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International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 4, Issue 2, February 2014)
Simulation studies on Energy Requirement, Work Input and
Grindability of Ball Mill
Abanti Sahoo
Chemical Engg. Dept., National Institute of Technology, Rourkela-769008,Odisha, India
The Bond grindability testing procedure has been
standardized to obtain the grindability values (i.e., Bond
work index (BWi)) on the same ore when tests are
performed in different laboratories or by different
operators. The results for the Bond work index grindability
thus obtained may differ substantially with different
standard Bond ball mills (Kaya et al., 2003). Therefore
there is need for development of a more general and
standard model to measure the performance of the ball mill.
Abstract— Energy requirement, work input and
grindability of ball mill have been studied by varying different
system parameters (viz. particle size & density of materials,
number of balls, time of grinding and speed of the mill).
Attempt has also been made to correlate the output with these
inputs on the basis of regression analysis. Different C++
programmings were written to compute energy requirement,
work input and grindability for the ball mill using the above
mentioned system parameters. Finally a comparison has been
made among experimentally observed values and the values
determined by other methods (i.e. C++ programming, and
regression analysis). For comparing the goodness of the fit,
the correlation coefficient and chi-square test have been used.
II. PREVIOUS WORK
Comminution in a mineral processing plant or, mill
involves a sequence of crushing and. grinding processes
(Prasher, C.L., 1987). Literature on impact crusher
performance in relation to machine configuration and
operational conditions, by experimental work and
mathematical modelling is given by Austin et al. (1979)
and Shi et al.(2003). The Bond grindability test has been
widely used for predictions of ball and rod mill energy
requirements and for selection of plant scale comminution
equipment (Babu and Cook,1973). It is known that the
efficiency of the mill can be increased by tuning the
rotation velocity so that the average collision velocity
becomes maximum. In this context attempt has been made
for a meticulous study for the effect of the various system
parameters on the performance of
ball mills. If the
peripheral speed of the mill is very high, it begins to act
like a centrifuge and the balls do not fall back, but stay on
the perimeter of the mill and that point is called the
"Critical Speed‖ (McCabe et al., 1993). Ball mills usually
operate at 65% to 75% of the critical speed and this is
calculated as under.
Keywords — Comminution, Bond’s Work Index,
Grindability,
Dimensional
analysis
approach,
C++programming, correlation coefficient and Chi-square test.
I. INTRODUCTION
Grinding is one of the most energy-intensive operations
in the preparation of ores and has a significant effect on the
economics of processing these raw materials. Therefore it
becomes especially important to select the correct
equipment for this operation. The choice of equipment
chosen for the comminution of ore is mainly based on the
Bond work index, which is usually determined by the
standard method of dry grinding. Traditional measurements
of ore grindability are the Bond Work Index. The Bond test
is still surprisingly popular, despite the advances in
modelling and computing power.
Grindability data, based on various techniques to
measure comminution characteristics, are used to evaluate
the crushing and grinding efficiency in mineral processing
operations. The importance of achieving improved
comminution efficiency, in terms of energy consumption,
has been emphasized by increase in the cost of electricity
(Horst and Bassarear, 1976). Bond’s grindability can be
empirically related to the energy required for comminution
and thus is useful for the design and selection of crushing
and grinding equipment (Deniz et al., 1996).
Ball milling is a wide spread milling technology,
particularly in mining; mainly because of its simple
construction and application.
nc 
1
2
g
Rr
(1)
Grindability (G):
Grindability is the number of net grams of screen
undersized product per revolution (Perry and Chilton,
1973). The main purpose of study of the grindability is to
evaluate the size and type of mill needed to produce a
specified tonnage of product and thereby the power
requirement for grinding.
592
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 4, Issue 2, February 2014)
The standard Bond grindability test is a closed-cycle dry
grinding and screening process, which is carried out until
steady state conditions are obtained (Bond, F., 1961, Yap et
al., 1982 and Magdalinovic, N., 1989). Meaningful
expressions between Bond index, grindability index and
friability value have been developed by Ozkahraman
(2005).
Wi  1.1





W  10Wi 


n




44.5
10 / P80  10 / F80

 

