A numerical parameter study on the impact of Marangoni convection on the mass transfer at buoyancy-driven single droplets

International Journal of Heat and Mass Transfer 71 (2014) 769–778

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International Journal of Heat and Mass Transfer

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A numerical parameter study on the impact of Marangoni convection onthe mass transfer at buoyancy-driven single droplets

0017-9310/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.11.027

⇑ Tel.: +49 30 314 23701.E-mail address: [email protected]

M. Wegener ⇑Chair of Chemical and Process Engineering, Ackerstraße 71-76, D-13355 Berlin, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 June 2013Received in revised form 11 November 2013Accepted 11 November 2013Available online 18 January 2014

Keywords:Interfacial instabilitiesComputational Fluid DynamicsMarangoni effectLiquid/liquid extraction

Based on earlier successful validations of 3D simulations with experiments in a Marangoni dominatedliquid/liquid system, this paper presents a numerical parameter study which investigates the effect ofdroplet diameter, kinematic viscosity ratio, interfacial tension gradient, and partition coefficient on themass transfer of a solute via the Marangoni induced flow patterns. Simulation results exhibit a stronginfluence of droplet size on the mass transfer performance and reasonable agreement with experimentsin the spherical droplet shape regime. In contrast to predictions based on the linear stability theory, theMarangoni convection was found to be unimpressed by a change in the kinematic viscosity ratio. For allkinematic viscosity ratios studied here, the mass transfer was significantly faster than in the system withconstant interfacial tension (i.e. no Marangoni convection). The same is true for the interfacial tensiongradient, which is the main driving force for Marangoni flow. Even if the gradient is decreased by a factorof 100, mass transfer enhancement factors >2 were found. Only an increase in the partition coefficient toa value of 10, and therewith a respective decrease in the concentration gradient at the interface, endedthe dominion of the Marangoni effect on the mass transfer rate.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The solutal Marangoni convection plays an important role forthe fluid dynamics and the mass transfer of a solute across theinterface of single droplets freely rising in an ambient liquid. TheMarangoni effect in moving droplets is a complex phenomenonwhich eludes stubbornly an analytical description, mostly due tothe fact that the Marangoni convection is a chaotic and inherentlythree-dimensional phenomenon which affects and is affected byvarious parameters and properties of the system under consider-ation. Moreover, the Marangoni convection is an utterly time-dependent and temporary process. It needs a certain time to devel-op its flow patterns, becomes dominant and eventually dies awayin the later stages of the process [9].

Fig. 1 shows the main stages of Marangoni convection patternsin and around rising droplets. The upper row shows schematics,whereas the lower row displays results from numerical simula-tions [22]. The convection originates from tangential shear stressesat the interface which in turn result from local gradients of interfa-cial tension around the droplet surface. In the early stage of masstransfer, the concentration gradient is large and the shear stressesat the interface are strong. The adjacent fluid layers are acceleratedby momentum transfer and result in vortex structures on a

sub-drop scale level and increasing radial displacement (or mixing)within the droplet (Fig. 1(a)). Strong Marangoni convection at theinterface increases the drag coefficient and reduces the drop risevelocity temporarily. The maximum level of reduction is reachedwhen the rise velocity fluctuates around the terminal velocity ofa comparable rigid sphere. Hence, Reynolds and Peclet numberare not constant and vary with time.

In the middle stage of the mass transfer process, the influence ofinterfacial tension induced shear stresses starts to weaken. Thedrag decreases, and eventually a more regular internal circulationestablishes in an asymmetric manner (Fig. 1(b)). In this stage,experiments showed that the rise velocity increases and the drop-let breaks out from its vertical path due to the asymmetric onset ofinternal circulation [25]. In the later stage of the process, the influ-ence of Marangoni convection becomes insignificant. The dropletshows the well known toroidal flow pattern which is now com-pletely axisymmetric (Fig. 1(c)). The drag reduces and the dropletapproaches its terminal velocity.

Is it of particular interest – especially from a practical point ofview – to be able to estimate the impact of Marangoni convectionon the mass transfer performance, e.g. in liquid/liquid extractionprocesses. For those processes, the single droplet is an importanttest case as it can be regarded as the smallest mass transfer unitin extraction columns, stirred tanks or likewise. Empirical masstransfer correlations developed to describe the complex dropletswarm behaviour in contact apparatuses are based on single

Nomenclature

Latin letterscs concentration of solute s, g/Lcs;0 initial concentration of solute s, g/Lc� related concentration of solute s c� ¼ cs=cs;0

dcr critical droplet diameter (Eq. 14), mdP droplet diameter, mDs;c diffusion coefficient of component s in continuous

phase, m2/sDs;d diffusion coefficient of component s in dispersed phase,

m2/sD� ratio of diffusion coefficients D� ¼ Ds;d=Ds;c

E enhancement factor, Eq. 13g gravitational acceleration, m2/sH� partition coefficientL characteristic length, mp pressure, Par radial coordinate, mR droplet radius, mt time, su velocity, m/s

