Dispersion properties for residence time distributions in tumbling ball mills

Powder Technology 222 (2012) 37–51

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Dispersion properties for residence time distributions in tumbling ball mills S. Nomura Hiro-Ohshingai 2-15-26, Kure, Hiroshima 737-0141, Japan

a r t i c l e

i n f o

Article history: Received 7 May 2011 Received in revised form 30 October 2011 Accepted 21 January 2012 Available online 28 January 2012 Keywords: Residence time distribution Dispersion coefficient Peclet number Continuous mill Tumbling ball mill Comminution kinetics

a b s t r a c t The present paper deals with a theoretical analysis of the dispersion properties, the dispersion coefficient and the Peclet number, of particulate material in a continuous ball mill. In the analysis, a dispersion zone where the brief dispersion of particles occurs, is postulated in the lower portion of an operated mill called the grinding zone. Consequently, the dispersion coefficient is derived to be a function of the size of the dispersion zone and 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 within reasonable agreement with reported data for dry and wet grinding operations, although minor variations are observed between theory and experiment. Additionally, the mill diameter and length are predicted to affect greatly the Peclet number, implying the importance of designing mill sizes for required product size distributions as the residence time distribution is dominated by the Peclet number. Further, a proportional relationship to predict the Peclet number is derived, which appears to be valid as confirmed with data regardless of the mill sizes tested. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Demands are growing to more precise control of the product size distributions when grinding particulate materials. Simultaneously, economical operations are always required as only a few percent of total energy applied to a system is used for pure grinding [1]. One way to reduce the grinding energy is to reduce the amount of over ground particles. In a continuous mill, reducing the amount of particles with relatively long residence times lessens the over grinding. To obtain a required fineness of product with minimum grinding energy consumed, the residence time distribution of particulate material is of great importance to be controlled. There have been a number of literatures reported with respect to the residence time distributions in continuous ball mills. Experimentally, for dry grinding, Mori et al. [2] and Swaroop et al. [3,4] examined either the dispersion coefficient (corresponding to the diffusion coefficient in molecule diffusion) or the Peclet number (originally defined in molecular diffusion) in relation to some operating conditions such as the ball filling, the feed rate, the speed of mill revolution and the ball size. For wet grinding, Kelsall et al. [5] obtained data for some operating parameters including the slurry density. Austin et al. [6] showed experimentally the validity of a constant Peclet number along the mill axis which was conventionally adopted as an assumption in theoretical derivations. Further, a method with short-lived

radioactive tracers was developed to apply the measurement to a pilot-plant scale ball mill in closed circuit [7]. From the theoretical point of view, an axial dispersion model was developed elsewhere based on a one dimensional axial diffusion equation of molecules and solutions of the residence time distribution were derived 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 of industrial mills. Kelsall et al. [5] and Furuya et al. [10] demonstrated a mill model composed of a plug-flow and a perfect mixing flow in series. The former used it to analyze experimental data. The latter defined a parameter indicating the degree of mixing and evaluated its effect on the product size distribution based on comminution kinetics. In spite of a number of experimental and theoretical findings reported, the dispersion properties have not been fully predictable in relation to operating conditions and mill sizes. Therefore, controlling the residence time distribution relies greatly on experiences or trial and error practices. The present study aims at clarifying the dispersion properties based on a model estimating the degree of dispersion as well as the mill charge model developed previously [11,12]. Data reported for dry and wet grinding operations are utilized to examine the developed theory. Also physical backgrounds are considered for trends of the dispersion properties varying with operating variables and mill sizes. 2. Theoretical 2.1. Residence time distribution and dispersion properties

E-mail address: [email protected]. 0032-5910/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2012.01.034

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 particulate material, the residence time distribution Φ(t) at the mill exit (x = L) is expressed by [2],

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

( ) L ðL−ut Þ2 Φðt Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi exp − 4Et 4πEt 3

2.2.1. Reason for adoption of simplified mill charge model As to the mill charge composed of balls and particles in a tumbling ball mill, a model whose cross sections are drawn schematically in Fig. 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 dissipation as to the mill charge were needed. However, because of the complexity, the average properties in relation to grinding conditions were hindered from being quantified. A simplified mill charge model was then proposed and enabled estimations of the circulating flow rate of the mill charge, the mean velocity of balls, the collision energy dissipated, the energy gained from the tumbling mill and so on. These properties were utilized to assess grinding characteristics, e.g., the grinding rate function called the selection function was derived in terms of operating conditions [11] and an empirical proportional relationship was proved to be valid between the grinding rate constant and the mill power drawn [12]. Further, based on the derived relationships, theoretical energy-size reduction relationships in batch grinding were clarified [14]. In each study, derived results were confirmed with experiments to verify the simplified mill charge model as well as other assumptions adopted. In the last two decades, computer simulations with DEM (Discrete Element Method) have been demonstrated aiming at accurate predictions of the motion and contour of the mill charge in tumbling mills [15–17]. In some cases, simulated impact forces and energy dissipated have been applied to the design of mill liners and lifters against the wear [17]. However, the DEM methods at present do not simulate the behavior of particles being ground and influences of the particles on the motion of balls and the grinding characteristics are unpredictable. Therefore, to simulate grinding characteristics with DEM, further developments are required such as converting computational outputs without grinding to those with grinding or relating computational outputs with breakage properties measured [16,17]. Under such circumstances, the simplified model is considered to be useful for estimating at least average properties of the mill charge in relation to the operating conditions.

ð1Þ

where E is the axial dispersion coefficient of particles corresponding to the diffusion coefficient in molecular diffusion and u is the axial mean velocity of material flowing in the mill. The mean residence time τ is obtained to be, ∞

τ ¼ ∫0 tΦðt Þdt ¼ L=u ¼ M H =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, ( ) 1 ð1−θÞ2 Φðt Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp − 4θ=Pe τ 4πθ3 =Pe

ð4Þ

where θ = t / τ, Pe = uL / E, θ is the dimensionless time variable and Pe is the Peclet number. The Peclet number is the ratio of mass flux resulting 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 plug flow (Pe equal to infinity means plug flow). Eq. (4) indicates that the residence time distribution is controlled by Pe. An analysis of E is conducted to clarify Pe in terms of operating conditions and mill sizes. From similarity between a one dimensional random walk and a one dimensional dispersion process, E is expressed by statistical parameters as [13], 2

E¼

‘ m 2t

ð5Þ

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

ð8Þ

2.2.2. States of balls and particles Balls 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 of the ball bed εb and so on. For instance, the value of U equal to unity means that particles charged occupy all of the interstices of the static ball bed with εb. The mill charge, when the mill is in operation, is conventionally divided into three zones, i.e., the grinding zone where balls collide each other to grind particles, the ascending zone where balls move upwards along the mill wall and the falling zone where balls cascade or cataract. The ‘grinding’ zone is named for the area where grinding of particles occurs and the ‘falling’ zone is named for balls falling (regardless of either cataracting or cascading) to release the potential energy. The case of particles under filled in the ball bed void is depicted in Fig. 1(a) and the overfilled case is shown in Fig. 1(b). The states of balls and particles in the grinding zone are represented by parameters denoted as Jo, Uo and εbo, the derivations of which are briefly explained in Appendix A.

