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A mechanistic state equation model for semiautogenous mills
ELSEVIER
lnt. J. Miner. Process. 44-45
(1996) 349-360
A mechanistic state equation model for semiautogenous mills R.. Amestica a, G.D. Gonzalez b, J. Menacho a, J. Barria a a Centro de InuestigucGn Minera y Metalrirgica, Santiago, Chile b Department of Electrical Engineering, University of Chile, Santiago, Chile
Abstract A dynamic model for a semiautogenous grinding mill is developed in the standard dynamic system form of state and state-output equations and the resulting model is analyzed with this perspective. The model is based on results obtained under stationary conditions using a pilot SAG mill and communicated in an earlier paper. From the point of view of the state equations, these results constitute sub-models representing grinding, classification, transport, and power draw, which are functions of the mill state and input variables. These sub-models may be easily changed in this state equation structure to accommodate new developments. Thanks to the form in which the model has been derived, it may be incorporated in a systematic way to grinding circuits. This model serves for the study by simulation of control strategies, either manual or automatic, as well as for testing estimators or soft-sensors for unmeasured variables. Results in accordance with real cases were obtained using the model for simulating some typical operating procedures.
1. Introduction In general it is very difficult to design and test control strategies and variable or parameter estimation schemes for SAG mills using direct on-line or off-line data taken from a SAG mill. The problem is that SAG mills are very complex systems involving important nonlinearities and unmeasured disturbances, some of which have fast rates of change as compared with the plant response time to the commands or manipulated variables. A previous step, then, using simulation is very convenient, and perhaps sometimes indispensable, in order find out which are the most promising designs, and only test these pre-selected designs in the actual plant. In designing models for SAG mills, or for any plant in general, the purpose for which the model is being developed is most important. In particular, if the purpose of the model is to design control systems or systems for the estimating mill variables, a dynamic model is needed that is qualitatively 0301-75 16/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 03’31-7516(95)00045-3
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J. Miner. Process. 44-45 (1996) 349-360
good, i.e., the relations between input, output and internal variables are well represented by the model regarding rates of change and directions of these changes. As for the magnitudes the demands may be relaxed, taking into account that the final adjustments by force require working with the actual plant, although now with preselected control strategies or estimators and soft-sensors, tested by simulation using the model. However, the closer the is model to being good also from a quantitative point of view, the easier shall be he transition from the results obtained by simulation, to the actual plant. The natural setting for a dynamic model is a system of state equations and state-output equations, because it is the intrinsic expression of the dynamic physical phenomena involved. Previous SAG mill models have been derived - and used for studying control strategies - which at most are acceptable from a qualitative point of view but have not been matched to any real SAG mill in a qualitative way (Gonzalez et al., 1987). The state equation form has been implicitly used there but no analysis from this stand point has been made. The model is mainly phenomenological, and this fact entails great complexity and a large set of parameters that is practically impossible to determine from experimental data. Other models are those that may by identified using data collected by on-line measurements in a plant (Ame’stica et al., 1993b; Herbst et al., 1989). In order to be able to do this, these models are relatively very simple and have few parameters. As a consequence, they are only valid around operating points and the parameters need to be continually updated. Although these models are very useful for estimating unmeasured variables in the SAG mil, they are clearly unsuitable for studying control systems or estimators and soft-sensors. In order to develop a model useful for these purposes empirical relations involving few parameters may be determined using the data collected in the experiments with the mill, giving rise to what is defined as a mechanistic model, although some phenomenological sub models may be retained if convenient. This is the approach that has been taken here, so that the model has a the structure of sate and state-output equations in which phenomenological or empirical relations linking internal variables with inputs and states take the form of static submodels. The static submodels used here are those to be found in Amestica et al. (1993a) and a power draw submodel developed by Austin (1990).
2. The state equation model The state equation model developed here is formed by two stages: (i> a dynamic balance model for the ore and water mill contents linking the various input, output and internal variables, and (ii) sub-models for grinding, classification, mass transport, and power draw, in terms of operating variables such as ore hold-up H,,,, mill power draw Mp, pulp solids percentage inside the mill, volumetric mill filing fraction J. These operating variables are in turn functions of the states and the input variables. 2.1. Flow and mill contents variables
Let (Fig. 1) wi = ore mass in size interval i, fi = feed mass ore flow to mill in size interval i, fi* = feed mass ore flow to grinding chamber in size interval i, pi = ore
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351
Mill Feed
Fig. 1.Blockdiagramof the
SAGmill showing the phenomena considered in the mill modeling.
