Stability of periodic bubble departures at a low frequency

Chemical Engineering Science 109 (2014) 171–182

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Stability of periodic bubble departures at a low frequency P. Dzienis, R. Mosdorf n Bialystok University of Technology, Faculty of Mechanical Engineering, Wiejska 45C, 15-351 Białystok, Poland

H I G H L I G H T S







Dynamics of bubble departures from glass nozzle has been studied. Stability loss of periodic bubble departures has been investigated. Stability of periodic bubble departures depends on the volume of air supply system. Non-linear gas compression is responsible for chaotic bubble departures.

art ic l e i nf o

a b s t r a c t

Article history: Received 28 June 2013 Received in revised form 24 January 2014 Accepted 2 February 2014 Available online 6 February 2014

The dynamics of bubble departures (at a frequency of f¼3 Hz) from a glass nozzle submerged in a tank filled with distilled water has been experimentally and theoretically studied. The volume of the system that supplies air to the nozzle (plenum chamber volume) and the air volume flow rate were changed in the experiment. The air pressure, bubble paths and liquid flow inside the nozzle were simultaneously recorded using a data acquisition system and a high-speed camera. It was shown that an increase in the plenum chamber volume leads to an increase in the intensity of the occurrences of chaotic changes in the subsequent waiting times. The analysis of the mechanism of the stability loss of the periodic bubble departures was based on changes in the time of the air pressure, the depth of the liquid penetration into the nozzle, the time of the bubble growth, the waiting time, and the bubble paths and their sizes, which is presented in this paper. The results of the analysis are compared with simulations that are based on the models of bubble growth and liquid flow inside the nozzle during the waiting time. It was shown that the air pressure rise, Δpl, during the waiting time is a non-linear function of the gas pressure after the bubble departure and the liquid velocity around the nozzle outlet. The nonlinearity of Δpl increases when the plenum chamber volume increases, and it decreases when the air volume flow rate increases. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Bubbles Bubble departures Plenum volume Bubble formation Bubble chains Chaotic bubble dynamics

1. Introduction There are a substantial number of papers that report the nonlinear behaviours of the bubbling process (Zun and Groselj, 1996; Kovalchuk et al., 1999; Zhang and Shoji, 2001; Mosdorf and Shoji, 2003; Cieslinski and Mosdorf, 2005; Vazquez et al., 2008; Ruzicka et al., 2009a; Stanovsky et al., 2011; Mosdorf and Wyszkowski, 2011). There are at least two groups of phenomena that are responsible for the occurrence of non-linear behaviours of bubbles. The first group is connected with the behaviour of the flow of bubbles in the liquid (Peebles and Garber, 1953; Hughes et al., 1955; Davidson and Amick, 1956; Davidson and Schüler, 1960; Kling, 1962; McCann and Prince, 1971; Zun and Groselj, 1996; Kyriakides et al., 1997). The second group is connected with

n

Corresponding author. Tel.: þ 48 85 746 92 00; fax: þ48 85 746 92 10. E-mail address: [email protected] (R. Mosdorf).

http://dx.doi.org/10.1016/j.ces.2014.02.001 0009-2509 & 2014 Elsevier Ltd. All rights reserved.

processes that directly concern the bubble departures and that occur in the gas supply system. The influence of the plate thickness, surface tension, the viscosity of the liquid and the height of the liquid column on the depth of the liquid penetration into the nozzle has been reported in the previous papers (Antoniadis et al., 1992; Dukhin et al., 1998a,b; Kovalchuk et al., 1999; Ruzicka et al., 2009a; Stanovsky et al., 2011). The impact of the chamber volume and the height of the liquid over the orifice outlet on the frequency of the bubble departures have been investigated. It has been observed that the increase in the chamber volume increases the time period between two subsequent bubbles (Antoniadis et al., 1992). The increase in the height of the liquid over the orifice outlet leads also to an increase in the time period between subsequent bubbles (Stanovsky et al., 2011). The time period between the departing bubbles decreases when the number of gas–liquid interface oscillations decreases (Ruzicka et al., 2009a). The phenomena of the liquid movement inside the orifice or nozzle have been experimentally investigated

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and modelled by many researchers (Dukhin et al., 1998a,b; Kovalchuk et al., 1999; Ruzicka et al., 2009b). Liquid weeping has been observed between subsequent bubble departures. The liquid weeping at orifices that have different diameters has been measured and modelled by Miyahara et al. (1984). The influence of the wake pressure on the liquid weeping at a single submerged orifice has been discussed in the paper (Zhang and Tan, 2000). It has been shown that the wake pressure at the orifice has a significant effect on the predicted values of the weeping flow rates and the weep points. The bubble motion above the nozzle, gas and liquid flow inside the nozzle, the evolution of gas pressure in the plenum, the dynamics of the bubble growth, its size and departure velocity are correlated with one another. Therefore, an explanation for the nature of the chaotic bubble departures requires considering the interactions between the phenomena that occur in the plenum chamber and the motion of bubbles over the nozzle outlet. In the experimental part of the present paper, the influence of the plenum chamber volume on the chaotic bubble behaviours has been analysed. A detailed analysis of the stability loss of the periodic bubble departures (at the mean frequency of 3 Hz) has been presented. In the other part of the paper, the model proposed by Ruzicka et al. (2009b) has been adopted for modelling the chaotic bubble departures. The mechanism by which the chaotic bubble departures appear in the model has been discussed and compared with experimental results.

