Minerals Engineering 24 (2011) 230–235

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Minerals Engineering

journal homepage: www.elsevier .com/ locate/mineng

Towards a mechanistic model for slurry transport in tumbling mills

I. Govender a,b,⇑, G.B. Tupper a,b, A.N. Mainza a

a Centre for Minerals Research, Department of Chemical Engineering, University of Cape Town, Cape Town, Western Cape 7701, South Africab Department of Physics, University of Cape Town, Cape Town, Western Cape 7701, South Africa

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

Article history:Available online 13 October 2010

Keywords:Slurry transportPEPTComminutionPressure

0892-6875/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.mineng.2010.08.010

⇑ Corresponding author at: Department of Physics, UTown, Western Cape 7701, South Africa. Tel.: +27 2121 021 650 5554/650 3342.

E-mail address: indresan.govender@uct.ac.za (I. Go

A new modelling approach to slurry transport in dynamic beds based upon combining space and time-averaged Navier–Stokes equations with a new type of cell model is described. The resulting Ergun-likeequation is used to correlate pressure drop with time-averaged distributions of the porosity, superficialfluid velocity and solids velocity for data derived from positron-emission-particle-tracking (PEPT) exper-iments in a scaled industrial tumbling mill fitted with lifter bars, pulp lifters and a discharge grate andrun with particles and re-circulating slurry.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Tumbling mills play a crucial role in the South African mineralsindustry. Given that SAG mills account for some 60% of operatingcosts, even incremental increases in efficiency hold the promise ofenormous savings. To achieve improved efficiency it is natural toseek a better understanding of the processes taking place insidethe mill. Considering the case of slurry transport which is of crucialimportance both because it is the means of extracting the mill prod-uct and because slurry pooling can seriously degrade grinding effi-ciency. Unfortunately, the lack of understanding of this complex,inefficient transport mechanism is the main bottleneck in trying toimprove grinding circuit efficiency (Songfack and Rajamani, 1999).

The development of positron-emission-particle-tracking (PEPT)has enabled the non-invasive observation of model mills underoperational conditions (Parker et al., 1993). Using particles labelledwith b+ emitters (the tracer), the subsequent annihilation of thepositron with an electron produces 511 keV back-to-back gammapairs. Detection of a few such pairs provides the raw data for track-ing the in situ flow field of the tracer through opaque environ-ments. Large particle tracking gives information about the chargewhile small particle tracking yields data about the fluid flow.Through PEPT one can build up time-averaged velocity, porosityand shear rate distributions – the key drivers to flow in tumblingmills. The obvious difficulty associated with measuring, and hencemodelling, these physical flow parameters have limited its value tothe minerals industry in favour of empirical models. Consequently,

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niversity of Cape Town, Cape021 3818/650 5554; fax: +27

vender).

many models are based on unrealistic assumptions. Hogg andRogovin (1982) assumed that transport is limited to the slurry pooland that the interstices in the charge are always saturated. Morrelland Stephenson (1996) constructed a more usable transport modelthrough both the slurry pool and charge using extensive measure-ments on a pilot scale ball mill fitted with discharge grates.Latchireddi (2002) extended the Morrell and Stephenson modelto include a wider range of discharge grates and added the pulplifter design. Both these models are built on extensive data basesand have thus gained wide usage in those operations falling withintheir window of design. However, both appear decoupled from thephysical parameters known to drive flow through porous networks– their extrapolation potential is thus unclear. Moys (1986) em-ployed a semi-mechanistic approach based on the Blake–Kozenyequation according to Bird et al. (1960). The use of empirical rela-tions to obviate complicated integration coupled with the uniformporosity and static bed assumptions make the model less applica-ble in cataracting environments where charge dynamics and dila-tion dominate.

A wide body of literature is also devoted to the understanding ofthe porosity and viscosity – the key parameters in any fundamen-tal fluid mechanics descriptions of transport through packed beds.

