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I. INTRODUCTION

Froth flotation is a complex physico-chemical process that utilizes natural and induces hydrophobicity by chemical reactors dosage to separate and collect valuable particles from slurry [2].

Flotation is the most widely used separation process in the mineral processing industry today. The importance of flotation in the global economy is significant. The quantity of mineral treated by flotation is about nine billions tons per day. The 95% of base metals are processed by this technique [1].

Control of the flotation process is an important aspect of plant optimization. The control and feedback typically is carried out by operators based on on-stream analyser and traditional controllers as PID controller for pulp level.

Base level flotation control is focused on maintaining primary variables at setpoints. These primary variables include: pulp level, air flowrate and reagent addition rate. This

This paragraph of the first footnote will contain the date on which you submitted your paper for review. It will also contain support information, including sponsor and financial support acknowledgment. For example, “This work was supported in part by the U.S. Department of Commerce under Grant BS123456”.

The next few paragraphs should contain the authors’ current affiliations, including current address and e-mail. For example, F. A. Author is with the National Institute of Standards and Technology, Boulder, CO 80305 USA (e-mail: author@ boulder.nist.gov).

S. B. Author, Jr., was with Rice University, Houston, TX 77005 USA. He is now with the Department of Physics, Colorado State University, Fort Collins, CO 80523 USA (e-mail: author@lamar.colostate.edu).

T. C. Author is with the Electrical Engineering Department, University of Colorado, Boulder, CO 80309 USA, on leave from the National Research Institute for Metals, Tsukuba, Japan (e-mail: author@nrim.go.jp).

is generally achieved through the usage of conventional SISO PID control; although more advanced methods are now commonly used in modern control strategies. Similarly, traditional base level flotation control was applied to single cells, although modern control strategies are now regularly applied to entire banks of cells (e.g. pulp level control) (BJ Shean, JJ cilliers areview of froth flotation control, international journal of mineral processing, Elsevier 2011 ed 100, p 57-71).

Considering that the flotation process is subject to a wide variety of disturbances, several regularity control strategies have been proposed, sucha as, adaptative control using lienar SISO models, multivariate models and nonlinear models.

Control of the plant dynamics can be achieved by proceeding the plant with an adaptative controller whose transfer function is the inverse of that plant [5]. This paper instead considers a technique called adaptive inverse control [6] [7] [8]—as reformulated by Widrow and Walach [9]— which does not require a precise initial plant model. Like feedback linearization, adaptive inverse control is based on the concept of dynamic inversion, but an inverse need not exist.

In previous work the laboratory of “Gestión en tecnologías de la información y comunicación” (GETIC) has worked with neural predistortion technique in order to linearize the power amplifier output for mobile satellite communications [10]. This technique widely applied to communication problems is applied to assess the control of a real dynamic process.

This paper shows the application of an adaptative pre distorter to control a flotation plant. The controller will be tested using a flotation model from fundamental equation as well as empirical [11].This paper is organized as it follows. First the actual process control and state of the art of flotation control and instrumentation is presented in section 2. Section 3 presents the main equation of the model and its parameters. In section 4 pre distortion technique is describe and how it deals with nonlinear plants. In section 5 implementation and simulation results are given.

II.BACKGROUND OF PROCESS CONTROL APLLIED TO FLOTATION BANKS

Flotation process is highly complex.Arbiter and harris (1962) estimating that there are approximatelu 100 variables that affect the flotation process. Moreover co interactions between variables further complicate control efforts. Increase air flowrate for example may result in a larger bubble size, wich will subsequently affect the bubble gas velocity, rate of

Controller based on predistortion for flotation bank

Fabián Seguel1, Nicolas Krommenacker,Ismael Soto1, Miguel Maldonado, MillarayCurilem2and Nestor Becerra3

