Chemical Engineering Science 63 (2008) 3141 -- 3151

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Chemical Engineering Science

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Numerical simulation of unsteadymass transfer by the level setmethod

Jianfeng Wanga,b, Ping Lua, Zhihui Wanga, Chao Yanga,c, Zai-Sha Maoa,∗aKey Laboratory of Green Process and Engineering, Institute of Process Engineering, Chinese Academy of Science, Zhongguancun Beíertiao 1, P.O. Box 353, Beijing 100080, ChinabDepartment of Chemical Engineering, Zhengzhou University, Zhengzhou 450052, ChinacJiangsu Institute of Marine Resource Exploitation, Lianyungang 222005, China

A R T I C L E I N F O A B S T R A C T

Article history:Received 26 December 2006Received in revised form 21 February 2008Accepted 12 March 2008Available online 18 March 2008

Keywords:Mass transferDropNumerical simulationExtractionLevel set approach

Unsteady mass transfer to/from a single drop in the continuous phase is formulated and numericallysimulated in a moving reference coordinate system by solving the motion and mass transfer equations ofan accelerating drop coupled with a level set equation for capturing the interface. Numerical simulationdemonstrates the evolution of mass transfer rate and average drop concentration. Numerical simula-tion of the flow field and the concentration field simultaneously in each time step is compared withexperimental data on single drop motion and mass transfer in two typical solvent extraction systems.The numerical predictions are found in good accord with the experimental measurements. The presentnumerical procedure in which the flow field is solved in a coupled way with the concentration field givesmore accurate prediction than the previous decoupling algorithm by the authors.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Mass transfer of a solute to/from a buoyancy driven drop is offundamental importance in liquid--liquid extraction and reaction. Ingeneral, advection and diffusion in both liquid phases show greatinfluences on the rate of interphase mass transfer. Since the dropdeforms and the mobility of interface varies along with mass trans-fer, the analysis and prediction of the mass transfer must be per-formed by considering the coupling with complex structures of fluidflow. In addition to experimental studies, many numerical inves-tigations have been performed, mostly devoted to studying singlespherical drops under steady-state motion. Petera and Weatherley(2001) simulated mass transfer from a falling drop by a modifiedLagrange--Galerkin finite element method, but a remeshing proce-dure had to be implemented to locate the new position of the in-terface. Mao et al. (2001) (Li et al., 2001, 2002, 2003a, b) simulatedsteady and transient mass transfer to/from single drops using a body-fitted orthogonal coordinate system proposed by Ryskin and Leal(1983). However, it is very difficult to construct the orthogonal curvi-linear coordinates for complicated and seriously deformed interfacesand extend the mapping procedure to three-dimensional drops withdeformation. Yang andMao (2005) later developed a level setmethodfor calculating the interphase mass transfer to/from single drops inan immiscible liquid with resistance in both phases. Their predicted

∗ Corresponding author. Tel.: +861062554558, 62573446;fax: +861062561822.

E-mail addresses: chaoyang@home.ipe.ac.cn (C. Yang), zsmao@home.ipe.ac.cn(Z.-S. Mao).

0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2008.03.018

results were in good agreement with the experimental data. Theirmethod is easy to be extended to a three-dimensional space, andstraightforward to other more challenging interphase mass transferto/from bubbles and drops with complicated and seriously de-formed free liquid--liquid or gas--liquid interfaces. But the masstransport equations therein are decoupled with the momentumtransport equations and the mass transfer in the acceleratingstage of rising/falling drops were not accounted for. Davidson andRudman (2002) developed a volume-of-fluid (VOF) method forcalculating mass transfer or heat transfer of a species within andbetween fluids with deforming interfaces. Davidson and Stevens(2005) also simulated mass transfer between a single drop and thesurrounding liquid phase in a spray column and compared with arange of mass transfer experimental data from Temos et al. (1996).Deshpande and Zimmerman (2006) simulated mass transfer acrossa moving droplet by adopting a two stage approach where theconvection--diffusion equations for mass transfer were coupled withthe governing equations of the level set method. They can infermass transfer coefficients without using any empirical correlations,but their simulation was not compared with the experimental data.Although unsteady-state effects can also play a very significant role,especially at the initial stage of the process, to date few numericalinvestigations were devoted to the case of a deforming drop underunsteady-state motion.

The main difficulty in numerical simulation is that the motion of adeformed drop with simultaneous mass transfer must be solved withthe unknown shape of free surface (Petera and Weatherley, 2001).Numerical methodmust ensure accurate estimation of the interfacial

3142 J. Wang et al. / Chemical Engineering Science 63 (2008) 3141 -- 3151

position and the concentration there so that the mass transfer cal-culation remains accurate. In recent years, several different methodsfor solving multiphase flow have been developed, e.g., the markerand cell (MAC) method (Harlow andWelch, 1965), VOF method (Hirtand Nichols, 1981), front tracking method (Unverdi and Tryggvason,1992), boundary integral method (Baker and Moore, 1989), etc. SinceOsher and Sethian (1988) introduced the level set method for mod-eling fronts, this method has been applied in many fields with theparticular advantages in easy and accurate capture of interface. Bythis method, the topology of the interface and its change are fullydescribed by the zero set of the level set function, which is initiallydefined in the whole flow field and is advanced along with the fluidflow. The method is stable, surface geometric parameters such ascurvature become easy to be calculated and three-dimensional cal-culation problems become easy to be implemented. The major short-coming of the level set approach is less accurate treatment of inter-face compared with the boundary-fitted coordinates method.

