Software: PFC: Examples

PFC2D: Granular Flow from a Hopper

PFC2D models the flow of material from a hopper that is defined by four walls that contain an initial compact assembly of particles, as is shown in the diagram below. The walls at each side of the hopper (walls AD and BC below) are fixed; they confine the flow of particles once the base wall (wall AB) is removed. The wall on top of the model (wall EF) is a servo-controlled wall that moves in the vertical direction; its velocity is adjusted automatically to maintain a specified value of stress (the stress on the servo-wall is computed as the resultant reaction force that the particles impose on the wall divided by the exposed wall area).

The analysis is performed twice, the only difference between the cases being the use of different frictional properties of particles and walls. For both analyses, the initial steps are the same: the hopper and assembly of particles are generated, the servo-mechanism is activated so that a specified target stress on the servo-controlled wall can be achieved. In the last stage, the base wall is removed and the particles flow downward by the action of gravity and the loading of the top wall. As particles escape the hopper, the servo-mechanism remains activated, maintaining the specified target stress on the wall. In the last stage of the first analysis, the friction coefficient is set to 1.0 for the particles and 0.0 for the walls. In the second analysis, the friction coefficient for both particles and walls is set to 1.78. It will be shown that the behavior of the system is dependent on the frictional properties of particles and walls.

After the particle assembly is generated and compacted, the model is allowed to "cycle" so that a certain value of stress (5 MPa) on the servo-controlled wall is obtained. Next, the base wall is deleted and the model is allowed to cycle with the servo-mechanism still activated. In the first analysis, after the wall is removed from the model the servo-controlled wall moves down as particles escape from the hopper (this sequence represents a run of 5000 cycles). An arch of contact forces (shown in black) develops in the upper part of the assembly, but it is not sufficiently stable to stop the flow of particles. Eventually, if cycling were continued, the hopper would become empty of particles. In the second analysis the higher values of friction specified in the model produce "locking" of the particles in the assembly. The arch that forms above the opening prevents more balls from escaping the hopper (this sequence represents a run of 10,000 cycles).

The figure below illustrates the stress on the servo-controlled wall. For both cases, the wall is removed at approximately one tenth of a second. The period preceding that records the modeling stage where the wall is brought to the target stress of 5 MPa. After the wall is removed, in the first case, the stress suddenly drops and oscillates around a mean value, just below the target stress of 5 MPa. The oscillation is an indication that the arch of contact forces seen in this case is not stable; the continuous flow of particles is continually disturbing the static equilibrium of the system. By comparison, in the second case the quasi-stable nature of the arch is reflected in the stress and velocity (not shown) of the servo-controlled wall, where the stress becomes fixed at the target stress and the velocity of the wall tends to zero.

The results suggest that the frictional properties for balls and walls specified in the two cases directly impact whether a quasi-stable arch of forces is formed, although in the second case a slight disturbance would probably break this stability and produce unstable flow.

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