Nonlinear analysis of characteristic frequencies and chaotic behavior of flow patterns during flow boiling of hydrocarbons in a small channel

International Journal of Multiphase Flow 114 (2019) 240–257

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International Journal of Multiphase Flow journal homepage: www.elsevier.com/locate/ijmulflow

Nonlinear analysis of characteristic frequencies and chaotic behavior of flow patterns during flow boiling of hydrocarbons in a small channel Jeferson Diehl de Oliveira a,∗, Jacqueline Biancon Copetti b, Maria Luiza Sperb Indrusiak a,b,c, Júlio César Passos c a

Department of Mechanical Engineering Graduate Program, Center of Innovation and Technology University Center of Serra Gaúcha, Os Dezoito do Forte St., 2366, Caxias do Sul, 95020-472, RS, Brazil Mechanical Engineering Graduate Program, LETEF, Laboratory of Thermal and Fluid Dynamic Studies, University of Vale do Rio dos Sinos, 93022-750, São Leopoldo, RS, Brazil c Department of Mechanical Engineering, LEPTEN, Laboratory of Process of Engineering and Energy Technology, Federal University of Santa Catarina, 88010-900, Florianópolis, SC, Brazil b

a r t i c l e

i n f o

Article history: Received 21 November 2018 Revised 19 March 2019 Accepted 19 March 2019 Available online 20 March 2019 Keywords: Flow patterns Flow boiling Hydrocarbons Frequency analysis Chaos

a b s t r a c t The dynamics of time series obtained from an optical sensor for analysis of flow patterns has been investigated during flow boiling of hydrocarbons in a horizontal tube with 1 mm inner diameter. The tests were performed with the refrigerants R-600a, R-290 and R-1270 at 17 °C saturation temperature, with mass flux ranging from 240 to 480 kg/(m²s), considering heat fluxes between 5 and 60 kW/m². Five flow patterns were identified, and their characteristic frequencies were investigated and related to dynamics of the liquid-vapor morphology over time. Results also show different long memory trends based on time series and nonlinear chaotic behavior has been observed. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Many experimental studies have been carried out to investigate characteristics of flow patterns due to their important influence on performance of steam generators, evaporators and condensers. For this reason, the use of nonlinear methods has an important role to determine the dynamics of flow patterns and their main characteristics. Investigations involving several geometries, heat and mass flux condition and flow direction have been reported in literature and many authors have focused their efforts on recognizing the nature of flow patterns through the use of sensors and the analysis of behavior of signals as well as through image analysis. Song et al. (1995) investigated the bubble-to-slug flow regime transition based on instabilities of the void fraction waves during air-water flow in vertical upwards. In their study, they used an impedance void meter to analyze characteristic and statistical properties such as void fraction, fluctuation of the signal, frequencies and power-spectral density function (PSDF) based on fast Fourier transform (FFT). As a result, they observed different predominant frequencies in bubble clusters and slug flow. They also

∗

Corresponding author. E-mail address: [email protected] (J.D. de Oliveira).

https://doi.org/10.1016/j.ijmultiphaseflow.2019.03.015 0301-9322/© 2019 Elsevier Ltd. All rights reserved.

concluded that intermittencies of the void fraction depend on size of bubble and on superficial velocities of the phases. A timefrequency analysis was used by Klein et al. (2004) to investigate air-water intermittent flow regimes in a horizontal test section of 60 mm inner diameter. The authors measured the void fraction using a conductivity probe and analyzed characteristic frequencies using Gabor transform, a particular short Fourier transform (SFT) method. The results indicated that the successive passage of tighter and looser trains of air plugs, characterized by intermittency frequency, was related to the transition between two different types of intermittent flow. Characteristic frequencies of flow pattern during flow boiling of R134a in horizontal tube of 0.5 mm inner diameter were analyzed by Revellin et al. (2006). The authors used a configuration of two laser beans and photodiodes and investigated the frequencies using FFT. As a result, the authors observed regime changes based on frequency analysis and concluded that the elongated bubble coalescence is proportional to the increases of mass flux. Nguyen et al. (2010) proposed a study on frequency analysis for two-phase flow pattern using continuous wavelet transform (CWT) technique in a vertical air-water flow with 80 mm inner diameter. The authors used a multi-channel impedance void meter to measure both void fraction and characteristic frequencies. It was found a correlation between the amplitude of the signal

