AN INVESTIGATION OFFINE COAL GRINDING KINETICS

by

_ Michael Duane Brown

Thesis submitted to the Faculty of theVirginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

in

Mining and Minerals Engineering

APPROVED:

AC &\ [R7 Ü ^

G. T. Adel, hairmanq R. H. Yoon/

so / ,„ /7*„" ""' °

,Ü.J. R. Lucas WM. E. Foreman

L1

June, 1986Blacksburg, Virginia „

AN INVESTIGATION OFFINE COAL GRINDING KINETICS

by

Michael Duane Brown

Commitee Chairman: Dr. Greg T. AdelMining and Minerals Engineering

(ABSTRACT)

In recent years, a great deal of interest has been

shown in developing methods for preparing super—clean

coal containing less than 2% ash and 0.5% sulfur. New

techniques for recovering fine coal, such as

micro—bubble flotation, can achieve the desired result

provided mineral matter is sufficiently liberated. To

achieve sufficient liberation, however, it is often

necessary to grind to a mean particle size finer than

10 microns. Since conventional ball mills are highly

inefficient in this fine size range, the stirred ball

mill has been proposed as a more suitable means for

ultrafine grinding.

A five inch diameter mill has been designed and

constructed to investigate the milling characteristics

of stirred ball mill grinding using an Elkhorn seam

coal. A study of particle size distributions produced

under various operating conditions and constant energy

input revealed that slow shaft speeds and small ball

sizes improved grinding efficiency. Feed percent

solids in the range of 20% to 60% had no effect on

grinding efficiency below 100 kwh/ton of energy input.

Breakage parameters were also determined in order

to model the ginding process using a population balance

technique. The grinding model was found to accuratly

predict particle size distribution for short grind

times but to deviate from actual data over longer

grinding periods.

ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to Dr.

Greg T. Adel and Dr. Roe-Hoan Yoon for their guidance and

support throughout this investigation. Special thanks

are also extended to Dr. J. Richard Lucas and Dr. William

E. Foreman for their patience and assistance in arranging

for the final completion of this thesis.

Final thanks are extended to my parents, who have

supported my interests and been an inspiration during the

entire term of my college career.

iv

TABLE OF CONTENTS

PageAbstract ........................................... iiAcknowledgments .................................... ivList of Figures .................................... viiList of Tables ..................................... xi

CHAPTER 1. Introduction.............................. 11.1 Need for Fine Coal Grinding................... 11.2 Grinding Techniques........................... 31.3 Related Studies............................... 6

1.3.1 Stirred Ball Mill Grinding............... 61.3.2 Grinding Kinetics........................ 101.3.3 Population Balance Model................. 12

1.4 Scope of Present Investigation................. 15

CHAPTER 2. Experimental Techniques................... 162.1 Coal Sample................................... 162.2 Grinding Equipment............................ 172.3 Sizing Equipment.............................. 232.4 Torque Measurement............................ 262.5 Experimental Procedure........................ 302.6 Product Sampling.............................. 31

CHAPTER 3. Experimental Results...................... 363.1 General Operating Characteristics............. 363.2 Equal Energy Testing.......................... 41

3.2.1 Percent Solids........................... 413.2.2 Dry Grinding............................. 463.2.3 Shaft Speed.............................. 493.2.4 Ball Size................................ 52

3.3 Analysis of Breakage Parameters............... 603.3.1 Selection Function....................... 623.3.2 Breakage Function........................ 65

CHAPTER 4. Discussion of Operating Conditions........ 694.1 Percent Solids...............·................. 694.2 Rotor Speed................................... 714.3 Ball Size ..................................... 744.4 Energy—Size Relationship...................... 79

CHAPTER 5. Grinding Simulation....................... 83

CHAPTER 6. General Conclusions and Suggestionsfor Future Work........................ 88

6.1 Conclusions................................... 88' 6.2 Recommendations............................... 89

REFERENCES............................................ 91Appendix A............................................ 94Appendix B............................................ 99

v

Appendix C............................................102

VITA..................................................108

vi

LIST OF FIGURES”

PageFigure 1.1 Comparison of different types of

grinding machines (Davis et al.,1980) .................. 5

Figure 1.2 Comparison of the effectiveness ofvarious grinding devices for theultrafine grinidng of a chalcopyriteconcentrate (Herbst and Sepulveda,1978) ................. . 7

Figure 2.1 Three dimensional schematic of batchstirred ball mill ............ 18

Figure 2.2 Front view of 5-inch diameter stirred‘

ball mill ................ 20

Figure 2.3 Top view of 5-inch diameter stirredball mill ................ 21

Figure 2.4 Schematic of mill mounted to drillpress (Mankosa, 1986) .......... 22

Figure 2.5 Schematic of size measuring techniqueemployed by Elzone 80-XY particle sizeanalyzer (Davis, 1986) ......... 25

Figure 2.6 Comparison of sampling techniques for1—minute grind ............. 33

Figure 3.1 Effect of media load on torquerequirement ............... 38

Figure 3.2 Effect of stirring speed on torquerequirement (Mankosa, 1986) ....... 39

Figure 3.3 Effect of ball diameter on torquerequirement ............... 40

Figure 3.4 Effect of percent solids on productsize distribution at 94 kwh/ton ..... 44

Figure 3.5 Effect of percent solids on productsize distribution at 280 kwh/ton .... 45

Figure 3.6 Effect of dry grinding on product sizedistribution .............. 48

vii

hPage

Figure 3.7 Effect of shaft speed on product sizedistribution .............. 50

Figure 3.8 Effect of shaft speed on median productdiameter at 63.5 kwh/ton ........ 51

Figure 3.9 Effect of ball size on product sizedistribution .............. 54

Figure 3.10 Effect of ball size on product sizedistribution after a 10-minute grind(100 x 150 mesh feed) .......... 56

Figure 3.11 Effect of ball size on product sizedistribution after a 1-minute grind(100 x 150 mesh feed) .......... 58

Figure 3.12 Effect of ball size on product sizedistribution after a 5-minute grind(100 x 150 mesh feed) .......... 59

Figure 3.13 Change in mediam product diameter over30-minute grind time for 1/16-, 3/32-and 1/8-inch balls ........... 61

Figure 3.14 Feed disappearance plots for varioussize fractions using 1/4-inch media .. . 63

Figure 3.15 Breakage rate function versus feedparticle diameter using 1/4-inch media . 64

Figure 3.16 Selection function versus geometricmean size using 1/4-inch media ..... 66

Figure 3.17 Cumulative breakage function versusnormalized size variable ........ 68

Figure 4.1 Diagram of parabolic depressionoccuring in the batch mill at highershaft speeds .............. 73

Figure 4.2 Feed disappearance plots for 100 x 150mesh feed using various ball sizes . .. 76

Figure 4.3 Breakage rate function versus balldiameter ................ 77

Figure 4.4 Charles Law plot for stirred ballmilling of Elkhorn seam coal ...... 80

viii

PageFigure 5.1 Comparison of experimental and

predicted size distributions after1-minute grind ............. 86

Figure 5.2 Comparison of experimental andpredicted size distributions after5—minute grind ............. 87

ix

LIST OF TABLES

Page

Table 1. Comparison of sampling techniques fora l—minute grinding period................. 34

Table 2. Standard milling conditions for comparisonof feed percent solids...................... 42

Table 3. Products produced at 64 kwh/ton andvaried shaft speeds........................ 53

x

CHAPTER 1.

Introduction

1.1 Need for Fine Coal Grinding

The primary purpose of grinding coal is the liberation

of finely dispersed shale particles which cannot be

separated by conventional cleaning techniques. It appears

that nearly complete liberation of coal and ash occurs wheno

the ground product has a mean size less than ten microns.

The needed mean size may, however, be as little as one or

two microns. The liberation provided by grinding to these

small sizes can assist in producing super—clean coals which

contain less than 1% ash and 1% sulfur. New separation

techniques, such as the microbubble flotation process, are

now being developed at Virginia Tech to produce these high

quality coals at a high recovery, provided the feed coal is

sufficiently liberated. The recent economic pressures for

maximizing coal recovery, and current concerns about

limiting SO2 emissions through government acid rain

legislation, make super-clean coals, and coal grinding,

particularly important.

Fine coal is also needed in many newly developed

processes such as the preparation and burning of coal-water

fuels and other liquid coal mixtures. Liquid—coa1

1

2

technology has advantages over current solid coal

processing and burning techniques since a liquid fuel can

be pumped, stored in tanks, transported via pipes, and

injected into furnaces and boilers. Furthermore, less

capital is required to convert a furnace from oil to a

pumpable coal—liquid fuel than is required to convert to a”

solid fuel. Finally, the use of coal—liquid fuels makes it

possible for utilities to become less dependent on foreign

raw materials, particularly oil.

It is important in transportation and storage of

liquid-coal fuels, that the mixture is both pumpable and

stable. Ultrafine coal enhances the stability of a liquid

coal mixture. Depending on the requirements of a

particular application, varying the degree of coal grinding

may be a useful technique for controling the stability of a

mixture. Department of Energy research has demonstrated

that a median particle size of about 15 microns will give

good slurry stability without the need for stabilizing

additives.

In processes in which coal is mixed with liquids and

burned in engines, a primary concern is wear on valves and

other internal machine parts. It appears that wear could

be reduced by grinding to ultrafine sizes (Atlantic

Research Corp.,1984). Fine coal may be less abrasive on

pumps and piping than larger sized coal. Provided it can

3

be ground inexpensively, ultrafine coal may prove

economically beneficial in reducing wear and extending the

life of coal burning machinery.

lt is apparent that fine coal grinding will be

important in producing superclean coals and coal mixtures.

