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Chemical Engineering and Processing 46 (2007) 1286–1291

Modelling continuous foam fractionation with reflux

Paul Stevenson ∗, Graeme J. JamesonCentre for Multiphase Processes, University of Newcastle, Callaghan, NSW 2308, Australia

Received 9 May 2006; received in revised form 15 August 2006; accepted 16 October 2006Available online 22 November 2006

bstract

An equilibrium stage approach is taken to modelling the performance of a continuous foam fractionation column with reflux. Such an approach

as been facilitated by recent developments in the understanding of pneumatic columns of foam that allow liquid rates within the rising column ofoam to be predicted with confidence. It is shown that the recovery of surfactant into the product stream increases monotonically with increasingeflux ratio but this is at the expense of reduced product rate.

2006 Elsevier B.V. All rights reserved.

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eywords: Foam; Fractionation; Reflux; Simulation

. Introduction

Foam fractionation is a process to enrich streams of surfactanthat relies on the fact that surfactant is preferentially adsorbednto bubble surfaces that rise in a pneumatic foam column. Theiquid recovered at the top of the column (known as the foamate),fter the foam is collapsed, is therefore richer in surfactant thanhe liquid at the base. Furthermore, foam fractionation can alsoe used as a cost-effective separation process for proteins andeptides, as well as a variety of pharmaceutical products fromroduction broths [1], so it is a technology worthy of develop-ent.Lemlich [2] described batchwise and continuous modes of

oam fractionation and showed how continuous columns canither be run as enrichers or as strippers. A generic methodf stagewise calculation of the column operation, that succes-ively utilised mass balances and an equilibrium relationship,as given and a graph that is analogous to the McCabe–Thielelot for conventional fractionation was drawn. However, sincehe upwards and downward streams were not defined, still lessalculated, Lemlich’s work is of limited utility to the engineer

anting to estimate the performance of a column.Neely et al. [3] conducted experiments on the batch foam

ractionation of an aqueous protein solution and suggested an

∗ Corresponding author.E-mail address: paul.stevenson@newcastle.edu.au (P. Stevenson).

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255-2701/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.cep.2006.10.010

quilibrium stage model similar to that of Lemlich’s. However,he model is highly empirical since the rates of the rising andalling streams were not calculated in a systematic manner basedpon the fluid mechanics of the froth.

Maruyama et al. [4] measured the axial concentration of sur-actant in a continuous fractionation column with no reflux, andave proposed a model. However, this model does not encapsu-ate the correct fluid mechanics of the pneumatic froth columnecause the authors note the complexity of existing models ofiquid drainage in foam.

Darton et al. [5] built a column of two distinct stages, inhich the foam was broken on each stage and reflux occurredn each stage. The column was operated at ‘total reflux’o no product stream was drawn from the top. The changen top and bottom concentration was measured and it washown that the solution at the top of the column was indeednriched with surfactant. However, the mass transfer betweeno-called rising and falling streams was not quantified so, again,he approach does not enable a priori estimates of perfor-

ance.In this work, a continuous enricher with reflux, such as that

hown schematically in Fig. 1, is considered. The calculationf flow rates by invoking recent work on pneumatic foams wille demonstrated, and an equilibrium stage approach taken to

alculate the enrichment and surfactant recovery. This representsmethod for calculating the performance of a foam fractionationolumn operating in continuous enriching mode. Other modesf operation are not considered, but the techniques are valid for

P. Stevenson, G.J. Jameson / Chemical Engineer

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ig. 1. A schematic diagram of a foam column operating in continuous enricherode, after Lemlich [2].

hese other modes and the calculation method will be found toe readily adaptable.

. Dynamics of pneumatic foam columns

A so-called pneumatic aqueous foam, such as those found inractionation and flotation columns, is formed by sparging gast a superficial rate (based on the column cross-sectional area)f jg into a pool of surfactant solution so that foam rises in theolumn with a liquid superficial velocity of jf. The liquid fractionf the foam is ε, and, in the absence of bubble coalescence,his is approximately constant up the column [6]. To begin tonderstand the dynamics of such foams we must first estimatehe liquid drainage in the foam. Consider a Lagrangian frame ofeference moving at the velocity of the bubbles, Vb, where:

b = jg

1 − ε(1)

iquid drains at a superficial rate of jd with respect to the bubbleelocity. There are a number of models that enable estimatesf jd. However, it has been shown, by invoking dimensionalrguments, that the drainage rate, non-dimensionalised as atokes-type number, can be expressed as a function of liquidraction, ε, only [7]. Further, it has been shown that, since nontrinsic value of surface viscosity can be known [8], and no

ethod of quantifying the viscous losses at the vertices in theoam exists [9], at least two adjustable constants are requiredo characterise the system. The following drainage expression,alid for foams that are spatially and temporally invariant (i.e.rainage rate is independent of surface tension), is therefore pro-osed:

k = mεn (2)

here m and n are the adjustable constants that are specific ton individual surfactant system at a particular concentration and

