6.2 Size Effects
419
Moreover, the similarity is also applicable to the case at R D = 0.5, which needs
a long time of simulation because of a large number of particles. First, the particle
discharge rate at R D = 0.5 and R L = 4.3 is 200 particles/s until 150s. After that, it
increases to 1,000 particles/s, which is called the state “A
. The non-dimensional
time τ is 40 at 150 s and 120 at 193s. Another discharge rate called the state “B
is 100
particles/s for all the time, where τ is 117.5 at 715 s. The mixing index and mixing
entropy of the two states are shown in Fig. 6.15c and d. The states “A
and “B
are
quite close to each other, which indicates the similarity of the flow characteristics
between different discharge rates. Therefore, the numerical results of the two-region
pebble bed under different discharge rates can be used for the following analysis.
6.2.4.3 Particle-Scale Mixing Index
In this section, let’s come to the PSMI analysis of particle mixing for all cases. The
PSMI is defined by the Lacey’s mixing index as follows [9]:
PSMI =
S
2
0 − S
2
S
2
0 − S
2
f
(6.5)
where S
2
0 and S
2
f are the particle-scale variances of the fully segregated and fully
mixed state respectively, i.e.
S
2
0 = P(1 − P),
(6.6)
S
2
f =
P(1−P)
N
(6.7)
where P is the overall particle number fraction of type 1. N is the average number
of particles for all sampling cells.
First, the PSMI of the two-region bed of all cases is calculated and presented in
Fig. 6.16. As shown in Fig. 6.16a, a bit intensive fluctuation exists at larger size ratios
(R D = 2.0). In the relatively steady state, PSMI increases gradually from 0.066 at
R D = 0.5 to 0.096 at R D = 1.0, 0.12 at R D = 1.5, and 0.15 at R D = 2.0 for R L =
3.0. The PSMIs under different region ratios are also different (Fig. 6.16b and c),
increasing from 0.15 at R L = 3.0 to 0.18 at R L = 3.5 and 0.21 at R L = 4.3 for R D =
2.0. The tendency of variation is similar for R D = 0.5. The variations of PSMI for
Case 1∼Case 12 are summarized in Fig. 6.16d. The effects of both size ratio and
region ratio on PSMI are remarkable in the two-region pebble bed. But the effect
of size ratio has more evident influence on the PSMI than the region ratio, which
means the mixing level is more prone to be intensified in the beds of large particle
size ratios rather than that of large region ratios.
The PSMI for all the two-region beds has the following limitation: (1) for a system
with the majority of particles being mixed even though the mixing states are not fully
developed, the PSMI is capable of providing a quantitative measure on the overall
degree of mixing. (2) for a system with only a limited number of particles being mixed
419
Moreover, the similarity is also applicable to the case at R D = 0.5, which needs
a long time of simulation because of a large number of particles. First, the particle
discharge rate at R D = 0.5 and R L = 4.3 is 200 particles/s until 150s. After that, it
increases to 1,000 particles/s, which is called the state “A
. The non-dimensional
time τ is 40 at 150 s and 120 at 193s. Another discharge rate called the state “B
is 100
particles/s for all the time, where τ is 117.5 at 715 s. The mixing index and mixing
entropy of the two states are shown in Fig. 6.15c and d. The states “A
and “B
are
quite close to each other, which indicates the similarity of the flow characteristics
between different discharge rates. Therefore, the numerical results of the two-region
pebble bed under different discharge rates can be used for the following analysis.
6.2.4.3 Particle-Scale Mixing Index
In this section, let’s come to the PSMI analysis of particle mixing for all cases. The
PSMI is defined by the Lacey’s mixing index as follows [9]:
PSMI =
S
2
0 − S
2
S
2
0 − S
2
f
(6.5)
where S
2
0 and S
2
f are the particle-scale variances of the fully segregated and fully
mixed state respectively, i.e.
S
2
0 = P(1 − P),
(6.6)
S
2
f =
P(1−P)
N
(6.7)
where P is the overall particle number fraction of type 1. N is the average number
of particles for all sampling cells.
First, the PSMI of the two-region bed of all cases is calculated and presented in
Fig. 6.16. As shown in Fig. 6.16a, a bit intensive fluctuation exists at larger size ratios
(R D = 2.0). In the relatively steady state, PSMI increases gradually from 0.066 at
R D = 0.5 to 0.096 at R D = 1.0, 0.12 at R D = 1.5, and 0.15 at R D = 2.0 for R L =
3.0. The PSMIs under different region ratios are also different (Fig. 6.16b and c),
increasing from 0.15 at R L = 3.0 to 0.18 at R L = 3.5 and 0.21 at R L = 4.3 for R D =
2.0. The tendency of variation is similar for R D = 0.5. The variations of PSMI for
Case 1∼Case 12 are summarized in Fig. 6.16d. The effects of both size ratio and
region ratio on PSMI are remarkable in the two-region pebble bed. But the effect
of size ratio has more evident influence on the PSMI than the region ratio, which
means the mixing level is more prone to be intensified in the beds of large particle
size ratios rather than that of large region ratios.
The PSMI for all the two-region beds has the following limitation: (1) for a system
with the majority of particles being mixed even though the mixing states are not fully
developed, the PSMI is capable of providing a quantitative measure on the overall
degree of mixing. (2) for a system with only a limited number of particles being mixed