Modeling the Multiphase Flow in Hydrocyclones Using the Coarse

Subscriber access provided by UNIV OF DURHAM

Separations

Modelling the multiphase flow in hydrocyclones using Coarse-grained VOF-DEM and Mixture-DEM approaches Li Ji, Kaiwei Chu, Shibo Kuang, Jiang Chen, and Aibing Yu Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b01699 • Publication Date (Web): 27 Jun 2018 Downloaded from http://pubs.acs.org on June 27, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Industrial & Engineering Chemistry Research

Modelling the multiphase flow in hydrocyclones using Coarse-grained VOF-DEM and Mixture-DEM approaches Li Ji1, Kaiwei Chu1,2*, Shibo Kuang1, Jiang Chen1,2, Aibing Yu1,2 1

ARC Research Hub for Computational Particle Technology, Department of Chemical Engineering, Monash University, Clayton, VIC 3800, Australia 2

SEU Monash Joint Graduate School, Suzhou 215123, China

* [email protected]

1 ACS Paragon Plus Environment

Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ABSTRACT Hydrocyclones are widely used in various industries to classify particles mainly by size. In this work, two numerical models are developed to model the multiphase flow in hydrocyclones: one is a combined approach of volume of fluid (VOF) model and discrete element method (DEM) with the concept of the coarse-grained (CG) particle (CG VOF-DEM); the other is a combined approach of the Mixture model and DEM model with the CG concept (CG Mixture-DEM). The simulation results show that the CG VOF-DEM model is quantitatively applicable to relatively dilute flows while only qualitatively applicable to dense flows. On the other hand, the CG Mixture-DEM model can be quantitatively applicable to both dilute and dense flows. The work suggests that the CG Mixture-DEM approach could be a useful tool to estimate the performance of hydrocyclones. Keywords: Hydrocyclone; multiphase flow; computational fluid dynamics; discrete element method; coarse-graining


Page 2 of 48

Page 3 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1. INTRODUCTION Hydrocyclones are widely used in many industries including mineral and chemical processing to classify particles mainly by size, because of their design simplicity, operation flexibility, high capacity, and low operation and maintenance costs.1 In spite of its simple design, the flow inside it is very complicated with the flow of multicomponent mixture (water, air, and solid particles), broad size and density distributions of the feed in many situations, anisotropic swirling turbulent vortex, and possible strong interactions between fluid phases and particles. The working mechanisms of hydrocyclones are well summarized in the literature.1,2 Specifically, the slurry is tangentially injected into the hydrocyclone to develop a centrifugal force. This force generates vortex motion to the fluid inside the hydrocyclone. The outer vortex, which forms near to the wall of the hydrocyclone, flows downward and carries coarse particles to the underflow. The inner vortex, which forms along the central axis, flows upward with the clean fluid or the fluid with fine particles.2 Due to the formation of vortex, a low-pressure region prevails along the axis and normally results in a free liquid surface in the hydrocyclone. This low-pressure region which communicates directly with the atmosphere through the underflow and overflow orifices becomes air filled.1 Thus, an air-core forms along the axis of the hydrocyclone. Although hydrocyclones are widely used, there are yet various challenging problems associated with such a separator. For example, the “fish-hook” phenomenon is still a debatable issue in hydrocyclones.3,4 In addition, the effects of particle size and density distributions on hydrocyclone performance are important but difficult to be comprehensively described due to the complexity. Previous studies are usually 3 ACS Paragon Plus Environment


conducted under certain simplified particle size and density distributions while only a few researchers tried to study the effects of the distributions.5,6 Moreover, the particle misplacement has always been an important issue in hydrocyclone studies,7 how to effectively mitigate this problem is not fully established. To better understand or solve the aforementioned challenging problems, further studies on hydrocyclones are necessary. Experimental work has been carried out to improve the understanding of hydrocyclones.8-11 Nonetheless, it is difficult to measure the internal flow structure and elucidate the underlying mechanics. Although empirical or analytic methods 12,13 are more convenient to perform, they are difficult to lead to a deep understanding of the underlying working mechanisms. Numerical methods have been increasingly used to study the multiphase flows in hydrocyclones in recent years. For the modelling of the fluid flow in hydrocyclones, it is shown in an early 2D numerical study 14 that the standard k − ε turbulence model is inadequate to simulate flows with swirling. Recently, 3D flow studies were carried out and suggest that RSM (Reynold Stress Model) can improve the accuracy of the numerical solution.15,16 As aforementioned, there is an air-core forming along the axis of the hydrocyclone. Actually, the air-core dimension is key to predicting the mass split to underflow and a large air-core diameter leads to a condition known as “roping”.17 Therefore, the air-core modelling plays an important role in numerical studies of hydrocyclone. Nevertheless, the nature of air-core was not considered in the early modelling studies but with simplified assumptions about its formation and behavior.18,19 Recently, Delgadillo and Rajamani

17

simulated the 3D flow field

distributions and air-core shape in a hydrocyclone using the RSM or LES (Large-eddy 4 ACS Paragon Plus Environment

Page 4 of 48

Page 5 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Simulation) and VOF (Volume of Fluid) models. The formation of an air-core was simulated in details by Wang et al. 20 by combining the RSM and VOF methods. Generally speaking, solid/particle phases can be modelled either as continuum medium 21-23 or as discrete entities.24-26 For hydrocyclones, the two-fluid model (TFM) model which treats both particles and fluid as continua, including its simplified versions such as the Mixture model, have been widely used.27-31 Note that the Mixture model is generally only applicable to flows with small Stokes number.32 The problem can be overcome by TFM model which is applicable to flows with large Stokes number.33 However, the TFM model treats each particle size/density as a separate Eulerian field 34 and the computational cost can be quite high when the solid particles have wide size and/or density distributions. The Lagrangian particle tracking (LPT) method has been combined with the VOF or Mixture model to offer a convenient way to model particles with a wide range of size and/or density distributions in hydrocyclones.20,35,36 However, LPT method is generally applicable only to dilute flows because of the ignorance of particle-particle interaction forces and interactions between the particles and the fluid phase. That is to say, the void fraction (ɛ) or porosity and particle-particle collisions are not included in the governing equations of LPT method used in the work of Wang et al.20 For the dense flows in hydrocyclones, e.g. feed solid concentrations can be up to 40% by volume,5 LPT model is not so applicable. Nonetheless, LPT method with particle interaction terms based on kinetic theory was found applicable for use for a wide range of feed solid concentrations.37 On the other hand, the combined approach of computational fluid dynamics (CFD) and discrete element method (DEM) (CFD-DEM) has been proved to be effective to 5 ACS Paragon Plus Environment


study the fundamentals of particle-fluid systems.24,38,39 CFD-DEM model, which is also applicable to particles of different sizes and densities, can overcome the deficiency of CFD-LPT approach, by considering particle-particle collisions and the interactions between the particles and the fluids. Different with the LPT-based model,20 the void fraction and particle-particle collisions are included in the governing equations of DEM-based method. The VOF-DEM model has been used to model particle-fluid flows

40-44

including gas cyclones

43

and conical hopper.22 Recently, the

mixture-DEM model has been used to study dense medium cyclones (DMCs) 45,46 and mixers.47 Even though the DEM-based model has been used to study different systems like gas cyclones and DMCs which are different from hydrocyclone in working mechanisms and geometrical, material and operational conditions, the DEM-based model has not been seriously used to study hydrocyclones. One of the main problems of DEM approach is its high computational cost since each individual particle is tracked by Newton’s second law. Therefore, it is computationally very demanding for the standard DEM model to simulate industryscale particulate systems. This situation is even worse for hydrocyclones where the particle size is fine or feed solid concentration may be high (up to 40% by volume). To overcome this problem, CFD-DEM model with different treatments such as DEM parcel in Ansys Fluent, parcel-particle concept 48, coarse-graining (CG) model, 45,49-54 and similar particle assembly models

55

or scaling laws

56

have been developed for

various particulate systems. Even though the treatments in those models may be different from each other in describing rotational motion, contact or cohesive forces 54, they are similar in principle as proposed by Patankar and Joseph.48 This work aims to propose a method to study hydrocyclones that handle particles with 6 ACS Paragon Plus Environment

Page 6 of 48

Page 7 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


a wide range of particle size/density distribution. To achieve this goal, both CG VOFDEM and CG Mixture-DEM models are applied to model the swirling multiphase flow in two different hydrocyclones. The difference between the two models lies in the modelling of the fine fraction of solid particles. In the CG VOF-DEM approach, all of the solid particles including the fine fraction are modelled by the CG DEM model. By contrast to this, in the CG Mixture-DEM approach, the fine fraction of particles are treated as continuum medium and modelled by the Mixture model.

