Mouhamadou Diop, Frédérick Gagnon, Li Min, and Mario Fafard. Correspondence should be addressed to Mouhamadou Diop;

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1 ISRN Materals Scence Volume 24, Artcle ID 45469, pages Research Artcle Gas Bubbles Expanson and Physcal Dependences n Alumnum Electrolyss Cell: From Mcro- to Macroscales Usng Lattce Boltzmann Method Mouhamadou Dop, Frédérck Gagnon, L Mn, and Maro Fafard NSERC/AlcoaIndustralResearchCharMACE 3 and Alumnum Research Centre (REGAL), Laval Unversty, Quebec, QC, Canada GV A6 Correspondence should be addressed to Mouhamadou Dop; mouhamadou-azz.dop.@ulaval.ca Receved 29 October 23; Accepted 5 January 24; Publshed 23 February 24 Academc Edtors: C. Carbonaro and A. O. Neto Copyrght 24 Mouhamadou Dop et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Ths paper llustrates the results obtaned from two-dmensonal numercal smulatons of multple gas bubbles growng under buoyancy and electromagnetc forces n a quescent ncompressble flud. A lattce Boltzmann method for two-phase mmscble fluds wth large densty dfference s proposed. The dffculty n the treatment of large densty dfference s resolved by usng nnevelocty partcles. The method can be appled to smulate flud wth the densty rato up to. To show the effcency of the method, we apply the method to the smulaton of bubbles formaton, growth, coalescence, and flows. The effects of the densty rato and the ntal bubbles confguraton on the flow feld nduced by growng bubbles and on the evoluton of bubbles shape durng ther coalescence are nvestgated. The nterdependences between gas bubbles and gas rate dssolved n flud are also smulated.. Introducton A steadly ncreasng computatonal power, new developments, and mprovement of numercal methods allow numercal smulaton to model more and more physcal phenomena. In ths sense, the n stu gas generated by alumna partcles and carbon engenders the formaton of gas bubbles n alumnum cells. These gas bubbles are produced by chemcal reacton of alumna wth the carbon, whch conssts of complex process of electrochemcal reducton of alumna nthecells.thecontnuouslyreleasedgasofthealumnareacton generates nucle underneath the surface of carbon anode anddetachestodssolventhecryoltewhchsubsequently growstobubbles.thebubblesarrangentoacellularstructure n the cryolte, whch s preserved by a rapd cyclc flow of gas. Amajoramofthesmulatonspresentedherestogan a better fundamental physcal understandng of gas bubbles because of the complexty of gas bubbles- lqud flow n chemcal system [, 2], snce t determnes largely the dynamc of the cryolte. The detaled understandng may help to reduce theenergyconsumptonandtomprovethepresentenergy effcency by controllng the dynamcs of bubbles n cells. In contrast wth actual alumnum electrolyss cell models, whch focus rather on specfc aspects and for the most part on gas bubbles system, a comprehensve and general gas bubbles model s needed to be developed here whch ncludes the wealth of phenomena occurrng n alumnum electrolyss cell process. The model wll elucdate the nfluence of gas bubbles as descrbed n [3, 4] andprocessng parameterssuchasthecryoltevscosty,thesurfacetenson, the dffuson coeffcent of the propellant n the cryolte, the ambent pressure, the heatng rate, and the cell geometry. Ths nformaton s then helpful for optmzng the gas bubbles structure. However, as an mtaton of the real world and due to the lmted computatonal speed, some physcal processes must be omtted or smplfed n the physcal model. The comprehensveness and generalty of the physcal model demandforanumercalsoluton.thechoceofthenumercal method s made n favor of the lattce Boltzmann model (LBM). LBM explores n partcular complex flow problems to

2 2 ISRN Materals Scence whch the bubbles clearly belong wth ts large, dynamcally movng gas/lqud nterfaces [5, 6]. Advantages of the method are the computatonal localty and parallelsm, the relatve ease of mplementaton, the possblty for the ncorporaton of addtonal features, and the strong physcal background as a knetc scheme. These advantages allowed developng a complete smulaton tool wthn the scope of ths work [7 2]. The man part of ths work s concerned wth the development and adaptaton of the new method called LBM to specfy the alumnum electrolyss process. As a man goal the complex gasbubblesevolutonnalumnumelectrolysscellssstuded wth an effcent method that s able to overcome dffcultes nthetreatmentofcomplexnterfaces[3 9]. For capturng the nterface, novel-free surface boundary condtons are developed. In consstency wth the hydrodynamcal problem, the LBM s utlzed for descrpton of the dffusve processes of the propellant. Furthermore, addtonal model features can be ncorporated wthn the framework of the LBM to tackle a multphyscal system of mllons of degrees of freedoms. 