12.121. The mechanics of particle movement through a fluid has re­ceived a great deal of engineering attention for many years. For example, scrubbers and devices for removing mists from chemical plants are common applications of fine particle dynamics. A first approach to describing par­ticle motion is to apply a force balance which includes an external force such as a gravitational force and a frictional or drag force. Thus, for a spherical particle having a diameter D and a density p falling through a fluid of density Pf with a velocity и, we can say that

Weight of sphere — Buoyant force = Drag force.

The drag force is expressed in terms of a drag coefficient CD according to the relation

Drag force = CD(ipfU2Ap),

where и is the constant settling velocity and AP is the projected area of the sphere, i. e., hrD2. Since the particle volume is zttD3, we have

(12.1)

In the particle diameter range of 3 to 100 pm, Stokes law applies and

24 _ 24jx

Re Dupf

Then simplifying, we have

For smaller particles, 0.1 to 3 (xm in diameter, an empirical correlation known as the Cunningham factor is applied to account for slip and free molecule effects. An average factor value of 1.8 may be assumed for es­timation purposes.

Example 12.2. Determine the settling velocity in air of a particle hav­ing a diameter of 10 jxm and a density of 3000 kg/m3. The air at 288 К and 1 atm has a viscosity of 1.8 x 10“5 Pa • s. Compare with the velocity of a particle having a diameter of 1 |xm.

gDp — P/) (9.81)(10 x 10“6)2(3000)

“ ~ 18(1 ~ 18(1.8 x 10“5)

= 9.1 x 10~3 m/s

For the l-|xm particle,

(1.8)(9.81)(1 x 1Q-6)2(30Q0)
18(1.8 x 10-5)

(A more accurate calculation yields 1.1 x 10 4 m/s.)