Mineral slurries exhibit non-Newtonian behaviour (see Shi andNapier-Munn, 1999; Klimpel, 1984). Beyond this conclusion, theliterature becomes inconsistent as to the nature of the slurry (seeKlimpel, 1984; Kawatra and Eisele, 1988). The combinatorial effectof solids concentration, temperature, chemical environment, parti-cle size and distribution and their response to particle density,interactions and composition has sustained the ambiguity on rheo-logical characterisation (Shi, 1994). The characterisation of viscos-ity is usually given by the slope of the experimentally constitutedrheogram (shear stress versus shear rate curve), however, very little

I. Govender et al. / Minerals Engineering 24 (2011) 230–235 231

is known on the operating shear rate range of the tumbling millsystem – a pertinent issue for the non-linear rheograms that typifymineral slurries. The experimental limitations associated withsettling, especially for dense mineral slurries, have relegated mostrheogram descriptions to high shear rate ranges. The authorshave found no quantitative information on mill operating shearrate ranges in the literature and conclude that much of the pub-lished work is derived for ranges dissimilar to tumbling mills; seeSection 4 for further details supporting this conclusion.

From the early work of Darcy (1856), on viscous flows to thestandard model of flow through unconsolidated static beds (Ergun,1952), there is no denying the overwhelming dependence of flowon porosity (Evans and Civan, 1994). In tumbling mills, the aggres-sive, opaque environment prevent apriori the use of direct mea-surement, favouring non-invasive imaging techniques like highspeed video (Santomaso et al., 2003), bi-planar X-ray imaging(Govender et al., 2004) and PEPT (Parker et al., 1993). The spatialdistribution of the porosity in tumbling mills has been investigatedboth experimentally by Parker et al. (1997) and numerically byYang et al. (2003, 2008). While the calculation from the numericalmodelling is based on explicit knowledge of each particles positionand size, and therefore clear, the PEPT technique uses the residencetime fractional distribution (Wildman et al., 2000; Wildman andParker, 2002), as a proxy for the packing density with the assump-tion that the time-averaged calculations from a single particle arerepresentative of the bed at steady state. Parker et al. (1997) foundthis distribution to be non-uniform for slowly rotating drums with-out lifter bars. The discrete element model used by Yang et al.(2003, 2008), in their numerical work also found a porosity distri-bution across the bed for slow moving mono-sized spheres. Multi-component mixtures and their influence on porosity distributionhave been indirectly investigated up to binary mixtures byYu and Standish (1987, 1991), Baumann et al. (1994), Dury andRistow (1999). However, to the best of our knowledge, no workhas been done on the porosity of multi-component mixtures ofsolid and liquid in tumbling mills.

On the theoretical side, a key issue for the construction of amechanistic model is the correlation of observables. A conven-tional starting point for the discussion of slurry transport is theErgun equation (Ergun, 1952). One must bear in mind, however,that the Ergun correlation between pressure drop Dp perunit length DL, porosity � and superficial flow velocity U for a(Newtonian) fluid with viscosity l and density q, and sphericalparticles of specific area av, was obtained by combining theCarmen–Kozeny and Burke–Plummer equations, and strictlyapplies to static packed beds; outside of that setting it comes withmany caveats (Zoltani, 1992).

Indeed, the solids bed in slurry transport is dynamic rather thanstatic. Yoon and Kunii (1970) in a simple experiment involvingdownward flow of solids at � = 0.37 against upwards gas flow,demonstrated that Dp/DL is a function of the so-called slip veloc-ity, Uslip = U � eV, where V is the solids velocity. Yoon and Kuniisuggested to merely replace U with Uslip in the Ergun equation,and in fact in the circumstance of fluidized beds the usual practice(see e.g. Mabrouk et al., 2007) is to follow this rule of thumb up toporosity � = 0.8.1 Yet there is no real foundation for such a shortcutsince the underlying capillary models are dubious at the higherporosities of eventual interest. Moreover, while an attempt canbe made to extend the derivation of the Carmen–Kozeny equationto dynamic beds by averaging over moving capillary segments,such approaches fail in that they lead to the wrong dependence

1 Fluidization commonly employs a different correlation at higher porosity (Wenand Yu, 1966); as 0.4 6 e 6 0.7 in the mill so there will be no occasion to use theWen–Yu equation.

on superficial and solid velocities. One is thus compelled to a dif-ferent approach.

