Annular flow is one of the most frequently observed flow patterns with gas–liquid two-phase flows in tubes or channels. In the annular flow pattern, a thin liquid film flows along the channel wall, while the gas flows in the center of the channel carrying liquid droplets in suspension. The fraction of the liquid flow rate that is transported as suspended droplets is quantified using the entrained liquid fraction (ELF), which is a key flow parameter in the analysis and modeling of annular flows. This review provides a critical assessment of ELF experimental data available in the open literature and of ELF prediction methods proposed to date. The experimental data assessment is carried out by means of a large ELF data bank collected from the literature (4175 data points from 53 literature studies; 10 fluids combinations; operating pressures from atmospheric to 20 MPa; experiments carried out with adiabatic, evaporating, and condensing flows through circular tubes, and non-circular channels with diameters from 3.02 to 155.7 mm), which is critically analyzed devoting special attention to important aspects not adequately addressed in previous studies, such as a cross-comparison between different ELF measuring techniques, and the analysis of flow development and gravity effects. The assessment of the ELF prediction methods focuses on 15 widely quoted methods, which are critically analyzed and whose prediction performance is evaluated against the measured data. The curated ELF experimental data bank is provided in full and usable form. Research gaps for further investigations are identified and discussed.

Among the various flow patterns that can be observed when a gas–liquid mixture flows in a tube or channel, annular flow is by far the most frequently encountered. Representative examples of two-phase flow systems where annular flows are commonplace include evaporators and steam generators, boiling water nuclear reactors, refrigeration and air conditioning units, chemical processing fluid systems, and oil and gas transportation pipelines. Smooth operation, large heat transfer coefficient, and low liquid inventory make annular flow a desirable flow pattern in thermal management applications in microelectronics, power electronics, and high-energy physics particle detectors. In annular flow, the gas or vapor (for simplicity, the term gas will be used henceforth to refer to the gas or vapor phase) flows in the center of the channel transporting part of the liquid in the form of suspended droplets, while the rest of the liquid gathers in a thin film that flows along the channel wall. It is worth noting that small bubbles can become entrapped in the liquid film or may result from wall nucleation in evaporating flows. Unfortunately, the aeration of the liquid film has not been extensively investigated to date, despite the potentially significant effect that the aeration may have on the liquid film fluid dynamics.1–3 

A distinctive feature of annular flows, in contrast with other two-phase flow patterns, is that the difference in average flow velocity (so-called slip) between the phases, and specifically between the gas core and the liquid film, can achieve substantial values. In fact, even though the liquid and gas phases are clearly intermixed, as droplets are entrained in the gas core and bubbles can be present in the liquid film, the gas core and the liquid film are physically segregated and can, therefore, flow with different average velocities. More specifically, the gas core normally flows faster than the liquid film, as a consequence of the no-slip boundary condition enforced onto the liquid film flow by the channel wall. The slip between the core flow and the liquid film, and the subsequent shear exerted by the core flow onto the liquid film are key features underpinning the physics of annular flows. The liquid film is, in fact, dragged along the conduit wall by the shearing action of the core flow, hence constituting a shear-driven thin liquid film flow. As a consequence of the shear, a variety of disturbances appear on the surface of the liquid film, ranging in size from small and localized ripples to so-called disturbance waves: large pockets of liquid that travel down the conduit by surfing along the liquid film, ejecting droplets that become entrained in the core flow. Once entrained, the droplets are accelerated by the core flow and eventually deposited back onto the liquid film, yielding a continuous exchange of liquid mass and linear momentum between the gas core and the liquid film. Despite their apparent morphological simplicity, annular flows are characterized by a rich physics, which combines wall-bounded and free-surface flow conditions, interface deformation, and liquid atomization, and further complicated by a tight mass and momentum coupling between the gas core and the liquid film resulting in a strongly coupled dynamics with several influencing factors at play simultaneously.

A key flow parameter in the analysis and modeling of annular flows is the entrained liquid fraction e (ELF), which is defined as the ratio of the entrained droplets mass flow rate m ̇ l e to the total liquid mass flow rate m ̇ l as follows:
e = m ̇ l e m ̇ l .
(1)
The ELF is dimensionless and bounded between 0 and 1 and quantifies how much of the liquid phase is transported along the conduit as suspended droplets, and how much is transported in the liquid film. More specifically, ELF values close to 0 correspond to annular flows with hardly any entrained droplets, where most of the liquid flows in the film. Though not often encountered in practical applications, annular flows with very low ELF may occur during off-normal operation of water-cooled nuclear reactor systems and heat transferring equipment or may be observed with downward annular flows when the shear of the core flow onto the liquid film is low, and the flow in the liquid film is, therefore, gravity-driven, as happens, for example, in liquid draining systems. The so-called ideal annular flow, characterized by ELF equal to 0 and hence a complete separation between liquid and gas, is sometimes employed in modeling studies as a first-order approximation. On the other hand, ELF values close to 1 indicate that most of the liquid is transported along the conduit as entrained droplets, which correspond to annular flows that are approaching the transition to dispersed mist flow. Annular flows with high ELF are important in the analysis of convective evaporation systems, particularly to predict the location along the channel where the liquid film loses its continuity (so-called dryout) as consequence of the shear of the core flow and of the evaporation process that depletes its mass inventory. Dryout is accompanied by a sharp reduction of the heat transfer coefficient and a consequent degradation of heat transfer effectiveness, which is of particular concern in the thermal management of imposed-heat-flux systems, such as nuclear reactors.

Notwithstanding, the extensive investigations carried out to date on annular flows, including various review papers recently published on this topic,4–7 a critical assessment of ELF experimental data available in the open literature, and a critical assessment of ELF prediction methods proposed to date are missing. The objective of the present study is to provide such an assessment. The ELF data bank collected from the literature to inform the present assessment counts 4175 data points and is, therefore, considerably larger and more diversified than the databanks considered in previous investigations. Moreover, the scope of the present assessment is broader and covers aspects only marginally considered or altogether neglected in previous studies, such as a critical comparison between various ELF measuring techniques, and a thorough assessment of inlet effects and gravity effects. Differently from previous studies, the curated ELF data bank is provided in full and usable form, as are the routines developed here to implement the ELF prediction methods.

The rest of this paper is organized as follows: the ELF experimental data bank collected from the open literature is presented and critically analyzed in Sec. II; ELF prediction methods are presented in Sec. III, while their performance is assessed in Sec. IV; concluding remarks and an outlook are finally provided in Sec. V.

The ELF experimental data bank collected from the open literature to inform the present assessment comprises 4175 data points gathered from 53 studies that have been published during the last six decades (the oldest study is dated 1960, while the most recent was published in 2020). The main details of the individual studies that comprise the data bank are summarized in Table I, while pie charts and histograms that collectively describe the data bank are provided in Figs. 1 and 2, respectively. The ELF data bank is critically analyzed later.

TABLE I.

ELF experimental data bank.

No. Reference Fluid P (MPa)/T(K) Typea d (mm)b Angle (°)c Inlet mixer-L/dd Mease
Owen33   H2O–air  0.17–0.38/286–299  31.8  90  #6–607  FS 
Wallis29   H2O–air  0.1/293  3.02, 3.94, 5.97, 8.25, 12.7, 15.6, 22.2  90, −90  #4,#7–29.3, 58.5, 78, 81.1, 83.1, 114.9, 117.1, 126.1, 174.2  CS 
Andreussi26   H2O–air  0.1/293  24.0  −90  #6–250  FS 
Assad et al.88   H2O–air  0.37/293  9.5  90  #6–440  FS 
Azzopardi and Zaidi16   H2O–air  0.1/293  38.0  90  #6–118.4  OP 
Al-Yarubi19   H2O–air  0.1/293  50.0  90  #1–40  CN 
Han et al.20   H2O–air  0.1/293  9.52  90  #2–166  CS+TR 
Lopez de Bertodano et al.35   R113  0.32–0.53/361–381  10.0  90  NA-350  FS 
Hinkle28   H2O–air  0.22–0.62/294  12.62  90  #6–11.6, 35.7, 58.9, 83, 119, 166, 215, 250, 262  CS 
10  Jagota et al.27   H2O–air  0.27–0.42/294  25.4  90  #3–153, 261  FS,TR 
11  Jepson et al.89   H2O–air, H2O–He  0.15/293  10.26  90  #6–295  FS 
12  Nakazatomi and Sekoguchi80   H2O–air  0.3–20/303  19.2  90  #6–474  CS 
13  Okawa et al.90   H2O–air  0.14–0.76/294–301  5.0  90  #6–320  FS 
14  Sawant et al.83   H2O–air  0.12–0.4/293  9.4  90  #6–210  FS 
15  Sawant et al.36   R113  0.28–0.85/355–404  10.2  90  #6–400  FS 
16  Schadel et al.91   H2O–air  0.1/293  25.4, 42, 57.2  90  #4–150  TR 
17  Steen and Wallis57   H2O–air, silicon–air  0.1–0.4/293  15.9  −90  NA-93  CS 
18  Wurtz11   H2 3–9/507–576  a/e  9, 10, 20  90  NR  FS 
19  Gill et al.92   H2O–air  0.1–0.15/293  31.8  90  #6–167  CS 
20  Gill et al.93   H2O–air  0.2–0.34/293  31.8  90  NA-520  NA 
21  Whalley et al.94   H2O–air, R140a–air  0.12–0.35/293  31.8  90  NA-590  NA 
22  Brown95   H2O–air  0.17–0.31/293  31.8  90  NA-480  NA 
23  Cousins et al.96   H2O–air  0.14–0.24/293  9.5  90  NA-480  NA 
24  Cousins and Hewitt97   H2O–air  0.14–0.24/293  9.53, 31.8  90  NA-230, 300  NA 
25  Minh and Huyghe98   H2O–air, ethanol–air  0.23–0.86/293  6, 12  90  NA-NA  NA 
26  Hewitt and Pulling99   H2 0.24–0.45/399–421  9.3  90  NA-390  NA 
27  Keeys et al.100   H2 3.45–6.89/515–558  12.6  90  NA-290  NA 
28  Nigmatulin et al.101   H2 1–10/453–584  13.3  90  NA-300  NA 
29  Singh et al.59   H2 6.89/558  12.5  90  #5–182  FS 
30  Adamsson and Anglart8   H2 7/559  14.0  90  NR  FS 
31  Milashenko et al.102   H2 7/559  13.1  90  NR  FS 
32  Ueda and Kim103   R113  0.33/361  10.0  90  NR  FS 
33  Langner and Mayinger17   R12  1.1/319  14.0  90  NR  OP 
34  Williams et al.104   H2O–air  0.12/293  95.3  #2–273  CS 
35  Dallman et al.32   H2O–air  0.11–0.23/293  25.4  #4–500  FS 
36  Paras and Karabelas15   H2O–air  0.11–0.20/293  50.8  #1–300  CS 
37  Ousaka and Kariyasaki14   H2O–air  0.10/290  26.0  NA-142  CS 
38  Geraci et al.105   H2O–air  0.14/293  38.0  0, 20, 45, 70, 85  #6–132  FS 
39  Feldhaus et al.12   H2O–air  0.24–0.27/293  9.35  90  #6–171  FS 
40  Asali et al.13   H2O–air  0.10/288  22.9  90  #4–333  CS 
41  Aliyu et al.18   H2O–air  0.16/293  101.6  90  NA-47  CN 
42  Sadatomi et al.74   H2O–air, H2O + PLE–air  0.10/304  5.0  90  #6–270  FS 
43  Kawahara et al.62   H2O–air  0.10/293  16.0  90  #5–137  FS 
44  Kawahara et al.63   H2O–air  0.10/293  16.0  90  #5–137  FS 
45  Laurinat106   H2O–air  0.10–0.21/283–294  50.8  #6–300  CS,FS 
46  Guevara and Gotham107   H2 0.05–0.16/355–386  38.1  NR  FS 
47  Dukler et al.108   H2O–air  0.10/293  50.8  90  #6–39, 79  FS 
48  Quandt9   H2O–air  0.10/287–330  11.72  90  #6–77.8  TR 
49  Wicks and Dukler71   H2O–air  0.12/293  26.0  #1, #2–223  CS 
50  Vuong et al.70   IsoparL–N2  1.48–2.86/300  155.7  NA-130  CS 
51  Schraub et al.10   H2O–air  0.47–0.63/292–307  12.6, 25.3  90  NA-119, 238  FS 
52  Magrini et al.25   H2O–air  0.10/293  76.2  0, 10, 20, 45, 60, 75, 90  NA-172  FS 
53  Rodrigues et al.69   IsoparL–N2  1.48–2.86/300  155.7  NA-130  CS 
No. Reference Fluid P (MPa)/T(K) Typea d (mm)b Angle (°)c Inlet mixer-L/dd Mease
Owen33   H2O–air  0.17–0.38/286–299  31.8  90  #6–607  FS 
Wallis29   H2O–air  0.1/293  3.02, 3.94, 5.97, 8.25, 12.7, 15.6, 22.2  90, −90  #4,#7–29.3, 58.5, 78, 81.1, 83.1, 114.9, 117.1, 126.1, 174.2  CS 
Andreussi26   H2O–air  0.1/293  24.0  −90  #6–250  FS 
Assad et al.88   H2O–air  0.37/293  9.5  90  #6–440  FS 
Azzopardi and Zaidi16   H2O–air  0.1/293  38.0  90  #6–118.4  OP 
Al-Yarubi19   H2O–air  0.1/293  50.0  90  #1–40  CN 
Han et al.20   H2O–air  0.1/293  9.52  90  #2–166  CS+TR 
Lopez de Bertodano et al.35   R113  0.32–0.53/361–381  10.0  90  NA-350  FS 
Hinkle28   H2O–air  0.22–0.62/294  12.62  90  #6–11.6, 35.7, 58.9, 83, 119, 166, 215, 250, 262  CS 
10  Jagota et al.27   H2O–air  0.27–0.42/294  25.4  90  #3–153, 261  FS,TR 
11  Jepson et al.89   H2O–air, H2O–He  0.15/293  10.26  90  #6–295  FS 
12  Nakazatomi and Sekoguchi80   H2O–air  0.3–20/303  19.2  90  #6–474  CS 
13  Okawa et al.90   H2O–air  0.14–0.76/294–301  5.0  90  #6–320  FS 
14  Sawant et al.83   H2O–air  0.12–0.4/293  9.4  90  #6–210  FS 
15  Sawant et al.36   R113  0.28–0.85/355–404  10.2  90  #6–400  FS 
16  Schadel et al.91   H2O–air  0.1/293  25.4, 42, 57.2  90  #4–150  TR 
17  Steen and Wallis57   H2O–air, silicon–air  0.1–0.4/293  15.9  −90  NA-93  CS 
18  Wurtz11   H2 3–9/507–576  a/e  9, 10, 20  90  NR  FS 
19  Gill et al.92   H2O–air  0.1–0.15/293  31.8  90  #6–167  CS 
20  Gill et al.93   H2O–air  0.2–0.34/293  31.8  90  NA-520  NA 
21  Whalley et al.94   H2O–air, R140a–air  0.12–0.35/293  31.8  90  NA-590  NA 
22  Brown95   H2O–air  0.17–0.31/293  31.8  90  NA-480  NA 
23  Cousins et al.96   H2O–air  0.14–0.24/293  9.5  90  NA-480  NA 
24  Cousins and Hewitt97   H2O–air  0.14–0.24/293  9.53, 31.8  90  NA-230, 300  NA 
25  Minh and Huyghe98   H2O–air, ethanol–air  0.23–0.86/293  6, 12  90  NA-NA  NA 
26  Hewitt and Pulling99   H2 0.24–0.45/399–421  9.3  90  NA-390  NA 
27  Keeys et al.100   H2 3.45–6.89/515–558  12.6  90  NA-290  NA 
28  Nigmatulin et al.101   H2 1–10/453–584  13.3  90  NA-300  NA 
29  Singh et al.59   H2 6.89/558  12.5  90  #5–182  FS 
30  Adamsson and Anglart8   H2 7/559  14.0  90  NR  FS 
31  Milashenko et al.102   H2 7/559  13.1  90  NR  FS 
32  Ueda and Kim103   R113  0.33/361  10.0  90  NR  FS 
33  Langner and Mayinger17   R12  1.1/319  14.0  90  NR  OP 
34  Williams et al.104   H2O–air  0.12/293  95.3  #2–273  CS 
35  Dallman et al.32   H2O–air  0.11–0.23/293  25.4  #4–500  FS 
36  Paras and Karabelas15   H2O–air  0.11–0.20/293  50.8  #1–300  CS 
37  Ousaka and Kariyasaki14   H2O–air  0.10/290  26.0  NA-142  CS 
38  Geraci et al.105   H2O–air  0.14/293  38.0  0, 20, 45, 70, 85  #6–132  FS 
39  Feldhaus et al.12   H2O–air  0.24–0.27/293  9.35  90  #6–171  FS 
40  Asali et al.13   H2O–air  0.10/288  22.9  90  #4–333  CS 
41  Aliyu et al.18   H2O–air  0.16/293  101.6  90  NA-47  CN 
42  Sadatomi et al.74   H2O–air, H2O + PLE–air  0.10/304  5.0  90  #6–270  FS 
43  Kawahara et al.62   H2O–air  0.10/293  16.0  90  #5–137  FS 
44  Kawahara et al.63   H2O–air  0.10/293  16.0  90  #5–137  FS 
45  Laurinat106   H2O–air  0.10–0.21/283–294  50.8  #6–300  CS,FS 
46  Guevara and Gotham107   H2 0.05–0.16/355–386  38.1  NR  FS 
47  Dukler et al.108   H2O–air  0.10/293  50.8  90  #6–39, 79  FS 
48  Quandt9   H2O–air  0.10/287–330  11.72  90  #6–77.8  TR 
49  Wicks and Dukler71   H2O–air  0.12/293  26.0  #1, #2–223  CS 
50  Vuong et al.70   IsoparL–N2  1.48–2.86/300  155.7  NA-130  CS 
51  Schraub et al.10   H2O–air  0.47–0.63/292–307  12.6, 25.3  90  NA-119, 238  FS 
52  Magrini et al.25   H2O–air  0.10/293  76.2  0, 10, 20, 45, 60, 75, 90  NA-172  FS 
53  Rodrigues et al.69   IsoparL–N2  1.48–2.86/300  155.7  NA-130  CS 
a

