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Dynamic model of a flotation column
Minerals Engineering, Vol. 13, No. 14-15, pp. 1465-1481, 2000
Pergamon
0892-6875(00)00131-X
DYNAMIC
MODEL
OF A FLOTATION
© 2000 Published by Elsevier Science Ltd All rights reserved 0892--6875/00/$- see front matter
COLUMN
M.A.M. PERSECHINI ~, F.G. JOTA t and A.E.C. PERES t § Programa de P6s Gradua~o em Engenharia E16trica. Federal University of Minas Gerais, Av. Ant6nio Carlos, 6627, Belo Horizonte, MG, 31270-901, Brazil Department of Electronics Engineering. Federal University of Minas Gerais, Av. Ant6nio Carlos, 6627, Belo Horizonte, MG, 31270-901, Brazil. E-mail: [email protected] t Department of Metallurgical and Materials Engineering, Federal University of Minas Gerais, R. Espfrito Santo, 35, Belo Horizonte, MG, 30160-030, Brazil (Received 24 January 2000; accepted 5 September 2000)
ABSTRACT A linear multivariable model for a flotation column is derived from process analysis and experimental data. The data have been taken from a pilot-scale column (height 720 cm; diameter 5.1 cm) with all necessary instrumentation installed. The flotation column was operated in a waterair system. The model has been derived aimed at utilisation in the design of the control system. The model describes the relationship between controlled variables (froth layer height, bias and air holdup in the recovery zone) and manipulated variables (wash water, air and non-floated fraction flowrates). It has been validated comparing the experimental data with those obtained from simulations. Finally, the conditions under which the model is applicable are presented. © 2000 Published by Elsevier Science Ltd. All rights reserved.
Keywords Column flotation; modelling; simulation.
~TRODUCTION In order to be useful for control design purposes, a model should incorporate the dominant characteristics of the modelled system. The model does not need to describe accurately the system dynamics since this will lead to an unnecessarily complicated controller. A good compromise is the development of a simple model capable of encompassing the dominant time constants, thus determining the relevant dynamic characteristics of the process and the correct steady-state gains between manipulated and controlled variables. In the case of the column flotation process, the main control objective is the optimisation of the metallurgical performance so as to guarantee that the column operation reaches the reference values necessary for the desired recovery and grade of the concentrate stream. To achieve this goal, it is first necessary to stabilise the process thus minimising the number and severity of the erratic operations. Under these conditions, one of the specific control objectives is to maintain at pre-specified values variables such as froth layer height, air holdup in the recovery zone and bias, by manipulating the air, the wash water and the non-floated flowrates. These variables are indirectly related to the recovery and to the concentrate grade.
1465
1466
M.A.M. Persechinietal.
Examples of models developed specifically for control purposes can be found in the literature. For example, to implement a predictive controller for a coal flotation column, Pu et al. (1991) developed a model in which the controlled variables are the froth layer height and the air holdup in the recovery zone, the air flowrate and the non-floated flowrate being the manipulated variables. Bergh and Yianatos (1995), describe a discrete time model for a pilot-scale column operating in a two-phase system. The model describes the relationships between the froth layer height, the air holdup in the recovery zone and the bias (defined as the difference between the non-floated flowrate and the pulp feeding flowrate) with the wash water flowrate and the percentage of aperture of the valves that control the non-floated flowrate and the air flowrate. However, the relationship between the froth layer height and the percentage of aperture of the air control valve has not been determined, del Villar et al. (1999) present a continuous time model relating the froth layer height and the bias with the non-floated flowrate and wash water flowrate. Their objective was to implement a multi-loop control strategy. In this work, a complete model describing all the interactions between the manipulated and the controlled variables is presented. The model, derived initially from the mass and energy balance equations of the process, has been specifically developed to emulate multivariable control strategies. In this case, the true physical variables are used in the modelling process and the model validation is accomplished using data gathered from a pilot-scale column equipped with sensors and actuators similar to the ones used in industrial plants.
PILOT PLANT DESCRIPTION The pilot flotation column is composed of a transparent acrylic tube with 5.1 cm internal diameter, 720 cm height and associated instruments. The column instrumentation is represented schematically in Figure 1.
~
........' ~
XC-O1 wallh water
xc-o2
i
":'"i'. '
f ...................
C-01
non-floated
pulp Fig. 1 Instrumentation of the pilot column.
The wash water flowrate is measured by means of an electromagnetic flowmeter (FT-03) with integral orifice assembly. The wash water flowrate is controlled by a pneumatic valve (XC-01). The air flowrate is measured by a mass flowmeter (Fr-04). The air flowrate is controlled manually using a flowmeter to set it at the desired value. The pulp feeding is controlled by means of a peristaltic pump with variable speed capability coupled with a frequency inverter. The non-floated flowrate is also controlled by a variablespeed peristaltic pump (SC-01) driven by a frequency inverter. The pressure measurements (PT-01 and PT02) are used to calculate the values of the air holdup in the recovery zone and of the froth layer height. These two meters are installed in the upper part of the column, respectively, at 230 cm and 350 cm from the top of the column.
Dynamic model of a flotation column
1467
The instruments are all connected to a data acquisition system (Smar Instrumentos, CD-600) which takes care of the analog-to-digital conversion of the output variables (namely, FT-03, FT-04, PT-01 and PT-02) and digital-to-analog conversion of the manipulated variables (XC-01 and SC-02). The data acquisition system is also connected via a serial RS-232 port to a microcomputer where a SCADA-like system runs (RealFlex 1.22 for QNX 2.2). In all tests presented here, the column has been fed with a water solution of a frother (Flotanol D14B, an etherpolyalkylethyleneglycol). The feed flowrate was 20 cm3/s (Jr=0.97 cm/s), with a frother concentration of 10 ppm. Controlled variables calculation The pilot column is equipped with two pressure gauges. The relationships between the pressure values and the process variables can be expressed by:
P~ = (H~ - h ) p ~ z g + hp yzg
(I)
P2 = ( H 2 - h)Pcz g + hPlz g
(2)
where Hz=230 cm e H2=350 cm. From Eqs. (1) and (2), the froth layer height is given by:
P1H2 - P2H1
h =
(P~ - P 2 ) + p l z g ( H , - H 2)
(3)
where the average value of the froth layer density,/~z, has to be estimated (as explained in sub-section "Froth layer density"). The air holdup in the recovery zone is calculated as: AP
Egcz = 1
(4)
P,,.l g A H
where zip is the pressure difference, AH the distance between the two pressure meters and P.,t is the pulp density. For water-air systems the value of P.,.I can be considered constant and approximately given by Pw =lg/cm 3 in the whole recovery zone. The bias is calculated from the mass balance equations:
A h p j z + A h Prz = m. w -. m B. + .m c s - m c
(5)
A(H,-h)pcz-A
(6)
pcz=mF+mB-mcs-mr
which, for the water-air system, may be represented as:
Ahpj~ + A h p ~ pw
= Qw - Q~ - Qc
(7)
1468
M.A.M. Persechiniet al.
A(H, -h)Pcz-AhPc z = QF + QB -
QT •
(8)
P~ Therefore
A(H,-h)Pcz-Ahpc Q8 =
z
+ QT. - QF
(9)
Pw
where Pcz is the average density for the recovery zone calculated by:
Pcz = (1 - c~c z )P.,.t ,
(10)
~gcz is the average air holdup for the entire zone which, in this case, is approximated by the air holdup calculated in the section delimited by the two pressure gauges (Eq. 4). In the present case, for the water-air system, p.,.~= Pw.
P I L O T PLANT M O D E L L I N G Care has to be taken in the development of the dynamic model of the column flotation process, in view of its utilisation for control design. It must be taken into consideration that, in water-air systems, the control strategy is generally restricted to controlling the froth layer height, the air holdup in the recovery zone and the bias. The manipulated variables are the flowrates of the non-floated fraction, air and wash water. The feed flowrate, though not directly manipulated, is explicitly considered in the model, and its variations are taken as disturbances. The modelling of the input-output relation requires, initially, the estimation of the floated and non-floated fractions flowrates and the froth layer density (parameters that are not measured). Then, the values of the bias, of the air holdup in the recovery zone and of the froth layer height might be determined. From these values, the model for the froth layer height, bias and air holdup in the recovery zone can be described, taking into consideration only the variations of air, wash water and non-floated fraction flowrates without knowing the pressure values. Non-floated fraction flowrate
Since the non-floated fraction flowrate is not directly measured, it must be inferred from the signal sent to the corresponding frequency inverter. The use of a peristaltic pump to control this flowrate requires special attention, due to the pressure variation at the pump inlet during the operation. For a given pump speed, changes in operating conditions cause changes in the value of the flowrate. The variables that most affect the flowrate are the level and the air holdup which reflect in a variation of the total mass inside the column. The non-floated flowrate is determined from the value of the pump command signal, Ur, using the linear function: Qr = 28"9UT
(11)
This relation represents a good compromise between accuracy and complexity of the model since, as explained before, the flowrate actually varies with the pressure at the pump head (which is not known). To get a better approximation, the pressure at the bottom of the column should be measured.