2. SIMULATION METHOD AND CONDITION 2.1

Simulation method

In the CG VOF-DEM approach, all of the solid particles including the fine fraction are modelled by a CG DEM model. The motion of particles is modelled as movement of discrete entities, by applying Newton’s laws of motion to individual particles, while the flow of water and air is treated as a continuous phase, described by the local averaged Navier-Stokes equations on a computational cell scale. Nonetheless, in the CG Mixture-DEM approach, the fine fraction of particles are treated as continuum medium and modelled by the Mixture model. Therefore, the major difference between the two models lies in the modelling of the fine fraction of solid particles. The mathematical formulation of the current VOF-DEM/Mixture-DEM model is in principle the same as that reported elsewhere.20,45,46 Therefore, only a brief description of the model is given in this work. In both the CG VOF-DEM and CG Mixture-DEM models, the modelling process is divided into two steps, as shown in Figures 1 and 2. In the CG VOF-DEM model, only air and water are considered in Step 1. The turbulence of water-air flow is 7 ACS Paragon Plus Environment


Page 8 of 48

modelled using RSM,57 and the interface between water and air-core is described by VOF model.20,46 In the CG Mixture-DEM model, in Step 1, besides water and air, additional phases are introduced to describe the behaviour of fine particles with different sizes. The multiphase model used in the Mixture-DEM approach is Mixture model instead of VOF model used in the VOF-DEM approach. The continuum fluid flow is calculated from the continuity and the Navier-Stokes equations based on the local mean variables defined over a computational cell. These are given by:

∂ (ρ f ε ) ∂t

+ ∇ ⋅ (ρ f εu ) = 0

∂ (ρ f εu ) ∂t

(1)

+ ∇ ⋅ (ρ f ε uu ) = −∇ P − Fp − f + ∇ ⋅ (ε τ ) + ρ f ε g + ∇ ⋅ ( − ρ f u ' u ' ) (2)

where ε , u , u ' , t , ρ f , P , Fp− f , τ , and g are, respectively, porosity, mean and fluctuating fluid velocities, time, fluid density, static pressure, volumetric fluidparticle interaction force, fluid viscous stress tensor, and acceleration due to gravity. Fp − f =

1 Vc

kc

∑f

p − f ,i

, where f p − f ,i is the total fluid force on particle i and kc is the

i =1

number of particles in a CFD cell of volume Vc . − ρ u' u' is the Reynolds stress term due to turbulence and modelled by RSM model, while the turbulence modification of fluid flow due to the presence of particles is not considered in this work. In VOF model, the fluid density in Eqs (1)-(2) is calculated according to the equation below (viscosity and other properties are computed in the same way as Eq. (3) 20):


Page 9 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


n

ρ f = ∑ αqρq

(3)

q =1

where n is the number of phases considered by the VOF model, ρq is the density of phase q, and αq is its corresponding volume fraction which varies between 0 and 1 and calculated by the equation 58 below:

∂α q ∂t

+ uj

∂α q ∂x j

=0

(4)

where uj is the velocity component in direction j. The interface between air and water by the use of VOF model can be tracked by Eq. (4) while surface tension is not considered following previous work.20,28,59 In the VOF model, a single momentum equation is solved throughout the domain, and the resulting velocity field is shared by the phases involved. This momentum equation is dependent on the volume fraction of a fluid phase.20 Unlike the VOF model, the phases in Mixture model are interpenetrating. Even though the same governing equations are used as VOF model, the drift velocity is introduced to allow phases to move at different velocities. In the Mixture model, a bubble size used to calculate drift velocity is set to 0.00001 m as previous work.28 The mixture properties by using the Mixture model can also be calculated in the same way as shown in Eq. (3). Note that the Mixture model and VOF model can give comparable water-air field under the same conditions as reported in the literature.28 The flows of particles are determined from the fluid flow patterns obtained above by using the DEM method.60 In DEM, a particle has two types of motion: translational and rotational, both obeying Newton’s second law of motion. During its movement, the particle may collide with its neighboring particles or with the wall and interact 9 ACS Paragon Plus Environment


Page 10 of 48

with the surrounding fluid, through which momentum is exchanged. At any time t, the equations governing the translational and rotational motions of particle i in this multiphase flow system are: ki dvi = f p − f,i + mi g + ∑ (fc,ij + fd,ij ) dt j =1

mi

(5)

and

Ii

dω i = dt

ki

∑ (T

c , ij

+ Tr ,ij )

(6)

j =1

where m i , Ii , ωi , and v i are, respectively, the mass, moment of inertia, rotational and translational velocities of particle i . The forces involved are: the particle-fluid interaction force f p − f ,i , gravitational force mi g , and interparticle forces between particles i and j . The torques include the interparticle torque Tc ,ij and rolling friction torque Tr ,ij . For multiple interactions, the interparticle forces and torques are summed for ki particles interacting with the particle i . f p − f ,i is the total particle-fluid interaction force, which is the sum of various particle-fluid forces. Researchers

61,62

proposed that the virtual mass force can be ignored for the case of a hydrocyclone. The lift force is usually not considered in the modelling of hydrocyclones 20,63 and its importance is still not clear although it has been considered in some studies.64,65 Basset force was suggested to be ignored for the case of hydrocyclones 61 or equal to zero in the steady movement.62 In this work, for simplicity, the virtual mass force, lift force, and Basset force are not included at this stage of model development. Therefore, the total particle-fluid interaction force f p − f ,i only includes the drag force and pressure gradient force in the current case. The fluid properties used to calculate the 10 ACS Paragon Plus Environment

Page 11 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


particle-fluid interaction forces are those relating to the properties of the mixture of water and air (and fine particles when the Mixture-DEM model is used). For simplicity, the lubrication effect on particle-particle interactions and particle dispersion due to turbulence are not considered. The equations used to calculate the forces on the original real particles can be found in Table 1. The governing equations of a CG CFD-DEM model are similar to those of a standard CFD-DEM model except that the particles are replaced with CG particles.45,48-51,53-56 In this work, the used coarse-graining strategy of keeping the identical CG particle size follows our previous work 45 and literature work.66 Note that the newly-proposed strategy of keeping the same size ratio α (size ratio α is the ratio of the diameter of the CG particle to the diameter of the original real particle which is represented by that CG particle)

67

can give more accurate results, however, at an expense of larger

computation cost. The material properties such as density, Young’s modulus, and friction and rolling coefficients for the calculation of interaction forces between original real particles are shown in Table 3. The drag force on the CG particle is scaled by α3 times of the drag force acting on the original real particle. For the contact forces between particle-particle and particle-wall, according to the linear momentum and impulse connection and assuming that the collision time between CG particles is α times of that between real particles, the normal contact and damping forces are 2 cg o = α 2f cn,ij scaled by α2 (i.e., f cn,ij and f dn,ij = α f dn,ij , where superscripts cg and o denote

cg

o

variables related to CG particle and original particle respectively) when compared with those of real particles to satisfy the conservation of linear momentum. At the same time, according to the angular momentum and impulse connection and assuming the same ratio of δ t / δ t ,max (particle-particle or particle-wall overlap/maximum overlap 11 ACS Paragon Plus Environment


Page 12 of 48

in tangential direction) for both CG particles and real particles, the tangential contact 2 cg o = α 2f ct,ij and f dt,ij = α f dt,ij ) when and damping forces are also scaled by α2 (i.e., f ct,ij

cg

o

compared with those of real particles to satisfy the conservation of angular momentum. Note that at present the common approach for the scaling of particleparticle interaction forces is to decrease the coefficient of restitution 52,53 or scale the contact force by α3.

