RATE OF BALL MILLING AND VIBRATION MILLING ON THE BASIS OF THE COMMINUTION LAW Probability Theorem K U N I O
T A M U R A ’ A N D
T A T S U O
T A N A K A
Department of Chemical Process Engineering, Hokkaido University, Sapporo, Japan The grinding mechanism of ball mill and vibration mill is expressed in terms of probabilities. Theoretical relations among operational variables derived from the final rate equations are confirmed by empirical facts, suggesting that the over-all treatment of milling mechanism in this paper is satisfactory. The relationship between the probabilities in this paper and the selection function or the breakage function in grinding kinetics is briefly discussed.
IN
RECENT years, a few papers have discussed the grinding kinetics involving the selection and breakage functions. I t is, however, not useful to define such functions, particularly the selection function, for the design and operation of mills, so long as these functions are only mathematical expressions without a clear physical picture. On the basis of a new concept proposed by Tanaka (1966), analysis is extended to the internal mechanics of batch ball milling and vibration milling-i.e., the grinding mechanism is expressed in terms of probabilities. Theoretical relationships among operational variables derived from the final rate equations are confirmed by various empirical facts, suggesting that the basic concept referred to in this paper can clarify the complicated interacting effects of various operating variables upon the ball milling and vibration milling mechanisms.
General Concept
As reported by Tanaka (1966), the production of new surface per impact per unit volume of particles, AS?,, is expressed by the equation:
where x is the diameter of the particle, P, is defined as the probability of collision between ball and particles, and K’ and p are constants. P, is the probability relating to the crushing strength, and in the case of aggregate crushing such as ball milling and vibration milling, it should be interpreted as an apparent probability, P,, , as in the following. Considering one ball, many particles are subject to its impact when it falls and some particles can be crushed, but others not. Since each of the crushed particles undergoes a different value of stress depending upon its location under the ball, we can define the probability, P,,, mean value of P, for each crushed particle, but for unbroken particles, Po = 0. Therefore, for all the particles impacted by one ball we have to take the average of these values, define P,,, and use it in place of Po in the basic equation. Furthermore, if the particles hit a ball ZAt times during the comparatively short interval of time, At, the specific surface increase, AS, (sq. cm./ cc.), is represented by &,,.Z.&,t, where 2 is the total
’ Present
address, Mitsui Toatsu Chemical Co., Chiba, Japan
number of collisions between the balls themselves or ball and mill shell per ball in unit time. The diameter x is assumed here to be represented by the average diameter of particles, interpreted more explicitly later. Hence,
I n the previous paper (Tanaka, 1966) AS, was written as dS/dE. However, the experiment and its analysis were performed under the condition of constant energy, so that the result itself is not wrong. At the same time, Pa, the probability relating to the crack tip propagation introduced previously, might better be omitted, because this effect can be explained by the variations in surface energy, due to the adsorption of particular gases or grinding aid molecules, which influences the compressive strength of material (Suzuki and Tanaka, 1968), and should be involved in P, as defined. I n any event, however, P, has nothing to do with the content of this paper, since milling is assumed to be carried out at atmospheric pressure. By replacing these relationships in Equation 1, considering that milling is a continuous process, a basic rate equation is obtained analogous to the grinding kinetics involving the selection function and the breakage function.
The selection function is defined as the fraction by weight of particles of given size x which are selected and broken per unit time, while the breakage function is defined as the fraction by weight of particles smaller than size x’ produced from breakage of size x. The selection function should be related to P,Z and the breakage function to K’(P,,)x’, conceptually. This basic concept will be applied to the analysis of the mechanism of ball milling and vibration milling. It is certain that in applying the basic model to the aggregate crushing like ball milling one has to bear in mind the particle size distribution of materials. However, the purpose of this paper is to account for the complicated interaction of various operating variables on the milling efficiency or milling capacity, a t least to the first approximation. There are a number of unsolved problems in Ind. Eng. Chern. Process Des. Develop., Vol. 9, NO.2, 1970
165
dealing with such complicated systems-for example, the mechanics of the particulate packed bed, as well as the mechanism of single particle fracture. Many investigators have used mathematical models without a clear physical background, but this sort of treatment will not lead to prediction of mill performance relating to the operational variables of milling equipment and the material properties. With this objective, knowing that the present treatment is rather crude, the authors have preferred to use the average value of particle size, and hence average particle properties. Emphasis has been put upon the confirmation of the developed theory from as many varieties of recognized empirical data or facts as possible.
