droplet population balance modelling—hydrodynamics and mass transfer

Chemical Engineering Science 61 (2006) 246–256www.elsevier.com/locate/ces

Droplet population balance modelling—hydrodynamics and mass transfer

Stephan A. Schmidt, Martin Simon, Menwer M. Attarakih, Luis Lagar G., Hans-Jörg Bart∗

Technische Universität Kaiserslautern, Lehrstuhl für Thermische Verfahrenstechnik, POB 3049, D-67653 Kaiserslautern, Germany

Received 1 June 2004; accepted 1 February 2005Available online 21 June 2005

Abstract

The hydrodynamic and mass transfer behavior of a rotating disc contactor extraction (RDC) column based on a bivariate populationbalance model is investigated using the generalized fixed-pivot technique for the discretization of droplet internal coordinate. Single-droplet and swarm-droplet studies in small lab-scale devices were used to evaluate breakage and coalescence parameters necessary forcolumn simulations. The breakage probability of single droplets was measured and a new correlation was developed, which also takesviscosity effects into account. Coalescence probability studies resulted in chemical system dependent parameters, which were obtainedby an inverse solution of a simplified balance model. In a final study, the hydrodynamic and mass transfer behavior of pilot plant RDCcolumns have been simulated based on the parameter set derived from the lab-scale units. The simulated mean Sauter diameter, hold-upvalues and concentration profiles were found to be well predicted at different operating conditions. The relative error for the simulatedmean Sauter diameters is about 15%, for the hold-up about 20% and for the concentration profiles about 20%.� 2005 Elsevier Ltd. All rights reserved.

Keywords:Population balance; Breakage; Coalescence; Single droplet; Simulation; Hydrodynamics; Mass transfer

1. Introduction

Liquid–liquid extraction processes are widely used inchemical and biochemical industries. The classical extrac-tion equipment used is mixer–settler cascades and differenttypes of counter-current extraction columns (Godfrey andSlater, 1994). The efficiency of an extraction column isgreatly affected by the dispersed phase hold-up and the par-ticle size distribution which determine the interfacial areaand the mean residence time of the dispersed phase. Thedispersion or back-mixing models usually applied cannotproperly describe the real hydrodynamic behavior becauseof its simplifications. The variation of dispersed phasehold-up and the droplet sizes along the column height dueto droplet interactions is neglected in these models. Thedroplet population balance model (DPBM) takes these intoaccount by considering buoyancy-driven motion of drops,

∗ Corresponding author. Tel.: +49 631 205 2414; fax: +49 631 205 2119.E-mail address:[email protected](H.-J. Bart)URL: http://www.uni-kl.de/LS-Bart/

0009-2509/$ - see front matter� 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2005.02.075

axial dispersion, mass transfer, and breakage and randominter-droplet coalescence.

Since the early sixties the DPBM, with its origin in crys-tallization (Hulburt and Katz, 1964), constituted a consider-able advance in the modelling of solvent extraction columns.The population balance approach has been used to modelthe complex hydrodynamic behavior of the dispersed phasein different types of agitated extraction columns (Godfreyand Slater, 1994; Al Khani et al., 1989; Kronberger et al.,1994). An extensive review of the modelling advances forsuch systems is given byAttarakih et al. (2003), wherethe fixed pivot technique was applied to solve the set ofalgebraic and integro-differential equations. However, lit-tle has been reported on solving the DPBM under masstransfer conditions (Attarakih et al., 2005; Gerstlauer, 1999;Zamponi et al., 1996). It is known that the direction of masstransfer notedly affects the breakage and coalescence behav-ior (Gourdon and Casamatta, 1991), but so far has not beenintroduced in DPBM. The scope of this paper is to evaluatethe DPBM parameters based on single-droplet experimentsin small lab-scale devices and then predict hold-up and con-centration profiles for different column geometries.

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S.A. Schmidt et al. / Chemical Engineering Science 61 (2006) 246–256 247

2. The droplet population balance model

The general population balance equation (SDPBE) fordescribing the coupled hydrodynamics and mass transferin liquid–liquid extraction columns in a one-dimensionaldomain could be written as

�nd,cy (�)

�t= − �[uynd,cy (�)]

�z+ �

�z

[Dy

�nd,cy (�)

�z

]

−2∑

i=1

�[�ind,cy (�)]��i

+ Qiny

Ac

niny (d, cy; t)�(z − zy) + Υ {�}. (1)