Economic conditions in today's coal industry require that

size reduction occur at the lowest possible cost. For this

reason, it is important to examine the coal grinding

process and to determine the operating characteristics

which influence its energy efficiency. It is equally

important to be able to describe mathematically the

breakage process in terms of the rate of size reduction and

the size distribution of the final product in order that

these fine grinding devices may be properly designed.

1.2 Grinding Techniques

There are several grinding techniques which may be

employed to produce fine and ultrafine products. Ball

milling is the most commonly used technique in the mineral

processing industry today. Although it is the most common,

it is not necessarily the most energy efficient method of

grinding. Conventional ball milling cannot achieve a minus

ten micron product without requiring extremely long

retention times and very large energy inputs. Other

4

approaches to ultrafine grinding include vibratory, impact,

ultrasonic, and attrition milling. Past studies conducted

by the United States Bureau of Mines indicate that of these

methods, attrition milling may be the most promising for

producing micronized material in a fast and energy

efficient manner (Davis,et.a1.,1980).

The Bureau—developed attrition process grinds through

intense agitation of a mixture consisting of the material

to be ground, a grinding media, and a suspending fluid.

This is also true of the stirred ball mill process

described by Herbst and Sepulveda (1978). Breakage in this

process is thought to occur by a shearing action as

particles are captured between balls, between balls and the

impeller, and between balls and the vessel wall.

The USBM data, shown in Figure 1.1, indicates that the

attrition process is far superior to other grinding methods

for reducing coarse kaolin to a product size below 2

microns. Studies done by Herbst and Sepulveda (1978) have

further demonstrated that a stirred ball mill, similar to

the USBM attrition mill, is also more efficient in particle

reduction than the conventional ball mill and the vibratory

mill. Their study demonstrates that particle size

reduction continues in a stirred ball mill after large

energy inputs, while conventional ball milling appears to

slow and eventually reach a limiting product size at which

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additional energy input has little effect on size

reduction. This is shown graphically in Figure 1.2.

With the reported success of these systems, it is

apparent that the stirred ball mill process should be

further quantified in terms of its energy requirements and

rate of size reduction for grinding of coal.

1.3 Related Studies

1.3.1 Attrition and Stirred Ball Mill Grinding

The first attrition grinder was adapted from a device

developed by the United States Bureau of Mines for

scrubbing glass sand to remove iron oxide stain (Dasher and

Ralston,1941). In the original scrubbing device, designed

by J.E. Norman and O.C. Ralston (1938), the slurried

material was scrubbed without grinding media. By simply

adding a granular grinding media, along with a suspending

liquid, the scrubber was modified to a grinding device.

Full introduction of the attrition grinder was‘

presented several years later by I.L Feld, T.N McVay, H.L.

Gilmore and B.H. Clemmons (1960). This work presented

diagrams showing the design characteristics of the mill and

data for grinding of coarse kaolin. All testing was done

in a batch mode using a 5 and a 10—inch diameter mill.

7

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Vibrotory

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LIJfl] 5U)

4ä— Conventdonol WETUJ

2 umStirred

1

0 100 200 300 400 500

SPECIFIC ENERGY INPUT (kwh/Ion)

Figure 1.2 - Comparison of the effectiveness of variousgrinding devices for the ultrafine grinidng ofa chalcopyrite concentrate (Herbst andSepulveda, 1978).

8

M.H. Stanczyck and I.L. Feld further studied the

operation of the attrition mill in an open circuit, a

closed circuit and a batch mode (1963;l965). Using a 5

inch diameter mill, they determined that the most important

operating characteristics in the attrition grinding of

kaolin were type, size and shape of grinding media,

grinding media to clay weight ratio, rotor speed, pulp

density, degree of pulp dispersion, and the angular

arrangement of rotor and stator bars. The reported

operating characteristics which had little effect on

grinding efficiency were pulp temperature, design of rotor

cage, rotor clearance, rotor—stator bar interval, addition

agents and chamber liners. Their report also indicated

that spherical media is preferable to sharp angular or long

bladed media with respect to media degradation, machine

abrasion and grinding efficiency (1968). The first

attempts to quantify the milling process were done while

studying the grinding of ceramic oxides in a 5 inch mill

(Stanley,et.al.,1974). Milling rate constants calculated

for 45x5O mesh dolomite were obtained under a variety of

operating conditions and reported to follow first—order

kinetics.

Continuing the initial mathematical study of the

attrition process, Sadler and Stanley (1975) further

verified the assumption of first-order kinetics for coarse

9

sizes using Ottowa sand as a grinding media. They

determined that for the size range studied, grinding

efficiency was constant regardless of power input to the

mill.

Another more recent study by the Bureau of Mines

(Davis,et.al.,1980) examines the attrition grinding process

for several minerals. An initial qualitative presentation

shows attrition milling to be superior to other methods of

grinding for producing -2 micron material. Preliminary

results are also reported on autogenous grinding of olevine

and mica. The majority of data presented in the literature

has concentrated on the basic ability of the attrition mill

to reduce material to fine sizes. Grinding performance, ,

therefore, is often stated in terms of the ability of the

mill to rapidly reduce a coarse material to extremely fine

sizes. Very little has been published which presents·

specific data describing energy requirements and breakage

rate values particulary for grinding of coal. Sepulveda

and Herbst (1982) have presented data for grinding of

several types of materials in a stirred ball mill similar

to that used in this research. Aside from their

verification of the high efficiency levels of the stirred

ball mill in comparison with more conventional grinding

methods, they demonstrated that a simplified modeling

approach using the Charles Equation produced good results

1O

in predicting the median diameter size of products for

different energy inputs to the mill. Mill operating

conditions were shown to have little effect on the energy

efficiency of the grinding process leading to the

conclusion that product size distributions are generally a

function of energy input alone. A generalized grinding

model based on the concept of distributed strength for the

material being ground was developed and an analytical

solution was obtained for batch grinding conditions.

1.3.2 Grinding Kinetics

The first attempts to relate breakage energy to the

performance of grinding devices were formulated as

empirical energy-size reduction relationships. These “laws

of comminution" proposed by Rittinger (1867), Kick (1885),

and Bond (1952) have been used for many years to estimate

mill capacity and power requirements. The laws are based

on work indices which are related to surface area, particle

volume or crack length. Since the relationships are

empirically based, they only fit experimental data for a

limited range of variables and in specific circumstances

(Hukki,1962). None of the laws is capable of defining the

ground product in terms of its particle size distribution.

In contrast to the empirically developed relationships of

11

Bond, Kick, and Rittinger, the population balance model isl

capable of predicting a product size distribution when the

initial feed distribution is known. Epstein (1947)

established the probabalistic approach to modeling the

comminution process. His major contribution to the

development of a population balance model was describing

the comminution process in terms of a selection function

and a breakage function. The selection function, S¤(y),

was defined as the probability that a particle of size y

would break during the nth step of the breakage process.

The breakage function B(x,y) was defined as the cumulative

weight distribution of particles appearing in or below size

class x as a result of breakage from a unit mass of size y,

Broadbent and Callcott (1956) used Epstein's selection

and breakage functions to introduce a matrix approach for

representation of the breakage process. The terms of the

feed matrix were obtained from either simple relationships

with particle size or from experimental data. The

selection and breakage approach was furthur modified by

Gardner and Austin (1962) who derived a size-mass balance

equation in differential terms to construct a breakage

model for batch grinding. A radioactive tracer technique

was employed to determine the breakage parameters and _

computer analysis provided an iterative solution to their

equation.

12

Using the techniques introduced by the previously

mentioned researchers, several investigators have been able

to present convincing cases for the use of population

balance models as an alternate to the empirical laws of

comminution (Herbst,et.al,1971,1972,1973;Austin,1973).

1.3.3 Population Balance Model

Several authors have discussed the formulation of

population balance models (Herbst,1971;Kim,1974;Reid,1965).

The most useful form of these models is generally agreed to

be the size-discretized, time continuous description. This

model completely describes the breakage process through

size- discretized selection and breakage functions which

may be obtained experimentally. The derivation of the

model is briefly described below.

The distribution of material in a batch mill can be

described in terms of incremental size intervals with a

maximum size, xi, and a minimum size x¤+i. The mass in the

size fraction, ml(i) will therefore be defined as the

fraction bounded above by xi and below by xi+l. Each size

increment can be defined through screening by establishing W

the xi to xi+i size ranges such that they correspond to the

geometric progression of a standard sieve series.

·To describe the time changes in the original

13

distribution of fractions m(i) through m(n) it is necessary

to track the material which is broken out of and into each

individual mass fraction. Most size—mass balance models

use the "first-order hypothesis" which states that the rate

of breakage of any size is proportional to the amount of

that size present in the mill. In this instance, the rate

of breakage is represented by the equation

Rate of Breakage = Si mi(t) H (1)

where Si is the fractional rate of breakage of size i

(time'l) and H is the total mass of material being ground.

The materiai which enters into a size fraction as a

result of the breakage of larger particles must be counted

to complete the mass balance. The breakage function, bij,

is defined as the fraction of primary breakage products of

material in the jth size interval which appears in the ith

size interval. Using this value, the rate of appearance fo

material in the ith interval from the jth interval can be

expressed as

biijSj mj(t) H (2)

To describe the full breakage of particles from the j

size into smaller intervals, a cumulative breakage function

can be defined as follows: _

14

i

B(i„J) = ;- bkk, (3)=n

j=l

By final mass balance of particles leaving and entering a

size interval, the basic form of the batch grinding equation

is achieved.

(4)dt _ 1J J J T l

3=l

The selection function may be dependent on the

operating characteristics of the mill at time t and is

thus said to be environment-dependent. The breakage

function, on the other hand, is thought to be a functionI

of the characteristics of the material being ground and

therefore environment-independent (Herbst and Fuerstenau,

1968).

