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Research Paper
Journal of Chemical Engineering of Japan, Vol. 50, No. 8, pp. 601–609, 2017
Hydrodynamic and Motion of Single and Conjunct Drops Rising in Glycerol–Water Solutions Li Rao1,2, Zhengming Gao1, Ao Nie1, Yuyun Bao1 and Ziqi Cai 1 State Key Laboratory of Chemical Resource Engineering, School of Chemical Engineering, Beijing University of Chemical Technology, Mailbox 230, Beijing 100029, China 2 Beijing Advanced Innovation Center for Imaging Technology, Capital Normal University, Beijing 100048, China 1
Keywords: Conjunct Drop, Oscillation, Fourier-Analysis, PIV A new predictive model for dual drops interactions has been formulated, covering four possible situations: namely coalescence, slip & separation, slip & conjunction and non-slip conjunction. The hydrodynamics and motion of a single drop or a conjunct drop rising in stagnant glycerol–water solutions were experimentally investigated. For Morton numbers ranging from 9.68×10−3 to 51.40, the terminal rising velocities, the transverse oscillation frequencies and the wake of the conjunct drop are similar to those for a single drop with the same spherical-equivalent diameter. The rising velocities of a single drop and a conjunct drop were quantified on the basis of the expression for the drag coefficient and the feature size of the drops. The surrounding liquid flow fields for a single drop and a conjunct drop were measured by means of 2-D Particle Image Velocimetry, in order to identify the regions of the wake behind drops, and the similarities and differences between the flow fields were also discussed.
Introduction The rising velocity of drops in another immiscible liquid is of central importance in a variety of liquid–liquid extraction processes. In these systems, one phase is dispersed into another one, and the mass transfer efficiency is closely related to the size of dispersed phase. Since drops occur in such systems in swarms with interactions instead of a single drop, the knowledge of the hydrodynamic aspects of the motion and interaction of drops is essential for the successful design and operation in such multiphase systems. The interaction between two drops usually results in either the drops rebounding from each other or coalescing to form a single drop (Yarin, 2006). Of practical significance is the case of coalescence results in a change in interfacial area which leads to hydrodynamic instabilities in the system (Ashgriz and Poo, 1990; Estrade et al., 1999). Recently, numerous experimental investigations have been carried out where individual flow-induced coalescence events between two drops have been studied under controlled conditions (Hayashi and Tomiyama, 2009; Bayareh and Mortazavi, 2011). The researchers have found that drop interaction was a complex phenomenon and numbers of factors played the dominant role in determining whether a pair of colliding drops would coalesce, or simply pass around one another and separate in the flow, such as the size difference of drops (Hsu, et al., 2008), relative drop positions (Zinchenko et al., Received on February 3, 2016; accepted on October 17, 2016 DOI: 10.1252/jcej.16we027 Correspondence concerning this article should be addressed to Y. Bao (E-mail address:
[email protected]) and Z. Cai (E-mail address:
[email protected]). Vol. 50 No.©8 2017 Copyright 2017The Society of Chemical Engineers, Japan
