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\begin{document}
\title{Effect of operation regime on bubble size and void fraction in a bubble
column with porous sparger}
\author[1]{Shahrouz Mohagheghian}%
\author[1]{Afshin J. Ghajar}%
\author[1]{Brian Elbing}%
\affil[1]{Oklahoma State University System}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\selectlanguage{english}
\begin{abstract}
Performance of bubble columns under transport processes is dependent on
bubble size distribution and void fraction. These multiphase parameters
are sensitive to the operation regime of a bubble column. The current
work presents a systematic study of bubble size and void fraction in a
batch bubble column within the homogeneous and heterogeneous regimes.
Effect of liquid viscosity and gas superficial velocity on bubble size
distribution, void fraction, and operation regime was investigated.
Results showed that increasing the viscosity accelerates the regime
transition. Bubble size distributions were statistically characterized
using probability density function and probability plots. It was shown
that bubble size distribution shifts from near-Gaussian in the
homogenous regime to lognormal (in parts) in the heterogeneous regime.
Dimensional reasoning was used to scale the bubble size and void
fraction with respect to the operation regime.%
\end{abstract}%
\sloppy
\emph{Transport Phenomena and Fluid Mechanics}
\textbf{Effect of operation regime on bubble size and void fraction in a
bubble column with porous sparger}
\textbf{Shahrouz Mohagheghian} \textsuperscript{a,}11Author to whom
correspondence should be addressed, \textbf{Afshin J. Ghajar}
\textsuperscript{b}, \& \textbf{Brian R. Elbing} \textsuperscript{c}
School of Mechanical and Aerospace Engineering, Oklahoma State
University
201 General Academic Building, Oklahoma State University, Stillwater, OK
74078
\textsuperscript{a}mohaghe@okstate.edu,\textsuperscript{b}afshin.ghajar@okstate.edu,
\&\textsuperscript{c}elbing@okstate.edu
\emph{Submitted to the AIChE Journal}
\section*{Abstract}
{\label{abstract}}
Performance of bubble columns under transport processes is dependent on
bubble size distribution and void fraction. These multiphase parameters
are sensitive to the operation regime of a bubble column. The current
work presents a systematic study of bubble size and void fraction in a
batch bubble column within the homogeneous and heterogeneous regimes.
Effect of liquid viscosity and gas superficial velocity on bubble size
distribution, void fraction, and operation regime was investigated.
Results showed that increasing the viscosity accelerates the regime
transition. Bubble size distributions were statistically characterized
using probability density function and probability plots. It was shown
that bubble size distribution shifts from near-Gaussian in the
homogenous regime to lognormal (in parts) in the heterogeneous regime.
Dimensional reasoning was used to scale the bubble size and void
fraction with respect to the operation regime.
\textbf{Keywords} : Operation Regime, Bubble Size Distribution, Bubble
Size Scaling, Void Fraction Scaling, Liquid Viscosity.
\section*{Introduction}
{\label{introduction}}
Bubble columns are commonly used as contact reactors in chemical
processing, bio-chemical, and metallurgical applications due to their
simplicity (e.g., no moving parts), low operation cost, and high
efficiency at heat and mass transfer. Understanding and modeling the
transport phenomena as well as hydrodynamics of bubble columns requires
a fundamental understanding of characteristics of the dispersed (gas)
phase (i.e. bubbles). Bubble size (\emph{d\textsubscript{b}} ),
population, and rise velocity (\emph{U\textsubscript{b}} ) significantly
influence the physical behavior of the bubbly
flow.\textsuperscript{1}Bubble size distribution (BSD) is a primary
aspect in the understanding of the physical behavior of the multiphase
flow and was studied in this work. Note that the bubble rise velocity is
a function of bubble size; therefore, any factor that effects the bubble
size effects the rise velocity, which in turn effects the void fraction
(\selectlanguage{greek}\emph{ε} \selectlanguage{english}). Both bubble size and void fraction are impacted by gas
superficial velocity, liquid properties, bubble column operation
condition, column geometry, and gas injection method. Current work
studies the effect of gas superficial velocity and liquid viscosity on
bubble size and void fraction.
Shah et al.\textsuperscript{2} showed that the void fraction is
predominately a function of the gas superficial velocity. The study of
bubble columns with different system characteristics showed that there
is a direct correlation between gas superficial velocity and void
fraction.\textsuperscript{3-11} Lockett and
Kirkpatrick\textsuperscript{12} and Kara et
al.\textsuperscript{13}showed that in the homogenous regime, void
fraction exhibits a linear increase with increasing gas superficial
velocity. However, in the heterogeneous regime the functional form
between gas superficial velocity and void fraction is less
apparent.\textsuperscript{13,14}Liquid properties effect the void
fraction by influencing the bubble formation as well as coalescence and
breakup processes.\textsuperscript{1} The bubble column literature
reports both increasing and decreasing void fraction with increasing
liquid viscosity.\textsuperscript{15-21} Besgni et
al.\textsuperscript{22}argues that viscosity has a dual effect on void
fraction. At low liquid viscosity, the coalescence is limited and
increasing the viscosity increases the drag force acting on bubbles and,
in turn, increases the bubble residence time and void fraction. However,
in more viscous liquids, viscosity increases the coalescence rate and,
consequently, produces larger bubbles with higher terminal velocity that
decrease the void fraction. Bubble column literature provides numerous
correlations for the prediction of the void fraction. Interested readers
are referred to Besagni et al.\textsuperscript{23} for a summary of
available correlations. Akita and Yoshida\textsuperscript{24} proposed a
well-known correlation for void fraction scaling based on dimensional
analysis. Their work suggests that the Froude number (\emph{Fr} ),
Archimedes number (\emph{Ar} ), and E\selectlanguage{ngerman}ötvös number (\emph{Eo} ) scale the
void fraction with a power law functional form, \selectlanguage{greek}\emph{ε/(1-
ε)\textsuperscript{4} =
CFr\textsuperscript{Χ}Ar\textsuperscript{Ψ}Eo\textsuperscript{Ω}} .
\selectlanguage{english}Here\emph{C} is a proportionality constant and \selectlanguage{greek}\emph{Χ,Ψ,Ω} \selectlanguage{english}are the
powers of each non-dimensional term. Similar functional forms are
reported in the bubble column literature.\textsuperscript{16,25-28}
Akita and Yoshida\textsuperscript{24} used the column diameter as a
characteristic length scale to calculate the aforementioned
dimensionless terms; however, in the present study using the bubble size
as the characteristic length scale seems more appropriate since the
bubble size is much smaller than the column diameter.
