Computers & Fluids 116 (2015) 118–128

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Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d

Simulations of filter media performances from microtomography-based computational domain. Experimental and analytical comparison P.-C. Gervais a,⇑, S. Bourrous a,b,c,d, F. Dany a, L. Bouilloux a, L. Ricciardi a a

Institut de Radioprotection et de Sûreté Nucléaire (IRSN), PSN-RES, SCA, Saclay, 91192 Gif-sur-Yvette, France Université de Lorraine, LRGP, UMR 7274, Nancy F-54001, France c CNRS, LRGP, UMR 7274, Nancy F-54000, France d Camfil Farr, Z.I. de Saint-Martin-Longueau, Pont-Sainte-Maxence F-60722, France b

a r t i c l e

i n f o

Article history: Received 27 August 2014 Received in revised form 18 February 2015 Accepted 26 April 2015 Available online 4 May 2015 Keywords: Gas filtration Fibrous media Synchrotron X-ray tomography Collection efficiency Permeability GeoDict

a b s t r a c t In this work, synchrotron X-ray microtomography was used to produce high spatial resolution images of one fibrous filter, made of binderless monodispersed fiberglass. Based on these images, representative computational domains were created using the import interface of the GeoDict software. Both flow and collection efficiency simulations were then carried out using the CFD modules of GeoDict. In parallel, permeability and collection efficiency measurements were performed on the same media, to provide an experimental comparison. A very good agreement was found between the experimental and simulated permeability values. However, simulated efficiency values tend to underestimate the experimental ones. In the second part of the paper, an image analysis program based on MatlabÒ was used to determine the structural properties of the fibrous structures, namely the thickness, the solid volume fraction and the fiber size distribution. These data were introduced into analytical models that successfully predict the permeability and collection efficiency values. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The use of fibrous media in areas related to air treatment dates back to antiquity [1] and it is very common nowadays. In addition to being easy to use and to maintain, they are one of the most efficient filtration techniques among the various existing devices. They are therefore implemented in many applications, such as engine intake, cleanrooms and in the nuclear industry to retain radioactive particles in normal operation or in accidental situations. The physical mechanisms involved in aerosol filtration by fibrous filters are laminar flow in porous media, particle transport and deposition by Brownian diffusion, interception and inertial impaction. These phenomena have been fully described in the literature since the 60s [2–6]. Nevertheless, the development of predictive models, used in design optimization or lifetime determination, is hardly achievable due to the wide range of operating conditions as well as aerosol and media characteristics. The initial performances of fibrous media are characterized by two important parameters: the filtration efficiency E and the pressure drop DP. Among the researches devoted to the modeling of such performances, numerical studies, consisting in designing fibrous ⇑ Corresponding author. Tel.: +33 (0)1 69 08 53 71. E-mail address: [email protected] (P.-C. Gervais). http://dx.doi.org/10.1016/j.compfluid.2015.04.019 0045-7930/Ó 2015 Elsevier Ltd. All rights reserved.

micro-geometries together with solving transport equations, seem to be relevant tools. Indeed, these approaches allow the influence of parameters like the fiber orientation [7], the fiber length [8], the solid volume fraction (SVF), the fiber diameter [9] or the fiber diameter ratio for bimodal fibrous media [10–12] to be investigated specifically. All of these parameters are the input data for the structure generation models and their determination involves assumptions. In almost all of the studies, anisotropy, fiber diameter and SVF are assumed as overall values. Moreover, in the case of 3D computations, fibers are designed as long straight cylinders inside the microstructure. In a study regarding the modeling of the initial pressure drop of fibrous filter media, Herman et al. [13] proposed a method to create random fibrous structures, based on log-normal fiber size distribution. Nevertheless, this work was limited to two-dimensional representations of media and the difference between experimental and simulated pressure drop values are significant. We believe that the more appropriate way to determine geometrical parameters is by 3D image analysis. Moreover, from the imaging technique point of view, fibrous structures do not yield sufficient image quality due to their thin thickness, as well as the low absorption contrast of the material. Recently, significant improvements in imaging and post-processing have allowed images to be created, imported and processed from computed-tomography, in order to compute flow and particle

