Analysis of holdups in continuous ball mills

Powder Technology 235 (2013) 443–453

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Analysis of holdups in continuous ball mills

Shinichiro NomuraHiro-Ohshingai 2-15-26, Kure, Hiroshima, 737-0141 Japan

E-mail address: [email protected].

0032-5910/$ – see front matter © 2012 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.powtec.2012.10.053

a b s t r a c t
a r t i c l e i n f o
Article history:Received 22 April 2012Received in revised form 25 October 2012Accepted 27 October 2012Available online 7 November 2012

Keywords:HoldupDischarge propertiesDead spaceResidence time distributionContinuous ball millFriction factor

A theoretical study is made on holdups of particulate material in continuous ball mills. The analysis is basedon a consideration that holdups are influenced by the discharge properties at the mill exit, the existence ofmedia balls interrupting the material to flow towards the mill exit and the dead space generated dependingon the exit configuration. As a result, the linear relation between holdup and feed rate obtained experimen-tally is derived to be valid under specific discharge properties. Further, the effects of other operating variablessuch as ball filling, ball diameter, mill speed and slurry density on holdups are clarified based on physicalbackgrounds deduced from the theory. The derived equation to predict holdups includes three constantsrelating to the discharge properties of mill exit, the resistance of media balls against material flow and thedead space, respectively. These constants are estimated by the least square regression to give a best fit todata of various operating conditions within reasonable convergence between calculated holdups and exper-imental ones. Also, the effect of material kind on holdup is discussed using existing data.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

To design mills or improve current grinding performances in con-tinuous ball mills, precise predictions of holdups of particulate mate-rial are of great importance. That is, holdups affect the residence timedistributions [1–6] and the grinding rates [7,8], both of which deter-mine the size distributions of ground product.

Holdups were measured in various ball mills, sometimes as one ofimportant factors controlling the residence time distributions. In drygrinding, Mori et al. [1] and Swaroop et al. [2,3] reported holdups inrelation to some operating variables such as ball filling, feed rate, millspeed and ball size. Of these, Swaroop et al. demonstrated linear rela-tions between holdup and feed rate. Wet mill data in past literatures[4,5,9–11] also showed an increase of holdup with increasing feedrate. The measurement techniques were also improved, e.g., Songfackand Rajamani [11] developed an apparatusweighing directly an operat-ed mill to enable the real time variation of the mass holdup to bemeasured.

For such experimental findings, there were some theoretical studiesreported. Tanaka [12] considered that the flow ofmaterial was governedby the resistance of the mill exit opening. He examined his theory withdata of Mori et al. [1] although some parameters were arbitrarily speci-fied. Gupta et al. [13] reported an empirical equation to predict holdups.They analyzed the data of four differentmaterials [9,10] that the effect ofmaterial kind on holdup was related to the shape of the size distributioncurve and the work index was used as the material characteristicparameter.

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Despite those experimental and theoretical studies, no predictivemethods useful for various mills and operating conditions have beendeveloped. Difficulty may exist in how to characterize the materialtransport in a mill in relation to operating conditions and the mill exitstructure and how to connect the characterized material transportwith holdups. Therefore, data of onemill have not been successfully cor-related to those of othermills with different conditions or exit openings.

This paper aims at establishing a theoretical basis to express holdupsin terms of operating parameters in a general form applicable to variousball mills. The effect of mill exit structure is not fully discussed as vari-ous exit arrangements are possible.

2. Theoretical

2.1. Relation between holdup and feed rate

The mass holdup of material denoted as MH is expressed as

MH ¼ ρLAJεbU ¼ ρLAH ð1Þ

where ρ is the density of material (the bulk density of particles ρpb fordry grinding or the density of slurry ρsl for wet grinding), L is the milllength, A is the cross sectional area of mill equal to (πD2/4), D is themill diameter, J is the fractional mill volume occupied by the ballbed, εb is the voidage of static ball bed, U is the fractional ball bedvoid filled with material in a static mill and AH is the cross sectionalarea of holdup.

To discharge continuously the ground material, various mill exitarrangements such as screen apertures or slit openings are adopted.The discharge properties may be represented by the discharge velocity

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D

Grinding zone

Ascending zone

Falling zone

Fig. 1. Schematic illustration of cross section of tumbling ball mill.

0.8

0.9

1.0

[ra

d]

Sand-0.5mm

Ball

n

R

uupb

u

Fig. 3. Forces acting on ball surface due to flow of particles.

444 S. Nomura / Powder Technology 235 (2013) 443–453

at the opening vop and the fractional cross sectional area opened for dis-chargeφs. A small difference in AH denoted as dAH leads to a small differ-ence in the discharge rate given as (ρvopφsdAH), resulting in a smalldifference in the feed rate denoted as dF at steady state, then,

ρvopφsdAH ¼ dF ð2Þ

Combining Eqs. (1) and (2) to eliminate dAH, a differential equa-tion is obtained to be

dMH ¼ ρLdAH ¼ ρLdF

ρvopφs¼ L

vopφs

!dF ð3Þ

Eq. (3) is integrated with the initial condition of MH=MHo, themass of material retained in a mill at F=0. For constant φs and vop,the integration is reduced to

MH ¼ Lφsvop

!F þMHo ¼ c1LF þMHo ð4Þ

where c1 is a constant expressing the discharge properties at the millexit. Eq. (4) reveals that the linear relation between MH and F is validwhen φs and vop are assumed to be constant anywhere on the crosssectional area effective for discharging.

2.2. Specification of MHo

Two causes to retain material in a ball mill are considered, one is theexistence of the ball bed to trap material and the other is a dead spacecreated depending on the mill exit structure. Denoting the mass ofmaterial trapped in the ball bed as MHb and that left in the dead spaceasMHd, MHo is given by

MHo ¼ MHb þMHd ð5Þ

These two masses are considered in the following sections.

d

dDD

HdA

bo

dr r

Ball bed with voidage of

Dead space

Fig. 2. Schematic illustration of dead space in an overflow or constricted ball mill.

2.2.1. Mass of material trapped in ball bed MHb

The mill charge, balls and particulate material, is divided into threezones [6,8] as depicted in Fig. 1, i.e., the grinding zone where balls col-lide each other to grind particles, the ascending zone where ballsmove upwards along the mill wall and the falling zone where balls cas-cade or cataract. When operated continuously, the material flowstowards the mill exit. Simultaneously, the material flow is interruptedby the ball bed formed in the grinding and ascending zones. Balls inthe falling zone are regarded not to be effective for interrupting thematerial flow.

The mass of material trapped in the ball bed MHb depends on thedegree of friction on the ball surface for material to flow and the sur-face area of balls interrupting the flow. Assume that MHb/ρ, the volu-metric one, is proportional to the friction factor f and the surface areaof balls in the grinding and ascending zones denoted as Abg, i.e.,

MHb

ρ¼ c2f Abg ¼ c2f

VRb Δtg þ Δta� �πd3b=6

πd2b� �2

435

¼ c2f6VRb Δtg þ Δta

� �db

24

35 ð6Þ

where c2 is a constant, VRb is the volumetric flow rate of balls in a cir-cumferential direction of the mill, Δtg and Δta are the mean residencetimes of balls in the grinding and ascending zones, respectively, anddb is the ball diameter. The derivations of VRb, Δtg and Δta are notedin Appendix A in the preceding paper [6].

