## Abstract

The filtration of aerosol particles using composites of nano- and microsized fibrous structures is a promising method for the effective separation of nanoparticles from gases. A multi-scale physical system describing the flow pattern and particle deposition at a non-steady-state condition requires an advanced method of modelling. The combination of lattice Boltzmann and Brownian dynamics was used for analysis of the particle deposition pattern in a fibrous system. The dendritic structures of deposits for neutral and charged fibres and particles are present. The efficiency of deposition, deposit morphology, porosity and fractal dimension were calculated for a selected operational condition of the process.

## 1. Introduction

Aerosol filtration in fibrous mats is one of the most efficient methods of solid–gas separation, especially in the case of submicrometre particles. The pattern of particle deposition in a fibrous filter is highly complex and intrinsically connected with the efficiency, loading capacity and permeability of the system. Therefore, rational design, optimization, operation, troubleshooting and innovation require in-depth understanding and reliable analysis of the deposition process and its effects on system variables. During an initial period, aerosol particles deposit on the collector surfaces, forming chain-like agglomerates—dendrites. This stage of filtration controls the entire process. Information about the geometry of dendrites, how they fill the void fraction of the filter structure, is crucial for proper design of the filter working at the non-steady-state condition.

## 2. Problem formulation

Let us consider a system of two cylindrical fibres of diameter *d*_{fm} (microsized) and *d*_{fn}(nanosized) immersed in a stream of monodispersed aerosol particles (figure 1). The configuration is defined by the distance between the centres of both fibres, *l*, and an orientation angle, *Θ*. These two parameters and fibre diameter define the filter structure. The aerosol flows normally to both fibres and the aerosol number concentration and velocity in an undisturbed region are *C*_{0} and *U*_{0}, respectively. We assume that the surfaces of both fibres are barriers of adsorption for aerosol particles.

For the system considered here, two different length scales are observed if one compares the mean-free path, *λ*, of the carrier gas molecule and the dimension of a particular fibre, *d*. For an ideal gas modelled as rigid spheres, the mean-free path of the molecules can be related to the temperature, *T*, and pressure, *P*, via
2.1
where *k* is the Boltzmann constant, *T* the absolute temperature, *P* the pressure and *σ*_{c} the diameter of the molecules.

The continuum assumption of the Navier–Stokes equations is valid provided the mean-free path of the molecules is smaller than the characteristic dimensions of the flow domain. If this condition is violated, the fluid will no longer be under local thermodynamic equilibrium and the linear relationship between the shear stress and rate of shear strain (Newton’s law of viscosity) cannot be applied. Velocity profiles, boundary-wall shear stresses, mass flow rates and pressure differences will then be influenced by non-continuum effects. In addition, the conventional no-slip boundary condition imposed at a solid–fluid interface will begin to break down even before the linear stress–strain relationship becomes invalid. The ratio between the mean-free path and the characteristic dimension of the flow geometry, *d*, is commonly referred to as the Knudsen number, *Kn*=*λ*/*d*. The value of the Knudsen number determines the degree of rarefaction of the gas and the validity of the continuum flow assumption. For *Kn*<10^{−2}, the continuum hypothesis is appropriate and the flow can be described by the Navier–Stokes equations using conventional no-slip boundary conditions. However, for 10^{−2}<*Kn*<10^{−1} (commonly referred to as the slip-flow regime) rarefaction effects start to influence the flow and the Navier–Stokes equations can only be employed provided tangential slip velocity boundary conditions are implemented along the walls of the flow domain. Beyond *Kn*=10^{−1} the continuum assumption of the Navier–Stokes equations begins to break down and alternative simulation technique approaches must be adopted. Finally, for *Kn*>10, the continuum approach breaks down completely and the regime can then be described as being a free molecular flow. Under such conditions, the mean-free path of the molecules is much greater than the characteristic length scale and consequently molecules reflected from a solid surface travel, on average, many length scales before colliding with other molecules.

The modelling of two-phase flow processes is an extremely difficult task within the classical hydrodynamics discipline owing mainly to the inherent free-boundary complication. The lattice Boltzman method has enjoyed rapid development and provides an interesting alternative to traditional numerical techniques for solving the Navier–Stokes equation.

