Utkarsh Sankrityayan-effect of Particle Size Distribution on Grinding Kinetics in Dry and Wet Ball Milling Operations

7/31/2019 Utkarsh Sankrityayan-effect of Particle Size Distribution on Grinding Kinetics in Dry and Wet Ball Milling Operations

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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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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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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


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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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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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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Kims Data (1974)

Set 2:


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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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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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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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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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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Utkarsh Sankrityayan-effect of Particle Size Distribution on Grinding Kinetics in Dry and Wet Ball Milling Operations