CHAPTER 10
MECHANICAL SEPARATIONS (cont'd)

THE VELOCITY OF PARTICLES MOVING IN A FLUID

Under a constant force, for example the force of gravity, particles in a liquid accelerate for a time and thereafter move at a uniform velocity. This maximum velocity which they reach is called their terminal velocity. The terminal velocity depends upon the size, density and shape of the particles, and upon the properties of the fluid.

When a particle moves steadily through a fluid, there are two principal forces acting upon it, the external force causing the motion and the drag force resisting motion which arises from frictional action of the fluid. The net external force on the moving particle is applied force less the reaction force exerted on the particle by the surrounding fluid, which is also subject to the applied force, so that

F_s = Va(r_p - r_f)

where F_s is the net external accelerating force on the particle, V is the volume of the particle, a is the acceleration which results from the external force, r_p is the density of the particle and r_f is the density of the fluid.

                         F_d = Cr_fv²A/2

where C is the coefficient known as the drag coefficient, r_f is the density of the fluid, v is the velocity of the particle and A the projected area of the particle at right angles to the direction of the motion.

If these forces are acting on a spherical particle so that V = pD³/6 and A = pD²/4, where D is the diameter of the particle, then equating F_s and F_d, in which case the velocity v becomes the terminal velocity v_m, we have:

(pD³/6) x a(r_p - r_f) = Cr_fv_m²pD²/8

It has been found, theoretically, that for the streamline motion of spheres, the coefficient of drag is given by the relationship:

                           C = 24/(Re) = 24m/Dv_mr_f

Substituting this value for C and rearranging, we arrive at the equation for the terminal velocity magnitude

                         v_m = D²a(r_p - r_f)/18m                                                                    (10.1)

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