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Minerals Engineering 20 (2007) 60–71

CFD simulation and experimental validation studies on hydrocyclone

K. Udaya Bhaskar a,*, Y. Rama Murthy a, M. Ravi Raju a, Sumit Tiwari a,J.K. Srivastava b, N. Ramakrishnan a

a Regional Research Laboratory, Hoshangabad Road, Bhopal 462 026, Indiab Ujjain Engineering College, Ujjain 456 001, India

Received 17 January 2006; accepted 25 April 2006Available online 19 June 2006

Abstract

Hydrocyclone is a key unit operation in mineral process industry and simulation of which using CFD techniques is gaining popularityin process design and optimization. The success of the simulation methodology depends primarily on how best the results are matchingwith the experimental values and the computational time it requires for obtaining such results. In the present investigation, attempts aremade to develop a methodology for simulating the performance of hydrocyclone. Initial work included comparison of experimental andsimulated results generated using different turbulence models i.e., standard k–e, k–e RNG and RSM in terms of water throughput andsplit with the help of suitably designed experiments. Among the three modeling methods, predictions using RSM model were found bet-ter in agreement with experimental results with a marginal error between 4% and 8%. Parametric studies have indicated that a decrease inthe spigot opening increased the upward vertical velocity of water more compared to a decrease in the downward vertical velocity. Anincrease in the inlet pressure has increased the axial velocities of water in both the upward and downward directions and increased themass flow rates through the cyclone. An increase in the inlet pressure has also increased the static pressure differential along the radiuswithin the cyclone body and hence more water split into overflow. Further, an increase in the inlet pressure has also increased the tan-gential velocities and reduced the cyclone cut size. The simulated particle distribution values generated using the particle injection tech-nique are found matching with the experimental results while achieving cut sizes between 4.9 and 14.0 lm.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Computational fluid dynamics; Modelling; Hydrocyclones; Partition numbers; Cyclone cut-size

1. Introduction

Hydrocyclones have been widely accepted in the area ofmineral processing due to several advantages, such as easeof operation, high throughput, less maintenance, less floorspace requirement etc. Despite its invention in late 18thcentury, thorough works related to understanding the prin-ciples began only in mid fifties (Kelsall, 1952). Kelsall stud-ies on the axial, radial and tangential velocity profilesformed the basis for subsequent research on hydrocy-clones. The complex phenomenon involved coupled withnon-availability of high-speed computational systems

0892-6875/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mineng.2006.04.012

* Corresponding author. Tel.: +91 755 2471956; fax: +91 755 2488323.E-mail address: kubhaskar2001@yahoo.com (K.U. Bhaskar).

restricted most of the research works focused on the empir-ical modeling till recent times (Lynch and Rao, 1975; Plitt,1976). However, the advent of high speed computationalsystems, in last couple of decades made researchers focusperformance simulations using Computational FluidDynamics (CFD) techniques (Pericleous and Rhodes,1986; Hsieh and Rajamani, 1988; Monredon et al., 1992;Rajamani and Milin, 1992; Dyakowski and Williams,1993; Dyakowski et al., 1994; Malhotra et al., 1994; Harg-reaves and Silvester, 1990; Devulapalli and Rajamani,1996; Griffths and Boysan, 1996; Slack and Wraith, 1997;Slack and Boysan, 1998; Stovin and Saul, 1998; Dyakowskiet al., 1999; Suasnabar and Fletcher, 1999; Slack et al.,2000; Ma et al., 2000; Nowakowski et al., 2000; Gradyet al., 2002; Slack et al., 2003; Nowakowski and Dyakow-ski, 2003; Grady et al., 2003; Schuetz et al., 2004; Cullivan

Nomenclature

~g gravitational accelerationt timeu, m velocity magnitude~m overall velocity vectorq density~s stress tensorl molecular viscosity of the fluidlt turbulent eddy viscosityj turbulent kinetic energye dissipation rate of the turbulent kinetic energyGk generation of turbulent kinetic energy due to the

mean velocity gradientGb generation of turbulent kinetic energy due to

buoyancyC1e, C2e and C3e constantsrk and re the turbulent Prandtl numbers for k and e

YM contribution of the fluctuating dilatation incompressible turbulence to the overall dissipa-tion rate

ak and ae the inverse effective Prandtl numbers for j ande

Sk and Se user-defined source termsCD drag coefficientqp density of the particleql density of the liquidDp diameter of the particlegx acceleration due to gravityFx additional force like buoyancy, etc. acting on

particleFD(up � u) drag forceX angular velocity/ equivalence ratio

Feed InletC

ylin

der

Vortex Finder

Over Flow

rust

um

K.U. Bhaskar et al. / Minerals Engineering 20 (2007) 60–71 61

et al., 2003, 2004; Nowakowski et al., 2004). A numericaltechnique needs development of suitable methodologyand thorough validation with the actual data prior toapplications in performance simulation and design.Though most of these studies have validated with waterflow characteristics, only few reported simulated resultsof solids separation behavior. The present study is meantfor establishing suitable methodology including turbulencemodel selection for simulating both water and solid distri-bution data on different geometries of cyclones. Commer-cially available CFD software ‘Fluent 6.1.22’ was used.The simulation results carried out on a 76 mm diameterhydrocyclone are validated with experimental data gener-ated in the laboratory in terms of cyclone throughout;water split and solids cut size.