discharge flow rate in size interval i,p,* = ore discharge flow rate from grinding chamber and size interval i, H,,, = mass of ore contents in mill (hold-up), F,,, = total ore feed flow rate to the mill, P,,, = total ore discharge flow rate from the mill, F * = total ore feed flow to the grinding chamber, P * = total ore discharge flow from grinding chamber, FA = water feed flow rate to the mill, PA = water discharge flow rate from the mill, and WA = water hold-up. The following column n-vectors, then, may be defined representing mill ore contents and flows at all size intervals (the superscript T indicating transposition): w = ( w , ,w2 ,..., wJT, mill ore contents; f= (f,,fi ,..., f,,jT, mill ore feed flow; f * = (f,’ ,f; ,...,f,,* >T, grinding chamber ore feed flow; PL =: (p; ,p; ,..a,p,* IT, grinding chamber ore discharge flow; and p = tp,,p*,..., pJT, mill ore discharge. Clearly, p=epi,
F=&, i= I
i=
F’=&,
1
P*=&;, i=
I
and H,,,= kwi i=
I
i=
1
(1)
Let the fine ore be defined as that which passes the discharge grate. Then the fine ore contents in the mill are given by: H, = kwi
(2)
i,
2.2. Sub-models The sub-models used here have been either taken from the literature (Austin, 1990) or derived in AmCstica et al. (1993a) sometimes using results published by other authors (Forssberg and Zhai, 1988; Gupta et al., 1981; Vanderbeek et al., 1985; Moys, 1986). 2.2.1. Grinding submodel The grinding mechanism used here is the one used by Ame’stica et al. (1993a), where the grinding takes place in the grinding chamber and is represented by the lower triangullar matrix (M,/H,,,)R‘K ER, where M,, is the mill power draw (see below), K,! is the specific grinding rate of the production of fines KE = diag(KB,KF,..., K:), below size interval i, and R is a lower triangular matrix filled with ones.
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2.2.2. Classification submodel The ore flows emerging from the grinding chamber represented by vector p* are
classified considering the effects of the grate and pulp lifters in such a way that the ore discharge flows from the mill are given by pi = (1 - ci)p:, where ci = (pi* - pi>/pi is the ore rejection coefficient of size i in the mill discharge system. It has been found that ci depends on the mill pulp solids percentage, defined as cr = H,(H, + IV,), after an analysis of the data collected from pilot mill already mentioned, so that ci = ci(cr). Fig. 2 gives this relation. Let C= diag(c,,c, ,..., c,,) be a diagonal matrix containing these coefficients, then, P= (I-
C(C,))P”
(3)
2.2.3. Mass transport submodel The ore contents H,,, in the grinding chamber are obtained by a dynamic balance between the feed flow to the chamber and the discharge flow from it, i.e.,
dfL -z=*-p dt
* -
-
It was assumed that the discharge flow from the grinding chamber, which is presented to classification is a function of its ore hold-up, P* = ?I’(H,)
(4)
In AmCstica et al. (1993a) it has been found that for a certain pilot mill P * = 296, which has turned out to have the familiar form associated with the discharge of liquids
Size, ----72.0
% -
micro
7 4.0 %
m. ***=*7 0.1 %
??
Fig. 2. Effect of the mill pulp solids percentage c, on the cj’s.
R. Amestica et al./lnt. J. Miner. Process. 44-45 (1996) 349-360
353
from vessels. The ore flow discharged from the grinding chamber and presented for classification at each size interval is P* pi* = HW’ m
P*
or p* = KW
(5)
m
This is SIDbecause the ore mass fraction in size interval j in the grinding chamber is given by wj/H,,, and, if perfect mixture is assumed, this proportion is kept in the discharge flow P *. Concerning the water contents H,, it was found from experimental data that the discharge flow of water from the mill may be expressed as (AmCstica et al., 1993a)
2.2.4. Power draw submodel The power draw model given by Austin (1990) and used here is M,,=kD2%(1
H,,, + W, + W,
-AJ)
v
m
(7)
f(4)
where D and L are the diameter and length of the mill, V, is the internal mill volume, J the volumetric mill fraction filling occupied by the total hold-up, W, is the mass hold-up of steel balls, fl+) is a function of the critical speed fraction 4 (Austin, 1990), while k and A are the experimental parameters of this model, determined by AmCstica et al. (1993a). 2.3. Decelopment of the SAG mill state equation 2.3.1. D,ynamic balance for the ore The change of ore mass content in the grinding chamber, dw = wdt, between instants t and t + dt, is produced by the difference between the input masses f *dt and the discharge masses p * d t and the changes in ore masses at the different size intervals due to grind:ing (M,/H,,,>R-‘KERwdt. Then the balance is dw=f
*dt-p*dt-
$R-‘KERwdt m
or
dw dr=f*-p*-H
From F:ig. 1, fi* =fi + ci(cf)pr, or f * = f + C(c,)p*. and using (3) and afterwards (2),
4: =f dw dt=
+ C(c,)p”
-p*
W --CP’ H,
- 2Re1KERw m
5R-‘KER~ m
(8)
Then, substituting this in (5)
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dw -=dt 2.3.2. Dynamic balance for the water The change in the mass of water contents dW, = W,dt dW, = F,dt - P,dt. Then, from (6), and dividing by dt,
(9)
in the mill is determined by
2.3.3. State model for the SAG mill Finally, from (9) and (lo), the state equation for the SAG mill is given by -=-
(lla) (lib)
