J. Shanghai Jiao Tong Univ. (Sci.), 2020 https://doi.org/10.1007/s12204-020-2251-7 Simulation of Bimodal Fiber Distribution Effect on Transient Accumulation of Particles During Filtration AKAMPUMUZA Obed1,3, WU Jiajun1 (���), QUAN Zhenzhen1,2 (���), QIN Xiaohong1∗ (���) (1. College of Textiles, Donghua University, Shanghai 201620, China; 2. Innovation Center for Textile Science and Technology, Donghua University, Shanghai 201620, China; 3. Uganda Industrial Research Institute, P.O. Box 7086, Kampala, Uganda) © Shanghai Jiao Tong University and Springer-Verlag GmbH Germany, part of Springer Nature 2020 Abstract: Modeling has become phenomenal in developing new products. In the case of filters, one of the most applied procedures is via the construction of idealized physical computational models bearing close semblance to real filter media. It is upon these that multi-physics tools were applied to analyze the flow of fluid and the resulting typical performance parameters. In this work, two 3D filter membranes were constructed with MATLAB; one had a random distribution of unimodal nanofibers, and the other, a novel modification, formed a bimodal distribution; both of them had similar dimensions and solid volume fractions. A comparison of their performance in a dust-loading environment was made by using computational fluid dynamic-discrete element method (CFD-DEM) coupling technique in STAR-CCM+. It was found that the bimodal nanofiber membrane greatly improved the particle capture efficiency. Whereas this increased the pressure drop, the gain was not too significant. Thus, overall, the results of the figure of merit proved that adopting a bimodal formation improved the filter’s quality. Key words: 3D virtual filters, computational fluid dynamic (CFD), discrete element method (DEM), coupling simulation, dust loading, STAR-CCM+ CLC number: TQ 021.1 Document code: A 0 Introduction Filtration is a multi-phase process in which the com- ponent phases interact in an intricate fashion. Parti- cles are deposited on the porous filter structure whereas the fluid that carries them there goes on to permeate through the filter’s pore network. During the initial stage of this process (also known as the clean filter phase), the quantity of the deposited particulate mat- ter is not too significant to affect the filter’s subsequent performance behavior. Due to the relative simplicity of this stage, most classical filtration theories were derived based on it[1-4]. However, filters deployed in their natu- ral operating environments experience a sustained flow of contaminant-laced fluids which builds the chains of particulate matter on their surfaces; this process was Received: 2019-12-12 Accepted: 2020-01-05 Foundation item: the Chang Jiang Youth Scholars Program of China (No. 51773037), the National Natural Science Foundation of China (Nos. 51803023 and 61771123), the Shanghai Sailing Program (No. 18YF1400400), the China Postdoctoral Science Foundation (No. 2018M640317), and the Funda- mental Research Funds for the Central Universities (No. 2232018A3-11) ∗E-mail: xhqin@dhu.edu.cn first observed by Watson[5]. Taking a case of simple air filters, the sustained exposure to dust laden air will have it accumulating mostly on fibers at the upstream side where various adhesion/cohesion mechanisms keep them mutually bonded together or onto the fibers to form clusters[6-9]. For ordinary electrically neutral fine aerosols/dust particles of less than 50 μm in diameter, Bowling[10] noted that van der Waals forces remain their primary force of adhesion onto dry surfaces. The availability of particle structures on the filter surface and the severity of their packing density greatly affect the filter’s subsequent performance raising both its in- herent pressure drop and efficiency[11-13]. The use of idealized physical models in which the fluid flow process is solved by using algorithms helps to boost experiments especially in instances where certain variables are too complex to deal with experimentally[14]. In this work, two 3D filter mem- branes dissimilar in their internal microstructures but of the same dimensions and solid volume fraction (SVF) were fabricated by using MATLAB software. Then, a comparison of their performance in a dust-loading environment was made by using computational fluid dynamic-discrete element method (CFD-DEM) cou- pling technique. J. Shanghai Jiao Tong Univ. (Sci.), 2020 1 Model Selection 1.1 Theory of Dendrite Formation In a transient filtration regime, the particles will stick to each other. When this process goes on for a while, this kind of interaction results in the particle clusters that have independent physical properties, which affect the general properties and performance of the filter[15]. An example of these is the filter packing density. Ac- cordingly, Thomas et al.