A novel method to capture mass transfer phenomena at free fluid–fluid interfaces

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Chemical Engineering and Processing 50 (2011) 68–76

Contents lists available at ScienceDirect

Chemical Engineering and Processing:Process Intensification

journa l homepage: www.e lsev ier .com/ locate /cep

novel method to capture mass transfer phenomena at free fluid–fluid interfaces

.Y. Keniga,∗, A.A. Ganguli a, T. Atmakidisb, P. Chasanisa

University of Paderborn, Chair of Fluid Process Engineering, Pohlweg 55, 33098 Paderborn, GermanyEXOTHERMIA SA, 55102 Thessaloniki, Greece

r t i c l e i n f o

rticle history:eceived 23 August 2010eceived in revised form7 November 2010ccepted 19 November 2010vailable online 1 December 2010

eywords:

a b s t r a c t

A rigorous Computational Fluid Dynamics (CFD) based approach incorporating interfacial mass transferat moving interfaces has been developed. The peculiarity of this approach is that it is able to govern mul-ticomponent systems as well as interfacial boundary conditions in an arbitrary form. This is important inorder to properly handle the typical concentration jump at the phase interface and to avoid an assumptionof a constant distribution coefficient which is seldom met in real separation units. As a test system, themovement of a rising toluene droplet in quiescence aqueous phase is studied, whereas acetone is used asthe transferring component. The level set method is applied for the description of the droplet movement.

ass transferFDluid–fluid interfaceevel set methodising droplet

To give a first validation to the developed CFD model, the toluene droplet is first assumed to be stag-nant and the mass transfer from/to the continuous aqueous phase is considered. A good agreement withanalytical results from the literature is found in terms of the asymptotic value of mass transfer rate. Fur-ther, simulations for rising droplet are performed to incorporate the coupling between the momentumand mass transfer. The method provides concentration fields that are in qualitative agreement with theavailable data from the literature. Another test accomplished with an increased distribution coefficientalso yields reasonable results.

© 2010 Elsevier B.V. All rights reserved.

. Introduction

Many industrial separation processes occur in fluid–fluid sys-ems including droplets/bubbles moving in a continuous phasee.g., solvent extraction, gas scrubbing, waste water treatment). To

ake these processes more efficient, the intensification of the inter-hase mass exchange is of crucial importance. In this respect, local

nterfacial mass transfer represents a dominant phenomenon. Toescribe it properly, a coupled problem including momentum andass transport at and around moving interfaces has to be solved.

uch problems are very difficult, since the deformation of the inter-ace influences interfacial mass transfer and vice versa, so that theraditional assumption that the velocity field is not affected by theoncentration field cannot be made.

For solving such complex problems, front capturing methods,.g., volume of fluid (VOF) and level set (LS) methods, have proveno be extremely useful [1–3]. The basic idea of the VOF method is the

efinition of a volume fraction function which takes values zero forhe first phase, one for the second phase, and between zero and oneor the cells containing the interface. A detailed review of the VOF

ethod is published in [4]. The main drawback of this method is the

∗ Corresponding author at: Tel.: +49 0 5251 60 2408; fax: +49 0 5251 60 3522.E-mail address: [email protected] (E.Y. Kenig).

255-2701/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.cep.2010.11.009

inherent numerical smearing. In the LS method, a function is usedto locate the interface, which takes positive and negative values ondifferent sides of the interface and zero directly at the interface.The interface is therefore called zero LS. An overview of differentLS methods is presented in [5]. This method is conceptually simpleand easy to implement. Its main drawback is the loss of mass (orvolume), especially for significantly deformed interfaces [6].

In many fluid–fluid systems, dispersed phase exists in form ofbubbles or droplets distributed in a continuous liquid phase. Heatand mass transfer phenomena occur within and outside the mov-ing fluid objects accompanied by the interface deformation andbubble/droplet movement. For example, in liquid-liquid extraction,moving droplets are the smallest representative elements in thesystem, and mass transfer takes place from these droplets to thecontinuous liquid phase or/and vice versa, whereas the dropletsdeform and move upward/downward. For such problems, bothmomentum and mass transfer equations need to be solved simul-taneously, because the movement and deformation of the interfacedepend on mass transfer.

