numerical simulation of interphase mass transfer with the level set approach

Chemical Engineering Science 60 (2005) 2643–2660

www.elsevier.com/locate/ces

Numerical simulation of interphase mass transfer with thelevel set approach

Chao Yang, Zai-Sha Mao∗

Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100080, China

Received 19 April 2004; received in revised form 18 October 2004; accepted 21 November 2004

Abstract

A level set approach is applied for simulating the interphase mass transfer of single drops in immiscible liquid with resistancein both phases. The control volume formulation with the SIMPLEC (semi-implicit method for pressure-linked equations consistent)algorithm incorporated is used to solve the governing equations of incompressible two-phase flow with deformable free interface on astaggered Eulerian grid. The solution of convective diffusion equation for interphase mass transfer is decoupled with the momentumequations. Different spatial discretization schemes including the fifth-order WENO (weighted essentially nonoscillatory), second-orderENO (essentially nonoscillatory) and power-law schemes, are tested for the solution of mass transfer to or from single drops. The conjugatecases with different equilibrium distribution coefficients are simulated successfully with the transformation of concentrations, moleculardiffusivities, mass transfer time and velocities. The predicted drop concentration, overall mass transfer coefficient and flow structure arecompared with the reported experimental data of a typical extraction system, i.e.,n-butanol–succinic acid–water, and good agreement isobserved.� 2005 Elsevier Ltd. All rights reserved.

Keywords:Mass transfer; Numerical simulation; Level set; Drop; Flow; Extraction

1. Introduction

Many industrial processes, especially solvent extractionand liquid–liquid reaction, are involved with mass transfer toand from drops moving in another liquid. Evaluation on themass transfer processes around and inside a drop is very im-portant to the optimization of operating characteristics andthe design and scale-up of separation and reaction appara-tus. In addition to the mass transfer process itself, the ac-companying deformation of free interface, drop coalescenceand breakage, etc., make the interphase mass transfer morecomplicated and particularly difficult to quantitatively un-derstand the mass transfer mechanism. Thus, many empiricaland theoretical equations for mass transfer coefficients for

∗ Corresponding author. Tel.: +86 10 62554558; fax: +86 10 62561822.E-mail addresses: [email protected] (C. Yang),

[email protected](Z.-S. Mao).

0009-2509/$ - see front matter� 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2004.11.054

drops dominated by external, internal or conjugate resistanceare proposed in the literature (e.g.,Clift et al., 1978; Steiner,1986; Steiner et al., 1990; Kumar and Hartland, 1999; Hen-schke and Pfennig, 2002; Favelukis and Mudunuri, 2003),mostly based on curve fitting with adjustable parameters orfor a few simple geometries and limiting range of operat-ing variables such as rigid or fluid spheres in creeping orpotential flow.

Uribe-Ramirez and Korchinsky (2000a,b)have derivedmore complicated analytical solution for prediction of singleor multicomponent mass transfer in drops at intermediateReynolds numbers(10�Re�250) for usual industrial op-eration conditions with the assumptions of spherical dropsand the mass transfer in bulk continuous phase ignored. Themass transfer induced turbulence inside a drop (Henschkeand Pfennig, 1999) and the contribution of the oscillation ofasymmetrical traveling drops to mass transfer (Al-Hassan etal., 1992) were considered to facilitate more accurate pre-diction. As even trace surfactants can significantly change

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2644 C. Yang, Z.-S. Mao / Chemical Engineering Science 60 (2005) 2643–2660

the drag and mass transfer rate, the effects of surface-activecontaminants on interphase mass transfer have also beenstudied extensively by experimental, theoretical or numericalmethods (Slater, 1995; Ramirez and Davis, 1999; Brodkorbet al., 2003; Lee, 2003; Li et al., 2003a).

However, experimental mass transfer coefficients are of-ten subject to significant error and the prediction of masstransfer rates is also particularly difficult, even in the caseof single drops moving through a quiescent liquid. With thedevelopment of a variety of numerical methods for simulat-ing deformable multiphase fluid interfaces, the perspectiveof direct numerical investigations of interphase mass trans-fer becomes brighter and brighter.

To date diversified numerical methods for simulating de-formable drops have been developed.Piarah et al. (2001)applied a commercial computational fluid dynamics (CFD)code to calculate the concentration fields inside and out-side of a spherical droplet without considering the deforma-tion of drops.Petera and Weatherley (2001)first simulatedthe motion of drops with a finite degree of deformation to-gether with simultaneous mass transfer in both phases by amodified Lagrange–Galerkin finite element method. But aremeshing routine has to be implemented in their algorithmto avoid the numerical instability caused by deformation ofmesh. The finite element method was also used byWaheedet al. (2002)to compute the mass transfer of single spheri-cal drops withRe�20 by free and forced convection, withthe equilibrium distribution coefficient set equal to unity andthe size of drop assumed constant.

Mao and coworkers (Mao et al., 2001; Li et al., 2001,2002; Li et al., 2003a,b) simulated the fluid flow of and masstransfer to single drops using a body-fitted orthogonal co-ordinate system proposed byRyskin and Leal (1983). TheRyskin–Leal boundary-fitted adaptive grid method was alsoused byPonoth and McLaughlin (2000)to calculate the masstransfer for bubbles in water with the effect of surfactantsand compared with the results obtained by the Lagrangianmarker method. However, it is very difficult to construct theorthogonal curvilinear coordinates for complicated and se-riously deformed interfaces and extend the mapping proce-dure to three-dimensional flow domains.

Sato et al. (2000)and Jung and Sato (2001)have sim-ulated the three-dimensional single droplet flow with masstransfer in continuous phase with different Schmidt numberby a front-capturing method (maker-density function) and afront-tracking method on unstructured grids. The predictedSherwood number agreed well with the empirical correla-tions, but for a high Schmidt number problem, further im-provements of numerical methods are needed.Davidson andRudman (2002)described a volume-of-fluid (VOF) basednumerical method for calculating advective and diffusivemass transfer of drops controlled by the dispersed phase,and provided an indirect comparison of the predicted andthe reported experimental concentration with correction ofterminal velocity for different column/drop diameter ratiowith experiments byTemos et al. (1996).

Although significant efforts have been made to model theinterphase mass transfer with multiphase flow, its numeri-cal simulation is not fully successful to compute highly de-formable or topologically changed interfaces by the above-mentioned methods. The level set approach firstly developedby Osher and Sethian (1988)shows its simplicity and ver-satility for simulating multiphase flow with propagating anddeforming interface and allows for large jump in density andviscosity across the interface as well as the irregular geome-try of interface typically using a delta function or the Heav-iside function without reconstructing the numerical grid.Many moving interface problems in bubble or drop motion,compressible fluid flow, crystal and thin film growth, etch-ing and etc. are attempted by this numerical scheme (Osherand Fedkiw, 2001; Fedkiw et al., 2003). Although it seemsthat the front-tracking type methods using the boundary-fitted coordinates give more accurate representation of theinterface shape, the front-capturing type methods such as thelevel set approach and the VOF method avoid the grid skewproblem and are more suitable for the multiphase flow withvery large deformation of the interface even with mergingand fragmentation. The mass conservation is well kept bythe VOF method, but the level set method can easily be ex-tended to three-dimensional cases and unstructured mesheswith the smooth scalar level set function to identify the ex-act location of the interface and calculate the interface cur-vature conveniently. After the fluid flow inside and outsidea drop has been resolved by the level set approach, it is de-sired to further extend the numerical algorithm to solutionof interphase mass transfer to a single drop.

Lakehal et al. (2002)presented some progress on the sim-ulation of heat and mass transfer in multiphase flow, but nodetails on interphase mass transfer was given. According tothe comparison of volume-of-fluid, level set and boundaryfitting methods, they observed that the level set method maybe somewhat difficult to conserve mass and the VOF ap-proach is more precise even with the disadvantage of diffi-cult extension to complex and three-dimensional topologicalchanges. In our previous work (Yang and Mao, 2002), thelevel set method was improved by applying a doubly finegrid and a coupling reinitialization procedure to suppress“parasitic” surface currents, maintain the level set functionas a distance function and guarantee the mass conservationof drops or bubbles. But up to now, the computation of si-multaneous interphase mass transfer by the level set methodhas not been reported, especially the case of mass transferresistance in both phases, which appears frequently in sol-vent extraction operations. The jump conditions appearingonly for the conjugate mass transfer have to be taken intoaccount, such as the discontinuity of concentration field be-cause the distribution coefficient of a solute is generally dif-ferent from unity (Petera and Weatherley, 2001), which isdifficult for the level set approach with the governing equa-tions based on the “one-fluid” formulation.

In this paper we attempt to utilize the level set method tocompute the interphase mass transfer of single drops moving

C. Yang, Z.-S. Mao / Chemical Engineering Science 60 (2005) 2643–2660 2645

in liquid in the laminar flow regime. The governing equa-tions of two phases flow and mass transfer are solved by afinite difference scheme on a stationary and regular Euleriangrid. The numerical results are validated with the reportedexperimental data (Li, 1998; Li et al., 2002).

2. Mathematical formulation

In this study, the typical unsteady mass transfer to a singledeformable drop falling or rising driven by buoyancy in animmiscible liquid is considered with following assumptions:(1) the fluids in both phases are Newtonian, viscous andincompressible, (2) the physical properties including massdiffusivities and surface tension are constant and not influ-enced by the concentration of solute, (3) the two-phase flowis axisymmetric and laminar, (4) no interface resistance tomass transfer, i.e., thermodynamic equilibrium always ex-ists for the solute between two phases at the interface. Be-sides, the interphase mass transfer does not affect the flowstructure of two phases, thus the solution of the motion andconvective diffusion equations is decoupled.

2.1. Governing equations

In the level set formulation, the velocityu(x, t) and thepressureP(x, t) inside and around the drop are governed bythe continuity and Navier–Stokes equations expressed as

∇ · u= 0, (1)

�(

�u�t

+ u · ∇u)

= −∇P + �g+ ∇ · � + ��n, (2)

where� is the stress tensor defined as

� = �(∇u+ (∇u)T). (3)

In the case of interphase mass transfer, there are fourrelations to be satisfied simultaneously. The mass transferin two bulk phases are governed by the general convectivediffusion equations:

�C1

�t+ u · ∇C1 =D1∇2C1, (4)

�C2

�t+ u · ∇C2 =D2∇2C2, (5)

subject to two interfacial conditions:

D1�C1

�n1=D2

�C2

�n2(flux continuity at the interface), (6)

C2 =mC1 (interfacial dissolution equilibrium). (7)

In the above equations subscript 1 indicates the continuousphase and 2 for the drop. The last is also understood as thecontinuity of the chemical potential across the interface.

2.2. Level set approach of fluid flow

A smooth scalar function denoted as� is introduced intothe formulation of multiphase flow and mass transfer sys-tems to define and capture the interface between two flu-ids, which is identified as the zero level set of the level setfunction� defined on the entire computational domain. Thefunction� is chosen as the signed algebraic distance to theinterface, being positive in the continuous fluid phase andnegative in the drop. The following Hamilton–Jacobi typeevolution equation (Osher and Sethian, 1988) can be usedto advance the level set function exactly as the drop moves

��t

+ ∇ · (u�)= ��t

+ (u · ∇)� = 0. (8)

After introducing the level set formulation, the motion oftwo separate domains for immiscible two fluid phases mayeasily be formulated as a single one.

In a two-dimensional coordinate system, Eqs. (1) and (2)for mass and momentum conservation coupled with the levelset function are thus written in terms of dimensionless vari-ables as

�u�x

+ 1

r

��y(rv)= 0, (9)

��(�u)+ �

�x

(�uu− �

Re

�u�x

)+ 1

r

��y

×(r�vu− r �

Re

�u�y

)

= −�p�x

+ 1

Fr�gx − 1

We�(�)��(�)

��x

+ 1

Re

��x

(�

�u�x

)+ 1

Re

1

r

��y

(r�

�v�x

), (10)

��(�v)+ �

�x

(�uv − �

Re

�v�x

)+ 1

r

��y

×(r�vv − r �

Re

�v�y

)

= −�p�y

+ 1

Fr�gy − 1

We�(�)��(�)

��y

+ 1

Re

��x

(�

�u�y

)+ 1

Re

1

r

��y

(r�

�v�y

)

−{

2

Re

�vr2

}, (11)

wherer ≡ 1 for Cartesian coordinates,r ≡ y for cylindricalcoordinates and curly brackets indicate the term presentsonly in cylindrical coordinates. The dimensionless groupsRe, Fr andWeare the Reynolds, Froude and Weber numbers,respectively

Re ≡ �1LV

�1= d1.5g0.5 �1

�1, (12)

Fr ≡ V 2

Lg, (13)


We ≡ �1V2L

�= �1d

2g

�, (14)

where the characteristic lengthL is chosen as the initial di-ameter of spherical drop(d) and the corresponding refer-ence velocity is defined asV = √

dg.The curvature of free surface,�(�), is expressed as

�(�)= ∇ · n = ∇ ·( ∇�

|∇�|). (15)

The regularized delta function��(�) is defined as

��(�)={ 1

2�(1 + cos(�/�)), if |�|< �,

0, otherwise,(16)

where � ≡ O(h) prescribes the finite “thickness” of theinterface. In this computation, we take�=h generally, whereh is equal to the dimensionless uniform mesh size near theinterface.