2. Physcs of Gas Bubbles Thssectongvesadetaledntroductontothetechnologcal aspects and the physcal phenomena of the gas bubbles process by n stu gas generaton. In partcular, the focus reles upon the anodc chemcal reacton of carbon. The propertes of the alumna, partcle sze and dstrbuton, atomzaton atmosphere, purty, and heat treatment are crucal for the alumnum electrolyss process. The alumna andthecryolteassumedaremxedhomogeneously.the subsequent nucleaton method needs to assure a hgh densfcaton n order to suppress gas loss by percolaton at the begnnng of the process. Optonally, the compacted materal can be cut nto near net-shape forms to study the phenomena of channelzaton and penetraton of fluds n fracture. The carbon anode consdered as compacted materal s consttuted of one or several peces of the compacted precursor. In the smelter there are flows from cell to cell n alumnum busbars whle, n each cell, the current flow (I) s setto25kaandrundownwardsthroughthecellcomposed of 8 anodes, cryolte, molten metalpad, and a cathode carbon block. In a matter of mnutes after the begnnng of the electrolyss process, gas bubbles nucleate and expand underneath the anodes n the cryolte and the process s nterrupted by removng the gas out of the cell n order to stablse the system. The fnal product conssts of a closed cell wth a cyclc formaton of gas bubbles generated from the perodcal reacton of alumna n the cryolte from feeders and surrounded by a fully dense skn called the top crust. The densty n the multphase flud flows ranges from.4 to.8 g/cm 3,whch corresponds to a relatve densty ρ r of 5 3%. Focusng on the actual electrolyss process, fve expanson stages, dependent on the temperature, are dentfed n Fgure: thermal expanson of the precursor (), bubble nucleaton under surface of anodes (I), expanson n the semsold and lqud range (II), expanson n the lqud temperature range (III), and fnally the gas bubbles collapse (IV). Expanson factor Oxde carbon release from Expanson of I Temperature ( C) Fgure : Schematc expanson of gas dvded nto fve regmes. Phase () depcts the thermal expanson of the precursor, phase (I) denotes the bubbles nucleaton and expanson under surface of anodes, the expanson n the semsold and lqud range s n phase (II), phase (III) s the expanson n the lqud temperature range, and fnally phase (IV) s the gas bubbles collapse. The begnnng of the gas formaton, regme II n Fgure, s marked by decomposton of the alumna, meltng theprecursorandnucleaton.thedecompostonofabout 6wt%upto96 Creleasesapproxmatelyagasvolumewhch exceeds tmes the precursor volume. The decomposton begns whle the alumna matrx s stll sold (at about 95 C). Partal pressures n the order of the matrx yeld strength may arse and tear open the powder compact precursor. The crack-lke openngs n the carbon anode are problematc because the CO 2 can percolate through these channels to the ambence beng lost for the multphase system. In addton to these cracks, the small pores over the anode serve as nucle when the alumna melts. In the melt, heterogeneous nucleaton may occur at nucleaton stes such as mpurtes or propellant partcles. The major gas bubbles expanson takes place whle the cryolte s lqud, regme III-IV n Fgure. The strong ncrease n expanson reles on the substantal oxde carbon release durng the consdered temperature nterval. The oxde carbon dssolves and dffuses nto numerous nucle, whch consequently expand. At a certan pont, the gas bubbles mpnge upon each other. They must gve up ther favorable shape. Thn flud flms separate the bubbles. The varatons of gas bubble szes and thus nteror pressure as well as precedng flm ruptures lead to bent and corrugated cell walls. The stablzaton of the gas bubbles flms s essental. Wthout stable cell walls, bubbles would mmedately undergo coalescence when they contact each other. By coalescence wth the atmosphere enforced by buoyancy, the bubbles and consequently massve gas portons from alumna are lost from the system. A typcal gas bubbles structure cannot evolve. The gas bubbles may owe stable flms to the presence of addtves such as oxdes. Snce the alumna possesses oxde, that oxde naturally resdes as a network n the precursor. The repulsve force s termed dsjonng pressure n the followng. Addtonally, network fragments make thn flms lesspenetrabletothemeltsothatthecryolteremansnthe II III IV

3 ISRN Materals Scence 3 flms. These effects resemble a drastcally enlarged vscosty. Ths assumpton s supported by the observed shape stablty of a once melted alumna precursor. Durng the gas bubbles expanson, flm ruptures, growth coalescence, capllary, gravtatonal dranage, and electromagnetc, nterbubble dffuson, gas bubbles flow and gas bubblescollapseoccur.theyaresketchednthefollowng. Flm ruptures are nevtable. One observes a mnmal flm thckness n gas bubbles of the order of 5 μm. Ths leads to the assumpton that a flm rupture event occurs when ths crtcal mnmal flm thckness dc s exceeded. A rupture event causes the collapse of the entre flm. Surface tenson forces wthdraw the flm remnants towards the surroundng structure. The structure tme t r, startng from the ntal defect n the flm to ts complete dsappearance, s a matter of seconds. One cause for the flm thnnng s smply the expanson of the bubbles. If gas bubbles pore nflates, ts cell wall thckness decreases due to mass conservaton. At a certan pont, d c s reached leadng to rupture and coalescence. Thereby, the gas bubbles system coarsens, that s, the mean pore dameters D ncrease. Consderng a representatve cell of the gas bubbles, the ratoof pore volume (D) to pore shell (d c ) s approxmately proportonal to /ρ r. Gas bubbles dranage s the flow of lqud through and out of the gas system under the nfluence of external and capllary forces. In the gravtatonal dranage, the cryolte n the plateau borders s drawn downwards by gravton so that the cryolte accumulates downwards at the cryolte layer. In the capllary dranage, the dfferent pressures n cell flms and plateau borders orgnatng from the surface tenson cause the cryolte to wthdraw from the flms. Naturally, these processes wll reduce the flm thckness n the gas bubbles system whch leadstocellwallruptures.estmatngthetmesscaleofgas bubbles expanson, t s found to be very short compared to usual bubble n flud. Only a drastcally enlarged vscosty and n partcular the statc