1-s2.0-S0032591012008169-main

re

onsodia

ct hiniresd bmedenxpeydrfollosicaiste

ciatedear expause any of th

of mucct. As thuidm

Powder Technology xxx (2013) xxxxxx

PTEC-09303; No of Pages 9

Contents lists available at SciVerse ScienceDirect

Powder Te

j ourna l homepage: www.e lstherefore, been proposed to evaluate the performance of hydrocyclonesbut the graphical method of representing the recoveries of each particlesize in hydrocyclone underow in relation to their availability in feed asa function of their respective sizes (usually in a log-normal scale) is themost popular one. The data thus plotted is popularly known as partitioncurve or Tromp curve. This is normally an S shaped curve and a varietyof quantitative expressions have been used to describe the shape of thecurve. However, partition curves do not always follow the conventionalS shaped pattern, rather they have sh-hook patterns as described in

have also proposed some empirical correlations to predict the shhookeffect in a hydrocyclone classier with reasonable accuracy. But thesemodels failed to explain whether this irregular behavior with ultrane particle sizes in a hydrocyclone is a characteristic phenomenonor not. Majumder et al. [5] have provided a mechanistic argumentsupporting the occurrence of sh-hook in all centrifugal separatorswhile treating ne and ultra ne particles. The basis of their argumentis that in a centrifugal force eld, there is a sudden drop in relativelycoarser particles settling velocities due to Reynolds number restrictions.literature [13]. This means that the recoveriecle sizes are initially higher than the coarser p

Corresponding author at: Universit de Toulouse, IGnie Chimique), 4 Alle Emile Monso, BP 44362, 3143

E-mail address: [email protected] (F

0032-5910/$ see front matter 2012 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.powtec.2012.12.017

Please cite this article as: F. Bourgeois, A.K(2013), http://dx.doi.org/10.1016/j.powtec.edium in dynamic condi-e size distributions, theize. Many criteria have,

Nageswararao [8] summarized that this is a random and sporadicoccurrence caused by the imprecision of measurement and it does notaffect the hydrocyclone performance. A few others [24] however,tions and usually the feed particles have widseparation is never perfect at any particle s1. Introduction

There are a number of issues assohydrocyclone that remain without clto some level of discord about their cleast surprising given the complexitplace inside this unit operation.

One issue that has been the subjectwhat is referred to as the sh-hook effein a hydrocyclone is performed using awith the operation of alanations, often leadingd effect. This is not thee separation that takes

h debate over the years ise particle size separation

up to a critical particle size and after that the recovery increases withincrease in particle sizes.

Although the sh-hook effect has been reported bymany for a verylong time, researchers and practitioners alike share different opinionsabout this phenomenon. There is one school of thought that believesthe sh-hook effect to be a real physical phenomenon [47], whereasa second one argues that it is measurement error related [8,3].

Flintoff et al. [3] opined that this sh-hook effect was due to poorexperimental procedures and/or agglomeration of ne particles. 2012 Elsevier B.V. All rights reserved.

dictory views behind the exIs the sh-hook effect in hydrocyclones a

Florent Bourgeois a,c,, Arun K. Majumder b

a Universit de Toulouse, INPT, UPS, LGC (Laboratoire de Gnie Chimique), 4 Alle Emile Mb Department of Mining Engineering Indian Institute of Technology, Kharagpur 721 302, Inc CNRS, Fdration de recherche FERMAT, Toulouse, France

a b s t r a c ta r t i c l e i n f o

Article history:Received 27 June 2012Received in revised form 15 November 2012Accepted 10 December 2012Available online xxxx

Keywords:HydrocycloneFish-hook effectClassicationFine particle processing

Although the sh-hook effealike share contradictory opothers opine that it is thesh-hook effect could indeein the raw size distributionpartition function and conerrors associated with the eusing a 100 mm diameter hof the partition functionssh-hook effect is a real phys of relatively ner parti-article sizes in underow

NPT, UPS, LGC (Laboratoire de2 Toulouse Cedex 4, France.. Bourgeois).

rights reserved.

. Majumder, Is the sh-hoo2012.12.017al phenomenon?

