Particulate,flow,modelling,in,a,spiral,separator,by,using,the,Eulerian,multi-fluid,VOF,approach

Lingguo Meng ,Shuling Go, *,Dezhou Wei, *,Qing Zho ,Boyu Cui ,Yni Shen ,Zhenguo Song

a School of Resources and Civil Engineering,Northeastern University,Shenyang 110819,China

b State Key Laboratory of Mineral Processing,Beijing 100160,China

Keywords: Spiral separator Computational fluid dynamics (CFD)Eulerian multi-fluid VOF model Bagnold effect Particulate flow

ABSTRACT The Euler-Euler model is less effective in capturing the free surface of flow film in the spiral separator,and thus a Eulerian multi-fluid volume of fluid(VOF)model was first proposed to describe the particulate flow in spiral separators.In order to improve the applicability of the model in the high solid concentration system,the Bagnold effect was incorporated into the modelling framework.The capability of the proposed model in terms of predicting the flow film shape in a LD9 spiral separator was evaluated via comparison with measured flow film thicknesses reported in literature.Results showed that sharp air-water and air-pulp interfaces can be obtained using the proposed model,and the shapes of the predicted flow films before and after particle addition were reasonably consistent with the observations reported in literature.Furthermore,the experimental and numerical simulation of the separation of quartz and hematite were performed in a laboratory-scale spiral separator.When the Bagnold lift force model was considered,predictions of the grade of iron and solid concentration by mass for different trough lengths were more consistent with experimental data.In the initial development stage,the quartz particles at the bottom of the flow layer were more possible to be lifted due to the Bagnold force.Thus,a better predicted vertical stratification between quartz and hematite particles was obtained,which provided favorable conditions for subsequent radial segregation.

As typical gravity concentration devices,spiral separators are widely used in mineral processing and coal preparation because of their excellent performance,such as efficient sorting,and low cost.Under the combined action of centrifugal and gravitational forces,the flow film in the spiral separator generates different flow patterns(primary and secondary flows)[1]and flow regimes(laminar,transitional,and turbulent flows)[2],resulting in the different radial distributions of light and heavy particles.Owing to their practicability and complexity,spiral separators have attracted considerable research attention;however,the mechanism of particulate flow behavior inside a spiral separator remains fundamentally unknown.

Until now,theoretical modeling,physical measurement,and numerical simulation methods have been applied to investigate the particulate flow behavior in spiral separators.Theoretical models can be developed based on an established scientific understanding of the processes and basic laws of multiphase hydrodynamics.Because of the complexity of the processes involved in a spiral separator,steady flow and thin-film assumptions are generally employed to simplify the governing equations[3].Therefore,the applicability of current theoretical models of spiral separators is limited to steady-state conditions.Physical measurements are considered a reliable method to obtain practical information regarding particulate flow.Although a spiral separator is an open system,the poor anti-interference ability of thin films renders the application of intrusive flow field measurement methods impossible.Fortunately,the positron emission particle tracking technique(PEPT),which allows the quantitative evaluation of flow phenomena in three dimensions in opaque media,was successfully applied to investigate the particle motion in a spiral separator [4].However,this technique is relatively expensive,which limits its use as a routine measurement method.In comparison,the superiority of numerical simulation lies in high efficiency and low-cost[5-8].In recent years,researchers have adopted various numerical methods for investigating the particulate flow in a spiral separator,such as computational fluid dynamics (CFD),discrete element method (DEM),and smoothed particle hydrodynamics (SPH).The applications of these models are summarized in Table 1.

Table 1 Summary of previous particulate flow models on spiral separator.

It can be seen from Table 1 that researchers are more inclined to use the Euler-Euler model based on CFD in the particulate flow simulation of spiral separators.The Euler-Euler model treats granular flows as continuous fluid flows and generally uses the kinetic theory of granular flow (KTGF) model to describe the granular interactions,which is the primary method for the prediction of large-scale particulate flow behavior.However,the clarity of the two-phase interface obtained by the Euler-Euler model is inferior to that of the volume of fluid (VOF) model.This is because that the basic property of the Euler-Euler model is a consequence of the temporal and spatial averaging,and each fluid as a continuum occupies the entire domain [14].Therefore,the Euler-Euler model shows poor accuracy in the prediction of the free surface of the flow films in spiral separators.To overcome this drawback,researchers modified the computational domain of spiral separator for particulate flow.Specifically,before building the computational domain for particulate flow [9,11,12],it is necessary to obtain the free surface position of the flow film by simulating the fluid flow.Then,the free surface profile is used as the upper boundary of the particulate flow computational domain,and remains fixed during the simulation process of particulate flow.Although this avoids the numerical diffusion at the air-water interface by removing the air computational domain,it is only applicable in the simulation of dilute particulate flow.Once the particle concentration increases,the flow film shape changes accordingly,rendering this method infeasible.Therefore,successfully predicting the change in the flow film free surface caused by a high solid concentration is a necessary step for the simulation of particulate flow in a spiral separator.

