Volume 5, Issue, Summer006 Approaches for Preicting Collection Efficiency of Fibrous Filters Q. Wang, B. Maze, H. Vahei Tafreshi, an B. Poureyhimi Nonwovens Cooperative esearch Center, North Carolina State niversity, aleigh, NC 7695-801 ABSTACT This paper escribes ifferent approaches for preicting collection efficiency of nonwoven fibrous filters. Traitionally, the flow fiel has been obtaine by analytically solving the Navier- Stokes equations insie over-simplifie geometries with the fibers place in regular arrays perpenicular to the flow irection. Our approach, on the other han, is to exploit a numerical metho base on the finite -volume technique to solve the flow insie - virtual webs generate base on the properties of real filters. Our results showe a goo qualitative agreement with previous works. Keywors: Permeability; Cell moel; CF; Filtration; Nonwoven. INTOCTION uring the past several years, there have been many pioneering stuies (avies, 197; Brown, 1998; Hins, 1999), which have helpe eveloping the filtration science an technology to its current level. This is because the rising awareness of environmental agencies an the general public for a clean environment together with emans of many avance inustries have urge the filtration inustry to investigate on ways to prove the inoor-air-quality. Fibrous filters, such as nonwoven meia, are wiely use to remove submicron particles owning to its low cost an easy implement. It is quite important to evaluate the filters performance of base on various particle an filter properties, which is generally characterize by collection efficiency an other parameters. In next section, we first introuce the traitional analytical metho to preict the collection efficiency. Section introuces our algorithm for generating nonwovens an then investigates on their nano-particle collection efficiency. In section 4, we compare our stuy with the well-known available - moel mentione in section. Analytical Moels: Cell Moel It is ifficult to analytically analysis the flow fiel aroun a fiber in the real fibrous filter. In orer to preict collection efficiency of fibrous filter, influence of a single fiber has been extensively investigate. Several results from calculation of efficiency of this smallest element in the fibrous filter have been obtaine base on ifferent flow moels. Potential flow an Corresponing author, email: hvtafres@ncsu.eu, telephone: 919-51-4778, fax: 919-515-4556 Article esignation: eferee 1 Volume 5, Issue,Summer 006
Lamb flow are two of those early moels escribing the flow aroun a cylinrical fiber (avies, 197), however because there is no neighboring fiber influence, they cannot present the realistic flow. The neighboring fiber interference was first aequately taken into account by Kuwabara (1959) an Happel (1959). Both theories iffer from the earlier isolate fiber moels, since they consier the fiber in a finite space instea of an infinite space (shown in Figure 1). For this reason, the effects of other fibers at the outlet coul be taken into account (Brown, 199). Figure 1: Cell Moel consiers a fiber in a finite space It has been known that the flow fiels near the surface of the cyliners without consiering the slipping effect coul be expresse in polar coorinates the equations by (Stechkina an Fuch, 1965; Happel, 1959; Kuwabara, 1959): = 1 ( 1 ln ) cosθ (1) = 1 θ ( 1 ln ) sin θ + () in which is the imensionless istance an an θ are the components of the which imensionless flow velocity 0 coul be represente by = 0 K () where is the flow velocity an K is the hyroynamic factor, which coul be varie by ifferent moels. K in Cell moel is Kuwabara hyroynamic factor Ku, which coul be represente by ln α α Ku = + α (Kuwabara, 4 4 1959; ao an Faghri, 1988). Collection efficiency of a fibrous filter E coul be obtaine by the single fiber efficiency by the following equation (Lee an Liu, 198 ;Hins, 1999): 4α t E = 1 exp( ) (4) π f in which α is the Soli volume fraction, is the air viscosity, t is the thickness of the filter, an f is the fiber iameter. Among all the mechanisms which will influence collection efficiency, interception, Brownian iffusion an inertial impaction are three of the most importance for pure aerosol filtration (avies, 197). Base on the stuies one by Hins (1999) an Lee et al (198), the arithmetic sum of efficiencies for these