, BP 44362, 31432 Toulouse Cedex 4, France

as been reported by many for a very long time, scientists and practitionersons about this phenomenon. While some believe that it is of physical origin,ult of measurement errors. This article investigates the possibility that thee measurement error related. Since all the experimental errors are embeddedasurements, the paper rst lays down the steps that lead to estimation of thece bounds, which are seldom reported in hydrocyclone literature, from therimental size distribution measurements. Using several data sets generatedocyclone operating under controlled dilute to dense regimes, careful analysiswing the developed methodology yields unambiguous evidence that thel phenomenon. An attempt is also made to reunite some of the major contra-nce of the sh-hook based on sound statistical arguments.

chnology

ev ie r .com/ locate /powtecAs scientists and practitioners alike share different opinions aboutthis phenomenon, an attempt has been made here to assess whethermeasurement errors could possibly be responsible, on their own ac-count, for the sh-hook effect. For the sake of clarity, it is emphasizedthat the sh-hook effect that is being investigated in this work is thatcaused by particle size only. This work is not concerned with otherfactors that can also affect classication, such as density variationswithin a particle size class [9].

k effect in hydrocyclones a real phenomenon? Powder Technology

2 F. Bourgeois, A.K. Majumder / Powder Technology xxx (2013) xxxxxx2. Accounting for particle size distribution measurement error in estimation of the partition function of the hydrocyclone

2.1. Mass balance solution with particle size distribution error

Mass balancing is an important topic in extractive metallurgy. This section recaps on the set of equations that can be used to reconcile themeasured particle size distributions around a hydrocyclone. The purpose is to give utter transparency to the approach that is used in this paperfor propagating the experimental particle size distributions' measurement errors right through to the estimation of the error associated with thepartition function of the hydrocyclone. One side value of this section is to give the practitioner a summarized account of results that can be readilyused to estimate the partition function from measurements. The results that are presented hereafter are largely based on the two-product massbalancing solution published by Bazin and Hodouin [10]. Notations follow closely those of these authors also so the reader can refer back to theiroriginal work without obfuscating change in notation.

The mass fractions in the feed, underow and overow streams are noted f, u and o respectively. If particle size distributions are discretizedinto n size classes, X and VX denote the column vector of the measured mass fractions and the corresponding covariance matrix.

X

f 1f 2f nu1u2uno1o2on

26666666666666666664

37777777777777777775

VX

var f 1 cov f 1; f 2 cov f 1; f n cov f 1; f 2 var f 2

cov f 1; f n var f n

var u1 cov u1;u2 cov u1;un cov u1;u2 var u2

cov u1;un var un

var o1 cov o1; o2 cov o1; on cov o1; o2 var o2

cov o1; on var on

26666666666666666664

377777777777777777751

In this work, the covariance terms between individual mass fractions, namely cov(fi,fj), cov(ui,uj) and cov(oi,oj) are all assumed to be equal to0, which will lead to minimum variance estimates from the mass balance calculations. This implies that individual mass fractions in the particlesize distributions are assumed to be uncorrelated.

The solid mass split to underow can be written as:

du fouo

2

Noting all estimators with the circumex ^ character, balanced mass fractions must satisfy the following independent constraints:

f^ dûu^ 1dû

o^ 0

f^ u^ o^ 1

3

This mass balancing problem is nonlinear. However, given a value of dû, dened as the estimate of the solid mass ow split to the underow,the mass balancing becomes a linear optimization problem. The mass balancing solution used here consists of minimizing the least-squares

criterion Jjdû XX^ t

V1X XX^

subject to the linear constraint AX^ B where A is a ((n+3)3n) matrix and B is a (n+3) vector fromEq. (4).

A

100100

10001

100001

010dû00

0100dû0

01000dû

001

1dû00

0010

1dû

00100

1dû

2666666664

3777777775

B

111000

2666666664

3777777775

4

The Lagrangian solution to the reconciliation problem has the following solution:

X^ jdû XVXAt AVXA

t 1

AXB

var X^

dû VXVXAt AVXAt

1AVX

5

Since dû can take values in the [0,1] range only, it is a simple matter to scan the [0,1] range and nd the value of dû that yields the lowest value

of the least-squares criterion.

