1. Faculty of Marine Technology, Tokyo University of Marine Science and Technology, 2-1-6 Etchujima, Koto, Tokyo 135-8533, Japan 2. School of Nuclear Engineering, Purdue University, 400 Central Drive, West Lafayette, IN 47907-2017, USA Abstract In order to extend a precise database on local two-phase flow parameters in mini pipes, experiments were conducted for adiabatic gas–liquid bubbly flows flowing down in vertical mini pipes with
inner diameters of 1.03, 3.00, and 5.00 mm. A stereo image-processing was applied to observe the phase distribution characteristics in pipe cross-section. The local flow parameters including profiles of void fraction, Sauter mean bubble diameter, and interfacial area concentration in pipe
cross-section were obtained at three axial locations in the test pipes with various flow conditions: superficial gas velocity of 0.00508–0.0834 m/s and superficial liquid velocity of 0.208–3.00 m/s. The axial developments of the local flow parameters were discussed in detail based on the obtained
data and the visual observation. It was confirmed that the core peak distributions were formed at low liquid flow rate conditions in which the buoyancy force dominated while the wall peak distributions were formed at high liquid flow rate conditions in which the body acceleration due
to the frictional pressure gradient dominated. The result indicated the existence of lift force pushing the bubbles towards the pipe wall even in vertical downward flows. The database obtained through the present experiment is expected to be useful in modeling the interfacial
area transport terms, the validation of the existing lift force models as well as the benchmarking of various CFD simulation codes.
Gas–liquid two-phase flow in mini pipes is often utilized in various heat transfer equipment such as high-power electronic devices, heat exchangers of refrigerators or air-conditioning systems, and so on. The thermal-hydraulic design of them has made it important to understand the basic flow characteristics of gas–liquid two-phase flow in the mini pipe. Based on the previous literature (e.g., Kandlikar, 2004), the hydraulic equivalent diameter of mini channels or mini pipes is reported to be ranged from several hundred microns to several millimeters. In the mini pipe, the effect of axial frictional pressure loss is remarkable and the capillary force often dominates the buoyancy force acting on the bubbles. For example, the rising velocity of bubbles in stagnant water becomes zero in a capillary tube (Gibson, 1913). It is therefore anticipated that gas–liquid two-phase flow will differ from that seen in a conventional pipe with a larger inner diameter, and this fact significantly affects boiling heat transfer (e.g.,
Mishima and Hibiki, 1996; Kandlikar, 2002; Serizawa et al., 2002; Qu and Mudawar, 2003; Ghiaasiaan, 2003; Zhang et al., 2006; Hibiki et al., 2007). In addition, the flow structure of two-phase flow in mini pipe changes continuously with flow development in the axial direction even under the adiabatic condition because the axial pressure change due to frictional pressure gradient is very large compared with that in conventional-size pipes. For example, assuming the turbulent single-phase flow, the increase rate of frictional pressure gradient due to decrease in pipe diameter is approximated with L S
1.25( / )D D using the Blasius’s friction factor, where D is the pipe diameter and the subscripts of L and S respectively mean the larger and smaller diameters. Remarkable axial pressure change acts as flow-induced body acceleration and causes rapid changes in the flow structure along the flow direction (Hibiki et al., 2007; Hazuku et al., 2010). It is therefore required to construct precise databases on local two-phase flow parameters in the axial flow direction and develop proper mechanistic models for the axial change of
two-phase flow parameters in mini pipes. On the other hand, flow analysis based on the two-fluid
model is useful basis for the practical design of heat removal systems (Ishii and Hibiki, 2010). In the two-fluid model formulation, accurate models or correlations for predicting two fundamental parameters of void fraction and interfacial area concentration are required. The void fraction expresses the phase distribution and is a required parameter for thermal-hydraulic design in various industrial processes. The drift-flux model (Zuber and Findlay, 1965) is often used to predict the void fraction from setting parameters such as superficial gas and liquid velocities. Hibiki et al. developed the constitutive equations of the distribution parameter and the drift velocity, which are applicable to the flows in mini pipes, by taking the effect of gravity and the frictional pressure loss gradient into account (Hibki and Ishii, 2003; Hibiki et al., 2006). The interfacial area concentration