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Figure 12-3 Particle in a centrifuge.

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Figure 12-3 Particle in a centrifuge.

When the particle reaches its terminal (radial) velocity, dVr/dt = 0, and Eq. (12-7) can be solved for Vrt, (the radial terminal velocity):

If NRe < 1, Stokes' law holds, and CD = 24/NRe, in which case Eq. (12-8) becomes

This shows that the terminal velocity is not a constant but increases with r, because the (centrifugal) driving force increases with r. Assuming that all of the fluid is rotating at the same speed as the centrifuge, integration of Eq. (12-9) gives

where t is the time required for the particle to travel a radial distance from rj to r2. The time available for this to occur is the residence time of the particle in the centrifuge, t = V/Q, where V is the volume of fluid in the centrifuge. If the region occupied by the fluid is cylindrical, then V = ^(r2 - r1)L. The smallest particle that will travel from the surface of the fluid (rj) to the wall (r2) in time t is d = /18mq ^M^2 (12-11)

Rearranging Eq. (12-11) to solve for Q gives

which can also be written

Here, Vt is the terminal velocity of the particle in a gravitational field and E is the cross-sectional area of the gravity settling tank that would be required to remove the same size particles as the centrifuge. This can be extremely large if the centrifuge operates at a speed corresponding to many g's.

This analysis is based on the assumption that Stokes' law applies, i.e., NRe < 1. This is frequently a bad assumption, because many industrial centrifuges operate under conditions where NRe > 1. If such is the case, an analytical solution to the problem is still possible by using the Dallavalle equation for CD, rearranged to solve for NRe as follows:

NRe = ^P dr = [(14 42 + i^lJ^^)1'2 - 3.797]2 (12-14)

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