Ellipsoidal air bubble motion in stagnant water through a vertical narrow rectangular channel

ELLIPSOIDAL AIR BUBBLE MOTION IN STAGNANT WATER

THROUGH A VERTICAL NARROW RECTANGULAR CHANNEL

Ö zd em ir, S .1. B ezd eg ü m eli, U .1, Y eşin , O .2

Turkish Atomic Energy Authority, Ankara, Turkey 2Middle East Technical University, Ankara, Turkey

ABSTRACT

This study presents motion o f ellipsoidal air bubble in stagnant water through a vertical narrow rectangular channel o f 2.1X66.5 mm cross section. Main analysis tool used in the study is image processing. This method appears to be promising for determining the governing mechanisms o f the bubble motion. The test section o f the facility is one to one scaled model o f single cooling channel o f TR-2 nuclear research reactor. The automatic bubble generation technique used in this experimental study is shown to have several advantages, e.g. the accuracy o f the bubble size, repeatability o f tests, and controlling the time between consecutive bubbles. This study focuses on the quantitative assessment o f ellipsoidal bubble size (perimeter, area, volume, width and height), shape, path and rising velocity. The range o f analyzed ellipsoidal bubbles is

2.1<dae< \5 3 mm. According to analyses, ellipsoidal bubbles follow sinusoidal path in narrow rectangular

channels. Same size ellipsoidal bubbles follow sinusoidal paths with the same amplitude and the same period. In addition, rise velocities o f ellipsoidal bubbles in the experimental runs are analyzed and compared with the rise velocity correlation o f ellipsoidal bubbles in infinite medium and a new empirical correlation is suggested for rising velocities o f two dimensional ellipsoidal bubbles.

1. INTRODUCTION

An important topic in fluid dynamics is two-phase flows. Two-phase flows can be found in many engineering fields, e.g. nuclear, mechanical, aerospace, biomedical, chemical, electrical, and environmental engineering. Adiabatic gas-liquid two-phase flow and vapor-liquid phase-change processes play an important role in a wide variety o f technological applications, specifically in nuclear reactor engineering and chemical engineering [1]. One o f the important topics in two phase flow is bubble motion. To be able to understand the bubble motion, all types o f bubbles should be investigated theoretically and experimentally.

Any improvement in understanding the bubble motion will be used not only to extend the currently available database o f empirical correlations, but more importantly, to assist in the development o f advanced computational tools. Once developed and validated against experimental results, these tools will be invaluable in aiding designers to develop new, more efficient, engineered systems [2]. The wide range o f differences in the velocity magnitude and orientation o f bubble must be captured by experimental or computational tools in order to be able to fully predict the behavior o f a single bubble rising in stagnant and flowing water.

The dynamics o f free rising single bubbles in stagnant liquids in infinite medium have been investigated by numerous authors. Most o f the studies up to 1978 were summarized by Clift et al. [3]. One o f the important studies performed on the determination o f shape o f the bubble is Grace’s study [4,5]. Grace classified bubbles using three dimensionless numbers, which are related to each other, i.e. the Reynolds number, the Eötvös number and the Morton number, in an experimentally determined diagram. The Grace’s diagram contains three main regimes in which the bubbles have different shapes: The spherical, ellipsoidal and spherical-cap regime.

Bubble motion is highly complex with an unsteady nature, showing wide variations in bubble shape. Although it has been extensively studied in the past; however, certain aspects o f bubble motion still remain poorly understood, due to lack o f experimental data in narrow rectangular channels under stagnant and flowing water conditions. In spite o f receiving much attention over recent decades, the present state o f knowledge for bubble motion in narrow channels is far from complete [6].

Image processing and analyzing techniques have been used in bubble motion investigation in recent years. Sadr-Kazemi and Cilliers [7] used an image processing algorithm for measurement o f bubble size and shape distributions. Donevski et al. [8] developed an image processing technique for the study o f bubble dynamics in sub-cooled flow boiling.