Dynamic model of a flotation column
1469
Froth layer density Knowledge of the average value of the froth layer density is necessary for the calculation of the froth layer height, according to Eq. (3), as in the calculation of the mass balance. Since this value is not measured, it may be inferred from the relation:
f p( z )dz Psz=
(12)
h
where p(z) is a function of the density variation along the axial direction z, represented by the longitudinal axis, with the origin at the top of the column. Tavera et al. (1998), Yianatos et al. (1986) and Yianatos et al. (1985) demonstrated that the air holdup varies in the cleaning zone (causing a variation in density: P(z) = (1 -eg(z)p.~l), but no equation for p(z) is presented. It is then necessary to determine experimentally a function that represents the average density, Pt~, in face of variations of the froth layer height and air and liquid surface rates. The pilot flotation column is composed of a transparent acrylic tube, which allows one to observe and record the actual interface position. Therefore, the froth layer height can be measured by the operator simultaneously along with the data acquisition. In the experiments in which the values of the froth layer height were measured simultaneously along with the pressure gauges, it was possible to calculate the average density with the help of Eqs. (1) and (2), so: P~ ( H 2 - h) - P2 (H~ - h)
p~z =
gh(H2 - H 1)
(13)
where the pressure values are determined as the average of the registered values around the point where the height was measured. Figure 2 shows plots that correlate the average froth layer density, calculated from Eq. (13), with the froth layer height (Figure 2(a)), wash water flowrate (Figure 2(b)) and air flowrate (Figure 2(c)). From Figure 2(a), it is observed that the average density increases with increasing heights. For instance, the density values stay between 0.19 g/cm 3 and 0.26 g/cm 3 for 30 cm froth layer height and between 0.30 g/cm 3 and 0.40 g/cm 3 for 100 cm layer. However, for different values of air and wash water flowrates, the froth layer density values are scattered with no clear indication of a tendency. It is possible to see, in Figures 2(b) and 2(c), that variations in the wash water flowrate (between 7.0 cm3/s and 10.6 cm3/s) and in the air flowrate (between 21.5 cm3/s and 39.0 cm3/s) do not alter the average froth layer density value, i.e., the average value remains constant at approximately 0.30 g/cm 3. This analysis suggests that the average froth layer density may be approximately taken as a function of the height: Pc = F(h). The selected structure for F(h) is a second order polynomial representing the integral in Eq. (12), divided by the height. The average froth layer density is then estimated as:
P tz
=
- 3 . 6 x 1 0 - 4 h 2 + 4.4x10-~ h - 5.64
h
(14)
where the coefficients have been adjusted using the least squares method, Ljung (1999). Figure 2(d) shows the adjusted experimental data (solid line).
1470
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Fig. 2
Average froth layer density. (a) Pfz x h (b) Pyz x Qw (c) Pfz x Qg. (d) Adjusted experimental data (solid line).
Floated
fraction
flowrate
The floated fraction flowrate is another variable that is not directly measured, but it may be calculated by means of the mass balance, Eq. (7). Alternatively, it may be estimated from the air flowrate and the air holdup at the top of the column, sgo, Finch and Dobby (1990): 1 -- 8g o
J c = Jg ~
(15) OCgo
Multiplying the cross sectional area of the column, A, by Eq. (15) and substituting the result in Eqs. (7) and (8), yields the steady-state condition: 1 O~go ag - -- QW + OF -- QT" -
(16)
OCgo
Table 1 shows some experimental data (measured values of J c , Jg and h) and the values of ~go calculated from Eq. (15). These data suggest that, as the average froth layer density in the cleaning zone varies with the height of the zone, the air holdup at the top of the column may also be represented as a function of the height. Nevertheless, the number of measurements is not sufficient for determining this function. Therefore, the floated fraction flowrate may be approximately given by a function of the air flowrate and the froth layer height: 1 -
AJ c = AJg ~
6g o
~ F ( J g , h ) = Q~ ( K h )
Ego
for a constant K to be determined.
(17)
Dynamicmodelof a flotationcolumn
1"471
TABLE 1 Variation of the air holdup at the top of the column (calculated according to Eq. 15) for some experimental data
Jc (cm/s)
Js (cm/s) 1.78 1.77 1.78
0.12 0.11 0.10
h (cm) 22 47 60
cgo (%) 93.7 94.2 94.6
Air holdup in the recovery zone To model the air holdup behaviour, it is necessary to know the input flowrates ( Q r , Qg and Qw ) that influence this variable. Finch and Dobby (1990) describe a method to determine the relationship between the air holdup and the air superficial rate from knowledge of the average bubble diameter and the liquid superficial rate. They also present an empirical relation between the bubble diameter and the air superficial rate, expressed by dh = C(Jg)n, where the constant C depends mainly on the type and concentration of reagents and the exponent n depends on the aeration characteristics of the system. So, for a certain column and the concentration of reagents kept constant, one has:
e~,cz = F(Jg, Ji)
(18)
Several experiments were performed and analysed, aimed at determining the transfer function between the air holdup in the recovery zone and the variation of the input flowrates. The model that best describes the input/output relationship has been approximated by a first order linear function, with negligible time delay, its continuous representation in the Laplace domain being: 7 . 7 8 x 1 0 -5
e~'cz(s) =
s + 7 . 8 1 x 1 0 -3
7 . 6 x 1 0 -5
ag
-~ s + 1.92x10 -2
(Qr -Qw - Q F )
(19)
where the coefficients in Eq. (19) were adjusted utilising the least squares method, Ljung (1999). Figures 3(b), 4(b), 5(b) and 6(b) show the values of the air holdup calculated using Eq. (4) (dotted line) and the simulated values Eq. (19) (solid line). In the experiment shown in Figure 3(a), a step change was applied to the input air flowrate, at time 200 s, keeping constant the wash water flowrate and the nonfloated flowrate. In this experiment, the froth height varied from 100 cm up to the overflow of the liquid phase. In the experiment shown in Figure 4(a), a step change in the air flowrate was applied, at time 80 s, but now the non-floated flowrate was controlled so as to keep the froth height constant at 54 cm. In both cases, by applying a positive step change in the air flowrate, the air holdup in the recovery zone starts increasing almost instantly (i.e., no apparent dead-time). The air holdup in the recovery zone exhibits a strong positive variation and eventually reaches steady state after nearly 400 s. Figure 5(a) shows the effect of a step change applied in the non-floated flowrate at time 500 s, keeping the air and wash water flowrates at constant mean values. Figure 5(b) shows that the air holdup increases slightly, from 17.1% to 17.2%, thus indicating a positive gain in steady state. In Figure 6(a), the wash water flowrate changed from 8.25 cm3/s to 7.55 cm3/s at time 200 s. It can be seen from Figure 6(b) that the air holdup mean value (dotted line) does not show any appreciable change as a result of the wash water flowrate change, but only a slightly negative gain in steady state. Nevertheless, all attempts at modelling the air holdup by considering only the air and non-floated flowrates did not produce good results. So, the wash water flowrate has been included in the model of the air holdup in the recovery zone.
1472
M . A . M . Persechini et al.
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, 4000 time (s)
, 6000
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8000
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4000 time ~ )
(a)
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8000
Effect of air flowrate change on the air holdup in the recovery zone keeping Qr constant (a) Input flowrates (b) measured (dotted line) and simulated (solid line) air holdup in the recovery zone.
Fig.3
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Effect of air flowrate change on the air holdup in the recovery zone keeping h constant (a) Input flowrates (b) measured (dotted line) and simulated (solid line) air holdup in the recovery zone.
Fig.4
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le
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Effect of non-floated fraction flowrate change on the air holdup in the recovery zone (a) Input flowrates (b) measured (dotted line) and simulated (solid line) air holdup in the recovery zone.
Dynamic
model
of a flotation
column
1473
13,1 13 ~ 381" v " J 0 34
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i
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w
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i
i ~
200 time (s)
300
400
(b)
Effect of wash water flowrate change on the air holdup in the recovery zone (a) Input flowrates (b) measured (dotted line) and simulated (solid line) air holdup in the recovery zone.
It is worth mentioning that, in the test in which an overflow of the liquid phase occurred, Figure 3, the final value of the simulated air holdup was 17%, different from the calculated value 15%. In the other tests satisfactory fitting of the model to the calculated data was observed.
Froth layer height The froth layer height is calculated by means of Eq. (3), by using the average density given by Eq. (14). Substituting Eq. (14) in Eq. (3), the froth layer height is then estimated by solving: (-3.6x10 -4 g ( H 2 - H~ ))h 2 + (4.4x10 ' g ( H z - H, ) + (P2 - P~))h - 5.64g(H z - H, ) - (P~H 2 - P2H, ) = 0 (20) The validity of this calculation method may be verified in Figure 7, where plots of height as a function of time are depicted, for several tests, comparing calculated values using Eq. (20) (solid line) with the values measured by the operator (marked with the + symbol). In Figure 7(a), a step change was applied to the nonfloated fraction flowrate at time = 500 s; in Figure 7(b) a step change was applied to the air flowrate at time = 1300 s, in Figure 7(c) a step change was applied to the wash water flowrate at time = 1700 s and in Figure 7(d) step changes were applied to the air flowrate at times = 3100 s and 4200 s. Since, in the experiments shown in Figure 7, the operating conditions vary significantly, Eq. (20) has been used to calculate the height of the froth layer for all experiments. Another way to verify the validity of this method is by the analysis of possible measurement errors. Errors 8P~ and 8P2, associated with the pressure gauges PT-001 and PT-002, and (~Pfzl associated with the estimation of the mean density of the froth layer, are related to the total error in height, 8h, in the froth layer as follows:
ah 6p ' +
Oh 6 P
+
O--~-h 6 p l z
(21)
Considering a typical situation, where h=100 cm, egcz=15% and pfz=O.35g/cm 3, an error of 1% in both PT01 and PT-02 pressure measurements and an error of 10% in ,Ot~, result, according to Eq. (21), in a total error of 9.58 cm in the froth layer height, which corresponds to nearly 10% of the height value. Equations (7) and (8), representing the mass balance, are added to provide the model of the froth layer height concerning the variations in the input flowrates, so:
1474.
