# Two-Phase Annular Flow in Vertical Pipes: A Critical Review of Current Research Techniques and Progress

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## Abstract

**:**

## 1. Introduction

## 2. Investigation Methodologies

#### 2.1. Visualization and Photography

#### 2.2. Laser-Induced Fluorescence (LIF)

#### 2.3. Particle Image Velocimetry (PIV)

#### 2.4. Laser Focus Displacement Meter (LFD)

#### 2.5. Ultrasonic Flow Meter and Near-Infrared Sensor

#### 2.6. Conductance Sensor

#### 2.7. Capacitance Sensor

#### 2.8. Wire-Mesh Sensor (WMS)

#### 2.9. Radiative Imaging

#### 2.10. Film Extraction

#### 2.11. Shadow Photography and Laser Diffraction

#### 2.12. Laser Doppler Anemometry (LDA)

#### 2.13. Other Mechanical Methods

#### 2.14. Numerical Simulation

#### 2.15. Challenges of Current Experimental Techniques

## 3. The Wavy Liquid Film

#### 3.1. Fundamental Understanding of the Liquid Film

_{L}= 211) and L/D=25 (Re

_{L}= 603). Prior to becoming fully developed, the film decelerates first to a local maximum thickness and then accelerates again to become thinner.

_{L}, while with increasing Re

_{L}, the film thickness was increasingly underpredicted by the theory, but with good agreement with Mudawwar and El-Masri’s semi-empirical turbulence model [210].

#### 3.2. Disturbance Wave Characteristics

#### 3.3. Correlations of the Film Thickness

#### 3.4. The Void Fraction of Annular Two-Phase Flow

## 4. The Entrained Droplets in the Central Gas Core

#### 4.1. Droplet Behaviour

#### 4.2. Correlation of Droplet Entrainment

## 5. Conclusions and Recommendations

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Nomenclature List | |||

Symbols | Greek Characters | ||

a | Void fraction | δ | Film thickness |

C | Friction factor | $\overline{\delta}$ | Time-averaged film thickness |

d | Diameter of the drop | ε | Entrainment rate |

D | Diameter of the pipe | λ | Wavelength |

E | Entrainment fraction | μ | Dynamic viscosity |

f | Friction factor | ν | Kinematic viscosity |

Fr | Froude number | ρ | Density |

g | acceleration of gravity | σ | Surface tension coefficient |

h | Disturbance wave height | τ | Shear stress |

j | Superficial velocity | Subscripts | |

k | Wave number | * | Friction |

Ka | Kapitza number | 32 | Sauter diameter |

L | Length | base | The base of the disturbance wave |

$\dot{m}$ | Mass flow rate | c | Gas core |

N_{u} | Viscosity number | DW | Disturbance wave |

N_{uf} | Non-dimensional viscosity number | e | Entrained |

Δp | Pressure difference | m | Modified |

Re | Reynolds number | max | Maximum condition |

St | Strouhal number | G | Gas |

u | Velocity | Gc | Critical gas state |

V | Volume | L | Liquid |

We | Weber number | La | Laplace length |

x | Vaper quality | lf | Liquid film |

X_{tt} | Lockhart-Martinelli parameter | lfc | Critical film flow |

${\left(\frac{dp}{dz}\right)}_{fric}$ | Pressure gradient due to friction loss | L, ref | Liquid at reference condition (at 20 °C) |

i | Interfacial | ||

v | Volume mean | ||

w | Wall |

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**Figure 1.**The visualized two-phase flow regime map, reprinted with permission from [14] 2017, Elsevier.