To address the limitations of the Euler-Euler model,Č erne et al.[14]first proposed a coupled model comprising the VOF model and the Euler-Euler model.The coupled model has been developed as a new multiphase flow model,the Eulerian multi-fluid VOF model,in the ANSYS Fluent software.The coupled model is designed to simulate the computational domain containing the area wherein the fluids are separated,which is evaluated using the VOF model,and the area wherein the fluids are mixed,which is evaluated using the Euler-Euler model.In particular,the Eulerian multi-fluid VOF model exhibits the advantages of the VOF model and the Euler-Euler model.The VOF model with higher interface tracking accuracy is used for flow regimes with obvious phase separation to avoid numerical diffusion of the interface.Moreover,the inclusion of the Euler-Euler model prevents the non-physical phenomenon of dispersed flow caused by the VOF model.Therefore,it could be theoretically considered that this coupled model is highly feasible for simultaneously investigating particle movement behavior and flow film shape changes.However,the application of this coupled model is mostly focused on gas-liquid flow in a mixed or transitional state such as churn flow,plunging jet flow and intermittent flow [15-17],the applicability of this coupled model to the gasliquid-solid three-phase system in a spiral separator remains to be tested.

In addition,a higher particle concentration not only changes the flow film shape but also leads to the enhancement of the Bagnold effect.The Bagnold effect was first proposed by Bagnold in 1954[18].The physical meaning of the Bagnold effect is as follows:When the particles in suspension are subjected to continuous shearing,a dispersion pressure will be generated perpendicular to the shear direction,which makes the particles have a tendency to expand along both sides.The dispersion pressure increases with the velocity gradient.When the velocity gradient is large enough to balance the dispersion pressure with the effective gravity of the particles in the medium,the particles will be suspended.The Bagnold effect significantly contributes to the vertical stratification of particles in the spiral separator[19].In fact,the particle concentration in a spiral separator gradually increases radially inward.Although the feed pulp concentration is lower than that in the normal operational conditions,the solid concentration in the inner trough region remains high[20].Therefore,the Bagnold effect will exist in most working conditions of the spiral separator and cannot be ignored.Das et al.[21] proposed an improved mathematical model,which incorporated Bagnold effect to predict the particle and flow behavior in a coal-washing spiral separator.Recently,Jain and Rayasam[22]proposed a simplified form of Bagnold’s original model in the particle-inertia region,which is of great significance for the quantitative study of the Bagnold effect.

In the present work,the optimization of the simulation modeling methods of the particulate flow in a spiral separator was attempted.The Eulerian multi-fluid VOF model with the lift force model considering Bagnold effect was first adopted to investigate the complex multiphase flow behavior in a spiral separator.The performance of the Eulerian multi-fluid VOF model with regard to predicting the change in the flow film shape was evaluated.Furthermore,an experimental test using mixed ore was carried out to verify the reliability of the proposed lift force model with Bagnold effect considered.Moreover,the mechanism of Bagnold effect on the vertical stratification of particles was further analyzed.

Systematic physical experiments were carried out in a laboratory-scale spiral separator to investigate the effect of trough length on the movement behavior of particles.The experimental data are used to verify the prediction accuracy of the particle separation performance using the simulation method.A schematic of the experimental setup of the spiral separator is shown in Fig.1.As shown in Fig.1,the spiral separator,pump,mixing tank,and feeding tank were assembled into a closed circuit arrangement.The spiral separator used in the experiment was laboratory-scale with parabolic cross-sectional geometry [23].The feed flow rate can be regulated by a control valve of the bypass.The feed box is mobile and can be placed at various heights to change the trough lengths.The outlet is divided into 3 regions(R1,R2,R3) for intercepting the products of the inner,middle,and outer troughs.The radial width of each region is shown in Fig.2.

Fig.1.Experimental setup of the laboratory-scale spiral separator.

Fig.2.Division of radial regions (R1-R3) at the outlet.

Quartz and hematite particles with narrow size ranges were prepared by pre-screening.The size distributions of the quartz and hematite particles were obtained using a laser particle size analyzer,and the measurement results are shown in Fig.3.

Fig.3.Particle size distribution of quartz and hematite particles.

It can be observed from Fig.3 that the median particle size(D50)of quartz and hematite particles are 75.6 and 86.7 μm,respectively.The mixed ore obtained by mixing quartz and hematite particles in a volume ratio of 1:1 was experimentally tested.Six trough lengths were investigated in the experiments: 0.33,1,2,3,4,and 5 turns,respectively.The experimental conditions in the laboratory-scale spiral separator are shown in Table 2.The test products were collected for a fixed time,dried,weighed,and analyzed for grade of iron and solid concentration by mass.The solid concentration by mass is defined as the percentage of the mass of the particles to the total mass of the pulp.Feed grade of iron is defined as the mass fraction of iron element in the feed materials.

