Essay: Simulation of the multiphase flow in dense medium cyclone with high orientation angle containing ‎ferrosilicon medium: a CFD study

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A computational fluid dynamics (CFD) numerical study of the multiphase flow in a dense medium ‎cyclone (DMC) have been conducted to predict the flow pattern and medium (defined as the ‎mixture of water and the ferrosilicon) segregation containing ferrosilicon. The volume of fluid ‎‎(VOF) model is used to show the initial shape and position of the air core (defined as the region ‎with volume fraction of air larger than 90%), static pressure and velocity distribution. These ‎results shown to be generally comparable to those reported in the literature. The mixture model is ‎used to show medium segregation. Medium density distribution is appropriate and it decreases from ‎wall to DMC center, but under-predicts the level of segregation compared to the data reported in the ‎literature. The turbulence is described by the Reynolds stress model (RSM). For both the VOF and the ‎mixture models, the velocity of entrance flow is different and it is determined based on the flow rate ‎of DMC pump. The effects of length of vortex finder (15-21 cm), medium density (2300-2640 kg⁄‎m^3 ) and orientation angle (10-25°) with horizontal level on flow pattern are studied. ‎
Keywords: smithsonite; ferrosilicon; orientation angle; dense medium cyclone (DMC); Reynolds stress ‎model (RSM); medium segregation ‎

‎1. Introduction‎

The dense medium cyclone (DMC) is a high-tonnage device that has been widely used to ‎upgrade the run-of-mine coal, in the 0.5-50 mm size range, in modern coal industry by ‎separating gangue from product coal (Narasimha et al., 2006). The DMC is also used in ‎other mineral processing plants for treating iron ore, dolomite, diamonds, potash, and lead–‎zinc ores (Chu et al., 2015). Normally, the feed slurry including water, magnetite, and ‎nonmagnetic particles is named “medium” (defined as the mixture of water and the ‎ferrosilicon) in practice. In the past decade or so, significant experimental research has been ‎carried out with hydrocyclones where dense medium is absent and where the measurements ‎are relatively easy to make (Wang et al., 2009). Hsieh, 1988; Hsieh and Rajaamani, 1991, ‎reported a comprehensive experimental study of the fluid flow pattern, pressure drop, and ‎solids motion in a 75 mm hydrocyclone using laser doppler velocimetry (LDV). The density ‎distribution of medium in a DMC has been reported by Galvin and Smitham, 1994 using X-‎ray tomography and by Subramanian, 2002 using gamma ray tomography (GRT).‎

