A three-phase model of froth flotation

International Journal o f Mineral Processing, 34 ( 1992 ) 261-273

261

Elsevier Science Publishers B.V., Amsterdam

A three-phase model of froth flotation G.S.

Hanumanth a and D.J.A. Williams b

aDepartment qf Materials Science and Engineering, McMaster University, 1280 Main Street West. Hamilton, Ontario, Canada L8S 4L 7 bDepartment of Chemical Engineering, University College of Swansea, Singleton Park, Swansea. Wales, UK CB2 3RA (Received 20 November 1990; accepted after revision 1 July 1991 )

ABSTRACT Hanumanth, G.S. and Williams, D.J.A., 1992. A three-phase model of froth flotation. Int. J. Miner. Process., 34:261-273. A three-phase model of froth flotation, consisting of a pulp and two distinct froth phases, is developed for describing mass transport in flotation cells. Methods are proposed for estimating the various rate coefficients contained in the model. A comparison of predictions with experimental data for china clay demonstrates that the model can successfully predict the pulp phase kinetics, as well as kinetics of solid drainage from froth and product recovery, over a wide range of froth depth, 0.12 m to 0.46 m.

INTRODUCTION

The multiphase approach to model flotation kinetics involves the division of the flotation cell contents into two or more well-mixed parts that combine to form a coherent system. Each of these integral parts is commonly referred to as a "phase" (Harris, 1978 ). The subprocesses occurring in each phase is collectively represented by an equivalent macroprocess, characterized by a rate coefficient. For example, the subprocesses of particle-bubble collision, attachment of particle to bubble followed in some cases by detachment, and entrainment of pulp by the rising bubble network are all collectively represented by one macroscopic process, characterized by a rate coefficient for the transfer of solids into the froth phase. A large body of work using such heuristic rate coefficients exists in the literature. Early modelling attempts dealt mainly with the pulp phase. For mechanical cells, Arbiter and Harris ( 1962 ) were among the first to take the froth phase into account. Their two-phase (pulp-froth) model of the flotation Correspondence to: G.S. Hanumanth, Department of Materials Science and Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4L7.

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G.S. HANUMANTH AND D.J.A. WILLIAMS

process was based on the assumption that both pulp and froth are well-mixed. Harris et al. (1963) and Harris and Rimmer (1966) carried out extensive testing of the two-phase model under both steady state and transient conditions. Lynch et al. (1974) have included essentially the same assumption in their model of froth behaviour. The pioneering work of Harris and coworkers at Columbia University on flotation modelling, and the contributions made by other researchers have been discussed by Harris ( 1978 ). As a theoretical improvement ot the well-mixed hypothesis some authors have attempted to describe froth behaviour by applying commonly used flow models. Plug flow with axial dispersion model was considered by, among others, Bushell ( 1962 ), Cooper ( 1966 ), and Sastry and Fuerstenau (1970). Cutting and Divinish ( 1975 ), and Moys ( 1978 ) also developed plug-flow models to describe the behaviour of deep froths in flotation cells. More recently, Moys (1984) examined the applicability of potential flow theory to froths. Residence time distribution measurements made by him using the conventional impulse-response technique in two-phase (liquid-gas) froths, showed good agreement with potential flow theory. In this paper we propose a new three-phase model to describe the kinetics of flotation process, including drainage of solids in the froth layer, and its dependence on froth height. In this model the cell contents are partitioned into a pulp phase, and two distinct froth phases in a manner that approximately resembles the observed structure of flotation froth, and allows the quantification of froth drainage rate and its dependence on froth height. Procedures are outlined to estimate the unknown parameters of the model, and the model is evaluated quantitatively by comparing its predictions with experimental results. THE MATHEMATICAL MODEL

The physical model is illustrated in Fig. 1. The following assumptions are made: 1. The pulp phase is well-mixed. The rate of entry of solids from the pulp phase into the froth phase is proportional to the instantaneous mass of solid present in the pulp. 2. The froth phase is divisible into two well-mixed layers. The first of these layers adjoins the pulp phase, and will be referred to as the primary froth phase (denoted by the subscript pf ). The remaining portion of the froth phase constitutes the secondary froth phase (denoted by the subscript sf). The physical characteristics of these froth layers gathered from observations are briefly discussed below. The depth of the primary froth phase is fixed for a given set of flotation conditions, and is independent of the total froth height. In this froth phase the gas volume fraction is relatively small, the resulting high liquid and solid

