Dynamic characterization of column flotation process laboratory case study

Minerals Engineering, Vol. 12, No. 11, pp. 1339-1346, 1999

Pergamon 0892-6875(99)00121-1

© 1999 Published by Elsevier Science Ltd All rights reserved 0892-6875/99/$ - see front matter

DYNAMIC CHARACTERIZATION OF COLUMN FLOTATION PROCESS LABORATORY CASE STUDY* T. CARVALHO, F. DURAO AND C. FERNANDES CVRM - Centro de Valoriza~lo de Recursos Minerais, Technical University of Lisbon, Portugal E-mail: [email protected]

(Received 17 November 1998; accepted 8 July 1999)

ABSTRACT

The objective of metallurgical column control is to achieve the economic optimum combination of concentrate grade and mineral recovery for any given feed. Metal prices and the cost of consumables will dictate the metallurgical concentrate and mineral recovery targets. Furthermore, experience has shown that collection zone height, air holdup and bias water flow rate are key parameters (controlled variables) that affect metallurgical performance. The collection zone height is related, mainly, with the residence time of the particles inside the collection zone, air holdup is correlated to the available surface area of air bubbles and bias water flow rate is an indirect measure of the cleaning action of the froth zone. However, these variables cannot be directly manipulated Instead, wash water, air and underflow flowrates are the directly manipulated variables [1]. Therefore, if dynamic relationships could be established between the three manipulated variables and the three controlled variables, column metallurgical control may be improved This study was an attempt to model these relationships using a tool known as System Identification, that includes Transient Analysis. The study consists in experimental tests, transient response analysis and identification of black box type models with cross validation. The experimental work was performed in a pilot scale laboratory flotation column of 3.2m high by 80ram of diameter. This flotation column operates in a plant with all the required instrumentation installed The study considers the operation of the two phase air-water system. © 1999 Published by Elsevier Science Ltd. All rights reserved

Keywords Column flotation; modeling; process control

INTRODUCTION Up to now, it was not possible to develop a phenomenological dynamic model of the column flotation process that can be used in its control, due to the following reasons [2]:

* Presented at Minerals Engineering ~9, Edinburgh, Scotland, September 1999

1339

T. Carvalhoet al.

1340 • • • • •

difficulty to describe heterogeneous particulate systems; lack of accurate knowledge of the physics and chemistry laws of the sub-processes involved; need to consider the axial position, leading to the formulation of distributed parameters models; need to solve a mathematical problem of a moving boundary (collection/froth zones interface); difficulty in obtaining accurate measures of most process variables.

To overcome the difficulties in the development of phenomenological models, it is common practice to develop empirical models, based on experimental data [3]. The procedure for empirical dynamic model's development is called System Identification encompassing the following steps [4]: • • • •

select suitable manipulated variables; collect experimental data by performing simple experiments (disturbance of the manipulated variables) and record the response of the controlled variables; transient analysis of the process responses (experimental data). This step allows one to determine how the variables affect each other, the time constants, pure time delays and static gains; identification of black-box type models, adjusting parametric linear models (with a pre-def'med structure) to the experimental data. Cross validation of the adjusted models.

Transient analysis The preliminary phase of identification is a structuring task, called transient analysis. This task (that can be the only one used in the case of very simple processes) permits the estimation of some parameters of transient response such as time delays, time constants and steady state gains and some global dynamic characteristics as oscillatory or underdamped behavior.

Model identification When the physical system is much too complex to be decomposed and analyzed, one of the most common approaches is to use linear ready-made models [5], such as ARX, ARMAX, Box & Jenkins models, etc. Assuming that two discrete time series, y(t) and u(t), t=T, 2T,...,nT t are related by a linear system, the input~output relationship description in the time domain between the two signals can be written according to Eq. (1).

y(t) = G(q) u(t - nk) + v(t)

(1)

where q is the shift operator, G(q) is the transfer function of the system, v(t) is an additive unmeasurable disturbance and nk specifies the time delay from input to output as a multiple of the sampling period T. The disturbance v(t) is usually described as a filtered white noise as follows (Eq. (2)):

v(t)=H(q)

e(t)

(2)

where H(q) is the filter and e(t) is a series of independent stochastic variables, white noise, with mean value 0 and variance ~.

