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Prediction of flow regimes transitions in bubble columns using passive acoustic measurements
Chemical Engineering and Processing 48 (2009) 101–110
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Chemical Engineering and Processing: Process Intensification journal homepage: www.elsevier.com/locate/cep
Prediction of flow regimes transitions in bubble columns using passive acoustic measurements A. Ajbar ∗ , W. Al-Masry, E. Ali Department of Chemical Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 19 September 2007 Received in revised form 13 February 2008 Accepted 14 February 2008 Available online 10 March 2008 Keywords: Bubble column Air–water system Acoustics Hydrophone Flow regimes Transitions Chaos
a b s t r a c t Passive acoustic sound, measured by a hydrophone in an air–water bubble column, is used to study the hydrodynamics of the unit. The recorded measurements taken at different superficial gas velocities are processed using both spectral and chaos-based techniques in order to characterize the column flow regimes and to predict the transitions points. These processing tools were supported by digital video imaging of bubbles motion inside the column. The results of data analysis indicate the applicability of passive sound measurements to identify flow regimes in the bubble column. In this regard the analysis of sound spectra gives a useful qualitative comparison of flow regimes. Chaos-based techniques, on the other hand, are more successful in predicting the transition points between the homogenous and the churn-turbulent flow regimes in the column. The critical gas velocities of the transition are associated with a marked change in some of the calculated chaotic invariants of sound pressures. The calculated superficial gas velocities of the critical points are also found to be consistent with experimental observations. Moreover, a useful visualization of the dynamics induced in the column by the alternation of small and large bubbles is possible through the inspection of phase-space trajectories reconstructed from time-series measurements. The shape and size of the trajectories are closely linked to the size distribution of bubbles, and they change as the flow moves from one regime to an other. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Owing to their simple construction and their good heat and mass transfer, bubble columns are widely used for gas–liquid reactions in such diverse fields as chemical, petrochemical, pharmaceutical and water treatment [1,2]. Notwithstanding their operational advantages, bubble column hydrodynamics are complex and are characterized by different flow patterns depending on gas flow rate, physico-chemical properties of the gas–liquid system, gas distributor design, column diameter, etc. In this regard, homogeneous (bubbly) flow, transition regime and heterogeneous (churn-turbulent) flow are successively observed with increasing superficial gas velocities [3]. Another flow pattern, the slug flow regime, was also reported for small diameter columns [2]. The substantial amount of experimental work in the last decades on bubble columns has allowed an accurate description of the cited flow regimes. The homogenous regime is encountered at low gas velocities when the gas from the sparger is uniformly distributed. This regime is characterized by a narrow bubble-size distribution and by a radially uniform gas hold-up. In this regime both the
∗ Corresponding author. Fax: +966 1 467 8770. E-mail address: [email protected] (A. Ajbar). 0255-2701/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2008.02.004
distributor and the properties of the gas–liquid system play a dominant role in bubble-size distribution and gas hold-up profile [4,5]. The heterogeneous regime, on the other hand, is encountered at high gas flow rates. Small bubbles coalesce to produce relatively larger bubbles. This regime is therefore characterized by a wide distribution of bubble size and increased radial gas hold-up profile. The hydrodynamics in this regime are therefore more influenced by the bubble induced turbulence than by the distributor [6]. The two flow regimes are separated by a transition regime characterized by the development of local liquid circulation patterns [7]. Since it is often desirable to operate under homogenous conditions, it is of importance to accurately determine the limits between the flow regimes. Various methods were used in the literature to determine the flow regime transitions. Gas hold-up measurements using simple manometers represents the simplest method. The curve representing the variations of gas hold-up with superficial gas velocity exhibits a pronounced maximum reflecting the transition between the homogenous and the heterogeneous regimes [8]. More advanced measurement techniques were also used in the literature. These include for instance wall pressure signals [9], local gas holdup fluctuations with optical or resistive probes [10], chordal void fraction fluctuations with absorption techniques [11], temperature fluctuations [12], optical measurements [13], ultrasonic methods [14] and the electrical capacitance tomographic method [15]. Paral-
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lel to these measurements techniques, various time-series analysis tools were used to characterize the flow regimes from the recorded experimental signals. These include statistical [16,17], spectral [7] and deterministic chaos [7–18] methods. These different processing tools are not necessary unrelated. van der Schaaf et al. [19], for instance, have shown that a direct relationship exists between chaos analysis (i.e. Kolmogorov entropy) and power spectral density of pressure drop fluctuations in a fluidized bed. Ruthiya et al. [20], on the other hand, showed that changes with gas velocity of both the coherent standard deviation and the average frequency of the pressure fluctuation data can be used to mark flow regime transitions in slurry bubble columns. They also showed that a direct relationship exists between the average frequency of pressure fluctuations and the Kolmogorov entropy. In this paper, we investigate the flow regimes and regime transitions in bubble columns from a different perspective. Passive sound measurements from an air–water bubble column are recorded and analyzed using a combination of spectral and chaos-based techniques. These techniques were proved to be effective in previous studies involving wall and local pressure drop measurements [7,17,18]. These processing tools are assisted by digital imaging of bubbles motion inside the column. The objectives of this work are two fold: the first objective is to test the applicability of sound pressures to characterize flow regimes in the column and to predict flow transitions. The second objective of this work is to evaluate the efficiency of these processing tools for the analysis of sound pressures. Given the large number of tools available in the literature for time-series analysis, the strategy followed in this paper is not to try to apply many of these techniques but rather to see if methods of different theoretical backgrounds can yield consistent conclusions on the characterization of flow regimes in the column. It should be noted that at this point of our research the choice of these processing tools rather than a direct analysis of pressure fluctuations (e.g. coherent standard deviation) is dictated by the known difficulty in extracting hydrodynamics information, such as bubble-size distribution and gas hold-up, from passive acoustic signals. This is probably one of the reasons why these methods are not commonly used in multi-phase systems. In the following, and before the details of the experiment and the results analysis are discussed, we present a brief overview of the theory of passive sounds in gas–liquid systems. 2. Passive acoustic emissions Gas bubbles entrained in a liquid are known to generate large sound pressures that are audible by the human ear i.e. passive acoustic emissions [21]. The sound is the result of oscillatory motion of the bubble wall due to its volume pulsations. These pulsations are excited by changes in the external or internal pressure on the bubble. The excitation can be due to many factors such as bubble formation, bubble coalescence or splitting, rising bubbles and flow past bodies [22]. Minnaert [21] was the first to study the sound of air bubbles formed at a nozzle. The author also derived the following relation that relates the natural frequency of oscillation to the bubble radius, f0 =
3P 0.5 1 0
2r
(1)
where f0 is the natural frequency of bubble oscillations, P0 is the pressure of the surrounding liquid, the liquid density, the ratio of specific heat capacities of the gas and r is the bubble radius. Plesset [23], later, showed that the cited equation should be corrected to include the effect of surface tension. However, for an air–water system, such as the one studied in this paper, the effect of surface tension can be generally neglected. This equation has been
exploited by a number of authors to study the oscillatory behavior of bubbles and to extract bubble-size distribution in systems other than bubble columns [24–31]. A review paper on the use of passive acoustic measurements in chemical engineering processes is also given by Boyd and Varley [30]. In bubble columns applications, Boyd and Varley [31] presented a study of the measurement of gas hold-up from low frequency acoustic emissions. Recently, AlMasry et al. [32] studied the sound pressures in a bubble column recorded by a miniature hydrophone. The authors used statistical analysis to estimate the bubble-size distribution and gas hold-up. Later the same authors [33] investigated the bubble distribution in the column when anti-foams were introduced in the air–water system. The authors showed that the addition of anti-foams increase the coalescence of bubbles and this was reflected in the sound pressures measurements and bubbles distribution. 3. Experimental An overview of the experimental set up is shown in Fig. 1. The test section of the column consisted of a transparent acrylic resin of 150 mm internal diameter and 1.5 m height. The gas distributor consisted of a ring sparger with six legs star-like cross with 85 holes and 1 mm diameter equally distributed. Compressed air cylinders at room temperature were used for a stable source of air supply. The gas flow rate was controlled using thermal mass flowmeters controllers (Omega FMA2613). A miniature hydrophone (Bruel & Kjaer, type 8103) was used to record the sound pressure fluctuations. The hydrophone was placed in the center of the column at a depth of 40 cm from the gas distributor. The hydrophone signals were pre-amplified by Bruel & Kjaer type 2635 charge amplifier. Acoustic pressures were digitized as voltages using Data Translation data acquisition system. In each experimental run the gas flow was set and the column was operated for several minutes to ensure that the flow has reached steady state before the acoustic signal was recorded. The superficial gas velocity was varied up to 6.6 cm/s. The sound measurements were collected at different gas flow rates using a sampling rate of 20 kHz and a capture time of 10 s. Since the bubble oscillation may last at the most 20 ms, the 10-s long data signal may contain several bubble pulsations. The recorded signal was later analyzed using a variety of tools including the Chaos Data Analyzer [34]. A total of 2 × 105 points were collected for each experimental run. However, not all this large recorded signal could be used for the computations, and the analysis was limited to 2 × 104 points. But in order to check the consistency of the results obtained, the original recorded signal was divided into different segments each one of 2 × 104 points and the calculations were repeated for some segments in order to check that they yield identical results. This deemed important especially for computing some chaotic invariants such as the Kolmogorov entropy and the correlation dimension. 4. Results and discussion We first present our observations of some of the main patterns seen in the column as the superficial gas velocity is increased. At a very low gas flow rate of around 1.0 l/min (Ug = 0.1 cm/s) the gas is not well sparged, and mainly bubble swarms are seen to form in the column. At the gas flow rate of 10 l/min (Ug = 1.0 cm/s), small bubbles of relatively uniform size rise in the column in a rather orderly way. When the superficial gas velocity increases to 1.8 cm/s, a slow circulation of the liquid can be seen. This circulation forces some of the bubbles to move downward. Most of bubbles in this stage have more or less a uniform size. When the gas velocity further increases above 2.8 cm/s and up to 4.7 cm/s,
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Fig. 1. Schematic diagram of the apparatus.