In order to avoid numerical instabilities at the interface,especially for large density ratio of two phases, in Eqs. (10)and (11) the dimensionless density and viscosity take theregularized forms:

��(�)= �2/�1 + (1 − �2/�1)H�(�), (17)

��(�)= �2/�1 + (1 − �2/�1)H�(�), (18)

where the regularized Heaviside functionH�(�) is definedas

H�(�)=

0, if �<− �,1

2

(1 + �

�+ sin(�/�)/

), if |�|��,

1, if �> �.

(19)

In addition, after some iteration steps� will no longerremain a distance function (i.e.,|∇�| �= 1) generally, evenif Eq. (8) advances the interface at correct velocities. Main-taining� as a distance function is very essential for accurateevaluation ofn and�(�). Therefore, a reinitialization proce-dure for resetting� as an exact distance function should beadopted to keep the interface thickness finite and preservemass conservation. A reinitialization method proposed bySussman et al. (1994)is often used, which is accomplishedby solving the following equation to steady state:

��

= sgn(�0)(1 − |∇�|), (20)

where�0 is a level set function at any computational time,i.e.,�0(X)=�(X, = 0), is the virtual time in a reinitial-ization step and sgn(�0) denotes the smoothed sign functionwith appropriate numerical smearing to avoid any numericaldifficulties. The formulation of fluid flow and the procedureof numerical solution have been presented previously by theauthors in detail (Yang and Mao, 2002), in which anotherarea-preserving reinitialization procedure for� was coupledwith Eq. (20) to guarantee the mass conservation by solving

a perturbed Hamilton–Jacobi equation proposed byZhanget al. (1998)to pseudo-steady state in each time step. Theimproved reinitialization procedure can maintain the levelset function as a distance function and guarantee the dropmass conservation.

2.3. Level set approach of interphase mass transfer

In the following, the “one-fluid” formulation coupled withthe level set function for interphase mass transfer is derived.When the distribution coefficient of a solute,m, is not unity,the discontinuity of solute concentration across the interfacemust be dealt in addition to the general discontinuity of thediffusion coefficient of solute in two phases. The simplestcase is thatm is equal to unity. In this case the concentrationis continuous across the interface. The only difficulty is thediscontinuity of molecular diffusion coefficients, which caneasily be handled by the level set function with the samesmoothed Heaviside function as used in Eqs. (17) and (18).Thus, Eqs. (4) and (5) for two phases may be expedientlysolved in a single domain by the level set approach similarto the solution of multiphase flow.

However, more general case is thatm is not equal tounity. In this case some measures must be taken to make theconcentration field become continuous across the interface,just as the continuity of fluid velocity at the interface. For thispurpose, it is proposed to the concentration transformationssuch asC1 = C1

√m and C2 = C2/

√m. Condition (7) is

cast intoC1 = C2 at the interface, andC1 and C2 becomecontinuous in the whole domain. Using these definitions, thetransformed Eqs. (4) and (5) remain in the same form

�Cj�t

+ u · ∇Cj =Dj∇2Cj (j = 1 or 2), (21)

where Cj denotesC1 or C2, andDj represents the cor-respondingD1 or D2. Then, condition (7) is turned intoC1 = C2 at the interface, andC1 andC2 become continuousin the whole domain.

The difficulty that remains is only condition (6). Whenthe transformations are utilized, Eq. (6) becomes

D1√m

�C1

�n1= √

mD2�C2

�n2. (22)

Therefore, at the interface, which is of finite thickness, thediffusion coefficients in both phases should be locally re-placed byD1/

√m and

√mD2 to satisfy the original condi-

tion of mass flux continuity. This would make the diffusivityin the interface region different from that in the bulk do-main and may results in unacceptable errors. In this paper, asimple transformation is applied to make the diffusivity be-ing equal throughout a fluid phase. For the continuous anddispersed phases, Eq. (21) can be rewritten separately as

�C1

�(√mt)

+ 1√mu · ∇C1 = 1√

mD1∇2C1, (23)


�C2

�(

1√mt) + √

mu · ∇C2 = √mD2∇2C2, (24)

and then can be put into a common equation over the wholedomain in the form analogous to the momentum equation(2)

�C

�t+ u · ∇C = ∇ · (D∇C) (25)

through different definition oft , D andu by the regularizedHeaviside functionH�(�):

t (�)={√

mt, if ��0,1√mt, if �<0,

(26)

D(�)= √mD2 +

(1√mD1 − √

mD2

)H�(�), (27)

u(�)= √mu+

(1√mu− √

mu)H�(�), (28)

whereu should be the velocity field in a frame of referencemoving with the drop.

The governing equations for interphase mass transfer arenondimensionalized and the expanded expressions in a two-dimensional axisymmetric coordinate system are

�C��

+ u �C�x

+ v �C�y

= 1

Pe1

(��x

(D

�C�x

)+ 1

r

��y

(rD

�C�y

)), (29)

where the dimensionless groupPe1 is the Peclet numberdenoted as

Pe1 ≡ LV

D1, (30)

C is the dimensionless concentration based on the referenceconcentration, usingC1,∞ for solute transportation from thecontinuous phase to the drop, butC0

2 for the inverse directionof solute transfer.D1 is chosen as the characteristic molecu-lar diffusivity, and the other characteristic variables of timeand velocity are the same as those for Eqs. (10) and (11).

Besides, the other transformation forms of concentration,such asC1 =mC1 andC2 =C2, or C1 =C1 andC2 =C2/m

are also tested. The overall interphase mass transfer can besolved in the similar way, and completely identical resultswere obtained. Therefore, the transformation form has noeffect on the numerical solutions of interphase mass transfer.

