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Analysis of the kinetics of coal breakage by wet grinding in the szego mill
287
Powder Technology. 43 (1985) 287 - 294
Analysis
.of the Kinetics
V. R. KOKA Department (Canada)
(Received
of Coal Breakage
by W&t Grinding
in the Szego
Mill
and 0. TRASS of Chemical
August
Engineering
and Applied
Chemistry.
20, 1984; in revised form December
SUMMARY This paper deals with the mechanism of breakage of Pittsburgh coal in the Szego mill, for the preparation of coal-grade 2 oil and coal-water slurries. Following a brief introduction to the miii and its operation, the kinetics of coal breakage is discussed_ The non-first-order breakage in the mill is due to materiaz and environment effects. A two-co.mponent mechanistic modeZ, considering the coal ?n a size interval to be composed of crushed feed particzes and fZaky particles, is used to expZain the breakage_ The selection functions for both components in various fi size intervals are estimated from experimental data using an optimization technique. Finally, the variation of the selection functions with particZe size is discussed.
University
of
Toronto,
Toronto,
Ont.
M5S
IA4
21, 1984)
particle size and breakage rate. In order to understand the mechanism of breakage in the mill, it is essential to know the breakage kinetics_ Also, the present population balance models [l - 71 primarily require the deter: mination of breakage kinetics in a mill. Rigorous studies [8, 91 utilizing opencircuit wet grinding have been made to determine the kinetics of breakage of coal in oil. Non-first-order breakage has been observed in both grade 2 and grade 4 fuel oils. Trass and Koka [9] presented the possible reasons for the non-first-order breakage occurring in the mill. Recently, kinetics of breakage of coal in water was determined, and the same behaviour was found. The objective of this study is to explain the mechanism of breakage of coal due to wet grinding in the Szego mill, by applying a two-component mechanistic model, which was first proposed by Gardner and Rogers [6] and further extended by Austin et al. 17, lo].
INTRODUCTION
The Szego mill is a recent introduction to the field of comminution and it falls in the category of ring-roller mills. It is mostly used for intermediate/secondary and fine grinding of material, in both wet and dry modes of operation. The mill is very effective and efficient in grinding materials of hardness 5 and below on the Moh scale. Preparation of coal-oil and coal-water mixtures, by continuous wet grinding in the Szego mill, is more efficient and economical when compared with other preparation techniques, especially dry grinding in other mills followed by dispersion in oil or water. Breakage kinetics essentially consists of (i) determining whether the breakage of a fl size interval of material in a mill is first order and (ii) finding the relationship between 0032-5910/35/$3.30
DESCRIPTION
OF THE MILL
The mill consists of a stationary cylindrical grinding surface (stator), in which is housed a rotor. The rotor consists of helically grooved rollers which are suspended on flexible wire rope shafts fixed to the rotor (see Fig. 1). The mill can be top driven or bottom driven. The stator is water-jacketed. The material to be ground is fed by gravity, or pumped to the top of the mill. The particles upon entering the grinding zone are repeatedly crushed between the rollers and the stator. The primary forces acting on the particles are the shearing and crushing forces, caused by rotational and centrifugal motion of the rollers. The helical grooves on the rollers aid in the transport of material through @ Elsevier Sequoia/Printed
in The Netherlands
288
Material ground
Fiexlble Shaft
Helical
to
be
Wire
Main
Shaft
Rope
Grooved
ROll.3
Grinding
Cylinder
Fig. 1. The Szego miII_
the mill; The mill has a number of design variables which can be utilized to meet specific product requirements.
THEORY
Since the establishment of a mechanistic approach to the breakage of solids, the grinding process has been conveniently represented by the breakage parameters, viz. the selection and breakage distribution functions_ Breakage of a given size fraction of material, in fully mixed batch mills [ll], is found to follow the first-order rate process, i.e. the rate of breakage of size i is given by
or In
[
wAt) -wiCO) 1=-_Sit
(1)
where w; is the weight fraction of material of size i, W-is the powder mass in the mill, Si is the selection function with units of fraction per unit time and t is the batch grinding time. Hence, a plot of Wi(t)/Wi(O) against time t on log-linear co-ordinates (often called a firstorder plot), would give a straight line with a slope of Si/2.3. There have been instances when the first-order breakage did not hold in the widely used tumbling mills due to the following effects. (a) When large particles are used as feed, the balls or rods are too small to nip the particles, and grinding becomes inefficient. (b) The grinding environment is changing due to the production of finer particles. This affects the breakage rate and is often called the environment effect. (c) If the size interval being studied is not narrow enough, some of the products of breakage will still remain in the original size interval. (d) The material in a size interval is heterogeneous in nature consisting of two or more components. This is normally called the material effect_ On breakage, particIes of a given size produce a set of primary daughter fragments which are mixed into the bulk of the powder, and then have a probability of being fractured. The set of primary daughter fragments from breakage of size j can be represented by the breakage distribution function bi,f where bi,i is the fraction of size j material which appears in size i on primary fracture. bisj is primarily a material property and is usually k --I. normalizable, Le. bi,i = bhvI when i-j= Using the selection and breakage distribution functions in the mass balance over a size interval, and assuming that breakage is first order gives the well-known equation [2] of batch grinding