the bubble movement increases with increasing gas velocity. The liquid circulation in the column also becomes important forcing bubbles to move in all directions. At this stage the size distribution of the bubbles is not uniform. When the gas flow rate exceeds the value of around 4.7 cm/s, a spiral liquid circulation can be seen and the bubbles travel with erratic movements throughout the column with the occurrence of frequent bubbles interactions. We conclude from these observations that there exist two flow regimes transitions roughly at the values of 2.8 and 4.7 cm/s. The transition may in fact be taking place in a larger range, so these critical values are chosen only as reference points. These critical points are also consistent with the results of statistical analysis carried out by the authors [32,33] on the same experimental data. 4.1. Time traces Time traces for a 0.3 s period are shown in Fig. 2(a and b) for values of superficial gas velocities, respectively, of 0.1 and 1.0 cm/s. Fig. 2a corresponds to very small gas flow rate (around 1 l/min) while Fig. 2b corresponds to a gas flow rate of 10 l/min. Comparing the two traces it can be seen that the oscillations at the very low gas flow rate are less dense. This corresponds to the signal dominated by low frequencies. From our visual observations, these fluctuations are the results mainly of bubble swarms, but may also include some background noise. When the gas flow rate increases to 10 l/s, the fluctuations (Fig. 2b) become more dense. It can be seen that superimposed onto the low frequency small fluctuation traces, are transient high frequency pressure pulses. An example of an individual decaying pulse not overlapping any other pulses is shown in Fig. 2c. It can be noted that at this stage no further easy information can be obtained from the time traces, and this represents one major weakness of sound pressure measurements. This is in contrast to
experiments in which pressure drop fluctuations are measured. For this case the average of pressure drop fluctuations could be used to yield a rough but useful estimate of the gas hold-up in the column. This, however, is not the case for sound pressure measurements. The intensity of the sound pressure signal cannot be easily related to gas hold-up, since it depends among other factors on the distance from the sound source at which it is measured. 4.2. Spectral analysis Next we analyze the sound pressure fluctuations in frequency domain. The classical Fourier transform (or power spectrum) indicates how the energy is distributed over the frequencies. Spectral analysis is particularly useful for sound pressure measurements. This is mainly due to Eq. (1) which provides a one to one relation between the frequency of the oscillation and the bubble size. In the power spectrum, the individual pressure peaks at any frequency are indicative of the number of bubbles oscillating at that frequency, whereas the frequency at which the pulses are observed is an indicative of the size of the bubbles. But it should be noted that the amplitudes of the spectra do not yield exactly the number of bubbles and may serve only as qualitative means for comparison. The power spectra was calculated using the software Chaos Data Analyzer [34]. The method uses maximum-entropy method. Unlike the fast Fourier transform that fits the power spectrum to a polynomial, the maximum-entropy method represents the data in terms of a finite number of complex poles of discrete frequency. This method is good in extracting sharp discrete lines from a noisy data record. Fig. 3a shows the power spectra at the very low gas flow rate of 1.0 l/min. There is a distinct low frequency regime, extending up to a frequency of 500 Hz. Applying Eq. (1), with P0 = 10, = 1000 kg/m3 and = 1.4, the frequencies below 500 Hz
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Fig. 4a shows that the intensity of the high frequencies signals has increased considerably. The low frequencies peaks are also becoming important. This is most probably due to the slow liquid circulation in the column. As the superficial gas velocity reaches 3.7 cm/s (Fig. 4b), the spectrum is relatively unchanged from Fig. 4a, with the exception that the intensity of some of the high frequencies has been reduced. When Ug is increased to 4.7 cm/s, the second transition point is crossed. Fig. 4c shows that the intensity of high frequency fluctuations (i.e. above 2000 Hz) are reduced substantially and the spectrum is dominated by frequencies below 2100 Hz. This is an indication that at this stage there is formation of many large bubbles in the column. Finally for relatively large superficial gas velocity of 6.6 cm/s (Fig. 4d), the power spectrum is basically unchanged from Fig. 4c. The only difference is that the intensity of low frequency regimes has increased substantially. This is indication of more turbulence in the flow and the formation of more large bubbles as result of coalescence of smaller bubbles. 4.3. Chaos-based analysis A number of time-series analysis methods are available in the literature that can supplement the power spectra analysis by pro-
Fig. 2. (a) Time trace for Ug = 0.1 cm/s. (b) Time trace for Ug = 1.0 cm/s. (c). Enlarged example of a typical pulse over a 3.3 ms period.
would correspond to bubble diameter above 13 mm, which was not visually observed in our experiments. So, consistently with the time trace of Fig. 2a, the low frequencies in the spectrum correspond necessarily to bubble swarms and low frequency apparatus noise. For the next figures, only the frequency range above 1000 Hz (i.e. diameter of bubble smaller than 6.5 mm) are displayed. As the superficial gas velocity increases to 1.0 cm/s (Fig. 3b), there is the appearance of a visible peak around the frequency of 1840 Hz. This corresponds, as we visually observed, to the formation of small bubbles for which the calculated diameter would be around 3.5 mm. The peak at this frequency is narrow indicating a relatively narrow size distribution of bubbles. In the same time there is a low intensity low frequency regime extending from 1000 to 1500 Hz. This may corresponds to low frequency sound due to some collective small bubble oscillations, as has been reported in some studies [35]. As the superficial gas velocity increases to 1.8 cm/s (Fig. 3c), the peak at the frequency of 1840 Hz is still visible, although its intensity has been somehow reduced. Most importantly there is the appearance of high frequency peaks extending from 2000 to 3000 Hz. These are due to the formation of more small bubbles in the column. As the superficial gas velocity increases to 2.8 cm/s, we enter the transition regime, as reported by our visual observations.
Fig. 3. (a) Power spectra for: (a) Ug = 0.10 cm/s; (b) Ug = 1.0 cm/s; (c). Ug = 1.8 cm/s.