Under Flow

F

Spigot

Fig. 1. Hydrocyclone geometry.

2. Model description

2.1. Geometry

The hydrocyclone geometry used for simulation and forexperimental studies is presented in Fig. 1. The cylindricalbody is 76 mm diameter and 85 mm in length with a closedend at the top surface and bottom face open. A frustumwith a larger diameter of 76 mm and smaller diameter of10 mm maintained at a cone angle of 10� is connected tothe main cylindrical body with the face having larger diam-eter. A cylindrical vortex finder with an inner diameter of25 mm and outer diameter of 40 mm protrudes into themain cylindrical body extending over a length of 60 mminside and 30 mm above the top closed surface. A rectangu-lar 20 · 10 mm tangential feed inlet opening is connected tothe cylindrical surface at a height of 15 mm below the topsurface. Studies were carried out on this geometry bychanging the bottom diameter of the frustum at 10, 15,20 and 25 mm at a constant cone angle of 10�.

2.2. Meshing scheme

Hydrocyclones truly cannot be modeled in a 2-D planedue to non-axisymmetric nature at the feed inlet opening.Earlier reports also indicate that the results using a 3-Dmodel are better matching with the experimental data com-pared to the results with axisymmetric geometry[www.psl.bc.ca/downloads/presentations/cyclone/cyclone.html]. Thus 3-D computational model was selected forthe study. Five different mesh densities between 50,000and 200,000 computational elements were attempted to

62 K.U. Bhaskar et al. / Minerals Engineering 20 (2007) 60–71

optimize the mesh density for a reasonable solution. Amesh density of 150,000 computational cells was opti-mized. Enormous computation time had become a con-straint though the results at higher mesh densities werefound slightly better. An unstructured grid based on tetra-hedral cells with T-grid meshing scheme pattern was usedwhile discretizing the geometry.

2.3. Boundary and initial conditions

The rectangular cyclone feed inlet face was defined aspressure inlet. The overflow and underflow outlets faceswere designated as pressure outlets. Radial pressure distri-bution from the cyclone axis to the edges is opted at boththe pressure outlets. Backflow direction was specified asnormal to the boundary zones and backflow turbulenceintensity was assigned a value of 10%. The primary waterphase with a density of 998.2 kg/m3and a viscosity valueof 1.003 · 10�6 kg/m s was allowed to enter the cycloneat inlet pressure values of 55 kPa and 83 kPa. The outletat the underflow was varied at 10 mm, 15 mm, 20 mmand 25 mm at each test run and water flow behavior wassimulated. Inert solid particles with density of 2650 kg/m3

and of different sizes varying between 1 and 25 lm wereinjected from the feed inlet boundary zone along the sur-face. The particles entering any of the pressure outlet zoneswere assigned to escape the vessel.

3. Simulation

The simulations carried out on hydrocyclone wereassumed to be operating without air core. Cartesian coor-dinate system was used for numerical simulations. Flowsimulation was carried out using a 3-D double precision,steady state, and segregated solver. In this method, the gov-erning Navier Stokes equations (Annexure) are solvedsequentially using iterative methods till the defined valuesof convergence are met. Initially, the properties of thewater were used along with the pressure and face massfluxes for calculating the momentum equations and furtherupdate the velocity field. PRESTO (Pressure staggeredoption), a pressure interpolation scheme which wasreported to be useful for predicting highly swirling flowcharacteristics prevailing inside the cyclone body (FluentEurope Ltd., 2002) was adopted. For turbulence calcula-tions k–e, k–e RNG, Reynolds stress model (RSM) (modelspresented in annexure) were independently used to evaluatethe comparative simulation results. To obtain the pressurefield inside the system, SIMPLE (Semi-Implicit PressureLinked Equations) algorithm scheme, which uses a combi-nation of continuity and momentum equations to derive anequation for pressure was used. Interpolation of field vari-ables from cell centers to faces of the control volumes wasopted with higher-order Quadratic Upwind Interpolation(QUICK) spatial discretisation scheme as it was reportedto be useful for swirling flows (Fluent Europe Ltd.,2002). Simulations were carried out for about 10,000 incre-

mental steps where in general a preset value of convergencecriteria 1 · 10�6 was achieved.

