Ind. Eng. Chem. Res. 1995,34, 3278-3288
3278
Spontaneous Motion of T w o Droplets Caused by Mass Transfer A. A. Golovin, A. Nir, and L. M. Pismen* Department of Chemical Engineering, Technion, Haifa 32000, Israel
Spontaneous motion of two droplets driven by surface tension gradients which result from mass transfer between the droplets and the continuous fluid phase is considered. It is shown that mass flux of a weak surfactant from the dispersed phase into the continuous one generates the droplet motion toward each other; the oppositely directed mass transfer makes the droplets move apart from each other. In the approximation of a quasistationary Stokes flow and a low Peclet number in the outer fluid, the velocities of the droplets’ spontaneous motion are calculated as functions of the separation distance between the drops and the Sherwood number characterizing the intensity of mass transfer. The mass transfer driven interaction of two drops significantly influences their motion and coalescence in the gravity field. The velocities of the droplets in this case are calculated as functions of the Marangoni and Archimedean numbers, as well as the ratios of the droplet radii. The results obtained account for a large number of experimental data on the influence of mass transfer direction on the rate of coalescence of drops and bubbles.
1. Introduction It is well-known that massheat transfer processes significantly influence the motion of drops and bubbles when dispersed in a continuous fluid. One of the main mechanisms of coupling between hydrodynamics and massheat transfer is the dependence of the interfacial tension on concentration and temperature. When concentration or temperature gradients are present in the continuous phase, they result in surface tension gradients at the interface of the drops and the bubbles which change the drag force acting on these particles. In the absence of body forces, these gradients lead to a drift of the particles. This phenomenon was first described by Young et al. (1959)and by Levich and Kuznetsov (1962). Thermocapillary migration of bubbles and drops has been attracting wide attention during the past 15 years, especially in application to material processing under conditions of microgravity (Subramanian, 1983;Rogers and Davis, 1990). Concentration and temperature gradients affecting the motion of drops and bubbles are not necessarily imposed externally. They can appear locally, in a fluid which is uniform far from the drop, and can be brought about by the motion itself, thus displaying the feedback between the transport processes and hydrodynamics. The most striking example is the surface motion of adsorbed surfactants induced by the fluid flow around a droplet. This mechanism, described by Levich (19621, is responsible for a large increase of the drag force acting on moving drops and bubbles compared to that predicted by the Rybchinsky-Hadamard formula. It plays an important role in liquid-liquid extraction processes (Skelland et al., 1987). Similar phenomena may be caused by temperature gradients, generated by flowinduced surface distributions of interfacial internal energy, which can also influence the motion of bubbles (Harper et al., 1967;Torres and Herbolzheimer, 1993). Another example of self-induced surface tension gradients was demonstrated by Ryazantsev (1985). He showed that the nonuniformity of fluid flow around a chemically reacting droplet generating, due to convective effects, a nonuniform heat flux at the droplet inter-
* To whom correspondence should be addressed. E-mail:
[email protected]. FAX: 972-4-230476.
@a
\ /
/ \
Figure 1. Schematic representation of the mechanism of interaction between two drops driven by mass transfer. Mass flux from the dispersed phase (dashed arrows) leads to a concentration rise in the gap between the drops. This produces surface tension gradients (bold arrows along the droplet surfaces) which, in turn, force the continuous fluid flow out of the interdroplet region (large curved arrows). This flow causes the drift of the drops toward each other. The direction of the flow corresponds to the transfer of surfactants when the surface tension decreases with concentration. In the opposite case the flow will be directed oppositely.