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

2.2.3. Dispersion zone Dispersion 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. However, all of them are not fully dispersed when particles are overfilled as depicted in Fig. 1(b). The balls must disperse more particles than

ηt ¼ mt s

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 is obtained to be E¼

  ‘2 m 1 ‘ 1 ¼ η‘ ¼ η‘vw 2mt s =η 2 ts 2

ð7Þ

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

Pe ¼

uL 2uL ¼ : E η‘vw

2.2. Specifying variables η, ℓ and vw

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

(a) Particles under filled

D

(b) Particles overfilled

Ras γ r

γ

Ras

θ

r

γ

bo

39

Ro

γ

o

θ

Ro bo

o

ε bo = εb

surface level of particles

ε bo > εb

Fig. 1. Cross sections of operating ball mill.

those in the under filled case and some particles possibly remain undispersed. Postulate a hypothetical ball bed whose voidage equal to εb as schematically drawn in Fig. 2. When all of the interstices of the ball bed with εb are filled with particles, the rest of the grinding zone particles are located above the hypothetical boundary and these particles are not to be dispersed. In other words, the dispersion zone is defined to have the same volume as that of the hypothetical ball bed with the voidage of εb. The whole grinding zone is the dispersion zone in the under 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 are left above the ball bed if no trembles are applied. The present study assumes that this possibility is ignored under the dynamic motion of the mill charge. A new parameter denoted as Mdis is introduced, expressing the mass ratio of particles in the dispersion zone to those in the grinding zone, i.e., M dis ¼ 1   M dis ¼ J os εb = α f f c

when under filled when overfilled

ð9Þ

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

particles in a mill and fc is the fraction of mill volume occupied by the bulk of particles charged. From the geometry shown in Fig. 2, Jos is given by, J os ¼

J o ð1−εbo Þ 2θbs − sinð2θbs Þ ¼ 2π ð1−εb Þ

ð10Þ

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 particles in the dispersion zone to that in the grinding zone. 2.2.4. Ratio of walking period to total period η The walking period of a random walker is equivalent to the period of particles in the dispersion zone, which is equal to (ΔtgpMdis) where Δtgp is the time period of particles in the grinding zone given by Eq. (A.4) in Appendix A. Then, η is expressed as the ratio of the time period in the dispersion zone to the total period in the mill, i.e., η¼

Δt gp M dis ¼ α f Mdis Δt gp þ Δt ap þ Δt fp

ð11Þ

where Δtap and Δtfp are the time periods of particles being in the ascending and falling zones given by Eqs. (A.17) and (A.19), respectively. 2.2.5. Length of one step ℓ The macroscopic motion of particles or the bulk flow of particles in the grinding zone is taken into account rather than the microscopic motion of individual particles. That is, the bulk flow of particles between colliding balls is regarded to correspond to the motion of a random walker and its direction is altered by the balls. The length of one step ℓ is assumed to be proportional to the distance of the bulk flow of particles without being interrupted by the balls whose mean free path is λb, i.e.,

θ

bs

θ

bo

Outside of dispersion zone Hypothetical boundary of dispersion zone ε bo = εb

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

‘∝λb ¼

db 6ð1−εbo Þ

ð12Þ

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

40

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

2.2.6. Velocity of walking vw The velocity of walking vw may correspond to the velocity of the bulk flow of particles in the grinding zone. For simplicity, the bulk flow velocity is assumed to be proportional to the mean velocity of balls in the grinding zone denoted as vs, i.e.,

2.3. Expressions of E and Pe with mill parameters

vw ∝vs :

E∝ηλb vs

ð16Þ

Pe ¼ uL=E∝uL=ðηλb vs Þ:

ð17Þ

ð13Þ

For dry grinding, the velocity of balls in the grinding zone is attenuated by the friction of surrounding particles. Then, vs is derived to be (see Appendix B1)   a λ vs ¼ a1 λb vb = e 1 b −1

ð14Þ

where a1 = Uoρpb(1 + μpb) / (dbρb) and μpb is the friction coefficient of particles. In the case of wet grinding, the drag force of fluid and the buoyancy force in addition to the friction of particles are taken into account to suppress the ball motion. Then, vs for wet grinding is expressed as (see Appendix B2),

a2 λb c

  vs ¼ ea2 λb ffi arctanðvb =cÞ− arccos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

ð15Þ

ðvb =cÞ þ1

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

Of the parameters used in Eq. (16), η expresses the size of the dispersion zone relative to the mill volume and λb and vs represent the mobility of balls in the grinding zone. That is, the dispersion of particles is enhanced by the enlargement of the dispersion zone or the magnification of the mobility of balls in the grinding zone. As the proportional constants in the above equations are not known at present, Eqs. (16) and (17) are rewritten by the normalized forms using E / Eo, Pe / Peo, η / ηo, λb / λbo, vs / vso, L / Lo and u / uo as follows,

E=Eo ¼ η=ηo ðλb =λbo Þðvs =vso Þ

ð18Þ

Pe=Peo ¼ ðL=Lo Þðu=uo ÞðEo =EÞ

ð19Þ

where the subscript o means the base case. When L / Lo and u / uo are constant, Pe / Peo is inversely proportional to E / Eo. 3. Results and discussion

where a2 = {Uo/(dbρb)} {ρpb(1+ μpb) + (3/4)Cfρw}, c ¼ Cf is the drag coefficient and ρw is the density of fluid.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U o ρw g=ðρb a2 Þ,

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

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

experiment J

F

-

kg/s

calculated 4

Holdup

U

db x100

fw

u x100

E x10^

E/Eo

Pe

Pe/Peo

η

λb

Vs

E/Eo

Pe/Peo

kg

-

cm

-

cm/s

cm2/s

-

-

-

-

m

m/s

-

-

(a) Pilot Mill (D = 0.545 m, L = 1.98 m) 0.1

0.094

41.50

1.52

3.0

0.534

0.450

1.61

0.57

55.20

3.80

0.16

0.010

0.758

0.61

3.54

0.2

0.107

50.60

0.93

3.0

0.534

0.417

2.44

0.86

33.87

2.33

0.32

0.008

0.629

0.85

2.37

0.3

0.107

65.70

0.80

3.0

0.534

0.321

1.60

0.56

39.83

2.74

0.42

0.008

0.543

0.95

1.64

0.4

0.100

95.70

0.87

3.0

0.534

0.208

2.83

1.00

14.53

1.00

0.52

0.008

0.457

1.00

1.00

0.1

0.097

42.94

1.57

4.0

0.534

0.446

3.09

1.09

28.58

1.97

0.16

0.013

0.834

0.89

2.40

0.2

0.095

47.29

0.86

4.0

0.534

0.398

5.74

2.03

13.72

0.94

0.32

0.011

0.695

1.25

1.54

0.3

0.100

61.81

0.75

4.0

0.534

0.320

4.35

1.54

14.59

1.00

0.42

0.011

0.599

1.39

1.11

0.4

0.107

92.24

0.84

4.0

0.534

0.229

4.54

1.60

10.01

0.69

0.52

0.011

0.504

1.47

0.75

0.4

0.069

83.80

0.77

3.0

0.534

0.163

2.43

0.86

13.27

0.91

0.52

0.008

0.460

1.01

0.78

0.4

0.100

95.70

0.87

3.0

0.534

0.208

2.83

1.00

14.53

1.00

0.52

0.008

0.457

1.00

1.00

0.4

0.136

110.10

1.01

3.0

0.534

0.244

2.87

1.01

16.85

1.16

0.52

0.008

0.454

0.99

1.19

Pe/Peo

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

U

db x100

fw

u x100

E x10^4

E/Eo

Pe

Pe/Peo

η

λb

Vs

E/Eo

kg

-

cm

-

cm/s

cm2/s

-

-

-

-

m

m/s

-

-

0.0020

2.09

0.70

1.6

0.30

0.048

0.51

0.71

4.71

1.92

0.43

0.004

0.333

1.00

1.36

0.2

0.0024

2.98

1.00

1.6

0.40

0.039

0.56

0.78

3.47

1.41

0.38

0.004

0.378

1.01

1.10

0.2

0.0022

3.02

1.02

1.6

0.50

0.036

0.72

1.00

2.46

1.00

0.33

0.004

0.430

1.00

1.00

0.2

0.0020

2.97

1.00

1.6

0.60

0.033

0.63

0.89

2.61

1.06

0.29

0.004

0.483

0.98

0.96

0.2

0.0019

3.06

1.03

1.6

0.70

0.031

0.65

0.91

2.39

0.97

0.25

0.004

0.540

0.93

0.94

J

F

-

kg/s

0.2

Steel ball: ρ = 7800 kg/m3, ε = 0.4, Limestone: ρ = 2700 kg/m3, and ρ = 1480 kg/m3 (ε = 0.45). b

