ro

ess

ycloookposct.law

effects are considered: the entrainment of ne particles in the boundary layer of the coarse settling par-

effect is primarily caused by ne particle entrainment, which is inuenced by the feed solid content and

e hyd

to chalves th

ity according to the Stokes formula, increases monotonically with d(see the dashed curve in Fig. 1b).

However, in many cases in the ne particle range, an increasedparticle removal can be observed (see the continuous curve inFig. 1b). This so-called sh-hook effect is subject of many investi-gations and discussions.

, or the varpe and ded by Duec

(2007).After analyzing the statistical properties of the measure

Bourgeois and Majumder (2013) came to the same conclusion thatthe shhook effect is a real physical phenomenon.

In several publications by Finch (1983), Del Villar and Finch(1992), and Kraipech et al. (2002), empirical correlations have beendeveloped to describe the sh-hook effect.

Schubert (2003, 2004) provided a qualitative explanation of thesh-hook effect using the buoyancy acting on the particles in anon-uniform rotational ow. The random motion of particles of

Corresponding author at: Friedrich-Alexander-Universitt, Erlangen-Nurem-berg, Germany. Tel.: +49 9131 85 23 200.

Minerals Engineering 62 (2014) 2530

Contents lists availab

n

elsE-mail address: johann.dueck@mbt.uni-erlangen.de (J. Dueck).acteristics in the processing zone of the apparatus, describes theinuence of various factors on the separation characteristics. Thetheoretical partition curve calculated using the free settling veloc-

agglomeration phenomena, measurement errorsin the particles size fractions relative to their sha

These doubts have been analyzed and refute0892-6875/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.mineng.2013.10.004

Downloaded from http://www.elearnica.iriationsnsity.k et al.

ments,each particle size d, which is discharged in the coarse product(underow). Schubert and Neesse (1980) demonstrated that thetypical S-shaped partition curve derives from the superpositionof the settling ow and a turbulent diffusion ow in the rotatinguid.

The so-called tapping model (Neesse et al. (1991), Schubert(2010)), which neglects the distribution of the hydrodynamic char-

Although the non-monotonic separation function was describedin the scientic literature (Finch (1983), many years ago, no con-sensus has developed regarding the physical basis of thisphenomenon.

Some researchers remain skeptical (Flintoff et al. (1987), Nage-swararao (2000)) of this effect, believing that it has no physical ba-sis and that the experimental observations are the result of1. Introduction

The fundamental scheme for thFig. 1a.

The partition curve (Fig. 1b) usedefciency of the hydrocyclones invothe feed particle size distribution. An approximated analytical solution for the partition curve is pre-sented for aRosinRammlerSperlingBennet (RRSB)-distributed feed. Experiments using 25-mm hydro-cyclone conrm the calculations.

2013 Elsevier Ltd. All rights reserved.

rocyclone is shown in

racterize the separatione mass fraction T(d) for

The phenomenon of increased ne particle removal through theunderow leads to practical consequences. For example, increasingthe removal of ne particles is benecial for water purication byremoving mechanical impurities. However, the sh-hook effect isdetrimental to ne particle classication because it reduces theseparation sharpness.ClassicationFish-hookFine particle separation

ticles, the hindered settling due to the increased effective density and viscosity of the uid, and the coun-ter ow of the displaced uid caused by the settling particles. The calculations indicate that the sh-hookThe theoretical partition curve of the hyd

Johann Dueck a,b,, Mohamed Farghaly c, Thomas Nea Friedrich-Alexander-Universitt, Erlangen-Nuremberg, Germanyb L.N. Gumilyov Eurasian National University, Kazakhstanc Faculty of Engineering, Al-Azhar University, Qena, Egypt

a r t i c l e i n f o

Article history:Available online 26 October 2013

Keywords:HydrocyclonePartition curve

a b s t r a c t

In many cases, the hydrocrange. The so-called sh-hpractical interest and has athe separation is less distinis derived considering the