Type of test—a: adiabatic flow; e: evaporating flow; c: condensing flow.

b

Pipe diameter if circular tube, hydraulic diameter (four times the flow area divided by the wetted perimeter) otherwise.

c

Flow inclination with respect to the horizontal (0°: horizontal flow; 90°: vertical upflow; −90°: vertical downflow).

d

Inlet mixer type according to the sketches in Fig. 5, and dimensionless distance L/d between mixer and ELF measuring location (NR: not relevant; NA: not available).

e

ELF measuring technique—CS: core sampling; FS: film suction; TR: tracer; OP: optical; CN: conductance; NA: not available.

FIG. 1.

Pie charts describing the experimental data bank in Table I. (a) Test fluids; (b) operating conditions; (c) test section geometry; (d) flow orientation; (e) ELF measuring technique; (f) inlet mixer type as per the sketches provided in Fig. 5 (NA: not available; NR: not relevant).

FIG. 1.

Pie charts describing the experimental data bank in Table I. (a) Test fluids; (b) operating conditions; (c) test section geometry; (d) flow orientation; (e) ELF measuring technique; (f) inlet mixer type as per the sketches provided in Fig. 5 (NA: not available; NR: not relevant).

Close modal
FIG. 2.

Normalized histograms describing the experimental data bank in Table I. (a) Operating pressure; (b) operating temperature; (c) mass flux; (d) vapor quality; (e) test section diameter or hydraulic diameter (if non-circular); (f) dimensionless distance between inlet mixer and ELF measuring location (when relevant). (g) void fraction, predicted according to Cioncolini and Thome;38 (h) void fraction, predicted according to Woldesemayat and Ghajar;39 (i) non-dimensional superficial gas velocity [Eq. (10)] for vertical upflow data; (j) non-dimensional gas superficial velocity [Eq. (10)] for vertical downflow data; (k) Mach number, Eq. (11); (l) liquid film Reynolds number [Eq. (14)] for vertical flow data.

FIG. 2.

Normalized histograms describing the experimental data bank in Table I. (a) Operating pressure; (b) operating temperature; (c) mass flux; (d) vapor quality; (e) test section diameter or hydraulic diameter (if non-circular); (f) dimensionless distance between inlet mixer and ELF measuring location (when relevant). (g) void fraction, predicted according to Cioncolini and Thome;38 (h) void fraction, predicted according to Woldesemayat and Ghajar;39 (i) non-dimensional superficial gas velocity [Eq. (10)] for vertical upflow data; (j) non-dimensional gas superficial velocity [Eq. (10)] for vertical downflow data; (k) Mach number, Eq. (11); (l) liquid film Reynolds number [Eq. (14)] for vertical flow data.

Close modal

The fluid combinations that have been used more frequently in the experiments [see Fig. 1(a)] are water–air (employed in 39 studies, accounting for 73% of the data points), saturated water (employed in eight studies, accounting for 13.8% of the data points), and Isopar-L and Nitrogen (employed in two studies, accounting for 6.2% of the data points). In particular, Isopar-L is a synthetic isoparaffin with properties similar to those of light condensate oils of interest in oil and gas transportation applications. Other fluid combinations (employed in nine studies that altogether account for 7% of the data points) include refrigerants R12 (saturated) and R113 (saturated), silicon–air, ethanol–air, refrigerant R140a–air, water–helium, and water with the addition of PLE (polyoxyethylene lauryl ether, a surfactant added to reduce the surface tension of water) tested with air.

The vast majority of the measurements (91.6% of the data points) have been carried out in adiabatic flow conditions [see Fig. 1(b)], and evaporating flow conditions have been explored in five studies (7.8% of the data points), while condensing flow conditions have been explored in only one experiment (corresponding to 0.6% the data points). Electrical heating with uniform axial power distribution along the test section has been used in all tests carried out with evaporating flows. Adamsson and Anglart8 also implemented non-uniform axial power distributions (inlet-peaked, middle-peaked, and outlet-peaked) to better approximate the heating conditions encountered in boiling water nuclear reactor cores.

Most tests have been carried out at relatively low operating pressures (below about 1 MPa) and operating temperatures close to ambient [see Figs. 2(a) and 2(b)]. The bins that can be noticed at a pressure of 7 MPa in Fig. 2(a) and at a temperature of 559 K in Fig. 2(b) correspond to tests carried out with saturated water at the operating conditions of boiling water nuclear reactors. As can be noticed in Figs. 2(c) and 2(d), the mass flux and vapor quality ranges explored in the experiments are wide enough to be informative of most practical applications, although mass fluxes in excess of about 2000 kg/m2s have been only sporadically investigated.

Circular tube test sections have been used in the greatest part of the experiments [97.4% of the data points, see Fig. 1(c)]. Other channel geometries have been only sporadically investigated (2.6% of the data points) and include the rectangular channel (76.2 mm long and 6.35 mm wide, with an aspect ratio of 12 and a hydraulic diameter of 11.72 mm) employed by Quandt,9 the annulus (rod/tube diameters of 12.7/25.3 mm and hydraulic diameter of 12.6 mm) employed by Schraub et al.,10 the annulus (rod/tube diameters of 17.0/26.0 mm and hydraulic diameter of 9.0 mm) employed by Wurtz,11 and the channel (hydraulic diameter of 9.35 mm, see the cross section schematics in Fig. 3) employed by Feldhaus et al.,12 which was conceived to be informative of water-cooled nuclear reactor fuel bundles.

FIG. 3.

Sketch of the cross section of the non-circular test section employed by Feldhaus et al.12 and designed to be informative of the flow through water-cooled nuclear reactor fuel bundles.

FIG. 3.

Sketch of the cross section of the non-circular test section employed by Feldhaus et al.12 and designed to be informative of the flow through water-cooled nuclear reactor fuel bundles.

Close modal

The diameters of the tubular test sections (or hydraulic diameter for non-circular channels, equal to four times the channel flow area divided by the wetted perimeter) vary from 3.02 to 155.7 mm [see Fig. 2(e)], hence covering a range wide enough to be informative of most practical applications, including comparatively large diameter pipes of interest in oil and gas transportation applications.

Most measurements (65.5% of the data points) have been carried out in vertical upflow conditions [see Fig. 1(d)], vertical downflow conditions have been explored in three studies (17.7% of the data points), while horizontal flow conditions have been explored in ten studies (9.8% of the data points). Other flow inclinations explored during the tests (2°, 10°, 20°, 45°, 60°, 70°, 75°, and 85°, where 0° corresponds to horizontal flow) altogether account for 7% of the data points.

Various different techniques [see Fig. 1(e)] have been developed and used to measure the ELF, including the core sampling method (employed in 15 studies, accounting for 42.4% of the data points), the film suction method (employed in 23 studies, accounting for 36.3% of the data points), the chemical tracer method (employed in four studies, accounting for 3.8% of the data points), and other techniques of less frequent use (employed in five studies that altogether account for 4.2% of the data points), which include optical methodologies and the conductance method. These measuring techniques are briefly reviewed later.

