ELSEVIER Powder Technology 81 (1994) 101-109

mwom TECHNOLOGY

The effect of mill power on the selection function for tumbling and vibration ball mills

S. Nomura a, T. Tanaka b, T.G. Callcott ’ ’ tire Research Laboratory, Babcock Hitachi KK., Takoramachi 3-36, Kure, Hiroshima, Japan

b Hokkaido Universi& Sapporo 060, Japan ’ Callcott Consulting Pty. Ltd., PO Box 122, Mayfield 2304 NSW, Australia

Received 23 October 1992; in revised form 3 August 1994

Abstract

In the present study, a proportional relationship is derived between the specific rate of grinding, called the selection function, and the net mill power for tumbling ball mills and vibration ball mills. The proportionality constant is a function of material properties and some mill operating parameters. Also, the net mill power is expressed in terms of mill size and operating parameters. A convex trend of the net mill power in relation to the ball filling, J with a peak at about J=O.4 is obtained for tumbling mills. Vibration mills have a so called double action showing an increase in mill power with increasing J even at J>OS. These results are confirmed to be in reasonable agreement with reported experiments.

Keywords Mill power; Tumbling ball mill; Vibration ball mill

1. Introduction

Tumbling ball mills and vibration ball mills, when in operation, have grinding zones in which particles are trapped by colliding balls. How the grinding zone is formed, depends upon the type of mill. It is generated in the lower portion of a tumbling mill whereas the entire region of a vibration mill becomes the grinding zone. In a previous paper [l], the grinding processes were modeled and the specific rate of breakage, known as the selection function, was derived in terms of operating parameters and material properties. This paper extends the theoretical consideration to the net mill power in relation not only to the selection function but also to mill size and operating conditions.

Mill power is measured with either a torque meter or a watt meter. The measured value includes the power dissipated by mechanical friction in power transmission elements, the power to drive a mill chamber and the power dissipated in the mill charge. The fractions relative to total power depend upon the mill type [2]. The present study deals only with the power dissipated in the mill charge. Such a net power is most significant when assessing milling performance and designing mills.

For design purposes, many equations describing mill power and grinding energy are used. These may be

classified into two relationships, viz. the relationship of power draft to mill size and the relation between energy and size reduced. The former correlates the power with the mill scales and operating parameters, examples being the dimensional analysis of Rose et al. [3,4]. The latter expresses the specific energy (energy per unit mass of product) required to produce a desired fineness by a comminution ‘law’ like the Bond ‘Third Theory’ [5]. The product of the specific energy and the mass rate of feed gives the power required.

However, use of such empirical relationships needs much experience to avoid or minimize design errors [6-S]. Therefore, sounder expressions are required. Major interest nowadays is in mathematical analyses based on a population balance with two probability functions called the selection and breakage functions.

Of the two functions, the selection function S(X) defined as the specific rate of grinding of particles of size x, provides a measure of the mill performance or grindability and thus relates closely with mill power or grinding energy. Experimentally, Herbst and Fuerstenau [9] found a proportional relationship between the se- lection function and the specific net mill power for a tumbling mill. The proportionality constant was in- dependent of the operating parameters they examined. However, a later experimental study [lo] showed a

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102 S. Nomura et al. I Powder Technology 81 (1994) 101-109

dependence of ball and powder filling on the propor- tionality constant. Clarifying this is of great importance for establishing theoretical procedures of mill designing.

This paper deals with the theoretical relationship between the selection function and the net mill power. It uses an earlier study [l]. Further, it develops the relation between net mill power and mill size and operating parameters.

2. Theoretical

2.1. Selection function versus net mill power

The selection function, S(X) is expressed empirically as

S(x) =Kr”Q(x) (1)

where x is the particle size, K is the rate constant, n is the distribution parameter and Q(X) is a function of x varying from one to zero with increasing x [ll-131. The validity of this empirical form has been proved for tumbling ball mills and vibration ball mills in our preceding paper [l]. The resultant form is as follows:

where

* = 6(1 -b&,)

cl=,‘,,.,l r” (2k&V,z) r P

where U’ is the ratio of the number of balls effectively trapping particles to the number of balls in the grinding zone, Nbo. In the case of tumbling ball mills, U’ is equal to U,, the fractional filling of ball void by powder in the grinding zone. However, U’ in vibration mills becomes U, for U, < 1 and unity for U, > 1. Note that in Eq. (2),A, is a constant showing the material strength and p and 17 are constants independent of material and operational conditions whereas U’ and A, the average distance between two adjacent balls, vary with operating parameters. Other symbols for the variables in the above equations are the same as those in the preceding paper [l]. They are given in Section 5.