Greek lettersb mass transfer coefficient, m/s# temperature, �Ch azimuth angle, radl dynamic viscosity, Pa s

l� ratio of dynamic viscosities l� ¼ ld=lcm kinematic viscosity, m2/sm� ratio of kinematic viscosities m� ¼ md=mc

q density, kg/m3

r surface tension, N/mu polar angle, rad

Subscripts0 initial, at t ¼ 01;2 phase 1, phase 2c continuous phasec! d mass transfer direction from continuous phase to drop-

letd dispersed phased! c mass transfer direction from droplet to continuous

phaseP droplets soluble componentTAW toluene/acetone/water

Dimensionless numbersMa Marangoni number, Ma ¼ ð@r=@cÞðDc LÞ=ðlDÞPe Peclet number, Pe ¼ Re � ScRe Reynolds number, Re ¼ udP qc=lcScc continuous phase Schmidt number, Scc ¼ mc=Ds;c

Scd dispersed phase Schmidt number, Scd ¼ md=Ds;dShd dispersed phase Sherwood number, Shd ¼ bdP=Ds;d

770 M. Wegener / International Journal of Heat and Mass Transfer 71 (2014) 769–778

droplet experiments. The Marangoni effect may increase theextraction efficiency substantially [15,19], whereas its absencemay lead to relatively lower mass transfer coefficients.

Consequently, the question arises under which conditions a sys-tem will exhibit noticeable Marangoni instabilities? This question

a b c

a b c

Fig. 1. Main stages of Marangoni convection flow pattern in a rising droplet. Upperrow: Schematics. Lower row: Concentration field obtained by numerical simulationof a acetone laden 2mm toluene droplet rising in water [22]. (a) Early stage in theprocess, large concentration gradients, strong Marangoni convection, enhancedradial mixing and fast mass transfer, (b) middle stage, decreasing strength ofMarangoni convection, asymmetric onset of internal circulation, (c) later stage, noMarangoni convection, symmetric toroidal internal flow field. Blue: low concen-tration, red: high concentration. (For interpretation of the references to colour inthis figure caption, the reader is referred to the web version of this article.)

is even more interesting for rising droplets as the local Sherwoodnumber (and hence the local concentration) changes with the polarangle [3,5] which means that concentration gradients (and henceinterfacial tension gradients) are always present. But it was ob-served that in some systems Marangoni effects are strong and lastlong, whereas in other systems, interfacial effects develop only fora short time or cannot be observed at all. Thus, a certain combina-tion of factors may lead to the promotion of Marangoni instabili-ties, whereas there may be others for which instabilities dieaway before they have been noticed. In case of noticeable Marang-oni instabilities, the next questions are: for how long and to whichextent do they exist?

Despite all efforts, to the author’s best knowledge, there seemsto be no reliable analytical description available to take the impactof the Marangoni convection on fluid dynamics and mass transferat single droplets into account. Hence, first and foremost carefullyconducted experimental investigations may elucidate the underly-ing phenomena in such liquid/liquid systems, but the number ofpossible experiments is usually limited due to time and budgetconstraints. In this regard, direct numerical simulations offer to ac-cess a broader parameter range and to investigate the dynamicinfluence of single parameters on the unsteady mass transfer withsimultaneous Marangoni convection.

Theoretical considerations by Sternling and Scriven [20] – con-sidering two immiscible semi-infinite quiescent fluid phases incontact along a plane surface – and later Sawistowski [14]identified four primary factors to be decisive for the appearanceof instabilities (a discussion of different aspects of Marangoniphenomena in solvent extraction can be found in [12]):

1. the mass transfer direction (a system can be stable for masstransfer in one direction but instable for the opposite direction),

2. the sign of the interfacial tension gradient @r=@c (which is usu-ally negative for liquid/liquid systems),

M. Wegener / International Journal of Heat and Mass Transfer 71 (2014) 769–778 771

3. the ratio of molecular diffusion coefficients D�, and4. the ratio of kinematic viscosities m�.

Other factors such as the partition coefficient, the absolute concen-tration level and absolute values of physical properties as secondarywhich only affect the intensity of the instability but not its appear-ance. Although the above stability analysis is strictly speaking notvalid for particulate systems, it has been applied to problems re-lated to mass transfer at bubbles, droplets, and falling films, leadingto mostly wrong or contradictory results [28], but in some casesalso to useful results [3].

A different approach is to combine decisive parameters into adimensionless number, the critical Marangoni number. A widevariety of different definitions can be found (see e.g.[2,11,17,18]), but up to now there is no general definition available[7]. A relatively popular expression is the following Eq. (1):

Ma ¼ @r@c

Dc LlD

ð1Þ

with the interfacial tension gradient @r=@c (sometimes with nega-tive sign, sometimes defined as absolute value), the driving concen-tration difference Dc (sometimes at t ¼ 0, sometimes reflecting theactual concentration difference), a characteristic length L (e.g. a filmthickness, a penetration depth, or simply the droplet radius), and inthe denominator viscosity and diffusion coefficient. Javadi et al. [7]suggest that more factors than those appearing in Eq. (1) shouldalso be taken into consideration, such as the density, or the liquidsurface visco-elasticity. Up to now, no critical Marangoni numberhas been presented for buoyancy-driven single droplets in a suc-cessful manner, and it seems still unclear which set of variableshas to be taken into account to capture the present problem.

The aim of the present paper is to elucidate the effect of some ofthe above mentioned influencing parameters (leaving the othersconstant, which is normally impossible in real systems) on thetransient mass transfer of a solute across the interface of a dropletinitially at rest accelerating in a quiescent ambient phase via theMarangoni induced flow patterns. The underlying question is:Which parameter is decisive for the temporal development ofMarangoni convection? The answer may help to identify the ingre-dients required to develop an appropriate model description and acritical Marangoni number.