describes the available area for the interfacial transfer of mass, momentum, and energy, and is a required parameter for a two-fluid model formulation. Measurements of local interfacial area concentration have been performed especially for bubbly and slug flows in conventional-size pipes over the past 20 years. Some correlations have been proposed to predict volume averaged interfacial area concentration (e.g., Delhaye and Bricard, 1994; Kocamustafaogullari et al., 1994; Millies et al., 1996; Hibiki and Ishii, 2001). As an advanced model of the interfacial transfer terms in the two-fluid model, the interfacial area transport equation has been proposed (Hibiki and Ishii, 2009). The successful development of the interfacial area transport equation is considered to have produced a significant improvement in the two-fluid model formulation and the prediction accuracy of system codes. The interfacial area concentration has thus been studied experimentally and theoretically over the past twenty years (Hibiki and Ishii, 2009; Lin and Hibiki, 2014; Chuang and Hibiki, 2015; Hibiki et al., 2018). At the first stage of the development of the interfacial area transport equation, the equation for the flow in conventional-size pipes was successfully developed by modeling the sink and source terms of the interfacial area concentration due to bubble coalescence and breakup (Liu and Hibiki, 2018; Shen and Hibiki, 2018). In order to extend this success to various pipe sizes, extensive efforts have been made to gather data for relatively large and small diameter pipes (Lin and Hibiki, 2014; Shen et al., 2018). However, very few data of the local two-phase flow parameters including void fraction and interfacial area concentration in mini pipes are available in spite of its significance in two-phase flow formulation. This is mainly due to the difficulty to measure the local two-phase flow parameters in mini pipes with a non-intrusive method.
In this context, authors constructed the precise databases on the axial development of the void fraction and the
interfacial area concentration of vertical upward and horizontal bubbly flows in mini pipes with pipe sizes of 0.55, 0.79, and 1.02 mm (Hazuku et al., 2010). The local two-phase flow parameters of vertical upward bubbly flows and bubbly flows under microgravity in 3.00, 5.00, and 9.00 mm diameter pipes were also compared through the previous experiments (Hazuku et al., 2012, 2015, 2016). The results showed significant differences between the flow characteristics in horizontal and vertical mini pipes as well as under microgravity. It is generally considered that the dependence of gravity on the two-phase flow structure in mini pipes is insignificant because the surface tension and the viscous force from the wall dominate rather than gravity in the mini pipe flow (e.g., Mishima and Hibiki, 1996). These results imply that the effect of gravity on the flow characteristics such as the phase distribution, relative motion between phases, axial changes of interfacial transfer terms and so on should be considered in the modeling for the mini pipe flow.
In order to extend the database on local two-phase flow parameters and to evaluate the gravity effect on them in the mini pipe, axial developments of void fraction and interfacial area concentration profiles in the vertical downward bubbly flow in mini pipes are experimentally evaluated in the present study. Since the buoyancy force in the downward bubbly flow acts on the bubbles in the opposite flow direction, the dominant factor among gravity, surface tension, inertia as well as the body acceleration due to frictional pressure gradient in characterizing the bubbly flow structures in the mini pipe will be found through the experiment changing the pipe diameter and the flow rates. The data are collected for adiabatic downward bubbly flows in mini pipes with pipe sizes of 1.03, 3.00, and 5.00 mm. A stereo image-processing was applied to observe the phase distribution characteristics in pipe cross-section. The local flow parameters including profiles of void fraction, Sauter mean bubble diameter, and interfacial area concentration in pipe cross-section are obtained at three axial locations in the test pipes. The axial developments of the local flow parameters are discussed based on the obtained data and the visual observation. Although considerable effort has been expended in the past to study the characteristics of two-phase flow in mini channels as described in the earlier, very few detailed measurements have been made of the phenomena occurring in a developing flow in mini pipes. The database obtained through the present experiment is expected to be useful in modeling the interfacial area transport terms, the validation of the existing lift force models as well as the benchmarking of various CFD simulation codes.
