# Modeling permeability of fibrous filters combining submicron and micron fibers in slip flow regime

Dominique Thomas, Elise Cabaset, Ming Wang, Augustin Charvet, Nathalie

Bardin-Monnier, Jean-Christophe Appert-Collin

► To cite this version:

Dominique Thomas, Elise Cabaset, Ming Wang, Augustin Charvet, Nathalie Bardin-Monnier, et al. Modeling permeability of fibrous filters combining submicron and micron fibers in slip flow regime. Chemical Engineering Science, 2026, 322, pp.123125. 10.1016/j.ces.2025.123125. hal-05425652

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# Modeling permeability of fibrous filters combining submicron and micron fibers in slip flow regime

Dominique Thomas* ID, Elise Cabaset, Ming Wang ID, Augustin Charvet ID,
Nathalie Bardin-Monnier, Jean-Christophe Appert-Collin

Université de Lorraine, CNRS, LRGP, F-54000 Nancy, France

## ARTICLE INFO

### Keywords:

- Air filtration

- Fibrous media

- Non-woven media

- Permeability

- Slip flow

- Submicron fiber

- Pressure drop

- Polymodal distribution

- Numerical simulation

## ABSTRACT

Fibrous media are essential in aerosol filtration due to their ability to balance filtration efficiency, pressure drop, and dust holding capacity. Their performance depends on structural parameters such as fiber diameter, packing density, and thickness and the most efficient filters often include submicron or nanofibers. However, predicting pressure drop remains challenging, particularly due to slip flow effects at small scales. Existing models rarely account for both slip effects and fiber size distribution. This study extends a model originally developed for micronic fibers to submicron fibers, incorporating both the influence of slip effects and fiber size distribution. In order to overcome uncertainties in determining the structural parameters of fibrous media, the model was initially validated on numerically generated fibrous structures.

This broader-scope model, when compared with other models from the literature, exhibits an improved predictive accuracy (within ± 10 %) for permeability values obtained from numerical simulations on both monomodal (219 cases) and polymodal (183 cases) fibrous structures, covering a packing density range from 0.1 % to 50 % and a fiber diameter range from 0.03 to 20 µm, leading to fiber Knudsen numbers between 0.007 and 4. An experimental validation was also performed on five commercial fibrous media and on data from the literature.

## 1. Introduction

According to Hutten (2015), “Two important goals of filtration are to protect humankind from the environment and the environment from humankind.” Indeed, filtration systems play a crucial role in safeguarding people in both living and working environments. These systems include air cleaners, mist eliminators and filters integrated into heating, ventilation and air conditioning (HVAC) systems. In industrial settings, baghouses are commonly used to reduce harmful emissions and limit environmental pollution. Another important application is the use of respiratory protection masks, which help shield individuals from harmful particles, airborne microorganisms and allergens. Recently, during the SARS-CoV-2 pandemic, surgical face masks or FFP2, FFP3 masks have demonstrated their efficiency (Bourrous et al., 2021; Charvet et al., 2022; Alcaraz et al., 2022).

Among the various filtration technologies available today, non-woven fibrous filter media are the most widely used for aerosol filtration, due to their low cost, ease of use and effective performance. The filter structure, including solid volume fraction (α), fiber size

distribution, and thickness (Z), significantly impacts the filtering performance, i.e. pressure drop, which affects operational costs (e.g., energy consumption for fan systems), and airborne particle removal efficiency. The presence of submicron fibers within the fibrous structure, or a layer of nanometric fibers, enhances collection efficiency (Leung and Hung, 2008; Hung and Leung, 2011), although this improvement comes at the cost of an increased pressure drop.

Numerous expressions exist for estimating the pressure drop of fibrous media (Spielman and Goren, 1968; Davies, 1973; Jackson and James, 1986), but none of them takes into account the fiber size distribution. Recently, Thomas et al. (2023) proposed a model that deals with the distribution of micron-sized fibers in non-woven media. Although extended to polymodal distributions by Cabaset et al. (2025), this model is not suitable for fibrous structures composed wholly or partially of submicron or even nanometric fibers. Indeed, when the diameter of the fibers approaches the mean free path (λ) of the gas molecules, the fluid around the fiber behaves like a rarefied gas, and the flow can no longer be considered as continuous. Consequently, the pressure drop is lower than that predicted by the model, as the drag

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: dominique.thomas@univ-lorraine.fr (D. Thomas).</footnote>
force exerted on nano- and submicron-sized fibers is significantly reduced. In air at 293 K and 1.013 bar, the mean free path ($\lambda$) is 66.5 nm (Baron and Willeke, 2001).

Devienne (1958) investigated the behavior of rarefied gas flow in pipes under laminar flow conditions and identified four distinct flow regimes, classified according to the Knudsen number—defined as the ratio of the gas’s mean free path to the pipe radius.

- $Kn < 10^{-3}$: Continuum flow regime. The Navier-Stokes equations can be applied.

- $10^{-3} < Kn < 0.25$: Slip flow regime. Gas molecules close to the wall in the so-called Knudsen layer interact with the fiber surface significantly more frequently than molecules in the bulk flow. The Navier-Stokes equations remain valid, provided that slip velocity at wall is accounted for.

- $0.25 < Kn < 10$: Transition regime

- $Kn > 10$: Free molecular regime. In this regime, intermolecular collisions are negligible compared with those between gas molecules and the pipe walls.

As highlighted by Colin (2005), Beskok et al. (1996) and Beskok and Karniadakis (1999), the threshold values typically used to define flow regimes based on the Knudsen number are only indicative and highly dependent on the characteristic length chosen in its calculation. In the case of fibrous media, the classification and limit values proposed by Pich (1971) and Davies (1973), who used the fiber diameter (df) as the characteristic length (i.e., $Kn_f = 2\lambda/df$), may be called into question. Indeed, if the average hydraulic pore diameter of the fibrous medium ($D_H = df(1-\alpha)/\alpha$) is instead considered as the characteristic length—bringing the definition closer to that originally proposed by Devienne (1958)—it can be shown that

$$Kn = Kn_f \frac{\alpha}{1 - \alpha} \qquad (1)$$

Based on this definition, the Knudsen number ($Kn$) and the regime become dependent of the solid volume fraction. For a fibrous medium with a packing density of 5 %, the slip flow regime limits ($Kn = 10^{-3}$ and $Kn = 0.25$) are reached for fibers with diameters of 7 µm and 28 nm, respectively. This suggests that, at ambient pressure, the flow through fibrous media occurs in most cases within the slip flow regime. Under such conditions, conventional models from the literature are no longer suitable for accurately predicting pressure drops, particularly when the fibrous media are composed of—or contain—submicron fibers.

