- Email: [email protected]
The effect of powder filling on selection and breakage functions in batch grinding
Powder Technology,
59 (1989) 275 - 283
275
The Effect of Powder Filling on Selection and Breakage Functions in Batch Grinding MING WE1 GAO and E. FORSSBERG Division of Mineral Processing, Lulea University of Technology,
S-951 87, Lulea (Sweden)
(Received April 27,1989)
SUMMARY
The effect of powder filling on selection and breakage functions is investigated in this paper. At three levels of powder filling, the selection functions are determined by onesize-fraction tests. The selection function is found to decrease rapidly with increasing powder filling. The breakage functions obtained by direct measurement are studied. When the powder filling is very low, the breakage function appears to be higher. Because of the non-normalization of the breakage function in this work, back-calculation method for determination of the breakage function is applied. It is found that for the non-normalized breakage function, if the selection function is known, it is possible to back-calculate a normalizable breakage function which gives the best simulation results. Finally, the effect of powder filling on product size distribution is studied. It is found that this effect can be totally compensated by adjusting the grinding time.
INTRODUCTION
The performance of a batch grinding process in a ball mill may be evaluated in terms of selection and breakage functions. These two concepts have led to the establishment of a mathematical model, known as the population balance model, for batch grinding:
d mitt)
~
dt
i-l
=-Simi(t)
+ x
b,iSjmj(t)
i=l
i>l
n>i>j>l
(1)
In eqn. (l), m,(t) is the mass fraction of particles in the ith size interval at grinding time t; oo32-5910/89/$3.50
Si is the selection function, or specific rate of breakage for the ith size interval with respect to grinding time; bij is the discretized breakage function which gives. the proportion of particles of size fraction j that reports to size fraction i after one breakage event. For the application of the population balance model, knowledge of the behaviour of the selection and breakage functions is of vital importance. The effect of mill powder filling on the selection and breakage functions is studied in this paper. By definition, the powder filling U is the fraction of the voids between the balls that is filled by ground material. It is represented by U = f,/O.U [l], f, and J being the fractional volume filling of the mill by powder and balls, respectively (based on a formal porosity of 0.4 for powder and ball bed). A number of researchers have reported their results for batch grinding nearly monosized materials at different powder fillings [ 2 - 51. Here, it was often found that the selection function decreased with increasing U. But the correlations proposed for describing the effect of powder filling on selection function were expressed differently. Shoji et al. [6] have shown that the slope of the linear part of the selection function was constant within a normal range of powder filling (for example from 0.3 to 2), and that the breakage function was essentially unaffected by the environment. However, in his work, the selection function for pure quartz was determined by the back-calculation technique. This is unreliable when the results were applied to study the selection function itself. Houillier [7] reported that as the powder charge increased, the breakage function was different. In his work, homogeneous quartz within the size interval -14 + 20 mesh was used. The choice of homogeneous materials @ Elsevier Sequoia/Printed in The Netherlands
276
in the work just mentioned apparently made it very difficult to understand the breakage behaviour of a complex ore. Furthermore, the conclusions drawn from testing one monosize material in most published work must be verified experimentally by using a sufficient number of size fractions, since the selection and breakage functions are both matrix quantities related to all the size intervals tested. Considering the problems mentioned above, a direct measurement procedure (called the one-size-fraction method) is used to determine the selection and breakage functions of a complex iron ore in this research. Different size intervals were tested at three levels of mill powder fillings.
EXPERIMENT
The mill used in this work had an internal diameter of 245 mm and a length of 380 mm. It was manufactured by the Outokumpu Company. The mill contains six welded steel liners of about 10 mm height. Steel balls of diameter 37 mm were used as the grinding medium and the mill was run at 56 rev/min, i.e., at 60.5% of the critical mill speed. The charge volume of the balls was chosen at 25% of the mill volume. The material used was an iron ore from Kirkenes concentrator, A/S Sydvaranger, Norway. The principal component of this iron ore is magnetite, with 20 - 30% of magnetic iron, and the main gangue is quartz. The experimental material was sieved into different size intervals from 13.33 mm down to 0.21 mm with an equally spaced interval of 0. The percentage of the monosize material in its corresponding fraction was at least over 85% to obtain reliable selection and breakage functions for that fraction. The standard experimental procedure for measuring the selection function is described in [S] . The monosize iron ore was batch ground and the remaining percentage of the initial size fraction was determined at different grinding times. This one-size-fraction test was conducted for different initial sizes until the S-values obtained at selected powder filling have contained a maximum value. Then the selection funtion for the whole range of the size distribution was deduced. To determine the breakage function experi-
mentally, the product size distribution for a short grinding time was obtained for each initial size fraction. During the experiment, the levels of powder filling studied were U = 0.1,0.5,1.0,1.7,respectively.
RESULTS
AND DISCUSSION
The effect of powder filling on selection function The breakage rate for size i can be defined by the equation Breakage rate to smaller size = Simi (t) W
(2)
where W is the total weight of powder. For the top size, i = 1 and eqn. (2) becomes
dm,(t) - dt = -Slm,(t) If S, is constant during the grinding process, i.e., the breakage is first order, then
h-h ml(t) = loglo w(O) Equation Figure obtained seen that grinding
2
(4)
(4) is linear on a linear-log scale. 1 shows the experimental results from one-size-fraction tests. It is within the range 0.1 < U < 1.7, the of Kirkenes iron ore with a size of
Feed size: -3.36+2.38
0
2
4 Grinding
6
8
mm
10
Time (minute)
Fig. 1. First-order plots of -3.36 + 2.38-mm monosize iron ore for U= 0.1, 0.5, 1.0 and 1.7 (0245 X 380smm mill, 37-mm steel balls, 25% charge volume, 60.5% of critical mill speed).