face, leads to surface tension gradients substantially affecting the drag force acting on the droplet. The effect of convection can be significantly strong and lead to the appearance of a tractive force and to a spontaneous drift of the droplet, even against body forces (Golovin et al., 1986;Golovin and Ryazantsev, 1990;Rednikov et al., 1994). Surface tension gradients causing the motion of drops can also appear in the course of massheat transfer as a result of ageornetric nonuniformity of the system. For instance, if there is a transfer of a weak surfactant from two adjacent droplets into the continuous phase, the concentration in the gap between them is higher and hence the surface tension is lower than in other regions on the droplet surfaces. The appearing surface tension gradients induce a motion of the surrounding fluid out of the region between the drops, thus forcing them to approach each other. When the transfer of the surfactants is from the continuous phase into the dispersed phase, the surface tension gradients are directed oppositely, and lead t o a repulsion of the droplets. In a similar way a drop interacts with a plane surface, either free or rigid. This mechanism of droplets interaction is presented schematicallyin Figure 1. It is responsible for the influence of the direction of mass transfer on the coalescence of drops in liquid-liquid extraction processes and on the attachment between droplets and solid particles in flotation, which has been extensively studied experimentally (Groothuis and Zuiderweg, 1960; Jef-
0888-588519512634-3278$09.0010 0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 3279 freys and Lawson, 1965;Gourdon and Casamatta, 1991; Skelland and Kanel, 1992; Finch and Lyman, 1976). It should be noted, however, that the effect of the direction of mass transfer is, as a matter of fact, coupled with the sign of the surface tension dependence on concentration (or temperature in case of heat transfer). Figure 1shows the case when surface tension decreases with concentration increase. There exist, however, solutes increasing interfacial tension of the solvent (e.g., inorganic salts, the so-called inactive substances) or fluids with surface tension increasing with temperature (e.g., high molecular weight alcohols). For such fluids and solutions, the direction of the described motion of droplets will be opposite. The spontaneous motion of two drops driven by mass/ heat transfer with the surrounding fluid can be considered, t o some extent, as a thermocapillary migration of each droplet in a nonuniform temperature field induced by its neighbor. Such an approach was applied by Rednikov and Ryazantsev (1990) to the modeling of thermocapillary interaction of two droplets with internal heat generation when the distance between them was much larger than the drops’ radii. They neglected hydrodynamic interaction and considered only the motion driven by a nonuniform surface tension, supposing that each droplet moves as if it was embedded in an external thermal gradient corresponding to the temperature field induced at its center by the other droplet. In the framework of this approximation, they showed that a heat source or sink inside the drops caused them to move toward or away from each other along their line of centers. Massheat transfer driven interaction of two drops decreases rapidly with the distance between them, and is more pronounced when the drops are adjacent. In this case, however, the approach applied by Rednikov and Ryazantsev (1990) is no longer valid, and rigorous methods should be used. These methods were developed for hydrodynamic interaction of drops and bubbles migrating in an externally imposed temperature gradient. In the case of two particles, a natural approach is to solve the relevant flow and concervation equations using the bispherical coordinate system (Meyyappan et al., 1983; Keh and Cheng, 19901, as was first proposed by Stimson and Jeffrey (1926). The method of reflections (Happel and Brenner, 1965)can be also effectively applied t o investigate interactions in two- and manyparticle systems (Anderson, 1985; Acrivos et al., 1990). Other methods combining analytical expansion of the flow and temperature fields in spherical harmonics and a subsequent numerical resolution of the boundary conditions were employed for a chain of particles (Keh and Chen, 1992; Wei and Subramanian, 1993) and an ensemble of drops and bubbles (Satrape, 1992; Zhang and Davis, 1992; Keh and Chen, 1993; Zhang et al., 1993) migrating in an externally imposed temperature gradient. When the effect of massheat transfer on the motion of two drops in close proximity is considered, the tangent sphere coordinates combined with a lubrication approximation are expected to be more effective to describe the evolution of a thin liquid film between the colliding droplets. Using this approach, Loewenberg and Davis (1993)showed recently that two unequal nonconducting drops migrating an external temperature gradient approach each other faster than those undergoing an ordinary gravity-induced motion, since the thermocapillary motion of the interface aids the withdrawal of continuous phase from the gap between the two drops.
