Bubble Behavior in Marine Applications of Bubble Columns: Case of

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Thermodynamics, Transport, and Fluid Mechanics

Bubble behavior in marine applications of bubble columns – Case of ellipsoidal bubbles in slanted and rolling columns Carsten Drechsler, Amir Motamed Dashliborun, Seyed Mohammad Taghavi, David W. Agar, and Faïçal Larachi Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b05673 • Publication Date (Web): 18 Jan 2019 Downloaded from http://pubs.acs.org on January 22, 2019

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Industrial & Engineering Chemistry Research

Bubble behavior in marine applications of bubble columns – Case of ellipsoidal bubbles in slanted and rolling columns Carsten Drechsler,1,2 Amir Motamed Dashliborun,1 Seyed Mohammad Taghavi,1 David W. Agar,2 Faïçal Larachi,1,* 1

Laval University, Department of Chemical Engineering, 1065, Avenue de la médecine, G1V

0A6 Québec, QC, Canada 2

TU-Dortmund University, Chair of Chemical Reaction Engineering, Emil-Figge-Straße 66,

44227 Dortmund, Germany * [email protected]

ACS Paragon Plus Environment

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ABSTRACT Using a hexapod ship motion emulator, the influence of floating vessel motion on the trajectory and velocity profile of single ellipsoidal bubbles is investigated from the images provided by a high-speed camera. Varying the amplitude and frequency of the vessel motion, relations between the column motion and variations in the bubbles’ trajectories are analyzed. Spectral analysis by means of Fast Fourier Transforms helps demarcating the factors influencing the bubbles’ zig-zag flow patterns, in terms of contributions stemming from column wall effects, column oscillation frequency and column angle of inclination. Although the column rolling motion is shown to have little effects on the rising velocity of the bubbles, its incidence on shaping the bubbles’ number density distribution and bubble flow patterns is remarkable, suggesting that these factors have to be accounted for in the marine context of floating vessels. KEYWORDS Hydrodynamics; Hexapod ship motion emulator; High-speed camera; Bubble column; Bubble rise velocity; Bubble trajectory


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1. Introduction Bubble columns are among the most important basic types of reactors that enable efficient realization of a multi-phase reaction system1, whereby a broad range of applications in various chemical as well as biochemical processes exists. Among these, the Fischer-Tropsch2 or methanol syntheses3,4 as well as fermentation processes5 might be exemplarily named. Their simplest design, which requires neither internals nor moving parts, involves low maintenance and operation costs, which go along with a high reactor durability.1,6 Moreover, they ensure high mass and heat transfer coefficients1, an easy addition or withdrawal of catalysts7 and good mixing characteristics of the two- or three-phase system8. Antagonistic to the relatively simple operation of bubble columns, their hydrodynamics is formidably complex, even at the level of a single-bubble scale which is often conjectured to represent a good proxy for deciphering the hydrodynamics of homogenous bubbly flow regimes.9 An impressive repository of information has thus far become available through various investigations on the bubble flow characteristics over the last century, making this contacting mode a well-researched unit operation.9,10 Nevertheless, recent trends for processing gas and oil from reservoirs in remote offshore areas are inciting for new applications of bubble columns to be installed and operated on mobile offshore platforms, e.g. floating production, storage and offloading (FPSO) units.11 In such settings, a large number of unit operations and storage devices are positioned on the same floating system.12 This allows a mobile complex to sail where necessary to exploit remote gas and oil reservoirs offering a high degree of flexibility while avoiding costly hyphenating pipelines to onshore processing/refining units.13,14Acknowledging that the use of FPSO ships would allow for various advantages, including cost-effective exploitation of fast-depleting reservoirs or their use in high-risk areas, e.g., iceberg traffic15, their operation is nonetheless


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bound to significant technical challenges. These latter are suspected to arise because the motion and inclination of the ship, triggered by wind and/or marine swell conditions, also has an impact on the process units onboard. Consequently, the operation of bubble columns needs to be further investigated under these new atypical constraints, which might be responsible for significant changes in the vessel performance.16,17 The hydrodynamics of static bubble columns has been subject to intensive research over the past several decades.9,10 Already in the work of Clift et al.,9 which summarizes the fundamentals of the hydrodynamics in a multi-phase system, a strong dependency between bubble shape and rise velocity has been recognized. Generating a bubble shape dependent map later confirmed by the studies of Mersmann et al.,18 the authors showed that the aforementioned dependency can be well expressed in terms of the system’s Reynolds, Eötvös and Morton numbers. Moreover, the authors remarked that not only the bubble’s rise velocity, but also its trajectory, significantly depend on the bubble shape and, thus, on its size. For instance, a transition from rectilinear trajectories reminiscent of small bubbles (equivalent diameter de  1.3 mm) to more complex paths at larger bubble sizes, such as the zig-zag or helical trajectories of ellipsoidal bubbles, has been acknowledged9,19 in the literature using state-of-art high-speed imaging techniques.20–22 Studies by Hassan et al.,22 which compare trajectories for different bubble sizes as a function of liquid-phase viscosity, ascribe the variations in bubble trajectory and velocity to a change in the bubble’s friction coefficient. Subsequent studies for bubble rise in deionized water by Yan et al.21 confirm that the non-random fluctuations in the rise velocity of ellipsoidal bubbles originate from coupled oscillations of bubble shape and bubble drag coefficient. Bubble’s shape and rising behavior are indeed strongly interrelated owing to friction and surface tension effects at the bubble interface as also evidenced by the surfactant addition studies of Kracht and Finch.23


