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Numerical Study of Particle-Fluid Flow in a Hydrocyclone B. Wang, K. W. Chu, and A. B. Yu* Centre for Simulation and Modeling of Particulate Systems, School of Materials Science and Engineering, The UniVersity of New South Wales, Sydney, NSW 2052, Australia
This paper presents a numerical study of the gas-powder-liquid flow in a standard hydrocyclone. In the approach, the turbulent fluid flow is described by the Reynolds stress model, the interface between the liquid and air core is modeled using the volume of fluid multiphase model, and the results of fluid flow are used in the simulation of particle flow described by the stochastic Lagrangian model. The flow features are examined in terms of flow field, pressure drop, volume split ratio reported to the underflow, particle trajectories, and separation efficiency. The validity of the proposed approach is verified by the good agreement between the measured and the predicted results. Discussion is then extended to other flow behavior in a hydrocyclone, including the origin of a short-circuiting flow, the structure of air core, and the motion of particles of different sizes. The model offers a convenient method to investigate the effects of variables related to geometrical and operational conditions on the performance of hydrocyclone. 1. Introduction
Table 1. Comparison of the Main Types of Cyclone in Industry
A cyclone is an important apparatus to separate dispersed solid particles from a fluid suspension by centrifugal and vortex action. In the past, various cyclones have been developed for different industrial needs. In essence, they can be categorized into three types as listed in Table 1. Gas cyclone is the first and the most common for the separation of solids from gases. In the past, it has been extensively studied, mainly by experimentation.1,2 In recent years, in line with the development of computer technology, computational fluid dynamics (CFD) has been found to be very useful in generating information for better fundamental understanding and process optimization. In particular, it has been reported that the Reynolds stress model (RSM)3-5 can describe the flow field in a gas cyclone accurately, and the stochastic Lagrangian model6 can be used to track the particle trajectories and evaluate the separation efficiency. Recently, Derksen7 and Wang et al.8 have used such approaches to investigate the performance of gas cyclones and shown that the computed velocity profiles, pressure drop, and separation efficiency are in good agreement with those measured. Hydrocyclone is also widely used in industry, particularly in mineral and chemical processing, because of its simplicity in design, high capacity, low maintenance and operational cost, and small physical size.9,10 The flow behavior in a hydrocyclone is much more complicated than that in a gas cyclone, although the overall separation mechanisms are similar. For example, the strong rotational flow of liquid in a hydrocyclone can create a low pressure axial core and a free liquid surface. Such a lowpressure core in a hydrocyclone may communicate directly with the atmosphere at outlets open to air. The air is inhaled through the apex and forms an air core, which often results in poorer performance as compared to a gas cyclone. Moreover, the suspension fed to a hydrocyclone is thickened partially by the effect of centrifugal forces. The slurry is withdrawn at the underflow with a high solid concentration, and the clarified liquid leaves the hydrocyclone by the overflow through the vortex finder. In this case, one more important parameter, in addition to those for gas cyclones, to describe the performance * To whom correspondence should be addressed. Phone: +61 2 93854429. Fax: + 61 2 93855956. E-mail:
[email protected].