Also recently, Mileva and Radoev (1993) studied the influence of soluble surfactants on the drag force acting on a solid particle approaching a free surface. Using the lubrication approximation, they showed that the drag force depends significantly on the direction of mass transfer of the surfactants adsorbing at the free surface. In the present paper, we shall study spontaneous motion of two drops driven by mass transfer between them and the surrounding fluid at an arbitrary nonzero separation distance. However all the results can obviously be applied to heat transfer as well. First, we study the spontaneous motion of two drops in the absence of buoyancy forces, when the motion is driven by the surface tension gradients only. Then we consider the effect of mass transfer on the hydrodynamic interaction of drops moving in a gravity field. Finally, we shall study spontaneous motion of two drops in the case when the surfactant transfers from one droplet into the other through the ambient fluid. Our analysis uses some earlier results on interaction of bubbles and drops migrating in an external temperature gradient (Meyyappan et al., 1983; Keh and Chen, 1990),as well as the theory of hydrodynamic interaction of two droplets by Haber et al. (1973). 2. Statement of the Problem Consider two droplets with radii a and b submerged in an immiscible unbounded fluid which is quiescent and has a uniform constant solute concentration at infinity. Due t o concentration nonequilibrium, there is a mass flux either from the dispersed phase into the continuous one, or in the opposite direction. The transferring substance is a weak surfactant (e.g., acetic acid, acetone, or alcohol),which means that the surfactant absorption a t the interface is relatively small and its hindering effect on the motion of the interface is negligible. In this section, we neglect the effect of gravity and study the motion of the droplets driven by interfacial tension gradients only. We suppose that at any moment the motion is sufficiently slow, inertia effects are absent, and the Stokes approximation holds. The velocities of the droplets, as yet unknown, should be found from the condition that the total force acting on each drop is zero. Since the motion of the drops is slow, we presume that the mass transfer in the outer fluid is controlled by diffusion (the Peclet number in the continuous phase is small), and is also quasistationary. In many cases there exists a considerable interfacial resistance to mass transfer in the vicinity of an interface in a dispersed phase. In these cases mass transfer inside the drops can be approximated by the Newton type mass flux between the continuous and the dispersed phase, so that one can introduce a mass transfer coefficient in the dispersed phase and presume that the concentration field at the interface satisfies the Robin type boundary condition. Two general cases should be, however, distinguished: when the interfacial resistance is caused by an adsorption-desorption barrier for the transfer of surfactants through liquid-liquid interface (England and Berg, 1971) or by interfacial chemical reactions, and when the diffusivity in the dispersed phase is much smaller than that in the outer fluid (e.g., if a carbon acid transfers from an oil droplet into water, DoIllDwakr 0.04 or alcohol desorbs from water droplet so that, though being small into air, DwakJDair in the outer fluid, the Peclet number inside the drops is large and the resistance t o mass transfer in the
-
-
3280 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
dispersed phase is thus concentrated in the vicinity of the interface in a thin diffusion boundary layer. In the former case, the mass transfer coefficient inside the drops is determined by the adsorption-desorption kinetic coefficients or by the interfacial reaction rate constants, and does not depend on the motion of the drops and their size. In the latter case, the mass transfer coefficient inside the drops is determined by the thickness of the diffusion boundary layer which is a function of the droplet velocity and of the position a t the interface. Thus, we assume further that the interfacial mass flux is described by the Robin type boundary condition, and that the mass transfer coefficient, KM,inside the drops can be introduced. Then, using the quasistationary approximation, we solve the stationary problem for the concentration field in the continuous phase only. We assume that the cores of the drops have the same concentration and that the concentration in the cores does not change significantly during the characteristic time of the drop motion. We shall discuss both of the described cases for the mass transfer coefficient. However, when considering the mass transfer coefficient determined by the diffusion boundary layer, we shall neglect its variation along the interface and take it equal to an average value dependent on the droplet size and velocity only. All physical properties of the fluids are assumed to be constant, except the interfacial tension, which is considered to depend linearly on concentration: (T = (TO (au/aC)(C- CO), where (TO is the interfacial tension corresponding to the concentration COfar away from the droplets, and aa/aC = constant. We suppose that the interfacial tension (TO is sufficientlylarge to preserve the spherical shape of the drops (the Weber numbers in both phases are small). We choose the following scaling: a for length, q-l(aa/ aC)(Cdz, - CO)for velocity; and we introduce the dimensionless concentration y = (C- Co)/(Cldke- Co), where a is the radius of the smaller drop, CIOis the concentration in its center, ke is the phase distribution coefficient (equal to the ratio of equilibrium concentrations inside and outside the drops), and q is the viscosity of the continuous fluid. The natural choice is the bipolar coordinate system connected with the two droplets (see Figure 2). It is linked with the cylindrical system by the following relations:
+
The interface of droplet 1is described by the coordinate surface 6 = a > 0 and the interface of the droplet 2 corresponds to 6 = -p < 0. Then c = sinh a, the ratio of the drops’ radii (i.e., the dimensionless radius of the rger drop) is r = b/a = sinh dsinh j3 L 1, and the separation distance between the drops is d = cosh a 1 + r(cosh p - 1). If the ratio of the radii and the separation distance are given, then cosh a =
(d
+ 1 + r)2+ 1 - r2, 2(d + 1 + r )
coshB= (d
+ 1 + r)2- 1 + r2 2r(d + 1 + r )
In the chosen coordinate system, the flow fields in the continuous fluid and inside the drops are described by the dimensionless Stokes equations
Figure 2. Bispherical coordinate system connected with the pair of drops.