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A further fact which cannot be ignored is that the streamwise bubble rise velocity goes hand in hand with its crosswise velocity component as has also been highlighted.24,25 At this point, a question legitimately arises as to whether or not velocity and trajectory properties, as just described regarding ellipsoidal bubbles, might be markedly altered by externally superimposed movements of the enclosing vessel such as in FPSO applications. More specifically, how the bubble responds to additional distortions of its trajectory due to liquid flow field secondary motions imparted by statically inclined or moving columns (e.g., rolling oscillation) is worthy of investigation. In this context, especially the analysis of ellipsoidal bubble flow is of interest, as in most thermo-physical systems, this flow regime applies for intermediate sized bubbles (de  1 – 15 mm)9, which are of high technical relevance, e.g., Fischer-Tropsch synthesis (1 – 5 mm small bubbles, 5 – 30 mm large bubbles).26 On the other hand, this regime is probably the one showing the richest, distinct flow behavior, including transitions in the secondary flow pattern from nearly rectilinear flow to helical and/or zig-zag motion.9 However, even if well documented information of the bubble’s terminal velocity9,18,27 and trajectory22,28–30 in dependence of its shape are available for single bubble rise in statically, noninclined columns, there seems to be a lack of data considering single bubble flow in statically or dynamically inclined columns. Only a few studies, exclusively dedicated to statically inclined bubble columns, can be found in the literature. Introducing a dissymmetry distributional coefficient, Hudson31 showed that the maldistribution of gas-liquid systems depends on the gas superficial velocity and on the angle of inclination of the column. Moreover, a secondary flow in the liquid phase, induced by the rising bubbles, with the liquid phase flowing upward with the bubbles on the one side of the column and downward at the column’s opposite side, has been reported, indicating a strong drag-based interaction in the hydrodynamics of the two-phase flow


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system. This latter observation is confirmed by Lu et al.32 who measured the influence of the inclination angle on the bubble rise velocity. These authors observed an influence of the inclination angle on the bubble rise velocity only in their multi-bubble flow tests. Whereas due to lack of bulk liquid circulation and in accordance with Masliyah et al.,33 no significant dependency on vessel slanting was observed for the single-bubble rise tests. Moreover, Masliyah et al.34 found that for small bubbles (Re < 300, We < 1.6), a unique drag coefficient master curve, embedding the inclination angle in the bubbles Reynolds and Eötvös numbers, can be established. For larger bubble sizes, on the contrary, investigations by the same authors33 indicate that a more complex relationship ties the drag coefficient to the traditional force-based dimensionless numbers. This results from the oscillatory behavior of the bubbles bouncing against the column wall. Considering the aforementioned aspects, it becomes obvious that the columns’ static inclination can have a significant effect on the hydrodynamics of gas-liquid flows. Furthermore, additional movement of the column would add more degrees of freedom to the system, which might influence the rise velocity of the bubbles, due to emerging secondary flows in the liquid. In addition, changes in the bouncing behavior of the bubble as a function of column motion might also occur. Studies on the influence of the vessel’s motion on bubble columns are very scarce. Using wiremesh capacitance imaging, Assima et al.17 measured the instantaneous gas holdup distribution in moving bubble columns mounted on a hexapod swell simulator subject to different oscillating behaviors, such as rolling, pitch, yaw, heave and sway at various frequencies. However, their study did not touch upon the behavior of single rising bubbles hosted in the same oscillating column motion to infer its resulting effects through bubble rise velocity and shape effects. Nevertheless, their study has shown that a strong relation exists between the column oscillation


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frequency and the instantaneous evolution of bubble distribution. For instance, under column rolling motion, the bubble swarm follows a periodical transverse zig-zag trajectory which dramatically differs from the nearly vertical bubble rise swarm prevailing under static vertical vessel conditions. Their observations reveal that the transverse frequency of bubble motion locks at diapason with the column motion frequency leading to a significant time-varying maldistribution in the bubbles density function. Moreover, the strong increase of bubbles rise velocity upon switching the vessel from statically inclined to dynamic conditions has only been described qualitatively on account of the difficulty of analyzing the multi-bubble system by wiremesh capacitance technique. Despite indications that the column motion has a significant influence on the hydrodynamics of the gas-liquid flow, observations at the single-bubble trajectory level were not possible. To the authors’ best knowledge and according to the above literature survey, there does currently not exist any literature on experimental investigations of the influence of vessel motion, corresponding to marine applications of bubble columns, particularly, on the single-bubble rise phenomena. Even if in principle the basic flow behavior of the bubbles might be captured by numerical simulations, e.g., minimally via inclusion of a Coriolis force in the bubble force balance equation, detailed experimental investigations on single bubble flow are crucial for understanding and validation of the basic hydrodynamics relevant to sea conditions as a basis on which more complex multi-bubble flow behaviors can be built upon. Based on statistical and spectral analyses of single-bubble trajectories under static and roll dynamic conditions in the hydrodynamically and technically most interesting ellipsoidal bubble shape regime, the present study evaluates the transition from the statically inclined to the dynamic column. Consequently,


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it allows for gaining detailed insights on the bubbles’ trajectory and terminal velocity and statistics as a function of inclination angle and roll frequency of the column. 2. Experimental Figure 1 illustrates the main elements of the experimental setting used to record the single bubble rise trajectories under emulated effects of ship motion in liquid phase batch operation mode. It consists of a Plexiglas column with an inner diameter (dc,i) of 5.7 cm filled with deionized water up to a height of 60 cm. An orifice (do = 1 mm) pierced at the center of the column’s bottom was connected via a long capillary tube to a mass flow controller through which bubble generation at constant gas flow rate conditions was ensured by a high pressure loss. A bubble detachment frequency (fb,detach) ca. ~1.7 Hz was achieved using pure nitrogen flow rate ( 1

) of ~1.7 mL min-

while generating ellipsoidal bubbles with a volume equivalent sphere diameter (de) of approx.

