3696 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
N(t) can be calculated by
N P tK 8^Pe =--1 ^th hf^thdt (19)
Given an initial condition, s t s
n
T
0 1 0 0 g,s =^^t ^t h h, h 6 @
substituting s t
0 ^h into eq. (11):
s t Pe P N N t0K
0 0 =+-1 ^^t h h@ 6 (20)
Rearranging equation 20, N can be obtained:
N P N t
0
0 =-K --1 ^Pe ^t0h hs^t h (21)
Substituting equation 21 into equation 16,
s Pe P P N N t 1
0 0
0 =-+K K --t -1 ^th ^Pe ^t ^th- hs^t h hA 7 #(22)
s Pe P N^t Pe P s t t 1
0 0
0 =-+K K --1 ^th ^N^t ^t ^t- h hh h h (23)
In order to avoid exceeding the computation power because
of the rapid growth of exponential functions with time, the
following method was further developed. That is, the time
zero point is not fixed and it moves forward progressively.
For every time interval, let t 0
n 1 =
-,t t
n n 1 x x =+=
-,
equation 23 can be rewritten as:
s Pe P N^0 Pe P s^0h 1 0hK =-+x K --1 ^x ^N ^xh ^x- h hh (24)
Since s s
n
=^t ^xh, h s s^t
n 1 =
-^0h h,
s Pe P N Pe P s
n n
1
1 =-+x xK K --1
-^t ^N^xh ^0hh ^t h h (25)
From N^ 0 0 =h ,i.e., if the mill is empty at the initial time,
s^ 0 0 =h ,equation 25 can be rewritten as:
s Pe P N^xh Pe P s^t
n n
1
1 =+x xK K --1
-^t h h (26)
The algorithm based on equation 26 is an analytical solu-
tions for the dynamic model for tumbling mills and will be
tested in this research.
RESULTS AND DISCUSSIONS
A Case Study of Comparison between the Analytical
Method and the Numerical Method
As mentioned previously, the mill products and contents,
product size distribution can be solved numerically using
the finite difference form of equation 2:
t
S S
f p AR R S ,,
,,,
j t j t
j t j t j t
1
1 1 1
-
=-+-+
+++^h (27)
Figure 5 shows the comparison between analytical solu-
tion and numerical solution. The feed rate has step changes
from 150 t/h to 125 t/h and then to 195 t/h.
It is apparent that the analytical method is more accu-
rate since it has no other input computational parameters
such as time step. With the numerical solution, if time step
increases from 0.01 hour to 0.1 hour and to 1 hour, the
numerical solution curve gradually moves away from the
analytical solution, indicating a dramatically increasing
error as shown in Figure 5c. Yet the choice of time step
length is arbitrary and thus the error caused by the tradi-
tional numerical method is inevitable to some extent. In
addition, iteration to converge the solution will always be
involved in numerical techniques, which lead to longer
computation time.
Second Case Study for the Analytical Solution
A simulation of a SAG mill has been used to test the ana-
lytical method in this research. The internal length of the
SAG mill is 5.3m and the internal diameter is 6.3m. The
grate size Xg is 22 mm, and the fine size Xm is 2.5 mm. The
SAG mill feed is zero initially and a constant feed of 453t/h
is added after the time 20 minute. After certain curve fit-
ting steps, the size-specific energy distribution is shown in
Figure 6a the discharge function is shown in Figure 6b the
transport function for each segment is shown in Figure 6c.
The feeds, products and the contents of each segment vary-
ing with time are shown in Figure 7a. The contents from
the first segment to the last segment along the length are
reducing, providing the pressure drop driving the slurry
flow (Figure 7a). In Figure 7b, it can be seen that as the ore
particles travel along the length of the mill they become
progressively finer. In Figure 7b, it also shows that product
size distribution (PSD) simulation result agrees well with
the plant data except with the critical particle size range.
