TimeâFrequency Analysis Based Flow Regime ... - ACS Publications

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Time−Frequency Analysis Based Flow Regime Identification Methods for Airlift Reactors Lijia Luo, Ying Yan, Yuanyuan Xu, and Jingqi Yuan* Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, China S Supporting Information *

ABSTRACT: The flow regime transitions in an airlift reactor were investigated based on pressure fluctuation signals. Two time−frequency analysis methods, i.e., Wigner−Ville distribution and wavelet transform, were used to extract flow regime characteristics from pressure signals. The main frequency derived from the smoothed pseudo-Wigner−Ville distribution of the pressure signal was used to quantify flow regime transitions in the reactor. Two flow regime transition points were successfully detected from the evolution of main frequencies of pressure signals. In addition, the local dynamic characteristics of the pressure signal at different frequency bands were analyzed by use of the wavelet transform. A new flow regime identification method based on the wavelet entropy of the pressure signal was proposed. This method was confirmed to be reliable and efficient to detect flow regime transitions in the reactor. fluctuations, and acoustic signals.8 Among different signals available, the pressure fluctuation signal is the most attractive option, because its measurement method is well developed, relatively cheap, highly robust, and particularly practicable in industrial conditions.8,9 Two kinds of pressure signals are commonly used in the literature, i.e., single-point absolute pressure and double-point differential pressure. The former almost contains dynamic information about all hydrodynamic behaviors in the reactor. The latter is more suitable for the analysis of hydrodynamic behaviors between two measurement ports. Numerous signal analysis techniques have been applied to extract flow regime characteristics from pressure fluctuation signals. Vial et al.10 proposed a new flow regime identification method based on the autocorrelation function of the pressure signal. Al-Masry et al.11 have applied statistical analysis to acoustic and differential pressure signals for flow regime identification. Other methods, for example, spectral analysis,11,12 chaotic analysis,13,14 higher-order statistics,15 and the S statistic test,16 are also regarded as powerful techniques for the analysis of the pressure signal. In particular, time−frequency analysis has been shown to give a deeper insight into the hydrodynamics and flow regime transitions in multiphase reactors, because it is able to reveal more detailed dynamic behaviors by analyzing signals in both time and frequency domains.17−25 Zhang et al.9,23 extracted the local bubble-induced pressure fluctuations from pressure signals using wavelet transform, and then identified flow regimes in an airlift reactor based on the evolution of the Hurst exponent, local energy ratio, and chaotic parameters of bubble-induced pressure signals. Briens and Ellis24 successfully detected flow regime transitions in three-phase fluidized bed systems

1. INTRODUCTION Airlift reactors have been widely used in the chemical industry and in biotechnological processes such as wastewater treatment, fermentation, and manufacture of pharmaceuticals.1−3 Compared with conventional stirred tanks or bubble columns, airlift reactors have many advantages, including low energy consumption, rapid mixing, homogeneous shear rate, and high mass transfer efficiency.2,3 Generally, airlift reactors are classified into two categories on the basis of their structure: external loop airlift reactor (ELALR) and internal loop airlift reactor (ILALR).4 A typical ILALR consists of two concentric cylinders, which divide the reactor into four main parts: riser, downcomer, bottom section (base), and top section (gas separator).3,5 The liquid circulation in the reactor is driven by the density difference between the riser and the downcomer. The reactor structure, operating conditions, and physical properties of the fluid have great influence on the hydrodynamics and mass transfer characteristics in the reactor. Moreover, the hydrodynamics in the reactor is characterized by different flow regimes depending on the superficial gas velocity.6−8 The hydrodynamic parameters, mass transfer, and mixing behaviors strongly depend on the flow structure and the corresponding flow regimes.7 It is therefore important to study the characteristics of different flow regimes and flow regime transitions for proper design, scaleup, operation, and control of the reactor. However, the determination of flow regimes and their transitions is still a difficult task, because the mechanism of multiphase flow is not yet fully understood. Traditional flow regime identification methods are based on visual observation, monitoring variations of hydrodynamic parameters or drift−flux analysis.8 The main drawback of these methods is their lack of universality and accuracy. Therefore, more general and accurate flow regime identification methods have been developed based on the analysis of dynamic fluctuation signals in multiphase reactors, such as pressure fluctuations, local gas holdup fluctuations, temperature © 2012 American Chemical Society