51

As pointed out by Lu et al.,49 it may not be necessary to

decrease the coefficient of restitution for liquid-solid systems where the particleparticle collision frequency could be much lower than that of gas-solid systems. Moreover, for current case, we have tested the effect of varying the contact force according to the work by Sakai et al.

54

in which the collision time between CG

particles should be similar to that between real particles. It was found that it does not provide significant better results for current case. Moreover, in CG DEM models, as the scaling ratio α for each CG particle can be different from each other, the contact forces which are in same magnitude in conventional DEM can be in different values between two colliding CG particles. The modelling of the solid flow by DEM is at an individual particle level, whilst the fluid flow by CFD is at a computational cell level. Their two-way coupling (i.e., fluid forces act on particles and particles react on fluid) is numerically achieved as follows. At each time step, DEM provides information, such as the positions and velocities of individual particles, for the evaluation of porosity and volumetric particle-fluid interaction force in a computational cell. CFD then uses these data to determine the fluid flow field, from which the particle-fluid interaction forces acting on individual particles are determined. The incorporation of the resulting forces into DEM produces information about the motion of individual particles for the next time step. 12 ACS Paragon Plus Environment

Page 13 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2.2

Simulation conditions

In order to investigate the applicability of the CG VOF-DEM and CG Mixture-DEM models under various conditions, numerical work is carried out for two hydrocyclones with different geometrical, operational, and material conditions. Figure 3 shows the geometry and mesh representations of the hydrocyclones with a body diameter of 75 mm. The mesh in Figure 3(b) contains 87,500 cells. Note that this hydrocyclone which is from a previous experimental work 9 has been widely employed to validate numerical results in the literature.20,28,31 Figure 4 shows the geometry (a) and mesh (b) representations of the 100 mm diameter hydrocyclone from literature.68 The mesh in Figure 4(b) contains 95,000 cells. For hydrocyclones with diameters at 75 mm and 100 mm, the mesh is finer in the vicinity of the wall and vortex finder, as shown in Figure 3(c) which takes A-A plane in 75 mm diameter hydrocyclone as an example. The current meshing approach has been successfully used in our previous works and the results can be regarded as independent of the mesh number.20,59 Note that a newlyproposed dual-grid approach 40 for VOF-DEM coupling is quite attractive and may be used in our future work. If the CG particle size is 1400 µm for the 75 mm diameter hydrocyclone, the maximum ratio (away from the wall and vortex finder) of grid size to CG particle size on A-A plane is 1.6, and the minimum (near the wall and vortex finder) is 0.86. At a similar location to A-A plane for the 100 mm diameter hydrocyclone, the maximum ratio of grid size to CG particle size is 1.1 and the minimum is 0.6 if the CG particle size is 2700 µm. For the case in which the CG particle is larger than the grid size, an approximation approach used previously

69

is

adopted in this work. This approach is to combine a group of small neighboring grids into one large grid which is larger than the size of the residing CG particle in the grids. In all of the simulations, a “velocity inlet” boundary condition is used at the cyclone 13 ACS Paragon Plus Environment


inlet and the “pressure-outlet” condition at the underflow and overflow. The pressure at the two outlets is 1 atm, i.e. the ambient atmospheric pressure. For the purpose of model validation, the geometrical and operational conditions, as shown in Tables 2 and 3, follow those in the corresponding experiments. The material properties used in the simulation are also summarized in Table 3. For the simulation work of the hydrocyclone with a body diameter of 75 mm, the inlet water and particle velocities are both 2.49 m/s. Limestone particles are respectively injected at three different feed solid concentrations, i.e., 1%, 4%, and 8%. Note that all of the feed solid concentrations referred in this work are by volume. For the hydrocyclone with a body diameter of 100 mm, the inlet water and particle velocities are both 4.1 m/s. Iron particles are injected at two different feed solid concentrations of 6% and 10%, respectively. The simulated size distributions for the two hydrocyclones are also consistent with the experimental ones, as shown in Figures 5 (a) and (b), respectively. The particle size range is 0.43-42 µm for the 75 mm diameter hydrocyclone and 1-841 µm for the 100 mm diameter hydrocyclone. In the CG VOF-DEM model, the sizes of coarse-grained particles are 1800 µm and 3000 µm respectively for the 75 mm diameter hydrocyclone at a feed solid concentration of 8% and the 100 mm diameter hydrocyclone at 10% , corresponding to maximum size ratios (CG particle diameter to real particle diameter) of 4186 and 3000 respectively. The two size ratios are much larger than the maximum size ratio 3 in the study by Sakai et al.,54 10 in that by Lu et al., 52 and 19 in that by Patankar and Joseph.48 Actually, the size ratio between the CG particle and real particle should be as small as possible to minimize error caused by the CG concept. Unfortunately, a large CG particle size has to be used in the case of 14 ACS Paragon Plus Environment

Page 14 of 48

Page 15 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


hydrocyclones given our available computation capability. As aforementioned, the CG Mixture-DEM model treats the fine fraction of solid particles as continuum phases. The critical particle size (CPS), below which particles are treated as continuum phases by the Mixture model, is 5.27 µm for the 75 mm diameter hydrocyclone and 19 µm for the 100 mm diameter hydrocyclone, as shown in Figure 5. In current work, the CPS is determined by trial simulations. Specifically, 1.69, 3.73, 5.27, and 10 µm are tried as CPS respectively for the 75 mm diameter hydrocyclone at the feed solid concentration of 8%, and the corresponding CG particle sizes are 1650, 1600, 1500, and 1400 µm. For the 100 mm diameter hydrocyclone, 10, 19, 38, and 100 µm are respectively tried as CPS at a feed solid concentration of 10%, and the corresponding CG particle sizes are 2800, 2700, 2600, and 2300 µm. The corresponding partition curves with different CPSs are shown in Figure 6 (a) and (b) for the 75 mm and 100 mm diameter hydrocyclone, respectively. It is found that, with the increase of CPS, the predicted separation performance increases first and then does not change obviously when the CPS is equal to or larger than 5.27 µm for the 75 mm diameter hydrocyclone. This indicates that the obtained calculated partition curve can be considered as independent of CPS when CPS increases to a certain value. Therefore, 5.27 µm is chosen as CPS of CG MixtureDEM model in the 75 mm diameter hydrocyclone. Similarly, 19 µm is chosen as CPS of CG Mixture-DEM model in the 100 mm diameter hydrocyclone. Notably, a larger CPS could also be used. However, the larger the CPS, the more particles have to be modelled by the Mixture model, which would increase the computational cost. To be general, an attempt is made to use Stokes number ( Stk ) as a criterion to determine CPS. Stokes number is defined as: 15 ACS Paragon Plus Environment


Stk =

Page 16 of 48

τp

(7)

ts

where τ p is the relaxation time of a particle, given by τ p =

ρ pd 2 . In the calculation 18 µ water

of Stokes number in hydrocyclone, the particle density is normally replaced by the density difference between the particle and liquid, i.e., ρ p − ρ water .70,71 In this work, the density difference is also adopted to calculate Stokes number. µ water is the water dynamic viscosity. ts is the characteristic time of water flow, given by ts = Dc ; here uwater

the characteristic length Dc is the hydrocyclone body diameter and the characteristic velocity u feed is the water feed velocity, as found previously on the calculation of Stokes number in hydrocyclone.70,71 Note that the size of real particles represented by different CG particles can be quite different from each other and thus different CG particles would have different Stokes number since the Stokes number is calculated by using the size of real particle represented by a CG particle. The particle size where Stokes number equals 0.1 is taken as CPS. It can be calculated that the CPSs in the 75 mm and 100 mm diameter hydrocyclones are 180 and 130 µm, respectively. For the 75 mm diameter hydrocyclone, the size range of fed particles is 0.43-42 µm which is smaller than the CPS of 180 µm calculated by Stokes number, which suggests that the 75 mm diameter cyclone can be just modelled by the Mixture model, as confirmed by previous studies.28,72 However, for the 100 mm diameter hydrocyclone, the Mixture model may not be suggested to be used for particles larger than 130 µm. That is to say, Stokes number may be used as the selection criterion of CPS when the trail simulations are not available.