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two steps: the first step consists in the reconstruction of attractors using the embedding space method. Specifically, a (d) dimensional pseudo-state trajectory is constructed from the one dimensional time-series data according to the rule I(t) = [I(t), I(t + k), I(t + 2k), . . . , I(t + (d − 1)k)]
(2)
where I(t) is the measured sound signal at time t, k is a chosen integral number of steps and d is the embedding dimension. I(t) is called then the embedded path. With suitable selection of time delay (k) and embedding dimension (d), the sequence of points I(t) that covers the length of the original time series can produce a map of the systems behavior in a d-dimensional called “phase-space trajectories”. Following the initial embedding, the second step is to use the principal component analysis (also called singular value decomposition) to determine the eigenvalues and eigenvectors associated with the data. There are at least two advantages of using the principal component analysis technique. The first advantage is that when the computed eigenvalues are normalized with respect to their sum, their relative magnitude can be used to determine the significance number of components i.e. those who contribute more than 0.01% of the variance. The number of significant eigenvalues is an indication of the complexity of the system [36–38]. A smaller number of significant eigenvalues indicates that the system has a small degree of freedom, and thus more organization is expected in its structure. The second advantage of the technique is that the eigenvectors may serve as new axis along which the trajectory structures can be viewed. It should be noted that the singular value decomposition decomposes the signal along the computed eigenvectors, similarly to what the Fourier series expansion does along the trigonometric functions. However, the virtue of the singular value decomposition is that the set of eigenvectors are obtained from the data itself and not imposed as, in Fourier expansion, from outside. Therefore, when plotting the phase trajectories along the computed eigenvectors coordinates, one expects that the shapes of these trajectories to be closely linked to the real bubble dynamics occurring in the column [36–38]. This idea has been recognized in the literature since the pioneering work of Daw et al. on air–solid fluidization [36,37] and was recently applied to bubble columns [18]. The computed eigenvectors could be seen roughly as approximating the smoothed time derivatives of the original time series. The first eigenvector is a component that is proportional to the measured signal. The second component is roughly proportional to the first time derivative and so on. Fig. 5 shows an example of a three dimensional plot of the reconstructed attractor for the superficial velocity of 2.8 cm/s. The axis (C1 , C2 , C3 ) represents the computed
Fig. 4. Power spectra for: (a) Ug = 2.8 cm/s; (b) Ug = 3.8 cm/s; (c) Ug = 4.7 cm/s; (d) Ug = 6.6 cm/s.
viding a quantification of the complexity of the behavior of each regime. Among them are some techniques that have their roots in nonlinear and chaos theory, thus the name of chaos-based techniques. Chaotic time-series analysis has been applied for the last two decades extensively to study multi-phase contacting devices, including bubble columns, using pressure drop measurements [7,18,36–39]. One objective of this paper is to see if these methods can provide similar results when applied to sound pressure fluctuations. The first method is the principal component analysis applied to embedded trajectories. The method proceeds in
Fig. 5. A three dimensional plot of the reconstructed attractor for Ug = 2.8 cm/s.
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eigenvectors. It can be seen that the trajectories appear as various sized loops. There are small and denser loops in the middle encircled by larger and less denser loops. The increased darkening represents the region of dynamic states most frequented by the system [18,36–38]. Each orbit represents the dynamic structure of an individual bubble or group of bubbles in the column [18,36–38]. For comparison purposes we find that it is more convenient to use the projected trajectories on a plane. Fig. 6a shows the projected trajectories for superficial gas velocity of 1.0 cm/s. We interpret the smaller and denser loops in the middle are associated with small bubbles formed in the column while the larger loops correspond to low frequency sound of bubble swarms, background noise and may include sound due to collective bubbles oscillations [35]. The idea that the loops in the phase-space trajectories are linked to the size of bubbles was also recognized in previous similar studies of hydrodynamics of air–solid fluidization [36–38]. When the superficial gas velocity is increased to 1.8 cm/s we enter the homogenous regime. Fig. 6b, shows that the larger loops have disappeared. The ‘size’ of the attractor, as measured from the x-axis and y-axis, has also diminished considerably. The dense loops should represent the dynamics of small bubbles formed in the system, since the sound emitted by the bubble swarms and background noise (large loops in Fig. 6a) is negligible compared to the sound of small bubble formed.
Fig. 6. Projected phase-space trajectories for: (a) Ug = 1.0 cm/s; (b) Ug = 1.8 cm/s.
Fig. 7. Digital image of the bubbles motion at the superficial velocity of 1.8 cm/s (homogenous regime). Small bubbles can be seen where some of them start to cluster.
Fig. 7 shows a digital image of the column at this superficial velocity. Small gas bubbles could be seen in the column with some of them starting already to cluster. These are the small bubbles that generate sound. The full video images taken in the experiment showed that the liquid moved upward in the center of the column and downward at the wall. As the superficial gas velocity is increased to 2.8 cm/s the hydrodynamics in the column enter the transition regime. Fig. 8a shows that larger loops appear again and the size of the attractor has also increased. However, these larger loops are of different nature than those of Fig. 6a. These large loops correspond in fact to large bubbles that form in the column as result of the coalescence of smaller bubbles. Some of these large loops may also be associated with the slow liquid circulation in the column. The inner denser loops, on the other hand, represent the dynamics of small bubbles. Fig. 9 shows a digital image of bubbles at this superficial velocity. The figure in fact shows clearly the formation of the first large bubbles in the column. Fig. 8b shows, on the other hand, the phase-space trajectories for a superficial velocity of 3.8 cm/s. At this velocity the column is in the transition regime. Fig. 8c shows that the large loops associated with the large bubbles increase evidently in size. Fig. 10 shows a digital picture of the bubbles in the column. The diameter and the frequency of occurrence of the large gas bubbles increase strongly in the transition regime. The liquid phase circulation is also more intense as we have seen from the full video images. Fig. 8c shows, on the other hand, the projected phase space for the second critical point (Ug = 4.7 cm/s) corresponding to the end of the transition regime. It can be seen that the size of the attractor has decreased, as it can be seen on the x-axis. In fact when we compared the different attractors, expect the one associated with the very small gas flow rate of 1.0 l/min, we find that the smallest attractors (in size) correspond respectively to the first transition point (Ug = 2.8 cm/s) followed by that of the second transition point (Ug = 4.7 cm/s). The reduction of the size of the attractors at these critical points represents, in our opinion, a sudden reorganization of the dynamics of the system, before the system moves to a more complex regime. These results are similar to observations made in the literature when the flow regimes in bubble columns were analyzed using pressure drop fluctuations [7]. Fig. 11 shows the corresponding digital images at the
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Fig. 9. Digital image of the bubbles motion at the superficial velocity of 2.8 cm/s (start of the transition regime). The formation of the first large bubbles could be seen.
Fig. 8. Projected phase-space trajectories for: (a) Ug = 2.8 cm/s; (b) Ug = 3.8 cm/s; (c) Ug = 4.7 cm/s.
second critical velocity of (Ug = 4.7 cm/s). It is difficult to extract much information about the single picture. However, the multiple images we have taken show that somehow bubbles reach almost their maximum size close to this critical velocity. Above the second critical point (Ug = 4.7 cm/s) the system enters the heterogeneous regime characterized by strong liquid circulation and multiple interacting large bubbles. Fig. 12 shows a digital image at the velocity of 6.6 cm/s. Before we conclude the description of phase-space trajectories, a note should be made about the complexity of the system as measured by its computed eigenvalues. Our results show that the maximum number of significant eigenvalues is three, even for the region above the second transition point. For example, for the superficial gas velocity of 6.6 cm/s, the eigenval-
Fig. 10. Digital image of the bubbles motion at the superficial velocity of 3.8 cm/s (transition regime). Large bubbles are interacting.
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Fig. 13. Variations of Kolmogorov entropy with superficial gas velocity. Points T1 and T2 are transition points.
Fig. 11. Digital image of the bubbles motion at the superficial velocity of 4.8 cm/s (second transition point). Bubbles are larger in size however they do have reached their maximum size.
Fig. 12. Digital image of the bubbles motion at the superficial velocity of 6.6 cm/s (heterogeneous regime). Multiple large bubbles are interacting.
ues () were computed using an embedding dimension d = 7 and a time lag of 1 i.e. 5 × 10−5 s. The first four normalized eigenvalues (/ 2 ) are respectively: 7.33 × 10−1 , 2.66 × 10−1 , 1.43 × 10−4 and 1.2 × 10−7 . It can be seen that only three eigenvalues are significant i.e. above 0.01% of the variance. Therefore, the dynamics of the system even in the churn-turbulence regime covered in this study (Ug < 6.6 cm/s) can be described by a low dimensional attractor. Next the embedded trajectories analysis is supplemented by computing some chaotic invariants of the system: namely the Kolmogorov entropy and the correlation dimension. Fig. 13 shows the variations of the Kolmogorov entropy with the superficial gas velocity. Kolmogorov entropy (in bits per cycle) drops continuously as the flow rate increases until it reaches a minimum value around Ug = 2.8 cm/s. An other less marked minimum point can also be observed in the diagram around Ug = 4.7 cm/s. The two minimum points correspond to the transitions points determined previously. The point corresponding to the end of homogenous regime is, however, more marked than the one corresponding to the beginning of the churn-flow regime. Both points corresponds to a decrease in the entropy and hence a decrease in the degrees of freedom of the dynamics of the flow. This is associated with some kind of self-organization in the flow before the system settles at a dynamically more complex regime.