3.22 mm (bubble volume Vb  0.0174 mL). The single-bubble trajectories were tracked and recorded by means of a high-speed camera (BASLER acA2040-90um35). Images were acquired at a frame rate (fframe) of 80 Hz for 15 s and 20 s, respectively, for static and column motion experiments. The suite of data post-processing steps included cutting the gray scale image to separate the region of interest, followed by a (subtractive) comparison with a reference image. The instantaneous coordinates of the bubble centroid were determined, after binarization of the difference image’s color map, using image processing functions (i.e., bwconncomp, regionprops) provided by MATLAB.36 Optical distortions by the cylindrical wall curvature and the difference in the refraction indices between fluids and column material were reduced to a minimum37 by immersing part of the column in a parallelepiped Plexiglas box filled with deionized water. This resulted in a corrected field of view corresponding to a measurement region of 40 cm in height starting 10 cm above the orifice. ACS Paragon Plus Environment

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Furthermore, use was made of diffusive layers to allow for scattering of light beams from the LED light source placed orthogonal to the camera’s field of view. Thus, the column being surrounded by these layers averted inhomogeneous illumination intensities of the field of view.

Figure 1. Experimental setting used to record the bubble trajectory. The hexapods’ three rotational degrees of freedom are shown. For reasons of clarity only one of the two diffusive layers which border the measurement region in

direction is shown. Furthermore, the device

which fixes the high-speed camera to the hexapod platform is not presented.38 Hexapod drawing adapted

with

permission

from

Symétrie

vendor

(Nîmes,

France,

http://www.symetrie.fr/en/applications-2/swell-simulator/) To emulate the influence of ship motion on the bubble trajectory, the column and measurement equipment have been attached to a hexapod ship motion emulator (Symétrie NOTUS


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Hexapod),38 which allows for six degrees of freedom in motion (translation along and rotation around the coordinate axis). Note that by doing so, all presented results are shown and analyzed with respect to the coordinate system of the moving frame, parallel to the column’s center line. For reason of clarity, only the rotation around the x-axis (roll motion of the ship), which is parallel to the cameras main axis, will be the subject of the presented study. Bubble trajectories under the influence of angles of inclination (Rx) up to 15° and column motion frequencies (fx) up to 0.150 Hz have been acquired and analyzed quantitatively to evaluate the influence of the angle and frequency of inclination on the bubble motion and gas distribution in the column. It is to be borne in mind that the presented experimental investigation focuses solely on liquid phase batch operation mode. Analysis of continuous operation of the liquid phase, e.g., co- or counter-current flow, which might come along with other influences on the drag forces, is left to subsequent studies. However, considering that liquid phase velocities (typically a few cm s-1) commonly encountered in continuous operation mode are circa one order of magnitude lower than the observed bubble rise velocities, effects of the liquid velocity field induced by externally-imposed advective flow can be expected to be only of secondary importance. 3. Results and Discussion 3.1. Inclined Column 3.1.1. Bubble Trajectories It is worth reminding that the following trajectories represent two-dimensional y-z paths of the centroids of the ellipsoidal bubbles being tracked. One might ask if a two-dimensional analysis is sufficient to capture the characteristics of the three-dimensional bubble flow. Taking into consideration that the bubble is injected in the center of a round (symmetric) column and that the investigated movement and inclination of the column are parallel to the camera’s field of view,


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one notices that the static and dynamic inclinations of the column, in first place, only give rise to asymmetries parallel to the cameras field of view. Thus, the main influence of the aforementioned effects might result in a change of bubble motion in the two dimensional y-z plane. To define a benchmark for the moving bubble column, the influence of static inclination on the bubble trajectory is analyzed in Figure 2 for inclination angles Rx of 0° (a1), 5° (a2) and 15° (a3). Regardless of Rx, all recorded bubbles exhibit the zig-zag motion typical of the bubble size under scrutiny.9,19–22 The bubbles propensity to build-up nearby the upper wall with increasing angle of inclination is discernable, where the “mean” trajectory, determined by the lift forces, follows nearly perfectly the behavior expectable from a rigid sphere drawn as a black dashed line in Figure 2.a2. However, in spite of the counteracting forces by the rigid wall, the bubbles still preserve the zig-zag character in their trajectories. Although it is noticed that the wall contact induced by the angle of inclination has a damping effect on the zig-zag motion, it is still clearly visible, even at high angles of inclination. The aforementioned observation can be attributed to a variation of the direction of the lift forces acting upon the bubble relative to its path along the rigid column wall. More precisely, with an increasing angle of inclination, the component of the decomposed lift force vector orthogonal to the column wall increases, pressing the bubble stronger against the column wall. Nevertheless, the magnitude of the gravitationally induced normal and friction forces at the column wall does not appear to be strong enough to suppress neither the bubbles secondary zig-zag motion, attributed to, e.g., vortex induced forces, 39

nor its rebounding from the column wall.30

Figure 2.b underlines the observed behavior by presenting the mean bubble density distribution during the experiments. One clearly notices that the bubble density at the wall increases with the angle of inclination. However, even at high angles of inclination, regions close by the wall


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significantly contribute to the overall density distribution, being rationalized by a bouncing bubble behavior.30