2 Experiment
Figure 1 is a schematic diagram of a flow loop. In this
Measurement of local two-phase flow parameters of downward bubbly flow in mini pipes
91
experiment, non-intrusive image-processing was used to measure the axial development of flow parameters. The test section was a round pipe made of fluorinated ethylene propylene (FEP) with an index of refraction of 1.34, similar to that of water (1.33) to avoid image distortion. Three pipes with nominal diameters of 1, 3, and 5 mm were used as test pipes. Because the tolerance of inner diameter of FEP pipe is ±0.1 mm based on manufacturer specifications, there is uncertainty of pipe diameter variation in axial direction. This uncertainty is not negligible in the experiment, especially for 1 mm diameter pipe. Thus the inner diameter should be evaluated accurately as the averaged value in pipe axial direction. It is, however, quite difficult to measure the local pipe diameter in axial direction directly because cutting plane of the FEP pipe is deformed due to its soft material. Therefore, the diameter of the test pipe with a nominal diameter of 1 mm was determined by using the analytical solution of the friction factor for laminar flow as follows (Mishima and Hibiki, 1996). The friction factor for laminar flow in a round pipe is given by the following well-known equation for Hagen–Poiseuille flow as
64
fRe
fN
=
The diameter was determined from the above equation by using the measured friction factor, measured flow velocity, and the fluid properties. The error in the diameter so obtained was estimated to be within ±1%. The evaluated value was 1.03 mm. The lengths of test pipes with diameters were 550 mm for 1.03 mm pipe and 800 mm for 3.00 and 5.00 mm pipes.
Fig. 1 Schematic diagram of flow loop.
Nitrogen gas was supplied from a nitrogen bottle and was introduced into a mixing chamber through a gas injector. As shown in Fig. 2, the gas injector consisted of a stainless needle as a bubble injection nozzle with inner and outer diameters of 0.1 and 0.25 mm, and a tapered acrylic cylinder. No significant swirl flow near the test section inlet was observed in video images of the trajectories of dispersed bubbles. The nitrogen gas and purified water were mixed in the mixing chamber and the mixture then flowed through the test section. Gas and liquid flow meters were installed at the upstream of the test section inlet to set the gas and liquid flow rates. Since the flow rates in the experiment using 1.03 mm diameter pipe is very low, the nitrogen gas and water were collected to measure their volumes after flowing through the test section. As shown in Fig. 1, the nitrogen gas and water were collected by a measuring cylinder placed in the gas–liquid separator, and then the flow rates of each phase were determined from the volume collected per unit time. It should be noted that the maximum solubility of nitrogen in water is negligible in these experimental conditions. The loop temperature was kept at 25 C within ±0.5 C. The difference between the inlet and outlet tem-peratures was within ±0.5 C. The pressure and differential pressure measurements were made with a pressure sensor and differential pressure cell, respectively. The pressure devices detected the system pressure and differential pressure through a tiny hole in the test section wall (diameter 0.2 mm). Because the locations of the pressure tap were away from the test section inlet and outlet, the pressure disturbance due to the inlet and outlet was not considered. Measurement accuracy was conservatively estimated to be 1%. An electrical conductivity meter was installed at the gas–liquid separator to monitor water quality. The experiment was performed at an electrical conductivity less than 1 μS/cm.
Fig. 2 Schematic diagram of mixing chamber.