2. Experimental setup, measurement techniques In the experiment, bubbles were generated in a tank (300 mm  150 mm  700 mm) that was filled with distilled water using a glass nozzle with an inner diameter of 1 mm. The length of the nozzle was 70 mm. The schema of the experimental setup is shown in Fig. 1.

Fig. 1. Experimental setup: 1 – glass tank, 2 – camera, 3 – light source, 4 – computer acquisition system, 5 – air pump, 6 – glass nozzle, 7 – air valve, 8 – plenum chamber with the possibility of changing the volume, 9, 11 – pressure sensors, 10 – flow metre, 12 – air tank, 13 – laser, 14 – phototransistor, and 15 – semi-transparent glass.

The nozzle was placed at the bottom of the tank. The temperature of the distilled water was controlled by the digital thermometer MAXIM DS18B20 (with an accuracy of 0.1 1C) and, during the experiment, it was 2071 1C. The air pressure fluctuations have been measured with the use of the silicon pressure sensor Frescale Semiconductor MPX12DP, whose sensitivity was 5.5 mV/kPa. The air volume flow rate (q) was measured using the flow metre (MEDSON s.c. Sho-Rate-Europe Rev D, P10412A). The accuracy of the flow metre was equal to 5%. The plenum chamber volume was changed (in the range from 0.45 ml to 10.45 ml) by changing the volume of the syringe (8). The accuracy of the volume measurement was 7 0.5 ml, and it was defined by a syringe scale. The air pressure was recorded using a data acquisition system Data Translation DT9804 with a sampling frequency of 2 kHz and 16 bits of resolution. Bubble departures and liquid movement inside the nozzle were recorded with a high speed camera, the CASIO EX FX 1. The data and video have been recorded after reaching a steady state by the system (approximately 15 min after the change in the air volume flow rate). The duration of each video was 20 s. The recorded videos (600 fps) in the grey scale have been divided into frames (432  192 pixels). An example of such frames is shown in Fig. 2. The depth of the liquid penetration inside the nozzle was measured using a computer programme, which analysed the subsequent frames. The programme was prepared by investigators with the use of the Lazarus environment. They counted the number of pixels that had a brightness of greater than a certain brightness threshold on the each frame, along the nozzle wall. The threshold was adjusted for each video by comparing the results of the computer calculations with visual observations of the depth of the liquid penetration into the nozzle in the video. The size of 1 pixel was estimated to be 0.14 mm and is based on the known diameter of the nozzle. Approximately 12,000 frames were used to reconstruct the time series for each steady state of bubble departures. The Sobel filter based on the convolution of the image with a small, integer-valued filter has been used to identify the bubbles on the frames (Hedengren, 1988). Because the Sobel algorithm identifies only the edge of the image of the bubble, the interior of the detected bubble was filled by black pixels. Finally, each bubble was visible in the frame as a set of black pixels. The path of each bubble was reconstructed by tracking the trajectory of mass centre of its 2D image in subsequent frames. The vertical position of the mass centre of the bubble image was reconstructed in the range of 5–35 mm over the nozzle outlet due to imaging and software limitations. The depth of the liquid penetration inside the nozzle and the air pressure changes were recorded with different sampling frequencies, which were equal to 600 Hz (for video) and 2 kHz (for the pressure measurement). For the synchronisation of these time series, the adjustment of the sampling frequency was required. The pressure signals were resampled to 600 Hz, and

Fig. 2. Liquid penetration into the nozzle and bubble departures for the air volume flow rate q¼ 0.00632 l/min. The time between the frames shown in figure is equal to 0.0183 s. In the recorded video, the time between the frames was equal to 0.001667 s.

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linear interpolation was performed for this downsampling. The error of the synchronisation was less than the time between the subsequent video frames (i.e., 1.666 ms). The size of the bubbles, Sb, was determined by counting the number of pixels that represent the image of the bubble at the moment of its departure. Calculation of the 2D image of the departing bubbles requires the identification of pixels that represent the bubble interior. To accomplish this goal, it is necessary to choose the threshold of the pixel brightness. Because of the low resolution of the images recorded by the high-speed camera, small changes in the lighting conditions in the laboratory as well as small changes in the automatic adjustment of the camera for different videos, it is impossible to compare the areas of 2D images that are obtained in different videos. Only the data obtained from the video recorded at a constant automatic adjustment of the camera can be compared. Therefore, the comparison of the size of the departing bubbles for different air volume flow rates and the plenum volumes has not been presented in this paper.

3. Data characteristics In the paper (Dzienis et al., 2012), the influence of the air flow rate on the chaotic nature of the bubble departures has been analysed. It has been shown that the increase in the air volume flow rate leads to increasing the frequency of the bubble departures. For all flow rates, the periodic and non-periodic bubble departures have been observed. It has been shown that for q in the range of 0.0063–0.0174 l/min, the increase in the air flow rate is accompanied by a decrease in the number of non-periodically departing bubbles and the maximum depth of the liquid penetration inside the nozzle decreases. Examples of the recorded the depth of liquid penetration into the nozzle and pressure fluctuations are shown in Fig. 3, where the data recorded for different plenum chamber volumes are presented. In the time series, the two time periods are distinguished. Examples of those periods are marked with I and II in Fig. 3a. In period I, the amplitude of the changes in the depth of the liquid penetration inside the nozzle is high. The difference between subsequent maximum depths of liquid penetration into the nozzle is less than 2%, (the bubbles depart periodically). In period II, the amplitude of the changes in the depth of the liquid penetration into the nozzle varies significantly for subsequently departing bubbles (the bubbles depart non-periodically). The obtained results show that an increase in the plenum chamber volume (V) causes changes in the frequency of the bubble departures. The maximum depth h(t) of the liquid penetration into the nozzle increases (in the volume range of 0.45–7.95 ml), but for V¼ 10.45 ml, it decreases somewhat in comparison with V¼ 7.95 ml. The depth h(t), becomes asymmetrical around its maximums. The significant increase in the time required for the liquid removal from the nozzle is visible in the range of 7.95– 10.45 ml. The increase in the plenum chamber volume causes also a change in the time periods, where the periodic and non-periodic bubble departures occur. The right side of Fig. 3 shows the system trajectory reconstruction in the space (p, h). The bubble growth creates a part of the trajectory where h¼0. The remaining part of the trajectory corresponds to the movement of liquid inside the nozzle. We can note that the increase in the depth of the liquid penetration increases the air pressure at the beginning of the bubble growth. During the periodic bubble departures, the subsequent trajectories are close to each other. Non-periodic bubble departures have a smaller depth of the liquid penetration and a lower initial value of the air pressure during the bubble growth.

