Recent advances in bias and froth depth control in flotation columns

Transcription

1 Recent advances in bias and froth depth control in flotation columns J. Bouchard a, A. Desbiens a, R. del Villar b, * LOOP (Laboratoire d observation et d optimisation des procédés) a Department of Electrical and Computer Engineering,Université Laval Québec, Qc, Canada b Department of Mining, Metallurgical and Materials Engineering,Université Laval, Québec, Qc, Canada * rene.delvillar@gmn.ulaval.ca Abstract This paper reviews recent work done at Laval University in the field of column flotation instrumentation and control. The presented control results rely on froth depth and bias sensors. This work establishes that flotation column control could be substantially improved by using different control methods, such as nonlinear, multivariable, and feedforward control. The emphasis is placed on the way the available information, from sensors and quantitative or even qualitative relationships, may be used to reach the control objectives. Laboratory and pilot-scale results illustrate the discussion. Keywords: column flotation, process control, process instrumentation, modelling, mineral processing. Introduction The metallurgical performance of the column flotation process is determined by the concentrate grade and recovery. Whereas the first one can be continuously monitored using an on-stream analyzer, the second one can only be estimated from a material balance calculation, assuming steady-state. Consequently, automatic control and optimization of flotation columns need to be hierarchically performed using process variables with a strong influence on the

2 metallurgical performance, such as froth depth (H), bias (J b ), gas hold-up (ε g ), or bubble surface area flux (S b ). Local flow rate controllers (regulating the feed, wash-water, tailings, air and reagents flow rates) are at the base of such a control structure where their set-points are the manipulated variables for the higher level of control, i.e. to regulate H, J b, ε g and S b. The ultimate level is given by the optimization of the metallurgical performance according to an economical criterion (e.g. net smelter return) in a cascade scheme using H, J b, ε g and S b set-points as independent variables. This paper reviews recent advances in the field of instrumentation and control of flotation columns using froth depth and bias. The first part of the paper is dedicated to the description of conductivity-based methods used for the on-line evaluation of bias and froth depth. Different approaches to model the process dynamics for controller tuning purposes are then described and their advantages and drawbacks are analyzed. The third part discusses the various ways the available information may be used to design an effective control strategy. Finally, some laboratory and pilot-scale results are shown to illustrate the achievable column flotation control using the different tools presented in the paper. On-line evaluation of froth depth and bias Background Froth depth (or pulp-froth interface position) determines the relative importance of the cleaning and collection zones, as shown in Figure 1. The most common techniques for froth depth measurement have been summarized by Finch and Dobby (1990). Recent developments are reported by Bergh and Yianatos (1993), and del Villar et al. (1995a, 1995b and 1999). All these methods are based on variations of specific gravity, temperature or conductivity between the two zones to locate the pulp-froth interface position.

3 Figure 1 - Flotation column Methods using either floats or pressure gauges are commonly used in industrial operations. Even though their accuracy is limited (due to assumptions of uniformity of the pulp and froth density and absence of solids accumulation on the float gauge), they are suited for routine process monitoring. More recently, techniques using temperature or conductivity profiles measurements along the column upper zone were developed. Aside from being quite accurate, the obtained information can also be used to infer the bias as indicated hereafter. Conductivity probes have been successfully tested by Gomez et al. (1989), Bergh et al. (1993), and del Villar et al. (1999). Further improvements have included a decrease of the conductivity profile scan time, from one minute (Gomez et al., 1989) to about one second (del Villar et al., 1999), and the determination of the profile inflection point, associated with the interface position (del Villar et al., 1999). The bias is another important variable for the optimization of column flotation due to its high correlation with the concentrate grade for a given reagent dosage and bubble surface area flux.