J.D. de Oliveira, J.B. Copetti and M.L.S. Indrusiak et al. / International Journal of Multiphase Flow 114 (2019) 240–257

and the maximum local wavelet energy coefficient, which results in a high precision method to characterize different flow regimes. A piezoelectric force sensor was used by Sim et al. (2010) to investigate the void fraction and flow patterns in a vertical air-water flow with a test section of 30 mm inner diameter. The predominant frequencies based on the PSDF was explored. According to the authors, the intensity of PSDF increases with increasing of superficial velocity. The results also demonstrated that the increase of superficial velocity increases the range of frequencies. A more complex methodology to relate predominant frequencies to flow patterns has been developed by Hanafizadeh et al. (2015) in vertical air-water two-phase flow. The authors studied characteristic frequencies of pressures measured throughout the test section and implemented information such as PSDF maximum and location of PDF in an artificial neural network to predict the flow pattern. The comparison with pictures of flow patterns indicated that their method has considerable accuracy. More recently, Zhai et al. (2017), performed tests in adiabatic condition with oil-water in a horizontal tube of 20 mm inner diameter, using a ring conductance array probe, a conductance cross-correlation velocity probe and mini-conductance probes. The authors investigated the signals of such probes, considering the influence of mean velocities of both phases. It was observed multi-scale characteristics of flow patterns, such as fluctuations of interfaces and movement of dispersed drops. Many other studies focused on the investigation of flow patterns, void fraction analysis and characteristic frequencies have been reported in literature (Costigan and Whalley (1997); Wang and Shoji (2002); Morsi et al. (2011); Monni et al. (2014); Saidj et al. (2014); Zeghloul et al. (2015); Dong et al. (2015)). Table 1 also presents some other works, including different methodologies applied in the investigations. To the best of the authors’ knowledge, there is a considerable lack of nonlinear analysis investigation on the nature of behavior intrinsic to flow patterns during flow boiling in small channels. The main contribution of the present study is to analyze experimental time series obtained from liquid-vapor interface with aid of an optical sensor. The flow patterns are identified using a highspeed camera. In turn, nonlinear analysis is implemented to investigate characteristic frequencies, persistence and chaotic dynamics of flow patterns according to the influence of parameters such as heat and mass fluxes.

241

pump, and also to assure that only subcooled liquid enters the flow meter. The pre-heater establishes the experimental conditions entering the test section immediately downstream. It consists of a horizontal stainless-steel tube with a 1.0 mm internal diameter, total length of 515 mm and heated length of 440 mm. This tube is uniformly heated by direct application of an electrical current in the wall (Joule effect), the intensity of which is controlled by the power supplied by a SORENSEN model DCS 8-125E, which provides both direct voltage and direct current measurements. The TS consists of the same tube and diameter with total length of 366 mm, heated length of 265 mm. As well as the PH, the TS is heated by Joule effect, the intensity of which is controlled by the power supply. The absolute internal roughness (Ra) of the tube was measured with a STARRETTTM roughness tester, model SR 200, and was found to be 1.48 μm. There is a visualization section downstream of the test section with a 136 mm length glass tube with the same test section internal diameter. Both PH and TS are thermally insulated. More details about the experimental apparatus can be seen in Oliveira et al. (2016). To analyze the characteristic frequencies of flow patterns, an optical sensor was built based on refraction and bright variation of light caused by vapor-liquid interface of two-phase flow (Oliveira, 2017). Such sensor is located on the beginning of visualization section to guarantee the same conditions obtained on the test section outlet, as presented in Fig. 2. The optical sensor is constituted, basically, by a high-brightness light emitting diode (LED), a light dependent resistor (LDR) and an electronic circuit responsible for converting electric resistance of LDR to resulting signal of electric voltage. The bright variation caused by the interface of two-phase flow changes the electric resistance of LDR. Such variation is transformed into a filtered-voltage signal through an electronic circuit formed by a low-pass filter of 4th order and an operational amplifier OPA 4227. In turn, the voltage signal is read by a data logger model NI 6009 managed by Labview 2013. Fig. 2 presents the scheme of sensor + filtering + acquisition system. 3. Methodology 3.1. On vapor quality and void fraction The heat flux applied to the test section was given by,

2. Experimental setup An experimental facility was developed to investigate the flow boiling, pressure drop and flow patterns in a horizontal small channel. The details of such set up are shown schematically in Fig. 1. The experimental system consists of a loop that provides controlled mass flux, and it was designed to test different fluids under a wide range of flow conditions. The main part of the loop has a pre-heater, a test section and a visualization section, represented by PH, TS and VS, respectively. The secondary part consists of a condenser, a refrigerant reservoir, a liquid refrigerant vessel, a dryer filter, a magnetic gear pump and a subcooler. The condenser and the subcooler have independent circuits and each one uses an ethylene-glycol/water solution as the secondary refrigerant, the temperature of which is controlled by a thermostatic bath. This set up controls the refrigerant saturation temperature. The refrigerant reservoir connected to the main circuit of the bench has small volume and is partially filled with refrigerant to assist in control of pressure fluctuations, which occur when there is a variation in heat flux conditions. A liquid refrigerant vessel maintains a constant static pressure at the pump suction. This procedure assures that the pump works uniformly and under immersion, avoiding cavitation. The subcooler is used to compensate the temperature rise, which usually occurs as the refrigerant passes the gear

qT S =

qT S ≡ Aheated

U ·I

π Di Lheated

(1)

where Di is the internal diameter, Lheated is the heated length and U and I represent the voltage and current applied by power supply in test section, respectively. The vapor quality in the test section inlet was calculated from energy balance in the pre-heater, according to Eq. (2).