There is a general scarcity in bubble size data reported in the bubble
column literature, partly because of the difficulties associated with
bubble size measurements. While Leonard et
al.\textsuperscript{29}outline the inconsistencies in the bubble size
distribution literature, there is a general consensus that in the
homogenous regime the bubble sizes increase with increasing the gas
superficial velocity while in the heterogeneous regime bubble size
decreases with increasing the gas superficial velocity. Li and
Prakash\textsuperscript{30} studied the spatial distribution of bubbles
and found that smaller bubbles dominate the near wall region, and larger
bubbles are more common in the central region of the column. In a highly
viscous liquid, the bubble surface is more stable, larger bubbles form
at the injector,\textsuperscript{31,32}and the coalescence rate is
larger than the breakage rate.\textsuperscript{2,33-35} The study of
bubble size distribution shows that in viscous liquids the probability
density function (PDF) of the BSD exhibits a bimodal
shape.\textsuperscript{15,21,36,37} In the bubble column literature,
scaling of the characteristic bubble length has been broadly approached
assuming the sizing is dominated by either a breakage
mechanism\textsuperscript{38} or bubble
formation.\textsuperscript{39,40} The former attempts to find a stable
bubble size under a given external (breakage) force in the heterogeneous
regime, and the latter aims to find a characteristic bubble length scale
in the homogenous regime using gravity, surface tension, and shear
forces acting on a bubble.
The goal of the current work is to study the bubble size and void
fraction in a batch bubble column with respect to operation regime and
contribute to the current understanding of these multiphase parameters.
This paper is organized as follows. Section 2~describes the experimental
setup including instrumentation used. In~Section 3,~the results are
presented for characterization and scaling of the bubble size and void
fraction. Finally, conclusions and remarks on the current work are given
in~Section 4.
\section*{Experimental Methods}
{\label{experimental-methods}}
\subsection*{2.1 Bubble column}
{\label{bubble-column}}
The bubble column was made from cast acrylic to achieve strength and
optical clarity; it was 1.2m in length with a 102mm internal diameter
(\emph{D} ). Figure 1 provides a schematic of the bubble column test
facility. Tap water was passed through a cartridge filter (W10-BC,
American Plumber, Pentair Residential Filtration, LCC) with 5\selectlanguage{greek}μ\selectlanguage{english}m nominal
filtration. Surface tension of the filtered water and other tested
liquids were measured with a force tesiometer (K6, Kr\selectlanguage{ngerman}üss GmbH) and
platinum ring (RI0111-282438, Krüss GmbH). Over several days the surface
tension of the filtered water was measured to be 72.6 ±0.4 mN/m, which
is comparable to the nominal surface tension of the pure water
(\textasciitilde{}72.8 mN/m). Liquid phase temperature was measured
using a thermocouple (HSTC-TT-K-20S-120-SMPW-CC, Omega Engineering).
Figure 1 also depicts the compressed airflow control panel. Airflow
passes through a cartridge filter (SGY-AIR9JH, Kobalt, Lowe's Companies,
Inc.) with 5\selectlanguage{greek}μ\selectlanguage{english}m nominal filtration. The mass flow of air was controlled
and monitored with a combination of a pressure regulator, rotameter
(EW-32461-50, Cole-Palmer), and a thermocouple (5SC-TT-K-40-39, Omega
Engineering). The rotameter measured the volumetric flow of air with an
accuracy of 2\% of the full scale (FS). The thermocouple measured the
air temperature immediately upstream of the rotameter with accuracy of
\selectlanguage{ngerman}±0.1°C. All tests were conducted with the air temperature between 20 °C
and 22 °C, and temperature difference between the air and liquid phase
was within ±2 °C. It is also worth mentioning that the height of liquid
in the bubble column was kept constant at 9\emph{D} following
recommendations from Besagni et al.\textsuperscript{41} for studying the
void fraction and bubble size independent of column aspect ratio.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}
\end{center}
\end{figure}
\textbf{Figure 1} . Schematic of the experimental setup including the
bubble column and airflow control and monitoring system.
The air sparger was comprised of a porous air stone covering
\textasciitilde{}90\% of the cross-section of the column that was
mounted on a cylindrical plenum. The porous air stone was fed from a 350
ml plenum, which used porous material identical to the air stone to
supply pressure drop for cross-sectional uniformity of air injection.
The sparger was designed to be pressurized up to 7 bar. A differential
pressure transducer (PX2300-DI, OMEGA) measured the pressure drop within
the line supplying the plenum. BSD depends heavily on the average pore
size in a homogenous bubbly flow. The average pore size
(\emph{r\textsubscript{p}} ) was calculated from\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(r_{p}=\frac{2\sigma}{{p}_{\text{cap}}}\), & (1)\tabularnewline
\bottomrule
\end{longtable}
where \selectlanguage{greek}\emph{σ} \selectlanguage{english}is the surface tension and \emph{[?]p\textsubscript{cap}}is
the differential pressure measured across the sparger at the onset of
bubbling. Equation (1) was adopted from Houghton et
al.,\textsuperscript{42} which explains that
the\emph{[?]p\textsubscript{cap}} measured in the aforementioned fashion
represents the average capillary pressure at the onset of bubbly. In the
current work the average pore size was 85\selectlanguage{greek}μ\selectlanguage{english}m \selectlanguage{ngerman}± 10\selectlanguage{greek}μ\selectlanguage{english}m.
The refraction index mismatch as well as the round geometry of the
acrylic column introduced a significant optical distortion. Thus a
refractive index matching box (water-box) was used to mitigate this
problem. The water-box was 0.2m \selectlanguage{ngerman}× 0.15m × 0.15m, made from casted
acrylic, and filled with water. Spatial calibration was performed with a
custom calibration plate, and the residual image distortion after
mounting the water-box was negligible relative to the bubble sizes
measured.
\subsection*{2.2 Bubble size measurement}
{\label{bubble-size-measurement}}
A camera (EOS 70D DSLR, Canon) was used to capture monochrome still
images of the bubbles. This camera had an APS-C CMOS image sensor
(22.5mm × 15mm) with a maximum resolution of 5472 × 3648 pixels. The
camera pixel size was 4.1\selectlanguage{greek}μ\selectlanguage{english}m \selectlanguage{ngerman}× 4.1\selectlanguage{greek}μ\selectlanguage{english}m with a 14-bit depth. The camera was
fitted with a 60 mm 1:2.8 lens (Canon) to produce a nominal
field-of-view of 120 mm by 80 mm. The column was backlit with an LED
panel (Daylight 1200, Fovigtec StudioPRO) that delivered up to
13,900-illumination flux (5600 K color temperature) at one meter.
