Two-phase Flow Induced Vibration in Piping Systems

Progress in Nuclear Energy 78 (2015) 270e284

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Progress in Nuclear Energy journal homepage: www.elsevier.com/locate/pnucene

Review

Two-phase flow induced vibration in piping systems Shuichiro Miwa a, *, Michitsugu Mori a, Takashi Hibiki b a b

Graduate School of Engineering, Hokkaido University, Kita 13 Nishi 8, Kita-ku, Sapporo 060-8628, Japan School of Nuclear Engineering, Purdue University, 400 Central Drive, West Lafayette, IN 47907-2017, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 17 July 2014 Received in revised form 29 September 2014 Accepted 9 October 2014 Available online 30 October 2014

Hydrodynamic force acting on the structures, pipes and various forms of objects can generate destructive vibrations, and could cause acoustic and noise problems in industrial machineries. Such phenomenon is known as Flow-Induced Vibration (FIV), and it can obstruct smooth operation of engineering devices and could potentially cause serious consequences like system failures. The subject has become increasingly important problem in engineering industry in recent years for both single-phase and multi-phase flow cases, as well as for various flow orientations including external and internal flows. Present review paper summarizes the historical background of FIV research and how the phenomenon has been classified in both industrial and academic fields, particularly focusing on the progress of two-phase FIV research. Special attention was paid to the subject of internal two-phase FIV generated at industrial piping systems two-phase flow regimes. Based on the extensive and comprehensive literature survey, most up-to-date progress of the research in the area of two-phase flow induced vibration in piping system are thoroughly reviewed and presented in this article. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Flow induced vibration Multiphase flow Elbow Fluid mechanics Momentum transfer

1. Introduction In recent years, two-phase flow induced vibration (FIV) has been given increasing attention in various engineering fields including petroleum pipelines, power and processing plants, heat exchangers, as well as in the nuclear power plant components (Altstadt et al., 1995; Anagnostopolous, P., 2002; Blevins, 1979, 1990; Cong et al., 2014; Fujita, 1990; Hara, 1975; JSME, 2003; Konno and Saito, 1985; Laggiard et al., 1995; Miwa et al., 2014a; Pettigrew and Taylor, 1994; Weaver et al., 2000). In these applications, knowledge of gaseliquid two-phase flow induced force fluctuation magnitudes and its predominant frequency is paramount for designing the safely operable engineering systems and avoiding structural damage that may be caused by fluidesolid interaction. The terminology, “FIV”, was first introduced by Robert Blevins in 1977. He was the first to classify the phenomena according to hydrodynamic and structural dynamics, and has developed a fundamental road map to analyze the problem (Blevins, 1990). Based on his classification, fluid dynamic mechanisms responsible for FIVs are categorized according to steady and unsteady conditions (Fig. 1). The categories of FIV phenomena are coupled with

* Corresponding author. E-mail address: [email protected] (S. Miwa). http://dx.doi.org/10.1016/j.pnucene.2014.10.003 0149-1970/© 2014 Elsevier Ltd. All rights reserved.

structural dynamics through hydrodynamic force acting on the structure. When the hydrodynamic force is acted on the structure surface, it will undergo deformation. The deformed structure will then react and apply the opposite force against fluid based on its material properties, such as elasticity, natural frequency, damping parameters, and so on. During the process, flow is disturbed and direction and magnitude of the hydrodynamic force may change considerably. Consequently, FIV is generated due to the linkage of force fluctuations between these two dynamic forces (Blevins, 1990). In order to predict FIV phenomena, separate models for fluid and structural dynamics can be developed, and they are coupled with hydrodynamic and structural force terms. Models available for structural dynamics are near-linear for most cases and can be modeled as linear-oscillator. In contrast, fluid dynamic models can be more complicated in general since it possesses inherently nonlinear and multi-degree-of-freedom behaviors. Due to such reason, fluid dynamic models of FIV must be always verified by the experiment. Models for structural response to fluid flow can be developed from the database of physical law and experimental data. Contrary to the single phase flow case, mechanisms of twophase flow-induced vibration can be quite different due to complex motions/interactions at phase boundary, differences in material properties (density, surface tension, viscosity etc.), and phase change process via energy transfer/generation. General

S. Miwa et al. / Progress in Nuclear Energy 78 (2015) 270e284

Nomenclature ai a A Aeff c cp cv C0 D DH DSm f fa F Fr G g i j l Lf Lg p R Re t

Interfacial area concentration [m1] Sound velocity [m/s] Cross section area [m2] Effective cross section area of liquid slug [m2] Speed of sound in liquid [m/s] Specific heat at constant pressure [kJ/kg K] Specific heat at constant volume [kJ/kg K] Distribution parameter [e] Pipe diameter [m] Hydraulic diameter [m] Sauter mean diameter [m] Frequency [Hz] Void propagation function Fluctuating force [N] Froude number [e] Mass flux [kg/s/m2] Gravity [m/s2] Imaginary number Superficial velocity [m/s] Elbow length [m] Liquid slug length [m] Bubble length [m] Pressure [Pa] Curvature radius [m] Reynolds number [e] Time [s]

classification of two-phase flow induced vibration, created based on JSME handbook (JSME, 2003) as well as available literature (Pettigrew and Taylor, 1994), is shown in Fig. 2. The first category, “momentum fluctuation”, includes FIV caused by the density difference between two phases, and large FIV is generated due to the change in flow direction and the impact force of two-phase mixture acted on piping component structure such as elbows and T-junctions (JSME, 2003). The second category,

u We x z

271

Velocity [m/s] Weber number [e] x-axis z-axis

Greek symbols Void fraction [e] Volumetric quality [e] Liquid film thickness [m] Elbow curvature angle [degree] Adiabatic index [e] Viscosity [e] Fraction of liquid film thickness per pipe ID [e] Density [kg/m3] Shear stress [kg/m s2] Surface tension [N/m]

a b d q k m x r t s

Subscripts f Liquid phase g Gas phase gs Group-1 in liquid slug unit k k-th phase in Test section inlet out Test section outlet t Two-phase 1 Group-1 2 Group-2 24 Two-phase

“thermal-hydraulic vibration associated with phase change”, induced from the nature of two-phase flow which involves phase change due to the energy transfer through interfacial boundary and/or energy generation within the phase. Such phenomena, including boiling and condensation, involve highly unstable and oscillatory behaviors, and easily promote FIV within two-phase flow systems. The third category, “bubble-induced vibration” is due to the dynamics of various shapes and sizes of bubbles that

Fig. 1. Classification of flow induced vibration (Blevins, 1990).

272


Fig. 2. Classification of two-phase flow induced vibration (JSME, 2003).

induce sloshing, fluctuations and disturbances within the flow fields. All of the above FIV categories are coupled with the fluctuating characteristics of two-phase flow, namely momentum, pressure and void fraction fluctuations. These phenomena are seen in various engineering fields depending on the flow orientations, such as axial flow, internal flow, and cross flow. 2. Existing works on two-phase flow induced vibration One of the challenging aspects of two-phase flow is its highly unstable and unsteady nature, which creates local fluctuations of phase fraction, density, velocity, pressure field, and momentum flux (Akagawa, 1974; Ishii, 1997; Ishii and Hibiki, 2011; Wallis, 1969). Unstable flow characteristics may generate periodic forces on piping structure or structure surfaces. Such phenomena can potentially cause fretting-wear or fatigue, or resonance when the predominant frequency coincides with the piping system's natural frequency. As mentioned in previous section, two-phase FIV can be classified according to flow orientations, namely the axial, cross, and internal flows (Figs. 2 and 3). Regardless of the flow orientations, it is analytically and experimentally confirmed that the effect of void fraction fluctuation is the most dominant and significant part of twophase FIV analysis (Hara, 1975; Liu et al., 2012; Miwa et al., 2014; Pettigrew and Knowles, 1997). Two-phase FIV due to each of these categories will be briefly reviewed in following subsections.

