Simulation of filtration processes in deformable media Part 3 . 2 : Interaction modelling and verification of a non-spherical dirt particle solver

A Lagrangian solver to realistically model large, non-spherical dirt particles and their behavior in the vicinity of deformable filtration fibres has been programmed. While this paper focuses on the realization of interaction effects and result verification, a related article, concerning basic solver concepts as well as drag force implementations, has been published, [3]. A digitally reconstructed, deformable filter fibre structure gives the framework for all particle interactions with their surroundings. Particle–fibre deposition effects, are being modelled in detail. Fibre forces, that act on a particle and confine it, can be applied on a user-defined basis. The collision module handles particle–particle interactions, detects exact particlesurface impact locations and calculates translationalas well as rotational collision effects. In order to validate Computational Fluid Dynamics results, like filter fibre efficiency and particle penetration depth, a semi-analytical verification approach for simplified fibre geometries has been devised. In addition to that, extensive, experimental test runs are currently conducted.


INTRODUCTION
The Open Source Computational Fluid Dynamics (CFD) toolbox OpenFOAM ® has served as a programming environment for the development of a novel, deterministic, micro scale, fluid-particle-fibre filtration solver [1], [2].This C++ based simulator is supposed to become an important tool for the CFD design of filter media for automotive lubrication.It was created to consider all physically relevant effects that go along with, or lead to a micro scale, dirt particle deposition in a realistically reconstructed filtration fibre geometry.
Concerning this subject, three articles, have been previously published.While [1] focuses on fibre deformation and fluid-structure interaction (FSI) effects, [2] describes the development of a large, spherical particle model for filtration applications.The third article [3] is a direct prequel to this work.A significant extension of the spherical dirt particle model, formulated in [2] is presented there.It describes the basic concepts and the essential drag force implementation method behind our novel, deterministic, Lagrangian, non-spherical particle model.This work however, primarily focuses on how particle interaction with its surroundings is handled.The second, essential topic deals with the creation of a simplified, semi-analytical approach to verify solver functionality and result quality.
The main aspects of the development effort behind this paper are: • Implementation of a detailed particle-fibre interaction-and deposition model.

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Creation of a non-spherical particle collision model, including exact impactpoint determination as well as the consideration of rotational collision effects.

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Programming of Python [11] based evaluation utilities, to extract essential data on result parameters (such as filter fibre efficiency and particle penetration depth) from OpenFoam ® text file output.• Device of a semi-analytical scheme to verify solver functionality and result quality, within the framework of simplified fibre geometries.

BASIC CONCEPTS OF THE NON-SPHERICAL FILTRATION SOLVER
The framework for CFD analysis of the filter region is given by digitally reconstructed fibre geometries that are transformed to structured grid meshes by Matlab ® utilities.A fluidstructure-interaction utility can handle fibre deformation effects on the basis of a stiff, explicitly coupled, single solver approach, [1], [10].
Because of the geometry range of ~200µm, the highly viscous, Newtonian oil current with a kinematic viscosity of η f ~10 -4 m 2 /s and relatively slow flow velocities (u f < 0.4m/s), the local Reynolds numbers Re in the fibre vicinity are expected to be mostly below 0.5, but surely below 1.0.Furthermore Knudsen numbers Kn range well below 0.015.Thus continuum equations are valid and the consideration of diffusion effects on particle motion becomes unnecessary, [14].Hence, a simple, incompressible, laminar and isothermal fluid solver can handle the fluid situation.
The accurate modelling of dirt particle motion and interaction, as well as fibre deposition is of main concern.It has been found that particle-shape effects play an important role in filtration applications.A direct comparison of spherical and non-spherical particle filtration behaviour leads to significant deviations in filter fibre efficiency.The reasons for that are varying drag-force-to-mass-ratios [3] as well as slip-and bulk effects (see chapter 1.2 and 1.3).This is why, in extension of the previously published, spherical particle model [2], a highly detailed, more sophisticated and more accurate, non-spherical particle model had to be created.
Regardless of their shape, all particles of equal mass feature the same diameter of a mass equivalent sphere D sph , which is defined as: (1) In [4] the shape parameter α ax was introduced to characterize the deviation of the three particle half axis of length a, b and c around D sph as: (2) The higher α ax , the further the particle is from sphericity.Direct comparison of the filtration behaviour of particles with equal D sph and varying α ax gives insight into the relevance of non-spherical particle modelling.So far, two main shape effects with influence on filter fibre efficiency E have been identified:

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The non-spherical particle slip effect.

