# Review of experimental models and findings on gas entrainment by vortex in hot plenum of sodium fast reactors

David Guenadou

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David Guenadou. Review of experimental models and findings on gas entrainment by vortex in hot plenum of sodium fast reactors. Nuclear Engineering and Design, 2025, 442, pp.114211. cea-05519760

HAL Id: cea-05519760

https://cea.hal.science/cea-05519760v1

Submitted on 19 Feb 2026

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<figure><img src="image_1.png" /></figure>

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Nomenclature

<table><tr><td>Cp</td><td>Caloric capacity, m<sup>2</sup>.s<sup>-2</sup>.K<sup>-1</sup> (J/K/kg)</td></tr><tr><td>g</td><td>Acceleration due to gravity, m.s<sup>-2</sup></td></tr><tr><td>H</td><td>Water head, m</td></tr><tr><td>L</td><td>Characteristic length, m</td></tr><tr><td>P</td><td>Pressure, kg.m<sup>-1</sup>.s<sup>-2</sup> (Pa)</td></tr><tr><td>Q</td><td>Flow rate, m<sup>3</sup>.s<sup>-1</sup></td></tr><tr><td>r</td><td>Radial component, m</td></tr><tr><td>T</td><td>Temperature, K</td></tr><tr><td>t</td><td>Time, s</td></tr><tr><td>V</td><td>Characteristic velocity, m.s<sup>-1</sup></td></tr><tr><td>v<sub>x</sub></td><td>Velocity component</td></tr><tr><td>β</td><td>Thermal expansion, K<sup>-1</sup></td></tr><tr><td>ρ</td><td>Density, kg.m<sup>-3</sup></td></tr></table>

<table><tr><td>v</td><td>Cinematic viscosity, m<sup>2</sup>.s<sup>-1</sup></td></tr><tr><td>σ</td><td>Surface tension, kg.s<sup>-2</sup> (N.m<sup>-1</sup>)</td></tr><tr><td>λ</td><td>Thermal conductivity, kg.m.s<sup>-3</sup>.K<sup>-1</sup> (W.m<sup>-1</sup>.K<sup>-1</sup>)</td></tr><tr><td>Γ</td><td>Circulation, m<sup>2</sup>.s<sup>-1</sup></td></tr><tr><td>ω</td><td>Vorticity, s<sup>-1</sup></td></tr></table>

Subscripts

<table><tr><td>r</td><td>Radial component</td></tr><tr><td>Θ</td><td>Tangential component</td></tr><tr><td>g</td><td>Gas</td></tr><tr><td>l</td><td>Liquid</td></tr></table>

Superscripts

* Ratio between model and reactor parameters

<figure><img src="image_2.png" /><figcaption>Fig. 1. Pool-type design of a NSSS SFR (Beao, 2009).</figcaption></figure>

(Chellapandi and Velusamy, 2015).

The presence of gas above the free surface in the hot plenum can have significant safety implications for the reactor. In 1989 and 1990, the PHENIX reactor experienced four emergency shutdowns, one of which was attributed to a gas bubble passing through the core (Guidez and Prèle, 2017). This is critical because the presence of gas affects the neutron properties of sodium. As a result, the safety authorities of that time recognized the need to investigate this issue, which had not been identified during the reactor's original design phase. Since then, numerous studies have been conducted globally to assess the risks associated with gas entrainment. To address this challenge, various experimental setups and models of different scales have been developed. While this article does not aim to provide an exhaustive overview, it highlights some of the most significant research efforts in the field of gas entrainment relevant to SFRs.

## 2. Gas entrainment issue and origin

To ensure the reactor operates under normal conditions, the presence of gas in the core must be avoided, as the passage of large gas pockets through the fissile zone poses significant nuclear safety risks. Indeed, an increase in void fraction results in a positive reactivity effect, which can lead to a power spike. Additionally, the heat transfer coefficient is drastically reduced, potentially causing inadequate core cooling, which could lead to fuel damage. Furthermore, any argon that passes through the fissile part of the core will become activated and radioactive due to the intense neutron flux in that area. This could affect the accuracy of detecting fission products in the primary coolant related to fuel assembly leaks. Finally, the presence of gas is detrimental to the operation of the primary pumps and can cause damage to them.

In normal operation, four main sources of gas entrainment have been identified (Guidez and Prèle, 2017):

* Sodium can become emulsified with argon around the shaft of the primary pumps (Khan et al., 2020). Due to the high temperatures, the lower bearing of the shaft is designed to be hydrostatic, lubricated by sodium. As a result, gas can be entrained at the rotating free surface.

* To avoid stress on the outer vessel, an overflow system is installed at the top of the cold plenum, as shown in Fig. 2. The system operates by circulating a flow along the vessel, fed by leaks at the base of the assemblies, while maintaining a constant level through continuous overflow. The sodium flow reversal at the overflow leads to gas entrainment in the cold plenum.

* The solubility of argon in sodium increases with temperature, meaning that argon dissolved in the hot plenum tends to be released in the cold plenum (Yamaguchi et al., 2011). This can result in the formation of argon pockets in the lower part of the reactor.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Overflow at the top of the cold plenum.</figcaption></figure>
* The hot plenum, with its free surface covered by argon, may entrain gas through the sodium intake at the IHXs, as illustrated in Fig. 3 (Chellapandi and Velusamy, 2015; Saha et al., 2013; Tenchine et al., 2014). After passing through the primary pumps, this gas may accumulate in the lower parts of the reactor, forming a gas pocket. Various gas entrainment mechanisms are possible depending on the condition of the free surface. Five main types are identified and sketched in Fig. 3. As illustrated in Sketch 1, shearing stress at the surface due to the difference in horizontal velocity between the sodium and argon may cause gas entrapment (Patwardhan et al., 2012). A related mechanism, in Sketch 5, induces gas entrainment due to the high velocity jets coming out from the ACS. In Sketch 3, bubbles trapped by the breaking of sodium ripples may be transported into the flow entering the IHXs (Patwardhan et al., 2012; Satpathy et al., 2013; Yao et al., 2021). On Sketch 4, a droplet impacting the free surface may trap gas bubbles upon impact (Castro et al., 2016). These four scenarios lead to a relatively slow accumulation of gas. On Sketch 2, surface vortices can form depending on the hydraulic conditions and in the most critical cases, create a gas column between the free surface and the IHX inlet, or cause bubble detachment from the vortex tips, as illustrated in Fig. 3.

Since the gas entrainment by vortex is the most severe, it is essential to thoroughly investigate its specific characteristics.

### 3. Phenomenology of the gas entrainment by vortices

The phenomenology of gas entrainment by vortices is relatively complex due to its two-phase nature. The feedback from experiments conducted at the CEA has shown that vortices only occur in certain high-risk areas where both vorticity and downward velocity are present. Vorticity is generated by the wake behind a component or by the interaction of two flows. In the hot plenum, sodium flows from the outer edge toward the center, bypassing the IHXs and various submerged components. Due to the vorticity created at the walls of these components, vortices are convected into the flow, as shown in Fig. 4. The convergence of flows circumventing the submerged components at the center of the hot plenum, close to the ACS, also generates vorticity. Conditions favoring downward velocity are particularly created at the IHXs due to pump suction and near the ACS due to the flow reversal along the wall toward the bottom. The characteristics of downward velocity and vorticity are highly dependent on the geometry of the hot plenum and the operational conditions of the reactor.

