XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 843
estimation of the length of travel during the cascading and
catracting phases. The utilization of the following equations
3 to 7 proves instrumental in determining these directional
coordinates, enhancing the ability to quantify and com-
prehend the intricate trajectories involved in the motion
of grinding media within the ball mill. These calculations
contribute to a more comprehensive understanding of the
spatial dynamics, aiding in precise assessments of media
behaviour during cascading and catracting motions in the
milling process.
*sini x
R d
2
2
cordinate
i
n
j
r
0 =
+
=
/
e o (3)
*%Y H Vcr
2
2R
100 cordinate
2 =-b b l l (4)
*sin x1
d
2
2R
cordinate
i
n
j
i
0 i =
+
=
/
e o (5)
y y a x b6x x x
0 0 0 0 -=---6 6x 6x @@-@@(6)
Y A Bx Cx 1cordinate 2 =++(7)
By simplifying the equation 6, a restructured quadratic
form of equation is generalized as mentioned in equation 7.
where,
,A ax bx B bx C 2
0 0 0
2
0 =--=+=-b 7y 6a A, @
and
,
*
*
tan
sec
a b V02
g
2 r
r
2
i
i
==
During the period of parabolic movement, the co-
ordinate of motion lies in two-dimensional direction viz
X- co-ordinate and Y- co-ordinate, the said grinding media
have uniform motion in X- co-ordinate, and does free fall
motion in Y- co-ordinate, which can be demonstrated by the
changes of velocities in X and Y directions. Subsequently,
a comprehensive model is devised to estimate media wear
induced by the cascading motion within the mill. The sim-
ulation incorporates a rigorous analysis of the total energy
balance at specific positions, namely the shoulder and toe,
leveraging kinetic and potential energy principles which
governing equations are given below.
Wear estimation method while Catracting Motion:
To estimate the mass loss of media resulting from catract-
ing motion, the energy conservation method is employed.
This involves incorporating the difference in total available
energy using a governing equation. Initially, the total energy
at the shoulder position is determined through the gov-
erning equation, considering potential and kinetic energy
stored in the media after time t. This accumulated energy
propels the media to strike the toe position. Subsequently,
a similar equation, encompassing kinetic and potential
energy, is utilized to estimate the net available energy at the
toe position. The representation of this process is encapsu-
lated in the following equation (equation 8–10 represent
energy stored at shoulder position and equation 11–13 rep-
resent energy available at toe position for respective media),
elucidating the energy dynamics governing media behav-
iour during catracting motion.
Total energy at Shoulder position:
E PE KEst
N
N nv
st
N
N nv
st
N
N nv
1 1 1
=+
=
=
=
=
=
=///(8)
PE g
x
3
4
N
N n
st
N
N n
b j
cordinate
cordinate 1 1
3
2
2
v v t r_d =
+Y
=
=
=
=//i (9)
KE v v
k
6
4
6
4
N
N n
st b j m s
b j
1
3 2 2
v t r_d
t r_d ~
=
+
=
=/i3
i
(10)
Total energy at Toe position:
E PE KE
,,,
N
N n
T t t
N
N n
T t t
N
N n
T t t
1 1 1
v v v =+
=
=
+
=
=
+
=
=
+
///(11)
PE g
x1
3
4
,
N
N n
T t t
N
N n
b j
cordinate
1 1
3
2
v v t r_d =+Y1cordinate2
=
=
+
=
=//i (12)
KE v v
k exh2
6
4
6
4
,
N
N n
T t t
N
N n
b j x z z
b j
1 1
2 2
v v t r_d ae
t r_d ~
=
+-
=
=
+
=
=//
^1 i3
i3
(13)
By calculating the ensuing loss in energy during catract-
ing motion, this study establishes a correlation between
energy reduction and a decrease in the net moment of
inertia experienced by individual media. Through iterative
analysis, the simulation predicts the cumulative mass loss
of media, reflecting the impact of wear during successive
iterations of motion and impact within the ball mill envi-
ronment which governing equation representing below as
mentioned in equation 14.
