What pipe friction loss is

When a fluid flows through a pipe, friction with the inner wall and flow turbulence progressively convert mechanical energy (pressure + velocity + elevation) into heat, dispersed into the environment. This dissipation manifests as a pressure drop between inlet and outlet: the friction loss, denoted ΔH_f (in metres of fluid column) or ΔP_f (in pascals). It is the key parameter for sizing water supply networks, heating systems, oil pipelines, gas pipelines, cooling loops, and for choosing the pump that must overcome these losses. Underestimating it results in flows below requirements; overestimating leads to oversized pumps with higher energy consumption. The reference formula is Darcy-Weisbach (1857), universally valid for incompressible Newtonian fluids in circular pipes.

The Darcy-Weisbach formula

The distributed head loss (along the pipe length) is expressed as:

ΔH_f = f · (L/D) · v² / (2g)  [m of fluid column]

or in pressure units: ΔP_f = ρ·g·ΔH_f = f·(L/D)·(ρv²/2) [Pa]. Terms: f Darcy friction factor (dimensionless, depends on regime and roughness), L pipe length (m), D internal diameter (m), v mean fluid velocity (m/s, v = Q/A with A = π·D²/4), g = 9.81 m/s². The v²/(2g) term is the velocity head (in metres of fluid), and f·L/D is the number of velocity heads dissipated. Example: water in DN100 pipe (D=100 mm), L=200 m, Q=10 L/s → v = 1.27 m/s, with f ≈ 0.020 → ΔH_f ≈ 3.3 m of water column (32 kPa).

Reynolds number and flow regimes

Reynolds number is the dimensionless parameter distinguishing flow type:

Re = ρ·v·D/μ = v·D/ν

with μ dynamic viscosity (Pa·s) and ν = μ/ρ kinematic viscosity (m²/s). For water at 20 °C: ν = 1.004·10⁻⁶ m²/s. For SAE 30 motor oil: ν ≈ 100·10⁻⁶ m²/s (100× more viscous). Three regimes:

• Laminar (Re < 2300): fluid flows in parallel layers without mixing. Parabolic velocity profile (max at centre, zero at walls). Friction factor known analytically: f = 64/Re (Poiseuille). Typical for oils, capillary tubes, low flows.
• Transition (2300 < Re < 4000): unstable flow alternating laminar and turbulent features. Indeterminate friction factor, best avoided in sizing.
• Turbulent (Re > 4000): vortices, 3D mixing, flattened velocity profile. Friction factor f depends on both Re and relative roughness ε/D. Typical of water networks, HVAC, gas pipelines.

Example: water v = 1.27 m/s in D = 100 mm pipe → Re = 126,800 (turbulent). Oil v = 0.3 m/s in D = 20 mm → Re = 64 (laminar).

Friction factor: from Colebrook to Swamee-Jain

In turbulent regime the friction factor is given by the implicit Colebrook-White equation (1939):

1/√f = −2·log₁₀[ε/(3.7·D) + 2.51/(Re·√f)]

This equation is transcendental (f appears on both sides) and requires iteration. In 1976 Swamee and Jain proposed an explicit approximation with 1-2% precision vs Colebrook:

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

valid for 5·10³ < Re < 10⁸ and ε/D < 0.05. The calculator uses this formulation. Absolute roughness ε depends on pipe material:

• New PVC: 0.007 mm.
• New copper: 0.0015 mm.
• New steel: 0.046 mm.
• Galvanised steel: 0.15 mm.
• Encrusted steel: 0.5-3 mm.
• New cast iron: 0.25 mm; old cast iron: 1-3 mm.
• Concrete: 0.3-3 mm.
• Glass: 0.001 mm (practically smooth).

Relative roughness ε/D is what matters in the calculation: a large pipe with ε = 0.5 mm can be "hydraulically smooth" if D is large (small ε/D).