Please cite this article as: F. Bourgeois, A.K. Majumder, Is the sh-hook effect in hydrocyclones a real phenomenon? Powder Technology(2013), http://dx.doi.org/10.1016/j.powtec.2012.12.017

Once the best solution to the mass balance problem has been found, the estimated values of the mass fractions in the feed, underow and

overow streams can be used to characterize the recovery curve to underow through its estimate R^ and variance var R^

. By denition, the

estimate of the recovery to underow is given by:

R^ U^F^

u^

f^ dû

u^

f^

! fô^

uô^

!u^

f^

!6

Propagation of the variance of the estimated mass fractions f^ ; u^ and o^ in all 3 streams yields the variance var R^

of the partition function.

var R^

uô^f^2uô^

2var f^

o^ fô^ f^ uô^ 2

0@

1A2 var u^ u^ fû^

f^ uô^ 2

0@

1A2 var o^ 2uô^2 fô^

f^ 3 uô^ 3 cov f^ ; u^

2u^2o^ fû^ f^ 3 uô^ 3 cov f^ ; o^

2uô^ fû^ fô^

f^ 2 uô^ 4 cov u^; o^

7

3F. Bourgeois, A.K. Majumder / Powder Technology xxx (2013) xxxxxxThe authors used Matlab to implement these equations and produce the results that are presented and discussed in the paper.

2.2. Particle size distribution error

Errors associated with the estimated performance of the hydrocyclone include the primary sampling of the feed, overow and underowstreams, the handling and sub-sampling stages that lead to the samples required for size analysis, and the size analysis itself. In this work, aswe are concerned with the sh-hook effect, which takes place in the ne particle range, we are dealing with particle size distributions measuredby laser diffraction.

The feed to a 100 mm diameter hydrocyclone was assayed using the same sampling procedure on 13 occasions. The experiments to whichthese measurements belong have been reported elsewhere [1113]. They concerned analysis of the ow pattern inside an operatinghydrocyclone. The feed sample was collected at the underow before every test, by running the hydrocyclone at a sufciently low mass owrate so that the entire feed stream would report to the underow. By accumulating test conditions, the experimental work led to collecting13 samples of the same hydrocyclone feed material. It is worth noting that the feed particles are composed of minus 200 m high purity silicaparticles. Details about the experimental set-up and operating conditions can be found in Davailles et al. [13].

Feed size distributions were measured with a Malvern Mastersizer 2000HydroS. Fig. 1 shows the cumulative feed size distributions for the 13measurements.

The mean values of the mass fractions fi and covariance matrix cov(fi

,fj) were calculated for every one of the n particle size classes from the13 feed size distributions measured with the laser size analyser (Fig. 1). The superscript is used to denote that it is a reference particle sizedistribution that will be used later for analysis of individual data sets. The average values f

i and corresponding 95% condence intervals are plot-

ted in Fig. 2, under Student's t-distribution assumption with 13-1=12 degrees of freedom.With the experiments used here, it is believed that our measurement conditions meet the requirements for a precise measurement of particle

size distribution by laser particle sizing as discussed by Xu and Di Guida [14]. Firstly, feed sampling and sub-sampling protocol is the same for all13 samples, so that it can be assumed that the masses of solid and water used for the laser size analyses are invariant. The feed particles are ne(minus 200 m) ground quartz particles (SIBELCO Millisil C6) that assay over 98.8% SiO2. To eliminate the possibility that measured partitionfunction variations may be affected by size-density variations, particle density was measured experimentally by Helium pycnometry as a func-tion of particle size. A 100 g feed sample was rst microsieved into 6 size fractions: 05 m, 5 m10 m, 10 m20 m, 20 m40 m,40 m63 m and 63200 m. Measured density and corresponding standard deviation for each feed size fraction are given in Table 1. Thedata conrm the invariance of particle density in the experiments. In addition, Fig. 3 shows an SEM photomicrograph of the silica particlesused in the hydrocyclone tests, which conrms that particle shape is relatively uniform across the particle size range.

Looking at the relative standard deviation (RSD) for the fi, which is plotted in Fig. 4 as a function of particle size, it appears that the RSD valuesare relatively uniformly distributed, with the exception of few extreme RSD values that occur on both ends, when the measured mass fractionsare insignicant, of the order of 108 or so. We can conclude that measurement error for individual mass fractions can be approximated by anaverage RSD value. Here, an average value of RSD of 8.4% gives a good description of the experimental RSD.