type gas cyclone function
separator
air core phases
no air-particles
separation foundation
sizes
hydrocyclone classifier or separator yes air-water-particles or air-waterfluid sizes or specific gravity
dense medium cyclone concentrator yes air-watermediumparticles specific gravity
of a hydrocyclone is the so-called volume split ratio defined as the ratio of underflow to overall flow. In the past, many efforts have been made to study experimentally the flow in a hydrocyclone. Knowles et al.11 used highspeed movies of anisole droplets moving through a hydrocyclone to determine the velocities of liquid flow. More recently a number of investigators reported their measurements using laser Doppler velocimetry (LDV) and electrical impedance tomography.12,13 It is noted that such an experimental method is technically difficult and expensive, largely limited to a dispersed liquid phase, and at present suitable only for laboratory scale studies rather than industrial design and application. In connection with the experimental efforts, different mathematical models, mainly analytical, have also been established for design and control of hydrocyclone operation.9,14-22 However, they have limited applicability because of the heavy empiricism involved. In view of these shortcomings, in the past two decades or so, various efforts have been made to develop a CFD model based on flow fundamentals. Boysan et al.23 developed one of the first CFD models and showed that the standard k- turbulence model was inadequate to simulate flows with swirl because it led to excessive turbulence viscosities and unrealistic tangential velocities. Recent studies suggest that RSM can improve the accuracy of the numerical solution.5,24-26 The Eulerian-Lagrangian model was also successfully applied to model the dilute flow phenomena in cyclones.6 However, the complexity caused by an air core is also found in the CFD study. For example, to consider the water/air interaction in a hydrocyclone, the coupling of a high-order accurate turbulence model with a multiphase model, like the coupling of the RSM model
10.1021/ie061625u CCC: $37.00 © 2007 American Chemical Society Published on Web 05/16/2007
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2. Model Description To describe the flow in a cyclone accurately, one has to deal with the problem of large velocity and pressure gradients in the radial direction. For the characterization of the rotating turbulent flow, an appropriate turbulent model must be applied. The present work is based on the RSM which can describe anisotropic turbulence. The RSM has been proven to be an appropriate turbulence model for cyclone flow, although it is computationally more expensive than other unresolved-eddy turbulence models.3,5 The governing equations for an incompressible fluid can thus be written as
∂F ∂ + (Fu ) ) 0 ∂t ∂xi i
(1)
[(
)]
∂ui ∂uj ∂ ∂ ∂p ∂ (Fui) + (Fuiuj) ) + µ + ∂t ∂xj ∂xi ∂xj ∂xj ∂xi Figure 1. Schematic and grid representation of the cyclone considered.
+
∂ (-Fu′iu′j) (2) ∂xj
where the velocity components are decomposed into the mean ui and fluctuating u′i velocities (i ) 1, 2, 3), which are related as given by
ui ) ui + u′i
(3)
where the Reynolds stress term -Fu′iu′j includes the turbulence closure, which must be modeled to close eq 2. Transport equations for the transport of the Reynolds stresses Fu′iu′j in the RSM are written as33
∂ ∂ (Fu′u′) + (Fuku′iu′j) ) DT,ij + Pij + φij + ij ∂t i j ∂xk Figure 2. Variation of the flow rate on the underflow with time.
and the volume of fluid (VOF) model, may lead to numerical divergence. In particular, because of the changed or fluctuated shape and surface of the air core, it is extremely difficult to determine the interface between water and air. Therefore, the nature of the air core in a hydrocyclone is often not considered in the previous CFD modeling but with simplified assumptions about its formation and behavior.27 For example, Pericleous28 assumed the underflow did not influence the air core. In the work of Monredon et al.,29 the diameter of the air core was assumed on the basis of experimental observation. Rajamani and Devulapalli30 thought there was a stationary, cylindrical air core in hydrocyclone. Suasnabar31 regarded the interface of water and air as a moving wall in a hydrocyclone. Schuetz et al.26 and Cullivan et al.24 published some successful numerical results of flow velocity distribution in hydrocyclone, but both just presented water-only predictions. Recently, Delgadillo and Rajamani32 obtained the flow field distribution and air core shape in a hydrocyclone by the use of the RSM or large-eddy simulation (LES) and VOF model. However, their results are largely preliminary and do not show detailed flow features. This paper reports our recent effort in modeling the gasliquid-solid flow in a hydrocyclone. It shows that the combined RSM, stochastic Lagrangian model and VOF can satisfactorily describe the flow and performance of a standard hydrocyclone, including the formation of an air core. The model should be useful to assess the performance of hydrocyclones under different geometrical, flow, and operational conditions.
