Powder Technology 222 (2012) 37–51

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Dispersion properties for residence time distributions in tumbling ball mills

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

E-mail address: nomurashin@frontier.hokudai.ac.jp.

0032-5910/$ – see front matter © 2012 Elsevier B.V. Alldoi:10.1016/j.powtec.2012.01.034

a b s t r a c t

a r t i c l e i n f o

Article history:Received 7 May 2011Received in revised form 30 October 2011Accepted 21 January 2012Available online 28 January 2012

Keywords:Residence time distributionDispersion coefficientPeclet numberContinuous millTumbling ball millComminution kinetics

The present paper deals with a theoretical analysis of the dispersion properties, the dispersion coefficient andthe Peclet number, of particulate material in a continuous ball mill. In the analysis, a dispersion zone wherethe brief dispersion of particles occurs, is postulated in the lower portion of an operated mill called the grind-ing zone. Consequently, the dispersion coefficient is derived to be a function of the size of the dispersion zoneand the mobility of balls in the grinding zone and the Peclet number is a function of the dispersion coefficient,the axial mean velocity of material flowing and the mill length. Results derived from the theory are withinreasonable agreement with reported data for dry and wet grinding operations, although minor variationsare observed between theory and experiment. Additionally, the mill diameter and length are predicted to af-fect greatly the Peclet number, implying the importance of designing mill sizes for required product size dis-tributions as the residence time distribution is dominated by the Peclet number. Further, a proportionalrelationship to predict the Peclet number is derived, which appears to be valid as confirmed with data regard-less of the mill sizes tested.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Demands are growing to more precise control of the product sizedistributions when grinding particulate materials. Simultaneously,economical operations are always required as only a few percent oftotal energy applied to a system is used for pure grinding [1]. Oneway to reduce the grinding energy is to reduce the amount of overground particles. In a continuous mill, reducing the amount of parti-cles with relatively long residence times lessens the over grinding.To obtain a required fineness of product with minimum grinding en-ergy consumed, the residence time distribution of particulate materi-al is of great importance to be controlled.

There have been a number of literatures reported with respect tothe residence time distributions in continuous ball mills. Experimen-tally, for dry grinding, Mori et al. [2] and Swaroop et al. [3,4] exam-ined either the dispersion coefficient (corresponding to the diffusioncoefficient in molecule diffusion) or the Peclet number (originally de-fined in molecular diffusion) in relation to some operating conditionssuch as the ball filling, the feed rate, the speed of mill revolution andthe ball size. For wet grinding, Kelsall et al. [5] obtained data for someoperating parameters including the slurry density. Austin et al. [6]showed experimentally the validity of a constant Peclet numberalong the mill axis which was conventionally adopted as an assump-tion in theoretical derivations. Further, a method with short-lived

rights reserved.

radioactive tracers was developed to apply the measurement to apilot-plant scale ball mill in closed circuit [7].

From the theoretical point of view, an axial dispersion model wasdeveloped elsewhere based on a one dimensional axial diffusion equa-tion of molecules and solutions of the residence time distribution werederived for sets of initial and boundary conditions [2,6,8]. Cho and Austin[9] reported that the solution of Mori et al. [2] fit very well to data ofindustrial mills. Kelsall et al. [5] and Furuya et al. [10] demonstrated amill model composed of a plug-flow and a perfect mixing flow in series.The former used it to analyze experimental data. The latter defined aparameter indicating the degree of mixing and evaluated its effect onthe product size distribution based on comminution kinetics.

In spite of a number of experimental and theoreticalfindings reported,the dispersion properties have not been fully predictable in relation tooperating conditions and mill sizes. Therefore, controlling the residencetime distribution relies greatly on experiences or trial and error practices.

The present study aims at clarifying the dispersion properties basedon a model estimating the degree of dispersion as well as the millcharge model developed previously [11,12]. Data reported for dry andwet grinding operations are utilized to examine the developed theory.Also physical backgrounds are considered for trends of the dispersionproperties varying with operating variables and mill sizes.

2. Theoretical

2.1. Residence time distribution and dispersion properties

Postulate a cylindrical mill with a length of L and a diameter of D.Particulate material enters from one end of the mill (x=0) and exits

38 S. Nomura / Powder Technology 222 (2012) 37–51

from the other end (x=L). Assuming the axial dispersion of particu-late material, the residence time distribution Φ(t) at the mill exit(x=L) is expressed by [2],

Φ tð Þ ¼ Lffiffiffiffiffiffiffiffiffiffiffiffiffi4πEt3

p exp − L−utð Þ24Et

( )ð1Þ

where E is the axial dispersion coefficient of particles correspondingto the diffusion coefficient in molecular diffusion and u is the axialmean velocity of material flowing in the mill. The mean residencetime τ is obtained to be,

τ ¼ ∫∞0 tΦ tð Þdt ¼ L=u ¼ MH=F ð2Þ

where MH is the mass holdup of particles and F is the mass feed rate.From Eq. (2), u is,

u ¼ LF=MH: ð3Þ

Using two dimensionless parameters, Eq. (1) is rewritten by,

Φ tð Þ ¼ 1

τffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4πθ3=Pe

q exp − 1−θð Þ24θ=Pe

( )ð4Þ

where θ= t /τ, Pe=uL/E, θ is the dimensionless time variable and Peis the Peclet number. The Peclet number is the ratio of mass fluxresulting from total bulk flow to mass flux resulting from dispersion.A greater value of L or u leads to a greater value of Pe towards plugflow (Pe equal to infinity means plug flow). Eq. (4) indicates thatthe residence time distribution is controlled by Pe. An analysis of Eis conducted to clarify Pe in terms of operating conditions and millsizes.

From similarity between a one dimensional random walk and aone dimensional dispersion process, E is expressed by statistical pa-rameters as [13],

E ¼ ‘2m2t

ð5Þ

where m is the number of steps of a random walker and ℓ is thelength of one step which is assumed to be constant. Note that thewalker does not always keep walking for a given period of t, i.e., hesometimes walks and sometimes rests. After walking and resting dur-ing the total period of t, he movesm steps. Denoting the period of onestep as ts and assuming ts is constant,

ηt ¼ mts ð6Þ

where η is the ratio of the walking period to the total period of t.Substituting the above equation into Eq. (5) to eliminate t, E isobtained to be

E ¼ ‘2m

2mts=η¼ 1

2η‘

‘

ts

� �¼ 1

2η‘vw ð7Þ

where vw is the velocity of walking. Then, Pe is expressed as,

Pe ¼ uLE

¼ 2uLη‘vw

: ð8Þ

2.2. Specifying variables η, ℓ and vw

This section explains briefly the mill charge model developed pre-viously [11,12] in Sections 2.2.1 and 2.2.2 and defines a dispersionzone in an operated mill in Section 2.2.3. Then, variables η, ℓ and vw

appeared in Eqs. (7) and (8) are specified in terms of operating con-ditions and mill sizes in Sections 2.2.4–2.2.6.

2.2.1. Reason for adoption of simplified mill charge modelAs to the mill charge composed of balls and particles in a tumbling

ball mill, a model whose cross sections are drawn schematically inFig. 1 is adopted in the present study. The reason is as follows.

To investigate grinding characteristics in previous papers [11,12],average properties representing the structure, flow and energy dissi-pation as to the mill charge were needed. However, because of thecomplexity, the average properties in relation to grinding conditionswere hindered from being quantified. A simplified mill chargemodel was then proposed and enabled estimations of the circulatingflow rate of the mill charge, the mean velocity of balls, the collisionenergy dissipated, the energy gained from the tumbling mill and soon. These properties were utilized to assess grinding characteristics,e.g., the grinding rate function called the selection function was de-rived in terms of operating conditions [11] and an empirical propor-tional relationship was proved to be valid between the grinding rateconstant and the mill power drawn [12]. Further, based on the de-rived relationships, theoretical energy-size reduction relationshipsin batch grinding were clarified [14]. In each study, derived resultswere confirmed with experiments to verify the simplified mill chargemodel as well as other assumptions adopted.

In the last two decades, computer simulations with DEM (DiscreteElement Method) have been demonstrated aiming at accurate predic-tions of the motion and contour of the mill charge in tumbling mills[15–17]. In some cases, simulated impact forces and energy dissipat-ed have been applied to the design of mill liners and lifters against thewear [17]. However, the DEMmethods at present do not simulate thebehavior of particles being ground and influences of the particles onthe motion of balls and the grinding characteristics are unpredictable.Therefore, to simulate grinding characteristics with DEM, further de-velopments are required such as converting computational outputswithout grinding to those with grinding or relating computationaloutputs with breakage properties measured [16,17]. Under such cir-cumstances, the simplified model is considered to be useful for esti-mating at least average properties of the mill charge in relation tothe operating conditions.

2.2.2. States of balls and particlesBalls and particles charged in a static mill are described by the ball

filling J, the fractional filling of ball void by powder U, the voidage ofthe ball bed εb and so on. For instance, the value of U equal to unitymeans that particles charged occupy all of the interstices of the staticball bed with εb.