Fig. 14. Variations of correlation dimension with superficial gas velocity.
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The correlation dimension (Dc ) represents an other chaotic invariant of the system. It characterizes the spatial correlation between measured points on the attractor, and it is one of the many so-called fractal dimensions of the attractor, should this latter exist. Fig. 14 shows the variations of (Dc ) with the superficial gas velocity. The overall trend of the figure is rather smooth and does not lend itself to any solid conclusions. We have found therefore that the correlation dimension is less effective than the Kolmogorov entropy in the detection of flow regime transitions. A similar observation has also been reached in [16] in the analysis of wall pressure drop fluctuations. One plausible reason is that the overall liquid circulation reduces the influence of local coherent structures that are responsible for changes in the chaotic invariants [17]. However, this justification does not explain why the Kolmogorov entropy, another chaotic invariant, is more effective in the detection of flow regime transitions.
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detect the different studied regimes. This is particularly important since passive sound measurements are intrusive techniques in nature and also because in the homogenous regime the distributor plays an important role in bubble-size distribution and gas hold-up profile. The second issue is to investigate whether simple analysis of sound pressure fluctuations such as the calculations of coherent standard deviation and the average frequency could be useful for the detection of transition points, in a similar way to previous studies carried out for pressure drop fluctuations. Acknowledgement This work was supported by a grant from the Research Center of College of Engineering at King Saud University. Appendix A. Nomenclature
5. Conclusions The difficulty in extracting hydrodynamics information such as bubble-size distribution and gas hold-up from passive acoustic signals is probably one of the reasons why these methods are not commonly used in multi-phase systems. In this paper, we have demonstrated the applicability of passive acoustic measurements to at least predict flow transitions in a lab scale air–water bubble column. The sound pressure signals taken at various superficial gas velocities were analyzed using a combination of spectral and chaos-based techniques. Spectral methods are particularly suitable for sound measurements since the frequency at which the pulses are observed is an indicative of the size of the bubble. The power spectra analysis managed to identify both the end of homogenous regime and the beginning of the churn-turbulent flow. The transition between the two regimes is characterized by an increase in the intensity of low frequency regimes, indicating the appearance of a combination of slow liquid circulation and large bubbles. Chaos-based techniques were used to supplement the power spectra analysis through the reconstruction of phasespace trajectories and the determination of Kolmogorov entropy and the correlation dimension, two commonly used chaotic invariants. The Kolmogorov entropy exhibits minimum values at the transition points indicating a sudden reorganization of the system, and the reduction in its degrees of freedom. However, the end of the bubbly regime is more marked than the beginning of the heterogeneous regime. Parallel to the Kolmogorov entropy the projected phase-space trajectories have allowed a useful visualization of the collective dynamics of bubbles formed in the column. It was shown that at the transition points the size of the attractors shrinks considerably, indicating a sudden reorganization in its structure. With the exception of very small flow rates, the projected phase-space trajectories are characterized with less dense outer loops surrounding denser and smaller loops. The larger loops correspond to large bubbles and liquid circulation while the denser loops are associated with the small bubbles formed in the column, as they are the most frequented states of the system. Moreover, it was shown that the dynamics of the various regimes, even at the churn-turbulent regime covered in this study, can be described by a maximum of three significant eigenvalues indicating a low dimensional attractor. The correlation dimension was not efficient in detecting the transition points. This observation which was also reported in previous studies involving pressure drop measurements deserves certainly further study. Two other issues also deserve further investigation. The first one concerns the effect of geometrical and other physical parameters of the column on the ability of sound pressure measurements to
Ci (i = 1,3) eigenvectors d dimension of the embedded attractor correlation dimension Dc f0 natural frequency of bubble oscillations (Hz) I(t + k) measured sound pressure at time t + k (Pa) I(t) vector of sound pressure k time delay pressure of liquid (N/m2 ) P0 r bubble radius (m) superficial gas velocity (cm/s) Ug Greek letters ratio of specific heat capacities of the gas liquid density (kg/m3 ) References [1] L.-S. Fan, Gas–Liquid–Solid Fluidization Engineering, Butterworth, MA, USA, 1989. [2] W.-D. Deckwer, Bubble Column Reactors, Wiley and Sons, Chichester, UK, 1992. [3] Y.T. Shah, B.G. Kelkar, S.P. Godbole, W.-D. Deckwer, Design parameters estimation for bubble column reactors, AIChE J. 28 (1982) 353–379. [4] E. Camarasa, C. Vial, S. Poncin, G. Wild, N. Midoux, J. Bouillard, Influence of the coalescence behaviour of the liquid and of the gas sparging on hydrodynamics and bubble characteristics in a bubble column, Chem. Eng. Proc. 38 (1999) 329–344. [5] M.C. Ruzicka, J. Zahradnik, J. Drahvs, N.H. Thomas, Homogenous–heterogenous regime transition in bubble columns, Chem. Eng. Sci. 56 (2001) 4609–4626. [6] C. Vial, R. laine, S. Poncin, N. Midoux, G. Wild, Influence of gas distribution and regime transitions on liquid velocity and turbulence in a 3-D bubble column, Chem. Eng. Sci. 56 (2001) 1085–1093. [7] H.M. Letzel, J.C. Schouten, R. Krishna, C.M. van den Bleek, Characterization of regimes and regime transitions in bubble columns by chaos analysis of pressure signal, Chem. Eng. Sci. 52 (1997) 4447–4459. [8] G.B. Wallis, One-dimensional Two-Phase Flow, McGraw-Hill, NY, USA, 1969. [9] J. Drahovs, J. Zahradnik, M. Puncochar, M. Fialova, F. Bradka, Effect of operating conditions on the characteristics of pressure fluctuations in a bubble column, Chem. Eng. Proc. 29 (1991) 107–115. [10] B.R. Bakshi, H. Zhong, P. Jiang, L.-S. Fan, Analysis of flow in gas–liquid bubble columns using multi-resolution methods, Trans. IChem E, A 73 (1995) 608–614. [11] R. Kikuchi, T. Yano, A. Tsutsumi, K. Yoshida, M. Puncochar, J. Drahovs, Diagnosis of chaotic dynamics of bubble motion in a bubble column, Chem. Eng. Sci. 52 (1997) 3741–3745. [12] P.R. Thimmapuram, N.S. Rao, S.C. Saxena, Characterization of hydrodynamics regimes in a bubble column, Chem. Eng. Sci. 47 (1992) 3355–3362. [13] J.S. Chang, G.D. Harvel, Determination of gas–liquid bubble column instantaneous interfacial area and void fraction by a real-time neutron radiography method, Chem. Eng. Sci. 47 (1992) 3639–3646. [14] A. Stravs, A. Pittet, U. Von Stockar, P.J. Reilly, Measurement of interfacial areas in aerobic fermentations by ultrasonic pulse transmission, Biotechnol. Bioeng. 28 (1986) 1302–1309. [15] F.M. Mousa, Experimental investigation of shallow bubble column hydrodynamics using electrical capacitance tomography, M.Sc. Thesis, King Saud University, Riyadh, Saudi Arabia, 2006.