2.1. Two-phase external flow 2.1.1. Two-phase axial flow Number of studies related to both single-phase and two-phase FIVs are reported for axial flow, particularly focusing on the applications related to nuclear thermal-hydraulic safety including boiling water reactor (BWR) fuel rods and tube bundle of the steam generators for pressurized water reactor (PWR) (Blevins, 1979; Pettigrew and Taylor, 1994; Pettigrew et al., 1998; Chen and Wambsganss, 1972; Chu et al., 2011; Feenstra et al., 2009; Fujita, 1990; Hong et al., 2007; Paidoussis, 1982; Schroder and Gelbe, 1999). Vibration mechanisms such as fluidelastic instability, hydraulic oscillations due to phase change, random turbulence excitation and acoustic noise are the primary vibration excitation mechanisms in axial two-phase FIV (Pettigrew et al., 1998; Taylor and Pettigrew, 2001). The vibration phenomena are also affected by the bundle geometry (Pettigrew et al., 2001). It is reported that the effect of void fraction fluctuations is the significant two-phase flow parameter for axial FIV phenomenon. Studies show that the damping effect becomes important for the void fraction less than 80%. However, the lack of experimental data is limiting the clear explanation for the linkage between two-phase flow regime and axial FIV (Carlucci, 1980; Carlucci and Brown, 1983; Pettigrew and Taylor, 1994). According to Pettigrew et al. (1998), key two-phase flow parameters affecting the excitation forces in axial two-phase

Fig. 3. Two-phase flow induced vibration flow patterns: (a) axial flow, (b) cross flow, and (c) internal flow.


flow are flow regime, void fraction, temperature, pressure, and mass flux or velocity. 2.1.2. Two-phase cross flow Two-phase FIV on cross flow exists in shell-and-tube heat exchangers (Kanizawa et al., 2012; Khushnood et al., 2004; Mitra et al., 2009; Nakamura et al., 1986, 1992; Pettigrew and Knowles, 1997; Sasakawa et al., 2005; Zhang et al., 2007), U-tube region in the pressurized water reactor (PWR) steam generators (Pettigrew et al., 1998), and offshore structures (JSME, 2003). Cross flow generally involves either isolated cylinder or bundles of cylinders. Primary causes of the FIV phenomena are due to the pressure fluctuations generated by Karman vortex, collisions of the bubbles against structure surface, and turbulent eddies interacting against structures (JSME, 2003). Pettigrew et al. (1998) also reported three key mechanisms for the cross flow FIV including fluidelastic instability, periodic wake shedding and random excitation due to turbulence. The occurrence of these phenomena will depend on the flow regime, which is associated with physical properties of fluids, inlet flow conditions, local void fraction, and flow channel geometries. Tube damping parameters for two-phase FIV on cross flow within bubbly flow regime (up to a ¼ 0.30) were reported by Pettigrew and Knowles (1997). 2.2. Two-phase internal flow Internal two-phase FIV is significantly important to secure the reliability and integrity of the piping systems in power/processing systems that involve piping structures. To date, there has been very few research performed on the subject of internal gaseliquid twophase flow induced vibration and force fluctuations, particularly. Majority of the two-phase FIV work has been focused on axial and cross flow. For internal two-phase FIV, flow turning elements are considered as one of the major sources for causing FIV by generating sudden change in momentum flux, pressure fields, or creating secondary vortices due to boundary layer separation (Belfroid et al., 2010; Cargnelutti et al., 2009, 2010; Chen, 1973; Crawford et al., 2007; Hayama and Matsumoto, 1986; Hiramatsu et al., 1984; Liu et al., 2012; Miwa et al., 2012a,b; Nakamura et al., 2005; Ortiz Vidal and Hernandez Rodriguez, 2011; Pontaza and Menon, 2011; Riverin et al., 2006; Riverin and Pettigrew, 2007; Tanaka and Fujita, 1987; Wang and Shoji, 2002; Weaver et al., 2000; Yamano et al., 2011). Response of the structure typically increases linearly (in time) when the dominant fluctuating frequency overlaps with natural frequency of the structure. Such phenomenon is known as the resonance, and must be avoided at the designing stage to ensure the safe system operation. Predictive capability of the dominant frequency and fluctuating force amplitudes are still under development for internal two-phase FIV. In this review paper, the mechanisms of internal two-phase flow induced vibration and assessment of the existing correlations and models are thoroughly reviewed. A comprehensive literature review is made in the open literature to collect the available experimental dataset and correlations. Fundamental work done on the internal two-phase flow with the most up-to-date models will be reviewed, and discussions including future direction of the research in this particular subject. 3. Internal two-phase flow induced vibration 3.1. Review of causes and mechanisms Internal two-phase FIV can be induced by various hydrodynamic phenomena. Depending on the geometrical configurations of flow channels and operating conditions, gaseliquid two-phase flow may create vibrations with various modes of amplitude and frequency

273

(JSME, 2003). In a slug flow regime, for instance, slug unit consist of slug bubble (Taylor bubble) and liquid slug flow intermittently. When the slug unit passes through the pipe turning element, force acting on the structure boundary significantly fluctuate due to the major difference in momentum, and consequently induces structural vibration. Generally, more pronounced structural vibration tends to occur in slug flow regime, compared to other flow regimes. Such trend was observed from reported studies where the maximum force fluctuations were measured at the void fraction corresponding to slug/churn flow regimes (Riverin and Pettigrew, 2007). This is caused by the large intermittent change in mixture momentum flux due to the fluctuations in mixture fluid density. Even when the designed flow regime in a piping system is set at stable annular flow, FIV can be induced due to the existence of slug or churn flow conditions during start-up, transient, shut-down, and low load operation. In heat exchanging systems, operational temperature during transient scenario can be much lower than the steady operating condition. As a result, void fraction tends to be lower than designed value which may result in the occurrence of slug or churn flow regime. Slug flow regime can be also generated during the low load operation condition in heated system as well. Two physical aspects should be considered as potential sources of flow induced vibration in slug/churn flow regimes. They are, (1) an agreement between the frequency of momentum/pressure fluctuations and the piping natural frequency, and (2) collision of liquid slug onto a structural surface transmitting a large excitation force. The frequency of passing liquid slugs depends on the length of Taylor bubbles and liquid slugs that are dependent on void fraction and volumetric flux. Such fluctuations are in the order of several Hz for internal two-phase flow. Usually, piping system in processing plants consists of the combinations of vertical, horizontal, and inclined pipes, and it is expected that frequency of passing slug flow is in the order of few Hz. Therefore, FIV in slug flow regime is expected to take place in relatively low frequency range, and collision of liquid slugs induces characteristic frequency vibration. FIV may occur even in relatively steady two-phase flow other than slug/churn flow regimes. Unlike single-phase flow, interaction of gas and liquid phases create significant local fluctuations including flow velocity and void fraction oscillations. Random motion of bubbles may also induce additional turbulence (bubble induced turbulence) and considerably disrupt the flow fields. Therefore, because of these fluctuating nature, force acting on the pipe turning element varies significantly, and consequently, induces vibration.

