INFLUENCE OF DRAG FORCES ON A SWARM OF BUBBLES IN ISOTHERMAL BUBBLY FLOW CONDITIONS

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1 Seveth Iteratoa Coferece o CF the Mera ad Proce Idtre CSIRO, Mebore, Atraa 9-11 eceber 9 INFLUENCE OF RAG FORCES ON A SWARM OF BUBBLES IN ISOTHERMAL BUBBLY FLOW CONITIONS Co LI 1, Shera C. P. CHEUNG 1, G.H. YEOH, Jya TU 1 1 Schoo of Aeropace, Mechaca ad Mafactr Eeer, RMIT Uverty, Vctora 383, Atraa Atraa Ncear Scece ad Techooy Oraato (ANSTO, PMB 1, Mea, NSW 34, Atraa *Correpod athor, E-a addre: jya.t@rt.ed.a ABSTRACT rve by the exteve dead of at hhy cocetrated a bbby fow ay eeer fed, erca tde have bee perfored to vetate the ehbor effect of a war of bbbe o the terfaca dra force. I th tdy, a ove dra coeffcet correato (Soet et a., 7 ter of oca vod fracto coped wth the popato baace ode baed o averae bbbe ber dety (ABN ha bee peeted ad copared wth Ih-Zber deey dtrbted fd partce dra ode. The predcted oca rada dtrbto of three prtve varabe: a vod fracto, Sater ea bbbe daeter, ad a veocty, are vadated aat the expereta data of Hbk et a. (1. I eera, atfactory areeet betwee predcted ad eared ret are acheved by both dra force ode. Wth addtoa coderato for coey packed bbbe, the atet coeffcet ode by Soet et a. (7 how coderaby better perforace captr the redcto of dra force crred by ehbor bbbe. NOMENCLATURE a f Iterfaca area cocetrato ra coeffcet C C ra coeffcet of oated bbbe Ier daeter of the ppe Bbbe Sater ea daeter E o Eötvo ber F ra force Averae ber dety of a phae (bbbe Re Mxtre Reyod ber Veocty of oated bbbe a qecet qd v Veocty vector Greek Sybo α Vod fracto Max pack vae α α oc σ RC φ Loca vod fracto Vcoty Mxtre vcoty ety Srface teo Coaecece rate de to rado coo TI φ Break-p rate de to trbece pact Sbcrpt Ga Idex of a/qd phae Lqd INTROUCTION Two-fd ode baed o the ter-peetrat cota approach probaby codered to oe of the practca ad accrate acrocopc forato of a-qd fow yte. Here, exchae that occr at the terface betwee the two phae are expcty accoted ad the dyac of the teracto are fy decrbed throh approprate cotttve reato over the terphae a, oet ad eery exchae. The extece of thee ter oe of the ot portat charactertc of the two-fd ode forato ce they detere ot oy the deree of echaca ad thera o-eqbr bt ao the rate of phae chae betwee phae. I the abece of heat ad a trafer, the copexty of the probe redce to coderato of oy the oet exchae ter, whch typca of othera bbby fow. The terfaca trafer the coervato oet eqato eeray vove the coderato of dra ad o-dra terfaca force dete. The terfaca dra force a ret of the hear ad for dra of the fd fow, whch deped o the dra coeffcet a we a the terfaca area cocetrato. For od partce, the dra coeffcet deped oy o the charactertc of the fow rrod the partce ad prary a fcto of the partce Reyod ber ad of the trbece tety of the coto phae (Bertoa et a., 4. For bbbe, the behavor are however copcated de to three portat apect. Frty, whe the qd pre eoh, t ha the pobty to p ao the rface of the bbbe. Th cotrat to fow pat rd (od bode where the op codto poed. Secody, aot a the erta cotaed the qd for bbby fow de to the reatve weak dety of bbbe copared to that of the qd; th erta dced hydrodyac force partcary portat the predcto of bbbe oto. Thrdy, bbbe have a tedecy to defor de to coaecece ad break-p; the chae of bbbe hape add ew deree of freedo to a aready copex probe (Maadet ad Eae,. Copyrht 9 CSIRO Atraa 1

