an improved model for simulation of flotation circuits -j.p ferreira

Pergamon Minerals Engineering, Vol. 13, No. 14-15, pp. 1441-1453, 2000

© 2000 Published by Elsevier Science Ltd All rights reserved

0892-6875(00)00129-1 0892-6875/00/$- see front matter

A N I M P R O V E D M O D E L F O R S I M U L A T I O N O F F L O T A T I O N C I R C U I T S

J.P. F E R R E I R A II a n d B.K. L O V E D A Y §

I[ School of Chemical Engineering, University of Natal, Durban, 4041, South Africa (now at Mintek, Private Bag X3015 Randburg, 2125, E-maih [email protected])

§ School of Chemical Engineering, University of Natal, South Africa. E-mail: [email protected] (Received 3 July 2000; accepted 31 August 2000)

ABSTRACT

Batch testing of pulp samples from various streams in a flotation plant has made it possible to model plant behaviour more accurately. A new and very flexible model is introduced for characterizing the floatability of plant feed in terms of a continuous distribution of rate constants. This model, which is the sum of two normal distributions, provides a more accurate means of modelling a complete set of data from a flotation circuit, particularly the middlings. A new froth model is also introduced which relates the recycle of particles from the froth to the pulp under plant conditions. The models were fitted to published data from Cominco' s Red Dog lead cleaning circuit and were compared to conventional discrete models. The recovery predicted by the models differed by up to 2%. ©2000 Published by Elsevier Science Ltd. All rights reserved.

Keywords Flotation kinetics; flotation froths

INTRODUCTION

There is an incentive for optimising flotation circuits because they process large tonnages of material on a daily basis and hence an increase in recovery of as little as 0.5% could be economically significant. The tools for quantifying such small improvements in recovery are not currently available and so even complex circuits are still designed by direct scale up of empirical laboratory and pilot-plant test work. Experimental techniques are limited in their application due to errors in flow and assay determinations, as well as variations in fee.d composition. This restricts the number of design variations that can be considered when designing a new flotation plant. Simulation provides a solution, provided that we can find mathematical models that are robust and accurate. This gives the designer the freedom of proposing many different circuit arrangements and flow rates without any great expense or loss of time. This paper introduces new continuous models for describing the pulp and froth phases respectively and compares them to the existing models.

MODEL TYPES

First order rate models

It is well known that particle/bubble attachment in a flotation cell is well described by a distribution of first order rate constants (Fichera, et al., 1992). Hence batch flotation tests produce a highly non-linear recovery

1441

1442 J.P. Ferreira and B. K. Loveday

vs. time profile that is independent of initial concentration (within limits, e.g. bubble overloading). Two main model types have been used to model distributions of first order rate kinetics, i.e. discrete and continuous models.

Discrete model

n

r e c = ~-~m i . ( 1 - e x p ( - k i . t ) ) i=1

where

(1)

rec n mi ki

is the cumulative recovery at time (t) is the number of floatability classes is the mass fraction of particles in floatability class i is the first order rate constant of particles in floatability class i

and

n

~-~m i = 1 i=1

(from the definition of a distribution)

Continuous model

kmax

r e c = f f ( k ) . ( 1 - e x p ( - k . t ) ) d k o

where

(2)

f(k)o dk is the mass fraction of particles with a rate constant between k and k + dk. Hencef(k) is a frequency function by mass.

k,,a~ is the maximum rate constant of the distribution.

and

kraax

~ f ( k ) . d k = l o

(from the definition of a distribution)

Figures 1, 2 and 3 illustrate continuous and discrete models fitted to recovery data as described later in the paper. For the discrete models n is the number of floatability classes.

Improved model for simulation of flotation circuits 1443

.£

=

t_ t L

18

16

14

12

10

8

6

4

2

0 i

0 0.1 '0.2 0.3 '0.4 Rate constant, k [min -1]

().5 0.6

0

&

0.7

0.6 . . . . . . .

0.4

0.3 - -

0.2

0.1

0 0

0.7

Fig. 1 Distribution of flotation rate constants in the feed for Pyrite.

0.037 I

0.163 Rate constant, k [min 4]

Fig. 2 Distribution of flotation rate constants in the feed for Pyrite n=3.

0.6

0.6

,- 0.5 0

0.4

0.3 . . . .

