Hongwei Li*,Junpeng Liu,Tao Li,Yunlong Zhou,Bin Sun

School of Energy and Power Engineering,Northeast Dianli University,Jilin 132012,China

Keywords:Small channel two-phase flow Flow pattern dynamics Phase space reconstruction Data reduction sub-frequency band wavelet

ABSTRACT A new method of nonlinear analysis is established by combining phase space reconstruction and data reduction sub-frequency band wavelet.This method is applied to two types of chaotic dynamic systems(Lorenz and R?ssler)to examine the anti-noise ability for complex systems.Results show that the nonlinear dynamic system analysis method resists noise and reveals the internal dynamics of a weak signal from noise pollution.On this basis,the vertical upward gas–liquid two-phase flow in a 2 mm × 0.81 mm small rectangular channel is investigated.The frequency and energy distributions of the main oscillation mode are revealed by analyzing the time–frequency spectra of the pressure signals of different flow patterns.The positive power spectral density of singular-value frequency entropy and the damping ratio are extracted to characterize the evolution of flow patterns and achieve accurate recognition of different vertical upward gas–liquid flow patterns(bubbly flow:100%,slug flow:92%,churn flow:96%,annular flow:100%).The proposed analysis method will enrich the dynamics theory of multi-phase flow in small channel.

1.Introduction

With the development of the micro electro-mechanical system,micro-chemical technology has become one of the development trends in the field of chemical engineering since 1990s[1].The tiny channels caused by scale miniaturization,the specific surface area,and the heat and mass transfer efficiencies relative to the conventional channel have been improved by two to three orders of magnitude.Special experimental research on the characteristics of two-phase flow in a non-circular small channel is necessary to improve the efficiencies of mass and heat transfer.Flow pattern is one of the most important parameters of two-phase flow；it not only affects the performance of the mixed- fluid flow characteristics and heat transfer but also is an important parameter of the two-phase flow control and forecast system.Therefore,the accurate identification of two-phase flow patterns,the understanding of its internal flow characteristics for twophase flow industrial system optimization design,and the dynamic monitoring of working conditions are of practical significance[2].

The two-phase flow model is a complex nonlinear dynamical system.Since 1990s,a trend has been growing in the study of flow pattern identification based on the method of chaos,fractal,complex networks,time and frequency domain analysis,etc.[3].Jin et al.[4–7]performed dynamics analysis on oil and gas–liquid two-phase flow conductance signal by using the multiple chaotic parameter index；this analysis was limited by the center of multiple gravity trajectory in phase space.Progress has been made in revealing the mechanism of two-phase flow patterns.Gao etal.[8,9]applied the complex network on the mechanicalcharacteristics of two-phase flow to evolution analysis,and revealed the detailed mechanism of interaction between gas and liquid.Sun et al.[10,11]provided the identification and analysis for the gas–liquid two-phase flow pattern and flow characteristics of a horizontal Venturi tube by adapting the optimal kernel time-frequency analysis method.Du et al.[12]analyzed the conductance signals of the gas–liquid two-phase flow by using the optimal kernel time-frequency characterization method.They accurately distinguished the different flow patterns by extracting the two time-frequency characteristic values(total energy and entropy).Manfredo[13]characterized flow properties by the phase density timefrequency distribution characteristics of gas–liquid two-phase flow image signal.Ommen et al.[14]reviewed two-phase time-frequency domain analysis methods and analyzed their advantages/disadvantages in the characterization of two-phase flow dynamics characteristics.

