An analytical method for selecting the optimal nozzle external geometry for. Nacional del Litoral CONICET, Güemes 3450, S3000GLN, Santa Fe, Argentina.

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1 An nlytcl metd f selectng te ptml nzzle extenl gemety f flud dynmc gugng J.M. Pelt,, Y.M.J. Ce, D.I. Wlsn Insttut de Desll Tecnlógc p l Indust Químc (INTEC), Unvesdd Ncnl del Ltl CONICET, Güemes 45, SGLN, Snt Fe, Agentn. Deptment f Cemcl Engneeng nd Btecnlgy, Unvesty f Cmbdge, Ne Museums Ste, Pembke Steet, Cmbdge, CB RA, UK Deptment f Cemcl Engneeng, Unvesty f Bt, Buldng 9 West, Clvetn Dn, Bt, BA 7AY, UK Abstct Flud dynmc gugng (FDG) s develped t mesue, n stu nd n el tme, te tckness f sft depst lye mmesed n lqud tut cntctng te sufce f te lye. An nlyss bsed n te lubctn ssumptn f te fl pttens n te spce beteen te nzzle nd te sufce beng guged yelded nlytcl expessns f te eltnsps beteen te mn fl vbles nd system pmetes. Nzzle spes f ptcul pessue, pessue gdent nd se stess pfles culd ten be dentfed. Te effect f fl te, nzzle gemety nd nzzle pstn n te pessue benet te nzzle nd se stess n te guged sufce sed vey gd geement t cmputtnl flud dynmcs (CFD) smultns. Cse studes pesented nclude nzzle spes f unfm pessue nd se stess pfles, c e useful f mesung te stengt f sft depst lyes Keyds: CFD; Clenng; Flud Dynmc Gugng; Flud mecncs; Fulng; Lmn fl. Cespndng ut. Tel.: ; fx: E-ml ddess: d@cm.c.uk (D.I. Wlsn).

2 Intductn Te depstn f flms n pcess sufces fm flng lqud ccus n mny ndustl mnufctung pcesses. Sme f tese flms e desed (e.g. cclte ctngs n bscuts) nd sme e unnted (e.g. fulng lyes n et excnges). In bt cses, t s ften mptnt t mesue te tckness nd stengt f tese flms. In te btecnlgy nd fd sects, tese flms e fequently sft nd/ fgle nd te stte s stngly dependent n te pesence f lqud, s tt mesuements f tckness nd stengt suld be detemned n stu nd n el tme ptcully ee te depst s evlvng. Ptble, pd, nn-cntct nd pecse tecnques e needed Flud dynmc gugng (FDG) s eltvely nvel tecnque tt s develped by Tuld et l. () t estmte n stu nd n el tme te tckness f sft depst lye mmesed n lqud tut tucng te sufce f ts lye. Te cncept cnssts f nzzle tt tds lqud fm te egn ne te depst sufce, s sn n Fgue. F cetn nge f clences beteen te nzzle nd te sufce (), te fl te ( m ) tug te nzzle s usefully senstve t (Fgue (b)). Te tckness f te depst, δ, cn be clculted fm te dffeence n clence beteen te nzzle nd te depst lye, (nfeed fm te fl te), nd te pstn f te nzzle eltve t te substte, nt, (estblsed ete by clbtn ndependent sensng). Te tecnque pvdes g ccucy tckness mesuement t eslutn f ± 5 µm t sensng tme f 5 s (Gdn et l., ) Studes suc s tt by Ce et l., (4) ve sn tt FDG culd ls be used t quntfy te stengt f sft depsts. Te guge emplys fls n te lmn egme,

3 llng cmputtnl flud dynmcs (CFD) t be used t gve elble estmtes f te fl feld nd stess dstbutn n te fl. Ce et l. quntfed te stengt f dffeent tmt pste lyes by mesung te defmtn f te flm fllng gugng t knn se stess exeted by te guge n te flm. Tckness mesuements ee mde t g clence (l se stess) nd fllng expsue f te flm t ge se stesses nduced by mvng te nzzle clse t te flm. Te stesses nduced n te flm by te gugng fl e detemned by te lqud fl te, clence nd ls te extenl gemety f te nzzle. Pelt et l. () demnstted te extenl spe f te nzzle culd ffect, vey ntcebly, te se stess ( ) nd pessue ( p ) pfles n guged sufce. Tey dentfed gemetes tt pduced nteestng nd p pfles suc s lne, peked bmdl dstbutns. Tese types f pfles e ttctve t FDG petn s tey ffe te pptunty t mnpulte te fces exeted n flm by smply cngng te nzzle. F exmple, n ppxmtely cnstnt esn f te flm cn be pduced unde te nzzle m en n even se stess 67 pfle s used. Altentely, nte spe culd yeld g senstvty t clence (mesuement pecsn) le mnmzng flud se. Pevus kes ve emplyed cmputtnl flud dynmcs (CFD) smultns t estmte te petng vbles ffect te mptnt fl vbles suc s te pfles f se stess nd pessue exeted n te depst (Ce et l., 4b; Gu et l., 9, ; Lste et l., ). Te defntn f te gemety nd mesng s stgtfd nd smultns tke fe mnutes t cnvege n mden PCs (Pelt et l., ). Explng nzzle gemetes s tme cnsumng, eve, s te mes must be e-defned f ec smultn. Ts ppe teefe exples te scpe f nlytclly-bsed ppces t