0.2

0.1

0 ~ 0.01 0.06

I Unfloatable fraction very small

0.27 Rate constants, k [min 11 0.6

Fig. 3 Distribution of flotation rate constants in the feed for Pyrite n=4.

Froth models

Probably the biggest stumbling block to the accurate modelling of flotation circuits is understanding what happens in the froth phase. It is known that recycle of material from the froth phase back to the pulp phase can be as high as 90%, i.e. this is a critical part of the overall flotation model.

Particle recycle between the froth phase and the pulp phase has been modeled by introducing a froth recovery factor (Ry). This factor is often assumed to be unity in laboratory batch tests and has the effect of decreasing the flotation rate constant for plant cells (Runge, et al., 1997). The implication is that the rate constant, k, for a flotation process may be expressed as a product of independent components, i.e.:

k = P . S b • R s (3)

where

P Sb

middlings.

is the mineral floatability is the bubble surface area flux is the froth recovery factor

M O D E L L I N G APPROACH

In recent years, significant progress has been made in the development of optimisation packages for flotation networks. These packages make use of distributed first order models, however they are usually of the discrete type, as calculations for flotation circuits are much easier using discrete models and the fit to batch data is just as good. Continuous models involve time consuming integration routines and difficult mathematical manipulations. Hence in the past, continuous models were limited to forms that were easy to integrate. However, the advent of more powerful computers has made it possible to explore more complicated mathematical forms. The ability of a model to reproduce the batch data is not a good enough criterion for ranking models, because a typical flotation circuit contains at least 3 stages with circulating loads that can exceed the flow of floatable material entering the flotation circuit (Loveday and Brouckaert, 1995). Large circulating loads cause a build up of middlings materials in the circuit and hence the influence of middlings is enhanced. Model parameters obtained by regression to simple batch data are insensitive to middling material since the fast floating material is dominant in these tests. Figure 1 shows the distribution that was obtained when fitting a continuous model to data for pyrite, in a circuit in which it is depressed. The overall recovery is shown in Figure 4 as a function of k. This is analogous to a Tromp curve. Figure 5 shows the flow in the recycle stream (cleaner tailings) as a function of k, showing a large recycle of

100 90 80

.-. 70 *~ 60 ~' 50 ~ 4o N 30

20 10 0

o

m

0.1


i i

0.'2 0.3 0.'4 0.5 Rate constant, k [rain .i]

400 350 300 250 200 150 100 50

06

FigA Recovery in the final concentrate.

k,o 0~1 0.'2 0.'3 Rate constant, k [min "~l

0.4 0.5

Fig.5 Flow in the recycle stream relative to feed flow.


The above complications have lead to an approach that makes flotation models more sensitive to middling materials. This was achieved by considering mineral fractionation within a flotation circuit. The flotation process may be viewed as being analogous to distillation, because the mineral composition of each stream in the flotation circuit represents the outcome of a fractionation process of the original feed to the circuit. Hence if the circuit is operating at steady state and the floatability of individual particles does not vary with time, then the floatability distribution of all the streams within the circuit can be related back to the feed or "parent" floatability distribution. This is best illustrated if we consider a flotation stage (e.g. rougher bank) at steady-state (Fig. 6).

>

Fig. 6 Flow diagram for a flotation stage.

If the rate constant distribution, ft(k), for the feed (stream 1) is known, we can calculate the distribution in any of the other streams as follows:

r,(k)

o

(4)

where f i (k) ri(k) km~

is the frequency function for particles with rate constant k in stream i is the recovery of particles with rate constant k to stream i is the maximum rate constant achieved by particles in the feed

This approach involves taking samples from the various streams around the circuit (streams 1 to 3 above) and performing batch tests on these samples to generate recovery vs. time data for each stream. The floatability distribution of the feed is characterised by assuming a model for fl(k). Fitting stream batch flotation data to equation (5) then identifies the model parameters. This produces a family of curves as in Figure 7. This approach was first developed and used by Marais and Valente (1994). Loveday and Raghubir (1995) and Runge, et al. (1997) have also reported work on this type of modelling.


==

100

90

80

70

60

50

40

30

20

1 0

0 I

0

f . / O . j / j . _ ~ j . . . . . .

• 1 exp.

- - 1 mod.

• 2 exp.

- - 2 mod.

x 3 exp.

- - 3 m o d .

- - 4 mod.

+ 4 exp.

- 5 exp.

- - 5 mod.