2 The ta teracto ao bbbe ca ao have a fcat fece o the dra force (Behzad et a., 4; Bertoa et a., 4; ad Soet et a., 7. Fre 1 trate the profe of the qd veocty pa throh the cae of t-bbbe coparo wth oe of a e bbbe wth for co fow. It deotrate that the profe of the qd veocty dtrbto ao rada poto ca be fcaty affected by add war of bbbe. Wth hher vod fracto, bbbe repace ad arrow ore pace of cro ecto for qd, whch creae a qd veocty de to qd a coervato. However, qd veocty coe to ppe wa ad bbbe rface coparatvey ow becae of the vco force. Th, reat qd veocty radet or eve a edde poby be prodced oe oca poto. O the other had, wth hh vod fracto, bbbe have tedecy to be coer ad t ay be fcaty affected by the preece of a ated wake behd other bbbe a t-bbbe yte, whch ay add copexty to erca ato of t-bbbe yte. Fay, thee etre o-for ad copcated qd behavor tr draatcay affect the ar-water p veocty dced dra force a we a the bbbe wake or hear dced ft force act o the two-phae fow. Over the year, evera ode have bee propoed to cacate the dra force for bbbe at hh vod fracto. Ih ad Zber (1979 have cateorzed the bbby fow behavor to dfferet ree. A xtre vcoty ode ha bee deveoped to obta each dra coeffcet correato for the dvda fow ree. CFX coerca oftware dra coeffcet ode baed o Ih ad Zber (1979 dra forato ha bee wdey apped ad ca yed reaoabe predcto for a rae of fow codto. May recet vetato by Rche ad Ia (, Behzad et a. (4 ad Soet et a. (7 have everthee foced o the ode of tabe dra coeffcet tper acro a wde rae of vod fracto ad dfferet fow ree. Throh th approach, the rato of the dra coeffcet to t e dpered eeet vae ca be ftted to a fcto of the phae fracto,.e. C / C f( α where C the dra coeffcet of a oated bbbe a fte ed. Soet et a. (7 have deveoped a ove dra coeffcet expreo by correat the dra coeffcet tper wth exhatve expereta data. The proved expreo to predct two-phae fow fro bbby to tratoa ree vetated the preet tdy. Wth th ree, bbbe are eeray fod to be hhy dtorted ad coey packed cotrat to the bbby fow ree of whch the bbbe are oray pherca hape ad aowed to ove freey (Hbk et a., 1 ad Che et a., 7. The prary a of th paper to copare the reatve ert ad capabte appy two dra coeffcet forato cooy ed Ih ad Zber (1979 ode ad recety propoed by Soet et a. (7 to evaate the dra force the coervato oet eqato. I order to predct the dyac chae of the terfaca trctre, the e of frt-order eqato to charactere the traport of terfaca area cocetrato (Hbk ad Ih,, Yao ad More, 4 or averaed bbbe ber dety (ABN (Che et a., 7 ha ffced. The ABN ode (Che et a. 7 th epoyed to predct the oca bbbe dtrbto a pward fow chae ad peca epha drected Se Bbbe Swar Bbbe Fre 1: Profe of qd veocty pa throh e bbbe ad t-bbbe toward vetato o bbby-to- tratoa ree (cap-bbby fow codto where hh vod fracto ad hh qd veocty ext. Nerca predcto throh thee two dra coeffcet correato are copared aat expereta data of othera aqd bbby fow a vertca ppe perfored by Hbk et a. (1. MATHMATICAL MOELS Two-Fd ad ABN ode The two-fd ode coervato eqato for a ad oet for bbby fow ca be wrtte a: ( α t ( α t v + ( α v + ( α v v T ( + ( e N α ] + F + F v v v α P + α + N where F the terfaca dra force dety whe F cot of the o-dra force cotrbto. For dpered othera bbby fow, a a e bbbe ze ve by the bbbe Sater ea daeter, the averae bbbe ber dety for bbby fow ca be defed a: (1 ( α (3 π / 6 3 S Ad the averae bbbe ber dety traport eqato take the for: RC TI ( φ + φ + v (4 t where v the ea a veocty. The pheoeooca echa of coaecece ad breakae are affected throh the orce ad k ter RC TI φ ad φ of whch they are de to rado coo, trbet dced breakae. The Yao ad More (4 ode adopted the preet tdy. More deta reard the ode ca be referred Che et a. (7. Iterfaca Moet Trafer de to ra The ter-phae oet trafer betwee a ad qd de to the dra force reted fro hear ad fro Copyrht 9 CSIRO Atraa

3 dra ad ca be odeed ter of the terfaca area cocetrato a ad p veocty v v f ( a: 1 v v v v F C a f ( (5 8 F It hod be oted that the above eqato depct the oet trafer fro the a phae to the qd phae. I the preet tdy, two correato of evera dtct Reyod ber reo for dvda bbbe propoed by Soet e a. (7 ad Ih ad Zber (1979 correato are epoyed to evaate the dra coeffcet C eqato (5. Ih ad Zber (1979 dra coeffcet correato take to accot for t-bbbe effect by coder dfferet bbbe hape ree; ch a: dee pherca partce ree, dee dtorted partce ree ad dee pherca cap ree. ee Spherca Partce Ree C ( phere ( Re (6 Re ee torted Partce Ree C ( epe Eo E (7 3 ee Spherca Cap Ree 8 C ( cap E (8 3 where Re xtre Reyod ber. More forato ca be fod Ih ad Zber (1979 I eqato (7, the dee dtorted partce ree dra coeffcet ode take the for of a tpy factor E, whch ve ter of the vod fracto a 1 + E Where f f ( f ( α 6 / 7 α (9 1 / ( α (1 α (1 Ad, Eo repreet the Eotvo ber whch defed by ( Eo (11 σ where σ the rface teo coeffcet. For dee pherca cap ree, the tpcato factor E take however the for: E ( 1 α (1 A peeted wth ANSYS CFX 11 (ANSYS, 5. CFX-11 Uer Maa the ree eecto baed o C( phere fc ( phere C ( epe C (13 ( C( epe, C ( cap fc ( phere C ( epe O the other had, throh recet expereta vetato by Soet et a. (7, a ove dra correato for pre ar-water yte ha bee propoed. It ca be wrtte a C C E (14 where C the dra coeffcet of a oated bbbe a fte ed, whch ca be obtaed throh the baace of boyacy, dra ad ravtatoa a 4 1 C (15 3 I the above eqato, repreet the veocty of a oated bbbe a qecet qd whch ca be cacated the correato of Jaaahad et a. (1994: b 1 b (16 + where b 1 b 1 b (17 σ (18 b + ( I eqato (14, the tpcato factor E accord to Soet et a. (7 α / E (1 α (1 α + (4.8 ] (19 1 α Where, et a vae of 5. The above odfcato vad for a wde rae of vod fracto ad acro dfferet fow ree. Iterfaca Moet Trafer de to No-ra Force Aode wth the dra force, other terfaca o-dra force codered the preet tdy cocde ft force, wa brcato force ad trbet dpero force. More deta ca be fod Che et a. (7. EXPERIMENTAL AN NUMERICAL ETAILS The erca ode ha bee vadated aat experet codcted by Hbk et a. (1. The tet ecto wa a rod tbe ade of acryc wth a er daeter 5.8 ad eth of 361. Loca fow eareet the dobe eor ad hot f aeoeter probe were perfored at three axa (heht ocato of z/6., 3.3 ad 53.5 ad 15 rada ocato of r/r to.95. Soto to the two et of baace eqato for a ad oet of each phae are oht. Rada yetry ha bee aed whch erca ato