PARTICLE LAYER
I
ANALYSIS OF BALL MILLING
MILL SHELL
Crushing Model
The mechanism of ball milling is assumed to be that materials are broken by the impact action due to the free-falling balls from some height within a mill. The particle size distribution of the crushed product should be introduced for exact analysis of the kinetics, but to avoid complexity and to the first approximation, it is assumed in this paper that the average particle size is sufficiently representative of the material being subject to crushing. Probability of Collision. To determine the value of probability of collision, P,, it is necessary to consider two items. The first is the ratio of number of particles coming under a ball, f l , to the total number of particles, n, per ball. Fraction f l may be regarded as a constant until U becomes unity, where U is the ratio of the apparent volume of particles in a mill to the total volume of space between balls. When U becomes greater than unity, this fraction increases, on the assumption that the excess particles tend to locate under the ball. These notions can be formulated as follows: f l
=
f, =
hi = const. hlnc + ( n - nc) n
when U 51 when U
>
(3)
1
where n, is the number of particles per ball when U = 1. When the void fractions t B and cP associated with balls and particles are, respectively, equal to 0.4, the geometric relationship between n and U is given by
n = 0.4 d 3 ~ - 3 U
(4)
where d is the diameter of the ball. The second item is the area of the impact zone in which particles can be nipped by a free-falling ball. This area is determined geometrically from the impact model as shown in Figure 1. Obviously, its magnitude increases with particle size until the maximum value calculated from the angle of nip is reached. The particles located outside this region cannot be nipped into but will jump away. Hence, the probability that particles located under a ball can be nipped may be calculated as the ratio of this area to the cross-sectional area of the ball.
- d'
4
166
I
/ / / / / / / / / / / / ///////I////I/ / ;
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970
Figure 1. Collision of a ball with particle layer in a ball mill
Since f l and f2 are mutually independent, the probability of collision, P,, is given by the product of f i and f 2 . From Equations 3, 4, and 5, the probability of collision, P,, is given by
P,=
4x 1*
when U 5 1
hi
Number of Collisions in Unit Time. The number of collisions per ball in unit time, Z, is given by
(7) where T , is the time needed for one revolution of the mill, T B is the time for the ball traverse, and N is the rotating speed of the mill. Considering the slipping effect of the ball, Equation 7 can be rewritten in the form
where T i is the time for the ball traverse without slip, V B is the peripheral speed of ball running adjacent to the mill shell, and V , is the speed of the mill shell. The degree of slip, s, between ball and mill shell is defined as
(9) or V B
-=l-s
Vll
I t is difficult to find literature dealing with the slip of balls in a rotating mill. However, the surging phenomenon in a ball mill is directly related to the slippingthat is, the more the slip, the more surging tends to occur. Fortunately, Rose and Blunt (1957) published their experimental results on surging, according to which a plot of J x fi on the ordinate and D / d on the abscissa showed clearly the region where surging takes place. The factor J is the fractional filling of the mill, p is the mean
coefficient of friction between ball and shell, and D is the mill diameter. When J x is specified, the surging, hence the slip, tends to occur as D/d decreases. Consequently, if the ordinate in their graph is assumed to represent the degree of dip, and its value is set equal to unity when D/d = 1,Figure 2 can be drawn by replotting Rose's data. According to this figure, we have
evenly divided by the total number of particles, nz, within a crushing zone. Thus
where m is the weight of the ball, u is the impact velocity of the ball, and g, is the gravitational conversion factor. If the energy, E , is completely converted to the strain energy of a particle, the stress, urn, generated within a particle is calculated from the formula:
On the other hand, regardless of rotating speed T W
-=
TS
1.45
is given by Rose and Sullivan (1958). The critical speed of the mill, N, (r.p.s.), is inversely proportional to the square root of the mill diameter.