4. Experimental

The experimental setup consisted a slurry tank of 200liters capacity mounted on a stable platform. A centrifugalpump with 3-phase, 5.5 kW motor was connected to theslurry tank at the bottom. Feed slurry consisting of claymaterial at 10% solids density was pumped into the cyclonethrough the pipeline connected to the pump. The other endof the pipeline was connected to the inlet opening of a76 mm diameter hydrocyclone. The pressure drop insidethe cyclone was maintained at required level with the helpof by-pass arrangement actuated through a control valveon the pipeline. The pressure drop in the cyclone was mea-sured with the help of a diaphragm type pressure gauge fit-ted near the feed inlet. The hydrocyclone was positionedupright above the slurry tank.

The experimental program was designed to achieve awide range of water splits into the overflow and underflowproducts suitably selecting the spigot opening and feedinlet pressures. Hydrocyclone main body was fixed to thetest-rig. Required opening spigot as per the experimentaldesign was fitted to the hydrocyclone bottom. Initially, dis-tribution studies were carried out by pumping water intothe cyclone at different spigot openings and feed pressures.Later, solids consistency of 10% by weight was maintainedin the slurry tank by mixing measured amount of clay andwater. Tests were carried out on the solid slurry by pump-ing into the cyclone body at desired feed inlet pressure andat different spigot openings. Samples of overflow andunderflow products were collected simultaneously for aspecific time in different containers. The underflow andoverflow products collected were filtered, dried andweighed. Particle size distribution of representative samplesof the dried products was analyzed using in Malvern laserparticle size analyzer. Distribution points based on reportof each size fraction in the feed to the underflow productwere generated.

5. Results and discussion

The simulated results of cyclone throughput (mass flowrate through overflow + mass flow rate through under-flow), and on the water-split (percent report of total waterentering the cyclone) into overflow product were used forvalidating the predictions at different test runs with theexperimental values. The simulation studies were carriedout using standard k–e, k–e RNG and RSM turbulencemodels, at three spigot openings, i.e. 10 mm, 15 mm,20 mm and 25 mm. All these simulations were carried outat a constant feed inlet pressure of 83 kPa. Initial discus-sions cover comparison of results obtained using the threeturbulence models specifically validating with the experi-mental data. Having identified the superior predictionswith RSM model, k–e and k–e (RNG), further simulation

Turbulence models

30

40

50

60

70

80

90

100

5 10 15 20 25 30Spigot opening (mm)

Rep

ort

to o

verf

low

(%

)

Experimental

RSMRNGK-e

Fig. 3. Experimental and simulated values of water split using differentturbulence models.

K.U. Bhaskar et al. / Minerals Engineering 20 (2007) 60–71 63

works were carried out using RSM turbulence model. Sim-ulated general flow patterns in the cyclone in terms ofvelocity and static pressures are discussed. Further, the dis-cussions cover comparison of simulated and experimentalsolid particles distribution curves. Finally the effects ofvariables on axial and tangential velocities at 55 kPa feedinlet pressure and on static pressures are discussed whilecorroborating the experimental results.

5.1. Validation results

The results obtained on the water throughput and watersplit under similar experimental and simulated conditionsusing different turbulence models is compared. The resultsare discussed in the following sections.

5.1.1. Water throughput

The results on the water throughput obtained during theexperimental work at different spigot openings are com-pared with the simulated values obtained using differentturbulence models in Fig. 2. The figure indicates that allthe three models predicted an increase in throughput withincrease in the spigot, which can also be observed fromthe experimental results. The results obtained with theRSM model are closer to the experimental results followedby the results obtained using k–e RNG turbulence model inturn followed by the results using standard k–e model. Thesimulated results match with the experimental results withmarginal error about 15–20% when k–e model was used10–15% with k–e RNG and 4–8% when RSM turbulencemodel was used.

5.1.2. Water split

Water split in hydrocyclone is the percentage of thewater reporting to over flow. The experimental and simu-lated results of water split using different turbulence modelsare presented in Fig. 3. Among the methods of simulations,the method using RSM model has better predictions over

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

5 10 15 20 25 30

Spigot opening (mm)

Wat

er t

hrou

ghpu

t (k

g/s)

Experimental

RSMRNGK-e

Fig. 2. Experimental and simulated data on water throughput usingdifferent turbulence models.

the entire range of spigot openings. It can also be observedfrom the figure that among the results obtained with RSMmodel, the prediction at widest spigot opening (25 mm) isslightly higher than the experimental values where as atnarrowest spigot opening the simulation prediction is lowerthan the experimental results. However, at intermediatespigot openings the data points indicated good match withthe experimental data. The dotted line enclosed the region,which corresponds to water split range of 45–87% in theoverflow product. In this region the simulated data agreeswell with the experimental results with a reasonable accu-racy. The general range of water splits in a normal operat-ing hydrocyclone will be within this region of goodprediction limits.