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The frequency of the bubble departures for each of the time series has been estimated using the FFT method (Chui, 1992). The power spectral analysis was used to determine the dominant frequency. The results are presented in Fig. 4. The results show that the changes in the plenum volume modify the frequency of the bubble departures. When the range of the plenum volume is (0.45–5.45 ml), an increase in the plenum volume decreases the bubble departure frequency. This result corresponds to the results obtained in the papers (Antoniadis et al., 1992; Stanovsky et al., 2011) for the plenum chamber located under the orifice. For a plenum volume that is greater than 5.45 ml, the mean frequency of the bubble departures increases together with the plenum volume, but the frequency modifications are small.

4. Mechanism of losing the stability of periodic bubble departures In Fig. 5, the part of the time series (a time interval equal to 6 s) that is shown in Fig. 3a and other parameters recorded in this time period have been presented. The following parameters are shown:


bubble paths (Fig. 5a),
bubble shapes at the ends of the paths (presented in Fig. 5a above the bubble paths),


h(t) – the depth of the liquid penetration into the nozzle (Fig. 5b),


Sb – the size of the area of the 2D bubble images (Fig. 5b –






points) at the moment of the bubble departure. Fig. 5b presents the percentage changes in the area of the 2D image of the departing bubbles – the maximum recorded size is equal to 100%, p – air pressure in the plenum chamber (Fig. 5c), tg – the time of the bubble growth (Fig. 5d), tw – the waiting time (Fig. 5d).

The intervals that are denoted by I and II in Fig. 5b correspond to the intervals that are shown in Fig. 3a. In interval I (Fig. 5b), the bubble growth time and waiting time are constant. There are two such intervals – at the beginning and at the end of the 6 s interval. In interval II (Fig. 5b), the successive times for the bubble growths and waiting times as well as the bubble sizes and bubble shapes vary in time. The lateral displacements of the bubbles at the ends of the bubble paths, shown in Fig. 5a, also vary for subsequent bubbles. The time interval (6 s) includes both the period in which the periodic bubble departures lose their stability, and then, the period in which the system returns to the periodic bubble departures. The subsequent cycles of the pressure growth and its drop are presented in Fig. 5c. The bubble growth begins when the air pressure in the system reaches its maximum value, pmax, and the bubble departure occurs when the air pressure reaches the minimum value, pmin. The example of the points of pmax and pmin has been shown in Fig. 5c. The pressure drop during the bubble growth has been denoted as Δpb and the pressure growth during the waiting time as – Δpl. Fig. 6 shows the changes in the values pmax, pmin, for the bubbles that precede the stability loss of the periodic bubble departures. The x-axis represents the subsequent numbers of bubbles. The bubbles that are numbered by positive numbers correspond to the bubbles shown in Fig. 5a, c, and d. The number 0 and the negative numbers represent the 10 bubbles that precede the bubbles shown in Fig. 5. The lines are trend lines. The positive slopes of the trend lines (Fig. 6) show that within the time interval

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Fig. 3. Time series of the depth of the liquid penetration into the nozzle and the air pressure in the plenum chamber for gas with a volume flow rate of q ¼0.00632 l/min and different volumes of the air supply system. (a) V ¼0.45 ml, (b) V¼ 2.95 ml, (c) V ¼ 5.45 ml, (d) V ¼ 7.95 ml, and (e) V ¼ 10.45 ml.

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of periodic bubble departures, both pressures pmax and pmin slowly increase. The understanding of the mutual relationships between the data presented in Figs. 5 and 6 requires considering the liquid flow around the nozzle outlet and the forces that act on the bubbles. Our interpretation of the relationships mentioned above is presented below.

Fig. 4. The mean frequency of the bubble departures, f, for different volumes of the air supply system, V, and air volume flow rates, q.

175

Periodic bubble departures induce liquid circulation around the chain of bubbles. The liquid flows up along the chain of bubbles, but close to the walls of the tank, it flows down. The liquid flow that is formed in such a way leads to the appearance of a vertical

Fig. 6. Minimum and maximum values of the air pressure in subsequent cycles of bubble growth, for q ¼0.00632 l/min and V¼ 0.45 ml, in the time period that precedes the stability loss of the periodic bubble departures.

Fig. 5. Summary of the recorded data for V ¼ 0.45 ml and the air volume flow rate of 0.00632 l/min: (a) successive paths of the bubbles, (b) depth of the liquid penetration into the nozzle, (c) changes in the gas pressure, and (d) times of the subsequent bubble growth and waiting times.

