Industriestrasse 1-3, Herzogenaurach, Germany c State Key Laboratory of Chemical Engineering, East China University of Science and Technology

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1 HEFAT 9 Inenaional Confeence on Hea Tansfe Flid Mecanics and Temodynamics 6 8 Jly Mala CFD modellin of ily vis olyme in film flow on a veically oaional disk aially immesed in liqid fo synesis of loyeyleneeala Md Salim Mia a Cisian Dassle b Xiaoan Yan *a Jianan H c a Insie fo As Science and Tecnoloy Glyndŵ nivesiy Wexam LL AW K b emobiliy Sysems Division Scaeffle Tecnoloies AG & Co. KG Indsiesasse Heoenaac Gemany c Sae Key Laboaoy of Cemical Enineein Eas Cina nivesiy of Science and Tecnoloy Meilon Road Sanai 7 P.R. Cina * Coesondin Ao s Addess: x.yan@lyndw.ac.k ABSTRACT Te esen sdy focses on liqid in film flows on veically oaional disk a is aially immesed in a liqid ba. Tis ae aims o invesiae e liqid in film flow on a oaional disk sin CFD modellin aoac and emloyin e maemaical model as oosed by Afanasiev e. al. [] and o define e sabiliy and saes of e in film ickness ofiles. Te dominan fa a deemine e film ickness ae idenified wi oosin a coelaion eqaion o edic e film ickness as a fncion of anla osiion adis oain seed visiy and sface ension. Te in film ickness vaiaion in e anla diecion and e film daed ino e liqid ae aiclaly invesiaed since ey ave been ovelooked in eviosly docmened eseaces. INTRODCTION Tin film flows ae enconeed in a wide ane of indsial alicaions []. An examle is e synesis of olyeyleneeala PET in olycondensaion ea in wic a seies of veically oaional disks ae aially immesed in ily vis olyme liqid s ickin and seadin e mel in e fom of in film on e sface of e disks. nlike e in film flow on a oionally oaional disk e in film flow on a veically oaional disk aially immesed in liqid is always associaed wi a meniscs eion wee e liqid is daed o by e disk movin and a secially oscillain eion wee e film fomed on e disk is daed ino e liqid. Wile a lo of sdies on e in flow on oional oaional disk ave been docmened in e oen lieae ee ave been limied sdies on veical se and e flid dynamical asecs ae sill no flly ndesood alo ee ae some discssion and eneal solions of film ickness ofiles fo e oblem of liqid da o [4 5]. In is eseac e oosed maemaical model by Afanasiev e.al. 8 is fe exended o define e film aen fomed on e veically oaional disk and Volme of Flid VOF meod is emloyed fo solvin is kind of oblem sin CFD code ANSYS Flen. Fo e oblem of e film fomaion on e veically oain disk aially immesed in liqid e foce balance is imoan as e sae and sabiliy of e in film is conolled by vaios foces acin on i incldin vis ineial sface ension cenifal coiolis and aviaional foces. Fo a veically oain disk e coiolis foce can be neleced a e leadin ode as e em of e coiolis foce as e same ode as e ems of e ineial foces de o e esicion of e lbicaion eoy [6 7 8]. NOMENCLATRE Ca [-] Caillay nmbe D [m] Immesion de d' [-] Dimensionless immesion de F [-] Fode nmbe G [m/s ] Gaviaional acceleaion H [m] Tin film ickness ' [-] Dimensionless film ickness R [m] Disk adis R' Re T [-] [-] [s] Dimensionless disk adis Reynolds nmbe Time We [-] Webbe nmbe Secial caaces [-] Asec aio of e flow [⁰] Anla coodinae Ω [m] Roain seed υ [m /s] Kinemaic visiy [k/m ] Densiy [Pa.s] Dynamic visiy ω Sbscis CFD CFL [N/m] [ad/s] Sface ension Anla velociy Comaional flid dynamics Coan-Fiedic-Lewy nmbe

