Detecting the Flow Pattern Transition in the Gas-Liquid Two-Phase Flow Using Multivariate Multi-Scale Entropy Analysis

2.1 Multivariate Coarse-Graining Algorithm

Given the multivariate time series, we employ the multi-scale coarse-graining algorithm proposed by Costa et al. [18] to implement the multivariate multi-scale calculation. The m-dimensional time series can be expressed as Xt={xti}i=1m(1≤t≤N), and the corresponding coarse-grained m-dimensional time series Yks={yi,ks}i=1m can be obtained by using the following formula:

where s refers to the scale factor. The length of time series of the coarse-graining m-dimensional time series in each dimension is equal to the length of primitive time series N divided by scale factor s in the corresponding dimension.

2.2 Multivariate Multi-Scale Weighted Permutation Entropy Algorithm

Given the m-dimensional primitive time series Xt={xti}i=1m with the length N of each dimension, we first implement the multivariate multi-scale calculation according to Formula (1), after which we introduce the embedding dimension d and delay time τ to realize phase space reconstruction. Then, we define the time series after coarse-graining and phase space reconstruction as Zi,ld,τ,s, whose formula is as follows:

By sorting the whole components of Zi,ld,τ,s in ascending order, the result can be expressed as follows:

If there exist components with an identical value in Zi,ld,τ,s, the permutation is conducted by comparing the values of n. Therefore, we can obtain a permutation of all the components in Zi,ld,τ,s as follows:

For a phase space with an embedding dimension equal to d, the total number of permutation is d!. We define the number of occurrences of the j^th permutation as N_j, where 1 ≤ j ≤ d!. Moreover, we, respectively define Z¯i,ld,τ,s and wi,ls as the arithmetic mean of Zi,ld,τ,s and the weight as follows:

The weighted probability pi,js of the j^th permutation can be expressed as

For the m-dimensional primitive time series, pi,js meets the equation of ∑i=1m∑j=1d!pi,js=1. The weighted probability p^.js of the j^th permutation in m-dimensional primitive time series can be expressed by the following formula:

Then the MWMPE can be expressed with the following form of Shannon entropy:

Based on the same principle, the definition of the SCWMPE can be obtained as follows:

2.3 Multivariate Multi-Scale Sample Entropy Algorithm

Given the m-dimensional primitive time series Xt={xti}i=1m with the length N of each dimension, we first implement the multivariate multi-scale calculation according to Formula (1), and then utilize the multivariate embedding theory [24] to realize phase space reconstruction. The resulting time series Zds(i) can be expressed as follows:

where D=[d1,d2,…,dm] is the embedding dimension vector, and τ=[τ1,τ2,…,τm] is the time delay vector. By defining a parameter n = max{D} × max{τ}, we can obtain the (N-n) composite delay vectors Zds(i), where i = 1, 2,…, N−n. For any two vectors, the distance between Zds(i) and Zds(j)is defined as follows:

Next, we set a threshold r, counting the occurrence number Q_i that satisfies d[Zds(i),Zds(j)]≤r(i≠j), thus obtaining the following formulas:

where B^d(r) is defined as the probability that any two phase-space vectors are close for embedding dimensions of d. Based on the same principle, we increase the embedding dimension of the multivariate series phase space vector from d to d + 1, thus obtaining the following formula:

The expression of MMSE is given as follows:

3.1 Experimental Facility

The experiment of the vertical upward gas-liquid two-phase flow was implemented in the lab of multiphase flow sensor system and fluid flow in Tianjin University. The schematic diagram of the experimental facility is shown in Figure 1. The experimental pipe is made up of acrylic material with an inner diameter of 20 mm and a length of 2820 mm. The experimental flow media are air and tap water. The industrial peristaltic pump is utilized to transport and control the flux of the liquid phase. The air compressor is utilized to generate the gas phase and the flux of gas phase is metered by the float flowmeter. The gas and liquid phase with a certain ratio are uniformly mixed in the mixer at the inlet and then injected into the experimental pipe. During the experiment, the gas superficial velocity U_sg is fixed at first, then we gradually change the liquid superficial velocity U_sl to carry out the experiment. After all the liquid superficial velocities are finished, we change the gas superficial velocity to the next value, and then the experiments are repeated in the above way. In the experiment, the gas superficial velocity U_sg varies from 0.0553 m/s to 0.6624 m/s, and the liquid superficial velocity U_sl varies from 0.0368 m/s to 1.1776 m/s. The flow conditions are shown in Table 1.

