MAPANGas-Liquid - Journal Two ofPhase Metrology Flow PatternSociety ofEvolution India, Vol. Characteristics 26, No. 3, 2011; Based pp. 255-265 on Detrended Fluctuation Analysis ORIGINAL ARTICLE

Gas-Liquid Two Phase Flow Pattern Evolution Characteristics Based on Detrended Fluctuation Analysis

LU-SHENG ZHAI, NING-DE JIN*, ZHONG-KE GAO, PING CHEN and HENG CHI

School of Electrical Engineering and Automation,Tianjin University,Tianjin 300072,China *e-mail: [email protected]

[Received: 27.12.2010 ; Accepted: 30.06.2011]

Abstract In this paper, we first extract a nonlinear time series from the Weierestrass function as a toy model and investigate the anti-noise ability of six different fractal scale algorithm. The results indicate that the fractal scales calculated from Detrended Fluctuation Analysis(DFA) are robust with respect to variation in noise level. Based on the conductance fluctuating signals measured from vertical gas-liquid two phase flow experiment, we calculate the fractal scales of five typical flow patterns. The results show that when the water superficial velocity ranging from 0.0453ms-1 to 0.226 ms-1 and the gas superficial velocity ranging from 0.0043ms-1 to 3.43 ms-1, the values of the fractal scale of bubble flow are lowest corresponding to the random complex dynamic behavior, while the values of slug flow are highest corresponding to the alternatively periodic motions between gas slug and liquid slug, and the values of churn flow lies between them indicating the relatively complex dynamic behavior. Our main result is that the fractal scales obtained from conductance fluctuating signals can not only effectively characterize the dynamic characteristics of gas-liquid two phase flow patterns, but also further provide valuable reference for understanding the transitions of different gas-liquid two phase flow patterns.

1. Introduction Saether et al. analyzed the distribution of the liquid slug length in horizontal slug flow and obtained the Gas-liquid two phase flow widely exists in relationship between the Hurst exponent and the production gas well, and the gas flow measurement length of the liquid slug [1]. Franca et al. used the fractal becoming more and more important with the method to investigate the flow pattern characterization increasing number of deep gas well in China oilfield. based on the measured differential pressure signals Because of the complex and unstable flow pattern in gas-liquid two phase flow [2]. Latifi et al. applied characteristics, the flow pattern has a great influence the concept of fractal to analyzed microelectrode time on the flow measurement sensor. Moreover, the flow series of gas-liquid two phase flow, and pointed out pattern identification based on the measurement the microelectrode time series was one kind of fractal signals from the well is very important for Brown motion [3]. They also studied the fractal establishing the flow measurement model. Since the dimension distribution of the microelectrode time 1990's, the fractal theory has been widely applied to series when the liquid flow rate was fixed and the gas analysis the flow structure in multi-phase flow system. flow rate was changing. Kozma et al. calculated the © Metrology Society of India, All rights reserved 2011. fractal dimension by using local temperature

