- Email: [email protected]
Nonlinear control of bubble size in a laboratory flotation column
13th Symposium on Automation in Mining, Mineral and Metal Processing Cape Town, South Africa, August 2-4, 2010
Nonlinear control of bubble size in a laboratory flotation column ´ Poulin ∗ , M. Maldonado ∗ , A. Desbiens ∗ , R. del Villar ∗∗ , E. ∗∗∗ A. Riquelme ∗
LOOP, Department of Electrical and Computer Engineering, Universit´e Laval, Qu´ebec City, Canada ∗∗ LOOP, Department of Mining, Metallurgical and Materials Engineering, Universit´e Laval, Qu´ebec City, Canada ∗∗∗ Department of Electrical Engineering, University of Concepcion, Chile Abstract: Gas dispersion properties have proven to be key variables of the flotation process. Among them, bubble surface area flux (BSAF) has been reported to linearly correlate with the flotation rate constant; therefore, it is a potential variable to achieve a desired metallurgical performance. BSAF can be represented as a combination of two other gas dispersion properties: superficial gas velocity and Sauter mean bubble diameter. Thus, controlling BSAF implies controlling bubble size and superficial gas velocity. This work focuses on the nonlinear control of the Sauter mean bubble diameter. Sauter bubble mean diameter was indirectly calculated from the bubble size distribution, estimated by using a Gaussian mixture model. To improve controllability, a so-called frit-and-sleeve sparger was installed to regulate bubble size independently from superficial gas velocity. With this device, the bubble size can be modified by manipulating the water flow rate circulating through the sleeve that surrounds the porous ring. A Wiener model is used to represent the dynamic relationship between the sleeve water flow rate and Sauter mean diameter. Wiener models consist of a linear system in series with a memory-less (static) nonlinear element. An IMC controller based on the identified Wiener model was implemented in a laboratory flotation column. Tracking performance and rejection of gas velocity and unmeasured frother concentration variations were then successfully evaluated. Keywords: Flotation column, Bubble surface area flux, control, Sauter mean diameter, Internal model control, Weiner model, Nonlinear control 1. INTRODUCTION
which have been widely applied (Barria and Valdebenito, 2008; Cortes et al., 2008; Moilanen and Remes, 2008), or froth appearance (Liu and MacGregor, 2008). The second approach aims at controlling some gas dispersion properties in the collection zone, such as gas hold-up and superficial gas velocity (Bergh and Yianatos, 1993; Carvalho and Dur˜ao, 2002; Persechini et al., 2004; Maldonado et al., 2009a). This approach has been motivated by the recent availability of industrial gas-dispersion sensors (Gomez and Finch, 2007; O’Keefe et al., 2007). This article extends previous works on control of some gas dispersion properties to the control of bubble size represented by the Sauter mean bubble diameter, a key step towards controlling the bubble surface area flux.
Flotation is a commonly used method for separating valuable minerals (metal containing) from useless mineral (gangue). Its performance is determined by the valuablemineral concentrate grade and recovery. Whereas the first of these two variables can be measured on-line using an X-ray on-stream analyzer (OSA), the latter must be estimated from steady-state material balance, which strongly limits its use for regulatory control purposes. Moreover, the long sampling times of these OSA devices, usually multiplexed, favour the use of a hierarchical control where secondary variables are controlled to reject the frequent disturbances occurring in this type of process. Recent studies (Gorain et al., 1997) have shown that the performance of a flotation device basically depends on three factors: the particle floatability, the froth recovery and the ”bubble surface area flux”, a combination of two other gas dispersion properties, superficial gas velocity and bubble size (Finch and Dobby, 1990). This finding suggests that for a well conditioned pulp, a given metallurgical performance can be achieved by modifying froth recovery and bubble surface area flux. Consequently, two control approaches have been proposed. The first focuses on controlling some froth characteristics, such as froth speed,
978-3-902661-73-9/10/$20.00 © 2010 IFAC
Bubble surface area flux can be mathematically expressed as follows:
Sb =
6Jg d32
(1)
where Jg is the superficial gas velocity and d32 is the Sauter mean bubble diameter. Since there is no uncertainty associated to equation 1, controlling BSAF implies controlling 19
10.3182/20100802-3-ZA-2014.00004
13th IFAC MMM Cape Town, South Africa, August 2-4, 2010
2. FLOTATION COLUMN SETUP
the ratio between superficial gas velocity and the Sauter mean bubble diameter.
The flotation column used in this work has three sections made of polycarbonate tubes for a total height of 5 m; the internal diameter of the bottom, intermediate and upper sections are respectively 15.24 cm, 10.16 cm, and 5.08 cm. Recirculation is provided by a peristaltic pump. A frit-andsleeve sparger is mounted in the bottom part of the column as shown in Figure 1. Gas flow rate is measured through a mass flow sensor/controller (Aalborg model GFC17). The mass flow sensor also provides an estimate of the volumetric flow based on a reference condition (21.1 o C and 101.3 kPa). Its readings must then be converted to the actual tests conditions (temperature and pressure) measured by sensors shown in Figure 1 using the following equation:
In flotation column operation, bubble size is affected by frother type and concentration, gas rate and sparging system. Frother dosage regulation is usually implemented using a ratio feed-forward control based on the actual tonnage processed. Nevertheless, because of the limitations of feed-forward control in the presence of modelling uncertainties and unknown disturbances, such as frother persistency, evaporation rate and most importantly, the effect of reprocessed water (still containing some residual frother), frother concentration in a given flotation machine is hard to assess. In this work, frother concentration is considered as an unmeasured disturbance. Although the authors have proposed a method to evaluate on-line the frother concentration (Maldonado et al., 2009b). Since superficial gas velocity also modifies bubble size (Finch and Dobby, 1990; Nesset et al., 2006), it directly affects BSFA through the numerator of equation (1) and indirectly by affecting bubble size (denominator). Whenever feasible, the sparger system adds another control degree of freedom to modify bubble size in flotation columns. In this work, a ’frit-and-sleeve’ sparger was implemented (Kracht et al., 2008). This type of sparger allows the modification of bubble size independently from gas velocity, thus improving BSAF controllability.