N , = 7.05/(D)1'2
(13)
Let 9 be the ratio of N to N,, so that
N = 7.054/ (0)'"
(14)
Substituting Equations 11, 12, and 14 into Equation 8, we have
Z = 0.664d-07D02
where A is the area, 1 is the length of material that undergoes the force, and Y is the modulus of elasticity of material. This assumption may not appear realistic, but Tamura and Tanaka will show that this is reasonably good. I n the case of a single spherical particle, A and 1 can conveniently be replaced by the cross-sectional area, (*/4)x2, and the diameter, x, of a particle, respectively. From the assumption that the free-falling height of the ball can roughly be replaced by the mill diameter, neglecting the effect of rotating speed, the impact velocity of the ball is given by
v = (2gD)'
(15)
Probability Relating to Crushing Strength. As the result of impact crushing tests and computations (Suzuki and Tanaka, 1968; Tanaka, 1966), the probability P a , relating to the crushing strength, used in Equation 1, was given by 1.5
P, = (1 - 2)
(19)
where g is the acceleration due to gravity. The total number of particles, nz, which are submitted to stress a t one impact of the ball is obtained by the product of P, and n. From Equations 4 and 6, we have
when U 5 1 when U > 1
nz = 1 . 6 k l d 2 ~ ~ 2 U I Z Z = 1.6d2C2(U + kl - 1)
(20)
Urn
Substituting above relationships into Equation 18, we have where U , is the compressive stress generated within a particle a t impact and uG is the compressive strength of a particle. This means that crushing can occur only when ( I , exceeds ua. Because of the stress distribution within a particle layer and the fact that some fraction of the particles does not reach the breakage point because of lack of breaking stress, it is very difficult to evaluate accurately an individual stress generated within a particle actually to be broken. Consequently it is assumed here, to the first approximation, that the energy, E , given to one particle actually to be broken is equal to the falling energy of the ball
1
5
10 D/d
50
100
[-I
urn = 0.91
[ ( U YpBdD + h1- l ) x
]I2
when U
>
1
The density of the ball, p B , is used in replacing m = (r/6)d3pBin Equation 18. By the considerations therefore, we have
Pa, = [ 1- 1.1u a
when (___ ypBdD)1z]1'5 klxU
U 5 1 (22)
Furthermore, it is necessary to consider the fact that not all the particles that received the impact action are broken. If the particle layer is deep enough, only particles existing in a limited range of depth, h', can be broken. Since Pa, is the apparent degree of breakage over the whole depth, h, of the particle layer and Pamis that of the actually broken particle layer, h', Po, and Pa, can be related in the following:
Figure 2. Slipping effect in relation to mill and ball size Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 2, 1970
167
Let us consider the value of h'. It is impossible, a t present, to estimate its accurate value because of the lack of a basic study. However, h' can be derived by consideration of the experimental data of Mori and Kojoh (1954) on the penetration of a free-falling ball into a particle bed. From their results as shown in Figure 3, the depth of layer crushed, h', can be correlated with the ball size and the falling height, replaced by the mill diameter, D, assuming that it is proportional to the depth of penetration. h' = k2d0.5D0.5 (24) where k2 is a constant. On the other hand, the value of h is obtained from geometrical considerations with the aid of Equation 4.
