Interactions between two in-line drops rising in pure glycerin☆

Li Rao,Zhengming Gao,Ziqi Cai*,Yuyun Bao*

State Key Laboratory of Chemical Resource Engineering,School of Chemical Engineering,Beijing University of Chemical Technology,Beijing 100029,China

1.Introduction

The prediction of coalescence of discrete fluid particles(DFPs)inside a continuum fluid is of high importance in many industrial processes.Taking liquid–liquid dispersion as an example,high coalescence rate is desirable for the process of extraction,however,low coalescence rate is usually required in the case when a uniform distribution of the well-dispersed phase is expected to enlarge the interfacial area and increase the mass transfer rate between phases.

It is generally regarded that the coalescence happens in three steps:the approaching(overtaking)process,the liquid film drainage,and the rupture of interface[1].The approaching process is defined as the period from two drops having initial deformation on their surfaces to the instant as two drops contacting.This time span is called the interaction time.The drop pair traps a liquid film between them that needs to be drained if coalescence is to take place.The time needed for this drainage is the drainage or coalescence time[2].While the film thickness is decreasing to a certain limit,the film ruptures and coalescence happens almost instantaneously.The rupture time is usually considered negligible compared to the interaction and drainage times[3].The most common criterion used for the occurrence of coalescence is that the film drainage time plus the film rupture time should be smaller than the interaction time[1].In this work,we are concerned with the drops overtaking process only,the estimation of the interaction time and the drop motions in the contact area.

The approaching(overtaking)process of two drops(or bubbles)has been the subject of considerable theoretical and experimental attention over the past years[4–8].Based on the pioneering works of Jeffrey[9]and van Wijngaarden[10],Kok[11]used the irrotational flow solution to derive the viscous drag force for two spherical bubbles in interaction,following the global kinetic energy balance proposed by Levich[4].Using the same approach,the case of growing bubbles rising in-line was considered by Harper[12].He showed that when a pair of bubbles rose in a quiescent liquid due to buoyancy,only two steady situations can be observed:the first one is two bubbles rising side by side in contact,with their center lines being perpendicular to the rise velocity;and the second one is an infinite separating distance.Zhang and Fan[13]used mathematical models to predict the velocity of an interactive spherical bubble rising in-line in liquids.The results indicated that both the wake effect[14]and the bubble acceleration effect should be taken into account in analyzing the motion of the trailing bubble in the far wake region of the leading bubble.However,most of the previous work on the approaching process of the fluid particles(drops or bubbles)has only concerned the case of two equal-sized particles or that of a particle to a fl at interface.

Due to the complexity of the mechanism and factors,very little work has been done on the interaction time for collisions between two buoyant drops in highly viscous Newtonian liquids;even though this case is frequently encountered in liquid–liquid extraction which eventually requires the coalescence of a drop with its homo phase,or the evolution of the dispersed phase volume fraction in multiphase flows common to industrial processing.

In this paper we study the hydrodynamic interactions between two buoyant drops in high viscosity Newtonian liquids.The purpose is to develop a fundamental interaction model for the approaching process suitable for the unequal-sized drops,where the buoyancy forces are much larger than the restoring forces due to the inter facial tension.In the interaction model,an energy balance is used to obtain the potential energy stored in the approaching process,the interaction time and contact area between two drops can be evaluated.The main advantage over the previous work(e.g.Jeelani and Hartland[7,8])is that the effect of the total particle system inertia is considered,which was assumed that all the initial kinetic energy of the relative motion of two particles(bubbles or drops)was available for surface deformation.Meanwhile,a new approach(the change of energy)to judge the interaction time between drops is developed,since it is more convenient than the geometric feature solution when used in other particles(e.g.rigid particles).The main macroscopic behavior such as the velocity,shape and trajectory of the drops,and thus the related drag coefficient and oscillation frequency are obtained.