One of the important assumptions, adopted previously [l], is that the rate of energy given to the charged balls through the mill motion, pb is equal to that

dissipated by the balls colliding in the grinding zone, pbc. The SpeCifiC fOITmS Of pb for tumbling and vibration ball mills are briefly explained in Appendices A and B, respectively. Pbc is the product of the kinetic energy of a collision and the number of collisions per unit time in the grinding zone. Using Pb =Pbc

(3)

where (VJA) gives the frequency of collisions per ball. Consider now the power, P, dissipated in the mill

charge composed of both balls and powder. Suppose the charged powder and balls move together in a mill. That is, the ratio P/P,, equals the ratio of the total mass of the mill charge to the mass of the balls only, denotes asf,. Then, P becomes as follows (see Appendix C):

p

b (4)

where U is the fraction of ball void occupied by powder in a mill at rest and the case of U= 1 and eb= l p= 0.4 was researched by Rose and Evans [4].

Using Eqs. (2), (3) and (4) and paying attention only to the mill operating parameters, the grinding rate constant K given in Eq. (1) reduces to (see Appendix D):

where K’ is a proportionality constant, M, is the mass of powder and (P/M,) is the specific net mill power. Eq. (5) reveals that the grinding rate constant K (or the selection function for a fixed X) is proportional to the specific net mill power (P/M,) and the propor- tionality constant K’ is composed of [U’l(f,h)], op- erating parameters, and [2@7pPl(Z4J], material prop- erties.

2.2. Power dissipated in the mill charge

2.2.1. Tumbling ball mill The dimensional analysis of Rose and Evans [4]

derived an equation valid for f,+( = N/N,) d 0.75, U = 1 and l b = l P = 0.4, which is given by

where constant K, was 3.13 for lifter mills and 3.66 for smooth mills. e5(./) was only given graphically in relation to J in their paper [4].

From Eq. (4) and Eqs. (Al)-(A3) and arranging like Eq. (6) Eq. (7) follows (see Appendix E):

S. Nomura et al. / Powder Technology 81 (1994) 101-109 103

where

(7)

43(J,fw)= $ (1--~bo)(.l-5iz)

I

Thus, the parameter KI c#J@) in Eq. (6) is now specified theoretically as &(J,fw) in Eq. (7).

2.2.2. Vibration ball mill One of the important characteristics of the vibration

mill is the double action (see Appendix B). Taking account of the double action effect, the net mill power P for vibration mills is obtained by substituting Eqs. (Bl) and (B3) into Eq. (4), so

c1 + 3fd)(Nd’Nb) [f (M j,, )]D 2 3 4T3 1 m b b v & (8)

where ~,(MbNr,)] is the total mass of balls and powder charged.

3. Results and discussion

Numerical values used for the calculations are listed in Table 1.

First of all, a new factor, CQ, is introduced in the calculations of U,,, l bo, J, and &, for tumbling ball mills (see Appendix F). CQ expresses the mass ratio of powder left in the grinding zone to the total powder charged. Further, C+ as given by Eq. (F6) is adopted.

With respect to tumbling mills, Herbst and Fuerstenau [9] obtained data of S(X) for various J (0.35-0.5), fw (0.53-0.9) and U (0.8-1.6). The measured S(x) was proportional to the specific net power (P/M,) and the proportionality constant was independent of the op- erational parameters they examined. On the other hand, Shoji et al. [lo] reported data of P/M,S(x), the mill power divided by the absolute mass rate of breakage, showing slightly dependent of both J and U.

