an evaluation of different models of water recovery in flotation-x.zheng

This article is also available online at:www.elsevier.com/locate/mineng

Minerals Engineering 19 (2006) 871–882

An evaluation of different models of water recovery in flotation

X. Zheng a,*, J.P. Franzidis b, N.W. Johnson c

a Newmont Golden Grove Operations, Department of Metallurgy, PMB 7, Geraldton, Western Australia 6530, Australiab Julius Kruttschnitt Mineral Research Centre, The University of Queensland, Brisbane, Australia

c Mineralurgy Pty Ltd., Brisbane, Australia

Received 9 March 2005; accepted 28 July 2005Available online 9 March 2006

Abstract

Water recovery is one of the key parameters in flotation modelling for the purposes of plant design and process control, as itdetermines the circulating flow and residence time in the individual process units in the plant and has a significant effect on entrain-ment and froth recovery. This paper reviews some of the water recovery models available in the literature, including both empiricaland fundamental models. The selected models are tested using the data obtained from the experimental work conducted in anOutokumpu 3 m3 tank cell at the Xstrata Mt Isa copper concentrator. It is found that all the models fit the experimental data rea-sonably well for a given flotation system. However, the empirical models are either unable to distinguish the effect of different celloperating conditions or required to determine the empirical model parameters to be derived in an existing flotation system. Themodel developed by [Neethling, S.J., Lee, H.T., Cilliers, J.J., 2003, Simple relationships for predicting the recovery of liquid fromflowing foams and froths. Minerals Engineering 16, 1123–1130] is based on fundamental understanding of the froth structure andtransfer of the water in the froth. It describes the water recovery as a function of the cell operating conditions and the froth prop-erties which can all be determined on-line. Hence, the fundamental model can be used for process control purposes in practice. Byincorporating additional models to relate the air recovery and surface bubble size directly to the cell operating conditions, the fun-damental model can also be used for prediction purposes.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Froth flotation; Flotation bubbles

1. Introduction

Water recovery is one of the key parameters in flota-tion plant design and operation. It determines to a largeextent the circulating load and residence time in the indi-vidual process units in the plant. Meanwhile, entrain-ment and froth recovery are strongly associated withwater recovery and influence both mineral recoveryand concentrate grade. Hence, the ability to model waterrecovery and to control it, especially in industrial scaleflotation cells has always been an important goal of both

0892-6875/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.mineng.2005.07.021

* Corresponding author. Tel.: +61 8 9956 4398; fax: +61 8 99564319.

E-mail address: [email protected] (X. Zheng).

researchers and plant operators. This paper reviewssome of the water recovery models available in the liter-ature. The selected models are evaluated using data ob-tained from the test work conducted in an Outokumpu3 m3 tank cell in the Xstrata Mt Isa copper con-centrator.

2. Definition of water recovery

Water recovery may be defined as the fraction of thewater entering the flotation cell that is recovered in theconcentrate. However, this definition of water recoveryis only one of several found in the literature owing tothe use of different flotation systems by different workers

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872 X. Zheng et al. / Minerals Engineering 19 (2006) 871–882

and use of the water recovery data for different pur-poses. For example, water recovery is sometimes calcu-lated as a fraction of the total water in the flotation cellthat is recovered to the concentrate. This definition ofwater recovery is commonly used for batch flotationtests.

It is also not unusual to use the water flow rate of theconcentrate directly instead of the fractional recovery ofthe feed water or the total water in the cell, especially ina batch flotation cell when the volume of the pulp variesduring the test. Some researchers avoid use of the frac-tional recovery of water because the concentrate waterflow rate is believed to be relatively independent of thefeed water flow rate or the amount of water in the pulpphase. If so, it would be misleading to compare the frac-tional water recovery for different operating conditionsor different sizes of flotation cells. In this instance, theconcentrate water flow rate would be a more properterm to use.

Finally, it should be noted that water recovery in aflotation cell is the result of a two-step transfer pro-cess—transfer of the water in the pulp phase to the frothphase and transfer of the water in the froth phase to theconcentrate launder. The amount of water entering thefroth phase from the pulp phase is closely related tothe pulp conditions such as the bubble surface area flux,the concentration of chemical reagents, the concentra-tion of suspended solids and the bubble loading condi-tion. Only a certain fraction of the water entering thefroth phase may be recovered into the concentrate.The rest returns to the pulp phase via drainage and/orwith the collapsed froth. These two steps of transferare not totally independent of each other. The amountof water entering the froth phase may to a large degreedetermine the characteristics of the froth (froth mobilityand stability and solids/liquid/air hold-up in the frothphase) and hence influence the recovery of the waterand solid particles in the froth phase to the concentratelaunder.

3. Modelling of water recovery

Research has been going for many years, trying tounderstand the froth behaviour and the mechanisms be-hind the water recovery in flotation. However, due tothe complicity of the froth system and the limitationsin measurement techniques, water recovery was mod-elled simply by related to an certain aspect of the cellperformance, commonly used the solids recovery (King,1973; Alford, 1990), in the early stage. An alternativeapproach was to find some statistically significant corre-lation between the water recovery and the cell operatingconditions. Such an approach was widely adopted as itprovided a relatively quick and simple solution (Savassi,1998; Gorain et al., 1998; Harris, 2000). Meanwhile,

attempt was also made to model the water recoverybased on fundamental understanding of the physics(Moys, 1984). In the recent years, a significant progresshas been made with the fundamental approach (Neeth-ling et al., 2003). Some of the models for water recoveryare now reviewed and evaluated with experimental dataobtained from the test work conducted in an Outok-umpu 3 m3 tank cell in the Xstrata Mt Isa copperconcentrator.