b

p

pb

p

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

41

with the open-end mill and those of J and F were with the constricted mill. The data are listed in Table 2 where the row of the base case is indicated by the shade for each mill. For dry grinding, the following equation is used to calculate U from the value of the mill holdup MH reported, i.e.,

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

3.1. Experimental data for comparison

  2 MH ¼ ρpb V M UJεb ¼ ρpb πD L=4 UJεb

3.1.1. Dry grinding data Mori 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 sets of data using the former mill and one set of data using the latter mill. In the former mill, two sets vary the ball filling J for two different ball diameters db of 0.03 m and 0.04 m and the rest varies the mass feed rate F. In the latter mill, the ratio of mill speed fw is varied. The calculated η, λb, vs, E / Eo and Pe / Peo values corresponding to the data are also in Table 1 in which the base case, distinguished by the shaded row for each mill, is determined by the one with J closer to 0.4 or otherwise the one with a middle value in a range of the variable tested. For db [m], u [m/s] and E [m 2/s], the values of db × 10 2 [cm], u × 10 2 [cm/s] and E × 10 4 [cm 2/s] are listed in the table for convenience. 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

ð20Þ

where ρpb is the bulk density of particles. 3.1.2. Wet grinding data Kelsall et al. [5] conducted tests for wet grinding with a laboratory mill (Ф0.305 m × 0.305 m), in which the distribution of residence time was approximated by that composed of a pure delay and a perfect mixer in series. The reported data were the dimensionless delay time δd (=to/τ) and the mean residence time τ appeared in the following equations. The fraction of impulse response remaining in a mill at time t, denoted as F′(t), is given by F ′ ðt Þ ¼ 1 F ′ ðt Þ ¼ exp½−K M ðt−t o Þ

for t≤t o for t≥t o

ð21Þ

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

experiment

calculated

J

F x10^3

-

g/s

Holdup kg

U

db x100

fw

u x100

E x10^4

E/Eo

Pe

Pe/Peo

η

λb

Vs

E/Eo

Pe/Peo

-

cm

-

cm/s

cm2/s

-

-

-

-

m

m/s

-

-

(a) Open-end mill (D = 0.127 m, L = 0.438 m) 0.24

2.80

0.440

0.59

1.9

0.67

0.279

1.26

0.89

9.68

1.40

0.33

0.0053

0.439

0.87

1.43

0.40

2.80

0.542

0.44

1.9

0.67

0.226

1.42

1.00

6.97

1.00

0.48

0.0053

0.347

1.00

1.01

0.52

2.80

0.635

0.39

1.9

0.67

0.193

1.31

0.92

6.45

0.93

0.63

0.0053

0.269

1.03

0.84

0.40

1.30

0.308

0.25

1.9

0.67

0.184

1.56

1.10

5.17

0.75

0.48

0.0053

0.350

1.01

0.81

0.40

1.99

0.417

0.34

1.9

0.67

0.209

1.73

1.21

5.30

0.76

0.48

0.0053

0.349

1.01

0.93

0.40

2.22

0.450

0.36

1.9

0.67

0.216

1.72

1.21

5.50

0.79

0.48

0.0053

0.348

1.00

0.96

0.40

2.78

0.542

0.44

1.9

0.67

0.225

1.42

1.00

6.93

1.00

0.48

0.0053

0.347

1.00

1.00

0.40

2.90

0.720

0.58

1.9

0.28

0.176

0.75

0.53

10.30

1.49

0.80

0.0053

0.121

0.58

1.35

0.40

2.90

0.620

0.50

1.9

0.40

0.205

1.07

0.75

8.38

1.21

0.71

0.0053

0.193

0.83

1.10

0.40

2.90

0.580

0.47

1.9

0.55

0.219

1.39

0.98

6.89

0.99

0.59

0.0053

0.272

0.97

1.01

0.40

2.90

0.530

0.43

1.9

0.67

0.240

1.57

1.10

6.68

0.96

0.48

0.0053

0.344

1.00

1.07

0.40

2.90

0.565

0.45

1.9

0.79

0.225

1.71

1.20

5.74

0.83

0.38

0.0053

0.423

0.97

1.03

0.40

2.90

0.770

0.62

1.9

0.97

0.165

1.18

0.83

6.13

0.88

0.25

0.0053

0.548

0.84

0.87

Pe/Peo

(b) Constricted-end mill (D = 0.08 m, L = 0.24 m) F x1000

Holdup

U

db x100

fw

u x100

E x10^4

E/Eo

Pe

Pe/Peo

η

λb

Vs

E/Eo

-

g/s

kg

-

cm

-

cm/s

cm2/s

-

-

-

-

m

m/s

-

-

0.14

1.97

0.235

2.45

1.3

0.60

0.201

0.22

0.67

21.87

1.35

0.13

0.0056

0.407

0.87

1.05

0.17

1.97

0.222

1.92

1.3

0.60

0.213

0.28

0.85

18.32

1.13

0.17

0.0048

0.386

0.93

1.03

0.20

1.97

0.213

1.60

1.3

0.60

0.222

0.33

1.00

16.18

1.00

0.21

0.0044

0.370

1.00

1.00

0.23

1.97

0.201

1.31

1.3

0.60

0.236

0.41

1.24

13.82

0.85

0.26

0.0040

0.355

1.09

0.97

0.26

1.97

0.204

1.18

1.3

0.60

0.232

0.41

1.24

13.58

0.84

0.31

0.0038

0.340

1.16

0.90

0.20

0.51

0.122

0.91

1.3

0.60

0.101

0.34

1.03

7.17

0.44

0.29

0.0036

0.374

1.15

0.40

J

0.20

0.94

0.153

1.15

1.3

0.60

0.148

0.40

1.21

8.91

0.55

0.27

0.0038

0.371

1.09

0.61

0.20

1.65

0.191

1.43

1.3

0.60

0.207

0.38

1.17

12.93

0.80

0.23

0.0042

0.369

1.03

0.91

0.20

1.80

0.201

1.51

1.3

0.60

0.215

0.37

1.13

13.91

0.86

0.22

0.0043

0.369

1.02

0.95

0.20

1.84

0.209

1.57

1.3

0.60

0.211

0.35

1.06

14.57

0.90

0.21

0.0044

0.368

1.01

0.94

0.20

1.97

0.213

1.60

1.3

0.60

0.222

0.33

1.01

16.09

0.99

0.21

0.0044

0.368

1.00

1.00

3

Lucite ball: ρ = 1250 kg/m3, ε = 0.4, Dolomite: ρp = 2860 kg/m3, and ρpb = 1400 kg/m (εp = 0.51). b

b

42

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

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

0.2 Present theory 0

Kelsall model ′

Ln(F'(t))