4 Defined by Finch and Dobby (1990) as the net downward flow of water through the froth, or by its equivalent the net difference of water flow between the tailings and feed (from a mass balance calculation around the collection zone), the bias can be qualitatively interpreted as the fraction of the wash-water flow used for froth cleaning. In practice, the easier-to-measure total wash-water flow rate is more often used for process control. However, the latter does not correlate well with the concentrate grade and recovery since it includes the water fraction shortcircuited to the concentrate, which does not contribute to froth cleaning. Accurate bias measurement, using flow meters and density meters, is difficult to achieve since it assumes steady-state operation. Moreover, Finch and Dobby (1990) have demonstrated that the error propagation resulting from the use of multiple measurement devices leads to high bias relative standard-deviations. These facts justify the development of a more practical method. Uribe-Salas et al. (1991) have suggested an approach based on a steady-state conductivity balance calculation. The final expression involves the knowledge of the water flow rate in the tailings (J' t ) and concentrate (J' c ) streams, as well as the conductivity of the wash-water (k w ), and the liquid conductivity of the feed (k' f ), tailings (k' t ), and concentrate (k' c ) streams: k' k' f t k' k c w J = J' J' b t c (1) k' k k' k f w f w Although this method is relatively accurate, it is limited to steady-state laboratory-scale trials on two-phase (water and air) systems for on-line applications. When used on a three-phase (minerals, water and air) system, the various conductivities must be measured off-line. Moreover, measuring the concentrate water flow rate J c is difficult as a result of its high air content.

5 Moys and Finch (1988) have reported a relationship between the bias and the temperature profile along the column. An equivalent relationship between the bias and conductivity profile was introduced by Xu et al. (1989) and later detailed by Uribe-Salas et al. (1991). Pérez and del Villar (1996) have proposed the use of a neural network model approach to obtain a mathematical representation of the relationship between bias and the conductivity profile. The method is discussed in this paper. Froth depth measurement The pulp-froth interface position is inferred from the conductivity profile along the upper part of the column, using a semi-analytical method developed by Grégoire (1997). The conductivity profile sensor is composed of eleven 10-cm spaced stainless electrode rings fitted directly to the laboratory column (5 cm internal diameter). As described by Desbiens et al. (1998) and del Villar et al. (1999), this approach eliminates the neural network determination of the conductivityprofile inflection point proposed by Pérez-Garibay (1996), thus eliminating the extensive experimentation required for the calibration of such models. The various electrode pairs (each corresponding to a conductivity cell) are sequentially activated with a 1 khz alternative current to avoid secondary currents and pulp polarization. The corresponding conductivity value is calculated through an electronic circuit with a total scan time of about one second. Grégoire s technique is based on the assumption that the resistance of the cell containing the pulp-froth interface can be approximated as a system of two resistances in series as shown in Figure 2. The resistance of the cell containing the interface (R) can be related to those of the froth (R froth ) and the pulp (R pulp ), as froth ( x ) Rpulp R = x R + 1 (2)

6 where x represents the distance between the interface and the upper electrode of the cell containing the interface. Figure 2 - Calculation of the interface position The measurement is achieved in two steps. First, an algorithm locates the cell containing the interface through an iterative procedure involving the largest conductivity change. Then, the actual froth depth is calculated from the conductivity and position of this cell combined with: the conductivity of the immediately adjacent cells (above and below), to evaluate the conductivity of the froth (k froth = R froth -1 ) and pulp (k pulp = R pulp -1 ), and the conductivity of the first and last two cells (k 1, k 2, k 9 and k 10 ), to evaluate the vertical component of the conductivity gradient through the froth and pulp. The latter information is used to calculate correction terms for the conductivity of the froth and the pulp within the cell containing the interface. This technique has been validated in a pilotscale flotation column using a mineral pulp feed (20-30 % solids) consisting of hematite and silica. A standard deviation of about 2 cm is obtained. Figure 3 compares values given by the

7 sensor (H measured ) with a visual measurement (transparent column) (H real ). The detailed algorithm can be found in Desbiens et al. (1998) and del Villar et al. (1999) H measured (cm) H real (cm) Figure 3 - Froth depth measurement precision Bias evaluation Bias evaluation can be achieved using the neural network modeling technique proposed by Pérez-Garibay and del Villar (1996). Different network structures have been successfully tested by Pérez-Garibay (1996), Vermette (1997), Grégoire (1997), Paquin (2001), and Aubé (2003), for a simplified two-phase system, and by Pérez-Garibay (1996) and Aubé (2003) for three-phase systems. Aubé has also demonstrated that a multilinear regression model could lead to similar results to those obtained with a neural network. In both cases, the inputs of the model are:

8 k 1 and k 2, the conductivities of the first two cells, k 9 and k 10, the conductivities of the last two cells, k f and k w, the feed and wash-water conductivities, and J w and J g, the wash-water and air superficial velocities, respectively. A comparison of the predictions made with a regression model (J model b ) and a reference bias value, calculated from a steady-state mass balance using reconciled data, is given in Figure 4. The experimental flow rates (mean values for a 10-minute steady-state observation window) and percents solids were reconciled using Bilmat 8.1 TM (Algosys). On average, the predictions are equal to the reference values. The tests were conducted using a mineral pulp feed (hematite and silica, 20-30% solids). Since the sensor calibration can only be made through steady-state bias values (water mass balances), the dynamic performance of the sensor could not be assessed.

9 Figure 4 - Bias evaluation Dynamic modelling and identification A dynamic model is a time-dependent description of a system where present outputs depend on past inputs. Because of their prediction capabilities, dynamic models are considered as a key to good controller design. Indeed, precise models result in more robust model-based controllers, and consequently to better performance over a wider range of operation. The dynamic behavior of a process can be described by either physical or empirical (black box) modeling, each with assets and drawbacks (Walter and Pronzato, 1997; Söderström and Stoica, 1988). Physical models are analytically obtained from basic physical laws, while

10 empirical modeling consists in adjusting parameters of a mathematical relation to fit available data. The main drawback of physical models is that some complex processes cannot be described by first principles only. Furthermore, it may be more difficult to design a model-based controller when the model is complex. Empirical models are much easier to obtain and use. They adequately represent the process only for conditions (operating points, types of inputs, etc.) similar to those found in the used data. The parameters of empirical models do not have any physical meaning and a priori available information is almost completely neglected. Therefore, a preferred approach is to combine both methods to obtain a more accurate grey box model, which remains simple enough for control purposes. Empirical models Two usual empirical model representations are transfer functions and state-space equations. A general structure for a discrete SISO (single input single output) transfer function is 1 ( z ) y ( t) 1 ( z ) 1 ( z ) 1 ( z ) ( z ) B d C = z u ( t) + e ( t) (2) F D A 1 where y(t) is the output (measurement), u(t) the input (manipulated variable) and e(t) a white noise generating an unknown stochastic disturbance. The polynomials are defined as follows: A B C D F 1 1 na ( z ) = 1 + a z a z 1 na 1 1 nb ( z ) = b z b z 1 nb 1 1 nc ( z ) = 1 + c1z cncz 1 1 ( z ) = 1 + d z d z 1 nd 1 1 nf ( z ) = 1 + f z f z 1 nf nd (3) where the parameters a i, b i, c i, d i and f i have to be estimated from the recorded data by using a prediction-error identification method, i.e. by minimizing a norm of the prediction-error sequence

11 (Ljung, 1999). When there is more than one output or input, a matrix of transfer functions must be built. The transfer function approach was used by del Villar et al. (1999) and Milot (2000) to find the relationships between the wash-water, tailings, and air flow rate set-points (the three manipulated variables), and the bias and froth depth (the two process outputs) (see the section Illustrations). To increase the precision of the interface position tails flow rate set-point model over a wider range of operation, Desbiens et al. (1998) have used a variable velocity-gain (K v ) that is function of the air flow rate. To further increase the range of validity, Milot et al. (2000) have modelled the bias wash-water flow rate set-point relationship at three different washwater flow rates. For an easier representation of MIMO (multiple input- multiple output), state-space equations are recommended: y ( t + ) = Ax ( t) + Bu ( t) + ω ( t) ( t) = C x ( t) + Du ( t) + E ω ( t) + ν ( t) x 1 (4) where x(t) is the state vector, u(t) is the manipulated variable vector, y(t) is the vector of output predictions. Stochastic disturbances are generated with ω(t) and ν(t), two independent white noises. A second important advantage of state-space identification over the transfer function approach becomes apparent: it does not require any hypothesis on the structure of the stochastic part of the model. The calibration objective consists in obtaining the matrices A, B, C, D and E to fit the dynamic experimental data. Milot (2000) calculated state-space models to explain the variations of the outputs (bias and froth depth) for input changes (wash-water, tailings, and air flow rate set-points). The model matrices were estimated using ADAPTx TM (Adaptics Inc.) (Larimore, 1999). The computational and theoretical basis of the method relies on the singular

