# Modeling of droplet entrainment in horizontal pipes at low pressure for system thermal-hydraulics codes

Philippe Fillion, Sofia Carnevali, Adrien Rodriguez

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Philippe Fillion, Sofia Carnevali, Adrien Rodriguez. Modeling of droplet entrainment in horizontal pipes at low pressure for system thermal-hydraulics codes. Nuclear Engineering and Design, 2025, 442, pp.114265. 10.1016/j.nucengdes.2025.114265. cea-05293633

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# Modeling of droplet entrainment in horizontal pipes at low pressure for system thermal-hydraulics codes

Philippe Fillion<sup>10</sup>*, Sofia Carnevali<sup>10</sup>, Adrien Rodriguez<sup>10</sup>

Université Paris-Saclay, CEA, Service de Thermo-hydraulique et de Mécanique des Fluides, 91191, Gif-sur-Yvette, France

## ARTICLE INFO

**Keywords:**

Horizontal flow

Dispersed flow

Droplet entrainment

System thermal-hydraulics code

## ABSTRACT

Accurately predicting complex thermal hydraulic transients is a primary goal of current systems codes. In the hot legs of the Pressurized Water Reactors, the flow regime may be dispersed or stratified with entrained droplets. Models have to account droplet entrainment due to the gas shearing over the liquid film and the deposition to these droplets due to gravity and turbulent diffusion mechanisms. To overcome models of the droplet entrainment implemented in the codes, adapted from correlations established in a vertical geometry, dedicated correlations for the entrainment rate and the deposition rate are being developed to better represent the droplet behavior within the horizontal pipes. In this study, these models are reviewed and compared against experimental data obtained in horizontal pipes with various diameters with various inlet gas and liquid superficial velocities in air-water conditions at low pressure. The analysis of the results allows to propose a new model derived from approach, both for the two-fluid model and the three-field model of system codes, accounting for the non equilibrium effects in terms of entrainment and deposition of the droplets appearing at the entrance of the pipe.

## 1. Introduction

System thermal-hydraulics codes have to deal with complex two-phase flows in Light Water Reactors, involving droplet entrainment in parts of the primary circuit due to accidental situations. This is the case of Loss of Heat Removal System during mid-loop operation or of transients involving a break, that is the Loss of Coolant Accidents (LOCA), where the flow regime in the hot legs may be stratified, and tends to be annular if the gas velocity increases, with entrained droplets by the vapor in both situations. The droplet entrainment in the hot leg has consequences on the core since it influences the reflooding rate and thus the clad temperatures. Modeling droplet entrainment is therefore a challenge to describe the reflooding phase of a LBLOCA (Large-Break LOCA). In order to best-estimate the quantity and the characteristics of the entrained droplets in the hot legs, or more generally in the horizontal parts of the hydraulic circuits, system thermal-hydraulics codes developed specific or adapted correlations usually initially dedicated for vertical flows, to the conditions encountered in these geometries.

First generations of the system codes, and still continuing now, are based on the two-fluid model, represented for example by the 6-equation model in ATHLET (Wielenberg and Bals, 2021), CATHARE (Emonot et al., 2011), RELAP5 (RELAP5 Development Team, 1995) and TRACE (USNRC, 2023) codes or by the 5-equation model in ATHLET considering a mixture momentum equation. The corresponding model

usually makes the assumption that continuous part and dispersed part of a phase are described with a single velocity and a single temperature. The influence of the droplets on the flow is representing by the entrainment fraction *E* defined as the ratio of the droplet flow rate over the total liquid flow rate. Assumption of fully developed flow with an equilibrium between the entrainment and the deposition processes is usually made. With an entrainment fraction correlation, two-phase models may encounter difficulties in such conditions at predicting the droplet behavior, namely at the entrance of the hot legs where the flow is not fully developed. Entrainment correlations may be adapted for accounting the entrance effect of the pipe, like in RELAP5.

System thermal-hydraulics codes adopt a one-dimensional (1D) nodalization of the hot legs. The challenge for the system codes, with such a 1D model of the pipe, is also to take into account the gravitational settling, leading to a strong stratification of the droplets at the bottom of the pipe. In the separate effect tests, the stratification of the droplets becomes significant when the diameter of the pipe is enlarged and should therefore be taken into account for the scalability of the corresponding model to horizontal legs of the integral test facilities having a diameter of around 10–20 cm and to hot legs of a Pressurized Water Reactor (PWR) with a diameter above 70 cm.

An advanced three-field model that share the liquid flow in a droplet and a continuous part, may solve some of these issues. The

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: philippe.fillion@cea.fr (P. Fillion).</footnote>
## Nomenclature

### Symbols

<table><tr><td>A</td><td>Cross-sectional area of the pipe (m<sup>2</sup>)</td></tr><tr><td>C</td><td>Droplet concentration (kg/m<sup>3</sup>)</td></tr><tr><td>C<sub>D</sub></td><td>Drag coefficient (-)</td></tr><tr><td>D</td><td>Pipe diameter (m)</td></tr><tr><td>d</td><td>Droplet diameter (m)</td></tr><tr><td>E</td><td>Entrainment fraction (-)</td></tr><tr><td>g</td><td>Gravity acceleration (m/s<sup>2</sup>)</td></tr><tr><td>h<sub>l</sub></td><td>Liquid height (m)</td></tr><tr><td>J</td><td>Superficial velocity (m/s)</td></tr><tr><td>k'<sub>A</sub></td><td>Atomization constant (-)</td></tr><tr><td>k<sub>D</sub></td><td>Deposition velocity (m/s)</td></tr><tr><td>k<sub>m</sub></td><td>Marginal wavenumber (m<sup>-1</sup>)</td></tr><tr><td>m<sub>D</sub></td><td>Deposition rate (kg m<sup>-2</sup> s<sup>-1</sup>)</td></tr><tr><td>m<sub>E</sub></td><td>Entrainment rate (kg m<sup>-2</sup> s<sup>-1</sup>)</td></tr><tr><td>P<sub>l</sub></td><td>Gas/film interfacial perimeter (m)</td></tr><tr><td>Q<sub>g</sub></td><td>Volumetric gas flowrate (m<sup>3</sup>/s)</td></tr><tr><td>V</td><td>Velocity (m/s)</td></tr><tr><td>W</td><td>Mass flowrate (kg/s)</td></tr><tr><td>z</td><td>Distance from inlet (m)</td></tr></table>

### Greek symbols

<table><tr><td>α</td><td>Volume fraction (-)</td></tr><tr><td>μ</td><td>Dynamic viscosity (Pa s)</td></tr><tr><td>ρ</td><td>Density (kg/m<sup>3</sup>)</td></tr><tr><td>σ</td><td>Surface tension (N/m)</td></tr><tr><td>τ</td><td>Relaxation time (s)</td></tr></table>

### Subscripts

<table><tr><td>∞</td><td>Equilibrium</td></tr><tr><td>B</td><td>Bulk</td></tr><tr><td>c</td><td>Critical</td></tr><tr><td>d, ld</td><td>Droplet, liquid dispersed</td></tr><tr><td>g</td><td>Gas</td></tr><tr><td>l</td><td>Liquid</td></tr><tr><td>lf</td><td>Liquid film</td></tr><tr><td>s</td><td>Superficial</td></tr><tr><td>W</td><td>Wall</td></tr></table>

volume fraction of droplets is determined by the balance of the entrainment rate $m_E$ and the deposition rate $m_D$, and additional momentum and energy equations relative to the dispersed field allow to predict the velocity and the temperature of the droplets. This approach is identified as a R&D direction for the future versions of the system codes (e.g. Bestion (2017)), and is more and more developed. In system codes adopting a three-field model dedicated to simulations in horizontal pipes (CATHARE 3 (Préa et al., 2020), SPACE (Ha et al., 2011)), the entrainment and deposition rates are given by semi-empirical correlations.

The next part aims at summarizing the current modeling of the droplet entrainment in horizontal pipes by several system codes, focusing on entrainment rate and deposition rate correlations included in the three-field models of SPACE and CATHARE 3 codes and the recent entrainment fraction correlation introduced in ATHLET code. Based on the results obtained from the considered models compared to experimental data in horizontal pipes, as well as the analysis of the experiments, a new model is proposed. This model, which applies to both the three-field model and the two-phase model with equilibrium

assumptions, is evaluated against the experimental data in stratified and annular flow regimes, with and without droplet entrainment. Extensions of the model accounting for the non-equilibrium effects at the entrance of the pipe are developed and assessed against experiments with droplet entrainment from stratified flows.

## 2. Models adopted in thermal-hydraulics system codes for the entrainment prediction

### 2.1. Entrainment fraction models

The 6-equation two-fluid models developed in the system thermal-hydraulics codes usually adopt a correlation to estimate the fraction of liquid flowing as entrained droplets. This is the case of CATHARE, RELAP5 and TRACE codes where the quantity of droplets is represented by the entrainment fraction $E$ defined as

$$E = \frac{W_{ld}}{W_{l}} \qquad (1)$$

where $W_{ld}$ is the mass flow of droplets in the gas and $W_{l}$ is the total mass flow of liquid.