The size and type of vortex significantly influence the consequences of gas entrainment, making it essential to establish a classification system for them. It will be helpful to determine in the experiments the domain of gas entrainment. Various classifications exist for vortices and, more broadly, for free surfaces, each varying in detail. In this study, we will focus on two classifications presented in Table 1, which are derived from the works of (Eguchi et al., 1994) and (Tenchine et al., 2014).

<figure><img src="image_4.png" /><figcaption>Fig. 3. Gas entrainment at the IHX primary sodium inlet.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4. Flow on the surface surrounding the IHX creating downstream vorticity.</figcaption></figure>

These two classifications are relatively similar. However, the one proposed by (Eguchi et al., 1994) is more detailed, as it focuses broadly on the state of the free surface, while (Tenchine et al., 2014) only consider vortices. The classification by (Eguchi et al., 1994) includes subcategories that show the process of free surface modification, whether caused by vorticity (subcategory 1), surface depression (sub-category 2), or wave breaking (subcategory 3). The classification is ordered according to the degree of surface modification and gas entrainment. Type 1 corresponds to a flat surface. In Type 2, the surface is simply deformed, while Type 3 involves downward suction, drawing bubbles or dye downward. However, this suction is limited to the surface area. (Tenchine et al., 2014) introduce a complementary vortex type, 3', compared to (Eguchi et al., 1994), where the downward suction is stronger, allowing particles to be conveyed downward, though still limited to the region near the surface. In Type 4, bubbles are transported from the surface to the inlet of the IHX, corresponding to a prohibited operating condition for SFRs. (Tenchine et al., 2014) also add the free surface type 5, representing the most critical case, where a gas core forms between the surface and the IHX inlet. In this extreme configuration, a large quantity of gas is directed into the lower parts of the reactor.

Research and experiments on gas entrainment primarily focus on vortex types 3 to 5, with type 3 representing the critical threshold that must not be exceeded for the safe operation of SFRs. Identifying the operating conditions that lead to the formation of this vortex type is essential for SFR development. Due to the complexities and challenges of conducting experiments with sodium, numerical simulations offer a promising alternative for advancing these studies.

### 4. Challenges of numerical modeling of vortices and the need for experimental studies

Numerical simulations allow for the study of a problem by understanding its phenomenology through the calculation of various fields, such as temperature, velocity, or other physical data related to the case under investigation. The parameters of the problem, including geometry and boundary conditions, can be adjusted to understand their impact and to find solutions to the issue at hand. However, to achieve these goals, the computational codes must incorporate appropriate and validated models to ensure consistent results. In the case of gas entrainment, particularly for SFRs, the major challenges for numerical simulation lie in modeling the liquid/gas interface within complex and large geometries. In order to capture small-scale phenomena, such as bubbles just a few millimeters in diameter, the mesh size in the computational domain must be of the same order of magnitude. However, since the computational domain can range from meters (in test models) to tens of meters (in reactors), this requires an extremely large number of mesh cells, which directly impacts computation time. Additionally, multiphase models are complex and may be misapplied, making it essential to critically analyze and scrutinize their results. For these reasons, fully simulating gas entrainment throughout the entire primary circuit of an SFR remains currently challenging to achieve. However, rapid progress in the field of simulation gives hope that this could be achieved in a few years. Numerous successful simulations have been conducted usually in smaller geometries as illustrated in Fig. 5 (Amar et al., 2024; Cristofano
<table><caption>**Table 1**
Classification of the free surface and vortices based on their characteristics.</caption><thead><tr><th>Type</th><th>Classification of (Eguchi et al., 1994)</th><th>Classification of (Tenchine et al., 2014)</th><th>Visual example</th></tr></thead><tbody><tr><td>1</td><td></td><td></td><td></td></tr><tr><td>2</td><td>1</td><td></td><td></td></tr><tr><td>2</td><td></td><td></td><td></td></tr><tr><td>3</td><td>1</td><td></td><td></td></tr><tr><td>2</td><td></td><td></td><td></td></tr><tr><td>3</td><td></td><td></td><td></td></tr><tr><td>3'</td><td>-</td><td></td><td></td></tr><tr><td>4</td><td>1</td><td></td><td></td></tr><tr><td>2</td><td></td><td></td><td></td></tr><tr><td>3</td><td></td><td></td><td></td></tr><tr><td>5</td><td>-</td><td></td><td></td></tr></tbody></table>

et al., 2014a; Lee et al., 2021; Lee and Park, 2022; Sakai et al., 2008; Yao et al., 2023, 2021). Most of them aim to simulate an actual experiment to assess the parameters of the calculation, such as the turbulence of the two-phase flow models. Generally, they successfully correlate experimental measurements with velocity, vortex locations, and depth. For phenomenological understanding, these simulations also help identify the key phenomena triggering vortex formation, enabling the development of simpler models that can be integrated into computational codes

to assess the risk of gas entrainment based on reactor operating conditions and geometries. Since experiments provide data for validating simulations, and simulations enhance the understanding of vortex formation, both complement each other.

An intermediate approach combining numerical modeling and mock-ups has been undertaken by the JAEA teams. Predicting gas entrainment using a general theoretical model does not seem feasible due to the complexity of the involved phenomena. Therefore, the authors of (Matsushita et al., 2025; Sakai et al., 2008) have developed a gas entrainment criterion based on the Burgers vortex model, with parameters determined experimentally. The methodology proposed by (Sakai et al., 2008) involves performing monophasic simulations with a slip boundary condition at the liquid/gas interface and applying the gas entrainment criterion to the numerically determined velocity fields. This approach significantly reduces computation time and allows for numerous studies. (Bhatia et al., 2021) implemented this methodology in TRIO_CFD, and the calculations showed quite good agreement regarding the locations of the vortex appearances.

Therefore, using physical models to study vortex-induced gas entrainment and obtain experimental results appears essential.

## 5. Water/sodium similitude

Although the state-of-the-art in gas entrainment experiments for SFRs includes work carried out with the actual coolant, i.e., liquid sodium, most studies have been conducted using a simulant fluid through a similarity-based approach. Indeed, experiments involving sodium are complex and costly due to the high temperatures required, as well as the risk of inflammation associated with the contact of liquid sodium droplets with air in case of leakage. Moreover, instrumentation and visualization of phenomena are difficult due to the opaque nature of sodium. In the past, many experiments were therefore conducted using water instead of sodium, as water is inexpensive, transparent for measurement purposes, and easy to handle. Part of the research has been carried out using water mock-ups. However, using a simulant fluid requires a rigorous approach. The application of similarity theory allows for studies to be conducted on scaled-down models using a simulant fluid. To ensure the representativeness of the phenomena and the validity of the studies, the dimensionless numbers specific to both the reactor and the model must be identical. The main dimensionless numbers that characterize fluid behavior in the primary circuit (Murakami et al., 2015; Padmakumar et al., 2007) are listed in Table 2. These numbers are derived from the application of the Buckingham Pi theorem, using the dimensions and characteristic physical quantities of the primary circuit.