dsjonng pressure n the flms hnder the gravtatonal and capllary dranage to cause gas bubbles collapse. When the stablzaton s sgnfcant, the dranage may even become neglgble. Dstnct gas bubble szes and consequently dstnct bubbles may lead to nterbubble dffuson of oxde carbon va the cell walls. Tny bubbles suffer from gas loss, whle large bubbles earn oxde carbon, smlar to Ostwald rpenng. A study of the dffuson flow of oxde carbon across cell borders demonstrates the large tme scales, compared to the bubbly tmes, of the gas transport n cryolte. Therefore, thenterbubbledffusonsneglgblenthecryoltebubbly evoluton. Durng the evoluton, gas bubbles are permanently exposed to mechancal forces due to the expanson tself or spatal restrcton by the cell desgn. Deformaton of pores, rearrangement or destructon of cell flms, and gas bubbles flow are the consequences. Topologcal rearrangements and flm ruptures occur often n an avalanche-lke manner. In the fnal stage of bubbles expanson, large bubbles are present compared to the gas bubbles system sze. Gas bubbles collapse atthecryoltesurfaceleadsnowtosevereoxdecarbonloss,n the system so that the gas system decays; that s, the collapse starts. 3. Physcal Model Revewng gas bubbles n alumnum electrolyss cell n Fgure 2, almost no theoretcal models can be found. Consderng the bubbly evoluton n lqud, t becomes apparent that specfc aspects of gas bubbles are commonly studed n solaton. In order to nclude many physcal phenomena of the electrolyss process, a comprehensve model s constructed based on fundamental physcal equatons. In order to obtan the full nformaton of gas bubbles structure, t s requred to drectly smulate gas bubbles structure n alumnum electrolyss process. Nevertheless, the model wll naturally contan smplfcatons because t s a realstc mtaton of the real ndustral problem and because of naccessblty n cells. The gas bubbles are stochastcally dstrbuted and formed smultaneously at the startng pont. Dssolved alumna partcles reactng wth the carbon anode dffuse nto the nucle over the surface of carbon anodes. The decomposton and dssoluton of alumna are modeled by a homogeneous volume source S n the cryolte doman Ω. The nucle grow over the undersurface of anodes to bubbles. Consequently, the surroundng cryolte s dsplaced. Interactons between bubbles are accounted for at frst over the anodes by a shortrange dsjonng pressure. Cell walls ruptures occur when a mnmal crtcal flm thckness s attaned. As a smplfcaton, the model uses a statc electromagnetc force term n the expresson of Naver-Stokes equaton. Furthermore, the startng pont of the model ndcates already that the meltng of the alumna precursor and an explct descrpton of the chemcal decomposton of the propellant are neglected. Smlarly, the fnal bubbly structure s acheved by smply stoppng the smulaton at certan pont of tme. In ths model the temperature remans constant at 96 Cnspaceandtme.Consequently, the materal parameters are constants. A temperature and heatng rate enters the model only mplctly by the homogeneous volume source term S of the dssolved oxde carbon. The solute oxde carbon c nthecryolteobeysthedffuson equaton n Ω: c t +u c α α D c =S(t), () α α where S(t) sthevolumesourceterm,d s the gas dffuson, c s the concentraton feld of oxde carbon, and t denotes the tme. The lqud doman contans slands, whch are the nucle or gas bubbles. At the gas-lqud nterface Ω, Severt slaw holds as follows: c=k S p g, (2) where the pressure P s dstnct for each bubble and k s s Severt s constant. Furthermore, the gradent at the nterface s parallelto theunt normaldrectonn of the nterface: ( x)c n. (3) y

4 4 ISRN Materals Scence Feeder Stckng contacts Vod doman Gas-lqud mxture x C Anodes x D x B x A Cathode block Fgure 2: Vertcal cut short of an alumnum electrolyss cell n whch the gas expands from lquds. The ponts x A, x B,andx C are the references ponts for the evoluton of gas n the cell. The dynamcs of cryolte flow caused predomnately by gas bubbles expanson s descrbed by the famous ncompressble Naver-Stokes equaton n Ω: u α α =, u α t + u β u α β α = P/ α + ] 2 β ρ u α +g α. Sncethevscousstressesofthegasareneglgble,freesurface boundary condtons are applcable at Ω: p 2ρ] u n n =p σκ Π, (5) u t n + u n =, (6) t where the subscrpts n and t denotethenormalandtangental component wth regard to Ω.ThegaspressuresofbubblesP, the surface tenson pressure σκ, and the dsjonng pressure Π enter the model va these boundary condtons. In the bubbles, the deal gas law s consdered: p = n RT, (7) V (4) where P s the gas pressure n bubble gven by the deal gas equaton, R denotes the deal gas constant and n denotes the amount of gas, T denotes the temperature, and V denotes the volume of bubble.thechangeofbubblevolumepertmes gven by the normal velocty u n of the bubble-lqud nterface: dv dt = u n dw at Ω. (8) Ω The gas exchange between cryolte and bubbles s gven by Frck s law so that the followng change of the amount of substance per tme apples: dn dt = D c dw. (9) Ω n A specal type of bubble s the atmosphere. It has a constant pressure P o at all tmes. Its oxde carbon partal pressure, enterng Severts law, can be chosen dfferently from p o. A heurstc approach condenses the flm stablzaton n a sngle functon, the dsjonng pressure Π. Π s a repulson experenced by bubble nthepresenceofanotherbubblej: Π={ k Π (d max d) d<d max, () d d max. The dstance d from bubble to bubble j s obtaned by movng along the normal drecton. When the neghborng