In the core sampling method, part of the core flow is extracted from the test section by means of a sampling tube, which is inserted in the test section pointing upstream with the opening facing the incoming flow. After separating the liquid and gas components in the sampled flow, the flow rate of the liquid phase is measured, and the reading is used to represent the mass flow rate of the entrained droplets, so that the ELF can be computed as indicated in Eq. (1). Sometimes, the sampling tube is mounted on a traverse, so that multiple core flow samplings can be taken in succession at different radial positions in the conduit, or multiple sampling tubes are installed at different radial positions in the conduit and used simultaneously, so as to improve the accuracy of the droplets flow rate measurement. The sampling probes are normally operated at (or close to) isokinetic conditions, where the velocity of the flow in the sampling tube matches (or closely approaches) in value the core flow velocity in the test section. As noted by various investigators,13–15 exactly matching isokinetic conditions during operation seems to be not strictly required, as reliable measurements are generated even when the isokinetic conditions are only approached. On the other hand, any misalignment of the sampling probe with respect to the tube axis in excess of a few degrees can compromise the accuracy of the readings.14 

In the film suction method, the liquid film flow and part of the gas flow are extracted from the test section by means of an aperture realized on the channel wall (typically a slot, a perforated section, or a porous section). While operating under seemingly identical testing conditions, the amount of flow extracted through the aperture is gradually and stepwise increased, and the liquid and gas flows are separated and individually measured at each step. When the measurements are plotted as the extracted liquid mass flow rate vs the extracted gas mass flow rate, the data typically display a plateaux, which is identified as the liquid film mass flow rate. The entrained droplets mass flow rate is computed by subtracting the liquid film mass flow rate from the total liquid mass flow rate, and then the ELF is computed as indicated in Eq. (1).

In the chemical tracer method, a chemical tracer (a dye or a sodium chloride solution) is introduced in the liquid film flow, and its concentration downstream of the injection point is monitored (by means of photocells for the dye concentration or with conductance probes for the sodium chloride concentration). These tracer concentration measurements, together with a mathematical model of the tracer transport in the flow, are then used to deduce the ELF.

With optical methodologies, the liquid film is removed from the test section by means of an aperture realized on the channel wall, so as to provide optical access to the core flow which can then be analyzed. In the methodology employed by Azzopardi and Zaidi,16 the entrained droplets have been measured using a commercial spray droplet size measuring system, whereas Langner and Mayinger17 measured the entrained droplets by means of fast video imaging and subsequent image processing. The entrained droplets mass flow rate is then used to compute the ELF as indicated in Eq. (1).

With the conductance method,18,19 a conductance probe is inserted into the liquid film to measure its time-average thickness and, hence, deduce the liquid film time-average flow area as the product of the liquid film thickness and the perimeter of the conduit. The signal of this probe is then cross-correlated with the signal generated by another conductance probe, located downstream of the first one, to deduce the average flow velocity of the liquid film. The liquid film mass flow rate is then computed as the product of the liquid film flow area, the liquid film average velocity, and the density of the liquid. The entrained droplets mass flow rate is computed by subtracting the liquid film mass flow rate from the total liquid mass flow rate, and then the ELF is computed as indicated in Eq. (1).

Finally, the method employed by Han et al.20 can be regarded as a combination of the core sampling and the chemical tracer techniques. Specifically, the liquid film is first removed from the test section by means of an aperture realized on the channel wall. Then, the entire core flow is sampled, and the mass flow rate of the entrained droplets measured and used with Eq. (1) to compute the ELF. Incomplete liquid film removal is quantified using a chemical tracer, which is injected into the liquid film flow to estimate the liquid film carry-over into the core flow and, correspondingly, to correct the ELF measurements.

Unfortunately, for nine studies in Table I (No. 20–28, accounting for 13.3% of the data points), details of the ELF measuring technique employed are not available (these cases are referred to as NA, corresponding to “Not Available,” in Table I). These datasets have been extracted from the Harwell data bank, which is a well-known experimental data bank on annular flow that was previously compiled at AERE Harwell.21 The Harwell data bank does not include details on the measuring techniques that have been used to generate the ELF data, and most of the original studies do not seem to be openly accessible. The Harwell data bank is described in detail by Oliemans et al.22 and has often been used in previous investigations,23–25 hence its inclusion here despite the missing information.

In summary, the ELF measuring techniques that have most frequently been used to date are the core sampling and the film suction methods, although none of these techniques seems to have achieved yet the status of a reference or standard methodology. Cross-comparisons between different ELF measuring techniques are unfortunately rather limited, and only Magrini et al.,25 Andreussi,26 and Jagota et al.27 seem to have addressed this matter; their results, which are restricted to two cross-comparisons involving the three main ELF measuring techniques (core sampling vs film suction, and film suction vs chemical tracer), are reproduced in Fig. 4 (note that the data in the usable form by Magrini et al.25 included in Table I have been generated with the film suction method, and these authors also implemented the core sampling technique to provide the cross-comparison reproduced in Fig. 4). As can be noted in Fig. 4(a), ELF measurements obtained with the core sampling method are systematically lower (by about 10%–20%) than those generated with the film suction technique, whereas in Fig. 4(b), the film suction and the chemical tracer techniques provide consistent measurements. When the core sampling method is used, the sampling tube is normally kept at a distance from the liquid film sufficient to ensure that the crests of the disturbance waves are not picked up by the probe, which would cause an overestimate of the ELF. This might result in a systematic underprediction of the ELF, on account of the fact that the region of the core flow close to the liquid film is not duly explored during the measurement, which could explain the trend in the data in Fig. 4(a). Considering that these ELF measuring techniques can be quite invasive, and taking into account the measuring errors that are on the order of 10%–15% or greater (see Sec. II H later on), the cross-comparisons in Fig. 4 would indicate that, when properly implemented, all the three main ELF measuring techniques seem capable of providing reliable measurements, although the scope of the available cross-comparisons is too limited to identify the best among these methods.

FIG. 4.

Cross-comparison between different ELF measuring techniques: (a) core sampling vs film suction (data of Magrini et al.25) (b) film suction vs tracer (data of Andreussi26 and Jagota et al.27) the dashed lines are ±15% error bands.

FIG. 4.

Cross-comparison between different ELF measuring techniques: (a) core sampling vs film suction (data of Magrini et al.25) (b) film suction vs tracer (data of Andreussi26 and Jagota et al.27) the dashed lines are ±15% error bands.

Close modal

In the greatest part of the studies collected in Table I, the liquid and gas phases have been merged at the inlet of the test section using a mixer. Various inlet mixers have been employed [see Fig. 5 for sketches of the mixer used, and Fig. 1(f) for their relative importance within the database], ranging from a simple tee connection [Mixers #1–2 in Figs. 5(a) and 5(b), with the liquid or the gas fed through the branch side] or cross connection [Mixer #3 in Fig. 5(c), with the liquid fed through the branch side] to more sophisticated designs where the liquid is gradually introduced all around the conduit periphery, so that an annular flow is created immediately downstream of the mixer [Mixers #4–6 in Figs. 5(d)–5(f), where the liquid is injected radially through an annular slot, multi-holes or a porous insert, respectively]. With the mixer design adopted by Wallis (1962) [Mixer #7 in Fig. 5(g)], the liquid and gas phases enter opposite sides of a tee connection and then flow together through a 180° bend before entering the test section. Despite the rather abrupt mixing of the phases at the tee connection, the centrifugal force induced by the curvature of the bend can be expected to separate the phases and, hence, yield an annular flow at the test section inlet, though likely distorted by the secondary flows originated in the bend.

FIG. 5.

Sketch of the inlet mixer designs employed in the studies in Table I: (a) tee connection with liquid injected through the branch side, (b) tee connection with gas injected through the branch side, (c) cross connection with liquid injected through the branch side, (d) liquid injected radially through annular slot, (e) liquid injected radially through multi-holes, (f) liquid injected radially through porous sinter, (f) liquid and gas enter opposite sides of tee connection linked to test section inlet via a 180° bend.

FIG. 5.

Sketch of the inlet mixer designs employed in the studies in Table I: (a) tee connection with liquid injected through the branch side, (b) tee connection with gas injected through the branch side, (c) cross connection with liquid injected through the branch side, (d) liquid injected radially through annular slot, (e) liquid injected radially through multi-holes, (f) liquid injected radially through porous sinter, (f) liquid and gas enter opposite sides of tee connection linked to test section inlet via a 180° bend.

Close modal

The most frequently used mixer in the experiments [see Fig. 1(f)] is Mixer #6 in Fig. 5(f), which has been employed in 17 studies (accounting for 33% of the data points). This is not surprising, considering that this mixer is the one that yields the smoothest injection of the liquid around the periphery of the conduit, and correspondingly the most gradual mixing of the phases. Despite having been employed in only one study, Mixer #7 in Fig. 5(g) accounts for 9.5% of the data point, while other mixer designs individually account for a few percent of the data points [see Fig. 1(f)].

When a mixer was used, its design and the length L of the flow calming section between the mixer and the ELF measuring location [included in dimensionless form as L / d in Table I and Fig. 2(f), where d is the test section diameter] are key factors when assessing the importance of inlet effects on the measurements. The most systematic and exhaustive investigation into inlet effects on the ELF has been provided by Hinkle,28 who measured the ELF with water–air vertical upflow (0.22–0.62 MPa pressure and ambient temperature) in a circular pipe (12.62 mm diameter) varying the dimensionless distance between the inlet mixer (Mixer #6) and the ELF measuring location from 11.6 up to 262 and exploring nine different L / d values in total. These measurements are reproduced in Fig. 6 as ELF displayed as a function of the dimensionless distance between the inlet mixer and the ELF measuring location, plotted for four different gas mass flow rates and four different liquid mass flow rates (one for each panel in Fig. 6). According to Hinkle,28 fully developed annular flow conditions are achieved in the conduit once the ELF variation along the channel becomes linear. In fact, the linear increase in the ELF with downstream distance can be attributed to the flow acceleration induced by the pressure reduction along the channel, which causes an expansion of the gas phase and a consequent acceleration of the core flow and increase in the shear that this latter exerts on the liquid film. As can be noted in Fig. 6, the ELF trend vs distance from the mixer becomes linear, hence indicating the attainment of fully developed annular flow conditions, at a downstream distance L / d that varies, depending on the flow conditions, from about 40 to about 120. Specifically, a shorter calming section (as short as L / d 40 50) suffices for large liquid flow rates and small gas flow rates, whereas a longer calming section (up to L / d 100 120) is required for small liquid flow rates and large gas flow rates. This indicates that a relatively thick liquid film (corresponding to a large liquid mass flow rate) exposed to mild shear (corresponding to a small gas flow rate) develops faster, whereas a thin liquid film (corresponding to a small liquid mass flow rate) exposed to intense shear (corresponding to a large gas flow rate) needs a longer calming section to develop fully. Further data from the same study, provided in Fig. 7, indicate no noticeable effect of the operating pressure on the length of the calming section required to develop the flow, at least within the pressure range explored.

FIG. 6.

ELF measurements vs dimensionless distance L / d between inlet mixer and ELF measuring location, for different gas mass flow rates (12.7, 19.0, 25.3, and 31.6 g/s) and liquid mass flow rates of (a) 12.8, (b) 25.1, (c) 37.7, and (d) 51.8 g/s (water–air vertical upflow, 12.62 mm diameter circular pipe, Mixer #6; data of Hinkle28).

FIG. 6.

ELF measurements vs dimensionless distance L / d between inlet mixer and ELF measuring location, for different gas mass flow rates (12.7, 19.0, 25.3, and 31.6 g/s) and liquid mass flow rates of (a) 12.8, (b) 25.1, (c) 37.7, and (d) 51.8 g/s (water–air vertical upflow, 12.62 mm diameter circular pipe, Mixer #6; data of Hinkle28).

Close modal
FIG. 7.

ELF measurements vs dimensionless distance L / d between inlet mixer and ELF measuring location, for different inlet pressures (water–air vertical upflow, 12.62 mm diameter circular pipe, Mixer #6; data of Hinkle28).

FIG. 7.

ELF measurements vs dimensionless distance L / d between inlet mixer and ELF measuring location, for different inlet pressures (water–air vertical upflow, 12.62 mm diameter circular pipe, Mixer #6; data of Hinkle28).

