void fraction plays a fundamental role in the modeling of two-phase flow pattern transitions, heat .... where t is the average liquid film thickness and d is the tube diameter. The liquid ..... vertical condenser tubes where the vapor phase flows upward but the shear it exerts on the ...... phase air-water flow in an inclined pipe.
John R. Thome and Andrea Cioncolini, 2015. Encyclopedia of Two-Phase Heat Transfer and Flow, Set 1: Fundamentals and Methods, Volume 3: Flow Boiling in Macro and Microchannels, World Scientific Publishing.
Chapter 4 Void Fraction
The void fraction is one of the most important parameters used to characterize two-phase flows in channels. It is required as input for determining numerous other key flow parameters, such as the two-phase density and viscosity or the cross sectional average velocities of the liquid and gas phases. Moreover, the void fraction plays a fundamental role in the modeling of two-phase flow pattern transitions, heat transfer and pressure drop. In this chapter, after defining the void fraction, discussing its practical relevance and describing the measuring techniques developed to date, available prediction methods are reviewed and critically analyzed.
1. Definitions and Practical Relevance The cross sectional void fraction ε, simply referred to as the void fraction in what follows, is a dimensionless geometric flow parameter defined as the ratio of the channel cross sectional area Ag occupied by the gas or vapor phase to the total cross sectional area A of the channel as follows:
ε=
Ag A
=
Ag Al + Ag
(1)
where Al is the channel cross sectional area occupied by the liquid phase, as shown schematically in Fig. 1 (top). As such, the void fraction is a flow parameter bounded between 0, corresponding to single-phase liquid flow, and 1 corresponding to single-phase gas flow. Values of the void fraction close to 0+ are characteristic of bubbly two-phase flow with most of the channel cross section occupied by the liquid phase and only a few gas bubbles entrained in the continuous liquid phase. On the contrary, values of the void fraction close to 1are typical of dispersed mist flow with most of the channel cross section occupied by the gas and only a few liquid droplets entrained in the continuous gas phase. More generally, the void fraction is related to the morphology of the two-phase flow: bubbly flow typically corresponds to void fraction values below 0.2–0.3, intermittent flows have void fraction values from about 0.2–0.3 up to 1
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0.7–0.8 while annular and mist flows are characterized by void fraction values above 0.7–0.8. Al Ag
Vl
Vg
Lg
Ll
Figure 1. Schematic representation of cross sectional void fraction for bubbly flow (top); schematic representation of volumetric void fraction (middle) and chordal void fraction (bottom).
It is worth noting that the void fraction ε, which is a geometric flow parameter, is profoundly different from the vapor quality x, which is a transport flow parameter. While the local value of the vapor quality is typically known from the process or from an energy/mass balance, the void fraction instead must be predicted from the flow variables.
Void Fraction
3
Besides being required as input for predicting the acceleration and gravitational pressure gradients and to derive the wall shear stress from measured pressure gradient data (see Chapter 6), the void fraction is crucial in most thermal-fluid simulations, such as coupled neutronics-thermal hydraulic calculations for nuclear power plants or prediction of flow/heat transport rates for two-phase natural circulation loops (thermosyphons). In addition to the (cross sectional) void fraction ε defined in Eq. (1) relative to the channel cross sectional area, refer to Fig. 1 (top), other void fraction definitions of frequent use are the volumetric void fraction εv, defined relative to the volume of a portion of the channel, and the chordal void fraction εch, defined relative to a line drawn through the channel cross section. The volumetric void fraction εv is defined as the ratio of the channel volume Vg occupied by the gas or vapor phase to the total volume V of the channel as follows:
εv =
Vg V
=
Vg Vl + Vg
(2)
where Vl is the channel volume occupied by the liquid phase, as shown schematically in Fig. 1 (middle). The chordal void fraction εch is defined as the ratio of the chord length Lg lying in the gas or vapor phase to the total length L of the chord as follows:
ε ch =
Lg L
=
Lg Ll + Lg
(3)
where Ll is the chord length lying in the liquid phase, as shown schematically in Fig. 1 (bottom). Although the most relevant void fraction definition for practical applications is the cross sectional one, the volumetric and the chordal void fractions are sometimes those that are experimentally accessible, as will be discussed in more detail in the following Section 2. It is therefore crucial to relate the cross sectional void fraction with the volumetric and chordal void fractions, in order to derive the cross sectional void fraction from volumetric and chordal measurements. Clearly, if the morphology of the two-phase flow is not varying appreciably along the channel, then the cross sectional void fraction is approximately constant along the channel and is numerically coincident with the volumetric void fraction. Relating the cross sectional and the chordal void fractions is not as easy, though, and normally requires a dedicated calibration of the experimental stand. A further void fraction definition of frequent use is the local void fraction εloc, defined in the neighborhood of a point inside the channel. This is actually a small
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volumetric void fraction, calculated in a small volume that envelopes the point of interest. It is worth noting, finally, that the complement to the void fraction, i.e. the flow parameter (1-ε), is referred to as the liquid holdup. Whilst this term is commonly used by the two-phase flow community, the flow parameter (1-ε) is actually the liquid fraction, since the liquid is flowing and not stopped or held up within the channel. 2. Measuring Techniques The most frequently used experimental techniques for measuring the void fraction are (i) the quick-closing valves method, (ii) the radiation attenuation method, (iii) the electrical impedance method and (iv) the processing of video images. These have been used in numerous studies and are discussed below.
Test Tube
Quick-closing valves
Flow Figure 2. Schematic representation of the quick-closing valve technique for volumetric void fraction measurement.
In the quick-closing valves method, a portion of the test tube is isolated using a pair of quick-closing valves which are tripped to close simultaneously, as illustrated schematically in Fig. 2. The liquid that is trapped between the valves is drained and its volume measured. Knowing the total volume of the test tube
Void Fraction
5
between the valves, the volumetric void fraction can then be calculated. The variation of the morphology of the two-phase flow along the test tube is normally neglected, so that the cross sectional void fraction can be assumed to be constant along the channel and numerically coincident with the measured volumetric void fraction. Clearly, the shorter the test tube, the smaller the variation of the morphology of the two-phase flow. In the radiation attenuation method, collimated beams of radiation (photons or neutrons) are directed through the test tube in the same cross sectional plane, as depicted in Fig. 3 for a three-beam photon configuration.
Radiation source Photomultiplier tubes
Figure 3. Schematic representation of a three-beam photon attenuation method for chordal void fraction measurement.
The beams are attenuated according to the chordal void fraction along their path, so that the measurement of the attenuation of each beam yields the chordal void fraction along the beam path. This technique requires a model to connect the cross sectional void fraction to the measured chordal void fractions, and this model is normally derived empirically from a preliminary calibration of the experimental stand. In the electrical impedance method, a pair of electrodes is connected to the test tube, as illustrated schematically in Fig. 4 for the case of half-cylinder electrodes positioned opposite to each other at the two sides of the test tube. Since the electric properties of the liquid and gas phases are normally different, measuring the electrical impedance of the two-phase flow provides the volumetric void fraction in the tube region enveloped by the electrodes. Depending on the electrical properties of the two phases and on the configuration of the electrodes, the measured impedance of the two-phase flow can be controlled by conductance, by capacitance or by both. The electrodes are typically a few channel diameters long, so that the cross sectional void fraction is assumed to be constant in the region between the electrodes and numerically coincident with the measured volumetric void fraction.
