XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 2077
Computation of Multivariate Tromp functions
In the present study, image data of the feed and of each of
the five concentrates (C1, C2, C3, C4, and C5) is available
for each mineral (calcite, fluorite, magnesite). In order to
evaluate the separation behavior of each particle system in
the separation experiments, the entirety of particle descrip-
tor vectors associated with particles in the feed material
(F) is used to model the mass-weighted probability density
f
m
F .Analogously, the probability densities f
m
i C are mod-
eled for each concentrate Ci with i =1,…,5. Then, the
mass-weighted probability density f
m
C of particle descrip-
tor vectors associated with particles in the entire concentrate
(C) are computed as a convex combination of ,f f
m m
1 5 f, C C
as described in Wilhelm et al. (2023). This enables to com-
pute the mass-weighted probability density f
m
T of particle
descriptor vectors associated with particles in the tailings
(T) from the convex combination of f
m
C and f
m
T resulting
in f
m
F with mixing parameter mC/(mC +mT), where mC is
the total mass of particles in (C) and mT the total mass of
particles in the tailing T.
In the following, the separation of each cyclone is ana-
lyzed by means of Tromp functions. Therefore, exemplary
for calcite, each cyclone is considered as an individual sepa-
ration experiment with its individual feed, concentrate and
tailings. More precisely, for each cyclone ,,,i 1 2 3 4,5, !"
probability densities f
m
i F ,f
m
i C and f
m
i T of particle descrip-
tor vectors associated with particles in the feed, concentrate
and tailings (T) of the i-th cyclone are determined, where
the last cyclone is considered as first. For this cyclone (i =
5), the probability density f
m
5 T equals to f
m
T and f
m
5 C
has already been modeled from image data as described
above. Then, the probability density f
m
5 F is computed as
a convex combination of f
m
5 T and f
m
5 C with the mixing
parameter /^m m m
C C T
5 5 5
+h, where m
C
5
is the total mass
of particles in the concentrate C
5 and m
T
5
the total mass
of particles in the tailing T
5 .For the next cyclone (i =4),
the probability density f
m
4 T equals to f
m
5 F .Again, f
m
4 C
is already modeled as described above. Analogously, to the
previous step f
m
4 F is computed as a convex combination of
f
m
4 T and f
m
4 C .This procedure is repeated for each cyclone,
where, for the first cyclone (i =1), f
m
1 F equals to f
m
F .This
enables the computation of bivariate Tromp functions T
i for each cyclone ,,,i 1 2 3 4,5, !"as the ratio of f
m
i C and
f
m
i F multiplied with the mass ratio of particles observed in
each concentrate and feed. Then, the values ,rh d T
i A ^of
the Tromp functions indicate the probability that a particle
with descriptor vector ,rh d
A ^is separated into the concen-
trate, as described in Wilhelm et al. (2023). With the same
procedure Tromp functions Ti are determined for each
cyclone ,,,i 1 2 3 4,5, !"for fluorite and magnesite.
In the present paper, Ti is determined for all
,,,i 1 2 3 4,5, !"and for each mineral. Furthermore, for
visualization purposes all mass-weighted probability densi-
ties are transformed to number-weighted probability densi-
ties, as described in Wilhelm et al. (2023).
RESULTS AND DISCUSSION
The results of the individual mass fractions that are recov-
ered in the different cyclones, plus the mass fraction of
ultrafine particles that is lost, are depicted in Figure 4. The
distribution of the mass fractions is similar for all three
minerals, with a decreasing trend from cyclone 1 to cyclone
5. The ultrafines and loss fraction, i.e., the particles that are
too fine and pass the last cyclone, make up the largest mass
fraction in all cases. The order of magnitude of the lost frac-
tion corresponds to the distribution of particle sizes of the
feed fraction, which is displayed in Figure 5. The highest
loss of 41 %is observed for magnesite, which also has the
largest amount of fines in the feed with a d50,feed of 4.6 µm.
Calcite and fluorite have a similar d50,feed of 15.0 µm and
16.4 µm, respectively, but due to its higher density fewer
fines are lost for the fluorite fraction.
In addition, Table 1 shows the cut size dT obtained
from partition curves using laser diffraction data, as well as
the separation efficiency k, calculated as the ratio of the d25
to the d75, where k values can range between 0 and 1, with
higher values indicating a sharper separation. Cut sizes are
similar for all minerals for each cyclone, but those for calcite
are generally slightly higher than for fluorite and magnesite.
For all three minerals, the highest separation efficiency, i.e.,
the largest k value, is obtained for the first cyclone. For the
second cyclone, the efficiency drops noticeably, while it
increases again for the following cyclones.