Based on Yang and Mao's (2005) work, the present study is aimedto simulate the mass transfer considering the initial rising/fallingstage of a drop, which is more close to the real situation. In our studythe drop interface evolution is modeled by the level set method, inwhich the interface is represented by the embedded set of the zerolevel of a scalar function defined in the whole computational domain.Numerical simulation of a single drop in a liquid--liquid extractionsystem is performed in an axisymmetric cylindrical reference framebased on the drop. The motion of a drop with a finite degree of defor-mation is coupled with simultaneous mass transfer for consideringthe effect of unsteady-state effects. The results are compared withthe present experimental data.

2. Mathematical formulation

In this study, we consider the axisymmetric flow of a liquid dropmoving under gravity through an immiscible liquid with the fol-lowing assumptions: (1) the two fluids are viscous, Newtonian andincompressible, (2) the flow is isothermal, (3) the interface tensionis taken as constant and there is no interfacial resistance to masstransfer, and (4) mass transfer is assumed to have no effect on thephysical properties of the system.

2.1. Governing equations

In a two-dimensional coordinate system, mass and momentumconservation with the level set approach incorporated are written interms of dimensionless variables as

�u

�x+ 1

r

�

�y(rv) = 0 (1)

�

��(�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

)

+ 1Re

1r

�

�y

(r�

�v

�x

)(2)

�

��(�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

)

+ 1Re

1r

�

�y

(r�

�v

�y

)−{

2Re

�v

r2

}(3)

where r ≡ 1 for Cartesian coordinates, r ≡ y for cylindrical coordi-nates and curly brackets indicate the term presents only in cylindri-cal coordinates.

The level set function � is introduced into the formulation ofmultiphase flow and mass transfer systems to define and capture theinterface between two fluids. The interface is defined as the zero levelset of the level set function � which is defined as the signed algebraicdistance of a node to the interface, being positive in the continuousfluid phase and negative in the drop. The following equation is usedto advance the level set function exactly as the drop moves:

��

��+ �

�x(u�) + 1

r

�

�y(rv�) = 0. (4)

�(�) is the curvature of drop surface defined as

�(�) = ∇ · n = ∇ ·( ∇�

|∇�|)

(5)

where n is the unit vector normal to the interface pointing towardsthe continuous phase and ��(�) the regularized delta function definedas

��(�) =⎧⎨⎩

12�

(1 + cos(�/�)) if |�| < �

0 otherwise(6)

where � prescribes the finite "half thickness'' of the interface. Wetake �=1.5�x, where �x is the dimensionless uniformmesh size nearthe interface.

H�(�) is the regularized Heaviside function expressed as

H�(�) =

⎧⎪⎪⎨⎪⎪⎩0 if � < − �12

(1 + �

�+ sin(�/�)/

)if |�|� �

1 if � > �

(7)

for defining the corresponding regularized (smoothed) density func-tion � and the regularized viscosity � as

��(�) = �2/�1 + (1 − �2/�1)H�(�) (8)

��(�) = �2/�1 + (1 − �2/�1)H�(�) (9)

The subscripts 1 and 2 denote the continuous phase and drop, re-spectively.

The dimensionless parameters in the above equations are theReynolds, Froude, Weber, and Peclet numbers, respectively,

Re = �12RV

�1, Fr = V2

2Rg, We = 2R�1V2

(10)

where �1, �1 are the density, viscosity of continuous phase. R is theinitial radius of a spherical drop and the characteristic velocity isdefined as V =√

2Rg.

2.2. Reinitialization of � as a distance function

Generally, � will no longer be a distance function (i.e., |∇�| �= 1)after some iterations, even if Eq. (4) moves the interface (� = 0) atthe correct velocity. Maintaining � as a distance function is essentialfor providing the interface with an invariant width and a sound basisfor estimating the surface curvature. This problem can be solved byadopting a reinitialization method proposed by Sussman et al. (1994)to solve the following initial problem to steady state:

��

��= sgn(�0)(1 − |∇�|) (11)

�(x,0) = �0(x) (12)

where � is the virtual time for reinitialization, �0(x) the level setfunction at any computational instant, and sgn(�0) the sign function

J. Wang et al. / Chemical Engineering Science 63 (2008) 3141 -- 3151 3143

needed for enforcing ∇�=1. Eq. (11) has the property that � remainsunchanged at the interface, therefore, the zero level set of �0 and� is the same. Away from the interface � will converge to |∇�| = 1,i.e., the actual distance function. In this paper, the area-preservingreinitialization procedure by Yang and Mao (2002) for � was coupledwith Eq. (11) to guarantee the mass conservation by solving a per-turbed Hamilton--Jacobi equation proposed by Zhang et al. (1998) topseudo-steady state in each time step. The improved reinitializationprocedure can maintain the level set function as a distance functionand guarantee the drop mass conservation.