Here q!~ is the stream function such that the velocity components
the superscript (0) corresponds to the continuous fluid, and the superscripts (a)and (B) denote the fluid inside the droplets ((a) for > a and (j3) for 6 < -@I. There should be no fluid flow at infinity and the velocities of the fluid inside the drops should be bounded, hence,
e
6 = 0, 5 = 0, q(O)/$
= 0;
g = fm,
q(aS)/$ < 00
(2) The velocities of the dispersed and the continuous phases should be equal at the drop interfaces. Besides, for a moving droplet with undeformable interface, the component of the drop velocity normal t o the drop’s interface and the normal component of the velocity of the ambient fluid must be equal a t the surface of the drop. This gives the following boundary conditions at the drops interfaces, corresponding to 6 = a and 6 =
-p: 6 = a,-B:
(3) (4)
where Val and Vas) are as yet unknown projections of the dimensionless velocities of the drops on the z-axis. The concentration field in the continuous phase, uniform far from the drops, is described by the Laplace equation with the appropriate boundary condition a t infinity
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3281
Ay = 0
(5)
E=O, c=o, y = o The mass transfer from the dispersed phase is described by the Robin type boundary conditions a t the interfaces of the drops
E = a: 6 = -/3:
+
h 3 Sh")(y - 1)= 0
aE
C;;,","@)are Gegenbauer polynomials, Pn+) are Legendre polynomials, and p = cos 5. First, the concentration field in the continuous phase should be found. Substituting (9) into the boundary conditions (61, we arrive a t the infinite system of linear equations for the coefficients E, and F,: e-; ( a n En-l ) -eP)E,
+ e y ) E n + l+ f?in)F,-, - +,""IF, + fF)F,+l= -cSh'")u;'
e?i'')En-l
- e P ' E , + ey'n)E,+l+ f!T'F,-l - f t " ) F , + ff3n)F,+1 = cShcB)ukB) (10)
h 9 - Sh'B'(y- 1)= 0
aE
(6)
where y is the dimensionless concentration, Sh(aJ) = are the mass transfer coefficients inside the droplets with D being the diffusivity of the surfactant in the outer fluid. The remaining boundary condition which links the concentration and the velocity fields is the balance of the tangential stresses a t the interfaces of the drops. This balance gives
K'$B'k,a/D are the Sherwood numbers, and
The coefficients e, f, and u are given in the Appendix. The coefficients E, and F, decrease with n. System (10) is solved as proposed by Meyyappan et al. (1983) and Keh and Chen (1990). We truncate the system at some large n = N setting E, = F, 0,n > N, we solve the finite system; then we choose some N 1 > Nand solve it again for n I N1 and repeat the procedure until the first coefficients with n less than a certain given number M < N as well as the values of the drop velocities become constant within a given degree of accuracy. The value of N depends on the desired accuracy and on the separation distance; e.g., for d 2 0.5, N = 20 is sufficient for the accuracy of 0.01%;however, the value of N grows rapidly a t d 0 where the calculations based on the bispherical coordinates fail. In order to find the coefficients in the expansions of the stream functions, we rewrite boundary conditions (7) using (3) and (4), in the following form:
-
E = a: 6 = -p: Here IItc is the tangential component of the viscous stress tensor,
q* is the ratio of the viscosities of the inner and the outer fluids.
3. Solution The solution of the problem (1)-(7) is analogous to that described in detail by Meyyappan et al. (1983) and Haber et al. (1973). General solutions of (11, (2), and ( 5 ) were found by Stimson and Jeffrey (1926):
c ca
q(i)= (cosh 6 - p)-3/2
w'~)(~)C;~'&.4, i = 0,a,,6
n=l
(8)
(11) = a, and the
where the upper sign corresponds to 6 lower sign relates to 5 = -/I. Substituting y , w'j', and w(:L'p)in ( l l ) , one finds the following linear system for the coefficients in the expansions of the stream functions:
,cn
(0) (0) 0) (0) (a) C(a)J(L'p) C(L'p))T. The where b n = (An 8, Pn An n n 2 n vectors tfB)depend on the coefficients E, and F, and the Sherwood numbers, and are given in the Appendix together with the matrix & and vectors vFB).Every coefficient (element of the vector b,) can be thus represented as the sum of four terms proportional to ~ 3 S h ( ~and $ ) c2VIa$),respectively: 7
b, = b2)c3Sh(a)+ b?)c3Sh@)- b(a)c2v(a) - bf)c2flB) n and
U, = E , cosh(n+l/2)6 + F, sinh(n+l/2)5
Finally, requiring the force acting on each droplet t o be zero, (see, e.g., Happel and Brenner (1965) and Meyyappan et al. (1983) for details)
3282 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
we obtain the following system for the dimensionless velocities Vu)and W):
V(”A; Va’A,
+ V(B’A,’ = cSh“’A; + cSh‘P’A$ + V(s’Aj = cSh‘u’A; + cSh”’A,
(14)
are given in the Appendix. where The A+’f@ so ution of the problem is thus reduced to the solution of two systems: system (10) for the concentration field and system (14) for the velocities of the drop motion. If the interfacial resistance to mass transfer is governed by an adsorption-desorption barrier or by an interfacial chemical reaction, we can consider Shea' = Sh(8)= constant. In this case systems (10) and (14) are decoupled, and we first solve system (10) and obtain the concentration field in the continuous phase. Then we solve system (14) which is in this case linear, and obtain the velocities of the drop motion as functions of the Sherwood number, the separation distance, and the ratios of the radii and viscosities. The solution is more complicated if the interfacial resistance to mass transfer is determined by the diffision boundary layers inside the drops when the diffisivity in the dispersed phase is much less than that in the continuous one, and the Peclet number in the dispersed phase is large. In this case, the Sherwood numbers Sh(a,P)are functions of the droplets’ velocities and sizes. Indeed, the mean thickness of the diffision boundary layer inside the drop with the radius a moving with the velocity U‘) is dg) d D 3 , where Dd is the diffusivity in the dispersed phase (here subscript “d”means “droplet”and subscript ”c”means “continuous phase”). From the mass balance at the interface,
Figure 3. Isoconcentration lines of the concentration field around two equal drops undergoing mass exchange with the surrounding fluid; Sh = 0.8, d = 0.4.