P.-C. Gervais et al. / Computers & Fluids 116 (2015) 118–128

transport on real structures. They were first applied to studies on cellulose materials [14–16] and recently carried out for fibrous filter media [17]. The main objective of this study is to use synchrotron X-ray microtomography as a way to create representative computational domains, in order to simulate filter media performances. Section 2 of this paper presents the methods used to acquire and process the images, as well as the simulation settings. The experimental set-up and the operating protocols to determine permeability and efficiency are presented in Section 3. In Section 4, we summarize the structural properties of the computational domains from the image analysis and we provide the experimental computations with the simulations. Finally, the analytical modeling and the comparison with the simulations are presented in the last part of this work. 2. Simulations The set of simulations was performed with the GeoDict software (from Math2Market GmbH, www.geodict.com). GeoDict is a voxel-based code dedicated to predicting material properties by solving transport equations for a virtual material. The mesh is formed of voxels, which can be empty (fluid) or filled (solid). Through an interface dedicated to image analysis, GeoDict also allows data to be imported and processed from computed tomography, in order to use a real structure as the computational domain. 2.1. Image acquisition and treatment The studied filter medium has been provided by the Bernard Dumas company (Creysse, France). It is made of binderless fiberglass. The nominal fiber diameter (df ) is given to as 2.6 lm. The given weight is between 72 and 74 g=m2 . The thickness (Z) and SVF are not given. Images were acquired on the ID19 beamline of the European Synchrotron Radiation Facility (ESRF, Grenoble, France). In addition to being non-invasive and non-destructive, synchrotron X-ray microtomography has a high spatial resolution, which can represent the diameter of a fiber by at least 9 voxels in our case. The experimental setup consists of a high resolution microtomograph with a sample-to-detector distance of 5 mm. The beamline energy was 18 keV. Data come from the attenuation of the X-ray beam when crossing the absorbent body. Images are produced from a 9 mm diameter sample. A FReLoN CCD camera with 2048  2048 pixel chip, associated with a 20 objective with 2.5 eye-piece were used, in order to obtain a pixel size of 0.28 lm. Five overlapped scans were carried out to visualize the

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entire sample thickness. Each of theses consisted of 1995 projections over 360 degrees, with an exposure time of 0.2 s. The method used for phase extraction is based on the single distance phase retrieval approach [18]. Fig. 1 shows an example of two successive tomograms reconstructed from a fibrous medium. The final 3D structure is composed of 4351 tomograms with an edge size of 573.44 lm. 2.2. Image import The image import protocol is shown in Fig. 2. The 4351 tomograms (a) are first read using the ImportGeo-Vol interface (b). 3D processing parameters are then applied to the images. The resizing step defines a 1024  1024 voxel square area in the center of the structure, avoiding edge effects (c). A median digital filter is used to reduce noise in images. The images are scanned and the filter modifies the value of a pixel with the median value of the neighboring pixel (d). The choice of the relevant threshold value, in order to binarize the images, is the final step of the import (e). Moreover, the structure is cleansed by reassigning the objects comprised under 500 voxels to the fluid zone. The resulting 3D microstructure (f) consists of a 1.2 mm thick fibrous core, with a filtration surface of 0.08 mm2 . 2.3. Computational domains From the original structure, four non-overlapping subvolumes (SV) were created and used as computational domains (Fig. 3). This choice was made in order to save memory when handling the computational domain and also to have reasonable computational time. The subvolumes considered in this paper are 512  512  4351 voxels with a resolution of 0.28 lm per voxel. The mesh number is therefore greater than 1.1  109 . To reflect the representativeness of the computational domains, the Brinkman screening length, which is given by the square root of the permeability, is used as a qualitative criterion. According to Clague and Phillips [19], the domain width must be at least equal to 14 times the Brinkman screening length to smooth out the local heterogeneities. Using the permeability experimental value (see Section 3), in this work, the computational domain widths are wider than 24 times the Brinkman screening length. 2.4. Flow simulation Explicit finite volume solver options are controlled with the FlowDict module. The air flow through the microstructure is

Fig. 1. Illustration of two successive tomograms reconstructed from a fibrous medium.

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Fig. 2. Illustration of the steps for structure import in GeoDict, based on microtomography images.

Table 1 Information regarding the flow simulations. SV

Processes

Iterations

Time (h)

1 2 3 4

32 32 28 28

5012 4936 5271 5018

37.7 46.8 55.8 55.4

computing time were set to 106 and 240 h respectively. Permeability was chosen as the stopping criterion. In each iteration, permeability is calculated from the current flow field using Darcy’s law [24]. The solver accuracy was set to 104 . To decide whether the accuracy has reached or not, the current permeability value is compared to the value obtained 100 iteration steps before. The physical parameters are chosen as follows; air den-

Fig. 3. An example of one of the computational domains created with GeoDict. The thickness of the medium is along the z direction, the ðxyÞ plane corresponds to the filtration surface.