" Jg =
1033.23 1033.23 + P
#"
T + 273.15 294.16
# (2)
where P is the absolute pressure measured in cmH2 O, T is the temperature in Celsius degrees and Jgref is the gas velocity calculated from the air mass flow meter lecture at reference conditions. A differential pressure transmitter DP (model ABB 264DS) was tapped between 250 cm and 320 cm above the sparger to measure gas holdup. For a two-phase system (air-water) this latter can be measured using the following relationship: ∆P εg (%) = 100 × (3) L where ∆P is the pressure differential in cmH2 O and L is the distance between taps (in this case 70 cm).
Section 2 describes the laboratory column and its instrumentation. The nonlinear Wiener model identification is detailed in section 3. The internal model control scheme is presented in section 4. Experimental results and discussion are found in sections 5 and 6. Windows XP® MatLab® Image J
Gap
Porous SS ring Sleeve
Video Sony frames camera Firewire
Ethernet / UDP
Windows XP® iFIX® MatLab®
DP
Jgref
Water inlet
TT
Frit and sleeve sparger
PT
RS232/ Modbus
Air inlet
I/O opto22
Fig. 2. Frit and sleeve sparger prototype Gear pump
FT FC Moore 353
Solution Tank 30L
The frit-and-sleeve sparger, depicted in Figure 2, consists of a porous ring surrounded by a sleeve forming a gap through which water is passed to produce shear and consequently modifying bubble size. This sparger provides another control degree-of-freedom to modify bubble size, the superficial water velocity (Jls ), i.e. the water flow rate through the gap divided by the cross sectional area of the gap: Qls Jls = (4) Agap
Water flow rate set point (Jls ref)
Air flow rate set point FC
Peristaltic pump FT
Air
Fig. 1. Laboratory column setup
where Qls is the volumetric water flow rate of liquid in the sparger and Agap is the cross sectional area of the gap. 20
13th IFAC MMM Cape Town, South Africa, August 2-4, 2010
The flow rate to the sparger is regulated by manipulating the speed of a gear pump with a PID controller (Moore 353). The set point of this feedback loop (Jls ref ) will be the manipulated variable to control the bubble size.
The Wiener model was identified in two steps; first, the static nonlinear element was determined by using steady-state information, then, the linear dynamic element was identified by using step response information. The technique used in either case is described hereafter.
To implement its feedback control, bubble size must be measured on-line. For diagnosis purposes, Sauter mean bubble diameter is usually calculated after a set of collected images have been processed by using the following equation: N X
d32 =
d3i
i=1 N X
To explore the static relationship between superficial water velocity in the sparger and Sauter mean bubble diameter, several experimental tests were conducted in the laboratory flotation column. Figure 4 contains three subplots, each for a given superficial gas velocity, i.e. 0.5, 0.8 and 1.1 cm/s. For a given superficial gas velocity, each subplot shows a monotone decreasing steady state relationship between superficial water velocity in the sparger and the Sauter mean bubble diameter for different frother concentrations. In general, it can be observed that the effect of frother concentration is to shift this decreasing relationship without significantly modifying its shape.
(5) d2i
i=1
where di is the equivalent diameter of bubble i and N is the number of counted bubbles. This calculation assumes a steady state condition, i.e. the data points (bubble sizes in this case) are sampled from a stationary bubble size density function and as such not suitable for control purposes. In this work an on-line estimation of the Sauter mean bubble diameter based on the bubble size density estimation is implemented (Maldonado et al., 2008).
Figure 5 shows the static gain (∆d32 /∆Jls ref ) corresponding to the nonlinear relationships shown in Figure 4 as a function of the superficial water velocity in the sparger. The selected nonlinear static model of the gain shown in Figure 5 corresponds to the following nonlinear steady-state relationship between Sauter mean diameter and superficial water velocity:
Data acquisition is performed by a HMI/SCADA software (iFIX) working under a Windows XP operating system. An Opto 22 I/O system is used to centralize sensor and actuator signals. A modified version of the McGill’s bubble viewer (Gomez and Finch, 2007) was implemented to measure bubble size. The image processing system is performed by a dedicated computer using Matlab’s Image Acquisition toolbox and ’Image J’ free license software which is used to process the collected images to obtain bubble sizes. A communication link between Matlab applications running on the control station and the image processing computer is implemented using Matlab’s Instrumentation toolbox under Ethernet UDP protocol.
−0.256 d32 = 3.706 · Jls − 0.226
Note that in steady-state Jls = Jls
Static gain ∆d32/∆Jls ref
0
Jls* Jls ref (t)
_ +
Jls ref (t)
G(s) G(0) = 1
v(t)
v(t) +
N( )
d32(t)
(6)
ref .