In the RELAP5 code, the calculation of the entrainment fraction is based on the Ishii and Mishima (1981) correlation, expressed as

$$E = \tanh(7.25 \times 10^{-7} \text{We}^{1.25} \text{Re}_{sl}^{0.25}) \qquad (2)$$

where We is an effective Weber number defined by

$$\text{We} = \frac{\rho_g J_g^2 D}{\sigma} \left( \frac{\Delta\rho}{\rho_g} \right)^{1/3} \qquad (3)$$

and $\text{Re}_{sl} = \rho_l J_l D / \mu_l$ is the total superficial liquid Reynolds number. This correlation is for the quasi-equilibrium condition regarding the entrainment and deposition mechanisms and was successfully validated by Ishii and Mishima against low pressure air-water flows in vertical configuration with diameter 0.95 cm < $D$ < 3.2 cm and 370 < $\text{Re}_{sl}$ < 6400. From this basis, the code calculates the fraction of the total liquid volume residing in the annular film region, with a corrective factor accounting for entrance effects, expressed as a function of the void fraction. In fully horizontal configuration, the onset of entrainment is estimated using a critical gas velocity

$$V_{g,c} = 0.5 \left[ \frac{(\rho_l - \rho_g)g\alpha_g A}{\rho_g D} \right]^{1/2} \qquad (4)$$

According to TRACE manual (USNRC, 2023), the application of the Ishii-Mishima model in the conditions encountered in the hot leg of a PWR leads to unrealistic values of $E$. Another correlation for horizontal flows in large diameter pipes is used in the TRACE code. The formulation is based on the Steen and Wallis work (Steen and Wallis, 1964; Wallis, 1968) obtained from experimental data in vertical pipes. $E$ can be expressed by

$$E = 1 - \exp(-0.125 \times 10^4 [\pi_2 - \pi_{2,c}]) \qquad (5)$$

where the $\pi_2$ parameter, defined by Steen and Wallis, is considered as a non-dimensional gas velocity

$$\pi_2 = J_g \frac{\mu_g}{\sigma} \left( \frac{\rho_g}{\rho_l} \right)^{1/2} \qquad (6)$$

The inception gas velocity, i.e. the critical velocity at which the entrainment begins, is expressed as $\pi_{2,c} = 2.46 \times 10^{-4}$. TRACE code uses the onset of entrainment criterion proposed by Steen and Wallis and a modified entrainment fraction formulation from these authors, for pipes of diameter greater than 3.2 cm:

$$E = 0.015 + 0.44 \log_{10} \left[ 0.9245 \left( \frac{\pi_2}{\pi_{2,c}} \right)^2 \right] \qquad (7)$$
<figure><img src="image_4.png" /><figcaption>Fig. 1. Comparison of the entrainment fraction E against $\pi_2$ for different variants of the Steen-Wallis correlation.</figcaption></figure>

The 6-equation model of the CATHARE code also uses a formulation derived from the Steen-Wallis correlation (Carnevali and Fillion, 2024). For high void fraction, the formulation is modified as

$$E = \left[ 1 - \min \left( 1, \frac{\pi_{2,c}}{\pi_2} \right) \right]^2, \qquad (8)$$

where the critical non-dimensional velocity $\pi_{2,c} = 2.1 \times 10^{-4}$.

The different variants of the Steen-Wallis correlation are compared on Fig. 1. CATHARE formulation is closer to the original Wallis formulation than TRACE one for the lowest velocities, but leads to entrainment fraction values much lower for the highest velocities, and $E = 1$ is asymptotically reaches.

ATHLET code uses a S-shaped entrainment correlation (Lee et al., 2022a) formulated as

$$E = 1 - \exp \left( 0.5 \frac{J_{g,c} - J_g}{J_{g,c}} \right) \qquad (9)$$

with

$$J_{g,c} = \max \left[ 0.6 \sqrt{\frac{\rho_l}{\rho_g}}, 0.5 \sqrt{\frac{(\rho_l - \rho_g)g\alpha A}{\rho_g D}} \right] \qquad (10)$$

These approaches are based on the correlation first developed for vertical flows in small diameter pipes. Due to their formulation, the original Steen-Wallis correlation or the modified Steen-Wallis correlations included in system codes only depend on the gas velocity and the thermodynamic conditions of the gas and liquid phases, and do not take into account the film velocity or the pipe diameter, parameters which may have complex effects on droplet entrainment. Pipe diameter only appears in the formulation of the critical gas velocity adopted in ATHLET code, but the study carried out by Lee et al. (2022a) concluded that the entrainment correlation implemented in the code leads to wrong results in horizontal pipes in air-water conditions.

## 2.2. Entrainment rate and deposition rate models

Entrainment rate and deposition rate models have been more recently introduced in system codes, firstly dedicated to the three-field models of the code, and then derived for the two-fluid models, aiming at replacing the first approaches described in the previous section for the estimation of the entrainment fraction in horizontal pipes. For the entrainment prediction in a horizontal pipe, Pan and Hanratty (2002) model for annular flows or derived models from the original formulation are the most recently used correlations by system codes. Other models have been assessed in previous versions or are already used in the current version of the codes.

### 2.2.1. Pan-Hanratty type and derived models

The entrainment model proposed by Pan and Hanratty (2002) for horizontal pipes is mainly based on approaches developed by Dallman et al. (1984) and Lopez de Bertodano et al. (1997) in vertical tubes, and adapted from Williams et al. (1996) where the entrainment rate $m_E$ in a horizontal pipe takes into account the fact that the liquid film is asymmetric in the pipe due to gravity. The local entrainment rate $m_E(\theta)$ varies around the circumference of the pipe and is expressed, for an azimuthal angle of the pipe $\theta$, as

$$m_E(\theta) = \begin{cases} k'_A \frac{V_g^2 (\rho_l \rho_g)^{1/2}}{\sigma} (\Gamma - \Gamma_c) & \text{if } \Gamma > \Gamma_c \\ 0 & \text{else.} \end{cases} \qquad (11)$$

In this model, $k'_A$ is the atomization constant deduced from experiments, $V_g$ is the gas velocity, $\rho_l$ and $\rho_g$ are the liquid and gas densities, $\Gamma$ is the film flowrate per pipe diameter and $\Gamma_c$ is the critical film flowrate per perimeter at which atomization occurs. The spatially averaged formulation of the entrainment rate, resulting to the integration of the local rate over the circumferential angle, is defined by

$$m_E = \langle m_E(\theta) \rangle = \frac{1}{\pi} \int_0^\pi m_E(\theta) d\theta, \qquad (12)$$

$\theta$ origin corresponds to the bottom of the pipe; the flow is assumed to be symmetrical with respect to the vertical axis linking the bottom and the top of the pipe and writes:

$$m_E = k'_A \frac{V_g^2 (\rho_l \rho_g)^{1/2}}{\sigma} \left( \frac{W_{lf}}{\pi D} - \Gamma_c^* \right) \qquad (13)$$

where $W_{lf}$ is the film flowrate, $D$ is the pipe diameter, and

$$\Gamma_c^* = \Gamma_c - \frac{1}{\pi} \int_{\theta_c}^{\pi} (\Gamma_c - \Gamma) d\theta \qquad (14)$$

In this expression, $\theta_c$ is the critical angle at which $\Gamma = \Gamma_c$.

In the Pan-Hanratty approach, the deposition rate also accounts for the non uniform concentration of the droplets within the pipe. The spatially averaged deposition rate is given by

$$m_D = \langle k_D(\theta) \frac{C_W}{C_B} \rangle \left( \frac{W_{ld}}{Q_g S} \right) \qquad (15)$$
<caption>**Table 1**</caption>

Experimental database description. Measurement locations, expressed in length-to-diameter ratio z/D, where z is the distance from the inlet, correspond to the locations where the droplet flow rate are measured.

<table><thead><tr><th>Experiment</th><th>Reference</th><th>D (mm)</th><th>P (bar)</th><th>T (°C)</th><th>J<sub>g</sub> (m/s)</th><th>J<sub>l</sub> (m/s)</th><th>No. of runs</th><th>Measurement locations (z/D)</th></tr></thead><tbody><tr><td>Magrini</td><td>Magrini et al. (2010)</td><td>76.2</td><td>1.17</td><td>20</td><td>40–80</td><td>0.0035–0.04</td><td>20</td><td>172</td></tr><tr><td>Mantilla 2-in</td><td>Mantilla (2008)</td><td>48.6</td><td>1.9–2.1</td><td>19–24</td><td>2–82</td><td>0.0035–0.1</td><td>49</td><td>253</td></tr><tr><td>Mantilla 6-in</td><td>*</td><td>153</td><td>1–1.7</td><td>15–23</td><td>2–20</td><td>0.004–0.1</td><td>39</td><td>53</td></tr><tr><td>PNU</td><td>Bae et al. (2019)</td><td>40</td><td>1.05</td><td>15</td><td>24–32</td><td>0.015–0.03</td><td>7</td><td>17.5 ; 55 ; 80 ; 105</td></tr><tr><td>REGARD</td><td>Henry (2016)</td><td>240</td><td>1.1</td><td>30</td><td>19.8–38.2</td><td>0.018 ; 0.036</td><td>8</td><td>3.8 ; 5.7 ; 7.7 ; 9.6 ; 11.6</td></tr><tr><td>Williams</td><td>Williams et al. (1996)</td><td>95.3</td><td>1.18</td><td>19.5</td><td>26–88</td><td>0.02–0.12</td><td>29</td><td>260</td></tr></tbody></table>

In this equation, $Q_g$ is the volumetric gas flowrate, $k_D(\theta)$ the local deposition velocity, $C_B$ the bulk concentration of the droplets, $C_W$ the concentration close to the wall and $\langle k_D(\theta) \frac{c_W}{C_B} \rangle$ results from the spatial averaging. $S = V_{ld}/V_g$ is the slip ratio between the droplet velocity and the gas velocity.

Under equilibrium conditions in terms of entrainment and deposition rate ($m_E = m_D$), the authors derived the following equation for the entrained fraction E:

$$ \frac{E/E_M}{1-(E/E_M)} = \frac{k'_A D V_g^3 S(\rho_g \rho_l)^{1/2}}{4 \left\langle k_D(\theta) \frac{c_W}{C_B} \right\rangle \sigma} \quad (16) $$

The maximum entrainment fraction $E_M$ is given by

$$ E_M = 1 - \frac{W_{lf,c}}{W_l} = 1 - \frac{\pi D \Gamma_c^*}{W_l} = 1 - \frac{\text{Re}_{lf,c}}{\text{Re}_{sl}} \quad (17) $$

where the superficial liquid Reynolds number $Re_{sl}$ is defined by

$$ \text{Re}_{sl} = \frac{\rho_l J_l D}{\mu_l} \quad (18) $$

The critical flow rate $W_{lf,c}$ is based on measurements of Andreussi et al. (1985) and is given from an expression of in term of the liquid film Reynolds number:

$$ \text{Re}_{lf,c} = 7.30(\log_{10} \omega)^3 + 44.2(\log_{10} \omega)^2 - 263(\log_{10} \omega) + 439 \quad (19) $$

Parameter $\omega$, valid in the range 1.8 – 28, only depends on the fluid properties:

$$ \omega = \frac{\mu_l}{\mu_g} \sqrt{\frac{\rho_g}{\rho_l}} \quad (20) $$

Pan and Hanratty considered flows where the gravitational settling has the main contribution for the deposition process, thus neglecting the turbulent diffusion mechanism, and modeled $\langle k_D(\theta)(C_W/C_B) \rangle$ by the terminal velocity $V_T$ of a spherical droplet:

$$ V_T^2 = \frac{4d_d g \rho_l}{3C_D \rho_g} \quad (21) $$

where $C_D$ is the drag coefficient depending on the droplet Reynolds number. Droplet size $d_d$ is represented by the Sauter mean diameter ($d_d = d_{32}$) given by the Tatterson et al. (1977) correlation:

$$ \left( \frac{\rho_g V_g^2 d_{32}}{\sigma} \right) \left( \frac{d_{32}}{D} \right) = 0.0091 \quad (22) $$

Pan and Hanratty also assumed that the slip ratio $S = 1$. Eq. (16) becomes

$$ \frac{E/E_M}{1-(E/E_M)} = A_2 \frac{D V_g^3 (\rho_g \rho_l)^{1/2}}{V_T \sigma} \quad (23) $$

where constant $A_2$ was set to $A_2 = 9 \times 10^{-8}$ from experimental data obtained in pipes of diameter in the range 2.31–9.53 cm.