3.1. Water recovery as a function of solids recovery

In this approach, concentrate water flow rate is linkeddirectly with the concentrate solids flow rate. This allowsthe water flows throughout the plant to be fixed un-iquely in flotation modelling (King, 1973). The modeldoes not require any information about the cell operat-ing conditions.

Based on the conservation of the volume and mass ofwater and solids in a given stream, the relationship be-tween the water and the solids flow rates can be ex-pressed as

Qw ¼SGs � SGp

SGp � SGw

� 1

SGs

� F s ð1Þ

or

Qw ¼1� X

X� 1

SGw

� F s ð2Þ

where Qw, volumetric flow rate of the water; Fs, massflow rate of the solids; X, mass fraction of the solids inthe stream; SGs, SGp and SGw, specific gravity of thesolids, slurry and water, respectively.

Generally, the solids and water specific gravities canbe assumed to be constant. Hence, to predict the waterflow from the solids flow is to predict the SG value ofthe concentrate slurry or the mass fraction of the solidsin the stream. In practice, for control purposes, thesevalues can be measured. The usefulness of the aboveequations in providing a means of accurate control overthe process relies on how sensitive the concentrate slurrySG and solids content are to the process variables whichare used to control the process.

In JKSimFloat (Alford, 1990), a commercial com-puter software package for flotation modelling and sim-ulation, the concentrate water flow rate is expressed as apower function of the concentrate solids flow rate:

Qw ¼ a � F bs ð3Þ

where a and b are empirically fitted parameters.Again, the usefulness of this model depends on the

extent to which the model parameters, a and b, remainconstant under different cell operating conditions or indifferent cells.

Table 1 summarises the concentrate water and solidsflow rates measured in an Outokumpu 3 m3 tank cell in

Table 1Calculated concentrate water flow as a function of solids flow

Air rate(l/min)

Frothheight(cm)

%Solids(%)

Fs

(ton/h)Qw

(m3/h)Qw

(cal-1)(m3/h)

Qw

(cal-2)(m3/h)

1085 9.6 36.66 1.69 2.92 3.37 2.951085 14.3 45.53 1.17 1.40 2.13 1.321085 19.0 48.27 0.92 0.99 1.58 0.781085 23.2 50.97 1.05 1.01 1.86 1.041380 9.1 28.42 2.20 5.54 4.69 5.241380 13.8 32.50 2.08 4.32 4.37 4.641380 18.5 37.33 1.62 2.72 3.20 2.691380 23.2 39.12 1.60 2.49 3.15 2.621674 10.5 26.13 4.55 12.86 11.59 11.591674 14.7 28.95 2.97 7.29 6.81 7.161674 19.4 28.90 2.61 6.42 5.80 6.191674 24.1 32.30 1.77 3.71 3.57 3.991968 10.5 28.43 5.83 14.68 15.79 15.341968 14.7 29.26 3.77 9.11 9.17 9.371968 19.4 28.80 2.78 6.87 6.27 6.641968 24.1 33.09 2.49 5.04 5.47 5.87

Note: (cal-1)—modelling fitting using all the data; (cal-2)—model fit-ting separately for different air rates.

X. Zheng et al. / Minerals Engineering 19 (2006) 871–882 873

the Xstrata Mt Isa copper concentrator. The experimen-tal cell was fed with the plant rougher feed ore and oper-ated at four different air rates and, at each air rate, fourdifferent froth heights. Also included in Table 1 are thefitted results using Eq. (3).

It can be clearly seen in Table 1 that the solids con-centration in the concentrate varied significantly withthe cell operating conditions. Hence, the assumption ofa constant concentrate %Solids value in order to use Eq.(2) for prediction purposes is not valid in practice. The%Solids value or slurry SG has to be measured if it isused for control purposes.

If all the experimental data are used in the model fit-ting regardless of cell operating conditions (cal-1), thefitted values of the constants a and b are:

a ¼ 1:75 and b ¼ 1:25

It can be seen in Fig. 1(a) that there is a significant dis-crepancy between the experimental and predicted water

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0 2 4 6 8 10 12 14 16 18Experimental water flow rate (m3/hr)

Cal

cula

ted

wat

er fl

ow ra

te (m

3 /hr)

Qa = 1085 l/minQa = 1380 l/minQa = 1674 l/minQa = 1968 l/min

R2 = 0.972

3

(a) (b

Fig. 1. Experimental versus fitted concentrate water flow rates using the wXstrata Mt Isa copper concentrator: (a) fitted with all data, (b) fitted accord

flow rates at the lower air rates. If the model is used to fitthe data separately according to the air rate, the fittedmodel parameters for the different cell operating condi-tions now become:

a ¼ 0:94 and b ¼ 2:18 at the lower air rates

a ¼ 2:09 and b ¼ 1:13 at the higher air rates

Fig. 1(b) shows that the fitting can be improved by treat-ing the data separately according to the air rate—thecorrelation coefficient improving from 0.986 to 0.993and the residual sum square errors of the regressionreducing from 6.154 to 3.205. However, it reveals thelimitation of applying the model to a wide range of oper-ating conditions.

Fig. 2 shows the concentrate water flow rates in eachof the three flotation cells of a rougher bank (Sala 80 lcells) in a platinum pilot plant (Harris, 2000). The datafrom the three cells were all fitted to Eq. (3) (Fig. 2(a))and then fitted separately, cell by cell (Fig. 2(b)).

Fig. 2(a) clearly shows that the model fitted concen-trate water flow rates in the first rougher cell fall in a dif-ferent line from those in the second and third cells. Thefitted constants a and b for the different cells are foundto be:

a ¼ 0:266 and b ¼ 1:08 for rougher cell 1

a ¼ 0:367 and b ¼ 1:05 for rougher cells 2 and 3

The significant difference in the model constant ‘‘a’’ sug-gests that the model derived from one flotation cellmight not be able to be used for other cells in the samecircuit. In practice, feed at the beginning of the flotationbank contains a significantly higher content of floatableparticles and a higher concentration of frother, whichlikely results in different froth properties from the cellsfurther down the bank. Therefore, it can also be ex-pected that the difference will remain in the cells of dif-ferent circuits.