**Figure 2.**(

**a**), A typical experimental setup for PIV/LIF measurement, (1)—(4) are the annular film in the front view, the targeting annular film full of tracer particles/dye that is illuminated by a laser sheet, the transparent wall, and water in the optical correction box, (

**b**), a visualized falling film, and (

**c**), a disturbance wave. (

**b**,

**c**) are adapted with permission from [48], 2014, Elsevier.

**Figure 3.**(

**a**). A typical raw PIV image; (

**b**), a processed raw image; (

**c**), a velocity vector field and (

**d**), a PTV velocity vector field [64].

**Figure 6.**(

**a**), An actual wire mesh sensor, adapted with permission from [144]. 2015, Elsevier, (

**b**), wire distribution of the 16 × 16 sensors, adapted with permission from [145]. 2015, Elsevier, (

**c**), a typical measurement frame showing the liquid phase (blue) and gas phase (white), adapted with permission from [145]. 2015, Elsevier.

**Figure 8.**Optical setup of the droplet size measurement, adapted with permission from [165], 2002, Elsevier.

**Figure 9.**Simulation of the droplet deposition process in an upwards annular two-phase flow (side view) [200].

**Figure 10.**A summary of the major research techniques used in the investigation of the annular two-phase flow in the recent two decades.

**Figure 11.**Structure of a typical two-phase annular flow in vertical pipe showing the disturbance waves, ripple waves, liquid film, and entrained droplets.

**Figure 12.**Film thickness distribution conducted from BBLIF [56].

**Figure 13.**The unsymmetrical circumferential film thickness determined using PLIF, adapted with permission from [54], 2019, John Wiley and Sons.

**Figure 14.**Velocity and thickness profiles of a falling film at a liquid Reynolds number of 5275, adapted with permission from [63], 1998, Elsevier.

**Figure 15.**PLIF images showing the droplet entrainment process: (

**a**) disturbance wave, (

**b**) wave undercut (

**c**) ligament break-up, (

**d**,

**e**) different stages of a bubble burst, (

**f**) liquid impingement, adapted with permission from [48], 2014, Elsevier and (

**g**) simulation of the droplet formation of undercutting zone, adapted with permission from [231], 2016, Elsevier.

Reference | Correlations of the Liquid Film Thickness |
---|---|

Ishii and Grolmes [217] (1975) | $\delta =0.347R{e}_{L}^{2/3}\sqrt{\frac{{\rho}_{L}}{{\tau}_{i}}}\frac{{\mu}_{L}}{{\rho}_{L}}$ |

Henstock and Hanratty [218] (1976) | $\delta =\frac{6.59F}{{\left(1+1400F\right)}^{0.5}}D$ $F=\frac{1}{\sqrt{2}R{e}_{G}^{0.4}}\frac{R{e}_{L}^{0.5}}{R{e}_{G}^{0.9}}\frac{{\mu}_{L}{\rho}_{G}^{0.5}}{{\mu}_{G}{\rho}_{L}^{0.5}}$ |

Tatterson et al. [219] (1977) | $\delta =\frac{6.59F}{{\left(1+1400F\right)}^{0.5}}D$ $F=\frac{\gamma \left(R{e}_{L}\right)}{R{e}_{G}^{0.9}}\frac{{\mu}_{L}{\rho}_{G}^{0.5}}{{\mu}_{G}{\rho}_{L}^{0.5}}$ $\gamma ={\left[{\left(0.707R{e}_{L}^{0.5}\right)}^{2.5}+{\left(0.0379R{e}_{L}^{0.9}\right)}^{2.5}\right]}^{0.4}$ |

Hori et al. [220,221] (1978) | $\delta =0.905R{e}_{G}^{-1.45}R{e}_{L}^{0.9}F{r}_{G}^{0.93}F{r}_{L}^{-0.68}{\left(\frac{{\mu}_{L}}{{\mu}_{L,ref}}\right)}^{1.06}D$ |

Ambrosini et al. [222] (1991) | $\frac{{\rho}_{L}\delta {u}^{*}}{{\mu}_{L}}=\{\begin{array}{c}0.34R{e}_{L}^{0.6}R{e}_{L}\le 1000\\ 0.0512R{e}_{L}^{0,875}R{e}_{L}1000\end{array},{u}^{*}=\sqrt{\frac{{\tau}_{i}}{{\rho}_{L}}}$ |