Model
-0.005
-0.01
-0.015
0
50
100
Nonlinear static gain
150 200 Jls ref (cm/s)
250
300
Fig. 5. Gain of the system vs superficial water velocity in the sparger
Fig. 3. Wiener model structure 3. NONLINEAR MODEL IDENTIFICATION
3.1 Static nonlinear element
A Wiener model is used to represent the dynamic relationship between superficial water velocity set point and Sauter mean diameter. Wiener models consist of a linear system in series with a nonlinear element, as shown in Figure 3. The linear system G(s) has a unitary static gain. The static gain of the system is characterized by a memory-less nonlinear function N (). Note that the operating point Jls ∗ is adequately subtracted and added thus allowing a linear transfer function to represent the dynamics of the system. For control design purposes, the nonlinear element must be selected in such a way that the nonlinearity function is invertible.
3.2 Linear dynamic element Figure 6 illustrates the identification procedure of the linear transfer function. The effect of the nonlinear static gain is removed and therefore, during the parameter estimation, the static gain of G(s) can be constrained to one. A simple first-order lag system with time delay was used to represent the dynamic behaviour between superficial water velocity in the sparger and the Sauter mean diameter. The following model was identified using step response information: 21
13th IFAC MMM Cape Town, South Africa, August 2-4, 2010
Fig. 4. - Static relationship between Sauter mean diameter and the superficial water velocity in the sparger for different frother concentration and superficial gas velocities. where n is selected to obtained a proper controller. The complimentary sensitivity function is given by:
Experimental data Jls ref (t)
d32(t)
T (s, λ) =
N( )-1 Jls*
+ + -
Jls ref (t)
v(t)
Identification
G(s)
Fig. 6. Identification of the linear dynamic element Q(s) =
−90s
G(s) =
e (90s + 1)
(7)
90s + 1 82.6s + 1
(11)
Obtaining Q(s) ≈ 1 is not a surprise. It means that the closed-loop dynamics is similar to the plant dynamics, which is usually a good trade-off between performance and robustness for industrial processes. For implementation purposes, this controller was discretized with a sampling time of 5 seconds.
Figure 7 compares the Wiener intermediate variable v(t), obtained by applying the inverse of the nonlinearity function to the measured d32 , and the estimated intermediate variable calculated with the above linear model. It can be seen that the Wiener model is able to capture reasonably well the nonlinear dynamic behaviour of the process.
5. EXPERIMENTAL RESULTS
4. IMC-BASED NONLINEAR CONTROL
To evaluate the control system performance, the laboratory flotation column was filled with a solution of 10 ppm frother concentration Dowfroth 250. Then, changes on superficial gas velocity and frother concentration were implemented to simulate disturbances affecting the bubble size control loop.
The IMC control structure is depicted in Figure 8. Assuming there is no uncertainty in the non-linear block, the static nonlinearity can then be completely removed by performing the inverse of the non-linearity. The classical internal model control approach leads to: ˜ −1 (s)M F (s) Q(s) = G
(10)
The maximum peak Mp tuning (Stryczek et al., 2000) is used to find the value of IMC filter λ for parametric uncertainty in the process model. The objective is to find the smallest IMC filter time-constant that assures that no closed-loop frequency response (complimentary sensitivity function) will have more than the specified maximum peak Mp = 1.05. Considering a 20% uncertainty in time delay and time constant and a 5% uncertainty in the static gain of the nominal process model, the following IMC controller was obtained:
v(t)
-
Q(s, λ) · G(s) ˜ 1 + Q(s, λ)[G(s) − G(s)]
Figure 9 shows the controlled variable, i.e, Sauter mean diameter, the manipulated variable, i.e, the superficial water velocity in the sparger, gas holdup and bubble surface area flux. A good tracking performance can be observed when the pump speed is not in a saturated condition. Approximately at 12000 seconds, a gas velocity step change from 0.5 to 0.8 cm/s was implemented. This originated an increase in bubble size, which was rapidly compensated by increasing superficial water velocity in
(8)
1 ˜ where G(s) = 90s+1 corresponds to the minimum phase part of the plant model and F (s) is a filter. It is usually selected as follows: 1 F (s) = (9) (λs + 1)n
22
13th IFAC MMM Cape Town, South Africa, August 2-4, 2010
Intermediate variable
350
Correlation coefficient: 0.983
300 250 200 150
v(t)estimated v(t)
100 50 0
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
Time (s)
Fig. 7. Validation of the model
d32 ref (t)
N( )-1
vref (t) +
Jls ref (t)
Jls ref (t)
Q(s)
+
d32(t)
Process
_
N( )-1
Jls* G(s)
_
+
v(t)
v(t)
Fig. 8. Nonlinear IMC based on Wiener model Setpoint
Gas velocity variation
Frother concentration variation
d32 (mm)
1 0.8
Jls ref (cm/s)
0.6 340 280 200 Saturation
100
εg (%)
10 8 6
Sb (s-1)
4 70 60 50 40 30 0
5000
10000
15000
20000
25000
Time (s)
Fig. 9. Feedback control results the sparger. Later on, at about 13000 seconds, the bubble size set-point was decreased to 0.6 mm, but this new set point was not achievable since the pump speed saturated. Finally, frother concentration was changed from 10 ppm to 20 ppm at around 16000 seconds. It can be seen that the bubble size was consequently affected (reduced), but the control system immediately reacted by decreasing the
superficial water velocity, which allowed the bubble size to return to its previous set point value. A significant correlation exists between gas hold-up and bubble surface area flux, initially suggesting that both variables carries similar information and consequently either variable could be used for control purposes.