In this study, the behavior o f ellipsoidal air bubble motion in water through a vertical narrow channel is investigated by using image processing and analyzing techniques. It focuses on the quantitative assessment of

some basic parameters o f bubbles such as size (perimeter, area, volume, width and height), shape, rising velocity and path.

Several experimental studies [4, 9, 10] conducted for bubbles moving in stagnant water showed that bubbles having 0.8 mm < dve <2 mm have an ellipsoid shape, dve <4.2 mm have an ellipsoid shape with surface oscillations and bubbles having dve >4.2 mm have a distorted bubble shape, where dve is volume equivalent air bubble diameter.

The term "ellipsoidal" is generally used to refer to bubbles which are oblate with a concave interface (viewed from inside) around the entire surface. It must be noted that actual shapes may differ considerably from true. Moreover, ellipsoidal bubbles commonly undergo periodic dilations or random wobbling motions that make characterization o f the shape particularly difficult [3]. In most systems, bubbles in the intermediate size range (typically between 1 mm < d ve< 15 mm) he in ellipsoidal regime [3].

2.THEORY

Bubble rising behavior in a liquid, depends on the physical properties o f the surrounding liquid. These are the density p, the dynamic viscosity p and the surface tension <r. The bubble rises due to the buoyancy force which is related to the gravitational acceleration g and the volume o f the bubble V [11].

A bubble rises through a denser liquid because o f its buoyancy. The velocity Ub with which a single bubble rises through stagnant liquid is governed by the interaction between buoyancy and the other forces acting on the bubble as a result o f its shape and motion. If the viscosity of the gas or vapor in the bubble is negligible, the only three forces besides buoyancy, which are important, are liquid inertial, viscous, and surface tension forces.

Theoretical studies on the bubble shape and motion extensively benefit from the experimental data that give the bubble shape as a function o f some dimensionless numbers. Determination of bubble shapes as a function o f the related dimensionless numbers has been studied experimentally for the last sixty years [3]. Reynolds, Weber and Eötvös are some o f these dimensionless numbers. Reynolds number is a measure o f the relative importance o f the inertial force compared to the viscous force. The Weber number is a measure o f the relative importance o f the dynamic pressure force compared to the surface tension force. When dynamic pressure force is dominant, the appropriate parameter to be considered in a study of bubble deformation is the Weber number. The Eötvös number is a measure of the importance o f the buoyancy force compared to the surface tension force. When buoyancy force is dominant, the appropriate parameter to be considered in a study o f bubble deformation is the Eötvös number. The Eötvös number is basically a measure of the bubble size, so that a functional relationship between a parameter and the Eötvös number describes how that parameter changes with the volume o f the bubble. Equations o f these dimensionless numbers are given below;

Refi =

PjU hd fl

p , u l d b E 0 = s d b2{ p , - p g)

P l O

where Ub is bubble velocity, db is bubble diameter, p is density, p is dynamic viscosity and o is surface tension.

In summary, for a given Reynolds number, two dimensionless numbers of interests for determining the gas bubble shape deformation are Weber number and Eötvös number.

3. EXPERIMENTAL FACILITY

An experimental facility was designed and constructed to make investigation on bubble motion in water through a vertical narrow rectangular channel with dimensions simulating a single cooling channel o f TR-2 nuclear research reactor in Istanbul [12]. The test section o f the facility was modeled as one to one scale o f the single cooling channel o f TR-2 nuclear research reactor. The designed test section was manufactured from plexiglas o f 1 cm thickness to obtain full optical access in the wide side and in the narrow side o f the channel. The test section, therefore allows the visualization o f the air bubble motion in stagnant water and in flowing water by using image processing techniques. Dimensions of the test section and coordinate system used in this study are shown in Figure 1. Air bubbles are introduced at a point, horizontally at the center o f a plate and vertically 200 mm from the bottom. At this specified point, a bubble injector is located to generate air bubbles.