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ing and Processing 46 (2007) 1286–1291 1287

he Stokes number is given by:

k = μjd

ρgr2b

(3)

here μ and ρ are the dynamic viscosity and density of thenterstitial liquid respectively, g the acceleration of gravity andb is the harmonic mean radius of the bubbles within the foam.q. (2) is valid for ε < (n − 1)/(n + 1) [10]. Stevenson et al. [11]ave shown that m = 0.016, n = 2 for foams stabilised by sodiumodecyl sulphate at a concentration of 2.92 g/l. These drainagexperiments employed a magnetic resonance imaging techniquehat is believed to be the first method of directly and non-nvasively measuring liquid drainage rate in foams. In addition,q. (2) has been fitted [8] to the forced drainage data of Neeth-

ing et al. [12] and shown that for solutions at an undisclosedoncentration of TTAB, m = 0.013, n = 1.78 and for solutionst an undisclosed concentration of SDS, m = 0.012, n = 1.74.he forced drainage experiment of Neethling et al. [12] proba-ly represents the most convenient and cost-effective method ofetermining m and n for a given surfactant system.

Stevenson [13] has presented a simple method of determininghe equilibrium liquid volume fraction of a rising column ofoam, if m and n are known. The liquid superficial velocity, bysing Eq. (2) for the drainage rate, is given by:

f = εjg

1 − ε− ρgr2

b

μmεn (4)

riefly, Eq. (4) is obtained by subtracting a liquid drainage termrom the product of bubble velocity, jg/(1 − ε), and liquid frac-ion since εjg/(1 − ε) would be the liquid flux if there was norainage (i.e. if the interstitial liquid were frozen). Eq. (4) is plot-ed in Fig. 2 for m = 0.016, n = 2, water as the interstitial liquid,ith jg = 5 mm/s, rb = 0.5 mm. It is seen that the plot of jf versusexhibits a maximum, and, by invoking stability arguments, it

s shown that this point represents the equilibrium state for a ris-ng froth. Thus, the equilibrium liquid fraction can be estimatedabout 0.074 in this example) and it is apparent that a rising frothdjusts so as to maximise the liquid rate, jf (to about 0.18 mm/sn this example). This approach has been experimentally verified

ig. 2. Plot of jf vs. ε for a rising foam showing a construction for estimatinghe enhanced volumetric liquid fraction of the foam.

1 ineering and Processing 46 (2007) 1286–1291

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288 P. Stevenson, G.J. Jameson / Chemical Eng

nd jf is determined by substituting the calculated ε intoq. (4).

Now that the liquid rate and liquid fraction in a pneumaticoam can be estimated, it is pertinent to consider how liquiddded to the top of the foam column changes the properties of theoam. Liquid is added to the top of foam columns in flotation asashwater, and in foam fractionation as reflux liquor. However,

he foam must follow the curve plotted in Fig. 2. Thus, if liquids added, as illustrated in Fig. 2, at about 0.09 mm/s, the foamdjusts to a liquid volume fraction of around 0.128, from theraph. However, it has been shown [13] that, no matter what theeflux rate, the superficial liquid rate recovered from the top ofhe column is jf.

Now consider a foam fractionation column in which a frac-ion R of the recovered liquid, or foamate, is used to reflux backo the top of column. The stationary observer (i.e. in an Eulerianeference frame) will observe a net flux of (1 − R)jf, i.e. a super-cial liquid rate jf travelling up the column and a superficial ratef Rjf travelling down. Moreover, assuming that the bubbles inhe foam have diameter d, the bubble surface area flux travellingp the column, Sb, i.e. the rate of bubble surface area passing aoint per unit column cross-sectional area, is given by:

b = 6jg

d(6)

he volume of gas per unit volume of foam is given by (1 − ε).aking the assumption of spherical bubbles, the volume of sin-

le bubble is πd3/6, so the number of bubble per unit volume ofoam is 6(1 − ε)/πd3. Now since the surface area of an individ-al bubble is πd2, the surface area of bubbles per unit volume ofoam is 6(1 − ε)/d. The absolute velocity of the bubbles, Vb, isg/(1 − ε), Therefore, we obtain the rate of surface per unit cross-ectional area of the column, Sb, by multiplying the absoluteubble velocity by the surface area of bubbles per unit volumef foam, which results in Eq. (6).

