Fundamentally Based Hydrocyclone Models

Early attempts at understanding the physical principles that govern size separation in hydrocyclones yielded theories based on equilibrium, residence time and crowding. More complete simulations in which fluid and particle motion is estimated from solution of the Navier-Stokes equations have been developed more recently.

Equilibrium orbit theory It can be postulated that particles will find an equilibrium orbit in the hydrocyclone where their terminal settling velocity radially outward is equal to the radial velocity of the liquid inward. A particle will report to the spigot if its equilibrium orbit is in the downward axial liquid flow and to the vortex finder if in the upward axial flow. The cut size is defined by particles that have an equilibrium orbit that coincides with the locus of zero vertical velocity and therefore have an equal probability of reporting to either product streams. An equilibrium orbit may not be achieved due to the short residence times and high solids concentrations in the hydrocyclone.

Residence time theory This theory determines whether the residence time in the hydrocyclone allows a particle entering the cyclone at the centre of the inlet to settle to the cyclone wall and enter the boundary layer flow to the underflow.

Crowding theory At higher feed concentrations, it is found that the separation size is primarily determined by the discharge capacity of the spigot and the feed size distribution. By controlling the outlet dimensions, it is thought that any cut size within the feed size distribution can be obtained.

Computational fluid dynamics (CFD) solutions

This is the preferred approach for fundamentally based modelling of hydrocyclone performance. Complete flow modelling of the hydrocyclone

Figure 5 Effect of feed solids concentration on hydrocyclone separation. Circles, 2.68 vol%; squares, 11.11 vol%; triangles, 17.54 vol%; diamonds, 23.75 vol%. (Reproduced with permission from Braun and Bohnet (1989). Copyright: Wiley-VCH.)

involves predicting the liquid-phase velocities, the slurry concentration profile, the turbulent viscosities and the slip velocities of particles with respect to the liquid phase for a range of particle sizes before predicting the partition curve. The solution is complex, because the governing fluid flow equations are nonlinear, simultaneous partial differential equations.

Chakraborti and Miller (1992) have published an extensive review of fluid flow modelling in hydrocyclones. They describe the flow models in detail, giving particular attention to models based on the Navier-Stokes equation and the treatment of fluid turbulence. They further discuss techniques for flow measurement and visualization and give a brief summary of pressure drop correlations and measurements. This paper is an essential reference for the fluid flow modelling approach.

The general approach to develop a complete CFD-based model of a hydrocyclone must include a wide range of components. If it is assumed that variations of local density and viscosity are small for dilute slurries and that particle-particle interactions are negligible, the fluid and particle modelling can be decoupled. Liquid velocities are predicted by combining the fluid transport equations for vorticity, stream function and angular spin velocity with a modified Prandtl mixing length model, which varies both radially and axially, for the turbulent viscosity. The set of simultaneous, nonlinear partial differential equations are solved by overlaying the hydrocyclone dimensions with a rectangular grid and using appropriate boundary conditions at the solid walls and liquid-air interface, to solve for conditions within each cell of the grid. By balancing all the forces on the particle, the particle motion with respect to the fluid can be computed. The particle trajectories are found by calculating axial and radial slip velocities with respect to the fluid. Size classification performance is determined by following a particle of a given size from the inlet until it exits. This computation is repeated for each particle size across the inlet diameter yielding the partition curve.

For concentrated slurries, liquid-phase velocities are affected by local density and viscosity, which in turn are affected by local solid concentration and particle size distribution. Since particle motion determines the concentration and size distribution at each location, this being determined from liquid velocities, an iterative solution is required so that local slurry property changes can be estimated and liquid-phase velocities recalculated.

Advances in CFD methods such as computation grid generation, numerical methods and computing resources are increasing the applicability of this modelling technique to improve designs.

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