where 1. the state variables of the mill are the masses of ore w,(i = l,...,n) and the mass of water WA 2. the inputs are: the ore flows given by vector f and the feed water flow given by WA, and the specific grinding rate matrix K E. (It is assumed that the values of a0 and A, in (1 lb) and the power draw model parameters k, L, D, V,,, and 4 are know from design values of the mill or analysis of experimental data.) That this is indeed a state equation is due to the following reasons. Assume an initial sate is known at a certain initial time to, and that f and FA are known in a time interval [ to,tf]. Then, if (11) is really a state equation it must be that the mill states may be determined in this time interval if f, FA, and K E are known in the same interval. By (1) H, is found in terms of the state variables. The mill power draw MP appearing in (7) may also be expressed in terms of the state variables. From (7) it may be seen that besides H,,, and WA there appears the volumetric mill fraction filling J. But also J may be expressed in terms of the state variables, as may be easily derived from AmCstica et al. (1993a). Clearly, from (2), cf is also a function of the state variables. In order to emphasize these facts, the state equations may be written as, dw -=G(w,W,)w+f dt
(124
d% -=h(w)W,+F, dt
( 12b)
Then, given f, F, and KE in a time interval [t,,f,], and an initial state given by and WA(tO), the state w(t), WA(t) may be found for all instants in [to, tf 1. Therefore (1 l>, or in its form (12), is truly a state equation. From this global point of w(t,>
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355
view, then, the model replicates what happens in reality, since if the actual mill begins to operate with a certain load of water, and a given ore, what happens thereafter depends on the way the ore and water feeds are manipulated. Since the state becomes known for every t in [tO,trl, also Mr, cf, J, p, P, and H,,,, among olher variables are also known for all t in this interval. In system theory these variables are known as the output variables, which in general are a function of the states and the input variables. The set of equations which determine these variables in terms of states and inputs are called state output equations. Cluss$kation of the state model inputs. For the purpose of studying control strategies using this sate model, the input variables may be classified into: (i) commands or manipulated variables, (ii) measured disturbances, and (iii) unmeasured disturbances. Of course this classification depends on the instrumentation and actuators available in a particular plant. For example, in a usual case it may be that: 1. F,,, and FA, i.e., the total ore and water feed flows are manipulated variables 2. f, i.e. the vector containing the ore feed flows at each size interval is an unmeasured disturbance, 3. KE, the matrix characterizing the ore grindability, is another unmeasured disturbance.
3. Results Two operating procedures are given here to show the behaviour of the SAG mill state model: (i) a change in the size distribution of the feed ore, and (ii) a change in the solids percentage in the feed ore flow. 3.1. Changes in feed size distribution
In order to perform this test, the 27 size intervals used in the model have been grouped in three size ranges, where the fine size range takes into account the fact that the grate opening is 12.5 mm: Fine: - 12.5 mm - Intermediate: + 12.5 to - 102 mm * Coarse: + 102 mm, fu) Decrease in intermediate size. Keeping the total feed ore constant the proportion of intemiediate size range was decreased and compensated with an equal increase of the coarse range (the fine ore flow remains unchanged). In this manner an increase in the grinding efficiency of the mill may be anticipated. That such is indeed the case may be observed in Fig. 3, where the ore discharge flow increases causing a the reduction of the mill ore contents shown in Fig. 4. The feed ore flow is then increased (Fig. 3) to bring up the mill load, as a result of which the mill throughput is increased. The water is seen to follow similar variations as the fine (discharged) ore. (b) Increase in intermediate size. Now the intermediate size feed ore is increased and the coarse ore is reduced by the same amount. Since the grinding media is thus reduced, one may expect that the efficiency of grinding decreases. That this is indeed the case may be seen in Fig. 5, where the mill filling J increases until the mill operating point ??
R. Amesiica et ul./lnt.
356
J. Miner. Process.
44-45 (1996) 349-360 -1.5
3.7-1.4 E (D I_ -1.3 _L z= I_ - 1.2 ,” _ S
3.3-
3.1r 0
6
4
2
8
time, r
-
Ore feed
-
--
1.1
10
hr -
Ore
Waterfeed
-----
Water
discharge
discharge
Fig. 3. Ore.andwater flow rates, in the feed and discharge of the mill, after a reduction of the intermediate ore feed flow rate and an equal increase of the course feed ore flow.
+ 0.35 zG 0.30 2 0.25 0.20 f 0.15 ‘r 0.10 c” 0.0s 0.00 I'0
0
2
4
6 time,
8
10
hr
Fig. 4. Hold-up of ore and water in the mill corresponding to the case of Fig. 3.
9.9 5
-0.12
9.0-
-0.38
. $ 9.70 g 9.6-
4
E
.*--
__
cc . . /
9.5-
-
-__
-0.34
- -__
--5_
-__
/ ,
9.4 f-/ 0
I 2
6
4 time, -Pwardraw
8
10
hr ---
HillfilUg
Fig, 5. Filling and power draw when coarse ore feed is reduced.
= O. ‘E ._s v 2
‘L .k ‘y
-0.30 2E ‘FZ S Lv 0.26
R. Amestica et al. / ht. J. Miner. Process. 44-45 (1996) 349-360 37
357 -1.50
T
/-------
e * --1.40 .
-______..--_---___
i --1.30
1
6 i:
32 -.
" --I20 s
3.1 7
1.10
$I’ 0
2
4
6
a
10
*#h ----oredschmp
I-ChfWd
--vvaerfeEd---werbs~
Fig, 6. Feed ore and water flow rates of the mill after a reduction of the coarse ore feed flow rate corresponding to Fig. 5..
goes the: unstable zone, i.e., the power decreases while the filling increases. The feed ore is then reduced and the recovery from this condition may be observed in the last part of the graph of Fig. 5. The corresponding ore and water feed and discharge flows are given in Fig. 6. As a result the mill throughput is reduced. 3.2. Chmge in feed solids percent In this test the water feed FA is rapidly increased in order to brimg the mill out of the unstable condition into which it is beginning to enter. In effect, in Fig. 8 the power is decreasing while the mill filling is increasing. The feed water is then increased so that the feed solids percentage drops from 0.78 to 0.74 (Fig. 7). As a result, the power undergoes a rapid increase matched by a fast decrease if the mill filling. Thereafter both
c 1.6
3.3 T
_---f t
e
*L 38 1 : 3.4
t
G 2
‘. .
