Definition of the Single Drop Breakup Event - Industrial & Engineering

Article pubs.acs.org/IECR

Definition of the Single Drop Breakup Event Jannike Solsvik,*,† Sebastian Maaß,*,‡ and Hugo A. Jakobsen† †

Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway SOPAT GmbH, Ackerstraße 76, 13355 Berlin, Germany

‡

ABSTRACT: Experiments are performed to record data of single drop breakup events in turbulent flow (stirred tank conditions) for improved understanding of the breakup phenomena. Different interpretations can be made of a multiple breakup event, in particular, either the initial breakup can be evaluated, or the entire breakup cascade can be considered. The examining of these two breakup definitions in the present study reveals that the binary breakup event, which is the most frequent used assumption for the number of daughter drops in the literature, is only a rough assessment of the physical truth. For a 1 mm toluene drop up to 47 daughter drops as a consequence of an entire breakup sequence were counted and the arithmetic average value was 11.7 with standard deviation of 7.7. The corresponding average value for the initial breakup was 2.39. By high-speed imaging it is possible to break down the breakup cascade into binary breakups. However, the isolation of these individual breakups is a strong simplification of the breakup process. The breakups in the cascade are triggered by the initial breakup and are not statistically independent. It is more meaningful to evaluate not only the initiation of a drop breakup but the whole sequence of the breakup cascade. The breakup time of the two breakup definitions appears significantly different. For a 1 mm toluene drop the difference in the arithmetic breakup time was 129 ms. The relative number distribution of the breakup time is wider for the breakup cascade than for the initial breakup as it is influence by the wider range and higher number of daughter drops. Resolution limitations of the camera used did not allow determination of the complete daughter size distribution of the final drop population of the breakup cascade with sufficient accuracy. Two different experimental set-ups were used to verify the general trends of the breakup definitions.

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INTRODUCTION

The transport equation of the number density function f n(d, r, t) of fluid particles can be given as1,3,4

In dispersed multiphase flows, one or more of the phases is composed of clearly identifiable discrete entities such as drops, bubbles, or solid particles. Many dispersed multiphase flow systems are polydisperse so that there is a distribution in the properties of the dispersed phase entities (e.g., size, shape, temperature and composition). Complex interphase mass, momentum, and energy transfer are often encountered through the interfaces between the dispersed and continuous phases. In a variety of industrial applications (e.g., chemical and nuclear reactors) the interphase transfer of mass, momentum, and energy can profoundly influence the overall process performance. The contact area between the phases is determined by the size distribution of the dispersed entities. For this reason, many population balance model investigations have been performed to determine the size distribution of the dispersed entities. Population Balance. In essence, a population balance of a dispersed system is a record for the number of dispersed entities. The record of these entities is dynamically dependent on the birth and death processes that create new entities and terminates existing entities within a finite or defined space.1 For fluid particles (i.e., bubbles and drops), coalescence and breakup events are sources for birth and death of the entities. A population balance equation is a continuity statement written in terms of a number density function. The number density function contains information about how the population constituted by the dispersed phase entities is distributed over certain characteristic properties of the entities (e.g., size, shape, temperature and composition) in space and time.2 © XXXX American Chemical Society

∂fn (d , r, t ) ∂t

+ ∇r ·[vr(d , r, t )fn (d , r, t )]

+ ∇d [vd(d , r, t )fn (d , r, t )] = − BD (d , r, t ) + BB(d , r, t ) − CD(d , r, t ) + CB(d , r, t )

(1)

Here, r and t are the physical space and time coordinates, d is the fluid particle diameter (internal/property coordinate), vr is the velocity in physical space, and vd is the growth velocity. The terms on the right-hand side (RHS) of eq 1 represent death and birth of fluid particles of size d due to breakup and coalescence events. In the present study, the focus is placed on the breakup processes. The death and birth of fluid particles due to breakup are given as, respectively BD (d , r, t ) = b(d)fn (d , r, t ) BB(d , r, t ) =

∫d

dmax

(2)

νP(d , d′)b(d′)fn (d′, r, t ) dd′

(3)

Here, b is the breakup frequency, ν represents the average number of daughter fragments generated in a breakup process, Received: February 12, 2016 Revised: February 23, 2016 Accepted: February 23, 2016