[16] derived an expression that defines its heavy clogging regime by considering the mass of all the collected particles (Mp) on the filter’s area (Af): αd = Mp Afρpzd , (1) where, αd is the packing density of the deposited par- ticles; ρp is the particle density; zd is the total density of the particle deposit. Kanaoka and Hiragi[17] indicated that the particles do not accumulate homogeneously on the fibers. In which regard, they established a function for the di- mensionless volume of collected particles (V ′): V ′ = 4MLF πρpd2 f , (2) where, MLF is the mass of the collected particles per unit length of the fiber; df is the fiber diameter. Also, Juda and Chrosciel[18] deduced a mathematical expression for the loaded filter’s characteristic packing density (β) as a function of clean filter packing density (β0), its volume (Vf) and V ′: β = β0 + V ′ Vf = β0 ( 1 + V ′ β0Vf ) . (3) On the other hand, Thomas et al.[19] did an exper- iment using particles of diameters: dp = 2.6 μm and dp = 0.15 μm as shown in Fig. 1(a). It was determined that at some point, these particles completely clogged the filter. Japuntich et al.[20] demonstrated that the ag- glomeration of particles as a result of particles loading on the filter surface, reduced the size of the available pores in the filter as shown in Fig. 1(b). This in turn increased the hydrodynamic drag experienced by the fluid as it went through the filter leading to a dynamic increase in pressure drop[21]. According to Bergman et al.[22], the dynamic loading of particles on the filter raised its pressure drop by a factor ΔPp: ΔP = ΔP0 + ΔPp, (4) where, ΔP is the overall pressure drop across a particle laden filter; ΔP0 is the pressure drop across a clean filter. In the same way, the increasing volume of particles piling up on the filter surface will offer a cumulative availability of surface upon which those particles that are still airborne can deposit. Accordingly, Kanaoka et al.[23] observed that the capture efficiency rising with the dynamic particle accumulation could be given by ηm/η0 = 1 + λm, (5) (a) Level of clogging attained with particles of diameter: dp=2.6 μm and dp=0.15 μm Fiber 1 A pore between two fibers (b) How the transient loading of particles reduces the available pore sizes of the membrane Flow Pa rti cle s Particles Fiber 2 Fig. 1 Effect of transient particle loading on filter properties J. Shanghai Jiao Tong Univ. (Sci.), 2020 where, ηm is the collection efficiency of a particle loaded fiber; η0 is the collection efficiency of a clean fiber; λm is the loaded filter coefficient (collection efficiency raising factor). 1.2 CFD-DEM Coupled Process Fluid and particle coupled systems consist of the con- tinuous and the discrete phases flowing together, and the Eulerian-Lagrangian approach was used in model- ing the transient accumulation of particulate matter. The continuous phase is characterized by pressure, ve- locity, and temperature while the discrete phase is de- fined by particulate concentration, velocity, tempera- ture and position. 1.3 Computational Fluid Dynamics (CFD) The above properties of the fluid streams were consol- idated into mass, momentum and energy conservation, and the Navier-Stokes governing equations were used for evaluation. The flow was transient and laminar. ∇ · U0 = 0, (6) ∂U0 ∂t + u0 · ∇U0 = ∇(p/ρ) + v∇ · ∇U0, (7) where, ∇ is the divergence operator; U0 is the superfi- cial velocity; t is the time; u0 is the fluid’s initial veloc- ity; p is the fluid pressure; ρ is the fluid density; v is the overall fluid velocity. 