4 77 78 nvestgte te mpct f nzzle spe n FDG pefmnce, t be used t dentfy lkely cnfgutns f fne tunng by smultn A key fetue f FDG petn (Fgue (b)) s tt te nzzle must be clse t te flm sufce n de t k ppely,.e. d t.5. Te dl dmensn unde te nzzle m s muc lge tn te gp beteen te nzzle nd te flm c, en cmbned t te lmn ntue f te fl, suggests tt nlyses bsed n te lubctn ppxmtn suld yeld useful esults. Ts ppc s emplyed by Ce et l. (5) nd lte by Gu et l. (9) t estmte te se stess mpsed n te sufce. In bt cses se stess dstbutns btned fm CFD smultns sed gd geement t te nlytcl slutn f dl fl beteen t pllel dsks btned by Mddlemn (998) usng lubctn tey. Ts ppe extends te ppc t cnsde te effect f nzzle gemety Teetcl ppc.. Equtns f cnge Te pyscl dmn f nteest s te gp beteen te nzzle nd te guged sufce (Fg. ). Te equtns f cnge, dpted t nclude te lubctn ppxmtn, e tten n cylndcl c-dntes fllng te pcedue pesented by Denn (98). Only te pncpl steps f te nlyss e pesented. Te fllng ssumptns e mde: () stedy stte; () Netnn flud; () te velcty cmpnent n te dectn s neglgbly smll; (v) xsymmety n ; nd (v) extenl fces e due t gvty nd ct nly n te z dectn. Tese ssumptns smplfy te cntnuty (Eq. ()) nd Nve Stkes (Eqs. () nd ()) equtns t 4

5 vz v z () v v v v p v z v z z () 7 v v p v v v v g z z z z z z z z z () 8 9 Equtns (- ) cn be mde nn-dmensnl usng te fllng denttes: (4) z z (5) v v U (6) v z v V z (7) 4 p p gzz (8) ee nd e te ntenl nd extenl dus f te nzzle, espectvely; s te clence beteen te nzzle nd te guged sufce t, U s te men velcty t te extenl dus clculted fm te fl te, nd V nd e te cctestc vlues f z-velcty nd pessue tt ll be defned lte. Te use f tese dmensnless vbles 5

6 sssts te de f mgntude nlyss dscussed bel nd lls te esults t be cndensed nt genel equtns nd smll numbe f fgues. Intducng Equtns (4-8) nt Equtns (-) yelds 4 V vz v U z (9) 5 6 U v V v p v vz U z UV v V z v z p v vz U z z z U v v z V vz v z () () Denn (98) pesented dmensnl nlyss f Eq. (9) c sed tt te mgntude f te fct multplyng te devtve n te secnd tem s O V U Dentng, ts gves O V O U.. Ts eltn, bsed n te cntnuty equtn nd dmensnl nlyss f te pblem, ss ntul y t defne V. Teefe, te defntn f V used ee ll be v v U z z. Eq. () becmes V U. Eq. (7) cn ten be tten s 4 5 v v p v Rev v v z z U z ee te Reynlds numbe s defned s Re U. () 6

7 6 7 8 Te petng mde f te guge eques te nzzle t be vey clse t te guged sufce suc tt s smll,.e.. Te fl s n te lmn egme (dscussed n Sectn.), s tt Re, nd Eq. () becmes: 9 p v U z () Fm te de f mgntude f te tems n Eq. () (Denn, 98), te pmete cn be defned s U. Rengng Eq. () gves 4 v v p v v z z z z z 4 z z Re v vz (4) Incptng te eltnsps cnge s Re 4, nd, gves te equtns f 46 vz v z (5) 47 p v z (6) p z (7) Tese equtns, c e te typcl set f expessns ppsed f lubctn fl n flud system (Denn, 98), ll be used t descbe te fl n te dmn beteen te nzzle nd te guged sufce Useful expessns deved fm te equtns f cnge Velcty dstbutns Te dmensnless velcty cmpnent n te dectn, v, cn be detemned fm Eq. (6). Gven tt te pessue s functn f lne (Eq. (7),.e. p f ), nd 7

8 57 58 mpsng bundy cndtns suc s v t z (guged sufce) nd v t z (nzzle sufce), ntegtn f Eq. (6) yelds: 59 dp z z v, z d (8) Nte tt ~ z / ~ s v ~ s negtve. In FDG petn te fl te tug te nzzle s set nd/ knn. Teefe n de t fnd te dmensnless pessue gdent n Eq. (8), te fllng expessn f te dmensnless fl te ll be used: Q v dz (9) ee Q Q U nd v ~ s te bslute vlue f v ~. Incptng Eq. (8) nt Eq. (9) nd ntegtng gves te dmensnless pessue gdent s: dp Q () d Eq. (8) ten becmes: 69 6Q z z v, z () 7 7 Te cespndng expessn f te z cmpnent f velcty s btned by ncptng Eq. () nt te dmensnless cntnuty equtn (Eq. (5)): 7 7 6Q d z z vz, z d An nteestng eltnsp s btned by dvdng Eqs. () nd (): () vz, z z d () v, z d Eq. () ss tt tee ll be nn-ze cmpnent f velcty n te z dectn f te 76 nzzle pfle s nt zntl. 8