2.3. Formulation of mass transfer

2.3.1. Transform of mass transfer equationThe quantitative simulation of interphase mass transfer is turned

out to be especially challenging, due to different diffusivities in twoliquid phases and the distribution coefficient (m) of a solute not equalto unity (the latter leads to the solute concentration discontinuousacross the interface, and the former results in the discontinuity ofconcentration gradient). Special care must be taken for resolvingthese discontinuities in the numerical simulation. For each phase themass transfer equation is as follows:

�C1�t

+ u · ∇C1 = D1∇2C1 (13)

�C2�t

+ u · ∇C2 = D2∇2C2 (14)

subject to two interfacial conditions:

D1�C1�n1

= D2�C2�n2

(flux continuity at the interface) (15)

C2 = mC1 (interfacial dissolution equilibrium). (16)

In order to keep the concentration continuous at the interfacefor the case of m �= 1, the variable transformations in Yang and Mao(2005) are adopted:

C1 = C1√

m, C2 = C2/√

m (17)

and Eq. (16) is turned into

C1 = C2. (18)

Thus, the concentration becomes continuous in the computationaldomain. Similarly, the boundary condition (15) becomes

D1√m

�C1�n1

= √mD2

�C2�n2

. (19)

Eqs. (13) and (14) are thus rewritten as

�C1�(

√mt)

+ 1√m

u · ∇C1 = D1√m

∇2C1 (20)

�C2�(t/

√m)

+ √mu · ∇C2 = √

mD2∇2C2. (21)

With the definitions of t, D, and u by the Heaviside function H�(�):

t(�) ={

t/√

m if � <0√mt if ��0

(22)

D(�) = √mD2 +

(1√m

D1 − √mD2

)H�(�) (23)

u(�) = √mu +

(1√m

u − √mu

)H�(�). (24)

Eqs. (20) and (21) can be rewritten as a unified equation over thewhole domain:

�C

�t+ u · ∇C = ∇ · (D∇C). (25)

By our numerical experience, the transformation has no adverse ef-fect on the numerical solutions of interphase mass transfer. Theabove transform method of mass transfer equations was also vali-dated by Kashid et al. (2007).

The equation of mass transport is similarly non-dimensionalizedto

�C

��+ u

�C

�x+ v

�C

�y= 1

Pe

(�2C

�x2+ 1

r

�

�y

(r�C

�y

))(26)

where the dimensionless group Pe is the Peclet number, C is thedimensionless concentration based on the reference concentration,using C1,∞ for solute transportation from the continuous phase tothe drop, but C2,0 for the inverse direction of solute transfer.

2.3.2. Mass transfer coefficientThe overall mass transfer coefficient kod may be calculated from

the simulated results of solute concentration with

kod = −Vd

S

1tout

lnC∗2 − C2,out

C∗2

(27)

so as to be compared easily with the measurements. In above, C2 isthe average concentration of the drop at any time in the computation.Vd and S are the volume and surface area of the drop, and for aspherical drop Vd/S = d/6. The corresponding Sherwood number is

Shod = d

D1kod. (28)

3. Computational scheme

3.1. Numerical solution method

Yang and Mao (2005) simulated the interphase mass transfer af-ter the steady state of flow was reached by the level set method, inwhich the mass transfer equation was decoupled from the fluid flow.However, in practical situations the mass transfer to/from a drop oc-curs normally before the motion of the drop approaches steady state.The algorithm in which the mass transfer equation is coupled withthe fluid flow is necessary. In this work, the governing equations aresolved in the reference coordinate system fixed on the drop. The ad-vantage of doing this is the assurance of the concentration field be-ing accurately resolved, which is the prerequisite for accurate simu-lation of mass transfer. The acceleration of the drop must be formu-lated in the governing equations in the moving reference coordinatesystem. Eq. (2) in the axial direction is now transformed to

�

��(�u) + �

�x

(�uu − �

Re�u

�x

)+ 1

r

�

�y

(r�vu − r

�

Re�u

�y

)

= −�p

�x+ 1

Fr�gx − �ra − 1

We�(�)��(�)

��

�x+ 1

Re�

�x

(�

�u

�x

)

+ 1Re

1r

�

�y

(r�

�v

�x

)(29)

which is subject to the upstream and downstream boundary condi-tions of

u = −U. (30)

3144 J. Wang et al. / Chemical Engineering Science 63 (2008) 3141 -- 3151

Here �r = �/�1, a is the acceleration of the drop and calculated ineach time step by

a = Un+1 − Un

��(31)

where Un+1 is the axial average velocity of the drop at �n+1 in thecoordinate system fixed on the earth, Un that at time �n, and �� thedimensionless time step.

In the numerical simulation of mass transfer from/to the drop,the effective velocity in mass transfer equation (26) should be thevelocity relative to the drop fixed in the reference coordinate system,namely, the velocity solved by Eq. (29). After the integration in eachtime step has converged, the drop velocity U would be updated byan increment of �U due to drop acceleration so that

Un+1 = Un + �U (32)

with the average velocity increment of the drop after the time stepevaluated as

�U =∫��0 un+1 d�∫

��0 d�(33)

where � is the surface of the drop at a current instant. Then, thewhole field of u is subtracted by �U to let the drop remain station-ary as required by the adoption of the moving coordinate systemattached on the drop. We also attempted to calculate the unsteadymass transfer coupled with the flow equations according to Eq. (25)in the fixed reference coordinate system, but the approach seemednot successful. The alternative is to solve the governing equations inthe moving reference coordinate system fixed on the drop and it isadopted in this work.

