Abstract

Accurate prediction of hydrocyclone flow field is essential for the determination of the separation efficiency of the device. This study demonstrates the importance of three-dimensional modeling in effecting an accurate prediction of separation performance. Three-dimensional modeling captures the details of the flow field near the flow inlet pipe where such details were neglected in standard axisymmetric modeling. The computational results are compared to experimental resluts for three hydrocyclone design variations. The results show accurate prediction for the separation performance for each hydrocyclone design.

Introduction

Hydrocyclones use the principle of centrifugal sedimentation to separate particulate matters based on size, shape, and density. They were originally designed for use in the pulp and paper industry to remove high specific gravity debris from paper stock. Many different designs have evolved and the range of application had extended to the removal of low specific gravity contaminants. In the pulp and paper industry, different types of hydrocyclones are commonly referred to as forward, reverse, and through-flow cleaners. The forward cleaners are the traditional hydrocyclones where high specific gravity rejects are released from the bottom tip of the cyclone while the accepts are collected from the top. The reverse and through-flow cleaners are for removing lower specific gravity contaminants. This study focuses on the traditional forward cleaner type hydrocyclones which are widely in use today.

The liquid stock that needs cleaning usually enters the hydrocyclone through an inlet pipe tangent to the inside wall near the hydrocyclone's top as shown in the figure. The swirling flow pattern imparted by this incoming fluid is the dominant flow feature of the cyclone. Several minor flow patterns also accompany this rotational flow and they are described in details in Bliss (1992) and Sevilla and Branion (1997).

Particles within a rotational flow field experience a centrifugal force which is opposed by a corresponding drag force. The centrifugal force is directly dependent on the circumferential velocity and the knowledge of such is vital to predicting the performance of the hydrocyclone. In addition, the minor flow patterns found in a hydrocyclone include small recirculation eddies which affect the trajectories of the particles. Hence, a detail prediction of the velocity distribution could lead to improvements in designs. Computational Fluid Dynamics (CFD) is a versatile means to predict the velocity profiles under a wide range of design and operating conditions. The use of CFD alleviates the problem of the usual engineering design approach based on correlation formulae established for a limited range of parameters.

CFD simulations are general and recent advances in computational methods and computer technology make CFD an efficient means to study the dynamics of many physical systems. Applications of numerical methods to flows in hydrocyclones have been examined by Hsieh and Rajamani (1991), Dyakowski and Williams (1993), Davidson (1994), and Malhotra et al. (1994). Different models for turbulence flows were examined in the above cited investigations, as well as different models for liquid-solid particle interaction. However, very little attention has been paid so far on the prediction of pulp fiber separation.

The present work is a continuation of the investigation carried out at the University of British Columbia to improve the quality of mechanical pulp. The current research is an extension of the numerical studies by Sevilla and Branion (1997) and He et al. (1997). A modified turbulence model that accounts for streamline curvature effects on turbulence intensity is used to compute the flow field. The motion of fibers within the hydrocyclone is computed using drag coefficient correlations specifically obtained for the flow of fibers. The results are expected to provide better insight on the fundamental principles which govern the separations that can be achieved with hydrocyclones.

To establish a suitable computational model, the study by Monredon et al. (1992) is repeated using three-dimensional CFD modeling as opposed to axi-symmetric modeling adopted by the initial investigators. Three-dimensional modeling gives a better account for the structure of the flow field near the inlet pipe. This extra detail is shown to be important for an accurate prediction of particle separation performance. Analysis is made to three hydrocyclone designs to cover a range of design considerations. Then a preliminary study is done to simulate the fractionation performance of a commercial Bauer cleaner for fibers.

Computational Modeling

For a dilute fluid suspension, the incompressible Navier-Stokes equations supplemented by a suitable turbulence model are appropriate for modeling the flow in a hydrocyclone. The most popular turbulence model in use for engineering applications is the k-e model where the scalar variables k and e represent the kinetic energy of turbulence and its dissipation rate, respectively. These equations describe the conservation of mass, momentum, and turbulence kinetic energy and its dissipation rate.