23
13th IFAC MMM Cape Town, South Africa, August 2-4, 2010
6. DISCUSSION
Carvalho, T. and Dur˜ao, F. (2002). Control of a flotation column using fuzzy logic inference. Fuzzy Sets and Systems, 125, pp. 121–133. Cortes, G., Verdugo, M., Fuenzalida, R., Cerda, J., and Cubillos, E. (2008). Rougher flotation multivariable predictive control; Concentrator A-1 division Codelco Norte. In R.Kuyvenhoven, C. Gomez, and A. Casali (eds.), Proceedings of the V International Mineral Processing Seminar, Procemin2008, 315–325. Santiago, Chile. Finch, J. and Dobby, G. (1990). Column Flotation. Pergamon Press, Oxford. Gomez, C. and Finch, J. (2007). Gas dispersion in flotation cells. International Journal of Mineral Processing, 84, pp. 51–58. Gorain, B., Franzidis, J-P., and Manlapig, E. (1997). Studies on impeller type, impeller speed and air flow rate in an industrial-scale flotation cell - Part 4: Effect of bubble surface area flux on flotation performance. Minerals Engineering, 10 (4), pp. 367–379. Kracht, W., Gomez, C., and Finch, J. (2008). Controlling bubble size using a frit and sleeve sparger. Minerals Engineering, 21, pp. 660–663. Liu, J. and MacGregor, J. (2008). Froth-based modeling and control of flotation processes. Minerals Engineering, 21, pp. 642–651. Maldonado, M., Desbiens, A., and del Villar, R. (2009a). Potential use of model predictive control in optimizing the column flotation process. International Journal of Mineral Processing, 93, 26–33. Maldonado, M., Desbiens, A., del Villar, R., and Aguilera, R. (2009b). On-line estimation of frother concentration in flotation processes. In C. Gomez (ed.), 48th Conference of Metallurgists COM’09, 135–146. Conference Proceedings, Sudbury, Canada. Maldonado, M., Desbiens, A., del Villar, R., Girgin, E., and Gomez, C. (2008). On-line estimation of bubble size distributions using Gaussian mixture models. In C.G. R.Kuyvenhoven and A. Casali (eds.), V International Mineral Processing Seminar, Procemin, 389–398. Conference Proceedings, Santiago, Chile. Moilanen, J. and Remes, A. (2008). Control of the flotation process. In R.Kuyvenhoven, C. Gomez, and A. Casali (eds.), Proceedings of the V International Mineral Processing Seminar, Procemin2008, 305–313. Santiago, Chile. Nesset, J., Hernandez-Aguilar, J., Acuna, C., Gomez, C., and Finch, J. (2006). Some gas dispersion characteristics of mechanical flotation machines. Minerals Engineering, 19, pp. 807–815. O’Keefe, C., Viega, J., and Fernald, M. (2007). Application of passive sonar technology to mineral processing and oil sands application. In Proceedings of the 39th Annual Meeting of the Canadian Mineral Processors, CIM, 429–457. Ottawa, Canada. Persechini, M., Jota, F., Peres, A., and Gon¸calves, F. (2004). Control strategy for a column flotation process. Control Engineering Practice, 12, pp. 963–976. Stryczek, K., Laiseca, M., Brosilow, C., and Leitman, M. (2000). Tuning and design of single input, single output control systems for parametric uncertainty. AIChE Journal, 46 (8), pp. 1616–1631.
It has been shown that for controlling BSAF, reference values for superficial gas velocity and Sauter mean diameter must be provided. Different combinations of gas velocity and Sauter mean diameter could generate the same BSAF, thus an optimal combination of these variables producing a desired BSAF must be determined based on the desired metallurgical performance. Something similar occurs with gas hold-up, since different conditions of gas velocity, frother concentration and bubble size can generate the same gas hold-up value. As it is widely accepted that bubble size must play a role on flotation performance, one possible control approach is to regulate bubble size by modifying the superficial water velocity in the sparger and to regulate bubble surface area flux (or gas hold-up) by manipulating gas velocity. The evaluation of bubble size via Sauter mean diameter discards all the available information regarding the shape of the bubble size distribution such as multi-modal and tailing behaviour. Indeed, it is possible to generate very different bubble size distributions having the same d32 . Of course, controlling d32 remains possible even if the distribution is not narrow and unimodal. However, in that situation, it is probably not meaningful to control d32 . To deal with this situation, it is necessary to consider explicitly the shape of distribution on the control synthesis leading to the branch of control of distributions. Note that multimodal distributions arise when operating at low frother concentrations. In this study, frother concentration was considered to be an unknown disturbance. However, since it has a strong effect on bubble size, its use as a manipulated variable deserves to be explored. Moreover, for mechanical cells where no spargers are used, it becomes a more relevant variable to modify bubble size. 7. CONCLUSIONS This work explores the control of Sauter mean diameter as a first step towards the control of bubble surface area flux. A frit-and-sleeve sparger was implemented to allow the modification of the bubble size by injecting water through a gap. A Wiener model-based was identified and an IMC was proposed to control the Sauter mean diameter by modifying the superficial water velocity through the frit-andsleeve sparger. Good tracking performance and rejection of unmeasured changes in superficial gas velocity and frother concentration were obtained in a laboratory flotation column. REFERENCES Barria, A. and Valdebenito, M. (2008). Implementation of rougher flotation control system at Codelco Chile, Andina division. In R.Kuyvenhoven, C. Gomez, and A. Casali (eds.), Proceedings of the V International Mineral Processing Seminar, Procemin2008, 215–220. Santiago, Chile. Bergh, L. and Yianatos, J. (1993). Control alternatives for flotation columns. Minerals Engineering, 6 (6), pp. 631–642. 24
Nonlinear control of bubble size in a laboratory flotation column ´ Poulin ∗ , M. Maldonado ∗ , A. Desbiens ∗ , R. del Villar ∗∗ , E. ∗∗∗ A. Riquelme ∗