2.Experimental

2.1.Experimental setup

The experimental equipment consists of a plexiglass column with 450×450×450 mm3,with which the wall thickness is 3 mm,and a drop-producing port and a discharge port set at the bottom of the column.Pure glycerol and soybean oil were used as the continuous and dispersion phase,respectively.The physical properties of these solutions were listed in Table 1.The rheological properties were measured with HAAKE Rheostress RS150 rheometer(HAAKE,Germany),and the surface tensions were measured with an automatic surface tension apparatus JYW-200B(Chengde Kecheng Testing Machine Co.,China).The motion of drops was recorded by a high speed CMOS camera(FASTCAM-ultima APX,Photron,Germany)at 250 fps(512×1024 pixels),with a high quality micro lens(Nikkor Micro 60 mm,F 2.8D).When the process of drops coalescence was recorded,the high speed camera covered the height of about 250 mm from bottom to top with an image resolution of 0.2441 mm per pixel.

Table 1 Properties of glycerol solution and soybean oil[(20± 1)°C]

2.2.Image processing

By using high-speed camera,the digital images with eight-order gray gradation were obtained,in which the information of gray scale and the position coordinates in each pixel were contained.After being captured,the images are clipped dynamically and processed with the Canny algorithm in MATLAB,which can detect the edge of the drop accurately even if image noise exists.Finally,the fundamental data including the center-of-mass coordinates and horizontal and vertical axes length of the drop can be obtained.The major source of the experimental error is the precision of the edge of the processed image.It is necessary to decrease the effect of the edge of the image out of focus by using an appropriate shutter speed.The typical original and processed images of drops are shown in Fig.1.

When two in-line drops rise,the coalescence process can be described as three steps: firstly,the overtaking process,from the moment when two drops have initial deformation on their surfaces as shown in Fig.1(a)to that while two drops contact in Fig.1(c);secondly,the trailing drop is“coated”and a liquid film forms between two drops,which is usually called liquid film thinning and drainage stage(see Fig.1(d));in the third stage,the coalescence finishes and one bigger drop forms after the rupture of interface,(see Fig.1(e)).In present research,we are only concerned with the first step covering the overtaking process from Fig.1(a)to(c).

3.Results and Discussions

3.1.Two methods to distinguish the approaching process

Interests in the coalescence process have been the primary motivation for the extensive study of the film drainage problem,and the interaction time between two drops is one of important parameters in the description of drop interaction.In the determination of coalescence time,various assumptions have been made for convenience.A common assumption used for the approaching process of bubbles or drops in a turbulent field[4,5]is that the interaction time is proportional to the characteristic life time of an eddy with size equal to the sum of the approaching particles.In fact,this hypothesis was initially made by Levich[4]based mainly on the dimensional analysis.Based on the parallel film concept for equal-sized particles with very small Weber numbers,Chesters[3],and Chesters and Hofman[6]discussed the coalescence of two unequal-sized bubbles by substituting an equivalent radius in the expressions for equal-sized particles,and by introducing an equivalent coefficient of virtual mass.However,this was only for the determination of the drainage or coalescence time,but not for the interaction time.The purpose of this section is to develop suitable approaches to distinguish the approaching process valid both for rigid particles like solids and non-rigid particles like drops or bubbles.The interaction time and contact area between two drops can be evaluated;moreover, the fraction of deformation ratio and energy loss available for the wake between two drops could also be determined.

3.1.1.Geometric feature change method during two drops approaching

In order to quantitatively define the shape of drops,we assumed that drop is axis-symmetrical,and named the long horizontal semiaxis as a,the upper and lower vertical semi axes as b1and b2,respectively(see Fig.2).We also define a deformation ratio E as the ratio of the sum of two vertical semiaxes of drop to the horizontal axis of drop,i.e.

Fig.2 shows that the shape of drops can be roughly classified into five types:sphereoblate ellipsoidE＜1),prolate ellipsoidsemi-ellipsoidand ellipsoidal cap

Due to the drops' internal rotation,the interface pressure gradient distribution,and the wake effect of adjacent drops[15],the deformation ratio of drop does not stay constant and shows a slight fluctuation even if the drop reaches the terminal velocity.

In the overtaking process of two drops,the trailing drop with a largersize rises at a higher velocity compared with the leading drop.When the trailing drop gets closer to the leading one,the flow field generated by the trailing drop will make the leading drop flattened into an oblate one.As the trailing drop almost“touch”the leading one,it becomes semi-ellipsoid and is likely to “coating”the trailing drop as mentioned by Chaudhari and Hofmann[1],and then coalescence happens.