(4)
1
P80

1
F80




& P  T *W
(5)
Based on this equation it is possible to calculate, the
specific energy requirement for a given grinding duty,
BBMWI, feed size and required product size. With the
knowledge on energy requirement it is possible to
determine the size of mill required and thereby the motor
power requirement can be determined. Bond has suggested
an intermediate course in which he postulated n to be -3/2
which leads to
(2)
Magdalinovic (1989) has stated simplified procedure for
a rapid determination of the work index by just two
grinding tests. The applicability of the simplified procedure
has been proved on samples of Cu ore, andesite and
limestone. The result by this method was not more than 7%
from the values obtained in the standard Bond test.
Yalcin et al., (2004) investigated the effect of various
parameters on the grindability of pure Sulfur and used the
obtained grinding data to establish mathematical models
and set up a computer simulation program. The established
mathematical model is as shown below.
 d 



  ln 2 

100 
 d5 0 
y 
1 e
k 

0.82
Gbg
In designing and optimizing a milling circuit using Bond
Ball Mill Work Index the most commonly used equation is
as follows (Bond, F., 1961).
Work Index:
Work index is defined as the gross energy required in
kilowatt-hours per ton of feed needed to reduce a very large
feed to such a size that 80% of the undersize passes through
100-μm screen [8]. The expression for the work index is as
given below
 1
1
W  0.3162  Wi 
 d 0.5  d 0.5
 P
F
Pi
0.23


1/ 2 
 100 
 1
1 
1 

1 
  10 Ei 
E  Ei 


 q1 / 2 
 L0.5 L0.5 
 L2 
1 
 2
(6)
Bond defines the quantity Ei as the amount of energy
required to reduce unit mass of the material from an
infinitely large particle size down to a particle size of 100
micro meters. It is also expressed in terms of q, the
reduction ratio.
Where, q= L1 /L2 ; L1 and L2 are the feed and product
size in micrometers respectively.
It is important to fix the point where the charge, as it is
carried upward, breaks away from the periphery of the
Mill. This is called as the "break point‖ or "angle of break"
because it is measured in degrees up the periphery of the
Mill from the horizontal. The most desirable angle of break
ranges from 50 to 60 degrees from the horizontal to
accomplish the cascading and sliding action for most
grinding and dispersing problems. The four factors
affecting the angle of break are speed of mill, amount of
grinding media, Amount of material and the consistency or
viscosity in wet grinding. Factors affecting the angle of
break and the product size from a ball mill are already
discussed (Sahoo and Roy, 2008). Fractional factorial
design and dimensionless analysis method have been used
to design the experiment using different factors against the
experimentally observed grindability data. In the present
study attempt has been made to develop a mathematical
expression by correlating the obtained grinding data with
the system parameters. Safonov et al., (2009) have
observed that the temperature, grinding conditions, and
other factors have a significant effect on the grinding of
bauxites.
(3)
Where, y is cumulative percent passing size d, d50 is the
50% passing size, n is distribution constant (The n values
ranged from 0.84 to 1.84), k is a correction factor (k values
ranged from 0.95 to 1.00).
By using the Bond method of grindability, Ipek et al.
(2005) have observed that less specific energy input is
required in separate grinding of ceramic raw materials than
grinding them in admixtures. The Bond work indices of the
admixtures containing softer component have been
observed greater than the weighted average of the work
indices of the individual components in the mixture. Deniz
and Ozdag (2003) have investigated the effect of elastic
parameters on grinding and examined the relationship
between them. The most widely known measure of
grindability is Bond’s work index which is defined as the
resistance of the material to grinding. The standard
equation used by them for the ball mill work index (Bond
work index) is as follows.
593
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 4, Issue 2, February 2014)
They have correlated this characteristic with the
mineralogical composition and textural-structural features
of bauxite.
The correlation coefficients or coefficients of
determination were found out to be 0.761 and 0.893 for
work input and grindability correlations respectively
indicating that the developed correlations fit closely to the
