Lattice Boltzmann Simulation of Non-Steady-State Particulate Matter Filtration Process in Woven Fiber

Abstract

To enhance the design process of high-performance woven fibers, it is vital to clarify the evolution of particle dendrites, the dynamic pressure drop, and the capture efficiency with respect to dust loading during the non-steady-state filtration process. A general element (orthogonal elliptical fibers) of woven filter cloths is numerically simulated using the 3D lattice Boltzmann-cell automation (LB-CA) method, where gas dynamics is solved by the LB method while the solid particle motion is described by the CA probabilistic approach. The dendrite morphologies are evaluated under various particle diameters, aspect ratios, packing densities, and inlet fluid velocities. For submicron particles in the “Greenfield gap” range, it is revealed that the normalized pressure drop is an exponential function of the mass of deposited particles, and the rate of increase is exactly proportional to the perimeter of the elliptical fibers. Moreover, the normalized capture efficiency is a linear function of the deposited mass. It is not advisable to increase the packing density too much, as this might simply increase the pressure drop rather than enhancing the normalized capture efficiency. It is also worth noting that the fitting slope is more likely to grow linearly once the aspect ratio exceeds 1.6, indicating that orthogonal elliptical woven fibers offer higher capture efficiency than normal orthogonal cylindrical woven fibers. The work is beneficial to gain insights into the angular distribution of particle dendrites, as well as the prediction of dynamic growth of pressure drop and capture efficiency of the elliptical fiber. These efforts could help to deepen the understanding and realize assistant designing for the filtration performance of woven fiber in the future.

1. Introduction

Fibrous filtration is increasingly important in many fields, including coal-fired power plants, the manufacturing industry, dust respirators, mining engineering, and indoor air purification [1,2]. Fibrous filtration has the outstanding advantages of high capture efficiency of fine particles and relative simplicity, but the filtration process of suspended particles from the airflow is rather complicated due to the multiple capture mechanisms of solid particles (e.g., Brownian diffusion, interception, and inertial impaction) and complicated particle–flow–fiber interactions [3].

Classical fibrous filtration theories have produced many idealized and (semi-)empirical models for computing the pressure drop and filtration efficiency, dominated by the multiple capture mechanisms of clean media [4,5,6]. These theories have led to a successful description of the steady-state filtration process [7]. However, the steady-state process only occurs in the very early stages of filtration. In real fibrous filtration, fine particles rest on the fiber surface, changing the shape of the filtration area and thus influencing the flow fields. The dendrite-like clusters significantly affect the filtration efficiency, pressure drop, and growth process. Thomas et al. [8] developed a model that accurately describes the pressure drop during filter clogging by dividing the filter into various slices, while Zhao et al. [9] experimentally investigated both the pressure drop and capture efficiency in dust-loaded electrostatic filters by replacing the fiber diameter and packing density with equivalent values. Based on previous research, many numerical simulations [10,11,12,13] and experimental measurements [14,15,16] have been conducted to study the dynamic dust-loaded processes.

Many numerical simulations have investigated different arrangements of fibrous filters and most considered a single-fiber model or a uniform arrangement. The real fibers used in fibrous filtration can be divided into two types: non-woven filter cloth and synthetic woven filter cloth. The true fibrous filtration of real non-woven filters has been intensively studied in recent years [17,18,19,20]. Lehmann et al. [17] obtained information about the lifetime capture efficiency, pressure drop, and dust loading of a non-woven oil filter. Huang et al. [19] presented quantitative prediction formulas for the normalized pressure drop and normalized capture efficiency of a single fiber, finding that they were exponential and linear functions of the deposited mass, respectively. Anda et al. [21] proposed that there always existed a trade-off between maximizing filtration efficiency and pressure drop for filter cloth, and the measurement and prediction of pressure drop and capture efficiency for many different fabrics could significantly help to make good filters from fabrics. However, different from non-woven filter, there has been very few fundamental filtration models of woven filters, and the underlying connection between pressure drop, capture efficiency, and deposited particles is still unclear.