The most important result in this study is Eq. (5), i.e. the rate constant K (or S(x) for a constant x) is proportional to (P/M,), supporting the finding of Herbst and Fuerstenau [9]. However, the proportionality con- stant K’ in theory is not only a material constant but also a function of some operating parameters. Specif- ically, the term of [U’/(f,A)] is affected mainly by the powder load U and the ball diameter d,.

Fig. 1 shows the comparison of the present theory with the experiment of Shoji et al., in which K’ divided by that at J=O.4 and U= 1 are shown in relation to U for some J values. Both theoretical and experimental K’ curves are convex with respect to U whereas their

104 S. Nomura et al. / Powder Technology 81 (1994) 101-109

Tumbling mill (D = 0.6 m db = 26 mm)

I

o.50 0.5 I

1 1.5 2

lJ 1-l

Fig. 1. K’ divided by that at J=O.4 and lJ=l in relation to powder

load in comparison with reported experiment.

variations with J are slightly different. Although the influence of .I on K’ is small, further theoretical and experimental investigation into this effect is needed.

Although no experimental evidence is available, it is worthwhile noting that the theoretical K' value is inversely proportional to the ball diameter d, as A is proportional to d,.

For vibration ball mills, past literature has not re- ported on the selection function in relation to the mill power. The following assessment is offered. Experi- mental data [3,14-171 correlating the rate of the specific surface increase, d&/d& with the amplitude diameter D, and the angular velocity w, show that dS,ldt a&l.5 lo 3%2 lo 3. Assuming that dS,/df aS(x) and PaDv2wv3 [3], the above experiments may result in S(x) a d&l& aD,1.5 to 3~V2 lo 3 = Dv2wv3 aP, i.e. S(x) is proportional to P. Also, there is an empirical comment [2] saying that the mill capacity of the vibration mill is proportional to the mill horse power (f,S(x) aP), that is, S(X) a P when fC is constant. Judging from these reports, although using severe assumptions, the pro- portionality relationship between S(X) and P may be valid for vibration mills as predicted in Eq. (5).

Next, we examine the derived net mill powers given by Eqs. (7) and (8). For tumbling mills, Eq. (7) is compared with the experiment of Rose and Evans [4] in Fig. 2. As thefw value is not stated in their experiment, the calculation assumes 0.7 for fW. The vertical axis of the experimental data is K,#3(J) given in Eq. (6). The calculated line is in reasonable agreement with the experiment in trend. Additionally, it is noted that Fig. 2 shows data for lifter and smooth mills although no explicit treatment on this property is made here. How- ever, implicitly, the present theory deals with lifter mills rather than smooth ones as it takes no account of slips of the mill charge on the inner mill wall.

Rose and Evans [4] did not note the effect of fW on Kl or +;(q in Eq. (6). The derived Eq. (7) indicates its effect on 43(J,fW) as shown in Fig. 3. The value of

0.2 0.4 0.6 0.6

J L-1 Fig. 2. Relationship between &(.I) and J [4]; (0) lifters, (0) no

lifters.

J I-1 Fig. 3. Effect of fW on ~(JJW)_

J 1-l Fig. 4. Relationship between the mill capacity f&x) and the ball

filling J.

43(J,fW) increases as the speed of revolution increases towards the critical one.

For vibration mills, the net mill power given by Eq. (8) is proportional to D:w,,~. Rose and Sullivan [3] obtained the same relation by dimensional analysis. Although some other expressions were reported [2], the exponents of 2 and 3 with respect to D, and o, seem to be reasonably accepted as the average values.

Next, consider the effect of ball filling .I on net mill power P. This leads to one of the interesting differences between vibration mills and tumbling mills.

The ordinate of Fig. 4 is the mill capacity fCS(x) instead of P since data are available only for mill capacity at present. Further, the mill capacity and the

S. Nomura et al. I Powder Technology RI 11994) 101-109 105

net mill power are identical because the relation of fcS(x) a S(x) aP is valid when operational parameters other than .I are constant. The experimental data are for calcite powder at U= 1 [l]. The critical .I value for the double action giving I$,,= 7r/2 is calculated to be 0.437. The calculated curve at .I> 0.437 is steeper than that at J<O.437 although the variation of the calcite data with .I is linear. However, if double action is not taken into account in the theory, the calculated curve would be in poorer agreement with the experiment, especially for J greater than 0.437.

