Chemical Engineering November 2013 - 52

Solids Processing
Breakage for CaO material - initial selectivity
Breakage for CaO material - final selectivity
100
100
10
20
30
40
50
60
70
80
90
2
Bin 1
3
4
Bin 2
5
6
7
Attrition particle size bin #
Bin 3
Bin 4
Bin 5
Figure 14. Breakage selectivity functions (B) for breakage
before 60 seconds
assumed that the rate constants are
not a function of time (t, in seconds)
and will remain constant. This simplifies
the analysis somewhat. However,
many other assumptions are
also possible [4].
The S and B functions are found
simultaneously from the solution of
Equations (3) and (4). Since mass
cannot be destroyed, the sum of the
B functions for any particle-size bin
must equal one. This solution of the
above equations is not a trivial task,
but it provides a great deal more
information concerning the breakage
phenomena experienced by bulk
materials. Choosing the particle-size
bins such that each adjacent bin
size consists of particles one-half the
size of the original bin suggests that
movement of material to an adjacent
bin would be interpreted as splitting
the original particles in half. If the B
values indicate a large fraction for
the next adjacent bin size, then fracture
of primary particles is occurring.
However, if the B values indicate that
one or more size bins adjacent to the
original size have low fractions followed
by large fractions distributed
in the finer size bins, then abrasion is
likely occurring, as abrasion creates
fine particles directly from largersized
particles.
The probability S function is a value
that identifies the likelihood that a
collision will result in a breakage of
particles. This probability is a function
of the size of the particles, and typically
shows an increasing functionality
between the particle size and the
Bin 1
Bin 2
Bin 3
Bin 4
Bin 5
10
20
30
40
50
60
70
80
90
2 3 4 5 6 7
Attrition particle size bin #
Bin 1
Bin 2
Bin 3
Bin 4
Bin 5
Figure 15. Breakage selectivity functions (B) for breakage
after 60 seconds
probability that a given size of particle
will break. The solution of Equations
(3) and (4) can be calculated numerically,
but can also be approximated by
Equation (5) if the rate constants are
assumed time invariant [3].
R t Ri
Gi = − +Si
H ∑ ( λ G Gi
i =
j
i
=
−
1
1
ij
j −
i ( ) = ( )exp G t H t2 2
R
R
∑
j
i
=
−
1
1
(
λ j
i
ij
i +
( )
( )
R
R
,
) ( )
j
i
( )
, λij
= S B 1 -S j
j+1
ij+
B
ij
(5)
Time sequence particle-size data for
CaO were used with Equation (5) to
determine the B(x,y) distribution matrix
and the S(x) breakage-rate function,
shown in Figures 13-15. For the
purposes of this article, B values are
most relevant, since they reveal how
much fracture or abrasion may occur
with the material. However, S values
can give an indication of fatigue. If the
breakage rates (S values) at the beginning
of the testing are low and then
increase, then the material is very
sensitive to fatigue events.
The CaO breakage analysis suggests
that lime material is best described
by a dual-rate breakage. Equation (5)
still applies, but must be implemented
in a piecewise linear fashion to combine
the solution of two constant-rate
equations. One equation set will consider
the first 60 s of breakage and use
the initial conditions at zero seconds.
The other rate equation will consider
breakage after 60 s and use as initial
conditions the particle-size informa52
ChemiCal engineering www.Che.Com november 2013
i
)
tion after 60 s of breakage. The initial
breakage rate is much higher than the
final breakage rate, as seen in Figure
13. This suggests that lime may not be
sensitive to fatigue.
B values were also computed for
both of these breakage rates, shown
in Figures 14 and 15. These data show
that about 38% of Bin Size 1 will initially
break in half to create Bin Size
2 particles; 54% of Bin Size 2 particles
will initially break in half to form Bin
Size 3 particles and 100% of Bin Size
3 and Bin Size 4 materials will break
in half to the next lowest bin size, indicating
that initial breakage of CaO is
due to significant fracture events.
However, the story changes after
breakage has occurred for a period of
time. Figure 15 shows the breakage
selectivity after 60 s of impact milling.
In this case, with the exception of Bin
Size 2, there is a shift to create particles
more by abrasion events. From
the Bin Size 1 material, 99% splits
into particles that are one fourth the
size of the original particles. All of
the Bin Size 2 material becomes Bin
Size 3 material. However, only 18%
of Bin Size 3 material shifts to become
Bin Size 4 material and 44% of
Bin Size 3 material becomes fine Bin
Size 7 material. The shift to abrasion
is also very evident in the Bin Size 4
and 5 particles.
It is possible to average the fractional
B values that shift one particle-size bin
to determine an overall fraction that is
sensitive to fracture. Likewise, the overall
fraction of material breaking into
particles one fourth the size or less can
Bin 5
Bin 4
Bin 3
Bin 2
Bin 1
Percent
Percent
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Chemical Engineering November 2013

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