In the laminar flow regime, pressure drop across a porous medium follows Darcy’s law:

$$\Delta P = \frac{Z}{\kappa} \mu U_f \qquad (2)$$

Here, $Z$, $\mu$ and $U_f$ are the filter thickness, the dynamic viscosity and the filtration face velocity, respectively. The permeability ($\kappa$) can be expressed as a function of an hydrodynamic factor (H), the fiber diameter (df) and the packing density ($\alpha$):

$$\kappa = \frac{H df^2}{16 \alpha} \qquad (3)$$

The hydrodynamic factor depends on the arrangement and orientation of the fibers in relation to the flow. Several authors (Spielman and Goren, 1968; Jackson and James, 1986; Thomas et al., 2023) have listed various expressions for the hydrodynamic factor as a function of the fiber volume fraction.

In non-continuous flow, the measured permeability is lower than the predicted values. As a result, several authors have proposed alternative expressions for the hydrodynamic factor, based on theoretical or experimental approaches. All of these expressions are formulated as a function of the fiber Knudsen number (Table 1) and some of them

involve the Kuwabara hydrodynamic factor ($H_{Ku}$):

$$H_{Ku} = \alpha - \frac{1}{2} \ln(\alpha) - \frac{1}{4} \alpha^2 - \frac{3}{4} \qquad (4)$$

Pich’s (1966) theoretical model is based on the calculation of the drag force acting on each fiber of the fibrous media, taking into account the slip effect at the fiber surface. To determine the velocity distribution inside the filter, he uses Kuwabara’s (1959) model, approximating the filter to an array of randomly distributed cylinders oriented perpendicular to the flow. Kwak et al. (2023) fitted the Pich’s model to the results of 3D numerical simulations of fibrous structures composed of monodisperse submicron fibers. Yeh and Liu (1974), adopting the same approach as Pich (1966) but with different boundary conditions, developed an alternative expression for the hydrodynamic factor. In the extreme case, when $Kn_f = 0$, the expressions by Pich (1966) and Yeh and Liu (1974) reduce to the Kuwabara hydrodynamic factor (Eq. (4)). In the hydrodynamic factor of Kirsch et al. (1973), also called $H_{Fan}$, $\Phi$ is a function dependent on fiber arrangement and should be considered as a corrective factor. Bian’s model is derived from experimental pressure drop data collected from electrospun nanofiber media. According to Kwak et al. (2023), the Bian’s model shows limited applicability because it is restricted to a specific range of packing densities.

These various models have never been systematically compared with experimental data spanning a broad range of fiber diameters or packing densities. To date, only the models proposed by Pich (1966) and Bian et al. (2018) have been evaluated by Kwak et al. (2023) against results from 3D numerical simulations, covering fiber diameters from 30 to 300 nm and solid volume fractions ranging from 0.5 % to 10 %. Moreover, none of these models account for the fiber size distribution of submicron fibers or at least for bimodal structures combining micronic and submicronic fibers. The present study addresses these limitations by proposing a new model, considering both the contribution of fiber size distribution and slip effects, to estimate the pressure drop across more realistic fibrous structures.

## 2. Numerical simulation

Developing a permeability model from experimental data would be time-consuming and difficult to carry out. Costly, because a large number of filter media with very different characteristics should have to

<table><caption>**Table 1**
Expressions for the hydrodynamic factor in the non-continuous flow regime.</caption><thead><tr><th>Authors</th><th>Hydrodynamic factor (H)</th><th>Comments</th></tr></thead><tbody><tr><td>Pich<br>(1966)</td><td>H<sub>Ku</sub> + 0.998 Kn<sub>f</sub> (1/2 - ln(α) + 1/2α<sup>2</sup>) / (1 + 1.996 Kn<sub>f</sub>)</td><td></td></tr><tr><td>Kirsch<br>et al.<br>(1973)</td><td>-0.5 ln(α) - 0.52 + 0.64 α + 1.43φ(1-α) Kn<sub>f</sub></td><td>0.02 &lt; α<br>&lt; 0.14<br>3 ≤ df ≤ 18<br>μm<br>Kn<sub>f</sub> ≤ 1.3</td></tr><tr><td>Yeh and<br>Liu<br>(1974)</td><td>α / (1 + Kn<sub>f</sub>) - ln(α) / 2 - α<sup>2</sup> / 4 + 3 / (4 + Kn<sub>f</sub>) + Kn<sub>f</sub>(2α - 1)<sup>2</sup> / (4 + Kn<sub>f</sub>)</td><td></td></tr><tr><td>Bian et al.<br>(2018)</td><td>16πα(1-α)Kn<sub>f</sub><sup>0.556</sup> / (1.364α<sup>0.158</sup>)</td><td>α not<br>precised<br>61 ≤ df ≤ 260 nm<br>0.5 ≤ Kn<sub>f</sub> ≤ 2.1</td></tr><tr><td>Kwak<br>et al.<br>(2023)</td><td>H<sub>Ku</sub> + 0.149 Kn<sub>f</sub> [H<sub>Ku</sub> + 18.35(1-α)<sup>2</sup> - 3.79(1-α)] / (1 + 0.149 Kn<sub>f</sub>)</td><td>5 10<sup>-3</sup> &lt; α<br>&lt; 0.1<br>30 ≤ df ≤ 300 nm<br>0.42 ≤ Kn<sub>f</sub> ≤ 4.2</td></tr></tbody></table>
be tested. Difficult because characterizing these different media in terms of packing density, thickness and fiber size distribution remains a challenge. Indeed, simply measuring the thickness of these compressible fibrous structures is complex and determining fiber size distribution requires the analysis of a very large number of Scanning Electron Microscopy (SEM) images. The electrospinning process can produce polymer nanofibers of relatively uniform diameter, but the resulting fibrous structure is not always homogeneous (Kim et al., 2020; Hung and Leung, 2011). Numerical tools such as GeoDict allow the construction of 3D fibrous structures with precisely defined characteristics, offering a wide variability in terms of packing density, fiber size distribution and thickness. This approach is therefore chosen to develop a pressure drop model, which will later be validated against experimental data from real media.

