Modeling of gas bubble breakup in liquid steel

Transcription

1 Moelng of gs bubble breup n lqu seel K. Wcherle SB-Techncl Unversy of Osrv 7833 Osrv Porub, Czech Republc , Fx: , e-l: l.wcherle@vsb.cz One porn process of seelng echnology s rgon-vcuu egssng. Bch ypclly erc ons of he ecrbonze ron s plce n lle, where xng s prove by gseous rgon njece usully hrough porous cercs brc he boo. Typcl flow re s, N 3 /n, power npu - W/ 3, whch s coprble wh coon eron recors. Bubbles re responsble for hree effecs of he process: - ss rnsfer - reovl of lue gses N, H, CO, - floon of scble prcles lqu or sol slg o he level. Slg s here prouc of recon of nor seel coponens P, S, S wh he refrcory gnese, wh e le power n conns lso precpe proucs of recons wh e reucon gens Al, C or CC. - r-lf effec - hoogenzon of he seel nely when specl lloys le snless seel re prouce Iporn wy of gs reovl s evcuon of he syse. Typcl hegh of he seel level s 5, vcuu s 5 P s pple. Bubble oveen uner hese conons s que specfc. Chnge fro he "ferrosc" pressure he boo.4 MP o hs vcuu ens h he bubble volue ncreses es urng s flong o he level. Accorng o, breup of ncresng bubbles s n he focus of neres. Drec observon of bubbles n lqu seel s que coplce; herefore oelng s n essenl wy of he process nvesgon. Experenl oelng by usng ore user-frenly lqu els le ercury, Woo el or n ws one n cern lborores. Checl engneerng slry nlyss enbles us o suy prculr feures of he gs-lqu els wh he sples r-wer syses. The proble of lrge bubble breup n low-vscosy lqus s essenl. Then, hs process ws nvesge experenlly wh lrge bubbles levng n ownsre wer flow. Ssc resuls were nrouce o he heorecl oel of rgon-vcuu seel egssng lle. Locl srbuon of he sze, velocy n nuber of bubbles ws obne. These resuls cn be pple o prec lqu crculon velocy n ss rnsfer coeffcen n he lle. Inroucon When he onogrph Bubbles, Drops n Prcles by Clf, Grce Weber [] ppere n 978, looe le h nohng ore woul be one hn polshng he presene es. Snce h e, every yer severl hunres of experenl n heorecl ppers on he subjec re presene. Prculr probles re ree n oern onogrphs on sngle bubbles [-8] n on her funcon n nusrl processes [9-]. Coprvely low s he frequency of ppers concernng eu sze bubbles n low vscosy lqus. Shpe n oveen of hese bubbles re resuls of synergec effec of fferen forces, nclung grvy, surfce enson, vscosy n ner. Usully, he bubble shpe s pulsng choclly n he bubble oon velocy chnges boh bsolue vlue n recon. Lrger bubbles re lso vulnerble o breup. Curren CFD ools enble nuercl reen of soe probles of wo-phse flow, however hey cnno equely nclue soe porn prculr effecs le surfce vscosy, ze poenl no he hecl oel. Few experenl cn be foun osly n poneer ppers. New experenl lborores probbly resgn o he possbly ecpher nyhng porn urng he shor pero of sngle bubble rse. Mos of experenl nvesgon hs been repeely evoe o he less problec cse of sll sphercl bubbles n vscous lqus, where he corresponence wh spler heory cn be