Close modal

Inlet effects have also been considered, though not exhaustively, by Wallis,29 who measured the ELF with water–air vertical downward flow (atmospheric pressure and ambient temperature) in circular pipes of various diameters (from 3.02 to 22.2 mm). Selected measurements are reproduced in Fig. 8 as ELF displayed as a function of the superficial gas velocity, for three different dimensionless distances between the inlet mixer (Mixer #4) and the ELF measuring location and for two different liquid flow rates (one for each panel in Fig. 8). As can be noted, for lower superficial gas velocities (below about 45–50 m/s), there is no noticeable effect of the dimensionless distance L / d on the ELF. Conversely, for higher superficial gas velocities, the ELF increases with the dimensionless distance L / d , and this increase appears more pronounced in Fig. 8(b), where the liquid flow rate is higher. Even though three different values of the dimensionless distance L / d are not sufficient to precisely deduce the attainment of fully developed annular flow conditions, the data in Fig. 8, nonetheless, provide valuable information. Specifically, since the shear that the core flow exerts onto the liquid film increases with increasing superficial gas velocity, these data indicate that, when the shear is mild (low superficial gas velocity), a calming section as short as L / d 30 is sufficient to fully develop the flow, while an annular flow with intense shear (large superficial gas velocity) requires a longer calming section to reach fully developed flow conditions. This confirms the conclusions inferred previously when analyzing inlet effects from the data of Hinkle.28 

FIG. 8.

ELF measurements vs superficial gas velocity for different dimensionless distances L / d between inlet mixer and ELF measuring location, and liquid volumetric flow rates of (a) 10.7 and (b) 21.3 cm3/s (water–air vertical downflow, 15.6 mm diameter circular pipe, Mixer #4; data of Wallis29).

FIG. 8.

ELF measurements vs superficial gas velocity for different dimensionless distances L / d between inlet mixer and ELF measuring location, and liquid volumetric flow rates of (a) 10.7 and (b) 21.3 cm3/s (water–air vertical downflow, 15.6 mm diameter circular pipe, Mixer #4; data of Wallis29).

Close modal

Cross-comparisons between different inlet mixer designs are unfortunately quite limited, and only Wallis29 and Han et al.20 seem to have addressed this matter; their results are reproduced in Figs. 9 and 10 (note that the data in the usable form by Han et al.20 included in Table I have been generated with Mixer #2, and these authors also swapped the liquid and gas feeds into their tee connection mixer to provide the cross-comparison reproduced in Fig. 10). As can be noted in Fig. 9 (Mixers #4 and #7 used, ELF measured at a downstream distance L / d of 81), there is no noticeable effect of the inlet mixer on the ELF at high liquid flow rates, though an effect is present at high superficial gas velocities and low liquid flow rates, indicating that an L / d of 81 is not sufficient to dissipate inlet effects at these conditions, which correspond to thin liquid films exposed to high shear and, therefore, confirm the previous observations. No noticeable inlet mixer effects are in Fig. 10, where Mixers #1 and #2 have been employed, and the ELF was measured at a downstream distance L / d of 166.

FIG. 9.

ELF measurements vs superficial gas velocity for different inlet mixer designs (Mixer #4 and Mixer #7) and liquid volumetric flow rates (water–air vertical downflow, 22.2 mm diameter circular pipe, L / d = 81.1; data of Wallis29).

FIG. 9.

ELF measurements vs superficial gas velocity for different inlet mixer designs (Mixer #4 and Mixer #7) and liquid volumetric flow rates (water–air vertical downflow, 22.2 mm diameter circular pipe, L / d = 81.1; data of Wallis29).

Close modal
FIG. 10.

ELF measurements vs superficial liquid velocity for different inlet mixer designs (Mixer #1 and Mixer #2) and superficial gas velocities (water–air vertical upflow, 9.5 mm diameter circular pipe, L / d = 166; data of Han et al.20).

FIG. 10.

ELF measurements vs superficial liquid velocity for different inlet mixer designs (Mixer #1 and Mixer #2) and superficial gas velocities (water–air vertical upflow, 9.5 mm diameter circular pipe, L / d = 166; data of Han et al.20).

Close modal

In summary, the length of the flow calming section L / d between the inlet mixer and the ELF measuring location required to dissipate inlet effects depends on the operating conditions and, plausibly, on the type of mixer employed, albeit the direct evidence available on this latter aspect is too limited to draw any definite conclusions. Specifically, the available data indicate that thick liquid films exposed to mild shear develop faster, and an L / d as short as 30–50 might suffice, whereas thin liquid films under intense shear take longer to reach fully developed flow conditions, and an L / d as long as 100–120 may be required.

Semi-empirical prediction methods that can be used to estimate the length of the calming section required to reach fully developed annular flow conditions have been developed by Ishii and Mishima30 [Eq. (2)] and by Kataoka et al.31 [Eq. (3)]:
L d 600 J g I R e l ,
(2)
L d 440 W e g 0.25 R e l 0.5 .
(3)
The dimensionless gas superficial velocity J g I, the Weber number W e g, and the liquid Reynolds number R e l are defined as follows:
J g I = J g σ g ρ l ρ g ρ g 2 ρ g ρ l ρ g 0.67 0.25 ,
(4)
W e g = ρ g J g 2 d σ ρ l ρ g ρ g 0.33 ,
(5)
R e l = ρ l J l d μ l ,
(6)
where ρ l and ρ g are the densities of the liquid and gas, respectively, μ l is the liquid viscosity, σ is the surface tension, g is the acceleration of gravity, d is the conduit diameter, and J l and J g are the liquid and gas superficial velocities, respectively,
J l = 1 x G ρ l ; J g = x G ρ g ,
(7)
where x is the vapor quality and G is the mass flux. The prediction methods in Eqs. (2) and (3) are applicable when the annular flow is generated by smoothly injecting the liquid around the perimeter of the conduit (Mixers #4–#6 in Fig. 5) and can be used as first approximations for other mixer designs. From inspecting Eqs. (2) and (3), it can be noticed that both methods predict that the calming section length L / d increases with the increasing gas flow rate, hence increasing shear (note the direct proportionality to J g I or W e g), and with the decreasing liquid flow rate, hence decreasing film thickness (note the inverse proportionality to R e l), so that the trends observed in the experiments are qualitatively captured by both prediction methods. Unfortunately, when applied to the data bank in Table I, these methods predict calming section lengths L / d varying from 3.4 to 631, which is clearly too wide range that includes predictions that are too short or too long to be physically plausible. Specifically, very short calming sections ( L / d < 10) are sometimes predicted for water–air flows through large diameter pipes and for fluids other than water–air, while very long calming sections ( L / d > 300) are sometimes predicted for tests,13,32,33 which, in principle, should be within the applicability range of these methods. Despite their success in qualitatively reproducing the observed trends, therefore, none of these prediction methods seems adequate when dealing with large and diversified databanks like the present one in Table I. The available data on inlet effects appear too restricted in scope to improve these methods, let alone developing new ones. As a preliminary rule-of-thumb inferred from the available data, an L / d on the order of 100–120 seems adequate to make inlet effects on the ELF minor or absent, irrespective of the operating conditions or the type of mixer used. Notably, this is less conservative than what reported in previous research, where calming section lengths up to 300 tube diameters were recommended for the ELF.34 As can be noted in Fig. 2(f), for most of the data in Table I, the calming section was long enough to make inlet effects minor or absent.

Inlet effects are clearly not relevant when the two-phase flow was generated by evaporating the liquid flow in a pre-heater or in a heated portion of the test section located upstream of the ELF measuring section because in these cases, a mixer was not employed (these cases are referred to as NR, corresponding to “Not Relevant,” in Table I). Note, however, that in some studies in Table I, the saturated two-phase flow was generated by mixing a saturated liquid flow with a saturated vapor flow,35,36 so that for these studies, inlet effects are, indeed, relevant.

Finally, for 17 studies in Table I (accounting for 31.3% of the data points), the mixer design is unfortunately not specified (these cases are referred to as NA, corresponding to Not Available, in Table I). However, for these studies, the calming section (see values included in Table I) was long enough to make inlet effects minor or absent.

When ELF measurements are performed using transparent test sections, the attainment of annular flow conditions can be confirmed by direct visual inspection of the flow, so that the gathered data only cover annular flow conditions, and any contamination from other flow patterns can be avoided. This is clearly not the case when the optical access is not available or the visual inspection of the flow is not feasible, and a contamination from other flow patterns is, therefore, possible. Specifically, the flow pattern that can contaminate annular flow measurements is intermittent flow and, in the case of experiments carried out with horizontal test sections, stratified flow.

Starting with the contamination from intermittent flow, the transition from intermittent flow to annular flow is normally associated with values of the void fraction on the order of 0.7–0.8.37 Therefore, a preliminary and effective way to assess whether the data bank in Table I is contaminated by intermittent flow is by estimating the value of the void fraction at the local operating conditions (note that the void fraction is not normally measured simultaneously with the ELF and, therefore, needs to be estimated). Among the several void fraction prediction methods available in the literature, two have been selected for use here: the method proposed by Cioncolini and Thome38 specifically for annular flows, and the well-known general-purpose method developed by Woldesemayat and Ghajar;39 the predictions of these two methods are provided in Figs. 2(g)–2(h). As can be noted, for the vast majority of the data points in Table I, the void fraction predicted according to both methods is large enough to ensure that the contamination from intermittent flow is minimal and restricted to a few percent of the data points at most.

As a further check for data bank contamination, the vertical upflow data from Table I are displayed in Fig. 11(a) on the flow pattern map proposed by Hewitt and Roberts,40 while the horizontal flow data are displayed in Fig. 11(b) on the flow pattern map developed by Taitel and Dukler41 and later extended to account for symmetric and asymmetric annular flows.42 These are the most widely quoted flow maps for vertical upflow and horizontal flow, hence their selection for use here among the several flow pattern maps proposed to date. In the Hewitt and Roberts40 flow map, the coordinates are computed from the mass flux, vapor quality, and the densities of the phases [see Fig. 11(a)], whereas in the map by Taitel and Dukler,41 the coordinates are the Martinelli parameter X and the densimetric Froude number F r, which are defined as follows:
X = d P / d z l d P / d z g ,
(8)
F r = x G ρ g ρ l ρ g g d ,
(9)
where d P / d z l and d P / d z g are the frictional pressure gradients computed for the liquid and gas phases when flowing alone in the conduit. Further details on the implementation of these flow pattern maps can be found in Refs. 43 and 44. As can be noted in Figs. 11(a) and 11(b), the contamination of the vertical upflow data from intermittent flow (churn flow in the terminology of the flow map) is minimal at most, and it is absent for the horizontal flow data, hence confirming the previous deduction based on the values of the void fraction.
FIG. 11.

Comparison of ELF data from Table I with flow pattern maps: (a) vertical upflow data displayed on the flow map by Hewitt and Roberts;40 (b) horizontal flow data displayed on the Taitel and Dukler41 flow map modified42 to account for symmetric and asymmetric annular flows.

FIG. 11.

Comparison of ELF data from Table I with flow pattern maps: (a) vertical upflow data displayed on the flow map by Hewitt and Roberts;40 (b) horizontal flow data displayed on the Taitel and Dukler41 flow map modified42 to account for symmetric and asymmetric annular flows.

Close modal

Regarding the horizontal flow data from Table I, it is evident from Fig. 11(b) that a minor contamination from stratified flow is likely. A stratified flow, where the liquid gathers in a rivulet that streams along the bottom of the channel, evolves into an annular flow when the local flow conditions promote the formation of a continuous liquid film around the entire channel perimeter. Since the effect of gravity is to drain the liquid in the film toward the bottom of the channel, the formation of a continuous liquid film necessitates that the transport of liquid toward the sides and the top of the channel is effective enough to counterbalance the draining effect of the gravity force. Various liquid transport mechanisms are at play simultaneously,42 including the spreading of the disturbance waves under the action of the gas core flow,45,46 the secondary flows that may originate in the core flow and that can further distribute the liquid in the channel cross section, and the entrained droplets dispersion due to the turbulence present in the core flow.47–49 The complexity of these mechanisms and the interplay among them50 have as yet precluded a complete mechanistic understanding of the stratified to annular flow transition, hence the usefulness of empirical predictions methods such as the flow pattern map employed herein.

In summary, the contamination of the data bank in Table I from intermittent flow is minimal at most for vertical flow conditions and absent for horizontal flow conditions, while a minor contamination of the horizontal flow data from stratified flow is likely.

With gas–liquid two-phase flows in general, when the flow velocity is large enough, the inertia of the flow overcomes the pull of the gravity force, whose effect on the flow correspondingly becomes minor or inconsequential. In the case of annular flows, when the flow velocity is large enough, the liquid film fluid dynamics is entirely controlled by the shear of the core flow, which increases in proportion to the flow velocity, and the effect of gravity on the flow becomes negligible. In this condition, the predominant mode of liquid entrainment is the atomization of the tips of the disturbance waves that form along the liquid film as a consequence of the shear exerted by the core flow. On the other hand, when the flow velocity is moderate, and the shear is correspondingly mild, the pull of gravity can affect the fluid dynamics of the liquid film, hence the liquid entrainment process. When present, the effect of gravity clearly depends on the direction of the flow with respect to the vertical. For the rest of the analysis, it is, therefore, convenient to consider vertical flows and horizontal flows separately.