Currently, woven filter cloth is deemed to be one of the most effective air filtration media of all fabric filters due to its mature manufacturing process, high strength, and homogeneous fiber size [22]. As a response to the current COVID-19 pandemic, woven cloth materials have been examined in terms of the protection they provide against the transmission of particles of certain sizes, particularly in the “Greenfield gap” from 0.25–0.5 μm [23,24,25]. To date, numerous experimental researchers have investigated the performance of woven cloth in face masks [26,27,28,29,30]. Christopher et al. [26] measured 32 available mask cloth for protection from the COVID-19 virus, and proposed that filtration efficiency and pressure were very relevant to those of single layer. Wang et al. [28] investigated the filtration performance of carbon woven fabric on the removal of PM 1.0 particles. Konda et al. [29] tested various common fabrics for use as cloth masks and presented the filtration efficiency data as a function of aerosol particle size. These experimental studies have examined many macro factors, i.e., particle size, textile parameters, porosity, and electrostatics, which influence the use of the mask and the optimization of the fiber filter. Nevertheless, they ignored the impact of angular distribution of particle dendrites deposited on a fiber. During the dynamic filtration process, the captured particles form branched dendrites, change the flow fluid, and further significantly affect the performance of filtration media in the so-called fiber-dendrite collection stage [19,31]. Gaining the insights of dendrites growth is beneficial to the in-depth descriptions of the dust-loaded process and further prediction of filtration performance of woven fiber.

Many simulations have been successfully conducted to explore the dust-loaded process and provide some details and predictions for the filtration performance of filter cloth [32,33,34]. Pachner et al. [32] derived mathematical equations for estimating the initial pressure drop (neglecting the dust-loading process) of woven screens. Ling et al. [34] proposed a simple modeling approach for the nanoparticle capture efficiency for personal protective woven fiber during clean filtration process. However, they paid attention to the steady-state filtration process (based on classical filtration theories), and very few simulations examined the non-steady-state dust-loaded process and the changed flow field influenced by particle dendrites during the fiber-dendrite collection stage [35,36,37]. Hund et al. [35] used direct numerical simulation to investigate the filter resistance on the single fabric layer, and found that the resistance on the first particle layer shows a maximum as particle size corresponds to the pore size. Vernikovskaya et al. [36] developed a one-dimensional mathematical model for the cake growth on the woven cloth by simply assuming particles accumulated in height, mainly concentrated on the fiber and pore diameters. Lantermann and HäNel [37] simulated the 3D structure of particles deposited on the woven fiber of two orthogonal cylindrical fibers. The authors reported that the changing boundary conditions had almost no influence on the filtration process for smaller particles, although a slight influence was observed for larger particles. In fact, the mass loading on the fiber exhibits a crucial impact on the pressure drop and filtration efficiency. This difference might be because there were too few particles in that study to form large dendritic structures. The dendritic structures on the fiber and the connection between pressure drop, capture efficiency, and particles dendrites have rarely been investigated, which is to be solved in this work.

The dynamic evolution of the capture efficiency, pressure drop, and morphologies of particle dendrites is vital in the design and optimization of filters. The aim of this work is to reveal these inherent features of the particulate matter removal process by a general element (two orthogonal fibers) of woven filter cloth. However, the mechanisms of fibrous filtration are rather complex due to the multiple capture mechanisms of solid particles. Fortunately, the lattice Boltzmann-cellular automata (LB-CA) probabilistic model provides a very promising approach for determining these mechanisms [38,39]. The solid particles are constrained to only move on the same regular lattices as the fluid particles, and their transport probabilities to neighboring nodes depend on the local fluid flow and other external forces subject to solid particles [40]. The LB-CA models can consider the effect of Brownian diffusion and the drag force due to their stochastic nature in treating particle motion, and the interaction between gas–solid flow and unsteady geometric boundary, such as the captured particles dendrites, can be easily realized, which have been successfully used to simulate the dynamic filtration process of a single fiber and multi-fibers with parallel and staggered fiber arrangements [19,40,41].

As seen in Figure 1, the woven filter cloth consists of many fibers arranged using different weaving methods (e.g., plain weave, twill weave, and satin weave), and various weaving patterns are commonly characterized by the same micro-structure formed by two orthogonal fibers [28,42]. Therefore, two (single-layer) orthogonal fibers can be recognized as a general element of three kinds of woven filter cloths. Hence, we use 3D LB-CA simulations to examine the filtration process under various conditions and to investigate the formation of particle dendrites. Since the branched dendrites change the flow fluid and significantly affect the filtration performance in the fiber-dendrite collection stage, we first investigate the angular distribution of particle dendrites. Then, the impacts of aspect ratio and packing density of elliptical fibers on the non-steady-state filtration process are examined; it is then proposed the formula that the logarithmic scaling of pressure drop (ΔP/ΔP₀) and the normalized capture efficiency (η/η₀) are linear functions of the mass of deposited particles (M). The results will help to provide the insights into the underlying connection between pressure drop, capture efficiency, and deposited particles, and make more accurate predictions of the pressure drop and collection efficiency in the future.

