2082 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
particles in cyclone 4 and cyclone 5. The bivariate probabil-
ity densities show that the classification is predominantly
influenced by the particle size, as coarser particles tend to
accumulate in the first cyclones, while finer particles are
found in the latter cyclones, regardless of the mineral under
investigation. Although, an effect of shape can be observed,
it is difficult to attribute this to the particle shape only, as
coarser particles also tend to be more round, whereas finer
particles have a wide range of roundness values.
In order to investigate the combined influence of par-
ticle size and shape on the separation of the three minerals
in a hydrocyclone cascade, bivariate Tromp functions have
been computed. The Tromp functions for each individual
cyclone for calcite, fluorite and magnesite are displayed in
Figures 7, 9 and 11, respectively. The color code gives infor-
mation on whether a particle reports to the correspond-
ing cyclone (high values of T) or if the particle reports to
the fines, i.e., to the following cyclone or to the losses after
cyclone 5 (low values of T). Note that particles, which are
separated in a previous cyclone, are not available for the fol-
lowing one and thus the Tromp function also shows values
of 0, although the particles do not participate in the process
anymore.
The Tromp functions for all three minerals show the
same trend regarding particle size, as the maximum value
of the Tromp functions shift from cyclone 1 to cyclone 5
down to finer particle sizes. The Tromp function for the
first cyclone has high values for very coarse and very round
particles, i.e., particles with these descriptor vectors are very
likely to be recovered in the first cyclone, while it decreases
significantly with decreasing particle size down to a value
of zero for all minerals below 40 µm. At the same time, the
value of the Tromp function also decreases with decreas-
ing roundness, however this effect is not as pronounced as
for particle size. Furthermore, as stated above, for the three
minerals of this study, it needs to be taken into account
that the particle shape is closely linked to the particle size,
as coarser particles have higher roundness values than finer
ones. The distribution of Tromp values for the second and
third cyclone for calcite and magnesite exhibit similar val-
ues, except for particles with roundness around 0.95, where
magnesite particles only have a probability of 60 %,whereas
the calcite particles with this property have an almost 100 %
probability of being recovered in the second cyclone. While
this is also observed for very round fluorite particles, the
probability declines with decreasing roundness, however it
increases again for particles with very low roundness values
of 0.1 – 0.4, which is not the case for calcite and mag-
nesite. For the last two cyclones there is a noticeable shift
observed for all three minerals, as the separation probability
increases significantly for fine particles while it drops for
coarse particles at the same time. Particles over a wide range
of roundness are recovered more preferably, as opposed to
the first cyclones, where the Tromp function values were
highest for particles with values close to a roundness of 1,
which is most likely a result of the classification by size.
This effect is most pronounced for the last cyclone, where
separation probabilities of around 90% are obtained for
particles below 30 µm with roundness values of 0.1–1.
CONCLUSION
This study investigates the combined effect of particle
size and shape, as areaequivalent diameter and roundness,
respectively, on the separation of individual fractions of
calcite, fluorite and magnesite in a hydrocyclone cascade.
Five fractions are obtained after classification from the indi-
vidual hydrocyclones, which are analyzed by laser diffrac-
tion for their particle size as well as dynamic image analysis,
from which particle discrete information on particle size
and shape is obtained. This particle discrete data is then
used to compute bivariate Tromp functions with respect to
particle descriptor vectors containing area-equivalent diam-
eter and roundness. The results show that there is a signifi-
cant influence of the particle size on the classification for
all three minerals. From the first to the last cyclone, there
is a clear shift in the particle size of the different particle
fractions, from coarser particles being recovered in the first
cyclone to finer particles in the last. The effect of round-
ness is not as pronounced as that of size, although, differ-
ences can be observed for the different particle fractions,
as the first cyclones have a high separation probability for
particles with very high roundness values, while fine par-
ticles over a large range of roundness report to the latter
cyclones. However, as the particle shape is closely linked
to the particle size, it is questionable whether the observed
behavior of the studied minerals is particle shape depended
or just a consequence of the classification by size, as coarser
particles are also more round than finer ones. Therefore,
future studies should include particle systems that dif-
fer more significantly in shape, e.g., spheres or flakes, to
reveal how these extreme variations of shape influence the
separation. Additionally, based on the results of the mass
fraction, an effect of density was observed, which should
also be addressed further in future studies. Unfortunately,
all particles that are too fine to be recovered in the last
cyclone, which is actually the largest mass fraction here, are
lost and so is the information on them. Hence, a possibil-
ity to recover that fraction needs to be found, in order to
obtain information on all of the particles taking part in the
process.
particles in cyclone 4 and cyclone 5. The bivariate probabil-
ity densities show that the classification is predominantly
influenced by the particle size, as coarser particles tend to
accumulate in the first cyclones, while finer particles are
found in the latter cyclones, regardless of the mineral under
investigation. Although, an effect of shape can be observed,
it is difficult to attribute this to the particle shape only, as
coarser particles also tend to be more round, whereas finer
particles have a wide range of roundness values.