dWi(t>
-
dt
i-l
=--SiWi(t)
f C
bi.jSjwj(t)
i=1
With known values of S, bi,i and Wi(O), the above equations can be solved for desired grind times and the size distributions can be predicted. Conversely, measured size distibutions at different grind times allow evaluation of the breakage parameters_
: 289 EXPERIMENTAL
PROCEDURE
Batch grinding of material in the Szego mill is not readily possible dtte to its configuration and the transporting action of the rollers. Grinding studies have, therefore, been made using the mill as an open-circuit grinder. The experimental procedure and size analysis techniques used are described in detail in references [8] and [9]_ The procedure involves passing the coal slurry a number of times through the mill, with the operating conditions (rotational speed and feed rate) maintained constant, and after each pass, samples were collected. The experimental procedure utilized can be considered to be equivalent to a number of identical mills in series with a fully mixed reservoir between each pair of mills. The coal slurry is ground semi-continuously, with the steady-state product from a mill being collected in a reservoir and then passed into the next mill. A 1’7 cm diameter, pilot-scale mill equipped with three 12 cm long rollers with helical, tapered grooves was used throughout this study_ The free volume in the mill was about 4.5 litres. The grade 2 oil had a viscosity of 0.0031 Pa-s at 38 “C, and a density of about 820 kg/m3. The operating conditions used in coal-grade 2 oil and coal-water grinding are given below. coalgrade 2 oil
coalwater
46%
45%
760
760
22
37
1.5
1.4
Coal concentration (wt./wt.) Rotational speed of rotor (i-pm) Hold-up (% of free mill volume) Slurry feed rate
seven equal fully mixed tanks in series,for ihe operating conditions. used in the present study. The mean residence time (MRT), obtained by dividing the hold-up by the feed rate, is a measure of the average time spent by the particles in the mill. Also, the same studies have-shown that for fixed operating conditions, the variation of the dispersion in the mill and MRT, due to changes in slurry properties (increase in the fraction of fines), is negligible. Since the operating conditions were kept constant for every pass in each run, the MRT for every pass was essentially constant. For breakage kinetics it is essential to make first-order plots from the acquired data. Replacing the batch grinding time with an equivalent time tes given by the product of the MRT and the pass number, some error is introduced since the particles entering the mill have a distribution of residence times. If the material flow in the mill is close to a plug flow, this error is negligible. Since the mill is equivalent to seven tanks in series, the error introduced can be considered to be negligible. Figure 2 shows the first-order plots for the breakage of various ,fl size intervals of coal in grade 2 fuel oil. A non-first-order behaviour is observed. The slope of the line is high initially, but it decreases with subsequent passes. This is consistently so v&h all size intervals. Also, the breakage rates are higher for size intervals having larger particles, and decrease for smaller sizes. This latter tenden-
(t/h) Except for two experiments, the single size method was used for the determination of breakage kinetics. The top size method was used in order to estimate the effect of the grinding environment on the rate of breakage of the top size. 0.01
RESULTS
AND
DISCUSSION
Residence time studies [12 J in the Szego mill have shown that the mill is equivalent to
I
0
1
4
!
I
1
1
8
Time, teq
12
16
set
Fig. 2. First-order plots for breakage of various fi size intervals of coal in grade 2 fuel oiI_
q
20x30
o
20x30+20% 20
O.Oll 0
Time, teq
cy is found in other mills, especially the tumbling mills_ Breakage of coal in water, Fig. 3, also shows the above-mentioned behaviour.
coals in the ball mill and Hardgrove mill, Austin et al. [13, 141 found a similar, non-first-order behaviour due to the environment effect. However, wet grinding of 40 - 45% by volume of coal in water slurry showed firstorder breakage. Klimpel [15] extended grinding of coal-water slurries up to 72% by weight coal concentration. Non-first-order breakage was found at the high concentration of 72%, but lower concentrations (56 to 67%) did not show this tendency. The increased slurry viscosity was presumed to be the cause. In the present case, there are two major possibilities for this non-first-order breakage, namely, (i) the effect of the changing grinding environment due to the production of finer particles, and (ii) the material in the size intervals is heterogeneous in nature, consisting of two or more components which break at different rates. To determine the importance of the environment effect, studies were made by mixing a 20 X 30 mesh size interval with different percentages of fines, and subjecting the slurry to breakage in the mill. Figure 4 shows the first-order plots drawn from the results obtained_ It can be seen that increasing the percentage of fines does slow down the For
dry
grinding
of
’ 6
set
Fig. 3. First-order plots for breakage of various d size intervals of coal in water_
non-first-order
’
different
’
’
16 Time, teq
’
x 30
fines +
40
24
X
32
fines
0
SeC
Fig. 4. First-order plots for breakage of 20 X 30 mesh particles in water, mixed with 0, 20 and 40 wt.% of fines (t63 pm). breakage,
but
very
slightly.
In Fig.
3 it is ob-
served that for the 20 X 30 mesh interval the slope of the line changes by a factor of 3.