The above model has successfully been applied to dry

ball milling systems but it may not be strictly applicable

to other systems. Since it is desirable to take advantage

of the mathematical simplicity of a linear model for

engineering applications, it is useful to study the

possible application of the population balance model to

other grinding systems.

15

1.4 Scope of the Present Investigation

Previous studies on the stirred ball mill process have

focused on the grinding of many different types of

materials. Little work has been done which specifically

concentrates on coal. In this investigation, the stirred

ball mill grinding process is examined as it relates to

producing ultrafine coal products with a mean particle

size less than 10 microns. A primary interest in this

investigation is to determine the effect of the

controllable variables in the milling process on grinding

rate and energy consumption. Contollable variables

include media size, weight percent solids of the slurry,

rotational speed of the central shaft, and grind time.

The population balance technique for grinding simulation

is also applied for the stirred milling process to

determine its ability to represent product distributions

over short and long grind periods.

CHAPTER 2

Experimental Techniques

The experimental portion of this research consisted

mainly of grinding experiments performed on one coal,

using one stirred ball mill, operated in a batch mode.

Standard sizing techniques were used on coarse material

while sub·sieve material was measured using a particle

size analysis system similar to the familiar Coulter

Counter. Power input to the mill was measured using a

torque sensor mounted on the central shaft of the stirred

ball mill.

2.1 Coal Sample

The coal used during this investigation was obtained

from United Coal Company's Wellmore operation in

Southwestern Virginia. The raw product was a bituminous

coal, typical of the Appalachian region, which was mined

from the Elkhorn seam. Approximately two hundred pounds

were removed from a raw coal stockpile and stored in

barrels at the lab.

The large coal was reduced below 20 mesh by passing it

through laboratory roll crushers. This product was then

cleaned in a heavy media Suspension at a specific gravity

16

17

of 1.30 to reduce the ash level to approximately 6.5%. A

large portion of the cleaned coal was hammer milled and

wet screened at various sizes to produce monosized feeds

for determining grinding rates. The sized products were

split into 350 gram lots and stored for grinding

experiments. Other portions of the cleaned coal were

passed through a small hammer mill and stored for

subsequent testing.

2.2 Grinding Equipment

All experiments conducted during this investigation

were performed in a batch mode using a stirred ball mill

designed and constructed specifically for this research.

The mill, shown in Figure 2.1, was designed for effective

swirling action of the media and material to be ground.

Rounded shafts, chambers, and pins were used to minimize

the the amount of internal wear and to provide for easy

construction.

In order to eliminate rust problems, the mill shell,

impeller pins, and impeller shaft were all constructed

from stainless steel. The mill shell consisted of a

1/8—inch thick stainless steel tube having inside

dimensions of 5 inches in diameter and 6 inches in length

1 8

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Q‘’

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Figure 2.1 — Three dimensional schematic of batch stirredball mill.

19

as shown in Figures 2.2 and 2.3. The shell was surrounded

by a larger tube to create a cooling jacket for the mill.

The sealed jacket was fitted with two small steel tubes

which were used to circulate cooling water across the

inner vessel as tests were conducted.

The central shaft consisted of a 3/4-inch diameter

stainless steel rod with six, 3/8-inch diameter pins to

provide stirring. The pins were 4 inches long and were

offset by 90 degrees at 1-inch increments along the length

of the shaft. The mill lid was allowed to rest on top of

the mill in most cases; however, for very high stirring

speeds the lid was strapped in place. A-teflon bearing

was used to provide a seal between the lid and the central

shaft.

The entire mill assembly was mounted on a variable

speed drill press with a 1 horsepower drive motor as shown

schematically in Figure 2.4. A series of belts and

pulleys enabled the stirring speed to be varied from 200

to 3500 rpm. A movable support platform allowed the mill

to be raised or lowered for easy loading and unloading of

the sample and media.

A Teledyne Model A-05 torque transducer with a

rotational speed pick-up was used to provide a measurement

of power input and rpm. The signal from the transducer was

amplified on a Brewer Engineering Model DDJ-335A/2 signal

20

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21

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22

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23

amplifier to provide a digital readout of torque in

inch—pounds, rotational speed in rpm, and energy input in

horsepower. The torque transducer was mounted above the

mill on the central shaft. Grinding media consisted of

l/4- inch, l/8—inch, 3/32-inch, and 1/16-inch steel

burnishing media obtained from Pioneer Steel Ball Company.

The media were formed from a hard carbon steel and were

thus prone to rusting. Stainless steel media are more

desirable if one can afford the added expense. The carbon

steel balls required a great deal of care to prevent

rusting. Especially in the small sizes, these balls

retained a significant amount of water and had to be dried

immediately after use.

2.3 Sizing Equipment

Sizing of the raw feed and the product was carried out

using 8" diameter Tyler series sieves ranging from 20 to

500 mesh. A mechanical sieve shaker, manufactured by

Ro—Tap, was used for preliminary separation of dry coal,

followed by manual wet screening to insure proper size

segregation. An Electrozone 8OXY particle size analyzer

manufactured by Particle Data Inc. was used to determine

the size distribution of sub·sieve particles appearing in

both the product and feed samples.

24

The Electrozone analyzer uses a measurement principle

in which particles suspended in an electrolyte flow

between two electrodes through a small orifice contained

in a glass tube. As particles pass through the orifice,

electrical pulses are generated at rates from a few

thousand to a few hundred per second, depending on flow

velocity, orifice size, and particle concentration. When

the particle concentration is kept low, individual

particles can be counted and sized. The amplitude of each

pulse is directly proportional to the volume of the

particle. By assuming the particle to be roughly

spherical, the volume can be converted to an "equivalent

spherical diameter" as is often used in sedimentation

measurements. Size distribution data obtained by

measuring particle pulses is digitized and stored in

computer memory. A schematic drawing of the particle

measuring circuit is shown in Figure 2.5.

Ground products from the stirred ball mill often had

size distributions which spanned the range from screenable

sizes to sub—sieve sizes. In all cases, the products from

the mill were carefully wet screened to capture any

material larger than 500 mesh. Two chambers, each

containing a different sized measuring orifice, were

available for sub—sieve size analysis. Each orifice

allowed acquisition of distribution data within a specific

25

I U

•

D

MANOMETER • • VACUUM

ELECTRODES ORIFICETUBE

STIRRER

ORIFICESAMPLE

8ELECTROLYTE

Figure 2.5 — Schematic of size measuring technique employedby Elzone 80-XY particle size analyzer (Davis,1986).

26

size range. Because the ground product size distributions

often occured across a broader size range than either

orifice was capable of counting, both orifice tubes were

required for complete analysis. The distribution data

from each tube was collected and stored through an on—line

link up with a Perkin Elmer mini-computer. A program was

written which combined the distributions from each tube

along with any size data which may have been obtained

through wet screening. The resulting solution was a table

showing the complete product size distribution by weight

over a chosen range of size classes. The program also

calculated a cumulative weight percent passing for each

size fraction. The complete program along with an example

of the computer generated size distribution curves appears

in Appendix A.

2.4 Torque Measurement

The torque applied to the central shaft of the batch

mill was measured to accurately define the power drawn by

the mill during grinding. The Teledyne Model A-05 torque

transducer, attached directly to the shaft, contained

strain gauges which measured torque deflection and

transmitted an output signal to an amplifier manufactured

by Brewer Engineering. The amplifier converted the analog

27

signal into a digital display of torque in inch—pounds.

The transducer also monotored rotational speed of the

shaft which was similarly displayed by the amplifier.

The displayed torque and shaft speed were used to

calculate the power applied to the shaft using the

following formula:

P = (2 $1 n) T. (5)

where

P = power input by the shaft (in—lb/min)

T = torque (in-lb.)

n = rotational Velocity (rpm).

By converting power into Kilowatts, the equation becomes

-5P = 1.182 x 10 Tn. (6)

If the specific energy of grinding, E, is defined as the

amount of energy consumed during grinding in Kwh/Ton, an

equation for specific energy can be derived from the power-

torque-relationship:

28

Pt (2 $1 n) Tt

E = ·——·—— = ——··—·——·—— (7)

W W

where

t = time (sec)

W = weight of material ground (tons).

Applying the conversions

51 ton = 9.0703 x 10 grams

and 1 hour = 3600 sec

and combining equation (5) with equation (7)

( 1.182 x 10 )(9.0703 x 10 ) Ttn

E = ——·———·——————·———·—·———·——··—————— (8)

3600 w

E = 2.978 x 10 Iäg-(9)

where

E = specific energy of grinding (kwh/ton)

29

t = time (sec)

T = torque (in—lb)

n = rotational Velocity (rpm)

w = weight of material being ground (grams).

The digital display of torque was generally constant

across the time period used for each test. Initial

readings on torque appeared slightly higher than readings

occurring later during the testing. These higher readings

can most likely be attributed to the torque sensing

equipment since they occur in the absence as well as the

presence of grinding media and material regardless of the

test conditions. Values displayed by the amplifier were _

observed regularly in 15 second intervals and recorded.

The values were numerically averaged to determine the

torque applied over the test period. This procedure is

similar to that used by SepulVeda(1982) after he

determined that the minor Variation in torque readings

could accurately be approximated by assuming a linear

relationship in torque change throughout the duration ofthe test.

30

2.5 Experimental Procedure

Initial tests on the 5—inch stirred ball mill were

conducted by charging the mill with media such that the

vessel was 50% filled by volume. An amount of coal slurry

which corresponded to 100 % filling of void spaces was

blended with the media as it was loaded in the mill. In

the tests involving very short grind times, it was

particularly critical to load the media and coal slurry so

as to obtain good blending and filling. This was

accomplished by carefully measuring the quantities of

coal, water, and media necessary for the test and then

layering the inner vessel alternately with the media,

coal, and water.

