Transformation of single drop breakup from binary to ternary and multiple in turbulent jet flows

Wenjun Liang,Dengfei Wang,Meijuan Qian,Ziqi Cai,Zhipeng Li,*,Zhengming Gao,*

1 Beijing Advanced Innovation Center for Soft Matter Science and Engineering,Beijing University of Chemical Technology,Beijing 100029,China

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

3 Daqing Petrochemical Research Center,Petrochemical Research Institute,China National Petroleum Corporation,Daqing 163714,China

Keywords: Drop Multiple breakup Jet flows PIV Breakup mechanism

ABSTRACT By releasing liquid drops in turbulent jet flows,we investigated the transformation of single drop breakup from binary to ternary and multiple.Silicone oil and deionized water were the dispersed phase and continuous phase,respectively.The probability of binary,ternary,and multiple breakup of oil drops in jet flows is a function of the jet Reynolds number.To address the underlying mechanisms of this transformation of drop breakup,we performed two-dimensional particle image velocimetry (PIV) experiments of single-phase jet flows.With the combination of drop breakup phenomenon and two-dimensional PIV results in a single-phase flow field,these transformation conditions can be estimated: the capillary number ranges from 0.17 to 0.27,and the Weber number ranges from 55 to 111.

1.Introduction

Liquid-liquid dispersions and emulsions are widely involved in a variety of industrial processes,such as emulsion production,chemical mixing and reaction,absorption,solvent extraction,and separation.Especially in mixing processes,drop breakup can affect the phase contact area,the mass and heat transfers in the system,and then the efficiency of the processes.

Drop breakup behavior was investigated in flows generated by different devices,including in stirred tanks [1],pump-mix mixer[2],rotor–stator mixers [3],pipe flow [4],and jets [5].Taylor [6]studied the deformation and breakup of single drops by the viscous force in a simple shear field.The breakup of single drops by viscous force mainly depends on the capillary number (Ca).The deformation of a drop increases with the increase ofCa.IfCaexceeds a critical value,the interfacial tension will no longer be able to maintain stability,and the drop will break into two or more fragments.The main conclusions of drop breakup for simple shear flow are summarized by Grace [7],Bentley and Leal [8],and de Bruijn [9].In the study area of mixing,many conclusions for complex flow fields are based on the typical results for simple shear fields.Many models for the mechanisms of drop breakup,such as the models of Coulaloglou and Tavlarides [10],Narsimhanet al.[11],Leeet al.[12,13],and Lehret al.[14],are based on turbulent fluctuations and collisions and focus on the breakup frequency and the distribution of daughter droplets.

Generally,the breakup mechanism can be expressed as a balance between the external stresses from the continuous phase,which attempt to destroy the fluid particle,and the surface stress of the particle and the viscous stress of the fluid inside it,both of which restore its form.The mechanisms of breakup can be classified into four main categories: (1) turbulent fluctuation and collision,(2) viscous shear stress,(3) shearing-off process,and (4)interfacial instability [15].Although liquid–liquid dispersions are widely used in industrial applications,the mechanisms of drop breakup are still incomplete.With the limitation of optical access,problems with spatial resolution due to the small length scale of the drops,issues with temporal resolution due to the short lifetime of turbulent vortices and drop deformation/breakup time [3],the measurement of single drop breakup is still a problem.Furthermore,the number of published experimental studies on single drop deformation and breakup in transitional and turbulent flows is still limited.It is important and necessary to strengthen the understanding of drop behavior to explain the mechanisms of drop breakup.

In our previous study[16],we constructed a well-defined liquid jet flow and investigated the characteristics of drop breakup in the jet flow,especially the critical conditions and breakup probability of binary breakup,breakup time,and daughter drop size distribution.Although we observed the process of breakup cascade,the probability of breakup cascade was relatively low.Many drops deflected horizontally without breakup.There is a lack of research on ternary and multiple breakup.The study of the transformation of drop breakup from binary to ternary and multiple is a bridge between basic research of binary drop breakup and drop dispersion in industry.The aim of this study is to investigate the characteristics of multiple drop breakups and the underlying mechanism of the breakups in a well-defined jet flow field.