We define T*as the starting time of the approaching process when the interaction between two drops becomes obvious because of the wake effect of the leading drop,corresponding to Fig.1(a).Both drops have larger velocities compared with the free rising drop:the trailing drop is accelerated by the “pulling”effect in the negative pressure area of the leading drop's wake,while the leading drop is accelerated by the “pushing”effect of the trailing drop.Due to the larger velocity of the trailing drop,the “pushing”effect is stronger in the approaching process.We define T0as the ending time when two drops “contact”and the leading drop becomes semi-ellipsoid with b2decreasing to 0,corresponding to Figs.1(c)and 2(d).

Fig.3shows the relationship between the deformation ratios for two drops with time.The deformation ratio of the leading drop decreases gradually as the interaction enhances,contrary to what happens on the trailing drop.

Fig.4 shows the relationship between the vertical semiaxes of the leading and trailing drops with time.It shows that b1and b2of the leading and trailing drops have slight fluctuations over time during their free rising,due to a superposition of the drop shape oscillation and the self-inner-rotation[16].For the leading drop,bL,2is near the wake and changes dramatically,especially when two drops get closer,so the enhanced “pushing”effect of trailing drops leads to a rapid decreasing of bL,2.The leading drop becomes half ellipsoid gradually,and bL,1is essentially stable due to the effect of interfacial tension.For the trailing drop,bT,1and bT,2increase gradually with the decrease of the centroid distance between two drops.Due to the effect of interfacial tension and wake,the trailing drop tends to form a sharp tail,leading to the increase of bT,2when two drops get together.

Fig.3.Relationship between deformation ratios for two drops with time.

Fig.4.Vertical axes of leading and trailing drops vs.time series.

For translational motions in an unbounded fluid,the shape of a single spherical drop is stable,however,the shape of a deformed drop is not stable,and it subsequently undergoes a continual deformation.If the restoring interfacial tension forces are not sufficiently large,prolate drops will develop long tails and oblate drops will develop cavities[17,18],which will deeply affect the rising velocity,trajectory and near flow field,that will be discussed later.

Fig.2.Shapes of a drop.

3.1.2.Energy change method for two drops approaching

The possible coalescence of two nearly contacting drops over long time interaction is controlled by the dynamics of thin film between drops.Luo and Svendsen[19]offered a rather different view on the coalescence process between two fluid particles.The film drainage was not taken into consideration at all.They determined the total collision time,but this time scale was questionable since no energy loss during the collision process was taken into account.

Kinetic energy change is the main factor to affect the energy dissipation.The diameter ratio of drops λ(DL/DT)is one of the important factors in the process of drop overtaking.Fig.5 shows the relationship between velocities of the leading and trailing drops and time,where T0represents the moment when two drops contact.

Fig.5.Transient change of drop velocity.

Fig.5(a)shows that given the sizes of two drops,the velocity of the leading drop increases with time due to the “pushing”effect of the trailing drop.Specifically,the velocity of the small leading drop of 9.76 mm increases by 45%over its original free rise velocity,and the velocity for the large leading drop of 11.56 mm increases by 37%.With the increasing size of the leading drops,the velocity of the leading drop also increases.

Bhaga and Weber[20]found that in order to guarantee the coalescence,the velocity of the trailing bubble would be greater than that of the leading one.Miyahara et al.[21]observed that if a large portion of the trailing bubble is outside the wake of the leading bubble,the trailing bubble splits into smaller ones due to the shear flow generated by the leading bubble.Cai et al.[22]measured the surrounding liquid flow field by instantaneous 2-D PIV to identify regions of increased velocity during the interaction stage of two bubbles.Two bubbles exhibited increased velocity at the top of the leading bubble and the side of the juncture with the trailing bubble.O take et al.[23]investigated the effect of the wake on bubble–bubble interactions.He found that within a critical initial distance which was larger than 3–4 times of the leading bubble diameter,the trailing bubble was accelerated in the wake of the leading bubble.The rise velocity is the sum of the trailing bubble's terminal velocity and the wake velocity.

