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EFFECT OF PARTICLE SIZE DISTRIBUTION ON
GRINDING KINETICS IN DRY AND WET BALL
MILLING OPERATIONS
Under the guidance ofProf. V.K. Gupta
Department of Fuel and Mineral Engineering
Indian School of MinesDhanbad, Jharkhand
Presented by:
UTKARSH SANKRITYAYAN (2008JE0428)
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INTRODUCTION
Breakage behaviourof particulate solids in ball mills isdependent on the mill charge, ball load, mill speed,pulp viscosity, and % solids in the pulp.
In the context of the size discretized batch grinding
kinetic model, grinding behaviour of particles ischaracterized by the breakage rate parameters, S, andbreakage distribution parameters, B.
Understanding the nature of variation of these two setsof parameters with various operating variables isimportant for simulation and control of operation ofproduction mills.
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INTRODUCTION
Recent studies have shown that the particlesize distribution also has considerableinfluence on the grinding behavior of
particles. This influence is more pronounced in case of
wet grinding as compared to dry grinding.
However, it is not known if only the breakagerate or the breakage distribution as well varieswith the particle size distribution.
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Frame Work for Analysis
The well known size-mass balance equation ofthe batch grinding equation
where, Mi(t) is mass fraction of the particulatesolids in the sieve size interval i, Si is fractional
rate of breakage for material in the size interval i,and bi,j is breakage distribution parameter whichdenotes the fraction of the material breaking outof size interval j that reports to size interval i.
1
,1
( )( ) ( )
i
i
i i i j j jj
dM tS M t b S M t
dt
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Wet Grinding
In the case of wet grinding, it is well known
that the grinding rate of coarse particles
increases with grinding time. This is due to the
fact that fine particles remain in suspension in
the slurry, leading to an increase in the
probability of coarser particles being ground
in the toe region of the mill
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Dry Grinding
Earlier, it was believed that the effect of particle size distribution ongrinding kinetics in ball mills is not significant in the case of drymode of grinding.
Observed variations in the grinding rate of particles in the dry modeof operation were attributed to factors such as: (i) interplay of
distributions of strength of particles and distribution of force (ii)cushioning effect of fine particles and variation in the shape ofparticles during breakage
Later, Gupta showed that inter-size particle-particle interactionsplay an important role in determining the breakage kinetics even inthe dry ball milling operation. It was demonstrated that: (i)
breakage rate of particles increases as the particle size distributionenvironment becomes finer, and (ii) particles of next smaller sizeinterval have maximum influence on the breakage rate of particlesof a given size.
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PAST RESEARCH WORK
Verma and Rajamani (1995) estimated breakage ratesfor particles of all size intervals using an indirectestimation method known as G-H method (Kapur andAgrawal, 1970; Kapur, 1982; Purker, Agrawal and Kapur,
1986).
However, it was assumed, without any experimentalevidence, that the breakage distribution function wasindependent of the particle size distribution
environment. It was perhaps for this reason that inseveral cases acceleration in breakage rate followed bydeceleration and acceleration was observed.
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Scope of Work
In this project, an attempt has been made to
track time variation of both the breakage rate
and breakage distribution parameters using
the functional form approach.
Various constants appearing in the functional
forms are assumed to vary linearly over short
time intervals.
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MODELLING APPROACH Relevant data available in the literature was analysed using the
well-known size discretized size-mass balance kinetic model
It has been found by several researchers (Klimpel and Austin, 1977;
Gupta et al, 1981; Gupta, Hodouin and Everell,1982; Gupta,
Hodouin and Spring, 1983; Austin, Klimpel and Luckie, 1984) that
the variation of Si and bi,j parameters with particle size can be
adequately described by the following two functional forms
S = Ax**Where x is particle size and A and are constants.
1
,
1
( )( ) ( )
ii
i i i j j j
j
dM tS M t b S M t
dt
x x i ib = ( ) + (1- )( )i,j x x
j j
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MODELLING APPROACH
In the current approach, it was thought that, as a firstapproximation, variation of S and b parameters withgrinding time can be approximated by a piecewise linearfunction in time. For example , over a short time interval ,variation of with grinding time t can be described by :
= a + b twhere, a and b are two constants. Thus, for estimation ofthe values of the parameters A, , , and , it requiredestimation of ten constants.
Best estimates of these ten constants were obtained using
the least squares fit criterion for Mi values. For thispurpose, an error function Er was defined as
m n 2
Er = [M (t ) - M (t )]i k i kk=1i=1
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MODELLING APPROACH
where
n : total number of size intervals andm : number of combinations of feed and
product size distributions.
The error function was minimized usingRosenbrook non-linear optimization algorithm
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RESULTS: Dry Ball Mill Grinding
Malghans Data
Ball mill diameter : 20 inch
Mill Speed : 60% Critical Speed
Ball Load: 0.3
Particle Load: 1.0(represented in terms of fractional filling of the mill volume, J, andvoid space of the static ball charge, U)
Data was available for grinding of 8/10 mesh single size fraction forgrinding time values 0.5, 1, 2, 3, 4 and 6 minutes.
