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New understanding of a hydrocyclone flow field and separation mechanism from computational fluid dynamics
Minerals Engineering 17 (2004) 651–660 This article is also available online at: www.elsevier.com/locate/mineng
New understanding of a hydrocyclone flow field and separation mechanism from computational fluid dynamics J.C. Cullivan a, R.A. Williams a
a,*
, T. Dyakowski b, C.R. Cross
c
School of Process, Environmental and Materials Engineering, Centre for Particle and Colloid Engineering, Leeds Institute of Particle Science, University of Leeds, Leeds, UK b Department of Chemical Engineering, UMIST, P.O. Box 88, Manchester, UK c Rio Tinto Technology, Castlemead, Lower Castle Street, Bristol, UK
Abstract The flow field of a 2 in. hydrocyclone is shown to be significantly asymmetric without precession, through both computational fluid dynamics (CFD) and experimental observation. Hence the application of full three-dimensional modelling is demonstrated to be essential. Further, CFD predicts that the axial pressure is not below atmospheric prior to development of the air core and that such development is not pressure driven. In fact, initial insight into a cause of instability of the air-core is identified from the CFD and supported through experimental observation. The predictions use the second-order differential-stress turbulence model which has previously been identified to represent a minimum model. Lastly, the inclusion of full three-dimensional modelling and highorder turbulence modelling leads to a new understanding of particle-separation classification within the hydrocyclone, including a significant stochastic component. 2004 Published by Elsevier Ltd. Keywords: Hydrocyclone; Computational fluid dynamics; Classification; Thickening
1. Introduction The hydrocyclone finds application in many industries ranging from pharmaceuticals to mineral processing. For the present research, a hydrocyclone design applied for the separation of china clay fines from grinding mill product has been investigated. In this instance, the china clay product is represented by the fine particle fraction (overflow, rp < 1 lm, r the radius). The course fraction (underflow) includes quartz and mica materials (rp > 1 lm). As depicted in Fig. 1, the hydrocyclone design under consideration (Appendix A), features a single tangential inlet to the hydrocyclone head (cylindrical), which generates an outer downward helical-flow towards the underflow and within this, an upward central helicalflow towards the overflow. The central flow develops as a result of both the converging conical geometry and due to the high swirl, a feature of which is axial flow reversal (Nissan and Bresan, 1961). To support the upward core-
*
Corresponding author. Tel.: +44-113-233-2789; fax: +44-113-2781. E-mail address: [email protected] (R.A. Williams).
0892-6875/$ - see front matter 2004 Published by Elsevier Ltd. doi:10.1016/j.mineng.2004.04.009
flow, a net inward radial-flow is required, which counteracts the centrifugal outward acceleration of particulates, providing a mechanism for differentiation of particulate trajectories as a function of their, relative size and density. The high swirl flow exhibits tangential velocity profiles of the ‘hat-type’ vortex (Alekseenko et al., 1999) with a pressure minimum existing along the vortical axis. For the hydrocyclone under consideration, which is operated with the outlets open to the atmosphere, the axial pressure is low enough for a central aircore to exist. From the application of a second-order differentialstress turbulence model (DSM) with fully three-dimensional computational fluid dynamic (CFD) modelling of the hydrocyclone, four key understandings of hydrocyclone operation have been challenged. These are • Symmetry. Although well recognised as exhibiting asymmetry, reported measurements have previously indicated particle trajectories to be virtually symmetric within the main conical body (except adjacent to the underflow), even in the case of a singular tangential inlet. Hence symmetry has often been assumed for the purpose of modelling. More recently, authors
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Fig. 1. Schematic of the hydrocyclone flow structure.
such as He et al. (1999) and Ma et al. (2000) have demonstrated such an assumption to be detrimental. • Air-core development mechanism. It has been understood that air-core presence along the hydrocyclone axis is primarily the result of a below atmospheric core pressure (due to swirl), such that a pressure differential drives air-core development and maintains it when the outlets are open to the atmosphere (specifically, the underflow open). In general the air-core is observed to develop from the underflow. • Air-core precession. Precession of the air core about the hydrocyclone axis is often observed and the existence of such phenomena within certain hydrocyclone designs and for certain operating conditions is certainly not contended. The present research indicates that air-core precession cannot be automatically assumed for the hydrocyclone and in fact this is neither predicted nor observed for the present research. • Particle separation mechanism. Classically, the particle separation is understood to be governed by a simple balance of the radially outward centrifugal force and the inward radial drag.
tured and structured modelling. For the present research, meshing was structured so that the pressure staggering option (PRESTO) could be applied, such that the pressure field is stored on a mesh staggered from that of the velocities. Subsequently, interpolation was not required to reconstruct the pressure field during iterations, the benefits of which, for recirculation and swirling flows are well recognised (Shyy, 1994). Therefore, a three-dimensional butterfly type mesh was constructed for the hydrocyclone including conformal meshing for the tangential inlet. A total of 99,776 mesh elements were used, a limitation dictated primarily by the computational resource available, a SUN Ultra 10 (500 Mb), coupled with the use of transient modelling. More details of the constructed mesh and modelling details are available from Cullivan et al., 2003). The flow field was modelled transiently to initiate the modelled flow field, providing numerical stability and also for investigation of the dynamics of hydrocyclone start-up and the mechanism of air-core inception and development. The high-order QUICK (Leonard, 1979) discretisation was employed for the velocity and turbulence variables with SIMPLEC for the pressure–velocity coupling. The boundary conditions were prescribed as a mass flow inlet (0.69 kg/s), outlet boundaries of fixed pressure (zero at centre, radially increasing to reflect local swirl prediction) and no-slip wall boundary conditions. Complex turbulence profiles are a feature of many high-swirl devices. For the hydrocyclone, increased accuracy of CFD predictions may be clearly observed through the application of Prantdl mixing-length models (Hsieh and Rajamani, 1988), two-equation models (Sevilla and Branion, 1997), DSM (Lu et al., 1999; Slack and Wraith, 1997) and large-eddy simulation (LES) (Slack et al., 2000). Only the later two turbulence models (DSM and LES) provide implicit accounting for local turbulence asymmetry and therefore it is only models like these that are able to accurately capture the detail of the hydrocyclone flow field, as demonstrated by Slack and Wraith (1997) and Slack et al. (2000). For this research the DSM is adopted such that Ui;i ¼ 0;
qUi;t þ qðUj Ui Þ;j þ P;i flðUi;j þ Uj;i Þ qsij gj si ¼ 0 ð2Þ where sij ¼ ui uj
2. Computational fluid dynamic modelling Modelling of the hydrocyclone has been undertaken with the commercial CFD code Fluent5. This code uses the finite volume method and provides both unstruc-
ð1Þ
ð3Þ
are the Reynolds stresses and P , si , Ui , ui , l and q are the mean pressure, source, mean and fluctuating components of velocity, molecular viscosity and density, respectively. The Reynolds stresses are modelled as (Hinze, 1975)
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Dsij ¼ sij;t þ Uk sij;k |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} Dt Cij
p 1 ¼ ðsik Uj;k þ sjk Ui;k Þ þ si uj þ sj ui þ ðui;j þ uj;i Þ |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} q q |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} Pij Gij
/ij
p 2mui;k uj;k ui uj uk þ ðuj dik þ ui djk Þ þ ðmsij;k Þk |fflfflfflffl{zfflfflfflffl} q ;k |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} eij dij
2xk ðsjm eikm þ sim ejkm Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Fij
ð4Þ where Cij , dij , Fij , Gij , Pij , eij and /ij are the convective transport, diffusion, production of turbulence by system rotation, production of turbulence by buoyancy, production of turbulence by mean strain, dissipation rate of turbulence and re-distribution of turbulence by the pressure strain, respectively. Also p, eijk , m and xi are the fluctuating pressure, permutation operator, kinematic viscosity and the angular velocity, respectively. The terms Gij , eij and /ij require modelling, of which the pressure–strain term is particularly important for highswirl flows for which the correlation between fluctuating pressure and velocity is sensitive. Therefore, this research uses the quadratic pressure–strain model of Fluent 5, which is based on the Speziale, Sarkar and Gatski model (Speziale et al., 1991). A previous paper (Cullivan et al., 2003) has compared predictions of the hydrocyclone flow field and clearly demonstrates the linear model to be inappropriate, as well as providing more detail of the modelling approach adopted. In that paper the quadratic model is identified as a lower bound for turbulence modelling of the hydrocyclone. The research presented here represents water-only predictions. A multi-phase water/air system is not considered for prediction of the mechanisms effecting aircore inception and dynamics. Although the present predictions are not strictly representative, the coupling of a high-order accurate turbulence model with a multiphase model such as volume-of-fluid lead to numerical divergence. In particular sufficient resolution of the aircore interface region was not achieved such that the span of the turbulence computational-stencil caused divergence. Factors such as, maintaining a structured mesh (PRESTO) and limiting computation time precluded adjustment for incorporation of two-fluid modelling, although this is a subject of future research.