-
Figure 4. Flow pattern (streamlines) generated by mass transfer driven interaction of equal drops; Sh = 0.8, d = 0.4,v* = 0.5.
the average mass transfer coefficient can be estimated as = Jva’Dd/a
and hence, the Sherwood numbers are
keJ(Dd/Dc)Map’/r (15) where r = bla is the ratio of the drops radii and Ma = (~u/aC>(C&, - Co)a/(yD,)is the Marangoni number. Thus, the concentration field in the continuous phase is now dependent on the velocities of the drop motion, and systems (10) and (14) are coupled and cannot be solved separately. Besides, the system (14) is now nonlinear since, in this case, the Sherwood numbers and the coefficients At,, depending on the Sherwood numbers, depend on the velocities V‘@)themselves. The two
coupled systems (10) and (14) can be then solved, e.g., by means of simple iterations. 4. Results and Discussion
4.1. Equal Drops. Two equal drops move with equal velocities. As expected, they approach each other if mass transfer of a weak surfactant goes into the continuous phase, and they move away one from the other if mass transfer is directed oppositely. A typical concentration field around the two droplets and the flow pattern are shown in Figures 3 and 4, respectively. The values of the surface tension gradients which govern the motion of the drops are determined by the intensity of the mass transfer characterized by the Sherwood number and by the separation distance between the drops. The dependence of the droplet velocities on the Sherwood number, when the latter is constant, is presented in Figure 5a for various values of the separation distance between the droplets. There exists some
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3283
-
to the Shenvood number when the latter is large. Thus, a t all separation distances V a Sh at Sh 0, and V Sh-l at Sh =, which can be seen in Figure 5a.
-
The velocity dependence on the separation distance between the drops is given in Figure 5b for the case of a constant Shenvood number. Again, there exists some “optimal” distance a t which the velocity is maximal. Indeed, the larger is the distance between the drops, the smaller is the concentration gradient driving their motion. On the other hand, when the gap between the droplets is very small, it becomes very difficult to squeeze the fluid out of a narrow interdroplet region. However, at d 0 the problem cannot be solved within the framework of the bispherical coordinates, and the lubrication approximation has to be applied in order to find the asymptotic behavior of the drops; this, however, is beyond the scope of the present paper. When the distance between the drops is very large, their hydrodynamical interaction is negligible and two distant drops move as if each of them was embedded in the external concentration gradient induced by the other droplet, and underwent a “thermocapillary drift”. Simple calculations analogous to those made by Young et al. (1959) and by Rednikov and Ryazantsev (1990) allow finding the following asymptotics for the dependence of the velocity of the drops spontaneous motion when d
-
ib \
.YSh=lO
\\‘\
-
-:
V=
.” 1
o-~
1o-2
on
lo-’
1
10’
10’
d
R
0.025
i
: : 0.0°5
t
:0
1
3
2
I 4
5
‘I’
Figure 5. Dimensionless velocity V of the spontaneous motion of two equal droplets caused by mass transfer: (a) as a function of the Shenvood number Sh, q* = 0.5; (b) as a function of the dimensionless separation distance d, v* = 0.5; dashed lines show asymptotical dependence (16); (c) as a function of viscosity ratio v* (d = 0.3, Sh = 2).
value of the Shenvood number at which the velocity is at maximum, since the concentration gradients generating the surface forces vanish both at very high and at very small transfer rates. Due to the linearity of the hydrodynamical problem, the velocity is proportional to the concentration gradients along the drop surfaces. As can be easily found from the boundary conditions (6), these gradients are proportional to the Shenvood number when the latter is small, and inversly proportional
(1
+
1 (3/2)q*)(Sh
+
Sh 1)(Sh
+ 2)d
-2
(16)
This asymptotic is shown in Figure 5b by the dashed lines. It should be noted, however, that in the case when the Shenvood number depends on the drop velocity, the velocity dependence on the separation distance will be qualitatively different and decrease with the distance more rapidly since the Shenvood number will also decrease with the increase of the separation distance. The effect of the viscosities of the drops on the velocity of their spontaneous motion is shown in Figure 5c. The velocity decreases monotonically with the viscosity of the dispersed phase, and is at maximum in the case of two bubbles. Let us now estimate the velocity U of the spontaneous motion of two drops of water-alcohol solution in air. In this case the diffusivity inside the droplets is much smaller than in the surrounding gas and if the Peclet number inside the drops is large there exists a diffusion boundary layer along the inner side of the interface governing the rate of mass transfer. The velocity of the drop motion should be thus found by solving the coupled nonlinear system (10) and (14) as described in the previous section. Besides, it is worth noting that in this case there is always enough “fuel” inside the drops to make them move. Indeed, the change of the concentration inside the drop (Ac)d during the characteristic time of the drop motion z alU is determined by the mass losses AM = 4na?t, where j = ShD,(AC),da is the mass flux at the droplet surface and (AC),o = C&, - COis the initial concentration difference in the continuous fluid. Hence,
-
where (AC)do is the concentration difference inside the drop across the boundary layer, CIO- k,Co. The physical parameters of the solution and of air at the room temperature can be found elsewhere (see, e.g.,
3284 Ind. Eng. Chem. Res., Vol. 34,No. 10,1995
:::\ 0.6 -
i
> 0.5'
0.21
0.1' 1
1 1.5
2
2.5
r
3
3.5
4
Figure 6. Dependence of velocity ratio u* = VWVa)on radii ratio r = b/a in the case of spontaneous motion of two different drops; v* = 0.3, Sh = 0.8.