Compared with that from regular channels,the information collected from tiny channels is sensitive to the change in experimental environment；and noise reduces more severely the amount of effective information.Therefore,revealing the inherent nonlinear dynamic characteristics of a tiny channel as easily as that in a regular channel seems difficult.In this case,an effective algorithm is required to prevent noise and acquire the essentials from signal.A sub-frequency band wavelet is selected in this study.The use of wavelet transform to analyze and identify model parameters is currently a popular research tool.Ruzzene et al.[15]utilized a band wavelet to analyze the free attenuation vibration response of structures and identify their natural frequency and damping ratio.Sun and Chang[16]proposed a wavelet model parameter identification method based on signal covariance to identify the modal parameters of a system with multi degrees of freedom under environmentalexcitation；they verified the effectiveness of the method experimentally.Min etal.[17,18]combined wavelettransform and singular-value decomposition model identification to effectively increase modal identification precision.Zhang et al.[19]proposed a recognition algorithm that combines data reduction and a sub-frequency band wavelet.The identification of structural model parameters with a wavelet analysis requires considerable computation time to ensure high accuracy.Phase space reconstruction must be applied to any component in the system to enable equivalent state space transformation as well as to apply a weak information sequence and obtain the intrinsic dynamic mechanisms.The system dynamic model can then be reconstructed.To this end,high-dimensional space can be utilized to obtain additional information.However,high-dimensional space entails a large amountof subsequent calculations.Data reduction and sub-frequency band wavelet are combined in this study to reduce this burden.The high-dimensional matrix obtained after phase space reconstruction is regarded as input.The dynamic characteristics of the gas–liquid two-phase flow pattern in a vertical,upward,tiny,rectangular pipe are analyzed.

2.Theoretical Basis

2.1.The phase space reconstruction

Phase space reconstruction was based on limited data to reconstruct an attractor and study the dynamic behavior of a system.The basic idea behind this process is as follows.The evolution of any componentin the system is determined by its interaction with other components；therefore,the related component information is implicit in the development of the component of interest.A component must be analyzed to reconstruct an equivalent state space,and the data measured at some fixed pointoftime delay mustbe regarded as a newdimension,which is identified as a point in multidimensional state space.This process is repeated,and the data at different values of the delay time are measured to produce many of these points,which can preserve many properties of attractors.The observation component of the system can be utilized to reconstruct the system dynamics model.

Forany time series signal z(it),i=1,2…,M(where t is the sampling intervaland M is the totalnumber ofpoints).After selecting appropriate embedding dimension m and delay time t,phase space reconstruction can be immediately conducted.The vector points of the reconstructed phase space can be expressed as

where k=1,2,…,N,N=M-(m-1)× τ/t is the totalnumber ofpoints in the phase space,τ being the delay time.If embedding dimension m is extremely small,a pointcannot be fully developed in the phase space；if m is too large,the kinetics ofthe phase space willbe polluted by noise.If delay time τ is extremely small,the phase space attractor will be compressed in line,not fully developed；if delay time τ is too large,the attractor dynamics will be split and no longer be continuous in the phase space.Therefore,embedding dimension m and time delay τ must be selected properly with a proper algorithm.The embedding dimension can be obtained with the FNN algorithm,whose advantage was discussed in Ref.[20].The C–C algorithm is commonly utilized to obtain delay time τ(refer to[21]for details).Compared with other delay time calculation methods(such as mutual information method-I,mutualinformation method-II,and autocorrelation function method),the C–C algorithm has a better capability to resist noise.Therefore,the C–C algorithm was employed in this study to compute delay time τ.Embedding dimension m was obtained with the FNN algorithm.

2.2.Determination of order time and frequency band

The SFBW method can determine the modal order Nmof the system and the approximate frequency range of each mode by decomposing the positive power spectrum density(PPSD)matrix.The lr covariance signal is reduced to an Nmsignal by the singular value decomposition(SVD)of the matrix.Finally,signal reduced by the wavelet analysis was used for the by-band modal parameter identification.The foremost advantage of the method is the ability to integrate multi-channel information for accurate positioning frequencies,damping ratios and major modes.The SFBW is very effective in signal identification.In this work,this method is adopted to analyze and recognize the differential pressurefluctuation signal of the two-phase flow.

Bart et al.[22]defined the positive cross-power spectral density Gij+as follows:

Positive power spectral density matrix is constituted as follows:

where l is the number of measuring points,and r is the number of reference points.

By using SVD for the PPSD singular value in each discrete frequency ω,the correlation can be obtained:

with

The vector B is constituted by the first singularvalue of each discrete frequency point as follows:

where Nωis number of frequency domain points.

The vector B exist a local peak in each order modal frequency point of the system.However,because of the existing problem of frequency resolution,the frequency range can only be determined approximately in each mode as follows:

2.3.Use of SVD to reduce covariance data

The covariance and cross covariance was calculated for l response signal.The r reference channel is selected,and the lr covariance signal is obtained.The covariance matrix is defined as follows:

SVD of R can then be obtained as follows:

where U=[u1u2…ulr], S=diag（ σ1σ2… σlr）, V=[v1v2…vN].

