Exploration of hydrocyclone designs using computational fluid dynamics

Int. J. Miner. Process. 84 (2007) 252 – 261 www.elsevier.com/locate/ijminpro

Exploration of hydrocyclone designs using computational fluid dynamics Jose A. Delgadillo ⁎, Raj K. Rajamani Department of Metallurgical Engineering, University of Utah, Salt Lake City, Utah, USA Received 9 May 2006; received in revised form 21 July 2006; accepted 26 July 2006 Available online 1 September 2006

Abstract The hydrocyclone has been widely used in the mineral industry for over hundred years, yet the standard geometry of hydrocyclones has remained almost unchanged. The exploration of new designs is time-consuming and costly to do by experimentation. In this paper, the computational fluid dynamics tool is used to explore alternative geometries in a way to manipulate the hydrodynamics to achieve the desired classification. Fluent™ 6.0 was used to solve the governing equations. The large-eddy simulation model was used for the turbulence closure and the Lagrangian particle-tracking method was used to predict the particle classification. Six new geometries are explored and compared with the standard design. The mass balance and the classification curve are the variables used to evaluate the performance of each of the novel designs. The results show that the modification of the geometry in designs #1 and #2 did not improve the classification performance, and they are not suitable for experimental validation. Designs #4 and #5 showed a similar performance achieved with the standard designs, whereas designs #3 and #6 were the most promising configurations to create a sharper particle classification. Designs #3 and #6 will be excellent candidates for further experimental validation. The principal contribution of this paper is that the computational fluid dynamics is the right tool to study and explore novel designs of hydrocyclones. © 2006 Elsevier B.V. All rights reserved. Keywords: Large-eddy simulation; Computational fluid dynamics; Hydrocyclone design

1. Introduction The standard design, cone-cylinder hydrocyclone with a single tangential inlet, shown in Fig. 1, is widely used in the mineral processing industry. Here, the cone angle is taken as the main design variable because it produces the upward spiraling flow (Campbell et al., 2005). The hydrocyclone suffers from two inherent deficiencies. The first one is the coarse particle by-pass ⁎ Corresponding author. E-mail addresses: [email protected] (J.A. Delgadillo), [email protected] (R.K. Rajamani). 0301-7516/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.minpro.2006.07.014

whereby coarse particles in the feed stream move along the boundary layer over the vortex finder and hence directly join the overflow stream. The second one is fine particle by-pass. This is unavoidable in the sense that very fine particles do not possess sufficient drag force to resist moving with the fluid medium, in this case water. Thus, the amount of fines reporting to the underflow is nearly equal to the fraction of feed water reporting to the underflow (Svarovsky, 1984). The only way to minimize the by-pass fraction is to reduce the amount of water passing through the underflow, which is accomplished by increasing the centrifugal force so that the underflow stream is highly concentrated with solids. Thus, in the design of hydrocyclone, the two by-pass

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Fig. 1. Standard hydrocyclone geometry with tangential inlet.

mechanisms must be minimized to realize higher performance efficiencies. In a hydrocyclone, the cylindrical part gives some residence time for the particles to experience centrifugal force, so that coarse particles can migrate toward the wall. The cone constricts the downward flow, thus creating a central spiraling upward flow. The division between upward and downward flow determines the midsection of the classification curve. The centrifugal force determines mostly the coarser side of the classification curve. Thus, it is possible to influence the classification by changing the dimensions of the cylinder and the cone. It is readily apparent that by choosing narrower cone angles the cut size can be decreased. However, this change comes at the expense of a higher pressure drop. Higher pressure drop means greater energy spent on the pump feeding the hydrocyclone. It would be ideal to improve classification without unduly increasing pressure drop. Hence, a lot of effort is still needed to improve the actual designs or explore novel designs to tune the performance for a specific operation (Rietema, 1961). Computational fluid dynamics (CFD) is a magnificent tool to explore novel designs (Olson and Van Ommen, 2004; Delgadillo and Rajamani, 2005a). In this paper, six different designs of hydrocyclones are tested with the methodology described by Delgadillo and Rajamani (2005a,b). The dimensions are held nearly equal to the standard 75-mm hydrocyclone so that the CFD computations do not steer too far away from the applicable large-eddy simulation (LES) regime. LES turbulence closure has been proved to be successful to predict