Minerals E

journal homepage: www.cyclone

e a

ne partition curve exhibits a non-monotonic course in the ne particleeffect indicates an increased separation of the ne fraction, which is ofitive effect on solid/liquid separation. However, for classication purposes,In this contribution an equation of a partition curve containing a sh-hooks of disturbed settling in dense, polydisperse suspensions. The following

le at ScienceDirect

gineering

evier .com/locate /mineng

Nomenclature

a centrifugal acceleration () entrainment constant ()cV total volume solid concentration ()d particle size (lm)dm characteristic particle size (lm)Dc diameter of the cylindrical portion of the hydrocyclone

(mm)Do overow diameter (mm)Din diameter of the inlet (mm)Dt coefcient of turbulent diffusion (m/s2)D(d) deceleration function for the disturbed settling ()E(d) acceleration function ()fe(d) entrainment function ()g(cV) function of solids content ()q(d) density of the particle size distribution (lm1)H depth of the sh-hook ()

n parameter of the distribution function (uin velocity of the suspension ow in the inlet (m/s)Dp inlet pressure (bar)s internal variable of integration (lm)S volume split ()T(d) partition functionVh(d) hindered settling velocity (m/s)VS(d) settling velocity (m/s)VSt,j Stokes velocity (m/s)wtan maximum tangential velocity (m/s)_Wo suspension throughput at overow (m3/s)_Wu suspension throughput at underow (m3/s)qf uid density (kg/m3)qp solid density (kg/m3)lf uid viscosity (kg/ms)C Gamma function ()

26 J. Dueck et al. /Minerals Engineering 62 (2014) 2530varying sizes in a turbulent environment was considered by Wangand Yu (2010). Majumder et al. (2003, 2007) attempted to explainthe origin of the sh-hook effect using a sudden decrease in thesettling velocity of the coarser particles due to the Reynoldsnumber restriction. Roldan-Villasana et al. (1993) introduced theconcept that a turbulent dispersion could inuence the motion ofne particles.

These concepts have not yet been applied in a systematiccalculation to determine which parametersthe hydrocyclone,the particulate material and/or the operating conditionscontrolthe characteristics of the sh-hook effect.

Kraipech et al. (2002) pointed to the mechanism of ne particleentrainment by larger particles, but did not offer an appropriatemathematical model. This was provided by Dueck et al. (2004),who explained the non-monotonic separation curves through theentrainment of ne particles caught in the boundary layer of thecoarse, rapidly settled particles. This model is based on experi-ments of Gerhart et al. (1999) and Kumar et al. (2000) and hasalready been implemented in the computations of Minkov andDueck (2012).

By varying several parameters, the computer simulationsrequire considerable effort.Inlet

Underflow

Overflow

Vortex finder

D

D

u

o

D

Din

c

0

0.25

0.5

0.75

1

0

Parti

tion

func

tion

T(d)

, -

(a)

Fig. 1. (a) Principal scheme for the hydrocycloneTherefore, this work focuses on the approximated analyticalcalculation of the separation and should be presented in a conve-nient form for analytical estimations that consider the collectiveeffects of disturbed settling in a dense polydisperse suspension.

2. Partition function

According to the tapping model of Schubert and Neesse (1980),the partition function T(d) as a function of the particle size d can beexpressed as follows:

Td 11 Sexp Dc2Dt Vsd

h i : 1In this equation, the volume split is represented by

S _Wo_Wu in which _Wo and _Wu are the suspensions ows of the over-ow and underow, respectively. The value of S can be determinedusing empirical formulas (Bradley (1965)).

Furthermore, Dc is the diameter of the cylindrical portion of thehydrocyclone, and Din is the diameter of the inlet.