mechanisms coul be consiere as an approximation of efficiency for the single fiber. = + + (5) I in which, an I are the single fiber efficiency ue to the interception, iffusion an inertial impaction, respectively. In the most penetrating particle size region (aroun 00 nm), work of Stechkina et al. (1969) has shown iffusion an interception are ominant filtration mechanism. Particularly, interception is ominant when is comparable to p f while Brownian motion is the most important mechanism when f >> p. The interception of a homogenous spherical particle by a cylinrical fiber coul be efine as a particle is intercepte by a fiber when the istance from the center of the particle mass to the fiber surface is equal or less than the raius of the particle (Hins, 1999). In the Kuwabara flow, the expression of the single fiber efficiency ue to the interception is (Lee an Liu, 198). Article esignation: eferee Volume 5, Issue,Summer 006
1 + 1 α α = [ ln( 1 + ) 1 + α + ( ) (1 ) (1 + ) ] (6) Ku 1 + In which is the fiber raius, an α is the soli volume fraction which equals to the ratio of fiber volume an filter volume. There exist several approximation forms of Equation 6 (Stechkina an Fuch, 1966; Lee an Liu, 198.), yet small iscrepancies from those approximation forms to the original expression cannot be avoie. The most wiely accepte approximation of the single fiber efficiency ue to the interception maybe is the simplifie form obtaine by Lee an Liu (198). 1 α = (7) Ku 1+ It is obvious to obtain from the above equation that will be high while is big, an low when is small. For aerosol filtration, particle may erivate from its original streamline when passing through a fibrous filter meia. The erivation is cause by the combination of particle inertia an iffusion ue to Brownian motion. erivation of Brownian motion governs the particle trajectory when the particle is extremely small compare to the fiber size (Hins 1999). Steckina an Fuchs (1966) are the first to use cell moel to eal with the efficiency cause by Brownian motion. They trie to get the particle trajectory ue to the Brownian motion by solving convective iffusion of inertial-less particles towars a cylinrical fiber in the imensionless coorinates (Stehina an Fuch, 1965). coul be calculate by the following equation: δ (1 + ) π n = ( ) θ (8) Ku 0 = 1+ The single-fiber efficiency for this mechanism ( ) achieve by the metho mentione above, coul be presente as follows: 1/ / 1 =.9Ku Pe + 0.64Pe (9) By using the metho of bounary layer theory, Lee an Liu (198) shows another preiction of iffusion by using Kuwabara cell moel 1 α 1/ / =.6( ) Pe. (10) Ku In which Pe is the Peclet Number. Lee an Liu claime with the inclusion of the factor 1 α in the theoretical expression, the results coul be applie over a wier range of the conition, especially for the case when α is high. Inertial impaction is another erivative eposit mechanism. It coul be efine as (Hins, 1999) as a particle, because of its inertia, is unable to ajust quickly enough to the abruptly changing streamlines near the fiber an crosses those streamlines to hit the fiber. The single fiber efficiency for impaction is given by (Lee an Liu, 198; Hins 1999) as, ( Stk) J I = (11) Ku 0.6.8 wherej = ( 9.6 8α ) 7.5 for < 0. 4 an Stk is the Stokes number, efine as the ratio of particle stopping istance to fiber iameter. It is obvious to see that collection efficiency of a fibrous filter is a function of a number of parameters of flow, particle an filter an coul be easily calculate base on the equations by using cell moel. However, a precise preiction of the collection efficiency requires the particle trajectory base on the flow to be solve for each fiber in the filtration meium, an to characterize the efficiency for single fiber by using the average result; In the cell moel the fibers insie the filter are treate the same way, even though there exist big ifference among fibers (Brown, 1998). Article esignation: eferee Volume 5, Issue,Summer 006