**SPACE code.** The Safety and Performance Analysis Code (SPACE) includes a three-field model, which is comprised of a gas, a continuous liquid, and a droplet field. The Pan-Hanratty model to predict the entrainment rate for a horizontal annular-mist flow was adopted in the first versions of the code (Ha et al., 2011).

Schimpf et al. (2018) developed new droplet entrainment and deposition models for mist-stratified and annular horizontal flows. They used Eq. (13) of the Pan-Hanratty model to estimate the entrainment rate, where the critical Reynolds is set to $Re_{lf,c} = 370$, from the expression (19), considering the thermodynamic properties at the REGARD experimental conditions (see Table 1). The authors deduced the atomization constant $k'_A = 3.8 \times 10^{-7}$, from these experiments. The authors assumed that droplet velocity is close to gas core velocity and thus set $S = 1$ as in the Pan-Hanratty approach. They also neglected the turbulent diffusion process for the deposition, and thus modeled the deposition rate as

$$ m_D = V_T \frac{W_{ld}}{Q_g} \quad (24) $$

The authors considered the volume median diameter $d_{vm}$ as a representative drop size $d_d$ in the terminal velocity and suggested a new droplet size model formulated as

$$ \left( \frac{\rho_g V_g^2 d_{vm}}{\sigma} \right)^{0.13} \left( \frac{d_{vm}}{D} \right)^{1.1} = 0.0013, \quad (25) $$

In the paper, the authors formulated, as Williams et al. (1996) did, the terminal velocity of the droplet following a Stokes law ($C_D = 24/Re_d$, where $Re_d = \rho_g V_T d_d / \mu_g$), as

$$ V_T = \frac{d_d^2 g \rho_l}{\mu_g} \quad (26) $$

but the correct formulation should be (from Eq. (21))

$$ V_T = \frac{1}{18} \frac{d_d^2 g \rho_l}{\mu_g} \quad (27) $$

The model was compared by the authors against fully developed flows (Williams and 2-inch Mantilla databases, see Table 1) and flows in non fully developed conditions (REGARD experiments).

**CATHARE 3 code.** First versions of the CATHARE three-field model used correlations for the estimation of the entrainment and deposition rates in the horizontal part of the primary circuit of the PWR, directly coming from those used in the core. These correlations were assessed against the reflooding phase of a LBLOCA simulated on integral test facility BETHSY, but the authors concluded that the correlations used in these simulations are inadequate to well represent the quantity of droplets flowing in the hot leg (Valette et al., 2011).

In order the better consider the droplet behavior in this part of the primary circuit, Henry et al. (2011) developed an original mechanistic model for the entrainment of droplet in horizontal stratified flow, aiming at its implementation in CATHARE. This work led to a set of models proposed by Henry (2016), dedicated to one-dimensional calculations in large diameter horizontal pipes, calibrated from REGARD data. Entrainment is supposed to be generated by the fragmentation of the waves produced by gas blowing at the film-gas interface. The entrainment rate depends on three sub-models: the entrained volume $V_E$ at the crest of the wave, the critical wavelength $\lambda_c$ and the characteristic time $\tau_c$, which corresponds to the time required for a wave to develop
until reaching the fragmenting amplitude. The entrainment rate is expressed as

$$m_E = \frac{\rho_{lc} V_E}{\lambda_c^2 \tau_c} \qquad (28)$$

Correlations are developed for $V_E$, $\lambda_c$ and $\tau_c$ from REGARD data. On this experiment, the model agrees well against the experimental entrainment rates. However, on the Williams runs in annular flows conditions, the predicted entrainment fractions are strongly underestimated, as the gas superficial velocity increases.

To overcome this misprediction, a model derived from the Pan-Hanratty approach was introduced in the three-field model of CATHARE 3 for the prediction of the entrainment rate in horizontal pipes having a large diameter for stratified and annular flows (Fillion, 2019). The estimation of the critical Reynolds number proposed by Pan and Hanratty (equation (19)) was used in these first simulations, limiting the values of $\omega$ given by expression (20) to its validity range. With the velocity deposition (see after), the atomization constant was set to $k'_A = 4 \times 10^{-7}$ for the REGARD experiments, but the author found that $k'_A = 4.5 \times 10^{-7}$ best fits the results if Williams and REGARD experiments are both considered.

In horizontal flow in large diameter pipes, two mechanisms acts on the droplet deposition: gravity and turbulent diffusion. The model suggested by Neiss (2013) for CATHARE, and retained by Fillion (2019), takes into account these two phenomena, following the works of several authors (e.g. Williams et al. (1996)), leading to a deposition rate written as:

$$m_D = k_D C = (k_{D,grav} + k_{D,turb})C \qquad (29)$$

where C is the droplet concentration, assuming a low droplet volume fraction, a void fraction value close to 1, and S = 1 (i.e. C = $W_{ld}/Q_g S \approx \rho_{ld} \alpha_{ld} / \alpha_g \approx \rho_{ld} \alpha_{ld}$). For vertical flows, Sugawara (1990) developed a correlation for the turbulent diffusion mechanism where the Schmidt number Sc = $v_g/D_g$ was used considering the heat transfer/mass transfer analogy:

$$\frac{k_{D,turb}}{V_g} = 9 \times 10^{-3} (C/\rho_g)^{-0.5} \text{Sc}^{-2/3} \text{Re}^{-0.2} \qquad (30)$$

This analogy was also retained by Neiss to model the deposition of the droplet within an horizontal pipe due the turbulent diffusion process. As in Sugawara paper, the corresponding correlation is based on the Nusselt number modeled using the Colburn correlation for the heat transfer. Neiss considered that the diffusion process is damped when the number of particles (droplets) increases. As the droplet volume fraction increases, the turbulence of the continuous phase no longer develops. This behavior is modeled by an apparent viscosity $\mu_m$ of the gas/droplet mixture which increases with the volume fraction of droplets. Consequently, the particles no longer diffuse in the flow, the deposit then decreases. Einstein's model (Einstein, 1906, 1911) is used to take into account this effect:

$$\mu_m = \mu_g(1 + 2.5\alpha_d) \qquad (31)$$

Moreover, since the correlation was established for dispersed flows where the density of the particles is close to the continuous phase, the correlation is modified by Neiss as

$$\mu_m = \mu_g(1 + 2.5\alpha_d \rho_d / \rho_g) \qquad (32)$$

Thus, the diffusion turbulent part is modeled as

$$\frac{k_{D,turb}}{V_g} = 0.023 \text{Sc}^{-2/3} \text{Re}_g^{-0.2} \frac{1}{1 + 2.5 \alpha_d \rho_d / \rho_g} \qquad (33)$$

Following the same approach of Pan and Hanratty and Williams et al. (1996), Neiss related the deposition constant to the terminal falling velocity of the droplet:

$$k_{D,grav} = g \tau \frac{S_{grav}}{S_{tot}}, \qquad (34)$$

where $S_{grav}$ and $S_{tot}$ are respectively deposition surface in the pipe and the total surface of the pipe, and $\tau$ is the relaxation time of the droplet:

$$V_T = g\tau. \qquad (35)$$

Neiss sets a constant relaxation time, with a value close to 0.1 s deduced from REGARD experiments corresponding to a droplet diameter of about 180 µm, and assumes the deposition due to the gravity only occurs on the lower part of the pipe, so $S_{grav}/S_{tot} = 1/2$. In the calculations presented by Fillion (2019) using CATHARE 3, the variation of the droplet diameter $d_d$ is taken into account in the calculation of the relaxation time

$$\tau = \frac{1}{18} \frac{d_d^2 \rho_d}{\mu_g} \qquad (36)$$

corresponding to a particle following the Stokes regime. In the first calculations, the Sauter mean diameter $d_{32}$ is used as a representative diameter for $d_d$. An empirical correlation of $d_{32}$ was developed (Fillion, 2019) from REGARD experiments for the largest pipe diameters. For the lowest pipe diameters, the correlation proposed by Berna et al. (2015) for the estimation of the volume median droplet diameter in horizontal flow is simplified for the CATHARE 3 code by choosing reference conditions for the liquid Reynolds number corresponding to liquid superficial velocity $J_{10} = 0.06 \text{ m/s}$ and near atmospheric conditions, with liquid density $\rho_{10} = 996 \text{ kg/m}^3$ and viscosity $\mu_{10} = 798 \times 10^{-6} \text{ Pa s}$. Using the corresponding reference scale $D_0 = \mu_{10}/(\rho_{10}J_{10})$, the proposed correlation in Fillion (2019) may be rewritten as

$$\frac{d_{vm}}{D} = 2.634 \text{We}_{sg}^{-0.23} \text{Re}_{sg}^{-0.54} \left(\frac{D}{D_0}\right)^{0.13} \qquad (37)$$

valid for diameters D in the range 0.025 – 0.096 m. The Sauter mean diameter is given by the simple expression

$$d_{32} = 0.6d_{vm}. \qquad (38)$$

Fitting the REGARD experimental data, the Sauter mean diameter is given by

$$\frac{d_{32}}{D} = 3.67 \times 10^{-3} \text{We}_{sg}^{-0.167} \qquad (39)$$

for $D \ge 0.24 \text{ m}$. For D comprise in 0.096 m and 0.24 m, a linear interpolation between the two equations is made.

In the REGARD experiments, due to the large diameter of the pipe, it exists a stratification of the flow. It results a strong concentration of the droplets in the lowest quarter of the pipe. This phenomenon is taken into account by Neiss (2013) considering an effective deposition velocity:

$$k_{D,eff} = f_{eff} \cdot k_D \qquad (40)$$

where $k_D$ is given by Eq. (29) and $f_{eff} = 4$.