sity: 1.204 kg=m3 and dynamic viscosity: 1.834  105 kg=ms. The results are obtained using a cluster with 512 GB of RAM and a 4-hexa-core AMD CPU with a speed of 3.0 GHz. The local computation is executed on a number of processors between 28 and 32. The information concerning the computations is given in Table 1. 2.5. Particle transport simulation

governed by the Stokes equations, which consist of the momentum balance equation:

rp ¼ lDu

ð1Þ

X

and the continuity equation:

ru¼0

Particle transport settings are controlled by means of the FilterDict module. Filtration simulations are performed using a Lagrangian description of the particle motion and are described by a force balance acting on each of them [20]:

ð2Þ

where u is the velocity vector and u; v and w are its components along the x; y and z directions respectively. p is the pressure and l is the dynamic viscosity. Velocity inlet and pressure outlet boundary conditions are used in the flow direction. In order to compare with the experiments, the mean velocity U was set to 4 cm=s. Periodic boundary conditions were imposed in tangential directions. The maximum number of iterations and the maximum

F¼ma

ð3Þ

where m is the particle mass and a is the acceleration. The particle motion results from inertia, friction with the fluid and Brownian diffusion. Electrostatic effects are not considered. Thus, Eq. (3) can be expressed as follows:

dx ¼ v dt

ð4Þ

dv ¼ cðv ðxÞ  uðxÞÞdt þ D  dWðtÞ

ð5Þ

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where t is the time, x is the position and v is the particle velocity. Brownian diffusion is calculated using the mass diffusivity D and a 3D Wiener process W. The latter is a mathematical model that describes a continuous-time stochastic process. Practically, in the time step discretization, a Gaussian distributed random value is added tho the equation. In the case of a Stokes flow and spherical particles, the friction coefficient is given by the following relation:

c¼

3pqldp Cu  m

ð6Þ

Cu, the Cunningham slip correction factor, is used to take the non-continuum effects into account when the drag force is calculated for a small particle:

"



Cu ¼ 1 þ Knp 1:17 þ 0:525e

1:56 Knp

# ð7Þ

The experimental coefficient values are given by the GeoDict user guide and come from Crowe [21]. Knp is the Knudsen number for particles in air. Here, it is defined as the ratio of the mean free path in air (k = 66 nm) to the considered particle radius. In order to compare the simulations with experimental data, simulated particles are created from the experimental soda-fluorescein aerosol distribution. The diameter size ranges from 0.0145 to 1.0833 lm, with 41 size classes. The count, as well as the mass particle distributions are shown in Fig. 4. As can be seen in Fig. 5, the particle diameter distribution can be modeled using a log-normal distribution. The count median diameter is 0.08 lm with a geometrical standard deviation rg ¼ 1:6. We consider that a particle is caught when its trajectory encounters a solid voxel. GeoDict does not directly simulate the particle distribution, but the particle tracks are found using a Lagrangian description. 1000 particles per size class are simulated, which allows a statistically reasonable efficiency to be calculated for the larger particles. Nevertheless, the interactions between particles are not taken into account and the loading of the filter cannot be simulated in this way. For each size class of particles, the fractional filtration efficiency is calculated from the balance between the upstream and downstream particle number:

Esize class i ¼ 1 

number of particles of size class i downstream number of particles of size class i upstream ð8Þ

Fig. 4. Frequency as well as cumulative particle size distributions, in count and mass, of the simulated aerosol.

Fig. 5. Frequency number particle size distributions of the simulated aerosol. The particle diameter distribution can be modeled using a log-normal (log-N) distribution.

The count percentage given for each size class of particles in the distribution (N i ) is then used to calculate the count efficiency of the filter:

Ecount ¼

X Ni  Esize class

i

ð9Þ

i

The mass efficiency of the filter is also determined as follows:

Emass ¼

X M i  Esize class

i

ð10Þ

i

where Mi is the mass percentage for each size class of particles. The material density is set to 1550 kg=m3 . To be representative, averaged results obtained with three initial positions of the particles are presented. Each simulation takes about 15 min to be performed. 3. Experimental measurements The experimental set-up (Fig. 6) is based on a ventilated cylindrical pipe (diameter 40 mm) made of stainless steel. It is composed of a soda-fluorescein aerosol generator able to produce a fluorescent aerosol, of which the count median diameter is around 0.08 lm with a geometrical standard deviation rg < 1:6 (mass median diameter < 0.15 lm). It consists of a nebulizer, supplied with compressed air in a 10 g/L solution of fluorescein. After leaving the sprayer, the aerosol passes through two serial separation stages, in order to filter out the largest particles. The generated volume flow rate is 10.8 m3 =h and the aerosol mass concentration is 2 mg=m3 . the pre-filtered airflow in the pipe is maintained by a diaphragm vacuum pump provided by KNF Neuberger (USA), associated with a mass air flow control (Brooks Instrument, USA). The airflow, the absolute pressure and the temperature are directly checked through a TSI mass flowmeter. Pressure tappings on both sides of the filter holder allow a differential pressure sensor to be connected, in order to monitor the filter pressure drop. For total efficiency determination, a sampling filter, classified as H14 by European Standard EN1822/2009 [22] that defines classes of filters by their retention at the Most Penetrating Particle Size (MPPS), is placed directly behind the tested filter. The distance between the tested and sampling filters is reduced, to minimize aerosol deposition on the walls. Efficiency is calculated from the concentrations measured on the tested filter, as well as on the sampling filter. For fractional efficiency determination, a Scanning Mobility Particle Sizer (SMPS), provided by Grimm technologies (USA), is used to determine the particle size distributions. It consists of a