 qPH m˙

x(i−T S) =



+ ii−PH − il il v

(2)

where il and ilv represent the liquid enthalpy and latent heat of vaporization as function of saturation pressure. In this case, the subscripts i-TS and i-PH represent inlet of test section and inlet of pre-heater, respectively. Considering the local enthalpy as a function of location z in test section, given by

i (z ) =

q

T S (z )

m˙

+ ii−T S



(3)

Consequently, the local vapor quality along the test section was calculated according to

x (z ) =

i ( z ) − il il v

(4)

242

Author

Hydraulic diameter (mm)

Working fluid

Parameter analyzed

Sensor

Method applied

Angeli and Hewitt (20 0 0)

24.3C, H

Oil – water

Void fraction

van Der Welle (1985)

–

Butrin and Tadrist (2004) Jana et al., (2006)

0.889R, V 25.4C, V

n-pentane Dyed kerosene – water

Inlet and outlet pressures Phase fraction

FFT Wavelet analysis

Flow boiling tests –

Wang et al., (2007)

0.186T, H

Water

Analysis of fluctuations

Mahvash and Ross (2008)

19C, V

Wall temperature, inlet and outlet pressure and temperature oscillations Void fraction

High frequency impedance probe Pressure transducer Parallel wire conductivity probe Pressure transducer and type-T thermocouple

Flow boiling in a test section of eight parallel microchannels –

Air – water C, H

Optical fiber probe

Schubring and Shedd (2009)

(8.8 and 15.1)

Jaworek and Kupra (2010)

(1, 2, 4 and 9.5)C, V

Isopropanol – water

Schembri and Bucolo (2011) Al-Wahaibi et al., (2012)

0.76R, H (19 and 25.4)C, H

Air – water Oil – water

Du et al., (2012)

120C, V

Air – water

Kanizawa and Ribatski (2012) Sun et al., (2012) Li et al., (2014) Oliveira et al., (2015)

15.9C, H 50 1.15R, V 50.8C, H

R-134a Air – water Nitrogen – water Air – water

Talley et al. (2015) Tan et al., (2015)

3.81C, H 50C, H

Air – water Oil – water

Void fraction Wave amplitude and superficial velocities Void fraction and superficial velocities Pressure drop Void fraction Differential pressure Void fraction and superficial velocities Void fraction Water fraction

Zhai et al., (2015)

20C, H

Oil – water

Phase fractions

Conductivity probe Capacitance and conductivity sensor Radial conductance probe

Rysak et al., (2016)

5C, H, V

Air – water

Void fraction

Video and laser probe

Air – water

C: circular; H: horizontal; R: rectangular; T: trapezoidal; V: vertical.

Superficial velocities and film thickness Phase shift and void fraction

LED/phototransistor probe Capacitance probe Photodiode probe High speed video camera Conductance probe Pressure transducer Pressure transducer Pressure transducer Infrared photo gate

Continuous hidden Markov model FFT Analysis of electric permittivity FFT and PDF gamma function Image treatment

Remarks

– – – –

Wigner-Ville and Choi-Williams distributions FFT Wavelet analysis Wavelet analysis Image analysis

– Flow boiling tests – – –

Data treatment Wavelet analysis

– –

Adaptive optimal Kernel time-frequency Lyapunov exponent and PDF

– –

J.D. de Oliveira, J.B. Copetti and M.L.S. Indrusiak et al. / International Journal of Multiphase Flow 114 (2019) 240–257

Table 1 Literature review of flow pattern studies.

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243

Fig. 1. Schematic of experimental setup.

Fig. 2. Schematic diagram of optical sensor: (1) independent DC power supply; (2) high bright LED; (3) LDR; (4) 4th order low-pass filter; (5) high frequency data acquisition system.

The void fraction was evaluated with the relation developed by Rouhani and Axelsson (1970) given by Eq. (5).

α=



x  1−x + ρv ρv ρl  0.25 −1 1.18(1 − x )[gσ (ρl − ρv )] + G ρl 0 . 5 x

[1 + 0.12(1 − x )]

(5)

where ρ l , ρ v , g, σ and G represent liquid density, vapor density, gravitational acceleration, surface tension and mass flux, respectively. The Rouhani and Axelsson model was developed for vertical flow, it takes important parameters into account that are related to two-phase flow in mini- and microchannels, based on liquid mass density ρ l, vapor mass density ρ v, gravitational acceleration g and surface tension coefficient σ . In addition, flow boiling in horizontal mini and micro channels have practically the same radial symmetry in interfacial topology that is found in vertical flow due to the dominance of capillarity. Moreover, this model has successfully been applied to horizontal flow boiling for hydrocarbons by Del Col et al. (2014). The visualization study done by Oliveira et al. (2016) and Oliveira et al. (2017) proves the hypothesis of radial symmetry. Thermodynamics and transport properties used in data reduction and pressure drop correlations were obtained from REFPROP v. 9.1 (Lemmon et al., 2013). Ex-

Fig. 3. Evolution of a small circle to an ellipsoid based on Lyapunov exponents.

perimental parameters related to the tests in this work, including reduced pressure PR , are presented in Table 2.