5 ISRN Materals Scence 5 bubble s too dstant; that s, d exceeds d max, no nteractons of the bubbles occur. At present, the functonal form of Π(d) s chosen arbtrarly. When the flm thckness d falls below a crtcal threshold dc, thecellwallssubjectedtorupture.thesecoalescence crtera are extracted n agreement wth numercal expermental data. To conclude, desred features, such as rheology or dranage, evolve automatcally n the model because of ts generalty. Furthermore, processng parameters, such as vscosty or ambent pressure, are ncorporated naturally and hence accessble to nvestgatons. 3.. Numercal Model. The lqud doman Ω n whch gas bubbles develop contans two domans: the gas Ω gas and the lquds Ω lq.thsdomanω canbeopenorclosedasshown n Fgure 2. One notes Γ Ωtheentrewallsofthedoman. We wll seek to establsh a system of equatons verfed by the velocty feld V and the pressure p n all the doman Ω, that s, n Ω gas as n Ω lq.theveloctyfeldandthepressureare defnednthelqudpartnthesamemannertheyarealso n the gaseous part, but ths tme takng nto account that p g =.Inthemxturegaslqud,oneapplestheconservaton laws. In addton, one apples a condton of stckng contact on walls Γ of the entre doman Ω. The last pont s dsputable and consttutes a frst approxmaton. Ths leads to the followng formulaton. Fnd the velocty V c 2 (Ω gas ), V Γ =, and the pressure p c (Ω gas ),suchas t ], θ[ Ω gas (t): [2η gas ((ε ( V)),W) ε ( V)] p=, V = dρ gas (W), ρ gas dt σn= over Ω vod Ω gas, () where W s the gaseous volume fracton n the mxture wthn the cryolte before the rse of gas bubbles over the surface of the lqud. W s a feld, dependng on the pont where we are n the gaseous doman and tme. The vscosty η gas depends on W, as well as the gas-lqud system. By neglectng the volume mass of gas (ρ gas ) before the volume mass of the lqud (about ρ lq = kg m 3 ), on can wrte ρ gas = ( W) ρ lq. (2) Thecryoltesassumedtobencompressble,sothat dρ gas = dw ρ gas dt W dt. (3) The mcro-macro-couplng s carred out then through the expanson rate of the gas dw/dt that remans to be determned. In the mcroscopc case, the expanson of gas bubbles s governed by the dfference of pressures between the nternal and the ambent, and the latter consttutes a macroscopc parameter of the problem, exteror of our mcroscopc model. Thus, t follows that one uses dw dt = W t + V W=f(W, p gas p), (4) where p gas denotes the nternal pressure of the gas and W denotes the expanson rate. Ths expanson rate s explctly dependent on tme t because t mples a gas creaton by chemcal reacton from the alumna. The tme appears explctly n the expresson of dw/dt. Fnally, the system n velocty pressure to solve wrtes as follows. Fnd the velocty V c 2 (Ω gas ), V Γ =and the pressure p c (Ω gas ),suchas t ], θ[ Ω gas (t): [2η gas ((ε ( V)),W) ε ( V)] p=, V = W f(w,p gas p), σn= over Ω vod Ω gas. (5) After the germnaton of gas bubbles and ther detachment from the undersurface of the anodes, the gas bubbles stll expand n the gas-lqud mxture and are assumed to be sphercal. They rse to the surface and start to reduce ther nteracton between them. At the surface of the lqud they occupy a small percentage of the total volume, to a structure of mxture gas and lqud wth deformed gas bubbles usng more than 85% of the total volume Mcro-Macro-Couplng. The mcro-macro-couplng n the alumnum electrolyss cell s carred out by closng (5) va the expanson rate dw/dt (see (4)). One proposes to determne ths expanson rate from smulatons usng a mcroscopc model. Ths model takes nto account the expanson of a representatve volume durng a moment of the macroscopc flow. Ths model takes as nput varables the ntal confguraton of gas as well as ts ntal pressure, and t ntervenes as a varable of macroscopc order, p ext,nthe ambent pressure at a consdered pont of the macroscopc flow. The expanson rate dw/dt s expressed frst when the nteractons of gas bubbles are not taken nto account. In ths study, a volume representatve n expanson, of volume Ω, contans a gaseous part Ω gas of volume Ω gas and a lqud part Ω lq of a constant Ω lq over tme. The gas rate W c Ω s defned by W c = Ω gas / Ω. One recovers the followng relaton between Ω gas and W c : Ω gas = W c W c Ω lq. (6) It s assumed that the gas n the cryolte s composed of n sphercal bubbles of volume Ω gas,, radus R,andthe pressure P g.whereasthecryoltesregardedasnewtonan and of consstency η o. The pressure s assumed to be constant n the lqud and equal to the external pressure that represents the constrant appled on the wall Ω by the external medum. Thus, gas bubbles grow by dfference of pressure, and ther nteractons are neglected here. Consequently, one fnds that n Ω gas = Ω g = 4π 3 = n = R 3. (7)

6 6 ISRN Materals Scence So that an algebrac transformaton wll gve Ω gas d Ω gas dt = n 3 n = R3 = RR 2. (8) Wthout the nteractons between bubbles, one apples over each bubble the expresson of the velocty (u(r) = RR 2 /r 2 ), pressure (P(r) = P ext ), and R/R = Δp/4η o.fnally,tgves Ω gas d Ω gas dt 3 = 4η o n = R3 n = (p gas p ext )R 3. (9) The above equaton now s smplfed because the bubbles aretakenwththesamepressurentheentrevolumeofthe doman Ω.One obtans Ω gas d Ω gas dt = 3 4η (p gas p ext ). (2) The dfferental of (6) and ts combnaton wth (2), n functon of pressure s dfference wrtes dw c dt = 3 4η (p gas p ext )W c ( W c ). (2) Fnally, one expresses the pressure of gas P gas n functon of W c. P g s supposed to be homogeneous n the gaseous doman, by usng the deal gas law for each bubble and by summng over all bubbles, are obtaned that p gas Ω gas = p gas, Ω gas,.usng(6), one can wrte that p gas = W c W c W c W c P gas,. (22) Thus, by neglectng the nteractons between