THREE-PHASE MODEL OF FROTH FLOTATION

263

PRODUCT

Ku,sf

SECONDARY FROTH PHASE

K ,pf

Kd,sf PRI MA2~A~0TH

Ku ,p

Kd ,pf

PULP PHASE FEED

Ql,st

Ct,st

Qo,sI

Co,sl

TAILINGS Fig. ]. A simp]ified illustration of the three-phase mode] of froth flotation wherein Q denotes volumetric flowrate, C concentration, and K rate coefficient.

content prevents the coalescence of bubbles to a large extent. There is a constant exchange of solid and liquid by a two-way mass transfer with the pulp and the adjacent froth phase. The rate of mass transfer from this phase is proportional to the instantaneous mass of solid present in the layer. On the contrary, the height of the secondary froth phase can be altered by relocating the froth recovery lip (Hanumanth and Williams, 1990). There is a two-way mass transfer across the interface with the primary froth phase, and the froth exits the flotation cell at the froth lip. Gas volume fraction is relatively high resulting in a greatly reduced solid and liquid content, as compared to the primary froth phase. Severe bubble coalescence is indicated by the presence of large bubbles, with diameters of the order of centimetres, as

264

G.S. HANUMANTH AND D.J.A. WILLIAMS

compared to bubble sizes of the order of millimetres in the primary froth phase. As before mass transfer rate is taken to be proportional to the instantaneous mass of solid present in the layer. Evidence for the assumption of a well-mixed pulp phase in a mechanically stirred cell is available in the literature, for example (Arbiter and Harris, 1962), and was confirmed in our experimental cell ( H a n u m a n t h and Williams, 1988). Further, in the case of the froth such an assumption can be somewhat justified by taking into account the different mixing processes occurring in the froth; for example, it has been demonstrated that both solid and liquid dispersion are important phenomena in flotation columns ( Dobby and Finch, 1985 ), mechanical froth removal processes induce froth flow which results in some measure of mixing, and bubble film rupture and coalescence lead to additional mixing. In view of the localized effect of some of these mixing processes, it is more realistic to divide the froth into two or more mixed phases. Two mixed froth phases are found to be adequate for our purpose, as results will below indicate. MASS T R A N S P O R T E Q U A T I O N S

The above assumptions lead to the following mass balance relationships: Pulp phase

dWF,

dt - K d ~ r W p f - Ku.p Wp - Qi.s, Ci.s, - Qo.~, Co.s,

( 1)

From the assumption that the pulp phase is well-mixed eq. ( l ) can be recast in the form: dWp dt

- Kd'pf W p f - -

a Wp "[- Qi,sl Ci,sl

(2)

where a=Ku,p + Q o . J V p

(3)

The physical significance of the quantity a is that it indicates the gross rate of depletion of solid in the pulp phase for a continuous flotation cell. Primary froth phase dWp dt - Ku.p Wp + Kd.sf Wsf -- ( Ku.pf + gd.pO Wpf

(4)

ATHREE-PHggEMODELOFFROTHFLOTATION

265

Secondaryfroth phase dWsf dt

--Ku'pfWPf--(gd'sf-Jrgusf) Wsf

(5)

In the above equations subscript u is used to represent upward mass transfer, and d the downward flow towards the pulp. Wp and Wpf were eliminated from eqs. ( 2 ) - ( 5 ) (by applying elementary differential operator calculus; see Wylie, 1979) to obtain a single third order ordinary differential equation in Wsf, the approximate roots of which were obtained numerically using the quotient/difference algorithm described by Burden et al. ( 1981 ). the approximate roots were then improved upon by Newton-Raphson iterations. Subsequent mathematical steps to yield W,f, Wp and Wpf are routine. ESTIMATION OF THE HEIGHT OF PRIMARY FROTH PHASE (hpf)

The primary froth phase is the layer adjacent to the pulp wherein solid concentration is uniform. The height of this layer is estimated empirically from semi-batch experiments (Hanumanth and Williams, 1990) as follows: 1. The volume of froth recovered in any given interval of time during the flotation of china clay was calculated by summing the individual volumes of 16,

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Fig. 2. Variation of solid concentration in froth as a function of time for different froth

heights hf.

