J. Cent. South Univ. (2018) 25: 1690-1700

DOI: https://doi.org/10.1007/s11771-018-3860-2

Application of time–frequency entropy from wake oscillation to gas–liquid flow pattern identification

HUANG Si-shi(黄思师), SUN Zhi-qiang(孙志强), ZHOU Tian(周天), ZHOU Jie-min(周孑民)

School of Energy Science and Engineering, Central South University, Changsha 410083, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract: Gas–liquid two-phase flow abounds in industrial processes and facilities. Identification of its flow pattern plays an essential role in the field of multiphase flow measurement. A bluff body was introduced in this study to recognize gas–liquid flow patterns by inducing fluid oscillation that enlarged differences between each flow pattern.Experiments with air–water mixtures were carried out in horizontal pipelines at ambient temperature and atmospheric pressure. Differential pressure signals from the bluff-body wake were obtained in bubble, bubble/plug transitional, plug, slug, and annular flows. Utilizing the adaptive ensemble empirical mode decomposition method and the Hilbert transform, the time–frequency entropy S of the differential pressure signals was obtained. By combining S and other flow parameters, such as the volumetric void fraction β, the dryness x, the ratio of density φ and the modified fluid coefficient ψ, a new flow pattern map was constructed which adopted S(1–x)φ and (1–β)ψ as the vertical and horizontal coordinates, respectively. The overall rate of classification of the map was verified to be 92.9% by the experimental data. It provides an effective and simple solution to the gas–liquid flow pattern identification problems.

Key words: gas–liquid two-phase flow; wake oscillation; flow pattern map; time–frequency entropy; ensemble empirical mode decomposition; Hilbert transform

Cite this article as: HUANG Si-shi, SUN Zhi-qiang, ZHOU Tian, ZHOU Jie-min. Application of time–frequency entropy from wake oscillation to gas–liquid flow pattern identification [J]. Journal of Central South University, 2018, 25(7): 1690–1700. DOI: https://doi.org/10.1007/s11771-018-3860-2.

1 Introduction

Gas–liquid two-phase flow is one of the most typical multiphase systems that are encountered in many industrial processes and facilities [1, 2]. Due to the random distribution of phase interfaces, identification of flow patterns is a difficult issue in gas–liquid flow measurement. In many cases, correct recognition of flow patterns is the prerequisite for investigating flow mechanism, heat/mass transfer performance, and measurement of other flow parameters [3, 4]. Therefore, effective methods for discerning gas–liquid two-phase flow patterns are of important theoretical and practical value.

In recent years, with the rapid development of the computer and information processing technology, the soft-sensing methods, especially the time–frequency based analysis, have been a focus in the field of flow pattern recognition [5]. As a powerful tool for analyzing time-varying and non-stationary signals, time–frequency analysis provides the distribution information of time- domain and frequency-domain, and depicts the relationship between the variation of time and frequency, which has been widely used in communications, automation, astronomy and etc.[6, 7]. Fourier transform (FT), wavelet transform (WT) and Hilbert–Huang transform (HHT) have been broadly applied in the characterization of gas–liquid two-phase flows. Although these methods make a great contribution to signal processing, they also have some limitations. FT is strictly restricted to linear systems and stationary data series. WT is capable of analyzing non- stationary signals; however, the decomposition results are not related to the signal itself but the given frequency, which is not adaptive and may produce energy leakage [8]. HHT becomes a focus research topic for the processing of non-stationary and nonlinear signals [9], but there is the mode-mixing problem in empirical mode decomposition (EMD), which leads to the intrinsic mode function (IMF) distortional. In this paper, the ensemble empirical mode decomposition (EEMD) is adopted to overcome the mode mixing of EMD, and then the Hilbert transform is applied to obtaining frequencies with physical meaning.

A method was put forward through constructing a new flow pattern map with the soft-sensing method, allowing not only to adopt the basic parameters acquired from the experimental tests, but also to extract some characteristic variables processed by the time–frequency analysis. During the experiments, a bluff body was placed in gas–liquid two-phase flows to induce fluid oscillations and to enlarge the differences between flow patterns [10]. Differential pressures across the bluff body were collected, which contained rich flow information [11, 12]. The adaptive EEMD and Hilbert transform were applied to the processing of differential pressure signals; thence the time– frequency entropies were extracted to construct the flow pattern map. Five typical kinds of gas–liquid two-phase flows were identified by the proposed method.