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Image processing of high definition photographs or videos taken with transparent test sections is also used to determine the volumetric or cross sectional void fraction, particularly in small channels where the other techniques are difficult to apply. Accounting properly for the refractive index and the internal/external tube curvature, image processing is also applicable to stratified wavy two-phase flow in horizontal large channels, as proven by Wojtan et al. (2004, 2005) and Ursenbacher et al. (2004).
Electrodes
Figure 4. Schematic representation of electrical impedance method for volumetric void fraction measurement.
d
t
Figure 5. Schematic representation of annular two-phase flow.
In the case of annular two-phase flow through circular pipes, as schematically depicted in Fig. 5, the void fraction can also be derived indirectly from measurement of the average thickness of the annular liquid film. Assuming
Void Fraction
7
cylindrical symmetry of the flow, the liquid film flow area Alf can be calculated as:
Alf = π t ( d − t )
(4)
where t is the average liquid film thickness and d is the tube diameter. The liquid film flow area Alf can also be calculated as a function of the void fraction ε as follows:
Alf = (1 − ε ) (1 − γ )
π 2 d 4
(5)
where γ is the liquid droplet hold-up, representing the fraction of the liquid phase cross sectional area occupied by the entrained droplets. Substituting Eq. (5) into Eq. (4) and rearranging yields:
(1 − ε ) (1 − γ ) =
4 t (d − t ) d2
(6)
A further relation that involves the void fraction ε and the droplet hold-up γ can be derived if the entrained liquid droplets and the carrier gas phase are assumed to travel at the same speed in the channel, i.e. if the slip between carrier gas and entrained liquid droplets is neglected:
γ
1− ε
ε
=e
1 − x ρg x ρl
(7)
where e is the entrained liquid fraction, representing the ratio of the entrained liquid mass flow rate to the total liquid mass flow rate (see Chapter 5 for more details), x is vapor quality and ρl and ρg are the liquid and gas densities. As can be seen, Eqs. (6) and (7) form a system of two nonlinear equations that can be solved for ε and γ, provided that the right-hand sides of both equations are known. Since the average liquid film thickness t is measured, the only parameter that remains to be determined is the entrained liquid fraction e that can be predicted with one of the techniques described in Chapter 5. If the liquid droplet hold-up is small (γ > t), then Eq. (6) yields:
ε ≈ 1− 4
t d
(8)
Equation (8) is explicit and can be used to provide a preliminary estimate of the void fraction in annular two-phase flow from measurements of the average thickness of the annular liquid film.
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J.R. Thome & A. Cioncolini
3. Experimental Studies A selection of studies that provide data on the void fraction in two-phase flow is summarized in Tables 1(a) and 1(b). Selected histograms that further describe these data are shown in Fig. 6. Table 1(a). Experimental void fraction databank. Reference Fluids d (mm) P (MPa) Anderson and Mantzouranis (1960) H2O-Air 10.8 0.11 Beggs (1972) H2O-Air 25.4; 38.1 0.55-0.68 Celata and Frazzoli (1981) H2O-Steam 21.2 4.9-6.9 Alia et al. (1965) H2O-Ar 15.0; 25.0 0.60-2.2 Alcohol-Ar Dejesus and Kawagi (1990) H2O-Air 25.4 0.15 Godbole et al. (2011) H2O-Air 12.7 0.11-0.26 Kaji and Azzopardi (2010) H2O-Air 19.0 0.15 Leung et al. (2005) H2O-Steam 13.4 2.0 Mukherjee (1979) Kerosene-Air 38.1 0.27-0.61 Spedding and Nguyen (1976) H2O-Air 45.5 0.11-0.12 Sujumnong (1998) H2O-Air 12.7 0.11-0.34 Ueda (1967) H2O-Air 19.4 0.11 Shedd (2010) R410a 1.05; 2.96 1.9-3.1 Silvestri et al. (1963) H2O-Ar 25.0 0.60-2.1 H2O+Alcohol-Ar Adorni et al. (1963) H2O-Ar 15.1 0.60-2.1 Casagrande et al. (1963) H2O-N2 25.0 0.29-2.4 H2O+Alcohol-Ar Cravarolo et al. (1964) H2O-Ar 15.1; 25.0 0.60-2.1 Alia et al. (1966) H2O-Ar 15.1; 25.0 0.60-2.1 Gill et al. (1964,1965) H2O-Air 31.8 0.11 Hall-Taylor et al. (1963) H2O-Air 31.8 0.11-0.16 Whalley et al. (1974) H2O-Air 31.8 0.12-0.35 Brown (1978) H2O-Air 31.8 0.17-0.31 Würtz (1978) H2O-Steam 20.0 7.0 Jones and Zuber (1975) H2O-Air 9.23 a 0.11 Hori et al. (1995) H2O-Steam 10.1 a 4.9-16.6 Takenaka and Asano (2005) H2O-Air 4.65 a 0.11 Morooka et al. (1989) H2O-Steam 12.1 a 0.49.0.98 Das et al. (2002) H2O-Air 25.4 a 0.11
G (kgm-2s-1) 16-1419 60-1617 170-300 306-3000 200-1320 78-1725 31-695 4500 23-2686 6-1057 55-8476 92-661 400-800 280-2900 312-3420 255-2880 266-2880 266-3290 24-555 34-76 78-789 158-316 500-2000 150-3057 556-4167 88-99 833-1389 139-491
The databank in Table 1 contains 3376 measurements of the void fraction collected from 29 different literature studies that cover almost exclusively vertical upflow conditions, 8 different gas-liquid and vapor-liquid combinations (both single-component saturated fluids such as water-steam and refrigerant R410a and two-component fluids, such as water-air, water-argon, water-nitrogen, water plus alcohol-air, alcohol-air and kerosene-air), operating pressures in the range of 0.1‒16.6 MPa, tube diameters from 1.05 mm to 45.5 mm, adiabatic and
Void Fraction
9
evaporating flow conditions (uniform and non-uniform heating) and circular and non-circular channels (annuli, rectangular channels and rod bundles). Table 1(b). Experimental void fraction databank with additional details. Reference x (1) (2) (3) (4) (5) Anderson and Mantzouranis (1960) 0.01-0.70 23.5 ↑ a 71 QCV Beggs (1972) 0.01-0.83 na ↑ a 14 QCV Celata and Frazzoli (1981) 0.16-0.98 na ↑ h 132 RA Alia et al. (1965) 0.01-0.81 60-157 ↑ a 336 RA Dejesus and Kawagi (1990) 0.01-0.12 66 ↑ a 54 RA Godbole et al. (2011) 0.01-0.32 50 ↑ a 148 QCV Kaji and Azzopardi (2010) 0.01-0.67 300 ↑ a 91 EI Leung et al. (2005) 0.01-0.11 na ↑ h 16 RA Mukherjee (1979) 0.02-0.76 na ↑ a 65 EI Spedding and Nguyen (1976) 0.01-0.93 na ↑ a 224 QCV Sujumnong (1998) 0.01-0.77 na ↑ a 104 QCV Ueda (1967) 0.01-0.05 68 ↑ a 68 QCV Shedd (2010) 0.02-0.96 90-114 → a 200 EI Silvestri et al. (1963) 0.06-0.84 60 ↑ a 436 FT Adorni et al. (1963) 0.06-0.82 99 ↑ a 121 FT Casagrande et al. (1963) 0.07-0.79 60 ↑ a 109 FT Cravarolo et al. (1964) 0.04-0.79 60-232 ↑ a 517 FT Alia et al. (1966) 0.04-0.90 140-233 ↑ a 136 FT Gill et al. (1964,1965) 0.09-0.94 64-171 ↑ a 147 FT Hall-Taylor et al. (1963) 0.41-0.78 184 ↑ a 18 FT Whalley et al. (1974) 0.10-0.90 590 ↑ a 139 FT Brown (1978) 0.33-0.66 420 ↑ a 30 FT Würtz (1978) 0.20-0.70 450 ↑ a 18 FT Jones and Zuber (1975) 0.01-0.12 264 ↑ a 15 RA Hori et al. (1995) 0.01-0.18 na ↑ h 37 RA Takenaka and Asano (2005) 0.03.0.12 20 ↑ a 7 RA Morooka et al. (1989) 0.01-0.12 na ↑ h 20 RA Das et al. (2002) 0.01-0.03 na ↑ a 90 QCV (1): Dimensionless distance L/d of test section inlet from mixer (2 component fluids, adiabatic tests only) (2): Flow direction: ↑ = vertical upflow; → = horizontal flow (3): Type of test: a = adiabatic; h = heated (4): Number of data points (5): Measuring technique: QCV = quick closing valves; RA = radiation attenuation; EI = electrical impedance; FT = annular film thickness a Hydraulic diameter (4 Aflow Pwet-1, Aflow = flow area, Pwet = wetted perimeter)