The mill charge, when the mill is in operation, is conventionallydivided into three zones, i.e., the grinding zone where balls collideeach other to grind particles, the ascending zone where balls moveupwards along the mill wall and the falling zone where balls cascadeor cataract. The ‘grinding’ zone is named for the area where grindingof particles occurs and the ‘falling’ zone is named for balls falling (re-gardless of either cataracting or cascading) to release the potentialenergy.

The case of particles under filled in the ball bed void is depicted inFig. 1(a) and the overfilled case is shown in Fig. 1(b). The states ofballs and particles in the grinding zone are represented by parametersdenoted as Jo, Uo and εbo, the derivations of which are brieflyexplained in Appendix A.

2.2.3. Dispersion zoneDispersion of particles is assumed to occur only in the grinding

zone in which the violent motion of balls disperses particles nearby.Particles in the grinding zone are all exposed to be dispersed. Howev-er, all of them are not fully dispersed when particles are overfilled asdepicted in Fig. 1(b). The balls must disperse more particles than

D

γ o

Ras

θ bo

γ

r γ o

Ras

θ bo

γ

rRoRo

(a) Particles under filled (b) Particles overfilled

surface level of particlesε bo = ε b

ε bo > ε b

Fig. 1. Cross sections of operating ball mill.

39S. Nomura / Powder Technology 222 (2012) 37–51

those in the under filled case and some particles possibly remain un-dispersed.

Postulate a hypothetical ball bed whose voidage equal to εb asschematically drawn in Fig. 2. When all of the interstices of the ballbed with εb are filled with particles, the rest of the grinding zone par-ticles are located above the hypothetical boundary and these particlesare not to be dispersed. In other words, the dispersion zone is definedto have the same volume as that of the hypothetical ball bed with thevoidage of εb. The whole grinding zone is the dispersion zone in theunder filled case. It may be possible, even if in the under filled case,that all of the particles are not filled in the interstices and some areleft above the ball bed if no trembles are applied. The present studyassumes that this possibility is ignored under the dynamic motionof the mill charge.

A new parameter denoted as Mdis is introduced, expressing themass ratio of particles in the dispersion zone to those in the grindingzone, i.e.,

Mdis ¼ 1 when under filledMdis ¼ Josεb= αf f c

� �when overfilled ð9Þ

where Jos is the fraction of mill volume occupied by the hypotheticalball bed, αf is the mass ratio of grinding zone particles to total

Hypothetical boundary of dispersion zone

θ bs Outside of dispersion zone

θ bo

ε bo = ε b

Fig. 2. Dispersion zone for particles overfilled in grinding zone.

particles in a mill and fc is the fraction of mill volume occupied bythe bulk of particles charged. From the geometry shown in Fig. 2, Josis given by,

Jos ¼Jo 1−εboð Þ1−εbð Þ ¼ 2θbs− sin 2θbsð Þ

2πð10Þ

where θbs is the angle for the surface level of the hypothetical ball bed.Note that Mdis is equivalent to the ratio of the time period of particlesin the dispersion zone to that in the grinding zone.

2.2.4. Ratio of walking period to total period ηThe walking period of a randomwalker is equivalent to the period

of particles in the dispersion zone, which is equal to (ΔtgpMdis) whereΔtgp is the time period of particles in the grinding zone given byEq. (A.4) in Appendix A. Then, η is expressed as the ratio of the timeperiod in the dispersion zone to the total period in the mill, i.e.,

η ¼ ΔtgpMdis

Δtgp þ Δtap þ Δtfp¼ αf Mdis ð11Þ

where Δtap and Δtfp are the time periods of particles being in the as-cending and falling zones given by Eqs. (A.17) and (A.19), respectively.

2.2.5. Length of one step ℓThemacroscopic motion of particles or the bulk flow of particles in

the grinding zone is taken into account rather than the microscopicmotion of individual particles. That is, the bulk flow of particles be-tween colliding balls is regarded to correspond to the motion of a ran-dom walker and its direction is altered by the balls. The length of onestep ℓ is assumed to be proportional to the distance of the bulk flowof particles without being interrupted by the balls whose mean freepath is λb, i.e.,

‘∝λb ¼ db6 1−εboð Þ ð12Þ

where db is the ball diameter and λb is derived as follows. The frequencyof collisions of a ball is given by vs/λb which is also equal to σvsn, wherevs is the mean velocity of balls, σ is the effective cross sectional area of aball for collision equal to (πdb2) and n is the number density of ballsequal to [(1−εbo)/(πdb3/6)]. The equation of vs/λb=σvsn is rewrittenas λb=1/(σn) to give Eq. (12).

40 S. Nomura / Powder Technology 222 (2012) 37–51

2.2.6. Velocity of walking vwThe velocity of walking vw may correspond to the velocity of the

bulk flow of particles in the grinding zone. For simplicity, the bulkflow velocity is assumed to be proportional to the mean velocity ofballs in the grinding zone denoted as vs, i.e.,

vw∝vs: ð13Þ

For dry grinding, the velocity of balls in the grinding zone is atten-uated by the friction of surrounding particles. Then, vs is derived to be(see Appendix B1)

vs ¼ a1λbvb= ea1λb−1� �

ð14Þ

where a1=Uoρpb(1+μpb)/(dbρb) and μpb is the friction coefficient ofparticles.

In the case of wet grinding, the drag force of fluid and the buoyan-cy force in addition to the friction of particles are taken into accountto suppress the ball motion. Then, vs for wet grinding is expressedas (see Appendix B2),

vs ¼a2λbc

arctan vb=cð Þ− arccos ea2λbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivb=cð Þ2þ1

p� � ð15Þ

where a2={Uo/(dbρb)}{ρpb(1+μpb)+(3/4)Cfρw}, c ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiUoρwg= ρba2ð Þp

,Cf is the drag coefficient and ρw is the density of fluid.

Table 1Data of Mori et al. [2] and calculated results for corresponding conditions.

(a) Pilot Mill (D = 0.545 m, L = 1.98 m)

experiment

J F Holdup U db x100 fw u x100 E x10^4 E

- kg/s kg - cm - cm/s cm2/s

0.1 0.094 41.50 1.52 3.0 0.534 0.450 1.61 0

0.2 0.107 50.60 0.93 3.0 0.534 0.417 2.44 0

0.3 0.107 65.70 0.80 3.0 0.534 0.321 1.60 0

0.4 0.100 95.70 0.87 3.0 0.534 0.208 2.83 1

0.1 0.097 42.94 1.57 4.0 0.534 0.446 3.09 1

0.2 0.095 47.29 0.86 4.0 0.534 0.398 5.74 2

0.3 0.100 61.81 0.75 4.0 0.534 0.320 4.35 1

0.4 0.107 92.24 0.84 4.0 0.534 0.229 4.54 1

0.4 0.069 83.80 0.77 3.0 0.534 0.163 2.43 0

0.4 0.100 95.70 0.87 3.0 0.534 0.208 2.83 1

0.4 0.136 110.10 1.01 3.0 0.534 0.244 2.87 1

J F Holdup U db x100 fw u x100 E x10^4 E

- kg/s kg - cm - cm/s cm2/s

0.2 0.0020 2.09 0.70 1.6 0.30 0.048 0.51 0

0.2 0.0024 2.98 1.00 1.6 0.40 0.039 0.56 0

0.2 0.0022 3.02 1.02 1.6 0.50 0.036 0.72 1

0.2 0.0020 2.97 1.00 1.6 0.60 0.033 0.63 0

0.2 0.0019 3.06 1.03 1.6 0.70 0.031 0.65 0

(b) Laboratory Mill (D = 0.254 m, L = 0.495 m)

Steel ball: ρb=7800 kg/m3, ε

b=0.4, Limestone: ρ

p=2700 kg/m3, and ρ

pb=1480 kg/m3 (ε

2.3. Expressions of E and Pe with mill parameters

Substituting Eqs. (12) and (13) into Eqs. (7) and (8), E and Pe areexpressed by parameters obtainable from operating conditions andmill sizes as

E∝ηλbvs ð16Þ

Pe ¼ uL=E∝uL= ηλbvsð Þ: ð17Þ

Of the parameters used in Eq. (16), η expresses the size of the dispersionzone relative to the mill volume and λb and vs represent the mobility ofballs in the grinding zone. That is, the dispersion of particles is enhancedby the enlargement of the dispersion zone or the magnification of themobility of balls in the grinding zone.

As the proportional constants in the above equations are notknown at present, Eqs. (16) and (17) are rewritten by the normalizedforms using E /Eo, Pe /Peo, η /ηo, λb /λbo, vs /vso, L /Lo and u /uo asfollows,

E=Eo ¼ η=ηo

λb=λboð Þ vs=vsoð Þ ð18Þ

Pe=Peo ¼ L=Loð Þ u=uoð Þ Eo=Eð Þ ð19Þ

where the subscript o means the base case. When L /Lo and u /uo areconstant, Pe /Peo is inversely proportional to E /Eo.

