## Falling Particles

Many engineering operations involve the separation of solid particles from fluids, in which the motion of the particles is a result of a gravitational (or other potential) force. To illustrate this, consider a spherical solid particle with diameter d and density ps, surrounded by a fluid of density p and viscosity which is released and begins to fall (in the x = —z direction) under the influence of gravity. A momentum balance on the particle is simply £Fx = max, where the forces include gravity acting on the solid (Fg), the buoyant force due to the fluid (Fb), and the drag exerted by the fluid (Fd). The inertial term involves the product of the acceleration (ax = dVx/dt) and the mass (m). The mass that is accelerated includes that of the solid (ms) as well as the "virtual mass" (mf) of the fluid that is displaced by the body as it accelerates. It can be shown that the latter is equal to one-half of the total mass of the displaced fluid, i.e., mf = 1 ms(p/ps). Thus the momentum balance becomes g(ps — p)^d3 CDp^d2 V2 3(ps + p/2) dV

At t = 0, V = 0 and the drag force is zero. As the particle accelerates, the drag force increases, which decreases the acceleration. This process continues until the acceleration drops to zero, at which time the particle falls at a constant velocity because of the balance of forces due to drag and gravity. This steady-state velocity is called the terminal velocity of the body and is given by the solution of Eq. (11-8) with the acceleration equal to zero:

where Ap = ps — p. It is evident that the velocity cannot be determined until the drag coefficient, which depends on the velocity, is known. If Stokes flow prevails, then CD = 24/NRe, and Eq. (11-9) becomes

However, the criterion for Stokes flow (NRe < 1) cannot be tested until Vt is known, and if it is not valid then Eq. (11-10) will be incorrect. This will be addressed shortly.

There are several types of problems that we may encounter with falling particles, depending upon what is known and what is to be found. All of these problems involve the two primary dimensionless variables CD and NRe. The former is determined, for gravitation-driven motion, by Eq. (11-9), i.e.,

and CD can be related to NRe by the Dallavalle equation [Eq. (11-5)] over the entire practical range of NRe. The following procedures for the various types of problems apply to Newtonian fluids under all flow conditions.

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