# Numerical Study on the Potential of Cavitation Damage in a Lead–Bismuth Eutectic Spallation Target

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Configuration of LBE Spallation Target Head

^{3}. To cool the irradiation samples and the sample holders, a rectification lattice with square apertures is installed in front of the samples. The size of the rectification lattice is 52 × 52 × 5 mm

^{3}, and the size of each square aperture is 4 × 4 mm

^{2}. Furthermore, to cool the side wall of the sample holder, a 2 mm-wide slit is arranged, adjacent to each edge of the rectification lattice, to direct the LBE flow. When the LBE flow returns from the BW to pass through the rectification lattice area, its flow path shrinks sharply. Therefore, the rectification lattice acts as an orifice.

## 3. Heat Deposition in LBE

^{3}, 55 J/cm

^{3}, and 82 J/cm

^{3}, for the beam current densities of 20 μA/cm

^{2}, 40 μA/cm

^{2}, and 60 μA/cm

^{2}, respectively. Considering that the Bragg-peak in the Y-direction appeared only 158 mm downstream of BW center, the length of the straight tube in the numerical simulation models was set to 300 mm, to ensure that the calculation of heat deposition in the LBE was not affected and to reduce the model size and calculation cost, to a great extent. It was expected that pressure waves would be generated in the LBE owing to the rapid depostion of large amounts of heat.

## 4. Numerical Simulation Models

#### 4.1. Model for Calculating Dynamics of Pressure Waves

^{−8}s, which was sufficiently small to ensure a stable explicit calculation. For the boundary condition, the maximum of the Y-direction was considered non-reflective. The solid liquid contact was set to an automatic contact type, according to which the sliding between the solid and the liquid was allowed, but not separation. Here the cut-off model was not applied to the boundary because the generated negative pressure was considerably small. For simplicity, the rectification lattice and irradiation samples were not included in the model. The rise in temperature of liquid metal due to the injection of proton beams, was assigned to the elements in the numerical simulation model, which could be calculated using Equation (1):

^{3}), $\rho $ is the density of liquid metal (kg/m

^{3}), and ${C}_{P}$ is the specific heat of liquid metal (J/ kg·K).

#### 4.2. Computational Fluid Dynamics (CFD) Analysis Model

^{4}under nominal operation condition of the LBE spallation target; hence, turbulent flow was formed in the flow channel. The high-Re k-epsilon (k-ε) turbulence model was adopted, which was considered to be suitable to simulate the main LBE flow features in the flow channel.

^{−5}s, so the Courant number was far less than 1, which guaranteed solution convergence. Mesh sensitivity tests were performed as well, and mesh-independent results were obtained, using the present mesh structures. The effects of gravity on LBE flow was considered in the simulations. A non-slip condition was applied at the wall surface with zero velocity.

^{−6}m

^{3}/s, at a temperature of 350 °C. The inlet velocity around the annular channel was assumed to be uniform. LBE at this flow rate was adequate to cool the BW, while avoiding severe erosion/corrosion damage to the target vessel, owing to the rapid LBE flow. The pressure at the outlet boundary was set to a constant value of zero. The outer surface condition of the BW was set to adiabatic, because a very small amount of heat is lost by conduction and radiation from the BW to the surrounding vacuum environment.

## 5. Results and Discussions

#### 5.1. Cavitation Damage due to Pressure Waves

#### 5.1.1. Time Response of Pressure

^{2}. The pressure distribution contour was obtained at 1 μs. After injection of the proton beam, positive pressure in the LBE target rose immediately. On the one hand, the pressure at maximum heat deposition increased as the temperature of the LBE increased. On the other hand, pressure waves were generated, owing to rapid expansion and spreading around of LBE, while the propagation of the pressure waves led to a decrease in pressure at the position of maximum heat deposition. Owing to the effects of the two above-mentioned factors, the pressure at the maximum heat deposition position increased to 0.6 MPa at 13 μs and then decreased subsequently. As the pulse duration was 0.5 ms, the pressure wave dynamics became very complex and pressure waves propagating from various directions or those reflected by the boundaries were superimposed on each other, at the detection position.

^{2}. The high-frequency components in the time responses could be attributed to an impulsive contact between the LBE and target vessel, during pressure wave propagation. The maximum negative pressure peak of ca. −0.03 MPa appeared at 1.6 ms, followed by a positive pressure of ca. 0.6 MPa at 1.8 ms.

#### 5.1.2. Threshold of Cavitation Damage Initialization

^{3}), $\sigma $ is surface tension (N/m), $\eta $ is viscosity of liquid metal (Pa·s),$\gamma $ is specific heat ratio, ${P}_{g}$ is pressure in the bubble (Pa), ${P}_{b}$ is pressure in the liquid surrounding the bubble (Pa), ${P}_{V}$ is vapor pressure (Pa), ${P}_{0}$ is static pressure in the liquid (Pa), and $P$ is pressure oscillation near the bubble (Pa).