The fit to the experimental data was found to be quite good.
The average error for Mi values, expressed in weight percent, variedbetween 0.3-0.6.
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RESULTS: Dry Ball Mill Grinding
0.01
0.1
1
0 1 2 3 4 5
1
2
4
6
8
10
12
Breakagerate(1/min)
Time, min
Fig 1 Variation of breakage rate parameters with grinding time for dry grinding of
limestone (Malghan, 1975).
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RESULTS: Dry Ball Mill Grinding
0.01
0.1
1
0.01
0.1
1
1234567891011
t=0.75
t=1.5
t=2.5
t=3.5
t=5
Size Class, i
Bi,1
Fig 2 Variation of breakage distribution parameters with grinding time for dry
grinding of limestone (Malghan, 1975).
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RESULTS: Wet Ball Mill Grinding
Kims (1974) Data
Ball mill diameter : 25.4cm
Two sets of data were analyzed. These are identified as Set 1 andSet 2.
SET 1
Mill Speed : 60% Critical Speed
Weight percent solid : 70%
Ball Load: 0.5
Particle Load: 1.15
(represented in terms of fractional filling of the mill volume, J, andvoid space of the static ball charge, U)
Data was available for grinding time values 0.5, 1, 2, 3, 4, 8 and12minutes.
The fit to the experimental data was found to be quite good.
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RESULTS: Wet Ball Mill Grinding
0.01
0.1
1
0 2 4 6 8 10 12
i=1
i=3
i=4
i=5
i=7
i=9
i=10
Time, min
brekagerate,
(1/min)
Fig 3 Variation of breakage rate parameters with grinding time for Set-1 data on wet grinding
of limestone (Kim, 1974).
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RESULTS: Wet Ball Mill Grinding
0.01
0.1
1
0.01
0.1
1
1234567891011
t=0.75
t=1.5
t=2.5
t=3.5,6,10
Size class, i
Bi,1
Fig 4 Variation of breakage distribution parameters with grinding time for Set-1 data on
wet grinding of limestone (Kim, 1974).
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RESULTS: Wet Ball Mill Grinding
Kims Data (1974)
Set 2:
Mill Speed : 50% Critical Speed
Weight percent solid : 60%
Ball Load: 0.5Particle Load: 1.0
(represented in terms of fractional filling of the millvolume, J, and void space of the static ball charge, U)
Data was available for grinding time values 0.5, 1, 2, 4, 6, 8and 12 minutes.
The fit to the experimental data was found to be quitegood.
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RESULTS: Wet Ball Mill Grinding
0.01
0.1
1
0 2 4 6 8 10 12
i=1
i=3
i=4
i=6
i=8
i=10
brekagerate,(
1/min)
Time, min
Fig 5 Variation of breakage rate parameters with grinding time for Set-2 data on wet
grinding of limestone (Kim, 1974).
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RESULTS: Wet Ball Mill Grinding
0.01
0.1
1
0.01
0.1
1
1234567891011
t=0.25
t=1.5
t=3
t=5
t=7
t=10
Bi,1
Size class, i
Fig 6 Variation of breakage distribution parameters with grinding time for Set-2 data on wet
grinding of limestone (Kim, 1974).
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RESULTS: Dry Rod Mill Grinding
Grandys (1970 Data)
Mill Charge: 5280 g of -7/8# Dolomite
Data was available for grinding time values .5,1, 1.5, 2, 3, 4, 6 and 8 minutes.
The fit to the experimental data was found to
be quite good.
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RESULTS: Dry Rod Mill Grinding
0.01
0.1
1
1 2 3 4 5 6 7 8
i=1
i=3
i=5
i=7
i=9
i=11brekagerate,
(1/min)
Time, min
Fig 7 Variation of breakage rate parameters with grinding time for Rod
Mill data on dry grinding of dolomite (Grandy, 1970).
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RESULTS: Dry Rod Mill Grinding
0.01
0.1
1
0.01
0.1
1
1234567891011
t=1.75
t=2.5
t=3.5
t=5
t=7
Bi,1
Size Class, i
Fig 4.12 Variation of breakage distribution parameters with grinding time
for Rod Mill data on dry grinding of dolomite (Grandy, 1970).
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Discussion and Conclusion
It is possible to track the variation of both thebreakage rate and breakage distributionparameters with grinding time using the piece
wise linearization approach At present it is not possible to fully explained
the variation of S and B parameters withgrinding time. It will require further thinkingand carrying out specially designedexperiments.
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Discussion and Conclusion
When B function was made time invariant, as
done by Rajamani et al., the overall fit to size
distribution data deteriorated significantly. For
example, the Err value for one set of wetgrinding data increased from 7.24 to 13.59.
This shows that the B parameters also varies
with grinding time ( Size Distribution ).
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