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Measurement of droplet breakage in a pump-mixer
Accepted Manuscript Measurement of droplet breakage in a pump-mixer Han Zhou, Xiong Yu, Shan Jing, Hao Zhou, Wenjie Lan, Shaowei Li PII: DOI: Reference:
S0009-2509(18)30812-1 https://doi.org/10.1016/j.ces.2018.11.035 CES 14623
To appear in:
Chemical Engineering Science
Received Date: Revised Date: Accepted Date:
8 June 2018 20 September 2018 14 November 2018
Please cite this article as: H. Zhou, X. Yu, S. Jing, H. Zhou, W. Lan, S. Li, Measurement of droplet breakage in a pump-mixer, Chemical Engineering Science (2018), doi: https://doi.org/10.1016/j.ces.2018.11.035
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Measurement of droplet breakage in a pump-mixer Han Zhoua, Xiong Yua, Shan Jinga, *, Hao Zhoua, Wenjie Lanb, Shaowei Lia, c,* a
Collaborative Innovation Center of Advanced Nuclear Energy Technology, Institute
of Nuclear and New Energy Technology, Tsinghua University, Beijing 100084, China b
State Key Laboratory of Heavy Oil Processing, China University of Petroleum (Beijing), Beijing 102249, China
c
State Key Laboratory of Chemical Engineering, Tsinghua University, Beijing 100084, China
Abstract: The direct experimental data for droplet breakage frequency and daughter droplet size distribution is urgently required for the application of the population balance model. To meet this requirement, a method to directly measure the droplet breakage in the pump-mixer was developed using a high speed online camera. Two typical breakup patterns, tensile breakup and revolving breakup were observed and the number of the breakup fragments was determined. The multiple breakage was treated as a series of binary breakups. In order to precisely describe the daughter size distribution, these binary breakups were divided into three different types, i.e. the original tensile breakup, intermediate tensile breakup, and revolving breakup. The influences of the rotating speed and interfacial tension on the droplet breakup frequency and daughter droplet size distribution were then quantitatively investigated.
*
Corresponding author, [email protected] (S. Jing), [email protected] (S. Li) 1
Empirical correlations were proposed and good agreement was found between the prediction results and the experimental data. Keywords: population balance model; pump-mixer; droplet breakup; daughter droplet size distribution
1. Introduction Solvent extraction is a powerful liquid-liquid separation technique and has extensive applications in many fields. In liquid-liquid extraction processes, one phase is usually dispersed into the other phase to form droplets. The droplet size distribution (DSD) determines the mass transfer area, making it a significant parameter for the extraction efficiency. In the extraction process, a narrow DSD is desired to optimize hydraulics and mass transfer performance (Kopriwa et al., 2012). It is also reported that the DSD depends on the droplet breakup and coalescence (Qin et al.,2016; Zaccone et al., 2007). Thus, an intensive understanding of the droplet breakup and coalescence is extremely important for the optimization of extraction process. A variety of industrial equipment have been designed for the extraction operation, such as centrifugal extractor (Ayyappa et al., 2016), extraction column (van Delden et al., 2006), mixer-settler (Thakur et al., 2003), etc. Pump-mixer, as a type of mixer in mixer-settler, has been widely used in nuclear fuel reprocessing and chemical industries. Compared to common mixers, pump-mixer can realize pumping and mixing simultaneously (Srilatha et al., 2008). It also enjoys the advantages of large capacity, high efficiency, reliable operation and strong adaptability. The hydraulic and
2
mass transfer performance of pump-mixer has been studied in the past several decades (Rao and Baird, 1984; Srilatha et al., 2008; Parvizi et al., 2016). It was found that complex interaction exists among the DSD, the breakage, and coalescence. In-depth exploring on it is of great significance for performance optimizing, precisely designing, and scaling-up of the mixer. Numerous studies on drop breakage and coalescence have been reported (Coulaloglou and Tavlarides, 1977; Luo and Svendsen, 1996; Alopaeus et al. 1999; Andersson and Andersson, 2006a, b). In these studies, the population balance model (PBM) is mostly considered due to its ability to predict the DSD by considering the effects of drop breakage and coalescence. In the framework of PBM, the influences of droplet breakage and coalescence is considered in the source term of the population balance equation (PBE). They are expressed in the form of three sub-models, i.e., breakage frequency, daughter droplet size distribution (DDSD), and coalescence rate models. The DSD can be predicted only if the three sub-models are determined. To precisely determine these three sub-models is a key challenge in employing PBM to calculate liquid-liquid two-phase flow. In this work, we will focus on the first two sub-models, which describe the droplet breakage behavior. Many efforts have been made to quantitatively describe the drop breakage with breakup frequency and DDSD. A comprehensive literature reviews have been made by Liao and Lucas (2009). They argue that turbulent particle breakup mechanisms can be mainly attribute to turbulent fluctuation and collision (Liao and Lucas, 2009). Furthermore, breakage mechanisms include viscous shear stress, turbulent shear, 3
interfacial instability and shearing-off may also cause breakup (Falzone et al., 2018). Thus, a variety of models have been proposed in the past few decades. Among those efforts, two classical models proposed by Coulaloglou and Tavlarides (Coulaloglou and Tavlarides, 1977), and Luo and Svendsen (Luo and Svendsen, 1996) are still widely used for the model extensions and simulation applications. In Coulaloglou and Tavlarides’ model, the breakage frequency is expressed as Eq. (1).
1 d 1/3 D c1 exp c2 2/3 d 2/3 D5/3 1 d D
2
(1)
This model assumes that the drop breakage is caused by the collision with the turbulent eddy. The breakage is dominated by the mother droplet size and the turbulent dissipation, while independent of the fluid viscosity. It is also assumed that only the binary breakages occurred, and the DDSD is a normal distribution as expressed in Eq. (2). 2 2 D3 Dm3 7.2 D 2 D, Dm exp 4.5 Dm3 Dm 6
( 2)
A survey of the modified and extended models based on Coulaloglou and Tavlarides’ model is provided by Solsvik et al. (Solsvik et al., 2014). However, these models are very sensitive to the adjustable parameters containing in them, making them suitable for a specific device and system. Luo and Svendsen proposed a new theoretical model of breakage frequency which containing no adjustable parameters. Based on the kinetic gas theory, they defined breakage frequency as a combined result of arrival or bombarding frequency
4
of eddies and breakage probability. The breakage frequency function and DDSD are expressed as Eq. (3) and Eq. (4).
D 0.462 1 d 2 D
1/3
f v v, v
2
v
max
min 1 max
0 min
1
m ax
0
min
1
11/3
2
12c f exp d df v (3) 2/3 5/3 11/3 0 c D
1 / exp 12c / 1 / exp 12c / 2
11/3
f
2
0
2/3
c
11/3
2/3
f
0
c
d df
D5/3 11/3 d
D
5/3 11/3
(4) v
This model is widely used in academic and many modified models have been proposed (Hagesaether et al., 2002; Andersson and Andersson, 2006b; Razzaghi and Shahraki, 2016). But researchers also find that these models are usually very sensitive to the upper limit of the integration ξmax, which defines the ratio between the sizes of the largest effective eddy and the droplet. In fact, Andersson and Andersson (Andersson and Andersson, 2006b) have pointed out that eddies with size up to three times of the droplets are still responsible for breakup. In addition to the theoretical study, single droplet experiments (Solsvik et al., 2015, 2016; Maaß and Kraume, 2012) also have been carried out to investigate drop breakage. Solsvik et al. (Solsvik et al., 2015, 2016) performed the single drop breakup studies and suggested that the drop size had a significant impact on the breakage frequency and the number of fragmentation during a single breakup event. Maaß et al. (Maaß and Kraume, 2012) investigated single drop breakup event under turbulent condition, which was comparable to those in a mixer. They improved the breakage time model by taking into account the influence of viscosity and interfacial tension. Some studies (Wang et al., 2005,2006; Ramachandran et al., 2009; Xing et al.,
5
2013; Vonka and Soos, 2015) have been carried out to introduce PBM into the gas-liquid or liquid-liquid dispersions. In this approach, PBM sub-models including breakup frequency and DDSD are normally taken into account. Nevertheless, the particular form and parameters in such models differ from each other due to the specificity of particular flow geometry, and even cannot be applied to a changed condition. Thus, more efforts are deserved to obtain universal PBM kernel functions with a great amount of experimental work. However, the direct experimental data for droplet breakage frequency and DDSD is very lacking in literature. The main purpose of this work is to develop an experimental method to measure the breakage frequency and DDSD and to systematically investigate the droplet breakage in a pump-mixer. Reference to the method in our previous work on a pulsed disc and doughnut column (Zhou et al., 2017; Fang et al., 2017), a high-speed online camera was involved for the experimental data collection in this work. Significant amount of videos of droplet breakup events were captured for data statistics. Drop breakup frequency and DDSD were then quantitatively calculated based on the statistical results. The influences of energy input and two-phase physical properties were investigated. Furthermore, empirical correlations for the breakup frequency and DDSD were proposed.
2. Experiments and methods 2.1 Experimental setup
6
Figure 1. The experimental setup (a) Photo of the experimental setup, (b) Schematic diagram of the experimental setup.
The experimental setup is shown in Fig. 1, where the pump-mixer is designed based on the Power-Gas industrial mixer (Reeve and Godfrey, 2002). The size of the mixing tank is 100 mm× 100 mm× 100 mm. The walls of the tank are made of quartz glass. There is a suction orifice at the bottom of the tank with a diameter of 32 mm. A 100 mm× 100 mm× 30 mm buffer chamber is connected to the mixing tank through the suction orifice. A peristaltic pump (Longer Pump WT600-2J) is used to drive the continuous phase (aqueous phase) into the mixer from the buffer chamber with a flow rate of 440-500 mL/min. The organic phase is pre-dispersed into droplets by the aqueous phase in a Y-junction channel, and then introduced to the mixer through a stainless steel tube with inner diameter of 3 mm. The organic and aqueous phases are introduced into the Y-junction channel by two peristaltic pumps (Longer Pump BT100-2J) with flow rates of 1.2-50 mL/min and 0-30 mL/min, respectively. In this study, both phases were fed continuously and the total flux was set as 500 mL/min. 7
Mixture of the two phases overflows from the top of the tank with flow rate of 500 mL/min. Table 2 show the flow ratios (Qo:Qa) under different conditions. The flow rate of the dispersed phase is sufficiently small to ensure a small holdup, so that the coalescence of the dispersed phase droplets can be ignored. Therefore, the results obtained in this work are only applicable for dilute dispersions.
Figure 2. Pump-mix impeller (a) schematic diagram of the impeller, (b) cutaway view of the impeller.
The impeller is made of stainless steel and its structure is shown in Fig.2. It has a diameter of 65 mm and contains eight equispaced, curved blades of 10 mm high and 2 mm thick. The distance between the tank bottom and the impeller is set as 5 mm. A motor (TaiBang Motor CV0.4KW) connected to the axle of the impeller is used for energy input. The rotating speed of the impeller is varied in the range of 360-600 rpm in our experiments. A microscope (Olympus SZ61) coupled with a high speed camera (Olympus i-Speed TR) is used to observe droplet behavior in the mixer. The frame rate of the camera is set as 5000 fps and the amplification factor of the microscope is fixed as 1.2. 8
The resolution of the captured image is 0.018 mm/pixel, so that droplets with size around 0.1 mm can still be clearly captured. More than 40 videos are recorded for each operation condition, and each video has a time duration of 1.267 s. The measurement of the droplet size and the breakup events are done manually with an image manipulation software, which was developed by our team based on Python. Solsvik et al. (Solsvik and Jakobsen, 2015) reported the results can be vague imprecise due to droplets overlap and the lack of three-dimensional observation. In this study, the holdup of the dispersed phase is dilute enough to ignore the influence of the overlap. Besides, this study was carried out under a steady state, most of the droplets in the mixer was in the near spherical shape. Thus It is reasonable to use two-dimensional image to determine the droplet size, thereby determining the DSD. For the determination of the droplet breakup events, the error is mainly attributed to the deformation of the mother or the daughter droplets. However, attributed to the lower number of fragments generated in a breakup event, this error can be avoided to the most extent by tracing the drops to a most optimum condition.
