Drag Coefficient and Dimensionless Numbers

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Swarm Effect

Anyway, the present investigation is not dealing with isolated bubbles, but with bubble swarms. Bubbles interact with eachother and can even coalesce or break-up. This makes it difficult to estimate the rising velocity, especially in the case of polydisperse bubble size distribution. (Batchelor, 1972), (Wijngaarden & Kapteyn, 1990) and (Koch, 1993) inves-tigated dispersed flows with void fraction under 5%. They concluded that for such dilute flows, bubble interactions are mainly binary and induce trajectory modification. (Hallez & Legendre, 2011) studied numerically the interactions between two bubbles ascend-ing side by side in stagnant liquid. The authors gave three possible contribution which are:
– potential effect.
– viscous correction (or Moore correction) – wake effects.
The wake effects were already investigated in detail by (Cartellier & Rivière, 2001) and (Cartellier et al., 2009). The authors reported a decreasing probability of the second bubble being in the wake of the first one due to lift force. Last one favors bubble ejection from the wake. However, (Wallis, 1969), (Garnier et al., 2002) and (Riboux et al., 2010) showed a decreasing bubble velocity with increasing void fraction. This result is in agreement with (Legendre et al., 2003) who investigated the infuence of the distance between two bubbles rising side by side via numerical simulations. They defined a critical distance under which the vorticity of the bubbles interact, modifying the pressure distribution and increasing the drag coefficient. This phenomenon is in competition with another one reported by (Bouche et al., 2012) who studied bubble interaction at high Reynolds numbers. They concluded that bubble induced agitation modifies the viscous dissipation behind the first bubble which entrains the following one. This effects the drag coefficient which decreases with increasing void fraction. A wake acceleration effect was also reported by (Krishna et al., 1999). The authors showed an increasement of the averaged gas velocity by a factor of 3 to 6 for large bubble swarms depending on column dimensions, bubble sizes and void fractions. The highest velocities were recorded in churnflow conditions. It seems that depending on bubble size and shape as well as on the void fraction, the drag coeffcient can increase or decrease which makes it difficult to estimate correct bubble swarm velocities, especially in the case of polydispersed bubble size distribution. All affecting parameters are driven by liquid properties and injection conditions. This is confirmed by (León-Becerril et al., 2002) and (Roghair et al., 2011) who explained the dispersion of results in literature by eccentricity and Eötvös number variations. However, two swarm velocity corrections from literature can be given for dilute bubbly flows.

Bubble Columns

The mixing ability of bubble columns depends mostly on the present flow structure induced by the injected gas bubbles. Depending on liquid properties, column geometry and injection conditions, different flow regimes can be observed. Furthermore, two bubble column categories can be distinguished. In literature, several authors like (Drahoš et al., 1991), (Zahradnik et al., 1997) or (Diaz et al., 2006) illustrated flow regime characteri-zation in terms of gas hold-up depending on the superficial gas velocity which is defined as: UGS = Qinlet (I.56).
In figure I.4 the above mentionned parameters are used to characterize bubble column types. The solid line indicates qualitatively different flow regimes as a function of the superficial gas velocity in the case of homogeneously aerated bubble columns. At low gas injection, the dispersed bubble regime is characterized by an uniformly rising bubble swarm with almost uniform gas hold-up in the bubble column cross section. With increasing superficial gas velocity heterogeneities in form of large eddies appear indicating the beginning of the transition regime. Finally, with further increasing superficial gas velocity a local minimum in the gas hold-up defines the beginning of the so called turbulent regime (I.4).
The second category, heterogeneously aerated bubble columns, are presented by the dashed line. The flow structure takes the shape of a bubble plume for higher aspect ratios than 2:25. For the whole range of superficial gas velocities, no regime transition characterized by the gas hold-up evolution can be observed. The present flow regime can be recognized by large ascending bubbles in the column middle and small descending bubbles along the column edges.
The present contribution focuses only on mixing abilities of bubble plumes generated in heterogeneously aerated bubble columns. The facility is used to study hydrodynam-ics of an isolated plume in order to better understand meso- and macro-scale mixing of large bioreactors. But even if such bubble columns show lower void fractions, it is still difficult to obtain local time resolved experimental data for both phases. For this reason we have chosen to work mainly with a pseudo two dimensional bubble column allowing the application of visual metrological methods like Particle Image Velocimetry (PIV) and Shadowgraphy. In addition, some complementary experiments were performed in a cylin-drical three dimensional bubble column as used in industries.
However, many authors were interested in the characterization of bubble plumes in pseudo two dimensional bubble columns in order to develop predictive tools like CFD simulation. Exact mechanisms and phenomena furthering mixing are still unclear since fluid dynam-ics depend on complex interactions between phases concerning mass, momentum, and energy transfer. Characteristics length and time scales of macromixing are unpredictible due to the lack of model and the lack of experimental informations. Especially effects of surface tension and viscosity are poorly or not at all investigated. Therefore, models with adequate closure terms have to be tested and compared to experimental data.
A large spectrum of investigations concerning cylindrical bubble columns can also be found in literature. Most authors were interested in the understand of the voidage to superficial gas velocity relation including regime transition and were looking for suitable flow pattern characterization ((Akita & Yoshida, 1973); (Deckwer, 1980); (Hikita et al., 1980); (Maruyama et al., 1981); (Zahradnik et al., 1997); (Vial et al., 2000); (Ruzicka et al., 2001); (Gourich et al., 2006) ; (León-Becerril et al., 2002)). All of them studied homogenously aerated columns showing uniform bubble swarms in most cases. Only few authors like (Simiano et al., 2006) or (Rensen & Roig, 2001) used 3D bubble columns with a small injection area in the column center, which can be considered as single spot injec-tion, to investigate bubble plumes. This configuration can also be modified to so called pseudo two dimensional bubble columns which allows the application of optical metro-logical methods (Sokolichin et al., 1997). The almost two dimensional column geometry damps three dimensional mesoscale instabilities which generates a quasi two dimensional bubble plume.

