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The effect of the flow field recalculation on fibrous filter loading: a numerical simulation Antonis Karadimos, Raffaella Ocone * School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK Received 18 November 2002; accepted 16 May 2003

Abstract We present a simple model for the loading process of aerosol particles on fibrous filters with the aim to show the influence that the flow recalculation around a single fibre has on the complete loading process, up to the final clogging of the filter. At each loading stage (i.e., at any given degree of particle deposition), the change of shape of the fibre is presented, and information on the flow field around a fibre and the filter efficiency are obtained. The effect of the flow recalculation on the filter efficiency is shown; the single fibre efficiency is proven to be overestimated when the effect of further deposition is neglected. To perform the study, we have developed a CFD code and we have combined it with particle trajectory simulations. D 2003 Elsevier B.V. All rights reserved. Keywords: Aerosol; Fibre; Filter

1. Introduction Fibrous filter materials are a widely used technology for the removal of fine particles from a gas stream. These materials are used in surgical face masks and other medical applications, vacuum cleaner exhausts, air conditioning units and automotive cabin intake filters in addition to industries which need to clean large volumes of air such as breweries and distilleries, speciality gas producers and the nuclear industry. The understanding of operability for such filters is essential in a number of environmental cleaning problems. Fibrous filtration is an efficient and low-pressure drop method used to remove fine ( < 10 Am diameter) aerosol particles from a gas stream. Fibrous filters consist of beds of fine fibres (the fibre diameter is usually smaller than 50 Am) usually aligned roughly perpendicularly to the gas flow. The beds are typically 95% porous and the resultant inter-fibre spacing is much larger than the material to be collected. Therefore, particles are not collected by any form of sieving process, but collection is rather due to inertia and/or interception consequent to the flow past a single fibre.

* Corresponding author. Fax: +44-131-451-3129. E-mail address: [email protected] (R. Ocone). 0032-5910/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0032-5910(03)00132-3

The depth filtration process allows fibrous filters to have, for the same degree of purification, minimal resistance as compared to surface filters. Collection by a single fibre, due to inertia and interception mechanisms, is not very efficient; nevertheless, as a particle passes through many layers of fibres, while penetrating the filter, the efficiency of the entire filter unit can approach 100%. In filtration theory, one aims to get information on the number of particles deposited per unit length of the fibre (for a given fluid flow rate, filter packing density and fibre diameter), which then furnishes the overall filter efficiency when integrated over the entire filter thickness. Calculation of the pressure drop across the filter is equally important, accounting for the operational cost and life expectancy of the filter. Theory for the performance of fibrous filters is well established for the case of clean filters and is excellently reviewed by Davies [1] and Kirsch and Stechkina [2]. The knowledge of the initial filter efficiency is a primary criterion for selecting the appropriate filter for a specific application. Determination of the two operational parameters, filter efficiency and pressure drop across the filter, is of major importance; however, as the deposition of particles proceeds, both pressure drop and efficiency change, and their evaluation becomes difficult. Particle deposition causes an increase of the flow resistance resulting in a rise

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of the pressure drop across the filter. As deposition evolves, many particles are collected by the fibre as well as by other deposited particles, which thus contributes to the increase of the filter efficiency. Experimental investigations of these features have been carried out for various operational parameters [3,4]; however, a universal model, able to predict the entire lifetime performance of filters, is not available yet. Likewise, the simulation work undertaken so far is limited to qualitatively verifying the experimentally observed behaviour. In formulating a universal predictive model, various factors need to be taken into consideration. The choice of how to solve the hydrodynamic field plays a major role in any study as it forms the basis for further particle – flow and particle –particle interaction to be incorporated into a model. There are three different approaches used for solving the fluid mechanics of filtration, as mentioned in the next paragraph. We chose to develop a two-dimensional Navier – Stokes solver (CFD code) to solve the fluid flow past the fibre, which we believe constitutes a good basis for further development. In this work, we aim to focus on the ability to recalculate the flow field at any arbitrary level of deposition and therefore to take into account the effect of particle accumulation both on the flow field and on further deposition. The result is an accurate prediction of the single fibre efficiency. The equations used in this model are based on the physics of the system and as such are independent of the experimental inputs. The present method qualifies as a deterministic approach of filter behaviour description, which we believe to be advantageous with respect to other parallel simulation techniques.