255 Lu-Sheng Zhai, Ning-De Jin, Zhong-Ke Gao, Ping Chen and Heng Chi fluctuating time series in two phase flow, and flow. The results indicate the fractal scales can observed that the fractal dimension that changes with effectively characterize the dynamic characteristics of different flow pattern could be used to distinguish the different gas-liquid two phase flow patterns. flow patterns [4]. Cai et al. calculated the fractal dimension using the correlation dimension to 2. Fractal Scale Algorithm distinguish the flow patterns in vertical rectangle pipe, and indicated that the gas-liquid flow in vertical The Hurst exponent (H) and fractal dimension rectangle pipe possess chaotic features [5]. Wu et al. (D) are two important parameters, and there exists found the dual-fractal characteristic of the fluctuating certain relationship between them. The Hurst pressure signals in the full development of gas-solid exponent is usually used to describe the long-range flow [6]. Jin et al. calculated the fractal dimension of correlations. When H=1/2, the past increment and time series in oil-water two phase flow, and indicated future increment of the fluctuating process are that fractal dimension regularly changed with the statistical independent. When H>1/2, the fluctuating variation of the total flow rate [7]. Bai et al. analyzed process are persistent (positive correlations). the pressure fluctuating signals of gas-water two Specifically, the past increase means the future phase flow in vertical pipe and discussed the impact increase. With the increasing of H, the data curve of the superficial liquid velocity on the fractal becomes more and more smooth, which means long- characteristic [8]. Zhao et al. analyzed the fractal range correlation of this process is better and the characteristic of the pressure fluctuating signals of the information is much clearer. When H<1/2, the fluidized bed in Wavelet decomposition scale, and fluctuating process are anti-persistent (negative found the fractal characteristic of the fluctuating correlations). Specifically, the past increase means the signals in micro-scale, medium-scale and future decrease. macroscopic-scale [9]. Obviously, fractal scale analysis 2.1 Rescaled Range Analysis (R/S ) has become an effective tool for characterizing the complex flow characteristics in multiphase flow The R/S method was proposed by Hurst in 1956 system. However, previous studies had limited the [10]. For uniform intervals time series x(t), (t=1,2, …,T), application of the fractal scale analysis in their the range R(t, τ) is the difference between the maximum respective research field, and a comprehensive and the minimum of the following formula: evaluation of different fractal scale algorithm was not discussed especially for the influence of the noise. * * u * * Moreover, fractal time series only consider the R(t,τ ) = max X (t + u) − X (t) − X (t + u) − X (t) 0≤u≤τ{ τ } similarity between the measured local signal and the * * u * * overall signal, and the dynamic mechanism of −min X (t + u) − X (t) − X (t + u) − X (t) physical system reflected by the fractal scale should 0≤u≤τ{ τ } be further exploited. (1)

In this paper, we first extract a nonlinear time The standard deviation S (t, τ): series from the Weierestrass function as a toy model and discuss the anti-noise ability of six different fractal 1 2 scale algorithm. Since the characterization value of  t+τ  t+τ  1 2 1 (2) the conductance signals directly reflect the phase S(t,τ ) =  ∑ x (u) −  ∑ x(u) τ u=t+1 τ u=t+1  holdup that is an important parameter for different flow patterns, the conductance signal is faithful t information for measuring different gas-liquid two where X*(t)= ∑x(u) , andτ is the delay time. Hurst phase flow patterns. Based on the evaluation of u=1 different fractal scale algorithms, we using DFA experimentally got the follow relationship from the investigate the measured conductance fluctuating measured time series: signals and indicate how the fractal scale changes with the parameter variation in gas-liquid two phase

256 Gas-Liquid Two Phase Flow Pattern Evolution Characteristics Based on Detrended Fluctuation Analysis

variogram in double logarithmic plot, and the slope R(t,τ ) ∝ τ H of the linear part is the fractal scale exponent. S(t,τ ) (3) 2.4 Power Spectral Density Method (PSD) where H is the Hurst exponent. The power spectrum density [13] of the time series 2.2 Detrended Fluctuation Analysis (DFA) is calculated by the modified period gram algorithm with window function and FFT analysis in this paper. The DFA method was initially proposed by Peng The relation between power spectrum density (PSD) et al. [11]. For time series X , i=l, 2,…, N, the average is i and frequency is: X . Firstly, computing the integrated and detrended

time series y(k), k=1, …,N, β S( f ) ∝1 f (8)

k y(k) = (X − X ) where β is the fractal scale. β can be converted into ∑ i (4) i=1 Hurst exponent H according to the following equations: y(k) is divided into N/n sub sequences with each interval length n. The local trend y (k) comes from β + 1 n H = (for fractal Gauss noise) (9) each interval with least square method, and the root- 2 mean-square fluctuation of this series is calculated by: β − 1 H = (for fractal Brown motion) (10) 2 1 N 2 = − F(n) ∑[y(k) yn (k)] (5) 2.5 Dispersional Analysis (DISP) N k=1 In the original algorithm [14], the x(t) series is A power law is expected as follows: divided into non-overlapping intervals of length n. Then the standard deviation (SD) of each interval is ∝ α (6) F(n) n computed by: where slope α is fractal scale in double logarithm n − 2 plot. For fractal Gauss noise series, the Hurst exponent ∑ = [x(t) x] SD = t 1 (11) H is α , and for fractal Brown motion series the Hurst n −1 exponent H is α -1. These computations are repeated over all possible 2.3 Semi-Variation method (SV) interval lengths. The SD is related to n by power law H-1 Temporal and spatial variability response is as SD ∞ n . The quantity (H-1) is expressed as the described with semivariogram in this method [12]. slope of the double logarithmic plot of SD as a function of n, and H-1 is the fractal scale of the time series. For the measured data Z(t1), Z(t2) ,…, Z(tn) at time t1, t2 ,…, t , the semi-variation γ (h) can reflect the n 2.6 Scaled Windowed Variance Method (SWV) relationship between the variable and time. For the SWV method [15], the time series x(t) is 2 1 N (h) divided into non-overlapping intervals of length n, =  + −  (h) ∑ Z (xi h) Z (xi ) (7) 2N(h) i=1 then the standard deviation SD is computed in each interval: where N(h) is the value of measured data with interval length h in temporal domain. We can get the semi-

