XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 3693
considered as a constant matrix during calculation inside a
short time interval, which can be denoted as H(t) in the fol-
lowing sections. H(t) will be iteratively calculated for each
time step.
Under the assumption of a perfect mixing condition,
the equation 1 can be written as equation 2, which has
been derived in the dynamic condition of a tumbling mill
using both appearance function and breakage rate function
(Whiten, 1974 Valery and Morrell, 1995 Valery, 1997).
Both appearance function and breakage rate function are
not varying with time.
dt
ds
f th p a r s th r s i
i i
j
i
ij j j i i
1
=-+-
=
/^th
^^th ^^th (2)
p t d
i i =^^thsi ^th h (3)
d th d c
max i i $=^^th (4)
where, si(t) is the mass of material in size class i at time t in
the mill fi(t) is the total flow rate of feed material in this size
class pi(t) is the total flow rate of discharge material in this
class ri is the rate at which particles in size class i break aij
is the breakage distribution or appearance function which
describes the fraction of material breaking into size class i
due to breakage of size class j di(t) is the discharge rate of
class size i. ci(t) is the classification function value for size
class i. The classification function is a continuous polyline
in the logarithmic coordinates in this model. dmax is the
maximum discharge rate.
Appearance Function
The appearance function, also known as the breakage dis-
tribution function, is used to describe the size distribution
after the collision. It is an important sub-model for many
mill models (Shi and Kojovic 2007 Kojovic et al., 2012
Yu, et al., 2016 2017).
JK Size-Dependent Appearance Function
The JK size-dependent appearance function is as follows,
in which a JK t10–tn family curve can be used to generate
appearance function (Kojovic et al. 2012, Shi and Kojovic
2007):
t M61 e X E
10
mat csh =-$$-F ^@(5)
where, M is maximum possible value of t10 in impact (per
cent), Fmat is the material property (kgJ–1m–1 )=f (X,p,q),
p and q are ore-specific constants, X is the average particle
size (mm), Ecs is the specific comminution energy (J/kg).
4D (Four-Dimensional) Appearance Function
Yu and co-workers developed a 4D appearance function
(Yu, et al., 2016 2017), in which the full product size dis-
tribution is related to the original sizes of ore X (mm), ore
characteristics, and input specific comminution energy Ecs
(kWh/ton) using the P80-m Weibull distribution.
,xh f X,E passing
cs =^(6)
*e`ln^0.2h` 100 1 passing P80
x m =-j j 7 A (7)
Based on their experimental data trends for Cadia ore and
the interpolated data, two empirical relationships for P80
and the parameter “m” in equation 7 were developed using
the multi-variable function regression and fitting tech-
niques (Yu et al., 2016):
80 *P Xtop X
X e *d
top
E
E
cstop
cs a=
b c
d
c m n (8)
m a
Xtop
X
E
E
b X
cstop
cs
top =+
+
+
(9)
where, α, β, γ, δ, a, b, c are fitting parameters. Xtop is the
largest size of the original sizes. Ecstop is the biggest Ecs value
among all the Ecs values at different X. Because of the non-
dimensional forms in equation 8 and 9, the 4D appearance
function model has a good scalable ability.
Breakage Rate Function (Selection Function)
Figure 2 shows a typical selection function curve. The shape
of this selection function is of typical features of a break-
age rate curve for a SAG mill (Napier-Munn et al., 1996)
and it is assumed that it does not change with time. The
breakage rate function (selection function) in JK models is
back-calculated using PBM method (Kojovic et al., 2012).
Discharge Function
The discharge function is closely related to the classification
function. Typically, a classification curve (e.g., Figure 3)
can be divided into three regions according to the particle
size (Kojovic et al., 2012). For particles smaller than Xm, it
is assumed that they behave like water and are subject to
no classification. Xg is the effective grate aperture size and
Xp is the effective pebble port aperture size, the classifica-
tion function ci is shown in Figure 3, where all the particles
greater than Xp, cannot be discharged.