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Fig. 6. Schematic diagram of the experimental setup.

Differential Mobility Analyzer (DMA) to select particles of known size, followed by a Condensation Particle Counter (CPC). The DMA is the Vienna type electrostatic classifier, which is able to select particles between 10 and 1100 nm depending on the applied voltage. A VKL 10 (Palas GmbH, Germany) is used upstream from the DMA, as a dilution system, in order to reduce the aerosol concentration. To respect isokinetic sampling, a nozzle with a suitable diameter is used in the aerosol sampling line. Commercial Grimm 1.35 software is used for temporal registration and particle size analysis. 3.1. Experimental permeability determination The permeability k is the ability of a material to be passed through by a fluid under a pressure gradient DP. For an incompressible fluid flow through a length Z of porous media in a stationary laminar flow, it is given by Darcy’s law [24]:

k¼l

Z U DP

ð11Þ

where l and U are the dynamic viscosity and the fluid face velocity respectively. For fibrous media, k depends on the solid volume fraction and on the fiber size, the shape and the arrangement. A set of pressure drop measurements for a flow rate between 1.2 and 16 L/min, which corresponds to filtration velocities between 2.3 and 27 cm/s, was performed on flat filter samples. The selected flow rate was randomized for each measurement to overcome experimental bias. The thickness of the media was determined by Scanning Electron Microscopy (SEM) visualization. A fixation method followed by sanding and polishing steps were performed in the same way as that used by Bourrous et al. [23] for the samples. 27 measurements performed on 6 images provide the averaged thickness value: Z exp ¼ 1185  91 lm. Darcy’s law [24] was then used to calculate the experimental value of permeability: kexp = 3.36  1011 m2 . 3.2. Experimental mass efficiency determination The experimental mass efficiency was determined by measury the aerosol concentration on both the tested filter and the sampling one. Experiments were performed by injecting the

soda-fluorescein aerosol upstream from the tested filter. The aerosol was collected on the tested filter as well as on the sampling filter. The collection time was five minutes, during which the pressure drop of the tested filter remained constant. The fluorescein was then extracted by washing, using ammonia water. The concentrations of the solutions were measured by fluorescence and the soda-fluorescein masses mtested and msampling were determined. Five measurements for a flow rate of 2.3 L/min, which corresponds to a filtration velocity of 3.9 cm/s, were performed on flat filter samples. Table 2 summarizes the data concerning the experimental efficiency measurements. The mass efficiency is finally given by the following expression:

Eexp

mass

¼1

msampling msampling þ mtested

ð12Þ

The experimental value of total mass efficiency was chosen as the average value from the measurements: Eexp mass ¼ 81:3%. 3.3. Fractional efficiency measurement Fractional efficiency was measured using two aerosol concentration measurements. The first one was performed without a filter in the holder, representing the upstream measurement; the second one was performed with the tested filter, which constitutes the downstream measurement. 2 upstream and 2 downstream scans were performed for a flow rate of 2.3 L/min, which corresponds to a filtration speed of 3.9 cm/s. The experimental results come from the sum of the number of particles, counted with the SMPS, on five fibrous samples. To obtain a statistically reasonable fractional efficiency, if the number of downstream particles is less than 500, it is not considered.

