Numerical simulation of the Marangoni e#ect on mass transfer to single slowly moving drops in the liquid-liquid system
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Chemical Engineering Science 59 (2004) 1815 – 1828
www.elsevier.com/locate/ces
Numerical simulation of the Marangoni e#ect on mass transfer to single
slowly moving drops in the liquid–liquid system
Zai-Sha Mao∗ , Jiayong Chen
Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100080, China
Received 11 September 2003; received in revised form 29 December 2003; accepted 31 January 2004
Abstract
The Marangoni e#ect is a frequently observed phenomenon of enhancement of interphase mass transfer in liquid–liquid systems. Such
an e#ect, originating from the hydrodynamic instability induced by surface tension sensitivity to surface concentration of transferred solute,
is mathematically formulated and numerically simulated for slowly moving single spherical drops in an axisymmetric boundary-3tted
coordinate system by solving coupled 4uid 4ow and solute mass transfer equations. Numerical simulation demonstrates the occurrence
of the Marangoni e#ect under typical conditions in liquid–liquid systems, and is in reasonable agreement with the classic theoretical
analysis. Su5cient spatial and temporal resolution in simulation reveals the multi-scale interaction of the drop-scale Marangoni e#ect
with the sub-drop-scale local interfacial convection. The e#ect of solute transfer direction, Peclet number, surface tension sensitivity
to solute concentration, and level of random perturbation on surface concentration are investigated numerically. It is shown that the
Marangoni e#ect occurs in the middle stage of a transient interphase mass transfer process, and the Marangoni convection at the
interface does not necessarily results in the Marangoni e#ect of mass transfer enhancement. Besides, the Marangoni e#ect occurs only
when the surface tension sensitivity to the solute concentration variation is above certain critical level. The present axisymmetric simulation
of the Marangoni e#ect provides necessary basis for further work on three-dimensional numerical analysis.
? 2004 Elsevier Ltd. All rights reserved.
Keywords: Marangoni e#ect; Spherical drop; Mass transfer; Numerical simulation; Solvent extraction; Instability
1. Introduction e#ect were mainly focused on the criteria to judge whether
the Marangoni e#ect would occur for speci3c liquid–liquid
It is known for many years that the local variation in so- systems. The classical analysis of Sternling and Scriven
lute concentration at the interface in liquid–liquid solvent (1959) presented such a basic criterion from the linear
extraction systems would cause local increase or decrease instability theory. Although several other propositions on
of interfacial tension, and thus induce additional convec- criteria were presented later, only partial qualitative success
tion at the interface (so-called interfacial turbulence). If this in prediction of critical conditions for occurrence of the
convection is localized and segmented, it would often gen- Marangoni e#ect was achieved. Agble and Mendes-Tatsis
erate local 4ow patterns on the sub-droplet scale, leading (2001) compared several stability criteria in predicting in-
to convection in the direction normal to the interface. It in terfacial Marangoni convection, and found that none of the
turn enhances interphase mass transfer. This phenomenon proposed criteria was accurate enough for the liquid–liquid
is known to chemical engineers as the Marangoni e#ect systems when a surfactant was present. They addressed
(Scriven and Sternling, 1960; Levich, 1962). Quite diversi- the system behavior in mass transfer, that was complicated
3ed forms of phenomena in interphase mass transfer were with small amount of surface-active agents. The mecha-
altogether termed as the Marangoni e#ect, such as drop nism and prediction of the Marangoni e#ect is surely a
pulsation, localized eruption, kicking, and surface rippling worthwhile subject in the fundamental research related to
(Sawistowski, 1971). Theoretical studies of the Marangoni solvent extraction, for most practical extraction systems
∗
are more or less contaminated by impurity surface active
Corresponding author. Tel.: +86-10-6255-4558;
fax: +86-10-6256-1822.
agents.
E-mail addresses: zsmao@home.ipe.ac.cn (Z.-S. Mao), To the authors’ knowledge, no numerical simulation
jychen@home.ipe.ac.cn (J. Chen). of the Marangoni e#ect on mass transfer of drops in
0009-2509/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ces.2004.01.0351816 Z.-S. Mao, J. Chen / Chemical Engineering Science 59 (2004) 1815 – 1828
liquid–liquid systems was reported. On the other hand,
numerical works on thermocapillary Marangoni e#ects, for
example, thermocapillary migration of bubbles (Nas and
Tryggvason 2003; Rother et al., 2002) and of drops (Rother
et al., 2002; Wang et al., 2004), liquid bridge instability
(Lappa et al., 2001; Tang and Hu, 1998; Li et al., 2003a),
have appeared quite frequently. As for simulation of the
Marangoni e#ect at gas-liquid surface, Lee et al. (2003)
reported the Marangoni convection induced by surfactant Fig. 1. Sketch of a spherical drop with the reference coordinate system
absorption into liquid falling 3lm. 3xed with the drop.
Although the Marangoni e#ect has been observed and
the related enhanced mass transfer quanti3ed usually on
the drop scale, plenty of experimental evidence has sug-
gested that the complex phenomenon coupling the 4uid liquid 4ow near the interface in both phases. Thus, the con-
4ow and mass transfer in two liquid phases might result in vective di#usion equation of solute transport must be solved
sub-drop-scale 4ow structures at the interface. Therefore, at the same time in a coupled way. A necessary link between
multi-scale consideration is necessary in approaching the the two sets of governing equations is the constitutive equa-
complexity of the Marangoni e#ect, either numerically or tion describing the in4uence of the solute concentration on
experimentally. In this case, su5cient spatial and/or tempo- the surface tension of the liquid–liquid system. The math-
ral resolution is required so as to reveal such sub-drop-scale ematical formulation will be stated in the following three
details. subsections.