9 Se stess n guged sufce F Netnn flud, te cmpnent f te se stess elevnt t te stess n te guged sufce z cn be expessed s (Bd et l., 7): 8 v vz z z (4) 8 Incptng te bve esults, nmely () V U ; () gves te 8 dmensnless fm: 84 z v z vz (5) 85 ee: U. z z Applyng te cndtn, Eqs. () nd () n Eq. (5), yelds te expessn f 88 te dmensnless se stess evluted t te guged sufce ( z ): 89 z z 6Q (6) 9 9 It s elpful t ntduce ne nmlzed dl c-dnte n de t smplfy te bve esults. Defnng 9 ' (7) scles te dl dmensn f te dmn fm t. Smlly, te fllng expessns f te dmensnless se stess nd pessue gdent e ntduced s tt te vbles e nly dependent f te gemetcl pmetes: (8) 6Q 9

10 97 98 dp d ' dp d ' Q Equtns () nd (6) cn n be expessed s: (9) 99 dp () d ' ' ' () Cmbnng Eqs. () nd () gves useful expessn f clcultng te ll se stess: dp d ' () 4 5 Mesung lcl se stesses s cllengng but t s eltvely stgtfd t mesue lcl pessue vlues unde te nzzle nd teefe te pessue gdent, s epted by Ce et l., (4) nd Pelt et l. () Equtns () - () e mptnt n cnsdeng nzzle gemety becuse tey elte te lcl se stess nd te pessue gdent t. By specfyng te pfle f ne f, 9 dp d ', te expessns f te est f te mn pmetes (.e. se stess, pessue gdent, extenl nzzle gemety, velcty cmpnents, etc.) cn be detemned Rnge f Teetcl Vldty f te Appc An mptnt cnsdetn s te blty t vefy te nge f vldty f te slutns. Vefctn s bsed n ceckng ete te mn ssumptns f te mdel e met. Tese ssumptns e: 6 () U m d Red Re 8 () 7, smply

11 8 Re d 8 (4) d 9 ee Re d 4m / d nd () (5) It s ntety tt te bve sttements epesent suffcent set f ceckng, becuse tese e less estctve cndtns tn tse ppeng n Eqs. () nd (4), suc s tse p n eltng t Re (p > ) nd (n > ). A typcl set f pmetes fm FDG mesuements usng te s te gugng flud epted by Ce et l. (4) e used s n llusttn f te ppc. Te dmensns 7 e =.5 mm, = 5 mm, = 5 mm, d =. m, fl te 5 g s -, gvng 8 9 Re d nd.4. Tese cndtns guntee tt f l fl tes, te ppc pesented n ts k cn be used t descbe te FDG system.. Mtels & Metds.. Cse studes Te metdlgy s tested by ppsng smple expessns f ne f, dp d ' nd clcultng te slutn f te emnng t pmetes. Te expessns tus btned ee cmped t numecl esults btned usng CFD smultns. A genel type f expessn s ssgned f ec pmete specfed: 7 f ' ' f f (6) 8 9 ee f ' s genec expessn f ete evluted t ' ' (ute dus)., dp d ', nd f s f '

12 CFD smultns Sme f te cse study scens ee evluted usng cmmecl CFD cde, n sml mnne t pevus studes epted by Pelt et l. (). Befly, D-xsymmetc cmputtnl dmn bsed n tt emplyed by Ce et l. (4) f mdellng qussttc FDG systems s used. Fgue ss epesenttve cmputtnl dmn emplyng cylndcl c-dntes nd te sscted bundy cndtns Te smultns ee pefmed usng dffeent vlues f Re d. Unless tese specfed ll smultns emplyed clence (clsest pnt beteen te nzzle nd te guged sufce) set t /d t =.5, c les tn te ncementl kng zne depcted n Fgue (b). Te fl t te ext f te dmn s ssumed t be lmn nd fully develped ng t te nge f Re d vlues used. Te gvenng equtns ee te Nve-Stkes nd cntnuty equtns. Te flud ppetes ee tken s tse f te t C Te dmn s dscetsed usng tngul mes. In znes ee t s mptnt t estmte te velcty gdents ccutely (e.g. te gp beteen te nzzle nd te guged sufce) ge densty mes s used (Fgue (b)). Te cmmecl fnte-element-bsed sfte COMSOL Multpyscs.5 nd 4 (COMSOL Ltd., Htfeld, Unted Kngdm) s used t pefm te CFD smultns n.4 GHz desktp PC equpped t 5 pcesss nd 4 GB f RAM. Ec smultn tk but 5 mn t cnvege. 6

13 Te CFD smultns ee vldted n te pevus study (Pelt et l., ). In bef, te vldtn cnssted f test f te ndependency f te studed vbles n te mes cnfgutn nd geement beteen pedcted vlues t expementl bsevtns. Te mes ndependency test s ced ut usng dffeent mes denstes ve te ente dmn. Mntnc cnvegence s bseved f meses t >, elements. Te secnd step s ced ut usng: () expementl vlues f pessue n te guged sufce, nd () teetcl vlues f se stess f selected petng cndtns bsed n te ssumptn tt te fl cn be ppxmted s dl beteen t pllel dsks Results & Dscussn Tble summses te expessns f te extenl nzzle gemety, pessue gdent unde te nzzle nd se stess n te guged sufce, bsed n te use f n bty functn f ' f ne f tese vbles. Te eltnsps e esy t mnpulte. Tese expessns ll be used n te fllng sectns t s sme exmples f specfed nzzle gemety, pessue gdent nd se stess Specfed nzzle gemety 4... Lne nzzle pfle A lne pfle s ne f te smplest extenl nzzle gemetes t mnufctue nd nlyse. Te nzzle spe cn be ete nml (.e. pllel t te guged sufce), cnvegent dvegent. F ts cse, te dmensnless nzzle pfle s gven by: f ' ' (7) ee s te slpe f te lne pfle f te extenl nzzle gemety. Wkng fm te fst n Tble, expessns f te pessue gdent nd se stess e:

14 88 89 dp d ' ' ' ' ' (8) (9) 9 Fg. 4 ss te functnlty btned f lne pfle f (Fg 4()) f dffeent nd 9 epesenttve vlues f. Te pfles n (Fg. 4(c)) s mked decese s ' nceses f pstve vlues f c s ccentuted t ge. Wt negtve, te pfles stll pesent cncve spe but t mnmum tn te nzzle egn. Ts fetue s dscussed n detl n Sectn Te pessue pfles n Fg. 4(d) ee btned by ntegtng te gdents n Fg. 4(b): p ' ' ln p ' ' ' Fgue 4(d) ss cnge n te pfles fm cnvex t cncve s (4) cnges fm 99 pstve t negtve vlues. Ts flls te cnge n te nzzle spe fm cnvegng t dvegng ne. As becmes me pstve, te extenl pt f te nzzle becmes me cnvegent, cncenttng te pessue dp ne te nne dus f te nzzle. Cnvesely, gly negtve vlues cncentte te pessue lss ne te ute m f te nzzle. Te sme nfmtn cn be extcted fm te pessue gdent (Fg. 4(b)) Fnlly, t s ntety tt en, Eq. (4) educes t te expessn f te dl fl beteen t pllel dsks (Mddlemn, 998): 4

15 7 8 p p (4) ln ' An mptnt pmete t quntfy s te e-veged se stess n te guged sufce, 9 s ts cn be elted t mtel ppetes en te nzzle defms te flm. Ts s defned s: ee ' ' ' d ' (4), nd m : s te se stess clculted f dl fl beteen pllel dsks t septn, evluted t (Mddlemn, 998). 4 Intducng Eq. (9) nt Eq. (4), gves: 5 ' ' (4) 6 Fgue 5() ss tt te vege se stess nceses t Re d : t suld be nted 7 tt tese esults ee ll clculted f fxed clence, s vyng te clence f 8 gven fl te ll ls cnge Re d nd. Te effect f nzzle spe n te veged 9 se stess s stnge en,.e. cnvegent nzzles, s te pstn ee te nzzle ppces te flm mst clsely s ee te ccumfeentl e s ls mnmum. Fgue 5(b) summses te effect f nzzle dt n tems f, te e-veged se 4 stess dvded by. Wen ', te dt s neglgble. Te plts s tt 5 nceses t nzzle dt en s pstve. Wen s nd t 6 cetn vlues f ' tee s mnmum n, lcted t 5

16 7 8 9 ' (44) Mnm e btned f.5. Te exstence f mnmum vlues f s f nteest f mesung depst tcknesses, ee l se stess s dvntgeus n de t vd defmng te depst. Wen cmpng te eductn n f nzzle t ' gven by Eq. (44), sutble efeence vlue s tt evluted t ' (dented ' ). Intducng Eq. (44) nt Eq. (4) nd dvdng by ' gves 4 mn ' (45) 4 5 Fgue 6 ss te esults cmputed f.5. Te eductn n se stess by usng te mnmum e-veged se stess exbts mxmum t '.5 (Fg. 6(b)). Te 6 plts s tt te eductn n cn be n te de f % f te nne dus f te 7 nzzle s mde s smll s pssble Te sddle pfle cse Pelt et l. () demnstted tt mldly dvegent nzzle t lne extenl sufce pfle nd ngle f ppxmtely -5 gve se stess pfle t t peks f te sme mgntude lcted t te pstns f te nne nd ute m. Ts s temed sddle pfle nd s cnsdeed fute usng te ppc pesented bve. F ts cse, ( ) cn be elted t te ngle ( ) by 45 tn 8 (46) 46 nd te se stess pfle n te guged sufce s estmted s: 6

17 47 tn 8 (47) Evlutng Eq. (47) t nd ssumng tt (.e. pek t ), gves te fllng expessn f te ngle tt ll pduce sddle pfle, sp : 5 8 sp ctn (48) 5 5 Expessng te tn functn s n nfnte sees (Bnsten et l., 7) nd ecllng tt ts gument s smll (becuse ), gves: 5 8 sp (49) Eq. (49) ndctes tt sp s () negtve; () pptnl t ; () ndependent f te fl cndtns, nd (v) dependent n nd, tt s, te sze f te nzzle. Te dstbutn f se stess n te guged sufce f selected vlues sp s cmped t CFD smultns n Fgue 7(). Te pfles exbt t peks, s expected, t vey gd geement n mgntudes nd lctns f te peks. 6 6 Fgue 7(b) ss te dmensnless se stess vlues evluted t ' (te nne dus f te nzzle) s functn f te slpe ngle f te nzzle m, f dffeent vlues f clence,. sp cespnds t te ngle ee nceses. Te pfles s stng dependency n ' ', nd becmes me negtve s, c ves t te 65 nzzle ngle. Te stng dependence f n n te nge clse t ' sp s ls 66 bseved by Pelt et l. (). Tese esults ndcte tt te desed effect, f se 7