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attachment, gas holdup, erc. This high number of variables makes the flotation control very complex.Wills and Napier 2006 suggest that if the process is running perfectly, variations in feed rate, pulp density, etc. may be compensated.To perform control on flotation banks information about disturbances, process operating parameters and product quality is required. There are two main manipulated variables in flotation control. The valve position for level control and the air flow rate. Stenlund and Medvedev (2000); Kämpjärvi and Jämsä-Jounela (2003); and Carr et al. (2009) report that flotation cells traditionally use feedback PI control to ensure pulp levels remain at desired set-points; although Wills and Napier-Munn (2006) add that feedforward control is also regularly integrated to account for flowrate variations upstream. This is achieved bymanipulating the tailings flow from the cell by adjustment of the slurry outlet valve. This technique is effective for the control of isolated cells, but is also commonly used to control a bank of cells in series. As such, sophisticated multivariable model-based control methods have been developed; Despite this, Carr et al. (2009) reports that these “more sophisticated methods… are rarely used in industrial processing plants.” Commercial control packages such as FloatStar Level Stabiliser by Mintek, or CellStation by Outotec, also aim to control all cell levels simultaneously using advanced control techniques. Both have reportedly been trialled successfully and installed on industrial plants (www.mintek.co.za; www.outotec.com).

Wills and Napier-Munn (2006) elaborate just how important and useful aeration control is; adding that “flotation generally responds faster to changes in aeration, than to changes in froth depth, and because of this aeration is often a more effective control variable”. Similarly, they infer (by use of an example) that in comparison to reagent addition, air is by far the “cheaper ‘reagent’ and leaves no residual concentration if used in excess.”

III. PLANT MODEL In order to perform the proposed controller a formulaltion of a model from fundamental equation as well as empirical relationships is developed. All the equations that will be descused corresponds to a single cell model. A cell bank is created by connecting individual cells models in serie.

The implementation of this model and model responses will be given.

Material enters to the cell in the pulp phase as feed and pulpo ut of the cell as tailings. Valuable mineral leaves the pulp phase by true flotation or entrainment. Gangue leaves the pulp phase just by entrainment.

In this case. The model involved two clases of mineral definded as

i=¿ (2) Gangue(1)Copper ¿

The componen balances for the minerals in the cell are shown in the equation XX

d M i , p

dt=mi , feed−mi ,tailings−mflotation−mentraiment+mdrop back [ kg

s ] Each component of the mass balance of the pulp is calculed by the following relationships

mi ,tailings=Ci , P Qtailings[ kgs ]

The pulp phase concentration as well as volumetric flow of tailings can be calculates from equations XX and XX respectively. In this case we asume that each cell is connected to the next by a control valve.

C i ,P=M i , P

V P [ kgm3 ]

Qtailings=K valve f ( x ) √g L2−L1[ m3

s ] The fraction of mineral leaving the pulp fase by true flotation is determined by

mi , flotation=V pr overall[ kgs ]

The volumen of the pulp phase is given by four components. Valuable mineral, gangue, wáter and air. The contribution of the first three component is given by the equation XX. The pulp level in the cell will be higher tan the level calculares from XX due the air holdup.

V P=∑i=1

n M i , p

ρi+

M w , p

ρw[m3 ]

V p , t=V p

1−∈g[m3 ]

Lp=V p ,t

Ac[ m ]

The pulp phase contains material leaving the pulp phase by flotation and re-entering material through drop back. It is posible conbine these terms and model net trate at wich material leaves the pulp phase. The collection zone constant is given by the equation XX. The flotation probability of wach class of material is given in the apendix A. It’s used to model the rate at wich particles attach to bubbles in the pulp phase. The overall flotation rate is given by the equation XX.

k overall=P Sb Rf [1s ]

roverall=koverall Ci , p[ kgm3 s ]

The fraction of each material recovered by entrainment can be determined using the equation XXmentrainment=kw CF M i , p (4.23)

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Rate recovery of water can be assumed to be proportional to the mass of water in the pulp phase (Lynch et al, 1981:74) The classification function is given by the equation XX.

CF= 11+w τ f

(4.21)

The following empirical relationship given by Gorain et al 1999 is used to calculate the bubble surface área flux.The values of the constants are contained in the apendix A.

Sb=a N sb(Q

A )c

A sd P80

e (4.17)

The relationshio given by Vera et all 2002 was used to predict thefroth recovery factor

R f ,i=e−β τ f+ (1−e−β τ f ) ( 11+wi τ f ) (4.18)

The correlation requieres froth retention time calculed by the level of froth in the cell and volumetric air flow.