266

G.S. HANUMANTH AND D.J.A. WILLIAMS

solid, liquid and air constituting the froth. The solid and liquid volumes were calculated gravimetrically as described elsewhere ( H a n u m a n t h and Williams, 1990), and the volume of air was taken to be equal to the total volume of air throughput to the cell during this time interval. This experimental procedure was repeated using cells containing different depths of froth in the range 0.02 m to 0.46 m. 2. The solid concentration of froth was calculated from the above information and plotted as a function of time for different froth heights (see Fig. 2). These results indicate that the concentration attains a value that is practically independent of height up to about 0.12 m; a relatively large drop in the concentration occurs when total froth height hf increases. Based on these results, as a conservative estimate, hpf is taken to be equal to 0.1 m. This implies that secondary froth phase develops only when the total froth height hf is greater than 0.1 m for the conditions of our study. ESTIMATION OF UPWARD RATE COEFFICIENTS

The value of K u , P is taken to be equal to the average value of the gammadistributed rate coefficient which was determined experimentally using a limiting froth height cell (hf=0.02 m, with negligible solid drainage) as described in H a n u m a n t h and Williams (1990). In order to calculate the upward transfer coefficients in the froth phases we make the approximation that froth volumetric flowrate is equal to the air volumetric flowrate. This approximation rests on the assumption that the degree of bubble breakage in the secondary froth is insignificant. Then, it can be shown that: Ku.pf =

Ug/hpf

Ku,sf= Ug/ h~f

(6)

(7 )

These upward rate coefficients represent the fraction of the total froth volume displaced per unit time. ESTIMATION OF DRAINAGE COEFFICIENTS

An empirical procedure is used to determine the drainage coefficients Kd.pr and K,t.sf as described below. For the china clay system, the drainage coefficients were estimated by measuring the drainage rate in a froth cell containing only primary froth ( for experimental details see H a n u m a n t h and Williams, 1990). Briefly, the experimental procedure consisted of measuring the pulp concentration during flotation in a shallow froth cell with negligible solid drainage, as a function of time. Then the procedure was repeated using a cell wherein a deep froth

A THREE-PHASE MODEL OF FROTH FLOTATION

267

(hf= 0.12 m ) permitted solids to drain into the pulp. In this cell, as results of Fig. 2 indicate, there exists practically a single froth layer of uniform concentration which, in our model corresponds to the primary froth phase. It follows that all the solid drainage to the pulp in this cell occurs from the primary froth phase. The drainage rate, determined from pulp concentration measurements, was found to 0.5 × 10 -3 kg/s in the first minute of experiment, further by a mass balance Wpf, the solid holdup in the primary froth phase, was found to be approximately 0.022 kg. Since Kd,pfin our model is equal to the proportionality constant between the drainage rate and solid holdup, a value of Kd.pf equal to 0.022 s - ~ was obtained. Starting with this value of Kd,pra procedure of trial and error was employed to curve fit the model to experimental results (see Fig. 3 ), and thereby Kd,pf and Kd,sf were estimated. For quick reference, the estimated values of upward and downward rate coefficients are presented in Table 1.

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Rate coefficients for the three-phase flotation model. Note: Ku,,f varies with h,,-, and can be calculated using eq. ( 7 )

Ku.p (s -~)

Ku,pf(s - l )

Ks,pf(S - I )

kd,~c(S I)