Received: Revised: Accepted: Published: 7104

August 10, 2011 February 8, 2012 March 14, 2012 March 14, 2012 dx.doi.org/10.1021/ie201774n | Ind. Eng. Chem. Res. 2012, 51, 7104−7112

Industrial & Engineering Chemistry Research

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according to standard deviations of wavelet coefficients of bubble probe signals. Shou and Leu25 applied the wavelet analysis to deal with pressure signals for identifying flow regimes in fluidized beds. Bakshi et al.26 employed multiresolution analysis to a local gas holdup signal to identify flow regimes in a bubble column. Despite these studies’ attempts to quantify flow regime transitions, firmly established criteria are not yet available. The application of time−frequency analysis for the flow regime identification is thus still limited, because it lacks quantitative information to characterize flow regimes and to accurately detect flow regimes transitions. Therefore, more applicable and efficient flow regime identification methods based on time−frequency analysis need further exploration and research. Moreover, most studies on the topic of flow regime identification were conducted in bubble columns or fluidized beds and there is a lack of data for airlift reactors. In recent years, wavelet entropy has been widely used to characterize dynamic processes.27 Successful applications of wavelet entropy in many different biological and physical fields, such as characterization of brain electrical signals,28 neuronal activity,29 solar activity,30 erythrocyte deformation,31 and fractional Brownian motion time series,32 have demonstrated its effectiveness. On the other hand, the Wigner−Ville distribution (WVD) also has gained much attention in the analysis of nonstationary signals.33 It provides a quadratic time−frequency energy distribution of the signal with excellent time and frequency resolution. The WVD has been successfully used in fault detection.34−36 Wavelet entropy and WVD may have great potential for flow regime identification in multiphase reactors. However, there are few studies on this topic. The objective of this work is to identify flow regimes in an internal loop airlift reactor using pressure fluctuation signals. Two time−frequency analysis methods, i.e.. Wigner−Ville distribution and wavelet transform, are used to extract dynamic characteristics from pressure signals. The main frequency derived from the Wigner−Ville distribution of the pressure signal and the wavelet entropy of the pressure signal are used as characteristic quantities to detect flow regime transitions in the reactor.

Figure 1. Schematic diagram of the experimental setup.

the downcomer above the bottom of the reactor (see Figure 1). Dynamic absolute pressure sensors with an accuracy of 0.5% and a maximal response frequency of 400 kHz were used to measure wall pressure fluctuations within the reactor. Pressure signals were sampled by a 12-bit A/D data acquisition system at a frequency of 400 Hz for an interval of 80 s at each superficial gas velocity. The static interference and electromagnetic radiated interference were restrained by good shielding and grounding. The 50 Hz power-line interference was eliminated by a band-pass filter. The wavelet threshold denoising algorithm37,38 was applied to reduce noise.

3. ANALYSIS METHODS 3.1. Wigner−Ville Distribution. The Wigner−Ville distribution (WVD) is a powerful time−frequency analysis tool for nonstationary signals due to its many desirable properties.33 The instantaneous energy and frequency, power spectral density, and group delay of the signal can be easily derived from the WVD. For a deterministic signal, x(t), {t:t ∈ (−∞,+∞)}, its WVD is defined as34

2. EXPERIMENTAL SETUP The schematic diagram of the experimental setup is shown in Figure 1. The airlift reactor consists of an external column with a height of 1300 mm and a 284 mm inside diameter, as well as an internal draft tube with a height of 820 mm and a 70 mm inside diameter. The draft tube was mounted in the center of the external column 40 mm above the bottom of the reactor. The air was dispersed into the reactor through a sparger located at the bottom of the annulus region between the draft tube and the external column. Thus, the annulus channel and the draft tube served as the riser and the downcomer, respectively. The sparger consists of four 1.8-mm-diameter orifices arranged in a symmetric configuration. The clearance between each orifice and the bottom of the reactor was 10 mm. The gas flow was controlled by a calibrated rotameter giving a range of superficial gas velocity from 0.7 × 10−3 to 3.1 × 10−3 m/s. The unaerated liquid level was controlled at 1032 mm, holding a top clearance (the distance between the liquid surface and the upper end of the draft tube) of 172 mm. All experiments were carried out at room temperature and atmospheric pressure. The reactor was equipped with several pressure sampling ports at 75 (PR1), 225 (PR2), 530 (PR3), and 855 (PR4) mm in the riser and 220 (PD1), 470 (PD2), and 750 (PD3) mm in

