Deposition of Submicron Charged Spherical Particles in the Trachea of the Human Airways.

Transcription

1 eposiion of Submicron Chrged Sphericl Pricles in he Trche of he Humn Airwys. eprmen of Engineering Sciences nd Mhemics ivision of Fluid nd Experimenl Mechnics Luleå Universiy of Technology Corresponding uhor: S Luleå, Sweden, Absrc: This pper presens numericl sudy of he deposiion of submicron chrged sphericl pricles cused by convecion, Brownin nd urbulen diffusion in pipe wih smooh wll nd wih crilginous ring wll srucure. The model is supposed o describe deposiion of chrged pricles in generion (rche) of he humn lung. The upper irwys of he humn lung re chrcerized by cerin wll srucure clled crilginous rings (fig.1) which re believed o increse he pricle deposiion when compred o n irwy wih smooh wll. The problem is defined by solving he fluid flow problem wih he id of low-reynolds number -epsilon model combined wih diffusion equion for he Brownin moion nd Poissons equion for he elecrosic field. The elecrosic field is genered by he spcechrge densiy of he pricles. Resuls re presened for generion, he rche of he humn irwys. eposiion resuls using Comsol muliphysics re compred wih n nlyic soluion for he cse of fully-developed urbulen flow in n irwy wih smooh wll. Keywords: Chrged nnopricles, Brownin nd urbulen diffusion, Elecrosics, eposiion, Humn respirory irwys. 1. Inroducion Sudies on pricle rnspor of submicron pricles re of high impornce in he nlysis of pricle deposiion in he respirory irwys, boh for ssessing helh effecs of inhled oxic mer nd for evluing he efficcy of drug delivery wih phrmceuicl erosols. Especilly he nowdys so populr pricles of nnomeer size for he developing of new improved sronger merils nd oher fscining echnicl pplicions re here of priculr ineres In his pper we nlyze how submicron pricles (dimeer<1µm) re deposied in he upper irwys of he upper humn lung irwys nd especilly we consider he effecs cused by chrged sphericl pricles. We lso include wll srucure clled crilginous rings, loced in he upper irwys of he lung. In previous wor (Åersed(1) 1,(11) ) we hve considered he lower irwys (generion>4) of he humn lung in which he flow is lminr. In he presen pper we consider he corresponding problem for he upper irwys, in priculr he uppermos irwy, he rche (generion ) in which he flow is usully urbulen. To find he rnspor nd deposiion in given flow geomery here re essenilly wo mehods. A Lgrngin pproch nd n Eulerin pproch. In he firs mehod he moion of he individul pricles in he flow field re rced. For he lminr cse he flow is genered by he Nvier-Soes equions while for he cse of urbulen flow some urbulence model is doped. The concep is hen o lunch lrge number N of pricles, derive heir individul moion nd compue pricle deposiion sisics from he frcion h his wll (Högberg e l. 3 ). To ge sisicl mesures of he deposiion lrge number of pricles (N) need o be simuled wih n error of he 1/ orderon ( ) This Lgrngin pproch is compliced if we for insnce wish o find he elecric field nd force field from concenrion of chrged pricles. Since we consider he effec of disribuion of chrged pricles, he Eulerin mehod is chosen. The concenrion is hen governed by convecive diffusion equion including effecs from Brownin moion nd migrion in elecric field. To include effecs from urbulen flow, urbulen diffusion is dded. For pricles of size smller hn bou 1µm, diffusion is he only effec from he urbulence. For lrger pricles, however, here is lso n effec from urbophoresis (Guh(1997) 4 ), which is fr more difficul problem o model, since nisoropic effecs hen become imporn, nd hus cnno be modeled by for insnce he simple urbulen - model considered in he presen pper.

