Bubble size distribution, void fraction and mass transfer coefficient were measured with a liquid (distilled water) height above the injector (

H) of 0.85 m. In a static column, gas holdup is virtually independent of column dimensions when the column aspect ratio is H/D >5 [

33]. Under vibration, Budzyński et al. [

22] recommend

H > 8

D to minimize the impact of the liquid surface deformation on void fraction measurements. Air was injected into the column at superficial gas velocities (

U_{SG}) of 1.0, 2.5, 5.0, 7.5 and 10.0 mm/s. Two sets of experiments were performed at these gas flow rates. The first set was conducted to validate the experimental setup against published research data, and the second to investigate the effects of large amplitude vibration. The first set was tested at vibration frequencies of 0, 10, 12.5, 15, 17.5, 20 and 22.5 Hz with an amplitude of 1.5 or 2.5 mm, which closely matches established data in the literature [

18,

19,

20]. The second set was tested at amplitudes of 4.5, 6.5 and 9.5 mm over a frequency range of 7.5–17.5 Hz, which broadens the parameter space available in the literature.

Column averaged void fraction <

ε> was estimated using Equation (2) with the change in the liquid-air interface height (

H). The average interfacial height was determined with still images of a foam float on the air-water interface. The change in the foam float weight between the beginning and end of testing was 0.02%, which justifies that the impact of the float on the surface was constant throughout testing. A sequence of images of the column head was acquired with ImageJ (1.49v, National Institute of Health, Bethesda, MD, USA) [

38,

39,

40] for each condition to determine the stagnant liquid height (

H) and float during injection (

H_{D}). The spatial calibration was determined for each image using a known reference distance in the image. Hence, each image used was individually calibrated to reduce focal length error. An average

H_{D} was determined from the processed image data.

#### 4.1. Bubble Size

Bubble imaging shows that vibration improves the interfacial area by altering the bubble size distribution from a poly-dispersed large bubble population to a more uniform distribution. It is worth mentioning that unless stated, the bubble size measurement was conducted 6

D downstream of the injector tube to eliminate any influence from the injection condition [

24]. Prior to analyzing the mean statistics, the temporal evolution of the bubble size is examined to determine steady state conditions.

Figure 3 shows a time trace of

d_{32} under vibration (

A = 6 mm,

f = 10 Hz) with

U_{SG} = 6.9 mm/s, which shows that the bubble size becomes nearly constant after ~10 s. The bubble size distribution is examined in

Figure 4 with a probability density function (PDF) of bubble size. For this condition, vibration modifies the bubble size distribution from a bimodal distribution (corresponding to pseudo-homogenous bubbly regime [

41,

42]) in static column to a unimodal distribution (corresponding to mono-dispersed homogenous bubbly regime [

41,

42]) due to bubble breakage. The phase-averaged results for the vibration condition can be produced by combining the known time history given the sample rate and tracking a reference point located on the column wall [

43]. It is interesting that while under vibration, the shape of the distribution appears independent of phase; the largest observed bubble sizes appear to have a significant phase dependence.

Figure 5 illustrates the bubble breakage along the column height and its dependence on the vibration amplitude. The vibration input power increases with increasing amplitude, which results in a reduction in the bubble size.

Figure 5 also shows that at lower vibration amplitudes (i.e.,

A = 1.5 and 2.5 mm) the size distribution remains significantly poly-dispersed with a bimodal distribution. Another interesting finding from

Figure 5 is that increasing the amplitude has a significant effect on bubble shape as well as hydrodynamic behavior of the system (operation regime). In the present work, at lower vibration amplitudes the bubble column operates at a pseudo-homogenous regime [

41,

42] with cap shape bubbles of various sizes (see

Figure 5,

A = 1.5–2.5 mm). Increasing the vibration amplitude breaks the aforementioned bubbles into smaller oblate spheroids (

Figure 5,

A = 4.5–9.5 mm) and shifts the operation regime from pseudo-homogenous to a mono-dispersed homogenous regime [

41,

42].