CONCLUSIONS
Dynamic simulations for tumbling mills based on an
analytical method have been developed and tested in this
research. Compared to numerical methods, the analytical
method is more accurate since it does not require the input
of a time step therefore, it overcomes inevitable errors of
the numerical methods coming from the choosing time
intervals. The analytical method does not require iterations,
which can be problematic of the numerical methods. Two
SAG mill operation cases with given different sub-models
was used to demonstrate the applicability of the analytical
solution method. It is proven that the analytical method
is robust, feasible and reliable, giving stable and straight-
forward results. The case studies using the analytical solu-
tion method showed that, as the classification function
gets finer, the content in the mill increases because a wider
size range of material is kept inside the SAG mill and the
dynamic response time is also longer. The transport func-
tions inside the mill play important role in the dynamic
response. In addition, the dynamic response time is linked
N(t) can be calculated by
N P tK 8^Pe =--1 ^th hf^thdt (19)
Given an initial condition, s t s
n
T
0 1 0 0 g,s =^^t ^t h h, h 6 @
substituting s t
0 ^h into eq. (11):
s t Pe P N N t0K
0 0 =+-1 ^^t h h@ 6 (20)
Rearranging equation 20, N can be obtained:
N P N t
0
0 =-K --1 ^Pe ^t0h hs^t h (21)
Substituting equation 21 into equation 16,
s Pe P P N N t 1
0 0
0 =-+K K --t -1 ^th ^Pe ^t ^th- hs^t h hA 7 #(22)
s Pe P N^t Pe P s t t 1
0 0
0 =-+K K --1 ^th ^N^t ^t ^t- h hh h h (23)
In order to avoid exceeding the computation power because
of the rapid growth of exponential functions with time, the
following method was further developed. That is, the time
zero point is not fixed and it moves forward progressively.
For every time interval, let t 0
n 1 =
-,t t
n n 1 x x =+=
-,
equation 23 can be rewritten as:
s Pe P N^0 Pe P s^0h 1 0hK =-+x K --1 ^x ^N ^xh ^x- h hh (24)
Since s s
n
=^t ^xh, h s s^t
n 1 =
-^0h h,
s Pe P N Pe P s
n n
1
1 =-+x xK K --1
-^t ^N^xh ^0hh ^t h h (25)
From N^ 0 0 =h ,i.e., if the mill is empty at the initial time,
s^ 0 0 =h ,equation 25 can be rewritten as:
s Pe P N^xh Pe P s^t
n n
1
1 =+x xK K --1
-^t h h (26)
The algorithm based on equation 26 is an analytical solu-
tions for the dynamic model for tumbling mills and will be
tested in this research.
RESULTS AND DISCUSSIONS
A Case Study of Comparison between the Analytical
Method and the Numerical Method
As mentioned previously, the mill products and contents,
product size distribution can be solved numerically using
the finite difference form of equation 2:
t
S S
f p AR R S ,,
,,,
j t j t
j t j t j t
1
1 1 1
-
=-+-+
+++^h (27)
Figure 5 shows the comparison between analytical solu-
tion and numerical solution. The feed rate has step changes
from 150 t/h to 125 t/h and then to 195 t/h.
It is apparent that the analytical method is more accu-
rate since it has no other input computational parameters
such as time step. With the numerical solution, if time step
increases from 0.01 hour to 0.1 hour and to 1 hour, the
numerical solution curve gradually moves away from the
analytical solution, indicating a dramatically increasing
error as shown in Figure 5c. Yet the choice of time step
length is arbitrary and thus the error caused by the tradi-
tional numerical method is inevitable to some extent. In
addition, iteration to converge the solution will always be
involved in numerical techniques, which lead to longer
computation time.
Second Case Study for the Analytical Solution
A simulation of a SAG mill has been used to test the ana-
lytical method in this research. The internal length of the
SAG mill is 5.3m and the internal diameter is 6.3m. The
grate size Xg is 22 mm, and the fine size Xm is 2.5 mm. The
SAG mill feed is zero initially and a constant feed of 453t/h
is added after the time 20 minute. After certain curve fit-
ting steps, the size-specific energy distribution is shown in
Figure 6a the discharge function is shown in Figure 6b the
transport function for each segment is shown in Figure 6c.
The feeds, products and the contents of each segment vary-
ing with time are shown in Figure 7a. The contents from
the first segment to the last segment along the length are
reducing, providing the pressure drop driving the slurry
flow (Figure 7a). In Figure 7b, it can be seen that as the ore
particles travel along the length of the mill they become
progressively finer. In Figure 7b, it also shows that product
size distribution (PSD) simulation result agrees well with
the plant data except with the critical particle size range.
CONCLUSIONS
Dynamic simulations for tumbling mills based on an
analytical method have been developed and tested in this
research. Compared to numerical methods, the analytical
method is more accurate since it does not require the input
of a time step therefore, it overcomes inevitable errors of
the numerical methods coming from the choosing time
intervals. The analytical method does not require iterations,
which can be problematic of the numerical methods. Two
SAG mill operation cases with given different sub-models
was used to demonstrate the applicability of the analytical
solution method. It is proven that the analytical method
is robust, feasible and reliable, giving stable and straight-
forward results. The case studies using the analytical solu-
tion method showed that, as the classification function
gets finer, the content in the mill increases because a wider
size range of material is kept inside the SAG mill and the
dynamic response time is also longer. The transport func-
tions inside the mill play important role in the dynamic
response. In addition, the dynamic response time is linked