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[16] C. Vial, E. Camarasa, S. Poncin, G. Wild, N. Midoux, J. Bouillard, Study of hydrodynamic behaviour in bubble columns and external loop airlift reactors though analysis of pressure fluctuations, Chem. Eng. Sci. 55 (2000) 2957–2973. [17] B. Gourich, C. Vial, A. Essadki, F. Allam, M.B. Soulami, M. Ziyad, Identification of flow regimes and transition points in a bubble column through analysis of differential pressure signal—Influence of the coalescence of the liquid phase, Chem. Eng. Proc. 45 (2006) 214–223. [18] T.-J. Lin, R.-C. Juang, Y.-C. Chen, C.-C. Chen, Predictions of flow transitions in a bubble column by chaotic time series analysis of pressure fluctuations signals, Chem. Eng. Sci. 56 (2001) 1057–1065. [19] J. van der Schaaf, J.R. van Ommen, F. Takens, J.C. Schouten, C.M. van den Bleek, Similarity between chaos analysis and frequency analysis of pressure fluctuations in fluidized beds, Chem. Eng. Sci. 59 (2004) 1829–1840. [20] K.C. Ruthiya, V.P. Chilekar, M.J.F. Warnier, J. van der Schaaf, B.F.M. Kuster, J.C. Schouten, Detecting regime transitions in slurry bubble columns using pressure time series, AIChE J. 51 (2005) 1951–1965. [21] M. Minnaert, On musical air bubbles and sounds of running water, Philos. Mag. 16 (1933) 235–248. [22] M. Strasberg, Gas bubbles as sources of sound in liquids, J. Acoust. Soc. Am. 28 (1956) 20–26. [23] M.S. Plesset, Bubble dynamics and cavitation erosion, J. Eng. Math. 2 (1974) 203–209. [24] L.A. Glasgow, J. Hua, T.-Y. Yiin, L.E. Erikson, Acoustic studies of interfacial effects in airlift reactors, Chem. Eng. Commun. 113 (1992) 151–181. [25] A.B. Pandit, J. Varely, R.B. Thorpe, J.F. Davidson, Measurement of bubble size distribution: an acoustic technique, Chem. Eng. Sci. 47 (1992) 1079–1089. [26] S. Prabhukumar, R. Duraiswami, G.L. Chahine, Acoustic measurement of bubble size distributions: theory & experiments, ASME Cavitation Multiphase Flow Forum 236 (1996) 509–514. [27] E.J. Terrill, K.W. Melville, A broadband acoustic technique for measuring bubble size and velocity, J. Atmos. Ocean. Technol. 17 (2000) 220–239.
[28] R. Manasseh, R.F. LaFontain, J. Davy, I.C. Shepherd, Y. Zhu, Passive acoustic bubble sizing in sparged systems, Exp. Fluids 30 (2001) 672–682. [29] J.W. Boyd, J. Varley, Acoustic emission measurement of low velocity plunging jets to monitor bubble size, Chem. Eng. J. 97 (2004) 11–25. [30] J. Boyd, J. Varley, The uses of passive measurement of acoustic emissions from chemical engineering processes, Chem. Eng. Sci. 56 (2001) 1749–1767. [31] J.W. Boyd, J. Varley, Measurement of gas hold-up in bubble columns from low frequency acoustic emissions, Chem. Eng. J. 88 (2002) 111–118. [32] W.A. Al-Masry, E.M. Ali, Y.M. Aqeel, Determination of bubble characteristics in bubble columns using statistical analysis of acoustic sound measurements, Chem. Eng. Res. Des. 83 (2005) 1196–1207. [33] W.A. Al-Masry, E.M. Ali, Y.M. Aqeel, Effect of antifoams agents on bubble characteristics in bubble columns based on acoustic sound measurements, Chem. Eng. Sci. 61 (2006) 3610–3622. [34] J.C. Sprott, Chaos Data Analyzer, University of Wisconsin, Madison, WI, 1998. [35] S.W. Yoon, S.W. Crum, A. Prosperetti, N.Q. Lu, An investigation of the collective oscillations in a bubble cloud, J. Acoust. Soc. Am. 89 (1991) 700– 706. [36] C. Daw, W. Lawkins, D. Downing, N. Clapp Jr., Chaotic characteristics of a complex gas–solid flow, Phys. Rev. A 41 (1990) 1179–1181. [37] C.S. Daw, J.S. Halow, Evaluation and control of fluidization quality through chaotic time series analysis of pressure drop measurements, AIChE Symp. Ser. 89 (1993) 103–120. [38] D.P. Skzycke, K. Nguyen, C.S. Daw, Characterization of the fluidization behavior of different solid types based on chaotic time series analysis of pressure signals, in: Proceedings of the 13th International Conference on Fluidized Bed Combustion, San Diego, ASME, May, 1993, pp. 9–13. [39] A. Ajbar, K. Alhumaizi, A. Ibrahim, M. Asif, Hydrodynamics of gas fluidized beds with mixture of group D and B particles, Can. J. Chem. Eng. 80 (2002) 281–288.
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Chemical Engineering and Processing: Process Intensification journal homepage: www.elsevier.com/locate/cep
Prediction of flow regimes transitions in bubble columns using passive acoustic measurements A. Ajbar ∗ , W. Al-Masry, E. Ali Department of Chemical Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 19 September 2007 Received in revised form 13 February 2008 Accepted 14 February 2008 Available online 10 March 2008 Keywords: Bubble column Air–water system Acoustics Hydrophone Flow regimes Transitions Chaos
a b s t r a c t Passive acoustic sound, measured by a hydrophone in an air–water bubble column, is used to study the hydrodynamics of the unit. The recorded measurements taken at different superficial gas velocities are processed using both spectral and chaos-based techniques in order to characterize the column flow regimes and to predict the transitions points. These processing tools were supported by digital video imaging of bubbles motion inside the column. The results of data analysis indicate the applicability of passive sound measurements to identify flow regimes in the bubble column. In this regard the analysis of sound spectra gives a useful qualitative comparison of flow regimes. Chaos-based techniques, on the other hand, are more successful in predicting the transition points between the homogenous and the churn-turbulent flow regimes in the column. The critical gas velocities of the transition are associated with a marked change in some of the calculated chaotic invariants of sound pressures. The calculated superficial gas velocities of the critical points are also found to be consistent with experimental observations. Moreover, a useful visualization of the dynamics induced in the column by the alternation of small and large bubbles is possible through the inspection of phase-space trajectories reconstructed from time-series measurements. The shape and size of the trajectories are closely linked to the size distribution of bubbles, and they change as the flow moves from one regime to an other. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Owing to their simple construction and their good heat and mass transfer, bubble columns are widely used for gas–liquid reactions in such diverse fields as chemical, petrochemical, pharmaceutical and water treatment [1,2]. Notwithstanding their operational advantages, bubble column hydrodynamics are complex and are characterized by different flow patterns depending on gas flow rate, physico-chemical properties of the gas–liquid system, gas distributor design, column diameter, etc. In this regard, homogeneous (bubbly) flow, transition regime and heterogeneous (churn-turbulent) flow are successively observed with increasing superficial gas velocities [3]. Another flow pattern, the slug flow regime, was also reported for small diameter columns [2]. The substantial amount of experimental work in the last decades on bubble columns has allowed an accurate description of the cited flow regimes. The homogenous regime is encountered at low gas velocities when the gas from the sparger is uniformly distributed. This regime is characterized by a narrow bubble-size distribution and by a radially uniform gas hold-up. In this regime both the
∗ Corresponding author. Fax: +966 1 467 8770. E-mail address: [email protected] (A. Ajbar). 0255-2701/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2008.02.004
distributor and the properties of the gas–liquid system play a dominant role in bubble-size distribution and gas hold-up profile [4,5]. The heterogeneous regime, on the other hand, is encountered at high gas flow rates. Small bubbles coalesce to produce relatively larger bubbles. This regime is therefore characterized by a wide distribution of bubble size and increased radial gas hold-up profile. The hydrodynamics in this regime are therefore more influenced by the bubble induced turbulence than by the distributor [6]. The two flow regimes are separated by a transition regime characterized by the development of local liquid circulation patterns [7]. Since it is often desirable to operate under homogenous conditions, it is of importance to accurately determine the limits between the flow regimes. Various methods were used in the literature to determine the flow regime transitions. Gas hold-up measurements using simple manometers represents the simplest method. The curve representing the variations of gas hold-up with superficial gas velocity exhibits a pronounced maximum reflecting the transition between the homogenous and the heterogeneous regimes [8]. More advanced measurement techniques were also used in the literature. These include for instance wall pressure signals [9], local gas holdup fluctuations with optical or resistive probes [10], chordal void fraction fluctuations with absorption techniques [11], temperature fluctuations [12], optical measurements [13], ultrasonic methods [14] and the electrical capacitance tomographic method [15]. Paral-
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lel to these measurements techniques, various time-series analysis tools were used to characterize the flow regimes from the recorded experimental signals. These include statistical [16,17], spectral [7] and deterministic chaos [7–18] methods. These different processing tools are not necessary unrelated. van der Schaaf et al. [19], for instance, have shown that a direct relationship exists between chaos analysis (i.e. Kolmogorov entropy) and power spectral density of pressure drop fluctuations in a fluidized bed. Ruthiya et al. [20], on the other hand, showed that changes with gas velocity of both the coherent standard deviation and the average frequency of the pressure fluctuation data can be used to mark flow regime transitions in slurry bubble columns. They also showed that a direct relationship exists between the average frequency of pressure fluctuations and the Kolmogorov entropy. In this paper, we investigate the flow regimes and regime transitions in bubble columns from a different perspective. Passive sound measurements from an air–water bubble column are recorded and analyzed using a combination of spectral and chaos-based techniques. These techniques were proved to be effective in previous studies involving wall and local pressure drop measurements [7,17,18]. These processing tools are assisted by digital imaging of bubbles motion inside the column. The objectives of this work are two fold: the first objective is to test the applicability of sound pressures to characterize flow regimes in the column and to predict flow transitions. The second objective of this work is to evaluate the efficiency of these processing tools for the analysis of sound pressures. Given the large number of tools available in the literature for time-series analysis, the strategy followed in this paper is not to try to apply many of these techniques but rather to see if methods of different theoretical backgrounds can yield consistent conclusions on the characterization of flow regimes in the column. It should be noted that at this point of our research the choice of these processing tools rather than a direct analysis of pressure fluctuations (e.g. coherent standard deviation) is dictated by the known difficulty in extracting hydrodynamics information, such as bubble-size distribution and gas hold-up, from passive acoustic signals. This is probably one of the reasons why these methods are not commonly used in multi-phase systems. In the following, and before the details of the experiment and the results analysis are discussed, we present a brief overview of the theory of passive sounds in gas–liquid systems. 2. Passive acoustic emissions Gas bubbles entrained in a liquid are known to generate large sound pressures that are audible by the human ear i.e. passive acoustic emissions [21]. The sound is the result of oscillatory motion of the bubble wall due to its volume pulsations. These pulsations are excited by changes in the external or internal pressure on the bubble. The excitation can be due to many factors such as bubble formation, bubble coalescence or splitting, rising bubbles and flow past bodies [22]. Minnaert [21] was the first to study the sound of air bubbles formed at a nozzle. The author also derived the following relation that relates the natural frequency of oscillation to the bubble radius, f0 =