By varying fiber diameters within the numerical structures and/or adjusting the gas mean free path (e.g., to simulate a change in filtration pressure), it is possible to explore a broad range of fiber Knudsen numbers—a key parameter in accounting for slip flow effect. In the GeoDict code, the slip velocity at the fiber surface is assumed to be proportional to the shear stress. The resolution of the Navier-Stokes equations in no-slip conditions leads to a quadratic Poiseuille-type velocity profile up to the pore-solid boundary, as one would expect for flow close to stationary walls. This approach has been experimentally compared and validated for nanofiber layers (Azimian et al., 2024).

Whatever the numerical fibrous structures (ranging from mono- to polymodal fiber size distribution), the voxel size is chosen to be one-tenth of the diameter of the finest fiber to ensure good resolution of the flow within the porous medium (Cabaset et al., 2025). Given the manufacturing processes of fibrous media (wet process, meltblown, etc.), the orientation of the fibers is not isotropic in the three dimensions. The fibers are preferentially oriented in the XY-plane perpendicular to the flow by adjusting the anisotropy parameters in GeoDict software (Fig. 1). A parameter study is systematically conducted to ensure that the size of the three-dimensional fibrous structures and the number of fibers are sufficient to obtain independence between the permeability results and the simulation domain size while limiting the calculation time. Inlet and outlet regions are added in the flow direction to ensure a stationary flow and a faster flow computation. The flow through each structure is simulated at 5 cm/s, a classical face velocity in filtration applications. In highly porous structures, such as fibrous media, the Left Identity Right (LIR) solver uses a very memory-efficient adaptive grid to compute flow (Linden et al., 2015). The voxel size is increased in the porous volume with low velocity gradients and is maintained at the user-specified size near the fiber surface. As the simulation domain represents a small part of a fibrous medium, periodic boundary conditions are applied to predict the pressure drop for a given fluid velocity.

<figure><img src="image_3.png" /><figcaption>Fig. 1. Example of a trimodal fibrous microstructure generated with GeoDict®.</figcaption></figure>

# 3. Results and discussion

## 3.1. Fibrous media with monodisperse fibers

In view to develop a model that takes into account the polydispersity of fibers and the slip effects in fibrous media, the first step consists in the development of a model including the slip effects at the surface of monodisperse fibers. Initially, only monodisperse fibrous structures are generated to enable a direct comparison between simulated permeabilities and theoretical values predicted by literature models (Table 1). In total, approximately 113 structures are generated, with packing densities ranging from 2 to 50 % and fiber diameters between 0.05 and 8 µm, covering a fiber Knudsen number range from 0.01 to 2.7. To these permeability values are added the 106 data points reported by Kwak et al. (2023), which span a fiber volume fraction of 0.5 % to 10 % and fiber diameters between 20 and 300 nm.

Fig. 2 illustrates the evolution of the permeabilities predicted by literature models as a function of the simulated values. These permeabilities are obtained using Equation (3) and the different formulations of the hydrodynamic factor summarized in Table 1.

The mean relative discrepancy (MRD), defined in Equation (5), is used to quantitatively compare the models.

$$MRD = \frac{1}{N} \sum_{i}^{N} \left| \frac{\kappa_{i}^{\text{Simulated}} - \kappa_{i}^{\text{Modeled}}}{\kappa_{i}^{\text{Simulated}}} \right| \quad (5)$$

Table 2 summarizes the MRD values for four existing models, based on permeability data from Kwak's simulations and our owns.

The models proposed by Pich and Yeh tend to underestimate the permeabilities derived from the simulations. As expected, Kwak's model shows good agreement (MRD = 0.042) with the dataset used for its development. However, it slightly overestimates our permeabilities of the present simulations for less resistant fibrous structures. The Kirsch model, which uses a correction coefficient (Φ) fixed here at 1, shows excellent correlation with all simulated permeability values (MRD = 0.055 for Kwak's data and MRD = 0.123 for data simulated in the present study) but tends to overestimate the permeability of fibrous structures with packing density greater than 40 %.

Since Kirsch's expression provides better predictions compared to the other models, efforts have been devoted to refining and further improving this formulation. Kirsch et al. (1973) extended the dimensionless drag force model $F_{Kn,f}$ (Eq. (6)), first developed for parallel cylinders (Kirsh et al., 1971) under slip-flow and low-Reynolds-number conditions, to fibrous media, resulting in the following empirical relationship:

$$\frac{1}{F_{Knf}} = \frac{1}{F_{Knf=0}} + \frac{C}{4\pi}(1 - \alpha)^n Kn_f \quad (6)$$

where, $F_{Knf}$ and $F_{Knf=0}$ represent the dimensionless drag forces acting on a unit length of fiber under slip and continuum flow regimes. In the Kirsch's model, C and n are, respectively, equal to 1.43 Φ and 1.Φ is a corrective factor that depends on the fiber arrangement.

Bao et al. (2016) and Choi et al. (2017a) experimentally confirmed the linear dependence of the reciprocal of the dimensionless drag force on $Knf$, for commercial fibrous media made of nanofibers.