3.2. Existing database and empirical correlations/models The main thrust of the past research has been towards the prediction of force magnitude and its dominant frequency at various two-phase flow regimes. Existing major works on the twophase flow induced force fluctuation modeling are summarized in Table 1. Few correlations and models that are applicable for given operation range are reported by several researchers (Tay and Thorpe, 2004; Riverin and Pettigrew, 2007; Cargnelutti et al., 2010). Introductory overview of the two-phase flow induced vibration phenomena are well summarized in Pettigrew and Taylor (1994). Even though these reports mainly focused on axial and cross flow oriented two-phase FIV, damping and vibration excitation mechanisms of two-phase flow are briefly mentioned, which can be applicable for some conditions in internal two-phase flow. In addition, these papers provide the basic road-map and overview of the subject and introduce available analysis methods of the twophase flow induced vibration problem.

Pressure transducers before/after the bend Conductance probes

20

60

0.17e1.25 L/s

0.2e0.7 m/s

Quartz force sensor (Kistler, 9167A1.5)

Unknown Optical probe

Unknown 0.17e1.25 L/s

Piezoelectric force sensor

Tri-axial Force Transducer (Tec Gihan) Tri-axial Force Transducer (Tec Gihan) Tri-axial Force Transducer (Tec Gihan) Piezoelectric force sensor Pressure Transducer (Honeywell-Yamatake) Pressure Transducer (HoneywellYamatake) Pressure Transducer (HoneywellYamatake) Unknown 0.610e2.310 36 m/s 0.610e2.310 36 m/s 0.610e2.310 m/s 19

Force sensor (B&K 8302) 3-128 kg/h

0.07e12.1 kg/h Water 0.10e18 m/s Water 0.10e18 m/s Water 0.10e18 m/s Water 0.1e10.4 L/s Water 0.1e10.4 L/s 0.38e2.87 Water, 5 wt% IPA, m/s 35 wt% glycerine Air

Air

20.6 mm ID R/d varied from 0.5 to 7.2 70 mm ID 105 mm Horizontal

Air

20.6 mm ID R/d ¼ 0.5

50.8 mm ID 76.2 mm

50.8 mm ID 76.2 mm

16.5 mm, 25 mm 50.8 mm ID 76.2 mm 6 mm ID

Cargnelluti et al. (2010) Liu et al. (2012) Miwa et al. (2014a) Miwa et al. (2014b) Riverin et al. (2006) Riverin et al. (2007) Tay and Thorpe (2004)

Air

Air

Vertical (UpwardHorizontal) Vertical (UpwardHorizontal) Vertical (HhorizontalDownward) Vertical (U-Bend and Tee) Vertical (U-Tube)

Air

Water Air Horizontal

Force sensor (B&K 8302)

Dynamic pressure transducer (Kulite XCE-093, HBM P3MB) Dynamic pressure transducer 90 0.1e70 m/s 25.4 mm ID unknown Belfroid et al. (2010)

Horizontal

Air

Water

0.05e2 m/s

Number of Void fraction data points measurement technique Superficial Superficial gas velocity liquid velocity Gas Liquid Flow direction Bend radius Geometry Investigators

Table 1 List of the internal two-phase FIV experiment data.

Optical probe (Thorlabs Inc. PPDA/LSD1) Optical sensor (Thorlabs Inc) Impedance Meter (in-house made) Impedance Meter (in-house made) Impedance Meter (in-house made) Optical probe


Pressure measurement technique Dynamic force measurement technique

274

Yih and Griffith (1968) were one of the first to conduct the experiment that related momentum flux fluctuations in two-phase flow to the FIV. Momentum flux fluctuations largely contributed to FIV particularly in the slug and annular flow regimes due to the oscillatory mixture density and flow behaviors. Such momentum fluctuation is caused by the density difference between gas and liquid phases. For annular flow, surface wave with random amplitudes of the liquid film interface, as well as the droplet deposition/ entrainment lead to the fluctuation. For slug flow regime, alternate passage of gas and liquid slugs across the test section induces the momentum flux fluctuation. The experiment conducted by Yih and Griffith covered wide range of superficial gas velocity (15e75 m/s) covering slug, churn and annular flow regimes. Three round tubes of 2.54, 1.59, and 0.64 cm, and various other flow channel shapes (annulus, rectangular, triangular and installation of spacer) were utilized to test the pipe size and channel geometry effect. Based on the measured data, power spectral density (PSD) curves of unsteadiness of the momentum fluxes for each operational condition were obtained. Effect of the momentum fluctuations with respect to channel geometry, volumetric quality (b), average flow velocity (V), and system pressure (P) were reported and they are introduced in next section. 3.3. Trend observed in internal two-phase FIV It was mentioned in the literature that the contribution of momentum flux fluctuation is significant only in low frequency range, which is below 50 Hz (Yih and Griffith, 1968). It is agreed by other researchers that important frequency range in two-phase flow induced vibration test lies in 0e50 Hz of range (Riverin and Pettigrew, 2006; Liu et al., 2012; Miwa et al., 2014a,b). When conducting FIV experiment, natural frequency of the supporting structure and piping materials should be set higher than the twophase flow fluctuation frequency range to avoid the resonance. Hence, selection of the appropriate parts and materials is extremely important in two-phase flow induced vibration experiment. 3.3.1. Channel geometry effect It is reported from the work of Yih and Griffith that high transverse vibrations of the test section pipe was seen for the rectangular pipe, which wasn't observed for the round tube. It was concluded that, due to the low natural frequency of rectangular pipe, the operation mode was close to its resonance at given fluctuation energy induced by momentum flux. In the case of annulus channel, PSD curve was more evenly distributed than other channel geometries, due to the increase in frictional pressure drop. No predominant momentum fluctuation frequency value was observed in annulus. When the spacer was placed between inner and outer surface of the annulus, large disturbances were greatly reduced. It was concluded that the spacer has an effect of breaking down the large disturbances into smaller ones, and consequently reducing the total fluctuation energy. Effect of the spacer or resistance/obstacles in the channel of the two-phase flow induced vibration, which is extremely important for LWR fuel design and safety, hasn't been reported other than Yih and Griffith. Pipe bend curvature has direct effect on phase separation at elbow section and alters void fraction and interfacial area concentration distribution in radial coordinate (Kim et al., 2007). However, since radius of curvature had little effect on momentum flux fluctuation in axial coordinate, no major effect on excitation force signal was reported (Riverin and Pettigrew, 2007). Although, diameter of pipes had little effect on the predominant frequency for some of the reported studies (Riverin and Pettigrew, 2007), Yih and Griffith's experimental result indicated that


unsteadiness of the momentum flux is larger for smaller pipe than that of larger pipes. It is speculated that such reduction in fluctuation is due to the enhanced two-phase mixing in larger pipe due to the secondary flow generated by cap shaped bubbles. In internal two-phase flow, when the pipe diameter reaches beyond the critical pipe diameter, D*, slug bubbles cannot be generated. Beyond this size, Taylor bubbles will break up due to the surface instability and cap bubbles are produced beyond bubbly flow regime. Critical pipe diameter is defined by Kataoka and Ishii (1987) as follows.

