Numerical Simulation of Particle Flow Motion in a Two-Dimensional Modular Pebble-Bed Reactor with Discrete Element Method

Modular pebble-bed nuclear reactor (MPBNR) technology is promising due to its attractive features such as high fuel performance and inherent safety. Particle motion of fuel and graphite pebbles is highly associated with the performance of pebbled-bed modular nuclear reactor. To understand the mechanism of pebble’s motion in the reactor, we numerically studied the influence of number ratio of fuel and graphite pebbles, funnel angle of the reactor, height of guide ring on the distribution of pebble position, and velocity by means of discrete element method (DEM) in a two-dimensional MPBNR. Velocity distributions at different areas of the reactor as well as mixing characteristics of fuel and graphite pebbles were investigated. Both fuel and graphite pebbles moved downward, and a uniform motion was formed in the column zone, while pebbles motion in the cone zone was accelerated due to the decrease of the cross sectional flow area. The number ratio of fuel and graphite pebbles and the height of guide ring had a minor influence on the velocity distribution of pebbles, while the variation of funnel angle had an obvious impact on the velocity distribution. Simulated results agreed well with the work in the literature.


Introduction
MPBNR technology is currently being focused on around the world, due to its preferable features such as high performance and safety.In the core of MPBNR, fuel and graphite pebbles drain slowly in a continuous process of refuel due to the gravity force on both fuel and graphite pebbles.Such a system has high temperature.Helium is usually used as the coolant in the way of flowing across the aperture of pebbles, raising fundamental questions about dense granular flow characteristics in the reactor.Up to now, dense granular flow of pebbles in MPBNR has been poorly understood and has not been easily accessible to experiments, and yet it has a major impact on reactor physics.
To address this problem, some researchers tried to use either experimental or numerical simulation methods to investigate the flow characteristics of dense granular flow.In the literature, Kadak and Bazant [1] conducted a series of one-to-ten-scale three dimensional experiments to assess the flow lines and the relative velocities of the pebbles with various distances from the center of the core.It was found that the mixing zone of pebbles could be effectively eliminated while maintaining the annular column during the recirculation process.Hassan and Dominguez-Ontiveros [2] presented the full velocity fields of pebbles and gas using particle image velocimetry technique (PIV) in combination with matched refractive index fluid.They obtained the vortex distribution between pebbles and delineated the complex flow structures.
Li and Ji [3] simulated the flow characteristics in the PBNR by coupling DEM and CFD methodologies.The results verified the principal phenomena of pebble landing in the experiment.It was also found that packing statistics did not change prominently during the dynamic recirculation process.They [4] also studied fast simulation of coupled pebble flow and coolant flow with DEM in pebbled bed reactor system by investigating relaxation of coupling frequency, and they got an optimal range.Rycroft et al. [5,6] numerically simulated the particle motion process in MPBNR, and various distributions of particle velocity along radial direction were obtained.They also developed a multiscale simulation method for the process of dense granular drainage based on their spot model.A comparison of the results of DEM simulation and spot model was conducted to validate the flexibility of spot simulation.It showed that the spot model could still create good results.Lately, they [7] numerically investigated the pebble flow of fuel and graphite in a scaled PBR, and they found that pebble flow and dust generation in a scaled-down facility might be significantly different.Li et al. [8] simulated the refueling process by means of the DEM and obtained the influence of frictional coefficients of particle-particle and particle-wall on the particle trajectory, as well as the residence time of spherical particles.Although much work had been done to investigate the pebble flow characteristics in the reactor by considering operating conditions and physical parameters, insufficient information of detailed pebble velocity distributions in the reactor under the influence of particle number ratio, height of guide ring, and so forth.had been obtained.Therefore, we investigated the effect of different parameters such as particle number ratio, height of guide ring as well as funnel angles, on the distribution of pebble velocity, and mixing characteristics of two kinds of pebbles, and discussed their implications for reactor design and the basic physics of granular flow.
With the development of computer and computational technology, numerical simulation method has been gradually regarded to be ideally suited to provide complete information in a granular flow.In this work, we presented DEM simulations of fuel and graphite pebbles which addressed amount of outstanding problems in reactor design.Our simulations were based on the MPBR geometry [9], consisting of spherical pebbles with 6 cm diameter in a cylindrical container.The container was at the height of approximately 10 m, in diameter of 3.84 m, and with bottom funnels angle of 30 ∘ or 60 ∘ .In such a design, there was a central column of guide ring to moderate reflector pebbles, which was surrounded by an annulus of fuel pebbles.Both of the pebble types were physically identical except that the fuel pebbles contained sand-sized uranium fuel particles.Pebble particles were continuously cycled to maintain a certain amount of pebbles in the reactor.