$F_{Knf}$ and $F_{Knf=0}$ can be related to the pressure drop of the fibrous medium by:

$$F_{KnforKnf=0} = \frac{\Delta P_{Knf} \text{ or } Knf=0 \pi df^2}{4\mu U_f \alpha Z} \quad (7)$$

From equations (2) and (3), equation (7) can be written as:

$$F_{KnforKnf=0} = \frac{4\pi}{H_{KnforKnf=0}} \quad (8)$$
<figure><img src="image_4.png" /><figcaption>Fig. 2. Comparison of predicted and simulated permeabilities. The generated structures covered packing density values from 0.5 % to 50 % and fiber diameters ranging from 30 nm to 8 µm, which correspond to fiber Knudsen numbers between 0.01 and 4.4.</figcaption></figure>

<table><caption>**Table 2**
Mean Relative Discrepancy for each model.</caption><thead><tr><th>Model</th><th>Pich</th><th>Yeh</th><th>Kwak</th><th>Kirsch</th></tr></thead><tbody><tr><td>Kwak's data</td><td>0.586</td><td>0.533</td><td>0.042</td><td>0.055</td></tr><tr><td>Present study</td><td>0.564</td><td>0.523</td><td>0.152</td><td>0.123</td></tr></tbody></table>

Equation (6) can be written from equation (8):

$$H_{Knf} = H_{Knf=0} + C(1 - \alpha)^n Knf \quad (9)$$

In a previous study on the permeability of fibrous media without slip effects, Thomas et al. (2023) have shown that the modified Happel hydrodynamic factor ($H_{Knf=0}$, Eq. (10)) could accurately (±10 %) predict the pressure drop for micronic fibrous structures. Accordingly, this expression of the hydrodynamic factor is preferred to that proposed by Kirsch.

$$H_{Knf=0} = 2.5^{1/4} \left( -\frac{1}{2} \ln(\alpha) + \frac{1}{2} \frac{\alpha^2 - 1}{\alpha^2 + 1} \right) \quad (10)$$

Permeability can be derived from equations (3) and (9) as follows:

$$\kappa_{Knf} = \kappa_{Knf=0} + \frac{df^2}{16\alpha} C(1 - \alpha)^n Knf \quad (11)$$

As defined by Klinkenberg (1941), $κ_{Knf=0}$ represents the intrinsic (or true) permeability, while $κ_{Knf}$ denotes the apparent permeability.

<figure><img src="image_5.png" /><figcaption>Fig. 3. Permeabilities calculated from the new model (Eq. (11)) where C = 1.33 and n = 1.5) as a function of the values obtained by simulation. Dotted lines represent a ± 10 % margin.</figcaption></figure>
Fitting the model (Eq. (11)) to the permeability data simulated by Kwak et al. (2023), along with a subset of our own simulated values, yielded the parameters C = 1.33 and n = 1.5. The model was subsequently tested against the second subset of our simulated data. Fig. 3 presents a parity graph comparing all simulated and calculated permeability values, showing that the model predicts the permeability with an accuracy of ± 10 %. This model gives MRD values equal to 0.041 and 0.035 for Kwak's data and our data, respectively.

This value of n differs with the value of 1 used in the model proposed by Kirsch et al. (1973), and later reported by Choi et al. (2017a). This discrepancy can be explained by the fact that the fiber volume fractions considered by both authors did not exceed 15 %, whereas the present study investigates values up to 50 %. Approximating (1 − α)1.5as (1 − α) introduces only a minor deviation—ranging from 3 % to 8 %—for packing density values of 5 % and 15 %, respectively. This low deviation may account for why those authors found n = 1. However, at higher fiber volume fractions, the deviation becomes significantly larger, ranging from 12 % to 41 % for solid volume fraction values between 20 % and 50 %.

The Fig. 4 shows the ratio of permeability with and without slip effects (Eq. (12)) as a function of the fiber Knudsen number for three different solid volume fractions.

$$ \frac{\kappa_{Knf}}{\kappa_{Knf=0}} = 1 + \frac{1.33(1-\alpha)^{3/2}}{2.5^{1/4} \left( -\frac{1}{2} \ln(\alpha) + \frac{1}{2} \frac{\alpha^2-1}{\alpha^2+1} \right)} K_{nf} \quad (12) $$

A strong correlation is observed between the simulated values and those predicted by the new model. As illustrated in this figure, neglecting slip effects when calculating the permeability can result in a significant underestimation of the permeability, and consequently, an overestimation of the pressure drop. This discrepancy becomes increasingly pronounced as the packing density increases. For α < 0.1, the deviation becomes greater than 10 % when Knf is greater than 0.08.

For comparison, Kirsch's et al. (1973) model was evaluated for a fibrous structure with a 10 % packing density. The model systematically overestimates the permeability ratio, with deviations becoming more pronounced as the fiber Knudsen number increases.

### 3.2. Fibrous media with polydisperse fibers

In practice, fibrous media are rarely composed exclusively of monodisperse fibers. They typically consist of a mixture of submicron and micron-sized fibers and this polydispersity has a significant impact on the pressure drop. Submicron fibers increase flow resistance and also enhance particle capture. In contrast, the presence of larger micron-

<figure><img src="image_6.png" /><figcaption>Fig. 4. Evolution of the permeability ratio with and without slip effect (Eq. (12)) versus the fiber Knudsen number for three different solid volume fractions (0.001, 0.05 and 0.1).</figcaption></figure>

sized fibers improves mechanical strength and structural integrity while helping to reduce the overall pressure drop.

In a previous study on mixture of micron-sized fibers, Cabaset et al. (2025) expressed the intrinsic permeability as follows:

$$ \kappa_{Knf=0} = \left[ \frac{1}{\sum f_i \left( \frac{16\alpha}{H_{Knf=0} - d_i^2} \right)^{3/4}} \right]^{4/3} \quad (13) $$

Where $f_i$ is the volume fraction of fibers with diameter $d_i$, and $H_{Knf=0}$ is the modified Happel hydrodynamic factor (Eq. (10)). Equation (13) estimates the permeability of fibrous media with multimodal distributions of micron-sized within ± 10 % of the simulated values, and within ± 20 % of the experimental values for homemade media. A major advantage of this relationship is that it accounts for the actual fiber size distribution, thereby eliminating the need to rely on an average or equivalent diameter.