The governing equations are discretized according to the controlvolume formulation with the power-law scheme as described byPatankar (1980). To ensure the difference accuracy, we use a fifth-order WENO method (Fedkiw et al., 1999) for the approximation ofthe convective terms of the mass transfer and � equations, a third-order TVD Runge--Kutta scheme (Chen et al., 1997) is adopted forthe time discretization of the WENO scheme.

3.2. Numerical procedure

The algorithm is summarized asStep 1: Initialize the flow field (u, v, and p), physical parameters

(�, �, , D, and C in each phase) and � as signed normal distance tothe interface.

Step 2: Solve Eqs. (3), (29) and (4) for one time step.Step 3: Solve the concentration field.Step 4: Construct a new distance function by solving Eqs. (12) and

(13) to steady state, and the steady-state solution is denoted as �n+1.Step 5: Advancing one time step. The zero level set of �n+1 gives

the new interface position. Renewing physical properties accordingto Eqs. (8) and (9), repeat steps 2, 3, and 4.

Special attention is taken to see that in the moving referencefixed on the drop, the upstream and downstream boundary condi-tions must be updated by Eq. (30) at every time step to keep thedrop stationary in the moving coordinate system. To satisfy the con-ditions is the most CPU cost part in the present numerical work. Therelaxation factors need to be adjusted for different physical param-eters to meet the correct boundary condition.

4. Experimental

Direct comparison between the experimental and the predictedmass transfer coefficient or concentration is very difficult for lack

Fig. 1. Sketch of the experimental setup 1, 2. dispersed phase tanks; 3. precisionmetering pump; 4. nozzle for dispersed phase; 5. moving drop; 6. extractor; 7.funnel; 8. precision adjusting valve; 9,10. continuous phase tanks.

of necessary parameters, especially at unsteady state. So the experi-ments of unsteady mass transfer were conducted based as suggestedin Li et al. (2001). A typical extraction system consisting of n-butanoland water with succinic acid as solute recommended by EFCE (theEuropean Confederation of Chemical Engineering) is adopted. Thepartition coefficient of succinic acid in n-butanol and water is 1.17,so the mass transfer resistance in both liquid phases is not negligible.

4.1. Experimental setup

The contactor was a glass pipe with 50mm internal diameter and700mm length comprising five taps with equal diameter beside thepipe for fitting the injection nozzles at different heights. As shownin Fig. 1, the equipment is basically the same as that of Li et al.(2001). The solvent drops were formed at needle tips at a constantrate which was controlled by a precision pump, and the drops areseparated by at least the distance of 40mm. Knowing the number ofdrops, the volume flow rate of the solvent and the time, the volume-equivalent drop size could be calculated. The drops were trapped ina small inverted funnel which just fitted on a standard glass tap holeto keep the funnel at the same horizontal line at the right heightfor collecting the drops. Drops were sucked into the neck of thefunnel during collection to minimize further mass transfer. In theinitial experiments the coalesced phase was removed by suction tomaintain a minimal interfacial area just below the neck of the funnel.

Four systems are investigated in experiment as listed in Table 1.Misek (1978) gave fairly complete data of the system of wa-ter/succinic acid/butanol. The physical properties of the four cases,such as the density, viscosity, the diffusivity of the continuous phaseand the dispersed phase, were obtained by a nonlinear regression.The partition data and the surface tension of these systems werecalculated by linear interpolation from the values of Misek (1978).The corresponding physical properties of the four cases are listed inTable 2.

J. Wang et al. / Chemical Engineering Science 63 (2008) 3141 -- 3151 3145

Table 1Experimental conditions

Case System continuous phase/solute/dispersed phase

Direction oftransfer

Direction ofdrop motion

1A Water/succinic acid/butanol c → d Rising1B Water/succinic acid/butanol d → c Rising2A Butanol/succinic acid/water c → d Falling2B Butanol/succinic acid/water d → c Falling

Table 2Physical property data and drop radius

System 1A 1B 2A 2B

�b (kgm−3) 855.78 864.600 994.0 996.1�c (kgm−3) 991.44 995.37 861.5 866.2�b × 103 (Pa s) 2.70 2.81 1.158 1.185�c × 103 (Pa s) 1.13 1.18 2.772 2.837Db × 1010 (m2 s−1) 3.24 2.81 7.4 7.2Dc × 1010 (m2 s−1) 7.58 7.27 3.2 3.1cb,0 (wt%) 0 5.17 0 4.79cc,0 (wt%) 2.92 0 4.49 0 × 103 (Nm−1) 1.12 1.0 0.9 1.0m 1.16 1.15 0.88 0.88R (mm) 0.52 0.64 0.53 0.59

Table 3The measurements of transient time t and solute concentration Cd at differentdistance h traveled by drops

Case h (cm) t (s) kod × 105

(m/s)Cd (%) Error of

Cd (%)Error oft (%)

Error ofkod (%)

1A 8 1.99 4.57 1.38 3.57 5.00 2.0012 3.23 3.23 1.53 8.80 2.73 3.6716 4.32 2.90 1.74 0.57 0.92 8.8620 7.24 1.99 1.91 2.06 0.83 5.05