00.10.20.30.40.50.60.70.80.9

1

1 10 100

Cum

ulat

ive

distr

ibut

ion

Particle size (m)Fig. 1. Cumulative feed size distributions, measured by laser diffractometry.


-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1 10 100Fr

eque

ncy

distr

ibut

ion

Particle size (m)

Fig. 2. Distribution of mass fractions fi with their 95% condence interval estimated from analysis of 13 hydrocyclone feed size distribution samples.

4 F. Bourgeois, A.K. Majumder / Powder Technology xxx (2013) xxxxxxFig. 5 zooms on the data from Fig. 4 in the domain that does not include the highest RSD values. Even though the average value of RSD is 8.4%,which may give a good description of the experimental RSD, there is still a noticeable variation of RSD as a function of particle size, which wenote RSD(fi). Moreover, this variation appears quite marked in the 10 m region where the sh-hook effect is known to occur.

To assess the level of quality of our data, we reviewed comparable data obtained by Xu and Di Guida [14] on laser diffractometry of glassspheres. The RSD values obtained by the authors, as shown in Table 2, lie in the range of those obtained in this work.

The match between their RSD values and ours indicates that the laser sizing measurement errors controlled the error in our measurements,which is an indirect conrmation of the quality of the sampling and handling steps used to sample the hydrocyclone feed.

The last issue that should be discussed is that of the calculation of the covariance matrix VX for the feed, underow and overow size distri-butions for one particular experiment, for which fi, ui and oi have been measured. Analysis of the 13 size distributions through Pearson productmoment correlation showed some degree of correlation between the measured mass fractions. These correlations were deemed low enoughhowever to assume that the measured mass fractions are only weakly correlated. This justies the setting of the covariance terms to zero asper Section 2.2.

The standard deviations of the n size fractions of the 3 streams are scaled directly from the size-dependent relative standard deviation (RSD)of the reference feed size distribution f using Eq. (8).

s f i f i

s ui ui

s oi oi

s f

i

f

i

for i; j 1;n 8

Although it is possible to scale the covariance terms in the same manner, it is recalled here that covariance terms in VX are set equal to zero.

2.3. Illustrative example of partition function estimation

The mass balance calculations that follow are done in 2 ways, in order to strengthen the analysis and assess its sensitivity to the properaccounting for the errors associated with the particle size distribution measurements: The rst set of calculations uses the average RSD of 8.4% calculated earlier, which is applied to all the mass fractions in all 3 stream sizedistributions irrespective of particle size.

The second set of calculations uses the size dependent RSD as per Eq. (8) and the size-dependent valuesRSD f i s f i

fidata from Fig. 4 instead.Fig. 3. SEM photomicrographs of silica particles.


The mass balance calculations were performed with the 2 methods above using a data set obtained with a 100 mm diameter HC100hydrocyclone from Neyrtec that was tted with an 18 mm diameter spigot and 33 mm diameter vortex nder. It was operated at a feed massow rate of 2.6103 m3 s1 and an operating pressure of 105 Pa. The feed to the hydrocyclone was characterized in the previous section.This illustrative example corresponds to a test with a feed solids concentration of 20 wt.%.

Key results from the mass balance calculations are given in Fig. 6.Fig. 6a and b show that differences in the mass balance outputs are marginal only between the two calculations, although mass balancing is

eraor

rage tom

alc

imarepth

Table 1Validation of the invariance of particle density with particle size by He-pycnometry.

Size class(m)

Mass analyzed(g)

Density(kg m3)

Standard deviation(kg m3)

05 4.8842 2631.8 2.9510 5.9525 2642.6 1.51020 6.6038 2641.4 0.92040 7.2606 2640.8 2.54063 8.6193 2641.2 0.563200 9.1402 2641.2 0.2

5F. Bourgeois, A.K. Majumder / Powder Technology xxx (2013) xxxxxxparticle size distribution measurement errors. For the sake of trans-parency, it is recalled that the same distribution RSD(fi) was appliedto all 3 streams as per Eq. (8), excluding the possibility that samplingand preparation errors may be different between the 3 streams. Thisis an assumption that could only be lifted by sampling all 3 streamsseveral times, which was not done here. However, given that thetests were performed in a controlled laboratory environment withrecirculation of the underow and overow streams to a feed sump, itis not expected that sampling errors would differ signicantly betweenthe streams.