2.2 Materials The experiments were performed at 20°C and at atmospheric pressure. The n-dodecane was selected as dispersed phase and distilled water was chosen as continuous phase. Different amount of Tri-Butyl-Phosphate (TBP) was added into the dispersed phase to adjust the interfacial tension between the two phases. The physical properties of the experimental systems are shown in Table 1. 9
Table 1. Physical properties of the experimental systems (20°C) System
Continuous
Dispersed
ρc
ρd
μc
μd
σ
No.
phase
phase
kg/m3
kg/m3
mPa·s
mPa·s
mN/m
1
Water
n-dodecane
997.0
750.1
1.12
1.64
34.61
2
Water
1vol%TBP/n-dodecane
997.0
755.0
1.12
1.65
22.94
3
Water
2vol%TBP/n-dodecane
997.0
757.0
1.12
1.65
19.58
4
Water
5vol%TBP/n-dodecane
997.0
764.0
1.12
1.66
15.07
5
Water
10vol%TBP/n-dodecane
997.0
775.0
1.12
1.69
12.61
6
Water
30vol%TBP/n-dodecane
997.0
850.0
1.12
1.80
9.78
Table 2. Operational rotating speeds and flow ratios Rotating
Rotating Qo:Qa
System No.
Qo:Qa
System No.
speed/rpm
speed/rpm
1
390
0.036
2
360
0.018
1
420
0.018
3
360
0.009
1
480
0.006
4
360
0.009
1
540
0.002
5
360
0.004
1
600
0.002
6
360
0.004
2.3 Method The breakup frequency function of droplets with specified size is defined as the breakup fraction of these droplets within unit time. It can be expressed by Eq. 5. 10
( D )
1 breakup fraction of 1 nb ( D)D t n( D)D tc droplets c
(5)
Herein D is the droplet size. n( D) is the droplet number density and thus n( D)D denotes the number of droplets in size of D to D D per unit volume. Correspondingly, nb ( D)D denotes the number of breakup events within these droplets. tc is the time duration within which the breakup events occur. The measurement of the droplet breakup frequency function is directly performed based on the above definition. Firstly, with certain size interval D ,
n( D)D is determined after measuring the size of enough droplets (no less than 2000 in this work) from the captured photos. The size of a droplet is measured on its major axis a and minor axis b in 2D images and calculated by D ab . Then, 1/2
nb ( D)D is determined by counting the breakup events in the video captured within time duration tc . Finally, the breakup frequency function is determined based on Eq. 5. The DDSD F fV , Dm , which is defined as the probability of generating a daughter droplet with volume ratio fV when a mother droplet with size Dm is broken-up, is experimentally determined using a similar method. The count number of the drop breakage is no less than 250 under each operating condition. Fig. 3 shows that when the count number is in excess of 250, the measuring error is less than ±5%.
11
Breakup frequency function /s-1
3.6 3.2
5%
2.8 2.4 2.0 0
50 100 150 200 250 300 Number of drop breakage events
Figure 3. The measuring error at different number of breakup events. The data points are measured at the conditions of D=0.55 mm, system No.4.
2.4 Determination of the turbulence energy dissipation 2.4.1 CFD simulation method The droplet breakup frequency was associated with the turbulence energy dissipation. Computational fluid dynamics (CFD) simulation was performed to provide the necessary results. Based on the experimental setup, the mixer was divided into two domains: the rotational region consisting of the impeller, and the stationary region of the remaining volume. Structured hex-grid was used as the meshing type. Fig. 4 shows the simulation domain and the grids generated in this study.
12
Fig. 4 CFD simulation domains and meshes (a) 1/4 geometry for simulation (b) grids details of the 1/4 stationary region, (c) grids details of the 1/4 rotating region. The rotating region consists of 0.58 million nodes, the stationary region has 0.94 million nodes. The maximum grid size was 0.8 mm for stationary region and 0.4 mm for the rotating region, the minimum grid size which exists in the interface of the two regions was 0.15 mm. The interface of the two regions was managed particularly, the grid nodes difference of the two sides was controlled within 20%. All CFD simulations were performed using CFX 16.1. Steady state was chosen as the analysis type. The Rotating Frames of Reference (RFR) approach was specified to the rotating region. No-slip conditions were used for the walls. The turbulence was modeled using the Standard k-ε model. The high resolution advection scheme was applied for the continuity, momentum, turbulence kinetic energy and turbulence energy dissipation. The convergence criteria were set with root mean square (RMS) below 10-6. 13
Table 3 shows the mesh independence tests of three groups of grids: the coarse mesh, the medium mesh (the chosen mesh), and the fine mesh. The power consumption and the average eddy dissipation in the observation domain were obtained for the comparison. The results demonstrate that the chosen mesh is acceptable.
Table 3. Results of grid independence tests under 600 rpm Number of nodes
Number of cells
maximum grid size Power
/Million
/ mm
/Million
domain
2
No.
/kg m rotating
stationary
rotating
stationary
rotating
/ m2s-3
stationary s-3
region
region
region
region
region
region
1
0.33
0.48
0.31
0.46
0.48
1.1
0.704
8.11
2
0.58
0.94
0.55
0.90
0.40
0.80
0.715
8.58
3
1.05
1.54
1.01
1.48
0.33
0.70
0.722
8.62
2.4.2 Verification of simulation approach Limited experimental data of similar setup in this study was published in the literature. Thus another simulation with the same modeling and meshing method was carried out in a baffled vessel with a Rushton turbine, which has a similar scale with our setup. The purpose is to verify the applicability of turbulence model and simulation approach.
14
(a)
0.40
0.36
0.34
0.34
z/T
0.38
0.36
z/T
0.38
0.32
0.32
0.30
0.30
r/T=0.183 simulation result Hartmann et al. (2004)
0.28 0.26 0.0
0.2
0.4
0.6
0.8
0.26 0.0
1.0
0.2
0.34
0.34
0.6
0.8
1.0
z/T
0.36
z/T
0.38
0.36
0.4
Vr / Vtip
0.40
0.38
0.32
0.32
0.30
0.30
r/T=0.25 simulation result Hartmann et al. (2004)
0.28 0.2
0.4
0.6
0.8
r/T=0.25 simulation result Hartmann et al. (2004)
0.28 0.26 0.0
1.0
Vc / Vtip
0.40
0.2
0.34
0.6
0.8
1.0
z/T
0.36
0.34
z/T
0.38
0.36
0.4
Vr / Vtip
0.40
0.38
0.32
0.32
0.30
r/T=0.317 simulation result Hartmann et al. (2004)
0.28 0.26 0.0
r/T=0.183 simulation result Hartmann et al. (2004)
0.28
Vc / Vtip
0.40
0.26 0.0
(b)
0.40
0.2
0.4
0.6
0.8
1.0
0.30
r/T=0.317 simulation result Hartmann et al. (2004)
0.28 0.26 0.0
Vc / Vtip
0.2
0.4
0.6
0.8
1.0
Vr / Vtip
Fig. 5. Axial profiles of the circumferential (a) and radial (b) velocity components at three radial locations r/T =0.183(left), r/T =0.25(middle) and r/T =0.317(right).
Appendix A shows the detailed description of this simulation. The simulation was performed in a standard configuration cylindrical vessel of diameter T = 100. The impeller was rotated at N=1082 rpm, and Vtip=1.89 m/s. Figure 5 shows axial profile of the dimensionless radial and circumferential velocity at three radial locations. Vc and Vr represents the radial and circumferential velocity separately, z / T represents 15
the dimensionless axial distance from the bottom to the plane. The figure shows that the simulation results agree well with the experimental results (Hartmann et al., 2004). Thus using the simulation approach in this study, the flow field should be predicted correctly. Moreover, the power number of the RT impeller can be predicted correctly and the average eddy dissipation show an underestimate of 15.9% of the simulation result (see Table A1). These results indicate that the simulation method in the section 2.4.1 can be applied for the following analysis.
3. Results and discussion 3.1 Description of drop breakup events In this study, more than 250 drop breakup events are recorded for each experimental condition. Two main breakup modes, breakup attribute to turbulent fluctuation and collision which is caused by turbulent eddies smaller than the droplet size (for convenience, the following is referred to as tensile breakup) and breakup due to turbulent shear stress of small droplets trapped into larger eddies (the following is referred to as revolving breakup), are observed, as is shown in Fig. 6. The tensile breakup is the most possible drop breakage mode. The droplet is first stretched to form a neck. The neck then shrinks and finally the droplet breaks up into two or more daughter droplets. The revolving breakup is also an important breakup mode in the pump-mixer. The droplet is first trapped into a vortex, and rotates with it. Then the breakup can be occurred due to the velocity gradient in the eddies and a certain part of the droplet which is relatively far away from rotating center is thrown 16
away to form several fragments.
Figure 6. The drop breakup process of the two breakage modes (a) tensile breakup, (b) revolving breakup
The number of the fragments is investigated with the definition of a single droplet breakup event illustrated by Solsvik et al. (Solsvik et al., 2016). The initial breakup is used in determining the number of daughter entities. Fig. 7 shows the probability of binary, ternary, quaternary and quaternary+ (one droplet breaks up into more than 4 daughter droplets) breakage reported by Andersson and Andersson (Andersson and Andersson, 2006a), Solsvik and Jakobsen (Solsvik and Jakobsen, 2015), and Zhou et al. (Zhou et al., 2017), Andersson and Solsvik carried out single droplet experiments and the results showed that the multiple breakup were more 17
common than binary breakup, with a proportion of more than 50%. In comparison, Zhou conducted the experiment under a fully developed droplet swarm flow, and the results showed that the binary breakage had a proportion of more than 50%. The difference is probably because the mother droplet size in the first two studies is much larger than the last one. Actually, Solsvik and Jakobsen (Solsvik and Jakobsen, 2015) has also pointed out that the diameter of the mother droplet is one of the most important factors determining the possible amount of fragments.
Andersson and Andersson, = 3.7 m2s-3 Andersson and Andersson, = 8.5 m2s-3 Solsvik et al., = 1.14 m2s-3 Dm < 2 mm
80
Probability (%)
Solsvik et al., = 1.14 m2s-3 Dm > 2 mm Zhou et al., = 0.042 m2s-3 Zhou et al., = 0.082 m2s-3
60
40
20
0
binary
ternary
quaternary quaternary +
Figure 7. The probability of different number of fragments reported in literatures. Andersson and Andersson (2006a) and Solsvik and Jakobsen (2015) performed single drop breakup experiments while Zhou et al. (2017) carried out their experiments with fully developed droplet swarm.
18
Fig. 8 shows the probability of different number of the fragments in this study. The experimental results show that the binary breakage has an occurrence probability of more than 50% for all conditions. Furthermore, Fig. 8a shows that the number of fragments decreases with the increase of the rotating speed. The reason is probably that the average droplet size gets smaller when increasing the rotating speed, which can be seen in the next section. Smaller droplets tend to be split into fewer segments because of the difficulty of deformation. The interfacial tension is also a main factor affecting droplet breakup. As is shown in Fig. 8b, the number of the fragments increases with the decrease of interfacial tension. The reason is that the surface stability of the droplets decreases with lower interfacial tension, thus the deformation is enhanced which tends to generate more fragments.
System No.1 420 rpm 480 rpm 540 rpm 600 rpm
Probability (%)
80
60
(b)
40
20
0
360 rpm System No. 2 3 4 5 6
80
Probability (%)
(a)
60
40
20
binary
ternary
0
quaternary quaternary +
binary
ternary
quaternary quaternary +
Figure 8. The probability of different number of fragments (a) for system 1 with different rotating speed. (b) for different systems with rotating speed of 360rpm.
To simplify the description of multiple droplet breakage, the assumption of binary breakup is widely used. A common approach is to treat the multiple fragmentation as a series of binary breakup, with an assumption that the larger 19
fragments are always generated firstly. Taking the quaternary breakage as an example, the treating approach is illustrated in Fig. 9. For ease of description, the breakup of the original droplet to form a droplet with size D1 and another imaginary droplet in Fig. 8b is defined as the original breakup while the breakup of the imaginary droplets is defined as the intermediate breakup. The above treating approach will be used in the data processing of this work.
Figure 9. Treatment of quaternary breakage as three binary breakages. (a) the entire quaternary breakage process, (b) a series of imaginary binary breakup processes.
Based on the descriptions above, the breakup processes were then quantitatively analyzed. For the convenience of the subsequent discussion, the occurrence possibility of three breakup processes, original tensile breakup, intermediate tensile breakup, and revolving breakup were firstly determined from the captured video, the results is
20
shown in Fig. 10. Furthermore, after dimensional analysis, the correlations of the occurrence possibility for the three breakup processes were proposed, the correlations are as Eqs. 6-8. Po (1 0.66WeD 0.19 4.2 10-4WeD -3.82 )1
(6)
Pi 0.66WeD 0.19 Po
(7)
Pr 1 ( Pi Po )
(8)
Where WeD
c D5/3 2/3 . Po represents the occurrence possibility of original tensile
breakup, while Pi and Pr represent the occurrence possibility of intermediate tensile breakup and revolving breakup respectively.