This model assumes that the turbulent diffusion coefcient Dt ofthe particle is independent of its size. Thus, the shape of the5 10 15 20

Particle size d, m

Partition function

Partition function(after Stokes)

H

(b)

and (b) partition curve of the hydrocyclone.

ity of a particle depends not only on its size, the medium proper-

D(d) are negligibly small, and the actual settling velocity VS (d) is

nginIf the particles size distribution is presented as a continuousfunction, q(d), such that

R10 qsds 1, then the settling rate

equation of a particle can be written as follows:

VSdVhd1d

2gcV febdd2cVZ 10s2gcV febsqsds 2

in which, according to Dueck et al. (2004) and Minkov and Dueck

(2005) fed R1bd s

6qsds1=3; gcV 94 c2=3V exp5cV , Vh VSt1 cV 4:5, VSt;j ad

2j qpqf 18lf

, and b 15(1 + 10, 5cV).The rst term on the right side of Eq. (2) corresponds to the hin-

dered sedimentation velocity of a particle (the Stokes velocity ac-counts for the impact of solid content). The second term, theacceleration function E d2gcV febd, reects the increase inthe particle velocity due to its entrainment by larger particles.The third term, the deceleration function D d2cV

R10

s2 gcV febsqsds, determines how the ow of the liquid dis-placed by the settling solid phase inuences the particle settlingvelocity.

In Eq. (2) the following designations are used: a centrifugalacceleration, entrainment constant, cV total solid volume con-centration, g(cV) function of the intensity of the entrainment onthe solid concentration, fe(d) entrainment function, q(d) densitypartition function is determined primarily by the settling velocity,Vs. According to Eq. (1), T(d) is a monotonous function of d if Vs(d) isalso a monotonous function of d. The Stokes formula for Vs yieldsthe monotonous S-shaped line of the partition function (Fig. 1b).The separation curve T(d) increases monotonically from T(0) < 1at d? 0 to T = 1 at d?1.

The partition function is typically characterized using the fol-lowing parameters:

(a) d50 the cut size with a 50% fractional recovery in theunderow (Eq. (1) indicates that, for d50, Vsd50 2Dt=Dc ln S).

(b) T0 the value of T(0).(c) Tmin the minimum value of the function. When the Stokes

formula is applied to obtain Vs, T0 = Tmin.

In many cases, the experimental determination of the partitioncurve demonstrates that the curve has a minimum value for parti-cle sizes below 10 lm (Fig. 1b). Such separation curve behavior iscalled the shhook effect. This phenomenon can result from dis-turbed particle settling due to particle interactions as described byseveral researchers (Roldan-Villasana et al. (1993), Kraipech et al.(2002), Dueck et al. (2004)).

3. Disturbed particle settling in a polydisperse suspension

Some experimental and theoretical results have been obtainedregarding the settling of dense suspensions (Gerhart et al. (1999),Gerhart (2001), Kumar et al. (2000), Dueck et al. (2004), Minkovand Dueck (2005)). These studies focused on the settling behaviorof polydisperse suspensions. The settling of polydispersesuspensions involves the following interparticle effects:

1. Hindered settling due to an increased effective density andviscosity of the uid.

2. Counter ow of the displaced uid caused by particle settling.3. Entrainment of ne particles in the surrounding coarse settling

particles.

J. Dueck et al. /Minerals Eof the particle size distribution, VSt(d) Stokes settling velocity,Vh(d) hindered settling velocity, qf uid density, qp soliddensity, lf uid viscosity in kg/ms.slightly lower than that of Stokes because the suspension has ahigher density and viscosity than water.

5. Experiments and calculationsties, and the solid-phase concentration in the suspension but alsoon the particle size distribution.

4. Approximation for the RRSB size distribution

In the present work, specic equations are derived for a typicalcase when the two-parameter RRSB (RosenRammlerBennetSperling) function for the particle size distribution is used:

qd ndm

ddm

n1exp d

dm

n 3

in which dm is the characteristic particle size and n characterizes thesteepness of the distribution function.