Backlighting was uniformly diffused using a 3 mm thick white acrylic
sheet. Homogenous backlighting simplifies image-processing as well as
improves the measurement accuracy. Bubble images were processed for
bubble size measurements using ImageJ (1.49v, National Institutes of
Health (NIH), Bethesda, MD, USA),\textsuperscript{43-46} a common open
access image-processing program. Within ImageJ, an edge detection
algorithm was used to sharpen the bubble edges, subtract the background,
and apply a grayscale threshold to convert the 14-bit images to binary
images. A subset of images from each condition were manually processed
and then used to determine the appropriate grayscale threshold. It is
worth mentioning that the bubble images become darker in background as
the number of bubbles per image increases. Therefore, a range of
acceptable threshold values were explored for each condition and
produced a 2\% variation in measured bubble size. Interested readers are
referred to the previous studies from the current research
group\textsuperscript{47-49} for more details on the image processing
scheme. Including uncertainty from the spatial calibration and image
processing procedures, the measurement uncertainty was less than 8\%. In
the current work, the imaging system and processing scheme could resolve
bubbles as small as 0.2 mm in diameter. Figure 2 provides an example of
a bubble image with the identified bubbles using the appropriate
threshold outlined. Figure 2 also depicts that the processing algorithm
can identify in-focus bubbles and exclude out-of-focus bubbles, which
minimizes the impact of out-of-plane bubble locations on the spatial
calibration. In addition, Figure 2 shows that, even with a proper
threshold, overlapping and defective bubbles can contaminate the size
distributions. Consequently, each image was manually inspected for the
aforementioned problems and impacted bubbles were removed from the
population sample.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}
\end{center}
\end{figure}
\textbf{Figure 2} . Sample of processed bubble-image using ImageJ for
bubble size measurements. Identified bubbles have contour lines around
them.
Bubbles were approximated by ellipsoids in shape, and Equation (2) was
used to determine the equivalent diameter (\emph{d\textsubscript{eq}} )
of a sphere with the same cross-sectional area
(\emph{A\textsubscript{b}} ), which was used as the bubble size
representative length. Here \emph{a} is the major bubble axis
and\emph{AR} is the aspect ratio of the bubble. The cross-sectional
area, bubble centroid location, and the aspect ratio were calculated for
each identified bubble in ImageJ. This equivalent bubble diameter was
used in Equation (3) to compute the Sauter mean diameter
(\emph{d\textsubscript{32}} ), which is the ratio of the representative
bubble volume to the bubble surface area and is a common measure of the
average bubble size.\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(d_{\text{eq}}=\sqrt{\frac{a^{2}}{\text{AR}}}\) & (2)\tabularnewline
\midrule
\endhead
\(d_{32}=\frac{\sum_{i=1}^{n}d_{i}^{3}}{\sum_{i=1}^{n}d_{i}^{2}}\) & (3)\tabularnewline
\bottomrule
\end{longtable}
\subsection*{2.3 Void fraction
measurement}
{\label{void-fraction-measurement}}
Void fraction is defined as the ratio of gas volume to the total volume
of the system. In the current work, void fraction was calculated from
the differential pressure (\emph{[?]p} ) along the column height during
operation. A differential pressure transducer (PX2300-DI, OMEGA) was
employed to obtain the hydrostatic pressure between two pressure taps
with a separation of 8\emph{D} along the column height (see Figure 1). A
data acquisition card (National Instruments, USB-6218 BNC) was used to
acquire the output signal from the pressure transducer and the signal
was recorded on a desktop computer (via LabVIEW 15.0.1). Here the
uncertainty associated with the pressure measurement was calculated to
be under 2\% of the measured value for all cases tested. Void fraction
was calculated using Equation (4), where \emph{[?]H} is the vertical
distance between the pressure taps, \selectlanguage{greek}\emph{ρ\textsubscript{L}}\selectlanguage{english}
and\selectlanguage{greek}\emph{ρ\textsubscript{G}}\selectlanguage{english} are the density of liquid and gas,
respectively, and \emph{g} is gravitational acceleration.\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\varepsilon=\frac{p}{(\rho_{L}-\rho_{G})gH}\) & (4)\tabularnewline
\bottomrule
\end{longtable}
\subsection*{2.4 Test matrix}
{\label{test-matrix}}
Table 1 provides the test matrix for examining the effect of liquid
properties as well as gas superficial velocity
(\selectlanguage{greek}\emph{U\textsubscript{SG}=4Q\textsubscript{G}/πD\textsuperscript{2}}\selectlanguage{english} )
on multiphase parameters (e.g. bubble size and void fraction). To
explore the effect of liquid properties on bubble size and void
fraction, aqueous solutions of glycerin with different concentrations
were tested. In the current work, the range of the liquid viscosity
tested was in excess of two orders of magnitude, while the surface
tension and density were changed by only about 10\% and 20\%,
respectively, relative to that of water.
\textbf{Table 1} . Test matrix for liquid phase properties (at 20 \selectlanguage{ngerman}°C)
and gas superficial velocities.\selectlanguage{english}
\begin{longtable}[]{@{}llllll@{}}
\toprule
\textbf{Index} & \textbf{Liquid phase} & \selectlanguage{greek}\emph{μ\textsubscript{L}}\selectlanguage{english}
\textbf{(Pa.s)} & \selectlanguage{greek}\emph{ρ\textsubscript{L}}\selectlanguage{english}
\textbf{(kg/m\textsuperscript{3})} & \selectlanguage{greek}\emph{σ} \selectlanguage{english}\textbf{(mN/m)} &
\emph{U\textsubscript{SG}} \textbf{(mm/s)}\tabularnewline
\midrule
\endhead
\emph{G1} & Glycerin 85\% & 0.161 & 1224 & 0.065 & 6.9, 13.8, 20.7,
27.6, 34.4, 41.4, 48.3, 55.2, 62, 69\tabularnewline
\emph{G}2 & Glycerin 79\% & 0.083 & 1208 & 0.065 &\tabularnewline
\emph{G3} & Glycerin 60\% & 0.016 & 1157 & 0.067 &\tabularnewline
\emph{W} & Water & 0.001 & 998 & 0.072 &\tabularnewline
\bottomrule
\end{longtable}
\section*{Results and Discussion}
{\label{results-and-discussion}}
\subsection*{3.1 Regime identification}
{\label{regime-identification}}
The objective was to find a threshold for the transient gas superficial
velocity and study the effect of viscosity on this threshold. BSD
characteristics and higher order statistics (skewness and kurtosis) were
used to identify the operation regimes. Figure 3 shows the probability
density function (PDF) of bubble size in aqueous solutions of glycerin
(G1 and G3, see Table 1) at various superficial gas velocities. Figure 3
shows that as the gas superficial velocity exceeds 27.6 mm/s in both
cases the PDF shape changes from a bell shape into a spike shape; bubble
column literature\textsuperscript{24,40} attributed the aforementioned
shift in PDF shape to regime alternation from homogenous to
heterogeneous regime. In addition, Figure 3a shows that the lower gas
superficial velocities (\emph{U\textsubscript{SG}} [?] 27.6 mm/s) exhibit
a bimodal shape in the PDF, this feature has been reported in studies of
the homogenous operation regime in the bubble column
literature.\textsuperscript{15,21,36,37} In homogenous bubbly flow with
no bubble breakage and coalescence events, the injection condition (i.e.