LOOP, Department of Electrical and Computer Engineering, Universit´e Laval, Qu´ebec City, Canada ∗∗ LOOP, Department of Mining, Metallurgical and Materials Engineering, Universit´e Laval, Qu´ebec City, Canada ∗∗∗ Department of Electrical Engineering, University of Concepcion, Chile Abstract: Gas dispersion properties have proven to be key variables of the flotation process. Among them, bubble surface area flux (BSAF) has been reported to linearly correlate with the flotation rate constant; therefore, it is a potential variable to achieve a desired metallurgical performance. BSAF can be represented as a combination of two other gas dispersion properties: superficial gas velocity and Sauter mean bubble diameter. Thus, controlling BSAF implies controlling bubble size and superficial gas velocity. This work focuses on the nonlinear control of the Sauter mean bubble diameter. Sauter bubble mean diameter was indirectly calculated from the bubble size distribution, estimated by using a Gaussian mixture model. To improve controllability, a so-called frit-and-sleeve sparger was installed to regulate bubble size independently from superficial gas velocity. With this device, the bubble size can be modified by manipulating the water flow rate circulating through the sleeve that surrounds the porous ring. A Wiener model is used to represent the dynamic relationship between the sleeve water flow rate and Sauter mean diameter. Wiener models consist of a linear system in series with a memory-less (static) nonlinear element. An IMC controller based on the identified Wiener model was implemented in a laboratory flotation column. Tracking performance and rejection of gas velocity and unmeasured frother concentration variations were then successfully evaluated. Keywords: Flotation column, Bubble surface area flux, control, Sauter mean diameter, Internal model control, Weiner model, Nonlinear control 1. INTRODUCTION
which have been widely applied (Barria and Valdebenito, 2008; Cortes et al., 2008; Moilanen and Remes, 2008), or froth appearance (Liu and MacGregor, 2008). The second approach aims at controlling some gas dispersion properties in the collection zone, such as gas hold-up and superficial gas velocity (Bergh and Yianatos, 1993; Carvalho and Dur˜ao, 2002; Persechini et al., 2004; Maldonado et al., 2009a). This approach has been motivated by the recent availability of industrial gas-dispersion sensors (Gomez and Finch, 2007; O’Keefe et al., 2007). This article extends previous works on control of some gas dispersion properties to the control of bubble size represented by the Sauter mean bubble diameter, a key step towards controlling the bubble surface area flux.
Flotation is a commonly used method for separating valuable minerals (metal containing) from useless mineral (gangue). Its performance is determined by the valuablemineral concentrate grade and recovery. Whereas the first of these two variables can be measured on-line using an X-ray on-stream analyzer (OSA), the latter must be estimated from steady-state material balance, which strongly limits its use for regulatory control purposes. Moreover, the long sampling times of these OSA devices, usually multiplexed, favour the use of a hierarchical control where secondary variables are controlled to reject the frequent disturbances occurring in this type of process. Recent studies (Gorain et al., 1997) have shown that the performance of a flotation device basically depends on three factors: the particle floatability, the froth recovery and the ”bubble surface area flux”, a combination of two other gas dispersion properties, superficial gas velocity and bubble size (Finch and Dobby, 1990). This finding suggests that for a well conditioned pulp, a given metallurgical performance can be achieved by modifying froth recovery and bubble surface area flux. Consequently, two control approaches have been proposed. The first focuses on controlling some froth characteristics, such as froth speed,
978-3-902661-73-9/10/$20.00 © 2010 IFAC
Bubble surface area flux can be mathematically expressed as follows:
Sb =
6Jg d32
(1)
where Jg is the superficial gas velocity and d32 is the Sauter mean bubble diameter. Since there is no uncertainty associated to equation 1, controlling BSAF implies controlling 19
10.3182/20100802-3-ZA-2014.00004
13th IFAC MMM Cape Town, South Africa, August 2-4, 2010
2. FLOTATION COLUMN SETUP
the ratio between superficial gas velocity and the Sauter mean bubble diameter.
The flotation column used in this work has three sections made of polycarbonate tubes for a total height of 5 m; the internal diameter of the bottom, intermediate and upper sections are respectively 15.24 cm, 10.16 cm, and 5.08 cm. Recirculation is provided by a peristaltic pump. A frit-andsleeve sparger is mounted in the bottom part of the column as shown in Figure 1. Gas flow rate is measured through a mass flow sensor/controller (Aalborg model GFC17). The mass flow sensor also provides an estimate of the volumetric flow based on a reference condition (21.1 o C and 101.3 kPa). Its readings must then be converted to the actual tests conditions (temperature and pressure) measured by sensors shown in Figure 1 using the following equation:
In flotation column operation, bubble size is affected by frother type and concentration, gas rate and sparging system. Frother dosage regulation is usually implemented using a ratio feed-forward control based on the actual tonnage processed. Nevertheless, because of the limitations of feed-forward control in the presence of modelling uncertainties and unknown disturbances, such as frother persistency, evaporation rate and most importantly, the effect of reprocessed water (still containing some residual frother), frother concentration in a given flotation machine is hard to assess. In this work, frother concentration is considered as an unmeasured disturbance. Although the authors have proposed a method to evaluate on-line the frother concentration (Maldonado et al., 2009b). Since superficial gas velocity also modifies bubble size (Finch and Dobby, 1990; Nesset et al., 2006), it directly affects BSFA through the numerator of equation (1) and indirectly by affecting bubble size (denominator). Whenever feasible, the sparger system adds another control degree of freedom to modify bubble size in flotation columns. In this work, a ’frit-and-sleeve’ sparger was implemented (Kracht et al., 2008). This type of sparger allows the modification of bubble size independently from gas velocity, thus improving BSAF controllability.
