Mathematical Problems in Engineering

Volume 2015, Article ID 169125, 7 pages

http://dx.doi.org/10.1155/2015/169125

## On the Accelerated Settling of Fine Particles in a Bidisperse Slurry

^{1}Tomsk State University, 36 Lenin Avenue, Tomsk 634050, Russia^{2}University of Erlangen-Nuremberg, 3 Paul-Gordan-Strasse, 91052 Erlangen, Germany^{3}Department of Natural Sciences, L.N. Gumilyov Eurasian National University, Munaitpasov Street 5, Astana 010008, Kazakhstan

Received 5 November 2014; Accepted 15 March 2015

Academic Editor: George S. Dulikravich

Copyright © 2015 Leonid L. Minkov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An estimation of increasing the volume average sedimentation velocity of fine particles in bidisperse suspension due to their capturing in the circulation zone formed in the laminar flow of incompressible viscous fluid around the spherical coarse particle is proposed. The estimation is important for an explanation of the nonmonotonic shape of the separation curve observed for hydrocyclones. The volume average sedimentation velocity is evaluated on the basis of a cellular model. The characteristic dimensions of the circulation zone are obtained on the basis of a numerical solution of Navier-Stokes equations. Furthermore, these calculations are used for modelling the fast sedimentation of fine particles during their cosedimentation in bidisperse suspension. It was found that the acceleration of sedimentation of fine particles is determined by the concentration of coarse particles in bidisperse suspension, and the sedimentation velocity of fine fraction is proportional to the square of the coarse and fine particle diameter ratio. The limitations of the proposed model are ascertained.

#### 1. Introduction

In a number of industries (mining, chemical, food, etc.) machines based on the principle of settling (sedimentation) of particles in a rotating fluid flow are used for the separation of particulate solids from air or liquid (air cyclones, hydrocyclones, centrifuges, decanters, etc.) [1].

Mathematical models for processes in such devices have long been developed that achieve impressive results. For example, Reynolds stress model [2–4] and Large Eddy Simulation approach [5–7] are employed to describe the turbulent flow field in cyclones. Particularities of solid phase motion are specified with discrete element model [2, 8], discrete phase model [4, 9], mixture model [10, 11], or Euler model [12, 13]. Even so, a number of effects are in principle incapable of being clearly mathematically described. Such effects include abnormal behavior of the so-called separation efficiency function, that is, the portion of the separated fraction of initial material depending on the particle size of the fraction.

This phenomenon is often a hindrance for engineers using hydrocyclones because the sharpness of fractionation often deteriorates due to the fine particles (in practice, particles smaller than 10 microns) contrary to expectations falling into the coarse product. Anomalous behavior of the separation function is particularly noticeable for the operation of small hydrocyclones, where the quality of the fine particle separation efficiency deteriorates, leading to an unexpected increase in the separation function with decreasing particle size (“fish-hook” effect) [1].

Abnormal growth of the separation function is the subject of lively debate. Various theories to explain this effect have been developed by now. One of them is based on a variable bypass to underflow of hydrocyclone [14, 15]; another is “mechanistic” model based on estimation of particle settling velocity depending on the number of Re [16]. Fluid mechanics based models, namely, the boundary layer model [17] and entrainment models [10, 18], which assign anomalous behaviour of the separate function to a hydrodynamic interaction of fine and coarse particles should be noted. A comprehensive critical analysis of existing models explaining “fish-hook” phenomenon has recently been published [19], where opposed to [20] the effect is attributed to measurement errors.

The most likely reason for this anomaly, in our opinion, is connected with the fact that there is an acceleration of sedimentation of fine particles under the influence of neighboring coarse particles in the polydisperse suspension. Due to an entrainment the fine particles have a much higher sedimentation velocity than expected and are intensively removed together with a coarse fraction from the hydrocyclone [18]. The mechanism of the accelerated sedimentation of fine particles has not been studied well enough.

To explain this effect, a model of the cell for bidisperse suspension has been proposed [21]. That model relies on the determination of the mean residence time of small particles in the cell surrounding the large ones and the subsequent determination of an average sedimentation velocity of small particles. Accelerated sedimentation of small particles occurs due to their retention in the flow boundary layer near the surface of the large particle. Herein the flow around the large particle is postulated identical to the Stokes flow that is valid for . An analysis of the effect of the Reynolds number on the sedimentation velocity of the particles and consequently on the hydrocyclone separation curve has been carried out [22], but it has not considered the particle-particle interaction phenomena which is so important in the range of the finest particles. Further development of the sedimentation model followed the path of propagation of the entrainment mechanism of small particles by the large ones for the case of polydisperse suspension, using it to evaluate the separation function of the hydrocyclone.

Thus, it was possible to calculate the separation function, including its nonmonotonic character [23]. At the same time it becomes clear that some large particles in suspension in a hydrocyclone can move at Re of the order of 10^{2}. It is known [24] that the boundary layer around the moving particles in a liquid is radically changed already at Re 25: instead of the smooth streamlines along the direction of particles motion a zone of reverse flows appears. They become more complicated for the increased Reynolds number. This requires further development of the entrainment model.

The logical explanation for the accelerated sedimentation of small particles in the presence of large ones can be considered as a capture of small particles entering the hydrodynamic wake generated by a large particle at the numbers of Re > 25.

The aim of this paper is to study the circulation zone that occurs in the laminar flow of an incompressible viscous fluid around a spherical particle and the determination of the volume average sedimentation velocity of small particles during the sedimentation of bidisperse suspension, based on a cellular model.

#### 2. Mathematical Formulation of the Problem

The system of Navier-Stokes equations describing the steady state laminar flow of an axisymmetric incompressible viscous fluid around a sphere in a cylindrical coordinate system is as follows [25]:where

The domain of integration of the system of (1), presented in Figure 1, is limited by the input boundary AB, the output boundary BC, and the axis of symmetry , as well as the contour of the sphere ED.