Weast (1986)). In order to estimate the velocity of the drops motion we recall that
(17) and apply the iteration procedure to systems (10)and (14),using (15)and (17). Taking two drops of 40 vol % water-alcohol solution placed in air, with 1 mm diameter and 1 mm separation distance ((au/K!d) = 0.224 g/(cm*s),q* = 154,D,= (dyn/cm)/vol %, q = 1.8 x 0.137cm2/s,Dd = 1.28 x cm2/s,k, = 1.27 x io4), we obtain U = 2.4 m d s . The Peclet numbers inside and outside the drops in this case are Ped = 925 and Pe, = 0.08, respectively. The Reynolds numbers are Red = 0.4 and Re, = 0.08, and the Weber numbers are Wed = and We, = The estimates show that the effect is rather strong and that the approximation made is reasonable. 4.2. Unequal Drops. When the sizes of the drops are not equal, the velocities of their spontaneous motion are different, but always directed oppositely, as in the case of two equal drops. Again, if the mass flux is directed out of the dispersed phase, the drops approach each other, and they repel each other in the opposite case. A larger drop moves with a smaller velocity. Figure 6 shows the dependence of the ratio of the velocities u* = VWVO)on the ratio of the drop radii r a t various separation distances for the case of a constant Shenvood number. As in the case of two equal drops, it can be found that the velocities of spontaneous motion of two drops with different sizes decay as d-2 at sufficiently large d. The flow pattern around two drops with unequal sizes differs from that in the case of two equal drops, by a flow separation appearing near the smaller drop; see Figure 7. The larger is the difference between the droplet radii, the closer is the ring of the flow stagnation to the smaller droplet. There is no flow stagnation, however, near the larger drop. 4.3. Combined Effect of Gravity and Capillarity. The mass transfer driven interaction of two droplets described above will substantially influence the rate of collision of two drops moving in a gravity field. We shall demonstrate this influence for the case of two drops moving in the gravity field along their line of centers. In order to find the drop velocities, one has to modify the conditions (13)and equate the viscous forces acting
Figure 7. Flow pattern (streamlines) generated by mass transfer when the sizes of the drops are different; Sh = 0.8, v* = 0.3,r = 1.8.
on the drops to the mass forces. It is more convenient in this case to choose another scaling for the velocity, namely, the characteristic velocity of the droplet motion driven by buoyancy: AQgu2/q,where AQ is the density difference between the dispersed and the continuous phases and g is the acceleration of gravity. As in the previous sections, we compute here quasistationary velocities a t a given distance between the drops. (In principal, since the problem is linear, the results of this section can be obtained simply by adding the velocity of the drop motion driven by surface tension gradients t o that caused by the gravity field, computed by Haber et al. (1973). However, we reproduce here the fresh solution of the whole problem to not make the reader refer to the cumbersome computations of the old problem.) The force balances for the dimensionless (in the new units) velocities give
m=
(ao/aC)(AC), -
--
Aem2
Ma
(Ar)(Sc)
is the parameter characterizing the influence of the
surface tension forces on the gravity-induced motion; Ma = u(aa/Z)(AC)d(/(rlo) is the Marangoni number, Ar = A ~ g a ~ / ( ris/ vthe ) Archimedean number, and Sc = v/D is the Schmidt number. The interaction of two drops caused by mass exchange with the surrounding fluid strongly affects their motion in a gravity field. The velocities of the drop motion as well as the concentration and the flow field can be found by solving the systems (18)and (lo),the solution being different for the constant Sherwood number and for the velocity-dependent one. Figure 8a shows the flow pattern when two drops descend in the outer fluid without mass transfer (m = 0). Figure 8b depicts the flow in the case when mass transfer goes from the drops into the outer fluid (rn -= 0) and the drops attract each
Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 3285 a
b
la
d
C
r 1.6r 1.6-
b
1.4-
1.21Le
0.8
Figure 8. Influence of mass transfer on flow in the case of two drops moving in the gravity field; r = 1.3, d = 0.4, Sh = 0.8,'I* = 0.3. (a) m = 0 (no interaction), Val= -0.546, VI) = -0.609; (b) m = -15, Va)= -0.894, Va) = -0.364 (thermocapillary interaction accelerates the upper drop and impedes the lower one); (c) m = -37.28, Pa)= -1.410, VI) = 0.000(the lower drop levitates in the gravity field); (d) m = -60, Val= -1.936, VS) = 0.371 (the lower droplet moves against gravity).