2 Experiments

Experiments adopting air and water as working fluids were conducted in a horizontal pipe at ambient temperature and atmospheric pressure, and the experimental system shown in Figure 1 has been described in detail in our previous papers [13, 14]. Air and water flowed smoothly in their respective pipelines, driven by the power equipment and undergone the pressure stabilizing devices. After the flow rate was measured by standard instruments, the working media flowed through the static mixer to ensure uniformity. Finally, the flow entered horizontally into the test section, a pipe with an inner diameter D=50 mm, installed with a transparent window. The cross section of the bluff body was a truncated isosceles triangle with a front face width w=14 mm and a blockage ratio b=w/D=0.28, as shown in Figure 2. Herein, the pressure fluctuations across the bluff body were detected by the duct-wall differential pressure method (DDPM) with a 10 s holding time [15], where the sampling rate was 1 kHz. Thus, 10000 points were contained in each data set. The two pressure tappings were located 1.0D upstream and 0.2D downstream of the front face of the bluff body so as to ensure the better reflection of the dynamic characteristics of the wake flow.

A series of experiments were carried out within the ranges of water flow rate 3.3–24 m³/h and air flow rate 20–180 m³/h. During each group of experimental process, the opening of water valve was kept unchanged, while gradually increasing the air flow rates. When a set of experiments were completed, the water valve opening was tuned to another value and remained fixed again, and the air flow rate was increased. The above processes were repeated until the whole range of water flow rates were covered. Accordingly, by varying the flow rates, five typical gas–liquid two-phase flow patterns were observed through the transparent window, and 329 sets of differential pressure signals were collected in total, which contained 79, 33, 48, 120, 49 sets of bubble, bubble/plug transitional, plug, slug, and annular flows, respectively.

Figure 1 Schematic of experimental setups:

Figure 2 Longitudinal-sectional view of bluff body:

Some other gas–liquid flow parameters were also obtained, such as the volumetric void fraction β, the dryness x, the ratio of density φ and the modified fluid coefficient ψ. The β and x reflected the proportion of air and water phases in the flow channel, which were calculated based on two-phase flow rates. The volumetric flow rate of water q_vL in the test section was measured directly by the electromagnetic flowmeter in front of the static mixture in view of the negligibility of fluid compressibility. According to the perfect gas equation, the local volumetric and mass flow rates of air in the test section were obtained:

          (1)

                          (2)

The total volumetric flow rate q_v,test was computed:

                         (3)

Then β and x were calculated as follows:

                                (4)

                       (5)

where ρ_G and ρ_L represent the density of gas and liquid phase, respectively.

In order to better describe the difference of the gas–liquid two-phase flow and consider the influence of adding air to water flow at a fixed opening of the valve, the ratio of density φ and the modified fluid coefficient ψ were introduced:

                                  (6)

                                 (7)

where q_mL is the mass flow rate of water, and q_mL0 is the original mass flow rate of water at a specific valve opening.

The acquired differential pressure signals across the bluff-body wake are plotted in Figure 3. The signals of different flow regimes manifested obvious differences. The dynamic pressure variations of the bubble, bubble/plug transitional, and slug flow fluctuated fairly stable. And the slug flow varied dramatically while the annular flow was relatively smooth. Therefore, the differential pressure signals were considered as a basis to distinguish the five kinds of flow patterns, namely it is feasible to realize flow pattern recognition based on the time–frequency characteristics.

3 EEMD and Hilbert transform

EMD method is an important part of HHT, and EEMD is an improved method for overcoming the mode mixing problem in EMD [16].

3.1 EEMD method

EMD is an adaptive technique that decomposes the original signal into a family of IMF components, which is based on time scale of the data itself. So EMD can deal with nonlinear and non-stationary signals efficiently [17]. Applying EMD, signals can be decomposed step by step [18]:

1) Identify all of the local maxima and local minima of the signal x(t). Then use cubic spline interpolation to define the upper envelope x_max(t) and the lower envelope x_min(t) of the original data series. The mean m₁(t) of the upper and lower envelopes is calculated as follows:

                     (8)

2) So, we can get:

                               (9)

Ideally, if h₁ satisfies the definitions of IMF, then h₁ is the first component of x(t). If not, treat h₁ as the original signal and repeat steps (1) and (2) until the presence of satisfied component. Then, define c₁ as the first IMF.