The remainder of this paper is organized as follows. In Section 2, the basic theory of 3D LB-CA gas–solid models is briefly introduced. The simulation details, including the boundary conditions, particle properties, and related characteristic parameters, are presented in Section 2.3. Section 2.4 shows the model verification, and the formation processes of particle dendrites resting on the woven fibers are described in Section 3.1. Section 3.2 shows the impacts of the aspect ratio (from 1–3), packing density (from 2.5–10%), and inlet fluid velocity (0.1–0.3 m/s) on the filtration processes, while Section 3.3 shows a short discussion. Finally, the conclusions obtained from this study are listed in Section 4.

2. Materials and Methods

2.1. LB Model for Fluid Flow

The LB-CA model was originally developed for simulating the filtration of a single fiber, as described in the previous publication [38,40]. In this work, the D3Q15 lattice model was used to simulate 3D flows. The state of each grid was denoted by the fluid particle distribution function $f_i\left(x,t\right)$, and the lattice Bhatnagar–Gross–Krook propagation scheme can be calculated from the classic Boltzmann equations. The discrete LB equation is described as:

$$f_i \left(x+c_i·\Delta t,t+\Delta t\right)-f_i \left(x,t\right)=\left[f_i^{eq} \left(x,t\right)-f_i \left(x,t\right)\right]/\tau$$

(1)

where $f_i^{eq}\left(x,t\right)$ represents the equilibrium distribution function, $c_i$ is the lattice velocities in the D3Q15 model, $\Delta t$ represents the time step, and $\tau$ is the dimensionless relaxation time. By decomposing the above discrete LB equation, the fictitious fluid particles on regular lattices experience two sequential sub-steps:

$$\mathrm{Collision}:f_i{(x,t)}^{\prime }=f_i\left(x,t\right)+\left[f_i^{eq}\left(x,t\right)-f_i\left(x,t\right)\right]/\tau$$

(2)

$$\mathrm{Streaming}:f_i\left(x+c_i·\Delta t,t+\Delta t\right)=f_i{(x,t)}^{\prime }$$

(3)

where $f_i{(x,t)}^{\prime }$ represents the post-collision distribution function. Note that the collision step is an absolutely local behavior and thus conducive to parallel computing. The equilibrium distribution function is given by:

$$f_i^{eq}=\rho {\alpha }_i[1+\frac{c_i·u}{c_s^2}+\frac{1}{2}(\frac{c_i·u}{c_s^2}{)}^2-\frac{u}{2c_s^2}]$$

(4)

where u is the macroscopic inlet fluid velocity, $\rho$ is the fluid density, ${\alpha }_i$ is a weight coefficient, and $c_s^{ }$ is the local speed of sound. The macroscopic density $\rho$ and momentum fluxes $\rho u$ are calculated as [43]:

$$\rho ={{\displaystyle \sum }}_{i=0}^{Q-1}{f}_i; \rho u={{\displaystyle \sum }}_{i=0}^{Q-1}{f}_i{c}_i$$

(5)

2.2. CA Model for Particle Movement

In the CA model, the particles are constrained to move on the regular lattices as fluid particles, and the transport probabilities ($p_i$) to neighboring nodes depend on the local fluid flow and other external forces. Let N(x_p, t) be the number of simulation particles at site x_p and time t, and N(x_p, t) can take any nonnegative value. The key idea of CA probabilistic model is that each of the N(x, t) particles jumps to a neighboring node at site x_p + c_iΔt with a probability p_i, which is proportional to the projection of the actual displacement of the particle in the direction c_i. Based on the D3Q15 model, the probability of one fluid particle traveling in the six adjacent directions (north, south, west, east, front, and back) can be calculated as:

$$p_i=\max \left(0,\frac{\Delta {\mathit{x}}_{\mathrm{p}}·{\mathit{c}}_i}{\Delta x}\right) \left(i=1~6\right)$$

(6)

The final particle position after each time step Δt is determined as follows:

$$x_{\mathrm{p}}^{\mathit{n}+1}=x_{\mathrm{p}}^{\mathit{n}}+{\mu }_1{c}_1\Delta t+{\mu }_2{c}_2\Delta t+{\mu }_3{c}_3\Delta t+{\mu }_4{c}_4\Delta t+{\mu }_5{c}_5\Delta t+{\mu }_6{c}_6\Delta t$$