2 . Governing equions The morphology of he respirory irwys of he humn lung is represened by bifurcing sysem of pipes where ech pipe belongs o cerin generion. The uppermos irwy (generion) is single pipe clled rche. The upper irwys hve cerin wll srucure clled crilginous rings, loced in he upper irwys of he lung see figure 1. In his pper we only consider resuls for his single pipe, he rche. For he nlysis we ssume pipe wih xil symmery. The min flow is hen in he x- direcion nd r is he coordine in he rdil direcion. Since he flow in rche is urbulen wih ypicl Reynolds number rnging from o 1 we pply low-reynolds number incompressible - model for he flow. Figure 1. Axisymmeric pipe geomery wih crilginous ring wll srucure. Rdius of Trche.8[m] lengh.96[m]. Ampliude of rings.8[m]. The equions for he fluid flow re hen T ρ ( u ) u ( pi ( µ µ )( u ( u) ) ρi) 3 u T µ ρ ( u ) µ u u µ u ( u) (( µ ) ) σ T ρ( u ) C1 µ u u C ρν u ( u) µ (( µ ) ) fc ρ σ µ ρ fc µ µ (.1) wih sndrd -epsilon vlues for he consns nd where f nd f µ re he low-reynolds number correcions. The convecive-diffusion equion including effecs from concenrion c of chrged pricles wih chrge q nd he elecrosic poenil φ is of he form ( u q q ) c ( c φ) c (( ) c) κt κ T (.) Here is he Brownin diffusion coefficien given by κt Cu 3πµ d λ d Cu 1 ( exp(.39 )) d λ where Cu is he Cunninghm fcor which is correcion fcor needed o bridge he gp beween he coninuum limi nd he free moleculr limi for he flow ps sphericl pricle. λ is he collision men free ph nd d is he pricle dimeer, κ is Bolzmnn s consn, T is he bsolue emperure nd µ is he dynmic viscosiy of he ir. The urbulen diffusion coefficien is reled o he urbulen viscosiy s µ ρ In he presenion of he resuls i is lso convenien o inroduce he dimensionless number α α 1 4 cq κt The elecrosic poenil is given by Poisson s equion qc φ (.3) which is he equion providing he connecion beween he chrge disribuion nd he elecrosic poenil. 3. Use of COMSOL Muliphysics The se of equions (.1-.3) re solved ogeher using Comsol Muliphysics 4.3. The differen physics inerfces used re he low-reynolds number - model for urbulen flow, Trnspor of dilued species nd Elecrosics.

3 A he inle x uniform velociy is chosen nd xl fluid flow oule condiions re pplied. On he x-xis xil symmery is chosen. For he rnspor of dilued species inerfce he concenrion he inle is uniform c c. The boundry condiion of he bsorbing wll is c nd he oule convecive flux is chosen. For he elecrosics inerfce, zero chrge is chosen he inle nd oule. Since he wll of he respirory irwys consiss of so clled mucus-lyer including minly wer, he wll is reed s good conducor so he wll poenil is chosen s zero. For he meshing seings fine physics-conrolled mesh is pplied. 4. lidion of Comsol model wih heory for smooh pipe To vlide he resuls using Comsol Muliphysics we firs consider some pproxime nlyicl resuls of he deposiion of chrged pricles in fully-developed urbulen flow in simple smooh pipe. We ssume h he concenrion is uniform equl o c excep in hin lyer close o he boundry. An pproxime soluion of Poisson s equion is hen simply cq φ ( r ) (4.1) 4 I is convenien o inroduce dimensionless quniies. The dimensionless disnce from he wll is inroduced s r y Re (1 ) where he Reynolds number Re bsed upon he fricion velociy is defined s u Re ν. Here u is he fricion velociy given by τ u w ρ, where τ w is he wll sher sress. For he fully developed cse he convecivediffusion equion (.) in he hin lyer pproximion hen becomes c c ( ) c α Re y y y (4.) where nd. Inegring his ν ν equion gives α c c c ( ) c Re dep y dep wheredep is he dimensionless deposiion u velociy. The soluion is Re cy ( ) c (1 exp( Zy ( ))) dep α α Re where Z( y ) y dy ( Y) (4.3) The deposiion velociy cn be obined by ing he limi y α dep α Re(1 exp( Z ( ))) Re (4.4) I is seen h he deposiion velociy is deermined once model for he eddy diffusiviy is nown. The lrges conribuion o he inegrl in (4.4) however comes from he region ner he wll nd herefore i is sufficien o now is behvior for smll y. The smll y behvior of he eddy diffusiviy is well nown from experimens s Ay y... The inegrl in(4.4) hen becomes (4.5) 3π Z( ) 1/3 /3 9 A (4.6) The resul for he deposiion velociy is hen