Fukano and Furukawa [94] (1998) | $\delta =0.0594exp\left(-0.34F{r}_{G}^{0.25}R{e}_{L}^{0.19}{\chi}^{0.6}\right)D$ $\chi =\frac{{j}_{G}{\rho}_{G}}{{j}_{G}{\rho}_{G}+{j}_{L}{\rho}_{L}}$ |

Okawa et al. [223] (2002) | $\delta \approx \frac{1}{4}\sqrt{\frac{{f}_{w}{\rho}_{L}}{{f}_{i}{\rho}_{G}}}\frac{\left(1-E\right){j}_{L}}{{j}_{G}}D$ ${f}_{i}=0.005\left(1+300\frac{\delta}{D}\right)$ ${f}_{w}=max\left(\frac{16}{R{e}_{L}},0.005\right)$ |

MacGillivray [224] (2004) | $\delta =39\frac{{\mu}_{L}}{{\rho}_{L}{j}_{L}}R{e}_{L}^{0.2}\frac{1-a}{a}{\left(\frac{{\rho}_{G}}{{\rho}_{L}}\right)}^{0.5}$ |

Hazuku et al. [68] (2008) | ${\delta}_{base}{\left(\mathrm{g}/{v}_{L}^{2}\right)}^{1/3}=0.977R{e}_{L}^{0.143}{\tau}_{i}^{*-0.117}$ ${\tau}_{i}^{*}=\frac{{\tau}_{i}}{{\rho}_{L}\mathrm{g}}{\left(\frac{g}{{v}_{L}^{2}}\right)}^{1/3}$ ${\tau}_{i}=\frac{D-2\overline{\delta}}{4}{\left(\frac{dp}{dz}\right)}_{fric}$ ${\tau}_{i}={f}_{i}{\rho}_{G}{j}_{G}^{2}/2$ |

Berna et al. [9] (2014) | $\delta =7.156R{e}_{G}^{-1.07}R{e}_{L}^{0.48}{\left(\frac{F{r}_{G}}{F{r}_{L}}\right)}^{0.24}D$ |

Pan et al. [39] (2015) | $\delta =2.03R{e}_{L}^{0.15}R{e}_{G}^{-0.6}D$ |

Almabrok et al. [148] (2016) | $\delta =1.4459R{e}_{L}^{0.3051}{\left(\frac{g}{{v}_{L}^{2}}\right)}^{-\frac{1}{3}}$ |

Rahman et al. [225] (2017) | $\delta =1.93\times {10}^{-3}R{e}_{G}^{-0.246}W{e}_{G}^{-0.161}F{r}_{L}^{0.15}{\left(\frac{{\dot{m}}_{L}}{{\dot{m}}_{G}}\right)}^{0.546}$ |

Ju et al. [226,227] (2019) | $\delta =0.071tanh\left(14.22W{e}_{L}^{0.24}W{e}_{G}^{-0.47}{N}_{{\mu}_{f}}^{0.21}\right)D$ ${\delta}_{base}=0.04tanh\left(4.31W{e}_{L}^{0.22}W{e}_{G}^{-0.44}\right)D$ ${N}_{\mu f}=\frac{{\mu}_{L}}{\sqrt{\left({\rho}_{L}\sigma \sqrt{\frac{\sigma}{g\mathrm{\u2206}\rho}}\right)}}$ $W{e}_{L}=\frac{{\rho}_{L}{j}_{L}{}^{2}D}{\sigma},W{e}_{G}=\frac{{\rho}_{G}{j}_{L}{}^{2}D}{\sigma}{\left(\frac{\mathrm{\u2206}\rho}{{\rho}_{G}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$ |

Rivera et al. [125] (2021) | $\delta =2.35R{e}_{G}^{-1.415}R{e}_{L}^{0.414}K{a}^{0.781}D$ |