3. Computational scheme

3.1. Numerical solution method and procedure

The control volume formulation with the SIMPLEC al-gorithm (Van Doormaal and Raithby, 1984) incorporated is

used to solve the governing equations including the abovemass transfer equations with continuity, momentum con-servation, and level set evolution equations. The power-lawscheme is adopted for discretization of the governing equa-tions (9)–(11) in a staggered non-uniform grid with moredense cells near the interface. In order to ensure variablesmore accurately interpolated and revolved, a double finegrid detailed byYang and Mao (2002)is also applied formass transfer. When the motion of a droplet is simulatedby a level set approach, the inevitable “parasitic” surfacecurrents arising from errors in the surface force calcula-tion are suppressed by this improvement. In a word, if theNx × Ny grid used for velocity and pressure solution, the2Nx × 2Ny one is used for�, �, �, D andC specificationor evolution. As the level set function, density, viscosity,diffusion coefficient and concentration are also calculatedon the fine grid instead of on the standard velocity-pressuregrid, errors in the discretized mass and momentum equationscan be reduced, which is essential in ensuring strict conser-vation of mass and momentum. Moreover, the deformableinterface is advanced for each time step starting from theinterface instead of the boundary of the computational do-main to improve the computational accuracy close to theinterface.

According to the assumptions in Section 2, the solution ofthe mass transfer equations can be decoupled from the solu-tion of two phase fluid flow. The momentum transfer prob-lem can thus be solved first and the resulting velocity fieldis utilized to formulate the time evolution of mass transferproblem. This assumption is reasonable for real processesof mass transfer to single drops, usually with sufficientlysmall transfer rates of solute in the laminar regime.Peteraand Weatherley (2001)also confirmed that drops attainedrapidly the terminal velocity by experimental observationsand calculations, so the velocity field in the procedure ofmass transfer is very close to that of a steady motion dropbefore mass transfer occurring.

Thus, the main calculation steps of this algorithm are out-lined as follows: (1) initialize the flow field (u andP) with thedrop at rest, physical parameters (density, viscosity, surfacetension and molecular diffusivity of solute in each phase),concentration, and� as the signed normal distance to theinterface, (2) compute the velocity and pressure to steadystate and update the level set function by the SIMPLEC al-gorithm, (3) compute the unsteady concentration field overthe whole domain with the steady state flow field, and cal-culate the corresponding overall mass transfer coefficientsand Sherwood numbers.

For the mass transfer to a single drop dominated by inter-nal resistance or controlled by both phases, the overall masstransfer coefficientkod may be evaluated from the overallsolute conservation based on the drop

kod(C∗2 − C2)S = Vd dC2

dt, (31)


whereC2 is the average concentration of the drop at anytime, that is almost the only available measure of soluteconcentration of drops in conventional experiments. If thetime interval tout − tin is chosen small enough,kod, S andVd may be taken approximately as constants. Then integratethe above equation gives

kod = −VdS

1

tout − tin ln

(C∗

2 − C2,out

C∗2 − C2,in

), (32)

in which the volumeVd and surface areaSare calculated bynumerical integration based on the location and shape de-termined by the level set function. The corresponding Sher-wood number is

Shod = d

D1kod. (33)

For the transient or steady mass transfer dominated by ex-ternal resistance, the average concentration of drops is neg-ligibly small. The overall mass transfer coefficient is calcu-lated using the local concentration gradient at the interface.The conservation of mass flux over the interface becomes

kod(C1,∞ − C1,�)S = �� Di�C1

�nd� (34)

or

kod = 1

S

1

C1,∞ − C1,�� Di

�C1

�nd�, (35)

where� is the surface of the drop, andDi is the moleculardiffusivity at the interface.

3.2. WENO scheme for discretization

The solution of time-dependent advective-diffusion trans-port equations, e.g., Eq. (21), is still elusive for cases inwhich sharp fronts in space and time develop, althoughdozens of numerical algorithms have appeared in the liter-ature. In practice, the diffusion term in advective diffusionequation is often smaller relative to advection, and some-times the effect of diffusion can be neglected and the equa-tion is reduced to a first-order hyperbolic partial differentialequation such as the Hamilton–Jacobi type evolution equa-tion (8) for the level set function.

It is well known that the advection dominated partialdifferential equations present serious numerical difficultiesdue to the moving steep fronts or shock discontinuitiesin solutions with degenerate diffusion, including someadditional difficulties such as strong coupling and nonlin-earity, strong heterogeneity of coefficients, and anisotropicdiffusion–dispersion in tensor form (Ewing and Wang,2001). Attention must be paid to discretization of the con-vection term, which often causes much numerical errorand instability in the presence of discontinuities or steepfronts. High resolution spatial discretization methods likethe WENO schemes must be assorted to for sufficientlyaccurate results.

Jiang and Shu (1996)derived a fifth-order WENO scheme(denoted as J-WENO), which has widely been implementedand is proved robust and efficient for solving conserva-tion equations, complicated shock and flow structure. Theartificial compression method (ACM) proposed byYang(1990)was also adopted to enhance the performance of theWENO scheme for sharpening the contact discontinuities(this scheme denoted as J-WENO+ACM), but the CPU costwas increased dramatically. Moreover, other more compli-cated and higher order WENO schemes were also reportedmainly for shock wave problems (e.g.,Balsara and Shu,2000; Qiu and Shu, 2002).

Farthing and Miller (2001)compared 14 different explicit-in-time finite-volume discretization methods for solving theadvective–diffusion equation applied in the water resourcefield. Two higher-order TVD (total variation diminishing)schemes performed well for their test problems. The WENOscheme was typically more accurate though it was slowerthan some other discretization methods.A level set algorithmand the fifth-order WENO scheme for solving efficiently hy-perbolic conservation equation and reducing the CPU timewere presented byAslam (2001)with a perfect discontinu-ity maintained while achieving high order pointwise con-vergence.Lim et al. (2002)also applied the high resolutionspatial discretization WENO schemes to the dynamic simu-lation of batch crystallization, including the correct estima-tion of parameters for agglomeration and breakage kinetics,and the crystal size distribution with good accuracy.

In this work, the fifth-order WENO scheme proposedby Fedkiw et al. (1999)(denoted as F-WENO) is adoptedfor spatial discretization of the reinitialization equationsand the advection equation for the level set function toachieve higher order accuracy. Different discretizationschemes, including power-law, ENO, WENO, and WENOwith ACM technique, are also tested for the solution ofthe advective–diffusion mass transfer equation. The power-law scheme with at most second order accuracy is fromPatankar (1980), and the detailed second-order ENO schemeis similar to that ofSussman et al. (1994). Two differentdiscretization methods for the fifth order WENO schemeare implemented for comparison. One is based on noncon-servative flux method, i.e., F-WENO proposed byFedkiwet al. (1999), for discretizing advection–diffusion equation(25). The other is a conservative flux difference method,i.e., J-WENO byJiang and Shu (1996), applied to masstransfer equation in the form of

�C

�t+ ∇ · (uC)= ∇ · (D∇C) (36)

since the fluids are incompressible (∇ ·u= 0). The concretediscretization formulations for the two WENO schemes andthe ACM technique are left out for conciseness (for detailsseeJiang and Shu, 1996; Fedkiw et al., 1999).