3. Experimental measurements In order to examine the validity of the CFD predictions and as part of an experimental campaign to
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examine the detail of hydrocyclone operation, a number of measurements have been undertaken. These have included high-speed video, radiography, ultrasound tomography (UST) and electrical impedance tomography (EIT). The former two modalities have provided immediate verification for the predicted mechanisms and flow field structure within the hydrocyclone, whilst the later two are being investigated further. For the high-speed video the air/water interface during start-up was observed within a 2 in. glass hydrocyclone of similar dimensions to that under investigation (Appendix A). Recordings were made with a Kodak Ektapro EM monochrome camera at 500 frames/s. For radiographic measurements, a medical X-ray camera of the University of London was applied providing a frame rate of 1/25 s and an exposure time of 1/10,000 s. More details of the experimental methods are provided in Cullivan et al. (2001).
4. Predicted flow field development The computational flow field was initiated as waterfilled with no-flow and at atmospheric pressure. Although this clearly does not allow for capture of the complete period of start-up as the hydrocyclone initially fills, it was observed from the high-speed video that the swirl flow field initially develops in a water-filled (coreless) hydrocyclone. On initiation of inlet flow into the empty hydrocyclone, a circumferential flow against the outer wall is initially established. The hydrocyclone then fills from the underflow upwards, the large central aircore being exhausted through the overflow. Eventually all the air is expelled through the overflow, after which a short time follows (2 s), during which the flow field continues to develop and before inception of air occurs at the underflow and subsequent air-core development occurs. Therefore, the pre-inception flow field, develops as a water-only flow and inference of the structure of this and of the mechanisms which drive inception may be drawn from the water-only CFD predictions. The presented hydrocyclone CFD predictions have provided a novel understanding of the development of the initial flow field. Alongside the expected tangential velocity profile (‘hat-type’ vorticity, Alekseenko et al., 1999)), novel predictions of asymmetry and of the transverse and axial velocity profiles have been obtained. 4.1. Asymmetry As noted above, the hydrocyclone flow field is axially asymmetric by virtue of the geometric inlet asymmetry, although the influence that such a geometric asymmetry has upon the eventual flow asymmetry throughout the hydrocyclone, is sometimes ignored. In fact, the present
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CFD research clearly indicates that structures developed within the head flow are critically influential throughout the remainder of the flow field. A feature of streamline curvature flow is the development of secondary flows, such that the secondary flow structure will continue to evolve through the hydrocyclone, particularly with the transition from the inlet region to the head and subsequently from the head to the body. Therefore, further investigation is required in order to quantify the influence that the precise inlet geometry has. As a wall bounded swirl-flow, secondary flows rapidly develop. Fig. 2 indicates the development of the secondary flows through the hydrocyclone. The my > 0 and mx > 0 velocity contours are shown for the x ¼ 0 and y ¼ 0 planes, respectively, superimposed upon mz > 0 contours. Notably, a secondary flow structure is well developed within the top of the head. As the streamlines pass over the tip of the vortex finder the radially inward flow is intensified (x-plane, y > 0), which combined with existing secondary flows, forms a constant transverse direction of flow across the hydrocyclone. The position of maximum axial-vorticity is offset from the axis by the expanding flow (transversely), passing over the vortex-finder tip, for which the downward axial outer-flow component is reduced (momentum conservation). As is evident from Fig. 2 the asymmetry developed at the vortex finder tip is maintained throughout the hydrocyclone such that alternating transverse velocities, through the geometrical axis, exist down through the hydrocyclone. Also, it is evident that the upper-body secondary flow structure evolves toward the more stable vortex structure of the lower hydrocyclone body. The asymmetry structure shown in Fig. 2 is predicted to be static in orientation and structure indicating a static air-core shape. Evidence for such has come from the radiography measurements. As shown in Fig. 3 the air-core orientations were constant over a range of feed pressure (1–4 105 Pa) for both 5 and 8 mm apex diameters. Therefore, the flow asymmetry is indicated to be a geometry controlled feature.
Fig. 3. Radiograph of the hydrocyclone and extracted air-core orientations. Air-core orientations correspond to operation with 5 and 8 mm spigot (underflow) diameters and with the inlet pressure set as 1.0, 1.5, 2.0, 2.5, 3.0, 3.5 and 4.0 · 105 Pa. The hydrocyclone was fed with kaolinite slurry at 6% v/v.
Notably, the fluid path lines do not exhibit the same helicity structure as for the asymmetry. In fact, as illustrated in Fig. 4 the ratio of axial to tangential velocity in the wall region remains of order )0.3 (in range )0.4 to )0.2) such that the frequency of flow revolution per asymmetry helical period increases down through the hydrocyclone. In contrast, a ratio of order of )0.4 would be required to match with the asymmetry helicity. As demonstrated from the research of Alekseenko et al. (1999), the axial velocity acts as an independent parameter to the vortex pitch and radius and hence the swirl number does not provide a unique characterisation of the flow field. Conservation of momentum results in variation of the axial to tangential velocity ratio, such that for the narrower cross-section
Fig. 2. Velocity contours of my > 0 and mx > 0 (blue/red) for the x ¼ 0 and y ¼ 0 planes, respectively, superimposed upon mz > 0 contours (green/red). (For interpretation of references in colour, the reader is referred to the web version of the article.)
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Fig. 5. Positive axial-velocity contours during hydrocyclone start-up. Fig. 4. Contours of ratio of axial velocity to tangential velocity. Thresholded from )0.3 (blue) to 0.5 (red). (For interpretation of references in colour, the reader is referred to the web version of the article.)
(core nearer to wall) the ratio is increased and vice versa and hence the range of )0.4 to )0.2 exists.