bubbles, and by assumng a homogenety of the pressure, the gas rate expanson n the representatve volume gves the expresson dw c /dt n functon of W c and P ext.now,followng,amacroscopc approach, the expanson rate of the gas W s a feld. One obtans then W t + V W= 3 [( W) 2 W P 4η W gas, PW ( W) ] n Ω gas. (23) To solve numercally (23), ts doman of valdty must be extended n all the doman Ω, namannertoretrevets second term null n the lqud doman Ω lq. 4. Numercal Implementaton Revewng the physcal model from a numercal pont of vew, the most challengng aspect s the parallel mplementaton of the large, complex, and dynamcally movng nterfaces between the cryolte and the gas bubble. Among the large varety of numercal methods, the lattce Boltzmann method (LBM) [2 23] s a promsng canddate for numercal nvestgatons n alumnum electrolyss cell. Sutable free surface boundary condtons need to be newly developed for the lattce Boltzmann method. In addton, the other features of the physcal model, for example, the dffuson equaton and the dsjonng pressure, need to be ncluded nto the LBM. At frst, the focus remans on the flud dynamcs. Snce the begnnng of the last century, t s known that the knetc Boltzmann equaton can approxmate the Naver-Stokes equaton. By numercally solvng the knetc Boltzmann equaton, the lattce Boltzmann model supples an approxmate soluton for the Naver-Stokes. The LBM s consequently a knetc scheme, based on partcle dynamcs. In detal, the space s dvded nto a 2D regular lattce. On each lattce ste, sngle-partcle dstrbuton functons f ( x, t) represent thenumberofpartclesatlattcestex and tme t n state. The dstnct states are defned by a set of nne dscrete veloctes: ξ = Δx Δt (, ) { =, (±, ), (, ±) { =,...,4, {(±, ±) =5,..., 8. (24) In each tme step Δt, the dstrbuton functons are advected along ther correspondng velocty and subsequently they undergo a collson operaton locally at each tme ste. In the smplest case, the collson step conssts of a relaxaton of the functon f ( x,t) towards an equlbrum functon f () ( x, t). Both steps are graphcally dsplayed n Fgure 8 and combned n the lattce Boltzmann equaton (LBE) [24]: f ( x+ξ,t+δt) f ( x, t) = Δt τ (f ( x, t) f () ( x, t))+δtf. (25) The macroscopc varables ρ and u are obtaned by the zeroth and frst moments: ρ ( x, t) = f ( x, t), (26) ρu ( x, t) = ξ f ( x, t). The collson operator s constructed to fulfll mass, momentum,andenergyconservatonateachlattcesteandtoensure the correct symmetres of the Naver-Stokes: f () =w ρ(+ ξ αu α c 2 s 4 9 { w = 9 { { 36 + (ξ αu α ) 2 2c 4 s =, =,...,4, =5,...,8, and c 2 s =/3sthe sound speed squared. u αu α 2cs 2 ), (27) (28)

7 ISRN Materals Scence 7 The force term ncludes the gravty g and electromagnetc force [23, 25, 26]: F =w ρ[ ξ α u α c 2 s + (ξ βu β )ξ α c 4 s ](g α + j B). (29) By Chapman-Enskog analyss, the LBM s proved to approxmate the Naver-Stokes n the ncompressblty, u c 2 s,and for small Reynolds numbers. The Chapman-Enskog analyss starts wth a separaton of the tme scales and expanson of the dstrbuton functons: f =f () + f () + 2 f (2) +O( 3 ), F = F () +O( 2 ), t = () t +ε 2 (2) t +O( 2 ), α = () α +O( 2 ), (3) where s a small number. The followng constrants are mposed on the expanded dstrbuton functons: [ ]f () ξ =ρ[ u ], [ ]f (n) ξ =ρ[ ] for n>. (3) Further, the left hand sde of the LBE s expanded by a second order Taylor seres and the resultng terms n (25)aresubsttuted by the above seres expansons. An equaton s derved contanng varous orders of epslon. Evaluatng the frst two moments of ths equaton results n the hydrodynamc equatons. The frst order of epslon provdes the Euler equatons. Includng the second order of epslon, the NSE s derved: ρ t + ρv α α =, ρv α t + ρv βv α β = P + Vρ α β β V α +ρg α +j α B β, (32) where V =(c 2 s Δt/2)(2τ/Δt ). The presented LBM s appled n the flud regon [27 29]. In the gas bubbles system, gas regons and the outer cell desgns are present as well. In the regular lattce, one may defne the dstnct regons by dstnct cells: Flud C F,bubble C B,nterfaceC F, and wall cells C W. The LBM s executed onthefludcells.inthenterfacecells,thefreesurface boundary condtons n (5) and need to be mplemented n (6). After the advecton step and before the collson takes place, the nterface cells obtan known dstrbuton functons from lqud regons and unknown ones from the gas regons. The mssng populatons are reconstructed by a frst order Chapman-Enskog approxmaton; see (3), where n f () all terms of O( u 2, 2 ) are neglected: f () =w τ( α (ρu α ) cs 2 ξ α ξ β α (ρu β )). (33) n Bubble t δ Interface Flud Fgure 3: Interface cell after advecton step wth unknowns functon (dotted lne) and known functon (full lne). In f (),(27), f () n (33). The macroscopc quanttes ρ, u, and α u β areunknown.thefreesurfaceboundarycondtons, (5)and(6), are utlsed to elmnate the unknowns n u n, t u n, and n u t. Therefore, the scalar products are expressed n the local coordnate system at the nterfacal consstng of the normal n and the vector tangental t as depcted n Fgure 3. Furthermore, the quadratc velocty terms n f () are neglected n order to obtan lnear system of equatons. Then, f =w (ρ + ξ αρu α c 2 s + τ (( ξ2 n cs 2 ) c2 s ρ PI j ) 2V +( ξ2 t cs 2 )ρ t u t ). (34) Here, ξ n =ξ α n α, ξ t =ξ α t α. P I j s the pressure at the nterface of the consdered gas bubble j ncludng surface tenson and dsjonng pressure. After reconstructon the usual collson operaton takes place. Yet, the nterface advecton remans to be dscussed. For ths reason, nterface cells contan addtonal nformaton whch ndcates the amount of flud nsde each cell. Ths lqud volume fracton Λ s updated accordng to the n- and outflow of the nterface cell. For more accuracy, the dfference between the ncomng and the outgong dstrbuton functons s calculated on each lnk and weghted by factor Φ whch depends on the poston of the nterface: Wth ΔΛ ( x) = Φ(f ( x+ e ) f ( x)). (35) x+ e C F, { Φ ( x) +Φ( x+ e Φ= ) x+ e { 2 C I, { x+ e C B, (36)