Rivera et al. [127] (2022) | $\delta =0.19R{e}_{L}^{0.54}\left(1-1.29\times {10}^{-5}R{e}_{G}^{0.93}\right){\left(\frac{{v}^{2}}{g}\right)}^{1/3}$ |

Pan et al. [39] | $\frac{{\delta}_{DW}}{D}=1400{\left(\frac{{u}_{c}}{{u}_{L}}\right)}^{-\frac{1}{3}}{\left[\frac{({\rho}_{L}-{\rho}_{c})g{\delta}^{2}}{\sigma}\right]}^{\frac{5}{8}}$ |

Ju et al. [226] | ${\delta}_{DW}=0.24tanh\left(4.22W{e}_{L}^{0.16}W{e}_{G}^{-0.46}\right)D$ |

Y. Rivera, et al. [125] | $\frac{{\delta}_{DW}}{D}=0.554\times {10}^{-3}R{e}_{G}^{-0.57}R{e}_{L}^{0.061}K{a}^{1.12}$ |

Reference | Correlations for the Void Fraction |
---|---|

Tandon et al. [229] (1985) | $a=\left\{1-1.928R{e}_{L}^{-0.315}{\left[F\left({X}_{tt}\right)\right]}^{-1}+0.9293R{e}_{L}^{-0.63}{\left[F\left({X}_{tt}\right)\right]}^{-2}\right\}50R{e}_{L}1125$ $a=\left\{1-0.38R{e}_{L}^{-0.088}{\left[F\left({X}_{tt}\right)\right]}^{-1}+0.0361R{e}_{L}^{-0.176}{\left[F\left({X}_{tt}\right)\right]}^{-2}\right\}R{e}_{L}1125$ $F\left({X}_{tt}\right)=0.15\left[{X}_{tt}^{-1}+2.85{X}_{tt}^{-0.476}\right]$ ${X}_{tt}={\left(\frac{{\mu}_{L}}{{\mu}_{G}}\right)}^{0.1}{\left(\frac{1-x}{x}\right)}^{0.9}{\left(\frac{{\rho}_{G}}{{\rho}_{L}}\right)}^{0.5}$ |

Usui and Sato [93] (1989) | ${\left(1-a\right)}^{23/7}-2{C}_{w}F{r}_{L}^{2}\left[1\pm \frac{{C}_{i}}{{C}_{w}}\mathrm{\xb7}\frac{{\left(1-a\right)}^{16/7}}{{a}^{5/2}}\mathrm{\xb7}\frac{{\rho}_{G}}{{\rho}_{L}}{\left(\frac{{j}_{G}}{{j}_{L}}\right)}^{2}\right]=0$ Free falling film, $a=1-{\left(2{C}_{w}F{r}_{L}^{2}\right)}^{7/23}$ |

Cioncolini and Thome [230] (2012) | $a=\frac{h{x}^{n}}{1+\left(h-1\right){x}^{n}}$ $(0<x<1,{10}^{-3}\frac{{\rho}_{G}}{{\rho}_{L}}1,0.7\epsilon 1)$ $h=-2.129+3.129{\left(\frac{{\rho}_{G}}{{\rho}_{L}}\right)}^{-0.2186}$ $n=0.3487+0.6513{\left(\frac{{\rho}_{G}}{{\rho}_{L}}\right)}^{0.515}$ |

Kumar et al. [194] (2017) | ${\left(1-a\right)}^{23/7}-2{C}_{w}F{r}_{L}^{2}\left[1\pm \frac{{C}_{i}}{{C}_{w}}\mathrm{\xb7}\frac{{\left(1-a\right)}^{16/7}}{{a}^{5/2}}\mathrm{\xb7}\frac{{\rho}_{c}}{{\rho}_{L}}{\left(\frac{{j}_{G}}{{j}_{L}}\right)}^{2}\right]=0$ Free falling film, $a=1-{\left(2{C}_{w}F{r}_{L}^{2}\right)}^{1/3}$ |