-
I X
/
d=2.45cm
d =3.17
0
d =3.96
/
0.6
0.8
1.0
6.0
4.0
2.0
FALLING HEIGHT OF BALL DIAMETW OF BALL
8.0
lM,
[-I
Figure 3. Depth of penetration of a steel ball as a function of falling height and ball size 1.0
I
0.2
I
h = apparent volume of particles under bottom of a ball/cross-sectional area of ball = [(void volume between balls per ball) (fractional filling of particles) (fractional volume of particles under bottom of ball to total volume of particles) ]/ (cross-sectional area of ball)
(-6 K
d3 --)
CB
1-eR
Uki
-&
4
= 0.44 k1dU
when U 5 1
I
d
- l)d
when U
>
1
M =
When the values of the probabilities obtained above are placed in the basic rate Equation 2, under the assumption that when U is smaller than unity h is equal to h', and when U is much greater than unity h is greater than h', we have the final equations of baIl milling involving various operational variables and material properties. For convenience, the specific surface per unit mass is used instead of S , (sq. cm./cc.), in the basic Equation 2, and S , is equal to S . p p , where pp is the density of a particle. Then when U 5 1,
=
dS dt
4
dS dt
= 0.19 p,D2LJU
l
l
600
+
U-'pP1(1- 1.1ga[
'
- = 1.1h,K'cpLJd-2.2D2.SXP (1 - 1.1ua[ +
=
E,
= 0.4 is assumed) pp
is the density
Discussion
Effect of Mill Diameter on Milling Capacity. Considering that the over-all rate of increase of specific surface is constant for a given continuous process, M dS/dt can be regarded as an expression of milling capacity. Thus,
(U + hl - 1)x YPBd D
l U 2 lS
11
( U + k l - 1)x ] YpadD
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970
(ea
(28)
where L is the length of the mill and of material crushed.
>1
I n these equations, M is the total mass of materials within a mill, given by 168
I
DZLJta(l- t,) Up,
or
M
1
4cO
5 cm.,pB = 7.8 g . m / ~ ~Y. ,= 1.9 x 1 0 8 g . f / ~ qcm. , U = 1, k, = 0.2
or
- = 6.0 k2K'dd
200 D [cml
x = 0.1 cm.,
(25)
Final Correlations
and when U
I
I
I
100
Figure 4. Relation between Psm and mill diameter
where e B = 0.4 is assumed. Similarly,
h = 0.44(U + kl
80
'/'
]
(27)
Equations 26 and 27 can be applied. I n these equations, M dS/dt can be reduced to a simple function of D , when other operational variables are kept constant. Values in parentheses-i.e., Po, -vary with mill diameter as shown in Figure 4. From this figure, it appears that Po,is roughly proportional to the 0.4 power of mill diameter for mills of industrial scale, though the exponent depends upon various factors. Hence, the effective power of D on the milling capacity should range from the minimum 2.6 to the maximum 3.1, according to Equations 26 and 27.
M - dS a dt
O2.6
-
3.1
10
T O8
Y
E
2
z
06
04
w
2;
l-
a
iij 0 2 cc
(29) 0
This result agrees well with the empirical facts that the milling capacity is reported to be proportional to the 2.6 or 3.0 power of the diameter of the ball mill or tube mill. If plug flow through a continuous tube mill is assumed, the discussion above, confined to a batch ball mill, can be directly applicable to data obtained from industrial tube mills. These mills usually are more than 10 meters long, and completely mixed flow cannot be expected. Figure 5 plots data from cement plants in Japan. The slope of these straight lines is about 2.7. Effect of Rotating Speed on Mill Capacity. Equation 27 shows that milling capacity is proportional to the rotating speed of the mill. Strictly, however, the motion of the ball depends upon the rotating speed. For example, a t low speeds, balls grind because of slipping and rolling each other and a t speeds higher than critical, all the balls adhere to the mill shell, so that it is impossible to expect the free-falling strike of balls in either case. Consequently, further consideration of these ranges of mill speed is clearly needed to analyze the milling mechanism more exactly. Nevertheless, the milling mechanism suggested by the present equations with respect to mill speed can be supposed to be fairly good, at least in a limited range, judging from the data of Gross (1938), as shown in Figure 6. Effect of Fractional Filling of Materials on Milling Capacity. The interrelationship between the milling capacity and 40
0
RAW MATERIAL
X COAL
0 CEMENT
20
m Y
Figure 5. Effect of mill diameter on milling
f
capacity
>
0
0.2
$ = N/NC
balls on milling capacity
10
05
u
10
[-I
Figure 6. Effect of fractional filling of
0
08
06
OL
15
20
1-1
Figure 7. Effect of fractional filling of particles on milling capacity
D =200cm.,d
= 5 c m . , p B =7.8g.,,,/cc., 1.9 X 10’ g.f /sq. cm., oa = 2.5 X 10‘ g.f /sq. cm., x = 0.1 cm., ki = 0.05
Y
=