Figure 3 Vortex-induced pressure differential fluctuation signals:

3) Separate c₁ from x(t) as follows:

                              (10)

4) Next, r₁ is regarded as the original signal, and steps (1), (2) and (3) are repeated. After n iterations, we could get:

                           (11)

The decomposition process can be stopped when r_n becomes a monotonic function from which no more IMF can be extracted. We finally obtain:

                          (12)

Adopting the EMD method, we can decompose a signal into a sum of several IMFs and a residue, but there exists a mode mixing problem which makes the physical meaning of the component unclear. The EEMD is a modified method of data analysis assisted by white noise. The EEMD method adds some white noise to the original signals as the data to be processed by EMD decomposition, and the IMF component groups are obtained. After that, the IMF component is averaged from the decomposed IMF groups, and is considered the IMF of the EEMD decomposition. The EEMD method uses the characteristic of uniform distribution of Gauss white noise to smooth the abnormal events such as pulse interference suppression and suppress mode mixing so that one can realize the accurate decomposition [19].

The main IMF of the five typical flow patterns is demonstrated in Figure 4. According to the collected amount of data points, the 12 IMFs were decomposed from the pressure fluctuation signal. Due to the low frequency and weak oscillation, only the preceding six IMFs c₁–c₆ sorted in frequency descending order are presented for the above five different flow patterns. Obviously, local properties containing flow patterns information differ from different IMFs, so as the corresponding vortex components.

Herein were significant effects on the vortex characteristics under the circumstances of the dispersion of gas within the liquid continuum. With the continuing increased air content, the vortex component became worse and instable, revealing the difficulty in vortex generation and shedding. But only the EEMD was not enough to separate the different flow pattern, further treatment was needed to extract the hidden flow characteristics.

3.2 Analysis of Hilbert spectrum

The Hilbert transform is applied to every IMF component c_i(t) to obtain a data series[20]:

                      (13)

An analytical signal z_i(t) can be constructed by:

              (14)

where the amplitude a_i(t) and the phase θ_i(t) are determined from:

                     (15)

                        (16)

The instantaneous frequency can be aquired as:

                 (17)

where the frequency and energy are the function of the time, and thus the energy distribution spectrum of the time–frequency plane, that is, the Hilbert spectrum can be presented [21]:

                (18)

Utilizing the EEMD and Hilbert transform, a meaningful instantaneous frequency and time–frequency spectrum can be obtained from the nonlinear and non-stationary time series. The spectrum is able to describe changes of amplitude (energy) with time and frequency accurately and effectively, and reflect the inherent characteristics of the signals as well.

With the help of the color legend, the Hilbert spectrum illustrates the amplitude of the IMFs in frequency–time–amplitude three-dimensional color graph. All types of Hilbert spectrum are shown in Figure 5. Obtained from the colorful energy bar, the distribution of energy was concerned with the time and frequency. As for various flow patterns, as seen from the Hilbert spectrum, each flow pattern had unique spectrum and the difference was quite obvious regarding time and frequency. For the bubble flow in Figure 5(a), energy was mainly concentrated in the low-frequency bands, especially dominated by stable frequency at approximately 25 Hz which probably corresponded to the fluid oscillation caused by the vortex shedding. It indicates that the maintained vortex street still existed in spite of the influence of relatively low gas dispersion within the liquid continuum.Figure 5(c) represents the Hilbert spectrum of plug flow, and compared with the bubble flow, the energy distribution form was basically similar, but due to the increasing containing gas phase, the energy-concentrated frequency fluctuated around 25 Hz which suggested that the vortex signal became unstable and sometimes was destroyed under the consequences of impact from increasing bubble. The bubble/plug transitional flow, which belonged to the transitional flow pattern, was between the bubble and plug flows, as shown in Figure 5(b). The plug flow could be seen from Figure 5(d), which was mainly concentrated in a lower frequency band with some energy dispersed in the middle frequency component. The gas phase played a leading role when annular flow turned up, and the rapid flow of gas speeded up the wave frequency reflecting high frequency component in the Figure 5(e). In the slug flow and annular flows, concentrated and stable frequency components disappeared, indicating that the vortex street has been completely destroyed, generation and shedding of the vortex did not occur any more.