3.2. On frequency analysis The fast Fourier transform (FFT) was used to convert the timedomain signal from the optical sensor (X (t)) to a frequency-

244

J.D. de Oliveira, J.B. Copetti and M.L.S. Indrusiak et al. / International Journal of Multiphase Flow 114 (2019) 240–257 Table 2 Experimental conditions. Parameter Working fluid [-] G [kg/(m2 s)] q”TS [kW/m2 ] Tsat [°C] Psat [kPa] (PR ) x( i-TS ) [-] Sample frequency of OS [kHz] Record length [n of samples]∗ ∗

R-600a, R-290 and R-1270 240, 320, 400 and 480 5, 10, 20, 40 and 60 17 275.4 (0.0757), 772.4 (0.1819) and 942.5 (0.202) 0.01–0.03 5 50,0 0 0

For each test.

Fig. 4. Characteristics obtained from sub-cooled liquid conditions of R-1270 for a time range of 1 s. (a) Signal of sensor (Vmax = 4,26 V). (b) PSD versus intrinsic frequencies of white noise (more intense fluctuations of the signal – circulated in red).

Fig. 5. Characteristics obtained in conditions of saturated vapor of R-1270 for a time range of 1 s. (a) Signal of sensor (Vmin = 0,0 V). (b) PSD versus intrinsic frequencies of white noise.

domain signal (X (f)) and it is defined by

X ( f ) = F {X (t )} =

N−1 1  X (t ) e− j2π f t t N

(6)

t=0

where N represents the total time of the analyzed signal, t is the interval between two consecutive data acquisitions, i.e., (ti+ 1 − ti ). The energy spectral density (ESD) which determinates the energy distribution of the signal from optical sensor is given by

ESD =

N−1  t=0

X |(t )|2 t

(7)

Since the power spectral density (PSD) establishes the time rate of the energy distributed along the signal, and is defined according to

P SD =

N−1 1 |X (t )|2 t T

(8)

t=0

Power spectrum (or Fourier spectrum) can be estimated through finite Fourier transform of the original signal. A usual procedure used to enhance quality of spectrum of a stationary signal is to estimate the spectral values for each of a collection of sample records, acquired sequentially and average the results. A time window (e.g. Hanning) is applied to each sample record to taper

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245

Fig. 6. Variation of vapor quality and void fraction in outlet of test section observed for the flow patterns, considering dimensionless number β for: R-600a ((a), (b)), R-290 ((c), (d)) and R-1270 ((e), (f)).

the data to provide a more gradual entrance and exit and avoid anomalies in the estimated spectrum (Bendat and Piersol, 1993). A smoothing can also be done in frequency by averaging over a proper spectral bandwidth. Welch (1967) established a method to obtain the power spectra using the fast Fourier transform and conveniently sectioning the original signal into several shorter ones, with superposition and a time window, which is the basis of the “welch” algorithm available in Matlab and used in this work. 3.3. On reconstruction of attractors The time evolution of a dynamic system can be defined in a phase space. In turn, a phase space indicates the behavior of the

system and existence of attractors presented during the evolution of time. Specially, in chaotic systems, strange attractors are observed in space phases. Considering a dynamic system obtained from time series X(t) = {χ 1, χ 2, χ 3, . . ., χ n }, we can construct the matrix trajectory by time delay coordinates method.

⎡ ⎤ χ1 χ2 ... χd ... χd+1 ⎥ ⎢χ2 χ3 X =⎢ . . .. .. ⎥ ⎣ .. .. ⎦ . . χm χm+1 . . . . χn

(9)

In this case, d represents the embedding dimension and m = n – d + 1 corresponds to the number of dots in d-dimension

246

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Fig. 7. Characteristic of plug flow with presence of dispersed bubbles with diameter smaller than 0.5 mm observed with R-600a. Red arrow indicates the direction of flow.

reconstruction using the method developed by Lei et al. (2002), considering the presence of intrinsic noise in the signal and using symplectic geometry.

The Hurst exponent H was used to classify time series into three different clusters, known as antipersistent, uncorrected and persistent time series:

•

•

0 < H < 0.5 can indicate a time series with long-term switching between low and high values in pairs. It means that a single low value will probably be followed by a high value and viceversa along the time into the future. H = 0.5 indicates that the time series has a similar behavior of a random walk. That is, it indicates that the time series is completely uncorrelated. 0.5 < H < 1 indicates that the time series has a long-term positive autocorrelation. It means, for example, that a low value in the time series will probably be followed by another low value and vice-versa.