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hydrodynamics in both the 75-mm and 250-mm hydrocyclones (Slack et al., 2000; Delgadillo and Rajamani, 2005a,b). By extension, it is presumed that the fluid dynamics and turbulent transport of momentum are similar in the novel designs explored here. In an effort to make the CFD tool easily accessible to anyone for hydrocyclone simulations, Slack et al. (2003) describe a custom interface for their CFD package. In fact, Slack et al. claim that such a tool would provide the capability to develop “designer cyclones” optimized for the specific operating conditions of a particular process. Thus, it is readily apparent that the CFD tool is much easier to use today than in the past. The coarse particle by-pass and fine particle by-pass are discussed in detail by Kawatra and Eisele (2005). As an illustration of the design modification, Campbell et al. (2005) studied by experimentation the desliming capacity. A 100-mm hydrocyclone was compared to a standard 250-mm hydrocyclone. In each device, the vortex finder and spigot dimensions were varied to achieve the best cut size and low partition imperfection. This work is a clear illustration where CFD study is capable of reducing the experimental work load, although it is not expected to make all experimentation unnecessary. 2. Model description In the description of the fluid dynamics of the hydrocyclone, the key component is the turbulence closure model. A number of models including the κ–ε model (Hargreves and Silvesters, 1990; Dyakowski and Williams, 1992; Malhotra et al., 1994; He et al., 1999) and the Reynolds stress model (Lu et al., 1999; Cullivan et al., 2004; Slack et al., 2000) have been tried in the past. In recent years, the availability of computing power has stepped up this analysis to three-dimensional computations which avoid a few assumptions inherent to two-dimensional analysis. It appears that large-eddy simulation is a leading candidate for the description of turbulence encountered here (Derksen and Van den Akker, 2000; Slack et al., 2000; Delgadillo and Rajamani, 2005a). In a recent manuscript (Delgadillo and Rajamani, 2005a), the present authors compared renormalization group (RNG) κ–ε model, the Reynolds stress model and the large-eddy simulation model, and showed that LES leads in the description of mass balance, velocity profile, air-core dimension and even root-mean-square velocity prediction. In a second paper (Delgadillo and Rajamani, 2005b), it was shown that LES predicts measured tangential and axial velocity profiles in 75-mm and 250-mm hydrocyclones. For

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these reasons, LES was chosen for the CFD solution presented here.

defined as H(x) = x for x ≥ 0 and 0 for x ≤ 0. The turbulent viscosity, μs, depends on the strain rate defined as:

2.1. Turbulence closure model

 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 S¯ij S¯ij ls ¼ CRNG V 1=3

In LES, velocity profiles are resolved by a filtering operation of the velocity field, and the smaller scales are modeled with the eddy viscosity model. In other words, the eddies up to the size of the computational mesh are solved and the eddies below the mesh size are modeled. Therefore, the velocity field is defined as the sum of the filtered velocity ūi and the residual component ū′,

ð6Þ

In Eq. (6), V is the volume of the computational cell and S¯ij is the filtered strain rate. A typical value of the constant CRNG is 0.157 (Yakhot et al., 1989). 2.2. Multiphase model

Eq. (1) is applied to the governing Navier–Stokes equations (Pope, 2000). From this transformation, a new term arises, containing the stress tensor of the residual motions τijsgs. The filtered Navier–Stokes equations are given in Eqs. (2) and (3).

The tangential acceleration applied to the flow creates a high centrifugal force that pushes the fluid to the wall, creating a low pressure in the central axis. The low pressure in the core of the hydrocyclone gives the right conditions to suck air into the system, forming an air-core. The interface is modeled with the volume-offluid model (VOF). The VOF model, whose model equations are shown below, tracks the location of such an interface at every time step.

