Implementation in the Software Package GeoDict

The software package GeoDict [108] offers modular tools for the multi-scale simula-tion of materials and fluid flows in diverse applicasimula-tions. The designasimula-tion GeoDict is composed ofGEOmetrical material designer and material property preDICTor.

Originally, the software package was developed to model the behavior of porous media and composite materials. For this purpose, GeoDict offers the possibility to virtually generate structures as well as to simulate multiphase flow physics in porous media. Hence, GeoDict is well-suited for simulating filtration procedures, composites, the oil and gas transport through digital rocks or electrochemical processes in fuel cell and battery media.

Simulating the separation efficiency of cabin air filter media with the software GeoDict demands four successive steps:

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1. the generation of a digital filter structure,

2. the calculation of the air flow field through this filter structure, 3. the derivation of the electrostatic field from the surface charges, and 4. the simulation of the particle trajectories.

The implementation of the four individual steps in GeoDict is described below.

Furthermore, the uni-directional coupling is explained.

Generation of the Filter Structure. GeoDict basically offers two methods for the creation of digital filter structures based on the provided ’ImportGeo’ and

’FiberGeo’ modules. With the first method, digital twins of real filter media can be obtained from x-Ray micro-computed tomography (xCT) scans. In order to convert the large series of two-dimensional images to a three-dimensional structure, the ’ImportGeo-Vol’ interface is used. The module is based on a segmentation of gray values to extract filter fibers from the background. In addition, it provides tools for image processing in order to properly prepare the structure for further analysis.

A detailed characterization of the digitized filter media provides a deep insight into the microscale structure. Details about the fiber size distribution and orientation, gradients in packing density amongst others can be obtained.

With the second method, three-dimensional fiber objects are modeled math-ematically. Statistical properties such as fiber parameters, packing density and thickness of filter media serve as input parameters to generate the structure. Thus, the method allows modifying individual parameters of the fibrous structure while leaving the rest untouched to a certain extent. The simulation of the respective impacts on filtration performance constitutes a great advantage over experimen-tal testings [62]. Furthermore, ’FiberGeo’ enables the generation of simplified structures such as the single fibers used in Chapter 4 or the wired weaves used in Chapter 6.

Hydrodynamics in GeoDict. Once the digital structure is generated, the actual simulation is started using the ’FilterDict’ module [110]. In order to solve the flow Eqs. (2.7) before, inside and after filter media, GeoDict uses an equidistant

2.3 Status Quo Simulation Approach

voxel mesh discretization. Since a stationary flow field is assumed in GeoDict, the time derivatives of the (Navier-) Stokes equations are neglected. The following three methods are implemented in GeoDict to iteratively solve the PDEs [65].

TheExplicit Jump (EJ)immersed interface method is based on a finite difference method on a regular grid [109]. The solver is limited to Stokes flow and is especially suitable for the simulation of highly porous media.

TheSIMPLE-FFTis an enhancement of the semi-implicit methods for pressure linked equations (SIMPLE) which uses a Fast Fourier transformation (FFT) as a fast solver for the pressure Poisson equation. The solver can be used to aquire a solution of the Stokes equations as well as the Navier-Stokes equations. By contrast to the EJ solver, the SIMPLE-FFT converges very fast for very dense filter structures.

TheLIR (Left Identity Right)solver uses a combination of Octrees and KD-trees for spatial partitioning. The mathematical structure of the LIR tree is based on the set of three symbolsA={L, I, R}. The LIR solver enables a local grid refinement in areas where the velocity or pressure gradient is high [65].

Due to its short computational runtime through the adaptivity and at the same time low memory requirements for porous media, only the LIR solver is deployed in this work.

Electrostatics in GeoDict. In the software GeoDict, the assigned surface charge densityξFis divided between the two adjacent voxels to the fiber surface.

Subsequently, the electrostatic potential and the electrostatic field are calculated according to Eq. (2.14) and Eq. (2.15). Periodic boundary conditions are speci-fied for the potentialΦperpendicular to the direction of flow. At the inlet and outlet position in flow direction, zero Dirichlet boundary conditions are applied.

These specifications lead to the fact that the constant component of the calu-lated potential depends on the inflow and outflow length. However, since the shape of the function remains the same, the electrostatic field, i.e., the gradient of the potential, is independent of the position of the zero Dirichlet boundaries [110].

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Particle Tracking in GeoDict. By means of the given fluid flow and elec-trostatic fields the particle trajectories through the filter media can be calculated in the next step according to the equation of motion (Eq. 2.10). This particle tracking is also part of the ‘FilterDict’ module in GeoDict: A batch of particles is added to the inflow area. Since particles do not interact with each other, particle concentration only influences the intensity of fluctuations in the filtration efficiency.

In order to make a precise statement about whether a particle with a certain diameter is collected or not, the particle trajectories of as many starting positions as possible must be considered. During particle tracking, GeoDict permanently searches for collisions of particles with the filter medium. The ratio between the number of captured particles and the number of added particles provides the filtration efficiency according to Eq. (2.3).

Coupled Simulation in GeoDict. By neglecting particle-particle interac-tions and the retroactive effect of the particle movement on the flow field, the uni-directional coupling method allows to calculate the individual components consecutively in separate steps. Both, the flow field and the electrostatic field are calculated in preceding steps and exported. The fields are subsequently read in again to calculate the particle trajectories.

For the simulation of a filter lifetime with GeoDict, the fiber structure is extended by the collected particles at regular intervals. Based on the updated geometry, a new flow field and a new electrostatic field are calculated and, subsequently, the next batch of particles is tracked and so forth. In this work, however, only initial collection efficiencies are simulated.