Table 2 Experimental conditions in the laboratory-scale spiral separator.

The experimental relative errors are mainly related to the uncertainty of flow field as well as particle movement.The multiphase flow behaviors in the spiral separator change all the time during the experimental process.If the sampling time is too short,the sample will be less representative.In order to avoid the adverse effects caused by the uncertainty of the flow field and particles,we adopted a sampling time of 20 s to ensure that enough samples were collected.In addition,the uncertainty of experimental equipment will also affect the accuracy of the experiment.After sampling,the pumping performance of the pump will change due to the reduction of the total amount of pulp,leading to the fluctuations of the feed flow rate.Therefore,the equal amount of pulp should be added in the mixing tank after each sampling to maintain a stable feed flow rate.Moreover,to reduce the error caused by artificial operation,each experiment needs to be repeated three times.Results of repeated experiments show that the relative standard deviation for the solid concentration by mass and the grade of iron is less than 5%,which confirms a good reproducibility.The final data is the average of 3 parallel experiments.

The Eulerian multi-fluid VOF model provides a framework to couple the VOF and Euler-Euler models,which allows the use of discretization schemes and options suited to both sharp and dispersed interface regimes.The basis of model coupling is that the color function for tracking the interface in the VOF model and the volume fraction variable in the Euler-Euler model have identical meanings.Furthermore,the VOF model and the Euler-Euler model can be converted by calculating the value of the void fraction gradient at adjacent cells.Meanwhile,the Eulerian multifluid VOF model overcomes the limitations of the VOF model that arise owing to the shared velocity and temperature formulation.On the other hand,the Eulerian multi-fluid VOF model prevents the interface diffusion of the Euler-Euler model because of the empirical closures required in the averaged equations.A detailed coupled process can be found in ˇC erne et al.[14].In a word,the Eulerian multi-fluid VOF model is suitable for both free surface and dispersed flow simulations and can capture the interface position.Specifically,when the Eulerian multi-fluid VOF model is used,the interaction between the air and water will be simulated by the VOF interface tracking algorithm since the air and the water in the spiral separator are distinctly stratified.Meanwhile,the interactionbetween water and particles will be simulated by the Euler-Euler model since the particles are dispersed in the water,which can be seen clearly from Fig.4.

3.1.Euler-Euler model

3.1.1.Governing equations

For the Euler-Euler model,the continuous and discrete phases are regarded as interpenetrating continua.Particles with different densities or sizes were treated as separate phases.The phaseaveraged continuity and momentum equations of each phase are given by

3.1.2.Drag force model

The Gidaspow drag force model was used to describe the drag force between the particles and water,which is expressed as follows [27]:

When α1>0.8,

When α1≤0.8,

where CDis the drag coefficient,expressed as follows:

Fig.4.Schematic diagram of computational domains for the VOF and Euler-Euler models.

where Re is the relative Reynolds number.

The symmetric drag model was used to describe the drag force between air and water,which can be expressed as follows [28]:

where,

The density and viscosity are calculated from the volumeaveraged properties,which are given as

and

The diameter in Eq.(7) is defined as

3.1.3.Bagnold lift force model

Bagnold considered that when the solid particles in the suspension are continuously subjected to shearing,the dispersion pressure is generated perpendicular to the shearing direction.If a particle is suspended under laminar flow conditions,the dispersion pressure is equal to the effective gravity of the particle per unit area.To distinguish the types of dispersion pressures,Bagnold proposed a dimensionless factor,Bagnold number (N),defined as the ratio of inertial stress to viscous stress,and it can be expressed as [22]

where λ is the linear concentration,expressed as follows [22]:

Bagnold defined two limiting flow regimes based on the value of N: the macro-viscous dominant region (N≤40) and the particle-inertia dominant region(N≥450)separated by a transition region (40

The dispersion pressure in the particle-inertia dominant region,P1,on a particle can be expressed as [19]

The dispersion pressure,P2,on a particle in the macro-viscous dominant region is given as [19]

Similar to the assumption by Jain and Rayasam [22],the particles were assumed to be inelastic spheres.Because the dispersion pressure develops across the plane of shear at right angles to the surface of shear,the corresponding Bagnold force can be considered as the product of the dispersion pressure and the vertical projected area of the particle.The Bagnold force in the particleinertia dominant region,FB,inertia,is given as

The Bagnold force in the macro-viscous dominant region,FB,vis-cosity,can be expressed as follows:

However,Bagnold did not explicitly express the dispersion pressure in the transition region,where the dispersion pressure in both the macro-viscous dominant region and the particleinertia dominant region contribute to particle suspension.In order to complete the Bagnold force equations,we assume a kind of Bagnold force in the transition region,which is given as

where N1=40 and N2=450.