The present study is based on earlier successful validations offull 3D simulations with experiments in the Marangoni dominatedtoluene/acetone/water system [21,22]. In the present study, thetoluene/acetone/water system shall represent the reference casefrom which the parameters, namely the droplet diameter dP , theratio of kinematic viscosities m�, the interfacial tension gradient@r=@c, and the partition coefficient H�, are changed in a systematicmanner to investigate their effect on the level of mass transferenhancement and on the strength of Marangoni convection.Although usually no comparisons with experimental data are pos-sible, a numerical parameter study gives qualitative insight of theMarangoni effect and helps to develop a deeper understanding.

2. Numerical simulation

The numerical method is the same as presented in detail in anearlier paper [23] which describes the validation of the numericalmethod for the Marangoni enhanced mass transfer and fluid dynam-ics in the toluene/acetone/water system. Hence, the method is onlybriefly recalled here for completeness. As in Wegener et al. [23], thecommercial Finite Volume code STAR-CD (v3.24) distributed by cd-adapco has been used. The computational domain representingdroplet and surrounding liquid is full 3D, hence without symmetryplanes. The droplet is fixed in the centre of the computational

domain, surrounded by a cylindrically shaped section of the ambientphase. The ratio between cylinder diameter and droplet diameter is4, and the cylinder height equals its diameter. The drop is sphericalat all times (fixed grid), and other than vertical movements were notpermitted. The mesh consists of over 135000 hexahedral cells,around 47000 represent the droplet. The resolution of the interfaceis again identical to [23]. The computational domain was distributedon 32 processors, and the code parallelised. All simulationswere carried out on the HLRN-I and HLRN-II systems (HLRN:North-German Supercomputing Alliance).

In the Finite Volume environment, mass is conserved exactly.The governing equations are solved separately in each phase (drop-let phase and surrounding phase), coupled via the boundary condi-tions at the interface, especially by the implementation of theMarangoni stress term in the shear stress balance. Flow field andconcentration field are solved simultaneously, the governing equa-tions are the continuity equation, the Navier–Stokes equations andthe mass balance equation, respectively:

r � u ¼ 0 ð2Þ

q@u@tþ qu � ruþrp� lr2u ¼ 0 ð3Þ

@cs

@tþ u � rcs � Dsr2cs ¼ 0 ð4Þ

As explained in more detail in [23], the gravity force has beenomitted in Eq. (3) due to a required coordinate transformation toreflect the moving reference frame. The gravity force appearsexplicitly in the force balance at the droplet surface. The dropletis fixed in the centre of the simulation domain. Its ‘‘movement’’is represented by the relative velocity between droplet and contin-uous phase. To transform Eqs. (2) and (3) from the droplet coordi-nate system to the reference coordinate system, an additionalsource term appears which forms – together with the pressureterm – a modified pressure. The modified pressure is used togetherwith the gravity, buoyancy and friction terms to calculate the forcebalance at the droplet surface according to Newton’s second law.The force balance is used to calculate the inlet velocity for the nexttime step. This explicit treatment is necessary as STAR-CD does notoffer to modify the momentum equations directly. The derivationof the modified pressure is given in detail in [23].

The other boundary conditions are as well identical to [23]. Thepartition coefficient H� relates the solute concentration cs of eachboundary cell of the drop phase d to the corresponding cell inthe continuous phase c:

H� ¼ cs;d

cs;c

����r¼R

ð5Þ

The partition coefficient needs to be determined experimentallyand was set constant to 0.63 for the toluene/acetone/water system[24] and for all other simulations, except for those mentioned inrows 5 and 6 in Table 2.

The mass flux across the interfacial cells obeys Fick’s law:

Ds;d@cs;d

@r

����r¼R

¼ Ds;c@cs;c

@r

����r¼R

ð6Þ

The radial component of the interfacial velocity is zero in eachphase due to the constant droplet shape, and the tangential compo-nents for both the polar and the azimuth angle are equal due to theno-slip condition:

ur;d

��r¼R ¼ ur;c

��r¼R ¼ 0 ð7Þ

ui;d

��r¼R¼ ui;c

��r¼R

i ¼ u; hð Þ ð8Þ

The Marangoni stress is implemented via the shear stress balance atthe interface [6], again for both angles in the spherical coordinatesystem:

Table 1Density and viscosity of the components toluene (dispersed phase d), water(continuous phase c) and acetone (solute s) at # ¼ 25 �C [10]. The penultimatecolumn shows the diffusion coefficient for acetone in toluene and water which washeld constant in the simulations. The last column shows the Schmidt numberSc ¼ m=D for the dispersed and the continuous phase.

q [kg/m3] l [mPas] Ds [10-9 m2/s] Sc [–]

Toluene (d) 862.3 0.552 2.9 220Water (c) 997.02 0.8903 1.25 714Acetone (s) 784.4 0.304 – –

Table 2Overview of performed simulations for the mass transfer direction d! c; cs;0 ¼ 30 g/L,and D� ¼ 2:3. Rows 1 and 2: different droplet diameters dP , rows 3 and 4: differentkinematic viscosities m, rows 5 and 6: different partition coefficients H� .