• 6 exp.

- - 6 m o d

5 10 15 20 25 30 35 40 45 50 Time [mini

Fig.7 Regression of the 7 parameter continuous model to batch test data for cumulative recovery of galena from plant samples taken at the positions indicated in Figure 8.

kmax(j)

Ri, j = R=u . ~f, , ( k ) . ( 1 - e x p ( - k . t ) ) . d k (5) o

where

Ri, j

R~ij

kmo~-)

is the recovery in stream i of mineral j at time t is the fractional recovery at infinite time in stream i from mineral j (i.e. 1 - R, ij is the unfloatable fraction) is the maximum rate constant for mineral j

The circuit

Data for this investigation was taken from a paper by Runge, et al. (1997), which analysed data from Cominco's Red Dog Mine in North America. Figure 8 shows the Red Dog circuit configuration at the time the test work was being conducted. The cleaning circuit consisted of a column unit and two conventional mechanical cells acting as a cleaner scavenger. The feed to the cleaning circuit was the combined rougher concentrate and the major minerals present were galena, sphalerite and pyrite. All other gangue minerals were grouped and referred to as non-sulphide gangue minerals.


1 Combined Pb

Rougher Cone.

.""" • ....... Node 2 "' "' -,

..'"

• " I PbCl0 0r I

k ' " ' . .

" " " , . , . . . . . . . . .

6 Pb Cleaner Scavenger

cone

all' 'qL

I

~ode I

. , . , , * . ' "

2_ Pb Column Feed

4 p Pb Column Tails

s Pb Cleaner Scavenger

Tail

. . , . . . . . . . . .

. . . ' " ' N o d e 3 ' " ' . . .

Column

".o .°

" ' . . . . . . . . . . . , , ' "

_3 Pb Column Concentrate._

v

Fig. 8 Flow diagram of the Red Dog lead cleaning circuit.

Circu i t model

The various flotation units were modeled, assuming perfect mixing in each cell. Various assumptions about mixing in the column cell were tested. Hence the recovery for a single rate constant (k) in each unit was expressed mathematically as:

R = 1 - - 1 (6)

where

n

z-

and

is the number of cells in series

is the total residence time in the bank of cells

k plan t = (~ "kbatc h (7)

where

is the scale up factor that accounts for the differences between the batch cell and the full-scale plant units.

The model parameters were then determined by simultaneous regression to all batch recovery data.


NEW MODELS

First order distributed rate model of the feed

The new model is continuous and takes the form of 2 superimposed normal distributions, for values of k ranging from zero to a maximum value (kmo~). The parameters kl and k2 indicate the value of k at the peaks.

(kl--kbatch) f ( k ) = b 1 . e x p (s,) 2 (k -- kbatc h

+b2"exp (s2)2 (8)

Parameters s~ and s 2 are the standard deviations of the 2 curves respectively and have the same units as k [min-~]. This model was selected because of its flexibility. For example, if the parameters kl or k2 are negative, the distribution takes the form of a steeply falling curve for values of k greater than zero. This shape is typical for gangue minerals.

The maximum value of k was also used as a parameter for regression, but it usually had little effect, as the curve was approaching zero anyway.

Froth model

As mentioned previously, particle recycle between the froth phase and the pulp phase was accounted for by the introduction of a froth recovery factor (Rf). This factor has the effect of decreasing the flotation rate constant (Runge, et al., 1997).

Recovery is always expressed in terms of the product of a rate constant (k) and time in a cell or cells. If it is assumed that the froth recovery factory (Rf) is constant for all minerals, it has the same effect as decreasing the residence time in the cell. The effective residence time is then defined as:

Teff = Rf''g'(gb)plant/(Sb)batch

and

k = kbatc h

No residence time or bubble surface area flux data was published by Runge et al. and hence the effective residence times are regressed empirical parameters.

Analysis of data from the Red Dog circuit showed that slow floating gangue minerals have lower effective residence times. Figures 9 and 10 show the relationship obtained when plotting the regressed effective residence times against the average rate constant for each of the 4 mineral groups present in the cleaner column and cleaner scavenger respectively. Clearly the froth factor is a function of particle floatability and it should not be changed for each mineral as an empirical factor.


3.5

E 2.5 ==

I'-

1.5

0.5

// o'.o o'.1 6. 0'.2 6. o'3 0'.4

kave [min "1]

Fig.9 Effective residence time vs. average rate constant for the Cleaner.