Unlike the static case that had a bubble size distribution partially resembling a Gaussian distribution, the vibration case is better approximated as a log-normal distribution. The bubble size scaled with the injector diameter (

d_{i}) distributions corresponding to the conditions shown in

Figure 5 are shown in

Figure 6, which exhibits a log-normal distribution. These results are similar to the size distribution observed with four-point optical probes in stationary columns [

44,

45].

The relative change in bubble size with vibration is shown in

Figure 7, which plots the change in the bubble size due to vibration (

d_{32} −

d_{0}) scaled by the Sauter mean diameter without vibration (

d_{0}). These results are plotted versus the vibration amplitude (

A) scaled with the injector tube diameter (

d_{i}). Different symbols denote the test frequency, and throughout this paper, the error bars represent plus-or-minus one standard deviation. The amplitude was scaled with the injector tube diameter since under static conditions, the bubble size at detachment scales with the injector tube diameter [

24]. These results show the general trend that bubble size reduction can be a significant percentage of the static condition; the general trend is increasing frequency/amplitude results in a decrease in bubble size and minimal variation in standard deviation with vibration amplitude or frequency. Simultaneous studies of the bubble size distribution (i.e.,

Figure 4) and Sauter mean diameter (i.e.,

Figure 7) under vibration suggest that a minimum bubble size limit exists (

d_{32}/

d_{i} ≈ 0.7 &

d_{32}/

d_{0} ≈ 0.2) based on the fluctuating pressure field and turbulent shear balance of the surface tension. Therefore, bubbles smaller than this minimum size are extremely rare (see

Figure 6).

It is known that bubble breakup improves mass transfer by increasing the interfacial area (

a). Our results indicate that increasing the vibration amplitude (at constant frequency) produces smaller bubbles. Therefore, one could expect an increasing trend in mass transfer. However, experimental measurement of mass transfer shows a contrary result; this will be discussed in

Section 4.3.

Waghmare et al. [

18] proposed a correlation (Equation (16)) following the work of Hinze [

31] to predict the maximum stable bubble size as a function of specific power input and the properties of the continuous phase (i.e., surface tension and density). In Equation (16) the proportionality coefficient (

k) is a function of critical Weber number (

We =

ρ_{L}U_{b}^{2}d/

σ) and depends on the bubble breakup mechanism [

19]. The proportionality constant has been reported as

k = 0.725 for isotropic turbulent [

31],

k = 1.67 for shear bubble breakup [

32],

k = 1.7 for bubble breakage in a (memberance) pulsing bubble column [

18,

19] and

k = 1.73 in a (piston) pulsing bubble column [

46]. Waghmare et al. [

18,

19] used a pulsing column that produces an oscillating shear flow from use of a membrane, which could explain the reported proportionality constant closely matching that of the shear breakup. The fit of Waghmare et al. [

18,

19] is compared with the current vibrating column results in

Figure 8. Vibrating the whole column produces an oscillatory pressure field with negligible shear, which is distinctly different from the shear breakup mechanisms. The current results in

Figure 8 demonstrate a minimum input power (

P_{m} ~ 0.54 W/kg) to reduce the bubble size, and below this threshold the bubble diameter remains nearly constant nominally at the static bubble column value. Once the threshold is exceeded, there is a decrease in bubble size with increasing input power consistent with Equation (16) when

k = 3.4. Note that close to the threshold level there is evidence that the vibration produces a slight increase in bubble size relative to the static case (

d_{0}), which is consistent with data from Waghmare et al. [

18,

19].

#### 4.2. Void Fraction

Void fraction was measured at

A = 1.5 and 2.5 mm over a range of frequencies (0–22.5 Hz) and superficial gas velocities (1.0–10.0 mm/s). The current void fraction measurements (

ε) at

U_{SG} = 2.5 mm/s and

A = 2.5 mm are presented in

Figure 9 scaled with the static column void fraction (

ε_{0}). Error bars in

Figure 9 represent the standard deviation of surface displacement fluctuations. These conditions were selected since they closely match that of Waghmare et al. [

18] (

A = 2.46 mm,

U_{SG} = 2.5 mm/s), which are included for comparison. There is excellent agreement between the current results and those of Waghmare et al. [

18]. These results also demonstrate a near step change in void fraction close to

f = 17.5 Hz and the chaotic oscillations at the free surface at these high frequencies (i.e., large error bars when

f > 15 Hz).