Page 17 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The simulations are all unsteady, undertaken by the unsteady solver in Ansys Fluent as a platform, and achieved by incorporating a CG DEM code into Fluent through its User Defined Functions (UDF). Note that Ansys Fluent also provides a similar parcel treatment for DEM. Nevertheless, our DEM house code has been extensively used and validated in past years 45,73 and thus we still use our house DEM code in this work. Actually, the computation cost of the CG VOF/Mixture-DEM model is still high even with the CG concept. Generally, on a single CPU server, two months are needed to reach 8 physical seconds in the hydrocyclone by the CG VOF-DEM calculation. The calculation by the CG Mixture-DEM model is more computationally demanding and three months are needed to reach 8 physical seconds in the hydrocyclone.

3. RESULTS AND DISCUSSIONS 3.1

Model validation

The validation of CG models is carried out by comparing the simulated results with experimental ones in this work. There are also alternative approaches to validate CG models as discussed in the literature 45,54,74 if experimental data are not available. One of the methods is to compare the agreement between results obtained from the CG model with those from original particle systems.45,54 However, this validation method becomes substantially impossible when the size ratio is high, thus the validation can only be shown efficiently by experiment.54 To overcome this, a method is newly proposed by extrapolating the trends found through the simulation of a series of smaller scale reactors to larger scale ones.

74

In current work, the validation of CG

models relies on the experimental data. The experimental data used to validate the CG VOF-DEM and CG Mixture-DEM 17 ACS Paragon Plus Environment


models are from three different studies. The experimental partition curve included in Figure 7 is from Hsieh’s experiments

9

which are widely used to validate numerical

models in the literature.20,28,31 The comparison indicates that partition curve predicted by the CG VOF-DEM approach agrees well with the experimental one at the feed solid concentration of 4%. Partition numbers simulated by a CFD-LPT model in the literature

20

are also given in Figure 7 to be compared with those predicted by CG

VOF-DEM model. It suggests that the CG VOF-DEM approach performs slightly better than the CFD-LPT model. The major reason should be that the CG VOF-DEM model considers particle-particle interactions and the reactions of particles on fluid flow which are ignored by the CFD-LPT model. Note that the CG VOF-DEM model is equal to CFD-LPT model for very dilute flows in which particle-particle interactions and the reactions of particles on fluid flow can be ignored.45 This experimental work conducted by Rajamani and Milan 75 is a further study under the dense flow condition based on Hsieh’s study.9 It can be seen from Figure 8 that the simulated partition numbers by the CG Mixture-DEM model are comparable with the experimental ones. Actually, CG VOF-DEM model is also employed to predict the partition curve at the feed solid concentration of 8%. However, at this feed, the CG VOF-DEM model is found to underestimate the separation performance. The similar issue happens to the simulation of 100 mm diameter hydrocyclones. Specifically, at a relatively low solid feed concentration (= 6%), both the CG VOFDEM and CG Mixture-DEM approaches give comparable results with experiments,68 as shown in Figure 9. However, at a relatively high solid feed concentration (= 10%), the CG VOF-DEM method underestimates partition numbers while the CG MixtureDEM model can still predict the experimental results reasonably well, as shown in Figure 9. Note the CG particle sizes of the CG Mixture-DEM model for the 100 mm 18 ACS Paragon Plus Environment

Page 18 of 48

Page 19 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


diameter hydrocyclones at feed solid concentrations of 6% and 10 % are 2200 and 2700 µm, respectively. The reason for the underestimation of CG VOF-DEM model at higher feed solid concentrations should be attributed to the extremely larger size ratio of CG particle to real particles (the size ratio is as high as 4,186 which means that 1 CG particle represents up to 6.9 billion real particles). Ideally, the size ratio should be reduced in the CG VOF-DEM model by decreasing the size of the CG particle, which, however, would make the simulation not be feasible given our current computational capacity. Compared with the CG VOF-DEM model, the CG Mixture-DEM model can reduce the maximum size ratio dramatically by treating fine particles as continuum phases. Specifically, the maximum size ratio for the 75 mm diameter hydrocyclone at the feed solid concentration of 8% can be dramatically decreased from 4186 to 285 after employing the CG Mixture-DEM model. Similarly, the maximum size ratio for the 100 mm diameter hydrocyclone at the feed solid concentration of 10% is reduced to 142 from 3000. Previous studies suggest that, as the size ratio increases, the prediction of particle-particle interactions in the CG DEM model is less accurate.45 Therefore, for dense flows, the size ratio should be kept as small as possible. 3.2

Particle-fluid flow features

Figures 10 (I-a) and (II-a) shows that in both the CG VOF-DEM and CG MixtureDEM models, coarse particles congregate at the wall and are mainly discharged from underflow, while fine particles are mainly discharged from the upper outlet. This is consistent with the basic working mechanisms of hydrocyclone. Nonetheless, differences are found in the prediction of time-averaged volume fraction by the two models. Figures 10 (I-b) and (II-b) show that the solid volume fraction at the spigot 19 ACS Paragon Plus Environment


region predicted by the CG VOF-DEM model is obviously higher than that predicted by the CG Mixture-DEM model. Actually, the accumulation of particles near spigot area deteriorates separation performance, as found in previous studies.29,76 This provides a reason why the CG VOF-DEM model underestimates the partition number. Figure 11 shows some representative results of the simulated flow field by the CG Mixture-DEM model for the 75 mm diameter hydrocyclone at the feed solid concentration of 8%. It can be clearly seen that the static pressure reduces radially from wall to centre in Figure 11 (a). As shown in Figure 11 (b), the tangential velocity increases first and then decreases after reaching a peak in the area near the centre. These features of flow field are consistent with previous experimental and numerical studies on hydrocyclones.9,20,28 Besides the information that can also be obtained by CFD-LPT and TFM models (including simplified VOF and Mixture models), DEMbased models can provide more information such as particle-particle and particle-wall interactions. The particle-particle and particle-wall interactions are quantified using the so-called Time Averaged Collision Intensity (TACI), as detailed in previous studies.77,78 Figures 11 (c) and (d) show that both time-averaged particle-particle and particle-wall interactions are stronger in the lower area of conical part and spigot. This distribution of particle-particle and particle-wall interactions is reasonable and consistent with the wear phenomenon of hydrocyclones in operating plants.79 Even though the CG VOF-DEM model quantitatively underestimates the partition number at higher feed solid concentrations, it is found that the model can still qualitatively predict the effect of feed solid concentration when the feed solid concentration is not very high (1%, 4%, and 8% respectively for the 75 mm hydrocyclone and 6% and 10% respectively for the 100 mm hydrocyclone in current 20 ACS Paragon Plus Environment

Page 20 of 48

Page 21 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


work). Note that an involute inlet is used for the 75 mm hydrocyclone in Figures 12, 13 and 14. From Figures 12 (a) and 9, it can be seen that the simulated separation efficiency reduces with the increase of feed solid concentration, while the cut size has the opposite trend. This is qualitatively consistent with the findings in previous experimental and numerical work.27,28,68,75 The results obtained by the CG MixtureDEM model for the 75 mm diameter hydrocyclone is shown in Figures 12 (b). The CG particle sizes of CG Mixture-DEM model at feed solid concentrations of 1, 4, and 8% are 800, 1250, and 1500 µm, respectively. It can be seen that the CG MixtureDEM model can also predict the effects of feed solid concentration on the hydrocyclone performance as predicted by the CG VOF-DEM model in Figure 12(a). Nevertheless, the two models are different in predicting quantitative results as discussed in Section 3.1. Figure 13 shows the effect of feed solid concentration on the distributions of timeaveraged solid volume fraction, particle-particle interactions, and particle-wall interactions in the 75 mm diameter hydrocyclone. Note that the simulated results of the 100 mm diameter hydrocyclone are qualitatively similar to those of the 75 mm diameter cyclone and thus not reported. Fig.13 (I) indicates that the time-averaged solid volume fraction in the spigot area increases as the feed solid concentration increases, which is consistent with the findings in the literature.28 It can be seen from Figure 13 (II-III) that, with the increase of feed solid concentration, time-averaged particle-particle and particle-wall collisions become more intensive. Apart from the increased collision intensity, the collision area is also enlarged from the lower conical part and spigot to the inlet area as the feed solid concentration increases. This suggests that the wear problem is more severe for hydrocylones operated under dense flow conditions. As a comparison, the corresponding results obtained by the CG 21 ACS Paragon Plus Environment


Mixture-DEM model are shown in Figure 14. It can be seen that the CG MixtureDEM model can also predict the effects of feed solid concentration as predicted by the CG VOF-DEM model in Figure 13. In addition, for the higher feed solid concentration of 8%, the predicted time-averaged solid volume fraction in the spigot area by the CG Mixture-DEM model is smaller than that by the CG VOF-DEM model, which means the CG VOF-DEM model overestimates the time-averaged solid volume fraction as aforementioned in Figure 10. With the decrease of feed solid concentration, the time-averaged solid volume fraction is less overestimated by the CG VOF-DEM model when compared with that of CG Mixture-DEM model. Overall, the predicted difference between the two models should be attributed to the effect of size ratio which is more significant when the feed solid concentration is high.