257 Lu-Sheng Zhai, Ning-De Jin, Zhong-Ke Gao, Ping Chen and Heng Chi

n 2 n x(t) − x n=+∞ (1 - cos(b t)) ∑ t=1[ ] W(t) = SD = (12) ∑ (2−D)n (13) n − 1 n=-∞ b

where b=2, D=1.1, 1.2,…,1.9, step ∆t = 0.1s. When the Then compute the average deviation SD of all precision of the Weierestrass function W(t) is 0.001, intervals of length n. This computation is repeated for the value of n is 10,000. According to the above over all possible interval lengths. For a fractal series parameters, we generate 9 groups of time series with SD is related to n by a power law SD ∞ n H . The fractal the length 10000. Figure 1 shows the 9 groups Weierestrass function at different fractal dimensions. scale H is expressed as the slope of the log-log plot of We can see the self-similarity characteristic in the SD as a function of n. figure, and the curve fluctuating trend of the time series increases with increasing of dimension D. 3. Fractal Scale Algorithm Evaluation In Fig. 2, we can find the relationship between the 3. 1 Fractal Scale Algorithm Evaluation with no Noise fractal scale of the six fractal scale algorithm and the dimension D of Weierestrass function. It is clear that Weierestrass function is designed as follows: the fractal scale of Dispersional analysis (DISP), Detrended Fluctuation Analysis (DFA), Semi-

Fig. 1. Weierestrass time series at different fractal dimension conditions

258 Gas-Liquid Two Phase Flow Pattern Evolution Characteristics Based on Detrended Fluctuation Analysis

1.004 2.0 R/S DFA 1.000 1.8

0.996 1.6

H 0.992 α1.4

0.988 1.2 0.984 1.0 1.2 1.4 1.6 1.8 2.0 1.0 1.0 1.2 1.4 1.6 1.8 2.0 D D

2.0 2.12 SV PSD 2.10 1.5 2.08

1.0 α β 2.06 2.04 0.5 2.02

0.0 2.00 1.0 1.2 1.4 1.6 1.8 2.0 1.0 1.2 1.4 1.6 1.8 2.0 D D

1.0 1.0 DISP SWV 0.8 0.8

0.6 0.6 H H -1 0.4 0.4

0.2 0.2

0.0 0.0 1.0 1.2 1.4 1.6 1.8 2.0 1.0 1.2 1.4 1.6 1.8 2.0 D D

Fig. 2. Relationship between fractal scale and dimension of Weierestrass function with no noise time series

Variation method (SV) and Scaled Windowed level Gauss noise to the original time series: Variance method (SWV) has a good linear relationship with the Weierestrass dimension. The fractal scale of Li = xi +ηεi (14) Power Spectral Density method (PSD) is somewhat less linear with Weierestrass dimension, while the where Li is the time series contaminated by noise, xi is fractal scale of Rescaled Range analysis(R/S) is not the original time series generated by Weierestrass linear with Weierestrass dimension. function, εi is Gauss random noise, η is the level of

noise. In this study, we choose η =1,3,5,7,10 to 3.2. Fractal scale algorithm evaluation with noise generate 5 groups Weierestrass time series.