5. Air-core development and dynamics The structure of the swirl flow field also has a critical influence upon inception, development and stability of the air core. During start-up, a central upward axial flow initially develops just below the vortex finder tip, where flow reverses and exits through the overflow. As the swirl flow develops, this region of upward flow broadens and penetrates further towards the underflow. Upon penetrating two thirds of the body length, an approximately cylindrical region of upward axial velocity develops, which then continues to penetrate close to the underflow (Fig. 5). Within this cylindrical region, the central axial-velocity is downwards towards the underflow and therefore opposes air-core development. Whilst the present CFD predictions represent that for water only, Fig. 6 shows the subsequent inception of an upward core at the underflow. Here the lowpressure core supports inception whilst the axial velocity structure and upper, central pressure profile (Fig. 7) restricts development of the incepted bubble into a full core.
The present predictions challenge the common understanding that the air-core develops as a result of a below atmospheric core-pressure such that the air-core is pressure driven. Rather, as shown in Fig. 7 the predicted axial pressure profile is only below atmospheric immediately adjacent to the underflow and is significantly greater than zero away from the underflow. This would indicate that air-core development would certainly not be pressure driven and therefore this must be transport driven. Although a full transient development of the upward core was not predicted for this research, comparison between CFD predictions and high-speed video observation provides support for this conclusion. Fig. 8 shows capture of a suspended bubble at the underflow for the hydrocyclone operated at a reduced feed pressure. This underflow bubble’s axial-position was observed to be stable as it rotated about the axis. The present CFD prediction provides an appropriate mechanism to support this observation. With increased feed pressure, air-core development was observed and an air-core was established. Whilst at full operating pressure the air-core was stable, this was not so at an intermediate pressure. Rather the air-core was then observed to pulse as shown in Fig. 9. The air core followed a cycle, which from a fully established air core, began with the breakage of air-core continuity between that within the main body and at the underflow. This restriction and then breakage occurred in the region of predicted secondary axial-flow reversal adjacent to the underflow (Fig. 5). The air core then began to
Fig. 6. Quadratic pressure–strain prediction of positive axial-velocity contours during inception of core over period of 12 ms.
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thin, indicating either an above atmospheric pressure, convective transport or both (both were predicted). Continuity was then re-established and an upward pulse observed which replenished the air-core, indicating a key role for convective transport in this process. The observation of Nissan and Bresan (1961), which has been replicated by the present authors, is also noteworthy. They observed a suspended bubble of air within the hydrocyclone, albeit for a short period of time. From the present predictions this may be explained as corresponding to the region just above that of secondary axial-flow reversal, where the upward axial-velocity is low whilst the pressure gradient would drive the bubble downwards, such that equilibrium may occur.
6. Particle separation and solids distribution
Fig. 7. Pressure profile prediction along hydrocyclone axis.
Fig. 8. Quasi-stable underflow air bubble. The air bubble was observed to maintain an almost constant penetration for the reduced operating pressure.
The predicted helical vortex flow field will clearly have an impact upon the particle separation performance and solids distribution within the hydrocyclone. Important new understandings of the particle behaviour within the hydrocyclone are provided from Lagrangian particle-tracking upon the Eulerian continuous phase prediction. The classical understanding held that the separation mechanism is deterministic with a balance between the inward radial particle-drag and outward centrifugal force determining the radial particulate motion. Recent research (Averous and Fuentes, 1997) has concluded that the separation mechanism is governed by radial turbulence fluctuations which project particles from the wall flow to the core flow. For that research the particulates were determined to be completely thrown against the outer wall on entry to the hydrocyclone. The present research indicates that both the deterministic and turbulence aspects play key roles for the separation mechanism. An important parameter for the setup of Lagrangian particle tracking is the integral length scale. Too large an
Fig. 9. High-speed video capture of pulsing air core in upper conical section, just below vortex-finder tip.
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integral length results in all particles being predicted as thrown against the outer wall on entry. Hence the length was reduced until the statistical destination of particles proved independent of the length scale (1 · 105 m). The resultant track data was collected for their coincidence with two orthogonal planes (x-plane and y-plane). The tracks were computed for 80 release positions over the inlet with 10 releases for each position and for particles of radius ½0:25; 0:5; 1; 1:5; 2; 2:5; 3; 3:5; 4; 4:5; 5 lm, (qp ¼ 2800 kg m3 ). Analysis of the particle trajectories demonstrates the existence of high solids concentration regions and the importance of turbulence and asymmetry for the separation mechanism. 6.1. Solids concentration The volume fraction predictions presented below represent a feed of solids concentration 0.05 v/v constituted as a Rosin–Rammler distribution such that Ff ðrp Þ ¼ 1 expð loge ð2Þðrp =4Þ2 Þ where the particle radius is in microns. As a result, the fractions of feed total solids represented by each size fraction (½0:25; 0:5; 1; 1:5; 2; 2:5; 3; 3:5; 4; 4:5; 5 lm) are ½4:4 106 ; 1:0 104 ; ½1:9; 8:8 103 ; ½2:5; 5:3; 9:2 102 ; ½1:4; 1:9; 2:3; 2:6 101 . Fig. 10 shows predicted volume fraction contours for the rp ¼ 0:25 lm and rp ¼ 2:5 lm classes of particles (inlet volume fractions of 2.2 · 107 and 2.6 · 103 , respectively). It is evident from this figure that regions of high concentration exist both at the underflow and within the upper body flow. In fact, the region of high concentration within the main body, appears to correspond with the marked region of recirculation in Fig. 2. This region of recirculation is predicted to become less concentrated and less significant as the particle size and hence probability of being away from the outer wall is decreased. An overall volume fraction distribution is shown in Fig. 11 where all the particle size contributions have been included. It is notable that the larger particle
Fig. 11. Volume fraction contours for complete particle size distribution, thresholded at 0.2. Axis are in meters.
sizes dominate the distribution generating a high wall and underflow concentration. For an inlet concentration of 0.05 v/v, a maximum solids volume fraction of 0.36 was predicted with the bulk at less than 0.2. Hence the present predictions are likely, only valid for inlet concentrations of the order 0.01 v/v or less. The issue of high-solids multiphase hydrocyclone modelling remains open for future research. The present predictions also provide evidence of a concentrated particle path along the hydrocyclone wall, as indicated in Fig. 10, which would naturally lead to scouring of a helical track into the hydrocyclone wall, as observed in practise. The high-solids region once established, remains throughout the hydrocyclone. 6.2. Particle separation mechanism The Lagrangian particle tracking also indicates a new understanding of the relative importance of deterministic and turbulent radial particulate transport between the wall and core flows (downward and upward regions,
Fig. 10. Volume fraction contours for rp ¼ 0:25 lm and 2.5 lm particles thresholded at 6 · 107 and 1 · 102 respectively. Axis are in meters.