Numerical Moels: - Virtual Web Moel As we coul see from section, most of the previous stuies have been limite to systems consisting of rows of fibers perpenicular to the flow irection in a two-imensional geometry. To our knowlege, there have been few attempts in realistically simulating the filter s isorere structure in a three-imensional geometry (Wang et. al 006). Moreover, the role of the filter structure an its relationship with performance of the meia has not been fully establishe. This is because of the ifficulties involve not only in generating - structures similar to a nonwoven meia, but in calculating the particle collection efficiency when the geometry is too complex as well. The current stuy is the first step to preict the collection efficiency base on - virtual webs (Wang et. al 006). Our virtual moels of these fiberwebs are basically similar with the real nonwoven meia. For simplicity, we assume that fibers lie horizontally in the plane of the web an o not ben at the crossovers. One of the most important features of our structure simulation is that it allows the orientation istribution of the fibers to be taken into consieration (as oppose to the cell moel which is base on a regular - fiber istribution) (Poureyhimi et. al 1996a; Poureyhimi et al. 1996b; Poureyhimi et al. 1997). To generate a - web, first a new fiber is generate at an altitue greater than the current thickness of the web. The fiber is then lowere until it reaches the fiber-webs. For more etails reaer are referee to our previous publication by Wang et al (006). The ifference existing between the longfiber web moel (such as Electrospinning, Meltblowing, an Spun-boning) an shortfiber web moel (such as Air-laying, Caring an Wet-laying) is that we use the so-calle µ-ranomness for generating continuous filaments an I-ranomness for short fibers (Poureyhimi et. al 1996a; Poureyhimi et al. 1996b; Poureyhimi et al. 1997). A steay state laminar incompressible moel has been aopte to escribe the flow regime insie our virtual geometry. The finite volume metho implemente in Fluent coe is exploite to solve the flow fiel. (a) (b) Figure : Nonwoven webs an their virtual moel: a) a typical long fiber web; b) a typical short fiber web. The governing equations incluing continuity, conservation of linear momentum, an energy coul be written in vectorial form as: + V t = 0 (1) V = p ( V) + 4 / ( V) (1) t T p c p = + Φ + k T (14) t t In above-mentione equations,, k, an c p represent air ensity, thermal Article esignation: eferee 4 Volume 5, Issue,Summer 006
conuctivity, an the specific heat of air, respectively. Symmetry bounary conition has been use for the sies of the computational box, even though there is no plane of symmetry in a isorere fibrous structure. This bounary conition is consiere for the simulations because there lacks of information regaring the flow velocity an/or pressure insie of the structure prior to the simulations. We assume a no-slip bounary conition for the air flow on the fiber surfaces. This is because for the air thermal conition an the fiber iameter consiere in this stuy, the continuum flow prevails, i.e., Kn f = λ / f << 1, where Kn f is the Knusen number of fiber, an λ is the mean free path of the air molecules (64.5 nm at STP). Once the particle-free flow fiel is obtaine, the airborne particulates, moele by rigi spheres of uniform ensity p = 1000kg/ m, are introuce into the flow omain. Particle trajectories are then tracke via the Lagrangian metho an their positions are monitore. Figure shows a typical trajectory of one particle insie flow omain. Figure : Particle trajectory of - virtual webs. Efficiency of a fibrous filter is etermine by the number of particles can remove from an aerosol flow: N in Nout E = (15) Nin where Nin an Nout are the number of entering an exiting particles, respectively. esults an iscussion Figure 4 shows the result of the simulation base on one of the above mentione - virtual web with ifferent operating flow velocity an particle size. It is obvious that the collection efficiency is higher for the nano particle uner with the low velocity. This is because for the nano particle, the ominant collection efficiency is Brownian iffusion. For low velocity, the particle will have more resience time insie Article esignation: eferee 5 the filter an higher opportunity to be capture by the fiber. Collection Efficiency (%) 50 45 40 5 0 5 0 15 10 5 0 v=0.1 m/s v=0.5 m/s 10 1 10 10 Particle iameter (nm) Figure 4: Collection efficiency is higher for the nano-particle at low velocity Volume 5, Issue,Summer 006