1.4 Discrete Element Method (DEM) The DEM used in the modeling of particle flows enables the tracking of individual particle motions, particle-particle interaction and fluid-particle interac- tion. Introduced by Cundall and Strack[24], it can con- veniently handle the particle dynamics of all shapes and sizes. Normally, the active forces depend on the particle characteristics and the carrier fluid properties, which can be categorized into fluid-particle external forces (fext) and particle-particle contact forces (fcon), and harmonized into one function by using the Newton’s second equation of motion. mp d2x dt2 = |fcon + fext| , (8) where, mp is the mass of a single particle; d2x/dt2 is the second derivative of the particle path with respect to time. DEM falls into two physics classifications, that is, soft sphere and hard sphere formulations. In the soft sphere method, bodies are considered deformable dur- ing collision, and deformation is taken into account via an overlap in force-displacement models. Forces acting between the two spheres are decomposed into normal and tangential components. In the hard sphere formu- lation, bodies are considered rigid with collisions taken instantaneously to register no deformations during the collision process. Soft sphere approach allows for multi- ple contacts and can be used to duly represent particle agglomerations experienced with particle loading. In this work, the major focus is on the agglomeration of particles on a filter surface, and soft sphere formula- tion is solved by a nonlinear viscoelastic model shown in Fig. 2 (fdiss is the viscous force following a collision; Particle i Particle j Initial contact point (a) Force-displacement model for contact between particles (b) Tangential particle contact process (c) Particle rolling (d) Particle sliding (e) Particle twisting δt δ n fdiss fel(k) ri rj Particle i Particle j ηs n Fig. 2 Particle-particle interactions J. Shanghai Jiao Tong Univ. (Sci.), 2020 fel(k) is the elastic collision force; k is the spring stiff- ness; ηs is the damping coefficient of the dashpot; n is the unit vector normal to the contact plane and point- ing from particle i towards particle j; ri is the radius of particle i; rj is the radius of particle j; δn is the overlap of the particles in the normal direction; δt is the overlap of the particles in the tangential direction). 2 Simulation 2.1 Geometry Design and Computational Do- main Set-up Cylinders have finite dimensions such as length and diameter, and bear a close semblance to fibers. This makes it practical to use them in the precise repre- sentation and computation of various filter parameters for instance packing density, thickness and solid vol- ume fraction. In this work, two different virtual filter membranes were built from a network of 3D cylinders generated via a MATLAB code. The first membrane was built from uniform fibers (1 μm in diameter) while in the second membrane, fibers that constituted 24% of its total SVF were reduced threefold to a diameter of 0.33μm. Thus, a bimodal structure was created in which majority of the fibers were 1 μm in diameter. Both of these structures were constructed by deposit- ing cylinders randomly along the x- and y-axes which allowed the membrane thicknesses to grow along the z- axis thus occupying a 3D space of dimensions (50 μm × 50 μm × 8 μm). Despite the internal microstructure of the membranes differing on the basis of fiber diameter, their thicknesses and SVFs were maintained uniform for both. The SVF was determined by summing up the entire volume of cylinders (fibers) with the respective fiber and membrane volumes given by Vcyl = πr2 f L, (9) Vmod = Vcyl1 + Vcyl2 + · · · + Vcyln , (10) where, Vcyl is the cylinder volume; Vmod is the total volume of all the fibers in the membrane; Vcyl1 , Vcyl2 , Vcyl3 and Vcyln are the volumes of respective cylinders in the membrane; L is the length of each cylinder; n is the number of cylinders in the membrane; rf is the cylinder radius. Uniform fibers of diameter (df = 1 μm) were used in the unimodal membrane shown in Fig. 3(a). Whereas, the bimodal membrane in Fig. 3(b) was a mixture of two fiber diameters (df1 = 1 μm and df2 = 0.333 μm). Both membranes had the same filter porosity (ε = 0.86). Before modeling them, appropriate computa- tional domains were set up as enclosures within which the membranes were subjected to an uninterrupted flow of air. Similar domains were used for both models and these had dimensions of 52 μm × 52 μm × 30 μm as shown in Figs. 3(c) and 3(d). The air volume that ex- tended beyond the upstream and downstream sides of the filter acted as the buffer zone to help establish the flow on both the upstream and downstream sides of the filter. Rebäı et al.