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component of the liquid velocity in the vicinity of the nozzle outlet. Such a liquid velocity modifies the drag force that acts on the bubbles (Zhang and Shoji, 2001). Bubbles depart (start to move upward) when the sum of the forces that act on the bubble become different from zero. The balance of forces just before the bubble departure is F AM þ F B þF M ¼ F s þ F d

ð1Þ

where F AM added  mass force : F AM ¼  ρl


 d dxc CM V b  vpp ; dt dt

F B buoyancy force : F B ¼ gðρl  ρg Þ U V b ; F M gas momentum : F M ¼ ρg

q2b π r 2o

where qb ¼

ð2Þ ð3Þ

π 8μg

!
 r 4n ðp  pb Þ; l ð4Þ

F s maximum value of the surface tension force : F s ¼ 2π r n s; when the cosine of the attachment angle is equal to 1; F d drag force : F d ¼ 0:5C d ρl π r 2




  dxc  dxc  vpp   vpp ; dt dt

ð5Þ ð6Þ

g is the gravitational acceleration (m/s2), to is the beginning of the bubble movement, s is the surface tension (N/m), rn is the nozzle diameter (m), ro is the bubble diameter (m), ρ is the density (kg/m3), μ is the viscosity (kg/ms), qb is the air flow rate supplied to the bubble (m3/s), Cd is the drag force coefficient, CM ¼ 0.5 is added-mass coefficient for a sphere, vpp is the velocity of the liquid around the growing bubble (m/s), and xc is the position of the bubble centre (m). The upward flow close to the nozzle outlet reduces the force of F d (6) and also leads to added-mass force FAM. Both these changes promote bubble departure and increase the value of pmin (Fig. 6). After the departure of bubble No. 4 (Figs. 5 and 6), a rapid growth of pmin occurred (the path of bubble No. 3 was rectilinear). The change in the time of the bubble growth (No. 4) was not recorded during the experiment because of the low measurement accuracy of the time (1/600 s). Only the increase in pmin for bubble No. 4 was recorded (Fig. 6). The increase in pmax (Fig. 6, bubble No. 4 leads to the increase in the air flow rate supplied to the bubble (qb) and, finally, to the growth of the force FM. This growth causes an earlier bubble departure and an increase in the pressure pmin (which is visible in Fig. 5 for the subsequent bubbles, which are numbered 5 and 6) and a decrease in the bubble size Sb (as shown in Fig. 5b). The increase of pressure pmin (pressure at the bubble departure) shortens the time (tw) of liquid motion inside the nozzle (Fig. 5d). This circumstance leads to a decrease in the pressure pmax (the liquid is removed in a shorter amount of time; therefore, the pressure pmax cannot be increased up to the previous value). This phenomenon is shown in Fig. 5c, where the grey lines show the trends in the pressure pmax, pmin changes. The smaller difference pmax  pmin is linked with a decrease in the size of the bubbles Sb. Smaller bubbles move faster – they are less deformed, and the paths of such bubbles turn, as shown in Fig. 5a. The increase in the liquid velocity vpp around the nozzle outlet reduces the drag force F d (preventing the detachment of the bubble) and increases the added-mass force FAM (promoting the detachment) which leads to a sooner detachment and hence increases pmin. When the pressure at which the bubble starts to grow, pmax, drops excessively, then the growing bubble departs at the very low pressure pmin (which is visible in Fig. 5c, bubble No. 8). This drop in the pressure (Δpb) increases the depth of the liquid

penetration into the nozzle (Fig. 5b, the liquid penetration after departing bubble No. 8) and leads to the increase in the time tw (as shown in Fig. 5d for bubble No. 9). The increase in the time tw causes the next bubble to start growing at the higher pressure (pmax). When the pressure drop Δpb (for bubble No. 8 in Fig. 5c) is sufficiently large, then the periodic bubble departures attempt to stabilise. However, when the pressure drop Δpb is not sufficiently large, then the stability is lost again (as can be seen in Fig. 5, for bubbles from No. 9 to No. 15). The pressure drop Δpb for the bubbles from No. 15 is sufficiently large; therefore, the periodic bubble departures are restored. The above discussion allows us to formulate the following scenario for the stability loss of the periodic bubble departures: The increase in the pressures pmin and pmax in periodic bubble departures (Fig. 6) is caused by the imbalance between the amount of gas that is supplied to the plenum chamber and the amount of gas that is consumed by bubbles. This imbalance occurs due to changes in the hydrodynamic conditions above the nozzle outlet. In non-periodic bubble departures, the relationship between pmin and pmax has a non-linear character. A small increase in the pressure pmin leads to a decrease in the pressure pmax. However, when pmax drops below a certain value, the bubble growth leads to a significant decrease in the pressure pmin, because of changing force balance. Low pmin allows a deep penetration of liquid to the nozzle. Afterward, the bubble departures become non-periodic. The changes in the relationships between the subsequent pmin and pmax are accompanied by changes in the bubble sizes and their paths. During the non-periodic bubble departures, the excess of gas accumulated in the plenum chamber in the periodic bubble departures is removed. For the air volume flow rate q ¼0.00632 l/min and the plenum chamber volume equal to 0.45 ml (Fig. 5), when the time of the bubble growth and the waiting time are constant for the subsequent bubbles, the frequency of the bubble departures is equal to 3.03 Hz. However, in the middle part of the time period shown in Fig. 6, where the bubbles depart non-periodically, the frequency of the bubble departures is equal to 3.77 Hz.

