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20STUDY OF MASS TRANSFER BETWEEN A DROPLET AND A CONTINUOUS LIQUID: PRELIMINARY EXPERIMENTAL RESULTS
I.E. Karpunin1, N.V. kozlov2, f. plouraboue3
1 Perm State Humanitarian Pedagogical University, 614990, Perm, Sibirskaya, 24
2 Institute of Continuous Media Mechanics UrB RAS, 614013, Perm, Acad. Korolev, 1
3 Institute of Fluid Mechanics of Toulouse (IMFT), Toulouse University, CNRS, INPT, UPS, Toulouse, France
Abstract. In the paper, a possibility of excitation of droplet inclusion oscillations in a continuous liquid is considered. For this, we propose to use a container with a simply connected cylindrical cavity that performs rotational oscillations relative to its axis. A theoretical formulation of the problem on the oscillating shear flow in the layer is given. Experimental techniques are developed. Preliminary experimental results demonstrate the possibility of periodic deformation of a liquid drop or an air bubble by means of shear oscillations of the surrounding liquid.
Key words: simply connected domain, coaxial layer, rotational oscillations, shear flow.
INTRODUCTION
The problem of mass transfer control at interfaces is one of the key in many chemical and biological technological processes. An example is the technology of liquid-liquid extraction, which consists in the transfer of a solute through the interface between two immiscible liquids to one of them containing a selective solvent. Liquid-liquid extraction is used to extract, separate and concentrate solutes and is one of the most com-
© Karpunin I.E., Kozlov N.V., Plouraboué F., 2019 DOI: 10.24411/2658-5421-2019-10914
mon processes in chemical technology. Extraction methods are used in the pharmaceutical and oil refining industries, as well as in nuclear-power engineering. The extraction of a certain component dissolved in the surrounding fluid occurs when the droplet moves, floats or sinks, due to buoyancy. The laws of droplet motion and mass transfer are determined by many parameters and attract the close attention of scientists; an analysis of results and a review of articles can be found in [1]. If droplet boundaries are mobile, then due to viscous friction in the moving droplet a toroidal vortex is excited, mixing the liquid in the droplet volume. Note that the rate of saturation of a drop with a reagent depends on the ratio of its area to its mass and decreases with the size. However, the boundaries of droplet inclusions lose their mobility as a result of surfactant accumulation, which leads to a decrease in the mass transfer rate [1]. In this case, the droplet moves in the surrounding fluid as a solid and the mass transfer is carried out only due to diffusion. Therefore, the intensification of mass transfer in a droplet with a surfactant-saturated surface is an important technological problem.
A solution to this problem can be the generation of flows that deliver reagents to the interface. An interesting technique is used in microfluid-ics, when flow inside a droplet is created as a result of its viscous friction against the channel walls, through which the droplet is pressed [2].
Another example is the excitation of averaged fluid motion (steady streaming) in the drop and beyond it as a result of its shape oscillations. Such a problem was considered theoretically in [3] in the high-frequency limit. The works [4-7] are devoted to a systematic experimental study of the structure and intensity of steady streaming in oscillating elastic containers, which simulate droplet inclusions with a surfactant-saturated (tangentially incompressible) interface. The studies were performed in elastic containers of various shapes with different types of vibrations in a wide range of dimensionless frequencies. They showed that the averaged flows are also generated in the region of moderate and low dimensionless frequencies and revealed general patterns of the flow structure transformation with the dimensionless frequency. The conducted studies testify in favor of the applicability of low-frequency oscillations for the excitation of flows in droplets several millimeters in size, characteristic for technological processes.
Various ways to provide steady streaming inside droplet are possible. According to [6], torsional oscillations of a deformable spheroid generate efficient fluid mixing in the container by steady streaming. The vibrational dynamics of various phase inclusions in a container with a simply connected cylindrical layer and the averaged lift forces acting on
them were studied experimentally and theoretically in [8, 9]. These works show that an oscillating azimuthal shear flow may be generated by rotational oscillations of a coaxial layer whose outer and inner cylindrical walls are connected with a partition extended along the cylinder axis. This shear flow may be used to excite the torsional oscillations of a droplet.
It is proposed to excite the oscillations of a drop using an oscillating shear fluid flow. The main goal is to study mass transfer at the boundary of the droplet (phase inclusion) under the conditions of steady flow generation as a result of oscillations that are excited by rotational vibrations of a cavity of a certain shape. The objective of this paper is to develop and debug an experimental setup and methods for studying the vibra-tional dynamics of fluid phase inclusions.
