Darcy Weisbach Equation

Overview

The Darcy-Weisbach equation relates pressure loss due to friction along a pipe to the flow velocity, pipe diameter, and friction factor. It is the most widely used equation for pipe pressure drop calculations.

Key Formulas

Darcy-Weisbach Equation

$\Delta P = f \cdot \frac{L}{D} \cdot \frac{\rho v^2}{2}$

Pressure drop due to friction in a pipe.

Head Loss Form

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

Head loss in terms of fluid column height.

Darcy Friction Factor (Laminar)

$f = \frac{64}{Re}$

For laminar flow (Re < 2300).

Variables

Darcy-Weisbach Pressure Drop Calculator

Notes

• The Darcy friction factor is 4× the Fanning friction factor — be careful with references
• For turbulent flow, use the Colebrook equation or Moody chart for friction factor
• For laminar flow (Re < 2300), f = 64/Re exactly

Head Loss Equations

The Darcy-Weisbach equation can also be expressed to calculate head loss, which is often more practical in hydraulic engineering.

Head Loss as Water Column: $\Delta h_{major\_loss,w} = \lambda \left( \frac{l}{d_h} \right) \left( \frac{\rho_f v^2}{2 \gamma_w} \right) = \lambda \left( \frac{l}{d_h} \right) \left( \frac{\rho_f}{\rho_w} \right) \left( \frac{v^2}{2 g} \right) \quad \text{(2)}$

Head Loss of Flowing Fluid (Simplified): When the fluid in the pipe and the reference fluid are the same (e.g., water flowing in a pipe), the equation simplifies to: $\Delta h_{major\_loss} = \lambda \left( \frac{l}{d_h} \right) \left( \frac{v^2}{2 g} \right) \quad \text{(2b)}$

Unit-Specific Conversions:

• For metric units (mm H₂O): $\Delta h_{major\_loss,w}\;(\mathrm{mm\ H_2O}) = \lambda \left( \frac{l}{d_h} \right) \left( \frac{\rho_f}{\rho_w} \right) \left( \frac{v^2}{2 g} \right) \times 1000 \quad \text{(2c)}$
• For imperial units (inches H₂O): $\Delta h_{major\_loss,w}\;(\mathrm{in\ H_2O}) = 12 \lambda \left( \frac{l}{d_h} \right) \left( \frac{\rho_f}{\rho_w} \right) \left( \frac{v^2}{2 g} \right) \quad \text{(2d)}$

Where:

• γ_w = specific weight of water (ρ_w * g, approx. 9807 N/m³ or 62.4 lbf/ft³)
• ρ_w = density of water (1000 kg/m³ or 62.425 lb/ft³)
• g = acceleration due to gravity (9.81 m/s² or 32.174 ft/s²)

Important Considerations

• Model Accuracy: The Darcy-Weisbach equation combined with the Moody diagram is considered the most accurate model for estimating frictional head loss in steady pipe flow.
• Alternative Methods: Due to the required trial-and-error iteration to find the friction factor (λ), less accurate empirical methods like the Hazen-Williams equation are sometimes preferred for simplicity.
• Reference Fluid: The head loss calculated by equations (2) - (2d) is expressed as a column of the reference fluid (commonly water). To find head loss in terms of another fluid (e.g., mercury), replace the density of water (ρ_w) with the density of that fluid.

References

• Original Source