2 PET Polyeylene eealae VOF Volme of flid MATHEMATICAL MODELLING Te ysical se fo e veically oain disk aially immesed in liqid is sown in Fie. A disk of adis R is oain a e anla velociy Ω abo is oional axis wic as a disance d o e liqid ba. Fo mos of in film flows on veically oain disks e flows can be eaed as incomessible. Fo e oblem of e veically oain disk cylindical coodinae sysem is emloyed fo convenience. Le e liqid velociy veco o be eesened by and ω denoes e anla velociy veco wi comonens Ω. Fie. Confiaion of a oain disk aially immesed in liqid Navie-Sokes eqaions wic ae sed o descibe e in film flow on a veically oaional disk can be exessed as: ν sin a ν b ν c wee ν and denoe esecively e densiy dynamic visiy kinemaic visiy and e esse of e liqid. Te only exenal foce acin on e in film liqid is aviy. Tin film flows on e oain disks like oe flows sold also saisfy e coniniy wic ives Fo e bonday condiion a e sface of e disk i.e. no-sli condiion is imosed and e disk sface is assmed o be imemeable. Ts Ω Te oal ae of cane of e in film ickness sold be eqal o eo wic esls in e followin kinemaic condiion 4 Ineaion of Eqaion and sbsiion ino Eqaion 4 by alyin Leibi ineaion le yields d d 5 A e in film sface e saic esse sold balance e sface ension foce wic eqies e nomal sess condiion o saisfy k n n 6 wile a e ineface of e fee sface if e ficion de o e indced ai flow is neleced e anenial sesses on e fee sface of e film sold disaea wic yields.. i n i 7 Te nomal and e anenial ve in adial and anla diecion can be fond if e ickness of e in film can be deemined based on n 8 wile e sess enso is defined and iven by Liqid d Ω

3 9 Fo mos of flids e sface ension is almos ncaned and can be assmed o be consan. Ten e mean cvae of e in film fee sface can be esimaed by k Sbsiion of eqaions 9 and ino eqaions ino 6 and 7 yields e bonday condiions fo e nomal sess wic can be wien as and e anenial sess condiion in adial diecion and e anenial sess condiion in anla diecion Diec solion of Eqaion oee wi e bonday condiions 6 and 7 coled wi Eqaion 5 is imossible since e eqaions involved ae ily non-linea. Howeve by inodcin easonable simlificaions and assmions is se of aial diffeenial eqaions can be solved a leas nmeically. Te followin secion will discss e exisin analysis fo e in film flows on a veically oaional disk Afanasiev e al. 8 o sc oblem wic will be sed fo idance fo CFD modellin of e in film flow on e veically oain disk. NONDIMENSIONAL ANALYSIS Fo e film fomaion on e veically oain disk e se of lbicaion aoximaion is feasible b e aoiae len scale sold be consideed. Obviosly e yical len scale sold be R e adis of e disk wile e anenial velociy of e oain disk may be e siable caaceisic velociy scale wic is iven by RΩ 4 and e ime scale is en as R T 5 Since e liqid film fomed on e sface of e oain disk is vey in a small aamee in e analysis can be inodced i.e. << R H 6 On is basis e followin dimensionless vaiables can be inodced: R H P T 7 Te dominan vis em is balanced wi aviaional em in e -momenm eqaion so a e caaceisic ei H can be scaled wic yields H 8 4

4 I is execed a e esse will comee wi e dominan vis em so a H P R 9 Becase e in film involves e fee sface e sface ension lays an imoan ole in conollin e liqid sface. I can be eclded a e followin scale can be sed o elae o e nomal sess bonday condiion i.e. R P H Eqain Eqaions 9 and ives e scale fo R as H R Wen e caillay nmbe Ca is inodced i can be sown a Ca is small fo e case of e in film flow on e veically oain disk. << R H Ca I is noed a is len scale is aoiae fo e in film eion away fom e liqid ba. Te len scale sold be econsideed fo e balance of aviy and sface ension foces. P R Te esse scale can be deemined by sface ension P 4 Fom eqaions and 4 e len scale nea e liqid ba can be obained R 5 We also define a Reynolds nmbe ReR/. Nondimensionalisin Eqaion yields sin Re 6 Re Re 8 Hee all vaiables ave been non-dimensionalised. Te bonday condiions a e disk ae Ω 9 Te bonday condiions a e fee liqid sface ae: Becase is vey small Eqaions 6 7 and 8 can be simlified as sin 5