One o f the important aspects o f the present study is to generate air bubbles o f desired size and shape during tests. In this system, a pneumatic solenoid valve and needle valve are used to control the bubble size. Control o f the pneumatic valve is provided by a computer and related software. Bubbles with the same shape and same volume can be obtained by using this system and it satisfies the repeatability requirement o f the tests. The imaging system consist o f a CCD digital camcorder, two halogen lamps, a light diffuser and a dark room, as shown in

F ig u re 2. The dark room was obtained by covering the environment o f the test set-up with thick black curtains so that it was shielded from the ambient light. The light, reflected from diffuser situated about 30 cm behind the test section, produced well-dispersed and uniform background illumination.

Figure 2. Illumination system

A Sony DCR-TRV461E DigitalS camcorder was used for image acquisition purposes. Digital images o f the front side of the test section are recorded with a digital camcorder at a rate o f 25 frames per second. The camera resolution is 720H x 576V pixels. The video images are stored in the DigitalS video cassette during the tests. Digital video records taken from the tests are transferred to the computer by Firewire-1394 interface. These interlaced PAL system digital video records are processed by Virtualdub freeware video processing software [13]. Frame rate o f recorded images is doubled to 50 Frames/s by using this software.

4. IMAGE PROCESSING AND ANALYZING TECHNIQUES

Bubble information such as its position, perimeter, area, volume, velocity and motion path can obtained by image processing and image analyzing techniques. Various techniques involved in image processing were applied to obtain accurate measurement results in this study.

Recognition o f the bubbles in image sequences by computer is a very complicated process and it needs some image manipulation operations to increase the accuracy, such as brightness/contrast adjustment, smoothing, sharpening, and some binary operations. After image enhancement process, edge detection, particle labeling and measurement techniques should be used to obtain required information in the image sequences. For this purpose ImageJ software is used as image analysis software. ImageJ is a public domain Java image processing program inspired by NIH Image [14]. Image processing steps are summarized in a flow chart (Figure 3). Image acquisition (camcorder) Image transfer Trimming the region of interest Frame Rate Conversion (25 to 50 F/s) Field separation Duplication of even and odd fields

Combination of new fields one one Cropping the region of interest RGB to 8 Bit color conversion Smoothing Background

subtraction _{Bubble information} obtained from each video

frame

Area Perimeter Position o f center o f mass

Position o f bubble nose Width Height Circularity Brightness/Contrast adjustment Binary threshold Spatial calibration Edge detection Analyzing bubbles

Figure 3. Flow chart o f image processing steps

Bubble area in 2D (A), bubble center of mass (Xm,Ym) bubble perimeter (P), upper comer o f bounding rectangle o f bubble (Xb,Yb), bubble width (w), bubble height (h), and circularity (C) are measured by using image processing.

To assess the bubble motion, additional information such as bubble velocities, related dimensionless numbers, bubble volume etc. is needed. Before starting the post-processing o f the image analysis results, vertical velocity component o f air bubble measured from the center o f mass(Ubvc), area equivalent bubble diameter (dae), bubble Reynolds number (Rebae) based on Ubvc and dae, Eötvös number (Eöae) based on dae and Weber number (Weae) based on dae were calculated by using the data obtained from video images.

5. RELIABILITY AND REPEATABILITY OF TESTS

To check the bubble generation system and the results o f image analysis process, a set o f tests were realized. Purpose o f these tests was to show the repeatability and reliability o f the experimental results. In this set o f tests, six different bubbles with the same size were generated with a time delay between each generation to eliminate the effect o f bubbles to one another. The recorded video images were processed and analyzed with the techniques explained above.

These analyzed bubbles are shown in Figure 4 as combined sequences o f images in series. Paths o f the bubbles are almost identical. As it can be seen from the images, it is possible to generate bubbles o f the same shape and size. If the results of six different tests are compared with each other, it is seen that the results of tests are satisfactory from the repeatability of tests and reliability o f image processing measurement points o f view (See the results in Figure 5).