Reference | Prediction of the Droplet Size |
---|---|

Kocamustafaogullari et al. [234] (1994) | $\frac{{d}_{32}}{D}=0.64{C}_{W}^{-4/15}W{e}_{m}^{-3/5}{\left(\frac{R{e}_{G}^{4}}{R{e}_{L}}\right)}^{4/15}{\left(\frac{{\rho}_{G}}{{\rho}_{L}}\right)}^{4/15}{\left(\frac{{\mu}_{G}}{{\mu}_{L}}\right)}^{4/15}$${C}_{w}=1/35.34{N}_{\mu}^{\frac{4}{5}}\left({N}_{\mu}\le 1/15\right)$ ${C}_{w}=0.25({N}_{\mu}1/15)$ |

Azzopardi [7] (1997) | $\frac{{d}_{32}}{D}=1.91R{e}_{G}^{0.1}W{e}_{G}^{-0.6}{\left(\frac{{\rho}_{G}}{{\rho}_{L}}\right)}^{0.6}+0.4\frac{E{j}_{L}}{{j}_{G}}$ |

Fore et al. [165] (2002) | $\frac{{d}_{v}}{D}=0.028W{e}_{G}^{-1}R{e}_{L}^{-1/6}R{e}_{G}^{2/3}{\left(\frac{{\rho}_{G}}{{\rho}_{L}}\right)}^{-1/3}{\left(\frac{{\mu}_{G}}{{\mu}_{L}}\right)}^{2/3}$ |

Azzopardi [237] (2006) | ${d}_{32}=\left[0.069{j}_{G}+0.0187{\left(\frac{{\rho}_{L}{j}_{L}}{{\rho}_{G}{j}_{G}}\right)}^{2}\right]\frac{\sigma}{{\rho}_{G}{j}_{G}}$ |

Berna et al. [8] (2015) | $\frac{{d}_{v}}{D}=0.11W{e}_{G}^{-0.68}R{e}_{G}^{0.33}R{e}_{L}^{0.11}{\left(\frac{{\rho}_{G}}{{\rho}_{L}}\right)}^{0.31}$ |

Wang et al. [180] (2020) | $\frac{{d}_{32}}{D}=0.022W{e}_{G}^{-0.545}W{e}_{L}^{0.214}R{e}_{L}^{-0.249}R{e}_{G}^{0.439}{\left(\frac{{\rho}_{G}}{{\rho}_{L}}\right)}^{0.117}$ |

Reference | Correlations of the Droplet Entrainment Rate |
---|---|

Bertodano et al. [170] (1997) | $\frac{\epsilon D}{{\mu}_{L}}=k{\left[W{e}_{G}{\left(\frac{\mathrm{\u2206}\rho}{{\rho}_{G}}\right)}^{1/2}\left(R{e}_{Lf}-R{e}_{Lfc}\right)\right]}^{0.925}{\left(\frac{{\mu}_{G}}{{\mu}_{L}}\right)}^{0.26}$ $R{e}_{Lf}=R{e}_{L}\left(1-E\right)$ |

Kataoka et al. [239] (2000) | $\frac{\epsilon D}{{\mu}_{L}}=6.6\times {10}^{-7}R{e}_{L}^{0.74}R{e}_{Lf}^{0.185}W{e}^{0.925}{\left(\frac{{\mu}_{G}}{{\mu}_{L}}\right)}^{0.26}$ |

Bertodano et al. [240] (2001) | $\frac{\epsilon D}{{\mu}_{L}}=\frac{3.8\times {10}^{-6}\sigma}{4}\left(R{e}_{Lf}-R{e}_{Lfc}\right)W{e}_{G}{\left(\frac{\mathrm{\u2206}\rho}{{\rho}_{G}}\right)}^{1/2}$ |