5. Modelling subsequent bubble departures 5.1. Model of bubble interactions Based on the previous discussion, it is assumed that the liquid flow in the vicinity of the nozzle outlet modifies the conditions of the bubble departures and the liquid penetration into the nozzle. The departing bubbles interact with each other by changing the field of the liquid velocity around the nozzle and the pressure in the plenum chamber. For modelling the interactions between the subsequently departing bubbles and the plenum chamber, we use the model of the bubble growth and liquid movement inside the nozzle, which has been proposed by Ruzicka et al. (2009b). Forces considered in the model of bubble growth have been supplemented by the drag and added-mass forces, which are dependent on the liquid velocity around the nozzle outlet (Zhang and Shoji, 2001) and the changes in the gas momentum (Zhang and Shoji, 2001; Vazquez et al., 2010). In the model, the dynamic pressure in the liquid over the nozzle outlet has also been considered. Fig. 7 shows a schematic model of the bubble growth and the liquid flow in the nozzle. In the model (Ruzicka et al., 2009b), the time period of the bubble growth was divided into two stages: an increase in the spherical bubble radius described by the Rayleigh–Plesset equation (Ruzicka et al., 2009b) and the vertical movement of the spherical bubble described by the motion equation of the bubble centre, which includes the added-mass force (Ruzicka et al., 2009b). The last phase of the bubble movement is shown

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Fig. 7. Schema of the model for the bubble growth and the liquid flow inside the nozzle: (a) bubble growth and (b) liquid flow in the nozzle.

in Fig. 7a. After the bubble departure, the liquid movement inside the nozzle starts. This movement is accompanied by the liquid flow above the nozzle outlet (Ruzicka et al., 2009b). It was assumed that the weight of this additional liquid corresponds to the mass of the liquid in the sphere of radius radd ¼2  rn. The radius of the sphere filled with liquid, radd, was determined as a result of a comparison between the calculation results and the experimental data. It is also assumed that the value of the initial liquid velocity is equal to the velocity of the departing bubble. In the model, the motion of the mass centre of the liquid in the coordinate system shown in Fig. 7b has been analysed. As the air pressure in the plenum chamber increases, the liquid inside the nozzle is removed from the nozzle at a relatively high velocity, which leads to the rapid formation of small air bubble at the nozzle outlet. It has been assumed that the initial bubble radius is 2  rn and that its initial growth velocity is equal to the velocity of the liquid removed from the nozzle. Thus, in the model, the phenomena that accompany the growth of the bubble with a radius of less than 2  rn and the phenomenon of breaking the bubble neck are omitted. Those phenomena influence the dynamics of the bubble departures; however, in the cases that are under consideration, the duration of the omitted phenomenon is much shorter than the waiting time and duration of the bubble growth. Therefore, it is assumed that the bubble growth rate and the dynamics of the liquid movement inside the nozzle have a decisive impact on the nature of the bubble departures. Eqs. (7)–(17) describe the bubble growth and the liquid movement inside the nozzle. These equations account for the additional forces that are responsible for the interactions between successively departing bubbles. An isothermal process is considered.

The bubble growth: !
 #
" dpb pb π r 4n dV ¼ ðpc  pb Þ  b dt Vb 8μg l dt

ð7Þ

where pb is the pressure of the air inside the bubble (Pa), Vb is the volume of the bubble (m3), μg is the dynamic viscosity of the gas (kg/ms), rn is the radius of the nozzle (m), l is the nozzle height (m), and pc is the air pressure in the plenum chamber (Pa): dpc kc pc ½q  ðπ =8μg Þðr 4n =lÞðpc  pb Þ ¼ dt Vc

ð8Þ

where q is the air flow rate (m3/s), and Vc is the plenum chamber volume (m3). !

   2 d r 3 dr 2 pb  ph  ð2s=rÞ  4μl ðdr=dtÞ=r ¼ ð9Þ r  2 dt ρl dt 2 where r is the radius of bubble (m), ph is the hydrostatic pressure (Pa), s is the surface tension (N/m), and ρl is the liquid density (kg/m3). When F B þ F M þ F AM F s  F d 4 0, then 
 d dxc ðρg þ C b ρl Þ UV b  vpp ð10Þ ¼ F B þ F M þ F AM  F s  F d dt dt otherwise, xc ¼ r vc ¼ dr=dt

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where the added-mass force FAM that helps to detach the bubble in the case of upward flow of the liquid (when vpp 4 vc) is  dV

b vc  vpp F AM ¼  ρg þ C b ρl ð11Þ dt Criterion for bubble departure: xc 4 ¼ r þ2r n

ð12Þ

The liquid movement:
 dpc pc dx ¼ q þ π r 2n l dt Vc dt d dt

  4 4 dx ρl π r 2n xl þ ρl π ð2r n Þ3 l ¼ F 1  F 2 3 3 dt

ð13Þ

ð14Þ

where


 s ρvpp jvpp j F 1 ¼  s Δp ¼  π r 2n pc  ph þ ρl gð2xl Þ þ 2  2 rn F 2 ¼ 2  8πμl xl

dxl dt

ð15Þ

ð16Þ


The vertical velocity of the liquid around the nozzle outlet is

Criterion for the end of the liquid movement: xl o0

Eqs. (7)–(17) has been solved using SCILAB. The ordinary differential equation solver (ODE) with root-finding capabilities has been used to make a determination of the time of the bubble growth and the waiting time. The relative and absolute tolerances were 10  5 and 10  7. The diagram in Fig. 8a shows the calculation algorithm, and Fig. 8b and c illustrates the comparison of the results of the simulations and experimental data. A good agreement of the simulation results with the measurement data has been obtained. The differences between the simulation results and the measurement data, according to the authors, are caused by omitting in the model the phenomena that accompany the growth of small bubbles and occur during the bubble detachment and measurement errors. The measurement errors reach the maximum value for the low depth of the liquid penetration into the nozzle, when the liquid velocity reaches a high value, and during the time of 1/600 s, the liquid–air interface displacement is significant. The diagram in Fig. 8a shows the method for the determination of the initial conditions in the simulation of subsequent bubble departures. The following has been assumed:

equal to the departure velocity of the previous bubble.