^{9}pulses of proton beam injections, and there was hardly any cavitation damage on the test specimens, when the MIMTM power was lower than 185 W [17,18,19]. In this case, the maximum cavitation bubble radius was approximately 250 μm, as given in Reference [20].

#### 5.1.3. Cavitation Bubble Expansion for LBE Spallation Target

^{2}. The responses are shown in the form of bubble expansion ratio, $R/{R}_{0}$. The bubble expansion ratio was smaller than 1.4 in all cases. The effects of beam current density on the bubble expansion ratio were studied as well, and the results are shown in Figure 9. The bubble core radius, ${R}_{0}$, was set to 20 μm. Although the beam current density increased from 20 μA/cm

^{2}to 60 μA/cm

^{2}, the maximum bubble expansion ratio did not exceed 1.4. This bubble expansion ratio was considerably lower than the threshold, which was discussed in the previous sub-section. Such a small expansion ratio of cavitation bubbles would not cause severe cavitation damage on the LBE vessel.

^{3}and the resulting temperature rise was 9.2 K; therefore, the maximum pressure generated was 43 MPa, according to Equations (1–2). Pulse duration of the proton beam was 1 μs, in the case of the mercury target, so the rate of pressure rise was 43 MPa/μs. By contrast, in the case of the LBE spallation target, when the heat deposition in the LBE due to one proton beam pulse was 34 J/cm

^{3}(20 μA/cm

^{2}), the maximum rise in the temperature of the LBE was 22.4 K, and the maximum pressure generated was 89.5 MPa. However, the pulse duration of the proton beam, in case of the LBE spallation target was 500 μs, so that the pressure rising rate was 0.18 MPa/μs. Therefore, the intensity of the pressure waves generated in the LBE was considerably weaker than that in mercury, leading to a considerably weaker and shorter duration of the negative pressure generated in the LBE adjacent to the BW. The resulting cavitation bubble expansion ratio was considerably smaller in the LBE.

#### 5.2. Cavitation Damage due to LBE Flow

#### 5.2.1. Steady-State Flow

^{−7}Pa, and it could be neglected. Therefore, to drive the growth of cavitation bubble cores, the inlet LBE flow speed should have been greater than 0.4 m/s, to generate a negative pressure adequate for overcoming the cover gas pressure, as can be deduced from the relationship shown in Figure 12. However, during normal operations of the LBE spallation target, LBE flow speed did not hit 0.4 m/s, as the maximum inlet flow speed was 0.25 m/s, considering the ability of the electro-magnetic pump (EMP).

^{4}[24]. The ${R}_{e}$ in the rectification lattice region of the LBE target had a similar value. Therefore, based on the value of the cavitation number, cavitation would not occur in the rectification lattice. To summarize, it was considered that the cavitation damage would likely not be caused by the turbulent flow of LBE, under a normal steady-state flow condition.

#### 5.2.2. Transient Flow

^{2}, according to the test performed using a mock-up loop of the target primary cooling system. Transient analysis was performed under this transient flow condition, as the first step. Figure 13 shows a snapshot of the averaged pressure distribution contour in the LBE; the time-point was 10 μs. Negative pressure with a maximum magnitude of approximately 0.047 MPa, was generated in the rectification lattice region, owing to the pressure change caused by a decrease of flow in velocity.

^{2}. However, when the initial inlet flow speed was 0.25 m/s, the expansion ratio increased abruptly to more than 100 and 600, when the rates of change in flow speed were 1.25 m/s

^{2}and 12.5 m/s

^{2}, respectively. Such a large expansion ratio was adequate for causing severe cavitation damage on the vessel wall. The above results suggest that cavitation damage would perhaps not have occurred under the nominal flow condition, even if the inlet flow speed decreased from 0.125 m/s to 0 m/s, in 0.01 s, which was apparently an assumed extreme abnormal state of target operation. However, in the case of 0.25 m/s, if the rate of change in inlet flow speed exceeded approximately 0.2 m/s

^{2}, the expansion ratio of the cavitation bubbles would be greater than 12.5 as a result; in such a case, the possibly of a cavitation damage could not have been avoided.

^{2}), and $b$, $c$, and $d$ are constants related to the material property.

^{2}and 0.00125 m/s

^{2}, respectively. The solid black line represent the initial pressure in the bubble. The negative pressure in the case of 1.25 m/s

^{2}was lower than that in the case of 0.00125 m/s

^{2}, and it was sustained for a longer time, under the bubble initial pressure line, which resulted in a considerably higher bubble expansion ratio, relative to that in the case of 0.00125 m/s

^{2}, as shown in Figure 15b. The amplitude of negative pressure and its duration were dominant factors governing the growth behavior of a cavitation bubble.