4. Conclusions

The theoretical consideration has been extended to the relationship between the selection function and the net mill power. For tumbling and vibration ball mills, the rate constant K is found to be proportional to (P/M,) and the proportionality constant K’ is a function not only of material properties but also of some operating parameters given in [U’l&A)]. Further, the net mill powers are expressed in terms of operating parameters. These theoretical results are in reasonable agreement with experiment.

The present study could establish a sounder basis for mill design by replacing conventional empirical relations.

5. List of symbols

All A, b

NY, 4 Cl CR

CY d, D

f

fco

fd

fin fw

$9

J

=&,V,2 (J) (2/5)C,(k~/2.8)~” (N m(5b’3)-2) constant with respect to material strength ( - ) breakage function (-)

[+WW,(l - +I& (m3-(5b/3’) number rate of balls ascending in the as- cending zone (s-l) [(9/8){(1- v,“)/Y,+ (1 - ~~,,‘)/y,)z]‘~ (Pa-‘“) diameter of a ball (m) mill diameter (m) amplitude diameter of oscillation (m) fractional volume filling of mill by powder

(-) fractional volume filling of mill by powder in the grinding zone (-) ratio of balls subjected to double action to Nd balls (-) mass ratio of mill charge to balls only (-) ratio of speed of revolution to critical one

(-) = (28--sin 20)/(27r) ( -) average falling distance (m) fractional volume filling of mill by balls at rest (-)

JO

K

Kl K’ k

L

Mb

M&J n

N

Nb NbO NC Nd

P

Pt7

PbC

4(X)

Q(x) r

RO

S(x) S” Ata

At, u

fractional volume filling of mill by balls in the grinding zone (-) grinding rate constant (m-1-(5b”) s-‘) constant ( - ) proportionality constant (kg m-‘-(5b/3) J-‘) constant with respect to material strength (N mbV2) mill length (m) mass of a ball (kg) mass of powder charged (kg) distribution parameter equal to [(5b/3) + l]

(-) speed of revolution (s-l) number of balls charged (-) number of balls in the grinding zone (-) critical speed of revolution (s-l) number of balls subjected to direct hit of wall

(-) power dissipated in mill charge (J s-‘) rate of energy given to balls in the grinding zone (J s-l) rate of energy dissipated by ball collisions in the grinding zone (J s-l) variable nearly equal to unity under normal operating conditions (-) function of x (-) radius of circular arc in the ascending zone

(m) radius of inner surface of the ascending zone

(m) selection function (s-l) specific surface on volume basis (m-l) residence time of balls in the ascending zone

(s) residence time of balls in the falling zone (s) fraction of static ball void filled by powder

(-) fractional filling of ball void by powder in the grinding zone ( -) ratio of the number of balls effectively trapping particles to the number of balls in the grinding zone (-) volume of a ball (m’) mill volume (m”) mean relative velocity of ball (m s-‘) mean velocity of oscillating mill wall (m s-‘) particle size (m) particle size to give a maximum S(x) value

(m) Young’s modulus of ball (Pa) Young’s modulus of particle (Pa) frequency of collisions per ball (s-l) frequency of collisions effective for crushing

(s-‘)

106 S. Nomura et al. I Powder Technology 81 (1994) 101-109

Greek letters

ratio of grinding zone powder to total powder charged ( - ) proportionality constant (-) void fraction of ball charge in a static mill

(-) void fraction of ball bed in the grinding zone

(-) void fraction of powder charged (-) a function of fw( =N/iVJ ( - ) a function of J and fw ( - ) a function of J ( - ) angle indicating bed surface level (rad) angle for ball bed surface in the grinding zone

(rad) cos-‘[cos &,/{l - (DJD)}] (rad) collision energy transfer coefficient ( - ) average distance between two adjacent balls

(m) Poisson’s ratio of ball ( - ) Poisson’s ratio of particle

R,D (-) density of ball (kg rne3) density of particle (kg m-“) angular velocity of mill revolution (rad s-‘) angular velocity of oscillation (rad s-l)

Acknowledgements

The paper is published with the permission of Bab- cock-Hitachi K.K.

