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Name: Kim
Status: Student
Grade: 9-12
Location: NY
Country: United States
Date: March 2008


Question:
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?



Replies:
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.

Vince Calder


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



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