The Particle-Based Method Utilization on The Various Obstacles Effect to The Two-Fluids Dam Break Simulation

The Molten Salt Fast Reactor (MSFR), a prominent Generation IV reactor design, offers improved safety by eliminating graphite as a moderator and incorporating a passive safety system. Understanding the fuel salt relocation mechanism is crucial in ensuring the reactor’s safety and performance. This study presents a particle-based approach using the Semi-Implicit Moving Particle (MPS) method, which accurately represents fluid as particles carrying essential physical properties. We validate the MPS method as an alternative tool for investigating fuel salt relocation in the MSFR, comparing its results with the well-established δ-SPH method for verification. The simulations encompass scenarios with and without obstacles, involving water-PEO fluids known for their distinct densities and propensity to induce splashing phenomena. The obstacles used are made in several size variations, such as semicircular, rectangular, and triangular obstacles. The excellent alignment of MPS method results with δ-SPH method outcomes confirms its effectiveness in predicting freeze plug melting and fuel salt relocation in the MSFR, providing valuable insights into the passive safety mechanisms. This research contributes to enhancing the safety and efficiency of advanced molten salt reactor designs. Where it can be concluded that the simulation results of the MPS method are almost the same as the simulation results of the δ-SPH method.


Introduction
The development of nuclear power plant can answer the global need for alternative new energy.However, several reactor accidents that have occurred in the past have caused pros and cons among the general public because nuclear waste can damage the surrounding environment.Thus, better understanding on how the accident occurs and progress is utterly essential.
The loss of coolant accident in Fukushima Daichi boosted the passive safety system development of the Gen IV nuclear reactor.One of the candidate occurs of Gen IV reactor is the Molten Salt Reactor (MSR).The freeze plug in Molten Salt Fast Reactor (MSFR) is one of the key feature [1][2][3][4].The freeze plug is relocated in the pipe connecting the reactor core to the safety tank.In this design, the plug will melt due to the temperature difference between the reactor core and the materials melting temperature.This allows the fuel to be drained in a safety tank in case of an emergency [5].There is a melting phenomena of fuel salt while relocate to the safety tank through the freeze plug.The phenomenon of two-fluids dam break with experiment and simulation can represent it.K. Soleimani and M.J. Ketabdari have researched two fluids dam break phenomena using the -SPH method [6].However, the conditions are incompressible, thus MPS method is way more suitable for these phenomena.
The application of advanced simulation methods holds great promise in addressing complex challenges within the nuclear field.Among these methods, the Moving Particle Semi-implicit (MPS) method, developed by Koshizuka and Oka, has emerged as a particularly promising approach [7].Unlike traditional computational methods, MPS is a particle-based technique well-suited to scenarios involving surface deformation, making it an asset in the realm of nuclear engineering.Specifically, in the context of Molten Salt Reactor (MSR) development, where precise modelling of intricate fluid dynamics and core behaviour is paramount, MPS is poised to make significant contributions.Several studies have shown that MPS has been used to simulate the freeze valve of MSR.The MPS method has been used to simulate the freeze valve with additional copper material in the valve [8].The study shows that thicker copper layer may accelerate the melting process.
The objective of this study is to simulate a dam break involving two fluids with varying obstacles using the MPS (Moving Particle Semi-implicit) method.This simulation aims to replicate the flow dynamics that occur when fuel salt is transferred from the MSR (Molten Salt Reactor) core to the drain tank, considering the presence of a freeze plug as a crucial component within this complex system.The results of this simulation are anticipated to provide valuable insights for the development and optimization of MSR systems, particularly in understanding the behaviour of fuel salt during its flow and the role of the freeze plug in ensuring safe and controlled operations.

Governing Equation
Governing Equation is used to govern the incompressible flow.Some of the main governing equations used in the MPS method are the equations of conservation of mass and momentum.
As for the equations for mass conservation: Where,  is the density of the fluid and  is time.Equation ( 1) is the mass conservation equation for creating the system.This equation states that for an incompressible fluid, the fluid density value will be constant, which means that the flow entering the system is the same as the flow leaving the system.So, the density will not change.
And for the law of conservation of momentum: Where   is the material derivative of the fluid velocity,  is the pressure,  is the dynamic viscosity and  ⃗ is velocity.Equation ( 2) is the Navier-Stokes equation which explains that fluid flow dynamics depend on pressure gradients, diffusivity caused by viscosity differences, and external forces in the form of gravity.