In this paper, the Hurst exponent was estimated by the classical (R/S)n method developed by Mandelbrot and Wallis (1969), based on the sums of both maximum and minimum fluctuations of the time series:



R S

n

=

1

σ

M axn

n  i=1

(χi − χ¯ ) − Minn

R S

3.4. On Hurst exponent

•

and σ represents the standard deviation of the time series. Such a result is assumed as part of the identity

n 



(χi − χ¯ )

(10)

i=1

where R is the range of maximum and minimum amplitudes of time series, S corresponds to standard deviation of such amplitudes

n

= C nH

(11)

where C is an arbitrary constant intrinsic to time series and n is the number of data points considered in each time series. Thus, by taking the log of both members, we have

R

l og

S

 

n

= l og(C ) + l og nH

(12)

Consequently, the Hurst exponent H is given by

H = [log(N )]

−1

R

log

SC

(13)

n

3.5. On largest Lyapunov exponent The most relevant characteristic of chaos is the unpredictability of the future events based on a deterministic time evolution. Another important feature of chaos behavior is its sensitive dependence on initial conditions SDIC. A way to determine quantitatively the degree of the SDIC of a dynamics system is by Lyapunov exponents. In fact, Lyapunov exponents present the rate of divergence of nearby trajectories in phase space. A graphical representation of such a rate can be seen in a two-dimensional case off the evolution of a small sphere of radius dr to an ellipsoid with axis l1 and l2 along the time t is presented in Fig. 3. The growing of the ith principal axis as a function of time t is described as

li (t ) ∼ dr · eλi t

(14)

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247

Fig. 8. Characteristics of plug flow and dispersed bubbles of R-290.

Consequently,

λi =

lim

dr→0,t→∞

1 li (t ) ln t dr

(15)

The Lyapunov exponents are listed according to the following order: λ1 ≥ λ2 ≥ λ3 ≥ … . Since λ1 is known as the largest Lyapunov exponent LLE and when it is positive, the dynamics of the system is characterized as chaotic. Considering a dynamic system obtained from a time series X, the LLE can be obtained according to

n−1

λ1 =

χ 

ln χi+1  i tn − t1

i=1

(16)

where χ i+1 and χ ’i represent the spacing between two trajectories at time ti+1 and ti , respectively. During years several methods have been developed to determine LLE in experimental time series. Wolf et al. (1985) method was the first one to purpose an algorithm to find the LLE and has been widely used. However, Wolf et al. (1985) method just assumes the existence of an exponential divergence, but it does not test it. In sequence, other authors have developed methods to establish LLE (Sano and Sawada, 1985; Eckmann et al., 1986; Rosenstein et al., 1993; Kantz, 1994). Two different methods were used to calculate the LLE of the experimental time series: Rosenstein et al. (1993) and Lai and Chen (1998). Since, the method purposed by Rosenstein et al. (1993) calculates directly the LLE and has the advantage of distinguishing chaos from noise, but such a

method can be used only for a short range of a time series. On the other hand, Lai and Chen (1998) method calculates Lyapunov exponents using Jacobian method, considering long ranges of time series. 4. Results In order to validate the optical sensor, as well as the methodology used, tests were carried out under conditions of sub-cooled liquid and saturated vapor to find intrinsic frequencies to the electronic system (white noise). Such tests were performed with and without isolation on visualization section to analyze the influence of light inside the laboratory and LED set on the high-speed camera as well as to evaluate the magnitude of “noise” found in the electronic circuit of the optical sensor itself. According to the tests, the absence of isolation around the visualization section does not influence the intrinsic frequencies of the system. Figs. 4(a) and 5(a) present the characteristics of electric signal obtained from optical sensor for sub-cooled liquid and saturated vapor cases, respectively, for R-1270. The maximum voltage of 4.26 V is indicated by the sensor in conditions without the presence of vapor phase. On the other hand, the minimum voltage obtained by the sensor is of 0 V and it is observed in conditions for vapor quality equal to 1. Since intrinsic frequencies for both conditions are presented in Figs. 4(b) and 5(b). It can be observed that white noise is presented and predominant from 100 Hz. In both cases, the PSD of white noises are in the order of magnitude of about 10−8 W/Hz. Both presence and absence of isolation on visu-

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Fig. 9. Phenomenon of coalescence observed with R-290 for G = 480 kg/(m2 s) e q”TS = 10 W/m2 .

alization section did not affect those the order of magnitudes, indicating that such frequencies originate from other electromagnetic source(s) than the artificial light in laboratory. The same maximum and minimum voltages, as well as the intrinsic frequencies, were also observed for R-600 and R-290. Signal amplitudes measured by optical sensor and the characteristic frequencies of flow patterns obtained from three fluids were described as function of experimental parameters q”TS , G, slip ratio in test section outlet S( o-TS ) and boiling number Bo, given by





x(o−T S) ρl 1 − α(o−T S) uv(o−T S)   S(o−T S) ≡ = ul (o−T S) α(o−T S) ρv 1 − x(o−T S) Bo =

q T S G il v

(17)

