Collection Efficiency for Filters with Staggered Parallel Y and Triple Y Fibers: A Numerical Study

Transcription

1 Colleton Effen for Flters wth Staggered Parallel Y and Trle Y Fbers: A Numeral Stud Huanng Zhu Juan P. Hnestroza College of Human Eolog Cornell Unverst Ithaa New York USA Corresondene to: Juan P. Hnestroza emal: jh433@ornell.edu ABSTRACT A numeral stud s erformed to determne the olleton effen of flters omosed of staggered arallel Y and trle Y fbers. B emlong the Latte Boltzmann method the Naver Stokes equatons are solved for flow felds n a twodmensonal doman wth erod boundar ondtons. Trajetores of artles wth dfferent dameters arrangng from. µm to 5 µm are then omuted b solvng the artle moton equaton hene obtanng the olleton effen of sngle Y and trle Y fbers. The effets of fber orentaton Brownan moton artulate sze and Renolds number on the olleton effen of these two tes of fbers are also evaluated numerall. Y and trle Y fbers have low akng denst and relatvel large area to volume rato and the latter roert hels enhane the Brownan dffuson olleton mehansm for smaller artles. The large vod zone of these fbers an aommodate more artles whh mroves the artle loadng aat. It s also found out that the flow feld and olleton effen of trle Y fber s less susetble to dfferent fber orentatons due to ts geometr smmetr. The numeral results obtaned n ths work rovde helful nformaton n desgnng hgh effen flters. INTRODUCTION Fbrous flters have been wdel used to remove artles from a gas flow. The effetveness of flters n aturng artles s manl aheved through the ombned effet of Brownan dffuson ntereton nertal maton and gravtatonal sedmentaton. Eletrostats ma also affet fbrous fltraton deendng on artle sze and eletro-roertes of the flter materal. Conventonal fbrous flters are made of randoml aked long lndral fbers however fbers wth nonrular ross-seton have great otental to make flters wth ossble enhanement n orost artle loadng aat and olleton effen. As a result researh n fbrous meda shows a growng nterest n fbers wth nonrular ross-setons n the ast deades [1-7]. For examle Fard and Lu [1-] develoed a numeral model for solvng the flow feld through a staggered arra of retangular fbers and then determned the sngle-fber olleton effen due to the mehanal olleton mehansms (nertal maton ntereton and dffuson). Based on Fard and Lu s aroah [1-] Chen et al. onduted a numeral stud to determne the olleton effen of a sngle retangular fber under dfferent fber aset rato flter akng denst artulate sze and Renolds [5]. Zhu et al. nvestgated the maton-domnated fltraton roess for flters wth retangular fbers [6]. B onsderng the eletrostat Cao et al. also develoed a numeral model to smulate the olleton effen of sngl harged or neutral artles n an eletret flter omosed of arra of retangular fbers [7]. Consderng that Y and trle Y fbers have large orost large surfae area to volume rato and large artle loadng aat we nvestgated the olleton effen of flters onsstng of staggered arra of arallel Y or trle Y fbers. We frst omuted the flow feld n a reresentatve ell ontanng a sngle or trle Y fber. To handle the omlex geometr and boundar ondtons of the flow roblem we hoose to use the Latte Boltzmann method (LBM) whh has been establshed as a owerful alternatve to the onventonal Naver-Stokes solvers n omutatonal flud dnams [8-11]. The LBM rovdes numerous advantages nludng lear hsal tures multsale smulaton aabltes and full arallel algorthms. In addton omlex boundar ondtons an be mlemented n LBM easl through a soalled boune-bak sheme. After the flow feld s obtaned trajetores of artles wth dfferent dameters arrangng from. µm to 5 µm were omuted b solvng the artle moton equaton n order to obtan the olleton effen of sngle Y and trle Y fbers. A detaled analss of the numeral results s resented n the results and dsussons seton whh hels understand the effet 16