The DMC is simple in design and geometry but the flow pattern within it is very complex ‎due to the presence of the turbulent vortex formed, the air-core, and the size and the density ‎distributions of the feed and process medium solids (Chu et al., 2009). Segregation of ‎medium and ore particles with the size and density to physically separate solid particles ‎using the strong centrifugal force field generated by the internal rotational motion is ‎another complicated phenomenon in DMC devices (Chu et al., 2009; Kuang et al., 2013; ‎Wang et al., 2009). The computational fluid dynamics (CFD) analysis of the DMC flow ‎using only magnetite medium was performed, and a mathematical model was proposed to ‎describe the multiphase flow in the DMC. In this model, the volume of fluid (VOF) ‎multiphase model is first used to determine the shape and position of the air core (defined ‎as the region with volume fraction of air larger than 90%), and the pressure and velocity ‎distributions. Then the mixture multiphase model was employed to describe the flow of the ‎dense medium and the density distribution, where the turbulence was described by the ‎Reynolds stress model (RSM) (Narasimha et al., 2006; Kuang et al., 2013; Wang et al., ‎‎2009). Of course, as related literature indicate that to simulate a complicated multiphase ‎flow a LES model is more adaptable than a RSM model but this model suffers from some ‎drawbacks, such as utilization of more resources and slower computation speed, and it may ‎need a finer grid (Narasimha et al., 2007a; Narasimha et al., 2007b; Shen et al., 2009).‎
All previous CFD investigations have only been done on coal in special density range ‎‎(1200-2200 kg⁄m^3 ) and medium has been created by magnetite with density 1500 kg⁄‎m^3 . ‎
The ferrosilicon medium rheology such as viscosity, stability and its other properties were ‎studied and were compared with magnetite medium. Ferrosilicon has a fast settling rate in ‎dense suspension, attributed to its very high solids density, coarser particle size distribution, ‎more spherical particle shape and low medium viscosity at the same medium density ‎‎(Dunglison et al., 2000; Svoboda, 2004; Shi, 2016). A characteristic curve between apparent ‎viscosity and medium density was established, which can be used in FeSi selection for ‎dense medium separation. Medium stability was determined from the FeSi sedimentation ‎rate measurement. It shows that medium stability was closely correlated with medium ‎viscosity (Shi, 2016).‎
Based on the CFD and CFD-DEM simulated data, a PC-based mathematical model is ‎formulated to predict the performance of DMCs under various conditions. Then, the effects ‎of some key variables related to DMC geometry (dimensions of different parts of the ‎DMC), operational conditions (medium to coal volumetric ratio and medium feed density ‎‎(RD)) and materials properties (Coal averaged relative density (RD)) were examined. With ‎increasing length of vortex finder the operational head was slightly decreased but with ‎increasing orientation angle, the operational head dropped further (Chen et al., 2012). The ‎PC-based model has been used to design and optimize DMC. In order to achieve high ‎capacity, the change of length of vortex finder had no effect, so standard Lv was used ‎‎(0.6*cyclone diameter) but in order to achieve high separation efficiency or lower Ep, ‎length of vortex finder should be decreased (Lv=0.4*cyclone diameter) (Chen et al., 2014). ‎Of course, the effects of these geometry parameters of length of vortex finder and ‎orientation angle on other variables of DMC performance such as the flow pattern and ‎medium segregation weren’t studied. A computational fluid dynamics (CFD) model is ‎proposed to describe the multiphase flow in a dense medium cyclone (DMC). The results ‎reveal that when vortex finder or spigot diameter is varied at the same U:O ratio, the offset ‎and medium split nearly remain the same, however, the coal feed rate and Ep are different ‎under the conditions considered. It is also shown that the fish-hook phenomenon is observed ‎when spigot diameter is equal to or slightly larger than vortex finder diameter, and a normal ‎operation becomes less stable with decreasing U:O ratio (Kuang et al., 2013). Different ‎designs of the outlet geometry of the vortex finder are used to achieve different purposes. ‎This phenomenon is studied numerically by use of a combined approach of CFD–DEM with ‎reference to the effect of the pressure at the vortex finder. It is shown that a relatively small ‎change of the vortex finder pressure can cause significant variations of both the medium-‎coal flow and DMC performance (Chu et al., 2012). The wear rate of DMC walls due to the ‎impact of coal particles is predicted by a combined computational fluid dynamics and ‎discrete element method (CFD-DEM) approach, using the Finnie wear model from the ‎literature. The numerical results show that the severe wear locations are generally the inside ‎wall of the spigot and the outside wall of the vortex finder. The specific wear pattern ‎depends on operational conditions and particle properties or particle type. The wear rate ‎generally increases with a decrease of M:C ratio. The wear of the inner spigot wall is found ‎to affect the DMC performance significantly. It is shown that as the life of the spigot wall ‎increases, operational head, medium split and cut density decrease, Ep increases, and ‎medium differential decreases to a minimum and then increases slightly (Chu et al., 2014).‎
All numerical studies of CFD and coupled CFD-DEM of the DMC, were done by magnetite ‎medium to preparation Coal with the orientation angle of DMC is equal to 10-15° (Chu et ‎al., 2015; Kuang et al., 2013; Wang et al., 2009).‎

In this work, the CFD study of the multiphase flow into the DMC used for pre-concentration ‎of Smithsonite ore. The VOF model to determine the shape and position of the air core and ‎the pressure and velocity distributions is used and the mixture model to describe the flow of ‎the dense medium and the density distribution is employed. To validate the results ‎achieved, the literatures is used. ‎
The density of employed ore in this work is 3090 kg⁄m^3 , it is more than coal density in ‎previous investigations, so the ferrosilicon medium with high density is used. A comparison ‎between magnetite and ferrosilicon is done that ferrosilicon medium has more stability and ‎lower viscosity than the magnetite. ‎
Finally, the effects of the length of vortex finder, high orientation angle of the DMC and ‎high density of the medium on flow pattern are studied.‎