These entrainment and deposition rate correlations were implemented into the three-field model of CATHARE 3, with a simplified version of Eq. (33) for the turbulent deposition velocity, considering a constant Schmidt number Sc ≈ 0.8, and assessed against REGARD and Williams databases (Fillion, 2019).

This model was later improved by Carnevali and Fillion (2024) in the frame of a benchmark concerning the estimation of the onset of entrainment criterion and the estimation of the entrainment rate within horizontal pipes, organized by FONESYS (Lanfredini et al., 2022). Mantilla tests were used for the comparisons, and the authors found the CATHARE 3 model underpredicted the entrainment fraction, and that the onset of entrainment given by Eq. (19) is delayed, regarding the experimental results at low liquid Reynolds number. For a better fitting of the experimental data, the author used the recommended value by Ishii and Grolmes (1975) for the onset of entrainment by the roll-wave shearing mechanism, and thus set

$$\text{Re}_{lf,c} = 160 \qquad (41)$$
Therefore, atomization constant appearing in Eq. (13) was reevaluated to

$$k'_{A} = 3.8 \times 10^{-7} \qquad (42)$$

to better fit the Mantilla, REGARD and Williams databases. This value is the same as those used by Schimpf et al. (2018) in their study (see SPACE code paragraph). Moreover, from the reappraisal of the weight of the turbulent diffusion and the gravity parts in the deposition process, the representative droplet diameter $d_d$, given by the Sauter mean diameter $d_{32}$ in Fillion (2019), was replaced by the volume median diameter $d_{vm}$ in the deposition model. Finally, the value of the effective deposition factor $f_{eff} = 4$ of Eq. (40) was replaced by $f_{eff} = 3.1$, from fitting of the model on the Mantilla 2-inch and 6-inch series, and REGARD and Williams databases.

**ATHLET code.** The ATHLET (Analysis of TThermal hydraulics of LEaks and Transients) code includes a 5-equation drift-flux model and a 6-equation two-fluid model (Wielenberg and Bals, 2021). A three-field version of the code is being recently developed and has been assessed against annular-dispersed vertical flows (von der Cron et al., 2022).

For flows in horizontal pipes and for the 6-equation model, a modification of the Pan-Hanratty model given by Eq. (23) was proposed and assessed in Lee et al. (2022a,b). The authors considered the work of Al-Sarkhi and Sarica (2011) showing that the legacy correlation of the code do not predict any entrainment for Mantilla tests at low liquid Reynolds number. From Mantilla (2008) criterion, the critical liquid Reynolds number as there is no entrainment is modified (Lee et al., 2022a) and then used in Lee et al. (2022b) for the ATHLET code, as:

$$\text{Re}_{lf,c} = \begin{cases} 2(h^+)^2 & \text{if } 0 \le h^+ < 5 \\ 4(12.514 + 5h^+ \ln(h^+) - 8.05h^+) & \text{if } 5 \le h^+ < 30 \\ 4(-56.09 + 2.5h^+ \ln(h^+) + 2.74h^+) & \text{if } 30 \le h^+ \end{cases} \qquad (43)$$

where $h^+$ is the dimensionless minimum film thickness

$$h^+ = \min[0.7\text{Re}_{sl}^{0.45}, 15]. \qquad (44)$$

This expression for $h^+$, established from Dallman (1978) experiments, excludes *de facto* the third case of the original Mantilla correlation (Eq. (43)). Using this sole modification for the critical liquid Reynolds number with respect to the Pan-Hanratty formulation, the authors obtained a good prediction of the 2-inch series of the Mantilla experiments.

### 2.2.2. Other models in system codes

In the SPACE code, the correlation developed by Lopez de Bertodano et al. (1997) was also implemented for the prediction of the entrainment rate for both vertical and horizontal annular flow conditions (Bae et al., 2018). The correlation reads

$$m_E = k_E \frac{\mu_l}{D_h} \left[ W e_g \left( \frac{\rho_l - \rho_g}{\rho_g} \right)^{1/2} (\text{Re}_{lf} - \text{Re}_{lf,c}) \right]^{0.925} \left( \frac{\mu_g}{\mu_l} \right)^{0.26} \qquad (45)$$

with $k_E = 4.47 \times 10^{-7}$. The film Reynolds number is defined by $\text{Re}_{lf} = \text{Re}_l(1 - E)$ and its critical value corresponding to the onset of entrainment $\text{Re}_{lf,c}$ was set to 80.

In the deposition rate model, calculated as $m_D = k_D C$, where $C = \rho_g W_{ld}/W_g$ is the droplet concentration, the deposition velocity is deduced from McCoy and Hanratty (1977) model, where the deposition velocity depends on the dimensionless relaxation time $\tau^+$ and the gas friction velocity $v^*$

$$\frac{k_D}{v^*} = \begin{cases} 3.25 \times 10^{-4}(\tau^+), & \tau^+ \le 22.67 \\ 0.17, & 22.87 < \tau^+ \le 8340 \\ \frac{15.525}{\sqrt{\tau^+}}, & \tau^+ > 8340 \end{cases} \qquad (46)$$

with

$$\tau^+ = \frac{d_d^2(v^*)^2 \rho_l \rho_g}{18 \mu_g^2} \qquad (47)$$

$$v^* = \sqrt{\frac{1}{2} V_g^2 f_l} \qquad (48)$$

and $f_l = 0.005(1+75\alpha_l)$. The droplet diameter is given by a correlation developed by Tatterson et al. (1977), formulated as

$$\frac{d_d}{D} \left( \frac{\rho_g V_g^2 f_s D}{2\sigma} \right)^{1/2} = 1.6 \times 10^{-2} \qquad (49)$$

where the friction factor $f_s = 0.046/\text{Re}_g^{0.2}$.

The Lopez de Bertodano et al. (1997) and McCoy and Hanratty (1977) correlations were assessed by Bae et al. (2018) in their study of horizontal annular flows. Since the original correlation does not fit the experimental data obtained by Tatterson in the considered flow regime, the authors reevaluated $k_E = 7.09 \times 10^{-7}$ and a new droplet deposition rate correlation was also developed from the McCoy-Hanratty model, using a horizontal annular air-water flow database. From their analysis of the tendencies of this deposition model against $J_g$, $J_l$ and $D$, the authors developed a correlation of the form

$$\frac{k_D}{v^*} = a \text{Re}_l^b(D^*)^c \text{Fr}_g^d \qquad (50)$$

where $D^* = D \sqrt{g \Delta \rho / \sigma}$ is the dimensionless pipe diameter ($\Delta \rho = \rho_l - \rho_g$). In this expression, the deposition velocity no longer depends on the droplet size $d_d$.

Another entrainment rate and deposition rate models were proposed by the same research teams (Bae et al., 2019) for horizontal stratified flows.

More recently, Khasawneh et al. (2021) developed two droplet size models, one for the stratified flow regime and one the annular flow regime. The authors substitute the droplet size model (equation (49)) due to Tatterson et al. (1977) used in the McCoy-Hanratty droplet deposition model implemented in the code. The proposed correlation for the stratified flow regime takes into account the fact that the flow is not established for the two considered experiments (REGARD and PNU (Bae et al., 2019) test sections), considering the effect of the developing length on droplet size, and thus depends on the length-to-diameter ratio. Using the Lopez de Bertodano et al. (1997) correlation for the entrainment rate, the authors compare the results obtained by the code against air-water experiment at low-pressure conditions, separating the experimental data in stratified and annular flow regimes.

## 3. Evaluation of entrainment models in system codes against a set of experiments

In order to verify the applicability of the entrainment models for system codes, we selected several databases corresponding to experiments performed in horizontal pipes at low pressure. One of the targeted conditions for a PWR corresponds to the reflooding phase during a LBLOCA. During this phase, the steam-water flow is at a pressure around 3 bar in the hot leg (Henry, 2016). Since any well-instrumented experiment in steam-water in these conditions is available in the literature for the development and the assessment of the entrainment model, in our knowledge, all of the selected experiments were carried out using an air-water mixture, in flow conditions from stratified to mist-annular regime. For each experimental database, test conditions and inner diameter of the pipe are summarized in Table 1. In the literature, these air-water experimental databases are considered sufficiently representative to develop or to validate models with the objective of their application to conditions occurred during a postulated LBLOCA. Indeed, Henry (2016) compares the fluid properties occurring in the REGARD test section with the targeted PWR conditions. The author points out that air-water at nearly atmospheric pressure presents a
density ratio and a surface tension close to the steam-water flow at the reflood phase during a LBLOCA. The author also remarks that the use of the air-water pair of fluid at constant temperature, without exchange between the two phases, remains representative since the temperature difference between the vapor and liquid is low during the reflood phase. In our study, REGARD and PNU experiments, equipped of several measurement devices along the test section, allow to assess the developing flow and thus simulate the entrance effect of the hot leg. Williams, Magrini and Mantilla 6-inch data are used to assess the entrainment fraction in equilibrium condition and in stratified to annular flow regimes. Mantilla 6-inch, having a large diameter, completes the database to assess the model where gravitational effects are dominant for the deposition, as in REGARD conditions. Moreover, Mantilla tests allows to validation the onset of droplet entrainment. As mentioned in the presentation of the models for the CATHARE 3 code, these tests were selected for a recent benchmark involving several system codes, focusing on the scalability of the closure laws, namely on the onset of droplet entrainment and entrainment fraction criteria (Lanfredini et al., 2022). Air-water REGARD, PNU, Mantilla and Williams data, included in our validation matrix, were also used by Bae et al. (2021) for the validation of a force balance model of the entrainment, with the aim at the application to a simulation of a LBLOCA. These authors calculated the droplet entrainment rate using the proposed force balance model for large pipes up to 1 m, and performed a sensitivity analysis for different pressure and temperature conditions. Khasawneh et al. (2021) used the PNU, REGARD, Mantilla 2-inch, Williams and Magrini experiments for the development of their models, with the objective of a “realistic prediction and understanding of the phenomena associated with droplets behavior in the PWR hot leg under LBLOCA reflood condition”.

