What is the Darcy-Weisbach equation?

The Darcy-Weisbach equation gives the major head loss due to friction in a pipe: h_f = f (L/D)(v^2/2g), where f is the Darcy friction factor, L the pipe length, D the diameter, v the mean velocity, and g gravity. It applies to laminar and turbulent flow.

How is the friction factor found?

For laminar flow (Re < 2000), f = 64/Re. For turbulent flow, f depends on the Reynolds number and relative roughness and is found from the Colebrook equation or read from the Moody chart. Approximations like the Swamee-Jain equation are common for direct calculation.

What is the difference between major and minor losses?

Major losses are friction losses along the length of pipe, given by Darcy-Weisbach. Minor losses occur at fittings, bends, valves, entrances and exits, expressed as h = K v^2/2g where K is a loss coefficient. In long pipelines major losses dominate; in short systems with many fittings, minor losses can be significant.

Darcy-Weisbach Equation and Head Loss in Pipes — with Worked Examples

Q: What is the difference between major and minor losses?

Major losses are friction losses along the length of pipe, given by Darcy-Weisbach. Minor losses occur at fittings, bends, valves, entrances and exits, expressed as h = K v^2/2g where K is a loss coefficient. In long pipelines major losses dominate; in short systems with many fittings, minor losses can be significant.

Whenever a fluid flows through a pipe it loses energy to friction, and that head loss determines the pump size, the pressure available, and the pipe diameter you must choose. The universal equation for friction loss is Darcy-Weisbach. This article explains it with worked examples.

The Darcy-Weisbach Equation

h_f = f · (L/D) · (v² / 2g)

Symbol	Meaning
h_f	Head loss due to friction (m)
f	Darcy friction factor (dimensionless)
L	Pipe length (m)
D	Pipe diameter (m)
v	Mean velocity (m/s)

Note: this uses the Darcy friction factor f. The Fanning friction factor f' = f/4 is used in some chemical-engineering texts — always check which one a chart refers to.

Finding the Friction Factor

Laminar (Re < 2000): f = 64/Re (exact, independent of roughness).
Turbulent: f from the Colebrook-White equation, which depends on Reynolds number and relative roughness ε/D, or read from the Moody chart.
Swamee-Jain gives an explicit approximation: f = 0.25 / [log₁₀(ε/3.7D + 5.74/Re⁰·⁹)]².

Worked Example 1 — Head Loss in a Pipeline

Water flows at 2 m/s through a 100 mm diameter pipe, 100 m long, with a friction factor f = 0.02. Find the friction head loss.

h_f = f (L/D)(v²/2g) = 0.02 × (100/0.1) × (2²/(2×9.81))
= 0.02 × 1000 × (4/19.62) = 20 × 0.2039 = 4.08 m

Worked Example 2 — Laminar Flow Loss

Oil (ν = 1×10⁻⁴ m²/s) flows at 0.5 m/s in a 50 mm pipe, 80 m long. Find the head loss.

Re = vD/ν = (0.5 × 0.05)/1×10⁻⁴ = 250 → laminar
f = 64/Re = 64/250 = 0.256
h_f = 0.256 × (80/0.05) × (0.5²/(2×9.81)) = 0.256 × 1600 × 0.01274 = 5.22 m

Minor Losses

Fittings, bends, valves, entrances and exits cause local losses:

h_minor = K · v² / 2g

Fitting	Typical K
Sharp pipe entrance	0.5
Pipe exit	1.0
90° elbow	0.9
Globe valve (open)	10
Gate valve (open)	0.2

Total head loss = major (Darcy-Weisbach) + sum of minor losses. In long transmission mains, friction dominates; in short, fitting-heavy systems, minor losses matter.

Worked Example 3 — Pump Head Requirement

For Example 1, if the pipe also has an entrance (K = 0.5), one elbow (K = 0.9) and an exit (K = 1.0), find the total head loss.

Velocity head v²/2g = 0.2039 m
Minor losses = (0.5 + 0.9 + 1.0) × 0.2039 = 2.4 × 0.2039 = 0.489 m
Total head loss = 4.08 + 0.49 = 4.57 m — the pump must supply at least this plus any elevation lift.

Common Mistakes

Confusing the Darcy and Fanning friction factors (factor of 4 error).
Using laminar f = 64/Re in turbulent flow.
Forgetting minor losses in short systems, or double-counting the exit loss.