3.3. Temporal evolution

The time step�� must satisfy the Courant–Friedrich–Lewyconditions and also the restrictions due to gravity, surfacetension and viscous terms to make the numerical proce-dure stable and convergent (Brackbill et al., 1992; Mulderet al., 1992; Sussman et al., 1994). While computing themultiphase flow and interphase mass transfer,

��s = h1.5√(�2/�1 + 1)We/4, (37)

��v = 3

14h2Re min

(��(�)��(�)

), (38)

��c = min

(h

|u|), (39)

��m = 3

14h2Pe1 min

(1

D�(�)

), (40)

�� = 0.5 min(��s ,��v,��c,��m). (41)

The restriction condition of��m in Eq. (40) for mass trans-fer is similar to ��v and derived in terms of the anal-ogy of mass and momentum transport. This time step isused for the computation of mass transfer, which is sta-ble and convergent for both fluid flow and interphase masstransfer.

In addition, in order to avoid any instability and diver-gence in the temporal integration of the level set functionevolution and reinitialization equations and the transientmass transfer equation, a third-order TVD Runge–Kuttascheme (Jiang and Shu, 1996; Yang and Mao, 2002) in realor virtual time is utilized for time discretization of WENOschemes.

4. Results and discussions

A buoyancy-driven drop with initial diameterd sur-rounded by an incompressible Newtonian fluid boundedin an adequately wide cylindrical column is consid-ered for validation of the applicability and robustnessof the above-mentioned numerical algorithm for inter-phase mass transfer. The free-slip boundary condition isimposed at the column wall. Typically we solve the lam-inar flow and mass transfer in a computational domain� = {(x, y)|0�x�20d,0�y�15d} to assure negligiblewall effect.

4.1. Convergence test

A number of numerical experiments with different ini-tial drop diameter, density, viscosity, surface tension, molec-ular diffusivity and distribution coefficient are carried outto test the level set approach for interphase mass transfer.Nonuniform grids of 50× 44, 74× 64, 99× 84, 123× 104,

147× 124, 183× 154 and 244× 204 (the last is referredto as the fine one) are applied for comparing the solutionof the two phases flow and mass transfer as a function oftime to ensure mesh independence of numerical results. Theeffect of grid on the calculated major and minor axes of anoblate ellipsoidal drop and the Reynolds number are shownin Table 1. Fig. 1presents the influence of mesh size on theoverall mass transfer coefficients. As the grid is refined, thepredicted drop shape, Reynolds number and interphase masstransfer coefficient decrease gradually to asymptotic values(Re∗ = 18.7, kod = 5.5× 10−6 m/s). A grid with 183× 154nodes is sufficient for spatial computational accuracy, andadopted for the subsequent simulations.

The present code is verified additionally for the masstransfer into or out of single drops with resistance only inthe continuous phase or in the dispersed phase against thecorresponding analytical results. For a spherical drop whichis internally stagnant and when the continuous phase veloc-ity is zero, i.e.,Pe → 0, the analytical result of the Sher-wood number isSh1 = 2.0. For another limiting diffusiondominated case of a stagnant spherical drop with negligi-ble external mass transfer resistance, the steady asymptoticvalue of the Sherwood number for long time isSh2 = 6.58derived by the Newman’s equation (Clift et al., 1978, p. 59).

The numerical time history of the Sherwood number re-sulting from the present algorithm for these two limitingproblems with no flow are shown inFigs. 2and3. The nu-merical solution gives results ofSh1 = 2.002 for the stag-nant continuous phase andSh2 = 6.507 for the stagnantdrop when the steady state is attained, with relative devi-ation about 0.1% and 1.1%, respectively. Such agreementsuggests the validity of the present method.

4.2. Comparison of spatial discretization schemes

To test different spatial discretization schemes for convec-tive terms in mass transfer equations (25), an axisymmetric,neutrally buoyant spherical drop moving at the same linearvelocity as the surrounding continuous liquid phase is con-sidered. The flow is further assumed one-dimensional withconstantu1=u2=1.1 (dimensionless value) andv1=v2=0,and the initial concentration field ofC1=1 (in the subdomainof ��0) andC2 = 0(�<0), where subscript 1 denotes thecontinuous phase and 2 the drop. Since the convective trans-port of mass is concerned, the viscosities and diffusivities ofboth phases are set to zero:�1 =�2 =0 andD1 =D2 =0. Inthis case, the initial concentration field with the concentra-tion jump at the drop surfaces must remain unchanged andmove downstream with the drop in advection. A good algo-rithm should generate the following exact solution:C2 = 0for all nodes in the drop andC1 = 1 for the other nodes(��0) at any computational time. In conducting this test,different schemes are applied to the convective terms in de-generate mass transfer equations and the Hamilton–Jacobievolution equation (8).


Table 1Influence of mesh size on axis size of an oblate ellipsoidal drop and predicted Reynolds number

Node number 50× 44 74× 64 99× 84 123× 104 147× 124 183× 154 244× 204

Major axis 1.008 1.002 1.010 1.020 1.021 1.025 1.024Minor axis 0.9260 0.9267 0.9330 0.9404 0.9408 0.9444 0.9436Reynolds number 18.92 18.47 18.54 18.69 18.66 18.69 18.66

d = 0.99 mm,�1 = 996.3 kg/m3, �2 = 832.5 kg/m3, �1 = 1.532× 10−3 Pa s,�2 = 3.714× 10−3 Pa s,� = 1.02× 10−3 N/m, D1 = 5.20× 10−10 m2/s,D2 = 2.30× 10−10 m2/s, m= 1.17, C0

1 = 2.94%,C02 = 0.

0 5 10 15 20 25 30 35

0

2

4

6

8

10

12 244X204 183X154 147X124 123X104 99X84 74X64 50X44

k od

(10

-5m

/s)

Time (s)

Fig. 1. Grid independence test: overall mass transfer coefficient versustime (the same simulation conditions as inTable 1).

0 50 100 150 200 250 3001

2

3

4

5

6

7

Predicted

Analytical

Sh 1

Dimensionless time

Fig. 2. Comparison of predicted transient external Sherwood number forstagnant continuous phase with analytical solution.