Alford (1990) applied this model in flotation col-umns. Uribe et al. (1999) studied the limitations ofapplying the model to flotation columns and used a

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0 2 4 6 8 10 12 14 16 18Experimental water flow rate (m3/hr)

Cal

cula

ted

wat

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ow ra

te (m

/hr)


R2 = 0.987

)

ater–solid model (Eq. (3)) for the Outokumpu 3 m3 tank cell data ating to air rate.

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0 5 10 15 20 25 30 35Experimental water flow rate (ml/sec)

Cal

cula

ted

wat

er fl

ow ra

te (m

l/sec

)

Rougher 1Rougher 2Rougher 3

R2 = 0.838S2 = 198.09

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0 5 10 15 20 25 30 35Experimental water flow rate (ml/sec)

Cal

cula

ted

wat

er fl

ow ra

te (m

l/sec

)

Rougher cell 1Rougher cell 2Rougher cell 3

R2 = 0.868S2 = 166.94

(a) (b)

Fig. 2. Experimental versus fitted concentrate water flow rates using the water–solid model (Eq. (3)) for platinum pilot plant data for Harris (2000):(a) fitted for all cells, (b) fitted for individual cells.


statistical technique to determine the main factors affect-ing the model results. Three more operating variableswere incorporated in the modified model, includingfroth height, wash water and column size:

J w ¼ ða � Rbs þ c � J bÞ � Hd

f � De ð4Þ

where Jw, superficial water velocity to the concentrate;Rs, fractional solids recovery to the concentrate; Jb, biaswater velocity; Hf, froth height; D, column diameter; a,b, c, d and e, constants.

It may be useful to extend the model modification ofUribe et al. (1999) to mechanical cells, if the interpreta-tion holds that froth height and flotation cell size (frothtransportation distance) can affect the relationship be-tween the concentrate water flow rate and the concen-trate solids flow rate.

In summary, the practical application of the concen-trate water flow rate model as a function of the solidsflow rate given in Eq. (3) is limited to a narrow rangeof cell operating conditions and to a narrow range offlotation cells operated with similar conditions, as thefroth properties and the solids content in the concen-trate depend strongly on the cell operating conditionsand duties.

3.2. Water recovery as a function of froth

residence time

Froth residence time is possibly the most frequentlyused variable in the modelling of froth recovery. Twodefinitions of mean froth residence time, based on theair rate and concentrate slurry flow rate, respectively,can be found in the literature (Gorain et al., 1998; Lynchet al., 1981):

sf ¼H f

J g

ð5Þ

sf ¼ð1� egÞ � V f

Qs

ð6Þ

where sf, mean froth residence time; Jg, superficial gasvelocity entering the froth phase; eg, mean air hold-upin the froth; Qs, slurry flow rate of the concentrate; Vf,effective volume of the froth zone (the volume occupiedby the moving froth towards the concentrate launder).

Eq. (5) is a simplified version of the following equa-tion, assuming that the horizontal cross-sectional areaof the froth zone does not change with level and thatthe water/solids hold-up in the froth phase is negligiblecompared with the volume occupied by the air:

sf ¼eg � V f

Qa

ð7Þ

where Qa is the rate of air flow passing through the frothzone.

It should be noted that the residence time of the airbubbles and attached particles in the froth phase usuallydiffers from that of the water and entrained particles inthe same froth but the two residence times are closely re-lated. It should also be noted that, in industrial flotationcells, there is always a significant portion of the frothburst on the surface of the cell and not reporting tothe concentrate launder (Barian et al., 2005; Zhenget al., 2005). As a result, the froth movement slows downand the residence time of the froth increases (Zheng andKnopjes, 2004). Therefore, Eqs. (5) and (7) should bemodified to more accurately reflect the true value ofthe mean froth residence time by taking into accountthe air recovery factor

sf ¼eg � V f

a � Qa

ð8Þ

where a is the fraction of the air leaving cell as unburstbubbles.

The main motivation of using Eq. (5) to estimate themean froth residence time in flotation modelling is thatEq. (5) includes two cell operating conditions only, frothheight and air rate, which can be either set or measuredeasily. It is important to point out that under normaloperating conditions, the top of the froth is above the


level of the concentrate launder weir. Until now, there isno information available in the literature on how theheight of the froth above the concentrate weir level is re-lated to the cell operating conditions or what is the typ-ical range of the froth height above the concentratelaunder lip in different flotation cells and/or when treat-ing different ores. In the rougher test work conductedusing the Outokumpu 3 m3 tank cell in the Xstrata MtIsa copper concentrator (Table 1), the height of the frothabove the concentrate launder lip level varied between3 cm and 4.5 cm while the pulp level was set to give afroth height between 6.2 cm and 20.2 cm (distance be-tween the pulp–froth interface and the concentrate laun-der lip). Generally, the froth is higher above theconcentrate launder lip at a deeper froth height setting.Therefore, care should be taken when using Eq. (5) topredict the mean froth residence time at a shallow frothand a low air rate.

Savassi (1998) proposed an empirical model for waterrecovery as a power function of the mean froth residencetime (calculated by Eq. (5)) based on the data from histest work in the lead rougher circuit at the Xstrata MtIsa lead/zinc concentrator.

Rw ¼ c � sdf ð9Þ

where Rw, fractional recovery of the feed water in theconcentrate; sf, mean froth residence time calculatedusing Eq. (5); c and d are constant.