As a preliminary step in the numerical approach to
the three-dimensional Marangoni e#ect in solvent ex- 2.1. Hydrodynamic formulation
traction systems, the mathematical model of the mass
transfer-induced Marangoni e#ect is formulated in this pa- In formulating the viscous 4ow both around and inside
per. On this basis, numerical simulation of a single drop a spherical drop, it is usually assumed that (1) the 4uids
slowly moving in a liquid–liquid system is conducted in an are viscous and Newtonian, (2) the 4ow 3eld is laminar
axisymmetric orthogonal reference frame. The drop is as- and axisymmetric, (3) the 4ow is isothermal, and (4) the
sumed to be spherical. For certain combination of physical physical properties of liquids (except the interfacial tension)
and operational parameters, numerical simulation reveals are constant despite of the change of solute concentration.
the occurrence of the Marangoni e#ect on the rate of solute Thus, the 4uid 4ow both in the external and internal regions
mass transport in agreement with the previous theoretical of a drop can be described by the equation of continuity and
prediction. The e#ect of solute transfer direction, Peclet the transient Navier–Stokes equations for incompressible
number, surface tension sensitivity to solute concentration, 4uid:
capillary number, disturbance level etc. are demonstrated
by numerical results. ∇ · u = 0; (1)
@u
+ u · ∇u = −∇p + ∇2 u: (2)
@t
2. Formulation of the Marangoni eect of mass transfer
Since the density of liquids is constant, the continuity
In a solvent extraction system, the extractant phase is remains to be the divergence-free condition, Eq. (1).
usually 3nely dispersed in the continuous phase and the The axisymmetrical 4ow in the continuous outer liquid
typical drop size is at the level of 0.1–2 mm diameter. phase may be described by a set of partial di#erential equa-
In such case, the Reynolds number of moving droplets is tions in terms of stream function and vorticity ! in a
mostly below 500 and the 4ow around drops and the circu- sectional plane (x; y) passing through the axis of symmetry
lation inside is presumed laminar. To simplify the numerical as sketched in Fig. 1. The governing equations of the only
analysis, spherical shape is assumed for drops undergoing non-zero component of vorticity ! and stream function
interphase mass transfer, even when locally varying surface in both phases in the orthogonal curvilinear coordinate sys-
tension gradient plays a signi3cant role and surface turbu- tem ( ; ) are (Dandy and Leal, 1989; Li and Mao, 2001;
lence breaks out. Li et al., 2003b, c)
To resolve the Marangoni e#ect underlain by local interfa- 2 1 @ 1 @ !1 @ 1 @ !1
1 L1 (y1 !1 ) − −
cial hydrodynamic disturbance, the Navier–Stokes equation h 1 h 1 @ 1 @ 1 y1 @ 1 @ 1 y1
and continuity equation are to be solved in an axisymmetri-
@!1
cal coordinate system. However, the possible local 4uctua- = ; (3)
tion of solute concentration at the drop surface would trigger @t
surface tension gradient, which will in turn alter the local L21 1 + !1 = 0; (4)Z.-S. Mao, J. Chen / Chemical Engineering Science 59 (2004) 1815 – 1828 1817
Fig. 2. The grid for numerical solution, the physical reference frame and the computational coordinate system: (a) external domain and (b) drop domain.
1 @ 2 @ !2 @ 2 @ !2 where = tUT =R and the relevant dimensionless parameters
2 L22 (y2 !2 ) + − are
h 2h 2 @ 2 @ 2 y2 @ 2 @ 1 y2
@!2 2RUT 1 2 2
= ; (5) Re1 = ; Re2 = Re1 ; = ; = :
@t 1 1 1
As depicted in Fig. 2,the external domain is a5xed to a
L22 2 + !2 = 0; (6) left-handed reference frame and hence the physical velocity
components are related to the stream function 1 by
where subscript 1 denotes the outer continuous phase, 2 the
drop, and the di#erential operator is 1 @1 1 @1
U 1 =− ; U1= ; (13)
Y1 H @ 1 Y1 H @ 1
1 @ f @ @ 1 @ 1
L2 = + : (7)
h h @ y @ @ fy @ while for the drop interior (in a right-handed reference
frame), the following de3nitions apply:
Eqs. (4) and (6) are actually the de3nition of ! in the present
orthogonal coordinate systems. 1 @2 1 @2
U2= ; U 2 =− : (14)
When being transformed non-dimensional in terms of the Y2 H @ 2 Y2 H 2
@ 2
following non-dimensional physical variables ( and )
de3ned by The distortion function is de3ned as the ratio of two scale
factors:
UT UT
!1 = 1 ; !2 = 2 ; Hi
R R fi ( ; ) = ; i = 1; 2: (15)
2 2
Hi
1 = R UT 1 ; 2 = R UT 2 ;
The boundary conditions at the drop surface and at the
the governing equations become axis of symmetry are routinely used:
On the axis of symmetry ( = 0; = 1),
Re1 1 @1 @ 1
L21 (Y1 1 ) −
2 H 1 H 1 @ 1 @ 1 Y1
1 = 2 = 1 = 2 = 0: (16)
@1 @ 1 Re1 @1
− = ; (8) At the drop surface ( = 1),
@ 1 @ 1 Y1 2 @
1 = 2 = 0 (impermeable interface); (17)
L21 1 + 1 = 0; (9)
Re2 1 @2 @ 2 U 1 =U 2
(kinematic continuity): (18)
L22 (Y2 2 )
+
2 H 2H 2 @ 2 @ 2 Y2
Special attention is given to the balance of tangential force
@2 @ 2 Re2 @2 along the interface, where the variation of interfacial tension
− = ; (10) also contributes to the balance, as was done by Li and Mao
@ 2 @ 1 Y2 2 @
(2001):
L22 2 + 2 = 0; (11)
1 (!1 − 2 ( ) u 1 ) − 2 (!2 − 2 ( ) u 2 )
1 @ f @ @ 1 @ 1 @!