‎2. Mathematical Modelling
Recognizing that the incompressible fluid flow in a DMC is quite complicated, the CFD ‎modelling was divided into two steps, as shown in Figure 1. These two steps are ‎devoted to solving the medium slurry flow. The continuum medium 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))/∂t+∂/(∂X_i ) (ρ_f u_i )=0 (1)‎
‎∂(ρ_f u_i )/∂t+∂/(∂X_j ) (ρ_f u_i u_j )=-∂p/(∂X_i )+∂/(∂X_j ) [μ((∂u_i)/(∂X_j )‎‎+(∂u_j)/(∂X_i ))]+∂/(∂X_j ) (〖-ρ〗_f (〖u^’〗_i 〖u^’〗_j ) ̅ ) (2)‎
where the velocity components are decomposed into the mean (u_i ) ̅ and fluctuating ‎‎〖u^’〗_i velocities (i =1, 2, 3). They are related, as given by
u_i=(u_i ) ̅+〖u^’〗_i (3) ‎
‎-ρ(u^’ u^’ ) ̅ is the Reynolds stress term in eq. (2) due to the turbulence and is modelled by ‎the Reynolds Stress Model (RSM) provided in the ANSYS-Fluent software (Wang et al., ‎‎2009). ‎
In the first step, only the air and the water are considered. In the VOF model is used to ‎describe the interface between the medium and the air core. The two-phases are modelled ‎by solving a single set of momentum equations and tracking the volume fraction of each of ‎the fluids throughout the domain. Also, the turbulence is modelled using the RSM model ‎‎(Wang et al., 2009). At this stage, the position of the air core and the initial velocity and ‎pressure distribution is obtained. The method is similar to that used for modelling ‎multiphase flow in hydro-cyclones (Wang et al., 2007). In step 2, one additional phase is ‎introduced to describe the behavior of ferrosilicon particles with average sizes. The ‎multiphase model is changed from the VOF to the Mixture model. Medium density ‎distribution is obtained at the end of this step. ‎
The details of the calculation of the dense medium flow can be found in Refs. (Chu et al., ‎‎2009; Wang et al., 2009; Wang et al., 2007). ‎

Figure 1. Steps used in the present simulation.‎

‎3. Operation conditions and details of numerical methods
‎3.1. Concentration circuit of Smithsonite mineral in DMC

The DMC is applied to concentrate a low-grade smithsonite ore using ferrosilicon medium, ‎as is shown in Figure 2. In this work, simulation data are used based on the industrial data. ‎Simulation’s parameters such as medium density and viscosity are also calculated. The ‎particles size distribution of smithsonite ore in this dense medium separation is equal to 0.5 ‎mm ≤ d≤20 mm, and the range of variation of the dense medium’s density is 2300 kg⁄‎m^3 ≤  ≤ 2640 kg⁄m^3 . The pulp specifications of the ferrosilicon medium are shown ‎in Table 1.‎

Figure 2. The DMC operated for low-grade Smithsonite ore that is used to simulate in ‎present study (Located in the Kane-Araie Aria Co., Zanjan, Iran).‎
Table 1. The pulp specifications of the ferrosilicon medium
The parameters of medium Units Value
FeSi Pump Qv m3/h‎ ‎97.2-194.4‎
Averaged FeSi Density kg/ m3‎ ‎6808‎
Pulp Density (max)‎ kg/ m3‎ ‎2640-2300‎
‎% solid (X): FeSi sump‎ ‎%‎ ‎72.8-66.3‎
Qmp: FeSi Pulp‎ t/h ‎256.6-223.6‎
Qms: FeSi Solid‎ t/h ‎186.9-148.1‎

‎3.2. Geometry and meshing

The details of geometry which is considered based on the industrial DMC, are shown in ‎Table 2. The solution geometry (the DMC’s internal volume) is used for the CFD ‎simulation. The DMC meshes were generated using the Hexahedral cells due to the ‎accuracy of the numerical results obtained based on these cells. In present study, the whole ‎computational domain is divided into 88881 hexahedron cells. A view of generated mesh in ‎the DMC is shown in Figure 3-A. In some parts of the DMC, such as the vicinity of the ‎walls and vortex finder, due to turbulent flow the grid is finer than its other parts. ‎
For better understanding the flow, such as, tangential velocity and axial velocity on entire ‎domain, the sections normal to Z-axis and Y-axis on the DMC are shown in Figure 3-B. The ‎DMC is operated at high orientation angle. In the present simulations, the orientation angle ‎‎(is defined as the angle between the DMC central axis and the horizontal plane) is 〖25〗‎‎^°.‎