ð17Þ


The initial value of the gas pressure, pb, in a small bubble

where xl is the height of the liquid penetration into the nozzle (m), s is the cross-sectional area of the nozzle (m2), and pc is the gas pressure in the plenum chamber (Pa). The air volume flow rate supplied to the bubble through the nozzle is determined by the Hagen–Poiseuille equation (Ruzicka et al., 2009b), and the pressure changes in the bubble are described by Eq. (7), in which the isothermal process is considered. The pressure changes in the air supply system are described by Eq. (8). The increase in the radius of a spherical bubble is described by the Rayleigh–Plesset equation in the form of (9). The bubble begins to move when F B þ F M þ F AM  F s  F d 4 0. In this case, the nozzle outlet is connected with the bubble by a thin neck (Fig. 7). The increase in the bubble radius is further described by the Rayleigh–Plesset Eq. (9), but its vertical motion is determined by Newton's second law (10). The equation of motion (10) considers the surface tension force (5), the buoyancy force (3), the drag force (6), added-mass force (2) and the change in the gas momentum (4). It has been assumed that the bubble is detached from the nozzle outlet when criterion (12) is fulfilled. The model of the liquid flow inside the nozzle was based on the equation of motion of the mass centre of the liquid that was involved in the flooding of the nozzle (14). The pressure changes in the air supply system are described by Eq. (13). The force F1 (15) is related to the pressure difference that occurs in the system (with accounting for the changes in the dynamic pressure), and the force F2 (16) is related to the resistance of the movement of the liquid in the nozzle. According to Ruzicka et al. (2009b), the mass of the liquid that is involved in the movement caused by flooding the nozzle is greater than the mass of the liquid inside the nozzle. The weight of the liquid that is involved in the movement has been described by the formula ð4=3Þρl π r 2n xl þ ρl ð4=3Þπ ð2r n Þ3 . The reduced mass of the moving liquid was calculated assuming that the kinetic energy of moving liquid (with parabolic velocity profile) is equal to kinetic energy of reduced mass. Therefore, the reduced mass of the moving liquid is equal to ð4=3Þρl π r 2n xl (14). The liquid flow stops when the mass centre of the liquid that is involved in the movement reaches zero, i.e., the criterion is fulfilled (17). In summary, the system of equations that describes the growth of the bubble has the following independent variables (pb, r, v, pc, xc, vc) and the parameter vpp. Equations that describe the liquid motion that is involved in flooding the nozzle have the following independent variables (xl, pc, vl) and the parameter vpp. The set of

(r ¼2  rn) and in the plenum chamber, pc, is equal to the gas pressure in the plenum chamber at the end of the liquid movement. The initial velocity of the bubble growth in Eq. (9) is equal to the liquid velocity at the end of the liquid movement. The initial value of the gas pressure, pc, for the liquid flow inside the nozzle is equal to the pressure of the gas in the plenum chamber during the last stage of the bubble growth. The initial value of the liquid velocity, vl, is equal to the value of the bubble departure velocity. In the simulation, this value is equal to vpp. (The departed bubble removes an amount of liquid from the vicinity of the nozzle outlet; therefore, the new liquid flows to the vicinity of the nozzle outlet with the same velocity and in the opposite direction).






Fig. 9a and b shows the simulation results of the variation in time of the mass centre position of the liquid for different air volume flow rates. For the low value of the air volume flow rate q¼ 0.00932 l/min, the changes in the mass centre position have a chaotic character (Fig. 9a), but for q ¼0.012 l/min, the position of the mass centre changes periodically (Fig. 9b). Fig. 9c shows the system trajectories in the space (p, xl), which were reconstructed based on the pressure and the mass centre position of the liquid in the nozzle (the picture consisted of 30,000 points). At the low air volume flow rate (0.0096 l/min) the mass centre position of the liquid involved in the flooding of the nozzle changes chaotically. Subsequent trajectories in the space (p, xl) do not coincide (Fig. 9c). The trajectory shape is similar to the shape of the trajectories that were reconstructed from the experimental data (Fig. 3). Fig. 9d shows the changes in the frequency of the bubble departures as a function of the air volume flow rate, and the results were compared with the experimental data. In the simulation, we obtained a small decrease in the bubble departure frequency together with an increase in the plenum chamber volume. This result is in accordance with the experimental results presented in Fig. 4. The average frequencies of the bubble detachments (in the simulation) are higher than those recorded in the experiment. The smallest differences occur at the low air volume flow rate. The differences between the experimental data and the simulation results are most likely caused by phenomena that are omitted in the model. The purpose of the construction of the model