For this case, the integrals in Eq. (2) can be estimated (Duecket al., 2010), which leads to the following expression for the sedi-mentation velocity of particles in a polydisperse suspension:

VSdVhd 1

dmd

2gcV 6=n 1C

26=n 1bddm

6nn 6=n 1C6=n 1

0BB@

1CCA

1=3

dmd

2cVC

2n 1

4

Eq. (4) contains an integral representation of the gamma func-tion: Cz 1 R10 tzezdt. For simple engineering calculations, ra-tional functions are convenient. Taking into account that theparameter n varies over a narrow range of 11.5, the followingapproximation can be applied:

Cz 1 2:6 103z6:8:Similarly, we can write g(cV) = 0.9c0.46.Thus, Eq. (4) can be presented as follows:

VSdVhd 1 E D 5

in which the entrainment function E and the deceleration functionD are

E dmd

20:9c0:46V

6:76 1066=n 1 6=n13:6

bddm6nn 2:6 1036=n 16=n6:8

0@

1A

1=3

6

D dmd

22:6 103cV 2n

6:8( )7

Fig. 2 illustrates the values calculated using Eqs. (5)(7) andindicates that for small particles the acceleration mechanism dom-inates, but for larger particles the deceleration effect is moreimportant.

The comparison with Vh (the Stokes velocity, corrected relativeto the solid content) demonstrates that small particles can settle atvelocities several orders of magnitude higher than that determinedusing the Stokes law. For large particles, both functions E(d) andAs demonstrated by Eq. (2), the predicted sedimentation veloc-

eering 62 (2014) 2530 27The experiments conducted by Gerhart (2001), taken for a com-parison with the calculations, are listed in Table 1.

1.0E+01

1.0E+03

1.0E+05

pa

n fu

nctio

ns,-

duri

28 J. Dueck et al. /Minerals EnginThe particle size distribution for the dispersed materials used inthe experiments can be approximated using Eq. (3) with theparameters dm = 6 lm and n = 1.23 provided in Table 1.

In Eq. (1), the volume split value (S) is derived from the exper-imental results with a value of S = 7.5.

The centrifugal acceleration (a) was determined using the for-mulas of Schubert et al. (1990) and Heiskanen (1993) as follows:

a w2tan=Dc 8

in which the maximum tangential velocity is

wtan 3:7DinDc uin 9

and the velocity of the suspension ow in the inlet is

Dof Dp !0:5

1.0E-05

1.0E-03

1.0E-01

0.01 0.1Relative

Sedi

men

tatio

Fig. 2. Disturbed settling functions (Eqs. (5)(7)) dening the interaction of particles(data used for calculation: n = 1.2, dm = 6 lm, cv = 0.04).uin 0:52Din qf: 10

For the diffusion coefcient Dt the following equation can beused (Schubert et al. (1990)):

Dt 16 104wtanDc: 11Applying these formulas to the parameters listed in Table 1, the

following values can be obtained:

uin 5:33 m=s; wtan 7:9 m=s; a 2490 m=s2;Dt 3:2 104 m2=s:Using these values, Eqs. (5)(7) can be applied to calculate the

settling velocities of particles of varying sizes.

Table 1Parameters of hydrocyclone experiments.

Hydrocyclone diameter Dc = 25 103 mInlet diameter Din = 10.5 103 mOverow diameter Dof = 1 102 mFeed pressure Dp = 105 PaParticle density qp = 2.6 g/cm3

Particle size distribution of ne material (Mf) dm = 6 lm, n = 1.2Particle size distribution of coarse material (Mc) dm = 11 lm, n = 1.36. Comparison of the calculated and measured partition curves

Using the values of S, Dc, Vs and Dt, the partition function in Eq.(1) can be determined. Fig. 3 presents the calculation of the settlingvelocities and the partition curves for two different conditions:rst, for the settling according to Stokes, and second, consideringthe disturbed settling in dense suspensions.

The settling velocity as a function of particle size is a non-monotonous function. As previously mentioned, the shape of thepartition curve under given operational conditions depends onlyon the settling velocity. Therefore, the partition curve may have ashape similar to that of the settling velocity curve versus the par-ticle size. Non-monotonous course of the sedimentation velocitycould be the reason for the so-called sh-hook effect, which, inpractice, often manifests itself as the measured curve. This resultis in agreement with the investigations of Gerhart (2001) and Due-ck et al. (2007).