The original ball charge at 50% mill filling was

weighed to establish the general weight of media which

would be used during all subsequent tests. The ball

charge consisted of an equal mixture of 1/4-inch, "

1/8—inch, and 3/32—inch balls and weighed 3610 grams.

Rather than constantly changing the amount of the

monosized media to correspond to 50% mill filling, the

media weight for each test was held constant for all ball

sizes to allow accurate comparison of torque and energy

requirements.

31

Similarly, the original calculations on void filling at

43% solids indicated that 150 grams of coal would be an

appropriate sample size for grinding. All subsequent

tests were conducted on the same weight of material,

unless otherwise noted, reqardless of particle size or

size distribution. The amount of water used during the

testing was adjusted to achieve the desired percent solids

of the feed. No compensation was made in the weight of

media or coal to account for the increased or decreased

volume of water in the grinding chamber.

2.6 Product Sampling

The first test results were obtained by charging the

mill, grinding, and then discharging the entire mill

contents for media separation and sizing. The ground

product was then discarded and the entire process repeated

to obtain product sizing at the next time increment. This

process required extensive sample preparation and a great

deal of lab time.

To reduce the time requirements, a product sampling

technique was developed. The technique involved halting

the grinding process at various time increments and

withdrawing a sample of the product. A scoop sampler was

used to withdraw approximately 20 grams of material from

32

the mill. The withdrawn material consisted of both ground

coal slurry and media with the slurry fraction weighing 4

to 5 grams. An equal weight of clean dry media was placed

back into the mill prior to additional grinding. This

process was repeated over the entire testing period; thus,

several data points could be obtained without discharging

the mill. Great care was taken to obtain a representative

sample by withdrawing material from several areas of the

mill at each time increment.

A comparison of size analysis data obtained by totally

discharging the mill and by using the sampling technique

shows good agreement between the product distributions.

Figure 2.6 shows a comparison of cumulative percent

passing versus size for a hammer mill feed ground for one

minute. Comparison of the actual point values of these

distributions, shown in Table l, demonstrates that there

is no more than 5 percent variance in the calculated

passing values at each size increment. After only one

minute of grinding, the potential for variance in the two

product distributions should be greater than most tests

done in this research. After longer periods of time, the

mill material becomes more fluid and homogeneous and the

sample is thus more likely to be representative of theu

entire mill contents. lt is important to remove both coal

and media in order to fully account for material which may

33

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be coated around the balls or trapped in void spaces. At

very high percent solids concentration, or during dry

grinding, it may be necessary to evaluate product size by

discharging the entire mill contents since there appears

to be severe coating of material which would be difficult

- to sample using the scoop technique.

CHAPTER 3

Experimental Results

Several operating variables have been examined in terms

of their effect on energy requirements and rate of size

reduction. These include media size, rotor speed, weight

percent solids, feed size and grinding time. Previous

investigations concentrated on extended grinding times and

very large energy inputs. The data presented in this

section represents coal grinding at grind times generally

less than 50 minutes and energy inputs below 100

kilowatt-hours per ton. For those tests conducted under

conditions of constant energy input, the time of grinding

was varied depending upon the measured torque values.

3.1 General Operating Characteristics

To establish a preliminary understanding of the energy

requirements for operating the stirred ball mill, torque

measurements were recorded under a variety of operating

conditions. In all cases, the feed coal had a natural

size distribution produced after hammer milling 1/4—inch

material. Typically, this coal was 60% passing 100 mesh

and contained 14% mineral matter. The torque

relationships are important since shaft torque is

’36

37

proportional to energy input during grinding. This data

is shown in Figures 3.1 through 3.3.

Contrary to conventional ball mill grinding, no maximum

is observed in the plot of torque vs. media load shown in

Figure 3.1. Instead, a continuous increase in torque can

be seen as media load is increased up to 100% filling (7.6

kg) at constant stirring speed. Since there is no

cascading or centrifuging of material in a stirred ball

mill, the torque is always seen to increase as the mill

load increases regardless of how that increased load is

achieved.

Figure 3.2 demonstrates the effect of increased shaft

speed on shaft torque using a constant weight of media.

It is observed that as the stirring speed increases the

torque value also increases. This relationship is true

for all ball sizes. The addition of coal to the mill has

little additional effect on the measured torque since this

extra load is only a fraction of the load created by the

grinding media. In fact, tests were conducted with up to

300 grams of coal at 43% solids with little effect on the

torque readings.

In examining the effect of ball size on measured

torque, the overall weight of material in the mill was

held constant. Figure 3.3 indicates that as the ball size

decreased the measured shaft torque also decreased. It

38

40

-1/6 u~.6ALLs-43% SOLIDS-350 RPM

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GEEE 20DOCZOI-

10 0 WITHOUT COAL•

100% INTERSTIALFILLING

.

0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

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Figure 3.1 — Effect of media load on torque requirement.

39

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appears that it is easier for the impeller to move through

the smaller balls than through the larger balls. This is

indicated by the observed fluctuations which occur on the

torque sensing equipment as well as by the noise created

during mill operation. During torque measurements using

the 1/4 —inch balls, the displayed values of torque changed

rapidly within 0.5 in—lbs of the mean. As the smaller

balls were tested, the degree of fluctuation about the

mean decreased. At the smallest size the fluctuation was

below the measurable limits of the sensing equipment. The

amount of noise produced from interaction between the

media, chamber walls and impeller followed a similar

pattern with the noise decreasing as the balls became

smaller.

3.2 Equal Energy Testing

3.2.1 Feed Percent Solids

To examine the effects of Variations in the feed

percent solids, the stirred ball mill was operated under

the conditions given in Table 2. During this series of

tests the initial weight of the coal sample remained the

same, but the amount of water was varied for each test to

achieve the desired feed slurry.

42

Table 2 - Standard Conditions for comparison of FeedPercent Solids

Sample Weight .................... 150 grams

Feed Size ........................ 60% -100 mesh

Rotor Speed ...................... 335 rpm

Media Weight ..................... 3800 grams

Media Size ....................... 1/4,1/8,3/34 mix

Coal Type ........................ Elkhorn Seam,13.7% Ash

43

The relationship between feed percent solids and

product size was investigated by grinding at solids

concentrations of 20, 43, and 60 percent by weight while

holding the energy input to the mill constant. This was

accomplished by adjusting the time of grinding based on

the measured torque on the central shaft during the

milling process. Within the range of solids

concentrations investigated, the 20% mixture required the

least amount of torque. The torque requirement increased

with an increase in solids concentration.

Figure 3.4 demonstrates the results of grinding at an

energy input of 94 kwh/ton. At this level of energy

input, the product distributions are identical with a

median size of 5.2 microns. This indicates that at low

energy levels, the grinding efficiency is unaffected by

solids concentration. The time requirements, on the other

hand, favor a higher solids concentration since it will

allow equal energy efficiency at higher mill throughput.

For example, the 60% solids mixture was ground for only 10

minutes while the 43% and 20% mixtures were ground for

14.2 and 20.5 minutes respectivly.

To further examine the effect of solids concentration,

grinding was continued with a larger energy input. Figure

3.5 shows the resulting product distributions obtained

after 280 kwh/ton of energy was input to the mill. After

44

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this considerably larger energy input, the 60% solids

mixture produced a product with a median particle diameter

of 7 microns and the 20% and 43% solids mixtures produced

a product with a median particle diameter of 3.8 microns.

This indicates that after an extended grinding period, the

grinding efficiency is influenced by solids concentration.

A higher solids concentration seems to adversly effect the

ability of the stirred ball mill to efficently reduce the

feed to extremely fine sizes. Again, the time of grinding

was significantly shorter for the higher percent solids

feeds.

It is interesting to note that an additional 186

kwh/ton energy input was required to further reduce the

5.2 micron product to 3.8 microns. This large amount of

additional energy which was necessary to reduce the end

product by only 1.4 microns in diameter demonstrates the

significance of a small change in the median particle size

of a ground product in the ultrafine size ranges.

3.2.2 Dry Grinding

As an extension of the investigation of feed percent

solids, the mill was charged with 150 grams of dry coal

and run using conditions similar to those used for wet

grinding. The feed consisted of the same Elkhorn seam

47

coal at 6.5% ash. The mill water jacket was used to cool

the inner vessel during the dry test since there was

considerable heating of the inner chamber. The mill was

again operated in a batch mode. There was no air sweeping

of the inner chamber during the dry grind to attempt to

remove ultrafine material.

Figure 3.6 demonstrates the product size distributions

attained for wet and dry grinding after an equal energy

input of 90 kwh/ton. The product distributions indicate

that the addition of water enhances the grinding

efficiency of the mill. A median product diameter of 8.5

microns was measured for the dry grind which compares to a

median product diameter of 5.2 microns in the wet grind.

It was apparent during this test that there may be some

error in the measured particle sizes of the dry

distribution. This is suspected since there was a heavy

coating of fine coal on the balls and central shaft. This

indicates that there may have been particle breakage

followed by melting and agglomeration of fine particles.

Since a particle agglomerate would be counted as a large

diameter particle in the size analyzer, the true particle

distribution may be misrepresented. Testing at higher

energy inputs seemed inadvisable after the results of the

initial testing.

48

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3.2.3 Shaft Speed

By varying the belts and pullies on the power drive,

it was possible to investigate the effects of rotor speed

on grinding efficiency. The lowest attainable speed was

216 rpm and the highest speed investigated was 830 rpm.

All mill conditions were held constant during the tests

and a ball mixture was used to minimize any potential

effects of ball size. The feed consisted of Elkhorn seam

coal containing 13.8% ash and having a natural size

distribution resulting from one pass through a hammer

mill.