## 6. Conclusions

- The intensity of pressure waves generated in the LBE was found to be weak due to the relatively long duration of the proton beam pulse. Therefore, the expansion ratio of the cavitation bubbles due to the pressure waves, was only 1.4, which was considerably lower than the threshold ratio that could lead to severe cavitation damage on the vessel.
- The magnitude of maximum negative pressure had a second power law relationship with the flow speed of the HLM. For the nominal inlet flow speed of 0.125 m/s, the negative pressure induced by the steady-state LBE flow was only −0.013 MPa, which was considerably smaller than the cover gas pressure of the LBE spallation target; therefore, this pressure could not drive the growth of the cavitation bubbles.
- For the transient LBE flow, negative pressure was generated in the LBE, due to a decrease in LBE flow velocity. Under normal target operation conditions, the duration of negative pressure was too short to drive the growth of adequately large cavitation bubbles. However, cavitation might have occurred under a few extreme flow variation conditions, for example, when the rate of change in inlet flow was higher than 0.2 m/s
^{2}, even as the initial inlet flow speed was 0.25 m/s. - The maximum cavitation bubble dynamics due to turbulent flow in an orifice could be classified into two stages. In the first stage, the maximum cavitation bubble expansion ratio shared a power law relationship with the inlet flow speed, but it was almost independent of the inlet flow speed change rate; in the second stage, the maximum cavitation bubble expansion ratio shared, a power law relationship with, both, the inlet flow speed and the rate of change in the inlet flow speed.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic drawing and dimensions of design of the Lead–Bismuth Eutectic (LBE) spallation target head: (

**a**) Schematic drawing of LBE spallation target head; and (

**b**) dimensions of the LBE spallation target head.

**Figure 2.**Energy deposition in LBE due to one pulse proton beam injection with various peak beam current densities: (

**a**) In radial-direction, Y = 15.8 cm; and (

**b**) in Y-direction, X = 0 cm.

**Figure 3.**Schematic drawing of half 3D model of the LBE spallation target: (

**a**) For calculating dynamics of pressure waves; and (

**b**) an example of the mesh structure.

**Figure 4.**Schematic drawing of a quarter 3D model of the LBE spallation target: (

**a**) For CFD analysis; and (

**b**) an example of the mesh structure of LBE.

**Figure 5.**(

**a**) Pressure distribution contour (at 1 μs); and (

**b**) time response of pressure at position of maximum heat deposition position in LBE (position B). Beam density = 20 μA/cm

^{2}.

**Figure 6.**Example of time response of pressure at position A (in Figure 3) and displacement of the Beam Window (BW) in Y-direction; beam density = 20 μA/cm

^{2}.

**Figure 8.**Time response of the expansion ratio of cavitation bubbles at Position A, under initial bubble core radii conditions; beam current density = 20 μA/cm

^{2}.

**Figure 9.**Time response of expansion ratio of cavitation bubbles at Position A, under various proton beam conditions; R

_{0}= 20 μm.

**Figure 10.**(

**a**) LBE flow velocity contour of the spallation target head; and (

**b**) an example of LBE flow velocity vector in rectification lattice area; inlet flow speed = 0.125 m/s.

**Figure 11.**(

**a**) Pressure distribution contour in LBE for the spallation target head; and (

**b**) an example of pressure distribution contour in the LBE for the rectification lattice area; inlet flow speed = 0.125 m/s.

**Figure 14.**Bubble expansion ratio as a function of rate of change in inlet flow speed for various inlet flow speeds.

**Figure 15.**(

**a**) Time response of pressure at position of maximum negative pressure; and (

**b**) time response of single bubble dynamics; inlet flow speed = 0.25 m/s.

Physical Properties | Symbol | Unit | 316 SS | LBE |
---|---|---|---|---|

Density | $\rho $ | kg/m^{3} | 7908 | 10450 |

Young’s modulus | $E$ | MPa | 1.742 × 10^{5} | 92.8 |

Poisson’s ratio | $\nu $ | - | 0.3153 | 0.4995 |

Thermal expansion coefficient | $\beta $ | K^{−}^{1} | - | 1.285 × 10^{−4} |

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**MDPI and ACS Style**

Wan, T.; Naoe, T.; Kogawa, H.; Futakawa, M.; Obayashi, H.; Sasa, T.
Numerical Study on the Potential of Cavitation Damage in a Lead–Bismuth Eutectic Spallation Target. *Materials* **2019**, *12*, 681.
https://doi.org/10.3390/ma12040681

**AMA Style**

Wan T, Naoe T, Kogawa H, Futakawa M, Obayashi H, Sasa T.
Numerical Study on the Potential of Cavitation Damage in a Lead–Bismuth Eutectic Spallation Target. *Materials*. 2019; 12(4):681.
https://doi.org/10.3390/ma12040681

**Chicago/Turabian Style**

Wan, Tao, Takashi Naoe, Hiroyuki Kogawa, Masatoshi Futakawa, Hironari Obayashi, and Toshinobu Sasa.
2019. "Numerical Study on the Potential of Cavitation Damage in a Lead–Bismuth Eutectic Spallation Target" *Materials* 12, no. 4: 681.
https://doi.org/10.3390/ma12040681