[25] noted that this buffer zone ought to be big enough to avoid interactions between the inlet and the outlet. The boundary conditions were a velocity inlet, a pres- sure outlet, symmetry sides and solid walls for the filter. Symmetry sides meant that the shear stress on these surfaces was zero. (a) Unimodal nanofiber membrane (b) Bimodal nanofiber membrane (c) Unimodal membrane within its computational domain df=1 μm ε=0.86 df1=1 μm df2=0.333 μm ε=0.86 (d) Bimodal membrane within its computational domain Domain outlet 30 µ m Domain inlet 52 μm Domain inlet Domain outlet 52 μm 30 μ m y zx O Fig. 3 Membrane models and computational domains of unimodal and bimodal nanofibers J. Shanghai Jiao Tong Univ. (Sci.), 2020 2.2 Meshing Process and Mesh Independence Test In simulation, meshing helps in geometry discretiza- tion, and its fineness influences both the precision of the final result and the calculation’s computational bur- den. Accordingly, a polyhedral mesh was administered and this aptly controlled the distribution of grid points, giving them a larger and sparse grid in the bulk of the domain where there is an uninterrupted flow while the mesh was denser and more clustered near the walls (Fig. 4). In addition, a “mesh independence test” tech- nique was applied to strike a balance between the ac- curacy and the computing resource requirement. (a) Polyhedral mesh in 2D (b) Polyhedral mesh in 3D Fig. 4 Details of mesh used on both membranes (represen- tation was done by using bimodal membrane) In this case, Ogorodnikov’s derivation[26] for the pres- sure drop (ΔP ) across a filter was used to study the mesh’s element size significance on the pressure drop results. ΔP = 4μU0αH r2 f (−0.5 lnα − 0.5) , (11) where, μ is the fluid viscosity; H is the membrane thick- ness. A unimodal nanofiber membrane structure is very close to what had been used in the equation’s formu- lation, therefore, it was adopted for the validation pro- cess. Accordingly, the following values were chosen for the variables involved: μ = 1.855 × 10−5, α = 01379, H = 8 μm, rf = 5 μm, U0 = 0.1, 0.2, 0.3, 0.4, 0.5m/s. The corresponding values of pressure drop were given in Table 1. Table 1 Face velocity and corresponding pressure drop from empirical formula Velocity/(m · s−1) ΔP/Pa 0.1 66.739 0.2 133.478 0.3 200.217 0.4 266.955 0.5 333.694 Pressure drop results corresponding with the num- ber of cells achieved at each meshing regime were jux- taposed with the pressure drop calculated using the empirical function in Eq. (9). Mesh settings used to achieve the minimum error between these two were then selected as presented in Table 2. Table 2 Results of mesh independence test to op- timize simulation results Test Cells Smallest cell/μm Simulation ΔP/Pa Empirical ΔP/Pa Error/Pa 1 343 865 0.45 267.402 333.694 66.294 2 451 131 0.39 297.356 333.694 36.338 3 657 176 0.27 315.000 333.694 18.694 4 680 659 0.25 322.790 333.694 10.904 5 1 224 476 0.21 330.000 333.694 3.694 2.3 Simulation of Fluid-Particle Coupling and Particle Adhesion Processes Coupling between these two involves an interactive exchange between their components during the flow. In a time-step dependent simulation, the results obtained in the current time-step become the starting point for the values at the next stage which helps to keep track of each individual particle’s instantaneous flow path. The CFD provides the fluid flow profiles over the fluid cells allowing for the tracking of particle flow patterns until the deposition. Spherical elastic particles considered here were introduced into the flow by a part injector located near the domain’s inlet, and the settings given in Table 3 were utilized in simulating the CFD-DEM coupling process. The density of the virtual fibrous materials corre- sponded with that of polyacrylonitrile (PAN), which is one of the most commonly used polymers in the production of electrospun nanofibrous membranes for air filtration. On the other hand, the properties of the simulation process such as memory requirements speed greatly rely on the elastic modulus of the par- ticles, where a very high elastic modulus often results in computationally expensive simulations. Accordingly, Hærvig et al.