Figure 1:

The schematic diagram of THE vertical upward gas-liquid two-phase flow experimental facility.

In this study, an eight-electrode rotating electric field conductance sensor [25] is used to capture the flow structure information of the gas-liquid two-phase flow from different directions of the pipe cross section. As stated in the descriptions in [25], the eight-electrode rotating electric field conductance sensor presents the high sensitivity and homogeneous sensitivity distribution under optimal geometric parameters, which means that the sensor can possess high resolution in water holdup variation caused by flow structural change. Moreover, the output of the sensor is a linear function of the conductivity; therefore, this sensor can well acquire flow structure information from different directions of the pipe cross-section with high resolution. As shown in Figure 1, the sensor consists of eight stainless steel electrodes that are flush-mounted on the inner wall of a 20 mm inner diameter pipe. In order to acquire the fully developed gas-liquid two-phase flow information, the eight-electrode rotating electric field conductance sensor is installed in the testing pipe with a distance of 2000 mm from the pipe inlet. The working principle of the eight-electrode rotating electric field conductance sensor is as follows: sinusoidal excitation signals with peak-to-peak value of 10 V, frequency of 20 kHz and fixed phase are applied to the electrodes (A+, B+, C+, D+, A−, B−, C−, D−) simultaneously, as shown in Figure 1, and the phase difference between adjacent excitation signals is 45°. Then, a rotating electric field in the sensor volume is obtained. By modulating the output voltage between A+ and A−, B+ and B−, C+ and C− and D+ and D−, the corresponding A, B, C and D four-channel signals, which can reflect the flow structure information of the gas-liquid two-phase flow, can be obtained. The four-channel signals are sampled by NI PXI-4472 synchronous acquisition card. The sampling time and sampling frequency are 30 s and 2 kHz, respectively.

The flow patterns can be directly observed and determined by the images of the gas-liquid two-phase flow captured by a high-speed camera. As shown in Figure 2, there are four typical flow patterns in the two-phase flow, namely, slug flow, churn flow, bubble flow and churn-bubble transitional flow. As shown in Figure 2a, the slug flow is mainly composed of the Taylor bubble, falling liquid film and liquid slug. The Taylor bubbles occupy most of the pipe section. The falling liquid films with a small number of gas bubbles exist between the pipe wall and the Taylor bubble. The liquid slugs contain a large number of gas bubbles with different sizes. The Taylor bubbles and liquid slugs appear as quasi-periodic alternating motion. The images of churn flow are shown in Figure 2b. As can be seen, the churn flow is characterized by the distortion of the interface between the gas phase and the liquid phase and the random flow phenomenon in which the gas phase and the liquid phase are oscillated up and down in the pipe. In the bubble flow as presented in Figure 2c, gas bubbles with different sizes are non-uniformly distributed in the liquid phase and their motions have strong randomness. In the churn-bubble transitional flow shown in Figure 2d, the distortion of the interface between the gas phase and the liquid phase as well as the non-uniform distribution of bubbles with different sizes in the liquid phase exist simultaneously.

Figure 2:

The four typical flow pattern images captured by THE high-speed camera. (a) Slug flow (U_sg = 0.0553 m/s, U_sl = 0.0736 m/s). (b) Churn flow (U_sg = 0.4416 m/s, U_sl = 0.8832 m/s). (c) Bubble flow (U_sg = 0.0736 m/s, U_sl = 1.1776 m/s). (d) Churn-bubble transitional flow (U_sg = 0.1472 m/s, U_sl = 0.8832 m/s).

3.2 Fluctuating Signals of the Rotating Electric Field Conductance Sensor

The four-channel fluctuating signals of the typical slug flow are shown in Figure 3a. As can be seen, the slug flow signals have a large fluctuation range. The peak value of the voltage signal can reach about 3.2 V, while the lowest value is about 0.8 V. When the liquid slugs pass through the conductance sensor, the conductance between the electrodes increases, then the fluctuating signals show a high voltage. Moreover, as the Taylor bubbles pass through the conductance sensor, they occupy most of the pipe section, resulting in a decrease of conductance between the electrodes, with the fluctuating signals showing a low voltage. The high and low voltages appear alternately, corresponding to the behavior of Taylor bubbles and liquid slugs appearing as quasi-periodic alternating motion.