+∞

WVD(t , f ) =

∫−∞

⎛ τ⎞ ⎛ τ⎞ z*⎜t − ⎟ z⎜t + ⎟e−j2πfτ dτ ⎝ ⎠ ⎝ 2 2⎠

(1)

where z(t) is the analytical signal corresponding to x(t) and z*(t) denotes the complex conjugate of z(t). The WVD possesses excellent time−frequency resolution, while it suffers from undesirable cross-term interference in analyzing a multicomponent signal that hampers the interpretation of the distribution.34,35 To suppress such cross-terms, two improved WVDs are proposed, i.e., pseudo-Wigner−Ville distribution (PWVD) and smoothed pseudo-Wigner−Ville distribution (SPWVD).34−36 The PWVD is a windowed version of the WVD:34 +∞

PWVD(t , f ) =

∫−∞

⎛ τ⎞ ⎛ τ⎞ h(τ ) z*⎜t − ⎟ z⎜t + ⎟e−j2πfτ dτ ⎝ ⎠ ⎝ 2 2⎠ (2)

where the time smoothing window h(·) is a real and symmetric function. The smoothing window can reduce the cross-terms present in the original WVD with the cost of losing the time− 7105

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frequency resolution.34 The effect of cross-term suppression depends on the length of the smoothing window. The suppression of cross-terms is better with a longer window, but it may accompany the undesirable smearing of instantaneous characteristics. The smoothing in both time and frequency domains results in the SPWVD:39

S = −∑ Gj ln Gj j

The wavelet entropy is a measure of the dynamic complexity of the signal. An ordered signal with a narrow frequency distribution of energy has low wavelet entropy, while a random signal with a broad frequency distribution of energy has high wavelet entropy. A high value of wavelet entropy indicates higher dynamic complexity, higher irregular behaviors, but low predictability.

+∞

SPWVD(t , f ) =

∫−∞

h(t − t ′) q(f − f ′) WVD(t ′, f ′)

dt ′ df ′

(3)

where h(·) and q(·) are time and frequency smoothing window functions (e.g., Hamming windows), respectively. The SPWVD results in a better suppression of cross-terms than the PWVD. 3.2. Wavelet Analysis. The wavelet transform is a powerful time−frequency analysis tool for analyzing nonstationary or transient signals. Different from the Fourier transform, which is precisely localized in the frequency domain but infinitely extended in the time domain, the wavelet transform is well localized in both time and frequency domains. The wavelet transform provides a better characterization of signal using multiresolution analysis with balanced resolution in time and frequency domains, which makes it known as the “mathematical microscope”. Wavelet transforms are mainly classified into the discrete wavelet transform (DWT) and the continuous wavelet transform (CWT). More detailed information about these two kinds of wavelet transforms is found in the literature.40 Practically, the discrete wavelet transform is used more widely because of its algorithmic properties; i.e., only a specific subset of scale and translation values is considered, which reduces computational costs. Therefore, only the one-dimensional discrete wavelet transform is performed in this study. For a discrete signal x(i), i = 1, 2, ..., N, the wavelet coefficients at dyadic scales j and displacement k are given by17 ψjk(i) = 2j /2ψ (2ji − k)