2 eonsre. Therefore, experenl wor n he re of eu sze bubbles n low vscosy lqus s essenl. Whn he scope of our prevous reserch projec suppore by grns GA 5/98/5 n 4//547, we evelope n equpen wh levng bubbles n ownsre flow [-] enblng rsng bubbles o be observe for long e pero. Recen resuls concernng bubble breup [3] n s epenence on he lqu properes [4] re use o prec ssc behvor of swr of rsng bubbles. Anlyss Le us conser gven volue of gs nfuse no lqu s lrge nuber of unfor sze bubbles of volue. Nuber of N /. The volue of gs s no consn when hyrosc pressure ecreses of ben lqu urng he bubble rse. Oher reson for he volue chnge y be e chnge of exernl pressure urng vcuu reen, herl expnson of gs, esorpon of ssolve gs ec. Anyhow, he resulng volue of gs s funcon. ccorngly, sngle bubble volue s / Equvlen bubble eer eer of sphere of encl volue s: / 3 6. π Bubbles y spl. Sll n eu sze ellpsol bubbles prove usully wo unfor sze ugher bubbles. Lrge, unsble bubbles relese choclly sller bubbles, neverheless he sze of grn-grn- -ugher bubbles sys ore n ore unfor n he sze srbuon of eu sze bubbles n resulng swr s que nrrow. Ths suon s consere s srng pon for he presene hecl oel. Accorngly, he volue of frs generon of ugher bubble s / 3 n for -h generon of ugher bubble s / Respecve nuber of bubbles, N, n he -h generon occupy he volue N 5 n ol gs volue s, 6 Then he volue frcon of -h generon bubbles s 7 Men Suer equvlen eer of ll bubbles cn be efne s 6 N S, 8 / 3 N A where A s surfce re of -h generon bubble. Averge bubble volue s efne s π 3 S S / 3 When he necs of bubble breup s nown, hese vlues cn be prece by he oel suggese n he presene pper. 4

3 Experenl suy of he bubble breup We hve sue experenlly he breup of sngle rsng bubble for se of eu sze bubbles. The eho n resuls were presene frs for wer [3], n recenly for vscous lqus [4]. When srng wh N bubbles, he nuber N of survvng bubbles ecys exponenlly N exp r N le he frs orer checl recon. The breup re cn be chrcerze by hlf-lfe, ln /, r of rsng bubbles. Apprenly, y epen les on ensy, grvy cceleron, vscosy n surfce enson. Densonl nlyss precs he relon of 3 ensonless groups, e.g.: Θ / feo, M, where he ensonless hlf-lfe ws nrouce by / 4 3/ 4 / ρ g Θ/, 3 / 4 σ Eövös n Moron nuber s efne n usul wy s: ρ g Eo, 4 σ 4 µ g M. 5 3 ρ σ Sple power funcon ws obne by he regresson of experenl Θ /.66 Eo -6.5 M -.4 R,93, 6 n pprenly he conrbuon of Moron nuber vscosy effec cn be neglece n he sue rnge M Then he coplee se of hs been nerpree by he sple relon 6 Eo Θ / 59 R,88. 7 s even n Fg.. Θ / Wer Glycerol 56% Glycerol 76% 5 Eo Fg. Densonless hlf-lfe s funcon of he bubble sze Experenl resuls n correlon 7 Ths resul cn be eploye o express he nec consn n s funcon of r, 8 where 4, 9

4 ln σ ρ π g g. 9b Ths consn becoes e.g. for wer he vlue w s - 55 c - s -. The csce of bubble breup Assue h he breup of bubbles n swr follows he relon 7 eerne for sngle bubble. I s plusble les for he syses wh low gs holup, where he bubble nercons re no essenl. By hs oel he volue frcon of oher bubbles ecreses by : r When he volue of gs n he syse ncreses, we hve. Frs generon of ugher bubbles cn spl s well n s blnce s ; b Generlly, for -h generon,,3 s. c We shll conser sple srng pon n for,,3. To splfy forlly followng foruls, we nrouce ensonless vrble s s τ, n -. 3 The hecl oel of he process s represene by se of ornry fferenl equons τ 4 n τ, for,,3 4b wh he conon for τ. 4c Ths syse hs soluon n he for of fne seres exp, j j j b τ 5 where + b b,, b, 6

5 n generlly j 3 b, j 6b Coplee soluon s le he concenron srbuon n consecuve recons of frs orer. For bubbles wh s ploe n Fg log τ Fg. Breup csce for equl bubble sze sr s funcon of ensonless e τ n ecc logrhe scle Respecve vlue of he Suer en eer by 8 s shown n Fg.3. Apprenly, here re wo sypoes, S / S for he sr-up pero τ <<., where breup cn be neglece n τ,,, τ Fg.3 Suer en eer of breng bubbles s funcon of e Asypoes for low n hgh τ S.64 for τ >>.. 8 / τ τ Ls relon cn be rewren n ensonl vrbles s.64 S Ψ 8b / where he correcon fcor s s Ψ 9