2011), system properties (Sajjadi et al., 2002), and external conditions (Bazhlekov et al., 2000). However, a less common result of the collision between drops is the formation of a single interpenetrating conjunct drop. This type of drop occurs more frequently in high viscosity liquid systems in which the favored morphology is not dominant in the continuous phase and, as a result, the favored morphology forms in the dispersion phase (Manga and Stone, 1993). Cai et al. (2012) studied the motion of conjunct bubbles rising in viscous liquids and found some similarities between the single and conjunct bubbles. Typically, a conjunct drop (or bubble) with regular shape may form after two drops (or bubbles) collide with each other in viscous liquids (Yarin, 2006) and it is interesting to understand the formation of a conjunct drop eventually. Previous studies about the behaviors of single drop were mainly conducted in the deformation, velocity and drag coefficient. References regarding the description and modeling of single drop motions have been presented during the last decades (Clift et al., 1978), involving different fluid dynamic properties (Zinchenko et al., 2011), such as the terminal velocity, the drag coefficient (Hu and Kinter, 1955; Klee and Treybal, 1956), the shape (Myint et al., 2007) and the oscillation of drops (Helenbrook and Edwards, 2002; Kelbaliyev and Ceylan, 2007). A number of correlations were proposed as suitable for drops (or bubbles) rising in Newtonian fluid. Table 1 presents some drag coefficient correlations which found to be valid for drops and bubbles having a wide range of Reynolds numbers. Rodrigue (2001) recommended a correlation for the terminal velocity (UT) of various bubble sizes by assuming that the liquid is clean (i.e. no impurities to generate Marangoni stresses), and the gas phase in a gas-liquid system having 601
Table 1 Drag coefficients for drops or bubbles Expression
1.68 1/4 , 0.1 ≤ Re ≤ 0.5 CD = Re 14.9 , 1 ≤ Re ≤ 10 Re 0.78
Equation
Reference
Particles
(2)
(Clift et al., 1978)
Bubbles and drops
(3)
(Rodrigue, 2001)
Bubbles
(4)
(Darton and Harrison, 1974)
Bubbles
(5)
(Grace, 1983)
Drops
(6)
(Oliver and Chung, 1987)
Bubbles and drops
(7)
(Polianin and Dil’man, 1994)
Drops
(8)
(Bozzano and Dente, 2001)
Bubbles
(9)
(Saboni and Alexandrova, 2002)
Drops
9/4
1/3 1 + 32 θ + 1 1+128 θ 2 2 1/3 K 1 1 CD = 1+128 θ + + 32 θ − Re 2 2 1/9 + 0.036 128 Re 8/9 MO1/9 3
7.7 ×10−12 < MO < 1.9 ×10−1 , K = 16 , and
θ = (0.0185)3 (2 / 3)1/3 Re 8/3 MO1/3
μd μc
CD =
8(3 κ + 2) 8 + Re ( κ +1) 3
CD =
1 24 4 14.9 + + κ 0.1 ≤ Re ≤ 10 κ +1 Re Re1/3 Re 0.78
CD =
8 3 κ +2 3 κ +2 +0.4 Re κ +1 κ +1
CD =
κ 1 16 32 24 (1 + 1.5 Re 0.678 ) 0 ≤ Re ≤ 100 + κ +1 Re ( Re + 32) ( κ +1) Re
CD =
48 Re
1+12 MO1/3 1/3 1+12 MO
κ=
2
0 < Re < 2
MO 3/2 + 0.9 1.4(1 + 30 MO1/3 )+ BO 3/2
24 3 κ +2 4 14.9 + 1/3 + 0.78 Re 2 + 40 +15 κ +10 κ Re Re Re Re CD = ( κ +1)( Re 2 + 5)
0.01 < Re < 400
negligible density and viscosity compared with the liquid phase. 1/3
σμ U T = 2 c2 ρc De
F 12(1+ 0.0185F )1/3
(1)
Here, F=g(ρc5De8/σμc4)1/3 is the flow number. Corresponding to Eq. (1), the correlation for the drag coefficient was proposed as shown in Eq. (3). As was demonstrated by numerous experimental data on the behavior of drops and bubbles, the drag coefficient depends on the Reynolds number (Re=ρUTDe/μ), Morton number (MO =gμc4Δρ/ρc2σ3), and Bond number (BO =ΔρgDe2/σ). The information about the dynamics of the single drop in Newtonian fluid is important to better understand the dynamics of the conjunct drop. To the knowledge of the authors to date, not much attention has been paid to the dynamic behavior of the conjunct drop in the literature. Study on the conjunct drop can contribute to deeper understanding of the interaction between drops, and the coalescence mechanism. The behaviors of the drop conjunction process in glycerol-water solutions were systematically investigated and 602
the effect of Bond Number (BO) and the equivalent diameter ratio (λ) on the formation of the conjunct drop were studied. The characteristic diameter and the terminal rising velocity of conjunct drop are compared with the existing correlations for bubble and drop motions in Newtonian liquids. The transversal oscillation motion of the conjunct drop was compared with that of a single drop with the same equivalent spherical diameter. Finally, particle image velocimetry (PIV) was used to visualize the flow field around the conjunct drop.