Handling of boundary conditions at the moving interface makes

numerical solution of such problems particularly difficult. Dueto the common assumption of the thermodynamic equilibrium,there is a concentration jump at the interface which is numeri-cally demanding and difficult to implement into the front capturingmethods [1]. In addition, the component flux continuity should also

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E.Y. Kenig et al. / Chemical Engineering

Nomenclature

C concentration (kmol m−3)C average concentration of acetone (kmol m−3)C transformed concentration (mass fraction) (wt.%)C∗

d equilibrium concentration (kmol m−3)Co Courant numberd droplet diameter (m)D binary diffusivity (m2 s−1)D transformed effective diffusivity of solute (m2 s−1)F� surface tension force introduced by Eq. (14)

(kg m s−2)f volume fractionfLS standard level set functiong gravity (9.81 ms−2)HD distribution coefficientH�(fLS(�x)) regularised Heaviside functionkod overall mass transfer coefficient (m s−1)�n unit vector normal to the interfacep pressure (kg m−1 s−2)Pr Prandtl numberRe Reynolds numberS droplet area (m2)Sh Sherwood numbert time (s)t transformed time (s)�t time step (s)�u velocity (m s−1)u transformed velocity vector (m s−1)Vd droplet volume (m3)Wj source term related to the reaction (kmol m−3 s−1)�x grid size (m)�x distance to the interface as in Eq. (8) (m)

Greek symbols˛1,˛2 parameters in Eqs. (27) and (28)� surface curvature� dynamic viscosity (kg m−1 s−1)� modified LS function� density (kg m−3)ε numerical parameter in Eq. (11)� surface tension (kg s−2)˝ symbol for cross-sectional area

Subscriptsave averagec continuous phased dispersed phaseD distribution coefficientG gasinit initialL liquidmax maximumI interfacej component indexR relativeT tolueneW water

bst

is captured based on Eq. (5). Such a movement causes changes of

0 initial value

e fulfilled at the interface. Up to now, just few groups have tried touggest solutions to this problem (s. [1–3]). In the following section,hese works are considered in more detail.

and Processing 50 (2011) 68–76 69

2. Governing equations and existing approaches

Let us consider a two-phase system comprising a dispersedphase (denoted with d) and a continuous phase (denoted with c).The governing continuity, momentum and mass transport equa-tions for the two-phase incompressible fluid flow of a binarymixture are as follows:

∇�u = 0 (1)

�∂�u∂t

+ �(�u · ∇)�u = ∇ · �[∇�u + (∇�u)T ] − ∇p + g� (2)

∂C

∂t+ �u · ∇C = ∇ · (D∇C) (3)

Eqs. (1)–(3) are valid for both contacting phases. The governingequation for the interface movement is given by

∂f

∂t+ ∇ · (�uf ) = 0 (4)

for the volume fraction function (VOF) or

∂fLS

∂t+ ∇ · (�ufLS) = 0 (5)

for the LS function.The interfacial boundary conditions for the momentum equa-

tions comprise common velocity and shear stress continuity (cf.[7]). For the mass transport equations (Eq. (3)), two interfacialboundary conditions have to be fulfilled. First, the interfacial con-centration jump is defined by the thermodynamic equilibriumassumption:

Cd = HDCc (6)

Second, the interfacial fluxes obey the continuity condition:

Dc∂Cc

∂n= Dd

∂Cd

∂n(7)

The standard LS function is defined as a distance function givenby∣∣fLS(�x)

∣∣ = minxI ∈ I

(∣∣�x − �xI

∣∣) (8)

where I is the interface, fLS(�x) > 0 at one side of the interface andfLS(�x) < 0 at another side of the interface.

The equation system given above requires a numerical solu-tion. The latter largely depends on the discretisation method andthe treatment of the discretised computational domain. To achievenumerical robustness, the moving interface is artificially smearedout using the following regularised Heaviside function [8]:

Hε(fLS(�x)) =

⎧⎨⎩

0, if fLS(�x) < −ε,12

(1 + fLS(�x)

ε+ sin(fLS(�x)/ε)/

), if

∣∣fLS(�x)∣∣ ≤ ε,

1, if fLS(�x) > ε,

(9)

where ε is a numerical parameter equal to a half of the thickness ofthe “smeared-out” interface.