In the case of the illustrative example of Section 2.3., a sh-hookeffect was observed (see Fig. 6a). The narrowness of the condence

cedure presented above for a test series that covered dilute to denseregime with the same hydrocyclone and feed size distribution asper Section 2. With the hydrocyclone set-up and operating conditionsreported previously, feed solids concentration was varied from 15 wt.%to 50 wt.% in 5 wt.% increments. Rawmeasurements of particle size dis-tributions canbe found in Appendix A of Davailles' PhDdissertation [11].

The estimated partition functions are plotted in Fig. 7a to h. Feedsolids concentration is reported below each gure.

Below a feed solids concentration of 10 wt.% and above 40 wt.%,the partition function does not appear to exhibit the sh-hook effectfor this particular hydrocyclone setting and operating conditions.Between 15 wt.% and 35 wt.%, the sh-hook effect is clearly visible.

120%

140%

160%

12%

14%

16%clearly more effective with the size dependent RSD(fi) than with the avclear minimum of the objective function with the secondmethod. It is wsolids split to underow in this example, namely 54.72% with the ave

There is no signicant difference in prediction of the recovery curvanalysis shows a slightly narrower condence interval with the 2ndbalancing perspective. Nevertheless, for all practical purposes, these cdone reliably with the average RSD value.

This section has also shown that it is a simple enough matter to esthence there is no justication for not estimating it in practice or notaround the partition function is invaluable information for analyzingused in Section 3 below to investigate the sh-hook effect.

3. Proof of the existence of the sh-hook effect

The purpose of the previous section was to present and validatethe mass balance calculation that yields estimation of the partitionfunction and associated condence interval, taking into account the0%

20%

40%

60%

80%

100%

1 10 100

RSD

(%)

Particle size (m)

Fig. 4. RSD (%) values for particle size distribution assays fi in all the measured sizeclasses.

Please cite this article as: F. Bourgeois, A.K. Majumder, Is the sh-hoo(2013), http://dx.doi.org/10.1016/j.powtec.2012.12.017ge RSD(fi) value of 8.4%. This is best seen in Fig. 6c, which shows a veryth noting that both approaches yield slightly different estimates of thee value of RSD against 52.96% with the size-dependent RSD.underow with the 2 approaches, as seen in Fig. 6a; however carefulethod, conrming that it clearly is the best approach from a massulations establish that mass balancing around a hydrocyclone can be

te the condence interval of the partition function of a hydrocyclone;orting it in written work. More importantly, estimation of the errore performance of a hydrocyclone. It is this very information that is

interval of the partition function around the dip and critical points,as dened by [6], excludes the possibility of size distributionmeasure-ment errors being the source of the sh-hook effect in this particularexperiment.

Partition functions were calculated using the mass balancing pro-0%

2%

4%

6%

8%

10%

1 10 100

RSD

(%)

Particle size (m)

Fig. 5. RSD (%) values for particle size distribution assays fi as a function of particle size.


To further conrm the signicance of the sh-hook effect, the valuesand 95% condence interval of the dip and critical points are plottedin Fig. 8 as a function of feed solids concentration.

Within the range of solids concentration where the sh-hook wasvisually observed, Table 3 gives the levels of signicance of the sh-hook effect as a function of feed solids concentration, which we deneas the level of signicance of the difference between the estimateddip and critical point values.

With the 20 wt.% and 25 wt.% feed solids concentration tests, thesh-hook effect is the most signicant at the condence level of99.8% and higher. At 15 wt.% and 30 wt.%, the signicance of thesh-hook effect is 66% and 62% respectively. At 35 wt.%, its signicanceis less than 3%, and above 35 wt.% it was not detected (See Fig. 7f, g andh).

Table 2Data calculated from laser diffraction measurements published by Xu and Di Guida(2003, Table 1 p. 147 and Table 2 p. 148).