6 becoes Ψ when he ol gs volue oes no chnge. Evenly, he en bubble sze oes no epen on nl conons oher bubble sze for lrger τ. Breup csce for bubble rsng n hyrosc ferrosc colun. Rsng bubbles n lrger colun expn ue o ecresng hyrosc pressure. In prculr, gs relese n eph H of lqu colun of ensy ρ sgnfcnly hgher hn he gs ensy, hen he gs volue epens on he bubble poson n on he exernl pressure he lqu level. I s nown [-8,] h rsng velocy, v, of eu sze bubbles n low vscosy lqus s que nepenen on Reynols n Eövös nuber n n wer s usully.-.3 /s. Conserng v o be consn, he chnge of volue of el gs wh e s gven by p + ρ g H 3 p + ρ g H ρ g v Le us rnsfor hs relon usng ensonless preer p B, 3 ρ g H whch chrcerzes he relon of exernl pressure o he hyrosc pressure, n ensonless coorne v X 3 H whch s ro of he bubble ph o he colun hegh. Then he correcon fcor Ψ n he relon 9 s expresse by + B X X Ψ 33 X B + In he sypoc cse n elsewhere when p >>ρ g H or B >> s he volue chnge neglgble n Ψ. Ψ B X Fg.4 Correcon fcor Ψ for 4 s n funcon of X for fferen B Oherwse,.64 S / [ ] Ψ X, B, 34 n he funcon Ψ X, B, rses wh X s ploe n Fg.4. Iporn preer B s very low for coon lborory experens n he ssupon of Ψ s ccepble n os cses here. Oher suon s n lrge sprge vessels le pchuc ns

7 for lechng of ores, where B.5. Jus exree cses occurs n ellurgy urng vcuu egssng of lqu seel, where p y be s low s P n H s hgh s 5 whch for ρ7 g/3 gves B.5. Due o he hgh vlue of Ψ, prculrly when he bubble pproches lqu level, s he expnson fs n bubble breup s slower hn he bubble expnson. In hs cse he forul 33 cn be splfe usng Ψ foro X n B<<. 35 B Sze of bubbles pprochng he level epens on he exernl pressure p. Afer csce of bubble breup ccorng 8b, en bubble volue s sply rele o he locl pressure, p, S ρ g v.64 p for X n B<<. 36 One of ovons of our reserch ws o unersn he bubble behvor n lqu seel. Gs concng wh lqu els s n porn process of oern ellurgy echnology [5-8]. lues of en bubble eers, ese by he presene oel for wer n for lqu seel re presene n Tb.. Exernl pressure P wh seel hegh 3-5, s coon n rgon vcuu seelng lles urng vcuu reovng of ssolve gses CO, H, N wh rgon servng s n ner n srrng eu. Whle for wer exernl pressure p P he en bubble sze bubbles he se level s S 9 by our oel, eer of bubbles levng wer o low pressure P spce s S 6. The eers os gs bubbles n lqu seel re pproxely wo es lrger hn n wer. Tb.. Sze of bubbles pprochng he level fro ep colun s funcon of exernl pressure Exernl pressure, p [P] Suer eer, wer S [] lqu seel Concluson Sze of bubbles rsng s swr n lrge colun where breup occurs cn be ese usng he evelope oel bse on experenl for sngle bubble rse n breup probbly uner consn pressure conons. Densonl nlyss gves resuls pplcble boh for snr lqus n for lqu els. Acnowlegeen We grefully cnowlege fnncl suppor by he grn No. 4/4/87 fro he Grn Agency of he Czech Republc.. Noon eprcl exponen of breup 8 bubble eer Eo Eövös nuber 4 ensonl preer 8 volue frcon of gven bubbles M Moron nuber 5 gs volue r necs consn of breup hlf-lfe of breup / sngle bubble volue preer 3 Θ / ensonless hlf-lfe of breup 3 µ ync vscosy of lqu ρ lqu ensy σ surfce enson τ ensonless e Ψ fcor of breup reron

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