M.A.
M,
Persechini e t al.
Ah Pyz + Ah Psz + A(Ht - h) Pc~- A h Pcz QF + Qw - Q c - Q~- -
(22)
Pw
As this is a non-linear equation, the model may be reduced to a linear function using Taylor series expansion, for a certain operating condition, since the variations of these values are small around this operation point, Seborg et al. (1989). Linearising Eq. (22) around an operation point with null derivatives ( h o = 0 , p czo= 0 and p fzo =0 ) and ho, Pczo and Py~o, values selected for the froth layer height and for the densities of the collection and cleaning zones, gives:
Ah o p ,~+ A(H, -ho) Pcz- Ah(pszo - Pczo) Q~ +Qw - Q c - Q r
-
(23)
Pw
75
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Froth layer height. Calculated values (solid line) and measured values (marked with the + symbol).
Equation (23) is a linearised model that describes the dynamic behaviour of the process around ho, Pczo and P/zo. To describe the relationship between the froth layer height and the input flowrates, the floated fraction flowrate, calculated from Eq. (17), is also linearised around the same operation point to give:
Qc = K ( q g o h + hoQg - qgoho)
(24)
where qgo is the mean value of the air flowrate around the chosen operating point. Replacing (24) in (23), and taking the Laplace transform, the continuous model of the froth layer height is described by:
Dynamic model of a flotation column
1475
OF (s) + aw (s) - Qr (s) - Khoag (s) - shoAD1z (s) - s ( n , - h o)ADcz (s)
h(s) =
sA(Pi~o
-
(25)
Pczo) + Kqgo
where Dyz and Dcz are the Laplace transform of,oy~ and Pcz. Equation (25) may be simplified, taking into consideration that the variations in densities have little effect on the behaviour of the froth layer height, if compared with the variations caused by the flowrates. Further, at steady state, the terms sh,,ADfz(s) and s(Ht-ho)ADcz(S), corresponding to the variations of the froth layer and collection zone densities, respectively, are zero. So:
h(s) =
1.029x10 -~s + 2.3x10 -s - 1.59x10 -4 s + 4.33x10 -7 (26) (s + 4.02xl 0 -4)(s + 1.92x10 -2) (QF (s) + Qw (s) - Qr (s)) + (s + 4.02x10 -4)(s + 7.98 lxl0 -3) Qg (s)
The model represented by Eq. (26) gives satisfactory results for variations in the non-floated fraction flowrate. This may be verified in the test shown in Figure 8, in which a step was applied at time = 550 s and in the test presented in Figure 9, in which, after time = 1000 s the non-floated fraction flowrate was kept constant. In both cases, the froth layer height starts changing at the same time events mentioned above (time = 550 s in Figure 8 and time = 1000 s in Figure 9), and the model correctly reproduces these changes. The model may also be considered satisfactory for changes in the wash water flowrate. In the test presented in Figure 9, the wash water flowrate was changed at 3100 s and 4200 s. At these same time instants the derivative of the froth layer height with respect to time changed its sign and the model fitted to this behaviour. The same effect may be observed in the test represented in Figure 10, where the step was applied at t = 1700 s.
~u~"40
.
.
.
.
100
.
35
90
0 27
500
1000
1500
2000
2500
3000
~-~
80
v
70
28.5
1
, ..;~,..... •
0
26/ 0
~,9.5
560 .
.
1000 .
1500 2000 . .
25'00
3000 ,.
~ 9 ~85
0
40
500
1000
1500 2000 time (s)
(a)
Fig.8
5o
2500
3000
3%
• oa
"
. ,,/.,,.-'."'
60
/ %,., ,.¢,",
560
lobo 1 ~
2~
25'00
tin~s)
Effect of non-floated flowrate on froth layer height (a) Input flowrates (b) measured (dotted line) and simulated (solid line) froth layer height.
In Figure 11, at time = 1300 s, the air flowrate was changed from 34.8 cm3/s to 20.5cm3/s causing, initially, an increase in the froth layer from 30 cm to 70 cm, in only 200 s. This increase may be explained by a decrease of the collection zone volume due to a reduction of the air flowrate. Nevertheless, as the air flowrate reduction causes a reduction of the floated fraction flowrate, the froth layer height decreases again. In the specific case of this test, the column did not reach steady state due to the overflow of the liquid phase after approximately t = 2000 s.
1476
M. A. M. Persechini et al.
A40
120
~ 3s h
IOO
83ol
1000 2900
3000
4000
6000
~" 8o
1000 2000
3000
4000 5000 6000
} 40
5000
~s ~ 2o 0 1o
8t" 0
+
2O
10C)0 20'00 3 0 ~ 4000 time (s)
5000
o
6000
+o~ 2o~o ~
60
1000
2000
3000
'°t
x :
+ / 2000
I000
~1
.,
.
.
.
1000
.
i.
3000
.
.
2000 time (a) (s)
.
~, 40
.
.
.
.
.
,
\'++
;~"
4000
l_ _
3000
101 0
4000
Fig.10 Effect of wash water flowrate on froth layer height a'nd simulated (solid line) froth layer height.
, 1000
, 2000
, 3000
~r~s)
4"000
(a) Input flowrates (b) measured (dotted line)
80
.
N 0
,
4000
~3o 5 201 10
moo
(a) Input flowrates (b)
Effect of wash water and non-floated flowrates on froth layer height measured (dotted line) and simulated (solid line) froth layer height.
0 40
4obo ~
tin~s)
(a)
Fig.9
+
v
6o 0 30.
500
1000
1500
2000
2500
3000
~" ,00 + t •
20L 0
~9.5.
500
08.~
~
l C)00 .
,
~
1500 .
.
2000
2500
3000
,+ '1
,%
I
t +
~ 20 J=
.
1~
time (s)
2 0 o 0 2500 ~
-20
500
10~ 1500
2000
2500
3000
(=)
Fig. 11 Effect of air flowrate (negative step change) on froth layer height (dotted line) and simulated (solid line) froth layer height.
(a) Input flowrates (b) measured
The same effect may be verified in the test represented in Figure 12, where an air flowrate step was applied at t = 200 s in the reverse direction with respect to the former. In this case, the froth Iayer height initially decreases, due to a larger displacement of liquid from the collection zone to the cleaning zone, reaching steady state at t = 6000 s, after increasing again up to 100 cm. A larger floated fraction flowrate was
Dynamic model of a flotation column
1477
observed as a consequence of a larger air flowrate, increasing the froth layer. In both cases the gain at steady state is positive and the dynamic behaviour of the height with respect to changes in the air flowrate presents a characteristic of a non minimum phase system.
24o I 0 3O
. 2000
120
,,~
4000
6000
IOOr 8000
~,9.5
g 8°t
, . t o I, ..i,
~ 6o
~28 026/
.p~, •
~ 40 o
2o'o0 .
4o~o .
6o~o .
8000
"
2C 0
~8.5
0
2000
4000 time (s)
6000
8000
-20
(a)
2000
4000 time (s)
6000
8000
(b)
Fig.12 Effect of air flowrate (positive step change) on froth layer height (a) Input flowrates (b) measured (dotted line) and simulated (solid line) froth layer height.
Bias The bias, defined as the net descending water flowrate through the froth layer, is calculated from the mass balance, Eq. (8), where the collection zone density is given by Eq. (10) and p.,.t= p~ for water-air system. So:
Q8 = - A ( H , - h) "c~,cz- A h(1 - ~ gcz ) + QT - QF
(27)
The Laplace transform has been used to derive the continuous time model after the linearisation of Eq. (27) around a specific operating point, ho and ego, with null derivatives, i.e., h,, =0 and g,o =0, representing an equilibrium point of the froth height and of the air holdup in the recovery zone. Under these conditions: Q8 (s) = Qr (s) - QF (s) + sggcz ( s ) A ( h o - H , ) + sh(s)A(,~ go - 1)
(28)
To determine the coefficients of Eq. (28), ho and 6go have been initially replaced by typical operation values (ho=100 cm and Cgo=15%) and, later, these values were adjusted by trial and error. Equation (28) may be then rewritten as: QB(s) = Q v ( s ) - Q F ( s ) - 3 . 8 4 x 1 0 5 s ~ g c z ( s ) - 4 . 8 x l O 4 s h ( s )
(29)
Figures 13 and 14 show the values of the bias, for different tests, calculated from Eq. (27) (dotted line) and simulated utilising the model described by Eq. (29) (solid line). In both cases, the model gives satisfactory outcomes if compared with the results obtained using the bias calculation, Eq. (25). It is worth noting in Figure 14 that the value of the non-floated fraction flowrate, therefore the difference between the feed and non-floated fraction flowrates, remained constant from t = 1100 s on. The variation of the bias is significant from this instant on, indicating that the process did not reach steady state.
1478
M.A.M.
~0.7
.
.
.
.
P e r s e c h i n i etal.
.
-
,
,
.
,
~.5[ .... ~,..... !, ..... !,..... !, ..... i---, 0.5
500
1000
1500
2000
2500
8_- - - - ~ . . . . . ~. . . . . :. . . . . ~ . . . . . ,--.
3000
0.46
~" 75• .....
/
~o.44
50.42
s6o
. ~ 0.16
.