4 α dep α 1/3 3 π Re (1 exp( )) 9 Re 1/3 A (4.7) In he limi of zero chrge α we find 9 A dep π 3 /3 1/3 (4.8) nd hence in he limi of α we find simply dep Re α (4.9) Nex he nlyicl resuls re o be compred wih he resul from Comsol Muliphysics. The clculion of he deposiion velociy using Comsol Muliphysics is done s follows. The pipe is chosen sufficienly long so h he flow becomes fully developed. A lengh of bou 8 imes he rdius is sufficien. The convecive pricle flux chds. ncflux is hen inegred wo close x-posiions, x1 nd x,in he fully developed region. The deposiion velociy is hen clculed from comsol dep ( chds. ncflux chds. ncflux )π rdr 1 ( x x ) π 1 For he comprison we chose Reynoldsnumber Re U men / ν 4 nd hree differen dimeers of he pricles, d1nm, d1nm nd d1nm. To find he effec of he chrged pricles we vry he elecrosic prmeer α 1 cq α 4 κ T As n exmple if he concenrion of 11 3 pricles inle is c 1 1 [1 / m ] nd he 19 chrge of he pricles is q [ C] he elecrosic prmeer isα 346. In figure he deposiion velociy clculed from he heory nd Comsol Muliphysics re compred. Figure. A comprison of deposiion velociy clculed from heory (solid line) nd clculed by Comsol Muliphysics (o). d1nm (blc), d1nm (red), d1nm (blue) From figure we noe h here is some disgreemen beween heory nd simulion especilly for lrge pricles (d1nm). For smller pricles here is beer greemen nd for he pricle size d1nm here is quie good greemen for smll α while for d1nm he greemen is good for lrger α. The reson for he filure using Comsol cn be deduced from he behvior of he eddy diffusiviy in he ner-wll region. The correc behvior is given by (4.5). In figure3 we hve comprison of he behvior of he eddy diffusiviy from heory nd Comsol. We noe h he correc power of 3 in heory is much lrger hn he corresponding power in Comsol which from he figure cn be esimed o be bou.6. Since he correc behvior of he eddy diffusiviy in he ner wll region is very imporn for he predicion of he correc deposiion velociy his explins he difference Figure 3. Behvior of eddy viscosiy in he ner wll region from heory (green) nd using Comsol (blue).

5 beween he resuls. The lrges disgreemen occurs for lrge pricles, which cn be undersood by he mgniude of he Brownin diffusion, which for lrge pricles is very smll. This ogeher wih he filure of he eddy diffusiviy gives n error in deposiion re up o lmos one order of mgniude. For smller pricles for which Brownin diffusion domines he sensiiviy of he error in he eddy diffusiviy is however no s imporn nd for pricles wih dimeer 1nm he greemen is quie good, les for lrge α. For pricles wih dimeer 1nm he greemen is quie good for smll o medium α bu no s good for lrge α. This disgreemen for lrge α cn be explined by he very hin boundry lyers obined in his limi, nd my herefore be difficul o resolve numericlly. As conclusion we cn sy h Comsol Muliphysics provides wih relible resuls les for smll pricles less hn 1nm nd cn herefore be pplied lso o he more complex geomery. 5. Pricle deposiion in rche wih crilginous ring wll srucure Here we consider deposiion resuls for he rche s pipe wih smooh wll nd wih he crilginous ring wll srucure seen in figure 1. Since he mos relible resuls using Comsol for smooh pipe re found for smll pricles we here only consider pricles wih dimeer 1nm. The men velociy is en o be 4 [m/s] wih uniform disribuion inle x. eposiion res re clculed using inegrion of he norml convecive flux chds. ncflux cross he inle nd oule s dep ( chds. ncflux chds. ncflux )π rdr ou chds. ncflux π rdr (5.1) The firs cse presened is for unchrged pricles. In figure 4 he sremlines of he velociy field nd he vriion in concenrion in in Figure 4. Flow in he neighborhood of he firs ring. Sremline: elociy field. Surfce: Concenrion. Ligher red mens lower concenrion. Figure 5. Sremline: elociy field. Surfce: Tol pricle flux mgniude. r red regions correspond o lrge flux nd dr blue regions correspond o smller flux. is presened in he region before nd fer he firs ring. Noe h he flow sepres fer he firs ring nd lso in smll region before he firs ring. In he sepred regions he concenrion is lower hn in he min srem. This does no men h he deposiion re or pricle flux in he sepred regions is lrger. The locl deposiion re is deermined from he locl norml concenrion grdiens he boundry. This cn be seen from figure 5 in which regions of dr red correspond o lrge pricle flux nd regions of dr blue o smller pricle flux. Nex he effec of chrged pricles is considered. As n exmple we consider he cse 11 of concenrion inle of c 1 1 pr/m 3, nd h he chrge of he pricles is 4 imes he elemenry chrge. This corresponds o vlue of he elecrosic prmeer α of 3459.