To evaluate the performance of different Figure 3 shows the influence of noise on the algorithms at the present of noises, we add different different fractal scale algorithms. We can see that the

259 Lu-Sheng Zhai, Ning-De Jin, Zhong-Ke Gao, Ping Chen and Heng Chi

1.14 2.0 η=0 R/S DFA η=0 η=1 1.11 1.8 η=1 η=3 η=3 η=5 1.6 η=5 1.08 η=7 η=7 η=10 α 1.4 η=10 H 1.05 1.2 1.02 1.0

0.99 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1.0 1.2 1.4 1.6 1.8 2.0 D D

2.4 2.0 η=0 PSD SV η=0 2.3 η=1 1.6 η=1 η=3 η=3 η=5 η=5 2.2 η=7 1.2 η=7 β η=10 H η=10 0.8 2.1

0.4 2.0

0.0 1.9 1.0 1.2 1.4 1.6 1.8 2.0 1.0 1.2 1.4 1.6 1.8 2.0 D D

1.0 1.0 SWV η=0 η=0 DISP η=1 η=1 0.8 0.8 η=3 η=3 η=5 η=5 0.6 η=7 0.6 η=7 η=10 η=10 H H -1 0.4 0.4

0.2 0.2

0.0 0.0 1.0 1.2 1.4 1.6 1.8 2.0 1.0 1.2 1.4 1.6 1.8 2.0 D D

Fig. 3. Relationship between fractal scale and dimension of Weierestrass function with noise time series

PSD analysis is sensitive to noise. The fractal scale presence of noise. In this paper, we adopt the DFA to calculated by SWV analysis decreases with the analyze the conductance fluctuating signal in gas- increasing of noise level and loses the linear relation liquid two phase flow. with the dimension D in the strong noise. The fractal scales calculated by DISP analysis and SV analysis 4. Conductance Fluctuation Signal Acquisition of are near linear relation with dimension D in lower Gas-liquid Two Phase Flow level noise and also lose the linear relation in strong noise, whereas the fractal scale computed by DFA The gas-liquid two phase flow dynamical method performs the best linear relation with experiment in vertical upward pipe is carried out in dimension D at different level noise. The DFA method oil-gas-water three phase flow laboratory of Tianjin performs good robustness and applicability to the University. The diameter of pipe is 125mm, and the

260 Gas-Liquid Two Phase Flow Pattern Evolution Characteristics Based on Detrended Fluctuation Analysis

Fig. 4. The experimental facility of gas-water two phase flow in vertical upward pipe and conductance sensor array at test section of flow loop vertical multi-electrodes array conductance sensor section is realized by LabVIEW7.1 which can perform locates in the measurement section of flow loop. As real time wave display, the real-time data storage and shown in Fig. 4, the sensor contains a pair of cross- analyzing on line. correlation velocity electrodes, a pair of phase volume fraction electrodes and a pair of exciting electrodes. The High speed dynamic camera used in the The experiment mediums are air and tap water. The experiment is developed by Weinberger company, experiment starts with passing over fixed water flow based on advanced CMOS technology. The parameters rate in pipe, and then gradually increases the gas in this experiment are in the following: resolution 640 phase flow rate. After completely forming steady gas- × 480, frame rate 200 f /s. The tricolor fluorescent lamp water two phase flow, we observe the flow pattern of with the color temperature 6500K is acted as gas-water two phase flow. In this experiment, the illumining source, which is bright without glitter. Four 3 range of the water phase flow rate Qw is from 0.1 m /h continuous dynamic evolution images of every flow 3 to 100 m /h and the range of the gas phase flow rate pattern are intercepted as showed in Fig. 5. The Usw Q is from 0.5 m3/h to 100 m3/h. The data sampling g denotes superficial water velocity and Usg denotes frequency is 400 Hz, and sample size of each group superficial gas velocity. Five typical flow patterns of flowing condition is 20400 points. This experiment gas-water two phase flow are observed in vertical has altogether gathered 80 groups of gas-water two upward pipe: Bubble flow, Bubble-Slug transitional phase flow signals. The measurement system consists flow, Slug flow, Slug-Churn transitional flow and of array conductance sensor, exciting signal Churn flow. Five typical fluctuating signals of different generating circuit, signal modulating module, data flow patterns are shown in Fig. 6. acquisition equipment and the data analysis software. The system is driven by 20 kHz constant voltage sine 5. Dynamic Characteristics Analysis of Gas-liquid wave source or constant current sine wave source. Two Phase Flow Pattern Evolution When the constant voltage source is applied, the virtual value of the exciting voltage is 1.4V. The signal Figure 7 indicates the steps for extracting fractal modulating module mainly constitutes with scale from conductance fluctuating signals by DFA differential amplifying, phase sensitive demodulation method. Firstly, the average of the original signals is and low pass filter. The data acquiring equipment is computed, and the differences of the original signals National Instrument Corporation's PXI 4472 data and the average are added together. Secondly, divide acquiring card, which is based on PXI bus technology, the signals calculated in Step 1 into non-overlapping equipped with eight data acquisition channels and intervals of length n and fit each interval linearly, such synchronous acquisition function. Data processing as Fig. 7(b). At last, calculate the root-mean-square