The equation uses the interpolation method and combines the traditional Bagnold force equation in the particle-inertia dominant region and macro-viscous dominant region.The advantage of this assumption is that it can qualitatively predict the variation trend of the two Bagnold forces(FB,inertiaand FB,viscosity)with the increase of N in the transition region.Specifically,we multiplied each Bagnold force by a scaling factor to estimate the Bagnold force in the transition region.When N is extremely close to 40,the Bagnold force in the transition region will be dominated by FB,viscosity.With the increase of N,the proportion of FB,viscositygradually decreases,while the proportion of FB,inertiaincreases accordingly.When N almost increases to 450,the Bagnold force in the transition region is nearly transformed into the FB,inertia.

Consequently,the complete Bagnold force expressed by Eqs.(16)-(18) was reproduced using the user-defined function code in the form of the lift coefficient.For the convenience of description,the proposed lift force model is called the Bagnold lift force model in the following sections.

3.2.VOF interface tracking model

The VOF interface tracking model is used to describe the gasliquid stratified flow pattern,the continuity and momentum equations for the incompressible two-phase flow can be expressed as[17]

The surface tension force between the gas and liquid phases is not considered in this paper.To capture the free surface interface of the flow film accurately,a compressive scheme was applied to discretize the convection terms in the volume fraction equation to maintain the sharpness of the interface.The compressive scheme is a second-order reconstruction scheme based on a slope limiter.Slope limiters are used in spatial discretization schemes to avoid spurious oscillations or wiggles that would otherwise occur with high-order spatial discretization schemes owing to sharp changes in the solution domain.The theory below is applicable to zonal discretization and phase-localized discretization,which uses the framework of the compressive scheme [28].

The slope limiter was constrained to values between 0 and 2(inclusive).For values less than 1,spatial discretization is represented by a low-resolution scheme.For values between 1 and 2,spatial discretization is represented by a high-resolution scheme.

3.3.RNG k-ε turbulence model

The RNG (renormalization group) k-ε turbulence model was used to model the turbulence problems in the spiral separator.The relevant transport equations of the RNG k-ε turbulence model are as follows [11]:

where k is the turbulence kinetic energy;ρ the density of fluid;vithe component of velocity in i direction;ε the dissipation rate of k;αkthe inverse effective Prandtl numbers for k;αεthe inverse effective Prandtl numbers for ε;Gkthe generation of turbulence kinetic energy owing to the mean velocity gradient;Rε the additional term;and μeffthe effective viscosity,which is given by

Constants in turbulence equations are C1ε=1.42,C2ε=1.68,Cμ=0.0845 and μ is the molecular viscosity.

3.4.Simulation strategy and simulation settings

Owing to the complexity of the multi-phase flow in the spiral separator,the simulation was divided into two steps to improve the computational stability.On the one hand,a preliminary flow field containing only the water and air phases in the spiral separator was simulated using the Eulerian multi-fluid VOF model.In the simulation,water was regarded as the primary phase while air was treated as the secondary phase.At the beginning of the simulation,the computing domain was completely occupied by air,and then water was injected through the inlet.Consequently,the fluid flow characteristics were obtained after the gas-liquid two-phase flow field was stable.On the other hand,based on the stable fluid flow field,a certain proportion of particles were released from the inlet.Subsequently,the particulate flow characteristics were investigated when the gas-liquid-solid three-phase flow field was stable.

CFD simulations were performed under identical operating conditions and structural parameters of the experiment using the laboratory-scale spiral separator.The particulate flow simulation settings in the laboratory-scale spiral separator are listed in Table 3.Considering that the particle size range(mainly ranging from 40 to 100 μm)of the material used in the experiment was narrow,it can be considered that the particles with the size of D50can represent the majority of particles.Therefore,the feed particle size adopted in the simulation was assumed to be the D50of the feed in the experiment for simplifying the study.

For evaluating the prediction accuracy of the Eulerian multifluid VOF model for the free surface of the flow film,CFD simulations of the liquid flow and the particulate flow were performedin a LD9 spiral separator.The simulated flow film thickness will be compared with the experimental measurement data by Holtham[1,20].In the Holtham’s works,he first measured the water flow film thickness in the LD9 spiral separator with a flow rate of 6 m3/h.Subsequently,he measured the pulp flow film thickness in the LD9 spiral separator with the same flow rate and the quartz mass fraction of 15%.The structural parameters and experimental conditions in the LD9 spiral separator are list in Table 4.

Table 3 Particulate flow simulation settings in the laboratory-scale spiral separator.