5 4 5 wi e condiions 4-4- and 4- o be simlified as Eqaions 4 and 5 can now be solved wi e bonday condiions 6 7 and 8 also coled wi e kinemaic condiion fo desciion of e in film fee sface wic is iven by d d 9 I can be seen fom Eqaion 5 a does no deend on. Wen velociy comonens and ae iven e film ickness can be fond o 9. Reaanin Eqaions and 4 ineain wice and alyin bonday condiions one can obain sin 4 4 Sbsiion 4 and 4 ino Eqaion 9 and ineaion yields Ω sin 4 Eqaion 4 is e sandad eqaion fo film ickness wee Ω Ω I sold be noed a bonday condiions 6 o 8 is only aoiae fa away fom e liqid ba. If e len scale is econsideed in e viciniy of e liqid ba as sown in Eqaion 5 based on e analysis by Afanasiev e al. 8 en bonday condiion 6 becomes 4 Sface ension foce as a sinifican inflence on film ickness ofiles and Eqaion 4 needs o be solved in conjncion wi Eqaion 4 wic can be wien as Ω k k sin 44 wee κ is cvae of fee sface. Diec solion of Eqaion -44 is no ossible as is kind of eqaion is ily nonlinea. Fo seady in film flow Eqaion -44 can be fe simlified as sin Ω k k 45 CFD modellin of e in film flow on a veically oaional disk Since e ovenin eqaions sed o descibe e in film flow oblem in cen sdy ae ily non-linea seekin fo fll analyical solions fo e oblem is imacical. Ts CFD modellin aoac as been adoed in is sdy [9]. Te se of a i qaliy mes and easonable bonday condiions wic ae incooaed ino CFD modellin is ccial o is sdy []. Fo e nmeical solion of e in film fee sface flow CFD code - Flen as been sed. Fo e flly ee dimensional oblem of e oain disk a cylindical coodinae sysem is emloyed wee e oain disk is laced in e middle of a cylindical vessel o oae veically abo e oional axis. Followin e wok done by Afanasiev e al. 8 e disk adis is assmed o be R7. mm. Te len of e cylindical vessel sold be lon eno and fa away fom e disk so a e bonday condiion / can be imosed wen e cylindical vessel is aly filled wi liqid. Te cylindical vessel as a diamee 6

Te oal nmbe of mes sed in e CFD modellin is 6974. Te se of a efined mes was also esed b e simlaion esls seem o be indeenden of e mes.