0.5

0.6

0.7

0.3 0.4 0.5 0.6

[-]

Fig. 4. Angle of repose in relation to voidage of particles for sand [14].

Table 1Data of Mori et al. [1] and calculated results.

(a) Pilot scale mill (D=0.545 m, L=1.98 m) with slit screen plate for discharge

Experiment Calculated

J F MH U db fw f Abg MHb εbo MHd MH

− kg/s kg − m − − m2 kg − kg kg

Case M-1 0.10 0.0180 17.70 0.65 0.03 0.53 0.655 4.04 1.17 0.40 14.80 19.300.10 0.0385 23.00 0.84 0.03 0.53 0.655 4.04 1.17 0.40 14.80 22.950.10 0.0685 31.60 1.16 0.03 0.53 0.655 4.04 1.17 0.43 16.93 30.710.10 0.0943 41.50 1.52 0.03 0.53 0.655 4.05 1.18 0.50 21.97 40.400.10 0.1257 57.10 2.09 0.03 0.53 0.655 4.07 1.18 0.57 30.03 54.10

Case M-2 0.20 0.0722 39.50 0.72 0.03 0.53 0.655 8.37 2.43 0.40 30.64 46.310.20 0.1067 50.60 0.93 0.03 0.53 0.655 8.37 2.43 0.40 30.64 52.530.20 0.1287 56.60 1.03 0.03 0.53 0.655 8.37 2.43 0.41 31.59 57.68

Case M-3 0.30 0.0680 56.00 0.68 0.03 0.53 0.655 13.00 3.77 0.40 47.58 63.850.30 0.1067 65.70 0.80 0.03 0.53 0.655 13.00 3.77 0.40 47.58 70.760.30 0.1272 73.30 0.89 0.03 0.53 0.655 13.00 3.77 0.40 47.58 74.63

Case M-4 0.40 0.0690 83.80 0.77 0.03 0.53 0.655 18.02 5.23 0.40 65.96 83.870.40 0.1003 95.70 0.87 0.03 0.53 0.655 18.01 5.23 0.40 65.96 89.510.40 0.1358 110.10 1.01 0.03 0.53 0.655 18.02 5.23 0.40 66.32 96.29

Case M-5 0.10 0.0138 19.38 0.71 0.04 0.53 0.655 3.03 0.88 0.40 14.80 18.230.10 0.0305 22.22 0.81 0.04 0.53 0.655 3.03 0.88 0.40 14.80 21.310.10 0.0523 29.40 1.08 0.04 0.53 0.655 3.03 0.88 0.42 15.82 26.280.10 0.0647 31.73 1.16 0.04 0.53 0.655 3.03 0.88 0.43 16.99 29.700.10 0.0967 42.94 1.57 0.04 0.53 0.655 3.04 0.88 0.51 22.71 41.290.10 0.1342 59.51 2.18 0.04 0.53 0.655 3.05 0.89 0.58 31.29 56.39

Case M-6 0.20 0.0737 39.02 0.71 0.04 0.53 0.655 6.28 1.82 0.40 30.64 45.950.20 0.0950 47.29 0.86 0.04 0.53 0.655 6.28 1.82 0.40 30.64 49.830.20 0.1335 57.35 1.05 0.04 0.53 0.655 6.28 1.82 0.41 31.97 58.18

Case M-7 0.30 0.0705 53.90 0.66 0.04 0.53 0.655 9.75 2.83 0.40 47.58 63.240.30 0.1000 61.81 0.75 0.04 0.53 0.655 9.75 2.83 0.40 47.58 68.700.30 0.1347 69.40 0.85 0.04 0.53 0.655 9.75 2.83 0.40 47.58 75.12

Case M-8 0.40 0.0605 82.94 0.76 0.04 0.53 0.655 13.51 3.92 0.40 65.96 80.930.40 0.1068 92.24 0.84 0.04 0.53 0.655 13.51 3.92 0.40 65.96 89.540.40 0.1355 100.44 0.92 0.04 0.53 0.655 13.51 3.92 0.40 65.96 95.08

Steel ball: ρb=7800 kg/m3, εb=0.4, Limestone: ρp=2700 kg/m3, ρpb=1480 kg/m3 (εp=0.45).

445S. Nomura / Powder Technology 235 (2013) 443–453

2.2.2. Mass of material left in dead space MHd

A dead space is generated depending on the mill exit configura-tion. Fig. 2 illustrates an overflow or constricted-end mill. The shadedregion is the dead space. Also, in a mill with a slit screen plate, theblocked surface area of the end wall may be wide enough to generatedead spaces inside the mill, e.g., the value of φs equal to 0.113 in themill tested by Mori et al. [1] means that almost 90% of the wall surfaceis blocked. The volume of material left in the dead space MHd/ρ isassumed to be proportional to the geometrically estimated volumeof the dead space whose cross sectional area is denoted as AHd, i.e.,

MHd

ρ¼ c3 LAHd½ �εbo ð7Þ

where c3 is a constant and εbo, the voidage of the ball bed in the grind-ing and ascending zones, is needed to subtract the volume of balls inthe dead space.

For an overflow or constricted-end mill, AHd is geometricallyobtained as (see Fig. 2)

AHd ¼ ∫D=2

Dd=2

2D2sinθ

� �dr ¼ A

2θd− sin2θd2π

� �ð8Þ

where θd, equal to arccos (Dd/D), is the angle designating the surfacelevel of the dead space and Dd is the diameter of the exit opening.

In the case of a slit screen type, AHd is given by the blocked surfacearea of the end wall covered by the grinding and ascending zones, i.e.,

AHd ¼VRb Δtg þ Δta� �1−εboð ÞL

24

35 1−φsð Þ ð9Þ

where the term in the square brackets is the cross sectional area ofthe grinding and ascending zones and (1−φs) is the fractional sur-face area blocked.

Note that some of the material trapped in the ball bed is alsocounted as the one left in the dead space. This double count is com-pensated by the values of c2 and c3 when estimated by the leastsquare regression based on holdup data, meaning that c2 is a constantdepending on the mill exit type or the dead space.

2.3. Friction factor f

Consider a number of hypothetical pipes with length L in the ballbed through which material flows at the velocity of u. The residencetime in the hypothetical pipes is equal to the mean residence timein the mill, i.e., L/u=MH/F. The friction factor f is given by

f ¼ τw1=2ð Þρu2 ð10Þ

where τw is the shear stress at the wall of the hypothetical pipes. Theforce exerted by material on the surface of a single ball due to friction,

Table 2Data of Swaroop et al. [2,3] and calculated results.