L2 = + ; (12) =− (shear stress balance): (19)
H H @ Y @ @ fY @ h1 @ 11818 Z.-S. Mao, J. Chen / Chemical Engineering Science 59 (2004) 1815 – 1828
When non-dimensionalized, it is turned into the following 4
form: '1 = (2 ( ) U 1 − 1 ); (24)
Re1
1 − 2 ( ) U 1 − (2 − 2 ( ) U 2 )
1 @X1 @2 Y1 @2 X1 @Y1
( ) = 3 − : (25)
!0 @! 1 @! H1 @ 1 @ 1 2 @ 21 @ 1
=− =− (20)
1 UT H 1
@ 1 CaH 1
@ 1 1 =1
2.2. Mass transfer formulation
with the surface tension variation due to variable solute con-
centration explicitly incorporated. Since surface tension is
2.2.1. Formulation of interphase mass transfer
variable, the normal stress balance across the interface is
The typical situation of mass transfer to a drop is that it
also di#erent from a system without solute mass transfer,
rises steadily at the terminal velocity UT in an immiscible
but the normal balance becomes irrelevant due to the present
liquid medium (Fig. 1). For a rigid spherical liquid drop
assumption of spherical shape despite the local Weber num-
with intermediate Reynolds number, the following reason-
ber is variable along the interface. ( ) is the dimensionless
able simplifying assumptions is generally justi3ed:
surface curvature in the plane passing through the axis of
symmetry, which is equal to 1 for a spherical drop as in (1) Except interfacial tension, the physical properties of
our case. In this formulation, the solute at the surface is as- 4uids are not in4uenced by the concentration of solute
sumed to behave in the same manner as in the bulk phase; to be transferred and remain constant;
therefore, no independent governing equation appears for (2) Thermodynamic equilibrium of solute between phases
the interface as a separate phase. exists at the interface (no interface resistance to mass
The link between the surface tension and the mass trans- transfer).
fer of solute is necessary for providing the surface gradient
appearing in Eq. (20). The relation of surface tension with The Marangoni e#ect is unsteady in nature, thus the tran-
the solute concentration is di#erent from system to system sient mass transfer either in the drop or in the continuous
and in general non-linear. The simplest model is the linear phase has to be resolved by numerical solution of the fol-
one: lowing governing convective di#usion equation. In terms of
vector notation it reads
! = 1 + $C1S (21)
@ci
with the surface tension sensitivity coe5cient $ as an index + ui · ∇ci = Di ∇2 ci ; i = 1; 2 (26)
@t
of the system to the e#ect of solute concentration. In most
cases, $ is negative, re4ecting the fact that a solute usually for both phases. In a general axisymmetric, orthogonal curvi-
decreases the interfacial tension. linear coordinate system, the expanded form of Eq. (26) is
Numerical solution of Eqs. (8)–(11) under suitable
boundary conditions provides the external and internal 4ow @ci u @ci u @ci
+ i
+ i
3elds and the 4ow parameters relevant for numerical sim- @t h i
@ i hi @ i
ulation of simultaneous interphase mass transfer. Limited
Di @ h i yi @ci @ h i yi @ci
by space, the details of numerical solution of 4uid 4ow are = + :
referred to the literature (Dandy and Leal, 1989; Li et al., h i h i yi @ i hi @ i @ i hi @ i
1999; Li and Mao, 2001). (27)
The drag coe5cient is calculated by the following surface
integral along the drop surface: Derivation of the above equation is referred to Mao
et al. (2001) and Li and Mao (2001). Using c1∞ , R and
@Y1 @X1
CD = 2 '1 − '1 Y1 d 1 ; (22) UT as the non-dimensionalizing factors, Eq. (27) may be
@ 1 @ 1
non-dimensionalized to
where '1 represents the normal stress of the outer phase
Pe1 @C1
adjacent to the interface, '1 is the tangential stress. The H 1 H 1 Y1
quantities involved in the above equation may be calculated 2 @
by
Pe1 @ @1 @ @1
+ − C1 + C1
2 @ 1 @ 1 @ 1 @ 1
4 f1 @ (Y1 1 ) d 1
'1 = U 21 +
Re1 Y1 @ 1 @ @C1 @ Y1 @C1
= f1 Y1 + ; (28)
@ 1 @ 1 @ 1 f1 @ 1
2 @
− (Y1 U 1 ) ; (23) Pe2 @C2
Y1 H @ 1 H 2 H 2 Y2
1 2 @Z.-S. Mao, J. Chen / Chemical Engineering Science 59 (2004) 1815 – 1828 1819
Pe2 @ @2 @ @2 solute 4ux continuity at the
+ C2 + − C2
2 @ 2 @ 2 @ 2 @ 2
drop surface; 1 = 2 = 1; (33)
@ @C2 @ Y2 @C2
= f2 Y2 + ; (29) C1 (0;
@ 2 @ 2 @ 2 f2 @ 2 1 ; ) = 1;
where Pe1 = 2RUT =D1 and Pe2 = 2RUT =D2 are the Peclet at the remote boundary; 1 = 0; (34)
numbers for the continuous phase and the drop respectively
@C1 @C2
and represent the relative strength of convection to molec- = 0; =0 on the axis of symmetry;
ular di#usion. Here, the dimensionless time = tUT =R, is @ 1 @ 2
based on the linear velocity of the drop. It is known from the 1 = 2 = 0; 1 = 2 = 1: (35)
numerical simulation of liquid 4ow for a drop that four other
non-dimensional parameters, Re1 , We, , and , govern the For the reverse d → c mass transfer from drop to con-
4uid 4ow inside and outside the drop with a free deformable tinuous phase, the initial drop concentration is the basis for
interface (Dandy and Leal, 1989; Li et al., 1999). In non-dimensionalization, and the conditions corresponding
the case of rigid liquid sphere, We is actually irrelevant to Eqs. (30), (31), and (34) become
since it appears only in the normal stress balance across the C1 ( 1 ; 1 ; 0) = 0; at = 0 (continuous phase); (36)
drop surface. The other three parameters exert in4uence on
the Marangoni e#ect of mass transfer through their e#ects C2 ( 2 ; 2 ; 0) = 1; at = 0 (drop phase); (37)
on 4ow structure, intensity of convection and shear near
the drop surface. C1 (0; 1 ; ) = 0; at the remote boundary; 1 = 0; (38)
Due to the constancy of physical properties of both bulk while Eqs. (32), (33), and (35) remain unchanged.