Simulation Model
Three-dimensional simulation is considered to be more realistic, however, it is also much more computing resource consumed compared with two-dimensional simulation, and the latter has a similar simulation results with three-dimensional simulation, thus two-dimensional pebble bed reactor is employed in this simulation.A scheme of the reactor with detailed parameters can be found in Figure 1.In this reactor, both fuel and graphite pebbles are padded in the reactor from the top of the bed with graphite pebbles in the center and fuel pebbles around graphite with the help of guide ring to prohibit the phase mixture at the top of the reactor.Both fuel and graphite pebbles exit from the outlet pipe at the bottom of the reactor due to the gravity force.In this process, effective pebbles can be recycled back to the top of the reactor and ineffective pebbles will be eliminated, while a new pebble of the same kind will be generated randomly at the top of the reactor to keep a continuum pebble flow in the core of the reactor.In the simulation, pebble flow was in the gravity circumstance.Impact of coolant gas on the motion of pebbles and variation of temperature and pressure are neglected.
A schematic model that just keeps the most important components of the reactor such as guide ring, column, cone, and outlet pipe are used in the simulation.Both of the fuel and graphite pebbles are assumed to be smooth and without surface faults.Detailed properties of both kinds of pebbles are listed in Table 1.
In the simulation of pebble flow, the DEM simulation is employed.This model is based on a two-dimensional soft sphere particle model [10,11], and particle collisions take finite time, and binary collision, many-body collision, and sustained contact among particles are possible, thus both collisional and frictional flows may be realized in the model.Therefore, the friction and rotation of particles are also taken into account.Particle motion in the MPBNR follows the movement function of particles, and the conservation of interaction force of particles obeys Newton's Third Law.In the DEM simulation, we calculate particle motion and displacement with the alternation of Newton's Second Law Frictional coefficient between fuel pebbles 0.7 Frictional coefficient between fuel and graphite pebbles 0.7 Frictional coefficient between graphite pebbles 0.7 Time step (s) 5 × 10 −7 and contact force calculation formula and carry out analyzing and coupling calculation of particle motion and collision with DEM program.This method realizes the calculation of macroscale motion of particles.
The motion of particles can be divided into translational and rotational movements.Thus, the movement of a particle with mass  and moment of inertia  can be described by Newton's law and the kinematic relation as where g is the acceleration due to gravity, F  is the total contact force on a particle, v is the translational velocity of the particle, and  is the rotational velocity of the particle.
In soft sphere model, the overlap between two particles is represented as a system of springs, dashpots, and slider in both normal and tangential directions.The contact point of two colliding particles is in the center of deforming area in Figure 2(a) can be described as where,   ,   are the positions of the center of deforming area and centroid of particle ,   is the radius of particle ,  is the total deformation quantity when particle  and particle  collide, and n is the unit vector from particle  to particle .When two particles collide, the spring causes the rebound off the colliding particles, and the dashpot mimics the dissipation of kinetic energy of particles due to inelastic collisions.The spring stiffness coefficients in the normal and tangential directions are   and   , respectively.Similarly, the dashpot damping coefficients in the normal and tangential directions are   and   , respectively.The spring stiffness and dashpot damping coefficients are essential properties of colliding particles.
The contact force F  can be divided into normal force F  and tangential force F  : When |F  | > |F  |, sliding friction occurs between colliding particles, and where  is the frictional coefficient.Generally speaking, the dashpot damping coefficient  is dependent on both the value of critical damping ratio of eigenvalue of viscous damping  and critical damping constant .Consider the following: where   is system mass, kg,

Results and Discussion
Both graphite pebbles and fuel pebbles flow in the center of the reactor.If they can follow an established route to move and mix complementarily, we can effectively control the reactivity of the reactor core to ensure safe operation.The behavior of pebbles is closely related to their velocity.Therefore, velocity distribution has an important effect on the distribution and mixing characteristic of pebbles, thus, has a powerful influence on the safety of operation.

Distribution of Pebbles in the
Reactor. Figure 3 shows the distribution of fuel and graphite pebbles in the reactor at different times without the help of guide ring.One can find that, with the increase of time steps, graphite pebbles flow downward from the top of the reactor to the bottom and exit from the outlet pipe in a way of slight oscillation.In such a process, a clear phase boundary can be found between fuel and graphite pebbles.Extra particles are filled into the reactor from the top of the bed to keep a certain amount of pebbles in the bed. Figure 4 shows the velocity distribution in the core of the reactor.From (a) and (b), we can see that vertical velocity of particles varies slightly at different heights at the column region of the reactor and graphite particles have a comparatively uniform velocity distribution in the center of the core, where fuel particle velocity varies at different heights and different radial positions, behaving an axial symmetry like distribution with slight differences near both sides of the wall.The cause of this phenomenon may be attributed to the fact that particles move downward with uniform velocity when they enter into the bed and the synchronous motion of particles at one side defers the particle velocity at the other side.From (c) we can find that, with the increase of particle velocity, vertical velocity of particles performs a slight variance at both sides of the core due to the friction of the wall at the cone region, however, particles show  a great change of vertical velocity in the center because of the continuous discharge of pebbles from the outlet pipe.Two factors may result in such a velocity distribution.One is the extrusion of particles from both sides of the cone, which defers the vertical velocity near the walls, and the other is the particle properties, such as frictional coefficient and density.One can also find that particles in the column region move randomly at radial direction from (d) and (e), but an overall distribution of radial velocity is that graphite pebbles have a smaller downward radial velocity compared with fuel pebbles.Radial velocity variance increases with the increase of distance from the center point, especially at the point where fuel and graphite pebbles contact each other, and then decreases.This phenomenon indicates that horizontal movement of particles is obvious at the mixing boundary.Figure 4(e) indicates that radial velocity of pebbles has an opposite direction at the center of cone region and increases with the decrease of height.Thus, the mixing characteristics of both fuel and graphite pebbles could be reflected by means of the variance of radial particle velocity.The vertical velocity distribution trend at the upper region of the column agrees with the work of Rycroft et al. [5] in Figure 5, where  is the diameter of the pebble and  = √/ = 0.078 s.