6 of 8 mm and a len of 4 mm. Since e film fomed on e oaional disk is vey in a fine mes is eqied in e viciniy of e disk. As sown in Fie a sced id wi exaedal elemens was eneaed wi ma face meses on e cicla sfaces exended by e cooe volme mes o avoid nmeical diffsion as mc as ossible. Te oal nmbe of mes sed in e CFD modellin is Te se of a efined mes was also esed b e simlaion esls seem o be indeenden of e mes. Volme of Flid meod VOF as been econised as an aoiae nmeical ecniqe fo ackin and locain e fee sface of wo o moe immiscible flids by calclain e volme facions in eac cell of a fixed Eleian id [4]. A volme facion aamee F fo eac of e Eleian id is defined in e VOF meod. A cell is assmed o be comleely filled wi liqid wen F wile emy wen F and i conains ineface of wo o moe ases if <F<. Sc fncion F can be ansoed by sin e advecion eqaion. Based on F vales e fee sface sae can be deemined sin a aicla ineolaion ecniqe.vof meod is emloyed in cen sdy fo ackin e locaion and e aen of e in film on e veically oain disk. One imoan feae of e VOF meod is ansien simlaion i.e. nseady simlaion. Ts e simlaion iself eqies a caefl selecion of e ime se so a e simlaion is sable. Te cieion sed fo deeminin e ime se is so-called Coan nmbe. In fac bo eal flid flow and nmeical simlaion of in film flow eqies a e fee sface fon advance canno exceed a mes ineval. Tis Coan-Fiedics-Lewy condiion CFL condiion is a necessay condiion fo conveence wile solvin ceain aial diffeenial eqaions nmeically. Te CFL condiion is exessed as. C 46 x wee is e liqid velociy is e ime se and x is e mes ineval o mes sie. Obviosly < x sold be ensed in all nseady simlaion o kee sabiliy. In is case a small coan nmbe as been sed wic ives a minimm ime se abo - s. NMERICAL RESLTS AND DISCSSION Te CFD simlaions ave been condced based on fo flids of diffeen visiies as sown in able. Te oeies of e es flid ave been aken e same as Afanasiev e. al. 8 fo comaison of e simlaion esls. Table. Diffeen flid oeies Pa.s N/m k/m Tes flid.77 PDMS PDMS PDMS In e simlaions e disk is assmed o oae a a consan anla velociy Ω wile i is alf immesed in liqid i.e. d' e immesion de d is nondimensionalised ee by dividin wi e adis of e disk. In Afanasiev s e. al. wok [] e film ickness ofile is only e fncion of. Howeve o CFD simlaion esls ave clealy indicaed a e acal film ickness is no only deenden on b also. Fo case of liqid bein daed o of a ba ool via a veically movin wads fla lae Landa and Levic 94 ave sowed a if e caillay nmbe Ca is small e film ickness can be esimaed by.9 Ca 47 6 wee is e film ickness is e velociy of e lae is densiy is aviaional foce and is e dynamic visiy of e liqid. Wilson 98 indicaed a e aoximae solion iven by Landa and Levic 94 is only valid wen e caillay nmbe aoaces o eo and e obained a eneal solion of e film ickness wen e fla lae is veically alined wic is exessed [.9458 Ca 6 ].685 Ca 48 By sin simila analysis fo in flow on veically oain disk Afanasiev e. al. [] obained a seady sae solion of e film ickness wic is iven by.9458 Ω 49 Fie. Gid eneaion fo e oain disk Te calclaed film ickness disibion a diffeen adial osiions is sown in Fie e dimensionless adis of e oain disk is aken as R'. Te comaisons ae sown a R' and. I can be seen fom Fie a e film fomaion can be caaceised by wo eions indicaed in Fie 4 <<9 as e eion fo e film da o and 9 <<8 as e eion fo e film o be daed in [5]. I was evealed fom e simlaion a e film is ick and 7

I as been indicaed fom e simlaion a e vis foce is dominan in e da-o eion wile e aviaional foce is dominan in e da-in eion. 5º 5º º 6º 9º º 5º 75º Fie 5.

7 nsable in e da-o oin and i adally becomes sable b e film as a endency of downwads and sinificanly affeced by all e foces acin on e film flow. Te foce balance may lay a leadin ole in conollin e film flow. Te foces sc as vis foce ineia aviaional foce and sface ension foce acin on e film flow on e oain disk ave been discssed by [6 7 8]. I as been indicaed fom e simlaion a e vis foce is dominan in e da-o eion wile e aviaional foce is dominan in e da-in eion. 5º 5º º 6º 9º º 5º 75º Fie 5. Da-o and da-in of in film a diffeen anla osiions Fie. Film ickness ofiles a diffeen adial osiions 9 Da in Da o 8 I sold be noed a exce fo e eions wee e film js sas o fom and e-enes e liqid e cane in e in film ickness is no emakable 5 <<7 o a sli vaiaion in film ickness is obseved as can be seen fom Fie-5. Fie 6 sows e vaiaions of film ickness a diffeen anla osiions fo iven adis. Noice a e film ickness is non dimensionalised by dividin e adis of e disk and mlilied by a faco of in all cases o conas moe clealy o eflec e sce of e film aens. In ode o coelae e simlaion esls dimensional analysis was condced. Te film ickness is deenden on a nmbe of aamees and a fncional elaionsi may be assmed. F Ω d 5 Fie 4. Da o and da in eion Vijayavan and Ga [9] obained a coelaion fo e film ickness fomed on a veically oain disk aially immesed in Newonian liqid based on ei exeimenal esls wic eads 7.99Ca Cas η R χ wee η is dimensionless sface ension nmbe R is dimensionless de χ is dimensionless aviaional nmbe and C as is modified caillay nmbe. I can be sees clealy fom eqaion 5 a e effec of anla osiion is no inclded in e coelaion. Howeve i as been evealed fom e simlaion a e in film ickness ofile does deend on anla osiion as sown in Fie 5. wee is e densiy of e flid is e dynamic visiy is e sface ension is e aviaional acceleaion Ω is e oain seed is e flow ime d is e immesion de of e liqid is e adis and is e anla osiion. Since evios sdies on in film flow ave clealy indicaed a e in film flow can be well caaceised by e followin dimensionless aamees like Caillay nmbe Ca Fode nmbe F Reynolds nmbe Re and Webbe nmbe We e fncional elaion 5 is assmed o be able o exessed as k k k k4 k Re Ca F We F F d 5 wee ' is e dimensionless film ickness k k k k k 4 ae emiical consans wic can be deemined fom e bes eession fiin o e simlaion esls. 8