Case no. J F MH U db fw f Abg MHb εbo MHd MH

− kg/s kg − m − − m2 kg − kg kg

(a) Open- end mill (D=0.127 m, L=0.438 m)Case O-1 0.24 0.0028 0.440 0.59 0.019 0.67 0.609 0.187 0.068 0.40 0.0 0.525

0.40 0.0028 0.542 0.44 0.019 0.67 0.609 0.330 0.119 0.40 0.0 0.5770.52 0.0028 0.635 0.39 0.019 0.67 0.609 0.455 0.164 0.40 0.0 0.622

Case O-2 0.40 0.0013 0.308 0.25 0.019 0.67 0.609 0.330 0.119 0.40 0.0 0.3310.40 0.0020 0.417 0.34 0.019 0.67 0.609 0.330 0.119 0.40 0.0 0.4450.40 0.0022 0.450 0.36 0.019 0.67 0.609 0.330 0.119 0.40 0.0 0.4830.40 0.0028 0.542 0.44 0.019 0.67 0.609 0.330 0.119 0.40 0.0 0.574

Case O-3 0.40 0.0029 0.720 0.58 0.019 0.28 0.609 0.410 0.148 0.40 0.0 0.6230.40 0.0029 0.620 0.50 0.019 0.40 0.609 0.388 0.140 0.40 0.0 0.6150.40 0.0029 0.580 0.47 0.019 0.55 0.609 0.358 0.129 0.40 0.0 0.6040.40 0.0029 0.530 0.43 0.019 0.67 0.609 0.331 0.120 0.40 0.0 0.5940.40 0.0029 0.565 0.45 0.019 0.79 0.609 0.306 0.111 0.40 0.0 0.5850.40 0.0029 0.770 0.62 0.019 0.97 0.609 0.278 0.100 0.40 0.0 0.575

(b) Constricted- end mill (D=0.08 m, L=0.24 m, Dd=0.045 m)Case C-1 0.14 0.0020 0.235 2.45 0.013 0.60 0.609 0.034 0.008 0.61 0.138 0.233

0.17 0.0020 0.222 1.92 0.013 0.60 0.609 0.042 0.009 0.55 0.125 0.2220.20 0.0020 0.213 1.60 0.013 0.60 0.609 0.048 0.011 0.51 0.115 0.2130.23 0.0020 0.201 1.31 0.013 0.60 0.609 0.056 0.012 0.46 0.104 0.2040.26 0.0020 0.204 1.18 0.013 0.60 0.609 0.064 0.014 0.44 0.098 0.200

Case C-2 0.20 0.0005 0.122 0.91 0.013 0.60 0.609 0.049 0.011 0.40 0.090 0.1240.20 0.0009 0.153 1.15 0.013 0.60 0.609 0.049 0.011 0.43 0.097 0.1500.20 0.0017 0.191 1.43 0.013 0.60 0.609 0.049 0.011 0.48 0.109 0.1930.20 0.0018 0.201 1.51 0.013 0.60 0.609 0.049 0.011 0.49 0.111 0.2030.20 0.0018 0.209 1.57 0.013 0.60 0.609 0.049 0.011 0.50 0.114 0.2070.20 0.0020 0.213 1.60 0.013 0.60 0.609 0.049 0.011 0.51 0.115 0.213

Lucite ball: ρb=1250 kg/m3, εb=0.4, Dolomite: ρp=2860 kg/m3, ρpb=1400 kg/m3 (εp=0.51).

Table 3Data of Kelsall et al. [4] and calculated results.

(a) Wet ball mill (D=0.3048 m) with 1/2-in. annular gap for discharge


J Fslurry MH slurry U L db fw aw ρsl f Abg MHb MH slurry

− kg/s kg − m m − − kg/m3 − m2 kg kg

Case K-1 0.18 0.010 1.47 0.52 0.3048 0.0254 0.78 0.667 1724 0.44 0.38 0.77 1.280.27 0.010 1.91 0.46 0.3048 0.0254 0.78 0.667 1724 0.44 0.58 1.15 1.670.41 0.010 2.05 0.33 0.3048 0.0254 0.78 0.667 1724 0.44 0.92 1.83 2.35

Case K-2 0.41 0.005 1.92 0.31 0.3048 0.0254 0.78 0.667 1724 0.44 0.92 1.83 2.090.41 0.010 2.07 0.33 0.3048 0.0254 0.78 0.667 1724 0.44 0.92 1.83 2.350.41 0.033 3.59 0.57 0.3048 0.0254 0.78 0.667 1724 0.44 0.92 1.83 3.54

Case K-3 0.41 0.010 1.12 0.36 0.1524 0.0254 0.78 0.667 1724 0.44 0.46 0.92 1.170.41 0.010 2.05 0.33 0.3048 0.0254 0.78 0.667 1724 0.44 0.92 1.83 2.350.41 0.010 3.51 0.37 0.4572 0.0254 0.78 0.667 1724 0.44 1.38 2.75 3.52

Case K-4 0.41 0.010 2.25 0.36 0.3048 0.0191 0.78 0.667 1724 0.44 1.23 2.44 2.960.41 0.010 2.08 0.33 0.3048 0.0254 0.78 0.667 1724 0.44 0.92 1.83 2.350.41 0.010 1.85 0.30 0.3048 0.0318 0.78 0.667 1724 0.44 0.74 1.46 1.980.41 0.010 1.82 0.29 0.3048 0.0381 0.78 0.667 1724 0.44 0.61 1.22 1.73

Case K-5 0.41 0.010 3.81 0.41 0.4572 0.0254 0.65 0.667 1724 0.44 1.47 2.94 3.710.41 0.010 3.51 0.37 0.4572 0.0254 0.78 0.667 1724 0.44 1.38 2.75 3.520.41 0.010 3.88 0.41 0.4572 0.0254 0.91 0.667 1724 0.44 1.30 2.58 3.36

Case K-6 0.41 0.012 1.59 0.29 0.3048 0.0254 0.78 0.550 1530 0.35 0.92 1.27 1.900.41 0.010 2.19 0.35 0.3048 0.0254 0.78 0.667 1724 0.44 0.92 1.83 2.350.41 0.009 4.04 0.59 0.3048 0.0254 0.78 0.750 1895 0.54 0.92 2.45 2.90

MH d=0.Steel ball: εb=0.4, ρb=7800 kg/m3, Calcite: ρp=2700 kg/m3, ρpb=1480 kg/m3 (εp=0.45), ρw=1000 kg/m3.


Table 4Data of Songfack and Rajamani [9] and calculated results.