H� ¼ 0:63; m� ¼ 0:72dP [mm] 1:0 1:5 2:0 2:5 3:0 3:5 4:0

H� ¼ 0:63; dP ¼ 2 mm; Scc � 3720 . . . 76m� [–] 0:14 0:36 0:72 1:5 7:0

m� ¼ 0:72; dP ¼ 2 mmH� [–] 0:1 0:63 1 10

772 M. Wegener / International Journal of Heat and Mass Transfer 71 (2014) 769–778

lc@uu;c

@r� uu;c

R

� �� ld

@uu;d

@r� uu;d

R

� �þ 1

R@r@u¼ 0 ð9Þ

lc@uh;c

@r� uh;c

R

� �� ld

@uh;d

@r� uh;d

R

� �þ 1

Rsinu@r@h¼ 0 ð10Þ

with the concentration dependent interfacial tension gradient@r csð Þ=@i which can be split in two separate expressions

@r csð Þ@i

¼ @r@cs� @cs

@ii ¼ u; hð Þ ð11Þ

where @cs=@i can be obtained from the CFD code. By contrast,@r=@cs has to be taken from experiments. For the toluene/acetone/water system, data published by Misek et al. [10] wereused to describe the sensitivity of the surface tension with thesolute concentration, see Fig. 2.

2.1. Overview of performed simulations

All geometry ratios and boundary conditions are identical to[23]. The physical properties of the toluene/acetone/water systemat # ¼ 25�C are given in Table 1 and are left unchanged throughoutthe study, except for the continuous phase viscosity to obtain thedifferent kinematic viscosity ratios as shown in rows 3 and 4 inTable 2. Five different kinematic viscosity ratios have been tested,ranging from 0.14 to 7.0 (which gives continuous phase Schmidtnumbers ranging from approx. 3720 to around 76 at constant dis-persed phase Schmidt number Scd ¼ 220). Seven different dropletdiameters have been investigated (from 1.0 to 4.0 in 0.5 mm steps),as can be seen from rows 1 and 2 in Table 2. Note that the Reynoldsnumber (and likewise the Peclet number) starts from zero (att ¼ 0) and increases as the droplet accelerates due to buoyancy.The Reynolds number is transient and adjusts itself according tothe force balance including the Marangoni shear stresses exertedat the droplet surface. For strong Marangoni convection, the Rey-nolds number is in the range of the value attained by a comparablerigid sphere, whereas for weak Marangoni convection it will bemore towards the case of a droplet with fully mobile surface. Aconstant Reynolds number is only attained in the case of noMarangoni convection after acceleration to the terminal velocity.For a 1 mm toluene droplet without Marangoni effect, the simu-lated Reynolds number at terminal velocity is around 40, whereasit is close to 1400 for a 4 mm droplet (note here that droplet defor-mation is not taken into account by the code, hence the terminalvelocity is higher than in reality; for simulations of deformable tol-uene droplets see e.g. [1]).

Fig. 2. Interfacial tension as a function of acetone concentration in toluene.Experimental data by Misek et al. [10]. The different lines represent the differentinterfacial tension gradients used in the simulations as shown in Table 3.

The partition coefficients cover the real system (H� ¼ 0:63),unity, and a tenfold increase and decrease compared to unity, seerows 5 and 6 in Table 2. The initial concentration of the solute inthe dispersed phase is set to cs;0 ¼ 30 g/L in all cases, as this con-centration showed vigorous Marangoni effects in the real systemleading to the characteristic two-step acceleration behaviour[25]. The mass transfer direction is from the droplet to the ambientphase (d! c) with the initial solute concentration in the continu-ous phase equal to zero. For each parameter set, both the case withMarangoni convection (variable interfacial tension) and withoutMarangoni convection (constant interfacial tension) was per-formed. The corresponding interfacial tension gradient @r=@cs

has been implemented in the dedicated user sub-routine inSTAR-CD and triggers the Marangoni convection. In order to inves-tigate the impact of the interfacial tension gradient on the masstransfer, different values for @r=@cs were tested, starting fromthe experimental values for the toluene/acetone/water system. Ascan be seen from Fig. 2, the decrease of the interfacial tension withthe solute concentration can be approximated with a linear func-tion in the concentration range <35 g/L. For the toluene/acetone/water case, the interfacial tension gradient can be described withthe constant value

@r@cs

� �TAW¼ �3:56 � 10�4 N=m

g=Lð12Þ

For the parameter study, four more cases were studied to investi-gate the impact of different levels of the interfacial tension gradient,as shown in Table 3: a gradient with the same value but with oppo-site sign, a threefold increase of the gradient, a fivefold decrease ofthe gradient (which corresponds to the gradient of interfacial ten-sion in the butyl acetate/acetone/water system [10]), and finally,

Table 3Performed simulations for different gradients of interfacial tension. The cases areexpressed as multiples of the linear approximation for the real toluene/acetone/watersystem @r=@csð ÞTAW . The other parameters were according to the standard toluene/acetone/water case, hence d! c; cs;0 ¼ 30 g/L, D� ¼ 2:3;H� ¼ 0:63; m� ¼ 0:72, anddP ¼ 2 mm.

@r@cs

� �TAW¼ �3:56 � 10�4 N=m

g=L

@r@cs

N=mg=L

h i� @r

@cs

� �TAW

3 � @r@cs

� �TAW

15 � @r

@cs

� �TAW

1100 � @r

@cs

� �TAW

M. Wegener / International Journal of Heat and Mass Transfer 71 (2014) 769–778 773

reflecting a very weak sensitivity of the interfacial tension on achange in solute concentration, a decrease by a factor of 100.