12 value decomposition. Three steps are performed (Larimore, 1999). First, a canonical variate analysis determines a linear combination of the process past to predict its future (i.e. the states). Then, the state order is obtained by minimizing the Akaike information criterion corrected for small data sets (Hurvich et al., 1990). Finally, the state space model parameters are computed by simple linear regression. Semi-physical models The system dynamics can also be explained by a set of algebraic and differential equations obtained from physics laws, mass and energy balance equations, etc. Modelling a complex system using such method is a considerable task. Therefore, empirical and physical models are often used together to yield a grey box model combining the robustness of a physical approach with the simplicity of an empirical model. Grey box models generally remain simple enough for controller design purposes. Some parameters from the physical and empirical parts of the model need to be identified from experimental data, usually by minimizing the prediction errors assuming an output-error model structure. Dumont et al. (2001) used this technique to propose two semi-physical models for a laboratory flotation column working with a two-phase system. The model output is the froth depth, while the inputs are the air, feed, and tailings flow rates. Control strategies and available information There are various ways to design a control strategy, even for the simplest SISO process. Generally, the control objectives guide the design of the control structure, the selection of control algorithms, the tuning of controllers, etc. However, the use of all available information is often neglected, even if it may substantially improve the controller performance beyond what a linear feedback SISO controller may achieve. Measurements provided by other sensors than those used

13 in the feedback control loop are obviously included in the available information, but quantitative and even qualitative relationships between process variables must also be considered in the control design. As presented hereafter, making use of all the available process information improves the control performance by decreasing the process output variability, reducing interactions between control loops, or by increasing the control system robustness and range of operation. To conveniently exploit the process knowledge, three different and complementary types of control techniques may be identified: the feedforward controller, the nonlinear controller and the multivariable controller. Feedforward control The main drawback of traditional feedback control schemes (for example PID controllers) is that the control action (u) can compensate the disturbances only a posteriori, when a significant output variation has been detected by the sensor. When the process behaviour exhibits important delays, slow dynamics, or disturbances with large magnitude, this may prevent tight respect of the set-points (y sp ) or even worst, full respect of security constraints. An efficient method to improve the control performance in these situations is to directly incorporate the measurable disturbances (d) into the controller as depicted in Figure 5. Feedforward Controller d y sp + - Controller Feedback + + u Process y Figure 5 - Feedforward control using a measurable disturbance

14 Such a control structure allows the controller to anticipate output variations, caused by the disturbance d, by making appropriate adjustments in the manipulated variable to prevent the disturbance from upsetting the controlled variable. Linear feedforward controllers are easy to implement within industrial control systems by using gain, lead-lag, and delay blocks. More details about this technique are presented by Deshpande and Ash (1988). Application in a model predictive control (MPC) framework is covered by Desbiens et al. (2000). Incorporating the effect of measurable disturbances in the model used for controller design is another way to perform feedforward control. Barrière et al. (2001) have used this method to render the froth depth independent of feed, air, and wash-water flow rates disturbances, using two semi-physical models developed by Dumont et al. (2001). Feedforward control should always be considered when disturbance measurements are available. It is a good and simple way to significantly increase regulation performances (i.e. decrease output variability) without tightening the feedback controller tuning and over wearing actuators.

