844 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
E
E
,
N
N n
sT, t t
N
N n
T t t
1
1
v
v
d ==
=
+
=
=
+
/
/(15)
where, E
sT, t
is loss in total energy occurred while bulk of
media impacted on toe position, whereas ‘δ’ is the colli-
sion factor. As, second moment of inertia (k) is function of
diameter, for each and every collision the radius will change
and due to this net energy will change also, the change in
radius for each event can be determined as follows:
k f_d
j
3
5 =i
*k f_d f_d
event j j
3
5
3
5 d =-i i
As the change in radius of gyration for respective media
occurs in in every event which directly relates to loss in
energy too. Hence, as per law of conservation of energy,
these losses will be encountered by decreases in diameter of
ball will be duto collision wear (as density of will never be
changed because of intrinsic metallurgical property), colli-
sion wear can be termed in SI unit as meter per hour and
can be formulate as follows:
*dt
n
new j j t
t t
1
3
5
3
5
3
5 d =-
+82^d 8 a_d _d h i i k
Total loss in mass m
c while catracting motion due to abra-
sion wear can be calculated as follows:
m
d
2 c
new
c
=-k
j ^h
Wear estimation method while Cascading Motion: In
furtherance, it is also important to calculate wear due to
cascading motion. To evaluate the wear kinetics while cas-
cading motion a convenient method of first order of wear
kinetics is used, where the governing equation of first order
wear kinetics is represented below.
f rh 4k 2 trr =+!^(19)
dt
d r 3
4
4rr
3
2
r tk
t -=!
a
^^Kr h h (20)
dr Kr dt
r
rnew
t
tn
0 0
=-!8 8 ^h (21)
Here wear rate f rh, ^can be represented by power function
of type r 2 +!,where !is kinetic order of wear law, t is
density of ball material, K is rate constant and r is ball/
media radius. The equation 21 represents linear wear rate of
the media, where r
0 represents initial radius of media and
r
new new radius of media after wear from time interval t
0 to t
n respectively. This said radius is function of mass and
density of media, as density of media is always constant
hence this radius has direct relationship with mass changes
due to abrasion m
a
for each iteration as represent in equa-
tion 22.
ma r
a new
=-k (22)
Rearranging and simplifying the equation 21 and 22, we
will get following differential equation:
dt f m, t a 2m =^h
Let initial condition are:
m t m
n 0
=^h
The computational formula of Runga-Kutta (RK) 4th
order method for solving the above differential equation is
following:
m m k k k 6
1 2 2k
n n 1 1 2 3 4 =-+++
+^h
,k hf m t
n n 1 =^h
,k hf mn h tn
k
2 2 2
1 =++a k
,k hf m h t
k
2 2 n n 3
2 =++a k
,k hf m h t k
n n 4 3 =++^h
In the context of grinding media wear rate estimation, the
first principle method entails a meticulous examination of
the weight loss incurred by the media during the dynamic
cascading and catracting motions within the confines of the
ball mill. This approach involves the summation of media
losses occurring in both cascading and catracting phases,
affording a real-time quantification of media wear through-
out the operational time span of the mill.
The total loss of media due to wear in terms of weight
m
t is represented in equation 23.
m m m
t a c
=+o (23)
The intricacies of this methodology are essential to unravel-
ling the dynamics of grinding media wear in practical mill
settings, providing a comprehensive understanding of the
wear rates under varying operational conditions.
(Note: Please see the Annexure for mathematical
notations)
RESULTS AND CONCLUSION
Results
Following the formulation of the mathematical method,
predictions for both catracting and cascading mass losses
of media due to wearing were derived based on all real time
input variables of ball mill operation. These prognostications
E
E
,
N
N n
sT, t t
N
N n
T t t
1
1
v
v
d ==
=
+
=
=
+
/
/(15)
where, E
sT, t
is loss in total energy occurred while bulk of
media impacted on toe position, whereas ‘δ’ is the colli-
sion factor. As, second moment of inertia (k) is function of
diameter, for each and every collision the radius will change
and due to this net energy will change also, the change in
radius for each event can be determined as follows:
k f_d
j
3
5 =i
*k f_d f_d
event j j
3
5
3
5 d =-i i
As the change in radius of gyration for respective media
occurs in in every event which directly relates to loss in
energy too. Hence, as per law of conservation of energy,
these losses will be encountered by decreases in diameter of
ball will be duto collision wear (as density of will never be
changed because of intrinsic metallurgical property), colli-
sion wear can be termed in SI unit as meter per hour and
can be formulate as follows:
*dt
n
new j j t
t t
1
3
5
3
5
3
5 d =-
+82^d 8 a_d _d h i i k
Total loss in mass m
c while catracting motion due to abra-
sion wear can be calculated as follows:
m
d
2 c
new
c
=-k
j ^h
Wear estimation method while Cascading Motion: In
furtherance, it is also important to calculate wear due to
cascading motion. To evaluate the wear kinetics while cas-
cading motion a convenient method of first order of wear
kinetics is used, where the governing equation of first order
wear kinetics is represented below.
f rh 4k 2 trr =+!^(19)
dt
d r 3
4
4rr
3
2
r tk
t -=!
a
^^Kr h h (20)
dr Kr dt
r
rnew
t
tn
0 0
=-!8 8 ^h (21)
Here wear rate f rh, ^can be represented by power function
of type r 2 +!,where !is kinetic order of wear law, t is
density of ball material, K is rate constant and r is ball/
media radius. The equation 21 represents linear wear rate of
the media, where r
0 represents initial radius of media and
r
new new radius of media after wear from time interval t
0 to t
n respectively. This said radius is function of mass and
density of media, as density of media is always constant
hence this radius has direct relationship with mass changes
due to abrasion m
a
for each iteration as represent in equa-
tion 22.
ma r
a new
=-k (22)
Rearranging and simplifying the equation 21 and 22, we
will get following differential equation:
dt f m, t a 2m =^h
Let initial condition are:
m t m
n 0
=^h
The computational formula of Runga-Kutta (RK) 4th
order method for solving the above differential equation is
following:
m m k k k 6
1 2 2k
n n 1 1 2 3 4 =-+++
+^h
,k hf m t
n n 1 =^h
,k hf mn h tn
k
2 2 2
1 =++a k
,k hf m h t
k
2 2 n n 3
2 =++a k
,k hf m h t k
n n 4 3 =++^h
In the context of grinding media wear rate estimation, the
first principle method entails a meticulous examination of
the weight loss incurred by the media during the dynamic
cascading and catracting motions within the confines of the
ball mill. This approach involves the summation of media
losses occurring in both cascading and catracting phases,
affording a real-time quantification of media wear through-
out the operational time span of the mill.
The total loss of media due to wear in terms of weight
m
t is represented in equation 23.
m m m
t a c
=+o (23)
The intricacies of this methodology are essential to unravel-
ling the dynamics of grinding media wear in practical mill
settings, providing a comprehensive understanding of the
wear rates under varying operational conditions.
(Note: Please see the Annexure for mathematical
notations)
RESULTS AND CONCLUSION
Results
Following the formulation of the mathematical method,
predictions for both catracting and cascading mass losses
of media due to wearing were derived based on all real time
input variables of ball mill operation. These prognostications