The state of any droplet is represented by a bivari-ate (joint) density function nd,cy (d, cy; t, z), wherend,cy (d, cy; t, z)�d�cy represents the number of dropletshaving droplet size and solute concentration in the range[d, d + �d] and[cy, cy + �cy] per unit volume of the con-tactor. So, the dispersed phase hold-up (volume fraction ofthe dispersed phase) is

�y =∫ ∞

0

∫ ∞

0v(d)nd,cy (d, cy; z, t)�d�cy , (2)

wherev(d) is the droplet volume.The components of the vector� = [d cy z t] are those

for the droplet internal coordinates (diameterd and soluteconcentrationcy), the external coordinate (column heightz)and the time,t , where the velocity vector along the inter-nal coordinates is given by� = [d cy]. The first and secondterm on the right-hand side of Eq. (1) describes the trans-port of droplets due to a droplet diameter-dependent risingvelocity uy and the axial dispersion. The next term consid-ers a change in solute content of droplet due to mass trans-fer and the penultimate terms represent the source term forthe solvent feed entering the extraction column at the heightzy with a volumetric flow rateQin

y . The source termΥ {�}appearing in Eq. (1) is more complex and is discussed byAttarakih et al. (2005)in detail. It consists of terms due tobreakage and a coalescence of drops. Breakage frequency�(d,�y), daughter droplet distribution function�n(d, d

′)and coalescence rate�(d, d ′,�y) appearing in the sourceterm are discussed below in detail.

The transport equations describing hydrodynamics andthe solute concentration,cx , in the continuous phase takesthe mass transfer from the continuous to the dispersed phaseinto account. The equation is coupled to the solute balance inthe continuous phase through the convective and the sourceterms of Eq. (1) and could be written as

�(�x)

�t= �

�z

(ux�x + Dx

�(�x)

�z

)+ Qin

x

Ac

�(z − zx), (3)

�(�xcx)

�t= �

�z

(ux�xcx + Dx

�(�xcx)

�z

)

+ Qinx c

inx

Ac

�(z − zx)

−∫ ∞

0

∫ cy,max

0cyv(d)nv,cy (�)�d�cy . (4)

Note that Eq. (3) describes the hydrodynamics of the contin-uous phase by analogy to the dispersed phase hydrodynam-ics (Casamatta, 1981; Kronberger, 1995). The dispersion co-efficients of the dispersed and continuous phases approacheach other as the energy dissipation is increased (Casamattaand Vogelpohl, 1985). Moreover, the assumption that is in-herent in these two equations is that the density of the con-tinuous phase is approximately constant; that is, the quantityof the solute transferred from the continuous to the dispersedphase is small, which is the usual case. Accordingly, the rateof mass transfer could be written in terms of an overall masstransfer coefficient, specific interfacial area and mass trans-fer driving force (Casamatta and Vogelpohl, 1985). Eq. (3)when combined with Eq. (1) integrated from zero to infinityafter multiplication byv(d), it results in an expression forthe continuous phase velocity,ux (Attarakih et al., 2005).The volume fraction of the continuous phase,�x , clearlysatisfies the physical constraint:�x + �y = 1.

The first and the second term on the right-hand side ofEqs. (3) and (4) are to be interpreted analogous to the re-spective terms in Eq. (1). The last term in Eq. (3) is thefeed of the continuous phase modelled by a point sourceat column levelz = zx . The last term appearing in Eq. (4)describes the total rate of solute transferred from the con-tinuous to the dispersed phase. For a detailed description ofthe spatial discretization and for the generalized fixed pivottechnique as extended to mass transfer, the interested readercould refer, respectively, toAttarakih et al. (2004a,b).

3. Experimental set-up

The single-drop experiments and swarm-droplet experi-ments were conducted in a geometrically similar lab-scaleRDC column with five compartments (Fig. 1). The heightof one compartment averages 0.03 m, the rotor disc di-ameter 0.09 m, the inner stator diameter 0.105 m and thecolumn diameter 0.15 m. For the determination of thebreakage probability of single droplets, only a single com-partment in the lab-scale device was used. Monodispersedorganic droplets were generated with a two-phase nozzle inthe range 2–5 mm. In the compartment, a mother dropletbreaks up with a certain breakage probabilityp(d) intotwo or more daughter droplets. All experiments were car-ried out with equilibrated standard test systemsn-butylacetate/water and toluene/water, recommend by the EFCE(Misek et al., 1985). The system isotridecanol/water wasinvestigated to characterize the influence of viscosity of thedispersed phase.Table 1depicts the main system properties

248 S.A. Schmidt et al. / Chemical Engineering Science 61 (2006) 246–256

Table 1System properties

System �y �x y x (mPa) (mPa) (kg m−3) (kg m−3) (mN m−1)