-0.2

F p ðt Þ ¼

d

∞ ∫t Φðt Þdt

¼

θ 1−∫0

-0.4

KM

Fp′(t) in Eq. (24) is equivalent to F′(t) in Eq. (23). Therefore, for the curve of F′(t) with the value of δd reported, the value of Pe to show a best fit curve of Fp′(t) is evaluated. Using the evaluated Pe with the aid of u and L, the value of E (=uL / Pe) is calculated. An example is depicted in Fig. 3 where the Kelsall model given by Eq. (23) uses δd = 0.23 and the present model of Eq. (24) gives a best fit for Pe = 3.03 and then E (=uL / Pe) to be 1.49 × 10 − 4 m 2/s with the aid of u = 0.149 × 10 − 2 m/s and L = 0.3048 m. Thus obtained Pe and E values are regarded as the experimental data listed in Table 3. For wet grinding, the following equation is used to calculate U from MH,

d)]

-0.8 d)

d

-1.2 0

ð24Þ

Pe = 3.03

1 -0.6

-1

" # 1 ð1−θÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp − dθ: 4θ=Pe 4πθ3 =Pe

50

100

150

200

250

time t [sec] Fig. 3. Kelsall model of a pure delay and perfect mixer in comparison with theoretical curve to give a best fit.

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

  2 MH ¼ aw ρsl V M UJεb ¼ aw ρsl πD L=4 UJε b

  ∞ ′ ∞ τ ¼ ∫0 t −dF ðt Þ=dt dt ¼ ∫t o t fK M exp½−K M ðt−t o Þgdt

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

¼ ð1=K M Þ þ t o :

ð22Þ

ð25Þ

3.2. Comparison between theory and experiment Substituting the above relation into Eq. (21) leads to F ′ ðt Þ ¼ 1 F ′ ðt Þ ¼ exp½−ðt−t o Þ=ðτ−t o Þ ¼ exp½−ðθ−δd Þ=ð1−δd Þ

3.2.1. Effect of ball filling Fig. 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 of Swaroop et al. are for two different mills and those of Kelsall et al. are for wet grinding. In each case, E / Eo increases with increasing J and Pe / Peo shows a reverse of this trend, in which the theory (full symbols) and the experiment (empty ones) are within reasonable agreement.

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

experiment

calculated 3

4

J

F x10^

Holdup

U

db x100

fw

u x100

aw

δd

τ

E x10^

E/Eo

Pe

Pe/Peo

η

λb

Vs

-

g/s

kg

-

cm

-

cm/s

-

-

s

cm2/s

-

-

-

-

m

m/s

(a) Wet Mill (D = 0.3048 m , L = 0.3048 m) 0.18 6.67 0.98 0.52 2.54

E/Eo

Pe/Peo

-

-

0.78

0.207

0.67

0.45

147

0.91

0.61

6.92

2.28

0.21

0.0071

0.733

0.77

1.82

0.27

6.67

1.27

0.46

2.54

0.78

0.160

0.67

0.25

191

1.50

1.00

3.24

1.07

0.26

0.0071

0.654

0.87

1.24

0.41

6.67

1.37

0.33

2.54

0.78

0.149

0.67

0.23

205

1.49

1.00

3.03

1.00

0.36

0.0071

0.548

1.00

1.00

0.41

3.33

1.28

0.31

2.54

0.78

0.079

0.67

0.16

384

1.00

0.67

2.43

0.80

0.36

0.0071

0.548

1.00

0.53

0.41

6.67

1.38

0.33

2.54

0.78

0.147

0.67

0.23

207

1.48

0.99

3.03

1.00

0.36

0.0071

0.548

1.00

0.99

0.41

22.17

2.39

0.57

2.54

0.78

0.282

0.67

0.30

108

2.23

1.49

3.85

1.27

0.36

0.0071

0.540

0.99

1.93

0.41

6.53

1.35

0.32

2.54

0.78

0.148

0.67

0.22

206

1.54

1.03

2.93

0.97

0.36

0.0071

0.548

1.00

0.99

0.41

6.53

1.77

0.43

2.54

0.78

0.112

0.67

0.19

271

1.28

0.86

2.67

0.88

0.36

0.0071

0.545

0.99

0.76

0.41

6.53

2.18

0.52

2.54

0.78

0.092

0.67

0.15

331

1.19

0.80

2.36

0.78

0.36

0.0071

0.542

0.99

0.63

0.41

6.53

7.63

1.83

2.54

0.78

0.026

0.67

0.09

1169

0.40

0.27

1.99

0.66

0.27

0.0094

0.464

0.84

0.21

0.41

6.67

1.42

0.34

1.91

0.78

0.144

0.67

0.21

212

1.54

1.03

2.84

0.94

0.36

0.0053

0.498

0.68

1.42

0.41

6.67

1.37

0.33

2.54

0.78

0.149

0.67

0.23

205

1.49

1.00

3.03

1.00

0.36

0.0071

0.548

1.00

1.00

0.41

6.67

1.27

0.31

3.18

0.78

0.160

0.67

0.23

191

1.61

1.08

3.03

1.00

0.36

0.0088

0.590

1.35

0.80

0.41

6.67

1.20

0.29

3.81

0.78

0.169

0.67

0.20

180

1.87

1.25

2.76

0.91

0.36

0.0106

0.626

1.71

0.66

0.41

6.67

2.54

0.41

2.54

0.65

0.120

0.67

0.23

381

1.81

1.21

3.03

1.00

0.45

0.0071

0.454

1.02

1.19

0.41

6.67

2.34

0.37

2.54

0.78

0.130

0.67

0.26

351

1.19

3.35

1.11

0.36

0.0071

0.546

1.00

1.32

0.41

6.67

2.59

0.41

2.54

0.91

0.118

0.67

0.26

388

1.78 1.61

1.08

3.35

1.11

0.29

0.0071

0.644

0.93

1.28

0.41

6.67

0.87

0.29

2.54

0.78

0.233

0.55

0.30

131

1.84

1.23

3.85

1.27

0.36

0.0071

0.552

1.01

1.56

0.41

6.67

1.46

0.35

2.54

0.78

0.139

0.67

0.22

220

1.44

0.97

2.93

0.97

0.36

0.0071

0.547

1.00

0.94

0.41

6.67

3.03

0.59

2.54

0.78

0.067

0.75

0.18

454

0.79

0.53

2.58

0.85

0.36

0.0071

0.536

0.98

0.46

3

3

3

Steel ball: εb = 0.4, ρb = 7800 kg/m , Calcite: ρp = 2700 kg/m , εp = 0.45, ρw = 1000 kg/m , and μw = 0.001 Pa s.

0

0

E/Eo [ - ]

2

0.2

0.4

0.6

0

0.2

0 0.6

0.4

4

Open-end Constricted-end

Open-end Constricted-end

Swaroop et.al.

2

1 0

2

0

0.2

0.4

0.6

0

0.2

0.4

0.6

E/Eo [ - ]

2

0

4 Kelsall et.al.

1 0

2

0

0.2

0.4

0.6

0

0.2

J [-]

0 0.6

0.4

Pe/Peo [ - ]

1

4

db 0.03m 0.04m

db 0.03m 0.04m

Mori et. al.

Pe/Peo [ - ]

E/Eo [ - ]

2

43

Pe/Peo [ - ]

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

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

1

1

0 0 2

E/Eo [ - ]

2 Mori et.al.

0.2

0.1

0.3

0

0.1

0.2

0.3

2

Open-end Constricted-end

Swaroop et.al.