_
I- 5. 1.5
----_____
I.4 j
1.3 E 6 12 d u
32
1.1 3 0 I
3$
0
2
1
6
4
a
10
tilm,hr
I-cxdeed-
--orubsdrarge
-wterfead
- - - - ~ifvdterdschagel
Fig. 7. Feed water flow increase to bring the mill out of an unstable operation.
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et al./
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Process.
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(1996)
349-360
-0.33 _---
- -‘\
.
‘.
-0.29
*. -N
0
2
--w_
--__
--__
6
L time,
- _ ___
8
0.25
10
hr
Fig. 8. Power draw and mill filling corresponding to the case of Fig. 7.
16
027 --_ _ + -
-
’
.-
---__
-_
0264
--_h__-_l.
0
2
6
4
8
11
10
tim,hr
I-FillECf0fractiOn ----(XedschargereteI Fig. 9. Ore discharge rate and fraction of fine ore inside the mill after the step increase of the water feed flow rate of Fig. 7.
rO.10
0.65,
I
I
/---
0 ,. 0.35z _,__----A I0.25 ! 0
: :.
----_
- -------__-___
’ 0.05 2
6
4 time.
0
10
hr
Fig. 10. Changes of the total ore and water hold-ups after reduction of the feed solids percentage.
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J. Miner. Process. 44-45 (1996) 349-360
359
power and mill filling decrease, showing that the stable region has been entered (Fig. 8). Fig. 9 shlows that after the feed water increase the ore discharge rate increases very fast due to the flushing effect of the water. Because of this, at the same time the fine ore size fraction in the mill decreases rapidly. After the abrupt changes caused by the feed water flow increase a new condition begins to develop in the mill as the grinding due to the coarse sizes produces a recovery of the fine size fraction (Fig. 9) inside the mill. Fig. 10 shows the changes and final condition of the total ore an water hold-up.
4. Conclusions By the use of dynamic balance of ore and water, together with submodels representing power draw, grinding, mass transport, and classification a state equation for a SAG mill has been developed in which the states are the mass of water and the ore masses, at different size intervals, contained in the mill. The submodels may be easily changed within tlhis model structure to suit new developments, whether in the phenomenological or empirical aspects. The model has shown a good behaviour when subject to changes in feed ore size distribution and feed percentage of solids, and the overload condition has appeared when it should be expected. Besides possible improvements in the submodels used, this state model also admits improvements in the state equation structure. Indeed, in order to treat varying grindabilities a variable combination of soft and hard ores may be represented, in which case the number of state variables representing the ore are doubled.
Acknowledgements Research leading to this paper has been funded by: PNUD project CHI 88/011, Technor!ogical development of comminution in the Chilean industry; Mining and Metallurgical Research Center (CIMM) and FONDECYT (Chile) project 1930885, Softsensors. Experimentation supporting this research was done at CODELCO-Chile, Chuquicamata Division.
References Ankstica, R., Gonz&lez, G.D., Bank, J., Magne, L., Menacho, J. and Castro, O., 1993a. A SAG mill circuit dynamic simulator based on a simplified mechanistic model. In: Proc. XVIII International Mineral Processing Congress, Australasian Inst. of Min. and Met., Sydney, Vol. 1, pp. 117-130. Amkstica, R., Gonzllez, G.D., Menacho, J. and Barria, J., 1993b. On-line estimation of fine and coarse ore, water, grinding rate, and discharge rates in SAG mill. In: Proc. XVIII International Mineral Processing Congress, The Australasian Inst. of Min. and Met., Sydney, Vol. 1, pp. 109- 115. Austin, L.G., 1990. A mill power equation for SAG mills. Min. Met. Process., (Feb.): 57-62. Forssbeq;, E. and Zhai, H., 1988. Mass transport behaviour in the full scale mills fractionally determined by the nN>velRTD methods. In: XVI International Mineral Processing Congress, Elsevier, pp. 1733- 1745.
360
R. Amestica et ul./lnt.
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(1996) 349-360
Gonzalez, G.D., Bamhona, C., Castelli, L,. Cipriano, A., Hemandez, L. and Yacher, L., 1987. Control strategy testing for a SAG mill using a qualitative SAG mill model designed for that purpose. In: Proc. Copper 87, Met. Sot. C.I.M., Chilean Inst. of Mining Eng., U. of Chile, Viiia de1 Mar, Vol. 2, pp. 409-421. Gupta, V.K., Hodouin, D. and Everell, M.D., 1981. The influence of feed rate and pulp composition on holdup weight and mean residence time of solids in grate discharge ball mill grinding. Int. J. Miner. Process., 8: 345-358. Herbst, J.A., Pate, W.T. and Oblad, A.E., 1989. Experiences in the use of model based expert control systems in autogenous and semiautogenous grinding circuits. In: SAG Milling Conference, Vancouver, pp. 669-686. Moys, M.H., 1986. The effect of grate design on the behaviour of grate-discharge grinding mills. Int. J. Miner. Process., 18: 85-105. Vanderbeek, J.L., Herbst, J.A., Hales, L.B. and Pennington, R.I., 1985. An investigation on the effects of mill volumetric load level, percent solids, and speed upon semiautogenous mill operation. Preprint.
