recent advances in bias and froth depth control in...
TRANSCRIPT
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
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
metallurgical performance, such as froth depth (H), bias (Jb), gas hold-up (εg), or bubble surface
area flux (Sb). 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, Jb, εg and Sb. 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, Jb, εg and Sb 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.
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.
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 short-
circuited 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 (kw), and
the liquid conductivity of the feed (k'f), tailings (k't), and concentrate (k'c) streams:
−
−−
−
−=
wf
wc
c
wf
tf
tb
k'k
k'k'J
k'k
'k'k'JJ (1)
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 Jc’ is difficult as a result of its high air content.
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 conductivity-
profile 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
(Rfroth) and the pulp (Rpulp), as
( )pulpfrothRxRxR −+= 1 (2)
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 (kfroth = Rfroth-1) and pulp (kpulp = Rpulp
-1), and
• the conductivity of the first and last two cells (k1, k2, k9 and k10), 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 pilot-
scale 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
sensor (Hmeasured) with a visual measurement (transparent column) (Hreal). The detailed algorithm
can be found in Desbiens et al. (1998) and del Villar et al. (1999).
30 35 40 45 50 55 60 65 7030
35
40
45
50
55
60
65
70
Hreal
(cm)
Hm
ea
su
red (
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:
• k1 and k2, the conductivities of the first two cells,
• k9 and k10, the conductivities of the last two cells,
• kf and kw, the feed and wash-water conductivities, and
• Jw and Jg, the wash-water and air superficial velocities, respectively.
A comparison of the predictions made with a regression model (Jbmodel) 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 8.1TM (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.
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
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
( ) ( )( )( )
( )( )( )
( )tezD
zCtuz
zF
zBtyzA
1
1
d
1
1
1
−
−
−
−
−
−
+= (2)
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:
( )( )( )( )( ) nf
nf
nd
nd
nc
nc
nb
nb
na
na
zfzfzF
zdzdzD
zczczC
zbzbzB
zazazA
−−−
−−−
−−−
−−−
−−−
+++=
+++=
+++=
++=
+++=
...1
...1
...1
...
...1
1
1
1
1
1
1
11
1
1
1
1
1
1
1
(3)
where the parameters ai, bi, ci, di and fi 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
(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 (Kv)
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 wash-
water flow rates.
For an easier representation of MIMO (multiple input- multiple output), state-space equations
are recommended:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )ttttt
tttt
ννννωωωω
ωωωω
+++=
++=+
EuDxCy
uBxAx 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 ωωωω(t) and νννν(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 ADAPTxTM (Adaptics Inc.)
(Larimore, 1999). The computational and theoretical basis of the method relies on the singular
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
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 (ysp) 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.
ProcessFeedbackController
d
yysp
u
FeedforwardController
+ -
+
+
Figure 5 - Feedforward control using a measurable disturbance
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.
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 (KP), integral (KI) and differential (KD) – 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.
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
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
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 on-
line 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
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)
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,
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).
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).
Column
flotation
process
Wash water flow
rate setpoint
Tail flow rate
setpoint
Bias
Froth
depth
Bias
setpoint
Froth depth
setpoint-
+
Bias model
Froth depth model
Bias feed forward models
Froth depth feed forward models
Bias tracking controller
Froth depth tracking controller
Bias feed forward controller
Froth depth feed forward controller
Bias feedback controller
Froth depth feedback controller
+
-
-
-
+
+
+-
-
+
-
FEED FORWARD
TRACKING
FEEDBACK
-
Measured disturbances
(feed and air flow rates)
Unmeasured
disturbances
Figure 9 - GlobPC structure applied to the column flotation process.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00
Jb
cm
/s
Bias set point
Linear control
0
0.05
0.1
0.15
0.2
0.25
0.3
0.00 16.00 32.00 48.00 64.00 80.00
Jb
cm
/s
Time (min)
Nonlinear control
Bias set point
0
0.05
0.1
0.15
0.2
0.25
0.3
0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00
Jb
cm
/s
Bias set point
Linear control
0
0.05
0.1
0.15
0.2
0.25
0.3
0.00 16.00 32.00 48.00 64.00 80.00
Jb
cm
/s
Time (min)
Nonlinear control
Bias set point
Figure 10 - Comparison of bias linear and nonlinear control
0 10 20 30 40 50 60 70 80 90
30
40
50
60
70
0 10 20 30 40 50 60 70 80 90
0.04
0.06
0.08
0.1
0.12
0.14
0 10 20 30 40 50 60 70 80 900
0.5
1
1.5
Tailings flow rate set point
Time (min)
Wash water flow rate set pointu (cm/s)
J b(cm/s)
H (cm)
0 10 20 30 40 50 60 70 80 90
30
40
50
60
70
0 10 20 30 40 50 60 70 80 90
0.04
0.06
0.08
0.1
0.12
0.14
0 10 20 30 40 50 60 70 80 900
0.5
1
1.5
Tailings flow rate set point
Time (min)
Wash water flow rate set pointu (cm/s)
J b(cm/s)
H (cm)
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
performance. A tighter respect of the set-point is made possible by the feedforward action, with a
gentle adjustment of the tailings flow rate.
PIBacksteppingMRNC
0 50 100 150 20040
42
44
46
0 50 100 150 2000.2
0.4
0.6
0.8
0 50 100 150 2000.2
0.4
0.6
Time [seconds]
Feed flowrate [cm/s]
Tailings flowrate [cm/s]
Froth depth [cm]
PIBacksteppingMRNC
0 50 100 150 20040
42
44
46
0 50 100 150 2000.2
0.4
0.6
0.8
0 50 100 150 2000.2
0.4
0.6
Time [seconds]
Feed flowrate [cm/s]
Tailings flowrate [cm/s]
Froth depth [cm]
Figure 12 - Froth depth nonlinear control (feed flow rate disturbance)
0 50 100 150 200 250
30
35
40
45
0 50 100 150 200 2500
0.5
1
0 50 100 150 200 2500.6
0.8
1
Time [seconds]
Air flowrate [cm/s]
Tailings flowrate [cm/s]
Froth depth [cm]
PIBacksteppingMRNC
0 50 100 150 200 250
30
35
40
45
0 50 100 150 200 2500
0.5
1
0 50 100 150 200 2500.6
0.8
1
Time [seconds]
Air flowrate [cm/s]
Tailings flowrate [cm/s]
Froth depth [cm]
PIBacksteppingMRNC
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
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, Jb, εg and Sb) 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).
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