# Characteristics and Mechanisms of Debris Bed Formation Behavior in Severe Accidents of Sodium-Cooled Fast Reactors: Experimental and Modeling Studies

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Investigations Focusing on Overall Accumulated-Bed Characteristics

#### 2.1. Investigations with Single-Sized and Single-Shaped Solid Particles

#### 2.1.1. Experimental Materials and Methods

_{in}), 1020 mm height) and an upper particle injection device followed by a cylindrical releasing nozzle. In total, 119 sets of experiments, including 43 sets using non-spherical particles, were conducted. To investigate the characteristics of accumulated particle beds, six key initial parameters, including particle density (ρ

_{p}), particle diameter (d

_{p}), particle shape (spherical or non-spherical), particle volume (V

_{p}, 3~10 L), nozzle diameter (d

_{n}, 20~40 mm), and nozzle height (H

_{n}, 0.47~0.72 m), were considered. As shown in Figure 4, five important quantities related to bed characteristics, including the final bed height (H

_{b}), final mound height (H

_{m}), final mound dimple area (A

_{dim}), final dimple volume (V

_{dim}), and final repose angle (θ

_{r}) of the bed mound, were noted.

#### 2.1.2. Experimental Results and Discussion

_{p}= 5 L, it was verified that, possibly due to the more significant impact of particle jets on the accumulated particle beds, more concave and flatter particle beds (i.e., particle beds with lower mound height, smaller repose angle, larger dimple area, and volume) could be attained when smaller-sized particles or a nozzle with larger diameter or higher placement were employed. In addition, it was confirmed that particle-shaped relevant characteristics (such as sphericity) could also effectively affect DBF behavior. Figure 5 shows the effect of particle shape. From this figure, it is observable that for non-spherical particles, potentially due to the smaller impacts of the falling particles on the accumulated particle beds and slower dissipation in lateral directions resulting from larger inter-particle friction [29], more particles accumulated on the bed center, leading to comparable higher particle mounds than those for the spherical cases. However, regarding the effect of particle density, the variations of H

_{m}, A

_{dim}, and V

_{dim}were found to be non-monotonous, although the repose angle seemed to decrease with the increase in particle density.

#### 2.1.3. Modeling Studies

_{b}, which is one key factor indicating bed shape characteristics, was empirically established by Shamsuzzaman et al. [30,39,61,73]. By using Buckingham’s π theorem [75], the empirical model was established with the consideration of various parameters (see Equation (1)).

_{1}, the first term uses the particle sphericity (ϕ), which was determined by Ergun’s equation (see Equation (2)) with the particle’s volume-equivalent diameter (d

_{ev}) and measured pressure drop (ΔP), to evaluate the effect of particle shape (esp. for non-spherical particles); the characteristic dimensional similarities are represented by the second, third, and fourth terms; buoyancy influence is considered in the fifth term; while the sixth term points out volumetric similarities; the seventh term signifies the particle-jet Reynolds number with particle terminal velocity V

_{T}[76]; and the eighth term represents particle falling intensity. Furthermore, Table 2 lists the empirical indices in Equation (1) obtained by regression.

_{f}, ρ

_{f}, and μ

_{f}are the fluid superficial velocity, density, and viscosity; ε stands for the bed voidage (=$\frac{{V}_{\mathrm{p}}-{m}_{\mathrm{p}}/{\rho}_{\mathrm{p}}}{{V}_{\mathrm{P}}}$).

_{b}could be observed as the particle density increased. Possibly, this is because a larger frictional particle-particle drag resisted the lateral sedimentation of falling particles in cases of larger-density fuel particles [29].

#### 2.2. Investigations with Mixed Solid Particles

#### 2.2.1. Experimental Materials and Methods

_{p}and H

_{n}were fixed at 5 L and 720 mm, respectively, and d

_{p}, d

_{n}, and the composition of particle mixtures were set as parameters. Al

_{2}O

_{3}, ZrO

_{2}, and SS particles of different sizes (2~6 mm) were utilized, respectively, to form the bicomponent or tricomponent mixed-sized particle mixture, and the volume of each component was the same. However, for composing mixed-density particle mixtures, Al

_{2}O

_{3}and SS particles of the same size (2~6 mm) were employed with a volume mixing ratio of 1:3 to 3:1. To study the effect of mixed particles on DBF behavior, the apparent surface area of each particle component was measured at four typical horizontal cross sections of particle beds (see Figure 8).

#### 2.2.2. Experimental Results and Discussion

#### 2.2.3. Modeling Studies

_{p}was related to ϕ, and d

_{p}was used. Additionally, the values of the empirical constant indices are given in Table 2. Figure 10 illuminates the accuracy of the predicted H

_{b}/d

_{n}results in accordance with Equation (4). Owing to this model, a good agreement between the experimental and predicted particle bed heights was achieved. In addition, for mixed-sized cases, higher mound heights can be reproduced, as observed in experiments through this model. Compared to the model proposed by Shamsuzzaman et al. (i.e., Equation (1)), this model can more reasonably predict the particle bed heights for non-spherical cases and mixed-sized cases [28].

## 3. Investigations Focusing on Flow Regime Characteristics

#### 3.1. Investigations with Single-Sized Spherical Particles

#### 3.1.1. Experimental Materials and Methods

_{n}= 110~130 cm), nozzle size (d

_{n}= 10~30 mm), and water depth (H

_{w}= 0~60 cm), were studied. Additionally, the influence of the thickness (W

_{tank}= 30~60 mm) of the tank was also investigated.

#### 3.1.2. Experimental Results and Discussion

_{p}, ρ

_{p}, d

_{n}, H

_{n}, and H

_{w}) that affect the prominence of pool convection and particle inertia on flow regime transition were investigated. The consistent trends with the 3D experiments that focused on accumulated-bed characteristics and the confirmation of no noticeable impact of gap thickness on flow regime characteristics and bed geometry indicated the reliability and effectiveness of the experimental results under 2D conditions for SFR safety analysis.

#### 3.1.3. Modeling Studies

_{convection}and I

_{inertia}are two dimensionless quantities that represent the prominences of pool convection and particle inertia, respectively; v

_{pr}stands for the average particle releasing rate estimated by ${\nu}_{\mathrm{pr}}=\frac{4{V}_{\mathrm{p}}}{t\pi {d}_{\mathrm{n}}^{2}}$, with the total particle volume V

_{p}, as well as the releasing time t, measured in accordance with a duration of releasing the first particle to the last one from the nozzle; ${\nu}_{\mathrm{c}}=\frac{{\mu}_{\mathrm{l}}}{{\rho}_{\mathrm{l}}{d}_{\mathrm{n}}}$ is the critical velocity for discharged particles to trigger the pool convection; a, b, c, d, and K

_{B}are empirical constants (see Table 3).

#### 3.2. Investigations with Single-Sized Non-Spherical Particles

#### 3.2.1. Experimental Materials and Methods

_{tank,}was fixed at 60 mm, while H

_{w}and d

_{n}varied from 60 to 80 cm and from 30 to 40 cm, respectively. In total, 46 experimental sets were conducted by employing non-spherical particles with different types (Al, Al

_{2}O

_{3}, ZrO

_{2}, SS, and Cu), sizes (d

_{ev}from 0.50 to 5.62 mm), and shapes (dish, triangular prism, cylinder, and irregular shape) [67,68]. To evaluate the particle-shaped characteristics, the particle sphericities (ϕ) of non-spherical particles were also approximated in accordance with Ergun’s equation (see Equation (2)).

