# CFD Simulation of the Filtration Performance of Fibrous Filter Considering Fiber Electric Potential Field

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## Abstract

Aiming at disclosing the quantitative effects of Coulomb forces on the filtration efficiency of aerosol particles, a three-dimensional random fiber model was established to describe the microstructure of fibrous filters. Then, computational models including the flow model, particle model, and electric field model were constructed to estimate the filtration efficiency using the Fluent custom user-defined function program, neglecting the non-uniformity of the fiber potential and the particle charge distribution. The simulation results using the established models agreed with the data in the literature. In particular, the electric field force was found to be one of the important factors required to improve the filtration efficiency estimation accuracy for the ultrafine particles. Moreover, the variation tendencies of the filtration efficiency and the pressure drop of fibrous filters were studied based on the influence factors of the fiber potential, particle charge-to-mass ratio, solid volume fraction, fiber diameter, and face velocity. The established models and estimated results will provide important guidance on the design of high-efficiency particulate air filters for aerosol particles.

## Keywords

Filtration efficiency Coulomb force Three-dimensional random fiber model Interception capture Brownian diffusion## List of Symbols

*C*_{C}Cunningham correction factor

*d*_{f}Fiber diameter (m)

*d*_{p}Particle diameter (m)

*E*Electric field strength (N/C)

*E*_{∑}Total filtration efficiency of the single fiber

*E*_{R}Interception efficiency

*E*_{D}Brownian diffusion efficiency

*E*_{E}Electrostatic force efficiency

*f*(*α*_{f})Dimensionless drag coefficient as a function of SVF

*F*Dimensionless drag coefficient by the method of Happel

*F*_{B}Brown force (N)

*F*_{D}Drag (N)

*F*_{E}Electric field force (N)

*k*_{B}Boltzmann constant (1.38×10

^{−23}m^{2}kg s^{−2}K^{−1})*Kn*_{f}Fiber Knudsen number

*Kn*_{p}Particle Knudsen number

*Ku*Kuwabara hydrodynamic factor

*L*Filter medium thickness (m)

*L*_{f}Total length of fiber per unit of media area (m)

*N*_{in}Total number of particles upstream

*N*_{out}Total number of particles downstream

*p*Pressure (Pa)

- Δ
*p* Pressure drop (Pa)

*Pe*Peclet number

*q*Particle charge (C)

*Q*/*M*Particle charge-to-mass ratio (C/kg)

- SVF
Solid volume fraction

*t*Time (s)

- Δ
*t* Time step (s)

*T*Thermodynamic temperature (K)

*u*Fluid velocity (m/s)

*u*_{p}Particle velocity (m/s)

## Greek Letters

*α*_{f}Filter medium fiber filling rate

*η*Filtration efficiency

*λ*Mean free path of the gas molecule (m)

*μ*Air viscosity (Pa s)

*ρ*_{p}Particle density (kg/m

^{3})*ϕ*Electrostatic potential (V)

*ξ*Zero mean, unit-variance independent Gaussian random numbers

## Introduction

Improving the capture efficiency of fine particles has attracted more attention because of the negative impact of fine particles on human health [1, 2, 3]. In particular, high-efficiency particulate air (HEPA) filters need to be designed to effectively capture ultrafine particles smaller than 300 nm [4, 5], which is important to produce clean breathable air and also to ensure safety in biosafety laboratories and the nuclear industry [6, 7, 8].

In recent years, many studies have been conducted on the synthesis of charged fibers with excellent fine particles filtration efficiency [9, 10]. Yeom and Shim [11] added boehmite nanoparticles as the electrostatic charging agent during the electrospinning of nylon-6 (PA6) fibers and found that the charged PA6 fibers provided a filtration efficiency of 27.8% toward 300 nm aerosol, which was much higher than that of uncharged fibers (6.3%), without changing the resistance. Tang et al. [12] studied the filtration efficiency of an electret filter for 3–500 nm particles and concluded that the collection efficiency of charged particles (50–100 nm) increased by over 20%. However, how to estimate the quantitative effect of electrostatic interaction on improving the filtration efficiency of fine particles remains unclear.