1

1 0 0

Open-end Constricted-end

1

2

3

0

1

2

x 10-3

3

0

x 10-3 2

2

E/Eo [ - ]

0

Kelsall et.al.

1

0 0

Pe/Peo [ - ]

2

1

10

20

30

0

10

20

-3

F [kg/s]

x 10

Fig. 5. Effects of F on E / Eo and Pe / Peo.

30 -3

F [kg/s]

x 10

0

Pe/Peo [ - ]

E/Eo [ - ]

3.2.2. Effect of feed rate When 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 than unity (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).

Pe/Peo [ - ]

For U greater than unity (or for F greater than that value), η / ηo decreases 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 to the increase of εbo. The value of vs / vso decreases slightly with increasing U (or F) in the whole range of U (or F) as more particles for greater U (or F) act on the ball surface due to the friction force to reduce the ball velocity. These effects are combined, resulting in a nearly constant trend of E / Eo against U (or F). For the trend of Pe / Peo, u is dominant under the nearly constant E / Eo, i.e., Pe / Peo increases with increasing F as u increases. The increase of u is explained from Eq. (3) that F / MH increases with increasing F because a linear relation is valid [3,4] between MH and F expressed 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 constant trends of E /Eo in the data of Mori et al. and those of Swaroop et al. are obtained within agreement with the theory. However, a slight difference

Theoretically, the increase of E / Eo is explained as follows. η / ηo increases with increasing J as the grinding zone or the dispersion zone is enlarged whereas vs / vso decreases due to the decrease of the energy given to a ball in the grinding zone. The trend of E / Eo depends on the magnitudes of the two conflicting effects. In Fig. 4, the magnitude of the increase of η / ηo is superior to that of the decrease of vs / vso. The decrease of Pe/ Peo with increasing J is derived as follows. Pe/ Peo is proportional to (Eo / E)(u / uo) under constant L, in which the decrease of Eo / E with increasing J is mentioned above and u is inversely proportional to J according to Eqs. (3) and (20) for constant F.

0

1 0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

Open-end

Swaroop et.al.

1

2

Open-end

1

1

0 0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

0 2

2

E/Eo [ - ]

0

Kelsall et.al.

1

1

0 0.2

0.4

0.6

0.8

1

0.2

0.4

fw [ - ]

0.6

0.8

1

0

Pe/Peo [ - ]

1

2

E/Eo [ - ]

2

Mori et.al.

Pe/Peo [ - ]

2

Pe/Peo [ - ]

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

E/Eo [ - ]

44

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

is observed in the case of Kelsall et al., i.e., the data of E /Eo increase with increasing F unlike the prediction mentioned above. The reason is not known at present and a possible cause will be discussed in Section 3.2.4. 3.2.3. Effect of mill speed Theoretically, when varying fw under the constant J, U, F and db, η / ηo decreases with increasing fw as the grinding zone becomes smaller whereas vs / vso increases due to the greater energy given to the grinding zone balls. These conflicting effects are combined to give rise to a convex curve of E / Eo with a peak at about fw = 0.6. The curve of Pe/ Peo shows an opposite to that of E / Eo because u / uo is constant. As depicted in Fig. 6, trends for E /Eo and Pe /Peo are within agreement between theory and experiment. In the case of Swaroop et al., the value of MH is also varied with fw, being concave (see Table 2), which is overlapped to emphasize the convex trend of E /Eo because vs /vso decreases with 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 fw are greater than 0.6 (being from 0.65 to 0.91). Although the data of Mori et al. show a minor increase of E / Eo with increasing fw from 0.3 to 0.5, the calculated E / Eo values corresponding to the data are nearly constant, which is also slightly different from the prediction mentioned above for fw less than 0.6. This is due to the reported value of MH (2.09 kg) for fw = 0.3 being smaller than those (about 3 kg) for other fw values for some reason. Basically,

an increase of fw from 0.3 to 0.5 under constant U leads to an increase of E. On the other hand, a simultaneous increase of MH from 2.09 kg to 3.02 kg for constant fw contributes to the decrease of E as vs / vso decreases with increasing MH (or U) as explained in Section 3.2.2. These two effects compensate each other, resulting in the calculated E / 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 a range 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 rate Theoretically, 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 E mentioned in Section 3.2.2 is also valid. For Pe / Peo, a relatively great decrease with increasing MH is predicted due to the inversely proportional 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 / Eo and Pe / Peo decrease with increasing MH. Slight differences in slope are observed between the theory and the experiment. A possible reason is discussed as follows. The two trends of E / Eo for the data of Kelsall et al. depicted in Figs. 5 and 7 are compared. These are opposite each other. When increasing either F in Fig. 5 or MH in Fig. 7, U increases in the both cases. The difference is that the value of u increases with increasing F but decreases 3

3 calculated

2.5

experiment

2

E/Eo [ - ]

2

1.5

1.5

1

1

0.5

0.5 0

0

2

4

6

MH [ kg ]

8

10

0

2

4

6

MH

8

[ kg ]

Fig. 7. Effects of MH on E / Eo and Pe / Peo in wet grinding mill (comparison with data of Kelsall et al. [5]).

10

0

Pe/Peo [ - ]

2.5

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

45

3

3 calculated

2.5

experiment

E/Eo [ - ]

2

2

1.5

1.5

1

1

0.5 0

Pe/Peo [ - ]

2.5

0.5

1

2

3

4

5 1

x

db [ m ]

2

3

10-2

db

4

5

x

[m]

0

10-2

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

with increasing MH. This difference, however, may not affect the value of E as u is sufficiently small compared to the mean velocity of balls vs. The conflicting trends in the data of E / Eo may be caused by an inaccuracy in estimating experimental Pe values, i.e., Pe values are estimated from the data of the pure delay and perfect mixer model. If the curve of experimental Pe/ Peo against F is steeper than the one in Fig. 5, the resultant E / Eo against F would be closer to the calculation. Also judging from the cases of the dry grindings in Fig. 5 showing minor variations of E / Eo against F, the nearly constant trend of E / Eo in the calculation for the case of Kelsall et al. seems plausible. The same consideration may 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, the trend of the experimental E / Eo would be closer to the calculated one. 3.2.5. Effect of ball diameter Theoretically, 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 proportional to λb1/3 (see Eq. (B.2)). This means that larger balls exhibit greater mobility to disperse particles. For Pe / Peo, the curve shows an opposite to that of E / Eo when u / uo is constant. In Fig. 4, two sets of data of Mori et al. are displayed for two ball diameters of 0.03 m and 0.04 m. The data exhibit that the larger ball diameter gives greater E / Eo values and smaller Pe / Peo values as explained theoretically. On the other hand, slight differences are observed in the case of Kelsall et al. plotted in Fig. 8. For E / Eo, the data show a minor increase with increasing db, in which three plots for db between 0.0191 m and

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

3.2.6. Effect of slurry density A 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 the constant J, fw and db and vs / vso decreases only slightly with increasing aw as more particles for greater aw act on the ball surface to suppress the ball motion due to the friction force. As for Pe / Peo, a relatively great decrease with increasing aw is predicted according to Eq. (19), in which u / uo decreases with increasing aw (or MH) under the nearly constant E / Eo. As shown in Fig. 9, a minor decrease of E / Eo with increasing aw is obtained in the data whereas the calculated E / Eo values are nearly constant. As for Pe / Peo, the slope of the data is gentler than the calculation. The difference may be again attributed to an inaccuracy in obtaining Pe values from the reported data. If the curve of the

3

3 calculated

2.5

experiment

2

E/Eo [ - ]

2

1.5

1.5

1

1 0.5 0 0.5

0.5

0.6

0.7

aw [ - ]

0.8

0.5

0.6

0.7

aw [ - ]

Fig. 9. Effects of aw on E / Eo and Pe / Peo in wet grinding mill (comparison with data of Kelsall et al. [5]).