τ f=A Lf

Q(4.19)

Lf =LC−Lp (4.20)

A mass flow can also be calculated for water as shown in equation XXd M w

dt=mw , feed−mw ,tailings−mw , entrainment (4.9)

Each componen of the mass balance is detailed next.The water mass that leaves the pulp in the tailings is determined by XXmw ,tailings=Cw, pQtailings (4.10)Concentration of water is determined by XX. The volumetric flow of tailings was described above.

Cw , p=M w , p

V p(4.13)

Water can only leave the pulp into concentrate by entrainment.mwentrainment=rw , entrainment V p (4.12)In this case it was assumed that k w is not a function of froth level.rw , entraiment=kw M w, p (4.11)The cell parameters are shown in the table 1 with their respective units.

ParameterSymbol Value Unit a

Mineral Flotability P1(mineral) 0,0025 Dimensionless

Mineral Flotability P2(ganga) 0 Dimensionless

Bubble surface área flux constant

a 137,5 Dimensionless

Bubble surface área flux constant

b 0,2905 Dimensionless

Bubble surface área flux constant

c 0,7278 Dimensionless

Bubble surface área flux constant

d 0,0685 Dimensionless

Bubble surface área e 0,3551 Dimensionless

flux constantDrainage Parameter w 1 [ 1

min ]Froth Parameter β 4 [1/min]Parameter in relationship for flow between cells

K flow 0,05 m4,5 skg

Valve constant K válvula enRougher0,12 Dimensionless

Valve constant K válvula enCleaner0,175 Dimensionless

Impeller aspect ratio A s 0,75 Dimensionless

Impeller perpheral speed

N s 0,75 [ms ]

80% Passing feed size P80 80 [ μm ]Cell cross section área A 1,875 [m2]Cell height h 1,75 [m]

IV. PREDISTORTION This paper introduces a neural network adaptive pre-distortion technique wich is commonly used in HPA to compensate nonlineatities. The technique adapts to changes in the nonlineality of the process.Several other solutions have been applied to the problem. These techniques tend to require a larger, heavier, more expensive and, be less efficient. The digital mapping techniques have been limited by the massive amount of storage required for a sufficiently accurate mapping to be stored. The use of neural networks was first investigated by the autors in [6] as a digital baseband predistortion wich utilizes a multilayer perceptron to approximate the inverse of any amplifier response.

Como utilizare la tecnica y la contribución.

A. Neural networks

Neural Network training and Kalman Filter

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B. Controller based on predistortion for flotation bank

V. SIMULATION AND RESULTS

Curvas de entrenamiento de la red neuronal, velocidad de convergencia y error en la aproximación Tablas de

Indice de adecuaciónError RMSRSD (relative standard deviation)

Gráficos de celda con ruido a la entrada sin control y su respuesta

Gráfico de respuesta con controlTablas de RSD con y sin control y comparación

(disminución de la perturbación)Tabla de aumento de recuperación con y sin control

(aumento de beneficio)

The controller C is designed to optimally drive the plant output on a referent trajectory given by an operator. A controller was built by using a neural network to generate an adaptative predistorter. The controllers has the following variables as inputs:

Controller: Desired output, Feed grade (%), Federate (tons/s) and collector addition (g/min).

For the training a signal with noise was used for collector addition is shown in the figure XX the control signal.

VI. CONCLUSIONS

Inverse control is very simple yet highly effective. It can be optimized by adapting a disturbance canceler for closing the loop

Results show the benefits of using the feed forward action to attenuate the effect of typical flotation feed disturbances. By using predistortion technique the noise power is reduced making the controlled response closer to the desired response than the uncontrolled plant.

The feed disturbances are commonly measured but the configuration of feed forward control can’t prevent the effect of unmeasured disturbances. A disturbance canceller can be adapted for its purpose.

ACKNOWLEDGMENT

The preferred spelling of the word “acknowledgment” in American English is without an “e” after the “g.” Use the singular heading even if you have many acknowledgments. Avoid expressions such as “One of us (S.B.A.) would like to thank ... .” Instead, write “F. A. Author thanks ... .” In most cases, sponsor and financial support acknowledgments are placed in the unnumbered footnote on the first page, not here.

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