Research Article Gas Bubbles Expansion and Physical

Research ArticleGas Bubbles Expansion and Physical Dependences inAluminum Electrolysis Cell From Micro- to MacroscalesUsing Lattice Boltzmann Method

Mouhamadou Diop Freacutedeacuterick Gagnon Li Min and Mario Fafard

NSERCAlcoa Industrial Research Chair MACE3 and Aluminium Research Centre (REGAL)Laval University Quebec QC Canada G1V 0A6

Correspondence should be addressed to Mouhamadou Diop mouhamadou-azizdiop1ulavalca

Received 29 October 2013 Accepted 5 January 2014 Published 23 February 2014

Academic Editors C Carbonaro and A O Neto

Copyright copy 2014 Mouhamadou Diop et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper illustrates the results obtained from two-dimensional numerical simulations of multiple gas bubbles growing underbuoyancy and electromagnetic forces in a quiescent incompressible fluid A lattice Boltzmann method for two-phase immisciblefluids with large density difference is proposed The difficulty in the treatment of large density difference is resolved by using nine-velocity particlesThemethod can be applied to simulate fluidwith the density ratio up to 1000 To show the efficiency of themethodwe apply the method to the simulation of bubbles formation growth coalescence and flows The effects of the density ratio andthe initial bubbles configuration on the flow field induced by growing bubbles and on the evolution of bubbles shape during theircoalescence are investigated The interdependencies between gas bubbles and gas rate dissolved in fluid are also simulated

1 Introduction

A steadily increasing computational power new devel-opments and improvement of numerical methods allownumerical simulation to model more and more physicalphenomena In this sense the in situ gas generated byalumina particles and carbon engenders the formation of gasbubbles in aluminumcellsThese gas bubbles are produced bychemical reaction of alumina with the carbon which consistsof complex process of electrochemical reduction of aluminain the cellsThe continuously released gas of the alumina reac-tion generates nuclei underneath the surface of carbon anodeand detaches to dissolve in the cryolite which subsequentlygrows to bubblesThebubbles arrange into a cellular structurein the cryolite which is preserved by a rapid cyclic flow of gas

A major aim of the simulations presented here is to gaina better fundamental physical understanding of gas bubblesbecause of the complexity of gas bubbles- liquid flow inchemical system [1 2] since it determines largely the dynamicof the cryoliteThedetailed understandingmayhelp to reduce

the energy consumption and to improve the present energyefficiency by controlling the dynamics of bubbles in cells

In contrast with actual aluminum electrolysis cell modelswhich focus rather on specific aspects and for the mostpart on gas bubbles system a comprehensive and generalgas bubbles model is needed to be developed here whichincludes the wealth of phenomena occurring in aluminumelectrolysis cell process The model will elucidate the influ-ence of gas bubbles as described in [3 4] and processingparameters such as the cryolite viscosity the surface tensionthe diffusion coefficient of the propellant in the cryolite theambient pressure the heating rate and the cell geometryThisinformation is then helpful for optimizing the gas bubblesstructure However as an imitation of the real world and dueto the limited computational speed some physical processesmust be omitted or simplified in the physical model Thecomprehensiveness and generality of the physical modeldemand for a numerical solutionThe choice of the numericalmethod is made in favor of the lattice Boltzmann model(LBM) LBM explores in particular complex flow problems to

Hindawi Publishing CorporationISRN Materials ScienceVolume 2014 Article ID 454691 11 pageshttpdxdoiorg1011552014454691