0.8

0

Pe/Peo [ - ]

2.5

46

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

3.0

2.5

Pe/Peo E/Eo

2.0

2.0

u/uo

1.5

1.5

1.0

1.0

0.5

0.5

o,

E/Eo , Pe/Peo [-]

2.5

λ b/λ bo, v s/v so, u/uo [-]

3.0

0.0 0.8

1.0

1.2

0.8

1.4

1.0

D/Do [-]

1.2

1.4

0.0

D/Do [-]

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

on E is minor, D is important as the residence time distribution Φ(t) is dominated by Pe. (b) Effect of mill length Only a slight decrease of E / Eo with increasing L / Lo is obtained in Fig. 11. The reason is that η / ηo and λb / λbo are constant under the constant J, U, fw and db and vs / vso decreases only slightly with increasing L / Lo as D decreases with increasing L under the constant VM causing a decrease of the potential energy of balls. Pe / Peo increases greatly with increasing L / Lo, as depicted in Fig. 11. The reason is that Pe / Peo is proportional to (L / Lo) 2, led by the proportional relationships between Pe and L (see Eq. (8)) and between u and L (see Eq. (3)) under the constant VM. Thus, L is also important as the residence time distribution is dominated by Pe.

experimental Pe / Peo is steeper than the one in Fig. 9, the data of E / Eo would be closer to the calculated ones. 3.2.7. Effects of mill sizes Although no data are available, sensitivities of mill sizes to E / Eo and Pe / Peo are examined theoretically. The pilot mill used by Mori et al. is arbitrary chosen as the base mill and the base conditions to normalize 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 the other is varying L under constant VM simulating the constant mean residence time when varying L / D. The case of varying L under constant 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 proportional relation between Pe and L is apparent (see Eq. (19)). (a) Effect of mill diameter Fig. 10 shows that E / Eo increases only slightly with increasing D / Do. The reason is that both η / ηo and λb / λbo are constant when J, U, fw and db are constant and vs / vso increases only slightly with increasing D / Do due to the increase of the potential energy of balls (see the average distance of balls to fall denoted as Hab given by Eq. (A.24)). On the other hand, Pe / Peo varies greatly, decreasing with increasing D / Do. The reason is that u / uo is inversely proportional to (D / Do) 2 under the constant J, U and F, resulting in Pe / Peo inversely proportional to (D / Do) 2 according to Eq. (19) in which E / Eo varies little and L / Lo is constant. Although the influence

3.3. Possibilities of predicting E and Pe For the data of E and Pe, the corresponding terms on the right hand sides of Eqs. (16) and (17) are calculated and plotted in Figs. 12 and 13, respectively. In the figures, the values of R 2 are also noted, representing the correlation coefficients for the proportionality between the 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 values from the original data as discussed in Section 3.2.4. In the cases of Mori et al. and Swaroop et al., possible reasons for the scatter are experimental errors or simplifications adopted in the theory.

3.0

2.5

Pe/Peo E/Eo

2.0

2.0

u/uo

1.5

1.5

1.0

1.0

0.5

0.5

o,

E/Eo , Pe/Peo [-]

2.5

λ b/λ bo, vs/vso, u/uo [-]

3.0

0.0 0.40.

60.

81.

01.

L/Lo [-]

21.

4

0.40.

60.

81.

01.

L/Lo [-]

Fig. 11. Effects of L / Lo on E / Eo and Pe / Peo under constant VM.

21.

4

0.0

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

8 Mori et.al.

x10-2

Swaroop et.al.

E (data) [m2/s]

6

Kelsall et.al.

4

2

R2 = 0.732 0 0

20

10

40

30

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

50

x10-2

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

47

properties for dry and wet grinding conditions although minor variations have been observed between the theory and the data, for which further investigations are required both theoretically and experimentally. Additionally, the theory has predicted significant influences of the mill length L and diameter D on Pe. This implies the importance of design of L and D to produce materials with desired size distributions as Pe determines the residence time distribution according to Eq. (1). Further, a proportional relationship to predict Pe has been derived as Eq. (17), which appears to be valid regardless of the mill sizes tested. For estimating the residence time distributions using Pe under given grinding conditions, the proportional relationship should be useful as a first approximation. In future, this theory will be tested against more data, e.g., data of relatively large scale mills, to establish a sound basis. Further development will be made to predict the size distributions of product in mills with residence time distributions. This will lead to a methodology which can be adopted to optimize operating conditions and mill sizes satisfying required finenesses with economical performances. List of symbols

The correlation for the plots of Pe in Fig. 13 is better than that for the 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 be close to those of u. Since both theory and experiment use the same equation, Eq. (3), to estimate u, the resultant theoretical and experimental trends for Pe tend to be close each other. Further, it is worthwhile to note that the proportional relationship expressed by Eq. (17) to predict Pe appears to be valid regardless of the mill sizes examined. 4. Conclusions The present study has developed a theory on the dispersion coefficient and the Peclet number in a continuous ball mill. Firstly, the dispersion coefficient has been expressed by statistical parameters of a random walk problem. Then, the statistical parameters have been specified by mill operating variables based on a dispersion model postulated in the grinding zone. The derived equations have revealed that the dispersion coefficient is a function of the size of the dispersion zone and the mobility of balls in the grinding zone represented by the mean velocity and the mean free path 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 have been within reasonable agreement with reported data of the dispersion

Fp′(t) fc fw G(ξ) g Hab Hap h(r)

60 Mori et.al. Swaroop et.al. Kelsall et.al.

Pe (data) [-]

a1 a2 aw awo Cf CR c D Do db dbo E Eo F Fb Fd Fn Ft F′(t)

40

J Jo Jos

20 2

R = 0.875 0

0

2

4 b

6

8

v s) (cal c.) [-]

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

KM L Lo ℓ Mb Mdis MH

constant defined in Eq. (B.8), m − 1 constant defined in Eq. (B.14), m − 1 mass fraction of solids in slurry, – mass fraction of solids in slurry of base case, – drag coefficient, – number rate of balls ascending in the ascending zone, s − 1 constant defined in Eq. (B.14), m/s mill diameter, m mill diameter of base case, m ball diameter, m ball diameter of base case, m dispersion coefficient of particles, m 2/s dispersion coefficient of particles of base case, m 2/s mass feed rate, kg/s buoyancy force acting on ball, N drag force of fluid acting on ball, N normal force of particles acting on ball surface, N tangential force of particles acting on ball surface, N fraction of impulse response remaining in a mill at time t given by Eq. (23), – fraction of impulse response remaining in a mill at time t given by Eq. (24), – fraction of mill volume occupied by bulk of particles charged, – ratio of angular velocity of mill revolution to critical one, – fraction defined in Eq. (A.14), – acceleration due to gravity, m/s 2 average falling distance of balls, m average falling distance of particles, m vertical distance from departing point along ascending circular arc with radius r to surface level of grinding zone, m fraction of mill volume occupied by ball bed, – fraction of mill volume occupied by grinding zone ball bed, – fraction of mill volume occupied by grinding zone ball bed with voidage equal to εb, – mixer time constant, s − 1 mill length, m mill length of base case, m length of one step of random walker, m mass of a ball, kg mass ratio of particles in dispersion zone to those in grinding zone, – mass holdup of particles, kg

