Relationship between settling velocity and particle size

1 Relationship between settling velocity and particle size 

When a particle falls under gravity in a viscous fluid three forces act upon it a gravitational force W acting downwards and a buoyant force U and a drag force acting upward. The resulting equation of motion is  

Small particles accelerate rapidly to a constant or terminal velocity (dw/dr = 0) and the motive force balances the drag force. 

For a sphere of diameter D and density falling in a fluid of density pp the equation of motion becomes  

Dimensionless analysis of the general problem of particle motion under equilibrium conditions gives a unique relationship between two dimensionless groups, drag coefficient (C, ) and Reynolds number Re) where  

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The simplest case to consider is the settling of a single homogeneous sphere, under gravity, in a fluid of infinite extent. Many experiments have been carried out to determine the relationship between settling velocity and particle size under these conditions and a unique relationship between drag factor (C ) and Reynolds number Re) has been found that reduces to a simple equation, known as the Stokes equation, at low Reynolds number. 

Both equations (6.4) and (6.5) contain D and u and need to be expressed in terms of a single variable in order to determine Z) if m is known or m if Z) is known. Stokes neglected the terms due to inertia and obtained a very simple relationship between settling velocity and particle size for particles settling with low velocities. Several attempts at theoretical solutions for the relationship between and Re at higher velocities have been made. Oseen [29] partially allowed for inertial effects to obtain ... 

Sedimentation is another classical particle classification and sizing method for liquid-bom particles. Sedimentation methods are based on the rate of settling of particles in a liquid at rest under a gravitational or centrifugal field. The relationship between settling velocity and particle size is reduced to the Stokes equation at low Reynolds numbers ... 

The underlying principle behind the gravity sedimentation method is Stokes law, which describes the relationship between the settling velocity of particles in a fluid medium of known density and viscosity [6], A number of instruments are available in which the settling velocity of particles is measured by x-ray absorption, light absorption, and density changes. One such instrument is the Sedigraph by the Micromeritics Corporation in which the particle size distribution is determined by x-ray absorption [6]. This instrument offers an excel-... 

At high velocities, due to the onset of turbulence, the drag on the sphere increases above that predicted by Stokes equation and particles settle more slowly than the law predicts. However, settling velocities can be related to particle diameters by applying Newton s equation which is available as an empirical relationship between C, and Re. The upper size limit for Stokes equation is limited, due to the onset of turbulence, to Reynolds number smaller than 0.25. 

The above equation makes a number of assumptions the particle is spherical, the concentration of suspended solids is low and the suspension is in a quiescent state. However, although this somewhat simplistic approach does deviate from the practical situation, it does identify that the relationship between particle size and settling velocity is exponential. Therefore, as particle size increases this has a significant impact on settling velocity. 

The relationship between particle density, particle diameter and fluid viscosity, governing settling velocity, holds true as long as the particle size is not so small that its motion is affected by Brownian movement. This is the net out-of-balance force of the collision of the molecules of the fluid on the particle. For large particles the collisions even out all round, but for very small ones they do not, and small irregular movements of the particle can be seen. Where Brownian movement is occurring, the terminal velocity can be substantially reduced and, at a certain particle size. Brownian movement is sufficient to keep the particles suspended indefinitely. This may be taken to be the case with particles of one micrometre or smaller in air. 

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