Minerals Engineering 24 (2011) 319–324

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Minerals Engineering

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

Power draw estimations in experimental tumbling mills using PEPT

L.S. Bbosa a,1, I. Govender a,b,⇑, A.N. Mainza a, M.S. Powell c

a Centre for Minerals Research, Department of Chemical Engineering, University of Cape Town, South Africab Department of Physics, University of Cape Town, South Africac Julius Kruttschnitt Mineral Research Centre, University of Queensland, Australia

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

Article history:Available online 15 December 2010

Keywords:Positron Emission Particle Tracking (PEPT)Power drawResidence time

0892-6875/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.mineng.2010.10.005

⇑ Corresponding author at: Centre for Minerals Resical Engineering, University of Cape Town, South Afric+27 21 021 650 5554.

E-mail addresses: lawrence.bbosa@uct.ac.za (L.S.uct.ac.za (I. Govender).

1 Tel.: +27 21 650 5520; fax: +27 21 650 5501.

Positron Emission Particle Tracking (PEPT) was employed to reconstruct the motion of mono-sized glassbeads in an experimental tumbling mill run in batch mode. In each case, the derived trajectory field of arepresentative tracer particle was used to determine the charge power draw at steady state operation.Two approaches for calculating power draw were considered: the torque of the centre of mass aboutthe mill centre, and the time averaged torque contribution per discrete grid cell summed over the volumeof the mill. Results were compared across different operating conditions and particle sizes to measuredpower.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The power draw of a tumbling mill is known to be an importantmeasure in determining its efficiency. Many models have beenderived to predict the power draw as a function of characteristics re-lated to charge motion (Harris and Schnock, 1985; Morell, 1992).While these models have been shown to calculate good approxima-tions of mill power, they have been observed to be limited to thescope under which they were defined (Govender et al., 2001a).

As in situ characterisation of charge motion has proved difficultdue to the aggressive internal environment of tumbling mills,many models have focused on using empirical relationships todescribe the distribution of power draw into the charge. Thus,the charge has often been simplified to a single bulk body over adefined region of the mill. It has been noted that in order to intro-duce more informative power draw functions, greater understand-ing of the fundamental mechanisms associated with charge motionis necessary (Govender et al., 2001b).

Positron Emission Particle Tracking (PEPT) offers a way ofstudying the internal environment of tumbling mills. PEPT is atechnique by which trajectory information of single particles intumbling mills can be obtained (Parker et al., 1997). With thismethod it has been shown that the bulk properties of a particularsize class can be ascertained from tracking the motion of singleparticles within that size class at steady state (Conway-Baker

ll rights reserved.

earch, Department of Chem-a. Tel.: +27 21 650 5554; fax:

Bbosa), indresan.govender@

et al., 2002). The unique value of this aspect is that data from PEPTcan be used to calculate charge properties for every size within agiven distribution including power draw.

2. Positron Emission Particle Tracking (PEPT)

Positron Emission Particle Tracking (PEPT) is a technique formeasuring the flow trajectory of a radioactive particle in a granularor fluid system such as a tumbling mill. This technique was originallyintroduced in the medical field as positron emission tomography(PET), and has been modified to suit engineering applications (Barleyet al., 2004). The premise of the method is the positron annihilationof a ‘‘tracer”, a particle tagged with a radionuclide. Positron-emittingtracers are normally labelled using radionuclides such as 18F, 64Cuand 68Ga. These radionuclides decay by emission of back to backgamma rays of 511 keV. Simultaneous detection of the two gammarays in an array of detectors (a PET ‘‘camera”) defines a straight linealong which the particle position lies. At a frequency of up to 250 Hz,the position of the particle can be triangulated in three dimensions.

The accuracy of the method depends on factors such as thespeed and activity of the particle, as well as the attenuation ofthe medium in which it travels. Fig. 1 shows a picture of an exper-imental mill in a parallel plate PEPT camera system (Positron Imag-ing Centre, University of Birmingham), along with a schematicdescribing the method used to detect and triangulate particlepositions.

3. Experimental methodology

Single particle tracking experiments using PEPT were conductedfor this study at the Positron Imaging Centre, University of

Fig. 1. PEPT camera in parallel plate configuration and schematic of its operation.

Fig. 2. Picture and schematic of 300 mm tumbling mill used for PEPT experiments.

320 L.S. Bbosa et al. / Minerals Engineering 24 (2011) 319–324

Birmingham. A 300 mm diameter mill with a variable speed drivewas designed for this purpose, whose picture and schematic is pro-vided in Fig. 2. A torque transducer was coupled to the drive shaftto measure the power draw of the mills. Spherical glass beads wereused as the dry charge. Tests were conducted using either 3 mm or5 mm charge. To determine the mass requirements for 31.25%

volumetric filling of the mill, the bulk density of the glass beadswas determined by assuming a packing ratio of 0.6.

Glass beads for either size were subjected to direct activationusing a 33 MeV 3He beam to produce the radioactive tracer parti-cles. The resulting positron emitter was 18F (which has a half lifeof 109 min). Experiments with each tracer particle were conducted

Table 1Summary of PEPT experiments investigated in this study.