A

DOI: 10.1021/acs.iecr.6b00591 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research Table 1. Previous Studies on Single Fluid Particle Breakup ExperimentsNumber of Daughter Fragments reference

experimental setup

system

mother drop size

Konno et al.6 Hancil and Rod7 Hesketh et al.8 Kuriyama et al.9 Andersson and Andersson11 Galinat et al.12 Galinat et al.13 Maaß et al.14 Maaß et al.15 Solsvik and Jakobsen18 Solsvik and Jakobsen17

stirred tank stirred tank pipe flow loop reactor/stirred tank static mixer pipe flow pipe flow stirred tank mimic stirred tank stirred tank stirred tank

liquid−liquid liquid−liquid gas-in-liquid liquid−liquid liquid−liquid liquid−liquid liquid−liquid liquid−liquid liquid−liquid gas-in-liquid liquid−liquid

d ∈ [0.26, 1] mm V ∈ [3, 39] mm3 d ∈ [2.5, 6] mm d ∈ [1, 3] mm d ∼ 1 mm d ∈ [2, 3] mm d ∈ [1.5, 3] mm d ∈ [0.56, 2] mm d ∈ [0.54, 3.1] mm d ∈ [2, 3.5] mm d ∈ [0.6, 4] mm

and P denotes the size distribution probability density function. A review of these functions has recently been performed by Solsvik et al.5 The different breakup models can give very different predictions as the models are based on different breakup criteria.5 In order to advance the development of breakup models for fluid particles, it is necessary to experimentally investigate the dynamic breakup mechanism in detail by single bubble/drop breakup experiments Single Fluid Particle Breakup Experiments. A number of fluid particle breakup experiments have been performed with the aim of examine the breakup time, breakup probability, number of daughter fragments and daughter size distribution as a function of parameters such as mother fluid particle size, chemical properties and turbulent characteristics, e.g., refs 6−18. In such experiments the breakup process is observed through high-speed imaging in a preferable characterized turbulent regime. Bubbles/drops are injected one at a time into a turbulent flow system and the eventual breakup event is recorded. Alternatively, breakup events are recorded from a dilute turbulent system. Different experimental setups have been used to generate turbulence, for example stirred tanks, pipe flow, static mixers, and nozzles. In the present work, the emphasis is placed on the determination of the number of daughter fragments produced upon breakup of a mother entity in single fluid particle breakup experiments under stirred tank conditions. Number of Fragments Produced upon Breakup. For fluid particles, the fragmentation of the mother entity into two daughter entities has been assumed in the majority of modeling investigations.1 Only a few models are proposed that allow for multiple breakup.6,19−22 Konno et al.6 observed drop breakup events in a dilute liquid−liquid stirred tank. The investigated mother drop sizes were in the range of 0.26−1.0 mm. The mean numbers of drops produced upon breakup were in the range of 2.6−4.4. Hancil and Rod7 carried out drop breakup experiments in a stirred tank. The average number of daughter drops (2−6) was an increasing function of the mother drop size (3−39 mm3). The authors stated that breakup of a given mother drop into a high number of daughter drops requires higher energy than that required for a low number. For a drop to break, it has to enter a region in the vessel where the energy dissipation is sufficiently high. The smaller the drop, the smaller is the volume in which the drop can break and so is the opportunity for its breakup. Thus, the small drops have less chance of breaking up into more daughter drops than the large ones. Hesketh et al.8 reported that only binary bubble breakup events occurred in their turbulent pipe flow system.

number of daughter drops ν̅ ν̅ ν ν̅ ν ν̅ ν̅ ν ν̅ ν ν

∈ [2.6, 4.4] ∈ [2.4, 6] =2 ∈ [4, 27] ∈ [2, 9] ∈ [∼2.6, ∼ 8.2] ∈ [∼2.0, ∼ 11.5] ∈ [2, 97] ∈ [2.02, 2.4] ∈ [2, 12] ∈ [2, 9+]

breakup definition unknown not explicitly defined not explicitly defined not explicitly defined unknown unknown unknown unknown initial breakup breakup cascade breakup cascade