1B 8 2.03 8.42 2.32 5.8 1.46 6.8512 3.29 6.28 1.96 4.0 2.69 3.7116 4.57 4.95 1.79 8.6 3.66 2.7320 7.09 3.91 1.41 1.41 5.00 2.00

2A 8 3.12 3.81 1.94 1.54 6.25 1.6012 4.23 3.02 2.04 0.98 5.4 1.8516 6.22 2.44 2.28 2.16 6.23 1.6120 7.57 2.10 2.35 2.1 0.61 1.6424 9.33 1.90 2.51 2.36 1.48 6.85

2B 8 2.31 3.23 3.28 3.29 0.86 4.9012 3.37 2.81 2.96 3.64 2.94 3.2816 5.13 2.59 2.44 2.83 5.66 1.3520 6.58 2.54 2.05 1.92 0.45 8.6824 8.12 2.36 1.81 3.28 0.50 7.11

4.2. Preparation for dispersed phase and continuous phase

The dispersed phase of 1000mL was mixed with 5000mL of con-tinuous phase, after being fully shaken. The two phases became sat-urated with each other after a few days. An appropriate amount ofthe solute was dissolved separately into the dispersed phase or thecontinuous phase so that the mass transfer experiment with differ-ent transfer direction can be conducted.

The concentration of the solute in the dispersed phase or con-tinuous phase was analyzed, generally in triplicate, by titration with0.1M NaOH using phenolphthalein as indicator.

4.3. Procedure

For rising drops, the funnel was fixed on the top of the extractioncolumn; the nozzle tip was inserted into the center of the extractioncolumn according to the desired height from the side tap of thecolumn. Then the column was filled with continuous phase. After

Fig. 2. Predicted and experimental concentrations and overall mass transfer coef-ficients at various times in case 1A: (a) overall mass transfer coefficient and (b)average drop concentration.

20min for reaching a steady state, the precision pump was thenstarted to produce the drop. If the interval between two drops is asfar as 20 times the diameter of a drop, a drop will not be influencedby the wake of the previous droplet. In general, the interval betweenthe drops is about 40--50mm, which is sufficiently large with respectto the maximum drop diameter of 1.28mm in this work. So thedroplets are believed not under the influence of the plume of thepreceding ones.

The temperature in all runs varied in the range of 25.28 ◦C.The drops were removed by siphon. The dispersed phase about

5mL was collected for compositional analysis. For the case of fallingdrops, the procedure was similar to that for the rising drops exceptthe position of the funnel was placed at the column bottom.

4.4. Analysis of the experimental data

For the mass transfer to/from a single drop with resistance inboth phases, the overall mass transfer coefficient kod from the ex-perimental measurements is calculated by the same Eq. (27) usingthe drop concentration measurements.

3146 J. Wang et al. / Chemical Engineering Science 63 (2008) 3141 -- 3151

Fig. 3. Predicted and experimental concentrations and overall mass transfer coef-ficients at various times in Case 1B: (a) overall mass transfer coefficient and (b)average drop concentration.

5. Results and discussions

5.1. Experimental results

All tests are focused on the effect of the initial stage of the ris-ing/falling drop on the mass transfer coefficient. Experiment in ev-ery system was repeated thrice for each value of the distance travelby drops, h. The average concentration of the collected drop and av-erage overall mass transfer coefficient together with their relativeerrors of the measured time and concentration of collected drop arelisted in Table 3. The largest relative error of 8.9% over all data isthought at the acceptable level. These data of kod will be comparedwith the present numerical simulation later in Figs. 2--5.

5.2. Numerical simulation

The single drops in the present experiment are consid-ered as buoyancy-driven ones surrounded by an incompress-ible Newtonian fluid bounded in a wide cylindrical column (� ={(x, y)|0�x�40R,0�y�15R} for half a drop) to validate the numer-ical algorithm for interphase mass transfer. The computational do-main is large enough for eliminating the boundary effect but smallerthan the practical domain (� = {(x, y)|0�x�1346R,0�y�96R})

Fig. 4. Predicted and experimental concentrations and overall mass transfer coef-ficients at various times in Case 2A: (a) overall mass transfer coefficient and (b)average drop concentration.

for saving the computer time. The free-slip boundary condition isadopted at the column wall.

Non-uniform grids of 109 × 46, 146 × 61, 188 × 76, 279 × 116(the last is referred to as the fine one) were tested for comparingthe solution of the two-phase flow and mass transfer as a functionof time to ensure mesh independence of numerical results. Fig. 6presents the influence of mesh size on the overall mass transfercoefficients, showing the mass transfer coefficient decreases grad-ually to the asymptotic value (kod = 2.30 × 10−5 m/s). A grid with182×76 nodes is considered sufficient for spatial computational ac-curacy, and adopted for the subsequent simulations. In the presentpaper, unsteady mass transfer to/from a single drop is simulated in amoving reference coordinate system by solving the motion and masstransfer equations of an accelerating drop coupled with a level setequation for capturing the interface. In every time step, the far fieldboundary conditions should be adjusted to assure the velocity of thedrop is equal to zero by iteration, which may produces some error inthe resultant drop velocity. We also compared the velocity of dropin the fixed reference (no fluctuations being observed) with that inthe moving reference coordinate system. It shows that the velocities

J. Wang et al. / Chemical Engineering Science 63 (2008) 3141 -- 3151 3147

Fig. 5. Predicted and experimental concentrations and overall mass transfer coef-ficients at various times in Case 2B: (a) overall mass transfer coefficient and (b)average drop concentration.