Aq A¯ ui þq ¼0 At Axi

ð2Þ

Aap Aui þ ap ¼0 At Axi

ð3Þ

Aðqui Þ Aðqui uj Þ Ap þ ¼− At Axi

Axj  A Aui AuTi þ l þ Axj Axj Axj þ qgi

ui ¼ ¯ui þ ¯ ui V

ui ¯ uj Þ A ¯ui Að ¯ 1A¯ p þ ¼− q Axi Axj At
 Assgs A A¯ ui ij þ l þ gi − Axj Axj Axj

ð1Þ

The new term that arises from the filtering operation, called the residual stress tensor, τijsgs, is modeled as the product of the eddy viscosity and the strain rate (Smagorinsky, 1963) as shown in Eq. (4). The eddy viscosity is resolved using the RNG model. The RNG model is very effective to model the transitional flows and nearwall regions where the molecular viscosity has more significance (Yakhot et al., 1989).
 uj A¯ ui A ¯ sgs sij ¼ −lt þ ð4Þ Axj Axi The turbulent viscosity μt is defined as the effective turbulent viscosity μs or molecular viscosity μ as shown here: 
2 1=3 ls leff −100 −l lt ¼ l 1 þ H l3

ð5Þ

where the effective viscosity, μeff, is defined as, μeff = μ[1 + H(x)]1/3 and H(x) is the Heaviside function,

ð7Þ

ð8Þ

Based on the local value of volume fraction αp, which ranges from 0 to 1. In the air-core region, the volume fraction αp is nearly zero. 2.3. Particle classification The additional model used in the hydrocyclone problem is the particle trajectory prediction. Particles are classified due to the drag force and diffusion of particles due to the fluid turbulence. The trajectory of particles is computed by Lagrangian tracking of particles upon the Eulerian continuous-phase predictions. The balance between the drag, gravity, buoyancy and centrifugal forces, shown in Eq. (9), determines the trajectory of the particles in the continuous phase (Haider and Levenspiel, 1989).

gx ðqp −qÞ dup ¼ FD u−up þ dt qp

ð9Þ

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The expected result is an improvement in the classification process.

where FD ¼

18l CD Re qp dp2 24

qdp jup −uj Re ¼ l

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ð10Þ

ð11Þ

where up is the particle velocity, u is the fluid velocity, ρ is the fluid density, ρp is the density of the particle, FD is the drag force, dp is the particle diameter and CD is the drag coefficient. 3. Description of novel designs The aim of this computational study is to optimize the classification performance through the modification of the standard geometry shown in Fig. 1. The standard geometry adopted by Hsieh and Rajamani (1988) was used as a reference to compare the performance of novel designs. The experiment chosen for comparison was the 75-mm hydrocyclone for which the flow rate of water in the inlet was 1.2 kg/s and the solid concentration was 4.8% by wt. Six novel designs are presented in this paper. 3.1. Novel design #1: modification of the cylinder in the standard hydrocyclone Design #1, shown in Fig. 2(a), is a modification of the standard cylindrical section. The cylinder flares out from the top and then contracts back. This modification is proposed to reduce the by-pass of coarse particles through the vortex finder. The coarse particles have a tendency to reach the vortex finder wall directly from the feed inlet. Then, they travel downward along the vortex finder wall and are caught in the upward flow through the vortex finder. Thus, coarse particles take a shorter path to the overflow. The expectation is that the widening cylinder would distance the coarse particles from the vortex finder. The dimensions are nearly identical to the standard geometry of the 75-mm diameter hydrocyclone.

3.3. Novel design #3: two-cone design The logic behind design #3 is to create the opposite effect created in designs #1 and #2. Design #3 is an exploration of the cone-angle effect on the classification process. The cylindrical section is replaced with a tapered cone of angle 30° (Rong and Napier-Munn, 2003). This modification changes the flow field and the classification process, creating a smooth classification and probably creating a lower turbulence in the fluid. The reduction of fluid turbulence improves the classification efficiency. The angle of the lower cone was kept to 20°. The geometry and dimensions proposed for design #3 are shown in Fig. 2(c). 3.4. Novel design #4: modification of cone angle The double-cone configuration used in design #3 depends on the combination of cone angles. Design #4, shown in Fig. 2(d), is an improvement of design #3 and an exploration of the cone-angle effect on the classification process. The angle of the cone tip was changed from 30° to 20° so that the main body is made up of a single cone. In this design there are no abrupt changes in the outer walls, and hence the velocity field transitions smoothly. The velocity field directly affects the classification efficiency. The angle of the lower cone was kept constant at 20°. 3.5. Novel design #5: modification of the conical section The standard geometry was modified in the cylindrical and conical sections of design #5. The cylinder length was reduced from 75 mm to 50 mm. Next, a cone angle of 30° was placed followed by a second cone of 10° angle. Here, the configurations of design #3 and the standard geometry are mixed to form a unique design. The dimensions and geometry are shown in Fig. 2(e). This modification will create a more aggressive classification process, and also the residence time will increase enormously.