T. Hazuku, T. Ihara, T. Hibiki
92
Viewing sections for recording bubble images in the test pipe were included at three axial locations, whose axial distances from the pipe inlet (z) normalized by the pipe diameter (D) were z/D = 15, 75, and 150 for 1.03 mm diameter pipe, z/D = 45, 75, and 150 for 3.00 mm diameter pipe, and z/D = 34, 72, and 110 for 5.00 mm diameter pipe. The water boxes were placed at the viewing sections to minimize the image distortion due to refraction. The local flow parameters of bubbly two-phase flow, including the void fraction, Sauter mean bubble diameter, and interfacial area concentration profiles, were measured by a stereo image-processing method (SIM) using two high-speed video cameras and two plate lights at each viewing section. The details of the SIM can be seen in the previous papers (Takamasa and Miyoshi, 1993; Takamasa et al., 2003; Hazuku et al., 2012). In the SIM procedure, bubble interfacial configurations, including the diameters and positions of bubbles in the pipe cross-section, were obtained by image-processing two backlit images taken at right angles to each other. The void fraction, interfacial area concentration, and bubble Sauter mean diameter profiles in pipe cross-section were calculated from the obtained images with an assumption of an ellipsoidal bubble.
More than 1000 bubbles were sampled to maintain similar statistics between the different combinations of experimental conditions. The void fractions measured by the image-processing method agreed with those obtained in a 1.09 mm round pipe by neutron radiography (Mishima and Hibiki, 1996) within the averaged relative deviation of 12.3%. The image-processing method for the interfacial area concentration measurement was also validated by a double-sensor conductivity probe method. A separate test was performed in a 25.4 mm round pipe, yielding good agreement for the interfacial area concentration measurement, within the averaged relative deviation of 6.95% (Hibiki et al., 1998). Since the measurement accuracy of the double sensor probe method is reported to be 7% (Wu and Ishii, 1999), the measurement accuracy of the image-processing method is within 15% by conservative estimate. The experimental conditions in each test pipe are tabulated in Tables 1–3.
Table 1 Experimental conditions in 1.03 mm pipe
Flow parameter #1 #2 #3
fj (m/s) 0.208 0.305 2.88
=g / 15z Dj (m/s) 0.00587 0.00554 0.0470
=g / 75z Dj (m/s) 0.00593 0.00556 0.0497
=g / 150z Dj (m/s) — — 0.0535
=/ 15z Dα (—) 0.0608 0.0168 0.0119
=Sm / 15z DD (mm) 0.583 0.404 0.314
i / 15z Da = (m−1) 625 249 228
Table 2 Experimental conditions in 3.00 mm pipe
Flow parameter #1 #2 #3 #4 #5
fj (m/s) 0.299 0.610 0.894 1.49 1.99
=g / 45z Dj (m/s) 0.00600 0.0121 0.0168 0.0280 0.0352
=g / 75z Dj (m/s) 0.00608 0.0126 0.0170 0.0255 0.0357
=g / 150z Dj (m/s) 0.00596 0.0119 0.0166 0.0276 0.0368
=/ 45z Dα (—) 0.0230 0.0116 0.0126 0.0164 0.0184
=Sm / 45z DD (mm) 0.914 0.689 0.791 0.795 0.797
i / 45z Da = (m−1) 151 101 95.6 123 139
Table 3 Experimental conditions in 5.00 mm pipe
Flow parameter #1 #2 #3 #4 #5
fj (m/s) 0.294 0.499 0.990 1.50 3.00
=g / 34z Dj (m/s) 0.00508 0.0134 0.0253 0.0395 0.0834
=g / 72z Dj (m/s) 0.00511 0.0124 0.0249 0.0393 —
=g / 110z Dj (m/s) 0.00501 0.0123 0.0249 0.0393 —
=/ 34z Dα (—) 0.0263 0.0201 0.023 0.0211 0.0198
=Sm / 34z DD (mm) 2.13 1.58 1.53 1.46 1.04
i / 34z Da = (m−1) 74.0 76.3 90.6 87.1 114
3 Results and discussion
3.1 Characteristics of local two-phase flow parameters
Figures 3, 4, and 5 respectively show the typical flow images along the test pipes in 1.03, 3.00, and 5.00 mm diameter pipes. As shown in the figure, bubble number decreases and bubble diameter increases due to bubble coalescence along the test pipe. It should be noted here that bubble breakup was not observed in the present experiment. For relatively low liquid flow rate conditions (for example, superficial liquid velocity, <jf> = 0.208 m/s in 1.03 mm diameter pipe, <jf> = 0.299 m/s in 3.00 mm pipe, and <jf> = 0.294 m/s in 5.00 mm pipe), majority of bubbles flow down continuously near the pipe center. At high flow rate conditions (for example, <jf> = 2.88 m/s in 1.03 mm pipe, <jf> = 1.99 m/s in 3.00 mm pipe, and <jf> = 1.50 m/s or 3.00 m/s in 5.00 mm diameter pipe), bubbles sliding on the pipe wall surface, namely the sliding bubbles, are formed.