Okawa and Kataoka [241] (2005) | $\epsilon ={\rho}_{L}min\left(0.0038{\pi}_{e1},0.0012{\pi}_{e1}^{0.5}\right)$ ${\pi}_{e1}=\frac{{f}_{i}{\rho}_{G}\left({j}_{G}^{2}-{j}_{Gc}^{2}\right)\delta}{\sigma}$ |

Ryu and Park [242] (2011) | $\epsilon ={\rho}_{L}{V}_{e}\frac{{u}_{G}-{u}_{DW}}{4h{\lambda}^{2}\sqrt{a}}$ |

Liu and Bai [243] (2017) | $\epsilon =4.347\times {10}^{-6}{\rho}_{L}R{e}_{L}^{0.584}{\left(\frac{{\rho}_{L}}{{\rho}_{G}}\right)}^{0.0561}{\left(\frac{{\tau}_{i}\delta}{\sigma}\right)}^{0.391}{\left(\frac{D}{\sqrt{\sigma /g\mathrm{\u2206}\rho}}\right)}^{-0.291}$ ${\tau}_{i}={f}_{i}{\rho}_{G}{j}_{G}^{2}/2$ |

Wang et al. [244] (2020) | $\epsilon ={\epsilon}_{max}\times tanh\left(3.56\times {10}^{-6}R{e}_{la}^{0.47}W{e}^{1.15}\right)$ $\frac{{\epsilon}_{max}\sqrt{\sigma /g\mathrm{\u2206}\rho}}{{\mu}_{L}}=\{\begin{array}{c}0.00515R{e}_{la}^{0.69}\left(R{e}_{la}\le 800\right)\\ 0.521(R{e}_{la}800)\end{array}$ $R{e}_{la}=\frac{{\rho}_{L}{j}_{L}\sqrt{\sigma /g\mathrm{\u2206}\rho}}{{\mu}_{L}}$ |

Reference | Correlations for the Entrainment Fraction |
---|---|

Oliemans et al. [245] (1986) | $\frac{E}{1-E}={10}^{-0.25}{\rho}_{L}^{1.08}{\rho}_{G}^{0.18}{\mu}_{L}^{0.27}{\mu}_{G}^{0.28}{\sigma}^{1.80}{D}^{1.72}{j}_{L}^{0.7}{j}_{G}^{1.44}{g}^{0.46}$ |

Ishii and Mishima [246] (1989) | $E=tanh\left(7.25\times {10}^{-7}W{e}_{G}^{1.25}R{e}_{L}^{0.25}\right)$ |

Utsono and Kaminanga [247] (1998) | $E=tanh\left(0.16R{e}_{L}^{0.16}W{e}_{G}^{0.08}-1.2\right)$ |

Petalas and Aziz [248] (2000) | $\frac{E}{1-E}=0.735{\left(\frac{{\mu}_{L}^{2}{j}_{G}^{2}{\rho}_{G}}{{\sigma}^{2}{\rho}_{L}}\right)}^{0.074}{\left(\frac{{j}_{L}}{{j}_{G}}\right)}^{0.2}$ |

Barbosa et al. [181] (2002) | $E=0.95+342.55\sqrt{\frac{{\rho}_{L}{\dot{m}}_{L}}{{\rho}_{G}{\dot{m}}_{G}}}{D}^{2}$ |

Pan and Hanratty [249] (2002) | $\frac{E/{E}_{M}}{1-E/{E}_{M}}=6\times {10}^{-5}{\left({u}_{G}-{u}_{Gc}\right)}^{2}\sqrt{{\rho}_{G}{\rho}_{L}}D/\sigma $ ${E}_{M}=1-\frac{{\dot{m}}_{lfc}}{{\dot{m}}_{l}}$ |

Sawant et al. [250] (2008) | $E=\left(1-\frac{250\mathrm{ln}R{e}_{L}-1265}{R{e}_{L}}\right)tanh\left(2.31\times {10}^{-4}R{e}_{L}^{-0.35}W{e}_{G}^{1.25}\right)$ |