(7)

where ${\mu }_1$ is a Boolean variable that is equal to 1 with a probability $p_i$, and superscripts n and n+1 represent the present and next moment, respectively. For example, the three probabilities p₁, p₄, and p₅ are positive. Thus, we create three independent random numbers (r₁, r₂, and r₃) from a uniform distribution in the interval [0, 1]. If p₁ > r₁, p₄ > r₂, and p₅ > r₃, then μ₁, μ₂, and μ₃ are set to 1, and so the particle will travel to position x_p + c₁Δt + c₄Δt + c₅Δt. The rules for the other possible directions can be obtained using the same principle. Note that the actual displacement of particles $\Delta {\mathit{x}}_{\mathrm{p}}$ is still unknown. The particle motion can thus be calculated as:

$$\frac{d{u}_{\mathrm{p}}}{dt}=F_{\mathrm{D}}+F_{\mathrm{B}}=\frac{u-u_{\mathrm{p}}}{{\tau }_{\mathrm{p}}}+\mathsf{\xi }\sqrt{\frac{216\mu k_{\mathrm{B}}T}{\mathsf{\pi }{\rho }_{\mathrm{p}}^2{d}_{\mathrm{p}}^5\Delta t}}$$

(8)

$$\frac{d{x}_{\mathrm{p}}}{dt}=u_{\mathrm{p}}$$

(9)

where u_p represents the particle velocity, ${\tau }_{\mathrm{p}}$ is the relaxation time of the particle, τ_p = Ccρ_pd_p²/(18μ), Cc is the Cunningham slip correction factor, μ is the dynamic viscosity of the fluid, $F_{\mathrm{D}}$ is the drag force, $F_{\mathrm{B}}$ is the Brownian force, $\mathsf{\xi }$ is a Gaussian random number, $d_{\mathrm{p}}$ is the particle diameter, $k_{\mathrm{B}}$ is the Boltzmann constant, and T is the fluid temperature. In general, the conventional finite difference method is applied to determine the particle displacement, and the particle velocity and displacement are thus calculated by:

$$u_{\mathrm{p}}^{\mathit{n}+1}=u_{\mathrm{p}}^{\mathit{n}}·\exp (-\frac{\Delta \mathit{t}}{{\tau }_{\mathrm{p}}})+\left(u+F_{\mathrm{B}}·{\tau }_{\mathrm{p}}\right)·(1-\exp (-\frac{\Delta \mathit{t}}{{\tau }_{\mathrm{p}}}))$$

(10)

$$x_{\mathrm{p}}^{\mathit{n}+1}=x_{\mathrm{p}}^{\mathit{n}}+\left(u_{\mathrm{p}}^{\mathit{n}}-u\right)\left(1-\exp \left(-\frac{\Delta \mathit{t}}{{\tau }_{\mathrm{p}}}\right)\right){\tau }_{\mathrm{p}}+u\Delta t+\left(\Delta t-\left(1-\exp \left(-\frac{\Delta \mathit{t}}{{\tau }_{\mathrm{p}}}\right)\right){\tau }_{\mathrm{p}}\right)F_{\mathrm{B}}·{\tau }_{\mathrm{p}}$$

(11)

Therefore, the actual displacement of particle within Δt is obtained from Equation (11), and the final particle position after each time step Δt is determined from Equation (7).

2.3. Simulation Details

In this study, the grid resolution (x-y-z) of 3D simulations was set to 128 × 64 × 64, and the grid length (dx) was set to 1 μm [19]. The orthogonal elliptic fibers were placed in the central region of the calculation domain, and the particle-loaded suspension flow followed the positive direction of the x-axis, perpendicular to the fibers. The inlet was set as a constant velocity boundary, while the outlet was set as a fully developed boundary, i.e., ∂u/∂x = ∂v/∂x = 0; the other four boundaries were assumed to be periodic. Non-equilibrium extrapolation scheme was used here to easily deal with inlet and outlet boundary conditions. Schematic diagrams of the calculation domain are shown in Figure 2, and the parameters of six simulation models are listed in

Table 1. Parameters of the simulation models.

Model	Packing Density	Aspect Ratio	Equivalent Diameter (μm)	Long Axis (μm)	Short Axis (μm)	Perimeter (μm)
a	5%	1	22.8	22.8	22.8	71.7
b	5%	2	22.8	32.3	16.1	133.8
c	5%	3	22.8	39.6	13.2	177.0
d	2.5%	2	16.1	22.8	11.4	94.6
e	7.5%	2	28.0	39.6	19.8	163.8
f	10%	2	32.3	45.7	22.8	189.2

