**3.2 Motion characteristics in the near-wall and central regions**

The particles initially fall vertically at a certain velocity, forming a stable accumulation structure in the pebble bed. The particle motion characteristics are different in the

**Figure 6.** *Diagram of pebble bed structure.*


#### **Table 1.**

*Parameters of the particle accumulation process.*

accumulation process of the different shapes of pebble beds. Due to the effect of gravity, particles may collide with the wall or other particles. If the particles collide, the velocity of the particles changes due to the effect of contact force. The average velocity and average velocity difference of particles in different regions are defined as follows:

$$v\_{\text{ave}} = \frac{1}{m} \sum\_{i=0}^{m} v\_i \tag{24}$$

$$\upsilon\_{\rm diff} = \frac{1}{m} \sum\_{i=0}^{m} \left( \upsilon\_i - \frac{1}{n} \sum\_{j=0}^{n} \upsilon\_j \right) \tag{25}$$

where *vave* is the average velocity of particles in the region, *vi* is the velocity of particle *i* in the region, *n* is the number of particles in the region. *vdiff* is the average velocity difference of particles in the region, *vj* is the velocity of particle *j* contacting particle *i*, and *m* is the total number of other particles in contact with particle *i*.

The difference in the average particle velocity between the central and near-wall regions at the bottom inclination of 45 degrees is shown in **Figure 7**. It can be seen that

#### **Figure 7.**

*(a) Average velocity of particles in the central region and near-wall region with the bottom inclination of 45 degrees; (b) average velocity difference of particles in the central region and near-wall region with the bottom inclination of 45 degrees.*

## *DEFEM Method and Its Application in Pebble Flows DOI: http://dx.doi.org/10.5772/intechopen.109347*

the horizontal velocity of particles is zero in the initial state. Moreover, the vertical velocity of particles increases steadily at the beginning of the accumulation process due to the effect of gravity. Then, it can be observed that some particles collide with the side wall, so the average velocity and average velocity difference of particles near the wall region fluctuate first.

In terms of vertical velocity, the peak velocity of particles in the central region is much larger than that in the near-wall region. This is because compared with the particles in the near-wall region, the particles in the central region can keep free falling motion for a longer time before colliding with the wall or other particles. The change in the vertical velocity of particles in the central region after the collision is more obvious; that is, the vertical velocity difference of central velocity particles is significantly greater than that of the near-wall region. In terms of the horizontal velocity, it is more about the impact of particles. Compared with the near-wall region, the probability of particle collision in the central region is greater. Therefore, the fluctuation of the average horizontal velocity and velocity difference in the central region is significantly greater than that in the near-wall region. Finally, the particles reach the state of stable accumulation; that is, the particles remain stationary (the average velocity and velocity difference of the particles are zero).

#### **3.3 Deformation of particles in the near-wall and central regions**

A major advantage of the DEFEM compared with the DEM is that it can calculate the deformation of particles. This part discusses the whole deformation of particles in the accumulation process of the pebble bed. The accumulation process of twodimensional particles is simulated in this paper, so the deformation of particles is represented by the change of area. The average coefficient of area ratio in the defined region is given as follows:

$$S\_{\text{ave}} = \frac{1}{m} \sum\_{i=0}^{m} \frac{S\_i}{S\_0} \tag{26}$$

where *Save* is the average coefficient of area ratio, *m* is the number of particles in the region, *Si* is the area of particles at the current time, and *S*<sup>0</sup> is the area of the initial moment of the particle (no deformation state). If the particle does not deform, the average coefficient of area ratio is equal to 1 (*Save* ¼ 1); if the particle is compressed, the average coefficient of area ratio is less than 1 (*Save* <1).

The average coefficient of area ratio in the central and near-wall regions with the bottom inclinations of 45 degrees and 15 degrees is shown in **Figure 8**. It can be seen that some particles are compressed (the average coefficient of area ratio is less than 1; *Save* <1) during the accumulation process. When the bottom inclination is 45 degrees, the absolute value of the peak about the average coefficient of area ratio is less than that of 15 degrees, and the changing trend is the same as that of the average stress of the particles. When the particle has severe deformation, the average coefficient of the area ratio and absolute internal value of stress also increases.

#### **3.4 Stress of particles in the near-wall and central regions**

Particles may have strain and stress in the accumulation process because they may collide with other particles or walls. In this paper, the von Mises stress is used, and its formula is as follows:

#### **Figure 8.**

*(a) The average coefficient of area ratio in the central region and near-wall region with the bottom inclination of 45 degrees; (b) the average coefficient of the area ratio in the central region and near-wall region with the bottom inclination of 15 degrees.*

$$\sigma = \sqrt{\frac{1}{2} \left[ \left( \sigma\_1 - \sigma\_2 \right)^2 + \left( \sigma\_2 - \sigma\_3 \right)^2 + \left( \sigma\_3 - \sigma\_1 \right)^2 \right]} \tag{27}$$

Where *σ* is the von Mises stress and *σ*1, *σ*2, and *σ*<sup>3</sup> are the first, second, and third principal stresses, respectively.

The average stress of particles in the region is defined as:

$$
\sigma\_{\text{ave}} = \frac{1}{m} \sum\_{i=0}^{m} \sigma\_i \tag{28}
$$

where *σave* is the average stress in the region, *σ<sup>i</sup>* is the stress of particle *i* in the region, and *m* is the number of particles in the region.

The average stress of particles in the central and near-wall regions with the bottom inclinations of 45 degrees and 15 degrees is shown in **Figure 9**.

[8] Different effects of bottom inclination.

#### **Figure 9.**

*(a) Average stress of particles with the bottom inclination of 45 degrees; (b) average stress of particles with the bottom inclination of 15 degrees.*
