# Characterization of Bubble Size Distributions within a Bubble Column

^{*}

## Abstract

**:**

_{mf}) and defined as the peak in the PDF. This length scale as well as the traditional Sauter mean diameter were used to assess the sensitivity of the BSD to gas injection rate, injector tube diameter, injection tube angle and column diameter. The d

_{mf}was relatively insensitive to most variation, which indicates these bubbles are produced by the turbulent wakes. In addition, the current work examines higher order statistics (standard deviation, skewness and kurtosis) and notes that there is evidence in support of using these statistics to quantify the influence of specific parameters on the flow-field as well as a potential indicator of regime transitions.

## 1. Introduction

_{32}), which fails to capture details of the size distribution. Thus, the current work aims to characterize the bubble size distribution (BSD) and its dependence on bubble column conditions via examination of the probability density function (PDF) and higher order statistics.

_{32}) is the most widely used characteristic length in bubble column studies (e.g., [2,3,4,5]). Sauter mean diameter,

_{i}is the number of bubbles with size d

_{i}. Sauter mean diameter is frequently used when the sizes are acquired using optical photography techniques. Here the bubble cross sectional area (A

_{proj}) is determined from the projected image. Then assuming that the bubbles are well approximated as ellipsoids (or more specifically an oblate spheroid), an equivalent bubble chord length,

_{i}in Equation (1).

_{32}is a probabilistic approach, which uses the mean of the PDF of the bubble chord length [5,6,7,8,9]. This method is most common when the measurements are acquired with electrical impedance/resistivity [10,11,12], wire mesh [13,14,15] or optical [16] point probes, which can only provide a single length scale but a relatively large sample size. These measurements are sensitive to the bubble size, velocity, shape and orientation as well as the sensor design (e.g., response from optical/impedance probes are unique to the sensor design and fluid properties). Consequently, these measurements are unable to provide details about the shape due to the required ad hoc assumptions to relate the signals to a bubble size. The current work uses bubble imaging of a large bubble population to produce PDFs that are not dependent on the assumption that the bubbles are spherical. These PDFs are then analyzed to identify an alternative length scale based on the peak in the PDF, which is then used along with the Sauter mean diameter to test sensitivity of the scales to operation conditions. In addition, higher order statistics from the PDFs are reported.

## 2. Experimental Methods

#### 2.1. Test Facility and Air Injection

#### 2.2. Bubble Size Measurement

_{proj}= 2 mm

^{2}was used to remove noise contamination from BSD and consequently the PDFs. Given the cross-sectional area, a nominal bubble size was determined using Equation (2). Note that not every image was processed because the sample rate (400 Hz) did not produce a sufficient duration for a new bubble population in each image. Consequently, the period between processed images was increased such that each processed image contained a new bubble population to ensure statistically independent bubble samples.

#### 2.3. Repeatability

_{m}) were selected that produced superficial gas velocities (volume averaged phase velocity; U

_{sg}= Q

_{c}/A

_{cs}) of 6.9 mm/s, 27.6 mm/s and 55.1 mm/s. Under these conditions, the bubble column was operating within the poly-dispersed homogenous regime [18], which is true throughout the current study. Each condition was repeated at least ten times with a minimum of 3000 bubbles sampled per condition. Results from these tests are shown in Figure 4 with the Sauter mean diameter (d

_{32}) plotted versus the vertical distance above the injection location (Z) scaled with the column diameter (D). Error bars represent the standard deviation of the mean for each condition. Similar to Akita and Yoshida [22], these results exhibit a decrease in d

_{32}with increasing gas flux for locations sufficiently far from the injection location. Note that increasing superficial velocity is known to increase or decrease [32,33] bubble size due to its complex role modifying bubble formation processes and liquid circulation. Figure 4 also indicates that beyond Z ~ 4D the bubble size remains constant within the measurement uncertainty. Consequently, the current work focuses on bubble measurements in the range of 4 < Z/D < 6 to minimize the influence of the injection method. It is noteworthy that the minimum height above the injector will be sensitive to the injection condition, which will be discussed subsequently. Furthermore, inspection of the images within the target height range showed minimal influence of bubble breakup and/or coalescence.