Eq.(8)is obtained from Eq.(7):

2.4.Break wavelet parameter identification

The wavelet coefficient matrix is obtained by the wavelet transform for Rvofeach row vectorwithin the range ofeach ordermode frequency:

where p is the number ofscales,N is the number oftime domain points,i is the number of signals,and j is the order.

Nmof the wavelet coefficient matrix obtains the regroup as follows:

where K is the current scale.

By using SVD on Ajk,the formula is expressed as follows:

with

The vector J is constructed by the first singular value σjk1of Ajk:

According to the maximum value in J that corresponds to scale ajm,the J order frequency,damping ratio,and virtual vibration mode can be obtained as follows:

The matrix Umand virtualvibration modesare reduced in Eq.(8).φjis defined as follows:

The first n rows of φjis the vibration type of the original data:

The observations of the algorithm process indicate that the covariance signal participating in the wavelet analysis changes from lr to Nmby SVD data reduction.lr is greater than Nm.Therefore,the proposed algorithm requires less data analysis than the originalalgorithm.The PPSD SVDdecomposes by responding to the data determined by the frequency range of order.The PPSD SVD avoids useless band calculations,and the algorithm computation load is further reduced.This algorithm mainly involves the numerical calculation of the SVD decomposition of the matrix and 1D continuous wavelet transform(1D convolution),but does not involve iteration.Hence,this algorithm has good numerical stability.

3.Evaluation Methods

Noise resistance was regarded as an evaluation index in this study to evaluate the method of dealing with a time series with strong noise pollution.The typical Lorenz and R?ssler chaotic time series were employed as examples for this purpose.The reason for selecting a chaotic sequence as the simulation signal in this study is that the signals collected from experiments usually have typical chaotic characteristics.Another reason is that chaotic systems exhibit typical dynamic behavior,and is both typical and classic.

3.1.Construction of chaotic time series

The Lorenz and R?ssler systems with different contents of noise composed of several groups of time series were produced under the following conditions.

(1)Lorenz chaotic signal by the Lorenz equation:

with the initial condition x=2,y=2,z=20,the iteration method is fourth-order Runge–Kutta,and the variable X taken as the simulation sequence.

(2)Lorenz+noise:Variable X is mixed with the SNR of 5 dB Gaussian white noise X*=awgn(X,5)；

(3)Lorenz+noise:Variable X*is directly mixed with the noise intensity of 15 dB Gaussian white noise X** =awgn(X,5)+wgn(length(X),15)；

(4)R?ssler system:

with the initial condition x=0.15,y=0.2,z=10,and the variable X taken as the simulation sequence.

(5)R?ssler+noise:The variable X is mixed with the SNR of 5 dB Gaussian white noise X*=awgn(X,5)；

(6)R?ssler+noise:The variable X*is directly mixed with the noise intensity of 15 dB Gaussian white noise X**=awgn(X,5)+wgn(length(X),15).

Fig.1 shows the two typical chaotic systems after phase space reconstruction.Two systems are typical chaotic dynamic system.

Fig.1.The attractors of two dynamics systems after phase space reconstruction.

3.2.Noise resistance analysis

Figs.2 and 3 show the positive power spectral density(PPSD)singular-value frequency figures and related noise signals of the two typicalchaotic systems.Signals I,IIand III in Fig.2 correspond to the signals(1),(2)and(3)in Section 3.1 respectively.Signals I,II and III in Fig.3 correspond to the signals(4),(5)and(6)in Section 3.1 respectively.The PPSD trends of the two systems exhibit no significant changes before and after noise addition；the effect of noise seems minimal.The result shows that the proposed method has strong noise resisting ability.

Fig.2.PPSDsingular value-frequency ofLorenz system afterphase space reconstruction(reconstruction parameters:τ=2Δt,m=4,N=3000,Δt is the time intervalbetween consecutive two points of time series).

Fig.3.PPSD singular value-frequency of R?ssler system after phase space reconstruction(reconstruction parameters:τ =2Δt,m=4,N=3000).