(18)

where uv ( o-TS ) and ul ( o-TS ) represent vapor and liquid mean velocities in two-phase flow, respectively. All images were captured by a high-speed camera with a range of 50 0 0 frames per second for a period of 10 s in each test. Thus, all results presented are based on comparisons between character-

istic frequencies and morphologic structures of flow patterns observed through the sequence of images. Five different flow patterns were identified during the tests: dispersed bubbles with plug, slug, churn and wavy-annular. Fig. 6 presents the regime maps as function of vapor quality and void fraction for all tests in the outlet of test section. In this case, all flow patterns are classified by different ranges of the nondimensional number β = S·Bo·10−5 , originally purposed by Oliveira et al., (2018) and it represents a simple method to predict flow pattern based on effects of superficial velocity of each phase, vapor quality and void fraction as function of both mass and heat fluxes. Its specific form was obtained in order to avoid overlapping of flow patterns when presented as function of vapor quality or void fraction. In fact, the presence of dispersed bubbles was found only with plugs during the tests involving low heat conditions. 4.1. Dispersed bubbles and plug flow For all conditions of mass fluxes and q”TS = 5 kW/m², plug flow is predominantly observed. In this case, such a pattern is obtained

J.D. de Oliveira, J.B. Copetti and M.L.S. Indrusiak et al. / International Journal of Multiphase Flow 114 (2019) 240–257

Fig. 10. Phase portraits observed from signals for plug flow with dispersed bubbles, considering τ = 4 · 10–3 s and d = 3.

Fig. 11. Characteristics of slug flow of R-1270.

249

250

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Fig. 12. Characteristics of slug flow of R-290.

Fig. 13. Phase portraits observed from signals for slug flow, considering τ = 4 · 10–3 s and d = 3.

only when slip ratio is higher than 1 and it is characterized by different sizes of plugs accompanied intermittently by dispersed bubbles. The results obtained for R-600a, according to Fig. 7, show that dispersed bubbles present a vast range of frequencies up to 20 Hz, including harmonic and subharmonics ones. However, the investigation of sequence of images indicates that plugs present, basically, two distinct frequencies due to differences of the sizes.

Plugs with length of approximately 1 mm present frequency equal to 4 Hz, whereas more elongated plugs (≥2 mm) have a characteristic frequency equal to 2 Hz. R-290 presented plug flow with dispersed bubbles for all mass fluxes with q”TS ≤ 10 kW/m². Fig. 8 presents the characteristic frequencies of plug flow observed in tests for q”TS = 10 kW/m² and G = 480 kg/(m²s). In this case, there are two distinct frequencies:

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251

Fig. 14. Characteristics of churn flow for R-600a.

4.8 Hz, due to plugs with length smaller than 1 mm, and 1.5 Hz for plugs bigger than 1 mm. The formation of more elongated bubbles can be explained by coalescence effect between smaller plugs, or between plug and dispersed bubbles along the flow. Fig. 9 presents the images obtained sequentially from the same conditions described in Fig. 8 and showing the process of coalescence between two dispersed bubbles and a small plug, forming a longer plug. For R-1270, plug flow with dispersed bubbles was also found for the same conditions of heat and mass fluxes with different ranges of frequencies caused by velocities of phases. In terms of heat transfer, Oliveira et al. (2018) has shown that such flow patterns are related to nucleate boiling dominance due to the low vapor quality conditions. The wide frequency ranges found in most of the observed flow patterns for the three refrigerants may be an indicative that variations caused by the liquid-vapor interface are governed by nonlinear dynamics. Based on this fact, the phase portraits obtained from signal fluctuations were analyzed for all experimental tests. Fig. 10 presents the strange attractor of signal fluctuations found in plug/dispersed bubbles for different mass flux conditions with heat flux equals to 5 kW/m². Such attractors are characterized by a presence of a central part formed by the passage of bubbles and external parts (petals) created from the passing of plugs. The flow pat-

terns related to Fig. 10(a) and (b) have Lyapunov exponents equal to 0.6613 and 0.782, indicating the existence of deterministic chaos in the signal. For R-290, Fig. 10(b) and (c), the largest Lyapunov exponents are equal to 2.038 and 2.891, respectively. Similar characteristics of both phase portrait and Lyapunov exponent are observed for R-1270 (Fig. 10(d) and (e)) comparing to R-290. In this case, LLEs are 2.576 and 2.905, respectively. In fact, such phase portraits present strange attractors from the chaotic nature of flow patterns. For R-600a, R-290 and R-1270, Hurst exponent ranges between 0.81–0.92, 0.80–0.84 and 0.67– 0.85, respectively, indicating intermittent variations in signal amplitude for that respective flow pattern. Such variations are related to the passage of vapor phase with different sizes along the flow caused by coalescence of bubbles. Although Talley et al. (2015) point out that the process of bubble coalescence can be considered random, the dynamic of this flow pattern, including union of bubbles, is deterministic. 4.2. Slug flow A typical case of slug flow for R-1270 is observed in Fig. 11 with the main characteristics of the flow pattern. In this case, the intermittence of liquid phase presents a predominant frequency of ap-

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Fig. 15. Characteristics of churn flow for R-1270.

Fig. 16. Liquid-vapor structure of churn flow for the refrigerants in different conditions of heat flux, considering G = 480 kg/(m²s).

