In the fiber medium, the Reynolds number was small, and thus, the inertial force was much smaller than the viscous force. The viscous force was dominant, while the inertial force was neglected, and the air followed the Stokes flow [29]. The finite volume method implemented by Fluent code was used to solve the gas flow field. The continuity equation and momentum equation of fluid flow are as follows:
$$\nabla \varvec{u} = 0$$
(1)
$$\nabla \varvec{p} = \mu \nabla^{2} \varvec{u}$$
(2)
where u is the fluid velocity (m/s); p is the pressure; and 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.
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.
In the Lagrangian method, the particle position is obtained by integrating the force balance equation of the particle:
$$\frac{{{\text{d}}u_{\text{p}} }}{{{\text{d}}t}} = F_{\text{D}} + F_{\text{B}} + F_{\text{E}} = \frac{18\mu }{{d_{\text{p}}^{2} \rho_{\text{p}} C_{\text{C}} }}\left( {u - u_{\text{p}} } \right) + \xi \sqrt {\frac{{216\mu k_{\text{B}} T}}{{\pi \rho_{\text{p}}^{2} d_{\text{p}}^{5} \Delta t}}} + qE$$
(3)
where FD, FB, and FE represent the drag, Brown force, and electric field force (N), respectively; up is the particle velocity (m/s); CC is the Cunningham correction factor that is calculated via CC = 1 + Knp (1.257 + 0.4e−1.1/Knp) [21, 30, 31]; dp is the particle diameter (μm); Knp = λ/dp is the Knudsen number; λ is the mean free path of the gas molecule (m); ρp is particle density (kg/m3, here 1000 kg/m3 is adopted); ξ represents a random number in a standard normal distribution with a zero mean and a variance of 1; kB is the Boltzmann constant; T is the thermodynamic temperature (K); Δt 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.
Figure 3 displays a typical standardized particle concentration contour. The normalized concentration of particles adjacent to the fiber was zero, indicating that the particles were deposited on the fiber, and the normalized concentration at the outlet was substantially zero, indicating that the particles were intercepted by the filter media.
Fig. 3Typical standardized particle concentration contour, with dp = 300 nm, df = 370 nm, and u = 0.05 m/s
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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/cm3 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.
The fiber electrostatic potential field was established by writing the user-defined scalar (UDS). A UDS was defined to represent the potential field, and three user-defined memory (UDM) locations were specified to store the field strength in 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).
$$\frac{{\partial^{2} \phi }}{{\partial x^{2} }} + \frac{{\partial^{2} \phi }}{{\partial y^{2} }} + \frac{{\partial^{2} \phi }}{{\partial z^{2} }} = 0$$
(4)
where \(\phi\) denotes electrostatic potential.
The electric field E (UDF) generated by the electrostatic potential is calculated using Eq. (5).
$$E = - \nabla \phi$$
(5)
The force exerted by the potential field on the particles is defined as
$$F = qE$$
(6)
where \(q\) denotes the amount of charge of the particles.
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.
Figure 4 compares the particle motion trajectory curves in the presence of a potential field with those in the absence of a potential field. Under the conditions of u = 0.05 m/s and dp = 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 [(Nin–Nout)/Nin] with and without considering the Coulomb force for the 300 nm particles, where Nin and Nout 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.
Fig. 4Motion trajectory curves of the particles: a in the absence of potential field; b in the presence of potential field
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Table 1 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 dp = 300 nm)Full size table
In the Fluent simulation calculation, the mesh density has a great influence on the numerical calculation results. Generally, the denser the mesh is, the more accurate the numerical results. Therefore, the numerical simulation results have practical significance only when the increase in the number of grids has little effect on the calculation results. We studied the relationship between the simulation results and the mesh density by increasing the number of grid points around the circular cross section of the fiber. As the number of grids around the cross section of the fiber increased, the pressure drop eventually reached a platform (Fig. 5). When the number of grid points around a single fiber was 46, the difference between the pressure drop data was negligible, compared with that when the number was 27. Thus, the number of grid points was set as 27 to ensure that the numerical simulation results were independent of the mesh density.
Fig. 5Relationship between pressure drop and the number of grid points around the fiber
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