Moving Particle Semi-Implicit (MPS) method
The MPS method is a particle-based method or meshless method for analyzing incompressible fluids with free surfaces based on Lagrangian calculations.The difference between two particles can be calculated as shown in Figure 1a.Th symbol of w(r) is a weight function as mentioned in Equation ( 3).The changes of particle's physical properties are influenced by the neighboring particle limited inside the effective radius re definition.A standard w(r) of MPS method is illustrated in Figure 1b.
The sum of all w(r) on particle i that are affected by particle j is called the particle density.Mathematically, the equation for particle density can be approximated by Equation (4).
Where  is the number of dimensions,  0 is the density of the number of particles suitable for the incompressed system, and λ is a coefficient that can be calculated with the following equation, λ is the selected parameter for the obtained result to be proportional to the analytical solution.

Simulation of two-fluid dam break
The dam break flow study was conducted using two types of fluids, water and polyethyleneoxid (PEO) with different densities and kinematic viscosities following simulation of the reference [6].The simulation was conducted with two main variations, with and without obstacle variation.All 2D geometry calculations were conducted with 0.005 s time step storage of simulation results.There might be a slight difference time of presented results with the previous studies in the range of 10 -3 s and considered insignificantly.While the fluid velocity is 1.08 m/s so that with a small time difference compared to fluid motion, it can be ignored because it is not significant.The specifications of two fluids used in the simulation are shown in Table 1.The geometry design of simulation is shown in Figure 2. The blue and right colored particles are water and PEO, respectively.In this study, simulations were carried out with a variety of obstacles, semi-circular, triangular and rectangular obstacles, as shown in Figure 3.The distance of all obstacle variations from the left wall is made the same.The obstacles sizes are shown in Table 2.   Figure 4 compares the results of the MPS method with the experiment and δ-SPH method of nonobstacle dam break simulation.At 0.13 s, the interface wave formed between both fluids with the rolled shape appeared at the end of the encountered position.No secondary wave formed until 0.261 s, although the water level near the wall had already decreased to 80% from the initial position.There followed a 10 cm increase in interface wave height or 12% of potential energy.Therefore, the loss of energy could be predicted as 8%.Both δ-SPH and MPS results showed the secondary wave formed at 0.330 s after the lowest position of interface particles impinged the PEO surface.The momentum transfer happened.Thus, the impinged particles vertically moved and presented the secondary wave.However, no secondary wave has appeared in the experimental result yet.It was caused by the existing gate, which lifted right before two fluids encountered.The lifting process might lead to the higher interface pressure with the opposite direction of two fluid waves, as shown in Figure 4 (b) at 0.131 s.Consequently, there was a wave delay formation in the experiment.The secondary wave of the experiment formed 0.065 s later than that of both simulations.However, at the end of the comparison, the secondary wave height of δ-SPH, experiment, and MPS results were 0.080 m, 0.098 m, and 0.075 m, respectively.It can be concluded that both simulations had close results to the experiment.

Simulation of two-fluid (Water-PEO) dam break with semi-circular obstacle
After comparing all methods, the conclusion was both numerical simulations represented well and had a good agreement with the experiment results.Therefore, in this subsection focused on the comparison of MPS calculation to δ-SPH related to the obstacle presence in the two fluid dam break simulation.Figure 5 compares two fluid dam break simulations with semi-circular obstacles of δ-SPH and MPS methods.At the beginning of the simulation, 0.131 s, the gate utilization in the δ-SPH simulation resulted in the backward press to the water in the x-negative direction.Some PEO particles relocated to be above water particles even it had higher density.Meanwhile, in the MPS simulation, the propagation wave tended to go to an x-positive direction due to the non-existent gate.At 0.260 s, a pressure impinges on the PEO particles when the wave already met the closest obstacle surface.It caused the momentum transfer to water particles, then the wave height increased, and the fluid splashed further until 0.395 s.After that, it all decreased at 0.460 s along with local stratification, in which water particles floated above PEO.Figure 6 shows compares two fluid dam break simulations with rectangular obstacles of δ-SPH and MPS methods.At the beginning of the simulation, 0.131 s, the gate utilization in the δ-SPH simulation resulted in the backward press to the water in the x-negative direction.Some PEO particles relocated to be above water particles even it had higher density.Meanwhile, in the MPS simulation, the propagation wave tended to go to an x-positive direction due to the non-existent gate.At 0.260 s, a pressure impinges on the PEO particles when the wave already met the closest obstacle surface.It caused the momentum transfer to water particles, then the wave height increased, and the fluid splashed further until 0.395 s.