, in which the packing density (α) refers to the solid volume fraction and the aspect ratio (ε) is calculated by dividing the long axis by the short axis of an elliptic cylinder. Models a, b, and c were constructed using different aspect ratios to evaluate the impact of the fiber shape (aspect ratio), whereas models a, d, e, and f were designed to evaluate the impact of the fiber size (packing density). The volume fraction occupied by particles in the fluid was determined to be as low as 6.5 × 10⁻⁶, and the particle-particle interactions in the fluid were not considered in this study [44]. The time step (Δt) was 10⁻⁶ s. The kinematic viscosity (ν) of the fluid (air) was set to 1.6 × 10⁻⁵ m²/s, and the density of the fluid (ρ₀) and the particles (ρ_p) were set to 1 kg/m³ and 1000 kg/m³, respectively. Note that the Reynolds number (Re = u₀d_f/ν, where d_f is the equivalent length of long axis) ranged from 0.14 to 0.86, and so the flow was approximately a Stokes flow. The filtration process could be divided into two parts: the simulation of the gas fields and, once the gas fields have reached the stable state, the simulation of gas–particle fields following the injection of particles. During the filtration process, the shape of the filtration medium (containing fibers and deposited particles) changed continuously due to the deposited particles. Once a particle was deposited on the fiber surface, a fluid node was converted to a boundary node, and particles could still move onto fluid and boundary nodes. If the accumulation volume of captured particles achieved 60%, the boundary node became a solid node, which influenced the flow field. Therefore, the flow field would be recalculated if a new solid node was formed.

To determine the effects of the fiber parameters,

Table 2. Parameters of the simulation cases.

Case	Packing Density	Aspect Ratio	dp (μm)	uin (m/s)	Perimeter (μm)	Equivalent Diameter (μm)
1	5%	2	0.2	0.1	133.8	22.8
2	5%	2	0.4	0.1	133.8	22.8
3	5%	2	1	0.1	133.8	22.8
4	5%	2	2	0.1	133.8	22.8
5	5%	1	0.4	0.1	71.7	22.8
6	5%	3	0.4	0.1	177.0	22.8
7	2.5%	2	0.4	0.1	94.6	16.1
8	7.5%	2	0.4	0.1	163.8	28.0
9	10%	2	0.4	0.1	189.2	32.3
10	5%	2	0.4	0.2	133.8	22.8
11	5%	2	0.4	0.3	133.8	22.8

summarizes the 11 typical simulation cases of woven fibers considered in this study. The effects of different particle diameters on the continuous formation process of particle dendrites were first investigated (Cases 1–4), and then the effects of the dynamic pressure drop and capture efficiency were examined under diverse operating conditions: (a) different aspect ratios of elliptical fibers (Cases 2, 5, and 6); (b) different packing densities of elliptical fibers (Cases 2, 7–9); and (c) different inlet flow velocities of the fluidizing gas (Cases 2, 10, and 11). When examining the effect of a particular parameter, all other operational parameters were held fixed, thereby allowing the differences to be accurately distinguished. Furthermore, note that Case 2 is considered to be the default condition, and used mesh is shown in Figure 3a, while the clean flow field around the fibers is shown in Figure 3b–d.

2.4. Model Verification

Figure 4a shows the simulation of sensitivity analysis, which includes the logarithmic scaling of the pressure drop of dust-loaded elliptical fibers for the different grid partitions. It is found that the curve of 192 × 96 × 96 is similar with that of 128 × 64 × 64 (the error is less than 2%), indicating that the mesh used in this study was adequate. Then, we simulated the Brownian-diffusion-dominated collection processes under different flow rates and particle diameters [37], where particle size is comparatively small. The simulations take into account the Brownian random diffusion, the van der Waals forces, and the drag forces. The fluid velocity was set as 0.2126 m/s for 1.0 L/min, and the diameter of the fiber was set as 65 μm. The particles can randomly deposit after colliding and contacting with the woven fiber. As shown in Figure 4b, the simulation results well correspond with the results of Lantermann et al. [37], proving the reliability of the simulations.

3. Results and Discussion

3.1. Particle Dendrites of the Woven Fibers

Figure 5 illustrates the growth process of the dendrite structures (green spots) of Case 2 during the whole filtration process. The diameter of the particles was set to 0.4 μm, which is in the Greenfield gap, and the dominant capture mechanism was Brownian diffusion. The dendrite growth exhibits three stages in this dynamic process, similar to previous results [19]. In the initial stage of nearly “clean” fibers [from Figure 5a,b], the predominant diffusion leads to an isotropic distribution of particles around the surface of the fibers. As the number of captured particles increases [Figure 5c], the particle dendrites develop many branches with high fractal dimension distributed in the radial direction, leading to changes in the flow field. Finally, the particle dendrites become so large that more and more particles become prone to being deposited on either end of the elliptical long axis [Figure 5d]. Additionally, the shape and number of dendrite-like clusters of two fibers are apparently analogous, and the fluid particles struggle to migrate to the leeward side of the fibers during the dust loading process.