Table 2 Data for the total experimental efficiency measurements. Set

mtested (lg)

mSampling (lg)

E (%)

1 2 3 4 5

18.7 21.3 21.5 15.7 17.6

3.65 4.97 4.69 3.94 4.50

83.7 81.1 82.1 79.9 79.6

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4. Results 4.1. Domain properties The properties of the fibrous structure acquired by microtomography were measured using an image analysis program based on MatlabÒ image processing. Based on 26 2D-slices in the ðxzÞ plane (see Fig. 3), the determination of the medium thickness Z was performed in two steps:  The first one consisted of the detection of the maximum medium thickness. The criterion to determine the thickness is the presence of at least one solid voxel in the domain. At this step, the minimal SVF and the maximal thickness were extracted. These values are not representative of the real medium, but they will be used as a basis for applying the second criterion.  The second step to evaluate the efficient characteristics of the real medium was the application of a more restrictive criterion. We estimated that the bulk of the medium begins when the local SVF corresponds to 5% of the mean minimal SVF of the medium. Using this criterion, the effective thickness and SVF were estimated. These two steps were necessary in order to avoid some effects due to the small surface acquired using microtomography, like single fibers appearing in the 3D volume. The fiber diameter distribution was measured on 9 images in the ðxyÞ plane, taking 400 2D-slices each into account, as shown in Fig. 7(a). After binarization and cleaning of each slice, pictures were structured by a spherical structuring element. On this basis, each picture was crossed by 6 horizontal lines, on which two points on the fiber edge were detected. From these two points, a third was found, also on the fiber edge on a vertical axis, then the fiber diameter was determined by trigonometric calculation. The mean diameter df , the standard deviation r and the overall distribution of the fibers were then calculated. As a hypothesis, we considered that the fibers are cylindrical and that the tortuosity of the fibers on a length equal to twice the diameter is negligible. This assumption was easily confirmed by the SEM pictures (see Fig. 7(b)) of the real medium, as well as the 3D structure. The properties of the 4 subvolumes resulting from the image analysis are summarized in Table 3.

Table 3 Properties of the 4 subvolumes resulting from image analysis. SV

SVF (%)

Z (lm)

df (lm)

r (lm)

1 2 3 4 Mean

3.66 3.34 3.46 3.39 3.46, r = 0.14

999 863 1002 970 959, r = 65

2.92 2.63 2.51 2.40 2.62

3.19 2.19 1.74 1.65 –

A difference of about 20% can be noted between the thickness determined by the analysis performed on the images acquired by microtomography and the averaged thickness value Z exp measured from the SEM pictures. This difference, widely acceptable, can be attributed to the choice of criterion in the analysis, as well as the restricted area treated using X-ray microtomography. Moreover, the fiber diameter values are very close to the value given by the manufacturer. Furthermore, the values of the standard deviation indicate that the fiber size polydispersity is not negligible. 4.2. Transport properties To quantitatively compare the simulation with experimental data, the error relative to the experimental data has been calculated. We used the average values of relative errors, called the Mean Relative Error (MRE), as a qualitative criterion [25]:

MREexp=sim ¼

N j yexp  ysim j 1X j j N j¼1 yexp j

ð13Þ

where y is the parameter of interest. 4.2.1. Permeability Fig. 8 allows the pressure field, computed with GeoDict, to be visualized in the case of one of the 4 subvolumes. It can be seen that the obstacles formed by the fibers create a pressure drop. This pressure drop is more pronounced in the middle of the computational domain, where larger fibers may be present. The pressure difference between the upstream and downstream parts of the computational domain is the DP value that was presented in Table 4. Due to the pressure drop dependence on the fluid velocity and sample thickness, it is easier to compare intrinsic quantities like the permeability. Based on the simulated pressure drop value,

Fig. 7. (a) View of the virtual medium built with GeoDict. The depth (color gradient) is built from 400 2D-slices. (b) SEM pictures of the real medium. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 9. Visualization of the filtered 0.3 lm-diameter particle trajectories (a), as well as the non-filtered ones (b) in one of the subvolumes.

Table 5 Summary of the collection efficiency simulation results.

Fig. 8. Visualization of the pressure field computed with GeoDict in one of the subvolumes.

Table 4 Summary of the flow simulation results. SV

1 2 3 4 Mean

DP (Pa) U ¼ 4 cm=s

ksim  1011 (m2 )

27.46 25.99 29.10 27.31 27.47, r = 1.27

3.25 3.44 3.07 3.27 3.26, r = 0.15

MREexp=sim ð%Þ kexp ¼ 3:36  1011 m2 3.2 2.3 8.7 2.7 4.2, r = 3.0

GeoDict applies Darcy’s law to calculate the permeability. Table 4 summarizes the flow simulation results. For each subvolume, the simulated pressure drop DP, the resulting permeability values ksim and the mean relative error are indicated. Based on the very low value of mean relative error average, we note a very good agreement between the experimental and simulated permeability values. 4.2.2. Collection efficiency To get a visual idea of the collection efficiency simulations, GeoDict allows the trajectories of all of the simulated particles to be represented. Focusing on the most penetrating particle size is proposed, in order to avoid image overload. Fig. 9 allows the filtered 0.3 lm-diameter particle trajectories (a) to be visualized, as well as the non-filtered ones (b), computed with GeoDict, in the case of one of the 4 subvolumes. Table 5 summarizes the efficiency simulation results. For each subvolume, the simulated values of the count efficiency (Esim count ), the mass efficiency (Esim mass ) and the mean relative error are indicated. Based on the values of the mean relative errors, we note a significant gap between the experimental and simulated values of mass efficiency. It can be noted that the simulated efficiencies underestimate the experimental ones. From our point of view, this behavior can be explained by the combination of several assumptions. First of all, potential electrostatic