A wider range of test conditions are shown in

Figure 10 with the void fraction (

ε) scaled with static column void fraction (

ε_{0}) and plotted versus the vibration amplitude scaled with the injection tube diameter (

d_{i}). These results were acquired at a single injection condition,

U_{SG} = 5 mm/s. The general trend is that increasing amplitude and/or frequency results in an increase in void fraction. Furthermore, the step change can be observed at each amplitude tested with the required frequency decreasing with increasing amplitude. These observations suggest that the void fraction should be proportional to a product of the frequency and amplitude, such as power input per unit mass (

P_{m}) or the transient buoyancy number (

M(

H)).

The current void fraction results are plotted versus

M(

H) and

P_{m} in

Figure 11 as well as a power-law fit for each plot. These results are consistent with Waghmare et al. [

18]. Also,

Figure 11 demonstrates that the void fraction is dependent on both of these parameters; however, it shows a stronger dependency on

M(

H). The correlation between

ε/

ε_{0} and

M(

H) supports the theory in Equation (15) that

M(

H) is a primary factor in scaling the void fraction.

The current void fraction results are compared with the predictive models proposed in the current work as well as Waghmare et al. [

18] in

Figure 12. Detailed investigations revealed that water is a low Morton number liquid (

Mo =

gμ_{L}^{4}/

ρ_{L}σ^{3} ~ 2.6 × 10

^{−10}) [

47]. Low Morton number liquids are characterized by a minimum in

C_{D,}_{∞} vs.

Re (Reynolds number) trend. Bubble Reynolds number (

Re =

U_{b}d/

ν_{L}; is based on bubble size (

d), rise velocity and, kinematic viscosity of the liquid (

ν_{L})) in this work ranges from 125 to 7000. Within the aforementioned range of Reynolds number and system properties (

Mo ~ 2.6 × 10

^{−10}) the drag coefficient exhibits exhibit a minimum (

C_{D,∞} = 0.15) at

Re = 440 and levels off when

Re > 4000 (

C_{D,∞} = 2.74). Therefore, using a proper Reynolds number based correlations to predict the drag coefficient is vital to produce an accurate model to predict void fraction. In the present work, experimental measurements of

C_{D,∞} from Brennen [

48] were used instead of using a correlation to calculate the drag coefficient on a single bubble.

Figure 12 illustrates the predictions of

ε/

ε_{0} from this work as well as that of Waghmare et al. [

18] in comparison with experimental measurements at various

M(

H)’s.

Figure 12 shows that Equation 19 has no success predicting the

ε/

ε_{0} accurately.

We used a trial and error approach to find a model that scales the void fraction over the entire test range of

M(

H). Starting with Equation (12) with

C_{D,∞} = 24/

Re a model for the void fraction is produced,

K_{ε} is assumed to be a constant related to experimental setup (here

K_{ε} = 50).

Figure 12 shows that Equation (29) offers an acceptable physics base prediction of

ε/

ε_{0} within the tested range. It is noteworthy that Equation (29) is only valid within the tested range and any predictions beyond the current parameter space must be verified against experimental data.

Example images from the recording of the free surface are shown in

Figure 13. Examination of these images reveals that the onset of air entrainment at the free surface occurs nominally at

M(

H) ~ 0.3. The chaotic oscillation of the liquid free surface captures large pockets of air, this phenomenon introduces an artificial increase in the measured void fraction. Surface entrainment is due to surface disintegration and free surface over turn at the wall [

49]. Surface disintegration happens at a wavy free surface when the wave crest evolves into a narrow-bottled neck fountain, which ultimately breaks into drop(s). Hashimoto and Sudo [

50] argued that the column aspect ratio (

H/

D) and vibration amplitude set the onset of surface disintegration. Air bubbles enter the column at the surface due to impact of the disintegrated drops. Note that on one hand increasing the vibration frequency reduces the size of the disintegrated drops [

51]. On the other hand, size of entrained air bubbles are proportional to the size of the drops and the wavelength of surface waves [

51]. Over turn of the wave crests at the column wall produces a thin film at the wall that also captures air bubbles into the column.