(4)
where F, ui, u′i, and xi are respectively liquid density, velocity, velocity fluctuation, and positional length. The two terms on the left are the local time derivative of the stress and convective transport term, respectively. The four terms in the right are
the turbulent diffusion term DT,ij ) -
∂ [Fu′u′u′ + p(δkju′i + δiku′j)] ∂xk i j k
the stress production term
(
∂uj ∂uj Pij ) -F u′iu′k + u′ju′k ∂xk ∂xk
)
the pressure strain term φij ) p
(
)
∂u′i ∂u′j + ∂xj ∂xi
the dissipation term ∂u′i ∂u′j ij ) -2µ ∂xk ∂xk where δ is the Kronecker factor and µ and p are respectively the molecular viscosity and pressure. In the modeling of the interface between the liquid and air core in a hydrocyclone, the so-called VOF free surface model is used. The VOF simulates two or more immiscible fluid phases, in which the position of the interface between fluids is
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Figure 3. Comparison between the measured (9) and predicted (b) tangential (left) and axial (right) velocities at different locations from the top wall: (a) 60 mm, (b) 120 mm, and (c) 170 mm.
of interest. In the VOF method, the fluids share a single set of momentum equations, and the volume fraction of each of the fluids involved is tracked throughout the computational domain. The tracking of the interface between the air core and the liquid in a hydrocyclone is achieved by solving the following equation:
∂Rq ∂Rq )0 + uj ∂t ∂xj
(5)
where Rq is the volume fraction of the qth phase which varies between 0 and 1 and uj is the velocity component in direction j.
A single momentum equation is solved throughout the domain, and the resulting velocity field is shared between the two phases. This momentum equation is dependent on the volume fraction of a fluid phase, given by
(
)
∂ ∂ ∂p ∂ ∂ui ∂uj Fui + Fuiuj ) + µ + + Fgi ∂t ∂xj ∂xi ∂xj ∂xj ∂xi
(6)
where the average density F is determined as
F)
∑RqFq
(7)
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Table 2. Geometry of the Hydrocyclone Considered parameter
symbol
dimension
diameter of the body diameter of inlet
Dc Di
diameter of vortex finder diameter of apex length of cylindrical part length of vortex finder included angle
Do Du Lc Lv a
75 mm 25 mm (same area quadrate is used) 25 mm 12.5 mm 75 mm 50 mm 20°
Viscosity and other properties are computed in the same way. In a computation, the implicit interface tracking algorithm in VOF is used. The motion of a particle is described by the stochastic Lagrangian multiphase flow model. The buoyancy force and liquid drag force on particles are calculated in a Lagrangian reference frame, given by
b g (Fp - F) du bp ) FD(u b-b u p) + dt Fp
Figure 4. Definitions of the sections used in this work.
(8)
where FD(u b-b up) is the drag force per unit particle mass and
FD )
18µ Rep CD dp2Fp 24
(9)
where b up is the particle velocity, b u is the velocity of the fluid phase, Fp is the density of the particle, and dp is the particle diameter. Rep is the particle Reynolds number, and CD is the drag coefficient. In the stochastic tracking, the turbulent dispersion of particles is predicted by integrating the trajectory equations for individual particles, using the instantaneous fluid velocity, uj + u′, along the particle path during the integration. The values of u′ that prevail during the lifetime of the turbulent eddy are sampled by assuming that they obey a Gaussian probability distribution:
x
u′ ) ζ u′2
(10)
where ζ is a normally distributed random number. Because the kinetic energy k of turbulence is known at each point in the flow, the values of the root mean square fluctuating components can be obtained (assuming isotropic) as
xu′2 ) xV′2 ) xw′2 ) x2k/3
(11)
Particle-eddy interaction time and dimension should not be larger than the lifetime and size of a random eddy. In this work, simulations were conducted using the Fluent CFD software package (version 6.0). The second-order upwinding and the SIMPLE pressure-velocity coupling algorithm were used. The convergence strategy used the unsteady solver. 3. Simulation and Experimental Conditions A standard hydrocyclone was used for the present modeling. Figure 1a shows the notations of the hydrocyclone dimensions, and Table 2 lists the geometrical parameters. The hydrocyclone was used by Hsieh34 in his experimental study. Therefore, his measurements can be used for model validation. Figure 1b shows the computational domain, containing 87 500 cells. The whole computational domain was divided by hexahedron grids. In the vicinity of the walls and vortex finder,