3P 0.5 1 0
2r
(1)
where f0 is the natural frequency of bubble oscillations, P0 is the pressure of the surrounding liquid, the liquid density, the ratio of specific heat capacities of the gas and r is the bubble radius. Plesset [23], later, showed that the cited equation should be corrected to include the effect of surface tension. However, for an air–water system, such as the one studied in this paper, the effect of surface tension can be generally neglected. This equation has been
exploited by a number of authors to study the oscillatory behavior of bubbles and to extract bubble-size distribution in systems other than bubble columns [24–31]. A review paper on the use of passive acoustic measurements in chemical engineering processes is also given by Boyd and Varley [30]. In bubble columns applications, Boyd and Varley [31] presented a study of the measurement of gas hold-up from low frequency acoustic emissions. Recently, AlMasry et al. [32] studied the sound pressures in a bubble column recorded by a miniature hydrophone. The authors used statistical analysis to estimate the bubble-size distribution and gas hold-up. Later the same authors [33] investigated the bubble distribution in the column when anti-foams were introduced in the air–water system. The authors showed that the addition of anti-foams increase the coalescence of bubbles and this was reflected in the sound pressures measurements and bubbles distribution. 3. Experimental An overview of the experimental set up is shown in Fig. 1. The test section of the column consisted of a transparent acrylic resin of 150 mm internal diameter and 1.5 m height. The gas distributor consisted of a ring sparger with six legs star-like cross with 85 holes and 1 mm diameter equally distributed. Compressed air cylinders at room temperature were used for a stable source of air supply. The gas flow rate was controlled using thermal mass flowmeters controllers (Omega FMA2613). A miniature hydrophone (Bruel & Kjaer, type 8103) was used to record the sound pressure fluctuations. The hydrophone was placed in the center of the column at a depth of 40 cm from the gas distributor. The hydrophone signals were pre-amplified by Bruel & Kjaer type 2635 charge amplifier. Acoustic pressures were digitized as voltages using Data Translation data acquisition system. In each experimental run the gas flow was set and the column was operated for several minutes to ensure that the flow has reached steady state before the acoustic signal was recorded. The superficial gas velocity was varied up to 6.6 cm/s. The sound measurements were collected at different gas flow rates using a sampling rate of 20 kHz and a capture time of 10 s. Since the bubble oscillation may last at the most 20 ms, the 10-s long data signal may contain several bubble pulsations. The recorded signal was later analyzed using a variety of tools including the Chaos Data Analyzer [34]. A total of 2 × 105 points were collected for each experimental run. However, not all this large recorded signal could be used for the computations, and the analysis was limited to 2 × 104 points. But in order to check the consistency of the results obtained, the original recorded signal was divided into different segments each one of 2 × 104 points and the calculations were repeated for some segments in order to check that they yield identical results. This deemed important especially for computing some chaotic invariants such as the Kolmogorov entropy and the correlation dimension. 4. Results and discussion We first present our observations of some of the main patterns seen in the column as the superficial gas velocity is increased. At a very low gas flow rate of around 1.0 l/min (Ug = 0.1 cm/s) the gas is not well sparged, and mainly bubble swarms are seen to form in the column. At the gas flow rate of 10 l/min (Ug = 1.0 cm/s), small bubbles of relatively uniform size rise in the column in a rather orderly way. When the superficial gas velocity increases to 1.8 cm/s, a slow circulation of the liquid can be seen. This circulation forces some of the bubbles to move downward. Most of bubbles in this stage have more or less a uniform size. When the gas velocity further increases above 2.8 cm/s and up to 4.7 cm/s,
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Fig. 1. Schematic diagram of the apparatus.
the bubble movement increases with increasing gas velocity. The liquid circulation in the column also becomes important forcing bubbles to move in all directions. At this stage the size distribution of the bubbles is not uniform. When the gas flow rate exceeds the value of around 4.7 cm/s, a spiral liquid circulation can be seen and the bubbles travel with erratic movements throughout the column with the occurrence of frequent bubbles interactions. We conclude from these observations that there exist two flow regimes transitions roughly at the values of 2.8 and 4.7 cm/s. The transition may in fact be taking place in a larger range, so these critical values are chosen only as reference points. These critical points are also consistent with the results of statistical analysis carried out by the authors [32,33] on the same experimental data. 4.1. Time traces Time traces for a 0.3 s period are shown in Fig. 2(a and b) for values of superficial gas velocities, respectively, of 0.1 and 1.0 cm/s. Fig. 2a corresponds to very small gas flow rate (around 1 l/min) while Fig. 2b corresponds to a gas flow rate of 10 l/min. Comparing the two traces it can be seen that the oscillations at the very low gas flow rate are less dense. This corresponds to the signal dominated by low frequencies. From our visual observations, these fluctuations are the results mainly of bubble swarms, but may also include some background noise. When the gas flow rate increases to 10 l/s, the fluctuations (Fig. 2b) become more dense. It can be seen that superimposed onto the low frequency small fluctuation traces, are transient high frequency pressure pulses. An example of an individual decaying pulse not overlapping any other pulses is shown in Fig. 2c. It can be noted that at this stage no further easy information can be obtained from the time traces, and this represents one major weakness of sound pressure measurements. This is in contrast to
experiments in which pressure drop fluctuations are measured. For this case the average of pressure drop fluctuations could be used to yield a rough but useful estimate of the gas hold-up in the column. This, however, is not the case for sound pressure measurements. The intensity of the sound pressure signal cannot be easily related to gas hold-up, since it depends among other factors on the distance from the sound source at which it is measured. 4.2. Spectral analysis Next we analyze the sound pressure fluctuations in frequency domain. The classical Fourier transform (or power spectrum) indicates how the energy is distributed over the frequencies. Spectral analysis is particularly useful for sound pressure measurements. This is mainly due to Eq. (1) which provides a one to one relation between the frequency of the oscillation and the bubble size. In the power spectrum, the individual pressure peaks at any frequency are indicative of the number of bubbles oscillating at that frequency, whereas the frequency at which the pulses are observed is an indicative of the size of the bubbles. But it should be noted that the amplitudes of the spectra do not yield exactly the number of bubbles and may serve only as qualitative means for comparison. The power spectra was calculated using the software Chaos Data Analyzer [34]. The method uses maximum-entropy method. Unlike the fast Fourier transform that fits the power spectrum to a polynomial, the maximum-entropy method represents the data in terms of a finite number of complex poles of discrete frequency. This method is good in extracting sharp discrete lines from a noisy data record. Fig. 3a shows the power spectra at the very low gas flow rate of 1.0 l/min. There is a distinct low frequency regime, extending up to a frequency of 500 Hz. Applying Eq. (1), with P0 = 10, = 1000 kg/m3 and = 1.4, the frequencies below 500 Hz
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Fig. 4a shows that the intensity of the high frequencies signals has increased considerably. The low frequencies peaks are also becoming important. This is most probably due to the slow liquid circulation in the column. As the superficial gas velocity reaches 3.7 cm/s (Fig. 4b), the spectrum is relatively unchanged from Fig. 4a, with the exception that the intensity of some of the high frequencies has been reduced. When Ug is increased to 4.7 cm/s, the second transition point is crossed. Fig. 4c shows that the intensity of high frequency fluctuations (i.e. above 2000 Hz) are reduced substantially and the spectrum is dominated by frequencies below 2100 Hz. This is an indication that at this stage there is formation of many large bubbles in the column. Finally for relatively large superficial gas velocity of 6.6 cm/s (Fig. 4d), the power spectrum is basically unchanged from Fig. 4c. The only difference is that the intensity of low frequency regimes has increased substantially. This is indication of more turbulence in the flow and the formation of more large bubbles as result of coalescence of smaller bubbles. 4.3. Chaos-based analysis A number of time-series analysis methods are available in the literature that can supplement the power spectra analysis by pro-
Fig. 2. (a) Time trace for Ug = 0.1 cm/s. (b) Time trace for Ug = 1.0 cm/s. (c). Enlarged example of a typical pulse over a 3.3 ms period.