0.7 0.6
Po Pi Pr
possibility
0.5 0.4 0.3 0.2 0.1 0.0 0.14
0.16
0.18
0.20
0.22
WeD
Figure 10. The occurrence possibility of the three breakup processes.
3.2 Number density function of droplets As expressed by Eq. 5, we need to determine the number density function n( D) before measuring the breakup frequency. The measured number density at different experimental conditions is shown in Fig. 11. The drop size distribution is similar to 21
normal distribution and the drop size is mainly ranging from 0.05 to 1.0 mm in this study. The effect of the agitation speed and interfacial tension on DSD can be observed in the figure. Higher agitation speed or lower interfacial tension leads to smaller droplet size and narrower DSD. The increasing of agitation speed will enhance the energy input while the decreasing of interfacial tension will reduce the inner energy resisting breakage. In both of the above conditions, the stability of the droplets is decreased and thus the DSD shifts to the smaller size dimension. (a)
(b) 16
System No.1 390 rpm 420 rpm 480 rpm 540 rpm 600 rpm
8 4
0
0.2
0.4 0.6 Diameter/mm
0.8
360 rpm System No. 2 3 4 5 6
4
n(D)/(1010Qa/Qo) m-4
n(D)/(1010Qa/Qo) m-4
12
3 2 1 0
1.0
0.2
0.4 0.6 Diameter/mm
0.8
1.0
Figure 11. Number density of the droplets (a) for system 1 with different rotating speed conditions, (b) for systems 2-6 with rotating speed of 360 rpm.
3.3 Drop breakup frequency function The influence of the dispersed phase flow rates on the drop breakup frequency function can be neglected due to the low holdup fraction. Therefore, for a certain dispersion system, the breakup frequency of droplets with specific size only depends on the agitation speed. As shown in Fig. 12a, the drop breakup frequency increases with the increasing of agitation speed. It can also be observed that the breakup frequency is higher for larger droplets. These results are similar to our previous study (Zhou et al., 2017). For tensile breakup, this can be explained by the breakup 22
mechanism of turbulent eddy collision. For a fixed rotation speed, the number of eddies colliding with the droplet is positively related with the droplet size (Wang et al., 2003). Furthermore, the interfacial energy per unit volume is lower for larger droplets. For revolving breakup, it can mainly attribution to the lower interfacial energy per unit volume and larger velocity gradient for larger droplets. Thus, the competition between larger external inputting energy and lower internal cohesive energy leads to higher breakup frequency for larger droplets. As the rotating speed increasing from 390 rpm to 600 rpm, the arriving eddies contain more energy, and the number density of the arriving eddies is enlarged, which generates much more effective collisions or interaction, thus leading the increase of breakup frequency. (b)
25 20 15
Breakup frequency function/s-1
Breakup frequency function/s-1
(a)
System No.1 390 rpm 420 rpm 480 rpm 540 rpm 600 rpm
10 5 0 0.0
0.4
0.8
1.2
1.6
Diameter/mm
10 360 rpm System No. 2 3 4 5 6
8 6 4 2 0
0.2
0.4
0.6
0.8
1.0
1.2
Diameter/mm
Figure 12. Breakup frequency of the droplets (a) for system 1 with different rotating speed conditions, (b) for systems 2-6 with rotating speed of 360 rpm. For different dispersion systems, the two-phase interfacial tension significantly affects the breakup frequency. Systems 2-6 in Table 1 with different interfacial tensions are used to investigate the influence of the interfacial tension on the drop breakage. The results are shown in Fig. 12b. The breakup frequency increases with the decrease of the interfacial tension. It is because the internal cohesive energy
23
resisting the droplet deformation is reduced and thus leads to the increase of the breakup frequency. Mathematic model is necessary to quantitatively describe the droplet breakup behavior. And it is essential to contain the parameters such as energy dissipation rate and physical properties in the model. The physical properties are listed in Table 1. The energy dissipation rate is obtained from the computational fluid dynamics (CFD) simulation. Fig. 13 shows the distribution of the simulated energy dissipation rate when the rotation rate is 600 rpm. z / T represents the dimensionless axial distance from the bottom to the plane. One can see that the maximum value of energy dissipation rate is located around the impeller, which is consistent with our observation that the droplet breakup events mostly occur at the same location.
Figure 13. The distribution of the energy dissipation rate under 600 rpm (a) z/T =0.17, (b) z/T=0.19, (c) z/T =0.21, (d) z/T =0.23.
24
Figure 14 shows axial profiles of normalized energy dissipation at three radial locations. The simulation results show a consistent profiles with different rotating speed, and it also shows a tight relation between the normalized energy dissipation and the dimensionless distance of r / T (radial) and z / T (axial).
(a) 0.30
0.20 0.15
480 rpm 540 rpm 600 rpm
0.10 0
1
(c) 0.30
2
r/T=0.35
0.25
360 rpm 390 rpm 420 rpm fitting line
z/T (-)
z/T (-)
0.25
3 2
/N3L (-)
4
5
360 rpm 390 rpm 420 rpm 480 rpm 540 rpm 600 rpm fitting line
0.20 0.15 0.10 0
1
2
3
/N3L2 (-)
4
5
r/T=0.38
0.25
z/T (-)
(b)0.30
r/T=0.33
360 rpm 390 rpm 420 rpm 480 rpm 540 rpm 600 rpm fitting line
0.20 0.15 0.10 0
1
2
3
/N3L2 (-)
4
5
Figure 14. Axial profiles of normalised energy dissipation (a) r/T =0.33, (b) r/T =0.35, (c) r/T =0.38.
Based on the simulation results, the correlation for the average energy dissipation rate can be expressed as Eq. 9.
N 3 L2
b( r / T , z / T )
(9)
where is the circumferential average energy dissipation rate. L is the dimeter of the impeller. N is the rotating speed, b(r / T , z / T ) is a dimensionless parameter related to the location.
25
In this study, the observation domain is in the range of r / T 0.325 0.38 and
z / T 0.103 0.25 . And the volume average value of b(r / T , z / T ) obtained from the simulation is 2.03. Thus correlation for the average energy dissipation rate in this study is established as expressed by Eq. 10.
domain 2.03N 3 L2
(10)
As is pointed out in the introduction section, in the model of Coulalooglou and Tavlarides and its modified models, the prediction of the drop breakup frequency is very sensitive to the adjustable parameters. Also, the model of Luo and Svendsen and its extensions are very sensitive to the choice of the integration limits. Besides, there is no uniform form to characterize the influence of the parameters such as energy dissipation rate, mother droplet size and interfacial tension. Thus, a more realistic empirical correlation equation should be proposed based on experimental data. After dimensional analysis, the empirical correlation equation proposed based on the experimental data is expressed as Eq. 11. ( D) 0.10
1/3 D
2/3
WeD 2.64 FrD 0.56 Re D 0.15
(11)
Where: WeD
d 2/3 D5/3
(12)
Re D
c 1/3 D 4/3 c
(13)
FrD
2/3
(14)
gD1/3
The comparison between the prediction data of Eq. 11 and the experimental data 26
is shown in Fig.15. A prediction error of ±20% can be observed from the figure, which is acceptable for the droplet size calculation with PBM.
Calculated data/s-1
20 10
+20%
5
-20%
2 1 0.5 0.1 0.1
0.5 1 2 5 10 20 Experimental data/s-1
Figure 15. Comparison between the experimental results and the results calculated from the empirical correlation equation.
3.4 Daughter droplet size distribution As is discussed in the previous section, the breakup events in the pump-mixer are divided into three different types, i.e., original tensile breakup, intermediate tensile breakup, and revolving breakup. The DDSD for these three breakup types are different because of the different breakup mechanisms. Therefore, we investigate the DDSD for these three situations separately.
3.4.1 DDSD for original tensile breakup
27
(a)
(b)
Dm= 0.40 mm
6
System No.1 480 rpm 540 rpm
5
Dm= 0.65 mm
5
System No.1 420 rpm 480 rpm
4
Fβo
Fβo
4
6
3
3
2
2
1
1
0
0.2
0.4 0.6 0.8 Volume ratio,fV
0
1.0
0.2
0.4 0.6 0.8 Volume ratio,fV
1.0
Figure 16. DDSD for original breakup under different rotating speed (a) Dm = 0.40 mm, (b). Dm = 0.65 mm.
(b)
6 5 4 3 2 1 0
Fβo
(c) 6 5 4 3 2 1 0
Dm = 0.2 mm System No. 5 6
0.2
0.4
0.6
0.8
Fβo
Fβo
(a)
1.0
Volume ratio,fV
6 5 4 3 2 1 0
Dm = 0.6 mm System No. 4 5 6
0.2
0.4
0.6
0.8
1.0
Volume ratio,fV
Dm = 1.0 mm System No. 2 4
0.2
0.4
0.6
0.8
1.0
Volume ratio,fV
Figure 17. DDSD for original breakup under different TBP concentration (a) Dm = 0.3 mm, (b) Dm = 0.6 mm, (c). Dm = 1.0 mm.
28
(a)
5
Fβo
Fβo
3
2
2
1
1 0.2
0.4
0.6
0.8
(c) 6
System No.1 480 rpm Dm = 0.40 mm
5
0
1.0
Volume ratio,fV
Fβo
2 1 0.6
0.8
0
1.0
Dm = 1.0 mm
0.2
2
1
1 0.4
0.6
0.8
Dm = 1.0 mm
0
1.0
0.2
Volume ratio,fV
(g) 6
Fβo
Fβo
3 2
1
1 0.4
0.6
1.0
0.8
Dm = 0.6 mm
4
2
0.2
0.8
System No.6 360 rpm Dm = 0.3 mm
5
3
0
0.6
(h) 6
Dm = 0.6 mm
4
0.4
Volume ratio,fV
System No.5 360 rpm Dm = 0.3 mm
5
1.0
3
2
0.2
0.8
System No.4 360 rpm Dm = 0.6 mm
6 4
3
0
0.6
5 Fβo
Fβo
(f)
Dm = 1.0 mm
4
0.4
Volume ratio,fV
System No.3 360 rpm Dm = 0.6 mm
5
1.0
System No.2 360 rpm Dm = 0.6 mm
Volume ratio,fV
(e) 6
0.8
3
1 0.4
0.6
4
2
0.2
0.4
Volume ratio,fV
5
3
0
0.2
(d) 6
Dm = 0.65 mm
4
Dm = 1.05 mm
4
3
0
System No.1 420 rpm Dm = 0.65 mm
6 5
Dm = 1.05 mm
4
Fβo
(b)
System No.1 390 rpm Dm = 0.65 mm
6
1.0
Volume ratio,fV
0
0.2
0.4
0.6
0.8
1.0
Volume ratio,fV
Figure 18. DDSD for original breakup with different mother droplet size (a) system No.1, 390 rpm, (b) system No.1, 420 rpm, (c) system No.1, 480 rpm, (d)-(h) system No.2-6, 360 rpm.
The DDSD for original tensile breakup in different experimental conditions are shown in Fig. 16, 17 and 18. One can see from the figures that the curves for DDSD are bell-shaped, which mean the equal sized breakup is more likely to occur for 29
original tensile breakup. Fig. 16 shows that the distributions broaden with the rotating speed increases, and the effect is more significant for the larger mother droplet. Fig. 17 shows that the distributions become wider as the interfacial tension decreases for large mother droplets, and the effect is not significant for small mother droplets. Fig. 18 shows the distributions widen when the size increases. We can conclude from the above results that the DDSD for original breakup is related to the breakup frequency. The curve of DDSD gets broader when the breakup frequency gets higher. To quantitatively describe the DDSD, an empirical correlation is proposed as expressed by Eq. 15. The comparison between the prediction results and the experimental data is shown in Fig. 19. The figure shows the relative error is less than ±20%. F o
f vWeD
0.5
0
f
0.38
WeD 0.38 v
, when f v 0.5
(15)
df v
5
Calculated data
+20% 4 -20%
3 2 1
+40% -40%
0
0
1
2
3
4
5
Experimental data
Figure 19. Comparison between the experimental results and the results calculated from the empirical equation 15.