So, in the column, the stationary observer sees a stream ofiquid of superficial velocity jf rising in the column, a streamf liquid Rjf descending, and a superficial surface rate of Sbdue to the upward motion of surfaces) rising in the column.hese concepts will be utilised in developing a mass balance forounter-current foam fractionation below.

. Equilibrium and material balance

The surfactant concentration in the bulk liquid, c, formsn equilibrium with the surface concentration (or surfacexcess). This equilibrium can be approximated by the Langmuirsotherm, i.e.:

= cΓsat

c + b(7)

here c is the concentration of surfactant in the bulk liquid, bconstant specific to an individual surfactant system that has

nits of concentration and Γ sat is the saturated surface excessuch that, if c is greater than the critical micelle concentration,he surface excess is approximately Γ sat. Now, consider a massalance around the foam fractionation column, see Fig. 3. Feed

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Fig. 3. A control volume on the entire unit.

olution is supplied to the column at a superficial rate (all super-cial rates are based upon the cross-sectional area of the column)

0 and at concentration c0. A bottoms stream is withdrawn fromhe pool at the base of the column at rate jb and concentration cb.ecause the bottom pool is assumed to be well-mixed it has con-entration cb also; further, because of the mixedness, the bottomool will be considered to be an equilibrium stage in the presentnalysis. Liquid is taken from the top of the column at a ratef and product concentration cp but a fraction R of this streams returned to the top of the column as reflux liquid. Thus theuperficial flow rate of product taken from the unit is (1 − R)jf.

Balances can be taken around the column on a volumetricasis:

0 = jb + (1 − R)jf (8)

nd based on the flow rates of surfactant:

0c0 = jbcb + (1 − R)jfcp. (9)

We proceed by considering that the height of the foam isade up of a discrete number of well-mixed stages such that

he surfaces leaving the stage are in equilibrium with the bulkiquid leaving. We do not know how many stages there are, bute consider a general case where there are Z equilibrium stages

n the foam to which we add the bottom pool, making a total of+ 1 equilibrium stages as shown in Fig. 4. Estimation of theumber of equilibrium stages in a specific column is discussedelow.

The foam rising from stage one is broken so we write a bal-nce on the surface active material as:

bΓ1 + jfc1 = (1 − R)jfcp (10)

f the reflux, R, and cp are specified, the equilibrium relation-hip can be invoked to calculate Γ 1 and c1 where these are theoncentration and surface excess of the stream leaving stage(streams leaving other stages are defined similarly). Next, a

alance around a general stage z is written as follows:

b(Γz+1 − Γz) + jf(cz+1 − cz) + Rjf(cz−1 − cz) = 0 (11)

ass balances and equilibrium are solved successively downhe column so that the concentration and surface excess of allnternal streams are calculated. A balance around the liquid pool

P. Stevenson, G.J. Jameson / Chemical Engineering and Processing 46 (2007) 1286–1291 1289

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Table 1Stream concentration and surface excesses for the illustrative example

Z cz (mol/m3) Γ z (mol/m2 × 10−6)

1 0.406 3.662 0.116 3.613 0.019 3.31B 0.010 3.02

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ig. 4. Concentrations and surfaces excesses of streams leaving the three equi-ibrium stages in the foam and the bottom pool.

the bottom stage B), in conjunction with Eqs. (8) and (9), yields0 and jb if the feed concentration, c0, is specified.

The question now arises as to how many equilibrium stagesre present within a foam column. Lemlich [2] suggests thathe height of a stage can be as low as a few centimetres, butlearly more work is required to enable confident prediction ofhe height of foam required.

. Simulation

As an illustrative example the stream concentrations for ahree-stage plus bottom pool system are calculated. Consider aoam created by sparging gas at a rate jg = 0.5 cm/s to a surfactantystem that has foam drainage parameters m = 0.016 and n = 211]. The average bubble radius, rb, is 0.5 mm. The dynamiciscosity and density of water at S.T.P. are used for the propertiesf the interstitial liquid. By using Eqs. (4) and (5) the liquidraction is found to be 0.074 and the liquid superficial velocitys 0.18 mm/s.