2 ISRNMaterials Science

which the bubbles clearly belong with its large dynamicallymoving gasliquid interfaces [5 6] Advantages of themethodare the computational locality and parallelism the relativeease of implementation the possibility for the incorporationof additional features and the strong physical backgroundas a kinetic scheme These advantages allowed developing acomplete simulation tool within the scope of this work [7ndash12]Themain part of thiswork is concernedwith the developmentand adaptation of the new method called LBM to specify thealuminum electrolysis process As a main goal the complexgas bubbles evolution in aluminumelectrolysis cells is studiedwith an efficient method that is able to overcome difficultiesin the treatment of complex interfaces [13ndash19] For capturingthe interface novel-free surface boundary conditions aredeveloped In consistency with the hydrodynamical problemthe LBM is utilized for description of the diffusive processesof the propellant Furthermore additionalmodel features canbe incorporated within the framework of the LBM to tackle amultiphysical system of millions of degrees of freedoms

2 Physics of Gas Bubbles

This section gives a detailed introduction to the technologicalaspects and the physical phenomena of the gas bubblesprocess by in situ gas generation In particular the focus reliesupon the anodic chemical reaction of carbon

The properties of the alumina particle size and distribu-tion atomization atmosphere purity and heat treatment arecrucial for the aluminum electrolysis process The aluminaand the cryolite assumed are mixed homogeneously Thesubsequent nucleation method needs to assure a high den-sification in order to suppress gas loss by percolation at thebeginning of the process Optionally the compacted materialcan be cut into near net-shape forms to study the phenomenaof channelization and penetration of fluids in fracture

The carbon anode considered as compacted materialis constituted of one or several pieces of the compactedprecursor In the smelter there are flows from cell to cell inaluminum busbars while in each cell the current flow (I) isset to 25 KA and run downwards through the cell composedof 8 anodes cryolite molten metalpad and a cathode carbonblock

In a matter of minutes after the beginning of the elec-trolysis process gas bubbles nucleate and expand underneaththe anodes in the cryolite and the process is interrupted byremoving the gas out of the cell in order to stabilise thesystemThe final product consists of a closed cell with a cyclicformation of gas bubbles generated from the periodical reac-tion of alumina in the cryolite from feeders and surroundedby a fully dense skin called the top crust The density in themultiphase fluid flows ranges from 04 to 08 gcm3 whichcorresponds to a relative density 120588

119903of 15ndash30

Focusing on the actual electrolysis process five expansionstages dependent on the temperature are identified inFigure 1 thermal expansion of the precursor (0) bubblenucleation under surface of anodes (I) expansion in thesemisolid and liquid range (II) expansion in the liquid tem-perature range (III) and finally the gas bubbles collapse (IV)

0

300 400 500 600 700 800

Expansion of

Oxide carbonrelease from

Expa

nsio

n fa

ctor

IV

III

II

I

Temperature (∘C)

Figure 1 Schematic expansion of gas divided into five regimesPhase (0) depicts the thermal expansion of the precursor phase (I)denotes the bubbles nucleation and expansion under surface ofanodes the expansion in the semisolid and liquid range is in phase(II) phase (III) is the expansion in the liquid temperature range andfinally phase (IV) is the gas bubbles collapse

The beginning of the gas formation regime 0ndashII inFigure 1 is marked by decomposition of the alumina meltingthe precursor and nucleation The decomposition of about6wtup to 960∘C releases approximately a gas volumewhichexceeds 10 times the precursor volume The decompositionbegins while the aluminamatrix is still solid (at about 950∘C)Partial pressures in the order of thematrix yield strengthmayarise and tear open the powder compact precursor

The crack-like openings in the carbon anode are problem-atic because the CO

2can percolate through these channels

to the ambience being lost for the multiphase system Inaddition to these cracks the small pores over the anode serveas nuclei when the alumina melts In the melt heterogeneousnucleationmay occur at nucleation sites such as impurities orpropellant particles

The major gas bubbles expansion takes place while thecryolite is liquid regime III-IV in Figure 1 The strongincrease in expansion relies on the substantial oxide carbonrelease during the considered temperature intervalThe oxidecarbon dissolves and diffuses into numerous nuclei whichconsequently expand At a certain point the gas bubblesimpinge upon each other They must give up their favorableshapeThin fluid films separate the bubblesThe variations ofgas bubble sizes and thus interior pressure aswell as precedingfilm ruptures lead to bent and corrugated cell walls

