Settling tanks: Stokes' law and surface overflow rate

Settling is the cheapest way to remove suspended solids, in grit chambers, primary clarifiers and water treatment plants alike.

Stokes' law

A small sphere settling slowly in water (laminar, Re below about 0.5) reaches a terminal velocity:

v_s = g (G − 1) d²18 ν

G is the particle's specific gravity, d its diameter and ν the kinematic viscosity of water (about 1.0 × 10⁻⁶ m²/s at 20 °C).

Since v_s grows with d², doubling particle size makes it settle four times faster.

The ideal settling tank

Water flows horizontally through a tank of plan area A at discharge Q. A particle is fully removed if it can drop the full depth before reaching the outlet. That works out to a simple condition:

v_s ≥ v_o = QA

v_o is the surface overflow rate. Depth cancels out. In theory, a shallow tank with the same plan area removes the same particles, which is the idea behind tube settlers.

Particles with v_s below v_o are removed in the fraction v_s / v_o.

Typical design values

Tank	Overflow rate (m³/m²/day)	Detention time
Plain sedimentation (water)	12 to 18	3 to 4 h
Clarifier after coagulation	20 to 40	2 to 2.5 h
Primary clarifier (sewage)	25 to 40	about 2 h

Worked example

Find the settling velocity of a sand particle, d = 0.05 mm, G = 2.65.

v_s = 9.81 × 1.65 × (5 × 10⁻⁵)²18 × 1.0 × 10⁻⁶ = 2.25 × 10⁻³ m/s

That is about 194 m/day. A tank designed for 30 m³/m²/day removes it completely.

Grit chambers

Grit chambers in sewage works are designed to remove sand of about 0.2 mm and above, while keeping a horizontal velocity near 0.3 m/s so lighter organic solids stay in suspension.