Now, some equilibrium properties need to be assumed; theollowing are equilibrium parameters for the surfactant Triton X-00 [5]: Γ sat = 3.677 × 10−6 mol/m2; b = 2.112 × 10−3 mol/m3.hese equilibrium parameters will be used in this illustrativexample; the critical micelle concentration for Triton X-100 is.3025 mol/m3. The column will be used to enrich feed of con-entration 0.05 mol/m3 to a product concentration of 1 mol/m3

i.e. an enrichment factor of 20). A reflux ratio, R, of 0.5 will besed.

Since one-half of the liquid recovered at the top of the columns returned to the top of the foam, the liquid fraction enhances tobout 0.128, as can be seen from Fig. 2. Note that Sb which in thisase is 30 s−1 from Eq. (6), is independent of the liquid fraction.

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Fig. 5. Recovery and product rate vs. R for the illustrative example.

he calculated surface excesses and stream concentrations arehown in Table 1.

The calculated product superficial velocity is 0.092 mm/s, theequired feed superficial velocity is 2.27 mm/s and the recoveryraction of surfactant is 0.82.

For the same process conditions as used in the illustrativexample above, the simulation is run to investigate the effectf changing the reflux fraction and the fractional recovery androduct superficial rate plotted as a function of R, see Fig. 5. It iseen, as expected, that the recovery rises monotonically reachingnity at total reflux where the product rate is zero. The minimumeflux to attain product specification is about 0.38.

If the simulation is repeated with an increased gas rate ofg = 0.6 mm/s we see that product rate increases from 0.074 to.11 mm/s but surfactant recovery decreases from 0.92 to 0.82.his is consistent with the experimental observation of Boonya-uwat et al. [14].

. Discussion

.1. Bubble coalescence

The effects of bubble coalescence have thus far been dis-ounted in the analysis. Bubble bursting at the top surface of theroth effectively enhances the reflux rate at the top of the columnince the liquid liberated as a bubble bursts does not overflowhe column but is re-applied to the top surface. In addition theiquid rate to the foamate is diminished. An analysis for the liq-id fraction enhancement and the decrease in foamate rate dueo bubble coalescence can be found in Stevenson [10].

Internal bubble coalescence within the froth causes the har-onic mean bubble size to increase as the froth rises in the

olumn and therefore Sk will decrease. This, in turn, will causespatial variation in liquid fraction and therefore the flow ratesithin the column will exhibit spatial variation; the methodol-gy described would require modification if internal coalescence

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290 P. Stevenson, G.J. Jameson / Chemical Eng

ere to be accounted for. In addition, foam coarsening (or Ost-ald ripening) has the effect of changing the harmonic meanubble size and the methodology would need further modifica-ion along similar lines to account for this.

.2. Determination of m and n

In order to proceed with the estimation of the performance offoam fractionation device, estimates of the relevant flow rates

n the column and the drainage parameters, m and n, are neces-ary. These parameters can be readily measured by performingforced drainage experiment as described by Stevenson [7]. It

hould be noted that the parameters are not only dependent onhe surfactant, but also on its concentration. In the simulationsescribed above it has been assumed that m and n are constanthereas, in fact, they change with concentration and thereforeill, in practise, vary up the column. The implication is that the

iquid fraction (and therefore the liquid flow rates) will vary uphe column, just as they will if internal coalescence or coarsen-ng occurs, rather than being constant as assumed in the analysisresented herein.

.3. Other modes of operation

Thus far, a foam fractionation column operating in enrichingode, with the feed being introduced to bottom pool, has been

onsidered. However, Lemlich [2] gives two other continuousodes of operation: Stripping, where the feed is added to the top

f the froth column; and combined operation, where the feed isdded within the column of rising foam in a process analogous toistillation for gas–liquid systems. The methodology presentedere may be adapted to these modes of continuous operation byodifying the appropriate mass balances; it should be recog-

ised that for the combined operation fractionation system, theiquid flow rates above and below the feed location will differ.

In addition, a semi-batch process (i.e. continuous with respecto the gas phase, batch with respect to the liquid) can be modelledy adopting the methodology proposed. However, it is importanto notice that m and n will exhibit temporal and spatial varia-ion as the bottom pool becomes depleted of surfactant, andhe coalescence rate will increase in time as the stability of theas–liquid interface diminishes.

