Modeling of granular flow in a silo using the moving particle semi-implicit method (MPS) with the exponential Herschel-Balky rheological model

Silos are widely used in many industrial and engineering processes, so discharge grain from the silos is considered as one of the most important issues. Due to the high cost of laboratory studies on different materials and conditions, computational methods are used as a substitute approach for much less cost. Because of the ability of nonlinear Lagrangian methods to model large deformations and discontinuities, this study develops and evaluates a mesh-free Lagrangian model based on a weakly-compressible MPS formulation to simulate of the discharge of the granular silo. In Lagrangian methods, unlike the Eulerian method, instead of networking the solution field and breaking the equations on the nodes, the solution field is divided into a number of particles and the broken equations are solved on these particles. In fact, the governing equations are transformed into particle interaction equations using different operators. In the meantime, the particles that are closer to the particle under study will have a greater effect on that particle. In such a way that the effect of relatively distant particles can be ignored in comparison with closer particles and the interaction between particles can be limited to a specific domain called the radius of effect. The effect of each particle on the calculated particle is measured by a weight function. In the WC-MPS method, the system is considered as a system with weakly compressibility and calculates the pressure of each particle using the state equation. In this study, the Tait's state equation is used, which is used for high pressure water flow. The MPS method uses particle density to track the free surface. Because there are no particles outside the free surface, the density of the particles on the free surface decreases sharply. A particle is known as a free surface particle whose density is somewhat lower than the standard particle density. The value of this limit may be selected from 80% to 99% depending on the problem Therefore, the pressure of this particle on the free surface will be set to zero in each time step and in the MPS method, and there is no need to apply any additional condition for the free surface. For solid (impermeable) boundaries, such as walls or beds, this boundary condition is applied. In the vicinity of solid boundaries, the particle density decreases, which can lead to computational disturbances. Therefore, a number of ghost particles are located outside the boundaries to prevent this density reduction.  In this method the granular material is considered as a non-Newtonian visco-plastic fluid and an exponentially Herschel-Balky (H-B) rheological model in combination with pressure-dependent yield criteria model is employed to model non-cohesive grain behavior. The ability of the developed numerical method to discharge grain from silos has been evaluated and compared with the experimental results and the DEM method. Comparison of the results of the developed numerical method for the discharge of grains from the silo with the available experimental measurements and the DEM method shows the capabilities of the proposed model to accurately predict the surface and velocity profiles in this sample problem.