2 of fber orentaton Brownan moton artulate sze and Renolds number on the olleton effen of these two tes of non-rular fbers. THEORY Comutaton of Flow Feld usng Latte Boltzmann Method In the Latte Boltzmann method the flud s reresented b artles wth dsrete velotes. The nteraton between the artles s desrbed b a lnearzed relaxaton-te ollson term. The Naver Stokes equatons follow from ths model as a low- Mah number lmt and the vsost of the lqud lnearl deends on the relaxaton rate. Among the Latte Boltzmann methods reorted n the lterature the most wdel used one s the latte Boltzmann Bhatnagar-Gross-Krook (BGK) model whh s abbrevated as LBGK [8 1]. The LBGK equaton utlzes a sngle relaxaton tme aroxmaton and s desrbed b the followng equaton f ( x + δ t t + δ t) f ( x t) 1 eq f ( x t) f ( x t) = τ [ ] = 1... q 1 (1) where f ( x t) s the denst dstrbuton funton along the dreton at latte ste x and tme t δ t s the tme ste s the dsrete velot and τ s the dmensonless sngle-relaxaton tme. Funton eq f the loal Maxwell te dstrbuton funton exressed as a Talor exanson to seond-order n flud velot. The denst ρ and bulk velot u an be determned from q 1 = ρ f () = q 1 = ρ u f (3) = resetvel. The relaxaton tme τ an be related to the knemat vsost ν b ν = s δ t ( τ. 5) (4) for a square latte where s s the sound seed of the latte flud whh s defned as = 1/ 3. s s FIGURE 1: Shemat lot of artle velot for the DQ9 model. An addtonal vetor wth zero omonents = ( ) s defned denotng a rest artle oulaton. In the oular nne-velot square latte model DQ9 the artle velotes are gven b Qan et al. [11] = = = 1 4 [ os (( 1) π / ) sn (( 1) π / ) ] = = 5 8. [ os (( 5) π / + π / 4) sn (( 5) π / + π / 4) ] (5) The equlbrum dstrbuton for the DQ9 model as stated b Qan et al. s: [11] f eq ω = = ρω ω = u+ = 1 4 ( u) 3 u u (6) 1 = ω = Through a mult-sale Chaman-Enskog analss of the LBGK model [81] t an be shown that the maroso varables ρ and u defned n equatons () and (3) obe the Naver Stokes equaton for a weakl omressble flud ( ρ u) = t ρ + (7) 1 t u + ( u ) u = - + ν u. (8) ρ x 17

3 And the ressure s gven b the equaton of state u s = ρ. (9) FIGURE : Geometr of staggered arra of Y fbers The flter s modeled as a staggered arra of nfntel long arallel Y or trle Y fbers fang to a horzontal flow dreton as shown n Fgure. The flow feld s assumed to develo a erod attern along the flow dreton therefore t an be omuted usng to a reresentatve ell ABEF. In the atual omutatons we emlo onl half of the orgnal ell b takng advantage of the smmetr roert of ths flow feld where the uer art ABCD s an nverson of the other half CDEF. The olleton effen s omuted under a stead flow feld and the Buoan effets are negleted. A no-sl boundar ondton s aled at the nterfae between flud and fber. Followng Chen et al. [5] the boundar ondtons for the omutaton doman ABCD are exressed as follows: along the uer boundar AB u = u = (1) x along the lower boundar CD u = u = (11) x A along the surfae of the fber u u = (1) x = X and long the left boundar AC B C D Y E F u u x ( x ) = u x ( x+ X Y / ) ( x ) = u ( x+ X Y / ). (13) Colleton Effen of a Sngle Fber A rgorous analss of the two-wa oulng of artle and flud nteratons s needed for a omlete understandng of artulate flows around fbers [1]. However for engneerng alatons of ver dlute artulate susensons wth artle volume onentratons less than.1 the effet of artles on the flow feld ma beome neglgble [6]. Under suh ondtons the flud and sold hases an be treated searatel. In the absene of external s suh as eletrostat and gravtatonal s the trajetor of ndvdual artles an be determned b solvng the followng equatons of artle moton: du = a d + dt a d a r 18μ = ρ d C ( u u ) (14) (15) where u s artle velot ρ and d are the denst and dameter of artles resetvel. The term a d n equaton (14) s the aeleraton of a artle due to the drag n whh the relatve sl between the artle and the flud s also onsdered. Varable C s the Stokes Cunnngham sl fator gven b Abuzed et al. [13] C 1.1d / ( e ) λ λ = 1+ (16) d where λ s the mean free ath of the gas hase. The term a r n equaton (14) s aeleraton due to the Brownan dffuson effets whh las an mortant role for submron artles. Followng Abuzed et al. [13] Chen et al. [5] and Cao et al. [6] the Brownan s modeled as a Gaussan random roess. The omonent of the Brownan for unt mass a r = x at an tme ste t s alulated as ( ) a r () t = Z π S Δt (17) where Z s a random number wth a zero mean and unt varane whle Δ t s the tme ste. The term S 18