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Fig. 17. Phase portraits observed from signals for slug flow, considering τ = 4 · 10–3 s and d = 3.

proximately 3.5 Hz for low mass flux conditions (G ≤ 320 kg/(m²s)) and 5 Hz for G ≥ 40 0 kg/(m²s). For R-60 0a and R-290 (Fig. 12), such a pattern presents practically the same characteristic frequencies though its predominance occurs only for q”TS = 10 kW/m2 . On the other hand, slug flow is also characterized by higher frequencies with smaller spectral energy due to the presence of small dispersed bubbles that come off the back of the slug. Such detachment is due to hydrodynamic instabilities caused by higher vaporphase velocity. Those frequencies are, generally, in the range between 8 and 18 Hz and they can vary according to the intensity of both heat and mass fluxes. Similar range of frequencies was found by Sardeshpande et al. (2016) in the investigation of pressure drop oscillations during flow boiling of R-718. According to their results, pressure fluctuations are caused by acceleration of liquid intermittencies due to the formation and growth of confined bubbles and vapor plugs. A chaotic behavior based on the largest Lyapunov exponents was also found and the trend of long-term positive correlation is still observed for signals obtained from this flow pattern. Results show that, for R-600a, the largest Lyapunov exponent, LLE, decreases smoothly with increasing of mass flux, but ranging positively from 2.39 to 2.18. On the other hand, the Hurst exponent, H, increases with increasing of mass flux from 0.68 to 0.82. For R-290, intermittent appearance of vapor slugs presents the LLE between 2.22 and 3.03 and H ranging from 0.62 to 0.7, showing that there is still existence of a long-time memory in the system. The time of stability indicated by Hurst exponents can also be stablished by the inverse of LLE (Grzybowski and Mosdorf (2014)), i.e., the time necessary to keep the trend of a high value followed by another high value, for example. For R-290, average Lyapunov time (1/λ) is 0.38 s and it means that a predominant frequency of approximately 2.63 Hz is found between intermittencies of liquid phase in this flow pattern. This predominant frequency can also be verified through the FFT according to Fig. 12. Examples of phase portraits found in slug flows are presented in Fig. 13. Similar strange attractors are observed between the patterns of R-290 and R-1270 (Figs. 13(b) and 13(c)). Such a similarity is characterized by a dense zone in the center of attractors and is related to predominant frequencies shown in Figs. 11 and 12, for example. In turn, the exis-

tence of dense zones in these cases can also be justified by intense presence of dispersed bubbles in liquid phase between slugs. On the other hand, due to higher surface tension of R-600a, dispersed bubbles were not easily found between slugs and causing lower density in the center of attractors, as shown in Fig. 13(a). 4.3. Churn flow The occurrence of churn flow as one of the main flow patterns found during the tests is intrinsically related to the large amount of heat required to change of flow pattern from slug flow to wavy-annular flow (Oliveira, 2017; Oliveira et al., 2018). The existence of churn flow is also explained by the radial symmetry found in small channels. Videos obtained from images collected were also analyzed to ensure the existence of such flow pattern. Churn flow has also been reported by Copetti et al., (2013) and Silveira et al. (2017) in experiments, using hydrocarbons and R134a. For R-600a, as shown in Fig. 14, churn flow presents a predominant frequency of 5 Hz for mass fluxes of 240 and 320 kg/(m2 s) and heat fluxes between 20 and 40 kW/m² that can be observed based on the counting of peaks of signal. Such a characteristic frequency is caused by the passage of vapor gaps caused by Kelvin–Helmholtz instabilities. However, with increasing of mass flux, it occurs a greater instability of interface of both phases, causing the appearance of frequencies ranging from 3 to 10 Hz. Oliveira et al. (2018) also showed that heat transfer effect depends on both mass and heat fluxes where slug and churn flow are predominant. Based on the typical frequencies found in churn flow, caused by such gaps, the effect of both superficial tension and reduced pressure are clearly observed for the three refrigerants. On the other hand, gaps of churn flow with R-290 and R-1270 (Fig. 15) present more dispersed characteristic frequencies, ranging between 2 and 10 Hz, according to conditions of both heat and mass fluxes. The behavior changes between persistent and antipersistent characteristics of signals were found in churn flow. For all mass fluxes and heat flux of 20 kW/m², the persistent characteristic is still observed, with Hurst exponents ranging between 0.596 and

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Fig. 18. Typical characteristics of wavy-annular flow of R-290.

0.5156. For cases with heat flux equal to 40 kW/m², the time series of churn flows are found to be antipersistent, indicated by Hurst exponents between 0.4094 and 0.3411. For R-600a, the largest Lyapunov exponents were found to be also positives ranging from 1.5666 and 4.3775 without any apparent dependence of mass or heat flux. For R-290 and R-1270, the chaotic behavior was also identified, indicated by the largest Lyapunov exponents varying from 1.7126 to 3.0324. In fact, flow presented the LLE for each condition of heat and mass flux and it is caused by the considerable turbulence and instability between both phases. The effect of superficial tension is also perceived when the morphology of liquid-vapor interface is observed, as seen in Fig. 16. For R-600a, churn flow is characterized by smooth oscillations on the interface. Even with the passing of gaps, the interface does not present considerable deformations in its structure. For R-290 and R-1270, the passing of gaps deforms considerable the interface, causing the appearance of more intense oscillations along the flow. For churn flow, phase portraits present wider orbits with distinct regions of concentration (Fig. 17). Such behavior can be explained by the increasing of Kelvin–Helmholtz instabilities caused by a greater vapor-phase velocity. On the other hand, the regions of concentration come from low predominant frequencies.

