Centrifuge terminal velocity

For very small particles or low density solids, the terminal velocity may be too low to enable separation by gravity settling in a reasonably sized tank. However, the separation can possibly be carried out in a centrifuge, which operates on the same principle as the gravity settler but employs the (radial) acceleration in a rotating system (o r) in place of the vertical gravitational... 

This shows that the terminal velocity is not a constant but increases with r, because the (centrifugal) driving force increases with r. Assuming that all of the fluid is rotating at the same speed as the centrifuge, integration of Eq. (12-9) gives... 

Here, Vt is the terminal velocity of the particle in a gravitational field and is the cross-sectional area of the gravity settling tank that would be required to remove the same size particles as the centrifuge. This can be extremely large if the centrifuge operates at a speed corresponding to many g s. 

In the case of a centrifugal separator (i.e., a centrifuge), the acceleration due to centrifugal force, which should be used in place ofg, is given as ra>, where r is the radial distance from the central rotating axis (m) and co is the angular velocity of rotation (radian s -). Thus, the terminal velocity Vj (ms ) is given as... 

Using a tubular-bowl centrifuge rotating at 3600 rpm, determine the terminal velocity of Escherichia coli in a saline solution (density pp = 1.0 gcm ) at a radial distance 10 cm from the axis. 

In order to separate the particles in a suspension, the maximum allowable flow rate through the tubular-bowl centrifuge, as shown schematically in Figure 9.3a, can be estimated as follows [1]. A suspension is fed to the bottom of the bowl at a volumetric flow rate of Q (m" s" ) and the clarified liquid is removed from the top. The sedimentation velocity of particles in the radial direction (v = dr/di ) can be given by Equations 9.7 and 9.8 with the use of the terminal velocity under gravitational force v (m s" ) and the gravitational acceleration (m s" ) ... 

When a spherical particle of diameter d settles in a viscous liquid under earth gravity g, the terminal velocity V, is determined by the weight of the particle-balancing buoyancy and the viscous drag on the particle in accordance to Stokes law. In a rotating flow, Stokes law is modified by the centrifugal gravity G = ffV, thus... 

In motion from a centrifugal force, the velocity depends on the radius, and the acceleration is not constant if the partide is in motion with respect to the fluid. In many practical uses of centrifugal force, however, du/dt is small in comparison with the other two terms in Eq. (7.32), and if du/dt is neglected, a terminal velocity at any given radius can be defined by the equation... 

Equation (7.40) is known as Stokes law, and it applies for particle Reynolds numbers less than 1.0. At = 1.0, Co = 26.5 instead of 24.0 from Eq. (7.38), and since the terminal velocity depends on the square root of the drag coefficient, Stokes law is about 5 percent in error at this point. Equation (7.40) can be modified to predict the velocity of a small sphere in a centrifugal field by substituting rco for g. 

Example 7-1. (a) Estimate the terminal velocity for 80-to-100-mesh particles of limestone (pp = 2800 kg/m ) falling in water at 30°C. (b) How much higher would the velocity be in a centrifugal separator where the acceleration is 50gl... 

PRINCIPLES OF CENTRIFUGAL SEDIMENTATION. In a sedimenting centrifuge a particle of given size is removed from the liquid if sufficient time is available for the particle to reach the wall of the separator bowl. If it is assumed that the particle is at all times moving radially at its terminal velocity, the diameter of the smallest particle that should just be removed can be calculated. 

During centrifugation, these forces come quickly into balance and the particle reaches a terminal velocity, V, described by the Svedberg equation ... 

Analogous to gravity settling, a particle will attain a limiting or terminal velocity when the sum of the buoyancy and frictional forces equals the centrifugal force ... 

Under gravity acceleration the distance covered by a particle in a given time is simply Ugt under centrifugal acceleration (as the motion is accelerated) the following expression is obtained from integration of the terminal velocity... 

In a centrifugal field, a particle sedimenting through a viscous medium also reaches a terminal velocity u. The centrifugal acceleration is rotational velocity in rad/s, and r is the distance from the center of rotation to where the measurement is made, called the analytical radius. In this case, the Stokes equation has the form ... 

Example 4-2 Solid particles (average density of 2800 kg/m ) are settling in water (30°C). What is the terminal velocity for the particles Also, what would be the velocity of the system in a centrifugal separator with an acceleration of 390 m/sec ... 

The terminal velocity in the centrifugal field of Example 3.1.5, case (2), is much larger than that due to the gravitational field. 

In a gas cyclone or swirl tube, the particles of interest are almost always moving relative to the gas at their terminal velocity, and the terminal velocity of a given particle determines whether it will be captured or lost. This terminal velocity is exactly analogous to that of a particle settling in the earth s gravitational field, g, under steady-state conditions except that, for a cyclone, the radially directed centrifugal force, mvg/r replaces the gravitational one. This will be discussed in detail later. 

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