(a) Overflow discharge ball mill (D=0.416 m, L=0.641 m)


Case no. J Fslurry MH slurry U Dd db* fw aw ρsl f Abg MHb εbo MHd MH slurry

− kg/s kg − m m − − kg/m3 − m2 kg − kg kg

Case S-1 0.30 0.0065 25.31 1.51 0.133 0.03 0.80 0.600 1607 0.38 2.17 19.53 0.50 9.72 30.020.30 0.0125 32.19 1.92 0.133 0.03 0.80 0.600 1607 0.38 2.20 19.78 0.56 10.89 32.160.30 0.0250 35.00 2.08 0.133 0.03 0.80 0.600 1607 0.38 2.21 19.90 0.58 11.30 34.190.30 0.0380 37.81 2.25 0.133 0.03 0.80 0.600 1607 0.38 2.23 20.02 0.60 11.67 36.240.30 0.0505 39.69 2.36 0.133 0.03 0.80 0.600 1607 0.38 2.24 20.12 0.61 11.91 38.070.30 0.0755 43.44 2.59 0.133 0.03 0.80 0.600 1607 0.38 2.26 20.32 0.63 12.35 41.69

Case S-2 0.30 0.0056 36.43 1.96 0.133 0.03 0.80 0.694 1776 0.47 2.20 26.82 0.56 12.16 39.650.30 0.0108 40.41 2.18 0.133 0.03 0.80 0.694 1776 0.47 2.22 27.03 0.59 12.72 41.040.30 0.0216 43.16 2.32 0.133 0.03 0.80 0.694 1776 0.47 2.24 27.19 0.61 13.08 42.850.30 0.0216 44.08 2.37 0.133 0.03 0.80 0.694 1776 0.47 2.24 27.25 0.61 13.19 43.020.30 0.0328 46.22 2.49 0.133 0.03 0.80 0.694 1776 0.47 2.25 27.38 0.62 13.44 44.750.30 0.0436 46.22 2.49 0.133 0.03 0.80 0.694 1776 0.47 2.25 27.38 0.62 13.44 46.050.30 0.0436 47.30 2.55 0.133 0.03 0.80 0.694 1776 0.47 2.26 27.45 0.63 13.57 46.240.30 0.0652 48.06 2.59 0.133 0.03 0.80 0.694 1776 0.47 2.26 27.51 0.63 13.65 48.970.30 0.0652 48.37 2.60 0.133 0.03 0.80 0.694 1776 0.47 2.26 27.53 0.63 13.68 49.020.30 0.0873 52.65 2.84 0.133 0.03 0.80 0.694 1776 0.47 2.29 27.85 0.65 14.14 52.430.30 0.1089 56.02 3.02 0.133 0.03 0.80 0.694 1776 0.47 2.31 28.13 0.67 14.46 55.63

Case S-3 0.35 0.0108 46.53 2.18 0.112 0.03 0.80 0.694 1776 0.47 2.63 31.97 0.59 14.07 47.330.35 0.0216 48.06 2.25 0.112 0.03 0.80 0.694 1776 0.47 2.64 32.10 0.60 14.27 48.960.35 0.0436 51.43 2.41 0.112 0.03 0.80 0.694 1776 0.47 2.67 32.42 0.62 14.68 52.320.35 0.0652 53.42 2.50 0.112 0.03 0.80 0.694 1776 0.47 2.68 32.62 0.63 14.92 55.350.35 0.1089 58.93 2.76 0.112 0.03 0.80 0.694 1776 0.47 2.74 33.29 0.65 15.53 61.85

Case S-4 0.30 0.0505 39.68 2.36 0.133 0.03 0.80 0.600 1607 0.38 2.24 20.12 0.61 11.91 38.060.30 0.0473 40.65 2.32 0.133 0.03 0.80 0.640 1675 0.42 2.24 22.76 0.61 12.32 40.740.30 0.0433 45.81 2.45 0.133 0.03 0.80 0.700 1788 0.47 2.25 27.90 0.62 13.45 46.530.30 0.0404 47.74 2.41 0.133 0.03 0.80 0.750 1895 0.54 2.24 33.32 0.61 14.16 52.310.30 0.0369 74.84 3.95 0.133 0.03 0.80 0.820 1811 0.60 2.51 39.88 0.74 16.28 60.57

Case S-5 0.30 0.0216 42.50 2.29 0.133 0.03 0.60 0.694 1776 0.47 2.49 30.30 0.60 12.88 45.770.30 0.0216 42.50 2.29 0.133 0.03 0.70 0.694 1776 0.47 2.36 28.65 0.60 12.94 44.180.30 0.0216 48.00 2.59 0.133 0.03 0.80 0.694 1776 0.47 2.26 27.50 0.63 13.64 43.730.30 0.0216 37.50 2.02 0.133 0.03 0.90 0.694 1776 0.47 2.11 25.65 0.57 12.38 40.61

Case S-6 0.30 0.0436 45.50 2.45 0.133 0.03 0.70 0.694 1776 0.47 2.37 28.89 0.62 13.30 47.410.30 0.0436 46.00 2.48 0.133 0.03 0.80 0.694 1776 0.47 2.25 27.37 0.62 13.42 46.010.30 0.0436 45.50 2.45 0.133 0.03 0.85 0.694 1776 0.47 2.19 26.64 0.62 13.39 45.250.30 0.0436 43.00 2.32 0.133 0.03 0.90 0.694 1776 0.47 2.13 25.89 0.61 13.12 44.23

*: assumed.Steel ball: εb=0.4, ρb=7800 kg/m3, Limestone: ρp=2700 kg/m3, ρpb=1480 kg/m3 (εp=0.45), ρw=1000 kg/m3.


denoted as Rmf, is related to τw as,

nh πdhLð Þτw ¼VRb Δtg þ Δta� �πd3b=6

24

35Rmf ð11Þ

where nh and dh are the number and diameter of the hypotheticalpipes. The term in the square brackets in Eq. (11) is the number ofballs in the grinding and ascending zones, which are effective forinterrupting the material flow. The surface area of the hypotheticalpipe walls is equal to the surface area of the balls, i.e.,

nh πdhLð Þ ¼VRb Δtg þ Δta� �πd3b=6

24

35 πd2b� �

ð12Þ

Combining Eqs. (10)–(12) leads to

Rmf ¼ 4fπ4d2b

� � 12ρu2

� �ð13Þ

where ρ is equal to ρpb for dry mill and equal to ρsl for wet mill. Thefollowing sections analyze Rmf for dry andwet mills to specify the fric-tion factors for these conditions.

2.3.1. Rmf for dry conditionRmf is the resistance force acting on a ball due to friction of particles

flowing at the velocity u as shown in Fig. 3. The rate of momentum ofparticles given by (ρpbu)u acts at every point on the ball surface. Thenormal stress denoted as σn acts perpendicularly to the surface andthe shear stress denoted as τR does tangentially. These are expressed by

σn ¼ ρpbuu� �

cosφ ð14Þ

τR ¼ μpbσn ¼ μpb ρpbuu� �

cosφh i

ð15Þ

where μpb is the friction coefficient of particles and φ is the angle desig-nating the force acting point on the ball surface (see Fig. 3). The horizon-tal components of the normal and shear stresses are σncosφ and τRsinφ,respectively. These local forces per unit area act on the small surfacearea given by (π/2)db2sinφdφ. Assuming the two forces act only on thesurface of the half sphere facing to the approaching material, the hori-zontal components multiplied by the small surface area are integratedover the surface of the half sphere. The resultant forces acting on thesphere surface due to the normal and shear stresses, denoted as Fn

Table 5Data of Kelly [9] and Hodouin et al. [10] and calculated results.

