Modeling the separation performance of depth filter considering tomographic data

Fibrous depth filters are frequently used for the purification of gas streams with low dust loadings, as well as processes where a high initial filtration efficiency is required (e.g., clean rooms for aseptic production). One tool suitable for supporting the development of optimized filter media is the use of numerical simulations. The drawback of this technique is the high computational resources required. In this work, a new and fast approach based on a one‐dimensional model was applied. Structural characteristics (e.g., porosity distribution and fiber diameter) of two different filter media were successfully determined using a novel X‐ray microscope. These characteristics were incorporated in the filtration model, and their influence on the calculations was evaluated. It was found that the porosity distribution does have an impact on local (microscopic) deposition rates, but only a minor influence on the macroscopic filtration efficiency (around 3%). Benefits of the model are the application of measured structural data and the low computational expense. Compared to experimental data (VDI 3926 / ISO 11057), the prediction of the filtration efficiency can be improved by incorporating the structural data in the model.


| INTRODUCTION
Due to the expected adverse health effects of particulate materials, the demand for effective and resource-saving separation systems for removing particles from fluids is rising sharply. [1][2][3] Filtration systems are used in several areas such as wastewater treatment or gas purification. [4][5][6][7] Due to their flexible handling and low investment costs, fibrous filters may be applied in various civil and industrial applications. For the purification of gas streams with low to medium dust loadings, or for applications where a high initial filtration efficiency is required (e.g., clean rooms in aseptic production), fibrous depth filters are the means of choice. In the context of filtration, a high filtration efficiency with a low pressure-drop is required. The optimization of the structural properties of the filter media (e.g., porosity and fiber diameter) with respect to the separation task at hand is vital. [8][9][10][11][12] In this field, simulations and predictive tools are an excellent tool for realizing this task. 11 Accounting for the large variation and complex geometry of the filter media in simulations is still challenging. Although the main physical processes involved in the particle separation process (diffusional deposition, interception, and impaction) have been described mathematically by numerous researchers since 1960, [13][14][15][16][17][18][19][20] predicting the influence of the filter media is challenging. The heterogeneous nature of filter media (e.g., large variation in porosity) is a particular challenge, and most of the models are unable to account for this. 9,21,22 In fact, most of the developed models were derived based on Kuwabara's flow field, 23 which makes assumptions such as parallel flow through a structured cylindrical array. Equations based on Kuwabara's flow are inadequate to predict filtration behavior 24 due to their over-simplification of the process.
Schweers et al. 21 developed one of the first numerical approaches that included structural heterogeneity of the filter media by dividing the filter into several sub-filter layers. Along with other parameters, variation in fiber diameter and porosity were considered as important 2 | MATERIALS AND METHODS

| Filter media
The following filters were used as received: a gradient filter comprised polyester fibers (TH 300-T20; T15/500, batch no.: 3200560/6, Afprofilter, Bönen, Germany), and a fine filter made of glass fibers donated by Trox GmbH (F7, batch no.: 1800655, Trox GmbH, Neunkirchen-Vluyn, Germany). Characteristic data of both filter media are summarized in Table 1 light by a scintillation crystal. The generated visible light was then focused by an optical system ( Figure 1).
Scans were carried out using an X-ray energy of 30 keV. The spatial resolution can be increased to 700 nm for an area less than 1 mm 2 . In the following examination, the region of interest is about 10 mm 2 , which limits the resolution to 30 μm.
Obtained images were evaluated using the open source software Fiji. Images were binarised by applying Fiji's maximum entropy threshold algorithm to the images. The porosity was then obtained by computing the ratio of white pixels (material) to black pixels (no material) in each image, which corresponds to the packing density, α, of the material. From the packing density, the porosity was calculated as Since the choice of correct threshold is a crucial step in image analysis, 39 obtained porosity data by XRM were averaged and compared to the average porosity determined by gravimetric analysis (ε grav ). To determine the average porosity, a filter sample with a known volume was weighed using a scale (Sartorius, Analytic AC 210S, Göttingen, Germany), and the porosity of the material was calculated as where V material is the volume of the fibers in the filter, and V sample is the bulk volume of the filter. V sample was calculated based on the cylindrical geometry of the filter samples (24 mm diameter for the glass filter and 26 mm for the polyester filter). The depth of the filter media was obtained from the XRM data. V material was calculated from the mass of the sample (m sample ) and the density of the material (ρ material ) as follows:

| Deposition efficiency
The measurement of the deposition efficiency was carried out according to the VDI 3926 using a certified setup (MMTC 2000, Palas, Karlsruhe, Germany). A schematic drawing of the setup is shown in Figure 2. The test dust was dispersed using pressurized air, and the pressure drop across the filter media was measured. The filtration per-

