Governing Formula
The Darcy–Weisbach equation relates the friction head loss in a pipe to its length, diameter, and dynamic head. For turbulent flow the friction factor f is obtained from the Colebrook–White equation, which couples the relative roughness ε/D with the Reynolds number.
h_f = f × (L/D) × (v² / 2g) Where:
-
h_f= Friction head loss over the pipe run [m] -
f= Darcy–Weisbach friction factor [—] -
L= Developed pipe length [m] -
D= Pipe internal diameter [m] -
v= Mean flow velocity [m/s] -
g= Gravitational acceleration (9.80665) [m/s²] -
ε= Absolute pipe wall roughness [mm]
Derived Equations:
Re = v·D/ν 1/√f = −2·log₁₀(ε/(3.7D) + 2.51/(Re·√f)) Δp = ρ·g·h_f How the Calculation Works
The engine first computes the Reynolds number from velocity, diameter, and the kinematic viscosity of water at the given temperature. The friction factor is then found by iterating the Colebrook–White equation to convergence, starting from a Swamee–Jain estimate:
1/√f = −2·log₁₀(ε/(3.7·D) + 2.51/(Re·√f))
The computed f feeds the Darcy–Weisbach head loss; the pressure drop follows from Δp = ρ·g·h_f. A per-100-metre gradient is reported so results scale to any pipeline length.
Worked Engineering Example
Design Scenario: Steel Line, 1.5 m/s in 100 mm Pipe over 100 m at 20 °C
- Reynolds number:
ν(20 °C) = 1.002×10⁻⁶ m²/s
Re = 1.5 × 0.1 / (1.002×10⁻⁶) = 149,738 → turbulent - Relative roughness:
ε/D = 0.045 / 100 = 0.00045 - Friction factor (Colebrook–White, iterated):
f ≈ 0.0191 - Head loss:
h_f = 0.0191 × (100/0.1) × (1.5² / (2 × 9.80665)) = 2.19 m - Pressure drop:
Δp = 1000 × 9.80665 × 2.19 / 10⁵ = 0.215 bar
Engineering Notes & Design Benchmarks
| Pipe Material | Absolute Roughness ε | Design Impact |
|---|---|---|
| PVC / HDPE | 0.007 mm | Lowest friction in modern water mains |
| Commercial steel | 0.045 mm | Standard value for carbon-steel piping |
| Ductile iron (cement lined) | 0.03 mm | Smooth lining; common in distribution |
| Galvanized steel | 0.15 mm | Zinc coating adds roughness |
| Concrete | 0.6 mm | High roughness; large diameters keep losses low |
The friction factor converges to the fully-rough plateau at high Reynolds numbers. Roughness values above follow tabulated engineering practice (Crane TP 410); for aged or scaled pipe, ε should be raised for conservative design.
Assumptions & Limitations
- Newtonian water at a uniform temperature; viscosity estimated with the Vogel equation.
- Fully developed, incompressible flow in a circular pipe of constant internal diameter.
- The Colebrook–White correlation applies to turbulent flow (Re > 4,000); laminar flow should use f = 64/Re.
- Fitting losses, bends, valves, and elevation changes are not included — they must be added separately.
Frequently Asked Questions
When should I use Darcy–Weisbach instead of Hazen–Williams?
Darcy–Weisbach is the physically correct method for incompressible flow and is preferred in modern design (e.g. AWWA guidance for larger mains). Hazen–Williams is an empirical shortcut that works best for water near 20 °C over the usual velocity range.
Why does the calculator need a Reynolds number input?
It is not an input — the engine computes it from velocity, diameter, and the viscosity of water at your temperature. The Reynolds number and relative roughness together determine the friction factor through the Colebrook–White equation.
How is the friction factor actually computed?
The engine starts from a Swamee–Jain estimate and iterates the Colebrook–White equation until the change in f is below 10⁻⁸. This avoids reading the Moody chart and gives a consistent, reproducible value.
Engineering Disclaimer
Engineering Note: This calculator provides preliminary friction and pressure-drop estimates for design and education. Final pipeline design must include fitting losses, elevation profile, pumping energy checks, and validation with manufacturer-recommended roughness data.
Technical References
- Crane Co., Flow of Fluids Through Valves, Fittings, and Pipe (TP 410), 2009.
- Moody, L. F., Friction Factors for Pipe Flow, Transactions of the ASME, Vol. 66, 1944.
- White, F. M., Fluid Mechanics, 8th ed., McGraw-Hill, 2016.
- Hydraulic Institute, System Efficiency and Design Standards (ANSI/HI).