3.2. Novel design #2: change on vortex finder length 3.6. Novel design #6: change of cone angles The geometry for design #2 is shown in Fig. 2(b). This design is a slight modification of design #1. The vortex finder length was reduced from 50 mm to 37.5 mm. This modification creates an increase in the tangential velocity at the top section. Hence, particles entering the hydrocyclone are immediately classified.

Novel design #6, shown in Fig. 2(f), is a modification of design #5. Here, the angles of the two cones were changed from 30° and 10° to 20° and 6°, respectively. The angle change is expected to produce a sharp classification process and an increase in the residence time.

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4. Computer simulations The flow problems for the novel designs were solved using three-dimensional unstructured meshes. The computations were run until a steady state was reached.

The residuals for momentum, velocity components and flow rate at the outlet streams were monitored for convergence. Fluent™ 6.0 code was used to perform the simulations with the large-eddy simulation model as turbulence closure.

Fig. 2. Dimensions for six new designs.

J.A. Delgadillo, R.K. Rajamani / Int. J. Miner. Process. 84 (2007) 252–261 Table 1 Cumulative size distributions for the standard design Size (μm)

Feed (%)

Overflow (%)

Underflow (%)

42.2 35.3 25.1 17.7 12.6 8.9 6.3 4.4 3.1 2.1 1.3 0.8 0.5

100 88 70.3 50.5 37.5 28.4 19.6 12.8 8.7 5.6 2.8 1 0

100 99.2 94.9 74.3 56.4 43 29.9 19.7 13.4 8.7 4.3 1.5 0

100 67.9 31.9 13 7.4 5.2 3.3 1.9 1.1 0.8 0.4 0.2 0

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overflow and underflow were calculated. The mass balance results are shown in Table 2. The air-core is another part of the performance evaluation. The prediction of the air-core profile is fundamental to the description of the flow dynamics. The tangential velocity pushes the water to the wall, creating the air-core. Hence, a reduction in the tangential velocity reduces the diameter of the air-core (Williams et al., 1994). 4.1. Results for design #1

Steady state was achieved after 1000 iterations, at which time the residuals and flow rates were constant. After the steady state was reached, a real time of 1 s was simulated with time steps of 0.0005 s. An average of over 1000 time steps were taken to record the velocity profiles, mass balance and flow rates. A sample of 1560 particles in each size fraction was injected at the inlet and tracked down the flow until each particle left through the outlets. The slurry used is one of 4.8% by wt of limestone ore (density 2700 kg/m3) with a particle size distribution shown in Table 1. From the particle trajectory the cumulative distributions were computed as the split ratio for each size class. The low solid content chosen here is in the very dilute flow regime. This regime is devoid of interparticle interactions and viscosity effects. Hence the simulations reveal totally the effect of the shapes of the cylinder and cone. The mass balance is primary in the analysis of the hydrocyclone performance. In the exploration of novel designs, these values determine the improvement of the process. The mass balance was computed from the particle trajectory for each size fraction. The split ratio of each size fraction was calculated by sampling the overflow and underflow streams. With the feed size distribution and solid split ratio, the size distributions of

A reduction in pressure drop of 36% was achieved with design #1. The pressure drop is closely related to the distribution of the centrifugal force. The reduction of tangential velocity results in an increase in the by-pass of coarse and fine particles in the overflow. This reduction is reflected in the modification of the water split. The water rate to the underflow increases by 76%, which creates more by-pass of fine particles. The mass balance shows that this new design is not much better than the standard design. The hydrocyclone efficiency is reflected by the classification curve, which is defined as the percentage of coarse particles reporting to the underflow stream. The predicted classification curve is compared with the standard design in Fig. 3. The modification in the geometry did not improve the classification performance. The by-pass of fine and coarse particles increases. This design will not be useful for further experimental validation. The weak point in design #1 is the lower tangential velocity in the vortex finder region due to the expansion of the cylinder. The weaker tangential velocity field reduces the upward spiraling stream, and hence the increase in underflow by-pass results. 4.2. Results for design #2 The reduction in the vortex finder length in design #2 creates a change in the water split ratio. The flow rate to the underflow is reduced from 18.7% to 9.6%. This