3. Results and discussion

Section 3.1 explains data for dry andwet grinding operations, whichare used for confirmation of the theory. These data are compared withthe theory in Section 3.2, in which the effects of mill sizes on E and Pe

calculated

/Eo Pe Pe/Peo η λb Vs E/Eo Pe/Peo

- - - - m m/s - -

.57 55.20 3.80 0.16 0.010 0.758 0.61 3.54

.86 33.87 2.33 0.32 0.008 0.629 0.85 2.37

.56 39.83 2.74 0.42 0.008 0.543 0.95 1.64

.00 14.53 1.00 0.52 0.008 0.457 1.00 1.00

.09 28.58 1.97 0.16 0.013 0.834 0.89 2.40

.03 13.72 0.94 0.32 0.011 0.695 1.25 1.54

.54 14.59 1.00 0.42 0.011 0.599 1.39 1.11

.60 10.01 0.69 0.52 0.011 0.504 1.47 0.75

.86 13.27 0.91 0.52 0.008 0.460 1.01 0.78

.00 14.53 1.00 0.52 0.008 0.457 1.00 1.00

.01 16.85 1.16 0.52 0.008 0.454 0.99 1.19

/Eo Pe Pe/Peo η λb Vs E/Eo Pe/Peo

- - - - m m/s - -

.71 4.71 1.92 0.43 0.004 0.333 1.00 1.36

.78 3.47 1.41 0.38 0.004 0.378 1.01 1.10

.00 2.46 1.00 0.33 0.004 0.430 1.00 1.00

.89 2.61 1.06 0.29 0.004 0.483 0.98 0.96

.91 2.39 0.97 0.25 0.004 0.540 0.93 0.94

p=0.45).

41S. Nomura / Powder Technology 222 (2012) 37–51

are also examined theoretically although no data are available. Possibil-ities to predict E and Pe are discussed in Section 3.3.

3.1. Experimental data for comparison

3.1.1. Dry grinding dataMori et al. [2] tested two mills, a pilot mill (Ф0.545 m×1.98 m)

and a laboratory mill (Ф0.254 m×0.495 m). Table 1 lists three setsof data using the former mill and one set of data using the lattermill. In the former mill, two sets vary the ball filling J for two differentball diameters db of 0.03 m and 0.04 m and the rest varies the massfeed rate F. In the latter mill, the ratio of mill speed fw is varied. Thecalculated η, λb, vs, E /Eo and Pe /Peo values corresponding to thedata are also in Table 1 in which the base case, distinguished by theshaded row for each mill, is determined by the one with J closer to0.4 or otherwise the one with a middle value in a range of the variabletested. For db [m], u [m/s] and E [m2/s], the values of db×102 [cm],u×102 [cm/s] and E×104 [cm2/s] are listed in the table forconvenience.

Two laboratory mills were used by Swaroop et al., an open-end mill(Ф0.127 m×0.438 m) [3] and a constricted-end mill (Ф0.08 m×0.24 m) [4]. The effects of J, F and fw on E/Eo and Pe/Peo were tested

Table 2Data of Swaroop et al. [3,4] and calculated results for corresponding conditions.

(a) Open-end mill (D = 0.127 m, L = 0.438 m)

experiment

J F x10^3 Holdup U db x100 fw u x100 E x10^4 E

- g/s kg - cm - cm/s cm2/s

0.24 2.80 0.440 0.59 1.9 0.67 0.279 1.26 0.

0.40 2.80 0.542 0.44 1.9 0.67 0.226 1.42 1.

0.52 2.80 0.635 0.39 1.9 0.67 0.193 1.31 0.

0.40 1.30 0.308 0.25 1.9 0.67 0.184 1.56 1.

0.40 1.99 0.417 0.34 1.9 0.67 0.209 1.73 1.

0.40 2.22 0.450 0.36 1.9 0.67 0.216 1.72 1.

0.40 2.78 0.542 0.44 1.9 0.67 0.225 1.42 1.

0.40 2.90 0.720 0.58 1.9 0.28 0.176 0.75 0.

0.40 2.90 0.620 0.50 1.9 0.40 0.205 1.07 0.

0.40 2.90 0.580 0.47 1.9 0.55 0.219 1.39 0.

0.40 2.90 0.530 0.43 1.9 0.67 0.240 1.57 1.

0.40 2.90 0.565 0.45 1.9 0.79 0.225 1.71 1.

0.40 2.90 0.770 0.62 1.9 0.97 0.165 1.18 0.

(b) Constricted-end mill (D = 0.08 m, L = 0.24 m)

J F x1000 Holdup U db x100 fw u x100 E x10^4 E/

- g/s kg - cm - cm/s cm2/s

0.14 1.97 0.235 2.45 1.3 0.60 0.201 0.22 0.

0.17 1.97 0.222 1.92 1.3 0.60 0.213 0.28 0.

0.20 1.97 0.213 1.60 1.3 0.60 0.222 0.33 1.

0.23 1.97 0.201 1.31 1.3 0.60 0.236 0.41 1.

0.26 1.97 0.204 1.18 1.3 0.60 0.232 0.41 1.

0.20 0.51 0.122 0.91 1.3 0.60 0.101 0.34 1.

0.20 0.94 0.153 1.15 1.3 0.60 0.148 0.40 1.

0.20 1.65 0.191 1.43 1.3 0.60 0.207 0.38 1.

0.20 1.80 0.201 1.51 1.3 0.60 0.215 0.37 1.

0.20 1.84 0.209 1.57 1.3 0.60 0.211 0.35 1.

0.20 1.97 0.213 1.60 1.3 0.60 0.222 0.33 1.

Lucite ball: ρb=1250 kg/m3, ε

b=0.4, Dolomite: ρp=2860 kg/m3, and ρpb=1400 kg/m3 (ε

with the open-end mill and those of J and F were with the constrictedmill. The data are listed in Table 2 where the row of the base case is indi-cated by the shade for each mill.

For dry grinding, the following equation is used to calculate Ufrom the value of the mill holdup MH reported, i.e.,

MH ¼ ρpbVMUJεb ¼ ρpb πD2L=4� �

UJεb ð20Þ

where ρpb is the bulk density of particles.

3.1.2. Wet grinding dataKelsall et al. [5] conducted tests for wet grinding with a laboratory

mill (Ф0.305 m×0.305 m), in which the distribution of residencetime was approximated by that composed of a pure delay and a per-fect mixer in series. The reported data were the dimensionless delaytime δd (=to/τ) and the mean residence time τ appeared in the fol-lowing equations. The fraction of impulse response remaining in amill at time t, denoted as F′(t), is given by

F ′ tð Þ ¼ 1 for t≤toF ′ tð Þ ¼ exp −KM t−toð Þ½ � for t≥to

ð21Þ

calculated

/Eo Pe Pe/Peo η λb Vs E/Eo Pe/Peo

- - - - m m/s - -

89 9.68 1.40 0.33 0.0053 0.439 0.87 1.43

00 6.97 1.00 0.48 0.0053 0.347 1.00 1.01

92 6.45 0.93 0.63 0.0053 0.269 1.03 0.84

10 5.17 0.75 0.48 0.0053 0.350 1.01 0.81

21 5.30 0.76 0.48 0.0053 0.349 1.01 0.93

21 5.50 0.79 0.48 0.0053 0.348 1.00 0.96

00 6.93 1.00 0.48 0.0053 0.347 1.00 1.00

53 10.30 1.49 0.80 0.0053 0.121 0.58 1.35

75 8.38 1.21 0.71 0.0053 0.193 0.83 1.10

98 6.89 0.99 0.59 0.0053 0.272 0.97 1.01

10 6.68 0.96 0.48 0.0053 0.344 1.00 1.07

20 5.74 0.83 0.38 0.0053 0.423 0.97 1.03

83 6.13 0.88 0.25 0.0053 0.548 0.84 0.87

Eo Pe Pe/Peo η λb Vs E/Eo Pe/Peo

- - - - m m/s - -

67 21.87 1.35 0.13 0.0056 0.407 0.87 1.05

85 18.32 1.13 0.17 0.0048 0.386 0.93 1.03

00 16.18 1.00 0.21 0.0044 0.370 1.00 1.00

24 13.82 0.85 0.26 0.0040 0.355 1.09 0.97

24 13.58 0.84 0.31 0.0038 0.340 1.16 0.90

03 7.17 0.44 0.29 0.0036 0.374 1.15 0.40

21 8.91 0.55 0.27 0.0038 0.371 1.09 0.61

17 12.93 0.80 0.23 0.0042 0.369 1.03 0.91

13 13.91 0.86 0.22 0.0043 0.369 1.02 0.95

06 14.57 0.90 0.21 0.0044 0.368 1.01 0.94

01 16.09 0.99 0.21 0.0044 0.368 1.00 1.00

p=0.51).

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0 50 100 150 200 250

time t [sec]

Ln(F

'(t))

Present theory

Kelsall model

KM

d

Pe = 3.031

d)

d)]

d

Fig. 3. Kelsall model of a pure delay and perfect mixer in comparison with theoreticalcurve to give a best fit.

42 S. Nomura / Powder Technology 222 (2012) 37–51

where to is the delay time and KM is the mixer time constant which isrelated to the mean residence time τ as follows.

