(79c) Numerical Modeling of Filling and Discharge Processes in Hoppers of Different Shapes

The flow of granular material in filling and discharge in hoppers of different shapes is analyzed in this paper. Discrete Element Method is applied to numerical modeling of filling and discharge. The granular material is represented by an assembly of 1980 particles. The filling process is simulated by compacting of the particles. The behavior of granular matter in filling was controlled by considering the evolution of total kinetic energy. Some numerical results of filling and discharge processes are presented. A detailed analysis of simulation results, distribution of the porosity field in the hopper after filling en masse and during discharge, evolution of discharge flow mass fraction and discharge rate velocity are presented. Finnaly, the tangential and the normal wall pressures on the left and the right wall of the model were computed. Other methods have a drawback of strongly depending on arbitrary assumptions on flow patterns during discharge and cannot consider the nature of granular materials. Existing theories of continuous mechanics or statistical physics have only limited applications for description of granular media and a unified theory encompassing all the granular phenomena is still missing [1]. The granular material is assumed to be composed of a set of discrete spherical particles with a specified grain size distribution and physical contact conditions. The motion of i-th particle is described by Newton's second law m_id²x_i(t)/dt²=F_i(t) I_id²θ_i(t)/dt²=T_i(t) where x_i and θ_i are vectors of position and orientation of the center of gravity, m_i is the mass and I_i is the inertia moment of the particle i (i = 1, N). The integration of above equations is performed by 5th order Gear predictor-corrector scheme from the applied forces and moments. Vectors F_i and T_i in the equations present the sum of gravity, inter-particle contact forces and torques, which act on the particle i. The evaluation of the inter-particle as well as particle-wall contact forces is performed by using a visco-elastic contact model [2]. The boundary conditions are determined by the walls of finite size which, in case of their contact with particles, are treated as the particles with infinite radius and mass. The characteristic dimension of the outlet for the analyzed hoppers is related to a minimal diameter d of the particles assuming to be a = 10d while the thickness of the hoppers at the orifice is assumed to be b = 5d. The characteristic dimension of the top hoppers edge is L = 33.3a, while the height of the hoppers is H = 28.3a. The geometry of the hopper and a transition dimension of the bottom edge L_b are defined by the angles of inclination θ_x and θ_y of generetrix to the horizontal. Hence, the value of the angles θ_x=68° and θ_y=90° corresponds to the plane-wedged hopper, while the angles θ_x=68° and θ_y=68° correspond to the space-wedged hoppers. The flat bottomed hopper is stated by assuming the angles to be θ_x=90° and θ_y=90° as well as L_b = L. The maximal angles are θ_x=68° and θ_y=68°, with the bottom edge L_b=a as practically accepted. Assuming the minimal diameter of the particle to be d = 0.06 m, the geometry of the hopper is defined as: b = 0.3 m, L = 2 m and H = 1.7 m and a = 0.6 m. The main geometric parameters of the hoppers are presented in Figure 1. Figure 1 The geometry of the hopper The values of the particle radii R_i ranging from 0.03 to 0.035m are defined randomly with uniform distribution. Inter-particle and particle-walls friction is defined by the same friction coefficient μ. Three types of material with three different μ values are considered. Total mass M of the material is fixed and is equal to M = 143.7 kg. The data of the visco-elastic particle are given below: Density ? ρ=500 kg/m³ Poisson's ratio ? ν=030 Elasticity modulus E=0.3∙10⁶ Pa Shear modulus G =0.11∙10⁶ Pa Normal viscous damping coefficient, γ_n=60.01/s Tangential viscous damping coefficient, γ_t=10.01/s Hence, the fill is simulated by compacting of the particles en masse i.e. all particles undergo the fall due to acting the force. The space above the hopper was divided into the cubic cells as an orthogonal and uniform grid with size of 0.1 m. Initially, at time t = 0, the particles are embedded into the centers of the cells and are free of contact. The initial velocity of the particles is defined randomly using uniform distribution ranging their magnitudes from 0 to 0.3 m/s. a)b)c) Figure 2 Particle velocity vectors time sequence in discharge: a-c) at time 0.6s The discharged mass fraction m as a normalized variable ranging between zero and unit (0≤ m ≤1) is computed as the ratio between the discharged mass M_d and the total mass M, while the discharge flow rate v(t) is defined as time-dependent discharge mass rate averaged in time t. A detailed analysis of simulation results is focused below on the behavior of granular material in the plane-wedged hopper (θ_x=68° and θ=90°). Functions for the shape of the hoppers presented above and different particle friction coefficient are presented in Figure 3. In Figure 4 the distribution of the porosity fields within the hopper after filling en masse and during the discharge is shown. a)b) Figure 3 Discharge flow mass fraction a) and discharge rate velocity b) Finally, the tangential and normal pressures on the walls of hopper were computed and are presented in Figure 5. The symbols p_n and p_t denote the normal and tangential pressures in Pascals, respectively. a)b) Figure 4 Porosity within the hopper. The end of filling: a) μ=0.6. Discharge process (μ=0.3): b) at 0.6s. a)b) Figure 5 Wall pressures: a) left wall, b) right wall 1. De Gennes P.G. Reflections on the mechanics of granular matter. Physica A 261, 1998, p. 267-293 2. Balevièius R., D?iugys A., Kaèianauskas R. Discrete element method and its application to the analysis of penetration into granular media. Journal of Civil Engineering and Management, 2004, 10(1), p. 3-14 3. Balevièius R, Kaèianauskas R, D?iugys A, Maknickas A, Vislavièius V. DEMMAT code for numerical simulation of multi-particle dynamics. Information Technology and Control, 2005, 34(1), p. 71-78 4. D?iugys A, Peters B.J. An Approach to Simulate the Motion of Spherical and Non-Spherical Fuel Particles in Combustion Chambers. Granular Material, 2001, 3(4), p. 231-266