Figure 6 shows the axial development of measured local void fraction profile, α in pipe cross-section in vertical downward bubbly flows. Open, triangle, and square symbols in each graph represent the data obtained at each axial location in the test pipes. The graphs displaying from the top to the bottom show in order of increase in liquid flow rate and the numbers indicated at right side correspond to those in Tables 1, 2, and 3. Left, middle, and right hand graphs respectively show the data in 1.03, 3.00, and 5.00 mm diameter pipes. In the same way, profiles of the local Sauter
Measurement of local two-phase flow parameters of downward bubbly flow in mini pipes
93
Fig. 3 Typical flow images in 1.03 mm diameter pipe.
Fig. 4 Typical flow images in 3.00 mm diameter pipe.
mean bubble diameter, DSm and the local interfacial area concentration, ai are shown in Figs. 7 and 8, respectively. Sauter mean bubble diameter was obtained from the local void fraction and the local interfacial area concentration with DSm = 6α/ai.
As shown by the void fraction profile, axial changes of the void fraction profiles are pronounced at low liquid flow rate conditions and core peak distributions which have sharp peak intensity near the pipe center are formed in all test
Fig. 5 Typical flow images in 5.00 mm diameter pipe.
pipes (Fig. 6, upper graphs). As the axial distance increases, the peak near the pipe center tends to gradually decrease in amplitude in 3.00 and 5.00 mm diameter pipes. This result is mainly caused by two reasons: increase in the bubble diameter due to bubble coalescence along the test pipe (Fig. 7, upper graphs), and decrease in gas volume due to increase in static pressure along the test pipe at such low liquid flow rate condition in which hydrostatic pressure loss dominates in total pressure loss.
As liquid flow rate increases, the void peak near the pipe center disappears and the profile tends to become the wall peak type with a broad peak between the pipe center and the wall. The value of void fraction near the pipe center tends to gradually increase with axial distance. This is mainly caused by an increase in bubble diameter due to bubble coalescence and pressure reduction due to increase in frictional pressure loss (Fig. 7, bottom graphs).
As for the interfacial area concentration profiles, the value near the pipe center gradually decreases with axial distance in 3.00 mm diameter pipe (Fig. 8, middle graphs in upper part). This is mainly caused by increase in the bubble diameter due to bubble coalescence (Fig. 7, middle graph in upper part). On the other hand, the axial changes of interfacial area concentration at low liquid flow rate condition in 1.03 and 5.00 mm diameter pipes are insignificant (Fig. 8, left and right hand graphs in upper part). As shown in Fig. 7, Sauter mean bubble diameters near the pipe center in 1.03 and 5.00 mm diameter pipes are respectively more than 0.75 and 3 mm, that means the bubble diameter covers the area more than half of pipe cross-section. This fact might cause restriction of relative motion between bubbles by presence of the wall and decrease in bubble collision frequency, resulting in the
T. Hazuku, T. Ihara, T. Hibiki
94
insignificant axial change of interfacial concentration profile in 1.03 and 5.00 mm diameter pipes.
At high liquid flow rate condition in which the wall peak distributions form, although the interfacial area concentration near the pipe wall tends to gradually decrease with axial distance due to bubble coalescence (Fig. 8, bottom graphs), the amount of change is less than that at low liquid flow rate condition. This may be because the residence time of two-phase flow in the length of the pipes tested and bubble coalescence efficiency are reduced at higher liquid flow rates, resulting in suppressed bubble coalescence.
