Level control strategies for flotation cells

Transcription

1 Minerals Engineering 16 (2003) This article is also available online at: Level control strategies for flotation cells P. K amj arvi, S.-L. J ams a-jounela * Laboratory of Process Control and Automation, Helsinki University of Technology, Kemistintie 1, Esoo, Finland Received 23 Aril 2003; acceted 2 June 2003 Abstract Flotation is a difficult rocess to run efficiently. One way to make flotation erformance better is to imrove cell level control. However, controlling ul levels in flotation cells is a comlex control task because of strong interactions between the levels in flotation cells. Therefore advanced controllers are needed to give good level control. This aer deals with a model of six flotation cells in series. Simulations are erformed to comare different control strategies. Four control strategies are considered: one SISO controller and three different MIMO controllers including a new multivariable controller. It is shown that level control erformances of the MIMO controllers are significantly better than that of the classical SISO controller. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Flotation machines; Froth flotation; Process control; Mineral rocessing 1. Introduction Level control of flotation cells is a very comlex task due to high interactions between the rocess variables. A control action imlemented at any oint in the flotation circuit tends to be transmitted to both ustream and downstream units, and sometimes with amlification. Large variations in the flow rate to the first cell and varying comosition of the raw ore also cause roblems. Flotation cells are conventionally controlled by isolated PI-controllers. PI control works well when the cell being controlled is isolated. However, in a flotation circuit where interactions are strong, PI control does not meet the requirements of high control erformance. Hence a considerable amount of research has been carried out over the last few years to develo better control techniques for flotation circuits (J ams a-jounela et al., 2001). Niemi et al. (197), Koivo and Cojocariu (1977) used a single cell model when develoing an otimal control algorithm via alications of the maximum rincile. Andersen et al. (1979) and Zargiza and Herbst (1987) reorted an alication of state feedback control and a Kalman filter for rougher flotation control. Hammoude and Smith (1981) used a linear model to develo a minimum-variance controller for recleaning. New advanced * Corresonding author. address: sirkka-l@hut.fi (S.-L. J ams a-jounela). control method has been also recommended for the control of flotation rocesses by (Stenlund and Medvedev, 2000). The aim of this research is to study and comare different control strategies from the oint of view of cell level control. In addition, a new control strategy is resented and imlemented. Its erformance is comared to three different strategies: one traditional SISO control strategy, and two MIMO control strategies. The strategies are comared by the means of secial erformance indices. In the following the mathematical model of a flotation cell is first develoed and a five cells in series cell configuration is constructed. The simulations are then erformed in order to determine suitable control arameters for controllers of the cell levels. Simulation was erformed with Matlab and its Simulink library. 2. Mathematical modeling of flotation cells in series In the flotation rocess the ul is fed into the first cell and the froth is collected in the launders. The feed can be measured by a flow measurement, and the remaining ul flows into the next cell. The magnitude of the flow deends on the ressure difference between two adjacent cells, the osition of control valves, and the viscosity and density of the ul. The magnitude of the ressure difference can be determined from the hysical /$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi: /j.mineng