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The non-spherical particle bulk effect.
1.2 THE NON-SPHERICAL PARTICLE SLIP EFFECT Longish, non-spherical particles with small, angular relaxation times τ ω show a tendency to easily align themselves along the streamlines of the surrounding fluid.If D P , D b , D c denote the axe diameters of an ellipsoid, so that D c ≤ D b ≤ D P , then the smallest possible, projected, frontal area of an ellipsoid is given by: (3) Particle alignment increases the likelihood of slipping through a pore of diameter D F in a direction, perpendicular to A p,min .Consequentially the two smaller axe diameters define the minimal pore size diameter D F,min , that an aligning non-spherical particle can theoretically slip through: (4) This means: (5) The diameter of mass equivalent, spherical particles D sph can also be written as: (6) Which leads to the relation: (7) Thus the hydrodynamic slip effect of non-spherical dirt particles increases particle penetration depth and decreases filter fibre efficiency as compared to the case of mass equivalent, spherical particles.
For filtration simulation this means that a representation of non-spheres by mass equivalent spheres with diameter D sph might lead to a considerable overestimation of the filter fibre efficiency.A representation by spheres of diameter D p , on the other hand, will not only lead to an overestimation of filtration efficiencies, but will also result in wrong particle masses and consequentially in wrong calculations of over all particle hydrodynamics.Figure 1 presents a basic sketch of the situation.

THE NON-SPHERICAL PARTICLE BULK EFFECT
Longish, non-spherical particles with larger, angular relaxation times τ ω show a tendency to hardly align themselves along the streamlines of the surrounding fluid.As a consequence the largest half axe diameter D P commonly defines the minimal pore size diameter D F,min that the particle can slip through: Since D p > D sph , a mass equivalent spherical particle might just slip through pores, that a bulky, non-spherical particle may not pass.Thus the hydrodynamic bulk effect of nonspherical dirt particles decreases particle penetration depth and increases filter fibre efficiency as compared to the case of mass equivalent, spherical particles.
In terms of filtration simulation this means that a representation of non-spheres by mass equivalent spheres with diameter D sph might lead to a considerable underestimation of the filter fibre efficiency.Figure 2 presents a basic sketch of the situation.The bottom line at this point is that a non-spherical particle model, capable of including particle shape effects as well as rotational particle alignment, gives a much more realistic insight into detailed interaction behaviour in the particle-fibre vicinity, than any spherical representation ever can.

PARTICLE INTERACTION EFFECTS -EVENT FORCES
Particle-fluid interaction through drag-and lift forces is an essential aspect of Lagrangian modelling.Two drag modules, to be used for either the free flow regime or the fibre vicinity domain, have been developed.They are thoroughly discussed in [3].
Forces with influence on particle motion, that occur due to individual impact events, are hereby called event or impact forces.They represent particle-wall, particle-particle and particle-fibre interaction effects.
All N impact forces with relevance for particle motion occur within infinitesimal time scales that are smaller than numeric fluid time steps ∆t f or particle time steps ∆t p .

PARTICLE-WALL INTERACTION
Particles of mass m p that hit a wall boundary patch and feature translational velocity u p as well as angular velocity ω p are reflected with user definable elasticity E w .The reflection is performed by the use of a wall event force F wall .Wall impact forces have already been introduced to handle spherical particle-wall interaction [2].Now the concept has to be extended in order to consider rotational effects, particle orientation, the moment of inertia I p and the exact (or at least approximated) point of impact at the particle surface.In Figure 3 the basic difference in spherical and non-spherical particle-wall impact is sketched out.To sense the impact location, the help points, with distance vector r i to the particle centre are used and F wall is modelled as: (9) Note that F wall always acts perpendicular to the wall boundary patch, whereas fibre forces F fibre (see chapter 2.2) also have components parallel to the boundary patch.Impact elasticity E w has decisive influence on the translational and rotational impact behaviour of the particle.In Figure 4 the situation before and after impact is sketched out.The difference between completely elastic (E w = 1) and completely plastic (E w = 0) impact scenario is shown.
Qualitative sketch of the influence of non-spherical impact elasticity on particle-wall impact situation.
Screenshots of a qualitative benchmark case are shown in Figure 5.Here a simplified, fibre-like object in the particle flow path has been prescribed the wall boundary condition and thus serves as obstacle.An oval shaped ellipsoid hits the obstacle and is affected by the wall impact force.Consequentially its translation and rotation is altered.

