XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 1015
components mass over the particles respectively, therefore
representing the particle size change. The following con-
clusions can be found in Figure 3. Perfect liberation does
not produce additional entropy as the additional distri-
bution entropy s(p00k) is compensated by the elimination
of the particle composition entropy s(c0
jk )as all particles
are liberated. Further comminution of the already liber-
ated particles leads to the scenario of over-grinding, yield-
ing an increase in the entropy contributions s(p00k) and
s(p0
jk ).The reference comminution scenario depicts a mix
of 50% liberation comminution and 50% random com-
minution—the reduction of the liberation contribution is
overcompensated by the increase in the distribution contri-
bution, and not all particles are perfectly liberated. If only
random breakage is used for the comminution model, the
reduction of compositional entropy is the lowest, with the
highest increase of the mass distribution entropy. Selective
breakage is best to keep the increase in entropy during com-
minution at a minimum with the ideal case of ideal libera-
tion, which does not produce any additional entropy. The
compositional entropy s(c0
j 0 )does not change in any of
the comminution scenarios as the bulk composition of the
system remains the same, independent of the disperse state
of the particles and components.
The reference comminution scenario can be combined
with a separation step to reflect a process chain in a plant.
As depicted in Figure 4, the separation produces addi-
tional entropy with the entropy contribution due to the
uncertainty into which product the mass of a feed batch
is recovered. However, with additional information—in
this case, the measurement of the mass split—this uncer-
tainty can be eliminated, and the system’s overall entropy
is reduced. The mean entropy contribution of the bulk
composition of the separation products s(cij0) is reduced
compared to the compositional entropy of the feed batch
s(c0j0) due to the separation of phases A and B. Reduced
by comminution, the entropy of the particle intergrowth
s(c0
jk )is not further reduced by separation s(cijk) as the
separation model induces no additional particle breakage.
Please note that additionally to the contribution s(ti00), in
other entropy aggregations, the contributions s(tij0) and
s(tijk) exist. Whereas s(tij0) reflects the uncertainty about the
mass split of the components of a system—which could
be measured by e.g., additional compositional analytics of
the products, s(tijk) represents the mean uncertainty of each
particle to be recovered in a respective product. The lat-
ter entropy contribution could only be resolved by tracking
each particle through the separation process or repeating a
test several times for a particular set of particles.
As statistical entropy represents a common measure
describing both comminution and separation, it enables the
combined optimization of these processes, helping to deter-
mine the best milling time for a specific separation scenario
and, therefore, increasing the overall process efficiency. The
relation of the entropy decrease with the milling time is
showcased in. After eight milling steps, the overall decrease
Figure 4. Entropy analysis of a flowsheet containing comminution and separation—figure adapted from (Schach et al., 2024)

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Extracted Text (may have errors)

XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 1015
components mass over the particles respectively, therefore
representing the particle size change. The following con-
clusions can be found in Figure 3. Perfect liberation does
not produce additional entropy as the additional distri-
bution entropy s(p00k) is compensated by the elimination
of the particle composition entropy s(c0
jk )as all particles
are liberated. Further comminution of the already liber-
ated particles leads to the scenario of over-grinding, yield-
ing an increase in the entropy contributions s(p00k) and
s(p0
jk ).The reference comminution scenario depicts a mix
of 50% liberation comminution and 50% random com-
minution—the reduction of the liberation contribution is
overcompensated by the increase in the distribution contri-
bution, and not all particles are perfectly liberated. If only
random breakage is used for the comminution model, the
reduction of compositional entropy is the lowest, with the
highest increase of the mass distribution entropy. Selective
breakage is best to keep the increase in entropy during com-
minution at a minimum with the ideal case of ideal libera-
tion, which does not produce any additional entropy. The
compositional entropy s(c0
j 0 )does not change in any of
the comminution scenarios as the bulk composition of the
system remains the same, independent of the disperse state
of the particles and components.
The reference comminution scenario can be combined
with a separation step to reflect a process chain in a plant.
As depicted in Figure 4, the separation produces addi-
tional entropy with the entropy contribution due to the
uncertainty into which product the mass of a feed batch
is recovered. However, with additional information—in
this case, the measurement of the mass split—this uncer-
tainty can be eliminated, and the system’s overall entropy
is reduced. The mean entropy contribution of the bulk
composition of the separation products s(cij0) is reduced
compared to the compositional entropy of the feed batch
s(c0j0) due to the separation of phases A and B. Reduced
by comminution, the entropy of the particle intergrowth
s(c0
jk )is not further reduced by separation s(cijk) as the
separation model induces no additional particle breakage.
Please note that additionally to the contribution s(ti00), in
other entropy aggregations, the contributions s(tij0) and
s(tijk) exist. Whereas s(tij0) reflects the uncertainty about the
mass split of the components of a system—which could
be measured by e.g., additional compositional analytics of
the products, s(tijk) represents the mean uncertainty of each
particle to be recovered in a respective product. The lat-
ter entropy contribution could only be resolved by tracking
each particle through the separation process or repeating a
test several times for a particular set of particles.
As statistical entropy represents a common measure
describing both comminution and separation, it enables the
combined optimization of these processes, helping to deter-
mine the best milling time for a specific separation scenario
and, therefore, increasing the overall process efficiency. The
relation of the entropy decrease with the milling time is
showcased in. After eight milling steps, the overall decrease
Figure 4. Entropy analysis of a flowsheet containing comminution and separation—figure adapted from (Schach et al., 2024)