4. RESULTS AND DISCUSSION 4.1. Flow Regimes in ILALRs. As described by Heijnen et al.,6 there exist three typical flow regimes in an ILALR: no gas entrainment in the downcomer (regime I), gas entrainment in the downcomer but no gas recirculation (regime II), and complete gas recirculation (regime III). Flow regime I presents at low superficial gas velocities when the liquid circulation velocity is insufficient to carry bubbles into the downcomer. As the superficial gas velocity increases, flow regime II occurs when the liquid velocity in the downcomer becomes equal to the slip velocity of air bubbles. In this flow regime, bubbles are entrained into the downcomer but finally stay “stationary” at certain positions. Thus, an upper bubble zone and a lower bubble free zone may be observed in the downcomer. The depth of the upper bubble zone will increase until it reaches the bottom edge of the downcomer with increasing superficial gas velocity. At higher superficial gas velocities, the liquid velocity in the downcomer is sufficiently high for the bubble circulation through the downcomer into the riser, leading to flow regime III. 4.2. Wigner−Ville Distribution of Pressure Signals. Figure 2 shows typical pressure fluctuation signals in the riser and downcomer, and the corresponding video snapshots at three different superficial gas velocities. As shown in Figure 2a, at a low superficial gas velocity of 0.7 × 10−3 m/s, bubbles almost rise through the riser in a straight line, and thus the interaction between bubbles is weak and the bubble size is relatively uniform. Meanwhile, there is no bubble entrainment in the downcomer. As a result, pressure fluctuations in both the riser and the downcomer are weak. Figure 2b indicates that, with an increase of superficial gas velocity, bubble coalescence and breakup emerge in the riser, and thus the amount of small bubbles increases and larger bubbles as well as bubble swarms start to appear, resulting in a nonuniform distribution of bubble size. Moreover, small bubbles are entrained into the downcomer due to the increase of liquid circulation velocity. Therefore, both in the riser and in the downcomer, the amplitudes of pressure fluctuations increase. At a high superficial gas velocity of 2.81 × 10−3 m/s, Figure 2c shows that a wide distribution of bubble size is formed by frequent bubble coalescence and breakup in the riser, and meanwhile complete bubble circulation has been established in the downcomer. Thus pressure fluctuations in the riser and in the downcomer become more intensive. Figure 2 indicates that hydrodynamic behaviors depend strongly on the superficial gas velocity. To quantify hydrodynamic characteristics at different superficial gas velocities, the SPWVD of the pressure signal was calculated using the MATLAB software, where Hamming windows with lengths of N/10 and N/4 (N is the length of the pressure signal) were used as time and frequency smoothing windows, respectively. Figures 3 and 4 show SPWVDs of pressure signals measured in

(4)

n

Wjk(x) =

∑ ψjk(i) x(i)

(5)

i=1 p

where j = 0, 1, ..., p and k = 1, ..., 2 with p = log N/log 2. A low j implies a fine scale, while a high j indicates a coarse scale. The signal can be decomposed into several lower resolution components by using the wavelet transform. After decomposition, high-scale and low frequency components of the signal are called approximations (a’s), while low-scale and high frequency components of the signal are called details (d’s).17 The wavelet energy at scale j is expressed as 2p − j

Ej =

∑ |Wjk|2 k=1

(6)

Then the normalized wavelet energy Gj at scale j is calculated by

Gj =

Ej Etotal

(8)

(7)

where Etotal = ∑pj=1 Ej is the summation of wavelet energy at all scales. The normalized wavelet energy reflects the probability distribution of wavelet energy at different scales. To quantify this probability distribution, the wavelet entropy is derived from the definition of Shannon entropy:41 7106



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Figure 2. Images captured between sampling positions of PR2 and PR3 and typical pressure fluctuation signals in the riser and downcomer: (a) Ug = 0.7 × 10−3 m/s; (b) Ug = 1.4 × 10−3 m/s; (c) Ug = 2.81 × 10−3 m/s.

the riser and downcomer, respectively. The color bar beside each plot denotes the time and frequency dependent energy. In the riser, at the superficial gas velocity of 0.7 × 10−3 m/s, the energy of the pressure signal concentrates mainly around 46 Hz and partly around 34 Hz as shown in Figure 3a. As the superficial gas velocity increases to 1.4 × 10−3 m/s, Figure 3b shows that the main energy of the pressure signal shifts to a low frequency around 20 Hz, and meanwhile the signal energy in the high frequency region (>50 Hz) increases, corresponding to the appearance of larger bubbles and bubble swarms as well as increases of small bubbles, respectively (see Figure 2b). At the superficial gas velocity of 2.81 × 10−3 m/s, as shown in Figure 3c, most of the signal energy concentrates at a high frequency of about 95 Hz, while little of the signal energy is present in the low frequency region below 50 Hz. The reason is that frequent bubble coalescence and breakup generate a larger number of small bubbles (see Figure 2c), and the movement of these small bubbles as well as the intensive liquid turbulence induces many high frequency pressure fluctuations.13 Figure 4 shows that, with increasing superficial gas velocity, the energy of the pressure signal in the downcomer shifts from the high