Figs.4 and 5 show the time–frequency spectrum and singular-value frequency charts calculated with the data obtained by the algorithm proposed in this study.Signals(a),(b)and(c)in Fig.4 correspond to signals(1),(2)and(3)in Section 3.1 respectively.Signals(a),(b)and(c)in Fig.5 correspond to the signals(4),(5)and(6)in Section 3.1 respectively.With differentamounts ofnoise pollution,the time spectrum and trend diagram of the two chaotic dynamic systems exhibitdifferent degrees of disturbance.Although the perturbation amplitude is not too large,the disturbance effect is obvious after adding noise disturbance,particularly in the R?ssler system.

The signal coupling oscillation frequency and damping ratio were extracted to evaluate quantitatively the advantages of the method in terms of noise suppression ability.The characteristics of the two chaotic systems are shown in Fig.6.No significant difference was observed in the characteristics of the two systems with different degrees of noise disturbance；thus,the method is proven to be suitable for chaotic dynamic systems with noise in fluence.

4.Flow Pattern Identification and Dynamic Mechanism Analysis

4.1.Experimental arrangements

The schematic ofthe testfacility was shown in Fig.7.The system consists ofgas–liquid two-phase flow,control,and data acquisition systems.

In the experiment,the nitrogen stream from a nitrogen cylinder was divided into two parts by a three-way valve.The first part flows into the experimental section through the flow meter and control valve；the other part enters the water tank to increase tank pressure to drive water flow into the experimentalsection via the flow meter and control valve.The tank pressure was maintained at 0.2 MPa by adjusting the control valve.Water and gas phases were fully mixed in the mixing chamber for the sake of two-phase flow stability.After passing the test sections,the two-phase mixtureflows into a separator.

The test section is made of translucent Plexiglas.Two pressuremeasuring holes with 0.5 mm diameters were located at a distance of 200 mm.A porous plate with 40 μm pore size was installed at the test section inlet to introduce the bubbles uniformly to the test section.The distance between the upstream hole and entrance was 100 mm,and the distance between the downstream hole and outlet was 80 mm.Rectangular channel width was 2 mm,and the height was 0.81 mm.The test section consisted of the gas–liquid two-phase mixing chamber,experimental channel,and gas–liquid two-phase separation chamber was shown in Fig.8.

Experimental data were acquired on single-phase pressure signal,single-phase liquid flow,and gas flow tests.Rosemount 3051C differential pressure transmitter(Rosemount,USA)with±0.075%accuracy,100 ms time update response time,0.124–13790 kPa pressure calibration ranges,and 256 Hz signalsampling frequency wasused to measure the differential pressure signal.

Fig.4.Typical data reduction sub-frequency band wavelet(DA SFBW)for the Lorenz systems.

Fig.5.Typical DA SFBW for R?ssler systems.

Fig.6.The frequency and damping ratio of two typical dynamic system and their disturbed signals.

Fig.7.Experimental system of nitrogen–water two-phase flow.

Fig.8.Test sections schematic diagram.

HW5 intelligent metal tube rotameter(Jiangsu Runda Co.,China)was used to measure liquid flow.This flow meter has a microprocessor core,and the accuracy of instantaneous and accumulated values is±0.5%.D-600MD digital gas flow meter with 0 L·min-1to 5 L·min-1range and±1%accuracy was used for gas flow measurement.

The experiment was performed under(25 ± 0.5)°C ambient temperature,0.1 m·s-1to 30 m·s-1super ficial gas velocity,and 0.01 m·s-1to 5 m·s-1super ficial liquid velocity.Agilent 34970A data acquisition device,which had resolution of 22 bits,basic direct current voltage accuracy of 0.004%and an extremely low readout noise,the scanning rate up to 250 channel per second,is applied to data acquisition.

4.2.Dynamic mechanism analysis

Fig.9.Typical nitrogen–water two-phase flow pattern in rectangular channel.