lnt. J. Miner. Process. 44-45
(1996) 349-360
A mechanistic state equation model for semiautogenous mills R.. Amestica a, G.D. Gonzalez b, J. Menacho a, J. Barria a a Centro de InuestigucGn Minera y Metalrirgica, Santiago, Chile b Department of Electrical Engineering, University of Chile, Santiago, Chile
Abstract A dynamic model for a semiautogenous grinding mill is developed in the standard dynamic system form of state and state-output equations and the resulting model is analyzed with this perspective. The model is based on results obtained under stationary conditions using a pilot SAG mill and communicated in an earlier paper. From the point of view of the state equations, these results constitute sub-models representing grinding, classification, transport, and power draw, which are functions of the mill state and input variables. These sub-models may be easily changed in this state equation structure to accommodate new developments. Thanks to the form in which the model has been derived, it may be incorporated in a systematic way to grinding circuits. This model serves for the study by simulation of control strategies, either manual or automatic, as well as for testing estimators or soft-sensors for unmeasured variables. Results in accordance with real cases were obtained using the model for simulating some typical operating procedures.
1. Introduction In general it is very difficult to design and test control strategies and variable or parameter estimation schemes for SAG mills using direct on-line or off-line data taken from a SAG mill. The problem is that SAG mills are very complex systems involving important nonlinearities and unmeasured disturbances, some of which have fast rates of change as compared with the plant response time to the commands or manipulated variables. A previous step, then, using simulation is very convenient, and perhaps sometimes indispensable, in order find out which are the most promising designs, and only test these pre-selected designs in the actual plant. In designing models for SAG mills, or for any plant in general, the purpose for which the model is being developed is most important. In particular, if the purpose of the model is to design control systems or systems for the estimating mill variables, a dynamic model is needed that is qualitatively 0301-75 16/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 03’31-7516(95)00045-3
350
R. Amestica et al./lnt.
J. Miner. Process. 44-45 (1996) 349-360
good, i.e., the relations between input, output and internal variables are well represented by the model regarding rates of change and directions of these changes. As for the magnitudes the demands may be relaxed, taking into account that the final adjustments by force require working with the actual plant, although now with preselected control strategies or estimators and soft-sensors, tested by simulation using the model. However, the closer the is model to being good also from a quantitative point of view, the easier shall be he transition from the results obtained by simulation, to the actual plant. The natural setting for a dynamic model is a system of state equations and state-output equations, because it is the intrinsic expression of the dynamic physical phenomena involved. Previous SAG mill models have been derived - and used for studying control strategies - which at most are acceptable from a qualitative point of view but have not been matched to any real SAG mill in a qualitative way (Gonzalez et al., 1987). The state equation form has been implicitly used there but no analysis from this stand point has been made. The model is mainly phenomenological, and this fact entails great complexity and a large set of parameters that is practically impossible to determine from experimental data. Other models are those that may by identified using data collected by on-line measurements in a plant (Ame’stica et al., 1993b; Herbst et al., 1989). In order to be able to do this, these models are relatively very simple and have few parameters. As a consequence, they are only valid around operating points and the parameters need to be continually updated. Although these models are very useful for estimating unmeasured variables in the SAG mil, they are clearly unsuitable for studying control systems or estimators and soft-sensors. In order to develop a model useful for these purposes empirical relations involving few parameters may be determined using the data collected in the experiments with the mill, giving rise to what is defined as a mechanistic model, although some phenomenological sub models may be retained if convenient. This is the approach that has been taken here, so that the model has a the structure of sate and state-output equations in which phenomenological or empirical relations linking internal variables with inputs and states take the form of static submodels. The static submodels used here are those to be found in Amestica et al. (1993a) and a power draw submodel developed by Austin (1990).
2. The state equation model The state equation model developed here is formed by two stages: (i> a dynamic balance model for the ore and water mill contents linking the various input, output and internal variables, and (ii) sub-models for grinding, classification, mass transport, and power draw, in terms of operating variables such as ore hold-up H,,,, mill power draw Mp, pulp solids percentage inside the mill, volumetric mill filing fraction J. These operating variables are in turn functions of the states and the input variables. 2.1. Flow and mill contents variables
Let (Fig. 1) wi = ore mass in size interval i, fi = feed mass ore flow to mill in size interval i, fi* = feed mass ore flow to grinding chamber in size interval i, pi = ore
R. Amestica et al./lnt.
J. Miner. Process. 44-4.5 (1996) 349-360
351
Mill Feed
Fig. 1.Blockdiagramof the
SAGmill showing the phenomena considered in the mill modeling.