τ ¼ ∫∞0 t −dF ′ tð Þ=dt� �

dt ¼ ∫∞tot KM exp −KM t−toð Þ½ �f gdt

¼ 1=KMð Þ þ to: ð22Þ

Substituting the above relation into Eq. (21) leads to

F ′ tð Þ ¼ 1 for θ≤δdF ′ tð Þ ¼ exp − t−toð Þ= τ−toð Þ½ � ¼ exp − θ−δdð Þ= 1−δdð Þ½ � for θ≥δd

ð23Þ

where θ= t /τ and δd= to /τ.For a set of δd and τ values, the corresponding Pe is estimated as

follows. The fraction of impulse response remaining in a mill at time

Table 3Data of Kelsall et al. [5] and calculated results for corresponding conditions.

(a) Wet Mill (D = 0.3048 m , L = 0.3048 m)

experiment

J F x10^3 Holdup U db x100 fw u x100 aw δd τ E

- g/s kg - cm - cm/s - - s

0.18 6.67 0.98 0.52 2.54 0.78 0.207 0.67 0.45 1470.27 6.67 1.27 0.46 2.54 0.78 0.160 0.67 0.25 1910.41 6.67 1.37 0.33 2.54 0.78 0.149 0.67 0.23 2050.41 3.33 1.28 0.31 2.54 0.78 0.079 0.67 0.16 3840.41 6.67 1.38 0.33 2.54 0.78 0.147 0.67 0.23 2070.41 22.17 2.39 0.57 2.54 0.78 0.282 0.67 0.30 1080.41 6.53 1.35 0.32 2.54 0.78 0.148 0.67 0.22 2060.41 6.53 1.77 0.43 2.54 0.78 0.112 0.67 0.19 2710.41 6.53 2.18 0.52 2.54 0.78 0.092 0.67 0.15 3310.41 6.53 7.63 1.83 2.54 0.78 0.026 0.67 0.09 1169

0.41 6.67 1.42 0.34 1.91 0.78 0.144 0.67 0.21 212

0.41 6.67 1.37 0.33 2.54 0.78 0.149 0.67 0.23 205

0.41 6.67 1.27 0.31 3.18 0.78 0.160 0.67 0.23 191

0.41 6.67 1.20 0.29 3.81 0.78 0.169 0.67 0.20 180

0.41 6.67 2.54 0.41 2.54 0.65 0.120 0.67 0.23 3810.41 6.67 2.34 0.37 2.54 0.78 0.130 0.67 0.26 3510.41 6.67 2.59 0.41 2.54 0.91 0.118 0.67 0.26 3880.41 6.67 0.87 0.29 2.54 0.78 0.233 0.55 0.30 1310.41 6.67 1.46 0.35 2.54 0.78 0.139 0.67 0.22 220

0.41 6.67 3.03 0.59 2.54 0.78 0.067 0.75 0.18 454

Steel ball: εb=0.4, ρ

b=7800 kg/m3, Calcite: ρ

p=2700 kg/m3, ε

p=0.45, ρ

w=1000 kg/m3,

t is derived from the present theory by integrating Eq. (4) from t toinfinity, which is denoted as Fp′(t), i.e.,

F ′p tð Þ ¼ ∫∞t Φ tð Þdt ¼ 1−∫θ

0

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4πθ3=Pe

q exp − 1−θð Þ24θ=Pe

" #dθ: ð24Þ

Fp′(t) in Eq. (24) is equivalent to F′(t) in Eq. (23). Therefore, for thecurve of F′(t) with the value of δd reported, the value of Pe to showa best fit curve of Fp′(t) is evaluated. Using the evaluated Pe withthe aid of u and L, the value of E (=uL /Pe) is calculated. An exampleis depicted in Fig. 3 where the Kelsall model given by Eq. (23) usesδd=0.23 and the present model of Eq. (24) gives a best fit forPe=3.03 and then E (=uL /Pe) to be 1.49×10−4 m2/s with the aidof u=0.149×10−2 m/s and L=0.3048 m. Thus obtained Pe and Evalues are regarded as the experimental data listed in Table 3.

For wet grinding, the following equation is used to calculate Ufrom MH,

MH ¼ awρslVMUJεb ¼ awρsl πD2L=4� �

UJεb ð25Þ

where ρsl is the density of slurry (see Appendix C).

3.2. Comparison between theory and experiment

3.2.1. Effect of ball fillingFig. 4 shows the effects of J on E /Eo and Pe /Peo, in which the data

of Mori et al. are for two ball diameters of 0.03 m and 0.04 m, those ofSwaroop et al. are for two different mills and those of Kelsall et al. arefor wet grinding. In each case, E /Eo increases with increasing J andPe /Peo shows a reverse of this trend, in which the theory (fullsymbols) and the experiment (empty ones) are within reasonableagreement.

calculated

x10^4 E/Eo Pe Pe/Peo η λb Vs E/Eo Pe/Peo

cm2/s - - - - m m/s - -

0.91 0.61 6.92 2.28 0.21 0.0071 0.733 0.77 1.821.50 1.00 3.24 1.07 0.26 0.0071 0.654 0.87 1.241.49 1.00 3.03 1.00 0.36 0.0071 0.548 1.00 1.001.00 0.67 2.43 0.80 0.36 0.0071 0.548 1.00 0.531.48 0.99 3.03 1.00 0.36 0.0071 0.548 1.00 0.992.23 1.49 3.85 1.27 0.36 0.0071 0.540 0.99 1.931.54 1.03 2.93 0.97 0.36 0.0071 0.548 1.00 0.991.28 0.86 2.67 0.88 0.36 0.0071 0.545 0.99 0.761.19 0.80 2.36 0.78 0.36 0.0071 0.542 0.99 0.630.40 0.27 1.99 0.66 0.27 0.0094 0.464 0.84 0.21

1.54 1.03 2.84 0.94 0.36 0.0053 0.498 0.68 1.42

1.49 1.00 3.03 1.00 0.36 0.0071 0.548 1.00 1.00

1.61 1.08 3.03 1.00 0.36 0.0088 0.590 1.35 0.80

1.87 1.25 2.76 0.91 0.36 0.0106 0.626 1.71 0.66

1.81 1.21 3.03 1.00 0.45 0.0071 0.454 1.02 1.191.78 1.19 3.35 1.11 0.36 0.0071 0.546 1.00 1.321.61 1.08 3.35 1.11 0.29 0.0071 0.644 0.93 1.281.84 1.23 3.85 1.27 0.36 0.0071 0.552 1.01 1.561.44 0.97 2.93 0.97 0.36 0.0071 0.547 1.00 0.94

0.79 0.53 2.58 0.85 0.36 0.0071 0.536 0.98 0.46

and μw=0.001 Pa s.

0

1

2

E/E

o [

- ] dbMori et. al.

0

1

2E

/Eo

[ -

] Open-end Constricted-end

Swaroop et.al.

0

1

2

J [ - ]

E/E

o [

- ] Kelsall et.al.

0 0.2 0.4 0.6 0 0.2 0.4 0.60

2

4

Pe/

Peo

[ -

]db

0 0.2 0.4 0.6 0 0.2 0.4 0.60

2

4

Pe/

Peo

[ -

]Open-end Constricted-end

0 0.2 0.4 0.6 0 0.2 0.4 0.6J [ - ]

0

2

4

Pe/

Peo

[ -

]

0.03m0.04m

0.03m0.04m

Fig. 4. Effects of J on E /Eo and Pe /Peo.

43S. Nomura / Powder Technology 222 (2012) 37–51

Theoretically, the increase of E /Eo is explained as follows. η /ηo in-creases with increasing J as the grinding zone or the dispersion zoneis enlarged whereas vs /vso decreases due to the decrease of the ener-gy given to a ball in the grinding zone. The trend of E /Eo depends onthe magnitudes of the two conflicting effects. In Fig. 4, the magnitudeof the increase of η /ηo is superior to that of the decrease of vs /vso.

The decrease of Pe/Peowith increasing J is derived as follows. Pe/Peois proportional to (Eo/E)(u /uo) under constant L, in which the decreaseof Eo/Ewith increasing J is mentioned above and u is inversely propor-tional to J according to Eqs. (3) and (20) for constant F.

3.2.2. Effect of feed rateWhen varying F, U (or MH) is varied. Therefore, the effect of F on E

is explained theoretically as the effect of U as follows. For U less thanunity (or for F less than a value to give U equal to unity), η /ηo and λb /λbo are constant (see Tables 1 and 2) according to Eqs. (11) and (12).

0

1

2

E/E

o [

- ]

Mori et.al.

0

1

2

E/E

o [

- ]

Open-end Constricted-end

x 10-3

x 10-3

Swaroop et.al.

0

1

2

F [kg/s]

E/E

o [

- ] Kelsall et.al.

0 0.1 0.2 0.3 0

0 1 2 3 0

0 10 20 30 0

Fig. 5. Effects of F on

For U greater than unity (or for F greater than that value), η /ηo de-creases slightly with increasing U (or F) due to the decrease of Mdis

given by Eq. (9) and a slight increase of λb /λbo is obtained due tothe increase of εbo. The value of vs /vso decreases slightly with increas-ing U (or F) in the whole range of U (or F) as more particles for greaterU (or F) act on the ball surface due to the friction force to reduce theball velocity. These effects are combined, resulting in a nearly con-stant trend of E /Eo against U (or F).