(a) Grate- discharge ball mill (D=0.4 m, L=0.4 m)


Case no. J Fslurry MH slurry U db* fw aw ρp ρsl f Abg MHb MH slurry

− kg/s kg − m − − kg/m3 kg/m3 − m2 kg kg

Limestone 0.48 0.0363 7.12 0.45 0.02 0.75 0.625 2700 1649 0.40 3.31 3.90 7.500.48 0.0363 6.99 0.44 0.02 0.75 0.624 2700 1647 0.40 3.31 3.88 7.49

WI, kWh/t=10.6 0.48 0.0334 8.78 0.52 0.02 0.75 0.678 2700 1745 0.45 3.31 4.62 7.940.48 0.0334 7.56 0.45 0.02 0.75 0.679 2700 1747 0.45 3.31 4.64 7.950.48 0.0335 7.93 0.47 0.02 0.75 0.676 2700 1741 0.45 3.31 4.59 7.920.48 0.0443 8.38 0.51 0.02 0.75 0.655 2700 1702 0.43 3.31 4.29 8.680.48 0.0565 9.73 0.61 0.02 0.75 0.625 2700 1649 0.40 3.31 3.90 9.510.48 0.0564 9.57 0.60 0.02 0.75 0.626 2700 1651 0.40 3.31 3.91 9.510.48 0.0564 9.28 0.58 0.02 0.75 0.626 2700 1651 0.40 3.31 3.91 9.510.48 0.0520 10.28 0.61 0.02 0.75 0.680 2700 1749 0.45 3.31 4.66 9.810.48 0.0523 9.47 0.56 0.02 0.75 0.676 2700 1741 0.45 3.31 4.59 9.780.48 0.0522 10.19 0.60 0.02 0.75 0.677 2700 1743 0.45 3.31 4.61 9.79

Quartz 0.48 0.0364 8.83 0.56 0.02 0.75 0.622 2670 1637 0.40 3.31 7.19 9.100.48 0.0363 8.86 0.56 0.02 0.75 0.625 2670 1642 0.40 3.31 7.26 9.16

WI, kWh/t=19.5 0.48 0.0363 9.01 0.57 0.02 0.75 0.625 2670 1642 0.40 3.31 7.26 9.160.48 0.0336 10.09 0.60 0.02 0.75 0.675 2670 1731 0.45 3.31 8.53 10.290.48 0.0333 10.49 0.62 0.02 0.75 0.680 2670 1740 0.46 3.31 8.67 10.420.48 0.0336 10.76 0.64 0.02 0.75 0.675 2670 1731 0.45 3.31 8.53 10.290.48 0.0446 9.85 0.60 0.02 0.75 0.650 2670 1685 0.43 3.31 7.86 10.200.48 0.0564 9.57 0.60 0.02 0.75 0.626 2670 1644 0.40 3.31 7.28 10.240.48 0.0566 11.06 0.70 0.02 0.75 0.624 2670 1640 0.40 3.31 7.24 10.200.48 0.0565 9.81 0.62 0.02 0.75 0.625 2670 1642 0.40 3.31 7.26 10.220.48 0.0527 12.27 0.74 0.02 0.75 0.670 2670 1721 0.45 3.31 8.39 11.150.48 0.0520 11.09 0.66 0.02 0.75 0.680 2670 1740 0.46 3.31 8.67 11.39

Feldspar 0.48 0.0252 7.70 0.48 0.02 0.75 0.648 2550 1650 0.43 3.31 6.56 8.120.48 0.0483 8.78 0.58 0.02 0.75 0.600 2550 1574 0.39 3.31 5.65 8.64

WI, kWh/t=14.2 0.48 0.0445 9.28 0.58 0.02 0.75 0.651 2550 1655 0.43 3.31 6.62 9.380.48 0.0445 9.91 0.62 0.02 0.75 0.651 2550 1655 0.43 3.31 6.62 9.380.48 0.0446 9.71 0.61 0.02 0.75 0.650 2550 1653 0.43 3.31 6.60 9.370.48 0.0448 9.04 0.57 0.02 0.75 0.648 2550 1650 0.43 3.31 6.56 9.330.48 0.0410 10.66 0.63 0.02 0.75 0.707 2550 1754 0.49 3.31 7.97 10.510.48 0.0641 10.20 0.64 0.02 0.75 0.650 2550 1653 0.43 3.31 6.60 10.57

Sulfide ore 0.48 0.0417 11.15 0.52 0.02 0.75 0.725 4200 2234 0.44 3.31 6.82 10.250.48 0.0417 10.12 0.47 0.02 0.75 0.725 4200 2234 0.44 3.31 6.82 10.250.48 0.0415 9.45 0.43 0.02 0.75 0.730 4200 2253 0.45 3.31 6.95 10.360.48 0.0417 9.63 0.44 0.02 0.75 0.725 4200 2234 0.44 3.31 6.82 10.250.48 0.0448 9.01 0.45 0.02 0.75 0.675 4200 2059 0.40 3.31 5.65 9.34

WI, kWh/t=12.5 0.48 0.0480 9.66 0.52 0.02 0.75 0.630 4200 1923 0.36 3.31 4.83 8.780.48 0.0480 7.76 0.42 0.02 0.75 0.630 4200 1923 0.36 3.31 4.83 8.780.48 0.0477 9.39 0.50 0.02 0.75 0.635 4200 1937 0.37 3.31 4.91 8.830.48 0.0278 8.87 0.41 0.02 0.75 0.725 4200 2234 0.44 3.31 6.82 9.100.48 0.0276 9.83 0.45 0.02 0.75 0.730 4200 2253 0.45 3.31 6.95 9.220.48 0.0306 7.86 0.40 0.02 0.75 0.660 4200 2011 0.38 3.31 5.36 7.870.48 0.0303 7.74 0.42 0.02 0.75 0.625 4200 1909 0.36 3.31 4.75 7.24

*: volume– surface mean diameter MH d=0.Steel ball: εb=0.4, ρb=7800 kg/m3, εp=0.45, ρw=1000 kg/m3.

0

40

80

120

0 0.1 0.2 0.3 0.4 0.5J [ - ]

MH [k

g]

experimentcalculated

d b = 0. 03 m

(a) Dry grinding (Mori et al.)

0

1

2

3

0 0.1 0.2 0.3 0.4 0.5

J [ - ]

MH

slur

ry [k

g]


case K-1

(b) Wet grinding (Kelsall et al.)

Fig. 5. Effects of J on MH for dry and wet mill conditions.


0

0.1

0.2

0.3

0.4

0.1 0.15 0.2 0.25 0.3

J [ - ]

MH [ k

g ]


case C -1

Dry grinding (Fuerstenau et al.)