Figure 3. Contour plots of SPWVDs of pressure signals measured at 530 mm in the riser for different superficial gas velocities: (a) Ug = 0.7 × 10−3 m/s; (b) Ug = 1.4 × 10−3 m/s; (c) Ug = 2.81 × 10−3 m/s.

frequency domain (>50 Hz) to the low frequency domain (below 30 Hz) because more and more bubbles enter the downcomer. All these results indicate that the SPWVD of the pressure signal is able to characterize hydrodynamic behaviors at different superficial gas velocities, and therefore it has the potential to identify flow regime transitions in the reactor. To detect accurate flow regime transition points in the reactor, derivation of a characteristic quantity from the SPWVD of the pressure signal that can be used as an indicator of flow regime transitions is required. For this purpose, the energy spectrum is defined from the SPWVD of pressure signal: 7107



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Figure 5. Energy spectrum of the pressure signal measured at 530 mm in the riser at Ug = 0.7 × 10−3 m/s.

further used as a characteristic quantity to identify flow regimes in the reactor. The main frequency is shown as a function of the superficial gas velocity in Figure 6. Both in the riser and in the

Figure 6. Evolution of the main frequency of the pressure signal as a function of the superficial gas velocity.

downcomer, the evolution of the main frequency may be divided into three different phases. At the beginning, main frequencies of pressure signals in the riser and downcomer are all relatively higher. As the superficial gas velocity increases, the main frequency of the pressure signal in the riser decreases due to the increase of the number of larger bubbles and the appearance of bubble swarms.42 The main frequency of the pressure signal in the downcomer also decreases because of the increase of the liquid circulation velocity. As the superficial gas velocity increases further and across 1.05 × 10−3 m/s, bubble coalescence and breakup emerge in the riser (see Figure 2b), which generate more high frequency pressure fluctuations, and thus the main frequency of the pressure signal turns to increase with superficial gas velocity. On the other hand, small bubbles begin to be entrained into the downcomer but without circulation. These small bubbles induce high frequency pressure fluctuations at positions where they pass, which slow down the decrease of the main frequency of the pressure signal. The complete gas circulation in the downcomer may be established

Figure 4. Contour plots of SPWVDs of pressure signals measured at 470 mm in the downcomer for different superficial gas velocities: (a) Ug = 0.7 × 10−3 m/s; (b) Ug = 1.4 × 10−3 m/s; (c) Ug = 2.81 × 10−3 m/s.

P(f ) =

∫t |SPWVD(t , f )| dt ∫f ∫t |SPWVD(t , f )| dt df

(9)

This energy spectrum provides a measure of the energy contribution from each frequency value. A typical energy spectrum of the pressure signal is depicted in Figure 5. Peaks at 34 and 46 Hz in Figure 5 agree well with the two high-energy zones in Figure 3a. Thus, the frequency value corresponding to the main peak in the energy spectrum of the pressure signal (such as 46 Hz in Figure 5), namely the main frequency, is 7108



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Figure 7. Discrete wavelet transform of the pressure signal measured at Ug = 1.4 × 10−3 m/s and 530 mm in the riser.

Table 1. Frequency Bands for Decomposed Pressure Signals at Different Scales scale freq band (Hz)