Figure 4 IMF figure:

Figure 5 Hilbert spectrum:

Therefore, the division of energy bar in Hilbert spectrum could be the judging criterion for the identification of different flow patterns. In other words, selecting time–frequency information as the identification parameter possessed a lot of theoretical and practical meanings.

4 Results and discussion

4.1 Acquisition of time–frequency entropy

The time–frequency distribution of the signal is the description of the energy variation in the sampling time, which for different flow patterns can be different as well. In order to quantitatively describe the change, the fluctuating pressure difference signals in the bluff body wake were processed by EEMD and Hilbert transform, resulting in the discrepancies over the time– frequency distribution. On the time–frequency plane, these discrepancies take the form of the energy distribution difference between N equal areas of small time frequency bands. For each energy band, the energy is W_i (i = 1, 2, …, N), and the total energy for the whole time frequency plane is E. So the normalized energy e_i can be worked out [22]:

                              (19)

where the e_i is the percent of the energy of each time–frequency block in the whole signal energy. Then, the calculation formula for the time– frequency entropy of the signal is:

                         (20)

The time–frequency entropy S of each flow pattern was calculated according to the above steps, the results are shown in Table 1. The values of the time–frequency entropy distributed in a reasonable range, while the slug and annular flow were relatively bigger than the others. Due to the similarity of the bubble and bubble/plug transitional flows, their ranges of time–frequency entropies were also similar.

4.2 Construction of flow pattern maps

The purpose of constructing the flow pattern map is to distinguish the flow patterns. The time– frequency entropy S contained time–frequency information of the gas–liquid mixture that flowed over a bluff body, so that it was reasonable to regard it as the basis to recognize the flow pattern.According to the 329 data points from the experiment, a flow pattern map was constructed eventually adopting the function S(1–x)φ as the vertical coordinate and (1–β)ψ as the horizontal coordinate, which could distinguish the bubble, plug, slug, and annular flows reasonably. And there was also a clear transition area between the bubble and plug flow, which was called the bubble/plug transitional flow. According to the product of the time–frequency characteristics and flow parameters of the flow pattern’s coordinates, the differences of the gas–liquid two-phase flow under different flow regimes were considered. In this map, the volumetric void fraction β and the dryness x represented the structure of the two-phase flow and reflected the proportion of air and water in the flow field. The density ratio φ and the modified fluid coefficient ψ reflected the compression effect of liquid imposed on gas as well as the exclusion effect of gas imposed on liquid under different flow patterns. Furthermore, the time–frequency entropy S depicted the time–frequency distribution information. All the parameters exhibited the difference among various flow regimes.

The construction processes of flow pattern maps were divided into three steps as follows [15]:

1) Plotting the data. Plot 329 points adopting S(1–x)φ as the vertical coordinate and (1–β)ψ as the horizontal coordinate. The distribution of the same pattern was more concentrated, and different patterns had rough boundaries, as illustrated in Figure 6(a).

2) Delineating the boundaries. Draw boundaries between different distributions of flow patterns as Figure 6(b). The boundaries were determined by the principle: If there was an overlap region between adjacent two flow patterns, the boundaries should discern these two patterns as far as possible. If there was a distance between adjacent two flow patterns, the boundary should pass through the middle of two flow patterns.

3) Shaping the map. After the delineation of all the boundaries between flow patterns, the construction data were removed, but the boundaries were reserved. Figure 7 shows the final map.

