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\begin{document}

\title{Breakup model of oscillating drops in turbulent flow field}



\author[1]{Han Zhou}%
\author[2]{Xiong Yu}%
\author[1]{Bo Wang}%
\author[1]{Shan Jing}%
\author[3]{Wenjie Lan}%
\author[1]{Shaowei Li}%
\affil[1]{Tsinghua University}%
\affil[2]{Tsinghua University Institute of Nuclear and New Energy Technology}%
\affil[3]{China University of Petroleum (Beijing)}%


\vspace{-1em}



  
  \date{May 26, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
Following the drop oscillation breakup mechanism, a theoretical model
for drop breakup probability is proposed based on the three-dimensional
Maxwell velocity distribution. The model considers both the interfacial
energy increase constraint and viscous energy increase constraint. The
model shows that for low-viscous drops, the breakup probability is
determined by the Weber number (\emph{We}\textsubscript{\emph{L}}), and
for intermediate or high viscous drops, the breakup probability is
determined by the combined influence of the Weber number
(\emph{We}\textsubscript{\emph{L}}) and the Ohnesorge number
(\emph{Oh}). By combining the theoretical model of drop breakup time
constructed in our previous work, the breakup frequency model is
obtained based on the statistical description framework. The accuracy
and generality of the model were then validated using the direct
experimental data. Moreover, effects of the drop diameter, turbulent
energy dissipation rate, and interfacial tension on the predicted drop
breakup frequency were analyzed in detail.%
\end{abstract}%



\sloppy


\textbf{Breakup model of oscillating drops in turbulent flow field}

Han Zhou\textsuperscript{a}, Xiong Yu\textsuperscript{a}, Bo
Wang\textsuperscript{a}, Shan Jing\textsuperscript{a,*} Wenjie
Lan\textsuperscript{b}, Shaowei Li\textsuperscript{a, c*}

\textsuperscript{a} Institute of Nuclear and New Energy Technology,
Tsinghua University, Beijing 100084, China

\textsuperscript{b} State Key Laboratory of Heavy Oil Processing, China
University of Petroleum (Beijing), Beijing 102249, China

\textsuperscript{c} State Key Laboratory of Chemical Engineering,
Tsinghua University, Beijing 100084, China

\textbf{*Corresponding authors.}

E-mail addresses: s-jing@tsinghua.edu.cn (\textbf{S. Jing} ),
lsw@tsinghua.edu.cn (\textbf{S. Li} ).

\textbf{Abstract:} Following the drop oscillation breakup mechanism, a
theoretical model for drop breakup probability is proposed based on the
three-dimensional Maxwell velocity distribution. The model considers
both the interfacial energy increase constraint and viscous energy
increase constraint. The model shows that for low-viscous drops, the
breakup probability is determined by the Weber number
(\emph{We\textsubscript{L}} ), and for intermediate or high viscous
drops, the breakup probability is determined by the combined influence
of the Weber number (\emph{We\textsubscript{L}} ) and the Ohnesorge
number (\emph{Oh} ). By combining the theoretical model of drop breakup
time constructed in our previous work, the breakup frequency model is
obtained based on the statistical description framework. The accuracy
and generality of the model were then validated using the direct
experimental data. Moreover, effects of the drop diameter, turbulent
energy dissipation rate, and interfacial tension on the predicted drop
breakup frequency were analyzed in detail.

\textbf{Keywords:} Oscillation; breakup model; Maxwell velocity
distribution; turbulent energy dissipation; interfacial tension

\subsection*{1 Introduction}

{\label{introduction}}

Turbulent dispersions of immiscible liquids are attracting and focused
topics in practical applications, such as solvent extraction, food
processing, chemical reaction, and so forth. Behaviors of drop swarms
such as drop breakup and coalescence are fundamentally important as the
understanding of those basic phenomena can improve the quantitative
analysis of two-phase dispersion characteristics, interphase mass
transfer, and interface reaction rate. Drop breakup refers to the
process in which a spherical drop is deformed and stretched by the
external flow and finally breaks up into several fragments. A complete
description of the drop breakup usually requires knowledge of the
following aspects, that is, the breakup probability of a drop with a
certain size, duration of the drop breakup process, and number and size
distribution of the fragments. The first two aspects are typically
described using the concept of drop breakup frequency. Over the years,
masses of breakup models for liquid drops were proposed by researchers,
however, the existing breakup models vary from each other in forms and
predicted results, implying a lack of understanding of the drop breakup
mechanism. To gain more insight into the drop breakup phenomenon, single
drop breakup experiments were carried out by some research
groups\textsuperscript{1--7}. Results of the single drop breakup
experiments verified that drop diameter, interfacial tension, and
external energy input are key parameters determining the drop breakup
behaviors. However, the direct experimental data concerning the
influence of various parameters on drop breakup, especially breakup time
and breakup frequency, are still very limited. Moreover, no unifying
drop breakup mechanism was found from the reported studies on drop
breakup. To obtain the direct experimental data of drop breakup
behaviors and analyze the mechanism of drop breakup in turbulent
dispersions, since 2016, our group has been engaged in the study of
breakup behaviors of drop swarms under turbulent
conditions.\textsuperscript{8--12} Through systematic experimental
studies in different apparatuses, we quantified the influences of
operating parameters and physical properties of the two phases on drop
breakup. The direct experimental data of drop breakup frequency, breakup
time, daughter drop size distribution, and so forth were obtained. In
our recent work\textsuperscript{13}, we found that the drop breakup
behavior is directly related to the second-order oscillation of the
drop, correspondingly, breakup models can be constructed based on the
drop oscillation breakup mechanism.

\subsection*{2 Breakup kernels for liquid
drops}

{\label{breakup-kernels-for-liquid-drops}}

To quantitatively describe the drop breakup behavior, models of drop
breakup frequency and daughter drop size distribution (DDSD) are
crucial. As for the DDSD, U-shaped\textsuperscript{14,15},
M-shaped\textsuperscript{16,17}, and inverted U-shaped
distributions\textsuperscript{18--20} are generally reported in the
literature. For binary breakup of liquid drops, the inverted U-shaped
distribution has been adopted by most researchers and also verified by
experimental data\textsuperscript{8,9,21,22}. There are many DDSD models
related to the inverted U-shaped distribution, and these models have
limited impact on the quantitative study of the drop breakup process
despite their differences. As such, researchers mainly focus on the
modeling of the drop breakup frequency. Although a vast number of drop
breakup frequency models exist in the literature, the vast majority of
them are constructed under a limited number of modeling frameworks. In
particular,frameworks based on the eddy-particle collision and
statistical description are generally adopted for modeling. According to
the eddy-particle collision framework, drop breakup frequency is the
product of the eddy-drop collision frequency and the breakup efficiency.
Prince and Blanch\textsuperscript{23} gives an expression in the
integral form based on the turbulent eddy size, then Luo and
Svendsen\textsuperscript{15}introduced the impact of the volume fraction
of fragments and proposed the modeling framework in the double integral
form, as is expressed in Eq..

\par\null

Where is the collision frequency between eddies and drops, is the
breakup efficiency for a mother drop with size \emph{d}. In the
statistical description framework, drop breakup frequency is counted as
the product of the inversed breakup time and breakup probability, which
is expressed as Eq..

\par\null

Where is the drop breakup time, is the breakup probability of a drop
with a diameter of \emph{d} , and are the number of breakup events and
total number of drops respectively. This framework was firstly proposed
by Coulaloglou and Tavlarides\textsuperscript{18} in 1977 and then
wildly adopted by researchers.

By comparing Eq. and Eq., it can be seen that the physical meanings of
the breakup probability (or called breakup efficiency) in the two
modeling frameworks are equivalent to each other. Therefore, the main
difference between the two modeling frameworks comes from the processing
of the time term, which is described by the breakup time in the
statistical description framework, and by the collision frequency in the
eddy-particle collision framework. In comparison, the physical meaning
of the breakup time is more explicit, which describes the time scale of
the drop breakup process. To model the drop breakup time, some
researchers\textsuperscript{5,18}, starting from the eddy turnover time,
assume that . According to our recent study\textsuperscript{24}, the
above correlation will be valid only if the turbulent stress is much
greater than the interfacial stress. However, such cases are less likely
to happen in liquid-liquid dispersions, especially under steady-state
operating conditions. To quantify the influence of each parameter on the
drop breakup time, our group performed systematic
experimental\textsuperscript{13} and theoretical
studies\textsuperscript{24} and proposed a novel breakup time model
based on the drop oscillation breakup mechanism:

\par\null

Where \emph{c\textsubscript{1}} and \emph{c\textsubscript{2}} are
constant parameters, \emph{c\textsubscript{1}} = 14.0
and\emph{c\textsubscript{2}} = 8.0. \emph{We\textsubscript{L}} has a
form of the Weber number and characterizes the relative magnitude of
turbulent stress and interfacial stress, \emph{We\textsubscript{L}} is
expressed as Eq..

\par\null

Where \emph{L} is the maximum scale of the local turbulent structure. In
practical calculations, \emph{L} values by the minimum length-scale of
the local runner\textsuperscript{24}.

For low-viscous drops, the influence of the drop viscosity on drop
breakup time is insignificant, thus, Eq. can be simplified to Eq.:

\par\null

Eq. and Eq. were validated by the systematic experimental data and can
be applied to Eq. directly. In combination with accurate modeling of the
drop breakup probability, an accurate description of drop breakup
frequency can be achieved.

This study aims to construct an accurate model of the drop breakup
frequency. Firstly, based on the three-dimensional Maxwell velocity
distribution and the drop oscillation breakup mechanism, the theoretical
model of the drop breakup probability is deduced. Further, the breakup
frequency model is obtained by combining the breakup probability model
with the breakup time models of Eq. and Eq.. Subsequently, the direct
experimental data of drop breakup frequency in different apparatus are
used to verify the accuracy and applicability of the constructed model
in this study. After that, drop breakup frequencies versus key
parameters are calculated, meanwhile, the impacting mechanisms of the
operating parameters and properties of two-phase on drop breakup
frequency are analyzed. The results of this study can be directly
applied to the population balance model, which further serves for the
accurate prediction of drop size distributions and dispersion
characteristics in liquid-liquid dispersion equipment.

\subsection*{3 Model development}

{\label{model-development}}

\subsubsection*{3.1 Theoretical model of drop breakup
probability}

{\label{theoretical-model-of-drop-breakup-probability}}

The drop breakup probability characterizes the instability of a drop
under the acting of the external force and can be statistically
expressed as the ratio of the number of broken drops to the total number
of mother drops, that is:

Various forms of drop breakup probability models exist in the
literature, and the majority of these models are constructed based on
Maxwell-Boltzmann velocity/energy distributions. However, due to
different selections of breakup constraints, there are significant
differences in the physical significance and predicted results of the
models constructed by different researchers. According to our recent
research\textsuperscript{13}, the drop breakup process is directly
related to the oscillation behaviors of the drop surface. This implies
that the breakup probability can be modeled based on the oscillation
mechanism.

To build a breakup probability model based on surface oscillation, we
adopt the following assumptions:

The turbulence is assumed to be locally isotropic.

This assumption is widely adopted in model constructions. Kolmogorov
pointed out that at sufficiently high Reynolds numbers, the fine-scale
structure of turbulent flows is statistically
characteristic.\textsuperscript{25} For devices such as stirred tanks
and extraction columns, experimental results also indicate that the
turbulent structure is locally
isotropic.\textsuperscript{26--28}Therefore, the theory of local
isotropy of turbulence can be used in the construction of the breakup
model.

Only binary breakup is considered.

Experimental studies showed that the binary breakup is dominant in
steady-state liquid-liquid dispersions, where the diameter of most drops
is close to the equilibrium size.\textsuperscript{8,9,21,29} However, as
the drop size deviates from the equilibrium size, the proportion of the
binary breakup decreases accordingly. \textsuperscript{2,3,9} According
to our previous studies \textsuperscript{8,21}, the tensile breakup is
the most common breakup pattern in turbulent dispersions. For multiple
tensile breakups, usually, two larger daughter drops and several
satellite drops are generated. Since the contribution of satellite drops
is relatively small (according to the description of Andersson and
Andersson \textsuperscript{6,30}, satellite drops account for about 8\%
volume of the mother drop), the effect of satellite drops can be
approximately neglected and the binary breakup assumption can still be
considered as a reasonable choice.

