Particle tracking
AddiDict simulates how particles move through a structure and interact with the solids. The movement of particles in the fluid is influenced by the following factors:
- Drag forces are the dominant forces in most cases, and particle trajectories are close to streamlines of the underlying flow field.
- Electric attraction. The particles and the surface of the solid parts can be electrically loaded, thus leading to attraction or bouncing effects.
- Diffusive (or Brownian) motion. Certain irregularities in the particle surface and collisions among particles lead to small random direction changes of particles.
AddiDict simulates particle movement by considering these three effects. All three effects can be switched off if it is known a-priori that they have no influence on the mass transport process.
Overall, the particle movement is governed by: Particle Momentum = Stokes Drag + External Forces
or, as formula:
In the friction coefficient:
the particle radius is optionally corrected by the Cunningham correction factor (Crowe, 2006):
to account for the reduced drag of very tiny particles that may easily pass between the molecules of the surrounding fluid. The particle diffusivity computes Brownian motion through:
The Brownian motion may be switched off by the user, and in that case the diffusivity is set to zero. External forces are the electrostatic force and an additional force field which is:
by default. The electrostatic field is set to zero if the user decides to switch off electrostatic effects.
The equations for friction, Cunningham correction, diffusivity, and external forces can be changed by the user through user defined functions, e.g., to include gravity, buoyancy forces in or a Cunningham correction with different parameters. See the appendix for details. The parameter for the Cunningham correction factor can be directly set in the AddiDict options.
The electrostatic charges are assumed as constant given forces on the surface of the solids. A constant charge density is assigned on all voxel walls that belong to this surface. The electric field is determined by solving the Poisson equation for the potential :
where is the surface charge density and F/m is the permittivity. Here, denotes the solid surface and is the Dirac distribution.
Here, is periodic in the tangential directions and should satisfy zero Dirichlet boundary conditions at and in flow direction. Numerically, infinity is replaced by and in Z-direction.
By construction, these boundaries lay away from the solid material and there is no conflict between singular forces on solid surfaces and these Dirichlet conditions. Due to the periodic boundary conditions, the potential feels a non-integrable amount of charges and tends to infinity in the solid material as the Dirichlet boundary is moved away from the solid. Thus, the potential depends on the position where the Dirichlet condition is located. However, only the electrical field is needed to determine the movement of the particles in the particle momentum equation and this remains almost unchanged from the location of the Dirichlet boundary as soon as this boundary is sufficiently far away from the solid material.
Used Variables and their Units