the value of U can be derived from Equations 26 and 27. These equations are numerically illustrated in Figure 7, from which it is clear that the maximum grinding capacity can be obtained when U is equal to unity. This qualitatively agrees with mill men’s experience that the most efficient milling condition is achieved when U = 1 or almost 1. Optimum Diameter of Ball in Relation to Particle Size. The smaller the particle size, the stronger its compressive strength, in general. Thus, if ua = a x b ( b 5 0) and = -0.7 for calcite, and these values are substituted in Equation 27 with all variables except d and x kept constant, Equation 27 becomes
0
210
5 2 8
where K” is constant and
5
g
c = 1.1 a
J 6 I
To maximize M d S / d t , differentiate Equation 30 with respect to x and set the derivative equal to zero, which gives
X X
4
1-1 ( U +YphBD
I
d0.5
1
2
D [ml
3
x=[
I’
1+ 5 (b + 0 . 5 ) ~
0’5’
Ind. Eng. Chern. Process Des. Develop., Vol. 9, No. 2, 1970
(32) 169
or
d = [I + 5 ( b + 0.5)c]'x2 *
+
(33)
O5
I n the case of crushing calcite, for example, its physical properties are that ua = 1.4 x lo6 x-OZ5 g.,/sq. cm. and Y = 1.9 x 10' g.,/sq. cm.-that is, a = 1.4 x lo6, b = -0.25. Under the following operational conditions, using steel balls in a mill of industrial scale, pB =
7.8 g.,/cc.
D = 200 cm. U=1 ( k l = 0.2 assumed) we have from Equation 33
d = 7x05 Coghill-deVaney's equation (Taggart, 1950) is d=9
-
(34)
13 xo5
(35) The constant in Equation 35 is reported to depend upon various factors. Considering this, the agreement between Equations 34 and 35 should be satisfactory. Selection Function in Grinding Kinetics. I n connection with grinding kinetics, the selection function P ( x ) should be related to P,Z, as pointed out above. Taking arbitrary units on the ordinate axis of P ( x ) , the selection function thus calculated can be represented by broken lines in Figure 8, as a function of the particle size for several ball sizes. Obviously, P ( x ) increases with particle size until a constant value is reached when the angle of nip, 8, is given as 30°. This result can be compared with the experimental data reported by Bowdish (1960), for example, in Figure 8. The selection function, P ( x ) , in a ball mill generally increases with particle size and reaches the maximum a t different values of x . These particle sizes in relation to ball sizes and the general form of the curves are supposed to be similar to the theoretical ones. We believe that this suggests that our consideration is not essentially far from the actual mechanism, even though various minor corrections should be made.
no doubt, however, that vibration milling is carried out by nipping the particles between grinding media. Hence, vibration milling may be analyzed in the same manner as ball milling. I n the present paper, the analysis is tried only for the case when U 2 1, since the fractional filling, J , of the mill is usually so large that the fractional filling, U , of particles cannot be more than unity in most industrial vibration mills. Probability of Collision. As described above, the probability of collision, P,,is expressed by the product of fi and f z . They are determined geometrically from the values of particle size x and ball size d, as illustrated in Figure 9. fi = k: (36) where ki is a constant, and
,,d
x,'
Thus, the probability
,d,',
P,,is given by 2' k1r Pc = d
(38)
Number of Collisions in Unit Time. A collision occurs whenever the distance between centers of balls becomes as small as d. Hence, the number of collisions, 2, per ball in unit time is obtained from the number of balls whose centers are located within a volume enveloped by the movement of the sphere of radius d , in unit time. The number of balls, ns, in unit volume of a mill is given by nB
= 1.15 Jd
(39)
-3
If the balls in the mill are assumed to be in random motion, the relative velocity, u, of two balls is given by
u = (2.4)"'af = 0.9
ao
(40)
where is the amplitude, f is the frequency, and w is the angular velocity of vibration of the mill body. In addition, for a more exact calculation we must take into account the fact that balls in a vibrating mill are not necessarily in flight a t all times. Hence, letting 8 equal the flying time of the ball in one cycle of vibration, (Y
ANALYSIS OF VIBRATION MILLING Crushing Model
I t is difficult to know completely the internal mechanics of vibration milling because of its complexity. There is
y103 6=32 cm
a ball vibra-
EXPERIMENTAL(BOWDISH)
t"' 0 .l
0 X
V 0.2
0.3
x
d = 5.1 cm d -3.0cm d =3.2 cm
0.4
[cml
Figure 8. Selection function in relation to particle size with ball size as a parameter
170
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 2, 1970
we obtain the number of collisions, Z, from the abovementioned relationships.