[27] found a way to ease on the time and resource requirements for DEM simulation by modify- ing the surface energy density (γ) of particles in the J. Shanghai Jiao Tong Univ. (Sci.), 2020 Table 3 Settings utilized in CFD-DEM coupling simulation CFD DEM Model dimensions 3D Particle type DEM particles Time Implicit unsteady Shape Spherical Material Gas Particle diameter/μm 0.5 Flow Coupled Material Solid Equation of state Constant density Equation of state Constant density Viscous regime Laminar Particle forces Drag, shear lift and spin lift forces Face velocity/(m · s−1) 0.5 Model 2-way coupling adhesive Johnson-Kendall-Roberts (JKR) contact and rolling model: γmod = γ Emod E , (12) where, γmod is the modified surface energy density of the particles; Emod is the modified elastic modulus; E is the normal elastic modulus. With this change, the stick/rebound remained the same but the collision process took place over longer time periods allowing for higher time step sizes. The fluid forces on the particles are dominated by the drag force (Fd) resulting from the relative velocities between the dispersed species and the fluid streamlines: Fd = vf − vp τp . (13) Schiller Naumann drag correlation (Cd) was deemed the most applicable for this coupling regime: Fd = mp(vf − vp) τpCd , (14) Cd = ⎧⎪⎨ ⎪⎩ 24 Rep (1 + 0.15Re0.687 p ), Rep � 103 0.44, Rep > 103 , (15) where, τp is the time required for the particle to re- spond to change in fluid velocity; vf is the fluid velocity; vp is the particle velocity; Rep represents the particle Reynolds’s number. The JKR particle adhesion model was adopted to account for the surface energy driven cohesive force re- sponsible for holding the particles together and onto fibers after collision. Fcohesion = RminWcohπF, (16) where, Rmin is the minimal radius of the surfaces in contact; Wcoh is the work of cohesion, J/m2; F is a multiplication model-blending factor with a value of 1.5 for the JKR model. Accordingly, the choice of the most appropriate work of cohesion value was guided by the Tabor factor: λT = ( 4Rγ2 E2D3 min )1/3 , (17) where, R is the effective particle radius; Dmin is the minimum separation distance between particles. Fibers were assigned a density of 1 190 kg/m3, an elastic modulus of 3 GPa, a Poisson’s ratio of 0.4, whereas particles were assigned a density of 1 500 kg/m3, a Poisson’s ratio of 0.3 and an elastic mod- ulus of 2×107 Pa. These settings were very instrumen- tal in particle-particle and particle-wall interactions. Overall, a work of cohesion of 0.000 5 J/m2 was also adopted. It should be noted that particle-particle and particle- wall interactions are characterized by the friction and the coefficient of restitution. A coefficient of restitution of 0.3 was used for both the impact of particles against particles and the particle-fiber impacts. Likewise, the friction played a role in the behavior of the particles during impact and adhesion processes. Thus, the values of 0.154 and 0.1 were adopted for the coefficients of static friction and rolling friction, respectively. 3 Results and Discussion A number of factors including particle phase, parti- cle size, and quantity of the deposited particles will in- fluence the precise pattern and the morphology of the dendrites. In this work, the fluid flow velocity and the architecture of the filter’s internal structure were con- centrated on and utilized to compare the performance of the two membranes. 3.1 Particle Deposition Visualization Filters comprise of several fiber layers, and each of them can potentially participate in capturing particles. However, in this case, the uppermost layer on the up- stream side of the flow played a dominant role in trap- ping the oncoming particles, and as a result, dendrites were predominantly formed here (Fig. 5). Relatively fewer particles penetrated deeper within the membrane. Two phenomenal explanations can be given for this pat- tern: in some instances, the impact of particles with fibers in the top most layer could have been too weak to reduce their kinetic energy and bring them to a stop, whereby they continued ahead to deposit deeper within the membrane; another reason could have been that, particles followed the fluid flow streamlines through the pores to deposit on those fibers in the heart of the J. Shanghai Jiao Tong Univ. (Sci.), 2020 (a) Surface deposits on the unimodal membrane and the details of particle-particle and particle-fiber contacts (b) Surface deposits on the bimodal membrane and the details of particle-particle and particle-fiber contacts Fig. 5 Particle dendrites formation on the filter membrane and the fluid streamlines outlining the disruption caused by this formation on the fluid flow membrane. These two points could also stand for par- ticles that penetrated through the membrane. Parti- cles were also observed to deposit mainly in the areas around the bigger membrane pores. This could be at- tributed to the conventional filter membrane’s fabrica- tion process, where fibers are randomly distributed to give an uneven pore distribution. Thus, the fluid flow through them tends to have more flux in bigger pores. Since fluid streamlines carry particles, deposition will be more concentrated in the areas of higher flow flux. All particle-particle interactions took place after depo- sition, and footprints of soft sphere contact are evident in the resulting particle contacts (Fig. 5). In all, the initial simulation stages had fewer particle- particle contacts compared to the particle-wall con- tacts. This demonstrates a dominant role that the fibers play in the capture process at this point. How- ever, a shift in this trend saw an exponential increase with time in the inter-particle contacts as manifested in Fig. 6. Another observation made was that, particle- particle and particle-wall contacts were more in num- ber than the total particles injected into the domain. It is clear that during the agglomeration process, par- ticles formed multiple contacts from various sides and a record of each of them was given, leading to a spike in the number of contacts. 1.6 1.2 0.8 0.4 0 P ar ti cl e co lli si on s× 10 − 4 Total particle injections Particle-wall contacts Particle-particle contacts 6 18 30 42 Time/ms (a) Unimodal membrane 54 66 78 6 18 30 42 Time/ms 54 66 78 2.0 1.6 1.2 0.8 0.4 0 P ar ti cl e co lli si on s× 10 − 4 Total particle injections Particle-wall contacts Particle-particle contacts (b) Bimodal membrane Fig. 6 Particle-particle and particle-wall interactions in relation to the total number of particles injected into the computa- tional domain J. Shanghai Jiao Tong Univ. (Sci.), 2020 3.2 Implication of Switching from Unimodal to Bimodal Structure on Pressure-Drop Pressure drop is one of the two pillars of a filtration process, and this may determine if a filter is feasible. In this case, the simulation was conducted at a face veloc- ity of 0.5m/s, and a work of cohesion of 0.025 J/m2 for particle-particle and particle-wall interactions. Prior to the particle loading regime, the unimodal nanofiber membrane and the bimodal membrane had a clean fil- ter pressure drop of 283Pa and 316.1Pa, respectively. During the dynamic loading regime, pressure drop in both filter membranes rose steadily in response to the particles dynamically accumulating on the filter, and this was represented by polynomial linearly fitted plots in Fig. 7. Whereas the two display the similar trend, a steeper gradient is observed with the bimodal mem- brane which indicates a relatively higher particle load- ing rate. A higher particle loading rate is closely linked with a faster increase in the dynamic pressure drop. Particle injections×10−3 2 4 (a) Umimodal membrane 6 8 Particle injections×10−3 20 4 (b) Bimodal membrane 6 8 294 290 286 282 Δ P /P a ΔP data Fitted curve ΔP data Fitted curve 332 328 324 320 316 Δ P /P a Fig. 7 Change in pressure drop of the unimodal and bimodal membranes with particle loading To determine the gradient of the plots, a fitted curve linear model of P1X + P2 was used, with P1 and P2 points along the fitting line. For the uni- modal membrane, the coefficients bound with 95% con- fidence were P1 = 0.001 275(0.001249, 0.001 302) and P2 = 282.4(282.3, 282.5). For the bimodal model, the coefficients with 95% confidence bounds were P1 = 0.001 634(0.00148, 0.001 788) and P2 = 316.1(315.5, 316.8). 