| Filtration model
To describe the filtration efficiency of the filter media, the filter was divided into several filter layers, called sub-filters ( Figure 3). Each subfilter had a defined depth and defined structural data such as porosity and fiber diameter.
The filtration efficiency of a sub-filter (T i ) was calculated as the ratio of the separated number of dust particles (n i − n i − 1 ) to the ingoing number of dust particles (n i − 1 ), and the total filtration efficiency (T) of whole filter media was calculated analogously to T i : T A B L E 2 Mass based particle size distribution statistics of the test dust Pural NF (av ± s, n = 5) The filtration efficiency depends on the fiber diameter (d f, i ), the porosity of the sub-filter (ε i ), the depth of the sub-filter (L f, i ), and the total deposition efficiency (φ i ) on a single fiber inside each sub-filter. These parameters were used to predict the filtration efficiency as follows: The deposition efficiency on a single fiber (φ i ) was calculated based on individual separation efficiencies for diffusional deposition (η D ), interception (η R ) and inertial impaction (η I ) assuming interaction between interception and diffusion only (η D+R,i ). Therefore, φ i was calculated as the sum of the individual mechanisms as follows: Due to the low gas velocity and the small particle size, it can be assumed that adhesion forces are predominant in this case. Therefore, particle bounce was not considered in the calculations.
For describing diffusional deposition and interception, the model obtained by Payet et al. 18 was chosen, where Here, Ku i is the Kuwabara factor, Pe i is the Peclet number (defined as the ratio of convectional transport to diffusive transport), C D, i is the slip correction factor, and C 0 D accounts for the interaction between the interception and diffusion mechanisms. The Kuwabara factor is given by and the Peclet number is calculated as The diffusivity (D) of each particle size was calculated from the Einstein-Stokes relation: Here, k B is the Boltzmann constant, ϑ (25 C) is the temperature, μ (1,824Á10 −5 Pa s) is the dynamic viscosity of air at ambient pressure at the chosen temperature, and d p is the particle diameter.
F I G U R E 2 Experimental setup for filter test rig according to the VDI 3926. FIR is flow indication and registration, PIC is pressure control with indication, KS is time switch with noncontinuous control, PDIRS+ is differential pressure indication and registration (switched when limit value is exceeded), PDIR is differential pressure indication and registration, TIR is temperature indication and registration, and MIR is moisture measurement indication 37 The velocity (u 0, i ) for estimating Pe i was calculated by multiplying the velocity in the previous sub-filter (u 0, i − 1 ) by the ratio of the porosity in the current sub-filter to the porosity in the previous sub-filter (ε i − 1 ): The two correction parameters, C D, i and C 0 D,i , are given by Particle separation by interception was taken into account and calculated as follows: F I G U R E 4 Three-dimensional visualization of glass filter (left) and polyester filter (right) F I G U R E 5 Raw cross-sectional image of the polyester filter (above, left) and after binarisation (above, right) and raw cross-sectional image of the glass filter (below, left) and after binarisation (below, right) Here, C R,i is a correction parameter for slip flow analogous to Equation (13), and R is the interception parameter defined as the ratio between fiber and particle diameters given by Separation based on inertial impaction was accounted for using the model proposed by Zhu et al. 40 Zhu developed an analytical model of the filtration of particles on cylindrical fibers based on Kuwabara's flow field: The inertial deposition depends on the Stokes number (Stk i ), which was calculated as

| Tomographic data
Both filter media were successfully measured by XRM, and their 3Dstructures were visualized (Figure 4). From these images, an overview of the structure can be observed. The spatial resolution of the tomograms in Figure 4 left is 3 μm, and right is 15 μm. The gravimetrical average porosity was calculated according to Equations (2) and (3). Average porosity obtained by XRM and gravimetric analysis data are summarized in Table 3. The gravimetric porosity agreed well with the average porosity obtained by XRM. The good agreement between the two methods supports the validity of the image analysis of the cross-sectional images of the two considered filter media.

