Orbit theory, hydrocyclones

Figure SJ,2 Variation of cut size with feed flow rate measured and equilitnium orbit theory. Hydrocyclone dimensians total length 273 nun conical length 221 nun diameter 42 mm inlet diameter 5 mm. Solid and liquid densities 2710 and 1000 kg m viscosity 0.001 Pa s. Flow splits in order 8.7,12,1 and 19.61 min" 0.138,0.104, and 0.082...

EquHibrium orbit theory is a useful means of correlating and eiq>kiiimg the relation between flow rate and hydrocyclone cut size. Its use as a predictive tool is limited, however, as tests must be conducted to determine several parameters required in the model notably the flow split and the eiqionent on the radius in the cyclone. It does not provide any information on the pressure drop required to perform the separation, or on the arpness of the cut. 

Each theory in this category offers a relatively simple correlation for the static pressure drop and the cut size of a hydrocyclone described by a few (but often not all) dimensions. The theories fall into two main groups the equilibrium orbit theory and the residence time theory. 

A number of physical models have been proposed for the separation process in a hydrocyclone (Driessen MG, 1951 Bradley and Pulling, 1959 Fahlstrom, 1960 Kelsall, 1952 Rietema, 1961 and Schubert and Neesse, 1980). Among these, different phenomenological approaches have led to the development of two basic theories the equilibrium orbit theory and the residence time theory. 

Equilibrium Orbit Theory. The general concept that particles of a given size reach an equilibrium radial orbit position in the hydrocyclone forms the basis of equilibrium orbit theory. The fine particles reach equilibrium at small radii where the flow is moving upwards and transports fines to the overflow, while the coarse particles find equilibrium position at large radii where the flow is moving downwards and carries these particles to the underflow outlet (apex). The dividing surface is the locus of zero vertical velocity (LZW). The size of the particles that find equilibrium radius on LZW will be the cut size that has an equal chance to finish in either overflow or underflow. 

In developing the equilibrium orbit theory, a key assumption made by Bradley and Pulling (1959) is the existence of a mantel in the hydrocyclone, which precludes inward radial velocity in the region immediately below the vortex finder. Furthermore, the LZW is assumed to be in the form of an imaginary cone whose apex coincides with the apex of the hydrocyclone and whose base is at the bottom of the mantle. Based on these assumptions, the equilibrium orbit theory has led to the development of empirical correlations for determining the cut size and pressure drop in hydrocyclone operation. 

The major deficiency of the equilibrium orbit theory lies in its lack of consideration of the effect of turbulence flow on particle separation and the residence time of the particles in the hydrocyclone (as not all particles are able to find equilibrium orbits within their residence time). In spite of such weaknesses, it proves to be a reasonable approach for determining the hydro-... 

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) ... 

Most studies of hydrocyclone performance for particle classification have been carried out at particle concentrations of about 1 per cent by volume. The simplest theory for the classification of particles is based on the concept that particles will tend to orbit at the radius at which the centrifugal force is exactly balanced by the fluid friction force on the particles. Thus, the orbits will be of increasing radius as the particle size increases. Unfortunately, there is scant information on how the radial velocity component varies with location. In general, a particle will be conveyed in the secondary vortex to the overflow, if its orbital radius is less than the radius of that vortex. Alternatively, if the orbital radius would have been greater than the diameter of the shell at a particular height, the particle will be deposited on the walls and will be drawn downwards to the bottom outlet. 

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