DH ffi 40 D*H ≡qffiffiffiffiffiffi s gDr

(1)

Here, DH, s, g, Dr are the hydraulic diameter, surface tension of the liquid phase, gravitational acceleration, and density difference between two phases, respectively. Therefore, separate flow regime for large diameter pipe must be utilized for the pipe size larger than D* (Schlegel et al., 2012). 3.3.2. Void fraction and volumetric quality effect Yih and Griffith observed a clear peak in power density spectrum at high volumetric quality condition, while multiple peaks were observed at low volumetric condition (b < 70%). The volumetric quality is defined as follows.

b¼

Qg Qg þ Qf

(2)

Here, Qg and Qf are the volumetric flow rate of gas phase and liquid phase, respectively. It was reported that at high b, significantly high amplitude of surface waves in annular flow regime, and/or the large momentum fluctuation induced by liquid slugs in slug flow regime, are the distinctive source of fluctuation, while at low b, magnitudes of these parameters aren't as effective. As a result, clear critical frequency, fc, was not observed for the low b two-phase flow. Similar analysis was conducted by Riverin and Pettigrew (2006, 2007). They conducted two-phase flow induced vibration experiments on the U-bend (two elbows) and tee configuration, respectively. In their experiment, 20.6 mm inner diameter PVC piping was utilized. Elbow section was modified to investigate the effect of bend radius (R/D) ranging from 0.5 to 7.2. Optical-probe was utilized to monitor the void fraction. Force transducer was installed at the elbow section to measure the time signal of the excitation force. Clear peak was observed at high volumetric quality region. They’ve also confirmed the direct trend between momentum flux fluctuations and excitation force signals, and magnitude of the excitation forces is largely influenced by the flow regime. Similarly, experimental work reported by Liu et al. (2012) showed that the largest void fraction fluctuations were observed in slug/churn flow regimes. Hence, volumetric quality, or void fraction, has definite effect on the excitation force. However, it must be mentioned that such effect is due to the flow regime rather than the void fraction value. Therefore, clear identification of two-phase flow regime is crucial for internal FIV analysis. 3.3.3. Average flow velocity effect It was observed that there exist critical frequency, fc, of the momentum flux fluctuation at given inlet velocity value (Yih and Griffith). High resolution of fc, where the peak is considerably sharp and dominant, was observed near the annular flow region. The maximum momentum flux fluctuation was observed at the boundary of slug to annular flow transition. Similar trend was observed by Riverin et al. (2006), Riverin and Pettigrew (2007) for the force acting on the U-tube bend, and Liu

275

et al. (2012) for the force acting on the elbow section. It is reported that the fluctuating force amplitude and peak frequency increased linearly with flow velocity. Root-mean-square (RMS) of the force and predominant frequency from the force spectrum was correlated with the superficial velocity (j). The relationship between inlet superficial velocity and excitation force can be explained by the form of F ¼ Cja. From the experimental results of U-tube bend, excitation force and total superficial velocity had dependency of j1.2, and operating condition at j ¼ 10 m/s corresponded to the largest RMS force signal, which is at churn flow regime. Liu et al. (2012) and Miwa et al. (2014a,b,c) reported monotonic increase in fluctuating force magnitude with increase in gas velocity for upward and horizontal two-phase flow. 3.3.4. System pressure effect It was observed by Yih and Griffith (1968) that the ratio between momentum flux fluctuation and stationary momentum flux value (PRMS/PST) decreases with increase in system pressure. Hence, unsteadiness of momentum fluxes tends to diminish at higher system pressure, and may suppress the internal FIV. It is due to the compressibility of gas phase, since the property of gas density is proportional to the system pressure. As the density of gas phase increases, both the fluctuating and steady components of momentum fluctuation of two-phase tend to be reduced. This finding, however, was based on the very limited range of the experimental data. Further investigation on the effect of gas phase density on momentum flux fluctuations should be considered. From these findings, Yih and Griffith attempted to develop the correlations to relate the unsteady momentum flux data with flow parameters. It was recommended to utilize the following dimensionless groups to correlate the PRMS/PST and fc data.

We0:4

fc

D V

PRMS ¼ XðbÞ PST

rffiffiffiffiffiffiffiffi Fr ¼ YðbÞ We

(3)

(4)

Here, fc, D, V, and b are the peak frequency, channel diameter, flow velocity, and volumetric quality, respectively. Usage of Froude number (Fr ¼ j/√(gD)) and Weber number (We ¼ rfj2D/s) into the dimensionless group was suggested to evaluate the effects of surface tension and gravity on the unsteadiness and predominant frequency of momentum fluxes. Proposed correlations were comparable to the experimental data obtained in the range specified by the authors. Study performed by Zhang et al. (2008) also compared the relationship between two-phase momentum flux and flow induced vibration. Authors conducted two-phase FIV experiment on cylindrical rod arrays placed within the rectangular flow channel. Drag and lift forces acting on 3 6 rod bundles were measured for various two-phase flow conditions. Measured forces were quasiperiodic, and it was related to momentum flux fluctuations seen in between the cylindrical rods. Authors investigated the relationship between drag force acting on the cylinder and dynamic characteristics of two-phase flow. A mathematical model was derived from the rod array geometry to correlate the void fraction fluctuation and dynamic drag force. Although, it was specifically modeled for the cross-flow application, their approach and methodology of FIV analysis can be applicable for internal two-phase flow scenario, since the proposed force fluctuation model was derived from the basic conservation equations for two-phase mixture that are dependent on void fraction value, and spectral analysis was conducted to investigate the characteristic frequencies responsible for two-phase FIV.

276


Direct comparison between RMS of the force exerted on pipe elbow and two-phase superficial velocity was done by Tay and Thorpe (2004). To simulate the oil-gas system in sub-sea flow lines, authors performed the experiment on gaseliquid slug flow in 70 mm inner diameter horizontal pipe with 90 elbow section. Various types of fluids were utilized for their experiment to study the effect of liquid viscosity and surface tension on the maximum forces acting on the elbow. The operational condition was mainly fixed within the range of slug flow regime. Surface tension of the glycerol solution was 70 mNm1, and viscosity was 2.60 mPa-s, respectively. Force signal was recorded from the load cell installed at the elbow section. Piston Flow Model (PFM) was proposed to predict the force exerted on elbow, and root-mean-square of the predicted force was comparable to the measurement at low flow condition. Their main finding was that the surface tension and viscosity had little effect on the internal two-phase FIV at elbow section. 3.3.5. Gravity, surface tension and viscosity effect As reported by Tay and Thorpe (2004), insignificancy of viscosity and surface tension on the force exertion on 90 bend was also confirmed by the empirical correlation proposed by Riverin et al. (2006). In the correlation, force fluctuation was related to surface tension term using Weber number as We0.4. Riverin also confirmed no major relationship between Reynolds number and excitation forces, which indicates small relation between fluid viscosity and excitation forces. Even when the liquid surface tension was reduced by 32%, no significant effect on forces acting on elbow was confirmed. However, reduced surface tension tends to diminish the momentum force due to the increase of gas hold-up in the liquid slug at high slug velocity. Tay and Thorpe's experiment mainly focused on the slug flow regime at relatively large pipe size. Wider range of flow regime and system scale needs to be analyzed to investigate the true effect of surface tension on two-phase flow induced vibration, particularly in bubbly flow regime and micro-, nano-scale internal two-phase flow. From Riverin and Pettigrew (2006, 2007)’s experimental work on U-bend (two elbows) and tee configuration, effect of fluid and dimensionless parameters was investigated by extending the work done by Yih and Griffith, and Tay and Thorpe. Dependence of superficial flow velocity, j, on force shows that the root-mean-square of force is dependent on the dimensionless variables such as, void fraction (a), We, Re, Fr, and mass ratio rL/rg. Hence, proposition of the dimensionless group reported by Yih and Griffith was modified as following.