The Moody diagram: the universal chart

The Moody diagram (Lewis Ferry Moody, 1944) plots f vs Re with parametric curves for different ε/D values. It is the traditional pre-calculator tool for reading f. Four zones are distinguished:

• Laminar zone (Re < 2300): -1 slope line in log-log scale, corresponding to f = 64/Re. Roughness curves coincide here because laminar flow does not "feel" the wall beyond the viscous sublayer.
• Transition zone: gap between Re 2300 and ~4000, dashed to indicate indeterminacy.
• Smooth turbulent zone: very small roughness curves (ε/D < 10⁻⁵) tend asymptotically to the Prandtl-Kármán curve, where friction factor depends on Re only: f = 0.316/Re^0.25 (Blasius, valid up to Re = 10⁵).
• Fully turbulent zone (large Re and non-negligible ε/D): curves become horizontal → f depends on ε/D only, not Re. Here f = [2·log₁₀(3.7·D/ε)]⁻² (Kármán-Nikuradse).

The calculator automates the Moody diagram reading with Swamee-Jain.

Recommended velocities and sizing criteria

Recommended velocities vary by application (UNI-EN 806-3 for sanitary water, ASHRAE for HVAC):

• Drinking water networks: 0.5-1.5 m/s; max 2 m/s for short runs.
• Pump suction: 0.5-1.5 m/s (avoids cavitation).
• Pump discharge: 1.5-3 m/s.
• DHW: 0.5-1.5 m/s (erosion/noise limits).
• Forced-circulation heating: 0.5-1.2 m/s (silence).
• Saturated steam: 15-40 m/s.
• Natural gas distribution: 5-20 m/s.
• Air in ventilation ducts: 2-8 m/s.

"Unit head loss" j = ΔH_f/L: recommended 0.005-0.03 m/m for cold water in networks. Above 0.05 m/m the pump consumes too much; above 0.15 m/m erosion and cavitation risks. Good economic compromise is optimum economic velocity, minimising pipe cost + pumping cost. Empirical formula: v_opt ≈ (Q/K)^0.2 with K = 3·10⁻⁴ m for water.

Local losses and pump head

Beyond distributed losses (calculated here with Darcy-Weisbach) there are local losses at bends, fittings, valves, filters, reductions. Expressed as:

ΔH_conc = ζ · v²/(2g)

with ζ tabulated coefficient per fitting: short-radius 90° elbow ζ ≈ 0.9; long-radius 90° elbow ζ ≈ 0.3; open ball valve ζ ≈ 0.05; partly-open control valve ζ = 1-100. In residential systems local losses can reach 30-50% of distributed. Total required pump head:

H_pump = ΔH_geo + Σ ΔH_f + Σ ΔH_conc + ΔH_utility

ΔH_geo geometric head, Σ ΔH_f sum of distributed losses in all sections, Σ ΔH_conc sum of local losses, ΔH_utility residual pressure required at delivery point (typically 10-15 m for domestic taps). Pump electrical power P_ele = ρ·g·Q·H_pump/η_pump, with η_pump 60-85% for residential centrifugal pumps.

How to use the calculator

Enter the internal diameter D in mm (not nominal DN, which is external: e.g. steel DN100 has D_internal ≈ 105 mm). Enter length L in metres. Enter flow rate Q in L/s (conversion: 1 m³/h = 0.278 L/s; 60 L/min = 1 L/s). Enter absolute roughness ε in mm (copper 0.0015; PVC 0.007; new steel 0.046; cast iron 0.25). Enter fluid density ρ (kg/m³, cold water 998; hot water 60°C 983; motor oil 850; 40% glycol 1060). Enter kinematic viscosity ν (m²/s: water 20°C 1.004·10⁻⁶; water 60°C 0.467·10⁻⁶; motor oil 100·10⁻⁶). The calculator returns velocity v, Reynolds number Re, friction factor f (laminar or turbulent with Swamee-Jain), head loss ΔH_f in metres, pressure drop ΔP_f in kPa, dissipated power in kW and qualitative assessment.