Sedimentation in a Centrifugal Field

A particle moving in a circular path continuously experiences a centrifugal force, Fc. This force acts in the plane described by the circular path and is directed away from the axis of rotation. The centrifugal force may be expressed as:

where m is the particle mass (g); a is the acceleration (cm s~2); w is the angular velocity (radians s_1 = 2n rpm/60); and x is the radial distance from the axis of rotation to the particle (cm).

Thus, centrifugal force is proportional to the square of the angular velocity and to the radial distance from the axis of rotation. The force generated during centrifugation can be compared to the gravitational force by the relative centrifugal force, RCF, often referred to as the g force:

Converting w to rpm and substituting values for the acceleration due to gravity, eqn [8] can be rewritten in a more convenient form as:

While RCF is a ratio, and therefore unitless, it is frequently expressed in units of g to indicate the number of times that the force of the applied centrifugal field is greater than the force of gravity.

The forces acting on a particle suspended in a liquid medium in a centrifugal field are illustrated in Figure 1. Within the centrifugal plane, the centrifugal force acts to move particles away from the axis of rotation, while the buoyancy and frictional forces oppose this movement. The effect of the Earth's gravity can generally be regarded as negligible. Analogous to the conditions for attaining terminal velocity in a gravitational field (eqn [5]), the particle will reach a limiting or terminal velocity in a centrifugal field when the sum of the frictional and buoyancy

Figure 1 Forces acting on a particle in a centrifugal field: Fb, buoyancy; Ff, frictional; Fc, centrifugal; and Fg, gravitational.

forces equals the centrifugal force:

Substituting eqns [2], [4] and [7] into eqn [10] gives:

Assuming a spherical particle and substituting frcr3 for volume gives:

Then solving for dx/dt:

Eqn [13] is more commonly expressed in terms of particle velocity, v, and particle diameter, d:

Eqn [14] may be integrated to determine the time required for a particle to traverse a radial distance from x0 to x1:

where xo is the initial position of the particle and x1 is the final position of the particle.

While modifications can be made to eqns [13]—[15] to account for specific rotor design, liquid—liquid, density-gradient separations, etc., these equations describe the relative impact of the more significant para meters that govern settling velocity. They show that the sedimentation rate (i.e. limiting velocity) of a particle in a centrifugal field:

• increases as the square of the particle diameter and rotor speed, i.e. doubling the speed or particle diameter will lessen the run time by a factor of four;

• increases proportionally with distance from the axis or rotation; and

• is inversely related to the viscosity of the carrier medium.

These are the fundamental premises that a practitioner must know in order to develop a rational approach to centrifugal separation.

Solar Panel Basics

Solar Panel Basics

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