According to the classical filtration theory, the filtration performance of fibers in a stable phase is associated with five capture mechanisms: interception effect, inertial collision, Brownian diffusion, electrostatic interaction, and gravity [13, 14, 15, 16]. Previous studies have reported numerical simulations of filtration performance considering Brownian diffusion, interception, and inertial impaction [17, 18, 19, 20, 21]. For example, a single fiber model has been established to simulate the collection efficiency, neglecting the interactions among the fibers [22, 23, 24]. Moreover, a parallel and staggered multi-fiber filtration model has been developed to simulate the collection efficiency [25]. As for the effect of electrostatic interactions on the collection efficiency, Wang [26] presented a comprehensive review of the theoretical and experimental studies of electrostatic forces in filtration. However, thus far, no report on the numerical simulation with electrostatic forces has been found in the literature.

Considering the three-dimensional (3D) porous microstructure of fibrous filters, in this study, we developed a software-coupling method, interconnecting MATLAB, AutoCAD, ICEM, and Fluent software, to generate a 3D random fiber model that approximately describes the microstructure of fibrous filters. Afterward, computational models considering fiber electric potential field were developed using the Fluent custom user-defined function (UDF) program, assuming a uniform fiber potential and neglecting the non-uniformity of the particles charge distribution. After the model verification, numerical simulations were performed to further clarify the regulations on improving the filtration efficiency of fibrous filters, which will provide important guidance on the design of HEPA filters.

## Generating Three-Dimensional Random Fiber Model

Shou et al. [27] suggested that a random fiber media model would provide more permeability, compared with the ordered fiber arrays. In this study, assuming a random fiber structure in the filter, we developed a coupling method, involving MATLAB R2014a (https://www.mathworks.cn), AutoCAD 2017 (https://www.autodesk.com.cn), ICEM 15.0 (https://www.ansys.com), and Fluent 15.0 (https://www.ansys.com/Products/Fluids), to generate a 3D random fiber model that approximately describes the porous microstructure of fibrous filters.

The first step was to generate the pillar model mimicking the filter fiber by coupling MATLAB and AutoCAD. We wrote a program via a randomized algorithm in MATLAB to generate a random point matrix. Then, the random point matrix was inputted into AutoCAD to form a set of line segments, and consequently, the pillar models were constructed with an average diameter on the basis of the centered line segments. For each line segment, the two terminal points should have the same *x* coordinate, to maintain all the fibers located horizontally.

*d*

_{f}) of 370 nm. The inlet and outlet of the calculation zone are assumed to be in the undisturbed flow field and placed, respectively, at distances of seven times

*d*

_{f}upstream and downstream of the filter medium.

*d*

_{f}and five times

*d*

_{f}upstream and downstream of the filter medium. Babaie et al. [28] set the length of the inlet and outlet media to 7 and 5 times the fiber diameter, respectively. In our work, the inlet and the outlet were first set at distances of 20 times

*d*

_{f}upstream and downstream of the simulation medium, and the corresponding velocity contour is shown in Fig. 2a. The distance was reduced to seven times

*d*

_{f}upstream and downstream of the medium, and the velocity contour displayed in Fig. 2b was obtained. The velocity flow field in Fig. 2b is similar to that obtained with the large distance of 20 times

*d*

_{f}(Fig. 2a). Therefore, we adopted the distance of seven times

*d*

_{f}in the simulation to save the computer’s running space.

## Computational Models and Simulation Method

### Flow Model

*is the fluid velocity (m/s);*

**u***is the pressure; and*

**p***μ*is air viscosity (Pa s). As shown in Fig. 1, the entrance of the computational domain was set as a velocity inlet, and a pressure boundary condition was applied to the outlet. The sides of the domain were set as symmetric boundary conditions.