The drop will break up when the surface oscillation velocity exceeds a
critical limit.

When a drop comes in contact with the turbulent eddy, energy is
transferred to the drop surface in the form of turbulent pulsations,
which cause the oscillation of the drop surface. According to the
breakup process captured by the high-speed camera, drop breakup is
mainly caused by the second-order oscillation, higher-order oscillations
are almost non-contributory to the drop breakup. In the second-order
oscillation, the drop moves along a central axis in a
stretching-contraction cycle, and the relative velocity between the two
poles of this axis is defined as the oscillation velocity, which is
positive for the stretching motion and negative for the contraction
motion, and the direction of the central axis is defined as the
direction of the oscillation velocity. The oscillation velocity defined
in this way is a vector, and its magnitude is defined as the oscillation
speed. When the oscillation speed is too small to break the drop, the
energy absorbed from the turbulent pulsation will be dissipated in
several oscillation periods, and the drop eventually reverts to the
original. On the contrary, when the oscillation speed is large, the drop
does not have enough time to dissipate the energy through surface
oscillations and eventually breaks up to form several fragments (only
the binary breakup is considered in this study). We refer to the minimum
oscillation speed at which the drop breaks up as the critical
oscillation speed, . (For simplification, the velocity and the speed are
both expressed by the same word `velocity')

The definition of the second-order oscillation velocity shows that it is
a three-dimensional random vector in the locally isotropic turbulence,
and therefore conforms to the three-dimensional Maxwell velocity
distribution. The three-dimensional Maxwell velocity distribution can be
expressed as:

\par\null

Herein is the root-mean-square velocity of the surface oscillation. is
the probability density function. Integrating Eq. and we obtain the
expression of the drop breakup probability:

\par\null

Solve Eq. and we obtain Eq.:

\par\null

As a result, the expression of the drop breakup probability based on the
three-dimensional Maxwell velocity distribution is obtained. Herein is
the upper incomplete gamma function. Next, we need to determine the
expressions for the root-mean-square oscillation velocity and the
critical oscillation velocity.

The oscillation of the drop surface is caused by the random pulsation of
the turbulence. The average oscillation kinetic energy per unit drop
volume can be expressed as follows:

The drop oscillation kinetic energy is equal to the turbulent kinetic
energy transferred to the drop surface, which, according to the energy
spectrum theory of turbulence, is expressed as:

\par\null

Where is the energy spectrum and\emph{k} is the wavenumber of the
turbulence. Kolmogorov's hypotheses give the universal form of the
energy spectrum in the inertial subrange:

\par\null

The minimum wavenumber in Eq. inversely proportional to the scale of the
maximum turbulent structure capable of transferring energy to the drop
surface, which can be calculated by Eq..

\par\null

Where \emph{L} is the maximum scale of the local turbulent structure
and\emph{d} is the drop diameter. \selectlanguage{greek}\emph{α} \selectlanguage{english}takes the value of 1.5.
Substituting Eq. and Eq. into Eq., the expression for the turbulent
kinetic energy is then obtained, as shown in Eq..

\par\null

Taking , then we obtained the expression for the root-mean-square
velocity :

\par\null

To determine the expression for the critical oscillation velocity , the
criteria for drop breakup need to be clarified. According to the
oscillation breakup mechanism raised above, external turbulent
pulsations acting on the drop surface cause the surface oscillations,
and the oscillation energy is continuously propagated and dissipated at
the surface in the form of capillary waves. When the oscillation energy
is small, the energy will be eventually dissipated after several
oscillation periods, conversely, when the energy exceeds a certain
value, the drop will break up. Therefore, drop breakup should satisfy
the energy constraint, that is, the surface oscillation energy is
greater than the critical value, . According to Andersson and
Helmi\textsuperscript{30}, there is a maximum interfacial energy
increase during the drop breakup (), which is larger than the
interfacial energy increase before and after the drop breakup (). Most
models in the literature only consider , thus causing an underestimation
of the energy constraint.\textsuperscript{30} In present work, we use
the maximum interfacial energy increase during the breakup process as
the energy constraint, that is:

\par\null

Meanwhile, we have:

\par\null

The parameter of \emph{c} characterizes the relative magnitude of the
energy potential of the breakup process, According to Andersson and
Helmi\textsuperscript{30}, the maximum interfacial energy increase has
to be approximately 1.5 times the interfacial energy increase before and
after drop breakup. According to the experimental results in our
previous studies\textsuperscript{8,9}, drops are more inclined to
undergo the equal breakup due to the turbulent energy can be better
utilized under such cases. For binary equal breakups, then
\emph{c}values as 0.39. For practical drop breakup processes, drops are
not ideally equal breakup, thereby leading to a smaller value of
\emph{c}than 0.39. By comparing with the experimental data, we find
that\emph{c} takes a value of 0.365 to better match the experimental
data, which will be discussed in detail in the following section.

From Eq. and Eq., it is obtained that:

\par\null

Combining Eq., Eq., and Eq., the expression for the drop breakup
probability is obtained as Eq..

\par\null

Where \emph{We\textsubscript{L}} is the same with Eq..

\subsubsection*{3.2 Theoretical model of drop breakup
frequency}

{\label{theoretical-model-of-drop-breakup-frequency}}

According to the statistical description framework of Eq., the drop
breakup frequency can be expressed as the product of the reciprocal of
the breakup time and the breakup probability. Substituting Eq. and Eq.
into Eq., the theoretical model of drop breakup frequency for
low-viscous drops is then obtained, as shown in Eq..