The stabilization of the gas bubbles films is essen-tial Without stable cell walls bubbles would immediatelyundergo coalescencewhen they contact each other By coales-cencewith the atmosphere enforced by buoyancy the bubblesand consequently massive gas portions from alumina arelost from the system A typical gas bubbles structure cannotevolve The gas bubbles may owe stable films to the presenceof additives such as oxides Since the alumina possesses oxidethat oxide naturally resides as a network in the precursorThe repulsive force is termed disjoining pressure in thefollowing Additionally network fragments make thin filmsless penetrable to the melt so that the cryolite remains in the

ISRNMaterials Science 3

films These effects resemble a drastically enlarged viscosityThis assumption is supported by the observed shape stabilityof a once melted alumina precursor

During the gas bubbles expansion film ruptures growthcoalescence capillary gravitational drainage and electro-magnetic interbubble diffusion gas bubbles flow and gasbubbles collapse occur They are sketched in the following

Film ruptures are inevitable One observes aminimal filmthickness in gas bubbles of the order of 50 120583m This leadsto the assumption that a film rupture event occurs whenthis critical minimal film thickness dc is exceeded A ruptureevent causes the collapse of the entire film Surface tensionforces withdraw the film remnants towards the surroundingstructure The structure time 119905

119903 starting from the initial

defect in the film to its complete disappearance is a matterof seconds One cause for the film thinning is simply theexpansion of the bubbles If gas bubbles pore inflates its cellwall thickness decreases due to mass conservation At a cer-tain point 119889

119888is reached leading to rupture and coalescence

Thereby the gas bubbles system coarsens that is the meanpore diameters 119863 increase Considering a representative cellof the gas bubbles the ratio of pore volume (119863) to pore shell(119889119888) is approximately proportional to 1120588

119903

Gas bubbles drainage is the flow of liquid through andout of the gas system under the influence of external andcapillary forces In the gravitational drainage the cryolite inthe plateau borders is drawn downwards by graviton so thatthe cryolite accumulates downwards at the cryolite layer Inthe capillary drainage the different pressures in cell films andplateau borders originating from the surface tension cause thecryolite towithdraw from the films Naturally these processeswill reduce the film thickness in the gas bubbles systemwhichleads to cell wall ruptures Estimating the times scale of gasbubbles expansion it is found to be very short compared tousual bubble in fluidOnly a drastically enlarged viscosity andin particular the static disjoining pressure in the films hinderthe gravitational and capillary drainage to cause gas bubblescollapse When the stabilization is significant the drainagemay even become negligible

Distinct gas bubble sizes and consequently distinct bub-bles may lead to interbubble diffusion of oxide carbon viathe cell walls Tiny bubbles suffer from gas loss while largebubbles earn oxide carbon similar to Ostwald ripeningA study of the diffusion flow of oxide carbon across cellborders demonstrates the large time scales compared to thebubbly times of the gas transport in cryolite Thereforethe interbubble diffusion is negligible in the cryolite bubblyevolution During the evolution gas bubbles are permanentlyexposed to mechanical forces due to the expansion itself orspatial restriction by the cell design Deformation of poresrearrangement or destruction of cell films and gas bubblesflow are the consequences Topological rearrangements andfilm ruptures occur often in an avalanche-like manner In thefinal stage of bubbles expansion large bubbles are presentcompared to the gas bubbles system size Gas bubbles collapseat the cryolite surface leads now to severe oxide carbon loss inthe system so that the gas system decays that is the collapsestarts

3 Physical Model

Reviewing gas bubbles in aluminum electrolysis cell inFigure 2 almost no theoretical models can be found Consid-ering the bubbly evolution in liquid it becomes apparent thatspecific aspects of gas bubbles are commonly studied in iso-lation In order to include many physical phenomena of theelectrolysis process a comprehensive model is constructedbased on fundamental physical equations In order to obtainthe full information of gas bubbles structure it is requiredto directly simulate gas bubbles structure in aluminumelectrolysis process Nevertheless the model will naturallycontain simplifications because it is a realistic imitation of thereal industrial problem and because of inaccessibility in cells