discharge flow rate in size interval i,p,* = ore discharge flow rate from grinding chamber and size interval i, H,,, = mass of ore contents in mill (hold-up), F,,, = total ore feed flow rate to the mill, P,,, = total ore discharge flow rate from the mill, F * = total ore feed flow to the grinding chamber, P * = total ore discharge flow from grinding chamber, FA = water feed flow rate to the mill, PA = water discharge flow rate from the mill, and WA = water hold-up. The following column n-vectors, then, may be defined representing mill ore contents and flows at all size intervals (the superscript T indicating transposition): w = ( w , ,w2 ,..., wJT, mill ore contents; f= (f,,fi ,..., f,,jT, mill ore feed flow; f * = (f,’ ,f; ,...,f,,* >T, grinding chamber ore feed flow; PL =: (p; ,p; ,..a,p,* IT, grinding chamber ore discharge flow; and p = tp,,p*,..., pJT, mill ore discharge. Clearly, p=epi,
F=&, i= I
i=
F’=&,
1
P*=&;, i=
I
and H,,,= kwi i=
I
i=
1
(1)
Let the fine ore be defined as that which passes the discharge grate. Then the fine ore contents in the mill are given by: H, = kwi
(2)
i,
2.2. Sub-models The sub-models used here have been either taken from the literature (Austin, 1990) or derived in AmCstica et al. (1993a) sometimes using results published by other authors (Forssberg and Zhai, 1988; Gupta et al., 1981; Vanderbeek et al., 1985; Moys, 1986). 2.2.1. Grinding submodel The grinding mechanism used here is the one used by Ame’stica et al. (1993a), where the grinding takes place in the grinding chamber and is represented by the lower triangullar matrix (M,/H,,,)R‘K ER, where M,, is the mill power draw (see below), K,! is the specific grinding rate of the production of fines KE = diag(KB,KF,..., K:), below size interval i, and R is a lower triangular matrix filled with ones.
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2.2.2. Classification submodel The ore flows emerging from the grinding chamber represented by vector p* are
classified considering the effects of the grate and pulp lifters in such a way that the ore discharge flows from the mill are given by pi = (1 - ci)p:, where ci = (pi* - pi>/pi is the ore rejection coefficient of size i in the mill discharge system. It has been found that ci depends on the mill pulp solids percentage, defined as cr = H,(H, + IV,), after an analysis of the data collected from pilot mill already mentioned, so that ci = ci(cr). Fig. 2 gives this relation. Let C= diag(c,,c, ,..., c,,) be a diagonal matrix containing these coefficients, then, P= (I-
C(C,))P”
(3)
2.2.3. Mass transport submodel The ore contents H,,, in the grinding chamber are obtained by a dynamic balance between the feed flow to the chamber and the discharge flow from it, i.e.,
dfL -z=*-p dt
* -
-
It was assumed that the discharge flow from the grinding chamber, which is presented to classification is a function of its ore hold-up, P* = ?I’(H,)
(4)
In AmCstica et al. (1993a) it has been found that for a certain pilot mill P * = 296, which has turned out to have the familiar form associated with the discharge of liquids
Size, ----72.0
% -
micro
7 4.0 %
m. ***=*7 0.1 %
??
Fig. 2. Effect of the mill pulp solids percentage c, on the cj’s.
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from vessels. The ore flow discharged from the grinding chamber and presented for classification at each size interval is P* pi* = HW’ m
P*
or p* = KW
(5)
m
This is SIDbecause the ore mass fraction in size interval j in the grinding chamber is given by wj/H,,, and, if perfect mixture is assumed, this proportion is kept in the discharge flow P *. Concerning the water contents H,, it was found from experimental data that the discharge flow of water from the mill may be expressed as (AmCstica et al., 1993a)
2.2.4. Power draw submodel The power draw model given by Austin (1990) and used here is M,,=kD2%(1
H,,, + W, + W,
-AJ)
v
m
(7)
f(4)
where D and L are the diameter and length of the mill, V, is the internal mill volume, J the volumetric mill fraction filling occupied by the total hold-up, W, is the mass hold-up of steel balls, fl+) is a function of the critical speed fraction 4 (Austin, 1990), while k and A are the experimental parameters of this model, determined by AmCstica et al. (1993a). 2.3. Decelopment of the SAG mill state equation 2.3.1. D,ynamic balance for the ore The change of ore mass content in the grinding chamber, dw = wdt, between instants t and t + dt, is produced by the difference between the input masses f *dt and the discharge masses p * d t and the changes in ore masses at the different size intervals due to grind:ing (M,/H,,,>R-‘KERwdt. Then the balance is dw=f
*dt-p*dt-
$R-‘KERwdt m
or
dw dr=f*-p*-H
From F:ig. 1, fi* =fi + ci(cf)pr, or f * = f + C(c,)p*. and using (3) and afterwards (2),
4: =f dw dt=
+ C(c,)p”
-p*
W --CP’ H,
- 2Re1KERw m
5R-‘KER~ m
(8)
Then, substituting this in (5)
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dw -=dt 2.3.2. Dynamic balance for the water The change in the mass of water contents dW, = W,dt dW, = F,dt - P,dt. Then, from (6), and dividing by dt,
(9)
in the mill is determined by
2.3.3. State model for the SAG mill Finally, from (9) and (lo), the state equation for the SAG mill is given by -=-
(lla) (lib)
