Discrete-Particle Settling Velocity (Stokes' Law)
The terminal settling velocity of a discrete spherical particle in still water by Stokes' law.
Example
You enter
- Particle diameter (mm) 0.05
- Particle specific gravity 2.65
- Water temperature (°F) 68
You get
- Settling velocity 2.247 mm/s
- Settling velocity (ft/min) 0.442
- Particle Reynolds number 0.112
No published worked example is on file for this calculation. It is checked against a hand-worked derivation of its method and an independent reference test.
Details, formula, and sources
ScopeVs = g(rho_p - rho_w)d^2/(18 mu) -- the workhorse of grit-chamber and Type I sedimentation-basin design. Water density and viscosity are computed from the temperature, so colder water settles grit more slowly. A 0.05 mm silt grain (SG 2.65) at 68 F settles about 2.25 mm/s (0.44 ft/min) at a particle Reynolds number of 0.11, well inside the Stokes regime. Stokes holds only while Re = rho_w Vs d / mu stays below about 1; a 0.5 mm sand grain hits Re ~ 112, which is flagged: there the transition/Newton law applies and the real velocity is lower. The overflow rate of an ideal clarifier equals this critical settling velocity. A discrete (Type I) settling screen; enter the solids by volume for the Richardson-Zaki hindered velocity (10% solids slows the example to 1.38 mm/s); flocculent and compression settling, particle shape, and short-circuiting are separate. Distinct from a rising oil droplet in an oil-water separator.
Formula or tablesettling_velocity Vs = g x (rho_p - rho_w) x d^2 / (18 mu) (Stokes, SI, reported in mm/s and ft/min); reynolds Re = rho_w x Vs x d / mu; Stokes valid for Re < ~1. Water mu from the Vogel correlation and rho_w from a table fit, both by temperature.; hindered (optional): v = vs (1 - c)^n, Richardson-Zaki n = 4.65 (Re < 0.2), 4.35 Re^-0.03 (0.2-1), 4.45 Re^-0.1 (1-500), 2.39 (> 500)
Source and editionStokes' law for discrete-particle (Type I) settling, first-principles; Davis & Cornwell, Introduction to Environmental Engineering, by name. The engineer governs the basin.
Free accessStokes' law and the particle Reynolds number are public physics; the water viscosity and density correlations are standard published fits; the particle size and density come from the grit or floc being settled.
What governsEstimate. Operator of record and primacy agency govern.
API field namesparticle_diameter_mm, particle_sg, water_temp_f, settling_velocity_mm_s, settling_velocity_ft_min, reynolds
- Stokes settling Vs = g(rho_p - rho_w)d^2/(18 mu) for a discrete sphere in the laminar (Re < ~1) regimeStokes' law / Davis & Cornwell
- Regime check Re = rho_w Vs d / mu; above ~1 the transition/Newton law applies and the tile flags itsedimentation practice