Figure 3:

The rotating electric field conductance sensor signals. (a) Slug flow; (b) churn flow; (c) bubble flow; (d) churn-bubble transitional flow.

The four-channel fluctuating signals of the typical churn flow are shown in Figure 3b. The fluctuating signals have a short duration of high voltage and low voltage and the fluctuation is random. Due to the increase of the gas and liquid superficial velocity, the large gas slug is crushed into a number of gas blocks with different sizes. The gas blocks and the liquid phases appear alternately and randomly, exhibiting disorderly motion characteristics.

The four-channel fluctuating signals of a typical bubble flow are shown in Figure 3c. The fluctuating range of measurement signals is in the range of 2.4–3.2 V. The highly intense fluctuation is consistent with the flow characteristics of the irregular random motion of a large number of bubbles with different sizes in the bubble flow.

The four-channel fluctuating signals of a typical churn-bubble transitional flow are shown in Figure 3d. The measurement signals appear in the form of irregular fluctuation with a small amplitude, thus conforming to the flow characteristics of the bubble flow. In addition, some signal fluctuations with large interval and large amplitude are consistent with the flow characteristics of the churn flow. Based on the above flow characteristics, the flow pattern conforms to the churn-bubble transitional flow pattern under this flow condition.

4.1 Multivariate Weighted Multi-Scale Permutation Entropy Analysis

In this section, the MWMPE is employed to analyze the gas-liquid two-phase flow fluctuation signals measured by the eight-electrode rotating electric field conductance sensor. First, we set the parameters during the MWMPE calculation process. The maximum coarse-graining scale is set to 20, which is sufficient for multi-scale analysis. The signal sequence length is set to 10,000, which can fully reflect the flow characteristics of the gas-liquid two-phase flow. The value of the embedding dimension d for all flow conditions is set to 6 using the method of the false nearest neighbors (FNN) [26]. According to descriptions in [16], the authors of the permutation entropy recommend setting the embedding dimension to 3, 4, …, 7, with the time delay τ assigned as 1. Hence, we set the embedding dimension d and the time delay τ as 6 and 1 for all flow conditions, respectively.

In order to further verify the feasibility of the selected embedding dimension d and time delay τ, we explore the calculation results of the MWMPE with different embedding dimensions and different delay times, as shown in Figure 4. We set the time delay τ as 1 and then change the embedding dimension from 3 to 7 to observe the sensibility of the MWMPE in one flow condition. The calculation results of the MWMPE using different d values are shown in Figure 4a. Under the same flow condition, with the increase of d, the calculation results of the MWMPE present an increasing trend. We find that when d is 6, the calculation results of the MWMPE present better increasing trend in the whole scales than other calculation results with different d values. Then, we set the embedding dimension as 6 and change the time delay from 1 to 5 to observe the sensibility of MWMPE in the same flow condition. The calculation results of the MWMPE in different τ values are shown in Figure 4b. Under the same flow condition, with the increase of τ, the calculation results of the MWMPE also present an increasing trend. We find that, when τ is 1, the calculation results of the MWMPE present better increasing trend in the whole scales than other calculation results with different τ values. Setting the embedding dimension as 6 and the time delay τ as 1 can effectively reflect the difference of entropy values at different scales.

Figure 4:

The calculation results of the MWMPE in different values of d and τ. (a) The calculation results of the MWMPE in different values of d. (b) The calculation results of the MWMPE in different values of τ.