6 In figure 6 he corresponding flow in he region behind he firs ring is presened. In he sepred region he concenrion is somewh smller hn for he cse wih unchrged pricles. The elecric field is herefore smller in hese regions compred wih he region close o he rings. The pricle flux due o elecric mobiliy is herefore smller nd deposiion res in fc become smller in hese res when compred o smooh pipe. In he region ner he rings he elecric field is lrger hn for he cse of smooh pipe. Tol deposiion is hen delice blnce of hese effecs. In figure 7 he ol deposiion is presened, clculed using (5.1) for smooh pipe nd pipe wih crilginous ring srucure. For unchrged pricles α, he effec of he crilginous rings is o increse deposiion while for lrger vlues of α he siuion is he opposie. Remrbly he effec of chrged pricles is he possibiliy o increse he moun of deposied pricles from 1% for unchrged pricles up o 5% for lrgeα. These resuls should be useful in n opiml design of herpeuic erosols. 6. Conclusions The deposiion of chrged submicron sphericl pricles in he rche of he humn irwys is invesiged. Especilly he effec of crilginous ring wll srucure is considered. Since he fluid flow in rche is in generl urbulen low-reynolds number - model is pplied. The effec of Brownin nd urbulen diffusion s well s migrion of he chrged pricles in he self-consisen elecric field is included. Figure 7. Tol deposiion s defined by (5.1). (o) smooh pipe. () pipe wih crilginous ring srucure. The Comsol model is vlided by compring i wih he heory for fully developed urbulen flow. The correc behvior of he eddy diffusiviy in he ner-wll region is shown o be crucil for correc compuion of he deposiion. eposiion for hree differen dimeers d1nm, d1nm nd d1nm re considered. For he smller pricles d<1nm he greemen beween heory is quie good. For he lrger pricles d1nm he greemen is poor. This filure of he Comsol model cn be deduced from n incorrec behvior of eddy diffusiviy in he ner-wll region. Since he Comsol model gives relible resuls for smll pricles he model is used o esime he deposiion of 1nm pricles in rche wih nd wihou he crilginous ring wll srucure. For unchrged pricles he crilginous rings increse ol deposiion, while for sufficienly lrge chrge he effec is he opposie. In generl chrged pricles enhnce he ol deposiion wih possibiliy o increse pricle deposiion from 1% for unchrged pricles up o 5% for chrged pricles. This resul should be of impornce in n opiml design of herpeuic erosols. 7. References Figure 6. Sremline: elociy field. Surfce: Concenrion. Arrow surfce: Elecric field 1. Åersed, H.O., The effec of crilginous rings on deposiion by convecion, Brownin diffusion nd elecrosics. Comsol conference, Pris (1)

7 . Åersed, H.O., eposiion of chrged nnopricles in he humn irwys including effecs from crilginous rings, Nurl Science, olume 3, ,(11) 3. Högberg S.M., Åersed H.O., Lundsröm T.S., Freund J., Respirory deposiion of fibers in he nonineril regime: developmen nd pplicion of semi-nlyicl model. Aerosol Science nd Technology. 44, (1) 4. Guh, A., A unified Eulerin heory of urbulen deposiion o smooh nd rough surfces, Journl of Aerosol Science, 8, (1997) 8. Acnowledgemens This wor is sponsored by he Swedish Agency for Economic nd Regionl Growh nd Cenre for Biomedicl Engineering nd Physics.