In this study, we evaluated in priority models validated or dedicated both in stratified and annular flow regimes. We thus selected the Steen-Wallis formulation included in the CATHARE code, the Pan-Hanratty-type models developed by Schimpf et al. (2018), Lee et al. (2022b) and Carnevali and Fillion (2024). This evaluation is completed by those of the model proposed by Khasawneh et al. (2021). As mentioned in Section 2.2.2, this model relies on two formulations for the droplet size depending on the flow regime (stratified and annular). Following the study of the authors, we evaluated the model against the PNU database for the stratified regime, and Magrini, Mantilla 2-inch, Williams tests in annular regime only. The author pointed out that the model for the entrainment rate is not relevant where the gravitational settling is dominant for the deposition, such as in REGARD experiment. For this reason, the Mantilla 6-inch tests are also not included in our evaluation for this model. We assessed the entrainment fraction E predicted by these models, assuming an equilibrium between the entrainment and the deposition processes, i.e. $m_E = m_D$, in order to compute the entrainment fraction (according to Eq. (16) for the Pan-Hanratty-type models). For Magrini, Mantilla and Williams experiments, where the measurement is located far from the entrance ($L/D > 50$) in the fully developed region, this assumption is valid, but in the case of the REGARD and PNU experiments, where the measurements are located throughout the test section, this assumption may lead to an overestimation of E for the measurement locations close to the inlet of the test section where $m_E > m_D$, so the comparison is performed at the farthest location from the entrance, at $z/D = 11.7$ and $z/D = 105$ respectively. Entrance effect will be discussed in Section 4.4.

Fig. 2 compares the entrainment fraction obtained by the considered models against the selected experiments. Performances of the models are also tested using several errors (MAE, MAPE and RMSE) reported in Table 2, defined by

$$RMSE = \sqrt{\sum_{i=1}^{N_{\text{exp}}} \frac{(E_i^{\text{calc}} - E_i^{\text{exp}})^2}{N_{\text{exp}}} } \quad (51)$$

$$MAE = \frac{1}{N_{\text{exp}}} \sum_{i=1}^{N_{\text{exp}}} |E_i^{\text{calc}} - E_i^{\text{exp}}| \quad (52)$$

$$MAPE = \frac{100}{N_{\text{exp}}} \sum_{i=1}^{N_{\text{exp}}} \frac{|E_i^{\text{calc}} - E_i^{\text{exp}}|}{E_i^{\text{exp}}} \quad (53)$$

where $N_{\text{exp}}$ is the number of experimental points, and $E_i^{\text{calc}}$ and $E_i^{\text{exp}}$ are the calculated and experimental entrainment fraction at point i, respectively. In this assessment, the published (Schimpf et al., 2018) model is modified by using Eq. (27) instead of (26) in the calculation of the terminal velocity of the droplet according to our remark in the description of the model for the SPACE code, but is referred as Schimpf et al. (2018) in the rest of the paper.

In general, the prediction of the entrainment fraction is underestimated by the Steen-Wallis formulation in CATHARE (Fig. 2a). For several tests of REGARD and for the tests of the Mantilla 6-inch series, the predicted entrainment fraction is null, due to the overestimation of the critical gas velocity compared to the experiment.

For the REGARD experiment, and the runs of the Williams database where $E < 0.1$, corresponding roughly to stratified conditions with atomization (dispersed-stratified regime), the Schimpf et al. (2018) and Carnevali and Fillion (2024) models give good results (error less than 20 %) or fairly results (error between 20 and 30%). However, the model proposed by Lee et al. (2022b) strongly overestimates the entrainment fraction for the REGARD tests.

For the Mantilla 2-inch test series, the entrainment fraction is underestimated by the Schimpf et al. (2018) model, and also at low values by the Carnevali and Fillion (2024) model but is fairly resolved from $E > 0.4$ by this model. Lee et al. (2022b) model gives the best results for this database. The Mantilla 6-inch runs correspond to conditions in stratified regime at low liquid and gas superficial velocity, and dispersed-stratified regime when the velocities increase. All the considered models fail to calculate the entrainment for the Mantilla 6-inch tests, predicting either no entrainment, due to the fact that the occurrence of the onset of entrainment is predicted by the model at higher gas velocity compared to the experiment, or an underestimation of the entrainment fraction.

The evaluated models also strongly underestimate the entrainment fraction for all the runs performed in the PNU experiment, with the exception of the model proposed by Khasawneh et al. (2021). The same behavior was highlighted by Khasawneh et al. (2021) in the analysis of the model included in the SPACE code, based on the Lopez de Bertodano et al. (1997) and McCoy and Hanratty (1977) correlations. Although the authors developed a new droplet diameter model based on the PNU data and included it in the McCoy and Hanratty (1977) correlation, the experimental droplet flowrate remains underestimated. Further analysis is still necessary to fully explain this discrepancy.

## 4. Suggestions for the improvement of the droplet entrainment prediction

As mentioned in the introduction, one of the objectives of this study is to propose a droplet entrainment model applicable both to the three-field models and to the two-fluid 6(or 5)-equation models implemented in system codes, and for stratified and annular flow regimes. According to the formulation of the models reviewed in the paper, the formulations derived from Pan-Hanratty approach or the variants of the Lopez de Bertodano/McCoy-Hanratty models of the SPACE code are relevant to meet this requirement. Indeed, it can be formulated as entrainment rate $m_E$ and deposition rate $m_D$ for the three-field model, and can be derived as an entrainment fraction model, under the equilibrium condition $m_E = m_D$. Moreover, if the kinematic equilibrium between the gas and the droplet velocity is reached, we can assumed that the slip ratio $S = 1$ in the Pan-Hanratty model. Otherwise, the model has to be supplemented by an algebraic model to determine the slip ratio. Under the two equilibrium conditions, the three-field model for the droplet entrainment relaxes to the two-phase model for fully developed flows.

Among the Pan-Hanratty-type formulations, only Schimpf et al. (2018) and Carnevali and Fillion (2024) models may be selected for
<figure><img src="image_5.png" /><figcaption>Fig. 2. Comparison of the calculated entrainment fraction by different models against the experimental value for the selected database.</figcaption></figure>

the aforementioned requirement, and better results are obtained by the last one. The model proposed by Lee et al. (2022b) cannot be selected because its formulation is only expressed as an entrainment fraction. The model developed by Khasawneh et al. (2021) based on the Lopez de Bertodano/McCoy-Hanratty models produces good results but has to be supplemented by a criteria for the stratified to annular transition to select the droplet diameter correlation, which is out of scope of the paper. In this section, different suggestions for the improvement of the Carnevali-Fillion model are thus assessed.

## 4.1. Onset of entrainment

The performance of the entrainment part of the Pan-Haranty-type models or the Lopez de Bertodano-type models depends on the critical liquid Reynolds number $Re_{lf,c}$ at which the droplet entrainment begins.

The attempt of the Carnevali-Fillion model to decrease $Re_{lf,c}$ from the correlation used in the Pan-Hanratty model, is not sufficient to give a good agreement with the experimental entrainment fraction obtained by Mantilla at low liquid Reynolds number. On the other hand, the model proposed by Lee et al. (2022a) for the onset of entrainment, from Mantilla's work, allows to correctly estimate the entrainment fraction in these conditions. We notice that, applying the Eqs. (43)–(44), the critical Reynolds number for $h^+ = 15$ corresponds to $Re_{lf,c} = 380$, a value close to $Re_{lf,c} = 370$ given by Eq. (19), used in the original Pan-Hanratty model, close to atmospheric pressure ($P = 1.1$ bar) and $T = 20$ °C.

The proposed modification applied to the Carnevali-Fillion model is thus to replace the critical value $Re_{lf,c} = 160$ for the onset of entrainment, coming from the Ishii-Grolmes criterion, with the model proposed by Lee et al. (2022a) for the critical film Reynolds number,
<table><caption>**Table 2**
Comparison of errors of the models for the entrainment fraction *E* against the experimental databases.</caption><thead><tr><th>Model</th><th>Database</th><th>RMSE (%)</th><th>MAE (%)</th><th>MAPE (%)</th></tr></thead><tbody><tr><td rowspan="6">Steen-Wallis CATHARE</td><td>Mantilla 2 in</td><td>11.2</td><td>6.0</td><td>40.7</td></tr><tr><td>Mantilla 6 in</td><td>1.5</td><td>0.8</td><td>100.0</td></tr><tr><td>Williams</td><td>14.2</td><td>10.8</td><td>38.4</td></tr><tr><td>Magrini</td><td>10.2</td><td>9.4</td><td>23.8</td></tr><tr><td>REGARD</td><td>3.5</td><td>3.0</td><td>81.7</td></tr><tr><td>PNU</td><td>26.5</td><td>24.7</td><td>88.9</td></tr><tr><td rowspan="4">Khasawneh et al. (2021)</td><td>Mantilla 2 in (annular)</td><td>18.0</td><td>15.2</td><td>30.1</td></tr><tr><td>Williams (annular)</td><td>11.7</td><td>9.5</td><td>18.9</td></tr><tr><td>Magrini (annular)</td><td>13.4</td><td>13.1</td><td>29.7</td></tr><tr><td>PNU</td><td>8.0</td><td>6.8</td><td>27.3</td></tr><tr><td rowspan="6">Schimpf et al. (2018)</td><td>Mantilla 2 in</td><td>13.4</td><td>7.9</td><td>51.7</td></tr><tr><td>Mantilla 6 in</td><td>1.4</td><td>0.7</td><td>79.6</td></tr><tr><td>Williams</td><td>11.7</td><td>9.3</td><td>30.6</td></tr><tr><td>Magrini</td><td>23.6</td><td>19.8</td><td>50.4</td></tr><tr><td>REGARD</td><td>0.8</td><td>0.6</td><td>15.2</td></tr><tr><td>PNU</td><td>25.3</td><td>23.5</td><td>83.3</td></tr><tr><td rowspan="6">Lee et al. (2022b)</td><td>Mantilla 2 in</td><td>4.4</td><td>2.4</td><td>25.6</td></tr><tr><td>Mantilla 6 in</td><td>1.4</td><td>0.7</td><td>86.9</td></tr><tr><td>Williams</td><td>7.5</td><td>5.1</td><td>15.9</td></tr><tr><td>Magrini</td><td>10.7</td><td>8.9</td><td>21.7</td></tr><tr><td>REGARD</td><td>7.0</td><td>5.0</td><td>81.1</td></tr><tr><td>PNU</td><td>25.8</td><td>24.1</td><td>87.0</td></tr><tr><td rowspan="6">Carnevali and Fillion (2024)</td><td>Mantilla 2 in</td><td>9.7</td><td>5.0</td><td>42.1</td></tr><tr><td>Mantilla 6 in</td><td>1.3</td><td>0.6</td><td>79.2</td></tr><tr><td>Williams</td><td>7.2</td><td>4.8</td><td>16.1</td></tr><tr><td>Magrini</td><td>12.6</td><td>10.9</td><td>27.2</td></tr><tr><td>REGARD</td><td>0.6</td><td>0.4</td><td>13.5</td></tr><tr><td>PNU</td><td>26.5</td><td>24.7</td><td>88.9</td></tr><tr><td rowspan="6">Proposed model</td><td>Mantilla 2 in</td><td>5.1</td><td>2.8</td><td>22.4</td></tr><tr><td>Mantilla 6 in</td><td>1.2</td><td>0.6</td><td>75.7</td></tr><tr><td>Williams</td><td>6.3</td><td>4.1</td><td>14.1</td></tr><tr><td>Magrini</td><td>8.7</td><td>7.6</td><td>18.4</td></tr><tr><td>REGARD</td><td>0.7</td><td>0.4</td><td>13.9</td></tr><tr><td>PNU</td><td>26.1</td><td>24.3</td><td>87.5</td></tr></tbody></table>