The standard k-e model was not optimized for strongly swirling flows found for example in hydrocyclones. Turbulence may be stabilized or destabilized in the parts of flow domain where strong streamline curvature is presence. The numerical studies carried out at UBC by He et al. (1997) reveal that the standard k-e model underpredicts the variation of the axial velocity profile across the radial direction and also underpredicts the intensity of the swirl component of the velocity. To compensate for this deficiency, the standard k-e model is modified by incorporating the following changes to the e-equation:

This modified k-e model was first proposed by Launder et al. (1977) for the prediction of wall bound flow with streamline curvature. The effect of the curvature on turbulence is made proportional to a Richardson number Ri defined above, which describes the ratio of apparent centrifugal force to inertial force. For stabilizing curvature, Ri is positive which leads to an increase in e as needed. Conversely, for destabilizing curvature, Ri is negative and e decreases. The parameter CC controls the effect of this curvature correction factor introduced through Ri. The numerical study by He et al. (1997) verified that CC=0.2 gives accurate prediction of velocity components in a large part of the flow field.

The computational method used is the finite volume discretization of the domain into computational cells where the conservation principle is enforced for each of the properties: mass, momentum, k and e. The computational cells can have arbitrary shapes to facilitate the fitting of general curvilinear geometries. A coordinate-independent numerical procedure is employed by the UBC-PSL research teams and forms part of the curvilinear coordinate equation solver. With proper boundary conditions set, the system of discretized equations are then solved by an iterative routine with a fast and robust convergence rate.

Boundary Conditions

A total of three hydrocyclones are studied. These cyclones were originally studied by Monredon et al. (1992) with minor geometric variations to study the effects on the flow field. The mass flow rate of the slurry suspension into the hydrocyclone is given and is constant at 67 kg/min for all cases. The inlet flow pipe has a diameter of 0.025 m and it intersects the cyclone tangentially. This inlet flow pipe was not simulated; instead, the intersection between the pipe and the cyclone was modeled as the flow inlet for the hydrocyclone. The angles of intersection at various intersecting points can be determined analytically and the respective tangential and radial velocity components are determined based on the mass flow and the angle of flow incidence. The inlet turbulence kinetic energy is determined from

while the dissipation rate is calculated from

In the above equations, 0.05 represents an estimation of the turbulence intensity level at the hydrocyclone inlet, Vin is the inlet flow velocity calculated from the mass flow rate, and lin is the length scale, estimated to be half of the inlet pipe diameter, for the turbulence structure at the hydrocyclone inlet.

The flow split ratios, defined as the ratio of overflow to inlet flow, were reported from the experiments and are used here to specify the mass flow rate out of the overflow (vortex finder) and underflow (spigot) openings. At these openings, uniform velocity values were imposed as the flow exit boundary conditions. An air core is present along the axis of the hydrocyclone. Radius of this air core varies in each case and are assumed to have the values recorded in Monredon (1990).

The effective opening areas for the overflow and underflow are thus the physical opening area minus the cross-sectional area of the air core. At the air core surface impermeability is assumed and the radial velocity component is zero. For the axial velocity component the assumption is

Particle Separation Prediction

Since the particle slip velocities in the hydrocyclone are usually small, it can be assumed that the particle-fluid momentum is absent. Thus the modeling can be separated into two parts: the liquid phase flow field to predict the liquid phase velocities, and the particle motion with respect to the fluid. Then the trajectories of the particles of each size can be calculated, from which the separation efficiency can be estimated. For dilute slurries, where the variations of local density and viscosity are small and the particle/particle interactions are neglected, the computation of the liquid-phase velocities and particle motion can be executed independently.

In hydrocyclone flow fields the time for a particle to reach its terminal settling velocity is usually very short. The usual assumption made is that the particle moves at its settling velocity Us, which is determined by a balance between the drag force FD and the combined gravitational and centrifugal forces acting on the particle.

The drag force is expressed as

while the combined gravitational and centrifugal forces are expressed component-wise as follows:

Axial component:
Radial component:
Circumferential component:

In the above expressions, V is the volume of the particle and g is the gravitational acceleration.