15 Nonlinear control Because a general theory is not available, nonlinear control is usually considered as a very complex academic solution to regulation problems. However, because linear controllers usually perform poorly when applied to highly nonlinear systems, or to moderately nonlinear systems operating over a wide range or conditions (Henson and Seborg, 1997), nonlinear control may sometimes be the only way to reach performance objectives when other conventional linear techniques fail. The design of nonlinear controllers must be considered on an individual basis. In some instances, the design is simple and can be handled with standard industrial controllers, while in others, custom-built software is required. The simplest way to take into account process nonlinearities is to use a qualitative knowledge of the process in the form of an empirical relationship describing the nonlinearities. For example, Desbiens et al. (1998) have defined the froth depth tailings flow rate set-point transfer function gain as a function of the air flow rate. PID controllers can then be adapted for nonlinear control purposes based on such models. Therefore, instead of remaining constant, each PID gain proportional (K P ), integral (K I ) and differential (K D ) can be replaced by a mathematical function obtained from an empirical nonlinear relationship (gain-scheduling). Thus, gain-scheduling is an efficient way to maintain control performances independent of operating conditions. The empirical basis of the technique makes however difficult to extrapolate the results outside the range covered by the empirical data used to develop the model. An example of this technique for the pulp level control in a laboratory flotation column, where the PI proportional gain varies according to the air flow rate, is presented in the next section.

16 Another method consists in using a combination of different models to calculate the control action, each one calibrated at a different operating point. The actual control action is obtained by interpolation of the control actions calculated with each model. This multi-model control scheme can easily be implemented in a predictive controller (see example in the next section). A more complex but also more robust way to take into account process nonlinearities is to directly use phenomenological or semi-physical models in the controller design. Nonlinear control techniques such as nonlinear predictive control (Kouvaritakis and Cannon, 2001), backstepping (Krstić et al., 1995), or model reference nonlinear control (MNRC) (Chidambaram, 1995), are then required. Nonlinear controllers are always limited to specific types of model structure and nonlinearities. In the section Illustrations, the design of backstepping and MNRC controllers, based on semi-physical models of the column, are described. Backstepping is a recursive control design consisting in a systematic construction of both the feedback law and the associated Lyapunov functions. The objective of MNRC is to obtain a desired error signal dynamics, by inverting the nonlinear model. Because the semi-physical model inputs include manipulated variable as well as measured disturbances, feedforward control is automatically part of the nonlinear design. Multivariable control Except for basic local loops, true SISO systems are not common in industry. Due to interactions between variables, the processes must be analyzed and controlled with a multivariable approach. This is another mean of using all available information. Multivariable processes are sometimes said to be difficult to handle, mainly because suitable tools have not been considered. For instance, when more than one variable have to be

17 controlled at the same time, blindly applying SISO control techniques often leads to poor performances. When multiple SISO controllers are jointly used, interactions between the loops must always be taken into account. For instance, closing a new SISO loop around a MIMO plant without further analysis could result in unacceptable control performances and improper conclusions about the process controllability. Directionality is another characteristic not found in SISO processes that must be analyzed when controlling a MIMO plant (Skogestad and Postlethwaite, 1996). Loop interactions and directionality must then be studied before selecting pairings (i.e. inputs-outputs combinations for each loop), specifications, and control techniques. Two particular cases of multivariable control are those of decentralized and decoupled controllers. A control is decentralized when all controllers are SISO. In such a case, the selection of the pairing is a crucial step. Furthermore, the tuning of each individual controller must be carefully calculated because of the interactions between the loops. In the presence of strong interactions, compromises must be made when tuning decentralized controllers. Every feedback channel of a multivariable controller may also be made independent or almost independent of all other channels via the addition of decouplers, i.e. by using feedforward controllers anticipating other control actions. Thus, the use of a decoupled controller is attractive because a set-point change or an output disturbance in a particular loop has little effect on other loops. Nevertheless, when a process is ill-conditioned (i.e. strongly directional) inverse-based controllers such as decouplers are sensible to input uncertainties (Skogestad and Postlethwaite, 1996). In this case, a decentralized scheme, leading to a more robust system, is