#### 3.2.2. Experimental Results and Discussion

#### 3.2.3. Modeling Studies

_{T}for non-spherical particles was suggested to be modified as follows [80]:

_{T}is a correction function associated with ϕ and the Reynolds number of particles determined by their d

_{ev}and V

_{T}[78,79]; V

_{TS}is the terminal velocity of spherical particles sharing the same d

_{ev}with non-spherical ones [68].

_{pr}in the base model. Moreover, considering the falling process of particles into the pool, it can be understood that such inter-particle interactions would restrict the intensity of pool convection or, in other words, result in a more significant effect for particle inertia. In order to take this influence into account for the base model of Equation (5), an extension scheme, K

_{NS}, was proposed [68]:

_{1}, s

_{2}, and s

_{3}are empirical constants (see Table 3); Re is the Reynolds number of falling particles determined by $Re=\frac{{\nu}_{\mathrm{pr}}{\rho}_{\mathrm{l}}{d}_{\mathrm{n}}}{{\mu}_{\mathrm{l}}}$. Concerning the reasonability of the proposed K

_{NS}, it can be understood that if particles are non-spherical (i.e., ϕ is smaller than 1.0), K

_{NS}becomes smaller than 1.0, leading to a smaller value of index I, indicating the possible emergence of a larger-numbered regime; while if particles are spherical (i.e., ϕ is 1.0), K

_{NS}becomes 1.0, and thus, the extended model will return to the base model developed in accordance with spherical particles (i.e., Equation (5)). As illustrated in Figure 15b, by using the extended model (i.e., Equation (7)), an extended regime map is attained for both spherical and non-spherical cases. According to the regime map depicted in Figure 15b, the reliability of the predicted results from the extended model is observed to be validated. On this basis, the potential for extending the applicability of the base model on DBF behavior under more realistic accidental situations was displayed to some degree.

#### 3.3. Investigations with Mixed-Sized Spherical Particles

#### 3.3.1. Experimental Materials and Methods

_{tank}= 60 mm and varying nozzle diameters (30 and 45 cm), as well as water depths (60 and 80 cm). In the experiments, glass, Al

_{2}O

_{3}, ZrO

_{2}, and SS particle mixtures possessing differences in mixed diameter ratios (from 6:2 to 2:0.4) and volumetric mixed ratios (from 1:3 to 3:1) were utilized. In total, 76 sets of experiments were carried out [69].

#### 3.3.2. Experimental Results and Discussion

_{w}and d

_{n}) can also be validated in mixed-sized experiments. While focusing on the mixed-sized effect, it was found that by using particle mixtures with a larger number of smaller particles (i.e., smaller volumetric mixed ratio), due to their smaller particle inertia, the flow regimes during the DBF process tended to shift from bigger-numbered to lower-numbered regimes.

#### 3.3.3. Modeling Studies

_{a}, and volume mean diameter (d

_{v})),

_{evp}) [81]:

_{p, j}and d

_{p, j}are, respectively, the number and diameter of the j-th sized particles for the particle mixtures. While in Equation (9), ε

_{j}and V

_{b, j}represent the porosity and volume of particle beds constituted by the j-th sized particles, respectively.

_{a}, overall, results can be better predicted and attained in contrast to other mean diameters.

_{MS}, with consideration of the flow regime characteristics in cases of mixed-sized particles. Therefore, the extended model was developed as [69]:

_{1}, p

_{2}, and p

_{3}are empirical constants (see Table 3); $\mathsf{\Omega}$ is the extent of convergence of particle size distribution determined by using Ergun’s equation with d

_{v}to characterize the effective sphericity of the particle mixture. Considering the rationale of the establishment of K

_{MS}, it can be easily deduced that in single-sized spherical cases, $\mathsf{\Omega}$ equals 1.0, so K

_{MS}becomes 1.0; while for mixed-size spherical particles, $\mathsf{\Omega}$ is smaller than 1.0, so K

_{MS}becomes larger than 1.0, namely diminishing the effective impact of particle inertia, which corresponds well with the experimental observations.

_{MS}, an extended regime map can be attained (see Figure 15d). In addition, to further validate the rationale of these two modeling approaches, the prediction results were also analyzed in accordance with the fundamental parametric effects (such as the effects of particle density and size and the effect of volumetric mixed ratio) on the flow regime transitions [69] observed in the experiments. Further, the extension potential of the model for application under more realistic accidental situations was again further confirmed.

#### 3.4. Investigations on the Effect of Coolant Boiling Caused by Accumulated Debris

#### 3.4.1. Experimental Studies Using the Gas-Injection Method

_{g}= 0~50 L/min) was injected from the bottom of a liquid pool through an airstone, which was installed to guarantee a relatively uniform distribution of gas. In order to concentrate on the influence of the injected gas (i.e., the sodium boiling), W

_{tank}, V

_{p}, d

_{n}, and H

_{w}were set at 60 mm, 10 L, 30 mm, and 60 cm, respectively. In total, 78 experimental sets were carried out by using various types of particles with different shapes, densities, and sizes.

_{g}) on DBF was analyzed for each main regime (including Regime I, Regime II, and Regime IV) separately. Figure 17 shows the typical transit DBF behavior in gas-injection cases, while Figure 18 shows the influence of Q

_{g}on DBF behavior for different flow regimes. In experiments for Regime I, as Q

_{g}increased, despite no noticeable variations in flow regime or bed shapes (see Figure 17a), an evident increase in the maximum height of particles suspended (see the illustration of the height of particles suspended in the fourth photo of Figure 17a), as shown in Figure 18a, was found due to the larger impetus that drives the upward particle movements (esp. for smaller and lighter particles). Moreover, from Figure 18a, it is also observable that with a similar Q

_{g}, due to the difficulty for the upward gas-liquid flow to suspend the larger-sized (i.e., greater-inertia) particles, the peak value for the height of the particles suspended seemed to decrease when the particle size became larger. In addition, the difference in the height of particles suspended between the cases with 0.18 mm and 0.25 mm particle diameters was observed to be greatly reduced in comparison to that with 0.125 mm and 0.18 mm ones, indicating the non-negligible role of particle inertia in those cases (esp. under conditions without gas injection) and a probable flow regime transition with a further increase in particle size.

_{g}, a lower-numbered flow regime may emerge as a result of the greatly enhanced pool convection and the effect of larger-rate flows on diminishing the dominance of particle inertia (see Figure 12b and Figure 17b). More quantitatively, as shown in Figure 18b, by measuring the center angle (illustrated in Figure 14a), a gradual increment of the center angle could be found as Q

_{g}increased, revealing the trends for Regime II transit to Regime I.

_{g}was employed due to the promoted intensity of pool convection. Furthermore, an additional point that can be found from Figure 18c is that as Q

_{g}increased because of the positive role of pool convection in pushing away the particles, the differences in the vertex angles for the cases sharing the same Q

_{g}would enlarge, revealing a flattening tendency for the particle bed (i.e., the potential flow regime transitioned to a smaller-numbered one) with a greatly enhanced pool-convection intensity caused by a large gas flow rate.