PARTICLE-FIBRE INTERACTION AND PARTICLE DEPOSITION MODEL
The modelling of particle interaction with filter fibres and deposition mechanisms is essential.Due to the non-stochastic nature of the particle solver, no simplifying deposition model as in [5] or [6] is applied.Each particle is treated as dynamically interacting individual, according to its momentum, acting hydrodynamic-or event forces as well as fibre forces, which restrain its movement.Thus every increase of hydrodynamic-or event forces can lead to a particle blow-off.
Supposedly a fibre material that shows any particle adhesion at all, can restrain any infinitesimally small particle (D sph <<) that hits the fibre.
It is plausible to assume that there is a maximum, possible amount of fibre force F fibre on the particle and that it is directly proportional to the contact area, which is again proportional to D sph 2 .Fibre forces eliminate particle momentum, which is proportional to D sph 3 and act against hydrodynamic drag which under Stokes flow conditions is roughly proportional to D sph 2 .As a consequence, particles will be harder to stop, as D sph gets larger.Above a certain ìsticking diameterî D stick , particles will deposit according to a probability density distribution function (PDDF) that can be derived from experiments.The task within solver design is to integrate any, user definable PDDF of the form seen in Equ.16 into the concept of deterministic particle-fibre interaction force deposition.
A particle-fibre interaction module has been created.It distinguishes between three modes of interaction: the impact-, the gliding-and the full stop-or deposition phase.

Impact phase
When a particle-fibre impact occurs, F fibre,impact acts just like F wall for completely plastic (E w = 0) impact and can be written as: (10) (10) F fibre,impact is designed to momentarily remove the particle help point velocity component perpendicular to the wall u p,i ⊥ .For the particle mass centre this means that the velocity component perpendicular to the wall u pi ⊥ is reduced and that a particle moment of impact M fibre,impact causes the particle to rotate around the impact point (see Figure 6).

Gliding phase
As long as the particle detects fibre contact and still retains |u p,|| | > 0, it glides along the fibre surface.The decelerating fibre counter force F fiber,gliding now acts in opposite direction of u p,|| and is proportional to D sph 2 , the amount |u p,|| | and a user definable material, fibre force constant D ff , so that: (11) The pre-and post impact situation is sketched out in Fig. 6.In Figure .6 ∆t p is the numerical particle sub-time step.
Sketch of a non-spherical particle-fibre impact/interaction situation.Ellipsoid just before (left) and after (middle, right) fibre impact.At impact particle is affected by F fibre,impact , rotation around impact point, towards fibre sets in (middle).Particle then decelerates (glides) along surface, being restrained by F fibre,gliding .

