6.2 Size Effects
415
For Case 3, N = 1513, 579, 221, 87, 32, and 12, respectively, and for Case 4,
N =1771, 601, 225, 84, 36, and 12 respectively. Thus, the fractal dimension D 0
increases from 1.316 for R D = 1.0 to 1.418 for R D = 0.5 through 1.393 for R D = 2.0,
which means the shape of the mixing region is more complex with larger particle size
difference. It is also noticed that D 0 for R D = 0.5 is slightly larger than that for R D
= 2.0. Due to the slope of the walls, the diffusion process of the particle from large
ones to small ones and that from small ones to large ones are not exactly equivalent.
Due to the existence of the sloped wall, the bulk movement of almost all particles
is from sidewall toward the central region, either slowly or rapidly. However, within
the mixing interface of different sized particles, diffusion takes place mainly from
the smaller-particle region to the larger-particle region since the packing structure of
larger particles has larger voids, and it is easier for smaller particles to diffuse into
the voids between larger particles. When R D = 2.0, the bulk movement and diffusion
direction are all from the sidewall to the central region, i.e., they are in the same
direction. However, when R D = 0.5, the bulk movement is still from the sidewall
toward the central region, whereas the diffusion direction is mainly from the central
region (smaller particle side) toward the sidewall (larger particle side), i.e., they are
in the counter direction. Thus, the counter-movements of bulk motion and particle
diffusion lead to a better mixing process and a more complicated mixing interface
than that under the same direction. That is why the D 0 for R D = 0.5 is slightly larger
than that for R D = 2.0.
6.2.3.2 Volume of the Mixing Region
The total particle volume in the mixing region is also a fundamental parameter for
quantifying the mixing process of the two-region pebble bed. Figure 6.12a shows
that the total particle volume in the mixing region increases gradually with time. It
takes about 100 s for R D > 1.0 and a much longer period (more than 300 s) for R D =
0.5 to reach a stable value. For R L = 4.3, the volume decreases from 2.0×10
−3 m3 at
R D = 0.5 to 1.7 ×10
−3 m3 for R D = 1.0 because of particle diffusion, and increases
rapidly for R D > 1.0, reaching 2.2×10
−3 m3 at R D = 1.5 and 3.2 ×10
−3 m3 at
R D = 2.0. However, as shown in Fig. 6.12b, different region ratios (R L = 3.0, 3.5,
and 4.3) do not bring noteworthy effect on the volume of mixing interface for R D >
1.0.
Moreover, let p i be the local number fraction of type-1 particles. If particle i is not
in the mixing region, p i is 0 or 1, and the local entropy is 0. A cumulative parameter,
i.e., mixing entropy, can be computed for mixing evaluation in the two-region pebble
bed. The trends of entropy for different cases are shown in Fig. 6.13. It is similar
to the total particle volume of the mixing interface. The particle size ratio brings in
much larger influence than the region ratio on the mixing entropy of the two-region
pebble bed. It is noticed that the mixing entropy ranges from about 0.035 for R D =
1.0 to 0.076 for R D = 2.0, which is far less than its maximum 0.693 for the fully
mixed state.
415
For Case 3, N = 1513, 579, 221, 87, 32, and 12, respectively, and for Case 4,
N =1771, 601, 225, 84, 36, and 12 respectively. Thus, the fractal dimension D 0
increases from 1.316 for R D = 1.0 to 1.418 for R D = 0.5 through 1.393 for R D = 2.0,
which means the shape of the mixing region is more complex with larger particle size
difference. It is also noticed that D 0 for R D = 0.5 is slightly larger than that for R D
= 2.0. Due to the slope of the walls, the diffusion process of the particle from large
ones to small ones and that from small ones to large ones are not exactly equivalent.
Due to the existence of the sloped wall, the bulk movement of almost all particles
is from sidewall toward the central region, either slowly or rapidly. However, within
the mixing interface of different sized particles, diffusion takes place mainly from
the smaller-particle region to the larger-particle region since the packing structure of
larger particles has larger voids, and it is easier for smaller particles to diffuse into
the voids between larger particles. When R D = 2.0, the bulk movement and diffusion
direction are all from the sidewall to the central region, i.e., they are in the same
direction. However, when R D = 0.5, the bulk movement is still from the sidewall
toward the central region, whereas the diffusion direction is mainly from the central
region (smaller particle side) toward the sidewall (larger particle side), i.e., they are
in the counter direction. Thus, the counter-movements of bulk motion and particle
diffusion lead to a better mixing process and a more complicated mixing interface
than that under the same direction. That is why the D 0 for R D = 0.5 is slightly larger
than that for R D = 2.0.
6.2.3.2 Volume of the Mixing Region
The total particle volume in the mixing region is also a fundamental parameter for
quantifying the mixing process of the two-region pebble bed. Figure 6.12a shows
that the total particle volume in the mixing region increases gradually with time. It
takes about 100 s for R D > 1.0 and a much longer period (more than 300 s) for R D =
0.5 to reach a stable value. For R L = 4.3, the volume decreases from 2.0×10
−3 m3 at
R D = 0.5 to 1.7 ×10
−3 m3 for R D = 1.0 because of particle diffusion, and increases
rapidly for R D > 1.0, reaching 2.2×10
−3 m3 at R D = 1.5 and 3.2 ×10
−3 m3 at
R D = 2.0. However, as shown in Fig. 6.12b, different region ratios (R L = 3.0, 3.5,
and 4.3) do not bring noteworthy effect on the volume of mixing interface for R D >
1.0.
Moreover, let p i be the local number fraction of type-1 particles. If particle i is not
in the mixing region, p i is 0 or 1, and the local entropy is 0. A cumulative parameter,
i.e., mixing entropy, can be computed for mixing evaluation in the two-region pebble
bed. The trends of entropy for different cases are shown in Fig. 6.13. It is similar
to the total particle volume of the mixing interface. The particle size ratio brings in
much larger influence than the region ratio on the mixing entropy of the two-region
pebble bed. It is noticed that the mixing entropy ranges from about 0.035 for R D =
1.0 to 0.076 for R D = 2.0, which is far less than its maximum 0.693 for the fully
mixed state.