2.Experimental

2.1.Drop breakup in jet flow experiments

The experimental setup for the drop breakup in a jet is shown in Fig.1.We designed a flow system to generate a stable and controllable jet flow field.This system contains a diaphragm pump,a damper,a buffer tank,a pressure gauge,two needle valves,and a back-pressure valve.The damper and the buffer tank were used to eliminate the periodic fluctuations in the flow rate,since the diaphragm pump is a positive displacement pump.The two needle valves were used for adjusting the flow rate.We adjusted the liquid pressure to up to about 250 kPa (indicated in red in Fig.1).The reading of the pressure gauge indicated the stability of the flow rate.Its fluctuations were less than 1%of the working pressure.The flow rate error in the measurements was ±0.5% within the experimental flow range.In our experiments,the jet flow rate was the only variable.

The tank measures approximately 300 mm long×100 mm wide×200 mm deep.A vertical partition was used for separating the oil and water and then recycling the deionized water.A lid over the tank was used for keeping the liquid level stable.The cross section of the jet pipe is a rectangle 16.00 mm long and 4.00 mm wide.The hydraulic diameter of the jet pipe isda=6˙40 mm,as calculated byda=4A/C,whereAandCare the cross-sectional area and perimeter,respectively.To ensure the full development of turbulence flow inside the jet pipe,we set the length of the pipe to 500 mm.The origin of the Cartesian coordinate system is located at the center of the cross section of the outlet of the jet pipe (see Fig.1).The Reynolds number of the jet flows is defined asRe=ρCa/μC,where μCis the viscosity of the deionized water,ρCis the density of the deionized water,is the average velocity of the jet flows,anddais the hydraulic diameter of the jet.The jet Reynolds numberReranged from 2727 to 4825.The vertical distance (in they-direction) from the center of the jet pipe to the tank bottom is 70 mm.

Fig.1.Experimental setup for drop breakup in jet flows: 1-diaphragm pump,2-damper,3-buffer tank,4-back pressure valve,5-pressure gauge,6-two needle valves,7-jet flow pipe,8-tank,9-needle,10-outlet,11-microinjection pump,12-camera,13-computer,and 14-light.Dimensions are in mm.

The dispersed phase used in our experiments was dimethyl silicone oil(from Guangzhou Batai Chemical Co.,Ltd,China).The continuous phase was deionized water.The physical parameters of the binary system silicone oil/water are consistent with those in our previous studies [16].The experimental temperature was controlled at (25.0 ± 1.0) °C.Therefore,μC=0˙883±0˙023（） mPa·s,μD=2˙206±0˙050（） mPa·s,and γ=33˙69±0˙42（） mN·m-1.The densities of the deionized water and the silicone oil are 997 kg·m-3and 837 kg·m-3,respectively,at 25.0°C.The error of density measurements is less than 0.1%.The physical properties of the silicone/water system are summarized in Table 1.

Table 1Physical parameters of binary system silicone oil/water at 25 °C

A 16 G (Birmingham Wire Gauge) needle was used to generate drops.The inner and the outer diameter of the needle are 1.3 mm and 1.6 mm,respectively.The needle outlet is flat.The center lines of the jet pipe and the needle are in the samex-yplane (atz=0).The horizontal and the vertical distance in thex- and y-directions between the outlet centers of the needle and the jet pipe are 13.66 mm and 20.55 mm,respectively (see Fig.1).

A high-speed camera(GO-5000M-USB,JAI,Denmark)was used to capture the deformation and breakup of the rising drop.The image resolution was 1280×1024 pixels,and the shooting frequency was 100 Hz.A fluorescent lamp was used to illuminate the drops.The exposure time was adjusted to 2000 μs to get a clear drop edge.

2.2.Drop detection

Detecting the drop edge is extremely important to the volume calculations of the mother and daughter drops.The Canny edge detection method [17]was used to identify the edge of the drops captured by high-speed camera.Two raw images of drops and the corresponding Canny edge detection results are shown in Fig.2.

The volume integral method was used to calculate the drop volumeVD=πR（y）2dy,wherey1andy2are the highest and lowest positions of the drop,respectively,andRis the drop radius as a function ofy.There are 630 drops in our current study.The average equivalent diameterdDis 5.77 mm (±2.26%).This deviation in diameter is larger than that in our previous study (0.52%) because the drops were slightly disturbed by the non-core region of turbulence when the drops were generated.