PEPT mill

Internal diameter (m) 0.3Internal length (m) 0.27% Filling by volume 31.25No. of lifters 20Speeds investigated (% mill critical speed) 50, 60, 75

Glass bead size (mm) Charge mass (kg)

Mono-size dry3 9.6625 9.662

iα__

-0.1 -0.05 0 0.05 0.1 0.15

-0.1

-0.05

0

0.05

0.1

0.15

θ

Fig. 3. Diagram illustrating power draw calculation using centre of mass approach.

L.S. Bbosa et al. / Minerals Engineering 24 (2011) 319–324 321

in 1 h durations. Table 1 summarises the experiments that wereconducted in this work.

4. Treatment of data

The Cartesian coordinates and logged time of the tracer particlewere imported into MATLAB (Mathworks, 2009a), which was usedto perform all the analyses required for this work. In order toexamine charge behaviour in the azimuthal plane of the mill, themill face was divided into a 50 � 50 set of discrete squares. Allaverage quantities per rectangular bin excluded data near the feedand discharge grate. Consequently, the charge was assumed to beaxially symmetric.

The velocity was calculated using a central difference approxi-mation scheme. The mass distribution was calculated using the to-tal charge mass, M, weighted by the normalized residence timefraction (RTF) in each bin; see Sichalwe et al. (2010) for a detailedexplanation.

θ

Fig. 4. Diagram illustrating power draw calculation using torque per bin approach.

1 The mean tracer position would only equal the centre of mass position if the PEPTsampling rate were very high (>1 � 10�4 s�1).

5. Power draw formulation

5.1. Measurement of power draw

A torque transducer coupled to the drive shaft provided voltagereadings directly proportional to the dynamic torque applied onthe shaft. Therefore, for each PEPT experiment the mean measuredpower draw (PM) was calculated by:

PM ¼ K � V �x; ð1Þ

where K = 2.1 was the calibration factor, V the average voltage atsteady state operation, and x was the angular velocity of the millin radians per second. The voltage readings from the torquesensor were observed to fluctuate in a sinusoidal motion with therotation of the mill. The amplitude of these fluctuations was�0.1 V (voltage ranged between 2 V and 4 V). Combining thisuncertainty with that of the angular velocity, the resulting propa-gated error in the measured power draw was determined usingthe following equation:

DPM ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDV

V

� �2

þ Dxx

� �2s

; ð2Þ

where DV was the standard deviation of the measured voltage,while Dx was 0.02 radians per second.

Two methods were investigated to calculate power draw fromPEPT data. These were named as follows:

� Centre of mass approach (PCOM).� Summed torque per bin method (PBIN).

5.2. Centre of mass approach

For the first approach, the power draw was calculated using avariation of the ‘‘torque arm” principle commonly used for tum-bling mills in comminution literature; for example Harris and Sch-nock, 1985. The centre of mass was based on the previouslycalculated residence time fraction distribution and not necessarilyequal to the mean position of the PEPT tracer coordinates.1 Theeffective power draw of the entire charge body (PCOM) – approxi-mated as a continuum – was thus the moment due to the centreof mass about the mill centre multiplied by the rotational speed ofthe mill,

PCOM ¼ M � g � R � cosðhÞ �x; ð3Þ

where M was the total mass of the charge, R the torque arm radiusfrom the mill centre to the centre of mass, g the acceleration due togravity (9.81 m/s2), and h the angle between the x-axis and the ra-dial arm R, as shown in Fig. 3.

Noting that R � cos(h) was simply the x-coordinate of the centreof mass in meters, the centre of mass power draw was recon-

Table 2Summary of power draw measurements and calculations.

Size (mm) Speed (% crit) PM (W) DPM PCOM (W) DPCOM PBIN (W) DPBIN

3 60 25.82 0.94 21.39 1.84 25.51 1.0775 32.31 0.50 29.36 1.40 32.34 1.07

5 50 19.69 0.44 17.21 1.36 19.98 1.0760 23.74 0.61 21.70 1.64 23.83 1.0775 29.83 1.11 28.89 1.85 29.70 1.07

Fig. 6. Plot of power draw values obtained for PEPT experiments (5 mm charge).

Fig. 5. Plot of power draw values obtained for PEPT experiments (3 mm charge).

Fig. 7. Plots of residence time fractional distribution (column 1), velocity field (column 2) and power draw distribution in Watts (column 3) for 3 mm charge at 60% and 75%critical speeds respectively.

322 L.S. Bbosa et al. / Minerals Engineering 24 (2011) 319–324

structed to incorporate the residence time distribution accordingto the methodology by Sichalwe et al. (2010). Thus the power draw

using the centre of mass approach could be determined from PEPTdata using:

L.S. Bbosa et al. / Minerals Engineering 24 (2011) 319–324 323

PCOM ¼M � g �x

T

Xn

i¼1

xi � ti; ð4Þ

where ti gave the time spent by the tracer in bin i, xi defined the xdistance to each bin, T gave the total time of the experiment andn the total number of bins.