A loop-reactor-type stirred tank was adopted by Kuriyama et al.9 to investigate the drop breakup mechanism. The results reported showed that the number of daughter drops was an increasing function of mother drop size (1−3 mm), drop viscosity, and impeller rotational speed. The maximum number of daughter drops generated in a breakup event was about 25. Martinez-Bazan et al.23 studied bubble breakup in a liquid jet. For low to moderate values of the turbulent Weber number, binary breakup was considered the most likely outcome. For larger values of the turbulent Weber number, the mean number of daughter bubbles increased. No further quantitative data was reported. Andersson and Andersson11 studied bubble and drop breakup in a static mixer. The analysis of the experimental data revealed that most bubble breakup events resulted in binary breakup. For drops, multiple breakup events occurred more frequent than the binary breakup event (average number of fragments was 3.2). Galinat et al.12,13 used a pipe flow system to study drop breakup. The mean number of daughter drops was reported to be an increasing function of the turbulent Weber number. The largest mean number of daughter drops reported was 12. Maaß et al.14 carried out single drop breakup experiments in a stirred liquid−liquid tank. For a mother drop of 0.56 mm, approximately 60% of the breakup events resulted in two daughter drops. The highest number of daughter drops observed was 97 for a 2 mm mother drop. Maaß et al.15 collected drop breakup data from turbulent conditions comparable to those in a stirred tank. The investigated mother drop sizes were in the range of 0.5−3.5 mm. The average number of daughter drops was in the range of 2.02−2.40. The authors reported a dependency of the frame rate of the high-speed camera on the observed number of daughter drops. Solsvik and Jakobsen17 investigated breakup of different oils in a stirred tank. The mother drop diameters were in the range of 0.6−4.0 mm. The authors reported a dependency on the number of daughter drops with the mother drop size: the probability of binary breakup increased with decreasing mother drop size. Overall, multiple breakup events occurred more frequent than binary breakups. The largest number of daughter drops was not quantified but was reported to be larger than 9. Solsvik and Jakobsen18 studied bubble breakup in a stirred tank. For mother bubble diameters in the range of 2.5−3.4 mm, the relative occurrence of multiple breakup increased with the average energy dissipation rate of the whole tank (0.5, 0.9, and 1.4 m2 s−3). The maximum number of daughter bubbles produced upon breakup was 12. B


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Industrial & Engineering Chemistry Research It is difficult to make a direct comparison of the literature data for the probable number of daughter drops produced in a breakup event. The outcome of the breakup experiments depends on the experimental arrangement, experimental conditions and physical properties of the chemicals. Another issue is the breakup event definition adopted. However, a compact overview is given in Table 1. Breakup Definition. Some authors describe the breakup process as a sequence of a number of binary breakups.19,24 In the study by Hancil and Rod7 they were not able to determine by visual observation whether the breakup process leading to multiple breakup was a sequence of binary breakups, or whether the daughter fragments were formed simultaneously. The authors argued that the first explanation could be accepted only with the assumption that the sequence of the binary breakup, triggered by the first one, is an extremely rapid one and hence the difference between the two descriptions of the breakup process is of no practical importance. Maaß and Kraume16 and Maaß et al.15 defined the breakup event to be the number of daughter fragments and the time taken up to the initial breakup. On the other hand, Solsvik and Jakobsen17,18 defined the breakup event as the final population of daughter fragments. The breakup definition adopted in single breakup experiments does not only affect the analysis of the number of daughter fragments, it also affects the determination of the breakup time and daughter size distribution. Present Study. The breakup definition is not well addressed in the majority of the experimental single fluid particle studies. For this reason, in the present study, the breakup mechanism is visualized through high-speed imaging and data is recorded using two definitions of the breakup event (if more than two fragments are produced): • Initial breakup: The breakup event ends at the initial breakup and further sequent fragmentation of the daughter fragments is not considered. An initial breakup thus refers to the first breakup of the mother drop and the movement prehistory of the drop is recorded. • Final population of a breakup cascade: The breakup event is considered consisting of a sequence of breakups. Deformed intermediates break and produce either daughter entities that do not undergo further breakup, or new intermediates that undergo further breakup. The breakup of a deformed intermediate is considered dependent on the previous breakup process. The two breakup definitions are illustrated in Figure 1. In the present study, data for the number of daughter fragments and breakup time obtained using these two breakup definitions have been compared. Advantages and disadvantages of the two breakup definitions are discussed and recommendations are given.