Po,, = P"m
(44)
Final Correlations
By replacing the probabilities obtained above in the basic grinding equation-i.e., Equation 2-the final rate equations for vibration milling are obtained as follows: The factor % is introduced so that each collision will not be counted twice. The value of 8 is theoretically given as a complicated function of vibration intensity, ao2/g, as shown in Figure 10 (Kohata et al., 1967) and its value can be nearly equal to unity in the range where aw2/g > 3.3, though it fluctuates periodically near unity. Probability Relating to Crushing Strength. The probability, P o , relating to the crushing strength is defined by 1.5
gf
P" = (1 - --)
1 - 2v =urn= 0.0043
3
112
( 'pdl:?')
(42)
where v is Poisson's ratio and its value is about 0.25 for stone. Thus the degree of breakage, Pam, is
I t may be considered that all the particles receiving the impact action are broken because the fractional filling of particles, U , in a vibration mill is not more than unity, since the ball filling is usually 80 to 90% of the mill volume and the layer of particle is thinner. Consequently, the apparent degree of breakage, PaaP,over the whole particle layer in the crushing zone is immediately equal to that of the actually broken particle layer without consideration of the depth of ball penetration as in the case of ball milling. 1.0
0.8
0.6
-I
0.4
02
_.I
1
2
3
or
M
I n the case of vibration milling, it is reasonable to assume UL as the tensile strength of material instead of the compressive strength, because the single particle tends to undergo the force, independently of other particles. As in ball milling, the value of tensile stress generated within a particle a t the moment of force application is calculated from the following equation, gh
dS dt
- -
4
5
6
7
dS dt
__ =
0 . 6 k i K ' J 2 L D 2 d - 2 ~ ~+ O ~ o
These equations are applicable only when U S 1. Discussion
Effect of Mill Diameter on Milling Capacity. From Equation 45, it is obvious that the milling capacity of a vibration mill is proportional to the square of the mill diameter, assuming that the over-all increment of specific surface is constant for a given process. This also means that the rate of milling, dS/dt, is independent of mill diameter. This is generally supported by recent empirical facts reported by Jimbo (1963). Effect of Amplitude and Angular Velocity of Vibration on Rate of Milling. When all the variables other than CY and w are constant, Equation 45 becomes
or
dS dt
- a arwq
(47)
where c' is a constant. Equation 46 suggests that the values of exponents q and r in Equation 47 vary with physical properties and operating conditions. Figure 11 shows the results of plotting P,,,, the term in parentheses in Equation 46, against CYW for a certain value of c'. From this figure the value of Po, is found to be proportional to a power of CYW varying from 1 to 2, depending upon the value of CYW in the range corresponding to ordinary industrial grinding conditions. For 20 cm. per second < CYW < 30 cm. per second, for example, values of exponents q and r in Equation 47 are equal to 3, considering 8 = 1. As described before, 8 is a function of aw'lg, so that q and r sometimes have different values in a limited range, which seems to vary from 1 to 3. These facts derived from the present equation agree well with empirical correlations listed below. Rose and Sullivan (1961) :