Having observed the better simulation results generatedusing RSM turbulence model for predicting the hydrocy-clone performance, the general flow patterns in terms ofvelocities and static pressures at different positions in verti-cal and horizontal planes, the simulation results obtainedusing this model are discussed.

5.2. General flow behavior

The general flow behavior in terms of velocity and pres-sure distributions at different planes inside the system arediscussed with a comparison of the published flow profilesof hydrocyclones (Williams et al., 1995; Dyakowski et al.,1999; Grady et al., 2003; Slack et al., 2003; Nowakowskiand Dyakowski, 2003; Nowakowski et al., 2004; Cullivanet al., 2003, 2004).

5.2.1. Axial velocityThe vertical velocity contours obtained in radial planes

at different axial heights are presented in Fig. 4A and inone axial plane corresponding to central radial positionare presented in Fig. 4B. It can be observed from the fig-ures that two kinds of vertical flows one traveling upwardindicated by values of positive axial velocity and the other

64 K.U. Bhaskar et al. / Minerals Engineering 20 (2007) 60–71

traveling downward indicated by negative axial velocityexist. It can be observed from Fig. 4A that concentric lay-ers of constant axial velocities exist at all the heights in thecyclone body. Further, it can be observed from Fig. 4B thatwith increasing radial distance from the cyclone axis, thepositive values of axial velocity decreases till it reaches zeroat some radial distance away from the cyclone axis. Nega-tive axial velocity begins beyond the radial distance of zerovertical velocity, and increases with increase in radial dis-tance. However, as the radial distance approaches cyclonewall, the values of negative vertical velocity again decrease.This can be due to higher friction between layers of water

Fig. 4A. Contours of axial velocity of water (m/s) in axial plane.

Fig. 4B. Contours of axial velocity of water (m/s) in radial planes.

body and cyclone wall than friction between individualwater layers, which are away from the wall.

The axial velocity profiles at different radial distancesand at different axial heights in the cyclone body are inte-grated and presented in Fig. 5. It can be observed fromthe figure that a peak positive axial velocity of 2.32 m/s isimmediately below the bottom of the vortex finder and aminimum value of positive axial velocity 0.25 m/s is at avertical distance of 275 mm from the top of the hydrocy-clone. The figure also indicates that at an axial height of325 mm, there is no positive vertical velocity at all theradial distances indicating no further classification of waterin this region. Further the figure indicates a negative axialvelocity about 0.23 m/s in magnitude near the walls ofcyclone body at an axial distance of 25 mm from the top.At a vertical distance of 325 mm from the top, a maximumvalue of �0.75 m/s can be observed.

The observations on the simulated results indicate thatwater split takes places to a maximum extent early in thecylindrical section below the vortex finder opening. At pro-gressively lower heights, smaller splits are allowed into theoverflow. Near the spigot opening the water split will beinsignificant.

5.2.2. Tangential velocity

The centrifugal force field necessary for classificationinside the system will be generated by the tangential-veloc-ity component of the continuous phase. The simulated

-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.5

-40 -30 -20 -10 0 10 20 30 40

Each division = 2.75m/s

Cyclone height in mm

Base line

25 mm

50 mm

75 mm

125 mm

175 mm

225 mm

275 mm

325 mm

Radial distance in mm

15 mm

-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.5

-40 -30 -20 -10 0 10 20 30 40

-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.5

-40 -30 -20 -10 0 10 20 30 40

Each division = 2.75m/s

Cyclone height in mm

Base line

25 mm

50 mm

75 mm

125 mm

175 mm

225 mm

275 mm

325 mm

Radial distance in mm

15 mm

Fig. 5. Axial velocity distribution of water at different vertical positionsinside the cyclone.

-40 -30 -20 -10 0 10 20 30 40

Each division = 4.35m/s

0.01.02.03.04.05.06.07.08.0

Cyclone height in mm

Base line

25 mm

50 mm

75 mm

125 mm

175 mm

225 mm

275 mm

325 mm

15 mm

Radial distance in mm

-40 -30 -20 -10 0 10 20 30 40

Each division = 4.35m/s

0.01.02.03.04.05.06.07.08.0

-40 -30 -20 -10 0 10 20 30 40

Each division = 4.35m/s

0.01.02.03.04.05.06.07.08.0

Cyclone height in mm

Base line

25 mm

50 mm

75 mm

125 mm

175 mm

225 mm

275 mm

325 mm

15 mm

Radial distance in mm

Fig. 6. Tangential velocity distribution of water at different verticalpositions inside the cyclone.