Starting with vertical flows, a simple way to assess gravity effects is by comparing ELF measurements taken during vertical upflow and vertical downflow and generated under otherwise comparable operating conditions (apart from the flow orientation). Differences between the ELF measurements can then be ascribed to the flow orientation, and similarity of the ELF measurements would indicate that gravity effects are inconsequential. Cross-comparisons between vertical upflow and downflow are unfortunately very limited, and only Wallis29 seems to have addressed this matter; his measurements (water–air at ambient conditions, vertical circular pipe of 12.7 mm diameter, Mixer #7 and L / d of 78) are reproduced in Fig. 12(a) as ELF plotted vs superficial gas velocity, for two different liquid flow rates. It is evident that, when the superficial gas velocity (and hence the shear) is large enough (above about 20–30 m/s), the ELF measurements taken during upflow and downflow tend to collapse on the same trend, indicating that the ELF does not depend on the flow orientation. At these conditions, therefore, the flow velocity and, consequently, the shear of the gas core onto the liquid film are large enough to make the effect of gravity on the flow negligible. On the other hand, as the superficial gas velocity (and hence the shear) is gradually reduced, the ELF measurements follow two different trends for the different flow directions, as a consequence of the effect of the gravity force on the flow. Specifically, in vertical upflow, the ELF goes through a minimum and then increases with decreasing superficial gas velocity, whereas in vertical downflow, the ELF decreases monotonically toward zero when the superficial gas velocity decreases. These trends can be qualitatively explained as follows. In the case of vertical upflow, gravity is opposing the flow: when the shear is mild, the pull of gravity can delay the rise of the liquid in the film, which consequently thickens, thereby increasing the entrainment. Further shear reduction will cause further film thickening and consequent increase in the ELF, hence explaining the increasing trend observed in Fig. 12(a). With vertical downflow, on the other hand, gravity is assisting the flow: as the shear is progressively reduced, the flow in the liquid film gradually switches from a shear-driven flow to a gravity-driven falling film flow, where the entrainment vanishes and the ELF correspondingly converges toward zero, as observed in Fig. 12(a).

FIG. 12.

ELF measurements vs (a) superficial gas velocity and (b) non-dimensional superficial gas velocity in Eq. (10), taken during upflow and downflow for different liquid flow rates (water–air at ambient conditions, vertical circular pipe of 12.7 mm diameter, Mixer #7 and L / d of 78; data of Wallis29).

FIG. 12.

ELF measurements vs (a) superficial gas velocity and (b) non-dimensional superficial gas velocity in Eq. (10), taken during upflow and downflow for different liquid flow rates (water–air at ambient conditions, vertical circular pipe of 12.7 mm diameter, Mixer #7 and L / d of 78; data of Wallis29).

Close modal
A simple criterion to assess the relative importance of the gravity force on vertical flow can be derived as follows. During vertical upflow, the annular flow pattern persists until the shear of the core flow becomes insufficient to drag along all the liquid in the film, and some of the liquid starts flowing downward under the pull of gravity—a condition referred to as flow reversal. A simple criterion for predicting flow reversal was proposed by Wallis51 and is based on the non-dimensional superficial gas velocity J g W , defined as follows:
J g W = J g ρ g g d ρ l ρ g .
(10)
According to this criterion, flow reversal occurs at J g W 1, while the annular flow remains co-current so long as J g W > 1. It is, therefore, possible to employ the value of the non-dimensional superficial gas velocity in Eq. (10) to gauge the relative importance of the gravity force on the annular flow during vertical upflow: the higher the value of J g W, the smaller the effect of gravity on the flow. Even though there is clearly no flow reversal during vertical downflow, J g W can still be computed, and its value used to gauge the relative importance of the gravity force on the annular flow. The ELF measurements previously discussed in Fig. 12(a) are reproduced in Fig. 12(b) using the non-dimensional superficial gas velocity in Eq. (10) as x axis coordinate. As can be noted, the ELF measurements taken during upflow and downflow collapse onto the same trend for J g W values above about 2–2.5, which can, therefore, be used as a threshold above which gravity effects become inconsequential during vertical flow. As can be noted in Figs. 2(i)–2(j), this is the case for the vast majority of the vertical upflow and downflow data collected in Table I, which, therefore for the most part, represent shear-driven annular flows, where the effect of gravity on the flow is minor or inconsequential. Despite its simplicity, the flow reversal criterion in Eq. (10) generally performs well,52 and it is, therefore, deemed appropriate for the present preliminary assessment. A more extensive account of flooding can be found in Richter.53 

In the case of horizontal flow, gravity is, in principle, neither assisting nor opposing the flow. However, when the flow velocity is moderate, and the shear is correspondingly mild, the gravity force can affect the distribution of the liquid mass in the channel cross section. Specifically, the liquid in the film tends to drain down the conduit wall under the action of the gravity force and, consequently, accumulates at the bottom of the channel, giving rise to an asymmetric liquid film that is thinner at the top of the conduit and thicker at the bottom. This clearly affects the liquid film atomization process, which becomes more intense at the bottom of the conduit where the liquid film is thicker and less intense or even absent at the top and along the sides of the conduit where the liquid film is thinner. As can be noted in Fig. 11(b), most of the horizontal flow data collected in Table I fall in the asymmetric annular flow region, indicating that these ELF measurements have been taken at operating conditions where the annular flow is affected by the gravity force, and, hence, the liquid film is not symmetric around the conduit perimeter.

Gravity effects may clearly be present also for the flow inclinations other than vertical or horizontal in Table I. However, the scope of the available data are too restricted to make any assessments.

The compressibility characteristics of the annular flow data in Table I can be assessed using the following simple form of the Mach number:
M a = V tpf a tpf V c a g = x G ρ g ϵ a g ,
(11)
where V tpf is the average two-phase flow velocity, which is approximated in Eq. (11) with the average velocity V c of the core flow computed by neglecting the slip between the gas and the entrained liquid droplets, ϵ is the void fraction (computed with the method by Cioncolini and Thome38) while a tpf is the sonic velocity for the two-phase flow, which is approximated with the gas sonic velocity a g. With two-phase flows, the value of the sonic velocity is intermediate between the values of the sonic velocities in the liquid and gas phases. Since sonic velocities in liquids are normally larger than those in gases, the sonic velocity in the gas can be employed as a simple lower bound estimate of the two-phase sonic velocity, particularly in the case of high void fraction two-phase flows such as annular flows.54 This is the approximation employed in Eq. (11), which, correspondingly, provides an upper-bound estimate of the Mach number. The threshold below which compressibility effects can be ignored is normally set at M a 0.3. As can be noted in Fig. 2(k), compressibility effects in the data bank in Table I are inconsequential.

Despite their practical importance, measurement errors are extensively discussed in only a few of the studies collected in Table I, and often the information provided is limited to the measuring errors of the liquid and gas flow rates (normally measured to within a few percent accuracy). As noted previously, the most frequently used ELF measuring techniques are quite invasive, and their implementation is not straightforward, so that inferring the ELF measurement error is not immediate, even when the measuring errors of the liquid and gas flow rates are provided. Based on the limited information provided in the studies in Table I, ELF measurement errors are typically on the order of ±10%–15% (or larger).

One aspect that has apparently been overlooked in previous investigations is that the film suction measuring technique becomes inaccurate when employed to measure low ELF values. With this technique, in fact, it is the liquid film mass flow rate m ̇ l f that is measured, so that the ELF is computed as follows:
e = m ̇ l e m ̇ l = m ̇ l m ̇ l f m ̇ l
(12)
and the corresponding measuring error is therefore
δ e e = 1 e e δ m ̇ l f m ̇ l f 2 + δ m ̇ l m ̇ l 2 .
(13)
It is evident from inspecting Eq. (13) that the relative error of the ELF diverges when e 0. For example, assuming that the liquid film and the liquid mass flow rates are individually measured with good accuracy, say to within ±2%, then according to Eq. (13), the error on the ELF would be of ±12%, ±17%, and ±27% for ELF values of 0.2, 0.15, and 0.1, respectively. This clearly indicates that the film suction technique should not be employed to measure low values of the ELF, and that ELF measurements shown in Table I with a value lower than about 0.1–0.2 that have been generated with the film suction technique should be used with caution.
Available evidence indicates that there might be a connection between the type of flow existing in the liquid film, that is, laminar or turbulent, and the intensity of liquid atomization process responsible for the entrainment. Specifically, the entrainment process seems to be active so long as the flow in the liquid film is turbulent, and when the liquid film laminarizes, the entrainment process becomes suppressed. During his experiments, Wallis29 noticed that the laminarization of the flow in the liquid film was accompanied by a reduction of the entrainment, hence indicating an increased stability of laminar liquid films as opposed to turbulent liquid films. Suppression of the liquid film atomization at low liquid film flow rates, when the flow in the film approaches laminarization, was also observed in subsequent studies.15,32 During their measurements, Miya et al.55 observed that the flow in the liquid film becomes laminar in the region between successive disturbance waves, so that these latter can be regarded as pockets of turbulent liquid flow that travel along the conduit by surfing along a laminar liquid film substrate. This suggests that any turbulence present in the liquid film flow might be localized within the disturbance waves. Since these latter are the main source of entrained droplets, then the entrainment process and the presence of turbulence in the liquid film flow would be linked: if the disturbance waves are not present, then the entrainment process becomes suppressed, and the flow in the liquid film becomes laminar. The existence of such a link can be checked on the present data bank as follows. A simple laminar to turbulent flow transition criterion for shear-driven annular liquid films, proposed by Cioncolini et al.,56 is based on the value of the liquid film Reynolds number R e l f, defined as follows:
R e l f = 1 e 1 x G d μ l .
(14)
According to this criterion, the flow in the liquid film is laminar for R e l f < 162, turbulent for R e l f > 2785, and transitional in between these values. Note that, in its original and stand-alone formulation, this criterion employs the prediction method of Cioncolini and Thome23 (see Sec. III I) to predict the value of the ELF that is required as input to Eq. (14). Since, in the present case, the measured ELF values are available, these latter were used instead. Note that this criterion is restricted in applicability to shear-driven annular liquid films; hence, it is not applicable to the horizontal flow data in Table I, which, as noted previously, are affected by gravity and, therefore, do not correspond to purely shear-driven annular flows. As can be noted in Fig. 2(l), the flow in the liquid film is transitional or turbulent for most of the data points in Table I taken during vertical flow, hence confirming the link between the entrainment process and the presence of turbulence in the liquid film flow.

In the tests carried out by Al-Yarubi19 (water–air at ambient conditions, vertical upflow, 50 mm diameter tube, Mixer #1), the calming section length was of L / d = 40: the author was aware that the calming section was likely too short to fully develop the flow; however, the main objective of this study was to demonstrate the effectiveness of the conductance probe technique to measure the ELF and not to generate data of archival value.

Since the original report by Steen and Wallis57 was not available, the main details on the tests and a selection of data in usable form were gathered from subsequent studies.30,58

In addition to the main details on the tests and the data provided in Singh et al.,59 additional details and further usable data from the same experiment were gathered from Ref. 60.

Adamsson and Anglart8 carried out ELF measurements with a setup conceived to generate data informative of boiling water nuclear reactors (water at 7 MPa pressure, vertical upflow, 14 mm diameter tube), exploring evaporating flow conditions with different heating power distributions along the test pipe: uniform, inlet-peaked, middle-peaked, and outlet-peaked; all details on the tests and the data are provided in Ref. 8, while the axial power profiles in tabular form can be found in Ref. 61.

The spacer grids employed for structural support in water-cooled nuclear reactor fuel bundles normally include mixing vanes designed to induce swirl in the coolant flow, hence droplet de-entrainment during annular two-phase flow: this delays the dryout thereby enhancing the fuel rod cooling effectiveness. Tests with representative spacer grids, with and without mixing vanes, have been conducted by Kawahara et al.62,63 and Feldhaus et al.12 The reduction of the ELF downstream of the mixing vanes can reach 20%–30%.23 

The present review is restricted to the entrained liquid fraction; hence, experimental data and prediction methods regarding the rates of liquid entrainment and droplets deposition are not considered.

