Dynamics of Particles Submerged in Fluids

If a particle moves relative to the fluid in which it is suspended, the force acting on the particle is the gravity force fg, while the forces opposing the motion are the buoyant force Fj, and the drag force Fq. The resultant of forces acting on the particle may be represented by 

The buoyancy force equals the mass of the displaced fluid due to the acceleration of the particle, based on the Archimedes principle, that is. 

The resultant force in Equation 10.1 equals the acceleration as the particle accelerates on its way down. Since acceleration du/dt is due to the mass of the particle, the resultant force can also be expressed as 

Substituting Equations 10.2 through 10.4 into Equation 10.5 and transposing 

In settling under influence of gravity g is constant while the drag always increases with velocity. Equation 10.6 shows that the acceleration decreases with time and approaches zero. The particle quickly reaches constant velocity, which is the maximum attainable, and which is called the terminal velocity. The equation for terminal velocity Mj is foimd, for settling under gravity, by taking du/dt = 0 and transposing for the particle-fluid relative velocity from Equation 10.6, so as to obtain  

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Within the context of particle technology, the most relevant application of centrifugation is centrifugal clarification since it follows theoretical principles of dynamics of particles submerged in fluids as discussed in this book. Those principles are used to describe centrifugal clarification as a unit operation. 

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