1 The observations are made at a fixed time interval T, the sampling period. Adopting T as the unit of time, one can simply write t=l,2 ..... n.

Dynamic characterizationof columnflotationprocess.Laboratorycase study

1341

Functions G(q) and H(q) can be described as rational functions of the delay operator q-l. One of the most common model structures is the ARX structure def'med by Eq. (3) or (4) 2

A(q) y(t)= B(q) u(t-nk) + e(t)

(3)

y(t)- B(q) u(t-nk) +

(4)

A(q)

1_~ e(t)

A(q)

Equation (6) is a full development of Eq. (3) presented as a difference equation.

y(~+atq-1y(t)+a2q-2y(t)+...+an°q-~°y(t)=b1q-1u(t-nk)+b2q-2u(t-nk)+...+bn~q-"~u(t-nk)+e(t)

(6)

As q-ly(t) represents y(t-1), one can write Eq. (7)

y(t)+a1y(t-~)+a2y(t-2)+...+a"°y(t-na)=bJu(t-nk-~)+b2u(t-nk-2)+...+b"~u(t-nk-nb)+e(t)

(7)

Other common models are given by Eq. (8) (ARMAX structure) and Eq. (9) (Box-Jenkins model).

y(t)- B(q) u(t-nk) + C(q) e(t)

(8)

y(t)- B(q) u(t-nk)+ C(q) e(t) F(q) D(q)

(9)

A(q}

A(q}

Model OE (output error), as given in Eq. (10), is a special case of the Box-Jenkins model, where noise is not modeled.

y (t) - B (q) u (t _ nk) + e (t) F(cl)

(10)

EXPERIMENTAL WORK Equipment description The experimental work was carried out in a laboratory scale pilot flotation column of 3.2m high by 80mm of diameter. The flotation column is equipped with variable speed peristaltic pumps (manipulation of underflow and feed flow rates) and control valves (manipulation of air and wash water flow rates). The underflow and feed flow rates are measured by electromagnetic flowmeters, the wash water flow rate by a turbine meter and the air flow rate by an orifice plate (see Figure 1).

2 The terms A(q), B(q) ,C(q) and F(q) are polynomials in the delay operator q-i (Eq. 5)).

A (cl)= l + al q'~ +... + a,o q "ha B (q) = bl + b2 q-I +... + bn~ q-n~+l

(5)

C(q)= l+ clq'l +... + cncq -no F ( q ) = l + f jq'l +... + f n:q "n: where na, nb, nc and nfare the orders of the respective polynomials and ai (/--1,2 ..... ha), bi (/--1,2 ..... nb), ci (/--1,2 ..... n,.) a n d f (/=-1,2..... nf) are the adjustable model parameters.

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T. Carvalhoet al.

The collection zone height is calculated or inferred by means of a soft sensor that uses the measurements of two pressure sensors. These measurements are also used to estimate the air holdup in the collection zone. The bias water flow rate is calculated as the difference between wash water and overflow flow rates. A programmable logic controller (PLC), to which all the instrumentation is connected, and a personal computer (PC), connected with the PLC, allow data acquisition and the real time manipulation of actuators. The system is operated with air and water. The frother concentration in the feed stream (10 ppm of AerofToth 65) was superimposed on all tests.

Wash Water

LF,GI~D

Air

Overflow Uaderflow

Fig. 1 Schematic diagram of the laboratory flotation column.

Experimental design Experimental tests were performed according to the experimental design presented in Table 1. In each experimental test, the column was first driven to the desired steady state conditions. Then, a step change was superimposed to one of the manipulated variables: underflow (Qu), air (QA) and washing water (Qw) flow rates. Transient responses of the controlled variables: air holdup in the collection zone (ec), collection zone height (H) and bias water flow rate (QB), were recorded.

