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# Application of Linear Additivity Principle in Ball Mill Grinding Processes.

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```Application of Linear Additivity Principle in Ball
Mill Grinding Processes
L. X. Liu
R & D Depatfment, Camer Transicold (S) f f e Ltd, SINGAPORE 2263
G. Q.Lu*
School of Mechanical and Production Engineering, Nanyang Technological
Universa SlNGAPORE 2263
This paper presents an experimental verification of the linear additivity principle of
mixed balls in ball mill grinding processes through batch grinding tests of pulverizedcoal slag from power plants. Based on an nth order model of the grinding process, the
theory is also applied to the calculation of suitable ball media composition f o r an
indurtrial continuous mill. The change in ball media composition as calculated using the
model, results in a 20% increase in production from an industrial mill.
Introduction
Grinding is a complex multi-variable process in which most of the operating
variables affect the operation through grinding media. The grinding medium usually
comprises mixed balls of different sizes, it is therefore desirable to study the grinding
mechanism of mixed balls. Austin et al. [ 11 developed a first-order kinetic model for
grinding processes and extended it to the case of mixed balls [21. Equation (1) is the
basic set of equations for first-order batch grinding with mixed balls:
where wl(t), w2(t), ... etc., are the size distributions at time t; Si is the overaIl
specific rate of breakage of size i material for a mixture of balls, and is equal to the
linear additivity of the specific rates of breakage for the single-sized balls, that is:
where 3 (df) is the specitk rate of breakage of size i material for single-sized balls of
size df ;6fis the weight fraction of balls of size df in the mixture of balls.
For a mixture of balls, Austin et al. [l] assumed that the breakage function bij
does not change with ball size. Viswanathan [3] proposed that a mixture of balls
behaves in the Same manner as if all the balls were of the Same "equivalent mean ball
size" (db) which is given by:
* Author f o r correspondence.
53
L.X.Liu and G. Q. Lu
However, there are no experimentalresults to verify the above assumptions.
In a previous study of grinding kinetics, it was shown [4]that the grinding
process is not completely in accord with the fmt-order kinetic equation. Kinetics of
nth order (n # 1) were proposed and verified with iron ore batch grinding, and can be
expressed as:
R(t) = R(0) exp(-k tn)
(4)
where R(t) is the cumulative weight percentage larger than a certain size at time t;
R(0) is the value at the beginning of grinding; k and n are the kinetic constant and
order, respectively.
Based on this nth order kinetic equation, the linear additivity principle of mixed
balls was developed and verified with grinding tests of iron ore [41. This study used
the batch grinding tests of pulverized-coal slag (the waste residue of a power plant),
and the linear additivity principle is verified further and applied to ball media
calculations of an industrial mill.
Formulation and Verification of the Linear Additivity Principle
The linear additivity of mixed balls states that the grinding effect of mixed balls is
equal to the linear additivity of the grinding effect of the single-sized balls.
Therefore, the product size distribution can be described by the kinetic parameters
for single-sized balls, written as:
5
Ri(t)=R i(0) 6 f exp (-kfi tnfi)
f= 1
where 61, 62, ...6f, ... are the weight fractions of the single-sized balls in a
mixture;
kfi and nfi are the kinetic parameters of size i using size f balls for grinding a feed
of distributed sizes.
In order to further verify the above model, batch dry-grinding tests with
pulverized-coal slag from a cement plant were carried out in a Bond Work Index
mill (305~305mm).Four different single-sized balls were used in the tests. The ball
diameters were 30mm, 38mm, 53mm and 60mm, respectively. The mixed balls used
in the experiments were composed of the above single-sized balls with the
corresponding weight fractions of 0.26 : 0.24 : 0.25 : 0.25. The weight of the ball
load was 26.94 kilograms with a ball filling (volume) ratio of 0.25. The interstitial
filling of the ball charge by the slag was 0.6. The slag load used for each test was
1.556 kg.
The dashed line in Figure 1 is the feed size distribution. For each f e d , grinding
tests were carried out with four different sized balls in order to determine the
grinding kinetic parameters. The grinding media and feed material were loaded into
the mill and grinding commenced. After grinding for a known time, the material was
discharged and size analysis was performed using standard sieves. After analysis, the
54
Application ofLinear Additivity Principle in Ball Mill Grinding Processes
material was re-loaded into the mill for the next run. From the product size
distributions at different grinding times, the kinetic parameters were obtained and are
given in Table 1.
Size (mm)
Figure I . Comparison of experimental and calculated size distributionsfor ball-mill
grinding.