where 1. the state variables of the mill are the masses of ore w,(i = l,...,n) and the mass of water WA 2. the inputs are: the ore flows given by vector f and the feed water flow given by WA, and the specific grinding rate matrix K E. (It is assumed that the values of a0 and A, in (1 lb) and the power draw model parameters k, L, D, V,,, and 4 are know from design values of the mill or analysis of experimental data.) That this is indeed a state equation is due to the following reasons. Assume an initial sate is known at a certain initial time to, and that f and FA are known in a time interval [ to,tf]. Then, if (11) is really a state equation it must be that the mill states may be determined in this time interval if f, FA, and K E are known in the same interval. By (1) H, is found in terms of the state variables. The mill power draw MP appearing in (7) may also be expressed in terms of the state variables. From (7) it may be seen that besides H,,, and WA there appears the volumetric mill fraction filling J. But also J may be expressed in terms of the state variables, as may be easily derived from AmCstica et al. (1993a). Clearly, from (2), cf is also a function of the state variables. In order to emphasize these facts, the state equations may be written as, dw -=G(w,W,)w+f dt
(124
d% -=h(w)W,+F, dt
( 12b)
Then, given f, F, and KE in a time interval [t,,f,], and an initial state given by and WA(tO), the state w(t), WA(t) may be found for all instants in [to, tf 1. Therefore (1 l>, or in its form (12), is truly a state equation. From this global point of w(t,>
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view, then, the model replicates what happens in reality, since if the actual mill begins to operate with a certain load of water, and a given ore, what happens thereafter depends on the way the ore and water feeds are manipulated. Since the state becomes known for every t in [tO,trl, also Mr, cf, J, p, P, and H,,,, among olher variables are also known for all t in this interval. In system theory these variables are known as the output variables, which in general are a function of the states and the input variables. The set of equations which determine these variables in terms of states and inputs are called state output equations. Cluss$kation of the state model inputs. For the purpose of studying control strategies using this sate model, the input variables may be classified into: (i) commands or manipulated variables, (ii) measured disturbances, and (iii) unmeasured disturbances. Of course this classification depends on the instrumentation and actuators available in a particular plant. For example, in a usual case it may be that: 1. F,,, and FA, i.e., the total ore and water feed flows are manipulated variables 2. f, i.e. the vector containing the ore feed flows at each size interval is an unmeasured disturbance, 3. KE, the matrix characterizing the ore grindability, is another unmeasured disturbance.
3. Results Two operating procedures are given here to show the behaviour of the SAG mill state model: (i) a change in the size distribution of the feed ore, and (ii) a change in the solids percentage in the feed ore flow. 3.1. Changes in feed size distribution
In order to perform this test, the 27 size intervals used in the model have been grouped in three size ranges, where the fine size range takes into account the fact that the grate opening is 12.5 mm: Fine: - 12.5 mm - Intermediate: + 12.5 to - 102 mm * Coarse: + 102 mm, fu) Decrease in intermediate size. Keeping the total feed ore constant the proportion of intemiediate size range was decreased and compensated with an equal increase of the coarse range (the fine ore flow remains unchanged). In this manner an increase in the grinding efficiency of the mill may be anticipated. That such is indeed the case may be observed in Fig. 3, where the ore discharge flow increases causing a the reduction of the mill ore contents shown in Fig. 4. The feed ore flow is then increased (Fig. 3) to bring up the mill load, as a result of which the mill throughput is increased. The water is seen to follow similar variations as the fine (discharged) ore. (b) Increase in intermediate size. Now the intermediate size feed ore is increased and the coarse ore is reduced by the same amount. Since the grinding media is thus reduced, one may expect that the efficiency of grinding decreases. That this is indeed the case may be seen in Fig. 5, where the mill filling J increases until the mill operating point ??
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3.7-1.4 E (D I_ -1.3 _L z= I_ - 1.2 ,” _ S
3.3-
3.1r 0
6
4
2
8
time, r
-
Ore feed
-
--
1.1
10
hr -
Ore
Waterfeed
-----
Water
discharge
discharge
Fig. 3. Ore.andwater flow rates, in the feed and discharge of the mill, after a reduction of the intermediate ore feed flow rate and an equal increase of the course feed ore flow.
+ 0.35 zG 0.30 2 0.25 0.20 f 0.15 ‘r 0.10 c” 0.0s 0.00 I'0
0
2
4
6 time,
8
10
hr
Fig. 4. Hold-up of ore and water in the mill corresponding to the case of Fig. 3.
9.9 5
-0.12
9.0-
-0.38
. $ 9.70 g 9.6-
4
E
.*--
__
cc . . /
9.5-
-
-__
-0.34
- -__
--5_
-__
/ ,
9.4 f-/ 0
I 2
6
4 time, -Pwardraw
8
10
hr ---
HillfilUg
Fig, 5. Filling and power draw when coarse ore feed is reduced.
= O. ‘E ._s v 2
‘L .k ‘y
-0.30 2E ‘FZ S Lv 0.26
R. Amestica et al. / ht. J. Miner. Process. 44-45 (1996) 349-360 37
357 -1.50
T
/-------
e * --1.40 .
-______..--_---___
i --1.30
1
6 i:
32 -.
" --I20 s
3.1 7
1.10
$I’ 0
2
4
6
a
10
*#h ----oredschmp
I-ChfWd
--vvaerfeEd---werbs~
Fig, 6. Feed ore and water flow rates of the mill after a reduction of the coarse ore feed flow rate corresponding to Fig. 5..
goes the: unstable zone, i.e., the power decreases while the filling increases. The feed ore is then reduced and the recovery from this condition may be observed in the last part of the graph of Fig. 5. The corresponding ore and water feed and discharge flows are given in Fig. 6. As a result the mill throughput is reduced. 3.2. Chmge in feed solids percent In this test the water feed FA is rapidly increased in order to brimg the mill out of the unstable condition into which it is beginning to enter. In effect, in Fig. 8 the power is decreasing while the mill filling is increasing. The feed water is then increased so that the feed solids percentage drops from 0.78 to 0.74 (Fig. 7). As a result, the power undergoes a rapid increase matched by a fast decrease if the mill filling. Thereafter both
c 1.6
3.3 T
_---f t
e
*L 38 1 : 3.4
t
G 2
‘. .