## 3. Digital fluid dynamics

Description of particle motion in a fluid requires the knowledge of the velocity field of fluid and particle position at any site of the space and moment of time. The classical approach to the flow phenomena is through partial differential equations (Navier–Stokes equations) that describe collective motion in a dissipative fluid. The kinetic theory models a fluid by using an atomic picture and imposing Newtonian mechanics on the motion of the atoms. Complete information on the statistical description of a gas at, or close to, the thermal equilibrium is assumed to be contained in the one-particle phase-space distribution function *f*(*x*,*t*,*α*) for the atomic constituents of the system. The variables *x* and *t* are the space and time coordinates of the atoms and *α* stands for all other phase-space coordinates, e.g. momentum flux or momentum. For an isolated gas with collisions the Liouville theorem is modified to the form
3.1
where *Ω*(*f*) is a function that models the rate of change of the distribution function. Since the collisions preserve conservation laws, by integration of equation (3.1) over *α*, the continuity equation and momentum tensor equation describing the macrodynamics of the system can be derived. To build the cellular-space picture with the dynamics of the collective motion predicted by the Navier–Stokes equation, a lattice on which particles move, collision rules and other restrictions characteristic for a chosen model should be defined. The lattice gas methods were pure cellular automata [1]. Models using this approach assume totally discrete physical space, time and the node state. The lattice gas algorithms are very stable and can handle complicated geometry and boundary conditions. The weakness of these methods is that they are ‘noisy’, requiring spatial and time averaging. The lattice Boltzmann model was the next step in developing such a description of the fluid dynamics problems [2]. In the case of the lattice Boltzmann approach, the node state is described by a continuous function.

The 19-speed cubic lattice was used for modelling the fluid flow. The function *f*_{i}(*x*,*t*) denotes the number of fluid particles entering the site *x* at time *t* with velocity *e*_{i}. Macroscopic quantities such as density *ρ* or momentum *ρu* are defined as
3.2
The evolution of the system is described by the expression
3.3
The outcome of a collision can be approximated by assuming that the momentum of interacting particles will be redistributed at some constant rate towards an equilibrium distribution [3]. This simplification is called the single-time-relaxation approximation and can be expressed by
3.4
The rate of change towards equilibrium is 1/*τ*, the inverse of relaxation time, and is chosen to produce the desired value of the fluid viscosity
3.5
The equilibrium distribution is given as follows:
3.6
where *α*_{i} are the model-dependent constants and *c*_{s} is the speed of sound.

In continuum flow analyses, a no-slip velocity constraint is enforced along all solid–fluid interfaces. In practice, the no-slip condition is found to be appropriate for *Kn*<10^{−2}. If the Knudsen number is increased beyond this value, rarefaction effects start to influence the flow and the molecular collision frequency per unit area becomes too small to ensure thermodynamic equilibrium. Under such conditions, a discontinuity in the tangential velocity will form at any solid–fluid interface.

In continuum regime the bounce-back boundary condition is used on the solid level. This means that when a fluid particle enters the solid site, it changes its direction of movement to the opposite one. This method naturally leads to zero velocity at the solid level. Our model involves two parameters *r* and *s*, representing the probability of a particle to be bounced back and slipped forward, respectively. The boundary kernel takes the form [4]
3.7
Obviously, the two parameters are not independent and must be chosen such that *r*+*s*=1. Assuming a second-order slip velocity one can write
3.8
The Knudsen number for the lattice is given by *ν*/(*c*_{s}*d*). Parameters *A* and *B* were given by Sbragaglia & Succi [5]:
3.9
Values of *r* can be estimated from experimental data [6]. A typical value of *r* is 0.59, which gives *A*=1.15 and *B*=3.

## 4. Particle motion

Determination of structures of deposited particles on the filter fibre requires the knowledge of the history of an individual particle and its position and velocity vectors. The Lagrangian method of analysis should be used for a description of the process. Particle trajectory is calculated for the generalized Basset–Boussinesq–Oseen equation, which in simplified form is reduced to the expression
4.1
where *m* is a particle mass, ** v** a particle velocity vector and

**F**

^{(R)}random, diffusional forces caused by collisions with gas molecules. The drag forces for small, spherical particles satisfying Stokes’ law can be expressed by 4.2 where

*C*

_{s}is the Cunningham factor. The external forces considered in this paper are represented by electric forces: Coulomb forces between particle and fibre 4.3 and between charged particles 4.4 polarization forces 4.5 and image forces 4.6 All mentioned external forces are acting on directions connecting centres of particles or centre of particle and axis of fibre.

*Q*

_{f}is electrical charge of a fibre,

*Q*

_{p}is charge of a particle,

*ε*is permittivity of free space,

*ε*

_{p}and

*ε*

_{f}are the relative dielectric constant of a particle and fibre, respectively,

*d*

_{f}is the fibre diameter and

*r*is the position of the centre of a particle.

