2.5 Particle Velocimetry Measurements
85
Table 2.2 Characteristic index for each bed base configuration
Categories
Arc shape with finite radius
Arc shape with infinite radius
R 1
R 2
R 3
R ∞,30 ◦
R ∞,45 ◦
R ∞,60 ◦
Characteristic
parameters
R 1 =
√
3es
3
R 2 = e s
R 1 =
es
√
7
R = ∞
R = ∞
R = ∞
β B = 30 ◦
β B = 45 ◦
β B = 60 ◦
Total mass
flow level
0.5407
0.6585
0.6391
0.6369
0.7217
1
θ in
77.6 ◦
73.4 ◦
75.5 ◦
75.6 ◦
73.8 ◦
66.4 ◦
σ (Δθ)
18.0 ◦
12.2 ◦
17.2 ◦
13.5 ◦
12.6 ◦
9.56 ◦
E(δ )
9.01
5.94
5.78
5.96
3.52
0
σ (δ )
7.29
4.80
4.40
4.52
3.18
0
β B is the angle of conical slope;
E(δ ) and σ (δ ) are the average and standard deviation of δ , respectively
be moved in the direction perpendicular to its normal vector away from the reference
plane in an inviscid fluid stream of velocity U to give the same flow rate as that
occurring between the surface and the reference plane in a real fluid. The displacement
thickness for incompressible flow can be defined based on the volumetric flow rate
as [43]:
δ
=
δ
0
1 −
u(y)
U
dy,
(2.13)
where U is not a constant. The component U in the tangential direction of the
wall surface normal to the boundary layer is chosen to calculate the displacement
thickness.
2.5.3.11 Particle Tracking Technique
The polished stainless black glass balls are chosen to achieve the necessary resolution in measuring the particle trajectories [27]. Under convenient illumination,
each particle reflects a small, bright, and well-defined spot, which allows for a very
precise position measurement (Fig. (2.26)). In the images, every pebble diameter
occupies about d=50 pixels. As the flow rate is about 2.5 particles per second within
almost 70,000 pebbles in the vessel, the rate of taking one image per second is
acceptable [44]. The trajectories of individual pebbles are focused on the Particle
Tracking Velocimetry (PTV) method. In contrast, the particle image velocimetry
(PIV) method can only obtain the velocity field without the movement information
of a given pebble [45].
85
Table 2.2 Characteristic index for each bed base configuration
Categories
Arc shape with finite radius
Arc shape with infinite radius
R 1
R 2
R 3
R ∞,30 ◦
R ∞,45 ◦
R ∞,60 ◦
Characteristic
parameters
R 1 =
√
3es
3
R 2 = e s
R 1 =
es
√
7
R = ∞
R = ∞
R = ∞
β B = 30 ◦
β B = 45 ◦
β B = 60 ◦
Total mass
flow level
0.5407
0.6585
0.6391
0.6369
0.7217
1
θ in
77.6 ◦
73.4 ◦
75.5 ◦
75.6 ◦
73.8 ◦
66.4 ◦
σ (Δθ)
18.0 ◦
12.2 ◦
17.2 ◦
13.5 ◦
12.6 ◦
9.56 ◦
E(δ )
9.01
5.94
5.78
5.96
3.52
0
σ (δ )
7.29
4.80
4.40
4.52
3.18
0
β B is the angle of conical slope;
E(δ ) and σ (δ ) are the average and standard deviation of δ , respectively
be moved in the direction perpendicular to its normal vector away from the reference
plane in an inviscid fluid stream of velocity U to give the same flow rate as that
occurring between the surface and the reference plane in a real fluid. The displacement
thickness for incompressible flow can be defined based on the volumetric flow rate
as [43]:
δ
=
δ
0
1 −
u(y)
U
dy,
(2.13)
where U is not a constant. The component U in the tangential direction of the
wall surface normal to the boundary layer is chosen to calculate the displacement
thickness.
2.5.3.11 Particle Tracking Technique
The polished stainless black glass balls are chosen to achieve the necessary resolution in measuring the particle trajectories [27]. Under convenient illumination,
each particle reflects a small, bright, and well-defined spot, which allows for a very
precise position measurement (Fig. (2.26)). In the images, every pebble diameter
occupies about d=50 pixels. As the flow rate is about 2.5 particles per second within
almost 70,000 pebbles in the vessel, the rate of taking one image per second is
acceptable [44]. The trajectories of individual pebbles are focused on the Particle
Tracking Velocimetry (PTV) method. In contrast, the particle image velocimetry
(PIV) method can only obtain the velocity field without the movement information
of a given pebble [45].