Figure 5 presents the calculation results of the MWMPE for the measurement signals from the eight-electrode rotating electric field conductance sensor under different flow conditions. As shown in Figure 5, the MWMPE values in four typical flow patterns have significant differences within the whole scales. The MWMPE values in bubble flow remained the highest, whereas the MWMPE values in the slug flow are the lowest. The MWMPE values in the churn flow and churn-bubble transitional flow are between the slug flow and bubble flow. Moreover, the churn-bubble transitional flow has the higher MWMPE values than churn flow. It should be noted that the increase of entropy values in small scales, which is defined as the growth rate of the MWMPE, varies significantly in different flow patterns. The growth rate of the MWMPE in the bubble flow is the highest, whereas the growth rate in the churn-bubble transitional flow is lower than that in the bubble flow. The churn flow presents a lower growth rate than the churn-bubble transitional flow. The growth rate of the MWMPE in the slug flow is the lowest. The above calculation results are related to the flow characteristics of the corresponding flow patterns.

Figure 5:

The calculation results of the MWMPE in the gas-liquid two-phase flow. (a) The calculation results of the MWMPE in slug flow, churn flow and bubble flow. (b) The calculation results of the MWMPE in slug flow, churn flow, churn-bubble transitional flow and bubble flow.

For the slug flow, the Taylor bubbles and the liquid slugs alternately rise and pass through the conductance sensor with a certain quasi-periodic movement characteristic; therefore, the flow characteristic of the slug flow are relatively stable, and the MWMPE values in the slug flow are low in the whole scales. As the increasing liquid superficial velocity enhances the overall turbulent energy, the Taylor bubbles are crushed into bubbles with different sizes and are randomly distributed in the liquid phase; the flow pattern then evolves to the bubble flow with the highest MWMPE values in the whole scales. When the gas superficial velocity and liquid superficial velocity both increase, the Taylor bubbles in the slug flow are broken into a number of irregular gas blocks with a turbulent movement, and the flow pattern gradually evolves to churn flow. The instability of the churn flow is higher than the slug flow, which presents a quasi-periodic motion behavior; at the same time, it is lower than the bubble flow with random motion behavior. Therefore, the MWMPE of the churn flow is between the slug flow and the bubble flow in the whole scales. The churn-bubble transitional flow forms in the process of churn flow evolving into bubble flow; therefore, the MWMPE of the churn-bubble transitional flow is higher than the churn flow and lower than the bubble flow. Moreover, under one flow pattern condition, with the increase of the total mixture velocity, the turbulent energy of the gas-liquid two-phase flow is enhanced, which then increases the flow instability of the gas-liquid two-phase flow. Therefore, the MWMPE in the whole scales shows an increasing trend. Moreover, the MWMPE can effectively reveal the flow instability of the gas-liquid two-phase flow during the flow pattern transition.

4.2 Single-Channel Weighted Multi-Scale Permutation Entropy Analysis

In order to assess the advantages of the MWMPE in the flow instability of the gas-liquid two-phase flow, in this section, the SCWMPE is employed to analyze the gas-liquid two-phase flow fluctuation signals measured by the eight-electrode rotating electric field conductance sensor. As shown in Figure 6, the calculation results of SCWMPE for the measurement fluctuating signals of channel A and channel C from the eight-electrode rotating electric field conductance sensor under different typical flow conditions are given.

Figure 6:

The calculation results of the SCWMPE in the gas-liquid two-phase flow. (a) The calculation results of the SCWMPE in channel A. (b) The calculation results of the SCWMPE in channel C. (c) The calculation results of the SCWMPE in channel A. (d) The calculation results of the SCWMPE in channel C.

Figure 6a,b are the calculation results of the SCWMPE in channel A and channel C, respectively, which can be compared with the calculation results of the MWMPE in Figure 5a. As shown in Figure 6a,b, the calculation results of the SCWMPE can reveal the instability of different flow patterns. However, its resolution in uncovering the instability of flow condition variation under the same flow pattern is limited, especially in the bubble flow and slug flow in large scales. Furthermore, the MWMPE values increase steadily with the increase of scales. However, the SCWMPE values are fluctuant in large scales. Similarly, the calculation results of SCWMPE in Figure 6c,d can be compared with the calculation results of MWMPE in Figure 5b. It can be found that the calculation results of SCWMPE in channel A shown in Figure 6c is similar to the calculation results of MWMPE in Figure 5b. However, the calculation results of SCWMPE in channel C present poor capacity in uncovering the instability of different flow patterns under different flow conditions. Single-channel signals only reflect the flow characteristics of the local measured section, so the weighted multi-scale permutation entropy based on the single-channel signals cannot accurately reflect the flow instability of the gas-liquid two-phase flow in detail. In summary, the MWMPE show its superiority in revealing the flow instability of gas-liquid two-phase flow compared with the SCWMPE.