The propagated error for this approach yielded Eq. (5), where D�twas the error associated with the average residence time, �t. Thiswas essentially the fitting error of the (smooth) arc length thatconnected consecutive PEPT data points passing through the binof interest (see Sichalwe et al. (2010) for more details), while Dxwas as before.

DPCOM ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD�t�t

� �2

þ Dxx

� �2s

: ð5Þ

Fig. 8. Plots of residence time fractional distribution (column 1), velocity field (column 2critical speeds respectively.

5.3. Torque per bin approach

This method was developed from the hypothesis that a moreaccurate estimate of the power draw would be to sum up the indi-vidual ‘‘torque arm” power contributions of every bin and use theaverage angular velocity in each bin rather than that of the mill.Again, the mass in each bin was computed by multiplying the res-idence time fraction by the total charge mass, where xi denoted thex-coordinate of the bin, while �ai denoted the average angularvelocity of the bin – calculated using the average tangential veloc-ity divided by the radial position of the bin as shown in Fig. 4.

By this method, the power draw was determined using the fol-lowing equation:

PBIN ¼M � g

T

Xn

i¼1

xi � ti � �ai; ð6Þ

) and power draw distribution in Watts (column 3) for 5 mm charge at 60% and 75%

324 L.S. Bbosa et al. / Minerals Engineering 24 (2011) 319–324

while the corresponding error propagation yielded:

DPBIN ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD�ti

�ti

� �2

þ D�ai

�ai

� �2svuut

; ð7Þ

D�ai was the standard deviation of the average angular velocity inthe ith bin and D�ti was as before.

6. Results and discussion

Table 2 is a summary of the power draw values and associatederrors for the two methods described. The measured power drawfor the smaller sized 3 mm bead charge was found to be higherat both speeds than the 5 mm charge. This was supported by thenormalized residence time distribution plots in Figs. 7 and 8, whichsuggested that the smaller, 3 mm size seemed, on average, to beradially further from the mill centre than the larger, 5 mm sizethereby producing a larger average torque arm and ultimately alarger power draw. Additionally, the 5 mm beads showed aclear tendency to concentrate around the centre of circulationwhile the 3 mm charge depicted a clear depletion of charge inthe same region. The mechanism for this segregation to distinctlydifferent regions was unclear. It was deemed that further investi-gation with a wider range of sizes would be required to determinewhether this behaviour was significant and to identify its underly-ing mechanism.

It was found that the centre of mass approach under estimatedthe measured power. This was attributed to the assumption thatthe entire charge was moving at the angular velocity of the mill.It has been shown that the charge body circulates at a faster ratethan the mill speed, becoming closer to the mill rotation rate asmill speed increases, until eventually converging to the mill rateof rotation when the entire charge centrifuges (Kallon et al.,2010). Fig. 5 – and particularly Fig. 6 – supported these observa-tions, although it was to be noted that more data would be neededto prove this conjecture. It was thus hypothesised that the centre ofmass power would need to factor in the circulation rate of thecharge in order to reflect the true power draw.

The torque per bin approach yielded power values that wereclosest to the measured power. This was to be expected as thismethod considered the individual power contributions of everycell in the grid. In Figs. 5 and 6, values of power draw obtainedfor the different methods were plotted with their error bars. Thesegraphs highlighted that the PBIN method was within statisticalagreement with the measured power draw for all experimentsand the PCOM method was not.

From the torque per bin method, the power draw distribution ofthe charge could be examined in greater detail. In Figs. 7 and 8,normalized distributions of the residence time fraction, velocityand power draw contribution for the conducted experiments wereplotted. These showed that the regions that drew the most powerwere in the rising charge where the particle spent the greatesttime. For velocity field plots the arrows pointed in the directionof motion while their length represented the speed at the given po-sition. Vital trends in the charge motion could be identified fromthese plots, such as the centre of circulation (McBride et al.,2003) – a key position about which the circulation rate hypothesisof power draw could be tested.

7. Conclusions

Two techniques were presented for calculating power drawfrom PEPT experiments. Using the torque arm approach (PCOM),the charge centre of mass was calculated as the moment fromthe mean position of tracer coordinates multiplied by the mill’srotational speed. For the second method, the mill profile was di-vided into discrete bins in which the individual torque and angularvelocity contributions were multiplied and cumulated to obtainthe power draw (PBIN).

It was found that the centre of mass approach under estimatedthe measured power, due to the assumption that the entire chargewas moving at the angular velocity of the mill. The lower powerdraw values were likely attributed to the angular velocity of thecharge being greater than that of the mill, particularly for lowerspeeds.

The torque per bin approach yielded power values that wereclosest to the measured power. Power draw values using thecumulated torque per bin method were within statistical agree-ment with the measured power draw for all the tests conducted.It was thus concluded to be the optimal method for determiningpower draw from tumbling mill tests using PEPT.

Acknowledgements

The author would like to thank the Positron Imaging Centre atthe University of Birmingham for the use of their facilities in carry-ing out PEPT experiments. Acknowledgements are also extended toAnglo Platinum and the Centre for Sustainable Resource Processing(CSRP) for funding this work.

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