pore size and gas superficial velocity) determines the bubble size. The
pore size distribution on the sparger is discrete and non-monodisperse,
which explains the deviation of bubble size distribution from a truly
Gaussian distribution. Higher order statistics (skewness and kurtosis)
of the BSD were obtained for further inspection of the operation regime
shift. Table 2 presents the Sauter mean diameter
(\emph{d\textsubscript{32}} ), (arithmetic) mean diameter
(\emph{d\textsubscript{10}} ), standard deviation (RMS), as well as
higher order statistics of BSD in G1 at the gas superficial velocities
tested. Skewness (\emph{S} ) and kurtosis (\selectlanguage{greek}\emph{κ} \selectlanguage{english}) of the bubble size
distribution in Table 2 show a significant deviation from a
Gaussian-distribution (\emph{S} \textasciitilde{}0, \selectlanguage{greek}κ\selectlanguage{english}\textasciitilde{}3)
when the gas superficial velocity exceeds \emph{U\textsubscript{SG}} =
27.6 mm/s.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}
\end{center}
\end{figure}
\textbf{Figure 3} . Probability density function (PDF) of bubble size in
(a) G1 and (b) G3.
\textbf{Table 2} . Bubble size and statistics at various gas superficial
velocities tested in G1.\selectlanguage{english}
\begin{longtable}[]{@{}llllll@{}}
\toprule
\emph{U\textsubscript{SG}} \textbf{(mm/s)} & \emph{d\textsubscript{32}}
\textbf{(mm)} & \emph{d\textsubscript{10}} \textbf{(mm)} &
\emph{RMS(d\textsubscript{b})} \textbf{(mm)} & S(d\textsubscript{b}) &
\selectlanguage{greek}κ\selectlanguage{english}(d\textsubscript{b})\tabularnewline
\midrule
\endhead
6.9 & 2.4 & 2.2 & 0.5 & 0.8 & 4.3\tabularnewline
20.7 & 2.3 & 2.1 & 0.5 & 0.5 & 3.4\tabularnewline
27.6 & 2.3 & 2.0 & 0.6 & 0.5 & 4.0\tabularnewline
41.4 & 2.2 & 1.8 & 0.6 & 1.4 & 6.4\tabularnewline
55.2 & 1.7 & 1.4 & 0.4 & 2.3 & 12.2\tabularnewline
62.1 & 1.8 & 1.4 & 0.4 & 3.3 & 20.0\tabularnewline
69.0 & 1.5 & 1.4 & 0.4 & 4.2 & 28.4\tabularnewline
\bottomrule
\end{longtable}
So far it was discussed that the homogenous operation regime exhibits
Gaussian-like characteristics and heterogeneous regime features a spike
shape distribution; this was inspected using PDF plots of BSD from the
range tested (see Table 1). Figure 4 shows the bubble size distribution
from G1 plotted on probability coordinates. Figure 4a shows that in
homogenous range (based on gas superficial velocity) the BSD exhibits
linearity on a normal distribution probability plot. The near-Gaussian
behavior in the homogenous regime was also discussed above by means of
the PDF shape and higher order statistics. BSD from heterogeneous
operation cases (41.4 mm/s [?] \emph{U\textsubscript{SG}} [?] 69 mm/s) in G1
was plotted on a lognormal probability plot (see Figure 4b). It is
interesting to see the strong linear behavior of BSD on a lognormal
probability plot. Figure 4b also shows that the range at which the BSD
exhibit linear trend on a lognormal probability plot starts at the mode
of the PDF (thick solid vertical line) and ends at the Sauter mean
diameter (\emph{d\textsubscript{32}} , vertical lines). The mode of the
PDF corresponds to the most frequent bubble
size,\emph{d\textsubscript{mf}} , interested reader can refer to
Mohagheghian and Elbing\textsuperscript{47} for analysis
of\emph{d\textsubscript{mf}} . The present study shows that only a
portion of the right leg of the BSD PDF within heterogeneous regime is
lognormal. The PDF exhibits a linear trend
between\emph{d\textsubscript{min}} and \emph{d\textsubscript{mf}} ;
furthermore, the PDF can be modeled with a second order polynomial curve
between\emph{d\textsubscript{32}} and \emph{d\textsubscript{max}} .\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}
\end{center}
\end{figure}
\textbf{Figure 4} . Probability plots of BSD with G1 in the (a)
homogenous regime; 6.9 mm/s [?]\emph{U\textsubscript{SG}} [?] 27.6 mm/s on a
normal probability plot and (b) heterogeneous regime; 41.4 mm/s [?]
\emph{U\textsubscript{SG}} [?] 69 mm/s on a lognormal probability plot.