4. CONCLUSIONS The complicated multiphase flows in hydrocyclones are modelled by a CG VOFDEM model and a CG Mixture-DEM model. The simulation results are compared favorably against experimental data, and the effect of feed solid concentration is investigated comparatively. Specifically, the following conclusions can be drawn: •

The CG VOF-DEM model is only applicable to relatively dilute flows (up to 6% by volume in this work). When the flow is dense, it underestimates partition numbers and overestimates the particle volume fraction at the spigot region. The problem of the CG VOF-DEM for the dense flow should be mainly caused by the extremely large size ratio of CG particle to real particle. Generally, as the size ratio increases, the prediction will be less accurate especially on the prediction of particle-particle interaction force.


Page 22 of 48

Page 23 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


•

The CG Mixture-DEM model is applicable to both dilute and dense flows but its computational cost is much higher than the CG VOF-DEM model. Moreover, extra efforts are needed to determine the critical particle size below which the particles can be modelled by the Mixture model. Current work suggests that the critical particle size can be determined by trial simulations and it is safe to model the particles by the Mixture model if the Stokes number of a particle is less than 0.1. As a general rule, the size ratio of the CG particle to the real particle should be minimized to reduce the error caused by the CG concept in the model.

•

The effects of feed solid concentration predicted by the CG VOF-DEM model qualitatively agree well with experimental data. With the increase of feed solid concentration, the separation efficiency decreases and cut size increases. Moreover, particle-particle and particle-wall interactions become more intensive under higher feed solid concentrations, which suggests that the wear rate will increase as well.

Finally, it should be pointed out that although the current work shows that the CG VOF-DEM and CG Mixture-DEM models can produce results which agree with experimental data either quantitatively or qualitatively, more efforts are necessary to further confirm the capability of the two models for hydrocyclones under various conditions. Acknowledgment The authors are grateful to the National Key R&D Program of China (2016YFB0600102), Nature Science Foundation of Jiangsu Province of China (SBK2017022643) and the Australian Research Council for the financial support of this work. 23 ACS Paragon Plus Environment



Page 24 of 48

Page 25 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


NOMENCLATURE

Dc

cyclone diameter, mm

Di

inlet diameter, mm

Do

vortex finder diameter, mm

Du

spigot diameter, mm

Lc

cylindrical length, mm

Lv

vortex finder length, mm

a

included angle, °

d

particle diameter, µm

R

particle radius, m

up

magnitude of inlet feed particle velocity, m/s

uair

magnitude of inlet feed air velocity, m/s

uwater magnitude of inlet feed water velocity, m/s qth

phase number of a fluid

αq

volume fraction of fluid qth in a computational cell

E

Young’s modulus, N/m2

fc

contact force, N

fd

damping force, N

f p− f

particle-fluid interaction force, N kc

F p− f

interaction forces between fluid and solid phases, equal to ∑f p− f ,i / ∆Vc , i=1

N/m3 g

gravity acceleration vector, 9.81 m/s2

G

gravity vector, N

I

moment of inertia of a particle, kg·m 25 ACS Paragon Plus Environment


kc

number of particles in a computational cell, dimensionless

ki

number of particles in contact with particle i , dimensionless

m

mass of a particle, kg

P

pressure, Pa

Poutle outlet pressure, atm ∆P

pressure drop, Pa

Re

Reynolds number, dimensionless

t

time, s

Stk

Stokes number, dimensionless

ts

characteristic time of water flow, s

T

driving friction torque, N·m

u

fluid velocity vector, m/s

u

magnitude of

ut

magnitude of tangential velocity, m/s

u'

fluctuating fluid velocity, m/s

V

volume, m3

v

particle velocity vector, m/s

∆Vc

volume of a computational cell, m3

c

Damping coefficient

u , m/s

Greek letters

α

size ratio of CG particle to original particle, dimensionless

β

empirical coefficient defined in Table 1, dimensionless

ε

porosity, dimensionless 26 ACS Paragon Plus Environment

Page 26 of 48

Page 27 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


αq

volume fraction of phase q in a computational cell

ρq

density of phase q in a computational cell, kg/m3

ρ

density, kg/m3

ρ water water density, kg/m3 ρ air

air density, kg/m3

δ

vector of the particle-particle or particle-wall overlap, m

δ

magnitude of δ , m

τ

viscous stress tensor, N/m3

ω

angular velocity, rad/s

ω

magnitude of angular velocity, rad/s

ωˆ

unit angular velocity

µq

viscosity of phase q in a computational cell, kg/m/s

µ

fluid viscosity, kg/m/s

µr

coefficient of rolling friction, m

µ water water viscosity, kg/m/s

µair

air viscosity, kg/m/s

µm

mixture viscosity, kg/m/s

τp

relaxation time of particle, s

µs

coefficient of sliding friction, dimensionless

ϕp

inlet feed particle concentration (by volume), %

ϕwater inlet feed water concentration (by volume), % ϕair

inlet feed air concentration (by volume), %

ν

Poisson’s ratio 27 ACS Paragon Plus Environment


Subscripts

c

contact

d

damping

D

drag

f

fluid phase

ij

between particle i and j

n

in normal direction

t

in tangential direction

max maximum p

particle

pg

pressure gradient

p− f

between particle and fluid

cg

coarse-grained particle

o

original particle


Page 28 of 48

Page 29 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


REFERENCES (1)

Bradley, D., The Hydrocyclone. Pergamon: London, 1965.

(2)

Rakesh, A.; Kumar Reddy, V. T. S. R.; Narasimha, M. Air-Core size measurement of operating hydrocyclone by electrical resistance tomography. Chem. Eng. Technol. 2014, 37, 795.

(3)

Bourgeois, F.; Majumder, A. K. Is the fish-hook effect in hydrocyclones a real phenomenon? Powder Technol. 2013, 237, 367.

(4)

Nageswararao, K.; Karri, B. Fish hook in classifier efficiency curves: An update. Sep. Sci. Technol. 2016, 158, 31.

(5)

Nageswararao, K. Further developments in the modelling and scale up industrial hydrocyclones. Ph.D. Thesis, University of Queensland, 1978.

(6)

Cui, B.; Zhang, C.; Wei, D.; Lu, S.; Feng, Y. Effects of feed size distribution on separation performance of hydrocyclones with different vortex finder diameters. Powder Technol. 2017, 322, 114.

(7)

Firth, B.; O'Brien, M.; Edward, D.; Clarkson, C. Fine coal classification. ACARP report, 1999, C3084.

(8)

Chu, L. Y.; Chen, W. M.; Lee, X. Z. Effects of geometric and operating parameters and feed characters on the motion of solid particles in hydrocyclones. Sep. Sci. Technol. 2002, 26, 237.

(9)

Hsieh, K. T. Phenomenological model of the hydrocyclone. Ph.D. Thesis, The University of Utah, 1988.