By defining the new, smoothed LS function [8]

� = Hε(fLS(�x)) (10)

it is possible to fix the interface smeared out over a small distanceequal to 2ε. The function defined by Eq. (10) takes the value of0.5 exactly at the interface and the values 0 or 1 away from theinterface.

As mentioned above, in the LS method, the interface movement

the � function and thus, the interface thickness would not remainconstant. In order to maintain the thickness constant during thesimulations and hence to limit the numerical error arising due tothe interface smearing, Eq. (5) is extended by a re-initialisation term

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0 E.Y. Kenig et al. / Chemical Engine

hat acts as an additional artificial compression of the interfacen the normal direction in regions where 0 ≤ � ≤ 1. The extendedquation including the re-initialisation term reads as follows [9]:

∂�

∂t+ ∇ · (��u) = �∇

(−�(1 − �)

∇�∣∣∇�∣∣ + ε∇�

)(11)

here � is the re-initialisation factor that should take values in theange of the maximum velocities.The density is a function of the LSunction defined by

= �c + (�d − �c)� (12)

nd the dynamic viscosity is

= �c + (�d − �c)� (13)

The surface tension description is based on the continuum sur-ace force model [10]. In this model, the surface tension force isepresented as a volume force:

�� = (−���n)∇� (14)

his term is implemented into the right side of the momentumquation (Eq. (2)). The normal vector �n and curvature � at theuid–fluid interface are determined in terms of the level set func-ion:

� = ∇�∣∣∇�∣∣∣∣∣∣∣�=0.5

(15)

= −∇ · �n (16)

A few approaches have recently been published to solve theoupled momentum and mass transfer equations for fluid–fluidystems. In [1], VOF method was used to model a gas–liquid sys-em (air bubbles rising in water with oxygen as the transferringomponent). To avoid the interfacial concentration jump, the mass-ransfer equation was transformed to

∂C

∂t+ �u · ∇C = D�C (17)

˜ ={

CLCG/HD

(18)

The interfacial flux continuity boundary condition transformsccordingly:

L∇CL · �n = HDDG∇CG · �n (19)

Another approach was suggested in [2]. Here, the LS methodas used to describe a rising droplet in a continuous liquid phase

water–succinic acid–butanol). The mass transfer equation wasodified in [2] to the following form:

∂C

∂t+ u · ∇C = ∇ · (D∇C) (20)

here C is transformed concentration (mass fraction), u is trans-ormed velocity vector, D is transformed effective diffusivity of theolute, t is transformed time. More explicitly:

(fLS) =

⎧⎨⎩√

HDt, if fLS ≥ 0,1√ t, if fLS < 0, (21)

HD

ˆ (fLS) =√

HDD2 +(

1√HD

D1 −√

HDD2

)Hε(fLS) (22)

and Processing 50 (2011) 68–76

u(fLS) =√

HD �u +(

1√HD

�u −√

HD �u)

Hε(fLS) (23)

C ={

Cc

Cd/HD(24)

and Hε(fLS) is a regularised Heaviside function given by Eq. (9).A further, VOF-based modelling approach has recently been sug-

gested for reactive absorption systems [3]. The authors considered aliquid film flowing along a corrugated surface counter-currently tothe laminar gas stream. The mass transport equation was modifiedto

∂Cj

∂t+ ∇ ·

(�uCj

)= ∇ ·

(Dj∇Cj + ˚j

)+ Wj (25)

where Wj is source term due to the reaction and

˚j =(

DjCj(1 − HD,j)

f + HD,j(1 − f )∇f

)(26)

According to the authors, Eqs. (25) and (26) account for the jumpconditions at the smeared-out interface, where the volume fractiongradients are usually very steep [3]. It is, however, difficult to judgewhether it is true based on this publication only.