Holtham[20]found a pronounced bulging of the pulp surface in the inner trough compared to the water flow film,implying that the addition of particles will have an important impact on the flow film shape.Therefore,the evaluation of the Eulerian multi-fluid VOF model can be achieved by examining whether it can accurately predict the changes of flow film shape.With reference to the measurement conditions by Holtham [1,20],the particulate flow simulation settings in the LD9 spiral separator are listed in Table 5.It is worth noting that the quartz used by Holtham has a wide particle size range,mainly ranging from 50 to 2000 μm with the D50of 530 μm.In fact,it is difficult to use Eulerian multi-fluid VOF model to model particles with wide size distributions since,theoretically,particles with different size values should be treated as separate phases,which will significantly increase the computational cost.Therefore,it is necessary to reasonably simplify the particle size to improve the computational efficiency.Holtham[20] further examined the particle size composition of quartz in the inner trough region of LD9 spiral separator,he found that the bulging of the pulp surface was mainly related to the quartz particles with the particle size around 560 μm,which is very close to the D50of the raw material.Consequently,the particle size of quartz was simplified to the D50in the simulation process.

Table 4 Structural parameters and experimental conditions in the LD9 spiral separator.

Table 5 Particulate flow simulation settings in the LD9 spiral separator.

3.5.Computational domain mesh and boundary conditions

The establishment of the computational domain mesh is carried out in the LD9 spiral separator and the laboratory-scale spiral separator.The mesh independence check is necessary for minimizing the errors and improving computational efficiency.In fact,thecomputational domain mesh of spiral separator is consistent for each turn.Therefore,it is sufficient to consider only one complete turn to carry out the mesh independence check.Three mesh schemes(145360,308416,and 474000 cells)in the LD9 spiral separator and three mesh schemes(78000,149760,and 303072 cells)in the laboratory-scale spiral separator were generated to conduct the mesh independence check from two aspects of liquid flow and particulate flow.For the liquid flow,it has been performed by predicting and comparing the surface tangential velocity of water at the end of the complete one turn with different mesh schemes,and the result can be found from Fig.5a and b.For the particulate flow,it has been investigated by predicting and comparing the distribution of the solid volume fraction along with the water depth at a radial position at the end of the complete one turn with different mesh schemes.The investigated radial positions of the two spiral separators should be selected in the region where particles are relatively enriched.Therefore,the r=150 mm in the LD9 spiral separator and r=80 mm in the laboratory-scale spiral separator are selected as the representative radial positions for the mesh independence test.Since the flow film thickness at each radial position is different,the relative water depth is used to characterize the vertical position of the fluid particle,which is defined as the ratio of the vertical distance between a fluid particle and the bottom wall to the total flow film thickness.Specifically,0 represents the bottom wall and 1 represents the free surface of the flow film.The results can be found from Fig.5c and d.The mesh independence test results show that the mesh size has little effect on the simulation results when the mesh number was beyond 308416 and 149760 for LD9 spiral separator and laboratory-scale spiral separator,respectively.

The complete computational domain meshes based on the second mesh scheme of the two types of spiral separators are presented in Fig.6a and b.In the two computational domain mesh of LD9 spiral separator and laboratory-scale spiral separator,the upper boundary of the computational domain should not impede the normal development of flow film.As shown in Fig.6c and d,the meshes were made substantially fine in the vicinity of the trough to capture a clear flow film surface.More specifically,the minimum mesh height at the trough boundary was set to 0.2 and 0.15 mm for the LD9 spiral separator and the laboratoryscale spiral separator,respectively.The least y+(non-dimensional wall distance) from the trough boundary for the first node was about 4 of the LD9 spiral separator and 2 of the laboratory-scale spiral separator,respectively.Consequently,the entire computational domain of the LD9 spiral separator with 6 turns was divided into 1.85×106hexahedral meshes and the laboratory-scale spiral separator with 5 turns was divided into 7.49×105hexahedral meshes.

Fig.5.Mesh independence test.

Fig.6.Computational domain mesh of two types of spiral separators.

The boundary of the computational domain of the two types of spiral separators is divided into the inlet plane,outlet plane,upper wall,and trough wall.A velocity inlet boundary condition was applied to the inlet plane.The feed fluid velocity direction was normal to the inlet boundary.The pressure outlet condition was applied to the outlet plane.The no-slip boundary condition was applied to the trough wall,and standard wall functions were adopted for the near-wall treatment.Meanwhile,the free-slip boundary condition was applied to the upper wall.

3.6.Numerical scheme

Simulations of the above mathematical model were based on finite volume method.The equations were discretized using the quadratic upwind interpolation (QUICK) scheme for momentum,turbulent kinetic energy,and turbulent dissipation rate.The compressive scheme was used for volume fraction,and the phase coupled SIMPLE scheme was chosen for pressure-velocity coupling.The time step used in the simulations was 0.001 s.Since it is not so easy to get full converged results of three phases flow in numerical simulations,residuals of all variables were restricted to 1×10-3.