Although the majority of these studies addressed low pressure, low mass flux adiabatic flow conditions, the databank in Table 1 includes also measurements carried out in evaporating flow conditions with water-steam (Celata and Frazzoli, 1981; Leung et al., 2005; Hori et al., 1995 and Morooka et al., 1989). In particular, Celata and Frazzoli (1981) used electrical heating and both uniform and outlet-peaked axial power distributions.
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J.R. Thome & A. Cioncolini
3000
Nr Data Pts
Nr Data Pts
2000 1500 1000 500 0 0
5
10
Pressure [MPa]
15
2000
1000
0 250
20
600 400 200 0 0
2000
4000
6000
8000
Mass Flux [kgm-2s-1]
450
500
550
600
650
400 200 0.2
0.4
0.6
0.8
1
0.4
0.6
0.8
1
Vapor Quality
600
Nr Data Pts
Nr Data Pts
400
Temperature [K]
600
0 0
10000
1500
1000
500
0 0
350
800
800
Nr Data Pts
Nr Data Pts
1000
300
10
20
30
Tube Diameter [mm]
40
50
400
200
0 0
0.2
Void Fraction
Figure 6. Selected histograms describing the void fraction databank in Table 1.
Moreover, as can be seen in Table 1, five studies addressed non-circular channels. In particular, Jones and Zuber (1975) measured the void fraction in a rectangular channel (4.98 mm side, 63.5 mm height, aspect ratio of 12.75 and hydraulic diameter of 9.23 mm) with water-air at low pressure. Hori et al. (1995) performed their measurements using water in evaporating flow conditions in a channel designed to mimic the inner subchannel among four fuel rods of a pressurized water nuclear reactor fuel bundle (hydraulic diameter of 10.1 mm). Takenaka and Asano (2005) tested adiabatic water-air at low pressure in a tight lattice rod bundle geometry (hydraulic diameter of 4.65 mm), while Morooka et al. (1989) used low pressure evaporating water in a rod bundle designed to simulate a boiling water nuclear reactor fuel assembly (hydraulic diameter of 12.1 mm). Finally, Das et al. (2002) performed their measurements with
Void Fraction
11
adiabatic water-air at low pressure in a concentric annulus (hydraulic diameter of 25.4 mm). For the annular flow void fraction data in Table 1 indirectly derived from the measurement of the average liquid film thickness (labeled ‘FT’ in Table 1), the method of Cioncolini and Thome (2012b) was used to predict the entrained liquid fraction required to calculate the void fraction from the measured liquid film thickness, as explained in the previous section. It is well known that adiabatic, two-component two-phase flows can be quite slow in approaching fully developed flow conditions and losing any memory effect of the mixing device. As can be seen in Table 1, although most of the adiabatic test rigs have been designed with calming sections long enough to significantly damp out any dependence on inlet conditions, a residual dependence on inlet effects might be present in some of the tests. Inlet effects, however, are not taken into account here. In order to get some clue regarding the relation that links the void fraction with the principal flow parameters, a selection of data from Table 1 is presented in Figs. 7 through 12, where the void fraction is plotted versus vapor quality in terms of operating pressure (Fig. 7), in terms of mass flux (Figs. 8, 9 and 10) and as a function of tube diameter (Figs. 11 and 12). As can be seen in Figs. 7 through 10, the void fraction increases with an increase of vapor quality, a decrease of operating pressure and an increase of mass flux. The trend of the void fraction with vapor quality, in particular, is growing and saturating as both vapor quality and void fraction approach unity. The effect of mass flux, however, is noticeable only when the mass flux variation is significant and it may be only a few percent, so that it might be regarded as a second order effect. Finally, there is no noticeable effect of tube diameter in Fig. 11, although microscale data in Fig. 12 show that the void fraction increases with a decrease of tube diameter. These observed trends of the void fraction can be interpreted physically as follows. An increase of vapor quality means that a bigger fraction of the total mass flow rate is transported as gas, so that more space in the channel is required to accommodate the higher gas mass flow rate and the void fraction correspondingly increases. An increase of the operating pressure reduces the gas specific volume, so that the gas phase occupies less space in the channel and the void fraction accordingly decreases. With saturated flows, an increase of the operating pressure is followed by an increase of the operating temperature, which triggers a reduction of the surface tension. This yields a finer dispersion of the phases, a behavior of the two-phase flow closer to a homogeneous mixture and a higher void fraction. Within the limits of the databank in Table 1, which is mostly two-component two-phase flow, the former effect of the operating pressure seems to be the dominant one. An increase of the mass flux yields a
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J.R. Thome & A. Cioncolini
more tight mass and linear momentum coupling between the phases, a behavior of the two-phase flow closer to a homogeneous mixture and correspondingly a higher void fraction. 1 0.9
Void Fraction
0.8 0.7 0.6 0.5 0.4 P=3.1 MPa P=1.9 MPa
0.3 0.2 0
0.1
0.2
0.3
0.4
0.5
0.6
Vapor Quality
0.7
0.8
0.9
1
Figure 7. Void fraction vs. vapor quality: effect of operating pressure (R410a data of Shedd (2010); Mass flux: 600 kgm-2s-1; Tube diameter: 2.96 mm).