Fig. 6. Grid independence test: overall mass transfer coefficient versus time(Case 2B).

Fig. 7. The drop velocity versus time in Case 2.

Table 4Comparison of the predicted and experimental rising height of the drop at unsteadymass transfer stage for Case 2A

t (s) Predicted h (cm) Measurement location h (cm) Relative error (%)

3.12 7.76 8.00 −3.004.23 10.90 12.00 −9.176.22 15.70 16.00 −1.887.57 19.20 20.00 −4.009.33 23.70 24.00 −1.25

become the same in the two sets of the coordinate systems after thevery short initial stage. Although in the first moments of the sim-ulation small fluctuations arise in the moving reference coordinatesystem, the small fluctuations don't influence the simulation resultsfor the latter period of interphase mass transfer. Fig. 6 shows thatthe fluctuations arise only in the first 0.34 s, during which the con-centration of the drop is difficult to be measured experimentally. Sothe fluctuations can be omitted safely.

In Fig. 7, the rising velocity of a drop in Case 2B is presented,which shows that the accelerating stage is equal to about 1 s andit coincides roughly with the period with velocity fluctuations ob-served. It is apparent that the acceleration lasts only a short timeand it gives little influence on later experimental measurements ofsolute concentration and mass transfer coefficient.

From the simulation starting from a just released drop at theneedle tip with zero rising velocity, the time instant for the drop topass the height h where the drops were collected can be obtainedand shown in Table 4 for drop 2A. Along with it, the flow fieldand the concentration field were also acquired, from the latter theinterphasemass transfer coefficient kod were calculated. As shown inTable 4, the predicted rising height h of drop 2A at the unsteady masstransfer stage for case 2A was compared with the experimental data.The predicted height is in good agreement with the experimentaldata. The mass transfer mechanism involved in the drop formationstage is very complicated and it combines simultaneous effects ofinterface creation, renewal, interfacial convection, etc. The effect ofmass transfer at the drop formation stage is not considered in thepresent simulation. That might be part of the reason for the averagerelative error of 3.9% in Table 4.

For the present simulation, the concentration Cd=0 is assumed forthe instant of t=0 when the drop starts to rise or fall for the directionof mass transfer from continuous phase to drop and Cd = C0

dfor the

direction of mass transfer from drop to continuous phase. Figs. 2--5

3148 J. Wang et al. / Chemical Engineering Science 63 (2008) 3141 -- 3151

Fig. 8. Predicted fractional solute concentration distribution and velocity vector field relative to the motion of the drop in Case 1A at various dimensionless times: (a)t = 0.02 s (vd = 1.18 cm/s), (b) t = 0.04 s, (vd = 1.65 cm/s) concentration, (c) t = 0.13 s (vd = 2.33 cm/s), and (d) t = 0.49 s (vd = 2.47 cm/s).

show the evolution of average drop concentration and overall masstransfer coefficients for mass transfer direction from the continuousphase to the drop or in the inverse direction. The predictions are ingood agreement with the corresponding experimental data.

Fig. 8 shows the predicted solute distribution in both phases andthe velocity vector field relative to the drop motion at selected di-mensionless times for Case 1A. (Note that only a section of the com-putational domain around the drop is shown). The results show theshape of the drop was gradually deformed into a spheroid fromits initial spherical shape, and a toroidal recirculation was formedwithin the drop. In the center part of the drop the concentrationof the solute progressively increased as it was swept to the down-stream direction, suggesting that convective mass transfer did playan important role. The convection at the drop nose promoted thetransport of solute to the drop.

Fig. 9 shows the predicted solute distribution in both phases andthe velocity vector field relative to the drop motion at different di-mensionless times for Case 2B (the direction of mass transfer: d →c). The results show that in the center part of the drop the soluteconcentration gradually decreased as it was swept by the recircula-tion towards the drop surface and then towards the rear of the drop

where it entered the continuous phase as a plume. The plume wasmore distinct at early time when the concentration of the drop washighest.

5.3. Discussion

In the present work, the effect of the formation stage of the dropon the mass transfer is not considered in the simulation. The testis performed on the effect of the initial extraction fraction of thedrop being assigned 5% as the initial condition in the simulation. Theresults with and without the estimated initial extraction fraction arecompared in Fig. 10. The effect of mass transfer in the drop formationstage seems to have little effect on the average concentration ofthe drop. The predicted overall transfer coefficients kod with theconsideration of the drop formation stage are in better agreementwith the experimental results comparing with those neglecting theeffect of the formation of the drop especially for the initial stage of themass transfer. But this effect seems not very significant in the laterstage. For simplicity, the mass transfer in the stage of formation ofthe drop is not considered when the accelerating and steady motionof single drops is the major concern.

J. Wang et al. / Chemical Engineering Science 63 (2008) 3141 -- 3151 3149

Fig. 9. Predicted fractional solute concentration distribution and velocity vector field relative to the motion of the drop in Case 2B at various dimensionless times: (a)t = 0.04 s (vd = 0.76 cm/s), (b) t = 0.14 s (vd = 1.75 cm/s) concentration, (c) t = 1.39 s (vd = 2.76 cm/s), and (d) t = 2.10 s (vd = 2.78 cm/s).