#### 3.4.2. Experimental Studies Using the Bottom-Heated Method

_{tank}and V

_{p}were fixed at 60 mm and 10 L, respectively; while d

_{n}, H

_{w}, and P varied within the ranges of 30~40 mm, 45~80 cm, and 0~2580 W, respectively. In the experiments, pre-heated particles at 383K comprising different materials (glass, Al

_{2}O

_{3}, and ZrO

_{2}), diameters (0.125~6.0 mm), and shapes (sphere, irregular shape, and cylinder) were employed. In total, there were 72 sets of experiments conducted.

_{w}on the characteristic angles for the particle beds was found to be inconsistent under boiling conditions in comparison to that under non-boiling conditions, which likely indicates two points concerning the mechanisms of DBF behavior in bubbling (including gas-injection and bottom-heated boiling) cases:

- (1)
- For the non-bubbling cases, pool convection is triggered by the falling particle jet and, thus, is limited within the center area. Therefore, the particles are driven away inside a relatively small region (such as around the two apexes of the bed). While for bubbling cases, due to the rather uniform distributions of bubbles in the pool, the influence area of pool convection is deemed to be wider.
- (2)
- Although increasing the water depth was found to generally enhance pool convection under non-bubbling conditions, for boiling conditions at a constant heating power, due to the larger water mass and the potentially enhanced heat dissipation resulting from larger heat-transfer areas within the environment, a higher water depth might also lead to reduced bubbling rate, thereby reducing the overall intensity of pool convection to some extent [71].

_{g}(10~20 L/min), revealing the probable existence of an equivalent quantity that can compare the two bubbling methods. Based on the temperature variations measured, owing to the thermocouples (see Figure 16b), and under the assumption that heat loss rate did not vary significantly within a fairly limited water temperature, the effective rate for bubbling, Q

_{g,eff}, was determined at nearly 18.49 L/min [71], which is within the range expected (i.e., 10~20 L/min) for P = 2580 W with H

_{w}= 60 cm. Therefore, though further investigations with more elaborate considerations may still be required, to a certain degree, the reliability of the gas-injection approach for simulating coolant boiling was validated by the bottom-heated experiments, and additional gas-injection experiments with a wider range of Q

_{g}are expectable to elucidate the impact of boiling intensity on flow regime transition.

#### 3.5. Investigations on the Effect of Coolant Boiling Caused by Falling Debris

#### 3.5.1. Experimental Materials and Methods

_{p}= 473~673 K). The water temperatures (358 K and 368 K) were selected to study the effect of liquid temperature. Other important parameters, including particle type (Alumina, Al

_{2}O

_{3}, ZrO

_{2}, and Steel), diameter (0.4~4 mm), and nozzle diameter (30~40 mm), were also considered.

#### 3.5.2. Experimental Results and Discussion

#### 3.5.3. Modeling Studies

_{eff}, was taken into account to heat the liquid and generate the boiling [72]:

_{g}is the effective power to generate bubbles (or gas) by heated particles from the overheated water, and P

_{w}stands for the effective power needed to heat the water to the boiling point from a subcooled status during particle discharging:

_{B}represents the liquid boiling point, T

_{w}is the initial water temperature, and C

_{w}is the water’s specific heat. Then, the effective bubble generation rate, Q

_{g,eff}, can be derived:

_{HP}, accounting for the boiling effect caused by the falling debris, was proposed for incorporation into the previous model:

_{g}is the gas Reynolds number accounting for the boiling effect on centralizing the particle accumulations, and r

_{1}, r

_{2}, and r

_{3}are the empirical constants given in Table 3. When ψ or Re

_{g}trends to zero (such as Q

_{g,eff}= 0 or v

_{pr}trends to ∞), indicating a negligible bubbling effect (e.g., minimal or non-intensive bubble generation or dominant particle-flow-induced convection), the value of K

_{HP}approaches 1, which restores the model to the base model that is suitable for non-bubbling cases. On the other hand, for cases where there is significant boiling during heated particle discharging, K

_{HP}is a positive value lower than 1 to consider the effects of boiling surrounding the falling particles on a regime transition to larger-numbered ones. Therefore, the proposed K

_{HP}was qualitatively reasonable.

_{HP}. It is seen that the model prediction and experimental observation show good agreement within the parameter ranges considered. While further analysis and validation may be required to examine the extrapolation and interpolation capabilities of the proposed K

_{HP}function under a wider range of experimental conditions, the introduction of the extension function suggests that the base model can be extended to predict the characteristics of DBF under more realistic situations for an accident to some extent.

## 4. Conclusions

## 5. Discussion of Future Prospects

- (1)
- Further analyses and validations of DBF characteristics in cases of multicomponent (i.e., more than three components) mixed-sized particles. As noted in Section 2 and Section 3, previous investigations under mixed-sized spherical particle conditions were carried out by only using bicomponent or a few tricomponent mixed-sized particle mixtures. To attain more reliable insights into DBF behavior under more realistic particulate situations, more experimental studies, as well as corresponding modeling verifications, with three (or even more) component mixed-size solid particles can be conducted.
- (2)
- Further experimental investigations on the characteristics of particle separation and stratification under mixed-size particle conditions. As mentioned in Section 3, from experiments focusing on flow regime characteristics, it was implied that the stratifications and separations of different particle components in the mixtures, which may affect the coolability of the debris beds, can possibly appear under some specific accident situations (e.g., significant sodium boiling) as a result of the differences in component inertia. Therefore, to study the characteristics of particle separations and stratifications, experiments using mixed-size particles can be carried out under the gas-injection method, which can effectively simulate the violent sodium boiling conditions.
- (3)
- Further experimental and modeling studies under mixed-density conditions. For the experimental studies, although fundamental mixed-density influences were preliminarily investigated through several experiments focusing on accumulated-bed characteristics, it is necessary to point out that due to the rather limited experimental parametric conditions and non-visual experimental processes, some potential valuable experimental evidence may have been missed. Therefore, mixed-density experiments should be continuously performed under more realistic parametric situations (e.g., smaller particle size and particle mixtures composed of components with densities comparable to the densities of debris in actual reactor accidents (such as MOX fuel and SS)) and in a visual quasi-2D water tank to study in detail the flow regime mechanism under mixed-density particle conditions. Again, since the particle inertia for mixed-density particle components is different, it is reasonable to imagine that particle separation and stratification phenomena may also be found and investigated in bubbling cases with mixed-density particles. Further, for the modeling studies, referring to the modeling developments regarding the molten-pool sloshing and debris bed self-leveling behaviors [81,82], some effective (or equivalent) density to estimate the overall density of the particle mixture can be tested by employing the base model. Further, the establishment of an extension scheme that can appropriately take the mixed-density effect into account can also be attempted.
- (4)
- Further modeling studies for bubbling conditions. To continuously extend the model’s predictability, an extension scheme can be developed for bubbling conditions. Preliminarily, considering the gas-injection cases, the modeling frameworks can be developed as follows:$$I={K}_{\mathrm{GI}}\left[{K}_{\mathrm{MS}}{K}_{\mathrm{NS}}{K}_{\mathrm{B}}\times \frac{{\left(\frac{{\nu}_{\mathrm{pr}}}{{\nu}_{\mathrm{c}}}\right)}^{a}{\left(\frac{{H}_{\mathrm{W}}^{2}}{LW}\right)}^{b}}{{\left(\frac{{\rho}_{\mathrm{p}}-{\rho}_{\mathrm{l}}}{{\rho}_{\mathrm{l}}}\right)}^{c}{\left(\frac{{\mu}_{\mathrm{l}}{V}_{\mathrm{T}}}{{\sigma}_{\mathrm{l}}}\right)}^{d}}\right]$$
_{GI}is a new extension scheme. Theoretically, K_{GI}is required to ensure two points: (1) K_{GI}should monotonously increase as Q_{g}becomes larger to represent the effects of the gas flow rate on promoting pool convection and diminishing particle inertial impact; and (2) K_{GI}should become 1.0 if no gas is injected (i.e., Q_{g}= 0 L/min) to guarantee the applicability of the base model for non-bubbling cases.Here, it is also noted that the gas flow rate considered in our previous investigations may not cover the possible range of vapor flow rates from the drastic boiling induced by the accumulated debris beds in an actual CDA of an SFR [59,83]. Therefore, further experiments with wider ranges of gas flow rates may also be necessary to ensure the predictability of the extended model under accidental conditions. - (5)
- Further investigations under large-scale 3D conditions. Although essentially consistent parametric effects on characteristics of DBF were confirmed in both 2D flow regime and 3D accumulated-bed experiments, it should be highlighted again that an insufficiently large range of parameters for the 3D accumulated-bed experiments in the comparison to that of actual accident conditions may cause the loss of valuable evidence (such as additional variation in particle bed shape). Through elaborately performing the large-scale 3D experiments, insights obtained from 2D flow-regime experiments are expectable to be further verified under the large-scale 3D conditions. It can be expected that a more applicable and dependable empirical model associating the flow regime and accumulated-bed characteristics can be developed for application in reactor safety assessments. In fact, referring to the previous modeling studies on debris bed self-leveling behavior [14,84], it can be expected that the flow regime boundary lines (or empirical constants) determined for the predictive model under 2D conditions can possibly vary to some degree for large-scale 3D predictions. Such a predictive model is supposed to be useful in the improved designs of in-vessel core catchers along with the developments and validations of SFR safety analysis codes.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A | area | ρ | density |