Table 1 Calculation results of flow patterns

Figure 6 Construction processes of flow pattern map:

Figure 7 Flow pattern map of gas–liquid two-phase flow

4.3 Testing of flow pattern map

In order to evaluate the validity and rationality of this method, a set of experimental data were chosen in this project, namely, 12 sets of bubble flow, 7 sets of plug flow, 18 sets of slug flow, 7 sets of annular flow and 5 sets of transition flow. The flow pattern verification is shown in Figure 8. If the checkpoint was situated in the incorrect identification region, this attempt was regarded as wrong and vice versa. Additionally, it would be half correct if the checkpoint fell on the border. Following this principal, the accuracy rate of the identification of the S(1–x)φ versus (1–β)ψ flow pattern map was calculated and shown in Table 2. This method achieved a fairly satisfactory result regarding the plug flow and annular flows, with a 100% identification rate. However, there were some acceptable errors when identifying the bubble, slug, and plug flows, where all the three flow patterns had one incorrect identification, plus a checkpoint falling on the border of bubble flow. So the accuracy rate of flow pattern identification of this method is 87.5% for the bubble flow, 85.7% for the plug flow and 94.4% for the slug flow.

It is illustrated that relatively high recognition rate was obtained, averaging at 92.9%. All the results mentioned above manifest that the S(1–x)φ versus (1–β)ψ flow pattern map was capable of recognizing flow regime effectively. Moreover, the map enabled the distinction of transitional flow regimes. However, due to the absence of obvious criterion during the transition of flow regimes, which were determined by a subjective visual observation method through a transparent window in the test section, some overlapping on the boundary of adjacent flow regimes was present and was considered acceptable.

Figure 8 Test results of flow pattern map

Table 2 Data for construction and identification rates of the flow-pattern maps

5 Conclusions

1) By utilizing the improved EEMD method and the Hilbert transform, the time–frequency entropies of typical gas–liquid two-phase flow patterns were obtained from the differential pressure fluctuations over a bluff body.

2) A new flow pattern map is constructed adopting S(1–x)φ as the vertical coordinate and (1–β)ψ as the horizontal coordinate, which distinguished the bubble, bubble/plug transitional, plug, slug, and annular flows with an overall identification rate of 92.9%.

3) The proposed method provides an effective way to identify gas–liquid two-phase flow patterns, and it can also be used to establish quantitative relationships among the variables in a gas–liquid flow system.

Nomenclature

a

Amplitude, kPa

b

Blockage ratio

D

Inner diameter of the test pipe, mm

e

Normalized energy of each time–frequency band

E

Total energy of the whole time frequency band

f

Frequency, Hz

p

Pressure, kPa

q

Flow rate, m³·h^–1

S

Time–frequency entropy

t

Temperature, K

w

front face width, mm

x

Dryness

Greek Symbols

β

Volumetric void fraction

ρ

Density

φ

Ratio of density

ψ

Modified fluid coefficient

θ

Phase

Subscripts

Test

Variables pertaining to the actual test

V

Variables pertaining to volume

M

Variables pertaining to mass

G

Variables pertaining to the gas phase

L

Variables pertaining to the liquid phase

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(Edited by HE Yun-bin)

中文导读

钝体尾迹波动时频熵在气液两相流流型识别中的应用

摘要：气液两相流广泛存在于工业过程及设备之中，其流型识别在多相流检测领域发挥着至关重要的作用。本文提出将钝体放置于流道增加扰动以放大各流型间特征的差异。以常温常压下不同比例混合的水和空气作为实验工质在水平管道中进行实验，获得了泡状流、泡塞流、塞状流、弹状流和环状流等流型下的钝体尾迹波动差压信号；利用自适应的集合经验模态分解方法和希尔伯特变换，得到了压差信号的时频熵S；通过结合S和其他流动参数，如体积含气率β、质量含气率x、密度比φ和液相修正系数ψ，采用S(1–x)φ和(1–β)ψ分别作为纵坐标和横坐标，构造了一个新型流型图。实验数据验证表明该流型图的识别率可达92.9%。本研究为气液两相流的流型识别提供了一种简单有效的解决方法。

关键词：气液两相流；尾迹波动；流型图；时频熵；集总经验模态分解；希尔伯特变换

Foundation item: Project(51576213) supported by the National Natural Science Foundation of China; Project(2015RS4015) supported by the Hunan Scientific Program, China; Project(2016zzts323) supported by the Innovation Project of Central South University, China

Received date: 2017-01-20; Accepted date: 2017-04-05

Corresponding author: SUN Zhi-qiang, PhD, Professor; Tel: +86-731-88879863; E-mail: zqsun@csu.edu.cn; ORCID: 0000-0003- 0518-3275