting

L_{eq} = \frac{K D}{f}

Converts a fitting's K factor into the length of straight pipe that would cause the same friction loss at the same friction factor.

Reynolds Number

Re = \frac{\rho v D}{\mu}

The dimensionless ratio of inertial to viscous forces that decides laminar versus turbulent flow.

Pipe Velocity from Flow and Diameter

v = \frac{4Q}{\pi D^{2}}

Average velocity in a full round pipe from volumetric flow and inside diameter — the first check on any piping design.

How they fit together

There are two families here and they answer the same question with different amounts of honesty. Darcy–Weisbach is dimensionally correct and works for any fluid at any temperature, but it needs a friction factor, which needs the Reynolds number and the relative roughness — f = 64/Re below Re ≈ 2300, and the Swamee–Jain explicit fit standing in for Colebrook above about 4000. Hazen–Williams skips all of that by burying roughness in a single C value, which is why it fits on a slide rule and why it is still the water industry's default.

Use Hazen–Williams for cold water in full pipes at ordinary velocities — that is exactly the regime the coefficient was fitted to, roughly 40 to 75 °F and 2 to 10 ft/s — and Darcy–Weisbach for anything else: glycol, hot water, oil, air, or any pipe where temperature matters. Running Hazen–Williams on 40% glycol at 20 °F is a real and common error that under-predicts loss badly, because viscosity does not appear in the equation at all. For fittings, K factors and equivalent lengths are two ways to say the same thing; pick one and stay with it, because adding both double-counts every elbow. And be honest about C: new steel is 130 to 140, but the same pipe at twenty years of tuberculation is closer to 80, which nearly doubles the calculated loss — assuming new-pipe C on an old system is why the retrofit pump comes up short.