In order to investigate the combined influence of par-
ticle size and shape on the separation of the three minerals
in a hydrocyclone cascade, bivariate Tromp functions have
been computed. The Tromp functions for each individual
cyclone for calcite, fluorite and magnesite are displayed in
Figures 7, 9 and 11, respectively. The color code gives infor-
mation on whether a particle reports to the correspond-
ing cyclone (high values of T) or if the particle reports to
the fines, i.e., to the following cyclone or to the losses after
cyclone 5 (low values of T). Note that particles, which are
separated in a previous cyclone, are not available for the fol-
lowing one and thus the Tromp function also shows values
of 0, although the particles do not participate in the process
anymore.
The Tromp functions for all three minerals show the
same trend regarding particle size, as the maximum value
of the Tromp functions shift from cyclone 1 to cyclone 5
down to finer particle sizes. The Tromp function for the
first cyclone has high values for very coarse and very round
particles, i.e., particles with these descriptor vectors are very
likely to be recovered in the first cyclone, while it decreases
significantly with decreasing particle size down to a value
of zero for all minerals below 40 µm. At the same time, the
value of the Tromp function also decreases with decreas-
ing roundness, however this effect is not as pronounced as
for particle size. Furthermore, as stated above, for the three
minerals of this study, it needs to be taken into account
that the particle shape is closely linked to the particle size,
as coarser particles have higher roundness values than finer
ones. The distribution of Tromp values for the second and
third cyclone for calcite and magnesite exhibit similar val-
ues, except for particles with roundness around 0.95, where
magnesite particles only have a probability of 60 %,whereas
the calcite particles with this property have an almost 100 %
probability of being recovered in the second cyclone. While
this is also observed for very round fluorite particles, the
probability declines with decreasing roundness, however it
increases again for particles with very low roundness values
of 0.1 – 0.4, which is not the case for calcite and mag-
nesite. For the last two cyclones there is a noticeable shift
observed for all three minerals, as the separation probability
increases significantly for fine particles while it drops for
coarse particles at the same time. Particles over a wide range
of roundness are recovered more preferably, as opposed to
the first cyclones, where the Tromp function values were
highest for particles with values close to a roundness of 1,
which is most likely a result of the classification by size.
This effect is most pronounced for the last cyclone, where
separation probabilities of around 90% are obtained for
particles below 30 µm with roundness values of 0.1–1.
CONCLUSION
This study investigates the combined effect of particle
size and shape, as areaequivalent diameter and roundness,
respectively, on the separation of individual fractions of
calcite, fluorite and magnesite in a hydrocyclone cascade.
Five fractions are obtained after classification from the indi-
vidual hydrocyclones, which are analyzed by laser diffrac-
tion for their particle size as well as dynamic image analysis,
from which particle discrete information on particle size
and shape is obtained. This particle discrete data is then
used to compute bivariate Tromp functions with respect to
particle descriptor vectors containing area-equivalent diam-
eter and roundness. The results show that there is a signifi-
cant influence of the particle size on the classification for
all three minerals. From the first to the last cyclone, there
is a clear shift in the particle size of the different particle
fractions, from coarser particles being recovered in the first
cyclone to finer particles in the last. The effect of round-
ness is not as pronounced as that of size, although, differ-
ences can be observed for the different particle fractions,
as the first cyclones have a high separation probability for
particles with very high roundness values, while fine par-
ticles over a large range of roundness report to the latter
cyclones. However, as the particle shape is closely linked
to the particle size, it is questionable whether the observed
behavior of the studied minerals is particle shape depended
or just a consequence of the classification by size, as coarser
particles are also more round than finer ones. Therefore,
future studies should include particle systems that dif-
fer more significantly in shape, e.g., spheres or flakes, to
reveal how these extreme variations of shape influence the
separation. Additionally, based on the results of the mass
fraction, an effect of density was observed, which should
also be addressed further in future studies. Unfortunately,
all particles that are too fine to be recovered in the last
cyclone, which is actually the largest mass fraction here, are
lost and so is the information on them. Hence, a possibil-
ity to recover that fraction needs to be found, in order to
obtain information on all of the particles taking part in the
process.