FIBRE
u p (t) Full stop-/deposition phase Particle acceleration into the opposite direction, due to F fiber,gliding would be unphysical.Therefore the reversal of velocity direction, or the deceleration below a user defined, very low, stop velocity u stop make a valid criterion for a particle being deposited or not.The reversal condition states that the ratio of numerical particle velocity-change due to fibre force |∆u p,|| | during ∆t p , and |u p,|| | equals 1, which means: A stop number N stop is introduced, that contains both, the reversal condition and a stop velocity condition so that: (13) Here the stop velocity condition u stop /|u p,|| | merely serves as break up condition for the convergence of particle velocity to zero.Therefore u stop should be set <<u f .
The final deposition condition is: Disregarding the stop velocity condition, this leads to the definition of the particle stop diameter D stop : A particle with D sph ≤ D stop will be stopped and deposited with deposition probability Φ d (D sph ) = 1, as soon as a fibre is touched.A particle with D stop <D sph <D siev yields higher momentum and larger hydrodynamic forces and therefore is deposited with a probability 0<Φ d <1.D siev marks a particle diameter below which sieving effects are still irrelevant for particle deposition.The PDDF Φ d (D sph ), given by Equ.16 is supposed to be user definable and is based upon a simple, Gaussian-Normal distribution [13], defined by the two parameters: average D stop and standard deviation σ d . (16) The standard deviation is given via a third, user definable deposition constant σ slope .The solver basically features three optional deposition modules: 1.) No adhesion-based deposition for D sph > D stop : (17) 2.) σ d depends directly on the user defined σ slope : (18) 3.) σ d depends not only on the user defined σ slope but also on fluid/particle properties and D ff : In order to decide whether the individual particle with D sph > D stop , featuring a deterministic deposition probability Φ d (D sph ), is actually deposited at fibre contact, the stop parameter P stop is defined by comparing Φ d to a randomized, rectangularily distributed variable Φ 01 . (20) The deposition condition is then: The bottom line at this point is: 1) Any particle that touches a fibre experiences F fibre,impact and looses its velocity component perpendicular to the fibre surface.2) Any particle that touches a fibre experiences F fibre,gliding which reduces its velocity parallel to the fibre surface.3) There are three ways for a particle to get deposited at fibre contact.
c) The stop number is smaller than 1 but the particle fulfils the stop parameter condition: Any particle which fulfils either condition 3a, 3b or 3c will be deposited and is subject to a fibre constraining force F fibre,constrain that brings the remaining movement of the particlefibre contact point to a halt: Here e up,i,|| is the base vector for the velocity of contact point i, u p,i,|| and r i is the distance vector of contact point i to the particle mass centre.
The occurrence of additional particle forces F i can still speed up the particle, so that condition 3b is void and F fibre,constrain =0 or so that F fibre,constrain still is >0, but is outmatched by the N-1 other acting forces.A blow off would be the consequence.It is up to the user to decide whether the consideration of blow off effects is relevant, or whether adhesional deposition probability should work strictly governed by Φ d .The latter choice requires a more rigid fibre constraining force F fibre,constrain,rigid : (26) Numerical evaluations of adhesion-based filter fibre efficiency curves, on simplified, single fibre geometries have been conducted as seen in Figure 7.The test cases already include all the important dynamic interactions between particles, fluid, walls and fibres.Results clearly show, that the task of embedding externally imposed PDDFs Φ d , to include experimentally studied particle-fibre interaction properties, into an otherwise discrete, deterministic system of force interactions, has been accomplished.Up to 100 mono disperse, non-spherical particles have been injected, just upstream of a simplified fibre geometry, seen in Figure 7. Injection position and Stokes number St of each particle are chosen such that the hitting of a fibre is assured.Otherwise injection is arbitrary.A deposition probability curve has been pre-defined as seen in Figure 8. Repeating the run with particles featuring varying, relevant diameters D sph , deposition efficiencies E for each particle class have been evaluated.As shown in Figure 8, the results match quite well.

PARTICLE-PARTICLE INTERACTION
Within the fibre vicinity, high particle cloud densities occur.As more and more particles get entangled in the fibre, their volume fraction increases and particle-particle interactions   As soon as a particle enters the fibre vicinity, its data (particle mass centre X p , orientation of particle x-and y axis, e x , e y , particle mass m p , and particle half axe diameters a, b, c) is stored in a dynamic collision list.It is now a potential collision partner.The help point method, [3] is used to actually realise accurate, ellipsoid collision calculation.
If any help point HP A of particle A detects an impact with particle B, the following procedure is initiated: Simulation of filtration processes in deformable media  1.) Calculation of impact point IP B on particle B (not necessarily a particle help point HP B ). IP B is determined as the intersection point of ellipsoid B and a straight line through HP A and X p B .The problem is transformed into the corotational coordinate system of particle B where X p B = 0, so that IP is: Note that the superscript ë stands for coordinates within the co-rotational particle coordinate system.The concept of the co-rotational particle coordinate system is thoroughly described in [3].
2.) Determination of collision vector n coll .Due to the help point discretization of the ellipsoid particle surface, an exact calculation of surface normal vectors at impact points of particle A and B, would not yield two vectors of the same direction.An exact determination of ellipsoid surface normal n B of particle B at IP B is necessary to give the impact an appropriate direction.The calculation is performed within the co-rotational coordinate system of particle B and looks like: (28) Here f Bxz and f Byz are: 3.) Modelling of the collision force F collision .The collision force is designed to either reverse (100% elastic impact) or eliminate (100% plastic impact) the relative velocity u p,rel , between HP A and IP B according to user-defined impact elasticity E coll .The direction of F coll is given by n coll .It obviously has to be reversed for collision force calculation for either one of the two collision partners.F coll for particle A then reads: (31) Here r HP,A stands for the distance vector of HP A to X P,A and r IP,B stands for the distance vector of IP B to X P,B .A similar collision force implementation is used by Shah & Megahed, [9].A sketch of the non-spherical impact situation is shown in Figure 9.