In Figure 3.7 the product size distributions for

grinding at 216, 521, and 830 rpm are plotted. Each

product was attained after grinding at a fixed energy

input of 63.5 kwh/ton. The test results indicate that for

low energy inputs, slower shaft speeds provide more energy

efficient grinding than faster shaft speeds. In this

particular instance, the 216 rpm test reduced the 65

micron feed to a median product diameter of 6 microns

while the products produced at 521 and 830 rpm had median

diameters of 7.4 and 9.0 microns respectively.

The relationship between shaft sppeed and median

particle diameter shown in Figure 3.8 indicates that there

is a linear relationship between the size produced and the

50

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shaft speed between 200 and 900 rpm at low energy inputs.

Although it appears beneficial in terms of energy

efficiency, the longer time requirements for grinding at

slower shaft speeds may be prohibitive. As can be seen in

Table 3, the grind time required at 216 rpm was over 10

times that of the 830 rpm test at equal energy input. By

extending the energy input over a longer period of time,

however, the product size was reduced by an additional 3

microns.

3.2.4 Ball Size

Product size comparisons were done using balls of 1/4-

inch, 1/8—inch, 3/32-inch, and 1/16-inch diameter at 216

rpm and 43% solids concentration. The energy input was

again held constant by adjusting the time of grinding

based on the torque values measured on the central shaft.

Since the smaller balls required less shaft torque, the

time of grinding during the tests with the small balls was

longer at equal energy input.

Figure 3.9 demonstrates the relationship between ball

size and product size distribution for grinding a coal

feed at an energy input of 90 kilowatt-hours per ton. It

is apparent from the diagram that at equal energy input

the smallest particle sizes are produced with the

53

Table 3 - Products produced at 64 kwh/ton and variedshaft speeds

Median ProductRPM Time (min) Diameter (microns)

216 23.4 6.0'

521 5.0 7.4

830 2.14 9.0

54

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1/16—inch balls. It was also found that as the media size

increased, the product size also increased. A ball

mixture consisting of equal weights of 1/8-inch,

3/32-inch, and 1/16-inch balls produced a product

distribution falling very near the average of the products

produced from the monosized media tests. Table 3 lists

the median particle sizes of the products corresponding to

each ball size after 90 kilowatt—hours per ton energy'

input as well as the grinding time required to input 90

kwh/ton.

Figure 3.10 demonstrates the product size

distributions attained when different ball sizes are used

to grind a 100x150 mesh monosized feed. In this test, the

time of grinding was held constant at 10 minutes while the

media size was varied. The size distributions of the

ground products appeared to be nearly indentical with a

median particle diameter of 8 microns. The energy input

required for grinding to this size varied with media

diameter. Again it was found that the most efficient

grinding occured when the smallest balls were used. This

test indicates that with 1/16—inch media it is possible to

reduce a feed with a median particle size of 128 microns

to a median size of 8 microns with an energy input of only

35.1 kwh/ton.

Grind times of 1 minute and 5 minutes were also

56

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examined using each ball charge. In the case of these

short grind times the product size distributions produced

with the different sized media were not the same. In

Figure 3.11 a 100x150 mesh monosized feed is shown after

grinding for 1 minute. This figure shows that the finest

product distribution is produced with the larger 1/8—inch

balls. A slightly larger product is produced as the balls

become smaller. As grinding progresses to 5 minutes, the

product distributions begin to become closer together as

can be seen in Figure 3.12. This seems particularly true

in the coarser ends of the product distributions. At this

stage of grinding it is apparent that the smallest balls

produced a slightly finer product than the larger

3/32-inch and 1/8—inch balls. The 1/8—inch balls have

nearly the same product size as the 1/16—inch balls, but

the mid sized 3/32—inch balls did not grind the coal as

fine as either of the other two sizes.

At the 10 minute stage of grinding, all distributions

fall together, as was previously indicated, with a median

size of 8 microns. This series of tests, conducted on a

100x150 mesh feed, presents evidence that there is a

relationship between the rate of particle size reduction

and ball size. The tests also indicate that particle size

of the feed infuences the rate of grinding.

To examine the effect of ball size on the rate of size

58

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59

E 5 EC C C-:-1

ELD (D N OQ ; Q ·-<•-1 gr) +—* •D Zw-O .¤"¤1*I ·:-I Q)

P CD+.><6-4U).,..;_g'U C!)D CDLO CD EN.,.4C)4UELD-1•gi_5, Q gw

E Q 0 woI LD -:-1 L OGum 5:-:II

QD .;-4*"‘/

' L U'} E‘”... ¤"¤|••

4 ELDR Q :.,..4

O N M aos~! 4 "*_,_, -I-«cn mf

•—‘ Z5Ü} < Q) ,-.4 ::-1 cd--40 Q?

¢ E <'•"I

am—‘—·‘ cE td

ca ~1 *"Q) L:

.:1 “-•“-·E US

IN-1CO"‘cuO O D <'I PD LO *'ISD-1

ÖUILSSQCI QUQOJBCI 8/\'§lE[NlUH[)E

60

reduction, the median product diameter was traced as it

changed through a 30 minute grind time. This plot appears

in Figure 3.13. As indicated by the product

distributions, the smaller balls do not grind the 100x150

mesh monosized feed as quickly as the larger balls over

the first 3 to 4 minutes of grinding. After 3.5 minutesA

of grinding, the median particle diameter produced using

the 1/16—inch balls is the same as that produced by the

3/32—inch balls. The median diameter produced by each

ball becomes equal after 10 minutes of grinding.

3.3 Analysis of Breakage Paramaters —

To quantify the process of particle size reduction in

greater detail it is necessary to determine the breakage

paramaters which were discussed in Chapter 2. There are

several techniques which may be employed to determine the

selection and breakage functions for use in the population

balance model. Two experimental techniques involve: 1)

grinding a series of monosized feeds, and 2) using

radioactive tracers. In this investigation, monosize

feeds were prepared and tested for those size ranges which

could be prepared through screening. The values for sizes

which were below 500 mesh could not be tested and thus

were projected through techniques developed by Herbst and

61

4: 1.: _:

In O N Ö Ü Q;D J-> ••• m U;

*4 M z: «-A·•-4 •—a

E GS• .¤Q‘CG J:

o m s-GJ ·—a

>< ua >'Q Ü O ®

o ~•-• \-•E

r-• L •—<

• • 13E nn ae "' g §an 01 m N,LL M V __I

I

07 +->_I¤_-} g ¤-:5O •

_„·°_.•' _•° _" L CD

-· E°"“‘

__••‘ _•' _•°C v-4

·¤Q-1 •P1

Eg-{

__.-'° _•°° ,•° C.,-4 „,-4

: :: :cx:

L• .· U bbQ S

-•‘°•:0

O O O O O O O O O O Ö O Ö O Ö EfV M OJ *4 O O1 G I\ LD LC) V O') CU *4 :Q'! V'! 71 I'! V"! bo

·¤-4

inJS‘l9lUE”§Q ÄÜÜDOJÖ UECDSW

62

Fuerstenau (1968).

3.3.1 Selection Function

The selection functions for the screenable size

fractions were determined by plotting the feed

disappearance curves of each monosize as it was ground at

335 rpm and 43% solids using l/4—inch media. A typical

disappearance plot obtained for a series of monosized

feeds is shown in Figure 3.14. Over the time period

plotted, first order breakage kinetics were observed.

This finding has also been reported elsewhere (Sadler,

Stanley and Brooks, 1975; Herbst and Sepulveda, 1978). In

this case it was necessary to use very short grinding

times since the feed disappeared so quickly in the coarse

size range.

A steeper slope on the feed disappearance plot

indicates a higher value for Siand thus a higher rate of

size reduction. Contrary to the typical disappearance

plots presented by prior investigators, the slopes do not

continually decrease with particle size. In this case,

the rate of breakage of the largest fraction, 28x35 mesh,

does not appear to be as fast as that for the next smaller

size fraction. This observation lead to the preparation

of Figure 3.15 which demonstrates the relationship between

63

o O OÜ

° ua>< O >< EJ >< G IZ -X- >< O >< I :Q ID G ln O OON m Q WJ-?

OGL4, Q-4

v-•cbN-„-I

U23O-„-JL-

LD cd. <• u >G CD

gl

Q) u-44-J

¤ EC o••"'*-« „' Z mc:

·«-4· ·• ¤ “"

- 22;%O 2 CEG.,-4L:

I- cduua Q2:E ä'“

äg as;¢.o1CDU}

Cu ZS

I

· VJ-J„ •

o o o °°O O') "•cp"' EbbÜU§U§ElI13[j 8Z'§S [388;] }0 1U8OJ8d g:

64

S-C)

4-*CI)EC'3

~-4'U

CD•—•U-•-4

4-*S-C6Q

U)"C5.„ „ E gg'¤«-·•

L „_,

ESS U.,-4 U)E$° Z g•

66: 8 -‘·

U')Q-• U') L gCD

-6-J cCD O C;.„-• __

40.,.6···• ääQ 5 E¤-1

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ct'! CQ)··-4$-4 CD

CQ 3

I

LO4-«

° gv

UOFÄCJUTH 6125 8ÜQ>|€8«JQ gbCr-

65

feed particle size and rate of size reduction using

1/4-inch balls. This figure shows that the rate of

grinding increases as the particle size increases to a

maximum at the 35x48 mesh size class. After this maximum,

there is a decrease in grinding rate as the particle size

becomes smaller.

The selection functions for sub-sieve size fractions

were determined by plotting the known values of S against

the normalized size function as has been previously done

by Herbst and Fuerstenau (1968). Figure 3.16 shows that

the experimentally determined selection values fall along

a straight line on a log-log plot. The sub-sieve

selection values were extrapolated from the curve under

the assumption of first order breakage kinetics. The

resulting values are presented in matrix form in Appendix

B.