I II ^ III IV V VI

Figure 4. Paths o f six different bubbles with same size under similar conditions

6

5

! 4

3

Average Bubble Properites

— — s — — —

dve

dae

2

3

4

5

Test Number

6. BUBBLE SHAPES AND TRAJECTORIES

A set o f tests were prepared and realized to obtain bubble shapes, paths and rise velocities at 21°C stagnant water conditions in the test section. In this set o f tests, bubbles with area equivalent bubble diameters between 2 mm and 15.3 mm were generated and tracked along the channel and recorded by the camcorder. As a result o f tests, it was seen that the behavior o f a single gas bubble rising in the channel depends on the size o f the bubble. When the bubble is small it has circular shape in 2D and rises along a vertical rectilinear path. Larger bubbles become ellipse and tend to rise along sinusoidal paths. Increase in bubble size transforms its shape into irregular deformed ellipse which is named as wobbling ellipse and it moves in a sinusoidal path. Further increase in the bubble size causes the bubble to be o f a cap shape and again to rise along a rectilinear path.

Bubble types, area and volume equivalent bubble diameter ranges, Reynolds, Eötvös and Weber Numbers with respect to area and volume equivalent bubble diameters, and bubble paths obtained in the study are given in Table 1. Dimensionless numbers with respect to the area and volume equivalent bubble diameters are given to be able to compare with the existing literature and to see two dimensional (2D) effects when compared with the three dimensional (3D) case.

<u

s H

The obtained paths o f the bubbles can be categorized as follows:

Bubbles having dae<2.1 mm move with a rectilinear motion if the Re number based on Ubvc and dae is less than 579. If the Reynolds number based on Ubvc and dae is greater than 579 and less than 3170 the bubbles are ellipse in shape and move on a sinusoidal path.

Figure 6a shows some of the bubble paths, where X is in lateral direction, and Y is in vertical direction, and

the starting point is at almost 20 cm above the bubble injection point. One can see that the bubbles follow similar sinusoidal paths. As an example to the sinusoidal path, a simple curve-fit to the bubble path was obtained and given in Figure 6b. When the bubble volume increases, the bubble area, amplitude and period

o f the sinusoidal path increase respectively. Same size ellipsoidal bubbles follow sinusoidal paths with the same amplitude and the same period as shown in Figure 6a.

Figure 6. a) Ellipsoidal bubble path, and b) The fitted sinusoidal curve to the ellipsoidal bubble path

7. RISING VELOCITIES OF AIR BUBBLES

After a bubble leaves the orifice o f the injector, it rises up through the liquid and eventually it reaches a constant speed, which is referred to as the terminal velocity [3]. According to Gaudin’s study ellipsoidal air bubble rising velocity distribution in infinite medium is approximated closely by

U T =

2 .1 4 ---+ 0 .5 0 5

Pd v,

M l

(1)

which is o f the form suggested by a wave analogy [3, 4, 15].

In this section, rising velocities o f ellipsoidal bubbles in the channel under stagnant water conditions are analyzed. The range o f analyzed ellipsoidal bubbles is 2.7<<7ae<8.5 mm without wobbling and 8.5<<7ae<15.3 mm with wobbling. The rising velocity o f the bubbles can be calculated from the relative positions of the bubbles at a time interval. The rise velocities were calculated using the obtained data from image processing. The results o f the ellipsoidal bubble tests and the correlation o f rise velocity o f ellipsoidal bubbles in 3D infinite medium are compared

Figure 7a). While area equivalent bubble diameter was less than 11.6 mm, side walls did not affect the bubble rise velocity significantly, so, it was assumed that ellipsoidal bubbles rise in infinite 2D medium. As it is shown in the

Figure 7, trend o f ellipsoidal bubble rise velocity in 2D is similar to the ellipsoidal bubbles in 3D infinite medium. Thus, using similar form o f equation, the following empirical correlation for 2D ellipsoidal bubble was obtained from the experimental data o f the present study (See

Figure 7b); u b = <7

- X

0 . 5 - — + 2.76

This equation can be used for ellipsoidal air bubbles rising in vertical narrow channels with about 2 mm channel gap under stagnant water conditions at ambient temperatures when narrow side walls effects are insignificant.