given by Eqs. (43)–(44). With this modification, the performance of the Carnevali–Fillion model is thus significantly improved for Mantilla 2-inch tests (see Fig. 3) at the lowest liquid superficial velocity ($J_f$ = 0.0035 m/s, corresponding to $Re_{sl}$ around 160–190, and a critical Reynolds number around 90–110), with other minor improvements on some Magrini runs. However, applying this modification, combined with the proposed deposition rate model described in Section 4.2, leads to lowest entrainment fraction for some other tests of the Mantilla 2-inch test section.

Fig. 4 shows the predicted results compared to the REGARD experiment. The modification of the critical Reynolds number from the Carnevali–Fillion model to the value calculated from Lee et al. correlation in the proposed model has a weak influence on the prediction of the entrainment rate $m_E$ against the measurements obtained in the REGARD test section, in the stratified regime with atomization. The experimental entrainment rate was deduced from the droplet mass flowrate measured at $z/D = 3.8$, with the assumption that the deposition can be neglected from the entrance of the test section to this location (Henry, 2016).

It is worth noting that the entrainment models derived from Pan-Hanratty model consider the entrainment limit only from the liquid velocity. Indeed, Pan and Hanratty point out that using $V_g - V_{g,c}$ instead of $V_g$ in Eq. (11), where $V_{g,c}$ is the critical gas velocity, is not physical in case of horizontal flows, since the critical velocity depends on gravitational settling, and they remark that its value is quite different from those observed in vertical flows. Thus, the onset of entrainment correlated to the critical gas velocity is not taken into account, contrary to Steen-Wallis correlations derived from experimental data in vertical pipes. Experimental observations performed in horizontal pipes (e.g. Mantilla tests) show it exists a limiting value of droplet entrainment due to the gas characteristics, allowing to determine the entrainment inception velocity, that is the critical gas velocity at which

the entrainment begins. Below this value, no entrainment takes place whatever the liquid film Reynolds number above its critical value. From experiments, models also combine liquid and gas conditions to determine the onset of entrainment. This is the case of the criteria proposed by Ishii and Grolmes (1975), used in ATHLET code (Lee et al., 2022a) for the investigation of the Mantilla database, or by Crowe (2005) (from Berna et al. (2014)). Fig. 5 compares these criteria against air-water Mantilla data obtained in the 2-inch and 6-inch test sections. The entrainment inception velocity is overestimated by the Ishii and Grolmes (1975) model. The Crowe (2005) criterion is slightly better for the highest liquid Reynolds number (in the rough turbulent regime) but the predicted critical gas velocity is about 1.5 time higher than the experimental ones for the 6-inch section. If the critical gas velocity is not far from the experiment for the 2-inch series, the onset of entrainment does not agree with the experiment at the lowest experimental Reynolds numbers. We also compare these criteria and data to a model used by Boulesteix (2010). In a horizontal pipe of 5 cm inner diameter and 6.5 m long, this author studied the critical conditions for the atomization for air-water flows. Boulesteix observed that the presence of roll-waves, due to the gas velocity, seems necessary to initiate the atomization, but is not sufficient to trigger the atomization, because it also depends on the liquid velocity. Assuming a flat film-gas interface at which the entrainment begins in the conducted experiments, Boulesteix compared the critical gas velocity based on Andritsos and Hanratty (1987) model against the obtained experimental data and found a good agreement at low liquid superficial velocity, but the predicted gas velocity tends to be overestimated when the liquid holdup increases. The Andritsos and Hanratty (1987) criterion actually corresponds to the initiation of large-amplitude waves and not directly to the initiation of the atomization. This criterion is based on the linear stability analysis of the liquid film, assuming the perturbation of the interface is due to
<figure><img src="image_6.png" /><figcaption>(a) $J_l = 0.0035$ m/s</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(b) $J_l = 0.018$ m/s</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(c) $J_l = 0.035$ m/s</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(d) $J_l = 0.01$ m/s</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(a) $J_l = 0.018$ m/s</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(b) $J_l = 0.036$ m/s</figcaption></figure>

<caption>Fig. 4. Comparison of predicted entrainment rate models against REGARD tests.</caption>

a Kelvin–Helmholtz (K-H) instability. The limit of instability caused by the gas velocity $V_{KH}$ is expressed as

$$V_{KH} = V_l + \sqrt{\left(\frac{k_m \sigma}{\rho_g} + \frac{\rho_l g}{\rho_g k_m}\right) \tanh(k_m [D - h_l])} \quad (54)$$

In this equation, $k_m$ represents the marginal wave number given by

$$k_m = \sqrt{\frac{\rho_l g}{\sigma}} \quad (55)$$

and $h_l$ is the liquid height. From this expression, the superficial gas velocity for the initiation of large-amplitude waves (LW) for air-water flows is correlated by the authors as

$$J_{g,\text{LW}} = 1.8V_{KH} \left( \frac{1}{\tanh(k_m h_l / 10)} \right)^{0.1} \frac{1}{\alpha} \quad (56)$$

In Boulesteix (2010), the expression used for the initiation of the atomization reads

$$J_{g,c} = 1.8\alpha_g V_{KH} \quad (57)$$
<figure><img src="image_12.png" /><figcaption>Fig. 5. Comparison of onset of entrainment models against Mantilla tests.</figcaption></figure>

As it shows in Fig. 5, the expression of the critical superficial gas velocity given by Eq. (57), deduced from K-H instability analysis, predicts reasonable results compared to the Mantilla experiments for the onset of entrainment, assuming a flat air-water surface at the onset of entrainment ($h_l$ in expressions (54)) is given by the experimental liquid height at the bottom of the pipe). Although the trend of the limit is not monotonic, that we attribute to measurement uncertainties for the experimental liquid holdup, the critical superficial gas velocity from Eq. (56) tends to decreases as the liquid Reynolds number increases, as the experimental results. At the highest liquid Reynolds numbers, this criterion gives a good prediction of the onset of entrainment for the 2-inch series, and is the best of the assessed criteria. For the 6-inch series, the criterion has also the best prediction. It qualitatively follows the trend of the experimental data, but tends to overestimate the critical gas velocity for all the considered liquid Reynolds numbers.

Despite the fact that this onset of entrainment model requires more validation and needs to be improved, it may be used in a system code and implemented as an approximation of the critical gas velocity. The height of the liquid can be estimated from the volume liquid fraction calculated by the code, assuming a flat interface between the film and the gas.

## 4.2. Deposition rate

The deposition model for the CATHARE 3 code is based on the superposition of the gravity and the turbulent mechanisms for the estimation of the deposition velocity. An effective velocity is calculated using a multiplying factor for both mechanisms, accounting the stratification of the concentration of the droplets at the bottom of the pipe, and then enhancing the deposition. In the previous models for the
<table><caption>**Table 3**
Summary of the proposed model.</caption><thead><tr><th>Submodel</th><th>Formulation</th></tr></thead><tbody><tr><td>Entrainment rate</td><td>m<sub>E</sub> = 3.8 &times; 10<sup>-7</sup> <sup>V<sub>g</sub><sup>2</sup> (&rho;<sub>l</sub>&rho;<sub>g</sub>)<sup>1/2</sup></sup>&frasl;<sub>&sigma;</sub> (<sup>W<sub>lf</sub></sup>&frasl;<sub>&pi;D</sub> - W<sub>lf,c</sub>), W<sub>lf,c</sub>/&pi;D = Re<sub>lf,c</sub>&mu;<sub>l</sub>/4,</td></tr><tr><td>Critical Reynolds number</td><td>Re<sub>lf,c</sub> given by Eqs. (43)-(44)</td></tr><tr><td>Deposition rate</td><td>m<sub>D</sub> = k<sub>D,eff</sub>C = 1.53(k<sub>D,grav</sub> + k<sub>D,turb</sub>)C</td></tr><tr><td>Droplet diameter</td><td>k<sub>D,grav</sub> = <sup>1</sup>&frasl;<sub>18</sub> <sup>d<sub>d</sub><sup>2</sup>&rho;<sub>d</sub></sup>&frasl;<sub>&mu;<sub>g</sub></sub><br>d<sub>d</sub> given by Eq. (59)<br><sup>k<sub>D,turb</sub></sup>&frasl;<sub>V<sub>g</sub></sub> = 0.023Sc<sup>-2/3</sup>Re<sub>g</sub><sup>-0.2</sup> <sup>1</sup>&frasl;<sub>1 + 2.5a<sub>d</sub>&rho;<sub>d</sub>/&rho;<sub>g</sub></sub>, Sc = 0.8</td></tr><tr><td>Entrainment fraction</td><td><sup>E<sub>&infin;</sub>/E<sub>M</sub></sup>&frasl;<sub>1 - (E<sub>&infin;</sub>/E<sub>M</sub>)</sub> = <sup>3.8 &times; 10<sup>-7</sup> DV<sub>g</sub><sup>3</sup>S(&rho;<sub>g</sub>&rho;<sub>l</sub>)<sup>1/2</sup></sup>&frasl;<sub>4k<sub>D</sub>&sigma;</sub>, equilibrium assumption m<sub>E</sub> = m<sub>D</sub></td></tr><tr><td>Maximum entrainment fraction</td><td>E<sub>M</sub> = 1 - <sup>W<sub>lf,c</sub></sup>&frasl;<sub>W<sub>l</sub></sub></td></tr><tr><td>Slip ratio</td><td>S = 1</td></tr><tr><td>Entrance effect</td><td>E(z) = [1 - exp(-2.24 &times; 10<sup>-6</sup>(D<sup>*</sup>)<sup>1.66</sup>&zeta;<sub>K</sub>(z)<sup>2</sup>)] E<sub>&infin;</sub><br>E(z) = A <sup>m<sub>E</sub>V<sub>g</sub></sup>&frasl;<sub>k<sub>D,c</sub>W<sub>l</sub></sub> [1 - exp(<sup>-P<sub>l</sub>k<sub>D,z</sub></sup>&frasl;<sub>V<sub>g</sub>A</sub>)]</td></tr></tbody></table>