Figure 1. Illustration of an initial breakup, a breakup cascade, and an independent breakup model.

based on the initial breakup were presented previously by Maaß et al.15 and Maaß and Kraume.16 A single blade representative for a section of a Rushton turbine was fixed in a rectangular channel. The relative velocity between the blade and the liquid was 1.5 m/s. Two oils were used as dispersed phase: toluene (99.98% purity) and petroleum (99.9% purity). Both oils were blended with a nonwater-soluble black dye, which decreases the interfacial tension between water and oil but increases the optical evaluation possibilities. The physical properties of the oils are listed in Table 2. Table 2. Properties of the Oils (Blended with Dye) oil

interfacial tension [mN/m]

viscosity [mPa·s]

density [kg/m3]

toluene petroleum n-dodecane

32 38.5 41.5

0.55 0.65 1.38

870 790 745

The mother drops were injected 1 m away from the blade. The diameter of the channel at this point was wide but decreased slowly toward the point of the fixed blade. The slow decrease in the channel diameter assured that no additional turbulent eddies was introduced. The wide diameter at the drop injection point gave a lower liquid velocity than at the breakup point. This allowed the injection of spherical drops with high accurate diameter. The mother drop diameter was between 0.54 and 3.1 mm (see Table 3). A highly accurate dosing pump produced mother drops with an standard deviation of the

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EXPERIMENTS Two experimental setups have been used to perform single drop breakup experiments (oil-in-water). A stirred tank and a breakup cell with turbulent conditions comparable to those in a stirred tank. Both set-ups have been used in previous studies.16,17 Fixed Blade in a Channel. In the present study the final drop population of the breakup cascade was analyzed (i.e data for the number of daughter drops and breakup time). Data

Table 3. Mother Drop Diameters

C

system

mother drop diameter [mm]

toluene/water petroleum/water

0.65, 1.0, 2.0, 3.0 0.54, 0.7, 1.0, 1.3, 1.9, 3.1 DOI: 10.1021/acs.iecr.6b00591 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

Figure 2. Evaluation of the statistics for the number of daughter drops produced from mother drops of specified sizes. The complete breakup cascade is considered when counting the number of daughter drops (i.e., the complete breakup cascade is considered a breakup event). Two oils are investigated: (a) petroleum and (b) toluene. The data is produced with the experimental setup using a fixed blade in a channel.

speed of the impeller (600, 650, or 700 rpm). After steady flow conditions was established, an oil drop was added to the tank through an injector. This drop was observed and the number of daughter drops generated upon breakup was detected manually. For each impeller rotational speed two set of data were collected from the image analysis as both the initial breakup and the complete breakup cascade were considered. Further details of the experiments are given by Solsvik and Jakobsen.17

diameter of less than 0.003 mm. The number of breakup event observations for each experimental condition (i.e., mother drop size and oil) ranges between 250 and 780 (see Figure 2). The number of experiments was verified through statistical analysis. A number of at least 750 breakup events for each experimental condition (i.e., mother drop size, oil, flow velocity, etc.) was the basis for the data previously reported by Maaß and coworkers15,16 for the initial breakup event. The breakup events were observed by pictures taken by a high-speed camera using a frame rate of 822 frames per second. An automated image processing program was used to deliver results for the number of daughter fragments and breakup times. Further details on the experiments are given by Maaß and Kraume.16 Stirred Tank. Baffles and a Rushton turbine were used to generate turbulent flow in a lab-scale tank. The average energy dissipation rate of the whole tank was between 0.9 and 1.4 m2 s−3. The breakup events were recorded with a high-speed camera taking 1000 frames per second. The dispersed phase, ndodecane oil, was blended with black dye to enable visualization of the breakup process. The physical properties of the oil are listed in Table 2. The tank was filled with the continuous liquid phase and the stirring was started at the desired rotational

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RESULTS AND DISCUSSION

In population balance modeling it is usually supposed that only binary breakup occurs. Recent experimental investigations do not support this assumption.17,18 For this reason the breakup event needs to be examined in detail. The results obtained from an examination by single fluid particle experiments are presented in the following. Experimental Setup Using a Fixed Blade in a Channel. The experimental data of the initial breakup presented in Figures 3−7 have previously been presented and discussed by Maaß et al.15 and Kraume.16 On the other hand, the experimental data considering the complete breakage cascade are new. D


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Figure 3. Number of fragments produced upon breakup as a function of petroleum mother drop diameter and breakup definition: drop population after the initial breakup15 and final drop population of the complete breakup cascade. The data is produced with the experimental setup using a fixed blade in a channel.

Figure 4. Number of fragments produced upon breakup as a function of toluene mother drop diameter and breakup definition: drop population after the initial breakup15 and final drop population of the complete breakup cascade. The data is produced with the experimental setup using a fixed blade in a channel.

E


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Figure 5. Average and maximum number of fragments produced upon breakup of (a) petroleum and (b) toluene oils. Two definitions of the breakup event were used: (i) initial breakup15 and (ii) final drop population of the breakup cascade. The data is produced with the experimental setup using a fixed blade in a channel.

