## Gravity Settling

Solid particles can be removed from a dilute suspension by passing the suspension through a vessel that is large enough that the vertical component of the fluid velocity is lower than the terminal velocity of the particles and the residence time is sufficiently long to allow the particles to settle out. A typical gravity settler is illustrated in Fig. 12-2. If the upward velocity of the liquid (Q/A) is less than the terminal velocity of the particles (Vt), the particles will settle to the bottom; otherwise, they will be carried out with the overflow. If Stokes flow is applicable (i.e., NRe < 1), the diameter of the smallest particle that will settle out is d = (^QY/2 (12-1)

If Stokes flow is not applicable (or even if it is), the Dallavalle equation in the form of Eq. (11-16) can be used to determine the Reynolds number, and hence the diameter, of the smallest setting particle:

1 / /C"\1/2 —= = 0.00433 + 0.208 /—^ J -0.0658 (12-2)

where

Cd 4^Apg J NReM

Alternatively, it may be necessary to determine the maximum capacity (e.g., flow rate, Q) at which particles of a given size, d, will (or will not) settle out. This can also be obtained directly from the Dallavalle equation in the form of Eq. (11-13), by solving for the unknown flow rate:

Dp where

III. CENTRIFUGAL SEPARATION A. Fluid-Solid Separation

For very small particles or low density solids, the terminal velocity may be too low to enable separation by gravity settling in a reasonably sized tank. However, the separation can possibly be carried out in a centrifuge, which operates on the same principle as the gravity settler but employs the (radial) acceleration in a rotating system (!2r) in place of the vertical gravitational acceleration as the driving force. Centrifuges can be designed to operate at very high rotating speeds, which may be equivalent to many g's of acceleration.

A simplified schematic of a particle in a centrifuge is illustrated in Fig. 12-3. It is assumed that any particle that impacts on the wall of the centrifuge (at r2) before reaching the outlet will be trapped, and all others won't. (It might seem that any particle that impacts the outlet weir barrier would be trapped. However, the fluid circulates around this outlet corner, setting up eddies that could sweep these particles out of the centrifuge.) It is thus necessary to determine how far the particle will travel in the radial direction while in the centrifuge. To do this, we start with a radial force (momentum) balance on the particle:

where Fcf is the centrifugal force on the particle, Fb is the buoyant force (equal to the centrifugal force acting on the displaced fluid), FDis the drag force, and me is the "effective" mass of the particle, which includes the solid particle and the "virtual mass" of the displaced fluid (i.e. half the actual mass of displaced fluid). Equation (12-6) thus becomes

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