Figure 5 presents the PDF of the bubble size at different viscosities
(see Table 1) illustrating the sensitivity of BSD to the viscosity of
the liquid phase. Once the gas superficial velocity is sufficiently high
(in this case \emph{U\textsubscript{SG}} = 27.6 mm/s), the viscosity
modifies the near-Gaussian distribution (in water) to a spike shape
distribution. It was discussed that the shift in the distribution shape
is an indication of operation regime change from homogenous to
heterogeneous. Manual inspections showed that, increasing the viscosity
reduces the bubble terminal velocity due to friction drag; moreover,
increasing the viscosity effects the bubble motion by creating planar
oscillations in the bubbles trajectory, these two effects in turn
enhance the bubble coalescence, this results in the formation of larger
bubbles that are more susceptible to shear breakage. At higher gas
superficial velocities, the number of coalescence and breakage events
increases; therefore, the BSD shape shifts towards a spike (lognormal
distribution) and the standard deviation decreases (i.e. distribution
narrows). In summary, increasing the viscosity modifies the BSD, and
increases bubble-wake interactions; these effects, alter the physical
structure of the bubbly flow from homogenous to heterogeneous.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}
\end{center}
\end{figure}
\par\null
\textbf{Figure 5} . Probability density function of bubble size at
different liquid viscosities and\emph{U\textsubscript{SG}} =27.6 mm/s.
Transport coefficients determine the performance and efficiency of a
bubble column and are sensitive to the bubble size and void fraction.
Bubble size and void fraction are heavily depended on the operation
regime, which sets the dominant fluidic mechanisms within a gas-liquid
system. The rest of this paper is structured as such to study the bubble
size and void fraction with respect to the operation regimes i.e.,
homogenous and heterogeneous.
\subsection*{3.2 Homogeneous regime}
{\label{homogeneous-regime}}
In this section bubble size and void fraction were studied in the
homogenous operation regime, which features a linear trend between the
void fraction and the gas superficial velocity as well as a direct
correlation between bubble size and gas superficial velocity.
Homogeneous bubbly flow is characterized by the absence of breakage and
coalescence and a Gaussian BSD; therefore, any attempt to scale the
bubble size should include the pore size (\emph{r\textsubscript{p}} )
and the gas superficial velocity (\emph{U\textsubscript{SG}} ) in the
parameter space. The present work also includes the liquid properties
(i.e. surface tension \selectlanguage{greek}\emph{σ} , \selectlanguage{english}liquid
viscosity\selectlanguage{greek}\emph{μ\textsubscript{L}}\selectlanguage{english} , and liquid
density\selectlanguage{greek}\emph{ρ\textsubscript{L}}\selectlanguage{english} ), and gravity (\emph{g} ) to scale the
bubble size in the homogenous regime. Using dimensional analysis, the
scaled bubble size was expected to be dependent on the Froude number
(\emph{Fr} ), Weber number (\emph{We} ), and Reynolds number (\emph{Re}
); see Equations (5)-(7).
Figure 6 validates the correlation for predicting bubble size
(\emph{d\textsubscript{32}} ) in homogeneous regime (see Equation 8)
against experimental bubble size data. Results show that in the
homogenous regime the proposed correlation predicts the bubble size. In
Equation (8) the power exponents were found following the recommendation
from Kazakis et al.,\textsuperscript{40} which argues that the sparger
material effects correlations of this type due to the sensitivity of
bubble size to pore dimensions in homogeneous bubbly flow. The power law
functional form between the aforementioned non-dimensional terms (see
Equations 5-7) was first suggested by Akita and
Yoshida;\textsuperscript{24} in addition, in Equations (5)-(7) the
exponents were obtained empirically.\textsuperscript{40}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}
\end{center}
\end{figure}
\par\null
\textbf{Figure 6} . Scaled bubble size in homogenous regime (water).\selectlanguage{english}
\begin{longtable}[]{@{}lll@{}}
\toprule
\(Fr=\frac{U_{\text{SG}}}{\sqrt{gd_{32}}}\) & (FroudeNumber) & (5)\tabularnewline
\midrule
\endhead
\(\text{We}=\frac{{U_{\text{SG}}}^{2}gd_{32}}{\sigma}\) & (Weber Number) & (6)\tabularnewline
\(\text{Re}=\frac{\rho_{L}d_{32}U_{\text{SG}}}{\mu_{L}}\) & (ReynoldsNumber) & (7)\tabularnewline
\(\frac{d_{32}}{{2r}_{p}}={6.4\left(\text{Fr}^{1.8}\text{We}^{-1.7}\text{Re}^{0.7}\right)}^{-0.132}\) & \(\frac{d_{32}}{{2r}_{p}}={6.4\left(\text{Fr}^{1.8}\text{We}^{-1.7}\text{Re}^{0.7}\right)}^{-0.132}\) & (8)\tabularnewline
\bottomrule
\end{longtable}
Figure 7 shows the void fraction from the tested conditions in the
present work, there is a direct correlation between void fraction and
gas superficial velocity. It was argued in the previous section that
regime shift from homogeneous to heterogeneous operation regime can be
identified from higher order statistics and probability density plots.
Here, it was attempted to investigate the regime change at similar gas
superficial velocity (\emph{USG} = 27.6 mm/s) using void fraction data.
Figure 7 shows that above \emph{USG} = 27.6 mm/s, void fraction
(\selectlanguage{greek}\emph{ε} \selectlanguage{english}) deviates from the linear trend with gas superficial
velocity, which indicates that the homogeneous regime was no longer
present. Figure 7 also shows that for the highest viscosity tested (G1)
the void fraction levels off after \emph{U\textsubscript{SG}} = 27.6
mm/s. Detailed observations of the bubble column showed that the void
fraction in G1 tests levels off due to a significant drop in gas
residence time (data not shown). In these cases, the relatively high gas
superficial velocity results in formation of slugs from bubble
coalescence near the sparger; these slugs are unstable and travel
significantly faster than bubbles and, consequently, the gas residence
time decreases.\textsuperscript{22} However, in the rest of the cases
tested in Figure 7 (i.e. G3 and water) the void fraction increases with
gas superficial velocity.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}
\end{center}
\end{figure}
\textbf{Figure 7} . Void fraction measurement in water and different
aqueous solutions of glycerin (G1 and G3).
A parameter space was identified via careful inspection of the
experimental setup to formulate a correlation to predict the void
fraction using dimensional analysis. It was concluded that the parameter
space should be comprised of liquid properties (i.e. surface tension,
viscosity, and density), external body force (i.e. gravity), bubble size
(\emph{d32} ), and the gas flow rate (i.e. gas superficial velocity).