70

60

5O

"~ 40

30

2oi 1° i

'. ' ' ' o'.2 ' ' ' d 0 0 0. 0.1 0. 0. 0.3 0. 4

kM [min'll

Fig. 10 Effective residence time vs. average rate constant for the Cleaner Scavenger.

This observed relationship gave rise to the development of a new empirical froth model:

R I =[1-al"exp(-a2"k~,ch)] (9)

This model takes a "black box" approach, instead of accounting for all the individual factors that impact on recovery in the froth phase. The parameters al and a2 characterize the froth condition in a particular bank of cells or column cell and apply to all minerals. Hence it will also include the effect of entrainment.

S IMULATION RESULTS

As mentioned previously, the model parameters were determined by simultaneous regression to batch flotation data for all the streams, as shown in Figure 7. The regression package was written in the Matlab programming environment and can deal with both the continuous and discrete cases. Tables 1, 2 and 3


summarize the parameters obtained for the first order rate model using continuous and discrete models respectively, and Table 4 summarizes the parameters obtained for the froth model and circuit model.

TABLE 1 Continuous model parameters (Equation 8)

Parameters

bl kl [min -1] Sl [min -1]

bE k 2 [min -1] S 2 [min -1]

km~, [min -1] (1-R~)

Galena

5.7551 0.0462 0.0085 1.3586 0.3632 0.4295 1.3271

0

Sphalerite

21.7417 0.0182 0.0136 2.7459 0.0327 0.1660 0.4806 0.0275

Pyrite

16.9294 0.0133 0.0392 1.1989

-0.1320 0.3251 0.5946

0

NSG

54.9168 -0.0246 0.0388 4.9060 -0.3723 0.3902 0.7183 0.0009

TABLE 2 Model parameters for the 3-fioatability class, discrete first order rate model

Parameters

ml m2 m3

k 1 [min -1] k2 [min- 1] k3 [min- 1]

Galena

0.0234 0.1813 0.7953

0 0.0607 0.4296

Sphalerite

0.2102 0.5015 0.2884

0.0319 0.1440

Pyrite NSG

0.2410 0.3433 0.6028 0.4698 0.1562 0.1869

0.0370 0.1608

0 0.0298 0.1624

TABLE 3 Model parameters for the 3-fioatability class, discrete first order rate model

Parameters Galena Sphalerite

m2

ml 0.0098 0.1239 0.1396 0.5470

m3 m4

k I [min -1] k2 [rain -1] k3 [rain -1]

0.4075 0.4431

0.0461 0.2143 0.7046 k4 [rain -1 ]

0.2949 0.0342

0.0239 0.1111 0.6458

Pyrite NSG

0.0010 0.0010 0.6194 0.7264 0.2998 0.0798

0.0161 0.0604 0.2754

0.1915 0.0811

0.0123 0.0745 0.3531

TABLE 4 Model parameters for froth and circuit models

Parameters

al a2

Cleaner Column Continuous Case

1.0 10.8231 1.7545

1.5

Scavenger Continuous Case

0.7582 17.6487

Cleaner Column Discrete

Case 1.0

11.1942 1.7180 2.8408

1.5

Scavenger Discrete

Case 0.6981 12.1373 2.9173

Improved model for simulation of flotation circuits 145 l

From the above the discrete model with 3 floatability classes is clearly unrealistic if we consider the large unfloatable fractions predicted for a rougher concentrate. The discrete model with 4 floatability classes and the continuous model both gave much improved fits to the batch data, of the order of 20 percent. Although these two models both fitted the data equally well this does not mean that they will give the same simulation results.

Having moved from a discrete model with 4 parameters to discrete and continuous models with 6 and 7 parameters respectively, the question of over-parameterisation needs to be addressed. Due to the large number of data points that were available from the circuit it was found that these models were very robust. Mass-balance smoothing was used to adjust the raw data prior to fitting the models. It should be noted that the data from the Red Dog cleaning circuit relates to only 2 stages and that most plants will have at least 3 stages. This would generate more data and hence the technique would become more robust.

The model parameters were used as inputs for a simulation package. The simulation package was also written in Matlab and predicts the recovery and grade of the various minerals over a wide range of residence times. The program also predicts the grade and recovery of the circuit for the addition of a re- cleaning stage. Figure 11 summarizes the simulation results obtained when varying the residence times around the circuit.

