92
2 Experiments in Pebble Flows
all particles remains higher for R (∞,60 ◦ ) situation. Furthermore, there is higher flow
uniformity and mass flow level in the radial direction for the case with a steeper
base bottom. A larger mass flow zone indicates that much more particles flow with a
relatively higher vertical velocity. Consequently, particles tend to drift horizontally
toward a zone with faster downward flows because they are likely to get more space
to move in the transverse direction. Meanwhile, particle flow in the pebble bed
follows approximately the equation of continuity. That is, the vertical mass flux in
one horizontal-stripe layer is equal to that in another layer, statistically. All bed
configurations retain the same recirculation flow rate (150/min) and the same pile
height of the pebbles in the bed, which all make the differences of the ¯
s y among these
cases small. In the arctangent function, the tilted angle θ in of the resultant movement
monotonically decreases with the reduction of the ratio of ¯
s y to ¯
s x . Thus, a steeper
bottom usually means a smaller tilted angle.
Moreover, there are smaller differences of θ in among the arc shapes with a finite
radius for the similar transverse velocity profile as shown in Fig. 2.27. R 1 , R 2 , and R 3
situations, all display smaller transverse dispersion in the base corner of the pebble
bed. Specifically, the R 3 bed configuration presents similar tilted angle (75.5
◦ ) with
the R (∞,30 ◦ ) case because of the slight base distinction between the two situations.
As aforementioned, higher mass level and flow uniformity will make a smaller tilted
angle. That is why the θ in of the R 1 case is a little larger than that of the R 2 situation.
The velocity vectors of all particles in the pebble bed are collected during the
experiment time of 3300 seconds. The angle between the velocity vector and the
radial direction for the particle i at time t is expressed as θ i (t), then the difference
between θ i (t) and θ in is
Δθ i (t) = θ i (t) − θ in .
(2.17)
More than 6,000,000 individual velocity vectors are enough to calculate the probability distribution of Δθ (Fig. 2.30) and the standard deviations σ (Δθ) (Table 2.2)
of Δθ for each bed configuration statistically.
From Fig. 2.30, the distribution of Δθ indicates some similar characteristics for
all bed configurations. Firstly, it is noticed that the number of particles with a positive
value of Δθ accounts for less than 50% of the total number of particles in the pebble
bed. That is, there are more than 50% of particles whose velocity vector angle θ is less
than θ in . The vertical velocity of particles with a larger θ is significantly higher than
those with a smaller θ . As a result, the ratio of S y to S x becomes more significant with
making the θ in higher than the average of the θ of all particles. So velocity vectors of
particles with larger θ contribute more to calculating θ in . Moreover, the Δθ presents
a left-skewed distribution profile, namely the tail on the left side of the probability
density function is longer or flatter than the right side. The positive Δθ displays a
concentrated distribution profile arranging from 0
◦ to 20
◦ , while the negative Δθ
presents a dispersed distribution. For instance, the Δθ reaches −50
◦ , which means
the particles move faster in the transverse direction than in the vertical direction for
the R 1 bed configuration.
2 Experiments in Pebble Flows
all particles remains higher for R (∞,60 ◦ ) situation. Furthermore, there is higher flow
uniformity and mass flow level in the radial direction for the case with a steeper
base bottom. A larger mass flow zone indicates that much more particles flow with a
relatively higher vertical velocity. Consequently, particles tend to drift horizontally
toward a zone with faster downward flows because they are likely to get more space
to move in the transverse direction. Meanwhile, particle flow in the pebble bed
follows approximately the equation of continuity. That is, the vertical mass flux in
one horizontal-stripe layer is equal to that in another layer, statistically. All bed
configurations retain the same recirculation flow rate (150/min) and the same pile
height of the pebbles in the bed, which all make the differences of the ¯
s y among these
cases small. In the arctangent function, the tilted angle θ in of the resultant movement
monotonically decreases with the reduction of the ratio of ¯
s y to ¯
s x . Thus, a steeper
bottom usually means a smaller tilted angle.
Moreover, there are smaller differences of θ in among the arc shapes with a finite
radius for the similar transverse velocity profile as shown in Fig. 2.27. R 1 , R 2 , and R 3
situations, all display smaller transverse dispersion in the base corner of the pebble
bed. Specifically, the R 3 bed configuration presents similar tilted angle (75.5
◦ ) with
the R (∞,30 ◦ ) case because of the slight base distinction between the two situations.
As aforementioned, higher mass level and flow uniformity will make a smaller tilted
angle. That is why the θ in of the R 1 case is a little larger than that of the R 2 situation.
The velocity vectors of all particles in the pebble bed are collected during the
experiment time of 3300 seconds. The angle between the velocity vector and the
radial direction for the particle i at time t is expressed as θ i (t), then the difference
between θ i (t) and θ in is
Δθ i (t) = θ i (t) − θ in .
(2.17)
More than 6,000,000 individual velocity vectors are enough to calculate the probability distribution of Δθ (Fig. 2.30) and the standard deviations σ (Δθ) (Table 2.2)
of Δθ for each bed configuration statistically.
From Fig. 2.30, the distribution of Δθ indicates some similar characteristics for
all bed configurations. Firstly, it is noticed that the number of particles with a positive
value of Δθ accounts for less than 50% of the total number of particles in the pebble
bed. That is, there are more than 50% of particles whose velocity vector angle θ is less
than θ in . The vertical velocity of particles with a larger θ is significantly higher than
those with a smaller θ . As a result, the ratio of S y to S x becomes more significant with
making the θ in higher than the average of the θ of all particles. So velocity vectors of
particles with larger θ contribute more to calculating θ in . Moreover, the Δθ presents
a left-skewed distribution profile, namely the tail on the left side of the probability
density function is longer or flatter than the right side. The positive Δθ displays a
concentrated distribution profile arranging from 0
◦ to 20
◦ , while the negative Δθ
presents a dispersed distribution. For instance, the Δθ reaches −50
◦ , which means
the particles move faster in the transverse direction than in the vertical direction for
the R 1 bed configuration.