Figure 2. Centroid trajectories (a) and contour plots of the bubble number-density distribution (b) for the column static inclination. Different colors of the trajectories (a) allow distinguishing between the trajectories of the individual bubble events tracked during multiple repeat-test experiments under the same conditions. The dashed line in a2) symbolizes the trajectory of an ideal rigid sphere. Inset in Figure a3 is a zoom-in of a time portion of one single bubble trajectory. 3.1.2. Axial and Radial Velocity Components To get a more detailed insight of the bubble behavior, the bubble’s axial and radial velocity components, relative to the inclined (moving) coordinate system of the column, at each recorded position of the trajectory are calculated as follows:


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v ,i ,diff  i 1  i   f frame ,

(1)

Where  denotes the respective direction or position (y,z) and i is the index of the frame being analyzed. The reader is carefully reminded that the definition of the axial (rise) and radial velocities is with respect to a moving coordinate system attached to the column. Hence, this leads to velocity vector components that differ from those traditionally expressed in a fixed coordinate reference system and whereby the term “rise velocity” is related to the bubble’s velocity component in the direction of gravity. Figure 3 summarizes the calculated axial profiles of axial and radial velocities. It shows that both of the axial and radial velocity components exhibit an oscillating behavior. While the radial velocity oscillation is mere translation of the zigzag path in the bubble trajectory, the oscillation with a non-zero mean in the axial velocity indicates a periodic acceleration and deceleration in the bubble rise velocity along the axial coordinate and may be puzzling at first sight.


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Figure 3. Axial profiles of axial (a) and radial (b) velocity components Eq.(1) for various bubbles trajectories and different angle of inclination (Rx). If the frequencies of the two observed oscillations appear to be similar, the question arises if they can be correlated. Introducing the scaled velocities:

v y ,i ,scaled 

v y ,i ,diff max v y , j ,diff

(2)

j

and

vz ,i ,scaled 

vz ,i ,diff  v z , j ,diff max vz ,k ,diff  v z , j ,diff

(3)

k

allows for a basic comparison of the two velocities as exemplified for one bubble trajectory in Figure 4. Note that for reasons of clarity -1·vz,scaled has been depicted instead of vz,scaled. The results shown in Figure 4 clearly indicate that there exists a strong correlation between the


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oscillating frequencies of the radial and axial velocity components. For instance, vz,scaled and

vy,scaled are in phase opposition, i.e., when one’s magnitude is maximum, the other’s becomes minimal and vice versa. In other words, the bubble rises faster at the turning points of the zig-zag motion than at positions between the turning points. These findings are in good agreement with the experimental findings of Liu et al.24 Moreover, non-stochastic oscillations in the bubble rise velocity have also been reported by other authors21,23,25,40 and have been attributed to a change of the bubbles drag coefficient, which might be rationalized as a change of the bubbles aspect ratio in the rise direction. Consequently, the oscillation in the axial velocity might be induced by the radial displacement of the bubble, i.e., by the orientation and aspect ratio of the bubble in the flow direction. Taking a closer look at the dependency of velocity components’ oscillations on the angle of inclination (Rx) (Figure 3), another interesting characteristic of the bubble rise behavior becomes obvious. One clearly notices that at high angles of inclination, where stronger wall effects are expected, the oscillatory behavior in both axial and radial directions goes unsuppressed even if the oscillations become blurred with an increased angle of inclination. Indeed, the latter observation indicates that by introducing asymmetric forces (e.g., friction or normal forces from the wall), the path instability of the bubbles seems to become more random, as small variations between the trajectories are amplified. This becomes particularly clear when comparing the trend of the radial velocity component (vy,diff) as a function of the axial position (Figure 3.b2). Up to a height of ca. 0.2 m, i.e., position where bubble gets in contact with the column wall, the evolution of the radial velocities of the recorded single bubble trajectories seems to follow a clear trend, whereas above this height the observed velocity profiles become less distinct.


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Figure 4. Evolution of scaled axial and radial velocity components along the axial coordinate . 3.1.3. Mean and RMS Axial and Radial Velocity Components After the previous qualitative depiction of the bubble rise behavior in an inclined column, quantitative measures, as illustrated in Figure 5, are required to allow for a more detailed analysis. The ensemble-averaged mean axial velocity components for all recorded bubble trajectories are first obtained for each inclination angle, Rx (Figure 5.a1) before to be compared to predictions from terminal velocity correlations in the literature. It must be borne in mind that these correlations are first modified by embedding Rx dependence to enable their extension to inclined geometries. This amendment merely assumes that only the vertical velocity projection is contributive, thus the cos(Rx) correction factor in the modified correlation of Clift et al.:9

vz 

2.14   s  0.505  g  d e  SF  cos  Rx  ,   de

(4)

and likewise the correlation given by Mendelson27 (validated by Krishna et al.41):


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vz 

2  s g  de   SF  cos  Rx    de 2

(5)

For small angular deviations from vertical, the experimentally determined rising velocities compare fairly well with correlation predictions (1.81 % for Eq.(4), -0.84 % for Eq.(5)). Nevertheless, the prediction error given by a simple vertical velocity projection progressively inflates with an increase in angle of inclination. In accordance with Masliyah et al.,33 this suggests that increased bubble-wall interactions at higher angle of inclination go totally uncaptured by current simple terminal velocity correlations. More precisely, due to the additional forces imposed upon the rising bubble by the rigid column wall, the observed axial velocity is smaller compared to the one predicted from a simple decomposition of the velocity vector. A more accurate prediction of the bubble rise velocity might be achieved by modifying the correlations of the bubble drag coefficient to account for the column’s inclination.34 However, bearing in mind that under marine operation conditions, a dynamic motion rather than a static inclination of the column is present, the angle of inclination as well as the bubble’s mean distance relative to the column wall will vary. Thus, experimental investigations on the influence of the column motion on the mean rise velocity compared to the static inclined case, as presented in the latter part of this study, can enlighten the aspect if correlations for statically inclined columns might be extended to the dynamic operation mode. Moreover, the observed decrease in the bubble velocity contradicts to some extent the findings for multi-bubble flow behavior,32 where a secondary circulation in the liquid phase leads to an increase in the bubble rise velocity with the angle of inclination. Thus, the latter effect seems to over-compensate the additional friction effects imposed by the rigid wall which become dominating in single-bubble experiments.