8 8 ISRN Materals Scence Φ s the area fracton obtaned through Φ=Λ/ρ, e =ξ Δt, and s related to va e = e.thetotalfluxnthenterface cell x s obtaned by summng ΔΛ over all. Empty or full nterface cells are detected when Λ exceeds ts lmts, < Λ < ρ. They are changed to gas or flud cells, respectvely. When cells are transformed, the closeness of the nterface needs to be guaranteed; that s, unknown dstrbuton functons must be advected nto nterface cells. Ths demands sometmes transformatons of cells n the close neghborhood of the ntally transformed nterface cell. Unknown dstrbuton functons n these cells are obtaned by lnear extrapolatons. The smplest wall boundary condtons are utlzed n the wall cells, whch represent the cell desgn. Bubble cells surrounded by nterface cells buld connected and closed areas of cells whch represent the bubbles. The volume and amount of gas n the bubbles need to be determned n each tme step. Intally, the bubble volume s known. Then, t s suffcent to calculate the change of volume per tme step. It s gven by the change of the volume fractons Λ n the nterface cells C Ij of the bubble j: ΔV j = ΔΛ ( x). x C Ij (37) Smlarly, only the oxde carbon dffuson across the nterface s recorded n order to obtan the amount of substance n the bubbles. In the lattce Boltzmann formalsm, the flux s gven by the exchange of dstrbuton functons across the nterface whch s approxmated by the cell borders between flud and nterface cells. 5. Numercal Valdaton These numercal mplementatons need to be valdated. For ths, comparsons between numercal and analytcal results gve nformaton on the valdty range of the parameters and the accuracy of ths scheme. Ths test wll merely focus on newly developed mplementaton ssues and on sngle relaxaton tme, whereas the nterface advecton s tested by streamng a gas bubble n unform flow feld. Dstortons of the round bubble shape and the small devatons from analytcally predetermned way of the bubble center appear after the advecton. Overall, the advecton appears to be very promsng results. The same holds for the curvature algorthm. The results of template sphere methods are better than those of standard fnte dfference methods (FDM) often appled to volume of flud methods (VOF). TheLaplacelawhasbeenusedforseveralparametersby consderng bubble n lqud flow. In Fgure 4,thenumercal result depcts a small devaton from the theoretcal expectaton marked by the dotted lnes. Durng these nvestgatons, small spurous currents occur. For the tested varables comprsed between. < σ <.5,.3 < V <.43,andρ= for the flud, are used all over the computatons. They reman small overall the experment. Dynamcally, thefreesurfaceboundarycondtonsarenvestgatedby standng capllary and external forces waves such as gravty and the electromagnetc force. These waves are damped by the vscosty of the lqud. Frequency and dampng rate obtaned numercally are n good agreement wth theoretcal expectatons. Good agreement s also depcted by a numercal nvestgaton on the free surface boundary condton as shownn(5) at the nterface of a rsng gas bubble. In addton, the gas bubble shapes and ther rsng velocty are observed. The later seems to decrease very fast below expectatons wth a shrnkng effect, when the surface tenson s much larger than the effects of the external forces mpact small or neglgble. In concluson, one can summarse that the numercal experment exhbts good agreements wth theoretcal expectatons n regard to parameter ranges of our numercal tests. 6. Results and Dscusson Havng provded and tested the physcal model and ts numercal mplementaton, the smulatons of gas bubbles expanson n alumnum cell process are fnally performed. The tests carred out show that the consstency of the lqud ntervenes as multplcatve factor n the expresson of the vscosty n the system bubbles lqud η gas.inthsstudy, one consders a system of n bubbles from 84 to 5, that s, havng dfferent denstes of gas bubbles. The choce of the structure of bubbles does not nfluence the vscosty of bubbles lqud and helps to keep the ntegralty of bubbles n the computatonal doman durng the bubbles expanson. Thus, the effects of edges are restrcted. In Fgure 5, one plottedthevaluesoftheratoλ = η gas /η n functon of gas rate, computed durng the expanson of bubbles. For a number of bubbles over 84, we have unquely represented the obtaned values before the effect of edges ntervenes (rate <7%), whose descrpton of bubbles-lqud structure s good. Comparng the curve of Fgure 5 wth the rght -Wplottedonthesamefgure,oneobservesanonlnear dependence of the vscosty to the gaseous fracton W. One dstngushes three dfferent areas n the curves of Fgure 5. On one hand, for a gas rate nferor to 2%, the more the densty of bubbles s mportant, the smaller the bubbles are. They offer less resstance to shearng. Ths explans the fact that the equvalent gas rate s hgh. As long as the nteracton between bubbles s neglgble, the gas fracton remans small. Fnally, when the gas fracton tends towards, the vscosty also tends asymptotcally towards zero. Thus, we have put n evdence, on one hand, the nonlnear dependence of the vscosty of mxture and, on the other hand, the dependence of the vscosty on the bubbles densty. One remarks that the densty of bubbles s consequent when the vscosty tends to an asymptotc curve. 6.. Dependence of Vscosty on Bubbles-Lqud Mxture. There are several numercal smulatons of bubbles-lqud structure by varyng the bubbles densty. The ntal gas fracton s the same for each smulaton, W c,aswell,and the number of bubbles consdered s mportant, more the szes of whose, gven at same gas rate are small. One used n ths study an ordnated structure (bubbles dspersed regularly on the grd) and bubbles havng dfferent szes and same ntal pressure n a manner to conserve the ntegralty of the structure n the computatonal doman, on another hand to