Fig. 3 plots the experimental water recoveries at dif-ferent cell operating conditions using the data obtainedduring the Outokumpu 3 m3 tank cell test work at theXstrata Mt Isa copper concentrator. Eq. (9) was usedto fit the experimental data for different air rates(Fig. 3(a)) and for different froth heights (Fig. 3(b)).

Eq. (9) appears to fit the experimental data well bothat different froth heights at a given air flow rate and atdifferent air rates at a given froth height. However, thewater recovery did not fall on the same curve at the dif-ferent air rates and different froth heights even at thesame froth residence time, especially at the lower airrates and shallower froth. Clearly, Eq. (9) using the

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0 5 10 15 20 25Mean froth residence time (sec)

Wat

er re

cove

ry (%

)


(a) (b

Fig. 3. Modelling of water recovery as a power function of mean froth resiconcentrator): (a) different air rates, (b) different froth heights.

mean froth residence time to determine the water recov-ery is inadequate to distinguish the effect of air rate andfroth height. Based on the experimental data shown inFig. 3, it can also be found that water recovery is moresensitive to froth height than to air rate. Neethling et al.(2000) demonstrated that water distributes unevenly in aflowing foam as the froth residence time varies fromlocation to location and the properties of the local frothdepends strongly on its residence time. A single value ofmean froth residence time does not adequately representthe true situation of froth transportation and cannot beused to determine the properties (i.e., water content) ofthe froth entering the concentrate launder. Based on theplant observations, the above phenomena cannot be ex-plained without carefully examining the change in frothcharacteristics, especially bubble size and froth stability,at different air rates and froth heights.

It should also be noted that the water recovery pro-posed by Savassi (1998) identified the froth residencetime is the key factor controlling the water recovery. Infact, there are two distinguished steps in the water recov-ery process, transfer of the water from the pulp phase tothe froth phase and transfer of the water in the frothphase to the concentrate launder. The recovery of thewater from the pulp phase to the froth phase is relativelyindependent of the events in the froth phase but depen-dent on the air rate, bubble size distribution in the pulpphase, bubble packing condition at the pulp–froth inter-face, and mineralisation of the bubble surface.

Meanwhile, Gorain et al. (1998) also used the meanfroth residence time in their model but to determinethe froth recovery only. Froth recovery is expressed asan exponential function of the mean froth residencetime:

Rf ¼ e�b�sf ð10Þ

where Rf, froth recovery; b, constant, related to the frothcharacteristics.

The overall recovery of the water is a result of therecovery of the water from the pulp phase to the froth

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45


Wat

er re

cove

ry (%

)

Hf = 6.2-7.2 cmHf = 10.4-11.8 cmHf = 14.6-16.0 cmHf = 18.8-20.2 cm

)

dence time (Outokumpu 3 m3 tank cell data at Xstrata Mt Isa copper

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45


Wat

er re

cove

ry (%

)Qa = 1085 l/minQa = 1380 l/minQa = 1674 l/minQa = 1968 l/min

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50

60


Wat

er re

cove

ry (%

)

Hf = 6.2-7.2 cmHf = 10.4-11.8 cmHf = 14.6-16.0 cmHf = 18.8-20.2 cm

(a) (b)

Fig. 4. Modelling of water recovery as an exponential function of mean froth residence time (Outokumpu 3 m3 tank cell data at Xstrata Mt Isacopper concentrator): (a) different air rates, (b) different froth heights.


phase and the recovery of the water from the froth phaseto the concentrate. The water returning from the frothphase to the pulp phase may be considered as an addi-tional feed to the flotation cell pulp. Hence, the overallwater recovery can be expressed by

Rw ¼Rcw � Rfw

1� Rcw þ Rcw � Rfw

ð11Þ

where Rcw, recovery of the water from the pulp phase tothe froth phase; Rfw, recovery of the water from thefroth phase to the concentrate, or froth recovery of thewater.

Combining Eqs. (10) and (11) gives:

Rw

1� Rw

¼ Rcw

1� Rcw

� expð�b � sfÞ ð12Þ

By plotting (Rw/(1 � Rw)) against sf, the values of Rcw

and b can be determined. Then, Rfw can be calculatedfrom the known b value. Fig. 4 shows the fitting usingEq. (12).

In comparison with the power function model (Eq.(9)), the exponential model (Eq. (12)) appears also tofit the experimental data well. However, the same limita-tion is still associated with the model by Gorain et al.(1998) as the one by Savassi (1998) due to use of a singlevalue of mean froth residence to represent the entirefroth transport behaviour in the cell, even though theapproach of Gorain et al. (1998) to modelling the waterrecovery does distinguish the two steps of water transferin the flotation cell.

Finally, it is worth mentioning that applying a simpleexponential function (Eq. (13)) would give a similar fit-ting result. Since the model of Gorain et al. (1998) is anempirical one, such a simplification is acceptable inpractice.

Rw ¼ Rcw � expð�b � sfÞ ð13Þ

In summary, it can be concluded that the mean frothresidence time is an important parameter for determin-ing the froth recovery of the water and the overall waterrecovery. Either power or exponential model fits theexperimental data well. However, it is clearly shown in

Figs. 3 and 4 that air rate and froth height affect thewater recovery in a different way and their effect onthe water recovery cannot simply be represented by asingle value of mean froth residence time.

3.3. First-order water recovery model (Harris, 2000)

Harris (2000) proposed a water recovery model whichwas built on the work of Gorain et al. (1998) and Harris(1998). In his model, water recovery from the pulp phaseto the froth phase and from the froth phase to the con-centrate launder are modelled separately. Water in thepulp phase is treated as a component in the same wayas the minerals in the system. Recovery of the waterfrom the pulp phase to the froth phase is assumed to fol-low a first-order kinetic equation (Scrimgeour et al.,1970). Hence, if the pulp phase is considered to be per-fectly mixed, then the water recovery can be expressedvia Eq. (14):

Rw ¼kw � s

1þ kw � sð14Þ

where s, mean residence time of the pulp slurry; kw, first-order rate constant for water, as a function of bubblesurface area flux in the pulp phase, and

kw ¼ P w � Sb � Rfw ð15Þwhere Pw, constant for a given flotation system; Sb, bub-ble surface area flux in the pulp phase, calculated fromthe superficial gas velocity and the bubble size.