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respectively). In particular, the present predictions indicate such radial transport to form regions within the hydrocyclone of preferential radial direction although these are attenuated by the turbulence component. For examination of the influence of turbulence effects the particle tracks have been reconstructed over a quarter revolution of the hydrocyclone (negative y-plane to positive x-plane). The deterministic particle motions between the two planes were reconstructed assuming linear velocity profiles to provide the stochastic component, represented as the difference between the actual (effective) distance and the deterministic reconstruction. Considering rp ¼ 0:25 lm and rp ¼ 2 lm cases, the predictions are briefly examined below in terms of radial transport. Three variables are considered, the deterministic (reconstructed) radial distance travelled as a result of the mean flow, drd , the effective (Lagrangian predicted) distance travelled, dre and the difference between these, Ddr (Ddr ¼ drd dre ). The ratio of the standard deviation of Ddr over drd is shown in Fig. 12. It is clear from this figure that the standard deviation is very significant for the wall and head regions of the hydrocyclone, including around the vortex finder tip. In these regions, turbulence transport even dominates convective transport. In fact, the ratio is only below 0.1 immediately adjacent to the vortex core and therefore
turbulence transport remains significant throughout the particulate separation region. It is of interest to note the component of turbulence transport as a function of the deterministic radial transport direction. Fig. 13 the effective distances as well as the sign multiple of the deterministic and difference distances for the rp ¼ 0:25 lm particles is shown. It is evident from this figure that a preferential structure of radial transport between the wall and core regions exists as a result of the flow asymmetry. The precise nature of the relationship between preferential transport and flow asymmetry remains to be determined, although an analysis similar to that of Alekseenko et al. (1999) could be considered. It is also evident from this figure that Ddr and dre are of equal sign along the wall region, which indicates that in a mean sense the turbulence contribution is negatively correlated with the radial convection direction, i.e. the turbulence damps the radial transport asymmetry. The precise influence of the turbulence for the core region is ignored for the present discussion since reconstruction of the tangential and radial velocity profiles has been based upon an axially centred system, which will lead to the positive correlation shown in Fig. 13. It is intended to make a more comprehensive analysis centred upon the vortex axis in the future.
Fig. 12. Contours of the standard deviation of the difference over the deterministic distance travelled for rp ¼ 0:25 lm and 2.5 lm particles, both thresholded at 2.0.
Fig. 13. Contours of the effective distance (thresholded at ±1 mm) and of the sign multiple of the effective and difference distances travelled for the rp ¼ 0:25 lm particles.
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7. Conclusions and future analysis The CFD and experimental research here presented for the hydrocyclone has challenged a number of key understandings of the hydrocyclone flow field and of the particle separation mechanism within. In particular, the present predictions indicate a clear and significant asymmetry throughout the hydrocyclone resultant from the single tangential inlet and wall bounded streamline curvature. Streamline expansion over the vortex-finder tip combined with existing secondary flow structure, established a globally static asymmetry throughout the hydrocyclone. Hence the air-core is predicted to exhibit a static asymmetry without precession, as was observed through high-speed video and radiography. The existence of such asymmetry throughout the hydrocyclone is predicted to play a key role in determining the particle separation mechanism, forming a structure of alternating radial particle-transport direction down through the hydrocyclone. The influence of turbulence upon this deterministic flow structure is indicated to be most significant in the wall, head and vortex finder regions, where the mean effect of turbulence is to dampen the asymmetry of the radial transport structure. The predicted pressure and velocity profiles have provided support for high-speed video observations of both a suspended bubble at the underflow and also of instability of the aircore at moderate operating feed pressures. As identified in the text, the present analysis for the balance of deterministic and turbulence transport effects is in error due to the use of a geometrically centred reference frame. This has been detrimental to the accuracy of reconstructed particle-motions near to the vortex core. A future analysis is intended, based upon a vortex centred coordinate frame, with an increased number of particle releases and an increased number of data collection planes, in order to reduce prediction uncertainty and the effects of the linear assumption. It is also necessary to gather experimental classification data for particulates with well characterised settling properties, such that a direct comparison may be made between the predicted and measured classification. Support of the presented flow phenomena and mechanisms within the hydrocyclone may be sought through direct flow field measurements, using laser induced fluorescence, for example. Further development of tomographic techniques such as electrical impedance and ultrasound is being pursued in order to provide non-intrusive tools for validation of the future high-solids predictions. Future analysis and measurement will likely include the predicted classification as well as the predicted residence time distribution (Stegowski and Leclerc, 2002), since this is expected to be particularly sensitive to the newly proposed separation mechanism as opposed to the classical understanding. In particular, the residence time is expected to be, in general greater, although with
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possibly a less significant peak at the cut size due to mixing effects for a broader range of particle sizes.
Acknowledgements The authors are grateful for the financial support provided by the University of Exeter Research Fund, University of Leeds, EPSRC (GR/L95472, GR/R22100/ 01) and by Rio Tinto Technology. The kind support of Dr M Slack of Fluent Europe Ltd. is also acknowledged.
Appendix A
Research hydrocyclone dimensions (not to scale).
References Alekseenko, S.V., Kuibin, P.A., Okulov, V.L., Shtork, S.I., 1999. Helical vortices in swirl flow. Journal of Fluid Mechanics 382, 195– 243. Averous, J., Fuentes, R., 1997. Advances in the numerical simulation of hydrocyclone classification. Canadian Metallurgical Quarterly 36 (5), 309–314.
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Cullivan, J.C., Williams, R.A., Cross, C.R., 2001. Verification of theoretical 3D-flow in a hydrocyclone using tomography. In: Fourth World Congress for Particle Technology, Sydney, pp. 1–9. Cullivan, J.C., Williams, R.A., Cross, C.R., 2003. Understanding the hydrocyclone separator through computational fluid dynamics. Transactions of the IChemE, Part A, Chemical Engineering, Research and Design 81 (A4), 455–466. He, P., Salcudean, M., Gartshore, I.S., 1999. A numerical simulation of hydrocyclones. Transactions of the IChemE, Part A, Chemical Engineering, Research and Design 77 (A5), 429–441. Hinze, J.O., 1975. Turbulence, 2nd ed. McGraw-Hill, New York. Hsieh, K.T., Rajamani, R.K., 1988. Model development and verification for single-phase flow. International Journal of Mineral Processing 22, 223–237. Leonard, B.P., 1979. A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Computer Methods in Applied Mechanics and Engineering 19, 58–98. Lu, Y.-J., Zhou, L.-X., Shen, X., 1999. Numerical simulation of strongly swirling turbulent flows in a liquid–liquid hydrocyclone using different turbulent models. In: Multiphase Flow and Heat Transfer, Proceedings of the Fourth International Symposium, pp. 654–663.
Ma, L., Ingham, D.B., Wen, X., 2000. Numerical modelling of the fluid and particle penetration through small sampling cyclones. Journal of Aerosol Science 31 (9), 1097–1119. Nissan, A.H., Bresan, V.P., 1961. Swirling flow in cylinders. American Institute of Chemical Engineers Journal 7 (4), 543–547. Sevilla, E.M., Branion, R.M.R., 1997. The fluid dynamics of hydrocyclones. Journal of Pulp and Paper Science 23 (2), 85–93. Shyy, W., 1994. Computational modeling for fluid flow and interfacial transport. Elsevier, Amsterdam, New York. Slack, M.D., Prasad, R.O., Bakker, A., Boysan, F., 2000. Advances in cyclone modelling using unstructured grids. Transactions of the Institute of Chemical Engineers 78 (A), 1098–1104. Slack, M.D., Wraith, A.E., 1997. Modelling the velocity distribution in a hydrocyclone. In: 4th International Colloquium on Process Simulation, pp. 65–83. Speziale, C.G., Sarkar, S., Gatski, T.B., 1991. Modelling the pressurestrain correlation of turbulence: an invariant dynamical systems approach. Journal of Fluid Mechanics 227, 245–272. Stegowski, Z., Leclerc, J.-P., 2002. Determination of the solid separation and residence time distributions in an industrial hydrocyclone using radioisotope tracer experiments. International Journal of Mineral Processing 66, 67–77.