## 3. Bubble Size Length Scales

_{32}) is widely used as the characteristic bubble length scale, bubble size distributions are often poly-dispersed, which makes a single length scale insufficient to characterize the distribution. Consequently, in the current work the PDF was examined to identify a length scale(s) that represents the size distribution. The PDFs generated from counting at least 10,000 bubbles per condition is provided in Figure 5a (PDF of conditions shown in Figure 4, though limited to 4 < Z/D < 6). Here there is a noticeable shift between the PDF peaks and d

_{32}. Consequently, the most frequent bubble size (d

_{mf}) was defined as the size corresponding to the peak in the PDF (mode). These representative conditions illustrate the different behavior between d

_{32}and d

_{mf}, with d

_{mf}being significantly smaller than d

_{32}over the range tested. In addition, while there is a noticeable dependence between d

_{32}and the volumetric gas flux, d

_{mf}appears to have negligible variation. It is worth mentioning that the high-pass filter forces the left leg of PDFs to be zero when A

_{proj}< 2 mm

^{2}. For a spherical bubble (b = 1; aspect ratios are nominally one for smallest bubbles), this minimum area translates into a minimum bubble size of d

_{b}< 1.6 mm.

_{32}and d

_{mf}. As seen in Equation (1), d

_{32}is a weighted average that is biased towards the largest bubbles generated due to the diameters being raised to powers before summing. Consequently, the influence of a large quantity of small bubbles has a weaker impact on d

_{32}than a few large bubbles. This can be seen in the cumulative density function (CDF) for these conditions provided in Figure 5b. The lowest flow rate exhibits significantly more large bubbles (e.g., 23% of bubbles are larger than 10 mm) than the highest injection flux (<5% of bubbles are larger than 10 mm), thus illustrating how these three conditions with nearly identical d

_{mf}values generate measurable deviations in d

_{32}.

_{32}and d

_{mf}is provided in Figure 6 with the most frequent bubble size plotted versus Sauter mean diameter for all test conditions. For reference, a dashed line corresponding to d

_{mf}= d

_{32}has been included, which shows that for all conditions d

_{mf}is smaller than d

_{32}. The majority of the data points collapse on a curve that appears to asymptote to d

_{mf}≈ 2 mm. The uniformity of these bubbles and insensitivity to the injection condition suggests that they are being generated by the flow-field, which the most likely mechanism would be the turbulent motions generated by the bubble wakes. This would suggest that d

_{mf}is a length scale associated with the velocity fluctuations within the flow-field. This conjecture is supported by the known Reynolds number dependence of bubble wakes. Bubble diameter (d

_{mf}) based Reynolds numbers (Re = V

_{b}·d

_{mf}/ν, where V

_{b}is the mean bubble rise velocity that is nominally U

_{sg}/α, α is the void fraction and ν is the kinematic viscosity) tested ranged between 590 and 11,000. Starting at a Reynolds number of ~500, vortices begin to be shed from bubbles and the flow-field becomes quite unsteady until ~1000. Starting at Re ~ 1000, a boundary layer forms on the bubble with a laminar near-wake region. However, the shear layer spreads resulting in a turbulent far-wake region. This behavior exits until Re ~ 3 × 10

^{5}, which is beyond the range of bubbles observed in the current study. Of note, a bimodal distribution is observed at lower Reynolds numbers (590 < Re < 2300), which is shown in Figure 7. This is a curious observation given that in this range the bubble wakes are unsteady with periodic shedding of vortex rings. The Strouhal number for Re ~ 1000 is ~0.3 [34], which the shedding from a 2.5 mm diameter bubble (nominal d

_{mf}for conditions in Figure 7) would produce an 8.3 mm long wavelength. This is comparable to the size of the second peak in the distribution.

