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9. Lu-Sheng Zhai.P65 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.
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