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G. Fragnière et al.
patterns and their impact on the product particle capturing probability are investigated. In the previous chapter only the grinding media contacts were studied, the
simulations described in this chapter describe the number of product particles being
stressed during these contacts.
The first simulation environment is set on the mesoscale, meaning only two grinding beads and their local environment around or in the gap between them are being
considered to determine the effect on fluid displacement. Data on the relative grinding bead motion is derived from stirred media mill simulations performed on the
macroscale [10]. The CFD domain around the beads is fully resolved, the simulation
of the grinding beads motion is possible due to mesh deformation. Since the mesh
deformation at the contact point would be too large it is not possible to simulate
the actual contact. The immersed boundary method is used for the coupling of CFD
and DEM. The grinding bead impact is set to be elastic, meaning below a critical
distance from the symmetry plane a complete reversal of grinding media motion
occurs. The grinding beads and product particles are assumed to be spherical. For
comparison an analytical method is applied to allow the direct comparison of the
normalized velocity of a collision of two grinding beads (normalized to the starting
velocity). The analytical model is the sum of the fluid displacement force and the
fluid resistance force. The fluid displacement for ball-ball and ball-wall contacts is
calculated according to Beinert et al. [10].
F dis,bb = −
3
2
πηv
r
h 0
(6)
F dis,bw = −6πηv
r
h 0
(7)
The fluid resistance force is calculated based on Kürten et al. [11] and depends
on the Reynolds number present in the system:
c w =
⎧
⎨
⎩
24/Re, if Re ≤ 0.25
21/Re + 6/
√
Re + 0.28, if 0.25 ≤ Re ≤ 4000
0.45, if Re ≥ 4000
(8)
For the following comparison of the numerical and analytical determination the
diameter of the grinding media is d GM = 375 μm, the grinding media density (ZrO 2 )
ρ GM = 6067 kg/m
3 and the surrounding fluid is water. The distance between the
grinding beads and the symmetry plane (0.25 * d GM ; 0.50 * d GM ; 1.00 * d GM ; 2.00
* d GM ) as well as the starting velocity (0.2; 1.0; 5 m/s) are varied. These three
starting velocities result in different Re numbers (8.4; 42.0; 210.0). The numerical
and analytical results for the normalized velocity in dependence of the normalized
distance between the grinding beads can be seen in Fig. 6 for the time before and
after the collision, with the lower values around 0.6 describing the velocity after the
collision:
G. Fragnière et al.
patterns and their impact on the product particle capturing probability are investigated. In the previous chapter only the grinding media contacts were studied, the
simulations described in this chapter describe the number of product particles being
stressed during these contacts.
The first simulation environment is set on the mesoscale, meaning only two grinding beads and their local environment around or in the gap between them are being
considered to determine the effect on fluid displacement. Data on the relative grinding bead motion is derived from stirred media mill simulations performed on the
macroscale [10]. The CFD domain around the beads is fully resolved, the simulation
of the grinding beads motion is possible due to mesh deformation. Since the mesh
deformation at the contact point would be too large it is not possible to simulate
the actual contact. The immersed boundary method is used for the coupling of CFD
and DEM. The grinding bead impact is set to be elastic, meaning below a critical
distance from the symmetry plane a complete reversal of grinding media motion
occurs. The grinding beads and product particles are assumed to be spherical. For
comparison an analytical method is applied to allow the direct comparison of the
normalized velocity of a collision of two grinding beads (normalized to the starting
velocity). The analytical model is the sum of the fluid displacement force and the
fluid resistance force. The fluid displacement for ball-ball and ball-wall contacts is
calculated according to Beinert et al. [10].
F dis,bb = −
3
2
πηv
r
h 0
(6)
F dis,bw = −6πηv
r
h 0
(7)
The fluid resistance force is calculated based on Kürten et al. [11] and depends
on the Reynolds number present in the system:
c w =
⎧
⎨
⎩
24/Re, if Re ≤ 0.25
21/Re + 6/
√
Re + 0.28, if 0.25 ≤ Re ≤ 4000
0.45, if Re ≥ 4000
(8)
For the following comparison of the numerical and analytical determination the
diameter of the grinding media is d GM = 375 μm, the grinding media density (ZrO 2 )
ρ GM = 6067 kg/m
3 and the surrounding fluid is water. The distance between the
grinding beads and the symmetry plane (0.25 * d GM ; 0.50 * d GM ; 1.00 * d GM ; 2.00
* d GM ) as well as the starting velocity (0.2; 1.0; 5 m/s) are varied. These three
starting velocities result in different Re numbers (8.4; 42.0; 210.0). The numerical
and analytical results for the normalized velocity in dependence of the normalized
distance between the grinding beads can be seen in Fig. 6 for the time before and
after the collision, with the lower values around 0.6 describing the velocity after the
collision: