Particle size by Stokes settling

(adapted from Soil Physics, 5^th ed., by Jury, Gardner, and Gardner)

Assume a single sphere is settling in a viscous fluid under the influence of gravity. The gravitational acceleration pulling the sphere downward is (recall that F = m a)

for m = mass, subscript s is for solid, ρ is density, R is radius, and g is gravitational acceleration.

The buoyancy force upward is based upon the difference between the densities of the particle and the fluid:

where the subscript f denotes the fluid.

Finally, according to Stokes’ law, the viscous drag on the sphere is given by

for η viscosity and V velocity.

Once the particle reaches terminal velocity – a dynamic steady state – the forces sum to zero:

Plugging in and solving for velocity, we can simplify to get

Since velocity = height (remember, movement is vertical) per unit time, we can solve for height to obtain

Or, solving for diameter d = 2R (particle radius), we find

Assumptions involved in applying this analysis to soils:

            Soil particles are spheres

            Particles act independently of each other (dilute and dispersed)

            Terminal velocity is attained instantaneously

            Flow is laminar

            The fluid is at rest

            The particles are uniformly dispersed throughout the fluid

            The measurement can be made at a plane of depth h