Pipe Flow (Darcy-Weisbach)

Pressure loss · Friction factor λ · Reynolds number · Flow velocity · Pipe sizing

Pipe Flow Calculator

Density ρ [kg/m³]
kin. viscosity ν [m²/s]
ε mm
DN100 = 100 mm | DN50 = 50 mm | measured inner diameter
Guide: drinking water 1–2 m/s | pressure line 1.5–3 m/s

Formulas & Fundamentals

Darcy-Weisbach – pressure loss:
Δp = λ · (L/D) · (ρ·v²)/2  [Pa]
λ = friction factor | L = length | D = diameter
ρ = density | v = velocity
Head loss (energy head):
h_f = λ · (L/D) · v²/(2·g)  [m]
g = 9.81 m/s² | Δp = ρ·g·h_f
Reynolds number:
Re = v·D / ν
ν = kinematic viscosity [m²/s]
Re < 2300: laminar | Re > 4000: turbulent
Friction factor λ:
laminar: λ = 64 / Re
turbulent (Colebrook-White):
1/√λ = −2·log₁₀( ε/(3.7·D) + 2.51/(Re·√λ) )
implicit – solved iteratively (Swamee-Jain initial value)
Flow rate & velocity:
Q = v · A = v · π·D²/4  [m³/s]
v = 4·Q / (π·D²)  [m/s]
Required diameter:
D = √(4·Q / (π·v_allow))  [m]
from flow rate Q and target velocity
Pipe roughness ε (typical values)
Materialε [mm]
Copper, glass (smooth)0.0015
PVC, plastic0.007
Steel new / seamless0.05
Galvanized steel0.15
Cast iron0.25
Concrete smooth0.3–3.0


Pipe Flow & Pressure Loss – Fundamentals of Hydraulics

What does the Darcy-Weisbach equation describe?

The Darcy-Weisbach equation is the fundamental relationship for calculating the pressure loss due to pipe friction in a flowing line. It is universally valid for laminar and turbulent flow as well as for all Newtonian fluids (water, oil, air), making it the standard in process engineering, water supply, heating and ventilation systems.

Δp = λ · (L/D) · (ρ·v²)/2
The term ρ·v²/2 is the dynamic (velocity) pressure; λ·(L/D) is the dimensionless resistance coefficient of the line.
Flow regimes
Reynolds numberFlow
Re < 2300laminar (layered flow)
2300 – 4000transition region
Re > 4000turbulent
Recommended velocities
Applicationv [m/s]
Drinking water service1.0–2.0
Pressure line (pump)1.5–3.0
Suction line0.7–1.5
Heating (circulation)0.5–1.5
Compressed air10–20

The friction factor λ – core of the calculation

The pipe friction factor λ (dimensionless) depends on the Reynolds number and the relative roughness ε/D. The Moody diagram presents this relationship graphically. Three regions:

1. Laminar flow (Re < 2300):
λ = 64 / Re
Independent of roughness – only viscosity determines the resistance.
2. Turbulent flow – smooth pipe (Blasius, Re < 10⁵):
λ = 0.3164 / Re0.25
3. Turbulent flow – general (Colebrook-White):
1/√λ = −2·log₁₀( ε/(3.7·D) + 2.51/(Re·√λ) )
Implicit equation – solved iteratively. Explicit approximation by Swamee-Jain:
λ = 0.25 / [ log₁₀( ε/(3.7·D) + 5.74/Re0.9 ) ]²

Worked example: drinking water pressure line

Problem:

Steel pipe DN100 (D = 100 mm), length L = 250 m, flow rate Q = 15 l/s, water 20 °C (ρ = 998 kg/m³, ν = 1.004·10⁻⁶ m²/s), roughness ε = 0.05 mm. Find: pressure loss Δp.

Solution:
  • Cross-section: A = π·0.1²/4 = 7.854·10⁻³ m²
  • Velocity: v = Q/A = 0.015/7.854·10⁻³ = 1.91 m/s
  • Reynolds: Re = v·D/ν = 1.91·0.1/1.004·10⁻⁶ = 190,200 → turbulent
  • Relative roughness: ε/D = 0.05/100 = 0.0005
  • Colebrook-White (iterative): λ ≈ 0.0193
  • Δp = 0.0193·(250/0.1)·(998·1.91²)/2 = 87,900 Pa ≈ 0.88 bar
  • Head loss: h_f = Δp/(ρ·g) = 87900/(998·9.81) = 8.98 m

Frequently Asked Questions

The pressure loss Δp is given in Pascal (Pa) or bar and is the actual pressure difference. The head loss h_f is given in metres of water column and is linked via h_f = Δp/(ρ·g). Both describe the same energy loss – in pump engineering the head in metres is usually used, in process engineering the pressure in bar.

In the Colebrook-White equation λ appears on both the left and right (implicit) and cannot be solved analytically. This calculator uses the explicit Swamee-Jain approximation as the starting value and then refines it through fixed-point iteration to high accuracy (typically < 0.1 % deviation). For practice, the Swamee-Jain formula alone is usually accurate enough.

The relative roughness ε/D relates the absolute wall roughness ε to the pipe diameter. It is dimensionless and determines the friction factor in the turbulent region. A small pipe with the same absolute roughness has a larger relative roughness and therefore higher friction. At very high Reynolds numbers (fully rough) λ depends only on ε/D, no longer on Re.

Local (minor) losses from fittings, bends and contractions are captured by resistance coefficients ζ (zeta): Δp_ζ = ζ·ρ·v²/2. The total pressure loss is the sum of pipe friction (Darcy-Weisbach) and all local losses: Δp_total = (λ·L/D + Σζ)·ρ·v²/2. Typical ζ values: 90° bend ≈ 0.3 | ball valve open ≈ 0.1 | gate valve open ≈ 0.2 | sharp-edged entry ≈ 0.5. This calculator covers pure pipe friction.

The economic velocity is a compromise: too small → large (expensive) pipe diameter; too large → high pressure loss and pumping energy cost plus erosion and noise problems. Typical design: pressure lines 1.5–2.5 m/s, suction lines 0.7–1.5 m/s. For drinking water, standards limit the velocity depending on the duration of use (peak up to 5 m/s, continuous operation ~2 m/s).

Summary

Pressure loss

Δp = λ·(L/D)·ρv²/2
h_f = Δp/(ρg)

Friction factor λ

laminar: 64/Re
turbulent: Colebrook-White

Reynolds number

Re = v·D/ν
limit: 2300

Typical applications
  • Water supply: sizing of drinking water and pressure lines, pump head
  • Heating & ventilation: pipe network calculation, hydraulic balancing
  • Process engineering: pipe sizing in plants, pump selection
  • Fire protection: sprinkler and hydrant lines, minimum pressure
  • Civil engineering: pressure pipelines, sewage pressure mains

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