The model for the froth recovery of water is ex-pressed as an exponential function of the mean froth res-idence time. Two forms of froth recovery models areadopted by Harris (2000):

Rfw ¼ ð1� awÞ � expð�bw � sfÞ þ aw ð16ÞRfw ¼ X � expðr � V f � v � sfÞ ð17Þwhere aw, bw, X, r and v are constant.

Eq. (16) was first proposed by Harris (1998). aw isgiven a physical meaning as the non-draining fractionof the froth. The modified form (Eq. (17)) excludes thenon-draining fraction factor but introduces a new coef-


ficient related to the fraction of stagnant plus collapsedfroth (X).

Combining Eqs. (14)–(16), the overall water recoverycan also be expressed by

Rw ¼ 1� 1

1þ P w � Sb � s � X � expðr � V f � v � sfÞð18Þ

The above model was reported to fit the same pilot plantdata (as shown in Fig. 2) well, and be able to predict thewater recovery with a reasonable accuracy (Harris,2000). However, if comparing the goodness of fittingusing the Harris (2000) approach (Fig. 5(a)) with the fit-ting results using Savassi (1998), Gorain et al. (1998)models (Fig. 5(b)) to the same data obtained in theOutokumpu 3 m3 tank cell data at the Xstrata Mt Isacopper concentrator, it is clear that, despite the attemptof Harris (2000) to develop the water recovery model astep further from Eq. (14) by incorporating a modelfor the recovery of the water from the pulp phase tothe froth phase using a first-order process, the resultsdo not seem to offer any improvement from the simplermodels of Savassi (1998) and Gorain et al. (1998).

It should be noted that, although the model of waterrecovery from the pulp phase to the froth phase has theform of a first-order process, it is not the author�s inten-tion to suggest the water recovery mechanism is indeedfirst-order: it is a purely empirical model which attemptsto correlate the water recovery with the cell operatingvariables. However, as mentioned at the beginning ofthe paper, it is still debatable whether the amount ofwater entering the froth phase is related to the feedwater flow rate. The results from the Outokumpu 3 m3

tank cell tests at the Xstrata Mt Isa copper concentratorwith different feed flow rates show a nearly unchangedconcentrate water flow rate at a given air rate and frothheight when the cell was operated as the first rougherand the bubble loading condition did not change signif-icantly with the different feed rates.

It should also be noted that Eq. (18) takes into ac-count not only the effect of froth residence time on froth

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0 5 10 15 20 25 30 35 40Experimental water recovery

Cal

cula

ted

wat

er re

cove

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Qa = 1085 litre/minQa = 1380 litre/minQa = 1674 litre/minQa = 1968 litre/min

R2=0.9663S2=56.91

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Cal

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ted

wat

er re

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)

(a) (b)

Fig. 5. Comparison of the model fitting results using the Outokumpu 3 m3

(2000), (b) Savassi (1998) and Gorain et al. (1998).

recovery but also the effect of froth volume. The addi-tional parameter in the model allows the effects of airrate and froth height to be considered and modelled sep-arately (to a certain degree). This is important especiallyin large industrial flotation cells where the froth rheolog-ical behaviour may differ significantly with location andstagnant froth may develop in certain areas of the frothphase.

In summary, the approach of Harris (2000) to model-ling the recovery of the water from the pulp phase tothe froth phase as a first-order process and modellingthe froth recovery of the water as a function of bothmean froth residence time and froth volume appearsto fit the Xstrata Mt Isa plant data well. However, thereare still questions to be answered:

• What is the relationship between the water recoveryfrom the pulp phase to the froth phase and the meanpulp residence time?

• How to distinguish the constant for water recoveryfrom the pulp to the froth (Pw) and the efficiency fac-tor for the froth volume and bubble bursting (X) (thetwo are indistinguishable in the model, Eq. (18)).

3.4. Drainage model for water recovery (Moys, 1984)

In the model of Moys (1984), water recovery is alsoconsidered as a two-step process. The pulp water entersthe froth phase in the film of the air bubbles. In the frothphase, water needs to survive the drainage process andbubble bursting on the surface before it can reach theconcentrate launder.

To determine the initial water flow rate from the pulpphase to the froth phase, it is assumed that the water iscarried by the air bubbles and that the film thickness isconstant in a given flotation system.

Qwð0Þ ¼ A � Sb � d ð19Þwhere Qw(0), water flow rate entering the froth phase atthe pulp–froth interface; A, cross-sectional area of the

0 5 10 15 20 25 30 35 40Experimental water recovery (%)

Savassi (1998), R2 = 0.9895, S2 = 18.01Gorain et al (1998), R2 = 0.9895, S2 = 16.68

tank cell data at the Xstrata Mt Isa copper concentrator: (a) Harris

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16Experimental concentrate water flow (ton/hr)

Cal

cula

ted

conc

entr

ate

wat

er fl

ow (t

on/h

r)

Qa = 1085 l/minQa = 1380 l/minQa= 1674 l/minQa = 1968 l/min

R2 = 0.9807S2 = 4.39

Fig. 6. Experimental versus fitted concentrate water flow rates usingMoys� model (1984) for the Outokumpu 3 m3 tank cell data fromXstrata Mt Isa copper concentrator.


pulp–froth interface; d, the volume of water per surfacearea of the air bubbles.

