^{1}

^{1}

^{2, 3}

^{1}

^{1}

^{2}

^{3}

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.

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.) [

Mathematical models for processes in such devices have long been developed that achieve impressive results. For example, Reynolds stress model [

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) [

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 [

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 [

To explain this effect, a model of the cell for bidisperse suspension has been proposed [

Thus, it was possible to calculate the separation function, including its nonmonotonic character [^{2}. It is known [

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.

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 [

The domain of integration of the system of (

Domain of integration.

The boundary conditions are specified as follows.

The solid wall (ED) is

The input boundary (AB) is

The output boundary (BC) is

The axis of symmetry

A solution of the system (

The implementation of this method was carried out using the software package ANSYS-Fluent. For the construction of the finite difference grid, the program Gambit was used.

The ratio of the radii of boundaries ABC and ED is set as a constant,

Special research found that sufficient accuracy of calculations (with an error of 2%) at Re = 100 is achieved by using a computational grid of 40,000 cells.

In particular, the experimental Schiller-Naumann correlation [

The difference in the results of the numerical solutions and the experimental data does not exceed 5% for

As was mentioned above when the Reynolds number is greater than 25 a circulation zone is formed at the rear of the sphere [

Below we limit ourselves to describing only those geometric characteristics of the circulation zone, which are needed for the later described model of the particle mixture sedimentation.

The dependence of the volume of the circulation zone related to the volume of the sphere on the Reynolds number is presented in Figure

Dependence of the circulation zone volume on the Reynolds number. 1: numerical experiment, 2: approximation curve, (

As follows from (

The size of the circulation zone can be characterized by its length

Figure

Dependence of the circulation zone length on the Reynolds number. 1: numerical experiment, 2: approximation curve, and 3: experimental data [

It is similarly possible to establish the dependence of another important parameter, namely, the radius of circulation zone (Figure

The results of experiments obtained by Taneda [

Thus, the relative length and the relative volume as functions of the Reynolds number can characterize the circulation zone at the rear of the sphere.

The sedimentation of a bidisperse suspension (a mixture of coarse particles with diameter

We suppose that, during the cosettling of coarse and fine particles, the latter trapped in the circulation zone formed at the rear of the coarse particle have the same sedimentation velocity as the coarse

Let us assume that the considered cell has a volume

The equilibrium equation of forces acting on a particle during the sedimentation allows one to obtain its velocity:

For fine particles which can be considered as the Stokes ones

Therefore,

As a volume fraction of the coarse particles in the suspension is expressed through the volume of the cell and the own volume of coarse particle by the relation

As can be seen from (

From (

An analysis of (

The limits of the applicability of formula (

As follows from (

Maximum possible volume fraction of coarse particles.

In the region of practical interest, Re is about 10^{2} and

Equations (

Figure

Maximum possible volume average sedimentation velocity of fine particles. 1—

It is noteworthy that the increase in the sedimentation velocity of fine particles by described mechanism is limited and should not be expected to be higher than 10 times.

Equation (

The circulation zone under operating conditions of a hydrocyclone (let us take ^{2}) with a quartz water-sand suspension is formed for particles larger than 88 microns and in the gravitational settling for particles larger than 400 microns. Thus, the described mechanism of accelerating the particles sedimentation in a suspension can probably be realized if the suspension includes particularly large particles.

Based on a numerical simulation of the laminar flow of incompressible viscous fluid around a spherical particle the geometric characteristics (volume and length) of the circulation zone formed in the rear of the sphere depending on the Reynolds number for 25 < Re < 1000 were obtained.

The formula for the volume average sedimentation velocity of fine particles settling in the presence of coarse particles (in the case of bidisperse suspension) under the assumption that the fine particles trapped in the circulation zone settle at a velocity of coarse ones was obtained. An expression for the maximum sedimentation velocity of fine particles shows nonmonotonic behavior depending on the Reynolds number.

Particle diameter

Drag function

Acceleration

Pressure

Time

Spatial coordinates

Drag coefficient

Relative length of circulation zone

Radius

Reynolds number

Reynolds number at which the recirculation zone appears (equal to 25)

Particle velocity

Fluid flow velocity

Relative volume of circulation zone

Volume fraction

Kinematic viscosity

Density

Volume.

Boundary ABC

Boundary ED

Coarse particle

Cell

Fine particle

Liquid

Maximum

Minimum

Radial direction

Solid

Sphere

Axial direction.

The authors declare that there is no conflict of interests regarding the publication of this paper.

Support for this work was provided by the Ministry of Education and Science of Russian Federation (agreement no. 10.1329.2014/K) and Tomsk State University Competitiveness Improvement Program.