SV

Esim

1 2 3 4 Mean

80.1 79.3 81.5 84.2 81.3, r = 2.0

count

Esim

ð%Þ

mass

ð%Þ

64.4 63.1 66.6 66.7 65.2, r = 1.6

MREexp=sim ð%Þ Eexp mass ¼ 81:3% 20.8 22.3 18.1 17.9 19.8, r = 2.2

effects, impossible to quantify in our set-up, due to the friction between the fluid and the fibrous medium, can have a significant impact on the underestimation of the efficiency by simulations. Interactions between collected particles and particles in the flow are not taken into account in the simulations. In reality, the initially collected particles act as a new collection area, which has the consequence of increasing efficiency. GeoDict allows this kind of simulation, treating the particles as batches, but it requires the flow to be recalculated after each batch, which is very time consuming. Lastly, given that both the filter holder and the airflow can mechanically act on the media, simulations have been carried out on 5% and 10% compressed structures in the flow direction. These simulations have shown no significant influence on the efficiency. 5. Analytical comparison The MRE is used to quantitatively compare the simulations with the data from analytical modeling: N j y  yj 1X j N j¼1 ymod j mod

MREmod=sim ¼

sim

j

ð14Þ

Table 6 Summary of the permeability modeling results. SV

r eq (lm)

kmod  1011 (m2 )

ksim  1011 (m2 )

MREmod=sim ð%Þ

1 2 3 4 Mean

2.18 1.84 1.68 1.62 1.84

4.25 3.30 2.73 2.62 3.29

3.25 3.44 3.07 3.27 3.26, r = 0.15

30.6 4.1 11.2 19.9 –

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5.1. Permeability The analytical modeling of the fibrous media permeability, based on the work of Miecret and Gustavson [26], is presented in Appendix A. It includes part of Section 2 of Gervais et al. [12]. The prediction of fibrous media permeability is reduced to finding

Fig. 10. Fiber radius class as a function of the number fraction for the 4 subvolumes.

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an equivalent radius from the fiber size distribution. In order to take the effective surface of the fibers into account, the area-weighted equivalent radius, req , is constructed from the root mean square of the fiber radii:

req ¼

qX ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n r2 i i i

ð15Þ

P where ni is the number fraction of a fiber radius class r i . i ni ¼ 1. The fiber size distribution of each subvolume is found from the image analysis (see Section 4.1). These are shown in Fig. 10. Table 6 summarizes the permeability modeling results. For each subvolume, the area-weighted equivalent radius, both the simulated and the predicted values of the permeability kmod , and the mean relative error are indicated. There is a significant gap for Subvolume 1. As seen from its fiber radius distribution, Subvolume 1 is the only one with at least one fiber present for the radius class between 4 and 7 lm. These coarse fibers affect, proportionally to the squared of their radius, the value of the area-weighted equivalent radius and consequently the permeability. Nevertheless, in the case of the mean structure, when taking into account the sum of the fibers that constitute the 4 subvolumes (over 350 fibers in total), the influence of coarse fibers is reduced and a more statistically reasonable equivalent radius is calculated. We note that the mean area-weighted equivalent radius leads to a mean area-weighted equivalent diameter equal to 3.7 lm, which is substantially greater than the value of the mean diameter df .

Fig. 11. Fractional collection efficiency as a function of particle diameter. Comparison between the results computed with GeoDict and the model of Miecret and Gustavson [26] in the case of each subvolume.

