Often, the grid resolution is not able to resolve all features of the filter material. In this case, not only fluid voxels or solid voxels exist, but some voxels model a porous media themselves. Small particles (with diameter much smaller than the voxel length) may pass through the voxel, but they may also be captured inside of the voxel.
In such unresolved simulations, the filter media is modeled as homogeneous material with a defined permeability. The interaction between particles and the surface cannot be modeled explicitly. Instead, the pass-through model describes the probability that a particle passes through the filter media. If a particle is filtered depends on the pass-through probability of the passed voxels, and the distance covered in these voxels. Particles can be trapped anywhere on their way through the medium.
Travel path of a particle inside the filter media and its crossings through the grid cells (voxels).
Assume that a filter media with a thickness has a rating (at a given flow velocity). Then, 1 in particles pass through. Thus, the probability for a particle to travel a distance without getting captured is
(251)
So, the probability that the particle travels through the whole media is
(252)
and the passing probability of passing the media times is
In this model, particles cannot enter the porous material. The material behaves like a solid material for the particles, but it is permeable for the fluid..
In the clogging model, it is assumed that the efficiency depends on the amount of deposited dust. A voxel representing a porous filter medium has in the clean state an initial fractional filtration efficiency of . During the simulation, dust may deposit inside the voxel, increasing the deposited dust volume fraction inside of the voxel from an initial fraction to an assumed maximal particle packing density . When is reached, the filtration efficiency reaches 100%:
(257)
The implemented Clogging model assumes a linear dependency between efficiency and local dust volume fraction . The passing probability is then again computed as
On the macro scale, a macroscopic equation for the concentration of particles, the Convection Diffusion-Reaction equation, can be adopted for particle transport (see Iliev et al., 2003 ,Dedering et al., 2008 ).
(259)
where is the concentration of particles, is the velocity, is the diffusivity coefficient; is the mass of the captured particles in the filter medium, and means the rate of deposition. When diffusion is negligible, the time variation of the concentration is solely governed by the absorption rate, and the 1D case is considered. Then, Equation (259) can be simplified to
(260)
Assuming that the amount of the deposited particles is small compared to the pore space of the filter medium, the mass of the deposited particles is considered to be proportional to the concentration of particles (see Brown et al., 1999)
(261)
where is the constant absorption rate. The analytic solution can then be derived for Equations (260) and (261) :
(262)
(263)
Then the -rating is given as
(264)
and the pass-through probability is
(265)
depending on the constant absorption rate and the velocity..
All Particles Pass