During the transfer process in the froth phase, waterdrains back to the pulp phase at a rate proportional tothe water concentration at the froth level of interest.

dQwðzÞdtf

¼ �kwd � QwðzÞ ð20Þ

where tf, time for the froth to rise from the pulp–frothinterface to the level of interest in the froth; kwd, drain-age rate constant of the water.

Since the froth residence time can be expressed as

dtf ¼efðzÞ � AðzÞ � dz

Qa

ð21Þ

where ef(z), air hold up at the froth level of interest; A(z),cross-sectional area of the froth phase at the level ofinterest.

Eq. (20) can be re-written as

dQwðzÞdz

¼ �kwd � efðzÞ � AðzÞ � QwðzÞ=Qa ð22Þ

If the froth cross-sectional area remains constantthroughout the entire froth phase and the change inair hold-up at different froth levels is negligible andthe drainage rate constant is assumed to be independentof froth level, then Eq. (22) can be integrated from thepulp–froth interface as the zero level to the top of thefroth phase, which gives the solution as

RfwðH fÞ ¼ expð�kwd � ef � A � H f=QaÞ ð23Þwhere Hf is the total froth height.

Because the (ef ÆA ÆHf/Qa) term can be replaced byfroth residence time, the froth recovery of the water atthe top of the froth phase becomes:

RfwðH fÞ ¼ expð�kwd � sfÞ ð24ÞNot all the froth reaching the top of the froth phase en-ters the concentrate. Some of the bubbles burst on thesurface. If the fraction of the froth on the surface thateventually reports the concentrate is a, then the frothrecovery of the water becomes:

Rfw ¼ a � expð�kwd � sfÞ ð25ÞThe froth recovery model given by Eq. (25) is identicalwith the one proposed by Gorain et al. (1998), whichwas obtained empirically (Eq. (10)).

Combining Eqs. (19) and (25), the concentrate waterflow rate can be determined:

Qw ¼ A � Sb � d � a � expð�kwd � sfÞ ð26ÞGenerally, the cross-sectional area of the pulp–frothinterface is known. The bubble surface area flux in theflotation cell can be measured. Moys suggests that theefficiency of froth removal be estimated by measuringthe volumetric flow rate of the concentrate including

both the air bubbles and the slurry and comparing itwith the total air introduced to the cell.

a ¼ Qca

Qa

ð27Þ

where Qca is the volumetric flow rate of the froth (theair) in the concentrate.

There are only two fitted parameters in Eq. (26)—thevolume of water per surface area of the air bubblesentering the froth phase and the drainage rate constantin the froth phase. Fig. 6 shows the results of fitting themodel to the data from the test work at the Xstrata MtIsa copper concentrator using the Outokumpu 3 m3

tank cell. It should be noted that model fitting is carriedout for one air rate at a time, i.e., the model fitting wasconducted independently for the different air rates. Ascan be seen, Moys� model (1984) fits the concentratewater flow rate very well.

Fig. 7 further examines the effect of air rate on thewater recovery from the pulp phase to the froth phaseand the effect of froth residence time on the froth recov-ery of the water using the same set of data obtained atthe Xstrata Mt Isa copper concentrator.

In general, there is a trend of increase in the waterflow into the froth phase from the pulp phase with in-crease in the air rate. However, the curve of the pulpwater recovery to the froth phase and air rate inFig. 7(a) shows an irregular shape. It is difficult to usethe water film theory to explain this phenomenon.Clearly, at a higher air rate, there are more air bubblesrising through the pulp–froth interface. If the bubblesize remains the same at different air rates, then theamount of water recovered from the pulp phase shouldbe proportional to the air rate. However, in reality,water is also carried into the froth phase in the Plateauborders. The amount of water in the Plateau borders de-pends not only on the bubble size and number but alsothe packing condition of the bubbles on the pulp–frothinterface. If the bubbles are closely packed at the

0

5

10

15

20

25

30

35

1000 1200 1400 1600 1800 2000 2200Air rate (l/min)

Wat

er fl

ow fr

om p

ulp

to fr

oth

(m3 /h

r)

y = 100.85e-0.0709x

R2 = 0.9997

y = 96.101e-0.1175x

R2 = 0.9896

0

10

20

30

40

50

60

0 5 10 15 20 25 30Froth residence time (sec)

Frot

h re

cove

ry o

f the

wat

er (%

) Qa = 1085 l/minQa = 1380 l/minQa = 1674 l/minQa = 1968 l/min

(a) (b)

Fig. 7. Moys model fitting results on Xstrata Mt Isa copper concentrator test work data: (a) effect of air rate on pulp water recovery, (b) effect offroth residence time on froth recovery.


pulp–froth interface, the liquid content at the interface isabout 0.26, while if the packing is random, the value isabout 0.36 (Neethling et al., 2003). Interestingly, the val-ues calculated from the Outokumpu 3 m3 tank cell datawere 0.26 and 0.30 at the air rates of 1968 l/min and1974 l/min, respectively.

It is also interesting to see that the froth recovery ofthe water calculated from the model falls on the samecurve at the two high air rates and again at the twolow air rates tested in the Outokumpu 3 m3 tank cell(Fig. 7(b)). The possible explanation is that the frothstructure and properties are similar within the two oper-ating air rate ranges. Further investigation is required.

In summary, Moys� model (1984) produces a good fitto the experimental data for the overall recovery of thewater, but the physical significance of the modelparameters, especially the water film thickness of theair bubbles, may not reflect the true water recoverymechanisms.

3.5. Fundamental approach to modelling of froth

recovery of water (Neethling et al., 2003)

Research into understanding the foam/froth struc-ture, developing measurement techniques and modellingthe water recovery in flotation has been advanced signif-icantly in the recent years, especially by a group ofresearchers at the UMIST (Neethling et al., 2000,2003; Neethling and Cilliers, 2003; Barian et al., 2005).The research outcomes have now become available forapplications in the mineral processing industry (Nee-thing and Cilliers, 2005).