0.0043

0.08

0.027

0.24

268

G.S. HANUMANTH AND D.J.A. WILLIAMS

RESULTS AND DISCUSSION

Semi-batch flotation tests Figures 4 to 6 show semi-batch flotation kinetics, wherein F(t), the residual mass of china clay in the pulp and froth phases, and R (t), the recovered mass, both normalized with respect to the initial mass of china clay in the batch of pulp, are plotted as functions of time for different froth heights. Experimental values, previously reported by us ( H a n u m a n t h and Williams, 1990), have been superimposed on these curves for the purpose of comparison. The model appears to overestimate the mass of residual solid in the pulp at the low froth heights, as compared to experimental values, the m a x i m u m deviation being about 15% which is observed when hf= 0.12 m. On the other hand, predicted recoveries show good agreement with experimental values until about 250 sec at which point the deviation starts to grow in low frothheight cells. However, the deviation tends to decrease with froth height. The lag observed in the experimental recovery data is due to the transient build-up of froth. But the model predicts a gradual accumulation of froth, and small recoveries during the transient build-up, as a result of the well-mixed assumption. The cumulative recovery after the end of the lag period is found .

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Fig. 4. Simulation of the kinetics of china clay flotation using the three-phase model. Experimental conditions: Initial mass = 0.75 kg, Vp= 0.0032 m 3, Ug-- 0.008 m/s, and hf= 0.12 m. Symbols denote experimentally determined values.

269

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TIME (s) Fig. 5. Simulation of the kinetics of china clay flotation using the three-phase model. Experimental conditions: Initial mass=0.75 kg, Vp= 0.0032 m 3, Ug=0.008 m/s, and hr--0.27 m. Symbols denote experimentally determined values.

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270

G.S. HANUMANTH AND D.J.A. WILLIAMS

to match the experimental value fairly closely. The accumulation of froth results in a relatively large difference in solid concentration between the primary and secondary froth phases, the primary froth phase being more concentrated. Further, it is observed that at early times in the experiment, in deep-froth cells, the total solid accumulation in the froth could amount to a substantial portion of the starting mass of solid in the pulp, for example it is nearly 15% in the case of a 0.46 m deep froth cell. The extent of solid drainage from the froth phases becomes clear from an examination of the variation of solid residue in the pulp. For a comparative study, if we regard hr=0.12 m as the reference experiment, there is an increase of about 22% in residual solid mass in the case of hf=0.27 m, and an even higher value of nearly 35% when hf=0.46 m. Further, the residue curves for hf=0.27 m and 0.46 m appear to diverge positively from the reference curve as a function of time in a nonlinear fashion, indicating that more solids drain back into the pulp with increasing time and froth depth. On the other hand, in the case of the recovery curves, the results are qualitatively reversed, recoveries decrease from a high of nearly 80% for the reference experiment to about 35% when hf increases to 0.46 m. The predicted trends show general agreement with experiments.

Time dependent continuous froth flotation In this section, the model is used to simulate transient behaviour of a flotation system in order to obtain useful insights into the flotation process, which are useful in planning control strategies in industrial flotation cells. As before, the system considered is china clay. The volumetric flowrates of both the feed and the tailings are arbitrarily chosen to be equal to 0.11 m i n - 1, any depletion of water due to froth removal is assumed to be compensated for fully by the inflow of makeup liquid, so that the pulp volume Vp remains constant. The variation of recovery rate is plotted as a function of time for different froth heights in Fig. 7. The results depict the manner in which the recovery rate approaches equilibrium when a perturbation, in the form of a step change in flowrate of feed and tailings ( Qi.s~= Oo,sl - - 0.1 1 m i n - l ), is introduced into the batch of pulp in the cell at time t = 0. The feed composition is assumed to be constant and equal to the original pulp concentration. The overall process is ultimately drawn to an equilibrium recovery rate by the interplay of flotation and drainage (Fig. 7 ). In moderately deep froths a peak in recovery rate is visible as a result of the competing flotation and drainage processes. As froth depth increases further, the accompanying increase in solid drainage rate drives the process to equilibrium faster, so much so that the peaks diminish to small value in sufficiently deep froth cells. For example, the ratio of peak value to steady state value of recovery rate decreases from a high of 1.75 for 0.12 m deep froth to a low of 1.26 for a froth

271

A THREE-PHASE MODEL OF FROTH FLOTATION

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depth of 0.46 m. This implies that the deeper the froth phase in flotation cells, the less severe the variation in recovery rate when there is an unforeseen increase in feed rate. In other words, the froth phase serves to damp the effect of feed rate fluctuations on recovery rate. CONCLUSIONS

