Results for hopper

Fig. 5.8 shows snapshots of the settling process. Only the half of the particle assembly is visualized in order to obtain an insight into the interior of the granular material. The same way of presentation is followed for all the cases studied. At the beginning of simulation, the particles (balls with the same diameter) with zero initial velocity are located regularly within the hopper. Due to the friction and damping forces the motion of particles (caused by the gravitational force) decays as the time increases. The total kinetic energy of the system as a function of time (Fig. 5.9) represents clearly this process. Although the system has a small kinetic energy at the end of the settling process (settling time= 1.5 s) practically the system of spheres (balls) can be considered to be at rest at t=1.5 s.

(a) (b)

Fig. 5.8. Position of particles during settling in the case of hopper: (a) at t=0s (initial configu-ration), and (b) at t=1.327 s [93]

0,00001 0,0001 0,001 0,01 0,1 1 10

0 0,5 1 1,5

Time [s]

Total kinetic energy [J]

Fig. 5.9. Total kinetic energy (translational and rotational) of the system as a function of time in the case of hopper [93]

At t= 1.5 s, the outlet is opened and the hopper discharge is started. To indicate the characteristic motion of the granular material one portion of the particles is shaded by dark gray which yields a pattern of bands. Fig. 5.10 shows snapshots of the granular material flow-ing out of the hopper. The motion of the particles can be followed easily through the move-ment of the interfaces between the bands. The interfaces deform across the whole cross sec-tion of the hopper and have a characteristic “V” shape within a secsec-tion located parallel to the

y-z plane. Distributions of the absolute velocity and the absolute angular velocity at t=2.45 s are shown in Figs. 5.11-5.12. The gray code of the individual particles indicates the magni-tude of the depicted quantity. Since all the particles have the same diameter and mass, the tendencies of the translational and rotational kinetic energy distributions were similar to Figs.

5.11-5.12. As anticipated, the most intensive flow appears near and above the outlet. Near the walls, the particles move slower than in the center region of the granular material.

(a) (b) (c)

Fig. 5.10. Flow patterns at different simulation times in the case of hopper: (a) at t=1.5s (stat-ic state at the end of the filling process), (b) at t=2.55 s, and (c) at t=4.5 s [93]

Fig. 5.11. Absolute velocity distribution at t=2.45 s in the case of hopper (in m/s);



 

v_abs v_x²v_y²v_z² [93]

Fig. 5.12. Absolute angular velocity distri-bution at t=2.45 s in the case of hopper (in 1/s) (the depicted quantity is derived in the

same manner as in Fig. 5.11) [93]

In the case of static state (t=1.5s), the distributions of normal wall forces and mean wall pressures along the walls as a function of height from the bottom of the hopper are shown in Fig. 5.13. The distributions of the normal wall forces are represented on the basis of the nor-mal component of particle/wall contact forces. In this study, the expression nornor-mal wall force is used as a synonym for the normal component of particle/wall contact force. The sum of the normal forces (resultant normal force) acting on the horizontal wall of Fig. 5.13 is F bot-tom=5.17N. For the right and the left wall of Fig. 5.13, these values are as follows:

Fright=34.17 N, Fleft=33.65 N. In case of granular materials, one part of the weight of the initial material head obtained at the end of the filling process is carried by the friction forces

arising on the walls. The contribution of friction forces can be determined from the

simula-tion shows that about 12% of the weight of the initial material head is carried by the fricsimula-tion forces arising on the walls. Additionally, it can also be concluded that mostly the inclined walls carry the weight of the granular material.

Fig. 5.13. Distributions of normal wall forces and mean wall pressures acting on the walls of the hopper at the end of the filling process (at t=1.5 s). The expression normal wall force is a synonym for the normal component of particle/wall contact force. [93] Contrary to Figs. 5.4 and 5.5 here not the width but the length of the lines representing contact forces is

proportion-al to the magnitude of the force.

The wall pressures are averaged on a wall segment of height dz (see Fig. 5.7a). In the present study, dz is taken to 2d. For each evaluation the segment boundary is moved by half particle diameter in vertical direction (z-direction). In the case of a given wall, the height dz belongs to a trapezoid-shaped wall segment. In order to compute the mean wall pressure, as a first step, the normal forces acting on the actual wall segment are summed. The mean wall pressure (moving average) is defined as the total force acting perpendicular to a wall segment divided by the area of the wall segment. Since the mean wall pressure distribution is comput-ed from the normal component of the particle/wall contact forces, it shows a fluctuation along the wall. It must be noted that the fluctuation is strongly affected by the averaging method applied to the pressure computation. It can be seen clearly that there is no significant devia-tion from linear pressure dependence in Fig. 5.13. As it is well known, the wall pressure act-ing on the walls of a container filled with any fluid decrease linearly with increasact-ing height because no shear forces are exerted between the fluid and the solid boundary if the fluid is at rest. Contrary to fluids, in the case of real granular material, the static wall pressures reach a constant, height-independent value at a sufficiently large depth below the free surface (pres-sure saturation) because the shear forces transfer one portion of the weight of the granular material to the walls. This tendency can hardly be identified in Fig. 5.13. The reason for this is that the system modeled is not sufficiently high to show unambiguously this tendency; so,

only the linear approximation of the wall pressure distribution valid in the neighborhood of the free surface is described. In order to determine the wall pressure distribution during the outflow, the pressures are averaged both on a wall segment of height dz and over 1000 time steps. In the case of hopper, the mean wall pressure distribution during discharge is shown in Fig. 5.14. The results reported here are in accordance with both the two-dimensional ones of Ristow and Herrmann [88] and the analytical calculation of Walters [94].

Fig. 5.14. Mean wall pressure distribution along the walls in the case of hopper outflow (at t=1.8 s) [93]