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Modelling of hydrocyclone performance based on

spray profile analysis

J.S.J. van Deventera,*, D. Fenga, K.R.P. Petersenb, C. Aldrichb

a

Department of Chemical Engineering, University of Melbourne, Melbourne, VIC 3010, AustraliabDepartment of Chemical Engineering, University of Stellenbosch, Matieland,

Stellenbosch 7602, South Africa

Received 26 February 2002; received in revised form 21 December 2002; accepted 31 December 2002

Abstract

Spray profile measurements can be used to calculate the underflow rate, and consequently, be

related to hydrocyclone performance. The flow geometry of the spray discharge is found to arise

from velocity patterns at the outlet orifice. Through a videographic example of an industrialhydrocyclone, it is shown that underflow profiles are typically parabolic, a feature which is

indicative of the velocity at which the fluid exits. The inclusion of gravity in this model clearly

highlights deficiencies in currently used models. Moreover, an intimate knowledge of factors

affecting the profiles of the underflow of a cyclone is essential for the correct interpretation of

videographic images. Subsequently, image data are used to estimate outlet velocities, which give an

excellent insight into various fluid mechanical phenomena that are not appreciated by analysing

basic operational variables. The exit velocities are used to calculate underflow rates, which are

related to mass recovery in the underflow.

D 2003 Elsevier Science B.V. All rights reserved.

Keywords:hydrocyclone; spray profile; modelling; image analysis; underflow

1. Introduction

Sensing technology applied to hydrocyclone monitoring is now a fruitful area of

research, while machine vision has been a relatively latecomer to the field. Petersen

(1993) investigated the use of electrical impedance tomography, where a current

0301-7516/03/$ - see front matterD 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0301-7516(03)00002-4

* Corresponding author. Tel.: +61-3-8344-6620; fax: +61-3-8344-4153.

E-mail address: [email protected] (J.S.J. van Deventer).

www.elsevier.com/locate/ijminpro

Int. J. Miner. Process. 70 (2003) 183203


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applied through the main body of a vessel gave rise to a resistance measurement. This

technique was applied to the outlet of an atomizer, producing accurate air core

diameter measurements. In a more sophisticated investigation, Van Latum (1992)

suggested the use of X-ray imaging to scan cross sections of a dense media cyclonebody to measure density profiles. Williams (1995) made use of two-dimensional

electrical impedance tomography to map physical objects in plane sections throughout

the main body of circular vessels. However, this system was hampered by low

resolving power so that typical features in the main body of a hydrocyclone could

not be mapped with any certainty. This technique was also adapted to estimate spray

angles (Williams et al., 1995). Computed air core areas could be related to feed flow

rate, which was in turn experimentally related to discharge angles. The use of the spray

angle in this investigation was to demonstrate that computed tomography could detect

malfunctions rather than using information from the spray angle itself. Although these

and other approaches have met with various degrees of success, adoption by industry

has not materialised at this stage. However, it is expected that nonintrusive sensor

technology will eventually play a major role in process monitoring, particularly with

improved robustness, which is necessary for the harsh industrial environment. Machine

vision has the advantage of being both nonintrusive and noncontact, and industrially

robust cameras are readily available.

While it is well known that the underflow implicitly characterises hydrocyclone

vortex flow and operation variables, little has been done to exploit this information for

performance estimation. Viljoen (1993) showed that the underflow discharge spray

angle correlates with inlet feed distribution size, and more importantly, applied thisinformation to predict a cut size parameter defined as percent 75 Am. Theunderflow meter has been developed into a fully operational system, and implemented

at various locations. The investigation by Del Villar et al. (1996) is similar to that of

Viljoen in that a soft-sensor approach was used to predict cut size. For purposes of

automatic control, a soft-sensor incorporating a neutral network was used to predict

the percent 45 Am of the hydrocyclone overflow. The major advantage of thisapproach is that existing on-line measurements are used as inputs for the soft-sensor.

The broader implication arising from this investigation is that soft-sensor-based

hydrocyclone control is a value-added technology that is quickly finding its place

in industry. It naturally follows from this investigation that visual sensing of hydro-cyclone underflows can play an important role in subsequent soft-sensor monitoring

and control strategies, based on the relationship between the underflow spray angle

and the classification characteristics of the hydrocyclone. The benefit arises from the

ability of such a system to provide an array of services on-line without the need for

several other sensors. Moreover, in contrast to Viljoens (1993) underflow meter,

noncontact measurement ensures that reliability and accuracy do not degrade as a

result of process conditions, a problem commonly experienced with most other contact

devices.

The aim of this paper is to mathematically describe the spray angle profile typically

observed in industrial images and to incorporate the effect of gravity on the profile. In thisway a robust description of the spray angle can be used to back-calculate exit velocities in

order to simulate hydrocyclone performance.

J.S.J. van Deventer et al. / Int. J. Miner. Process. 70 (2003) 183203184


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2. Theoretical background

One of the first equations for the atomizer spray angle was presented by Taylor

(1948),

cosh wU

xffiffiffi

2p

yz2

1z23=21

2

1

z21

lnz

1

where w = axial velocity at outlet, U= overall head velocity, z= air core radius/outlet

radius at the outlet, x = axial velocity/overall head velocity at the outlet, and y = angular

velocity/(outlet radius

overall head velocity).

This equality for the spray angle is derived under the assumption that the cosine ofthe spray angle is a simple relationship between the overall velocity and the axial

velocity. As will be shown later, this is only approximately true at the immediate outlet,

but in general incorrect, as vortex fluid flow through an orifice is complicated by

torsional geometry. Dombrowski and Hasson (1969) transformed the above equation to

show that the spray angle is independent of inlet conditions (velocity and pressure) and

dependent only upon vessel dimensions. However, this is only the case for a vessel

which is being operated at near maximum inlet pressure conditions (a maximum spray

angle reaches).

This phenomenon (spray angle is independent of inlet pressures after certain levels)

was verified more recently by Dumouchel et al. (1993) who theoretically showed that

after a certain point (inlet pressure), an increase in initial fluid spin (tangential/radial

velocity ratio) results in a small or negligible increase in the tangential velocity at the

Fig. 1. Hydrocyclone underflow showing the fluid flow in a predominately tangential direction.

J.S.J. van Deventer et al. / Int. J. Miner. Process. 70 (2003) 183203 185


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atomizer outlet. An integral part of their investigation was to predict spray angles from

theoretical analysis and to compare to experimental results. They used the most widely

accepted equation for swirling fluid through an outlet, which is

h2arctan vw

2

where v and w are the tangential and axial velocities, respectively. Intuitively, this

equation is acceptable, considering that the two major features contributing to the

resulting angle, the axial and tangential velocities, are represented. However, as with

the atomizer, the predicted spray angle assumes that the tangential velocity is in a

direction perpendicular to the outlet. Visual inspection of hydrocyclone operation

reveals that pulp flow at the underflow exits in a predominantly tangential orientation.

Fig. 1 clearly supports this, showing the streamlines in a tangential orientation (due topoor reproduction quality, this is indicated by the solid black lines).

3. Spray angle model

3.1. Model development

This section will set up a three-component vector system that refers to individual points

on the outer spray edge. Combining all reference points constitutes the observed spray angle

profile. Fig. 2 represents a quarter section of the hydrocyclone/atomizer outlet, with theobserver looking from the bottom and up into the vessel. Consider a stream of fluid

originating from point #1 which is directed in a tangential path with respect to the outlet. Due

to a radial velocity component, the fluid flow is now directed at an angle x, from the original

tangential direction along the vector labelled with componentsm and c. If the spray profile is

observed tangentially at point #2, fluid from #1 will make a contribution to this profile. Now,

Fig. 2. Geometry of fluid flow as seen at the outlet of a hydrocyclone or atomizer. Fluid stream is expelled in a

tangential and radial direction.



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considering that this contribution from point #1 is elevated toward the reader, then the

elevation of vectorq +p will describe the angle from the outlet edge to the point on the spray

discharge surface. This calculation is performed at all of the angular positions, b, which will

be affected by this particular fluid stream. Effectively, this is the same as calculating theprofile from a single point on the underflow outlet circumference and describing the

contribution to the spray profile of fluid flow streams at an angle b away. As indicated in Fig.

2, the fluid stream from point #1 will make no contribution to the profile at 90j(they2-axis)

since they2-axis is parallel with the fluid streamline, whenx = 0.

All of the labelled vectors are found as follows fromFig. 2.At points #1 and #2, the y1-

and y2-coordinates arey1 = r, y2 = 0 and y1 = rcosb, y2 = rsinb, respectively.

Therefore, dis given by

d

r

rcosb

r

1

cosb

3

Now, the vectorq can be found from the vectordas follows.

cosb = d/q, therefore

q r1cosbcosb

4

Now,

q2 s2 d2 5

s2 r21cosb2

cos2b r21cosb2

s2 r21cosb2 1cos2b

1

s

r

1

cosb

1

cos2b

1 1=2

r

1

cosb

tanb

Vectorm

mcosxrsinbr1cosbtanbrtanb 6

m rtanbcosx

Vector f

msinxf rtanbtanxVectors p and c



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Inspecting the region bounded by the vectorsp,c and fa little more closely as shown in

Fig. 3, a right angle triangle can be completed by using the vectors e and l.

psinbland ccosxlZp ccosx

sinb

Now, to find c,

pcosbfertanbtanxcsinx

c rcosx

tan2btanx

1tanbtanx

7

Therefore,

p rcosb

tanbtanx

1tanbtanx

8

The vertical height from the plane of the outlet is given by the vectorzas shown inFig. 4.

z mctanurtanucosx

tanb

1tanbtanx

9

q

p

r

1cosbcosb

tanbtanx

cosbsinbtanx 10The two vectors zand q +p, Eqs. (9) and (10), give sufficient description to formulate an

equation for the discharge spray angle. The form of the equation is given by

h2arctan BqpBz

11

where h is the tangent angle to the outer surface of the spray discharge measured along

with vector (p + q) and the differential of (q +p) with respect tozgives the gradient of the

outer surface of the spray discharge. Now, both vectors are independent of each other and

Fig. 3. Section ofFig. 2with added vectors e and l.



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are only dependent on the angular distance, b. Therefore, the following computation is

performed to find this differential.

BqpBz

BqpBb

Bb

Bz

Now,

Bz

Bbrtanu

cosxcosbsinbtanx2

ZBb

Bz cosx

rtanucosbsinbtanx2 12

and

BqpBb

rK2cos2b

Ksinbsecbtanx 13

Therefore, the combined result becomes

BqpBz

cosxtanu

Ksinbsecbtanx vw

sinbuv

secb1sin2btanbh i

14

where the equalities K=(1 tanb tanx), tanu=x/vx

, tanx = u/v, cosx= v/vx

have been

used to incorporate the three-dimensional velocity components, u, w, vx

, which are the

radial, axial and along the line of flow velocities, respectively. The final form of the

equation for the discharge spray angle is given by

h2arctan vw

sinbuv

secb1sin2btanbh i

15

Fig. 4.The two sections of the fluid flow geometry. The left-hand diagram is the elevation of the fluid stream in

Fig. 2.The right-hand diagram shows the angle X that is made from the plane of the outlet to the elevated fluid

stream.



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changing with position due to the effect of gravity. Incorporating a gravity term requires

the expression for the vertical height, z, to be altered.

Horizontal vector component:

Sx

Uxt

Ucosut

Vertical vector component:

SyzUyt12

agt2 Usinut1

2agt

2

SyztanuSx12

agS2x

U2cos2u

where Uis the overall velocity with corresponding horizontal and vertical components of

Ux and Uy, and Sx and Syare the horizontal and vertical vector components, respectively.

With reference toFig. 4(left), the horizontal component,Sx, can be replaced by the m + cvector, which gives

Syz rtanutanbKcosx

T r2tan2b

K2cos2x

whereK= 1 tanbtanxand T=(1/2)(ag/U2cos2x).Therefore, the two reference vectors forthe spray edge are given by

z rtanutanbKcosx

T r2tan2b

K2cos2x 16

qpr 1cosbcosb

tanbtanxcosbsinbtanx

17

Fig. 6. Typical industrial hydrocyclone underflow. The curved effect of the underflow is clearly shown on the left

spray edge.



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Following the same steps as in the previous section, the vectors z and q +p can be

manipulated to find the discharge spray angle. The final result becomes

BbBz

K3

cos2

xrKtanusec2b2Tr2tanbsec2b 18

and

BqpBb

rK2cos2b

Ksinbsecbtanx 19

Therefore,

B

q

p

Bz Kcos2

x

Ksinb

secbtanx

Ktanucosx2Trtanb 20

BqpBz

v2 uvtanb"h

1uv

tanbi

sinbuv

secb

wvuwtanb agrtanb

# 21

where the equalities tanu= w/vx

, tanx= u/v and cosx= v/vx

have been substituted to

reveal the axial, tangential, fluid flow projection, radial and fluid stream velocity,w,v,vx

,

u and U, respectively. Finally, the discharge spray angle is given by the relation

h2arctan vutanb vutanbsinbusecbwvuwtanb agrtanb 22

This equation gives a general description of the discharge phenomenon. If the gravity

term,ag, is set to zero, Eq. (22) reduces to Eq. (15) as required. Fig. 7shows a comparison

between the two results derived thus far. Using velocity values ofu = 0 m/s,v= 0.5 m/s and

w = 0.5 m/s (radial component omitted), outlet radius of 1 cm and a total ofb = 85jvalues,

Fig. 7. The effect of including gravity forces in the spray angle calculation (outlet radius is 1 cm).



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the upper line shows the model without the gravity effect and the lower line shows the

model with the gravity correction included.Fig. 7reveals significant gravity effects within

the 40 12 cm cross section.The gravity affected profile inFig. 7can be classified into three separate regions. The

initial upward curved region is the first. The second is the relatively flat, downward

sloping region after the initial curve. Thirdly, the second region then connects to a second

downward curving region where gravity forces are dominant. These three stages are shown

schematically in Fig. 8. As defined by Dombrowski and Hasson (1969), the discharge

spray angle is defined in this flat, second region.

The model can now be used for theoretical spray angle prediction and compared to

experimental data.

3.3. Features of the new model

The equations presented above give a detailed, robust description of the spray discharge

geometry. These results have the potential to be used widely in vortex simulation

investigations where the exiting discharge spray angle can be used to predict the three

components of fluid velocity.

For all practical purposes, the radial component of (Eqs. (16), (17) and (22) is notrequired for use in hydrocyclone spray angle calculation. The essential aspects of these

equations that could benefit most studies in vortex fluid flow through orifices would be

gravity effects and the tangential and axial velocity components. Therefore, upon

simplification, (Eqs. (16), (17) and (22) become

xr 1cosbcosb

23

z

rwtanb

v

agr2tan2b

2v2

24

h2arctan v

2sinb

wvagrtanb

25

Fig. 8. Conceptual spray profile: defining flat region and maximum b value.



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3.3.1. Spray angle uniqueness

The ability ofthe model (Eqs. (23)(25)) to find unique values for the exit velocities is

demonstrated inFig. 9where the x- and z-coordinates, i.e. spray profile (from Eqs. (23)

and (24)) are plotted for r=0.1 m, v= 2.5 m/s, w = 1.5 m/s and forv=5 m/s, w = 3 m/s.Even though both profiles have the same v/w ratio, they become increasingly different

further from the outlet. Moreover, it is not possible to adjust the axial velocity (of either

profile) such that both curves lie on top of each other. Evidently, the signature of gravity

facilitates the estimation of outlet velocities from spray angle profiles. However, some care

must be taken when applying this technique. At relatively high exit velocities, the task of

finding a unique set of outlet values becomes more difficult, since the effect of gravity is

less significant in the near vicinity of the outlet. This problem could be overcome by using

a greater extent of the spray profile to the point where gravity forces begin to dominate.

However, in most systems, liquid sheet breakup also occurs, and consequently causes a

deviation of the fluid trajectory due to changes in momentum.

3.3.2. Comparing new model with Eq. (2)

The apparent simplicity involved in predicting spray angle profiles belies the complex-

ity of the process, especially considering that accurate data are difficult to obtain and there

is an absence of literature for comparative purposes. In spite of these limitations, a

comparison of Eq. (25) to the most widely used equation (Eq. (2)) using a range of typical

operating values will serve for comparative purposes.

Using an axial velocity ofw = 2 m/s, a tangential velocity range ofv= 0.5 m/s tov= 10

m/s is used to calculate spray angle values for Eqs. (2) and (25). Fig. 10shows the resultswhere two outlet radii,r= 0.05 m andr= 0.005 m, have been used in order to show effects

in vessels of differing scale, viz. hydrocyclones and atomizers.

Fig. 10clearly shows that when the outlet radius, gravity effects and the geometry of

the outlet flow are considered (Eq. (25)), calculated spray angles can considerably differ to

Fig. 9. Two examples of a spray profile (model: Eqs. (23) and (24)) using different values for tangential and axial

velocities, but having the same tangential/axial velocity ratio. This shows that profiles have unique sets of exit

velocities.



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those obtained with Eq. (2). At a tangential velocity of 2 m/s and an outlet radius of 0.05

m, Eq. (2) indicates that spray angles will be over predicted by approximately 20j. The

other conclusion is that discrepancies between the two models become more significant

when using larger systems, such as a hydrocyclone. In the case of the 0.005-m outlet,

pressure values and outlet velocities are relatively high, and therefore, under certaincircumstances, Eq. (2) may be sufficient for spray angle calculations.

3.4. Experimental verification of spray angle curvature

As discussed above, the profile of typical spray angles will be difficult to detect under

high pressure conditions and for small sized vessels. The experimental investigation on

industrial hydrocyclones in Leeudoorn (South Africa) Gold Mine offers such an oppor-

tunity. The milled product is pumped through the primary cyclone where the coarse

underflow is recycled to the mill. The overflow stream is fed to a secondary cyclone, the

underflow of which is fed back to the primary sump, where it is mixed with the effluent ofa SAG mill before being recycled to the primary cyclone. The overflow of the secondary

cyclone is withdrawn from the circuit. In this investigation, the camera was placed

approximately 50 cm from the underflow outlet of the secondary cyclone in order to assess

the robustness of the equipment, as well as to monitor the underflow of the cyclone.

Variation was induced in the system by changing the dilution of the pulp in the sumps

feeding the hydrocyclone. This affected the feed distribution, inlet pressure and flow rate,

as reflected in the spray angle of the secondary cyclone portrayed in Fig. 11. A typical

example of this industrial data is shown in Fig. 6 where a small upward curvature is

visible. In order to verify this effect, the data inFig. 11(spray angle change over time) will

be investigated. The important features in this data are (1) dilution change beginning att= 157 min and ending att= 184 min, (2) primary sump: 200 160 m3/h and (3) secondary

sump: 300250 m3/h.

Fig. 10. Comparison of spray angle models for an axial velocity of 2 m/s and tangential velocities from 0.5 to

10 m/s. Eq. (2) is calculated without a knowledge of the outlet radius. The curve for Eq. (25) shows that the

outlet radius, gravity effects and other geometrical properties can have a significant effect on calculated spray

angles.