0

-0.2

-0.4

-o.6

~ -0.8

~, -1

~ -1.2

~ -1.4

-1.6

-1.8

-2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 I T I I I I I I f

Relative residence time

Fig. 11 Comparison between the simulated galena recoveries for the continuous and 4 floatability class discrete models.

A relative residence time of 1 represents the operating conditions of the plant. Hence a relative residence time of 0.5 means that both the cleaner and scavenger are being simulated at 50% of their operational residence times. From this simulation it is clear that the models are sensitive to a reduction in residence time and their predicted recoveries differ by up to 2%. These discrepancies become important when trying to determine whether a proposed plant modification would be economically viable.

The second stage of the simulation involved changing the circuit configuration by addition of a re-cleaner stage. Any modifications to the circuit will chance the shape of the separation curve that characterizes the system (Figure 12) as well as the quantity of circulating material in the system (Figure 13).


2 Stage 90 L circuit/ / 80 / / / 3.Stage

~ 7° f ~ 60 / . _ / circuit

50 . . . . . . . .

40

20 I J

10 ' i

0 0 -0.1 012 013 014 Rate constant, k [min "i]

0.=5

1200

1000

O

Fig. 12 Recovery in the final concentrate.

800

600

400

200

0 0.'1 0~2 0.'3 Rate constant, k [rain "=]

0 .'4 0 .'5

Fig. 13 Flow in the recycle stream relative to feed flow.

From Figure 12 it is apparent that the addition of re-cleaner causes a significant shift in the cut point (ks0) of the circuit separation curve. The shape of the distributions for galena, sphalerite, pyrite and gangue in the vicinity of the cut point (ks0) determines how recovery of these minerals will change. Figure 13 shows that changes in circuit configuration can have a significant impact on the quantity of middlings circulating around the system. All these factors emphasize the sensitivity of the circuit to the middlings and hence the need to model this material accurately. Referring back to Figures 2 and 3, it is clear that discrete kinetic rate distribution models are not able to model this critical range very well. Hence it is suggested that simulation results obtained from discrete models, will always be suspect, particularly if the cut-point is altered. The new flexible continuous model appears to be capable of describing this critical range of rate constants more effectively.

Table 5 summarizes the predicted recoveries obtained from the various models when simulating a 3-stage circuit. Of particular interest also are the large discrepancies between the discrete models with 3 and 4 floatability classes respectively.


TABLE 5 Comparison between the simulated recoveries of the various models for a 3-stage circuit

Mineral Continuous A 4 Floatability class A 3 Floatability class Model Discrete Model Discrete Model

Galena 83.86 - 1.10 84.96 3.42 81.54

Sphalerite 24.14 - 2.08 26.22 -0.1 26.32

Pyrite 13.55 1.82 11.73 -3.6 15.33

NSG 15.96 - 1.92 17.88 0.1 17.78

CONCLUSIONS

Modelling and simulation exercises conducted using published data from batch tests on various streams in the Red Dog cleaning circuit have shown that:

The method of using batch test data from streams around the circuit to determine the distribution of rate constants in the feed is a very powerful tool. This is because it takes into account the importance of middlings and recycling material.

• Discrete models do not model the middlings well enough to be considered robust simulation tools. Hence the need for flexible and robust continuous models.

• Batch test data from plant streams provide sufficient information to warrant the use of 6 or 7 parameters in modeling the distribution of rate constants in the feed.

• Froth recovery (Ry) is not constant for all minerals but is a function of the rate constant with which the particles float (in a batch cell).

REFERENCES

Loveday, B.K. and Brouckaert, C.J., An analysis of flotation circuit design principles. The Chemical EngineerJ., 1995, 59, 115-21.

Loveday, B.K. and Raghubir, S., Design and optimisation of flotation circuits using simulation. In Proceedings of SAIMM Colloquim - Interactions between comminution and downstream processing, Mintek, Randburg, 1995.

Marias, P. and Valente, M., Personal communications, 1994. Runge, K.C., Manlapig, E.V., Frew, J.A. and Harris, M.C., Floatability of streams around the Cominco Red

Dog lead cleaning circuit. In Proceedings of the Sixth Mill Operators' Conference, Mandang, Papua, New Guinea, AuslMM, Publication Series 3/97, 1997, 149-155.

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