48

m n Pe Peo Ras Ro r t t1 Δta Δtap Δtf Δtfp Δtg Δtgp to ts U Uo u uo VM VRb VRp Vw v vb vs vso vw x

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

number of steps of random walk, – number density of balls in grinding zone, m − 3 uL / E, Peclet number, – uoLo / Eo, Peclet number of base case, – radius of inner surface of ball bed in ascending zone, m radius of inner surface of ascending zone, m radial distance from mill center, m time variable, s time period of a ball traveling a distance of λb, s mean residence time of balls in ascending zone, s mean residence time of particles in ascending zone, s mean residence time of balls in falling zone, s mean residence time of particles in falling zone, s mean residence time of balls in grinding zone, s mean residence time of particles in grinding zone, s delay time, s period of one step, s fraction of ball void filled by bulk of particles in static mill, – fraction of ball void filled by bulk of particles in grinding zone, – mean axial velocity of material flowing in mill, m/s mean velocity of material flowing in mill of base case, m/s volume of a mill, m 3 volumetric flow rate of balls circulating, m 3/s volumetric flow rate of particles circulating, m 3/s volume of water added per unit volume of dry particles, – velocity of a ball, m/s mean velocity of balls in grinding zone with no particles, m/ s mean velocity of balls in grinding zone, m/s mean velocity of balls in grinding zone of base case, m/s velocity of one step walk, m/s axial distance from mill inlet, m

Greek letters ratio of grinding zone particles to total particles charged to αf mill, – γ arccos(−rωr / g), rad γo arccos(Ro / r), rad δd 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, rad θbo angle for surface level of grinding zone, rad θbs angle for surface level of hypothetical ball bed with voidage of εb, rad λb average distance between two adjacent balls, m λbo λb of base case, m μpb 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/m 3 ρp density of particle, kg/m 3 ρpb bulk density of particles, kg/m 3 ρsl density of slurry, kg/m 3 ρw density of water, kg/m 3 σ effective cross sectional area of a ball for collision, m 2

σn τ τR Φ(t) Φr φ ωr

normal stress acting perpendicularly to ball surface, N/m 2 L / u, mean residence time, s shear stress acting tangentially on ball surface, N/m 2 impulse response function of residence time distribution at mill exit (x = L), s − 1 angle of repose, rad angle defined in Fig. B.1, rad angular velocity of mill revolution, rad/s

Acknowledgments The author would like to thank Dr. T.G. Callcott (Callcott Consulting, 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 rotating mill. Fig. 1(a) draws the case of particles under filled in the interstices of the ball bed and Fig. 1(b) is that of particles overfilled. The grinding zone is defined as the portion lower than the surface level with the angle of θbo. The area above the surface level where balls and particles move upwards along the mill wall is called the ascending 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 are considered, i.e., V M J ð1−εb Þ ¼ V Rb Δt g þ V Rb Δt a þ V Rb Δt f

ðA:1Þ

V M f c ¼ V Rp Δt gp þ V Rp Δt ap þ V Rp Δt fp

ðA:2Þ

where VRb and VRp are the volumetric flow rates of balls and bulk of particles circulating, fc is the fraction of mill volume occupied by the bulk of particles charged and variables denoted as Δt with subscripts are the mean residence times of balls and particles in the three zones. These two equations are solved simultaneously for given static conditions of J, U and εb to evaluate the dynamic states given by Jo, Uo and εbo. The calculations require equations noted below for the three zones. A.1. Grinding zone This section specifies the first terms on the right hand sides of Eqs. (A.1) and (A.2). Variables Δtg, Δtgp, VRb and VRp are expressed as follows, Δt g ¼ V M J o ð1−εbo Þ=V Rb

ðA:3Þ

Δt gp ¼ V M J o εbo U o =V Rp

ðA:4Þ

  D=2 2 V Rb ¼ ∫Ro ð1−εbo ÞLωr rdr ¼ ð1−εbo Þωr ðV M =πÞ 1−ξi =2

ðA:5Þ

  D=2 2 V Rp ¼ U o ∫Ro εbo Lωr rdr ¼ U o εbo ωr ðV M =πÞ 1−ξi =2

ðA:6Þ

where ξi = Ro / (D / 2), ωr is the angular velocity of mill revolution and parameters Jo, Uo and εbo are obtained as follows. Of these, Jo is expressed geometrically as, J o ¼ ð2θbo − sin2θbo Þ=ð2πÞ:

ðA:7Þ

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

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

A.1.1. Case of particles under filled Uo 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, U o ¼ α f f c =ðJ o εbo Þ≤1

ðA:8Þ

where αf is the ratio of the grinding zone particles to the total particles charged, given by   α f ¼ V Rp Δt gp =ðV M f c Þ ¼ Δt gp = Δt gp þ Δt ap þ Δt fp :

ðA:9Þ

49

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),

Δt a ¼ ð2=ωr ÞGðξs Þ

Δt ap ¼

ðA:16Þ

2   ½Gðξi Þ−ð1−εb ÞGðξs Þ: ε bo ωr 1−ξ2i

ðA:17Þ

A.3. Falling zone

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

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,

ε bo ¼ εb :

Δt f ¼ ð2H ab =g Þ

ðA:18Þ

 1=2 Δt fp ¼ 2H ap =g

ðA:19Þ

ðA:10Þ

A.1.2. Case of particles overfilled Balls and particles in the grinding zone are assumed to be completely mixed. In other words, the ball bed is enlarged until the overfilled particles are taken in the interstices, i.e., Uo is equal to unity, U o ¼ α f f c =ðJ o εbo Þ ¼ 1:

ðA:12Þ

ðA:13Þ

i

  2 2 hðr Þ ¼ ð−r cosγ Þ þ ðr cosγ o Þ ¼ ðD=2Þ f w ξ þ cosθbo :

where GðξÞ ¼ ∫1ξ ðγ−γo Þξdξ; cosγ ¼ −rωr =g ¼ −f w 2 ξ; cosγo ¼ Ro =r ¼ ξi =ξ, ξ = r / (D / 2), ξi = Ro / (D / 2), ξs = Ras / (D / 2) and Ras is the distance between the ball bed surface and the mill center in the ascending 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 balls and particles under the centrifugal force field until the voidage of the ball bed being equal to that of the static state εb. Then, VRb (already given by Eq. (A.5)) is again expressed using Ras (or ξs) like this, ðA:5′Þ

Equating Eqs. (A.5) and (A.5′) to give     2 2 ð1−εb Þ 1−ξs ¼ ð1−εbo Þ 1−ξi :

ðA:15Þ

ðA:22Þ

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

ðA:14Þ

  D=2 2 V Rb ¼ ∫Ras ð1−εb ÞLωr rdr ¼ ð1−εb Þωr ðV M =πÞ 1−ξs =2:

ðA:21Þ

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

h   ih   i 1 2 2 2 ∫ξ hðr Þξdξ ¼ ðD=2Þ 1−ξ =2 ð1=2Þf w 1 þ ξ þ cosθbo :

h i ξ V Rp Δt ap ¼ U o ∫1ξs ðγ−γ o Þεb LðD=2Þ2 ξdξ þ ∫ξs ðγ−γ o ÞLðD=2Þ2 ξdξ ¼ U o ðV M =πÞ½Gðξi Þ−ð1−εb ÞGðξs Þ

1

h i ξ V Rp Hap ¼ U o ∫1ξs hðr Þεb Lωr ðD=2Þ2 ξdξ þ ∫ξs hðr ÞLωr ðD=2Þ2 ξdξ i h i ¼ U o ωr ðV M =πÞ ∫1ξi hðr Þξdξ−ð1−εb Þ∫1ξs hðr Þξdξ