New understanding of a hydrocyclone flow field and separation mechanism from computational fluid dynamics J.C. Cullivan a, R.A. Williams a
a,*
, T. Dyakowski b, C.R. Cross
c
School of Process, Environmental and Materials Engineering, Centre for Particle and Colloid Engineering, Leeds Institute of Particle Science, University of Leeds, Leeds, UK b Department of Chemical Engineering, UMIST, P.O. Box 88, Manchester, UK c Rio Tinto Technology, Castlemead, Lower Castle Street, Bristol, UK
Abstract The flow field of a 2 in. hydrocyclone is shown to be significantly asymmetric without precession, through both computational fluid dynamics (CFD) and experimental observation. Hence the application of full three-dimensional modelling is demonstrated to be essential. Further, CFD predicts that the axial pressure is not below atmospheric prior to development of the air core and that such development is not pressure driven. In fact, initial insight into a cause of instability of the air-core is identified from the CFD and supported through experimental observation. The predictions use the second-order differential-stress turbulence model which has previously been identified to represent a minimum model. Lastly, the inclusion of full three-dimensional modelling and highorder turbulence modelling leads to a new understanding of particle-separation classification within the hydrocyclone, including a significant stochastic component. 2004 Published by Elsevier Ltd. Keywords: Hydrocyclone; Computational fluid dynamics; Classification; Thickening
1. Introduction The hydrocyclone finds application in many industries ranging from pharmaceuticals to mineral processing. For the present research, a hydrocyclone design applied for the separation of china clay fines from grinding mill product has been investigated. In this instance, the china clay product is represented by the fine particle fraction (overflow, rp < 1 lm, r the radius). The course fraction (underflow) includes quartz and mica materials (rp > 1 lm). As depicted in Fig. 1, the hydrocyclone design under consideration (Appendix A), features a single tangential inlet to the hydrocyclone head (cylindrical), which generates an outer downward helical-flow towards the underflow and within this, an upward central helicalflow towards the overflow. The central flow develops as a result of both the converging conical geometry and due to the high swirl, a feature of which is axial flow reversal (Nissan and Bresan, 1961). To support the upward core-
*
Corresponding author. Tel.: +44-113-233-2789; fax: +44-113-2781. E-mail address: [email protected] (R.A. Williams).
0892-6875/$ - see front matter 2004 Published by Elsevier Ltd. doi:10.1016/j.mineng.2004.04.009
flow, a net inward radial-flow is required, which counteracts the centrifugal outward acceleration of particulates, providing a mechanism for differentiation of particulate trajectories as a function of their, relative size and density. The high swirl flow exhibits tangential velocity profiles of the ‘hat-type’ vortex (Alekseenko et al., 1999) with a pressure minimum existing along the vortical axis. For the hydrocyclone under consideration, which is operated with the outlets open to the atmosphere, the axial pressure is low enough for a central aircore to exist. From the application of a second-order differentialstress turbulence model (DSM) with fully three-dimensional computational fluid dynamic (CFD) modelling of the hydrocyclone, four key understandings of hydrocyclone operation have been challenged. These are • Symmetry. Although well recognised as exhibiting asymmetry, reported measurements have previously indicated particle trajectories to be virtually symmetric within the main conical body (except adjacent to the underflow), even in the case of a singular tangential inlet. Hence symmetry has often been assumed for the purpose of modelling. More recently, authors
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Fig. 1. Schematic of the hydrocyclone flow structure.
such as He et al. (1999) and Ma et al. (2000) have demonstrated such an assumption to be detrimental. • Air-core development mechanism. It has been understood that air-core presence along the hydrocyclone axis is primarily the result of a below atmospheric core pressure (due to swirl), such that a pressure differential drives air-core development and maintains it when the outlets are open to the atmosphere (specifically, the underflow open). In general the air-core is observed to develop from the underflow. • Air-core precession. Precession of the air core about the hydrocyclone axis is often observed and the existence of such phenomena within certain hydrocyclone designs and for certain operating conditions is certainly not contended. The present research indicates that air-core precession cannot be automatically assumed for the hydrocyclone and in fact this is neither predicted nor observed for the present research. • Particle separation mechanism. Classically, the particle separation is understood to be governed by a simple balance of the radially outward centrifugal force and the inward radial drag.
tured and structured modelling. For the present research, meshing was structured so that the pressure staggering option (PRESTO) could be applied, such that the pressure field is stored on a mesh staggered from that of the velocities. Subsequently, interpolation was not required to reconstruct the pressure field during iterations, the benefits of which, for recirculation and swirling flows are well recognised (Shyy, 1994). Therefore, a three-dimensional butterfly type mesh was constructed for the hydrocyclone including conformal meshing for the tangential inlet. A total of 99,776 mesh elements were used, a limitation dictated primarily by the computational resource available, a SUN Ultra 10 (500 Mb), coupled with the use of transient modelling. More details of the constructed mesh and modelling details are available from Cullivan et al., 2003). The flow field was modelled transiently to initiate the modelled flow field, providing numerical stability and also for investigation of the dynamics of hydrocyclone start-up and the mechanism of air-core inception and development. The high-order QUICK (Leonard, 1979) discretisation was employed for the velocity and turbulence variables with SIMPLEC for the pressure–velocity coupling. The boundary conditions were prescribed as a mass flow inlet (0.69 kg/s), outlet boundaries of fixed pressure (zero at centre, radially increasing to reflect local swirl prediction) and no-slip wall boundary conditions. Complex turbulence profiles are a feature of many high-swirl devices. For the hydrocyclone, increased accuracy of CFD predictions may be clearly observed through the application of Prantdl mixing-length models (Hsieh and Rajamani, 1988), two-equation models (Sevilla and Branion, 1997), DSM (Lu et al., 1999; Slack and Wraith, 1997) and large-eddy simulation (LES) (Slack et al., 2000). Only the later two turbulence models (DSM and LES) provide implicit accounting for local turbulence asymmetry and therefore it is only models like these that are able to accurately capture the detail of the hydrocyclone flow field, as demonstrated by Slack and Wraith (1997) and Slack et al. (2000). For this research the DSM is adopted such that Ui;i ¼ 0;
qUi;t þ qðUj Ui Þ;j þ P;i flðUi;j þ Uj;i Þ qsij gj si ¼ 0 ð2Þ where sij ¼ ui uj
2. Computational fluid dynamic modelling Modelling of the hydrocyclone has been undertaken with the commercial CFD code Fluent5. This code uses the finite volume method and provides both unstruc-
ð1Þ
ð3Þ
are the Reynolds stresses and P , si , Ui , ui , l and q are the mean pressure, source, mean and fluctuating components of velocity, molecular viscosity and density, respectively. The Reynolds stresses are modelled as (Hinze, 1975)
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Dsij ¼ sij;t þ Uk sij;k |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} Dt Cij
p 1 ¼ ðsik Uj;k þ sjk Ui;k Þ þ si uj þ sj ui þ ðui;j þ uj;i Þ |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} q q |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} Pij Gij
/ij
p 2mui;k uj;k ui uj uk þ ðuj dik þ ui djk Þ þ ðmsij;k Þk |fflfflfflffl{zfflfflfflffl} q ;k |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} eij dij
2xk ðsjm eikm þ sim ejkm Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Fij
ð4Þ where Cij , dij , Fij , Gij , Pij , eij and /ij are the convective transport, diffusion, production of turbulence by system rotation, production of turbulence by buoyancy, production of turbulence by mean strain, dissipation rate of turbulence and re-distribution of turbulence by the pressure strain, respectively. Also p, eijk , m and xi are the fluctuating pressure, permutation operator, kinematic viscosity and the angular velocity, respectively. The terms Gij , eij and /ij require modelling, of which the pressure–strain term is particularly important for highswirl flows for which the correlation between fluctuating pressure and velocity is sensitive. Therefore, this research uses the quadratic pressure–strain model of Fluent 5, which is based on the Speziale, Sarkar and Gatski model (Speziale et al., 1991). A previous paper (Cullivan et al., 2003) has compared predictions of the hydrocyclone flow field and clearly demonstrates the linear model to be inappropriate, as well as providing more detail of the modelling approach adopted. In that paper the quadratic model is identified as a lower bound for turbulence modelling of the hydrocyclone. The research presented here represents water-only predictions. A multi-phase water/air system is not considered for prediction of the mechanisms effecting aircore inception and dynamics. Although the present predictions are not strictly representative, the coupling of a high-order accurate turbulence model with a multiphase model such as volume-of-fluid lead to numerical divergence. In particular sufficient resolution of the aircore interface region was not achieved such that the span of the turbulence computational-stencil caused divergence. Factors such as, maintaining a structured mesh (PRESTO) and limiting computation time precluded adjustment for incorporation of two-fluid modelling, although this is a subject of future research.