TABLE 1 Experimental design Negative Disturbances

Positive Disturbances To

From

To

54

42

30

120

120

100

240

240

180

From

To

From

Qww(Vh)

30

42

42

QtJ (l/h)

100

QA (I/h)

180

[

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The tests were performed with different steady state (initial) conditions in terms of air holdup, collection zone height and bias water flow rate. The study involved 29 experimental tests with positive (increase in the manipulated variable) and negative disturbances and different magnitudes. Figure 2 shows an example of a positive step disturbance introduced, in this case, on wash water flow rate, with a magnitude of 12 1/h. For the case of wash water flow rate, two different step magnitudes were performed. These magnitudes were chosen in a preliminary study. Q ~

SO

45

.

4O

.

.

.

(I/h) .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

................. i.......

35

J't

3O

~_~,

.,-.,,,,,

=--=

V..

=h ~

v~'," r-,v

...... ! ................ ::...............

25

20 0

100

200

4OO

300

t (s)

Fig.2 Step disturbance of washing water flow rate.

RESULTS AND DISCUSSION Transient

response

analysis

Transient response analysis was the first step performed. Different SISO models, corresponding to the 9 possible controlled/manipulated variable pairings, were considered. The output variables affected by the input step disturbances were identified. Time delays, time constants and steady state gains were estimated. The relationships which depend upon initial conditions were isolated. Table 2 shows a summary of the results obtained. In this table, the terms k, 0 and • represent, respectively, steady state gain, time constant and time delay. Table 3 shows some examples of experimental results obtained. TABLE 2 Transient analysis

results

I CONTROLLED

f'i

VARIABLES

%

i

- positive or n e g a t i v e Q,.,

N o effect

MANIPULATED

VARIABLES

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0.9
- dependenceon initial process 270<~<340 (s) conditions

(~h)

Qv

H~

No effect

-

sometimes:inverseresponse

-

Negative

- Weak

5<0<15 (s)

- unstable -

sometimes:inverse

response

(~h) 4.3
- unstable

QA

23
- inverse r e s p o n s e

(vh)

9 < 0 < 1 5 ~S)

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T. Carvalho et al.

TABLE 3 Some examples of controlled variables responses to positive disturbances of the manipulated variables CONTROLLED VARIABLES

MANIP~A~D

Qw~(i/h)

No effect

No satisfactory model

Pure integrator

VARIABLES

Oo (l~)

No effect

No satisfactory model

Pure integrator

QA (L/h)

Model 1

Model 2

No satisfa~ory model

The main conclusions drawn in the transient analysis phase are: 1. Air flow rate affects strongly the three controlled variables; 2. Air holdup in the collection zone is not sensitive to the disturbance of underflow and wash water flow rates, being, however, strongly sensitive to the air flow rate. This effect does not depend on the initial conditions of the process; 3. Bias water flow rate is strongly influenced by all the manipulated variables; 4. The response of bias water flow rate to perturbation of wash water flow rate depends on the process initial conditions. It presents in some tests an inverse response; 5. Response of bias water flow rate to underflow flow rate disturbance depends on the initial conditions of the process; 6.The collection zone height is affected by the three manipulated variables; 7. Generally, after disturbance of wash water flow rate the level of the interface doesn't stabilize, due to the unbalance of the flow rates. In some cases, it stabilizes, being the response of collection zone height modeled by a steady state gain and a time delay; 8. Collection zone height responds linearly to a disturbance superimposed in the underflow flow rate. It does not stabilize atter disturbance, showing a slight inverse response; 9.Collection zone height presents an inverse response to the air flow rate change. After disturbance, new equilibrium conditions are not achieved.