4 s the setral ntenst of a whte nose roess [14-15] gven b S 16μ kt = (18) 5 π d ρ C where T s the flud absolute temerature and k s the Boltzmann onstant. A large number of artles are requred for statstal relablt of the omutatonal results. Postons of the aroahng artles on the nlet lane are randoml hosen [5 7] and for eah artle the trajetor s obtaned b numerall ntegratng equaton (14). The number of artles whh are olleted b the fber or leave the omutaton ell s ounted. The olleton effen of artles for a sngle fber s onsequentl defned as the rato of the number of artles olleted on the fber to the number of artles enterng the omutatonal doman. RESULTS AND DISCUSSIONS In the resent stud the sze of the reresentatve ell (ABEF n Fgures ) s μm tmes μm and wthn the ell Y or trle Y fbers are stuated n the enter. fber s omosed of three equal length (33.3μm) branhes whh jon at the enter of the doman. The wdth of these fber branhes s 4μm and the angles these branhes make wth ostve x-axs are 1 and 4 degree resetvel. As for the Y fber wth reversed orentaton the angles these three branhes make wth the ostve x-axs are 6 18 and 3 degree resetvel. The trle Y fbers onsst of three smaller sze Y fbers that jon at the enter of the doman (Fgures 3). The length of the branhes of the smaller Y fbers s 16.7μm. Trle Y fber wth reversed orentaton s also onsdered (Fgures 3d). It s noted that all these four fbers have the same ntereton rato 8.9 erendular to the flow dreton.e. ostve x dreton. The fber volumetr akng denst s 1. for the Y fbers and 1.5 for the trle Y fbers. Partle denst hosen s kg/m 3 and the gas vsost s kg/m s. Two Renolds numbers 1 and 1 of the flow felds are onsdered n the resent stud whh are omuted based on the length of the reresentatve ell Vx/V Theoretal soluton Numeral data /Y FIGURE 4: Velot rofles of lane Poseulle flow. FIGURE 3: Illustraton of geometr of Y and trle Y fbers: (a) Y fber (b) Y fber n reversed orentaton () trle Y fber and (d) trle Y fber n reversed orentaton. As a verfaton of our LBM algorthm we frst erformed a numeral smulaton of a standard lane Poseulle flow and omare numeral results wth analtal solutons. Fgure 4 shows a good quanttatve agreement between the numeral results and the analtal soluton. For eah Y or trle Y fber we also onsder ts reversed orentaton; therefore there are totall four tes of geometr onfguratons Y fber and Y fber wth reversed orentaton trle Y fber and trle Y fber wth reversed orentaton (Fgures 3a-d). The Y 19

5 FIGURE 5a: Streamlne around a Y fber Re =1 FIGURE 5d: Streamlne around a trle Y fber n reversed orentaton Re =1. To obtan a relable flow feld a onvergene test s erformed b refnng latte denstes untl a 1 tolerane for velot s reahed. The dsrete latte emloed for the omutatonal ell ABCD was b 1 n x and dreton resetvel. Fgures 5a-d show the streamlnes for Y and trle Y fbers wth dfferent orentatons. The re-rulaton zones an be found at dfferent loatons deendng on the fber geometr and flow dreton. It s also observed that the two flow atterns for the flow domans ontanng trle Y fbers (Fgures 5-5d) are loser than those n the doman ontanng Y fbers (Fgure. 5a-5b). Ths s manl due to the fat that the sx end onts of the outer branhes of the trle Y fber have the same sae oordnates as those of the trle Y fber wth reversed orentaton. In other words the trle Y fbers have a more sotro geometr onfguraton whh has an mat on the orresondng flow feld. FIGURE 5b: Streamlne around a Y fber n reversed orentaton Re =1. FIGURE 6a: Partle trajetor d =. μ m Re =1. FIGURE 5: Streamlne around a trle Y fber Re =1.

6 FIGURE 6b: Partle trajetor d = 1μ m Re =1. artle trajetor llustrates the domnant Brownan dffuson effets. For a artle of d = 1μ m under the same Renolds number Re = 1 (Fgure 6b) the Brownan moton effets beomes less mortant and the artle trajetor aroxmatel follows the streamlne. Under suh rumstanes the man olleton mehansm beomes ntereton. As the artle sze nreases and Renolds number beomes larger the artle trajetores devate the streamlne and the major olleton mehansm beomes nertal maton. Fgure 6 shows the nertal maton effet for a artle wth d = 5 μ m under the flow of Re =1. Fgure 6d llustrates the effet of Brownan moton. FIGURE 6: Partle trajetor d = 5 μ m Re =1. Colleton effen w/o Brownan w Brownan FIGURE 7a: Colleton effen of a Y fber for artles wth dfferent dameters Re =1. FIGURE 6d: Partle trajetor d =. μ m Re =1. Fgures 6a to 6d show the trajetores of artles wth dfferent szes whh onsder Brownan dffuson. Sefall Fgure 6a shows the trajetor of a artle wth d =. μ m under the flow ondton of Re = 1. The zgzag ath of the Colleton effen w/o Brownan w Brownan FIGURE 7b: Colleton effen of a Y fber wth reversed orentaton for artles wth dfferent dameters Re =1. 1