Table 1Design details of hydrocyclone used for simulation

Dimensions (mm) Cyclone 1 Cyclone 2 Cyclone 3 Cyclone 4

CD 76 76 76 76CyL 85 85 85 85VFD 25 25 25 25VFL 90 90 90 90FI (l · w) 20 · 10 20 · 10 20 · 10 20 · 10CA* 10� 10� 10� 10�SPD 10 15 20 25

CD: cyclone diameter; CyL: cylindrical length; VFD: vortex finderdiameter; VFL: vortex finder length; FI: feed inlet dimensions(length · width); CA*: cone angle in degrees; SPD: Spigot diameter.

0

25

50

75

100

0 5 10 15 20 25Particle size (microns)

Per

cent

age

repo

rt in

UF

Sim. 20mm spi.

Exp. 20mm spi.

Sim. 15mm spi.

Exp. 15mm spi.

Sim. 10mm spi.

Exp. 10mm spi.

Fig. 7. Experimental and simulated partition numbers at different spigotopenings and at 55 K Pascal feed pressure.

0

25

50

75

100

0 5 10 15 20 25Particle size (microns)

Per

cent

age

repo

rt in

UF

Sim. 20mm spi.

Exp. 20mm spi

Sim.15mm spi.

Exp. 15mm spi

Sim. 10mm spi.

Exp. 10mm spi

Fig. 8. Experimental and simulated partition numbers at different spigotopenings and at 83 K Pascal feed pressure.

K.U. Bhaskar et al. / Minerals Engineering 20 (2007) 60–71 65

results of tangential velocity at different vertical heights areplotted in Fig. 6. It can be observed from the figure that ini-tially with increasing radial distance from the axis the tan-gential velocity increases. The tangential velocity valuesafter achieving a maximum value at some radial positiondecrease with a further increase in the radius approachingtowards the walls. The observations made are similar tothe reports of earlier workers in the literature (Slack andWraith, 1997). The profiles of tangential velocity remainsimilar at different axial heights in the cyclone body. Max-imum values of tangential velocities are observed in thecylindrical portion. It can also be noted that the maximumvalues of tangential velocity decrease with decrease in axialheight. For example, at an axial height of 125 mm, themaximum tangential velocity observed is 5.70 m/s and atan axial height of 275 mm the maximum value is 4.99 m/s. The observation infers that at lower cyclone heightstowards the spigot opening relatively lower centrifugalfields of force are generated. This phenomenon in hydrocy-clone re-orients the entrapped fine size fractions for pro-gressive report into the vertical axial flow while treatingsolid materials (Cullivan et al., 2004).

5.3. Particle separation behavior

The performance of hydrocyclone is represented interms of distribution or partition curves (Wills, 1997).

The partition curve represents the report of any size mate-rial in the feed to the underflow product (this number isalso referred as distribution point or partition number).

Particle distribution behaviour inside the cyclone is sim-ulated using Lagrangian particle tracking approach uponthe Eulerian continuous phase predictions. Particle-track-ing approach, which is reported to be useful technique

Table 2Experimental and simulated cyclone cut size values at different test runs

Testrun

FP(kPa)

SPD(mm)

Throughput(kg/s)

Splitof (kg/s)

Cut size d50

(lm)

Exp. Sim.

1 55 10 0.932 94.8 11.4 14.02 55 15 0.947 79.2 10.8 12.03 55 20 1.0333 55.9 5.8 6.34 83 10 1.161 95.7 9.5 10.05 83 15 1.174 81.5 8.3 8.96 83 20 1.306 58.7 4.5 4.9

FP: feed pressure; SPD: Spigot diameter; Exp.: experimental; Sim.:simulated.

66 K.U. Bhaskar et al. / Minerals Engineering 20 (2007) 60–71

for simulation purposes when particle concentration isbelow 10% by weight (Stovin and Saul, 1998), is used forachieving the distribution behavior of different sizes. Parti-cles are injected through the feed inlet into the cyclonebody after achieving the convergence. Sample group of1000 particles of a defined sizes within a selected range wereinjected into the cyclone body along the face of the inletopening. The density of the material is maintained constantat 2650 kg/m3which corresponds to the density of claymaterial used in the experimental studies. Each time 10

Axial Height 175mm

-

-1

-0.5

0

0.5

1

1.5

2

2.5

-40 -30 -20 -10 0 10 20 30 40

12psi, 10spd12psi, 15spd12psi, 20spd

-

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-40 -30 -20 -10 0 10 20 30 40

12psi, 10spd12psi, 15spd12psi, 20spd

Axial Height 75mm

Fig. 9. Effect of spigot openings on the axial velocity distribution

sample runs were carried out and the report of particlesinto the overflow and underflow outlet streams were aver-aged. The partition numbers were obtained for each sizeand geometry in consideration (Table 1).