Figure 6. Relative number distribution of breakup time for (i) initial breakup16 and (ii) final drop population of the breakup cascade. The arithmetic breakup times are also indicated. 1 mm toluene drop. The data is produced with the experimental setup using a fixed blade in a channel.

High-Speed Image Analysis. The presentation in Figures 8 and 9 contain the original breakage sequence after image processing and a model representation of these breakage events which assume the restoration of spherical drops after each breakage events. This restoration is equal to the assumption that all deformations or oscillations are dissipated after the first breakage and that they are not relevant for further events. Figures 8 and 9 clearly show that the assumption of restoration is rough and that the deformations or oscillations significantly influence the breakups in the cascade. The description of breakages does not necessarily need to track every single deformation and its consequence on the breakage. A single breakup event could be interpreted as the sum of the individual events in a breakup cascade. In such, only the initial drop is seen as a sphere and also the sum of the volume of the occurring daughter drops of the complete breakup cascade is seen as spherical. That this is a more precise assumption can be seen on the last image comparison in Figures 8 and 9. After several breakage events, the remaining fluid particles can be assumed as spherical drops with fair approximation.

Figure 7. Arithmetic breakup times of toluene and petroleum oil drops. Measurements based on two definitions: (i) initial breakup16 and (ii) final drop population of the breakup cascade. The data is produced with the experimental setup using a fixed blade in a channel.

Number of Daughter Drops. An oil drop of known size was inserted in the flow of water and accelerated into a region with a fixed blade of a Rushton turbine miming the impeller region in a stirred tank (see section Fixed Blade in a Channel). The number of daughter drops produced from the initial breakup was analyzed in a previous study by Maaß et al.15 using the same experimental setup. In their framework, the description of the breakup event stops at the initial breakup and the existing deformations caused by the initial breakup event are assumed to fully dissipate before a breakup of an intermediate daughter drop take place. Thus, the model representation of the breakup event assumes restoration of spherical drops after each breakup event. This restoration is equal to the assumption that all deformations are dissipated after the initial breakup and are not F


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among sequential breakup events is far from the real situation. [Hagesaether25 discussed the dependency of the breakups in the breakup cascade.] That is, the continued deformation of an intermediate drop have a significant influence on the following breakup as reshape to spherical geometry in general do not take place within a single “eddy-collision” [For example, Ghasempour26 and Ghasempour et al.27 interpret the breakup event as due to interactions with one single vortex.] and deformed drops break more easily into further entities. Of course, not every largely deformed drop breaks but may retract from its elongation. It is a matter of probability of breakup. An alternative is to interpret the single breakup event as the sum of the final entities produced in the sequence of breakup events triggered by the initial or previous breakup. In this framework the initial drop as well as the final population of drops can be seen as spheres. An illustration of an initial breakup event, a breakup cascade and an independent breakup model is given in Figure 1. In Figures 3 (petroleum) and 4 (toluene) the data for the number of daughter drops produced in a series of coherent breakups are presented and compared to the results by Maaß et al.15 based on the initial breakup. Each subfigure represents the distribution in the number of daughter drops for a given mother drop size (Table 3). As a general trend, the number of daughter drops increases with increasing mother drop size. For petroleum oil, the relative numbers for producing two daughter drops in the initial breakup were 95, 89, 80, 77, 72, and 76% for mother drop diameters of 0.54, 0.7, 1.0, 1.3, 1.9, and 3.1 mm, respectively. Similarly, for toluene oil the binary outcomes were 89, 74, 75, and 72% for mother drop diameters of 0.65, 1.0, 2.0, and 3.0 mm, respectively. In the other framework where the final drop population of a breakup cascade is considered, much higher numbers of daughter drops are achieved for all the mother drop sizes studied. In the breakup cascade analysis, the relative numbers of binary breakups decreased significantly compared to the occurrence of two daughter drops in the initial breakup analysis. For petroleum, the relative occurrence of binary breakup events was 38, 19, 12, 6, 4, and 6% for mother drop diameters of 0.54, 0.7, 1.0, 1.3, 1.9, and 3.1 mm, respectively. Similarly, for toluene the outcome was 7, 8, 3, and 2% for mother drop diameters of 0.65, 1.0, 2.0, and 3.0 mm, respectively. A frame rate of 822 fps was used to collect the data in Figures 3 and 4. The used frame rate allows a detailed timely resolution of the single drop breakup events, while keeping the computer memory usage and computational costs reasonable. For 1 mm toluene drops at given flow condition, Maaß et al.15 investigated the influence of frame rate on the average number of daughter drops. With increasing frame rate the average number of daughter drops approach a value of two (see Table 4). Thus, if the time resolution is high enough, every single drop breakup event can be differentiated into a series of binary breakups. The

Figure 8. Breakup of a 3 mm toluene mother drop: original image processing images (left) and model interpretation of the breakup process assuming spherical drops (right). The mother drop, daughter drops of the initial breakup, and the breakup cascade are indicated in the figure.