Figure 5 shows the average collection efficiency of our virtual filter versus particle iameter along with the preictions of the Kuwabara s cell moel. Monoisperse aerosols having a particle size ranging of 100 nm to 400 nm have been introuce to the simulation omain an tracke insie the structure. The inlet face velocity of was consiere to be 0.1 m/s. It can be seen that the filter collection efficiency ecreases by increasing the particle size in the range of 100 nm to 400 nm. The collection efficiency preicte by our simulations follows a tren very similar to that of the Kuwabara s cell moel. However, there seems to be a slight ifference between the two preictions. Collection Efficiency (%) 10 8 6 4 0 Present Work Cell Moel - 0 100 00 00 400 500 Particle iameter (nm) Figure 5: A comparison between collection efficiencies from CF an cell moel. Conclusions In this work, we simulate the filtration process of nanoparticle via ifferent categories of nonwoven meia. The numerical simulations presente here are obtaine via solving the Navier-Stokes equations insie virtual - filter meia constructe in accorance with the real features of the nonwoven filters. We emonstrate that the collection efficiencies are higher for smaller face velocities. Our simulate collection efficiencies were compare with the preictions of the Kuwabara s cell moel an similar tren has been foun. Acknowlegement The current work is supporte by the Nonwovens Cooperative esearch Center. Their support is gratefully acknowlege. EFEENCE: Brown,.C., 1998. Airflow through filtersbeyon single-fiber theory. In: Spurny, K.., (e) Avances in Aerosol Filtration. Lewis Publishers. avies, C.N., 197. Air Filtration. Acaemic Press, Lonon. Happel, J., 1959.Viscous flow relative to arrays of cyliners., Aiche Journal 5(),174-177. Hins, W.C., 1999. Aerosol technology: properties, behavior, an measurement of airborne particles, n en. Wiley, New York. Kuwabara, S., 1959. The forces experience by ranomly istribute parallel circular cyliners of spheres in a viscous flow at small eynols number. Journal of The Physical Society of Japan 14 (4), 57-5. Lee, K.W., Liu, B.Y.H., 198. Theoretical stuy of aerosol filtration by fibrous filters 1(), 147-161. Poureyhimi, B., amanathan,., ent,., 1996a. Measuring fiber orientation in nonwovens.1. Simulation. Textile esearch Journal 66 (11), 71-7. Poureyhimi, B., amanathan,., ent, P., 1996b.Measuring fiber orientation in nonwovens.. irect tracking. Textile esearch Journal66 (1), 747-75. Poureyhimi, B., ent,., avis, H., 1997. Measuring fiber orientation in nonwovens.. Fourier transform. Textile esearch Journal 67 (), 14-151. Article esignation: eferee 6 Volume 5, Issue,Summer 006
ao, N., Faghri, M., 1988. Computer moeling of aerosol filtration by fibrous filters. Aerosol Science an Technology 8 (), 1-56. Stechkina, I.B., Fuchs, N.A., 1965. Stuies on fibrous aerosol filters-i. Calculation of iffusional eposition of aerosols in fibrous filters. Annual of Occupational Hygiene 9, 59-64. Stechkina, I.B., 1966. iffusion precipitation of aerosols in fibrous filter. oklay Akaemi Nauk SSS 167 (6), 17-. Stechkina, I.B., Kirsh, A.A., Fuchs, N.A., 1969. Investigations of fibrous filters for aerosols calculation of aerosol eposition in moel filters in regions of maximum particle break through. Colloi Journal- SS, 97-. Wang, Q., Maze, B., Vahei Tafreshi, H., Poureyhimi, B. 006. Acase stuy of simulating submicron aerosol filtration via lightweight. Chemical Engineering Science 61, 4871-488 APPENIX Notation: c p Specific heat of air f Fiber iameter p Particle iameter E Collection efficiency J Parameter relate with inertial impaction,where J 0.6.8 = ( 9.6 8α ) 7.5 for < 0.4 k K Thermal conuctivity Hyroynamic factor in Cell moel Kn f Knusen number of fiber Ku Hyroynamic factor in Kuwabara Cell moel, where ln α α Ku = + α 4 4 N in Number of entering particles of the filters N out Number of exiting particles of the filters Pe Peclet number Fiber raius Stk Stokes number (the ratio of particle stopping istance to fiber iameter). t α Thickness of the filter Flow velocity Soli volume fraction Single fiber efficiency Single fiber efficiency ue to the interception Single fiber efficiency ue to the iffusion I Single fiber efficiency ue to the inertial impaction λ The mean free path of the air molecules (64.5 nm at STP) µ Air viscosity Air ensity p Particle ensity? Viscous issipation Article esignation: eferee 7 Volume 5, Issue,Summer 006