Particle settling basics

Particle settling velocity and time

The time required for a particle to settle can be derived from the general particle equation of motion, as derived by e.g. Maxey and Riley (https://doi.org/10.1063/1.864230). Simplifying the equation to only account for the Stokes drag and gravitational force, the equilibrium free-fall velocity $v$ can be derived from:

Here, $\rho_p$ and $\rho_f$ are the particle and fluid densities, $d_p$ is the particle diameter and $\mu$ is the dynamic viscosity of the fluid. Solving for $v$ and assuming that the particle density is much greater than the fluid density yields:

$$Using this velocity, the time required for a particle to fall to the ground from a given height can be calculated. All this is under the assumption that the air is still and that other than gravity and Stokes drag no other forces act on the particle. It is also only valid (in air) for particles greater than about 1 $\mu m$.

Numerical example

First, load unit support:

import Pkg

Pkg.add("Unitful")

using Unitful

using Unitful: μm, m, s, °C, Pa, kg, mm

We consider a water droplet in air at 20°C:

μ = 1.82e-5kg/(m*s) # Air viscosity at 20°C

ρ_p = 1000kg/m^3 # Particle density

g = 9.81m/s^2 # Gravitational acceleration;

This function yields the settling velocity in mm/s, for a given diameter:

# Velocity as a function of particle diameter

v(d) = uconvert(mm/s,g*ρ_p*d^2/(18*μ))

For a 5 micron particle we get:

v(5μm)

Let's compute how long it takes to fall from 1.5 m:

tfall(d) = uconvert(s, 1.5m/v(d))

Since the velocity depends on the square of the diameter, the settling time goes up rather quickly as the diameter decreases:

uconvert(u"minute", tfall(10μm))

uconvert(u"minute", tfall(5μm))

uconvert(u"hr", tfall(1μm))