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1014 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
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A particle system was simulated by generating inde-
pendent, uniform random numbers to showcase the appli-
cation of the presented entropy notation. The particles
contain two different components (A and B), and 5% of
the particles were already liberated, containing only one of
the two phases. To simulate comminution, two breakage
mechanisms are introduced: liberation breakage—causing
a multiphase particle to break up precisely into particles
consisting of the respective phases and leaving already lib-
erated particles unmodified and random breakage, which
distributes each component of a particle over p new par-
ticles whereby p is a randomly generated number using a
discrete uniform distribution with values between 2 and
5. The components of the original particle are distributed
over the particles created by comminution using a p-part
Dirichlet distribution. The two breakage mechanisms can
be combined, adjusting the percentage of particles undergo-
ing liberation breakage and random breakage accordingly.
A function based on the model of Rogers was imple-
mented to simulate separation processes. The model con-
tains a parameter α, which represents the sharpness of the
resulting partition curve. The ideal case of perfect separa-
tion is represented as a step function at the threshold value.
In contrast, perfect splitting is represented by a partition
value of 0.5 over the whole range of particle property val-
ues. Comminution over various milling times is simulated
by taking only 5% of the number of particles into account
for each comminution step. The remaining particles and
the child particles produced by the comminution step are
fed to the next step. An increasing number of comminution
steps, therefore, simulates longer milling times. The cor-
responding separation scenario remains the same for each
number of comminution steps.
RESULTS
Figure 3 depicts the entropy results for different commi-
nution scenarios. The change of intergrowth is represented
by the entropy contribution of s(c0
jk )which can be under-
stood as the mean composition entropy of all particles in
the system, whereas s(c0
j0 )gives the entropy contribution
of the bulk composition. The contributions s(p00k) and
s(p0
jk )represent the distributions of the systems mass and
Figure 3. Overall entropy values for different comminution scenarios for two aggregation orders: c
0j0 .p
0jk —first aggregating
over the bulk composition of the batch and second over the distribution of the mass of the components over particles p
00k .
c
0jk —First aggregation over the distribution of the system mass over particles and then over the composition of each particle.
Note: we are not considering the dimension of partition yet, as we are only focusing on comminution—figure adapted from
(Schach et al., 2024)

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Extracted Text (may have errors)

1014 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
log^t
log^c
log^pijkhW
s t c p s
R
t
t c
t cij0 p
i
I
j
J
k
K
i ij0 ijk ijk
i i00
i ij0 ij0
i ijk
i
I
i
I
j
J
i
I
j
J
k
K
000 1 1 1 00
1
00
1
00
1
1
00
1 1
=
+-
===
=
==
===
///
/
//
///
h
h
R
T
S-
S
S
S
S
S
S
S-
S
V
X
W
W
W
W
W
W
W
W
A particle system was simulated by generating inde-
pendent, uniform random numbers to showcase the appli-
cation of the presented entropy notation. The particles
contain two different components (A and B), and 5% of
the particles were already liberated, containing only one of
the two phases. To simulate comminution, two breakage
mechanisms are introduced: liberation breakage—causing
a multiphase particle to break up precisely into particles
consisting of the respective phases and leaving already lib-
erated particles unmodified and random breakage, which
distributes each component of a particle over p new par-
ticles whereby p is a randomly generated number using a
discrete uniform distribution with values between 2 and
5. The components of the original particle are distributed
over the particles created by comminution using a p-part
Dirichlet distribution. The two breakage mechanisms can
be combined, adjusting the percentage of particles undergo-
ing liberation breakage and random breakage accordingly.
A function based on the model of Rogers was imple-
mented to simulate separation processes. The model con-
tains a parameter α, which represents the sharpness of the
resulting partition curve. The ideal case of perfect separa-
tion is represented as a step function at the threshold value.
In contrast, perfect splitting is represented by a partition
value of 0.5 over the whole range of particle property val-
ues. Comminution over various milling times is simulated
by taking only 5% of the number of particles into account
for each comminution step. The remaining particles and
the child particles produced by the comminution step are
fed to the next step. An increasing number of comminution
steps, therefore, simulates longer milling times. The cor-
responding separation scenario remains the same for each
number of comminution steps.
RESULTS
Figure 3 depicts the entropy results for different commi-
nution scenarios. The change of intergrowth is represented
by the entropy contribution of s(c0
jk )which can be under-
stood as the mean composition entropy of all particles in
the system, whereas s(c0
j0 )gives the entropy contribution
of the bulk composition. The contributions s(p00k) and
s(p0
jk )represent the distributions of the systems mass and
Figure 3. Overall entropy values for different comminution scenarios for two aggregation orders: c
0j0 .p
0jk —first aggregating
over the bulk composition of the batch and second over the distribution of the mass of the components over particles p
00k .
c
0jk —First aggregation over the distribution of the system mass over particles and then over the composition of each particle.
Note: we are not considering the dimension of partition yet, as we are only focusing on comminution—figure adapted from
(Schach et al., 2024)