2.3.Single-phase 2D PIV

The single-phase 2D PIV(TSI,USA)technique was used to measure the velocity of the jet flow field.The rectangular area captured by the charge-coupled device (CCD) camera is approximately 90 mm×60 mm (the red frame in Fig.1).The time interval Δtbetween each pair of images is set to 200 μs.The size of the interrogation window is 32×32 pixels with 50%overlap.The PIV image resolution was 0.03327 mm·pixel-1,and the velocity vector resolution was 0.53 mm.At eachRe,500 PIV images were captured at a frequency of 1 Hz.We used the median filtering method [18]to delete the erroneous instantaneous vectors and the linear interpolation method [19]to fill in the missing vectors.

The Kolmogorov length scale of the jet flows was 0.011–0.017 mm,as estimated by η=daRe-3/4[20],whereReranges from2727 to 4825.The PIV velocity vector resolution is much larger than the Kolmogorov length scale.For the calculation of the shear rater′=（ε/υ）1/2[21],where υ is the kinematic viscosity of the continuous phase,the dissipation rate of turbulent kinetic energy(ε)should be estimated.LimitedbythePIV resolution,a largeeddymethod[22]was used to estimate ε from the PIV data:〈ε〉=23/2Δ2〈S3〉,whereCSis 0.19 for a 50%overlap of interrogation window[23],Δ is the size of an interrogation window,andSis（∑i，jSijSij）1/2with the strain rate tensorassociated with the velocity field resolved by the PIV.The term〈S3〉is estimated by〈S3〉=〈S2〉3/2andaccording to Meneveau and Lund[24]and the isotropy assumptions[21,25].

3.Results and Discussion

3.1.Drop deformation and breakup in jet flows

With the experimental setup shown in Fig.1,we changed the flow rate by adjusting the needle valve.The flow rate was calibrated by weighing the mass of outlet liquid per unit time after the pressure gauge value was stable.After a drop was released by a microinjection pump,it would rise and hit the jet flow.Deformation and even breakup will happen under different conditions.All behavior of drops was captured by a high-speed camera.

Fig.2.Raw image of drop (a) attached to outlet of needle and (b) shortly after detaching from needle.The time separation between the images is Δt=0˙01 s.The black lines in panels(c)and(d)are the detected edges of the drops in panels(a)and(b),respectively.The red asterisks are the calculated centroids of the drop.In panel(c), y1 and y2 are the highest and lowest positions of the drop,respectively; x1 (y)and x2 (y) are the left and right boundaries of the drop,respectively; and R is the radius of the drop as a function of y.

In the current study,we distinguish the types of drop breakup as binary,ternary,and multiple breakup.As shown in Figs.3 and 4,in a typical ternary and a typical multiple breakup atRe=3941,a mother droplet breaks up into 3 and 4 daughter drops,respectively.The phenomenon that a drop breaks into only two daughter drops occurs under a narrow flow condition [16],and only one ‘‘neck” forms and then breaks.But the ranges of the jetRefrom binary to multiple breakups are relatively wide,see Fig.5.The drop deforms in a complicated way as well,and several‘‘necks”often form at the same time.And then the necks break and three or more drops are generated.The probability of binary,ternary,and multiple breakup under differentReconditions is shown in Fig.5.As the Reynolds number increases fromRe=2727 to 4825,the probability of binary breakup decreases but the probability of multiple breakup increases.Within the range ofRestudied,the probability of ternary breakup first increases,reaches a maximum,and then decreases,but the overall change is slight.It should be noted that the probability of the three types of breakup is approximately the same whenReis 4000.We define the expected number of daughter drops(Enum),which is the number of daughter drops divided by the number of mother drops.Fig.6 showsEnumas a function of the jet Reynolds numberRe.The growth ofEnumis obviously related to the ternary and the multiple breakup of mother drops.