The experimental plan and operating conditions are based on the conservation of dimensionless numbers calculated from the phenomena involved and the geometry. Considering that the hot plenum is almost isothermal in the areas affected by gas entrainment, i.e., near the IHXs, thermal effects can be neglected. As a result, the Richardson and Peclet dimensionless numbers do not influence the flow, and only the Reynolds, Froude, Euler, and Weber numbers need to be considered under these conditions. However, it is generally impossible to preserve all these dimensionless numbers simultaneously, as their expressions differ (Grewal and Gluekler, 1982). For instance, the Reynolds number scales linearly with length, whereas the Froude number is inversely proportional to the square root of the length. The dimensionless numbers to be preserved must therefore be carefully selected. This choice is based on expert judgment or on the analysis of the governing equations for the phenomenon under investigation. Regarding the Reynolds number, it is assumed to be maintained if the flow in both the model and the reactor share the same regime (either laminar or turbulent). In studies of free surfaces, (Eguchi et al., 1994) have demonstrated that preserving only the Froude and Weber numbers is sufficient to accurately capture the phenomenology. However, this imposes a model scale of 5/8, which is relatively large and leads to expensive and complex experiments. To
<figure><img src="image_6.png" /><figcaption>Fig. 5. Comparison of numerical and experimental results in a channel setup (Lee and Park, 2022).</figcaption></figure>

<table><caption>**Table 2**
Inventory of the dimensionless numbers characteristic of the primary circuit.</caption><thead><tr><th>Dimensionless number</th><th>Expression</th><th>Interpretation</th></tr></thead><tbody><tr><td>Reynolds</td><td>Re = <sup>VL</sup>&frasl;<sub>ν</sub></td><td>Inertia forces</td></tr><tr><td>Weber</td><td>We = <sup>ρV<sup>2</sup>L</sup>&frasl;<sub>σ</sub></td><td>Viscous forces</td></tr><tr><td>Froude</td><td>Fr = <sup>V</sup>&frasl;<sub>&radic;(gL)</sub></td><td>Inertia forces</td></tr><tr><td>Richardson</td><td>Ri = <sup>gρΔTL</sup>&frasl;<sub>V<sup>2</sup></sub></td><td>Surface tension forces</td></tr><tr><td>Euler</td><td>Eu = <sup>ΔP</sup>&frasl;<sub>ρV<sup>2</sup></sub></td><td>Inertia forces</td></tr><tr><td>Peclet</td><td>Pe = <sup>ρC<sub>p</sub>VL</sup>&frasl;<sub>λ</sub></td><td>Gravity forces</td></tr><tr><td></td><td></td><td>Inertia forces</td></tr><tr><td></td><td></td><td>Pressure forces</td></tr><tr><td></td><td></td><td>Inertia forces</td></tr><tr><td></td><td></td><td>Convection effects</td></tr><tr><td></td><td></td><td>Conduction effects</td></tr></tbody></table>

reduce costs, many studies opt to conserve only the Froude number and allow for some distortion in the Weber number by reducing the size of the model, while still keeping it at a sufficiently large scale. This approach minimizes surface tension effects, which tend to dominate in smaller-scale models (Banerjee et al., 2013).

## 6. Methodology of the experimental study of the gas entrainment

The gas entrainment issue can be studied experimentally through two complementary approaches, depending on the type of model used. The first approach focuses on investigating the phenomenology in small-scale setups specifically designed to study gas entrainment. These setups do not replicate the actual reactor geometry but are optimized to facilitate experimentation, particularly with respect to boundary conditions and instrumentation. This allows for more precise measurements and the ability to conduct detailed parametric studies.

On the other hand, integral models are designed for broader, multipurpose studies that include gas entrainment as one of the key aspects. These models are typically constructed as part of a reactor prototype development program to validate the chosen technological concepts and to provide a comprehensive database of results. These data are used to validate simulation models and calculation codes, particularly in relation to safety demonstrations. These models are either representative of the entire reactor vessel or, in some cases, just the hot collector section.

The following sections present the current state of research and key findings from both of these experimental approaches.

## 7. Phenomenological studies on analytical models

Analytical models include setups that focus on the phenomenology of gas entrainment or vortex formation. The CEA implemented the VEDETTE and BANGA models (Moudjed et al., 2016) during the EFR and ASTRID projects, but many other installations exist internationally. These setups generally consist of a channel or cylinder with suction at the bottom of the tank as illustrated in Fig. 6. Most experiments have been conducted using water as a simulating fluid (Baum and Cook, 1975; Caruso et al., 2014; Cristofano et al., 2014b; Cristofano et al., 2016; Ezure et al., 2008, 2011, 2019; Kimura et al., 2010; Li et al., 2008; Moudjed et al., 2016; Patwardhan et al., 2012; Takahashi et al., 1988; Taştan and Yildirim, 2010), but some were carried out with sodium (Baum and Cook, 1975; Kimura et al., 2010), or even Freon, White Spirit (Baum and Cook, 1975), or a water/xylene mixture (Patwardhan et al., 2012) to assess the influence of physical properties. The initiation of gas entrainment is achieved in different ways depending on the models. The simplest method involves rotating the liquid with a propeller (Patwardhan et al., 2012), an example of which is shown in Fig. 7a. This allows results to be obtained in a simple system but one that is not very representative of the configuration of SFRs. Another technique involves drawing liquid from the bottom of a tank (see Fig. 7b). Some setups feature a bypass of the suction, ensuring a constant level and the ability to vary suction velocity conditions. Others use multiple fluid inlets to create asymmetrical conditions in the tank. These models are representative in cases of gas entrainment by surface ripples or surface depression (subcategories 1 and 2 of the (Eguchi et al., 1994) classification from Table 2), but they cannot properly reproduce vortices due to

<figure><img src="image_7.png" /><figcaption>Fig. 6. BANGA analytical model.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 7. Different types of analytical models for the study of gas entrainment –a. by rotating a propeller (Patwardhan et al., 2012) – b. by injecting liquid from the bottom of the tank (Cristofano et al., 2014b) – c. in a straight channel (Moudjed et al., 2016).</figcaption></figure>

the lack of control over surface vorticity. The third method uses a straight channel with a free surface, featuring suction at the bottom (see Fig. 7c) and water supply on half of one side to create a sheared flow. The inlet has a wall that generates and controls vorticity by using different wall shapes to represent the flow downstream the IHXs. This type of analytical model better accounts for the phenomena involved in vortex formation in sodium-cooled fast reactors (SFRs).

The models using water are made of transparent materials, allowing most results to be obtained through visual observation of the vortex shapes according to classifications that are more or less aligned with those presented in Table 2. The studies by (Li et al., 2008) and (Ezure et al., 2011) are based on a duration of 10 min to obtain a satisfactory number of vortices for statistical analysis. Some works utilize high-speed cameras, and their results are analyzed using imaging techniques (Guenadou et al., 2021; Taştan and Yildirim, 2010). This allows for the acquisition of more detailed data, such as the size of the entrained bubbles, the volume of gas entrained, or the displacement of the vortex centers. Most of the time, authors consider that gas entrainment occurs when bubbles or gas reach the outlet of the model. In the case of experiments using sodium, where visual observations at the outlet are not possible, the identification of gas entrainment is based on an electromagnetic process.

