Background Information

A lot of times, a murky water-based mixture that appears homogeneous will eventually separate into distinct layers of sediment and water. This process is called, surprisingly, sedimentation. It occurs if the particles that make the water murky (dirt particles are a typical example) are actually denser than the water around them. In accordance with Archimedes' principle, these denser particles sink in the direction of the gravitational pull [Bartlett]. This is the principle behind gold panning. The gold is more dense than the sand and grit that it is embedded in, and with enough sloshing around the gold eventually makes its way to the bottom of the pan, where it is easily recovered.

There are a number of factors that affect the rate that sedimentation takes place. Since, in most practical situations, sedimentation is facilitated by means of a centrifuge, the closed formula for the velocity of sedimentation of a given particle in a centrifuge is given as

$$v=\omega ^{2}rm\frac{\left( 1-\overline{\nu }\rho \right) }{f},$$

where is the angular acceleration of the centrifuge, r is the radius of the centrifuge, m is the mass of the particle, is the ``partial specific volume'' of the particle (given as the inverse of the particle's density), is the density of the solution, and f is the frictional coefficient of the particle, determined empirically [Paule]. Obviously, there are quite a few parameters that go into this formula, and of course it is only approximate. The above formula can be transformed to give the the velocity of sedimentation outside of a centrifuge by noticing that is just the centripetal acceleration of an object rotating around a central point, Since all accelerations are alike, both gravitation and centripetal, that term can be replaced by g, the gravitational acceleration on the surface of the earth, where g=9.8 m/s². This gives the desired result,

$$v=gm\frac{\left( 1-\overline{\nu }\rho \right) }{f}.$$