liquid phases, the coupling between the mass transfer and the
4uid 4ow develops only from the change of surface proper- 2.2.3. Mass transfer coe3cient
ties. The local 4uctuation in the surface concentration gen- To evaluate the mass transfer coe5cient and the
erates additional tangential stress at the surface, which may Sherwood number for c → d mass transfer, it is necessary
grow and then in4uence the boundary condition for solv- to calculate 3rst the local di#usive 4ux Nloc and de3ne the
ing the 4uid 4ow. Therefore, coupling is solely expressed in local mass transfer coe5cient kloc :
the boundary condition Eq. (20). Thus, the numerical sim-
D1 @c1
ulation for each time step must consist of a few iterations Nloc = −
h @ 1 1 =1
of solution procedure for 4uid 4ow and mass transfer, so 1
that the PDEs and the coupling boundary conditions become cL2 D2 @c2
compatible to each other. = kloc c1∞ − = ; (39)
m h 2 @ 2 2 =1
2.2.2. Initial and boundary conditions where the remote boundary concentration c1∞ , and the
For completeness of the mathematical formulation, both only available measurement of the drop concentration cL2
the initial and boundary conditions are speci3ed to the gov- (the average over the whole drop) is used to de3ne the driv-
erning Eqs. (28) and (29). The concentration of solute in ing force and mass transfer coe5cient. The latter may be
the continuous phase is assumed to be c1∞ at = 0, which expressed in terms of dimensionless concentration gradient
is in turn used as non-dimensionalizing factor for the ex- as:
ternal and internal concentration 3elds. Thus, the following D1 @C1
kloc = −
initial and boundary condition may be assumed for mass RH (1 − CL =m) @ 1
1 2 1 =1
transfer from the continuous phase to drops (mass transfer
direction c → d): D2 @C2
= :
RH 2 (1 − CL 2 =m) @ 2 2 =1
C1 ( 1 ; 1 ; 0) = 1; at = 0 (continuous phase); (30)
The local Sherwood number is then
C2 ( 2 ; 2Rkloc 2 @C1
2 ; 0) = 0; at = 0 (drop phase); (31) Shloc = =−
D1 H 1 (1 − CL 2 =m) @ 1 1 =1
C2 (1;
2 ; ) 2Pe1 @C2
= :
=mC1 (1; 1 ; ); solute dissolution equilibrium at Pe2 H 2 (1 − CL 2 =m) @ 2 2 =1
the drop surface; = = 1; (32) and the drop area average Sh based on the continuous phase
1 2
reads
1
1 @C1 1 @C2 2 (fY1 =(1 − CL 2 =m))(@C1 =@ 1 )| 1 =1 d 1
− = Shoc = − 0 : (40)
Pe1 H 1
@ 1 1 =1
Pe2 H 2
@ 2 2 =1
1
Y H 1d 1
0 11820 Z.-S. Mao, J. Chen / Chemical Engineering Science 59 (2004) 1815 – 1828
If based on the drops Eq. (40) would become outside the drop is carried out by solving Eqs. (8)–(11) with
1 pertinent boundary conditions. This can be either done by
2 0
(fY1 =(m − CL 2 ))(@C1 =@ 1 )| 1 =1
d 1 numerical solution of steady Navier–Stokes equations with
Shod = − 1
0
Y1 H 1 d 1 necessary under-relaxation, or by solution of time-dependent
Navier–Stokes equations for a long enough time period so
= mShoc : (41) that the 4ow approaches the steady state.
(2) Utilizing the solved velocity 3eld (or the stream func-
For d → c mass transfer, the de3nition equation (39)
tion) as the starting initial 4ow conditions, simultaneous so-
reads
lution of transient mass transfer of a solute into the drop and
D1 @c1 cL2 D2 @c2 the 4uid 4ow under in4uence of the changing surface tension
Nloc = − = kloc −0 =
h 1 @ 1 1 =1 m h 2 @ 2 2 =1 due to non-uniform concentration pro3le on the drop sur-
face. This stage of simulation is carried out in the real-time
for the initial concentration is zero in the continuous phase. domain. Exact solution of the 4ow and concentration 3elds
Therefore, Eqs. (40) and (41) become must be achieved for each time step, so that the correct mass
1 transfer Shoc can be evaluated by Eq. (40) or (42). In each
2m 0
(fY1 = CL 2 )(@C1 =@ 1 )| 1 =1 d 1 time step, it is necessary to solve the stream function , the
Shoc = − 1
; (42)
0 1
Y H 1d 1 vorticity and the concentration of solvent C and to re-
fresh the surface parameters involved in Eq. (20) in a few
1
2 0
(fY1 = CL 2 )(@C1 =@ 1 )| 1 =1 d 1 Shoc iterations so that these 3elds become compatible with one
Shod = − = : (43)
1
Y H 1d 1 m another at the end of a time step.
0 1
The governing equations were discretized according to the
Control Volume Formulation with the Power-Law Scheme
2.2.4. Remarks on assumptions adopted as described by Patankar (1980). The numerical details can
As suggested by Eq. (19), the concentration gradient be referred to our previous works (Mao and Chen, 1997; Li
along the interface a#ects the tangential shear stress balance et al., 1999; Mao et al., 2001). The 3nite external domain
via its e#ect on the interfacial tension. In this case, the linear with the outer radius of 100R was veri3ed to be su5ciently
rising velocity of a drop would be a#ected even for a drop large for getting rid of the in4uence of physical domain
with speci3ed diameter having reached the steady state of size. The computational grid was speci3ed analytically as
buoyancy-driven motion before the start of mass transfer. similarly described in Mao (2002), but the mesh size in
This factor makes the drop behavior more complex and the the external domain in the direction was designated as
numerical simulation overwhelmingly di5cult. This factor an algebraic series to assure better numerical accuracy in
is temporarily ignored in this work and the drop is assumed the region close to the drop surface. The distortion function
to rise at the steady terminal velocity of UT . de3ned by Eq. (15) was calculated subsequently from the
The 4ow and mass transfer are assumed axisymmet- coordinates (x; y) of the grid nodes.