18 recommended. Skogestad and Postlethwaite (1996) review a complete methodology for the design and analysis of multivariable control systems. In summary, when building a MIMO control system, making use of the available information implies a complete analysis of the process interactions and directionality. The reward is a design with guaranteed control performances. Illustrations Generally, industrial flotation columns do not benefit from sophisticated control systems mainly because froth depth is the only critical process variable that can presently be measured online using commercially available instruments. Developments of new on-line sensors for bias, gas hold-up, and bubble surface area flux, are now offering new possibilities for flotation column control such as those previously discussed. The next paragraphs present some experimental results for froth depth control and from control strategies involving both bias and froth depth. Del Villar et al. (1999) have applied a decentralized PI control strategy to a two-phase system in a laboratory-scale column. The bias and froth depth were controlled by manipulating the voltage at the wash-water pump and tailings pump terminals, respectively. PI controllers were used since they are simple, and well accepted by plant operators. The previously described drawbacks of a decentralized PI structure are not an issue in this case. Indeed, the variable pairing is obvious since changes of the tailings flow rate have practically no effect on the bias (del Villar et al., 1999). The coupling is therefore weak and SISO tuning methods can be used. Other limitations are the neglect of process nonlinearities (e.g. froth depth behaviour depending on air

19 flow rate, as shown later on) for the design and the absence of feedforward control to anticipate air and feed flow rate variations. Recently, a similar strategy was implemented on a pilot scale flotation column using a mineral pulp feed (Bouchard, 2004). Two local PI control loops were used for the wash-water and tailings flow rates. Their setpoints were supervised by the froth depth and bias controllers in a cascade scheme. Figure 6 shows a setpoint step change for the froth depth (Figure 6a) and for the bias (Figure 6b), while Figure 7 presents two tests for the evaluation of the closed-loop performance in the presence of air and feed flow rate disturbances. Figure 6 - Tracking performance (Decentralized PI)

20 These results indicate that the control performances are satisfactory for the nominal range of operation. For small froth depth and bias set-point changes, the interaction between both control loops is rather weak. However, a more exhaustive investigation is necessary to get more information about the behaviour outside the nominal operating region for the three-phase system. Figure 7 - Regulation performance (Decentralized PI) Gain-scheduled control of froth depth Desbiens et al. (1998) have implemented a very simple and effective nonlinear froth depth PI controller on a two-phase system, where the proportional gain is function of the air flow rate. Figure 8 shows how the closed-loop behaviour remains constant even when the air flow rate operating point is changed. The first part of the graph depicts two froth depth set-point step changes (40 to 60 cm and 60 to 40 cm) for a constant air flow rate of 1.25 cm/s. At about 800 s,

21 the air flow rate is increased to 1.8 cm/s and the PI proportional gain is adjusted accordingly. Note that the gain-scheduled PI does not have a feedforward action based on the air flow rate and therefore, does not allow anticipating for the sudden change in air flow rate. However, it takes into account the change of process velocity gain according to air flow rate variations and makes a proper adjustment of the PI gain to maintain the same froth depth dynamics. When the froth depth is brought back to 40 cm, two other set-point step changes are made (40 to 60 cm and 60 to 40 cm), and despite a 26% decrease in the process gain, the dynamics are similar to those obtained before the air flow rate change. The gain-scheduling technique has led to control performances practically independent of the air flow rate over a broad range of operating points. This is an interesting feature, since the air flow rate is often used to adjust the flotation column metallurgical performance. Multivariable nonlinear predictive control Milot et al. (2000) have tested a multivariable nonlinear GlobPC controller (Desbiens et al., 2000), illustrated in Figure 9 (two-phase system). The interaction between the bias and froth depth control loops is eliminated using feedforward (decoupling). A multi-model scheme is used. Linear models explaining the dynamics between the bias and the wash-water flow rate set-point model were identified for three different wash-water flow rates. The actual control action is calculated as a weighted sum of the control action obtained from calculations on each of the three models. The value of the wash-water flow rate determines the appropriate weights. As shown in Figure 10, the multi-model controller, unlike a linear controller, maintains good performances regardless of the bias set-point, and consequently of the value of the wash-water flow rate. The linear controller is identical to the nonlinear one except for the use of a single model (the second of the three models).