This paper describes the first efforts to obtain a correlation fordynamical beds. In this the focus will be on viscous flow since thereone has better theoretical control (the turbulence picture implicitin the Burke–Plummer term of Ergun’s equation is subject to con-siderable doubt (Bear, 1972)). For the same reason a restriction toNewtonian fluids is made.

Section 2 describes a fresh attack on the dynamic bed problem:spatial averaging, which is a long standing approach to porousmedia (Whitaker, 1999), is combined with time averaging as famil-iar in the treatment of turbulence (albeit here the motivation is dif-ferent). This leads to a formal relationship between pressure dropand drag on the (moving) solids in the flow.

To realise the drag relation in a practical way, we use a new cellmodel based upon cell averaging. This approach has the advantageof automatically giving the correct dependence on superficial fluidand solid velocities, as well as being free from arbitrary adjustableparameters and applicable at all porosities.

In Section 3, a description of the PEPT experimental program isgiven while the methodology for obtaining key inputs to the newcell averaging model is presented in Section 4. Section 4 also givesa quantitative comparison between the new model and the Ergunapproach. Finally our conclusions and directions for future investi-gations are done in Section 5.

2. Pressure drop model

As a matter of principle all information regarding fluid flow maybe had from the solution of the incompressible Navier–Stokesequations. In practice this is rather too much detail. Moreoverthe information concerning the influence of the solids on the flowis only implicit in the no-slip boundary conditions at the solidsurfaces.

In turbulence theory one smears over detail via time averagingon a scale T:

~uðt;~rÞ ¼ 1T

Z tþT

tdt0~uðt0;~rÞ: ð1Þ

On the other hand purely spatial averaging is a frequently used ap-proach to stationary flow through static packed beds (see Whitaker,1999, and references therein). Writing the fluid volume spatialaverage as

h~ui ¼ �Vf

ZVf

d~r0~u; ð2Þ

and noting that in the case of the dynamic bed one is not concernedwith short time-scale fluctuations so that it is useful to average overboth space and time, we can define

U!� h~ui ð3Þ

and similarly P � hpi. We emphasize that such definitions are par-ticularly appropriate to PEPT experiments, since in the interpreta-tion of PEPT data one invokes the ergodic hypothesis for thetrajectories of individual tracer particles.

In averaging to the continuity equation for an incompressiblefluid, ~r �~u ¼ 0, one encounters a term d�/dt, where � is the averageporosity. This source term represents that in the dynamic bed a netflow of solids into or out of the averaging volume V during the timeinterval T is possible. On sufficiently long time scales one expectsthe system to become stationary. One can then omit the sourceterm, yielding

~r � U!¼ 0 ð4Þ

232 I. Govender et al. / Minerals Engineering 24 (2011) 230–235

Dropping the source and the inertial terms leaves the averagedStokes equation as

q@

@tU!¼ �D

!�~rP þ l~r2 U

!: ð5Þ

The first term on the right hand side of Eq. (5) is of special note:

D!¼ 1

V~FD ð6Þ

wherein ~FD is the time-averaged drag force on the solid particles inthe volume V. In particular, this supports the Yoon–Kunii drag pic-ture for the dynamic bed.

As such, Eq. (5) is only a formal expression since the evaluationof D!

requires a full solution to the fluid dynamic problem. One can,however, make use of Eq. (6): on average D

!is given by the drag

force on a single representative of the N particles divided by theaverage particle volume V/N. This is the essence behind the cellmodel picture; a spherical solid particle of radius r0 is surroundedby a fluid envelope extending to radius r1 such that (r0/r1)3 = 1 � e.Since the situation is stationary and inertial effects are neglected,one only needs the general solution of Stokes equations (Happeland Bremmer, 1965), which involves four constants a, b, c and d.The associated drag force only depends on a; the others are deter-mined by the boundary conditions imposed.