3.2 Effect of flow parameters on phase distribution characteristics
Next, the phase distribution characteristics in vertical downward bubbly flows are quantitatively evaluated using a normalized peak void fraction NαP and a normalized void peak position rN aP , as defined by the following equations.
( )º -P P C PNα α α α (1)
rN r Ra aºP P (2)
where αP, αC, rαP, and R are void fraction at the peak, void
Fig. 6 Void fraction profile in pipe cross-section.
Measurement of local two-phase flow parameters of downward bubbly flow in mini pipes
95
fraction at the pipe center, radial void peak position in pipe cross-section, and the pipe radius, respectively. NαP is an index related to the ratio of void peak intensity to the void fraction at the pipe center. NαP = 0 and 1 indicate a very sharp core peak and no core peak, respectively. NrαP is an index related to the position of the void peak in the pipe cross-section. Increasing NrαP from 0 to 1 means the radial position of void peak moves from the pipe center to the wall.
Figures 9 and 10 show, respectively, the values of NαP and NrαP. Items (a), (b), and (c) in each figure indicate the dependencies of the liquid Reynolds number Ref , the
Sauter mean bubble diameter normalized by the pipe diameter <DSm>/D, and the ratio of frictional pressure gradient to buoyancy MF/Δρg on the normalized parameters, respectively. The open circle, triangle, and square symbols represent the results in 1.03, 3.00, and 5.00 mm diameter pipes, respectively.
With respect to the Reynolds number (Figs. 9(a) and 10(a)), both values of NαP and NrαP are null at relatively low liquid flow rates (Ref ≤ 2000). These results indicate that core peak distributions with strong peak intensity are formed at low liquid flow rates. At relatively high liquid flow rates like Ref ≥ 3000, the void peak position tends to move towards
Fig. 7 Sauter mean bubble diameter profile in pipe cross-section.
T. Hazuku, T. Ihara, T. Hibiki
96
the pipe wall and this transition of the peak position occurs in higher range of liquid Reynolds numbers as the pipe diameter becomes larger.
With respect to the Sauter mean bubble diameter (Figs. 9(b) and 10(b)), core peak distributions are formed regardless of the pipe diameter when <DSm>/D > 0.4, while dependency of the Sauter mean bubble diameter is insignificant when <DSm>/D < 0.4.
With respect to the MF/Δρg (Figs. 9(c) and 10(c)), both values of NαP and NrαP in the range of MF/Δρg < 1 become almost null, i.e., the core peak distribution, regardless of
the pipe diameter. In contrast, the values of NαP tend to gradually increase and the values of rN aP become over 0.5 in the range of MF/Δρg > 1. These results indicate that core peak distribution disappears and the distribution tends to gradually transit to wall peak distribution at MF/Δρg 1.
Since the buoyancy in the downward bubbly flow acts on the bubbles in the opposite flow direction, it is considered that the direction of lift force changes towards the pipe center and only core peak distribution pattern can be formed. However, the present result clearly indicates the existing of lift force pushing the bubbles towards the pipe wall causes
Fig. 8 Interfacial area concentration profile in pipe cross-section.
Measurement of local two-phase flow parameters of downward bubbly flow in mini pipes
97
the formation of wall peak distribution even in the downward bubbly flow. The formation of the wall peak distribution in the downward bubbly flow may be caused by the relative velocity due to the body acceleration which is generated by large frictional pressure gradient, as expressed by following equation (Tomiyama et al., 1998; Hibiki and Ishii, 2003; Hibiki et al., 2006, 2009).
[ ]= - +br r F
D f
8 Δ (1 )3rv v ρg α MC ρ
(3)
where rv , br , DC , fρ , Δρ , and g are, respectively, the relative velocity between phases, bubble radius, drag coefficient, liquid density, density difference between phases, and gravitational acceleration. A mathematical symbol of < > means the area-averaged value.