..~
lo'oo lgoo .
.
7 -
~
:.i .....
:.....
i_.._~_~. ,
'
-,
,
I1
I
0
500
1000
1500 time (s)
2000
2500
3000
5.5
II
~
:',
0
tl k
I
:
'
.
.
.
,
0 0.51
2000
.
,
lq~
/
8000
10000 12000
o.,l"l'VVvl'! . . . . 0 2000 4000 6000 8000
10000 12000
-L
"----;
~; !' !1
~ o.,~ g ~0.15
f"
o.i~
2obo 4 ~
i;i
i
I
II
I
i
l 3000
I
TrVr"'~
6000
t
(a) Input flowrates (b) measured (dotted line) and simulated (solid
.
,"
4000
I
time(s)
8,
~ o.551
, i
(b)
Fig.13 Effect of air flowrate on bias line) bias. 0.65
I
5bo adoo ~500 20'00 2~0
(a)
~
.,
. i
0~0.14
,
2goo 30o0
2obo
.
-;i .....
......
f,obo 8 ~
l
.....
i
',. . . .
" .....
i
" ....
i :_
,,,amutdi~
.
~o~o n ~
2000
4000
6000
8000
10000 12000
time(s)
time (s)
(a)
(b)
Fig. 14 Effect of wash water flowrate on bias (solid line) bias.
(a) Input flowrates (b) measured (dotted line) and simulated
PILOT PLANT MODEL For control design purposes, it is convenient to write the pilot plant model in matrix form. This facilitates, for instance, the correlation analysis of controlled and manipulated variables. The pilot plant model which has been obtained for operation in a two phase (water-air) system can then be rewritten as:
rhl[ Egcz
[_QB J
=
o G21
s(K~G,~ + KzGzl )
G22
s(K~G~2+ KzGz2 )
G23
1+
Q~
s(K~G]3 + KzG23 ) Qr - Qv
where the transfer functions
Glt
- (1.029x10-3 s + 2 . 3 x 1 0 -5) (s + 4 . 0 2 x 1 0 - 4 ) ( s + 1 . 9 2 x 1 0 -2)
(31)
(30)
Dynamicmodel of a flotationcolumn
1479
- 1.59x10-4 s + 4 . 3 3 x 1 0 -7 G12 -
(s + 4.02x10-4)(s + 7.98 l x l 0 -3)
(32)
( 1 . 0 2 9 x 1 0 -3 s + 2 . 3 x 1 0 -5)
G~3
(s + 4 . 0 2 x l 0 - 4 ) ( s + 1.92X10 -2 )
(33)
are related to the froth layer height model (Eq. 26), whereas
G~ 1 =
7.6x10 -5 s + 1.92x10 -2
(34)
7 . 7 8 x 1 0 -5 G22 =
s + 7.8 l x l 0 -3
(35)
- 7 . 6 x 1 0 -5 G23 = s + l . 9 2 x 1 0
-2
(36)
are related to the air holdup model in the recovery zone (Eq. 19). In Eq. (30), the relation between the bias and the manipulated variables (wash water, air and non-floated fraction flowrates) is obtained by substituting Eqs. (19) and (26) in Eq. (29), and from Eq. (29) -4.8x104=KI, and -3.84x105=K2. This model is valid under the following conditions: • • •
•
•
feed flowrate constant and equal to 20 cm3/s at 10 ppm frother dosage; floated fraction flowrate greater than zero; non-floated fraction flowrate greater than zero; froth layer height between lower and higher limits (the lower limit is 20 cm, corresponding to the level where the wash water is introduced; the higher limit is 140 cm, corresponding to the minimum value for which the pressure gauge PT-01 was calibrated); column operating in a bubbly flow regime.
To achieve these conditions, the variations of the input flowrates must be constrained to certain limits. Initially, the air flowrate should not be varied stepwise to avoid transient effects on the froth layer height, as shown in Figures ll(b) and 12(b).The constraints imposed to the non-floated fraction, wash water and feed flowrates are:
Qw +QF-QT Qw + QF -
>o
Q r < 0.25Qg
(37) (38)
The limit imposed by Eq. (37) is necessary for the flow of the floated fraction, or else the column will empty. On the other hand, the sum of flows should not be too large to prevent the froth layer from quickly disappearing, causing a liquid phase overflow. The maximum value of the left hand side in Eq. (37) depends on the floated fraction flow capacity, which is directly related to the air flowrate. According to Finch and Dobby (1990), the air holdup at the top of a column may reach 80% or even more. Taking this observation into account and using Eq. (15), the higher limit for the flowrates is determined by Eq. (38).
M.A.M. Persechiniet al.
1480
CONCLUSIONS From dynamic mass balances and the analysis of the data gathered from a pilot-scale column (both for the air-water system), a simplified representation of the process dynamics has been obtained through a linear, multivariable model. This model was derived aimed at its utilisation in control. The manipulated variables are the flowrates of the air, the wash water and the non-floated fraction; the controlled variables are the froth layer height, the air holdup in the recovery zone and the bias. The model derived in the present work may be considered adequate to represent the process with respect to steady-state gains between manipulated and controlled variables and the main time constants. Nevertheless, it is still necessary to consider some important aspects, mainly in the description of the froth layer height behaviour. In particular, the initial response of the froth layer height to stepwise changes of the air flowrate is inverse to the steady state gain, characterising a non minimum phase system. Provided that the conditions under which the model was derived are respected, the representation obtained may be utilised as a tool for studies and behaviour simulations of the process variables and as an aid to the determination of a control strategy adequate for column flotation. Finally, the same methodology used in this work for obtaining the dynamic model of the flotation column, operating in a water-air system, has been used to develop a model considering the effects of solids. Aiming at this goal, the mass balance equations were rewritten taking into account the different densities for each of the process flowrates (QF, Qr, Qc and Qw), and experimental data are now being gathered for further analysis.
ACKNOWLEDGEMENTS The authors acknowledge the technical support provided by Setor de Tecnologia Mineral da Supervisdo de Processos do Centro de Desenvolvimento da Tecnologia Nuclear (CDTN) responsible for the design, construction and operation of the pilot scale flotation column. We are also thankful to ATAN Automation Systems Ltd. for providing the software tools: QNX 2.2, RealFlex 1.22 and the communication driver for the CD-600. Nomenclature A D G H J P Q U
Column section Density Transfer function Height Superficial rate Pressure Flowrate Signal command
V g h m A 6 ~ p
Volume Gravity acceleration B Froth layer height C Mass F Difference T Error W Holdup cz Density fz
SUBSCRIPTS Bias Floated fraction Feed Non-floated fraction Wash water Collection zone Cleaning zone
g l o s
Air Liquid Operating point Solid sl Pulp t Total height w Water
REFERENCES Bergh, L. G. and Yianatos, J. B., Dynamic simulation of operating variables in flotation columns. Minerals Engineering, 1995, 8(6), 603-613. Finch, J. A. and Dobby, G. S., Column flotation, 1st edn. 1990, Pergamon Press, Oxford. Ljung, L., System identification, 2nd edn. 1999, Prentice Hall, New Jersey. Pu, M., Gupta, Y. P. and A1 Taweel, A. M., Model predictive control of flotation columns. In Proc. Column'91. Sudbury, Canada, 1991, pp. 467-479 Seborg, D. E., Edgar, T. F. and Mellichamp, D. A., Process dynamics and control, 1989, John Wiley & Sons, New York Tavera, F. J., Gomez, C. O. and Finch, J. A., Estimation of gas holdup in froths by electrical conductivity: Application of the standard addition method. Minerals Engineering, 1998, 11(10), 941-947.
Dynamic model of a flotation column
1481
del Villar, R., Gr6goire, M. and Pomerleau, A., Automatic control of a laboratory flotation column. Minerals Engineering, 1999, 12(3), 291-308. Yianatos, J. B., Finch, J. A. and Laplante, A. R., Holdup profile and bubble size distribution of flotation column froths. Canadian Metallurgical Quartely, 1986, 25(1), 23-29. Yianatos, J. B, Laplante, A. R. and Finch, J. A., Estimation of local holdup in the bubbling and froth zones of a gas-liquid. Chemical Engineering Science, •985, 40(10), 1965-1968.
Correspondence on papers published in Minerals Engineering is invited by e-mail to bwills @ m i n - e n g . c o m
Pergamon
0892-6875(00)00131-X
DYNAMIC
MODEL
OF A FLOTATION
© 2000 Published by Elsevier Science Ltd All rights reserved 0892--6875/00/$- see front matter
COLUMN
M.A.M. PERSECHINI ~, F.G. JOTA t and A.E.C. PERES t § Programa de P6s Gradua~o em Engenharia E16trica. Federal University of Minas Gerais, Av. Ant6nio Carlos, 6627, Belo Horizonte, MG, 31270-901, Brazil Department of Electronics Engineering. Federal University of Minas Gerais, Av. Ant6nio Carlos, 6627, Belo Horizonte, MG, 31270-901, Brazil. E-mail: [email protected] t Department of Metallurgical and Materials Engineering, Federal University of Minas Gerais, R. Espfrito Santo, 35, Belo Horizonte, MG, 30160-030, Brazil (Received 24 January 2000; accepted 5 September 2000)
ABSTRACT A linear multivariable model for a flotation column is derived from process analysis and experimental data. The data have been taken from a pilot-scale column (height 720 cm; diameter 5.1 cm) with all necessary instrumentation installed. The flotation column was operated in a waterair system. The model has been derived aimed at utilisation in the design of the control system. The model describes the relationship between controlled variables (froth layer height, bias and air holdup in the recovery zone) and manipulated variables (wash water, air and non-floated fraction flowrates). It has been validated comparing the experimental data with those obtained from simulations. Finally, the conditions under which the model is applicable are presented. © 2000 Published by Elsevier Science Ltd. All rights reserved.
Keywords Column flotation; modelling; simulation.
~TRODUCTION In order to be useful for control design purposes, a model should incorporate the dominant characteristics of the modelled system. The model does not need to describe accurately the system dynamics since this will lead to an unnecessarily complicated controller. A good compromise is the development of a simple model capable of encompassing the dominant time constants, thus determining the relevant dynamic characteristics of the process and the correct steady-state gains between manipulated and controlled variables. In the case of the column flotation process, the main control objective is the optimisation of the metallurgical performance so as to guarantee that the column operation reaches the reference values necessary for the desired recovery and grade of the concentrate stream. To achieve this goal, it is first necessary to stabilise the process thus minimising the number and severity of the erratic operations. Under these conditions, one of the specific control objectives is to maintain at pre-specified values variables such as froth layer height, air holdup in the recovery zone and bias, by manipulating the air, the wash water and the non-floated flowrates. These variables are indirectly related to the recovery and to the concentrate grade.
1465
1466
M.A.M. Persechinietal.
Examples of models developed specifically for control purposes can be found in the literature. For example, to implement a predictive controller for a coal flotation column, Pu et al. (1991) developed a model in which the controlled variables are the froth layer height and the air holdup in the recovery zone, the air flowrate and the non-floated flowrate being the manipulated variables. Bergh and Yianatos (1995), describe a discrete time model for a pilot-scale column operating in a two-phase system. The model describes the relationships between the froth layer height, the air holdup in the recovery zone and the bias (defined as the difference between the non-floated flowrate and the pulp feeding flowrate) with the wash water flowrate and the percentage of aperture of the valves that control the non-floated flowrate and the air flowrate. However, the relationship between the froth layer height and the percentage of aperture of the air control valve has not been determined, del Villar et al. (1999) present a continuous time model relating the froth layer height and the bias with the non-floated flowrate and wash water flowrate. Their objective was to implement a multi-loop control strategy. In this work, a complete model describing all the interactions between the manipulated and the controlled variables is presented. The model, derived initially from the mass and energy balance equations of the process, has been specifically developed to emulate multivariable control strategies. In this case, the true physical variables are used in the modelling process and the model validation is accomplished using data gathered from a pilot-scale column equipped with sensors and actuators similar to the ones used in industrial plants.
PILOT PLANT DESCRIPTION The pilot flotation column is composed of a transparent acrylic tube with 5.1 cm internal diameter, 720 cm height and associated instruments. The column instrumentation is represented schematically in Figure 1.
~
........' ~
XC-O1 wallh water
xc-o2
i
":'"i'. '
f ...................
C-01
non-floated
pulp Fig. 1 Instrumentation of the pilot column.
The wash water flowrate is measured by means of an electromagnetic flowmeter (FT-03) with integral orifice assembly. The wash water flowrate is controlled by a pneumatic valve (XC-01). The air flowrate is measured by a mass flowmeter (Fr-04). The air flowrate is controlled manually using a flowmeter to set it at the desired value. The pulp feeding is controlled by means of a peristaltic pump with variable speed capability coupled with a frequency inverter. The non-floated flowrate is also controlled by a variablespeed peristaltic pump (SC-01) driven by a frequency inverter. The pressure measurements (PT-01 and PT02) are used to calculate the values of the air holdup in the recovery zone and of the froth layer height. These two meters are installed in the upper part of the column, respectively, at 230 cm and 350 cm from the top of the column.
Dynamic model of a flotation column
1467
The instruments are all connected to a data acquisition system (Smar Instrumentos, CD-600) which takes care of the analog-to-digital conversion of the output variables (namely, FT-03, FT-04, PT-01 and PT-02) and digital-to-analog conversion of the manipulated variables (XC-01 and SC-02). The data acquisition system is also connected via a serial RS-232 port to a microcomputer where a SCADA-like system runs (RealFlex 1.22 for QNX 2.2). In all tests presented here, the column has been fed with a water solution of a frother (Flotanol D14B, an etherpolyalkylethyleneglycol). The feed flowrate was 20 cm3/s (Jr=0.97 cm/s), with a frother concentration of 10 ppm. Controlled variables calculation The pilot column is equipped with two pressure gauges. The relationships between the pressure values and the process variables can be expressed by:
P~ = (H~ - h ) p ~ z g + hp yzg
(I)
P2 = ( H 2 - h)Pcz g + hPlz g
(2)
where Hz=230 cm e H2=350 cm. From Eqs. (1) and (2), the froth layer height is given by:
P1H2 - P2H1
h =
(P~ - P 2 ) + p l z g ( H , - H 2)
(3)
where the average value of the froth layer density,/~z, has to be estimated (as explained in sub-section "Froth layer density"). The air holdup in the recovery zone is calculated as: AP
Egcz = 1
(4)
P,,.l g A H
where zip is the pressure difference, AH the distance between the two pressure meters and P.,t is the pulp density. For water-air systems the value of P.,.I can be considered constant and approximately given by Pw =lg/cm 3 in the whole recovery zone. The bias is calculated from the mass balance equations:
A h p j z + A h Prz = m. w -. m B. + .m c s - m c
(5)
A(H,-h)pcz-A
(6)
pcz=mF+mB-mcs-mr
which, for the water-air system, may be represented as:
Ahpj~ + A h p ~ pw
= Qw - Q~ - Qc
(7)
1468
M.A.M. Persechiniet al.
A(H, -h)Pcz-AhPc z = QF + QB -
QT •
(8)
P~ Therefore
A(H,-h)Pcz-Ahpc Q8 =
z
+ QT. - QF
(9)
Pw
where Pcz is the average density for the recovery zone calculated by:
Pcz = (1 - c~c z )P.,.t ,
(10)
~gcz is the average air holdup for the entire zone which, in this case, is approximated by the air holdup calculated in the section delimited by the two pressure gauges (Eq. 4). In the present case, for the water-air system, p.,.~= Pw.
P I L O T PLANT M O D E L L I N G Care has to be taken in the development of the dynamic model of the column flotation process, in view of its utilisation for control design. It must be taken into consideration that, in water-air systems, the control strategy is generally restricted to controlling the froth layer height, the air holdup in the recovery zone and the bias. The manipulated variables are the flowrates of the non-floated fraction, air and wash water. The feed flowrate, though not directly manipulated, is explicitly considered in the model, and its variations are taken as disturbances. The modelling of the input-output relation requires, initially, the estimation of the floated and non-floated fractions flowrates and the froth layer density (parameters that are not measured). Then, the values of the bias, of the air holdup in the recovery zone and of the froth layer height might be determined. From these values, the model for the froth layer height, bias and air holdup in the recovery zone can be described, taking into consideration only the variations of air, wash water and non-floated fraction flowrates without knowing the pressure values. Non-floated fraction flowrate
Since the non-floated fraction flowrate is not directly measured, it must be inferred from the signal sent to the corresponding frequency inverter. The use of a peristaltic pump to control this flowrate requires special attention, due to the pressure variation at the pump inlet during the operation. For a given pump speed, changes in operating conditions cause changes in the value of the flowrate. The variables that most affect the flowrate are the level and the air holdup which reflect in a variation of the total mass inside the column. The non-floated flowrate is determined from the value of the pump command signal, Ur, using the linear function: Qr = 28"9UT
(11)
This relation represents a good compromise between accuracy and complexity of the model since, as explained before, the flowrate actually varies with the pressure at the pump head (which is not known). To get a better approximation, the pressure at the bottom of the column should be measured.
Dynamic model of a flotation column
1469
Froth layer density Knowledge of the average value of the froth layer density is necessary for the calculation of the froth layer height, according to Eq. (3), as in the calculation of the mass balance. Since this value is not measured, it may be inferred from the relation:
f p( z )dz Psz=
(12)
h
where p(z) is a function of the density variation along the axial direction z, represented by the longitudinal axis, with the origin at the top of the column. Tavera et al. (1998), Yianatos et al. (1986) and Yianatos et al. (1985) demonstrated that the air holdup varies in the cleaning zone (causing a variation in density: P(z) = (1 -eg(z)p.~l), but no equation for p(z) is presented. It is then necessary to determine experimentally a function that represents the average density, Pt~, in face of variations of the froth layer height and air and liquid surface rates. The pilot flotation column is composed of a transparent acrylic tube, which allows one to observe and record the actual interface position. Therefore, the froth layer height can be measured by the operator simultaneously along with the data acquisition. In the experiments in which the values of the froth layer height were measured simultaneously along with the pressure gauges, it was possible to calculate the average density with the help of Eqs. (1) and (2), so: P~ ( H 2 - h) - P2 (H~ - h)
p~z =
gh(H2 - H 1)
(13)
where the pressure values are determined as the average of the registered values around the point where the height was measured. Figure 2 shows plots that correlate the average froth layer density, calculated from Eq. (13), with the froth layer height (Figure 2(a)), wash water flowrate (Figure 2(b)) and air flowrate (Figure 2(c)). From Figure 2(a), it is observed that the average density increases with increasing heights. For instance, the density values stay between 0.19 g/cm 3 and 0.26 g/cm 3 for 30 cm froth layer height and between 0.30 g/cm 3 and 0.40 g/cm 3 for 100 cm layer. However, for different values of air and wash water flowrates, the froth layer density values are scattered with no clear indication of a tendency. It is possible to see, in Figures 2(b) and 2(c), that variations in the wash water flowrate (between 7.0 cm3/s and 10.6 cm3/s) and in the air flowrate (between 21.5 cm3/s and 39.0 cm3/s) do not alter the average froth layer density value, i.e., the average value remains constant at approximately 0.30 g/cm 3. This analysis suggests that the average froth layer density may be approximately taken as a function of the height: Pc = F(h). The selected structure for F(h) is a second order polynomial representing the integral in Eq. (12), divided by the height. The average froth layer density is then estimated as:
P tz
=
- 3 . 6 x 1 0 - 4 h 2 + 4.4x10-~ h - 5.64
h
(14)
where the coefficients have been adjusted using the least squares method, Ljung (1999). Figure 2(d) shows the adjusted experimental data (solid line).