In brief, Neethling and Cilliers (2003) stated thatmost of the froth water is held and moves in the Plateauborders, which present in the froth phase as a networkof interconnected, nearly randomly distributed drainagechannels. In a froth phase that is at steady state and hasa relatively low gas velocity, the cross-sectional area ofthe Plateau borders decreases rapidly with height abovethe pulp–froth interface before reaching the concentrateoverflow level. For most heights of the froth, any de-

crease in water content is caused predominantly by coa-lescence, i.e. increase in bubble radii and therefore adecrease in the total length of Plateau borders. Testwork and measurement were conducted both in a two-phase foam in the laboratory and in a three-phase frothin the industrial flotation cell, and a froth recoverymodel for was developed which calculates the lengthof the Plateau borders and the cross-sectional area ofthe Plateau borders.

The numerical solution for the length of Plateau bor-ders per volume of the froth is based on the dodecahe-dron geometry of the froth:

k ¼ 5ffiffiffi3p

p � u � r2or k ¼ 6:81

d2f

ð28Þ

where k, length of Plateau borders per volume of thefroth; r, bubble radius; u, Golden ratio ð¼ 1þ

ffiffi5p

2Þ; df,

bubble diameter.The cross-sectional area of the Plateau borders in the

froth phase is determined through a balance of forcesacting on the water in the Plateau borders. Three mainforces are considered: gravitational forces, capillaryforces and forces due to viscous dissipation. For thefroth on the top of the cell, capillary force may be ne-glected. Hence, the solution for the cross-sectional areaof the Plateau borders on the top of the froth is given as

At ¼Qa

W � D � ð1� aÞ � g � 150

q � g ð29Þ

where At, cross-sectional area of the Plateau borders ontop of the froth; W and D, dimensions of the top of thecell; a, fraction of the air leaving cell as unburst bubbles;g, water viscosity in the froth, q, water density; g, grav-itational constant.

The Qa

W �D� �

term is in fact the superficial gas velocity,Jg. Combining Eqs. (28) and (29) gives the volumetricfraction of the water at the top of the cell:

ew ¼ At � k ¼255:6

r2� J g � ð1� aÞ � g

q � g ð30Þ


where ew is the water hold-up at the top of the frothsurface.

It should be taken into account that only a fraction ofthe air bubbles remaining unburst and reporting to theconcentrate launder:

Qac ¼ a � Qa ð31ÞThe a value can be estimated through measurement ofthe velocity of the froth flowing over the lip, the heightof the froth above the concentrate launder lip and thevolumetric flow rate of the air into the cell:

a ¼ vf � hf � lQa

ð32Þ

where vf, velocity of the froth flowing over the lip; hf,froth height above the concentrate launder lip; l, thetotal lip length of the concentrate launder.

Hence, the concentrate water flow rate can beobtained as

Qw ¼ a � Qa �ew

1� ew

¼ a � Qa �1

1� 255:6r2 � J g � ð1� aÞ � g

q�g� 1

!ð33Þ

It should be noted that the above equation ignores thewater in the lamellae and takes no account of the capil-lary term. In addition, variation in the froth propertiesacross the horizontal direction, i.e., bubble size andfroth viscosity, is also ignored. The key parameters arethe bubble size and the fractional recovery of the airbubbles in the concentrate launder. Fig. 8 shows themeasured bubble size at the different froth heights atthe different air rates.

In general, the size of the bubbles on the cell surfaceincreased with increase in the froth height in an approx-imately linear manner. However, it should be noted thatusing image analysis to determine the surface bubble sizein two dimensions may not truly represent the bubbleshape in three dimensions, which is especially important

0

2

4

6

8

10

12

14

0 5 10 15 20 25 30Froth Height (cm)

Bub

ble

Dia

met

er (m

m)

Qa =1085 l/minQa = 1380 l/minQa = 1674 l/minQa = 1968 l/min

Fig. 8. Measured mean bubble diameter at different froth heights atdifferent air rates in the Outokumpu 3 m3 tank cell data at the XstrataMt Isa copper concentrator.

for the calculation of the length of the Plateau borders.In addition, a large portion of the froth entering the con-centrate is coming from the froth flow beneath the sur-face layer. The actual bubble size could be significantlysmaller than that measured on the surface. Therefore,for the purposes of model fitting, it is convenient to as-sume that the size of the surface bubbles increases line-arly with increase in the froth height:

r ¼ r0 þ a � H f ð34Þwhere r0, radius of the air bubble initially entering thefroth phase; a, constant.

The measured bubble size on the cell surface is alsoaffected by the froth stability. When the froth is unstableand bubble bursting occurs rapidly, fine bubbles are ex-posed between the large bubbles. This often happens ina high air rate or at a low bubble loading condition.Fig. 9 is the image of the surface froth at four differentheights (increase from the top image to the bottomone) at two air rates tested in the Outokumpu 3 m3 tankcell.

Accurate measurement of the air recovery is also dif-ficult. To collect the unburst bubbles in the concentratelaunder is almost impossible. The image processingmethod (Eq. (32)) depends strongly on how representa-tive the measured froth velocity and froth height abovethe concentrate launder lip are of the average values(Neethling et al., 2003). Hence, again for modelling pur-poses, the fraction of unburst bubbles on the surface ofthe cell is considered to be linearly proportional to thefroth height:

Fig. 9. Froth images at different froth heights and air rates in theOutokumpu 3 m3 tank cell at the Xstrata Mt Isa copper concentrator(same scale for all the images): (a) air rate = 1085 l/min, (b) airrate = 1674 l/min.