3. Results and discussion

3.1. Variation of droplet diameter

Drop sizes in liquid/liquid extraction systems are usually in themillimetre range, apart from processes in which emulsification oc-curs. The drop size distribution shows a certain spread to a more orlesser extent, consequently the impact of Marangoni convection onthe mass transfer of droplets of different sizes is investigated.According to Table 2, seven different droplet diameters weresimulated. Experimental data for the toluene/acetone/water sys-tem are available from earlier works [22,27] and were used forcomparison. For all cases, both the case with and the case withoutMarangoni term was simulated to compare the effect of Marangoniinduced flow patterns on the mass transfer, which is of course notpossible in the real system.

Fig. 3 shows 3D simulation results of the related concentrationinside the droplet c� ¼ csðtÞ=cs;0 as a function of time for differentdroplet diameters in the toluene/acetone/water system withMarangoni convection (the cases without Marangoni term arenot shown here for clarity). Generally, the concentration decreasesfaster for smaller droplets. In case without Marangoni, the knowntoroidal structures develop and mass transfer is mainly of diffusivenature within the droplet. Convection is important in the begin-ning of the process only [13] which means that the higher drop risevelocity (=higher Peclet number) of larger droplets does not lead todisproportionately high mass transfer coefficients. Due to the dif-fusion limitation, the length of diffusion paths to reach the inter-face is of major importance.

These distances are shorter for smaller droplets, consequentlymass transfer proceeds faster in smaller droplets. Likewise, thearea/volume ratio is larger for smaller droplets. In case of Marang-oni convection, the third term on the left hand side in Eqs. (9) and(10) comes into play. These terms represent the additional tangen-tial shear stress which in turn leads to a tangential interfacialvelocity. This velocity is on average larger for smaller droplets.Overall, radial mixing becomes more important than diffusion,and similarly, concentration decreases faster in smaller dropletsthan in larger droplets. Fig. 4 quantifies the differences betweenthe cases with and without Marangoni term. The figure displays,from left to right, the time at which 40% (or c� ¼ 0:6), 50%

Fig. 3. Related concentration c� ¼ csðtÞ=cs;0 as a function of time for differentdroplet diameters. 3D simulation results for the toluene/acetone/water system withMarangoni convection. Initial droplet solute concentration cs;0 ¼ 30 g/L, masstransfer direction d! c.

(c� ¼ 0:5), and 60% (c� ¼ 0:4) mass transfer is completed. On theabscissa, the droplet diameter is plotted. In all diagrams, the uppercurve represents the slower mass transfer cases without Marang-oni effect, the lower curve the corresponding simulations withMarangoni term.

The ratio of times at x% completion of mass transfer betweenwithout and with Marangoni effect for each droplet diameter is de-fined as the enhancement factor E

Ex% ¼tx%;noMAC

tx%;MACð13Þ

and is shown in Fig. 5. For the 4 mm droplets, E ¼ 1:45 . . . 1:48 at allthree times, which indicates that mass transfer with Marangoniconvection is about 50% faster. The enhancement factor increasesquickly if the droplet diameter decreases.

It reaches the value 2 at around 3 mm, and for the 1 mm drop-lets finally 3:45;3:76, and 4:07 for t40%; t50%, and t60%, respectively.An enhancement factor of 4 is a remarkable acceleration in themass transfer rate which is clearly caused by the vigorous and cha-otic Marangoni flow patterns which promote the radial displace-ment within the droplet.

Experimental data from [22,27] are available for 2 mm dropletsonwards, see the comparison in Fig. 6. Again, the times t40%; t50%,and t60% are shown from left to right. First, the diameter range inthe non-oscillating regime, i.e. for droplets smaller than the criticaldiameter as defined by Klee and Treybal [8]

dcr ¼ 0:33q�0:14c Dq�0:43l0:3

c r0:24 ð14Þ

with ½q� ¼ g=cm3; ½l� ¼ Poise;½r� ¼ dyn=cm, and ½dcr � ¼ cm(dcr ¼ 4:45 mm in the toluene/water system [26]) is discussed. Forall three diagrams, the completion time increases with the dropletdiameter. The simulation data exhibit quadratic growth in all threecases in Fig. 6 with increasing coefficients from left to right. There isa remarkable agreement between simulation and experiment in thenon-oscillating regime, with the preponderant part of the datawithin �20% tolerance. The mean deviation of all data points is ex-actly 20%. The attribute ‘‘remarkable’’ was given since the agree-ment is satisfactory despite the fact that the initial conditions insimulation and experiment are quite different. In the simulation,the droplet formation stage is not included, the droplet rises fromt ¼ 0 with a homogeneous concentration distribution equal to cs;0.In the experiments, this value applies to the dispersed phase beforedrop formation. In the drop formation stage, the mass transfer isheavily affected by Marangoni convection since the concentrationgradient is highest [24]. Anyhow, the concentration distributionwithin the droplet directly after completion of droplet formationis unknown and experimentally not accessible. Hence, it wasdecided to use the initial solute concentration as a start value inthe simulations.

No attempts were made to carry out simulations in the oscillat-ing regime since the CFD code was designed for spherical dropletsonly. As can be seen in Fig. 6, there is a distinct break in the exper-imental curve beyond dcr , caused by the more complex fluid dy-namic effects in deformed and/or oscillating droplets incombination with the Marangoni effect. The involved phenomenaare beyond the scope of the present study, and to the author’sknowledge, only a few groups are developing at present numericalcodes to simulate fluid dynamics and mass transfer at singledeformable and/or oscillating droplets with simultaneous Marang-oni convection in 3D.