Aside from the no-slip condition at r0, the existing cell modelsdiffer by the conditions at the outer boundary which representthe influence of the other particles. In the Happel cell model(Happel, 1958), the outer boundary is taken to be a free surfacehaving vanishing tangential stress and normal flow. Such condi-tions are not very appropriate for describing flow past a fixedsphere – indeed what is done there is to consider a particle movingwith velocity V

!¼ �U

!to compute the drag then transform the

result to a frame where V!¼ 0. Inevitably, such a construction leads

to D!/ Urel

!¼ U!�V!

– Uslip

!which is unacceptable. The Kuwabara

cell model (Kuwabara, 1959), imposes that the vorticity vanish atr1 and that the radial fluid velocity there is equal to the value faraway. In effect Kuwabara’s model enforces the usual conditionsat infinity for Stokes flow around an isolated sphere at a finiteradius r1. The shortcoming is that there is no obvious correspon-dence between the radial fluid velocity and the superficial velocity(Table 1).

What suggests itself as more appropriate is to retain the condi-tion of zero vorticity at r1 but in the new model impose that theaverage fluid velocity in the envelope be equal to V

!. One finds that

in this ‘cell averaging model’ FD / Uslip automatically. The final re-sult is most conveniently expressed, dropping the time derivativeand inhomogeneous Brinkman terms in Eq. (5), as a modified formof Darcy’s law:

Table 1The milling configurations employed in PEPT experiments.

Description 5 mm glass + slurry

Internal mill diameter (m) 0.3Internal mill length (m) 0.285Volume fill (glass beads) (%) 31.25Slurry solids concentration by volume (%) 8Slurry size fraction (lm) �75 + 53Mass of 5 mm glass beads (kg) 10.2Uncertainty, bead diameter (mm) 0.3Mass of water in mill (kg) 2.3Mass of slurry particles (kg) 0.6Density of glass beads (kg m�3) 2700Density of slurry particles (kg m�3) 2800Density of pulp (kg m�3) 1150Mill speeds (rpm) 46,357.9Volumetric flow rate in (m3 s�1) 8.3 � 10�5

Volumetric flow rate out (m3 s�1) 8.2 � 10�5

�~rP ¼ lKðU!�eV

!Þ: ð7Þ

The permeability K(�) is a function of porosity given by

a2vKðeÞ ¼ 2

ð1� eÞ ½2� e� 1:8ð1� eÞ1=3 � 0:2ð1� eÞ2�: ð8Þ

Of particular note is that this result is free of arbitrary parameters,unlike the capillary model.

By contrast, the Carmen–Kozeny truncation of Ergun’s equationwould yield

�~rP ¼ lKE

U!; a2

vKEðeÞ ¼ 0:24e3

ð1� eÞ2: ð9Þ

These two expressions for the permeability are compared in Fig. 1; atthe small porosity typical of a packed bed there is little difference,however as e exceeds 0.5 the two curves diverge and at large poros-ity the Ergun equation seriously overestimates the permeability.This is readily understood: once the porosity is larger than that ofsimple cubic packing the picture of tortuous capillaries that under-lies the Ergun equation ceases to make sense. In contrast, as e ? 1the cell model tends to the reasonable limit of Stokes flow.

3. Experimental

The experimental rig, Fig. 2, consisted of a mill constructed fromHigh Density Polyethylene (HDPE), a DC drive with step-down gearbox, torque sensor and a reticulating pump for wet experimentsthat re-circulated the fluid. The mill shell and pulp lifters weremanufactured from HDPE (specific density of 0.95) while the lifterswere made from aluminium; see Fig. 3 for lifter geometry.