H2O/n-butyl acetate 0.73 1.34 881.5 998.9 14.0H2O/toluene 0.59 1.00 865.6 997.7 36.1H2O/isotridecanol 45.9 1.06 846.9 995.6 15.2

1- aqueous storage tank

2- organic storage tank

3- overflow

4- photoelectrical probe

5- motor

6- distributor

7- pump

8- flow control

9- 5 RDC-compartments

3

9 4

5

1 2

7

6

8

Fig. 1. Experimental set-up for coalescence measurements.

of the investigated systems. The droplet swarm coalescenceinvestigations were performed with five compartments ofthe same RDC geometry (seeFig. 1). For different opera-tional conditions, droplet size distributions were measuredat the inlet and the outlet of the column using a photoelec-trical suction probe according toPilhofer and Miller (1972).The total volumetric phase flow throughput was in the range1.4× 10−5–2.8× 10−5 m3 s−1. The rotational speed variedfrom 2.5 to 5 s−1.

4. Modelling and parameter estimation

4.1. Column hydrodynamics

The droplet transport is based on the classical slip velocityconcept ofGayler et al. (1953):

uy(d,�) = ur(d,�) − ux . (5)

Hereuy is the mean velocity of droplets with diameterdandux the continuous phase velocity. The influence of thedispersed phase hold-up on the relative velocity,ur , alsonamed the slip velocity, is taken into account by the follow-ing empirical expression:

ur(d,�y) = u∗r (d) · (1 − �y)

m

= kV (d) · ut (d) · (1 − �y)m. (6)

In this case, the characteristic velocity of a single dropletu∗r (d) is corrected to take into account the interactions with

other droplets of the swarm;m is a function of the parti-cle Reynolds number (Bailes et al., 1986). Furthermore, the

characteristic velocity is related to the single droplet termi-nal velocity,ut , and to a slowing factorkV , which takes intoaccount the effect of the column internal geometry on thedroplet terminal velocity (per definition 0<kV =1). Dropletsize-dependent parameters were used according toGarthe(2004). The axial dispersion coefficient of the continuousphase was taken fromSommeregger et al. (1980)and thatfor dispersed phase droplet (size-dependent) from a correla-tion of Modes (2000). Generally, dispersion coefficients ap-proach similar values for both phases and become indepen-dent on droplet size at high specific throughputs and powerinput (Gourdon et al., 1994), which simplifies calculations.

4.2. Mass transfer

The dispersed phase individual mass transfer coefficientis found, depending on the behavior of the single droplet,whether it is rigid, circulating or oscillating (Steiner, 1986;Kumar and Hartland, 1999). In the present work, the modelof Handlos and Baron (1957)was used assuming turbulentinner mixing of the droplets, which is generally true withdroplet sizes and solute concentrations found in extractioncolumns. This is in contrast to mixer–settler where, due to thehigher power input, smaller and thus rigid droplets withoutany motion in them are produced. The equilibria and phys-ical properties involved were calculated according to corre-lations recommended by the EFCE (Misek et al., 1985). Formass transfer coefficient in the continuous phase, the modelrecommended byHeertjes et al. (1954)was used. The over-all mass transfer coefficient can be obtained from the indi-vidual mass transfer coefficients based on the two-resistancetheory.

4.3. Drop breakage

In the case of stirred extraction columns, the breakage ofdroplets is predominantly determined by the turbulent struc-ture of the continuous phase. Basically, turbulence from thebulk continuous phase affects the droplet surface. If the en-ergy of a turbulent eddy exceeds a critical value, the interfacebecomes unstable and the droplet finally breaks up. A funda-mental work on single-droplet breakage in a RDC has beenreleased byBahmanyar and Slater (1991)andCauwenberget al. (1997). Cauwenberg et al. formulated a correlation forthe breakage probability of single droplets using a modifiedWeber numberWemod based on the shearing action of a ro-tating disc.