3.2. Variation of kinematic viscosity ratio

Schulze [16] investigated in decolourisation experiments themass transfer in different droplet systems, amongst others

Fig. 4. Time when 40% (left), 50% (middle), and 60% (right) of mass transfer is completed as a function of droplet diameter for the case with and without Marangoniconvection. The other parameters are equal to those in Fig. 3.

Fig. 5. Enhancement factor as a function of droplet diameter for the three timest40%; t50% , and t60% . Other parameters as in Fig. 3. The enhancement factor is definedas the ratio of the time of x% completion of mass transfer in case withoutMarangoni convection to the corresponding time in case with Marangoni convec-tion (Eq. 13).

774 M. Wegener / International Journal of Heat and Mass Transfer 71 (2014) 769–778

toluene/acetone/sodium hydroxide and cyclohexanol/acetic acid/sodium hydroxide. In the toluene system strong Marangoni struc-tures evolved, whereas no such instabilities were visible in thecyclohexanol system. Taking the physical parameters into account,a presumably easy explanation is the difference in the ratio of kine-matic viscosities, assuming that this ratio plays a key role in the(in)stability behaviour of a system if one applies the findings fromSternling and Scriven’s theory [20] to droplets. A system is stable ifthe diffusivity ratio and the ratio of kinematic viscosities are equalto unity, in other cases the stability depends on the mass transferdirection. For toluene/acetone/water with toluene as dispersed

Fig. 6. Time when 40% (left), 50% (middle), and 60% (right) of mass transfer is completedby [22,27]. dcr denotes the critical diameter as defined in Eq. 14 for the toluene/water s

phase, D� > 1 and m� < 1, and following [20], instability is predictedfor the mass transfer direction c! d, but a stable configuration orat least oscillatory instability is predicted for d! c. However,according to experimental [21,24,27] and numerical [23] findings,in both directions Marangoni instabilities are strong in the toluene/acetone/water system. Hence, the influence of the ratio of kine-matic viscosities was investigated numerically in this studyaccording to Table 2 (rows 3 and 4), for dP ¼ 2 mm;cs;0 ¼ 30 g=L;D� ¼ 2:3, and the mass transfer direction d! c. Besides the valuefor the toluene/acetone/water system, m� ¼ 0:72, two smaller andtwo larger values were simulated. Since the kinematic viscosity ra-tio affects the terminal drop rise velocity (or the Peclet number) viathe drag coefficient, the case without Marangoni effect was alsosimulated for each m�.

Fig. 7 shows the simulated Sherwood number as function oftime for the different ratios of the kinematic viscosity, for boththe cases with and those without Marangoni term. One can seeimmediately that the curves with Marangoni convection are wellabove the corresponding curves without Marangoni convection,irrespective of whether m� < 1 or m� > 1. The curves withoutMarangoni separate from the common curve at Sherwood numbers�60, whereas in the case with Marangoni, the curves separate theearlier (note the logarithmic scale on the abscissa) the higher thekinematic viscosity ratio is, with the Sherwood number increasingwith m� (Sh ¼ 90 . . . 130). The latter finding is due to the fact thatthe drop rise velocity increases with increasing m� which also in-creases the interfacial velocities. Therefore, the overall resistanceagainst mass transfer decreases due to the higher convection atthe continuous phase boundary layer. As a restriction, one mustsay that the influence of the kinematic viscosity ratio on the masstransfer is relatively small, as an increase in m� by a factor of 50%

increases mass transfer by 30% only.The enhancement factor as defined in Eq. (13) at 50% comple-

tion of mass transfer is displayed in Fig. 8. It shows that for

as a function of droplet diameter. Comparison between simulation and experimentsystem dividing the non-oscillating from the oscillating regime.

Fig. 7. Sherwood number Shd ¼ bdP=Ds;d as a function of time for different ratios ofkinematic viscosity m� ¼ md=mc with and without Marangoni term. The mass transferdirection is from the dispersed phase to the continuous phase, the initial dropletsolute concentration is cs;0 ¼ 30 g/L, the droplet diameter is 2 mm, the diffusivityratio D� ¼ 2:3, and the partition coefficient H� ¼ 0:63.

Fig. 8. Enhancement factor as defined in Eq. 13 at c� ¼ 0:5 (or t50%) as a function ofthe ratio of kinematic viscosity m� .

Fig. 9. Related concentration c� ¼ csðtÞ=cs;0 as a function of time for different valuesof the interfacial tension gradient. The values given at the curves represent theprefactors with respect to the linearised value for the toluene/acetone/water (TAW)system, see Table 3. Initial solute concentration cs;0 ¼ 30 g/L, mass transfer directiond! c, droplet diameter dP ¼ 2 mm, diffusivity ratio D� ¼ 2:3, kinematic viscosityratio m� ¼ 0:72, and partition coefficient H� ¼ 0:63. The case for no Marangoniconvection is also given for comparison.

M. Wegener / International Journal of Heat and Mass Transfer 71 (2014) 769–778 775

m� < 1, the enhancement factor is around >2, whereas for m� > 1 itshows values of around 1.5, but no clear asymptotic behaviourwhich may explain a systematic trend in damping the Marangoniconvection patterns is found. This result suggests that mass trans-fer is significantly enhanced by Marangoni effects, regardless of thefact that mass transfer is directed towards the phase with higher orlower kinematic viscosity.