2. The model The modelling aspect of particle capture on a fibre has advanced consequently to the achievements in the hydrodynamics of low Reynolds number; this paved the way to the quantitative calculation of the filter efficiency. To calculate particle deposition, the flow field in the proximity of the fibre needs be known in detail. The approximation of Stokes flow regime for the flow around an isolated cylinder [5] has represented the basis of a number of simulations of particle deposition on a single fibre [6– 8]. Simulation of flow trajectories can satisfactorily predict the change of the filter performance during loading, as observed in experiments [4,8]. Another study by Payatakes and Tien [9] simulates the grow the process of the dendritic structure of particles depositing on the fibre using the approximation of Stokes flow. The study furnishes a realistic image of particle accumulation upon the fibre. The above simulations (Monte Carlo simulations) are based on the Kuwabara flow field and as such suffer from two drawbacks. Firstly, the solution is only an approximate

approach to the Stokes regime; secondly and more importantly, the flow field is steady and cannot be combined with particle flow interactions to account for the effect on the flow of the deposition. The body of studies mentioned above proves that the simulation of the filtration process qualitatively predicts the changes of operational parameters of the filter, such as pressure drop and single fibre efficiency, at the later stages of loading. Nevertheless, for a quantitative description of these changes, we still rely on experimental findings. The particle – flow simulation by lattice-gas models is a parallel approach, which has recently been developed [10]. These models have the ability to incorporate the deposition effects on the flow field and thus to calculate the loading of the fibre more accurately. A further advantage consists in the relative simplicity of application and computational requirements. The different approach between the above models and the CFD approach lies mainly in the probabilistic nature of the former. Despite the fact that these models are still under development and their success under evaluation, they seem to be very promising in the process of loading behaviour prediction. The present study is based on the equations of the physical motion of the system’s components and consists of a numerical simulation of the Navier –Stokes and an Eulerian simulation of the particle’s equations of motion combined together to show the effect of the particle – flow interaction. The main advantage of this approach is the ability to calculate the flow field at any arbitrary loading level of the fibre, thus taking into account how the dendrite growth affects the flow. Recalculation of the flow around the deposited particles (the dendrite) is thus performed at various loading stages (defined by prescribed number of particles released into the system). A disadvantage of such a deterministic approach is the numerical complexity and the computational requirements to reach the solution. On the other hand, the preliminary findings reported here verify the effect of the deposition on the flow and on the single fibre efficiency and are in good accordance with the results of the lattice gas models. A two-dimensional finite difference Navier – Stokes solver has been developed for simulating the carrier fluid flow around a fibre. The flow is assumed to be laminar (with a Reynolds number closely above 1 for the specific test case), which is consistent with the filtration theory [11]. The obtained flow field, whilst similar to the Kuwabara field, does not show the exact symmetry of the flow pattern upstream and downstream the fibre. Indeed, as expected, a small wake, in the region downstream the fibre, is detected correctly by our simulations. This is due to the value of the Reynolds number, which in our case exceeds the value of 1, representing a low Reynolds laminar flow pattern. The Stokes flow, approximated by the Kuwabara flow field, corresponds to

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Reynolds numbers in the area just below the value of 1. A non-orthogonal curvilinear system of coordinates has been chosen to easily follow the streamlines and to accurately predict the flow in the near vicinity of the fibre. The vectorial form (independent of coordinate system) of the Navier – Stokes equations, expressing the conservation of mass and momentum, respectively, is given by:

into account the effect of the viscous drag on the free molecular flow regime for very fine particles (aerodynamic slip); in our case, it has a value of 1.16. The methodologies on which particle and fluid flow simulations are based are outlined in the next two paragraphs.