7 Colleton effen w/o Brownan w Brownan FIGURE 7: Colleton effen of a trle Y fber for artles wth dfferent dameters Re =1. Colleton effen w/o Brownan w Brownan FIGURE 7d: Colleton effen of a trle Y fber wth reversed orentaton for artles wth dfferent dameters Re =1. Fgures 7a-d show the smulated sngle fber olleton effenes for artles wth sze rangng from. μm to 5 μm under the flow feld of Re =1. Wthout Brownan dffuson effets the olleton effen s a monotonall nreasng funton of artle sze; whle onsderng Brownan effets the olleton effen frst dereases as artle sze nreases untl mnmum value s reahed and then nreases wth further nreases n artle sze. It s found that the values of olleton effen are sgnfantl hgher for artles smaller than.1μm when the Brownan dffuson effet of artles s a domnant olleton mehansm. As the artle sze beomes larger than 1μm the nertal maton olleton beomes more mortant resultng n hgher olleton effen. 3. B omarng the olleton effen between the Y and trle Y fbers we also fnd that for smaller artles the olleton effen values of the trle Y fbers are hgher than those of Y fbers. Obvousl the larger surfae area of trle Y fbers s n favor of Brownan dffuson effet. Colleton effen w/o Brownan w Brownan FIGURE 8a: Colleton effen of a Y fber for artles wth dfferent dameters Re =1. Colleton effen w/o Brownan w Brownan FIGURE 8b: Colleton effen of a Y fber wth reversed orentaton for artles wth dfferent dameters Re =1. Fgures 8a-d show the sngle fber olleton effenes for dfferent sze of artles under the flow feld of Re =1. As the artle sze nreases the olleton effen nreases sgnfantl beause at hgh Renolds number and large artle sze the nertal maton domnates the other mehansms. It s also noted that at a hgher Renolds number of Re =1 the Brownan dffuson

8 Colleton effen w/o Brownan w Brownan FIGURE 8: Colleton effen of a trle Y fber for artles wth dfferent dameters Re =1. Colleton effen w/o Brownan w Brownan FIGURE 8d: Colleton effen of a trle Y fber wth reversed orentaton for artles wth dfferent dameters Re =1. surfae area of trle Y fbers omared wth that of Y fbers. Colleton effen Y fber Y fber wth reversed orentaton FIGURE 9a: Comarson of olleton effen of Y fbers wth dfferent orentatons Re =1. Colleton effen Y fber 3Y fber wth reversed orentaton FIGURE 9b: Comarson of olleton effen of trle Y fbers wth dfferent orentatons Re =1. effet on olleton effen s less romnent. As a result the olleton effen dereases at hgher Renolds number for smaller artles whh s n aordane wth the results of Chen et al. [5] and Ramarao et al. [16] For large artles ( d μ m ) the olleton effenes are muh hgher than those under smaller Renolds number a onsequene of the domnaton of nertal maton mehansm. Though at hgher Renolds number the Brownan dffuson effet on olleton effen s less romnent we an stll observe that for smaller artles the olleton effen of trle Y fbers s noteabl hgher for the ases onsderng Brownan effets. Ths dfferene an hardl be observed for Y fbers. Agan ths an be asrbed to the larger Colleton effen Y fber Y fber wth reversed orentaton FIGURE 9: Comarson of olleton effen of Y fbers wth dfferent orentatons Re =1. 3