3. Experimental measurements In order to examine the validity of the CFD predictions and as part of an experimental campaign to
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examine the detail of hydrocyclone operation, a number of measurements have been undertaken. These have included high-speed video, radiography, ultrasound tomography (UST) and electrical impedance tomography (EIT). The former two modalities have provided immediate verification for the predicted mechanisms and flow field structure within the hydrocyclone, whilst the later two are being investigated further. For the high-speed video the air/water interface during start-up was observed within a 2 in. glass hydrocyclone of similar dimensions to that under investigation (Appendix A). Recordings were made with a Kodak Ektapro EM monochrome camera at 500 frames/s. For radiographic measurements, a medical X-ray camera of the University of London was applied providing a frame rate of 1/25 s and an exposure time of 1/10,000 s. More details of the experimental methods are provided in Cullivan et al. (2001).
4. Predicted flow field development The computational flow field was initiated as waterfilled with no-flow and at atmospheric pressure. Although this clearly does not allow for capture of the complete period of start-up as the hydrocyclone initially fills, it was observed from the high-speed video that the swirl flow field initially develops in a water-filled (coreless) hydrocyclone. On initiation of inlet flow into the empty hydrocyclone, a circumferential flow against the outer wall is initially established. The hydrocyclone then fills from the underflow upwards, the large central aircore being exhausted through the overflow. Eventually all the air is expelled through the overflow, after which a short time follows (2 s), during which the flow field continues to develop and before inception of air occurs at the underflow and subsequent air-core development occurs. Therefore, the pre-inception flow field, develops as a water-only flow and inference of the structure of this and of the mechanisms which drive inception may be drawn from the water-only CFD predictions. The presented hydrocyclone CFD predictions have provided a novel understanding of the development of the initial flow field. Alongside the expected tangential velocity profile (‘hat-type’ vorticity, Alekseenko et al., 1999)), novel predictions of asymmetry and of the transverse and axial velocity profiles have been obtained. 4.1. Asymmetry As noted above, the hydrocyclone flow field is axially asymmetric by virtue of the geometric inlet asymmetry, although the influence that such a geometric asymmetry has upon the eventual flow asymmetry throughout the hydrocyclone, is sometimes ignored. In fact, the present
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CFD research clearly indicates that structures developed within the head flow are critically influential throughout the remainder of the flow field. A feature of streamline curvature flow is the development of secondary flows, such that the secondary flow structure will continue to evolve through the hydrocyclone, particularly with the transition from the inlet region to the head and subsequently from the head to the body. Therefore, further investigation is required in order to quantify the influence that the precise inlet geometry has. As a wall bounded swirl-flow, secondary flows rapidly develop. Fig. 2 indicates the development of the secondary flows through the hydrocyclone. The my > 0 and mx > 0 velocity contours are shown for the x ¼ 0 and y ¼ 0 planes, respectively, superimposed upon mz > 0 contours. Notably, a secondary flow structure is well developed within the top of the head. As the streamlines pass over the tip of the vortex finder the radially inward flow is intensified (x-plane, y > 0), which combined with existing secondary flows, forms a constant transverse direction of flow across the hydrocyclone. The position of maximum axial-vorticity is offset from the axis by the expanding flow (transversely), passing over the vortex-finder tip, for which the downward axial outer-flow component is reduced (momentum conservation). As is evident from Fig. 2 the asymmetry developed at the vortex finder tip is maintained throughout the hydrocyclone such that alternating transverse velocities, through the geometrical axis, exist down through the hydrocyclone. Also, it is evident that the upper-body secondary flow structure evolves toward the more stable vortex structure of the lower hydrocyclone body. The asymmetry structure shown in Fig. 2 is predicted to be static in orientation and structure indicating a static air-core shape. Evidence for such has come from the radiography measurements. As shown in Fig. 3 the air-core orientations were constant over a range of feed pressure (1–4 105 Pa) for both 5 and 8 mm apex diameters. Therefore, the flow asymmetry is indicated to be a geometry controlled feature.
Fig. 3. Radiograph of the hydrocyclone and extracted air-core orientations. Air-core orientations correspond to operation with 5 and 8 mm spigot (underflow) diameters and with the inlet pressure set as 1.0, 1.5, 2.0, 2.5, 3.0, 3.5 and 4.0 · 105 Pa. The hydrocyclone was fed with kaolinite slurry at 6% v/v.
Notably, the fluid path lines do not exhibit the same helicity structure as for the asymmetry. In fact, as illustrated in Fig. 4 the ratio of axial to tangential velocity in the wall region remains of order )0.3 (in range )0.4 to )0.2) such that the frequency of flow revolution per asymmetry helical period increases down through the hydrocyclone. In contrast, a ratio of order of )0.4 would be required to match with the asymmetry helicity. As demonstrated from the research of Alekseenko et al. (1999), the axial velocity acts as an independent parameter to the vortex pitch and radius and hence the swirl number does not provide a unique characterisation of the flow field. Conservation of momentum results in variation of the axial to tangential velocity ratio, such that for the narrower cross-section
Fig. 2. Velocity contours of my > 0 and mx > 0 (blue/red) for the x ¼ 0 and y ¼ 0 planes, respectively, superimposed upon mz > 0 contours (green/red). (For interpretation of references in colour, the reader is referred to the web version of the article.)
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Fig. 5. Positive axial-velocity contours during hydrocyclone start-up. Fig. 4. Contours of ratio of axial velocity to tangential velocity. Thresholded from )0.3 (blue) to 0.5 (red). (For interpretation of references in colour, the reader is referred to the web version of the article.)
(core nearer to wall) the ratio is increased and vice versa and hence the range of )0.4 to )0.2 exists.