Figs. 4–8depict the non-dimensional concentration pro-files from the numerical solution of the convective masstransfer equation for different spatial discretization schemesafter an evolution time long enough. The corresponding pre-dicted average concentration of the drop changed with the

0 3000 6000 9000 12000 150004

8

12

16

20

24

28

32

36

40

44

48

Predicted

Analytical

Sh 2

Dimensionless time

Fig. 3. Comparison of predicted transient internal Sherwood number forstagnant drop with analytical solution.

increase of mass transfer time is shown inFig. 9. The nu-merical result of the power-law scheme with the SIMPLECalgorithm is the poorest and also the most CPU cost, butthe fifth-order WENO coupled with Yang’s ACM techniqueperforms the best among all tested schemes and followed bythe J-WENO scheme. As shown inFigs. 10and11, the pre-dicted time history of the overall mass transfer coefficientand average velocity of the drop by the WENO schemes arecloser to the experimental data (Li, 1998; or Li et al., 2002)with the similar trend as shown by the comparison inFigs.4–8. The predicted value of overall mass transfer coefficientby the power-law scheme is about two times that by otherschemes.

Although the artificial compression technique ofYang(1990) can enhance the performance of WENO schemesat contact discontinuities, the CPU cost is increased obvi-ously according to the present simulation and the reportedresults (Jiang and Shu, 1996). Since the accuracy of theJ-WENO scheme is adequate, this scheme is adopted forall the subsequent solution of the convective–diffusionequation.

Moreover, we have tested the effect of the above dis-cretization schemes on the solution of the level set func-tion evolution and reinitialization equations (Yang and Mao,2002), the fifth-order WENO schemes also give the best per-formance and are adopted for all computations. But for the


-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0Distance to drop center

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Con

cent

ratio

n

� <0

rising drop

u

-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0

Distance to drop center

�= 5.21

�= 1.30

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Con

cent

ratio

n

�<0

rising drop

u

(a)

(b)

Fig. 4. Concentration distribution at symmetry axis for power-law scheme(u0 = 1.1, v0 = 0, C0

1 = 1, C02 = 0 , m= 1.17, grid: 123× 104).

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0


0.0

0.2

0.4

0.6

0.8

1.0

1.2

Con

cent

ratio

n

� <0

rising drop

u

Fig. 5. Concentration distribution at symmetry axis for ENO scheme at� = 5.21 (the same simulation conditions as inFig. 4).

level set function equations no significant difference wasfound between the conservative flux based J-WENO schemeand the nonconservative flux based F-WENO scheme byFedkiw et al. (1999).

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0


0.0

0.2

0.4

0.6

0.8

1.0

1.2

Con

cent

ratio

n

� <0

rising drop

u

Fig. 6. Concentration distribution at symmetry axis for F-WENO schemeat � = 5.21 (the same simulation conditions as inFig. 4).

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0


0.0

0.2

0.4

0.6

0.8

1.0

1.2

Con

cent

ratio

n

� <0

rising drop

u

Fig. 7. Concentration distribution at symmetry axis for J-WENO schemeat � = 5.21 (the same simulation conditions as inFig. 4).

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0Distance to drop center

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Con

cent

ratio

n

� <0

rising drop

u

Fig. 8. Concentration distribution at symmetry axis for J-WENO+ ACMscheme at� = 5.21 (the same simulation conditions as inFig. 4).

4.3. Comparison with experimental data

Although in the past decades many experimental studieswere focused on the motion and mass transfer of single


0.0 4.0 8.0 12.0 16.0 20.0

Time (s)

0.01

0.10

1.00

Ave

rage

Con

cent

ratio

n

Power-lawENOF-WENOJ-WENOJ-WENO+ACM

Fig. 9. Influence of spatial discretization schemes on predicted transientaverage concentration of drop (the same simulation conditions as inFig. 4).

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0

Time (s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Power-lawENOF-WENOJ-WENOJ-WENO+ACMExperimental

C2

(wt%

)

Fig. 10. Comparison of predicted average drop concentrations for differentspatial discretization schemes with experimental data (the same conditionsas inTable 1).

drops and drop swarms, direct comparison between theexperimental and the predicted mass transfer coefficientor concentration is very difficult for lack of necessary pa-rameters (Davidson and Rudman, 2002). Hence, works inour group (Li, 1998; Li et al., 2001, 2002; Mao et al.,2001) were devoted to the single drop extraction experi-ments to check the applicability of the modified numericalmethod in a boundary-fitted orthogonal coordinate system.In this paper, such experimental data are used for vali-dating the present level set approach for interphase masstransfer.

A typical extraction system ofn-butanol–succinicacid–water recommended by EFCE (the European Confed-eration of Chemical Engineering) is adopted for compari-son. In this system, the drop is composed ofn-butanol and

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Time (s)

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

k od

(10-5

m/s

)

13.2

13.6

14.0

14.4

14.8

15.2

15.6

ExperimentalF-WENOJ-WENOJ-WENO+ACMENOPower-law

Fig. 11. Comparison of predicted overall mass transfer coefficients fordifferent spatial discretization schemes with experimental data (the sameconditions as inTable 1).

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0Time (s)

Average drop concentration

Overall mass transfer coefficient

0.0

0.5

1.0

1.5

2.0

2.5

3.0

ExperimentalPredicted

C2

(wt%

)

0.0 5.0 10.0 15.0 20.0 25.0 30.0Time (s)

0.0

1.0

2.0

3.0

4.0

5.0

k od

(10-5

m/s

)


(a)

(b)

Fig. 12. Predicted and experimental concentrations and overall masstransfer coefficients as a function of mass transfer time for serial no.BC4-1: (a) average drop concentration, (b) overall mass transfer coefficient(d=0.99 mm,�1=996.3 kg/m3, �2=832.5 kg/m3, �1=1.532×10−3 Pa s,�2 = 3.714× 10−3 Pa s,�= 1.02× 10−3 N/m, D1 = 5.20× 10−10 m2/s,D2 = 2.30× 10−10 m2/s,m= 1.17,C0

1 = 2.94%,C02 = 0; solute transfer

direction: from continuous phase to drop).


0.0 5.0 10.0 15.0 20.0 25.0Time (s)

0.00

0.50

1.00

1.50

2.00

2.50

3.00


C2

(wt%

)

0.0 5.0 10.0 15.0 20.0 25.0

Time (s)

0.0

2.0

4.0

6.0

8.0

k od

(10-5

m/s

)


(a)

(b)



Fig. 13. Predicted and experimental concentrations and overall masstransfer coefficients as a function of mass transfer time for serial no.BC9-1: (a) average drop concentration, (b) overall mass transfer coeffi-cient (d = 1.38 mm, other conditions the same as inFig. 12).

the succinic acid is used as solute. The distribution coeffi-cient is not equal to unity and the mass transfer process iscontrolled by resistance in both phases. The possible con-tamination to the extraction system has been carefully ex-cluded. Some measures, such as guaranteeing constant in-terface area, were also taken to minimize the mass transfer“end effects” occurring during droplet formation and coa-lescence (Li et al., 2001). The mass transfer mechanism in-volved in the drop formation stage is very complicated andit combines simultaneous effects of interface creation, re-newal, interfacial convection and etc. Besides, the time formass transfer to approach steady state is sufficiently longerthan that for attaining the terminal velocity, so that the con-tribution in the stages of drop formation and accelerated mo-tion can be eliminated by subtracting the mass transfer atthe first sampling location and the solution of the convectivediffusion equation can be decoupled with the momentumequation.