3.3.2 The Breakage Function

The cumulative breakage distribution functions were

estimated from a relationship between the breakage

function and the selection function presented by Herbst

and Fuerstenau (1968). The relationship is presented as

follows:

66

bbC

·»-IU}Z

Q)

EO

U)Q E

~ ä<—•

LE + Eo

gw X ~«-¤-¤

·f'\ Le. 4.>

ESEX Q)

COC; E··‘•fDL QQ -

G)Äää c> °° °°~—•·<rm -• N

U? U)

r: S3CO >CDE 3

O g S.‘ c3LO L gw"’

E3ä EO C

·-ICDO-S

LD e.>CU·r4

m mg,E:

I

$3O LD O ID •

-

•

-

• cf)

CUv‘• <—* O Q)

saC

um 'g ‘u0;10un;4 uorqoayag ¤„-I-ICu

67

where Fi is the fines production rate which can be

determined from the slope of a plot of the cumulative

fraction finer than Xiversus grinding time. Several

investigators have reported that the breakage distribution

function is normalizable for many different

materials(Herbst and Fuerstenau, 1968; Klimpel 1964;

Austin et al., 1971). This means that particles of all

sizes for a particular material will break down to the

same fractional distribution. Thus, a l0g—log plot of QJ

against the relative size Xi/%j%+qshould result in a

straight line. This type of plot, shown in Figure 3.17

did not result in a single straight line for the data

prepared in this investigation. Instead, a series of

lines can be drawn through the data points with each

successively smaller size fraction following a line which

is closer to the previous line. It is suspected that the

relationship demonstrated in Figure 3.17 can be related to

the possible inaccurate measurement of the finest sizes by

the particle size measurement technique. Nevertheless, an

assumption was made that the breakage functions for the

sub-sieve sizes were normalizable along the line produced

by the lowest experimentally determined breakage values.6

The complete breakage function is given in Appendix B.

68

an °{ g¤¤ 2 ° ~—· ‘ N 3D O ; O

U)V la I ¤ ; ID:

“E

“·¤0Q N• ·•-I‘=„

*1 E°• \E0 ·x¥°··Li~ <1 *-\-. 0••.•.r\‘\=\‘‘\

Q-:

X >\"•

\ •*.U. gg g\‘\‘°•

\'¤> S3°‘.

X > Q°•· \ •|¤|

4-* ¤>-_X GU > 'pq ·•-I CD

¤> ‘;E°°•_ ‘s‘ E p—• cg

5E "s‘ Z5 GS0 >•· \

°'. \s |°• \

:~v-I«-• .• C"}o an *1 ° Q,«-•

O:;0~r-(cz.

UOT§1OUH;{ BÖEXEBJQ S/\§T|„9[|'ll.IH']Q

CHAPTER 4

Discussion of Operating Conditions

4.1 Percent Solids

Tests conducted in this research indicate that there

is no change in coal grinding efficiency between 20 and

60% solids for energy inputs of 94 kwh/ton. This tends to

agree with the findings of Sepulveda (1980) for grinding

of several different materials between 20 and 50% solids.

For a higher specific energy input (280 kwh/ton), however,

a coarser size distribution is produced at 60% solids -

while the distributions at 20% and 43% are, once again, .

identical.

As the coal becomes finer, the surface area of the

entire mass increases. The increased surface area will

possibly result in increased adsorption of water and thus

create a highly viscous mass. Sepulveda showed, through

grinding with heavy oils, that increasing the viscosity of

the mill material is detrimental to efficient grinding.

This effect seems to apply in the case of high percent

solids as well. During high solids testing the inner

vessel walls become heavily coated with a pasty layer of

media and fine coal. This action renders the balls

motionless and thus they are unable to make contact with

69

70

fine particles and other media.

Provided the energy input is not excessive, it is

possible to reduce the time required for size reduction

* and increase mill throughput by grinding at higher solids

concentrations. This is true at 94 kwh/ton between 20

and 60% solids where the higher concentration results in a

50% reduction in residence time.

It may not be valid to assume that grinding

efficiency is independent of solids concentrations below

94 kwh/ton. For example, below 20% solids, it is expected

that fine coal particles would be effected by ·

hydrodynamics in the grinding chamber. Since the fine

coal has a specific gravity very near to that of water,

particles would seem to travel with the grinding fluid and

thus have a lower chance of impacting directly with the

balls or between the balls and the grinding chamber.

If the size or type of media changes, the conclusion

of equal energy efficiency at varying percent solids may

become invalid. Very small or light media is more likely

to be influenced by the amount of water which is added to

the mill. At very low solids concentrations, for example,

the light media may flow too easily with the mill

material.

71

4.2 Rotor Speed

The chief observation made during the examination of

various stirring speeds was that lower speeds provided

finer grinding than higher speeds at equal energy input.

The tradeoff, however, is that a significantly longer

period of grinding is necessary to achieve an equal energy

input as the shaft speed decreases. Since the minimum

allowable shaft speed in this research was 216 rpm, it is

not known to what minimum speed the ball mixture will

continue to grind with increasing efficiency. At some

point the rotor speed will become too slow to provide

enough kinetic energy for the media to'effectively shear

or crush the material being ground.

From observations of the grinding process during

numerous tests it was observed that the increased

efficiency at lower speeds may be the result of a more

fluid swirling action of the media in a lower breakage

zone. The slower speeds ground the product with much less

noise and less fluctuation in torque readings. More noise

indicates that there is increased media interaction,

increased turbulence in the rotating mass, and accelerated

mill and media wear. None of these actions contribute to

efficient grinding.

Further observation of the shaft pins indicated

that there was a parabolically—shaped grinding zone which

72

appeared in the mill at high speeds. Upon emptying the

mill after one experiment it was observed that the coating

of coal on the top stirring pin existed only on the

outside edges. For each succeeding pin below, the coating

extended closer toward the central shaft. At slower

speeds, the coating may not even appear on the uppermost

pin and the angle of the cone becomes smaller. The

majority of media and coal during slow speed tests seems

to reside in the lower portion of the mill. As the shaft

speed increases, media is thrown to a higher region in

which less contact can be made with coal particles. This

is shown schematically in Figure 26.

Other investigators (Sadler,et.al.,l974; Nguyen,l981)

have demonstrated an advantage for high shaft speed in

terms of energy efficiency. In these cases however, the

rotor speeds were varied at speeds far greater than those

tested during this research. The increase in grinding

efficiency with slower rotor speed as found in this

investigation, may indicate that there is a different

breakage process occuring between very low and very high

shaft speeds. It may be possible that when the shaft

speed increases to a particular level there is turbulent

action in the mill. Increasing the rotation to a higher

speed may cause centrifugal forces which lead to uniform

73

ID

R\\\\\\\\\& &\\\\\\\\\\§

¤ *QI III

\ ' / II ~ / II.‘~_. __„_’ II I „ I

~

\\ 1/

~I \‘ • / §I I I~

—

~

Figure 4.1 — Diagram of parabolic depression occurringin the batch mill at higher Shaft speeds.

74

circulation of media around the inner wall of the grinding

chamber. This action may change the mechanism of breakage

to one similar to low speed ranges and thus increase the

grinding efficiency.

4.3 Ball Size

Based on equal energy input, small balls produced a

finer product distribution than larger balls. This

observation is consistent with the findings of other

researchers who have found a similar relationship for

grinding of coal in attrition and stirred ball mills

(Stanczyk and Feld,l972;Herbst and Sepulveda,l978).

From simple observation of the grinding tests, one

might conclude that the energy input to the mill is being

used more effectively with smaller balls. The mill

operates with much less noise and much smaller variance in

the measured torque values when grinding with the

l/l6—inch balls. This observation indicates that there is

more uniform swirling of mill material in the grinding

chamber. For larger sized balls, there is less uniform

flow and a more noise. Energy is apparently lost to ball

motion in regions outside of the breakage zone.

Since there is a larger number of small balls than

large balls at equal weights, it is possible that

75

grinding efficiency is improved because of the increased

number of points of media contact. In addition, the

smaller interstitial spaces between the smaller diameter

balls may provide less opportunity for very fine particles

to escape contact. The probability of contact may be

related to the rate of particle breakage and is likely to

be a function of both the diameter of the media and the

average particle diameter of the feed.

It was shown during the determination of the grinding

rate constants that a 35x48 mesh feed had an initial

grinding rate which was faster than the size fractions

both immediatly above and immediately below when it was

ground using l/4-inch balls. It is not proper to assume

from this data that l/4-inch balls are optimum for the

35x48 mesh feed since a different ball size might result

in a higher breakage rate. In order to determine optimum

ball size for various feed sizes, breakage rates can be

measured using several ball sizes on different monosized

feeds. Figure 4.2 shows the feed disappearance plots for

a lOOxl5O mesh feed ground with varying media size. Closer

evaluation of Figure 4.2 reveals that the 1/8-inch balls

provide a faster rate of size reduction than l/l6—inch,

and l/4-inch balls. This relationship is more easily seen

in the plot of the breakage rate function vs. ball

diameter shown in Figure 4.3. Apparently, ball size is

76

C C ec2 2 E *5·•·~

„ E. 2N 0 N E1 <1 -x- Q,—· <*· Q 1 2§ « •• °"‘

.:><GJE

I Qu

G -•LOI-«

><

3•-•LQ L,

m N O2 E

‘ "“.E-• O U) ww

IQO Q) +->Q>LU] Qwq

Emu -5 "";*vg;) C ¤.UßU- '•"* CD,-«E ää

° ¤> ;„E ¤>:s•*'* CLOI- ¤.·«-4

EV;.,.l$um

LD¤ -x- <1

“fEE2 S32

I

Nq• 4

c o o fbO M

-• _„_,«-a · CN

ÖU§U§9LU8H SZIQS D88:_{ }0 1U80•JSd

77

lf] .s N ;

Q +->GJEcf

Eüf Z¤ E

2“°

2·•-• m U)Ev-• C s.:QD L) Q)L G) uu C >mg «-• •«-1mm -

¤"' ¤> - Eä 4->

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ä <+-«(Uvr ·•—• Ea o ¤ g

•—1

"' 22CD

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UO§Z}3U|'l;] S19};-| 8ÖQ>|Q8«JQ

78

important to the initial rate of particle size reduction.

