XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 3833
b s p t M th
j
i
ijk jk jk
1
1
=
-/^^h
minus the discharge rate due to breakage to finer particle
sizes sikM(t)pik(t). Win and Wout are the overall mass flow-
rates of feed and discharge streams and pik and pin,ik are
the mass fractions of particles in size class i of component k
inside the mill and in its feed, respectively. Meanwhile, sik
and bijk represent the size-discretized selection and breakage
functions, of component k respectively.
The selection function (sj) represents the specific rate of
breakage of particles of size dpi, which varies as a function
of the design and operating conditions of the mill, besides
of the material being comminuted (King, 2001). The func-
tion is represented by
s
C1n
dp
s dp
C C C C
1
i
i
i 1
2 3 4 5 =
+
a
K
d n
(2)
where s1, a, µ and Λ are material parameters estimated
empirically. C1 to C5 are scaling parameters (Austin et al.,
1984).
The breakage function bij represents the material frac-
tion of particles in size class i that is comminuted into
particles of size class j. The result is a matrix, since when
a particle of a particular size class is broken, its fragments
report to finer size classes. Its modeling in cumulative form
(Bij) is given by Equation 3 (King, 2001), referred to as
the non-normalizable breakage function, used for materials
where the behavior of the function depends not only on the
size ratio (King, 2001):
*
*
D
D
D
D
D
D
Di D
D
Di
D
Di
Di D
for
for
*ij
i
j
i
j
i
ij
j j
1
$
U Uhd
U Uhd
=+-
=+-
~c b
c b
c
d
d
^1
^1
n
m n
n
n
Z
[
\
]B
]
]
]
]]B
]
]
(3)
where D represents the particle size classes, D*, ω, Φ, β and
γ are material parameters. A more convenient way to imple-
ment this function is in its distributed form, given by bij =
Bi–1,j – Bi,j and bij =0. In this work, it is assumed that selec-
tion and breakage function parameters remain unchanged
for each component in the presence of another component
in the mill.
Additionally, a global mass balance is carried out in
each grinding chamber, where the material output flow
Wout =α(M(t)/M*)cis a function of the hold-up, based on a
power-law relationship, where α and c are parameters that
depend on the operating and design conditions of the mill,
and M* is a reference mass. Thus, the mass balance inside a
grinding chamber is given by
dt
dM W
M
M
*in
c
ad =-^th
^th
n (4)
where M is the mass of solids inside the chamber and Win is
the mass flow rate of the mil feed stream.
The dynamic separator is modelled considering a
sequence of steady states, termed pseudo-dynamics, where
the operation is described by a combination of algebraic
equations, and the input parameters are subject to dis-
turbances. This approach is common in units with rapid
dynamics (Muanpaopong et al., 2023). This approach can
be considered valid since, due to the rapid flow of material,
the residence time of particles can be neglected, unlike the
approach used for the ball mill, where the material stays a
substantial period of time. The purpose of this operation
is to separate the material into coarse (underflow) and fine
(overflow) fractions. It is modeled based on the Whiten
partition model (Napier-Munn et al., 1996), modified
to capture the fishhook effect often observed in partition
curves in dry classification. It is given by
exp exp^a
c C
X
X
1
2
1h
*
*
c c c
c c c
b
b b
=-
+-
+-
^dp
_1
_a
h i
i^exp^a h
h
H (5)
where c(dp) is the fraction of feed particles of size dp
directed to the oversize, C is the fraction of feed subjected
to classification, αc is a model parameter, and X =dp/d50c,
where dp is the average particle size, and d50c is the particle
size with the same probability of reporting to either of the
streams. βc e βc* are fitting parameters.
The effect of operating conditions on the curve param-
eters was modeled based on expressions proposed by Benzer
et al. (2001), where the main parameters in Equation 5
depend on the feed rate (Win, in t/h), rotor speed of the
classifier (Vr, in m/s) and air velocity (Va, in m/s). In the
multicomponent version proposed in the present work,
the effect of classification of multicomponent feeds was
described through a modification of the original equa-
tion, including the effect of density of each component l
by incorporating a fitting parameter that also depends on
its mass fraction in the feed mixture to the classifier. These
effects are described by Equations 6 to 8:
.,p .031 d c Vr
Va 50 0 1395f 0
l l l t =-^h (6)
..59 ln` C Win
Va 19 122 115 =+j (7)
b s p t M th
j
i
ijk jk jk
1
1
=
-/^^h
minus the discharge rate due to breakage to finer particle
sizes sikM(t)pik(t). Win and Wout are the overall mass flow-
rates of feed and discharge streams and pik and pin,ik are
the mass fractions of particles in size class i of component k
inside the mill and in its feed, respectively. Meanwhile, sik
and bijk represent the size-discretized selection and breakage
functions, of component k respectively.