other due to surface tension forces. As a result, the velocity of the lower droplet decreases and the velocity of the upper one increases. Figure 8c presents the situation when droplet interaction induced by mass transfer completely stops the lower droplet which becomes levitating in the surrounding fluid. The capillary interaction of the drops can be so strong that it forces the lower droplet to move up, against the mass forces, as shown in Figure 8d. The effect of mass transfer on the collision rate of the two drops can be measured by the relative change of their relative velocity, AV/(Av)o, where (AVO is the relative velocity of two drops moving in the gravity field in the absence of the surface tension forces. In the case of the constant Sherwood numbers, system (18) gives for this quantity
+
+
AT(A, A i ) - A;(Ai AB+) 3c d(Sh,d,r) = -Sh (19) r3(A: fi Ai) - (Ai Ai) Figure 9a presents the dependence of 6 on r a t a fxed Sherwood number and viscosity ratio, and for various separation distances between the drops. This function tends to infinity when r 1 since two drops move in the gravity field with equal velocities (Haber et al., 1973) and their relative velocity (AI90 is zero. This means that the effect of mass transfer on drop collision is much more pronounced when the sizes of the drops are close to each other. It should be noted that both 6 and rn can change their signs and, thus, drops which would never coalesce if there was no mass transfer, will collide
-
0.60.40.20 0.2;
0.5
1
1.5
2
3
2.5
3.5
I
4
Sh
2t IC 1.5-
"0
\
'-1 r4.2
0.5
1
1.5
2
2.5
3
3.5
d
-A' - 1 md(Sh,d,r) (Am,
+ +
-
+
Figure 9. Parameter 6 determined by (17) and describing the relative effect of the interaction brought about by mass transfer, on the drop motion in the gravity field: (a) as a hnction of the drop radii ratio r; (b) as a function of the Sherwood number Sh; (c) as a function of the dimensionless separation distance d (q* = 0.3, Sh = 0.8).
if the transfer is intensive enough; on the contrary when directed oppositely, mass transfer can prevent coalescence of two drops which would collide if they moved in the absence of the surface tension gradients caused by mass transfer. Parts b and c of Figure 9 give the dependence of 6 on Sh and on d, respectively, a t fixed separation distance and various radii ratios. As could be expected from the results of section 4.1, the dependence 6(Sh) has a maximum, and the influence of mass transfer on the droplet coalescence vanishes when the transfer is either
3286 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
Figure 11. Streamlines of the flow driven by the interparticle mass transfer when two equal drops move with equal velocities in the direction of the droplet 4 t h higher concentration: 1, corotating vortices around each droplet; 2, large vortex enveloping vortices 1; 3, narrow counterrotating vortex. Figure 10. Isoconcentration lines around two equal drops in the case when the surfactant transfers from one drop into the other through the ambient fluid. The absolute values of the concentration differences between the drops and the surrounding fluid are equal; Sh = 0.8, d = 0.4.
very weak or very intensive; 6 also decreases with the growing separation distance between the drops. 4.4. Interparticle Mass Transfer. In this section we consider another type of droplet interaction driven by mass transfer induced capillary forces. It occurs when the surfactant transfers from the one droplet into the other through the continuous phase (or, in case of heat transfer, when one droplet is hotter and the other is colder than the surrounding fluid) and the resulting surface tension gradients force the drops to move in the same direction. For the sake of brevity, we consider here only a particular case of two equal drops with equal absolute values of the concentration differences between them and the outer fluid. In the absence of the gravity field, such drops will drift as a whole with equal velocities in the direction of the droplet with a higher concentration as if they were submerged in an externally imposed concentration gradient. This problem is described by the same equations and boundary conditions (1)-(7), however, the mass transfer balance (6)should read
6 = a: 6 = -a:
10"
'
"""'
'
'
'
"""
'
'
" " "
'
'
"""'
'
''7
'
'
'
' '1o2, A
! a 1o.2 :
t 1o
>
.~
. 1o
-~
1
10.61 1oQ
'
'
'
"'
1o.2
'
""""
'
" " " "
10"
1 oo
'
" " " "
10'
d
h Z! + Sh(y - 1) = 0 86 h
$ - Sh(y + 1)= 0.