The results of this study show that the three-phase model of froth flotation described herein can be used to predict the pulp phase and recovery kinetics of froth flotation. Furthermore, it is possible, by using the model, to estimate froth solid drainage rate, its variation with froth depth, and its influence on pulp phase and recovery kinetics over a wide range of froth depths. From simulation studies of continuous china clay flotation it emerged that the froth phase is likely to damp out the effect of small unforeseen transients in the feed rate on recovery rate, the extent of damping increasing with froth depth. The deviations of predictions from experiment could be due in part to an inherent distribution in the values of the rate coefficients. Under these circumstances, a rate distributed multiphase model is likely to give improved predictions.

272

G.S. HANUMANTH AND D.J.A. WILLIAMS

NOTATION a

C F(t) h K Q R(t) 1

U V W

rate coefficient ( s- ~) concentration ( k g / m 3 ) residual mass fraction height ( m ) rate coefficient (s-1 ) volumetric flowrate (m3/s) mass fraction recovered time (s) superficial velocity ( m / s ) volume (m 3 ) mass (kg)

Subscripts d f g i o P pf sf sl

downward whole froth gas inlet outlet pulp primary froth secondary froth slurry

ACKNOWLEDGEMENTS

One of the authors (G.S.H.) is grateful to English China Clays (ECC) Int. Ltd., Cornwall, U K for financial and material support. Thanks are also due to the research staff at ECC Int. Ltd., Dr. R. Bryant and Prof. J.F. Richardson of Chemical Engineering Department, University College, Swansea for helpful discussions. The valuable comments of the anonymous referees on an earlier version of the paper which helped to improve it are gratefully acknowledged.

REFERENCES Arbiter, N. and Harris, C.C., 1962. Flotation kinetics. In: D.W. Fuerstenau (Editor), Froth Flotation. AIME, New York, NY, pp. 215-262. Burden, R.L., Faires, J.D. and Reynolds, A.C., 1981. Numerical Analysis, 2nd edition. Prindle, Weber and Schmidt, Boston. Bushell, C.H.G., 1962. Kinetics of flotation. Trans. Amer. Inst. Min. Eng., 223: 226-273.

A THREE-PHASE MODEL OF FROTH FLOTATION

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Cooper, H.R., 1966. Feedback Process Control of Mineral Flotation, 1. Development of a model for froth flotation. Trans. Amer. Inst. Min. Eng., 235: 439-446. Cutting, G.W. and Devinish, M., 1975. A steady-state model of flotation froth structures. Soc. Min. Eng., AIME, Annu. Meeting, New York, Feb. 16-20, 1975. Preprint 75-B.56. Dobby, G.S. and Finch, J.A., 1985. Mixing characteristics of industrial flotation columns, Chem. Eng. Sci., 40: 1061-1068. Hanumanth, G.S. and Williams, D.J.A., 1988. Design and operation characteristics of an improved laboratory flotation cell. Miner. Eng., 1: 177-188. Hanumanth, G.S. and Williams, D.J.A., 1990. An experimental study of the effects of froth height on flotation of china clay. Powder Technol., 60:131-144. Harris, C.C., 1978. Multiphase models of flotation machine behaviour. Int. J. Miner. Process., 5: 107-129. Harris, C.C., Jowett, A. and Ghosh, S.K., 1963. Analysis of data from continuous flotation testing. Trans. Amer. Inst. Min. Eng., 226: 444-447. Harris, C.C. and Rimmer, H.W., 1966. Study of two-phase model of the flotation process. Trans. Inst. Min. Metall., 75: C153-162. Lynch, A.J., Johnson, N.W., McKee, D.J. and Thorne, G.C., 1974. The behaviour of minerals in sulphide flotation processes with reference to simulation and control. J. S. Aft. Inst. Min. Metall., 74: 349-361. Moys, M.H., 1978. A study of plug-flow model for flotation froth behaviour. Int. J. Miner. Process., 5: 21-38. Moys, M.H., 1984. Residence time distribution and mass transport in the froth phase of the flotation process. Int. J. Miner. Process., 13:117-142. Wylie, C.R., 1979. Differential Equations. McGraw-Hill, New York.