! FRMS r ¼ B a; L ; We; Re; Fr FST rg

(5)

Eq. (5) can be further simplified. Force exerted on the bend is affected by the superficial velocity in a format of j1.2. Function B in Eq. (5) could be expressed using the dimensionless numbers to make the superficial velocity term j1.2, since FST term contains j2 term. Based on the less significance of surface tension and viscosity effect, as reported by Tay and Thorpe, usage of We0.4 is reasonable, and Re dependency may not be significant in Eq. (5). Above considerations led to the modification of Eq. (5) as follows.

FRMS ¼

FRMS B We0:4 ¼ CWe0:4 ¼ 1b FST

(6)

Here, b is the volumetric quality. Applicability of Eq. (6) was assessed with the experimental data including that of Yih and Griffith (1968), and Tay and Thorpe (2004). From their analysis, the constant C is suggested by authors to be 10.

From the experimental work performed by Cargnelutti et al. (2009), force fluctuation on internal two-phase flow on 6 mm ID bend was reported. Force level was measured for various flow regimes, and it was reported that the slug flow regime had the highest excitation force level. They attempted to correlate the experimental data with Weber number alone, as proposed by Riverin in Eq. (6), but reasonable agreement wasn't obtained. As suggested by Yih and other researcher, force on the bend ought to be correlated with the combination of dimensionless numbers like We, Fr, and Re for developing empirical correlations. This indicates that inclusion of gravitational effect, surface tension, and viscosity into the Frms correlation may be necessary, even though they appear to be insignificant. 3.3.6. Mass damping effect As can be seen from the work mentioned above, characterization of the relationship between surface tension and exerted force is not so simple. Based on the previously reported work, surface tension and flow regime play important role on the two-phase mass damping for internal two-phase flow. According to Pettigrew et al. (1998) and Weaver et al. (2000), two-phase mass damping was found to increase with surface tension at certain flow condition, but the behavior wasn't consistent for all the flow regimes. Therefore, effect of surface tension on damping at the specific flow regime should be investigated further for the future research. Beguin et al. (2009) reported the maximum two-phase damping was observed at the transition between bubbly to slug flow regime, and begins to decrease as void fraction tends to increase further. It was also reported that the mass damping is lower for the smaller diameter tubes. Authors pointed out the direct correlation between two-phase mass damping and interfacial area concentration. Further investigation on this topic is recommended in the future, since interfacial area concentration measurement technique has been developing considerably in recent years (Ishii and Hibiki, 2011). 3.3.7. Spectral behavior of internal two-phase FIV Spectral analysis is a vital technique to investigate the FIV phenomena. Typically, FFT or PSD methods are commonly employed to analyze the periodic nature of the force and momentum flux fluctuation signal. In addition to Yih and Griffith, several other studies on the spectral analysis for internal two-phase flow induced vibration are reported, including Wang and Shoji (2002), Nakamura et al. (2005), Riverin and Pettigrew (2007), and Liu et al. (2012). Wang and Shoji (2002) investigated the fluctuation characteristics of two-phase flow at a T-junction. They focused on ChurnTurbulent flow regime, which induces the highest fluctuation phenomenon compared to other flow regimes. They analyzed the differential pressure signals at varying gas and liquid flow rate at an impacting T-junction to investigate the pressure fluctuation. The dominant frequency on the PSD map was compared for different air mass flow rate to liquid mass flow rate ratio (Wair/Wwater ¼ W3/W1). It was found that dominant frequency stayed around 0.18 Hz at W3/ W1 ¼ 0.2e0.7, which implies that the fluctuation is periodic. Beyond W3/W1 ¼ 0.3, the peak frequency shifted to 1.6 Hz. Nakamura et al. (2005) conducted a single-phase FIV experiment that simulates the hot-leg and cold-leg sections of the sodium-cooled fast reactor. Effect of pressure fluctuation at high Reynolds number was studied by installing multiple pressure transducers near and around the elbow piping. Although their operational condition was a single phase flow in turbulent condition, the presented method of analysis on pipe elbow can be applicable to two-phase flow, especially for bubbly flow case since


the flow field in that regime is governed by the liquid phase, which creates pressure fluctuations due to secondary flow and Dean vortex generations. From their flow visualization study, performed on the acrylic test section, existence of the flow separation and reattachment zones near the downstream of the elbow section was confirmed. Frequency analysis was carried out and peak frequency was observed at the Strouhal number (¼fD/U) of 0.45. From the PSD analysis, peak frequency was confirmed at the separation region. Hence, excitation force of the elbow is affected by the flow separation at the down-stream region of the elbow, and the maximum flow-induced random vibration force in the pipe were observed in the region of flow separation downstream of the elbow. Riverin and Pettigrew (2007) carried out a spectral analysis of the measured forces and observed the linear trends between amplitude of force and predominant frequency. It was suggested that Power Spectral Density (PSD) and frequency to be nondimensionalized as follows:

PSD ¼

PSD

(7)

ðGDÞ2

f ¼

277

fD j

(8)

Here, G is defined as the mass flux of the two-phase mixture and can be expressed as,

G ¼ arg þ 1 a rL j

(9)

They characterized the dimensionless PSD output and frequency as the triangular shaped lines. Authors proposed that linear interpolation of the dimensionless PSD and frequency diagram would provide prediction for the peak frequency at given void fraction condition, which is given as follows.

2 3 m 1 PSDðf Þ 6 7 PSD f ¼ 4 m01 5 f ;f f0 f0

(10a)

Table 2 List of the internal two-phase flow induced vibration correlations and models. Investigators

Presented Correlation/Model

Major assumption

Remarks

Cargnelluti et al. (2010)

Slug flow regime: F ¼ 2rf u2Slug A Annular and stratified flow regime: F ¼ 2rm u2m A

-Considers momentum change alone -Taylor bubble diameter is equivalent to pipe diameter -Friction and gravity effects are neglected -Assumes “smooth” transport of two-phase mixture through pipe bend

-Model developed for small tube (6 mm) -Local probe was utilized to measured void fraction database

-Liquid slug is a homogeneous/isothermal two-phase mixture

-Requires time signals of void fraction and elbow inlet/outlet boundary pressure as model input -Applicable for verticalupward, and horizontallydownward elbow orientations -Needs to obtain empirical constant C for specific test section geometry and flow regime -Model developed for the Utube, but transducer is attached to one-side only. -Measurement of stiffness (k), natural frequency (fn), and damping ratio (z) for the test section is needed

2

Liu et al. (2012)

f p2 R i

3

i2pjf ðL2 x0 Þ 4i2pRfjt e jt þj2t 5 t 4p2 R2 f 2 þj2t

F FSx ðf Þ ¼ Arf j2t A f ðx0 ; f Þe

P out ðf ÞA

A is the liquid faction spectrum defined as:

i2pjf x0 t F a jft A f ðx0 ; f Þ ¼ j1t e Miwa et al. (2014a,b)

Included impact force effect of the slug flow regime as an additional term for the model proposed in Liu et al. (2012). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffi kP0 2P0 Lg FIMPACt ¼ p1ffiffiffi r2f Aeff að1aÞr r Lf 2

Riverin et al. (2006)

f

2f

FRMS ¼ rFuRMS ¼ CWe0:4 ; C ¼ 10 and C ¼ 3.51 for slug flow 2 A f

m

regime in 6 mm ID test loop Riverin and Pettigrew (2007)

Force PSD: PSDðf Þ ¼ PSDðfm01 Þðf Þm1 ; f f0 ðf0 Þ

PSDðf Þ ¼ PSDðfm20 Þðf Þm2 ; f f0 ðf0 Þ

Tay and Thorpe (2004)