1. PROBLEM FORMULATION
We consider the coaxial liquid layer 1 separated radially by a divider 2 (Fig. 1). The volume is filled with liquid, and a droplet 3 of the second liquid, immiscible with the first one, is introduced inside. The droplet has smaller density than the continuous liquid. The container performs rotational oscillations as a whole with the zero average rotation. At this, the droplet is brought in motion mainly under the action of inertia and viscous forces. We study the droplet dynamics and its impact on the steady streaming inside and outside the droplet, and finally the rate of mass transfer between the dispersed and the continuous phases by steady streaming.
Fig. 1. The cavity schematic: 1 - container, 2 - divider, 3 - inclusion
1.1. Theoretical formulation. The Reynolds number of the configuration under study is defined by Re = Ro2Wv / n , which is also related to the Womersley number Wo2 = Re . The Reynolds number range of the presented experiments are between 2.5 -103 -1.6 -105 for water and 1.3 -102 - 6.6 -103 for silicone oil. Inside a classical Taylor - Couette configuration, when differential rotation is applied and without divider in the inertialess regime, a circular oscillatory Couette flow is expected between the two cylinders. In such Taylor - Couette device, a transition from circular Couette flow (CCF), to Taylor vortex flow (TVF) and wavy vortex flow (WVF) is expected at Rec = 120 and 151, respectively [10]. Here we consider the common rotation of both cylinders, with a divider, a distinct configuration to classical Taylor-Couette. Adding the divider between the two cylinders drastically changes the flow field solution from the classical Taylor - Couette flow. In the oscillatory frame, the flow field can be analyzed as the one produced inside a fixed and closed cavity subjected to an oscillating inertial force field. As such, it resembles the second Stokes problem arising between two parallel plates from an imposed oscillating pressure gradient. In such configuration the Stokes layers are produced by inertia effects giving rise to a viscous boundary layer, the thickness of which is dS =y]2n / Wv . Let us now
analyze the main feature of the resulting axi-symmetric azimuthal flow in this configuration. We consider an angular velocity Qr(t) = j0Wv sin(Wvt)k whose rotation direction k is parallel to the cylinder axis, and whose center of rotation is placed at the origin of cylindrical coordinate system. In this theoretical section we consider the case of small amplitude vibrations j0 ^ 1.
Following [8, 11] the Navier - Stokes equations in the rotating frame
is:
P\ + v'Vv + 2^r x v + x(^r xr) + xr | = -Vp + mDv (1.1)
^ at at J
Where 2fir x v is the Coriolis acceleration, -piir x (ftr x r) the centrifugal force and /dt)xr , the tangential acceleration. We now use the following dimensionless choice: the oscillation period to build di-mensionless time Wvt = t , the external radius as a reference length-scale r = Ro r , the rotational velocity of the external wall of the cylinder
v = joWv Ro v exp(iQvi) and the inertial pressure p = j0r(RoWv)2 p exp(iQvi), so as to obtain the dimensionless, complexified, quasi-steady formulation of (1.1)
— v + v•Vv + 2kx v + kx(kxr) + — kxr = —— Vp+ —Av . (1 2)
jo jo jo Re
Then, seeking for an asymptotic expansion for small amplitude oscillation,
p = p o + jo pi +...
~ ~ ~ (1.3)
v = v o +jo vi + ...,
and using (1.3) in (1.2) whilst collecting each order, one gets at leading order
iv o + ik x r = -V po,
showing that the acceleration of the fluid and the centrifugal acceleration are absorbed by the leading order oscillating pressure, built along the azimuthal direction only. At next order
iv 1 + vo • Vvo + 2k x vo + k x (k x r) = -Vp1 DDvo.
Re
Using again the pressure, as in static gravity, to absorb for the extra centripetal acceleration field, one gets
iv1 + k x (k x r) = iv1 + rer =-V p>1,
building, this time, a pressure gradient correction in the radial direction, so that, the velocity field fulfills
vo •'Vvo + 2kxvo = -1 Dvo. (1.4)
Re
In this contribution we propose, as a first step, so as to be able to match some experimental observation, to analyze the predominant azimuthal contribution of the flow. At this stage, it is also interesting to realize that, from the presence of the divider, this approximation does not hold everywhere. The flow cannot fulfill the azimuthal invariance and does indeed depend on j nearby the divider. There is also a need for an extra radial velocity component associated with re-circulation nearby the divider, to expel the fluid out from the dead-end. Nevertheless, these flow complexities are localized nearby the divider, inside boundary layers, and are not the main interest of the study. Hence, it is interesting to provide an approximate expression for the azimuthal velocity field v out of complex boundary layer, either near both inner or outer cylinder radius, or nearby the divider.