For the trend of Pe /Peo, u is dominant under the nearly constantE /Eo, i.e., Pe /Peo increases with increasing F as u increases. The in-crease of u is explained from Eq. (3) that F /MH increases with increas-ing F because a linear relation is valid [3,4] between MH and Fexpressed by MH=aF+b where a and b are positive constants.

Fig. 5 shows the effects of F on E/Eo and Pe/Peo. Nearly constanttrends of E/Eo in the data of Mori et al. and those of Swaroop et al. areobtainedwithin agreementwith the theory. However, a slight difference

0

1

2

Pe/

Peo

[ -

]

x 10-3

x 10-3

0

1

2

Pe/

Peo

[ -

]Open-end Constricted-end

0.1 0.2 0.3

1 2 3

10 20 30

F [kg/s]

0

1

2

Pe/

Peo

[ -

]

E /Eo and Pe /Peo.

0

1

2

Pe/

Peo

[ -

]

0

1

2

E/E

o [

- ]

Mori et.al.

0

1

2E

/Eo

[ -

] Open-end Swaroop et.al.

0

1

2

Pe/

Peo

[ -

]Open-end

0

1

2

fw [ - ] fw [ - ]

E/E

o [

- ] Kelsall et.al.

0.2 0.4 0.6 0.8 10.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 10

1

2

Pe/

Peo

[ -

]

Fig. 6. Effects of fw on E /Eo and Pe /Peo.

44 S. Nomura / Powder Technology 222 (2012) 37–51

is observed in the case of Kelsall et al., i.e., the data of E/Eo increase withincreasing F unlike the prediction mentioned above. The reason is notknown at present and a possible cause will be discussed in Section 3.2.4.

3.2.3. Effect of mill speedTheoretically, when varying fw under the constant J, U, F and db, η /ηo

decreases with increasing fw as the grinding zone becomes smallerwhereas vs/vso increases due to the greater energy given to the grindingzone balls. These conflicting effects are combined to give rise to a con-vex curve of E/Eo with a peak at about fw=0.6. The curve of Pe/Peoshows an opposite to that of E/Eo because u /uo is constant.

As depicted in Fig. 6, trends for E/Eo and Pe/Peo arewithin agreementbetween theory and experiment. In the case of Swaroop et al., the valueof MH is also varied with fw, being concave (see Table 2), which is over-lapped to emphasize the convex trend of E/Eo because vs/vso decreaseswith increasing MH (or U) as explained in Section 3.2.2. The curve of E/Eo in the case of Kelsall et al. shows a minor decrease as the values of fware greater than 0.6 (being from 0.65 to 0.91).

Although the data of Mori et al. show a minor increase of E /Eowith increasing fw from 0.3 to 0.5, the calculated E /Eo values corre-sponding to the data are nearly constant, which is also slightly differ-ent from the prediction mentioned above for fw less than 0.6. This isdue to the reported value of MH (2.09 kg) for fw=0.3 being smallerthan those (about 3 kg) for other fw values for some reason. Basically,

0

0.5

1

1.5

2

2.5

3

MH [ kg ]

E/E

o [

- ]

calculated

experiment

0 2 4 6 8 10 0

Fig. 7. Effects of MH on E /Eo and Pe /Peo in wet grindin

an increase of fw from 0.3 to 0.5 under constant U leads to an increaseof E. On the other hand, a simultaneous increase ofMH from 2.09 kg to3.02 kg for constant fw contributes to the decrease of E as vs /vso de-creases with increasing MH (or U) as explained in Section 3.2.2.These two effects compensate each other, resulting in the calculatedE /Eo nearly constant in the case of Mori et al.

Since variations of both E /Eo and Pe /Peo are found to be minor in arange of practical operations for fw being about between 0.6 and 0.8[1], fw is considered to be less important for E /Eo and Pe /Peo.

3.2.4. Effect of mill holdup under constant feed rateTheoretically, a nearly constant trend (only a slight decrease) of E /

Eo against MH (or U) is obtained. The reason for the effect of U on Ementioned in Section 3.2.2 is also valid. For Pe /Peo, a relatively greatdecrease with increasingMH is predicted due to the inversely propor-tional relation between u and MH as seen in Eq. (3).

The data of Kelsall et al. are shown in Fig. 7, in which both E /Eoand Pe /Peo decrease with increasing MH. Slight differences in slopeare observed between the theory and the experiment. A possible rea-son is discussed as follows.

The two trends of E/Eo for the data of Kelsall et al. depicted in Figs. 5and 7 are compared. These are opposite each other.When increasing ei-ther F in Fig. 5 or MH in Fig. 7, U increases in the both cases. The differ-ence is that the value of u increases with increasing F but decreases

MH [ kg ]

2 4 6 8 100

0.5

1

1.5

2

2.5

3

Pe/

Peo

[ -

]

g mill (comparison with data of Kelsall et al. [5]).

0

0.5

1

1.5

2

2.5

3

db [ m ] db [ m ]

E/E

o [

- ]

calculated

experiment

x 10-2 x 10-2 1 2 3 4 5 1 2 3 4 5

0

0.5

1

1.5

2

2.5

3

Pe/

Peo

[ -

]

Fig. 8. Effects of db on E /Eo and Pe /Peo in wet grinding mill (comparison with data of Kelsall et al. [5]).

45S. Nomura / Powder Technology 222 (2012) 37–51

with increasing MH. This difference, however, may not affect the valueof E as u is sufficiently small compared to the mean velocity of balls vs.The conflicting trends in the data of E/Eomay be caused by an inaccura-cy in estimating experimental Pe values, i.e., Pe values are estimatedfrom the data of the pure delay and perfect mixer model. If the curveof experimental Pe/Peo against F is steeper than the one in Fig. 5, the re-sultant E/Eo against F would be closer to the calculation. Also judgingfrom the cases of the dry grindings in Fig. 5 showing minor variationsof E/Eo against F, the nearly constant trend of E/Eo in the calculationfor the case of Kelsall et al. seems plausible. The same considerationmay be possible for the curve of the experimental Pe /Peo against MH

in Fig. 7. If the curve of Pe/Peo is steeper than the one in Fig. 7, thetrend of the experimental E/Eo would be closer to the calculated one.

3.2.5. Effect of ball diameterTheoretically, E/Eo increases with increasing db as λb /λbo increases

according to Eq. (12) and vs/vso also increases as vb in Eq. (14) is propor-tional to λb1/3 (see Eq. (B.2)). This means that larger balls exhibit greatermobility to disperse particles. For Pe /Peo, the curve shows an oppositeto that of E/Eo when u/uo is constant.

In Fig. 4, two sets of data of Mori et al. are displayed for two balldiameters of 0.03 m and 0.04 m. The data exhibit that the larger balldiameter gives greater E /Eo values and smaller Pe /Peo values asexplained theoretically.

On the other hand, slight differences are observed in the case ofKelsall et al. plotted in Fig. 8. For E /Eo, the data show a minor increasewith increasing db, in which three plots for db between 0.0191 m and

0

0.5

1

1.5

2

2.5

3

aw [ - ]

E/E

o [

- ]

calculated

experiment

0.5 0.6 0.7 0.8 0.

Fig. 9. Effects of aw on E /Eo and Pe /Peo in wet grindin

0.0318 m are nearly constant. For Pe /Peo, the variation of the data isminor. The nearly constant three E /Eo values may not be understoodfor the two reasons. One is that larger balls should exhibit greater Evalues as explained theoretically. The other is that MH decreaseswith increasing db in the experiment (see Table 3), i.e., the decreaseof MH must contribute to an increase of E /Eo (although only a slightincrease) as noted in Section 3.2.4. The experimental trend of E /Eomay be again caused by an inaccuracy in obtaining Pe from thereported data.

3.2.6. Effect of slurry densityA typical property of wet grinding is the weight fraction of solids

in slurry aw. Its effects on E /Eo and Pe /Peo are examined.Theoretically, E /Eo varies little against aw (see Table 3). The reason

is that η /ηo and λb /λbo are constant for U less than unity under theconstant J, fw and db and vs /vso decreases only slightly with increasingaw as more particles for greater aw act on the ball surface to suppressthe ball motion due to the friction force. As for Pe /Peo, a relativelygreat decrease with increasing aw is predicted according to Eq. (19),in which u /uo decreases with increasing aw (or MH) under the nearlyconstant E /Eo.

As shown in Fig. 9, a minor decrease of E /Eo with increasing aw isobtained in the data whereas the calculated E /Eo values are nearlyconstant. As for Pe /Peo, the slope of the data is gentler than the calcu-lation. The difference may be again attributed to an inaccuracy inobtaining Pe values from the reported data. If the curve of the

aw [ - ]5 0.6 0.7 0.8

0

0.5

1

1.5

2

2.5

3

Pe/

Peo

[ -

]

g mill (comparison with data of Kelsall et al. [5]).