3. Numerical simulation of the flow field

! divðqUf Þ ¼ Sm ! ! divðqUf Uf T Þ ¼ SV

ð1Þ

! where q is the fluid density, Uf is the velocity vector, T is the stress tensor of the fluid and Sm, SV are the source and sink terms for the mass and velocity, respectively. The particle trajectory simulation starts from writing the equations of motion for a single particle, eventually solved in conjunction with the flow field obtained through the flow solver. The equations of motions for the particles represent a relationship between the fluid drag, the particle momentum and the gravity, according to: ! qp pdp2 ! ! dUp ! ! qp ¼ CD ðUp Uf Þ AUp Uf A 6 dt 2Cn 4

pdp3

111

pdp3 6

! qp g

ð2Þ

! ! where g is the acceleration of gravity, Up is the velocity of the particle, CD is the coefficient of aerodynamic resistance and Cn is the Cunningham correction factor, which takes

The flow field investigation inside the filter starts by considering the two-dimensional flow around the cross section of a single fibre lying within a staggered arrangement of neighbouring fibres. With an inlet gas velocity of 10 m/s and a particle diameter of 1 Am, this corresponds to a Reynolds number of 3.8 (transitional regime between Stokes and laminar flow condition the Kuwabara flow field assumes Stokes flow around the fibre). Under this assumption, the steady 2D Navier – Stokes equations for an incompressible fluid (Eq. (1)) are solved with a finite difference methodology based on the semi-implicit method for pressure linked equations, the so called SIMPLE algorithm. The appropriate form of the momentum balance, deriving from Eq. (1), simplifies the numerical procedure, while achieving at the same time the most accurate possible solution. A generalised curvilinear system of coordinates is chosen in accordance with the geometry of the system under consideration. In Fig. 1, the initial grid is shown, whereby the generalised coordinates coincide with the direction of the gridlines in every gridpoint; the fixed base reference system is the Cartesian coordinate system. According to the transformation yi(x j), where yi are the Cartesian coordinates and

Fig. 1. Initial numerical grid for the flow field simulation around a single fibre.

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x j are the generalised curvilinear coordinates, the expression for the equations of continuity and of momentum transfer takes the following form [12]: B B ½qðu1 b11 þ u2 b12 Þ þ 2 ½qðu1 b21 þ u2 b22 Þ ¼ JSm Bx1 Bx

ð3Þ

and l Buk 1 1 1 1 qðu1 b1 þ u2 b2 Þuk ðb b þ b12 b12 Þ J Bx1 1 1 Buk 2 1 l 1 Bu1 1 Bu1 2 2 1 b þ 2 ðb1 b1 þ b2 b2 Þ b þ b J 1 Bx1 k Bx2 k Bx Bu2 1 Bu2 2 1 b þ b þb12 þ pb k Bx1 k Bx2 k B l Buk 1 2 þ 2 qðu1 b21 þ u2 b22 Þuk ðb b þ b12 b22 Þ Bx J Bx1 1 1 Buk l 2 Bu1 1 Bu1 2 b1 þ 2 ðb21 b11 þ b22 b12 Þ b þ b J Bx Bx1 k Bx2 k Bu2 1 Bu2 2 2 b þ b ð4Þ þb22 þ pb ¼ JSuk k Bx1 k Bx2 k

B Bx1

where J is the Jacobean matrix of the transformation including bmj , the partial derivatives of the transformation which are defined as: bmj ¼ J

Bxi By j

ð5Þ

The above equations are discretised by integration over the control area of the cells (finite volume methodology) giving a system of algebraic equations applied on the points of the computational grid. The final assembled form of the algebraic equation to obtain the unknown velocity (UP), as expressed for the gridnode P in terms of the neighbouring nodes, is given by Patankar and Spalding [13]: X X Ap Up ¼ ðACn þ ADN IkDC n ÞUn þ Suu þ Sup Up þ n

Ap ¼

k

X ðACn þ ADN n Þ Sm ; Sm ¼ Ce Cw þ Cn Cs

ð6Þ

n

where n = E, W, N, S; k = e, w, n, s. The generic variable (UP) represents the two components of velocity, u and v. The notations E, W, N and S stand for the east, west, north and south neighbouring nodes of P, while e, w, n and s are the respective cell faces. The A-terms represent the total flux due to convection and diffusion, while the C-terms refer to the respective cell face’s convective contribution to the total flux. The I-terms are fluxes due to the cross-diffusive terms and are included into the source terms (represented by S). Eq. (6) constitutes a linear system of n equations with n unknowns, which can be solved by a