Hence, as a first step, we then seek for an approximate solution when neglecting the boundary layer effects at large Reynolds number. Seeking
for an azimuthal axi-symetrical velocity vo = (0,v0(v),0), in (1.4) and using the standard decomposition v0-Vv0 =V(v0 • v0)/2 + Vx v0xv0 leads to the following non-linear azimuthal momentum problem
V ^ - 2V0 = 0 dr
whose non-trivial solution is given by v0 = 2r + Const This outer solution automatically matches the perfect fluid boundary condition v • n = 0 at r = Rj / Ro ° Rj (where we have introduced the obvious notation R = Ri / Ro) and r = 1. The constant Const is determined by the zero azimuthal net flux condition, since the divider closes the cavity, including the conditions of boundary layers. So, the velocity field far from the divider resembles a Couette flow, having a spatially uniform, oscillating shear, the only non-zero component being, at leading order, srv = 2 jW exp(iWvt).
1.2. Experimental techniques. The container 1 is made of acrylic glass (Fig. 2), its parts are assembled and sealed hermetically. The front flange 2 is used for observation and video registration. The rear flange 3 contains three openings: 4 - an inlet used to fill the container with the working liquid, 5 - an outlet for the air, 6 - a nozzle used for introducing a droplet in the working liquid.
Fig. 2. Experimental setup
The dimensions of the annulus are the following: inner radius R = 3o.omm, outer radius Ro = 5o.omm, height h = 21.omm. The angle between the divider and the nozzle is p / 2 rad. The inner diameter of the nozzle d » o.8 mm. The divider thickness is 4.4 mm. The working liquids were water (p1 = 1.ogr/cm3, n1 = 1.ocSt) and silicone oil
(p2 = 0.91gr/cm3, n2 = 5.1 cSt). Also, industrial oil I-20A (p2 = 0.87 gr/cm3, n2 = 58 cSt) was used with the air bubble as a phase inclusion. Before an experiment, the container was filled with water (industrial oil) through the inlet and the air bubbles were chased through the outlet. Then the outlet was hermetically closed. An oil droplet (air bubble) was injected through the nozzle, and when the container started vibrating the droplet (bubble) detached from the nozzle and floated towards the upper boundary. The volume of the droplet (bubble) was about 0.1 ml.
The container is mounted on a rotating platform 7 that is installed on a shaft with bearings 8. A flexible coupling 9 transmits the rotation from a stepper motor SL86STH118-6004A 10 to the container. The motor is controlled by a OSM-88U unit and fed by a LRS-350-48 DC source. The controller is programmed for periodic rotation with angular frequency Wr = j0Wv sin(Wvt). Here j0 and Wv are the angular amplitude and frequency of rotational vibrations that are varied in experiments in the ranges 0 < j0 < p / 2 and 1 < Wv < 65.
The program for the motor controller contains 48 paired instantaneous values of the rotation speed (in steps per second) and the angular displacement (in steps). These values are calculated from the sine function with given frequency and amplitude and are distributed over the half of oscillation period with equal time step. They are repeated with the opposite sign (direction) over the second half of the period, which in total provides a sinusoidal signal discretized with 96 points per period. The motor positioning accuracy is set at 3200 steps per revolution, which is equivalent to 0.11 degrees per step.
For video registration an Optronis CL600x2 camera is used. The frames of 800x800 pixels are taken at a framerate 100 fv, where fv = Wv / 2p is the linear frequency of rotational vibration. In order to determine the amplitude of container oscillations, the camera view is set along the container axis, in a way that the latter is seen in the frame center. The captured video frames are exported and processed in ImageJ software. The character of container motion is controlled by tracking the angle of the divider. The fast Fourier transform (FFT) is used to obtain the container oscillation frequency out of the trajectory oscillogram. In the case when the motion of a phase inclusion is being registered, the frame center is placed at the average position of the inclusion. At that, the container axis is seen at the bottom of the frame on the vertical center-line. Then, in order to measure the container angular coordinate, a
radial black line traced on the flange is used as a marker. The oscillation amplitude of the inclusion is found from the trajectory of its geometric center. In all cases, the container axis is used as one of the reference points during the measurements of the angle. In order to measure the aspect ratio of the deformed inclusion, a bounding rectangle is traced around it in such a way that one of its sides is tangential to the container wall, and the other is, consequently, parallel to the container radius.
1.3. Apparatus performance. Test runs were conducted, when the annulus was filled with only one liquid - water. The container motion was registered with the high-speed camera at a frame rate 100 fv. The coordinate of a point on the container wall was tracked frame-to-frame using ImageJ software. In the parameter range of fv = 0.25 -10 (Hz) and j0 = (0.1 - 0.5)^ (rad), the container performs regular rotational oscillations with a constant frequency and amplitude (Fig. 3). At smaller amplitudes, the apparatus shows some deviation of the motion trajectory from the harmonic signal due to the growth of discretization error.