COMBINED NON-SPHERICAL FILTRATION SOLVER
Along with the results, published in [1]- [4], the solver module can be combined to a unified filtration simulator.
Figure 10 shows an exemplary screenshot of a benchmark case, which includes a dense cloud of non-spherical dirt particles, getting entangled in a realistically reconstructed, deforming filter fibre geometry.The screenshot gives a qualitative impression of what the solver can do.
Figure 9 Sketch of impact situation with non-spherical particles A and B, before (left), during (middle) and after (right) collision.Post impact situation: The translational velocity of the smaller particle A is more strongly impaired and is smaller than that of particle B. The angular velocity of particle A is higher than that of particle B.

SOLVER VERIFICATION AND RESULTS
Solver verification can only occur by comparison of statistically averaged results.Crucial process parameters such as pressure drop, filter fibre efficiency and penetration depth are the key to comparing calculations and experiment.

NUMERICAL AND SEMI-ANALYTICAL FILTER FIBRE EFFICIENCY CURVE
A first, important step in validating qualitative aspects of solver functionality can be taken by comparing CFD calculations and semi-analytical results for artificially created, simplified fibre geometries.The process parameter to be chosen for benchmarking is filter fibre efficiency E. It is defined by: Here E i is the fractional filter fibre efficiency of size class i; n i is the total number of dirt particles per size class i and n s,i is the number of dirt particles retained by the filter.

Simplified geometry
In order to be able to establish a well defined, semi-analytically derived filter fibre efficiency curve over the entire, relevant regime of particle sizes (2µm-200µm), a simplified fibre geometry has to be created.The main reasons for that are:

Semi-analytical approach
The set up of a suitable, semi-analytical model for filter fibre efficiency calculation requires the integration of all relevant filtration aspects.The most relevant effects to be considered over a dirt particle size-range from 2µm to 200µm and particle Reynolds numbers Re p well below 1 are:

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Inertial impact • Particle-fibre adhesion • Blow off due to particle momentum or interactions • Sieving The total particle deposition probability P Dep of any particle (which is equivalent to E) can be calculated as a function of various "single effect" deposition contributions (see Equ.33).Hereby the inertial impact contribution is described by the hitting channel probability P HCh and by the inertial impact probability P Stokes which can be quantified via the Stokes Number St (see Equ.35).The particle-fibre adhesion as well as blow off effects can be considered via the particle sticking probability P Stick .Sieving contributions to overall filter fibre efficiency are finally quantified over the pore size dependent, flow path blocking probability P Block . (33)