\par\null

\subsection*{4 Results and discussion}

{\label{results-and-discussion}}

\subsubsection*{4.1 Validation}

{\label{validation}}

Recently, our group measures the drop breakup frequency in a pulsed disc
and doughnut column\textsuperscript{9,21} and in a
pump-mixer\textsuperscript{8,10,13}. The experimental data can be
directly used for the model validation. To compare the experimental data
of different researchers conveniently and to show the results
intuitively, the dimensionless abscissa and ordinate were adopted.
The\emph{We\textsubscript{L}} is used as the abscissa and drop breakup
probability is used as the ordinate. It should be noted that only the
experimental data of the drop breakup frequency is provided in our
previous work, therefore, it needs to be further processed in
conjunction with the breakup time model to obtain the data of the
breakup probability, that is . The above processing is still essentially
a direct experimental data validation of drop breakup frequency rather
than drop breakup probability. Another point to note is the value
of\emph{L} , which characterizes the maximum scale of the local
turbulent structure. \emph{L} depends on equipment parameters and can be
valued by the minimum length-scale of the local
runner\textsuperscript{24}. For the pulsed disc and doughnut column used
by Zhou et al.\textsuperscript{9,21}, \emph{L} values as the distance
between the adjacent disc and doughnut plate, \emph{L} = 30 mm. For the
stirred tank, \emph{L} values as the height of the blade, that is
\emph{L} = 10 mm (pump-impeller in Zhou et al.\textsuperscript{8,10})
and \emph{L} = 8.8 mm (Rushton Turbine impeller in Zhou et
al.\textsuperscript{13}).

In our previous study, the multiple breakup is artificially treated as
the cascaded binary breakup. Although this treatment does not affect the
application of the breakup model to the population balance model (as the
cascaded binary breakup assumption is also applied in the construction
of the daughter drop size distribution model). However, from a practical
physical point of view, the cascaded binary breakup assumption leads to
an overestimation of the breakup probability of small-sized drops. To
avoid the impact of the cascaded binary breakup assumption on the model
validation, experimental data of the initial drop breakup are adopted
and are applied to compare with the predicted results of the breakup
model proposed in this work, as shown in Figure 1.

It can be seen that the experimental data and the predicted results of
the model almost converge on a single curve, implying that the two are
in good agreement, which also verifies the accuracy and applicability of
the breakup model constructed in this study.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image41/image41}

\end{center}
\end{figure}

Figure 1 Comparison of the predicted results with the initial breakup
experimental data (Hao Zhou (2017, 2019)\textsuperscript{9,21}, Han Zhou
(2019)\textsuperscript{8,10}, Han Zhou (2021)\textsuperscript{13})

To further verify the generalizability of the breakup model constructed
in the present work, experimental data of drop breakup frequency which
are obtained by single drop breakup experiments were chosen. Andersson
and Andersson \textsuperscript{6} measured the drop breakup frequency of
the kerosene-water system and three valid data points were provided in
the reported literature. This work was carried out in a static mixer,
the maximum scale of the local turbulent structure \emph{L} was taken as
the local flow channel width, \emph{L} = 3 mm according to the
information provided by Bouaifi et al. \textsuperscript{31} Ashar et
al.\textsuperscript{3} measured the drop breakup frequency of the
rapeseed oil-water system in a rotor-stator mixer, and \emph{L} was also
taken as the local flow channel width (stator opening width), \emph{L} =
6 mm. Maa\selectlanguage{ngerman}ß and Kraume\textsuperscript{1} determined the drop breakup
probability of the toluene-water and petroleum-water systems in a mimic
stirred tank, the value of \emph{L} is taken as the blade
height,\emph{L} = 14.5 mm. It should be noted that the value of the
turbulent energy dissipation rate was not directly given in their
original work, but only the apparent fluid flow speed in the channel was
provided as 1.5 m/s. According to the work of Maaß et
al.\textsuperscript{32}, the blade tip speed \emph{v\textsubscript{tip}}
in a stirred tank corresponding to this speed is
\emph{v\textsubscript{tip}} = 1.5/0.7 m/s = 2.14 m/s. Accordingly, the
turbulent energy dissipation rate in the vicinity of the impeller can be
calculated as \textsuperscript{33,34}:\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image43/image43}

\end{center}
\end{figure}

Figure 2 Comparison of the predicted results with experimental data in
the literature\textsuperscript{1,3,6,35}

Combining the above parameters with the physical data provided in the
literature, we nondimensionalized the experimental data and plotted them
in Figure 2. It can be seen that the predicted results of the model are
in good agreement with the experimental data of Andersson and Maa\selectlanguage{ngerman}ß,
which further validates the accuracy and generalizability of the
theoretical model. However, Figure 2 shows that the predicted results of
the model are in relatively poor agreement with the experimental data of
Ashar et al., especially at the lower \emph{We\textsubscript{L}} , which
we analyze as a result of the higher drop viscosity of the experimental
system in Ashar et al. Moreover, the high spatial inhomogeneity of the
turbulent energy dissipation rate in their experimental device can also
cause deviations between the model predictions and experimental results.

\subsubsection*{4.2 Comparison of drop breakup frequency
models}

{\label{comparison-of-drop-breakup-frequency-models}}

As is discussed in Section 2, most breakup frequency models in the
literature are based on limited modeling frameworks.
Particularly\selectlanguage{english},frameworks based on the eddy-particle collision and based
on the statistical description are mostly adopted by researchers.
Breakup frequency models proposed by Coulaloglou and
Tavlarides\textsuperscript{18} (CT model, as shown in Eq.) and proposed
by Luo and Svendsen\textsuperscript{15} (Luo model, as shown in Eq.) are
the most representative and well-known models under the above two
frameworks, respectively.