Fig. 6. Effect of J on MH for a constricted-end mill.


and Ft, respectively, are reduced to

Fn ¼ ∫π=20 σn cosφð Þ π

2d2b

� �sinφdφ ¼ 1

3π2d2b

� �ρpbu

2� �

ð16Þ

Ft ¼ ∫π=20 τR sinφð Þ π

2d2b

� �sinφdφ ¼ 1

3μpb

π2d2b

� �ρpbu

2� �

ð17Þ

Then, the resistance force of the ball Rmf is given by

Rmf ¼ Fn þ Ft ¼43

1þ μpb

� � π4d2b

� � 12ρpbu

2� �

ð18Þ

Comparison of Eq. (18) with Eq. (13) results in f to be

f ¼ 13

1þ μpb

� �ð19Þ

2.3.2. Rmf for wet conditionSlurry is regarded as a heterogeneous mixture of particles and

water. The particles and the water act individually on the ball surface,i.e., the former exerts the resistance force given by Eq. (18) and thelatter does the drag force denoted as Fd expressed as,

Fd ¼ Cfπ4d2b

� � 12ρwu

2� �

ð20Þ

where ρw is the density of water and Cf is the drag coefficient. Adding

(a) Dry grinding (Swaroop et al.)

0.0

0.2

0.4

0.6

F [ kg/s]

MH

[

kg ]

case O-2

case C-2

0 0.001 0.002 0.00

Fig. 7. Relation betw

the force given by Eq. (20) into Eq. (18), Rmf for slurry is obtained to be

Rmf ¼43

1þ μpb

� � π4d2b

� � 12ρpbu

2� �

þ Cfπ4d2b

� � 12ρwu

2� �

¼43

1þ μpb

� �ρpb þ Cfρw

ρsl

264

375 π

4d2b

� � 12ρslu

2� � ð21Þ

where ρsl is the density of slurry (see Appendix C in Ref. [6]). Compari-son of Eq. (21) with Eq. (13) leads to f to be

f ¼43 1þ μpb

� �ρpb þ Cfρw

4ρsl¼ 1

31þ μpb

� �aw þ 1

4Cf ρw=ρslð Þ ð22Þ

where aw, equal to ρpb/ρsl, is themass fraction of solids in slurry and Cf isassumed to be 0.44 in a range of high Reynolds numbers for balls in vi-olent motion in tumbling mills.

2.3.3. Friction coefficient of particles μpbThe coefficient of internal friction or repose is adopted as the value

of μpb. Fig. 4 shows the data for sand with the mean diameter of0.5 mm [14], in which the relation between the angle of repose Φr

and the voidage of particles εp is assumed to be linear, i.e.,

μpb ¼ tanΦr ¼ tan 1:36−1:31εp� �

ð23Þ

Since μpb for arbitrary materials are not known at present, thosegiven by Eq. (23) are assumed as the average ones in the present cal-culations. For wet grinding, Eq. (23) is also used, in which εp isreplaced by εsl, the voidage of particles in slurry (see Appendix C inRef. [6]).

3. Results and discussion

Dry mill data of Mori et al. [1] and those of Swaroop et al. [2,3] arelisted in Tables 1 and 2. Wet mill data of Kelsall et al. [4] and those ofSongfack and Rajamani [11] are in Tables 3 and 4. Further, wet milldata of four different materials reported by Kelly [9] and Hodouin et al.[10] are listed in Table 5. The calculated values of f, Abg, MHb, εbo andMHd corresponding to experimental conditions are also in those tables.

To evaluate the three unknown constants, c1, c2 and c3, introducedin the present theory, Eq. (4) is used, which is rewritten with the aidof Eqs. (5)–(7) to be

MH ¼ c1 LF½ � þ c2 ρf Abg

h iþ c3 ρLAHdεbo½ � ð24Þ

(b) Wet grinding (Songfack and Rajamani)

20

40

60

80

0 0.04 0.08 0.12

Fslurry [ kg/s ]

MH

slu

rry

[ k

g ]

case S-1

case S-2

case S-3

een MH and F.

30

40

50

60

0.5 0.6 0.7 0.8 0.9 1

fw [ - ]

MH

slur

ry [ k

g ]


case S-6

(a) Dry grinding (Swaroop et al.) (b) Wet grinding (Songfack and Rajamani)

0.4

0.6

0.8

1.0

1.2

0 0.2 0.4 0.6 0.8 1

fw [ - ]

MH

[ kg ]


case O-3

Fig. 8. Effect of fw on MH.


Specifically, the least square regression is applied to give a best fit tothe data of MH as the terms in the square brackets in Eq. (24) are es-timated using the operating conditions. For each mill and for eachmaterial, c1, c2 and c3 are evaluated as these constants depend onthe mill exit structure and the kind of material from the followingreason. As defined in Eq. (4), c1 is affected by the mill exit structurevia φs and the kind of material via vop. The values of c2 and c3depending on the mill exit structure as noted in Section 2.2.2, arealso affected by the kind of material in the present calculation. Thisis because μpb for a given material, followed by the friction factor f,is assumed by the average value obtained from Eq. (23). The differ-ence between the real μpb and the average one is compensated bythe value of c2 resulted from the least square regression, e.g., materialwith μpb smaller than the average gives a smaller c2 value.

The difference between calculated MH values and experimentalones for each test mill is expressed by the root mean square of rela-tive errors (RMSre) given as

RMSre ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1n

Xni¼1

MHD;i−MHC;i

MHD;i

!2vuut ð25Þ

where MHD and MHC are the experimental and calculated MH values,respectively.

In the following sections, the effects of operating variables on MH

are analyzed in Section 3.1 and the difference between calculatedMH and experimental ones are examined in Section 3.2. Also in

0

1

2

3

4

0.01 0.02 0.03 0.04 0.05

db [m]

MH

slur

ry [ k

g ]


case K-4

Wet grinding (Kelsall et al.)

Fig. 9. Effect of db on MH.

Section 3.2, the influence of material kind on holdup is discussedbased on existing data.

3.1. Effects of operating variables on MH

3.1.1. Ball filling JThe effects of J on MH under the constant F [1,4] are shown in

Fig. 5, in which the full and blanc symbols denote calculated andexperimental ones, respectively. In the case of Mori et al. [1](Fig. 5(a)), the MH value at F being between 0.09 kg/s and 0.11 kg/sis selected for each J. For both the dry and wet conditions, theoryand experiment reveal that MH increases with increasing J as thenumber of balls interrupting the material flow increases in Eq. (6).

On the other hand, the constricted-end mill data [3] plotted inFig. 6 show an opposite trend of MH decreasing with increasing J.The reason was mentioned in Ref. [3] that charging more balls inthe mill, more particulate material below the discharge lip wasreplaced by the charged balls. Theoretically, for U greater than unityas depicted in Fig. 2, when more balls are filled in the dead space,the value of εbo is decreased (see case C-1 in Table 2). This leads tothe decrease of material retained in the dead space according toEq. (7).

3.1.2. Feed rate FSwaroop et al. [2,3] reported linear relations between MH and F in

the open-end and constricted-end mills under dry conditions, asdepicted in Fig. 7(a). According to Eq. (4), the linearity is valid

20

40

60

80

0.5 0.6 0.7 0.8 0.9

aw [m]

MH

slur

ry [ k

g ]


case S-4

Wet grinding (Songfack and Rajamani)

Fig. 10. Effect of aw on MH.

2 4MH (data) [kg]

MH

(ca

lc)

[kg]

600

2

4

case K-1case K-2case K-3case K-4case K-5case K-6

6

Fig. 13. Comparison of calculated MH with data of Kelsall et al. [4].