deposition (Neiss, 2013; Fillion, 2019; Carnevali and Fillion, 2024), the hypothesis is made that the gravity process only acts on the bottom half of the pipe, and the corresponding deposition velocity is divided by 2. In this suggestion, we assume that the effective factor, which is link to the radial profile of the concentration of droplets, takes into account the fact that the gravity process only acts on the bottom half of the pipe. The gravity deposition model is thus modified with

$$k_{D,grav} = g\tau \qquad (58)$$

The droplet diameter $d_d$ in the expression of $\tau$ is also modified, allowing to reflect the influence of the liquid velocity. The original formulation of Berna et al. (2015) of horizontal flows in used for the lowest diameters of the pipe in place of Eq. (37), whereas the REGARD fitting is kept:

$$d_d = d_{vm} = \begin{cases} 2.634We_{sg}Re^{-0.54}Re^{0.13}D & D \le 0.096 \text{ m} \\ 1.67d_{32} \quad (\text{Eq. (39)}) & D \ge 0.24 \text{ m} \\ \text{linear interpolation} & 0.096 < D < 0.24 \text{ m} \end{cases} \qquad (59)$$

In the considered experiments, any measurement of the deposition rate is available. The deposition rate is deduced from the entrainment rate, using the equilibrium assumption ($m_D = m_E$). As a consequence, the effective factor appearing in Eq. (40) is modified. The value $f_{eff} = 1.53$ is determined from the experiments, minimizing the global error between the calculated entrainment fraction and the experimental entrainment fraction, under the assumption $m_D = m_E$. During this step, data of PNU were excluded, due to the fact that models assessed against these data underestimate the entrainment fraction. Data of REGARD close to the entrance, where the entrainment and deposition rates are not at the equilibrium, are also excluded for this calibration.

### 4.3. Results on the experimental database

Fig. 2(f) shows the entrainment fraction obtained by the proposed model, summarized in Table 3, including the suggestions (onset of entrainment and deposition model), compared to the considered experimental databases. For these comparisons, assumption of equilibrium is made as in Section 3. Compared to the Carnevali–Fillion model, the prediction is improved mostly at moderate entrainment fraction (0.2–0.4), or at the beginning of the entrainment due to the modification of the onset of entrainment. The performance of the entrainment fraction model is quantified using the MAE (Mean Absolute Error) and MAPE (Mean Absolute Percentage Error) for each database (Table 2), showing a significant improvement for Mantilla 2-inch and Magrini tests, whereas the performance of the proposed model is slightly improved compared to the Carnevali–Fillion model for the other tests. On the

REGARD database, the proposed model gives satisfactory results, with a similar error compared to the Carnevali–Fillion model and the Schimpf et al. model. For the Mantilla, Williams and Magrini experiments, the performance of the proposed model is equivalent to the Lee et al. model, and the obtained results are similar. All the assessed models strongly underestimate the entrainment fraction obtained in the PNU experiment. These obtained results for the PNU database require further investigation.

### 4.4. Non fully developed flows – Entrance effect

In the case of the hot leg of the PWR, the equilibrium of the entrainment and the deposition processes is not reached due to the short length-to-diameter ratio. In this case, the entrainment fraction model based on the equilibrium assumption may lead to errors close to the entrance of the pipe. Two methods are studied in this section in order to take into account the non-equilibrium effect close to the inlet of the pipe.

#### 4.4.1. Method 1: non-equilibrium effect model from droplet mass balance equation

These non equilibrium effects may be derived from the entrainment rate and deposition rate, related in the one-dimensional mass conservation equation for the droplets, without phase change:

$$\frac{\partial W_{ld}}{\partial z} = A \frac{\partial \rho_l J_l E}{\partial z} = P_l(m_E - m_D) \qquad (60)$$

where $z$ is the coordinate along the pipe and $P_l$ is the interfacial perimeter between the film and the gas. In the case of wavy stratified flow, the modeling of the interfacial area is complex. A sensitivity analysis on this parameter to the droplet flowrate has been carried out by Schimpf et al. (2018) against the REGARD data. The authors found that an area corresponding to an annular flow better reproduces the development of the entrainment than a flat interface for horizontal stratified flow. With our proposed models for $m_E$ and $m_D$, the developing length of the entrainment fraction $E(z)$ along the pipe assuming this choice of the interfacial perimeter (annular) is underestimated, i.e. the perimeter is too wide (see Fig. 6). In the REGARD conditions, the interface corresponds to a stratified wavy flow. The interface is highly agitated and a curvature is created when the gas velocity increases. In such situation, Bae et al. (2019) used a method proposed by Ahn et al. (2018) for the estimation of the length of the interface. The method is based on empirical correlation for the calculation of wetted perimeter of the pipe. From this calculation, the gas/film perimeter can be computed. The proposed entrainment rate and deposition rate calibrated from REGARD and PNU data, coupled with this model for the interfacial
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 6. Proposed entrainment fraction model. Influence of the interfacial perimeter on the entrainment fraction profile along the REGARD test section.</figcaption></figure>

perimeter allows to these authors to well estimate the development of the droplet flowrate for these experiments. Calculation of the interfacial perimeter relies on the hypothesis of a smooth interface, without taking into account the deformation due to the waves. Wave oscillations are also a parameter in the calculation of the gas/film interfacial area from REGARD experiments by Henry (2016), through the wavelength. Such approach requires the calculation or the wave characteristics, with is out of scope of our study. Fig. 6 shows the entrainment fraction calculated along the REGARD test section using four models for the interfacial perimeter: (i) perimeter corresponding to a flat interface (ii) perimeter corresponding to a curved interface using the Bae et al. (2019) model (iii) perimeter corresponding to an annular flow (iv) perimeter corresponding to an annular flow in the lower part of the pipe (“half-annular” perimeter): indeed, in the case of wavy stratified flows, considering the deposition due to the gravity only, half of the pipe may be affected and, although the choice is subject to debate, we consider an interface corresponding to this perimeter. For the purpose

of comparison only, the interfacial perimeter is thus approximated by

$$P_l \approx 0.5 \times \pi \sqrt{1 - \alpha_{lf} D} \qquad (61)$$

The “half-annular” perimeter model gives the best results regarding the experimental points. Model of a curved interface tends to underestimated the development of the entrainment for the lowest superficial liquid velocity but gives fair results when the superficial liquid velocity increases. The entrainment fraction is always strongly underestimated assuming a flat interface between the gas and the film.

Under some assumptions, an analytical expression of E(z) can be deduced from the mass conservation equation: Neiss (2013) estimates the droplet mass flowrate along the pipe from Eq. (60), due to the fact that $m_D = k_D C$ and $C = \alpha_{ld} \rho_l$, as

$$\frac{\partial W_{ld}}{\partial z} = m_E P_l - k_D \frac{W_{ld}}{V_{ld} A} P_l \qquad (62)$$
<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 7. Proposed entrainment fraction model. Influence of the entrance effect factor (with n = 1.15 in Eq. (71); m = 1.66 in Eq. (72) or (73) gives very similar results) compared to entrainment fraction calculated from entrainment and deposition rate balance, and to compared to REGARD experimental measurements.</figcaption></figure>

Neiss assumes that the entrainment rate $m_E$ and the deposition velocity $k_D$ are constant along the pipe. We also assume that the perimeter $P_i$ is constant. Under these assumptions, the droplet flowrate is deduced by integration of Eq. (62). For a droplet flowrate null at the inlet of the pipe, the droplet flowrate profile reads

$$W_{ld}(z) = A \frac{m_E V_{ld}}{k_D} \left[ 1 - \exp \left( \frac{-P_i k_D z}{V_{ld} A} \right) \right] \quad (63)$$

In our proposed model, $m_E$, $k_D$ are not constant because they depend on the droplet volume fraction. A simple way to apply Eq. (63) is to consider the entrainment fraction and deposition velocity under equilibrium conditions. From the entrainment fraction $E_\infty$, the equilibrium values $m_{E,\infty}$, $k_{D,\infty}$ and $P_{i,\infty}$ can be computed, and the entrainment fraction profile becomes

$$E(z) = A \frac{m_{E,\infty} V_{ld}}{k_{D,\infty} W_i} \left[ 1 - \exp \left( \frac{-P_{i,\infty} k_{D,\infty} z}{V_{ld} A} \right) \right] \quad (64)$$

Using the proposed models in the previous section for the entrainment rate $m_E$ and the deposition velocity $k_D$, this modified Neiss model is tested against the REGARD tests (Fig. 7) with the hypothesis of $V_{ld} = V_g$, i.e. $S = 1$. It gives very close results compared to the entrainment fraction calculated by the mass conservation equation. The advantage of this method is to use of the entrainment and deposition rate models, but an assumption of the interfacial perimeter between the film and the gas is necessary. In the original formulation of Neiss, this perimeter was set to the pipe perimeter, and we replaced this value with the generic value $P_i$ in our approach.