Ar | Archimedes number | σ | surface tension |

C | specific heat | ϕ | particle sphericity |

D | diameter of experimental device | ψ | quantities for characterizing |

d | diameter | bubbling impact on restricting the | |

$\overline{d}$ | mean diameter | particle-flow induced pool convection | |

Fr | Froude number | Ω | degree of convergence in particle |

g | gravitational acceleration | size distribution | |

H | height | ||

h_{lg} | latent heat of liquid | Subscripts | |

L | length of experimental device | a | stands for an area mean term |

m | mass | B | boiling |

P | heating power | b | particle bed |

ΔP | pressure-drop | c | critical value |

Q | flow rate | dim | dimple |

Re | Reynolds number | ev | stands for a volume-equivalent term |

T | temperature | f | fluid |

t | the time span between the first and last particles | g | gas |

that flows out of the particle-releasing nozzle | in | inner container | |

U | superficial velocity | ip | represents an initial value of particle |

V | volume | j | j-th size particles |

v | velocity | L | left |

V_{T} | particle terminal velocity | l | liquid |

V_{TS} | terminal velocity of a spherical particle with the | m | particle mound |

volume-equivalent diameter of non-spherical particle | n | nozzle | |

W | width of experimental device | p | particle |

pr | particle releasing | ||

Greek symbols | R | right | |

ε | bed voidage | r | repose |

θ | angle | tank | water tank |

λ | thermal conductivity | v | stands for a mean volume term |

μ | viscosity | w | water |

Abbreviation | |||

CDA | Core Disruptive Accident | PAHR | Post-Accident Heat Removal |

DBF | Debris Bed Formation | PAMR | Post-Accident Material Relocation |

IGCAR | Indira Gandhi Centre for Atomic Research | SFR | Sodium-cooled Fast Reactor |