1
Void Fraction
0.95
0.9
0.85 G=500 kgm-2s-1
0.8
G=1000 kgm-2s-1 G=2000 kgm-2s-1
0.75 0.3
0.35
0.4
0.45
0.5
Vapor Quality
0.55
0.6
0.65
0.7
Figure 8. Void fraction vs. vapor quality: effect of mass flux (H2O data of Würtz (1978); Pressure: 7.0 MPa; Tube diameter: 20.0 mm).
Finally, there seems to be no effect of the tube diameter except for very small tubes. Slender tubes better confine the flow, preventing the phases from developing a slip and yielding a behavior of the two-phase flow closer to a homogeneous mixture with a higher void fraction. It is worth highlighting, however, that the microscale data of Shedd (2010) presented in Fig. 12 are the first reliable microscale data that appear in the literature and have not been confirmed yet by independent measurements.
Void Fraction
13
1 0.95
Void Fraction
0.9 0.85 0.8 0.75 G=1000 kgm-2s-1
0.7 0.65 0.1
G=2000 kgm-2s-1 0.2
0.3
0.4
0.5
Vapor Quality
0.6
0.7
0.8
0.9
Figure 9. Void fraction vs. vapor quality: effect of mass flux (H2O-Argon data of Alia et al. (1965); Pressure: 2.2 MPa; Tube diameter: 25.0 mm).
1 0.9
Void Fraction
0.8 0.7 0.6 0.5
G=400 kgm-2s-1 G=600 kgm-2s-1
0.4
G=800 kgm-2s-1 0
0.1
0.2
0.3
0.4
0.5
0.6
Vapor Quality
0.7
0.8
0.9
1
Figure 10. Void fraction vs. vapor quality: effect of mass flux (R410a data of Shedd (2010); Pressure: 1.9 MPa; Tube diameter: 2.96 mm).
Additional measurements of the void fraction in microchannels have been provided by Triplett et al. (1999), Serizawa et al. (2002), Chung and Kawaji (2004) and Kawahara et al. (2005), covering five fluids (water-steam, water-air, water-nitrogen, water plus ethanol-nitrogen and ethanol-nitrogen) and tube diameters from 0.020 mm to 0.53 mm. In these studies, image processing of high definition photographs taken with transparent test sections was used to deduce the volumetric void fraction. In particular, these authors processed their images as follows: in bubbly flows, bubbles were assumed to be either spherical or ellipsoidal; in elongated bubble flow, individual bubbles were assumed to be cylindrical with hemispherical caps; in churn flow, a void fraction value of 0.5
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J.R. Thome & A. Cioncolini
was arbitrarily assumed for the dispersed gas segments in the images; while slugannular flows were not analyzed due to the difficulty in processing the images. 1 0.95
Void Fraction
0.9 0.85 0.8 0.75 0.7 d=25 mm d=15 mm
0.65 0
0.1
0.2
0.3
0.4
0.5
Vapor Quality
0.6
0.7
0.8
0.9
Figure 11. Void fraction vs. vapor quality: effect of tube diameter (H2O-Argon data of Alia et al. (1965); Pressure: 2.2 MPa; Mass flux: 1000 kgm-2s-1).
1 0.9
Void Fraction
0.8 0.7 0.6 0.5 0.4 d=2.96 mm d=1.19 mm
0.3 0.2 0
0.1
0.2
0.3
0.4
0.5
0.6
Vapor Quality
0.7
0.8
0.9
1
Figure 12. Void fraction vs. vapor quality: effect of tube diameter (R410a data of Shedd (2010); Pressure: 2.0 MPa; Mass flux: 800 kgm-2s-1).
Although valuable, these microscale void fraction data should be regarded as more qualitative than quantitative, due to the optical distortion of the images and the several arbitrary assumptions made in the successive image processing. As such, these microscale data were not included in the databank in Table 1.
Void Fraction
15
4. Prediction Methods Void fraction prediction methods can be classified into four main groups: slip ratio methods, so-called Kεh methods, drift-flux correlations and finally wholly empirical correlations. In what follows, these four groups of void fraction prediction methods are presented and selected correlations for each group described. More details can be found in the recent assessments of void fraction prediction methods of Vijayan et al. (2000), Coddington and Macian (2002), Woldesemayat and Ghajar (2007) and Godbole et al. (2011). Asymptotic consistency of a void fraction prediction method requires that ε → 0+ for x → 0+; ε → 1- for x → 1- and ε → x+ in the limit of ρg ρl-1 → 1-, i.e. when the densities of the two phases become similar. As can be seen, although the void fraction and the vapor quality are different flow parameters, geometric the former and transport-related the latter, they tend to coincide when the density ratio tends to unity. This happens, for example, with saturated two-phase flows at operating pressures close to the critical pressure. 4.1 Slip ratio methods These prediction methods specify an empirical relationship for predicting the slip between the phases. In particular, in one-dimensional, segregated two-phase flow approximation, the mass flow rates of the liquid Γl and gas Γg phases in the channel can be expressed as follows:
Γl = (1 − x) Γ = ρl Vl Al = ρl Vl (1 − ε ) A
(9)
Γg = x Γ = ρ g Vg Ag = ρ g Vg ε A
(10)
where Γ is the total mass flow rate flowing in the channel while Vl and Vg are the average one-dimensional velocities of the liquid and gas phases. Taking the ratio of Eq. (9) to Eq. (10) yields:
1 − x ρl Vl 1 − ε = x ρ g Vg ε
(11)
This last equation links the vapor quality, the densities of the phases, their average velocities and the void fraction. Solving for the void fraction results in:
⎛ 1 − x ρg ⎞ ⎟⎟ ε = ⎜⎜1 + S gl x ρ l ⎠ ⎝
−1
(12)
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J.R. Thome & A. Cioncolini
where the slip ratio Sgl is a dimensionless flow parameter defined as the ratio of the average gas velocity to the average liquid velocity:
S gl =
Vg Vl
(13)
The slip ratio is an unbounded flow parameter that can take on positive as well as negative values. In most applications its value is above unity, indicating that the phases are separated in the channel and are flowing in the same direction (upward, downward or horizontally) with the gas traveling faster than the liquid. Positive values below unity can be found in downward annular flows at very low mass fluxes, where the pull of gravity accelerates the liquid to flow faster than the gas. Negative values of the slip ratio, finally, can be found in annular flows in vertical condenser tubes where the vapor phase flows upward but the shear it exerts on the liquid condensate film is not large enough to overcome the downward pull of gravity, so that the condensate drains downwards. Since the vapor quality and the densities of the phases are normally known, Eq. (12) can be used to predict the void fraction provided that the slip ratio can be estimated from the flow variables. Slip ratio methods then specify an empirical relation to predict the slip ratio and, in turn, the void fraction using Eq. (12). The simplest slip ratio method is the so-called homogeneous model: the liquid and gas phases are assumed to flow in the channel at the same velocity, so that the slip ratio is one and the void fraction is correspondingly given as: −1
⎛ 1 − x ρg ⎞ x ⎟⎟ = ε h = ⎜⎜1 + x ρl ⎠ x + (1 − x) ρ g ρl−1 ⎝
(14)
which is called homogeneous void fraction εh. As can be seen, Eq. (14) is asymptotically consistent for x → 0+, x → 1- and ρg ρl-1 → 1-. The homogeneous model is in general appropriate for unseparated flows, such as bubbly flow and mist flow, when gravity is not significantly affecting the flow. For separated flows with slip ratios above unity, the homogeneous model overpredicts the void fraction, but still providing a simple and useful upperbound to the actual value of the void fraction, a benchmark frequently used to check the consistency of measured data or predictions. The trends of the homogeneous void fraction versus vapor quality and density ratio ρg ρl-1 are shown in Fig. 13. As can be seen, the predicted void fraction profiles are growing and saturating with vapor quality at a rate of growth modulated by the density ratio.