ric so that they can be formulated in a two-dimensional In choosing the proper spatial grid 3neness and time step
axisymmetric orthogonal coordinate system. Therefore, size to assure the numerical simulation be grid-independent,
the roll cells of di#erent size formed at the drop surface the results in a series of our previous papers were referred.
are toroidal. In fact, the interfacial turbulence induced by In compromise of numerical accuracy and computational
the Marangoni e#ect is three-dimensional in nature. As e5ciency, a 41 × 81 spatial grid was found su5ciently 3ne
might be expected from the 3rst principles of 4uid mechan- and used later throughout in the numerical solution of 4uid
ics, nearly equi-dimensional roll cells are most likely to 4ow and mass transfer. For the simulation of transient mass
appear at the 3rst stage at the drop surface, since the random transfer of a drop with an intermediate Reynolds number
disturbance is localized rather than toroidal. Probably many around 50, the dimensionless N below 0.01 was enough
roll cells with suitable location and circulation direction to achieve the independence of solution from the time step
only would develop synergistically into a toroidal roll cell. size (Mao et al., 2001; Li and Mao, 2001). But for the
It is thus anticipated that the real Marangoni e#ect would present case of simulating a slowly rising drop (Re = 0:1),
occur under more general conditions than suggested by the it was found that N as low as 2:5 × 10−5 gave satisfactory
present axisymmetric time-dependent simulation. Direct simulation results.
numerical simulation of three-dimensional Marangoni e#ect
induced by mass transfer is currently under consideration.
4. Simulation results
3. Numerical procedure 4.1. Typical Marangoni e7ect of a spherical drop
The numerical simulation proceeds in the following steps: To be closer to pendent drops in most experimental stud-
(1) Numerical simulation of steady 4uid 4ow inside and ies, the object of the present simulation is a drop withZ.-S. Mao, J. Chen / Chemical Engineering Science 59 (2004) 1815 – 1828 1821
Re1 =0:1. It is believed that the toroidal Marangoni roll cells
on a slowly moving drop resulted from the present simula-
tion are more rational than that for a stationary drop, because
the toroidal structure is more consistent with the situation
of the 4ow and transport dominated by external axial 4ow
in the continuous phase.
Fig. 3 is the illustration of development of surface in-
stability with the solute transfer from the continuous phase
into a blank drop. In this case, the surface concentration was
disturbed arti3cially in each time step:
C̃ S1 = C1S (1 + $p 0); (44)
where 0 is a pseudo-random number in the range from −1 to
1, and $p is a constant to designate the level of perturbation
to the surface solute concentration. At = 0:05, no surface
turbulence appears, the 4ow 3eld is similar to the steady
4ow without mass transfer, the concentration contour lines
are concentrically distributed at two sides of the drop sur-
face, because the interfacial turbulence needs certain time
for itself to develop. The surface Marangoni e#ect develops
to a signi3cant level only when the normal gradient of solute
concentration at the surface decreases to a lower level, as
indicated for = 0:10. At this moment the Marangoni con-
vection patterns much smaller than the drop scale appear in
many isolated locations on the surface. When they develop
larger in size so that they may bridge two ends of the con-
centration boundary layer, the convective transport in the
normal direction to the interface becomes e#ective and dis-
plays its signi3cant role in enhancing the interphase mass
transfer. From the concentration contour plots at = 0:10,
the sub-drop-scale Marangoni convection patterns are quite
apparent. It is conjectured that these small patterns are more
e#ective in mass transfer promotion than the large vortexes
on the drop scale as occurring at = 0:15, 0.20, and 0.30,
since the large patterns cover mostly a wide area with C1
very close to unity, where the convection contributes little to
the overall mass transfer. The streamline maps also demon-
strate that the surface turbulence is dynamic and transient
in nature, the patterns change from time to time, and con-
ceivably are swept downstream by the external bulk 4ow.
As the mass transfer is concerned, up to = 0:30 the mass
transfer enhancement remains dominant since there exist
sub-drop-scale interfacial convection patterns at the drop
surface, besides the large vortexes.
At the beginning of mass transfer, the overall mass trans-
fer Sherwood number drops in general as mass transfer pro-
ceeds and the driving force of mass transfer decreases, as il-
lustrated in Fig. 4. No matter whether disturbed or not, Shoc Fig. 3. Marangoni e#ect induced evolution of 4ow patterns (left) and
follows the same route of gradual decrease. If $ = 0 (sur- concentration contours (right) with time for a spherical drop with Re1 =0:1,
face tension not sensitive to the solute concentration), the = 0:5, = 0:91, Pe1 = 100, Pe2 = 40, Ca = 2 × 10−4 , $ = −1, m = 2,
Sherwood number drops monotonously as always. But for concentration perturbation $p = 0:0001, c → d mass transfer. The left
cases with $ ¡ 0, signi3cant Marangoni e#ect may occur plots are streamline maps with the bulk 4ow in the continuous phase from
left to right, and on the right are corresponding concentration contours.
sooner or later. As the concentration boundary layer grows
thick enough (this giving the surface turbulence a chance to
make apparent enhancement on the mass transfer in the nor-
mal direction), the disturbance to the solute surface concen-1822 Z.-S. Mao, J. Chen / Chemical Engineering Science 59 (2004) 1815 – 1828
80 Table 1
=0 Cases analyzed by Sternling and Scriven (1959)
=-1, p=0.0001
Case A B C D
60 =-1, p=0.00001
=-1, p=0 2 =1 0.5 0.5 0.5 0.5
D2 =D1 2 1 0.667 0.25
Sh oc
40 Note: 1 = 0:001 Pa s, D1 = 2 × 10−9 m2 =s, m = 1, 2 =1 = 2, d!=dc =
−10−6 N m2 =kg.