1. Experimental 1.1 Apparatus and system The experimental equipment consisted of a plexiglass column which is 450×450×450 mm3, a drop-producing port set at the center of the bottom of the column the diameter was 30 mm, and a discharge port set at the corner of the bottom of the column with a diameter 30 mm. The motion of the drops was recorded by a high speed CMOS camera (FASTCAM-ultima APX, Photron) at 500 fps (512×1024 pixels), with a high quality micro lens (Nikkor Micro Journal of Chemical Engineering of Japan
Table 2 Properties of glycerol–water solutions and oil (20±1°C) Continuous phase
Solution Glycerol mass fraction Density ρ, [kg/m3] Viscosity μ, [Pa·s] Surface tension σ, [N/m] Interfacial tension σ, [N/m] κ=μd/μc MO Re for single drop Re for conjunct drop
S1
S2
S3
S4
S5
S6
Soybean oil
1.00 1264 1.11 0.0612 0.0308 0.251 51.40 0.01–0.06 0.01–0.12
0.99 1262 0.945 0.0607 0.0281 0.295 27.72 0.01–0.09 0.02–0.55
0.97 1256 0.714 0.0592 0.0271 0.391 9.783 0.01–0.17 0.02–0.46
0.95 1250 0.407 0.0601 0.0232 0.688 0.9920 0.02–0.4 0.08–1.07
0.90 1237 0.279 0.0593 0.0222 1.00 0.2304 0.06–1.56 0.36–2.83
0.85 1219 0.132 0.0632 0.0218 2.12 0.009679 0.2–3.62 0.61–9.01
0.00 889 0.279 0.0292 — —
60 mm, F 2.8D). The high speed camera system recorded the whole process of drop movement to the continuous phase surface from the drop generation hole with an image resolution of about 0.3515 mm/pixel. The viscosity ratio plays an important role in droplet morphology, unstable droplet behavior, and terminal droplet characteristics (Ohta et al., 2010). Six glycerol–water solutions and soybean oil were used as the continuous and dispersion phases, respectively. The physical properties of the solutions are listed in Table 2. The diameter for the leading drops ranged from 1.10 to 12.6 mm and for trailing drops from 3.20 to 15.8 mm. The rheological properties were measured with a HAAKE Rheostress RS150 rheometer (Haake, Germany), and the surface tensions were measured with an automatic surface tension apparatus JYW-200B (Cheng de Kecheng Testing Machine Co., Ltd., China). 1.2 Image processing By using a high-speed camera, digital images with eightorder grey gradation were obtained, in which the information of grey value and the position coordinates in each pixel were contained. After being captured, the images were clipped dynamically and processed with Canny algorithm in MATLAB, which can detect the edge of the drop accurately even with the existence of image noise. The typical original images and the processed ones for drops are shown in Figure 1. The oils were blended with a non-water soluble dye, which increases the optical evaluation possibilities. That is, coloring of the oil generated the contrast between the continuous and dispersed phases to allow the drops to be recognized in the video images. The program first isolates the area of drop which needs to be identified (Figures 1(a) to (b)). Then, it detects the edge of the drop from the raw image (Figures 1(b) to (c)). Finally, the fundamental data including the peak velocity and horizontal and vertical axes length of drop can be obtained (Figure 1 (d)).