B a l l size (mm)
Particle
3o
38
60
54
size (mm)
k
1 .o
0.3
n
k
n
k
n
k
*
-
-
-
-
n
-
1.185
0.087
0.510
1.450
1.278
0.114
0.059
1.208
0.082
0.154
1.391 0.089
0.044 1.210 0.043
1.058
0.042
1.092
0.076
0.024
1.168 0.022
1.273
0.032
1.027
0.023
1.046
-
1.241
55
L. X.Liu and G.Q.Lu
The grinding kinetic parameters were used to calculate the product size
distribution for mixed balls of different weight fractions. Comparison of the
calculated and experimental product size distributions are shown in Figure 1. The
calculated results show good agreement with the experimental results, and indicate
that the principle of linear additivity of mixed balls is useful for predicting the
product size distribution with mixed balls.
Application of the Linear Additivity Model in an Industrial Mill
The linear additivity principle can be used for the optimization and control of
grinding circuits. For a certain characteristic material and its corresponding feed
size distribution, product size distributions can be calculated by using Equation (5)
for mixed balls of any weight fraction. For a desired product size distribution, a
suitable weight hction of mixed ball can be calculated.
The kinetic parameters obtained from the laboratory mill (Table 1) are used to
calculate a suitable ball media composition for a continuous mill in a small cement
plant. The coal slag from a power plant is ground to a product of 90% passing 200
mesh, as used for cement production. The mill is 1.5m diameter and 5.8m long with
two chambers. The grinding media in the chambers are mixed steel balls. From the
slag feed rate and the interstitial filling rate of slag in the ball media, the mean
residence time of 8 minutes was calculated. Using the kinetic parameters in Table 1
and the feed size distribution in Figure 1, the product size distribution at 8 minutes of
grinding for mixed balls of various weight fractions were calculated using Equation
(5). Table 2 summarizes the results.
From Table 2, by using single-sized balls of 38mm, the weight percentage of
particles coarser than 0.3mm is at a minimum, and that of particles less than 200
mesh is at a maximum value. This is because the weight percentage of particles
coarser than 0.3mm in the coal slag feed is more than 90%. Therefore, the calculated
ball charge should generate the highest reduction rate for particles coarser than
0.3mm. Also the single-sized balls of 38mm (from all the four single-sized balls)
have the strongest grinding effect on particles coarser than 0.3mm, as can be Seen
from the kinetic parameters.
Table 2. Calculated product size distributions for mixed balls of different weight
fractionsafter 8 mk. of grinding.
Weight fraction of
mixed balls
Product size distribution (“A)
Particle size (mm)
(30mm : 38mm : 54mm : 60mm)
0
0
0.3
0.5
0.22
0.3
0
56
: 0.25
: 0
: 0.4
: 0.5
: 0.28
: 0.7
: 1
:
:
0.5
:
0.5
0.3
: 0.5
: 0
: 0
:
:
0
:
:
:
0.28
0
0
:
:
0.25
0.22
0
: o
22.66
26.78
1827
16.80
21.40
15.96
14.68
57.69
60.79
54.79
54.13
57.11
53.36
52.2
75.75
77.59
73.81
73.51
75.31
72.95
72.10
24.45
22.41
26.19
26.49
24.69
27.05
27.90
Application of Linear Additivity Principle in Ball Mill Grinding Processes
In the cement plant, mixed balls of 70mm, 60mm, 50mm and 40mm diameter
were used (weight fractions 0.24 :0.26 : 0.26 : 0.24). From the above analysis, it was
proposed that the plant change the ball charge for the mill. Since there are
occasionally some coarser particles in the feed and the ball diameters will reduce due
to wear and tear, a mixture of balls of 50mm and 40mm with a weight fraction of 0.5
:0.5 was adopted. As a result, the mill production then increased by 20% (from 2.0
t/h to 2.4 t/h) without any change in the operating conditions, and without any
increase in the mill power consumption.
Concluding Remarks
A kinetic model for ball mill grinding processes, i.e. the linear additivity principle of
mixed balls has been elucidated and experimentally verified by laboratory batch
grinding tests. The kinetic parameters obtained from the tesrs were used to calculate
a suitable ball media composition for a continuous industrial mill. A 20% increase in
production in this mill is achieved by changing the ball media composition. The
linear additivity principle of mixed balls is of industrial significance for ball mill
grinding processes, and may be used for computer simulations of closed-circuit
grinding processes.
References
1. Austin, L.G., Bagga, P. and Celik, M. 1981. Breakage properties of some materials in a
2
laboratmy ball mill.Powder Technol.,28,235-243.
Austin, L. G. and Brarne, K. 1983. Comparison of the Bond method for sizing wet
tumbling ball mills with a size-mass balance simulation model. Powder Technol., 34.261274.
3. Viswanathan. K. 1986. Computer based models for grinding w d industrial case studies.
Aujbereit Tech.,lO.560-572.
4. Liu. LX., Chen. B. and Liu, Q. 1988. A study of grinding kinetics and its applications to
the choice and calculation of ball media Proceedings of 16th Internalwnul Congress on
Mineral Processing. Stockholm, Sweden, 5-10 June, 245-255.
Received 27 February 1993; Accepted after revision: 3 August 1993.
57
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