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Fig. 8. Scheme of the simulation for subsequent bubble departures and a comparison between the results of the simulation and the experimental data: (a) the scheme of a simulation, where vpp is the vertical velocity of the liquid around the nozzle, (b) comparison between the simulation results of the bubble growth and the experimental data, and (c) comparison between the results of a simulation of the liquid flow and the experimental data for the liquid penetration (experiment and simulation: q¼ 0.00632 l/min; V ¼0.45 ml; H¼ 0.45 m. The initial conditions in the simulation: vol ¼ 0.21 m/s; po ¼4725, vob ¼0.05 m/s; ro ¼0.0005 m, pob ¼ 5730 Pa).

presented in the paper was to identify the mechanism of the appearance of chaotic changes in the frequency of the bubble departures. The model has a qualitative character; therefore, the obtained results of the simulations shown in Figs. 8 and 9 can be treated as having good agreement with the measurements. 5.2. Mechanism of chaos appearance in the model Fig. 10 presents the changes in air pressure in the plenum chamber (Fig. 10a), the velocity of the liquid around the nozzle outlet (Fig. 10b) and the mass centre position of the liquid in the nozzle (Fig. 10c). Subsequent liquid penetrations into the nozzle and bubble growths are accompanied by the changes in the liquid velocity of vpp, which is shown in Fig. 10b. Fig. 10a shows the five points that correspond to subsequent phases of the growth and departure of two bubbles. The points have been marked with black dots. The process under consideration begins at the point that is marked with B (Fig. 10a), where the process of the flooding of the nozzle starts at the pressure pmin. Then, the air pressure increases, reaching the value of pmax ¼pmin þ Δpl at the moment of liquid removal from the nozzle. This point is the pressure at which the bubble starts to grow (pmax). During the bubble growth, the pressure in the plenum chamber is reduced reaching the value of p ¼pmax  Δpb at the time of bubble departure. In the next period, the gas pressure increases during the waiting time, and after that, it decreases during the bubble growth. Finally, point E is reached. During the waiting time, after point E, the depth of the liquid penetration into the nozzle reaches a large value (Fig. 10c). This process is accompanied by an increase in the pressure in the

plenum chamber. A similar situation appears before point B. The cycle between points B and E is unstable. A similar character of pressure changes was observed in the experiment. In Figs. 5c and 10a, the grey lines have been used to mark the trend lines of the pressures pmax and pmin. To understand the mechanism of the pressure changes, as shown in Fig. 10a, we analysed the influence of the small pressure changes pmax and pmin and the small changes in the liquid velocity vpp on the pressure changes Δpb (pmax, vpp, …) and Δpl (pmin, vpp, …). The ranges in the pressure and liquid velocity changes in the simulations are similar to the ranges that appear in the simulation of non-periodic bubble departures. The air volume flow rate is equal to 0.0096 l/min. In the analysis of the changes in Δpl (pmin, vpp, …), it has been assumed that vol ¼vpp. Numerical calculations show that the function of Δpb (pmax, vpp, …), at a constant velocity vpp and volume of the plenum chamber, is a linear function of pmax, with the slope equal to approximately 1.06 (Fig. 11a). The change in the volume of the plenum chamber does not change the character of the function Δpb (pmax, vpp, …). A more complex situation occurs for the function Δpl (pmin, vpp, …). The function Δpl (pmin, vpp, …) is a non-linear function of pmin. The changes in the volume of the plenum chamber modify the shape of the function Δpl (pmin, vpp, …). The derivative ∂½Δpl ðpmin ; vpp ; :::Þ=∂pmin is shown in Fig. 11c. The increase in the plenum volume and the decrease in the air flow rate enhance its nonlinearity. The non-linear character of the function Δpl (pmin, vpp, …) (Fig. 11c) is responsible for the large sensitivity of pmax, which

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Fig. 9. Simulation results: (a) changes in the mass centre position of the liquid for q¼ 0.0078 l/min, V ¼0.5105 ml, (b) changes in the mass centre position of the liquid for q¼ 0.012 l/min, V¼ 0.5105 ml, (c) trajectories in the space (p, xl) for q¼ 0.0078 l/min, and (d) changes in the average frequency of the bubble departures and comparison with the experimental data.

significantly reduces (due to reduced liquid penetration in the nozzle) when pmin slightly increases. Due to the linear character of the dependence Δpb (pmax) (Fig. 11a), a reduction of pmax decreases Δpb, which brings the pressure to return to a value that is close to the previous pmin. However, because the slope of Δpb (pmax, …) is slightly larger than 1, Δpb does not completely compensate the reduction of pmax, and pmin slightly increases. Because of large sensitivity of pmax to pmin, pmax decreases strongly during the next cycle. This process is shown in Fig. 10a, where the grey lines show the trends of the changes in the pressures pmin and pmax. This behaviour of the pressure changes is similar to the behaviour of the pressure changes in the experiment (Fig. 5c). The changes in Δpb and Δpl as a function of vpp are shown in Fig. 11b. The small increase in the liquid velocity vpp leads to an increase in the values of Δpb and an increase in the values of Δpl. However, the magnitude of the changes in Δpl is much greater than the change in Δpb. A further increase in vpp (when vpp is greater than 0.1 m/s) leads to a decrease in Δpl. The appearance of a maximum of Δpl (vpp) in the range of the liquid velocity shown in Fig. 11b, is connected with the existence of two ways, how vpp affects the flow inside the nozzle. On the one hand, the initial velocity of the liquid flow inside the nozzle, vol increases with vpp, leading to the initial increase of Δpl at small values of vpp (Fig. 11b). On the other hand, the liquid pressure over the liquid–