Data digitizing was carried out with the web-based free tool WebPlotDigitizer,64 while data processing was carried out with the free software GNU Octave version 5.2.0.65,66 Fluids thermo-physical properties were computed with NIST-REFPROP67 with the exception of Ethanol, whose properties were obtained from Hewitt,68 and Isopar-L whose properties are provided in the studies69,70 where this liquid was employed.

The curated ELF data bank is freely available through the open access repository Figshare (dx.doi.org/10.6084/m9.figshare.24002586). Specifically, the data bank is provided as a single matrix of numerical data, counting 4175 rows (one for each data point) and 21 columns whose entries are explained in Table II.

TABLE II.

Entries in the ELF data bank as available through Figshare.

No. Column Field Values
Experiment identifier: integer that identifies the reference study where the data are communicated; possible values: 1–53  Same values as the first column of Table I  
Fluids identifier: integer that identifies the fluid combination used in the experiment; possible values: 1–10  1: H2O–air 
2: H2
3: R12 
4: R113 
5: Silicon–air 
6: Ethanol–air 
7: R140a–air 
8: H2O–He 
9: H2O + PLE–air 
10: IsoparL–N2 
Test identifier: integer that identifies the type of test; possible values: −1, 0 1  −1: Condensing flow 
0: Adiabatic flow 
1: Evaporating flow 
Test section geometry identifier: integer that identifies the geometry of the test section; possible values: 1–4  1: Circular tube 
2: Annulus 
3: Channel employed by Feldhaus et al.,12 see Fig. 3  
4: Rectangular channel employed by Quandt9  
ELF measuring technique identifier: integer that identifies the ELF measuring technique employed in the tests; possible values: 0–7  0: Not available (NA) 
1: Film suction (FS) 
2: Core sampling (CS) 
3: Chemical tracer (TR) 
4: Optical (OP) method employed by Azzopardi and Zaidi16  
5: Conductance (CN) 
6: Combination of core sampling (CS) and chemical tracer (TR) methods employed by Han et al.20  
7: Optical (OP) method employed by Langner and Mayinger17  
Mixer identifier: integer that identifies the type of mixer used to merge the liquid and gas phases at the inlet of the test section; possible values: 0–8  0: Not relevant (NR) 
1: Mixer #1 (see Fig. 5
2: Mixer #2 (see Fig. 5
3: Mixer #3 (see Fig. 5
4: Mixer #4 (see Fig. 5
5: Mixer #5 (see Fig. 5
6: Mixer #6 (see Fig. 5
7: Mixer #7 (see Fig. 5
8: Not available (NA) 
Spacer grid and mixing vanes identifier: integer that specifies the type of spacer grids and flow vanes used during the tests; possible values: 0–7  0: Spacer grids not present 
1: EggCrate spacer grid, ELF measured upstream12  
2: EggCrate spacer grid, ELF measured downstream12  
3: UltraFlow spacer grid, ELF measured upstream12  
4: UltraFlow spacer grid, ELF measured downstream12  
5: Bare spacer grid without mixing vanes62,63 
6: Spacer grid with mixing vanes at 30° inclination62,63 
7: Spacer grid with mixing vanes at 20° inclination62,63 
Operating pressure (MPa)   
Operating temperature (K)   
10  Liquid density (kg/m3  
11  Gas density (kg/m3  
12  Liquid viscosity (kg/ms)   
13  Gas viscosity (kg/ms)   
14  Surface tension (N/m)   
15  Speed of sound in gas (m/s)   
16  Test section diameter (m) if circular tube, hydraulic diameter (m) otherwise   
17  Flow inclination (°) with respect to the horizontal (0°: horizontal flow; 90°: vertical upflow; −90°: vertical downflow)   
18  Test section L / d (-)  Reset to zero if not relevant 
19  Liquid mass flow rate (kg/s)   
20  Gas mass flow rate (kg/s)   
21  ELF (-)   
No. Column Field Values
Experiment identifier: integer that identifies the reference study where the data are communicated; possible values: 1–53  Same values as the first column of Table I  
Fluids identifier: integer that identifies the fluid combination used in the experiment; possible values: 1–10  1: H2O–air 
2: H2
3: R12 
4: R113 
5: Silicon–air 
6: Ethanol–air 
7: R140a–air 
8: H2O–He 
9: H2O + PLE–air 
10: IsoparL–N2 
Test identifier: integer that identifies the type of test; possible values: −1, 0 1  −1: Condensing flow 
0: Adiabatic flow 
1: Evaporating flow 
Test section geometry identifier: integer that identifies the geometry of the test section; possible values: 1–4  1: Circular tube 
2: Annulus 
3: Channel employed by Feldhaus et al.,12 see Fig. 3  
4: Rectangular channel employed by Quandt9  
ELF measuring technique identifier: integer that identifies the ELF measuring technique employed in the tests; possible values: 0–7  0: Not available (NA) 
1: Film suction (FS) 
2: Core sampling (CS) 
3: Chemical tracer (TR) 
4: Optical (OP) method employed by Azzopardi and Zaidi16  
5: Conductance (CN) 
6: Combination of core sampling (CS) and chemical tracer (TR) methods employed by Han et al.20  
7: Optical (OP) method employed by Langner and Mayinger17  
Mixer identifier: integer that identifies the type of mixer used to merge the liquid and gas phases at the inlet of the test section; possible values: 0–8  0: Not relevant (NR) 
1: Mixer #1 (see Fig. 5
2: Mixer #2 (see Fig. 5
3: Mixer #3 (see Fig. 5
4: Mixer #4 (see Fig. 5
5: Mixer #5 (see Fig. 5
6: Mixer #6 (see Fig. 5
7: Mixer #7 (see Fig. 5
8: Not available (NA) 
Spacer grid and mixing vanes identifier: integer that specifies the type of spacer grids and flow vanes used during the tests; possible values: 0–7  0: Spacer grids not present 
1: EggCrate spacer grid, ELF measured upstream12  
2: EggCrate spacer grid, ELF measured downstream12  
3: UltraFlow spacer grid, ELF measured upstream12  
4: UltraFlow spacer grid, ELF measured downstream12  
5: Bare spacer grid without mixing vanes62,63 
6: Spacer grid with mixing vanes at 30° inclination62,63 
7: Spacer grid with mixing vanes at 20° inclination62,63 
Operating pressure (MPa)   
Operating temperature (K)   
10  Liquid density (kg/m3  
11  Gas density (kg/m3  
12  Liquid viscosity (kg/ms)   
13  Gas viscosity (kg/ms)   
14  Surface tension (N/m)   
15  Speed of sound in gas (m/s)   
16  Test section diameter (m) if circular tube, hydraulic diameter (m) otherwise   
17  Flow inclination (°) with respect to the horizontal (0°: horizontal flow; 90°: vertical upflow; −90°: vertical downflow)   
18  Test section L / d (-)  Reset to zero if not relevant 
19  Liquid mass flow rate (kg/s)   
20  Gas mass flow rate (kg/s)   
21  ELF (-)   

Numerous empirical or semi-empirical prediction methods have been developed to predict the ELF for practical applications. In the following, fifteen ELF prediction methods widely quoted in the literature are presented and described in detail. As already noted, the value of the ELF is bounded between 0 and 1: when a prediction method is not bounded in this range, this is explicitly highlighted. Similarly, prediction methods of implicit formulation [i.e., of the form e = f e], hence requiring numerical solution, are explicitly highlighted.

Prediction methods that incorporate empirical fits based on limited datasets, hence not applicable or difficult to extrapolate to other flow conditions,32,71 or prediction methods of limited scope specifically developed for particular applications72–74 are not considered here. Prediction methods based on artificial intelligence, hence not of immediate implementation, are also not considered.75 

Aside from the correlations discussed later, the ELF can also be predicted by employing a mechanistic model for annular flow.21,31,76–78 Mechanistic models provide estimates of all the annular flow parameters of interest in practical applications (such as the average liquid film thickness, the wall shear stress and associated pressure gradient, the heat transfer coefficient, and so on), including the ELF that is normally computed as the result of the two competing effects of liquid entrainment and droplets deposition, which are separately modeled. Despite being more plausible than correlations from a physical point of view, existing comparisons indicate that empirical or semi-empirical correlations normally outperform mechanistic models,22,44 chiefly on the account of the practical difficulties in procuring the detailed experimental data needed for calibrating these latter. Mechanistic models are, therefore, not considered here.

The prediction method proposed by Paleev and Filippovich79 reads as follows (where “log” stands for the logarithm with base 10):
e = 0.015 + 0.44 log 10 4 ρ g ρ l 1 + e 1 x x μ l J g σ 2 .
(15)
The experimental data bank that underlies the development of Eq. (15) is not described in detail by the authors; however, the range of flow parameters covered corresponds to a variation of the argument of the logarithm in the range of 1–100. As can be noted in Eq. (15), the method is of the form e = f e , and, hence, it is implicit in e, and it is not bounded in 0 , 1.
According to the graphical prediction method proposed by Wallis58 and reproduced in Fig. 13, the ELF is predicted as a function of the following non-dimensional group:
Π w = ρ g ρ l μ g J g σ .
(16)
FIG. 13.

ELF graphical prediction method proposed by Wallis:58 discrete points (digitized from the original fitting line) vs non-dimensional group in Eq. (16), together with fitting relation in Eq. (17).

FIG. 13.

ELF graphical prediction method proposed by Wallis:58 discrete points (digitized from the original fitting line) vs non-dimensional group in Eq. (16), together with fitting relation in Eq. (17).

Close modal

The similarity between this non-dimensional group and the argument of the logarithmic term in the method of Paleev and Filippovich79 in Eq. (15) is not accidental. In fact, the author derived the graphical correlation in Fig. 13 as a modified version of the method by Paleev and Filippovich,79 which was specifically conceived to provide a better account of liquid viscosity and liquid film flow rate effects, besides also being explicit and hence more convenient to use.

In addition to the data used by Paleev and Filippovich,79 the underling experimental data bank includes ELF measurements for vertical downflow through a circular tube (15.9 mm diameter) of water–air and silicon–air at operating pressures close to atmospheric.

As can be noted in Fig. 13, in its original formulation, the graphical correlation is limited to values of the ELF within the range of 0–0.8. For use here, the graphical correlation has been fitted with the generalized sigmoid function provided in the following equation:
e = 1 + 1.275 10 6 Π w 1.242 60.44 .
(17)
As can be noted in Fig. 13, the fitting relation in Eq. (17) covers the entire range of ELF values from 0 to 1, thereby extending the applicability range of the graphical correlation.
In the method developed by Oliemans et al.,22 the ELF is predicted as follows:
e 1 e = 10 b 0 ρ l b 1 ρ g b 2 μ l b 3 μ g b 4 σ b 5 d b 6 J l b 7 J g b 8 g b 9 .
(18)
The exponents b 0 b 9, whose numerical values have been derived to make the right-hand side of Eq. (18) a non-dimensional group, are summarized in Table III as a function of the liquid film Reynolds number R e l f , defined in Eq. (14). Since this latter depends on the ELF, the prediction method in Eq. (18) is implicit and has to be used iteratively. Operatively, a first-approximation ELF estimate is obtained by employing the exponents from the first row in Table III, which are applicable for any value of the liquid film Reynolds number. This latter can then be calculated, and the estimate of the ELF can correspondingly be refined by picking the appropriate values of the exponents in Table III, according to the value of the liquid film Reynolds number. Iterating further is clearly possible; however, the predictions are not continuous across the transition values of the liquid film Reynolds number; hence, further iterations may cause numerical oscillations, should the ELF value fall close to one of the transition points.
TABLE III.

Parameters for Oliemans et al.22 ELF prediction method [Eq. (18)].