2. Results and Discussion 2.1 Formation of the conjunct drop We observed that the collisional interactions between two buoyancy-driven drops could lead to four possible results, Vol. 50 No. 8 2017
Dispersion phase
Fig. 1 Original and processed images of conjunct drop
Fig. 2 Illustration of interactions between two drops
namely, coalescence (I in Figure 2), slip and separation (II in Figure 2), slip and conjunction (III in Figure 2) and nonslip conjunction (IV in Figure 2). Experimental studies with different viscosity liquids have shown that the following independent variables are important to characterize a dual drop collision process: (i) De is the equivalent spherical diameter and De is given by De =[6(VL+VT)/π]1/3 =DT[1+λ3]1/3 for two interaction drops, where VL and VT are the volumes of the lead603
Fig. 4 Formation of a slip and conjunct drop (S1: DL =5.84 mm, DT =9.73 mm, De =10.4 mm)
Fig. 3 Schematic of various drop collision regimes in different glycerol–water solutions
ing and trailing drops, respectively. The equivalent diameter ratio λ is defined as the ratio of DL and DT. (ii) Bond number (BO), representing the ratio of buoyancy forces to interfacial tension forces, is defined as ΔρgDe2/σ, where Δρ is the density difference between the drop and the continuous phase; g is the gravitational acceleration; and σ is the interfacial tension. Four different possible results of two interactive drops, as defined above, can be divided into four corresponding regimes based on the relationship between Bond number and the equivalent diameter ratio (BO−λ), as shown in Figure 3. When the diameter ratio is large (λ>0.55), coalescence takes place at high Bond numbers (BO >15), as shown in regime I of Figure 3. Two drops form a combined mass that oscillates and remains a single drop. In the slip and separation regime II, two drops get closer to each other due to the velocity difference then the smaller drop is swept around the larger one, but finally two drops separate and remain in two separate drops. Both regime III and IV refer to the conjunct drop, but formed in two different ways. Two drops get closer and contact with each other, then (i) the smaller drop slips around the larger one and is sucked in from behind (regime III of Figure 3), and (ii) the smaller drop conjuncts with the large one without slip or coalescence (regime IV of Figure 3). The Bond numbers for conjunct regimes are lower than those corresponding to the coalescence regime, so that the kinetic energy of drops cannot expel the intervening liquid film. Finally, two drops are connected and rise together. In this study, we focused on the slip and conjunct drop. Figure 4 illustrates the formation of a slip and conjunct drop with a 9.73 mm diameter larger drop getting closer to a 5.84 mm smaller one, then the smaller drop slipping around the larger one, and finally forming the slip and conjunct drop. Figure 5 presents typical images of a conjunct drop formed in six different solutions. In images a, b and c, the smaller drop was chased and slipped below the larger one, then deformed into a drop with an elongated vertical axis, but the large drop was still in a regular spherical shape. With the decrease MO, the larger drop becomes the spherical cap in shape with the impact of the smaller drop. 604
Fig. 5 Typical conjunct drop profiles in six solutions
Images d, e and f show the conjunct drop in smaller MO solutions, while the smaller drop actually penetrating into the base of the larger one. The shape of the combined drops was almost a spherical cap, but with a distinct annular step to a pointed trailing tail. As for images e and f, with the increasing drop size, the smaller drop penetrates into the larger one deeper. The tapered tail, comprising the larger drop, becomes much shorter and wider with a pointed apex. The shape of the conjunct drop looks like a “mushroom”. 2.2 Drop rising velocity In creeping flow, the Stokes equation for a solid sphere can be used to predict the terminal velocity (UT) of a fully immobile-surface drop (Clift et al., 1978). UT =
4 g ( ρ l − ρd ) D e 3 ρ lCD
(10)
Here, CD is the drag coefficient. From Eq. (10), we can conclude that there is a quantitative relationship between the rising velocity and the drag coefficient of a drop, so the terminal velocity of the drop in the rising process can also be Journal of Chemical Engineering of Japan
Fig. 6 Measured and modeled U versue De Note: (○) Uexp for single drops; (●) Uexp for conjunct drop; ( bined with Eq. (4); ( ) Umod Eq. (12) combined with Eq. (9)