The experiments are characterised by re-circulating slurry flow-ing through the dynamic, porous network of 5 mm glass beads atsteady state. The slurry was pumped from a sump, through themain chamber of the mill and discharge grate, into the pulp cham-ber and back into the sump. An iterative procedure was thenapplied to ensure standard operating conditions: The pump speedwas adjusted and the mill ‘‘crash stopped” until visual inspectionof the slurry level matched that of the stationary charge, i.e. theinterstices of the stationary bed were completely filled by the slur-ry. Using this observation and the measured volumetric flow ratesat the inlet and outlet, the fluid in the mill at steady state can beestimated. The glass beads off course remain in the mill betweenthe feed end and the discharge grate as they are too large to passthrough the 3 mm diameter grate holes. In each of these experi-ments, a small bluestone particle, roughly 1 mm in size and havingthe same density as the slurry, was employed to represent the flow

Fig. 1. Permeability multiplied by specific surface area squared versus porosity forthe cell model, Eq. (8), and the truncated Ergun equation (9).

Fig. 2. Experimental mill employed in positron-emission-particle-tracking experiments. The right image illustrates the mill between the two detectors of the PEPT unit.

Fig. 3. Lifter geometry: 10 mm high, 12 mm wide with a 51� leading face angle. Fig. 4. Porosity distribution of 5 mm glass beads at 75% of critical speed. Mill wasfilled to 31.25% with the glass beads.

I. Govender et al. / Minerals Engineering 24 (2011) 230–235 233

of the slurry. The limitations of fine particle activation using theBirmingham cyclotron prevented the use of a typical slurry particle(�75 lm + 53 lm).

4. Data analysis

Key inputs to the cell averaging model proposed in Eqs. (7) and(8) are superficial velocity (U), solids velocity (V), viscosity (l) andporosity (�) of which the latter is clearly dominant. Bearing in mindthat the proposed model is an averaged model, it is reasonable totreat the time-averaged velocity of the fluid tracer (1 mm rockfragment) and glass bead as equivalent to U and V respectively atgiven regions in space.

The determination of the porosity distribution is based on themethodology of Wildman et al. (2000): The number density in asegment x of the mill is given by:

nðxÞ ¼ NFðxÞVs

ð10Þ

where N is the number of particles, F(x) is the residence time frac-tion of the tracer in segment x, and Vs is the volume of the segment.Residence time fraction – or the fraction of total observation time aparticle spends in a particular segment – is given by:

F ¼ tT

ð11Þ

where t is the time spent by the particle in a given element and T isthe total system observation time. The packing density (or packingfraction) in segment x is then given by:

gðxÞ ¼ nðxÞpd3

6ð12Þ

where d is the average diameter of a (spherical) particle in the sys-tem. The porosity is then simply

eðxÞ ¼ 1� gðxÞ ð13Þ

Fig. 4 shows the time-averaged porosity distribution of the 5 mmglass beads at 75% of critical speed. It should be noted that the plotis truncated to exclude porosities greater than 0.99. Consequently,the velocity inputs to the pressure drop calculations are also ex-cluded at the corresponding segments.

Recent work by the authors on rheological characterisation ofslurries (Mangesana et al., 2008), identified the difficulty in charac-terising slurry viscosity at low shear rates and/or high solids con-centrations: At solids concentrations greater than 40% by weightthe tube viscometer blocked while at low shear rates the particlessettled out of the slurry mixture at all solids concentrations. Mosttube viscometers are designed to operate at high shear rates withrelatively small solids (�50 lm in size) in order to avoid settlingissues. Unfortunately, typical milling environments operate at rela-tively low shear rates. To illustrate this, the velocity profile alonga diametrical line of the time-averaged velocity field of the 1 mmPEPT tracer particle is shown in Fig. 5 (left). Using the velocitiesand corresponding positions along the diametrical line, a modelfor the shear rate was developed; the reader is referred toGovender et al. (2010) for details of this model. Fig. 5 (right) is a plotof the shear rate along the diametrical line, from which the maxi-mum value is a little above 20 s�1. This is an order of magnitudelower than the comfortable range employed in tube viscometers.