30
3.4.2 DDSD for intermediate tensile breakup
(a) 6
3
4 3
2
2
1
1
0
0.2
0.4 0.6 0.8 Volume ratio,fV
360 rpm System No. 2 3 4 5
5
Fβi
4
Dm = 0.6 mm
6
System No.1 390 rpm 420 rpm 480 rpm 540 rpm 600 rpm
5
Fβi
(b)
Dm = 0.6 mm
1.0
0
0.2
0.4 0.6 0.8 Volume ratio,fV
1.0
Figure 20. DDSD for intermediate tensile breakup (a) with different rotating speed, (b) with different interfacial tension. Fig. 20a and 20b show DDSD of the intermediate tensile breakup with different rotating speed and interfacial tension respectively. We can see from the figure that the influence of the rotating speed and the interfacial tension on the DDSD is slight for the intermediate tensile breakup. Moreover, the curves of the DDSD are U-shaped, which means that the intermediate breakup of a multiple breakup mostly generates unequal segments. The reason is that satellite droplets are usually generated when a multiple tensile breakage occurs. A satellite droplet must be one of the daughter droplets belong to a certain intermediate breakup. The size of the satellite droplet is usually much smaller than the other daughter droplets, leading to a U-shaped DDSD. An empirical correlation for the U-shaped DDSD is obtained based on the experimental data as expressed by Eq. 16. The comparison between the calculated results and the experimental data is shown in Fig. 21 and a relative error of no more than ±20% is also observed.
31
F i
f v 0.368
0.5
0
f v 0.368 df v
, when f v 0.5
(16)
5
Calculated data
+20% 4 -20%
3 2 1 0
0
1
2
3
4
5
Experimental data
Figure 21. Comparison between the experimental results and the results calculated from the empirical equation 16. 3.4.3 DDSD for revolving breakup Fig. 22a and 22b show DDSD for the revolving breakup with different rotating speed and interfacial tension respectively. The figures indicate that the rotating speed and interfacial tension have little influence on the DDSD. M-shaped distributions are observed for the revolving breakup. This is consistent with the phenomena shown in Fig. 3b, the revolving breakup process is more likely to create two fragments with large volume difference. The smaller fragment was thrown away from the end far from rotating center, and the volume ratio of smaller fragment to the mother droplet is mostly in the range of 0.1-0.2.
32
(a)
4
3
3
2
2
1
1
0
0.2
0.4 0.6 Volume ratio,fV
360 rpm System No. 2 3 4 5 6
5
Fβr
4
Dm = 0.6 mm
6
System No.1 390 rpm 420 rpm 480 rpm 540 rpm
5
Fβr
(b)
Dm = 0.6 mm
6
0.8
0
1.0
0.2
0.4 0.6 Volume ratio,fV
0.8
1.0
Figure 22. Daughter droplet size distribution with different conditions (a) with different rotating speed, (b) with different interfacial tension. An empirical correlation for the M-shaped DDSD is also suggested based on the experimental data of the revolving breakup process. The correlation is expressed as Eq. 17. F r
(1.55 f v )18.42 f v1.58
0.5
0
(1.55 f v )18.42 f v1.58
, when f v 0.5
(17)
The comparison between the experimental data and the prediction results is shown in Fig. 23. The prediction error for most of the data points is within ±20%.
5
Calculated data
+20% 4 -20%
3 2 1 0
0
1
2
3
4
5
Experimental data
Figure 23. Comparison between the experimental results and the results calculated from the empirical equation 17. 33
The proportion of the three breakup types has been determined in the previous section, expressed by Eq. 6, 7 and 8. Thus the DDSD for the droplet breakup in the pump-mixer can be expressed as F Po F o PF i i Pr F r
(18)
This is the final form for the DDSD which can be used in the PBM calculation.
4. Conclusions The direct experimental data for droplet breakage frequency and DDSD is very lacking in literature. In this study, the drop breakup in the pump-mixer was investigated by using the high speed online camera. Two typical breakup patterns, tensile breakup and revolving breakup were observed from the captured video. Binary breakup occupied 50% to 80% of all the breakup events in our experiments. The statistical data of the droplet number density indicated that the drop size was in the range of 0.05-1.0 mm obeying a normal distribution. The average size decreased with the increasing of the stirring intensity and decreasing of the interfacial tension. The breakup frequency was directly measured based on its definition. The results showed that the breakup frequency increased with the increasing of the stirring intensity and drop size or the decreasing of the interfacial tension. The DDSD for three different breakup types, original tensile breakup, intermediate tensile breakup, and revolving breakup were separately determined. For the original tensile breakup, the DDSD was a bell-shaped distribution, which broadened with the increasing of stirring intensity and mother droplet size or the decreasing of the interfacial tension. The DDSD was a
34
U-shaped distribution for the intermediate tensile breakup, and was an M-shaped distribution for the revolving breakup. The U-shaped and M-shaped distributions were little influenced by the stirring intensity and interfacial tension. To quantitatively describe the drop breakup, the empirical correlations of drop breakup frequency and DDSD were proposed based on the experimental data, the prediction results of those correlations and the experimental data were then compared and showed good agreements.
Acknowledgments We gratefully acknowledge the support of the National Natural Science Foundation of China (21576147, 21776151, 21376132).
Notation Latin symbols b
dimensionless parameter related to the location
cf
coefficient of surface area
D
diameter of the droplet, mm
fv
volume ratio of the daughter droplet to the mother droplet
Fβi
daughter droplet size distribution function of intermediate tensile breakup
Fβo
daughter droplet size distribution function of original tensile breakup
Fβr
daughter droplet size distribution function of revolving breakup
FrD
droplet dimeter based Froude number, FrD 2/3 / ( gD1/3 )
L
dimeter of the imlpeller, m 35
n(D)
number density function of the droplets, m-4
N
rotating speed of the impeller, r/s
Pi
occurrence possibility of middle tensile breakup
Po
occurrence possibility of first tensile breakup
Pr
occurrence possibility of revolving breakup
r
radial coordinate, m
ReD
droplet dimeter based Reynolds number, ReD c 1/3 D4/3 / c
t
times, s
T
dimeter of the mixer, m
tc
time duration, s
v
volume, m3
WeD
droplet dimeter based Weber number, WeD d 2/3 D5/3 /
z
axial coordinate, m
Greek letter
0
numerical constant
D breakup frequency function, s-1
energy dissipation rate, m2/s3
domain
average energy dissipation rate in the research domain, m2/s3
vessel
average energy dissipation rate in the vessel, m2/s3
circumferential average energy dissipation rate, m2 /s3
size of eddies, m
c
viscosity of the continuous phase, Pa s 36
d
viscosity of the dispersed phase, Pa s
c
continuous phase density, kg/m3
d
dispersion phase density, kg/m3
interfacial tension, N/m
d
dispersed phase volume fraction
Abbreviations CFD
Computational fluid dynamics
DDSD
daughter droplet size distribution
DSD
droplet size distribution
PBE
population balance equation
PBM
population balance model
RFR
Rotating Frames of Reference
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Ayyappa, S.V.N., Balamurugan, M., Kumar, S., Mudali, U.K., 2016. Mass transfer and hydrodynamic studies in a 50 mm diameter centrifugal extractor. Chem. Eng. Process. 105, 30-37. Baldi, S., Yianneskis, M., 2004. On the quantification of energy dissipation in the impeller stream of a stirred vessel from fluctuating velocity gradient measurements. Chem. Eng. Sci. 59, 2659-2671. Coulaloglou, C., Tavlarides, L., 1977. Description of interaction processes in agitated liquid-liquid dispersions. Chem. Eng. Sci. 32, 1289-1297.Falzone, S., Buffo, A., Vanni, M., Marchisio, D.L., 2018. Simulation of turbulent coalescence and breakage of bubbles and droplets in the presence of surfactants, salts, and contaminants. Advances in Chemical Engineering. 52, 125-188. Fang, Q., Jing, D., Zhou, H., Li, S.W., 2017. Population balance of droplets in a pulsed disc and doughnut column with wettable internals. Chem. Eng. Sci. 161, 274-287. Hagesaether, L., Jakobsen, H.A., Svendsen, H.F., 2002. A model for turbulent binary breakup of dispersed fluid particles. Chem. Eng. Sci. 57, 3251-3267. Hartmann, H., Derksen, J.J., Montavon, C., Pearson, J., Hamill, I.S., van den Akker, H.E.A., 2004. Assessment of large eddy and RANS stirred tank simulations by means of LDA. Chem. Eng. Sci. 59, 2419-2432. Kopriwa, N., Buchbender, F., Ayesterán, J., Kalem, M., Pfennig, A., 2012. A critical review of the application of drop-population balances for the design of solvent extraction columns: I. Concept of solving drop-population balances and modelling breakage and coalescence. Solvent Extr. Ion Exch. 30, 683-723. Liao, Y., Lucas, D., 2009. A literature review of theoretical models for drop and bubble breakup in turbulent dispersions. Chem. Eng. Sci. 64, 3389-3406.
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Liu, H., Jing, S., Fang, Q., Li, S., 2016. Droplet breakup in a square-sectioned pulsed disc and doughnut column. Ind. Eng. Chem. Res. 55, 2242-2251. Luo, H., Svendsen, H.F., 1996. Theoretical model for drop and bubble breakup in turbulent dispersions. AIChE J. 42, 1225-1233. Maaß, S., Kraume, M., 2012. Determination of breakage rates using single drop experiments. Chem. Eng. Sci. 70, 146-164. Parvizi, S., Keshavarz Alamdari, E., Hashemabadi, S.H., Kavousi, M., Sattari, A., 2016. Investigating factors affecting on the efficiency of dynamic mixers. Miner. Process. Extr. Metall. Rev. 37, 342-368. Qin, C.P., Chen, C., Xiao, Q., Yang, N., Yuan, C.S., Kunkelmann, C., Cetinkaya, M., Mulheims, K., 2016. CFD-PBM simulation of droplets size distribution in rotor-stator mixing devices. Chem. Eng. Sci. 155, 16-26. Ramachandran, R., Immanuel, C.D., Stepanek, F., Litster, J.D., Doyle, F.J., 2009. A mechanistic model for breakage in population balances of granulation: Theoretical kernel development and experimental validation. Chem. Eng. Res. Des. 87, 598-614. Rao, N.V.R., Baird, M.H.I., 1984. Davy mckee combined mixer settler: Some laboratory studies. Can. J. Chem. Eng. 62, 497-506. Razzaghi, K., Shahraki, F., 2016. Theoretical model for multiple breakup of fluid particles in turbulent flow field. AIChE J. 62, 4508-4525. Reeve, R.N., Godfrey, J.C., 2002. Phase inversion during liquid–liquid mixing in continuous flow, pump–mix, agitated tanks. Chem. Eng. Res. Des. 80, 864-871. Solsvik, J., Borka, Z., Becker, P.J., Sheibat-Othman, N., Jakobsen, H.A., 2014. Evaluation of
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breakage kernels for liquid-liquid systems: Solution of the population balance equation by the least-squares method. Can. J. Chem. Eng. 92, 234-249. Solsvik, J., Jakobsen, H.A., 2015. Single drop breakup experiments in stirred liquid–liquid tank. Chem. Eng. Sci. 131, 219-234. Solsvik, J., Maaß, S., Jakobsen, H.A., 2016. Definition of the single drop breakup event. Ind. Eng. Chem. Res. 55, 2872-2882. Srilatha, C., Savant, A.R., Patwardhan, A.W., Ghosh, S.K., 2008. Head–flow characteristics of pump-mix mixers. Chem. Eng. Process. 47, 1678-1692. Thakur, R.K., Vial, C., Nigam, K.D.P., Nauman, E.B., Djelveh, G., 2003. Static mixers in the process industries - A review. Chem. Eng. Res. Des. 81, 787-826. van Delden, M.L., Vos, G.S., Kuipers, N.J.M., de Haan, A.B., 2006. Extraction of caprolactam with toluene in a pulsed disc and doughnut column - Part II: Experimental evaluation of the hydraulic characteristics. Solvent Extr. Ion Exch. 24, 519-538. Vonka, M., Soos, M., 2015. Characterization of liquid-liquid dispersions with variable viscosity by coupled computational fluid dynamics and population balances. AIChE J. 61, 2403-2414. Wang, T., Wang, J., Jin, Y., 2003. A novel theoretical breakup kernel function for bubbles/droplets in a turbulent flow. Chem. Eng. Sci. 58, 4629-4637. Wang, T., Wang, J., Jin, Y., 2005. Population balance model for gas-liquid flows: influence of bubble coalescence and breakup models. Ind. Eng. Chem. Res. 44, 7540-7549. Wang, T., Wang, J., Jin, Y., 2006. A CFD–PBM coupled model for gas–liquid flows. AIChE J. 52, 125-140. Xing, C., Wang, T., Wang, J., 2013. Experimental study and numerical simulation with a coupled
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41
Appendix A: Verification of simulation methods by comparing of experimental data and simulation results in a baffled vessel with a Rushton turbine.