The particle dendrites are dependent on the particle size. Figure 6 exhibits the morphologies (five views) of the dendrite structures for particles of different diameters (0.2–2 μm). The dendrite structures resting on the woven fibers are similar to those on a single fiber for particles in the Greenfield gap. Generally, the dominant capture mechanism is diffusion when the particle diameter is less than 0.5 μm [31]. As shown in Figure 6a,b, in terms of small particles (i.e., d_p = 0.2 μm or 0.4 μm), the particles are relatively uniformly distributed and rest on any part of the fiber surface. With rising particle diameter, the diffusion mechanism becomes weaker, and hence interception and inertia impaction become more significant, resulting in an increasing number of particles being deposited on the lateral sides and front sides of the fibers. When the particle diameter reaches 1 μm, the Brownian force is rather weak, and the structures of particle deposition are a combination of the dendrites expected from interception particles [Figure 6c]. This means that particles with a larger size led to a less uniform angular distribution. As for the largest particles (d_p = 2 μm), which are in the inertial impaction dominant regime, most are deposited on the windward side of the fiber, producing slender clusters that grow in the opposite direction to the flow [Figure 6d]. This might be because large particles do not fully comply with the streamlines and instead tend to move in a straight path, and so relatively few particles of large size will reach the rear side of the horizontal fiber.

3.2. Factors That Impact the Filtration Process

3.2.1. Aspect Ratio

To determine the effect of the aspect ratio of elliptical fibers, particle sizes in the range of the Greenfield gap were chosen. In the case of small particles (d_p = 0.4 μm), Cases 5 (ԑ = 1), 2 (ԑ = 2), and 6 (ԑ = 3) are considered. As shown in Figure 7, the particle dendrites of these three cases possess the same mass load per unit area of woven fibers, consisting of 36,508, 53,504, and 70,712 particles on the surfaces, respectively. Cylindrical fibers have an aspect ratio of exactly 1. With regard to such fibers, submicron particles can be deposited on any side, irrespective of the front or back arrangement of two fibers. As the aspect ratio increases, an increasing number of particles are liable to be deposited on the fiber’s lateral and front sides (as indicated by Figure 7b,c), and the branched dendrites grow larger, indicating that the interception and impaction become more important as the cross-sectional area of the fiber media increases.

Both the pressure drop and capture efficiency are known to be significant in the filtration process, and these two parameters vary dynamically during non-steady-state filtration. Huang et al. [19] proposed exponential and linear functions of the deposited mass to describe the normalized pressure drop (ΔP/ΔP₀) and normalized capture efficiency (η/η₀) of non-woven fibers, respectively. As depicted in Figure 8, similar patterns can be observed in the woven fibers. As shown in Figure 8a, there is an exponential relationship between the normalized pressure drop and the total deposited mass; this can be written as ln(ΔP/ΔP₀) = kM + b. The fitting parameters listed in

Table 3. Fitting parameters of the simulation cases.

Case	Packing Density	Aspect Ratio	k	b	λ	γ
5	5%	1	0.88	−0.32	0.19	1.00
2	5%	2	3.36	−1.02	0.53	1.00
6	5%	3	4.94	−0.82	1.02	1.01
7	2.5%	2	1.93	−0.52	0.36	0.99
8	7.5%	2	4.53	−1.43	0.57	1.00
9	10%	2	6.78	−2.31	0.58	1.01
10	5%	2	3.37	−1.01	0.53	1.00
11	5%	2	3.33	−1.04	0.53	1.00

produce coefficients of determination (R²) greater than 0.997. The normalized pressure drop ΔP/ΔP₀ can be modeled as ln(ΔP/ΔP₀) = 0.88M − 0.32 (Case 5), ln(ΔP/ΔP₀) = 3.36M − 1.02 (Case 2), and ln(ΔP/ΔP₀) = 4.94M − 2.31 (Case 6). The cylindrical fibers give the lowest rate of increase in ΔP/ΔP₀, and the rate of increase in ΔP/ΔP₀ is positively correlated with the fiber shape (aspect ratio). This is due to the significant growth of branched dendrites on the fiber’s lateral sides (as indicated by the front view in Figure 7). Namely, the slope k is linearly dependent on the perimeter of the fiber cross-sectional area, and the linear relation has the form k = 0.0384perimeter − 1.7885 (71.7 < perimeter < 177), where R² > 0.995, which mainly comes from the remarkable reduction in fluid cross-sectional area.