So, to account for slip effects on submicron fibers, equation (13) is modified by replacing the hydrodynamic factor without slip effect by Equation (9), with C and n equal to 1.33 and 3/2, respectively.

$$ \kappa_{Knf} = \left[ \frac{1}{\sum f_i \left( \frac{16\alpha}{(H_{Knf=0} + 1.33(1-\alpha)^{3/2}Knf_i)d_i^2} \right)^{3/4}} \right]^{4/3} \quad (14) $$

Here, Knf is equal to 2 λ/di, and $H_{Knf=0}$ is calculated using Equation (10).

The general expression (Eq. (14)) is applicable for calculating permeability in all cases, including micron-sized, submicron, and mixed fiber size distributions. For a mixture of micron-sized fibers, Knf tends toward zero, reducing equation (14) to equation (13). Furthermore, if all micron-sized fibers have the same diameter (df), equation (14) simplifies to equation (3). Fig. 5 shows a parity plot comparing the simulated and the calculated permeability values (Eq.14) for numerically generated fibrous media composed of submicron and micron fiber blends with bi-, tri-, and quadrimodal diameter distributions. The key characteristics of the simulated media are summarized in Table 3. The structures generated for the simulations were constructed by selecting the fiber diameters and the packing density values within the ranges specified in Table 3. Using the model, the majority of simulated permeabilities are predicted with an accuracy of ± 10 %. The MRD values are equal to 0.039, 0.025, and 0.049 for bimodal, trimodal, and quadrimodal distributions, respectively.

For certain applications, the dust holding capacity of the media is a priority. In such cases, particle penetration into the fibrous structure should be promoted, which implies aiming for a low single-fiber collection efficiency by using fibers with larger diameters. The overall filtration efficiency can then be achieved by adjusting the medium thickness, which will also determine the initial pressure drop. In this approach, the permeability estimation is a key parameter for identifying the optimal fibrous structure.

Fig. 6 illustrates the evolution of the permeability determined by numerical simulation for three fibrous media with identical packing density (α = 0.1) and a bimodal distribution of fiber diameters (d1 and d2), as a function of the volume fraction of the coarser fiber (d1).

All other factors being equal, increasing the volume fraction of the coarser fiber leads to a higher permeability and, consequently, to a lower pressure drop for a given medium thickness.

A good agreement can be observed between the simulated values and those predicted by the permeability model given by Equation (14) (solid line). Neglecting the slip effect leads to an underestimation of
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Permeabilities calculated from equation (14) as a function of the values obtained by simulation for a) bi-, b) tri-, and c) quadrimodal diameter distributions. Dotted lines represent a ± 10 % margin.</figcaption></figure>

<table><caption>**Table 3**
Key characteristics of the simulated media.</caption><thead><tr><th>Fiber distribution</th><th>Bimodal</th><th>Trimodal</th><th>Quadrimodal</th></tr></thead><tbody><tr><td>Fiber diameter (d<sub>i</sub>) (μm)</td><td>[0.05, 2]</td><td>[0.075, 20]</td><td>[0.1, 20]</td></tr><tr><td>Packing density (α) (%)</td><td>[3, 15]</td><td>[5, 20]</td><td>[5, 10]</td></tr><tr><td>Volume fraction (f<sub>i</sub>)</td><td>variable</td><td>variable</td><td>variable</td></tr><tr><td>Total number of simulations performed</td><td>90</td><td>37</td><td>56</td></tr></tbody></table>

permeability, as shown by the dashed line (Eq. (13)).

Fig. 6 also highlights that multiple distinct media structures with the same permeability (0.4 µm² in this example) can be obtained by adjusting the volume fraction of the coarser fibers (here 0.58, 0.85, and 0.95 depending on the structure). For a given medium thickness, all three structures will exhibit the same pressure drop, but not the same collection efficiency. According to the aerosol filtration theory (Davies, 1973; Thomas et al., 2016), the fibrous structure containing the finest fibers will be the most efficient—in this case, the one composed of fibers with diameters of 1 and 0.2 µm.

### 3.3. Model validation using experimental data

To assess the robustness of the permeability model, it must be compared with experimental data. As a first step, the established model (Eq. (2) and Eq. (11)) is compared with experimental pressure drop data for monodisperse fibrous media, based on studies conducted by Lee et al., 2023, Pan et al. (2021) and Wang et al. (2008), as reviewed by Kwak et al. (2023) and supplemented with data from Hung and Leung (2011), Mei et al. (2013) and He et al. (2016) (see Table A1). All these media exhibit a packing density below 15 %, with fiber diameters ranging from 94 to 1290 nm.

It should be noted that the model predicts most of the experimental data within ± 50 % (Fig. 7). Kwak's (2023) model, tested on the same dataset, gives the experimental pressure drop with the same level of uncertainty (± 50 %). These significant discrepancies can be explained in particular by the strong assumption of a single fiber diameter. Due to the manufacturing process, the fiber size distribution of the experimentally tested media is necessarily polydisperse, even multimodal. Moreover, the difficulty of measuring the thickness of compressible
<figure><img src="image_8.png" /><figcaption>Fig. 6. Evolution of the permeability determined by numerical simulation for three fibrous media with identical packing density (α = 0.1) and a bimodal distribution of fiber diameters (d₁ and d₂), as a function of the volume fraction of the coarser fiber (d₁).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. Experiment pressure drop as a function of the pressure drop calculated from equations (2) and (11) (where C = 1.33 and n = 1.5) for monomodal fibrous structure. Dotted lines represent a ± 50 % margin.</figcaption></figure>

fibrous media and therefore their packing density is also a source of uncertainties.

The Fig. 8 shows the experiment pressure drop a function of the pressure drops calculated using Equations (2) and (14) on fibrous media with either a bimodal fiber size distribution (Choi et al., 2017a; Mei et al., 2013 – see Table A2.1) or a monomodal but polydisperse distribution following a log-normal law (Zhang and Liu, 1992– see Table A2.2).