IVR | In-Vessel Retention | SS | Stainless Steel |

JAEA | Japan Atomic Energy Agency | SYSU | Sun Yat-Sen University |

## References

- OECD Nuclear Energy Agency. Technology Roadmap Update for Generation IV Nuclear Energy Systems; Generation IV International Forum (GIF); OECD Nuclear Energy Agency: Paris, France, 2014. [Google Scholar]
- Raj, B.; Chellapandi, P.; Rao, P.V. Sodium Fast Reactors with Closed Fuel Cycle; CRC Press: Boca Raton, FL, USA, 2015. [Google Scholar]
- Xu, R.; Cheng, S. Review of the molten-pool sloshing motion in case of Core Disruptive Accident: Experimental and modeling studies. Prog. Nucl. Energy
**2021**, 133, 103647. [Google Scholar] [CrossRef] - Marchaterre, J.F. Overview of core disruptive accidents. Nucl. Eng. Des.
**1977**, 42, 11–17. [Google Scholar] [CrossRef] - Maschek, W.; Li, R.; Matzerath Boccaccini, C.; Gabrielli, F.; Morita, K. Investigation on upper bounds of recriticality energetics of hypothetical core disruptive accidents in sodium cooled fast reactors. Nucl. Eng. Des.
**2018**, 326, 392–402. [Google Scholar] [CrossRef] - Yamano, H. Thermal-Hydraulic Phenomena Contributing to Reactivity Mitigation in Core Disruptive Accidents of Fast Reactors. Ph.D. Thesis, Kyushu University, Fukuoka, Japan, 2009. [Google Scholar]
- Ohshima, H.; Kubo, S. Handbook of Generation IV Nuclear Reactors; Woodhead Publishing: Manchester, UK, 2016. [Google Scholar]
- Yamano, H.; Sato, I.; Tobita, Y. Development of technical basis in the initiating and transition phases of unprotected events for Level-2 PSA methodology in sodium-cooled fast reactors. Nucl. Eng. Des.
**2012**, 249, 212–227. [Google Scholar] [CrossRef] - Sato, I.; Yamano, H.; Tobita, Y. Development of Severe Accident Evaluation Technology (Level 2 PSA) for Sodium-Cooled Fast Reactors—Identification of Dominant Factors in Initiating Phase of Unprotected Events. In Proceedings of the 2009 International Congress on Advances in Nuclear Power Plants (ICAPP’09), Tokyo, Japan, 10–14 May 2009. [Google Scholar]
- Yamano, H.; Tobita, Y.; Fujita, S. A three-dimensional neutronics-thermohydraulics simulation of core disruptive accident in sodium-cooled fast reactor. Nucl. Eng. Des.
**2009**, 239, 1673–1681. [Google Scholar] [CrossRef] - Morita, K.; Matsumoto, T.; Emura, Y.; Abe, T.; Tatewaki, I.; Endo, H. Investigation on Sloshing Response of Liquid in a 2D Pool against Hydraulic Disturbance. In Proceedings of the Ninth Korea-Japan Symposium on Nuclear Thermal Hydraulics and Safety (NTHAS-9), Buyeo, Republic of Korea, 16–19 November 2014. [Google Scholar]
- Xu, R.; Cheng, S. Experimental and numerical investigations into molten-pool sloshing motion for severe accident analysis of sodium-cooled fast reactors: A review. Front. Energy Res.
**2022**, 10, 893048. [Google Scholar] [CrossRef] - Tatewaki, I.; Morita, K.; Endo, H. A Study on Characteristics of Molten Pool Sloshing in Core Disruptive Accidents of Fast Reactors. In Proceedings of the 23rd International Conference on Nuclear Engineering: Nuclear Power-Reliable Global Energy (ICONE-23), Chiba, Japan, 17–21 May 2015. [Google Scholar]
- Xu, R.; Cheng, S. Debris Bed Self-Leveling Mechanism and Characteristics for Core Disruptive Accident of Sodium-Cooled Fast Reactor: Review of Experimental and Modeling Investigations. Sci. Technol. Nucl. Install.
**2022**, 2022, 2755471. [Google Scholar] [CrossRef] - Xu, Y.; Xu, R.; Cheng, H.; Liu, X.; Cheng, S. Numerical simulation of jet breakup phenomenon during severe accident of sodium-cooled fast reactor using MPS method. Ann. Nucl. Energy
**2022**, 172, 109087. [Google Scholar] [CrossRef] - Cheng, S.; Yamano, H.; Suzuki, T.; Tobita, Y.; Nakamura, Y.; Zhang, B.; Matsumoto, T.; Morita, K. Characteristics of self-leveling behavior of debris beds in a series of experiments. Nucl. Eng. Technol.
**2013**, 45, 323–334. [Google Scholar] [CrossRef] - Fauske, H.K.; Koyama, K. Assessment of fuel coolant interactions (FCIs) in the FBR core disruptive accident (CDA). J. Nucl. Sci. Technol.
**2002**, 39, 608–614. [Google Scholar] [CrossRef] - Koyama, K.; Yamada, Y.; Hayakawa, S.; Watanabe, M.; Watanabe, O. Development of severe accident evaluation technology (level 2 PSA) for sodium-cooled fast reactors; dentification of Dominant Factors in Core Material Relocation and Heat Removal Phases. In Proceedings of the International Congress on Advances in Nuclear Power Plants (ICAPP’09), Tokyo, Japan, 10–14 May 2009. [Google Scholar]
- Suzuki, T.; Tobita, Y.; Nakai, R. Evaluation of recriticality behavior in the material-relocation phase for Japan sodium-cooled fast reactor. J. Nucl. Sci. Technol.
**2015**, 52, 1448–1459. [Google Scholar] [CrossRef] - Bala Sundaram, G.; Velusamy, K. Effect of debris material composition on post accidental heat removal in a sodium cooled fast reactor. Nucl. Eng. Des.
**2021**, 375, 111065. [Google Scholar] [CrossRef] - Lipinski, R.J. A Coolability Model for Postaccident Nuclear Reactor Debris. Nucl. Technol.
**1984**, 65, 53–66. [Google Scholar] [CrossRef] - Lipinski, R.J.; Gronager, J.E.; Schwarz, M. Particle Bed Heat Removal with Subcooled Sodium: D-4 Results and Analysis. Nucl. Technol.
**1982**, 58, 369–378. [Google Scholar] [CrossRef] - Shamsuzzaman, M.; Horie, T.; Fuke, F.; Kai, T.; Zhang, B.; Matsumoto, T.; Morita, K.; Tagami, H.; Suzuki, T.; Tobita, Y. Experimental Evaluation of Debris Bed Characteristics in Particulate Debris Sedimentation Behaviour. In Proceedings of the 21st International Conference on Nuclear Engineering, Chengdu, China, 29 July–2 August 2013. [Google Scholar]
- Lin, S.; Cheng, S.; Jiang, G.; Pan, Z.; Lin, H.; Wang, S.; Wang, L.; Zhang, X.; Wang, B. A two-dimensional experimental investigation on debris bed formation behavior. Prog. Nucl. Energy
**2017**, 96, 118–132. [Google Scholar] [CrossRef] - Matsuba, K.; Isozaki, M.; Kamiyama, K.; Tobita, Y. Distance for fragmentation of a simulated molten-core material discharged into a sodium pool. J. Nucl. Sci. Technol.
**2016**, 53, 707–712. [Google Scholar] [CrossRef] - Tentner, A.; Parma, E.; Wei, T.; Wigeland, R. Severe Accident Approach-Final Report; Evaluation of Design Measures for Severe Accident Prevention and Consequence Mitigation; Argonne National Laboratory: Argonne, IL, USA, 2010. [Google Scholar]
- Suzuki, T.; Kamiyama, K.; Yamano, H.; Kubo, S.; Tobita, Y.; Nakai, R.; Koyama, K. A scenario of core disruptive accident for Japan sodium-cooled fast reactor to achieve in-vessel retention. J. Nucl. Sci. Technol.