Void Fraction
17
Homogeneous Void Fraction
1
0.8
0.6
0.4 Density Ratio=10-3 Density Ratio=10-2
0.2
0 0
Density Ratio=10-1 Density Ratio=1 0.1
0.2
0.3
0.4
0.5
0.6
Vapor Quality
0.7
0.8
0.9
1
Figure 13. Homogeneous void fraction vs. vapor quality and density ratio.
In the method proposed by Fauske (1962), the slip ratio is predicted as: 1
⎛ρ S gl = ⎜ l ⎜ρ ⎝ g
⎞2 ⎟ ⎟ ⎠
(15)
Correspondingly, the void fraction is:
⎛ 1 − x ⎛ ρg ⎜ ε = ⎜1 + ⎜ x ⎜⎝ ρl ⎝
12
⎞ ⎟⎟ ⎠
−1
⎞ x ⎟ = ⎟ x + (1 − x)(ρ g ρl−1 )1 2 ⎠
(16)
As can be seen, this method is asymptotically consistent for x → 0+, x → 1- and ρg ρl-1 → 1-. According to Moody (1966), the slip ratio is predicted as: 1
⎛ρ S gl = ⎜ l ⎜ρ ⎝ g
⎞3 ⎟ ⎟ ⎠
(17)
Correspondingly, the void fraction is:
⎛ 1 − x ⎛ ρg ⎜ ε = ⎜1 + ⎜ x ⎜⎝ ρl ⎝
⎞ ⎟⎟ ⎠
2 3 −1
⎞ x ⎟ = ⎟ x + (1 − x)(ρ g ρl−1 ) 2 3 ⎠
(18)
18
J.R. Thome & A. Cioncolini
As can be seen, this method is also asymptotically consistent for x → 0+, x → 1and ρg ρl-1 → 1-. The methods of Fauske (1962) and Moody (1966) have been proposed to predict two-phase critical flows in nuclear reactor safety analysis. A critical or choked flow is obtained when a fluid system is discharging mass in the form of a single-phase or two-phase flow and a further reduction in downstream pressure does not increase the discharged mass flow rate that therefore attains a maximum or critical value. Critical flow rate calculations are relevant in general for the design of equipment involving two-phase flow, such as safety blowdown valves and fittings, and are of paramount importance in nuclear reactor safety for loss of coolant accident studies. Fauske (1962), in particular, assumed the critical mass flow rate to correspond to the maximum of the momentum density ρm (see Chapter 6 for more details) defined as:
⎧ x2 ⎫ ⎪ (1 − x) 2 ⎪ ρm = ⎨ + ⎬ ⎪ ⎩ ρl (1 − ε ) ρ g ε ⎪ ⎭
−1
(19)
Moody (1966), on the other hand, assumed the critical mass flow rate to correspond to the maximum of the kinetic density ρk defined as: −1 2
3 ⎧ x3 ⎫ ⎪ (1 − x) ⎪ ρk = ⎨ 2 + 2 2 2⎬ ρ g ε ⎪⎭ ⎪ ⎩ ρl (1 − ε )
(20)
The methods of Fauske (1962) and Moody (1966) are best suited for watersteam two-phase flows at high mass flux conditions. It is worth noting that the same slip ratio equation used by Moody (1966), Eq. (17), was also proposed by Zivi (1964). 4.2 Kεh methods In these methods, the void fraction is predicted by multiplying the homogeneous void fraction εh in Eq. (14) with an empirically derived correction factor K as follows:
ε = K εh
(21)
These prediction methods are not particularly accurate and only rarely used, and are included here for completeness and historical interest.