Balancing the drag force with the external forces acting on the particle gives

Define

Then the settling velocity Us can be written, for the case of spherical particles, as

First, the relative velocity or the slip velocity between the fluid and the particle is calculated. Then the particle Reynolds number is computed as

A drag force on the particle is then calculated using this particle Reynolds number, and the following drag law expression taken from Clift et al. (1978):

In each of the simulations, 200 particles having a certain diameter are randomly located at the inflow patch and their trajectories are traced until they exit the hydrocyclone either through the overflow or underflow orifice. Then a different particle size is assigned and repeated for a range of particle sizes. The results yield a particle separation efficiency curve.

The Three cases studied

The geometric dimensions of the three hydrocyclones studied are listed in the following table:

Dimensions

(in mm)

Cyclone 1

(Monredon case 1)

Cyclone 2

(Monredon case 2)

Cyclone 3

(Monredon case 4)

Cyclone Diameter 75 75 75
Inlet Diameter 25 25 25
Cylindrical Length 75 75 75
Vortex Finder
Diameter
25 25 22
Vortex Finder
Length
50 50 50
Spigot Diameter 11 15 11
Cone Angle 20 20 20

Cyclone 1 is considered the base case.
Cyclone 2 is studied for variation on the spigot diameter.
Cyclone 3 is studied for variation in the vortex finder diameter.

Experimental results: Flow characteristics

Monredon (1990) states that, for the flow field, the experimental results show no specific trend can be noticed for the tangential velocity when the spigot diameter is increased, while the water split factor changed from 4.9% for Cyclone 1 to 21.4% for Cyclone 2. The upward axial velocity decreases when the spigot diameter is increased. The spigot diameter can be varied to control the underflow rate. Only negative axial- velocity values at 200-mm from the top while Cyclone 1 still exhibits a flow split between the underflow and the overflow streams.

From Cyclone 1 to Cyclone 3, the pressure drop has to be increased by 54% in order to keep the same volumetric flow rate. This increase can be explained by the decrease of the total outflow area which drops from 585 sq. mm to 475 sq. mm. For Cyclone 3, a net increase in the tangential velocities in the cylindrical part as well as in the conical part was observed. Also the air-core diameter increased when the vortex finder diameter was decreased. The governing factor in this case is the difference in the pressure drop. The fluid flow rate going down was observed to be much larger than the throughput capacity of the underflow. Hence a significant fraction of the overflow stream arose from an ejection of fluid from the conical part in the spigot region. Therefore higher turbulence may exist near the spigot region for Cyclone 3 which would affect the particle size classification.

Experimental results: Particle Separation

Monredon (1990) states that, at a fixed slurry concentration, when the spigot diameter was increased, the efficiency dropped for both the coarse and fine particles. The drop in efficiency for the fine particles is due to the increase in the water split ratio. When spigot diameter is enlarged a greater portion of the inflow fluid reports to the underflow, which carries with it more of the particles in each size class. In addition to the fluid split effect, the crowding effect plays a role in the classification of coarse particles. Crowding effect refers to the crowding of particles between the air core and the conical wall especially near the spigot region. For Cyclone 2, because of the increase in the tangential velocities, the flow of solids along the conical wall increases. The bulk volume of solids in the spigot region becomes greater than the solids capacity of the spigot pipe, hence a portion of the coarse particle is carried upward to the overflow.

This saturation in the spigot region was responsible for the decrease in the efficiency and the corresponding increase in the cut size. When the vortex finder diameter was decreased from 25-mm (Cyclone 1) to 22-mm (Cyclone 3), the cyclone exhibited a slight increase in cut size and a decrease in efficiency for both fine and coarse particles. The drop in efficiency for the coarse particles was very pronounced and can be explained for the potential high turbulence flow existed near the spigot, which led to imperfect classification. This turbulence was caused by a drastic flow split takes place in that region.

The New 3D CFD Results

Computational Mesh:

The computational mesh is constructed using a union of cylindrical grids for the radial-circumferential planes with curvilinear grids for the radial-axial planes. This design is to take advantage of the simplicity of the cylindrical coordinate system to allow for a simple yet accurate calculation of geometric quantities such as the grid cell area and arc-length. The grid accurately captures the major geometrical features of the hydrocyclone.