Two types of studies can be distinguished regarding the analytical models. The first focuses on determining a mapping of gas entrainment based on the physical and geometric parameters of the system, aiming to establish correlations on the thresholds of gas entrainment from the characteristic data of the system (Baum and Cook, 1975; Caruso et al., 2014; Cristofano et al., 2014b; Ezure et al., 2011; Kimura et al., 2010; Moudjed et al., 2016; Patwardhan et al., 2012; Takahashi et al., 1988; Taştan and Yildirim, 2010). The other type aims to analyze the flow around the vortices to construct predictive models for the onset of gas entrainment (Cristofano et al., 2016; Ezure et al., 2019, 2011, 2008; Li et al., 2008; Moudjed et al., 2016).

Mappings are established based on the parameters of the system, namely the operating and geometric conditions. The limits of gas entrainment are determined according to these parameters, but the mappings are most often represented using the characteristic dimensionless numbers of the system. In the case of the study of gas entrain-

ment, as mentioned in the previous chapter, the dimensionless numbers to consider are the Froude (Fr) and Weber (We) numbers (Baum and Cook, 1975; Caruso et al., 2014; Patwardhan et al., 2012; Taştan and Yildirim, 2010). Some studies include the Reynolds number (Caruso et al., 2014; Taştan and Yildirim, 2010) to characterize the influence of turbulence or the Strouhal number, which defines the frequency of vortex shedding downstream an obstacle. When the model imposes a rotation of the fluid, some authors additionally introduce the rotational Froude number (Ro) (Takahashi et al., 1988), which is defined based on circulation and whose expression is presented in Equation (1). The circulation $\Gamma$ can be calculated from the vorticity field using Equation (2), where the surface S is defined by the closed contour of path C.

$$Ro = \frac{\Gamma}{gH^3} \qquad (1)$$

$$\Gamma = \int_C Vdl = \int_S \omega ds \qquad (2)$$

The importance of the rotational Froude number in studying vortices is also highlighted in the theoretical approach by (Holtorff, 1964). By integrating the vortex model from (Einstein and Li, 1955) into the Navier-Stokes equations, the vortex shape can be calculated. (Holtorff, 1964) provides a dimensionless solution that incorporates the key parameters defining the vortex profile. This leads to the conclusion that both the Froude and Reynolds numbers of the rotational movement, as defined respectively in Equation (3) and (4), must be preserved. Although the formulations of Ro and Fr are slightly different, both account for the tangential velocity component in the Froude number calculation. Therefore, in a similarity-based approach, conserving these numbers seems more appropriate. Furthermore, the authors emphasize that it is sufficient to preserve only the rotational Froude number (Fr) in the vortex's outer region, whereas both Fr and the rotational Reynolds number (Re) must be conserved at the vortex center. This is consistent with the assumption of negligible viscosity in the outer region of the vortex (Einstein and Li, 1955).

$$Fr_r = \frac{v_\theta}{\sqrt{gr}} \qquad (3)$$
$$Re_r = \frac{v_r r}{\nu} \qquad (4)$$

The work of (Taştan and Yildirim, 2010) establishes a threshold value for the Reynolds (Re), Weber (We), and Froude (Fr) numbers beyond which they no longer have an influence on gas entrainment. These results were obtained in a channel-type model, as shown in Fig. 7c, by varying the water depth and velocity conditions. At a fixed ratio of inlet velocity to suction velocity, an increase beyond a certain threshold of Re, We, or Fr has no effect on the water height triggering the gas entrainment. A related finding, obtained by the authors of (Moudjed et al., 2016), confirms this trend. The size of the vortices varies depending on the inlet/suction conditions until it reaches a plateau. The study by (Taştan and Yildirim, 2010) indicates that the threshold value for We, Fr, and Re is solely dependent on velocity, meaning that when this threshold is exceeded, inertial effects dominate over viscosity and surface tension effects in the gas entrainment mechanism. This is typically observed for the Reynolds number, where at a specific value, the turbulence effects overcome the molecular viscosity.

The mappings used to identify the limits of gas entrainment and differentiate between the various types of vortices appearing at the surface (Caruso et al., 2014; Cristofano et al., 2014b) can be instrumental in establishing correlations regarding the onset of gas entrainment based on system parameters. Numerous studies propose correlations to calculate the critical water depth for the transition to gas entrainment as a function of dimensionless numbers or system characteristics, referred to as the “critical height.” For instance, the work of (Takahashi et al., 1988), conducted in a cylinder with vertical suction, establishes two semi-empirical correlations based on the rotational number (Ro) or the Froude number (Fr). These correlations consider either the center of the vortex reaching the suction inlet (derived from the equation for vortex shape) or the scenario where bubbles achieve sufficient velocity (through a force balance) to descend to the suction. Remarkably, these semi-empirical correlations align closely with experimental results, allowing for a precise relationship to be proposed, accurate to ± 40%, linking the critical height with Ro and Fr. Moreover, the influence of the fluid’s physical properties is examined by (Baum and Cook, 1975) in a cylindrical model with suction at the bottom, akin to that shown in Fig. 7b. They establish an empirical correlation based on the Weber (We) and Froude (Fr) numbers, utilizing experimental results that vary the flow rate, water height, and fluid nature. Further contributions from (Caruso et al., 2014) and (Cristofano et al., 2014b) lead to the development of correlations defining the transition boundaries between different types of vortices at the surface, in accordance with the classification outlined in Table 2. These studies were conducted in the model depicted in Fig. 7b, with suction at the bottom, and the resulting relationships are expressed in terms of the Reynolds (Re), Weber (We), and Froude (Fr) numbers. Notably, at high Re, surface disturbances have a significant influence on the reduction of gas entrainment. Some authors also propose correlations for the critical height that rely solely on the Froude number (Li et al., 2008; Patwardhan et al., 2012). However, many studies emphasize that the established correlations are specific to the characteristics of the model, particularly its geometry (Caruso et al., 2014; Cristofano et al., 2014b). Consequently, these correlations are only applicable within the context of a specific system, and this methodology does not allow for the formulation of a general relationship concerning gas entrainment applicable to reactor scenarios. Despite this limitation, these analytical models are valuable for investigating the phenomenology of surface flow, enabling the identification of parameters that influence gas entrainment and facilitating the construction of predictive models. Since one of the mechanisms generating vortices is linked to the creation of vorticity, several authors focus on examining the influence of walls (Moudjed et al., 2016; Taştan and Yildirim, 2010). These studies are conducted in a straight channel with vertical suction at the bottom (see Fig. 7c). (Taştan and Yildirim, 2010) demonstrate that when a wall is positioned near the suction tube, friction against it tends