R e l f b o b 1 b 2 b 3 b 4 b 5 b 6 b 7 b 8 b 9
All values  −2.52  1.08  0.18  0.27  0.28  −1.80  1.72  0.70  1.44  0.46 
102–3 × 102  −0.69  0.63  0.96  −0.80  0.09  −0.88  2.45  0.91  −0.16  0.86 
3 × 102–103  −1.73  0.94  0.62  −0.63  0.50  −1.42  2.04  1.05  0.96  0.48 
103–3 × 103  −3.31  1.15  0.40  −1.02  0.46  −1.00  1.97  0.95  0.78  0.41 
3 × 103–104  −8.27  0.77  0.71  −0.13  −1.18  −0.17  1.16  0.83  1.45  −0.32 
104–3 × 104  −6.38  0.89  0.70  −0.17  −0.55  −0.87  1.67  1.04  1.27  0.07 
3 × 104–105  −0.12  0.45  0.25  0.86  −0.05  −1.51  0.91  1.08  0.71  0.21 
R e l f b o b 1 b 2 b 3 b 4 b 5 b 6 b 7 b 8 b 9
All values  −2.52  1.08  0.18  0.27  0.28  −1.80  1.72  0.70  1.44  0.46 
102–3 × 102  −0.69  0.63  0.96  −0.80  0.09  −0.88  2.45  0.91  −0.16  0.86 
3 × 102–103  −1.73  0.94  0.62  −0.63  0.50  −1.42  2.04  1.05  0.96  0.48 
103–3 × 103  −3.31  1.15  0.40  −1.02  0.46  −1.00  1.97  0.95  0.78  0.41 
3 × 103–104  −8.27  0.77  0.71  −0.13  −1.18  −0.17  1.16  0.83  1.45  −0.32 
104–3 × 104  −6.38  0.89  0.70  −0.17  −0.55  −0.87  1.67  1.04  1.27  0.07 
3 × 104–105  −0.12  0.45  0.25  0.86  −0.05  −1.51  0.91  1.08  0.71  0.21 

The experimental data used to develop Eq. (18) were taken from the Harwell data bank,21 which contains 728 ELF data points generated with vertical upflow through circular tubes (diameters from 6 to 31.8 mm) of four fluid combinations (water, water–air, R140a–air, and ethanol–air) covering operating pressures in the range of 0.1–9 MPa.

Even though the right-hand side of Eq. (18) is a non-dimensional group, its formulation and the different values of the exponents make any physical interpretation not straightforward. As noted by the authors, among the parameters included in Eq. (18), those exerting the strongest influence on the ELF are the superficial gas velocity, the tube diameter, and the surface tension. Specifically, the ELF is directly proportional to the former two and inversely proportional to the latter, hence pointing to shear-driven liquid film atomization as the main source of entrainment. According to the authors, attempts to improve the prediction method in Eq. (18) by identifying simpler non-dimensional groups of more immediate physical interpretation were not successful.

In the method proposed by Ishii and Mishima,30 the ELF is predicted as follows:
e = tanh 7.25 10 7 W e 1.25 R e l 0.25 ,
(19)
where the Weber number W e and the liquid Reynolds number R e l are
W e = ρ g J g 2 d σ ρ l ρ g ρ g 0.33 ; R e l = ρ l J l d μ l .
(20)
The underling experimental data bank includes upflow and downflow through circular tubes (diameters from 9.5 to 32 mm) of water–air covering operating pressures in the range of 0.1–0.4 MPa.

This prediction method has been quite influential and has been used as starting point of various subsequent correlations.

Nakazatomi and Sekoguchi80 developed two different correlations: one applicable for low operating pressures (0.1–5 MPa),
e = tanh 10 4 C W e a F r 1.5 n b
(21)
and another applicable for high operating pressures (7–20 MPa),
e = exp 0.35 m W e 0.14 m F r 1.5 N 2 + 0.2 ,
(22)
where the Weber number W e and the Froude number F r are defined as
W e = ρ l J l 2 d σ ; F r = J g g d ;
(23)
while the parameters C , N , n , a , b , and m are computed as
C = N 1.75 + 10 N 0.8 + N 1.75 ; N = 100 ρ g ρ l ; n = ρ l ρ g ; a = 0.68 m 1 m 2 ; b = 2.5 exp m ; m = l n ρ l ρ g .
(24)
The underling experimental data bank includes upflow through a circular tube (19.2 mm diameter) of water–air covering operating pressures in the range of 0.1–20 MPa.

Operatively, in order to also cover the pressure range of 5–7 MPa, Eq. (21) can be extrapolated up to 6 MPa, while Eq. (22) can be extrapolated down to 6 MPa; however, the respective predictions at 6 MPa do not match. This aspect was apparently not considered by the authors.

The prediction method proposed by Utsuno and Kaminaga81 reads as follows:
e = tanh 0.16 W e 0.08 R e l 0.16 1.2 ,
(25)
where the Weber number W e and the liquid Reynolds number R e l are calculated as indicated in Eq. (20). The recommended application range covers Weber number values within 260–83k and liquid Reynolds number values within 5.4k–350k. This correlation was conceived as a modified version of the previous method by Ishii and Mishima30 and was derived employing water data in vertical circular tubes covering operating pressures of 3–9 MPa specifically targeting boiling water nuclear reactor applications.
According to Pan and Hanratty,82 the ELF is predicted as follows:
e / e max 1 e / e max = 6 10 5 W e .
(26)
The novel idea of this prediction method is the existence of an upper-bound for the ELF, whose maximum possible value e max is expressed as
e max = 1 m ̇ l c m ̇ l ,
(27)
where m ̇ l is the liquid mass flow rate, while m ̇ l c is the critical liquid film mass flow rate, below which entrainment does not occur. This latter can be calculated from the critical liquid Reynolds number R e l c, defined as follows:
R e l c = 4 m ̇ l c π μ l d = 7.3 log ω 3 + 44.2 log ω 2 263 log ω + 439 ; ω = μ l μ g ρ g ρ l .
(28)
The Weber number W e in Eq. (26) is defined as follows:
W e = ρ l ρ g J g J g c 2 d σ ; J g c = 40 σ d ρ l ρ g .
(29)
The underling experimental data bank includes upflow and downflow through circular tubes (diameters from 9.5 to 52.7 mm) of four fluid combinations (water–air, water–helium, R140a–air, and R113) covering operating pressures in the range of 0.1–0.5 MPa.

This prediction method has been quite influential, particularly the idea of a maximum value for the ELF, which has been incorporated into various subsequent correlations. The related idea of the existence of a critical liquid film mass flow rate below which entrainment does not occur provides support to the observation that the liquid film flow laminarization suppresses the entrainment process, as discussed in Sec. II I.

The prediction method proposed by Sawant et al.83 reads as follows:
e e max = tanh 2.31 10 4 W e 1.25 R e l 0.35 ,
(30)
where the maximum possible value of the ELF is computed as
e max = 1 250 l n R e l 1265 R e l .
(31)
The liquid Reynolds number R e l is calculated as indicated in Eq. (20), while the Weber number W e is defined as follows (formally analogous to the expression in Eq. (20), apart from the value of the exponent on the density ratio term):
W e = ρ g J g 2 d σ ρ l ρ g ρ g 0.25 .
(32)
The underling experimental data bank includes upflow through circular tubes (9.4 mm diameter) of water–air at operating pressures in the range of 0.1–0.6 MPa.
An improved version of this prediction method was proposed in a subsequent publication by the same authors:36 the ELF is still predicted as indicated in Eq. (30); however, the maximum possible value of the ELF is now computed as follows:
e max = 1 13 N μ 0.5 + 0.3 R e l 13 N μ 0.5 0.95 R e l ,
(33)
where the viscosity number N μ is defined as
N μ = μ l ρ l σ σ g ρ l ρ g 0.5 .
(34)
The underling experimental data bank includes upflow through circular tubes (diameters from 9.4 to 32 mm) of five fluid combinations (water, water–air, R113, R140a–air, and water–helium), covering operating pressures in the range of 0.1–7 MPa.

These prediction methods are modified versions of the previous method by Ishii and Mishima,30 which also incorporate the idea of the upper-bound for the ELF proposed by Pan and Hanratty.82 

In the method developed by Cioncolini and Thome,24 the ELF is predicted as follows:
e = 1 + 13.18 W e c 0.655 10.77 ,
(35)
where the core flow Weber number W e c is defined as
W e c = ρ c V c 2 d c σ .
(36)
The core flow density ρ c, average velocity V c , and equivalent diameter d c are computed as follows:
ρ c = x + e 1 x x ρ g + e 1 x ρ l ; V c = x G ϵ ρ g ; d c = d ϵ x ρ l + e 1 x ρ g x ρ l .
(37)
The void fraction required as input to Eq. (37) is predicted with the method by Woldesemayat and Ghajar.39 

The underlying experimental data bank contains 1504 data points generated with vertical upflow through circular tubes (diameters from 5 to 57.1 mm) of eight fluid combinations (water, water–air, R12, R113, R140a–air, ethanol–air, water–helium, and silicon–air), covering operating pressures in the range of 0.1–10 MPa.

An improved version of this prediction method, which was proposed in a subsequent publication by the same authors,23 reads as follows:
e = 1 + 279.6 W e c 0.8395 2.209 ,
(38)
where the core flow Weber number W e c is now defined as
W e c = ρ c J g 2 d σ .
(39)
The core flow density ρ c is computed as indicated in Eq. (37).

The underlying experimental data bank contains 2293 data points generated with vertical upflow and downflow through circular tubes (diameters from 5 to 57.1 mm) of eight fluid combinations (water, water–air, R12, R113, R140a–air, ethanol–air, water–helium, and silicon–air), covering operating pressures in the range of 0.1–20 MPa.

As can be noted, both methods in Eqs. (35) and (38) are of the form e = f e and, hence, are implicit in e. As discussed by the authors, Eq. (38) can be solved numerically with a two-step predictor-corrector scheme.

The prediction method proposed by Al-Sarkhi et al.84 is a modified version of the previous method of Sawant et al.:83 the ELF is predicted as indicated in Eq. (30), while the maximum possible value of the ELF is now computed as follows:
e max = 1 exp R e l 1400 0.6 ,
(40)
where the liquid Reynolds number R e l is calculated as indicated in Eq. (20).

The experimental data bank that underlies the development of Eq. (40) is not described in detail by the authors.

The prediction method proposed by Bhagwat and Ghajar,85 which is a modified version of the previous correlation of Cioncolini and Thome,23 reads as follows:
e = 1 + C W e c 0.8395 2.209 ,
(41)
where the core flow Weber number W e c is computed as indicated in Eq. (39), while the parameter C is calculated according to the operating pressure P sys as follows:
C = 280 + 120 cos θ 2 0.1 P sys 10 MPa , C = 4637.8 P sys 1.6 + 120 cos θ 2 P sys 10 MPa ,
(42)
where θ is the flow inclination with respect to the horizontal (i.e., θ = 0 for horizontal flow, and positive/negative angles indicate upflow/downflow) and P sys is to be entered in MPa. This prediction method was conceived as an improved version of the method of Cioncolini and Thome,23 with better accuracy at high pressure and with non-vertical flows.
In the prediction method proposed by Berna et al.,4 the ELF is predicted as follows:
e 1 e = 5.51 10 7 W e 2.68 R e g 2.62 R e l 0.34 ρ g ρ l 0.37 μ g μ l 3.71 C w 4.24 ,
(43)
where the Weber number W e and the gas Reynolds number R e g are
W e = ρ g J g 2 d σ ; R e g = ρ g J g d μ g .
(44)
The liquid Reynolds number R e l is computed as indicated in Eq. (20), while the parameter C w is calculated as
C w = 0.0280 N μ 0.8 N μ 0.0667 , C w = 0.25 N μ > 0.0667 ,
(45)
where N μ is the viscosity number defined in Eq. (34).

The underlying experimental data bank covers vertical upflow and horizontal flow through circular tubes (diameters from 9.53 to 152 mm) of three fluid combinations (water–air, water–air plus butanol, and water–air plus glycerine) and operating pressures in the range of 0.1–0.2 MPa.

Aliyu et al.18 developed two different correlations: one applicable for low superficial gas velocities ( J g≤ 40 m/s) and another applicable for high superficial gas velocities ( J g> 40 m/s),
e 1 e = 1.25 10 3 W e 0.15 R e g 0.2 R e l 0.23 J g 40 m / s ,
(46)
e 1 e = 2 10 3 W e 0.5 R e l 0.29 J g > 40 m / s ,
(47)
where the Weber number W e is computed as indicated in Eq. (32), the gas Reynolds number R e g is computed as indicated in Eq. (44), and the liquid Reynolds number R e l is computed as indicated in Eq. (20).

The underlying experimental data bank contains 1391 data points generated with vertical upflow/downflow and horizontal flow through circular tubes (diameters from 5 to 127 mm) of eight fluid combinations (water, water–air, R12, R113, glycerine–air, ethanol–air, water–helium, and silicon–air), covering operating pressures in the range of 0.1–9 MPa.

Even though the prediction methods presented earlier differ widely in terms of their formulation, the most frequently employed dimensionless groups include various forms of Weber, Reynolds, and capillary numbers. These dimensionless numbers encapsulate the effects of inertia, viscous dissipation, and surface tension on the flow, so that their inclusion into ELF prediction methods appears justified. In fact, as noted previously, the key factor underpinning the entrainment process is the shear of the gas core onto the liquid film, which, in turn, depends on the inertia of these latter. Additional influencing factors include the liquid viscosity, which opposes the liquid film acceleration, and the surface tension, which resists the liquid film atomization.

For use herein, the fifteen ELF prediction methods discussed earlier have been implemented as stand-alone Octave function scripts (employing the built-in function “fzero” to handle those methods of implicit formulation that require numerical solution); the Octave scripts (fully commented) are freely available through the open access repository Figshare (dx.doi.org/10.6084/m9.figshare.24037548).