TDMA-routine (tridiagonal matrix), using a standard algorithm for stepwise Gaussian elimination [14]. The SIMPLE algorithm [13] is chosen for the solution of the system of equations. According to this methodology, the solution for the flow field is obtained by the following iterative procedure: (i) an initial pressure field is chosen either by estimation or by experimental input (ii) a new flow field is obtained based on the initial input pressure field (iii) the mass balance over the entire field gives the pressure correction formula according to which corrected values for the pressure field can be obtained (iv) the process is repeated, starting from (ii) until the pressure and velocity fields reach convergence (negligible mass residuals over the entire field). In order to express the equations in terms of the unknown variable UP at point P, the neighbouring nodal values of the variables are used (E, W, N, S) in order to interpolate the variables on the cell faces (e, w, n, s). The hybrid-differencing scheme is adopted for combining numerical accuracy and stability at all Peclet numbers. The value /e (east cell face) is given, for instance, as a function of the Peclet number according to the following relation [15]: 8 1 > > Up ; Pee > > > > 1 f1P > > < 1 Pee < Ue ¼ Ue ; > f 1P > > > > 1 1 > > : f1P Up þ ð1 f1P ÞUE ; V Pee V f1P 1 f1P

ð7Þ

where f1P is a weighting factor and the Peclet number is defined as the ratio of convective transport to diffusion for each direction. The geometrical grid used is dictated by the transformation adopted and has the advantage of easily following the curvilinear surface of the fibre. Numerical diffusion, which is inevitably associated with the use of non-orthogonal grids, is accounted for by the above mentioned hybrid scheme. The technique of underrelaxation is employed to smooth any abrupt changes in the values of the variables and improve the stability of the solution process. Appropriate conditions are applied at the computational domain boundaries; the velocity profile is assumed to be uniform at the inlet and its derivative null at the outlet. Symmetry conditions employ an additional zero diffusion flux on the particular boundaries. The case of an impermeable wall is finally the most complicated boundary condition. Apart from the no slip condition applied for the velocities, a universal profile (‘‘law of the wall’’) is imposed with the introduction of appropriate wall functions in the source terms of the equations, to account for the shear effect

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of the boundary on the flow. The code has been validated in the past for various cases of flow simulations [15,16].

4. Generation of the geometrical grid The geometrical non-orthogonal curvilinear grid used is bounded to the variable transformation adopted and an algebraic method is used to construct the grid. According to such methodology, a distribution of points is given in the periphery of the initial Cartesian grid. Then specific points are projected to follow the circular surface of the fibre. The inertial domain points are produced by interpolation of the peripheral points and an iterative technique is used to smooth their distribution. Finally, to achieve the desired geometry, the initial grid is symmetrically expanded in both x- and y-directions. In the case reported here, the grid consists of 139 lines in x-direction and 149 in y-direction, a total of 20,711 points. A non-orthogonal curvilinear grid has the advantage of following easily a circular surface (the fibre); however, it might introduce numerical diffusion in the numerical procedure. As already mentioned, underrelaxation of the variables, as well as the use of the hybrid differencing scheme incorporated in the flow solver, reduce the risk and the iterative procedure has proven to have a stable convergence. The recalculation of the flow field after a certain degree of deposition can be implemented by simply considering the no-slip boundary condition at the external surface of the particle dendrite. The perimeter line between the areas of particle deposition and the remaining ‘free’ flow areas are then represented by a rectangular line surrounding the grid (‘step approach’). All points within this line are ‘blocked’, i.e. they change from being points of the main flow to points of a wall boundary. The new flow calculation is thus based on the new grid and as such bypasses the areas of deposited particles. A certain degree of inaccuracy in determining the final position of particles attaching on other particles is due to the rectangular shape of the perimeter line that does not match entirely with the shape of the particles. A technique is being developed to transform the grid line to follow the exact shape of the particles; however, the inaccuracy introduced by considering the rectangular shape, which we adopt here, is not severe.