The kinematic viscosity of cyclohexanol is around 0.18, quiteclose to the simulated value m� ¼ 0:14. In the simulations, rela-tively strong Marangoni flows were found, in contrast to the exper-iment. Hence, the kinematic viscosity ratio seems not to be themain determining factor for system instability (which not excludesthat a higher viscosity in the dispersed phase may possibly dampenMarangoni flow patterns in combination with a different set ofother influencing physical parameters).

3.3. Variation of interfacial tension gradient

As can be seen from Eqs. (9)–(11), the concentration dependentinterfacial tension gradient is the driving force for the Marangoniconvection. A finite value exists even for very small concentrations.It is very difficult if not impossible to change the interfacial tensiongradient alone in experiments. Schulze [16] presented measure-

ments with different interfacial tension gradients by addition ofanother component, methyl ethyl ketone, in the toluene/acetone/water system. Measurements of the interfacial tension showedthat the interfacial tension gradient decreases for small acetoneconcentrations. As expected, the mass transfer was slower for thesystem with lower interfacial tension gradient, and the author as-sumed that with even lower gradients the mass transfer rate wouldtend towards the system without Marangoni convection. Strictlyspeaking, and as also mentioned by the author, this cannot be pro-ven by experiments since higher concentrations of the fourth com-ponent changes other physical parameters such as the distributioncoefficient, viscosities, etc. making a valid comparison impossible.

However, in a numerical simulation, the effect of the interfacialtension gradient on the mass transfer rate can be investigated sep-arately. Fig. 9 shows the temporal evolution of the related concen-tration c� ¼ csðtÞ=cs;0 of a 2 mm droplet for five different gradientsof interfacial tension, as defined in Table 3, plus the case for noMarangoni convection ð@r=@cs ¼ 0Þ. All other values were equalto the standard toluene/acetone/water case.

First, there is no difference between the case þð@r=@csÞTAW and�ð@r=@csÞTAW . This means that the sign in Eqs. (9) and (10) does notseem to have any influence on the mass transfer. The result sug-gests that only the absolute value of the gradient is important,not its sign. As expected, the concentration decreases faster withtime if the TAW-gradient is increased by a factor of 3. Marangoniconvection is stronger in this case. Accordingly, the mass transferperformance gets worse if the gradient is decreased by a factor of5. But one has to note the steeper slope of this curve which resultsfrom the higher drop rise velocity (or Peclet number). Hence, thiscurve intersects the standard TAW-case at later times.

The decrease in the mass transfer rate is even more apparentwhen the gradient is decreased by a factor of 100. But comparedto the case without Marangoni convection, there is still a signifi-cant enhancement. For example, at c� ¼ 0:5 or 50% completion ofmass transfer, the decrease in concentration is twice as fast.According to these results, it can be concluded that if Marangoniconvection occurs, mass transfer is significantly enhanced,although the gradient is apparently small. However, nothing canbe said for smaller gradients which could not be tested due to alimitation in the available overall computing time on the HLRN-complex. So the question remains under which circumstancesMarangoni convection cannot evolve.

Fig. 10. Related concentration c� ¼ csðtÞ=cs;0 as a function of time for differentvalues of the partition coefficient H� . Initial solute concentration cs;0 ¼ 30 g/L, masstransfer direction d! c, droplet diameter dP ¼ 2 mm, diffusivity ratio D� ¼ 2:3,kinematic viscosity ratio m� ¼ 0:72. Continuous lines: with Marangoni convection,dashed lines: no Marangoni convection.

Fig. 11. Enhancement factor E at t50% as a function of the partition coefficient.

776 M. Wegener / International Journal of Heat and Mass Transfer 71 (2014) 769–778

3.4. Variation of partition coefficient

The last set of simulations deals with the change in the partitioncoefficient, according to rows 5 and 6 in Table 2. As long as themass transfer resistance is not completely in the dispersed phase,the Sherwood number (or the mass transfer coefficient) depends

Fig. 12. Sectional view of the simulated droplet concentration for different partition coParameters as in Fig. 10. The droplets are moving upwards. Note the different concentr

on the partition coefficient [4]. The larger the partition coefficient,the smaller is the relevant concentration difference and the largeris the mass transfer resistance in the continuous phase. Vice versa,with smaller partition coefficients, the concentration jump at theinterface becomes larger, the gradient increases and likewise themass transfer is increased. To the author’s knowledge, there areno publications available which explore the influence of the parti-tion coefficient in droplet systems dominated by Marangoniconvection.

Fig. 10 shows the related acetone concentration in the dropletas a function of time for different partition coefficients. For all par-tition coefficients, both the case with and the one without Marang-oni term have been simulated. For larger distribution coefficients,the mass transfer becomes very slow, and accordingly one singlesimulation becomes relatively expensive in terms of computingtime. Therefore, in Fig. 10, the simulations were only carried outuntil c� ¼ 0:5 was reached. The results show, according to theabove mentioned considerations, that the acetone concentrationdecreases faster for lower partition coefficients. This is true forthe cases with and the cases without Marangoni convection. Apartfrom H� ¼ 10, a significant enhancement can be stated for allMarangoni cases compared to their corresponding counterpartwithout Marangoni convection. For H� ¼ 10, both curves, withand without Marangoni convection, are virtually the same. Theenhancement factor at c� ¼ 0:5 (or t50%) is depicted in Fig. 11 as afunction of the partition coefficient. For H� ¼ 0:1;0:63 and 1, Et50%

is close to 2, whereas for H� ¼ 10; E decreases drastically andreaches only 1 (which corresponds to no mass transfer enhance-ment). Only the concentration gradient is the main driving force,fluid dynamic issues do not need to be considered here, since theReynolds numbers of all cases are basically identical at the timecorresponding to c� ¼ 0:5. This last statement is interesting byitself.

