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16 / June 2020 powderbulk.com α (measured from horizontal), and the wall friction angle. The parti- cles' acceleration, a, is measured as follows a = g(sinα − cosα tan) The particles will continue to accelerate — provided that the cone angle (α) is greater than the wall friction angle (') — and the par- ticles' velocity upon reaching the cone's end, V 2 , is determined by V 2 = √V 1 2 + 2as where s, the distance the particles traveled, is calculated from s 2 = D 2 + H 2 2 where D is the cone's diameter, which is smaller than the ring's diameter, and H is the cone's height, as shown in Figure 3. The height can be calculated from the diameter and cone angle by H = Dtanα 2 The particles continue to accel- erate after leaving the cone. The distance the particles travel hori- zontally and then vertically, marked as X and Y respectively on Figure 3, over time, t, can be calculated from X = (V 2 cosα)t and Y = (V 2 sinα)t + g t 2 2 where g is the particles' accelera- tion due to gravity, X is equal to the difference between the radii of the cone and the ring, and Y, which is the vertical distance the particles travel after leaving the cone and the cone's and how the cone sits just within the ring's top. This helps to facilitate the dispersion dynamics. A dispersion cone and ring can be designed using simple physics. If the particles fall freely when they're dropped onto the cone, their veloc- ity before impact, V 0 , on average is their free-fall velocity V 0 = √2gh where g is the acceleration due to gravity and h is the drop height, as shown in Figure 3. Other measure- ments pertaining to dispersion cone schematics are also shown in Figure 3 and will be covered later. Upon hitting the cone, the parti- cles will start to descend down the cone. From a momentum balance, V 1 , which is the particles' velocity after impact, as shown in Figure 4, is determined by V 1 = V 0 (cosθ − sinθ tan) where θ is the impact angle and ' is the angle of wall friction. The angle of wall friction is the inverse tangent of the friction coef cient and can be determined from shear cell testing. While sliding on a straight surface, the particles accelerate or decelerate, depending on the rela- tive magnitudes of the cone angle, FIGURE 3 Dispersion cone and ring dimensions D X h H α s Y FIGURE 4 Particles' different velocity point changes V 0 θ V 1 V 2 FIGURE 2 Dispersion cone schematic Top view Cone Ring Side view