3.2.Jet flow field

Limited by the unmatched refractive indices of oil and water and the low PIV shooting frequency,the PIV experiments were performed in a single-phase flow.The flow is turbulent in the range ofRein the current study.A sample of instantaneous velocity fields measured by 2D PIV at different Reynolds numbers is shown in Fig.7.We can see that the turbulence intensifies asReincreases from the comparison of Fig.7a and b.

In a Reynolds decomposition,the instantaneous velocity can be divided into the mean velocity and the fluctuating velocity:ui=+ui′.The mean velocities at different Reynolds numbers are shown in Fig.8.The mean velocity distribution atX=13˙66 mm is approximately a parabola.The width of the mainstream is about 10 mm,which is 2.5 times the width of the jet pipe.The mean velocity in the core area ranges from 0.58 m·s-1to 0.82 m·s-1whenReranges from 3322 to 4781.The energy dissipation rate in the average flow 〈ε〉avat different Reynolds numbers is shown in Fig.9.The energy dissipation rate tends to double peaks atX=13˙66 mm.A comparison of different energy dissipation rate components atRe=4487 is shown in Fig.10.The maximum dissipation rate in the average flow is 37% higher than the maximum dissipation rate in the fluctuating flow atRe=4487 atX=13˙66 mm.The maximum total energy dissipation rate is 33% higher than the sum of 〈ε〉avand 〈ε〉fluct.

3.3.Mechanism of drop breakup

The breakup of drops changes from binary to ternary and multiple asReincreases from 2727 to 4825.The transformation of drop breakup is governed by the capillary number and/or the Weber numbers:Ca=μCdDr′/2γ and,where μCis the dynamic viscosity of deionized water,dDis the diameter of a drop,r′is the shear rate calculated byr′=（ε/υ）1/2,γ is the interfacial tension,ρCis the density of deionized water,andis the average velocity of the flow.

Fig.3.Typical process of ternary breakup at Re=3941.The time interval between two frames is Δt=0˙02 s.The white area on the left is the exit of the jet pipe.

Fig.4.Typical process of multi-breakup at Re=3941.The time interval between two frames is Δt=0˙02 s.The white area on the left is the exit of the jet pipe.

Typical drop breakup visualizations are combined with capillary number contours and Weber number contours in Figs.11 and 12,respectively,in which the Reynolds numbers of the drop visualization experiments (Re1) and those of the PIV experiments(Re2) are closely matched.The minor difference between the two Reynolds numbers is due to the fact that we cannot perform the two experiments simultaneously.The drop starts deforming when it moves to the lower influential region of the jet,where the Capillary number has peak levels(see Fig.11)while the Weber number is locally minimum at the same location (see Fig.12).When the drop continues rising around the center line of the jet,the inertial effect dominates with peak levels in Weber number and local minimum value in Capillary number,and then drop breakup (see Fig.12a) or breakup cascade (see Fig.12b) occurs.All the breakup or breakup cascade happens inside the influential region of the jet flow in this work.In our previous work regarding pure binary breakup [16],however,the breakup process occurred during the drop was going through the influential region of the jet flow.

Fig.5.Probability (P) of binary,ternary,and multiple breakup of oil drops in jet flows as a function of jet Reynolds number Re.

Fig.6. Enum as a function of jet Reynolds number Re.

Fig.7.Instantaneous velocity fields measured by 2D PIV at different Reynolds numbers: (a) Re=3322 and (b) Re=4487.

Fig.8.Mean velocity from PIV at (a) Re=3322; (b)Re=4478; (c) Y=0 mm; (d) X=13˙66 mm.

Fig.9.Energy dissipation rate in average flow 〈ε〉av at (a) Re=3322; (b) Re=3904; (c) Re=4487; (d) X=13˙66 mm.

AsRegoes from 3332 to 4781,atX=13˙66 mm,the maximumCa(Fig.11c and d)ranges from 0.17 to 0.27,and the maximumWe(Fig.12 d) from 55 to 111.In all cases of drop breakup in the current study,the inertial effect (We) largely deforms the drops,and the viscous effect (Ca) shears off the drops.Therefore,Figs.11 and 12 provide an estimation of the transformation conditions for drop breakup from binary to ternary and multiple:Cafrom 0.17 to 0.27 andWefrom 55 to 111.