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For each image of the hydrocyclone underflow, five sets ofx- andz-coordinates from the

image matrix were measured along the spray profile (z-coordinates were chosen approx-

imately 20 pixels apart), from which a set of four gradients were calculated, as given by

Spray profile section gradient

xi1xizi1zi

26

where i = 1, 2, 3 and 4. The underflow images are classified according to pre-dilution, during

dilution and post-dilution phases. For each phase of operation, average values of each of the

four measured gradients are calculated. Fig. 12 shows the averaged data with the

Fig. 11. Variation in spray angle of the underflow of the secondary cyclone in response to a dilution change

beginning att= 157 min and ending att= 184 min (200 160 m3/h for the primary sump and 300250 m3/h for

the secondary sump).

Fig. 12. Calculation of spray angle gradient along four sections of the underflow. Data are grouped into pre-

dilution, during dilution and after dilution stages. Average spray angle gradients of each stage are shown. Errors

for pre- and post-dilution are included.



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corresponding error bars for pre- and post-dilution. Qualitatively, Fig. 12shows that the pre-

dilution phase is well separated from the other two phases of operation. To confirm this

observation, a series oft-tests were performed.

In identifying differences between the three phases of the data, the t-test results indicatethat there is a definite difference in the underflow angle gradients for data taken during the

pre-dilution stage and for data during the dilution (a certainty typically greater than

95.5%). Similarly, a difference exists between pre- and post-dilution phases where the

certainty increases slightly to greater than 97.5%. In contrast, dilution and post-dilution

phases produce a nullresult. It can be concluded that there has been a real change in the

spray angle(Fig. 11)and that the gradient observed along the spray profile also changes

upon dilution.

4. Prediction of outlet velocities

4.1. Theoretical modelling

This section is to predict hydrocyclone flow rate conditions from image spray angle

data. Firstly, the model equations (Eqs. (23) and (24)) must be modified such that they are

amenable for comparison to image measurements. The unknowns in these two equations

areb and the exit velocities,vand w. The parameterb can be removed from the equations

by rearranging Eq. (23), giving

bcos1 rxr

27

which can then be substituted into Eq. (24). Consequently, if the outlet radius, rand the

exit velocities are known, and the measurements ofx-coordinate of the spray profile are

taken, then it is possible to predict the z-coordinate values. However, since the exit

velocities are unknown, it is more appropriate to take z- and x-coordinate measurements

from videographic images and then back-calculate the exit velocities. Coordinates of the

spray profile are extracted for the spray angle measurements(Petersen, 1998).

Eq. (24) is linear in form and can be represented by a simple addition of linear terms as

z= m1x1 + m2x2 + c where

m1w=v x1rtanb

m21=v2 x20:5agr2tan2b

c0 28

Now, given thatx, zand rare known through experiment, multi-linear regression can beused to find values form1 and m2, which effectively solves for the tangential and axial

velocities.



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In order to verify the model, a laboratory-scale hydrocyclone was used to demonstrate the

flow properties by analysing spray angles under different feed loading conditions. The

dimensions of the hydrocyclones used were inlet dimensions 8 13 mm, underflowdiameter 9 mm, overflow diameter 12 mm, and cyclone diameter 36 mm. For a range of

inlet pressures, spray angles were measured for a series of feed pulp concentrations (1%, 2%,

4%, 7%, 10%, 15%, 17%, 19%, 20%, 22% and 24% by mass; 600lm).Fig. 13showsspray angle measurements for inlet pressures of 40, 50 and 60 kPa, respectively. Fig. 14

Fig. 13. Spray angle measurements for a range of feed concentrations at varied inlet pressures. Trends indicate

that the higher pressure leads to a lower spray angle at higher feed concentrations.

Fig. 14. Three discharge profiles, data and predictions. For water-only conditions, pressures of 40, 50 and 60 kPa

were used, yielding calculated axial velocities of 10, 14 and 20 m/s, respectively. Image resolution is an important

aspect of detecting small, but crucial profile variations. Solid linesmodel.



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shows three examples of fitting the model to the laboratory data from Fig. 13(40, 50 and 60

kPa for water only). Each spray profile has been separated along the x-axis by 5 pixels for

demonstration purposes. The differences between each profile are not easily discernible by

visual inspection, indicating the highly sensitive diagnostic nature of the spray angle. It alsosuggests that high resolution data acquisition is paramount. Also shown by the thick lines are

the calculated profiles where axial and tangential velocities were found through the multi-

linear regression fitting procedure, from which z-coordinates were calculated using the x-

coordinate data. The corresponding tangential and axial velocities are given, demonstrating

that with an increase in the pressure (for water only), the exit velocities also increase. It is

Fig. 15. Calculated velocities from laboratory-scale image data. Differences with respect to pressure are evident.

Crossover point at approximately 7% concentration due to higher pressures eventually leading to increased

recovery. (a) Axial; (b) tangential.



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also interesting to note that the axial velocity is consistently higher than that of the tangential

velocity. This is due to the design of the cyclone cone section, which in this case is relatively

long and has an acute angle (approximately 15j). Although not shown here, the industrial-

scale results are reversed, where the tangential velocity is consistently higher than the axialvelocity, a system with a much less acute conical section.

Upon applying the exit velocity calculation procedure to the complete set of laboratory

spray profile images, Fig. 15 shows the resulting axial and tangential velocities. The

most interesting feature is that all three sets of data, for both tangential and axial

velocity, decrease noticeably from a concentration of approximately 2% onwards,

indicating that the underflow is sensitive to changes in concentration, i.e. viscosity

changes. In contrast, the corresponding underflow flow rate data are relatively constant

(slight progressive increase) until approximately 10%. It is only by expecting that the air

core radius decreases, could this happen, which further suggests a very sensitive

viscosity dependence.

The other important feature inFig. 15is that similar to the spray angle data, there is a

cross-over point at a concentration of 7%. This indicates a two-flow regime where under

dilute conditions, high inlet pressure results in a high transferral of momentum to the

underflow, whereas under high loading conditions, high pressure leads to greater material

recovery at the underflow.

Revisiting Fig. 13, it is evident in all three examples that the model slightly under-

predicts the data very close to where the spray initially exists the outlet such that the

upward curvature is overemphasised.Fig. 16shows a plot of the z-coordinate data versus

the calculatedz-coordinates, where the thick parity line highlights the slight departure ofthe model close to the outlet. The R2 correlation for the entire data set is 0.93, whereas if

only the coordinates greater than 30 (pixels) from the outlet are considered, then the R2drops to 0.82. The reason for this could be twofold: (a) The laboratory cyclone exhibits a

Fig. 16. Calculated z-coordinates versus measured z-coordinates for the data shown in Fig. 11.The correlation

between the data sets is high, but there is an evident discrepancy atz-coordinates close to the outlet.



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very small spray profile curvature due to the presence of high exit velocities in a relatively

small sized outlet. Consequently, it is essential that high resolution spray profile coordinate

extraction is performed, a factor which is largely the result of image capture quality. (b)

The spray profile model is developed on the assumption that the material projected fromthe underflow outlet is massless, and without a sheet thickness. A rigorous development of

force balances may be required, especially the centrifugal force, which could be re-

sponsible for a radical velocity component.

In comparison to the laboratory cyclone, the industrial cyclone discussed above displays

a relatively extended spray profile curvature. Applying the same correlation analysis to the

pre-dilution data, theR2 correlations for all the data and the data close to the outlet (> 40pixels) are 0.93 and 0.78, respectively. The correlations are consistent with those for the

laboratory data, suggesting that an account of fluid flow forces would provide the necessary

further refinement to the geometry considerations. In spite of the small discrepancies for the

laboratory system, the high correlations show that the spray profile model provides a more

than adequate description of the underflow structure for inferral purposes.

5. Summary and conclusions

The most significant conclusions for this investigation on hydrocyclone discharge spray

angles and the development of a soft-sensor are as follows.

The assumption that the exiting tangential fluid from an orifice is directed tangentiallyfrom the plane of the outlet has been used to formulate a spray angle model. The spray angle model accurately simulates the typical profile features of rotating fluid

through an outlet. In particular, the model predicts that the discharge spray profile

curves upwards after it immediately exits the outlet, indicating that the spray angle is

continually increasing. This result is supported by videographic evidence from an

industrial hydrocyclone. Gravity modifies the spray profile by causing a downward turn. The point along the

spray profile at which the gradient goes through a minimum is the effective spray angle.