were perfored o a 6 deree rada ector of the ppe wth yetry bodary codto at both vertca de. A three-deoa eh cota 18, hexaoa eeet eerated over the etre ppe doa. At the et of the tet ecto, fory dtrbted perfca qd ad a veocte, vod fracto are pecfed accordace wth the fow codto decrbed. A the daeter of the jected bbbe kow, et bbbe ze of 3 adopted baed o expereta forato at ocato of z/6.. The ta bbbe ber dety wa cacated Eqato (3. eta of the bodary codto have bee ared Tabe 1. The Shear Stre Traport (SST trbet ode epoyed the preet tdy. A depcted F., for fow codto epoyed for vadato are the bbby-to- tratoa ree, partcary cap bbbe are oberved fow codto of <j f >.986 / ad <j >.31 /. A deotrated Ho ad Yeoh (5, coaecece of capped bbbe ad t teracto wth bbby fow xtre ay becoe fcat bbby-to- tratoa fow ree Copyrht 9 CSIRO Atraa 3

4 ca otceabe dcrepacy betwee predcted ret ad eared data. Tabe 1: Bbby fow codto ad t et bodary codto epoyed the preet tdy. Sperfca qd veocty (/ j f Hbk et a. (1 experet.491 α z /. S z /..986 (%] α z /. S z /. (] (%] (] Sperfca a veocty j (/ b Cap bbbe were expereta oberved th fow codto Fre : Map of tbe fow ree ad trato fow codto tded the preet tdy RESULT AN ISCUSSION Sato of the othera pward bbby fow a tbe have bee carred ot for for dfferet operato, teded to cover a rae of a ad qd veocty. By coparo of ato ret wth expereta data, vae of the varo adjtabe paraeter the ode are detered, o a to ve reaoabe ret over a operat codto. I the preet tdy, the breakae ad coaecece cabrato factor are et a.6 ad.1 repectvey to baace whoe yte bbbe ber dety. The ar cabrato factor adjtet were epoyed reearch of Che et a. (5 ad Oo et a. (1. Vod fracto dtrbto Fre 3 how the vod fracto dtrbto obtaed fro Soet et a. (7 ad Ih ad Zber (1979 dra coeffcet correato copar wth the expereta data at the deoe axa poto Z/ I the preet reearch, the frt three fow codto (Fre.3a, b, c charactered a wa peak phae dtrbto are recorded ad have bee.19.] 3.].4.] 3.].19 5.] 3.].31 b 5.] 3.] captred very we by both dra coeffcet ode. Te Averaed Ga Vod Fracto -] Te Averaed Ga Vod Fracto -] Te Averaed Ga Vod Fracto -] Te Averaed Ga Vod Fracto -] Rada Poto -] (a Ih ad Zber Soet et a (b Rada Poto -] Ih ad Zber Soet et a. Ih ad Zber Soet et a. Ih ad Zber Soet et a. <j f >.491 / <j >.19 / <j f >.491 / <j >.19 / <j f >.986 / <j >.4 / Rada Poto -] (c.5 <j f >.986 / <j >.31 / (d Rada Poto -] Fre 3: Predcted rada vod fracto dtrbto ad experet data of Hbk et a. (1. However, the two dra coeffcet correato both how the der-predcted vod fracto at the core (F.3d for fow codto of <j f >.986 / ad <j >.31 /, whch trato phae dtrbto ad cap bbbe have bee oberved. Sce atera ft force preet the rato for bbbe toward the ppe cetre or wa, t becoe the doat factor to over vod fracto dtrbto. I th reearch, Toyaa (1998 ft coeffcet correato baed o coderato of bbbe deforato ha bee epoyed. However, th coeffcet wa deveoped oy fro expereta data of e bbbe r throh a fte taat qd a vertca ppe. The atet ft force correato propoed by Copyrht 9 CSIRO Atraa 4