In all these works, the distribution coefficient HD is used as atransformation parameter and thus assumed to be constant. Thisrepresents a significant limitation, because, in reality, the dis-tribution coefficient depends on state variables and may suffersignificant changes. Thus, a comprehensive mathematical modeltaking account of the interface deformation, interfacial concentra-tion jump as well as the distribution coefficient variation is required.

3. The new approach

In this work, we suggest a general approach capable of handlingmass transfer at moving interfaces. For the sake of simplicity, herewe consider the fluid–fluid systems under the following assump-tions:

• Newtonian incompressible fluids.• Laminar flow of both phases.• Isothermal system.• No chemical reaction in the system.• Absence of surface active contaminants.

The method is, however, not limited by these assumptions andcan certainly govern more general systems.

The main idea of the present approach is to incorporate theinterfacial boundary conditions (e.g., Eqs. (6) and (7)) into the masstransfer equations themselves (e.g., in Eq. (3)). It has to be done insuch a way that the boundary conditions are fulfilled only in theregion very close to the interface, whereas, outside this region, themass transfer equations are valid.

More exactly, the interfacial mass transfer related boundarycondition is directly implemented into mass transfer equations (Eq.(3)) as an additional source term. As we have two interfacial bound-ary conditions to be fulfilled, mass transfer equations for both thecontacting phases should be solved for the whole computationaldomain. For instance, if we take the boundary conditions given byEqs. (6) and (7), the following extended equations are obtained:( )

∂Cd

∂t+ �u · ∇Cd = ∇ · (Dd∇Cd) + ˛1 Dc

∂Cc

∂n− Dd

∂Cd

∂n(27)

∂Cc

∂t+ �u · ∇Cc = ∇ · (Dc∇Cc) + ˛2

(Cc − Cd

HD

)(28)

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E.Y. Kenig et al. / Chemical Engineering and Processing 50 (2011) 68–76 71

case:

(Ite

aFt

4

cog

bIIotcC

C

w�l

To

�

Bit

mairsv

liquid phases. In the considered case, this interface has a circularform. The initial interfacial boundary condition, as it is available inCOMSOL 3.5a, should be related to the LS function. It was done herein such a way that the LS value of 0.5 was allocated to all points ofthe circular interface, whereas the LS values of zero and one were

2.06

2.065

2.07

2.075

2.08

2.085

2.09

SH

ER

WO

OD

NU

MB

ER

; S

h (

-)

Fig. 1. Grid used for the study of a stagnant droplet

At the interface, the values of ˛1 and ˛2 are set sufficiently highe.g., 104), so that the boundary conditions are fulfilled only there.n the rest of the computational domain, ˛1 and ˛2 are equal to zero,hus transforming Eqs. (27) and (28) to the common mass transferquation (cf. Eq. (3)).

The suggested method can be used for any fluid–fluid systemsnd permits HD to variate throughout the computational domain.urthermore, the method is not limited to systems with binary massransfer; it can readily be extended to multicomponent systems.

. Implementation

In this work, the new approach is illustrated with a case studyomprising a rising toluene droplet in a stagnant continuous aque-us phase, with acetone as the transferring component. The basicoverning equations, Eqs. (1)–(4), have to be solved simultaneously.

To solve the equation system, the commercial finite-element-ased solver COMSOL Multiphysics 3.5a by COMSOL AB is applied.

t includes the LS method for capturing free interface movement.n our work, we chose the value ε (cf. Eq. (9)) equal to the sizef one mesh element for all simulations. The solution stability ofransient free-interface problems can be controlled by two criteriaonsidering the selected time step. The first one is related to theourant number which is defined as

o = umax �t

�x(29)

here umax is the maximum velocity in the computational domain,x is spatial step and �t is time step. For a two-dimensional prob-

em, the first stability criterion reads as follows:

u · �t

�x+ v · �t

�y< Co (30)

he second stability criterion is referred to the explicit treatmentf the surface tension; it is given as

t ≤√

(�T + �W)(�x)3

4�(31)

oth stability criteria should be satisfied, and, in this way, the min-mum time step obtained by these criteria is selected. In our work,he value of the time step of 0.0001 s was used.