particular set of experimental data.
The values of the grindability calculated as per Eq-(8)
based on dimensional analysis was again used in Eq-(4) to
determine the Bond Work Index of the ball mill. The values
of Bond Work Index thus obtained were used in Eq-(5) to
determine the work input of the ball mill. Finally the
calculated values of the work input calculated as per Eq-5
and Eq-7 are compared with each other and also against the
experimental values. The comparison plot is shown in Fig.4. The standard deviation, mean deviation and chi square
(χ2) obtained from various comparisons are shown in
Table-2. It is observed that values of the work input
calculated by both the methods i.e. from Eq-(5) and (7)
agreeing well with each other and with the experimentally
measured values in most of the cases.
The developed correlations were verified over a wide
range of parameters. For each parameter several values
were assumed and checked for the output. MATLAB
coding was developed to verify each parameter over a wide
range. Finally the effect of each parameter on output with
theoretical data has been compared with the same with the
experimental data. Flowchart for the MATLAB coding has
been shown in Fig.-5. It is further observed that the system
parameters namely particle size, particle density, speed of
the ball mill and time of grinding have opposite effects on
grindability in comparison to the Bond Work Index.
Grindability increases with the decrease in each of these
parameters whereas the Bond Work Index increases with
the increase in each of these parameters. A sample plot of
comparison with respect to particle size is shown in Fig.-6
which has been shown for both grindability and Bond’s
work input for the ball mill. This observation is justified by
Eq (4) as work index is inversely proportional to G 0.82.
Only the number of balls has the similar effects on both the
type of the performances of the mill i.e. for grindability as
well as for the work index. Both, the work input and the
grindability values increase with the increase in the number
of balls (Fig-7). For the most efficient results, the mill
should be at least half filled with grinding media. The
grinding efficiency of the one-half charge is considerably
greater than for the one-third and, therefore it can be
expected that power consumption per gallon output will
actually be less than with the smaller charge MIKRONS
(2011).
III. EXPERIMENTATION
A ball mill of 36.6 cm diameter and 50 cm length was
used in the laboratory for experimentation. The material of
construction of the grinding media was mild steel. The steel
balls each of 5.41 cm diameter and density 7.85kg/m3 were
used for the experiments. The mill was made to revolve at
different speeds (0-110 rpm) to grind various materials like
dolomite, iron ore, coal and limestone by using a variac
which was set at different supplied voltages for each type
of material. The effects of various system parameters (viz.
particle size, material density, speed of the mill, time of
grinding and the number of balls) on the performance of
the ball mill were studied. The product was sieved and
measured. The amounts of undersize or fines were found
out by sieving with a 120-mesh size. Bond Index and
grindability of the mill were calculated using the amount of
fines per revolution. Scope of the experiment is given in
Table 1 and the experimental set up is shown in Fig. 1(A).
Mechanism of operation in a ball mill is shown in Fig.1(B). Energy consumed for grinding the given feed sample
into fines was noted down from the energy meter reading.
The same procedure was repeated with different parameters
as mentioned in the scope of the experiment. Each time 1.0
kg of material was taken as feed for running the ball mill
and bond index was calculated from the energy meter
reading observed for getting around 800gms of fines
passing through the 100 micron mesh size from 1kg of
feed.
IV. RESULTS AND DISCUSSION
Development of the Correlations:
Values of the work index and grindability have been
calculated as per their definitions. In the present work,
attempt has been made to develop expressions for the Bond
Work input and grindability of the ball mill on the basis of
dimensional analysis by correlating the experimentally
observed values of the work index and the grindability
(calculated as per the definitions) against the various
system parameters. The correlation plots for work index
and the grindability are shown in Fig.- 2 and 3 respectively.
The developed correlations are given as under.