5 Hbk ad Ih (7 t bbbe yte ha ao bee teted, however o fcat proveet ha bee oberved. Accord to the reearch by Hbk ad Ih (7, the ft force t-bbbe yte affected by ay factor, ch a reatve veocty betwee two phae, hear rate of qd, bbbe rotatoa peed, rface bodary codto, vod fracto. I th tdy, Ih ad Zber (1979 dra coeffcet correato ee to ve better predcto vod fracto. It ay cotrbte fro other factor rather tha reatve veocty. Sater ea bbbe daeter Fre 4 trate the predcted ad eared Sater ea bbbe daeter dtrbto. Except for the fow codto of <j f >.986 / ad <j >.31 /, the Sater ea bbbe daeter appeared aot for ao the rada drecto. Ow to the tedecy of a bbbe rat toward the wa, hhy cocetred bbbe ear the wa have reater pobty to cood for hty arer bbbe. For cae of fow codto of <j f >.986 / ad <j >.4 / (Fre 4c, both Soet et a. (7 ad Ih ad Zber (1979 dra coeffcet ve reaoaby ood predcto copared wth expereta data. For other two wa peak fow cae (Fre 4a, b, hty over-predcto are ve by both ode, however Soet et a. (7 dra correato eeray ve better erca predcto copared wth Ih ad Zber (1979 dra coeffcet. For tded fow codto of <j f >.986 / ad <j >.31 /, cap-bbbe wth wde rae of bbbe ze ao rada poto were oberved experet (Fre 4d. It ay be dffct to catch by ABN ode ce th popato baace ode oy preet behavor of averae bbbe daeter ad oray ve fat bbbe ze dtrbto predcto. Frtherore, or orce ter of wake etraet by Yao ad More (4 ode ed th tdy poby aother reao for dcrepacy betwee erca predcto ad expereta obervato ce cap bbbe ha troer wake etraet to ck the foow a bbbe. Te-averaed a veocty The oca rada a veocty profe fro the two dra coeffcet correato ad the expereta data at the ear tato are how Fre 5. Wth addtoa coderato of the ehbor bbbe, the Soet et a. (7 dra coeffcet correato baed o the trodcto of oca vod fracto ha the tedecy to ve coparatvey better predcto of the a veocty profe a for fow codto, copared to the Ih ad Zber (1979 correato. However, for the tdy cae of <j f >.986 / ad <j >.31 /, the Soet et a. (7 dra coeffcet correato t der predcted oca rada a veocty at the cetre of ppe. Geeray vertca bbby fow, the dety dfferece boyat force rearded a the drv force for bbby dpered phae r. For core peak fow codto of <j f >.986 / ad <j >.31 /, hh vod fracto cocetrate wa oberved the cetre of ppe whch ay ead troer drv force (dety dfferece boyat force per t voe for bbbe dpered phae. O the other had, hhy cocetrated vod fracto ha tedecy that bbbe are protected der ther ehbor wake area ad poby have weaker dra fece. Frtherore, the copex bbbe coaecece/breakae, ophtcated bbbe defor ad tro trbet fow x ay be dffcty to be ercay ated a cotttve eqato theoretcay at th tae. Te Averaed Sater Mea Bb.. ] Te Averaed Sater Mea Bb.. ] Te Averaed Sater Mea Bb.. ] Te Averaed Sater Mea Bb.. ].8.7 <j Ih ad Zber f >.491 / Soet et a..6 <j >.19 / (a Rada Poto -].8.7 <j f >.491 / Ih ad Zber Soet et a. <j >.19 / (b Rada Poto -].8.7 <j f >.986 / Ih ad Zber Soet et a. <j >.4 / (c Rada Poto -].8.7 <j f >.986 / Ih ad Zber Soet et a. <j.6 >.31 / (d Rada Poto -] Fre 4: Predcted Sater ea bbbe daeter dtrbto ad experet data of Hbk et a. (1. CONCLUSIONS A averaed oe-rop popato baace approach, the averae bbbe ber dety (ABN traport eqato, coped wth the Eera-Eera two-fd ode preeted th paper to hade the a-qd bbby fow der othera codto, partcary der bbby-to- trato ree. Baed o ABN ode, dra echa repectvey deveoped by Soet et a. (7 ad Ih ad Zber (1979 are copared aat expereta data codcted by Hbk et a. (1 der varo fow codto. Loca rada dtrbto of three prtve varabe: vod fracto, Sater ea bbbe daeter ad a veocty, are copared. I eera, both of the dra coeffcet Copyrht 9 CSIRO Atraa 5