Fig. 3. Container oscillations at fv = 2.7 Hz and j = 0.3p rad. Here, A is the angular coordinate of a point on the container wall, t is time
2. EXPERIMENTAL RESULTS
2.1. Dynamics of a droplet. First experiments demonstrated that the interaction of the droplet and the container wall leads to the irregular motion at small vibration amplitudes, while at higher amplitudes the droplet breaks in several parts. Nevertheless, it was possible to successfully achieve the periodic deformation of the droplet (Fig. 4). The experimental technique of investigation of droplet dynamics needs to be im-
proved, meanwhile in the present experiments we will consider the dynamics of an air bubble.
a b
Fig. 4. Photographs of the droplet in different oscillation phases at
o
fv = 0.25 Hz, j = 18.3 : the container is rotating clock-wise (a), it is immobile in the extreme right position (b). Point B moves periodically between points A and A '
2.2. Dynamics of an air bubble. A system similar to that of the droplet is a bubble immersed in continuous liquid. Its dynamics proved to be more smooth and regular, while the oscillations of the shape retain the same basic features (Fig. 5). For this reason, a system of the air bubble immersed in oil was studied in more detail.
a b
Fig. 5. Photographs of the bubble in different oscillation phases at
o
fv = 2.1 Hz, jo = 31.6 : the container is in the extreme right position (a),
it is rotating counter clock-wise (b). Points A and A ' mark the extreme positions of point B
The rotational vibrations applied to the container lead to the oscillations of the bubble. Its geometric center moves along a sinusoidal trajectory, which is determined by the container oscillations (Fig. 6). Bubble's center oscillation frequency, fb , equals to the excitation frequency, fv , as demonstrated by the spectra in Fig. 6. At small fv , the phase differ-
ence between the bubble and the container is clearly pronounced (Fig. 6a), and with the increase in fv, the two motions have a tendency to synchronize (Fig. 6 b). At the same time, the ratio between the bubble's motion amplitude and the container oscillation amplitude, Ab / A, increases with the frequency of vibration.
2
f, Hz
t, s
20
f, Hz
40 b
Fig. 6. Oscillograms and spectra of oscillations: fv = 0.35 Hz,
j0 = 35.4° (a), fv = 3.52 Hz, j0 = 35.7° (b). Points: ° - coordinate of the container wall, x - coordinate of the bubble's geometric center
The oscillatory motion of the bubble is accompanied by its deformation. To quantify the shape we use the aspect ratio X calculated from azimuthal and radial sides of the bounding rectangle to the bubble, X = dp /dr. Bubble's aspect ratio oscillates with the double frequency,
2fv (Fig. 7). Visual observation reveals that the minima of X correspond to the moments when the bubble changes the direction of its mo-
40
20
A
0
20
40
0
4
0
5
10
15
t, s
a
40
20
A
A
0
20
40
0
tion. Comparison of figures 7 a and 7 b shows that with the increase in fv the average aspect ratio Xav decreases. This means that the interface
between the bubble and the viscous boundary layer on the wall reduces and the bubble's geometric center moves away from the wall.
Fig. 7. Oscillograms at: fv = 0.35 Hz, j0 = 35.4° , Xav = 1-45 (a); fv = 1.76 Hz, j0 = 35.5°, Xav = 1.29 (b). Points: ° - coordinate of the container wall (left axis), ◊ - aspect ratio X (right axis)
Conclusion. The possibility of excitation of droplet shape oscillations by oscillating shear flow has been considered. A theoretical formulation and an experimental approach have been proposed that consist in using a coaxial layer whose walls form a simply connected surface. It is demonstrated that by means of rotational vibrations of such layer it is possible to achieve periodic deformations of a fluid inclusion immersed in surrounding liquid. Such characteristics as the coordinate of the inclusion and its aspect ratio have been measured from the captured images. The
preliminary experimental results prove that the formulated problem is interesting from the points of view of both academic research and application.
The research was supported by the Government of Perm Region (Programs for the support of International Research Teams, grant C-26/174.9).
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ИССЛЕДОВАНИЕ МАССОПЕРЕНОСА МЕЖДУ КАПЛЕЙ И ОКРУЖАЮЩЕЙ ЖИДКОСТЬЮ: ПРЕДВАРИТЕЛЬНЫЕ ЭКСПЕРИМЕНТАЛЬНЫЕ РЕЗУЛЬТАТЫ
И.Э. Карпунин, Н.В. Козлов, Ф. Плурабуэ
В работе рассматривается возможность возбуждения колебаний капельного включения в сплошной жидкости. Для этого предложено использовать контейнер с односвязной цилиндрической полостью, совершающий вращательные колебания относительно собственной оси. Дана теоретическая постановка задачи об осциллирующем сдвиговом течении в слое. Разработана методика эксперимента. Предварительные экспериментальные результаты продемонстрировали возможность периодической деформации капли жидкости или пузырька воздуха посредством сдвиговых колебаний окружающей жидкости.
Ключевые слова: односвязная область, коаксиальный слой, вращательные колебания, сдвиговое течение.