8 Eqaion 5 as clealy indicaed a e film ickness is affeced by oaional seed wic as been confimed in o simlaions as can be seen fom Fie 8. Te simlaions wee also comaed wi e asymoic solions of Afanasiev e al.[] and e CFD esls ae consisen wi e solions as obained by Afanasiev e al.[] wic as been sown in Fie 9. Fie 6. Film ickness a diffeen adial and anla osiions Te CFD simlaion esls as sown in Fie 6 can be cvefied sin e leas sqae ecniqe o minimise e oal eo. I as been fond o e ials a a combinaion of exonenial and olynomial cve fiin can delive e bes cve fi wic is iven by ae b b b b b ce 5 x 4 x 4 wee e coefficiens x x a b b b b b 4 and c ave been obained fom e CFD simlaion esls fo any iven adis. Te ediced film ickness a diffeen anla osiions fo iven adis is now well exessed based on exession 5. Fie 8. Film ickness a diffeen oaional seed Fie 9. Comaison of e film ickness ofile Fie 7. Film ickness a diffeen adial diecion I can be seen fom Fie 7 a e film ickness deceases fom e cenal coe owads e fine of e disk. Tis is de o e aviaion foce acion wic dives e film downwad. Howeve as menioned ealie e film ickness vaies alon e cicmfeenial diecion. Based on o simlaion esls e followin fied exession fo a iven adis R'7 is obained wic is iven by.6.48e * e I can be seen fom Fie 9 a ee exis diffeence beween e film ickness ofile ediced by CFD modellin and a obained by Afanasiev e al. [] in e da-o and da-in eions. Te eason fo is diffeence is sill nclea b vey likely e solion as obained by Afanasiev e al.[] does no flly eflec e inflence of e sface ension wile o CFD modellin as emloyed e fll Navie-Sokes eqaions fo e oblem. Tis eqies fe invesiaion. Te inflence of immesion de on e fomed film ickness was also assessed in o CFD simlaions. Fie sows e film ickness disibions fo ee diffeen immesion des. I is ineesin o noe ee a exce fo e eions of da-o and da in; e film ickness only slily canes fo diffeen immesion des wic ae in conadicion o e esls as obained by Afanasiev e al. 8. One of 9

9 exlanaions is a e immesion de only affecs e da o bonday condiions b e film flow on e a of e oaional disk is only affeced by e oveall acin foce balance. Ts e calclaed film ickness can be esimaed by Re.4 F Ca We Fie sows e ediced film ickness ofiles sin coelaion elaionsi 56. I can be seen fom Fie a e film ickness sinificanly canes and becomes icke wen e liqid visiy inceases. Tis can be seen clealy fom eqaion 49 since e film ickness is ooional o e visiy o owe /. Fie. Film ickness a diffeen immesion des Te film fomaion on veically oain disk is obviosly ime deenden. Fie sows e CFD esls fo e film ickness vaiaion alon e cicmfeenial diecion a iven adis R 7. I can be seen fom Fie a e film ickness ofile is almos same fo diffeen flow ime of e liqid. Wen flow ime is lon eno e film flow becomes seady. Ts fo a iven immesion de coelaion 5 may be simlified as k k k k4 k Re Ca F We 55 Fie. Film ickness a diffeen flow ime In ode o assess e inflence of visiy on e film flow beavio fo diffeen flids of diffeen visiies ave been seleced in e CFD modellin. Te densiy fo PDMS- PDMS- and PDMS- is 975 k/m. By sin coelaion 55 fo bes fiin o e simlaions i was fond a k k k k k 4 ake e followin vales. k.8 9 k.4 k -4.9 k 6.56 k Fie. Effec of visiy on film ickness ofiles CONCLSION Te oblem of e film fomaion on e veically oaional disk aially immesed in liqid as been solved nmeically o obain e in film ickness disibions. Based on CFD simlaion esls a coelaion is oosed o descibe e film ickness disibion by definin e dominain fa conollin e in film flows. I was fond fom e simlaion a e wo main dominan fa conollin e film ickness ofile ae e visiy and oaional anla velociy. Te film ickness inceases sinificanly wi incease of e anla velociy of e disk consisen wi e findins as eoed in e blised lieae []. Te cane in e film ickness and e sae of e in film can be caaceied by e Fode nmbe. Te simlaion also indicaes a an incease in e visiy cases e aveae film ickness o incease indicain Caillay nmbe and Reynolds nmbe o be e imoan dimensionless aamees in in film flow analysis. In addiion e effec of e sface ension on e film ickness ofile can be indicaed by incooain e Webbe nmbe. Te coelaion wic maces e simlaion esls eqies inodcion of a osiion faco o accon fo e cane in e film ickness ofile in diffeen anla osiions of e oain disk. 4