Void Fraction
19
In the method proposed by Armand and Treschev (1947), the correction factor K is predicted as:
K = 0.833 + 0.167 x
(22)
Correspondingly, the void fraction is:
⎛ 1 − x ρg ⎞ ⎟ ε = (0.833 + 0.167 x )⎜⎜1 + x ρl ⎟⎠ ⎝
−1
(23)
This method is not asymptotically consistent for ρg ρl-1 → 1- and is often misquoted without the ‘+ 0.167 x’ term. According to Bankoff (1960), the correction factor K is given as:
K = 0.71 + 0.0145 P
(24)
where P is the operating pressure in MPa, so that the correction factor is not dimensionless. Correspondingly, the void fraction is:
⎛ 1 − x ρg ⎞ ⎟ ε = (0.71 + 0.0145 P )⎜⎜1 + x ρl ⎟⎠ ⎝
−1
(25)
This method is not asymptotically consistent for x → 1- and ρg ρl-1 → 1-. 4.3 Drift-flux correlations Drift-flux correlations are based on the Zuber and Findlay (1965) drift-flux model, a type of two-phase flow model that relies on the superficial velocities (or volumetric fluxes) of the phases. In particular, the Zuber and Findlay void equation is as follows:
ε=
Jg C0 ( J l + J g ) + Vdrift
(26)
where Jl and Jg are the superficial velocities of the liquid and gas phases:
Jl =
(1 − x) G
ρl
; Jg =
xG
ρg
(27)
while G is the total mass flux of the two-phase flow. The distribution parameter C0 accounts for the shape of the void fraction profile in the channel cross section, while the drift velocity Vdrift accounts for the relative velocity between the liquid
20
J.R. Thome & A. Cioncolini
and vapor phases. Available drift-flux correlations require two empirical relations, one to predict the distribution parameter and the second for the drift velocity, which in general are functions of the flow pattern. Drift-flux methods are always asymptotically consistent for x → 0+, provided that the denominator in Eq. (26) does not vanish in this limit, while the consistency for x → 1- and ρg ρl-1 → 1- depends on the asymptotical behaviors of the distribution parameter and the drift velocity. In particular, if:
x → 1− ⇒ C0 +
ρg G
Vdrift → 1+
(28)
then the method is asymptotically consistent for x → 1-, while if:
ρ g ρl−1 → 1− ⇒ C0 +
ρ Vdrift → 1− G
(29)
then the method is asymptotically consistent for ρg ρl-1 → 1-, where ρ is the common value that the densities ρl and ρg approach in this limit. Drift-flux models are particularly effective with distributed and unseparated two-phase flows, where one phase is continuous, the other phase is dispersed and the dominant relative motion between the two phases is caused by an external force field such as gravity. Most separated two-phase flows, on the other hand, cannot be effectively handled by a drift-flux model since in these flows the relative motion is strongly dependent on the pressure and velocity gradients existing in the two phases (Beattie and Sugawara, 1986; Brennen, 2005). Nonetheless, the drift-flux model framework has been quite successful in developing both flow pattern specific and general purpose void fraction prediction methods, the latter capable of handling two-phase flows regardless of the flow pattern. In order to deduce the values of the distribution parameter and of the drift velocity from measured void fraction data, the Zuber and Findlay void equation Eq. (26) is recast as follows:
Vg =
Jg
ε
= C0 ( J l + J g ) + Vdrift = C0 J + Vdrift
(30)
where Vg is the gas average velocity and J is the total superficial velocity. If the distribution parameter C0 and the drift velocity Vdrift are assumed constants for a given flow pattern, a given fluid and fixed operating pressure, then according to Eq. (30) the gas average velocity is a linear function of the total superficial velocity. A plot of measured data as gas average velocity versus total superficial velocity yields a Zuber and Findlay diagram, as shown in Fig. 14, which provides
Void Fraction
21
the distribution parameter from the slope of the line drawn through the points and the drift velocity from its intercept on the vertical axis.
-1
Average Gas Velocity [ms ]
2.5
2
1.5
C =1.20; V 0
=0.35 ms-1
drift
1
0.5
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
-1
1.6
1.8
2
Total Superficial Velocity [ms ]
Figure 14. Zuber and Findlay diagram for bubbly flow data of Spedding and Nguyen (1976) (H2OAir; Pressure: 0.11 MPa; Tube diameter: 45.5 mm; Void fraction: 0.04-0.30).
It can be noted that for high vapor quality and high void fraction values the average gas velocity and the total superficial velocity tend to become similar and correlated, highlighting the inadequacy of drift flux models for these flow conditions. The main general purpose, drift-flux void fraction correlations are described below. According to Rouhani and Axelsson (1970), the distribution parameter and the drift velocity are predicted as:
C0 = 1 + 0.2 (1 − x)
(31) 0.25
Vdrift
⎧ g σ ( ρl − ρ g ) ⎫ = 1.18 ⎨ ⎬ ρl2 ⎩ ⎭
(32)
where g is the acceleration of gravity and σ is the surface tension. The experimental databank used by the authors covers water-steam data in evaporating upflow conditions in non-circular channels (rectangular, annulus, and rod bundle), operating pressures in the range 1.9‒13.8 MPa and vapor qualities from 0+ to 0.18, corresponding to bubbly flow and slug/intermittent flow. In its original formulation, the distribution parameter was C0 = 1.54 for G < 200 kgm-2s-1 and C0 = 1.12 for G > 200 kgm-2s-1. The smooth and more accurate
22
J.R. Thome & A. Cioncolini
expression for the distribution parameter in Eq. (31) was provided by Steiner (1993), which also extended this correlation to horizontal flows. This method is not asymptotically consistent for x → 1- and ρg ρl-1 → 1-, although the discrepancy is typically negligible for engineering applications. In the so-called DIX model (Chexal et al., 1986), the distribution parameter and the drift velocity are calculated as follows: 0.1 ⎧ ⎛ J ⎞n ⎫ ⎛ ρg ⎞ ⎪ ⎜ l⎟ ⎪ C0 = ⎨1 + ⎬; n = ⎜⎜ ⎟⎟ J l + J g ⎪ ⎜⎝ J g ⎟⎠ ⎪ ⎝ ρl ⎠ ⎩ ⎭
Jg
(33)
0.25
Vdrift
⎧ g σ ( ρl − ρ g ) ⎫ = 2.9 ⎨ ⎬ ρl2 ⎩ ⎭
(34)
The DIX model was explicitly derived for the analysis of boiling water nuclear reactors. This method is asymptotically consistent for ρg ρl-1 → 1- but not for x → 1-, although the discrepancy is typically negligible for engineering applications. In the drift-flux method proposed by Woldesemayat and Ghajar (2007), the distribution parameter is predicted as indicated in Eq. (33), while the drift velocity is correlated as: 0.25
Vdrift
⎧ g σ d (1 + cosϑ ) ( ρl − ρ g ) ⎫ = 2.9 ⎨ ⎬ ρl2 ⎩ ⎭
(1.22 + 1.22 sin ϑ )
Patm P
(35)
The numerical constant 2.9 appearing in Eq. (35) has the dimension m-0.25, ϑ is the channel inclination angle (ϑ=0 for horizontal flow), Patm is atmospheric pressure and P is the system pressure. The experimental databank used to derive Eq. (35) contained 2845 data points for horizontal, inclined and vertical upflow conditions, 3 fluids (water-air, water-natural gas and kerosene-air) and tube diameters from 12.7 mm to 102.3 mm. This method is asymptotically consistent for ρg ρl-1 → 1- but not for x → 1-, although the discrepancy is typically negligible for engineering applications. 4.4 Miscellaneous empirical correlations Miscellaneous correlations are purely empirical methods that do not fit into any of the other groups. Selected examples are discussed below.
