XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 3695
ln ln
ln
ln ln
ln
ln ln
ln
ln ln
ln
c
x Xm
X X
x cgh
cg X X
X cgh
X x Xg
X X
xcg
X X
c X
X x X
x X
1
1
0
i
m g m g
g
m
g p g p
g p
g p
p
1
1 1
1 1
2
=
-
-
+--
-
---
^^1
Z
[
\
]
]
]
]]
]
]
]
]]
(11)
where, Di is the discharge rate of size class i ci is the classifi-
cation function value for size class i (describing the classifi-
cation effect of grate) Dmax is the maximum discharge rate.
Dmax is calculated iteratively using the mass transfer
function that relates the slurry hold-up to the normal-
ized volumetric flowrate of slurry discharged from the mill
(Kojovic et al., 2012):
L m1(F/V) m
2 =(12)
where, L is the fraction of mill volume occupied by below
grate size material F is the normalised flowrate through the
mill (in equivalent mill fills per minute) V is the normalised
volumetric flowrate of the slurry. (F/V) means the volumet-
ric discharge rate. m1: constant linked to grate design, grate
open area and mill speed m2=0.5.
Transport Function
The transport function is used to describe how the ore mate-
rials transport through the mill. It is also known as residence
time distribution (RTD) function that is defined as a ratio
of the mass of ore material in a mill to the feed rate. The
transport function is very important for a dynamic grind-
ing progress. It has a significant influence on the particle
size distribution, slurry hold-up, and throughput and other
parameters. As a reasonable trial to introduce a transport
function along the mill, the mill is treated as a combination
of a series of segments as shown in Figure 4a. Every segment
is perfectly mixed. After each segment, a simple transport
function is introduced in Figure 3b. This follows the form
of the discharge function, but with a bypass to allow a frac-
tion of all sizes to move freely along the mill.
Energy Model
The energy model is used to calculate the available energy
input to the comminution process, which is based on the
mill power draw.
Morrell Power Model
An energy model is based on Morrell power model (Morrell,
1996):
KD L Gross power NoLoad power .5
e c
2 t adi =+_(13)
.D Lh@0.82 1 68 NoLoad power .05 2 z^0.667Ld =+6 (14)
where, D is the diameter of cylindrical section of mill inside
liners, (m) K is the lumped parameter used in calibration
of model. L is the length of cylindrical section of mill inside
liners, (m) Ld is the length of cone-end, (m) Le is the effec-
tive grinding length, (m) α is an empirical parameter ϕ is
the fraction of critical speed ρc is the density of total charge
(t/m3).
AN ANALYTICAL SOLUTION
According to the mathematical theory (Ju and Ge, 1996),
the solution of the first-order nonhomogeneous linear dif-
ferential equations for the above Equation 1 is:
s th e N N tH =+^^th@ ^th 6 (15)
s th Pe P N N tK =+-1 ^^th@ 6 (16)
where N is a constant vector, N(t) is a variable vector and
is a function of time. P is an invertible matrix. Matrix H(t)
can be diagonalized:
,,g,m P HP diag^m
n 1 2 m K ==-1 h (17)
λi are the eigenvalues of the matrix H(t) which can be
derived from the determinant of (H(t) – λI):
det^H^th 0 mIh -=(18)
Figure 4. (a) Multi-segment mill assumption (b) An example of transport function
ln ln
ln
ln ln
ln
ln ln
ln
ln ln
ln
c
x Xm
X X
x cgh
cg X X
X cgh
X x Xg
X X
xcg
X X
c X
X x X
x X
1
1
0
i
m g m g
g
m
g p g p
g p
g p
p
1
1 1
1 1
2
=
-
-
+--
-
---
^^1
Z
[
\
]
]
]
]]
]
]
]
]]
(11)
where, Di is the discharge rate of size class i ci is the classifi-
cation function value for size class i (describing the classifi-
cation effect of grate) Dmax is the maximum discharge rate.
Dmax is calculated iteratively using the mass transfer
function that relates the slurry hold-up to the normal-
ized volumetric flowrate of slurry discharged from the mill
(Kojovic et al., 2012):
L m1(F/V) m
2 =(12)
where, L is the fraction of mill volume occupied by below
grate size material F is the normalised flowrate through the
mill (in equivalent mill fills per minute) V is the normalised
volumetric flowrate of the slurry. (F/V) means the volumet-
ric discharge rate. m1: constant linked to grate design, grate
open area and mill speed m2=0.5.
Transport Function
The transport function is used to describe how the ore mate-
rials transport through the mill. It is also known as residence
time distribution (RTD) function that is defined as a ratio
of the mass of ore material in a mill to the feed rate. The
transport function is very important for a dynamic grind-
ing progress. It has a significant influence on the particle
size distribution, slurry hold-up, and throughput and other
parameters. As a reasonable trial to introduce a transport
function along the mill, the mill is treated as a combination
of a series of segments as shown in Figure 4a. Every segment
is perfectly mixed. After each segment, a simple transport
function is introduced in Figure 3b. This follows the form
of the discharge function, but with a bypass to allow a frac-
tion of all sizes to move freely along the mill.
Energy Model
The energy model is used to calculate the available energy
input to the comminution process, which is based on the
mill power draw.
Morrell Power Model
An energy model is based on Morrell power model (Morrell,
1996):
KD L Gross power NoLoad power .5
e c
2 t adi =+_(13)
.D Lh@0.82 1 68 NoLoad power .05 2 z^0.667Ld =+6 (14)
where, D is the diameter of cylindrical section of mill inside
liners, (m) K is the lumped parameter used in calibration
of model. L is the length of cylindrical section of mill inside
liners, (m) Ld is the length of cone-end, (m) Le is the effec-
tive grinding length, (m) α is an empirical parameter ϕ is
the fraction of critical speed ρc is the density of total charge
(t/m3).
AN ANALYTICAL SOLUTION
According to the mathematical theory (Ju and Ge, 1996),
the solution of the first-order nonhomogeneous linear dif-
ferential equations for the above Equation 1 is:
s th e N N tH =+^^th@ ^th 6 (15)
s th Pe P N N tK =+-1 ^^th@ 6 (16)
where N is a constant vector, N(t) is a variable vector and
is a function of time. P is an invertible matrix. Matrix H(t)
can be diagonalized:
,,g,m P HP diag^m
n 1 2 m K ==-1 h (17)
λi are the eigenvalues of the matrix H(t) which can be
derived from the determinant of (H(t) – λI):
det^H^th 0 mIh -=(18)
Figure 4. (a) Multi-segment mill assumption (b) An example of transport function