6 Te Averaed Ga Veocty /] Te Averaed Ga Veocty /] Te Averaed Ga Veocty /] Te Averaed Ga Veocty /] Ih ad Zber Soet et a. (a Rada Poto -] <j f >.491 <j >.19 /.4 <j f >.491 /. <j Ih ad Zber >.19 / Soet et a (b Rada Poto -] <j f >.986 / Ih ad Zber.4 <j >.4 / Soet et a (c Rada Poto -] <j f >.986 /.4 <j >.31 / Ih ad Zber Soet et a (d Rada Poto -] Fre 5: Predcted rada a veocty profe ad experet data of Hbk et a. (1. correato predcto yed far areeet wth expereta ret. e to dra force copared wth other terfaca force ha coer reatohp wth qd/a veocty; Soet et a. (7 ode whch accot addtoa coderato of ehbor bbbe preet reatvey better predcto a veocty. Sch fd acerta that the effect of ehbor bbbe cod be feta hh vod fracto fow codto where bbbe are coey packed. Athoh addtoa coderato were attepted to corporate to the dra coeffcet, oe otabe dcrepace betwee the erca ad expereta ret t rea et ore -depth vetato o the dra coeffcet of a war of bbbe hod be carred ot ftre. ACKNOWLEGMENTS The faca pport provded by the Atraa Reearch Coc (ARC-covery P ratefy ackoweded ad the frt athor ao pported by RMIT Uverty throh Iteratoa Pot Gradate Reearch Schoarhp (IPRS. REFERENCES ANSYS, 5. CFX-1 Uer Maa. ANSYS CFX. BEHZAI, A., ISSA, R. I., RUSCHE, H., (4. Mode of dpered bbbe ad dropet fow at hh phae fracto, Che. E. Sc., 59, BERTOLA, F., BALI, G., MARCHISIO,., VANNI, M., (4. Moet trafer a war of bbbe: etate fro fd-dyac ato, Che. E. Sc., 59, CHEN, p., SANYAL, J., UUKOVIC, M. P., (5. Nerca ato of bbbe co fow: effect of dfferet breakp ad coaecece core, Che. E. Sc., 6, CHEUNG, S.C.P., YEOH G.H., TU, J.Y., (7. O the erca tdy of othera vertca bbby fow two popato baace approache, Che. E. Sc., 6, HIBIKI, T., ISHII, M., (. eveopet of oerop terfaca area traport eqato bbby fow yte, It. J. Heat Ma Tra., 45, HIBIKI, T., ISHII, M., Xao Z., (1. Axa terfaca area traport of vertca bbbe fow, It. J. Heat Ma Tra., 44, HIBIKI, T., ISHII, M., (7. Lft force bbby fow yte, Che. E. Sc., 6, HO, M.K., YEOH, G.H., (5. Pheoeooca vetato of a-qd fow, Tra. Aerca Nc. Soc., 93, ISHII, M., ZUBER, N. (1979. ra coeffcet ad reatve veocty bbby, dropet or partcate fow, A.I.Ch.E. Jora, 5, JAMIALAHMAI, M., BRANCH, C., ULLERSTEIN- HAGEN, H., (1994. Tera bbbe re veocty qd, Tra. I. Che. E., 7, MAGNAUET, J., EAMES, I. (. The oto of hh-reyod-nber Bbbe Ihooeeo fow, A. Rev. Fd Mech., 3, OLMOS, E., GENTRIC, C., VIAL, Ch., WIL, G., MIOUR, N. (1. Nerca ato of tphae fow bbbe co. Ifece of bbbe coaecece ad break-p, Che. E. Sc., 56, RUSCHE, H., ISSA, R. I. (. The effect of vodae o the dra force o partce, dropet ad bbbe dpered two-phae fow, I Japaee Eropea Two- Phae Fow Meet, Thkba, Japa. SIMONNET, M., GENTRIC, C., OLMOS, E., MIOUX, N., (7. Expereta deterato of the dra coeffcet a war of bbbe, Che. E. Sc., 6, TOMIYAMA A. (1998. Stre wth coptatoa bbbe dyac, Proceed of the Thrd Iteratoa Coferece o Mtphae Fow. Lyo, Frace. YAO, W., MOREL, C., (4. Voetrc terfaca area predcto pward bbby two-phae fow, It. J. Heat Ma Tra., 47, Copyrht 9 CSIRO Atraa 6