2.2 Model of multi-fiber filters

2.2.1 Geometric model and simulation method of multi-fiber filters

The calculation model and boundary conditions are presented in Fig.2-1.The fibrous model was constructed using Gambit software,with a filtration domain length,width and height denoted by L, D and H respectively.The fibrous arrangements were either parallel or staggered.An addition 0.5L upstream and downstream of the filtration domain was constructed where the airflow and particles were assumed to be in an undisturbed flow field.In this simulation, L, D and H were 392μm,488μm and 448μm respectively.The reference SVF (α) was defined as 5.6%^[27].

The air flow was assumed to enter into the filtration domain through a velocity inlet and to leave with a pressure outlet boundary condition.The periodic boundary condition was considered for the sides of the computational domain ^[31].For the fibrous surfaces,a no-slip boundary condition was assumed ^[2].The air flow through the fibrous media was usually considered as a steady-state laminar flow ^[7,20,31].The FVM by Patankar ^[34] was used to solve the air-flow equations.The continuity equation and the conservation of momentum equation are written as follows ^[20].

where ρ, v, p and μ are the fluid density,velocity,pressure and viscosity,respectively.

After the particle-free flow field was obtained,airborne particles were then introduced into the solution domain.The volume fraction of particles was less than 10%,which was regarded as the discrete phase.Combined with the obtained flow field,the force situation and trajectory of each particle were calculated by the Lagrange method of the DPM solver,which was often used to solve multiphase flow problems ^[35].The dominant forces acting on a particle are the air drag force,the Brownian forces and the gravity.The equation of motion for a spherical particle in air flow can be written as

whereis the velocity vector of the flow filed,is the velocity vector of the particle, C_c=1+Kn_p(1.257+) is the Cunningham correction factor,where Kn_p=2λ/d_pis the particle Knudsen number, d_pis the particle diameter,ρ_pis the density of the particle,ρis the density of the flow field, G_iis a random number chosen from a normal distribution with a zero mean and a unite variance, S₀ is the,andis the acceleration of gravity.

This chapter used the DPM model for particle motion in the fibrous filters.The DPM model supposed that particles were trapped when they touched the fiber surface,and deposited particles would not affect the flow field.So this chapter only considered the clean filtration stage.A more accurate model for particle motion including trapping,rebounding and interaction among particles will be considered in our future work.Finally,particles trapped by the fibers were counted to gain the filtration efficiency.In this simulation,all of the particles were modeled as rigid spheres with a uniform density of 1,000 kg/m^{3 [37]}.They were uniformly distributed in the inlet plane and would be released from the inlet plane to the flow field.

2.2.2 Model validation

To validate the model,the number of mesh nodes around the fiber was increased from 10 to 40 to ensure that the calculation results were mesh-independent.A fiber diameter of 28μm was chosen.It was a staggered model and the face velocity was 1.0 m/s.The corresponding results of mesh density analysis are shown in Fig.2-2.It is found that,beyond 25,the number of nodes does not affect the results of the pressure drop of the filter.For other fiber diameters in our simulation,the number of mesh nodes can be chosen proportionally according to the above results.For incompressible flow at very low Re (Reynolds number) the air flow in the filter follows Darcy's law ^[20,37],which means that pressure drop is proportional to the face velocity of the entrance.Liu and Wang ^[38] found that the ratio of the pressure drop to face velocity is not a constant at higher Re,but rises with face velocity since a wake region is formed and additional energy is dissipated.The dimensionless drag force acting on the fibers,corresponding spectral intensity of the noise given by [36]:S₀=216μkT/

which is related to the pressure drop,was expressed by Liu and Wang ^[38]

where Z is the thickness of the fiber,the SVF is calculated as: α=/4lh, V is the face velocity andΔp is the actual pressure drop.

Kuwabara ^[39] presented the dimensionless drag force in the low Reynolds number Re for the staggered model as

It should be noted that the above equations are only applicable for low Re.The Re is defined as:Re=ρVL_c/µ^[40],whereρis the air density, V is the face velocity, L_cis a characteristic length,and µ is the air viscosity.The air density is 1.225 kg/m³ and the air viscosity is 1.789×10^-5 Pa·s at standard sea level conditions ^[41].Fig.2-3 shows that simulation results between the dimensionless drag force and the Re.It is found that Re=1.0 is the turning point for the dimensionless drag force.The drag force remains mostly unchanged when Re＜1.0,while it increases with Re when Re＞1.0.Our results are in good agreement with results from the Lattice Boltzmann-cellular Automata (LB-CA) probabilistic model by Wang et al ^[27].These data also show that our simulation results are in line with the predictions of Kuwabara's model when Re＜1.0 ^[39].It is worth noting that Re of the following simulation cases are all smaller than 1.0,indicating that inertial effects are negligible ^[20].Taken together,these results validate our simulation model.

The filtration efficiencyηof clean filter medium is closely linked to the single fiber filtration efficiency η_s and the total filter thickness Z.The filtration efficiency of multi-fiber is usually calculated by Hinds ^[42]

The single fiber filtration efficiencyη_sdepends on the trapping mechanisms involved in the filtration process.The mechanical single fiber efficiency can be calculated as follows ^[37]

where η_R^[38] is the filtration efficiency due to interception,_Iη^[43] is the filtration efficiency due to inertial impaction, η_D^[44] is the filtration efficiency due to Brownian diffusion.

Fig.2-4 compares the filtration efficiency of the parallel and staggered designs with others'results.Here we use the same model of Wang's ^[27].The SVF is chosen as 17.2%.The face velocity is 0.1 m/s.It is found that the filtration efficiency decreases with d_f(fiber diameter) when d_f＜1μm,while it increases with d_fwhen d_f＞1μm.And the staggered design performs higher filtration efficiency especially for the large particles.Fig.2-4 also shows that the tendency of our results from the DPM model is coherent with the results of Wang ^[27] and the analytical expression ^[27,42] on the whole.Although having some deviations for the parallel design,the filtration efficiency of present results is close to the results of Wang ^[27] for the staggered design.So it verifies the feasibility of our simulation method.