Monomodal glass spheres Bimodal glass spheres

d^ 65:27m d^1 46:47m d^2 96:43mRSD=7.7% RSD=13.6% RSD=8.4%

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100

Rec

over

y to

und

erflo

w

particle size (m)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100

Rec

over

y to

und

erflo

w

particle size (m)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1 10 100

Solid

split

du

Particle size (m)

a)

b)

10000c)

6 F. Bourgeois, A.K. Majumder / Powder Technology xxx (2013) xxxxxx100

10001

10

0 0.2 0.4 0.6

Solid split du

Fig. 6. Comparison between the mass balancing outputs using

Please cite this article as: F. Bourgeois, A.K. Majumder, Is the sh-hoo(2013), http://dx.doi.org/10.1016/j.powtec.2012.12.0170.8 1the average RSD of 8.415% and the size-dependent RSD.


We have shown in this section that, under some operating condi-tions, the sh-hook effect can occur at a signicance level that excludesthe possibility of measurement errors being a probable cause. In otherwords, the sh-hook effect is a real phenomenon whose origin goesback to the physics of particle transport inside the hydrocyclone.

4. Discussion

4.1. Origin of the sh-hook

The conditions that will yield the sh-hook phenomenon to besignicant can be numerous, and it should not be inferred from thedata that were presented in the previous section that the sh-hookis most signicant in the 2030 wt.% region only. Indeed, many con-gurations and operating conditions can lead to the sh-hook effect.Further discussion on this is outside the scope of this article.

4.2. Appearance of randomness and sporadicity of the sh-hook effect

By refuting the possibility thatmeasurement errors could be the rea-son beyond the sh-hook effect, we have now established beyonddoubt that the sh-hook effect is not a random effect. What couldthen possibly explain that a number of researchers, like Nagaswarareo[8] claim it is random and sporadic? One possible way of answeringthis is to ask ourselves whether it could be possible for the sh-hookeffect to be present and for the observer not to observe it, giving animpression of randomness and sporadicity.

In this paper, we have demonstrated the signicance of the sh-hook effect based on analysis of the error of the estimate of the recov-ery curve. It is often the case that recovery curves are calculated with-out any indication of their condence bounds. Fig. 9 illustrates theidea that it is possible, in principle, to obtain a high degree of variabil-ity in sh-hook measurement if one does not account for the con-dence bounds of the partition function. Starting with the earlier

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100

Rec

over

y to

und

erflo

w

particle size (m)

a) 15wt%

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100

Rec

over

y to

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erflo

w

particle size (m)

b) 20wt%

0.8

0.9

1c) 25wt%

0.8

0.9

1d) 30wt%

7F. Bourgeois, A.K. Majumder / Powder Technology xxx (2013) xxxxxx0

0.1

1 10 100particle size (m)0.2

0.3

0.4

0.5

0.6

0.7

Rec

over

y to

und

erflo

wFig. 7. Estimated partition functions for a 100 mm diameter hydrocyclone with fe

Please cite this article as: F. Bourgeois, A.K. Majumder, Is the sh-hoo(2013), http://dx.doi.org/10.1016/j.powtec.2012.12.0170

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 10 100

Rec

over

y to

und

erflo

w

particle size (m)ed solids concentration in the range 15 wt.% to 50 wt.% in 5 wt.% increments.


0.4

0.5

0.6

0.7

0.8

0.9

1

very

to u

nder

flow

e) 35wt%

0.8

0.9

1f) 40wt%

8 F. Bourgeois, A.K. Majumder / Powder Technology xxx (2013) xxxxxx0.2

0.3Rec

opartition function whose sh-hook effect is signicant at more than60% level, Fig. 9 shows that it is possible to draw a partition functionthat shows a sharp sh-hook or no sh-hook inside the bounds of the

0

0.1

1 10 100particle size (m)

0

0.1

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1

1 10 100

Rec

over

y to

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erflo

w

particle size (m)

g) 45wt% h

Fig. 7 (cont

0.00

0.05

0.10

0.15

0.20

0.25

0.30

10% 15% 20% 25% 30% 35% 40% 45% 50%

Rec

over

y to

und

erflo

w

Feed solids concentration (wt%)

dip point (Pd)Critical point (Pc)

no fish-hookfish-hook region

Fig. 8. Comparison between dip point and critical point measurements.