Table 2 Computed mass balance, pressure drop and air-core diameter for the six designs Design

Split ratio to underflow, water (%) Split ratio to underflow, solids (%) Pressure drop (kPa) Air-core diameter (mm) Inlet pressure (kPa)

Standard

#1

#2

#3

#4

#5

#6

4.46 39.34 39.70 11.0 141.0

18.66 27.77 29.07 8.0 130.4

9.55 48.21 30.11 9.0 131.4

7.41 53.92 59.64 10.0 161.0

4.64 33.27 40.09 10.0 141.4

11.61 42.39 37.63 11.0 138.9

13.57 44.13 27.03 8.4 128.3

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fore, with these simulation results, it can be concluded that there is no more room for improving the flaredcylinder design. 4.3. Results for design #3

decrease in the flow rate is a signal that the velocity profiles have changed. Furthermore, the reduction of water reporting to the underflow reduces the by-pass of fine particles. The modification of the vortex finder length improves the performance of this design. The computed air-core profile is expected to be smaller than the standard air-core diameter. The reduction in the tangential velocity reduces the centrifugal force needed to maintain a larger diameter. The air-core diameter for design #2 is 2 mm less in diameter than in the standard mean hydrocyclone. The modification of the cylindrical section is the focus of this effort because most of the by-pass occurs in this region of the hydrocyclone body. The classification curve is shown in Fig. 4. The increase in the tangential velocity leads to a better classification. The by-pass of coarse particles is reduced. There is a slight improvement over the standard design. Design #2 exhibits some improvement in the size classification, but it is not an overall improvement in the classification process. There-

The conical design of design #3 increases the pressure drop compared to the standard design. The water reporting to the underflow in this design is 7.4%, compared to the 4.5% in the standard design. The by-pass of fines is very closely related to the amount of water in the underflow. The computed conditions show that the bypass of solids increases in the underflow. The rise in the pressure drop can lead to a better classification process, particularly in the coarse sizes, which are driven to the wall at higher centrifugal forces. The tangential velocity profile at 120 mm from the top wall is compared with the velocity profiles of the standard design in Fig. 5. The increase in the tangential velocity creates the centrifugal force needed to classify the coarse particles and remove medium size particles from the overflow stream. The analysis of the flow field show that this profile will lead to a better classification process with a finer cut size and a reduction of the bypass of coarse particles. Fig. 6 shows the classification curve compared with the standard design values. There is a small increment in the by-pass of the very fine sizes, but the classification of coarse particles is greatly improved. The overall classification process has been improved with the novel design. However, it should be noted that the pressure drop increased by 50% over the standard design, which means more energy per unit mass of slurry is spent on this design. The fine cut size allows a better classification for mineral-processing circuits with low mineralliberation sizes.

Fig. 4. Design #2: Size classification compared with standard design at 4.8% wt slurry and inlet flow rate of 1.12 l/s.

Fig. 5. Design #3: Tangential velocity predictions at 120 mm from the top.

Fig. 3. Design #1: Size classification compared with standard design at 4.8% wt slurry and inlet flow rate of 1.12 l/s.

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Fig. 6. Design #3: Size classification compared with standard design at 4.8% wt slurry and inlet flow rate of 1.12 l/s.

Fig. 8. Design #5: Size classification compared with standard design at 4.8% wt slurry and inlet flow rate of 1.12 l/s.