ciw2/g [ -1
Figure 10. Relation between 0 and vibration intensity
g dt
cc
N 3w 3
when
(
2
4
ug;
" ) 0 5
> 2.5
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 2, 1970
(48) 171
c
I
I I
--- CALCULATED
5L / W
EXPERIMENTAL
4 L - A
\
(ROSE 8 SULLIVAN)
\
----
W
2 2
5 5 W [r W [r
I
0
1
l
1
l 0.5
0
u
I
01 20
d
=
ui =
k;
=
1
l
i
l
40 60 otw [ c m / s e c l
l
80
1
l
l
1 10
[-I
Figure 12. Effect of fractional filling of particles on milling rate a = 0.3 cm., w = 152 rad./sec., d = 1.5 cm.,
l
100
pB = 7.8 g.,,,/cc.,
Y = 1.9 X 10” g.f/sq. cm.,
u: = 7.1 X 104g.,/sq. cm., x = 0.1 cm.,
k(
= 0.07
Figure 11. Relation between Porn and ayw Y = 1.9 X lo8 g . f / s q . cm.,
1.5 cm., P H = 7.8 g.,,,/cc., 7.1 X 1 0 g . f / r q . cm., x 0.02
= 0.1 cm.,
U
= 1,
Jimbo (1963) roughly obtained from the graph reported by Bate1 (1958)
Nishitake (1967):
dS
- maw
dt
kao (wherein ( 7-S ) k = constant)
Masuda (1967):
2 dt
o:
J
- 2,2
2.5
Effect of Fractional Ball Filling on Rate of Milling. According to Equation 45, the rate of milling, dSldt, is proportional to the mill filling, J . However, a closer observation reveals that a t low mill filling balls travel not over the whole space of the mill but within a limited region. Hence, the actual number of collisions of balls is more than that calculated from Equation 41, which was derived from the assumption that balls travel over the whole mill. As a result, it seems that the rate of milling at this range of mill filling is higher than the calculated one. On the contrary, a t high mill filling, say J > 0.9, the mobility of balls is so restricted that the rate of milling decreases rapidly to zero. This also agrees with the data of Rose and Sullivan (1961). Effect of Fractional Filling of Particles on Rate of Milling. Equation 45 is plotted in Figure 12 to indicate the effect of fractional filling, U , of particles on the rate of milling, which compares well with the data of Rose and Sullivan (1961). Comparison of Milling Rates of Ball Mill and Vibration Mill. Equations 27 and 45, which represent the rates of ball milling and vibration milling, respectively, are solved by a numerical method of integration to give graphs of S us. t, remembering that x is inversely proportional to 172
Ind. Eng. Chern. Process Des. Develop., Vol. 9, No. 2, 1970
Figure 13. Comparison of milling rate of ball mill and vibration mill
S. Figure 13 shows the results thus calculated, under industrial milling conditions. Operating conditions are: For ball mill, D = 200 cm., d = 5 cm., p B = 7.8 g , m / cc., 4 = 0.7, and U = 1. For vibration mill, d = 1.5 cm., a = 0.3 cm., o = 152 rad./sec. (24 cyclesisec.), p B = 7.8 g.,/cc., J = 0.75, and U = 1. Physical properties are: Y = 1.9 x loa g.,/sq. cm., = 2.5 x lo4 x - l g.,/sq. cm., d = 7.7 x 10’ x - ’ g.f/ sq. cm., and pp = 2.7 g.,/cc. Furthermore, p = -0.5, Izl = 0.2 (ball mill), and hi = 0.03 (vibration mill) are assumed, considering the difference of particle dispersion in each type of mill. I n this case, the rate of vibration milling is about 10 times that of ball milling. This value and the shape of curves are acceptable from general experience (Rose and Sullivan, 1961). Conclusions
A theoretical analysis of the mechanism of ball and vibration milling used probabilities and was confirmed by various empirical facts.