W  4E  07  d F 

G  36815  d F 
0.662
0.524
n1.711N 0.207t 0.172F 1.364
(7)
n1.975N 0.711t 0.542  F 1.284  (8)
594
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 4, Issue 2, February 2014)
[2 ] Babu, S.P. and Cook, D.S., 1973. Breaking, crushing and
grindingSME, Mining Engineering Handbook, vol. 2. AIMMPE,
Inc., New York.
[3 ] Bond, F., 1961. Crushing and grinding calculations. Brit. Chem.
Eng. 6, 543–548.
[4 ] Deniz, V. and Ozdag, A new approach to Bond grindability and
work index: dynamic elastic parameters, Mineral Engineering, Vol.
16, Issue-3, March-2003, p 211-217.
[5 ] Deniz, V., Balta, G. and Yamik, A. The interrelationships between
Bond grindability of coals and impact strength index (ISI), point
load index (Is) and Friability index ( FD), in: M. Kemal, et al (Eds.),
Changing Scopes in Mineral Processing, A.A. Balkema, Rotterdam,
Netherlands, 1996, pp. 15–19.
[6 ] Horst, W.E. and Bassarear, J.H., Use of simplified ore grindability
technique to evaluate plant
performance, Trans. Soc. Min. Eng.
AIME, 260 (1976) 348–351.
[7 ] Ipek, H., Ucbas, Y. and Hosten,C., The bond work index of mixture
of ceramic raw materials, Mineral Engineering, Vol. 18, (2005), p
981-983.
[8 ] Kaya, E.; Fletcher P. C. an Thompson P., Minerals & metallurgical
processing, vol. 20, (3), 2003, p 140-142.
[9 ] Magdalinovic, N. 1989(a), Calculation of energy required for
grinding in a ball mill, International Journal of Mineral Processing,
Vol. 25, (1-2), p 41-46.
[10 ] Magdalinovic, N., 1989(b), A procedure for rapid determination of
the Bond work index. Int. J. Mineral Processing, 27, 125–132.
[11 ] McCabe, W. L., Smith, J. C. and Harriot, P., Unit Operation in
Chemical Engineering, (Fifth Edition), McGraw-Hill, Inc,
Singapore, 1993, p-980.
[12 ] MIKRONS,
Design
Highlights
http://www.ballmill.in/ballmill_design.html, Copyright © 2011
MIKRONS.
[13 ] Ozkahraman, H. T., A meaningful expression between bond work
index, grindability index and friability value, Minerals Engineering
18 (2005) 1057–1059.
[14 ] Perry, R. H. and Chilton, C. H., Chemical Engineers’ Handbook ,
(5th edition), McGraw-Hill, p- 8-25.
[15 ] Prasher, C.L., 1987. Crushing and grinding process handbook.
Consultant to chemical and mechanical engineering industry, Linora
Technical Associates, John Wiley & Sons Limited, Chichester, New
York.
[16 ] Safonov, A. I., Suss, A. G., Panov, A. V., Luk’yanov, I. V.,
Kuznetsova, N. V. and Damaskin, A. A.; Effect of process
parameters on the Grindability and bond index of bauxites And
alumina-bearing ores, Metallurgist, Vol. 53, Nos. 1–2, 2009.
[17 ] Sahoo, A. and Roy, G. K., ―Correlations For The Grindability Of
The Ball Mill as a measure of its Performance‖, Asia Pacific Journal
of Chemical Engineering, vol- 3, Issue – 2, June – 2008, p 230-235.
[18 ] Shi, F.N., Kojovic, T., Esterle, J.S. and David, D., 2003. An energy
based model for swing hammer mills. Int. J. Miner. Process. 71,
147–166.
[19 ] Yalcin, T., Idusuyi, E., Johnson, R. and Sturgess, C. A simulation
study of sulphur grindability in a batch ball mill, Powder
Technology, Vol. 146, Issue 3, 8 September 2004, p 193-199.
V. CONCLUSION
In today’s industrial scenario, ball mill is widely used in
multifarious industries as size reduction process is energy
inefficient, it is necessary to optimize the operation so as to
reduce cost to some extent. As it has been explicitly seen
that the parameters influencing the performance of ball mill
cannot be ignored, the expressions correlating all these
variables can considerably be used to optimize the
operation of a ball mill in general over a wide range of
parameters which has been justified with correlation
coefficient (R2) and Chi-square-test (χ2) values. Thus these
results can be used as the basis of calculation to determine
the design criteria or the ranges of various parameters to be
used for a specific process. The chi-square test indicates
good correlation fit for both the type of performances. The
bond work index and grindability calculations are the
determining factors for the design of ball mill and size
reduction of ores as the power consumption will indicate
directly about the cost benefit too.
Nomencature
d : Particle diameter (size)
mm
F80 : 80% passing size of feed
µm
g : Gravitational constant, 981
gm/cm2
G : Grindability of the mill
g/rev
Gbg :
Bond’s std ball mill grindability
g/rev
n : Speed of ball mill, rpm
N : Number of balls
P : Power draw
kW
Pi :
screen size for performing the test
µm
P80 : 80% passing size of product
µm
r : radius of grinding balls
cm
R : Radius of the ball mill
cm
t : Time of grinding
min
T : Throughput of new feed
t/h
W : Work input
kW-hr/t
Wi : Work index of the material
kW-hr/t
 : Density of material
kg/m3
Subscripts:
c : for critical condition
f : for feed particles
p : for product particles
REFERENCES
[1 ] Austin, L.G., Jindal, V.K. and Gotsis, C., 1979. A model for
continuous grinding in a laboratory hammer mill. Powder Tech. 22,
199–204.
595
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 4, Issue 2, February 2014)
Figure Caption:
Fig.-1: Experimental set-up
Fig.-2: Correlation plot for Bond Work Index against system parameters
by dimensional analysis.
Fig.-3: Correlation plot for grindability against system parameters by
dimensional analysis.
Fig.-4: Comparison plot of calculated values of Bond
Work Index by
both the approaches against the experimental values
TABLE-I:
SCOPE OF THE EXPERIMENT
Fig.-1(B) : Mechanism of operation in a Ball Mill
Fig. 2 : Correlation plot for the Work Index of the ball mill
TABLE-II
COMPARISON OF THE PERFORMANCE OF THE BALL MILL
BY VARIOUS METHODS
Fig. 3 : Correlation plot for the Work Index of the ball mill
Fig. 4 : Comparison of work index of the ball mill by various methods
Fig.-1(A) : Experimental set-up
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Fig.-7 : Comparison between theoretical data and experimental data
on effect of number of Balls for both grindability and Bond’s work
input of the ball mill.
Fig.-5 : Flow chart for MATLAB Coding for verification of developed
correlation
Fig.-6 : Comparison plot between theoretical data and experimental
data on particle size for both grindability and Bond’s work input of
the ball mill.
597