Table 4. Influence of Frame Rate on the Average Number of Daughter Drops at Initial Breakup as Measured by Maaß et al.15

Figure 9. Breakup of a 0.65 mm toluene mother drop: original image processing images (left) and model interpretation of the breakup process assuming spherical drops (right). The mother drop, daughter drops of the initial breakup, and the breakup cascade are indicated in the figure.

relevant for further breakup events. Observations of breakup events by video imaging with high temporal and spatial resolutions revealed that the assumption of independence G

frame rate (fps)

average number at initial breakup

125 500 822 1450

3.82 2.83 2.39 2.15 DOI: 10.1021/acs.iecr.6b00591 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Breakup Time. Using the experimental setup described in section Fixed Blade in a Channel, Maaß and Kraume16 determined the necessary time to deform the mother drop until it broke for the first time. That is, the breakup time for initial breakup is assumed to be determined by the drop deformation time. These data are compared to the breakup time of the breakup cascade. For 1 mm toluene mother drops, Figure 6 holds the relative number distribution of the breakup time. As expected, a wider distribution in the breakup time appears for the breakup cascade in comparison to the distribution of the initial breakup time. The most frequent breakup time of the breakup cascade was 68 ms whereas the most frequent breakup time of the initial breakup was 3.7 ms. The arithmetic average breakup times of the breakup cascade and initial breakup were 141 and 12.4 ms, respectively. In Figure 7, the arithmetic average breakup time is given as a function of the mother drop size for petroleum and toluene oils. The comparisons in Figures 6 and 7 show clearly that the initial breakup time is significantly smaller than the time from the first deformation until the final drop population of the cascade is achieved. As the breakup cascade results in breakup events with very different breakup times (i.e., wide distribution), the breakup times should not only be given as a function of the mother drop size but also as a function of the number of daughter drops. This is left for further studies. The two possible interpretations of the breakup event provides very different breakup times. Choosing the breakup cascade interpretation requires that the underlying functions in the breakup models must be redefined. This might be necessary in order to consider all the breakup phenomena taking place. In particular, choosing the initial breakup interpretation the successive cascade phenomena, which may be important, will be lost. For this reason, the breakup cascade time might be more suitable for determining the breakup rate. Hence, the current breakup rate (and daughter size distribution function) models might need to be modified. Stirred TankExperimental Setup Using a Rotating Blade. The results presented previously were obtained in a particular breakup cell with flow conditions comparable to those in a stirred tank (see section Fixed Blade in a Channel). It is expected that the general trends observed are comparable to similar applications. To test this hypothesis, single breakup experiments were also performed in a conventional stirred tank (see section Stirred Tank). The previous breakup data presented by Solsvik and Jakobsen17 consider the complete breakup cascade only. In the present study, a new set of breakup data was generated in order to compare the breakage outcome using the two alternative breakup event definitions (i.e., initial breakup and breakup cascade). The results of the stirred tank are presented in Figure 10 and Table 5. The results of the number of daughter drops should not be compared in detail with the results obtained in the experimental setup described in section Fixed Blade in a Channel. The reason for this is several, e.g., different influence of the shear around the blade, different turbulent conditions, the injection of the mother drop gives a distribution in the size in the stirred tank whereas the mother drop size could be controlled in the experimental setup using a fixed blade in a channel, different chemicals, etc. However, a direct comparison between the two experimental set-ups was not the intention by performing additional breakup experiments in the stirred tank configuration; rather we are looking for trends in the breakup

