Large-scale numerical simulations of polydisperse particle flow in a silo

Abstract

Very recently, we have examined experimentally and numerically the micro-mechanical details of monodisperse particle flows through an orifice placed at the bottom of a silo (Rubio-Largo et al. in Phys Rev Lett 114:238002, 2015). Our findings disentangled the paradoxical ideas associated to the free-fall arch concept, which has historically served to justify the dependence of the flow rate on the outlet size. In this work, we generalize those findings examining large-scale polydisperse particle flows in silos. In the range of studied apertures, both velocity and density profiles at the aperture are self-similar, and the obtained scaling functions confirm that the relevant scale of the problem is the size of the aperture. Moreover, we find that the contact stress monotonically decreases when the particles approach the exit and vanish at the outlet. The behavior of this magnitude is practically independent of the size of the orifice. However, the total and partial kinetic stress profiles suggest that the outlet size controls the propagation of the velocity fluctuations inside the silo. Examining this magnitude, we conclusively argue that indeed there is a well-defined transition region where the particle flow changes its nature. The general trend of the partial kinetic pressure profiles and the location of the transition region results the same for all particle types. We find that the partial kinetic stress is larger for bigger particles. However, the small particles carry a higher fraction of kinetic stress respect to their concentration, which suggest that the small particles have larger velocity fluctuations than the large ones and showing lower strength of correlation with the global flow. Our outcomes explain why the free-fall arch picture has served to describe the polydisperse flow rate in the discharge of silos.

Appendix: Discrete element modeling implementation (DEM)

The DEM implementation is a polydisperse generalization of a monodisperse hybrid CPU / GPU algorithm that allows us to efficiently evaluate the dynamics of several hundred thousand particles [37, 41]. For each particle $i=1 \ldots N$, the DEM simulation includes three translational degrees of freedom and the rotational movement is described by a quaternion formalism. We have used a linear spring approach; thus, the normal interaction force ${\mathbf {F}}_{ij}^{n}$ between the particles i with radius $r_i$ and j with radius $r_j$ depends linearly on the particles’ overlap distance $\delta = r_i + r_j - \Vert \Delta R_{ij}\Vert$, where $\Vert \Delta R_{ij}\Vert$ is the relative distance between the particles. Moreover, the local dissipation is introduced by a non-linear viscous damping term, which depends on the normal relative velocity ${\mathbf {v}}_{rel}^{n}$. Hence, the total normal force reads as

$$\begin{aligned} {\mathbf {F}}_{ij}^{n} =- k_n~\delta ~{\hat{n}} - \gamma _n m_{eff} {\mathbf {v}}_{rel}^{n}, \end{aligned}$$

(9)

where

$$\begin{aligned} k_n = \frac{16.0}{15.0}\sqrt{R_{eff}} Y \left( \frac{15 m_{eff} V_c^2}{16\sqrt{R_{eff}}Y} \right) ^{1/5} \end{aligned}$$

(10)

and

$$\begin{aligned} \gamma _n = \sqrt{ \frac{4 m_{eff} k_n}{1+ \left( \frac{\pi }{\ln (e_n)} \right) ^2}} \end{aligned}$$

(11)

represent the damping coefficient, $m_{eff} = m_im_j/(m_i+m_j)$, $R_{eff} = r_ir_j/(r_i+r_j)$, and Y is the particles Young’s modulus. The tangential component $F_{ij}^{t}$ also includes an elastic term and a viscous term,

$$\begin{aligned} {\mathbf {F}}_{ij}^{t} =-k_t {\mathbf {\xi }}- \gamma _t m_{eff} {\mathbf {v}}_{rel}^{t}, \end{aligned}$$

(12)

where $\gamma _t$ is a damping coefficient and ${\mathbf {v}}_{rel}^{t}$ is the tangential relative velocity of the overlapping pair. The variable $|{\mathbf {\xi }}|$ represents the elongation of an imaginary spring with elastic constant $k_t$. As long as there is an overlap between the interacting particles, ${\mathbf {\xi }}$ increases as $d{\mathbf {\xi }}/dt = {\mathbf {v}}_{rel}^t$ [13]. The elastic tangential elongation ${\mathbf {\xi }}$ is kept orthogonal to the normal vector [15] and it is truncated as necessary to satisfy the Coulomb constraint $|{\mathbf {F}}_{ij}^{t}|<\mu | {\mathbf {F}}_{ij}^{n}|$, where $\mu$ is the friction coefficient. The equations of motion are integrated using Fincham’s leap-frog algorithm (rotational) [42] and a Verlet Velocity algorithm (translational) [43].

In all the simulations reported here, the used contact parameters correspond to particles with a Young’s modulus $Y = 120~GPa$, normal restitution coefficient $e_n=0.88$, density $\rho _p=7520$ kg/m$^3$, and friction $\mu =0.5$. We set $\frac{k_t}{k_n} = \frac{2}{7}$ and $\frac{\gamma _t}{\gamma _n}=0.1$, and gravitational acceleration $g=10~m/s^2$. In all cases, the molecular dynamics time step was set as $\Delta t = \frac{t_c}{50} = 6.3 \times 10^{-7}~s$, where $t_c$ is approximately equal to time of contact between two small particles $t_c = \pi \sqrt{\frac{m_{eff}}{2k_n}}$.