1. u : 0.5 2. " = 1.0 3. u : 1.7 Feed size: -0.295+0.21mm
0
2
4
6 8 10 Grinding Time (minute)
12
Fig. 2. Non-first-order breakage of -0.295 + 0.21mm monosize iron ore at U = 0.5,l.O and 1.7 (9245 X 380-mm mill, 37-mm steel balls, 25% charge volume, 60.5% of critical mill speed).
-3.36 + 2.38 mm follows a first-order process. However, the plots in Fig. 2 indicate that for a much finer initial size fraction such as -0.295 + 0.21 mm, the breakage process is no longer first order for short grinding times. It appears that the higher the powder filling, the more obvious the non-first-order breakage. This may be because the -0.295 + 0.21-mm feed is too fine to be ground efficiently by 37-mm grinding balls. When the ball drops into a bed of such a fine powder, little breakage occurs due to a cushioning effect. The appearance of non-first-order breakage at the begining of grinding may also be due partly to the heterogeneity of the material. The soft components or easily broken particles are quickly ground out of the top size interval in a short grinding period, and the breakage rate then decreases. To study the transition of the specific rate of breakage from a first-order to a non-firstorder process, Fig. 3 shows the results of batch grinding for six monosize fractions of the iron ore, ranging from 0.21 to 13.33 mm. Here it is seen that too fine or too coarse a particle size will both be broken down in nonfirst-order process. However, if the ratio of ball size to top feed size is appropriate around 11 - 22 in this case), the first-order breakage process dominates. In the solution
0
5
10
15
20
2
Grinding Time (minute)
Fig. 3. Feed disappearance plots for different monosizes of iron ore (0245 X 380-mm mill, 37-mm steel balls, 25% charge volume, 60.5% of critical mill speed).
to eqn. (1) [9], the selection function is required to be time-independent. The nonfirst-order breakage of heterogeneous materials such as the iron ore studied must, therefore, be avoided. In such a situation, a pseudo time-independent selection function determined by linear regression may be applied, since the correlation coefficients for the plots in Fig. 3 all exceed 0.99, as can be seen from Table 1. Thus, the specific rates of TABLE
1
Specific rates of breakage at U = 1 .O for those size intervals in Fig. 3 (0245 X 380-mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed)
Size interval (mm)
Specific rate of breakage (min-‘)
Correlation coefficient
-13.33 + 9.5 -6.7 + 4.75 -4.75 + 3.36 -1.68 + 1.19 -0.595 + 0.42 -0.295 + 0.21
0.0439 0.1733 0.2459 0.3001 0.1419 0.0762
-0.9817 -0.9992 -0.9998 -0.9996 -0.9995 -0.9980
1.
-3.36+2.38
2.
-0.595+0.42
mm
3.
-0.295+0.21
mm
0
00
mm
Experimental Estimated
0
0
0
Measured
I
I
I
I
0 Powder 0.01
I 0.01
I
I 11111~
I
I I lllll~
0.1
I 1.0
Partuzle
Sire
I I I lrl 1
(mm)
Fig. 4. Selection functions for different powder fillings (8245 x 380-mm mill, 37-mm steel balls, 25% charge volume, 60.5% of critical mill speed).
breakage measured by one-size-fraction tests will be treated as a constant in the following discussion. Figure 4 illustrates the selection functions obtained at powder fillings of 0.5, 1.0 and 1.7. It is seen that the increase in selection function with increasing powder filling is almost linear for the entire particle size range, and the slope of the linear part of each plot remains constant. This agrees with Shoji et al.3 result [6]. It can also be seen from Fig. 4 that the particle size at which the maximum value of the selection function occurs does not change appreciably with the powder filling. For all of three levels of U studied, it remains at around 1.8 - 2.5 mm. According to the form of the equation for the selection function proposed by Austin et al. [8], the following formula was found to describe the variation in the selection function with powder filling from U = 0.5 to 1.7: si = 0.291 u- 1.22 XiO.SS
(5)
I
I
I
Filling
2 I_
where Xi is the lower limit of size interval i in millimetres. Figure 5 demonstrates a more clear relationship between selection function and powder filling. It is seen that when U is reduced to a very low value, such as 0.1, the increase in specific rate of breakage is so large that it appears that it will become unbounded. By regression, an empirical equation can be found to fit the plots in Fig. 5. This may be written in the form Si = Xi U+i
(minute))’
(6)
where X and K are sizedependent parameters. Their values for several particle sizes are given in Table 2. The relationship between the absolute rate of breakage SW (g/min) uersus powder filling has been studied in [7] and [lo]. In general, it was concluded that the absolute rate of TABLE
2
Values of h and
-3.36 -0.595 -0.295
(minute)-’
I
Fig. 5. Variation in specific rate of breakage with powder filling for different size intervals (0245 x 380mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
Particle
1
1 1
K
in eqn. (6)
size + 2.38 mm + 0.42 mm + 0.21 mm
h
K
0.310 0.149 0.087
0.722 1.045 0.940
219
The effect of powder filling on breakage function The breakage function is defined as the primary progeny fragment distribution. Having obtained the S-values for other size intervals smaller than the top size, the breakage function can be determined by the zero-order production law of fines in comminution [ 111, The zero-order production relationship for each particle size interval in a mill discharge may be written dPi(t) dt
0
2
1 Powder
Filling
U
Fig. 6. Variation in absolute rate of breakage with powder filling for different size intervals (0245 X 380. mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
& Fi
(7)
where Pi(t) is the cumulative mass fraction sm_allerthan the size interval i at grinding time t. Fi is the constant zero-order rate for the production of fines smaller than size interval i. It was found that for short grinding times, the selection and breakage functions may be related in the equation BijSj = Fi
(8)
Thus, the breakage function Bij can be culated from the equation breakage SW has a maximum corresponding to the powder filling at which all of the grinding zone within a mill is utilized. It will decrease either below or beyond this optimum powder filling. However, the results shown in Fig. 6 demonstrate that only the coarse particles behave in the manner just mentioned, such as -3.36 + 2.38 mm and -1.69 + 1.19 mm. As the top size of grinding is reduced, the plot of absolute breakage rate SW versus U becomes progressively flatter, and, finally, the absolute rate of breakage appears to be inversely proportional to the powder filling for the -0.295 + 0.21-mm size fraction. The results presented here reveal that an optimum value of the absolute rate of breakage only exists for coarse particle sizes. For fine particle sizes, the relationship between the absolute rate of breakage and powder filling becomes inversely proportional, i.e., the lower the powder filling, the higher the absolute rate of breakage. The reason for this different behaviour may be related to the effect of particle size. For coarse particles, the absolute rate of breakage is dominated by the fraction of total grinding zone in a mill that is utilized. However, for fine particles, this may be determined mainly by the ball-ore contact frequency.