Fig. A1 Simulation domain: half of the mixing vessel with a Rushton turbine. (a) 1/2 geometry for simulation (b) grids details of the 1/2 stationary region, (c) grids details of the 1/2 rotating region. The purpose of this CFD simulation is to verify the applicability of the k model and the simulation method applied in this study. In this section, we carried out the simulation in a baffled vessel of diameter T = 100 mm stirred by a Rushton turbine of diameter L = T/3. The scale is similar to our experimental setup. The geometric structure is identical to the experimental setup used by Baldi and Yianneski (Baldi and Yianneskis, 2004). Fig. A1 shows the simulation domain and the grids generated in this study. The simulation was performed using the same simulation approach as is
42
described in section 2.4.1. Table A1. Results of grid independence tests under 1082 rpm, Re=20000. Experimental results (Baldi and Number of nodes /Million
Simulation results Yianneskis, 2004)
No.
1
rotating
stationary
Power
vessel /
Power
region
region
number
m2s-3
number
0.41
2.30
3.90
1.00 4.1
2
0.74
2.65
3.92
vessel / m2s-3
1.26
1.06
Table A1 shows the mesh independence tests of two groups of grids. For No.1, the maximum grid size was 0.8 mm in stationary region (the grid size of the interface was 0.3 mm) and 0.3 mm in rotating region. For No.2, the maximum grid size was 0.8 mm in stationary region (the grid size of the interface was 0.2 mm) and 0.2 mm in rotating region. The power number and the average eddy dissipation were obtained for the comparison. The simulation results were verified with experimental results by Baldi and Yianneski (Baldi and Yianneskis, 2004) and Hartmann et al. (Hartmann et al., 2004). It shows that power number agree with the Baldi and Yianneskis’ result well under the error below 5%, the vessel average eddy dissipation agree with the experimental result under the error below 20%. The flow field in the vessel predict by the simulation was also verified by comparing the radial and circumferential velocity at three radial locations with the results of Hartmann et al., and a great matched-degree achieved, as shown in Figure 4. The results show that meshes and the simulation approach applied in this study is acceptable. 43
Highlights •
Two typical breakup patterns, tensile breakup and revolving breakup were observed.
•
Droplet breakage frequency and daughter droplet size distribution (DDSD) in pump-mixer were experimentally measured.
•
The turbulence energy dissipation was determined using CFD simulation.
•
Correlations of droplet breakage frequency and DDSD were proposed.
44
S0009-2509(18)30812-1 https://doi.org/10.1016/j.ces.2018.11.035 CES 14623
To appear in:
Chemical Engineering Science
Received Date: Revised Date: Accepted Date:
8 June 2018 20 September 2018 14 November 2018
Please cite this article as: H. Zhou, X. Yu, S. Jing, H. Zhou, W. Lan, S. Li, Measurement of droplet breakage in a pump-mixer, Chemical Engineering Science (2018), doi: https://doi.org/10.1016/j.ces.2018.11.035
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Measurement of droplet breakage in a pump-mixer Han Zhoua, Xiong Yua, Shan Jinga, *, Hao Zhoua, Wenjie Lanb, Shaowei Lia, c,* a
Collaborative Innovation Center of Advanced Nuclear Energy Technology, Institute
of Nuclear and New Energy Technology, Tsinghua University, Beijing 100084, China b
State Key Laboratory of Heavy Oil Processing, China University of Petroleum (Beijing), Beijing 102249, China
c
State Key Laboratory of Chemical Engineering, Tsinghua University, Beijing 100084, China
Abstract: The direct experimental data for droplet breakage frequency and daughter droplet size distribution is urgently required for the application of the population balance model. To meet this requirement, a method to directly measure the droplet breakage in the pump-mixer was developed using a high speed online camera. Two typical breakup patterns, tensile breakup and revolving breakup were observed and the number of the breakup fragments was determined. The multiple breakage was treated as a series of binary breakups. In order to precisely describe the daughter size distribution, these binary breakups were divided into three different types, i.e. the original tensile breakup, intermediate tensile breakup, and revolving breakup. The influences of the rotating speed and interfacial tension on the droplet breakup frequency and daughter droplet size distribution were then quantitatively investigated.
*
Corresponding author, [email protected] (S. Jing), [email protected] (S. Li) 1
Empirical correlations were proposed and good agreement was found between the prediction results and the experimental data. Keywords: population balance model; pump-mixer; droplet breakup; daughter droplet size distribution
1. Introduction Solvent extraction is a powerful liquid-liquid separation technique and has extensive applications in many fields. In liquid-liquid extraction processes, one phase is usually dispersed into the other phase to form droplets. The droplet size distribution (DSD) determines the mass transfer area, making it a significant parameter for the extraction efficiency. In the extraction process, a narrow DSD is desired to optimize hydraulics and mass transfer performance (Kopriwa et al., 2012). It is also reported that the DSD depends on the droplet breakup and coalescence (Qin et al.,2016; Zaccone et al., 2007). Thus, an intensive understanding of the droplet breakup and coalescence is extremely important for the optimization of extraction process. A variety of industrial equipment have been designed for the extraction operation, such as centrifugal extractor (Ayyappa et al., 2016), extraction column (van Delden et al., 2006), mixer-settler (Thakur et al., 2003), etc. Pump-mixer, as a type of mixer in mixer-settler, has been widely used in nuclear fuel reprocessing and chemical industries. Compared to common mixers, pump-mixer can realize pumping and mixing simultaneously (Srilatha et al., 2008). It also enjoys the advantages of large capacity, high efficiency, reliable operation and strong adaptability. The hydraulic and
2
mass transfer performance of pump-mixer has been studied in the past several decades (Rao and Baird, 1984; Srilatha et al., 2008; Parvizi et al., 2016). It was found that complex interaction exists among the DSD, the breakage, and coalescence. In-depth exploring on it is of great significance for performance optimizing, precisely designing, and scaling-up of the mixer. Numerous studies on drop breakage and coalescence have been reported (Coulaloglou and Tavlarides, 1977; Luo and Svendsen, 1996; Alopaeus et al. 1999; Andersson and Andersson, 2006a, b). In these studies, the population balance model (PBM) is mostly considered due to its ability to predict the DSD by considering the effects of drop breakage and coalescence. In the framework of PBM, the influences of droplet breakage and coalescence is considered in the source term of the population balance equation (PBE). They are expressed in the form of three sub-models, i.e., breakage frequency, daughter droplet size distribution (DDSD), and coalescence rate models. The DSD can be predicted only if the three sub-models are determined. To precisely determine these three sub-models is a key challenge in employing PBM to calculate liquid-liquid two-phase flow. In this work, we will focus on the first two sub-models, which describe the droplet breakage behavior. Many efforts have been made to quantitatively describe the drop breakage with breakup frequency and DDSD. A comprehensive literature reviews have been made by Liao and Lucas (2009). They argue that turbulent particle breakup mechanisms can be mainly attribute to turbulent fluctuation and collision (Liao and Lucas, 2009). Furthermore, breakage mechanisms include viscous shear stress, turbulent shear, 3
interfacial instability and shearing-off may also cause breakup (Falzone et al., 2018). Thus, a variety of models have been proposed in the past few decades. Among those efforts, two classical models proposed by Coulaloglou and Tavlarides (Coulaloglou and Tavlarides, 1977), and Luo and Svendsen (Luo and Svendsen, 1996) are still widely used for the model extensions and simulation applications. In Coulaloglou and Tavlarides’ model, the breakage frequency is expressed as Eq. (1).
1 d 1/3 D c1 exp c2 2/3 d 2/3 D5/3 1 d D
2
(1)
This model assumes that the drop breakage is caused by the collision with the turbulent eddy. The breakage is dominated by the mother droplet size and the turbulent dissipation, while independent of the fluid viscosity. It is also assumed that only the binary breakages occurred, and the DDSD is a normal distribution as expressed in Eq. (2). 2 2 D3 Dm3 7.2 D 2 D, Dm exp 4.5 Dm3 Dm 6
( 2)
A survey of the modified and extended models based on Coulaloglou and Tavlarides’ model is provided by Solsvik et al. (Solsvik et al., 2014). However, these models are very sensitive to the adjustable parameters containing in them, making them suitable for a specific device and system. Luo and Svendsen proposed a new theoretical model of breakage frequency which containing no adjustable parameters. Based on the kinetic gas theory, they defined breakage frequency as a combined result of arrival or bombarding frequency
4
of eddies and breakage probability. The breakage frequency function and DDSD are expressed as Eq. (3) and Eq. (4).
D 0.462 1 d 2 D
1/3
f v v, v
2
v
max
min 1 max
0 min
1
m ax
0
min
1
11/3
2
12c f exp d df v (3) 2/3 5/3 11/3 0 c D
1 / exp 12c / 1 / exp 12c / 2
11/3
f
2
0
2/3
c
11/3
2/3
f
0
c
d df
D5/3 11/3 d
D
5/3 11/3
(4) v
This model is widely used in academic and many modified models have been proposed (Hagesaether et al., 2002; Andersson and Andersson, 2006b; Razzaghi and Shahraki, 2016). But researchers also find that these models are usually very sensitive to the upper limit of the integration ξmax, which defines the ratio between the sizes of the largest effective eddy and the droplet. In fact, Andersson and Andersson (Andersson and Andersson, 2006b) have pointed out that eddies with size up to three times of the droplets are still responsible for breakup. In addition to the theoretical study, single droplet experiments (Solsvik et al., 2015, 2016; Maaß and Kraume, 2012) also have been carried out to investigate drop breakage. Solsvik et al. (Solsvik et al., 2015, 2016) performed the single drop breakup studies and suggested that the drop size had a significant impact on the breakage frequency and the number of fragmentation during a single breakup event. Maaß et al. (Maaß and Kraume, 2012) investigated single drop breakup event under turbulent condition, which was comparable to those in a mixer. They improved the breakage time model by taking into account the influence of viscosity and interfacial tension. Some studies (Wang et al., 2005,2006; Ramachandran et al., 2009; Xing et al.,
5
2013; Vonka and Soos, 2015) have been carried out to introduce PBM into the gas-liquid or liquid-liquid dispersions. In this approach, PBM sub-models including breakup frequency and DDSD are normally taken into account. Nevertheless, the particular form and parameters in such models differ from each other due to the specificity of particular flow geometry, and even cannot be applied to a changed condition. Thus, more efforts are deserved to obtain universal PBM kernel functions with a great amount of experimental work. However, the direct experimental data for droplet breakage frequency and DDSD is very lacking in literature. The main purpose of this work is to develop an experimental method to measure the breakage frequency and DDSD and to systematically investigate the droplet breakage in a pump-mixer. Reference to the method in our previous work on a pulsed disc and doughnut column (Zhou et al., 2017; Fang et al., 2017), a high-speed online camera was involved for the experimental data collection in this work. Significant amount of videos of droplet breakup events were captured for data statistics. Drop breakup frequency and DDSD were then quantitatively calculated based on the statistical results. The influences of energy input and two-phase physical properties were investigated. Furthermore, empirical correlations for the breakup frequency and DDSD were proposed.
2. Experiments and methods 2.1 Experimental setup
6
Figure 1. The experimental setup (a) Photo of the experimental setup, (b) Schematic diagram of the experimental setup.
The experimental setup is shown in Fig. 1, where the pump-mixer is designed based on the Power-Gas industrial mixer (Reeve and Godfrey, 2002). The size of the mixing tank is 100 mm× 100 mm× 100 mm. The walls of the tank are made of quartz glass. There is a suction orifice at the bottom of the tank with a diameter of 32 mm. A 100 mm× 100 mm× 30 mm buffer chamber is connected to the mixing tank through the suction orifice. A peristaltic pump (Longer Pump WT600-2J) is used to drive the continuous phase (aqueous phase) into the mixer from the buffer chamber with a flow rate of 440-500 mL/min. The organic phase is pre-dispersed into droplets by the aqueous phase in a Y-junction channel, and then introduced to the mixer through a stainless steel tube with inner diameter of 3 mm. The organic and aqueous phases are introduced into the Y-junction channel by two peristaltic pumps (Longer Pump BT100-2J) with flow rates of 1.2-50 mL/min and 0-30 mL/min, respectively. In this study, both phases were fed continuously and the total flux was set as 500 mL/min. 7
Mixture of the two phases overflows from the top of the tank with flow rate of 500 mL/min. Table 2 show the flow ratios (Qo:Qa) under different conditions. The flow rate of the dispersed phase is sufficiently small to ensure a small holdup, so that the coalescence of the dispersed phase droplets can be ignored. Therefore, the results obtained in this work are only applicable for dilute dispersions.
Figure 2. Pump-mix impeller (a) schematic diagram of the impeller, (b) cutaway view of the impeller.