Vibration prediction: !2 Z ∞ 1=k y2 ¼ PSDðf ÞðGDÞ2 df 0 1 ðf =fn Þ2 i½2zðf =fn Þ y: mean square value of a response D: pipe diameter fn: natural frequency G: two-phase mixture mass flux k: stiffness z: damping ratio R R R R F ¼ ð in PdA out PdAÞ vtv V urdV S uru ,dA The slug velocity is utilized for velocity u,p which ffiffiffiffiffiffi is defined as: uslug ¼ 1:2ðjf þ jg Þ þ ð0:54 1:76Eo0:56 Þð gDÞ D 2 Here, EoD ¼ rf gD =s

-f0 , PSDðf0 Þ,m1 and m2 are determined from curve-fitting. -Considers two-phase as “mixture” and disregards slip between phases

-Applicable for horizontal slug flow regime -Utilizes liquid density in the model calculation (r ¼ rf )

-Requires time signals of void fraction and elbow inlet/outlet boundary pressure as model input

-“Piston Flow Model”, developed for horizontally oriented pipe bend. -Experimental database developed from oil-gas test section system -No flow regime identification was performed, but slug flow structure was assumed from the liquid height measurement.

278

2 3 m 2 PSDðf Þ 6 7 PSD f ¼ 4 m02 5 f ;f f0 f0


(10b)

Variables and coefficients shown in above correlation, m1, m2, f0 , PSDðf0 Þ can be obtained from the least squares method. This technique may be useful for evaluating the critical frequency at a given operating inlet condition on the U-tube bend. Authors reported the coefficients value at each void fraction condition to regenerate the triangular-lines on the dimensionless PSD vs. frequency domain. 3.3.8. Summary of the existing work Existing internal FIV database and correlations/models are summarized in Tables 1 and 2. Although these correlations and models were developed for variable fluid types, local void fractions, pressure condition at the given flow turning element geometry, there have been few, if any, attempts to consider the effect of twophase flow structure. In fact, previously reported correlations and models are developed from the experimental databases that were performed under severe vibrational condition. As pointed out by Hibiki and Ishii (1998), internal gaseliquid two-phase flow structure, including void fraction and interfacial area concentration profile, is largely influenced by the presence of external vibration. Hence, in order to analyze the two-phase flow induced force fluctuation from fluid dynamics standpoint, it is crucial to avoid resonance mode by separating the characteristic frequencies of twophase flow parameters and natural frequency of the test section. Furthermore, most of the previously developed FIV database utilizes insufficient data-acquisition techniques including single-point local probe for void fraction measurement, which neglects the areaaveraged characteristics of two-phase flow parameters. In order to overcome such shortcomings, Liu et al. (2012) performed two-phase FIV experiment on rigidly supported 50.8 mm ID stainless pipe bend at vertical-to-horizontal flow orientation (Fig. 4). The stainless steel test section was rigidly installed on the steel I-beam and its structural natural frequency far exceeded that of two-phase flow characteristic frequencies to avoid resonance. To develop a reliable database, state-of-the-art instrumentations were utilized including, arch and ring type impedance probes for areaaveraged void fraction (Mi et al., 1998), tri-axial force transducers for dynamic force signal, tri-axial accelerometer for structural acceleration, and twelve digital pressure transmitters for the

simultaneous local pressure measurements at the elbow surface. Developed database covered entire flow regimes that are observable in vertical two-phase flow including, bubbly, slug, churnturbulent and annular flow from Mishima-Ishii’s flow regime map (1984). Then, by conducting a control volume analysis over the elbow section, force fluctuation model was developed from a local instantaneous formulation of two-fluid model. The model developed in the study evaluates the two-phase flow induced force fluctuations that are due to momentum and pressure fluctuations alone, but it was concluded that the additional force term was necessary for the excitation force characterization of slug flow regime. It was then suggested that the impact force caused by the collision of liquid slug against structure surface needs to be included into the model to improve the predictive capability of the force fluctuation model. Similar conclusion was drawn for the internal FIV experiment performed on horizontal two-phase flow, which covered the flow regimes reported by Mandhane et al. (1974) by Miwa et al. (2014b). For its practical application, as was pointed by Tay and Thorpe (2004), it is crucial to identify the maximum possible force induced by liquid slug at designing stage to prevent FIV in piping structure. For that reason, it is essential to develop an impact force term for slug flow regime at pipe bend for vertical two-phase flow to facilitate the development and validation of the two-phase flow induced force fluctuation model. In the next section, the latest modeling work done on the internal two-phase force fluctuation on slug flow regime is reviewed. 4. Impact force modeling for slug and churn flow regimes (Miwa et al., 2014a, 2014b, 2014c) Miwa et al. (2014a and 2014b) developed the impact force term due to the collision of liquid slug against piping wall, which can be included into the force balance equation for both upwardhorizontal (Miwa et al., 2014a) and horizontal-downward (Miwa et al., 2014b, 2014c) configurations. Its detailed description and derivations can be explained as follows. To develop a physical model for the impact force in slug flow regime, first, let us consider the impact force induced by the collision of liquid plug (or slug) against a wall boundary. Fig. 5 depicts the liquid slug of length Lf being pushed by the external pressure P0 in positive x-direction. Taylor bubble of length Lg is placed in between liquid slug and wall boundary, which is being compressed while liquid slug moves

Fig. 4. Internal two-phase FIV experimental facility of.Liu et al. (2012)


Fig. 5. Schematic of the liquid slug model within pipe (Miwa et al., 2014a).

towards the wall boundary. Let us assume that compression of Taylor bubble takes place rapidly so that pressure of gas phases reaches to zero instantaneously. To find the acceleration of the liquid slug in this scenario, one can apply the equation of motion for the liquid slug (Fujii et al., 1999).

d2 x rf Lf A þ pDLf tw AP0 ¼ 0 2 dt

(11)

From the first term of the left hand side, inertial term, frictional resistance exerted by the wall surface, and the force exerted by the external pressure P0, respectively. tw is the wall shear stress, which can be expressed in terms of the slug velocity as follows:

tw ¼

8m dx D dt

d2 x þ dt 2

32m rf D2

!

dx P 0 ¼0 dt rf Lf

(13)

To supply initial and boundary conditions to solve above equation, let us assume that liquid slug is initially at complete rest at the position x ¼ 0. That is.

dx ¼0 dt

when t ¼ 0

(14)

Then, above equation yields following relations.

dx D2 P0 ¼ uf ¼ dt 32Lf mf

" 1 exp 32

mf rf D2

#!

sffiffiffiffiffiffiffiffiffiffiffiffiffi" # 2P0 Lg 16 2L Lg 1 2 f rf Lf D rf P0

(18)

One obtains the expression for the pressure rise due to the impact, known as the water-hammer pressure, by dividing above terms by the cross sectional area. For gaseliquid two-phase flow through pipe elbow operated under the slug flow regime, it is expected that the collision of liquid slug against the wall boundary instantaneously creates a sharp pressure increase that is similar to water-hammer like phenomena. Since the liquid slug observed in slug flow regime in gaseliquid two-phase flow contains dispersed bubbles, it should be treated as a two-phase mixture. With that in mind, les us now substitute the liquid slug velocity defined in Eq. (17) into Eq. (18) and replace the speed of sound, c, with two-phase sound velocity, a24. Then, the impact force due to liquid slug in slug flow regime is expressed as:

FIF ¼ r2f a2f A

(16)

(17)

Let us now consider the pressure increase during the impact of liquid slug. According to the Guided Acoustic Shock Theory (Lesser, 1995), impact of liquid slug against the wall boundary creates a

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2P0 Lg r2f Lf

(19)

Slug length, Lg, can be calculated by relating it to the Sauter mean diameter of the Taylor bubble. Here, Sauter mean diameter is defined as:

DSm ¼

6a ai

(20)

In gaseliquid two-phase flow, various sizes bubbles coexist within the flow channel. First group is classified as Group-1, which consist of the small scale spherical bubbles. A second group, named as Group-2, consists of the relatively large bubbles including distorted bubbles, cap bubbles, and Taylor bubbles (Ishii and Hibiki, 2011). From the definition of area-averaging.