The different signals of the four typical flow patterns were collected and are shown in Fig.7.The pictures of four typical flow patterns are shown in Fig.9 with jGand jLstanding for the super ficial velocity of gas and liquid.For the same flow pattern,the different flow conditions would lead to different pressure oscillation modes.However,their internaldynamics are similar.The reconstruction attractors and the corresponding original signals are shown in Fig.9.Based on the structure of the attractor diagram,the dynamics of gas–liquid two-phase flow are analyzed as follows.Air bubbles with different small sizes formed in the bubbly flow and dispensed randomly in the upward liquid phase.No obvious bubble coalescence was observed.Bubbles scattered around the attractor,and the unstable periodic orbit(UPO)of the attracting feature was unobvious.In the slug flow,the bubbles coalesced and formed a large air plug with a diameter close to the inner diameter of the pipe.The air plugs have the stable shape and have a certain regular interval.Bubble coalescence is an important mechanism of gas–liquid two-phase flow structure evolution and plays an important role in structural dynamics.This phenomenon also reflects the attracting features of the corresponding signal in the phase space UPO.An obvious cluster was formed in the phase space attractor,and regular-shaped periodic orbits,which can generate the stable periodic rules of slug flow,occurred around the cluster.The air plug size had a weak impact on the phase space reconstruction attractors,mainly depended on the plug frequency.In the churn flow pattern,the large bubble slug collided,smashed,deformed,and then mixed with liquid as a turbulent mixture with unstable up and down rolling as the bubble velocity increased.Owing to the minimal bubble coalescence and the existence of large bubbles in the churn flow pattern,the chaotic characteristic was made obvious；however,periodicity and cluster formation were unobvious.In the annular flow,gas phase existed in the pipeline in the form of a gas column,while liquid attached to the inner wall of the pipe that constantly shook with the high flow speed of the gas column.No obvious mechanisms of bubble coalescence and smash were observed.The attractor winding effect appeared to be long,and no significant cluster was formed.After phase space reconstruction,the attractor can not only reflect the attracting features of the internal system UPOs but can also reveal the mechanisms of bubble coalescence and smash in the gas–liquid two-phase flow(Fig.10).

Fig.10.The differential pressure signals and the phase space reconstruction attractors of typical two-phase flow patterns(reconstruction parameters:τ=2Δt,m=4,N=256).

Fig.11.PPSD singular value-frequency of different flow conditions.

Fig.11 presents the PPSD singular-value frequency trend chart under different flow conditions.Different flow patterns are presented as the super ficial velocity of the two phases changes.The intrinsic dynamic mechanismof two-phase flow pattern formation can be determined based on the change in the position of the main frequency.The arrangementin Fig.11(a)to(l)correspondsto dispersed bubbly,bubbly slug transition,slug,slug-churn transition,churn,churn-annular transition,and annular flows.The peak frequency was initially set at very low frequency,and gradually moved to 1.75 Hz.After lingering at this point,the peak begins to move to the left direction,diverges,and finally forms a peak with a scattered band.The bubble flow[Fig.11(a)to(c)]is featured with small bubbles unevenly dispersed in the liquid phase and has no obvious period.The peak position is partially located on the low frequency side without a stable oscillation frequency.The bubbleslug transition flow[Fig.11(d)and(e)]begins to form a stable main frequency with increasing amplitude in the middle frequency.In the slug flow[Fig.11(f)and(g)],the peak value in the middle frequency becomes gradually stable and the small peak on the low frequency side disappears completely.With the increase in gas super ficialvelocity,the gas slug breaks down and the frequency value begins to abate and then move to the low frequency side.This flow pattern is called slugchurn transition flow[Fig.11(h)].After the frequency peaks disappear,the churn flow pattern[Fig.11(i)and(k)]is formed；however,the main peak value of the churn flow fluctuation is insignificant.Intense oscillation occurs as the gas-phase velocity increases when the liquid membrane is driven by high-speed flow.Fig.11(l)shows a scattered distribution,which is called annular flow.

The entireflow transition process can be further validated in Fig.12 The gradual flow pattern changes is accompanied by the change in frequency position and decided by the flow conditions.Two main frequencies were observed in the slug-churn transition flow[Fig.12(e)]；however,they are unobvious and develop gradually.Fig.12(f)shows a stable churn flow pattern.A gas core begins to form and causes a chaos jitter in the liquid membrane at high gas super ficial velocity[Fig.12(h)and(i)].

4.3.Flow pattern identification

To investigate quantitatively the DASFBWofdifferent flow patterns,we define two characteristic parameters,i.e.,singular value-frequency entropy(S-F entropy)and damping ratio.

The uncertainty ofa systemcan be characterized in terms ofentropy.We divide the PS-F plane(Eq.(5))into N parts of equal area.The time–frequency entropy can be defined as

where Piis the energy of each part,and E is the total energy of the PS-F plane.

The energy of a matrix can be defined as

where pi,jis the element of the matrix.

A higher uncertainty in the gas bubble distribution in the pipe leads to bigger time–frequency entropy.The bubble,slug,churn,and annular flows appear sequentially at the same liquid super ficial velocity with increasing gas super ficial velocity.From the entropy value distribution of different flow pattern view(Fig.13(a)),the bubbly flow shows high entropy,indicating strong random bubble distribution in this flow pattern.However,the slug flow entropy value is always low,indicating that the bubble distribution regularity is obvious and the flow has good periodicity.Churn flow and annular flow also display big differences in entropy distribution.The PS-F entropy of annular flow is also low.From a macro point of view,the gas core occupying the center of the pipe is continuous,while the liquid film which attached to the wall does not stir much the gas flow.

The size of the damping ratio depends on the anti-jamming capability ofthe signalitself.Asmallerdamping ratio demonstrated the weaker anti-interference ability of signals；thus,small perturbations will lead to considerable changes.The slug and annular flows have the smallest and largest damping ratios,respectively(Fig.13b).Annular flow has the strongestability to maintain the status quo ante.The flow pattern ofannular flow is difficult to change with varying gas and liquid phases.This phenomenon can be seen from the relatively broad variation ranges of gassuper ficialvelocity in the annular flow.Given thata gas slug iseasily broken by high-speed gas flow in the slug flow,the dispersed small bubbles of the bubble flow can easily be thawed by gradually increasing the gas flow and merging the gas slug into slightly larger gas bubbles or gas plugs.Thus,both damping ratios were relatively small.

Fig.12.Time–frequency spectrum and singularvalue vs.frequency diagram ofdifferent flow conditions:(a)j L=0.896 m·s-1,j G=0.442 m·s-1；(b)j L=0.583 m·s-1,j G=0.512 m·s-1；(c)j L=0.525 m·s-1,j G=0.579 m·s-1；(d)j L=0.504 m·s-1,j G=0.645 m·s-1；(e)j L=0.459 m·s-1,j G=0.711 m·s-1；(f)j L=0.765 m·s-1,j G=2.034 m·s-1；(g)j L=0.0.911 m·s-1,j G=2.833 m·s-1；(h)j L=0.668 m·s-1,j G=3.505 m·s-1；(i)j L=0.479 m·s-1,j G=4.973 m·s-1.

We divide the plane of two parameters into four zones to identify different flow patterns(Fig.14).The damping ratio and PS-F entropy plane have a good classification performance when the four flow patterns are clearly distinguished.This result is caused by the combination ofadvantages ofdamping ratio and PS-F entropy.The sample identification results are as follows:for all 35 data points,the recognition rate of bubbly and annular flow up to 100%,churn flow is 96%and slug flow is 92%.

The phase space reconstruction combined with the data reduction sub-frequency band wavelet method has been applied to a small circularchannel(diameteris 1.15 mm,length is 380 mm)and also has a good classification performance for flow patterns.This method has not been applied to the two-phase flow of regular size pipe.The applicability is needed to verify.

Fig.13.Characteristic value distribution of nitrogen gas–water flow in the transformation process:(a)energy entropy；(b)damping ratio.

5.Conclusions

Phase space reconstruction combined with DA SFBW analysis was developed in this study.This method can effectively block noise interference,completely retain the effective original signal information,and reveal the inherent dynamic characteristics of signals heavily polluted by noise.The method is based on the time-frequency spectrum versus power spectral density map.It was applied to two chaotic dynamical systems(Lorenz and R?ssler)to verify its applicability.The results indicate that the method is effective in analyzing complex nonlinear dynamic systems and exhibits good anti-noise performance.

The internal characteristics of flow patterns and the transition patterns among them can be determined by applying the proposed method to the pressure signals of small nitrogen–water flow channels.For further quantitative analysis,the DA SWBF of PS-F entropies and damping ratio are extracted to analysis dynamic characteristics and distinguish flow pattern.The results indicate thatthe PS-F vs.damping ratio entropy map are efficient at distinguishing two-phase flow patterns,and the recognition rate is better than 92%,even up to 100%for bubbly and annular flows.

Fig.14.Flow pattern identification based on PS-F entropy and damping ratio.