Inertial impact effects
In order for any particle to impact on a fibre surface, two conditions will have to be satisfied: 1.) Particle must be on collision course.2.) Particle inertia must be high enough to "break out" of fluid stream lines around fibre.
In the context of this work a particle is defined to be on collision course with a fibre as long as at least the fraction n of D sph overlaps with a part of the projected, crossectional, fibre-covered area A Fibre of the entire flow channel A FCh .With knowledge of the projected, crossectional, free flow area due to pores A Pores and the number of pores n Pores , the hitting channel probability can be calculated as: Even if a particle is on collision course, impact is still not assured due to fluid deviation around the fibre.Larger particle inertia increases the likelihood of impacting anyway.The inertial impact probability P Stokes can be characterized via the Stokes number [12], that gives a dimension less relation between inertial and viscous forces on the particle: Here D k stands for the diameter of the collector, which in our case is the average fibre diameter D fibre , ρ p and ρ f are the particle-and fluid density respectively and η f is the kinematic fluid viscosity.
The minimal, critical, particle-fibre Stokes number St min is defined as the level below which any particle, even though on collision course, is deviated around the fibre (see Figure 12a).The ( ) ) P Block maximum, critical Stokes number St max , on the other hand, is hereby defined as the level above which any particle, whose course overlaps with at least the fraction n of D sph with a fibrecovered area, impacts on the fibre structure (see Figure 12b).Using those definitions, the inertial impact probability is in this work calculated as: (36) In order to obtain values for St min and St max a simple, numerical simulation where particles are set on a collision course with fibres of diameter D fibre is used (equivalent to sketches in Figure 12a and 12b.)The parameter n is set to 1.0%.The overall hitting probability P hit can now be semi-analytically determined and is: For "small" particles with D sph < D stop , the hitting probability is equivalent to the deposition probability.For "larger" particles with D sph > D stop , other effects have to be taken into account as well.

Particle-fibre adhesion and blow off due to particle momentum or interactions
The user-definable, adhesional deposition probability for ìmedium sizedì particles with D stop < D sph < D siev is deeply incorporated into the force interaction scheme of the CFD solver (see chapter 2.2).For the sake of semi-analytical verification of solver functionality, the underlying PDDF for particle-adhesional sticking probability P stick is chosen.The same parameters (D stop , σ d ) that are used in the CFD run are inserted, thus P stick is calculated as: (38) Sieving due to pore sizes For "large" particles with D sph > D siev the sieving effect is the most relevant reason for deposition.It is simply based upon the relation of particle diameter to pore size distribution σ pores.Underlying a Gauss-Normal distribution [13] of pore sizes, and using the geometrical parameters medium pore size D pores and standard deviation σ pores , the distribution function Φ pores is given by: ( The cumulative pore size distribution function is equivalent to the flow path blocking probability P Block that can be calculated by use of the error-function erf: (40) Comparing CFD and analytical results By using Equ.33 through Equ.40, the whole semi-analytical approach to describe the particle deposition probability within a simplified fibre geometry can be summarized as: (41) By inserting corresponding material-, and geometrical parameters into Equ.41, and into the input dictionaries of the CFD model, two comparable filter fibre efficiency curves are obtained.The results of the comparison are shown in Figure13.Semi-analytical results and CFD results match qualitatively and quantitatively.A verification of solver functionality is hereby considered to be achieved.

PRESSURE DROP
An experimental facility has been devised to test filter fibre samples in terms of essential process parameters.Pressure drop over filter life time and against flow rate variations can thus be monitored.The test results can then be directly compared to CFD runs, conducted on digitally reconstructed geometries of the same filter fibre material.So far an accordance of

APPLICATION AND (NON-) SPHERICAL RESULTS
In practice the CFD tool will rather be applied on realistically reconstructed, digitalized fibre geometries (as seen in Figure 14) than on simplified models.The means of result evaluation are manifold.Important insight into qualitative and quantitative aspects of process parameter behaviour, over filter life time can be gained.Two of the most interesting results are filter fibre efficiency-and particle penetration depth curves.
A typical example of solver application would be to take several, microscopic, digitally reconstructed portions out of a filter fibre sample and to conduct deposition-and penetration analysis for the entire range of relevant particle sizes.The results would then be averaged in order to obtain an idea of the basic particle deposition characteristic of the fibre.
All relevant solver output data (such as pressure-/velocity field information, particle positions and orientation) is given in text files.Using Python [11] scripting, several evaluation modules have been created.Those modules use the text-based output data to extract result parameters (such as filter fibre efficiency and particle penetration depth).
To give an impression of what the solver can do as of now, three examples shall be considered.

Comparison of material with/without adhesional effects
The first example is supposed to point out the potential of the solver in terms of material design.A digitally reconstructed fibre geometry (see Figure 14) has been tested in interaction with clouds of arbitrarily shaped (non-spherical) particles with a half-axe standard deviation of α ax = .Filter fibre efficiency curves have been evaluated for two cases of fibre quality.Case a) features fibres that do not show any particle-fibre interaction, while case b) has the very same fibre topology, but includes particle-fibre adhesion.The results, seen in Figure 15, show a dramatic increase in filter efficiency (at equivalent pressure drop) if adhesional effects (and not just pore size distributions) are considered in material selection.