20
Table 2
Parameters for current numerical simulation of Marangoni e#ect for a
0 spherical drop corresponding to Table 1
0.0 0.1 0.2 0.3
Case A B C D
= 2 =1 2 2 2 2
Fig. 4. Temporal evolution of average Shoc of a spherical drop as in4u- = 2 =1 4 4 4 4
enced by the concentration perturbation level (Re1 = 0:1, = 0:5, = 0:91, Re1 0.1 0.1 0.1 0.1
Pe1 = 100, Pe2 = 40, Ca = 2 × 10−4 , m = 2). Re2 0.2 0.2 0.2 0.2
Pe1 500 500 500 500
Pe2 250 500 750 2000
tration might trigger the Marangoni e#ect promoting inter- Note: D1 = 2 × 10−9 m2 =s, m = 1, Ca = 5 × 10−4 , $ = −0:1.
phase mass transfer. Di#erent level of perturbation on sur-
face solute concentration dictates the time instant when the
Marangoni e#ect starts to develop. Comparing three curves
with $p equal to 1 × 10−4 , 1 × 10−5 , and 0, respectively, it parameters in Table 2, that were needed for conducting sim-
is evident that arti3cial disturbance to surface concentration ulation of a speci3c case, from the original Table 4 of S&S.
helps the intrinsic Marangoni e#ect to overcome hydrody- In our formulation, the values of $ and Ca are to be pro-
namic or thermodynamic barriers to display itself. The ar- vided in Eq. (20) to express the strength of mass transfer to
ti3cial disturbance is not the necessary condition, the curve induce the Marangoni e#ect.
with $p =0 suggests that even the numerical truncation error In S&S, the normal concentration gradient was speci3ed
can trigger the Marangoni e#ect when the physico-chemical a constant, but the gradient is certainly a variable in the
parameters fall right in the region of instability. All subse- cases of drop mass transfer and it decreases monotonously
quent simulation was then conducted with $p = 0 to empha- since the capacity of a drop to extract solute is 3nite. In
size the intrinsic susceptivity of the system to the Marangoni this case, we presumed reasonable values typical for liquid–
instability. liquid extraction systems, in addition to these used in S&S.
The following values are taken: d = 10−3 –10−4 m, c∞ =
4.2. Comparison with the linear instability theory 104 mol=m3 , 1 =10−3 Pa s, !0 =0:01 N=m. From Re1 =0:1,
it is deduced UT = 10−3 –10−4 m=s, so
Sternling and Scriven (1959, hereafter the paper abbrevi-
1 UT 10−3 (10−4 –10−3 )
ated as S&S) used the linear instability approach to analyze Ca = = = 10−5 –10−4 :
the induction of the Marangoni e#ect due to interphase mass !0 0:01
transfer and discussed in detail four cases typical to solvent
From d!=dc = −10−6 N m2 =kg in S&S, we get d!=dc of
extraction systems with either hydrodynamic stability or in-
about −10−7 N m2 =mol, namely,
stability concluded. To demonstrate the applicability of the
present numerical approach, these four cases were simulated d! c∞ 104
$= = −10−7 = −10−7 = −0:1:
in this study. The original speci3cations of the cases studied dC !0 0:01
are listed in Table 1, and the corresponding non-dimensional
parameters to start the numerical simulation are in Table 2. Also from Eq. (20), it is recognized that the far right term
In the present nomenclature, we designate label (1) to four is actually
cases of mass transfer from phase A to phase B in S&S and 1 d! 1 d! dC1 $ dC1
(2) to the corresponding reverse mass transfer from phase − =− =−
CaH 1
d 1 CaH 1
dC1 d 1 CaH 1
d 1
B to phase A in S&S. In the present simulation, phases A
and B correspond to the continuous phase and drop respec- and the ratio of $ over Ca is to be speci3ed, instead of $
tively. So eight cases in total are to be simulated: cases A1, and Ca separately. Thus, from the range for Ca, we select
B1, C1, and D1 for c → d mass transfer and cases A2, B2, Ca = 5 × 10−4 , together with all other parameters needed
C2, and D2 for d → c transfer. It is easy to 3gure out the for the simulation in Table 2.Z.-S. Mao, J. Chen / Chemical Engineering Science 59 (2004) 1815 – 1828 1823
80
=-0.1 =-0.1
=0 60 =0
60
40
Sh oc
40
Sh oc
20
20
0 0
0.0 0.1 0.2 0.3 0.4 0.5 0.6
0.0 0.2 0.4 0.6
θ θ
Case A1 Case B1
30 30
=-0.1
=-0.1 =0
=0
20 20
Sh oc
Sh oc
10 10
0 0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6
θ θ
Case C1 Case D1
Fig. 5. Sherwood number for mass transfer inwards into a spherical drop (c-to-d) as in4uenced by the Marangoni e#ect (Re1 = 0:1, m = 1, other
parameters in Table 2).
Fig. 5 presents the results of continuous phase to drop in our case, but the one-dimensional rectilinear mass trans-
mass transfer (c-to-d transfer), and Fig. 6 shows the coun- port was analyzed in S&S; (3) the mass transfer is a tran-
terpart for mass transfer from the drop to the continuous sient process since the limited capacity of drop extraction
phase (d-to-c transfer). The comparison of the simulation in our case so that the mass transfer driving force decreases
with S&S is summarized in Table 3. Cases A1, B1, C1, gradually, but the overall concentration gradient across the
and D1 (with digit 1 indexing c-to-d transfer) present ob- interface was constant in S&S. Therefore, the results from
vious Marangoni e#ect on mass transfer, in agreement with the current simulation is not fully in correspondence to that
the analysis in S&S, but the numerical simulation for cases of S&S.