Figures 6, 8 and 10 present the bivariate probability
densities of particle shape, as roundness, and particle size, as
areaequivalent diameter, for the feed and cyclone fractions
of calcite, fluorite and magnesite, respectively. All three
minerals have similar probability densities for their feed as
well as the same trend for the individual cyclones. Particles
in cyclone 1 are rather coarse and round with highest fre-
quencies of particle sizes around 70 µm and roundness val-
ues around 0.9. Particles in cyclone 2 still show high values
of roundness, but the dominant particle size is now in the
range of 40 µm and 60 µm. This reduction in particle size is
continued for cyclone 3, 4 and 5, with particle size fractions
around 35 µm, 25 µm and 20 µm, respectively. However,
while the particles in cyclone 3 mainly exhibit roundness
values from around 0.8 to 1 with few exceptions, for all
three minerals, an increasing number of particles with a
wide range of roundness values down to 0.2 is observed for
Computation of Multivariate Tromp functions
In the present study, image data of the feed and of each of
the five concentrates (C1, C2, C3, C4, and C5) is available
for each mineral (calcite, fluorite, magnesite). In order to
evaluate the separation behavior of each particle system in
the separation experiments, the entirety of particle descrip-
tor vectors associated with particles in the feed material
(F) is used to model the mass-weighted probability density
f
m
F .Analogously, the probability densities f
m
i C are mod-
eled for each concentrate Ci with i =1,…,5. Then, the
mass-weighted probability density f
m
C of particle descrip-
tor vectors associated with particles in the entire concentrate
(C) are computed as a convex combination of ,f f
m m
1 5 f, C C
as described in Wilhelm et al. (2023). This enables to com-
pute the mass-weighted probability density f
m
T of particle
descriptor vectors associated with particles in the tailings
(T) from the convex combination of f
m
C and f
m
T resulting
in f
m
F with mixing parameter mC/(mC +mT), where mC is
the total mass of particles in (C) and mT the total mass of
particles in the tailing T.
In the following, the separation of each cyclone is ana-
lyzed by means of Tromp functions. Therefore, exemplary
for calcite, each cyclone is considered as an individual sepa-
ration experiment with its individual feed, concentrate and
tailings. More precisely, for each cyclone ,,,i 1 2 3 4,5, !"
probability densities f
m
i F ,f
m
i C and f
m
i T of particle descrip-
tor vectors associated with particles in the feed, concentrate
and tailings (T) of the i-th cyclone are determined, where
the last cyclone is considered as first. For this cyclone (i =
5), the probability density f
m
5 T equals to f
m
T and f
m
5 C
has already been modeled from image data as described
above. Then, the probability density f
m
5 F is computed as
a convex combination of f
m
5 T and f
m
5 C with the mixing
parameter /^m m m
C C T
5 5 5
+h, where m
C
5
is the total mass
of particles in the concentrate C
5 and m
T
5
the total mass
of particles in the tailing T
5 .For the next cyclone (i =4),
the probability density f
m
4 T equals to f
m
5 F .Again, f
m
4 C
is already modeled as described above. Analogously, to the
previous step f
m
4 F is computed as a convex combination of
f
m
4 T and f
m
4 C .This procedure is repeated for each cyclone,
where, for the first cyclone (i =1), f
m
1 F equals to f
m
F .This
enables the computation of bivariate Tromp functions T
i for each cyclone ,,,i 1 2 3 4,5, !"as the ratio of f
m
i C and
f
m
i F multiplied with the mass ratio of particles observed in
each concentrate and feed. Then, the values ,rh d T
i A ^of
the Tromp functions indicate the probability that a particle
with descriptor vector ,rh d
A ^is separated into the concen-
trate, as described in Wilhelm et al. (2023). With the same
procedure Tromp functions Ti are determined for each
cyclone ,,,i 1 2 3 4,5, !"for fluorite and magnesite.
In the present paper, Ti is determined for all
,,,i 1 2 3 4,5, !"and for each mineral. Furthermore, for
visualization purposes all mass-weighted probability densi-
ties are transformed to number-weighted probability densi-
ties, as described in Wilhelm et al. (2023).
RESULTS AND DISCUSSION
The results of the individual mass fractions that are recov-
ered in the different cyclones, plus the mass fraction of
ultrafine particles that is lost, are depicted in Figure 4. The
distribution of the mass fractions is similar for all three
minerals, with a decreasing trend from cyclone 1 to cyclone
5. The ultrafines and loss fraction, i.e., the particles that are
too fine and pass the last cyclone, make up the largest mass
fraction in all cases. The order of magnitude of the lost frac-
tion corresponds to the distribution of particle sizes of the
feed fraction, which is displayed in Figure 5. The highest
loss of 41 %is observed for magnesite, which also has the
largest amount of fines in the feed with a d50,feed of 4.6 µm.
Calcite and fluorite have a similar d50,feed of 15.0 µm and
16.4 µm, respectively, but due to its higher density fewer
fines are lost for the fluorite fraction.
In addition, Table 1 shows the cut size dT obtained
from partition curves using laser diffraction data, as well as
the separation efficiency k, calculated as the ratio of the d25
to the d75, where k values can range between 0 and 1, with
higher values indicating a sharper separation. Cut sizes are
similar for all minerals for each cyclone, but those for calcite
are generally slightly higher than for fluorite and magnesite.
For all three minerals, the highest separation efficiency, i.e.,
the largest k value, is obtained for the first cyclone. For the
second cyclone, the efficiency drops noticeably, while it
increases again for the following cyclones.
Figures 6, 8 and 10 present the bivariate probability
densities of particle shape, as roundness, and particle size, as
areaequivalent diameter, for the feed and cyclone fractions
of calcite, fluorite and magnesite, respectively. All three
minerals have similar probability densities for their feed as
well as the same trend for the individual cyclones. Particles
in cyclone 1 are rather coarse and round with highest fre-
quencies of particle sizes around 70 µm and roundness val-
ues around 0.9. Particles in cyclone 2 still show high values
of roundness, but the dominant particle size is now in the
range of 40 µm and 60 µm. This reduction in particle size is
continued for cyclone 3, 4 and 5, with particle size fractions
around 35 µm, 25 µm and 20 µm, respectively. However,
while the particles in cyclone 3 mainly exhibit roundness
values from around 0.8 to 1 with few exceptions, for all
three minerals, an increasing number of particles with a
wide range of roundness values down to 0.2 is observed for