The treatment described should lead not only to the development of grinding kinetics involving operational variables, but also to direct application of the comminution law to the design, development, and selection of a particular type of mill. Nomenclature
a A b c, c‘ d
D E
f f l
fi
g
gc gi gm
h
= constant = area of material undergoing force, sq. cm. = constant = constants = diameter of ball, cm. = diameter of mill, cm. = crushing energy to one particle, g., cm. = frequency of vibration, rad./sec. = ratio of number of particles under a ball to total number of particles per ball, dimensionless = ratio of number of particles nipped between ball and mill shell or balls themselves to particles under ball, dimensionless = acceleration due to gravity, cm./sq. sec. = gravitational conversion factor, g., cm./g., sq. sec. = grams force = grams mass = depth of layer between ball and mill shell or balls themselves, cm. depth of layer actually crushed, cm. fractional ball filling of mill, dimensionless constants
nZ =
constants length of material undergoing force, cm. length of mill, cm. weight of ball, g., total mass of particles in mill, g., number of particles per ball, dimensionless number of balls in unit volume of mill, cc.-’ number of particles per ball when U = 1, dimensionless number of particles per ball which undergo the force, dimensionless rotating speed of mill, set.-' critical speed of mill, set.-' selection function, set.-' probability of collision, dimensionless
N = N, = P(x) = P, = Pa, Pa,, Po, = probabilities relating to crushing strength, 4 =
r = s =
s=
S”= AS =
As” = As,’ = t = At =
TB = TBf = T, =
u= u =
dimensionless constant constant degree of slip, dimensionless specific surface of particles, sq. cm./g., specific surface of particles, sq. cm./cc. specific surface increment, sq. cm./g., specific surface increment, sq. cm./cc. specific surface increment per impact, sq. cm./cc. time, sec. short interval of time, sec. time for ball traverse, sec. time for ball traverse without slip, sec. time of one revolution of mill, sec. fractional filling of particles, dimensionless impact velocity of ball, cm./sec.
VB = peripheral speed of ball running along mill shell, cm./sec. peripheral speed of mill shell, cm./sec. x , xf = particle sizes, cm. Y = modulus of elasticity of materials, g.,/sq. cm. number of collisions per ball in unit time, sec. -‘
v, =
z=
GREEKLETTERS = d = t = 0 = e = (Y
A,
=
u = p ua
=
d
=
=
u,,, =
UL = = w
=
amplitude of vibration, cm. constant void fraction, dimensionless angle of nip, degree ratio of flying time of ball to one cycle of vibration, dimensionless mean coefficient of friction between ball and mill shell, dimensionless Poisson’s ratio of materials, dimensionless density, g.,/cc. compressive strength of particle, g.,/sq. cm. tensile strength of particle, g.,/sq. cm. compressive stress generated within a particle a t impact of ball, g.,/sq. cm. tensile stress generated within a particle at impact of ball, g.,/sq. cm. ratio of mill speed to critical one, dimensionless angular velocity of vibration, rad./sec.
SUBSCRIPTS
B
= ball p = particle
Literature Cited
Batel, W., Chem.-1ng.-Tech. 30, 567 (1958). Bowdish, F., Trans. A I M E 217, 194 (1960). Gross, J., “Crushing and Grinding,” U. S. Bur. Mines Tech. Rept. 402 (1938). Jimbo, G., “Recent Aspects of Chemical Engineering,” p. 150, Maruzen and Co., Tokyo, 1963. Kohata, T., Gotoh, K., Tanaka, T., Chem. Eng. (Japan) 31, 55 (1967). Masuda, T., Preprint, Session 8, Chemical Plant Conference, Tokyo, 1967. Mori, Y., Kojoh, K. Chem. Eng. (Japan) 18, 172 (1954). Nishitake, S., J . Soc. Material Sei. (Japan) 16, 337 (1967). Rose, H. E., Blunt, G. D., Proc. Inst. Mech. Engrs. (London) 171, 993 (1957). Rose, H. E., Sullivan, R. M. E., “Treatise on the Internal Mechanics of Ball, Tube and Rod Mills,” p. 49, Constable and Co., London, 1958. Rose, H. E., Sullivan, R. M. E., “Vibration Mills and 72, Constable and Co., Vibration Milling,” pp. 65 London, 1961. Suzuki, A., Tanaka, T., IND.ENG.CHEM.,PROCESS DESIGN 7, 161 (1968). DEVELOP. Taggart, A. F., “Handbook of Mineral Dressing,” 4th ed., pp. 5-27, Wiley, New York, 1950. Tamura, K., Tanaka, T., unpublished paper, 1969. Tanaka, T., IND.ENG.CHEM.PROCESS DESIGNDEVELOP. 5 , 3 5 3 (1966).
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RECEIVED for review October 20,1967 ACCEPTED May 24,1969
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 2, 1970
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