The partition curves generated at 10 mm, 15 mm, and20 mm spigot openings at a feed pressure of 55 K Pascalusing different partition numbers, are presented in Fig. 7.Also presented in the figure are the experimental valuesobtained under similar conditions. It can be observed thatthe simulated data points are reasonably matching with theexperimental values at all the test conditions. Similar obser-vations can also be made at other set of test conditions car-ried out at 83 K Pascal (Fig. 8). Further, the cut size of thecyclone (d50), (the size of the particle having a partitionvalue of 50%) of the simulated and experimental conditionsare presented in Table 2. The data indicates that the pre-dicted values are closely matching with the experimentalresults.

5.4. Parametric studies

The effect of important parameters like spigot openingand feed inlet pressure on the simulated results of axial

Axial Height 125mm

Axial Height 225mm

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

40 -30 -20 -10 0 10 20 30 40

12psi, 10spd12psi, 15spd12psi, 20spd

-1

-0.5

0

0.5

1

1.5

2

2.5

40 -30 -20 -10 0 10 20 30 40

12psi, 10spd12psi, 15spd12psi, 20spd

of water at different radial points (feed pressure 83 K Pascal).

K.U. Bhaskar et al. / Minerals Engineering 20 (2007) 60–71 67

and tangential velocities, static pressures are discussed asfollows.

5.4.1. Effect of spigot opening

The effect of change in spigot opening from 10 to20 mm on the axial velocities at 83 kPa feed inlet feedpressure, is presented in Fig. 9. It can be observed fromthe figure that the maximum values of positive axial veloc-ities indicating an upward vertical flow around the cycloneaxis is higher at lower spigot opening at all the axial posi-tions. On the other hand, the negative axial velocities indi-cating a downward flow near the cyclone walls are smallerat smaller spigot opening. It may also be observed that themagnitude of increase in upward vertical velocity is morecompared to the magnitude of decrease in downwardvelocity. For instance, at 125 mm axial height, a decreasein the spigot opening from 20 mm to 10 mm has increasedthe positive axial velocity by 0.47 m/s from a value of1.37 m/s to 1.84 m/s. A similar decrease in the spigotopening has decreased the negative axial velocity by

-1

-0.5

0

0.5

1

1.5

2

2.5

3

-40 -30 -20 -10 0 10 20 30 40

8 psi, 10spd

10psi, 10spd

12psi, 10spd

-1

-0.5

0

0.5

1

1.5

2

2.5

3

-40 -30 -20 -10 0 10 20 30 40

8psi, 10spd10psi, 10spd

12psi, 10spd

Axial Height 75mm

Axial Height 175mm

Fig. 10. Effect of feed inlet pressure on the axial velocity distribut

0.27 m/s from �0.63 m/s to �0.90 m/s. Similar observa-tion can be made at all the axial heights. Due to increasein constriction for water flow at smaller spigot opening,the kinetic energy within the water body is dissipated forincrease in velocities in the axially upward direction andfurther increase the water split to overflow. This can bewell observed with the experimental values presented inFig. 4. This increase in mass flow through the vortex finderincreases the chances of relatively coarser particles reportinto overflow, which is well validated, by an increase inthe cut size of the cyclone at lower spigot opening (Table2).

5.4.2. Effect of feed inlet pressureThe simulated data on the axial velocities at different

radial distances and axial heights along the central cycloneplane (results obtained at 10 spigot opening) are presentedin Fig. 10. It can be observed from the figure that anincrease in feed inlet pressure from 55 kPa to 83 kPaincreases the rising axial velocity near the central region

-1

-0.5

0

0.5

1

1.5

2

2.5

3

-40 -30 -20 -10 0 10 20 30 40

8psi, 10spd

10psi, 10spd12psi, 10spd

-1

-0.5

0

0.5

1

1.5

2

2.5

3

-40 -30 -20 -10 0 10 20 30 40

8psi, 10spd10psi, 10spd

12psi, 10spd

Axial Height 125mm

Axial Height 225mm

ion of water at different radial points (10 mm spigot opening).

68 K.U. Bhaskar et al. / Minerals Engineering 20 (2007) 60–71

around the axis and also increases the maximum values ofnegative axial velocity at radial distances near the walls. Anincrease in upward and downward axial flow rates athigher feed inlet pressures increases the total mass flow ofwater through the cyclone. This is in agreement with theexperimental results (Table 2).