As shown in Figure 8b, during the initial dust-loading process, the rate of increase in the normalized capture efficiency (η/η₀) decreases as the mass load (M) increases and then reaches a constant value (a linear rate of increase), which is similar to the conclusions of previous studies [19]. The normalized capture efficiency of the woven fibers can be expressed as η/η₀ = λM + γ, in which a larger value of λ represents a faster rate of increase in the non-steady-state capture efficiency. Table 3 suggests that the normalized efficiency can be modeled as η/η₀ = 0.19M + 1.00 (Case 5), η/η₀ = 0.53M + 1.00 (Case 2), and η/η₀ = 1.02M + 1.01 (Case 6). Apparently, the rate of increase (λ) of elliptical fibers is somewhat higher than that of cylindrical fibers, indicating that elliptical woven fibers have a higher capture efficiency than normal cylindrical woven fibers. To accurately determine the relationship between the rate of increase (λ) and the fiber shape (aspect ratio), we performed multiple computations with 11 aspect ratios from 1–3 and present the slope λ and the error bars in Figure 9. Note that λ tends to grow slowly as the aspect ratio increases from 1 to 1.6. Once the aspect ratio exceeds 1.6, however, λ displays a fixed linear growth. Actually, this linear rate can be fitted as λ = 0.50ԑ − 0.47, where R² > 0.997. This expression might be useful in determining effective mathematical equations for predicting the dynamic capture efficiency during the dust-loading process.

3.2.2. Packing Density

Figure 10 depicts the dendrite morphologies of these four cases under the same mass load per unit area. The predominant diffusion leads to an isotropic distribution of particles around the fiber surface. However, it is apparent that for a fixed fiber shape (aspect ratio), the dendrite structures exhibit similar morphologies for all packing densities. This suggests that the fiber shape (aspect ratio) plays a more significant role in the growth of dendrite structures, while the fiber size (packing density) has no significant impact.

Figure 11a also indicates an exponential relationship between ΔP/ΔP₀ and M, and fibers with a high packing density produce a sharp increase in ΔP/ΔP₀. That is, there is a strong positive correlation between the slope k and the fiber perimeter, and the linear equation (k = 0.0384perimeter − 1.7885) again describes the slope k, except in Case 9, which might be because the perimeter is too large and beyond the scope of application (71.7 < perimeter < 177).

During the dust-loading process, the deposited particles more easily become part of the filtration medium when there is a higher packing density, and then the fibers’ surface geometry changes due to the larger cross-sectional area. Namely, the capture efficiency rises as the particle capture mechanism proceeds. As shown in Figure 12, λ cannot continue to rise as the packing density increases, indicating that there must be a critical maximum of λ. Specifically, as the packing density increases from 2.5–6%, the non-steady-state capture efficiency increases relatively rapidly, reaching a critical maximum (approximately 0.58) when the packing density is greater than 6%. This implies that, in the design of filtration media, it is not advisable to increase the packing density too much, as this will increase the pressure drop, rather than enhance the normalized capture efficiency.

3.2.3. Inlet Fluid Velocity

Three inlet fluid velocities were examined (Case 10: u_in = 0.1 m/s, Case 2: u_in = 0.2 m/s, and Case 11: u_in = 0.3 m/s). As shown in Figure 13, the three kinds of woven fibers exhibit almost the same patterns of deposited particles (namely, the same mass load per unit area), and the front views show that slightly more particulate matter is deposited on the windward side of the orthogonal fibers with a higher inlet fluid velocity. This might be because the increased fluid velocity greatly improves the drag force of the particle–flow–fiber interactions, weakening the diffusion ability of fine particles and strengthening the particle deposition ability of fibers to different extents.

Table 3 implies that the parameter k is positively related to the fiber perimeter, and the rate of increase in the normalized pressure drop is independent of the inlet fluid velocity. In addition, the front views in Figure 13 indicate that fibers subjected to a lower inlet fluid velocity have slightly more particulate matter deposited on their lateral side, producing significant branched dendrites, which might well explain the slight differences in the slope k (3.33–3.37) between the three cases. Figure 14b and Table 3 demonstrate that the normalized efficiency of the three cases can be written as η/η₀ = 0.53M + 1.00, indicating that the inlet fluid velocity has no impact on the rate of increase in the normalized capture efficiency. Nevertheless, it must be noted that the clean capture efficiency is rather different and decreases as the inlet fluid velocity increases, and so the non-steady-state capture efficiency is only related to the initial capture efficiency during the particulate matter filtration process.