The effect of gas superficial velocity, gravity, and liquid properties
were accounted for using Froude number (Equation 5), Archimedes number
(Equation 9), and E\selectlanguage{ngerman}ötvös number (Equation 10) for scaling the void
fraction. Equation (11) shows the resulting correlation for scaling the
void fraction, where \emph{G} ( ) is an unknown function. Following
Akita and Yoshida,\textsuperscript{24} Mouza et
al.,\textsuperscript{39}Kazakis et al.,\textsuperscript{40} and
Anastasiou et al.,\textsuperscript{50} a power law functional form was
considered for the unknown function \emph{G} . Figure 8 validates the
correlation for predicting the void fraction (\selectlanguage{greek}\emph{ε} \selectlanguage{english}) in homogeneous
regime against experimental data showing that Equation (12) successfully
predicts the void fraction within \selectlanguage{ngerman}±5\% accuracy of the current
measurements.\selectlanguage{english}
\begin{longtable}[]{@{}lll@{}}
\toprule
\(\text{Ar}=\frac{{d_{32}}^{3}\rho_{L}^{2}g}{\mu_{L}^{2}}\) & (Archimedes Number) & (9)\tabularnewline
\midrule
\endhead
\(\text{Eo}=\frac{{d_{32}}^{2}\rho_{L}g}{\sigma}\) & (E\selectlanguage{ngerman}ötvös Number) & (10)\tabularnewline
\(\varepsilon=G(\text{Fr},\ \text{Ar},\ \text{Eo})\ \) & \(\varepsilon=G(\text{Fr},\ \text{Ar},\ \text{Eo})\ \) & (11)\tabularnewline
\(\varepsilon=0.0278\left(\text{Fr}^{1.117}\text{Ar}^{0.1}\text{Eo}^{-0.032}\right)^{0.4959}\) & \(\varepsilon=0.0278\left(\text{Fr}^{1.117}\text{Ar}^{0.1}\text{Eo}^{-0.032}\right)^{0.4959}\) & (12)\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}
\end{center}
\end{figure}
\textbf{Figure 8} . A correlation for scaling the void fraction in the
homogenous regime (water).
\subsection*{3.3 Heterogeneous regime}
{\label{heterogeneous-regime}}
The heterogeneous operation regime features frequent breakup and
coalescence events. Coalescence produces larger bubbles, which are more
susceptible to deformation and breakage. Generally, coalescence
increases the number of large bubbles and breakage increases the number
of small bubbles; therefore, in a statistically stationary bubble size
population the coalescence skews the PDF negatively (towards the right
tail) and breakage skews the PDF positively (towards the left tail).
This explains the shift in PDF shape from a bell (hump) shape to a
positively skewed spike shape when the operation regime changes from
homogenous to heterogeneous regime. To approach the physical scaling of
the bubble size characteristic length scale, it was hypothesized that in
heterogeneous regime the interfacial momentum transfer sets the stable
bubble size. Therefore, the energy supplied to the liquid phase from the
injection of the gas phase is expected to power the interfacial momentum
transfer. In the current work, statistically stationary samples of
bubble size were used to test this hypothesis. Sauter mean diameter was
measured according to the test matrix in Table 1 to test the
relationship between bubble size and specific input power per unit mass
(\emph{P\textsubscript{m}} = \emph{gU\textsubscript{SG}} ).
Hinze\textsuperscript{38} studied the breakage of drops and recommended
using the maximum stable drop size (\emph{d\textsubscript{95}} ) under
shear breakage for scaling and argues that \emph{d\textsubscript{95}} is
the characteristic length that constrains 95\% of the dispersed phase
volume. Alves et al.\textsuperscript{51} argue that the Sauter mean
diameter is proportional to the maximum stable bubble size; therefore,
in the present work \emph{d\textsubscript{32}} was used as the bubble
size characteristic length scale for bubble size scaling. Figure 9 shows
the measured \emph{d\textsubscript{32}} at
various\emph{P\textsubscript{m}} levels, which shows that for the
glycerin conditions (G1-G3) the Sauter mean diameter decreases with
increasing specific input power. Hinze\textsuperscript{38} proposed a
correlation (Equation 13) to predict the maximum stable bubble size as a
function of specific power input, surface tension, and density of the
continuous phase. In Equation (13), the proportionality coefficient
(\emph{k} ) is a function of the critical Weber number
(\emph{We\textsubscript{cr}} ). It has been demonstrated that the
proportionality constant corresponds to different mechanisms, including
\emph{k} = 0.725 for isotropic turbulent\textsuperscript{38} and
\emph{k} \textasciitilde{} 1.7 for shear bubble
breakup.\textsuperscript{52,53} Figure 9 compares the predicted bubble
size from Equation (13) (\emph{k} = 0.45) with the measured bubble size
(Sauter mean diameter) from all cases tested in the present study.\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(d_{32}=k\frac{\left(\frac{\sigma}{\rho_{L}}\right)^{3/5}}{{P_{m}}^{2/5}}\) & (13)\tabularnewline
\bottomrule
\end{longtable}
The non-dimensional form of Equation (13) was produced using dimensional
analysis on the given parameter space, which results in a single
non-dimensional term equal to proportionality constant (\emph{k} ). If
this is repeated with the addition of viscosity
(\selectlanguage{greek}\emph{μ\textsubscript{L}}\selectlanguage{english} ) into the parameter space, then Equation
(14) relates the bubble size (\emph{d\textsubscript{32}} ) with the
input power (\emph{P\textsubscript{m}} = \emph{gU\textsubscript{SG}} )
and liquid properties (surface tension, liquid viscosity, and liquid
density). Equation (14) suggests that the unknown functional
form\emph{f} () needs to be found experimentally from bubble size
(\emph{d\textsubscript{32}} ) data. Detailed inspections show that at
lower specific input powers the bubble column is still operating in the
homogenous regime; consequently, in the absence of shear breakage bubble
size cannot be predicted from Equation (13). Figure 9 also shows that
the \emph{d\textsubscript{32}} from conditions tested in water increase
with increasing gas superficial velocity (specific input power), this is
due to homogenous operation regime. The non-dimensional terms in
Equation (14) are well established dimensionless terms; the scaled
bubble size (left hand side) is the Ohnesorge number (\emph{Oh} ), which
is the ratio of the product of the inertia and surface tension forces to
viscous forces. The scaled specific input power, which is related to the
shear breakage, is the product of the Morton number\(\left(Mo=\frac{g\mu_{L}^{4}}{\rho_{L}\sigma^{3}}\right)\)
and the Capillary number\(\left(Ca=\frac{\mu_{L}U_{\text{SG}}}{\sigma}\right)\); here the scaled
\emph{P\textsubscript{m}} term is a combination of viscous, inertia,
surface tension, and gravitational forces.\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\frac{\rho_{L}d_{32}\sigma}{\mu_{L}^{2}}=f\left(\frac{P_{m}\mu_{L}^{5}}{{\rho_{L}\sigma}^{4}}\right)\) & (14)\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image9/image9}
\end{center}
\end{figure}
\textbf{Figure 9} . Bubble size (Sauter mean diameter) measurement in
water and different aqueous solutions of glycerin.