22 Figure 8 - Control performance of froth depth nonlinear control. The control performance of the predictive controller, for froth depth and bias set-point step changes, is shown in Figure 11. Feedforward leads to very good decoupling. Because of its flexibility, MPC is an advantageous technique for industrial applications. It can easily manage operating and safety constraints, the multivariable case is a simple extension of SISO control, and there are various ways of taking into account process nonlinearities (e.g. with multiple linear models or with empirical or phenomenological models).

23 FEED FORWARD Bias feed forward controller Froth depth feed forward controller Measured disturbances + (feed and air flow rates) + Froth depth feed forward models Bias feed forward models Unmeasured disturbances Bias setpoint Froth depth setpoint TRACKING Bias tracking controller Froth depth tracking controller Wash water flow rate setpoint Tail flow rate setpoint Column flotation process Bias model Bias Froth depth Froth depth model Bias feedback controller Froth depth feedback controller FEEDBACK Figure 9 - GlobPC structure applied to the column flotation process. J b cm/s J b cm/s Linear control Bias set point Nonlinear control Bias set point Time (min) Figure 10 - Comparison of bias linear and nonlinear control

24 u (cm/s) J b (cm/s) H (cm) Tailings flow rate set point Wash water flow rate set point Time (min) Figure 11 - Control performance multivariable MPC Froth depth nonlinear control based on semi-physical models Two semi-physical representations of the froth depth dynamics were proposed by Dumont et al. (2001) for a two-phase system. Essentially, both nonlinear models are based on simple physical phenomena, such as Newton's second law and Archimedes' principle, to predict the froth depth. The inputs are the air, feed, and tailings flow rates. The non-measured concentrate flow rate is predicted by an empirical approach. Based on these semi-physical models, Barrière et al. (2001) have proposed MRNC and backstepping controllers for froth depth. Figure 12 and Figure 13 compare the behavior of these two nonlinear controllers with that of a standard PI controller, when feed and air flow rates disturbances occur. Table 1 gives the ISE criteria (integral of the square of the errors) for each controller. Including physics into the controller algorithm, significantly improves the

25 performance. A tighter respect of the set-point is made possible by the feedforward action, with a gentle adjustment of the tailings flow rate PI 42 Backstepping MRNC Froth depth [cm] Tailings flowrate [cm/s] Feed flowrate [cm/s] Time [seconds] Figure 12 - Froth depth nonlinear control (feed flow rate disturbance)

26 PI 30 Backstepping MRNC Froth depth [cm] Tailings flowrate [cm/s] Air flowrate [cm/s] Time [seconds] Figure 13 - Froth depth nonlinear control (air flow rate disturbance) Table I - ISE criteria Controller ISE criteria Feed flow rate disturbance (Figure 12) PI 847 Backstepping 334 MNRC 69 Air flow rate disturbance (Figure 13) PI 2574 Backstepping 590 MNRC 1158 Conclusion Over the past few years, substantial amount of work has been accomplished by LOOP researchers to improve flotation column control. Laboratory and pilot-plant results indicate that

27 integrating knowledge of the process and newly available measurements is necessary to reach the control objectives. Most of the work had dealt however with a simplified water-air system, but more recently, promising results have been obtained for a control structure implemented in a pilot-scale column processing a mineral-pulp feed. Flotation column control and optimization should benefit from the following future developments: consideration of gas hold-up and/or bubble surface area flux on-line measurement for control purposes; improvement of the bias sensor allowing for dynamic measures; industrial validation of new sensors and control strategies; investigation of relationships between process variables (H, J b, ε g and S b ) and metallurgical performance. Acknowledgements The authors would like to acknowledge the support of FQRNT (Fonds Québécois de la Recherche sur la Nature et les Technologies), NSERC (Natural Science and Engineering Research Council), La Compagnie Minière Québec Cartier and COREM (Consortium en Recherche Minérale). References Aubé, V., Validation semi-industrielle des capteurs de profondeur d écume et de différentiel d eau dans une colonne de flottation, M.Sc. thesis, Département de génie des mines, de la métallurgie et des matériaux, Université Laval, 2003.

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