B,,
=
II
1 dPi(t) Si
dt
d-
(9)
Using eqn. (9), the breakage functions with an initial size interval of -3.36 + 2.38 mm are calculated at different powder fillings. The results are plotted in Fig. 7. It is seen that as the powder filling decreases, the breakage function increases. However, at higher powder fillings, such as from 1.0 to 1.7, the breakage function from grinding -3.36 + 2.38mm size iron ore changes very little with changing powder filling. It may be inferred from these results that the dramatic change in the grinding mechanism will alter the breakage function. In the case of high powder filling, there is sufficient material between the grinding balls to prevent them from contacting each other more directly. The grinding mechanism may include the abrasion between the particles and the impact given by the balls. But when the powder filling is reduced to a very low level, such as 0.5 or 0.1, the particles are ground by strong impact from direct contact between the balls. A large quantity of fragments is therefore produced during the primary breakage event. Figure 8 shows the breakage functions with a much smaller initial size of -0.595 + 0.42
280
Initial Sue:
0.01
I 0.01
I I Illll~
-3.36+2.38 mm
I 0.1
I I IIIIII
I 1.0
I I I III 10.0
Particle Size (mm)
Fig. 7. Breakage functions of Kirkenes iron ore for U=0.1,0.5,1.0and1.7withaninitialsizeof-3.36+ 2.38 mm (0245 X 380-mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
mm. The difference between the breakage functions is clearly reduced, and the breakage functions obtained here are much higher than those in Fig. 7. These results indicate that the fine and coarse particles of Kirkenes iron ore have different breakage behaviours. We can see this more clearly from Figs. 9,10, and 11. Apparently, the breakage function for Kirkenes iron ore is non-normalized. The breakage function changes with particle size. This may be due to heterogeneity of the material. A bigger iron ore particle has more flaws and cracks in it and when it undergoes a primary breakage event, the quantity of the daughter products produced by coarse particles will be determined principally by these flaws and cracks. However, for the smaller sizes, such as -0.595 + 0.42 mm, there are far fewer flaws and cracks available for breakage, and the grain boundary or crushed grains will dominate the generation of daughter products. Once the smaller particles are broken, a higher breakage function will therefore be obtained. Apart from a time-independent selection function, the solution to eqn. (1) also requires a normalized breakage function, i.e., B,j = However due to the heterogeneity of Bi-j+l,l* 1.0
Initial Size 1. -.5.7+4.75 mm 2. -3.36+2.38 mm 3. -1.68+1.19 mm initial Size: -0.595+0.42 mm
0.1
1.0
4. -0.595+0.42 mm
10.0
Particle Size (mm)
Fig. 8. Breakage functions of Kirkenes iron ore for U = 0.5, 1.0 and 1.7 with an initial size of -0.595 + 0.42 mm (9245 X 380-mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
0.1 Relative Particle Size
1.0 Xi/Xl
Fig. 9. Breakage functions with different initial sizes of Kirkenes iron ore for U = 0.5 (0245 x 380-mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
Initial Initial 7
-
1.
-6.7+4.75
mm
2.
-3.36+2.x
mm
3.
-1.60+1.19
mm
4.
-0.595+0.42
mm
o,ol I
I I Illll~
I
I I lllll(
I
Relative
Particle
Size
1
I
0.01
I 0.01
,
/
1.
-6.7+4.75
2
2.
-3.Jb2.30
mm
3.
-1.68+1.19
mm
4.
-0.595+0.42
I
II11111
mm
I 1.0
0.1 Relative
mm
I111111
Particle
Sire
1.
-6.1+4.75
2.
-3.36+2.30
mm mm
3.