would correspond to bubble diameter above 13 mm, which was not visually observed in our experiments. So, consistently with the time trace of Fig. 2a, the low frequencies in the spectrum correspond necessarily to bubble swarms and low frequency apparatus noise. For the next figures, only the frequency range above 1000 Hz (i.e. diameter of bubble smaller than 6.5 mm) are displayed. As the superficial gas velocity increases to 1.0 cm/s (Fig. 3b), there is the appearance of a visible peak around the frequency of 1840 Hz. This corresponds, as we visually observed, to the formation of small bubbles for which the calculated diameter would be around 3.5 mm. The peak at this frequency is narrow indicating a relatively narrow size distribution of bubbles. In the same time there is a low intensity low frequency regime extending from 1000 to 1500 Hz. This may corresponds to low frequency sound due to some collective small bubble oscillations, as has been reported in some studies [35]. As the superficial gas velocity increases to 1.8 cm/s (Fig. 3c), the peak at the frequency of 1840 Hz is still visible, although its intensity has been somehow reduced. Most importantly there is the appearance of high frequency peaks extending from 2000 to 3000 Hz. These are due to the formation of more small bubbles in the column. As the superficial gas velocity increases to 2.8 cm/s, we enter the transition regime, as reported by our visual observations.
Fig. 3. (a) Power spectra for: (a) Ug = 0.10 cm/s; (b) Ug = 1.0 cm/s; (c). Ug = 1.8 cm/s.
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two steps: the first step consists in the reconstruction of attractors using the embedding space method. Specifically, a (d) dimensional pseudo-state trajectory is constructed from the one dimensional time-series data according to the rule I(t) = [I(t), I(t + k), I(t + 2k), . . . , I(t + (d − 1)k)]
(2)
where I(t) is the measured sound signal at time t, k is a chosen integral number of steps and d is the embedding dimension. I(t) is called then the embedded path. With suitable selection of time delay (k) and embedding dimension (d), the sequence of points I(t) that covers the length of the original time series can produce a map of the systems behavior in a d-dimensional called “phase-space trajectories”. Following the initial embedding, the second step is to use the principal component analysis (also called singular value decomposition) to determine the eigenvalues and eigenvectors associated with the data. There are at least two advantages of using the principal component analysis technique. The first advantage is that when the computed eigenvalues are normalized with respect to their sum, their relative magnitude can be used to determine the significance number of components i.e. those who contribute more than 0.01% of the variance. The number of significant eigenvalues is an indication of the complexity of the system [36–38]. A smaller number of significant eigenvalues indicates that the system has a small degree of freedom, and thus more organization is expected in its structure. The second advantage of the technique is that the eigenvectors may serve as new axis along which the trajectory structures can be viewed. It should be noted that the singular value decomposition decomposes the signal along the computed eigenvectors, similarly to what the Fourier series expansion does along the trigonometric functions. However, the virtue of the singular value decomposition is that the set of eigenvectors are obtained from the data itself and not imposed as, in Fourier expansion, from outside. Therefore, when plotting the phase trajectories along the computed eigenvectors coordinates, one expects that the shapes of these trajectories to be closely linked to the real bubble dynamics occurring in the column [36–38]. This idea has been recognized in the literature since the pioneering work of Daw et al. on air–solid fluidization [36,37] and was recently applied to bubble columns [18]. The computed eigenvectors could be seen roughly as approximating the smoothed time derivatives of the original time series. The first eigenvector is a component that is proportional to the measured signal. The second component is roughly proportional to the first time derivative and so on. Fig. 5 shows an example of a three dimensional plot of the reconstructed attractor for the superficial velocity of 2.8 cm/s. The axis (C1 , C2 , C3 ) represents the computed
Fig. 4. Power spectra for: (a) Ug = 2.8 cm/s; (b) Ug = 3.8 cm/s; (c) Ug = 4.7 cm/s; (d) Ug = 6.6 cm/s.
viding a quantification of the complexity of the behavior of each regime. Among them are some techniques that have their roots in nonlinear and chaos theory, thus the name of chaos-based techniques. Chaotic time-series analysis has been applied for the last two decades extensively to study multi-phase contacting devices, including bubble columns, using pressure drop measurements [7,18,36–39]. One objective of this paper is to see if these methods can provide similar results when applied to sound pressure fluctuations. The first method is the principal component analysis applied to embedded trajectories. The method proceeds in
Fig. 5. A three dimensional plot of the reconstructed attractor for Ug = 2.8 cm/s.
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eigenvectors. It can be seen that the trajectories appear as various sized loops. There are small and denser loops in the middle encircled by larger and less denser loops. The increased darkening represents the region of dynamic states most frequented by the system [18,36–38]. Each orbit represents the dynamic structure of an individual bubble or group of bubbles in the column [18,36–38]. For comparison purposes we find that it is more convenient to use the projected trajectories on a plane. Fig. 6a shows the projected trajectories for superficial gas velocity of 1.0 cm/s. We interpret the smaller and denser loops in the middle are associated with small bubbles formed in the column while the larger loops correspond to low frequency sound of bubble swarms, background noise and may include sound due to collective bubbles oscillations [35]. The idea that the loops in the phase-space trajectories are linked to the size of bubbles was also recognized in previous similar studies of hydrodynamics of air–solid fluidization [36–38]. When the superficial gas velocity is increased to 1.8 cm/s we enter the homogenous regime. Fig. 6b, shows that the larger loops have disappeared. The ‘size’ of the attractor, as measured from the x-axis and y-axis, has also diminished considerably. The dense loops should represent the dynamics of small bubbles formed in the system, since the sound emitted by the bubble swarms and background noise (large loops in Fig. 6a) is negligible compared to the sound of small bubble formed.
Fig. 6. Projected phase-space trajectories for: (a) Ug = 1.0 cm/s; (b) Ug = 1.8 cm/s.
Fig. 7. Digital image of the bubbles motion at the superficial velocity of 1.8 cm/s (homogenous regime). Small bubbles can be seen where some of them start to cluster.
