Reynolds Number and Pipe Pressure Loss: Darcy–Weisbach Worked Example
A water-pipe example that calculates velocity, Reynolds number, friction factor, head loss, and pressure loss using Darcy–Weisbach.
Why this calculation matters
Pipe pressure loss is not determined by flow rate alone. Diameter strongly affects velocity, Reynolds number, and friction loss, while roughness and fittings add further resistance.
This example uses water in a 25 mm internal-diameter pipe and deliberately separates straight-pipe Darcy loss from minor losses so each assumption is visible.
What you will calculate
- Convert volumetric flow to mean velocity.
- Use Reynolds number to identify the flow regime.
- Estimate a Darcy friction factor for turbulent flow.
- Convert head loss into pressure drop and identify omitted minor losses.
Given values
- Water at approximately 20 °C: density ρ = 998 kg/m³, dynamic viscosity μ = 1.002 mPa·s
- Internal diameter D = 0.025 m
- Flow rate Q = 0.001 m³/s (60 L/min)
- Straight pipe length L = 10 m
- Absolute roughness ε = 1.5 μm for the example
Governing equations
Pipe area
A = πD²/4Use internal, not nominal, diameter.
Mean velocity
v = Q/AContinuity for incompressible flow.
Reynolds number
Re = ρvD/μIndicates laminar, transitional, or turbulent behavior.
Darcy–Weisbach head loss
hf = f(L/D)v²/(2g)f is the Darcy friction factor, not the Fanning factor.
Pressure loss
Δp = ρghfConverts fluid head to pressure for a constant-density fluid.
Worked solution
1. Calculate area and velocity
A 25 mm pipe has only about 4.91×10⁻⁴ m² of flow area. At 0.001 m³/s, the mean velocity is therefore about 2.04 m/s.
A ≈ 4.909×10⁻⁴ m²; v ≈ 2.037 m/s2. Calculate Reynolds number
The Reynolds number is about 50,700, clearly in the turbulent regime for ordinary internal pipe flow. A laminar 64/Re friction-factor equation would therefore be inappropriate.
Re ≈ 998×2.037×0.025/(1.002×10⁻³) ≈ 5.07×10⁴3. Estimate the Darcy friction factor
Using a smooth-pipe-style explicit turbulent correlation with ε = 1.5 μm gives a Darcy friction factor of roughly 0.021. A Moody chart or Colebrook solution should be close for the same relative roughness.
f ≈ 0.02094. Calculate straight-pipe loss
For 10 m of straight pipe, the head loss is about 1.77 m of water. That corresponds to about 17.3 kPa of pressure drop. Fittings, valves, entrances, exits, elevation changes, and equipment losses are not included.
hf ≈ 0.0209×(10/0.025)×2.037²/(2×9.80665) ≈ 1.77 m; Δp ≈ 17.3 kPaEngineering interpretation
At 60 L/min through a 25 mm ID, 10 m pipe, the example predicts approximately 2.04 m/s velocity, Re ≈ 50,700, f ≈ 0.021, 1.77 m head loss, and 17.3 kPa straight-pipe pressure loss.
The result is a hydraulic model for the stated diameter, fluid properties, roughness, and straight length. Real systems need fitting losses, elevation, pump/system-curve interaction, and current pipe data.
Sanity checks
- A smaller diameter at the same Q must increase velocity sharply.
- For the same geometry, pressure loss should increase strongly as flow rises.
- Switching accidentally between Darcy and Fanning friction factors introduces a factor-of-four error.
- At very low Reynolds number, a turbulent correlation should not be used.
Common mistakes
- Using nominal pipe size instead of actual internal diameter.
- Using kinematic viscosity in an equation written for dynamic viscosity without conversion.
- Ignoring fittings when the straight run is short.
- Adding elevation head to friction loss with the wrong sign.
References and model boundaries
- Darcy–Weisbach and Reynolds-number relationships from standard fluid-mechanics references.
- Fluid properties and pipe roughness should be updated for the actual temperature, material, and product.
For safety-critical, regulated, production, or otherwise consequential work, independently verify the result using the governing standard, current manufacturer data, and qualified engineering review. See the site methodology and engineering disclaimer.