the grid was refined. Trial numerical results demonstrated that the solution is independent of the characteristics of the mesh size. The pressure at the two outlets (vortex finder and apex) was 1 atm. Unless otherwise specified, the inlet water velocity and the particle velocity were both 2.25 m/s. Limestone particles with a density of 2700 kg/m3 were injected at the inlet. The particle size distribution was according to Hsieh’s experimental data. Because the volume percentage of solid phase was less than 10%, it was reasonable to ignore the effect of solid phase on liquid phase and the interaction between particles. This treatment has been widely accepted for dilute flows.6,35 For dense flows, the so-called combined continuumdiscrete method can apply, as demonstrated in the study of gas fluidization.36-38 In this work the time step was chosen as 10-4-10-2 s with most of computations conducted using small time steps. Figure 2 shows the flow rate on the underflow with time in the simulation under different time steps, which indicates that the results are not sensitive to the time step in this range. The results also show the flow field of the simulated hydrocyclone do not change significantly after 3 s. Therefore, unless otherwise specified, the results presented in this paper are those when time ) 10 s. 4. Results and Discussion 4.1. Model Validation. Figure 3 compares the experimental and predicted velocity profiles at three different axial positions located at 60, 120, and 170 mm from the top of the hydrocyclone. They show that the simulated results are in good agreement with the experimental ones, particularly for the axial velocities. The tangential velocity distribution is a Rankine vortex with a quasi-free vortex in the outer and a quasi-forced vortex in the inner part.39 Moreover, the comparison between the experimental and the numerical results has also been made for other variables in the hydrocyclone. The predicted pressure drop and the volume split ratio are respectively 43 638 Pa and 4.692%, compared with those measured, that is, 46 700 Pa and 4.811%. The errors are less than 10%. Good qualitative agreement between measured and predicted results can also be observed in the discussion of the flow fields and hydrocyclone performance, as given in the following sub-section. 4.2. Fluid Flow Field. To explore the inner flow in the hydrocyclone, different sections were used in this work. Figure
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Figure 5. Snapshots (at section 1) showing the development of flow field in the hydrocyclone.
4 shows the positions of six sections. Section 1 is parallel to the inlet, and section 2 is perpendicular to the inlet. Sections 3-6 are located at the heights of 10, 40, 70, and 120 mm from the top wall situated on the roof of the cyclone. These sections are used in Figures 5-11. Figure 5 shows the distribution of fluid density as a function of time. The results demonstrate how the flow field in the hydrocyclone is developed. They are very similar to the description of Cullivan et al.,25 based on the observation by high-speed video. The flow field is initiated as air-filled at atmospheric pressure. On initiation of inlet flow into the empty hydrocyclone, a circumferential flow against the outer wall is established. The hydrocyclone then fills from the underflow upward, the large central air core being exhausted through the overflow. Eventually all the air is expelled through the overflow, and after a short time (∼2 s), the air core becomes relatively stable. Finally air core fluctuates slightly in the cyclone. In the following, the flow of liquid (water) is described first. Then, discussion will be extended to the flow pattern of air core, which is quite different from that of liquid.