The present model provides a reasonably accurate estimate of the experimental data, with the exception of a portion of Mei's dataset, which is significantly overestimated. For the data from Zhang and Liu (1992), the relative deviation between experimental and calculated values is approximately 22 % when considering the full fiber size distribution. In contrast, the Kwak's model—based solely on the effective or the median diameter—yields a deviation of around 500 % (not shown here). These results highlight the critical importance of considering the

<figure><img src="image_10.png" /><figcaption>Fig. 8. Experiment pressure drop a function of the pressure drops calculated from equations (2) and (14) for various polydisperse fibrous structures. Dotted lines represent a ± 50 % margin.</figcaption></figure>

complete fiber size distribution when predicting the pressure drop.

In addition to literature data, laboratory permeation tests are performed at atmospheric and reduced pressures to modulate the air mean free path. Permeation tests were conducted at room temperature on five commercial glass fiber media, whose characteristics are detailed in the Table 4. The medium thickness was measured under a compressive stress of 0.5 kPa. The fiber size distribution is determined through the analysis of multiple SEM images taken from each type of commercial media (see Appendix B). Depending on the medium, between 500 and 1,400 individual fibers were analyzed.

The tests are performed under varying pressure conditions, ranging from 300 to 1000 mbar. For a given pressure, the linear relationship between the media pressure drop and the face velocity allows the determination of media permeability using Darcy's law (Eq. (2)). The Fig. 9 illustrates the linear variation of apparent permeability as a function of the mean free path, calculated using the Willeke (1976) equation (Eq. (15)), which depends on pressure (P) and temperature (T).

$$ \lambda(nm) = 66.5 \frac{Po}{P} \frac{T}{To} \frac{1 + 110/To}{1 + 110/T} \quad (15) $$

With Po = 101.3 kPa and To = 293 K. A decrease in pressure (i.e., an increase in the mean free path) leads to a linear increase in the permeability, and consequently, a reduction in the pressure drop for a given face velocity. This linear behavior is consistent with the trend predicted by Equation (11).

The Fig. 10 presents the ratio of experimental permeability to the modeled permeability (Eq. (14), which accounts for the fiber diameter distribution.

While the prediction is satisfactory for media M4 and M5, the model tends to underestimate the experimental permeability for media M1 to

<table><caption>**Table 4**
Key characteristics of the five commercial glass fiber media.</caption><thead><tr><th>Media</th><th>M1</th><th>M2</th><th>M3</th><th>M4</th><th>M5</th></tr></thead><tbody><tr><td>Z (0.5 kPa)<br>(μm)</td><td>492 ± 3</td><td>494 ± 2</td><td>504 ± 3</td><td>474 ± 4</td><td>499 ± 24</td></tr><tr><td>α (%)</td><td>5.77 ± 0.04</td><td>5.59 ± 0.13</td><td>5.54 ± 0.08</td><td>6.05 ± 0.12</td><td>6.44 ± 0.30</td></tr></tbody></table>
<figure><img src="image_11.png" /><figcaption>Fig. 9. Experimental permeability according to the free mean path for the five commercial fibrous media.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. Variation of the $κ/κ_{Mod}$ ratio with respect to the applied pressure for the five commercial fibrous media (solid symbols: same fiber length – empty symbols: different fiber lengths).</figcaption></figure>

M3. The observed discrepancies likely arise from the inherent difficulty of characterizing real fibrous media. Owing to their compressible nature, the thickness measurements are uncertain; a 10 % error in thickness translates into an equivalent uncertainty in experimental permeability. Moreover, the fiber size distribution determined from SEM surface images may not accurately represent the internal fiber distribution. In addition, commercial fibrous media often contain binders that enhance mechanical strength but are not included in the model. Local structural heterogeneities in the tested samples may further contribute to these differences.

Moreover, commercial glass fibers consist of a mixture of different batches of fibers in varying proportions. The diameter and length of the fibers vary from one batch to another. Due to the lack of available information, all fibers in the model are assumed to have the same length for calculation of the volume fraction corresponding to each fiber diameter $d_i$. Assuming that, for media M1, M2, and M3 — originating from the same supplier — fibers with diameters greater than 2 µm (accounting for 2.5 %, 2.6 %, and 1.2 % of the total fiber population in M1, M2, and M3, respectively) are 1.66 times longer than the remaining fibers, the ratio $κ/κ_{Mod}$ is found to be below 1.2 for all three media over
the entire pressure range (Fig. 10, open symbols). Moreover, for the media M1, the inclusion of just three fibers with a diameter of 5.8 µm—in addition to the 1,414 fibers analyzed—induces a significant drop in the $K/K_{Mod}$ ratio, from 1.41 to 1.04, at 1000 mbar.

These examples highlight the high sensitivity of the model and underscore the importance of accurately characterizing the fiber size distribution in fibrous media.

## 4. Conclusions

A permeability model (Thomas et al., 2023; Cabaset et al., 2025), originally developed for micrometric fibers and accounting for the fiber diameter distribution, has been extended to submicron and even nanometric fibers by incorporating slip flow effects. This broader-scope model, when compared with other models from the literature, demonstrates an improved predictive accuracy (within ± 10 %) for permeability values obtained from numerical simulations on both monomodal (219 cases) and polymodal (183 cases) fibrous structures, covering a packing density range from 0.1 % to 50 % and a fiber diameter range from 0.03 to 20 µm (or $7 \times 10^{-3} < Kn_f < 4$). This model stands out by accounting for the fiber diameter distribution into the estimation of fibrous media permeability.

Compared to experimental values, the model shows a satisfactory agreement (within ± 50 %) when considering experimental uncertainties related to the measurement of compressible fibrous media thickness, packing density and fiber diameter distribution. This level of accuracy also reflects the underlying assumptions of the model: a homogeneous media structure, fibers of uniform length and no influence

from binders (approximately 5 % by mass) typically present in commercial fiber glass media.

This new model can provide valuable guidance for manufacturers in selecting fiber blends based on the target permeability for a given packing density. Since filtration performance is not solely determined by the pressure drop, the next step will be to incorporate fiber size distribution into collection efficiency models. Combining permeability and efficiency models will enable the optimization of fiber structures adapted to specific application requirements, such as protection level, clogging behavior, retention capacity and filter lifespan.