Appendix A: tumbling ball mill

Fig. Al illustrates a cross section of an operating tumbling ball mill. The mill charge may be divided into three zones, grinding (lower part), ascending and falling. Balls gain potential energy due to the mill tumbling. Power is expended for lifting the mill charge to a definite height at a required mass rate. The rate of energy given only to the balls, Pb, is derived as

Pb = c,ikf,gH (Al)

where CR, the number rate of balls ascending in the ascending zone or projecting into the falling zone, and H, the average height for the ball fall, are given by

c

R (Aa

Fig. Al. Schematic drawing of operating tumbling ball mill.

H=(D/2) ; fw2(l+5i2)+cos %, 1 w9 ,$ = m,/D = ‘OS ebo for e,, G n-12

( - cos eJR/fw for eb> d2

where symbols for the variables are as given in Section 5.

Appendix B: vibration ball mill

A horizontal vibration mill is considered. The mill has a circular oscillation whose amplitude diameter and angular velocity are D, and w,, respectively. The mill wall oscillates between the two dotted circles drawn in Fig. Bl(a) and transfers kinetic energy by directly hitting balls within the two dotted circles.

Unlike a tumbling mill, the grinding zone is the entire region of the mill charge. Further, a double action as noted by Rose and Sullivan [3] occurs in a vibration mill, which is explained as follows.

Assume that the mill body vibrates only upward and downward for simplicity and the level of the mill charge is expressed by 0, as illustrated in Fig. Bl. When, as in Fig. Bl(a), &, is less than the critical value &= 42 (or 50% of the mill volume), only the upward motion of the mill shell projects the mill charge upwards. While the shell moves downwards, free falling is un- impeded by the upper wall of the container. On the other hand, when filling greater than the critical as in Fig. Bl(b), the mill charge, even during the downward motion, is forced downwards instead of falling freely. Both upward and downward motion of the mill, the so called double action, transfers energy to the mill charge. In a mill with circular vibration, balls in the shaded regions A and B in Fig. Bl(b) are subjected to the double action.

The resultant power to the ball charge is given by

S. Nomura et al. / Powder Technology 81 (1994) 101-109 107

Fig. Bl. (a) Motion of vibration mill and (b) its double action region.

031)

fd= O 1 if (O,, G 7r/2) 2 - (ri+&) if (& > 7r/2

where Nd is the number of balls hit directly by the mill wall, fd is the fraction of N,, balls undertaken by the double action and V, is the mean velocity of mill wall towards the mill center. N,, and V, are specified as follows:

(B2)

where f( 0) = (20-sin 20)/(2r) and cos 0, = cos e,/ {I- (Q/D%

Appendix C: derivation of Eq. (4)

The mass ratio of the balls and powder charged to the balls only, fm, is represented by

fm= MN

MbNb+M, =I+ P A4

b b Mb% (Cl)

where &Nb and MP are the masses of the balls and the powder, respectively, which are expressed as follows:

MbNb=~r&-Eb)Pb PI

~p=w-c(~-~p)Pp (6)

where V, is the mill volume and J and fc are the fractional volume filling of mill by balls and that by powder, respectively. As J, fc and U are in relation of u=fc/(./cb), Eq. (Cl) is reduced to

Appendix D: derivation of Eq. (5)

VW

The grinding rate constant K is expressed by the first term of the brackets in the right hand side of Eq. (2). That is,

Z,G K= f,V,h

(D1)

=

fcVh4h The above equation is rearranged to

(“2v:)( ‘zPp)( F)

K= VhJ.fc(l - 4Pp

( w where the first term of the brackets in the numerator is Pb according to Eq. (3) and the denominator is equal to MP according to Eq. (C3). Then,

K=(“T)($($)

Further, using Eq. (4), Eq. (5) is derived.