4.4. Results for design #4

4.5. Results for design #5

The pressure drop for design #4 is very similar to the pressure drop for the standard design. The reduction in the pressure drop is attributed to the straight cone design from the tip to the bottom, which reduces turbulence transitions due to contracting walls. The flow rate remains very similar to the standard design values in all the streams with a small change in the solid concentration at the underflow. The change in the mass balance shows that top cone has very little effect on the classification process. The computed classification curve, shown in Fig. 7, is very similar compared with the classification curve for the standard design. There is a slight increase in coarse particle by-pass. This by-pass can be attributed to the lack of a high tangential velocity field. Hence, design #4 is not an improvement of the classification process. It should be noted that this design required the same pressure drop as the standard design, yet it produces slightly inferior size classification.

Predicted operational conditions for design #5 are very similar to those for the standard design operation. The main difference is the split to the underflow. The increase in the water flow rate in the underflow increases by-pass of fine particles. The pressure drop shows a small reduction, which results in the increase of coarse particle by-pass. Fig. 8 shows the classification curve that predicted for design #5 and the standard design. The classification curves are very similar. The only difference is the increase of fine particles to the underflow. This new design performs very similarly to the standard design. The combination of different cone angles could make a difference in the classification process.

Fig. 7. Design #4: Size classification compared with standard design at 4.8% wt slurry and inlet flow rate of 1.12 l/s.

Fig. 9. Design #6: Size classification compared with standard design at 4.8% wt slurry and inlet flow rate of 1.12 l/s.

4.6. Results for design #6 A reduction of the pressure drop in design #6 was expected due to the reduction of diameter in the lower

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cone. This characteristic is an advantage because more slurry can be processed at the same pressure drop as in the standard design. The increase in water reporting to the underflow is also due to the pressure drop. If the flow rate at the inlet is increased, the amount water reporting to the underflow is reduced. The air-core diameter was reduced because the fluid is not moving as fast as in the standard design. The predicted classification curve, shown in Fig. 9, shows that the slope of the classification curve is steeper than that for the standard design. Since the pressure drop is lower, the flow rate to this design can be increased at the same level of pumping energy. Thus, when the pressure drop is increased, one expects more improvement in size classification. This design is a real improvement in the classification process, and can be further optimized by focusing on the combination of different cone angles and inlet flow rate. In particular, the location of the second cone would influence this design greatly. 5. Conclusions The optimization of the classification performance through the modification of the standard geometry is easily achieved with the CFD methodology shown here. The modification in the geometry in designs #1 and #2 did not improve the classification performance. The bypass of fine and coarse particles increases. These designs will not be useful for further experimental validation. The weak point in these designs is the lower tangential velocity in the vortex finder region. With the modifications in designs #3 and #4, there is a small increase in the by-pass of the very fine sizes, but the classification of coarse particles is greatly improved. The overall classification process was improved in these two novel designs. However, it should be noted that the pressure drop increased over the standard design, which means more energy per unit mass of slurry is spent with this design. The fine cut size allows a better classification for mineral-processing circuits with low mineral liberation sizes. In designs #5 and #6, the classification curves show that even with a lesser tangential velocity field a slightly better efficiency was achieved. The slope of the classification curve is steeper than that of the standard classification curve. Since the computed pressure drop is lower, the flow rate can be increased to match the pressure drop of the standard hydrocyclone. Thus, when the pressure drop is increased, one expects more improvement in size classification. The design of new geometries opens a new chapter in the hydrocyclone optimization. It is possible to evaluate

changes in the standard geometry to manipulate the dynamics achieving the desired classification. The proposed designs showed that a double cone is the solution to improve classification. Designs #3 and #6 are promising configurations to create a sharper particle classification. These designs must be tested in experimental work. The principal contribution of this paper is that CFD is the right tool to study and explore the fluid dynamics of novel designs of hydrocyclones. Nomenclature CD Drag coefficient dp Particle diameter (m) FD Drag force (N) gi Gravitational force component (m/s) H(x) Heaviside function p Pressure (Pa) Re Reynolds number Sij Strain rate (m2/s) t Time (s) ui Velocity component (m/s) ū Filtered velocity (m/s) ū′ Subgrid-scale velocity (m/s) up Particle velocity (m/s) V Volume cell (m3) xi Dimensional component (m) αp Volume fraction μ Molecular viscosity (P) μeff Effective viscosity (P) μs Subgrid scale viscosity (P) μt Turbulent viscosity (P) ρ Density (kg/m3) sgs τij Subgrid stress tensor (Pa)

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