" Jg =
1033.23 1033.23 + P
#"
T + 273.15 294.16
# (2)
where P is the absolute pressure measured in cmH2 O, T is the temperature in Celsius degrees and Jgref is the gas velocity calculated from the air mass flow meter lecture at reference conditions. A differential pressure transmitter DP (model ABB 264DS) was tapped between 250 cm and 320 cm above the sparger to measure gas holdup. For a two-phase system (air-water) this latter can be measured using the following relationship: ∆P εg (%) = 100 × (3) L where ∆P is the pressure differential in cmH2 O and L is the distance between taps (in this case 70 cm).
Section 2 describes the laboratory column and its instrumentation. The nonlinear Wiener model identification is detailed in section 3. The internal model control scheme is presented in section 4. Experimental results and discussion are found in sections 5 and 6. Windows XP® MatLab® Image J
Gap
Porous SS ring Sleeve
Video Sony frames camera Firewire
Ethernet / UDP
Windows XP® iFIX® MatLab®
DP
Jgref
Water inlet
TT
Frit and sleeve sparger
PT
RS232/ Modbus
Air inlet
I/O opto22
Fig. 2. Frit and sleeve sparger prototype Gear pump
FT FC Moore 353
Solution Tank 30L
The frit-and-sleeve sparger, depicted in Figure 2, consists of a porous ring surrounded by a sleeve forming a gap through which water is passed to produce shear and consequently modifying bubble size. This sparger provides another control degree-of-freedom to modify bubble size, the superficial water velocity (Jls ), i.e. the water flow rate through the gap divided by the cross sectional area of the gap: Qls Jls = (4) Agap
Water flow rate set point (Jls ref)
Air flow rate set point FC
Peristaltic pump FT
Air
Fig. 1. Laboratory column setup
where Qls is the volumetric water flow rate of liquid in the sparger and Agap is the cross sectional area of the gap. 20
13th IFAC MMM Cape Town, South Africa, August 2-4, 2010
The flow rate to the sparger is regulated by manipulating the speed of a gear pump with a PID controller (Moore 353). The set point of this feedback loop (Jls ref ) will be the manipulated variable to control the bubble size.
The Wiener model was identified in two steps; first, the static nonlinear element was determined by using steady-state information, then, the linear dynamic element was identified by using step response information. The technique used in either case is described hereafter.
To implement its feedback control, bubble size must be measured on-line. For diagnosis purposes, Sauter mean bubble diameter is usually calculated after a set of collected images have been processed by using the following equation: N X
d32 =
d3i
i=1 N X
To explore the static relationship between superficial water velocity in the sparger and Sauter mean bubble diameter, several experimental tests were conducted in the laboratory flotation column. Figure 4 contains three subplots, each for a given superficial gas velocity, i.e. 0.5, 0.8 and 1.1 cm/s. For a given superficial gas velocity, each subplot shows a monotone decreasing steady state relationship between superficial water velocity in the sparger and the Sauter mean bubble diameter for different frother concentrations. In general, it can be observed that the effect of frother concentration is to shift this decreasing relationship without significantly modifying its shape.
(5) d2i
i=1
where di is the equivalent diameter of bubble i and N is the number of counted bubbles. This calculation assumes a steady state condition, i.e. the data points (bubble sizes in this case) are sampled from a stationary bubble size density function and as such not suitable for control purposes. In this work an on-line estimation of the Sauter mean bubble diameter based on the bubble size density estimation is implemented (Maldonado et al., 2008).
Figure 5 shows the static gain (∆d32 /∆Jls ref ) corresponding to the nonlinear relationships shown in Figure 4 as a function of the superficial water velocity in the sparger. The selected nonlinear static model of the gain shown in Figure 5 corresponds to the following nonlinear steady-state relationship between Sauter mean diameter and superficial water velocity:
Data acquisition is performed by a HMI/SCADA software (iFIX) working under a Windows XP operating system. An Opto 22 I/O system is used to centralize sensor and actuator signals. A modified version of the McGill’s bubble viewer (Gomez and Finch, 2007) was implemented to measure bubble size. The image processing system is performed by a dedicated computer using Matlab’s Image Acquisition toolbox and ’Image J’ free license software which is used to process the collected images to obtain bubble sizes. A communication link between Matlab applications running on the control station and the image processing computer is implemented using Matlab’s Instrumentation toolbox under Ethernet UDP protocol.
−0.256 d32 = 3.706 · Jls − 0.226
Note that in steady-state Jls = Jls
Static gain ∆d32/∆Jls ref
0
Jls* Jls ref (t)
_ +
Jls ref (t)
G(s) G(0) = 1
v(t)
v(t) +
N( )
d32(t)
(6)
ref .