18 67 68 stess dstbutn clse t unfm, s nly cevble t n nge f petng cndtns f gven nzzle, s tt ltentve gemetes suld be nvestgted Nn-lne nzzle pfles Te flexblty f mden fbctn tecnques mens tt tee e fe lmts n te spes vlble f FDG nzzles. Sme gemetes e ese t mnufctue tn tes, t exmples beng tdl nd pblc-tdl pfles (Fgue 8). Te nn-dmensnl expessns f tese pfles e: ' ' ' (tdl) (5) 76 4m ' ' ' ' ' (pblc-tdl) (5) 77 ee m s te vlue f t te pstn f te mnmum dstnce beteen te nzzle nd 78 te guged sufce. Te se stess pfles evluted f te bve gemetes e 79 8 cmped t CFD smultns n Fgue 8 f epesenttve set f petng cndtns. Gd geement s btned, supptng te use f te nlytcl ppc Specfed pessue gdent Te smplest pfle tt epesents ls n nteestng scen s tt f lne vtn n pessue gdent. Te pessue gdent s tten s dp f pg ' pg ' (5) d ' 86 ee pg s te slpe. Fm Tble, te fllng expessns f te extenl nzzle 87 pfle nd te se stess n te guged sufce e btned: 88 ' pg ' (5) 8

19 ' pg (54) ' Fgue 9 ss te pfles f ec pmete f dffeent nd epesenttve vlues f pg. In genel, te unfm pessue gdent eques t decese s ' nceses f, nd ts effect s ccentuted s pg becmes me negtve. Ts bevu pg pessts even f pg. F sme pstve vlues f pg, mnmum s bseved n. Wen pg, te pfle cnges fm cnvex t cncve t mnmum ne ' Te cespndng se stess pfles n Fg. 9(b) s ncesng vlues s deceses f pg, nd mnmum en pg. Te pesence f te mnmum ndctes tt sddle pfle (n tems f se stess n te guged sufce) cn be btned n ts cse. Te fgue ls ss tt cnvex type f pfle cn be expected f bt pstve nd negtves vlues f pg. ' Te pessue pfles n Fgue 9(c) ee evluted usng pg p ' pg ' p (55) 44 Te plts s cnge n te pfles fm cncve t cnvex en pg cnges fm pstve t negtve. Te pessue vlues n ts cse e le tn tse n te pevus sectn becuse te gdents e lne Te e-veged se stess cn be clculted, s n te cse f lne, usng Eqs. (4) nd (54), vz. 9

20 4 4 5 F, ; ; ' pg ' 4 pg pg pg ' 4 pg pg pg 4 5 pg pg ' pg F, ; ; ' pg (56) 4 ee F, b; c; x s te Guss ypegemetc functn (Gspe nd Rmn, 4). Te 4 dependence f n Re d nd ' s pltted n Fgue. Te pfles exbt stnge 4 effect f Re d n cmped t Fgue 5 ( lne) f ptcul ' vlue nd te educed senstvty f t te mgntude f pg vlues s ' s decesed Lne pessue pfle: dp d ', pg 47 A secnd cse f nteest s ee te pessue ves lnely css te nzzle m,.e. 48 nd Eq. (5) gves pg dp d '. Te dmensnl pessue pfle evluted t te 49 guged sufce ( z ) s ten gven by: 4 d Red p p (57) 4 ee p s te pessue t. Cmpsns beteen Eq. (57) nd CFD smultns f 4 4 dffeent vlues f Re d n Fgue s vey gd geement up t Re d 4. At te gest Re d vlue cnsdeed, Red 6, te men pecentge e s und 6.64% Specfed se stess pfle Te scen ee te se stess n te guged sufce ves lnely s n cnsdeed. Te bss functn s:

21 f ' ' (58) s s ee s s te gdent n te se stess pfle. Te cespndng fms f te nzzle spe nd pessue gdent e: 4 ' s ' (59) 4 4 dp ' s ' d ' (6) Te lc f ec pmete evluted f dffeent nd epesenttve vlues f s e 44 pesented n Fgue. Fgue (c) ss tt deceses s ' nceses f, s cespndng t ecessed nzzle t te pnt f clsest ppc t te lye lcted t te ute m. Wen s, tee s mnmum n s ' nceses. Te pessue gdent pfles n Fg. (d) s stngly nn-lne bevu nd e me senstve t s. Te cespndng pessue pfles e ll mntnc n s, gven by 49 5 s 5 ' p F, ; ; ' F, ; ; p s s s s (6) 44 ee F, b; c; x s te Guss ypegemetc functn (Gspe nd Rmn, 4) Cnstnt A specl cse f Eq. (6) ses en s nd te se stess n te sufce beng studed s unfm. Ts scen s desble f FDG mesuements f depst stengt defmtn. A numbe f scens ee evluted nd cmped t CFD smultns. Fgue ss cmpsn t CFD nd nlytcl pedctns f pessue nd se stess n te guged sufce f dffeent vlues f Re d, ee te lcl pessue s gven by Eq. (6), n dmensnl fm:

22 449 d Re d p p (6) Fgue () ss tt te geement s gd n ll cses, t men pecentge e < 6%. Abslute pessues e cnsdeed ee s tese cn be mesued t esnble ccucy f vefctn pupses. Fg. (b-d) ss te cespndng se stess pfles: te smultns gee t Eq. (58) ell undenet te nzzle lp, t pek t te nne nd ute m lctns Fnlly, te e-veged se stess s gven by: 457 s ' s ' (6) 458 Te effect f Re d nd ' n summzed n Fgue 4 ss sml bevu t te esults btned f lne 4.4. Cmpste nzzles dp d ' Te nlytcl expessns develped ee cn be used t nvestgte ptentl cmbntns f fetues. F exmple, te nzzle spe culd be specfed t explt t spects f FDG ctn, ne f c s senstve t nzzle spe ne te nne m nd secnd c s senstve t spe ne te ute m. By y f exmple, Fgue 5 ss nzzle gemety t n ute zne f cnstnt nd n nne zne t cnstnt. Te Fgue ls ss gd geement beteen te dmensnl se stess clculted f ptcul Re d nd te cmpste, nlytcl mdel. Ts nzzle ffds bette senstvty t clence (dt nt sn), mptnt n lctng te sufce n tckness mesuements, nd esnbly unfm se stess. Te scpe f desgnng nzzles f ptcul pplctns