-1.68+1.19
mm
I I
I lllln
0.1 Relative
,
1
2
1 ,,:-;.:,‘1,:[: 0.01
Xi/Xl
Fig. 1G. Breakage functions with different initial sizes of Kirkenes iron ore for U = 1.0 (g245 X 380-mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
1.0
I
I lllll
110
0.1
1
Size
Size
I Illlrl 10.0
Xi/Xl
Fig. 11. Breakage functions with different initial sizes of Kirkenes iron ore for U = 1.7 (9245 X 380-mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
1.0 Particle
Size
II .O
Xl/Xl
Fig. 12. Measured and back-calculated breakage functions of Kirkenes iron ore for U = 1.0 (0245 x 380mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
the iron ore, this requirement is not satisfied here. It is ‘not realistic to measure the breakage functions for all the size intervals. Back-calculation of the breakage function therefore appears to be necessary for simulation of the experimental product size distributions. After many simulation trials, it was found that it is always possible to determine one normalized breakage function by backcalculation if the selection function is known. Figure 12 shows the normalizable breakage function back-calculated at a powder filling of 1.0. The back-calculation was performed by forcing the simulated product size distributions to fit those obtained experimentally. It is seen that the back-calculated breakage function is approximately equal to the average of the breakage functions of all the size intervals studied. The simulation results thus obtained are demonstrated in Fig. 13. The effect of powder filling on product size distribution Another interesting phenomenon was found by comparing the results obtained from two batch grinding tests performed at U = 0.7 and 1.4. In Fig. 14, the similarity between the product size distributions obtained at the
Feed .
11
I I Illll~ 0.01
I
Simulated .
s Experimental
I I lllll~
0.1
I
I Ill1 10.0
1.0 Particle Size (mm)
Fig. 13. Simulated and measured product size distributions of batch grinding tests for U = 1.0 (0245 X 380-mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
1. -
2
3
8_8
1 min
kO.7
++
2 min
k1.4
+r3
3 min
u.o.7
+-I-
6 ml"
kl.4
5 min
u=o.7
10 min
lk1.4
M 4
I
1
0.01
++
I
1111111
Feed
I I lllll~
0.1
I 1.0
I I III 10.0
Particle Sure (mm)
Fig. 14. Product size distributions of batch grinding tests for U = 0.7 and 1.4 (8245 X 380-mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
same ratio of grinding time to powder filling is appealing. It appears that the effect of powder filling on the product size distribution can be compensated by the grinding time. Low powder fillings and short grinding times will produce product size distributions which closely resemble those obtained at high powder fillings and long grinding times, so long as the ratio of t to U is maintained constant. This will greatly facilitate evaluation of the effect of powder filling on the grinding process since only grinding time is related to this. In the case of continuous grinding, powder filling and grinding time may be related to the feed rate and residence time. It can therefore be inferred that to obtain a stable product size which is prescribed in advance, it is the optimum ratio of feed rate to residence time that must be always controlled, rather than these parameters individually.
CONCLUSIONS (1) In the range of 0.1< U < 1.7, the firstorder breakage process occurs for the coarse iron ore particle sizes. However, for finer sizes, non-fir&order breakage occurs at short grinding times. The transition from first-order to non-first order breakage depends mainly on the ratio of the ball size to the top particle size in feed. (2) The selection function decreases rapidly with an increase in powder filling. However, the slope of the linear part of the selection function appears to remain constant. (3) The effect of powder filling U on the absolute rate of breakage varies with particle size. For coarse particles, there exists a value of U at which the absolute rate of breakage reaches a maximum value. However, for very fine particles, an inversely proportional relation exists between the absolute rate of breakage and powder filling. (4) The breakage function for Kirkenes iron ore with a coarse initial size increases with decreasing powder filling. But for smaller size intervals, such as -0.595 + 0.42 mm considered in this study, the breakage function is nearly independent of powder filling. (5) The breakage function for a heterogeneous material is difficult to determine by directly experimental methods. The employ-
283
ment of the back-calculation method appears to be necessary. For non-normalized breakage functions, a normalizable breakage function for all the sizes (which gives the best simulation results) can be determined by forcing the simulated mill discharge to fit the experimental results. (6) The effect of powder filling on product size distribution can be compensated by adjusting grinding time in batch grinding. In the case of a high powder filling, a longer grinding time will produce the same size distribution as that obtained at a low powder filling and a short grinding time so long as the ratio of grinding time to powder filling is maintained constant.
ACKNOWLEDGEMENT
This work was conducted under the financial support from the Swedish Mineral Processing Research Foundation. A/S Sydvaranger
is thanked material.
for supplying
the experimental
REFERENCES 1 L. G. Austin, R. R. Klimpel and P. T. Luckie, Process Engineering of Size Reduction: Ball Milling, AIME, New York, 1984, p. 18. T. S. Mika, L. M. Berlioz and D. W. Fuerstenau, Dechema Monograph., 57 (1967) 205. D. F. Kelsall, K. J. Reid and C. J. Restarick, Powder Technol., 2 (1968/69) 162. V. K. Gupta and P. C. Kapur, Powder Technoi., 10 (1974) 217. R. L. Houillier and J. C. Marchand, Powder Technol., 14 (1976) 71. K. Shoji, S. Lohrasb and L. G. Austin, Powder Technol., 25 (1980) 109. R. L. Houillier, A. V. Neste and J. C. Marchand, Powder Technol., 16 (1977) 7. L. G. Austin, R. R. Klimpel and P. T. Luckie, Process Engineering of Size Reduction: Ball Milling, AIME, New York, 1984, p. 181. 9 K. J. Reid, Chem. Eng. Sci., 20 (1965) 953. 10 K. Shoji, L. G. Austin, F. Smaila, K. Brame and P. T. Luckie, Powder Technol., 31 (1982) 121. 11 J. A. Herbst and D. W. Fuerestenau, Trans. AIME, 241 (1968) 538.