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5.2. Collection efficiency The analytical modeling chosen for the calculation of the fibrous media collection efficiency is presented in Appendix B. For each size class, the fractional efficiency is calculated using the relation (B.3) and can be compared with the simulated Esize class i (see Eq. (8)). The particle diameter ranges from 0.0145 to 1.0833 lm with 41 size classes and the filtration velocity is 4 cm/s. Fig. 11 shows the fractional collection efficiency computed with GeoDict compared to the model of Miecret and Gustavson [26] for each subvolume. For each subvolume, the mean relative error is calculated using the relation (14) and taking into account all of the fractional efficiencies by class. Based on the low value of the mean relative errors, we note a satisfying agreement between the simulated and predicted values of the collection efficiency by the model. Although the Most Penetrating Particle Size seems to be correctly predicted, it can be noted that the predicted efficiencies rather underestimate the simulated ones in the inertial regime. From the fractional collection efficiency values predicted by the model, the count and the mass efficiencies of the filter can be calculated using the count and the mass percentage for each size class of particles. Table 7 summarizes the collection efficiency modeling results. For each subvolume, the efficiencies values predicted by the model, in count and mass, are presented. The mean relative errors associated are calculated in relation to the simulated efficiency values of Table 5. Fig. 12 shows the fractional collection efficiency, averaged from the 4 subvolumes, compared to the model of Miecret and Gustavson [26] calculated using mean values of the structural properties (see Table 3). The experimental fractional efficiency is also represented. Both the simulated efficiency and the efficiency

Table 7 Summary of the collection efficiency modeling results. SV

Emod ð%Þ

1 2 3 4 Mean

73.5 71.7 79.6 80.4 77.2

count

MREmod=sim ð%Þ 9.0 10.6 2.4 4.7 5.3

count

Emod ð%Þ 57.5 55.8 65.5 66.6 62.3

mass

MREmod=sim ð%Þ

mass

12.0 13.1 1.7 0.2 4.7

Fig. 12. Fractional collection efficiency as a function of particle diameter. Comparison between the experiments, the averaged results computed with GeoDict and the model of Miecret and Gustavson [26].

predicted by the model are consistent, but not close to the experimental value. This highlights the fact that there are one or more effects, not considered in the simulation and modeling, that can substantially increase the efficiency of the filter. This is in agreement with the comments concerning the underestimation of the experimental value of the mas efficiency by the simulated ones (see Section 5.2).

6. Conclusion The development of predictive models of the pressure drop and collection efficiency for fibrous media is of great interest in many industrial fields, such as the nuclear safety field. In fact, it allows the design of cleaning systems to be improved, as well as allowing the filter element behavior to be simulated in accidental situation scenarios. The numerical approach is the ideal tool to use, in order to take into account the wide range of operating conditions, as well as aerosol and media characteristics. Nevertheless, parametric simulations require a code validation step to achieve credible results. The computational domain is the basis for the simulation, and virtual fibrous geometry design involves a lot of assumptions, even if the structural parameters are well-known. In this study, dealing with GeoDict code, we used synchrotron X-ray microtomography to produce images of one particular type of fibrous media, with a spatial resolution of 0.17 lm. Four computational domains were then created from theses images. Both airflow and collection efficiency simulations were carried out. In parallel, some experimental measurements were performed on the same media, providing an experimental confrontation for the medium used in the design of the secondary filters. A very good agreement was found between the experimental value of permeability and the simulated results from the microtomography-based computational domain. An average difference of less than 5% is observed, which allows us to validate the representativeness of the subvolume. Nevertheless, a significant gap is found between the experimental and simulated values of efficiency. From our point of view, this is due to not taking into account the electrostatic effects in our simulations. The problem is to experimentally quantify and to accurately model these effects in GeoDict. At present, GeoDict may take into account an overall surface charge for the aerosol and for the fibrous media. The first simulations of this type showed limitations of this kind of modeling. Further work will be to use neutralized aerosol and conductor media, such as metallic fibers, to be free of any charge effect during the experiments. Another reason considered could be the interactions between particles in the flow. If the pressure drop is constant during all the experiments, initially collected particles act as a new collection area, which has the consequence of increasing efficiency. Work is in progress in that respect but, even though the flow field can be reused, a new one must be calculated from the new structure, taking into account the particles caught on the fibers. In addition, the porosity of the deposit must be taken into account using the Stokes–Brinkman solver, which is more time consuming than the Stokes solver. This limits the reproducibility tests and the parametric simulations. In the second part of the paper, an image analysis program based on MatlabÒ was used to determine the structural properties of the fibrous structures, namely the thickness, the solid volume fraction and the fiber size distribution. These data were introduced into analytical models that successfully predict the permeability and collection efficiency values. The comparison between the simulated data and the analytical model is quite encouraging. Parametric simulations are envisaged to expand the validity range of the model. In the future, synchrotron X-ray nanotomography will be considered for producing images from H13 and H14 media, used for the design of High Efficiency Particulate Air filters and constituted of much finer