4.1.Prediction of flow film shape changes

The flow film shape at the end of the sixth turn of the LD9 spiral separator before and after particle addition with the Eulerian multi-fluid VOF model are presented in this section.In addition,the simulation results of the liquid flow with the Euler-Euler model based on two numerical schemes for the volume fraction equations(QUICK and modified high-resolution interface capturing(modified HRIC))were also compared.The performance of the Eulerian multifluid VOF model and the Euler-Euler model for capturing the free surface of the flow film is shown in Fig.7.

As shown in Fig.7,the flow film region in the liquid flow is characterized by the volume fraction of the water phase,and the flow film region in the particulate flow can be observed by subtracting the air volume fraction from the total volume fraction of 1.It can be observed from Fig.7a and b that the sharp free surfaces of the liquid flow and the particulate flow can be explicitly tracked by the Eulerian multi-fluid VOF model.Based on the Eulerian multifluid VOF model,pronounced bulging of the flow film surface in the inner trough can be observed after the addition of particles,as shown in Fig.7b.In comparison,it can be seen from Fig.7c that the Euler-Euler model based on the numerical scheme of QUICK can hardly explicitly capture the free surface due to the largescale interfacial diffusion.As shown in Fig.7d,the Euler-Euler model based on the numerical scheme of modified HRIC shows a better performance in capturing the free surface in the inner and middle troughs than that of QUICK,while the interfacial diffusion still occurs at a small part of the outer trough.

Fig.7.Performance of the Eulerian multi-fluid VOF model and the Euler-Euler model for capturing the free surface of the flow film.

The comparisons between the simulation results based on the Eulerian multi-fluid VOF model and the experimental data in terms of the flow film thickness are shown in Fig.8.It is worth noting that the flow film thickness is defined as the axial distance between the free surface of flow film and bottom wall.As shown in Fig.8,the predicted values of the flow film thickness in the liquid flow and the particulate flow both showed good consistency with the experimental data.Therefore,the Eulerian multi-fluid VOF model exhibited a more realistic interface corresponding to the liquid flow and particulate flow.It can eliminate the limitation on the assumed free surface of the computational domain in the simulation of particulate flow using the Euler-Euler model[9,11,12].Furthermore,this implies that high-concentration particulate flow can be further investigated based on the Eulerian multifluid VOF model.Note that the interaction effect between the particles and fluids becomes more significant as the solid concentration increases.On the one hand,the enrichment of particles will affect the flow film shape.On the other hand,the change in the flow film shape affects the vertical stratification and the radial segregation of particles.Therefore,it is crucial to accurately predict the separation behavior of the particles in the flow film.

4.2.Prediction of particle separation performance

4.2.1.Validation of the proposed Bagnold lift force model

The prediction performance of the proposed Bagnold lift force model was evaluated in this section.Based on the working conditions of the experimental test in the laboratory-scale separator,the effect of the Bagnold lift force model on the particulate flow was simulated and investigated.In addition,considering that the Saffman lift force (FS) may have a potential impact on the particle behavior in the spiral separator,the simulation of the particulate flow based on the Saffman-mei lift force model was also carried out [29].The simulated grade of iron and solid concentration by mass with different trough lengths were compared with the experimental data.The results are shown in Figs.9 and 10.

Fig.8.Comparison results of flow film thickness between the simulation (SIM) using Eulerian multi-fluid VOF model and the experimental data (EXP).

It can be observed from Figs.9 and 10 that the trends of the grade of iron and solid concentration by mass along with different trough lengths obtained from the simulation and experiment are generally consistent.Note that the simulated results without considering lift force model and with the Saffman-mei lift force model considered both under-and over-predict the grade of iron and solid concentration by mass in regions R1and R2,respectively.When the Bagnold lift force model was considered,the agreement between the experimental data and simulated results was significantly improved.It seems that the predicted particle segregation process,such as vertical stratification and radial segregation,will be more realistic after considering the Bagnold lift force model.Moreover,the vertical stratification of particles is a prerequisite for radial segregation,and the Bagnold force plays an important role in vertical stratification.Therefore,the difference in the simulation results will be analyzed in terms of the vertical stratification of particles.

4.2.2.Effect of Bagnold lift force model on the particle vertical stratification in the initial development stage

The initial development stage of the flow film is an important period for vertical particle stratification.The volume fraction distributions of hematite and quartz particles on the trough surface at the first turn were investigated,as shown in Fig.11.The solid volume fraction on the trough surface represents the proportion of particles exhibiting preferential settling;thus,it can reflect the stratification performance to a certain extent.

