Particle Size and Settling Rate
Country: United States
Date: March 2008
I have learned in Earth Science that larger, more dense, spherical
particles settle first in still water. However, I was wondering why this happens?
How do density, size, and shape of an object affect its settling rate?
A complete answer to your question is beyond the scope of a short answer forum
such as this. The simplified equation for spherical particle (called Stokes'
Law) will give you a feel for the variables that are important: v(the
sedimentation rate in units of distance/time) =
2/9 x R^2 x (D -d) x g / n where: R = the radius of the spherical particle,
"D" and "d" are the density of the particle and surrounding fluid respectively,
g = gravitational constant (this value may be very different if the particles
are in a centrifuge), and "n" is the viscosity of the fluid.
Sedimentation occurs because the mass of the sedimenting particles are more
strongly affected by gravity. Note, however, that "v" is zero if the density
of the particles and surrounding fluid are the same.
The general equation can be solved for particles of different shapes, but
that only affects the numerical coefficient. So for "globular" particles the
difference is not very large. It is also important to realize that "R" for
very small particles may not be identical to the "size" of the particle,
since the particles may drag along loosely associated molecules of the fluid.
This is called the "friction factor".
Your question does not have a simple answer because all of the variables can
be affected by factors not specifically included in the model.
Settling rate is a result of several forces acting on a particle inside a fluid.
The main forces to consider are the downward force of gravity, the upward force of
buoyancy, and the force of drag which opposes the particle’s motion through the
fluid. Gravity and buoyancy are “static” forces in that they are always the same
for a given particle regardless of how fast it is moving. The force of drag
depends on the particle’s speed relative to the fluid, and also on its shape and
cross-sectional area. I hope you are getting an idea of how density, size, and
shape affect an object’s settling rate.
The settling rate is the speed at which the viscous drag acting on the settling
particle exactly opposes the downward force on the particle. That downward force
is the interaction between the downward force of gravity on the particle and the
upward buoyancy force. (I have grouped the forces in this way because the force of
gravity and the force of buoyancy are the same at any speed the particle might
move. The force of viscous drag, however, changes with speed.)
The larger the object is, and the denser it is, the greater the gravitational force
acting downward on it. The force of buoyancy on the object is how much the water
pushes it upward. The strength of this buoyancy force is proportional to the
volume of the object. (It is actually equal to the weight of the water displaced
by the particle, which is equal to the weight of a volume of water equal to the
volume of the particle.) If the particle is more dense than water, its
gravitational force exceeds its buoyancy force and the net force on the particle
is downward. If the particle is less dense than water, its buoyancy force is
stronger than the gravitational force and the net force on the particle is upward.
So, first let’s assume that the particle is denser than water, so that it eventually
will settle rather then float. If it begins at rest in the water, the downward
force of gravity (minus buoyancy) will accelerate it downward. The greater this
downward force, the faster it will accelerate. As it accelerates, however, the
force of drag opposing its motion through the water increases. Eventually, it will
be moving so fast that the force of drag exactly cancels the downward force, so it
no longer accelerates, but continues moving downward at that speed.
The greater the mass of the particle, the faster this speed will be at which drag
strops further acceleration.
The larger the particle, the greater the force of drag at a given speed.
These two effects work somewhat at cross-purposes, as a larger object will have a
greater mass. But the force of gravity increases as the mass of the object, which
scales with the cube of its size (say, its diameter for a spherical particle),
while the force of drag increases as the cross-sectional area of the object, which
scales with the square of its size. So if two objects have the same density but
different sized, the larger one settles faster. If two objects have the same size
but different densities, the denser one settles faster.
Richard Barrans, Ph.D., M.Ed.
Department of Physics and Astronomy
University of Wyoming
Click here to return to the Environmental and Earth Science Archives
Update: June 2012