Further, it can be observed that the increase in the mag-nitude of upward vertical velocity is more compared to theincrease in downward velocity magnitude. For instance, at175 mm axial height, an increase in the feed pressure from55 kPa to 83 kPa increased the positive axial velocity by0.22 m/s from a value of 1.03 m/s to 1.25 m/s. A similarincrease in the feed inlet pressure has increased the negativeaxial velocity only by 0.08 m/s from �0.53 m/s to �0.61 m/s. The observation indicates an increased flow in the verti-cal direction and increased split to overflow. This observa-tion is found similar to the experimental split valuespresented in Table 2.

The effect of feed inlet pressure on the static pressurevalues along the radial distances at different axial heightsare presented in Fig. 11. It can be observed from the figurethat an increase in the feed inlet pressure increases the sta-tic pressure values dominantly near the walls of the cyclone

-10000

0

10000

20000

30000

40000

50000

60000

-40 -30 -20 -10 0 10 20 30 40

8psi10psi12psi

-

0

10000

20000

30000

40000

50000

60000

-40 -30 -20 -10 0 10 20 30 40

8psi10psi12psi

Axial Height 75mm

Axial Height 175mm

Fig. 11. Effect of feed inlet pressure on the pressure distrib

and minimum near the radial position near to the axis ofthe cyclone. The observation indicates that higher-pressuredifferential between radial positions near the walls andtowards the cyclone center is generated at higher feed inletpressures. This increase in differential feed pressureincreases the inward radial flow of water eventually report-ing to the overflow product.

The effect of feed inlet pressure on the tangential veloc-ity distribution at different radial positions and at differentaxial positions is presented in Fig. 12. The figure indicatesthat an increase in feed pressure increases the tangentialvelocity at all the radial positions being maximum nearthe walls to minimum near the cyclone center. At any pointon the radial position, an increase in tangential velocity willincrease the centrifugal forces on suspended solid particlesand hence relatively finer particles also get centrifugedtowards the cyclone walls to report into the underflow.The remnant finer fractions at shorter radial distancesfrom the cyclone axis will be affected by the drag to reportinto the overflow. Thus at higher feed pressures, a decreasein solids cut size is expected. The results obtained throughexperiments are matching with the expected increase in thecut size at higher feed pressures (Table 2).

0

10000

20000

30000

40000

50000

60000

40 -30 -20 -10 0 10 20 30 40

8psi10psi12psi

0

10000

20000

30000

40000

50000

60000

-40 -30 -20 -10 0 10 20 30 40

8psi10psi12psi

Axial Height 125mm

Axial Height 225mm

ution at different radial points (10 mm spigot opening).

0

1

2

3

4

5

6

-40 -30 -20 -10 0 10 20 30 40

8psi10psi12psi

-1

0

1

2

3

4

5

6

-40 -30 -20 -10 0 10 20 30 40

8psi10psi12psi

0

1

2

3

4

5

6

-40 -30 -20 -10 0 10 20 30 40

8psi10psi12psi

0

1

2

3

4

5

6

-40 -30 -20 -10 0 10 20 30 40

8psi10psi12psi

Axial Height 175mm

Axial Height 225mm

Axial Height 125mm

Axial Height 75mm

Fig. 12. Effect of feed inlet pressure on the tangential velocity distribution of water at different radial points (10 mm spigot opening).

K.U. Bhaskar et al. / Minerals Engineering 20 (2007) 60–71 69

6. Summary

A CFD simulation and supporting validation study on76 mm hydrocyclone have demonstrated the applicabilityin the analysis of the water throughput, splits and cyclonecut sizes for various test conditions. Among the turbulencemodels studied, the simulation results adopting RSMmodel is found to have better predictions with experimen-tal results. The solid particle distribution simulationthrough discrete phase modeling technique was foundmatching with the experimental results at 10% solids byweight with reasonable accuracy. Parametric results haveindicated that the feed inlet pressure has major influenceon mass flow through the cyclone and spigot opening onpercent split into the overflow. The study has indicated thatthe methodology adopted can be useful for predicting theperformance of hydro- cyclones.

Appendix

The model used for flow simulation solves the conserva-tion equations for mass (or continuity) and momentum.The turbulence in the system is solved through transportequations. Navier–Stokes equations for incompressibleflows along with appropriate turbulence model are adopted

for flow predictions. Under steady state conditions, theequations for mass and momentum in a general form areas follows:

r � qm ¼ 0; ð1Þr � ðq~m~mÞ ¼ rp þr � ð~sÞ þ q~g; ð2Þ

where p is a static pressure, q~g is the gravitational bodyforce~s is the stress tensor given by

~s ¼ leffective ðrmÞ � 2

3r:q~m2

� �; ð3Þ

where leffective = l + lt.