(20)
In the considered symmetric case V(O)is even and y is odd, while in the case of equal drops studied in section 4.1 v ( O ) is odd and y is even. The concentration field is shown in Figure 10, and the streamlines of the flow generated by the pair of drifting droplets linked by mass transfer are presented in Figure 11. Besides two corotating vortices around each drop (l), there exists a third large vortex (2), envelopingthe vortices around each droplet. This large vortex separates the two vortices (1)from the third, narrow counterortating vortex (3), between the two vortices (1). The location of vortex (3) depends on the separation distance between the drops. The larger is
10-51 1o-=
10-2
lo-'
1op
10'
1o2
I
1os
sh
Figure 12. Drift velocity of two equal drops in the case of interparticle mass transfer: (a) as a function of the dimensionless separation distance d ; (b)as a function of the Shenvood number; (7 = 0.5).
the separation, the closer is this vortex to the droplet line of centers. The drift velocity of such a droplet pair decreases with the increase of the separation distance. In the case of a constant Sherwood number, this dependence is monotonic, and the velocity is maximal in case of two
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3287 Table 1. Matrix & in Eq 12
0
0
0
0
0
0
0
0
0
0
-r]*n'_e,
-9'n:e:
0
-
touching drops, Figure 12a. When d the velocity dependence on the separation distance and the Sherwood number is also described by formula (16). The dependence of the driff velocity on the Sherwood number, when the latter does not depend on the droplet velocity, is depicted in Figure 12b and appears to be qualitatively the same as in the case of two equal drops studied in section 4.1. OQ,
-n+ea+
O
0
-q*n2_e,
0
Appendix The coefficients in the first of eqs 10 are
e?)
= (sinh a
+ 2 ~ S h ' ~cosh(n+l/2)a ') + (2n
Nomenclature a = radius of the smaller droplet Ar = ( @ d - ec)ga3/(~v) = Archimedean number b = radius of the larger droplet c = sinh a C = concentration of the transferred surfactant CO= concentration in the continuous phase at infinity Clo = concentration in the core of the drops d = dimensionless separation distance D = diffusivity g = acceleration of gravity h = (cosh 5 - COS C)/C k, = phase distribution constant KM= mass transfer coefficient in the dispersed phase Ma = (au/X)(Cldke- Co)a/(vD)= Marangoni number r = b/a = ratio of drops radii Re = Ua/v = Reynolds number Sc = v/D = Schmidt number Sh = KMk,a/D = Sherwood number U = velocity of the drop spontaneous motion V = dimensionless velocity We = 2a~U21u= Weber number
Greek Symbols y = (C - Co)/(Cldk,- CO) = dimensionless concentration 6 = G(Sh,d,r) = function defined in eq 18 77 = viscosity v* = q d / v c = ratio of the viscosities v = kinematic viscosity 8 = density; radial cylindrical coordinate u = surface tension 6, 5 = bispherical coordinates = a = surface of the smaller drop 6 = -j3 = surface of the larger drop = stream function
e
-q*n:ea+
Indices c = continuous fluid d = droplet a = smaller drop j3 = larger drop
Acknowledgment A.A.G and L.M.P. acknowledge the support by The Israel Science Foundation. A.A.G. acknowledges the support of the Ministry for Immigrant Absorption.
-n-ep
f>n'
= (sinh a
+ 2cSh'")
+ 1) cosh a sinh(nSll2)a
+
sinh(n+l/2)a (2n 1)cosh a cosh(n+l/2)a
+
fp"' = (n + 1) cosh(n+3/2)a and
The coefficients and. f p s n ) can be obtained by replacing a with -/? and Sh(") with -Sh@) in the expressions for e(a+)and Matrix Ai$ in eq 12 is given in Table 1. Here n- = n - 1/2, n+ = n + 312, (c,s)& = (cosh, sinh)n*(a,B), and = e-n+W). The vectors tjP'@' and vjP'@) are paln).
t:) = (0,0,0,0,0,0,~~',0) tf) = (0,0,0,0,0,0,0,-~')
3288 Ind. Eng. Chem. Res., Vol. 34,No. 10,1995
where
e)n(n + =
1)[ 1 (-4&-(n+l/a)a 2 sinh a 4cSh'a' E,-,s, + Fn-,c, - E,+p; - Fn+l~;) +
+
E, cosh(n+l/2)a
+ Fn sinh(n+l/2)a
+ 1)[ -1 (-4J&-(n+1/2)8 2 sinh B + 4cSh'p' E,-,sj + Fn-,c; + E,+,s; - Fn+,c;> +
= n(n
+
1
-
I
E , cosh(n+l/2)p - Fn sinh(n+l/2)p
The solution of the system (12)can be represented in the form
bn = c3Sh(a$(ta) + c3Sh(B$(b) - C2V(a$(a) - ,2V@b:B) n n n where b!)IaJl = bFP) = (a#). 41t n(ad% n Vn Finally, using (13) and (4) the linear system (14) is obtained with the following coefficients: ca
n=l ca
m
n=l m
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Abstract published in Advance ACS Abstracts, August 15, 1995. @
