What's Stokes' law and when does it apply?

Stokes' law gives terminal settling velocity for a sphere in laminar flow (Re < 1). Valid for small particles in water — fine silt, clay, sludge flocs. For sand and larger, use the transition or Newton regime.

What's a typical settling velocity for grit?

0.21 mm sand at 20°C: v_s ≈ 0.07 ft/s (2.1 cm/s). This is the design particle for aerated grit chambers. Loading rate is sized so the particle settles before exiting.

Why does temperature matter?

Kinematic viscosity of water roughly halves between 0°C and 30°C. Cold water (winter) doubles laminar drag, halving Stokes settling velocity for fine particles. Design with winter conditions for size-conservative results.

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Particle Settling Velocity

Terminal settling velocity of a discrete spherical particle in still fluid. Auto-detects flow regime (Stokes, transition, or Newton) by iterating on Reynolds number and the drag coefficient. Used for grit-chamber design, primary clarifier sizing, sediment basin analysis, and lab particle settling.

Units

Particle diameter, dmm

Particle specific gravity, s— (sand: 2.65; sludge: 1.05; coal: 1.4)

Fluid temperature, T°C

Settling velocity, v_s—m/s

Reynolds number——

Flow regime——

Drag coefficient, C_D——

Defaults: 0.2 mm sand grain (SG = 2.65) at 20°C — common grit-chamber design particle. Stokes is valid only for Re < 1; Newton's law for Re > 1000. Transition zone uses C_D = 24/Re + 3/√Re + 0.34.

Force balance, terminal velocity:

$$ v_s = \sqrt{\frac{4 g (s - 1) d}{3 C_D}} $$

Stokes' law (Re < 1):

$$ v_s = \frac{g (s - 1) d^2}{18 \nu} \quad (C_D = 24/Re) $$

Newton's law (Re > 1000):

$$ v_s = \sqrt{3.33 \, g (s - 1) d} \quad (C_D \approx 0.4) $$

v_s terminal settling velocity · g 9.81 m/s² · s particle specific gravity · d particle diameter · ν kinematic viscosity of water (≈ 1.0 × 10⁻⁶ m²/s at 20°C) · C_D drag coefficient · Re = v_s d / ν.

Pick a regime, then check Re

The most common error is using Stokes for a particle that's actually outside its valid range. Stokes assumes laminar flow around the particle (Re < 1), which limits diameter to about 0.06 mm for sand at 20°C. Anything coarser than fine silt sits in the transition or Newton regime, where Stokes overpredicts settling velocity by 2× to 5×.

This calculator iterates: it assumes a regime, computes v_s, computes Re from v_s, and re-checks which formula's range it lands in. The reported velocity always corresponds to the regime that satisfies the consistency check.

Why settling velocity matters

Grit chamber design. Aerated grit chambers are sized so the smallest particle to be captured (typically 0.21 mm sand) settles to the bottom before the flow exits. Surface loading rate must be ≤ v_s for that particle.
Primary clarifier. Overflow rate (gpd/ft² or m³/m²/d) must be less than the settling velocity of the design particle. Typical primary clarifier OFR: 600 to 1200 gpd/ft² (24 to 49 m³/m²/d) for plain sedimentation.
Sediment basin (construction stormwater). NRCS / EPA Phase II requires a 0.02 mm silt particle as the design particle for construction site sediment basins. Stokes' law applies (laminar regime).
Stokes' Pond & sediment forebay. Particle removal efficiency depends on basin surface area, not depth — for ideal sedimentation. Doubling depth doesn't help; doubling surface area doubles capacity (Hazen 1904).

Real particles aren't spheres

The drag coefficient assumes a smooth sphere. Real silt and sand are angular, leading to 20% to 40% lower settling velocity than the formula predicts. For sludge flocs, the discrepancy is larger because flocs are porous, irregular, and grow / break apart with shear. Hindered settling at high concentration (greater than ~3000 mg/L) further reduces v_s by interaction effects.

Temperature dependence

Kinematic viscosity of water roughly halves between 0°C and 30°C. Cold water (winter) doubles the laminar drag, halving Stokes settling velocity for fine particles. Warm water settles faster. Design typically uses winter temperature for size-conservative design.

Reference: Davis, M.L. (2010). Introduction to Environmental Engineering, 4th ed., McGraw-Hill, ch. 6. Crittenden, J.C., et al. (2012). MWH's Water Treatment, 3rd ed., Wiley, ch. 9.

Reference: settling velocity of common particles

Terminal settling velocity in still water at 20°C, S_s = 2.65 (quartz)
Particle class	d (mm)	v_s (mm/s)	Re	Regime
Clay	0.002	0.0036	<0.01	Stokes
Fine silt	0.010	0.089	<0.01	Stokes
Medium silt	0.020	0.35	0.01	Stokes
Coarse silt	0.050	2.15	0.11	Stokes
Very fine sand	0.10	7.98	0.79	Stokes
Fine sand	0.20	26.3	5.3	Transition
Medium sand	0.50	90.4	45	Transition
Coarse sand	1.0	175	174	Transition
Very coarse sand	2.0	292	582	Transition
Fine gravel	5.0	514	2560	Newton

Solved iteratively with C_d = 24/Re + 3/√Re + 0.34 (Fair & Geyer; Metcalf & Eddy), using ρ and μ from the water properties card. These are discrete (Type I) settling velocities for spheres — angular natural grains settle roughly 10–30% slower, and flocculent particles do not follow this table at all.