-
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: [email protected] (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.
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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