to reduce vortex formation by diminishing the circulation value at the free surface. Additionally, the authors note that turbulence also contributes to a decrease in the occurrence of gas entrainment. The investigations by (Moudjed et al., 2016) focus on the impact of a fixed obstacle on a wall situated upstream of the suction tube in the channel (see Fig. 7c). Their findings indicate an increase in vortex formation and the volume of gas entrained when the obstacle is sufficiently large, thereby affirming the importance of model geometry on gas entrainment dynamics. Furthermore, it can be deduced from these studies that vorticity plays an important role in the vortex formation process, as it is favored by wall shear. An important finding from the work of (Baum and Cook, 1975) and (Kimura et al., 2010) reveals that the onset of gas entrainment occurs earlier in experiments using water compared to those using sodium. This indicates that the conditions for vortex formation in water models are enveloping, allowing for the assurance of operating conditions without gas entrainment when sodium is used, based on the results obtained with water. This experimental observation can be explained using the dimensionless numbers characteristic of the system. As explained in Section 5, only the Froude and Weber numbers impact gas entrainment. Since the Froude number is independent of the fluid properties, only the Weber number needs to be considered. By using the values of the density and surface tension of water and sodium, we can write that $We_{water} \approx 2.7 We_{Na}$. This means that, with water, the inertia effects, which tend to enhance the vortex size, are higher, or the surface tension effects, which tend to flatten the vortex tip, are lower. Both lead to a deeper vortex. This justifies, in analytical experiments, the use of water as a simulant fluid for studies on vortices in SFRs.

Several studies (Cristofano et al., 2016; Ezure et al., 2011; Hirakawa et al., 2018; Li et al., 2008) have investigated the measurement of the velocity field around different types of vortices according to the classification presented in Table 2. These measurements were conducted using Particle Image Velocimetry (PIV) in horizontal planes located 10 mm below the surface for the work of (Ezure et al., 2011) and between 25 and 45 mm for (Cristofano et al., 2016). Since the position of the vortex is not constant within the PIV measurement plane, (Cristofano et al., 2016) calculated the time-averaged velocities by realigning the center of the vortex at each measurement instance. (Cristofano et al., 2016) also observed measurement artifacts near the center of the vortex due to lens effects resulting from the curvature of the liquid-gas interface. In a cylindrical coordinate system with the vortex center as the origin, all studies show that the tangential velocity increases linearly until it reaches a maximum value, then decreases, with this maximum located at the vortex radius, which is approximately equal to the radius of the suction nozzle. Regardless of the type of vortex, whether with or without gas entrainment, the profile of the tangential velocity retains a similar shape. As for the radial velocity, it is directed towards the center of the vortex and also exhibits a maximum, just like the tangential component. Its profile is consistent across different types of vortices, but it is dependent on the measurement depth, becoming rather flat at the surface. According to (Cristofano et al., 2016), theoretically, in a vortex, circulation increases linearly up to a certain radius and then stabilizes to a constant value in the outer regions. The calculation of circulation based on the velocity measurements from (Cristofano et al., 2016) aligns with theoretical values for planes close to the surface but shows a decreasing profile in the outer regions as measurements approach the suction nozzle. (Cristofano et al., 2016) propose an approximation of the tangential velocity profile using the Rankine model (Rankine, 1877; Rott, 1958) and the radial velocity profile by the solution of a potential flow in the simplified case of a siphon. These models correlate well with experimental results in the outer regions of the vortex (far from the center) and in areas close to the surface for tangential velocity, as well as at greater depths for the radial component.

In light of these studies, it is evident that the results obtained from analytical models are geometry-dependent and cannot be directly applied to nuclear reactor safety studies. These models are primarily useful for understanding the phenomena involved in gas entrainment.
However, reduced-scale models that represent the actual geometry are essential for studying gas entrainment and validating design options.

## 8. Integral models

Before the 2000 s, experimentation was highly developed because the capabilities of numerical tools to account for all the phenomena present in the hot collector were limited. For this reason, many models were primarily developed as part of European, Japanese, Russian, and Indian nuclear programs from the late 1980 s onwards. These models are representative of the upper plenum or a sector of this one.

The design rules for experiments using simulant fluids are explained by (Bomelburg, 1968). The choice of scale is crucial, as it involves a balance between maintaining the prototype's geometry and the economic cost of manufacturing and operating the model. If the scale is too small, errors may arise due to more pronounced boundary layer effects. (Tenchine, 2010) suggests using a scale larger than 1/5 and, if possible, conducting experiments on a section of the geometry based on its symmetries, if they exist. For such sector models, the IGCAR model (Lijukrishnan, 2022), shown in Fig. 8, is an example, built at a 5/8 scale and representing a 90° sector of the actual geometry. It aimed at assessing the efficacy of mitigation devices regarding gas entrainment. Other authors, such as (Kimura et al., 2008), have also developed models at this scale with a 90° sector. However, the velocity conditions at the outer walls closing the sector were imposed based on extrapolated results from a smaller-scale model. This approach enables a more realistic representation of the flow, as it ensures that the boundary conditions more accurately reflect the expected flow behavior in the integral system. The scale of the model must also be large enough to avoid distorting the system's characteristic dimensionless numbers. However, the literature reports small-scale models that allow for preliminary studies at a lower cost. For all these reasons, integral models require considerable design effort and are expensive to build and operate.

Most models are made of transparent materials to visualize the flow (Eguchi et al., 1994; Kimura et al., 2008), but some have been constructed from steel due to their large size, for structural strength considerations, and to reduce manufacturing costs (Banerjee et al., 2013, 2007). However, they still include windows to allow for optical measurements. Examples of such models include the MICAS model, built

from PMMA, and an IGCAR facility, made of steel, as shown in Fig. 8.

Certain components with highly complex geometries can be challenging to represent accurately at reduced scale. In such cases, it may be necessary to simulate these components by replicating only their effects on the flow, rather than their exact geometry. This approach is particularly relevant for two key components in the reactor vessel: the core and the ACS (Control Rod Drive Mechanism). In the reactor, the core consists of hexagonal fuel assemblies, which are separated by gaps that allow sodium to flow between them. However, at a smaller scale, accurately representing these gaps becomes problematic, as they would become too small, and the hydraulic conditions of the flow would no longer be properly maintained. For studies focused solely on the flow characteristics within the vessel, several researchers simplify the core outlet by using a perforated plate with cylindrical holes that match the layout of different zones based on the type of assembly. The core is generally divided into three or four sections to represent the assemblies mappings, with the entire system being fed by a high-flow hydraulic loop. The flow rates in the different core zones are independently controlled using valves regulated by flow meters, or by employing pre-calibrated orifices. As for the ACS, scaling down the model complicates the representation of the control rods inserted into guide tubes. The pressure losses induced by these tubes, which regulate the flow through the ACS, can be replicated in reduced-scale models by incorporating appropriately designed orifices. These orifices simulate the flow conditions and pressure drop, ensuring that the hydraulic behavior of the ACS is accurately reproduced even at a smaller scale.