Sawant et al. [172] (2009) | $E=\left[1-\frac{13{N}_{\mu f}^{-0.5}+0.3{\left(R{e}_{L}-13{N}_{\mu f}^{-0.5}\right)}^{0.95}}{R{e}_{L}}\right]tanh\left(2.31\times {10}^{-4}R{e}_{L}^{-0.35}W{e}_{G}^{1.25}\right)$ ${N}_{\mu f}=\frac{{\mu}_{L}}{\sqrt{\left({\rho}_{L}\sigma \sqrt{\frac{\sigma}{g\mathrm{\u2206}\rho}}\right)}}$ |

Cioncolini and Thome [251] (2010) | $E={\left(1+13.18W{e}_{c}^{-0.655}\right)}^{-10.77}$ $W{e}_{c}=\frac{{\rho}_{c}{u}_{c}^{2}{D}_{c}}{\sigma}$ |

Cioncolini and Thome [252] (2012) | $E={\left(1+279.6W{e}_{c}^{-0.8395}\right)}^{-2.209}$ $W{e}_{c}=\frac{{\rho}_{c}{j}_{G}^{2}{D}_{c}}{\sigma}$ |

Berna et al. [8] (2015) | $\frac{E}{1-E}=5.51\times {10}^{-7}W{e}_{G}^{2.68}R{e}_{G}^{-2.62}R{e}_{L}^{0.34}{\left(\frac{{\rho}_{G}}{{\rho}_{L}}\right)}^{-0.37}{\left(\frac{{\mu}_{G}}{{\mu}_{L}}\right)}^{-3.71}{C}_{w}^{4.24}$ ${C}_{w}=\{\begin{array}{c}0.028{N}_{\mu}^{-0.8}\left({N}_{\mu}\le 1/15\right)\\ 0.25({N}_{\mu}1/15)\end{array}$ |

Aliyu et al. [152] (2017) | $E=\{\begin{array}{c}\frac{1\times {10}^{-2}W{e}^{0.33}R{e}_{L}^{0.27}}{1+1\times {10}^{-2}W{e}^{0.33}R{e}_{L}^{0.27}}\\ \frac{1.25\times {10}^{-3}W{e}^{0.15}R{e}_{G}^{0.2}R{e}_{L}^{0.23}}{1+1.25\times {10}^{-3}W{e}^{0.15}R{e}_{G}^{0.2}R{e}_{L}^{0.23}}\end{array}\begin{array}{c}({j}_{G}40m/s)\\ ({j}_{G}\ll 40m/s)\end{array}$ |

Zhang et al. [253] (2020) | $E\approx 0.075\frac{D{\rho}_{L}{\rho}_{G}^{2}{\delta}^{2}{u}_{*}^{8}}{{j}_{L}{\mu}_{G}^{\frac{2}{3}}{\sigma}^{\frac{7}{3}}{u}_{G}^{\frac{5}{3}}}{\left(\frac{\delta}{D}\right)}^{\frac{1}{3}}$ |

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**MDPI and ACS Style**

Xue, Y.; Stewart, C.; Kelly, D.; Campbell, D.; Gormley, M. Two-Phase Annular Flow in Vertical Pipes: A Critical Review of Current Research Techniques and Progress. *Water* **2022**, *14*, 3496.
https://doi.org/10.3390/w14213496

**AMA Style**

Xue Y, Stewart C, Kelly D, Campbell D, Gormley M. Two-Phase Annular Flow in Vertical Pipes: A Critical Review of Current Research Techniques and Progress. *Water*. 2022; 14(21):3496.
https://doi.org/10.3390/w14213496

**Chicago/Turabian Style**

Xue, Yunpeng, Colin Stewart, David Kelly, David Campbell, and Michael Gormley. 2022. "Two-Phase Annular Flow in Vertical Pipes: A Critical Review of Current Research Techniques and Progress" *Water* 14, no. 21: 3496.
https://doi.org/10.3390/w14213496