Fig. 7 shows a digital image of the column at this superficial velocity. Small gas bubbles could be seen in the column with some of them starting already to cluster. These are the small bubbles that generate sound. The full video images taken in the experiment showed that the liquid moved upward in the center of the column and downward at the wall. As the superficial gas velocity is increased to 2.8 cm/s the hydrodynamics in the column enter the transition regime. Fig. 8a shows that larger loops appear again and the size of the attractor has also increased. However, these larger loops are of different nature than those of Fig. 6a. These large loops correspond in fact to large bubbles that form in the column as result of the coalescence of smaller bubbles. Some of these large loops may also be associated with the slow liquid circulation in the column. The inner denser loops, on the other hand, represent the dynamics of small bubbles. Fig. 9 shows a digital image of bubbles at this superficial velocity. The figure in fact shows clearly the formation of the first large bubbles in the column. Fig. 8b shows, on the other hand, the phase-space trajectories for a superficial velocity of 3.8 cm/s. At this velocity the column is in the transition regime. Fig. 8c shows that the large loops associated with the large bubbles increase evidently in size. Fig. 10 shows a digital picture of the bubbles in the column. The diameter and the frequency of occurrence of the large gas bubbles increase strongly in the transition regime. The liquid phase circulation is also more intense as we have seen from the full video images. Fig. 8c shows, on the other hand, the projected phase space for the second critical point (Ug = 4.7 cm/s) corresponding to the end of the transition regime. It can be seen that the size of the attractor has decreased, as it can be seen on the x-axis. In fact when we compared the different attractors, expect the one associated with the very small gas flow rate of 1.0 l/min, we find that the smallest attractors (in size) correspond respectively to the first transition point (Ug = 2.8 cm/s) followed by that of the second transition point (Ug = 4.7 cm/s). The reduction of the size of the attractors at these critical points represents, in our opinion, a sudden reorganization of the dynamics of the system, before the system moves to a more complex regime. These results are similar to observations made in the literature when the flow regimes in bubble columns were analyzed using pressure drop fluctuations [7]. Fig. 11 shows the corresponding digital images at the
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Fig. 9. Digital image of the bubbles motion at the superficial velocity of 2.8 cm/s (start of the transition regime). The formation of the first large bubbles could be seen.
Fig. 8. Projected phase-space trajectories for: (a) Ug = 2.8 cm/s; (b) Ug = 3.8 cm/s; (c) Ug = 4.7 cm/s.
second critical velocity of (Ug = 4.7 cm/s). It is difficult to extract much information about the single picture. However, the multiple images we have taken show that somehow bubbles reach almost their maximum size close to this critical velocity. Above the second critical point (Ug = 4.7 cm/s) the system enters the heterogeneous regime characterized by strong liquid circulation and multiple interacting large bubbles. Fig. 12 shows a digital image at the velocity of 6.6 cm/s. Before we conclude the description of phase-space trajectories, a note should be made about the complexity of the system as measured by its computed eigenvalues. Our results show that the maximum number of significant eigenvalues is three, even for the region above the second transition point. For example, for the superficial gas velocity of 6.6 cm/s, the eigenval-
Fig. 10. Digital image of the bubbles motion at the superficial velocity of 3.8 cm/s (transition regime). Large bubbles are interacting.
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Fig. 13. Variations of Kolmogorov entropy with superficial gas velocity. Points T1 and T2 are transition points.
Fig. 11. Digital image of the bubbles motion at the superficial velocity of 4.8 cm/s (second transition point). Bubbles are larger in size however they do have reached their maximum size.
Fig. 12. Digital image of the bubbles motion at the superficial velocity of 6.6 cm/s (heterogeneous regime). Multiple large bubbles are interacting.
ues () were computed using an embedding dimension d = 7 and a time lag of 1 i.e. 5 × 10−5 s. The first four normalized eigenvalues (/ 2 ) are respectively: 7.33 × 10−1 , 2.66 × 10−1 , 1.43 × 10−4 and 1.2 × 10−7 . It can be seen that only three eigenvalues are significant i.e. above 0.01% of the variance. Therefore, the dynamics of the system even in the churn-turbulence regime covered in this study (Ug < 6.6 cm/s) can be described by a low dimensional attractor. Next the embedded trajectories analysis is supplemented by computing some chaotic invariants of the system: namely the Kolmogorov entropy and the correlation dimension. Fig. 13 shows the variations of the Kolmogorov entropy with the superficial gas velocity. Kolmogorov entropy (in bits per cycle) drops continuously as the flow rate increases until it reaches a minimum value around Ug = 2.8 cm/s. An other less marked minimum point can also be observed in the diagram around Ug = 4.7 cm/s. The two minimum points correspond to the transitions points determined previously. The point corresponding to the end of homogenous regime is, however, more marked than the one corresponding to the beginning of the churn-flow regime. Both points corresponds to a decrease in the entropy and hence a decrease in the degrees of freedom of the dynamics of the flow. This is associated with some kind of self-organization in the flow before the system settles at a dynamically more complex regime.
Fig. 14. Variations of correlation dimension with superficial gas velocity.
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The correlation dimension (Dc ) represents an other chaotic invariant of the system. It characterizes the spatial correlation between measured points on the attractor, and it is one of the many so-called fractal dimensions of the attractor, should this latter exist. Fig. 14 shows the variations of (Dc ) with the superficial gas velocity. The overall trend of the figure is rather smooth and does not lend itself to any solid conclusions. We have found therefore that the correlation dimension is less effective than the Kolmogorov entropy in the detection of flow regime transitions. A similar observation has also been reached in [16] in the analysis of wall pressure drop fluctuations. One plausible reason is that the overall liquid circulation reduces the influence of local coherent structures that are responsible for changes in the chaotic invariants [17]. However, this justification does not explain why the Kolmogorov entropy, another chaotic invariant, is more effective in the detection of flow regime transitions.
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detect the different studied regimes. This is particularly important since passive sound measurements are intrusive techniques in nature and also because in the homogenous regime the distributor plays an important role in bubble-size distribution and gas hold-up profile. The second issue is to investigate whether simple analysis of sound pressure fluctuations such as the calculations of coherent standard deviation and the average frequency could be useful for the detection of transition points, in a similar way to previous studies carried out for pressure drop fluctuations. Acknowledgement This work was supported by a grant from the Research Center of College of Engineering at King Saud University. Appendix A. Nomenclature
5. Conclusions The difficulty in extracting hydrodynamics information such as bubble-size distribution and gas hold-up from passive acoustic signals is probably one of the reasons why these methods are not commonly used in multi-phase systems. In this paper, we have demonstrated the applicability of passive acoustic measurements to at least predict flow transitions in a lab scale air–water bubble column. The sound pressure signals taken at various superficial gas velocities were analyzed using a combination of spectral and chaos-based techniques. Spectral methods are particularly suitable for sound measurements since the frequency at which the pulses are observed is an indicative of the size of the bubble. The power spectra analysis managed to identify both the end of homogenous regime and the beginning of the churn-turbulent flow. The transition between the two regimes is characterized by an increase in the intensity of low frequency regimes, indicating the appearance of a combination of slow liquid circulation and large bubbles. Chaos-based techniques were used to supplement the power spectra analysis through the reconstruction of phasespace trajectories and the determination of Kolmogorov entropy and the correlation dimension, two commonly used chaotic invariants. The Kolmogorov entropy exhibits minimum values at the transition points indicating a sudden reorganization of the system, and the reduction in its degrees of freedom. However, the end of the bubbly regime is more marked than the beginning of the heterogeneous regime. Parallel to the Kolmogorov entropy the projected phase-space trajectories have allowed a useful visualization of the collective dynamics of bubbles formed in the column. It was shown that at the transition points the size of the attractors shrinks considerably, indicating a sudden reorganization in its structure. With the exception of very small flow rates, the projected phase-space trajectories are characterized with less dense outer loops surrounding denser and smaller loops. The larger loops correspond to large bubbles and liquid circulation while the denser loops are associated with the small bubbles formed in the column, as they are the most frequented states of the system. Moreover, it was shown that the dynamics of the various regimes, even at the churn-turbulent regime covered in this study, can be described by a maximum of three significant eigenvalues indicating a low dimensional attractor. The correlation dimension was not efficient in detecting the transition points. This observation which was also reported in previous studies involving pressure drop measurements deserves certainly further study. Two other issues also deserve further investigation. The first one concerns the effect of geometrical and other physical parameters of the column on the ability of sound pressure measurements to
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