Power draw estimations in experimental tumbling mills using PEPT

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    Tracumclelatine avomp

    2010 Elsevier Ltd. All rights reserved.

    The power draw of a tumbling mill is known to be an importantmeasure in determining its efciency. M

    a functSchnocto calobserv(Gove

    ge motironmng em

    method 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

    given distribution including power draw.

    the 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

    Corresponding author at: Centre for Minerals Research, Department of Chem-ical Engineering, University of Cape Town, South Africa. Tel.: +27 21 650 5554; fax:+27 21 021 650 5554.

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

    Minerals Engineering 24 (2011) 319324

    Contents lists availab

    Minerals En

    journal homepage: www.els1 Tel.: +27 21 650 5520; fax: +27 21 650 5501.describe the distribution of power draw into the charge. Thus,the charge has often been simplied to a single bulk body over adened 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 this

    et al., 2004). The premise of the method is the positron annihilationof a tracer, a particle taggedwith 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) denes a straight linealongwhich 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 ofderived to predict the power drawaslated to charge motion (Harris andWhile these models have been showntions of mill power, they have beenscope under which they were dened

    As in situ characterisation of chardue to the aggressive internal envmany models have focused on usi0892-6875/$ - see front matter 2010 Elsevier Ltd. Adoi:10.1016/j.mineng.2010.10.005any models have beenion of characteristics re-k, 1985; Morell, 1992).culate good approxima-ed to be limited to thender et al., 2001a).ion has proved difcultent of tumbling mills,pirical relationships to

    2. Positron Emission Particle Tracking (PEPT)

    Positron Emission Particle Tracking (PEPT) is a technique formeasuring the ow trajectory of a radioactive particle in a granularoruid systemsuchas a tumblingmill. This techniquewasoriginallyintroduced in the medical eld as positron emission tomography(PET), andhasbeenmodied to suit engineeringapplications (Barley1. Introduction et al., 2002). The unique value of this aspect is that data from PEPTcan be used to calculate charge properties for every size within aPower draw estimations in experimental

    L.S. Bbosa a,1, I. Govender a,b,, A.N. Mainza a, M.S. PoaCentre for Minerals Research, Department of Chemical Engineering, University of CapebDepartment 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

    Article history:Available online 15 December 2010

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

    a b s t r a c t

    Positron Emission Particlebeads in an experimental trepresentative tracer partiTwo approaches for calcuthe mill centre, and the timof the mill. Results were cpower.ll rights reserved.mbling mills using PEPT

    ll c

    n, South Africa

    king (PEPT) was employed to reconstruct the motion of mono-sized glassbling mill run in batch mode. In each case, the derived trajectory eld of awas used to determine the charge power draw at steady state operation.g power draw were considered: the torque of the centre of mass abouteraged torque contribution per discrete grid cell summed over the volumeared across different operating conditions and particle sizes to measured

    evier .com/locate /minengle at ScienceDirect

    gineering

  • ngin320 L.S. Bbosa et al. /Minerals EBirmingham. 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%

    Fig. 1. PEPT camera in parallel plate cong

    Fig. 2. Picture and schematic of 300 mm tumeering 24 (2011) 319324volumetric lling 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

    uration and schematic of its operation.

    bling mill used for PEPT experiments.

  • The velocity was calculated using a central difference approxi-

    5.2. Centre of mass approach

    For the rst 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 previously

    i__

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

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

    ngineering 24 (2011) 319324 321mation 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.

    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 K V x; 1where 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 torquein 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.

    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

    L.S. Bbosa et al. /Minerals Esensor were observed to uctuate in a sinusoidal motion with therotation of the mill. The amplitude of these uctuations was0.1 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 DV

    V

    2 Dx

    x

    2s; 2

    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).-0.1 -0.05 0 0.05 0.1 0.15

    -0.1

    -0.05

    0

    0.05

    0.1

    0.15

    calculated 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 M g R cosh x; 3where 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-

    1 The mean tracer position would only equal the centre of mass position if the PEPTsampling rate were very high (>1 104 s1).

  • ngin322 L.S. Bbosa et al. /Minerals Estructed to incorporate the residence time distribution accordingto the methodology by Sichalwe et al. (2010). Thus the power draw

    Table 2Summary of power draw measurements and calculations.

    Size (mm) Speed (% crit) PM (W) DPM

    3 60 25.82 0.9475 32.31 0.50

    5 50 19.69 0.4460 23.74 0.6175 29.83 1.11

    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 eld (column 2critical speeds respectively.eering 24 (2011) 319324using the centre of mass approach could be determined from PEPTdata using:

    PCOM (W) DPCOM PBIN (W) DPBIN

    21.39 1.84 25.51 1.0729.36 1.40 32.34 1.07

    17.21 1.36 19.98 1.0721.70 1.64 23.83 1.0728.89 1.85 29.70 1.07

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

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

  • PCOM M g xTXni1

    xi ti; 4

    where ti gave the time spent by the tracer in bin i, xi dened 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 Dtwas the error associated with the average residence time, t. Thiswas essentially the tting 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 Dtt

    2 Dx

    x

    2s: 5

    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 M gTXni1

    xi ti ai; 6

    L.S. Bbosa et al. /Minerals Engineering 24 (2011) 319324 323Fig. 8. Plots of residence time fractional distribution (column 1), velocity eld (column 2critical speeds respectively.) and power draw distribution in Watts (column 3) for 5 mm charge at 60% and 75%

  • while the corresponding error propagation yielded:

    DPBIN

    Xni1

    Dtiti

    2 Dai

    ai

    2svuut; 7

    sition. Vital trends in the charge motion could be identied from

    7. Conclusions

    Two techniques were presented for calculating power draw

    324 L.S. Bbosa et al. /Minerals Engineering 24 (2011) 319324these 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.Dai was the standard deviation of the average angular velocity inthe ith bin and Dti 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 signicant 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 reect 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 gre...

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