1532 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
MULTI-DIMENSIONAL PARTICLE-
DISCRETE PROPERTY DISTRIBUTIONS
3D Particle-Discrete Data
The well-known particle size distribution (PSD) represents
a one-dimensional distribution, beginning with details
about each particle size, specifically its equivalent diameter.
The data are then categorized into various size classes and
weighted based on factors like number, cord length, surface
area, or volume.
Conventional measurement methods for particle char-
acterization do not describe the individual particle at all,
for example in the case of laser diffraction, or only partially
and incompletely, for example in static 2D image analysis.
Although there are methods to deal with this (Frank 2019)
only 3D particle-discrete data is the ideal starting point for
an overall description of a particle originating from a par-
ticle collective with distributed properties. In this context
particle-discrete means, that there is an information vector
for each individual particle containing the required infor-
mation on size, shape, composition and others. An example
for 3D experimental data is given in Fig. 1 with particle
data coming from optical scanning and X-ray tomographic
measurements.
From the 3D measurements it is possible to character-
ize each particle by an information vector that contains all
relevant individual geometrical data about its properties.
The collection of all vectors from all particles can now be
used to investigate various multi-dimensional correlations.
Multi-dimensional Characterization of Ore Particle
Systems
Multi-dimensional implies that some of the particle-dis-
crete properties cannot be treated separately or sequentially.
For example, both fine and coarse fractions are examined
concurrently. This simultaneous consideration of multiple
characteristics (such as size, shape, or perhaps other proper-
ties like density or chemical composition) is what makes
the analysis multi-dimensional. In this context, multi-
dimensional means that the analysis is not limited to just
one dimension or property at a time, but rather it encom-
passes multiple dimensions or properties in a single process.
Each dimension (e.g., size, shape, composition) offers
a different perspective and provides unique information,
but when combined, they give a more comprehensive
understanding of the particle characteristics. This multi-
dimensional approach (Buchwald 2024) can lead to more
accurate, detailed, and insightful results than would be pos-
sible by examining each property isolated.
The Multi-Dimensional Particle Property Distribution
As in the one-dimensional case, property classes are defined.
In the multi-dimensional case, a property class address
more than one property (see Figure 2). For the two-dimen-
sional example, with the properties x (equivalent diameter
/particle size) and y (shape factor), each class contains the
fraction of all particles, which fulfill the specifications:
• Equivalent diameter: x x xi
i 1 1 #
-• Shape factor: y y y
j j 1 1 #
-
The concept of a two-dimensional particle density dis-
tribution uses a similar definition to the one-dimensional
case. To illustrate this, let’s consider an example involving
a dataset of particles for which the volume of each particle
is known. If we operate under the premise that the density
of the solids is uniform* across all particles, it’s possible to
compute a distribution weighted by mass. This approach
in two dimensions is fundamentally aligned with the
*In the case of intergrown ore particles the assumption of
constant density is not valid, therefore particle-discrete data
containing the material information is needed to set up a
reliable mass distribution.
Figure 1. 3D representatives of particles of different materials from which geometrical data can be retrieved, determined by 3D
scanning (left) and by 3D X-ray tomography measurements (middle and right)
MULTI-DIMENSIONAL PARTICLE-
DISCRETE PROPERTY DISTRIBUTIONS
3D Particle-Discrete Data
The well-known particle size distribution (PSD) represents
a one-dimensional distribution, beginning with details
about each particle size, specifically its equivalent diameter.
The data are then categorized into various size classes and
weighted based on factors like number, cord length, surface
area, or volume.
Conventional measurement methods for particle char-
acterization do not describe the individual particle at all,
for example in the case of laser diffraction, or only partially
and incompletely, for example in static 2D image analysis.
Although there are methods to deal with this (Frank 2019)
only 3D particle-discrete data is the ideal starting point for
an overall description of a particle originating from a par-
ticle collective with distributed properties. In this context
particle-discrete means, that there is an information vector
for each individual particle containing the required infor-
mation on size, shape, composition and others. An example
for 3D experimental data is given in Fig. 1 with particle
data coming from optical scanning and X-ray tomographic
measurements.
From the 3D measurements it is possible to character-
ize each particle by an information vector that contains all
relevant individual geometrical data about its properties.
The collection of all vectors from all particles can now be
used to investigate various multi-dimensional correlations.
Multi-dimensional Characterization of Ore Particle
Systems
Multi-dimensional implies that some of the particle-dis-
crete properties cannot be treated separately or sequentially.
For example, both fine and coarse fractions are examined
concurrently. This simultaneous consideration of multiple
characteristics (such as size, shape, or perhaps other proper-
ties like density or chemical composition) is what makes
the analysis multi-dimensional. In this context, multi-
dimensional means that the analysis is not limited to just
one dimension or property at a time, but rather it encom-
passes multiple dimensions or properties in a single process.
Each dimension (e.g., size, shape, composition) offers
a different perspective and provides unique information,
but when combined, they give a more comprehensive
understanding of the particle characteristics. This multi-
dimensional approach (Buchwald 2024) can lead to more
accurate, detailed, and insightful results than would be pos-
sible by examining each property isolated.
The Multi-Dimensional Particle Property Distribution
As in the one-dimensional case, property classes are defined.
In the multi-dimensional case, a property class address
more than one property (see Figure 2). For the two-dimen-
sional example, with the properties x (equivalent diameter
/particle size) and y (shape factor), each class contains the
fraction of all particles, which fulfill the specifications:
• Equivalent diameter: x x xi
i 1 1 #
-• Shape factor: y y y
j j 1 1 #
-
The concept of a two-dimensional particle density dis-
tribution uses a similar definition to the one-dimensional
case. To illustrate this, let’s consider an example involving
a dataset of particles for which the volume of each particle
is known. If we operate under the premise that the density
of the solids is uniform* across all particles, it’s possible to
compute a distribution weighted by mass. This approach
in two dimensions is fundamentally aligned with the
*In the case of intergrown ore particles the assumption of
constant density is not valid, therefore particle-discrete data
containing the material information is needed to set up a
reliable mass distribution.
Figure 1. 3D representatives of particles of different materials from which geometrical data can be retrieved, determined by 3D
scanning (left) and by 3D X-ray tomography measurements (middle and right)