Numerical methodology in the CFD-DEM model

In the CFD-DEM model, the flow in the entire flow region is calculated from the continuity and Navier-Stokes equations, with a porosity term and an additional body-force term used to account for the presence of particles in the fluid. These are given by [34, 35](1)(2)(3)where , , , are the volume factor of the gas phase (porosity), density, velocity, and pressure, respectively; and are apparent volume and total particle volume. is the viscosity. is the interphase force exerted by the pebble bed in the lattice on the helium; the main consideration here is the drag force. For gas–particle interactions, the ratio of buoyancy to gravity is on the order of 10^-3; therefore, buoyancy can usually be ignored.

The porosity and body force of the fluid mesh elements are determined using the DEM. The equations of motion for the particles include an additional force term that considers their interaction with the fluid. The equations [36, 37] are(4)(5)where , , are the particle mass, velocity, and angular velocity, respectively; is the fluid-particle interaction force; and are the contact force and moment between particle i and j, respectively; is the body force, including gravity and drag force; and is the particle moment inertia.

The force that occurs during the interaction between a particle and a fluid can be characterized by a combination of drag, pressure gradient, viscous tensor gradient, and various other forces. When considering a system comprising gas and particles, the predominant factor influencing the interaction between the fluid and pebble bed is the drag force, which has a significant impact on the precision of the coupled CFD-DEM model. Consequently, achieving an accurate representation of the interphase forces experienced by particles and fluid is essential for CFD-DEM simulations. The interphase force, denoted by , can be mathematically described as [38](6)(7)where V_cell is the volume of the grid, n_p represents the number of particles in the pebble beds, denotes the drag force of helium on each particle, A_i is the projected area of the particle along the direction of helium flow, and C_d is the coefficient of drag force. In this study, the Gidaspow model [39] was used to describe the drag coefficient, which can be regarded as a combination of the Wen-Yu model [40] and the Ergun equation [41]. The Gidaspow drag model has a wide range of applications and high calculation accuracy, especially for a dense phase gas–solid pebble bed. C_d takes the following form:(8)(9)Two-way coupling between the DEM and CFD is numerically achieved by solving Eqs. (1)–(3) using the CFD code, and Eqs. (4)–(9) using the DEM code. The DEM determines the porosity and interaction force, which are then normalized by the volume obtained from CFD. CFD calculates the fluid velocity and pressure for each element and subsequently shares these data with the DEM during each exchange. The computational time steps used for the fluid and powders are 6.0×10^-6 s and 6.0×10^-8 s, respectively.

For the particle phase, Li et al. [42] suggested a critical time step based on the time required for a Rayleigh wave to propagate along the smallest particle:(10)where is the time step in DEM, is the minimum particle size, and E and are the Young’s modulus and Poisson’s ratio of the particles, respectively. In general, the critical time step in DEM is much smaller than that in CFD. Therefore, the total stability condition is dominated by the DEM simulation. In the proposed model, is set as 100 times that of and the exchange of information between the DEM and CFD is performed at each step of CFD calculation.

Figure 1 shows the coupling procedures adopted in the CFD-DEM simulations. The detailed procedure during the calculation was as follows: First, DEM data, including positions and velocities, were transferred into the CFD solver; then, a translation from the Lagrangian field to the Euler field was performed by the CFD solver, and the porosity and the interphase force between the fluid and particle were updated according to the exchanged DEM data. Second, the velocity of the fluid U_f and pressure p were calculated using the continuity and Navier–Stokes equations. Third, U_f was modified by considering the influence of particles on the fluid, and the modified U_f was used to calculate the redistributed pressure. Finally, the drag force applied to the particle body was calculated using all updated parameters, and the CFD working step was temporarily paused, waiting for the DEM calculation step and data exchange. The drag force obtained in CFD was transferred to the DEM, and the resultant force of the particle was updated and used to calculate the new position and velocity of the particle. The coupling of CFD with the DEM was achieved using the commercial software Fluent for CFD and EDEM for the DEM [43].

Fig. 1Full-screen View

(Color online) Scheme diagram of CFD-DEM coupling procedures

Simulation study setup

The type and size of tritium breeders significantly affect the performance (e.g., porosity) of the solid blankets in fusion reactors. In most solid blanket designs, 1 mm diameter Li₄SiO₄ particles are used as tritium breeders. A numerical sample of the Li₄SiO₄ pebble bed was prepared using the DEM code. Table 1 lists the physical properties of the Li₄SiO₄ particles and the key simulation parameters [44]. The boundary conditions [45] and the powder numbers are listed in Table 2.