9 ISRN Materals Scence 9 (P gas P lq ).E.E 2.E 3.E 4.E 5.. Curvature K δ =.5 δ =. δ =.5 δ =. Gas rate Tme (s) 84 bubbles 52 bubbles 789 bubbles bubbles 5 bubbles bubbles Fgure 4: Numercal tests of Laplace law. The dashed lnes represent the theoretcal equaton. Fgure 6: Evoluton of gas rate over smulatons of volume expanson of bubbles lqud wth 84 to 5 bubbles. Vscosty of mxture/consstency of lqud Gas rate Parameter Number of bubbles n functon of parameter 84 bubbles 52 bubbles 789 bubbles bubbles 5 bubbles bubbles Fgure 5: Functon λ=η gas /η obtaned by numercal smulaton n functon of the number of gas bubbles. satsfy the hypothess of the model. Because the nteracton among bubbles of same pressure s absent, the gas rate W c model predcts an accurate profle for 84, 52, 789, and 5 bubbles, comparng wth the analytcal expresson of gas rate. The results show that the gas rate and the expanson rate are strongly nfluenced by the densty of bubbles. The former wll decrease wth the ncrease of latter. Ths reflects the nteractons between bubbles and the constrants exerted by the ones to the others when the expanson of 84 bubbles s consdered. As long as the gas fracton does not exceed 45%, ts evoluton remans smlar to that obtaned wthout nteracton. The dvergence shown between the two fractons beyond 5% explans the fact that the gas goes out of the computatonal volume causng a decrease of the amount of lqud present n ths volume. Consequently, beyond a certan lmt, the expanson rate of gas fracton s overestmated comparedtotherealty.thsapproachsapplcableforeach smulaton. Fgure 7: Evoluton of gas rate durng expanson of bubbles-lqud structure. The ponts correspond to the obtaned values by computaton of the value of parameter. The curves are the approxmatons estmated from (23). Fgure 6 showsthatthemorethenumberofconsdered bubbles s mportant, the more these effects of edges are neglgble, compared to whole bubble remanng nsde the system. We choose to express the nteracton of bubbles n the expresson of gas rate by ntroducng smply n (23), a parameter Υ determned from the curves n Fgure 6.Itsvalue s n functon of bubbles densty (Fgure 7) and then depends also on the radus of these bubbles. The curves correspondng to (23) related to the parameter Υ are plotted n Fgure 7. Fnally, a mcro-macro-couplng s used to express the expanson rate of the mxture bubbles lqud n whch the lqud s Newtonan and of consstency η. 6.2.FreeExpansonofGasBubblesnCell. The computatonal doman Ω s regarded as an open system; that s, the gaseous matter can freely go out of the boundary part Ω located at a certan heght on whch one apples a constrant null such as (σn=) as ndcated n Fgure 8 depctng the alumnum electrolysscell.wehavechosen,onfgure 8, theevoluton

10 ISRN Materals Scence Gas fracton Tme (s) Vscosty of mxture/consstence of lqud Tme (s) Ste A Ste B Ste C Ste D Ste A Ste B Ste C Ste D (a) (b) 35. Velocty (mm/s) Pressure (Pa) Tme (s) Tme (s) Ste A Ste B Ste C Ste D Ste A Ste B Ste C Ste D (c) (d) Fgure 8: Evoluton of gas fracton, vscosty, velocty, and of the pressure at dfferent ponts n the cell durng the gas expanson. of the gas fracton, vscosty, and of the pressure at dfferent pontsa,b,c,andd.thepressureslowerthantheprevous one n Fgure 8(d),partcularlyatD,duetotheoutgongface on whch σn=; the velocty at the end of the expanson s tself hgher than the prevous one n Fgure 8(c). The mcro-macro-couplng used n ths study s based on the fact that the smulaton of the expanson of the representatve volume needs a macroscopc varable, that s, the ambent pressure. The expanson happens through the fluctuatons of pressure Δp n the mxture bubbles-lqud. Thus, heterogenetes of gas fracton and then the sze of bubble (or the densty of bubbles) appear n the mxture. The expanson of the mxture depends on ts structure, va the vscosty and the volume mass. 7. Concluson In ths study, a new scheme has been developed to smulate the expanson of gas bubbles n alumnum electrolyss cell n order to descrbe the evoluton of the vscosty and the volume mass of a representatve volume of bubbles-lqud mxture durng the expanson of the gas. It s found that the dependence of these two parameters s on the bubbles densty n the mxture. The nfluence of gas bubbles nteractons on these parameters has been revealed. From these results, we have carred out a mcro-macro-couplng to express the evoluton of the mxture volume mass durng expanson n functon of pressures fluctuaton n the mxture. Ths approach helps to take nto account the electrolyss process and also the nfluence of the mxture structure durng the smulatons of expanson. These smulatons provde the varatons of gas fracton n the cell. The smulatons also show that, close to the cooled edges, the gas expanson s slower and even stopped. Thus, a fne surface skn s created on the cryolte. Conflct of Interests The authors declare that there s no conflct of nterests regardng the publcaton of ths paper.