Wemod = 0.8x �0.2

x dD1.6R (2�)1.8(N1.8 − N1.8

crit)

. (7)

They investigated different chemical systems with high andlow interfacial tension but only at a low dynamic viscosityof the dispersed phase. In high viscous systems, it is alsonecessary to take the resisting shear stress force in the dropletinto account (Hinze, 1955). The total local shear stress,�,


imposed by the continuous phase, acts to deform a drop andto break it if the counterbalancing surface tension forces andviscous stresses inside the fluid particle are overcome. Thecondition for breakage is then (Clift et al., 1978)

�>( + �y√

�/y)/d (8)

and a new breakage correlation based on the modified Webernumber has been derived which is also valid for high viscoussystems (Schmidt, 2004):

p(d)

1 − p(d)

= 1.3 × 10−6

Wemod

1 + 0.33�y√dd

[Wemod]0.5

2.78

. (9)

Eq. (9) is dependent on geometrical and physical propertiesas well as the energy input. The critical rotor speed,Ncrit,in Eq. (7), characterizes the rotor speed, below which nobreakage occurs. For each droplet size and chemical system,a specific energy input is necessary to induce the rupture ofa droplet. The critical rotor speed can be derived from thecorrelation ofCalabrese et al. (1986), which was developedfor batch-stirred tank contactors:

x 2/3d

5/3crit

= a

1 + b

(x

y

)1/2�y

1/3d1/3crit

. (10)

For a specific energy input per unit mass, one can calculatethe critical drop diameterdcrit, for which breakage will justoccur. Eq. (10) can also be formulated in terms of a criticalrotor speedNcrit for a specific drop size, above which thedroplets will break up. After some algebraic manipulations,the following expression for the critical rotor speed adaptedfor the RDC geometry has been obtained:

Ncrit = 0.016D

−2/3R �yd

−4/3

(xy)1/2 +

(0.008

D−2/3R �yd

−4/3

(xy)1/2

)2

+ 0.127

xD4/3R d5/3

0.5

. (11)

The breakage probability can be described by its dependenceon the energy input of the RDC stirrer and the physical prop-erties of the system with Eqs. (9) and (11). For the investi-gated single-droplet systems (Fig. 1, with one compartment),the breakage probability could be explicitly determined andrelated to the power input of one compartment. As depictedin Fig. 2, it can be seen for a single droplet(d ′ =3 mm) thatat a certain critical rotor speed the drops starts to break up.Furthermore, if the viscosity of the disperse phase and thesurface tension are decreased, higher breakage probabilitiesare obtained.

0

0.25

0.5

0.75

1

0 2 4 6 8 10 12 14

rotor speed N [1/s]

brea

kage

pro

babi

lity

p(d)

[-]

water / toluenewater / isotridecanolwater / n-butyl acetatecorrelation

d'= 3 mm

Fig. 2. Breakage probability for different systems with respect to rotationalspeedN .

Following the approach ofGourdon et al. (1994), thebreakage frequency,�(d,�), required in the DPBM equa-tion can be calculated from the breakage probability and thedroplet residence time in the compartment:

�(d,�) = p(d)uy(d,�)

Hc

= p(d)

�m. (12)

The daughter droplet distribution is assumed to follow thebeta distribution, which is given byBahmanyar and Slater(1991):

�(d, d ′) = 3xm(xm − 1)

[1 −

(d

d ′

)3](xm−2)

d5

d ′6 , (13)

wherexm is the average number of daughter droplets pro-duced by the breakage of a mother droplet with diameterd ′. The number of daughter droplets,xm, is correlated inEq. (14) with mother droplet diameter,d ′, and the criti-cal droplet diameter,dcrit, and takes a value equal to 2 andhigher:

xm = 2 + 1.77× 10−4[(

d ′

dcrit

)− 1

]3.19

. (14)

The correlations were obtained for droplet size range from 2to 5 mm, a broad viscosity and surface tension range (Table1) at practically relevant power input (up to a rotationalspeed of 8.3 s−1) with about 140 single-droplet experimentsusing the set-up shown inFig. 1. The correlations yieldedon an overall relative error of about 22%

4.4. Coalescence

Preliminary experiments in a venturi-tube have shown thatthe coalescence probability of droplets strongly depends onthe droplet size, the hold-up and system properties, such aspH and ionic strength (Simon et al., 2003) and also on the


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6drop size [mm]

cum

ula

tive

vol.

dis

trib

utio

nS 3[-

]

N= 3.3 s-1

N= 4.2 s-1

N= 5.0 s-1

sim. distribution 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6drop size [mm]

cum

ula

tive

vol.

dis

trib

utio

nS 3[-

]

N= 3.3 s-1

N= 4.2 s-1

N= 5.0 s-1

sim. distribution

Fig. 3. Cumulative outlet distribution for toluene/water at 273 K(Qc = Qd = 2.78× 10−5 m3 s−1).