5. Particle trajectory simulation The prediction of the performance of a filter requires knowledge of the trajectories of the particles passing through the filter medium. To describe the motion of an aerosol, it is necessary first to compute the velocity field of the carrier gas and then to compute the motion of the particles based on this field. The velocities of the carrier gas are obtained via the use of the CFD code according to

113

the outlines of the previous paragraph. For any particle trajectory simulation to be linked to the actual flow of the particles through the filter, the following two assumptions need to be satisfied: (a) the particles do not interact with each other and (b) their flow does not influence the flow field. These assumptions are valid if particles do not collide with each other and if they do not pass from each other’s wake. A useful rule of thumb, that quantifies these two assumptions for the case of a monodisperse aerosol, is that the average distance between particles is at least 10 times the particle diameter [17]. An upper limit of particle number thus exists beyond which their influence on the flow is of considerable significance; in our case with particles of a diameter dp equal to 1 Am, this upper limit is of order of 8 1015 particles. Our calculations involve a significantly smaller amount of particles and consequently the motion of the carrier gas is considered independent of the motion of the particles. Hence, we can use the flow field obtained from our CFD code as a base for the numerical integration of the particle trajectory equations. The filtration test case examined here involves a single fibre in a staggered arrangement of neighbouring fibres with a 5 Am diameter. A mono-disperse stream of 1 Am diameter particles is injected by releasing one particle at a time with an initial velocity of 0.75 m/s upstream of the fibre at a randomly generated y-position. The particle flows towards the fibre under the influence of the carrier gas and gravity, the latter acting in the x-direction (filter considered in an upstream flow path), and deposition is assumed when the particle contacts a fibre. The prescribed conditions of limitsize particle diameter, the relative high initial particle velocity and the carrier gas flow conditions that clearly exceed the Reynolds numbers of Stokes flow regime all suggest that inertial impaction and interception are the dominant deposition mechanisms. For the case studied, the effect of gravitational settling is proven to be negligible ! and the term g is kept in the equations only for reasons of completeness. A particle dendrite expanding upstream the fibre has been chosen intentionally to test the concept of flow recalculation in conditions of maximum ‘flow disturbance’; this is obtained when the impaction and interception mechanisms predominate. On the contrary, in the case where the diffusion mechanism is also important, a deposition pattern uniformly distributed around the fibre would be observed. Moreover, the particle size chosen in our calculations (1 Am) is well above the size limit proposed by Davies [1] and Walsh [18], who suggested that particle Brownian motion contributes to the deposition mechanism only when the particle size is below 0.2 –0.5 Am. A further assumption concerns the strength of the attracting forces between particles, which is assumed to be sufficient to keep them from rebounding and re-entrainment (due to fluid drag) into the flow stream after deposition. The latter assumption becomes less valid in the later stages of the fibres loading. Since the particle density is of order of 103

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times greater than that of the carrier air the effect of the buoyancy can be neglected. The equations of motion for the particles become then a relationship between the fluid drag, the particle momentum and the gravity, as shown in Eq. (2). Projection of this equation along the x- and y-directions (2D formulation) produces the equations: 2

g

ð8Þ BUpy 3qCD ðUf y Upy Þ½ðUf x Upx Þ2 þ ðUf y Upy Þ2 1=2 ¼ Bt 4qp dp Cn ð9Þ The drag coefficient CD in the above equation is obtained as a function of Reynolds number from the relation: 24 ½1 þ 0:15Re0:687

p Rep

ð10Þ

whereby the Reynolds number is based on the relative velocity: ! ! A Up Uf Adp qp Rep ¼ l