### Particle Model

In Fluent, the Euler method and the Lagrangian method are generally used to simulate the gas–solid two-phase flow process, whereby the Euler method is applied to a mixture with a higher dispersion volume fraction, while the Lagrangian method has a good simulation effect for a mixture with a dilute dispersed phase. Here we adopted the Lagrangian method to simulate the fiber filtration process.

*F*

_{D},

*F*

_{B}, and

*F*

_{E}represent the drag, Brown force, and electric field force (N), respectively;

*u*

_{p}is the particle velocity (m/s);

*C*

_{C}is the Cunningham correction factor that is calculated via

*C*

_{C}= 1 +

*Kn*

_{p}(1.257 + 0.4e

^{−1.1/Kn}

_{p}) [21, 30, 31];

*d*

_{p}is the particle diameter (μm);

*Kn*

_{p}=

*λ*/

*d*

_{p}is the Knudsen number;

*λ*is the mean free path of the gas molecule (m);

*ρ*

_{p}is particle density (kg/m

^{3}, here 1000 kg/m

^{3}is adopted);

*ξ*represents a random number in a standard normal distribution with a zero mean and a variance of 1;

*k*

_{B}is the Boltzmann constant;

*T*is the thermodynamic temperature (K); Δ

*t*is the time step (s);

*q*is the particle charge (C); and

*E*is the electric field strength (N/C).

The discrete phase model (DPM) in Fluent was used to solve the force balance equation of the particle. The boundary conditions of the calculation domain inlet and outlet were set as the “escape”; the boundary condition of the fiber surface was set as the “trap”; and the boundary condition of the wall was set as the “reflect”. The unidirectional coupling method was used to deal with the interaction force between air and particles, ignoring the interaction force among the particles. The standard DPM model treats particles as mass points and only considers the trapping of particles; however, it neglects the effect of inertia on nanoparticles. In air filtration, the main factors that determine the capture efficiency are the interception effect and the diffusion effect. Therefore, the UDF was programmed to improve the DPM model, and the distance between the particle center trajectory and the fiber surface was monitored in real time to introduce the interception effect on the particle. As this distance was shorter than the particle radius, the interception effect was significant to capture the particle. In addition, the Brownian force numerical formula [21] was introduced by the UDF to calculate the influence of the Brownian diffusion motion on the particle interception effect.

The present work is associated with the treatment of indoor air in biosafety laboratories containing a low concentration of ultrafine particles, whereby a total removal of the ultrafine particles is required. During the simulation, the change in geometry of the domain was neglected as the particles were captured by the fibers; that is, if the particles were captured by the fibers, they would be deleted from the domain. Such an assumption has also been adopted in previous studies. Hosseini and Tafreshi [32] established a model in disordered 2D domains for the filtration of particles smaller than 500 nm, assuming that the particles captured by the fibers would be removed from the domain. Jin et al. [33] modeled the filtration of particles larger than 500 nm, assuming that the particles would be removed from the domain when they were captured by the fiber.

### Electric Field Model

Electrostatic forces occur when particles and fibers are charged or an external electric field is applied to the medium. In theory, the electrostatic forces include Coulomb force, image force, and polarization force; however, in the literature, each force is usually considered according to proper assumptions. For instance, Nielsen and Hill [34] studied the collection efficiency of particles by a single spherical collector in wet scrubbers, considering the Coulomb forces, electrical image forces, and external electric field force. They concluded that the effect of the charged-particle and charged-collector image forces on the particle trajectory was much smaller than that of Coulomb forces. D’Addio et al. [35] established a particle scavenging model that considered Coulomb forces in the wet scrubbing of submicron particles (100–450 nm) and successfully described the scavenging coefficient of wet electrostatic scrubbing with charged particles and droplets. Hamaguchi and Farouki [36] estimated the magnitude of the polarization force under typical glow discharge conditions and high particulate density. Kanaoka et al. [37] simulated the agglomerative deposition process of fine aerosol particles (390 nm) under a dust-laden condition and concluded that the Coulomb force essentially determined the trajectory of charged particles. Zuo et al. [38] studied the removal of particulate matter in polluted air, quantitatively analyzed the forces exerted on the moving particles (10 μm), and concluded that the image force could be neglected, compared with the Coulomb force. Therefore, we focused on the Coulomb force to study the effect of electrostatic force on the filtration efficiency for low-concentrated ultrafine particles.