_{mf}and d

_{32}are explored in more detail in the following section with a parametric study to assess the sensitivity to individual control parameters. Of note, over the conditions explored d

_{mf}(mode of PDF) is similar to d

_{10}(mean of PDF; defined using Equation (1) with the powers changed from 3.2 to 1.0). Given that the PDFs are skewed to larger bubbles, d

_{10}is generally larger than d

_{mf}and smaller than d

_{32}. While the behaviors are similar, they carry distinctly different physical information. While not explored in the current study, if the Reynolds number based on bubble diameter decreased below ~500, it is expected that d

_{mf}> d

_{10}. This is contrary to the current work where d

_{mf}< d

_{10}for all conditions.

## 4. Parametric Studies

#### 4.1. Gas Injection Rate

_{c}) is determined from the mass flowrate into the column (ṁ), column pressure (P

_{c}) and column temperature (T

_{c}). In the current experiment, the column temperature and pressure were held nearly constant at T

_{c}= 21 ± 1 °C and atmospheric pressure (plus hydrostatic pressure), respectively. Consequently, the mass flow rate was the only parameter varied, which was controlled with a combination of meter pressure (P

_{m}) and metered volumetric flow rate (Q

_{m}). Figure 8 compares d

_{32}and d

_{mf}dependence on the superficial velocity (U

_{sg}). Four different meter gauge pressures (P

_{m}= 40, 260, 400 and 600 kPa) were used to achieve 1.4 ≤ U

_{sg}≤ 55 mm/s. Sauter mean diameter shows good collapse over most of the test conditions, but there is some deviation observed with the P

_{m}= 40 kPa condition. Conversely, d

_{mf}collapses at lower superficial velocities but show some deviation at higher fluxes with P

_{m}= 600 kPa.

_{m}= 40 kPa with U

_{sg}= 11.1 mm/s condition. Images at the injection location compare this condition with other low mass flux conditions in Figure 9. Here it is apparent that the initial bubble size distribution is significantly different compared to the other low mass flux conditions. The Reynolds number based on the injector tube diameter for the outlier condition is 4800, which is at the transition between laminar and turbulent flow in a pipe. This makes the airflow at this superficial gas velocity transitional, which transitional flows are extremely sensitive to the operating condition. The data suggests that the lower metering pressure makes the initial bubble formation more sensitive to the inlet airflow condition. The metering pressure could impact bubble detachment from the injection tube since the upstream pressure could modify the bubble shape during expansion (especially with transitional flow). In addition, the initial bubble size distribution as well as breakup and coalescence behaviors are sensitive to the density of the gas [35].

#### 4.2. Injector Tube Angle

_{inj}= 0.8 mm. Results for both d

_{mf}and d

_{32}are provided in Figure 10 at each injector tube angle. These results show that d

_{mf}has negligible variation even with the significant misalignment. Conversely, d

_{32}has a measurable decrease at 45° relative to the 90° condition. There are two potential mechanisms responsible for this deviation; (i) the misalignment between gravity (buoyancy force) and the bubble wake where the turbulent production is located and/or (ii) increased influence of wall effects as the initial bubbles were directed into the column wall where the stress distribution will deviate from the core of the column. The wall effects are mostly likely for the current work since the decrease in bubble size suggests a higher shear stress.

#### 4.3. Injection Tube Diameter

_{inj}= 0.8 and 1.6 mm). Based on past observations [40], it is expected that increasing the injector tube diameter will increase the bubble size. Results for both d

_{mf}and d

_{32}are provided in Figure 11. The most frequent bubble size shows negligible variation between the injector tube diameters. This is consistent with the turbulent scales within the wakes setting d

_{mf}. The Sauter mean diameter trend is nearly identical between tube diameters, but the curve for the smaller tube is shifted downward slightly. This supports previous observations since it exhibits a dependence on the tube diameter, but the tube diameter was not varied by an order of magnitude resulting in the bubble size having a relatively small variation.