1470
M.
0.45
i i
i i
i i
~ *
i i
÷o*****
% * ,
A,
M.
Persechini
et al.
0.4 ~0.35
,
....
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a
0.2
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0.3
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.
.
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.
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.
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60 froth
I00 (cm)
010 Ilyer
height
I'70 "
140
0'17
II wash
9 water
0.451_r . . . . I
'
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(b)
(a) 0.45 . . . . .
flow
i
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(cm3/s)
~035
|
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t
....
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,
-11 ~
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,•
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"
~= 1 -
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i t
40
0
45
(c)
|
"120
40
60 80 100 froth layer height (cm)
t
120
140
(d)
Fig. 2
Average froth layer density. (a) Pfz x h (b) Pyz x Qw (c) Pfz x Qg. (d) Adjusted experimental data (solid line).
Floated
fraction
flowrate
The floated fraction flowrate is another variable that is not directly measured, but it may be calculated by means of the mass balance, Eq. (7). Alternatively, it may be estimated from the air flowrate and the air holdup at the top of the column, sgo, Finch and Dobby (1990): 1 -- 8g o
J c = Jg ~
(15) OCgo
Multiplying the cross sectional area of the column, A, by Eq. (15) and substituting the result in Eqs. (7) and (8), yields the steady-state condition: 1 O~go ag - -- QW + OF -- QT" -
(16)
OCgo
Table 1 shows some experimental data (measured values of J c , Jg and h) and the values of ~go calculated from Eq. (15). These data suggest that, as the average froth layer density in the cleaning zone varies with the height of the zone, the air holdup at the top of the column may also be represented as a function of the height. Nevertheless, the number of measurements is not sufficient for determining this function. Therefore, the floated fraction flowrate may be approximately given by a function of the air flowrate and the froth layer height: 1 -
AJ c = AJg ~
6g o
~ F ( J g , h ) = Q~ ( K h )
Ego
for a constant K to be determined.
(17)
Dynamicmodelof a flotationcolumn
1"471
TABLE 1 Variation of the air holdup at the top of the column (calculated according to Eq. 15) for some experimental data
Jc (cm/s)
Js (cm/s) 1.78 1.77 1.78
0.12 0.11 0.10
h (cm) 22 47 60
cgo (%) 93.7 94.2 94.6
Air holdup in the recovery zone To model the air holdup behaviour, it is necessary to know the input flowrates ( Q r , Qg and Qw ) that influence this variable. Finch and Dobby (1990) describe a method to determine the relationship between the air holdup and the air superficial rate from knowledge of the average bubble diameter and the liquid superficial rate. They also present an empirical relation between the bubble diameter and the air superficial rate, expressed by dh = C(Jg)n, where the constant C depends mainly on the type and concentration of reagents and the exponent n depends on the aeration characteristics of the system. So, for a certain column and the concentration of reagents kept constant, one has:
e~,cz = F(Jg, Ji)
(18)
Several experiments were performed and analysed, aimed at determining the transfer function between the air holdup in the recovery zone and the variation of the input flowrates. The model that best describes the input/output relationship has been approximated by a first order linear function, with negligible time delay, its continuous representation in the Laplace domain being: 7 . 7 8 x 1 0 -5
e~'cz(s) =
s + 7 . 8 1 x 1 0 -3
7 . 6 x 1 0 -5
ag
-~ s + 1.92x10 -2
(Qr -Qw - Q F )
(19)
where the coefficients in Eq. (19) were adjusted utilising the least squares method, Ljung (1999). Figures 3(b), 4(b), 5(b) and 6(b) show the values of the air holdup calculated using Eq. (4) (dotted line) and the simulated values Eq. (19) (solid line). In the experiment shown in Figure 3(a), a step change was applied to the input air flowrate, at time 200 s, keeping constant the wash water flowrate and the nonfloated flowrate. In this experiment, the froth height varied from 100 cm up to the overflow of the liquid phase. In the experiment shown in Figure 4(a), a step change in the air flowrate was applied, at time 80 s, but now the non-floated flowrate was controlled so as to keep the froth height constant at 54 cm. In both cases, by applying a positive step change in the air flowrate, the air holdup in the recovery zone starts increasing almost instantly (i.e., no apparent dead-time). The air holdup in the recovery zone exhibits a strong positive variation and eventually reaches steady state after nearly 400 s. Figure 5(a) shows the effect of a step change applied in the non-floated flowrate at time 500 s, keeping the air and wash water flowrates at constant mean values. Figure 5(b) shows that the air holdup increases slightly, from 17.1% to 17.2%, thus indicating a positive gain in steady state. In Figure 6(a), the wash water flowrate changed from 8.25 cm3/s to 7.55 cm3/s at time 200 s. It can be seen from Figure 6(b) that the air holdup mean value (dotted line) does not show any appreciable change as a result of the wash water flowrate change, but only a slightly negative gain in steady state. Nevertheless, all attempts at modelling the air holdup by considering only the air and non-floated flowrates did not produce good results. So, the wash water flowrate has been included in the model of the air holdup in the recovery zone.
1472
M . A . M . Persechini et al.
40~
.
.
18
.
17 1~201 0 30.
, 2000
, 4000
, 6000
16
6000
g15
~ 261
20bo
o
~. 9.5
40bo
.
.
60bo .
eooo
14 13
I
12 I~ 6.51 0
, 2000
, 4000 time (s)
, 6000
11 0
8000
20'00
4000 time ~ )
(a)
6(~X)
8000
Effect of air flowrate change on the air holdup in the recovery zone keeping Qr constant (a) Input flowrates (b) measured (dotted line) and simulated (solid line) air holdup in the recovery zone.
Fig.3
18
16
0
200 .
.
0 9.5
200 .
400 . .
60
400 .
.
600
800
1000
600
800
1000
12 .
10 v
0
200
400 600 time (s)
800
8
1000
2~
0
4~
tin~s)6 ~
(a)
s~
Effect of air flowrate change on the air holdup in the recovery zone keeping h constant (a) Input flowrates (b) measured (dotted line) and simulated (solid line) air holdup in the recovery zone.
Fig.4
17.6
83010
# 27,t_ ~
26/.
o
, 1000
500
I 1~o
s~o
6.5! o
soo
lOOO time (s)
(a}
i
17.4 I
1500
t
~
17.2
lSOO
^ ;A
~11
1500 16.8
Fig.5
1~0
16.6
o
,
I
%?;,
ii
i
le
, ,A
I I
le
I
i L'
s~o
lO'OO
lSOO
time (s)
(b)
Effect of non-floated fraction flowrate change on the air holdup in the recovery zone (a) Input flowrates (b) measured (dotted line) and simulated (solid line) air holdup in the recovery zone.
Dynamic
model
of a flotation
column
1473
13,1 13 ~ 381" v " J 0 34
~
' 100
, 200
'v,,'
, " "-' 300
12.9
400
o~" 12.8 [~ 32
g
0 30
~
~.8.5
0
160 .
360
2bo ,
4o0
12.7
Fig.6
1O0
a;S)
200 time
300
400
jl
I~
,,, 12.6
12.5 0
~
12.4 0
i
. '
,, ~
=lp L, l a I'
100
i
i
,; L
,
j
w
,,
i
i ~
200 time (s)
300
400
(b)
Effect of wash water flowrate change on the air holdup in the recovery zone (a) Input flowrates (b) measured (dotted line) and simulated (solid line) air holdup in the recovery zone.
It is worth mentioning that, in the test in which an overflow of the liquid phase occurred, Figure 3, the final value of the simulated air holdup was 17%, different from the calculated value 15%. In the other tests satisfactory fitting of the model to the calculated data was observed.