It was attempted to find the functional form \emph{f} () between the
scaled bubble size and scaled specific input power in Equation (14).
Hinze\textsuperscript{38} suggested a power law correlation (Equation
15) between the scaled bubble size and scaled specific input power. The
current study found a power law correlation with similar power (slopes)
to that of Hinze\textsuperscript{38} (see Equation 16). Bubble size
measurements from Mohagheghian and Elbing\textsuperscript{47} were used
for further validation of Equation (16). Note that Mohagheghian and
Elbing\textsuperscript{47} measurements were carried out in the same
test facility; however, a single point air injection method was used for
bubbling the column. Figure 10 shows that data from the present study,
Hinze,\textsuperscript{38} and Mohagheghian and
Elbing\textsuperscript{47} collapse on Equation (16) (dashed black
line). To further examine the present correlation for scaling the bubble
size, similar studies\textsuperscript{39,40} were used to check validity
of the present correlation.\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\frac{\rho_{L}d_{32}\sigma}{\mu_{L}^{2}}=0.725\left(\frac{P_{m}\mu_{L}^{5}}{{\rho_{L}\sigma}^{4}}\right)^{-0.4}\) & (15)\tabularnewline
\midrule
\endhead
\(\frac{\rho_{L}d_{32}\sigma}{\mu_{L}^{2}}=0.2477\left(\frac{P_{m}\mu_{L}^{5}}{{\rho_{L}\sigma}^{4}}\right)^{-0.4}\) & (16)\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image10/image10}
\end{center}
\end{figure}
\textbf{Figure 10} . Scaled bubble size versus scaled specific input
power using results from the literature in addition to the current
study.
In the rest of this section void fraction measurements and scaling in
heterogeneous regime are discussed. The same parameter space for
producing Equation (11) was employed for scaling the void fraction and
finding the function form of \emph{G()} . Here, it was assumed that the
bubbles are traveling at terminal velocity (see Figure 11); therefore,
the drag force (\emph{F\textsubscript{D}}
\(\propto\)\selectlanguage{greek}\emph{ρ\textsubscript{L}d\textsubscript{32}\textsuperscript{2}U\textsubscript{b}\textsuperscript{2}}\selectlanguage{english}
) was balanced with buoyancy force (\selectlanguage{greek}\emph{F\textsubscript{B} =
ρ\textsubscript{L}gd\textsubscript{32}\textsuperscript{3}}\selectlanguage{english} ). This
assumption establishes a relationship between bubble size and bubble
velocity (\emph{U\textsubscript{b}\textsuperscript{2} \textasciitilde{}
gd\textsubscript{32}\textsuperscript{3}} ). It is known that the void
fraction is the ratio of gas superficial velocity to the bubble velocity
(\selectlanguage{greek}\emph{ε = U\textsubscript{SG}/U\textsubscript{b}}\selectlanguage{english} ); therefore, the
void fraction is proportional to bubble Froude number (\emph{Fr =
U\textsubscript{SG}/{[}gd\textsubscript{32}{]}\textsuperscript{0.5}} ).
Assuming that void fraction scales as a power law function of Froude
number (Equation 5), Archimedes number (Equation 9), and E\selectlanguage{ngerman}ötvös number
(Equation 10), then Equation (17) gives the general form of \emph{G()} .
The exponents in Equation (17) (i.e. \selectlanguage{greek}\emph{Χ} , \selectlanguage{english}\selectlanguage{greek}\emph{Ψ} , \selectlanguage{english}and \selectlanguage{greek}\emph{Ω}
\selectlanguage{english}) were calculated from Equation (18) (\selectlanguage{greek}\emph{Χ=}\selectlanguage{english} 1.117, \selectlanguage{greek}\emph{Ψ=}\selectlanguage{english} 0.1,
and\selectlanguage{greek}\emph{Ω=}\selectlanguage{english} -0.032). Figure 12 shows that the proposed coordinates (see
Equation 19) were able to successfully scale the void fraction within
the heterogeneous regime. Equation (19) successfully predicts the void
fraction within \selectlanguage{ngerman}±25\% accuracy for the current data.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image11/image11}
\end{center}
\end{figure}
\textbf{Figure 11} . Schematic of the primary acting forces on a single
bubble at terminal velocity.\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\varepsilon=A\text{Fr}\text{Ar}^{\Psi}\text{Eo}^{\Omega}\) & (17)\tabularnewline
\midrule
\endhead
\(\frac{U_{\text{SG}}}{\sqrt{gd_{32}}}\ \cong\ \text{Fr}\text{Ar}^{\Psi}\text{Eo}^{\Omega}\) & (18)\tabularnewline
\(\varepsilon=0.035\left(\text{Fr}^{1.117}\text{Ar}^{0.1}\text{Eo}^{-0.032}\right)^{0.75}\) & (19)\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image12/image12}
\end{center}
\end{figure}
\textbf{Figure 12} . A correlation for scaling the void fraction in the
heterogeneous regime.
\section*{Conclusions}
{\label{conclusions}}
A systematic study of bubble size and void fraction in a batch bubble
column with a porous sparger was carried out. The measurements (i.e.
bubble size and void fraction) were carried out in homogenous and
heterogeneous operation regimes. Bubble size measurements were performed
using optical photography of large populations of bubbles (2400 and
more). Void fraction was measured from the differential pressure across
the bubble column height. Water and aqueous solutions of glycerin were
used to test the effect of viscosity on the operation regime, bubble
size, and void fraction. Gas superficial velocity was tested in the
range of 6.9 mm/s \textless{} \emph{U\textsubscript{SG}} \textless{} 69
mm/s using compressed air. Regime transition corresponds to the change
of physical behavior of the gas liquid system in bubble columns;
therefore, it is appropriate to present any measurements with
consideration of the operation regime. Current work uses PDF as well as
probability plots to characterize the bubble size distribution in
homogenous and heterogeneous operation regimes.
Results showed that in the homogenous regime, the bubble size
distribution is poly-dispersed and the PDF exhibits Gaussian
characteristics. In the heterogeneous regime, bubble coalescence events
and shear breakage modified the bubble size distribution, results in the
distribution approaching mono-dispersed as indicated by the PDF having a
``spike'' shape with a lognormal right leg.