_{sg}= 6.9 mm/s produced PDFs with and without an apparent second peak) demonstrate that the PDFs are nearly identical between the two injectors. This explains why d

_{mf}is nearly identical between the two injector diameters, but not the shift in d

_{32}. The difference between the PDFs is that the larger injector tube diameters produced larger maximum sized bubbles (i.e., larger tube diameter produces a longer tail in the PDFs). Maximum measured bubble sizes (d

_{max}) for U

_{sg}= 6.9, 27.6 and 55.1 mm/s are provided in Table 1. This shows that the smaller bubble tube diameter produces significantly smaller d

_{max}(up to 40% smaller than the large tube). This supports the comments that both length scales are important since while d

_{mf}is insensitive to these changes, d

_{32}is modified because of these larger bubbles. While d

_{32}is sensitive to these variations, higher order statistics (particularly skewness, a measure the asymmetry of a distribution) should be more sensitive to these variations.

#### 4.4. Column Diameter

## 5. Higher Order Statistics

_{inj}= 0.8 mm and the closed symbols are d

_{inj}= 1.6 mm. Thus focusing on the large column (D = 102 mm) and P

_{m}= 600 kPa, the smaller injector tube diameter results in a smaller skewness at a given U

_{sg}. The kurtosis (a measure of “tailedness” of a distribution) is provided in Figure 14d, which for all conditions the kurtosis is greater than that of a normal distribution (κ = 3). The relatively high kurtosis values indicate the presence of infrequent excessive deviations from the mean. Furthermore, use of the skewness and kurtosis can provide a quantitative measure of the bimodality of the distribution (e.g., Sarle’s bimodality coefficient). There is a peak in this bimodality coefficient at a Reynolds number based on the d

_{mf}at ~1000. This supports the previous observations that the bimodality could be the product of the transition from the unsteady flow-field between 500 < Re < 1000 and the turbulent far-wake with Strouhal shedding above 1000. Thus, the higher order statistics are a potential means for identifying regime transitions within the column.

## 6. Conclusions

_{inj}= 0.8 or 1.6 mm) and the superficial gas flux (1.4 < U

_{sg}< 55 mm/s) were varied during testing. The range of superficial gas fluxes was controlled via a combination of pressure and volumetric flux measured/controlled upstream of the injection tube. However, the temperature both at the metering location as well as within the column were held nearly constant throughout testing. The large sampling of bubbles were used to generate PDFs for each test condition. The maximum peak in the PDFs was used to identify a new bubble length scale, which was termed the most frequent bubble size (d

_{mf}). This bubble length scale was compared with the traditional Sauter mean diameter (d

_{32}), which is a weighted average. Both were applied to a parametric study to determine the information that each length scale provides. In general, Sauter mean diameter is more sensitive to the largest bubbles within the flow while d

_{mf}is related to the turbulent structures created in the bubble wakes. Consequently, the difference between d

_{32}and d

_{mf}is a nominal range of bubble sizes expected within a given flow.

_{32}is biased towards the largest bubbles within the flow, it is expected that there would be a dependence of d

_{32}on injector tube diameter.

_{mf}.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic of the experimental setup including the bubble column, gas injection system and instrumentation to monitor and control flowrates. (

**b**) Top view of the column showing the camera and lighting configuration for bubble imaging.

**Figure 2.**(

**a**) Effect of column curvature on spatial calibration, ΔX = 5 mm, D = 102 mm. (

**b**) Spatial variation of the calibration coefficient across the column mid-plane.

**Figure 3.**Example image of bubble identification (identified bubbles are outlined). Note that out-of-focus bubbles are not identified due to the blurred edges.

**Figure 4.**Sauter mean diameter (d

_{32}) plotted versus the scaled vertical distance above the injection location. Each data point is the average of 10 repetitions, and the error bars are the standard deviation of the mean (P

_{m}= 600 kPa, T

_{c}= 21 ± 1 °C, D = 102 mm, d

_{inj}= 1.6 mm).

**Figure 5.**(

**a**) Probability density functions (PDF) and (

**b**) cumulative density function (CDF) of bubble size (d

_{b}) for the same conditions shown in Figure 4. The PDF/CDF for each U

_{sg}was determined from counting at least 10,000 bubbles. Dashed lines in (

**a**) correspond to the d

_{32}values for each condition (P

_{m}= 600 kPa, T

_{c}= 21 ± 1 °C, D = 102 mm, d

_{inj}= 1.6 mm).