Fig.11b and d present the volume fractions of hematite and quartz particles on the trough surface when the Bagnold lift force model was not considered.It can be seen from the figures that the hematite particles tend to concentrate in the middle trough,while the volume fraction of hematite particles in the inner trough is relatively low.In comparison,the volume fraction of quartz particles on the trough surface was lower than that of the hematite particles.As expected,some of the quartz particles gathered in the outer trough.However,a small amount of quartz particles settled in the inner trough.This can be explained as follows.Because the lift force is not introduced,the sunken quartz particles cannot rise to the fast-flowing layers and move outward,resulting in the quartz particles remaining at the bottom of the flow film.When the Bagnold lift force model is considered,it can be seen from Fig.11a and c that the volume fraction of hematite particles in the inner trough increases significantly,while the volume fraction of quartz particles in the inner trough decreases accordingly.It can be considered that more hematite particles preferentially sink to the bottom of the flow film compared with quartz particles.

Fig.9.Comparison of the simulated and measured grade of iron with different trough lengths.

Fig.10.Comparison of the simulated and measured solid concentration by mass with different trough lengths.

To further illustrate the particle stratification in the flow film,the volume fraction distribution of hematite and quartz particles in the 0.25 turns was investigated,as shown in Fig.12.The flow film at 0.25 turns was not fully developed,where the flow film thickness was relatively uniform radially and the radial distribution range of particles was relatively wider.Therefore,the 0.25 turns were chosen to investigate the stratification performance of the particles.

Fig.12a and c show that significant vertical segregation can be observed between hematite and quartz particles in the middle and inner troughs when the Bagnold lift force model is considered.In comparison,hematite and quartz particles are highly mixed in the middle and inner troughs when the Bagnold lift force model is not considered,as shown in Fig.12b and d.The predicted differences in particle vertical stratification performance can be observed more clearly in Fig.13,where the solid volume fraction distributions along with the water depth at the 0.25 turns with two radial positions (r=60 mm and r=90 mm) are plotted.It can be observed that the volume fraction of hematite particles is higher than that of quartz particles at a lower water depth and lower than that of quartz particles at a higher water depth.When the Bagnold lift force model is considered,the differences in the volume fraction between quartz and hematite particles become more significant.It can be concluded that the Bagnold lift force model proposed herein can improve the prediction performance of the particle vertical stratification in the inner and middle troughs.A better particle stratification performance means that more light particles in the upper fast-flowing layer move outward under the action of centrifugal force.Meanwhile,there will be more heavy particles in the lower slow-flowing layer moving inward,driven by the radial pressure gradiant force.

4.2.3.Mechanism of Bagnold effect on the particle vertical stratification

To further determine the dominant Bagnold flow regime,the distribution of the Bagnold number in the flow film at 0.25 turns was investigated.The results are presented in Fig.14.

As shown in Fig.14,the Bagnold numbers of both hematite and quartz particles in the upper layer of the flow film are very small,which is mainly related to the lower solid concentration and shearing rate.As the water depth decreased,the Bagnold number of hematite and quartz particles gradually increased.For hematite particles,the Bagnold number of the red part on the bottom layer exceeds 40,and can reach a maximum value of 81.This means that the types of Bagnold force applied on hematite particles are in the macro-viscous regime and the transitional regime.For quartz particles,the Bagnold number of the bottom flow film was less than 40,and the maximum value was 33.Therefore,the type of Bagnold force exerted on quartz particles is only in the macro-viscous regime.The maximum Bagnold numbers of both hematite and quartz particles are much less than 450;therefore,it can be considered that the contribution of the Bagnold force in the particleinertia regime to particle suspension is negligible,which is mainly related to the particle size.Bagnold [18] considered that it is unlikely that the dispersive force effect in the particle-inertia regime would exist when the particle size is less than 200 μm.

Fig.11.Solid volume fraction distributions on the trough surface at the first turn.

Fig.12.Solid volume fraction distributions in the flow film at the 0.25 turns.

In a laminar flow system,the Bagnold force is an important factor for particle suspension,which counteracts the effective gravity force of the particle.The effective gravity force is defined as the difference between the gravitational and buoyancy forces acting on a particle in the medium.The effect mechanism of the Bagnold force on the particle vertical stratification was analyzed by taking the ratio of the Bagnold force to the effective gravity force.

The effective gravity force Fgon a particle can be expressed as follows:

The ratio of the Bagnold force in the macro-viscous regime to an effective gravity force can be expressed as follows:

The ratio of the Bagnold force in the transitional regime to an effective gravity force is given as

Fig.13.Solid volume fraction distributions along with the relative water depth at the 0.25 turns.

Fig.14.Distribution of Bagnold number in the flow film at the 0.25 turns.

The ratio of the Bagnold force to the effective gravity force along with the relative water depth at the 0.25 turns is presented in Fig.15.