4.Conclusions

The transformation of drop breakup from binary to ternary and multiple and the expected daughter drop number were investigated in a well-defined turbulent jet flow.The dispersed phase and continuous phase were silicone oil and deionized water,respectively.The process of drop breakup was captured by a high-speed camera,and the flow field was investigated by a 2D PIV system.

In the drop breakup experiments,in the range of jet flowRe2727 to 4825,the probability of binary breakup shows a downward trend.Ternary breakup begins to occur atRe=2727 and its probability reaches a maximum of 33.3% atRe=3498.The trend of multi-breakup probability is opposite to that of binary breakup probability.The expected of daughter drop number ranges from 3.59 to 7.65 within the range ofRestudied.To further understand the transformation conditions of drop breakup from binary to ternary and multiple,the PIV technique was used to quantify the jet flow field characteristics.With the combination of drop breakup results and single-phase 2D PIV data,the following transformation conditions can be estimated:Cafrom 0.17 to 0.27 andWefrom 55 to 111.

Fig.10.Energy dissipation rate derived from PIV at Re=4487 (a) in average flow 〈ε〉av; (b) in fluctuating flow 〈ε〉fluct.(c) Total energy dissipation rate 〈ε〉total calculated without Reynolds decomposition.(d) Comparison of different energy dissipation rate components at X=13˙66 mm.

Fig.11.Visualization of drop breakup combined with Ca contours derived from PIV at various jet Reynolds numbers:(a)Re1=3336,Re2=3322;(b)Re1=3941,Re2=3904;(c)Re1=4501,Re2=4487.Re1 and Re2 are the Reynolds numbers in drop breakup experiments and single-phase PIV experiments,respectively.The alternating solid and dotdash black lines indicate the drop circumferences 0.02 s apart.(d) Comparison of Ca at different jet Reynolds numbers at X=13˙66 mm.

Limited by the different refractive indices of the dispersed phase and continuous phase as well as the low frequency of the PIV system,the PIV experiments were performed separately from the drop breakup experiments.The refractive index matching of the two phases [26,27]and a high frequency PIV system will be the solution for the coupling of drops and flow field in the future.Another option is two-phase direct numerical simulation [28].

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors gratefully acknowledge the financial supports from the National Key Research and Development Program of China(2016YFB0302801),National Natural Science Foundation of China(21676007),Fundamental Research Funds for the Central Universities (XK1802-1),and Scientific Research and Technology Development Projects of China National Petroleum Corporation (2016B-2605).

Nomenclature

Across-sectional area of jet pipe,m2

Cperimeter of jet pipe,m

CSSmagorinsky-Lilly constant

Fig.12.Visualization of drop breakup combined with Weber number contours derived from PIV at various jet Reynolds numbers:(a)Re1=3336,Re2=3322;(b)Re1=4501,Re2=4487. Re1 and Re2 are the Reynolds numbers in drop breakup experiments and single-phase PIV experiments,respectively.The alternating solid and dot-dash black lines indicate drop circumferences 0.02 s apart.Comparison of We at different jet Reynolds numbers at (c) Y=0 mm and (d) X=13˙66 mm.

CaCapillary number

dahydraulic diameter,m

dDequivalent diameter of drop,m

Enumthe expected number of daughter drops

ReReynolds number of jet flow

r′shear rate,s-1

Sstrain rate tensor,s-1

Ttemperature,°C

Δttime interval,s

uiinstantaneous velocity of flow,m·s-1

VDvolume of drop,m3

WeWeber number

γ interfacial tension,N·m-1

Δ size of interrogation window,m

ε dissipation rate of turbulent kinetic energy,m2·s-3

〈ε〉avdissipation rate in average flow,m2·s-3

〈ε〉fluctdissipation rate in fluctuating flow,m2·s-3

〈ε〉totaltotal dissipation rate in instantaneous flow,m2·s-3

η Kolmogorov scale,m

μCviscosity of continuous phase,Pa·s

μDviscosity of dispersed phase,Pa·s

ρCdensity of continuous phase,kg·m-3

ρDdensity of dispersed phase,kg·m-3

Δρ density difference,kg·m-3

υ kinematic viscosity of continuous phase,m2·s-1