(10)

Dai, G. Q.; Chen, W. M.; Li, J. M.; Chu, L. Y. Experimental study of solidliquid two-phase flow in a hydrocyclone. Chem. Eng. J. 1999, 74, 211.

(11)

Zhang, Y. H.; Qian, P.; Liu, Y.; Wang, H. L. Experimental study of hydrocyclone flow field with different feed concentration. Ind. Eng. Chem. Res. 2011, 50, 8176.

(12)

Plitt A mathematical model of the hydrocyclone classifier. Miner. Process.

1976, 114. (13)

Schubert, H. A hydrocyclone separation model in consideration of the turbulent multi-phase flow. Particul. Sci. Technol. 2007, 3, 1.

(14)

Boysan, F.; Ayers, W. H.; Swithenbank, J. Fundamental mathematical modelling approach to cyclone design. Trans. Inst. Chem. Eng. 1982, 60, 222.



(15)

Cullivan, J. C.; Williams, R. A.; Dyakowski, T.; Cross, C. R. New understanding of a hydrocyclone flow field and separation mechanism from computational fluid dynamics. Miner. Eng. 2004, 17, 651.

(16)

Schuetz, S.; Mayer, G.; Bierdel, M.; Piesche, M. Investigations on the flow and separation behaviour of hydrocyclones using computational fluid dynamics Int. J. Miner. Process. 2004, 73, 229.

(17)

Delgadillo, J. A.; Rajamani, R. K. A comparative study of three turbulenceclosure models for the hydrocyclone problem. Int. J. Miner. Process. 2005, 77, 217.

(18)

Nowakowski, A. F.; Cullivan, J. C.; Williams, R. A.; Dyakowski, T. Application of CFD to modelling of the flow in hydrocyclones. Is this a realizable option or still a research challenge? Miner. Eng. 2004, 17, 661.

(19)

Monredon, T. C.; Hsieh, K. T.; Rajamani, R. K. Fluid flow model of the hydrocyclone: an investigation of device dimensions. Int. J. Miner. Process.

1992, 1–2, 65. (20)

Wang, B.; Chu, K. W.; Yu, A. B. Numerical study of particle-fluid flow in a hydrocyclone. Ind. Eng. Chem. Res. 2007, 46, 4695.

(21)

Gidaspow, D., Multiphase flow and fluidization. 1st edition ed.; Academic Press: 1994.

(22)

Zheng, Q. J.; Xia, B. S.; Pan, R. H.; Yu, A. B. Prediction of mass discharge rate in conical hoppers using elastoplastic model. Powder Technol. 2017, 307, 63.

(23)

Anderson, T. B.; Jackson, R. A fluid mechanical description of fluidized beds. Ind. Eng. Chem. Fundam. 1967, 6, 527.

(24)

Zhu, H. P.; Zhou, Z. Y.; Yang, R. Y.; Yu, A. B. Discrete particle simulation of particulate systems: Theoretical developments. Chem. Eng. Sci. 2007, 62, 3378.

(25)

Monaghan, J. J. Smoothed particle hydrodynamics and its diverse applications. Annu. Rev. Fluid Mech. 2012, 44, 323.

(26)

Chen, S.; Doolen, G. D. Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 1998, 30, 329.

(27)

Davailles, A.; Climent, E.; Bourgeois, F.; Majumder, A. K. Analysis of swirling flow in hydrocyclones operating under dense regime. Miner. Eng.

2012, 31, 32. 30 ACS Paragon Plus Environment

Page 30 of 48

Page 31 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(28)

Kuang, S. B.; Chu, K. W.; Yu, A. B.; Vince, A. Numerical study of liquid– gas–solid flow in classifying hydrocyclones: Effect of feed solids concentration. Miner. Eng. 2012, 31, 17.

(29)

Ji, L.; Kuang, S.; Qi, Z.; Wang, Y.; Chen, J.; Yu, A. Computational analysis and optimization of hydrocyclone size to mitigate adverse effect of particle density. Sep. Sci. Technol. 2017, 174, 251.

(30)

Nowakowski, A. F.; Kraipech, W.; Williams, R. A.; Dyakowski, T. The hydrodynamics of a hydrocyclone based on a three-dimensional multicontinuum model. Chem. Eng. J. 2000, 80, 275.

(31)

Brennan, M. S.; Narasimha, M.; Holtham, P. N. Multiphase modelling of hydrocyclones - prediction of cut-size. Miner. Eng. 2007, 20, 395.

(32)

Manninen, M. T., V.; Kallio, S., On the mixture model for multiphase flow. Technical Research Centre of Finland: VTT Publications: 1996.

(33)

Chu, K. W.; Kuang, S. B.; Zhou, Z. Y.; Yu, A. B. Model A vs. Model B in the modelling of particle-fluid flow. Powder Technol. 2018, 329, 47.

(34)

Loth, E., Particles, drops and bubbles: fluid dynamics and numerical methods. Cambridge University Press: London, 2010.

(35)

Murthy, Y. R.; Bhaskar, K. U. Parametric CFD studies on hydrocyclone. Powder Technol. 2012, 230, 36.

(36)

Xu, Y. X.; Song, X. F.; Sun, Z.; Lu, G. M.; Li, P.; Yu, J. G. Simulation analysis of multiphase flow and performance of hydrocyclones at different atmospheric pressures. Ind. Eng. Chem. Res. 2012, 51, 443.

(37)

Zhou, Q.; Wang, C. S.; Wang, H. B.; Wang, J. W. Eulerian-Lagrangian study of dense liquid-solid flow in an industrial-scale cylindrical hydrocyclone. Int. J. Miner. Process. 2016, 151, 40.

(38)

Xu, B. H.; Yu, A. B. Numerical simulation of the gas-solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics. Chem. Eng. Sci. 1997, 52, 2785.

(39)

Tsuji, Y.; Tanaka, T.; Ishida, T. Lagrangian numerical-simulation of plug flow of cohesionless particles in a horizontal pipe. Powder Technol. 1992, 71, 239.

(40)

Pozzetti, G.; Peters, B. A multiscale DEM-VOF method for the simulation of three-phase flows. Int. J. Multiphas. Flow 2018, 99, 186.



(41)

Jing, L.; Kwok, C. Y.; Leung, Y. F.; Sobral, Y. D. Extended CFD-DEM for free-surface flow with multi-size granules. Int. J. Numer. Anal. Met. 2016, 40, 62.

(42)

Pozzetti, G.; Peters, B. On the choice of a phase interchange strategy for a multiscale DEM-VOF method. 2017: AIP Conference Proceedings, 1863.

(43)

Chu, K. W.; Wang, B.; Xu, D. L.; Chen, Y. X.; Yu, A. B. CFD–DEM simulation of the gas–solid flow in a cyclone separator. Chem. Eng. Sci. 2011, 66, 834.

(44)

Sun, X.; Sakai, M. Three-dimensional simulation of gas–solid–liquid flows using the DEM–VOF method. Chem. Eng. Sci. 2015, 134, 531.

(45)

Chu, K.; Chen, J.; Yu, A. Applicability of a coarse-grained CFD–DEM model on dense medium cyclone. Miner. Eng. 2016, 90, 43.

(46)

Chu, K. W.; Wang, B.; Yu, A. B.; Vince, A. CFD-DEM modelling of multiphase flow in dense medium cyclones. Powder Technol. 2009, 193, 235.

(47)

Pezo, M.; Pezo, L.; Jovanović, A.; Lončar, B.; Čolović, R. DEM/CFD approach for modeling granular flow in the revolving static mixer. Chem. Eng. Res. Des. 2016, 109, 317.

(48)

Patankar, N. A.; Joseph, D. D. Lagrangian numerical simulation of particulate flows. Int. J. Multiphas. Flow 2001, 27, 1685.

(49)

Lu, L.; Yoo, K.; Benyahia, S. Coarse-grained-particle method for simulation of liquid–solids reacting flows. Ind. Eng. Chem. Res. 2016, 55, 10477.

(50)

Chu, K. W.; Chen, J.; Wang, B.; Yu, A. B.; Vince, A.; Barnett, G. D.; Barnett, P. J. Understand solids loading effects in a dense medium cyclone: Effect of particle size by a CFD-DEM method. Powder Technol. 2017, 320, 594.