CT model:

\par\null

Luo model:

\par\null

In such a scenario, the two models are also analyzed in the present
work. To intuitively compare the experimental data of drop breakup
frequency obtained by different researchers, the method recommended by
Lehr et al. \textsuperscript{36} to dimensionless the drop size and drop
breakup frequency is introduced, that is:

Where \emph{L} \textsuperscript{*} and \emph{T} \textsuperscript{*} have
length and time dimensions respectively:

\par\null\par\null

In the original model of Coulaloglou and Tavlarides\textsuperscript{18},
the drop breakup frequency depends on the dispersed phase density, that
is, the density \selectlanguage{greek}\emph{ρ\textsubscript{f}}\selectlanguage{english} in Eq. and Eq. takes the value
of the dispersed phase density \selectlanguage{greek}\emph{ρ\textsubscript{d}}\selectlanguage{english} . Firstly, we
compared the predicted results of the CT model with the experimental
data, as shown in Figure 3a. It can be seen that by changing the
specific values of the adjustable parameters, the original CT model can
conform to the experimental data under different experimental
facilities, but the accurate prediction of the breakup frequency under
different facilities cannot be achieved by a fixed combination of the
parameters. From the original literature of Coulaloglou and Tavlarides,
it can be seen that the density term comes from the estimation of the
turbulent stress. According to the current cognition of most
researchers, the magnitude of the turbulent stress should be related to
the continuous phase density, which means that the dispersed phase
density in the CT model should be replaced by the continuous phase
density, and thus, the density \selectlanguage{greek}\emph{ρ\textsubscript{f}}\selectlanguage{english} in Eq. and Eq.
is taken as the continuous phase density \selectlanguage{greek}\emph{ρ\textsubscript{c}}\selectlanguage{english} in
the following analysis. And then, we compare the predicted results of
the density-corrected CT model with the experimental data, and the
result is shown in Figure 3b. It can be seen that by adjusting the
adjustable parameters combination, the modified CT model can better
match the experimental data of different researchers relative to the
original model. Also, we plot the predicted results of the Luo model in
Figure 3b, and it shows that the Luo model overestimates the actual
magnitude of the breakup frequency.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image50/image50}

\end{center}
\end{figure}

Figure 3 Comparison of the predicted results with experimental data. (a)
based on the dispersed phase density; (b) based on the continuous phase
density

For the CT model, the adjustable parameters can be optimized to fit the
experimental data of different researchers, while the predictive
capability of the CT model is still not comparable to the breakup
frequency model constructed in this study. Moreover, it should be noted
that the breakup time in the CT model is assumed to be linearly related
to the eddy turnover time, . As is discussed in our previous
study\textsuperscript{13,24}, this is only applicable in the case
of\emph{We\textsubscript{L}} [?] 1, which is difficult to be satisfied for
most liquid-liquid dispersion systems under normal operating conditions.

\subsubsection*{4.3 Sensitivity analysis of model
parameters}

{\label{sensitivity-analysis-of-model-parameters}}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image52/image52}

\end{center}
\end{figure}

Figure 4 Sensitivity analysis of model parameters. (a) prediction
results of the breakup probability versus \emph{We\textsubscript{L}} ;
(b) prediction results of the breakup probability versus

In the constructed breakup model, \emph{L} characterizes the maximum
scale of the local turbulence structure and is directly related to the
parameters of the apparatus. For practical liquid-liquid dispersion
equipment, \emph{L} values as the minimum scale of the local runner. For
example, in a pulsed disc and doughnut column extraction column, the
value is taken as the distance between plates. For mixing equipment,
since drop breakup occurs mainly in the region of the blade
runner,\emph{L} is taken as the blade height. For static mixers,
\emph{L} is counted as the runner width of the minimum mixing unit, and
so forth. Based on the above criteria, analysis in Section 4.1 shows
that the predicted results of the model are in good agreement with the
experimental data, which indicates the accuracy and applicability of the
criteria valuing \emph{L} . However, due to the diversity and complexity
of the practical apparatus, the minimum scale of the local runner may
deviate from the maximum scale of the local turbulence structure,
thereby influencing the predictive accuracy of the proposed breakup
model. In such a scenario, the sensitivity of the value of \emph{L} to
the predicted breakup probability is analyzed in this section, as shown
in Figure 4. Herein, denotes the value of the maximum scale of the local
turbulence structure and \emph{L} denotes the actual value used for the
model calculation. Figure 4a shows the prediction results of the breakup
probability versus \emph{We\textsubscript{L}} , it indicates that the
larger the value of , the larger the predicted result of the breakup
probability, which is because the larger the value of means the higher
the turbulent stress and thus the drop is more prone to break up.
Besides, Figure 4b shows that the deviation of the predicted results of
the breakup probability model is approximately within \selectlanguage{ngerman}±30\% when the
value of fluctuates between 0.5 and 2.0.

\subsubsection*{4.4 Influence of the drop diameter on drop breakup
frequency}

{\label{influence-of-the-drop-diameter-on-drop-breakup-frequency}}

Figure 5 shows the calculated drop breakup frequency versus drop
diameter. It indicates that the drop breakup frequency firstly increases
with the increase of the drop diameter and then reaches it's maximum,
further increase the drop diameter will decrease the predicted drop
breakup frequency. The definition of the breakup frequency (see Eq. in
Section 2) indicates that it is the product of the reciprocal of the
breakup time and the breakup probability. As both the drop breakup
probability and breakup time increase with the increase of drop diameter
when the drop diameter is relatively small, with the increase of the
drop diameter, the breakup probability increases more quickly than the
breakup time, and the overall performance is that drop breakup frequency
increases with the increase of drop diameter. However, when the drop
diameter exceeds a certain value, the effect of drop diameter on the
breakup time is more significant than that of the breakup probability,
and then the performance is that the breakup frequency gradually
decreases with the increase of drop diameter. Moreover, Figure 5 shows
that as the turbulent energy dissipation rate increases or the
interfacial tension decreases, the peak value of the breakup frequency
gradually increases and moves toward smaller drop diameters, indicating
that higher turbulent kinetic energy dissipation rate and lower
interfacial tension are more favorable for the drop to break up, which
is consistent with the experimental conclusions in our previous
studies\textsuperscript{8,9}.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image58/image58}

\end{center}
\end{figure}

Figure 5 Influence of the drop diameter on drop breakup frequency. (a)
with different turbulent energy dissipation rates, \selectlanguage{greek}\emph{σ} \selectlanguage{english}= 35 mN/m;
(b) with different interfacial tensions, \selectlanguage{greek}\emph{ε} \selectlanguage{english}= 3
m\textsuperscript{2}s\textsuperscript{-3}