Comparison of Fig.5(b)with Fig.5(a)shows that the velocities of the trailing drops are greater than those of the leading ones.Due to the effect of leading drop's wake,the velocity increase of the trailing drop decreases with the trailing drop size.For the trailing drop of 13.15 mm,its velocity increases only 26%at the contacting moment and for the drop of 14.48 mm,its velocity increases about 30%.

In this work,the image sequence of two drops in the stage of interaction is captured and shown in Fig.6.

Fig.6.Sequence of photographs showing the interaction stage between two drops in pure glycerin.(Note:DL=11.8 mm,DT=14.3 mm.Drops:soybean oil).

An energy balance is used to find the potential energy stored in the approaching process.The energy balance of kinetic(ΔEk),surface(ΔEσ)and gravitational potential(ΔEp)on a drop can be expressed as

Fig.7 shows the energy of drops changing with time during the process that the trailing drop is smoothly accelerated until it coalesces with the leading drop.It can be seen that the starting and ending time of energy changing are the same as those of the drop deformation discussed in Fig.3.With the decrease of the centroid distance between two drops,energy loss is constantly increasing.In the moment of Tε0,the energy dissipation(εDs)of drops reaches the maximum.

Fig.7.The energy change of two drops in approaching process.

The wake between two drops appears to be closely related with the energy dissipation.When the trailing drop is in the wake of the leading drop,the increasing of the trailing drop's rise velocity expends its potential energy in a higher rate compared with a relatively slower rising free single drop.Therefore,the drops pair in the wake dissipates more energy than the situation without interaction.The dissipated energy appears as an increase of the internal energy.Stewart[24]argues that the energy in the wake often causes the trailing bubble to break up spontaneously or upon collision with the leading drop.Tsao and Koch[25]analyzed the energy change of a bubble toward a horizontal wall.It seems that most of the energy is lost when the bubble reaches its closest proximity to the wall and the potential energy transfers to kinetic energy with a larger deformation due to surface tension and the boundary layer interference.Bhaga and Weber[20]determined the fluid motion in wakes using hydrogen bubble tracers.Closed wakes are shown to contain a toroidal vortex with its core in the horizontal plane where the wake has its widest cross section.From the stand point of momentum,the ‘tail’behind the closed wake which we observed in this experiment is formed due to a velocity defect.So the change of the kinetic energy is the main factor to affect the energy dissipation.

In present experiments,the diameters of the leading drops and trailing drops range from 9.51 mm to 12.6 mm and from 12.7 mm to 15.8 mm,respectively.The interaction time(ΔT=T0? T*)between two drops ranges from 1.32 s to 1.48 s for different sized drops.However,the starting and the ending times determined by the change of leading drops' deformation or by the change of energy for two drops are consistent with each other,as shown in Fig.8 with three pairs of drops.

Fig.8.Interaction times of two in-line rising drops.(Note:T=0s is the moment when two drops rise into the shooting area).

In conclusion,both methods can be used to determine the interaction time of two in-line rising drops based on the change of geometric feature or the change of energy.The former one is simple and intuitive,but it is restricted by the properties of particles,not suitable for the rigid particles.The latter one is not restricted by the particle properties but it requires the models and the energy dissipation values of the external flow interaction,obtained from the wake flow of the continuous phase.

3.2.Rise velocity and drag coefficient of drops

Drag coefficient is another important parameter in the description of drop dynamics in fluid,reflecting the resistance that drop suffers.When a single drop rises in fluid,the drag coefficient is inversely proportional to the Reynolds number.The drag coefficient of a drop is constant while rising in fluid with stable shape at the terminal velocity.However,during the approaching process,the drag coefficient always changes with two drops' relative position.

Fig.9 shows a simplified illustration of two in-line rising drops.The total force F(t)elongates the trailing drop(prolate distortion)owing to the viscous stresses associated with the convergence of streamlines in the flow produced by the leading drop and meanwhile the leading drop becomes flattened(oblate distortion).This model assumes that the shape of both drops only depends on the total force F(t)outside the interaction region.It is necessary to provide the boundary condition to govern the deformation of drops during interaction.

Fig.9.Effect of the force on deformation and accelerated movement of drops.

The force balance on a drop can be expressed as below:

whereis the mass of drop;Udis the drop's rising velocity;is the resultant force of drop's gravity and buoyancy.is the drag force acting on the leading drop,whereis the drag force acting area.Crab tree and Bridgewater[26]proposed a widely used correlation for the wake velocity Uwof the leading drop downstream:as shown in Fig.9;X is the relative distance of the two drops;FAis the additional mass force:

Fig.10(a)and(b)shows the relationships between the drag coefficient for the leading drop with time versus Re,respectively.Given the size of the trailing drop,from T*to T0,CD,Ldeclines continuously with the approaching of two drops,mainly due to the effect of the trailing drop.

Fig.10.(a)Drag coefficient of leading drop vs.time(DL=9.76–11.56mm,DT=13.4mm).(b)Drag coefficient of leading drop vs.Reynolds number(DL=9.76–11.56 mm,DT=13.4 mm).

Fig.10(b)shows that the drag coefficient of the leading drop declines with the Reynolds number and the trend is like asymptotic of which the coefficient is affected by the drop diameter ratio λ.Given the size of the trailing drop DT,the size of the leading drop DLaffects the decrement of CD,Lduring approaching process;CD,Ldecreases with the decreasing DL.The smaller DLis,the more its drag coefficient is reduced.The curve of CDfor single drop rising in stagnant liquid(Gao et al.[16])is compared with the situation in this work.The drag coefficient for a free rising single drop is higher than the cases with a trailing drop behind.

With the influence of the wake of the leading drop,the drag force of the trailing drop is described by

Here,is the wake area of the leading drop;andis the projected drag force area of the trailing drop considering the wake effect of the leading drop.As shown in Fig.9,WLis the width of leading drop;and Lwis the wake length induced by the leading drop.Nevers and Wu[28]suggested that the wake length Lwin conical wake could be about seven times of the leading bubble half width in glycerin coalescence,i.e.Lw=7WL/2 and Dwis the wake affected area radius:

Fig.11 shows the relationship between the drag coefficient of the trailing drop CD,Tand time.The variation of CD,Tcan be explained by the existence of an equilibrium distance between two interactive drops.Legendre et al.[29]showed that for distances larger than an equilibrium distance,the bubbles are attracted due to the Venturi effect in the separating gap between the bubbles as predicted by the potential theory.When the separating distance decreases,the vortices produced on each drop enters an equilibrium distance and then generates a blocking effect for the flow in the gap,resulting in an increase in the pressure and the drops being repelled from each other.In our experiment,when the trailing drop enters the wake region of the leading one,the trailing drop deforms rapidly,and CD,Tdecreases and reaches a stable value from 0.4 s to 1.1 s.When two drops get close to each other,the drag coefficient CD,Tincreases rapidly due to the blocking effect.We observed the range of the equilibrium distance for 15.13 to 15.87 mm and the drag coefficient will remain stable in this distance.

Fig.11.Drag coefficient of trailing drop vs.time(DL=10.3 mm,DT=13.15–14.48 mm).

3.3.Dynamics of the drop motion

Whether the coalescence would occur was dependent on the drop rising velocity,shape and even the wake.A single rising drop shows a rocking motion and follows a zigzag or spiral trajectory.This phenomenon is associated with the wake-shedding,i.e.,a drop undergoing lateral oscillation tends to have a vertical terminal velocity less than that calculated from the drag on a vertically rising spherical drop.This retardation appears to become more significant as the density ratio is reduced[30].The reason for the lateral oscillation may come from the instantaneous unbalanced forces acting on the deformed drop.For the drops with different sizes,the frequency spectra of the lateral oscillation are obtained through Fourier analysis,as shown in Fig.12.Fig.12 shows that the response on the spectra shifts to the low-frequency range when the equivalent diameter of the drop increases.The oscillations can be associated with the resonance frequency effect.

Fig.12.Spectra of a single drop lateral oscillation as analyzed by Fast Fourier Transform.

Fig.13 shows the response shifts on diameter more clearly,presenting the relationship between the maximal response frequencies(fmax)and the equivalent drop diameter.The correlation of the fmaxversus Deis

The frequency analysis also helps us to understand the dynamic behavior of drops during their approaching process.The spectra of two drops with different sizes in the approaching process are obtained by using the same method and the results are presented in Fig.14.It can be observed that the transverse oscillation frequencies of two drops have the same trend during the approaching process.

Fig.13.Maximal response oscillation frequency versus drop diameter.

For a single rising drop, the frequency of oscillation changes with the diameter of drops and can be associated with the equivalent diameter effect.However,for two interactive drops,the maximal response oscillation frequency becomes almost independent of the drop size,as shown in Fig.15.For the equivalent diameter of trailing drops DT=15.79 mm,the average maximal response frequency roughly equals to that for a free rising single drop with same equivalent diameter,implying that the larger the trailing drop,the greater effect of the oscillation frequency between the interactions.

Fig.16 shows the tendency of the average maximal response frequencies versus the drop diameter ratio λ.It can be seen that the average maximal response frequency of the two drops decreases gradually with the increase of the equivalent diameter ratio.

We believe that the onset of the oscillatory motion is due to a superposition of the drop shape oscillation and the drop internal circulation[16].The existence of deformation will change the effective area of the drag force,which causes an unbalanced force couple on the drop,makes the drop rotation and the further deformation,and finally leads to an oscillation.In the internal circulation measurements some results have been obtained for large liquid skirted drops using tracer particles,providing a qualitative picture of the internal motion as shown in Fig.17[30].

Deformation at the rear of an axisymmetric drop results in the closed streamlines which leave and re-enter the drop,defining a vortex shown in Fig.17(a).Thus,a neutrally buoyant fluid element in the region behind the leading drop may be advected into the cavity which develops behind the deforming leading drop.In Fig.17(b),we show the streamlines for axisymmetric two-drop geometry.The effect of the trailing drop is to increase the extent of the vortex outside the leading drop.A sufficiently deformable drop partly located in the vertical region tends to be entrained.

Owing to the deformation,each drop has a horizontal component of velocity.It is noticed that the streamlines leave the drop and travel through the developing cavity at the rear of the drop.Thus, fluid elements in this cavity will be entrained into the drop wake.The effect of the trailing drop is to further extend the vortex outside the leading drop.

The drop deformation could significantly affect the changes in the drag force of the trailing drop,which in turn affects the collision or separation process of the drops.The drop heat and mass transfer rates may be greatly influenced by the drop shape;for example,the presence of drop lateral oscillation may affect the drop ignition time and burning rate and cause incomplete reaction and pollutant emission in a reaction or combustion system.The investigation of drop deformation effects on the lateral oscillation behavior in the approaching process is thus of importance from both fundamental and applied aspects.

Fig.14.Spectra of the lateral oscillation analyzed by Fast Fourier Transform.(The blue dashed lines show the oscillation frequency of single drop with same diameter).

4.Conclusions

The velocity,shape,drag coefficient and the lateral oscillation between two in-line rising drops during the approaching process were systematically measured and analyzed.The geometric feature method and energy change method were proposed to judge the starting and ending times of the approaching process.The former is simple and intuitive,suitable for the soft particles.The latter requires the energy dissipation information of the continuous phase wake flow,but it can be extended to the rigid particle systems.

A conical wake model was applied to describe the wake shape behind the leading drop[28],the motive interaction force between two drops comes from the leading drop wake and the pressure change in front of the trailing drop.An analytical expression for the hydrodynamic force on the deformed drops in the approaching process has been obtained.The numerical solution of this model presents the variation of the drag coefficients with time,where the drag coefficient of the trailing drop will first decreases,then remains unchanged and finally increases due to the existence of an equilibrium distance.And the drag coefficient of the leading drop decreases with the increase of time or Reynolds number.

Fig.15.Comparison of maximal response oscillation frequencies between a free rising single drop and two interactive drops.

Fig.16.Relationship between average maximal response oscillation frequency and drop diameter ratio λ(Note:Inside the brackets is the equivalent diameters(in mm)of leading and trailing drops in the approaching process).

Fig.17.Streamlines relative to the translating drops[30].

By analyzing the rising trajectories of two in-line drops, the frequency spectra of lateral oscillation are obtained.The transverse oscillation frequencies of two drops have the same trend during the approaching process.With the decrease of the two drop equivalent diameter ratio,the average maximal response frequencies of two drops decrease gradually,owing to a superposition of the drop shape oscillation and the drop internal circulation.

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