ð11Þ

Eq. (10) furnishes, for the initial velocity of the particular flow field, a Stokes number of 8.78. Eqs. (8) and (9) constitute a system of coupled differential equations. The particle trajectories are obtained by solving the system with a forward marching finite difference method. The initial conditions at t = 0 are known and the particles are moving into the x –y space as time elapses. The time step dt is chosen such as the forward spatial step of the particles in every iteration to be of the same order of magnitude with the average spatial distance between the nodes of the numerical grid used for the flow field calculation. This is critical to avoid ‘uncoupling’ between the particle trajectories and the flow field. The velocity values of the flow field, which are used for the determination of the particle relative velocity, are obtained by interpolation of the known values at the neighbouring (to the particle) grid nodes. Then by rearrangement of Eqs. (8) and (9), the particle velocities at the arbitrary time step t n + 1 are obtained from the following equations: nþ1 n ¼ Upx þ f dt Upx nþ1 n Upy ¼ Upy þ f Vdt

n dt x nþ1 ¼ x n þ Upx

n dt y nþ1 ¼ y n þ Upy

ð13Þ

2 1=2

BUpx 3qCD ðUf x Upx Þ½ðUf x Upx Þ þ ðUf y Upy Þ

¼ Bt 4qp dp Cn

CD ¼

velocity after discretisation and rearrangement of the terms):

ð12Þ

where f is a function of the parameters involved in Eq. (8): f = f (Rep, CD, U, V, dp, g) and f V the respective function derived from Eq. (9) (the difference being the gravity term g). The positions of the particles at the same time are given by the recurrent formula (derived from the definition of the

Since the initial conditions of the system at x = 0 and t = 0 0 0 0 0 are known as [(x0,y0), (Upx ,Upy ) and (Ufx ,Ufy )], the algorithm for the determination of the particle trajectories can be described in the following steps. (i) At t1 = t 0 + dt, the new positions of the particle x1, y1 0 0 are computed from (x0,y0) and (Upx ,Upy ) according to Eq. (13) [assuming that the particle travels from (x0,y0) to (x1,y1) in the time interval t0 ! t0 + dt with the constant velocity 0 0 (Upx ,Upy )]. (ii) In the new position (x1,y1), or in the position (x n + 1,y n + 1), the relative velocity of the particle is calculated 0 0 n n by use of the (Upx ,Upy ) [or (Upx ,Upy )] particle velocity component of previous time step and by interpolation of the fluid velocity component from the neighbouring (to the particle) grid points of the known flow field. If (xi,yi) and (xi + 1,yi + 1) are the consecutive grid nodes coordinates between which the particle is positioned (on the x-axis) then the fluid velocity (x-component) at the new particle position is obtained by an interpolation equation of the general form: Uf x ðx nþ1 Þ ¼

Uf x ðxiþ1 Þ ðx nþ1 xi Þ þ Uf x ðxi Þ ðxiþ1 x nþ1 Þ ðxiþ1 xi Þ ð14Þ

In the above equation, the lower index (i) stands for the numbering of the grid coordinates, while the upper index (n) is a notation for the time step of the particle position coordinate. The y-velocity component of the fluid in the same particle position is given by: Uf y ðy nþ1 Þ ¼

Uf y ðyiþ1 Þ ðy nþ1 yi Þ þ Uf y ðyi Þ ðyiþ1 y nþ1 Þ ðyiþ1 yi Þ ð15Þ

(iii) After the fluid velocities have been determined in the latest position of the particle, the relative velocity can be calculated; based upon such velocity, the Reynolds number and the Stokes number are computed from Eqs. (9) and (10), respectively. The functions f and f Vcan then be obtained if the particle and fluid properties are used. (iv) The new velocity components of the particle are obtained from Eq. (12) and the new positions of the particle can be calculated via Eq. (13) like in step (i). The method is then repeated to give the entire trajectory of the particle in several hundreds of consecutive time steps. The coordinates of deposition are marked in the program and a grid of ‘‘blocked’’ space is created around the fibre so that the next particle can be captured by other deposited particles

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115

Fig. 2. Flow field around a single fibre in a staggered arrangement of fibres. Fig. 5. Deposition pattern after 150 released particles.

as well. If no deposition occurs, the particle escapes the region under scrutiny. After the particle under investigation becomes either captured or escapes the area, the next particle is released and the whole process is repeated. The efficiency of the single fibre is calculated by direct numbering of the particles according to their final position. The computational program is written in Fortran language, as the CFD code, and a Pentium III class computer is used for the numerical solution.

