ENV-02

Stokes settling velocity

vs = g (ρs−ρ) d² /(18 μ) for a laminar sphere.

Reading speed

SedimentationStokes 1851Metcalf & Eddy

Governing equation

v_s=\dfrac{g(\rho_s-\rho)d^2}{18\mu}

where

\rho_s: Particle density (kg/m³)
\rho: Fluid density (kg/m³)
d: Diameter (µm)
\mu: Viscosity (Pa·s)
v_s: Settling velocity (mm/s)

Lecture brief

Historical brief

Streeter–Phelps (1925) oxygen sag, settling theory and Guldberg–Waage kinetics made water and air quality a rate problem. The lab computes sag, overflow and a snapshot of reactor mass balance. This sheet (ENV-02 — Stokes settling velocity) is the form associated with Stokes 1851 · Metcalf & Eddy. Working symbols: $\rho_s$ , $\rho$ , $d$ , $\mu$ $\rightarrow$ $v_s$ . Equating buoyant weight to Stokes drag 3π μ d vs yields the quadratic diameter law used in clarifier design.

Purpose

Purpose: compute

v_s

from

\rho_s

\rho

d

\mu

in Environmental via

v_s=\dfrac{g(\rho_s-\rho)d^2}{18\mu}

vs = g (ρs−ρ) d² /(18 μ) for a laminar sphere. Use it when a real environmental question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\rho_s = 2650.000\,\mathrm{kg/m^{3}}

\rho = 1000.000\,\mathrm{kg/m^{3}}

d = 50.000\,\mathrm{\mu m}

\mu = 0.001\,\mathrm{Pa·s}

, the governing relation

v_s=\dfrac{g(\rho_s-\rho)d^2}{18\mu}

yields

v_s = 2.2481\,\mathrm{mm/s}

. Re < 1, isolated sphere, Newtonian fluid. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Particle density \rho_s2650 kg/m³Fluid density \rho1000 kg/m³Diameter d50 µmViscosity \mu0.001 Pa·s

Outputs

Settling velocity v_s2.2481 mm/s

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

Re < 1, isolated sphere, Newtonian fluid.

Equating buoyant weight to Stokes drag 3π μ d vs yields the quadratic diameter law used in clarifier design.

Reading speed

Watch on YouTube