2 1062 P. K amj arvi, S.-L. J ams a-jounela / Minerals Engineering 16 (2003) height difference of the cells. The ul level in a cell is measured and controlled by adjusting the control valve. In the following a flotation cell is considered as a tank of erfectly mixed ul. Since the ul is erfectly mixed, the density is the same throughout the cell, i.e. there are no satial density gradients in the cell. As the froth flow is small comared with the ul flow, it is ignored in the outgoing flows. The imact of the air feed on the ul level is also ignored. The cells under study do not have the roerties of ideal tanks because the cross-sectional area of the cell is not constant. A mathematical model for the hysical roerties will be develoed and discussed next Single cells in series In a flotation rocess several single cells are connected in series as shown in Fig. 1. For the first cell in the series: ov 1 ot ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ðq F 1 Þ¼q KC v ðu 1 Þ y 1 y 2 þ h 1 ð1þ where q ¼ feed rate to the first cell, y 1 ¼ ul level in the first cell, y 2 ¼ ul level in the second cell, h n ¼ hysical difference in height between the cells, u 1 ¼ control signal, K ¼ constant coefficient, F 1 ¼ outflow from the first cell, and C v ¼ valve coefficient. The equations for cells 2, 3, and are resectively (i ¼ 2, 3, and ) ov i ot ¼ðF ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 1 F i Þ¼KC v ðu i 1 Þ y i 1 y i þ h i 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi KC v ðu i Þ y i y iþ1 þ h i ð2þ The equation for the last cell (n ¼ 5) ov n ot ¼ðF n 1 F n Þ¼KC v ðu n 1 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi KC v ðu n Þ y n þ h n ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y n 1 y n þ h n 1 ð3þ In the case of an ideal tank, the cross-section of a cell is assumed to be constant. The ul levels in the cells can therefore be written as oy 1 ot ¼ ðq F 1Þ A 1 ¼ q KC vðu 1 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y 1 y 2 þ h 1 A 1 ðþ oy i ot ¼ K ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 1C v ðu i 1 Þ y i 1 y i þ h i 1 A i KC ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vðu i Þ y i y iþ1 þ h i A i where i ¼ 2, 3, and. oy n ot ¼ KC ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vðu n 1 Þ y n 1 y n þ h n 1 A n KC ffiffiffiffiffiffiffiffiffiffiffiffiffiffi vðu n Þ y n þ h n where n ¼ 5. A n 2.2. Double cells in series ð5þ ð6þ Mathematical models for the double cells can be derived in a similar manner (J ams a-jounela et al., 2003). The rincile differences comared with the mathematic models of single cells in series are that, in a double cell, both ul levels are controlled by maniulating a control valve in the second cell outflow, as can be seen from Fig. 2. The cells are hysically on the same level. The ul level in the second cell is measured and comared with the set oint. This signal is used as inut to a PI-controller. The outut signal of the controller is the desired valve osition as denoted by u 1. The cells in a double cell are connected via a flange. The flange limits the maximum flow between the cells. The ressure difference in the cells is the only driving force for the flow, and deends on the density of the ul. The velocity of outflow from the first cell can be derived using Bernoull s equation, resulting in v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2gðy 1 y 2 Þ ð7þ q y 1 y 2 y 3 F 1 F 2 u 1 Fig. 2. Double flotation cell series. F 3 Fig. 1. Schematic diagram of flotation cells.

3 P. K amj arvi, S.-L. J ams a-jounela / Minerals Engineering 16 (2003) where g ¼ gravity, y 1 ¼ ul level in the first cell, and y 2 ¼ ul level in the second cell. The volume flow across the flange can be calculated as F 1 ¼ va flange ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2gðy 1 y 2 Þl l h l ð8þ where A flange ¼ cross-sectional area of the flange, l l ¼ length of the flange, and h l ¼ height of the flange. The ressure dro due to flowing resistances is assumed to be negligible and can be ignored. The change of the ul volume in the first cell with resect to time can be written as ov 1 ¼ q F ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2gðy 1 y 2 Þl l h l ð9þ ot The outflow of the second double cell is similar to that of a single cell, and the flow into the second cell is the outflow of the first. Therefore the mathematical model is ov 2 ot ¼ F 1 F ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2gðy 1 y 2 Þl l h l ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 2 C v ðu 1 Þ y 2 y 3 þ h Modelling the Outokumu flotation cells ð10þ Fig. 3 shows the cross-section of the flotation cells under study. The ul level changes as a function of the cell volume. The change is linear from the bottom of the cell to the starting level of the launders and boosters, denoted by H lowerart in Fig. 3. The cross-section of the ul subsequently decreases on moving uwards because the launders and Ôboosters reduce the volume of the cell. Calculation of the ul volume can be divided into two sections: the ul level below and above the level of H lowerart. When the volume is smaller than the volume of the cylinder s H lowerart, the ul level is determined as h tot ¼ V ð11þ R 2 tot where R tot is the radius of the cell. If the ul volume is equal or greater than H lowerart, the volume is written as V ¼ V lowerart þ V uerart ðhþ V B ðhþ V R ðhþ ð12þ where V lowerart ¼ volume of the cell below the level of H lowerart, V lowerart ¼ volume of the cylinder above H lowerart, V B ¼ volume of the boosters, and V R ¼ volume of the launders. A level variable, the zero oint of which is equal to H lowerart is denoted by h. The volumes of the launder and booster are obtained by geometric relations, and substituting them in Eq. 12 results in h 3 3 K2 B þ h 2 ðk B R total ÞþhððK R R 2 total ÞÞ þðv V lowerart Þ¼ 0 ð13þ where K B ¼ constant coefficient of the booster dynamics and K R ¼ constant coefficient of the launder dynamics. The third order equation has a solution in ½H lowerart H total Š, where H total is the total height of the cell. K B, K R and V lowerart are constants and secific for each cell size. The total level of ul can be determined by summarizing h and H lowerart. The effect of ul level non-linearity can be seen in Fig., in which the ul level is resented in ideal tank conditions as a function of the ul volume. The cell tye is TC-50, the maximum volume of which is 50 m Valve sizing and characteristic curve of the valves Valve sizing is based on the C v value, which is calculated according to the ISA standard as follows: R r,out L C L B L R r C r 2 h r B H C a H B.6..2 TC50, linear and nonlinear ul level 3.8 level H lowerart R total volume Fig. 3. Cross-section of a flotation cell of Outokumu Mintec. Fig.. Pul level as a function of ul volume.