5. Air-core development and dynamics The structure of the swirl flow field also has a critical influence upon inception, development and stability of the air core. During start-up, a central upward axial flow initially develops just below the vortex finder tip, where flow reverses and exits through the overflow. As the swirl flow develops, this region of upward flow broadens and penetrates further towards the underflow. Upon penetrating two thirds of the body length, an approximately cylindrical region of upward axial velocity develops, which then continues to penetrate close to the underflow (Fig. 5). Within this cylindrical region, the central axial-velocity is downwards towards the underflow and therefore opposes air-core development. Whilst the present CFD predictions represent that for water only, Fig. 6 shows the subsequent inception of an upward core at the underflow. Here the lowpressure core supports inception whilst the axial velocity structure and upper, central pressure profile (Fig. 7) restricts development of the incepted bubble into a full core.
The present predictions challenge the common understanding that the air-core develops as a result of a below atmospheric core-pressure such that the air-core is pressure driven. Rather, as shown in Fig. 7 the predicted axial pressure profile is only below atmospheric immediately adjacent to the underflow and is significantly greater than zero away from the underflow. This would indicate that air-core development would certainly not be pressure driven and therefore this must be transport driven. Although a full transient development of the upward core was not predicted for this research, comparison between CFD predictions and high-speed video observation provides support for this conclusion. Fig. 8 shows capture of a suspended bubble at the underflow for the hydrocyclone operated at a reduced feed pressure. This underflow bubble’s axial-position was observed to be stable as it rotated about the axis. The present CFD prediction provides an appropriate mechanism to support this observation. With increased feed pressure, air-core development was observed and an air-core was established. Whilst at full operating pressure the air-core was stable, this was not so at an intermediate pressure. Rather the air-core was then observed to pulse as shown in Fig. 9. The air core followed a cycle, which from a fully established air core, began with the breakage of air-core continuity between that within the main body and at the underflow. This restriction and then breakage occurred in the region of predicted secondary axial-flow reversal adjacent to the underflow (Fig. 5). The air core then began to
Fig. 6. Quadratic pressure–strain prediction of positive axial-velocity contours during inception of core over period of 12 ms.
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thin, indicating either an above atmospheric pressure, convective transport or both (both were predicted). Continuity was then re-established and an upward pulse observed which replenished the air-core, indicating a key role for convective transport in this process. The observation of Nissan and Bresan (1961), which has been replicated by the present authors, is also noteworthy. They observed a suspended bubble of air within the hydrocyclone, albeit for a short period of time. From the present predictions this may be explained as corresponding to the region just above that of secondary axial-flow reversal, where the upward axial-velocity is low whilst the pressure gradient would drive the bubble downwards, such that equilibrium may occur.
6. Particle separation and solids distribution
Fig. 7. Pressure profile prediction along hydrocyclone axis.
Fig. 8. Quasi-stable underflow air bubble. The air bubble was observed to maintain an almost constant penetration for the reduced operating pressure.
The predicted helical vortex flow field will clearly have an impact upon the particle separation performance and solids distribution within the hydrocyclone. Important new understandings of the particle behaviour within the hydrocyclone are provided from Lagrangian particle-tracking upon the Eulerian continuous phase prediction. The classical understanding held that the separation mechanism is deterministic with a balance between the inward radial particle-drag and outward centrifugal force determining the radial particulate motion. Recent research (Averous and Fuentes, 1997) has concluded that the separation mechanism is governed by radial turbulence fluctuations which project particles from the wall flow to the core flow. For that research the particulates were determined to be completely thrown against the outer wall on entry to the hydrocyclone. The present research indicates that both the deterministic and turbulence aspects play key roles for the separation mechanism. An important parameter for the setup of Lagrangian particle tracking is the integral length scale. Too large an
Fig. 9. High-speed video capture of pulsing air core in upper conical section, just below vortex-finder tip.
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integral length results in all particles being predicted as thrown against the outer wall on entry. Hence the length was reduced until the statistical destination of particles proved independent of the length scale (1 · 105 m). The resultant track data was collected for their coincidence with two orthogonal planes (x-plane and y-plane). The tracks were computed for 80 release positions over the inlet with 10 releases for each position and for particles of radius ½0:25; 0:5; 1; 1:5; 2; 2:5; 3; 3:5; 4; 4:5; 5 lm, (qp ¼ 2800 kg m3 ). Analysis of the particle trajectories demonstrates the existence of high solids concentration regions and the importance of turbulence and asymmetry for the separation mechanism. 6.1. Solids concentration The volume fraction predictions presented below represent a feed of solids concentration 0.05 v/v constituted as a Rosin–Rammler distribution such that Ff ðrp Þ ¼ 1 expð loge ð2Þðrp =4Þ2 Þ where the particle radius is in microns. As a result, the fractions of feed total solids represented by each size fraction (½0:25; 0:5; 1; 1:5; 2; 2:5; 3; 3:5; 4; 4:5; 5 lm) are ½4:4 106 ; 1:0 104 ; ½1:9; 8:8 103 ; ½2:5; 5:3; 9:2 102 ; ½1:4; 1:9; 2:3; 2:6 101 . Fig. 10 shows predicted volume fraction contours for the rp ¼ 0:25 lm and rp ¼ 2:5 lm classes of particles (inlet volume fractions of 2.2 · 107 and 2.6 · 103 , respectively). It is evident from this figure that regions of high concentration exist both at the underflow and within the upper body flow. In fact, the region of high concentration within the main body, appears to correspond with the marked region of recirculation in Fig. 2. This region of recirculation is predicted to become less concentrated and less significant as the particle size and hence probability of being away from the outer wall is decreased. An overall volume fraction distribution is shown in Fig. 11 where all the particle size contributions have been included. It is notable that the larger particle
Fig. 11. Volume fraction contours for complete particle size distribution, thresholded at 0.2. Axis are in meters.
sizes dominate the distribution generating a high wall and underflow concentration. For an inlet concentration of 0.05 v/v, a maximum solids volume fraction of 0.36 was predicted with the bulk at less than 0.2. Hence the present predictions are likely, only valid for inlet concentrations of the order 0.01 v/v or less. The issue of high-solids multiphase hydrocyclone modelling remains open for future research. The present predictions also provide evidence of a concentrated particle path along the hydrocyclone wall, as indicated in Fig. 10, which would naturally lead to scouring of a helical track into the hydrocyclone wall, as observed in practise. The high-solids region once established, remains throughout the hydrocyclone. 6.2. Particle separation mechanism The Lagrangian particle tracking also indicates a new understanding of the relative importance of deterministic and turbulent radial particulate transport between the wall and core flows (downward and upward regions,
Fig. 10. Volume fraction contours for rp ¼ 0:25 lm and 2.5 lm particles thresholded at 6 · 107 and 1 · 102 respectively. Axis are in meters.