Please cite this article as: F. Bourgeois, A.K. Majumder, Is the sh-hoo(2013), http://dx.doi.org/10.1016/j.powtec.2012.12.0170.2

0.3

0.4

0.5

0.6

0.7

Rec

over

y to

und

erflo

w95% condence interval of the partition function. It is this error asso-ciated with measurement of the partition function which the authorsbelieve can in part explain the appearance of randomness andsporadicity of the sh-hook effect that has been reported by someauthors.

This possible explanation for the apparent randomness andsporadicity appears even more when considering that these effectscould be obtained with a partition function that is measured in thelaboratory, under well controlled operating conditions. If we nowconsider the same measurements in the plant, where operating

0

0.1


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100

Rec

over

y to

und

erflo

w

particle size (m)

) 50wt%

inued).

Table 3Signicance of thesh-hook effect as a function of solids concentration in the conditions ofthe experiment.

Feed solids concentration 15 wt.% 20 wt.% 25 wt.% 30 wt.% 35 wt.%

Signicance level 66.23% 99.99% 99.80% 62.30% 2.85%


conclusions may be drawn:

NotationsSymbolsdu Solid mass ow split to the underowf Vector of mass fractions in the feed streamu Vector of mass fractions in the underow streamo Vector of mass fractions in the overow streamn Number of particle size classes

Subscriptsi,j Indices of particle size class

Superscripts^ Circumex symbol denotes the estimate of a random variablet Transpose of a vector or matrix

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1R

ecov

ery

to u

nder

flow

9F. Bourgeois, A.K. Majumder / Powder Technology xxx (2013) xxxxxx Estimation of condence bounds of the partition function is ofsignicant value for analysis of the performance of a hydrocyclone.conditions may uctuate, the possibility of missing the sh-hookeffect for a particular measurement can be greater.

5. Conclusions

Based on the aforementioned results and discussion the following

0

0.1


Fig. 9. Illustration of the possibility of missing the sh-hook effect. Partition functions estimated from carefully generated experimentaldata set in a 100 mm diameter hydrocyclone followed by rigorousstatistical analyses of the data have completely ruled out the possibil-ity of measurement errors as the cause of the observed sh-hookeffect. This conclusion implies that the sh-hook phenomenon is ofphysical origin.

Statistical arguments have also been put forward to explain why insome situations the sh-hook effect may appear to be random andsporadic.

Please cite this article as: F. Bourgeois, A.K. Majumder, Is the sh-hoo(2013), http://dx.doi.org/10.1016/j.powtec.2012.12.017References

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[3] B.C. Flintoff, L.R. Plitt, A.A. Turak, Cyclonemodeling a reviewof present technology,CIM Bulletin 69 (1987) 114123.

[4] W. Kraipech, W. Chen, F.J. Parma, T. Dyakowski, Modelling the sh-hook effect ofthe ow within hydrocyclones, International Journal of Mineral Processing 66(2002) 4965.

[5] A.K. Majumder, P. Yerriswamy, J.P. Barnwal, The sh hook phenomenon incentrifugal separation of ne particles, Minerals Engineering 16 (2003) 10051007.

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

[8] K. Nageswararao, A critical analysis of thesh hook effect in hydrocyclone classiers,Chemical Engineering Journal 80 (2000) 251256.

[9] S.K. Kawatra, T.C. Eisele, Causes and signicance of inections in hydrocycloneefciency curves, in: Advances in Comminution, The Society for Mining, Metallurgyand Exploration, Littleton, CO, 2006, pp. 131147.

[10] C. Bazin, D. Hodouin, Importance of covariance in mass balancing of particle sizedistribution data, Minerals Engineering 14 (8) (2001) 851860.

[11] Davailles, A., 2011. Effet de la concentration en solide sur les performancesde separation d'un hydrocyclone (simulations numriques et experiences dereferences), Ph.D. dissertation, The University of Toulouse, 197 pp.

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Is the fish-hook effect in hydrocyclones a real phenomenon?1. Introduction2. Accounting for particle size distribution measurement error in estimation of the partition function of the hydrocyclone2.1. Mass balance solution with particle size distribution error2.2. Particle size distribution error2.3. Illustrative example of partition function estimation

3. Proof of the existence of the fish-hook effect4. Discussion4.1. Origin of the fish-hook4.2. Appearance of randomness and sporadicity of the fish-hook effect

5. ConclusionsNotationsReferences

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