6. Results and discussion

Fig. 3. Deposition pattern after 50 released particles.

The case study presented here considers the inertial and the interceptional deposition as the dominant mechanisms

Fig. 4. Numerical grid for the computation of the flow around the dendrite of Fig. 3.

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for particle capture. The aim is to show the effect on the loading process of the velocity field recalculation around the dendrites. This could be achieved by using the developed numerical CFD tool, which showed high modelling flexibility in the flow simulation of the gradually changing filtration geometry. The aerosol particles are released at a random height and with respect to their final position (derived from the trajectory simulation), the single fibre efficiency can be calculated by direct particle counting. The efficiency is given as a function of the number of particles released in the fibre region. The change of shape of the fibre is taken into account and after a certain degree of deposition the computational grid is rearranged around the attached particles (by altering the boundary condition of the gridpoints corresponding to the deposited dendrite) and a new flow field is calculated. The next particle trajectory simulation is based on the new flow field. The process is repeated and the interaction of particles and flow field simulation produces the gradually shaping fibre-dendrite up to the complete filling of the region with particles. The present approach is more accurate than the simulations based on a constant flow field since it takes directly into account the influence of the flow on the particles. The input data for the simulation were based on the assumptions associated with the above mentioned aims of the study. A relative high velocity of 10 m/s was chosen, but compatible with the requirement of laminar/viscous flow for the carrier gas flow. The particle diameter dp was chosen to be 1 Am to retain the low inertia for a small particle and yet excluding Brownian motion and diffusional deposition [1,18]. The size of the fibre (5 Am) was chosen to be small with respect to the particle diameter dp, in order to reduce the

overall number of particles released in the system. The region dimensions were calculated to furnish a filter packing density of 2%, which is a common value for air filters. The initial velocity of the particles, when injected into the region, produces an initial Stokes number of 4.36. This leads to a deposition pattern of particles being captured in an area expanding in both x- and y-directions, which is characteristic of the mixed influence of both inertial and interceptional collection mechanisms. The calculation of the flow field was repeated every time 50 particles had been injected and the final point of calculation was estimated after 250 particles were released. The shape of the final dendrite and the number of deposited particles were compared with the values from a dendrite formed in a ‘‘once-through particle simulation’’ without any changes of the flow field. The single fibre efficiencies were calculated for the two cases and compared to show the influence of the flow on the loading of the filter. A slower increase with the single fibre efficiency was observed time when the interaction between particles and flow was taken into account. The predicted behaviour of the filter is consistent with filtration theory [11]. Similar conclusions have been drawn by other studies, where the effect of the deposition on the flow was again taken into account even if neither the Kuwabara flow field nor a CFD computer code were used for the computation of the flow field [19]. In Fig. 1, the initial grid, used to produce the flow field of Fig. 2, is shown. The first dendrite formed is shown in Fig. 3 (50 particles released) and that corresponds to the grid of Fig. 4 used to calculate the next flow field. The result obtained after 150 particles have been injected is shown in Fig. 5. Different colours correspond to particles captured at different successively recalculated flow fields The grid generated for successive calculations is reported in Fig. 6 and the next

Fig. 6. Numerical grid for the computation of the flow around the dendrite of Fig. 5.

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Fig. 7. Dendritic deposition after 200 released particles.

117

Fig. 10. Deposition pattern after 250 released particles based on the initial flow field of Fig. 2.

pattern for the dendrite is shown in Fig. 7, corresponding to a number of 200 particles released in the region. Fig. 8 shows a zoom on the upper side of the dendrite overlapped by a vector plot of the flow field, indicating how the carrier gas bypasses the deposited particles. Finally, Figs. 9 and 10 represent the final calculations of

Fig. 8. Close view of the dendrite of Fig. 7 overlapped by the new flow field, which bypasses the deposited particles. Fig. 11. Number of captured particles: effect of the flow field re-calculation.