Appendix E: derivation of Eq. (7)

Starting with the following equation

P fmpb -=-= fmtcRMbgH) p,D5N3 pbPN3 pJFN3

w

(El)

where Eq. (4) and (Al) are used. Substituting Eqs. (A2) and (A3) into Eq. (El)

x 4 {o.5fw2(i + 6:) + cos e,} 1 WI

Then, as Mb =,$,vb, w, = 2rN and 27rN= = (2g/D)ln, Eq. (E2) leads to

108 S. Nomura et al. I Powder Technology 81 (1994) 101-109

X {0.5f,2( 1+ 6,‘) + cos ebO}]

x f (l-&)(1-{;)

x {0.5fw2( 1+ 4”) + cos 6,,} 1 (E3)

Substitution of Eq. (C4) into Eq. (E3) gives Eq. (7).

Appendix F: a factor, (Y” in a tumbling ball mill

Variables Jo, l3,,, U, and E,,,, indicate the state of the grinding zone and their derivations are made using a new factor, (Ye, introduced in this paper.

For a given ebo, Jo and 13,~ are calculated by solving simultaneously the following two equations [l]

Jo = (213,- sin 2&,,)/(2r) (FI)

Nb = Nbo + C,(At, + At,) (F2)

where NbO, CRAtt, and C,At, the numbers of balls in the grinding, ascending and falling zones, resptictively, are expressed by ebO, Jo, O,,, and some operating pa- rameters.

To determine lJ, and +,,,, a new factor, (Ye, expressing the ratio of the grinding zone powder to total powder charged, is used. Then, U, is given by

U, =fdJo 4 = dcl(Jo 4 (W Assuming the powder and balls to be well mixed in the grinding zone, U, does not exceed unity. Conse- quently, l bo is determined from the equation of 1 = a&J (J,,E& unless l bo is less than E,,. When U, is less than unity, ebO is equal to E,,. That is,

U,=l E bo = dclJo

for a&/J, > E,,

uo = dif,l(Jo%) Ebo = Eb

for cqfc/Jo < $,

WI

(F5)

With respect to (Ye, when fft=NbolNb, as adopted previously [l], balls and powder are completely mixed and the mass ratios of powder to balls in the three zones of grinding ascending and falling are identical. When (Ye is equal to unity, all of the charged powder remains in the grinding zone. In practice, CX~ would be between Nbo/Nb and unity as separation of powder from the balls occurs to some extent due to slipping of the

Tumbling ball mill (D = 0.6 m db= 26 mm) 1.51

: Present theory ; _ _ _ _ _ _ _ _ _ i Experiment of Sljoji et al.

i i

0.1 0.2

fc i-l Fig. Fl. Dimensionless mill capacity in relation to powder filling for

tumbling ball mill in the case of (Y~=N~/(N~+C&$

Tumbling ball mill (D = 0.6 m db= 26 mm)

- ! Present theory / _ _ _ _ _ _ _ _ _ 2 / Experiment of Shoji et al

1 1

0.1 0.2

fc f-1 Fig. F2. Dimensionless mill capacity in relation to powder filling for

tumbling ball mill in the case of q=NJN,,.

powder between the interstices of the ball bed during ascending. Therefore, powder tends to concentrate in the ascending and grinding zones rather than in the falling zone. As an intermediate state, it is considered that all of the powder exists within the ascending and grinding zones. Then, CQ is equal to the ratio of the balls in the grinding zone to those in the grinding and ascending zones, i.e.

Nbo af = Nbo + &At,

To examine Eq. (F6), the mill capacities, f&r) are calculated based on Eq. (F6) and (Yf=Nbo/N,,, which are shown in Figs. Fl and F2, respectively. Particular attention is paid to U ( =fclebJ) to give the maximum mill capacity. This U value varies slightly from 0.7 to 0.8 with J from 0.2 to 0.5 in Fig. Fl whereas the U value is unity in Fig. F2, for which the experimental peak of the mill capacity appears at U=O.83 [lo]. Consequently, the calculation using Eq. (F6) gives better agreement with the experiment than that using cu, = NbO/ Nb as far as the U value for the maximum mill capacity is concerned.

S. Nomura et al. I Powder Technology 81 (1994) 101-109

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