0

40

80

120

0 40 80 120

MH (data) [kg]

MH (

calc

) [

kg]

case M-1

case M-2

case M-3

case M-4case M-5

case M-6

case M-7

case M-8

Fig. 11. Comparison of calculated MH with data of Mori et al. [1].


when the discharge properties, φs and vop, are assumed to be constantanywhere on the cross sectional area effective for discharging. Thosemill end structures seem to allow this assumption.

Data of awet overflowmill [11] also reveal linear trends except plotsfor smaller F values in Fig. 7(b). According to Ref. [11], the slurry levelsfor the smaller F values were below the lip of the overflow opening andno significant discharges by overflowing the opening were observedexcept the discharges by sticking to the lifters and flowing by gravityfrom the top of the discharge end-plate through the overflow opening.Theoretically, the apparent values of φs for the smaller F values aresmaller than those for other greater F values. According to Eq. (3),smaller values ofφs give greater values of (dMH/dF), resulting in steeperslopes obtained in Fig. 7 (b).

3.1.3. Mill speed fwIn Fig. 8(a), data of a dry mill [3] show that MH decreases with

increasing fw up to about fw=0.7 and then increases. However, thecalculated MH decreases monotonically with increasing fw as Abg

decreases (see case O-3 in Table 2). The increase of MH for fw>0.7in the experiment may be led by more material sticking on the insidemill wall due to centrifugal force for greater fw, which is not taken intoaccount in the present theory. Further, the mass fractions of stickingmaterial are magnified for relatively small holdups, e.g., U beingabout 0.5 (see case O-3 in Table 2).

On the other hand, wet mill data [11] in Fig. 8(b) reveal a slightdecrease of MH even for fw>0.7. In this case, the mass fractions ofsticking material may not be magnified as the holdups are relatively

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0

MH (data) [kg]

MH

(ca

lc)

[kg

]

case O-1case O-2case O-3case C-1case C-2

Fig. 12. Comparison of calculated MH with data of Swaroop et al. [2,3].

great, e.g., U greater than 2.0 (see case S-6 in Table 4). The influenceof sticking material may be minor in a range of practical operationswith U being about 1.0 and fw being about between 0.6 and 0.8.

3.1.4. Ball diameter dbFig. 9 shows the effect of db onMH for a wet mill [4]. Both theory and

experiment reveal that MH decreases with increasing db. According toEq. (6), MHb decreases with increasing db as Abg decreases.

3.1.5. Mass fraction of solids in slurry awTheoretically, MH increases with increasing aw under the constant

solid feed rate (awF) asMHb increases. This is due to the friction factorf increasing with increasing aw according to Eq. (22) (see also case S-4in Table 4). The data of Songfack and Rajamani [11] indicate the sametrend as depicted in Fig. 10.

3.2. Comparisons of calculated MH with data

Two cases of dry grinding are discussed in Figs. 11 and 12 pottingcalculated MH values against corresponding data of MH and threecases of wet grinding are in Figs. 13–15. Of these, Fig. 15 deals withholdups of four different materials.

Note that for mills with relatively large opened areas such as anopen end mill [2], an annular gap mill [4] and a grate discharge endmill [9,10], the dead spaces generated are considered to be minor.The least square regressions based on Eq. (24) are not converged forthese mills with arbitrary constant values of φs (b1) assumed in

0

20

40

60

80

100

0 20 40 60 80 100

MH (data) [kg]

MH

(ca

lc)

[kg

]

case S-1

case S-2

case S-3case S-4

case S-5

case S-6

Fig. 14. Comparison of calculated MH with data of Songfack and Rajamani [11].

5

10

15

5 10 15

MH (data) [kg]

MH

(ca

lc)

[kg]

Limestone

Quartz

Feldspar

Sulfide ore

Fig. 15. Comparison of calculated MH with data of Kelly [9] and Hodouin et al. [10].

60

120

180

240

300

0.001 0.002 0.003 0.004

c2 [m]

c 1 [

s/m

]

5

10

15

20

25

WI [

kWh/

t]

c 1 c 2 = 0.434 s

Fig. 16. Correlations between c1 and c2 and between WI and c2 for four differentmaterials.


Eq. (9), indicating that the third term in the right hand side of Eq. (24)is insignificant for these mills. Thus, the dead spaces are ignored forthese mills. The values of c1, c2 and c3 evaluated from the regressionanalysis are listed in Table 6 with the RMSre values.

In Fig. 11 for the case of Mori et al. [1], the convergence is fairlywell in a wide range of the holdup although some scatters areobserved. A reason for the scatters may be that accurate measure-ments were not easy in such a large pilot-scale mill, causing experi-mental errors.

A satisfactory convergence is obtained for the case of Swaroop etal. [2,3] in Fig. 12 except two plots for fw equal to 0.97 and 0.28 incase O-3, the highest and lowest mill speeds. A high mill speed nearcritical may cause a relatively great amount of material sticking onthe inside wall due to centrifugal force, which is not taken intoaccount in the present theory. An extremely low mill speed mayworsen the axial flow of material, leaving more material in the millthan expected. Such high and low mill speeds are not important forpractical operations with fw being between 0.6 and 0.8.

Next, wet mill data of Kelsall et al. [4] and those of Songfack andRajamani [11] are shown in Figs. 13 and 14, respectively. In Fig. 13,some plots are apart from the straight lines, for a high mill speed,i.e., fw=0.91 and for a high solid fraction in slurry, i.e., aw=0.75.The discrepancies in these cases may be caused again by relativelygreat amounts of material sticking on the inside walls. Fig. 14 revealsa reasonable convergence except one plot for aw=0.82. The samereason noted above is considered for this discrepancy.

Table 6Estimated values of c1, c2 and c3 for each mill and each material.

Researchers Material c1s/m

c2m

c3−

RMSre−

Dischargeconfiguration

Mori et al. [1] Limestone 92.5 2.99E−04 0.837 0.084 Slit screen

Swaroop et al.[2]

Dolomite 373.4 4.24E−04 – 0.112 Open-end

Fuerstenau et al.[3]

Dolomite 186.2 2.58E−04 0.825 0.012 Constricted

Kelsall et al. [4] Calcite 169.0 2.62E−03 – 0.138 Annular gap

Songfack et al.[11]

Limestone 186.7 1.46E−02 0.465 0.060 Overflow

Kelly [9] Limestone 248.2 1.77E−03 – 0.047 GrateQuartz 131.0 3.31E−03 – 0.048Feldspar 154.8 2.79E−03 – 0.035 Overall

RMSre=Hodouin et al.[10]

Sulfideore

205.4 2.09E−03 – 0.070 0.053

Holdup data of four different materials [9,10] are plotted againstthe calculated ones in Fig. 15, in which a reasonable convergence isobserved. Between c1 and c2 for the four materials (see Table 6), a cor-relation seems likely as depicted in Fig. 16. As mentioned already inthe beginning of this chapter, μpb for each material is assumed bythe average one given by Eq. (23) and the resultant c2 value varieswith the friction property of the material. Also, c1 depends on thekind of material via vop. If vop is influenced by the friction property,both c1 and c2 are to be affected by the same factor, leading to a cor-relation between c1 and c2. An inversely proportional relation isapproximately obtained between c1 and c2 in Fig. 16.