10 REFERENCES [] Afanasiev K. Mnc A. and Wane B. Tin film dynamics on a veically oain disk aially immesed in a liqid ba A. Ma. Modelin Vol [] Pama N.H. Timkdl M.S. and Hinc E.J. Coain flow of vis Newonian liqids on a oain veical disk Pysics of Flids Vol. 9. [] Danis M. Sama R.K. and Ali S. Gas absoion wi fis ode cemical eacion in a lamina fallin film ove a eacin solid wall A. Ma. Modellin Vol [4] Landa L. and Levic B. Dain of a liqid by a movin lae Aca Pysicocim.RSS Vol [5] Wilson S.D.R. Te da-o oblem in film coain eoy J. En. Ma Vol [6] Emslie A.G. Bonne F.T. and Peck L.G. Flow of a vis liqid on a oain disk J. Al. Pys. Vol [7] Myes T.G. and Cain J.P.F. Te effec of e Coiolis foce on axisymmeic oain in film flows In. J. of Non-Linea Mecanics Vol [8] Myes T.G. and Lombe M. Te imoance of e Coiolis foce on axisymmeic oional oain in film flows Cem. En. and Pocessin Vol [9] Lan H. Fiedic M. Amaly B.F. and Dallmeie J.A. Simlaion and measemen of D sea-diven in liqid film flow in a dc In. J. of Hea and Flid Flow Vol [] Abin J. Flece D.F. and Xeeb C. Desin of mico mixes sin CFD modellin Cem. En. Science Vol [] Hasan N. and Nase J. Deeminin e ickness of liqid film in lamina condiion on a oain dm sface sin CFD Cem. En. Science Vol [] Kim J. Adaive Mes Refinemen fo Tin Film Eqaions Jonal of e Koean Pysical sociey Vol [] Gao D. Moley N.B. and Di V. Nmeical simlaion of wavy fallin film flow sin VOF meod J of Com. Pysics Vol [4] Haon Y. Leende D. and Raynal L. Volme of flid meod fo inefacial eacive mass ansfe: Alicaion o sable liqid film Cem. En. Science Vol [5] Kesi H. S. Kisle S. F. and Sciven L. E. Risin and fallin film flows: viewed fom a fis-ode aoximaion Cem. En. Science Vol [6] Y S.H. Lee K.S. and Yook S.J. Film flow aond a fas oain olle In. J. of Hea and Flid Flow Vol [7] Maa O.K. and Lawence C.J. Te effec of sfacan on e flow of a in liqid film ove a sinnin disc Cem. En. Science Vol [8] Kecenikove R. and Homsy G. M. Sfacan effecs in e Landa Levic oblem J. Flid Mec. Vol [9] Vijayavan K. and Ga J.P. Tickness of e film on a veically oain disk aially immesed in Newonian liqid Ind. En. Cem. Fndam Vol

Chapter Finite Difference Method for Ordinary Differential Equations

Chapter Finite Difference Method for Ordinary Differential Equations Chape 8.7 Finie Diffeence Mehod fo Odinay Diffeenial Eqaions Afe eading his chape, yo shold be able o. Undesand wha he finie diffeence mehod is and how o se i o solve poblems. Wha is he finie diffeence