_
I- 5. 1.5
----_____
I.4 j
1.3 E 6 12 d u
32
1.1 3 0 I
3$
0
2
1
6
4
a
10
tilm,hr
I-cxdeed-
--orubsdrarge
-wterfead
- - - - ~ifvdterdschagel
Fig. 7. Feed water flow increase to bring the mill out of an unstable operation.
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-0.33 _---
- -‘\
.
‘.
-0.29
*. -N
0
2
--w_
--__
--__
6
L time,
- _ ___
8
0.25
10
hr
Fig. 8. Power draw and mill filling corresponding to the case of Fig. 7.
16
027 --_ _ + -
-
’
.-
---__
-_
0264
--_h__-_l.
0
2
6
4
8
11
10
tim,hr
I-FillECf0fractiOn ----(XedschargereteI Fig. 9. Ore discharge rate and fraction of fine ore inside the mill after the step increase of the water feed flow rate of Fig. 7.
rO.10
0.65,
I
I
/---
0 ,. 0.35z _,__----A I0.25 ! 0
: :.
----_
- -------__-___
’ 0.05 2
6
4 time.
0
10
hr
Fig. 10. Changes of the total ore and water hold-ups after reduction of the feed solids percentage.
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power and mill filling decrease, showing that the stable region has been entered (Fig. 8). Fig. 9 shlows that after the feed water increase the ore discharge rate increases very fast due to the flushing effect of the water. Because of this, at the same time the fine ore size fraction in the mill decreases rapidly. After the abrupt changes caused by the feed water flow increase a new condition begins to develop in the mill as the grinding due to the coarse sizes produces a recovery of the fine size fraction (Fig. 9) inside the mill. Fig. 10 shows the changes and final condition of the total ore an water hold-up.
4. Conclusions By the use of dynamic balance of ore and water, together with submodels representing power draw, grinding, mass transport, and classification a state equation for a SAG mill has been developed in which the states are the mass of water and the ore masses, at different size intervals, contained in the mill. The submodels may be easily changed within tlhis model structure to suit new developments, whether in the phenomenological or empirical aspects. The model has shown a good behaviour when subject to changes in feed ore size distribution and feed percentage of solids, and the overload condition has appeared when it should be expected. Besides possible improvements in the submodels used, this state model also admits improvements in the state equation structure. Indeed, in order to treat varying grindabilities a variable combination of soft and hard ores may be represented, in which case the number of state variables representing the ore are doubled.
Acknowledgements Research leading to this paper has been funded by: PNUD project CHI 88/011, Technor!ogical development of comminution in the Chilean industry; Mining and Metallurgical Research Center (CIMM) and FONDECYT (Chile) project 1930885, Softsensors. Experimentation supporting this research was done at CODELCO-Chile, Chuquicamata Division.
References Ankstica, R., Gonz&lez, G.D., Bank, J., Magne, L., Menacho, J. and Castro, O., 1993a. A SAG mill circuit dynamic simulator based on a simplified mechanistic model. In: Proc. XVIII International Mineral Processing Congress, Australasian Inst. of Min. and Met., Sydney, Vol. 1, pp. 117-130. Amkstica, R., Gonzllez, G.D., Menacho, J. and Barria, J., 1993b. On-line estimation of fine and coarse ore, water, grinding rate, and discharge rates in SAG mill. In: Proc. XVIII International Mineral Processing Congress, The Australasian Inst. of Min. and Met., Sydney, Vol. 1, pp. 109- 115. Austin, L.G., 1990. A mill power equation for SAG mills. Min. Met. Process., (Feb.): 57-62. Forssbeq;, E. and Zhai, H., 1988. Mass transport behaviour in the full scale mills fractionally determined by the nN>velRTD methods. In: XVI International Mineral Processing Congress, Elsevier, pp. 1733- 1745.
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Gonzalez, G.D., Bamhona, C., Castelli, L,. Cipriano, A., Hemandez, L. and Yacher, L., 1987. Control strategy testing for a SAG mill using a qualitative SAG mill model designed for that purpose. In: Proc. Copper 87, Met. Sot. C.I.M., Chilean Inst. of Mining Eng., U. of Chile, Viiia de1 Mar, Vol. 2, pp. 409-421. Gupta, V.K., Hodouin, D. and Everell, M.D., 1981. The influence of feed rate and pulp composition on holdup weight and mean residence time of solids in grate discharge ball mill grinding. Int. J. Miner. Process., 8: 345-358. Herbst, J.A., Pate, W.T. and Oblad, A.E., 1989. Experiences in the use of model based expert control systems in autogenous and semiautogenous grinding circuits. In: SAG Milling Conference, Vancouver, pp. 669-686. Moys, M.H., 1986. The effect of grate design on the behaviour of grate-discharge grinding mills. Int. J. Miner. Process., 18: 85-105. Vanderbeek, J.L., Herbst, J.A., Hales, L.B. and Pennington, R.I., 1985. An investigation on the effects of mill volumetric load level, percent solids, and speed upon semiautogenous mill operation. Preprint.