Hence, it can be concluded that there is some other cause which is actually effecting the non-first-order breakage. Pictures of the top fl size interval in the samples collected after every pass through the mill were taken. From Figs. 5(a) and 5(b), which are pictures of 20 X 30 mesh particles in the initial feed to the mill and in the product after eight passes through the mill, it is obvious that the size interval after eight passes consists of mostly flaky particles. The production of flakes can be expected due to the shearing action in the grinding zone. These flakes are, of course, smaller in volume or weight than the feed particles, but by sieve analysis they report in the same size range. This clearly proves that the material in the size interval is heterogeneous. In each size interval we have two types of material: type A (non-flaky) and type B (flaky), but having identical chemical composition. Types A and B have different breakage rates, even though they are of the same size. Assuming that each component breaks in a first-order manner, the end result will be a non-first-order breakage for each size interval, and this is mathematically illustrated below. The following is a treatment of the nonfirst-order breakage (Figs. 2 and 3) using a two-component mechanistic model. Since the
291
interval. When type B breaks, the products of breakage will contain smaller A and B types, but there cannot be any type A material produced which will remain in the original size interval. Consider only the breakage of a size interval i, and not its production from larger sizes. Assuming that breakage of each component in the size interval follows the firstorder eqn. (I), we have
dw,itt)
= -SAiW,4i(t)
dt
dwm(t)
~
dt
wi(t)
=
= -SafWzi(t) w_4i(t)
+
+ ABiiSAiWAi(t)
(4) (5)
WBi(f)
Here, WAi, WBi and Sni, Szi are the weight fractions and selection functions of components A and B respectively in size interval i. ABii is the fraction of the products of breakage of A which goes to B, and still remains in the interval i. Solution [IO J of these equations gives wi(t)
=
WAi(O)C(1
+
Qi
-
Qi)
exP(-S’git)l
exPC_-S&l
+
wBi(“)
exP(--SBif) (6)
(b) Fig. 5. (a), 20 X 30 mesh particles in the initial feed to the mill; (b), 20 x 30 mesh particles after 8 passes through the mill.
where Qi = ABiiS_ai/(S_hi -Szi)Since the fraction wgi(O) of type B material in the feed was negligible, wai(o) was taken as zero and we have ~~~(0) = Wi(O), and eqn. (6) reduces to wi(t)
environment effect is not considerable we assume that it is negligible, and it is only the heterogeneous nature of the material which is causing the non-first-order breakage. Gardner and Rogers [6] were the first to propose and develop a model for this kind of breakage. They assumed that the material is made up of hard and soft breaking material, and hard breaks into hard material and soft breaks to soft. Austin et aZ_ [7, lo] extended this model to incorporate the possibility of hard breaking to soft and vice versa. The Austin et al. model will be applicable to the present situation, since from Figs. 5(a) and 5(b) we are aware that, when type A material breaks, a fraction of the products of breakage goes to type B material, and still remains in the original size
wi(o>
= (1 -
Qi)
eXp(-S*it)
+
$i exp(-SBit)
Eqn. (7) is the working equation for the breakage kinetics. Determination
of SAi, Sgi and ABii
Looking at eqn. (7), we can see that if values of Wi(t)/wi(O) are known experimentally at different grind times, it is possible to determine the values of SAP, Ssi and Qi- This can be done by using an optimization procedure, the purpose of which is to select values of SAi, Sgi and Qi, which will make the values calculated from eqn. (7), Of Cwi(t)lwi(o)lcal~ agree best with the experimental values The objective function to be Cwi(t)lwi(o)Jexpminimized is the summed squared error in
292
the values wi(t)/wi(0),
of calculated i.e.
and it can be seen how well they experimental plots.
and experimental
fit the
Variation of SAi, SBi and ABii with particle size In tumbling mills the relationship between the selection function S and the particle size is of the form
(8) where n is the number of experimental wi(t)/wi(0) values taken. The optimization technique selected for use in this study is a direct search method proposed by Luus and Jaakola [16] which can optimize a continuous non-linear objective function_ This method is simple to apply and very effective_ The search procedure utilizes pseudo-random numbers over a region, and after each iteration the size of the region is reduced so that the optimum can be found as accurately as desired. In order to use the optimization technique to calculate SAi, Sai and Qi values of each size interval, initial estimates of the parameters and their regions of search have to be supplied_ The computation time can be reduced if these values are very close to the actual values_ The initial and final slopes of the first-order plot are good approximations to Sxi and SBi, and the slopes were calculated from the first two and the last two experimental data points. The initial regions of search were given as initial estimate *l-O SC’ for Semi,and initial estimate 20.2 S-I for Safe Looking at the first-order plots, it was concluded that the value of @i would range from 0 to 1.0, hence the initial estimate of $i was 0.5 and its region 0 to 1.0. The Table shows the optimum S,i, Ssi and ABii values for various fl size intervals of coal ground in water and grade 2 fuel oil, estimated using the above optimization procedure. First-order plots (Figs. 2 and 3) were made using these values and eqn. (7),
i=l,2,3
Si = AXiu
,...,
n-1
(91
or logSi=logA
+alOgXi
where Xi is the reduced size of size interval i, that is, the mean size of size interval i divided by the mean size of the top size interval, i.e. interval 1. Normally, a geometric series of size ratio R is used (in the present case R = O-707), thenXi = l.O,Xa=R,Xs=R*, --.,.Xi=Ri-‘It should be noted that A = S1. To see whether the above relationship holds for breakage in the Szego mill, plots of SAi and Sai against Xi were made on log-log co-ordinates (see Figs. 6 and 7). For coalwater grinding (Fig. 6), eqn. (9) applies very well for both S,i and Ssi. The slopes of the lines give ar, = 0.85 and 0~s = 0.80. In the case of coal-grade 2 oil grinding this relationship does not hold. .The probable reason for this could be that the ,/% size intervals used were in a different range (600 to 2380 pm), when compared with the range (149 to 841 Pm) used in coal-water grinding_ Since the width of the ridge on the roller was 2.5 mm, the particles which are big (about 2 mm) cannot be nipped easily and they can escape from the grinding zone. If still larger sizes are used, the selection function value may decrease. It is interesting to note that the abovementioned behaviour is observed even in tumbling mills; there is a maximum in the S
TABLE Estimated Coal
values
of the parameters
ABii
in water s_Ai
20 30 40 50 70
0.7656 O-7037 0.4772 0.3916 0.2406
30 40 50 $0 100
from
experimental Coal
Size interval (mesh) x x x x x
S-hi, SE< and
(Se11
SBi
(S-l)
0.0992 0.0860 0.0696 0.0501 0.0369
in grade
data,
using
2 fuel
the optimization
oiI
ABii
Size interval (mesh)
SAi
0.3696 0.4856 0.4660 0.4762 0.5098
8 I.2 16 20
1.4527 1.3729 1.0804 0.4829
12 16 x 20 x 30 x
x
procedure
(5-l)
SBi
(s-l)
0.1364 0.1414 0.1293 0.0742
ABii
0.0671 0.1529 0.2411 0.1757
.u
1
flaky&/.’