59 (1989) 275 - 283
275
The Effect of Powder Filling on Selection and Breakage Functions in Batch Grinding MING WE1 GAO and E. FORSSBERG Division of Mineral Processing, Lulea University of Technology,
S-951 87, Lulea (Sweden)
(Received April 27,1989)
SUMMARY
The effect of powder filling on selection and breakage functions is investigated in this paper. At three levels of powder filling, the selection functions are determined by onesize-fraction tests. The selection function is found to decrease rapidly with increasing powder filling. The breakage functions obtained by direct measurement are studied. When the powder filling is very low, the breakage function appears to be higher. Because of the non-normalization of the breakage function in this work, back-calculation method for determination of the breakage function is applied. It is found that for the non-normalized breakage function, if the selection function is known, it is possible to back-calculate a normalizable breakage function which gives the best simulation results. Finally, the effect of powder filling on product size distribution is studied. It is found that this effect can be totally compensated by adjusting the grinding time.
INTRODUCTION
The performance of a batch grinding process in a ball mill may be evaluated in terms of selection and breakage functions. These two concepts have led to the establishment of a mathematical model, known as the population balance model, for batch grinding:
d mitt)
~
dt
i-l
=-Simi(t)
+ x
b,iSjmj(t)
i=l
i>l
n>i>j>l
(1)
In eqn. (l), m,(t) is the mass fraction of particles in the ith size interval at grinding time t; oo32-5910/89/$3.50
Si is the selection function, or specific rate of breakage for the ith size interval with respect to grinding time; bij is the discretized breakage function which gives. the proportion of particles of size fraction j that reports to size fraction i after one breakage event. For the application of the population balance model, knowledge of the behaviour of the selection and breakage functions is of vital importance. The effect of mill powder filling on the selection and breakage functions is studied in this paper. By definition, the powder filling U is the fraction of the voids between the balls that is filled by ground material. It is represented by U = f,/O.U [l], f, and J being the fractional volume filling of the mill by powder and balls, respectively (based on a formal porosity of 0.4 for powder and ball bed). A number of researchers have reported their results for batch grinding nearly monosized materials at different powder fillings [ 2 - 51. Here, it was often found that the selection function decreased with increasing U. But the correlations proposed for describing the effect of powder filling on selection function were expressed differently. Shoji et al. [6] have shown that the slope of the linear part of the selection function was constant within a normal range of powder filling (for example from 0.3 to 2), and that the breakage function was essentially unaffected by the environment. However, in his work, the selection function for pure quartz was determined by the back-calculation technique. This is unreliable when the results were applied to study the selection function itself. Houillier [7] reported that as the powder charge increased, the breakage function was different. In his work, homogeneous quartz within the size interval -14 + 20 mesh was used. The choice of homogeneous materials @ Elsevier Sequoia/Printed in The Netherlands
276
in the work just mentioned apparently made it very difficult to understand the breakage behaviour of a complex ore. Furthermore, the conclusions drawn from testing one monosize material in most published work must be verified experimentally by using a sufficient number of size fractions, since the selection and breakage functions are both matrix quantities related to all the size intervals tested. Considering the problems mentioned above, a direct measurement procedure (called the one-size-fraction method) is used to determine the selection and breakage functions of a complex iron ore in this research. Different size intervals were tested at three levels of mill powder fillings.
EXPERIMENT
The mill used in this work had an internal diameter of 245 mm and a length of 380 mm. It was manufactured by the Outokumpu Company. The mill contains six welded steel liners of about 10 mm height. Steel balls of diameter 37 mm were used as the grinding medium and the mill was run at 56 rev/min, i.e., at 60.5% of the critical mill speed. The charge volume of the balls was chosen at 25% of the mill volume. The material used was an iron ore from Kirkenes concentrator, A/S Sydvaranger, Norway. The principal component of this iron ore is magnetite, with 20 - 30% of magnetic iron, and the main gangue is quartz. The experimental material was sieved into different size intervals from 13.33 mm down to 0.21 mm with an equally spaced interval of 0. The percentage of the monosize material in its corresponding fraction was at least over 85% to obtain reliable selection and breakage functions for that fraction. The standard experimental procedure for measuring the selection function is described in [S] . The monosize iron ore was batch ground and the remaining percentage of the initial size fraction was determined at different grinding times. This one-size-fraction test was conducted for different initial sizes until the S-values obtained at selected powder filling have contained a maximum value. Then the selection funtion for the whole range of the size distribution was deduced. To determine the breakage function experi-
mentally, the product size distribution for a short grinding time was obtained for each initial size fraction. During the experiment, the levels of powder filling studied were U = 0.1,0.5,1.0,1.7,respectively.
RESULTS
AND DISCUSSION
The effect of powder filling on selection function The breakage rate for size i can be defined by the equation Breakage rate to smaller size = Simi (t) W
(2)
where W is the total weight of powder. For the top size, i = 1 and eqn. (2) becomes
dm,(t) - dt = -Slm,(t) If S, is constant during the grinding process, i.e., the breakage is first order, then
h-h ml(t) = loglo w(O) Equation Figure obtained seen that grinding
2
(4)
(4) is linear on a linear-log scale. 1 shows the experimental results from one-size-fraction tests. It is within the range 0.1 < U < 1.7, the of Kirkenes iron ore with a size of
Feed size: -3.36+2.38
0
2
4 Grinding
6
8
mm
10
Time (minute)
Fig. 1. First-order plots of -3.36 + 2.38-mm monosize iron ore for U= 0.1, 0.5, 1.0 and 1.7 (0245 X 380smm mill, 37-mm steel balls, 25% charge volume, 60.5% of critical mill speed).