It has been suggested that the electric field is determined only by the fiber potential when the particle charge is small [39]. As for the particle charge distribution, Hoppel [40, 41] studied the charge distribution of aerosols using ion-aerosol balance equations and concluded that the Boltzmann statistics could provide an accurate estimation of the charge distribution for aerosol particles larger than 0.5 μm, whereas for smaller particles, the accuracy was reduced. Dhanorkar and Kamra [42] studied the particle charge distribution in a polluted atmosphere with a total aerosol concentration of 1000 particles/cm^{3} and found that the particle charge distribution depended on the aerosol concentration and ionization rate. In the present work, we focused on the removal of ultrafine particles with low concentration in the atmosphere of biosafety laboratories. It is reasonable to neglect the charge distribution of ultrafine particles and assume that the charged particles behavior does not affect the fiber potential field.

*x*,

*y*, and

*z*directions. The force acting on the particle by the electric field was proportional to the spatial gradient of the electrostatic potential, and the Laplace equation (UDF) of the electrostatic potential is expressed as Eq. (4).

*E*(UDF) generated by the electrostatic potential is calculated using Eq. (5).

The zeta potential of Whatman EPM 2000 glass fibers was measured at different pH environments, as shown in Fig. S1 in the supporting information. Assuming that the surface potentials of the fibers were uniform, the fiber potential was set as − 0.3 V, which corresponds to the value at neutral pH.

The potential values on the inlet and the outlet of the calculation domain were set as zero; the potentials on the other boundaries of the calculation domain were also set as zero; and the fiber potential was set as − 0.3 V, which corresponds to the zeta potential measured at pH 7.0.

D’Addio et al. [35] measured the charge of an aerosol with a particle size distribution of 100–450 nm and reported a particle charge-to-mass ratio (*Q*/*M)* of 0.075–0.1 C/kg. In the present study, neglecting the non-uniformities of the charge distribution of the particles, the particle *Q*/*M* ratio was set as 0.085 C/kg. Since the fiber surface was electrically identical to the particles, an electrostatic repulsion force was generated to cause the particle trajectory to wrap around, so that a portion of the particles was confined to the filter medium region by the interception.

The UDS contour represents the distribution of the potential field. The model overall potential contour of the constructed electric field is shown in Fig. S2. Since the potential of − 0.3 V was applied only to the fiber surface in the entire calculation domain, it can be seen that the intermediate filter medium region had a lower overall potential value.

The UDM contour illustrates the model field strength distribution of the constructed electric field, as shown in Fig. S3. The field strength is defined by the negative gradient of the potential, indicating that the faster the potential drops, the larger the gradient and the stronger the field strength. In the filter medium region, the potential value of the wall was 0 on both sides, and the potential value applied to the surface of the fiber was − 0.3 V; thus, the potential gradient on both sides of the filter medium was extremely large, whereas it was not strong enough in the intermediate fiber aggregation area.

*u*= 0.05 m/s and

*d*

_{p}= 300 nm, without a potential field, most of the particles could not be captured by the fiber surface (Fig. 4a). However, in the presence of a potential field, a large number of particles changed the motion trajectory due to the electrostatic force and were captured by the fiber surface (Fig. 4b).In addition, we compared the retention ratio [(

*N*

_{in}–

*N*

_{out})/

*N*

_{in}] with and without considering the Coulomb force for the 300 nm particles, where

*N*

_{in}and

*N*

_{out}are the total numbers of particles at the inlet and outlet, respectively. As listed in Table 1, without considering the electrostatic force, the retention ratio was 0.0023, which is much lower than that when the electrostatic force was considered (0.388). This confirms that the Coulomb force plays an important role in the filtration efficiency.

