872 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
assuming discrete size classes and first order grinding kinet-
ics is given by (Austin et al., 1984):
,…n
dt
dm
S m t b S m
i 1 2
i
i i
j
i
ij j j
1
1 =-+
=
=
-/^th
^^th h
(1)
where, mi(t) is particle mass fraction in the ith size class
at time t bij depicts ratio of particles that move from size
interval i to j due to breakage to the total broken particles
Si represents fraction of particles that are broken from the
ith size interval, defined as:
S x
Ac x
x
1
i
i
i
0
n
=
+a
m
a
m
k
(2)
.8
D
D
D D
D
1 6.6J
1 6.6J
3
1 6.6J
1 6.6J
.5
.3
.3
.5 .3
.3
.3
T
T
U
T
T T
T
UTh
0
2
2 1 1.32^U
0 0
2
2 1 1.32^U
Th
A
AT
=+
+
+
+
--
--
a
a
a k
k
k
Z
[
\
]
]
]
]]A
]
(3)
dT
d .2
T
1 n n =c m (4)
where, α is a characteristic of the material and is indepen-
dent of the mill conditions such as rotational speed or ball
diameter µ and λ describe the size at which bend–over hap-
pens and how quickly it falls to the right of the maximum
of Si value in selection function vs feed size curve. While D
represents the mill diameter, the d represents the ball diam-
eter. The xi is the size fraction and x0 is the standard size.
In the present study, it is assumed that the standard size,
x0 =1 mm. Equation 3 and Equation 4 are used for scaling
up the parameter A and μ from initial condition to test
condition, denoted with subscript T. These two parameters
are further used in Equation 2 for calculation of Si.
The breakage function, Bi,j is defined as:
B x
x
x
x
,j
j
i
j
i
1 1
z zhc =+-
c b
--
c ^1 m m ,i j (5)
b B B
,,j j i 1,j =-
+,i j (6)
b B
,,j n j n =,j =n (7)
Parameters φ, β and γ are characteristics of the material.
For simplicity, Equation 1 is converted into a matrix form
as shown below:
dt
dm
BS IShm i
i =-
^th
^^th (8)
where, B is a (M × M )lower triangular matrix, I is an iden-
tity matrix of size M, S is a diagonal matrix of size (M × M)
and mi is a vector (M × 1) of particle mass fractions. The
Figure 1. Schematic of the comprehensive modelling framework for grinding
assuming discrete size classes and first order grinding kinet-
ics is given by (Austin et al., 1984):
,…n
dt
dm
S m t b S m
i 1 2
i
i i
j
i
ij j j
1
1 =-+
=
=
-/^th
^^th h
(1)
where, mi(t) is particle mass fraction in the ith size class
at time t bij depicts ratio of particles that move from size
interval i to j due to breakage to the total broken particles
Si represents fraction of particles that are broken from the
ith size interval, defined as:
S x
Ac x
x
1
i
i
i
0
n
=
+a
m
a
m
k
(2)
.8
D
D
D D
D
1 6.6J
1 6.6J
3
1 6.6J
1 6.6J
.5
.3
.3
.5 .3
.3
.3
T
T
U
T
T T
T
UTh
0
2
2 1 1.32^U
0 0
2
2 1 1.32^U
Th
A
AT
=+
+
+
+
--
--
a
a
a k
k
k
Z
[
\
]
]
]
]]A
]
(3)
dT
d .2
T
1 n n =c m (4)
where, α is a characteristic of the material and is indepen-
dent of the mill conditions such as rotational speed or ball
diameter µ and λ describe the size at which bend–over hap-
pens and how quickly it falls to the right of the maximum
of Si value in selection function vs feed size curve. While D
represents the mill diameter, the d represents the ball diam-
eter. The xi is the size fraction and x0 is the standard size.
In the present study, it is assumed that the standard size,
x0 =1 mm. Equation 3 and Equation 4 are used for scaling
up the parameter A and μ from initial condition to test
condition, denoted with subscript T. These two parameters
are further used in Equation 2 for calculation of Si.
The breakage function, Bi,j is defined as:
B x
x
x
x
,j
j
i
j
i
1 1
z zhc =+-
c b
--
c ^1 m m ,i j (5)
b B B
,,j j i 1,j =-
+,i j (6)
b B
,,j n j n =,j =n (7)
Parameters φ, β and γ are characteristics of the material.
For simplicity, Equation 1 is converted into a matrix form
as shown below:
dt
dm
BS IShm i
i =-
^th
^^th (8)
where, B is a (M × M )lower triangular matrix, I is an iden-
tity matrix of size M, S is a diagonal matrix of size (M × M)
and mi is a vector (M × 1) of particle mass fractions. The
Figure 1. Schematic of the comprehensive modelling framework for grinding