assumption that all multiple breakup events can be dissected into a series of binary breakups is a rough description. The sequence of binary breakups happens extremely rapid and the breakup events are strongly connected with each other. Every binary breakup event in the sequence is triggered by the previous breakups. The average and maximum values for the number of daughter drops are shown as a function of the mother drop size in Figure 5. As expected, much higher numbers of daughter drops are achieved in the breakup cascade than in the initial breakup. The largest number of daughter drops produced in the breakup cascades of petroleum oil was 14 for a mother drop diameter of 0.54 mm and 159 for a mother drop diameter of 3.1 mm. The corresponding maximum numbers of daughter drops of the initial breakup are respectively 3 and 7. Considering the final drop population of the breakup cascades the average number of daughter drops is larger than two: 3.1 and 33.8 for mother drop diameters of 0.54 and 3.1 mm, respectively. On the other hand, the corresponding average values of the initial breakup are respectively 2.02 and 2.34. Similar trends are observed for toluene. Only daughter drops with a diameter larger than 0.1−0.2 mm could be detected. Thus, the reported number of daughter drops might be higher if very small and not detectable fragments are produced upon breakup. Figure 2 presents the reliability of the achieved experimental number of daughter drops of the breakup cascade. For each mother drop size, the cumulative standard deviation of the number of daughter drops over the number of investigated events is given along with the cumulative arithmetic average value of the number of daughter drops. A wide distribution for the number of daughter drops was measured from the breakup cascade. This results in large values for the standard deviations, that is, between 40 and 88%. However, a statistically confident interpretation of the average number of daughter drops is achieved for each mother drop size as all the data sets in Figure 2 approach a constant average value. The statistical analyses of the experimental data for the initial breakup have previously been presented by Maaß et al.15 and Maaß and Kraume.16 Breakup Probability. The breakup probability is defined as the ratio between the number of broken drops and the total number of injected drops under the same experimental conditions. The observed breakup probability of the single drop analysis remains uninfluenced by the breakup definition since the statement whether a drop breaks is detached from the question of how many fragments are generated by the breakup event. Data for the breakup probability using the experimental setup described in section Fixed Blade in a Channel is given by Maaß and Kraume.16 Daughter Drop Size Distribution. Determining the daughter size distribution based on single drop breakup experiments was first applied by Cabassud et al.28 Using the experimental setup as described in section Fixed Blade in a Channel, Maaß et al.15 investigated the daughter drop size distribution of the initial breakup. The breakage of a mother drop into a very small and a very large daughter drop was observed as most probable. Due to the resolution of the camera, only daughter drops with a diameter larger than 0.1−0.2 mm could be detected. In the present study, the resolution of the used high-speed camera did not allow a precise determination of the sizes of the final population of daughter drops produced in the breakup cascades. The tiny daughter drops resulting at the end of the breakup cascades were too small. H


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behavior due to the two breakup definitions. The main trend for the number of daughter drops is the same in the two experimental setups: a higher average number of daughter drops is achieved in the complete breakup cascade compared to the initial breakup. The results presented in Figure 10 span a wide range of mother drop sizes (see Table 5). From previous results, e.g. Figure 5, it is known that the mother drop size influences on the number of daughter drops. With better image quality (resolution and light) it is believed that a higher fraction of the multiple breakup events of the initial breakup would have been considered binary. Considering the two experimental setups, the configuration with a fixed blade in a channel is recommended above the conventional stirred tank in terms of precision (e.g., reproducible mother drop size and larger variance in the turbulence due to the rotation of the blade) and simplicity (e.g., frequent cleaning of the stirred tank due to accumulation of daughter drops in the tank). Modeling. On the basis of the results achieved in the present study, it is reasonable to argue that the breakup of a drop should be considered as the complete cascade of breakup events. The sequence of breakup events initiated from a single mother drop are strongly connected to each other. Up to now, this is not considered in population balance models. Using the historical initial breakup event interpretation, all the breakup events are assumed to be independent of each other as required by the kinetic theory modeling approach. However, any subsequent breakup cascade are then neglected which might lead to erroneous estimates of the overall breakage rate. On the other hand, considering the whole breakup cascade as one breakup event, the breakup events might also be treated as independent of each other. Nevertheless, it is clear that within the breakup cascade each of the breakage subevents might not be independent of each other. However, the advantage of breakup cascade interpretation is that all the breakup phenomena of the cascade are taken into account in the breakup model. A further step in population balance modeling would be the implementation, not only of a fixed value for the number of daughter drops, but rather a phenomenological model, which describes the number of expected daughter drops as a function of mother drop size, physical properties of the chemicals and turbulence parameters. This development would automatically influence the description of the resulting daughter size distribution. An experimental evaluation of the daughter drop sizes although recorded was not possible in this study due to the limited resolution of the available camera. All the underlying functions in the breakup birth and death terms must be consistent, meaning that all functions are associated with the same breakup event definition. Hence, the choice of the number of daughter entities produced in a breakup event, the daughter size distribution function and the breakup frequency directly influence on each other based on the breakup event definition. An inconsistent combination of breakup functions derived from different breakup event definitions must thus be avoided as such a model will not produce physical outcomes of the breakup events. The breakup probability, on the other hand, is independent of the breakup event definition.