Figure 6 shows the calculated tangential velocity distribution in detail. The flow field in the cyclone indicates the expected forced/free combination of the Rankine vortex. The value of the tangential velocity equals zero on the wall and the center of the flow field (air core is a forced vortex). From section 3, it can be seen that the flow enters the inlet and accelerates up to 1.5-2.0 times of the inlet velocity at point A. Then the velocity decreases as the flow spins down along the wall. Before it goes below the vortex finder, the fluid flow collides with the follow-up flow and forms a chaotic flow close to the vortex finder outside wall (point B). At point B, the flow velocity decreases sharply and may even change its flow direction. This behavior would increase the loss of energy and the pressure drop in the hydrocyclone. This is probably the main cause of the short circuit flow. Figure 7 shows the calculated axial velocity distribution. The upward flow in the hydrocyclone is a helical twisted cylinder and not completely symmetric axially, especially in the conical section. The black line in the figure is the dividing line between the upward flow and the downward flow, which is called the locus of zero vertical velocity.9 The diameter of upward flow
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Figure 6. Distribution of tangential velocity (m/s) in the hydrocyclone (anticlockwise is positive and clockwise is negative), with (a-f) corresponding to sections 1-6 shown in Figure 4.
Figure 7. Distribution of axial velocity (m/s) in the hydrocyclone (upward is positive and downward is negative, the axial velocity in the black line equals zero), with figure layout the same as in Figure 6.
is larger than that of the vortex finder. Because the upward flow swarms into the vortex finder and the short circuit flow goes out from the vortex finder, the axial velocity reaches a peak value under the vortex finder. Furthermore, the secondary flows are present in the axial velocity distribution. A proportion of the feed liquid passes directly across the cyclone roof and down the outside wall of the vortex finder to join the overflow stream within the vortex finder, which is a short circuit flow (point A). At point B, a vertical flow can exist in the region between the outer wall of vortex finder and the wall of body, which is called the eddy flow.9 It exists in the form of a recirculating eddy or eddies. Figure 8 shows the calculated radial velocity distribution. The distribution is like a helical twisted cylinder. The axis of the forced vortex does not coincide with the geometrical axis of the hydrocyclone and is curved rather than straight. The distribution of radial velocity in the central core vortex is eccentric with respect to the central axial line. The value of one side is positive, and the other is negative. So, the
combination of flow source and sink is distributed near the axis of the cyclone, forming a pattern of flow dipoles. The orientation of a dipole was observed to locate upwardly along the cyclone central line. The main reason for this phenomenon is that the liquids squeeze each other, which may cause instability in a hydrocyclone. There is a zone under the vortex finder, at point A (sections 1 and 2), where fluid directly flows into the vortex finder rather than moving down to the conical part and then flowing upward. Moreover, at point B (section 3), the radial velocity becomes negative again and directs to the center because of the collision of fluid streams. It results in a short circuit flow and reduces hydrocyclone performance. Figure 9 shows the fluid velocity vectors in the upper part of the hydrocyclone. The velocity of air is much higher than that of the liquid phase. Once a particle moves into the air core, it could be dragged to the overflow very quickly by the air flow. That is why there are no particles in the air core. Furthermore, the collision of air streams from the overflow and underflow
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Figure 8. Distribution of radial velocity (m/s) in the hydrocyclone (outward is positive and inward is negative), with figure layout the same as in Figure 6.
Figure 9. Fluid velocity vectors at section 1 (for the top part only).
can be seen at the bottom of the vortex finder. The upward air flow in the center from the bottom is squeezed to the edge of the vortex finder by the air from the top, escapes along the wall, and drives the liquid flow quickly to overflow. Therefore, the axial velocity of liquid flow in the vortex finder is often very high. Satisfactory prediction of the air core is very important to the prediction of volume split ratio to underflow. Figure 10 presents the predicted shape and position of the air core in the hydrocyclone. The predicted average air core diameter is 9.8 mm, close to the experimental value of 10 mm.34 It can be seen that the streamline in the external flow follows a circle, which means the vortex in the external downward flow is a free vortex. On the contrary, the streamlines in the inner flow converge at one point indicating a forced vortex. So the flow field in hydrocyclone is the expected forced/free combination of the Rankine vortex. At section 6, the forced vortex in the lower cone part does not develop sufficiently, as the streamlines do not converge at one point and the center is eccentric.