..
Fig. 8, Flaky and non-flaky. &rticles region.
_ .-.,
_..
Reduced
I.”
Size,
Xi
Fig. 6. Coal-in-water breakage: Variation of selection functions SAi and Sgi with reduced size Xi.
t
flakes
p,-_ 0.01
L
0.1 Reduced
Fig. 7. Coal-in-grade selection functions
Size,
at the grinding
:
I
V.Y.
-.
When the roller approaches the particles in the slurry, the fIaky particles would align themselves as shown in Fig. 8. Then. the probability of a flake being selected for. breakage would be much smaller than that for a granular particle oPthe same diameter. It would be expected to be equal to, or slightly greater than, that for a non-flaky particle of diameter equal to the thickness of the flake. The thi&neBs of the flake is -about 8 to 10 times sm‘&er than its diameter. Taking a diameter over thickness ratio of 10, and assuming that a flake behaves like a non-flaky particle of diameter equal to its thickness, then the reduced size Xi (which we can call the modified reduced size) would be smaller by a factor of 10. With this in mind a plot of Sai against the modified reduced size for the flakes was made (dashed line in Fig. 6); we can see that it lines up well with the straight line for non-flaky particles. A close observation of the Table shows a certain trend between ABii and reduced size Xi. ABii is found to decrease with increase in particle size. We can conclude from this that, when larger non-flaky particles are subjected to breakage, a smaller fraction of the products of breakage goes to flaky particles which still remain in the origins size interv~.
Xi
2 fuel oil breakage: Variation of SAi and SB~ with reduced size Xi.
curve as a function of particle size. An interesting observation to make is the difference between SAi and Sai for various size iXIbSNa&L SBi is smaller than SAi by a factor of 6 to 10, which implies that flaky particles are more difficult to break and they behave like hardbreaking materials. This behaviour can be expected to be just the opposite in tumbling mills, where flaky particles are easier to break. In order to explain this, we have to lock at the grinding zone.
CONCLUSIONS
The following conclusions can be readily drawn on the breakage of coal in the Szego mill: (1) Non-first-order breakage in the mill is mainly due to the heterogeneous nature of the material in the size inte~als. (2) The two-component model describes the breakage very well, giving a good insight into the grinding mechanism_ (3) Flaky particles are considerably harder to break and they behave as non-flaky particles of smaller size, having a diameter equal to the thickness of the flake.
294 ACKNOWLEDGEMENT
The authors gratefully acknowledge financial assistance from the Natural Sciences and Engineering Research Council of Canada under the Strategic Grants Program. Ontario Hydro has kindly supplied the Pittsburgh coal used in this study. The authors also wish to thank General Comminution Inc. for providing the required equipment and help whenever needed _
5 T_ S. Mika, Insf. ~Min_.MetaH Trans., Sec. C., 84 (1975) 239. 6 R. P. Gardner and R. S. Rogers, Powder Techiol. 12 (1975) 247_ 7 L. G. Austin, K. Shoji and D. Bell, Powder TechnoL. 31 (1982) 127. 8 V_ R. Koka, M_A.Sc_ Thesis, Univ. Toronto,
Canada (1982). 9 0. Trass and V. R. Koka, Advances in Particulate Madras, India (1982). 10
T. Trimarchi and N. P. Weymont, 17 (1977) 109_ L. G. Austin, K. Shoji, V. Bhatia, V. Jindai, K. Savage and R. R. Kiimpel, Ind. Eng. Chem.
Proc. 12
REFERENCES
13 1 K. J_ Reid, Chem. Eng. Sci_. 20 (1965) 953_ 2 L. G. Austin, Powder TechnoL, 5 (1971/72) 1. 3 J. A. Herbst and D. W. Fuerstenau, Trans. SME/ AlME,, 4
24I
(1968)
538.
D. F. Kelsall and K. J. Reid, AICRE-I_ Symp. Series, 4 (1965) 14.
Chem.
E.
14 15 16
Symp. on Recent and Technology,
L. G. Austin,
Powder
11
Intl.
Science
Technol.,
Des.
Develop.,
15 (1976)
187.
E. R. Vasquez, M.A.Sc. Thesis. Univ. Toronto, Canada (1982). L. G. Austin, P. Bagga and M. Celik, Powder TechnoL. 28 (1981) 235. L. G. Austin, J. Shah, J. Wang, E. GaiIagher and P. T. Luckie, Powder Technol, 29 (1981) 263. R. Klimpel, Powder Technol.. 32 (1982) 267_ R. Luus and T. H. I_ Jaakola, AIChE J_. 19 (1973) 760.