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16

Experimental concentrate water flow rate (m3/hr)

Cal

cula

ted

conc

entr

ate

wat

er fl

ow ra

te (m

3 /hr)

R2 = 0.9668S2 = 7.82

Fig. 10. Experimental versus fitted concentrate water flow rates usingNeethling et al.�s model (2003) based on the data obtained in theOutokumpu 3 m3 tank cell at the Xstrata Mt Isa copper concentrator.


a ¼ 1� H f

H fðmaxÞ ð35Þ

where Hf (max) is the maximum froth height that can beobtained under the given operating conditions. At thismaximum froth height, all bubbles burst.

Fig. 10 shows the fitted water recovery using Eqs.(33)–(35) in comparison with the experimentally mea-sured values.

The fitted results using the model developed byNeethling et al. (2003) match the experimental ones sat-isfactorily. Fig. 11 compares the measured bubble sizeon the surface with the fitted value.

Although there is a significant difference between themeasured bubble size and the model fitted one, there isclearly a strong correlation between them. It is an areain the future to improve the measurement techniqueand/or develop a model for calibration.

The air recovery calculated is between 25% and 71%in the Outokumpu 3 m3 tank cell, higher than the valuesreported for the other operations (Barian et al., 2005;Zheng et al., 2005). This may attribute largely to the

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0 0.2 0.4 0.6 0.8 1 1.2 1.4Measured bubble size (cm)

Cal

cula

ted

bubb

le s

ize

(cm

)

Qa =1085 l/minQa = 1380 l/minQa = 1674 l/minQa = 1968 l/min

Fig. 11. Comparison of the measured and model fitted bubble size onthe cell surface based on the data obtained in the Outokumpu 3 m3

tank cell at the Xstrata Mt Isa copper concentrator.

use of a large froth crowder (1 m in diameter) in the testcell which has a diameter of 1.3 m at the top of the cell.

The most significant feature of the fundamentalmodel (Eq. (33)) is that all the variables in the modelhave physical meaning. They can be determined experi-mentally, i.e. a and r from a froth monitoring device,and Qa and Jg from an air flow meter. Hence, the modelis ready for use in practice for process control purposes.Another advantage of the model (Neethling et al., 2003)over the other empirical models in this paper is that itfits all the experimental data well at the same timeregardless of the cell operating conditions. By incorpo-rating additional models to relate the air recovery andbubble size on the surface to the cell operating condi-tions directly, the fundamental model will also be ableto use for prediction purposes.

4. Summary

No matter whether they are empirically or fundamen-tally based, the mathematical models for water recoveryor concentrate water flow rate reviewed in this papertend to fit the experimental data reasonably well for agiven flotation system. However, the empirical modelswhich relate the water recovery to the solids recovery(Alford, 1990) or describe the water recovery as a func-tion of the mean froth residence time (Savassi, 1998) aregenerally unable to distinguish the effect of different celloperating conditions such as froth height and air rate.Hence, these models can only be used within a relativelynarrow range of cell operating conditions. The empiricalmodels which consider the water recovery as a two-steptransfer (Gorain et al., 1998; Harris, 2000) may be usedfor prediction purposes if the model parameters can bederived beforehand. However, often these parametershave no physical meaning and have to be derived inan existing system. The fundamental approach adoptedby Neethling et al. (2003) resulted in a water recoverymodel with all the variables directly related to the phys-ical conditions in the flotation process. The variables canalso be determined experimentally. Hence, the model isready for use in practice for process control purposes.Once the bubble size on the surface and the air recoveryin the flotation system can be modelled, the water recov-ery model developed by Neethling et al. (2003) can alsobe used for prediction purposes.

Acknowledgements

The authors would like to thank the sponsors of theAMIRA P9 project for the funding which made thiswork possible. The authors would also like to thankOutokumpu for the loan of the 3 m3 tank cell used inthe test work. Support from the staff at the Xstrata Mt


Isa copper concentrator, especially Mr. David Carr, dur-ing the on-site test work is also gratefully acknowledged.

References

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Barian, N., Hadler, K., Ventura-Medina, E., Cilliers, J.J., 2005. Thefroth stability column: linking froth stability and flotation perfor-mance. Mineral Engineering 18, 317–324.

Gorain, B.K., Harris, M.C., Franzidis, J.-P., Manlapig, E.V., 1998.The effect of froth residence time on the kinetics of flotation.Minerals Engineering 11 (7), 627–638.

Harris, M.C., 1998. The use of flotation plant data to simulateflotation circuits, SAIMM Conference.

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Neethling, S.J., Lee, H.T., Cilliers, J.J., 2003. Simple relationships forpredicting the recovery of liquid from flowing foams and froths.Minerals Engineering 16, 1123–1130.

Neethling, S.J., Cilliers, J.J., 2003. Modelling flotation froths. Inter-national Journal of Mineral Processing 72, 267–287.

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Savassi, O.N., 1998. Direct estimation of the degree of entrain-ment and the froth recovery of attached particles in industrialflotation cell, Ph.D. thesis (unpublished), University of Queens-land.

Scrimgeour, J.H.C., Hamilton, R.E., Toong, T., 1970. The use ofmathematical modeling in developing advanced control system formining industry processes. Transactions of the Canadian Instituteof Mining 73, 305–314.

Uribe, A.S., Vazuez, D.V., Perez, R.G., Nava, F.A., 1999. A statisticalmodel for the concentrate water in flotation columns. MineralsEngineering 12 (8), 937–948.

Zheng, X., Knopjes, L., 2004. Modelling of froth transportation inindustrial flotation cells Part II: Modelling of froth transportationin an Outokumpu tank flotation cell at the Anglo PlatinumBafokeng—Rasimone Platinum Mine (BRPM) concentrator. Min-eral Engineering 17/9-10, 989–1000.

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an evaluation of different models of water recovery in flotation-x.zheng

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