Centrifugal separation equation

The terminal settling velocity u, for a single spherical particle in a centrifugal separator can be calculated from equation 9.5 with the centripetal acceleration rto2 replacing the gravitational acceleration g to give... 

A very small particle may still be in laminar flow in a centrifugal separator. In this case rco2 may be written in place of g in equation 9.8 to give... 

In centrifugal separation, g in Stokes equation is replaced by centrifugal force, 03 R, where co is the centrifugal speed in radianss" (27t radians = 360°) and R is the distance (cm) of the particle from the axis of rotation. 

In the centrifugal separation, the centrifugal acceleration in a circular motion can be represented by equation 2.8 ... 

The rotating flow is the combination of a solid-body rotation flow and a vortex flow. As will be seen in Chapter 17, this case corresponds to many configurations for which centrifugal separation is implemented. Theoretically speaking, such flows verify the properly AS = 0. This properly greatly simplifies the BBOT equations. It also makes it easier to identify the terms and mechanisms responsible for centrifugal separation. 

Centrifugal separation of solid particles in a fluid, but also of non-miscible droplets in another liquid or of gas bubbles in a liquid, is a frequently employed process. Its principle has already been described in the previous chapter on the basis of the BBOT equations. 

The similarities between gravitational separation and centrifugal separation are strong. Equation [17.44] shows that the action of rotation is quantified by the difference between the density of the particle and that of the fluid. If ps> Pf, the particle is centrifuged. In the reverse case, the particle experiences a movement toward the rotation axis (this is the case for air bubbles in a liquid, for example). The driving force is the difference between the centrifugal force exerted on the particle... 

Relation [17.71] gives the separation condition for any particle of size d introduced into the hydrocyclone. The equality determines the maximiun size Jmax of the particles so that all of them flow out through the spigot. By comparison with the results obtained for centrifuge decanters (equation [17.52]) and separators (equation [17.59]), it is found that the larger the flow rate is, the more effective the separation performed by a hydrocyclone will be. In the previous two cases, the flow rate should not exceed a certain value. This remarkable result, which seems smprising at first glance, since the residence time diminishes when the flow rate increases, is explained by the fact that the intensity of the vortex increases with the squared flow rate. 

The different cases of centrifugal separation treated in sections 17.4-17.7 of this chapter implement three steps (i) determination of the velocity field of the fluid, (ii) determination of the velocity of a particle by incorporating the centrifugation term (equation [17.44]), and (iii) formulation of a physical criterion ensuring that all particles entering the apparatus are separated. It is sought to identify the particle entering the apparatus that will be the most difficult to separate. The mathematical... 

Factors Influencing Centrifugal Sedimentation. The sedimentation velocity of a particle is defined by equations I and 2. Each of the terms therein effects separation. 

The ratio of these two equations gives the radial separation afforded by the gas centrifuge under equiUbrium conditions, that is, for no internal gas circulation. An equiUbrium separation factor between gas at the axis of the centrifuge and gas at the periphery is therefore given by ... 

It should be noted that the separation factor for the centrifuge process is a function of the difference in the mol wts of the components being separated rather than, as is the case in gaseous diffusion, a function of their ratio. The gas centrifuge process would therefore be expected to be relatively more suitable for the separation of heavy molecules. As an example of the equiUbrium separation factor of a gas centrifuge, consider the Zippe centrifuge, operating at 60°C with a peripheral velocity of 350 m/s. From equation 68, OC is calculated to be 1.0686 for uranium isotopes in the form of UF. ... 

The separation parameters have been calculated for a centrifuge in which the behavior of the circulating gas is described by Martin s equation. The flow pattern efficiency is shown in Figure 15(b) as a function of the dimensionless parameter M, where M is equal to (ME /2RT). In this case the maximum flow pattern efficiency attainable is 0.956. 

Consider a thin layer solid bowl centrifuge as shown in Figure 4.20. In this device, particles are flung to the wall of the vessel by centrifugal force while liquor either remains stationary in batch operation or overflows a weir in continuous operation. Separation of solid from liquid will be a function of several quantities including particle and fluid densities, particle size, flowrate of slurry, and machine size and design (speed, diameter, separation distance, etc.). A relationship between them can be derived using the transport equations that were derived in Chapter 3, as follows. 

The filtration equation (5.3-21) must be adopted if a centrifuge is used for separation. During centrifugation, the surface of the filter cake accessible for flow decreases since the cake builds up toward the axis of the centrifuge. Therefore, must be substituted by the product AcAim where is the average flow area and Aim is the logarithmic mean area through the cake. This yields ... 

The dipole moment is a fundamental property of a molecule (or any dipole unit) in which two opposite charges are separated by a distance . This entity is commonly measured in debye units (symbolized by D), equal to 3.33564 X 10 coulomb-meters, in SI units). Since the net dipole moment of a molecule is equal to the vectorial sum of the individual bond moments, the dipole moment provides valuable information on the structure and electrical properties of that molecule. The dipole moment can be determined by use of the Debye equation for total polarization. Examples of dipole moments (in the gas phase) are water (1.854 D), ammonia (1.471 D), nitromethane (3.46 D), imidazole (3.8 D), toluene (0.375 D), and pyrimidine (2.334 D). Even symmetrical molecules will have a small, but measurable dipole moment, due to centrifugal distortion effects. Methane " for example, has a value of about 5.4 X 10 D. 

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