4 106 P. K amj arvi, S.-L. J ams a-jounela / Minerals Engineering 16 (2003) rffiffiffiffiffiffi q C v ¼ 1:17Q ð1þ D where Q is the flow rate (m 3 /h), C v the valve caacity coefficient, q the ul density (kg/m 3 ), and D the ressure difference over the valve. The flow rate (m 3 /h) through the cell is calculated as Q ¼ 1:2 V cell ð15þ s=60 where V cell is the cell volume (m 3 ) and s the ul retention time in the cell. The valves in the models are sized for a flow which has a retention time of 1.5 min in one cell. The flow rate to the first cell is also calculated using this retention time value. Characteristic curves of the control valves are roduced by Larox Flowsys and were used in the Simulink models in order to cause realistic and non-linear behavior for the valves. Fin Feedforward Measurement y set - e F in ṃ PI controller u Valve F y m Measurement Flotation cell dynamics Fig. 5. Control diagram of feed-forward controller. m ( s ) 1 H ( ) m 2 ( s) 11 s H ( ) 21 s H ( ) 12 s H ( ) 22 s y y ( ) 1 s y ( s 2 ) 3. Control strategies The different control strategies are discussed and described in the following sections. These strategies are selected because they can be used with basic PI-controllers and without any additional instrumentation. Traditionally in flotation cell series there is only one flow measurement in the beginning of the series and level PIcontrollers in every cell Feed-forward controller A flow feed-forward controller monitors disturbances in the inflow to the first cell and uses roortional action to close or oen the valves of the cell in order to comensate for disturbances. Comensation is linearly deendent on the difference between the current inflow and the normal inflow. The measurement signal is filtered in order to revent the feed-forward control from reacting to random variation in the flow. However, this kind of controller does not rovide any extra erformance imrovement in the event of disturbances occurring somewhere else in the cell series. The model of the feedforward controller is shown in Fig Decouling controller A decouling controller is based on differential equations (1) (3). The urose of the decouling controller is to eliminate the crosswise effects of control loos, and hence the stability of a single control circuit deends only on its own stability features. The basic model of the decouling controller is shown in Fig. 6. Fig. 6. Control diagram of a basic decouling controller. The mathematical criterion to be fulfilled for decouling a tank i will be DF i in DF i out ¼ 0 ð16þ where DF i in is a change of inflow to tank i. Using the valve functions from Eqs. (1) (3) the equation can be written as follows ffiffiffiffiffiffiffiffiffiffiffi F i ¼ K i C vi u i ðdh i Þ ð17þ where Dh i is the level difference over the valve. Substituting in Eq. (16), it becomes ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K i 1 C i 1 ðu i 1 þdu i 1 Þ ðdh i 1 þ DDh ð i 1 ÞÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K i C i ðu i þdu i Þ ðdh i þ DDh ð i ÞÞ¼ 0 ð18þ Solving this equation for the change in the control signal gives qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Du i ¼ ðk i 1 C i 1 =K i C i Þu 0 i 1 ðdh 0 i 1Þ= ðdh 0 iþ u i ¼ f u 0 i 1 ; Dh0 i 1 ; Dh0 i ð19þ Eventually, the control signal for a tank i becomes u i ¼ u PI þ fðu i 1 ; Dh i 1 ; Dh i Þ ð20þ where u PI is the control signal from a PI-controller. In order to handle the variations from the inflow, the feedforward is attached to the first tank Multivariable controller similar to Floatstar TM A multivariable controller (Schubert et al., 1995) controls the total inventory of material in the ustream tanks. In this control strategy, controlling a valve is