Void Fraction
23
Lockhart and Martinelli (1949) provided a graphical and tabular correlation for the void fraction expressed as a function of the so-called Lockhart-Martinelli parameter X, defined as the square root of the ratio of the single-phase frictional pressure gradients of the liquid and gas phases flowing alone in the channel: 0.5
⎛ dP ⎞ X =⎜ ⎟ ⎝ dz ⎠lo
0.5
− 0.5 ⎛ f ⎞ ⎛ 1 − x ⎞ ⎛ ρg ⎞ ⎛ dP ⎞ ⎜ ⎟ = ⎜⎜ lo ⎟⎟ ⎜ ⎟⎜ ⎟ ⎝ dz ⎠ go ⎝ f go ⎠ ⎝ x ⎠ ⎜⎝ ρl ⎟⎠
0.5
(36)
This parameter can be interpreted as a measure of how much the two-phase flow behaves as a liquid rather than as a gas. The single-phase frictional pressure gradients for the liquid and vapor phases flowing alone in the channel are calculated as follows:
G 2 (1 − x) 2 ⎛ dP ⎞ G 2 x2 ⎛ dP ⎞ ; ⎜ ⎟ = 2 f go ⎜ ⎟ = 2 flo ρl d ρg d ⎝ dz ⎠lo ⎝ dz ⎠ go
(37)
where the single-phase only-liquid and only-gas Fanning friction factors for the liquid flo and the gas fgo are calculated with the relations of Hagen‒Poiseuille and McAdams (1954):
f = 16 Re−1 for Re < 1500;
f = 0.046 Re−0.2 for Re > 1500
(38)
Lockhart and Martinelli (1949) originally assumed laminar flow for Re ≤ 1000, turbulent flow for Re ≥ 2000 and did not provide any clear indication for the transition region 1000 < Re < 2000. The threshold of Re = 1500 proposed in Eq. (38) simplifies the calculation and provides a smooth transition between the Hagen‒Poiseuille and McAdams (1954) friction factor relations, as the values calculated with these two expressions match at Re = 1500. The single-phase Reynolds numbers for the only-liquid Relo and the only-gas Rego flows for use with Eq. (38) are defined as:
Relo =
G (1 − x) d
µl
; Re go =
G xd
µg
(39)
where µl and µg are the liquid and gas viscosities. The correlation is reported in tabular form in Table 2 and is reproduced in Fig. 15, together with the following simple fitting equation proposed by Butterworth (1975):
ε = (1 + 0.28 X 0.71 )−1
(40)
The experimental databank used by the authors covers two-phase flows of air and different liquids (water, benzene, kerosene and different oils), operating
24
J.R. Thome & A. Cioncolini
pressures in the range of 0.1‒0.4 MPa and tube diameters from 1.5 mm to 25.8 mm. X ε X ε
Table 2. Lockhart and Martinelli (1949) correlation. 100 70 40 20 10 7 4 0.1 0.16 0.24 0.34 0.47 0.52 0.60 2 1 0.7 0.4 0.2 0.1 0.07 0.69 0.77 0.81 0.86 0.91 0.95 0.96
As can be seen in Table 2, this method is limited to void fraction values in the range of 0.1‒0.96, although the fitting Eq. (40) can be extrapolated beyond these limits. 0
Void Fraction
10
Data Points from Table 2 Fitting Equation, Eq. (40) -1
10 -2 10
10
-1
10
0
Lockhart-Martinelli Parameter
10
1
10
2
Figure 15. Lockhart and Martinelli (1949) void fraction correlation.
Cioncolini and Thome (2012a) proposed a void fraction correlation specifically designed for annular two-phase flow as:
h xn ε= ; 0 < x < 1, 10 −3 < ρ g ρl−1 < 1, 0.7 < ε < 1 n 1 + (h − 1) x
(41)
where the parameters h and n are:
h = −2.129 + 3.129 ( ρ g ρl−1 ) −0.2186 n = 0.3487 + 0.6513 ( ρ g ρl−1 )0.5150
(42)
The void fraction in annular flows is predicted in terms of only the vapor quality and the density ratio. The experimental databank used to derive Eqs. (41) and (42) contains 2673 data points for annular flow that cover 8 fluids (water-
Void Fraction
25
Void Fraction: Experimental
Void Fraction: Experimental
Void Fraction: Experimental
steam, R410a, water-air, water-argon, water-nitrogen, water plus alcohol-air, alcohol-air and kerosene-air), operating pressures in the range of 0.1‒7.0 MPa, tube diameters from 1.05 mm to 45.5 mm and both circular and non-circular channels in adiabatic and evaporating flow conditions. As can be seen, this method is asymptotically consistent for ρg ρl-1 → 1- and for x → 1-. Woldesemayat and Ghajar (2007) 1 0.8 0.6 0.4 0.2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Void Fraction: Predicted
0.8
0.9
1
0.8
0.9
1
0.8
0.9
1
Rouhani and Axelsson (1970) 1 0.8 0.6 0.4 0.2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Void Fraction: Predicted Chexal et al. (1989)
1 0.8 0.6 0.4 0.2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Void Fraction: Predicted
Figure 16. Void fraction: experimental data of Table 1 vs. predictions of Woldesemayat and Ghajar (2007) correlation, Eqs. (26), (33) and (35) [top], predictions of Rouhani and Axelsson (1970) correlation, Eqs. (26), (31) and (32) [middle] and predictions of the DIX method (Chexal et al., 1989), Eqs. (26), (33) and (34) [bottom].
26
J.R. Thome & A. Cioncolini
4.5 Comparison of above methods to experimental database With respect to the databank in Table 1, the best prediction methods for the void fraction among those presented above are the drift-flux methods of Woldesemayat and Ghajar (2007), Rouhani and Axelsson (1970) and the DIX model (Chexal et al., 1986). For annular flows, the best predictions are given by the Cioncolini and Thome (2012a) correlation. Measured data from Table 1 are compared with the predictions of these methods in Fig. 16. Typical mean absolute percentage errors are on the order of 6‒7 %. Based on available data, these prediction methods can be extrapolated to non-circular channels using the hydraulic diameter (4 Aflow Pwet-1) in place of the tube diameter. The axial power (heat flux) profile in evaporating flow conditions seems to be of second order importance on the value of the void fraction. 5. Nomenclature A, Aflow Ag Al d e g G Jg Jl Pwet Re t x Γ Γl Γg γ µg µl ε ρg ρl
channel cross section flow area (m2) channel cross sectional area occupied by the gas phase (m2) channel cross sectional area occupied by the liquid phase (m2) tube diameter (m) entrained liquid fraction (-) acceleration of gravity (ms-2) mass flux (kgm-2s-1) superficial gas velocity (ms-1) superficial liquid velocity (ms-1) channel wetted perimeter (m) Reynolds number (-) average liquid film thickness in annular flow (m) vapor quality (-) total mass flow rate (kgs-1) liquid mass flow rate (kgs-1) gas mass flow rate (kgs-1) liquid droplet hold-up in annular flow gas viscosity (kgm-1s-1) liquid viscosity (kgm-1s-1) cross sectional void fraction (-) vapor density (kgm-3) liquid density (kgm-3)
Void Fraction
σ
27
surface tension (kgs-2)