Table 2. Geometry dimensions of the DMCs considered ‎
Parameter symbol value
Diameter of the body Dc ‎34 cm‎
Type of inlet ‎-‎ Tangential
Shape of inlet ‎-‎ Square
Side length of inlet Li ‎9 cm‎
Side width of inlet W ‎6 cm‎
Hydraulic diameter of inlet (4A/P)‎ Dhyd.‎ ‎7.2 cm
Length of vortex finder Lv ‎50-56 cm
Diameter of vortex finder ‎ Do ‎20 cm‎
Length of cylindrical part Lc ‎50 cm‎
Height of conical part Lp ‎78 cm
Spigot diameter Du ‎10 cm‎
Orientation angle  ‎25 degrees

Figure 3. A) a view of generated mesh of the DMC and B) the sections normal to Z-axis and ‎Y-axis on the DMC.‎

‎3.3. Steady state and convergence criteria

It is assumed that the steady-state operating condition is achieved when the variables ‎variation such as, the pulp flow rate over time should be negligible on entire domain of the ‎DMC. Required simulation time must be determined in order to guarantee the convergence ‎to a reliable solution. During the two-phases CFD simulation with the VOF model, the ‎instantaneous variation of the volume flow rate and velocity through these cross sections on ‎entire domain are checked. When the variations versus time approached zero, the steady ‎state conditions have been achieved. The time needed to achieve the steady state for the ‎DMC is 3-4s. ‎

‎3.4. Selection of the suitable media‎
‎3.4.1. Media types and sizes

The most common medium for separation is a suspension in water of fine Magnetite, ‎ferrosilicon (FeSi), or a mixture of the two, depending on the separation density required. ‎Ferrosilicon alone for 2900 to 3700 kg⁄m^3 (Svoboda, 2004). In all previous papers, ‎Magnetite has been used in coal preparation but in this work, Smithsonite ore is ‎concentrated and the range averaged density of ore different particles is 2890 – 3090 kg⁄‎m^3 , so ferrosilicon medium is used. Ferrosilicon is invariably used, for instance, for the ‎concentration of diamonds, Lead & Zinc ores, for the recovery of aluminum in recycling ‎plants, and for beneficiation of iron and manganese ores. Ferrosilicon containing between ‎‎14% to 16% silicon has a density ranging from 6700 to 7100 kg⁄m^3 . It is non-rusting ‎and strongly magnetic. With concentrations of silicon greater than 22% the material is ‎feebly magnetic, while with concentrations of silicon lower than 15%, it is prone to rusting ‎‎(Svoboda, 2004). In this work, particle size distribution of used ferrosilicon and its other ‎characterizations in Figure 4 and Table 3 are shown, respectively.‎

Figure 4. Particle size distribution of used ferrosilicon medium.‎

Table 3. Ferrosilicon characterizations used in this work.‎
d, ‎
micron Retained, ‎
G Retained, %‎ Density, ‎kg⁄m^3 ‎ Dynamic ‎viscosity, ‎kg/m/s ‎‎(Pa.s)‎ Fe, ‎
‎%‎
‎75‎ ‎48.4‎ ‎9.7‎ ‎6812‎ ‎0.004287‎ ‎55.33‎
‎45‎ ‎60.4‎ ‎12.1‎ ‎6937‎ ‎0.004362‎ ‎66.74‎
‎38‎ ‎43‎ ‎8.6‎ ‎7105‎ ‎0.004463‎ ‎74.83‎
‎25‎ ‎36‎ ‎7.2‎ ‎6958‎ ‎0.004375‎ ‎78.09‎
‎-25‎ ‎309.4‎ ‎62.2‎ ‎6723‎ ‎0.004234‎ ‎78.38‎
Total ‎ ‎497.2‎ ‎100‎ ‎-‎ ‎-‎ ‎-‎
Averaged ‎ ‎-‎ ‎-‎ ‎6808‎ ‎0.004285‎ ‎74.39‎