3.3. Discussions

The impacts of the particle sizes (from 0.2–2 μm), aspect ratio (from 1–3), packing density (from 2.5–10%), and inlet fluid velocity (0.1–0.3 m/s) are well determined during the fiber-dendrite collection processes. Figure 15 shows the number distribution of deposited particles on the leeward and windward sides of fiber surfaces under 11 different cases (Table 2). It is found that with the increase in particle diameter, more particles are deposited on the windward sides of fiber surfaces (Cases 1–4). It is because the morphologies of particle dendrites depend on the dominant filtration mechanism [19]. Brownian diffusion is important to the dendrites structure of fine particles (i.e., d_p = 0.2 μm, windward particle proportion is 0.55), while the interception effect is enhanced for larger particles (i.e., d_p = 2 μm, windward particle proportion is 0.70). As for the different aspect ratios of elliptical fibers (Cases 2, 5, and 6), it can be seen in Figure 15 that fine particles are peculiarly prone to collide and make contact with the woven fibers with higher aspect ratio. It is in consistent with the particle distribution in Figure 7 that more particles deposit on the end of elliptical long axis. The growth of dendrites structures significantly decreases the cross-sectional area of fiber media, leading to the rapid increase in pressure drop and capture efficiency during the fiber-dendrite collection stage. In terms of the woven fibers with same aspect ratio but different packing densities (Cases 2, 7–9), the windward particle proportion is slightly increased from 0.57 to 0.62 due to the decrease in cross-sectional area of the fiber media. Figure 10 shows that dendrites structures on windward sides are nearly the same, which cannot significantly change the flow field. The higher slope k of fibers with larger packing density mainly comes from the remarkable reduction in fluid cross-sectional area as dendrites grow. Finally, the similar particle distributions under different inlet flow velocities of the fluidizing gas (Cases 2, 10, and 11) well explain the same trend of dynamic pressure drop and capture efficiency in Section 3.2.3.

It can be concluded that the normalized pressure drop (ΔP/ΔP₀) is an exponential function of the mass of deposited particles (M), taking the form ln(ΔP/ΔP₀) = kM + b, which is in agreement with the results of single fiber [19]. Our results imply that the slope k is positively correlated to the fiber perimeter but is independent of the inlet fluid velocity. That is, the fiber shape and size have a vital impact on the normalized pressure drop, and so woven fibers with a larger aspect ratio or packing density will lead to a higher rate of increase in the normalized pressure drop. This can be explained by the reduced cross-sectional area of flow fluid, which is caused by different particle dendrites on the woven fiber. As for the capture efficiency, it is found that the normalized capture efficiency (η/η₀) is a linear function of M that can be expressed as η/η₀ = λM + γ. Fibers with a larger aspect ratio or packing density will lead to a larger rate of increase in the normalized collection efficiency, whereas the inlet fluid velocity again has no obvious impact. Note that the slope λ cannot keep rising as the packing density increases, and so it is not advisable to increase the packing density too much in the design of filtration media. Furthermore, once the aspect ratio of woven fibers exceeds 1.6, λ becomes more likely to grow linearly. Figure 15 shows that the windward particle proportions for case 2, 5, and 6 are 0.54, 0.57, and 0.63, respectively. Different increments indicate that particles are easily deposited on the windward sides as aspect ratio of woven fibers is low, and λ is able to grow rapidly due to the change in windward area (i.e., more than 1.6). This might be helpful in determining effective mathematical equations for predicting the dynamic filtration performance of real woven fibers.

Finally, particular attention in this paper is mainly paid on the angular distribution of particle dendrites as well as the dynamic growth of the increase rate of pressure drop and capture efficiency, and future work could look at different fiber sizes, particle sizes, and fiber orientations, as well as the mathematical formula between single layer and multi-layer arrangement. These efforts could be beneficial to the understanding and aided design for the filtration performance of woven fiber.

4. Conclusions

In this work, the filtration processes of a general element (two orthogonal elliptical fibers) of woven filters have been quantitatively evaluated using the LB-CA model. The formation and growth of dendrites under different conditions were explored in detail. We have investigated the dynamic growth of pressure drop and capture efficiency, and the universal mathematical formula adopted for more conditions might need further investigations. The major conclusions are as follows:

• The particle diameter and fiber shape (aspect ratio) have a significant influence on the angular distribution of particle dendrites, whereas the fiber size (packing density) has no obvious impact on the dendrite morphologies of submicron particles.
• The normalized pressure drop (ΔP/ΔP₀) is an exponential function of the mass of deposited particles (M), taking the form ln(ΔP/ΔP₀) = kM + b, where k is positively correlated to the fiber perimeter. Woven fibers with a larger aspect ratio or packing density will lead to a higher rate of increase in the normalized pressure drop.
• The normalized capture efficiency (η/η₀) is a linear function of the mass of deposited particles, which can be expressed as η/η₀ = λM + γ.
• The slope λ cannot keep rising as the packing density increases, and so it is not advisable to increase the packing density too much in the design of filtration media.