NUMERICAL MODEL BASED ON A BERNOULLI INTEGRAL EXTENSION FOR PRESSURE ANALYZE ON A MILLIMETER DEVICE IN A TURBULENT FLOW

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1 591 TECHNICAL UNIVERSITY OF CLUJ-NAPOCA ACTA TECHNICA NAPOCENSIS Srs: Ald Mahmacs, Mchancs, and Engnrng Vol. 6, Issu IV, Novmbr, 17 NUMERICAL MODEL BASED ON A BERNOULLI INTEGRAL EXTENSION FOR PRESSURE ANALYZE ON A MILLIMETER DEVICE IN A TURBULENT FLOW Ghorgh NISTOR, Adran ZAFIU, Dnl POPA Absrac: Thr massv calculus rsourcs ar ndd for comung a RANS flud moon and a zolcrc modl, as w alrady nocd n our rvous modls. Th machd and vrfd hyohss abou h dvc rsons and qualav bhavor ma a nw aroach o b ncssary. Thrfor, som horcal and fld obsrvaons on PDEs oghr wh h ald gncally algorhms mmrg for h roblm s convrgn soluon. Th am of h ar s o valda a modl comud wh a flud rssur dndng on flud sd and dvc' body dmnsons for omal lcrc rsons. Ky words: urbulnc flow, Brnoull ngraon nson, rssur analyz 1. INTRODUCTION W obsrvd h dndnc of h rssur ald on h dvc body facs wh h squar sd of h flud. Th ald rssur s n charg of h zo-rsons hrough s nrnal srss nsor, whch furhr ouus an lcrcal sgnal [1],[],[]. Hr h dvc s a dcor of a flud sd,flud ha can hav dffrn vscoss. Th mahmacal modl has hr subsysms, such as h urbuln flud moon, h sold mchanc and h zolcrc comonns, machd by h boundary and nal condons. Th hyscal dvc consss n a zolcrc rob and a foam mound bwn wo mal lcrods [4],[6], wch allows h zolcrc onal o b masurd, as w rsnd n our rvous ar[7]. So w sarch for a mos smlfd numrcal modl o nvsga h qualav bhavor of h zolcrc dvc and h urbuln flow. Ths modl has o b valdad by comarson wh h rsuls from our rvous rsarch, o furhr nvsga anohr roosd dvc.. A BERNOULLI INTEGRAL EXTENSION W assum ha h rssur ald on h mllmr dvc dnds wh h squar sd of h flud. L s frs rov hs assumon: W sarch for h rssur quaon of h modl bul on our rvous sudy [7]. As a gnral rul, h roduc u can b wrn as h addon bwn a gradn and a vcor roduc among a roor and h vcor flud fld: 1 r u = ( u u ) [( rou) u], In h lar assumon h vlocy fld s an rroaonal fld ( rou r = ), so w oband h rlaonsh for h rssur. 1 = u u ν ( ) ν δ ν ( ) ( ν ρ) δ cons Hnc h rsson abov suggss ha h rssur dnds on h sd s squar.

2 59. MATHEMATICAL SUPPORT Th quaons w ar loong for ar h classc consrvaon quaon [5]: ρ u ( u ) = [ δ ( ) ν ν ( ν ν ) δ ρδ, (1) whch by comonns bcoms: ( ) ρu = δ ν ν, () ( ν ν ) δ ρδ and quvaln ( ) ρu = δ ν ν ( ν ν ) ( ν ν ) δ ( ν ν ) δ ρ δ, = 1, and h quaon gvng h roduc rm P, () u u u P = ν, (4) ρ whr h noaons ar: u -vlocy flud fld s comonns (SI un: - m s ), - ρ -dnsy of h flud (SI un: g m ), - - rssur of h flud (SI un: Pa ), - - nc nrgy (SI un: m s ), - ε - s ra of dssaon (SI un: m s ), - ν - dynamc vscosy (SI un: g ( m s) ), ν - - urbuln vscosy (SI un: Pa s ), - ν ff - ffcv vscosy (SI un: Pa s ), s[7]. 4. THE INPUT DATA AND THE INTERACTION BETWEEN THE DEVICE BODY AND THE FLUID MOTION W consdr a mllmr dvc (fg.1., a, fg.1., b and fg.1., c) Fg. 1., a. Dvc body. Th szs wll vary as blow,as wll as h flud sd (ab.1). W fd h lms of h dvc body sz and h focal ara of h flud sd, consdrng a low vscosy urbuln flud moon. Tab. 1. Inal daa ( 4 = 19 sas) Wdh [mm] Dh [mm] Hgh [mm] Flow sd [m/s] Fg.1. b. Dvc body n h volum un.

3 Fg.1. c. Dvc body. 5. THE NUMERICAL ALGORITHM W follow h acon of h flud moon vrsus h dvc and h urbuln bhavor of h flow. Ths wll hav a drc ffc on h dvc lcrc rsons (fg. ). In ordr o undrsand h moranc of h rssur masurmn on ach sd of h mllmr dvc, h sram lns of h urbuln moon ar smulad n fg.. fac fac( D, Dy, Dz, S) S S S ( D, Dy, Dz, S ) c S c1 S c Dz Dz Dz ( D, Dy, Dz, S ) c Dz c1 Dz c Dy Dy Dy ( D, Dy, Dz, S ) c Dy c1 Dy c D D D ( D, D, D, S ) c D c D c 59 =, (5) =, (6) =, (7) =, (8) y z 1 From (6), (7), (8), (9) rsul: l,,, l= =, (9) l = c D D D S. (1) fac 4 Thr ar coffcns for ach rssur. Th oal numbr of coffcns ar 4, 81 for ach fac. Runnng h algorhm w oban h followng coffcns (ab ): Tab.. Th valuad rror s abou 1.% c l l y z Th ffc of h rssur w analyz blow. 6. CONCLUSIONS Fg.. Dvc body confrond wh h sram lns of h urbuln moon - h connuous currn nrac wh h dvc. W wll now dfn h rssur funcons for ach sd (fron, lf,rgh, o, bac) and hr ndndn varabls. Consdrng ha h rssur of ach fac s dffrn from anohr and dnds of h dmnsons and h flud sd, w hav dmnsonlss rlaons: Th rssur on h dvc body dnds of: - h facs of h body dvc (fron, o, bac, lf and rgh), - h dmnsons of h body dvc(wdh, dh and hgh ), - h flud sd n h mannr w rovd. L' s hs dndnc of h rssur of ach varabl as w wll show blow (fg., fg. 4, fg. 5 and fg. 6). W r o mrov h sha of h dvc and h omal oson for h zo comonn, n ordr o mrov h lcrc rsons, hngs w wll dvlo lswhr. W noc ha h urbulnc bhavor n no nglgbl, so ha mus b an no consdraon vn f w wor wh ar or war.