Following material parameter determination, pebble-bed particles were generated using EDEM software. Slight compaction by the plate ensured close interactions between the breeding particles. As shown in Fig. 2, the CFD-DEM model measured 12 mm in length, 12 mm in width, and 15 mm in height. The velocity inlet was set as 5 mm above the pebble bed surface to simulate the purge gas. A particle factory or a broken particle source was incorporated within an area measuring 8 mm × 8 mm × 1 mm at the top of the pebble beds. In the CFD-DEM model, the powders attempt to percolate through a fixed, nondeformable pebble bed under the influence of the purge gas.

Fig. 2Full-screen View

(Color online) CFD-DEM model

The sizes of the powders produced by crushing Li₄SiO₄ pebbles cannot be uniformly distributed, and the powder flowability may change with the particle size (d_p) distribution. Table 3 lists the powder gradations and properties obtained from previous experimental and numerical studies [46]. In this study, three types of powder distributions were identified: fine, coarse, and well-graded particles. The powder grading table shows the mass percentages of various powder sizes. All powders were spherical, and the simulation settings and boundary conditions were consistent with those used for breeding particles. In EDEM software, each particle size gradation is depicted in four colors representing different sizes. For example, fine powder sizes ranging from 0.08 mm to 0.16 mm are color-coded as green, yellow, pink, or red in the software. Powders smaller than 0.08 mm were excluded from the simulation owing to their small size and the extensive time required for generation.

Table 4 outlines the various cases and parameters: Cases 1–3 and 6 examined the purge velocity; Cases 1, 4, and 5 focused on the powder size distribution; Cases 1, 7, and 8 investigated the coefficient of friction; and Cases 1 and 9 compared the two breeder orientations. In this study, two ITER-relevant volumes were considered, as shown in Fig. 3. These volumes differed in their configurations according to the direction of gravity. Because of their similarity, we employed generic coordinate systems (χ, ζ). The ζ configuration represents the TBM orientation of the EU [47], whereas the χ configuration aligns with several current TBM designs at ITER [48, 49].

Fig. 3Full-screen View

Sketches of the two breeder orientations: (a) χ-configuration, (b) ζ-configuration

Validation

2.3.1

Verification of the pebble bed model

The specificity of a pebble bed mainly depends on the influence of the wall effect [30]. The classic mode of Klerk [50] was adopted to verify the validity of the porosity distribution along the x-axis for the pebble beds in this study. Local porosity is expressed as(11)(12)where R represents the dimensionless wall distance, d is the particle diameter, L is the box length, and is the porosity at the center of the pebble beds. Figure 4 shows the porosity along the x-axis direction of the DEM static pebble beds, calculated and compared using Eq. (11). The results demonstrate the distinct oscillation characteristics of porosity within the pebble beds near the wall. The oscillation amplitude decreased along the radial direction and agreed closely with the experimental results. Therefore, the pebble-bed model obtained using DEM was considered valid for this study.

Fig. 4Full-screen View

(Color online) Porosity distribution of particles along the x-axis

2.3.2

Verification of the CFD-DEM model

The Ergun equation [51] has been extensively employed to predict the pressure drops within pebble beds. This equation describes the relationship between the pressure drop and average velocity for the flow in the pebble beds:(13)where is the pressure drop from inlet to outlet, is the height of the pebble bed, and V_in is the inlet gas velocity. To verify the constructed CFD-DEM model, the calculated results were compared with those of the Ergun equation. The comparison is shown in Fig. 5. A close agreement was observed, indicating that the proposed CFD-DEM model can accurately capture the behavior of particles and fluid flow.

Fig. 5Full-screen View

(Color online) Comparison of the results of pressure drop in pebble beds

Law of powder transport and clogging in pebble beds

Figure 6 illustrates the migration and clogging processes of the powders with varying particle size distributions within the pebble beds. As shown in Fig. 6(a), for Case 1, the surface-accumulated powders moved downward, driven by the purging gas, and their number gradually decreased. A large number of particles accumulated in the upper pore structure of the pebble beds after running for 2 s. However, the distribution of powder in the pore structure decreased rapidly as the height of the pebble beds decreased. The distribution range of powder in the pebble beds was wide. Figure 6(b) shows that in Case 4, the smaller-sized surface-accumulated particles gradually migrated inward, driven by the purge gas. However, their migration distance was significantly shorter than that in Case 1, with most powder deposited at the upper part of the pebble beds and a smaller spread range. Figure 6(c) indicates that in Case 5, there was negligible vertical migration of surface-accumulated particles due to the purge gas, leading to significant accumulation at the upper part of the pore structure of the pebble beds.