1 10rticle size, (d/dm)

Entrainment function E(d)

Counter Current function D(d)

Total effect 1+E(d)+D(d)

ng settling in an RRSB-distributed suspension, depending on the relative particle size

eering 62 (2014) 2530A comparison between the calculated values and the experi-mental results under the conditions listed in Table 1 is presentedin Fig. 4.

The separation model indicates that there is sufcient con-dence in the explanation of the sh-hook effect. No further accor-dance can be expected for the deviation between the experimentaland calculated values given the extensive simplications in theow model.

Experimental partition functions demonstrate the sh-hook ef-fect, which can be characterized by the depth H (the difference be-tween the value of partition function at d = 0 and the minimumvalue of the separation curve) as indicated in Fig. 1b. The calculatedand measured values of H, depending on the solid content (cv) forthe materials provided in Table 1, are plotted in Fig. 5.

The sh-hook depth (H) presents a non-monotonic curve versusthe solid content cv as predicted by the disturbed settling. Fig. 5also indicates that the values of T(0) vary with cV in a manner sim-ilar to that of H.

Using this fact, the dependence of T0 on the suspension param-eters can be analyzed as follows:

In Eq. (5), D can completely neglected relatively to E if d tendstoward zero as demonstrated in Fig. 2. Considering the denomina-tor of Eq. (6), in the function E, the term containing d can beneglected.

After the transformations, the settling velocity of the smallestparticles, Vs(0) (d tends toward zero), can be obtained:

ngin1.0E+08

J. Dueck et al. /Minerals EVS0 7:07VStdmc0:46V 1 cV 4:5n2:26 12

in which VSt(dm) is the Stokes sedimentation velocity for a particleof size dm.

1.0E+00

1.0E+02

1.0E+04

1.0E+06

0.1 1

Relative partic

Settl

ing

velo

city

, m

/s

Fig. 3. Calculated partition curves and settling velocities for a 25-mm hydrocyclone usinTable 1.

0

0.2

0.4

0.6

0.8

1

0.1 1 10 100

Relative particle size, m

Part

ition

func

tion,

-

Mf (calc) Mc (calc)

Mf (exp) Mc (exp)

Fig. 4. Calculated and measured partition functions for a 25-mm Hydrocycloneusing a solid content cV = 0.04, a ne particle suspension Mf and a coarse particlesuspension Mc (parameter are listed in Table 1).

0

0.2

0.4

0.6

0.8

0 0.1 0.2 0.3 0.4

Solid volume concentration cV,-

Fish

-Hoo

k de

pth

H,-

0

0.2

0.4

0.6

T(0)

, -

Mc (exp)

Mc (calc)

Mf (exp)

Mf (calc)

T(0) for Mc

T(0) for Mf

Fig. 5. Comparison of calculated and measured sh-hook depths as a function ofthe solid concentration.10 100

le size d/dm, -

0

0.25

0.5

0.75

1

Part

ition

func

tion,

-

Stokes velocity

Settling velocity

Partition function(Stokes velocity)

Partition function(disturbed settling)

g a ne particle suspension and solid content cV = 0.04. The parameters are listed in

0.6

0.8

1

nctio

n, - dm=6 m

dm=7 m

eering 62 (2014) 2530 29The maximum value of VS(0) in Fig. 5 occurs at the concentra-tion cV = 0.09, which is higher than the experimental value ofapproximately 0.04.

In addition, VS(0) explicitly depends on the parameters dm and nfrom the size distribution in Eq. (12): VS0 / d2m and VS(0) / n2.26.

Given VS(0), T0 can be easily estimated based on Eq. (1).The calculated and experimental curves of H are qualitatively

similar, but quantitative differences can arise for various rea-sonsthe simplications included in Eq. (5), for example. Speci-cally, these variations may be caused by the difference betweenthe inlet solid concentration used for the calculations and the ac-tual cV values inside the hydrocyclone.

The physically reasonable model appears to adequately de-scribe some of the effects observed in the experiments. A paramet-ric study using the particle size distribution Eq. (3) was performedto clarify the effect of the constants in the equation on the value ofthe sh-hook.

In Fig. 6, each curve is drawn by changing one variable only (dm)with all other parameters held constant. The increase in dm causesa marked increase in T0 and smooth increases in the Tmin values,leading to an increased depth of the sh-hook effect (H), whichcan be interpreted as follows: the coarser the particles, the greaterthe chance for small particles to enter the boundary layer of a largeparticle and be captured by it.

This phenomenon is conrmed by the experiments of Gerhart(2001) in which small and coarse materials were mixed in variousproportions. In these experiments, the value of H increased steadilywith the proportion of the coarse material.

0

0.2

0.4

0.1 1 10 100

Relative particle size, m

Part

ition

fu

dm=8 m

dm=9 m

T(d) afterStokes

Fig. 6. Partition curves for different dm values at n = 0.23 (all other parameters areprovided in Table 1).

Bradley, D., 1965. The Hydrocyclone. Pergamon Press, London.Del Villar, R., Finch, J.A., 1992. Modelling the cyclone performance with a size

dependent entrainment factor. Minerals Engineering 5 (6), 661669.Dueck, J., Neesse, T., Minkov, L., Kilimnik, D., Hararah, M., 2004. Theoretical and

experimental investigation of disturbed settling in a polydisperse suspension.In: Matsumoto, Y., Hishida, K., Tomiyama, A., Mishima, K., Hosokawa, S. (Eds.),Proc. of ICMF-2004. Fifth Int. Conf. on Multiphase Flow, 30 May4 June 2004,Yokohama (Japan). Paper No. 106, pp. 18.

Dueck, J., Minkov, L., Pikutchak, E., 2007. Modeling of the sh-hook-effect in a0.4

0.6

0.8

1

ion

func

tion,

-

n=1.1

n=1.2

n=1.3

30 J. Dueck et al. /Minerals Engineering 62 (2014) 2530Fig. 7 illustrates that the growth of parameter n leads to a weak-ening of the sh-hook effect.

Thus, the theory predicts that the effect should be particularlysignicant for a suspension with at distribution functions andmore coarse fractions.

7. Conclusions

The presented separation model rst enables the rst approxi-mated calculation of the non-monotonous course of the hydrocy-clone partition curve. The model indicates the importance of thedisturbed settling of the particles. Even given the excessive simpli-cations of the complicated three-dimensional turbulent ow in-side the cyclone, the separation can be satisfactorily simulated byconsidering the particle interactions.

The entrainment of the ne particles by the settling of thecoarse particles is primarily responsible for the sh-hook effect.Consequently, the parameters of the feed size distribution andthe feed solid content were introduced into the equation for thepartition curve, providing a new element in the separation model.Although the experimental database remains relatively small, theexperiments with a 25-mm cyclone largely conrm the calcula-tions. One can conclude that the hydrocyclone separation in thene particle range is primarily limited by the sh-hook effect,which can be explained physically. The approximated partitionfunction also indicates the factors inuencing the sh-hook effect.

0

0.2

0.1 1 10 100

Relative particle size, m

Part

it n=1.4

T(d) afterStokes

Fig. 7. Partition curves for different n values using dm = 8 lm (all other parametersare provided in Table 1).These factors can be controlled using known methods: dilution ofthe feed and/or changing the feed size distribution using a multi-stage separation. The opposite is true for thickening and success-fully removing the nest fractions in which high sh-hookfractions would be advantageous, and the addition of coarseparticles for that purpose is less practicable.

References

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particle transport in a hydrocyclone. In: Proceedings of the 21st InternationalSymposium on Transport Phenomena November 25, 2010, Kaohsiung City,Taiwan.

Finch, J.A., 1983. Modelling a sh-hook in hydrocyclone selectivity curves. PowderTechnology 36, 127129.

Flintoff, B.C., Plitt, L.R., Turak, A.A., 1987. Cyclonemodeling a review of presenttechnology. CIM Bulletin 69, 114123.