Model
-0.005
-0.01
-0.015
0
50
100
Nonlinear static gain
150 200 Jls ref (cm/s)
250
300
Fig. 5. Gain of the system vs superficial water velocity in the sparger
Fig. 3. Wiener model structure 3. NONLINEAR MODEL IDENTIFICATION
3.1 Static nonlinear element
A Wiener model is used to represent the dynamic relationship between superficial water velocity set point and Sauter mean diameter. Wiener models consist of a linear system in series with a nonlinear element, as shown in Figure 3. The linear system G(s) has a unitary static gain. The static gain of the system is characterized by a memory-less nonlinear function N (). Note that the operating point Jls ∗ is adequately subtracted and added thus allowing a linear transfer function to represent the dynamics of the system. For control design purposes, the nonlinear element must be selected in such a way that the nonlinearity function is invertible.
3.2 Linear dynamic element Figure 6 illustrates the identification procedure of the linear transfer function. The effect of the nonlinear static gain is removed and therefore, during the parameter estimation, the static gain of G(s) can be constrained to one. A simple first-order lag system with time delay was used to represent the dynamic behaviour between superficial water velocity in the sparger and the Sauter mean diameter. The following model was identified using step response information: 21
13th IFAC MMM Cape Town, South Africa, August 2-4, 2010
Fig. 4. - Static relationship between Sauter mean diameter and the superficial water velocity in the sparger for different frother concentration and superficial gas velocities. where n is selected to obtained a proper controller. The complimentary sensitivity function is given by:
Experimental data Jls ref (t)
d32(t)
T (s, λ) =
N( )-1 Jls*
+ + -
Jls ref (t)
v(t)
Identification
G(s)
Fig. 6. Identification of the linear dynamic element Q(s) =
−90s
G(s) =
e (90s + 1)
(7)
90s + 1 82.6s + 1
(11)
Obtaining Q(s) ≈ 1 is not a surprise. It means that the closed-loop dynamics is similar to the plant dynamics, which is usually a good trade-off between performance and robustness for industrial processes. For implementation purposes, this controller was discretized with a sampling time of 5 seconds.
Figure 7 compares the Wiener intermediate variable v(t), obtained by applying the inverse of the nonlinearity function to the measured d32 , and the estimated intermediate variable calculated with the above linear model. It can be seen that the Wiener model is able to capture reasonably well the nonlinear dynamic behaviour of the process.
5. EXPERIMENTAL RESULTS
4. IMC-BASED NONLINEAR CONTROL
To evaluate the control system performance, the laboratory flotation column was filled with a solution of 10 ppm frother concentration Dowfroth 250. Then, changes on superficial gas velocity and frother concentration were implemented to simulate disturbances affecting the bubble size control loop.
The IMC control structure is depicted in Figure 8. Assuming there is no uncertainty in the non-linear block, the static nonlinearity can then be completely removed by performing the inverse of the non-linearity. The classical internal model control approach leads to: ˜ −1 (s)M F (s) Q(s) = G
(10)
The maximum peak Mp tuning (Stryczek et al., 2000) is used to find the value of IMC filter λ for parametric uncertainty in the process model. The objective is to find the smallest IMC filter time-constant that assures that no closed-loop frequency response (complimentary sensitivity function) will have more than the specified maximum peak Mp = 1.05. Considering a 20% uncertainty in time delay and time constant and a 5% uncertainty in the static gain of the nominal process model, the following IMC controller was obtained:
v(t)
-
Q(s, λ) · G(s) ˜ 1 + Q(s, λ)[G(s) − G(s)]
Figure 9 shows the controlled variable, i.e, Sauter mean diameter, the manipulated variable, i.e, the superficial water velocity in the sparger, gas holdup and bubble surface area flux. A good tracking performance can be observed when the pump speed is not in a saturated condition. Approximately at 12000 seconds, a gas velocity step change from 0.5 to 0.8 cm/s was implemented. This originated an increase in bubble size, which was rapidly compensated by increasing superficial water velocity in
(8)
1 ˜ where G(s) = 90s+1 corresponds to the minimum phase part of the plant model and F (s) is a filter. It is usually selected as follows: 1 F (s) = (9) (λs + 1)n
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13th IFAC MMM Cape Town, South Africa, August 2-4, 2010
Intermediate variable
350
Correlation coefficient: 0.983
300 250 200 150
v(t)estimated v(t)
100 50 0
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
Time (s)
Fig. 7. Validation of the model
d32 ref (t)
N( )-1
vref (t) +
Jls ref (t)
Jls ref (t)
Q(s)
+
d32(t)
Process
_
N( )-1
Jls* G(s)
_
+
v(t)
v(t)
Fig. 8. Nonlinear IMC based on Wiener model Setpoint
Gas velocity variation
Frother concentration variation
d32 (mm)
1 0.8
Jls ref (cm/s)
0.6 340 280 200 Saturation
100
εg (%)
10 8 6
Sb (s-1)
4 70 60 50 40 30 0
5000
10000
15000
20000
25000
Time (s)
Fig. 9. Feedback control results the sparger. Later on, at about 13000 seconds, the bubble size set-point was decreased to 0.6 mm, but this new set point was not achievable since the pump speed saturated. Finally, frother concentration was changed from 10 ppm to 20 ppm at around 16000 seconds. It can be seen that the bubble size was consequently affected (reduced), but the control system immediately reacted by decreasing the
superficial water velocity, which allowed the bubble size to return to its previous set point value. A significant correlation exists between gas hold-up and bubble surface area flux, initially suggesting that both variables carries similar information and consequently either variable could be used for control purposes.
