XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 1013
an umbrella term for various definitions, and the precise
definition depends on the application of SEA. A compo-
nent might represent a chemical substance or material fol-
lowing the definition of Brunner and Rechberger (Brunner
and Rechberger, 2003), a functional component of a prod-
uct as described by Parchomenko et al. (Parchomenko et
al., 2020), a mineral phase or mineral group as described
in, e.g., (Buchmann et al., 2018), or a slag- or metal phase
from metallurgical products. Particles represent the third
aggregation level. The system’s total mass and components
are distributed over K particles. Due to the property of the
statistical entropy, the disorder of a system increases mono-
tonically with an increasing number of events—in this case,
the number of particles. Including the disperse particle
properties allows for the description of comminution and
liberation. From those three levels—batches, components
and particles—different orders to aggregate a system mass
can be derived, which are depicted in Figure 2, wherein the
intensive properties p—distributions, t—partitions, and
c—compositions are introduced as fractions/ratios of the
overall extensive system mass N.
Based on the intensive properties p, t, and c, we can
derive six different ways of expressing the overall system
entropy, always following the form of a product. The fol-
lowing example shows the calculation of the system entropy
based on the aggregation: Nijk/N000 =tioocij0pijk. Whereby
the first term containing sijk represents the intrinsic system
entropy, which is usually constant. This assumption is valid
as long as no phase transitions are considered, as they might
be relevant for metallurgical processes such as smelting.
Figure 2. Aggregation of a system’s total mass N000 over different dimensions batches (I), components (J) and particles (K),
always keeping one dimension constant—figure adapted from (Schach et al., 2024)
an umbrella term for various definitions, and the precise
definition depends on the application of SEA. A compo-
nent might represent a chemical substance or material fol-
lowing the definition of Brunner and Rechberger (Brunner
and Rechberger, 2003), a functional component of a prod-
uct as described by Parchomenko et al. (Parchomenko et
al., 2020), a mineral phase or mineral group as described
in, e.g., (Buchmann et al., 2018), or a slag- or metal phase
from metallurgical products. Particles represent the third
aggregation level. The system’s total mass and components
are distributed over K particles. Due to the property of the
statistical entropy, the disorder of a system increases mono-
tonically with an increasing number of events—in this case,
the number of particles. Including the disperse particle
properties allows for the description of comminution and
liberation. From those three levels—batches, components
and particles—different orders to aggregate a system mass
can be derived, which are depicted in Figure 2, wherein the
intensive properties p—distributions, t—partitions, and
c—compositions are introduced as fractions/ratios of the
overall extensive system mass N.
Based on the intensive properties p, t, and c, we can
derive six different ways of expressing the overall system
entropy, always following the form of a product. The fol-
lowing example shows the calculation of the system entropy
based on the aggregation: Nijk/N000 =tioocij0pijk. Whereby
the first term containing sijk represents the intrinsic system
entropy, which is usually constant. This assumption is valid
as long as no phase transitions are considered, as they might
be relevant for metallurgical processes such as smelting.
Figure 2. Aggregation of a system’s total mass N000 over different dimensions batches (I), components (J) and particles (K),
always keeping one dimension constant—figure adapted from (Schach et al., 2024)