(51)

Sakai, M.; Koshizuka, S. Large-scale discrete element modeling in pneumatic conveying. Chem. Eng. Sci. 2009, 64, 533.

(52)

Lu, L.; Xu, J.; Ge, W.; Yue, Y.; Liu, X.; Li, J. EMMS-based discrete particle method (EMMS–DPM) for simulation of gas–solid flows. Chem. Eng. Sci.

2014, 120, 67. (53)

Benyahia, S.; Galvin, J. E. Estimation of numerical errors eelated to eome basic assumptions in discrete particle methods. Ind. Eng. Chem. Res. 2010, 49, 10588.

(54)

Sakai, M.; Abe, M.; Shigeto, Y.; Mizutani, S.; Takahashi, H.; Vire, A.; Percival, J. R.; Xiang, J. S.; Pain, C. C. Verification and validation of a coarse 32 ACS Paragon Plus Environment

Page 32 of 48

Page 33 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


grain model of the DEM in a bubbling fluidized bed. Chem. Eng. J. 2014, 244, 33. (55)

Mokhtar, M. A.; Kuwagi, K.; Takami, T.; Hirano, H.; Horio, M. Validation of the similar particle assembly (SPA) model for the fluidization of Geldart's group A and D particles. AICHE J. 2012, 58, 87.

(56)

Thakur, S. C.; Ooi, J. Y.; Ahmadian, H. Scaling of discrete element model parameters for cohesionless and cohesive solid. Powder Technol. 2016, 293, 130.

(57)

Launder, B. E.; Spalding, D. B., Lectures in mathematical models of turbulence. Academic Press: London, England, 1972.

(58)

Hirt, C. W.; Nichols, B. D. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comp. Phys. 1981, 39, 201.

(59)

Wang, B.; Yu, A. B. Numerical study of particle–fluid flow in hydrocyclones with different body dimensions. Miner. Eng. 2006, 19, 1022.

(60)

Cundall, P. A.; Strack, O. D. L. Discrete numerical-model for granular assemblies. Geotechnique 1979, 29, 47.

(61)

A.C.Hoffmann; L.E.Stein, Gas cyclones and swirl tubes principles, design and operation. Second edition. Springer Science & Business Media: New York, 2007.

(62)

Marthinussen, S. A. The effect of fluid viscosity on hydrocyclone performance. Master Thesis, University of Bergen, 2011.

(63)

Swain, S.; Mohanty, S. A 3-dimensional Eulerian–Eulerian CFD simulation of a hydrocyclone. Appl. Math. Model. 2013, 37, 2921.

(64)

Mousavian, S. M.; Najafi, A. F. Influence of geometry on separation efficiency in a hydrocyclone. Arch. Appl. Mech. 2008, 79, 1033.

(65)

Kraipech, W.; Nowakowski, A.; Dyakowski, T.; Suksangpanomrung, A. An investigation of the effect of the particle–fluid and particle–particle interactions on the flow within a hydrocyclone. Chem. Eng. J. 2005, 111, 189.

(66)

Verma, V.; Li, T.; De Wilde, J. Coarse-grained discrete particle simulations of particle segregation in rotating fluidized beds in vortex chambers. Powder Technol. 2017, 318, 282.

(67)

Lu, L.; Xu, Y.; Li, T.; Benyahia, S. Assessment of different coarse graining strategies to simulate polydisperse gas-solids flow. Chem. Eng. Sci. 2018, 179, 53. 33 ACS Paragon Plus Environment


(68)

Muzanenhamo, P. Assessing the effect of cone ratio, feed solids concentration and viscosity on hydrocyclone performance. Master Thesis, University of Cape Town, 2014.

(69)

Wahyudi, H.; Chu, K.; Yu, A. 3D particle-scale modeling of gas–solids flow and heat transfer in fluidized beds with an immersed tube. Int. J. Heat Mass Tran. 2016, 97, 521.

(70)

Svarovsky, L., Hdrocyclones. Holt, Rinehart and Winston: London, 1984.

(71)

Castilho, L. R.; Medronho, R. A. A simple procedure for design and performance prediction of Bradley and Rietema hydrocyclones. Miner. Eng.

2000, 13, 183. (72)

Ghodrat, M.; Kuang, S. B.; Yu, A. B.; Vince, A.; Barnett, G. D.; Barnett, P. J. Computational study of

the multiphase flow and performance of

hydrocyclones: effects of cyclone size and spigot diameter. Ind. Eng. Chem. Res. 2013, 52, 16019. (73)

Chu, K. W.; Yu, A. B. Numerical simulation of complex particle-fluid flows. Powder Technol. 2008, 179, 104.

(74)

Lu, L.; Benyahia, S. Method to estimate uncertainty associated with parcel size in coarse discrete particle simulation. AICHE J. 2018.

(75)

Rajamani, R. K.; Milan, L., Fluid-flow model of the hydrocyclone for concentratated slurry classfication, in Hydrocyclone: Analysis and Application, Svarovsky, L.; Thew, M. T., Editors. 1992, Springer Netherlands: Dordrecht. p. 95.

(76)

Ghodrat, M.; Qi, Z.; Kuang, S. B.; Ji, L.; Yu, A. B. Computational investigation of the effect of particle density on the multiphase flows and performance of hydrocyclone. Miner. Eng. 2016, 90, 55.

(77)

Chu, K. W.; Yu, A. B. Numerical simulation of the gas-solid flow in threedimensional pneumatic conveying bends. Ind. Eng. Chem. Res. 2008, 47, 7058.

(78)

Chu, K. W.; Wang, B.; Yu, A. B.; Vince, A.; Barnett, G. D.; Barnett, P. J. CFD-DEM study of the effect of particle density distribution on the multiphase flow and performance of dense medium cyclone. Miner. Eng. 2009, 22, 893.

(79)

Olson, T. J.; Van Ommen, R. Optimizing hydrocyclone design using advanced CFD model. Miner. Eng. 2004, 17, 713. 34 ACS Paragon Plus Environment

Page 34 of 48

Page 35 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(80)

Di Felice, R. The voidage function for fluid-particle interaction systems. Int. J. Multiphas. Flow 1994, 20, 153.

(81)

Rong, L. W.; Dong, K. J.; Yu, A. B. Lattice-Boltzmann simulation of fluid flow through packed beds of uniform spheres: Effect of porosity. Chem. Eng. Sci. 2013, 99, 44.



Page 36 of 48

Table 1 Equations used to calculate the forces and torques acting on particle i . Forces and torques

Symbols

Equations

−

fcn,ij

Contact Normal

E 2 Ri δ n3 / 2n 2 3(1 − v ) 1/2

forces

Damping

Contact

fdn,ij

 3m E  i  v −cn  R δ  2 (1 − v 2 ) i n  n ,ij  

fct,ij

3/ 2 µ s f cn ,ij   min {δ t , δ t ,max }   δ t 1 − 1 − −   δ t   δ t ,max 



Tangential forces



 1 − δ t / δ t ,max − c t  6m i µ s f cn ,ij  δ t , max 

Damping

fdt,ij

Rolling

Tij

R i × (f ct ,ij + f dt ,ij )

Friction

Mij

ˆi − µrfcn,ijω

Gravity

Gi

mi g

   

1/ 2

v t ,ij

Torque

Body force

Viscous drag Particle-fluid

force

  ρ u − v i (u i − v i ) πd i2 − β  0.63 + 4.80.5  f i εi  Re p ,i  2 4  2

f d ,i

interaction force

Pressure gradient force where: n =

Ri Ri

V p ,i ∇P

f pg,i

(

)

, vij = v j − vi + ωj × R j − ωi × Ri ， vn,ij = vij • n • n ，

ˆi = v t,ij = (v ij × n )× n ， ω

d i ρ f ε i ui − v i ωi ， Re p ,i = ， µf ωi

 (1.5 − log Re p ,i )  β = 2.65(ε + 1) − (5.3 − 3.5ε )ε exp −  2  

kc

2

2


81

，ε = 1 −

∑V

i

i =1

∆Vc

80

Page 37 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 2 Geometrical conditions for the 75 mm and 100 mm diameter hydrocyclones. Symbol Units