If Eq. (3) is applicable to the downward flow in mini pipe, the relative velocity, rv takes a negative value at MF/Δρg < 1 and a positive value at MF/Δρg > 1. It is confirmed that a threshold value at the phase distribution transition was MF/Δρg 1 in the present analysis. Thus, the mechanisms behind the lift force formation due to the relative velocity between both phases may be reasonably explained by Eq. (3).
Although validation by simultaneous measurements with local relative velocity between phases would be required in order to evaluate validity of the above explained mechanism, the expression by Eq. (3) might become useful in various two-phase flow modeling.
4 Conclusions
Local two-phase flow parameters including profiles of void fraction, Sauter mean bubble diameter, and interfacial area concentration in the vertical downward bubbly flow in mini pipes with inner diameters of 1.03, 3.00, and 5.00 mm were measured using a stereo image-processing method. The axial developments of the flow parameters and the gravity effect on them were discussed in detail based on the obtained data and the visual observation. It was confirmed that the core peak distributions were formed at low liquid flow rate conditions in which the buoyancy force dominated while the wall peak distributions were formed at high liquid flow rate conditions in which the body acceleration due to the frictional pressure gradient dominated. The result indicated the existence of lift force pushing the bubbles
Fig. 9 Normalized peak void fraction.
Fig. 10 Radial void peak position.
T. Hazuku, T. Ihara, T. Hibiki
98
towards the pipe wall even in vertical downward flows. The database obtained through the present experiment will be useful in the validation of the existing flow models or the benchmarking of various CFD simulation codes.
Acknowledgements
The authors are thankful to Professor T. Takamasa and Mrs. Y. Ohkubo of Tokyo University of Marine Science and Technology for their assistance in conducting the experiment.
References
Chuang, T. J., Hibiki, T. 2015. Vertical upward two-phase flow CFD using interfacial area transport equation. Prog Nucl Energ, 85: 415–427.
Delhaye, J. M., Bricard, P. 1994. Interfacial area in bubbly flow: Experimental data and correlations. Nucl Eng Des, 151: 65–77.
Ghiaasiaan, S. M. 2003. Gas–liquid two-phase flow and boiling in mini and microchannels. Multiphase Sci Tech, 15: 323–334.
Gibson, A. H. 1913. LXXXIV. On the motion of long air-bubbles in a vertical tube. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 26: 952–965.
Hazuku, T., Hibiki, T., Takamasa, T. 2016. Interfacial area transport due to shear collision of bubbly flow in small-diameter pipes. Nucl Eng Des, 310: 592–603.
Hazuku, T., Takamasa, T., Hibiki T. 2012. Characteristics of developing vertical bubbly flow under normal and microgravity conditions. Int J Multiphase Flow, 38: 53–66.
Hazuku, T., Takamasa, T., Hibiki, T. 2010. Interfacial-area transport of vertical upward bubbly flow in mini pipe. Int J Microscale and Nanoscale Thermal and Fluid Transport Phenomena, 1: 59–84.
Hazuku, T., Takamasa, T., Hibiki, T. 2015. Phase distribution characteristics of bubbly flow in mini pipes under normal and microgravity conditions. Microgravity Sci Tech, 27: 75–96.
Hibiki, T., Hazuku, T., Takamasa, T., Ishii, M. 2007. Some characteristics of developing bubbly flow in a vertical mini pipe. Int J Heat Fluid Fl, 28: 1034–1048.
Hibiki, T., Hazuku, T., Takamasa, T., Ishii, M. 2009. Interfacial-area transport equation at reduced-gravity conditions. AIAA J, 47: 1123–1131.
Hibiki, T., Hogsett, S., Ishii, M. 1998. Local measurement of interfacial area, interfacial velocity and liquid turbulence in two-phase flow. Nucl Eng Des, 184: 287–304.
Hibiki, T., Ishii, M. 2001. Interfacial area concentration in steady fully-developed bubbly flow. Int J Heat Mass Tran, 44: 3443–3461.
Hibiki, T., Ishii, M. 2003. One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase flow regimes. Int J Heat Mass Tran, 46: 4935–4948.