‎3.4.2. Viscosity of dense medium suspensions ‎

Magnetite media are characterized by high apparent viscosities, extremely sensitive to rate ‎of shear. Viscosities of the magnetite medium ranging from 700 cP (0.7 Pa×s) at the shear ‎rate of 1000 s-1 to 2740 cP (2.74 Pa×s) at 1000 s-1 were reported at the operating density of ‎‎2800 kg⁄m^3 . Viscosities of ferrosilicon under similar conditions ranged from 84 cP ‎‎(0.084 Pa×s) to 200 cP (0.2 Pa×s). Similar results were obtained for various grades of ‎ferrosilicon (Svoboda, 2004). Therefore, according to the above description about magnetite ‎and ferrosilicon, FeSi has greater solids density, coarser particle size distribution, more ‎spherical particle shape and smaller medium viscosity at the same medium density, so, in ‎this work ferrosilicon medium is used. Medium viscosity is an important parameter ‎describing the behavior of the medium flow and the separation of particles in DMCs. Its ‎determination is complicated, because it depends on many variables, such as ferrosilicon ‎particle size distribution, particle shape, medium density, medium contamination, and so on ‎‎(Svoboda, 2004). Medium dynamic viscosity in Inlet boundary conditions, is calculated by ‎the solid fraction of ferrosilicon on proposed model by Ishii and Mishima, 1984 that was ‎corrected by Wang et al., 2009 (eq. 4). ‎
μ_m=3.8μ_c (1-α_d/0.62)^(-1.55) (4)‎
Ferrosilicon medium dynamic viscosity (μ_m) has been reduced for increasing the density ‎to 2640 kg⁄m^3 and grinding type of ferrosilicon particles to milled (coarse).‎

‎3.4.3. Calculation of “ferrosilicon particles” Viscosity as fluid phase

Based on simulation approach (Figure 1), the first and second stages are related to CFD ‎simulation, so ferrosilicon particles are considered as fluid phase. The ferrosilicon powder ‎dynamic viscosity was measured based on equation derived from linear trendline of ‎dynamic viscosity of air, water and magnetite fluids vs. density (Table 4 and Figure 5). ‎

Table 4. The viscosity and density of fluids.‎
Dynamic Viscosity,‎
Kg/m/s Density,‎
kg‎⁄m^3 ‎ Type of ‎fluids
‎0.000017894‎ ‎1.225‎ Air ‎
‎0.001003‎ ‎998.2‎ Water ‎
‎0.0033‎ ‎4945‎ Magnetite

Figure 5. The relationship between the fluid dynamic viscosity vs. density.‎

According to the linear relationship in Figure 5. and ferrosilicon powder density (6808 kg⁄‎m^3 ), ferrosilicon dynamic viscosity is gotten 0.004285 Pa.s. Also, air is considered as ‎primary continuous phase and water and ferrosilicon as secondary continuous phases. The ‎characterization of used fluids in DMCs considered is shown in Table 5.‎

Table 5. The characterization of used fluids in DMCs considered.‎
Phase Parameters ‎ Symbol(Units)‎ Value
Air Density ρ_g (kg⁄‎m^3 )‎ ‎1.225‎
Dynamic Viscosity ‎ μ_g ‎‎(kg/m/s)‎ ‎0.000017894‎
Velocity at inlet U_ig (m/s)‎ ‎0‎
Water ‎ Density ρ_w (kg⁄‎m^3 )‎ ‎998.2‎
Dynamic Viscosity ‎ μ_w ‎‎(kg/m/s)‎ ‎0.001003‎
Velocity at inlet U_iw (m/s)‎ ‎5 – 12‎
Volume of fraction ‎at inlet ‎%‎ ‎71.8 and 22.4‎
Ferrosilicon ‎ Density ρ_m (kg⁄‎m^3 )‎ ‎6808‎
Dynamic viscosity ‎ μ_m‎(kg/m/s)‎ ‎0.004285‎
Velocity at inlet U_im(m/s)‎ ‎5 -12‎
Size ‎ d_m(μm) ‎ ‎50 ‎
Volume of fractions ‎
at inlet ‎(%)‎ ‎28.2 and 22.4‎
Medium ‎ Density (and volume ‎of fractions of ‎ferrosilicon into ‎medium)‎ ρ_m(kg⁄‎m^3 )‎ ‎2640 (28.2%), 2300 (22.4%)‎
Density (dynamic ‎viscosity) ‎ μ_m ‎‎(kg/m/s)‎ ‎2640 (0.009775), 2300 (0.007632)‎
Surface ‎tension ‎stress ‎‎(sigma)‎ Water – Air ‎0.072‎ ‎-‎
Particles – Air ‎0.072‎ ‎-‎
Water – Particles ‎0‎ ‎-‎

‎3.4.4. Media losses

In DMS plants that use ferrosilicon, the loss amount is from 20% to 40% of the DMS plant ‎operating costs and used range may widely be, from 0.1 to 2.8 kg/t, although in modern ‎plants they do not exceed 0.25 kg⁄t (Svoboda, 2004). In this DMC plant of Zinc Carbonate ‎ore, the ferrosilicon losses are equal to 0.4-0.6kg⁄t.‎

‎3.4.5 Simulation and Boundary Conditions (BC)‎

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