The impeller is made of stainless steel and its structure is shown in Fig.2. It has a diameter of 65 mm and contains eight equispaced, curved blades of 10 mm high and 2 mm thick. The distance between the tank bottom and the impeller is set as 5 mm. A motor (TaiBang Motor CV0.4KW) connected to the axle of the impeller is used for energy input. The rotating speed of the impeller is varied in the range of 360-600 rpm in our experiments. A microscope (Olympus SZ61) coupled with a high speed camera (Olympus i-Speed TR) is used to observe droplet behavior in the mixer. The frame rate of the camera is set as 5000 fps and the amplification factor of the microscope is fixed as 1.2. 8
The resolution of the captured image is 0.018 mm/pixel, so that droplets with size around 0.1 mm can still be clearly captured. More than 40 videos are recorded for each operation condition, and each video has a time duration of 1.267 s. The measurement of the droplet size and the breakup events are done manually with an image manipulation software, which was developed by our team based on Python. Solsvik et al. (Solsvik and Jakobsen, 2015) reported the results can be vague imprecise due to droplets overlap and the lack of three-dimensional observation. In this study, the holdup of the dispersed phase is dilute enough to ignore the influence of the overlap. Besides, this study was carried out under a steady state, most of the droplets in the mixer was in the near spherical shape. Thus It is reasonable to use two-dimensional image to determine the droplet size, thereby determining the DSD. For the determination of the droplet breakup events, the error is mainly attributed to the deformation of the mother or the daughter droplets. However, attributed to the lower number of fragments generated in a breakup event, this error can be avoided to the most extent by tracing the drops to a most optimum condition.
2.2 Materials The experiments were performed at 20°C and at atmospheric pressure. The n-dodecane was selected as dispersed phase and distilled water was chosen as continuous phase. Different amount of Tri-Butyl-Phosphate (TBP) was added into the dispersed phase to adjust the interfacial tension between the two phases. The physical properties of the experimental systems are shown in Table 1. 9
Table 1. Physical properties of the experimental systems (20°C) System
Continuous
Dispersed
ρc
ρd
μc
μd
σ
No.
phase
phase
kg/m3
kg/m3
mPa·s
mPa·s
mN/m
1
Water
n-dodecane
997.0
750.1
1.12
1.64
34.61
2
Water
1vol%TBP/n-dodecane
997.0
755.0
1.12
1.65
22.94
3
Water
2vol%TBP/n-dodecane
997.0
757.0
1.12
1.65
19.58
4
Water
5vol%TBP/n-dodecane
997.0
764.0
1.12
1.66
15.07
5
Water
10vol%TBP/n-dodecane
997.0
775.0
1.12
1.69
12.61
6
Water
30vol%TBP/n-dodecane
997.0
850.0
1.12
1.80
9.78
Table 2. Operational rotating speeds and flow ratios Rotating
Rotating Qo:Qa
System No.
Qo:Qa
System No.
speed/rpm
speed/rpm
1
390
0.036
2
360
0.018
1
420
0.018
3
360
0.009
1
480
0.006
4
360
0.009
1
540
0.002
5
360
0.004
1
600
0.002
6
360
0.004
2.3 Method The breakup frequency function of droplets with specified size is defined as the breakup fraction of these droplets within unit time. It can be expressed by Eq. 5. 10
( D )
1 breakup fraction of 1 nb ( D)D t n( D)D tc droplets c
(5)
Herein D is the droplet size. n( D) is the droplet number density and thus n( D)D denotes the number of droplets in size of D to D D per unit volume. Correspondingly, nb ( D)D denotes the number of breakup events within these droplets. tc is the time duration within which the breakup events occur. The measurement of the droplet breakup frequency function is directly performed based on the above definition. Firstly, with certain size interval D ,
n( D)D is determined after measuring the size of enough droplets (no less than 2000 in this work) from the captured photos. The size of a droplet is measured on its major axis a and minor axis b in 2D images and calculated by D ab . Then, 1/2
nb ( D)D is determined by counting the breakup events in the video captured within time duration tc . Finally, the breakup frequency function is determined based on Eq. 5. The DDSD F fV , Dm , which is defined as the probability of generating a daughter droplet with volume ratio fV when a mother droplet with size Dm is broken-up, is experimentally determined using a similar method. The count number of the drop breakage is no less than 250 under each operating condition. Fig. 3 shows that when the count number is in excess of 250, the measuring error is less than ±5%.
11
Breakup frequency function /s-1
3.6 3.2
5%
2.8 2.4 2.0 0
50 100 150 200 250 300 Number of drop breakage events
Figure 3. The measuring error at different number of breakup events. The data points are measured at the conditions of D=0.55 mm, system No.4.
2.4 Determination of the turbulence energy dissipation 2.4.1 CFD simulation method The droplet breakup frequency was associated with the turbulence energy dissipation. Computational fluid dynamics (CFD) simulation was performed to provide the necessary results. Based on the experimental setup, the mixer was divided into two domains: the rotational region consisting of the impeller, and the stationary region of the remaining volume. Structured hex-grid was used as the meshing type. Fig. 4 shows the simulation domain and the grids generated in this study.
12
Fig. 4 CFD simulation domains and meshes (a) 1/4 geometry for simulation (b) grids details of the 1/4 stationary region, (c) grids details of the 1/4 rotating region. The rotating region consists of 0.58 million nodes, the stationary region has 0.94 million nodes. The maximum grid size was 0.8 mm for stationary region and 0.4 mm for the rotating region, the minimum grid size which exists in the interface of the two regions was 0.15 mm. The interface of the two regions was managed particularly, the grid nodes difference of the two sides was controlled within 20%. All CFD simulations were performed using CFX 16.1. Steady state was chosen as the analysis type. The Rotating Frames of Reference (RFR) approach was specified to the rotating region. No-slip conditions were used for the walls. The turbulence was modeled using the Standard k-ε model. The high resolution advection scheme was applied for the continuity, momentum, turbulence kinetic energy and turbulence energy dissipation. The convergence criteria were set with root mean square (RMS) below 10-6. 13
Table 3 shows the mesh independence tests of three groups of grids: the coarse mesh, the medium mesh (the chosen mesh), and the fine mesh. The power consumption and the average eddy dissipation in the observation domain were obtained for the comparison. The results demonstrate that the chosen mesh is acceptable.
Table 3. Results of grid independence tests under 600 rpm Number of nodes
Number of cells
maximum grid size Power
/Million
/ mm
/Million
domain
2
No.
/kg m rotating
stationary
rotating
stationary
rotating
/ m2s-3
stationary s-3
region
region
region
region
region
region
1
0.33
0.48
0.31
0.46
0.48
1.1
0.704
8.11
2
0.58
0.94
0.55
0.90
0.40
0.80
0.715
8.58
3
1.05
1.54
1.01
1.48
0.33
0.70
0.722
8.62
2.4.2 Verification of simulation approach Limited experimental data of similar setup in this study was published in the literature. Thus another simulation with the same modeling and meshing method was carried out in a baffled vessel with a Rushton turbine, which has a similar scale with our setup. The purpose is to verify the applicability of turbulence model and simulation approach.
14
(a)
0.40
0.36
0.34
0.34
z/T
0.38
0.36
z/T
0.38
0.32
0.32
0.30
0.30
r/T=0.183 simulation result Hartmann et al. (2004)
0.28 0.26 0.0
0.2
0.4
0.6
0.8
0.26 0.0
1.0
0.2
0.34
0.34
0.6
0.8
1.0
z/T
0.36
z/T
0.38
0.36
0.4
Vr / Vtip
0.40
0.38
0.32
0.32
0.30
0.30
r/T=0.25 simulation result Hartmann et al. (2004)
0.28 0.2
0.4
0.6
0.8
r/T=0.25 simulation result Hartmann et al. (2004)
0.28 0.26 0.0
1.0
Vc / Vtip
0.40
0.2
0.34
0.6
0.8
1.0
z/T
0.36
0.34
z/T
0.38
0.36
0.4
Vr / Vtip
0.40
0.38
0.32
0.32
0.30
r/T=0.317 simulation result Hartmann et al. (2004)
0.28 0.26 0.0
r/T=0.183 simulation result Hartmann et al. (2004)
0.28
Vc / Vtip
0.40
0.26 0.0
(b)
0.40
0.2
0.4
0.6
0.8
1.0
0.30
r/T=0.317 simulation result Hartmann et al. (2004)
0.28 0.26 0.0
Vc / Vtip
0.2
0.4
0.6
0.8
1.0
Vr / Vtip
Fig. 5. Axial profiles of the circumferential (a) and radial (b) velocity components at three radial locations r/T =0.183(left), r/T =0.25(middle) and r/T =0.317(right).
Appendix A shows the detailed description of this simulation. The simulation was performed in a standard configuration cylindrical vessel of diameter T = 100. The impeller was rotated at N=1082 rpm, and Vtip=1.89 m/s. Figure 5 shows axial profile of the dimensionless radial and circumferential velocity at three radial locations. Vc and Vr represents the radial and circumferential velocity separately, z / T represents 15
the dimensionless axial distance from the bottom to the plane. The figure shows that the simulation results agree well with the experimental results (Hartmann et al., 2004). Thus using the simulation approach in this study, the flow field should be predicted correctly. Moreover, the power number of the RT impeller can be predicted correctly and the average eddy dissipation show an underestimate of 15.9% of the simulation result (see Table A1). These results indicate that the simulation method in the section 2.4.1 can be applied for the following analysis.
3. Results and discussion 3.1 Description of drop breakup events In this study, more than 250 drop breakup events are recorded for each experimental condition. Two main breakup modes, breakup attribute to turbulent fluctuation and collision which is caused by turbulent eddies smaller than the droplet size (for convenience, the following is referred to as tensile breakup) and breakup due to turbulent shear stress of small droplets trapped into larger eddies (the following is referred to as revolving breakup), are observed, as is shown in Fig. 6. The tensile breakup is the most possible drop breakage mode. The droplet is first stretched to form a neck. The neck then shrinks and finally the droplet breaks up into two or more daughter droplets. The revolving breakup is also an important breakup mode in the pump-mixer. The droplet is first trapped into a vortex, and rotates with it. Then the breakup can be occurred due to the velocity gradient in the eddies and a certain part of the droplet which is relatively far away from rotating center is thrown 16
away to form several fragments.
Figure 6. The drop breakup process of the two breakage modes (a) tensile breakup, (b) revolving breakup
The number of the fragments is investigated with the definition of a single droplet breakup event illustrated by Solsvik et al. (Solsvik et al., 2016). The initial breakup is used in determining the number of daughter entities. Fig. 7 shows the probability of binary, ternary, quaternary and quaternary+ (one droplet breaks up into more than 4 daughter droplets) breakage reported by Andersson and Andersson (Andersson and Andersson, 2006a), Solsvik and Jakobsen (Solsvik and Jakobsen, 2015), and Zhou et al. (Zhou et al., 2017), Andersson and Solsvik carried out single droplet experiments and the results showed that the multiple breakup were more 17
common than binary breakup, with a proportion of more than 50%. In comparison, Zhou conducted the experiment under a fully developed droplet swarm flow, and the results showed that the binary breakage had a proportion of more than 50%. The difference is probably because the mother droplet size in the first two studies is much larger than the last one. Actually, Solsvik and Jakobsen (Solsvik and Jakobsen, 2015) has also pointed out that the diameter of the mother droplet is one of the most important factors determining the possible amount of fragments.
Andersson and Andersson, = 3.7 m2s-3 Andersson and Andersson, = 8.5 m2s-3 Solsvik et al., = 1.14 m2s-3 Dm < 2 mm
80
Probability (%)
Solsvik et al., = 1.14 m2s-3 Dm > 2 mm Zhou et al., = 0.042 m2s-3 Zhou et al., = 0.082 m2s-3
60
40
20
0
binary
ternary
quaternary quaternary +
Figure 7. The probability of different number of fragments reported in literatures. Andersson and Andersson (2006a) and Solsvik and Jakobsen (2015) performed single drop breakup experiments while Zhou et al. (2017) carried out their experiments with fully developed droplet swarm.
18
Fig. 8 shows the probability of different number of the fragments in this study. The experimental results show that the binary breakage has an occurrence probability of more than 50% for all conditions. Furthermore, Fig. 8a shows that the number of fragments decreases with the increase of the rotating speed. The reason is probably that the average droplet size gets smaller when increasing the rotating speed, which can be seen in the next section. Smaller droplets tend to be split into fewer segments because of the difficulty of deformation. The interfacial tension is also a main factor affecting droplet breakup. As is shown in Fig. 8b, the number of the fragments increases with the decrease of interfacial tension. The reason is that the surface stability of the droplets decreases with lower interfacial tension, thus the deformation is enhanced which tends to generate more fragments.