23 47 47 s teefe demnstted. In tese cmbntns, t s mptnt t nte tt Eqs. (4) nd (5) must ld f te ndvdul smple systems Cnclusns An nlyss f te fl ptten unde te gugng nzzle s been develped usng te lubctn ppxmtn t btn expessns f te mn fl vbles n n FDG expement. A set f genel equtns f te extenl nzzle gemety, pessue gdent unde te nzzle nd se stess n te guged sufce ee btned. Cse studes ee pesented cnsdeng te smple cmmn scens (.e. lne pfles f ll vbles studed). Cmputtnl flud dynmcs smultns ee used t vldte te mdel usng epesenttve cses nd sed gd geement. expessns ly n te nge estmted by te mdel. Te nge f pplcblty f te Tese esults nt nly pvde tl t nvestgte te effect f te extenl nzzle gemety n pmetes ffectng te sufce beng studed by te FDG tecnque, n desgn, but ls ll te effect f cnges n petng vbles n te pefmnce f n exstng FDG nzzle t be evluted,.e. ssessng peblty. Te tls ll ntl cnfgutns f nzzle spe f specfed petng cndtns t be dentfed, f ptmztn by CFD smultns nd eventully n vv by expements Fute develpments culd nclude extensn f te nlyss f systems t ge Reynlds numbes (.e. sgnfcnt netl effects), nn-netnn fluds, nd pus sufces (e.g. membnes)

24 Refeences Bd, R.B., Stet, W.E., Lgtft, E.N., 7. Tnspt Penmen, secnd ed. Jn Wley & Sns, Inc., Ne Yk, USA. Bnsten, I. N., Semendyyev, K. A., Musl, G., Muelg, H. Hndbk f mtemtcs. 5 t Ed. Spnge-Velg. Beln. Gemny. Ce, Y. M. J., Cds, S. S. S., Ptesn, W. R., Wlsn, D. I. 4b. CFD studes f dynmc gugng. Cemcl Engneeng Scence 59(6), Ce, Y. M. J., Höflng, V., Augustn, W., Ptesn, W. R., Wlsn, D. I. 5. A metd f mesung te stengt f scle depsts n et tnsfe sufces. Develpments n Cemcl Engneeng nd Mnel Pcessng (-), -. Ce, Y. M. J., Ptesn, W. R., Wlsn, D. I. 4. Flud dynmc gugng f mesung te stengt f sft depsts. Junl f Fd Engneeng 65(), Denn, M. 98. Pcess Flud Mecncs. Pentce-Hll, Lndn, UK. Gspe, G., Rmn, M. 4. Bsc ypegemetc sees. Cmbdge Unvesty Pess. Cmbdge. U.K. Gdn, P. W. Bke, A. D. M., Ce, Y. M. J., Wlsn, D. I., Yk, D. W.. A scnnng flud dynmc gugng tecnque f pbng sufce lyes. Mesuement Scence nd Tecnlgy (8), t numbe: 85. Gu, T., Albet, F., Augustn, W., Ce, Y. M. J., Mye, M., Ptesn, W. R., Scll, S., Sek, I., Wng, K., Wlsn, D. I.. Applctn f flud dynmc gugng t nnul test pptuses f studyng fulng nd clenng. Expementl Teml nd Flud Scence 5(), Gu, T., Ce, Y. M. J., Ptesn, W. R., Wlsn, D. I. 9. Expementl nd CFD studes f flud dynmc gugng n duct fls. Cemcl Engneeng Scence 64(), 9-7. Lste, V. Y., Lucs, C., Gdn, P. W., Ce, Y. M. J., Wlsn, D. I.. Pessue mde flud dynmc gugng f studyng cke buld-up n css-fl mcflttn. Junl f Membne Scence 66(-), 4-. Mddlemn, S An Intductn t Flud Dynmcs. Pncples f Anlyss nd Desgn. Jn Wley & Sns. Ne Yk. Pelt, J. M., Ce, Y. M. J., Wlsn, D. I.. Effect f te nzzle extenl gemety n te pessue nd se stess exeted n te sufce beng guged n flud dynmc gugng. Sent f publctn. Tuld, T. R., Ptesn, W. R., Mcled, N., Wlsn, D. I.. Develpment f nvel nn-cntct pxmty guge f tckness mesuement f sft depsts nd ts pplctn n fulng studes. Cndn Junl f Cemcl Engneeng 78(5),

25 Nmencltue pmete f Eqs. (7), (5) nd (58), =, pg, s. d d t dmete f te tube, m nzzle tt dmete, m f dl pfle functn f specfed vble, =, pg, s. g z stndd gvty n z dectn (-9.8), m s - nt clence beteen te nzzle nd te guged sufce, m pstn f te nzzle eltve t te substte, m dmensnless clence ( ) m mnmum dmensnless clence m mss fl te, kg s - p pessue, P p dmensnless pessue defned by Eq. (8) Q fl te, m s - Q dmensnless fl te ( dl pstn, m Q U ) dmensnless dl pstn defned by Eq. (4) ' dmensnless dl pstn defned n Eq. (7) R dus f te tube, m Re Reynlds numbe ( U ) 55 Re d Reynlds numbe bsed n d U men flud velcty t te extenl dus clculted fm te fl te, m s - v velcty cmpnent n dectn, m s - v z velcty cmpnent n z dectn, m s - v dmensnless velcty cmpnent n dectn defned by Eq. (6) v z dmensnless velcty cmpnent n z dectn defned by Eq. (7) z lengt f te nzzle m, m xl pstn, m z dmensnless xl pstn defned by Eq. (5) 5