### CRediT authorship contribution statement

**Dominique Thomas:** Writing – review & editing, Writing – original draft, Validation, Supervision, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Elise Cabaset:** Software, Investigation, Formal analysis, Data curation. **Ming Wang:** Investigation, Formal analysis, Data curation. **Augustin Charvet:** Writing – review & editing, Supervision, Software, Methodology, Investigation, Data curation. **Nathalie Bardin-Monnier:** Writing – review & editing, Supervision, Software, Methodology, Investigation. **Jean-Christophe Appert-Collin:** Writing – review & editing, Visualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Appendix A

<table><caption>**Table A1**

Characteristics of monodisperse fibrous media.</caption><thead><tr><th>Author</th><th>Fiber diameter (nm)</th><th>SVF (–)</th><th>Thickness (µm)</th><th>Face velocity (cm/s)</th><th>Operating Pressure (kPa)</th><th>Pressure drop (Pa)</th></tr></thead><tbody><tr><td rowspan="10">Lee et al. (2023)</td><td>280</td><td>0.028</td><td>9.3</td><td>5.3</td><td>a. p.</td><td>33</td></tr><tr><td>373</td><td>0.042</td><td>33.2</td><td>5.3</td><td>a. p.</td><td>88</td></tr><tr><td>446</td><td>0.037</td><td>110.2</td><td>5.3</td><td>a. p.</td><td>207</td></tr><tr><td>510</td><td>0.036</td><td>14.4</td><td>5.3</td><td>a. p.</td><td>18</td></tr><tr><td>534</td><td>0.055</td><td>72.2</td><td>5.3</td><td>a. p.</td><td>151</td></tr><tr><td>616</td><td>0.074</td><td>19.1</td><td>5.3</td><td>a. p.</td><td>53</td></tr><tr><td>779</td><td>0.045</td><td>29.8</td><td>5.3</td><td>a. p.</td><td>32</td></tr><tr><td>858</td><td>0.025</td><td>105.8</td><td>5.3</td><td>a. p.</td><td>33</td></tr><tr><td>270.65</td><td>0.041</td><td>3.82</td><td>5</td><td>a. p.</td><td>5.7</td></tr><tr><td rowspan="2">Pan et al. (2021)</td><td>270.65</td><td>0.063</td><td>6.42</td><td>5</td><td>a. p.</td><td>28.2</td></tr><tr><td>270.65</td><td>0.070</td><td>7.9</td><td>5</td><td>a. p.</td><td>40.8</td></tr><tr><td rowspan="4">Wang et al. (2008)</td><td>150</td><td>0.134</td><td>0.15</td><td>10</td><td>a. p.</td><td>20.4</td></tr><tr><td>150</td><td>0.104</td><td>0.15</td><td>10</td><td>a. p.</td><td>13.7</td></tr><tr><td>150</td><td>0.059</td><td>0.15</td><td>10</td><td>a. p.</td><td>4.6</td></tr><tr><td>150</td><td>0.034</td><td>0.15</td><td>10</td><td>a. p.</td><td>2.3</td></tr><tr><td rowspan="6">Hung and Leung (2011)</td><td>94</td><td>0.006</td><td>6.1</td><td>5</td><td>a. p.</td><td>20.9</td></tr><tr><td>185</td><td>0.006</td><td>6.2</td><td>5</td><td>a. p.</td><td>5.4</td></tr><tr><td>185</td><td>0.011</td><td>6.9</td><td>5</td><td>a. p.</td><td>14.3</td></tr><tr><td>185</td><td>0.018</td><td>8.4</td><td>5</td><td>a. p.</td><td>36</td></tr><tr><td>185</td><td>0.020</td><td>15.7</td><td>5</td><td>a. p.</td><td>74</td></tr><tr><td>220</td><td>0.023</td><td>15</td><td>5</td><td>a. p.</td><td>62.2</td></tr><tr><td rowspan="3">He et al. (2016)</td><td>1240</td><td>0.079</td><td>330</td><td>2.071</td><td>101.3</td><td>270</td></tr><tr><td>1240</td><td>0.079</td><td>330</td><td>14.91</td><td>101.3</td><td>1300</td></tr><tr><td>1240</td><td>0.079</td><td>330</td><td>23.85</td><td>101.3</td><td>2140</td></tr></tbody></table>

(continued on next page)
<table><caption>Table A1 (continued)</caption><thead><tr><th>Author</th><th>Fiber diameter (nm)</th><th>SVF (–)</th><th>Thickness (µm)</th><th>Face velocity (cm/s)</th><th>Operating Pressure (kPa)</th><th>Pressure drop (Pa)</th></tr></thead><tbody><tr><td rowspan="30">Mei et al. (2013)</td><td>1240</td><td>0.079</td><td>330</td><td>2.71</td><td>57.3</td><td>160</td></tr><tr><td>1240</td><td>0.079</td><td>330</td><td>4.96</td><td>57.3</td><td>330</td></tr><tr><td>1240</td><td>0.079</td><td>330</td><td>2.71</td><td>15.5</td><td>120</td></tr><tr><td>1240</td><td>0.079</td><td>330</td><td>10.84</td><td>15.5</td><td>470</td></tr><tr><td>1240</td><td>0.079</td><td>330</td><td>21.19</td><td>15.5</td><td>860</td></tr><tr><td>1240</td><td>0.079</td><td>330</td><td>2.71</td><td>8.7</td><td>90</td></tr><tr><td>1240</td><td>0.079</td><td>330</td><td>5.28</td><td>8.7</td><td>180</td></tr><tr><td>1240</td><td>0.079</td><td>330</td><td>21.27</td><td>8.7</td><td>650</td></tr><tr><td>1240</td><td>0.079</td><td>330</td><td>4.61</td><td>4.5</td><td>110</td></tr><tr><td>1240</td><td>0.079</td><td>330</td><td>10.19</td><td>4.5</td><td>230</td></tr><tr><td>358</td><td>0.062</td><td>10.75</td><td>5.3</td><td>a. p.</td><td>65</td></tr><tr><td>425</td><td>0.066</td><td>10.25</td><td>5.3</td><td>a. p.</td><td>50</td></tr><tr><td>501</td><td>0.061</td><td>10.75</td><td>5.3</td><td>a. p.</td><td>34</td></tr><tr><td>660</td><td>0.059</td><td>11.5</td><td>5.3</td><td>a. p.</td><td>19</td></tr><tr><td>880</td><td>0.058</td><td>11.25</td><td>5.3</td><td>a. p.</td><td>10</td></tr><tr><td>1290</td><td>0.059</td><td>11.5</td><td>5.3</td><td>a. p.</td><td>5</td></tr><tr><td>358</td><td>0.068</td><td>18.5</td><td>5.3</td><td>a. p.</td><td>128</td></tr><tr><td>358</td><td>0.074</td><td>27.75</td><td>5.3</td><td>a. p.</td><td>220</td></tr><tr><td>358</td><td>0.082</td><td>40</td><td>5.3</td><td>a. p.</td><td>365</td></tr></tbody></table>