| Calculation of filtration efficiency
In the previous section, XRM was used to obtain tomographic data such as the porosity distribution along the depth of the two filters.
These data were used in the filter model presented in Section 3.1.1. carried out using average porosity and porosity distribution data from XRM. In this work, the medium fiber diameter (given in Table 1) was assumed to be constant in each sub-filter, and the mean value was used for both filter media. Results of the calculation are shown in Figure 7 (left). The graph of filtration efficiency as a function of particle diameter exhibited a minimum in filtration efficiency for particle sizes around 1 μm, as expected. Taking into account, the obtained microstructural porosity distribution inside of the investigated filter media led to an increase of predicted filtration efficiency for both media. In particular, inertial impaction and interception were observed to have a more significant impact on efficiency than diffusional deposition. Similar tendencies were observed for both filter media when accounting for porosity distribution data. In order to study the plausibility of the method, the face velocity was increased to 0.1 m/s (Figure 7, right). Expected results were obtained. Increasing the gas velocity led to an increase of inertial dominated filtration and to a decrease of diffusional deposition. Based on the particle size of the test dust in these calculations, it was concluded that the dust particles were separated mainly by inertial forces (gray marked regions in Figure 7). Here, it also became evident that both filter media were F I G U R E 7 Calculated filtration efficiency of both filter media as a function of particle diameter using averaged porosity and local porosity obtained by X-ray microscope (left), and calculated filtration efficiency using local porosity data and varying face velocity (right). The gray areas indicate particle sizes of the used test dust Pural NF [Color figure can be viewed at wileyonlinelibrary.com] effective at separating the majority of the dust particles (more than 50% of the mass of particles in the dust, see Table 2).
In order to study the influence of local structural inhomogeneity and to gain a deeper understanding of mechanisms inside the material, the local filtration efficiencies of the particle sizes along the filter depth were calculated. It was possible to make this calculation for each observed particle size. Results are shown for three sample particle sizes, one in the diffusional dominated region, one in the filtration minimum, and one in the inertial dominated region (particle diameter of 0.01, 1.5 and 3 μm, Figure 8). It is shown, and was expected, that the calculated local efficiencies are highly linked to the porosity distribution. Nevertheless, a deeper insight into the physics of separation within these two materials has been gained. In the case of the glass filter, low efficiencies at the surface layer of the material were calculated, and a relatively constant efficiency inside the bulk of the material, corresponding to the porosity distribution, was found. Similarly, small filtration efficiencies at the surface layer were observed for the polyester filter.
Due to the different structure of the polyester filter media, the values and the distribution of local filtration efficiencies differ from the glass filter media. In general, the filtration efficiency of the polyester is about one order of magnitude smaller, which is compensated partially by the larger filter depth of the polyester filter. The filtration efficiency reaching higher values in the deeper layers of the media is a result of the composite structure of the media. This composite structure is usually manufactured to generate a more uniform loading of the material with particles.
In Figure 8, the local fractional separation efficiencies for chosen particle sizes along the filter depth were examined. It can be emphasized that the particle separation efficiency is inversely related to the porosity curve shown in Figure 6, which is due to the dependency of the separation on the porosity. An almost constant separation efficiency of the particles inside the glass filter and an increased particle separation in the second part of the polyester filter was observed.
The order of the level of local separation efficiency as a function of particle diameter for the glass filter (T 3μm ) (T 1.5μm ≈ T 0.01μm )) and for the polyester filter (T 0.01μm > T 3μm > T 1.5μm ) follows the calculated macroscopic filtration efficiencies shown in Figure 7. This order can be explained by the influence of the filter media structure and the operating conditions on the separation mechanisms. In general, the filtration efficiency increases for larger particle diameter due to higher inertial forces, and also for smaller particles where the deposition is mainly dominated by the diffusion. This behavior is shown in Figure 7.
Due to the structural properties of the polyester filter media at the investigated operating conditions (gas velocity), particles of 1.5 μm size do not appear to be efficiently deposited either by diffusion or by inertia. It can explain that why this particle size causes the lowest separation efficiency. This is consistent with the results in Figure 7. In the case of the glass filter, separation by inertia appears to be the dominate separation mechanism, which means that the largest considered particle size is separated most efficiently. In contrast, the smallest particle diameter considered is the most efficiently separated particle size in the polyester filter. This can be explained by the differences in porosity. Due to the higher porosity of the polyester filter, the gas velocities in the filter are lower, which supports the separation by diffusion.
It is shown that the local filtration efficiency differs between the two filters by one order of magnitude ( Figure 8). This is not reflected in the macroscopic filtration efficiency of both filters in Figure 7, as this is (partly) compensated by the higher depth of the polyester filter.
The calculated local filtration efficiencies for all particle sizes of the considered test dust (data in Table 1) were then used to calculate the deposited mass in the initial (clean) state along the filter depth. This was done using the particle size distribution of test dust and the The influence of the particle size distribution on the deposition pattern was also determined. A finer test dust probably led to a different deposition profile, that is, a more uniform loading, since entrapment of a smaller amount in the first layer occurred leading to deeper penetration of the particles. In the glass filter, the low porosity and the comparably small fiber diameter are responsible for the high filtration efficiency.
For the first experimental validation of the applied method, the total separated mass was measured (Section 2.2.3). These data were also calculated with the developed model using data presented in