11 ISRN Materals Scence Acknowledgments The authors would lke to acknowledge the fnancal support of the Natural Scences and Engneerng Research Councl of CanadaandAlcoa.Apartoftheresearchpresentednths paper was fnanced by the Fonds de Recherche du Quebec- Nature et Technologes (FRQ-NT) by the ntermedary of the Alumnum Research Center-REGAL. The authors wsh also to thank Dr. Donald Zegler from the Alcoa Canada Prmary Metals for the scentfc dscussons. References [] R. Keunngs, Advances n the computer modelng of the flow of polymerc lquds, Computatonal Flud Dynamcs Journal, vol. 9, pp , 2. [2] L. Lefebvre and R. Keunngs, Numercal smulaton of chemcally-reactng polymer flows, n Proceedngs of the st European Computatonal Flud Dynamcs Conference, vol. 2, pp , Brussels, Belgum, September 992. [3] A. Arefmanesh and S. G. Advan, Dffuson-nduced growth of a gas bubble n a vscoelastc flud, Rheologca Acta,vol.3,no. 3, pp , 99. [4] D. F. Baldwn, C. B. Park, and N. P. Suh, Mcrocellular sheet extruson system process desgn models for shapng and cell growth control, Polymer Engneerng & Scence, vol.38,no.4, pp , 998. [5] R. Benz, S. Succ, and M. Vergassola, The lattce Boltzmann equaton: theory and applcatons, Physcs Report,vol. 222,no. 3, pp , 992. [6] P.L.Bhatnagar,E.P.Gross,andM.Krook, Amodelforcollson processes n gases. I. Small ampltude processes n charged and neutral one-component systems, Physcal Revew, vol. 94, no. 3, pp , 954. [7] J.H.HanandC.D.Han, Bubblenucleatonnpolymerclquds. I. Bubble nucleaton n concentrated polymer solutons, Polymer Scence B,vol.28,no.5,pp.7 74,99. [8] J.H.HanandC.D.Han, Bubblenucleatonnpolymerclquds. II. Theoretcal consderatons, Polymer Scence B,vol.28,no.5,pp ,99. [9]S.Hutzler,D.Weare,andF.Bolton, TheeffectsofPlateau borders n the two-dmensonal soap froth. III. Further results, Phlosophcal Magazne B,vol.7,no.3,pp ,995. [] C. Isenberg, The Scence of Flms and Soap Bubbles,Dover,New York, NY, USA, 992. [] D. Weare and R. Phelan, A counter-example to Kelvn s conjecture on mnmal surfaces, Phlosophcal Magazne Letters, vol. 69,no.2,pp.7,994. [2] J.U.Brackbll,D.B.Kothe,andC.Zemach, Acontnuummethod for modelng surface tenson, Computatonal Physcs,vol.,no.2,pp ,992. [3] N. S. Ramesh, D. H. Rasmussen, and G. A. Campbell, Numercal and expermental studes of bubble growth durng the mcrocellular foamng process, Polymer Engneerng & Scence, vol. 3, no. 23, pp , 99. [4] L. E. Scrven, On the dynamcs of phase growth, Chemcal Engneerng Scence,vol.,no.-2,pp. 3,959. [5] D. L. Weare and S. Hutzler, The Physcs of Foams, Oxford Unversty Press, New York, NY, USA, 999. [6] R.Clft,J.R.Grace,andM.Weber,Bubbles, Drops and Partcles, Academc Press, New York, NY, USA, 22. [7] F. Bolton and D. Weare, The effects of Plateau borders n the two-dmensonal soap froth I. Decoraton lemma and dffuson theorem, Phlosophcal Magazne B,vol.63,no.4,pp , 99. [8] J. M. Buck, Lattce Boltzmann methods n nterfacal wave modellng [Ph.D. thess], Unversty of Ednburgh, Ednburgh, Scotland, 997. [9] J. W. Bullard, E. J. Garbocz, W. C. Carter, and E. R. Fuller Jr., Numercal methods for computng nterfacal mean curvature, Computatonal Materals Scence,vol.4,no. 2, pp.3 6, 995. [2] N. Cao, S. Chen, S. Jn, and D. Martnez, Physcal symmetry and lattce symmetry n the lattce Boltzmann method, Physcal Revew E,vol.55,no.,pp.R2 R28,997. [2] X. He and L.-S. Luo, A pror dervaton of the lattce Boltzmann equaton, Physcal Revew E, vol. 55, no. 6, pp. R6333 R6336, 997. [22] X. He and L.-S. Luo, Theory of the lattce Boltzmann method: from the Boltzmann equaton to the lattce Boltzmann equaton, Physcal Revew E,vol.56,no.6,pp ,997. [23] Z. Guo, C. Zheng, and B. Sh, Dscrete lattce effects on the forcng term n the lattce Boltzmann method, Physcal Revew E,vol.65,no.4,ArtcleID4638,6pages,22. [24] J. M. Buck and C. A. Created, Gravty n a lattce Boltzmann model, Physcal Revew E,vol.6,no.5,pp , 2. [25] S. Chen, D. Martínez, and R. Me, On boundary condtons n lattce Boltzmann methods, Physcs of Fluds, vol. 8, no. 9, pp , 996. [26] M. R. Swft, E. Orlandn, W. R. Osborn, and J. M. Yeomans, Lattce Boltzmann smulatons of lqud-gas and bnary flud systems, Physcal Revew E, vol. 54, no. 5, pp , 996. [27] A.K.Gunstensen,D.H.Rothman,S.Zalesk,andG.Zanett, Lattce Boltzmann model of mmscble fluds, Physcal Revew A,vol.43,no.8,pp ,99. [28] D. Grunau, S. Chen, and K. Eggert, A lattce Boltzmann model for multphase flud flows, Physcs of Fluds A,vol.5,no.,pp , 993. [29] M. Do-Quang, E. Aurell, and M. Vergassola, An nventory of lattce Boltzmann models of multphase flows, Tech. Rep., Parallel and Scentfc Computng Insttute, Department of Scentfc Computng, Uppsala Unversty, Uppsala, Sweden, 2.

12 Nanotechnology Volume 24 Internatonal Internatonal Corroson Polymer Scence Volume 24 Volume 24 Smart Materals Research Compostes Volume 24 Volume 24 Metallurgy BoMed Research Internatonal Volume 24 Nanomaterals Volume 24 Submt your manuscrpts at Materals Volume 24 Nanopartcles Volume 24 Nanomaterals Advances n Materals Scence and Engneerng Volume 24 Volume 24 Nanoscence Scentfca Volume 24 Coatngs Volume 24 Crystallography Volume 24 Volume 24 The Scentfc World Journal Volume 24 Volume 24 Textles Ceramcs Internatonal Bomaterals Volume 24 Volume 24

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