94
2 Experiments in Pebble Flows
is the distance from the wall to a point where the flow velocity has essentially reached
the point of 99% “free stream” velocity. However, the pebble flow is very slow and
confined by the friction force from neighbors and walls, and the traditional boundary
layer thickness of the pebble flow is so thicker than the so-called “free-stream” zone
(where velocity is more significant than 99% of U ) which is much smaller. Instead,
considering both transverse and vertical movement for different bed configurations,
the funnel flow zone can be referred to the boundary zone. At each height, the velocity
of the outer boundary of the mass flow zone is regarded as U while calculating the
thickness of displacement in the boundary funnel flow zone. A smaller δ
is required
in the real pebble-bed reactor design to make the pebbles in the corners spend shorter
time for flowing out of the vessel. In this way, the δ
of R (∞,60 ◦ ) bed configuration
will be equal to zero so that there is no funnel flow zone in this situation.
The formation of the boundary layer concerns many terms, for example, the bed
configuration, the friction, and so on. The pebbles in the lower corner of the pebble
bed flow very slowly and are nearly stagnant, which is called the stagnant zone. The
pebbles between the stagnant zone and the central zone are easier to move under a
more inclined base slope, because a large slope of inclination provides a larger driving
force along the wall. Therefore, the enlarged base cone angle significantly decreases
the scale of the stagnant zone, up to complete disappearance, just as presented in the
case of a 60
◦ cone angle. Due to the existence of the stagnant zone, the flow field
becomes non-uniform. Pebbles will form rugged and lumpy borders and hinder the
flowing of their neighboring pebbles significantly.
Meanwhile, such resistance will transmit within a certain range toward the bed
center, and finally produce higher non-uniformity of the velocity distribution. In
other words, the velocity distribution will be more uniform if there is no stagnant
zone. Because of the non-uniformity, the averaged vertical velocity decreases continuously from the central zone to the near-wall zone. In this way, the enlarged base
cone angle significantly reduces the scale of the stagnant zone, even up to complete
disappearance just as presented in the case of a 60
◦ cone angle. In other words, the
enlarged base cone angle increases the uniformity of the vertical velocity throughout
the whole flow field. Thus, the mass flow level of the case 60
◦ is 1, which means
that the vertical velocity in the pebble bed is higher than 0.3v
0 (the vertical velocity
of the particle at the central position that stands for the highest vertical velocity). In
this way, the boundary layer with a smaller velocity almost does not exist in the 60
◦ case according to the definition of the boundary layer in this section.
The thickness of displacement for each bed configuration is shown in Fig. 2.31a.
The thickness of most curves presents the dropping tendency in the lower part after
a slow increase when the height decreases. As is well known, in fluid mechanics,
the thickness of the boundary layer becomes thicker along the flow direction. As
a result, the thickness of the displacement in the flow field will be thicker when
the boundary layer thickness increases. With the pebbles moving down toward the
outlet, the increasing of δ
in the higher part seems to be caused by the influence of
“viscosity” as known in fluid mechanics. The “viscosity” in the pebble flow is more
complicated than fluids because of many-body interactions, non-thermal fluctuations,
and other reasons (Liu and Nagel 1998, Corwin, Jaeger et al. 2005). However, from
2 Experiments in Pebble Flows
is the distance from the wall to a point where the flow velocity has essentially reached
the point of 99% “free stream” velocity. However, the pebble flow is very slow and
confined by the friction force from neighbors and walls, and the traditional boundary
layer thickness of the pebble flow is so thicker than the so-called “free-stream” zone
(where velocity is more significant than 99% of U ) which is much smaller. Instead,
considering both transverse and vertical movement for different bed configurations,
the funnel flow zone can be referred to the boundary zone. At each height, the velocity
of the outer boundary of the mass flow zone is regarded as U while calculating the
thickness of displacement in the boundary funnel flow zone. A smaller δ
is required
in the real pebble-bed reactor design to make the pebbles in the corners spend shorter
time for flowing out of the vessel. In this way, the δ
of R (∞,60 ◦ ) bed configuration
will be equal to zero so that there is no funnel flow zone in this situation.
The formation of the boundary layer concerns many terms, for example, the bed
configuration, the friction, and so on. The pebbles in the lower corner of the pebble
bed flow very slowly and are nearly stagnant, which is called the stagnant zone. The
pebbles between the stagnant zone and the central zone are easier to move under a
more inclined base slope, because a large slope of inclination provides a larger driving
force along the wall. Therefore, the enlarged base cone angle significantly decreases
the scale of the stagnant zone, up to complete disappearance, just as presented in the
case of a 60
◦ cone angle. Due to the existence of the stagnant zone, the flow field
becomes non-uniform. Pebbles will form rugged and lumpy borders and hinder the
flowing of their neighboring pebbles significantly.
Meanwhile, such resistance will transmit within a certain range toward the bed
center, and finally produce higher non-uniformity of the velocity distribution. In
other words, the velocity distribution will be more uniform if there is no stagnant
zone. Because of the non-uniformity, the averaged vertical velocity decreases continuously from the central zone to the near-wall zone. In this way, the enlarged base
cone angle significantly reduces the scale of the stagnant zone, even up to complete
disappearance just as presented in the case of a 60
◦ cone angle. In other words, the
enlarged base cone angle increases the uniformity of the vertical velocity throughout
the whole flow field. Thus, the mass flow level of the case 60
◦ is 1, which means
that the vertical velocity in the pebble bed is higher than 0.3v
0 (the vertical velocity
of the particle at the central position that stands for the highest vertical velocity). In
this way, the boundary layer with a smaller velocity almost does not exist in the 60
◦ case according to the definition of the boundary layer in this section.
The thickness of displacement for each bed configuration is shown in Fig. 2.31a.
The thickness of most curves presents the dropping tendency in the lower part after
a slow increase when the height decreases. As is well known, in fluid mechanics,
the thickness of the boundary layer becomes thicker along the flow direction. As
a result, the thickness of the displacement in the flow field will be thicker when
the boundary layer thickness increases. With the pebbles moving down toward the
outlet, the increasing of δ
in the higher part seems to be caused by the influence of
“viscosity” as known in fluid mechanics. The “viscosity” in the pebble flow is more
complicated than fluids because of many-body interactions, non-thermal fluctuations,
and other reasons (Liu and Nagel 1998, Corwin, Jaeger et al. 2005). However, from