The gas bubbles are stochastically distributed and formedsimultaneously at the starting point Dissolved aluminaparticles reacting with the carbon anode diffuse into thenuclei over the surface of carbon anodes The decompositionand dissolution of alumina are modeled by a homogeneousvolume source 119878 in the cryolite domain Ω The nuclei growover the undersurface of anodes to bubbles Consequentlythe surrounding cryolite is displaced Interactions betweenbubbles are accounted for at first over the anodes by a short-range disjoining pressure Cell walls ruptures occur when aminimal critical film thickness is attained As a simplificationthe model uses a static electromagnetic force term in theexpression ofNavier-Stokes equation Furthermore the start-ing point of themodel indicates already that themelting of thealumina precursor and an explicit description of the chemicaldecomposition of the propellant are neglected Similarly thefinal bubbly structure is achieved by simply stopping the sim-ulation at certain point of time In thismodel the temperatureremains constant at 960∘C in space and time Consequentlythe material parameters are constants A temperature andheating rate enters the model only implicitly by the homo-geneous volume source term 119878 of the dissolved oxide carbon

The solute oxide carbon 119888 in the cryolite obeys the diffu-sion equation inΩ

120597119888

120597119905+ 119906120572

120597119888

120597120572minus120597119863

120597120572

120597119888

120597120572= 119878 (119905) (1)

where 119878(119905) is the volume source term 119863 is the gas diffusion119888 is the concentration field of oxide carbon and 119905 denotes thetime

The liquid domain contains islands which are the nucleior gas bubbles At the gas-liquid interface 120597Ω Sievertrsquos lawholds as follows

119888 = 119896119878radic119901119892119894

(2)

where the pressure 119875119894is distinct for each bubble 119894 and 119896

119904is

Sievertrsquos constant Furthermore the gradient at the interfaceis parallel to the unit normal direction 119899 of the interface

(120597119909

120597119910

) 119888 prop 119899 (3)


Sticking Void domain

Gas-liquidmixture

Cathodeblock

Anodes

contacts

Feeder

xD

xA

xB

xC

Figure 2 Vertical cut short of an aluminumelectrolysis cell inwhich the gas expands from liquidsThe points119909119860119909119861 and119909

119862are the references

points for the evolution of gas in the cell

The dynamics of cryolite flow caused predominately by gasbubbles expansion is described by the famous incompressibleNavier-Stokes equation in Ω

120597119906120572

120597120572= 0

120597119906120572

120597119905+120597119906120573

120597120573

120597119906120572

120597120572= minus

120597119875120597120572

120588+ ]1205972120573119906120572+ 119892120572

(4)

Since the viscous stresses of the gas are negligible free surfaceboundary conditions are applicable at 120597Ω

119901 minus 2120588]120597119906119899

120597119899= 119901119894minus 120590120581 minus Π (5)

120597119906119905

120597119899+120597119906119899

120597119905= 0 (6)

where the subscripts 119899 and 119905 denote the normal and tangentialcomponentwith regard to 120597ΩThe gas pressures of bubbles119875

119894

the surface tension pressure 120590120581 and the disjoining pressureΠ enter the model via these boundary conditions In thebubbles the ideal gas law is considered

119901119894=119899119894119877119879

119881119894

(7)

where 119875119894is the gas pressure in bubble 119894 given by the ideal gas

equation 119877 denotes the ideal gas constant and 119899119894denotes the

amount of gas 119879 denotes the temperature and119881119894denotes the

volume of bubble 119894 The change of bubble volume per time isgiven by the normal velocity 119906

119899of the bubble-liquid interface

119889119881119894

119889119905= minusint120597Ω119894

119906119899119889119908 at 120597Ω

119894 (8)