. Conclusions

It has been shown that, because flow rates in pneumaticolumns of foam can be predicted, an equilibrium stage approacho the transfer of surfactant from bulk liquid to surfaces cane invoked to calculate the performance of a foam fractiona-ion column operated in enriching mode. The analysis is readilydaptable for columns operated in stripping mode, or for com-ined operation where the feed is introduced within the frotholumn.

The occurrence of internal and surface coalescence, andoam coarsening complicate the estimation of column perfor-ance, although the presented methodology remains conceptu-

lly sound. [

ing and Processing 46 (2007) 1286–1291

This paper provides, for the first time, a method for estimatinghe flow rates that are required in the theory of Lemlich [2], toescribe batchwise and continuous modes of foam fractionationith reflux. The prediction of the performance of a continuous

oam fractionation column with reflux is now enabled.

ppendix A. Nomenclature

a constant in Langmuir’s isotherm [mol/m3]surfactant concentration in the bulk of the liquid[mol/m3]bubble diameter [m]acceleration due to gravity [m/s2]

0 superficial liquid velocity of the feed [m/s]d liquid superficial drainage velocity [m/s]f liquid superficial velocity up the column [m/s]g gas superficial velocity [m/s]

dimensionless number used in Eq. (2)dimensionless index used in Eq. (2)

b bubble radius (≡0.5d) [m]reflux ratio

b rate of surfaces per unit cross-sectional area of column[s−1]

k Stokes number (≡ jdμ/(r2bgρ))

b absolute bubble velocity [m/s]running stage numbernumber of equilibrium stages not including bottompool

reek symbolssurface excess [mol/m2]

sat saturation surface excess [mol/m2]volumetric liquid fraction in the foamliquid dynamic viscosity [Pa s]liquid density [kg/m3]

eferences

[1] C.E. Lockwood, P.M. Bummer, M. Jay, Purification of proteins using foamfractionation, Pharm. Res. 14 (1997) 1511–1515.

[2] R. Lemlich, Principles of foam fractionation and drainage, in: R. Lemlich(Ed.), Adsorptive Bubble Separation Techniques, Academic Press, NY,1972, pp. 33–51.

[3] C.B. Neely, J. Eiamwat, L. Du, V. Loha, A. Prokop, R.D. Tanner, Modelinga batch foam fractionation process, Biol. Bratislava 56 (2001) 583–589.

[4] H. Maruyama, A. Suzuki, H. Seki, N. Inoue, Enrichment in axial directionof aqueous foam in continuous foam separation, Biochem. Eng. J. 30 (2006)253–259.

[5] R.C. Darton, S. Supino, K.J. Sweeting, Development of a multistaged foamfractionation column, Chem. Eng. Proc. 43 (2004) 477–482.

[6] P. Stevenson, C. Stevanov, Effect of rheology and interfacial rigidity on liq-uid recovery from rising froth, Ind. Eng. Chem. Res. 43 (2004) 6187–6194.

[7] P. Stevenson, Dimensional analysis of foam drainage, Chem. Eng. Sci. 61(2006) 4503–4510.

[8] P. Stevenson, Remarks on the shear viscosity of surfaces stabilised with

soluble surfactants, J. Colloid Interface Sci. 290 (2005) 603–606.

[9] S.A. Koehler, S. Hilgenfeldt, H.A. Stone, Liquid flow through aqueousfoams: the node-dominated foam drainage equation, Phys. Rev. Lett. 82(1999) 4232–4235.

10] P. Stevenson, Hydrodynamic theory of rising froth, Miner. Eng., in press.

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P. Stevenson, G.J. Jameson / Chemical Eng

11] P. Stevenson, M.D. Mantle, A.J. Sederman, L.F. Gladden, Quantitativemeasurements of liquid hold-up and drainage velocity in foam using nuclearmagnetic resonance imaging, AIChE J., in press.

12] S.J. Neethling, H.T. Lee, J.J. Cilliers, A foam drainage equation generalizedfor all liquid contents, J. Phys.: Condens. Matter 14 (2002) 331–342.

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13] P. Stevenson, The wetness of a rising foam, Ind. Eng. Chem. Res. 45 (2006)803–807.

14] S. Boonyasuwat, S. Chavadej, P. Malakul, J.F. Scamehorn, Surfactantrecovery from water using a multistage foam fractionator. Part I, Sep. Sci.Technol. 40 (2005) 1835–1853.