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Fluid Mechanics

Pipe flow calculations: pressure drop, flow regime, energy balance and valve sizing. Click a calculator to open it. The theory is shown right next to it.

Pipe Pressure Drop

Darcy-Weisbach with Colebrook friction factor and minor losses

Flow rate Q [m³/h]

Inside diameter d [mm]

Pipe length L [m]

Pipe roughness k [mm]

Density ρ [kg/m³]

Kinematic viscosity ν [10⁻⁶ m²/s]

Sum of loss coefficients Σζ [-] (bends, valves …)

Flow velocity v–

Reynolds number Re–

Darcy friction factor λ–

Δp pipe friction–

Δp minor losses–

Total pressure drop Δp–

Guide value: water lines are usually sized for v = 1–3 m/s.

Theory

The pressure drop is the sum of pipe friction and minor losses:

Δp = ( λ · Ld + Σζ ) · ρ2 · v²

The Darcy friction factor λ depends on the flow regime:

laminar: λ = 64Re

turbulent: Colebrook equation (iterative)

v = Q/A: velocity in the pipe
Re = v·d/ν: Reynolds number
k: pipe roughness (new steel pipe ≈ 0.05 mm, plastic ≈ 0.007 mm)
ζ: loss coefficients of fittings and valves

Reynolds Number & Flow Regime

Laminar or turbulent?

Velocity v [m/s]

Inside diameter d [mm]

Kinematic viscosity ν [10⁻⁶ m²/s] (water 20 °C: 1.0 · air: 15.3 · oil VG46: 46)

Reynolds number Re–

Flow regime–

Theory

The Reynolds number is the ratio of inertial to viscous forces:

Re = v · dν

Re < 2320: laminar flow (layered, low losses)
Re 2320–4000: transitional
Re > 4000: fully turbulent flow

In practice, pipe flows are almost always turbulent.

Bernoulli Equation

Pressure at point 2 from the energy balance

Point 1

Pressure p₁ [bar abs]

Elevation h₁ [m]

Velocity v₁ [m/s]

Point 2

Elevation h₂ [m]

Velocity v₂ [m/s]

Density ρ [kg/m³]

Loss Δp_L [bar]

Dynamic pressure term ρ/2·(v₁²−v₂²)–

Elevation term ρ·g·(h₁−h₂)–

Pressure p₂–

Theory

Conservation of energy along a streamline (incompressible):

p₁ + ρ·g·h₁ + ρ2·v₁² = p₂ + ρ·g·h₂ + ρ2·v₂² + Δp_L

The three terms are static pressure, hydrostatic (elevation) pressure and dynamic pressure. Δp_L accounts for friction losses between the two points.

Applications: nozzles, diffusers, Venturi tubes, tank discharge
g = 9.81 m/s²

Valve Kv Value

Flow coefficient for liquids

Flow rate Q [m³/h]

Pressure drop Δp [bar]

Density ρ [kg/m³]

Kv value–

Recommended Kvs (×1.2)–

The Kvs value of the selected valve should be about 10–30 % above the calculated Kv.

Theory

The Kv value is the water flow rate in m³/h at Δp = 1 bar:

K_v = Q · √ρ / 1000Δp

Q: operating flow rate [m³/h]
Δp: pressure drop across the valve [bar]
ρ: density of the fluid [kg/m³]

Conversion: Cv (US) = 1.156 · Kv. The Kvs value is the Kv at 100 % valve travel.

Tank Discharge

Torricelli with discharge coefficient

Liquid level h [m]

Orifice diameter d [mm]

Discharge coefficient μ [-] (sharp-edged 0.62 · rounded 0.97)

Theoretical velocity v–

Flow rate Q–

Q–

Theory

According to Torricelli, the discharge velocity equals the free-fall velocity from height h:

v = 2 · g · h

The actual flow rate is corrected with the discharge coefficient μ (friction and jet contraction):

Q = μ · A · 2 · g · h

sharp-edged orifice: μ ≈ 0.60–0.64
well-rounded nozzle: μ ≈ 0.95–0.99
short pipe outlet: μ ≈ 0.82

Rough estimate for preliminary design. Does not replace a code-compliant design calculation.