0.0

0.5

1.0

1.5

2.0

2.5

3.0

D/Do [-] D/Do [-]

E/E

o , P

e/P

eo [

-]Pe/Peo

E/Eo

0.8 1.0 1.2 1.4 0.8 1.0 1.2 1.40.0

0.5

1.0

1.5

2.0

2.5

3.0

o, λ

b/λ

bo,

v s

/v s

o, u

/uo

[-]

u/uo

Fig. 10. Effects of D /Do on E /Eo and Pe /Peo under constant L.

46 S. Nomura / Powder Technology 222 (2012) 37–51

experimental Pe /Peo is steeper than the one in Fig. 9, the data of E /Eowould be closer to the calculated ones.

3.2.7. Effects of mill sizesAlthough no data are available, sensitivities of mill sizes to E /Eo

and Pe /Peo are examined theoretically. The pilot mill used by Moriet al. is arbitrary chosen as the base mill and the base conditions tonormalize the variables in Eqs. (18) and (19) are J=0.4, U=1.0,fw=0.7, db=0.04 m, D=0.545 m and L=1.98 m.

Two cases are studied, one is varying D under constant L and theother is varying L under constant VM simulating the constant meanresidence time when varying L /D. The case of varying L under con-stant D (VM is varied) is not examined as the values of η /ηo, λb /λbo,vs /vso, u /uo and thus E /Eo are constant except Pe /Peo and the propor-tional relation between Pe and L is apparent (see Eq. (19)).

(a) Effect of mill diameterFig. 10 shows that E /Eo increases only slightly with increasingD /Do. The reason is that both η /ηo and λb /λbo are constantwhen J, U, fw and db are constant and vs /vso increases onlyslightly with increasing D /Do due to the increase of the poten-tial energy of balls (see the average distance of balls to falldenoted as Hab given by Eq. (A.24)).On the other hand, Pe /Peo varies greatly, decreasing with in-creasing D /Do. The reason is that u /uo is inversely proportionalto (D /Do)2 under the constant J, U and F, resulting in Pe /Peo in-versely proportional to (D /Do)2 according to Eq. (19) in whichE /Eo varies little and L /Lo is constant. Although the influence

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.40. 60. 81. 01. 21. 4

L/Lo [-]

E/E

o , P

e/P

eo [

-]

Pe/Peo

E/Eo

Fig. 11. Effects of L /Lo on E /Eo an

on E is minor, D is important as the residence time distributionΦ(t) is dominated by Pe.

(b) Effect of mill lengthOnly a slight decrease of E /Eo with increasing L /Lo is obtainedin Fig. 11. The reason is that η /ηo and λb /λbo are constantunder the constant J, U, fw and db and vs /vso decreases onlyslightly with increasing L /Lo as D decreases with increasing Lunder the constant VM causing a decrease of the potential ener-gy of balls.Pe /Peo increases greatly with increasing L /Lo, as depicted inFig. 11. The reason is that Pe /Peo is proportional to (L /Lo)2,led by the proportional relationships between Pe and L (seeEq. (8)) and between u and L (see Eq. (3)) under the constantVM. Thus, L is also important as the residence time distributionis dominated by Pe.

3.3. Possibilities of predicting E and Pe

For the data of E and Pe, the corresponding terms on the right handsides of Eqs. (16) and (17) are calculated and plotted in Figs. 12 and13, respectively. In the figures, the values of R2 are also noted, repre-senting the correlation coefficients for the proportionality betweenthe vertical axis and the horizontal one.

Some plots are scattered in both figures. In the case of Kelsall et al.,the scatter may be caused by an inaccuracy in estimating Pe valuesfrom the original data as discussed in Section 3.2.4. In the cases ofMori et al. and Swaroop et al., possible reasons for the scatter are ex-perimental errors or simplifications adopted in the theory.

L/Lo [-]

0.40. 60. 81. 01. 21. 40.0

0.5

1.0

1.5

2.0

2.5

3.0

u/uo

o, λ

b/λ

bo,

v s/

v so,

u/u

o [-

]

d Pe /Peo under constant VM.

0

2

4

6

8

0

b vs (calc.) [m 2/s]

E (

data

) [m

2 /s]

Mori et.al.

Swaroop et.al.

Kelsall et.al.

x10-2

x10-2

R2 = 0.732

5040302010

Fig. 12. Relationship between experimental E values and calculated (ηλbvs).

47S. Nomura / Powder Technology 222 (2012) 37–51

The correlation for the plots of Pe in Fig. 13 is better than that forthe plots of E in Fig. 12. This is due to some operating parameters,when varied, affecting u rather than E, i.e., trends of Pe tend to beclose to those of u. Since both theory and experiment use the sameequation, Eq. (3), to estimate u, the resultant theoretical and experi-mental trends for Pe tend to be close each other.

Further, it is worthwhile to note that the proportional relationshipexpressed by Eq. (17) to predict Pe appears to be valid regardless ofthe mill sizes examined.

4. Conclusions

The present study has developed a theory on the dispersion coef-ficient and the Peclet number in a continuous ball mill. Firstly, the dis-persion coefficient has been expressed by statistical parameters of arandom walk problem. Then, the statistical parameters have beenspecified bymill operating variables based on a dispersion model pos-tulated in the grinding zone.

The derived equations have revealed that the dispersion coeffi-cient is a function of the size of the dispersion zone and the mobil-ity of balls in the grinding zone represented by the mean velocityand the mean free path and the Peclet number is a function ofthe dispersion coefficient, the axial mean velocity of material flow-ing and the mill length. Results derived from the theory have beenwithin reasonable agreement with reported data of the dispersion

0

20

40

60

0 2 4 6 8

b vs) (calc.) [-]

Pe

(dat

a) [

-]

Mori et.al.

Swaroop et.al.

Kelsall et.al.

R2 = 0.875

Fig. 13. Relationship between experimental Pe values and calculated uL /(ηλbvs).

properties for dry and wet grinding conditions although minorvariations have been observed between the theory and the data,for which further investigations are required both theoreticallyand experimentally.

Additionally, the theory has predicted significant influences of themill length L and diameter D on Pe. This implies the importance of de-sign of L and D to produce materials with desired size distributions asPe determines the residence time distribution according to Eq. (1).

Further, a proportional relationship to predict Pe has been derivedas Eq. (17), which appears to be valid regardless of the mill sizes test-ed. For estimating the residence time distributions using Pe undergiven grinding conditions, the proportional relationship should beuseful as a first approximation.

In future, this theory will be tested against more data, e.g., data ofrelatively large scale mills, to establish a sound basis. Further devel-opment will be made to predict the size distributions of product inmills with residence time distributions. This will lead to a methodol-ogy which can be adopted to optimize operating conditions and millsizes satisfying required finenesses with economical performances.

List of symbols

a1 constant defined in Eq. (B.8), m−1

a2 constant defined in Eq. (B.14), m−1

aw mass fraction of solids in slurry, –awo mass fraction of solids in slurry of base case, –Cf drag coefficient, –CR number rate of balls ascending in the ascending zone, s−1

c constant defined in Eq. (B.14), m/sD mill diameter, mDo mill diameter of base case, mdb ball diameter, mdbo ball diameter of base case, mE dispersion coefficient of particles, m2/sEo dispersion coefficient of particles of base case, m2/sF mass feed rate, kg/sFb buoyancy force acting on ball, NFd drag force of fluid acting on ball, NFn normal force of particles acting on ball surface, NFt tangential force of particles acting on ball surface, NF′(t) fraction of impulse response remaining in a mill at time t

given by Eq. (23), –Fp′(t) fraction of impulse response remaining in a mill at time t

given by Eq. (24), –fc fraction of mill volume occupied by bulk of particles

charged, –fw ratio of angular velocity of mill revolution to critical one, –G(ξ) fraction defined in Eq. (A.14), –g acceleration due to gravity, m/s2

Hab average falling distance of balls, mHap average falling distance of particles, mh(r) vertical distance from departing point along ascending cir-

cular arc with radius r to surface level of grinding zone, mJ fraction of mill volume occupied by ball bed, –Jo fraction of mill volume occupied by grinding zone ball bed,

–

Jos fraction of mill volume occupied by grinding zone ball bedwith voidage equal to εb, –

KM mixer time constant, s−1

L mill length, mLo mill length of base case, mℓ length of one step of random walker, mMb mass of a ball, kgMdis mass ratio of particles in dispersion zone to those in grind-

ing zone, –MH mass holdup of particles, kg

48 S. Nomura / Powder Technology 222 (2012) 37–51

m number of steps of random walk, –n number density of balls in grinding zone, m−3

Pe uL /E, Peclet number, –Peo uoLo /Eo, Peclet number of base case, –Ras radius of inner surface of ball bed in ascending zone, mRo radius of inner surface of ascending zone, mr radial distance from mill center, mt time variable, st1 time period of a ball traveling a distance of λb, sΔta mean residence time of balls in ascending zone, sΔtap mean residence time of particles in ascending zone, sΔtf mean residence time of balls in falling zone, sΔtfp mean residence time of particles in falling zone, sΔtg mean residence time of balls in grinding zone, sΔtgp mean residence time of particles in grinding zone, sto delay time, sts period of one step, sU fraction of ball void filled by bulk of particles in static mill, –Uo fraction of ball void filled by bulk of particles in grinding

zone, –u mean axial velocity of material flowing in mill, m/suo mean velocity of material flowing in mill of base case, m/sVM volume of a mill, m3

