ENV-21

Stokes settling velocity

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

Reading speed

SedimentationStokes 1851

Governing equation

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

$d \in \left[1, 400\right]$

where

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

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

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

$d \in \left[1, 400\right]$

$d = 1$ $v_s = 8.99\times 10^{-4}$

Along the curve

②

Domain of the variable — blue

d_{\min} = 1,\quad d_{\max} = 400

$d \in \left[1, 400\right]$

$d = 1$ $d = 1$

Along the curve

③

Response curve — green

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

$d \in \left[1, 400\right]$

$d = 1$ $v_s = 8.99\times 10^{-4}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}v_s}{\mathrm{d}d}

$d \in \left[1, 400\right]$

$d = 3.509$ $v_s^{\prime} = 0.006$

Along the curve

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-21 — Stokes settling velocity) is the form associated with Stokes 1851. Working symbols: $\rho_s$ , $\rho$ , $d$ , $\mu$ $\rightarrow$ $v_s$ . Buoyant weight equals Stokes drag 3π μ d vs. Valid for Re < 1.

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 = 1.000\,\mathrm{mPa·s}

, the governing relation

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

yields

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

. A grain, a viscous fluid, a falling arrow. 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 \mu1 mPa·s

Outputs

Settling velocity v_s2.2481 mm/s

Reading speed

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Narration of this film

A grain, a viscous fluid, a falling arrow.

Buoyant weight equals Stokes drag 3π μ d vs. Valid for Re < 1.

Reading speed

Watch on YouTube