Figure 10. Shape and position of the air core and the streamline in the hydrocyclone, with figure layout the same as that for sections 3-6 in Figure 6.
It is of interest to note that the streamlines of sections 3 and 4 are different. In section 3, the streamline points out, which is the source from which the fluid flows out. In section 4, the streamline points in, which is the sink into which the fluid flows. So the air core in the upper part and that in the lower part have different properties. To verify this consideration, further simulations were carried out. The development of the air core in such a hydrocyclone was examined when the underflow was sealed with water, while the other conditions were the same. As shown in Figure 11, the lower part of the air core disappears, and only the upper part remains. Then the underflow is opened to air again, and after 0.1 s, it can be seen that the air core grows up from the underflow. Finally, the air from the underflow joins the air from the overflow and a full air core is established. So the air core is indeed composed of two parts. Air from the underflow goes upward and through
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Figure 11. Shape and position of the air core (air volume fraction) in the hydrocyclone (at section 1): Left, the underflow sealed with water; middle, the underflow opened to air for 0.1s ; and right, the underflow opened to air for 1.0 s and the full air core maintained.
Figure 12. Separation efficiency curve for the hydrocyclone considered.
the whole cyclone and air from overflow just keeps cone-shaped in the vortex finder. 4.3. Particle Flow Pattern. Figure 12 shows the calculated separation efficiency curve. The numerical results are again comparable to the experimental data obtained by Hsieh.34 The cut size d50 (the particle size at which the hydrocyclone efficiency is 50%) has been calculated from the simulation results to be approximately 11.5 µm in this case.
Figure 13. Typical trajectories of particles with different diameters.
In Hsieh’s experiment, five typical particles of different diameters were chosen to track their trajectories in the hydrocyclone. As shown in Figure 13, large particles are mainly collected while small particles escape from the cyclone. The particles with smallest diameter cannot move outward to the wall of cyclone because the centrifugal force on them is not greater than the fluid drag force on particles. A particle of cut size is initially dragged down by the external downward flow. At the same time, the radial velocity pointing to the air core causes the particle to be dragged inward. In this situation, because the particle size is small respectively, the inward drag force is higher than the centrifugal force according to eq 9. Consequently, in some part of the conical section, such a particle is caught by the upward inner flow and escapes through the vortex finder or follows the external downward flow to be collected at the underflow. Such a particle is shown to be very unstable, and it has a larger residence time than other sized particles. The trajectory of coarse particles was also tested. The numerical result illustrates that coarse particles mainly remain on the conical cyclone wall when their size exceeds a critical value. This phenomenon would cause erosion in cyclone operation, observed by Bradley.9 The reasonable explanation would be that oversize particles are held by a centrifugal force against the wall when the upward and downward forces on the particles are in balance. There are three forces on a particle on the conical wall: a supporting force N, a liquid drag force FD,
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Figure 14. Schematic diagram showing the forces on a particle in the conical section.
Figure 16. Spatial distribution of particles of different sizes in the hydrocyclone.
the wall. Under this condition, the forces acting on a single particle can be written as
G + FDz ) Nz
(12a)
or
upt2 tgθ r
mg + 3πµdp(ugz - upz) ) m
(12b)
Re-arranging the equation gives 2
g+
Figure 15. Snapshots showing the flow of particles of different diameters, where red, orange, green, cyan, and blue represent five diameters of particles which are respectively 2, 11.5, 21, 30.5, and 40 µm.