Particles which are larger than a specific size (35x48

mesh in the case of 1/4 inch balls) may not be easily

broken because they are not being contacted in the same

manner in which a smaller particle is being contacted.

Tracking the median particle size as it changes over

time, as was done in Figure 3.13, further demonstrates

that there is a relationship between the rate of grinding

of a product and the size of the media. Although the

initial rates of size reduction using the 1/16 and

3/32—inch balls was less than that of the 1/8—inch balls,

the products eventually became equal in size indicating

that the long term grinding rate becomes equal after 10

minutes. This indication however, may not be valid since

subsequent testing revealed that some fine material may

not have been accounted for in the sizing distributions

after long periods of grinding.

Careful examination of Figure 3.13 shows that size

reduction begins immediately for the 1/8-inch and 3/32

inch media and continues at a constant rate until roughly

5 minutes of grinding. The 1/16¥inch balls do not begin

rapid size reduction until nearly 45 seconds after

beginning the milling process. After that time, size

reduction occurs at a very rapid pace until it slows at 5

minutes with the other balls. The rate of size reduction

79

of the median particle size appears to be greater for the

smaller balls than for the larger balls in the range of

100 to 10 microns. This is likely to result from the

ability of the smaller balls to contact coal particles

when larger balls cannot as was previously discussed.

4.4 Energy—Size Relationship

The results of most of the grinding tests presented

in the preceeding chapters can be represented on a single

log-log plot in terms of specific energy input vs. median

particle diameter of the product. Herbst and Sepulveda

(1978) have previously shown that stirred ball mill

grinding can be represented by Charles Equation. By

assuming that the feed size is much larger than the

resulting product size, this equation reduces to the

following form·

E=A<dp>'°‘

<11>l

where E is the specific energy input, A is the Charles

Constant, o< is the slope of the equation when plotted

on l0g—log paper, and dp is the median product size.

A plot of Charles Equation for the data collected in

this research appears in Figure 4.4. Through linear

SO ‘

1000

500CO

-6->\.:3 „.¥

>ZCD

E 100 CCLU

2 OO„_ 50-«-1UGJQ.U)

1010 50 100

Median Product Diameter, Microns

Figure 4.4 — Charles Law plot for stirred ball milling ofElkhorn seam coal.

81

regression analysis, the Charles constant was determined

to be 5896 and the slope was 2.33. These values are near

those reported for stirred ball mill grinding of Lower

Freeport and Montana Rosebud coal by Herbst and Sepulveda

(1978). As was previously reported, it appears to be more

difficult to grind coal in the stirred ball milling

process than it is to grind other industrial materials.

It may be possible that the tendency for coal to

plastically deform makes it more difficult to grind than

harder, more brittle materials.

Sepulveda (1980) concluded that although there is

some minor variation in grinding efficiency which can be

attributed to the operating characteristics in the mill,

stirred ball mill behavior is to a good first

approximation only dependent on the energy input per unit

mass and not the manner in which the energy is input. The

data presented in Figure 4.4 seems to indicate that the

effect of mill operating conditions may infact be more

pronounced than was previously reported by Sepulveda. The

three darkened data points which tend away from the

straight line plot result from the tests of the influence

of shaft speed at equal energy input. As the shaft speed.

increased, the product particle size also increased. This

relationship seems to indicate that if all the plotted

. points on the Charles plot were run at a higher speed the

82

line would be shifted to the right effectively raising the

value of the Charles constant. Coincidentally, all of the

data points available for plotting the energy-size

relationship were obtained while milling at 335 rpm.

On the lower portion of the graph, three points are

plotted which lie in a vertical progression. These points

demonstrate the difference in energy required to produce a

12 micron product by varying the media size. Again, there

may be a slight change in the movement of the Charles plot

depending on the size of the media used in grinding.

CHAPTER 5

Grinding Simulation

The population balance model, described in Chapter l,

can be symbolically expressed as follows:

i—ldlm1<t>Hl'"'‘°'‘'‘" =‘dt

jtl

where S5 and Bqj are experimentally determined parameters

defined as the selection function and breakage function

respectively. Although the above equation has no analytical

solution, it can be simplified to one with an exact solution

if Si and Bij are assumed to be independent of changes in

size distribution in the mill. Under these circumstances,

the size distribution occuring after one breakage event can

be viewed as the feed distribution for the following

breakage event and so-forth. The differential mass balance

equation can thus be solved numerically through matrix

multiplication and computer iteration.

The Crude Euler technique has been used to solve the

population balance equation for stirred ball mill grinding

of coal. The numerical expression for this technique is as

follows:

Yi ‘ Y1-1=^X f(Xi—l ·Y1-1)

83

84

A computer program was written to compute the

particle size distribution resulting from stirred ball

mill grinding of coal when given an initial feed size

distribution and a grinding time. Input data consisted of

experimentally determined selection and breakage functions

‘as described in Chapter 3.

Over very short grinding periods, the computer model

was found to accurately predict the product particle size

distribution of the coarse size classes. In the fine

sizes however, there is a slight deviation from the

experimentally determined product distribution. The model

predicts that a larger quantity of material will be broken

out of the finer size range than was actually measured by

the particle size analyzer. This is demonstrated in

Figure 5.1 after only 1-minute of grinding. Figure 5.2

shows that after a longer grinding time, 5- minutes, the

difference in computed and experimentally determined size

distributions becomes more pronounced.

The increase in product fineness predicted by the

computer model after 1-minute may in fact be more

representative of the true product distribution than the

experimentally determined distribution. The computer

generated distribution indicates that there is a

A possibility that fine particles are not being counted in

the product measuring process. This error is likely since

85

it was not possible to obtain a single orifice tube for

use in the size analyzer which was capable of detecting

the entire particle distribution in the product,

particularly in the finest size ranges.

Because the feed particles disappear so quickly in

the stirred milling process it was expected that the

computer model would begin to deviate from experimentally

determined size data after short grinding times. As has

been observed in the modeling of ball mill grinding, the

first order assumption for breakage kinetics seems to

apply over short grind times but to become invalid after

longer grind times. In the case of stirred ball milling,

the rate of initial size reduction is very rapid and thus

non-linear breakage seems to occur very early in the

grinding process. Non-linearity can be associated with

the fact that the size—discretized selection functions are

not time independent. In addition, overestimation of

product fineness resulted from prediction of the particle

breakage rates for the very small size classes based on

the measured values for large particles.

A complete list of the computer program, written in

basic, is included in Appendix C.

S6

•·—•t¤ .

C 4-: :°·‘u :E _„ .+1 .¤ ;

L. U g3+ c. :>< ¤* Z

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.,-4 Q)-:->-» 2 E:CD äm

Q C1-1 N ><O• -r··1·••

.°. ¤: °€• '—I

CLI • \® L.) Op•-,-1 (DCD•

·«-J--JE ;”~=VU ¤.<1>CL ENQ.-:

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ÖUI§SSEd ÜLUSOJSCI 8A{Z}E[Hl1IÜQ

87

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ÖUTQSSE.;1 I],U3I]L|Bd 3/\T§]E’[VllUNQ

CHAPTER 6

General Conclusions and Suggestions for Future Work

6.1 Conclusions

1. Stirred ball milling was shown to be effective in

producing fine coal products with a median size

between 3.8 and 10 microns.

2. A scoop sampling technique was shown to accurately

represent the product size distribution when

compaired with fully dicharging the mill contents.

3. Variations in slurry percent solids between 20 and

43% were found to have no effect on grinding

efficiency for energy inputs up to 280 kwh/ton.

Below 94 kwh/ton, there was no effect on grinding

efficiency between 20% and 60% solids.

4. Between 840 and 216 rpm, the grinding efficiency of

the stirred milling process was found to increase

with decreasing shaft speed.

5. Tests using monosized ball feeds of 1/4", 1/8",

3/32" and 1/16" diameter showed increasing grinding

efficiency as ball size decreased.

6. The energy required for particle size reduction was

found to be closly approximated by Charles Equation

but dependent on operating characteristics of the

88

89

mill for particle grinding to 5 microns.

7. Initial rate of particle breakage was shown to be

dependent on the relative size of the feed particles

and the media diameter.

8. A computer model of the breakage process, using the

population balance approach, showed good predictivel

capability at very short grind times but deviated

from experimental data after 5 minutes.

6.2 Recommendations

Based on the observations and data obtained in this

investigation, the following are areas recommended for

future research:

1. Since it is desirable to achieve ultrafine product

sizes at minimal cost, additional research should be

conducted to determine the optimum combination of

operating conditions for minimizing energy

consumption.

'2. Grinding test should be conducted using particle

dispersion agents to determine if it is feasible to

reduce the viscosity effect seen at high percent

solids. Such studies could allow lower mill

residence times and higher throughput levels.

3. Studies should be conducted to determine the amount

90

of media wear which occurs in the milling process.

4. The effect of different ball materials should be

investigated as a possible means for reducing mill

energy requirements while achieving fine particle

distributions.

5. The observed relationship between ball size and

particle size in determing the initial rate of coal

breakage should be further quantified through

additional grinding of monosized material. The

effect of ball mixtures should also be investigated

using monosized feeds.