The resulting gravity model is able to predict the two-dimensional coordinates of the

spray surface and the corresponding angle. However, the exit axial and tangentialvelocities must be known. These can be determined by matching the model to profile

coordinates extracted from image data. The exit velocities can subsequently be used to calculate the underflow flow rate

conditions. However, this can only be implemented by knowing the air core ratio,

which is found by back-calculating from the experimental spray angle and flow rate

data. The air core calculations are fitted with an empirical equation. The result forms

part of a soft-sensor based on spray angle and feed pressure acquisition.

Nomenclature

a air core radiusag acceleration of gravity

d vector parallel withy1-axis extending from circumference



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d particle size

k1, k2, k3 constants

K constants

n tangential fluid flow vectorq +p perpendicular distance from outlet radius to tangential fluid stream in the outlet

plane

qu underflow flow rate

r classification sharpness

R underflow outlet radius

s vector which closesq and d into a triangle

u radial exit velocity [m/s]

v tangential exit velocity [m/s]

w axial exit velocity [m/s]

x the vector (q +p) with surface tension effects

y1, y2 axes of outlet plane

z z-coordinate

Greek letters

a classification sharpness

b angle measured at outlet radius

/ rescaled concentration

c complement to u

g total efficiencygu(X) fractional efficiency to underflow

u tangential velocity angle measured from horizontal

h observed discharge profile spray angle

h* reference angle to discharge profile from vertical

x angle from tangential direction caused by radial flow action

X complement to h*

References

Del Villar, R.G., Thibault, J., Del Villar, R., 1996. Development of a softsensor for particle size monitoring.

Minerals Engineering 9 (1), 5572.

Dombrowski, N., Hasson, D., 1969. The flow characteristics of swirl (centrifugal) spray pressure nozzles with

low viscosity liquids. AIChE Journal 15 (4), 604611.

Dumouchel, C., Bloor, M.I.G., Dombrowski, N., Ingham, D.B., Ledoux, M., 1993. Viscous flow in a swirl

atomiser. Chemical Engineering Science 48 (1), 8187.

Petersen, K.R.P., 1993. Vortex chamber air-core prediction for the simulation of hydrocyclone operation. MSc

thesis, University of Queensland, Australia.

Petersen, K.R.P., 1998. The monitoring of hydrocyclone underflows and mill feed systems using image process-

ing techniques. PhD thesis, University of Stellenbosch, South Africa.

Taylor, G.I., 1948. The mechanics of swirl atomisers. Proceedings of the 7th International Congress of Applied

Mechanics, vol. 2, pp. 280285.

Van Latum, L., 1992. Computed tomographic imaging of a dense media cyclone. Applied Research in the



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Minerals Industry (JKMRC Postgraduate Students Conference). JKMRC, The University of Queensland,

Queensland, Australia, pp. 6573.

Viljoen, T., 1993. Recent developments in instrumentation. SAIMM School: Process Simulation, Control and

Optimisation, The South African Institute of Mining and Metallurgy, Randburg, 1820 August.

Williams, R.A., 1995. Measurement and modelling of slurry mixing using resistance tomography. Proceed-

ings of the XIX International Mineral Processing Congress, Comminution and Simulation and Control,

San Francisco, USA. Society for Mining, Metallurgy, and Exploration, Inc. (AIME), Littleton, Co., USA,

pp. 251 256.

Williams, R.A., Dickin, F.J., Gutierrez, J.A., Dyakowski, T., Beck, M.S., 1995. Using electrical impedance

tomography for controlling hydrocyclone underflow discharge. Proceedings of the 8th IFAC International

Symposium on Automation in Mining, Mineral and Metal Processing, Sun City, South Africa. Pergamon

Press Inc., NY, USA, pp. 239242.


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