Table 2RDC column geometries

Wolschner Brodkorb Schmidt Garthe(1980) (1999) (2004) (2004)

Column diameter (m) 0.1 0.152 0.15 0.08Stator diameter (m) 0.074 0.111 0.105 0.05Rotor diameter (m) 0.06 0.101 1.09 0.045ompartment height (m) 0.036 0.76 0.03 0.047Active height (m) 1.20 1.82 1.65 2.85

0

1

2

3

4

5

0 1 2 3 4 5simulated mean Sauter diameter d32 [mm]

exp

. m

ea

n S

au

ter

dia

me

ter

d32

[m

m]

hh

h

exp. toluene/water (Schmidt, 2004)exp. n-butyl acetate/water (Schmidt, 2004)exp. toluene/acetone/water (Wolschner, 1980)exp. toluene/acetone/water (Wolschner, 1980)exp. toluene/acetone/water (Garthe, 2004)

ba

b

bb

15%

15%

Fig. 4. Comparison of experimental and simulated mean Sauter diametersd32 ((a) without and (b) with coalescence).

mass transfer direction (Gourdon and Casamatta, 1991). Itis believed that droplet coalescence occurs if the randomcontact time between any two coalescing droplets exceedsthe time required for the complete intervening film drainageand rupture (Coulaloglou and Tavlarides, 1977). In stirredliquid–liquid extraction columns, coalescence mainly oc-curs in weak turbulence zones of the column, which directlycorrespond in a RDC to the zones below the stator plates(Kentish et al., 1998). Coulaloglou and Tavlarides (1977)developed a model for a stirred vessel, which is based on

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 1 1.5 2 2.5 3 3.5 4

mea

n S

aute

r dia

met

er d

32 [m

m]

exp. toluene/acetone/water (Garthe 2004)

N= 3.3 s-1

Qc = 1.1∗ 10-5 m³ /s

Qd = 1.3∗ 10-5 m³ /s

column height [m]

simulated

Fig. 5. Comparison of simulated and experimental mean Sauter diameterprofile.

the kinetic theory of gases and the drainage film theory.The authors expressed the coalescence frequency as a prod-uct of collision rateh(d, d ′,�) and coalescence efficiency�(d, d ′,�):

�(d, d ′,�) = h(d, d ′,�)�(d, d ′,�). (15)

The authors considered collision between two droplets to besimilar to the collision between two gas molecules in kinetictheory of gases. Thus

h(d, d ′,�) = c1 1/3

1 + �(d + d ′)2(d2/3 + d ′2/3)1/2. (16)

Using the film drainage theory, the authors have devel-oped a correlation to describe the coalescence efficiency,�(d, d ′,�), which depends on the system properties, thedroplet diameter and the energy input. Here, for a success-ful collision between two droplets, the contact time mustexceed the drainage time of the continuous phase betweentwo droplets:

�(d, d ′,�) = exp

(− c2�xx

2(1 + �)3

(dd ′

d + d ′

)4)

. (17)

Despite the fact that the model exhibits some deficiencies(e.g.Kumar et al. (1993)criticised among others the modelfor the critical film thickness) it describes the experimentalinvestigation in an empty venturi-tube satisfactorily (Simonet al., 2003).

In a stirred extraction column, the breakage and coales-cence process take place simultaneously and it is not possi-ble to estimate the coalescence parameters independent frombreakage. This is in contrast to breakage, which can be eas-ily correlated to a Weber number for one column compart-ment. Because a predictive and reliable correlation for coa-lescence occurrences is presently not available we decidedto solve the inverse problem. In contrast to approaches fromliterature, where breakage and coalescence are optimized


0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35

ho

ld u

p,e

xpe

rime

nta

l [%

]d

exp. toluene/acetone/water (Garthe, 2004)exp. n-butyl acetate/water (Schmidt, 2004)exp. toluene/water (Schmidt, 2004)

20%

20%

hold up, simulated [%]

Fig. 6. Comparison of experimental and simulated hold-up values.

Table 3Experimental and simulated hold-up values for toluene/acetone/water(Garthe, 2004)

Qd Qc N Column Hold-up Hold-upheight exp. sim.

(m3 s−1)×10−5 (m3 s−1)×10−5 (s−1) (m) (%) (%)

1.11 1.33 3.33 3.43 6.62 6.722.33 7.07 6.681.23 8.00 7.12

1.41 1.70 3.33 3.43 8.72 9.212.33 8.72 9.081.23 12.82 9.67

1.70 2.07 3.33 3.43 9.39 11.922.33 10.62 11.681.23 11.69 12.34

2.00 2.41 3.33 3.43 12.92 15.432.33 13.23 14.991.23 25.00 15.73

1.11 1.33 6.67 3.43 6.94 7.682.33 7.54 7.491.23 8.45 7.57

1.41 1.70 6.67 3.43 9.23 11.102.33 9.54 10.851.23 12.62 10.97

1.70 2.06 6.67 3.43 15.20 14.232.33 15.70 13.801.23 20.62 14.16

2.00 2.41 6.67 3.43 14.15 20.482.33 17.23 19.611.23 29.37 20.25

together, only the coalescence parameters then need to beestimated. To evaluate the unknown coalescence parametersc1 andc2 in Eqs. (16) and (17), an optimization-simulation

Table 4Experimental and simulated hold-up values (Schmidt, 2004)

Qd Qc N Column Hold-up Hold-upheight exp. sim.

