Darcy-Weisbach Equation
Learn the Darcy-Weisbach equation for calculating pressure drop due to friction in pipes, with worked examples and tables.
The Formula
ΔP = f · (L / D) · (ρv² / 2)
The Darcy-Weisbach equation is the fundamental formula for calculating frictional pressure loss in pipes and ducts carrying fluids. It was developed through the combined work of Henry Darcy and Julius Weisbach in the 19th century in France and Germany, respectively. The equation relates the head loss (or pressure drop) to the pipe's physical properties and the flow conditions.
The first form of the equation expresses the result as head loss hf, measured in meters (or feet) of fluid column. The second form gives the pressure drop ΔP directly in Pascals (or pounds per square foot). Both forms are mathematically equivalent, related by ΔP = ρghf.
The Darcy friction factor f is the most complex part of the equation. For laminar flow (Reynolds number below 2300), the friction factor is simply f = 64/Re, where Re is the Reynolds number. For turbulent flow, the friction factor depends on both the Reynolds number and the relative roughness of the pipe wall. Engineers typically use the Moody chart or the Colebrook-White equation to determine f for turbulent flow.
The Colebrook-White equation for turbulent flow is: 1/√f = −2 log10(ε/(3.7D) + 2.51/(Re√f)), where ε is the pipe surface roughness. Since f appears on both sides, it must be solved iteratively or approximated using explicit formulas such as the Swamee-Jain equation.
This equation is essential in the design of piping systems for water supply, oil and gas transport, HVAC systems, and chemical processing. Engineers use it to size pumps, select pipe diameters, and estimate energy losses in fluid transport networks. The equation applies to any Newtonian fluid in fully developed flow through a circular pipe.
Variables
| Symbol | Meaning |
|---|---|
| hf | Head loss due to friction (m or ft) |
| ΔP | Pressure drop (Pa or lb/ft²) |
| f | Darcy friction factor (dimensionless) |
| L | Length of the pipe (m or ft) |
| D | Internal diameter of the pipe (m or ft) |
| v | Average flow velocity (m/s or ft/s) |
| g | Gravitational acceleration (9.81 m/s² or 32.2 ft/s²) |
| ρ | Fluid density (kg/m³ or slugs/ft³) |
Example 1
Water flows through a 50 m long pipe with diameter 0.1 m at 2 m/s. The friction factor is 0.02. Find the head loss.
hf = f · (L/D) · (v²/2g)
hf = 0.02 × (50/0.1) × (2² / (2 × 9.81))
hf = 0.02 × 500 × 0.2039 = 2.04 m
The frictional head loss is approximately 2.04 meters of water column.
Example 2
Oil (ρ = 850 kg/m³) flows through a 200 ft (61 m) pipe of 6 inches (0.152 m) diameter at 1.5 m/s with f = 0.03. Find the pressure drop.
ΔP = f · (L/D) · (ρv²/2)
ΔP = 0.03 × (61/0.152) × (850 × 1.5² / 2)
ΔP = 0.03 × 401.3 × 956.25 = 11,510 Pa
The pressure drop is approximately 11,510 Pa (about 11.5 kPa or 1.67 psi).
When to Use It
The Darcy-Weisbach equation is the standard method for computing friction losses in pipe systems.
- Designing water distribution and supply networks
- Sizing pumps by calculating total system head loss
- Oil and gas pipeline design for transport over long distances
- HVAC duct sizing and pressure drop calculations
- Chemical plant piping design for process fluids
- Irrigation system planning and sprinkler system design