Comparison of (non-)spherical particle filter fibre efficiency
The second example shows the importance of considering non-spherical particle shape effects.Filter fibre efficiency curves, using spherical and non-spherical particles on one and the same, non-adhesional filter fibre medium (see Figure 14), have been evaluated.The ellipsoids feature a half-axe standard deviation of α ax = and are always compared to their mass equivalent, spherical counterparts.The results, seen in Figure 16, point out the importance of considering particle shape effects.Relevant deviations in fibre efficiency are apparent.Fibre efficiency results, shown in Figure 16, reveal that smaller ellipsoids (D sph < 17.5µm) are deposited less easily than mass-equivalent spheres, whereas larger (D sph > 17.5µm), nonspherical particles rather get stuck.This behaviour is both plausible and to be expected.Ellipsoids with smaller, angular relaxation times experience the slip effect.They align easily along current stream lines, which enables them to slip through pores, that mass-equivalent spheres get stuck in (see Figure 1 and chapter 1.2).Larger ellipsoids with larger, angular relaxation times do not align as readily and experience the bulk effect.They get stuck in pores that their spherical counterparts just fit through (see Figure 2 and chapter 1.3).Between 30µm and 37µm spherical filter fibre efficiency decreases.This phenomenon can be attributed to increasing particle-particle interactions due to higher particle-particle hitting probability.Momentum transfer leads to blow off effects.

Comparison of (non-)spherical particle penetration depth
The third example gives an evaluation of particle penetration depth for the spherical and nonspherical case shown in chapter 4.3.2(both without adhesional effects) and for the nonspherical case, featuring adhesional effects, that is shown in chapter 4.3.1.
Hereby the particle penetration depth is represented via the relative penetration measure P that attributes the value 1 to particles slipping through the fibre mesh, and 0 to particles caught right at entry into the fibre vicinity.
Figure 17 Comparison of particle penetration depth curves for one and the same fibre geometry (see Figure 14).Case a) (purple) is calculated with spherical particles and no adhesion, case b) (yellow) is calculated with non-spherical particles and no adhesion and case c) (orange) is calculated with non-spherical particles and with adhesion.Test case parameters match those, given for Figure 15.
As expected, the adhesional case (orange) shows remarkably lower P-values over the entire, relevant particle size regime, than the non-adhesional cases (yellow purple).The relationship between particle penetration depth and filter fibre efficiency is demonstrated by comparing spherical and non-spherical, non-adhesional cases in both Figure 16 and 17.For D sph >20µm, non-spherical particles yield smaller penetration values than mass equivalent spheres.This corresponds with the fact, that filter fibre efficiencies for larger ellipsoids are higher.