Effect of Coulomb force on the retention ratio (fiber potential = − 0.3 V, *Q*/*M* value = − 0.085 C/kg, *u* = 0.05 m/s, and *d*_{p} = 300 nm)

| | ( | |
---|---|---|---|

Without Coulomb force | 431 | 430 | 0.0023 |

With Coulomb force | 431 | 264 | 0.388 |

### Mesh Independence Test

## Model Verification

First, we performed the steady-state calculation of the single gas-phase flow in the laminar flow state. After the calculation results converged, the stable air flow field was obtained, and then the inert particles with a density of 1050 kg/m^{3} were introduced through a DPM. The presence of the particle phase caused the perturbation of the continuous field, and the motion trajectory of each particle was tracked by the Lagrangian method, and their positions were monitored to perform the operation of the gas–solid two-phase flow field.

### Pressure Drop

Pressure drop is an important property of fibrous filters. Kuwabara [29] first derived a formula to estimate the pressure drop across fibrous filters. Since then, a series of empirical equations have also been developed to estimate the pressure drop using parameters involving the fiber media thickness, SVF, air viscosity, gas velocity, and fiber diameter [43, 44]. In the present study, we compared our simulation results with those of two popular models, by Davies [45] and Happel [46].

*p*is the pressure drop;

*α*

_{f}is the filter medium fiber filling rate;

*L*is the filter medium thickness; and

*f*(

*α*

_{f}) is the dimensionless drag coefficient as a function of SVF.

*F*by Happel’s method [46].

### Filtration Efficiency

*N*

_{in}and

*N*

_{out}are the total numbers of particles at the inlet and outlet, respectively.

*η*) is estimated using Eq. (14) [49].

*E*

_{∑}is the total filtration efficiency of the single fiber model.

*E*

_{R},

*E*

_{D}, and

*E*

_{E}are the capture efficiencies due to the interception effect, Brownian diffusion, and electric field force, respectively.

At lower flow rates, the diffusion and Coulomb forces synergistically capture the particles due to a longer residence time. As the gas flow rate increases, the contribution of the diffusion mechanism to the media filtration efficiency becomes smaller, and the speed-independent interception mechanism becomes dominant.

*E*

_{D}. On the basis of the above parameters, including a fiber diameter of 370 nm and SVF of 3% for particles 10–500 nm, the Brownian force efficiency was calculated accordingly using the equations in Table 2, and the results are displayed in Fig. 7, together with the simulated results in this work. The

*E*

_{D}profile simulated in this work approximates that calculated by Stechkina and Fuchs’s formula [52] with a particle size of 10–400 nm, and it was close to that calculated by Liu and Rubow’s formula [53] when the particle size was 400–500 nm.

Equations to calculate the Brownian-capture efficiency (*E*_{D})

Expression | Remark | Reference |
---|---|---|

\(E_{\text{D}} = 2.9\,{Ku}^{{ - \frac{1}{3}}} Pe^{{ - \frac{2}{3}}} + 0.62Pe^{ - 1} \quad \left( {16} \right)\) |
Analysis of boundary layer | Stechkina and Fuchs [52] |

\(E_{\text{D}} = 1.6\left( {\frac{1 - \alpha }{Ku}} \right)^{{\frac{1}{3}}} Pe^{{-\frac{2}{3}}} C_{\text{d}} \quad (17)\) | \(C_{\text{d}} = 1 + 0.388Kn_{f} \left( {\frac{{\left( {1 - \alpha } \right)Pe}}{\text{Ku}}} \right)^{{\frac{1}{3}}}\) 10 Slip flow regime | Liu and Rubow [53] |

\(E_{\text{D}} = 1.6\left( {\frac{1 - \alpha }{Ku}} \right)^{{\frac{1}{3}}} Pe^{{ - \frac{2}{3}}} C_{\text{d}} C_{\text{d}}^{'} \quad (18)\) | \(C_{\text{d}}^{'} = \frac{1}{{1 + \left( {E_{\text{D}} } \right)_{{{\text{Liu and Rubow}}\left( {1990} \right)}} }}\) 10 0.02 μm < | Payet et al. [54] |