The solver uses an affine invariant form of the damped Newtonethod as described in [11]. The convergence criterion is based on

weighted Euclidean norm of the estimated relative error [12]. The

terations are terminated when the relative tolerance exceeds theelative error. In our simulations the tolerence convergence waselected to be 0.0001 with respect to all variables, viz. pressure,elocity, LS function and concentration.

computational domain (a) and magnified view (b).

5. Illustrative examples

Despite that our method is developed to handle mass transferin systems with moving interfaces, we include here a preliminarystudy of a stagnant droplet in order to ensure that, for pure masstransfer problems, the method works properly.

5.1. Stagnant droplet

5.1.1. Geometry and meshThe two-dimensional computational domain 10 mm × 12 mm

was selected for all governing equations. The droplet with the diam-eter 2 mm was placed at the centre of the computational domain. Anon-uniform grid with triangular elements was used, keeping themesh size along the droplet circumference very small (100 �m).The grid used in this case study is shown in Fig. 1. To check forgrid independence, four different grids were tested. Fig. 2 showsthe variation of Sherwood number (Sh = kodd/D) with the number ofmesh elements. The convergence can be ascertained, and, therefore,totally 9635 mesh elements were used, with the smallest elementof 100 �m and the largest element of 500 �m.

5.1.2. Boundary conditionsThe no-slip boundary condition was used at the walls. The

boundary of the droplet is in essence the interface between the two

2.055

0 5000 10000 15000 20000 25000

NUMBER OF MESH ELEMENTS (-)

Fig. 2. Variation of Sherwood number against grid refinement.

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72 E.Y. Kenig et al. / Chemical Engineering and Processing 50 (2011) 68–76

F agnip

aLs

5

•

•

•

5

5

uad

ig. 3. Grid used for study of a rising droplet case: overall view of the whole grid (a); mrocedure for higher times of the droplet rise (c).

ssigned to both sides of the interface. The smooth variation of theS function, however, has to be performed during the initialisationtep (see Section 5.1.3).

.1.3. Solution procedureThe main steps of the solution procedure are as follows:

The physical properties (density �, viscosity �, surface tension �and diffusivity D), the concentration C in each phase and the LSfunction are set.The LS function � has to be initialised. It should vary smoothlyacross the interface from zero to one. To obtain �, Eq. (11) (herewithout term ∇ · (��u), as �u = 0) is solved until t ≈ 5ε/�. As theinitial condition, the values of � equal to zero on one side andone on the other side of the interface are applied.Eqs. (11), (27) and (28) are solved iteratively. The advection termin Eqs. (27) and (28) is set to zero.

.2. Rising droplet

.2.1. Geometry and meshFor the rising droplet case, the same domain was used. A non-

niform grid with triangular elements was selected, with mesh sizelong the droplet circumference being equal to 55 �m. Above theroplet, the mesh size distribution is similar to that of the droplet,

fied view of the mesh in and around the droplet (b); schematic of the re-initialisation

as shown in Fig. 3. Again, four grid sizes were tested, and coarsergrid was chosen for the rest of the computational domain. Based onthe qualitative and quantitative results, we decided to use the gridshown in Fig. 3, with a total of 31,912 mesh elements, the largestelement being of the size of 1 mm.

The length of the domain is limited to keep computational timeand memory reasonable. However, the relaxation time for masstransfer is much higher than the time of droplet rising withinthe domain. For this reason, the simulation is interrupted whenthe droplet approaches the upper boundary of the computationaldomain. Afterwards, the computational domain is moved upwardsby 5 mm and re-initialised. The velocity and concentration fields inthe the overlapping part of the two domains, as shown in Fig. 3c,are used as new initial conditions to initialise the second simula-tion step. The simulations are carried out until the droplet rise timeof 1 s is reached.

5.2.2. Boundary conditionsThe boundary conditions are similar to those described in Sec-

tion 5.1.2.

