Stokes hydrocyclones

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

A similar approach to the analysis of hydrocyclones was presented by Svarovsky (1984, 1990). He deduced that the system can be described in terms of three dimensionless groups in addition to various dimensionless geometric parameters. These groups are the Stokes number,... 

Gravity and centrifugal sedimentation can be combined for the same sample in order to directly determine Stokes diameter for a wide range of particle sizes. In such a way conversion are avoided and a mass distributions, applicable to processes where gravimetric efficiencies are relevant, can be properly derived. Ortega-Rivas and Svarovsky (1994) determined particle sizes distributions of fines powders using a combined Andreasen Pipette-pipette centrifuge method. They derive relations useful to model hydrocyclone separations, which were later employed to describe apple juice clarification. 

As was stated in the introduction, Stokes diameter jcst is usually used to characterize particle size in those applications where it is the behaviour of particles in liquids that determines the separation efficiency and other operational characteristics of the separators (e.g. in sedimentation, centrifugation and hydrocyclones). Methods that measure Stokes diameter, such as sedimentation or fluid classification, have therefore been used extensively in this field. Although preference is naturally given to wet methods, air classification is also widely used. 

Another theoretical approach to cut size prediction that can be classified as another version of the residence-time theory is that of Trawinski. In direct analogy with gravity settling Trawinski used Stokes law, an effective clarification area and an average acceleration in a hydrocyclone to derive an expression for the cut size. The same author also proposed a rather simplistic correlation for the pressure drop-flow rate relationship. 

Despite the very different approaches and assumptions, the forms of correlations obtained by the equih-brium orbit theory and residence time theory are similar. For specific hydrocyclone designs, both theories provide their respective empirical equations for determining the cut size and pressure drop in terms of three dimensionless groups, the Stokes number at cut size, Stkso, the Euler number, Eu, and the Reynolds number, Re (see discussions in Sec. 5.4 below) ... 

A second, very classical rotating flow is the vortex flow, of which a practical example in hydrocyclones will be seen in section 17.2.3. For now, let us seek a simple soluhon of the Navier-Stokes equations, in the form given by [17.5], which is steady and irrotational. The general solution of [17.9b] for the azimuthal component of the velocity can be written as ... 

Oil droplet size (at fixed concentrations). Efficiency generally decreases as the oil droplet size is reduced. This is consistent with Stokes law, where a smaller droplet will move less rapidly toward the hydrocyclone core. Droplets below a certain size (about 30 pm) are not captured by the hydrocyclone and, therefore, as the median feed oil droplet size decreases, more of the smaller droplets escape and the efficiency drops. Restrictions (valves, fittings, etc.) and pumps causing droplet shearing in the incoming flow should be avoided. 

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