a7 0−1.06

d7 1.06−3.125

d6 3.125−6.25

d5 6.25−12.5

at superficial gas velocities higher than 2.1 × 10−3 m/s. In this case, bubble coalescence and breakup as well as liquid turbulences in the riser also tend to be stable. Therefore, main frequencies of pressure signals in both the riser and the downcomer change little at high superficial gas velocities. All these results indicate that the transition point from flow regime I to flow regime II may be marked by the superficial gas velocity around 1.05 × 10−3 m/s, and the transition point from flow regime II to flow regime III may correspond to the superficial gas velocity around 2.1 × 10−3 m/s. 4.3. Wavelet Analysis of Pressure Signals. The pressure fluctuation signals in the reactor are further analyzed using the wavelet transform. The discrete wavelet decomposition is performed using the Daubechies 10 wavelet. Typical decomposed pressure signals at seven scales of details (d1−d7) and one approximation (a7) are shown in Figure 7. The frequency band corresponding to each scale is listed in Table 1. The approximation (a7) is neglected in the following analysis because it mainly represents the tendency of the pressure signal. Figure 7 indicates that fluctuations of low-scale details (d1−d4) are more intensive than those of high-scale details (d5−d7). To quantify the differences among d1−d7, the normalized wavelet energies at different scales are compared in Figure 8. It is clear that the wavelet energies at low scales (d1−d4) are much larger than those at high scales (d5−d7), which implies that the energy of the pressure signal concentrates mainly in the frequency region larger than 12 Hz. This result is in good agreement with the SPWVD of the pressure signal as shown in Figure 3b. Although the wavelet transform reveals local dynamic characteristics of the pressure signal at different frequency bands, it cannot be directly used to detect flow regime transitions in the reactor. Therefore, the wavelet entropy is

d4 12.5−25

d3 25−50

d2 50−100

d1 100−200

Figure 8. Comparison of the normalized wavelet energy at d1−d7.

calculated from the probability density distribution of the wavelet energy at different scales of details according to eq 8. Figure 9 shows the variations of wavelet entropies of pressure signals with the superficial gas velocity. Both in the riser and in the downcomer, the evolution of the wavelet entropy exhibits three different phases. At first, the wavelet entropy of the pressure signal in the riser increases with superficial gas velocity. This is because the number of bubbles increases with superficial gas velocity, resulting in the increase of the complexity of pressure fluctuations. However, in the downcomer, the liquid circulation reduces the dynamic complexity of pressure fluctuations, and thus the wavelet entropy of the pressure signal decreases with increasing superficial gas velocity. As the superficial gas velocity increases further and across 1.05 × 10−3 m/s, the wavelet entropy of the pressure signal in the riser 7109



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Table 2. Comparison of Different Techniques Used in the Present Work and Those Used by Luo et al.15 riser analysis methods SPWVD wavelet entropy HOS15 WT15

downcomer

firsta secondb

first

second

main frequency wavelet entropy

++c ++

++ ++

= ++

++ +

average bispectrum generalized average frequency

++ ++

++ ++

++ =

+ +

characteristic quantities

a Refers to the transition from flow regime I to flow regime II. bRefers to the transition from flow regime II to flow regime III. c++, sharp transition; +, transition can be detected; =, blurred transition.

order of these analysis methods is represented as wavelet entropy > HOS > SPWVD > WT. Actually, wavelet transform, SPWVD, and WT are all belong to time−frequency analysis methods, while they have different characteristics. The main advantage of wavelet transform is that it can decompose pressure signals in different frequency bands, and thus unfold the detailed local characteristics of pressure signals.24−26 The other advantage of wavelet transform is that its algorithm is mature and has a high computational efficiency. The wavelet entropy of the pressure signal represents the complexity of the energy distribution in different frequency bands and thus can characterize flow regimes. The SPWVD and WT are able to reveal the energy distribution of the pressure signal in the time−frequency domain. Compared with wavelet transform, they possess extremely high time and frequency resolution without losing the amplitude and phase information of the signal. However, they both suffer from the interference of cross-terms, and thus the smoothing window is required to reduce the influence of cross-terms. Moreover, due to lack of efficient algorithms, the calculations of WT and SPWVD are relatively time-consuming. The main frequency or the generalized average frequency derived from SPWVD or WT represents the aggregation of signal energy at a certain frequency, and thus can be used to quantify flow regime transitions. In a word, the time−frequency analysis techniques, such as wavelet transform, SPWVD, and WT, are able to extract flow regime characteristics from pressure signals, while SPWVD and WT are fit to analyze a short data sequence and wavelet transform is better for a long data sequence.