System No.1 420 rpm 480 rpm 540 rpm 600 rpm
Probability (%)
80
60
(b)
40
20
0
360 rpm System No. 2 3 4 5 6
80
Probability (%)
(a)
60
40
20
binary
ternary
0
quaternary quaternary +
binary
ternary
quaternary quaternary +
Figure 8. The probability of different number of fragments (a) for system 1 with different rotating speed. (b) for different systems with rotating speed of 360rpm.
To simplify the description of multiple droplet breakage, the assumption of binary breakup is widely used. A common approach is to treat the multiple fragmentation as a series of binary breakup, with an assumption that the larger 19
fragments are always generated firstly. Taking the quaternary breakage as an example, the treating approach is illustrated in Fig. 9. For ease of description, the breakup of the original droplet to form a droplet with size D1 and another imaginary droplet in Fig. 8b is defined as the original breakup while the breakup of the imaginary droplets is defined as the intermediate breakup. The above treating approach will be used in the data processing of this work.
Figure 9. Treatment of quaternary breakage as three binary breakages. (a) the entire quaternary breakage process, (b) a series of imaginary binary breakup processes.
Based on the descriptions above, the breakup processes were then quantitatively analyzed. For the convenience of the subsequent discussion, the occurrence possibility of three breakup processes, original tensile breakup, intermediate tensile breakup, and revolving breakup were firstly determined from the captured video, the results is
20
shown in Fig. 10. Furthermore, after dimensional analysis, the correlations of the occurrence possibility for the three breakup processes were proposed, the correlations are as Eqs. 6-8. Po (1 0.66WeD 0.19 4.2 10-4WeD -3.82 )1
(6)
Pi 0.66WeD 0.19 Po
(7)
Pr 1 ( Pi Po )
(8)
Where WeD
c D5/3 2/3 . Po represents the occurrence possibility of original tensile
breakup, while Pi and Pr represent the occurrence possibility of intermediate tensile breakup and revolving breakup respectively.
0.7 0.6
Po Pi Pr
possibility
0.5 0.4 0.3 0.2 0.1 0.0 0.14
0.16
0.18
0.20
0.22
WeD
Figure 10. The occurrence possibility of the three breakup processes.
3.2 Number density function of droplets As expressed by Eq. 5, we need to determine the number density function n( D) before measuring the breakup frequency. The measured number density at different experimental conditions is shown in Fig. 11. The drop size distribution is similar to 21
normal distribution and the drop size is mainly ranging from 0.05 to 1.0 mm in this study. The effect of the agitation speed and interfacial tension on DSD can be observed in the figure. Higher agitation speed or lower interfacial tension leads to smaller droplet size and narrower DSD. The increasing of agitation speed will enhance the energy input while the decreasing of interfacial tension will reduce the inner energy resisting breakage. In both of the above conditions, the stability of the droplets is decreased and thus the DSD shifts to the smaller size dimension. (a)
(b) 16
System No.1 390 rpm 420 rpm 480 rpm 540 rpm 600 rpm
8 4
0
0.2
0.4 0.6 Diameter/mm
0.8
360 rpm System No. 2 3 4 5 6
4
n(D)/(1010Qa/Qo) m-4
n(D)/(1010Qa/Qo) m-4
12
3 2 1 0
1.0
0.2
0.4 0.6 Diameter/mm
0.8
1.0
Figure 11. Number density of the droplets (a) for system 1 with different rotating speed conditions, (b) for systems 2-6 with rotating speed of 360 rpm.
3.3 Drop breakup frequency function The influence of the dispersed phase flow rates on the drop breakup frequency function can be neglected due to the low holdup fraction. Therefore, for a certain dispersion system, the breakup frequency of droplets with specific size only depends on the agitation speed. As shown in Fig. 12a, the drop breakup frequency increases with the increasing of agitation speed. It can also be observed that the breakup frequency is higher for larger droplets. These results are similar to our previous study (Zhou et al., 2017). For tensile breakup, this can be explained by the breakup 22
mechanism of turbulent eddy collision. For a fixed rotation speed, the number of eddies colliding with the droplet is positively related with the droplet size (Wang et al., 2003). Furthermore, the interfacial energy per unit volume is lower for larger droplets. For revolving breakup, it can mainly attribution to the lower interfacial energy per unit volume and larger velocity gradient for larger droplets. Thus, the competition between larger external inputting energy and lower internal cohesive energy leads to higher breakup frequency for larger droplets. As the rotating speed increasing from 390 rpm to 600 rpm, the arriving eddies contain more energy, and the number density of the arriving eddies is enlarged, which generates much more effective collisions or interaction, thus leading the increase of breakup frequency. (b)
25 20 15
Breakup frequency function/s-1
Breakup frequency function/s-1
(a)
System No.1 390 rpm 420 rpm 480 rpm 540 rpm 600 rpm
10 5 0 0.0
0.4
0.8
1.2
1.6
Diameter/mm
10 360 rpm System No. 2 3 4 5 6
8 6 4 2 0
0.2
0.4
0.6
0.8
1.0
1.2
Diameter/mm
Figure 12. Breakup frequency of the droplets (a) for system 1 with different rotating speed conditions, (b) for systems 2-6 with rotating speed of 360 rpm. For different dispersion systems, the two-phase interfacial tension significantly affects the breakup frequency. Systems 2-6 in Table 1 with different interfacial tensions are used to investigate the influence of the interfacial tension on the drop breakage. The results are shown in Fig. 12b. The breakup frequency increases with the decrease of the interfacial tension. It is because the internal cohesive energy
23
resisting the droplet deformation is reduced and thus leads to the increase of the breakup frequency. Mathematic model is necessary to quantitatively describe the droplet breakup behavior. And it is essential to contain the parameters such as energy dissipation rate and physical properties in the model. The physical properties are listed in Table 1. The energy dissipation rate is obtained from the computational fluid dynamics (CFD) simulation. Fig. 13 shows the distribution of the simulated energy dissipation rate when the rotation rate is 600 rpm. z / T represents the dimensionless axial distance from the bottom to the plane. One can see that the maximum value of energy dissipation rate is located around the impeller, which is consistent with our observation that the droplet breakup events mostly occur at the same location.
Figure 13. The distribution of the energy dissipation rate under 600 rpm (a) z/T =0.17, (b) z/T=0.19, (c) z/T =0.21, (d) z/T =0.23.
24
Figure 14 shows axial profiles of normalized energy dissipation at three radial locations. The simulation results show a consistent profiles with different rotating speed, and it also shows a tight relation between the normalized energy dissipation and the dimensionless distance of r / T (radial) and z / T (axial).
(a) 0.30
0.20 0.15
480 rpm 540 rpm 600 rpm
0.10 0
1
(c) 0.30
2
r/T=0.35
0.25
360 rpm 390 rpm 420 rpm fitting line
z/T (-)
z/T (-)
0.25
3 2
/N3L (-)
4
5
360 rpm 390 rpm 420 rpm 480 rpm 540 rpm 600 rpm fitting line
0.20 0.15 0.10 0
1
2
3
/N3L2 (-)
4
5
r/T=0.38
0.25
z/T (-)
(b)0.30
r/T=0.33
360 rpm 390 rpm 420 rpm 480 rpm 540 rpm 600 rpm fitting line
0.20 0.15 0.10 0
1
2
3
/N3L2 (-)
4
5
Figure 14. Axial profiles of normalised energy dissipation (a) r/T =0.33, (b) r/T =0.35, (c) r/T =0.38.
Based on the simulation results, the correlation for the average energy dissipation rate can be expressed as Eq. 9.
N 3 L2
b( r / T , z / T )
(9)
where is the circumferential average energy dissipation rate. L is the dimeter of the impeller. N is the rotating speed, b(r / T , z / T ) is a dimensionless parameter related to the location.
25
In this study, the observation domain is in the range of r / T 0.325 0.38 and
z / T 0.103 0.25 . And the volume average value of b(r / T , z / T ) obtained from the simulation is 2.03. Thus correlation for the average energy dissipation rate in this study is established as expressed by Eq. 10.
domain 2.03N 3 L2
(10)
As is pointed out in the introduction section, in the model of Coulalooglou and Tavlarides and its modified models, the prediction of the drop breakup frequency is very sensitive to the adjustable parameters. Also, the model of Luo and Svendsen and its extensions are very sensitive to the choice of the integration limits. Besides, there is no uniform form to characterize the influence of the parameters such as energy dissipation rate, mother droplet size and interfacial tension. Thus, a more realistic empirical correlation equation should be proposed based on experimental data. After dimensional analysis, the empirical correlation equation proposed based on the experimental data is expressed as Eq. 11. ( D) 0.10
1/3 D
2/3
WeD 2.64 FrD 0.56 Re D 0.15
(11)
Where: WeD
d 2/3 D5/3
(12)
Re D
c 1/3 D 4/3 c
(13)
FrD
2/3
(14)
gD1/3
The comparison between the prediction data of Eq. 11 and the experimental data 26
is shown in Fig.15. A prediction error of ±20% can be observed from the figure, which is acceptable for the droplet size calculation with PBM.
Calculated data/s-1
20 10
+20%
5
-20%
2 1 0.5 0.1 0.1
0.5 1 2 5 10 20 Experimental data/s-1
Figure 15. Comparison between the experimental results and the results calculated from the empirical correlation equation.
3.4 Daughter droplet size distribution As is discussed in the previous section, the breakup events in the pump-mixer are divided into three different types, i.e., original tensile breakup, intermediate tensile breakup, and revolving breakup. The DDSD for these three breakup types are different because of the different breakup mechanisms. Therefore, we investigate the DDSD for these three situations separately.
3.4.1 DDSD for original tensile breakup
27
(a)
(b)
Dm= 0.40 mm
6
System No.1 480 rpm 540 rpm
5
Dm= 0.65 mm
5
System No.1 420 rpm 480 rpm
4
Fβo
Fβo
4
6
3
3
2
2
1
1
0
0.2
0.4 0.6 0.8 Volume ratio,fV
0
1.0
0.2
0.4 0.6 0.8 Volume ratio,fV
1.0
Figure 16. DDSD for original breakup under different rotating speed (a) Dm = 0.40 mm, (b). Dm = 0.65 mm.
(b)
6 5 4 3 2 1 0
Fβo
(c) 6 5 4 3 2 1 0
Dm = 0.2 mm System No. 5 6
0.2
0.4
0.6
0.8
Fβo
Fβo
(a)
1.0
Volume ratio,fV
6 5 4 3 2 1 0
Dm = 0.6 mm System No. 4 5 6
0.2
0.4
0.6
0.8
1.0
Volume ratio,fV
Dm = 1.0 mm System No. 2 4
0.2
0.4
0.6
0.8
1.0
Volume ratio,fV
Figure 17. DDSD for original breakup under different TBP concentration (a) Dm = 0.3 mm, (b) Dm = 0.6 mm, (c). Dm = 1.0 mm.
28
(a)
5
Fβo
Fβo
3
2
2
1
1 0.2
0.4
0.6
0.8
(c) 6
System No.1 480 rpm Dm = 0.40 mm
5
0
1.0
Volume ratio,fV
Fβo
2 1 0.6
0.8
0
1.0
Dm = 1.0 mm
0.2
2
1
1 0.4
0.6
0.8
Dm = 1.0 mm
0
1.0
0.2
Volume ratio,fV
(g) 6
Fβo
Fβo
3 2
1
1 0.4
0.6
1.0
0.8
Dm = 0.6 mm
4
2
0.2
0.8
System No.6 360 rpm Dm = 0.3 mm
5
3
0
0.6
(h) 6
Dm = 0.6 mm
4
0.4
Volume ratio,fV
System No.5 360 rpm Dm = 0.3 mm
5
1.0
3
2
0.2
0.8
System No.4 360 rpm Dm = 0.6 mm
6 4
3
0
0.6
5 Fβo
Fβo
(f)
Dm = 1.0 mm
4
0.4
Volume ratio,fV
System No.3 360 rpm Dm = 0.6 mm
5
1.0
System No.2 360 rpm Dm = 0.6 mm
Volume ratio,fV
(e) 6
0.8
3
1 0.4
0.6
4
2
0.2
0.4
Volume ratio,fV
5
3
0
0.2
(d) 6
Dm = 0.65 mm
4
Dm = 1.05 mm
4
3
0
System No.1 420 rpm Dm = 0.65 mm
6 5
Dm = 1.05 mm
4
Fβo
(b)
System No.1 390 rpm Dm = 0.65 mm
6
1.0
Volume ratio,fV
0
0.2
0.4
0.6
0.8
1.0
Volume ratio,fV
Figure 18. DDSD for original breakup with different mother droplet size (a) system No.1, 390 rpm, (b) system No.1, 420 rpm, (c) system No.1, 480 rpm, (d)-(h) system No.2-6, 360 rpm.