4 594 Th massv calculus rsourcs ar no longr rqusd on hs aroach, hng ha allow us o sarch for ohr omal rsons. W fnd ha h lcrc sgnal dvc s basd on h rssur fld caurd by h surfac srss omal convrd. Fg.. Prssur vs. flud flow sd. Fg.4. Prssur vs. dvc dh.

5 595 Fg. 5. Prssur vs. dvc wdh. Fg. 6. Prssur vs. dvc hgh. 6. ACKNOLEDGEMENT Th auhors, flng mos oblgd, wsh o acnowldg all hos who conrbud o h rnng of hs Procdngs. REFERENCES [1] Ma, R.-H., Wang, Y.-H., L, C.-Y, Wrlss Rmo Wahr Monorng Sysm Basd on MEMS Tchnologs. Snsors, 11, 11(): [] Pros, A., Alsr H., Barbara, W., Yaong, Z., Thohl, G., Rbcca

6 596 Chung, Fabrcaon and characrzaon of a wnd snsor for ngraon wh a nuron crcu, Mcrolcronc Engnrng, Volum 84, Issus 5-8, Procdngs of h nd Inrnaonal Confrnc on Mcroand Nano-Engnrng, May-Augus 7, Pags , ISSN , DOI: 1.116/, m [] Kazuho, A., Yoshsugu, K., Tauzo, I., Ingrad dsgn of zolcrc damng sysm for flbl srucur, Ald Acouscs, Volum 65, Issu, March 4, Pags 9-1, ISSN -68X, DOI: 1.116/.aacous,.8.5. [4] Arhur, G.G., McKon, B.J., Darng, S.S., Morrson, J.F., Cu, Z., Manufacur of mcro-snsors and acuaors for flow conrol, Mcrolcronc Engnrng, Volum 8, Issus 4-9, Mcro- and Nano- Engnrng MNE 5, Arl-Smbr 6, Pags 15-18, ISSN , DOI: 1.116/.m [5] Carafol, E., Consannscu, V-N., Dnamca fludlor comrsbl, Bucurş, Edura Acadm Român, [6] Css, D., Crcu ngra analogc. Crcu fundamnal, Edura Casa Cărţ d Şnţă, 7. [7] Zafu, A., Nsor, G., Paul, S., MEMS wnd sd snsor: From urbulnc flud flow and zolcrc mahmacal modl o numrcal smulaon dvc rsons, 11 Inrnaonal Smconducor Confrnc (CAS), Volum 1, [8] John, M-C., Lnar Vlocy Masurmn, Pnn Sa Unvrsy, 1. [9] LDuff all, Vlocy masurmn n a flud low cos snsor and sgnal rocssng dsgn, IEE, Snsors, 1. MODEL NUMERIC BAZAT PE O EXTENDERE A INTEGRALEI BERNOULI PENTRU ANALIZA PRESIUNII PE UN DISPOZITIE DE ORDINUL MILLIMETRILOR ÎN FLUX TURBULENT Absrac: Aşa cum s-a drmna înr-o lucrar anroară sun ncsar rsurs d calcul masv nru a calcula o mşcar fludă RANS ş un modl zolcrc. Ioza folosă ş vrfcaă rvnd răsunsul dsozvulu ş comoramnul calav drmnă o nouă abordar. Prn urmar, unl obsrvaţ orc ş rmnal asura PDE-urlor, îmrună cu algorm gnc alcaţ, drmnă soluţa convrgnă a roblm. Scoul lucrăr s d a valda un modl d calcul c rsuun o rsun a fludulu în funcţ d vza fludulu, d dmnsunl ş lml corulu dsozvulu în vdra obţnr unu răsuns lcrc omza. Ghorgh NISTOR, Ș. l. dr. ng., Unvrsaa dn Pș, Daramnul d Mamaca Informaca, -mal: gh.nsor@yahoo.com, Offc Phon: , Hom Addrss: sr. Mha Vazul, bl. D5, sc. A, a.1, Pș, Argş, Româna, Hom Phon Adran ZAFIU, Ș. l. dr. ng., Unvrsaa dn Pș, Daramnul d Elcronca, -mal: adran.zafu@sara.ro, Offc Phon: , Hom Addrss: sr. Ngru Voda nr 18, Pș, Argş, Româna, Hom Phon Dnl POPA, rof. unv. dr. ng. Unvrsaa dn Pş, Unvrsaa dn Pș, Daramnul d Auovhcul ș Transorur, -mal: dnl.oa@u.ro, dnl_oa@yahoo.com, Offc Phon: , Hom Addrss: sr. Smur, nr. 9, bl. PS 8, sc. A, a. 9, Pș, Argş, Româna, cod 1146, Hom Phon