The calculation of operating conditions is based on the conservation of one or more dimensionless numbers relative to the real case. Most experiments conducted to study the free surface and vortices use the Froude number, sometimes s supplemented by the Weber number. It has been established in section 7 that this methodology is not the most suitable, as the rotational Froude number is better adapted for studying vortices specifically. This methodology is suitable for analytical model because the flow behavior can be tailored to fit the required tangential velocity. However, in practice, this number is rarely used in studies involving integral models. Since it takes the tangential velocity around the vortices as the reference parameter, it is very difficult to determine this value prior to experimentation in order to establish the operating conditions. Moreover, the rotational Froude number ($Fr_r$) pertains

<figure><img src="image_9.png" /><figcaption>**Fig. 8.** Example of models made from different materials – MICAS on the left in PMMA (Guenadou et al., 2021), and an IGCAR facility on the right in steel (Vinod, 2022).</figcaption></figure>
specifically to vortex characteristics and does not account for other phenomena present in integral models, such as wall friction or shear flow. For these reasons, the conventional Froude number is typically used in integral models to study free surface behavior and gas entrainment. By analyzing the phenomena involved in gas entrainment in integral models, we can attempt to justify the appropriate use of the conventional Froude number. Based on a simple force balance on a bubble with a given volume, it is assumed that gas entrainment occurs when the momentum force exceeds the buoyancy force. In terms of energy, this implies that the kinetic energy available at the inlet of the IHX must be greater than the potential energy required to move the gas bubble from the surface to the IHX inlet, as expressed by Equation (5).

$$K \cdot \rho_l V_{\text{IHX}}^2 > (\rho_l - \rho_g) g H \quad (5)$$

With $V_{\text{IHX}}$, the average entry velocity at the IHX window, and H, the distance between the free surface and the top of the IHX window, with K as a dimensionless constant. This coefficient reflects the fact that not all the kinetic energy at the IHX inlet is transferred to the bubble, as the velocity of the bubble is inferior to or equal to the velocity at the IHX inlet. The parameter K may be interpreted as accounting the dragging effect. Considering that water has a much higher density than air, the gas density can be neglected in the right term of Eq. (5). Based on this, gas entrainment can be expressed by Equation (6).

$$1/\sqrt{K} < V_{\text{IHX}}/\sqrt{gH} = Fr \quad (6)$$

In this form, Equation (6) states that gas entrainment begins once a threshold value of the Froude number is reached, calculated based on the geometric and operational data of the model. The operational conditions which preserve the same free surface and vortex phenomenology, are determined by maintaining the same Froude number as in the reactor case, as expressed by Equation (7).

$$Fr^* = V_{\text{IHX}}^* / \sqrt{H^*} = Fr_{\text{MODEL}} / Fr_{\text{REACT}} = 1 \quad (7)$$

The velocity can be expressed in terms of the flow rate and the inlet surface area of the IHX. Given that all length ratios between the model and the reactor are equal, as the models are typically constructed with geometric similarity, the experimental conditions can be formulated by Equation (8).

$$Q_{\text{MODEL}} = Q_{\text{REACT}} L^{2.5} \quad (8)$$

Studying the flow behavior is essential to understand the phenomenology of the gas entrainment. This requires measuring accurately the velocity in various locations of the experimental model. However, instrumentation is primarily challenging due to the increased geometric complexity, which reduces the accessibility for placing sensors. However, it remains significantly simpler than in experiments involving sodium, as a wide range of measurement techniques are available for water and are generally more precise than those used for sodium (Bomelburg, 1968). The key parameters to measure when studying flow within the hot plenum are fluid velocity and temperature.

Larger-scale models facilitate the installation of instrumentation. When optical access is available, as in models made from transparent materials, velocity is typically measured using laser velocimetry systems such as Laser Doppler Anemometry (LDA) or Particle Image Velocimetry (PIV) (Eguchi et al., 1994; Kimura et al., 2008). PIV is particularly effective for capturing the velocity around vortices by taking measurements in horizontal planes just below the surface, similar to what is done in analytical models (Guenadou et al., 2021). In steel models, velocity measurements are performed using ultrasonic techniques or microturbines (Banerjee et al., 2013, 2007), though these are less effective due to lower precision and intrusiveness, respectively. Vortex characterization at the free surface is conducted using various methods, ranging from simple to advanced. The most basic method involves

operators counting vortices over a set period (usually 10 min) (Banerjee et al., 2013; Eguchi et al., 1994). However, this method lacks precision, as it depends on human observation, which can vary between or even during experiments. Another technique uses an optical probe to measure vortex occurrence and depth, relying on changes in the refractive index between air and water. However, this method only provides localized measurements along a line. Finally, in some models, installation of windows at the outlet of the hydraulic circuit is useful for assessing the gas entrainment onset (Banerjee et al., 2013, 2007). This review highlights that the current methods used to analyze surface phenomena exhibit one or more limitations. In response, new developments are underway at CEA to accurately measure vortex shape and behavior using an array of cameras.

In most studies, gas entrainment and vortex formation are characterized based on the evolution of the Froude number (Fr). However, it is important to note that the definition of the characteristic length used to calculate this parameter is not consistent across different studies. Some studies use the water height in the vessel as the characteristic length (Kimura et al., 2008), while others use the diameter of the fluid inlet (Eguchi et al., 1994). All studies investigating the influence of the Froude number on the number of surface vortices report an increase in vortex occurrence as the Fr increases (Kimura et al., 2008). This indicates that an increase in velocity or a decrease in water height (if used as the characteristic dimension) amplifies vortex formation. Several studies have confirmed a threshold for surface velocity that must not be exceeded to avoid gas entrainment (Banerjee et al., 2013, 2007; Kimura et al., 2008). As surface flow becomes higher, vorticity, which influences vortex formation, becomes more pronounced. Consequently, several vortex mitigation systems have been proposed, including the use of submerged vertical plates to reduce surface velocity. These systems, tested in various models, have proven relatively effective but raise issues regarding the inspectability of the vessel using ultrasonic methods. Regarding water height, (Kimura et al., 2008) showed that reducing it aggravates gas entrainment. Gas entrainment appears when the ratio of velocity to water height exceeds a certain threshold, corresponding to a downward velocity gradient. This observation is consistent with the gas entrainment models (Ito et al., 2010; Sakai et al., 2008), where the downward velocity gradient plays a key role.

Vortices are primarily generated in two locations as sketch on Fig. 9:

* At the interface of two shear flows (Kimura et al., 2008),

* Downstream submerged structures due to Von Karman-type instability (Eguchi et al., 1994; Kimura et al., 2008).

These locations, characterized by high vorticity, combined with downward velocity, lead to the formation of vortices with the potential for gas entrainment. It is important to note that the position of the vortices is not stable over time, and the variation in their shape results in bubbles detaching from their tips, causing intermittent gas entrainment (Kimura et al., 2008).