9 Colleton effen Y fber 3Y fber wth reversed orentaton FIGURE 9d: Comarson of olleton effen of trle Y fbers wth dfferent orentatons Re =1. The omarsons of olleton effen of Y and trle Y fbers wth dfferent orentatons and Renolds numbers are shown n Fgures 9a-d. The olleton effen for Y fber s hgher than that of Y fber wth reversed orentaton; whle for trle Y fbers t s true the other wa around. In addton for flow felds wth the same Renolds number the dfferenes of olleton effen values between the two trle Y fbers are smaller than that of the two Y fbers. As we have dsussed earler n ths seton trle Y fbers have a more sotro geometr onfguraton than that of Y fbers whh results n loser flow felds and hene olleton effen values. CONCLUSIONS Colleton effen of mro and sub-mron szed artles flow through flters onsstng of staggered arra of Y and trle Y fbers was nvestgated b onsderng Brownan dffuson ntereton and nertal omaton. The effet of Renolds number and fber orentaton was also studed. It s onluded that at low Renolds numbers the Brownan dffuson la a major role on olleton effen eseall for artles wth dameter smaller than.1 μ m. The smulaton results also onfrm that artle sze rangng from.1 to 1 μ m are harder to ature n other words artles n these ranges enetrate the flter most. Other olleton mehansms suh as eletrostats mght have to be onsdered n olletng artles n ths range. If the artle dameter further nreases larger olleton effen an be aheved eseall under hgher Renolds numbers where nertal mat beomes the domnant mehansm. The above onlusons are also generall true for aerosol flters wth other te of fbers suh as retangular or rular ross-seton n shae. What s new for the Y and trle Y fbers n the resent stud s that the have lower akng denst large area to volume rato and large vod volume. On one hand ths nrease olleton effen for smaller artles when Brownan s domnate; on the other hand those vod zones an aommodate more artles whh wll otentall extend the serve lfe of these flters. In addton we fond out that the flow feld and olleton effen of trle Y fbers are less senstve to fber orentatons due to ther geometr smmetr onfguraton. Ths s an amenable roert n that no matter what dreton the ar flow through the flter we are able to aheve smlar olleton effen. ACKNOWLEDGMENT Ths work made use of the Researh Comutng falt of the Cornell Center for Materals Researh (CCMR) wth suort from the Natonal Sene Foundaton Materals Researh Sene and Engneerng Centers (MRSEC) rogram (DMR 544). REFERENCES [1] Fard B. Lu B. Y. H.; Flow feld and ressure dro of flters wth retangular fbers. Aerosol Sene and Tehnolog [] Fard B. Lu B. Y. H.; Effen of fbrous flters wth retangular fbers. Aerosol Sene and Tehnolog [3] Ushe Z.; n: Leung W. W. Ed.. Advanes n Fltraton and Searaton Tehnolog [4] Mng O Y.; Lu B. Y. H.; Analtal soluton of flow feld and ressure dro for flters wth retangular fbers. J. Aerosol Sene [5] Chen S. Cheung C. S. Chan C. K.; Zhu C.; Numeral Smulaton of Aerosol Colleton n Flters wth Staggered Parallel Retangular Fbers Comutatonal Mehans [6] Zhu C.; Ln C. H.; Cheung C. S.; Inertal maton domnated fbrous fltraton wth retangular or lndral fbers. Powder Tehnolog [7] Cao Y. H.; Cheung C. S.; Yan Z. D.; Numeral Stud of an Eletret Flter 4

10 Comosed of an Arra of Staggered Parallel Retangular Slt-Te Fbers Aerosol Sene and Tehnolog [8] Chen S.; Doolen G. D.; Latte Boltzmann method for flud flow Ann. Rev. Flud Meh [9] Lallemand P.; Luo L.-S.; Theor of the latte-boltzmann method: dserson dssaton sotro Gallean nvarane and stablt Phs. Rev. E [1] Chen S.; Chen H.; Martnez D.; Matthaeus W.; Latte Boltzmann model for smulaton of magnetohdrodnams Phs. Rev. Lett [11] Qan Y.; d Hum`eres D.; Lallemand P.; Latte BGK Models for Naver-Stokes Equaton Eurohs. Lett [1] Fan L. S.; Zhu C.; Prnles of Gas Sold Flows Cambrdge Unv. Press [13] Abuzed S.; Busnana A. A.; Ahmad G.; Wall deoston of aerosol artles n a turbulent hannel flow. J. Aerosol S [14] Iwan D. W.; Mason B. A. Jr.; Equvalent lnearzaton for sstems subjeted to nonstatonar random extaton. Int. J. Non- Lnear Meh [15] Orab I. I.; Ahmad G.; Nonstatonar resonse analss of a duffng osllator b the Wener Hermte exanson method. J. Al. Meh [16] Ramarao B.V. Ten C. Mohan S.; Calulaton of sngle fber effenes for ntereton and maton wth suerosed Brownan moton. J. Aerosol S AUTHORS ADDRESS Huanng Zhu; Juan P Hnestroza College of Human Eolog 4 M Van Rensselaer Hall Cornell Unverst Ithaa NY USA 5