Next, a possibility is considered how to relate c1 or c2 with mate-rial properties other than μpb. According to Gupta et al. [13], eachmaterial showed a certain characteristic shape in the size distributionregardless of the operating conditions, to which the work index wasfound to correspond. Or according to Kelly [9], the particle size oflimestone was much finer than those of quarts and feldspar. A previ-ous study by the authors [14] indicated that finer particles tended toreveal less frictions due to rounder shapes. Such information implies apossibility that finer materials, or materials with smaller work indi-ces, may have less frictions leading to smaller c2 values. In Fig. 16, acorrelation is observed between c2 and the work index WI althoughfurther studies are needed to acquire sounder backgrounds.

5

10

15

5 10 15

MH (data) [kg]

MH

(ca

lc)

[kg]

LimestoneQuartzFeldsparSulfide ore

Gupta et al. [13]RMS re = 0.059 -

Fig. 17. Comparison of MH calculated from Gupta's equation with data of Kelly [9] andHodouin et al. [10].


Incidentally, the regression analysis of Gupta et al. [13] led to anempirical equation as,

awMH ¼ 4:020−0:176WIð Þ 60awFð Þ þ 0:040þ 0:01237WIð Þ 100awð Þ− 4:970þ 0:395WIð Þ

ð26Þ

where WI is the work index in unit of kWh/t. Their calculated resultsfrom Eq. (26) are plotted against corresponding data in Fig. 17. Asfar as RMSre is concerned, the present theory provides the valueof RMSre (see the overall one in Table 6) similar to that noted inFig. 17. More importantly, the present theory is valid for various oper-ating conditions.

4. Conclusions

This paper has dealt with holdups of particulate material in ballmills operated continuously. The analysis has been based on a consid-eration that holdups are influenced by the discharge properties at themill exit, the existence of balls interrupting the material to flowtowards the mill exit and the dead space generated depending onthe exit configuration. As a result, the linear relation between holdupand feed rate observed experimentally has been proved to be validunder specific discharge properties. Further, effects of operating vari-ables such as ball filling, ball diameter, mill speed and slurry densityon holdups have been clarified based on physical backgroundsdeduced from the theory.

The derived equation includes three constants, c1, c2 and c3, relat-ing to the discharge properties of mill exit, the resistance of mediaballs against material flow and the dead space, respectively. Theseconstants have been estimated from the least square regression with-in reasonable convergence between calculated holdups and data.Also, possible correlations have been discussed between c1 and c2and between c2 and the work index to describe the effect of materialkind on holdup.

Although further studies are required both theoretically andexperimentally, the present theory would provide useful informationto predict holdups, followed by the residence time distributions andthe grinding rates, heading for reliable mill design or optimizationof operating conditions to be achieved.

List of symbolsA cross sectional area of mill, m2

Abg surface area of balls in grinding and ascending zones, m2

AH cross sectional area of holdup, m2

AHd cross sectional area of dead space, m2

aw mass fraction of solids in slurry, -Cf drag coefficient, -c1 constant relating to mill exit discharge properties, s/mc2 constant relating to media balls interruptingmaterial flow, mc3 constant relating to dead space, −D mill diameter, mDd diameter of exit opening, mdb ball diameter, mdh diameter of hypothetical pipes in ball bed, mF mass feed rate, kg/sFd drag force of fluid acting on a ball, NFn horizontal force acting on ball surface due to normal stress

of particles, NFt horizontal force acting on ball surface due to shear stress of

particles, Nf friction factor, −fw ratio of angular velocity of mill revolution to critical one, −J fractional mill volume occupied by ball bed, −

L mill length, mMH mass holdup of material, kgMHb mass of material trapped in ball bed, kgMHC calculated mass holdup, kgMHD experimental mass holdup, kgMHd mass of material retained in dead space, kgMHo mass of material retained in mill at F=0, kgnh number of hypothetical pipes in ball bed, −RMSre root mean square of relative errors, −Rmf force exerted bymaterial on surface of a ball due to friction, Nr radius defined in Fig. 2, mΔta mean residence time of balls in ascending zone, sΔtg mean residence time of balls in grinding zone, sU fractional ball bed void filled with material in static mill, −u mean axial velocity of material flowing in mill, m/sVRb volumetric flow rate of balls in circumferential direction of

mill, m3/svop discharge velocity of material at mill opening, m/sWI Bond work index, kWh/t

Greek lettersεb voidage of static ball bed, −εbo voidage of ball bed in grinding zone, −εp voidage of bulk of particles, −εsl fraction of slurry volume occupied by either air or water, −θ angle defined in Fig. 2, radθd angle for surface level of dead space, radμpb friction coefficient of bulk of particles, −ρ density of fluid, kg/m3

ρp density of particle, kg/m3

ρpb bulk density of particles, kg/m3

ρsl density of slurry, kg/m3

ρw density of water, kg/m3

σn normal stress acting perpendicularly to ball surface, N/m2

τR shear stress acting tangentially on ball surface, N/m2

τw shear stress acting tangentially on wall of hypothetical pipe,N/m2

Φr angle of repose, radφ angle defined in Fig. 3, radφs fractional cross sectional area opened for discharge at mill

exit, −

References

[1] Y. Mori, G. Jimbo, M. Yamazaki, Kagaku Kogaku 28 (1964) 204–213.[2] S.H.R. Swaroop, A.-Z.M. Abouzeid, D.W. Fuerstenau, Powder Technology 28

(1981) 253–260.[3] D.W. Fuerstenau, A.-Z.M. Abouzeid, S.H.R. Swaroop, Powder Technology 46

(1986) 273–279.[4] D.F. Kelsall, K.J. Reid, C.J. Restarick, Powder Technology 3 (1969/70) 170–178.[5] R.S.C. Rogers, D.G. Bell, A.M. Hukki, Powder Technology 32 (1982) 245–252.[6] S. Nomura, Powder Technology 222 (2012) 37–51.[7] K. Shoji, L.G. Austin, F. Smaila, K. Brame, P.T. Luckie, Powder Technology 31

(1982) 121–126.[8] S. Nomura, K. Hosoda, T. Tanaka, Powder Technology 68 (1991) 1–12.[9] F.J. Kelly, CIM Bulletin 63 (1970) 573–581.

[10] D. Hodouin, M.A. Berube, M.D. Everell, Research Report, CANMET, Energy, Minesand Resources, Ottawa, Canada, August 1977, pp. 1–90.

[11] P. Songfack, R. Rajamani, International Journal of Mineral Processing 57 (1999)105–123.

[12] T. Tanaka, Journal of Chemical Engineering Japan 5 (1972) 425–426.[13] V.K. Gupta, D. Hodouin, M.D. Everell, International Journal of Mineral Processing

8 (1981) 345–358.[14] S. Nomura, F. Itoh, T. Tanaka, Journal of the Society of Powder Technology, Japan

27 (1990) 680–685.

Documents

Analysis of holdups in continuous ball mills