Gerhart, C., 2001. Untersuchung zum Trennverhalten von Hydrozyklonen niedrigerTrennkorngren. Diss., Tech. Fak., Uni Erlangen, Germany.

Gerhart, Ch., Dck, J., Neee, Th., 1999. Fundamental Study of the PolydisperseSuspensions in Hydraulic Classication Grundlagenuntersuchungen zurbehinderten Sedimentation polydisperser Suspensionen bei derHydrostromklassierung, Aufbereitungs-Technik/Mineral Processing, vol. 40,no. 7, pp. 328334.

Heiskanen, K., 1993. Particle Classication. Chapman and Hall, LondonGlasgowNew YorkTokyoMelbourneMadras, p. 321.

Kraipech, W., Chen, W., Parma, F.J., Dyakowski, T., 2002. Modelling the sh-hookeffect of the ow within hydrocyclones. International Journal of MineralsProcessing 66, 4965.

Kumar, S., Pirog, Th.W., Ramkrishna, D., 2000. A new method for estimatinghindered creaming/settling velocity of particles in polydisperse system.Chemical Engineering Science 55, 18931904.

Majumder, A.K., Yerriswamy, P., Barnwal, J.P., 2003. The sh hookphenomenon in centrifugal separation of ne particles. MineralsEngineering 16, 10051007.

Majumder, A.K., Shah, H., Shukla, P., Barnwal, J.P., 2007. Effect of operating variableon shape of sh-hook curves in hydrocyclones. Minerals Engineering 20, 204206.

Minkov, L., Dueck, J., 2005. Collective effects by settling of polydisperse densesuspension. Eurasian PhysicalTechnical Journal 2 (1(3)), 4763.

Minkov, L.L., Dueck, J.H., 2012. Numerical modelling of a nonmonotonic separationhydrocyclone curve. Journal of Engineering Physics and Thermophysics 85 (6),13171326.

Nageswararao, K., 2000. A critical analysis of the sh hook effect in hydrocycloneclassiers. Chemical Engineering Journal 80, 251256.

Neesse, Th., Schubert, H., Graichen, K., 1991. Practical and theoretical aspects ofdense-ow classication. Aufbereitungstechnik-Technik/Mineral Processing 32(9), 459472.

Roldan-Villasana, E.J., Williams, R.A., Dyakowski, T., 1993. The origin of the sh hook effect in hydrocyclone separators. Powder Technology 77, 245250.

Schubert, H., 2004. On the origin of anomalous shapes of the separation curve inhydrocyclone separation of ne particles. Particulate Science and Technology22, 219234.

Schubert, H., 2010. Which demands should and can meet a separation model forhydrocyclone classication? International Journal of Mineral Processing 96, 1426.

Schubert, H., Neesse, T., 1980. A hydrocyclone separation model in consideration ofthe multi-phase ow. In: Preprints Int. Conf. Hydrocyclones, Cambridge, BHRAFluid Engineering, Pap. 3, pp. 2336.

Schubert, H., Heidenreich, E., Liepe, F., Neee, T., 1990. MechanischeVerfahrenstechnik. vol. 3. Au., Deutscher Verlag fuer Grundstofndustrie,Leipzig, p. 407.

Schubert H., 2003. On the origin of anomalous shapes of the separation curve inhydrocyclone separation of ne particles above all on the so-called sh-hookeffect (Zu den ursachen anomaler verlufe der trennkurve bei derfeinstkornklassierung in hydrozyklonen - Insbesondere zum so genanntensh-hook-effekt). Aufbereitungs-Technik/Mineral Processing, vol. 44, N. 2, pp.517.

Wang, B., Yu, A.B., 2010. Computational investigation of the mechanisms of particleseparation and sh-hook phenomenon in hydrocyclones. AICHE Journal 56(7), 17031715.

The theoretical partition curve of the hydrocyclone1 Introduction2 Partition function3 Disturbed particle settling in a polydisperse suspension4 Approximation for the RRSB size distribution5 Experiments and calculations6 Comparison of the calculated and measured partition curves7 ConclusionsReferences