The effects of modeling (use of a simulant fluid) and scaling are critical points when interpreting results and applying them to real-world cases. Since the scales of the models can vary over a wide range (those in this bibliography range from 1/10 to 5/8), their results must be analyzed with this variability in mind. To compare vortex counting measurements, studies, conducted at CEA on the setups HIPPO and OREILLETTE, use the same time dimension based on the Froude number, following Equation (9). Using this formulation, they observed an invariance in the total number of vortices counted over a period of 800 s, regardless of the scale. However, larger-scale models exhibit deeper vortices in dimensionless terms.

$$t^* = L^{1/2} \cdot Fr^{1/2} \quad (9)$$

The length of vortices is a critical parameter in triggering gas entrainment, and the scale of the model affects this phenomenon (Kimura et al.,
<figure><img src="image_10.png" /><figcaption>Fig. 9. Locations of the generation of the vorticity by shear flows (left sketch) or downstream an obstacle (right sketch).</figcaption></figure>

2008). The study conducted by (Eguchi et al., 1994) on four models of different scales, ranging from 1/10 to 5/8, confirms that the critical Froude number (Fr)—defined as the Fr value at the operating parameters that trigger gas entrainment—is scale-dependent, as shown in Equation (10). Consequently, some authors (Kimura et al., 2008) suggest that gas entrainment studies should be performed on models with a minimum scale of 1/3. In fact, the comparison between measurements on the COLCHIX 4 and JESSICA models, with respective scales of 1/8 and 1/3, shows a much more agitated free surface in the larger model, implying that small-scale tests are not conservative regarding gas entrainment. However, the work of (Kimura et al., 2008) demonstrates that the vortex formation mechanism remains unchanged, with vortices and gas entrainment still occurring downstream of submerged structures. As a result, models smaller than 1/3 scale remain valuable in a phenomenological approach.

$$Fr_c^* \propto L^{-0.5} \qquad (10)$$

The study of the scale effect by (Eguchi et al., 1994) shows that the gas entrainment is triggered at the same velocity regardless of the model scale. This suggests that in Equation (8), which defines the flow rate conditions in the model, the exponent of the parameter L* should be reduced by a value of 0.5 to achieve the same velocity conditions. This recommendation is also suggested by other authors, with a value ranging between 0.3 and 0.5 (Eguchi et al., 1994).

From the perspective of the effects of using water as a simulant fluid instead of sodium, since these two fluids do not have exactly the same physical properties, several studies have focused on evaluating the impact of variations in viscosity and surface tension on gas entrainment. In the case of surface tension, it seems to play a role only in small-scale models, as the Weber number approaches that of the real case when the scale increases. Furthermore, studies by (Eguchi et al., 1994) show that the critical Froude number (Fr) decreases when surface tension decreases, making results obtained on water models conservative since the surface tension of sodium is greater than that of water. Regarding the effects of viscosity, a decrease in viscosity results in a lower critical Fr, which does not allow for conservativeness in this case. However, since the influence of viscosity is reflected in the Reynolds number (Re), and its impact is minimal on vortices leading to gas entrainment once a certain threshold is exceeded (Eguchi et al., 1994), the model effects are negligible if all areas of the experimental model that may contain vortices exhibit a sufficiently high Re.

(Guenadou et al., 2021) carried out investigation of the velocity around the vortex in the MICAS model. It aimed at building a gas entrainment criterion that can be implemented in numerical tools, as those ones are not able to simulate the vortices behavior. The strategy for evaluating the risk of gas entrainment using simulation tools will focus on simpler, single-phase calculations that are easier to implement and offer reasonable computation times. Initially, the free surface is considered flat, and the influence of vortices on its shape is neglected. The velocity fields calculated with this method will then help identify areas where vortices are likely to develop, based on the gas entrainment criterion developed by (Sakai et al., 2008). This criterion states that gas entrainment occurs if the length of the vortex matches the distance between the free surface and the inlet of the IHX, the size of the vortex

being estimated using the Burgers vortex model. This one is built from the downward velocity gradient, the circulation and one corrective coefficient. Downward velocity gradient and circulation can be determined from the velocity fields calculated by the computational codes, making this criterion easily implementable in simulation tools. The corrective coefficient must be determined from the experimental velocity fields measured around vortices in conditions of gas entrainment or non-entrainment. Experimental velocity fields measured by PIV 20 mm below the surface show good agreement with various vortex models, as shown in Fig. 10. The bundle of green curves, corresponding to velocity measurements at different angles around the center of the vortex, matches well with the Burgers model, whose parameters were calculated using different methods. The first one indicates that the location of the absolute maximum velocity is at the radius of the vortex, while the parameters of the vortex model in the second one are calculated to fit the curve using a least squares method. The experimentally determined correction factor for the gas entrainment criterion, based on these measurements, significantly differs from the value mentioned by (Sakai et al., 2008). This discrepancy with the initial criterion developed by (Sakai et al., 2008) may be attributed either to the specific geometry of the MICAS setup or to the fact that turbulence is neglected in the theoretical model, both of which could influence the accuracy of the predictions. More advanced criteria account for turbulence and surface tension (Ito et al., 2012, 2010). Both factors tend to reduce the vortex length and correlate better with results obtained from analytical experiments. However, no comparison with measurements in integral models has been conducted.

## 9. Future directions for R & D

From the past studies and experiments presented in the previous sections, the results obtained permitted to understand the basis of the phenomena involved in the gas entrainment by vortex. However, there appear to be specific areas where there is room for improvement and further work is required. It would be useful in the development and design of the SFR with regard to its safety.

Experimental studies necessitate reliable instrumentation. Without accurate measurement methods, experimental data becomes useless. A review of past campaigns reveals a significant challenge in characterizing vortices at the free surface, particularly in integral models. There is considerable scope for advancement in this area, aiming to simultaneously determine the occurrence, location, depth, and diameter of vortices.

One promising approach employs ultrasonic transducers to identify the shape and location of vortices. Ultrasonic waves are transmitted into the measurement area and reflect off the free surface, including those formed by gas cores within vortices. This technique offers the advantage of applicability in both sodium and simulant fluid experiments, and it does not require optical access, making it feasible for steel models. However, its main drawback is the limited inspection domain. This technique has been successfully implemented in in-service inspections (Baqué et al., 2015) and for bubble detection (Cavaro et al., 2013). Nevertheless, further adaptations and calibrations are needed to address gas entrainment by vortices.
<figure><img src="image_11.png" /><figcaption>**Fig. 10.** Tangential and radial velocities from the vortex center – at various angles (green curves), mean value (blue curve), Burgers model (red curve) parameters determined assuming $r_0$ at maximal absolute velocity.</figcaption></figure>

Another approach involves the use of an array of cameras positioned around the model at various angles relative to the free surface. As illustrated in Fig. 11, a camera located above the surface tracks the occurrence, location, and movement of vortices, while strategically placed cameras measure their shapes and depth. By cross-referencing images from different cameras, vortices at the surface can be identified. With a known vortex location and a pixel/mm calibration accounting for depth of field, vortex characteristics can be calculated. This technique is limited to water experiments with full optical access and requires specific lighting conditions. It is currently under development, and further work is needed to fully implement the measurement method.

Another area of progress involves measuring the velocity around vortices, which is crucial for constructing vortex models and validating computational tools. Traditional methods, such as Particle Image Velocimetry (PIV), allow velocity measurements in narrow domains, like the plane created by a laser sheet. Recent advancements in particle tracking using multiple cameras overcome this limitation, enabling three-dimensional velocity measurements around vortices. These commercial devices require no further development. However, their application is currently limited to simple geometries, such as analytical models, due to the need for extensive optical access. Despite this limitation, the technique shows promise, and ongoing studies are testing its feasibility in the BANGA loop (see Fig. 6).