### 4.4.2. Method 2: entrance effect factors

Entrance effects may also be modeled by the modification of the entrainment fraction $E_\infty$ deduced from equilibrium conditions, using a multiplier $f$ called entrance factor depending on the distance from the
inlet of the pipe:

$$E(z) = f(z)E_{\infty} \qquad (65)$$

From data obtained in the case of annular flows in vertical tubes with a null entrainment fraction at the inlet, Ishii and Mishima (1981) correlated the entrance effect as

$$E(z) = [1 - \exp(-10^{-5} \zeta_{\text{IM}}(z)^2)] E_{\infty} \qquad (66)$$

In this equation, $\zeta_{\text{IM}}$ is the dimensionless distance given by

$$\zeta_{\text{IM}}(z) = \frac{z}{D} \sqrt{\frac{\text{Re}_{\zeta sl}}{J_g^*}} \qquad (67)$$

and $J_g^*$ a dimensionless gas flux defined as

$$J_g^* = \frac{J_g}{\left[ \frac{\sigma_g \Delta \rho}{\rho_g^2 \left( \frac{\rho_g}{\Delta \rho} \right)^{2/3}} \right]^{1/4}} \qquad (68)$$

Kataoka et al. (2000) proposed another model accounting for entrance effects, as

$$E(z) = [1 - \exp(-1.87 \times 10^{-5} \zeta_{\text{K}}(z)^2)] E_{\infty} \qquad (69)$$

where the Weber number, given by Eq. (3), is used for the definition of the dimensionless distance

$$\zeta_{\text{K}}(z) = \frac{z}{D} \frac{\text{Re}_{sl}^{0.5}}{\text{We}^{0.25}} \qquad (70)$$

The entrance factors proposed by Ishii and Mishima or by Kataoka, applied to the REGARD conditions in stratified regime, lead to wrong results, with an important overestimation of the developing length. We attribute this result to the fact that the pipe diameter of REGARD ($D = 0.24$ m) is much larger to the tube diameter ($D = 0.0095$ m) of the experiments studied by Ishii and Mishima. At least, parameter $10^{-5}$ or $1.87 \times 10^{-5}$ appearing in Eq. (66) or (69), fitting from experimental data, should be likely reevaluated. We thus modified this expression to account the pipe diameter as

$$E(z) = [1 - \exp(-10^{-5}(D/D_0)^n \zeta_{\text{IM}}(z)^2)] E_{\infty} \qquad (71)$$

where $D_0 = 0.0095$ m. Exponent $n \ge 1$ has to be determined from experimental data. Similar expression is given for the other entrance factor

$$E(z) = [1 - \exp(-1.87 \times 10^{-5}(D/D_0)^m \zeta_{\text{K}}(z)^2)] E_{\infty} \qquad (72)$$

with $m \ge 1$. In our study, the diameter effect for the estimation of the developing length is also given by $D^* = D\sqrt{g\Delta\rho/\sigma}$ as in Wang et al. (2020). Expression (72) is replaced with

$$E(z) = [1 - \exp(-2.24 \times 10^{-6}(D^*)^m \zeta_{\text{K}}(z)^2)] E_{\infty} \qquad (73)$$

Using these expressions, we evaluated the entrainment fraction model accounting the entrance effect on the REGARD experiment (Fig. 7). Applying these factors to the calculated $E_{\infty}$ given by the proposed model for $m_E$ and $m_D$ assuming $m_E = m_D$ at $z/D = 11.6$, the entrainment fraction is compared to the entrainment fraction calculated by the entrainment rate and deposition rate balance, using the mass conservation Eq. (60). The entrance factor is also tested against the PNU database (Fig. 8). Since the proposed model for the entrainment fraction strongly underpredicts the experimental entrainment fraction obtained in the PNU test section (see Section 4.3), we set as $E_{\infty}$ the experimental value at the farthest location from the entrance ($z/D = 107$). Parameter $n = 1.15$ appearing the proposed entrance factor (equation (71)) and $m = 1.66$ in Eqs. (72) and (73) is set both for REGARD ($D = 0.24$ m) and PNU ($D = 0.04$ m) data.

For REGARD experimental conditions, the flow is almost fully developed around 6 or 8 diameters from the inlet, depending on the superficial gas velocity. The proposed entrance factor allows to qualitatively reproduce the entrainment fraction profile calculated from Eq.

(60), assuming a “half-annular” interfacial perimeter, and to better fit the experimental entrainment fraction at the locations $z/D = 3.8$ and $z/D = 5.7$ close to the inlet of the test section. Same trend is also obtained for the PNU experiment at the closest location ($z/D = 17.5$) from the entrance, regarding the experimental measurements. The development for the flow in terms of droplet entrainment, reached around at $z/D = 70$, is also well captured by the model.

The advantage of this approach is it may be applied for any entrainment model and do not require any model or assumption for the film/gas interface, but has some discrepancy at the beginning of the entrance with respect to the mass conservation Eq. (60) or the modified Neiss method (64).

## 5. Conclusions

Several models for the prediction of the droplet entrainment in horizontal pipe developed or validated for the system thermalhydraulics codes ATHLET, CATHARE, RELAP5, TRACE, and SPACE are reviewed. The empirical models (Steen-Wallis, Ishii-Mishima) for the prediction of the entrainment fraction for the two-fluid 6-equation models are challenged by a semi-empirical approach used by the three-field models, mostly derived from the Pan and Hanratty (2002), Lopez de Bertodano et al. (1997) and McCoy and Hanratty (1977) works for the estimation of the entrainment rate or the deposition rate. This approach can be adapted for the two-fluid model, such as in ATHLET code, under the assumption of equilibrium of the entrainment and depositions processes.

Some of these models are evaluated using this assumption against several experimental databases with different diameter and test conditions, from stratified flow regime without entrainment to annular flow regime with a important quantity of droplets. Since any separate effect tests in steam-water conditions corresponding to the reflooding phase during a LBLOCA exist in the literature, representative air-water experiments at low pressure have been selected. From the obtained results, three modifications for the improvement for the Carnevali and Fillion (2024) model, dedicated to the CATHARE code, are proposed. The two first modifications concern the deposition rate where the contribution of the gravity part of the model is modified, and the droplet diameter is governed by the Berna et al. (2015) correlation for the lowest pipe diameters. The entrainment rate model, based on the Pan-Hanratty approach using the maximum entrainment fraction concept, does not well predict the entrainment at low liquid Reynolds number. The maximum entrainment fraction relies on the onset of entrainment model. Among different models for its prediction, the correlation proposed by Lee et al. (2022b) is evaluated, allowing to significantly improve the calculated entrainment fraction for the Mantilla 2-inch tests. Different onset of entrainment models are evaluated against the Mantilla tests, showing that a model based on the Andritsos and Hanratty (1987) criterion used to predict the initiation of the large-amplitude waves may be adapted for the prediction of the initiation of the atomization. These new entrainment rate and deposition rate models based on the Carnevali-Fillion formulations, including the onset of entrainment model from Lee et al. and the modification of the deposition rate are evaluated against the experimental database. Performances of the Carnevali-Fillion model are improved against Mantilla 2-inch, Magrini, Williams data in stratified/annular flow conditions and REGARD data in stratified flow regime. However, entrainment fraction remains underestimated against Mantilla 6-inch and PNU experiments in stratified flow regime. This point requires further investigations in order to improve the predictions.

The proposed entrainment model, summarized in Table 3, dedicated to stratified and annular flow regimes in horizontal pipe at low pressure may be used in system code, both for the two-fluid model and the three-field model. The two-fluid formulation of the model, expressed as an entrainment fraction is directly derived from the models for
<figure><img src="image_17.png" /><figcaption>Fig. 8. Entrance effect factor (with $n = 1.15$ in Eq. (71); $m = 1.66$ in Eq. (72) or (73) gives very similar results) applied to PNU experiment.</figcaption></figure>

the estimation of the entrainment rate and deposition rate using a Pan-Hanratty approach, assuming equilibrium conditions.

Non-equilibrium effects appearing at the entrance of the pipe, where the flow is not fully developed, are taking into account using two methods. First method accounting entrance effects is derived from Neiss (2013) work, using directly the proposed models for the deposition velocity and the entrainment rate, and the droplet mass balance equation. On the REGARD experiment, this model gives results very close to those produced by the entrainment rate and deposition rate balance equation, under the assumption of droplet/gas slip ratio equal to 1. To complete the calculation of the droplet flowrate in the developing length, a good estimation of the gas/film interface is necessary. The second method is based on an entrance factor, modifying the entrainment fraction at equilibrium condition, adapted from Ishii and Mishima (1981) and Kataoka et al. (2000) formulations. The proposed entrance factors allow to reproduce the behavior of the experimental entrainment fraction profiles obtained in REGARD and PNU test sections, and qualitatively approximate the entrainment fraction deduced from the droplet mass balance equation and the first method in the case of REGARD flow conditions. This entrance effect model assumes conditions without any entrainment at the inlet of the considered pipe but sufficient to generate droplets. A more generic model should be developed and validated for any entrainment condition at the inlet of the pipe.

Several steps are planned or could be envisaged for the application of LOCA transients. An intermediate step is to validate the model against several transients performed in integral effect tests (IET), in steam-water conditions. The validation matrix may include BETHSY 6.7c test, corresponding to the reflood phase during a LBLOCA. BETHSY facility is a scaled-down model of a three-loop PWR 2775 MWth, at full scale for elevation at a scale 1/100 for volume and for power. The hot leg diameter *D* is set to 118 mm, following the Froude number criterion ($D = D_{\text{Reactor}}/100^{2/5}$), aiming at preserving “as possible phenomena relevant to countercurrent flow, transient regimes and stratified flow in horizontal parts” (BETHSY team, 1990; Petelin et al., 2007). BETHSY 6.7c transient was already simulated using the 3-field model implemented in CATHARE 3, but with older entrainment rate and deposition rate models based on vertical geometries (Valette et al., 2011). The IET validation may be completed by some of the UPTF transients (Weiss et al., 1994) dedicated to the simulation of the reflood phase. The UPTF facility is a real reactor geometry and is challenging regarding the diameter effect, since the proposed model is only validated for smaller pipe diameters, even if data of REGARD helped to improve the reliability regarding the diameter effects. However, in

our knowledge, droplets characteristics remains poorly known in very large diameter horizontal pipes and in steam-water conditions. If more and more mechanisms or effects are taken into account by the models, a part still relies on an empirical nature which poses an inherent and unavoidable scalability challenge. A dedicated experiment may help at completing the development, validation and the transposition of the models to reactor conditions.

### CRediT authorship contribution statement

**Philippe Fillion:** Writing – original draft, Visualization, Validation, Supervision, Software, Methodology, Formal analysis, Conceptualization. **Sofia Carnevali:** Writing – review & editing, Methodology, Conceptualization. **Adrien Rodriguez:** Writing – review & editing, Methodology, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

This work has been carried out in the framework of the Energies Division simulation program of the French Alternative Energies and Atomic Energy Commission (CEA).

### Data availability

The authors do not have permission to share data.

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