Surface entrainment elevates the free surface and increases the measured void fraction. One could now see the reason why models without considering the effect of surface entrainment will fail at predicting the void fraction.

#### 4.3. Volumetric Mass Transfer Coefficient

Particular frequencies and vibration modes correspond to significant intensification of mass transfer and void fraction [

12,

13,

14,

15,

16,

17,

18,

19,

20,

21,

22]. The current mass transfer measurements were validated with comparison to a subset of conditions tested in Waghmare et al. [

19]. Results from both studies are provided in

Figure 14 with the volumetric mass transfer coefficient (

k_{L}a) plotted versus the vibration frequency for a single injection rate (

U_{SG} = 2.5 mm/s) and vibration amplitude (

A = 2.5 mm). While the two results partially differ in the magnitude (with worth case of 35% difference at 20 Hz), overall they are consistent in trend as well as location of excitation modes (

f = 17.5 Hz). Both results show a steady increase in the mass transfer coefficient from ~10 Hz to 17.5 Hz, and at 20 Hz a slight drop creates a local maximum. It is noteworthy that at

f = 20 Hz a repeatable response of the setup was observed that could correspond to the natural frequency of the shaker table. Detailed investigations in the vicinity of

f = 20 Hz showed that

k_{L}a at

f = 19 Hz and

f = 21 Hz lies within the uncertainty range of Waghmare et al. [

19]. The decay mechanism at

f = 20 Hz was not investigated here; however, one can see the same behavior in data from Waghmare et al. [

19] as well. From

Figure 14 one could draw a conclusion that vibration modes corresponding to mass transfer intensification are independent of vibration method since both piston pulsing from Waghmare et al. [

18,

19] and column shaking in the present work produced almost identical results. However, more comprehensive data would be needed for both vibration methods to fully understand the difference and the impact of the vibration methods on

k_{L}a and void fraction results. Note that chaotic surface disintegration at higher frequencies causes larger error bars on the present data in

Figure 14 due to unintended surface entrainment (see

Figure 9).

Table 1 summarizes the measurements of

k_{L}a/

k_{L}a_{0} in the current work; here,

k_{L}a_{0} is the volumetric mass transfer coefficient in a static column. As expected increasing the vibration amplitude increases the

k_{L}a; however,

A = 2.5 mm and

f = 17.5 Hz corresponds to an agitation mode at which the

k_{L}a trend exhibits a significant peak. Bubble size is smaller at

A = 2.5 mm and

f = 20 Hz, the peak in

k_{L}a at

A = 2.5 mm and

f = 17.5 Hz shows that the phase interfacial area is not the only contributing factor in improving the mass transfer. In other words, although

d_{32} continues to decrease in

Figure 7,

k_{L}a/

k_{L}a_{0} exhibits an optimum at

A/

d_{i} = 4.25 (

Figure 15). Additionally, (

ε −

ε_{0})/

ε_{0} exhibits a larger increase and does not show a reduction after

A/

d_{i} > 4.25 (see

Figure 10) comparing with (

k_{L}a −

k_{L}a_{0})/

k_{L}a_{0}. This steady rise of void fraction agrees with a steady reduction in bubble size (see

Figure 7). However, the presence of a diminishing (

k_{L}a −

k_{L}a_{0})/

k_{L}a_{0} values for

A/

di > 4.25 (

Figure 15) indicates that improving

k_{L}a is not simply due to increased specific interfacial area (

a), but rather an improvement in liquid mass transfer coefficient (

k_{L}) dependent upon the frequency/amplitude combinations leading to a tuned column. This independent effect of frequency on

k_{L} has been reported in [

1].