Foundations of Brownian dynamics (BD) were established by Chandrasekhar [7] for a Stokesian particle in a stationary fluid and for a force-free field. In this work, extension of BD for the case of moving fluid in the presence of the external forces derived by Podgórski [8] was used. Integration of equation (4.1) for the time interval Δ*t* (same as used for lattice Boltzmann computations), small enough that the host fluid velocity *u*_{i} and external force may be assumed constant over (*t*,*t*+Δ*t*), gives the following bivariate normal density probability distribution functions *φ*(Δ*v*_{i},Δ*L*_{i}) such that during time interval Δ*t* the particle will change its *i*th component of velocity by Δ*v*_{i} and it will be displaced by the distance Δ*L*_{i} in *i*th direction:
4.7
The expected values of particle velocity change 〈Δ*v*_{i}〉 and the linear displacement 〈Δ*L*_{i}〉 are expressed as
4.8
and
4.9
where *τ*_{p} is particle relaxation time. The standard deviations *σ*_{vi} and *σ*_{Li} are as follows:
4.10
and
4.11
The coefficient of correlation is given by
4.12
We can therefore formulate the following generalized algorithm for the BD. For a given initial particle position and its initial velocity components, *v*_{i}, at a moment *t*, we calculate the local fluid velocity, *u*_{i}, and the external forces, . Then, one calculates the expected values 〈Δ*v*_{i}〉 and 〈Δ*L*_{i}〉 from equations (4.8) and (4.9) and the correlation coefficient, *ρ*_{c}, from equation (4.12). Next, we generate two independent random values *G*_{Li} and *G*_{vi}, having a Gaussian distribution with zero mean and unit variance. Finally, we calculate the change of particle velocity, Δ*v*_{i}, and the particle linear displacement, Δ*L*_{i}, during time step Δ*t* from the expressions accounting for deterministic and stochastic motion
4.13
All the steps are repeated for each coordinate *i*=1,2,3. Having determined the increments Δ*v*_{i} and Δ*L*_{i} the new particle velocity at the moment *t*+Δ*t* is obtained as *v*_{i}(*t*+Δ*t*)=*v*_{i}+Δ*v*_{i}, and in the same manner the new particle position is calculated. After completing one time step of simulations, the next step is performed in the same way.

## 5. Results, discussion and conclusions

The models described in the previous sections were used for simulation of deposition of diffusional particles in the system of two cylindrical collectors. A cylinder of diameter *d*_{fm} was placed centrally in the computational domain. The lattice had dimensions of 256×256×32 nodes. The diameter of the microfibre of the filter was equal to 32 μm and nanofibre and particle diameters were 500 nm. The distance between the centres of the fibres was assumed to be 64 μm. The orientation angle *Θ* was selected from the range 0^{°}–90^{°}. In the fluid inlet a plug flow distribution was assumed, no-gradient velocity condition was applied in the outlet and no stress on the boundaries parallel to the main gas flow. The calculations were carried out for two values of fluid flow, namely 2 and 20 cm s^{−1}. The charge density on fibres was equal to 500 nA s m^{−2}. The aerosol particles were singly charged for the two systems of electrically neutral fibres and particles, or charged fibres and particles. The calculations of particle deposition efficiency, *E*, were carried out using the collection efficiency definition from classical filtration theory for the microfibre (the fraction of particles flowing towards the collector that makes contact with the collector or particles that have been already deposited). The results of calculations shown below were selected for two limiting cases from the set of obtained data. They correspond, first, to the case of the neutral particles and fibres when the efficiency of collection is the smallest and, second, to the case of the charged particles and fibres (Coulomb effect) when the collection is the highest.

Figures 2–5 present the patterns of deposited particles for the selected orientation angle *Θ*=45^{°} for different values of air velocity and different cases of fibre and particle electrostatic status. For neutral fibres and particles (figures 2 and 4), when the velocity of the gas is small, the diffusion is a predominant mechanism of particle deposition. Particles more uniformly cover the nano- and microfibre in comparison to the case of higher gas velocity, when the convective effect causes the location of the particles in the front of fibres. The nanofibre is a strong attractor for the collection of particles. Incorporation of Coulombic interaction between particles and fibres results in a much higher (more than 200%) deposition efficiency of particles of the bifibrous system, and more uniform deposition of particles on both fibres in comparison to the neutral system. The attractive Coulomb force for singly charged nanoparticles is much higher than polarization or image forces. Only near the fibre surface are polarization and image forces significant and may balance the repulsive force from previously deposited particles. For neutral particles and charged fibres the increase of collection efficiency of approximately 20–30% is expected from the calculations. These results are consistent with experimental results for electret microfibre filters [9]. The detailed analysis of the morphology of the deposits indicates their similar average porosity from the range 0.25–0.3 and fractal dimension of dendrites from the range 2.66–2.81 for all operational conditions used in the calculation.

The combination of the lattice Boltzmann and BD models is a useful tool in describing the growth dynamics of structures of deposited matter during the flow of nanoparticles through a system of micro- and nanosized fibres. This method can be effective in cases when the continuum approach breaks down and boundary conditions for the solution of the Navier–Stokes equation describing the local and temporary velocity distributions change significantly due to the development of a complicated structure of deposits. The modelling provides important information about the morphology of deposits useful in designing composite fibrous filters.

## Footnotes

One contribution of 25 to a Theme Issue ‘Discrete simulation of fluid dynamics: applications’.

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