D D c
max i i =(10)
considered as a constant matrix during calculation inside a
short time interval, which can be denoted as H(t) in the fol-
lowing sections. H(t) will be iteratively calculated for each
time step.
Under the assumption of a perfect mixing condition,
the equation 1 can be written as equation 2, which has
been derived in the dynamic condition of a tumbling mill
using both appearance function and breakage rate function
(Whiten, 1974 Valery and Morrell, 1995 Valery, 1997).
Both appearance function and breakage rate function are
not varying with time.
dt
ds
f th p a r s th r s i
i i
j
i
ij j j i i
1
=-+-
=
/^th
^^th ^^th (2)
p t d
i i =^^thsi ^th h (3)
d th d c
max i i $=^^th (4)
where, si(t) is the mass of material in size class i at time t in
the mill fi(t) is the total flow rate of feed material in this size
class pi(t) is the total flow rate of discharge material in this
class ri is the rate at which particles in size class i break aij
is the breakage distribution or appearance function which
describes the fraction of material breaking into size class i
due to breakage of size class j di(t) is the discharge rate of
class size i. ci(t) is the classification function value for size
class i. The classification function is a continuous polyline
in the logarithmic coordinates in this model. dmax is the
maximum discharge rate.
Appearance Function
The appearance function, also known as the breakage dis-
tribution function, is used to describe the size distribution
after the collision. It is an important sub-model for many
mill models (Shi and Kojovic 2007 Kojovic et al., 2012
Yu, et al., 2016 2017).
JK Size-Dependent Appearance Function
The JK size-dependent appearance function is as follows,
in which a JK t10–tn family curve can be used to generate
appearance function (Kojovic et al. 2012, Shi and Kojovic
2007):
t M61 e X E
10
mat csh =-$$-F ^@(5)
where, M is maximum possible value of t10 in impact (per
cent), Fmat is the material property (kgJ–1m–1 )=f (X,p,q),
p and q are ore-specific constants, X is the average particle
size (mm), Ecs is the specific comminution energy (J/kg).
4D (Four-Dimensional) Appearance Function
Yu and co-workers developed a 4D appearance function
(Yu, et al., 2016 2017), in which the full product size dis-
tribution is related to the original sizes of ore X (mm), ore
characteristics, and input specific comminution energy Ecs
(kWh/ton) using the P80-m Weibull distribution.
,xh f X,E passing
cs =^(6)
*e`ln^0.2h` 100 1 passing P80
x m =-j j 7 A (7)
Based on their experimental data trends for Cadia ore and
the interpolated data, two empirical relationships for P80
and the parameter “m” in equation 7 were developed using
the multi-variable function regression and fitting tech-
niques (Yu et al., 2016):
80 *P Xtop X
X e *d
top
E
E
cstop
cs a=
b c
d
c m n (8)
m a
Xtop
X
E
E
b X
cstop
cs
top =+
+
+
(9)
where, α, β, γ, δ, a, b, c are fitting parameters. Xtop is the
largest size of the original sizes. Ecstop is the biggest Ecs value
among all the Ecs values at different X. Because of the non-
dimensional forms in equation 8 and 9, the 4D appearance
function model has a good scalable ability.
Breakage Rate Function (Selection Function)
Figure 2 shows a typical selection function curve. The shape
of this selection function is of typical features of a break-
age rate curve for a SAG mill (Napier-Munn et al., 1996)
and it is assumed that it does not change with time. The
breakage rate function (selection function) in JK models is
back-calculated using PBM method (Kojovic et al., 2012).
Discharge Function
The discharge function is closely related to the classification
function. Typically, a classification curve (e.g., Figure 3)
can be divided into three regions according to the particle
size (Kojovic et al., 2012). For particles smaller than Xm, it
is assumed that they behave like water and are subject to
no classification. Xg is the effective grate aperture size and
Xp is the effective pebble port aperture size, the classifica-
tion function ci is shown in Figure 3, where all the particles
greater than Xp, cannot be discharged.
D D c
max i i =(10)