2074 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
experiencing hindered settling due to particle swam effects,
thus creating challenges for the development of separation
models (Schubert, 2010). Adding to this, is the complexity
of the particle system, as particles usually vary in size, shape
and composition. Typically, these challenges are addressed
by model simplification or by including a mean parameter
for describing the whole particle system (Schubert, 2003
Schubert, 2010).
In order to integrate the particle properties into the
model, a better understanding of how certain particle prop-
erties influence the process is crucial. Whereas the influence
of the particle size and density is rather well investigated,
there are only very few studies found in literature on the
impact of particle shape on the classification in a hydro-
cyclone (Endoh et al., 1994 Kashiwaya et al., 2012).
Kashiwaya et al. (2012) investigated the behavior of plate-
like, blockshaped and spherical particles in a hydrocyclone
as well as in a hydrocyclone cascade. They showed that the
separation outcome is significantly influenced by the par-
ticle shape, as particles that were expected to be recovered
in the underflow product on the basis of their size, were
actually recovered in the overflow product, which they pre-
sumed to be a result of their platelike shape.
Additionally, since the particle separation behavior
is typically governed by multiple properties rather than a
single one, it is crucial to investigate their combined effect,
i.e., considering not only the effect of size, but the combi-
nation of size and density or size and shape of individual
particles at the same time. Rather than using the commonly
known univariate partition curves, also known as univari-
ate Tromp functions, that display the separation function
based on a single particle property, bivariate Tromp func-
tions provide a more detailed insight into the influence of
two particle descriptors at the same time and thus offer a
more detailed look into their complex interactions. In order
to compute bivariate Tromp functions, particle discrete
information on the individual material streams is needed.
One way of obtaining this information is via automated
mineralogy (MLA) or dynamic image analysis. While MLA
allows the analysis of different phases, thus being more suit-
able for complex mineral ores, dynamic image analysis pro-
vides information on the particle shape and size, but not
on the particle composition and is therefore more suitable
for single mineral analysis (Li &Iskander, 2020 Schulz et
al., 2020). Several studies have shown that multidimen-
sional Tromp functions can be computed based on different
approaches. Schach et al. (2019), for example, used kernel
density estimators to determine bivariate Tromp functions
for the separation in a Falcon separator according to par-
ticle size and mass density. Wilhelm et al. (2023), on the
other hand, investigate the influence of particle shape and
size for particles with different levels of wettability on their
separation by flotation, using bivariate Tromp functions
computed by using a parametric modeling approach.
Within this study, individual particle fractions of cal-
cite, fluorite and magnesite are processed in a turbulent
crossflow separator cascade Cyclosizer, from which five dif-
ferent product fractions are obtained. Since the particle sys-
tems used for this study consist of one material only and are
processed individually, dynamic image analysis is chosen as
the characterization technique of the product fractions, as
it is less time-consuming than automated mineralogy and
still provides sufficient information on the relevant par-
ticle property information, here size and shape. Based on
the particle discrete data obtained from image measure-
ments, bivariate Tromp functions are computed regarding
roundness and areaequivalent diameter. These functions
are determined using kernel density estimators. In this
way, the influence of particle size and shape on the par-
ticle separation behavior in a turbulent crossflow cascade is
investigated.
MATERIAL AND METHODS
Material
Calcite (ρ =2.71 g/cm3), Fluorite (ρ =3.18 g/cm3) and
Magnesite (ρ =3.05 g/cm3), all purchased from Krantz
Rheinisches Mineralien-Kontor, Germany, are used as
materials in this study. Suitable size fractions of –71 µm
are obtained by dry sieving prior to the separation tests.
Each particle system is represented by a set of particle
descriptor vectors, where each vector contains a size and
shape descriptor characterizing a single particle. The size
of a particle is described by its area-equivalent diameter
with respect to Eq.1, whereas the shape is described by its
roundness with respect to Eq. 2. Note that for each particle
both descriptors are computed from its projection obtained
from dynamic image analysis. These sets of particle descrip-
tor vectors are modeled by bivariate probability densities,
which are displayed in Figure 1.
area equivalent diameter d
projected area
2
A r -=(1)
roundness r perimeter
4rarea
2 =(2)
Turbulent Cross-Flow Separator Cascade
The separation tests are carried out using a Cyclosizer M16
from MARC Technologies Pty Ltd, Australia, which is dis-
played in Figure 2. It consists of 5 inverse cyclones that
are placed in a series, where the overflow of one cyclone