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Figure 5. Evolution of the mean axial (a1) and radial (a2) velocities and their corresponding mean standard deviation (or root-mean-square, rms) velocities (b1, b2) along a bubble trajectory as a function of the angle of inclination (Rx). The error bars express the 1- confidence interval of the mean bubble velocity (a1, a2), respectively of the rms velocities (b1, b2). The predictions from amended mean bubble rise velocity correlations by Clift et al.9 Eq.(4) and Mendelson27 Eq.(5) are also shown for a bubble diameter (de = 3.22 mm) with their corresponding  5 % envelopes (de = 3.22  0.16 mm). Inspection of the angular profile of the mean radial velocity component (Figure 5.a2) reveals a convex shape with a maximum nearby 4°. This maximum is likely a function of column diameter and also of orifice location on the injection plate, and can be rationalized as follows. In the ensemble-averaging over a region where the bubble flows through the liquid when the column is inclined, either the mean radial velocity differs from zero for far-from-wall trajectory events, or it drops due to bubble proximity to the wall. In the limit of bubble-wall contact, a zero mean


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radial velocity is expected. Consequently, in the case of an infinite aspect ratio (large column diameter), no maximum would be observed. Instead, the radial velocity would gradually increase with the angle of inclination until its maximal value, which equals the value of the axial velocity in the non-inclined case, is reached for the limiting case of a 90° inclination. Furthermore, after bubble release, the first bubble-wall contact occurs at an axial position which depends on the vessel inclination angle. This axial position which divides the bubble flow in regimes with zero mean and non-zero mean radial velocity, could be determined from the flow pattern of a rigid sphere (Figure 2.a2), that is, the position where the rigid sphere would hit the wall. The latter is in accordance with the direction of the gravity induced lift force vector in reference to the column’s inclined coordinate system. Variability in the axial and radial motion along the bubble trajectory for the ensemble of acquired trajectories is cast in terms of mean standard deviations of the velocity axial and radial components as a function of the angle of inclination (Figure 5.b) which can be thought of as the classical root-mean-square (rms) velocities used in turbulence descriptions. Here, the expression “mean” refers to double-averaging for one defined angle of inclination over time for each trajectory and then over the ensemble of trajectories. Regarding the mean standard deviation of axial velocity component (Figure 5.b1), three distinct regions can be observed. At low angles of inclination up to 2°, the free bubble flow in the liquid is recognizable owing to a plateau at a low mean deviation. For angles of inclination above approx. 2°, the standard deviation gradually increases up until 5°. Finally, a plateau above angles higher than 5° appears which encompasses the bubble behavior under the influence of wall effects. Thus, increased contact which occurs between bubbles and wall at high angles of inclination translates into a significant increase in the variability of the axial velocity along the bubble trajectory.


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Interestingly, a totally different picture is rendered by the evolution of the mean standard deviation of the radial velocity component (Figure 5.b2). Here, contributions from far-from-wall (unconstrained, i.e., non-zero mean radial velocity) and close-to-wall (constrained, i.e., zero mean radial velocity) zig-zagging bubble trajectories contribute differently depending on the prevailing inclination angle. At low angles of inclination (Rx  2°), the mean standard deviation remains nearly constant before slightly increasing due to the variation in the velocity induced by the transition between free flow and wall-constrained contact. Unlike the trend by the axial velocity standard deviation (Figure 5.b1), a significant decrease in radial velocity standard deviation can be noticed the higher the angles of inclination. This evolution is expectable on account of the filtering/damping effect prompted by the bubble-wall contacts. This becomes significantly visible when comparing mean standard deviations at free flow conditions (Rx = 0°) and at Rx = 15°. 3.2. Rolling Column 3.2.1. Bubble Trajectories To investigate the influence of the column motion on the bubble trajectory, amplitudes of the column’s inclination Rx of 3°, 5°, 10° and 15° have been investigated at motion frequencies (fx) between 25 mHz and 150 mHz. Showing the recorded several bubble trajectories for Rx = 3° and

Rx = 15° at three different motion frequencies, Figure 6 illustrates the impact of the column’s motion on the bubble rise behavior. Three main observations can be made. First, especially at low angles of inclination, an increase in the frequency of motion leads to a shift of the bubble trajectories towards the column wall, hence progressively depopulating the column centerline and core from their bubbles. Second, the influence of the column motion frequency on the bubble motion is less prominent at higher angles of inclination. Dominance of this latter on the


