XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 1533
principles used in one-dimensional density distribution
analysis. The mass of particles in class i,j,
,j Dm ,is divided
by the overall mass m
total and the size of the class, which
in the two-dimensional case involves two dimensions, x
i D
for the property x and y
j D for the property y .
,q x y m x îy
,,j
i j
total i j
j
i j
3
3,
D Dy
Dm
Dx
DQ
==^h
The width of the property classes is the difference
between the upper and the lower limit for each property:
x x x
i i i 1 D =-
-
y y y
i j j 1 D =-
-
The cumulative sum is calculated via summing up (class
wise definition of the distribution) or integrating (smooth
distribution function) the density distribution from the
minimal values x
min and y
min up to a point (,x y).
Q Q3,v,n q xvyn
,j ,n
j
v
i j
v
i
3, 1 1 1 1 3,v ==
n n ====
////
A multi-dimensional separation is characterized as
a separation process based on or influenced by multiple
properties. Regardless of the complexity of the initial mix-
ture, the multi-dimensional separation yields two distinct
product fractions, each differing in several characteris-
tics. Unlike separations solely based on size, this method
employs the terms concentrate (c) and reject (r) to describe
the two resulting product fractions. The concentrate con-
tains the desired particles, while the reject consists of parti-
cles that are of lesser interest. Additionally, the term supply
material, as per ISO 9276-4, is denoted by the index s. In
the context of a two-dimensional separation, the mass bal-
ance of the distribution densities can be expressed using the
respective mass fractions of these components.
m
m
m
m
c r 1
S
C
S
R =+=+
q x,yh cq rq
S C R =^^x,yh+ ^x,yh
If one defines the two-dimensional separation function
on the basis of the associated two-dimensional density dis-
tributions of feed material and concentrate and the mass
balance, it leads to the separation function TSC(x, y):
T q
cq
SC
S
C =^x,yh ^x,yh
^x,yh
MODEL CASE OF MULTI-DIMENSIONAL
SEPARATION
The two-dimensional separation function can be plotted
as heat-plot. The color code defines the probability of a
particle with given properties to end up in the concentrate
(yellow: 0 %red: 50 %black: 100 %).The marginal distri-
butions are showing the functions of the one-dimensional
separation according one of the parameters either T^xvh or
T^w/lh. The reject is made up of all particles that do not
meet the criteria of the two different thresholds.
In mineral processing, it is common to employ mul-
tiple sequential process steps to generate the required speci-
fications of the particle i.e., ore system. When separations
based on size and shape are conducted in series, the result-
ing concentrate typically comprises particles that not only
exceed a specified size but also possess an aspect ratio above
Figure. 2. Schematic representation of the two-dimensional definition of classes for the properties x and y
principles used in one-dimensional density distribution
analysis. The mass of particles in class i,j,
,j Dm ,is divided
by the overall mass m
total and the size of the class, which
in the two-dimensional case involves two dimensions, x
i D
for the property x and y
j D for the property y .
,q x y m x îy
,,j
i j
total i j
j
i j
3
3,
D Dy
Dm
Dx
DQ
==^h
The width of the property classes is the difference
between the upper and the lower limit for each property:
x x x
i i i 1 D =-
-
y y y
i j j 1 D =-
-
The cumulative sum is calculated via summing up (class
wise definition of the distribution) or integrating (smooth
distribution function) the density distribution from the
minimal values x
min and y
min up to a point (,x y).
Q Q3,v,n q xvyn
,j ,n
j
v
i j
v
i
3, 1 1 1 1 3,v ==
n n ====
////
A multi-dimensional separation is characterized as
a separation process based on or influenced by multiple
properties. Regardless of the complexity of the initial mix-
ture, the multi-dimensional separation yields two distinct
product fractions, each differing in several characteris-
tics. Unlike separations solely based on size, this method
employs the terms concentrate (c) and reject (r) to describe
the two resulting product fractions. The concentrate con-
tains the desired particles, while the reject consists of parti-
cles that are of lesser interest. Additionally, the term supply
material, as per ISO 9276-4, is denoted by the index s. In
the context of a two-dimensional separation, the mass bal-
ance of the distribution densities can be expressed using the
respective mass fractions of these components.
m
m
m
m
c r 1
S
C
S
R =+=+
q x,yh cq rq
S C R =^^x,yh+ ^x,yh
If one defines the two-dimensional separation function
on the basis of the associated two-dimensional density dis-
tributions of feed material and concentrate and the mass
balance, it leads to the separation function TSC(x, y):
T q
cq
SC
S
C =^x,yh ^x,yh
^x,yh
MODEL CASE OF MULTI-DIMENSIONAL
SEPARATION
The two-dimensional separation function can be plotted
as heat-plot. The color code defines the probability of a
particle with given properties to end up in the concentrate
(yellow: 0 %red: 50 %black: 100 %).The marginal distri-
butions are showing the functions of the one-dimensional
separation according one of the parameters either T^xvh or
T^w/lh. The reject is made up of all particles that do not
meet the criteria of the two different thresholds.
In mineral processing, it is common to employ mul-
tiple sequential process steps to generate the required speci-
fications of the particle i.e., ore system. When separations
based on size and shape are conducted in series, the result-
ing concentrate typically comprises particles that not only
exceed a specified size but also possess an aspect ratio above
Figure. 2. Schematic representation of the two-dimensional definition of classes for the properties x and y