261 Lu-Sheng Zhai, Ning-De Jin, Zhong-Ke Gao, Ping Chen and Heng Chi

(a)

(b)

(c)

(d)

Figure contd.

262 Gas-Liquid Two Phase Flow Pattern Evolution Characteristics Based on Detrended Fluctuation Analysis

(e)

Fig. 5 (a). The dynamic images of bubble flow (Usw= 0.091 m/s,Usg= 0.005 m/s); (b) bubble-slug transitional flow (Usw= 0.091 m/s,Usg= 0.018 m/s); (c) slug flow (Usw= 0.091 m/s,Usg= 0.315m/s); (d) slug-churn transitional flow

(Usw= 0.091 m/s,Usg= 0. 566 m/s) and (e) churn flow (Usw= 0.091 m/s,Usg= 1. 517 m/s)

Fig. 6. the conductance fluctuating signals of five typical gas-water two phase flow patterns fluctuation of this series, and the slope is the calculated Slug flow. The fractal scale increases to around 1.8. fractal scale in double-logarithm plot. We can see, in The fractal scale of Slug flow distributes between 1.8 Fig. 7(c), the fractal scale 1.71 is the slope of the fitting and 1.9. With further increase of the gas superficial line. velocity, the fractal scale of Churn flow decreases with the increasing of the superficial velocity of gas, but Figure 8 shows the relationship between the the overall level of the fractal scale is still higher than fractal scale and the superficial velocity when the Bubble flow. Moreover, the fractal scale of bubble flow -1 superficial velocity of water varies from 0.0453ms to is the lowest, and shows fluctuating trend with the -1 0.226 ms and the superficial velocity of gas varies increasing of the superficial velocity of gas phase. This -1 -1 from 0.0043ms to 3.43ms . The results show that: result shows that Bubble flow possesses complicated when the superficial velocity of gas is low, the fractal and random dynamic behaviors. Slug flow with the scales of Bubble flow varies from 1.6 to 1.7. With the highest fractal scale corresponding to the alternatively increasing of the superficial velocity of gas, the flow periodic motion between gas slug and liquid slug, pattern translates into Bubble-Slug transitional flow indicating the dynamic behavior of this flow pattern which is the transitional pattern from Bubble flow to is simple and possesses long-range correlation. If the

263 Lu-Sheng Zhai, Ning-De Jin, Zhong-Ke Gao, Ping Chen and Heng Chi

0.50 12 -1 Churn flow Vsw=0.136ms Vsg=2.93ms-1 8 0.25

) 4 volt k / 0.00 ( V y 0

-0.25 -4 (a) (b) -0.50 -8 0 400 800 1200 1600 2000 0 400 800 1200 1600 2000 t/sec k

7(a) 7(b)

0.5 1.950 α =1.71 0.0 1.885

) 1.820

n -0.5 ( F

10 -1.0 α 1.755 lo g -1.5 1.690

-2.0 1.625 Bubble flow (c) Bubble-slug transitional flow 1.560 Slug flow -2.5 Slug-churn transitional flow 0.4 0.8 1.2 1.6 2.0 Churn flow

log 10 n 0.01 0.1 1 23 -1 Vsg (m.s ) 7(c)

Fig. 7. A fractal scale processing example of churn Fig. 8. Fractal scale versus superficial gas velocity flow using the DFA method; (a) Measured conductance at different flow patterns fluctuating signal; (b)Integrated and detrended time

series of y(k) an (c) Plot of log10 F(n) versus log10 n

superficial velocity of the gas phase still increase, the noise level. In comparison, the anti-noise ability of fractal scale of Churn flow gradually decreases, but DISP and SV analysis is much worse than DFA the value of the fractal scale is still higher than fractal method. PSD, SWV and R/S analysis cannot work scale of Bubble flow. This result indicates that the out the right fractal scale exponent when the noise dynamic behavior of Churn flow is much more exists. The fractal scale calculated from the complicated, but it still has long-rang correlation. conductance fluctuating signals can reflect the flow pattern characteristics of gas-liquid two phase flow. 6. Conclusion The relationship between the fractal scale and the gas superficial velocity can provide valuable reference for The influence of noises on the fractal scale understanding the gas-liquid flow pattern evolution calculated by DFA is the least especially for lower dynamic characteristics.