Fig. 6Full-screen View

(Color online) Migration and clogging process of powders of different grades in pebble beds. (a) Case 1, (b) Case 4, (c) Case 5

Gerber et al. [52] observed that a decrease in the particle size ratio (d_p/d) resulted in clogging of the particles closer to the filler surface. A smaller ratio accelerated surface deposition and increased accumulation. With a smaller particle size ratio, the particles encounter similar-sized pores more frequently during transport, leading to earlier deposition and shallower clog depths. Continuous particle accumulation at these sites leads to clogging at shallower depths. The small particle size ratio caused the fine particles to clog only a thin layer of gravel pores near the surface of the pebble beds, hindering further permeation. The pebble-bed model features pores composed of larger pore bodies and smaller pore throats. Powders are captured in pores larger than or equal to their volume, irrespective of the particle size. Clogging can be distinguished by the presence of larger powder particles, which divide the clogged areas into zones A and B, as shown in Fig. 6(a). Zone A contains large powder particles, whereas Zone B does not. The clogging mechanism of Zone A was: Red or pink particles initially settled within the pebble bed, predominantly in a shallow 5 mm layer. After forming a pore skeleton in the shallow pores of the pebble beds, the particles could not easily migrate again. This narrowing of the channels in the shallow layer facilitated the rapid accumulation of smaller-sized deposits. The clogging mechanism of Zone B was: Smaller powder particles, such as green particles, migrated and permeated inward through voids under the sweeping gas and gravity, eventually forming local deposits alongside larger particles.

Based on the two-dimensional images, Fig. 7 presents an analysis of the powder retention rate within the pebble beds. Considering the pebble-bed surface as the zero horizontal plane (H = 0), the distribution of the accumulated powder retention rate above depth H is defined as R_H(14)where C_H is the powder content (g) in the part above depth H, and C_t is the initial total powder mass (g). Figure 7(a) shows the distribution of the accumulated retention rates of the powders for different running times. Within 1 s, the vertical accumulation retention rate of the powder in the surface layer (1-5 mm) of the ball bed decreased, signifying gradual powder migration into the porous medium owing to purging. Beyond 1 s, the retention rate of the powder at the surface layer remained constant, indicating the stabilization of powder deposition in this area. From 1–2 s, a noticeable difference in the retention rate occurred in the middle layer (6-12 mm) of the pebble beds, but beyond 2 s, only minor changes were observed in the lower layer (12-15 mm), with the retention rate largely unchanged elsewhere. This result indicates that the powder deposition stabilized within the pebble beds, with clogging development progressing from top to bottom. Initially, the powder stabilized at the top of the pebble beds, followed by stable deposition in the middle and bottom regions.

Fig. 7Full-screen View

(Color online) Distribution of accumulated retention rate of powders. (a) Different time points; (b) Different size gradations

Figure 7(b) shows the size grades. The disparity between well-graded and coarse powders was minimal. Both exhibited substantial deposition in the surface layer, with retention rates rapidly reaching 100%. In contrast, the fine powders demonstrated notable differences, characterized by deeper penetration. This minimal difference occurred because both well-graded and coarse powders contain particles larger than 0.16 mm. At equilibrium, these large particles clogged the surface pores of the pebble beds, hindering smaller particles in the well-graded powders from penetrating the surface, thus limiting any further increase in the retention rate. This result further confirms the two different clogging mechanisms mentioned in the analysis: One involves a powder with a large particle size and the other does not.

Effect of factors on powder migration development

3.2.1

Influence of powder size grading

Figure 8 illustrates the variation in the powder transport speed with size grading, representing the average speed of all powders. This result indicates a gradual decrease in powder transport speed over time. Ultimately, the powders became static in the pebble beds, with fine powders taking longer to settle than the well-graded and coarse powders. Furthermore, the powder transport speed increased with decreasing particle size, because smaller powders are more easily mobilized by fluid forces. The average speed of fine powders was 80% higher than that of well-graded and coarse powders, highlighting the significant impact of large powders on the powder flow velocity in pebble beds. The results of dynamic migration and static clogging of the powder were similar; that is, the presence or absence of a large particle size had a significant influence on the dynamic migration process.

