Flow in pipes

The volume flow through a pipe is Q = v × A: average velocity times the cross-section area. A pipe 0.1 m across has an area π × 0.05² = 0.00785 m², so 0.02 m³/s through it moves at 2.55 m/s. Typical liquid velocities in plant pipes are 1 to 3 m/s.

What goes in must come out, so with a liquid the flow rate is the same all along a pipe (continuity): v₁A₁ = v₂A₂. Area goes with diameter squared, so halve the diameter and the velocity goes up four times. That's why a thumb over a hose makes a jet.

Viscosity, μ (mu), is a fluid's resistance to flowing: honey is thick, water thin. Water's viscosity is about 0.001 Pa·s at 20 °C; it drops as liquids warm up.

Whether flow is smooth or chaotic depends on one number, the Reynolds number: Re = ρvD ÷ μ, with D the pipe diameter. It compares the push of the fluid's momentum against viscosity's drag, and has no units. In a pipe, below about 2100 the flow is laminar: smooth layers sliding past each other. Above about 4000 it's turbulent: swirling eddies. In between it can go either way.

Water at 2.55 m/s in that 0.1 m pipe: Re = 1000 × 2.55 × 0.1 ÷ 0.001 = 255,000, well turbulent. Most plant flows are. Turbulence mixes and transfers heat much better, but costs more pumping energy. Laminar flow shows up with thick fluids (oils, polymers) and in very small channels.