Froth layer height The froth layer height is calculated by means of Eq. (3), by using the average density given by Eq. (14). Substituting Eq. (14) in Eq. (3), the froth layer height is then estimated by solving: (-3.6x10 -4 g ( H 2 - H~ ))h 2 + (4.4x10 ' g ( H z - H, ) + (P2 - P~))h - 5.64g(H z - H, ) - (P~H 2 - P2H, ) = 0 (20) The validity of this calculation method may be verified in Figure 7, where plots of height as a function of time are depicted, for several tests, comparing calculated values using Eq. (20) (solid line) with the values measured by the operator (marked with the + symbol). In Figure 7(a), a step change was applied to the nonfloated fraction flowrate at time = 500 s; in Figure 7(b) a step change was applied to the air flowrate at time = 1300 s, in Figure 7(c) a step change was applied to the wash water flowrate at time = 1700 s and in Figure 7(d) step changes were applied to the air flowrate at times = 3100 s and 4200 s. Since, in the experiments shown in Figure 7, the operating conditions vary significantly, Eq. (20) has been used to calculate the height of the froth layer for all experiments. Another way to verify the validity of this method is by the analysis of possible measurement errors. Errors 8P~ and 8P2, associated with the pressure gauges PT-001 and PT-002, and (~Pfzl associated with the estimation of the mean density of the froth layer, are related to the total error in height, 8h, in the froth layer as follows:
ah 6p ' +
Oh 6 P
+
O--~-h 6 p l z
(21)
Considering a typical situation, where h=100 cm, egcz=15% and pfz=O.35g/cm 3, an error of 1% in both PT01 and PT-02 pressure measurements and an error of 10% in ,Ot~, result, according to Eq. (21), in a total error of 9.58 cm in the froth layer height, which corresponds to nearly 10% of the height value. Equations (7) and (8), representing the mass balance, are added to provide the model of the froth layer height concerning the variations in the input flowrates, so:
1474.
M.A.
M,
Persechini e t al.
Ah Pyz + Ah Psz + A(Ht - h) Pc~- A h Pcz QF + Qw - Q c - Q~- -
(22)
Pw
As this is a non-linear equation, the model may be reduced to a linear function using Taylor series expansion, for a certain operating condition, since the variations of these values are small around this operation point, Seborg et al. (1989). Linearising Eq. (22) around an operation point with null derivatives ( h o = 0 , p czo= 0 and p fzo =0 ) and ho, Pczo and Py~o, values selected for the froth layer height and for the densities of the collection and cleaning zones, gives:
Ah o p ,~+ A(H, -ho) Pcz- Ah(pszo - Pczo) Q~ +Qw - Q c - Q r
-
(23)
Pw
75
7O
70, . . . . .
t. . . . .
L
. . . .
.t
i
i F . . . .
i
. . . .
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. . . . . . . . . .
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.
. . . .
.
.
, ---;
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!--
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i
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.....
....
.
i
i
~, . . . .
. . . .
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:
:
~ ..... J
i
i
,
_;
....
• .....
i i ,
411 3,5
i ~'~
~
I
0
500
i
1000
i
~
I
i
i i
lO
1500 2o00 time (s)
i
2500
5OO
3000
IOO0 1500 time Is)
(b)
Co)
120 t
i
-
-~-
~o
_1,5
......
: .......
i. . . . . . .
......
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Fig.7
loo . . . . . ) ..... ) . . . .
_:;/_
t~r~ Is)
i
~
8o . . . . .
~4o o
i
i
....
i
i
~, . . . .
~, . . . .
i
i
....
....
,
:
'
1~
~oo
~x~
~, . . . .
~
....
.~
'
~o
~o
+
-
-'
6ooo
'Tg'
Froth layer height. Calculated values (solid line) and measured values (marked with the + symbol).
Equation (23) is a linearised model that describes the dynamic behaviour of the process around ho, Pczo and P/zo. To describe the relationship between the froth layer height and the input flowrates, the floated fraction flowrate, calculated from Eq. (17), is also linearised around the same operation point to give:
Qc = K ( q g o h + hoQg - qgoho)
(24)
where qgo is the mean value of the air flowrate around the chosen operating point. Replacing (24) in (23), and taking the Laplace transform, the continuous model of the froth layer height is described by:
Dynamic model of a flotation column
1475
OF (s) + aw (s) - Qr (s) - Khoag (s) - shoAD1z (s) - s ( n , - h o)ADcz (s)
h(s) =
sA(Pi~o
-
(25)
Pczo) + Kqgo
where Dyz and Dcz are the Laplace transform of,oy~ and Pcz. Equation (25) may be simplified, taking into consideration that the variations in densities have little effect on the behaviour of the froth layer height, if compared with the variations caused by the flowrates. Further, at steady state, the terms sh,,ADfz(s) and s(Ht-ho)ADcz(S), corresponding to the variations of the froth layer and collection zone densities, respectively, are zero. So:
h(s) =
1.029x10 -~s + 2.3x10 -s - 1.59x10 -4 s + 4.33x10 -7 (26) (s + 4.02xl 0 -4)(s + 1.92x10 -2) (QF (s) + Qw (s) - Qr (s)) + (s + 4.02x10 -4)(s + 7.98 lxl0 -3) Qg (s)
The model represented by Eq. (26) gives satisfactory results for variations in the non-floated fraction flowrate. This may be verified in the test shown in Figure 8, in which a step was applied at time = 550 s and in the test presented in Figure 9, in which, after time = 1000 s the non-floated fraction flowrate was kept constant. In both cases, the froth layer height starts changing at the same time events mentioned above (time = 550 s in Figure 8 and time = 1000 s in Figure 9), and the model correctly reproduces these changes. The model may also be considered satisfactory for changes in the wash water flowrate. In the test presented in Figure 9, the wash water flowrate was changed at 3100 s and 4200 s. At these same time instants the derivative of the froth layer height with respect to time changed its sign and the model fitted to this behaviour. The same effect may be observed in the test represented in Figure 10, where the step was applied at t = 1700 s.
~u~"40
.
.
.
.
100
.
35
90
0 27
500
1000
1500
2000
2500
3000
~-~
80
v
70
28.5
1
, ..;~,..... •
0
26/ 0
~,9.5
560 .
.
1000 .
1500 2000 . .
25'00
3000 ,.
~ 9 ~85
0
40
500
1000
1500 2000 time (s)
(a)
Fig.8
5o
2500
3000
3%
• oa
"
. ,,/.,,.-'."'
60
/ %,., ,.¢,",
560
lobo 1 ~
2~
25'00
tin~s)
Effect of non-floated flowrate on froth layer height (a) Input flowrates (b) measured (dotted line) and simulated (solid line) froth layer height.
In Figure 11, at time = 1300 s, the air flowrate was changed from 34.8 cm3/s to 20.5cm3/s causing, initially, an increase in the froth layer from 30 cm to 70 cm, in only 200 s. This increase may be explained by a decrease of the collection zone volume due to a reduction of the air flowrate. Nevertheless, as the air flowrate reduction causes a reduction of the floated fraction flowrate, the froth layer height decreases again. In the specific case of this test, the column did not reach steady state due to the overflow of the liquid phase after approximately t = 2000 s.
1476
M. A. M. Persechini et al.
A40
120
~ 3s h
IOO
83ol
1000 2900
3000
4000
6000
~" 8o
1000 2000
3000
4000 5000 6000
} 40
5000
~s ~ 2o 0 1o
8t" 0
+
2O
10C)0 20'00 3 0 ~ 4000 time (s)
5000
o
6000
+o~ 2o~o ~
60
1000
2000
3000
'°t
x :
+ / 2000
I000
~1
.,
.
.
.
1000
.
i.
3000
.
.
2000 time (a) (s)
.
~, 40
.
.
.
.
.
,
\'++
;~"
4000
l_ _
3000
101 0
4000
Fig.10 Effect of wash water flowrate on froth layer height a'nd simulated (solid line) froth layer height.
, 1000
, 2000
, 3000
~r~s)
4"000
(a) Input flowrates (b) measured (dotted line)
80
.
N 0
,
4000
~3o 5 201 10
moo
(a) Input flowrates (b)
Effect of wash water and non-floated flowrates on froth layer height measured (dotted line) and simulated (solid line) froth layer height.
0 40
4obo ~
tin~s)
(a)
Fig.9
+
v
6o 0 30.
500
1000
1500
2000
2500
3000
~" ,00 + t •
20L 0
~9.5.
500
08.~
~
l C)00 .
,
~
1500 .
.
2000
2500
3000
,+ '1
,%
I
t +
~ 20 J=
.
1~
time (s)
2 0 o 0 2500 ~
-20
500
10~ 1500
2000
2500
3000
(=)
Fig. 11 Effect of air flowrate (negative step change) on froth layer height (dotted line) and simulated (solid line) froth layer height.
(a) Input flowrates (b) measured
The same effect may be verified in the test represented in Figure 12, where an air flowrate step was applied at t = 200 s in the reverse direction with respect to the former. In this case, the froth Iayer height initially decreases, due to a larger displacement of liquid from the collection zone to the cleaning zone, reaching steady state at t = 6000 s, after increasing again up to 100 cm. A larger floated fraction flowrate was
Dynamic model of a flotation column
1477
observed as a consequence of a larger air flowrate, increasing the froth layer. In both cases the gain at steady state is positive and the dynamic behaviour of the height with respect to changes in the air flowrate presents a characteristic of a non minimum phase system.
24o I 0 3O
. 2000
120
,,~
4000
6000
IOOr 8000
~,9.5
g 8°t
, . t o I, ..i,
~ 6o
~28 026/
.p~, •
~ 40 o
2o'o0 .
4o~o .
6o~o .
8000
"
2C 0
~8.5
0
2000
4000 time (s)
6000
8000
-20
(a)
2000
4000 time (s)
6000
8000
(b)
Fig.12 Effect of air flowrate (positive step change) on froth layer height (a) Input flowrates (b) measured (dotted line) and simulated (solid line) froth layer height.