250 released particles. In Fig. 9, the flow has been recalculated, as described above, at every 50 particles, while in Fig. 10 the initial flow field has been kept constant in a once through simulation of the 250 particles. The two dendrites reflect the influence of the flow field on

Fig. 9. Final deposition pattern (‘clogging’) after 250 released particles.

Fig. 12. Single fibre efficiency: effect of the flow field re-calculation.

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the deposition process. The deposited particles force the fluid and the oncoming particles to deviate from their initial course; they then either deposit on the dendrites formed on the neighbouring downstream fibres or escape the area under scrutiny. This has a clear effect on the efficiency as can be seen in Figs. 11 and 12. The number of collected particles on the central fibre is reduced and thus the predicted single fibre efficiency is lower here than in simulations that do not take into account the effect of the changing geometry on the flow. The overall capture efficiencies of the region (due to central plus downstream fibres) have also been calculated and they show a smaller change with respect to the previous condition. This is quite an expected result since particles bypassing the central dendrite are captured more easily by the neighbouring dendrites. Efficiencies have been calculated by direct particle counting and have been averaged on a base of 25 particle increments. In Fig. 11, the net number of collected particles is shown and the behaviour of the curves is consistent with filtration theory and experiments. The results show that good prediction of the particle deposition due to inertia-interception mechanisms is achieved. A novel element is the consideration of the effect of the deposition on the flow and thus indirectly on the evolution of the filter efficiency in time. The degree of achieved accuracy and the modelling flexibility justify in our opinion the mathematical complexity and the higher demand of computational time of this method.

f1P f fV g I J Pe Re Rep Sm St Sv T ! Uf ui ! Up xi xj xn yi

7. Conclusions The effect of the fluid flow recalculation on particle deposition in a fibrous filter has been investigated solving both the flow of the carrier gas and the flow of the particles past the fibre. The results are combined to show the effect of the dendrites on the flow and to consider the influence this has on the efficiency of the filter. The experimentally observed increase of efficiency, as loading increases, is predicted. A slower increase of particle capture and single fibre efficiency is observed with time; this can be attributed to the interaction between particles and gas. We are currently extending the results to include the pressure change. Nomenclature A generic terms of total flux due to convection and diffusion (kg s 1) CD aerodynamic resistance coefficient (dimensionless) Cn aerodynamic slip coefficient (dimensionless) C generic terms of convection contributions to flux (kg s 1) df fibre diameter (Am) dp particle diameter (Am) E, W, N, S east, west, north and south neighbouring gridnodes e, w, n, s east, west, north and south neighbouring cell faces

yi yn bji l q qp Up

weighting function in hybrid-differencing scheme (dimensionless) function of all constant parameters in particle equation of motion (x-direction) (m s 2) function of all constant parameters in particle equation of motion ( y-direction) (m s 2) acceleration of gravity (m s 2) generic terms of cross-diffusive contributions to flux (kg s 1) Jacobean matrix of the coordinate transformation Peclet number (dimensionless) Reynolds number of the flow (dimensionless) Reynolds number of the particle (dimensionless) source term of mass in momentum equations of the flow (kg m 3 s 1) Stokes number of the flow (dimensionless) velocity source term in momentum equations of the flow (kg m 2 s 2) stress tensor in momentum equations of the flow (kg m 2 s 1) fluid velocity vector (m s 1) fluid velocity components in local coordinates (m s 1) particle velocity vector (m s 1) gridnode x-coordinate at the nth time step of particle trajectory (m) generalised curvilinear coordinates of the flow solver (m) x-coordinate of the particle in the nth time step (m) y-coordinate of the gridnode at the nth time step of particle trajectory (m) Cartesian coordinates of the flow solver (m) y-coordinate of the particle in the nth time step (m) partial derivatives of coordinate transformation of flow solver (dimensionless) fluid viscosity (N s m 2) fluid density (kg m 3) particle density (kg m 3) generic variable for the unknown velocity of point P in the algebraic form of the momentum equations (m s 1)

Acknowledgements Support from EPSRC (Grant GR/L77614) is greatly acknowledged.

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