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fibers, with a wider fiber size distribution. Using the GeoDict code, the permeability, collection efficiency and penetration profile will be studied. Acknowledgements This work is part of the LIMA joint research program (on media-aerosol interactions) between the French Institut de Radioprotection et de Sûreté Nucléaire (IRSN) and the Reactions and Chemical Engineering Laboratory (LRGP) of the French National Center for Scientific Research (CNRS). The authors are grateful to the NOVITOM company, in Grenoble, France (www.novitom.com), for the acquisition and reconstruction of the microtomography. The authors would like to thank B. Marcillaud and F.-X. Ouf (IRSN) for their helpful technical support, with the SEM characterization and the SMPS measurements respectively. Appendix A. Analytical model of permeability for fibrous media The pressure drop, DP, for an incompressible fluid flow through a length Z of porous media in a stationary laminar flow is given by Darcy’s law [24]:

DP 1 ¼ lU Z k

ðA:1Þ

where l and U are the dynamic viscosity and fluid face velocity respectively. For fibrous media, the permeability k depends on the solid volume fraction and on the fiber size, shape and arrangement. In the case of a unimodal distribution of fibers, the permeability is assumed to be a function of the solid volume fraction of the media, a, and the fiber radius, r:

k ¼ r2  f ðaÞ

1 3 2

16a ð1 þ 56a3 Þ

ðA:3Þ

This correlation, often used by the air filtration community, was obtained experimentally by testing fibrous filters with diameters between 1.6 and 80 micrometers and solid volume fraction lower than 30%. It was further validated by Werner and Clarenburg [28] in this range of media characteristics.

 Brownian diffusion (D),  interception (R),  inertial impaction (I), In the work of Miecret and Gustavson [26], the single fiber capture efficiency is regarded as the sum of the single fiber capture efficiencies associated with each mechanism. Another term is used to take into account the interaction between diffusion and interception:

gðdp ; df Þ ¼ gD þ gR þ gI þ gDR

ðB:4Þ

The model of the single fiber capture efficiency due to Brownian diffusion is given by Davies [29]:

gD ¼ 1:5Pe2=3

ðB:5Þ

where Pe, the Péclet number, is the ratio between the convective and the diffusive transport:

Pe ¼

df  us D

ðB:6Þ

where us and D are the superficial velocity and the diffusion coefficient, respectively. The model of the single fiber capture efficiency due to interception is given by Stechkina and Fuchs [3]:

gR ¼ 2:4a1=3

 1:75 dp df

ðB:7Þ

The model of the single fiber capture efficiency due to inertial impaction is given by Suneja and Lee [30]:

"

2

1:53  0:23  ln Ref þ 0:0167  ln Ref gI ¼ 1 þ Stk

#2 ðB:8Þ

where Ref is the Reynolds number of the fibers:

Ref ¼ Appendix B. Analytical model of filtration efficiency by fibrous media

ðB:3Þ

f

The single fiber capture efficiency depends on the physical mechanisms that govern the particle collection. Without an external force field and neglecting the electrostatic effects, the main mechanisms recognized in aerosol filtration are:

ðA:2Þ

For fibrous media with high porosity, Davies proposed an empirical expression for f ðaÞ, the dimensionless permeability [27]:

f ðaÞ ¼

a  Z 4gðdp ;df Þ1 a pd

Eðdp ; df Þ ¼ 1  e

q  us  df l

ðB:9Þ

and St is the Stokes number: 2

It is possible, under certain assumptions, to calculate the filtration efficiency of fibrous media from the single fiber capture efficiency, gðdp ; df Þ. The modeling takes into account the air flow and particle characteristics as well as the structural properties of the filter. Calculation results from the particle concentration (C p ) balance on a thickness element dZ of the media:

dC p ¼ C p ðZÞ  gðdp ; df Þ  df  Lv  dZ

ðB:1Þ

where Lv , the specific length, is the total length of the fiber by the bulk volume of the fibrous media. Knowing that the filtration efficiency results from the upstream and downstream particle concentration ratio:

E¼1

C p ðupÞ C p ðdownÞ

ðB:2Þ

the integration between 0 and Z of (B.1) gives the following expression for the filtration efficiency:

St ¼

Cu  qp  dp  us 18  qf  df

ðB:10Þ

Cu is the Cunningham correction factor for air:

"



Cu ¼ 1 þ Knp 1:17 þ 0:525e

1:56 Knp

# ðB:11Þ

Knp is the Knudsen number for particles in air:

Knp ¼

2k dp

ðB:12Þ

where k is the mean free path of the air. The term used to take into account the interaction between diffusion and interception is given by Miecret and Gustavson [26]: 1=2 gDR ¼ 1:24HKu Pe1=2

 2=3 dp df

ðB:13Þ

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where HKu is the hydrodynamic factor determined by Kuwabara [31] in the case of a flow model around a set of cylinders in a continuum regime:

HKu ¼ a 

1 1 3 lnðaÞ  lnða2 Þ  2 4 4

ðB:14Þ

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