*Corresponding authors.
E-mail addresses: s-jing@tsinghua.edu.cn (S. Jing ),
lsw@tsinghua.edu.cn (S. Li ).
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
(WeL ), and for intermediate or high viscous
drops, the breakup probability is determined by the combined influence
of the Weber number (WeL ) and the Ohnesorge
number (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.
Keywords: Oscillation; breakup model; Maxwell velocity
distribution; turbulent energy dissipation; interfacial tension
1 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
groups1–7. 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.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 work13, 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.
2 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-shaped14,15,
M-shaped16,17, and inverted U-shaped
distributions18–20 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 data8,9,21,22. 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 Blanch23 gives an expression in the integral form based on
the turbulent eddy size, then Luo and Svendsen15introduced the impact of the volume fraction of fragments and proposed
the modeling framework in the double integral form, as is expressed in
Eq..
Where is the collision frequency between eddies and drops, is the
breakup efficiency for a mother drop with size 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..
Where is the drop breakup time, is the breakup probability of a drop
with a diameter of d , and are the number of breakup events and
total number of drops respectively. This framework was firstly proposed
by Coulaloglou and Tavlarides18 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
researchers5,18, starting from the eddy turnover time,
assume that . According to our recent study24, 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
experimental13 and theoretical
studies24 and proposed a novel breakup time model
based on the drop oscillation breakup mechanism:
Where c1 and c2 are
constant parameters, c1 = 14.0 andc2 = 8.0. WeL has a form
of the Weber number and characterizes the relative magnitude of
turbulent stress and interfacial stress, WeL is
expressed as Eq..
Where L is the maximum scale of the local turbulent structure. In
practical calculations, L values by the minimum length-scale of
the local runner24.
For low-viscous drops, the influence of the drop viscosity on drop
breakup time is insignificant, thus, Eq. can be simplified to Eq.:
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.
3 Model development
3.1 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
research13, 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.25 For devices such as stirred tanks
and extraction columns, experimental results also indicate that the
turbulent structure is locally isotropic.26–28Therefore, 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.8,9,21,29 However, as
the drop size deviates from the equilibrium size, the proportion of the
binary breakup decreases accordingly. 2,3,9 According
to our previous studies 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 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:
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:
Solve Eq. and we obtain Eq.:
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:
Where is the energy spectrum andk is the wavenumber of the
turbulence. Kolmogorov’s hypotheses give the universal form of the
energy spectrum in the inertial subrange:
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..
Where L is the maximum scale of the local turbulent structure andd is the drop diameter. α 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..
Taking , then we obtained the expression for the root-mean-square
velocity ：
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
Helmi30, 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.30 In present work, we use
the maximum interfacial energy increase during the breakup process as
the energy constraint, that is:
Meanwhile, we have:
The parameter of c characterizes the relative magnitude of the
energy potential of the breakup process, According to Andersson and
Helmi30, 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 studies8,9, 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 cvalues as 0.39. For practical drop breakup processes, drops are not
ideally equal breakup, thereby leading to a smaller value of cthan 0.39. By comparing with the experimental data, we find thatc 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:
Combining Eq., Eq., and Eq., the expression for the drop breakup
probability is obtained as Eq..
Where WeL is the same with Eq..
3.2 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..
4 Results and discussion
4.1 Validation
Recently, our group measures the drop breakup frequency in a pulsed disc
and doughnut column9,21 and in a
pump-mixer8,10,13. 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. TheWeL 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 ofL , which characterizes the maximum scale of the local turbulent
structure. L depends on equipment parameters and can be valued by
the minimum length-scale of the local runner24. For
the pulsed disc and doughnut column used by Zhou et al.9,21, L values as the distance between the
adjacent disc and doughnut plate, L = 30 mm. For the stirred
tank, L values as the height of the blade, that is L = 10
mm (pump-impeller in Zhou et al.8,10) and L =
8.8 mm (Rushton Turbine impeller in Zhou et al.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.