26 Geek symbls α ngle f te ntenl dvegent zne f te nzzle, deg δ tckness f te depst, m dmensnl eltn ( zmutl c-dnte, - ) ngle f te extenl sufce f te nzzle, - sp ngle f te extenl sufce f te nzzle tt gves sddle se stess pfle, - λ lengt f nzzle ext, m dynmc vscsty, P cctestc pessue, defned s U / densty, kg m se stess, P dmensnless se stess ( U ), P dmensnless se stess defned n Eq. (8) Subscpts t te ute exteme f te nzzle t te nne exteme f te nzzle t te guged sufce Acknledgments JMP ses t cknledge Ds Susn Zll nd Amel Rubl f te nvluble suppt dung s pstdctl vst t gup t Cmbdge, nd fnncl suppt fm te Cnsej Ncnl de Investgcnes Centífcs y Técncs (CONICET) nd Unvesdd Ncnl del Ltl f Agentn. A Ryl Acdemy f Engneeng/EPSRC Resec Fellsp f YMJC s gtefully cknledged

27 Fgue cptns Fgue. Flud dynmc gugng pncples. () Scemtc f nzzle, t nset sng key dmensns; (b) clbtn cuves sng eltnsp beteen mss fl te, m, nd dmensnless clence, d. t Fgue. Scemtc f te pyscl dmn nvestgted n ts k. Fgue. Smultn gemety () sng bundy cndtns nd dmn dmensns, nd (b) mes used ( g mes densty s emplyed unde te nzzle m, n te tt, nd lng te bse). Fgue 4. Dmensnless pfles f (): nzzle extenl gemety ( ), (b): pessue gdent ( dp d ' ), (c) se stess ( ), nd (d) pessue ( p ) s functn f te dmensnless dl pstn ( ' ) f dffeent vlues f, f te lne nzzle spe Fgue 5. () Effect f Re d n e-veged se stess nd (b) dmensnless e- 66 veged se stess s functn f ', f dffeent vlues f f te cse f 67 lne nzzle spe Fgue 6. () Dmensnless e-veged se stess s functn f ' f selected 6 negtve vlues f sng te exstence f mnmum : lc f mnm dented mn by te dsed lne. (b) Evlutn f Eq. (45) sng te effect f n lc pstn nd mgntude. Fgue 7. () Cmpsn f teetcl (Eq. (47), sld lc) nd CFD smulted (dsed lc) dmensnless se stess pfles f selected vlues f sp. (b) Dmensnless se stess evluted t te nne dus f te nzzle ( ' ) s functn f te ngle f te nzzle m f dffeent vlues f. 68 7

28 Fgue 8. Cmpsn beteen se stess pfles btned nlytclly (sld lc) nd fm CFD smultns (dsed lc) f () tdl nzzle, (Eq. (5)) nd (b) pblc- tdl pfle f ((Eq. (5)). CFD cndtns: Re d, =.5 mm, =.5 mm nd = 5 mm. Fgue 9. Dmensnless pfles f (): pessue gdent ( dp d ' ), (b) se stess ( ), (c) pessue ( p ), nd (d): nzzle extenl gemety ( ) s functn f te dmensnless dl pstn ( ' ) f dffeent vlues f pfle. Fgue. () Effect f Re d n e-veged se stess veged se stess pessue gdent pfle cse. pg, f te cse f lne pessue gdent s functn f ', f dffeent vlues f nd (b) dmensnless e- pg f te lne Fgue. Cmpsn beteen teetcl (Eq. 57, sld lc) nd CFD smultn (symbls) f pessue s functn f ' f dffeent Re d f te cse f lne pessue pfle. Symbls: () Re d =, () Re d = 5, () Re d =, () Re d =, () Re d = 4, nd (+) Re d = 6. F ec smultn: =.5 mm, =.5 mm nd = 5 mm Fgue. Dmensnless pfles f () se stess ( ), (b) pessue ( p ), (c): nzzle 69 extenl gemety ( ), nd (d): pessue gdent ( dp d ' ) s functn f te dmensnless dl pstn ( ' ) f dffeent vlues f stess pfle. s, f te cse f lne se Fgue. Cmpsn beteen CFD smultn (dsed lc) nd nlytcl pedctns (sld lc) f () pessue t (Eq. 6) nd se stess vlues (Eq. (58)), f (b).7, (c) nd (d).5, s functn f ' f dffeent Re d. Cndtns: =.5 mm, =.5 mm nd = 5 mm

29 648 Fgue 4. () Effect f Re d n e-veged se stess nd (b) dmensnless e- 649 veged se stess s functn f ', f dffeent vlues f s f te lne se stess pfle cse. Fgue 5. Se stess vlues s functn f ' f te exmple f cmpste nzzle t cnstnt t te ute pt f te nzzle nd cnstnt t te nne pt ( Re d =, =.5 mm). Lc s te nlytcl mdel pedctns nd ccles te vlues btned fm CFD smultn. 9

30 Fl p b () p b R Incementl (kng) zne m Asympttc zne Δp Δp Depst Nzzle p Depst nt λ d t α δ = nt - p Nzzle Fl /d t.5 (b) Δp > Δp /d t Δp = p p b Fgue. Flud dynmc gugng pncples. () Scemtc f nzzle, t nset sng key dmensns; (b) clbtn cuves sng eltnsp beteen mss fl te, m, nd dmensnless clence, d. t 66

31 66 Fl Nzzle bdy z Nzzle () Guged sufce Nzzle z Pyscl dmn Fgue. Scemtc f te pyscl dmn nvestgted n ts k.