**2014**, 51, 493–513. [Google Scholar] [CrossRef] - Sheikh, M.; Son, E.; Kamiyama, M.; Morioka, T.; Matsumoto, T.; Morita, K.; Matsuba, K.; Kamiyama, K.; Suzuki, T. Sedimentation behavior of mixed solid particles. J. Nucl. Sci. Technol.
**2018**, 55, 623–633. [Google Scholar] [CrossRef] - Shamsuzzaman, M.; Horie, T.; Fuke, F.; Kamiyama, M.; Morioka, T.; Matsumoto, T.; Morita, K.; Tagami, H.; Suzuki, T.; Tobita, Y. Experimental study on debris bed characteristics for the sedimentation behavior of solid particles used as simulant debris. Ann. Nucl. Energy
**2018**, 111, 474–486. [Google Scholar] [CrossRef] - Park, S.; Park, H.S.; Jeun, G.; Cho, B.J. Three-Dimensional Modeling of Debris Mixing and Sedimentation in Severe Accidents Using the Moving Particle Semi-Implicit Method Coupled with Rigid Body Dynamics. Nucl. Technol.
**2013**, 181, 227–239. [Google Scholar] [CrossRef] - Vasilyev, B.A.; Shepelev, S.F.; Ashirmetov, M.R.; Poplavsky, V.M. BN-1200 Reactor Power Unit Design Development. In Proceedings of the International Conference on Fast Reactors and Related Fuel Cycles: Safe Technologies and Sustainable Scenarios (FR13), Paris, France, 4–7 March 2013. [Google Scholar]
- Harvey, J.; Narayanan, K.S.; Das, S.K.; Rao, E.V.H.M.; Lydia, G.; Malarvizhi, B.; Murthy, S.S.; Kumaresan, M.; Kasinathan, N.; Rajan, M. Assessment of Debris Bed Formation Characteristics Following Core Melt Down Scenario with Simulant System. In Proceedings of the 16th International Conference on Nuclear Engineering, Orlando, FL, USA, 11–15 May 2008. [Google Scholar]
- Ma, W.; Dinh, T.-N.; Buck, M.; Buerger, M. Analysis of the Effect of Bed Inhomogeneity on Debris Coolability. In Proceedings of the 15th International Conference on Nuclear Engineering (ICONE15), Nagoya, Japan, 22–26 April 2007. [Google Scholar]
- Yakush, S.; Kudinov, P.; Dinh, T.N. Modeling of Two-Phase Natural Convection Flows in a Water Pool with a Decay-Heated Debris Bed. In Proceedings of the 2008 International Congress on Advances in Nuclear Power Plants (ICAPP’08), Anaheim, CA, USA, 8–12 June 2008. [Google Scholar]
- Xiong, Z.; Cheng, S.; Xu, R.; Tan, Y.; Zhang, H.; Xu, Y. Experimental study on eutectic reaction between fuel debris and reactor structure using simulant materials. Ann. Nucl. Energy
**2020**, 139, 107284. [Google Scholar] [CrossRef] - Hotta, A.; Akiba, M.; Morita, A.; Konovalenko, A.; Villanueva, W.; Bechta, S.; Komlev, A.; Thakre, S.; Hoseyni, S.M.; Sköld, P.; et al. Experimental and Analytical Investigation of Formation and Cooling Phenomena in High Temperature Debris Bed. J. Nucl. Sci. Technol.
**2020**, 57, 353–369. [Google Scholar] [CrossRef] - Alvarez, D.; Amblard, M. Fuel Levelling. In Proceedings of the 5th Information Exchange Meeting on Post Accident Debris Cooling, Karlsruhe, Germany, 28–30 July 1982. [Google Scholar]
- Bürger, M.; Buck, M.; Schmidt, W.; Widmann, W. Validation and application of the WABE code: Investigations of constitutive laws and 2D effects on debris coolability. Nucl. Eng. Des.
**2006**, 236, 2164–2188. [Google Scholar] [CrossRef] - Yakush, S.; Kudinov, P. Simulation of Ex-Vessel Debris Bed Formation and Coolability in a LWR Severe Accident. In Proceedings of the ISAMM-2009, Böttstein, Switzerland, 25–28 October 2009. [Google Scholar]
- Ma, W.; Dinh, T.-N. The effects of debris bed’s prototypical characteristics on corium coolability in a LWR severe accident. Nucl. Eng. Des.
**2010**, 240, 598–608. [Google Scholar] [CrossRef] - Karbojian, A.; Ma, W.M.; Kudinov, P.; Dinh, T. A scoping study of debris bed formation in the DEFOR test facility. Nucl. Eng. Des.
**2009**, 239, 1653–1659. [Google Scholar] [CrossRef] - Zhang, B.; Harada, T.; Hirahara, D.; Matsumoto, T.; Morita, K.; Fukuda, K.; Yamano, H.; Suzuki, T.; Tobita, Y. Experimental investigation on self-leveling behavior in debris beds. Nucl. Eng. Des.
**2011**, 241, 366–377. [Google Scholar] [CrossRef] - Cheng, S.; Tagami, H.; Yamano, H.; Suzuki, T.; Tobita, Y.; Zhang, B.; Matsumoto, T.; Morita, K. Evaluation of debris bed self-leveling behavior: A simple empirical approach and its validations. Ann. Nucl. Energy
**2014**, 63, 188–198. [Google Scholar] [CrossRef] - Cheng, S.; Tagami, H.; Yamano, H.; Suzuki, T.; Tobita, Y.; Taketa, S.; Nishi, S.; Nishikido, T.; Zhang, B.; Matsumoto, T.; et al. An investigation on debris bed self-leveling behavior with non-spherical particles. J. Nucl. Sci. Technol.
**2014**, 51, 1096–1106. [Google Scholar] [CrossRef] - Takasuo, E. An experimental study of the coolability of debris beds with geometry variations. Ann. Nucl. Energy
**2016**, 92, 251–261. [Google Scholar] [CrossRef] - Basso, S.; Konovalenko, A.; Yakush, S.E.; Kudinov, P. The effect of self-leveling on debris bed coolability under severe accident condi-tions. Nucl. Eng. Des.
**2016**, 305, 246–259. [Google Scholar] [CrossRef] - Cheng, S.; Hirahara, D.; Tanaka, Y.; Gondai, Y.; Zhang, B.; Matsumoto, T.; Morita, K.; Fukuda, K.; Yamano, H.; Suzuki, T.; et al. Experimental investigation of bubbling in particle beds with high solid holdup. Exp. Therm. Fluid Sci.
**2011**, 35, 405–415. [Google Scholar] [CrossRef] - Guo, L.; Morita, K.; Tobita, Y. Numerical simulations on self-leveling behaviors with cylindrical debris bed. Nucl. Eng. Des.
**2017**, 315, 61–68. [Google Scholar] [CrossRef] - Tagami, H.; Cheng, S.; Tobita, Y.; Morita, K. Model for particle behavior in debris bed. Nucl. Eng. Des.
**2018**, 328, 95–106. [Google Scholar] [CrossRef] - Li, C.-Y.; Wang, K.; Pellegrini, M.; Erkan, N.; Okamoto, K. Numerical Simulation and Validation of Debris Bed Self-Leveling Behavior with Mixed-Density Particles Using CFD-DEM Coupling Algorithm. Nucl. Technol.
**2022**, 208, 843–859. [Google Scholar] [CrossRef] - Pohlner, G.; Vujic, Z.; Bürger, M.; Lohnert, G. Simulation of melt jet breakup and debris bed formation in water pools with IKE-JET/IKEMIX. Nucl. Eng. Des.
**2006**, 236, 2026–2048. [Google Scholar] [CrossRef] - Sun, R.; Wu, L.; Ding, W.; Chen, R.; Tian, W.; Qiu, S.; Su, G. From melt jet break-up to debris bed formation: A review of melt evolution model during fuel-coolant interaction. Ann. Nucl. Energy
**2022**, 165, 108642. [Google Scholar] [CrossRef] - Xiang, Y.; Thakre, S.; Ma, W.; Bechta, S. A scoping study on debris bed formation from metallic melt coolant interactions. Nucl. Eng. Des.
**2021**, 385, 111533. [Google Scholar] [CrossRef] - Xiang, Y.; Deng, Y.; Fang, D.; Zhao, N.; Ma, W. Experimental investigation on ex-vessel debris bed formation using low melting-point melt of binary metals. Prog. Nucl. Energy
**2023**, 157. [Google Scholar] [CrossRef] - Iwasawa, Y.; Sugiyama, T.; Abe, Y. Experiments of melt jet-breakup for agglomerated debris formation using a metallic melt. Nucl. Eng. Des.
**2022**, 386, 111575. [Google Scholar] [CrossRef] - Modak, M.; Nirgude, V.V.; Park, H.S.; Choi, Y.J.; Seo, M.R. DAVINCI-SP tests for debris bed formation and spreading in a pool with center conical structure under two-phase condition. Nucl. Eng. Des.