〈a〉 ¼

The last term of the right hand side of Eq. (16) is relatively small, so it can be neglected. As a result, liquid slug velocity during the collision is expressed as follows.

sffiffiffiffiffiffiffiffiffiffiffiffiffi 2P0 Lg uf ¼ rf Lf

F ¼ rf cuf A

(15)

After some algebraic manipulation and simplification of Eq. (15), liquid slug velocity can be expressed as follows.

dx ¼ uf ¼ dt

sharp pulse, or shock, which travels through the liquid slug at the speed of sound, c. Most significant part of a liquidesolid impact occurs in a time frame of 106 to 103 s of order after the initial contact. As depicted in the Fig. 5, let us consider the liquid slug length Lf of the cross section A is traveling at the velocity uf, and is about to collide against the wall boundary. Immediately after the collision, amount of time it would take for a shock wave to travel across the liquid slug is, Lf/c. Change in linear momentum due to the collision is simply the product of mass and change in velocity of the liquid slug. Then, the force exerted on the wall-boundary due to the liquid slug collision is:

(12)

Substituting Eq. (12) into Eq. (11), equation of motion of liquid slug can be expressed as a second order ordinary differential equation.

279

1 A

Z A

adA;

(21)

sum of the Group-1 and Group-2 area-averaged void fractions should add up to the total area averaged void fraction, thus the relation below must hold.

〈a〉 ¼ 〈a1 〉 þ 〈a2 〉

(22)

Then, area-averaged Sauter mean diameter of Group-2 bubbles can be defined as follows.

DSm;2 ¼

6〈a2 〉 6 ¼ ð〈a〉 〈a1 〉Þ 〈ai2 〉 〈ai2 〉

(23)

According to Ishii and Mishima (1980), interfacial area concentration for slug and churn-turbulent flow regime can be obtained from the following relation.

280


〈ai 〉 ¼ 〈ai1 〉 þ 〈ai2 〉 ¼

6〈ags 〉 1 〈a〉 4:5Cct 〈a〉 〈ags 〉 þ DH DSm;1 1 〈ags 〉 1 〈ags 〉 (24)

Cct is known as the roughness parameter, which takes into account the effect of irregular interface at churn-turbulent flow regime and is set to 1.0 for the slug flow regime. ags is the Group-1 void fraction within liquid slug, which excludes a1 in the liquid film surrounding the Taylor bubble.

〈ags 〉 ¼ 1

1 〈a〉 1 〈a〉 þ 〈a1 〉

(25)

The area averaged Group-1 void fraction term, can be obtained from the correlation proposed by Ozar et al. (2012) from the knowledge of the total area averaged void fraction, and liquid superficial velocity, jf, and it is expressed as follows.

8 〈a〉 > > >

> > 〈a1;max 〉 〈a1;base 〉 〈a〉 〈a1;max 〉 > > < 〈a1;max 〉 þ 〈a 1;max 〉 〈a〉 〈acrit 〉 〉 〈a 〉 〈a 1;max crit 〈a1 〉 ¼ > > > > 〈a〉 〈a1;max 〉 〈acrit 〉 〈a〉 > > > : 〈a1;base 〉 (26) Maximum Group-1 void fraction is observed at the transition from bubbly to slug flow regime, where Group-1 bubbles begin to coalesce and form Group-2 bubbles. Parameters listed in Eq. (26) can be obtained from a following relation.

( 〈a1;max 〉 ¼ ( 〈acrit 〉 ¼

0:235 þ 0:011〈j*f 〉 for 〈j*f 〉 6:1

0:325 0:004〈j*f 〉 for 〈j*f 〉 6:1 0:511 þ 0:006〈j*f 〉 for 〈j*f 〉 6:1

0:645 0:015〈j*f 〉 for 〈j*f 〉 6:1 (

〈a1;base 〉 ¼

0:099

0:009〈j*f 〉

0:054

for

〈j*f 〉

6:1

for 〈j*f 〉 6:1

〈jf 〉 !1=4

pD2 4

L1 þ L2

L2 ¼ (29)

(30)

pðxDÞ2 4 pD2 4

þ pxDL2 L1 þ L2

(32)

(28)

Let us now consider the slug unit shown in Fig. 6. As can be seen, typical slug unit in vertical gaseliquid two-phase flow mainly consists of two components, namely (1) Taylor bubble (Group-2) surrounded by the liquid film and Group-1 bubbles, and (2) liquid slug containing Group-1 bubbles. Suppose that such slug unit passes through the flow channel with diameter D. If we assume that maximum width of Taylor bubble is expressed as the fraction of diameter, xD, thickness of the liquid film can be expressed as, D (1x)/2. Let us name the lengths of Group-1 and Group-2 bubbles as, L1 and L2, respectively, then, volume averaged void fraction of Group-2 can be calculated from a following. pðxDÞ2 L 2 4

ai2 ¼

2

By substituting Eqs. (31) and (32) into Eq. (23), Group-2 Sauter mean diameter can be expressed as a function of length scales and it is now possible to solve for L2 in terms of DSm, 2.

sgDr r2f

a2 ¼

Similarly, interfacial area concentration (¼available bubble surface area/total volume of the mixture) of the Group-2 bubble can be expressed from the slug unit as:

(27)

Here, non-dimensional liquid superficial velocity is defined as follows.

〈j*f 〉 ¼

Fig. 6. Schematic of the slug unit seen in vertical slug flow regime (Miwa et al., 2014a).

(31)

xDDSm;2 1 ¼ D 3xD 2DSm;2 3 2DSm;2 Sm;2 xD

(33)

Length of the liquid slug (Lf ¼ L1) can be determined from Lg (¼L2). If we can obtain the area averaged void fraction time series from experimental measurement, its mean (amean), maximum (a2) and minimum (ags) values can be extracted from the signal. If we assume that slug flow is a repetition of the unit shown in the Fig. 6, minimum and maximum void signal should correspond to the passage of liquid slug and Taylor bubble at the measurement location, respectively. Mean void signal is identical to the averaged void fraction of the slug unit, consists of Taylor bubble and liquid slug (Fig. 6). Considering the condition mentioned above, liquid slug length can be obtained from the following relation.

Lg ða amean Þ Lf ¼ 2 amean ags

(34)

Consequently, length scale needed to calculate the impact force term became a function of void fraction values, ags, amean, and a2. Mean void fraction can be determined from regime-dependent correlation of Drift flux model, or from the experimental measurement. In order to determine appropriate ratio between Taylor bubble diameter and channel diameter, parameter x was calculated from the minimum film thickness value (dmin) of the slug unit and the


local void fraction value calculated at the minimum film thickness (asb) using Mishima-Ishii’s model (1984), as shown in a following formulation.