5.2.3. Solution procedureThe main steps of the solution procedure are as follows:

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E.Y. Kenig et al. / Chemical Engineering and Processing 50 (2011) 68–76 73

Fi

•

•

•

6

d(tmuc

¯ ∫

ig. 4. Variation of Sherwood number with time: mass transfer with negligiblenternal resistance (a); mass transfer with negligible external resistance (b).

The physical properties (density �, viscosity �, surface tension �and diffusivity D), the concentration C in each phase and � as thedistance function are set.The LS function is initialised similarly as in the previous case (cf.Section 5.1.3).Eqs. (1), (2), (11), (27) and (28) are solved iteratively, in such a waythat the horizontal and vertical velocity components are providedas an input to the mass transfer equations. The interfacial bound-ary condition, Eqs. (6) and (7), should be fulfilled in a region veryclose to the interface. Test simulations were performed with vary-ing � limits, and it was found that the simulations for 0.4 ≤ � ≤ 0.6yeild stable results. Therefore, the region 0.4 ≤ � ≤ 0.6 was chosenfor all simulations in this work.

. Results and discussion

Simulation results obtained with the new method were vali-ated using the analytical solutions of two well-known cases, i.e.a) mass transfer from a stagnant organic-phase droplet to a con-

inuous aqueous phase, with negligible internal resistance, and (b)

ass transfer to a stagnant organic-phase droplet from a contin-ous aqueous phase, with negligible external resistance. In bothases, the organic-phase droplet was considered to be suspended

Fig. 5. Concentration contours of acetone for stagnant droplet case: t = 0 s (a); 300 s(b); 1000 s (c).

in the aqueous continuous phase. The transferring component wasacetone. Afterwards, simulations for the case of rising droplet wereperformed.

6.1. Model validation for a stagnant droplet

The overall mass transfer coefficient was calculated as follows[7]:

kod = −Vd

S

1t0 − t

lnC∗

d − Cd

C∗d − Cd,0

(32)

where Cd is the average concentration of acetone in the droplet atan arbitrary time instant given by∫

�≥0.5

Cd d˝

Cd =

�≥0.5

d˝

(33)

C∗d is equilibrium concentration.

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74 E.Y. Kenig et al. / Chemical Engineering and Processing 50 (2011) 68–76

F with H

6

n

S

Is2iv

ig. 6. Concentration contours of acetone and relative velocity vectors for a system

.1.1. Mass transfer with negligible internal resistanceThe Sherwood number as a function of Reynolds and Schmidt

umbers for a spherical droplet is given by [13]:

h = 2 + 0.6 Re1/2Pr1/3 (34)

nserting Re = 0, one obtains from Eq. (34): Sh = 2. We performedimulations starting with an initial acetone concentration of0 mol%. The concentration of acetone in the droplet was mon-

tored with the time interval of 10 s and the corresponding kodalue was calculated according to Eq. (32). Fig. 4a shows that, after

D = 0.63: t = 100 ms (a); 120 ms (b); 200 ms (c); 500 ms (d); 750 ms (e); 1000 ms (f).

1000 s, the value of Sherwood number reaches 2 and then remainsconstant.

6.1.2. Mass transfer with negligible external resistanceIf external resistance is negligible, Sherwood number for a stag-

nant droplet is given by [13]:

Sh = 22

3

∑∞n=1exp(−n22t)∑∞

n=1(1/n2)exp(−n22t)(35)

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E.Y. Kenig et al. / Chemical Engineering and Processing 50 (2011) 68–76 75

F with

F

S

Nho

ig. 7. Concentration contours of acetone and relative velocity vectors for a system

or infinite time, t = ∞, Sherwood number reduces to

22

h =

3= 6.58

umerical simulations for this case are illustrated in Fig. 4b. Alsoere we can observe a good agreement with the asymptotic valuef Sh = 6.58 that is reached after approximately 200 s.

HD = 20: t = 100 ms (a); 120 ms (b); 200 ms (c); 500 ms (d); 750 ms (e); 1000 ms (f).