(m3 s−1)×10−5 (m3 s−1)×10−5 (s−1) (m) (%) (%)

3.11a 2.78 5.00 1.55 16.01 14.140.81 14.54 14.14

4.17 1.55 11.97 12.420.81 10.57 12.42

3.33 1.55 9.67 10.850.81 9.52 10.85

2.50 1.55 8.33 9.660.81 8.78 9.67

1.67 1.55 7.90 9.390.81 8.40 9.39

1.56a 2.78 5.00 1.55 11.47 6.280.81 10.89 6.28

4.17 1.55 6.52 5.630.81 6.21 5.63

3.33 1.55 5.39 4.970.81 4.55 4.97

2.50 1.55 4.38 4.410.81 5.03 4.43

1.67 1.55 5.43 4.200.81 4.37 4.23

3.33b 3.33 5.83 1.55 14.27 14.890.81 15.77 14.90

3.33b 3.33 4.17 1.55 10.89 11.110.81 11.56 11.32

3.11b 2.78 5.83 1.55 13.60 13.030.81 10.78 13.03

1.56b 2.78 7.50 1.55 13.40 7.000.81 10.54 7.00

1.56b 2.78 5.83 1.55 7.85 5.780.81 6.57 5.77

1.56b 2.78 5.00 1.55 7.20 5.160.81 6.30 5.12

an-butyl acetate/water.btoluene/water.

algorithm was developed based on a simplified stage-wisepopulation balance model for a short segment of a column(neglecting the axial dispersion). The program calculates theunknown coalescence parameters within the chosen coales-cence model minimizing the square sum of errors accordingto the following objective function (Attarakih et al., 2004a):

Fobj =NDC∑k=1

[Sexp3,k − Ssim

3,k ]2. (18)

In Eq. (18), the simulated steady-state outlet cumulative vol-ume distributionS3 is fitted to the experimental one usingthe Rosenbrock method (Raman, 1985). Fig. 3 depicts thesimulated outlet distributions for three operating conditionsas compared to the measured ones.


0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

activ

e he

ignt

[m]

activ

e he

ignt

[m]

activ

e he

ignt

[m]

activ

e he

ignt

[m]

N= 10 s-1

Qc=1.7 *10-5 m³/s

Qd=1.7 *10-5 m³/s

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2

N= 11 s-1

Qc=1.7 ∗ 10-5 m³/s

Qd=1.7 *10-5 m³/s

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6

N= 9 s-1

Qc=1.7 ∗ 10-5 m³/s

Qd=2.8 *10-5 m³/s

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4

N= 9 s-1

Qc=2.8 ∗ 10-5 m³/s

Qd=1.7 *10-5 m³/s

concentration [kg/kg%] concentration [kg/kg%]

concentration [kg/kg%] concentration [kg/kg%]

Fig. 7. Comparison of experimental (open symbol continuous phase, full symbol dispersed phase) and simulated (solid lines) concentration profiles atdifferent operating conditions for toluene/acetone/water (Wolschner, 1980).

For investigated systems (toluene/water,n-butylacetate/water) a parameter set is obtained, where the parameters arenot influenced by the operational conditions (rotor speed,volumetric flow rates) but are specific for each chemicalsystem. For the system toluene/water the parameters arec1=9.859×10−2 andc2 =1.646×1011 m−2 and for the systemn-butylacetate/water, they arec1 = 3.600× 10−2 andc2 =1.152× 1010 m−2. The parameters should be, in principle,of the same order of magnitude. However, further work isrequired considering chemical different systems to improvethe coalescence model.

The coalescence phenomenon is more complex if masstransfer is taken into account. In the case of solute transferfrom the dispersed to the continuous phase, experiments

have shown that the coalescence rate is enhanced, whilefor the reverse direction it is strongly suppressed. This hasbeen reported by several authors (Godfrey and Slater, 1994)and in the case of mass transfer from the continuous to thedispersed phase, coalescence can mostly be neglected.