F

FSx ðf Þ

¼ Arf j2t A f ðx0 Z þ

asb ¼

∞ ∞

i2p f ðL2 jt ; f Þe

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# u 1 kP0 2P0 Lg i2pft u pffiffiffi r2f ðtÞAeff t dt: e r 2 2f t Lf a t 1 a t rf

D 2dmin D

(35)

Here, asb is defined as,

asb

. m . 1=ðm2Þ rf DrgD j þ ð1 asb Þ3=ð2mÞ 3Cf D yf rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ . ffi C0 j þ 0:35 DrgD rf (36)

Considering the turbulent flow condition for slug flow regime, Cf and m are given as, 0.046 and 0.2, respectively. C0 is the distribution parameter and can be obtained for the vertical flow in a circular channel (Ishii and Hibiki, 2011). From the void fraction value asb, x value is estimated to be in between 0.8 and 0.9. In order to determine the best x value for each slug flow regime, it was correlated with mean void fraction value. Physically, when the flow channel is filled with the Taylor bubble (a ¼ 1), x should also equal to 1 since it implies a single-phase flow condition. From the database obtained from the experiment (Liu et al., 2012), x is expressed in terms ofa as follows.

x ¼ 0:28a þ 0:72:

(37)

Additionally, two-phase sound speed (a24) relation is necessary to complete the impact force term. As the liquid slug collides against the wall boundary, shock wave propagates at the two-phase sound velocity. In the case where the heat capacity of continuous phase exceeds the dispersed phase, isentropic process can be assumed during the collision when propagation sound wave possesses rather high frequency (JSME, 2003). Then, two-phase sound velocity can be approximated with the form shown below:

a2f

direction force spectra without major modifications. Complete formulation to determine the x-direction force frequency spectra is expressed as following (Miwa et al., 2014a,b,c).

2 2 3 if p R x0 Þ i2pRf jt e jt þ j2 t5 4 P out ðf ÞA 4p2 R2 f 2 þ j2t

"

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u kP0 u ¼t a 1 a rf

281

(40)

The final term shown in the right hand side indicates the Fourier transform operation and can be easily computed by the FFT algorithm to obtain the force spectra. Currently derived impact force term is calculated based on the Taylor bubble length and Group-2 void fraction, which is based on Ozar's Group-1 void fraction correlation. As a result, impact force term is automatically turned off whenever the Group-2 bubble is absent within the given twophase flow condition. Therefore, the newly developed model can be utilized for all the internal gaseliquid two-phase flow regimes without having special treatment to include or exclude impact force term. Force fluctuations predicted by the proposed model is compared with the FIV experiment performed on upward-horizontal orientation for slug and churn flow regimes (Figs. 7 and 8). Model output without the impact force term is plotted in the blue dashed line and output with impact force term is plotted in the red dashed line. As can be seen, addition of impact force term significantly improves the magnitude of the dominant peak. Clearly, amplitude of dominant peak becomes closer to experimentally obtained signal whenever the Group-2 bubble is present in the system. Note that some measured frequency peaks are not captured in the model, which is possibly due the multidimensional effect that are not captured in the current model, such as Dean vortices. However, as it is evident from the plots, magnitude of such force fluctuation is extremely small compared to the momentum fluctuation and impact force. It is important to note here that the impact force term is automatically turned off at non-slug bubble condition shown in Fig. 9, because this flow condition is at the transition between bubbly to slug flow regime. This implies that whenever Group-2 bubble is not formed in the system, impact force term is

(38)

Here, k is the heat capacity ratio, or adiabatic index, which is equivalent to cp/cv. Then, final form of the impact force term can be expressed by the following formulation.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 kP0 2P0 Lg u FIF ðtÞ ¼ pffiffiffi r2f ðtÞAeff t r t Lf 2 2f a t 1 a t rf

(39)

The factor square root of 2 arises to account for the force decomposition into x- and z-coordinates at the elbow section. Therefore, impact force term can be added to both x- and z-

Fig. 7. Comparison of the force spectrum obtained from experiment and the model at slug flow regime (Miwa et al., 2014a).

282


Fig. 8. Comparison of the force spectrum obtained from experiment and the model at churn flow regime (Miwa et al., 2014a).

automatically turned off. Near the slug to annular flow transition, width of the peak is slightly unmatched, but the dominant frequency is well predicted by the model. Model performances in all the slug flow conditions are depicted in Figs. 10 and 11. As can be seen, currently developed model is capable of predicting the force fluctuation of dominant frequency within 30% of accuracy and average error was found to be close to 30%. Some of the points that are close to the ±30% error lines are the conditions near the flow regime transition boundaries. It is also apparent from the plot that inclusion of the impact force term significantly improves the model accuracy, and it should be always included for the force fluctuation calculation in slug flow regime. It was also found that the model is sensitive to the selection of film thickness value, x, thus, it is crucial to utilize Eq. (37) to consider the scale effect of Taylor bubble thickness. 5. Conclusions and future perspectives In this review article, historical background of two-phase FIV research and its classification was briefly introduced. Reported studies on internal two-phase flow induced vibration were extensively reviewed. Much progress has been accomplished to understand the FIV mechanisms over past few decades, particularly for external two-phase flow induced vibration. However, it is evident from the literature survey that internal two-phase flow induced vibration requires further attention. As pointed out by Hibiki and Ishii (1998), the vibration of flow channel significantly influences the two-phase flow structure. Hence, in order to investigate the

Fig. 9. Comparison of the force spectrum obtained from experiment and the model without the presence of Taylor bubble (Miwa et al., 2014b).

Fig. 10. Peak frequency predictive capability of the model (Miwa et al., 2014a).

force fluctuations created by two-phase flow dynamics alone, structure vibration of the experimental loop should be minimized and natural frequency of the test loop must be determined beforehand. It is also clear that majority of the observed phenomena are flow regime specific. Based on the review, it is agreed by the authors that slug and churn flow regimes have the strongest fluctuation compared to other flow regimes, in general. Effect of twophase flow regime on the force needs to be investigated further, particularly for large diameter pipe two-phase flow and annular two-phase flow regime, where almost no database is available as of now. In order to minimize internal two-phase FIV, following geometric and operation conditions can be considered. First, natural frequency of the piping structure should be set much higher than the two-phase characteristic frequency range (0e50 Hz) to avoid resonance mode. In order to increase the piping natural frequency, solid fixtures/supports should be mounted against the wall or rigid beam-structure. Secondary, it is important to minimize the occurrence of liquid slug, when operating under two-phase flow

Fig. 11. Peak force fluctuation predictive capability of the model (Miwa et al., 2014a).


condition. Impact force of liquid slug against piping wall is considerably large, resembling the water-hammer-like phenomena. Sharp-turning elbow section should be avoided in piping system, in order to diminish the collision effect due to impact force. Effect of surface tension and viscosity are reported to be very small, however, further flow regime specific investigation is necessary to assess the FIV effect based on those parameters. It is possible that surface tension effect is not negligible at micro-, nano-scale twophase flow, as well as for the internal two-phase flow in pressurized system. Additionally, two-phase FIV in micro-, mini-scale channel, and annular two-phase flow should be investigated further by measuring the important parameters such as film thickness and droplet fraction (Sawant et al., 2008). Effect of the droplet impingement and film dynamics observed in annular flow are expected to be closely connected to FIV phenomena and such investigation is necessary to understand the force fluctuations created by entire two-phase flow regimes. Furthermore, investigation of the possible two-phase flow fluctuations occurrence due to rotating flow field in annular spacing, known as TaylorCouette flow (Hajmohammadi and Nourazar, 2014; Hajmohammadi et al., 2014), is suggested to enhance the understanding of two-phase FIV. Extensive and comprehensive literature review on above mentioned subjects is strongly encouraged.

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Two-phase Flow Induced Vibration in Piping Systems

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