Parameter

Value (75mm

Value (100mm

hydrocyclone)

hydrocyclone)

Diameter of the body

Dc

mm

75

100

Diameter of inlet

Di

mm

25

30

Diameter of vortex finder

Do

mm

25

41

Diameter of apex

Du

mm

12.5

23

Length of cylindrical part

Lc

mm

75

28

Length of vortex finder

Lv

mm

50

65

Included angle

a

°

20

15



Page 38 of 48

Table 3 Operational conditions and material properties for the 75 mm and 100 mm diameter hydrocyclones. Phase

Symb

Parameter

ol

Units

Density

ρp

kg/m

Particle diameter

d

Inlet feed particle concentration

Value (75mm Value (100mm hydrocyclone) hydrocyclone)

3

2700

3600

µm

0.43-42

1-841

ϕp

%

1,4, and 8

6 and 10

Inlet feed particle velocity

up

m/s

2.5

4.1

Rolling friction coefficient

µr

mm

0.005

Sliding friction coefficient

µs

--

0.3

Poisson’s ratio

ν

--

0.3

Young’s modulus

E

N/m2

1 × 108

DEM time step

∆τ

s

5 × 10-7

Damping coefficient

c

--

0.3

Density

ρair

kg/m3

1.225

Solid

Viscosity Air

Water

1.8 × 10-5

µ air kg/m/s

Inlet feed air volume fraction

ϕair

%

0

Outlet pressure

Poutlet

atm

1

Inlet feed gas velocity

µ air

m/s

2.5

4.1

Density

ρwater kg/m3

998.2

Viscosity

µ water kg/m/s

0.001

Inlet feed water volume fraction ϕwater Inlet feed water velocity

uwater

%

99, 96, and 92

94 and 90

m/s

2.5

4.1


Page 39 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Step 1

Step 2

Two-way coupling

Water Air

RSM VOF

Aircore Velocity distribution

RSM DEM

Partition curve Split ratio and so on

Particles

Figure 1 Modelling strategy for the CG VOF-DEM model

Figure2 Modelling strategy for the CG Mixture-DEM model



(a)

(b)

Page 40 of 48

(c)

Figure 3 (a) Geometry, (b) mesh representation, and (c) A-A plane mesh representation of the 75 mm diameter hydrocyclone.

(a)

(b)

Figure 4 (a) Geometry and (b) mesh representation of the 100 mm diameter hydrocyclone.


Page 41 of 48

100

100

80

Feed passing (%)

Feed passing (%)

Size distribution

60 40 20

Size distribution

80 60 40 20

0

Mixture DEM 5.27

1

0

10

Mixture DEM

100

1

Particle size (µm)

10 19

100

1000

Particle size (µm)

(a)

(b)

Figure 5 Size distributions of solid particles for (a) 75 mm diameter and (b) 100 mm diameter hydrocyclones. 100

Partition number (%)

100


80 60 Exe. CPS 1.69µm CPS 3.73µm CPS 5.27µm CPS 10µm

40 20 0 0

10

20

30

80 60 Exe. CPS 10µm CPS 19µm CPS 38µm CPS 100µm

40 20 0

40

0

20

40

60

80

100

Particle size (µm)

Particle size (µm)

(a)

(b)

Figure 6 Trail simulations to determine CPSs for (a) 75 mm diameter and at a feed solid concentration of 8% and (b) 100 mm diameter hydrocyclones at a feed solid concentration of 10%. 100


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


80 60 40 Con. 4% Exe. CG VOF-DEM CFD-LPT

20 0 0

10

20

30

Particle size (µm)

40

50

Figure 7 Experimental (Hsieh, 1988) and simulated partition curves for the 75 mm diameter hydrocyclone at a feed solid concentration of 4%.




100 80 60 40 Con. 8% Exe. CG Mixture-DEM

20 0 0

10

20

30

40

50

Particle size (µm)

Figure 8 Comparison of experimental (Rajamani and Milan, 1992) and simulated partition curves for the 75 mm diameter hydrocyclone at a feed solid concentration of 8%.

100


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

80 60 40 6%

20

10%

Exe. CG VOF-DEM CG Mixture-DEM

0 0

20

40

60

80

100

Particle size (µm) Figure 9 Comparison of experimental (Muzanenhamo, 2014) and simulated partition curves for the 100 mm diameter hydrocyclone.


Page 42 of 48

Page 43 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(I)

AveCon 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

(II)

AveCon 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

(a)

(b)

Figure 10 Predicted (a) flow pattern at t=2.4 s and (b) distributions of time-averaged solid volume fraction by the (I) CG VOF-DEM and (II) CG Mixture-DEM models in the 75 mm diameter hydrocyclone at a feed solid concentration 8%.



Particle-particle interaction (N/m³/s)

Particle-wall interaction (N/m²/s)

1.6E+08 1.45455E+08 1.30909E+08 1.16364E+08 1.01819E+08 8.72736E+07 7.27284E+07 5.81831E+07 4.36378E+07 2.90925E+07 1.45473E+07 2000

(a)

(b)

240000 218364 196727 175091 153455 131818 110182 88545.5 66909.1 45272.7 23636.4 2000

(c)

(d)

Figure 11 Predicted distribution of (a) pressure, (b) tangential velocity, (c) timeaveraged particle-particle, and (d) time-averaged particle-wall interactions by the CG Mixture-DEM model in the 75 mm diameter hydrocyclone at a feed solid concentration of 8%.

100

100



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 44 of 48

80 60 40 20

1%

4%

7%

CG VOF-DEM

0 0

10

20

30

80 60 40 20 1%

40

50

60

4%

7%

CG Mixture-DEM

0 0

10

20

30

40

50

60

Particle size (µm)

Particle size (µm)

(a)

(b)

Figure 12 Simulated partition curves at different feed solid concentrations for 75 mm diameter hydrocyclone by (a) the CG VOF-DEM model and (b) the CG MixtureDEM model.


Page 45 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(I) AveCon 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02


(II)

1.6E+08 1.45455E+08 1.30909E+08 1.16364E+08 1.01819E+08 8.72736E+07 7.27284E+07 5.81831E+07 4.36378E+07 2.90925E+07 1.45473E+07 2000

Particle-wall interaction (N/m²/s) 240000 218364 196727 175091 153455 131818 110182 88545.5 66909.1 45272.7 23636.4 2000

(III)

(a)

(b)

(c)

Figure 13 Predicted distribution of (I) time-averaged solid volume fraction, (II) timeaveraged particle-particle, and (III) time-averaged particle-wall interaction by CG VOF-DEM model in the 75 mm diameter hydrocyclone at feed solid concentrations of (a) 1%, (b) 4% and (c) 8%. 45 ACS Paragon Plus Environment


Page 46 of 48

(I) AveCon 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02


(II)

1.6E+08 1.45455E+08 1.30909E+08 1.16364E+08 1.01819E+08 8.72736E+07 7.27284E+07 5.81831E+07 4.36378E+07 2.90925E+07 1.45473E+07 2000

Particle-wall interaction (N/m²/s)

(III)

240000 218364 196727 175091 153455 131818 110182 88545.5 66909.1 45272.7 23636.4 2000

(a)

(b)

(c)

Figure 14 Predicted distribution of (I) time-averaged solid volume fraction, (II) timeaveraged particle-particle, and (III) time-averaged particle-wall interaction by CG Mixture-DEM model in the 75 mm diameter hydrocyclone at feed solid concentrations of (a) 1%, (b) 4% and (c) 8%. 46 ACS Paragon Plus Environment

Page 47 of 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Highlights:

•

Two CFD models based on coarse-grained (CG) DEM are developed for hydrocyclones.

•

The CG VOF-DEM model is quantitatively applicable to dilute flows but only qualitatively to dense flows.

•

The CG Mixture-DEM model can be quantitatively applicable to both dilute and dense flows.

•

The effect of feed solid concentration is investigated using both models.



Graphic abstract


Page 48 of 48

Modeling the Multiphase Flow in Hydrocyclones Using the Coarse

Recommend Documents