a. p.: ambient pressure.

<caption>Table A2.1</caption>

Characteristics of bimodal fibrous media.

<table><thead><tr><th>Author</th><th>d1 (nm)</th><th>f1</th><th>d2 (nm)</th><th>f2</th><th>SVF (–)</th><th>Thickness (µm)</th><th>Face velocity (cm/s)</th><th>Operating Pressure (kPa)</th><th>Pressure drop (Pa)</th></tr></thead><tbody><tr><td rowspan="8">Mei et al. (2013)</td><td>358</td><td>0.5</td><td>1290</td><td>0.5</td><td>0.061</td><td>11.25</td><td>5.3</td><td>a. p.</td><td>55</td></tr><tr><td>358</td><td>0.325</td><td>1290</td><td>0.675</td><td>0.062</td><td>11</td><td>5.3</td><td>a. p.</td><td>44</td></tr><tr><td>358</td><td>0.25</td><td>1290</td><td>0.75</td><td>0.062</td><td>10.75</td><td>5.3</td><td>a. p.</td><td>32</td></tr><tr><td>425</td><td>0.50</td><td>880</td><td>0.5</td><td>0.060</td><td>11</td><td>5.3</td><td>a. p.</td><td>47</td></tr><tr><td>425</td><td>0.325</td><td>880</td><td>0.675</td><td>0.059</td><td>11.75</td><td>5.3</td><td>a. p.</td><td>34</td></tr><tr><td>425</td><td>0.25</td><td>880</td><td>0.75</td><td>0.060</td><td>11.5</td><td>5.3</td><td>a. p.</td><td>26</td></tr><tr><td>501</td><td>0.5</td><td>660</td><td>0.5</td><td>0.061</td><td>11.25</td><td>5.3</td><td>a. p.</td><td>31</td></tr><tr><td>501</td><td>0.325</td><td>660</td><td>0.675</td><td>0.060</td><td>11</td><td>5.3</td><td>a. p.</td><td>27</td></tr><tr><td>501</td><td>0.25</td><td>660</td><td>0.75</td><td>0.058</td><td>11.5</td><td>5.3</td><td>a. p.</td><td>25</td></tr><tr><td rowspan="4">Choi et al. (2017b)</td><td>780</td><td>0.05</td><td>11,200</td><td>0.95</td><td>0.152</td><td>276</td><td>20</td><td>a. p.</td><td>76</td></tr><tr><td>780</td><td>0.1</td><td>11,200</td><td>0.9</td><td>0.172</td><td>237</td><td>20</td><td>a. p.</td><td>221</td></tr><tr><td>780</td><td>0.2</td><td>11,200</td><td>0.8</td><td>0.186</td><td>235</td><td>20</td><td>a. p.</td><td>350</td></tr><tr><td>780</td><td>0.3</td><td>11,200</td><td>0.7</td><td>0.195</td><td>239</td><td>20</td><td>a. p.</td><td>1121</td></tr></tbody></table>

a. p.: ambient pressure.

<caption>Table A2.2</caption>

Characteristics of polydisperse fibrous medium.

<table><thead><tr><th>Author</th><th>Effective fiber diameter (nm)</th><th>Mean Fiber diameter (nm)</th><th>GSD</th><th>SVF (–)</th><th>Thickness (µm)</th><th>Face velocity (cm/s)</th><th>Operating Pressure (kPa)</th><th>Pressure drop (Pa)</th></tr></thead><tbody><tr><td rowspan="7">Zhang and Liu (1992)</td><td>830</td><td>720</td><td>2.38</td><td>0.078</td><td>406</td><td>6</td><td>a. p.</td><td>93.2</td></tr><tr><td>830</td><td>720</td><td>2.38</td><td>0.078</td><td>406</td><td>11</td><td>a. p.</td><td>175.5</td></tr><tr><td>490</td><td>420</td><td>2.20</td><td>0.08</td><td>400</td><td>6</td><td>a. p.</td><td>329.5</td></tr><tr><td>490</td><td>420</td><td>2.20</td><td>0.08</td><td>400</td><td>11</td><td>a. p.</td><td>624.7</td></tr><tr><td>490</td><td>420</td><td>2.20</td><td>0.08</td><td>400</td><td>19</td><td>a. p.</td><td>1123</td></tr><tr><td>250</td><td>210</td><td>2.34</td><td>0.053</td><td>300</td><td>6</td><td>a. p.</td><td>378.5</td></tr><tr><td>250</td><td>210</td><td>2.34</td><td>0.053</td><td>300</td><td>11</td><td>a. p.</td><td>708</td></tr><tr><td>250</td><td>210</td><td>2.34</td><td>0.053</td><td>300</td><td>19</td><td>a. p.</td><td>1310</td></tr></tbody></table>

a. p.: ambient pressure.
## Appendix B

<figure><img src="image_13.png" /><figcaption>Fig. B1. Fiber count fraction as a function of fiber diameter for the five commercial media evaluated.</figcaption></figure>

## Data availability

Data will be made available on request.

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