5 P. K amj arvi, S.-L. J ams a-jounela / Minerals Engineering 16 (2003) e1 P1 y set P2 e2 P3 e3 e u F PI Valve controller - y m Measurement Flotation cell dynamics Fig. 7. Control diagram of a multivariable controller similar to Floatstar TM (tank ). y influenced not only by the difference between a set oint and the measured level in the tank, but also the differences between set oints and the measured levels in all the tanks in ustream. These variables are summed and fed to the PI-controller of the cell. Furthermore, the variables are scaled by a suitable factor deending on the valve size, osition and rocess. In this strategy each control valve can be regarded as a sluice gate of a dam. When a damned inventory is too high in ustream, the valves are oened more than usual, even when there seems to be no need to take such an action on the basis of the levels in the neighboring vessels. The control diagram is shown in Fig New multivariable controller When some disturbance occurs in tank, it has an effect to level in the revious tank in ustream. Previously e1 P1 e2 e3 P2 P3 Filter P5 Feedforward e5 F in,m F in Measurement y set - e u F PI controller Valve Flotation cell dynamics y y m Measurement Fig. 8. The control diagram of MV controller (tank ). Fig. 9. Simulink model of the flow dynamics in a flotation cell.

6 1066 P. K amj arvi, S.-L. J ams a-jounela / Minerals Engineering 16 (2003) described multivariable control does not take disturbances of this kind into account. Also strategy does not take into account disturbances arising from ul feed. Therefore a flow feed-forward has been added to the system and the difference between set oint and measured level in next tank is also added to the controller of revious tank. The control diagram is shown in Fig. 8.. Simulations In the simulations a configuration of six TC-50 cells in series was studied in accordance with the ideal tank assumtion. Therefore the effects of boosters and launders were not considered. The valves were 100% oversized according to the ISA standard, and the retention time in each cell was 1.5 min. Control strategies included conventional PI-controllers with feed-forward control, decouling controller, a multivariable controller similar to Floatstar TM and a feed-forward multivariable controller. The simulation results of a 3 cm change in the set oints of the cell levels at times 100, 150, 200, 250, 300 and 350 s are resented in the following. Making ±20% change in the feed to the first flotation cell was also simulated with different strategies. The set oint of the cell level is lowest in the first cell, and the set oint values increase on moving towards the last cell in the series, where the oerating range of the level controller is smaller. The simulation schemes were constructed with Matlab Simulink software. The Simulink model of the flow dynamics in the flotation cell is resented in Fig. 9. The controllers were tuned and comared using the following indices. The IAE index (integral of the absolute value of the error) integrates the absolute value of errors, and even-handedly weights all the deviations. ISE (integral of the square error) gives more weight to big deviations from the set oint. ISE ¼ IAE ¼ Z t2 t¼t 1 ðyðtþ y s ðtþþ 2 dt Z t2 t¼t 1 jyðtþ y s ðtþjdt 5. Simulation results ð21þ ð22þ The simulations of the configurations of six TC-50 cells in series resulted in arameters for the PI-controllers. Integration times in the traditional system with a feed-forward controller were between 15 and 50 s and roortional gains between 0.8 and 1.2. Because MIMO control strategies resond better to disturbances, the PI arameters were set faster. Integration times in all the PI-controllers were set to 15 s and gain to 1. In the de (a) (b) Fig. 10. Feed-forward controller. On the left the resonse to a )20% change in ul feed, and on the right the resonse to set oint changes Fig. 11. Decouling controller. On the left the resonse to a )20% change in ul feed, and on the right the resonse to set oint changes.