5. Results and discussion

The hydrodynamic and mass transfer behavior of geomet-rically different RDC columns was simulated based on onemodel parameter set from single-droplet and droplet-swarmexperiments. Simulations of mass transfer were comparedwith 24 experimental data from the literature (Wolschner,


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 2 4 6 8 10 12concentration [kg/m³]




N= 2.25 s-1

Qc=1.13 ∗ 10-4 m³/s

Qd=3.75 *10-5 m³/s

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8N= 2.25 s-1

Qc=6.25 ∗ 10-5 m³ /s

Qd=3.75 *10-5 m³/s

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

act

ive

he

ign

t [m

]a

ctiv

e h

eig

nt

[m]

act

ive

he

ign

t [m

]a

ctiv

e h

eig

nt

[m]

N= 2.25 s-1

Qc=3.75 *10-5 m³/s

Qd=3.75 *10-5 m³/s

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8N= 4.5 s-1

Qc =3.75 ∗ 10-5 m³/s

Qd =3.75 *10-5 m³/s

Fig. 8. Comparison of experimental (open symbol continuous phase, full symbol dispersed phase) and simulated (solid lines) concentration profiles atdifferent operating conditions for toluene/acetone/water (Brodkorb, 1999).

1980; Brodkorb, 1999; Garthe, 2004). Table 2shows the di-mensions of the RDC columns investigated. The mass trans-fer direction was, in all cases, from the continuous to thedispersed phase, with acetone in water as solute. The soluteconcentration of the feed at the inlet averages between 10and 60 kg m−3 and the solute concentration in the incomingsolvent is less than 3 kg m−3. The total throughput variedbetween 2.8 × 10−3 and 9× 10−3 m3 m−2 s with phase ra-tios (Qy/Qx) from 0.3 to 1.7. The rotor speed ranged be-tween 1.17 and 11 s−1. In the present case study, the impellerReynolds numbersRe were up to 6700, within a fully tur-bulent regime and the mean droplet Reynolds numberRedvaried from 60 to 300. All simulations were carried out ona personal computer with 1.7 MHz leading to effective realsimulation times of less than 10 min.

Fig. 4shows the simulated mean Sauter diametersd32 withcomparison with the experimental data ofWolschner (1980)andGarthe (2004), with mass transfer, and that ofSchmidt(2004), without mass transfer. The mean relative error be-tween simulated and experimental values averages about15%. Initially, the data of Wolschner (1980) was stimulatedneglecting the coalescence yielding a low mean relative errorbetween simulated and experimental values of about 10%.However, the experimental data ofGarthe (2004)undersimilar conditions revealed that the coalescence (mass trans-fer from continuous to dispersed) could not be neglected.Fig. 5depicts the experimental mean Sauter diameter alongthe column height for one experiment from Garthe in com-parison to the simulated one considering coalescence, wherean increase in droplet diameter due to coalescence can be no-


0

0.5

1

1.5

2

2.5

0 2 4 6 8concentration [kg/kg%]

act

ive

he

ign

t [m

]

act

ive

he

ign

t [m

]

N= 3.3 s-1

Qc=1.1 ∗ 10-5 m³/s

Qd=1.3 *10-5 m³/s

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6concentration [kg/kg%]

N= 6.7 s-1

Qc =1.1 *10-5 m³/s

Qd=1.3 *10-5 m³/s

Fig. 9. Comparison of experimental (open symbol continuous phase, full symbol dispersed phase) and simulated (solid lines) concentration profiles atdifferent operating conditions for toluene/acetone/water (Garthe, 2004).

ticed. For the system toluene/acetone/water the coalescenceparameters withc1 =9.859×10−2 andc2 =1.0×1012 m−2

were found adequate to predict the mean Sauter diametersfor the data of Garthe. Inclusion of coalescence yields moreaccurate description of the mean Sauter diameters (seeFig. 4) for the data of Wolschner (average deviationabout 8%).

Fig. 6 shows the simulated hold-up values in comparisonto the experimental ones for the data ofGarthe (2004)andSchmidt (2004). Tables 3and4 depict the measured and thesimulated hold-up values for different operational conditionsand column heights. Here also a good prediction of the ex-perimental data with a mean relative error of about 16% wasfound. Unfortunately, the other authors did not report exper-imental values for the dispersed phase hold-up under masstransfer conditions. Consequently, the adequate predictionof the hold-up and the droplet size, allows as demonstratedhere, to determinate the specific interfacial area required forthe calculation of mass transfer.