and the gravity G. As shown in Figure 14, when the upward force, that is, the axial component force of the supporting force (Nz), is equal to the sum of the gravity and the axial component of liquid drag force (FDz), the particle will keep spinning on
upt 18µ (ugz - Vpz) ) tgθ 2 r d p Fg
(13)
Equation 13 indicates that as particle diameter dp decreases, the radius of the orbit r will decrease. When the particle diameter is smaller than a critical value and at the same time r is smaller than the radius of the apex radius, a particle will be collected at the bottom. When a particle is larger than this critical diameter, it will be held on the wall. Figure 15 shows the change of location with time for 35 000 particles of the five diameters mentioned above over a period of 2 s (particles were only fed into the cyclone during the first second). There are no clear particle clusters in the hydrocyclone compared with the gas cyclone separator,8 because water has a much higher viscosity than air, and the drag force on particles in a hydrocyclone is higher than in a gas cyclone separator. Moreover, it can be seen from this figure that larger particles concentrate in the lower region of the hydrocyclone, while smaller particles concentrate in the upper region. At the end of the process, particles remaining in the hydrocyclone are of about the cut size, which have a relatively longer residence time. Figure 16 shows the spatial size distribution of particles in the hydrocyclone, which is stratified along the radial direction. At the first place, there are no particles in the air core. Then the smallest particles (blue) concentrate around the air core and
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will escape from the vortex finder along the upward flow. The particles of around cut size (cyan) move around the locus of zero vertical velocity (see Figure 7). Finally the biggest particles are accumulated on the wall, especially on the conical wall. Note that the large particles are not stable on the wall in real operation. They will flow downward and be collected at the underflow because of the interaction with other particles. This effect is not considered in the present work focused on the dilute flow in a hydrocyclone. Effort is now being made to extend the cyclone modeling involving dense flows by means of the combined continuum-discrete method.40 5. Conclusions A computer model has been developed to simulate the gasliquid-solid flow in a hydrocyclone, where RSM is used to describe the anisotropic turbulence flow and VOF to determine the interface between air and fluid. The applicability of this approach has been verified by the good agreement between the calculated and the measured flow fields. On this basis, a stochastic Lagrangian model has been used to predict the flow pattern of particles in the hydrocyclone, and its validity is confirmed by comparing the predicted and measured collection efficiency. The following conclusions can be drawn from the present study: (i) There are short circuit flows in a hydrocyclone, as reported elsewhere.9,10 The analysis of the liquid flow pattern suggests that the collision between water streams after running about a circle with that just entering is probably the main reason for short circuit flow. (ii) The combination of flow source and sink distributes near the axis of cyclone, forming a pattern of flow dipole. The orientation of a dipole is observed to locate upward along the cyclone central line so that the forced vortex in the cyclone is a twisted cylinder. (iii) The air core can be modeled by the combined RSM and VOF approach and is shown to be composed of two parts. Air from underflow goes upward and through the whole cyclone and air from overflow just keeps cone-shaped in the vortex finder. (iv) Particles with size exceeding a critical diameter, which depends on the geometry of cyclone and flow condition, may be retained on the conical wall of the cyclone as a result of balanced forces, such as the supporting force from the wall, the liquid drag force, and the gravity force. This may lead to the unstable flow of particles and the formation of erosion on the conical part in a hydrocyclone. (v) The spatial distribution of particles of different sizes in a hydrocyclone is stratified along the radial direction, with the smallest close to the air core and the largest close to the wall. Acknowledgment The authors would like to thank the Australian Research Council for the financial support of this work. Literature Cited (1) Cresswell, C. R. Notes on Air Pollution Control; H. K. Lewis and Co.: London, U.K., 1974. (2) Rietema, K.; Verver, C. G. Cyclones in Industry; Elsevier: Amsterdam, 1961. (3) Hoekstra, A. J.; Derksen, J. J.; Van Den Akker, H. E. A. An experimental and numerical study of turbulent swirling flow in gas cyclones. Chem. Eng. Science 1999, 54, 2055-2065. (4) Pant, K.; Crowe, C. T.; Irving, P. On the design of miniature cyclones for the collection of bioaerosols. Powder Technol. 2002, 125, 260-265.
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ReceiVed for reView December 18, 2006 ReVised manuscript receiVed April 6, 2007 Accepted April 13, 2007 IE061625U