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bubble trajectory prevails on the column’s motion frequency. Third, independent of the amplitude or frequency of column motion, the bubble trajectory conserves a zig-zagging shape, indicating that viscous drag and surface tension effects imposed by secondary flows of the liquid phase are also compounded in the overall oscillatory behavior of the bubbles. Here, a more detailed analysis reveals that the column motion frequency has no significant influence on the typical relation of the bubbles’ axial and radial velocity components discussed above. The observable slightly bended trajectories of the bubbles under column roll motion can be reasoned in the time dependent variation of the direction of the lift (gravitational) force vector relative to the columns moving coordinate system and might be incorporated in the general equation of bubble motion via Coriolis forces. Additionally, one notices that at lower angles of inclination virtually no bubble ventures across the centerline of the column, whereas higher inclinations and motion frequencies of the column do not hamper centerline cross-over. Nevertheless, the fraction of bubble events crossing the columns’ center line still remains small, even at high frequencies and angles of inclination. This behavior reflects the fact that bubbles attached to the column wall will stay close to it until the radial component of the lift force vector changes its orientation. Consequently, only bubbles for which the sign of the aforementioned force vector changes are able to cross the column centerline. However, as the ratio between the bubbles’ residence time in the observation region and the cycle time of column is comparably small (ca. 1/4) the majority of bubbles will keep sticking on one side of the column.


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Figure 6. Bubble trajectories in case of dynamic motion of the column for maximum angular amplitudes (Rx) of 3° (a) and 15° (b) at three different column motion frequencies (fx) of 0.025 Hz, 0.075 Hz and 0.150 Hz. To allow for more insightful analyses of above tendencies with view on their possible consequences for bubble columns in marine applications, column-height averaged bubble number density distributions are obtained from the measurement region-of-interest as a function of inclination angle and motion frequency (Figure 7). Compared to the static inclined case at equal angular amplitude, one would expect that the motion of the column will tend to homogenize the spatial distribution of bubbles upon increasing rolling frequency. As such, one might envision the action from column rolling as tantamount to that of a gas-liquid mixer (or stirrer) whose function is to attenuate gradients of gas volume fraction in a multiphase system. The actual picture inferred from Figure 7 nuances this reductionist view. Indeed, an increase in column rolling frequency significantly reduces the bubble number density at the center of the


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column in the case of low to moderate inclined columns. Although only single-bubble events are dealt with in an ensemble sense here, this degradation is believed to reflect in an unfavorable maldistribution of the two phases in actual many-bubble bubble column operation, also indicated by the findings of Assima et al.17 for many-bubble flow under column roll motion. Only at very large inclination angles that faster rolling movements favor accomplishment of more even bubble number distributions. Nevertheless, this effect remains very small and seems to be overcompensated by the strong effect of the large angle of inclination which presses the bubbles very close to the column wall. Note that due to the bubbles’ zig-zag trajectory, even at high angles of inclination, a comparably high bubble number density exists over a relatively broad region parallel to the column wall. The overall observed tendencies point in the direction that the column motion could lead to a decrease of bubble columns’ performance due to uneven distribution of the two phases. One might then speculate on possible applications of low amplitude – low frequency moving bubble columns to break with the just-described congestion at the column wall to decrease transport resistances in the liquid phase’s hydrodynamic (diffusive) boundary layer or to reduce a possible formation of biofilms (fouling effects), e.g., in membrane applications or bio-reactors.


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Figure 7. Contour plots of column-height averaged radial bubble number density distributions as a function of column motion frequency (fx) and angles of inclination (Rx). 3.2.2. Mean and RMS Axial and Radial Velocity Components In view of the fate to be inherited by rolling bubble columns on floating platforms, the variation of the mean bubble rise velocity component (Figure 8.a1) as compared to the static case is of interest. Comparing the evolution of the axial velocity component as a function of column rolling frequency with its sibling for a static inclined vessel (fx = 0), no significant influence of frequency fx can be noticed. Vessel rolling has little effect on the mean rising velocity of the bubbles. Upon increasing column frequency, a decrease of mean radial velocity towards a value close to zero is observed (Figure 8.a2). Note that phase-lag issues between the camera recording frequency and the hexapod oscillating frequency might lead to small statistical scatters reflecting


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in negative-side or positive-side convergence towards zero mean radial velocity as shown by the rolling frequencies 25 mHz, 75 mHz and 125 mHz versus those of 50 mHz, 100 mHz and 150 mHz. However, the observed deviations are very small and do not significantly influence any of the presented results.

Figure 8. Evolution of the mean axial (a1) and radial (a2) velocities and corresponding mean standard deviations (b1,b2) along a bubble trajectory as a function of rolling frequency (fx) and vessel inclination (Rx). The error bars indicate the 1- confidence interval of the mean bubble velocity (a1, a2), respectively of the rms velocities (b1, b2). Changes in rolling frequency reflect in featureless variations of the axial rms velocity component (Figure 8.b1). This holds true also for the incidence of inclination. For low angles of inclination, an increase of the column’s frequency is not able to shift the bubble trajectory from the center to the wall where the axial velocity fluctuations would be significantly influenced. Likewise, the column movement is not strong enough to allow bubbles at the wall to migrate towards the ACS Paragon Plus Environment

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column core so that the free-flow flow oscillations could be promoted. The latter aspect might also be the reason why no distinct trend is detected at high inclination angles for the axial velocity’s standard deviation. The evolution of the mean standard deviation of the radial velocity component (Figure 8.b2) unveils no clear-cut variation with rolling frequency in addition to the non-monotonic impact of inclination angle. Consequently, increasing the rolling frequency does not seem to have a distinct influence on the bubbles’ zig-zag behavior purported by the oscillating radial velocity. Nevertheless, comparing the evolution of the standard deviation for an amplitude Rx of 15° for static (fx = 0 mHz) versus dynamic (fx = 25 mHz) case, an increase of the standard deviation of the radial velocity seems to exist. This could vouch for a stronger variation of the radial velocity associated with intensified zig-zagging upon changing frequency from 0 to 25 mHz, due to a lesser bubble-wall interaction. 3.3. Frequency spectrum analysis of the bubble trajectory The bubble flow zig-zag motion, as just seen above, might be altered differently for static inclined versus rolling column. Therefore, a more quantitative analysis of these observations is attempted in this section by investigating the bubble trajectory’s frequency spectra obtained by means of Fast Fourier Transformation (FFT).