7 P. K amj arvi, S.-L. J ams a-jounela / Minerals Engineering 16 (2003) couling controller the PI-arameters were between s and 1 1. s, corresondingly. The resonses of the feed-forward controller to disturbances in ul feed and to set oint changes are resented in Fig. 10. As can be seen, the )20% change in ul feed is affecting all the cell levels in the series. The set oint changes in a cell also have undesirable effects on the adjacent cells. There is always a considerably large erturbation in the level of the next cell every time a set oint change is made in the system. The resonses of the decouling controller are illustrated in Fig. 11. The decouling controller is a MIMO controller, and it also takes into account the interactions between cells. As can been seen from the grahs, the decouling controller effectively eliminates disturbances arising from changes in the ul feed. Furthermore, set oint changes in the cells do not affect to the other cells. The resonses of configurations in which a controller similar to Floatstar TM and the new multivariable controller are used are shown in Figs. 12 and 13. The new multivariable controller seems to be more robust than the other controller, esecially during changes in ul feed. The IAE and ISE indices, which deict the erformance of controllers, are shown in Tables 1. As was to be exected, the traditional SISO control with flow feedforward had the oorest figures in all cases Fig. 12. Multivariable controller similar to Floatstar TM. On the left the resonse to a )20% change in ul feed, and on the right the resonse to set oint changes Fig. 13. New multivariable controller. On the left the resonse to a )20% change in ul feed, and on the right the resonse to set oint changes. Table 1 The erformance indices for the feed-forward controller Feed-forward ISE(20%) IAE(20%) ISE()20%) IAE()20%) ISE(s..c.) IAE(s..c.) controller

8 1068 P. K amj arvi, S.-L. J ams a-jounela / Minerals Engineering 16 (2003) Table 2 The erformance indices for the decouling controller Decouling controller ISE(20%) IAE(20%) ISE()20%) IAE()20%) ISE(s..c.) IAE(s..c.) Table 3 The erformance indices for the multivariable controller similar to Floatstar TM MV controller ISE(20%) IAE(20%) ISE( 20%) IAE( 20%) ISE(s..c.) IAE(s..c.) Table The erformance indices for the new multivariable controller New MV controller ISE(20%) IAE(20%) ISE()20%) IAE()20%) ISE(s..c.) IAE(s..c.) Conclusions All the simulated configurations were successfully tuned. It is noticeable that the classical SISO strategy with feed-forward controller cannot even aroach the erformances of the MIMO controllers. This is due to high interactions between the control loos, which SISO systems cannot take into account. The differences between different MIMO systems are somewhat smaller. All the controllers erformed robustly to disturbances in ul feed and to set oint changes. The decouling controller had the best IAE and IDE indices. However, the decouling controller is sensitive to model uncertainties (Skogestad and Postelwaite, 1996). This also means that rocess changes can strongly degrade the control erformance. References Andersen, R., Gronli, B., Olsen, T., Kaggernd, I., Ramslo, K., Sandvik, K., An otimal control system of the rougher flotation at the Folldal Verk concentrator, Norway. In: Proceedings of the 13th International Mineral Processing Congress. New York, USA, Hammoude, A., Smith, H., Exeriments with self-tuning control of flotation. In: Proceedings of the 3rd IFAC Symosium on Automation in Mining, Mineral and Metal Processing. Oxford, UK, J ams a-jounela, S.-L., Laurila, H., Karesvuori, J., Timeri, J., Evaluation of the future automation trends in control and fault diagnostics a case study in flotation lant. In: 10th IFAC Symosium on Automation in Mining, Mineral and Metal Processing. J ams a-jounela, S.-L., Dietrich, M., Halmevaara, K., Tiili, O., Control of ul levels in flotation cells. Control Engineering Practise, Koivo, H., Cojocariu, R., An otimal control for a flotation circuit. Automatica 13 (1), Niemi, A., Maijanen, J., Nihtil a, M., 197. Singular otimal feed forward control of flotation. In: IFAC/IFORS Symosium on Otimization Methods Alied Asects. Varna, Bulgaria, Schubert, J.H., Henning, R.G.D., Hulbert, D., Craig, I.K., Flotation control a multivariable stabilizer. In: XIXth IMPC, San Fransisco, vol. 3, Skogestad, S., Postelwaite, I., Multivariable Feedback Control: Analysis and Design. John Wiley & Sons. Stenlund, B., Medvedev, A., Level control of cascade couled flotation tanks. Future trends in automation in mineral and metal rocessing. In: J ams a-jounela, S.-L., Vaaavuori, E. (Eds.), IFAC Worksho 2000, Helsinki, Finland, Zargiza, R., Herbst, J.A., A model based feed forward control scheme for flotation lants. In: 116th AIME annual meeting. Denver, CO, USA,