Chemical Engineering November 2013 - 48

Solids Processing
Crack will
open under
loading
port elastic behavior and must
deform plastically, resulting in
a permanent deformation of
the material. The energy corresponding
to the onset of plastic
deformation equals the area
under the stress-strain curve up
to the elastic limit and is called
the resilience (R), as seen in Figure
1. Straining the material far
enough will result in the failure
of the material and cause breakage.
The energy associated with this breakage
equals the area under the stressstrain
curve up to the rupture strength
and is called toughness (T) (Figure 2).
We can define a brittleness number
(BR) based on the ratio of the resilience
to the toughness as described
in Equation (1).
BR =
R
T
(1)
A brittleness number close to 1 suggests
that the material will break at
strain conditions very near the elastic
limit. This denotes a brittle failure,
suggesting that the material may be
subject to breakage by impact conditions.
If the brittleness number is
much less than 1, then particles break
after significant plastic deformations
and require significant shear to induce
breakage. These particles will
likely be insensitive to breakage by
fracture. Particles with a low brittleness
number may experience fatigue
and can break after repeated stressstrain
or impact events. Material can
experience repeated stress events and
these tend to open or close cracks in
the particle, depending on the orientation
of the crack relative to the direction
of impact or application of stress.
Figure 3 illustrates this phenomenon.
If the impact is perpendicular to the
crack, then the crack closes during impact.
Conversely, the crack opens if the
impact is in line with the crack.
During every cycle, the region
near the tip of the crack experiences
stress values in excess of the elastic
limit, causing the crack to grow incrementally.
There is typically little
to no control over the direction of the
impact or stress application relative
to the direction of the crack, so only
a portion of the impacts or stressloadings
actually lead to crack growth
and breakage events. As long as the
stress local to the crack tip exceeds
the elastic limit, then breakage over
time can occur. This suggests that
repeated stress applications that do
not exceed the ultimate strength, but
are above the elastic limit, can cause
particle breakage.
Traditionally, charts showing stress
applied versus number of repeated cycles
(called S-N stress fatigue curves)
have been used to determine the number
of life cycles at a given stress a
pure material can withstand. Figure
4 shows an example of an S-N fatigue
curve. If the maximum yield-stress
value is applied, then one cycle would
suffice to break the material. However,
if the stress is lower, then many
load cycles are required before the material
would break. The measurement
of S-N curves is relatively simple for
pure materials, but is much more difficult
for particles. For this reason,
fatigue characteristics are usually determined
using a population-balance
model, which will be explained later
in this article. Although some experimental
work does exist for being able
to measure particle-breakage and
stress-strain behavior on the particle
level, the more common method of
dealing with fatigue events is through
a population-balance model.
Ranking the mills
Mills can be evaluated based on their
48 CHEMICAL ENGINEERING WWW.CHE.COM NOVEMBER 2013
usefulness to induce fracture, fatigue,
abrasion or shear-based breakage.
The remainder of this article will deal
with ranking and scoring various mill
types. A method to measure a particular
material's sensitivity to key breakage
mechanisms is also presented.
Combining the knowledge of millling
action with the material's sensitivity
will help engineers select the proper
mill for their process. Below are some
general mill categories, along with
each mill's evaluation.
Jaw crusher. A jaw crusher is a type
of mill unit that is used as a primary
crusher. It works by receiving particles
from a feeder and pinching them
between two jaw-plates positioned to
form a converging-plane flow hopper.
One jaw-plate is fixed and the other
jaw-plate pivots about a fixed point,
moving in a circular cam-like motion
about an eccentric bearing (Figure 5).
Generally, a drive-wheel attached to
the fly-wheel aids crushing. The jaw
breaks the particle into two or three
smaller particles. These particles dislodge
from the jaw and fall until they
are caught by the converging set of
jaws and the pinching process is repeated.
The pinching process continues
until particle fragments are small
enough to exit the gap between the
jaws. The final particle size produced
by the jaw crusher is controlled by this
gap. Typically, this type of crusher can
handle particles as large as 1,000 mm
10
20
30
40
50
60
70
80
Crack will close
under loading
FIGURE 3. The orientation of a crack
is important in determining material
breakage behavior
10
100
1,000
10,000 100,000
Number of cycles
FIGURE 4. S-N fatigue curves are used to determine
the stress a pure material can withstand
TABLE 1. SCORING FOR
JAW CRUSHER
Jaw crusher
Rank for different types of materials
Fracture
Fatigue
Abrasion
Shear
Score on a scale from 1 to 10 (10 best)
9
3
1
1
TABLE 2. SCORING FOR
CONE CRUSHER
Cone crusher
Rank for different types of materials
Fracture
Fatigue
Abrasion
Shear
Score on a scale from 1 to 10 (10 best)
9
3
2
1
Stress applied, MPa
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Chemical Engineering November 2013

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