The ELF data bank gathered herein, being larger and more diversified than those considered previously, offers interesting opportunities to assess the performance of the prediction methods discussed in Sec. III. The assessment provided herein, in particular, is restricted to vertical flow data. Several other assessments, with different data subsets extracted from the ELF data bank, would clearly be informative—way too many to fit within the present space limitations. As already explained, the ELF data bank is freely accessible, and so are the Octave scripts that implement the ELF prediction methods: readers can, therefore, carry out any further assessment as per their interest.

The performance of the prediction methods from Sec. III is assessed here by means of two accuracy metrics and two bias metrics, which have been selected among the various measures of performance proposed to date. In particular, the two accuracy metrics are the mean absolute percentage error (MAPE) [Eq. (48)] and the median symmetric accuracy (MSA) [Eq. (49)] while the two bias metrics are the mean absolute error (MAE) [Eq. (50)] and the symmetric signed percentage error (SSPE) [Eq. (51)]
MAPE = 100 n k = 1 n Q k 1 ,
(48)
MSA = 100 exp median l n Q 1 ,
(49)
MAE = 100 n k = 1 n Q k 1 ,
(50)
SSPE = 100 s gn median l n Q e xp median l n Q 1 ;
(51)
where Q = ELF pred / ELF meas is the accuracy ratio, that is, the array formed with the ratios of the predicted ELF values to the measured ones, with Q k representing one single entry of the array and n indicating the size of the array, and sgn denotes the signum function.

Being easy to interpret, the MAPE and the MAE are two of the most commonly used metrics to assess modeling and forecasting quality. However, they penalize overprediction more heavily than underprediction, and they are not robust to outliers.86,87 The MSA and the SSPE are alternative accuracy and bias metrics that have being conceived86 to retain the simple interpretability of the MAPE and the MAE while, at the same time, mitigating their drawbacks. Specifically, the MSA and the SSPE equally penalize overprediction and underprediction, are robust to outliers, and can, therefore, be regarded as improved versions of the MAPE and MAE, respectively. In addition to the accuracy and bias metrics summarized earlier, following common practice in the two-phase flow literature, the accuracy of the prediction methods is also assessed by computing the percentages of predictions that fall within the ±15%, ±30%, and ±50% error bands. As noted previously, not all ELF prediction methods are bounded between 0 and 1: depending on the data subset being considered, therefore, nonphysical predictions (such as a negative ELF value or a value above 1) might occur. The fraction of the dataset against which each prediction method generates a physically meaningful prediction (that is, a prediction bounded between 0 and 1) is also included in the assessment.

The assessment is restricted to ELF data (1692 data points in total from Table I) generated with vertical flows, both upflow and downflow, through circular tubes under adiabatic flow conditions, without any contamination from intermittent flow (void fraction above 0.7, as discussed in Sec. II E), and where inlet effects and gravity effects on the flow are inconsequential ( L / d above 100 when relevant, and J g W above 2.5, as discussed in Secs. II D and II F). Very low ELF measurements (below 0.05) are also excluded. This dataset covers conditions that fall within the operating range of all the ELF prediction methods considered, and it is sufficiently large and diversified to provide an informative performance assessment. Performance metrics are summarized in Table IV, while parity plots are provided in Fig. 14. Among the methods capable of generating a physically meaningful prediction against the entire dataset, those with the best metrics are the methods by Oliemans et al.,22 Cioncolini and Thome,23 Al-Sarkhi et al.,84 and Bhagwat and Ghajar.85 The method by Sawant et al.83 has also good performance metrics, despite the fact that it generated unphysical predictions a couple of times. The performance of these prediction methods is further inspected in Fig. 15, where the respective accuracy ratios are displayed in the form of boxplots. Overall, taking into account the performance metrics together with the accuracy ratio distributions and parity plots, the best performing ELF prediction methods against the vertical flow data considered here appear to be those by Oliemans et al.,22 Cioncolini and Thome,23 and Bhagwat and Ghajar,85 with an accuracy (measured by the MSA) on the order of 25% and a bias (measured by the SSPB) within a few percent.

TABLE IV.

Performance assessment of ELF prediction methods against vertical flow data (1692 data points from Table I).

MAPE (%) MSA (%) MPE (%) SSPB (%) Within 15% (%) Within 30% (%) Within 50% (%) a
Paleev and Filippovich79   127  83.8  101  38.6  15.0  31.6  43.8  0.938 
Wallis58   80.2  59.2  59.6  23.7  18.9  34.9  56.0  1.0 
Oliemans et al.22   42.7  24.2  12.8  −2.8  39.0  57.1  73.7  1.0 
Ishii and Mishima30   41.7  30.8  5.9  2.3  30.3  53.7  72.3  0.992 
Nakazatomi and Sokoguchi80   153  91.7  126  51.3  8.7  20.7  40.2  1.0 
Utsuno and Kaminaga81   61.7  176  −51.1  −175  15.2  18.8  30.1  0.430 
Pan and Hanratty82   34.8  31.0  −28.3  −29.2  30.4  58.9  76.6  0.987 
Sawant et al.83   35.4  26.0  −5.1  −5.4  37.8  58.2  76.3  0.999 
Sawant et al.36   32.8  27.7  −22.3  −21.4  35.2  57.6  73.9  0.996 
Cioncolini and Thome24   37.2  29.8  8.3  −2.8  32.6  60.0  78.5  1.0 
Cioncolini and Thome23   34.6  26.7  3.6  −2.5  32.8  60.1  81.3  1.0 
Al-Sarkhi et al.84   34.1  26.3  −3.4  −0.2  35.3  57.1  76.4  1.0 
Bhagwat and Ghajar85   31.5  24.0  −1.0  −7.5  39.6  65.1  82.3  1.0 
Berna et al.4   86.8  116  14.6  −30.6  8.7  18.6  34.5  1.0 
Aliyu et al.18   66.6  57.2  33.6  8.0  21.9  35.9  54.8  1.0 
MAPE (%) MSA (%) MPE (%) SSPB (%) Within 15% (%) Within 30% (%) Within 50% (%) a
Paleev and Filippovich79   127  83.8  101  38.6  15.0  31.6  43.8  0.938 
Wallis58   80.2  59.2  59.6  23.7  18.9  34.9  56.0  1.0 
Oliemans et al.22   42.7  24.2  12.8  −2.8  39.0  57.1  73.7  1.0 
Ishii and Mishima30   41.7  30.8  5.9  2.3  30.3  53.7  72.3  0.992 
Nakazatomi and Sokoguchi80   153  91.7  126  51.3  8.7  20.7  40.2  1.0 
Utsuno and Kaminaga81   61.7  176  −51.1  −175  15.2  18.8  30.1  0.430 
Pan and Hanratty82   34.8  31.0  −28.3  −29.2  30.4  58.9  76.6  0.987 
Sawant et al.83   35.4  26.0  −5.1  −5.4  37.8  58.2  76.3  0.999 
Sawant et al.36   32.8  27.7  −22.3  −21.4  35.2  57.6  73.9  0.996 
Cioncolini and Thome24   37.2  29.8  8.3  −2.8  32.6  60.0  78.5  1.0 
Cioncolini and Thome23   34.6  26.7  3.6  −2.5  32.8  60.1  81.3  1.0 
Al-Sarkhi et al.84   34.1  26.3  −3.4  −0.2  35.3  57.1  76.4  1.0 
Bhagwat and Ghajar85   31.5  24.0  −1.0  −7.5  39.6  65.1  82.3  1.0 
Berna et al.4   86.8  116  14.6  −30.6  8.7  18.6  34.5  1.0 
Aliyu et al.18   66.6  57.2  33.6  8.0  21.9  35.9  54.8  1.0 
a

Fraction of the dataset against which the method provides a meaningful prediction (i.e., bounded between 0 and 1).

FIG. 14.

Parity plots with comparisons between ELF predictions and ELF measurements (1692 data points from Table I for vertical flow).

FIG. 14.

Parity plots with comparisons between ELF predictions and ELF measurements (1692 data points from Table I for vertical flow).

Close modal
FIG. 15.

Accuracy ratio of ELF prediction methods tested against vertical flow data (1692 data points from Table I) displayed in the form of boxplot (following common practice, for each box, the red line corresponds to the median, the box sides to the lower and upper quartiles, and the whiskers length is 1.5 times the interquartile range).

FIG. 15.

Accuracy ratio of ELF prediction methods tested against vertical flow data (1692 data points from Table I) displayed in the form of boxplot (following common practice, for each box, the red line corresponds to the median, the box sides to the lower and upper quartiles, and the whiskers length is 1.5 times the interquartile range).

Close modal

This review study provided critical assessments of experimental data and prediction methods for the entrained liquid fraction (ELF), a key flow parameter in the analysis and modeling of annular gas–liquid two-phase flows. The assessment of the experimental data relied upon a large ELF data bank gathered from the literature (4175 data points from 53 separate studies), which was critically analyzed paying special attention to important aspects not adequately considered in previous investigations. The assessment of the ELF prediction methods covered fifteen widely quoted correlations, which were analyzed, and whose prediction performance was assessed against a subset of vertical flow data (1692 data points) extracted from the ELF data bank. The main conclusions of this study and an outlook on research avenues for future investigations can be summarized as follows.

Despite the numerous experimental investigations carried out to date, most of the available ELF data have been generated with adiabatic flows through vertical circular tubes, employing water–air as testing fluids and exploring comparatively low operating pressures. Therefore, there appears to be ample scope to carry out further experimental studies. Areas only marginally explored or completely unexplored, which should be considered in future experimental investigations, include the following:

  • Non-circular channels, such as square/rectangular channels and particularly rod bundle geometries (with/without spacer grids and mixing vanes to control the entrainment process), which are relevant for water-cooled nuclear reactor systems;

  • Evaporating and condensing flow conditions: non-adiabatic flows have been only sporadically investigated, and any effect of the heat transfer on the entrainment process is still largely unclear;

  • Tests at comparatively high operating pressure (indicatively, above about 1 MPa), and/or tests with fluids other than water–air, to better cover operating conditions and fluids (such as refrigerants for example) of relevance for final applications. Notably, the variation in thermo-physical properties associated with testing different fluids and/or operating conditions would allow extending the range of values presently covered for the annular flow dimensionless parameters. In turn, this holds the promise of better elucidating the underlying physics of these flows and would consequently create the conditions for designing better ELF prediction methods;

  • Experiments on horizontal and inclined flows, in order to better resolve the effect of the gravity force on the liquid entrainment process;

  • Combined experiments where the entrainment process and the turbulent structure in the liquid film are simultaneously resolved, in order to better clarify the possible link between atomization and turbulence.

When properly implemented, all available ELF measuring techniques appear capable of providing reliable measurements, provided that the liquid film suction technique is avoided when measuring low ELF values. Whenever a mixer is employed, an L / d on the order of 100–120 seems adequate to make inlet effects on the ELF measurements minor or absent. Measuring the local pressure, and possibly also the pressure gradient, nearby the ELF measurement location is advisable, particularly in the case of non-adiabatic flows where flow acceleration (in the case of evaporating flows) or deceleration (in the case of condensing flows) effects can be particularly pronounced.

Comparing the accuracy of the best ELF prediction methods, which is on the order of 25% as per the assessment documented herein, with the typical ELF measuring errors that are on the order of 10%–15%, there appears to be scope to develop more accurate ELF prediction methods for final applications. Specifically, future investigations should focus on the following aspects:

  • Develop a better fundamental understanding of the liquid entrainment process and of the physics of annular two-phase flows, which, as already explained, includes various influencing factors, a strong mass and momentum coupling between the liquid film and the gas core flows, and associated non-linearities. Further experimental studies along the lines outlined earlier would clearly be instrumental in this respect;

  • Taking advantage from a deeper fundamental understanding of the physics of annular flows, develop more accurate ELF prediction methods, possibly also incorporating gravity effects and heat transfer effects, which are presently not included in the available prediction methods. Further experimental studies along the lines outlined earlier would clearly be instrumental in this respect.

The curated ELF data bank is freely available through the open access repository Figshare (dx.doi.org/10.6084/m9.figshare.24002586) and so are the Octave stand-alone scripts that implement the ELF prediction methods (dx.doi.org/10.6084/m9.figshare.24037548).

The author has no conflict to disclose.

Andrea Cioncolini: Conceptualization (lead); Investigation (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are openly available in Figshare at dx.doi.org/10.6084/m9.figshare.24002586, Ref. 109, and dx.doi.org/10.6084/m9.figshare.24037548, Ref. 110.

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