The standard k–e model

The standard k–e model is a semi-empirical model basedon model transport equations for the turbulent kineticenergy (k) and its dissipation rate (e), and are given by

o

otðqkÞ þ o

oxiðqkuiÞ ¼

o

oxjlþ li

rk

� �okoxj

� �þGk � qe; ð4Þ

o

otðqeÞ þ o

oxiðqeuiÞ ¼

o

oxjlþ lt

re

� �okoxj

� �þC1e

ekðGkÞ �C2eq

e2

k;

ð5Þ

70 K.U. Bhaskar et al. / Minerals Engineering 20 (2007) 60–71

Gk ¼�qu0iu0jouj

oxi: ð6Þ

In these equations, Gk represents the generation of turbu-lent kinetic energy due to the mean velocity gradients,C1e, C2e and C3e are constants. rk and re are the turbulentPrandtl numbers for k and e, respectively. The ‘eddy’’ orturbulent viscosity, gt defined in Eqs. (2) and (3) can becomputed by combining k and e as follows:

lt ¼ qCgk2

e; ð7Þ

where Cg is a constant.The model constants C1e, C2e, Cg, rk and re were

assumed to have the following values: C1e = 1.44,C2e = 1.92, Cg = 0.09, rk = 1.0, re = 1.3.

The RNG k–e model

The renormalization group (RNG) k–e model is similarin form to the standard k–e model but includes an addi-tional terms for dissipation rate e development that signif-icantly improve the accuracy, especially for rapidly strainedflows. The effect of swirl on turbulence is included in theRNG model, enhancing accuracy for swirling flows.

The RNG k–e model has a similar form to the standardk–e model:

o

otðqkÞ þ o

oxiðqkuiÞ ¼

o

oxjðakleff

ok

oxjÞ þ Gk

þ Gb � qe� Y M þ SK ð8Þ

and

o

otðqkÞ þ o

oxiðqeuiÞ ¼

o

oxjðaeleff

oe

oxjÞ þ C1e

ekðGK þ C3eGbÞ

� C2eqe2

k� Re þ Se: ð9Þ

In these equations, Gk represents the generation of turbu-lence kinetic energy due to the mean velocity gradients.Gb is the generation of turbulence kinetic energy due tobuoyancy, YM represents the contribution of the fluctuat-ing dilatation in compressible turbulence to the overall dis-sipation rate. The quantities ak and ae are the inverseeffective Prandtl numbers for k and e, respectively. Sk

and Se are user-defined source terms.

Reynolds stress model (RSM)

For steady-state, the Reynolds-Stress Model (RSM)adopted, uses the following transport equations for theReynolds stresses:

o

otðqu0iu

0jÞ þ

o

oxkðqu0ku0iu

0jÞ ¼ P ijþ F ijþDTij þ/ij � eij; ð10Þ

P ij ðstress productionÞ ¼ �q u0iu0k

ouj

oxkþ u0ju

0k

oui

oxk

� �; ð11Þ

F ij ðrotation productionÞ ¼ �2qXkðu0ju0meikm þ u0iu0mejkmÞ; ð12Þ

DTi0 ij ðturbulent diffusionÞ ¼ � o

oxkqu0iu

0ju0k þ pðdkju0i þ diku0jÞ

h i;

ð13Þ

/ij ðpressure strainÞ ¼ þ ou0ioxjþ

ou0joxi

!; ð14Þ

eij ðdissipationÞ ¼ �2lou0ioxk

ou0joxk

: ð15Þ

More details on the underlying equations in this methodare provided in the literature (Fluent Europe Ltd., 2002)

Equations of motion for particles

In hydrocyclone systems operating at dilute concentra-tion of solids (typically below a value of 10% solids byweight), discrete phase modeling technique can be adoptedfor identifying the particle positions inside the system. Theparticles introduced into the system are simulated in aLagrangian frame of reference using the definitions formaterial parameters like particle size, specific gravity, andinitial position etc. assuming solids spherical particles.

The trajectory of the discrete phase particle is obtainedby integrating the force balance on the particle. This forcebalance equates the particle inertia with the other forcesacting on the particle, and can be written (for the x direc-tion in Cartesian coordinates) as

dup=dt ¼ F Dðu� upÞ þ ðgxðqp � qlÞ=qpÞ þ F x; ð16Þ

where FD(u � up) is the drag force per unit particle massand

F D ¼ ð18l=qpÞ � D2p � ðCDRe=24Þ: ð17Þ

Here u is the fluid phase velocity, up is the particle veloc-ity,l is the molecular viscosity of the fluid, ql is the fluiddensity, qp is the density of the particle and Dp is the par-ticle diameter, Re is the relative Reynolds number whichis defined as

Re ¼ q � Dpðup � uÞ=l: ð18Þ

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