Consequently, a reasonable measure of the viscosity, even at arelatively low solids concentration by volume (u = 0.08) was notpossible at typical tumbling mill shear rates. However, an estimateof the average viscosity can be obtained at low solids concentrationby using the modified Einstein model according to Kunitz (1926):

Fig. 5. Velocity profile (left) and corresponding shear rate (right) along a diametrical line passing through the mean centre of circulation.

Fig. 6. (From left to right) slurry velocity, slip velocity, cell averaged model pressure drop, and difference between the cell model and Ergun pressure drops, in the azimuthaldirection for the mill operating at 60% of critical (top) and 75% of critical (bottom).

Fig. 7. (From left to right) slurry velocity, slip velocity, cell model pressure drop, and difference between the cell model and Ergun pressure drops, in the axial direction for themill operating at 60% of critical (top) and 75% of critical (bottom).

234 I. Govender et al. / Minerals Engineering 24 (2011) 230–235

gs ¼ gw1þ 0:5uð1�uÞ4

" #ð14Þ

where gs and gw is the absolute viscosity of the slurry mixture andwater respectively, and u is the volume fraction of solids to slurrymixture (solids + water). Hence using the known viscosity for water(gs = 5 � 104 Pa s at 50 �C), the slurry viscosity is 7.3 � 10�4 Pa s.

The results are shown in Figs. 6 and 7, for the azimuthal andaxial directions respectively, with the later being an average persegment across the entire length of the main milling chamber(0.285 m). In each case we have checked that including the Brink-man term has a negligible effect on the pressure drop. It should benoted that the magnitudes are truncated to ensure discernablevisual differences in the colour plots. Therefore, the hottest regionsin the plot do in fact span higher values.

In the azimuthal direction there is a clearly visible distinctionbetween the superficial slurry velocity and the slip velocity. Thisis to be expected since the slip velocity includes the charge motionwhich is large in this direction. Curiously, however, there is littledifference between the pressure drop calculated from the cell aver-aged model, Eqs. (7) and (8), and using the simple truncated Ergunequation (9). One can understand the latter as follows: in the re-gions where Ergun overestimates the permeability, and so wouldunderestimate the pressure drop, there is a compensating effectthat the superficial velocity is larger than the slip velocity.

In contrast, for the axial direction there is no average chargemotion so the superficial and slip velocities coincide. Consequentlythere is a marked difference between the pressure drops calculatedfrom the cell averaged model and the truncated Ergun equation.This is entirely due to the difference in the two expressions forthe permeability.

I. Govender et al. / Minerals Engineering 24 (2011) 230–235 235

5. Conclusions

The first steps towards a mechanistic model have been taken: anew pressure drop correlation for the dynamic bed is proposedwhich does not rely upon artificial concepts (such as tortuous cap-illaries), and employs the physical slip velocity combined with apermeability that is free of arbitrary parameters. Results obtainedusing PEPT data, Figs. 6 and 7, show clear differences betweenour correlation and the Ergun approach, with the (truncated) Ergunequation giving a lower axial pressure drop. For the observedporosity distribution, Fig. 4, a pressure distribution is seen in themill. Since the limitations of the tube viscometer prevented ameaningful characterisation of the viscosity distribution an aver-age viscosity was used according to Kunitz (1926).

The future work will include a smaller (<100 lm) slurry tracer,an improved viscosity measurement and uncertainty estimates ofthe pressure drop that will quantitatively determine the statisticalvalidity of the qualitative differences observed here. The axial aver-aging will also be done in smaller segments to explore the pressuredrop as a function of axial position. With more realistic solids con-centrations (40–60% solids by mass) the cell model must be revisedto deal with the non-Newtonian slurry.

Acknowledgement

The authors acknowledge the Positron Imaging Centre,University of Birmingham for use of the PET scanner in conductingthese experiments.

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