The selection function (sj) represents the specific rate of
breakage of particles of size dpi, which varies as a function
of the design and operating conditions of the mill, besides
of the material being comminuted (King, 2001). The func-
tion is represented by
s
C1n
dp
s dp
C C C C
1
i
i
i 1
2 3 4 5 =
+
a
K
d n
(2)
where s1, a, µ and Λ are material parameters estimated
empirically. C1 to C5 are scaling parameters (Austin et al.,
1984).
The breakage function bij represents the material frac-
tion of particles in size class i that is comminuted into
particles of size class j. The result is a matrix, since when
a particle of a particular size class is broken, its fragments
report to finer size classes. Its modeling in cumulative form
(Bij) is given by Equation 3 (King, 2001), referred to as
the non-normalizable breakage function, used for materials
where the behavior of the function depends not only on the
size ratio (King, 2001):
*
*
D
D
D
D
D
D
Di D
D
Di
D
Di
Di D
for
for
*ij
i
j
i
j
i
ij
j j
1
$
U Uhd
U Uhd
=+-
=+-
~c b
c b
c
d
d
^1
^1
n
m n
n
n
Z
[
\
]B
]
]
]
]]B
]
]
(3)
where D represents the particle size classes, D*, ω, Φ, β and
γ are material parameters. A more convenient way to imple-
ment this function is in its distributed form, given by bij =
Bi–1,j – Bi,j and bij =0. In this work, it is assumed that selec-
tion and breakage function parameters remain unchanged
for each component in the presence of another component
in the mill.
Additionally, a global mass balance is carried out in
each grinding chamber, where the material output flow
Wout =α(M(t)/M*)cis a function of the hold-up, based on a
power-law relationship, where α and c are parameters that
depend on the operating and design conditions of the mill,
and M* is a reference mass. Thus, the mass balance inside a
grinding chamber is given by
dt
dM W
M
M
*in
c
ad =-^th
^th
n (4)
where M is the mass of solids inside the chamber and Win is
the mass flow rate of the mil feed stream.
The dynamic separator is modelled considering a
sequence of steady states, termed pseudo-dynamics, where
the operation is described by a combination of algebraic
equations, and the input parameters are subject to dis-
turbances. This approach is common in units with rapid
dynamics (Muanpaopong et al., 2023). This approach can
be considered valid since, due to the rapid flow of material,
the residence time of particles can be neglected, unlike the
approach used for the ball mill, where the material stays a
substantial period of time. The purpose of this operation
is to separate the material into coarse (underflow) and fine
(overflow) fractions. It is modeled based on the Whiten
partition model (Napier-Munn et al., 1996), modified
to capture the fishhook effect often observed in partition
curves in dry classification. It is given by
exp exp^a
c C
X
X
1
2
1h
*
*
c c c
c c c
b
b b
=-
+-
+-
^dp
_1
_a
h i
i^exp^a h
h
H (5)
where c(dp) is the fraction of feed particles of size dp
directed to the oversize, C is the fraction of feed subjected
to classification, αc is a model parameter, and X =dp/d50c,
where dp is the average particle size, and d50c is the particle
size with the same probability of reporting to either of the
streams. βc e βc* are fitting parameters.
The effect of operating conditions on the curve param-
eters was modeled based on expressions proposed by Benzer
et al. (2001), where the main parameters in Equation 5
depend on the feed rate (Win, in t/h), rotor speed of the
classifier (Vr, in m/s) and air velocity (Va, in m/s). In the
multicomponent version proposed in the present work,
the effect of classification of multicomponent feeds was
described through a modification of the original equa-
tion, including the effect of density of each component l
by incorporating a fitting parameter that also depends on
its mass fraction in the feed mixture to the classifier. These
effects are described by Equations 6 to 8:
.,p .031 d c Vr
Va 50 0 1395f 0
l l l t =-^h (6)
..59 ln` C Win
Va 19 122 115 =+j (7)