4.3. DEM03: Two stacked, compressed particles

This case serves to verify the MFIX-DEM linear spring-dashpot collision model through analysis of a multi-particle, enduring collision. This test case is based on the work of Chen et al. [4] and the MFIX-DEM test case was originally reported in Garg et al. [8].

4.3.1. Description

Two particles of equal radius, \(r_{p}\), are stacked between two fixed walls such that the particles are compressed. The lower and upper walls are located at \(y_{l} = 0.0\) and \(y_{w} = 3.6r_{p}\) and the particle centers are initially located at \(y_{10} = 0.25y_{w}\) and \(y_{20} = 0.75y_{w}\). This configuration, illustrated in Fig. 4.7, ensures that the particles remain in contact and compressed.

../_images/dem03-setup.png

Fig. 4.7 Two smooth spherical particles stacked between two fixed walls so that the system is always under compression. A sketch of the problem mechanics is provided along with force balances for the lower and upper particles.

An expression for the acceleration of the lower particle (particle 1) is

(4.12)\[\begin{split}\frac{d^{2}y_{1}}{dt^{2}} =& - g - \frac{k_{\text{nw}}}{m_{1}}\left( y_{1} - r_{p} \right) - \frac{\eta_{n1w}}{m_{1}}\frac{dy_{1}}{\text{dt}}\\ & -\frac{k_{n12}}{m_{1}}\left( 2r_{p} - \left( y_{2} - y_{1} \right) \right) - \frac{\eta_{n12}}{m_{1}}\left( \frac{dy_{1}}{\text{dt}} - \frac{dy_{2}}{\text{dt}} \right)\end{split}\]

where \(y_{1}\) and \(y_{2}\) are the particle center positions measured from the lower wall, \(g\) is the acceleration due to gravity, \(k_{\text{nw}}\) and \(k_{n12}\) are the particle-wall and particle-particle spring coefficients, \(\eta_{n1w}\) and \(\eta_{n12}\) are the particle-wall and particle-particle damping coefficients, and \(m_{1}\) is the mass of particle 1. Similarly, acceleration of the upper particle (particle 2) is given by

(4.13)\[\begin{split}\frac{d^{2}y_{2}}{dt^{2}} =& - g - \frac{k_{\text{nw}}}{m_{2}}\left( r_{p} - \left( y_{w} - y_{2} \right) \right) - \frac{\eta_{n2w}}{m_{2}}\frac{dy_{2}}{\text{dt}}\\ & +\frac{k_{n12}}{m_{2}}\left( 2r_{p} - \left( y_{2} - y_{1} \right) \right) + \frac{\eta_{n12}}{m_{2}}\left( \frac{dy_{1}}{\text{dt}} - \frac{dy_{2}}{\text{dt}} \right)\end{split}\]

where \(\eta_{n2w}\) is the particle-wall damping coefficient for the upper particle, and \(m_{2}\) is the mass of the upper particle.

4.3.2. Setup

Table 4.4 DEM-03 Setup, Initial and Boundary Conditions.

Computational/Physical model

1D, Transient

Granular flow (no gas)

Gravity

Thermal energy equation is not solved

Geometry

Coordinate system

Cartesian

x-length

1.0

(m)

y-length

0.0018

(m)

z-length

0.0010

(m)

Solids Properties

Normal spring coefficient, \(k_{n}\)

103

(N·m-1)

Restitution coefficient, \(e_{n}\)

varied

( )

Friction coefficient, \(\mu\)

0.0

( )

Solids 1 Type

DEM

Diameter, \(d_{p}\)

0.001

(m)

Density, \(\rho_{s}\)

20000

(kg·m-3)

Solids 2 Type

DEM

Diameter, \(d_{p}\)

0.001

(m)

Density, \(\rho_{s}\)

10000

(kg·m-3)

Boundary Conditions

All boundaries

Solid walls

4.3.3. Results

Analytical solutions to equations Eq.4.12 and Eq.4.13 describing the motion of the particles are readily obtainable for perfectly elastic \(\left( \eta_{n12} = \eta_{n1w} = \eta_{n2w} = 1.0 \right)\) particles of equal mass \(\left( m_{1} = m_{2} \right)\). This is not the case for inelastic particles of different mass, therefore a fourth-order Runge-Kutta method is used to calculate a secondary numerical solution which is considered to be the analytical solution during the analysis.

Simulations were conducted for six friction coefficients, [0.5, 0.6, 0.7, 0.8, 0.9, 1.0], using the Adams-Bashforth time-stepping method. Fig. 4.8 shows the motion of the lower (left) and upper (right) particles as well as the absolute value of the relative error for a restitution coefficient of 1. The percent relative difference in results remains below 0.1% for this case. This is the largest observed difference across all cases with the difference in relative error decreasing with decreasing restitution coefficient.

../_images/image541.png

Fig. 4.8 Comparison between the fourth-order Runge-Kutta solution (solid line) and MFIX-DEM simulation (open symbols) for the center position of two stacked particles compressed between fixed walls for a restitution coefficient of 1. The absolute percent relative errors are shown as dashed lines.