1. u : 0.5 2. " = 1.0 3. u : 1.7 Feed size: -0.295+0.21mm
0
2
4
6 8 10 Grinding Time (minute)
12
Fig. 2. Non-first-order breakage of -0.295 + 0.21mm monosize iron ore at U = 0.5,l.O and 1.7 (9245 X 380-mm mill, 37-mm steel balls, 25% charge volume, 60.5% of critical mill speed).
-3.36 + 2.38 mm follows a first-order process. However, the plots in Fig. 2 indicate that for a much finer initial size fraction such as -0.295 + 0.21 mm, the breakage process is no longer first order for short grinding times. It appears that the higher the powder filling, the more obvious the non-first-order breakage. This may be because the -0.295 + 0.21-mm feed is too fine to be ground efficiently by 37-mm grinding balls. When the ball drops into a bed of such a fine powder, little breakage occurs due to a cushioning effect. The appearance of non-first-order breakage at the begining of grinding may also be due partly to the heterogeneity of the material. The soft components or easily broken particles are quickly ground out of the top size interval in a short grinding period, and the breakage rate then decreases. To study the transition of the specific rate of breakage from a first-order to a non-firstorder process, Fig. 3 shows the results of batch grinding for six monosize fractions of the iron ore, ranging from 0.21 to 13.33 mm. Here it is seen that too fine or too coarse a particle size will both be broken down in nonfirst-order process. However, if the ratio of ball size to top feed size is appropriate around 11 - 22 in this case), the first-order breakage process dominates. In the solution
0
5
10
15
20
2
Grinding Time (minute)
Fig. 3. Feed disappearance plots for different monosizes of iron ore (0245 X 380-mm mill, 37-mm steel balls, 25% charge volume, 60.5% of critical mill speed).
to eqn. (1) [9], the selection function is required to be time-independent. The nonfirst-order breakage of heterogeneous materials such as the iron ore studied must, therefore, be avoided. In such a situation, a pseudo time-independent selection function determined by linear regression may be applied, since the correlation coefficients for the plots in Fig. 3 all exceed 0.99, as can be seen from Table 1. Thus, the specific rates of TABLE
1
Specific rates of breakage at U = 1 .O for those size intervals in Fig. 3 (0245 X 380-mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed)
Size interval (mm)
Specific rate of breakage (min-‘)
Correlation coefficient
-13.33 + 9.5 -6.7 + 4.75 -4.75 + 3.36 -1.68 + 1.19 -0.595 + 0.42 -0.295 + 0.21
0.0439 0.1733 0.2459 0.3001 0.1419 0.0762
-0.9817 -0.9992 -0.9998 -0.9996 -0.9995 -0.9980
1.
-3.36+2.38
2.
-0.595+0.42
mm
3.
-0.295+0.21
mm
0
00
mm
Experimental Estimated
0
0
0
Measured
I
I
I
I
0 Powder 0.01
I 0.01
I
I 11111~
I
I I lllll~
0.1
I 1.0
Partuzle
Sire
I I I lrl 1
(mm)
Fig. 4. Selection functions for different powder fillings (8245 x 380-mm mill, 37-mm steel balls, 25% charge volume, 60.5% of critical mill speed).
breakage measured by one-size-fraction tests will be treated as a constant in the following discussion. Figure 4 illustrates the selection functions obtained at powder fillings of 0.5, 1.0 and 1.7. It is seen that the increase in selection function with increasing powder filling is almost linear for the entire particle size range, and the slope of the linear part of each plot remains constant. This agrees with Shoji et al.3 result [6]. It can also be seen from Fig. 4 that the particle size at which the maximum value of the selection function occurs does not change appreciably with the powder filling. For all of three levels of U studied, it remains at around 1.8 - 2.5 mm. According to the form of the equation for the selection function proposed by Austin et al. [8], the following formula was found to describe the variation in the selection function with powder filling from U = 0.5 to 1.7: si = 0.291 u- 1.22 XiO.SS
(5)
I
I
I
Filling
2 I_
where Xi is the lower limit of size interval i in millimetres. Figure 5 demonstrates a more clear relationship between selection function and powder filling. It is seen that when U is reduced to a very low value, such as 0.1, the increase in specific rate of breakage is so large that it appears that it will become unbounded. By regression, an empirical equation can be found to fit the plots in Fig. 5. This may be written in the form Si = Xi U+i
(minute))’
(6)
where X and K are sizedependent parameters. Their values for several particle sizes are given in Table 2. The relationship between the absolute rate of breakage SW (g/min) uersus powder filling has been studied in [7] and [lo]. In general, it was concluded that the absolute rate of TABLE
2
Values of h and
-3.36 -0.595 -0.295
(minute)-’
I
Fig. 5. Variation in specific rate of breakage with powder filling for different size intervals (0245 x 380mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
Particle
1
1 1
K
in eqn. (6)
size + 2.38 mm + 0.42 mm + 0.21 mm
h
K
0.310 0.149 0.087
0.722 1.045 0.940
219
The effect of powder filling on breakage function The breakage function is defined as the primary progeny fragment distribution. Having obtained the S-values for other size intervals smaller than the top size, the breakage function can be determined by the zero-order production law of fines in comminution [ 111, The zero-order production relationship for each particle size interval in a mill discharge may be written dPi(t) dt
0
2
1 Powder
Filling
U
Fig. 6. Variation in absolute rate of breakage with powder filling for different size intervals (0245 X 380. mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
& Fi
(7)
where Pi(t) is the cumulative mass fraction sm_allerthan the size interval i at grinding time t. Fi is the constant zero-order rate for the production of fines smaller than size interval i. It was found that for short grinding times, the selection and breakage functions may be related in the equation BijSj = Fi
(8)
Thus, the breakage function Bij can be culated from the equation breakage SW has a maximum corresponding to the powder filling at which all of the grinding zone within a mill is utilized. It will decrease either below or beyond this optimum powder filling. However, the results shown in Fig. 6 demonstrate that only the coarse particles behave in the manner just mentioned, such as -3.36 + 2.38 mm and -1.69 + 1.19 mm. As the top size of grinding is reduced, the plot of absolute breakage rate SW versus U becomes progressively flatter, and, finally, the absolute rate of breakage appears to be inversely proportional to the powder filling for the -0.295 + 0.21-mm size fraction. The results presented here reveal that an optimum value of the absolute rate of breakage only exists for coarse particle sizes. For fine particle sizes, the relationship between the absolute rate of breakage and powder filling becomes inversely proportional, i.e., the lower the powder filling, the higher the absolute rate of breakage. The reason for this different behaviour may be related to the effect of particle size. For coarse particles, the absolute rate of breakage is dominated by the fraction of total grinding zone in a mill that is utilized. However, for fine particles, this may be determined mainly by the ball-ore contact frequency.
B,,
=
II
1 dPi(t) Si
dt
d-
(9)
Using eqn. (9), the breakage functions with an initial size interval of -3.36 + 2.38 mm are calculated at different powder fillings. The results are plotted in Fig. 7. It is seen that as the powder filling decreases, the breakage function increases. However, at higher powder fillings, such as from 1.0 to 1.7, the breakage function from grinding -3.36 + 2.38mm size iron ore changes very little with changing powder filling. It may be inferred from these results that the dramatic change in the grinding mechanism will alter the breakage function. In the case of high powder filling, there is sufficient material between the grinding balls to prevent them from contacting each other more directly. The grinding mechanism may include the abrasion between the particles and the impact given by the balls. But when the powder filling is reduced to a very low level, such as 0.5 or 0.1, the particles are ground by strong impact from direct contact between the balls. A large quantity of fragments is therefore produced during the primary breakage event. Figure 8 shows the breakage functions with a much smaller initial size of -0.595 + 0.42
280
Initial Sue:
0.01
I 0.01
I I Illll~
-3.36+2.38 mm
I 0.1
I I IIIIII
I 1.0
I I I III 10.0
Particle Size (mm)
Fig. 7. Breakage functions of Kirkenes iron ore for U=0.1,0.5,1.0and1.7withaninitialsizeof-3.36+ 2.38 mm (0245 X 380-mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
mm. The difference between the breakage functions is clearly reduced, and the breakage functions obtained here are much higher than those in Fig. 7. These results indicate that the fine and coarse particles of Kirkenes iron ore have different breakage behaviours. We can see this more clearly from Figs. 9,10, and 11. Apparently, the breakage function for Kirkenes iron ore is non-normalized. The breakage function changes with particle size. This may be due to heterogeneity of the material. A bigger iron ore particle has more flaws and cracks in it and when it undergoes a primary breakage event, the quantity of the daughter products produced by coarse particles will be determined principally by these flaws and cracks. However, for the smaller sizes, such as -0.595 + 0.42 mm, there are far fewer flaws and cracks available for breakage, and the grain boundary or crushed grains will dominate the generation of daughter products. Once the smaller particles are broken, a higher breakage function will therefore be obtained. Apart from a time-independent selection function, the solution to eqn. (1) also requires a normalized breakage function, i.e., B,j = However due to the heterogeneity of Bi-j+l,l* 1.0
Initial Size 1. -.5.7+4.75 mm 2. -3.36+2.38 mm 3. -1.68+1.19 mm initial Size: -0.595+0.42 mm
0.1
1.0
4. -0.595+0.42 mm
10.0
Particle Size (mm)
Fig. 8. Breakage functions of Kirkenes iron ore for U = 0.5, 1.0 and 1.7 with an initial size of -0.595 + 0.42 mm (9245 X 380-mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
0.1 Relative Particle Size
1.0 Xi/Xl
Fig. 9. Breakage functions with different initial sizes of Kirkenes iron ore for U = 0.5 (0245 x 380-mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
Initial Initial 7
-
1.
-6.7+4.75
mm
2.
-3.36+2.x
mm
3.
-1.60+1.19
mm
4.
-0.595+0.42
mm
o,ol I
I I Illll~
I
I I lllll(
I
Relative
Particle
Size
1
I
0.01
I 0.01
,
/
1.
-6.7+4.75
2
2.
-3.Jb2.30
mm
3.
-1.68+1.19
mm
4.
-0.595+0.42
I
II11111
mm
I 1.0
0.1 Relative
mm
I111111
Particle
Sire
1.
-6.1+4.75
2.
-3.36+2.30
mm mm
3.