Figure 7. a) Rise velocities of ellipsoidal bubbles in 2D and 3D, b) Curve-fit to the experimental data

8. CONCLUSIONS

Ellipsoidal bubble paths in stagnant water through a narrow rectangular channel were obtained. Bubble rising velocities versus area equivalent bubble diameter were calculated and the results are given in terms of Reynolds, Eötvös and Weber Numbers based on the area equivalent bubble diameters.

Velocities o f ellipsoidal bubbles rising in stagnant water through the channel were analyzed in detail. The results o f the ellipsoidal bubble tests and the rising velocity correlation of ellipsoidal bubbles in 3D infinite medium were compared. As a result, an empirical correlation for 2D ellipsoidal bubbles is suggested from the experimental data.

The data collected by the image analysis and the empirical correlation obtained from the experimental results can be incorporated into the new models and computer codes currently under development.

9. REFERENCES

1. Q. Bi, "Characteristics o f Two-Phase Flow and Boiling Heat Transfer in M iniature Non-Circular

Channels ”, Ph.D. Thesis, The Hong Kong University o f Science & Technology, 2000

2. D.R. Todd, "M ethodologies fo r Analyzing P IV and S IV Results fro m a Two-Phase Air/W ater

E xperim ent”, Ph.D. Thesis, Texas A&M University, 2002

3. R. Clift, J.R. Grace, M. E. Weber, "Bubbles, Drops, and P articles”, Academic Press, London, 1978 4. I.S. Lioumbas, A.A. Mouza, S.V. Paras, "Local Velocities Inside the Gas Phase During Counter­

Current

Two-Phase Flow in a Narrow Vertical C hannel”, Trans. IChemE, Vol 80, Part A, pp. 667-673,

September 2002

5. J.R. Grace, D. Harrison “The Influence o f Bubble Shape on the Rising Velocities o f Large Bubbles ”, Chemical Engineering Science, Vol. 22, pp. 1337-1347, 1967

6. I. Zun, J. Groselj, “The Structure o f Bubble Non-Equilibrium Movement in Free-Rise and Agitated-Rise C onditions”, Nuclear Engineering and Design 163, pp. 99-115, 1996

7. N. Sadr-Kazemi, J.J. Cilliers, “An Image Processing Algorithm fo r M easurement o f Flotation Froth

Bubble Size and Shape D istributions”, Mineral Engineering, Vol. 10, pp. 1075-1083, 1997 8. B. Donevski, T. Saga, T. Kobayashi, S. Segawa, “A Study o f Bubble Dynamics in Subcooled Flow

Boiling Using Image Processing Technique”, Proceeding o f the 4th World Conference on Experimental

9. E.T. White, R.H. Beardmore, “The Velocity o f Rise o f Single Cylindrical A ir Bubbles through Liquids

Contained in Vertical Tubes ”, Chemical Engineering Science, Vol. 17, pp. 351-361, 1962

10. G.P. Celata, M. Cumo, F. D Annibale, A. Tomiyama, “Bubble Rising Velocity in Saturated Liquid up to

Critical P ressu re”, Experimental Heat Transfer, Fluid Mechanics, and Thermodynamics, pp. 1319­

1328,2001

11. A.W.G. de Vries, “Path and Wake o f a Rising B u b b le”, Ph.D. Thesis, University o f Twente, 2001 12. S. Ozdemir, “Investigation o f A ir Bubble Motion in Water Through a Vertical Narrow Rectangular

Channel by Using Image Processing Techniques”, Ph.D. Thesis, Middle East Technical University,

Ankara, Turkey, 2005

13. Virtualdub Software, “Virtualdub.org”, www.virtualdub.org, January, 2004

14. National Institute o f Health, “Im a g eJ”, http://rsb.info.nih.gov/ij/, 17 November 2004

15. R. Krishna, M.I. Urseanu, J.M. van Baten, J. Ellenberger, “Wall Effects on the Rise o f Single Gas