The shading regions in Figure 5 are where the drop breakup frequency can
be accurately predicted by the proposed model in this study. It is noted
that the breakup model constructed in the present work is based on the
binary breakup assumption and the breakup time model adopted in this
study is also valid for the breakup process with a small number of
fragments generated (generally, no more than 4 fragments). According to
our previous study\textsuperscript{24}, the predicted results of the
breakup time are in good agreement with the experimental data
when\emph{We\textsubscript{L}} is lower than 0.7 (corresponding to a
breakup probability of approximately 0.5). Therefore, we can
approximate\emph{We\textsubscript{L}} = 0.7 as the criterion,
when\emph{We\textsubscript{L}} \textless{} 0.7, the model can accurately
predict the breakup frequency (corresponding to the shaded regions), and
when \emph{We\textsubscript{L}} \textgreater{} 0.7, the predictive
accuracy of the breakup frequency prediction will be limited, and the
effect of multiple breakups needs to be quantified. Noteworthy,
according to our previous experimental
results\textsuperscript{8--10,12,21}, drop breakups are dominated by the
low-fragment breakups (especially, the proportion of the binary breakup
is more than 50\%, and the breakups with the fragments number lower than
4 take the proportion of more than 90\%) under steady-state operating
conditions. Thus, the breakup model constructed in this study can be
accurately applied to predict the drop breakup frequency in the
practical apparatus.

\subsubsection*{4.5 Influence of the turbulent energy dissipation rate on
drop breakup
frequency}

{\label{influence-of-the-turbulent-energy-dissipation-rate-on-drop-breakup-frequency}}

The turbulent energy dissipation rate is one of the most important
parameters affecting the drop breakup frequency. The larger the value of
the turbulent energy dissipation rate, the more favorable the energy
transfer from the eddies to the drop surface. Figure 6 showed that the
drop breakup frequency gradually increases with the increase of the
turbulent energy dissipation rate, which is consistent with the above
analysis. Moreover, Figure 6a indicates that the breakup frequency
versus the turbulent energy dissipation rate varies greatly for
different drop diameters. The intersections of the gray straight line
with each line in Figure 6a characterize values of drop breakup
frequency at \emph{We\textsubscript{L}} = 0.7 for each case, and region
below the intersection (or the shading region in Figure 6b) represents
the applicable scope of the breakup model. Meanwhile, for the upper side
of the intersection, the effect of the multiple breakups needs to be
considered.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image59/image59}

\end{center}
\end{figure}

Figure 6 Influence of the turbulent energy dissipation rate on drop
breakup frequency. (a) with different drop sizes, \selectlanguage{greek}\emph{σ} \selectlanguage{english}= 35 mN/m;
(b) with different interfacial tensions, \emph{d} = 1 mm

\subsubsection*{4.6 Influence of the interfacial tension on drop breakup
frequency}

{\label{influence-of-the-interfacial-tension-on-drop-breakup-frequency}}

Figure 7 shows the drop breakup frequency versus the interfacial
tension. It can be seen that drop breakup frequency increases with
increasing the interfacial tension at very small interfacial tensions,
implying that the effect of the interfacial tension on drop breakup time
is more significant compared with the effect on drop breakup
probability. As the interfacial tension exceeds a critical value, Figure
7 shows that the breakup frequency decreases with the increase of the
interfacial tension, implying that the effect of interfacial tension on
drop breakup probability is more significant at this moment. The lower
side of the intersection of the gray line with each line in Figure 7a
(or the shading region in Figure 7b) also represents the accurate
prediction ranges of the breakup model\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image60/image60}

\end{center}
\end{figure}

Figure 7 Influence of the interfacial tension on drop breakup frequency.
(a) with different drop sizes, \selectlanguage{greek}\emph{ε} \selectlanguage{english}= 3
m\textsuperscript{2}s\textsuperscript{-3}; (b) with different turbulent
energy dissipation rates, \emph{d} = 1 mm

\subsubsection*{4.7 Extension of drop breakup frequency model by
considering the effect of drop
viscosity}

{\label{extension-of-drop-breakup-frequency-model-by-considering-the-effect-of-drop-viscosity}}

For systems with high viscosity, the effect of drop viscosity on drop
breakup behaviors needs to be considered.\textsuperscript{10,37} The
breakup probability and breakup frequency model constructed in Section 3
is only applicable to describe the breakup behavior of low-viscous
drops, thus it is necessary to extend the model by considering the
effect of drop viscosity. According to the drop oscillation breakup
mechanism, the influence of the drop viscosity is mainly reflected in
two aspects:

1) The viscous hamper increases with the increase of drop viscosity,
which is manifested by the decrease of the oscillation amplitude with
increasing the viscosity and thus leads to the increase of drop breakup
time. The breakup time considering the effect of drop viscosity can be
modeled by Eq. in Section 2.

2) The occurrence of the drop breakup should satisfy the energy
constraint, in addition to overcoming the increase in the maximum
interfacial energy during the breakup process, it is necessary to
consider the effect of the increase in the drop viscous energy.

For a droplet with a size of \emph{d} , the viscous energy it contains
can be expressed as:

Where \emph{p\textsubscript{vis}} is the viscous energy per unit drop
volume and can be calculated by Eq.:

\par\null

Therefore, the increase in viscous energy during the drop breakup
process is expressed as:

Based on the above analysis, the breakup constraint is expressed as Eq.:

\par\null

Then it is obtained that:

\par\null

Thus, we obtain the extension of the drop breakup probability model by
considering the effect of drop viscosity, as shown in Eq..

\par\null

Where \emph{Oh} is the Ohnesorge number, which characterizes the
relative magnitude of the viscous and interfacial stresses:

Combined with the breakup time model of Eq. in Section 2, the extended
breakup frequency model can be expressed as:

\par\null

In our previous work\textsuperscript{10}, we investigated the effect of
dispersed phase viscosity on drop breakup frequency and obtained the
direct experimental data, which is applied to verify the accuracy of
Eq.. Figure 8 shows the comparison of the predicted results with the
initial breakup experimental data. Compared with the breakup model of
Eq. which is constructed for low-viscous drops, Eq. is in better
agreement with the experimental data. Moreover, Figure 8 shows that the
drop breakup probability gradually decreases as \emph{Oh} increases.