Number Ratio of Fuel and Graphite
Pebbles.In the running process, pebble number ratio varies with the change of the operating conditions to satisfy the different power demanding.Meanwhile, parameters such as shape and size of pebble flower region as well as mixing situation will change.Vertical velocity distributions of pebbles with number ratio of   :   = 1 : 1 are shown in Figure 6.At different particle number ratios, vertical velocity distribution indicates the same pattern in the bed.
Figure 7 shows the horizontal velocity distributions of pebbles at different number ratios.Horizontal velocity of particles increases with the increase of the particle number ratio, indicating that transverse interactions of fuel and graphite pebbles become active and phase mixing turns to be more serious.

Funnel Angle of the Reactor.
When pebbles flow in the bed, an unreasonable design of the bed structure will make pebbles form a detention region, and thus it can hinder the recirculation of pebbles in the bed.A number of parameters such as funnel angle, ratio of outlet pipe diameter, bed diameter, and frictional coefficient between pebbles and walls may have an impact on pebble flow characteristics in the bed.Among all these parameters, funnel angle has a more prominent effect [5].With the comparison of Figures 4 and 8, it can be observed that funnel angle has an obvious influence on the velocity distribution of pebbles in the bed.Vertical velocity of particles decreases obviously when the funnel angle equals 60 ∘ , and variance of vertical velocity differs a little at different heights between the column and cone region.Radial velocity distribution indicates that increase of the funnel angle will decrease the radial velocity of particles substantially, thus it can improve the mixing situation.

Height of Guide
Ring.Generally, with the help of guide ring at the top of the reactor core, mixing phenomenon of fuel and graphite pebbles will be hindered in the process of adding new pebbles, thus it can help to keep a clear phase boundary for both pebbles [5].Vertical and horizontal velocity distributions are shown at guide ring height of 5.5 m and 6.0 m in Figures 9 and 10, respectively.One can find that the variance of the height of guide ring has a minor influence on the particle velocity distribution in the bed.Particle velocity at vertical direction near the walls in the column differs violently but velocity in the core varies smoothly.And vertical velocity of particles in the core of the cone increases with the decrease of the height, especially near the region of phase boundary, where particle velocity variations increase obviously.However, from the point of radial velocity distribution, increase of the height of guide ring will improve the phase mixing of both fuel and graphite pebbles, since the radial velocity of particles decreases with the increase of the height of guide ring.

Conclusion
We noted a number of favorable investigations in our simulation with the discrete element method, which provided some valid information, since the simulations probed the system at a level of detail not easily attained in experiments.Our simulation acquired particle velocity distributions at different regions of the reactor as well as mixing characteristics of fuel and graphite pebbles under different conditions, such as variation of particle number ratio, funnel angle, and height of guide ring.Our DEM simulations predicted a flow pattern of uniform and stable particle flow in the center of column region and an increase particle velocity in center of cone region.The number ratio of fuel and graphite pebbles and the height of guide ring had a minor influence on the velocity distribution of pebbles, while increase of funnel angle would defer the downward flow of particles and thus had a major  impact on the particle velocity distribution.The mixing phenomenon of fuel and graphite pebbles increased with the increase of particle number ratio at phase boundary, however, decreased with the increase of funnel angle and the height of guide ring, among which increase of funnel angle had a great improvement of the phase mixing effect.We would further combine our DEM model for the pebble flow with experiments and computational approaches to reactor core physics in order to implement a more accurate study of the interaction mechanism of gas flow and the pebble flow.

Figure 2 :
Figure 2: Models of contact force in soft sphere approach.

Figure 3 :
Figure 3: Distribution of fuel and graphite pebbles in the reactor at different time steps with funnel angle of 30 ∘ and particle number ratio of   :   = 2 : 3.

0Figure 5 :
Figure 5: Comparison of velocity profiles for simulations without wall friction for two different heights by Rycroft et al. [5].

Figure 7 :
Figure 7: Horizontal velocity distributions of pebbles at different number ratios.