air interface decreases with vpp Eq. (14). This relationship causes a decrease in Δpl. The changes of nonlinearity of Δpl (vpp), when the plenum volume and air flow rate are changed, are shown in Fig. 11b. The nonlinearity of the gas compression in the plenum increases with the plenum volume and is responsible for the appearance of deterministic chaos in the simulations. The above mechanism of creating non-periodic bubble departures refers to the model and is a consequence of the model assumptions. The key assumption is that the liquid velocity around the nozzle modifies both the liquid pressure and the initial velocity of the liquid flow inside the nozzle. On the basis of the present experimental results, it is difficult to estimate the actual relationship between those two mechanisms. However, the character of the changes in pressure and the depth of the liquid penetration is similar for both experimental and model results. This similarity leads to the conclusion that the nonlinearity of the gas compression in the plenum plays an important role in the generation of the non-periodic bubble departures. The similarity between the system trajectories in the experiment (Fig. 3) and in the simulation (Fig. 9c) in the space (p, xl) confirms this conclusion. The experiments presented in this paper are affected by inaccuracies in the measuring instruments. Those errors make it difficult to identify the shape of the functions that are responsible for the appearance of the non-periodic bubble departures (e.g., the

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Fig. 10. Changes in the air pressure, liquid velocity and depth of liquid penetration into the nozzle, for V ¼0.5105 ml and q ¼0.0078 l/min: (a) pressure changes, (b) liquid velocity changes, and (c) depth of liquid penetration into the nozzle pmin – air pressure in the initial phase of a liquid penetration into the nozzle, pmax – air pressure in the initial phase of bubble growth, Δpl – change in the air pressure during the waiting time, Δpb – change in the air pressure during the bubble growth.

return map of the subsequent waiting times). Such an analysis could be useful for the final validation of the model assumptions. Such a verification would require more accurate measuring instruments, and acquiring large amount of data.

6. Conclusions In this paper, the dynamics of the bubble departures (at a frequency equal to 3 Hz) from a glass nozzle that is submerged in a tank filled with distilled water has been experimentally and theoretically studied. The air pressure, bubble paths and liquid movement inside the nozzle were simultaneously recorded when using the data acquisition system and a high-speed camera. The analysis of the recorded data shows that the slow growth of the pressure pmax (Fig. 6) during the periodic bubble departures leads to a decrease in the bubble growth time – the bubbles become smaller and, thus, the pressure pmin increases. This arrangement is responsible for a decrease in the duration of the liquid penetration into the nozzle and a drop in the pressure of pmax. When the pressure pmax, at which the bubble starts to grow, drops too low, then the growing bubble departs at the very low pressure of pmin. This pressure drop increases again the depth and duration of the liquid penetration into the nozzle, which stabilises the bubble departures. During the non-periodic bubble departures, the excess of air accumulated in the

Fig. 11. Dependence of Δpb and Δpl on (a) pmax, (b) vpp, and (c) the derivatives ∂Δpl/∂pmin.

system during the periodic bubble departures is released, and periodic bubble departures are established again. It has been shown that an increase in the plenum chamber volume leads to an increase in the intensity of the occurrence of chaotic changes in the subsequent waiting times. During the non-periodic bubble departures, the changes in the waiting times were many times greater than the changes in the bubble growth time. Therefore, in the model, the initial boundary conditions for the phenomena that occur in the waiting time were associated with the departure velocity of the previous bubble. The models of bubble growth and liquid movement inside the nozzle that were proposed in the paper (Ruzicka et al., 2009b) have been used. The forces that were considered in the model of bubble growth have been supplemented by the drag and added-mass forces (which are dependent on the liquid velocity around the nozzle outlet) and the changes in the gas momentum. In the model of liquid flow in the nozzle, the liquid velocity over the nozzle outlet has been considered. This model allows us to simulate the non-periodic bubble departures. The analysis that is

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conducted when using the model of subsequent bubble departures shows the following:


The non-linear character of the pressure increase Δpl (pmin,





vpp, …) during the waiting time amplifies the fluctuations in the pressure pmin. The nonlinearity of the function Δpl (pmin, vpp, …) increases when the volume of the plenum chamber increases. Therefore, the increase in the volume of the plenum chamber leads to chaotic bubble departures. The nonlinearity of the function Δpl (pmin, vpp, …) decreases when the air volume flow rate increases. Therefore, the increase in the air volume flow rate leads to the disappearance of the chaotic bubble departures.

The nature of the time series of the depth of the liquid penetration into the nozzle and the plenum pressure fluctuations in the simulations were notably similar to those recorded in the experiment. Additionally, the shapes of the reconstructed trajectories in the space (p, xl) are also similar. It can be concluded that the model assumptions qualitatively correspond with phenomena that occur in the actual system. However, the experimental results are not the final verification of the model assumptions and further experimental study with more accurate instrumentation is required. Acknowledgements The authors would like to express their appreciation to MSc Tomasz Wyszkowski at the Bialystok University of Technology for his outstanding help in performing the experiment. References Antoniadis, D., Matzavinos, D., Stamatoudis, M., 1992. Effect of chamber volume and diameter on bubble formation at plated orifices. Chem. Eng. Res. Des. 70, 161–165. Chui, Ch.K., 1992. An Introduction to Wavelets. Academic Press. Cieslinski, J.T., Mosdorf, R., 2005. Gas bubble dynamics experiment and fractal analysis. Int. J. Heat Mass Transfer 48 (9), 1808–1818. Davidson, J.F., Schüler, B.O.G., 1960. Bubble formation at an orifice in an in viscid liquid. Trans. Inst. Chem. Eng. 38, 335–345.

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