Loss in energy while travelling of media from shoulder
to toe position can be determine as follows:
E E E
,
N
N n
sT, t
N
N n
st
N
N n
T t t
1 1 1
v v v =-
=
=
=
=
=
=
+
///(14)
estimation of the length of travel during the cascading and
catracting phases. The utilization of the following equations
3 to 7 proves instrumental in determining these directional
coordinates, enhancing the ability to quantify and com-
prehend the intricate trajectories involved in the motion
of grinding media within the ball mill. These calculations
contribute to a more comprehensive understanding of the
spatial dynamics, aiding in precise assessments of media
behaviour during cascading and catracting motions in the
milling process.
*sini x
R d
2
2
cordinate
i
n
j
r
0 =
+
=
/
e o (3)
*%Y H Vcr
2
2R
100 cordinate
2 =-b b l l (4)
*sin x1
d
2
2R
cordinate
i
n
j
i
0 i =
+
=
/
e o (5)
y y a x b6x x x
0 0 0 0 -=---6 6x 6x @@-@@(6)
Y A Bx Cx 1cordinate 2 =++(7)
By simplifying the equation 6, a restructured quadratic
form of equation is generalized as mentioned in equation 7.
where,
,A ax bx B bx C 2
0 0 0
2
0 =--=+=-b 7y 6a A, @
and
,
*
*
tan
sec
a b V02
g
2 r
r
2
i
i
==
During the period of parabolic movement, the co-
ordinate of motion lies in two-dimensional direction viz
X- co-ordinate and Y- co-ordinate, the said grinding media
have uniform motion in X- co-ordinate, and does free fall
motion in Y- co-ordinate, which can be demonstrated by the
changes of velocities in X and Y directions. Subsequently,
a comprehensive model is devised to estimate media wear
induced by the cascading motion within the mill. The sim-
ulation incorporates a rigorous analysis of the total energy
balance at specific positions, namely the shoulder and toe,
leveraging kinetic and potential energy principles which
governing equations are given below.
Wear estimation method while Catracting Motion:
To estimate the mass loss of media resulting from catract-
ing motion, the energy conservation method is employed.
This involves incorporating the difference in total available
energy using a governing equation. Initially, the total energy
at the shoulder position is determined through the gov-
erning equation, considering potential and kinetic energy
stored in the media after time t. This accumulated energy
propels the media to strike the toe position. Subsequently,
a similar equation, encompassing kinetic and potential
energy, is utilized to estimate the net available energy at the
toe position. The representation of this process is encapsu-
lated in the following equation (equation 8–10 represent
energy stored at shoulder position and equation 11–13 rep-
resent energy available at toe position for respective media),
elucidating the energy dynamics governing media behav-
iour during catracting motion.
Total energy at Shoulder position:
E PE KEst
N
N nv
st
N
N nv
st
N
N nv
1 1 1
=+
=
=
=
=
=
=///(8)
PE g
x
3
4
N
N n
st
N
N n
b j
cordinate
cordinate 1 1
3
2
2
v v t r_d =
+Y
=
=
=
=//i (9)
KE v v
k
6
4
6
4
N
N n
st b j m s
b j
1
3 2 2
v t r_d
t r_d ~
=
+
=
=/i3
i
(10)
Total energy at Toe position:
E PE KE
,,,
N
N n
T t t
N
N n
T t t
N
N n
T t t
1 1 1
v v v =+
=
=
+
=
=
+
=
=
+
///(11)
PE g
x1
3
4
,
N
N n
T t t
N
N n
b j
cordinate
1 1
3
2
v v t r_d =+Y1cordinate2
=
=
+
=
=//i (12)
KE v v
k exh2
6
4
6
4
,
N
N n
T t t
N
N n
b j x z z
b j
1 1
2 2
v v t r_d ae
t r_d ~
=
+-
=
=
+
=
=//
^1 i3
i3
(13)
By calculating the ensuing loss in energy during catract-
ing motion, this study establishes a correlation between
energy reduction and a decrease in the net moment of
inertia experienced by individual media. Through iterative
analysis, the simulation predicts the cumulative mass loss
of media, reflecting the impact of wear during successive
iterations of motion and impact within the ball mill envi-
ronment which governing equation representing below as
mentioned in equation 14.
Loss in energy while travelling of media from shoulder
to toe position can be determine as follows:
E E E
,
N
N n
sT, t
N
N n
st
N
N n
T t t
1 1 1
v v v =-
=
=
=
=
=
=
+
///(14)