Pseudo-2D Bubble Columns

In the case of pseudo two dimensional bubble columns a sinusoidal trajectory of ascending bubbles predominates the column flow regime while a helical trajectory predominates in the case of cylindrical bubble columns. In both column types, bubble plumes show large ascending bubbles in the column middle and small descending bubbles at the column edges captured by liquid recirculation. All acting mixing mechanisms are the same, which explains the interest of pseudo-2D columns.
Becker et al. (Becker et al., 1994) gave the state of art of modelling of gas-liquid flows in bubble columns and confronted results to experimental observations. They were able to reproduce experimental results with numerical simulations based on a dynamic laminar two-dimensional two-phase Euler-Euler model. A strong influence of the gas distribution system which will be discused later, was also mentionned. (Delnoij et al., 1997) proposed an Eulerian/Lagrangian model for a 2D-BP to model the flow pattern that was confronted to experimental results. Further, the effect of aspect ratio, relation between column height and column width, from 1 to 11 has been studied. First, they observed flow transition for aspect ratios in a range from 1 to 3. Secondly, they revealed that at 7.7 and higher aspect ratios, flow structure was found to consist of two different regions. In the upper part bubbles are dispersed over the entire cross section of the bubble column and vortices do not appear. In the lower part a clear bubble plume was observed. Some years later Diaz et al. (Diaz et al., 2006) also investigated flow transitions for low aspect ratios and proposed figure I.5 to illustrate different flow regimes depending on the superficial gas velocity and aspect ratio. They completed the work of (Delnoij et al., 1997) and defined three different flow regimes which were observed during their work.
For small aspect ratios (WH 1:5) two pseudo steady flow structures can be seen. At low superficial gas velocities a Single Cell Bubbly Flow (SCBF) characterized by convex bubble trajectory and only one vortex cell, appears. The second one is called Double Cell Transition Flow (DCTF) and has the same shape than a vertical vortex dipole, with large upward moving bubbles in the column middle and two vortex cells, one on each side generating downward moving bubbles on the column edges. For aspect ratios larger than WH = 2 an unsteady Vortical Flow (VF) is formed. This regime can be recognized by its sinusoidal trajectory of ascending bubbles. At the column edges and close to the free surface vortical cells are formed and their number increases with increasing aspect ratio. Plus, bubble plumes (or vortical flows) show a low-frequency oscillating behaviour. The focus of the present investigation will be on this phenomenon in aim to better understand its contribution to column mixing.