CONCLUSION
A non-spherical dirt particle model for filtration applications has been programmed using the Open Source CFD tool box OpenFOAM ® .While a previously published article [3] has presented the most important, underlying concepts behind the new solver.This work has focused on the modelling of particle interaction events, the verification of solver functionality and the presentation of some initial results as well as solver capabilities.
Event force concepts to simulate particle-particle, particle-wall and particle-fibre interaction have been devised.Now a full, four-way coupling (particles-fluid-particles) can be realised.Moreover a controlled, user definable particle-fibre deposition scheme, in accordance with the force modelling concept [3] is now in place.Based on the geometrically versatile shape of ellipsoids, dirt particles, that feature 6 degrees of motional freedom, can now be modelled as they make their way through digitally reconstructed, realistic filter fibre geometries.A wide variety of particle shapes, -anything from plates to sticks to simple spheres -, can be approximated by this concept.The particles can interact with the fibre structure, the fluid and among each other.Special, Python [11] based evaluation tools have been created to extract relevant result parameters from the OpenFOAM ® text file output.Thus new insight into the progression of important filtration process parameters, such as pressure drop, filtration efficiency and particle penetration depth can now be gained.The entire, computational development effort, as presented in [3], [4] and here, has entailed several important sub steps.They are summarized in Figure 18.Article [3]: "Basic concepts and fluidparticle force implementation."Design of an explicit, force-and torque vector model, which reduces the modelling to the mere formulation of single force effects.Device of a surface help point scheme to account for large particle effects in terms of fluid particle, particle-fibre, particle-wall and particle-particle interaction.
Implementation and description of an efficient particle-fluid, two-way coupling method [2].
A novel drag force implementation that uses a combination of nonspherical, semi-empirical drag force formulas [7], a panel method to consider free flow swirling effects, a plugging method to include inter particle and particle-fibre hydrodynamics and a simple adaptation of basic concepts known from the immersed boundary method [8].
Article [4]: "Adaptive time stepping." Introduction of an adaptive time stepping scheme for explicit Euler discretization of the PME.This article: "Interaction modelling and validation."Development of an exact and faster, non-spherical particleparticle collision scheme.
Implementation of a stable, versatile particle-wall interaction model.Implementation of a detailed particle-fibre interaction and deposition model.
Due to the object oriented nature of the underlying programming language C++, it is easy to include the fully developed particle module into any laminar, continuum based, Eulerian fluid solver.Additional particle features, like electro-magnetic interaction or diffusion influence on particle motion could be added with comparably little effort.Plans like the integration of a fully versatile chemistry model are harder to realise but not impossible.
Of course validation is important too.Solver functionality as well as qualitative and quantitative result-related aspects have been tested.Two kinds of result-verification were used: comparison with semi-analytical calculations and comparison with experimental results.While a scheme of analytical validation has been devised and extensively discussed in this work, the ongoing efforts, concerning experimental verification have not yet been published.
In the future, this CFD tool will enable the virtual, purely digital pre-design of filter fibre materials.It will then be possible to upload artificially created fibre structures, conduct the CFD analysis and to consequentially use the results as a good estimate on the expected performance of the newly designed fibre material.

Figure 1 Figure 2
Figure 1 Non-spherical particle slip effect.Representation by spherical particles of diameter D p (left) means overestimation of filtration efficiencies and particle mass as compared to the more realistic, non-spherical representation (right).The use of spheres with D sph still leads to overestimation of filtration efficiencies because of Equ.7.

Figure 5
Figure 5 Screen shot of benchmark case.Ellipsoid particle just before (left) and after (right) hitting a wall boundary patch.Particle experiences rotational and translational acceleration.
diameter of a volume equivalent sphere is below the stop diameter:(22) b) Due to the effect of F fibre,gliding the particle velocity falls below the stop velocity:

Figure 7
Figure 7 Benchmark case to test particle-fibre interaction module.Simple fibre geometry with ellipsoids being deposited due to particle-fibre adhesion.

Figure 8
Figure 8 Adhesion-based, filter fibre efficiency for the simplified fibre structure case seen in Figure 7. Pre-defined PDDF for adhesion based deposition (blue) compared to numerical results (purple).

Figure 10
Figure 10 Screen shot of benchmark case.Dense cloud of non-spherical particles in realistic, deforming filter fibre geometry.Fibre vicinity drag module [3] is active, four-way coupling is engaged, deposition mechanisms initiated and FSI utility [1] is switched on.

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Figure11shows an exemplary, simplified, 30.000 cell, fibre geometry, where all relevant geometrical parameters are easily determinable.

Figure 11
Figure 11 Exemplary image of simplified fibre geometry.Easy determination of geometrical parameters: A pores , n pores , D pores , σ pores , D fibre .

Figure 12b Figure 12a
Figure 12bDefinition of St max .The smallest particle that just collides with the fibre, even though its collision course is just such that the n th fraction of D sph overlaps with a fibre, defines St max .
10% between lab tests and CFD runs has been achieved.More detailed results on pressure drop analysis will be published in the future.

Figure 16
Figure 16 Comparison of filter fibre efficiency curves for one and the same fibre geometry without adhesional effects (equivalent to chapter 4.3.1,case b)).Case a) (green) is calculated with spherical particles and case b) (red) is calculated with non-spherical particles of mass equivalent spheres.Test case parameters match those, given for Figure 15.D sph [lm]

Figure 18
Figure 18 Overview of the entire computational development effort in direct relation with this paper.
Figure 3 Sketch of spherical (left) and non-spherical (right) particle-wall impact.Non-spheres experience rotational as well as translational acceleration at impact.