4.4. Wavy-annular flow The wavy-annular flow is characterized by the presence of a liquid front wave sliding on the liquid-vapor interface in annular flow. Such a flow patterns was found for all mass fluxes with higher heat fluxes q”TS ≥ 40 kW/m² for G = 240 kg/(m²s) and q”TS = 60 kW/m² for G >240 kg/(m²s). Surface tension and reduced pressure effects are also observed in wavy-annular flow. For, R-290, according to Oliveira et al. (2018), such a flow pattern is characterized by the convective boiling predominance and presents the highest heat transfer coefficients. That characteristics can be explained by the reduction in the thickness of the liquid film layer and it causes the thermal resistance of the liquid phase to decrease, thereby reducing the difference between the internal wall temperature and the saturation temperature. For both R-290 (Fig. 18) and R-1270 (Fig. 19), wave fronts (characteristics of such a flow pattern) have frequencies between 1.5 Hz and 4.5 Hz. Nevertheless, higher frequencies with lower energies appear and are generated by faster intermittent instabilities in the interface of two-phase flow, varying between 6 and 12 Hz for both fluids. On the other hand, R-600a presents only one predominant frequency for both wavy- and smooth-annular flow. In this case, wave

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Fig. 19. Typical characteristics of wavy-annular flow of R-1270.

fronts present predominantly 5 Hz for G = 240 and 320 kg/(m2 s) with q”TS = 40 kW/m2 and, for other conditions, a frequency of 7.3 Hz. As well as to churn flow observed with q”TS = 40 kW/m², antipersistent time series were found in wavy-annular. In this case, Hurst exponent presented the lowest values for this flow pattern, ranging between 0.2871–0.3088, 0.3101–0.3668 and 0.2327–0.3437 for R-600a, R-290 and R-1270, respectively. On the other hand, the largest Lyapunov exponents decreased for this flow pattern when compared to churn flow for R-600a. According to data analysis, R-600a presented the largest Lyapunov exponents varying from 1.2915 (for G = 480 kg/(m²s) and q”TS = 60 kW/m²) to 2.2966 (for G = 240 kg/(m²s) and q”TS = 60 kW/m²). For R-290 and R-1270, this trend was not observed and the largest Lyapunov exponents ranged between 2.1326 and 3.3188, and 0.8434 and 1.4800, respectively. Fig. 20 shows examples of phase portraits for wavy-annular flow. The tendency of spreading orbits is still observed. For R-600a (Fig. 20(a)), it can be seen the existence of a strange attractor (indicated in figure) caused by the greater proximity among the characteristic frequencies. For R-290, (Fig. 20(b)) three denser regions are found and related to the distinct predominant frequencies previously mentioned. On the other hand, an apparently

unique denser region is observed for R-1270 but, in fact, such region consists of attractor close together. 5. Conclusions The main aim of the present investigation was to analyze time series provided by signals of void fraction during flow boiling of R600a, R-290 and R-1270 for a saturation temperature of 17 °C performed in a horizontal tube having an internal diameter of 1.0 mm. A method based on fast Fourier transform was used to establish characteristic frequencies of flow patterns. The following main conclusions can be drawn from the results obtained: •

•

The use of an optical probe to measure light fluctuations caused by the dynamic of liquid-vapor interface has been effective to recognize characteristics of flow patterns for different conditions of G and q”. Both vapor quality and void fraction also have considerable influence on flow patterns since the affect the velocity of each phase.

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Fig. 20. Phase portraits observed from signals for wavy-annular flow, considering τ = 4 · 10–3 s and d = 3.

•

•

•

•

Five different flow patterns were identified on the basis of images, including dispersed bubbles flow, plug flow, slug flow, churn flow and wavy-annular flow. The predominant frequencies of each flow pattern were found to be related to its morphology and regime characteristics such as heat and mass fluxes. The existence of chaos in all-time series was investigated based on the largest Lyapunov exponent LLE. For all cases, positive LLEs were found, indicating a deterministic chaotic behavior of liquid-vapor interface of flow patterns. The LLE increases with increasing the heat flux during the development of flow pattern from dispersed bubbles flow to churn flow and decrease in wavy-annular flow. The dynamics of time series was observed, using phase portraits to recognize the presence of strange attractors. Similar phase portraits were found for each type of flow pattern, indicating, consequently, the existence of similar intrinsic characteristics of the signals over time. A test to detect long memory in time series of flow pattern was implemented based on Hurst exponent. For a fixed mass flux condition, Hurst exponent decrease with increasing heat flux. And for this reason, bubbly, plug and slug flows are characterized by persistent time series. On the other hand, churn and wavy-annular flows are represent by antipersistent time series. And, apparently, none mass flux effect was identified on both long-memory and the LLE.

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