32 () Outlet bundy v z = vz, ve - R (b) Axl symmety R.5R. R.5R R Wll bundy (Nn slp cndtn).5r Wll bundy (Slp cndtn) z 5R Open bundy (p = p ) Tngles 4R 667 Wll bundy (Nn slp cndtn) Fgue. Smultn gemety () sng bundy cndtns nd dmn dmensns, nd (b) mes used ( g mes densty s emplyed unde te nzzle m, n te tt, nd lng te bse).

33 tu tu PG ' ' c dp d ' p b ' d ' Fgue 4. Dmensnless pfles f (): nzzle extenl gemety ( ), (b): pessue gdent ( dp d ' ), (c) se stess ( ), nd (d) pessue ( p ) s functn f te dmensnless dl pstn ( ' ) f dffeent vlues f, f te lne nzzle spe. 678

34 P Re d b Fgue 5. () Effect f Re d n e-veged se stess nd (b) dmensnless e- 68 veged se stess s functn f ', f dffeent vlues f f te cse f 68 lne nzzle spe

35 ' ' mn '.98 Eq. (45) mn ' '.9.9 b Fgue 6. () Dmensnless e-veged se stess s functn f ' f selected 689 negtve vlues f sng te exstence f mnmum : lc f mnm dented mn by te dsed lne. (b) Evlutn f Eq. (45) sng te effect f n lc pstn nd mgntude. 5

36 = -.9 sp ' b ' mm deg Fgue 7. () Cmpsn f teetcl (Eq. (47), sld lc) nd CFD smulted (dsed lc) dmensnless se stess pfles f selected vlues f sp. (b) Dmensnless se stess evluted t te nne dus f te nzzle ( ' ) s functn f te ngle f te nzzle m f dffeent vlues f. 7 6

37 7 z..5. P R ' z b.5.5 P.5.5 m ' Fgue 8. Cmpsn beteen se stess pfles btned nlytclly (sld lc) nd fm CFD smultns (dsed lc) f () tdl nzzle, (Eq. (5)) nd (b) pblc- tdl pfle f ((Eq. (5)). CFD cndtns: Re d, =.5 mm, =.5 mm nd = 5 mm. 7

38 79 dp d ' 5 4 pg pg b ' ' p pg c pg d ' ' Fgue 9. Dmensnless pfles f (): pessue gdent ( dp d ' ), (b) se stess ( ), (c) pessue ( p ), nd (d): nzzle extenl gemety ( ) s functn f te dmensnless dl pstn ( ' ) f dffeent vlues f pfle. pg, f te cse f lne pessue gdent 8

39 77 P pg Re d pg b ' 79 Fgue. () Effect f Re d n e-veged se stess nd (b) dmensnless e- 7 veged se stess s functn f ', f dffeent vlues f pg f te lne 7 7 pessue gdent pfle cse. 9

40 p [P] ' Fgue. Cmpsn beteen teetcl (Eq. 57, sld lc) nd CFD smultn (symbls) f pessue s functn f ' f dffeent Re d f te cse f lne pessue pfle. Symbls: () Re d =, () Re d = 5, () Re d =, () Re d =, () Re d = 4, nd (+) Re d = 6. F ec smultn: =.5 mm, =.5 mm nd = 5 mm. 79 4

41 s p -4 s b ' ' 5 4 s.8 c 5 4 s d ' dp d ' ' 7 7 Fgue. Dmensnless pfles f () se stess ( ), (b) pessue ( p ), (c): nzzle 74 extenl gemety ( ), nd (d): pessue gdent ( dp d ' ) s functn f te dmensnless dl pstn ( ' ) f dffeent vlues f stess pfle. s, f te cse f lne se 4

42 b p [P] Re d = ' [P] 4 Re d = ' c 6 5 d [P] Re d = ' 5 [P] Re d = ' Fgue. Cmpsn beteen CFD smultn (dsed lc) nd nlytcl pedctns (sld lc) f () pessue t (Eq. 6) nd se stess vlues (Eq. (58)), f (b).7, (c) nd (d).5, s functn f ' f dffeent Re d. Cndtns: =.5 mm, =.5 mm nd = 5 mm

43 746 P s.8.4 Re d s b ' 749 Fgue 4. () Effect f Re d n e-veged se stess nd (b) dmensnless e veged se stess stess pfle cse. s functn f ', f dffeent vlues f s f te lne se 4

44 ' [P] Fgue 5. Se stess vlues s functn f ' f te exmple f cmpste nzzle t cnstnt t te ute pt f te nzzle nd cnstnt t te nne pt ( Re d =, =.5 mm). Lc s te nlytcl mdel pedctns nd ccles te vlues btned fm CFD smultn. 44

45 76 Tble. Expessns f, dp d ' nd usng genel expessn f te type f ' 76 (tn ec f te tble) f ete, dp d '. 76 dp d ' f ' ' f pg ' ' ' f ' ' f f ' f ' pg pg ' ' f ' f ' ' s fs ' s