Fig. 8Full-screen View

(Color online) Transport speed of powders in different size grading

3.2.2

Influence of powder friction coefficient

Figure 9 shows the relationship between the average vertical velocity of the fine powders and changes in the friction coefficient. The changes in the average vertical velocity and transport speed were similar. As the friction coefficient increased, the average vertical velocity of the fine powder decreased significantly. According to the friction coefficient, ranging from low to high, the average vertical speeds were 1.09 mm/s, 0.43 mm/s, and 0.32 mm/s for the three cases. For powders with varying friction coefficients, the migration velocity followed a consistent pattern over time; it first decreased rapidly and then gradually approached a static state.

Fig. 9Full-screen View

(Color online) Vertical velocity of powders for different friction coefficients

3.2.3

Influence of purge velocity

Figure 10 illustrates how the vertical velocity of the powders changed with the purge velocity, which was adjusted by controlling the outlet velocity. Figure 10(a) shows that the average vertical migration velocity of the powders tended to increase with increasing purge velocity. This effect is less pronounced than that of the size grading or friction coefficient because the purge velocity has a limited impact range of 0.07-0.2 mm/s, which is a factor of 4 smaller in magnitude. Furthermore, Fig. 10(b) demonstrates that the migration of the coarse powder was less affected by the purge velocity than that of the fine powder. Despite higher purge velocities, it is challenging to mobilize large particles that cause clogging, indicating clogging stability. In summary, the impact of the purge velocity on powder migration in pebble beds was limited, and this effect varied significantly with different particle size ratios.

Fig. 10Full-screen View

(Color online) Vertical velocity of powders for different purge velocities: (a) Fine powders, (b) coarse powders

Study on stabilized powder distribution in clogged pebble beds

3.3.1

Distribution frequency of clogging powders in pebble beds

The powder clogging data from the final stability state were selected for analysis. Different colors, each corresponding to a unique powder size distribution, were extracted. The depth was segmented into 1 mm layers for statistical analysis. The ratio of the clogging powder in each layer to the total clogging powder was calculated. Figure 11 shows the dimensionless ratio N/N₀ normalized by N₀. This result illustrates the distribution of powders with a specific size grading, where N₀ is the total number of clogging powders and N represents the number of clogging particles in each layer. Powders of different sizes within the same layer were stacked. Figure 11(a) illustrates how the distribution frequency of fine powder changed across different layers during clogging. The red powder with d_p = 0.14-0.16 mm was distributed within the 0-4 mm range of the pebble beds. Pink powder, d_p = 0.12-0.14 mm, exhibited peak frequencies in the 0-3 mm range in the pebble beds, with most powders concentrated within 5 mm. The void structure of the surface layer of the spherical bed was blocked by large particles, and green and yellow powders (d_p = 0.08-0.12 mm) were primarily distributed in the 0-5 mm range of the pebble beds. However, the green and yellow powders still penetrated the interior and even the bottom of the pebble beds. Powder with a small particle size was more widely distributed in the pebble beds.

Fig. 11Full-screen View

(Color online) The distribution frequency of clogging powders in pebble beds. (a) Case 1, (b) Case 4, (c) Case 5, (d) Case 2, (e) Case 7, (f) Case 9

As shown in Fig. 11(a)-(c), the coarse powder seldom invaded the middle part of the pebble-bed layers, whereas the fine powder penetrated the bottom. The fine powder was distributed more evenly at depth because the coarse powder prevents downward clogging. However, a comparison of well-graded and coarse powders revealed no significant differences in the concentration or invasion depth. A comparison between Fig. 11(a), (d) shows that under varying purge velocities, larger powder particles (> 0.14 mm) primarily settled in the surface layer of the pebble beds, whereas smaller particles (< 0.12 mm) were distributed more evenly throughout the middle and lower sections. From the superposition of the frequency distribution, with a purge velocity of 0.1 m/s, the clogging powder in the 0-3 mm region of the pebble beds constituted 76% of the total powder. At a purge velocity of 0.2 m/s, the clogging powder in the 0-3 mm area represented 81% of the total powder, with pink powder (d_p = 0.12-0.14 mm) present in the middle part of the pebble beds. The results indicate that the powder penetration depth and quantity into the pores increased with increasing purge velocity. Comparing Fig. 11(a) and (e) shows that as with increasing friction coefficient, the peak distribution frequency of the clogging powder also increased. In Case 1, 63% of the powder was clogged in the surface layer of the pebble beds (0-2 mm), and in Case 7, this value increased to 69%. Powder with larger particles predominantly clogged the surface layer, whereas smaller particles penetrated the bottom layer. This result demonstrates that the friction coefficient significantly affects powders with larger surface areas, with a minimal effect on smaller powders.

3.3.2

Effect of breeder orientation on powder migration and clogging

Figure 12 illustrates the impact of various proliferation orientations on powder migration and clogging. Figure 12(b) reveals that in the ζ configuration, the powder near the upper wall (Y=-6 mm) moved downward away from the top wall, with significant deposits forming as powders from the middle rolled down to the lower wall area (Y = 6 mm). In contrast, the pebble beds in the χ configuration exhibited distinct powder migration and sedimentation patterns. Figure 12(a) shows that the powder near the wall settled in the same post-migration area, which is a trend observed in other areas of the pebble beds. When the breeder particles in the pebble beds are crushed, they fall owing to gravity and fluid force, yet maintain their general direction. Comparing Fig. 11(f) and (a) shows that the powder in the ζ configuration predominantly settled in the 0-5 mm region in the upper part of the pebble beds, and the quantity of deposited powder sharply decreased with decreasing height.

Fig. 12Full-screen View

(Color online) Three-dimensional distribution of clogging powders with different breeder orientations. (a) Case 1, (b) Case 9

Characterization and developmental stages of migration and clogging

The migration and clogging of powder in pebble beds significantly impacts the system, and is influenced by various factors. Complex dynamics of the powder behavior in the pebble beds were observed. The influence of different parameters on the dynamic behavior of the powder varied. To characterize the powder behavior, including the migration distance and velocity, we defined the migration efficiency v^*(l, v). From prior analysis and with the purging gas parameters in the fixed ball bed established, the factors affecting v^*(l, v), in descending order of significance, were the powder size distribution S, breeder orientation of the pebble beds g, friction coefficient f, and average purge velocity V. Furthermore, parameters not investigated in this study, such as the particle size and temperature, may also affect powder migration. These effects were uniformly classified as other factors (oth). Thus, the migration efficiency v^*(l, v) was characterized as(15)By analyzing the average transport velocity at various times, we observed the behavior of powder migration within the 0.2 to 5.0 s range. Powder migration velocity in the pebble beds between 0.2 and 5.0 s was normalized. Figure 13(a) illustrates that the powder migration velocity can be categorized into three phases: rapid decay, linear decay, and stability. During the 0.2–5.0 s period, the average velocity of powder decreased rapidly (rapid decay stage) first, followed by a linear decrease in the rate of change (linear decay stage), culminating in stabilization. Stages 1, 2, and 3 were modeled using polynomial and linear analyses:

Fig. 13Full-screen View

(Color online) Normalized analysis of migration and clogging development. (a) Normalized velocity; (b) Powder retention rate

Stage 1:(16)Stage 2:(17)Stage 3:(18)In this paper, v_n1 represents the normalized migration velocity during Phase 1, v_n2 during Phase 2, and v_n3 during Phase 3. The parameters in Eqs. (16) and (17) vary according to their boundary conditions. Taking Case 1 as an example, under conditions such as a purge velocity of 0.1 m/s, a friction coefficient of 0.1, and the fine-powder size distribution, the values of a₁, b₁, c₁, d₁, and e₁ were respectively 1.1024, -2.2963, 1.3778, -0.0786, and 0.2935.

Following powder stabilization , we analyzed the powder clogging trends at different locations within the pebble beds. Figure 13(b) shows that the 0-4 mm depth was the predominant area for powder retention. Clogging can be classified into two categories: extensive superficial and uniform internal. Initially, the retention rate in the 0-4 mm range increased quickly, indicating extensive surface deposition blockage. This stage was followed by a gradual and consistent change in the retention rate, suggesting uniform internal deposition of fine-grained powder. Types 1 and 2 were fitted with polynomial and logarithmic analyses, respectively:

Type 1:(19)Type 2:(20)The parameters in Eqs. (19) and (20) changed according to the boundary conditions. For Case 1, with a purge velocity of 0.1 m/s, friction coefficient of 0.1, and fine-powder size distribution, the values of a₂, b₂, c₂, d₂, and e₂ were -9.085, 66.267, -32.37, 0.5646, and 88.098, respectively.