264 Gas-Liquid Two Phase Flow Pattern Evolution Characteristics Based on Detrended Fluctuation Analysis

Acknowledgement of Oil-water Two Phase Flow Patterns in Vertical Upward Flow Pipes, Journal of This work is supported by National Natural Chemical Industry and Engineering Science Foundation of China (Grant No. 50974095) (in Chinese), 52 (2001) 907-915. and National Science and Technology Major Projects [8] Bai Bofeng, Guo Liejin and Chen Xuejun, (Grant No. 2011ZX05020-006). Pressure Fluctuation for Air-water Two-phase References Flow, Journal of Hydrodynamics (In Chinese), Ser. A, 18 (2003) 476-482. [1] G. Saether, K. Bendilsen, J. Müller and E. [9] Zhao Guibing and Yang Yongrong, Frøland, The Fractal Statistics of Liquid Slug Characteristics of Multi-scale and Multi-fractal Lengths, International Journal of Multiphase of Pressure Signals in a Bubbling Fluidized Bed, Flow, 16 (1990) 1117-1126. Journal of Chemical Engineering of Chinese [2] F. Franca, M. Acikgoz, R.T. Lahey and A. Universities (in Chinese), 17 (2003) 648-655. Clausse , The Use of Fractal Techniques for Flow [10] E.H. Hurst, Methods of Using Long-term Storage Regime Identification, International Journal of in Reservoirs: Proceedings of the Institute of Multiphase Flow, 17 (1991) 545-552. Civil Engineers, Part 1, 5 (1956) 519-591. [3] M.A. Latifi, A. Naderifar and N. Midoux, [11] C.K. Peng, S. Havlin, H.E. Stanley and A.L. Stochastic Analysis of the Local Velocity Goldberger, Quantification of Scaling Gradient in a Trickle-Bed Reactor, Chemical Exponents and Crossover Phenomena in Non- Engineering Science, 49 (1994) 5281-5289. stationary Heartbeat Time Series, Chaos, 5 [4] R. Kozma, H. Kok, M. Sakuma, D.D. Djainal and (1995) 82-87. M. Kitamura, Characterization of Two-phase [12] R. Webster, Quantitative Spatial Analysis of Soil Flows Using Fractal Analysis of Local in the Field, Advance of Soil Science, 3(1985) 1-70. Temperature Fluctuations, International [13] B.B. Mandelbrot and J.W. Van Ness, Fractional Journal of Multiphase Flow, 22 (1996) 953-968 Brownian Motions, Fractional Noises and [5] Y. Cai, M.W. Wambsganss and J.A. Applications, SIAM Review, 10 (1968) 422-437. Jendrzejczyk, Application of Chaos Theory in [14] J.B. Bassingthwaighte, Physiological Identification of Two Phase Flow Patterns and Heterogeneity, Fractals Link Determinism and Transitions in a Small Horizontal, Rectangular, Randomness in Structure and Function, News Channel, ASME Journal of Fluids Engineering, in Physiological Sciences, 3 (1988) 5-10. 118 (1996) 383-39. [15] M.J. Cannon, D.B. Percival, D.C. Caccia, G.M. [6] B.L. Wu, J. Briens and X. Zhu, Multi-scale Flow Raymond and J.B. Bassingthwaighte, Behavior in Gas-solids Two-phase Flow Evaluating Scaled Window Variance Methods Systems, Chemical Engineering Journal, 117 for Estimating the Hurst Coefficient of Time (2006) 187-195. Series, Physica A, 241 (1997) 606-626. [7] Jin Ningde, Ning Yingnan, Wang Weiwei, Liu Xingbin and Tian Shuxiang, Characterization

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