As shown inFigs. 12–19, the predicted drop average con-centrations and overall mass transfer coefficients for masstransfer direction from the continuous phase to the drop or in

0.0 4.0 8.0 12.0 16.0 20.0

Time (s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0


C2

(wt%

)

0.0 4.0 8.0 12.0 16.0 20.0

Time (s)

0.0

2.0

4.0

6.0

8.0

10.0

k od

(10-5

m/s

)


(a)

(b)




the inverse direction are in reasonably good agreement withthe corresponding experimental data. InTable 2, althoughbetter agreement between computational and experimentalReynolds numbers is observed, most predicted overall masstransfer coefficients, including the result in a boundary-fittedorthogonal coordinate system, are slightly higher than theexperiments.Li et al. (2002)conjectured the experimentalsystem might not be very pure, though serious care wastaken in conducting experiments. Despite the fact that theboundary-fitted orthogonal coordinate method offers moreaccurate numerical results, it is very difficult to construct or-thogonal curvilinear coordinates for highly deformed dropsand be extended to the simulation of three-dimensional prob-lems.

4.4. Effect of physical parameters

For the sake of comparison of the present method withother approach, an example of interphase mass transfer withthe similar conditions as inFig. 10 reported byLi et al.(2002)was also simulated. As shown inFig. 20, the concen-tration contour maps at different time are very similar to the


0.0 4.0 8.0 12.0 16.0 20.0Time (s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0


C2

(wt%

)

0.0 4.0 8.0 12.0 16.0 20.0

Time (s)

0.0

2.0

4.0

6.0

8.0

k od

(10-5

m/s

)


(a)

(b)




numerical results by the body-fitted orthogonal coordinatemethod (Li et al., 2002, p. 8).

Figs. 21–23show three typical examples of the transientevolution of concentration profiles for mass transfer resis-tance both in the external and internal domains of a singledrop with differentRe, We and Pe. The computed streamfunction contours inside and outside the drop at the flowsteady state are also depicted inFigs. 21–23(a). The streamfunction contours at the rear of drop become more flexu-ous with the increase of Reynolds number, which indicatesthe circulating wake will appear if the Reynolds number isincreased further.

It is obviously observed that the distortion of the concen-tration contour lines develops gradually to form a series ofclosed contours (called Kronig–Brink vortex) (Kronig andBrink, 1950) along with the increase of mass transfer time.As the Reynolds number increases further, the concentra-tion contour lines are swept to the downstream. The con-vection at the drop nose promotes the transport of solute tothe drop, making the contour lines over there compressedspatially.

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Time (s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0Experimental

Predicted

C2

(wt%

)

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Time (s)

0.0

1.0

2.0

3.0

4.0

5.0

k od

(10-5

m/s

)


(a)

(b)



Fig. 16. Predicted and experimental concentrations and overall masstransfer coefficients as a function of mass transfer time for serial no.BD4-1: (a) average drop concentration, (b) overall mass transfer coefficient(d=0.98 mm,�1=988.2 kg/m3, �2=841.1 kg/m3, �1=1.44×10−3 Pa s,�2 = 3.34× 10−3 Pa s,� = 1.00× 10−3 N/m, D1 = 5.5 × 10−10 m2/s,D2 = 2.1 × 10−10 m2/s,m= 1.16, C0

1 = 0, C02 = 5.66%; solute transfer

direction: from drop to continuous phase).

Figs. 24and 25 show the influence of the Peclet num-ber on the concentration profiles under the condition of thesame flow structure as inFig. 22 (i.e., with a fixed valuesof ReandWe). With the decrease of the Peclet number, theconcentration contour lines become less distorted and theconcentration wake at the rear of drop closer to the drop. Forthe mass transfer case dominated by the molecular diffusion(for small Pe), the profiles will become a set of concentriccircles. The concentration profiles become unsymmetricalasPe increases. It seems that the larger the value ofReorPe, the more severely distorted are the concentration pro-files, suggesting that convective mass transfer does play animportant role.

5. Conclusions and future works

A level set approach for numerical simulation of in-terphase mass transfer from or to a deformable droplet


0.0 5.0 10.0 15.0 20.0 25.0Time (s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0Experimental

Predicted

C2

(wt%

)

0.0 4.0 8.0 12.0 16.0 20.0

Time (s)

0.0

2.0

4.0

6.0

8.0

10.0

k od

(10-5

m/s

)


(a)

(b)



Fig. 17. Predicted and experimental concentrations and overall masstransfer coefficients as a function of mass transfer time for serial no.BD9-1: (a) average drop concentration, (b) overall mass transfer coeffi-cient (d = 1.35 mm, other conditions the same as inFig. 16).

moving in a continuous immiscible liquid in the laminarflow regime is presented. For the conjugate mass transfercases with resistance in both drop and continuous phase andequilibrium distribution coefficients differing from unity,transformations of concentrations, molecular diffusivities,mass transfer time and velocities in both phases accordingto the interface boundary conditions of mass flux conti-nuity and dissolution equilibrium have to be implementedfor solving the convective–diffusion equation in a singleEulerian domain accurately and successfully.

The control volume formulation with the SIMPLEC algo-rithm incorporated is used to solve the governing equationsof incompressible two-phase flow with deformable free in-terface on a staggered Eulerian grid. The mass transfer equa-tions decoupled from that of two phases fluid flow is solvednumerically after the velocity field is resolved. Based on thecomparison of different spatial discretization schemes suchas the power-law, ENO, WENO and WENO coupled withACM technique for computing the governing equations ofmass transfer, the fifth-order WENO scheme performs betterand is applied for the present simulation including the solu-

0.0 5.0 10.0 15.0 20.0 25.0Time (s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0ExperimentalPredicted

C2

(wt%

)

0.0 4.0 8.0 12.0 16.0 20.0

Time (s)

0.0

3.0

6.0

9.0

12.0

k od

(10-5

m/s

)



Overall mass transfer coefficient(b)

(a)


tion of the evolution and reinitialization equations of levelset function.