Powder Technology. 43 (1985) 287 - 294
Analysis
.of the Kinetics
V. R. KOKA Department (Canada)
(Received
of Coal Breakage
by W&t Grinding
in the Szego
Mill
and 0. TRASS of Chemical
August
Engineering
and Applied
Chemistry.
20, 1984; in revised form December
SUMMARY This paper deals with the mechanism of breakage of Pittsburgh coal in the Szego mill, for the preparation of coal-grade 2 oil and coal-water slurries. Following a brief introduction to the miii and its operation, the kinetics of coal breakage is discussed_ The non-first-order breakage in the mill is due to materiaz and environment effects. A two-co.mponent mechanistic modeZ, considering the coal ?n a size interval to be composed of crushed feed particzes and fZaky particles, is used to expZain the breakage_ The selection functions for both components in various fi size intervals are estimated from experimental data using an optimization technique. Finally, the variation of the selection functions with particZe size is discussed.
University
of
Toronto,
Toronto,
Ont.
M5S
IA4
21, 1984)
particle size and breakage rate. In order to understand the mechanism of breakage in the mill, it is essential to know the breakage kinetics_ Also, the present population balance models [l - 71 primarily require the deter: mination of breakage kinetics in a mill. Rigorous studies [8, 91 utilizing opencircuit wet grinding have been made to determine the kinetics of breakage of coal in oil. Non-first-order breakage has been observed in both grade 2 and grade 4 fuel oils. Trass and Koka [9] presented the possible reasons for the non-first-order breakage occurring in the mill. Recently, kinetics of breakage of coal in water was determined, and the same behaviour was found. The objective of this study is to explain the mechanism of breakage of coal due to wet grinding in the Szego mill, by applying a two-component mechanistic model, which was first proposed by Gardner and Rogers [6] and further extended by Austin et al. 17, lo].
INTRODUCTION
The Szego mill is a recent introduction to the field of comminution and it falls in the category of ring-roller mills. It is mostly used for intermediate/secondary and fine grinding of material, in both wet and dry modes of operation. The mill is very effective and efficient in grinding materials of hardness 5 and below on the Moh scale. Preparation of coal-oil and coal-water mixtures, by continuous wet grinding in the Szego mill, is more efficient and economical when compared with other preparation techniques, especially dry grinding in other mills followed by dispersion in oil or water. Breakage kinetics essentially consists of (i) determining whether the breakage of a fl size interval of material in a mill is first order and (ii) finding the relationship between 0032-5910/35/$3.30
DESCRIPTION
OF THE MILL
The mill consists of a stationary cylindrical grinding surface (stator), in which is housed a rotor. The rotor consists of helically grooved rollers which are suspended on flexible wire rope shafts fixed to the rotor (see Fig. 1). The mill can be top driven or bottom driven. The stator is water-jacketed. The material to be ground is fed by gravity, or pumped to the top of the mill. The particles upon entering the grinding zone are repeatedly crushed between the rollers and the stator. The primary forces acting on the particles are the shearing and crushing forces, caused by rotational and centrifugal motion of the rollers. The helical grooves on the rollers aid in the transport of material through @ Elsevier Sequoia/Printed
in The Netherlands
288
Material ground
Fiexlble Shaft
Helical
to
be
Wire
Main
Shaft
Rope
Grooved
ROll.3
Grinding
Cylinder
Fig. 1. The Szego miII_
the mill; The mill has a number of design variables which can be utilized to meet specific product requirements.
THEORY
Since the establishment of a mechanistic approach to the breakage of solids, the grinding process has been conveniently represented by the breakage parameters, viz. the selection and breakage distribution functions_ Breakage of a given size fraction of material, in fully mixed batch mills [ll], is found to follow the first-order rate process, i.e. the rate of breakage of size i is given by
or In
[
wAt) -wiCO) 1=-_Sit
(1)
where w; is the weight fraction of material of size i, W-is the powder mass in the mill, Si is the selection function with units of fraction per unit time and t is the batch grinding time. Hence, a plot of Wi(t)/Wi(O) against time t on log-linear co-ordinates (often called a firstorder plot), would give a straight line with a slope of Si/2.3. There have been instances when the first-order breakage did not hold in the widely used tumbling mills due to the following effects. (a) When large particles are used as feed, the balls or rods are too small to nip the particles, and grinding becomes inefficient. (b) The grinding environment is changing due to the production of finer particles. This affects the breakage rate and is often called the environment effect. (c) If the size interval being studied is not narrow enough, some of the products of breakage will still remain in the original size interval. (d) The material in a size interval is heterogeneous in nature consisting of two or more components. This is normally called the material effect_ On breakage, particIes of a given size produce a set of primary daughter fragments which are mixed into the bulk of the powder, and then have a probability of being fractured. The set of primary daughter fragments from breakage of size j can be represented by the breakage distribution function bi,f where bi,i is the fraction of size j material which appears in size i on primary fracture. bisj is primarily a material property and is usually k --I. normalizable, Le. bi,i = bhvI when i-j= Using the selection and breakage distribution functions in the mass balance over a size interval, and assuming that breakage is first order gives the well-known equation [2] of batch grinding