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Figure 9. Bubble trajectory in spatial y(z) (a) and time-parameterized y(t) (b) domains, and in frequency domain (c) obtained by FFT for the static non-inclined column. As measurement frequency of trajectory is known, the axial coordinate is transformed to the time domain. The marked maxima in the frequency spectrum (c) indicate the detected leading frequency. Figure 9 exemplifies the transformation of the bubble’s trajectory into a frequency spectrum for the vertical column (Rx = 0). Three main regions emerge from the resulting frequency spectrum (Figure 9.c). The low-frequency narrow band around 1-2 Hz with high-amplitude dynamic features reveals no distinct information about the bubble’s oscillation or trajectory. Due to the limited measurement time, determined by the bubble’s residence time in the observational field of view, more accurate determination of this frequency band was not achievable. Nevertheless, a dominating frequency at about 6 Hz with an amplitude ca. 0.3 cm clearly protrudes in the Fourier spectrum. Comparing this contribution to the frequency spectrum with the bubble trajectory (Figure 9.a, Figure 9.b), it becomes obvious that this oscillation is associated to the zig-zag motion of the bubble. Additionally, it seems that for the presented case, the frequency spectrum becomes featureless in the higher frequency range.


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Figure 10. Heat maps of the leading bubble frequencies (fb,y) in the Fourier analysis for the statically inclined (a) and dynamic case (b) as a function of the static angle of inclination (Rx), or respectively, of the column frequency (fx). Only frequencies smaller 15 Hz are shown as amplitudes beyond this frequency become negligibly small. Aiming for an analysis of a possible influence of the column motion and angle of inclination on the bubble frequency, the contribution of frequencies with amplitude larger than 0.7 mm (cut-off amplitude) to the frequency spectrum are presented in Figure 10. Only the frequency values at maximal amplitude are considered. By also including frequency information results for the static inclined vessel at fx = 0 (Figure 10.a), one clearly distinguishes transition from single-frequency spectral signature to a more blurred spectrum in which more than one governing frequency might be detected, indicating a transition from a comparably simple oscillating pattern to a more complex bubble secondary motion. Here, a single-frequency at approximately 6.5 Hz is associated with a bubble rise under free flow conditions.20 On the contrary, inception of two or


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more frequencies, where the most prominent modes according to Figure 10 are present around

ca. 1.5 Hz and 9.5 Hz is reminiscent of a bubble trajectory subject to strong influences imposed by the column wall. A more detailed analysis of the bubble trajectories indicates that the detected more complex Fourier spectrum under wall influence might be attributed to asymmetric zig-zag motion due to bubble rebounds on the column wall. Moreover, the frequency of the bubble zigzag motion seems to increase close by the column wall whereby such motion is sometimes accompanied by a secondary motion of similar amplitude but at significantly lower frequency. Analyzing the dependency of bubble frequency (fb,y) against column rolling frequency (fx) for different angular amplitudes (Rx) (Figure 10.b) unveils one governing bubble frequency at work at low angles of inclination. Interestingly, the column rolling frequency has virtually no influence on the main bubble frequency. This state of affairs may be attributed to the fact that most of the observed bubbles are not getting in contact with the vessel’s wall. However, in the case of high angles of inclination (Figure 10.b3 and Figure 10.b4), the bubble leading frequencies tend to evolve from multiple leading frequencies to one leading frequency as column rolling frequency increases. The interpretation of this observation antagonizes the previous one by indicating a transition from bubble behavior under strong wall effects to bubble behavior in free bubble flow. Especially when comparing the statically inclined with dynamically inclined column, a clear tendency of shifting the leading bubble motion frequency towards an amplitude in the range of ca. 6 Hz can be recognized. Even if it is conceded that the features borne in Figure 10 contour plots are tributary of the selected cut-off amplitude, a more detailed analysis reveals, on the contrary, that the overall tendencies remain unaffected by such selection.


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4. Conclusion New experimental results on the influence of static inclination and column rolling motion on the flow pattern of single ellipsoidal bubbles have been obtained to emulate the special hydrodynamics taking place in bubble columns onboard floating platforms of interest to the marine application of multiphase chemical reactors. By attaching a bubble column and highspeed camera to a hexapod ship motion emulator, single-bubble trajectories of a typical zig-zag shape were captured and analyzed with a special focus on the evolution of bubble’s trajectory and corresponding radial and axial velocity profiles as a function of vessel static inclinations, and rolling frequencies and amplitudes. Our findings highlighted a strong relation between the oscillating bubble’s axial and radial velocity components although without qualitative alterations of the trajectory type. Investigating the column motion frequencies ranging from 0 to 15 mHz and angles of inclination up to 15° unveiled a strong dependency of the bubble motion and velocity profiles to the angle of inclination. However, the influence of column motion frequency appeared to mostly impact the constraining role imposed by the column wall. The influence of column rolling frequency was also unveiled by application of FFT analysis to the individual trajectory time series and after ensemble averaging the single-trajectory frequency spectra. This analysis identified a transition between the bubble’s zig-zag motion in free flow and under the influence of wall effects. By shedding light on the influence of column roll motion (inclination and frequency) on single ellipsoidal bubble characteristics, our findings are offered as a first step for better design of FPSO bubble column settings. Nevertheless, even if changes in the bubble trajectory could be quantitatively and qualitatively analyzed, more detailed research work is required to link the observed characteristics to changes in bubble shape and to derive ad-hoc correlations and