Figure 10. n-Dodecane. Number of fragments produced upon breakup as a function of stirring rate and breakup definition: initial and final drop population. The data is produced with the experimental setup using a rotating blade in a tank.

Table 5. Mother Drop Diameter and Number of Breakup Events for n-Dodecane Oil 600 rpm

650 rpm

Mother Drop Diameter [mm] minimum 0.8 0.8 maximum 4.4 3.8 average 2.5 2.4 median 2.5 2.5 Number of Breakup Observations dm ≤ 2.0 mm 44 48 dm ∈ ⟨2.0, 3.0] mm 93 66 dm ∈ ⟨3.0, 4.0] mm 28 48

700 rpm 0.8 4.2 2.3 2.4 58 63 35

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CONCLUSION Single fluid particle experiments provide the necessary information that is needed within the population balance equations for proper system characterization. In the present study, such experiments were performed to investigate the assumption of binary breakup, which is commonly assumed in population balance modeling. The assumption of binary breakup in population balance modeling is fair if only the initial breakup is analyzed. With the interpretation of the breakup event as the complete sequence of breakup events in the breakup cascade, the average number of daughter drops increases. For petroleum drops of 1.0 and 3.1 mm, the arithmetic average numbers of daughter drops were 7.11 and 33.82, respectively. Correspondingly, these numbers were 2.23 and 2.34 for the initial breakup. The standard deviation for the number of daughter drops of a given mother drop size is very high in the breakup cascade, i.e. 40−88%. This large variation in the number of daughter drops results in a wide number distribution of breakup times. The average breakup times are significantly longer in the complete breakup cascade compared to the time from the deformation to the initial breakup. The breakup process in the complete breakup cascade is complex. A breakup in the sequence is triggered by the previous breakup(s) as surface deformation energy is transferred from a breaking drop to its daughter drops. On the basis of the analysis of the single drop breakup experiments, we state the following: • Only independent breakup events should be countered and considered. That is, a new breakup process should start from a spherical entity and no deformation energy should be passed on from one breakup event to the next. If deformation energy is passed on and influences on the succeeding breakup cascade, the breakup events are not independent, which is a requirement for a statistical treatment of the breakup process. In this view the whole breakup cascade should be considered a single breakup event. • The breakup cascade should not be considered a series of binary breakup events due to the difference in energy states for the different breakage events within the cascade. If it is required that all the breakup events are independent, the entities must be spherical and have no memory of the previous breakup event (i.e., no energy transferred to the entity from the previous fragmentation). • Information of the complete breakup process is lost if only the initial breakup is analyzed. Further population balance modeling should not only use a higher value for the number of daughter drops, which currently is treated as a single value might be changed into a model function instead in the future. These models should reflect the influence of the mother drop size, physical properties of the chemicals, turbulent flow parameters, etc. Such a development would required that an appropriate daughter size distribution function is applied along with it a population balance model. The functions for the number and sizes of the daughter drops are also strongly correlated with the breakup time function. Thus, significant work remains extending the present population balance equation framework to consider the complete breakup cascade and not only the initial breakup. This seem to require that a more physical realistic model framework must be developed.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (J.S.). *E-mail: [email protected] (S.M.). (www.sopat.eu) Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The financial support by the Norwegian Research Council through the GASSMAKS program (J.S.) and the Max-Buchner Forschungsstiftung (S.M.) is gratefully appreciated.

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NOMENCLATURE BB = birth due to breakage [1/(m3·m1·s)] BD = death due to breakage [1/(m3·m1·s)] b = breakage frequency [1/s] CB = birth due to coalescence [1/(m3·m1·s)] CD = death due to coalescence [1/(m3·m1·s)] d = diameter [mm] dm = mother drop diameter [mm] f n = number density function [1/(m3·m)] P = size distribution probability density function [1/m] r = space coordinate [m] t = time [s] vr = velocity in physical space [m/s] vd = velocity in property space, growth velocity [m/s] V = volume [m3] ν = number of fragments/daughter particles ν̅ = average number of fragments/daughter particles REFERENCES

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Definition of the Single Drop Breakup Event - Industrial & Engineering

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