The present algorithm was compared with Yang and Mao (2005).As listed in Table 5, the relative deviation between the predictedand experimental by our algorithm is far less than those by Yangand Mao (2005). The maximum deviation of Yang and Mao's (2005)work is 57.2%, but in this study 25.5%. The minimum error in Yangand Mao's (2005) work is almost equal to the maximum error inthe present study, even a little larger. Therefore, the predicted inter-phase mass transfer results by the present algorithm considering theinitial accelerating stage of the drop coincide better with the realis-tic situation. Although the acceleration lasts only a short time and itgives some influence on later experimental measurements of soluteconcentration and mass transfer coefficient, the effect of the initialaccelerating stage of the drop on the mass transfer in the simulationshouldn't be omitted. It should be noted that the algorithm that themass transfer equation is solved by coupling with the momentumequation in the present study is a further development from Yangand Mao (2005), in which the mass transfer equation is decoupledwith the momentum equation.

We have also simulated one case (No. 1 in their Table 3) of Peteraand Weatherley (2001) to test our algorithm. The flow structureand the concentration field are in qualitative agreement with Peteraand Weatherley's results. At 13.0 s, our predicted terminal velocity is5.98 cm/s, 10.3% higher than the experimental result in their paper

and the flux is 3.39 (in the same unit), about 13% lower than Peteraand Weatherley's prediction. Considering quite obvious difference inrespective works, the agreement is thought reasonable.

6. Conclusions and future work

A new level set method for numerical simulation of the masstransfer from/to a drop considering the initial unsteady acceleratingstage of a falling/rising drop is developed. The mass transfer equa-tions coupled with two phase fluid flow equations are solved nu-merically in a moving reference coordinate fixed on the drop forthe case with resistance in both drop and continuous phase and thedistribution coefficient differing from unity. Although in the mov-ing reference coordinates fixed on the drop, the CPU cost increasesa little, it is worthy to consider the drop accelerating stage to get amore realistic solution. The results are in better agreement with theexperimental data than that by the method of simulating only thestage of steady motion of drops. The present study provides neces-sary basis for further work on three-dimensional numerical analysisof the Marangoni effect of a drop induced by surface tension gradientcaused by interphase mass transfer. The simulation of the Marangonieffect of a drop considering the initial stage of a falling/rising dropbased on the present algorithm is currently under way.

3150 J. Wang et al. / Chemical Engineering Science 63 (2008) 3141 -- 3151

Fig. 10. Predicted and experimental concentrations and overall mass transfer coef-ficients at various times in Case 2B: (a) overall mass transfer coefficient and (b)average drop concentration.

Table 5Comparison of the relative deviations of predicted and experimental average overall mass transfer coefficients between Yang and Mao (2005) and the present study

Method Case Experimental average Predicted average Deviationkod × 105 (m/s) kod × 105 (m/s) (%)

Yang and Mao (2005) BC4-1 0.449 0.709 36.67BC9-1 0.589 1.126 47.69BC12-1 0.897 1.400 35.93BC16-1 2.566 1.900 25.95BD4-1 0.374 0.728 48.62BD9-1 0.752 1.275 41.01BD12-1 0.661 1.543 57.16BD16-1 1.108 1.739 36.29

The present simulation 1A 4.57 2.16 16.413.23 2.75 11.992.90 3.67 5.171.99 3.82 7.87

1B 8.42 6.34 24.206.28 4.93 21.504.95 4.14 16.193.91 3.21 17.90

2A 3.81 3.57 6.303.02 3.05 0.982.44 2.57 5.062.10 2.35 10.641.90 2.17 12.44

2B 3.23 4.31 25.532.81 3.51 19.942.59 2.80 7.52.54 2.53 0.392.36 2.31 2.12

Note: The experimental run numbers of BC4-1, BC9-1, BC12-1, BC16-1, BD4-1, BD9-1, BD12-1, BD16-1 are the same as in Yang and Mao (2005).

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%Cd average concentration of solute in the drop, wt%d diameter of drop, mD molecular diffusivity, m2/sD transformation of solute molecular diffusivity, m2/sFr Froude number, V2/(2Rg)

g acceleration vector of gravity, m/s2

h distance travel by drops, mk mass transfer coefficient, m/sp dimensionless pressurePe Peclet number, Pe = 2RU/D1r dimensionless radial coordinateR volume equivalent drop radius, mRe Reynolds number, Re = �12RV/�1S surface area of drop, m2

Sh Sherwood number, tkod/D1t time, st transformed time, su velocity vector, m/su dimensionless axial velocity non-dimensionalized

with V

u transformation form of velocity vector, m/sU dimensionless velocity of the drop in the refer-

ence coordinate system fixed on the earth (non-dimensionalized with V)

v dimensionless radial velocity non-dimensionalizedwith V

V characteristic velocity, V =√2gR, m/s

Vd volume of drop, m3

J. Wang et al. / Chemical Engineering Science 63 (2008) 3141 -- 3151 3151

We Weber number, We = 2R�1V2/x dimensionless axial coordinatey dimensionless radial coordinate

Greek letters

�x mesh size of velocity--pressure grid� "thickness'' of interface� dimensionless time, tV/(2R)

� Gaussian curvature� viscosity, Pa s�� dimensionless viscosity� density, kg/m3

�r relative density �i/�1�� dimensionless density surface tension, N/m� virtual time� level set function

Subscripts

in first measurement locationn time stepod overallout second measurement locationr relative0 initial time1 continuous phase2 dispersed phase∞ remote boundaryx axial directiony radial or transverse direction

Acknowledgments

Financial support from the National Natural Science Foundationof China (Nos. 20490206, 20236050, 20576133), PetroChina andthe National Basic Research Program of China (Nos. 2004CB217604,2004CB619203) is gratefully acknowledged.