Based on the analysis in the above sections, we can conclude that for
low-viscous drops, drop breakup probability is mainly determined by the
Weber number (\emph{We\textsubscript{L}} ), and for intermediate to high
viscous drops, the breakup probability is determined by the combined
influence of the Weber number (\emph{We\textsubscript{L}} ) and the
Ohnesorge number (\emph{Oh} ).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image69/image69}

\end{center}
\end{figure}

Figure 8 Comparison of the predicted results with the initial breakup
experimental data, \emph{c\textsubscript{vis}} = 3.0

\subsection*{5 Conclusion}

{\label{conclusion}}

The construction of the accurate drop breakup model is crucial for the
application of the population balance model and the study of
liquid-liquid dispersion characteristics. Based on the drop oscillation
breakup mechanism, in the present work, we proposed a theoretical model
of drop breakup probability by adopting the three-dimensional Maxwell
velocity distribution. The model considers the breakup constraints of
both the maximum increase of the interfacial energy and the increase of
drop viscous energy during the breakup process. The results show that
for low-viscous drops, drop breakup probability is only related to the
Weber number (\emph{We\textsubscript{L}} ), which characterizes the
relative magnitude of the external turbulent stress and the interfacial
stress. When the drop viscosity exceeds a certain limit, the effect of
drop viscosity needs to be considered and the Ohnesorge number
(\emph{Oh} ), which characterizes the relative magnitude of the viscous
stress of the drop and the interfacial stress, is introduced.

By combining the theoretical model of drop breakup time constructed in
our previous work, the breakup frequency model is obtained based on the
statistical description framework. The accuracy and generality of the
model were then validated using the direct experimental data. Moreover,
we systematically analyzed the effects of the drop diameter, turbulent
energy dissipation rate, and interfacial tension on the predicted drop
breakup frequency. Results showed that the breakup frequency increases
monotonically with increasing turbulent energy dissipation rate.
Differently, with the increase of the drop diameter or the interfacial
tension, the breakup frequency firstly increases and then decreases. The
reason for the above results is related to the coupling effect of the
parameters on the breakup time and breakup probability. The results of
this study are applicable to predict drop breakup frequency for drop
breakups with low fragments generated. Combined with the results in our
previous studies, the proportion of the breakups with the number of
fragments lower than 4 accounts for more than 90\% under steady-state
operating conditions. Therefore, the model constructed in the present
work can directly serve for the quantitative characterization of drop
breakup behaviors under steady-state operating conditions.

\subsection*{Acknowledgments}

{\label{acknowledgments}}

We gratefully acknowledge the support of the National Natural Science
Foundation of China (21776151, 21576147).

\subsection*{Notation}

{\label{notation}}

\emph{c\textsubscript{f}} coefficient of surface area,

\emph{d} drop diameter, m

\emph{D} the diameter of the impeller, m

\emph{E} (\emph{k} ) the energy spectrum of turbulence,
m\textsuperscript{3}/s\textsuperscript{2}

the energy constraint, J

\emph{k} the wavenumber of the turbulence, 1/m

\emph{L} the maximum scale of the local turbulent structure, m

\emph{N} the rotating speed of the impeller, 1/s

\emph{N} (\emph{d} ) the number of mother drops

\emph{Oh} the Ohnesorge number

\emph{p\textsubscript{os}} the average oscillation kinetic energy per
unit drop volume, J/m\textsuperscript{3}

\emph{p\textsubscript{t}} the turbulent stress, Pa

\emph{p\textsubscript{vis}} the viscous energy per unit drop volume,
J/m\textsuperscript{3}

\emph{P\textsubscript{b}} (\emph{d} ) drop breakup probability

\emph{t\textsubscript{b}} drop breakup time, s

the critical oscillation velocity, m/s

the root-mean-square velocity of the surface oscillation, m/s

\emph{We\textsubscript{L}} the Weber number

Greek letter

drop breakup frequency, 1/s

the interfacial energy increase before and after drop breakup, J

the maximum interfacial energy increase during the drop breakup, J

the increase in viscous energy during the drop breakup process, J

the number of the broken drops

turbulent energy dissipation rate,
m\textsuperscript{2}/s\textsuperscript{3}

eddy size, m

\selectlanguage{greek}\emph{μ\textsubscript{d}}\selectlanguage{english} dispersed phase viscosity, Pa s

continuous phase density, kg/m\textsuperscript{3}

dispersed phase density, kg/m\textsuperscript{3}

\selectlanguage{greek}\emph{σ} \selectlanguage{english}interfacial tension, N/m

dispersed phase volume fraction

collision frequency between eddies and drops, 1/(m\textsuperscript{4}s)

the size ratio between an eddy and a droplet

\subsection*{Reference}

{\label{reference}}

1. Maa\selectlanguage{ngerman}ß S, Kraume M. Determination of breakage rates using single drop
experiments. \emph{Chem Eng Sci} . 2012;70:146-164.

2. Solsvik J, Jakobsen HA. Single drop breakup experiments in stirred
liquid--liquid tank. \emph{Chem Eng Sci} . 2015;131:219-234.

3. Ashar M, Arlov D, Carlsson F, Innings F, Andersson R. Single droplet
breakup in a rotor-stator mixer. \emph{Chem Eng Sci} . 2018;181:186-198.

4. Herø EH, La Forgia N, Solsvik J, Jakobsen HA. Single drop breakage in
turbulent flow: Statistical data analysis. \emph{Chem Eng Sci X} .
2020;8:100082.

5. Vankova N, Tcholakova S, Denkov ND, Vulchev VD, Danner T.
Emulsification in turbulent flow: 2. Breakage rate constants. \emph{J
Colloid Interface Sci} . 2007;313(2):612-629.

6. Andersson R, Andersson B. On the breakup of fluid particles in
turbulent flows. \emph{AIChE J} . 2006;52(6):2020-2030.

7. Galinat S, Masbernat O, Guiraud P, Dalmazzone C, No\selectlanguage{polish}ı\selectlanguage{ngerman}¨k C. Drop
break-up in turbulent pipe flow downstream of a restriction. \emph{Chem
Eng Sci} . 2005;60(23):6511-6528.