1016 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
of the system entropy is at its maximum. With more than ~
17 milling steps, the entropy reduction becomes negative,
reflecting an increase in entropy, meaning that the entropy
reduction of the separation process cannot compensate for
the increase in entropy by the comminution of the particles.
SUMMARY AND OUTLOOK
Statistical entropy analysis is already used in various fields
of the raw material sector, including waste management
and primary and secondary resources. Until now, the
analysis was based only on one of the three dimensions
of concentration of components, partition in separation
processes or disperse properties as size and liberation.
With this contribution, we introduce a uniform frame-
work that can consider all of these dimensions, increasing
the precision of entropy analysis. Further, using statistical
entropy to describe disperse particle properties enables for
the first time to describe comminution. Using simulated
particles, we showed that an ideal liberation breakage does
not increase the system’s entropy, as the additional disor-
der due to an increased number of particles is compensated
by liberation. Separation increases the entropy of a system
due to the uncertainty in which product the system’s mass,
the component or the particle is recovered. However, this
uncertainty can be eliminated by additional information,
such as measuring the mass split with a balance or analyzing
the product phases. If the separation entropy surpasses the
entropy produced by comminution, the entropy of a mate-
rial stream can be reduced. A couple of comminution and
separation are optimized by choosing the number of mill-
ing steps with the highest decrease in entropy for a fixed
separation scenario.
The introduced entropy notation can be easily extended,
including additional dimensions. One example might be
to include the microstructure of particles, considering the
number of grains in one mineral phase in a particle. As a
next step, the entropy concept should be applied to actual
data from a process plant rather than using simulated data.
REFERENCES
Ballantyne, G.R., Powell, M.S., 2014. Benchmarking com-
minution energy consumption for the processing of
copper and gold ores. Miner. Eng. 65, 109–114. doi:
10.1016/j.mineng.2014.05.017.
Brunner, P.H., Rechberger, H., 2003. Practical Handbook of
Material Flow Analysis.
Buchmann, M., Schach, E., Leißner, T., Kern, M., Mütze,
T., Rudolph, M., Peuker, U.A., Tolosana-Delgado, R.,
2020. Multidimensional characterization of separa-
tion processes—Part 2: Comparability of separation
efficiency. Miner. Eng. 150, 106284. doi: 10.1016
/j.mineng.2020.106284.
Figure 5. Entropy decrease by separation in dependency on the actual milling cycles—figure adapted from (Schach et al.,
2024)

Previous Page Next Page

Extracted Text (may have errors)

1016 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
of the system entropy is at its maximum. With more than ~
17 milling steps, the entropy reduction becomes negative,
reflecting an increase in entropy, meaning that the entropy
reduction of the separation process cannot compensate for
the increase in entropy by the comminution of the particles.
SUMMARY AND OUTLOOK
Statistical entropy analysis is already used in various fields
of the raw material sector, including waste management
and primary and secondary resources. Until now, the
analysis was based only on one of the three dimensions
of concentration of components, partition in separation
processes or disperse properties as size and liberation.
With this contribution, we introduce a uniform frame-
work that can consider all of these dimensions, increasing
the precision of entropy analysis. Further, using statistical
entropy to describe disperse particle properties enables for
the first time to describe comminution. Using simulated
particles, we showed that an ideal liberation breakage does
not increase the system’s entropy, as the additional disor-
der due to an increased number of particles is compensated
by liberation. Separation increases the entropy of a system
due to the uncertainty in which product the system’s mass,
the component or the particle is recovered. However, this
uncertainty can be eliminated by additional information,
such as measuring the mass split with a balance or analyzing
the product phases. If the separation entropy surpasses the
entropy produced by comminution, the entropy of a mate-
rial stream can be reduced. A couple of comminution and
separation are optimized by choosing the number of mill-
ing steps with the highest decrease in entropy for a fixed
separation scenario.
The introduced entropy notation can be easily extended,
including additional dimensions. One example might be
to include the microstructure of particles, considering the
number of grains in one mineral phase in a particle. As a
next step, the entropy concept should be applied to actual
data from a process plant rather than using simulated data.
REFERENCES
Ballantyne, G.R., Powell, M.S., 2014. Benchmarking com-
minution energy consumption for the processing of
copper and gold ores. Miner. Eng. 65, 109–114. doi:
10.1016/j.mineng.2014.05.017.
Brunner, P.H., Rechberger, H., 2003. Practical Handbook of
Material Flow Analysis.
Buchmann, M., Schach, E., Leißner, T., Kern, M., Mütze,
T., Rudolph, M., Peuker, U.A., Tolosana-Delgado, R.,
2020. Multidimensional characterization of separa-
tion processes—Part 2: Comparability of separation
efficiency. Miner. Eng. 150, 106284. doi: 10.1016
/j.mineng.2020.106284.
Figure 5. Entropy decrease by separation in dependency on the actual milling cycles—figure adapted from (Schach et al.,
2024)