**2023**, 405. [Google Scholar] [CrossRef] - Kondo, S.; Konishi, K.; Isozaki, M.; Imahori, S.; Furutani, A.; Brear, D. Experimental study on simulated molten jet-coolant interactions. Nucl. Eng. Des.
**1995**, 155, 73–84. [Google Scholar] [CrossRef] - Magallon, D.; Hohmann, H.; Schins, H. Pouring of 100-kg-Scale Molten UO2 into Sodium. Nucl. Technol.
**1992**, 98, 79–90. [Google Scholar] [CrossRef] - Tobita, Y.; Kamiyama, K.; Tagami, H.; Matsuba, K.-I.; Suzuki, T.; Isozaki, M.; Yamano, H.; Morita, K.; Guo, L.; Zhang, B. Development of the evaluation methodology for the material relocation behavior in the core disruptive accident of sodium-cooled fast reactors. J. Nucl. Sci. Technol.
**2016**, 53, 698–706. [Google Scholar] [CrossRef] - Mathai, A.M.; Sharma, A.K.; Anandan, J.; Malarvizhi, B.; Das, S.K.; Nashine, B.; Chellapandi, P. Investigation of fragmentation phenomena and debris bed formation during core meltdown accident in SFR using simulated experiments. Nucl. Eng. Des.
**2015**, 292, 87–97. [Google Scholar] [CrossRef] - Yakush, S.; Kudinov, P.; Dinh, T.N. Multiscale Simulations of Self-Organization Phenomena in the Formation and Coolability of Corium Debris Bed. In Proceedings of the 13th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-13), Kanazawa, Japan, 27 September–2 October 2009. [Google Scholar]
- Shamsuzzaman, M.; Horie, T.; Fuke, F.; Kai, T.; Zhang, B.; Matsumoto, T.; Morita, K.; Tagami, H.; Suzuki, T.; Tobita, Y. Experimental Investigation of Debris Sedimentation Behaviour on Bed Formation Characteristics. In Proceedings of the Eighth Japan-Korea Symposium on Nuclear Thermal Hydraulics and Safety (NTHAS8), Beppu, Japan, 9–12 December 2012. [Google Scholar]
- Shamsuzzaman, M.; Matsumoto, T.; Kamiyama, M.; Morioka, T.; Morita, K.; Tagami, H.; Suzuki, T.; Tobita, Y. Experimental study on sedimentation behavior of core debris. In Proceedings of the Ninth Korea-Japan Symposium on Nuclear Thermal Hydraulics and Safety (NTHAS9), Buyeo, Republic of Korea, 16–19 November 2014. [Google Scholar]
- Sheikh, M.; Son, E.; Kamiyama, M.; Morioka, K.; Matsumoto, T.; Morita, K.; Matsuba, K.; Kamiyama, K.; Suzuki, T. Experimental Investigation on Characteristics of Mixed Particle Debris in Sedimentation and Bed Formation Behavior. In Proceedings of the 11th International Topical Meeting on Nuclear Reactor Thermal Hydraulics, Operation and Safety (NUTHOS-11), Gyeongju, Republic of Korea, 9–13 October 2016. [Google Scholar]
- Sudha, A.J.; Murthy, S.; Kumaresan, M.; Lydia, G.; Nashine, B.; Chellapandi, P. Experimental analysis of heaping and self-levelling phenomena in core debris using lead spheres. Exp. Therm. Fluid Sci.
**2015**, 68, 239–246. [Google Scholar] [CrossRef] - Cheng, S.; Zhang, T.; Cui, J.; Gong, P.; Qian, Y. Insight from Recent Experimental and Empirical-Model Studies on Flow-Regime Charac-teristics in Debris Bed Formation Behavior. J. Nucl. Eng. Radiat. Sci.
**2018**, 4, 031003. [Google Scholar] [CrossRef] - Cheng, S.; Wang, S.; Jiang, G.; Yu, J.; Qian, Y.; Gong, P.; Cui, J. Development and analysis of a regime map for predicting debris bed formation behavior. Ann. Nucl. Energy
**2017**, 109, 658–666. [Google Scholar] [CrossRef] - Cheng, S.; Gong, P.; Wang, S.; Cui, J.; Qian, Y.; Zhang, T.; Jiang, G. Investigation of flow regime in debris bed formation behavior with nonspherical particles. Nucl. Eng. Technol.
**2018**, 50, 43–53. [Google Scholar] [CrossRef] - Cheng, S.; He, L.; Zhu, F.; Wang, J.; Xu, R.; Zhang, H.; Tan, Y.; Xu, Y. Experimental study on flow regimes in debris bed formation behavior with mixed-size particles. Ann. Nucl. Energy
**2019**, 133, 283–296. [Google Scholar] [CrossRef] - Cheng, S.; Cui, J.; Qian, Y.; Gong, P.; Zhang, T.; Wang, S.; Jiang, G. An experimental investigation on flow-regime characteristics in debris bed formation behavior using gas-injection. Ann. Nucl. Energy
**2018**, 112, 856–868. [Google Scholar] [CrossRef] - Cheng, S.; He, L.; Wang, J.; Zhu, F.; Cui, J. An experimental study on debris bed formation behavior at bottom-heated boiling condition. Ann. Nucl. Energy
**2019**, 124, 150–163. [Google Scholar] [CrossRef] - Xu, R.; Cheng, S.; Xu, Y.; Tan, Y.; Zhang, H. Investigations on flow-regime characteristics during debris bed formation behavior in sodium-cooled fast reactor by releasing high-temperature particles. Nucl. Eng. Des.
**2022**, 395, 111866. [Google Scholar] [CrossRef] - Dinh, N.; Ma, W.; Karbojian, A.; Kudinov, P.; Tran, C.T.; Hansson, C.R. Ex-Vessel Corium Coolability and Steam Explosion Energetics in Nordic Light Water Reactors; NKS-160; KTH-NPS-SARAM-071001; Royal Institute of Technology (KTH): Stockholm, Sweden, 2008. [Google Scholar]
- Shamsuzzaman, M.; Zhang, B.; Horie, T.; Fuke, F.; Matsumoto, T.; Morita, K.; Tagami, H.; Suzuki, T.; Tobita, Y. Numerical study on sedimentation behavior of solid particles used as simulant fuel debris. J. Nucl. Sci. Technol.
**2014**, 51, 681–699. [Google Scholar] [CrossRef] - Buckingham, E. On Physically Similar Systems; Illustrations of the Use of Dimensional Equations. Phys. Rev.
**1914**, 4, 345–376. [Google Scholar] [CrossRef] - Fan, L.-S.; Zhu, C. Principles of Gas-Solid Flows; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Chen, Y.; Ma, W. Development and application of a surrogate model for quick estimation of ex-vessel debris bed coolability. Nucl. Eng. Des.
**2020**, 370, 110898. [Google Scholar] [CrossRef] - Geldart, D. Gas Fluidization Technology; John Wiley & Sons: Chichester, UK, 1986. [Google Scholar]
- Geldart, D. Estimation of basic particle properties for use in fluid—Particle process calculations. Powder Technol.
**1990**, 60, 1–13. [Google Scholar] [CrossRef] - Cheng, S.; Li, X.; Liang, F.; Li, S.; Li, K. Study on sloshing motion in a liquid pool with non-spherical particles. Prog. Nucl. Energy
**2019**, 117, 103086. [Google Scholar] [CrossRef] - Phan, L.H.S.; Ngo, P.M.; Miura, R.; Tasaki, Y.; Matsumoto, T.; Liu, W.; Morita, K. Self-leveling behavior of mixed solid particles in cylindrical bed using gas-injection method. J. Nucl. Sci. Technol.
**2019**, 56, 111–122. [Google Scholar] [CrossRef] - Xu, R.; Cheng, S.; Li, S.; Cheng, H. Knowledge from recent investigations on sloshing motion in a liquid pool with solid particles for severe accident analyses of sodium-cooled fast reactor. Nucl. Eng. Technol.
**2022**, 54, 589–600. [Google Scholar] [CrossRef] - Tagami, H.; Tobita, Y. Numerical Simulation for Debris Bed Behavior in Sodium Cooled Fast Reactor. In Proceedings of the 10th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-10), Okinawa, Japan, 14–18 December 2014. [Google Scholar]
- Cheng, S.; Yamano, H.; Suzuki, T.; Tobita, Y.; Nakamura, Y.; Zhang, B.; Matsumoto, T.; Morita, K. Empirical correlations for predicting the self-leveling behavior of debris bed. Nucl. Sci. Tech.
**2013**, 24, 010602. [Google Scholar]

