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