474
7 Applications: Pebble Flow Optimizations
distributions at t = 10 s, 20s, 50s, and 150s for the R ∞ , R 1 , R 2 , and R 3 configurations are shown in Fig. 7.26.
For the R ∞ configuration (Fig. 7.26a), the pebbles move nearly uniformly in the
central region of the bed, except the near-wall regions. Some pebbles are stagnated
near the wall, moving fairly slowly and then stay around the bottom corner of the
bed. It is true that, as aforementioned, the formation of stagnant zones is not suitable
for real reactors, although the pebbles may be finally drained from the orifice after a
sufficiently long time.
For the R 1 configuration (Fig. 7.26b), the situation seems to be much worse than
the former. A larger stagnant zone composed of stagnant pebbles is observed around
the corner. The stagnant zone has a flat bottom, which would be the main reason
for the stagnation zone formation. When the pebbles contact the flat bottom, the
driven force of gravity is counteracted completely, and the pebbles have to move
horizontally until they get the drainage orifice. A flat configuration cannot exist in
real reactors.
For the R 3 configuration (Fig. 7.26d), the situation seems still not to be very
good. It is noticed that the R 3 configuration is quite near the R ∞ configuration,
since the circumcircle is very large, and the arc curve in R 3 configuration is quite
near the straight-line edge in R ∞ configuration (Fig. 7.25). Thus, there are still some
pebbles that stagnated around the corner. Moreover, the configuration transition from
the vertical wall to the arc shape is not smooth. Therefore, the pebbles can pile up
near the corner (e.g., the right bottom region, Fig. 7.26d, at t = 50s). There, the
pebble lattice is nearly regular (more rigid than the irregular lattice of piling) and
cannot be discharged as easily as that with the irregular lattice of a pebble. Thus, the
pebble distribution is not symmetrical, and some pebbles are still stagnated around
the corner.
For the R 2 configuration (Fig. 7.26c), the results seem to be better than any of the
former cases. The pebble distributions are almost uniform at any time point in most
regions of the bed, except a very narrow region near the wall. Finally, at t = 150 s,
the stagnant regions with stagnated pebbles at the bottom are very small.
The bed configurations mentioned above belong to the series of arcs R n , (n =
1, 2, ..., ∞) whose centers are on the perpendicular bisector of the inclined edge. As
the case of n → ∞ is not suitable for flow uniformity and so is the case of n → 1,
there must be an optimal choice of R n,opt (1< n < ∞), which has the most uniform
pebble flow. Figure 7.26c shows a possible option for the best R n,opt when n = 2.
At least, it is quite near the best configuration with almost no stagnated pebbles.
Moreover, Fig. 7.27 shows the results under the brachistochrone-shaped bed configuration. For the nature of the brachistochrone shape, it is assumed to be a good
configuration since it is the best trajectory on which the pebbles can move the fastest
7 Applications: Pebble Flow Optimizations
distributions at t = 10 s, 20s, 50s, and 150s for the R ∞ , R 1 , R 2 , and R 3 configurations are shown in Fig. 7.26.
For the R ∞ configuration (Fig. 7.26a), the pebbles move nearly uniformly in the
central region of the bed, except the near-wall regions. Some pebbles are stagnated
near the wall, moving fairly slowly and then stay around the bottom corner of the
bed. It is true that, as aforementioned, the formation of stagnant zones is not suitable
for real reactors, although the pebbles may be finally drained from the orifice after a
sufficiently long time.
For the R 1 configuration (Fig. 7.26b), the situation seems to be much worse than
the former. A larger stagnant zone composed of stagnant pebbles is observed around
the corner. The stagnant zone has a flat bottom, which would be the main reason
for the stagnation zone formation. When the pebbles contact the flat bottom, the
driven force of gravity is counteracted completely, and the pebbles have to move
horizontally until they get the drainage orifice. A flat configuration cannot exist in
real reactors.
For the R 3 configuration (Fig. 7.26d), the situation seems still not to be very
good. It is noticed that the R 3 configuration is quite near the R ∞ configuration,
since the circumcircle is very large, and the arc curve in R 3 configuration is quite
near the straight-line edge in R ∞ configuration (Fig. 7.25). Thus, there are still some
pebbles that stagnated around the corner. Moreover, the configuration transition from
the vertical wall to the arc shape is not smooth. Therefore, the pebbles can pile up
near the corner (e.g., the right bottom region, Fig. 7.26d, at t = 50s). There, the
pebble lattice is nearly regular (more rigid than the irregular lattice of piling) and
cannot be discharged as easily as that with the irregular lattice of a pebble. Thus, the
pebble distribution is not symmetrical, and some pebbles are still stagnated around
the corner.
For the R 2 configuration (Fig. 7.26c), the results seem to be better than any of the
former cases. The pebble distributions are almost uniform at any time point in most
regions of the bed, except a very narrow region near the wall. Finally, at t = 150 s,
the stagnant regions with stagnated pebbles at the bottom are very small.
The bed configurations mentioned above belong to the series of arcs R n , (n =
1, 2, ..., ∞) whose centers are on the perpendicular bisector of the inclined edge. As
the case of n → ∞ is not suitable for flow uniformity and so is the case of n → 1,
there must be an optimal choice of R n,opt (1< n < ∞), which has the most uniform
pebble flow. Figure 7.26c shows a possible option for the best R n,opt when n = 2.
At least, it is quite near the best configuration with almost no stagnated pebbles.
Moreover, Fig. 7.27 shows the results under the brachistochrone-shaped bed configuration. For the nature of the brachistochrone shape, it is assumed to be a good
configuration since it is the best trajectory on which the pebbles can move the fastest