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Oscillating Bubble Plume

Several authors like (Delnoij et al., 1997), (Rensen & Roig, 2001), (Buwa & Ranade, 2003) and (Diaz et al., 2006) took a closer look to the low frequency of the wandering bubble plume. For example (Mudde & Simonin, 1999) were able to reproduce bubble plume oscillations numerically. Their results showed comparable time scales than experimental data.
Anyway, from this moment we only consider bubble columns for aspect ratios from 3 to 7. In this way one makes sure that the aspect ratio can be left out of consideration. (Delnoij et al., 1997) studied oscillation frequencies for superficial gas velocities in the range from 2 mm/s to 6 mm/s and showed a strong relation between these two parameters.
With increasing superficial gas velocity, plume oscillation frequency increases as well. In our laboratory, periodic plume time scales were already investigated via optical probe measurements by (Aouinet, H., 2016). Rensen & Roig (Rensen & Roig, 2001) found that this frequency is persistent over the whole column height. They concluded that the horizontal density gradient and the velocity profile of the entrained liquid are the driving terms of the unstable bubble plume behavior. One should mention that the ascending bubble plume naturally disperses which, after a certain height, is damped by the confinement of column walls.
However, in the current investigation our focus is on the impact of fluid properties. (Buwa & Ranade, 2003) were the first ones who used other fluids than pure water in pseudo-2D bubble columns. They pointed out that the oscillation period does not change by the addition of saturated NaCl. This is in disagreement with (Cachaza et al., 2011) who used NaCl to modify surface tension in order to analyze their impact on flow patterns. It is probable, that (Buwa & Ranade, 2003) did not notice any differences because of the small superficial gas velocity range (no flow regime modification) and the use of NaCl as tracer to measure mixing times and not no modify fluid properties.
To our knowledge (Cachaza et al., 2011) are the only ones who investigated the influence of surface tension on flow patterns in such pseudo-2D configurations. Fig-ure I.6 shows flow structures for three liquids with different surface tensions, but with almost same density and viscosity (see figure I.1) at two different superficial gas velocities.

Metrological Methods

In the case of 2D-BP five different metrological methods are applied in order to study hydrodynamic structures as well as oxygen mass transfer through the bubble interface. Pressure sensors, oxygen probes and PIV measurements are used to analyse the liquid phase. The bubbles or dispersed gas phase is examined by using shadowgraphy. Plus, absorbance measurements were realized to determine mixing times.

Particle Image Velocity

The particle image velocity (PIV) measurement technique is an optical method to obtain flow visualizations in form of two dimensional velocity fields.

2D Bubble Plume (2D-BP)

The idea is as follows: a flat laser sheet illuminates a fluid charged with seeding particles. In orthogonal direction to the acquisition window (or to the laserstheet) a camera takes image pairs with short time-interval depending on fluid velocity. The images are divided in a certain number of submatrix or interrogation areas. Next, spatial intercorrelation is applied to every submatrix of image pairs, giving a velocity vector. Depending on image and submatrix size, resolution can change significantly. For more details of PIV application in multiphase flows, one refers to the article from (Lindken & Merzkirch, 2002). In our case, an adaptive PIV method which is included in the image treatment software DynamicStudio from Dantec, is used. This method iteratively optimize the size and shape of each interrogation area in order to adapt to local flow gradients and seeding densities. In this way, field resolution can be improved. The acquisition window was fixed to 1600 840 pixels2 which corresponds to 167 87 mm2 covering half of the column width. The right top angle of the interrogation window is just next to the pressure sensor on the right side. The intercorrelation matrix size could be decreased to 16 16 pixels, the time between images was fixed to 2ms and the acquisition frequency to 15Hz. Rhodamin-B colored particles with size range from 1 m to 20 m are used as seeding particles because of their light spectrum. Indeed, the highest light absorption is in the green range with a peak at 550nm while the highest emission is in the orange and red range with peak at 590nm. Hence, a green (532nm) laser (Skylight from Dantec) is used as lightsource for Rhodamin-B particle excitation. Plus, a highpass light filter (over 570nm) is installed to the camera in order to only register the emitted red light from the particles. In this way, light reflections from bubble interfaces could be avoided on PIV-images. In order to obtain a good statistic of at least 20 oscillation periods, 10000 image pairs are taken during every experiment.