-1.68+1.19
mm
I I
I lllln
0.1 Relative
,
1
2
1 ,,:-;.:,‘1,:[: 0.01
Xi/Xl
Fig. 1G. Breakage functions with different initial sizes of Kirkenes iron ore for U = 1.0 (g245 X 380-mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
1.0
I
I lllll
110
0.1
1
Size
Size
I Illlrl 10.0
Xi/Xl
Fig. 11. Breakage functions with different initial sizes of Kirkenes iron ore for U = 1.7 (9245 X 380-mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
1.0 Particle
Size
II .O
Xl/Xl
Fig. 12. Measured and back-calculated breakage functions of Kirkenes iron ore for U = 1.0 (0245 x 380mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
the iron ore, this requirement is not satisfied here. It is ‘not realistic to measure the breakage functions for all the size intervals. Back-calculation of the breakage function therefore appears to be necessary for simulation of the experimental product size distributions. After many simulation trials, it was found that it is always possible to determine one normalized breakage function by backcalculation if the selection function is known. Figure 12 shows the normalizable breakage function back-calculated at a powder filling of 1.0. The back-calculation was performed by forcing the simulated product size distributions to fit those obtained experimentally. It is seen that the back-calculated breakage function is approximately equal to the average of the breakage functions of all the size intervals studied. The simulation results thus obtained are demonstrated in Fig. 13. The effect of powder filling on product size distribution Another interesting phenomenon was found by comparing the results obtained from two batch grinding tests performed at U = 0.7 and 1.4. In Fig. 14, the similarity between the product size distributions obtained at the
Feed .
11
I I Illll~ 0.01
I
Simulated .
s Experimental
I I lllll~
0.1
I
I Ill1 10.0
1.0 Particle Size (mm)
Fig. 13. Simulated and measured product size distributions of batch grinding tests for U = 1.0 (0245 X 380-mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
1. -
2
3
8_8
1 min
kO.7
++
2 min
k1.4
+r3
3 min
u.o.7
+-I-
6 ml"
kl.4
5 min
u=o.7
10 min
lk1.4
M 4
I
1
0.01
++
I
1111111
Feed
I I lllll~
0.1
I 1.0
I I III 10.0
Particle Sure (mm)
Fig. 14. Product size distributions of batch grinding tests for U = 0.7 and 1.4 (8245 X 380-mm mill, 25% charge volume, 37-mm steel balls, 60.5% of critical mill speed).
same ratio of grinding time to powder filling is appealing. It appears that the effect of powder filling on the product size distribution can be compensated by the grinding time. Low powder fillings and short grinding times will produce product size distributions which closely resemble those obtained at high powder fillings and long grinding times, so long as the ratio of t to U is maintained constant. This will greatly facilitate evaluation of the effect of powder filling on the grinding process since only grinding time is related to this. In the case of continuous grinding, powder filling and grinding time may be related to the feed rate and residence time. It can therefore be inferred that to obtain a stable product size which is prescribed in advance, it is the optimum ratio of feed rate to residence time that must be always controlled, rather than these parameters individually.
CONCLUSIONS (1) In the range of 0.1< U < 1.7, the firstorder breakage process occurs for the coarse iron ore particle sizes. However, for finer sizes, non-fir&order breakage occurs at short grinding times. The transition from first-order to non-first order breakage depends mainly on the ratio of the ball size to the top particle size in feed. (2) The selection function decreases rapidly with an increase in powder filling. However, the slope of the linear part of the selection function appears to remain constant. (3) The effect of powder filling U on the absolute rate of breakage varies with particle size. For coarse particles, there exists a value of U at which the absolute rate of breakage reaches a maximum value. However, for very fine particles, an inversely proportional relation exists between the absolute rate of breakage and powder filling. (4) The breakage function for Kirkenes iron ore with a coarse initial size increases with decreasing powder filling. But for smaller size intervals, such as -0.595 + 0.42 mm considered in this study, the breakage function is nearly independent of powder filling. (5) The breakage function for a heterogeneous material is difficult to determine by directly experimental methods. The employ-
283
ment of the back-calculation method appears to be necessary. For non-normalized breakage functions, a normalizable breakage function for all the sizes (which gives the best simulation results) can be determined by forcing the simulated mill discharge to fit the experimental results. (6) The effect of powder filling on product size distribution can be compensated by adjusting grinding time in batch grinding. In the case of a high powder filling, a longer grinding time will produce the same size distribution as that obtained at a low powder filling and a short grinding time so long as the ratio of grinding time to powder filling is maintained constant.
ACKNOWLEDGEMENT
This work was conducted under the financial support from the Swedish Mineral Processing Research Foundation. A/S Sydvaranger
is thanked material.
for supplying
the experimental
REFERENCES 1 L. G. Austin, R. R. Klimpel and P. T. Luckie, Process Engineering of Size Reduction: Ball Milling, AIME, New York, 1984, p. 18. T. S. Mika, L. M. Berlioz and D. W. Fuerstenau, Dechema Monograph., 57 (1967) 205. D. F. Kelsall, K. J. Reid and C. J. Restarick, Powder Technol., 2 (1968/69) 162. V. K. Gupta and P. C. Kapur, Powder Technoi., 10 (1974) 217. R. L. Houillier and J. C. Marchand, Powder Technol., 14 (1976) 71. K. Shoji, S. Lohrasb and L. G. Austin, Powder Technol., 25 (1980) 109. R. L. Houillier, A. V. Neste and J. C. Marchand, Powder Technol., 16 (1977) 7. L. G. Austin, R. R. Klimpel and P. T. Luckie, Process Engineering of Size Reduction: Ball Milling, AIME, New York, 1984, p. 181. 9 K. J. Reid, Chem. Eng. Sci., 20 (1965) 953. 10 K. Shoji, L. G. Austin, F. Smaila, K. Brame and P. T. Luckie, Powder Technol., 31 (1982) 121. 11 J. A. Herbst and D. W. Fuerestenau, Trans. AIME, 241 (1968) 538.