8. Zhou H, Yu X, Jing S, Zhou H, Lan W, Li S. Measurement of droplet
breakage in a pump-mixer. \emph{Chem Eng Sci} . 2019;195:23-38.

9. Zhou H, Jing S, Yu X, Zhou H, Lan W, Li S. Study of droplet breakage
in a pulsed disc and doughnut column-Part I: Experiments and
correlations. \emph{Chem Eng Sci} . 2019;197:172-183.

10. Zhou H, Yang J, Jing S, Lan W, Zheng Q, Li S. Influence of
Dispersed-Phase Viscosity on Droplet Breakup in a Continuous
Pump-Mixer.\emph{Ind Eng Chem Res} . 2019;58(51):23458-23467.

11. Liu H, Jing S, Fang Q, Li S. Droplet Breakup in a Square-Sectioned
Pulsed Disc and Doughnut Column. \emph{Ind Eng Chem Res} .
2016;55(7):2242-2251.

12. Fang Q, Jing D, Zhou H, Li S. Population balance of droplets in a
pulsed disc and doughnut column with wettable internals. \emph{Chem Eng
Sci} . 2017;161:274-287.

13. Zhou H, Yu X, Wang B, Jing S, Lan W, Li S. Experimental study on
drop breakup time and breakup rate with drop swarms in a stirred
tank.\emph{AIChE J} . 2021;67(1):e17065.

14. Tsouris C, Tavlarides LL. Breakage and coalescence models for drops
in turbulent dispersions. \emph{AIChE J} . 1994;40(3):395-406.

15. Luo H, Svendsen HF. Theoretical model for drop and bubble breakup in
turbulent dispersions. \emph{AIChE J} . 1996;42(5):1225-1233.

16. Wang T, Wang J, Jin Y. A novel theoretical breakup kernel function
for bubbles/droplets in a turbulent flow. \emph{Chem Eng Sci} .
2003;58(20):4629-4637.

17. Zhao H, Ge W. A theoretical bubble breakup model for slurry beds or
three-phase fluidized beds under high pressure. \emph{Chem Eng Sci} .
2007;62(1):109-115.

18. Coulaloglou CA, Tavlarides LL. Description of interaction processes
in agitated liquid-liquid dispersions. \emph{Chem Eng Sci} .
1977;32(11):1289-1297.

19. Han L, Luo H, Liu Y. A theoretical model for droplet breakup in
turbulent dispersions. \emph{Chem Eng Sci} . 2011;66(4):766-776.

20. Solsvik J, Skjervold VT, Han L, Luo H, Jakobsen HA. A theoretical
study on drop breakup modeling in turbulent flows: The inertial subrange
versus the entire spectrum of isotropic turbulence. \emph{Chem Eng Sci}
. 2016;149:249-265.

21. Zhou H, Jing S, Fang Q, Li S, Lan W. Direct measurement of droplet
breakage in a pulsed disc and doughnut column. \emph{AIChE J} .
2017;63(9):4188-4200.

22. Wang B, Zhou H, Yu X, et al. Determination of dynamic interfacial
tension in a pulsed column under mass transfer condition. \emph{AIChE J}
. 2020;66(8):e16257.

23. Prince MJ, Blanch HW. Bubble coalescence and break\selectlanguage{english}-up in air\selectlanguage{english}-sparged
bubble columns. \emph{AIChE J} . 1990;36(10):1485-1499.

24. Zhou H, Yu X, Wang B, Jing S, Lan W, Li S. Modeling study on drop
breakup time in turbulent dispersions. \emph{Chem Eng Sci} .
2021;238:116599.

25. Kolmogorov AN. The local structure of turbulence in incompressible
viscous fluid for very large Reynolds numbers. \emph{Cr Acad Sci URSS} .
1941;30:301-305.

26. Brodkey RS. \emph{The Phenomena of Fluid Motions} . Courier
Corporation; 1995.

27. Sreenivasan KR. On the scaling of the turbulence energy dissipation
rate. \emph{Phys Fluids} . 1984;27(5):1048-1051.

28. Batchelor GK. \emph{The Theory of Homogeneous Turbulence} .
Cambridge University Press; 1956.

29. Hesketh RP, Etchells AW, Russell TWF. Experimental observations of
bubble breakage in turbulent flow. \emph{Ind Eng Chem Res} .
1991;30(5):835-841.

30. Andersson R, Helmi A. Computational fluid dynamics simulation of
fluid particle fragmentation in turbulent flows. \emph{Appl Math Model}
. 2014;38(17):4186-4196.

31. Bouaifi M, Mortensen M, Andersson R, et al. Experimental and
Numerical Investigations of a Jet Mixing in a Multifunctional Channel
Reactor: Passive and Reactive Systems. \emph{Chem Eng Res Des} .
2004;82(2):274-283.

32. Maass S, Gabler A, Zaccone A, Paschedag AR, Kraume M. Experimental
Investigations and Modelling of Breakage Phenomena in Stirred
Liquid/Liquid Systems. \emph{Chem Eng Res Des} . 2007;85(5):703-709.

33. Placek J, Tavlarides LL, Smith GW, Fo\selectlanguage{polish}ř\selectlanguage{english}t I. Turbulent flow in stirred
tanks. Part II: A two-scale model of turbulence. \emph{AIChE J} .
1986;32(11):1771-1786.

34. Placek J, Tavlarides LL. Turbulent flow in stirred tanks. Part I:
Turbulent flow in the turbine impeller region. \emph{AIChE J} .
1985;31(7):1113-1120.

35. Karimi M, Andersson R. An exploratory study on fluid particles
breakup rate models for the entire spectrum of turbulent
energy.\emph{Chem Eng Sci} . 2018;192:850-863.

36. Lehr F, Millies M, Mewes D. Bubble-Size distributions and flow
fields in bubble columns. \emph{AIChE J} . 2002;48(11):2426-2443.

37. Hert SCD, Rodgers TL. On the steady-state drop size distribution in
stirred vessels. Part I: Effect of dispersed phase viscosity.\emph{AIChE
J} . 2018;64(9):3293-3302.

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