calculated by the drag coefficient. Cai et al. (2012) studied a conjunct bubble rising in viscous liquids, and selected the representative expression Eq. (3) to calculate the drag coefficient of conjunct bubble. Based on the assumption of an immobile interface (K=24) and replacement of De with the projected diameter DA, they proposed the equation of the terminal velocity of the conjunct bubble as follows. UT =
4 gDA 3CD
(11)
In order to investigate whether the conjunct drop has a common feature with the conjunct bubble, we use the rising velocities measured by experiment Uexp, choosing the equation of Darton and Harrison (1974) (Eq. (4)), equation of Saboni and Alexandrova (2002) (Eq. (9)) and the equation of Rodrigue (2001) (Eq. (3)) as the coefficient calculation equations, iteration to solve the rising velocities of the drops Umod. On condition of the fixed experimental parameter, we Vol. 50 No. 8 2017
) Umod Eq. (12) combined with Eq. (3) K=20; (
) Umod Eq. (12) com-
assume that the equation for the rising velocities of drops as follows. U mod =
4 g ( ρ l − ρd ) D ′ 3 ρ lCD
(12)
Here, the velocity was mainly related to drag coefficient CD and diameter factor D′, the results being shown in Figure 6. In Figure 6, the measured drop rising velocity, Uexp, and modeled velocity Umod are reported as a function of equivalent diameter, De in six different solutions (S1–S6 see Table 2). With the prediction based on Eq. (12), making use of the CD expression Eqs. (3), (4) and (9) and assuming D′=De2/DA, the rising velocities of the single and the conjunct drop becomes Eq. (13). U mod =
4 g ( ρ l − ρd )De2 3 ρ lCD DA
(13)
It can be seen that in all six liquids, the rising velocities obtained by experiments for both the single and conjunct 605
drops are similar. The rising velocity of drops is determined by the interaction of the buoyancy force and average resistance. The buoyancy force is estimated by the density and volume of the drop, while the average resistance is affected by the drag coefficient and the shape of drops. Thus, the rising velocity of drops depends on the volume, drag coefficient and shape of drops with the same density. However, Helenbrook and Edwards (2002) indicated that the average drag coefficient for a slightly deformed spherical drop was equal to that for the sphere with the same volume. Happel and Brenner (2012) also showed that sharp edges had little effect on the terminal velocity. Wairegi and Grace (1976) reported that the terminal velocity of an irregularly shaped drop was derived by only considering the shape and motion near the nose. In our experiment, the shape and motion near the nose of the single drop are approximate to those for the conjunct drop, and the volumes for the single and conjunct drops are the same, though the conjunct drop has the tapered tail. Therefore, the single and conjunct drops with the same equivalent diameter have similar rising velocity. Figure 6 shows that Eq. (12) combined with Eqs. (4) and (9) give under-predictions of the rising velocity for a given De in solutions S1–S3, but give over-predicted results in solutions S4–S6, as shown in blue and green lines. The reason may be due to the change of the viscosity ratio and Morton number (Ohta et al., 2010). Equation (12) combined with Eq. (3) for K=20, which is different from the K value of 16 reported by Rodrigue (2001), shows very good agreement with the experimental results. There are some reasons for the difference in K values which can estimate whether the interface of a drop is mobile or immobile (K=16 for mobile and K=24 for immobile). The model developed by Rodrigue (2001) was based on the assumption that the particles had a mobile drop interface and in the completely unpolluted water. As is well known, the slip velocity of drop surface is inversely proportional to the viscosity of continuous phase. The motions of drop with immobile interface in high viscous continuous phase were studied in our experiment; therefore, the assumption with K=16 cannot fit the experimental conditions, whereas the drag coefficient of drops increases with the increasing viscosity of continuous phase and the value of K=20 is used in accordance with all data. Equation (13) reflects the influence of diameter factor D′=De2/DA, which is a function of the drop equivalent diameter De and the projected diameter DA, in determining the rising velocities of the single and the conjunct drop in our experiment. Both for single and conjunct drops, the value of projected diameter DA is equal to the value of the equivalent diameter De when De