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

D=2

V Rb H ab ¼ ∫Ras hðrÞð1−εb ÞLωr rdr ¼ ð1−ε b Þωr ðV M =πÞ∫ξs hðr Þξdξ ðA:20Þ

A.2. Ascending zone

V Rb Δt a ¼ ∫Ras ðγ−γ o Þð1−ε b ÞLrdr ¼ ð1−εb ÞðV M =πÞGðξs Þ

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

ðA:11Þ

Rearranging the above equation leads to εbo as follows; εbo ¼ α f f c =J o ≥εb :

1=2

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

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

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

h   i 2 2 H ab =ðD=2Þ ¼ ð1=2Þf w 1 þ ξs þ cosθbo

  σ n ¼ ρpb vv cosφ

ðA:24Þ

h   i H ap =ðD=2Þ ¼ ð1=εbo Þ ð1=2Þf 2w 1 þ ξ2i þ cosθbo h   i 2 2 þ ½ð1−εbo Þ=εbo  ð1=2Þf w 1 þ ξs þ cosθbo :

ðA:25Þ

Appendix B. Mean velocity of balls in the grinding zone Firstly, consider the velocity of balls in the grinding zone with no influence of particles nearby. In the ascending zone, balls at the number rate denoted as CR gain the potential energy with the height of Hab due to the mill tumbling. This energy is transferred to the kinetic energy of balls with the mean velocity of vb in the grinding zone. Multiplying the frequency of collision of balls in the grinding zone per unit time, the energy balance is expressed as, h i 2 C R Mb gH ab ¼ ð1=2ÞM b vb ½ð1=2Þðvb =λb ÞNbo 

ðB:1Þ

where CR = VRb / (Mb / ρb), Nbo = VMJo(1 − εbo) / (Mb / ρb) and VRb is already given by Eq. (A.5) in Appendix A. Then, vb is obtained to be  h  i1=3 2 vb ¼ 4 1−ξi ωr gHab λb =ð2πJ o Þ :

ðB:2Þ

Under the existence of particulate material, the motion of balls in the grinding zone is affected by the resistance forces exerted by solid particles in dry grinding and by slurry in wet grinding. These forces are taken into account in the derivations of the mean velocities of balls as follows. B.1. Dry grinding As depicted in Fig. B.1, a ball moves at a velocity v within the bulk of particles whose bulk density is ρpb and the force is exerted by the particles on the ball surface to reduce its velocity. The rate of momentum of the bulk of particles denoted as (ρpbv)v acts at every point on the ball surface. Consider two forces per unit area, one is the normal stress denoted as σn acting perpendicularly to the surface and the

τR ¼ μ pb σ n ¼ μ pb

ðB:3Þ

h  i ρpb vv cosφ

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 area act on the small surface area given by (π / 2)db2sinφdφ. Assuming the two forces act only on the surface of the half sphere facing to the approaching particles, the horizontal components multiplied by the small surface area are integrated over the surface of the half sphere to get the resultant forces of the particles acting on the ball as h i π=2 2 2 2 F n ¼ ∫0 −ðσ n cosφÞðπ=2Þdb sinφdφ ¼ −ðπ=2Þdb ð1=3Þρpb v ðB:5Þ h i π=2 2 2 2 F t ¼ ∫0 −ðτ R sinφÞðπ=2Þdb sinφdφ ¼ −ðπ=2Þdb ð1=3Þμ pb ρpb v : ðB:6Þ Then, the equation of motion of the ball is given by   2 2 Mb ðdv=dt Þ ¼ U o ðF n þ F t Þ ¼ −U o ðπ=6Þdb ρpb 1 þ μ pb v

ðB:7Þ

where the fraction of the ball surface given by Uo is apparently surrounded by particles and exerted by the resistance forces. Solving the above differential equation with the initial condition of v = vb at t = 0, v¼

1 a1 t þ ð1=vb 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 =t 1 :

ðB:9Þ

The value of t1 is obtained from the following equation; t

λb ¼ ∫01 vdt

ðB:10Þ

Substituting Eq. (B.8) into v in Eq. (B.10), t1 is obtained. Thus obtained t1 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 internal friction or repose. Fig. B.2 shows experimental data of the angle of repose in relation to the void fraction of sand with the mean diameter of 0.5 mm obtained by the authors [18]. The straight line relation

1.0 Sand-0.5mm

[rad]

0.9

(a) Water under filled

0.8

(b) Water overfilled

0.7

water level 0.6

0.5 0.3

surface of particles

surface of particles

water level 0.4

0.5

p

0.6

[-]

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

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

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

between Φr and εp is assumed for simplicity in the present calculations. Then,

μ pb

  ¼ tanΦr ¼ tan 1:36−1:31εp :

ðB:11Þ

51

where ρw is the water density and Vw is the volume of water per unit volume of dry particles. Rewriting Eq. (C.1), Vw is given as    V w ¼ ρp =ρw 1−εp ð1−aw Þ=aw :

ðC:2Þ

In Fig. C.1(a), the water level is below the surface of particles (Vw b εp) 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.,

B.2. Wet grinding In the case of wet grinding, in addition to the resistance forces of the bulk of particles given by Eqs. (B.5) and (B.6), the drag force of fluid denoted as Fd acts on the ball. Further, the buoyancy force denoted as Fb is taken into account as the vertical motion of balls may be dominated in the grinding zone when the balls fall. The sum of the drag force and the buoyancy force is expressed as follows,

εsl ¼ εp :

   2 2 3 F d þ F b ¼ −C f πdb =4 ρw v =2 −ðπ=6Þdb ρw g

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 is equal to the water level. Then, the slurry void fraction εsl is given as

ðB:12Þ

where Cf is the drag coefficient assumed to be 0.44 in a range of high Reynolds numbers. As the fraction of the ball surface (or volume) equal to Uo is apparently surrounded by slurry, the equation of motion of the ball given by Eq. (B.7) is modified to be M b ðdv=dt Þ ¼ U o ðF n þ F t þ F d þ F b Þ   2 2 2 2 ¼ −U o ½ðπ=6Þdb ρpb 1 þ μ pb v þ C f ðπ=8Þdb ρw v þ

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 above differential equation for v = vb at t = 0 is given as ðB:14Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where a2 =[Uo/(dbρb)][ρpb(1+μ pb)+(3/4)Cf ρw] andc ¼ U o ρw g=ðρb a2 Þ . 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 for wet 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 by εsl, the slurry void fraction defined as the fraction of slurry volume occupied by either air or water. Also, the density of slurry ρsl is used in Eq. (21). These variables are estimated as follows. Consider a unit volume of dry particles with water to fill the void as schematically depicted in Fig. C.1. The mass fraction of solids in slurry, denoted as aw, is expressed as   ρp 1−εp  aw ¼  ρp 1−εp þ ρw V w

In this case, the slurry density ρsl is expressed as     ρsl ¼ ρp 1−εp þ ρw V w ¼ ρp 1−εp =aw :

ε sl ¼ 

ρp ð1−aw Þ Vw  ¼ : ρ ð 1−a p w Þ þ ρw aw 1−εp þ V w

ðC:4Þ

ðC:5Þ

As to ρsl, the following equation is used.

ðπ=6Þd3b ρw g

arctanðv=cÞ ¼ arctanðvb =cÞ−a2 ct

ðC:3Þ

ðC:1Þ

  ρp 1−εp þ ρw V w ρp ρw  ¼ : ρsl ¼  ρ ð 1−a p w Þ þ ρw a w 1−εp þ V w

ðC:6Þ

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