6. Additional studies should be conducted to describe

the mechanism of particle breakage at Very high and

Very low shaft speeds. This work could include the

effects of Variations in the design of the impellerand the chamber walls. ’7. Continuous studies, and large scale studies should

be conducted to verify the observations made using

the small batch mill constructed for this research. ‘

The focus of this additional work should be to

determine the potential for practical application on

and industrial scale.

REFERENCES

' 9 l

92

REFERENCES

Atlantic Research Corporation, 1984. Arc-Coal Coal-WaterFuel, Information Circular 0584, 5390 CherokeeAvenue, Alexandria, Virginia 22312.

Austin, LAL, 1973, "Understanding Ball Mill Sizing",Ind, Eng. Chem. Process Des, Deyelop,, vol. 12, no. 2,p. 121.

Bond, F.C., 1952. "Third Theory of Communition," Trans,AIME, vol. 193, p. 484.

Broadbent, S.R. and Calcott, TLG., 1956. "Coal BreakageProcess - 1 and 2f'Q, Inst, Fuel, vol. 29, p. 524.

Broadbent, S.R. and Calcott, T., 1956. "A Matrix Analysis ofProcesses Involving Particle Assemblies," Phil, Trans,Soc, Lond, (A), vol. 249, p. 99.

Brown, M.D., Adel, G.T. and Yoon, R.H., 1984. "A LaboratoryStirred Ball Mill For Fine Grinding of Coal," 3rdAnnual Collegiate Assoc. for Mining Engrg. Education,Lexington, KY, October 2-4.

Davies,E.G„ Collins,E.W.and Feld,I.Lg l973,"Large-Scale Continuous Attrition Grinding of Coarse Kaolin",USBM R, I, 7771.

Davis,E.G., Hansen, J.P. and Sullivan, G.V. in P.Somasundaran,ed., Fine Particles Processing, vol. 1,AIME, New York, 1980, p. 74.

Epstein, B., 1947, "The Mathematical Description of CertainBreakage Mechanisms Leading to the Logarithmico-NormalDistribution", Q, Frank. lnst,, vol 244, p. 471.

Feld, IA„, McVay, TAL, Gilmore, HA„ and Clemmons, B.H„1960, "Paper—Coating Clay from Coarse Kaolin by a NewAttrition—Grinding Process", USBM R, I. 5697.

Gardner, R.P„ Austin, LAL and Walker, PAL, Jr„ 1961,“A

Phenomenological Approach to the Batch Grinding ofCoals", Special Research Report Submitted to the CoalResearch Board of the Commonwealth of Pennsylvania,Number SR-23.

Herbst, JLA„ 1971, Batch Ball Mill Simulation: An Approachfor Wet Systems", D.Eng. Dissertation, University ofCalifornia, Berkeley, California.

93

Herbst,J.A.and Fuerstenau,D.W„ 1968,"The Zero OrderProduction of Fine Sizes in Comminution and itsImplications in Simulation", AlME Trans., vol. 241,December, p.538.

Herbst, J.A., Grandy, G.A. and Mika, T.S., 1971, "On theDevelopment and Use of Lumped Parameter Model forcontinuous Open- and Closed-Circuit Grinding Systems",Trans. IHM, vol. 80, p. Cl93.

Herbst, J;A. and Sepulveda, J.L. 1978, Fundamentals ofFine and Ultrafine Grinding in a Stirred Ball Mill",Proceedings og International Powderand Bulk Solids Handling, pp. 452-470.

Hukki, R., 1961. "Proposal for a Solominic SettlementBetween the Theories of von Rittinger, Kick and Bond,"Trans, AIMH, vol.220, p.403.

Kick, F., 1885. "Das Gesets der Proportionalen WinderStande und Seine Anwendunger," Arthur Felix Leipzig

Kim, JJL, 1974, "A Normalized Model for Wet Batch BallMilling", PhJL Dissertation, University of Utah, SaltLake City, Utah.

Klimpel, R„ Pomder Technology, 32, (1982) 267.

Reid, K., 1965. "A Solution to the Batch Grinding Equation,“Chemical Hhgrg. §ci,, vol. 20, p.953.

Sadler, L.Y., lll, Stanley, D.A. and Brooks, D.R., 1975,"Attrition Mill Operating Characteristics", PowderTechnology, vol. 12, p. 19.

Sepulveda, JJ„, 1981, "A Detailed Study on Stirred BallMill Grinding," PhJL Thesis, Department of Metallurgyand Metallurgical Engineering, University of Utah,Salt Lake City, Utah.

Siddique, M„ 1977.“A

Kinetic Approach to Ball Mill sca1é—up for Dry and Wet Systems," M.S.Thesis, Dept.Mining, Metalurgical and Fuels Engrg, Univ. of Utah,Salt Lake City, Utah.

· Stanczyk, M.H. and Feld, I.L., 1963, "Continuous AttritionGrinding of Coarse Kaolin. Part I. Open-CircuitTests", HSHH_R, I, 6327, 28 pp.

94

Stanczyk, M.H. and Feld, I.L., 1965, "Continuous AttritionGrinding of Coarse Kaolin. Part II. Closed—CircuitTests", QSBM R. I, 6694.

Stanczyk, M.H. and Feld, I.L., 1968, "Investigation ofOperating Variables in the Attrition Grinding Process",USBM R. I, 7168.

Stanczyk, M.H. and Feld, I.L., 1972, "Ultrafine Grinding ofSeveral Industrial Minerals by the Attrition—GrindingProcess", QSBM R, I, 7641.

Stanley, D.A., Sadler, L.Y., III, Brooks, D.R. and Schwartz,M.A., 1974, "Attrition Milling of Ceramic Oxides",Ceramic Bulletin, vol. 53, no. 11, p. 813.

Szegvari, A., 1956, “Treatment of Liquid Systems andApparatus Therefor", U.S. Patent 2,764,359.

APPEND I X A

95

96

PROGRAM FOR BLENDING SIZE DISTRIBUTION DATAMICHEAL D. BROWNAPRIL 1984

DIMENSION P(30),RETAN(30),PASS(31)OPEN (1,FILE='INPUTWN=0DO 17 J=1,20P(J)=0

17 CONTINUEREAD(1,*)(P(I),I=1,5)

14 CONTINUEREAD(1,*)WGHT,TSIZE,NCLASS,NUMBRWRITE(*,11)NCLASS,NUMBR

11 FORMAT(5X,'DISTRIBUTION FROM CLASS ',I2,' TO CLASS ',I2I=NCLASSOLDY=1.0

15 CONTINUEREAD(1,*,END=50)X,YIF(X.GE.TSIZE) GOTO 100IF(I-NUMBR) 202,203,204

203 PRINT*,'THERE IS SOME AMOUNT OF MATERIAL PASSING THE& SMALLEST SCREEN SIZE IN THE PREVIOUS DISTRIBUTION.& YOU SHOULD DEFINE A WIDER RANGE OF SIZESH

204 CONTINUE202 CONTINUE

WRITE(*,20l)I,TSIZE201 FORMAT(2X,'CLASS ',I2,' =+',F6.2,' MICRONS')

TSIZE=TSIZE/1.4142I=I+1

100 CONTINUEDR.Y=1 SOBY=Y/100.0PCT=OLDY-YP(I)=PCT*WGHT+P(I)OLDY=YIF(I.LE.NUMBR) GOTO 15

50 CONTINUEN=N+1IF(N.EQ.2) GOTO 300CLOSE (1)OPEN(1,FILE='PUTIN°)GOTO 14

300 CONTINUETOTAL=0DO 110 I=1,NUMBRTOTAL=TOTAL+P(I)

110 CONTINUEl

TRETAN=0

97

DO 120 I=1,NUMBRRETAN(I)=P(I)/TOTAL*100TRETAN=TRETAN+TRETAN(I)PASS(I)=100.0—TRETAN

120 CONTINUEWRITE(*,130)

130 FORMAT(/3X,'SIZE CLASS',3X,‘WEIGHT',3X,'% RETAINED',&3X,'CUM % PASSING')

DO 200 M=l,NUMBRWRITE(*,140)M,P(M),RETAN(M),PASS(M)

140 FORMAT(6X,I2,SX,F8.2,3X,F8.2,5X,F8.2)200 CONTINUE

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APPENDIX B

l00

101

Selection Function

The selection function values for each size class asdiscussed in Chapter 3 are as followsz

Size Class Size Mesh Si

1 28 x 35 2.83

2 35 x 48 2.263 48 x 65 1.794 65 x 100 1.45

5 100 x 150 1.156 150 x 200 .94

7 200 x 270 .76

8 270 x 400 .60

9 400 x 500 .41

Size Microns

10 23 .33

11 16.3 .26

12 11.5 .22

13 8.1 .17

14l

5.75 .14

15 4.10 .11

16 2.90 .09

17 2.0 .07

18 1.45 .04

102

Breakage Function

The breakage function described in Chapter 3 can be

represented as a lower triangular matrix with values as

followsz

b(1,1)= b(2,2) = b(3,3) = b(n,n) = 0b(2,1) = b(3,2) = b(4,3) = b(n+1,n)b(i„J) = b(i+l„J+l)

b(l,l)= 0.000_ b(2,1)= 0.268

b(3,1)= 0.200b(4,l)= 0.150b(5,l)= 0.112b(6,l)= 0.082b(7,l)= 0.058b(8,l)= 0.041b(9,1)= 0.028b(10,l)= 0.020b(11,l)=

‘ 0.013b(12,l)= 0.009b(l3,l)= 0.006b(l4,1)= 0.004b(15,1)= 0.003b(l6,1)= 0.002b(17,1)= 0.002 .b(l8,l)= 0.002b(l9,1)= 0.000

Class 19 = 1 - SUM b(l,j) to b(18,j)

APPENDIX C

103

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