Bias The bias, defined as the net descending water flowrate through the froth layer, is calculated from the mass balance, Eq. (8), where the collection zone density is given by Eq. (10) and p.,.t= p~ for water-air system. So:
Q8 = - A ( H , - h) "c~,cz- A h(1 - ~ gcz ) + QT - QF
(27)
The Laplace transform has been used to derive the continuous time model after the linearisation of Eq. (27) around a specific operating point, ho and ego, with null derivatives, i.e., h,, =0 and g,o =0, representing an equilibrium point of the froth height and of the air holdup in the recovery zone. Under these conditions: Q8 (s) = Qr (s) - QF (s) + sggcz ( s ) A ( h o - H , ) + sh(s)A(,~ go - 1)
(28)
To determine the coefficients of Eq. (28), ho and 6go have been initially replaced by typical operation values (ho=100 cm and Cgo=15%) and, later, these values were adjusted by trial and error. Equation (28) may be then rewritten as: QB(s) = Q v ( s ) - Q F ( s ) - 3 . 8 4 x 1 0 5 s ~ g c z ( s ) - 4 . 8 x l O 4 s h ( s )
(29)
Figures 13 and 14 show the values of the bias, for different tests, calculated from Eq. (27) (dotted line) and simulated utilising the model described by Eq. (29) (solid line). In both cases, the model gives satisfactory outcomes if compared with the results obtained using the bias calculation, Eq. (25). It is worth noting in Figure 14 that the value of the non-floated fraction flowrate, therefore the difference between the feed and non-floated fraction flowrates, remained constant from t = 1100 s on. The variation of the bias is significant from this instant on, indicating that the process did not reach steady state.
1478
M.A.M.
~0.7
.
.
.
.
P e r s e c h i n i etal.
.
-
,
,
.
,
~.5[ .... ~,..... !, ..... !,..... !, ..... i---, 0.5
500
1000
1500
2000
2500
8_- - - - ~ . . . . . ~. . . . . :. . . . . ~ . . . . . ,--.
3000
0.46
~" 75• .....
/
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50.42
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.
7 -
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:.....
i_.._~_~. ,
'
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,
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0
500
1000
1500 time (s)
2000
2500
3000
5.5
II
~
:',
0
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'
.
.
.
,
0 0.51
2000
.
,
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/
8000
10000 12000
o.,l"l'VVvl'! . . . . 0 2000 4000 6000 8000
10000 12000
-L
"----;
~; !' !1
~ o.,~ g ~0.15
f"
o.i~
2obo 4 ~
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i
l 3000
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6000
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(a) Input flowrates (b) measured (dotted line) and simulated (solid
.
,"
4000
I
time(s)
8,
~ o.551
, i
(b)
Fig.13 Effect of air flowrate on bias line) bias. 0.65
I
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(a)
~
.,
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,
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4000
6000
8000
10000 12000
time(s)
time (s)
(a)
(b)
Fig. 14 Effect of wash water flowrate on bias (solid line) bias.
(a) Input flowrates (b) measured (dotted line) and simulated
PILOT PLANT MODEL For control design purposes, it is convenient to write the pilot plant model in matrix form. This facilitates, for instance, the correlation analysis of controlled and manipulated variables. The pilot plant model which has been obtained for operation in a two phase (water-air) system can then be rewritten as:
rhl[ Egcz
[_QB J
=
o G21
s(K~G,~ + KzGzl )
G22
s(K~G~2+ KzGz2 )
G23
1+
Q~
s(K~G]3 + KzG23 ) Qr - Qv
where the transfer functions
Glt
- (1.029x10-3 s + 2 . 3 x 1 0 -5) (s + 4 . 0 2 x 1 0 - 4 ) ( s + 1 . 9 2 x 1 0 -2)
(31)
(30)
Dynamicmodel of a flotationcolumn
1479
- 1.59x10-4 s + 4 . 3 3 x 1 0 -7 G12 -
(s + 4.02x10-4)(s + 7.98 l x l 0 -3)
(32)
( 1 . 0 2 9 x 1 0 -3 s + 2 . 3 x 1 0 -5)
G~3
(s + 4 . 0 2 x l 0 - 4 ) ( s + 1.92X10 -2 )
(33)
are related to the froth layer height model (Eq. 26), whereas
G~ 1 =
7.6x10 -5 s + 1.92x10 -2
(34)
7 . 7 8 x 1 0 -5 G22 =
s + 7.8 l x l 0 -3
(35)
- 7 . 6 x 1 0 -5 G23 = s + l . 9 2 x 1 0
-2
(36)
are related to the air holdup model in the recovery zone (Eq. 19). In Eq. (30), the relation between the bias and the manipulated variables (wash water, air and non-floated fraction flowrates) is obtained by substituting Eqs. (19) and (26) in Eq. (29), and from Eq. (29) -4.8x104=KI, and -3.84x105=K2. This model is valid under the following conditions: • • •
•
•
feed flowrate constant and equal to 20 cm3/s at 10 ppm frother dosage; floated fraction flowrate greater than zero; non-floated fraction flowrate greater than zero; froth layer height between lower and higher limits (the lower limit is 20 cm, corresponding to the level where the wash water is introduced; the higher limit is 140 cm, corresponding to the minimum value for which the pressure gauge PT-01 was calibrated); column operating in a bubbly flow regime.
To achieve these conditions, the variations of the input flowrates must be constrained to certain limits. Initially, the air flowrate should not be varied stepwise to avoid transient effects on the froth layer height, as shown in Figures ll(b) and 12(b).The constraints imposed to the non-floated fraction, wash water and feed flowrates are:
Qw +QF-QT Qw + QF -
>o
Q r < 0.25Qg
(37) (38)
The limit imposed by Eq. (37) is necessary for the flow of the floated fraction, or else the column will empty. On the other hand, the sum of flows should not be too large to prevent the froth layer from quickly disappearing, causing a liquid phase overflow. The maximum value of the left hand side in Eq. (37) depends on the floated fraction flow capacity, which is directly related to the air flowrate. According to Finch and Dobby (1990), the air holdup at the top of a column may reach 80% or even more. Taking this observation into account and using Eq. (15), the higher limit for the flowrates is determined by Eq. (38).
M.A.M. Persechiniet al.
1480
CONCLUSIONS From dynamic mass balances and the analysis of the data gathered from a pilot-scale column (both for the air-water system), a simplified representation of the process dynamics has been obtained through a linear, multivariable model. This model was derived aimed at its utilisation in control. The manipulated variables are the flowrates of the air, the wash water and the non-floated fraction; the controlled variables are the froth layer height, the air holdup in the recovery zone and the bias. The model derived in the present work may be considered adequate to represent the process with respect to steady-state gains between manipulated and controlled variables and the main time constants. Nevertheless, it is still necessary to consider some important aspects, mainly in the description of the froth layer height behaviour. In particular, the initial response of the froth layer height to stepwise changes of the air flowrate is inverse to the steady state gain, characterising a non minimum phase system. Provided that the conditions under which the model was derived are respected, the representation obtained may be utilised as a tool for studies and behaviour simulations of the process variables and as an aid to the determination of a control strategy adequate for column flotation. Finally, the same methodology used in this work for obtaining the dynamic model of the flotation column, operating in a water-air system, has been used to develop a model considering the effects of solids. Aiming at this goal, the mass balance equations were rewritten taking into account the different densities for each of the process flowrates (QF, Qr, Qc and Qw), and experimental data are now being gathered for further analysis.
ACKNOWLEDGEMENTS The authors acknowledge the technical support provided by Setor de Tecnologia Mineral da Supervisdo de Processos do Centro de Desenvolvimento da Tecnologia Nuclear (CDTN) responsible for the design, construction and operation of the pilot scale flotation column. We are also thankful to ATAN Automation Systems Ltd. for providing the software tools: QNX 2.2, RealFlex 1.22 and the communication driver for the CD-600. Nomenclature A D G H J P Q U
Column section Density Transfer function Height Superficial rate Pressure Flowrate Signal command
V g h m A 6 ~ p
Volume Gravity acceleration B Froth layer height C Mass F Difference T Error W Holdup cz Density fz
SUBSCRIPTS Bias Floated fraction Feed Non-floated fraction Wash water Collection zone Cleaning zone
g l o s
Air Liquid Operating point Solid sl Pulp t Total height w Water
REFERENCES Bergh, L. G. and Yianatos, J. B., Dynamic simulation of operating variables in flotation columns. Minerals Engineering, 1995, 8(6), 603-613. Finch, J. A. and Dobby, G. S., Column flotation, 1st edn. 1990, Pergamon Press, Oxford. Ljung, L., System identification, 2nd edn. 1999, Prentice Hall, New Jersey. Pu, M., Gupta, Y. P. and A1 Taweel, A. M., Model predictive control of flotation columns. In Proc. Column'91. Sudbury, Canada, 1991, pp. 467-479 Seborg, D. E., Edgar, T. F. and Mellichamp, D. A., Process dynamics and control, 1989, John Wiley & Sons, New York Tavera, F. J., Gomez, C. O. and Finch, J. A., Estimation of gas holdup in froths by electrical conductivity: Application of the standard addition method. Minerals Engineering, 1998, 11(10), 941-947.
Dynamic model of a flotation column
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del Villar, R., Gr6goire, M. and Pomerleau, A., Automatic control of a laboratory flotation column. Minerals Engineering, 1999, 12(3), 291-308. Yianatos, J. B., Finch, J. A. and Laplante, A. R., Holdup profile and bubble size distribution of flotation column froths. Canadian Metallurgical Quartely, 1986, 25(1), 23-29. Yianatos, J. B, Laplante, A. R. and Finch, J. A., Estimation of local holdup in the bubbling and froth zones of a gas-liquid. Chemical Engineering Science, •985, 40(10), 1965-1968.
Correspondence on papers published in Minerals Engineering is invited by e-mail to bwills @ m i n - e n g . c o m