For the validation of the column performance under masstransfer conditions, the measured concentration profiles re-ported byWolschner (1980)were compared to the simulatedones inFig. 7.Here, the lines represent the simulated con-centrations profiles of the continuous phase and dispersedphase and the dots represent the experimental values. Forall experiments, the solute concentrations along the columnheight are well predicted for each considered phase, with amean relative error of about 13%.Fig. 8 depicts simulatedconcentrations profiles in comparison to the experimentalvalues given byBrodkorb (1999)for different operating con-ditions. Here also a good prediction of the concentrationsprofiles is obtained, with a mean relative error of about 20%for all experiments. The higher error results from a simu-

lated higher mass transfer rate. On the other hand, this maybe related to minor chemical impurities during the experi-ments, which is well known to suppress the mass transferrate. Since the mean droplet Reynolds numbers averages be-tween 150 and 300, the model ofHandlos and Baron (1957)seems to adequately describe the mass transfer.Fig. 9 de-picts simulated concentrations profiles in comparison to theexperimental values given byGarthe (2004)for two dif-ferent operational conditions. Here again a good predictionof the mass transfer performance in the RDC column wasachieved, with a mean relative error about 20%. Finally, itcan be summarized that based on single-droplet experimentsin connection with the DPBM the mass transfer performanceof RDC extraction columns at different operating conditionscan be predicted within a good accuracy.

6. Conclusions

Single and swarm-droplet experiments have been per-formed in lab-scale equipment in order to characterizethe breakage and coalescence phenomena of droplets inliquid–liquid extraction systems. A new correlation for thebreakage probability has been developed based on data in abreakage dominated regime. Besides the energy input andthe surface tension, the viscosity of the dispersed phaseis also a determining factor affecting the rupture of singledroplets, which is considered in the correlation for dropbreakage. The droplet swarm coalescence in a column can-not be determined independent from breakage and dependson hold-up, droplet size, energy input and system proper-ties, such as ionic strength and pH-value, and even on masstransfer direction. The inverse problem was solved with an


optimization algorithm after Rosenbrock considering thenew breakage correlation derived. The parameters derivedare valid for different column geometries and operatingconditions but dependent on the chemical system used, re-quiring further improvement of the coalescence model inuse. The final column simulations for the experimental datafrom the literature were performed on a 1.7 MHz PC in lessthan 10 min solving the balance equations using the gen-eralized fixed pivot technique for the discretization of thedroplet internal coordinates. The mean Sauter diameters,hold-up and concentration profiles could be well predicted,based on parameters derived in small-scale laboratory units.This approach consumes minimum time and matter, whichpromotes the use of DPBM models for further applicationsin industrial scale.

Notation

a, b constants [Eq. (10)], dimensionlessAc column cross-sectional area, m2

B total phase load, m3/m2 sc solute concentration in the organic phase,

kg/kg%, kg m−3

c1 parameter [Eq. (16)], dimensionlessc2 parameter [Eq. (17)], m−2

d droplet diameter, md32 Sauter mean droplet diameter, mdcrit critical droplet diameter, mD axial dispersion coefficient, m2 s−1

DR rotor diameter of RDC, mh collision frequency, m3 s−1

HC compartment height, mm swarm exponent, dimensionlessnd,cy�d�cy the number of droplets withd and cy ∈

[d ± �d] × [cy ± �cy] per unit volume of thecontactor, m−3

N rotor speed, s−1

Ncrit critical rotor speed, s−1

NDC number of droplet classesp(d) breakage probability, dimensionlessQ flow rate, m3 s−1

Re impeller Reynolds numbers(cND2r /�c)

Red droplet class Reynolds number(cdu/�c)S3 cumulative volume distribution, dimension-

lesst time, su droplet velocity, m s−1

v droplet volume, m3

xm average number of daughter dropletsproduced

z space coordinate, m

Greek letters

� daughter droplet volume probability density,m−1

� breakage frequency, s−1

energy dissipation, m2 s−3

� dynamic viscosity, Pa� coalescence efficiency, m3 s−1

density, kg m−3

interfacial tension, N m−1

� phase hold-up, dimensionless� internal and external coordinates vector

([d cy z t])� coalescence frequency, m3 s−1

Subscripts

r relative velocityx, y continuous and dispersed phases, respec-

tively

Superscripts

· derivative with respect to timein inlet

Acknowledgements

The authors wish to thank the AiF (Arbeitsgemeinschaftindustrieller Forschungsvereinigungen “Otto von Guericke”e.V.), the BMWA (Bundesministerium für Wirtschaft undArbeit) and the DFG (Deutsche Forschungsgemeinschaft)for financial support.

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