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constitutive equations accounting for bubble’s trajectory, velocity profile and secondary motions. Finally, an extension of the presented results to two and multi bubble flow would allow for an investigation of the transition between single and multi-bubble scenarios, giving rise to a deeper understanding of bubble-bubble interactions in complex flow conditions. Ç AUTHOR INFORMATION Corresponding Author Email: [email protected] ; Phone +1-418-656-3566 ORCID C. Drechsler 0000-0003-2642-3303 S.M. Taghavi 0000-0003-2263-0460 F. Larachi: 0000-0002-0127-4738 Present Address Department of Chemical Engineering, Université Laval, 1065 Avenue de la Médecine, Québec, Québec G1V 0A6 Canada Notes The authors declare no competing financial interest. ACKNOWLEDGMENTS The authors gratefully acknowledge the Chair of Sustainable Energy Processes and Materials (University Laval), the Laboratory of Complex Fluids Research (University Laval) and the Chair of Chemical Reaction Engineering (TU-Dortmund, University) for their support. ABBREVIATIONS


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FFT, Fast Fourier Transformation; FPSO, Floating production, storage and offloading; rms, Root-mean-square; std, Standard deviation NOMENCLATURE Variables and parameters

a d f fx fframe g ntraj R SF t v V y z

  s 

/m /m /s-1 / s-1 / s-1 /m s-2 //° //s /m s-1 /m3 /m3 s-1 /m /m

Amplitude Diameter Frequency Column motion frequency Framerate Gravitational constant Number of data points taken for a single bubble trajectory Angle of inclination Scale factor Time coordinate Velocity Volume Volume flow Radial coordinate Axial coordinate

/kg m-3

Density Standard deviation Surface tension Spatial coordinate

/kg s-2 /m

Subscripts b c detach diff e i j k N2 o scaled x /m y /m z /m

Bubble Column Detachment Differential Equivalent Index ∈ 1, or frame number Index ∈ 1, Index ∈ 1, Nitrogen Orifice Scaled x-Axis Radial coordinate Axial coordinate


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Dimensionless groups Eo Eötvös number M Morton number Re Reynolds number We Weber number


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(30) Vries, A.W.G. de; Biesheuvel, A.; van Wijngaarden, L. Notes on the path and wake of a gas bubble rising in pure water. Int. J. Multiphase Flow 2002, 28, 1823–1835, DOI: 10.1016/S0301-9322(02)00036-8. (31) Hudson, C. L. Effect of Inclination on Hydrodynamics in Bubble Columns and Fluidized Beds. Dissertation, University of Western Ontario, London, Ontario, Canada, 1996. (32) Lu, G. Z.; Bittorf, K.; Thompson, B. G.; Gray, M. R. Liquid circulation and mixing in an inclined bubble column. Can. J. Chem. Eng. 1997, 75, 290–298, DOI: 10.1002/cjce.5450750203. (33) Masliyah, J. H.; Jauhari, R.; Gray, M. R. Rise velocities of air bubbles along an inclined surface. In Mixed-Flow Hydrodynamics; Cheremisinoff, N. P., Ed.; Gulf Publishing: Houston, Texas, 1996. (34) Masliyah, J.; Jauhari, R.; Gray, M. Drag coefficients for air bubbles rising along an inclined surface. Chem. Eng. Sci. 1994, 49, 1905–1911, DOI: 10.1016/0009-2509(94)80075-8. (35) Basler, I. acA2040-90um - Basler ace. https://www.baslerweb.com/en/products/cameras/area-scan-cameras/ace/aca2040-90um/ (accessed October 2, 2018). (36) MATLAB; The MathWorks, Inc. Natick, Massachusetts, United States, 2016. (37) Gadallah, A. H.; Siddiqui, K. Bubble breakup in co-current upward flowing liquid using honeycomb monolith breaker. Chem. Eng. Sci. 2015, 131, 22–40, DOI: 10.1016/j.ces.2015.03.028. (38) Symétrie. Symétrie NOTUS Hexapod. http://www.symetrie.fr/en/products/motionhexapods/notus/ (accessed October 5, 2018). (39) Mathai, V.; Zhu, X.; Sun, C.; Lohse, D. Mass and Moment of Inertia Govern the Transition in the Dynamics and Wakes of Freely Rising and Falling Cylinders. Phys. Rev. Lett. 2017, 119, 54501, DOI: 10.1103/PhysRevLett.119.054501. (40) Di Marco, P.; Grassi, W.; Memoli, G. Experimental study on rising velocity of nitrogen bubbles in FC-72. Int. J. Therm. Sci. 2003, 42, 435–446, DOI: 10.1016/S1290-0729(02)00044-3. (41) Krishna, R.; Urseanu, M. I.; van Baten, J. M.; Ellenberger, J. Wall effects on the rise of single gas bubbles in liquids. Int. Commun. Heat Mass Transfer 1999, 26, 781–790, DOI: 10.1016/S0735-1933(99)00066-4.


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GRAPHICAL ABSTRACT

Hexapod drawing adapted with permission (http://www.symetrie.fr/en/applications-2/swell-simulator/)

from

Symétrie


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Bubble Behavior in Marine Applications of Bubble Columns: Case of

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