Linear parametric models identification The transfer function matrix was built up by identifying the different SISO models corresponding to the 9 possible controlled/manipulated variable pairings. Model identification and parameter estimation were performed using the System Identification Toolbox, of L. Ljung, for MATLAB (Windows) [5]. The statistical quality of the models (standard deviation of the parameter estimation errors) and the adjustment of the models to the observed time series were evaluated. Output Error (OE) type models describe the transfer functions of all pairings that could be modeled. For identification purposes, the variable increment of collection zone height (AH), calculated as the difference of collection zone height in contiguous sampling times was considered. The validity of the linear time invariant models was evaluated. Table 4 illustrates some experimental results and the correspondent results of the simulation step (experimental results represented in grey line and simulated ones in black harline). The following conclusions could be produced: 1.The dynamic relationship between air holdup and air flow rate is well described by model Eq. (11);

O. 005 q-i ~ c (t) = 1_ 0. 9 4 q-1Q A (t) + e (t)

(11)

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1345

2. In the model identification of bias water flow rate response to wash water flow rate disturbance, models Eq. (12) and Eq. (13) were identified. Equation (12) adjusts very well to some validation data sets while model Eq. (13) adjusts very well only to other validation data sets, showing that the responses are very sensitive to small changes of the parameters (small differences in equation parameters lead to different

(12)

Q8 (t) - 1.19 q-J - 1.18 q.2 Q ~ / (t) + e (t) 1 - o. 98 q~ responses);

Q B (t) =

1.09 q-I. 1.10 q-2 1_ 0. 9 4 q. 1 Q .~e (t) + e (t)

(13)

3. It was not possible to find a model or sets of models that suitably describe the dynamics of bias water flow rate response to underflow flow rate step changes;

QB (t)

=

- O. 0006 q-1 1_1.77q-l+O.14q-2+1.09q-3_O.45q

(14)

-4QA (t) + e(t)

4. Air flow rate affects bias water flow rate according to the model Eq. (14); 5. The effect of underflow flow rate and wash water flow rate on collection zone height could be described by pure integrators models. 6. The response of collection zone height to a step change of air flow rate is complex. It was not possible to obtain an adequate linear model that could describe the response. TABLE 4 Some illustrative experimental results and correspondent simulated responses

s©(%) o . (1~)

Qw (~) t(.} t(.)

t (.)

(%) Q. (vn~

::iiiiiiiiiii:i:::::-i

q~

...... ........ .

(lib) t (s) t Is)

t (.)

o.(~) •

-i .......

~. . . . . . . -~. . . . . . . . ,~. . . . . . .

(I,~) g

m

m t(s)

. t (s)

t (s)

1346

T. Carvalhoet al. CONCLUSIONS

In order to overcome the difficulty in the development of a phenomenological model of column flotation process, experimental work was carried out to build up empirical models for process control purposes. In this study only some relationships between controlled and manipulated variables could be identified using linear ready made models. This was the case of the effect of air flow rate on both air holdup and bias water flow rate. It was also established that the effect of wash water and underflow flow rates on the collection zone height are described by pure integrator models, while these operational variables don't affect the air holdup. The identification of the relationships between wash water and underflow flow rates with bias water flow rate and between air flow rate and collection zone height was not successful due, very likely, to the non linear dynamic relationships between the variables involved. Work using other modeling tools (as Artificial Neural Networks and Fuzzy Inference Systems) is the next step. With these alternative modeling paradigms we hope to cope with the non linear dynamic model for the colunm flotation process. REFERENCES .

2.

,

4. 5.

Finch, J. A. and Dobby, J. S., Column Flotation, 1990, Pergamon Press, Oxford. Carvalho, M. T., Sousa, J. M., Dur~to, F. O., and Martins, P. M., Real Time Fuzzy Control of Column Flotation Process. In Proc. Symposium of Artificial Intelligence in Real Time Control, Valencia, Spain, 1994, pp. 87-92. Bergh, L. G. and Yianatos, J. B., Experimental Studies on Flotation Column Dynamics. Minerals Engineering, 1994, 7 (2/3) pp. 345-355. Ljung, L., Modeling of Dynamic Systems, 1994, Prentice Hall, Englewood Cliffs. Ljung, L., System Identification Toolbox, for use with Matlab. User's Guide, 1993, The Mathworks Inc.

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