The Reynolds number and time history of concentrationand mass transfer coefficient are well predicted by this algo-rithm as verified against the reported experimental data ofthen-butanol–succinic acid–water system recommended byEFCE. For two limiting cases of mass transfer into or out ofsingle drops with resistance dominated only in the stagnantcontinuous phase or in the stagnant spherical drop, the nu-merical results are almost identical to the corresponding ana-lytical solutions. The good agreement between the results ofnumerical simulation and those from experiments and theo-retical analysis for the transient interphase mass transfer ofa drop in this paper also confirm the correct jump conditionsat the interface are satisfied by the necessary transformationmaking the concentration continuous. These numerical testsindicate the present level set approach is simple, robust andeffective in simulating the interphase mass transfer of singledrops.

The proposed level set approach is easy to be extendedto a three-dimensional space, and straightforward to other


0.0 5.0 10.0 15.0 20.0 25.0Time (s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

Experimental

Predicted

C2

(wt%

)

0.0 4.0 8.0 12.0 16.0 20.0

Time (s)

0.0

5.0

10.0

15.0

20.0

k od

(10-5

m/s

)


(a)

(b)




more challenging interphase mass transfer to or from bubblesand drops with complicated and seriously deformed freeliquid–liquid or gas–liquid interfaces involved. However, itis worthy to carefully consider how to take the advantagesof the level set method and meanwhile to maintain

Table 2Comparison of predicted and experimental Reynolds numbers and average overall mass transfer coefficients forn-butanol–succinic acid–water system(the same conditions as inFigs. 12–19)

No. Experimental Predicted Reynolds Experimental average Predicted average Predicted averageReynolds number number kod × 105(m/s) kod × 105(m/s) kod × 105(m/s) by Li (1998)

BC4-1 18.9 19.0 0.449 0.709 0.370BC9-1 37.7 38.4 0.589 1.126 0.807BC12-1 49.5 48.5 0.897 1.400 1.278BC16-1 61.9 59.6 2.566 1.900 2.130BD4-1 21.2 18.8 0.374 0.728 0.408BD9-1 39.0 37.3 0.752 1.275 0.907BD12-1 48.1 50.6 0.661 1.543 1.137BD16-1 55.2 56.9 1.108 1.739 2.340

Fig. 20. Influence of mass transfer time on concentration profiles: (a)concentration contour map at� = 16, (b) concentration contour map at�=20, (c) concentration contour map at�=34, (d) concentration contourmap at�= 50, (e) concentration contour map at�= 1003. (Re∗ = 10.18,We∗ = 4.03, �2/�1 = 2.0, �2/�1 = 5.0, Pe∗1 = 1.018× 105).

feasibility of the simulation for the problem of largenumber of drops with more nodes needed, especially fordrop swarm systems with coalescence and breakage. The


Fig. 21. Concentration and stream function contours of BC4-1 at differ-ent time: (a) steam function contour map at steady state, (b) concentra-tion contour map at� = 71, (c) concentration contour map at� = 83,(d) concentration contour map at� = 94 (Re∗ = 19.04, We∗ = 0.8455,Pe∗1 = 5.630× 104, other conditions the same as inFig. 12).

Fig. 22. Concentration and stream function contours of BC9-1 at differ-ent time: (a) steam function contour map at steady state, (b) concentra-tion contour map at� = 49, (c) concentration contour map at� = 68,(d) concentration contour map at� = 87 (Re∗ = 38.38, We∗ = 2.465,Pe∗1 = 1.135× 105, other conditions the same as inFig. 13).

Fig. 23. Concentration and stream function contours of BC16-1 at dif-ferent time: (a) steam function contour map at steady state, (b) concen-tration contour map at� = 57, (c) concentration contour map at� = 84(Re∗ = 59.62, We∗ = 4.664, Pe∗1 = 1.763× 105, other conditions thesame as inFig. 15).

Fig. 24. Concentration profiles for the case of 10 times molecular dif-fusivity of BC9-1 under same flow structure: (a) concentration contourmap at�=49, (b) concentration contour map at�=68, (c) concentrationcontour map at� = 87 (Pe∗1 = 1.135× 104, other conditions the same asin Fig. 22).

mathematical modeling of mass transfer under the effect ofsurface-active contaminants, especially the Marangoni ef-fect and the induced surface convection, are currently underway.


Fig. 25. Concentration profiles for the case of 100 times molecular dif-fusivity of BC9-1 under same flow structure: (a) concentration contourmap at�=49, (b) concentration contour map at�=68, (c) concentrationcontour map at� = 87 (Pe∗1 = 1.135× 103, other conditions the same asin Fig. 22).

Notation

C concentration of solute, wt%C∗ concentration in equilibrium with other phase, wt%C transformation form of concentration, wt%C average concentration of solute, wt%d volume-equivalent diameter of drop, mmD molecular diffusivity of solute, m2/sD transformation form molecular diffusivity of solute,

m2/sFr Froude number,V 2/(Lg)

g acceleration vector of gravity, m/s2

g gravitational acceleration, m/s2

h mesh size of velocity–pressure gridk mass transfer coefficient, m/sL characteristic length(L= d), mm distribution coefficientn unit vector normal to a surfacen normal coordinate, m

p dimensionless pressureP pressure, PaPe Peclet number,LV /DPe∗ predicted real Peclet number,dU/Dr radial coordinate, mRe Reynolds number,�1LV /�1Re∗ predicted real Reynolds number,�1dU/�1S surface area of drop, m2

Sh Sherwood number,dkod/D

t time, s

u velocity vector, m/su transformation form of velocity vector, m/su axial velocity component, m/sU terminal velocity, m/sv radial or transverse velocity component, m/sV reference velocity

√2Rg, m/s

Vd volume of drop, m3

We Weber number,�1V2L/�

We∗ predicted real Weber number,�1U2d/�

X spatial position vector(x, y), mx axial coordinate, my radial or transverse coordinate, m

Greek letters

� surface area of drop� “thickness” of interface� dimensionless time,tV /L� Gaussian curvature� viscosity, Pa s� density, kg/m3

� surface tension, N/m virtual time� level set function

Subscripts

1 continuous phase2 dropi interfacein first measurement locationod overallout second measurement locationx axial directiony radial or transverse direction� surface of drop∞ remote boundary

Superscript

0 initial time

Acknowledgements

The financial support from the National Natural ScienceFoundation of China (Nos. 20236050, 20106016, 20490206)is gratefully acknowledged. Authors wish to thank Prof. Ji-ayong Chen in our institute, for his valuable advice and con-tinuous encouragement.

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numerical simulation of interphase mass transfer with the level set approach

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