Figure 9. Variations of wavelet entropies of pressure signals measured in the riser and downcomer.

starts to decrease because the appearance of bubble coalescence and breakup (see Figure 2b) increases the information loss rate and thus decreases the dynamic complexity of pressure fluctuations. However, in the downcomer, the wavelet entropy of the pressure signal begins to increase because the entrainment of small bubbles makes pressure fluctuations become more complicated. At superficial gas velocities larger than 2.1 × 10−3 m/s, the wavelet entropy of the pressure signal in the riser tends to be stable, and the increase of the wavelet entropy of the pressure signal in the downcomer also slows down. The reason may be that the completed gas circulation has been established in the reactor because of the high liquid circulation velocity, and therefore the pressure fluctuations in both the riser and the downcomer tend to be stable. Thus, the three flow regimes in the reactor are successfully identified from the evolution of the wavelet entropy of the pressure signal. The superficial gas velocity about 1.05 × 10−3 m/s corresponds to the transition point from regime I to regime II, and that about 2.1 × 10−3 m/s marks the transition point from regime II to regime III. These results agree well with those obtained from the SPWVD of the pressure signal. 4.4. Comparative Study. The flow regime transition points obtained in this study agree well with those in our previous work,15 where higher-order statistics (HOS) and the Wigner trispectrum (WT) have been applied to analyze pressure fluctuation signals and the average bispectrum and the generalized average frequency of pressure signals have been used to quantify flow regime transitions. The methods used in the present work have been compared with those used in our previous work.15 The results are summarized in Table 2. Although the wavelet entropy of the pressure signal provides results similar to those of the average bispectrum, its calculation is simpler than the average bispectrum. The main frequency based on SPWVD and the generalized average frequency based on WT all present a low sensitivity for the first flow regime transition point in the downcomer, while the second flow regime transition point in the downcomer obtained from the main frequency is clearer than that indicated by the generalized average frequency. Additionally, the calculation of WT imposes a heavy computational burden, while the SPWVD is able to reduce computational costs and meanwhile maintains a similar degree of output information as that of WT. As a result, according to effectiveness of the flow regime identification, the

5. CONCLUSIONS The flow regime identification in an internal loop airlift reactor was investigated. The time−frequency analysis methods were applied to analyze pressure fluctuation signals within the reactor. The SPWVD was found to be a powerful tool to unfold the time−frequency characteristics of pressure signals. The main frequency derived from the SPWVD of the pressure signal was used as a characteristic quantity to identify flow regime transitions in the reactor. Two flow regime transition points were successfully detected according to the evolution of the main frequency of the pressure signal. In addition, a new flow regime identification approach based on the wavelet entropy of the pressure signal was proposed. This method consists of two procedures: the pressure signal was first decomposed at several scales corresponding to different frequency bands by applying the wavelet transform, and then the wavelet entropy was calculated from the probability density distribution of the 7110



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wavelet energy at different scales. The wavelet coefficients at different scales obtained by the wavelet transform were helpful for characterizing the pressure signal in both localized time and frequency domains. The wavelet entropy represents the uniformity of the energy distribution of the pressure signal in the frequency domain, and thus it is able to characterize flow regimes. The wavelet entropy analysis indicated the same transition points as those obtained from the SPWVD of the pressure signal. In a word, time−frequency analysis methods, such as the Wigner−Ville distribution and wavelet transform, have great potential for the flow regime identification in multiphase reactors.

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ASSOCIATED CONTENT

S Supporting Information *

Calculated main frequencies and wavelet entropies of pressure signals at different superficial gas velocities; MATLAB source code for SPWVD analysis and wavelet analysis. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel./fax: +86-21-34204055. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This study is supported by the National High Technology Research and Development Program of China (Grant No. 2009AA04Z162) and the China Postdoctoral Science Foundation (Grant No. 2011M500777).

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NOMENCLATURE Ej = wavelet energy at the scale j f = frequency, Hz Gj = normalized wavelet energy at the scale j i = time index j = wavelet scale k = displacement parameter N = length of the signal P(f) = energy spectrum S = wavelet entropy t = time, s Ug = superficial gas velocity, m s−1 W(x) = wavelet coefficient of x(t) x(t) = pressure signal z(t) = analytical signal corresponding to x(t) z*(t) = complex conjugate of z(t) h(·) = time smoothing window q(·) = frequency smoothing window

Greek Symbols

ψ(i) = wavelet function τ = time delay, s

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REFERENCES

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