dWi(t>
-
dt
i-l
=--SiWi(t)
f C
bi.jSjwj(t)
i=1
With known values of S, bi,i and Wi(O), the above equations can be solved for desired grind times and the size distributions can be predicted. Conversely, measured size distibutions at different grind times allow evaluation of the breakage parameters_
: 289 EXPERIMENTAL
PROCEDURE
Batch grinding of material in the Szego mill is not readily possible dtte to its configuration and the transporting action of the rollers. Grinding studies have, therefore, been made using the mill as an open-circuit grinder. The experimental procedure and size analysis techniques used are described in detail in references [8] and [9]_ The procedure involves passing the coal slurry a number of times through the mill, with the operating conditions (rotational speed and feed rate) maintained constant, and after each pass, samples were collected. The experimental procedure utilized can be considered to be equivalent to a number of identical mills in series with a fully mixed reservoir between each pair of mills. The coal slurry is ground semi-continuously, with the steady-state product from a mill being collected in a reservoir and then passed into the next mill. A 1’7 cm diameter, pilot-scale mill equipped with three 12 cm long rollers with helical, tapered grooves was used throughout this study_ The free volume in the mill was about 4.5 litres. The grade 2 oil had a viscosity of 0.0031 Pa-s at 38 “C, and a density of about 820 kg/m3. The operating conditions used in coal-grade 2 oil and coal-water grinding are given below. coalgrade 2 oil
coalwater
46%
45%
760
760
22
37
1.5
1.4
Coal concentration (wt./wt.) Rotational speed of rotor (i-pm) Hold-up (% of free mill volume) Slurry feed rate
seven equal fully mixed tanks in series,for ihe operating conditions. used in the present study. The mean residence time (MRT), obtained by dividing the hold-up by the feed rate, is a measure of the average time spent by the particles in the mill. Also, the same studies have-shown that for fixed operating conditions, the variation of the dispersion in the mill and MRT, due to changes in slurry properties (increase in the fraction of fines), is negligible. Since the operating conditions were kept constant for every pass in each run, the MRT for every pass was essentially constant. For breakage kinetics it is essential to make first-order plots from the acquired data. Replacing the batch grinding time with an equivalent time tes given by the product of the MRT and the pass number, some error is introduced since the particles entering the mill have a distribution of residence times. If the material flow in the mill is close to a plug flow, this error is negligible. Since the mill is equivalent to seven tanks in series, the error introduced can be considered to be negligible. Figure 2 shows the first-order plots for the breakage of various ,fl size intervals of coal in grade 2 fuel oil. A non-first-order behaviour is observed. The slope of the line is high initially, but it decreases with subsequent passes. This is consistently so v&h all size intervals. Also, the breakage rates are higher for size intervals having larger particles, and decrease for smaller sizes. This latter tenden-
(t/h) Except for two experiments, the single size method was used for the determination of breakage kinetics. The top size method was used in order to estimate the effect of the grinding environment on the rate of breakage of the top size. 0.01
RESULTS
AND
DISCUSSION
Residence time studies [12 J in the Szego mill have shown that the mill is equivalent to
I
0
1
4
!
I
1
1
8
Time, teq
12
16
set
Fig. 2. First-order plots for breakage of various fi size intervals of coal in grade 2 fuel oiI_
q
20x30
o
20x30+20% 20
O.Oll 0
Time, teq
cy is found in other mills, especially the tumbling mills_ Breakage of coal in water, Fig. 3, also shows the above-mentioned behaviour.
coals in the ball mill and Hardgrove mill, Austin et al. [13, 141 found a similar, non-first-order behaviour due to the environment effect. However, wet grinding of 40 - 45% by volume of coal in water slurry showed firstorder breakage. Klimpel [15] extended grinding of coal-water slurries up to 72% by weight coal concentration. Non-first-order breakage was found at the high concentration of 72%, but lower concentrations (56 to 67%) did not show this tendency. The increased slurry viscosity was presumed to be the cause. In the present case, there are two major possibilities for this non-first-order breakage, namely, (i) the effect of the changing grinding environment due to the production of finer particles, and (ii) the material in the size intervals is heterogeneous in nature, consisting of two or more components which break at different rates. To determine the importance of the environment effect, studies were made by mixing a 20 X 30 mesh size interval with different percentages of fines, and subjecting the slurry to breakage in the mill. Figure 4 shows the first-order plots drawn from the results obtained_ It can be seen that increasing the percentage of fines does slow down the For
dry
grinding
of
’ 6
set
Fig. 3. First-order plots for breakage of various d size intervals of coal in water_
non-first-order
’
different
’
’
16 Time, teq
’
x 30
fines +
40
24
X
32
fines
0
SeC
Fig. 4. First-order plots for breakage of 20 X 30 mesh particles in water, mixed with 0, 20 and 40 wt.% of fines (t63 pm). breakage,
but
very
slightly.
In Fig.
3 it is ob-
served that for the 20 X 30 mesh interval the slope of the line changes by a factor of 3.