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respectively). In particular, the present predictions indicate such radial transport to form regions within the hydrocyclone of preferential radial direction although these are attenuated by the turbulence component. For examination of the influence of turbulence effects the particle tracks have been reconstructed over a quarter revolution of the hydrocyclone (negative y-plane to positive x-plane). The deterministic particle motions between the two planes were reconstructed assuming linear velocity profiles to provide the stochastic component, represented as the difference between the actual (effective) distance and the deterministic reconstruction. Considering rp ¼ 0:25 lm and rp ¼ 2 lm cases, the predictions are briefly examined below in terms of radial transport. Three variables are considered, the deterministic (reconstructed) radial distance travelled as a result of the mean flow, drd , the effective (Lagrangian predicted) distance travelled, dre and the difference between these, Ddr (Ddr ¼ drd dre ). The ratio of the standard deviation of Ddr over drd is shown in Fig. 12. It is clear from this figure that the standard deviation is very significant for the wall and head regions of the hydrocyclone, including around the vortex finder tip. In these regions, turbulence transport even dominates convective transport. In fact, the ratio is only below 0.1 immediately adjacent to the vortex core and therefore
turbulence transport remains significant throughout the particulate separation region. It is of interest to note the component of turbulence transport as a function of the deterministic radial transport direction. Fig. 13 the effective distances as well as the sign multiple of the deterministic and difference distances for the rp ¼ 0:25 lm particles is shown. It is evident from this figure that a preferential structure of radial transport between the wall and core regions exists as a result of the flow asymmetry. The precise nature of the relationship between preferential transport and flow asymmetry remains to be determined, although an analysis similar to that of Alekseenko et al. (1999) could be considered. It is also evident from this figure that Ddr and dre are of equal sign along the wall region, which indicates that in a mean sense the turbulence contribution is negatively correlated with the radial convection direction, i.e. the turbulence damps the radial transport asymmetry. The precise influence of the turbulence for the core region is ignored for the present discussion since reconstruction of the tangential and radial velocity profiles has been based upon an axially centred system, which will lead to the positive correlation shown in Fig. 13. It is intended to make a more comprehensive analysis centred upon the vortex axis in the future.
Fig. 12. Contours of the standard deviation of the difference over the deterministic distance travelled for rp ¼ 0:25 lm and 2.5 lm particles, both thresholded at 2.0.
Fig. 13. Contours of the effective distance (thresholded at ±1 mm) and of the sign multiple of the effective and difference distances travelled for the rp ¼ 0:25 lm particles.
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7. Conclusions and future analysis The CFD and experimental research here presented for the hydrocyclone has challenged a number of key understandings of the hydrocyclone flow field and of the particle separation mechanism within. In particular, the present predictions indicate a clear and significant asymmetry throughout the hydrocyclone resultant from the single tangential inlet and wall bounded streamline curvature. Streamline expansion over the vortex-finder tip combined with existing secondary flow structure, established a globally static asymmetry throughout the hydrocyclone. Hence the air-core is predicted to exhibit a static asymmetry without precession, as was observed through high-speed video and radiography. The existence of such asymmetry throughout the hydrocyclone is predicted to play a key role in determining the particle separation mechanism, forming a structure of alternating radial particle-transport direction down through the hydrocyclone. The influence of turbulence upon this deterministic flow structure is indicated to be most significant in the wall, head and vortex finder regions, where the mean effect of turbulence is to dampen the asymmetry of the radial transport structure. The predicted pressure and velocity profiles have provided support for high-speed video observations of both a suspended bubble at the underflow and also of instability of the aircore at moderate operating feed pressures. As identified in the text, the present analysis for the balance of deterministic and turbulence transport effects is in error due to the use of a geometrically centred reference frame. This has been detrimental to the accuracy of reconstructed particle-motions near to the vortex core. A future analysis is intended, based upon a vortex centred coordinate frame, with an increased number of particle releases and an increased number of data collection planes, in order to reduce prediction uncertainty and the effects of the linear assumption. It is also necessary to gather experimental classification data for particulates with well characterised settling properties, such that a direct comparison may be made between the predicted and measured classification. Support of the presented flow phenomena and mechanisms within the hydrocyclone may be sought through direct flow field measurements, using laser induced fluorescence, for example. Further development of tomographic techniques such as electrical impedance and ultrasound is being pursued in order to provide non-intrusive tools for validation of the future high-solids predictions. Future analysis and measurement will likely include the predicted classification as well as the predicted residence time distribution (Stegowski and Leclerc, 2002), since this is expected to be particularly sensitive to the newly proposed separation mechanism as opposed to the classical understanding. In particular, the residence time is expected to be, in general greater, although with
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possibly a less significant peak at the cut size due to mixing effects for a broader range of particle sizes.
Acknowledgements The authors are grateful for the financial support provided by the University of Exeter Research Fund, University of Leeds, EPSRC (GR/L95472, GR/R22100/ 01) and by Rio Tinto Technology. The kind support of Dr M Slack of Fluent Europe Ltd. is also acknowledged.
Appendix A
Research hydrocyclone dimensions (not to scale).
References Alekseenko, S.V., Kuibin, P.A., Okulov, V.L., Shtork, S.I., 1999. Helical vortices in swirl flow. Journal of Fluid Mechanics 382, 195– 243. Averous, J., Fuentes, R., 1997. Advances in the numerical simulation of hydrocyclone classification. Canadian Metallurgical Quarterly 36 (5), 309–314.
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Cullivan, J.C., Williams, R.A., Cross, C.R., 2001. Verification of theoretical 3D-flow in a hydrocyclone using tomography. In: Fourth World Congress for Particle Technology, Sydney, pp. 1–9. Cullivan, J.C., Williams, R.A., Cross, C.R., 2003. Understanding the hydrocyclone separator through computational fluid dynamics. Transactions of the IChemE, Part A, Chemical Engineering, Research and Design 81 (A4), 455–466. He, P., Salcudean, M., Gartshore, I.S., 1999. A numerical simulation of hydrocyclones. Transactions of the IChemE, Part A, Chemical Engineering, Research and Design 77 (A5), 429–441. Hinze, J.O., 1975. Turbulence, 2nd ed. McGraw-Hill, New York. Hsieh, K.T., Rajamani, R.K., 1988. Model development and verification for single-phase flow. International Journal of Mineral Processing 22, 223–237. Leonard, B.P., 1979. A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Computer Methods in Applied Mechanics and Engineering 19, 58–98. Lu, Y.-J., Zhou, L.-X., Shen, X., 1999. Numerical simulation of strongly swirling turbulent flows in a liquid–liquid hydrocyclone using different turbulent models. In: Multiphase Flow and Heat Transfer, Proceedings of the Fourth International Symposium, pp. 654–663.
Ma, L., Ingham, D.B., Wen, X., 2000. Numerical modelling of the fluid and particle penetration through small sampling cyclones. Journal of Aerosol Science 31 (9), 1097–1119. Nissan, A.H., Bresan, V.P., 1961. Swirling flow in cylinders. American Institute of Chemical Engineers Journal 7 (4), 543–547. Sevilla, E.M., Branion, R.M.R., 1997. The fluid dynamics of hydrocyclones. Journal of Pulp and Paper Science 23 (2), 85–93. Shyy, W., 1994. Computational modeling for fluid flow and interfacial transport. Elsevier, Amsterdam, New York. Slack, M.D., Prasad, R.O., Bakker, A., Boysan, F., 2000. Advances in cyclone modelling using unstructured grids. Transactions of the Institute of Chemical Engineers 78 (A), 1098–1104. Slack, M.D., Wraith, A.E., 1997. Modelling the velocity distribution in a hydrocyclone. In: 4th International Colloquium on Process Simulation, pp. 65–83. Speziale, C.G., Sarkar, S., Gatski, T.B., 1991. Modelling the pressurestrain correlation of turbulence: an invariant dynamical systems approach. Journal of Fluid Mechanics 227, 245–272. Stegowski, Z., Leclerc, J.-P., 2002. Determination of the solid separation and residence time distributions in an industrial hydrocyclone using radioisotope tracer experiments. International Journal of Mineral Processing 66, 67–77.