Hibiki, T., Ishii, M. 2009. Interfacial area transport equations for gas– liquid flow. The Journal of Computational Multiphase Flows, 1: 1–22.
Hibiki, T., Ozaki, T., Shen, X. Z., Miwa, S., Kinoshita, I., Hazuku, T., Rassame, S. 2018. Constitutive equations for vertical upward two-
phase flow in rod bundle. Int J Heat Mass Tran, 127: 1252–1266. Hibiki, T., Takamasa, T., Ishii, M., Gabriel, K. S. 2006. One-dimensional
drift-flux model at reduced gravity conditions. AIAA J, 44: 1635–1642.
Ishii, M., Hibiki, T. 2010. Thermo-Fluid Dynamics of Two-Phase Flow, 2nd edn. Springer Science & Business Media.
Kandlikar, S. G. 2002. Two-phase flow patterns, pressure drop, and heat transfer during boiling in minichannel flow passages of compact evaporators. Heat Transfer Eng, 23: 5–23.
Kandlikar, S. G. 2004. Heat transfer mechanisms during flow boiling in microchannels. J Heat Transfer, 126: 8–16.
Kocamustafaogullari, G., Huang, W. D., Razi, J. 1994. Measurement and modeling of average void fraction, bubble size and interfacial area. Nucl Eng Des, 148: 437–453.
Lin, C. H., Hibiki, T. 2014. Databases of interfacial area concentration in gas–liquid two-phase flow. Prog Nucl Energ, 74: 91–102.
Liu, H., Hibiki, T. 2018. Bubble breakup and coalescence models for bubbly flow simulation using interfacial area transport equation. Int J Heat Mass Tran, 126: 128–146.
Milliest, M., Drew, D. A., Lahey, R. T. Jr. 1996. A first order relaxation model for the prediction of the local interfacial area density in two-phase flows. Int J Multiphase Flow, 22: 1073–1104.
Mishima, K., Hibiki, T. 1996. Some characteristics of air–water two- phase flow in small diameter vertical tubes. Int J Multiphase Flow, 22: 703–712.
Qu, W. L., Mudawar, I. 2003. Measurement and prediction of pressure drop in two-phase micro-channel heat sinks. Int J Heat Mass Tran, 46: 2737–2753.
Serizawa, A., Feng, Z. P., Kawara, Z. 2002. Two-phase flow in microchannels. Exp Therm Fluid Sci, 26: 703–714.
Shen, X. Z., Hibiki, T. 2018. Bubble coalescence and breakup model evaluation and development for two-phase bubbly flows. Int J Multiphase Flow, 109: 131–149.
Shen, X. Z., Schlegel, J. P., Hibiki, T., Nakamura, H. 2018. Some characteristics of gas–liquid two-phase flow in vertical large- diameter channels. Nucl Eng Des, 333: 87–98.
Takamasa, T., Goto, T., Hibiki, T., Ishii, M. 2003. Experimental study of interfacial area transport of bubbly flow in small-diameter tube. Int J Multiphase Flow, 29: 395–409.
Takamasa, T., Miyoshi, N. 1993. Measurements of bubble interface configurations in vertical bubbly flow using image-processing method. Transactions of the Japan Society of Mechanical Engineers Series B, 59: 2403–2409.
Tomiyama, A., Kataoka, I., Zun, I., Sakaguchi, T. 1998. Drag coefficients of single bubbles under normal and micro gravity conditions. JSME International Journal Series B, 41: 472–479.
Wu, Q., Ishii, M. 1999. Sensitivity study on double-sensor conductivity probe for the measurement of interfacial area concentration in bubbly flow. Int J Multiphase Flow, 25: 155–173.
Zhang, W., Hibiki, T., Mishima, K., Mi, Y. 2006. Correlation of critical heat flux for flow boiling of water in mini-channels. Int J Heat Mass Tran, 49: 1058–1072.
Zuber, N., Findlay, J. A. 1965. Average volumetric concentration in two-phase flow systems. J Heat Transf, 87: 453–468.