3.3 Significance of Velocity on the Dynamic Loading Efficiency with Increasing Flow Velocity Magnitude The flow properties of the carrier fluid strongly in- fluence the particle flow and deposition. By consider- ing a range of face velocity magnitudes (0.3, 0.5, 0.7, 1.0, 1.5, 2.0m/s), the difference in performance of the two structures in a particle loading mode was investi- gated. In each of the cases, a full simulation run was performed over a number of time steps to achieve a fully loaded filter. Then the trend of particles’ capture effi- ciency over this timeframe was examined. In addition, a laminar flow environment that upholds Darcy’s law principles was adopted for all the studies. Convention- ally, filtration efficiency of filters is determined by using multi-pass test (ISO 4548): a procedure corresponding to this method was applied to determine the dynamic particles’ capture efficiency of the virtual filters used. This was achieved by putting provisions for counting particles both on the upstream and downstream sides of the filter. Then, the particle penetration was com- puted by K = N/N0, (18) From this, efficiency of the filter could be got from η = 1 − K, (19) where, N is the number of particles going through the filter; N0 is the number of particles injected into the filter; η is the membrane’s particle capture efficiency. Both membranes showed similar trends at the prevailing particle-particle and particle-wall cohe- sion/adhesion parameters used. Increasing the face ve- locity from 0.3m/s to 0.7m/s saw an increase in the capture efficiency of all the membranes (Fig. 8). How- ever, as the fluid flow velocity increased to 1m/s and above, this trend declined. Increasing this velocity be- yond a certain threshold led to the detachment of the loosely held particles. At higher velocities, when loose contacts between particles and the fibers are broken, it lowers the efficiency of the membrane. A change in microstructure from the unimodal to the bimodal J. Shanghai Jiao Tong Univ. (Sci.), 2020 Particle injections×10−3 (a) Umimodal membrane 95 90 85 80 75 70 0.3 m/s 0.5 m/s 0.7 m/s 1.0 m/s 2.0 m/s η/ % η/ % 2 4 6 8 Particle injections×10−3 (b) Bimodal membrane 95 90 85 80 75 70 0.3 m/s 0.5 m/s 0.7 m/s 1.0 m/s 2.0 m/s 2 4 6 8 Fig. 8 Increase of filtration efficiency with fiber loading nanofiber membrane structure improved the dynamic efficiency of the fibrous media. 3.4 Significance of Change in Membrane Mi- crostructure on Efficiency Besides pressure drop, the other crucial element of filtration process is efficiency. This defines the percent- age of total number of particles that a filter is able to intercept. The efficiency of the two media was com- pared and the better performing filter was determined by using the figure of merit (FOM) for each of the re- spective filter membranes during the dynamic particle accumulation phase (Fig. 9). The FOM is a benefit to cost ratio factor given by FOM = − ln(1/K) ΔP . (20) Figure 9(a) shows that the bimodal membrane ex- hibited a much better filtration efficiency compared to the to the unimodal membrane. The improvement in the particle’s capture efficiency can be attributed to the increased surface area that resulted from reducing the size of some fibers. However, this increased surface area increased the pressure drop of the bimodal mem- brane, which resulted in a relative drop in the FOM of the bimodal membrane. Despite this, overall, the Particle injections×10−3 (a) Capture efficiency against the increasing number of particle injections (b) FOM against the increasing number of particle injections 95 90 85 80 75 70 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 Unimodal Bimodal Unimodal Bimodal η/ % 2 4 6 8 Particle injections×10−3 2 4 6 8 F O M × 10 3 Fig. 9 Particle capture efficiency and FOM with filter clog- gings bimodal membrane showed better filtration properties (Fig. 9(b)). 4 Conclusion Filters operate in a transient environment character- ized by time dependent accumulation of particles. How- ever, most filter performance studies only deal with the clean filtration stage. In this work, a practical effort was made to investigate the performance of a filter during the dynamic loading process. Furthermore, regarding the performance of unimodal nanofiber membrane and a bimodal nanofiber membrane under dynamic particle loading conditions, it was observed that: (1) CFD-DEM can be used to successfully model the dynamic particle growth on filters in a transient en- vironment and give reliable data for the properties of these growths. (2) Reducing the size of some of the fibers to create a bimodal arrangement tremendously improved the filter efficiency FOM of bimodal membranes. It can be concluded that, using bimodal membranes improves the filter performance. J. Shanghai