VRb volumetric flow rate of balls circulating, m3/sVRp volumetric flow rate of particles circulating, m3/sVw volume of water added per unit volume of dry particles, –v velocity of a ball, m/svb mean velocity of balls in grinding zone with no particles, m/

svs mean velocity of balls in grinding zone, m/svso mean velocity of balls in grinding zone of base case, m/svw velocity of one step walk, m/sx axial distance from mill inlet, m

Greek lettersαf ratio of grinding zone particles to total particles charged to

mill, –γ arccos(−rωr /g), radγo arccos(Ro / r), radδd dimensionless delay time, –εb 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, –η ratio of period of walking to total period or ratio of period of

particles in dispersion zone to total period in mill, –ηo ratio of period of particles in dispersion zone to total period

in mill of base case, –θ t /τ, dimensionless time variable, –θb angle for surface level of static ball bed, radθbo angle for surface level of grinding zone, radθbs angle for surface level of hypothetical ball bed with voidage

of εb, radλb average distance between two adjacent balls, mλbo λb of base case, mμpb friction coefficient of bulk of particles, –ξ r /(D /2) dimensionless radius, –ξi Ro /(D /2), –ξs Ras /(D /2), –ρb density of ball, 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

σ effective cross sectional area of a ball for collision, m2

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

τ L /u, mean residence time, sτR shear stress acting tangentially on ball surface, N/m2

Φ(t) impulse response function of residence time distribution atmill exit (x=L), s−1

Φr angle of repose, radφ angle defined in Fig. B.1, radωr angular velocity of mill revolution, rad/s

Acknowledgments

The author would like to thank Dr. T.G. Callcott (Callcott Consult-ing, N.S.W., Australia) for his kind advice on this publication.

Appendix A. States of balls and particles in operated mill

Fig. 1 shows two schematic drawings of the cross section of a ro-tating mill. Fig. 1(a) draws the case of particles under filled in the in-terstices of the ball bed and Fig. 1(b) is that of particles overfilled. Thegrinding zone is defined as the portion lower than the surface levelwith the angle of θbo. The area above the surface level where ballsand particles move upwards along the mill wall is called the ascend-ing zone. The space where balls cascade or cataract is the falling zone.

To analyze the fractions of balls and particles in the three zones,the volumetric balances of balls and particles in an operated mill areconsidered, i.e.,

VMJ 1−εbð Þ ¼ VRbΔtg þ VRbΔta þ VRbΔtf ðA:1Þ

VMf c ¼ VRpΔtgp þ VRpΔtap þ VRpΔtfp ðA:2Þ

where VRb and VRp are the volumetric flow rates of balls and bulk ofparticles circulating, fc is the fraction of mill volume occupied by thebulk of particles charged and variables denoted as Δt with subscriptsare the mean residence times of balls and particles in the threezones. These two equations are solved simultaneously for given staticconditions of J, U and εb to evaluate the dynamic states given by Jo, Uo

and εbo. The calculations require equations noted below for the threezones.

A.1. Grinding zone

This section specifies the first terms on the right hand sides ofEqs. (A.1) and (A.2). Variables Δtg, Δtgp, VRb and VRp are expressed asfollows,

Δtg ¼ VMJo 1−εboð Þ=VRb ðA:3Þ

Δtgp ¼ VMJoεboUo=VRp ðA:4Þ

VRb ¼ ∫D=2Ro

1−εboð ÞLωrrdr ¼ 1−εboð Þωr VM=πð Þ 1−ξ2i� �

=2 ðA:5Þ

VRp ¼ Uo∫D=2Ro

εboLωrrdr ¼ Uoεboωr VM=πð Þ 1−ξ2i� �

=2 ðA:6Þ

where ξi=Ro /(D /2), ωr is the angular velocity of mill revolution andparameters Jo, Uo and εbo are obtained as follows. Of these, Jo isexpressed geometrically as,

Jo ¼ 2θbo− sin2θboð Þ= 2πð Þ: ðA:7Þ

As to Uo and εbo, the two cases, Fig. 1(a) and (b), give differentequations noted below.

Fig. B.1. Forces acting on ball surface.

49S. Nomura / Powder Technology 222 (2012) 37–51

A.1.1. Case of particles under filledUo is less than unity. From the definition of Uo, i.e., the fraction of

the ball bed void filled by the bulk of particles in the grinding zone,

Uo ¼ αf f c= Joεboð Þ≤1 ðA:8Þ

where αf is the ratio of the grinding zone particles to the total particlescharged, given by

αf ¼ VRpΔtgp= VMf cð Þ ¼ Δtgp= Δtgp þ Δtap þ Δtfp� �

: ðA:9Þ

In this case, εbo is assumed to be equal to that of the static ball bed εb, i.e.,

εbo ¼ εb: ðA:10Þ

A.1.2. Case of particles overfilledBalls and particles in the grinding zone are assumed to be completely

mixed. In other words, the ball bed is enlarged until the overfilled parti-cles are taken in the interstices, i.e., Uo is equal to unity,

Uo ¼ αf f c= Joεboð Þ ¼ 1: ðA:11Þ

Rearranging the above equation leads to εbo as follows;

εbo ¼ αf f c=Jo≥εb: ðA:12Þ

A.2. Ascending zone

VRbΔta and VRpΔtap, the second terms on the right hand sides ofEqs. (A.1) and (A.2), are specified. These are the volumetric holdupsof balls and bulk of particles in the ascending zone, obtained fromthe following equations as,

VRbΔta ¼ ∫D=2Ras

γ−γoð Þ 1−εbð ÞLrdr ¼ 1−εbð Þ VM=πð ÞG ξsð Þ ðA:13Þ

VRpΔtap ¼ Uo ∫1ξs γ−γoð ÞεbL D=2ð Þ2ξdξþ ∫ξs

ξiγ−γoð ÞL D=2ð Þ2ξdξ

h i¼ Uo VM=πð Þ G ξið Þ− 1−εbð ÞG ξsð Þ½ �

ðA:14Þ

where G ξð Þ ¼ ∫1ξ γ−γoð Þξdξ; cosγ ¼ −rωr=g ¼ −f w

2ξ; cosγo ¼ Ro=r ¼ξi=ξ, ξ=r /(D /2), ξi=Ro /(D /2), ξs=Ras /(D /2) and Ras is the dis-tance between the ball bed surface and the mill center in the ascend-ing zone.

Ras (or ξs) is estimated as follows. In the under filled case of Fig. 1(a), Ras is equal to Ro (or ξs is equal to ξi). In the overfilled case of Fig. 1(b), balls in the ascending zone tend to shift toward the mill wall, i.e.,segregation may occur due to the difference in density between ballsand particles under the centrifugal force field until the voidage of theball bed being equal to that of the static state εb. Then, VRb (alreadygiven by Eq. (A.5)) is again expressed using Ras (or ξs) like this,

VRb ¼ ∫D=2Ras

1−εbð ÞLωrrdr ¼ 1−εbð Þωr VM=πð Þ 1−ξ2s� �

=2: ðA:5′Þ

Equating Eqs. (A.5) and (A.5′) to give

1−εbð Þ 1−ξ2s� �

¼ 1−εboð Þ 1−ξ2i� �

: ðA:15Þ

Using Eq. (A.15), ξs for the case of particles overfilled is obtained.Substituting Eq. (A.5′) into Eq. (A.13) (or Eq. (A.6) into Eq. (A.14),

Δta ¼ 2=ωrð ÞG ξsð Þ ðA:16Þ

Δtap ¼ 2

εboωr 1−ξ2i� � G ξið Þ− 1−εbð ÞG ξsð Þ½ �: ðA:17Þ

A.3. Falling zone

To estimate Δtf and Δtfp given in the third terms of Eqs. (A.1) and(A.2), free falls of balls and particles are assumed. Then,

Δtf ¼ 2Hab=gð Þ1=2 ðA:18Þ

Δtfp ¼ 2Hap=g� �1=2 ðA:19Þ

where Hab and Hap are the average falling distances of balls and parti-cles, respectively, which are derived as follows.