As described in earlier papers [21,23,25], the drop rise velocityis reduced to the value of the rigid sphere for strong Marangoniconvection due to an increase in the drag coefficient induced bythe additional Marangoni shear stresses. So for all investigated par-tition coefficients investigated here, the drop rise velocity was thatof the rigid sphere at c� ¼ 0:5 (around 60 mm/s for a 2 mm toluenedroplet rising in water), even for H� ¼ 10 where basically no differ-ence in the mass transfer performance between the cases with andwithout Marangoni convection can be stated. Obviously, the driv-ing concentration difference plays a major but ambivalent rolewhen it comes to the development of Marangoni structures. Forlarge partition coefficients, the gradient at the interface is smalland the solute concentration decreases slowly. Nevertheless, thedrag coefficient is reduced significantly to the same level as incases of lower partition coefficients which leads to the assumptionthat tangential gradients can still be substantial as relatively strongshear stresses develop.

efficients (left: H� ¼ 0:1, middle: H� ¼ 1, right: H� ¼ 10) at the time when c� ¼ 0:5.ation scale in the right hand figure.

M. Wegener / International Journal of Heat and Mass Transfer 71 (2014) 769–778 777

In Fig. 12, a cross-sectional view of the concentration fields forthree different partition coefficients (H� ¼ 0:1;1; and 10) are plot-ted at c� ¼ 0:5, hence the mean concentration is equal in all cases.The absolute value of the solute concentration is displayed to-gether with absolute time, and local concentrations on the righthand side of each droplet. The concentration field is strongly influ-enced by Marangoni convection. There are no toroidal structures,and the concentration varies strongly in radial direction. ForH� ¼ 0:1, the concentration is relatively high in the droplet centre(around 24 g/L), whereas it is quite low at the interface (4–5 g/L).Directly adjacent to the interface, regions with larger concentra-tions can be seen, indicating the large concentration gradient.The concentration is lower at the rear of the droplet compared tothe front. The concentration difference between centre and inter-face gets smaller for H� ¼ 1 (reduction from �18 g/L to 11 g/L)and for H� ¼ 10 (between 15 and 14 g/L). But even in the last case,as expected from the reduced drop rise velocity, Marangoni struc-tures instead of toroidal flow structures can be seen. So althoughlarge partition coefficients prevent the enhancement of masstransfer, they allow Marangoni instabilities to develop.

4. Summary and conclusion

A detailed numerical parameter study of the influence of drop-let diameter, viscosity ratio, interfacial tension gradient, and parti-tion coefficient on the temporal evolution of the solutal Marangoniconvection in freely single rising droplets has been presented inthis paper. The commercial code STAR-CD has been used in a sim-ilar way as presented in an earlier study [23], i.e. full three-dimen-sional simulations (hence no symmetry planes), but restricted to aspherical droplet shape.

A comparison with experimental data in the toluene/acetone/water system was possible for the simulations of different dropletdiameters, and data by [21,22,27] were used to validate thenumerical results. The results revealed that the mass transferenhancement due to Marangoni convection is more prominentfor smaller droplets. Comparison with experimental data in thespherical droplet regime showed satisfactory agreement within�20%. No simulations were performed in the non-sphericalregime, since experimental data confirmed more complex masstransfer characteristics for droplets with sizes exactly beyond thecritical diameter as defined by Klee and Treybal [8], Eq. (14).

The main question in the introduction was whether there is adecisive parameter which may be able to control the strength ofMarangoni convection. The answer to that question may give hintsto how a critical Marangoni number could possibly look like. Aconclusion from the instability analysis of planar interfaces bySternling and Scriven [20] was that the ratio of kinematic viscositiesand the diffusivity ratio play a significant role in the stability ofthe liquid/liquid system. In the present numerical investigationon droplets (curved interfaces), no stable configuration was foundfor different kinematic viscosity ratios, which is in contrast toSternling and Scriven’s prediction. In other words, in the investi-gated range of kinematic viscosity ratios, Marangoni convectionalways occurred and mass transfer was always enhanced comparedto the case without Marangoni convection. The effect of thediffusivity ratio was not investigated due to time restrictions inthe project.

Even a reduction of the interfacial tension gradient by a factor of100 still showed mass transfer enhancement factors >2. This re-veals that even smallest gradients may trigger Marangoni effectsand contribute significantly to a better transfer of solute acrossthe interface. A remaining open question is below which value ofthe interfacial tension gradient Marangoni convection will be tooweak to be of dominant importance. Eventually, only a change inthe partition coefficient showed damping of Marangoni convec-

tion. If the partition coefficient is increased to 10, virtually no masstransfer enhancement could be stated, which can be attributed tothe relatively small concentration gradient at the interface for highpartition coefficients. This result implies that the partition coeffi-cient plays a significant role in the formation of Marangoni struc-tures and should be investigated in detail in future experimentaland numerical experiments taking also different mass transferdirections into account.

Acknowledgements

To A.R. Paschedag, M. Kraume and E. Bänsch. This one is for you.Thank you for all your support.

The author also likes to thank the German Research Foundation(DFG) for financial support. The simulations were performed on theIBM pSeries 690 Supercomputer provided by The North-GermanSupercomputing Alliance (HLRN) and on the SGI-System (HLRNII). The great support of the staff members is highly acknowledged.

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