The DDSD for original tensile breakup in different experimental conditions are shown in Fig. 16, 17 and 18. One can see from the figures that the curves for DDSD are bell-shaped, which mean the equal sized breakup is more likely to occur for 29
original tensile breakup. Fig. 16 shows that the distributions broaden with the rotating speed increases, and the effect is more significant for the larger mother droplet. Fig. 17 shows that the distributions become wider as the interfacial tension decreases for large mother droplets, and the effect is not significant for small mother droplets. Fig. 18 shows the distributions widen when the size increases. We can conclude from the above results that the DDSD for original breakup is related to the breakup frequency. The curve of DDSD gets broader when the breakup frequency gets higher. To quantitatively describe the DDSD, an empirical correlation is proposed as expressed by Eq. 15. The comparison between the prediction results and the experimental data is shown in Fig. 19. The figure shows the relative error is less than ±20%. F o
f vWeD
0.5
0
f
0.38
WeD 0.38 v
, when f v 0.5
(15)
df v
5
Calculated data
+20% 4 -20%
3 2 1
+40% -40%
0
0
1
2
3
4
5
Experimental data
Figure 19. Comparison between the experimental results and the results calculated from the empirical equation 15.
30
3.4.2 DDSD for intermediate tensile breakup
(a) 6
3
4 3
2
2
1
1
0
0.2
0.4 0.6 0.8 Volume ratio,fV
360 rpm System No. 2 3 4 5
5
Fβi
4
Dm = 0.6 mm
6
System No.1 390 rpm 420 rpm 480 rpm 540 rpm 600 rpm
5
Fβi
(b)
Dm = 0.6 mm
1.0
0
0.2
0.4 0.6 0.8 Volume ratio,fV
1.0
Figure 20. DDSD for intermediate tensile breakup (a) with different rotating speed, (b) with different interfacial tension. Fig. 20a and 20b show DDSD of the intermediate tensile breakup with different rotating speed and interfacial tension respectively. We can see from the figure that the influence of the rotating speed and the interfacial tension on the DDSD is slight for the intermediate tensile breakup. Moreover, the curves of the DDSD are U-shaped, which means that the intermediate breakup of a multiple breakup mostly generates unequal segments. The reason is that satellite droplets are usually generated when a multiple tensile breakage occurs. A satellite droplet must be one of the daughter droplets belong to a certain intermediate breakup. The size of the satellite droplet is usually much smaller than the other daughter droplets, leading to a U-shaped DDSD. An empirical correlation for the U-shaped DDSD is obtained based on the experimental data as expressed by Eq. 16. The comparison between the calculated results and the experimental data is shown in Fig. 21 and a relative error of no more than ±20% is also observed.
31
F i
f v 0.368
0.5
0
f v 0.368 df v
, when f v 0.5
(16)
5
Calculated data
+20% 4 -20%
3 2 1 0
0
1
2
3
4
5
Experimental data
Figure 21. Comparison between the experimental results and the results calculated from the empirical equation 16. 3.4.3 DDSD for revolving breakup Fig. 22a and 22b show DDSD for the revolving breakup with different rotating speed and interfacial tension respectively. The figures indicate that the rotating speed and interfacial tension have little influence on the DDSD. M-shaped distributions are observed for the revolving breakup. This is consistent with the phenomena shown in Fig. 3b, the revolving breakup process is more likely to create two fragments with large volume difference. The smaller fragment was thrown away from the end far from rotating center, and the volume ratio of smaller fragment to the mother droplet is mostly in the range of 0.1-0.2.
32
(a)
4
3
3
2
2
1
1
0
0.2
0.4 0.6 Volume ratio,fV
360 rpm System No. 2 3 4 5 6
5
Fβr
4
Dm = 0.6 mm
6
System No.1 390 rpm 420 rpm 480 rpm 540 rpm
5
Fβr
(b)
Dm = 0.6 mm
6
0.8
0
1.0
0.2
0.4 0.6 Volume ratio,fV
0.8
1.0
Figure 22. Daughter droplet size distribution with different conditions (a) with different rotating speed, (b) with different interfacial tension. An empirical correlation for the M-shaped DDSD is also suggested based on the experimental data of the revolving breakup process. The correlation is expressed as Eq. 17. F r
(1.55 f v )18.42 f v1.58
0.5
0
(1.55 f v )18.42 f v1.58
, when f v 0.5
(17)
The comparison between the experimental data and the prediction results is shown in Fig. 23. The prediction error for most of the data points is within ±20%.
5
Calculated data
+20% 4 -20%
3 2 1 0
0
1
2
3
4
5
Experimental data
Figure 23. Comparison between the experimental results and the results calculated from the empirical equation 17. 33
The proportion of the three breakup types has been determined in the previous section, expressed by Eq. 6, 7 and 8. Thus the DDSD for the droplet breakup in the pump-mixer can be expressed as F Po F o PF i i Pr F r
(18)
This is the final form for the DDSD which can be used in the PBM calculation.
4. Conclusions The direct experimental data for droplet breakage frequency and DDSD is very lacking in literature. In this study, the drop breakup in the pump-mixer was investigated by using the high speed online camera. Two typical breakup patterns, tensile breakup and revolving breakup were observed from the captured video. Binary breakup occupied 50% to 80% of all the breakup events in our experiments. The statistical data of the droplet number density indicated that the drop size was in the range of 0.05-1.0 mm obeying a normal distribution. The average size decreased with the increasing of the stirring intensity and decreasing of the interfacial tension. The breakup frequency was directly measured based on its definition. The results showed that the breakup frequency increased with the increasing of the stirring intensity and drop size or the decreasing of the interfacial tension. The DDSD for three different breakup types, original tensile breakup, intermediate tensile breakup, and revolving breakup were separately determined. For the original tensile breakup, the DDSD was a bell-shaped distribution, which broadened with the increasing of stirring intensity and mother droplet size or the decreasing of the interfacial tension. The DDSD was a
34
U-shaped distribution for the intermediate tensile breakup, and was an M-shaped distribution for the revolving breakup. The U-shaped and M-shaped distributions were little influenced by the stirring intensity and interfacial tension. To quantitatively describe the drop breakup, the empirical correlations of drop breakup frequency and DDSD were proposed based on the experimental data, the prediction results of those correlations and the experimental data were then compared and showed good agreements.
Acknowledgments We gratefully acknowledge the support of the National Natural Science Foundation of China (21576147, 21776151, 21376132).
Notation Latin symbols b
dimensionless parameter related to the location
cf
coefficient of surface area
D
diameter of the droplet, mm
fv
volume ratio of the daughter droplet to the mother droplet
Fβi
daughter droplet size distribution function of intermediate tensile breakup
Fβo
daughter droplet size distribution function of original tensile breakup
Fβr
daughter droplet size distribution function of revolving breakup
FrD
droplet dimeter based Froude number, FrD 2/3 / ( gD1/3 )
L
dimeter of the imlpeller, m 35
n(D)
number density function of the droplets, m-4
N
rotating speed of the impeller, r/s
Pi
occurrence possibility of middle tensile breakup
Po
occurrence possibility of first tensile breakup
Pr
occurrence possibility of revolving breakup
r
radial coordinate, m
ReD
droplet dimeter based Reynolds number, ReD c 1/3 D4/3 / c
t
times, s
T
dimeter of the mixer, m
tc
time duration, s
v
volume, m3
WeD
droplet dimeter based Weber number, WeD d 2/3 D5/3 /
z
axial coordinate, m
Greek letter
0
numerical constant
D breakup frequency function, s-1
energy dissipation rate, m2/s3
domain
average energy dissipation rate in the research domain, m2/s3
vessel
average energy dissipation rate in the vessel, m2/s3
circumferential average energy dissipation rate, m2 /s3
size of eddies, m
c
viscosity of the continuous phase, Pa s 36
d
viscosity of the dispersed phase, Pa s
c
continuous phase density, kg/m3
d
dispersion phase density, kg/m3
interfacial tension, N/m
d
dispersed phase volume fraction
Abbreviations CFD
Computational fluid dynamics
DDSD
daughter droplet size distribution
DSD
droplet size distribution
PBE
population balance equation
PBM
population balance model
RFR
Rotating Frames of Reference
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breakage kernels for liquid-liquid systems: Solution of the population balance equation by the least-squares method. Can. J. Chem. Eng. 92, 234-249. Solsvik, J., Jakobsen, H.A., 2015. Single drop breakup experiments in stirred liquid–liquid tank. Chem. Eng. Sci. 131, 219-234. Solsvik, J., Maaß, S., Jakobsen, H.A., 2016. Definition of the single drop breakup event. Ind. Eng. Chem. Res. 55, 2872-2882. Srilatha, C., Savant, A.R., Patwardhan, A.W., Ghosh, S.K., 2008. Head–flow characteristics of pump-mix mixers. Chem. Eng. Process. 47, 1678-1692. Thakur, R.K., Vial, C., Nigam, K.D.P., Nauman, E.B., Djelveh, G., 2003. Static mixers in the process industries - A review. Chem. Eng. Res. Des. 81, 787-826. van Delden, M.L., Vos, G.S., Kuipers, N.J.M., de Haan, A.B., 2006. Extraction of caprolactam with toluene in a pulsed disc and doughnut column - Part II: Experimental evaluation of the hydraulic characteristics. Solvent Extr. Ion Exch. 24, 519-538. Vonka, M., Soos, M., 2015. Characterization of liquid-liquid dispersions with variable viscosity by coupled computational fluid dynamics and population balances. AIChE J. 61, 2403-2414. Wang, T., Wang, J., Jin, Y., 2003. A novel theoretical breakup kernel function for bubbles/droplets in a turbulent flow. Chem. Eng. Sci. 58, 4629-4637. Wang, T., Wang, J., Jin, Y., 2005. Population balance model for gas-liquid flows: influence of bubble coalescence and breakup models. Ind. Eng. Chem. Res. 44, 7540-7549. Wang, T., Wang, J., Jin, Y., 2006. A CFD–PBM coupled model for gas–liquid flows. AIChE J. 52, 125-140. Xing, C., Wang, T., Wang, J., 2013. Experimental study and numerical simulation with a coupled
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41
Appendix A: Verification of simulation methods by comparing of experimental data and simulation results in a baffled vessel with a Rushton turbine.
Fig. A1 Simulation domain: half of the mixing vessel with a Rushton turbine. (a) 1/2 geometry for simulation (b) grids details of the 1/2 stationary region, (c) grids details of the 1/2 rotating region. The purpose of this CFD simulation is to verify the applicability of the k model and the simulation method applied in this study. In this section, we carried out the simulation in a baffled vessel of diameter T = 100 mm stirred by a Rushton turbine of diameter L = T/3. The scale is similar to our experimental setup. The geometric structure is identical to the experimental setup used by Baldi and Yianneski (Baldi and Yianneskis, 2004). Fig. A1 shows the simulation domain and the grids generated in this study. The simulation was performed using the same simulation approach as is
42
described in section 2.4.1. Table A1. Results of grid independence tests under 1082 rpm, Re=20000. Experimental results (Baldi and Number of nodes /Million
Simulation results Yianneskis, 2004)
No.
1
rotating
stationary
Power
vessel /
Power
region
region
number
m2s-3
number
0.41
2.30
3.90
1.00 4.1
2
0.74
2.65
3.92
vessel / m2s-3
1.26
1.06
Table A1 shows the mesh independence tests of two groups of grids. For No.1, the maximum grid size was 0.8 mm in stationary region (the grid size of the interface was 0.3 mm) and 0.3 mm in rotating region. For No.2, the maximum grid size was 0.8 mm in stationary region (the grid size of the interface was 0.2 mm) and 0.2 mm in rotating region. The power number and the average eddy dissipation were obtained for the comparison. The simulation results were verified with experimental results by Baldi and Yianneski (Baldi and Yianneskis, 2004) and Hartmann et al. (Hartmann et al., 2004). It shows that power number agree with the Baldi and Yianneskis’ result well under the error below 5%, the vessel average eddy dissipation agree with the experimental result under the error below 20%. The flow field in the vessel predict by the simulation was also verified by comparing the radial and circumferential velocity at three radial locations with the results of Hartmann et al., and a great matched-degree achieved, as shown in Figure 4. The results show that meshes and the simulation approach applied in this study is acceptable. 43
Highlights •
Two typical breakup patterns, tensile breakup and revolving breakup were observed.
•
Droplet breakage frequency and daughter droplet size distribution (DDSD) in pump-mixer were experimentally measured.
•
The turbulence energy dissipation was determined using CFD simulation.
•
Correlations of droplet breakage frequency and DDSD were proposed.
44