Simulation Results for Velocities

To demonstrate the simulation results obtained from our three-dimensional computation, the flow field for Cyclone 1 along selected planes is displayed in the following graphs.

Velocity field near the inlet:

The figures below shows the velocity field at the horizontal level where the inlet pipe enters the hydrocyclone. The graphs indicate the deviation from axisymmetry in this region that is shown clearly in the contour graph for the swirl intensity. The swirl contours are the magnitude of the circumferential component of the velocity and it exhibits significant deviation form axisymmetry. These results indicate the importance of a complete three-dimensional calculation of the hydrocyclone flow fields.

Velocity field along the center-plane:

Shown below are the velocity distributions along the center-plane which bisects the hydrocyclone vertically across its inlet opening. The velocity vector plot displays some of the minor flow patterns in the hydrocyclone flow field. A recirculation zone exists beneath the inlet region is seen clearly in the velocity vector plot. The fluid is seen flowing downward along the hydrocyclone wall and the flow direction reverses along the lower part of the conical section. The upward fluid flow is moving rapidly near the central air-core. The contour graph for the circumferential velocity component shows high swirl levels both at the outer and inner rims of the hydrocyclone. The swirl movements are reversed between the outer and inner zones.

Velocity vectors in the center axial-radial plane Tangential velocity contours in the center axial-radial plane

Comparison with Experimental Data

Experimental data gathered by Monredon (1990) are available at selected locations. The data were collected using Laser-Doppler velocimetry and the axial and tangential (circumferential) velocity components were measured. the radial component of the velocity has small magnitude and is difficult to measure accurately.

velocity Components:

Cyclone 1

The comparison shown above indicates our numerical model is generally quite accurate for the prediction of hydrocyclone flow field. At the axial position 120-mm from the hydrocyclone top, the largest discrepancy between the computed and measured results occurs near the air-core interface. This is expected owing to the simplistic boundary conditions we assumed for the velocity components at the interface. The axial velocity component is under-predicted while the tangential component is over-predicted. Nevertheless, the comparison is quite excellent beyond the inner core region and it is expected that a more accurate modeling of the air-core interface would improve the comparison. At the axial position 200-mm from the hydrocyclone top, the agreement is surprisingly good despite the fact that at such a position so near the spigot opening, the uncertainties in the boundary conditions for both the air-core interface and the underflow exit could severely affect the numerical results.

Cyclone 2

The agreement is generally quite good at both the 60-mm and 120-mm levels. There are some discrepancies for the axial component: at 60-mm there is over-prediction near the hydrocyclone perimeter and at 120-mm there is under-prediction near the air-core. These discrepancies are likely caused by the specification of boundary conditions at both the wall and air-core. We acknowledge the use of the simple wall function model is inadequate for the highly swirling flow found in a hydrocyclone. At the 200-mm location, the numerical results over-predict the axial velocity component, especially in the inner region where the numerical results suggest an upward turning of the flow is present. This discrepancy can be attributed to the uncertainties in the boundary condition that should be specified at the underflow exit. The imprecision for the air-core interface in this lower region of the cyclone also leads to disagreement for the results.

Cyclone 3

For Cyclone 3, the quality of agreement is generally the same as for the previous two cases. The largest discrepancies are near the air-core interface, while the overall agreement is deemed to be acceptable for engineering applications.

Particle Separation:

Accurate prediction for the particle separation performance is the ultimate goal in a hydrocyclone modeling study. Three-dimensional calculation is better suited for particle separation prediction than axisymmetric calculation as done by Monredon et al. (1992) because the inlet velocity components are more accurately represented in a three-dimensional calculation. In axisymmetric calculations for hydrocyclones (see for example Malhotra et al. (1994)), the radial inlet velocity needs to be artificially reduced to maintain the same inlet mass flow rate since the radial velocity is specified around the entire perimeter of the hydrocyclone. This artificial reduction causes the injected particles to have lesser radial momentum and hence incorrect trajectories.

thus, a three-dimensional calculation should give a better representation of the particle trajectory since the input radial momentum is more accurately accounted for. In the following calculations, two hundred particles are initially distributed randomly at the flow inlet and are transported by the flow field until they either leave through the overflow or the underflow opening.

