Darcy–Weisbach vs. Hazen–Williams in Pipe Head Loss Calculations
Two methods for the same question — how much head does a pipe consume for a given flow? This guide explains what each one actually computes, where they agree, and the situations in which you should not reach for Hazen–Williams. The formulas below are exactly those used in the pipe flow and Darcy–Weisbach calculators.
1. Darcy–Weisbach: physical, general, temperature-aware
The Darcy–Weisbach equation relates head loss to a dimensionless friction factor f:
where L is pipe length, D internal diameter, v mean velocity, and g gravity. The friction factor is not a constant — it depends on the Reynolds number Re = vD/ν and on the relative roughness ε/D. Turbulent f is traditionally taken from the implicit Colebrook–White equation; the explicit Swamee–Jain approximation gives the same result without iteration:
The engines use the Swamee–Jain value as the initial guess for a Colebrook–White iteration converged to |Δf| < 10−8. Critically, kinematic viscosity ν enters through Re — the interaction uses the Vogel equation for water viscosity (μ = 2.414×10−5 × 10247.8/(T+273.15−140), valid ≈0–100 °C), so cold water produces higher f and more head loss than warm water at the same velocity.
2. Hazen–Williams: empirical, water-turbulent, C-based
The Hazen–Williams expression is an empirical index formula calibrated for clean water in the fully turbulent range. In SI units it reads:
with Q in m³/s, D in m, L in m. The coefficient C is the pipe condition factor — typically 80–150 (the calculators reject C above 150): smooth PVC and HDPE sit near 140–150, ductile iron and concrete around 120–130, and older or corroded steel can fall to 80–100. C is a single curve-fit number: it folds roughness and the viscosity assumptions of the calibration into one value, which is why the method is simple but has no explicit temperature term.
3. Worked comparison
Take a 150 mm PVC line carrying 30 m³/h over 100 m at 20 °C, C = 140 for the Hazen–Williams side and rock-like smoothness (ε ≈ 0.0015 mm) for the Darcy–Weisbach side. Q = 30 m³/h = 8.33 × 10−3 m³/s; area = 0.01767 m² so v = 0.472 m/s and Re ≈ 70,000 (fully turbulent).
| Method | Key inputs | Head loss (m / 100 m) |
|---|---|---|
| Hazen–Williams | C = 140, SI formula | ≈ 0.17 |
| Darcy–Weisbach | ε = 0.0015 mm, 20 °C, Re ≈ 70,000, f ≈ 0.023 | ≈ 0.18 |
For this case the two methods nearly coincide — that is the norm in well-designed turbulent water lines, which is exactly why Hazen–Williams earned its keep in municipal practice. The divergence appears off-design. Repeat the same calculation on a 25 mm line at the same velocity in the flow transition zone (Re ≈ 4,000): the Colebrook side shows a friction factor around 0.04 and a higher loss, while Hazen–Williams with its fixed log-slope exponent under-reads the loss because it was never calibrated for laminar or transitional regimes.
4. Choosing between them
- Hazen–Williams: quick sizing in pressurized water networks with clean water, moderate velocities, diameters above ~50 mm, and established pipe materials where C is defensible. Simple to solve by hand and standard in network software for potable systems.
- Darcy–Weisbach: when temperature or viscosity matters (chilled water, hot water, high-viscosity fluids), for roughness-driven calculations where you have ε rather than an index, for pipes outside water service, and for any small-diameter or low-velocity line that may leave the fully turbulent range. This is warranted pumping duty.
- Never apply Hazen–Williams to laminar flow (Re < 2,000), partially full conduits, or fluids other than water — its exponents are purely empirical and do not extrapolate.