Jiao Tong Univ. (Sci.), 2020 References [1] SPIELMAN L, GOREN S L. Model for predicting pressure drop and filtration efficiency in fibrous media [J]. Environmental Science & Technology, 1968, 2(4): 279-287. [2] KIRSCH A A, FUCHS N A. Studies on fibrous aerosol filters. II. Pressure drops in systems of parallel cylin- ders [J]. The Annals of Occupational Hygiene, 1967, 10(1): 23-30. [3] HAPPEL J. Viscous flow relative to arrays of cylinders [J]. AIChE Journal, 1959, 5(2): 174-177. [4] KUWABARA S. The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers [J]. Journal of the Physical Society of Japan, 1959, 14(4): 527-532. [5] WATSON J H L. Filmless sample mounting for the electron microscope [J]. Journal of Applied Physics, 1946, 17(2): 121-127. [6] DALLAVALLE J M, ORR C, HINKLE B L. The ag- gregation of aerosols [J]. British Journal of Applied Physics, 1954, 5(S3): S198-S206. [7] ESMEN N A, ZIEGLER P, WHITFIELD R. The adhe- sion of particles upon impaction [J]. Journal of Aerosol Science, 1978, 9(6): 547-556. [8] STENHOUSE J I T, LLOYD P J, BUXTON R E. The retention of large particles (> 2 μm) in fibrous filters [J]. American Industrial Hygiene Association Journal, 1976, 37(7): 432-436. [9] JORDAN D W. The adhesion of dust particles [J]. British Journal of Applied Physics, 1954, 5(S3): S194- S197. [10] BOWLING R A. A theoretical review of particle adhesion[M]//Particles on surfaces 1. Boston, USA: Springer, 1988: 129-142. [11] THOMAS D, PENICOT P, CONTAL P, et al. Clog- ging of fibrous filters by solid aerosol particles experi- mental and modelling study [J]. Chemical Engineering Science, 2001, 56(11): 3549-3561. [12] LETOURNEAU P, VENDEL J, RENAUDIN V. Ef- fects of the particle penetration inside the filter medium on the HEPA filter pressure drop [C]//22nd DOE/NRC Nuclear Air Cleaning and Treatment Con- ference. Denver, USA: IAEA, 1992. [13] QIAN F, HUANG N, LU J, et al. CFD-DEM simu- lation of the filtration performance for fibrous media based on the mimic structure [J]. Computers & Chem- ical Engineering, 2014, 71: 478-488. [14] PAYATAKES A C. Model of transient aerosol particle deposition in fibrous media with dendritic pattern [J]. AIChE Journal, 1977, 23(2): 192-202. [15] TIEN C, WANG C S, BAROT D T. Chainlike forma- tion of particle deposits in fluid-particle separation [J]. Science, 1977, 196(4293): 983-985. [16] THOMAS D, CHARVET A, BARDIN-MONNIER N, et al. Aerosol filtration [M]. Oxford, UK: Elsevier, 2016. [17] KANAOKA C, HIRAGI S. Pressure drop of air filter with dust load [J]. Journal of Aerosol Science, 1990, 21(1): 127-137. [18] JUDA J, CHROSCIEL S. A theoretical model of pres- sure loss increase during the filtration process [J]. Staub-Reinhaltung der Luft, 1970, 30(5): 12-15 (in German). [19] THOMAS D, CONTAL P, RENAUDIN V, et al. Mod- elling pressure drop in HEPA filters during dynamic filtration [J]. Journal of Aerosol Science, 1999, 30(2): 235-246. [20] JAPUNTICH D A, STENHOUSE J I T, LIU B Y H. Effective pore diameter and monodisperse particle clogging of fibrous filters [J]. Journal of Aerosol Sci- ence, 1997, 28(1): 147-158. [21] PAYATAKES A C. Model of the dynamic behavior of a fibrous filter. Application to case of pure intercep- tion during period of unhindered growth [J]. Powder Technology, 1976, 14(2): 267-278. [22] BERGMAN W, TAYLOR R D, MILLER H H, et al. Enhanced filtration program at LLL: A progress re- port [C]//17th DOE Nuclear Air Cleaning Conference. Boston, USA: OSTI, 1978. [23] KANAOKA C, EMI H, MYOJO T. Simulation of the growing process of a particle dendrite and evaluation of a single fiber collection efficiency with dust load [J]. Journal of Aerosol Science, 1980, 11(4): 377-389. [24] CUNDALL P A, STRACK O D L. A discrete numer- ical model for granular assemblies [J]. Geotechnique, 1979, 29(1): 47-65. [25] REBAÏM, DROLET F, VIDAL D, et al. A Lattice Boltzmann approach for predicting the capture effi- ciency of random fibrous media [J]. Asia-Pacific Jour- nal of Chemical Engineering, 2011, 6(1): 29-37. [26] BUDYKA A K, OGORODNIKOV B I. Aerosol filtra- tion (aerosol sampling by fibrous filters)[M]//Aerosols handbook. Boca Raton, USA: CRC Press, 2004: 541- 556. [27] HæRVIG J, KLEINHANS U, WIELAND C, et al. On the adhesive JKR contact and rolling models for re- duced particle stiffness discrete element simulations [J]. Powder Technology, 2017, 319: 472-482.