Darcy-Weisbach Equation Explained: Formula, Friction Factor, and Examples

The Darcy-Weisbach Equation

The Darcy-Weisbach equation is the most accurate and universally applicable formula for calculating friction head loss in pipes. Unlike Hazen-Williams (which only works for water), Darcy-Weisbach works for any fluid at any temperature.

hf = f × (L/D) × (V²/2g)

Where:

• hf = friction head loss (m)
• f = Darcy friction factor (dimensionless)
• L = pipe length (m)
• D = pipe internal diameter (m)
• V = mean fluid velocity (m/s)
• g = gravitational acceleration (9.81 m/s²)

The challenge is determining f (the friction factor).

The Friction Factor (f)

Laminar Flow (Re < 2000)

f = 64 / Re

Exact solution. No approximation needed.

Turbulent Flow (Re > 4000)

The Colebrook-White equation (implicit — requires iteration):

1/√f = -2 × log10(ε/(3.7D) + 2.51/(Re×√f))

The Swamee-Jain approximation (explicit — no iteration):

f = 0.25 / [log10(ε/(3.7D) + 5.74/Re^0.9)]²

Swamee-Jain is accurate to within 1% of Colebrook-White for Re between 5000 and 10⁸ and ε/D between 10⁻⁶ and 0.05. This covers virtually all practical engineering applications.

Pipe Roughness (ε)

Example 1: Water in Steel Pipe

Data: commercial steel pipe, D = 100 mm (0.1 m), L = 200 m, Q = 10 L/s, water at 20°C.

Step 1: Velocity

A = π × 0.1² / 4 = 0.00785 m²
V = 0.010 / 0.00785 = 1.274 m/s

Step 2: Reynolds number

Re = V × D / ν = 1.274 × 0.1 / (1.004×10⁻⁶) = 126,900

Step 3: Friction factor (Swamee-Jain)

ε/D = 0.000045 / 0.1 = 0.00045
f = 0.25 / [log10(0.00045/3.7 + 5.74/126900^0.9)]²
f = 0.25 / [log10(0.0001216 + 0.0000959)]²
f = 0.25 / [-3.637]²
f = 0.0189

Step 4: Head loss

hf = 0.0189 × (200/0.1) × (1.274²/(2×9.81))
hf = 0.0189 × 2000 × 0.0827
hf = 3.13 m

Result: 3.13 m of friction loss over 200 m of steel pipe at 10 L/s.

Example 2: PVC Pipe (Low Roughness)

Same conditions but PVC pipe (ε = 0.0015 mm):

ε/D = 0.0000015 / 0.1 = 0.000015
f = 0.25 / [log10(0.000015/3.7 + 5.74/126900^0.9)]²
f = 0.0172
hf = 0.0172 × 2000 × 0.0827 = 2.84 m

PVC gives 10% less friction than steel — consistent with its smoother surface.

Darcy-Weisbach vs Hazen-Williams

Bottom line: Darcy-Weisbach is the gold standard. Use Hazen-Williams only for quick estimates with water at ambient temperature.

Automate the Calculation

HydroApp Pro calculates friction losses using both methods. It automatically determines the Reynolds number, applies Swamee-Jain for the friction factor, and computes head losses per segment. Supports mixed pipe materials and diameters in the same network.

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