6. References Adorni, N., Casagrande, I., Cravarolo, L., Hassid, A., Pedrocchi, E. and Silvestri, M., (1963). Further investigations in adiabatic dispersed flow: pressure drop and film thickness measurements with different channel geometries-analysis of the influence of geometrical and physical parameters. CISE Report R‒53, Segrate, Italy. Alia, P., Cravarolo, L., Hassid, A. and Pedrocchi, E., (1965). Liquid volume fraction in adiabatic two-phase vertical upflow – round conduit. CISE Report R‒105, Segrate, Italy. Alia, P., Cravarolo, L., Hassid, A. and Pedrocchi, E., (1966). Phase and velocity distribution in two-phase adiabatic annular dispersed flow. CISE Report R‒109, Segrate, Italy. Anderson, G.H. and Mantzouranis, B.G., (1960). Two-phase (gas-liquid) flow phenomena – I: pressure drop and hold-up for two-phase flow in vertical tubes. Chem. Eng. Sci. 12, pp. 109‒ 126. Armand, A.A. and Treschev, G.G., (1947). Investigation of the resistance during the movement of steam-water mixtures in heated boiler pipes at high pressure. Izv. Ves. Teplotekh. Inst. 4, pp. 1‒5. Bankoff, S.G., (1960). A variable density single fluid model for two-phase flow with particular reference to steam-water flow. ASME J. Heat Transfer 82, pp. 265‒272. Beattie, D.R.H. and Sugawara, S., (1986). Steam-water void fraction for vertical upflow in a 73.9 mm pipe. Int. J. Multiphase Flow 12, pp. 641‒653. Beggs, H.D., (1972). An experimental study of two-phase flow in inclined pipes. Ph.D. Thesis, University of Tulsa, Oklahoma, USA. Brennen, C.E., (2005). Fundamentals of Multiphase Flow. Cambridge University Press, New York, USA. Brown D.J., (1978). Disequilibrium annular flow. Ph.D. Thesis, University of Oxford, England. Butterworth, D., (1975). A comparison of some void fraction relationships for co-current gas-liquid flow. Int. J. Multiphase Flow 1, pp. 845‒850. Casagrande, I., Cravarolo, L., Hassid, A. and Pedrocchi, E., (1963). Adiabatic dispersed two-phase flow: further results on the influence of physical properties on pressure drop and film thickness. CISE Report R‒73, Segrate, Italy. Celata, G.P. and Frazzoli, F.V., (1981). High void fraction measurements in full-scale heat transfer tests. Nucl. Technol. 54, pp. 422‒425. Chexal, B., Horowitz, J. and Lellouche, G., (1986). An assessment of eight void fraction models for vertical flows. EPRI Report NSAC‒107, Palo Alto, USA. Chung, P.M.Y. and Kawaji, M., (2004). The effect of channel diameter on diabatic two-phase flow characteristics in microchannels, Int. J. Multiphase Flow 30, pp. 735‒761. Cioncolini, A. and Thome, J.R., (2012a). Void fraction prediction in annular two-phase flow. Int. J. Multiphase Flow 43, pp. 72‒84. Cioncolini, A. and Thome, J.R., (2012b). Entrained liquid fraction prediction in adiabatic and evaporating annular two-phase flow. Nucl. Eng. Des. 243, pp. 200‒213. Coddington, P. and Macian, R., (2002). A study of the performance of void fraction correlations used in the context of drift-flux two-phase flow models. Nucl. Eng. Des. 215, pp. 199‒216.
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J.R. Thome & A. Cioncolini
Cravarolo, L., Hassid, A. and Pedrocchi, E., (1964). Further investigation on two-phase adiabatic annular-dispersed flow: effect of length and some inlet conditions on flow parameters. CISE Report R‒93, Segrate, Italy. Das, G., Das, P.K., Purohit, N.K. and Mitra, A.K., (2002). Liquid holdup in concentric annuli during cocurrent gas-liquid upflow. Canadian J. Chem Eng. 80, pp. 153‒157. Dejesus, J.M. and Kawagi, M., (1990). Investigation of interfacial area and void fraction in upward cocurrent gas-liquid flow. Canadian J. Chem. Eng. 68, pp. 904‒912. Fauske, H.K., (1962). Contribution to the theory of two-phase, one component critical flow. Argonne National Laboratory Report ANL‒6633. Gill, L.E., Hewitt, G.F. and Lacey, P.M.C., (1964). Sampling probe studies of the gas core in annular two-phase flow-II: Studies of the effect of phase flow rates on phase and velocity distribution. Chem. Eng. Sci. 19, pp. 665‒682. Gill, L.E., Hewitt, G.F. and Lacey, P.M.C., (1965). Data on the upwards annular flow of air-water mixtures. Chem. Eng. Sci. 20, pp. 71‒88. Godbole, P.V., Tang, C.C. and Ghajar, A.J., (2011). Comparison of void fraction correlations for different flow patterns in upward vertical two-phase flow. Heat Transfer Eng. 32, pp. 843‒ 860. Hall-Taylor, N., Hewitt, G.F. and Lacey, P.M.C., (1963). The motion and frequency of large disturbance waves in annular two-phase flow of air-water mixtures. Chem. Eng. Sci. 18, pp. 537‒552. Hori, K., Akiyama, Y., Miyazaki, K., Kurosu, T. and Sugiyama, S. (1995). Void fraction in a single channel simulating one subchannel of a PWR fuel assembly. In eds. Celata, G.P. and Shah, R.K, Two-Phase Flow Modeling and Experimentation, ETS, Italy, pp. 1013‒1027. Godbole, P.V., Tang, C.C. and Ghajar, A.J., (2011). Comparison of void fraction correlations for different flow patterns in upward vertical two-phase flow. Heat Transfer Eng. 32, pp. 843‒ 860. Jones, O.C. and Zuber, N., (1975). The interrelation between void fraction fluctuations and flow patterns in two-phase flow. Int. J. Multiphase Flow 2, pp. 273‒306. Kaji, R. and Azzopardi, B.J., (2010). The effect of pipe diameter on the structure of gas/liquid flow in vertical pipes. Int. J. Multiphase Flow 36, pp. 303‒313. Kawahara, A., Sadatomi, M., Okayama, K., Kawaji, M. and Chung, P.M.Y., (2005). Effects of channel diameter and liquid properties on void fraction in adiabatic two-phase flow through microchannels. Heat Transfer Eng. 26, pp. 13‒19. Leung, L.K.H., Groeneveld, D.C., Teyssedou, A. and Aubé, F., (2005). Pressure drop for steam and water flow in heated tubes. Nucl. Eng. Des. 235, pp. 53‒65. Lockhart, R.W. and Martinelli, R.C., (1949). Proposed correlation of data for isothermal two-phase two-component flow in pipes. Chem. Eng. Progr. 45, pp. 39‒48. McAdams, W.H., (1954). Heat Transmission. McGraw Hill, New York, USA. Moody, F.J., (1966). Maximum two-phase vessel blowdown from pipes. ASME J. Heat Transfer 87, pp. 285‒295. Morooka, S., Ishizuka, T., Iizuka, M. and Yoshimura, K., (1989). Experimental study on void fraction in a simulated BWR fuel assembly. Nucl. Eng. Des. 114, pp. 91‒98. Mukherjee, H., (1979). An experimental study of inclined two-phase flow. Ph.D. Thesis, University of Tulsa, Oklahoma, USA. Rouhani, S.Z. and Axelsson, E., (1970). Calculation of void volume fraction in the subcooled and quality boiling regions. Int. J. Heat Mass Transfer 13, pp. 383‒393.
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