The state of the art in vortex studies primarily involves two approaches: integral models, which replicate specific reactor designs, and analytical setups, designed to investigate the phenomena leading to

vortex formation. Both approaches are complementary, with their own advantages and drawbacks. Integral models offer a realistic representation of reactor geometries but can pose challenges in terms of complex measurements. In contrast, analytical setups provide a controlled environment for studying vortex formation but may lack representativeness due to simplified geometries, potentially affecting the understanding of gas entrainment phenomena. To bridge this gap, an intermediate approach has been employed by some researchers (Kimura et al., 2008; Lee and Park, 2022). Typically, these models consist of a section of the actual geometry, selected based on symmetrical considerations. This symmetrical design aims to decrease the size and simplify measurements, as the model features planar walls. As a result, a higher scale can be chosen compared to an integral model. However, designing these models is more challenging than integral or analytical setups, as they must respect the underlying phenomenology and boundary conditions, such as velocity, while allowing for geometrical simplifications. Due to their smaller size compared to integral models, these intermediate models enable simulations of the entire phenomenon, including vortex formation, as demonstrated in recent work by (Lee and Park, 2022). Experimental results from these models are invaluable for validating numerical simulations. In summary, this intermediate approach offers a balanced and effective means of studying vortex dynamics in complex geometries, bridging the gap between integral and analytical models. This intermediate approach shows promise for future studies, as it combines the strengths of both integral and analytical methods, aiming to provide a more comprehensive understanding of vortex dynamics in

<figure><img src="image_12.png" /><figcaption>**Fig. 11.** Measurement device to identify the characteristics of the vortices using an array of cameras.</figcaption></figure>
complex geometries.

In the section discussing the Na/water similarity, it was noted that it is challenging to simultaneously satisfy the Froude and Weber numbers. Consequently, many studies have based the design of the model and the calculation of operating conditions solely on the Froude number. However, this approach may have consequences on the results regarding vortex characteristics, particularly when the model is at a small scale. To address this issue, one potential solution involves modifying the physical properties of water. The surface tension of water is approximately two times less than that of sodium. While it is possible to slightly increase the surface tension of water by 3 to 5 % by, for example, adding chemical substances, this approach may not be ideal as the scale of the model plays a significant role in the similarity. By setting the Weber and Froude numbers equal in both the model and the reactor, we can express the required surface tension ratio as Equation (11).

$$ \sigma^* = \rho^* \cdot L^*^2 \qquad (11) $$

Fig. 12 illustrates the required water surface tension as a function of the model scale, calculated using Equation (11). This graph demonstrates that the surface tension must be reduced to simultaneously satisfy the Froude and Weber numbers. This can be achieved by adding chemical substances such as surfactants to the water (Pabon and Corpart, 2002). (Eguchi et al., 1994) previously employed this method to modify the surface tension and study its impact on vortex behavior. By using the most effective surfactants, it is possible to reach a surface tension of approximately 10 mN/m. The required surfactant concentration (<0.1 g/L) is so low that it does not significantly alter the water density. Reducing the water surface tension to its lowest possible value (10 mN/m) allows for simultaneous Weber and Froude similarity in the grey area of Fig. 12. This means that, using this approach, experiments at a 1/4th scale while maintaining the Froude and Weber numbers are feasible. It is worth noting that, with raw water, the required scale is 5/8, which is significantly larger. However, some work needs to be done on the practical implications of using water with surfactants. For instance, this may lead to the formation of foam and could perturb the measurements. This methodology can be combined with the intermediate model approach presented in the previous paragraph, as it allows for the construction of models at a larger scale compared to an integral model.

## 10. Conclusion

The presence of gas in the core of SFRs, which poses significant safety concerns, must be avoided. One of the sources in hot pool of this gas is the entrainment of argon at the free surface by vortices within the IHXs. Due to the complexity of the physical and geometrical factors involved, it is challenging to simulate this phenomenon accurately with computational codes. Therefore, physical models are necessary to study gas entrainment.

To tackle the issue, two parallel approaches are employed. The first focuses on the phenomenological aspects of vortex formation, using

<figure><img src="image_13.png" /><figcaption>Fig. 12. Evolution of the required surface tension versus the model scale to simultaneously satisfy the Froude and Weber numbers.</figcaption></figure>

simplified experimental setups designed to replicate the conditions that lead to gas entrainment. The second approach, involving integral model studies, aims to validate design choices that minimize vortex formation at the free surface. These complementary methods allow for a deeper understanding of both the fundamental mechanisms and the practical design solutions needed to mitigate gas entrainment risks.

Most of the experiments have been conducted with water instead of sodium using a similitude approach. With respect of the dimensionless numbers characterizing the process of formation of the vortices, it is possible to study the gas entrainment at a lower scale with a simulant fluid, i.e., water in most of the works. However, this method requires some precautions, as it is difficult, even impossible, to keep all the characteristic number at the same time. The effects of distortions on the experimental results have to be considered. It is important to note that the limited experiments conducted with sodium have shown that the results obtained using water as a simulant fluid are conservative.

Analytical models feature simple geometries, such a cylinder or a channel, makes it possible to extensive measurements. Numerous studies established a map of the gas entrainment depending on system's dimensionless numbers and geometry. In some cases, these contributed to correlations that predict vortex formation. However, even those are geometry dependent and their application is limited, some phenomena trends can be established. From those works, it can be stated that, when a specific threshold value of the characteristic dimensionless numbers of the system is exceeded, only inertia and viscosity affect the behavior of the vortices. Thanks to the convenient optical access provided by analytical setups, some studies have focused on measuring velocity fields around vortices. These measurements are intended to develop vortex models that can be accurately simulated using numerical tools. The goal is to improve the representation of vortex formation and behavior in simulations, enhancing predictions of gas entrainment and other flow phenomena in reactor systems.

From another perspective, integral models aim to replicate the reactor geometry as accurately as possible, albeit at a reduced scale. The scale is chosen as a balance between economic feasibility and adherence to the key dimensionless parameters governing the system. These setups are generally constructed from transparent materials, enabling optical measurement techniques. Operating conditions are derived based on Froude number conservation, and in certain cases, the Weber number is also considered to account for surface tension effects. Two primary types of studies have been conducted using these integral models. The first examines vortex occurrence as a function of geometry and operational conditions, demonstrating that gas entrainment correlates with the local vorticity and the downward velocity gradient. Other studies focus on mapping the velocity fields around vortices to develop gas entrainment models similar to those used in analytical models.

While this article presents a broad array of studies, substantial work remains to accurately model the gas entrainment mechanism. Further advancements in measurement techniques are needed, especially for accurately evaluating the amount of vortices and capturing the surrounding velocity field. To this end, research at CEA is ongoing, focusing on the in-depth study of vortex behavior. A new multi-camera setup will be installed on an intermediate model designed to closely reflect the gas entrainment phenomena. This setup aims to provide improved representativeness, enhancing our understanding and ability to simulate the entrainment process. The influence of surface tension will also be taken into account by the addition of surfactants.

### CRediT authorship contribution statement

D. Guenadou: Writing – original draft.

### 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.

## Data availability

No data was used for the research described in the article.

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