Shadowgraphy

A second camera with the exact same interrogation window (as the first one for PIV) is used for shadowgraphy in order to analyze the gas phase. Informations about bubble size, shape, spatial distribution and velocity are extracted. Plus, a second acquisition window over the whole column width is used in order to perform complementary void fraction measurements. The second window is fixed to 2048 1280 pixels corresponding to 28:7 17:9 cm2. One must mention that shadowgraphy is a two dimensional measurement. Because of the high image depth of camera objectives, the whole column depth is taken into account. This is particulary important, when it comes to void fraction calculations. A homogeneous LED-pannel is installed behind the transparent column. The camera registers projected shadows of each bubble on the interrogation window. Bubbles appear as dark shadows because of the light refraction on the interface as illustrated in figure II.3.
The frequency is the same as for PIV measurements (15Hz) and image pairs are taken with a time interval of 2ms as well. Both cameras are perfectly synchronized. In our investigation a blue LED-pannel (at 480nm) and a bandpass light filter (around 480nm) are used. Hence, the second camera captures only the gas phase without any light perturbation from the PIV laser. In this way, one could make sure to focus on just one fluid at the time per camera in order to consider both phases separately. In case of shadowgraphy 10000 images pairs are taken for the same reason than for PIV.

Image processing

Authors like (Ferreira et al., 2012), (Mikaelian et al., 2015) and others used shadowgraphy to characterize bubble morphology in terms of size distribution and shapes in more or less dense bubbly flows. In most cases, a bubble size based filter is applied that one wanted to avoid here. Our data treatment is supposed to be applicable to all kinds of bubble sizes and shapes, even in bubbly flows with polydispersed size distribution. The idea is to isolate well identified bubbles (WIB) in order to extract informations like bubble eccentricity as a function of bubble size. The second aim is to characterize bubble interactions well enough to calculate void fractions as close as possible to reality. Hence, an algorithm in Matlab was developed to detect all kinds of objects on shadowgraphy images and identify their nature such as overlapping, coalescing, upbreaking and deformed single bubbles. First of all, non homogeneities of the background light are removed by applying a so called flat on every image. The flat is nothing else than a shadowgraphy picture of the same acquisition window without any bubbles. A light filter is used to further intensify object contours as it can be seen in figure II.4.

Table of contents :

I Bubbly Flow 
I.1 Bubble Dynamics
I.1.1 Size & Shape
I.1.2 Mass Transfer
I.1.3 Bubble Rising Velocity
I.1.3.1 Terminal Velocity
I.1.3.2 Drag Coefficient and Dimensionless Numbers
I.1.3.3 Contamination
I.1.3.4 Swarm Effect
I.2 Bubble Columns
I.3 Pseudo-2D Bubble Columns
I.4 Oscillating Bubble Plume
I.5 Conclusion
II Experimental Methods & Analyzes 
II.1 2D Bubble Plume (2D-BP)
II.1.1 Setup
II.1.2 Fluid properties
II.1.3 Metrological Methods
II.1.3.1 Particle Image Velocity
II.1.3.2 Shadowgraphy
II.1.3.3 Image processing
II.1.3.4 Bubble Image Velocimetry (BIV)
II.1.3.5 Mixing Time
II.1.3.6 Pressure & Oxygen sensors
II.1.4 3D Bubble Plume (3D-BP)
II.1.4.1 Setup
II.1.4.2 Wire-Mesh
II.2 Time Series Analysis
II.2.1 Spectral Analysis
II.2.2 Autocorrelation
II.2.3 Proper Orthogonal Decomposition
IIIExperimental Results 
III.1 Oscillation Frequency
III.1.1 Water
III.1.2 Liquid Property Influences
III.1.2.1 Surface tension
III.1.2.2 Viscosity
III.1.3 Dimensionless numbers
III.2 Bubble Characterisation
III.3 Bubble Dispersion
III.3.1 Water
III.3.2 Liquid Properties
III.4 Liquid Velocity
III.4.1 Water
III.4.1.1 Vertical Direction
III.4.1.2 Horizontal Direction
III.4.2 Liquid Properties
III.4.2.1 Vertical Direction
III.4.2.2 Horizontal Direction
III.5 Bubble Swarm Velocity
III.5.1 Water
III.5.2 Liquid Properties
III.6 Mean Velocity Difference
III.7 Mixing Time
III.8 Mass Transfer
III.9 Comparisons with 3D-BP
III.9.1 Oscillation Frequency
III.9.2 Dimensionless Numbers
III.9.3 Bubble Dispersion
III.9.4 Conclusion
IV CFD 
IV.1 Modelling
IV.1.1 Two Fluid Model
IV.1.2 Closing Terms
IV.2 Test Case
IV.2.1 Mesh
IV.2.2 Simulated Conditions
IV.3 Results
Conclusion and Perspectives
Shadowgraphy Images in Different Fluids and for Both Spargers
References

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