As shown in Fig.15,the dominant force of the particles in the vertical direction gradually transforms from the Bagnold force to the effective gravity force as the relative water depth increases.For the two radial positions,the balance points between the Bagnold force and effective gravity force of quartz and hematite particles are approximately 0.4 and 0.3,respectively.The higher Bagnold force at the bottom of the flow film is mainly related to the higher shear rate and solid concentration.The ratio of the Bagnold force to the effective gravity force of quartz particles is greater than that of hematite particles at each relative water depth,which indicates that the Bagnold force realized on the quartz particles has a stronger resistance against its effective gravity force.

It can be concluded from Fig.15 that the Bagnold force is beneficial to the rapid vertical segregation between quartz and hematite particles in the initial development stage.After the pulp was fed from the inlet,the distribution state of the particles in the pulp was mixed.Subsequently,the Bagnold effect increased with the sedimentation of particles and an increase in pulp velocity.Because the effect of the Bagnold force is greater on quartz particles,the quartz particles mixed at the bottom of the flow film will have a greater chance of being lifted than the hematite particles.

4.2.4.Effect of Bagnold lift force model on the subsequent radial segregation

The vertical stratification of particles during the initial development stage plays an important role in the subsequent radial segregation.Fig.16 compares the volume fraction distribution of hematite and quartz particles in the complete trough surface with and without the Bagnold lift force model.

Fig.15.Ratio of Bagnold force to effective gravity force along with the relative water depth.

Fig.16.Solid volume fraction distributions in the trough surface.

It can be clearly seen from Fig.16 that significant settling bands of quartz and hematite particles are formed in the trough surface based on the trough length.In general,the settling band of quartz particles is more outward than that of hematite particles.However,when the Bagnold lift force model is not considered,a small part of the quartz particle settling band can be observed in the inner trough,as shown in Fig.16d,and it gradually shrinks inward with an increase in trough length.Combined with Fig.11d and 16d,it can be considered that the quartz particles appearing in the inner trough can be attributed to the poor particle vertical stratification.In the initial development stage,the quartz particles mixed at the bottom of the flow film lose the chance to be lifted owing to the absence of lift force.Furthermore,it will stay at the bottom of the flow film continuously and gradually move inward under the effect of the inward flow,leading to the entrainment of quartz particles in the inner trough.Fig.17 displays the recovery of quartz and hematite particles in the R1region with different trough lengths.As shown in Fig.17,when the Bagnold lift force model was considered,more hematite particles were recovered in the R1region,while fewer quartz particles were observed in the R1region,which means that better separation performance occurred in the inner trough.This is mainly related to the particle vertical stratification and secondary flow.

Overall,the proposed Bagnold lift force model can be considered effective for predicting particle segregation in a spiral separator.It can be concluded that satisfactory particle vertical stratification in the initial development stage is conducive to subsequent radial segregation.Nevertheless,the impact of the secondary flow on the radial segregation of particles is complicated,which is related to the stability of the flow field.Therefore,a more detailed investigation of the radial segregation of particles will be conducted in the future.

In this work,we attempted to improve the simulation modeling methods of particulate flow in the spiral separators.The Eulerian multi-fluid VOF model considering the Bagnold effect was adopted to investigate the complex multiphase flow behavior in the spiral separators.The primary findings of this study are summarized as follows:

(1) Compared with the Euler-Euler model,the Eulerian multifluid VOF model effectively avoids the interfacial diffusion and gives a better description of the sharp interfaces of air-water and air-pulp in the LD9 spiral separator.

(2) The predicted change in flow film thickness before and after addition of particles in the LD9 spiral separator based on the Eulerian multi-fluid VOF model agrees well with the reported values from literature.It implies that this model can effectively overcome the limitation of the computational domain with an assumed upper boundary based on the Euler-Euler model.

(3) The Bagnold lift force model,which is applied in the macroviscous region,transitional region,and particle-inertia region was proposed.The equation of FB,transitionuses the interpolation method and combines the equation of FB,inertiaand FB,viscosity.This treatment method has certain rationality from a qualitative point of view and has room for improvement and refinement.

(4) The proposed Bagnold lift force model can be considered effective for predicting particle segregation in a spiral separator.When the Bagnold lift force model was not considered,the volume fraction of quartz particles in the bottom film increased,which moved inward under the action of the inward flow,resulting in under-prediction of the grade of iron and solid concentration by mass in the inner trough.When the Bagnold lift force model was used,a better predicted vertical stratification between quartz and hematite particles was obtained,which provided favorable conditions for subsequent radial segregation.

Fig.17.Recovery of quartz and hematite in the R1 region with different trough lengths.

Acknowledgements

This work was supported by the National Natural Science Foundation of China(Nos.51974065 and 52274257),the Open Foundation of State Key Laboratory of Mineral Processing (No.BGRIMMKJSKL-2020-13),and the Fundamental Research Funds for the Central Universities (Nos.N2201008 and N2201004).