Simulation of two-fluid (Water-PEO) dam break with rectangular obstacle
After that, it all decreased at 0.460 s along with local stratification, in which water particles floated above PEO.In addition, a cavity is formed between the first wave and the second wave.Figure 7 shows compares two fluid dam break simulations with triangular obstacles of δ-SPH and MPS methods.At the beginning of the simulation, 0.131 s, the gate utilization in the δ-SPH simulation resulted in the backward press to the water in the x-negative direction.Some PEO particles relocated to be above water particles even it had higher density.Meanwhile, in the MPS simulation, the propagation wave tended to go to an x-positive direction due to the non-existent gate.At 0.260 s, a pressure impinges on the PEO particles when the wave already met the closest obstacle surface.It caused the momentum transfer to water particles, then the wave height increased, and the fluid splashed further until 0.395 s.After that, it all decreased at 0.460 s along with local stratification, in which water particles floated above PEO.

Discussion
The result for the case of of two-fluid (water-PEO) dam break without obstacle and gate from Figure 4 shows that the simulation results with the MPS method are the same as the experimental and simulation results of the -SPH method.The wave movement is already the same in the direction of the PEO fluid.
For case of of two-fluid (water-PEO) dam break with obstacles variation of semi-circular, rectangular and triangular can be seen as Figure 5 to Figure 7.The MPS and -SPH methods produce different results because the SPH method uses a gate to separate water-PEO, but if reviewed again in the experimental results by [9], it can be seen that the movement of the water-PEO wavefront approaches the PEO fluid.It can be seen that the sharper the edge of the obstacle the higher the height of the jet formed.So in the triangular obstacle, the jet formed is higher than the other obstacles.The spark that occurs starting at time 0.260 s looks different in each case of obstacle.There are also visible voids between the fluid particles and the obstacle wall.This is due to the varying fluid-obstacle contact angles of 109°, 90° and 120° for semicircular, rectangle and triangle respectively as illustrated in Figure 8. Fluid meeting an obstacle with a large contact angle gets greater momentum transfer.This is based on the analysis of the energy equation  =  . (cos ), where  is the contact angle or the projection angle of a given force  along  .This energy is represented as an impulse to the fluid from the obstacle so that a spark is seen.The greater the contact angle, the higher the spark that occurs.This is also reinforced by the graph in Figure 9. Figure 9 shows that the height of the splashing when it starts to hit the surface of the obstacle.Fluid hitting the surface of a triangular obstacle produces the highest spark compared to other obstacles.However, the flow slipped on the obstacle and a long jet was formed that eventually collapsed with the loss of energy.Where the spark that hits the semicircular obstacle collapses faster due to greater energy loss.Figure 10 shows that the front propagation of water-PEO solution interface for case without obstacle is sightly bigger than with obstacles variation.Then for the case with obstacles variation, case with rectangular obstacle is sightly smaller than the triangular and semicircular obstacle.

Conclusion
Simulation of the MPS method with the case of dam break flow of two types of fluids, there are water-PEO without obstacle and with variations in obstacles, validates that this method can be used to simulate cases of freeze plug melting and fuel salt removal during an accident case at MSFR.Comparison of the simulation results of the MPS method with the -SPH method showed a difference in the first two times due to the gate between water and PEO in the -SPH method with varying obstacles.However, if reviewed again the existence of the gate does not really affect the mixing of the two fluids because the gate withdrawal is done very fast.The simulation results revealed that the sharper the edge of the obstacle the higher the height of the jet formed.

Acknowledgments
The author would like thank to ITB.This research is financially supported by Bandung Institute of Technology (ITB) under the 2023 Program Riset ITB, Grant Number: 324H/IT1.C02/KU.
(a) Difference between two particles.(b) Weight function.

Figure 1 .
Figure 1.Visualization of calculating the difference between two particles and the graph for the weight function [7].

Figure 2 .
Figure 2. Design of dam break geometry with two fluids without obstacle [6].

Figure 8 .
Figure 8. Illustration of the contact angle between the fluid and the obstacle wall.

Figure 9 .
Figure 9.The height of the splashing when it starts to hit the surface of the obstacle.

Figure 10 .
Figure 10.Front propagation of water-PEO solution interface versus non-dimensional removal period for case with and without obstacle.

Table 2 .
Size of Obstacles Variation