Fig. 5 gives a qualitative illustration of mass transfer from thedispersed to the continuous phase. Fig. 5a shows the initial state:the droplet in the quiescent continuous phase at time t = 0. The

initial volume fraction of acetone in toluene (dispersed phase) is20 mol% (red colour), while in water (continuous phase), it is zero(dark blue colour). Fig. 5b shows the acetone concentration distri-bution after t = 300 s. It is obvious that acetone concentration in thedroplet decreases considerably. In Fig. 5c, it is seen that, at t = 1000 s,

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7 ering

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attNtt

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tbcttaca(

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6 E.Y. Kenig et al. / Chemical Engine

he volume fraction of acetone in dispersed phase approachesero.

.2. Rising droplet

A toluene droplet with diameter 2 mm rising in a quiescentqueous phase was considered. Two different values of the dis-ribution coefficient HD (low and high) were used in order to testhe robustness and qualitatively evaluate the model performance.umerical simulations were carried out till t = 1000 ms. Fig. 6 shows

he concentration distribution of acetone in both phases. In addi-ion, internal circulations are shown using the relative velocity

�R = �u − �uave (37)

here �uave is average droplet velocity at arbitrary time instantefined in terms of the level set function as:

�ave =∫

�≥0.5�u d˝∫

�≥0.5d˝

(38)

nd �u is local velocity.The concentration contours at different time instants during

he droplet rise are shown in Fig. 6a–f. In this case, the distri-ution coefficient HD = 0.63 (low value). The decrease in acetoneoncentration inside the droplet can be observed. It starts fromhe centre (Fig. 6a). As the droplet rises, the acetone flux fromhe droplet to the continuous phase results in the acetone tracesround the droplet interface (Fig. 6b and c). An interplay of theonvective flow and diffusion in the continuous phase brings aboutstreak of acetone directed downwards from the droplet bottom

Fig. 6d–f).Another test simulation was performed with a higher value of

he distribution coefficient, namely HD = 20. The relevant concen-ration contours are shown in Fig. 7 for the same time instants as inig. 6 to facilitate the comparison. Obviously, till t = 120 ms, bothest cases give the concentration contours with a similar trend.owever, it is clearly seen that acetone concentration shows higheralues in the area beneath the droplet in Fig. 7c–f than in Fig. 6c–f.his is as expected, since the higher HD value results in faster ace-one mass transfer. The acetone concentration inside the dropleteeps on decreasing with time, whereas the streak follows theroplet rise path (Fig. 7e and f). These simulation results show aertain similarity to the experimental observations published byück et al. [14] for a gas bubble rising in a liquid. For this system,

he distribution coefficient is also high (HD = 33). Unfortunately, noata on concentration contours in relevant liquid–liquid systemsould be found.

The internal circulations inside the droplet are visible in bothgures. They are especially pronounced after 200 ms. Furthermore,he concentration distributions inside the droplet follow the course

f circulation. This can be recognised via the vortex-like struc-ures (Fig. 6c–f and Fig. 7c–f). These tendencies are in agreementith those obtained numerically in [15]. On the overall, the first

imulation results appear reasonable and the new method can beonsidered as qualitatively validated.

[

[

and Processing 50 (2011) 68–76

7. Conclusions

In this work, a novel method is suggested that enables therigorous treatment of coupled mass transport boundary condi-tions (thermodynamic equilibrium, flux continuity) at the movingfluid–fluid interface. The method is implemented into COMSOLMultiphysics 3.3a and tested for both a stagnant and a risingdroplet in a continuous liquid phase. Simulation results per-formed for a toluene–acetone–water system are compared withtwo well-known analytical solutions for a stagnant droplet (neg-ligible internal resistance and negligible external resistance), anda good agreement between numerical and analytical solutions isobtained for both cases.

A further example is a rising droplet in a continuous liquid phase.Here, a good qualitative agreement is achieved with the results pub-lished in the literature. The suggested method is applicable over awide range of distribution coefficients (0.7 ≤ HD ≤ 20) and not lim-ited by the systems and examples considered. With respect to themass transfer related interfacial boundary conditions, it hardly hasany limitations, since the expressions in the source terms can begiven in an arbitrary way. Thus, the method can readily be extendedto cover multicomponent systems and different fluid–fluid pro-cesses. In the subsequent work, the capacity and robustness of thenew method will further be tested.

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