\(E_{D} = 1.6\left( {\frac{1 - \alpha }{Ku}} \right)^{{\frac{1}{3}}} Pe^{{-\frac{2}{3}}} \quad (19)\) | 0.029 < Theoretical model | Lee and Liu [55] |

*E*

_{R}) caused by the interception effect. We also calculated the

*E*

_{R}value, with a fiber diameter of 370 nm and SVF of 3%, for the 10–500 nm particles according to these equations and plotted

*E*

_{R}profiles (Fig. 8) to compare with the

*E*

_{R}profile simulated in this work. The simulated

*E*

_{R}profile approximates that calculated by the formula proposed by Pich [56] as the particle size was in the range of 10–200 nm, while for large particles (200–500 nm), our simulated results are close to that calculated by the formula proposed by Lee and Gieseke [57].

Equations to calculate the interception capture efficiency (*E*_{R})

Expression | Remark | References |
---|---|---|

\(E_{\text{R}} = 0.6\frac{1 - \alpha }{Ku}\frac{{R^{2} }}{1 + R}\quad (20)\) | 0.01 m/s < 0.05 μm < 0.0086 < In good agreement for particles | Lee and Liu [55] |

\(E_{\text{R}} = \frac{{\left( {1 + R} \right)^{ - 1} - \left( {1 + R} \right) + 2\left( {1 + 1.996{Kn}} \right)\left( {1 + R} \right){ \ln }\left( {1 + R} \right)}}{{2\left( { - 0.75 - 0.5{ \ln }\alpha_{\text{f}} } \right) + 1.996{Kn}\left( { - 0.5 - { \ln }\alpha_{\text{f}} } \right)}}\quad (21)\) | No slip flow at gas–fiber interface In good agreement for particles | Pich [56] |

\(E_{\text{R}} = \frac{1 - \alpha }{Ku}\frac{{R^{2} }}{{\left( {1 + R} \right)^{m} }}\quad (22)\) | \(m = \frac{2}{{3\left( {1 - \alpha } \right)}}\) | Lee and Gieseke [57] |

Some deviations still exist between the experimental data and the simulated values when considering the fiber potential value (Fig. 9). Fotovati et al. [48] established a series of 3D fibrous geometries with an SVF of 7.5% and fiber diameter of 10 μm to study the effect of fiber orientations on the filtration efficiency and concluded that the in-plane and the through-plane orientations of fibers had distinct effects on the filtration efficiency for the particles with different sizes. In this work, we established a random fiber media model; we neglected the fibers orientation and assumed a uniform fiber potential, which is probably the major cause of the deviation.

Furthermore, using our models, we simulated the pressure drop and filtration efficiency for particles with a broad size distribution (50–500 nm), following the experimental parameters in the work of Hung and Leung [59]. Hung and Leung [59] prepared filters by electrospinning nylon-6 nanofibers on the substrates and tested the filtration efficiency for NaCl aerosol particles (50–500 nm at a speed of 5 cm/s) neutralized by an aerosol neutralizer (Po-210). The structural parameters of the filters were a mean fiber diameter of 185 nm, fiber volume fraction of 0.0182, and thickness of 8.4 μm. Their experimental data indicated that the filtration efficiency was 44.1% for the most penetrating particles of 100 nm and 76.2% for the 500 nm particles.