Results also showed that increasing the viscosity accelerates the regime
transition from homogenous to heterogeneous by allowing the formation of
larger bubbles as well as bubble interaction (i.e. breakage and
coalescence). Bubble size measurements were carried out in both
operation regimes. In the homogenous regime, the characteristic bubble
size (i.e. Sauter mean diameter) shows strong dependence on the sparger
characteristics and injection condition due to the absence of breakage
and coalescence. In the heterogeneous regime, experimental data exhibits
a strong correlation between the Sauter mean diameter and specific input
power (per unit mass). Dimensional analysis was used to propose a
correlation between the scaled bubble size and the scaled specific input
power. This correlation was validated against experimental data in
literature both from static and vibrating bubble column studies. Void
fraction was also measured in both the homogenous and heterogeneous
regimes. As expected, the trend between void fraction and gas
superficial velocity was dependent on the operation regime. Using
dimensional analysis correlations for scaling the void fraction in
homogenous and heterogeneous regimes were proposed and validated against
experimental data.
\section*{Acknowledgments}
{\label{acknowledgments}}
The authors would like to thank Adam Still for the initial design and
fabrication of the facility as well as assistance with the initial
operation of it. This research was funded, in part, by B.R.E.'s
Halliburton Faculty Fellowship endowed professorship.
\section*{Nomenclature}
{\label{nomenclature}}\selectlanguage{english}
\begin{longtable}[]{@{}lll@{}}
\toprule
\textbf{Symbol} & \textbf{Description} & \textbf{Unit}\tabularnewline
\midrule
\endhead
\emph{a} & Bubble major axis & {[}mm{]}\tabularnewline
\emph{A} & Cross-sectional area &
{[}mm\textsuperscript{2}{]}\tabularnewline
\emph{AR} & Bubble aspect ratio (ratio of the major to minor axis) &
{[}-{]}\tabularnewline
\emph{Ar} & Archimedes number & {[}-{]}\tabularnewline
\emph{BSD} & Bubble size distribution & {[}-{]}\tabularnewline
\emph{C} & Proportionality coefficient & {[}-{]}\tabularnewline
\emph{Ca} & Capillary number & {[}-{]}\tabularnewline
\emph{d} & Bubble diameter & {[}mm{]}\tabularnewline
\emph{D} & Column diameter & {[}mm{]}\tabularnewline
\emph{Eo} & E\selectlanguage{ngerman}ötvös number & {[}-{]}\tabularnewline
\emph{F} & Force & {[}kgms\textsuperscript{-2}{]}\tabularnewline
\emph{Fr} & Froude number & {[}-{]}\tabularnewline
\emph{g} & Gravitational acceleration &
{[}ms\textsuperscript{-2}{]}\tabularnewline
\emph{k} & Proportionality coefficient & {[}-{]}\tabularnewline
\emph{Mo} & Morton number & {[}-{]}\tabularnewline
\emph{n} & Number of bubbles in a sample population &
{[}-{]}\tabularnewline
\emph{Oh} & Ohnesorge number & {[}-{]}\tabularnewline
\emph{P} & Input power from gas injection &
{[}kgm\textsuperscript{2}s\textsuperscript{-3}{]}\tabularnewline
\emph{PDF} & Probability density function & {[}-{]}\tabularnewline
\emph{Q} & Volumetric flow rate &
{[}m\textsuperscript{3}s\textsuperscript{-1}{]}\tabularnewline
\emph{r} & Average pore radius & {[}\selectlanguage{greek}μ\selectlanguage{english}m{]}\tabularnewline
\emph{Re} & Reynolds number & {[}-{]}\tabularnewline
\emph{RMS} & Standard deviation of bubble size distribution &
{[}mm{]}\tabularnewline
\emph{S(d\textsubscript{b})} & Skewness of bubble size distribution &
{[}mm{]}\tabularnewline
\emph{U} & Velocity & {[}mms\textsuperscript{-1}{]}\tabularnewline
\emph{We} & Weber number & {[}-{]}\tabularnewline
\textbf{Greek letters and symbols} & \textbf{Greek letters and symbols}
& \textbf{Greek letters and symbols}\tabularnewline
\textbf{Symbol} & \textbf{Description} & \textbf{Unit}\tabularnewline
\emph{[?]H} & Vertical distance between two pressure taps &
{[}m{]}\tabularnewline
\emph{[?]p} & Differential pressure & {[}kg
m\textsuperscript{-1}s\textsuperscript{-2}{]}\tabularnewline
\selectlanguage{greek}\emph{ε} & Void fraction & {[}-{]}\tabularnewline
\selectlanguage{greek}\emph{κ(d\textsubscript{b})}\selectlanguage{english} & Kurtosis of bubble size distribution &
{[}-{]}\tabularnewline
\selectlanguage{greek}\emph{μ} & Viscosity &
{[}kgm\textsuperscript{-1}s\textsuperscript{-1}{]}\tabularnewline
\selectlanguage{greek}\emph{ρ} & Density & {[}kgm\textsuperscript{-3}{]}\tabularnewline
\selectlanguage{greek}\emph{σ} & Surface tension &
{[}kgs\textsuperscript{-2}{]}\tabularnewline
\selectlanguage{greek}Χ & Froude number exponent in Equation (17) & {[}-{]}\tabularnewline
\selectlanguage{greek}Ψ & Archimedes number exponent in Equation (17) & {[}-{]}\tabularnewline
\selectlanguage{greek}Ω & E\selectlanguage{ngerman}ötvös number exponent in Equation (17) & {[}-{]}\tabularnewline
\textbf{Subscripts} & \textbf{Subscripts} &
\textbf{Subscripts}\tabularnewline
10 & Arithmetic mean diameter &\tabularnewline
32 & Suater mean diameter &\tabularnewline
95 & Maximum stable bubble size &\tabularnewline
b & Bubble &\tabularnewline
B & Buoyancy force &\tabularnewline
cap & Capillary &\tabularnewline
cr & Critical &\tabularnewline
D & Drag force &\tabularnewline
eq & Equivalent &\tabularnewline
G & Gas phase &\tabularnewline
L & Liquid (phase) &\tabularnewline
m & Specific value &\tabularnewline
max & Maximum &\tabularnewline
min & Minimum &\tabularnewline
mf & Most frequent &\tabularnewline
p & Porous sparger &\tabularnewline
SG & Superficial gas &\tabularnewline
\bottomrule
\end{longtable}
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{\label{references}}
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