Hence, it can be concluded that there is some other cause which is actually effecting the non-first-order breakage. Pictures of the top fl size interval in the samples collected after every pass through the mill were taken. From Figs. 5(a) and 5(b), which are pictures of 20 X 30 mesh particles in the initial feed to the mill and in the product after eight passes through the mill, it is obvious that the size interval after eight passes consists of mostly flaky particles. The production of flakes can be expected due to the shearing action in the grinding zone. These flakes are, of course, smaller in volume or weight than the feed particles, but by sieve analysis they report in the same size range. This clearly proves that the material in the size interval is heterogeneous. In each size interval we have two types of material: type A (non-flaky) and type B (flaky), but having identical chemical composition. Types A and B have different breakage rates, even though they are of the same size. Assuming that each component breaks in a first-order manner, the end result will be a non-first-order breakage for each size interval, and this is mathematically illustrated below. The following is a treatment of the nonfirst-order breakage (Figs. 2 and 3) using a two-component mechanistic model. Since the
291
interval. When type B breaks, the products of breakage will contain smaller A and B types, but there cannot be any type A material produced which will remain in the original size interval. Consider only the breakage of a size interval i, and not its production from larger sizes. Assuming that breakage of each component in the size interval follows the firstorder eqn. (I), we have
dw,itt)
= -SAiW,4i(t)
dt
dwm(t)
~
dt
wi(t)
=
= -SafWzi(t) w_4i(t)
+
+ ABiiSAiWAi(t)
(4) (5)
WBi(f)
Here, WAi, WBi and Sni, Szi are the weight fractions and selection functions of components A and B respectively in size interval i. ABii is the fraction of the products of breakage of A which goes to B, and still remains in the interval i. Solution [IO J of these equations gives wi(t)
=
WAi(O)C(1
+
Qi
-
Qi)
exP(-S’git)l
exPC_-S&l
+
wBi(“)
exP(--SBif) (6)
(b) Fig. 5. (a), 20 X 30 mesh particles in the initial feed to the mill; (b), 20 x 30 mesh particles after 8 passes through the mill.
where Qi = ABiiS_ai/(S_hi -Szi)Since the fraction wgi(O) of type B material in the feed was negligible, wai(o) was taken as zero and we have ~~~(0) = Wi(O), and eqn. (6) reduces to wi(t)
environment effect is not considerable we assume that it is negligible, and it is only the heterogeneous nature of the material which is causing the non-first-order breakage. Gardner and Rogers [6] were the first to propose and develop a model for this kind of breakage. They assumed that the material is made up of hard and soft breaking material, and hard breaks into hard material and soft breaks to soft. Austin et aZ_ [7, lo] extended this model to incorporate the possibility of hard breaking to soft and vice versa. The Austin et al. model will be applicable to the present situation, since from Figs. 5(a) and 5(b) we are aware that, when type A material breaks, a fraction of the products of breakage goes to type B material, and still remains in the original size
wi(o>
= (1 -
Qi)
eXp(-S*it)
+
$i exp(-SBit)
Eqn. (7) is the working equation for the breakage kinetics. Determination
of SAi, Sgi and ABii
Looking at eqn. (7), we can see that if values of Wi(t)/wi(O) are known experimentally at different grind times, it is possible to determine the values of SAP, Ssi and Qi- This can be done by using an optimization procedure, the purpose of which is to select values of SAi, Sgi and Qi, which will make the values calculated from eqn. (7), Of Cwi(t)lwi(o)lcal~ agree best with the experimental values The objective function to be Cwi(t)lwi(o)Jexpminimized is the summed squared error in
292
the values wi(t)/wi(0),
of calculated i.e.
and it can be seen how well they experimental plots.
and experimental
fit the
Variation of SAi, SBi and ABii with particle size In tumbling mills the relationship between the selection function S and the particle size is of the form
(8) where n is the number of experimental wi(t)/wi(0) values taken. The optimization technique selected for use in this study is a direct search method proposed by Luus and Jaakola [16] which can optimize a continuous non-linear objective function_ This method is simple to apply and very effective_ The search procedure utilizes pseudo-random numbers over a region, and after each iteration the size of the region is reduced so that the optimum can be found as accurately as desired. In order to use the optimization technique to calculate SAi, Sai and Qi values of each size interval, initial estimates of the parameters and their regions of search have to be supplied_ The computation time can be reduced if these values are very close to the actual values_ The initial and final slopes of the first-order plot are good approximations to Sxi and SBi, and the slopes were calculated from the first two and the last two experimental data points. The initial regions of search were given as initial estimate *l-O SC’ for Semi,and initial estimate 20.2 S-I for Safe Looking at the first-order plots, it was concluded that the value of @i would range from 0 to 1.0, hence the initial estimate of $i was 0.5 and its region 0 to 1.0. The Table shows the optimum S,i, Ssi and ABii values for various fl size intervals of coal ground in water and grade 2 fuel oil, estimated using the above optimization procedure. First-order plots (Figs. 2 and 3) were made using these values and eqn. (7),
i=l,2,3
Si = AXiu
,...,
n-1
(91
or logSi=logA
+alOgXi
where Xi is the reduced size of size interval i, that is, the mean size of size interval i divided by the mean size of the top size interval, i.e. interval 1. Normally, a geometric series of size ratio R is used (in the present case R = O-707), thenXi = l.O,Xa=R,Xs=R*, --.,.Xi=Ri-‘It should be noted that A = S1. To see whether the above relationship holds for breakage in the Szego mill, plots of SAi and Sai against Xi were made on log-log co-ordinates (see Figs. 6 and 7). For coalwater grinding (Fig. 6), eqn. (9) applies very well for both S,i and Ssi. The slopes of the lines give ar, = 0.85 and 0~s = 0.80. In the case of coal-grade 2 oil grinding this relationship does not hold. .The probable reason for this could be that the ,/% size intervals used were in a different range (600 to 2380 pm), when compared with the range (149 to 841 Pm) used in coal-water grinding_ Since the width of the ridge on the roller was 2.5 mm, the particles which are big (about 2 mm) cannot be nipped easily and they can escape from the grinding zone. If still larger sizes are used, the selection function value may decrease. It is interesting to note that the abovementioned behaviour is observed even in tumbling mills; there is a maximum in the S
TABLE Estimated Coal
values
of the parameters
ABii
in water s_Ai
20 30 40 50 70
0.7656 O-7037 0.4772 0.3916 0.2406
30 40 50 $0 100
from
experimental Coal
Size interval (mesh) x x x x x
S-hi, SE< and
(Se11
SBi
(S-l)
0.0992 0.0860 0.0696 0.0501 0.0369
in grade
data,
using
2 fuel
the optimization
oiI
ABii
Size interval (mesh)
SAi
0.3696 0.4856 0.4660 0.4762 0.5098
8 I.2 16 20
1.4527 1.3729 1.0804 0.4829
12 16 x 20 x 30 x
x
procedure
(5-l)
SBi
(s-l)
0.1364 0.1414 0.1293 0.0742
ABii
0.0671 0.1529 0.2411 0.1757
.u
1
flaky&/.’