**Figure 6.**Comparison between the most frequent bubble size (d

_{mf}) and the Sauter mean diameter (d

_{32}). The dashed line corresponds to d

_{mf}= d

_{32}. Open and closed symbols correspond d

_{inj}= 0.8 and 1.6 mm, respectively.

**Figure 7.**PDFs from bimodal conditions (U

_{sg}= 1.4, 3.5, 4.9 and 6.9 mm/s). While the d

_{mf}is still determined from the smaller bubbles, there is a second weaker peak near 10 mm (D = 102 mm; d

_{inj}= 1.6 mm).

**Figure 8.**(

**a**) Sauter mean diameter and (

**b**) most frequent bubble size plotted versus the superficial gas velocity. Error bars represent the standard deviation for the given condition (D = 102 mm; d

_{inj}= 1.6 mm; T

_{c}= 21 ± 1 °C).

**Figure 9.**Still frames in the D = 102 mm column with d

_{inj}= 1.6 mm with an injection condition of (

**a**) P

_{m}= 260 kPa, U

_{sg}= 3.5 mm/s; (

**b**) P

_{m}= 600 kPa, U

_{sg}= 6.9 mm/s and (

**c**) P

_{m}= 40 kPa, U

_{sg}= 11.1 mm/s.

**Figure 10.**Bubble sizes (d

_{mf}and d

_{32}) plotted versus the superficial gas velocity with the injector tube angle either 45° or 90° from horizontal (see insert sketch) (D = 102 mm; d

_{inj}= 0.8 mm; P

_{m}= 600 kPa).

**Figure 11.**Bubble sizes (d

_{mf}and d

_{32}) plotted versus the superficial gas velocity varying the injector tube diameter (D = 102 mm; P

_{m}= 600 kPa).

**Figure 12.**Bubble size PDF for the two injector tube diameters (d

_{inj}= 0.8 or 1.6 mm) tested at (

**a**) U

_{sg}= 6.9 mm/s and (

**b**) U

_{sg}= 27.6 mm/s (D = 102 mm; P

_{m}= 600 kPa; T

_{c}= 21 ± 1 °C).

**Figure 13.**Bubble sizes (d

_{mf}and d

_{32}) plotted versus the superficial gas velocity with different column diameters (d

_{inj}= 0.8 mm; P

_{m}= 600 kPa).

**Figure 14.**Higher order statistics from the PDFs including (

**a**) unweighted mean, (

**b**) standard deviation σ, (

**c**) skewness S and (

**d**) kurtosis κ of the bubble diameter. Dashed line on the kurtosis plot at κ(d

_{b}) = 3 corresponds to the kurtosis value of a normal distribution. The same legend is used for all plots. Open and closed symbols correspond to d

_{inj}= 0.8 and 1.6 mm, respectively.

**Table 1.**The maximum measured bubbles size (d

_{max}) spanning the flow rates tested with both injector tube diameters.

U_{sg} (mm/s) | Maximum Measured Bubble Size (mm) | |
---|---|---|

d_{inj} = 0.8 mm | d_{inj} = 1.6 mm | |

6.9 | 10.2 | 11.7 |

27.6 | 9.9 | 16.7 |

55.1 | 9.4 | 15.8 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Mohagheghian, S.; Elbing, B.R. Characterization of Bubble Size Distributions within a Bubble Column. *Fluids* **2018**, *3*, 13.
https://doi.org/10.3390/fluids3010013

**AMA Style**

Mohagheghian S, Elbing BR. Characterization of Bubble Size Distributions within a Bubble Column. *Fluids*. 2018; 3(1):13.
https://doi.org/10.3390/fluids3010013

**Chicago/Turabian Style**

Mohagheghian, Shahrouz, and Brian R. Elbing. 2018. "Characterization of Bubble Size Distributions within a Bubble Column" *Fluids* 3, no. 1: 13.
https://doi.org/10.3390/fluids3010013