Considering the effect of Coulomb forces, we again adopted a fiber potential of − 0.3 V; however, Hung and Leung [59] did not measure the charge of their NaCl aerosol particles after neutralization. Tsai et al. [60] measured the charge distribution of NaCl particles neutralized by Po-210. They found that after Po-210 neutralization, the NaCl aerosol particles (0.25–0.70 μm) reached the Boltzmann charge equilibrium, with a negative charge of about 0.85–2 elementary units of charge, while the NaCl particles (0.01–0.10 μm) were negatively charged with 0.1–1.0 elementary unit of charge. Thus, the particle *Q*/*M* ratio was set as − 0.02 C/kg or − 0.085 C/kg in the simulation. As shown in Fig. 10b, the simulated results with a *Q*/*M* value of − 0.02 C/kg provided a filtration efficiency of 41.9% for the most penetrating particles of 100 nm, while with a *Q*/*M* value of − 0.085 C/kg, the estimated filtration efficiency was 47.2% for the 100 nm particles. The average deviations of estimated filtration efficiency were 2.5% and 4.2% at *Q*/*M* values of − 0.02 C/kg and − 0.085 C/kg, respectively (Fig. 10b). The results illustrate that the established models considering Coulomb forces can enhance the filtration performance simulation accuracy.

## Results and Discussion

Based on the above established models considering the mechanisms of Brownian diffusion and interception capture and the Coulomb force, we studied the variation tendency of filtration efficiency and pressure drop of fibrous filters, based on factors such as fiber potential, particle *Q*/*M* ratio, SVF, fiber diameter, and face velocity.

*Q*/

*M*of 0.00139–0.1 C/kg [35, 61, 62]. According to the potential feature of fibers as mentioned in the above section, we adopted − 0.3, − 0.2, and − 0.1 V as the potential of fiber surfaces to calculate the filtration efficiency for the 10–500 nm particles with different

*Q*/

*M*ratios, ranging from − 0.01 to − 0.085 C/kg, maintaining the SVF value of 3% and fiber diameter of 370 nm. Figure 11a shows that at a high fiber potential (− 0.3 V), under a

*Q*/

*M*value of − 0.085 C/kg, the filtration efficiency increases with the particle size; the filtration efficiency is close to 10% for the 200 nm particles, while it approximates 70% for the 500 nm particles. When the fiber potential is lower (− 0.2 and − 0.1 V), the filtration efficiency tends to increase with the particle size, but the value decreases to some extent. Under lower

*Q*/

*M*values (− 0.01, − 0.02, − 0.05 C/kg), the filtration efficiency reduces greatly, although it varies similarly with the particle size. As shown in Fig. S4, for the 500 nm particles, the filtration efficiency is about 46.40% at

*Q*/

*M*value of − 0.05 C/kg, whereas it decreases to 12.06% at

*Q*/

*M*value of − 0.01 C/kg, even when the fiber potential is maintained at − 0.3 V. Figure 11b shows the filtration efficiency for the 300 nm particles. The filtration efficiency increases with the absolute values of

*Q*/

*M*and the fiber potential. Givehchi et al. [63] found that when the particle size was increased from 10 nm to 100 nm, the efficiency caused by Coulomb force increased from almost negligible to 30%. According to the study by Wang [26], particle trapping caused by Coulomb force increased as the particle charge and fiber potential increased.

*Q*/

*M*ratio of − 0.085 C/kg. As shown in Fig. 12, the pressure drop increased almost linearly with the SVF, which is due to the positive correlation expressed by Eqs. (7) and (8). However, the filtration efficiency for the particles showed no significant increase at a high SVF value. For the 400 nm particles, the filtration efficiency was about 99.871% at an SVF of 2% and 99.991% at an SVF of 3%. It exhibited a similar tendency for the 300 nm particles. In the study by Yun et al. [64], when the fiber diameter of a polyacrylonitrile filter was fixed at 270 nm, as the SVF increased from 11.2 to 15.2%, the pressure drop increased from 10062 to 14750 Pa/mm, and the filtration efficiency also increased. Li et al. [65] simulated eight flow fields of filter media with different SVFs and found that the pressure drop increased nonlinearly with the SVF. Soltani et al. [66] studied the effect of 3D fiber SVF on permeability and concluded that SVFs were positively correlated with filtration efficiency. Hosseini and Tafreshi [21] studied the effects of six different fiber diameters (100–1000 nm) on filtration efficiency under three different SVFs (2.5%, 5%, 7.5%). Our simulated results are consistent with these tendencies in the literature.