**Figure 1.**Formation of the debris beds [24].

**Figure 4.**Schematic illustrations of key quantities for representing the accumulated-bed characteristics. (

**a**) Particle mound/bed height and repose angle; (

**b**) dimple area; and (

**c**) dimple volume [29].

**Figure 7.**Extrapolated data of the predicted bed heights with spherical fuel particles. (

**a**) H

_{n}= 720 mm, d

_{n}= 30 mm, and (

**b**) H

_{n}= 720 mm, d

_{n}= 40 mm [63].

**Figure 8.**Positions of measured apparent surface areas of beds (Position 1: particle bed top; Position 2: middle of particle mound; Position 3: bottom of particle mound; Position 4: bottom of particle bed) [64].

**Figure 9.**Distribution of mixed particles for particle beds (d

_{n}= 40 mm). (

**a**) Al

_{2}O

_{3}mixed-sized particles (d

_{p}= 2 and 6 mm, mixing ratio = 1:1), and (

**b**) Al

_{2}O

_{3}and SS mixed-density particles (d

_{p}= 2 mm, mixing ratio = 1:1) [64].

**Figure 10.**Experimental and predicted H

_{b}/d

_{n}values (“Exp.” and “Pred.” mean experimental and predicted results, respectively; Symbols in the figure show the data of experimental cases performed) [28].

**Figure 11.**Experimental Visualization system for investigations focusing on flow regime characteristics. (

**a**) Schematic view of experimental system, and (

**b**) detailed view of the main apparatus [24].

**Figure 12.**Typical flow-regime characteristics (glass spherical particles, H

_{n}= 110 cm, d

_{n}= 30 mm, H

_{w}= 60 cm, and W

_{tank}= 60 mm) (arrows represent the flow directions); (

**a**) d

_{p}= 0.25 mm; (

**b**) d

_{p}= 0.5 mm; (

**c**) d

_{p}= 2 mm; and (

**d**) d

_{p}= 8 mm [69].

**Figure 14.**Illustrations for the base angle, vertex angle, and center angle. (

**a**) Pool-convection dominant regime (Regime II), and (

**b**) particle-inertia dominant regime (Regime IV) [24].

**Figure 15.**Development of regime maps in accordance with the empirical model focusing on flow regime characteristics. (

**a**) Single-sized spherical particles [67]; (

**b**) single-sized and non-spherical particles with the extension scheme coupled [68]; (

**c**) single-sized and mixed-sized spherical particles with the base model using d

_{a}[69]; (

**d**) single-sized and mixed-sized spherical particles with the extended model [69]; and (

**e**) boiling condition with releasing heated particles [72].

**Figure 17.**Transit gas-injection DBF behavior for different flow regimes (glass spheres) (Arrows representing the visually-observed flow directions). (

**a**) Regime I (d

_{p}= 0.125 mm, Q

_{g}= 20 L/min); (

**b**) Regime II (d

_{p}= 0.5 mm, Q

_{g}= 10 L/min); and (

**c**) Regime IV (d

_{p}= 6 mm, Q

_{g}= 50 L/min) [70].

**Figure 18.**Influence of gas flow rate on DBF behavior in cases of different flow regimes. (

**a**) Regime I; (

**b**) Regime II; and (

**c**) Regime IV [70].

**Figure 20.**DBF flow-regime characteristics with boiling liquid surrounding the falling particles (Al

_{2}O

_{3}spheres, d

_{p}= 0.5 mm, d

_{n}= 30 mm, and T

_{w}= 368 K) (Arrows representing the visually-observed flow directions). (

**a**) Regime II (T

_{p}= 473 K); (

**b**) Regime III (T

_{p}= 523 K); and (

**c**) Regime IV (T

_{p}= 673 K) [72].

Physical Quantities | Condition | ||
---|---|---|---|

Reactor Accident | Experiments Focusing on Overall Accumulated-Bed Characteristics | Experiments Focusing on Flow-Regime Characteristics | |

Material of debris | Mixture of MOX fuel and SS | Al_{2}O_{3}, ZrO_{2}, and SS | Glass, Al_{2}O_{3}, ZrO_{2}, SS, Cu, and Pb |

Density of debris (kg/m^{3}) | 7620 (SS)∼10,800 (MOX fuel) at 1000 K | 3600 (Al_{2}O_{3}), 6000 (ZrO_{2}), 7800 (SS) at 298 K | 2600 (Glass), 3600 (Al_{2}O_{3}), 6000 (ZrO_{2}), 7900 (SS), 8900 (Cu), and 11,340 (Pb) at 298 K |

Diameter of debris | 0.1 mm to several millimeters | 1.1~6.0 mm | 0.125~8.0 mm |

Liquid coolant | Sodium | Water | Water |

Coolant density (kg/m^{3}) | 830 | 997 | 997 |

Viscosity (Pa.s) | 2.40 × 10^{−4} | 8.91 × 10^{−4} | 8.91 × 10^{−4} |

Modeling for Single-Sized and Single-Shaped Solid Particle Cases | Modeling for Mixed Solid Particle Cases | ||
---|---|---|---|

Empirical Constant Index | Value | Empirical Constant Index | Value |

a_{1} | −0.425 | a_{2} | 1.36 |

b_{1} | −0.103 | b_{2} | 3.76 |

c_{1} | −0.407 | c_{2} | 0.589 |

d_{1} | −0.161 | d_{2} | −0.661 |

e_{1} | −0.585 | e_{2} | −1.60 |

f_{1} | 0.456 | f_{2} | −0.545 |

g_{1} | 0.035 | g_{2} | 0.490 |

h_{1} | −0.137 | k_{2} | 162,755 |

k_{1} | 0.2049 |

Empirical Constant | Value |
---|---|

a | 1.20 |

b | 0.60 |

c | 0.90 |

d | 1.05 |

K_{B} | 1.01 × 10^{−8} |

s_{1} | 65.5 |

s_{2} | 0.263 |

s_{3} | 0.441 |

p_{1} | 122.451 |

p_{2} | 0.555 |

p_{3} | 0.625 |

r_{1} | 0.016 |

r_{2} | 1.295 |

r_{3} | 0.012 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xu, R.; Cheng, S.
Characteristics and Mechanisms of Debris Bed Formation Behavior in Severe Accidents of Sodium-Cooled Fast Reactors: Experimental and Modeling Studies. *Appl. Sci.* **2023**, *13*, 6329.
https://doi.org/10.3390/app13116329

**AMA Style**

Xu R, Cheng S.
Characteristics and Mechanisms of Debris Bed Formation Behavior in Severe Accidents of Sodium-Cooled Fast Reactors: Experimental and Modeling Studies. *Applied Sciences*. 2023; 13(11):6329.
https://doi.org/10.3390/app13116329

**Chicago/Turabian Style**

Xu, Ruicong, and Songbai Cheng.
2023. "Characteristics and Mechanisms of Debris Bed Formation Behavior in Severe Accidents of Sodium-Cooled Fast Reactors: Experimental and Modeling Studies" *Applied Sciences* 13, no. 11: 6329.
https://doi.org/10.3390/app13116329