VRbHab ¼ ∫D=2Ras

h rð Þ 1−εbð ÞLωrrdr ¼ 1−εbð Þωr VM=πð Þ∫1ξs h rð Þξdξ ðA:20Þ

VRpHap ¼ Uo ∫1ξs h rð ÞεbLωr D=2ð Þ2ξdξþ ∫ξs

ξih rð ÞLωr D=2ð Þ2ξdξ

h i¼ Uoωr VM=πð Þ ∫1

ξi h rð Þξdξ− 1−εbð Þ∫1ξs h rð Þξdξ

h i ðA:21Þ

where h(r) is the vertical distance from the point of departure along theascending circular arc with the radius r to the surface level of the grind-ing zone, given by,

h rð Þ ¼ −r cosγð Þ þ r cosγoð Þ ¼ D=2ð Þ f 2wξ2 þ cosθbo

� �: ðA:22Þ

Then, the integration appeared in Eqs. (A.20) and (A.21) is solvedas follows,

∫1ξh rð Þξdξ ¼ D=2ð Þ 1−ξ2

� �=2

h i1=2ð Þf 2w 1þ ξ2

� �þ cosθbo

h i: ðA:23Þ

50 S. Nomura / Powder Technology 222 (2012) 37–51

Substituting Eqs. (A.5′) and (A.23) into Eq. (A.20) (or Eqs. (A.6)and (A.23) into Eq. (A.21)), Hab and Hap are derived respectively as

Hab= D=2ð Þ ¼ 1=2ð Þf 2w 1þ ξ2s� �

þ cosθboh i

ðA:24Þ

Hap= D=2ð Þ ¼ 1=εboð Þ 1=2ð Þf 2w 1þ ξ2i� �

þ cosθboh i

þ 1−εboð Þ=εbo½ � 1=2ð Þf 2w 1þ ξ2s� �

þ cosθboh i

:

ðA:25Þ

Appendix B. Mean velocity of balls in the grinding zone

Firstly, consider the velocity of balls in the grinding zone with noinfluence of particles nearby. In the ascending zone, balls at the num-ber rate denoted as CR gain the potential energy with the height of Hab

due to the mill tumbling. This energy is transferred to the kineticenergy of balls with the mean velocity of vb in the grinding zone. Mul-tiplying the frequency of collision of balls in the grinding zone perunit time, the energy balance is expressed as,

CRMbgHab ¼ 1=2ð ÞMbv2b

h i1=2ð Þ vb=λbð ÞNbo½ � ðB:1Þ

where CR=VRb /(Mb /ρb), Nbo=VMJo(1−εbo)/(Mb /ρb) and VRb isalready given by Eq. (A.5) in Appendix A. Then, vb is obtained to be

vb ¼ 4 1−ξ2i� �

ωrgHabλb= 2πJoð Þh i1=3

: ðB:2Þ

Under the existence of particulate material, the motion of balls inthe grinding zone is affected by the resistance forces exerted by solidparticles in dry grinding and by slurry in wet grinding. These forcesare taken into account in the derivations of the mean velocities ofballs as follows.

B.1. Dry grinding

As depicted in Fig. B.1, a ball moves at a velocity v within the bulkof particles whose bulk density is ρpb and the force is exerted by theparticles on the ball surface to reduce its velocity. The rate of momen-tum of the bulk of particles denoted as (ρpbv)v acts at every point onthe ball surface. Consider two forces per unit area, one is the normalstress denoted as σn acting perpendicularly to the surface and the

0.5

0.6

0.7

0.8

0.9

1.0

0.3 0.4 0.5 0.6

p [-]

[rad

]

Sand-0.5mm

Fig. B.2. Angle of repose in relation to void fraction for sand [18].

other is the shear stress denoted as τR acting tangentially. These areexpressed by

σn ¼ ρpbvv� �

cosφ ðB:3Þ

τR ¼ μpbσn ¼ μpb ρpbvv� �

cosφh i

ðB:4Þ

where μpb is the friction coefficient of particles.The horizontal components of the normal and shear stresses are

−σncosφ and −τRsinφ, respectively. These local forces per unit areaact on the small surface area given by (π /2)db2sinφdφ. Assuming thetwo forces act only on the surface of the half sphere facing to theapproaching particles, the horizontal components multiplied by thesmall surface area are integrated over the surface of the half sphereto get the resultant forces of the particles acting on the ball as

Fn ¼ ∫π=20 − σn cosφð Þ π=2ð Þd2b sinφdφ ¼ − π=2ð Þd2b 1=3ð Þρpbv

2h i

ðB:5Þ

Ft ¼ ∫π=20 − τR sinφð Þ π=2ð Þd2b sinφdφ ¼ − π=2ð Þd2b 1=3ð Þμpbρpbv

2h i

: ðB:6Þ

Then, the equation of motion of the ball is given by

Mb dv=dtð Þ ¼ Uo Fn þ Ftð Þ ¼ −Uo π=6ð Þd2bρpb 1þ μpb

� �v2 ðB:7Þ

where the fraction of the ball surface given by Uo is apparently sur-rounded by particles and exerted by the resistance forces. Solvingthe above differential equation with the initial condition of v=vb att=0,

v ¼ 1a1t þ 1=vbð Þ ðB:8Þ

where a1=Uoρpb(1+μpb)/(dbρb).The mean velocity of the ball denoted as vs traveling the distance

between two balls denoted as λb in the period of t1 is given by

vs ¼ λb=t1: ðB:9Þ

The value of t1 is obtained from the following equation;

λb ¼ ∫t10vdt ðB:10Þ

Substituting Eq. (B.8) into v in Eq. (B.10), t1 is obtained. Thus obtainedt1 is substituted into Eq. (B.9) to derive Eq. (14) for vs in dry grinding.

Note that μpb used in Eq. (B.4) is assumed by the coefficient of in-ternal friction or repose. Fig. B.2 shows experimental data of the angleof repose in relation to the void fraction of sand with the mean diam-eter of 0.5 mm obtained by the authors [18]. The straight line relation

water level

water level

surface of particles surface of particles

(a) Water under filled (b) Water overfilled

Fig. C.1. Schematic drawings of slurry (solid particles with water).

51S. Nomura / Powder Technology 222 (2012) 37–51

between Φr and εp is assumed for simplicity in the present calcula-tions. Then,

μpb ¼ tanΦr ¼ tan 1:36−1:31εp� �

: ðB:11Þ

B.2. Wet grinding

In the case of wet grinding, in addition to the resistance forces ofthe bulk of particles given by Eqs. (B.5) and (B.6), the drag force offluid denoted as Fd acts on the ball. Further, the buoyancy forcedenoted as Fb is taken into account as the vertical motion of ballsmay be dominated in the grinding zone when the balls fall. The sumof the drag force and the buoyancy force is expressed as follows,

Fd þ Fb ¼ −Cf πd2b=4� �

ρwv2=2

� �− π=6ð Þd3bρwg ðB:12Þ

where Cf is the drag coefficient assumed to be 0.44 in a range of highReynolds numbers. As the fraction of the ball surface (or volume)equal to Uo is apparently surrounded by slurry, the equation of mo-tion of the ball given by Eq. (B.7) is modified to be

Mb dv=dtð Þ ¼ Uo Fn þ Ft þ Fd þ Fbð Þ

¼ −Uo½ π=6ð Þd2bρpb 1þ μpb

� �v2 þ Cf π=8ð Þd2bρwv

2

þ π=6ð Þd3bρwg�

ðB:13Þ

where μpb is estimated using Eq. (B.11) in which εp is replaced by εsl,the slurry void fraction (see Appendix C). The solution of the abovedifferential equation for v=vb at t=0 is given as

arctan v=cð Þ ¼ arctan vb=cð Þ−a2ct ðB:14Þ

where a2=[Uo/(dbρb)][ρpb(1+μpb)+(3/4)Cfρw] andc¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiUoρwg= ρba2ð Þp

.Likewise for the dry grinding, substituting Eq. (B.14) into Eq. (B.10),

t1 is obtained. Thus obtained t1 is substituted into Eq. (B.9) to give vs forwet grinding as Eq. (15).

Appendix C. Properties of slurry in wet grinding

To evaluate μpb using Eq. (B.11) for wet grinding, εp is replaced byεsl, the slurry void fraction defined as the fraction of slurry volume oc-cupied by either air or water. Also, the density of slurry ρsl is used inEq. (21). These variables are estimated as follows.

Consider a unit volume of dry particles with water to fill the voidas schematically depicted in Fig. C.1. The mass fraction of solids inslurry, denoted as aw, is expressed as

aw ¼ρp 1−εp

� �ρp 1−εp

� �þ ρwVw

ðC:1Þ

where ρw is the water density and Vw is the volume of water per unitvolume of dry particles. Rewriting Eq. (C.1), Vw is given as

Vw ¼ ρp=ρw

� �1−εp

� �1−awð Þ=aw: ðC:2Þ

In Fig. C.1(a), the water level is below the surface of particles(Vwbεp) and the slurry level is equal to the surface of particles.Then, the slurry void fraction εsl is equal to that of dry particles, i.e.,

εsl ¼ εp: ðC:3Þ

In this case, the slurry density ρsl is expressed as

ρsl ¼ ρp 1−εp� �

þ ρwVw ¼ ρp 1−εp� �

=aw: ðC:4Þ

In the case of the water level above the surface of particles(Vw≥εp) as schematically drawn in Fig. C.1(b), the slurry level isequal to the water level. Then, the slurry void fraction εsl is given as

εsl ¼Vw

1−εp� �

þ Vw

¼ ρp 1−awð Þρp 1−awð Þ þ ρwaw

: ðC:5Þ

As to ρsl, the following equation is used.

ρsl ¼ρp 1−εp

� �þ ρwVw

1−εp� �

þ Vw

¼ ρpρw

ρp 1−awð Þ þ ρwaw: ðC:6Þ

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