The results are compared to experimental data reported in Monredon (1990) where the concentration of the particles was measured for each case and an estimation was made for the effective viscosity for the slurry mixture which was used in the calculation of the drag force on each particle. The separation efficiency curves predicted by Monredon (1990) based on his axisymmetric flow field results are also plotted for comparison.

Cyclone 1:

Compared to the prediction based on an axisymmetric flow field, the separation efficiency curve predicted by the three-dimensional method gives better predictions especially for particles with large diameters. The axisymmetric calculation predicts too high a tendency for large (hence heavier) particles to be separated downward. The three-dimensional calculation allows the heavier particles to have a higher chance to be injected towards the center region where they can be short-circuited out of the hydrocyclone through the vortex finder.

Cyclone 2:

The three-dimensional results predict a more gentle rise for the separation efficiency curve which agrees with the experimental results. However, both three-dimensional and axisymmetric results over-predict the particle separation characteristics for small particles. Small particles have less inertia and are more easily perturbed by turbulence structures in the flow. Effects of turbulence structures on particle trajectories are not accounted for in the presence model and may be responsible for the discrepancies.

Cyclone 3:

This case demonstrates clearly the superiority of three-dimensional results over axisymmetric results. the three-dimensional results give very accurate prediction for the particle separation for both the small and large diameter limits. The slope of the separation efficiency curve is predicted more accurately by the particle trajectory calculation based on the three-dimensional flow field.

Conclusions

The numerical schemes developed jointly by UBC and PSL are capable of accurate predictions of both the flow dynamics and particle separations in hydrocyclones. The CFD technique developed is applicable to a variety of hydrocyclone designs and the solution method is efficient and robust. The accurate comparisons shown here for mineral processing applications give confidence in applying our techniques to the pulp and paper industry.

References

Bliss, T. (1992), "Centrifugal Cleaning", Chapter XIII in "Pulp and Paper Manufacture, Volume 6, Stock Preparation", editors, R.W. Hagemeyer and D.W. Manson, pp.248-261.

Clift, R., Grace, J.R. and Weber, M.E. (1978), "Bubbles, drops, and particles", Academic Press, New York.

Davidson, M.R. (1994), "A Numerical Model of Liquid-Solid Flow in a Hydrocyclone with High Solids Fraction", FED-Vol.185, Numerical Methods in Multiphase Flows, pp.29-38.

Dyakowski, T. and Williams, R.A. (1993), "Modelling Turbulent Flow within a Small-Diameter Hydrocyclone", Chemical Engineering Science, Vol.48, No.6, pp.1143-1152.

He, P., Salcudean, M., Branion, R. and Gartshore, I. (1997), "Mathematical Modelling of Hydrocyclones", FEDSM97-3315, ASME Fluids Engineering Division Summer Meeting.

Hsieh, K.T. and Rajamani, R.K. (1991), "Mathematical Model of the Hydrocyclone Based on Physics of Fluid Flow", AIChE Journal, Vol.37, No.5, pp.735-746.

Launder, B.E., Priddin, C.H. and Sharma, B.I. (1977), "The Calculation of Turbulent Boundary Layers on Spining and Curved Surfaces", ASME Journal of Fluids Engineering, Vol.99, pp.231-239.

Malhotra, A., Branion, R.M.R. and Hauptmann, E.G. (1994), "Modelling the Flows in a Hydrocyclone", The Canadian Journal of Chemical Engineering, Vol.72, pp.953-960.

Monredon, T.C. (1990), "Hydrocyclone: Investigation of the Fluid-Flow Model", Thesis, Master of Science in Metallurgy, Department of Metallurgical Engineering, University of Utah.

Monredon, T.C., Hsieh, K.T. and Rajamani, R.K. (1992), "Fluid flow model of the hydrocyclone: an investigation of device dimensions", International Journal of Mineral Processing, Vol.35, pp.65-83.

Sevilla, E.M. and Branion, R.M.R. (1997), "The Fluid Dynamics of Hydrocyclones", Journal of Pulp and Paper Science, Vol.23, No.2, pp.J85-J93.