*Q*/

*M*of − 0.085 C/kg. As shown in Fig. 13, under the same SVF value, the pressure drop decreased as the fiber diameter rose from 200 to 500 nm, and the pressure drop value was independent of particle size. When the fiber diameter was 500 nm, the filtration efficiency decreased with an increase in the particle size from 50 to 250 nm and then increased with an increase in the particle size from 250 to 500 nm, achieving a filtration efficiency of 99.66% for the 500 nm particles and the minimum value of 98.37% for the 250 nm particles. For the case of thin fiber diameter (370 nm), the filtration efficiency had a minimum value of 99.31% for certain particles around 100 nm and then reached about 100% for the particles larger than 200 nm. When the fiber diameter was decreased to 200 nm, the filtration efficiency was about 100% for all the particles (10–500 nm). When the fiber diameter was 200 nm, 370 nm, and 500 nm, the most penetrating particle sizes (MPPSs) were 50 nm, 100 nm, and 250 nm, respectively. Previous works have reported a similar relationship between the fiber diameter and the MPPS. Podgórski et al. [67] found that when the fiber diameter was increased from 100 nm to 10 μm, the MPPS increased from 54 nm to 366 nm. Balgis et al. [68] used monodisperse aerosol particles with a diameter of 100 nm to perform a particle penetration test and pressure drop measurement through an air filter composed of fibers of different diameters and found that as the fiber diameter increased, the filtration efficiency decreased but the pressure drop increased.

*Q*/

*M*of − 0.085 C/kg, adopting different fiber diameters. As shown in Fig. 14, the pressure drop increased with the velocity, while the relationship between the filtration efficiency and the velocity was significantly associated with the fiber diameter. With a large fiber diameter (500 nm), the filtration efficiency decreased from 98.95% at the velocity of 0.05 m/s to 91.17% at 0.45 m/s. When the fiber diameter was reduced to 370 nm, the filtration efficiency was 98.14% at 0.45 m/s. For a much thinner fiber (200 nm), at 0.45 m/s the filtration efficiency was 99.99%. This indicates that the thinner the filter fiber, the higher the filtration efficiency. Similarly, it has been reported that when the velocity increased from 0.05 m/s to 0.1 m/s, the filtration efficiency of a commercial HEPA filter for 300 nm particles reduced from 99.99 to 99.98% and the pressure drop increased from 250 to 500 Pa [69].

## Conclusion

In this study, computational models were established and verified to determine the quantitative effects of Coulomb forces on the filtration efficiency of aerosol particles, while considering the interception capture and Brownian diffusion. First, a software-coupling method was developed by interconnecting MATLAB, AutoCAD, ICEM, and Fluent software to generate a 3D random fiber model to describe the microstructure of fibrous filters. Then, computational models including the flow model, particle model, and electric field model were developed using the Fluent custom UDF program, assuming a uniform fiber potential and neglecting the non-uniformity of particle charge distribution. The simulated results of the pressure drop and the filtration efficiency were compared with those estimated using modeling equations in the literature. The Coulomb force was found to be the key factor required to estimate the filtration efficiency of ultrafine particles.

Furthermore, using the established computational models, the variation tendencies of the filtration efficiency and the pressure drop of fibrous filters were studied based on the influence factors of the fiber potential, particle *Q*/*M* ratio, SVF, fiber diameter, and face velocity. By comparing with the experimental data in the literature, the filtration efficiency was found to increase with the particle size; however, the value is greatly associated with the particle *Q*/*M* ratio. The pressure drop increased almost linearly with the SVF, while the filtration efficiency for the particles exhibited no significant increase at a high SVF. The filtration efficiency became larger as the fiber diameter reduced. The established models and the estimated results will provide important guidance on the design of HEPA filters for aerosol particles.

## Notes

### Acknowledgements

This work was supported by the National Key Research and Development Program (No. 2016YFC1201503), Natural Science Foundation of China (No. 21576206) and the Program for Changjiang Scholars and Innovative Research Team in University (No. IRT_15R46).

## Supplementary material

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