A2, B2, C2, and D2 (with digit 2 indexing d-to-c transfer) It is observed from Fig. 5 that for all cases with mass trans-
di#ers from results of Sternling and Scriven (1959). It is fer from the continuous phase into the drop (A1, B1, C1,
now di5cult to address the di#erence between the results, and D1), the mass transfer rate as expressed in terms of Shoc
because the objects for analysis are somewhat di#erent in decreases gradually before is large enough. Two factors
several aspects: (1) since the aim of the present paper is fo- are possibly responsible for no mass transfer enhancement
cused on liquid drops in slow motion (Re1 = 0:1), the inter- in this period: (1) the Marangoni convection needs time to
face between two phases is spherical, in contrast to the 4at develop itself; (2) at the beginning the solute concentration
interface considered in S&S; (2) the 4ow and mass transfer gradient is steep at the interface in both the continuous and
to a spherical drop are two-dimensional and axisymmetric drop phases so that the 4uctuation in surface concentration1824 Z.-S. Mao, J. Chen / Chemical Engineering Science 59 (2004) 1815 – 1828
80
100 =-0.1, Ca=5×10-4
=-0.1
=-0.1, Ca=2.5× 10-3
=0
80 60
60
Shoc
Shoc
40
40
=0
20
20
0 0
0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6
Case A2 Case B2
60 40
=-0.1
=-0.1
=0
=0
40
Shoc
Shoc
20
20
0 0
0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6
Case C2 Case D2
Fig. 6. Sherwood number for mass transfer outwards from a spherical drop (d-to-c) as in4uenced by the Marangoni e#ect (Re1 = 0:1, m = 1, other
parameters in Table 2, Ca = 5 × 10−4 is a constant except for case A2).
Table 3
Comparison of the mass transport instability predicted by numerical simulation with that by Sternling and Scriven (1959)
Case A B C D
1. Transfer direction: c-to-d
This work MC: yes MC: yes MC: yes MC: yes
MT: enhanced MT: enhanced MT: enhanced MT: enhanced
Sternling and Scriven instability instability instability instability
(di#usion-limited) (4ow-limited) (4ow-limited) (4ow-limited)
2. Transfer direction: d-to-c
This work MC: yes MC: yes MC: yes MC: yes
MT: enhanced MT: enhanced MT: enhanced MT: no
Sternling and Scriven stable stable instability instability
(di#usion-limited) (di#usion-limited)
Note: (1) Phenomenon identi3ed from simulation without arti3cial perturbation on the surface concentration of transferred solute. (2) MC stands for
Marangoni convection, MT: mass transfer.Z.-S. Mao, J. Chen / Chemical Engineering Science 59 (2004) 1815 – 1828 1825 might be suppressed by the high rate of solute di#usion in surface and adds to the mass transfer resistance. This gives combination of the surface convection for drop motion at rise to a factor contributing adversely to interphase mass Re1 = 0:1. Another feature of mass transfer enhancement is transfer. For case D2, the large vortex almost covers the the extent of enhancement 4uctuates, and this is believed to whole drop surface after = 0:15, and small 4ow patterns be the re4ection of the transient nature of mass transfer in- of interfacial turbulence are obviously suppressed. The vo- duced hydrodynamic instability. The transient nature is also luminous toroidal pattern even retards the mass transfer as re4ected by the temporal development of the 4ow patterns is greater than 0.3. As mentioned before, only sub-drop and the concentration contours similar to that demonstrated scale Marangoni convection promotes the interphase mass in Fig. 3. These patterns are conceived to be swept from the transfer e#ectively. As the Peclet number Pe2 increases in left hand side to the right hand side along the direction of the order of A2, B2, C2, and D2, the enhancement extent bulk 4ow in the continuous phase. and 4uctuation in Shoc curves subject to the Marangoni ef- By the way, the Sh curve with the Marangoni e#ect will fect decrease gradually, which is found in good correspon- eventually collapse to its baseline without surface tension dence of the trend of size increase of formed toroidal vortex sensitivity to solute concentration when becomes large at the drop surface. enough and concentration gradient drops low enough, in the similar trend of decaying Marangoni e#ect demonstrated in 4.3. In8uence of Ca on the Marangoni e7ect many experimental study (Agble and Mendes-Tatsis, 2000). When becomes large, the solute extraction approaches the Since the capillary number Ca is the key parameter to equilibrium, and the overall concentration di#erence over induction of the Marangoni e#ect, a parameter sensitivity the drop surface decreases to a low level. At this moment test would be interesting. The case A2 in Fig. 6 shows that the surface tension gradient along the drop surface would the Shoc curve changes with the value of Ca, and therefore be too small to maintain the local Marangoni convection. further numerical tests are performed. When Ca takes 2:5 × Thus, the Marangoni e#ect would die away eventually as the 10−3 or lower, the Marangoni e#ect occurs, but when it is interphase mass transfer proceeds long enough. Therefore, assigned to be 3 × 10−3 , 5 × 10−3 , 1 × 10−2 with $ = −0:1, the Marangoni e#ect occurs only in the middle stage of no Marangoni e#ect on mass transfer is observed and the interphase mass transfer for a drop has 3nite capacity for Sherwood number drops to the level of the base case with solute extraction. $ = 0. It is evident that there is a critical value of −$=Ca It is noticed that for cases A2 and B2 (d → c transfer), below which no Marangoni convection and enhancement Sternling and Scriven (1959) predicted stable mass transfer, of interphase mass transfer would be observed. It is also but the present simulation predicts signi3cant Marangoni observed in coincidence with our intuitive guess that the convection and obvious Marangoni e#ect of enhancing in- larger the absolute value of −$=Ca, the earlier the Marangoni terphase mass transfer. e#ect occurs. For cases C2 and D2, the Marangoni convection also Although no mass transfer enhancement is observed for evolves and develops into a large, well de3ned 4ow struc- the large Ca, the di#erence exists in the strength of the ture (as indicated in Fig. 7 for D2). However, only moderate Marangoni convection. Fig. 8 shows the di#erent Marangoni enhancement of interfacial mass transfer rate is observed for convection patterns at = 0:30 and the corresponding so- case C2 (refer to case C2 in Fig. 6) and even a little nega- lute concentration contours. It is observed that the solute tive e#ect on the mass transfer is observed for case