The maxima in

Figure 15 suggest that within the tested frequency range an optimum amplitude exists. Even with the suggestion of an optimum amplitude, nearly equivalent mass transfer coefficients can be found at other frequency/amplitude combinations. An example is presented to illustrate the necessity to optimize the frequency/amplitude combination for

k_{L}a. Both

f = 12.5 Hz,

A = 4.5 mm and

f = 22.5 Hz,

A = 2.5 mm produce nearly equivalent

k_{L}a, which represents an 80% increase in amplitude traded for a 44% reduction in frequency. This trade gives a 44% reduction of vibration power

P_{m}, where

P_{m} is the specific input vibration power taken as the mass specific integral of force times the velocity over a quarter period,

Ultimately, this example serves to emphasize the need for a model that includes unifying parameters composed of frequency/amplitude combinations that can predict and optimize mass transfer in a vibrating bubble column.

Figure 16 demonstrates the dependence of

k_{L}a/

k_{L}a_{0} on

M(

H) and

P_{m} as well as a logarithmic fit. Given the importance of both power input (

P_{m}) and transient buoyancy number (

M(

H)) in governing the physical behavior of the system,

Figure 16 shows that

k_{L}a/

k_{L}a_{0} exhibits a stronger dependency on

M(

H) in comparison with

P_{m}.

Figure 17 illustrates the predictions of

k_{L}a/

k_{L}a_{0} from this work as well as that of Waghmare et al. [

18] in comparison with experimental measurements at various

M(

H)’s. We used the same approach as Waghmare et al. [

18] to build a correlation to predict mass transfer using a better estimation of void fraction. When

M(

H) < 0.3 both models predict the volumetric mass transfer coefficient relatively well; however, onset of surface entrainment attributes to the model failure. Due to scarcity of models for prediction of mass transfer in vibrating bubble columns, we decided to provide a correlation to predict mass transfer and test it within the range of our experimental data as well as that of Waghmare et al. [

18].

Previous works [

10,

16,

17,

18,

19] have contributed a physics-based model that supports the theory that specific power

P_{m} and superficial gas velocity

U_{SG} are the primary factors in mass transfer. However, Waghmare et al. [

18,

19] have been supported in part by limited experimental evidence. The model is given by the expression:

Regression of the data from the present research shows that

M(

H) is a stronger factor than

P_{m}. It is noteworthy that

Bj was selected for regression instead of

M due to the fractional difference between

Bj and

M and repeated use of

Bj in vibrating bubble column literature. Upon further investigation, data from Waghmare et al. [

19] also show the same result. A simple hypothesis test (

t-test) on the exponents of the

P_{m} terms compared to the theoretical exponential value of 0.8 found in Equation (31) indicates that none of the regression results using data from Waghmare et al. [

19] are statistically equal to the exponent; see

Figure 18. Taken together with the

U_{SG} data, as in Equation (32), regression of the data gives:

Using Minitab, linear regressions and ANOVA analyses were performed on both data from Waghmare et al. [

19] and the present research. Void fraction and volumetric mass transfer coefficient models from Waghmare et al. [

19] are given in Equations (33) and (34), here,

U_{SG},

P_{m} and

Bj can be analyzed separately for independent influences on mass transfer coefficient. Assuming

U_{SG} has a linear contribution to the results

P_{m} and

Bj can be analyzed independently. Analysis over the present work and that of Waghmare et al. [

19] shows

Bj scales the

k_{L}a more prominently in comparison with

P_{m} (see

Figure 19). Independent linear regression of both data sets as a function of

Bj gives exponential values that are statistically equivalent. Therefore,

P_{m} may not be as significant a factor as initially proposed.

A linear ANOVA analysis over

U_{SG},

P_{m} and

Bj showed that

Bj is the prominent term in scaling the mass transfer, and results of the present work as well as those from Waghmare et al. [

19] are in agreement with correlation of Equation (33) as shown in

Figure 20,

Here, K is a constant term related to the experimental conditions and fluid properties, here, K = 0.015.