..
Fig. 8, Flaky and non-flaky. &rticles region.
_ .-.,
_..
Reduced
I.”
Size,
Xi
Fig. 6. Coal-in-water breakage: Variation of selection functions SAi and Sgi with reduced size Xi.
t
flakes
p,-_ 0.01
L
0.1 Reduced
Fig. 7. Coal-in-grade selection functions
Size,
at the grinding
:
I
V.Y.
-.
When the roller approaches the particles in the slurry, the fIaky particles would align themselves as shown in Fig. 8. Then. the probability of a flake being selected for. breakage would be much smaller than that for a granular particle oPthe same diameter. It would be expected to be equal to, or slightly greater than, that for a non-flaky particle of diameter equal to the thickness of the flake. The thi&neBs of the flake is -about 8 to 10 times sm‘&er than its diameter. Taking a diameter over thickness ratio of 10, and assuming that a flake behaves like a non-flaky particle of diameter equal to its thickness, then the reduced size Xi (which we can call the modified reduced size) would be smaller by a factor of 10. With this in mind a plot of Sai against the modified reduced size for the flakes was made (dashed line in Fig. 6); we can see that it lines up well with the straight line for non-flaky particles. A close observation of the Table shows a certain trend between ABii and reduced size Xi. ABii is found to decrease with increase in particle size. We can conclude from this that, when larger non-flaky particles are subjected to breakage, a smaller fraction of the products of breakage goes to flaky particles which still remain in the origins size interv~.
Xi
2 fuel oil breakage: Variation of SAi and SB~ with reduced size Xi.
curve as a function of particle size. An interesting observation to make is the difference between SAi and Sai for various size iXIbSNa&L SBi is smaller than SAi by a factor of 6 to 10, which implies that flaky particles are more difficult to break and they behave like hardbreaking materials. This behaviour can be expected to be just the opposite in tumbling mills, where flaky particles are easier to break. In order to explain this, we have to lock at the grinding zone.
CONCLUSIONS
The following conclusions can be readily drawn on the breakage of coal in the Szego mill: (1) Non-first-order breakage in the mill is mainly due to the heterogeneous nature of the material in the size inte~als. (2) The two-component model describes the breakage very well, giving a good insight into the grinding mechanism_ (3) Flaky particles are considerably harder to break and they behave as non-flaky particles of smaller size, having a diameter equal to the thickness of the flake.
294 ACKNOWLEDGEMENT
The authors gratefully acknowledge financial assistance from the Natural Sciences and Engineering Research Council of Canada under the Strategic Grants Program. Ontario Hydro has kindly supplied the Pittsburgh coal used in this study. The authors also wish to thank General Comminution Inc. for providing the required equipment and help whenever needed _
5 T_ S. Mika, Insf. ~Min_.MetaH Trans., Sec. C., 84 (1975) 239. 6 R. P. Gardner and R. S. Rogers, Powder Techiol. 12 (1975) 247_ 7 L. G. Austin, K. Shoji and D. Bell, Powder TechnoL. 31 (1982) 127. 8 V_ R. Koka, M_A.Sc_ Thesis, Univ. Toronto,
Canada (1982). 9 0. Trass and V. R. Koka, Advances in Particulate Madras, India (1982). 10
T. Trimarchi and N. P. Weymont, 17 (1977) 109_ L. G. Austin, K. Shoji, V. Bhatia, V. Jindai, K. Savage and R. R. Kiimpel, Ind. Eng. Chem.
Proc. 12
REFERENCES
13 1 K. J_ Reid, Chem. Eng. Sci_. 20 (1965) 953_ 2 L. G. Austin, Powder TechnoL, 5 (1971/72) 1. 3 J. A. Herbst and D. W. Fuerstenau, Trans. SME/ AlME,, 4
24I
(1968)
538.
D. F. Kelsall and K. J. Reid, AICRE-I_ Symp. Series, 4 (1965) 14.
Chem.
E.
14 15 16
Symp. on Recent and Technology,
L. G. Austin,
Powder
11
Intl.
Science
Technol.,
Des.
Develop.,
15 (1976)
187.
E. R. Vasquez, M.A.Sc. Thesis. Univ. Toronto, Canada (1982). L. G. Austin, P. Bagga and M. Celik, Powder TechnoL. 28 (1981) 235. L. G. Austin, J. Shah, J. Wang, E. GaiIagher and P. T. Luckie, Powder Technol, 29 (1981) 263. R. Klimpel, Powder Technol.. 32 (1982) 267_ R. Luus and T. H. I_ Jaakola, AIChE J_. 19 (1973) 760.