D2. This concentration contour maps consist of almost concentric cir- seems to suggest that the Marangoni e#ect does not nec- cles, indicating the normal concentration gradient being little essarily lead to the Marangoni e#ect promoting interphase disturbed by possible surface concentration 4uctuation. Al- mass transfer. Here we tentatively suggest to reserve the though when Ca = 3 × 10−3 , the Marangoni convection pat- Marangoni e#ect solely for the mass transfer enhancement tern is quite signi3cant, but it wraps the drop and contributes due to interfacial tension gradient, while use the Marangoni little to interphase mass transfer rate. As Ca increases grad- convection speci3cally for the 4ow structure induced by the ually, the surface tension stress becomes less sensitive to surface tension gradient. the solute concentration, leading to gradual vanish of the An intuitive explanation to the negligible mass transfer Marangoni convection pattern. When Ca = 1 × 10−2 , the enhancement is suggested as follows. The surface concen- pattern is marginally observable, and the streamline map is tration of solute along the drop surface from the front nose to now almost the same as that in the case without solute mass the wake increases as a general trend, for a surface element transfer. will be enriched in solute as it undergoes the mass transfer while moving from the nose to the wake. This gives rise to a strong tangential stress to counteract the normal surface 5. Conclusions 4ow caused by the external bulk 4ow, and it even reverses sometime the surface 4ow direction into toward the drop Axisymmetric numerical simulation of typical spherical nose, thus a developing toroidal vortex is induced, which drops in liquid–liquid systems based on 3rst principles largely isolates the bulk continuous 4ow from the drop demonstrated the occurrence of the Marangoni convection
1826 Z.-S. Mao, J. Chen / Chemical Engineering Science 59 (2004) 1815 – 1828 Fig. 7. Evolution of the Marangoni convection (left) and concentration contours (right) induced by interphase mass transfer at the surface of a spherical drop in case D2.
Z.-S. Mao, J. Chen / Chemical Engineering Science 59 (2004) 1815 – 1828 1827
the solute but do not coalesce into large patterns to be-
come the insulation to the interphase mass transfer.
Since the pattern coalescence relies closely on dimension-
ality of the Marangoni convection, the present axisymmet-
Ca=3×10-3
ric simulation provides insight to the Marangoni e#ect in
liquid–liquid systems in the qualitative context only. The
three-dimensional numerical simulation of the Marangoni
e#ect with respect to deformable drops is believed to of-
fer more accurate interpretation and to facilitate its practi-
Ca=5×10-3
cal application in various sectors in process industry. More-
over, the Marangoni e#ect on the drop scale originates from
sub-drop-scale 4ow structure and localized mass transfer at
the interface, and a general methodology integrating the in-
vestigations on di#erent scales seems quite necessary to ef-
fectively approach the complexity of the Marangoni e#ect,
Ca=1×10-2 either numerically or experimentally.
Fig. 8. Variation of the Marangoni convection as in4uenced by Ca at
= 0:30 for case A2 when no mass transfer enhancement observed
($ = −0:1).
Notation
and enhancement of interphase mass transfer. The following c concentration, mol=m3
tentative conclusions may be drawn: C dimensionless concentration
Ca dimensionless capillary number, 1 UT =!0
1. The Marangoni e#ect as a drop-scale phenomenon is CD total drag coe5cient, dimensionless
the result of the mechanismic coupling between the liq- d drop diameter, m
uid 4ow and interphase mass transfer, and the localized D molecular di#usivity, m2 =s
sub-drop-scale circulation patterns play an important role f( ; ) distortion function
in promoting the convective mass transfer of solute across h ;h scaling factor, m
the interface. The hydrodynamic instability in the form of H ;H scaling factor, dimensionless
interfacial turbulence originates from the surface tension k mass transfer coe5cient, m/s
sensitivity to solute concentration and it occurs only when m partition coe5cient, c2S =c1S
the surface tension is su5ciently sensitive (the quotient p pressure, Pa
of −$=Ca above a critical value). This sensitivity is also Pe Peclet number, dUT =D
dependent on the magnitude of the mass transfer driving R drop radius, m
force, since $ is proportional to the solute concentration Re Reynolds number, dUT 1 =1
in the solute-rich phase. Sc Schmidt number, =D
2. The Marangoni e#ect occurs only when the concentration Sh Sherwood number, kd=D
boundary layer becomes thick enough (after the initial t time, s
mass transfer period with high mass transfer coe5cient) u velocity component, m/s
so that the Marangoni convection in the direction normal u velocity vector, m/s
to the interface may play a signi3cant role in convective U dimensionless velocity
transport of the solute. When the mass transfer driving UT terminal velocity, m/s
force drops to a very low level during the 3nal period, We Weber number, dUT2 1 =!0
the Marangoni e#ect vanishes gradually, as many exper- x; y coordinate in physical plane, m
imental studies have suggested. X; Y dimensionless coordinate in physical plane
3. It is necessary to distinguish between the Marangoni con- (X = x=R; Y = y=R)
vection and the thus induced enhancement of mass trans-
fer. Numerical simulation demonstrates that the situation Greek letters
exists that the Marangoni convection is signi3cant but
no enhancement of interphase mass transfer is observed. ! dimensionless surface tension, !=!0
Apparently this is related to the scale of the Marangoni !0 surface tension of pure system, N/m
convection patterns. The promotion by roll cells on the $ surface tension sensitivity coe5cient
sub-drop scale is the most e#ective: they penetrate into $p perturbation level
the continuous phase far enough to e#ectively transport 0 random number in [ − 1; 1]1828 Z.-S. Mao, J. Chen / Chemical Engineering Science 59 (2004) 1815 – 1828
( ) surface curvature in the plane passing the axis microgravity. International Journal of Heat and Mass Transfer 44,
of symmetry 1983–2003.
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' stress tensor component, dimensionless function on the accuracy of numerical simulation of the motion of a
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Subscripts Marangoni convection of silicon melt by a non-contaminating method.
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