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6. DISCUSSION
Carvalho, T. and Dur˜ao, F. (2002). Control of a flotation column using fuzzy logic inference. Fuzzy Sets and Systems, 125, pp. 121–133. Cortes, G., Verdugo, M., Fuenzalida, R., Cerda, J., and Cubillos, E. (2008). Rougher flotation multivariable predictive control; Concentrator A-1 division Codelco Norte. In R.Kuyvenhoven, C. Gomez, and A. Casali (eds.), Proceedings of the V International Mineral Processing Seminar, Procemin2008, 315–325. Santiago, Chile. Finch, J. and Dobby, G. (1990). Column Flotation. Pergamon Press, Oxford. Gomez, C. and Finch, J. (2007). Gas dispersion in flotation cells. International Journal of Mineral Processing, 84, pp. 51–58. Gorain, B., Franzidis, J-P., and Manlapig, E. (1997). Studies on impeller type, impeller speed and air flow rate in an industrial-scale flotation cell - Part 4: Effect of bubble surface area flux on flotation performance. Minerals Engineering, 10 (4), pp. 367–379. Kracht, W., Gomez, C., and Finch, J. (2008). Controlling bubble size using a frit and sleeve sparger. Minerals Engineering, 21, pp. 660–663. Liu, J. and MacGregor, J. (2008). Froth-based modeling and control of flotation processes. Minerals Engineering, 21, pp. 642–651. Maldonado, M., Desbiens, A., and del Villar, R. (2009a). Potential use of model predictive control in optimizing the column flotation process. International Journal of Mineral Processing, 93, 26–33. Maldonado, M., Desbiens, A., del Villar, R., and Aguilera, R. (2009b). On-line estimation of frother concentration in flotation processes. In C. Gomez (ed.), 48th Conference of Metallurgists COM’09, 135–146. Conference Proceedings, Sudbury, Canada. Maldonado, M., Desbiens, A., del Villar, R., Girgin, E., and Gomez, C. (2008). On-line estimation of bubble size distributions using Gaussian mixture models. In C.G. R.Kuyvenhoven and A. Casali (eds.), V International Mineral Processing Seminar, Procemin, 389–398. Conference Proceedings, Santiago, Chile. Moilanen, J. and Remes, A. (2008). Control of the flotation process. In R.Kuyvenhoven, C. Gomez, and A. Casali (eds.), Proceedings of the V International Mineral Processing Seminar, Procemin2008, 305–313. Santiago, Chile. Nesset, J., Hernandez-Aguilar, J., Acuna, C., Gomez, C., and Finch, J. (2006). Some gas dispersion characteristics of mechanical flotation machines. Minerals Engineering, 19, pp. 807–815. O’Keefe, C., Viega, J., and Fernald, M. (2007). Application of passive sonar technology to mineral processing and oil sands application. In Proceedings of the 39th Annual Meeting of the Canadian Mineral Processors, CIM, 429–457. Ottawa, Canada. Persechini, M., Jota, F., Peres, A., and Gon¸calves, F. (2004). Control strategy for a column flotation process. Control Engineering Practice, 12, pp. 963–976. Stryczek, K., Laiseca, M., Brosilow, C., and Leitman, M. (2000). Tuning and design of single input, single output control systems for parametric uncertainty. AIChE Journal, 46 (8), pp. 1616–1631.
It has been shown that for controlling BSAF, reference values for superficial gas velocity and Sauter mean diameter must be provided. Different combinations of gas velocity and Sauter mean diameter could generate the same BSAF, thus an optimal combination of these variables producing a desired BSAF must be determined based on the desired metallurgical performance. Something similar occurs with gas hold-up, since different conditions of gas velocity, frother concentration and bubble size can generate the same gas hold-up value. As it is widely accepted that bubble size must play a role on flotation performance, one possible control approach is to regulate bubble size by modifying the superficial water velocity in the sparger and to regulate bubble surface area flux (or gas hold-up) by manipulating gas velocity. The evaluation of bubble size via Sauter mean diameter discards all the available information regarding the shape of the bubble size distribution such as multi-modal and tailing behaviour. Indeed, it is possible to generate very different bubble size distributions having the same d32 . Of course, controlling d32 remains possible even if the distribution is not narrow and unimodal. However, in that situation, it is probably not meaningful to control d32 . To deal with this situation, it is necessary to consider explicitly the shape of distribution on the control synthesis leading to the branch of control of distributions. Note that multimodal distributions arise when operating at low frother concentrations. In this study, frother concentration was considered to be an unknown disturbance. However, since it has a strong effect on bubble size, its use as a manipulated variable deserves to be explored. Moreover, for mechanical cells where no spargers are used, it becomes a more relevant variable to modify bubble size. 7. CONCLUSIONS This work explores the control of Sauter mean diameter as a first step towards the control of bubble surface area flux. A frit-and-sleeve sparger was implemented to allow the modification of the bubble size by injecting water through a gap. A Wiener model-based was identified and an IMC was proposed to control the Sauter mean diameter by modifying the superficial water velocity through the frit-andsleeve sparger. Good tracking performance and rejection of unmeasured changes in superficial gas velocity and frother concentration were obtained in a laboratory flotation column. REFERENCES Barria, A. and Valdebenito, M. (2008). Implementation of rougher flotation control system at Codelco Chile, Andina division. In R.Kuyvenhoven, C. Gomez, and A. Casali (eds.), Proceedings of the V International Mineral Processing Seminar, Procemin2008, 215–220. Santiago, Chile. Bergh, L. and Yianatos, J. (1993). Control alternatives for flotation columns. Minerals Engineering, 6 (6), pp. 631–642. 24