References

Baker, G.R., Moore, D.W., 1989. The rise and distortion of a two-dimensional gasbubble in an inviscid liquid. Physics of Fluids A 1, 1451--1459.

Chen, S., Merriman, B., Osher, S., Smereka, P.J., 1997. A simple level set method forsolving Stefan problems. Journal of Computational Physics 135, 8--29.

Davidson, M.R., Rudman, M., 2002. Volume-of-fluid calculation of heat or masstransfer across deforming interfaces in two-fluid flow. Numerical Heat TransferPart B---Fundamentals 41, 291--308.

Davidson, M.R., Stevens G.W., 2005. Paper presented at International SolventExtraction Conference (ISEC'2005), September, Beijing, China.

Deshpande, K.B., Zimmerman, W.B., 2006. Simulation of interfacial mass transfer bydroplet dynamics using the level set method. Chemical Engineering Science 61,6486--6498.

Fedkiw, R., Aslam, T., Merriman, B., Osher, S., 1999. A non-oscillatory Eulerianapproach to interfaces in multimaterial flows (the ghost fluid method). Journalof Computational Physics 152, 457--492.

Harlow, F.H., Welch, J.E., 1965. Numerical calculation of time-dependent viscousincompressible flow of fluid with free surface. Physics of Fluids 8, 2182--2189.

Hirt, C.W., Nichols, B.D., 1981. Volume of fluid (VOF) method for dynamics of freeboundaries. Journal of Computational Physics 39, 201--255.

Kashid, M.N., Agar, D.W., Turek, S., 2007. CFD modelling of mass transfer with andwithout chemical reaction in the liquid--liquid slug flow microreactor. ChemicalEngineering Science 62, 5102--5109.

Li, T.W., Mao, Z.-S., Chen, J.Y., 2001. Terminal effect of drop coalescence on singledrop mass transfer measurements and its minimization. Chinese Journal ofChemical Engineering 9, 204--207.

Li, T.W., Mao, Z.S., Chen, J.Y., Fei, W.Y., 2002. Experimental and numericalinvestigation of single drop mass transfer in solvent extraction systems withresistance in both phases. Chinese Journal of Chemical Engineering 10, 1--14.

Li, X.J., Mao, Z.-S., Fei, W.Y., 2003a. Effects of surface-active agents on mass transferof a solute into single buoyancy driven drops in solvent extraction systems.Chemical Engineering Science 58, 3793--3806.

Li, X.J., Mao, Z.-S., Fei, W.Y., 2003b. Unsteady motion of a single droplet in surfactantsolution. Chinese Journal of Chemical Engineering 11, 715--725.

Mao, Z.-S., Li, T.W., Chen, J.Y., 2001. Numerical simulation of steady and transientmass transfer to a single drop dominated by external resistance. InternationalJournal of Heat and Mass Transfer 44, 1235--1247.

Misek, T., 1978. Recommended Systems for Liquid Extraction Studies, EPCEPublications Series. Institute of Chemical Engineers, Rugby, UK.

Osher, S., Sethian, J., 1988. Front propagating with curvature-dependent speed:algorithm based on Hamilton-Jacobi formulations. Journal of ComputationalPhysics 79, 12--49.

Patankar, S.V., 1980. Numerical Heat Transfer and Fluid Flow. Hemisphere,Washington.

Petera, J., Weatherley, L.R., 2001. Modelling of mass transfer from falling drops.Chemical Engineering Science 56, 4929--4947.

Ryskin, G., Leal, L.G., 1983. Orthogonal mapping. Journal of Computational Physics50, 71--100.

Sussman, M., Smereka, P., Osher, S., 1994. A level set approach for computingsolutions to incompressible two-phase flow. Journal of Computational Physics114, 146--159.

Temos, J., Pratt, H.R.C., Stevens, G., 1996. Mass transfer to freely-moving drops.Chemical Engineering of Science 51, 27--36.

Unverdi, O.S., Tryggvason, G., 1992. A front-tracking method for viscous,incompressible, multi-fluid flows. Journal of Computational Physics 100, 25--37.

Yang, C., Mao, Z.-S., 2002. An improved level set approach to the simulation of dropand bubble motion. Chinese Journal of Chemical Engineering 10, 263--272.

Yang, C., Mao, Z.-S., 2005. Numerical simulation of interphase mass transfer withthe level set approach. Chemical Engineering Science 60, 2643--2660.

Zhang, H., Zheng, L.L., Prasad, V., Hou, T.Y., 1998. A curvilinear level set formulationfor highly deformable free surface problems with application to solidification.Numerical Heat Transfer, Part B 34, 1--20.