4.3 Multivariate Multi-Scale Sample Entropy Analysis

In light of the descriptions in [23], the MWMPE demonstrated anti-noise performance in relation to some typical time series and its superior ability to reveal the flow instability of the vertical upward oil-water two-phase flow. In addition, we find that the MMSE also presents satisfactory performance. Hence, we choose the MMSE analysis as a comparator to the MWMPE. Figure 7a,b illustrate the calculation results of the MMSE in typical flow pattern conditions. It can be seen from Figure 7a that the MMSE analysis can identify three typical flow patterns through the entropy values, but the MMSE values are fluctuant in large scales, especially in the bubble flow and churn flow. This is because the multi-scale sample entropy has an inherent defect in that the multi-scale sample entropy may yield an inaccurate estimation of entropy as the coarse-graining procedure reduces considerably the length of a time series in large scales [27]. The resolution of the MMSE in uncovering the instability of the flow condition variation is limited in the bubble flow, slug flow and transitional flow. In addition, it can also be seen from Figure 7b that the MMSE cannot identify churn flow and churn-bubble transitional flow. Therefore, compared with the MMSE, the MWMPE analysis presents better performance against the flow instability of the vertical upward gas-liquid two-phase flow.

Figure 7:

The calculation results of the MMSE in the gas-liquid two-phase flow. (a) The calculation results of the MMSE in the slug flow, churn flow and bubble flow. (b) The calculation results of the MMSE in the slug flow, churn flow, churn-bubble transitional flow and bubble flow.

4.4 Joint Distribution of the Normalized Mean Value and Normalized Entropy Rate in the MWMPE

To further characterize the flow instability of the flow pattern transition in the gas-liquid two-phase flow, we extract the mean value of MWMPE in whole scales and the entropy rate of MWMPE in small scales under the whole flow conditions. The mean value of the MWMPE is determined by calculating the MWMPE values in the whole scales. As for the determination of the entropy rate, we define the slope obtained by linearly fitting the MWMPE calculation results in the scale range of 1–5 as the entropy rate. As stated in Section 4.1, the MWMPE values present a high increasing rate with the increase of scale in small scales; moreover, the increasing rates vary among different flow patterns. Therefore, the MWMPE range in the small scales can be considered as a sensitive area for the gas-liquid two-phase flow pattern transition. Thus, we carry out the normalization process in the mean value and entropy rate of the MWMPE using the following formula and then build the graph of joint distribution as shown in Figure 8.

Figure 8:

The joint distribution of the normalized mean value and normalized entropy rate in the MWMPE.

In the formula above, z is the mean value or the entropy rate of the MWMPE, and z^∗ is the normalized mean value (NM) or the normalized entropy rate (NR) of the MWMPE. As shown in Figure 8, the normalized mean value and normalized entropy rate of the slug flow are the smallest, indicating that the flow characteristic is the most stable. In comparison, the normalized mean value and normalized entropy rate of the bubble flow are the highest, indicating that the flow instability is the most obvious. The normalized mean value and normalized entropy rate of the churn flow and churn-bubble transitional flow are between the slug flow and bubble flow, which are consistent with their dynamic flow instability.

The joint distribution of the normalized mean value and normalized entropy rate in the MWMPE can satisfactorily indicate flow instability boundaries between different flow patterns. As shown in Figure 8, the five boundaries are NM = 0.21, NM = 0.38, NM = 0.72, NR = 0.25 and NR = 0.67, respectively. The flow instability of the bubble flow is maximal and it mainly lies in NM ≥ 0.72 and NR ≥ 0.67. The main region of the churn-bubble transitional flow is 0.38 ≤ NM ≤ 0.72, 0.25 ≤ NR ≤ 0.67, churn flow is 0.21 ≤ NM ≤ 0.38 and slug flow is NM ≤ 0.21. In conclusion, the MWMPE can satisfactorily detect flow pattern transition in the gas-liquid two-phase flow using the multivariate multi-scale entropy analysis.