The gas exchange between cryolite and bubbles is given byFrickrsquos law so that the following change of the amount ofsubstance per time applies

119889119899119894

119889119905= minusint120597Ω119894

119863120597119888

120597119899119889119908 (9)

A special type of bubble is the atmosphere It has a constantpressure 119875

119900at all times Its oxide carbon partial pressure

entering Sievertsrsquo law can be chosen differently from 119901119900

A heuristic approach condenses the film stabilization ina single function the disjoining pressure Π Π is a repulsionexperienced by bubble 119894 in the presence of another bubble 119895

Π = 119896Π(119889max minus 119889) 119889 lt 119889max

0 119889 ge 119889max(10)

The distance 119889 from bubble 119894 to bubble 119895 is obtained bymoving along the normal direction When the neighboring


bubble is too distant that is 119889 exceeds 119889max no interactionsof the bubbles occur At present the functional form of Π(119889)is chosen arbitrarily

When the film thickness 119889 falls below a critical threshold119889119888 the cell wall is subjected to rupture These coalescencecriteria are extracted in agreement with numerical experi-mental data To conclude desired features such as rheologyor drainage evolve automatically in the model because ofits generality Furthermore processing parameters such asviscosity or ambient pressure are incorporated naturally andhence accessible to investigations

31 Numerical Model The liquid domain Ω in which gasbubbles develop contains two domains the gas Ωgas and theliquids Ωliq This domain Ω can be open or closed as shownin Figure 2 One notes Γ sub 120597Ω the entire walls of the domainWe will seek to establish a system of equations verified by thevelocity field V and the pressure 119901 in all the domain Ω thatis in Ωgas as in Ωliq The velocity field and the pressure aredefined in the liquid part in the same manner they are alsoin the gaseous part but this time taking into account that119901119892= 0 In themixture gas liquid one applies the conservation

laws In addition one applies a condition of sticking contacton walls Γ of the entire domainΩ

The last point is disputable and constitutes a first approx-imation This leads to the following formulation

Find the velocity V isin 1198882(Ωgas) V|Γ = 0 and the pressure119901 isin 1198881(Ωgas) such as forall119905 isin]0 120579[isin Ωgas(119905)

nabla sdot [2120578gas ((120576 (V)) 119882) 120576 (V)] minus nabla119901 = 0

nabla sdot V = minus1

120588gas

119889120588gas

119889119905(119882)

120590 119899 = 0 over 120597Ωvoid cap 120597Ωgas

(11)

where119882 is the gaseous volume fraction in themixture withinthe cryolite before the rise of gas bubbles over the surface ofthe liquid119882 is a field depending on the point where we arein the gaseous domain and timeThe viscosity 120578gas depends on119882 as well as the gas-liquid system By neglecting the volumemass of gas (120588gas) before the volumemass of the liquid (about120588liq = 1000 kg sdotm

minus3) on can write

120588gas = (1 minus119882) 120588liq (12)The cryolite is assumed to be incompressible so that

1

120588gas

119889120588gas

119889119905= minus

1

1 minus119882

119889119882

119889119905 (13)

The micro-macro-coupling is carried out then through theexpansion rate of the gas 119889119882119889119905 that remains to be deter-mined In the microscopic case the expansion of gas bubblesis governed by the difference of pressures between the internaland the ambient and the latter constitutes a macroscopicparameter of the problem exterior of ourmicroscopicmodelThus it follows that one uses

119889119882

119889119905=120597119882

120597119905+ V sdot nabla119882 = 119891 (119882 119901gas minus 119901) (14)

where 119901gas denotes the internal pressure of the gas and 119882denotes the expansion rate

This expansion rate is explicitly dependent on time 119905because it implies a gas creation by chemical reaction fromthe alumina The time appears explicitly in the expression of119889119882119889119905 Finally the system in velocity pressure to solve writesas follows




1 minus119882119891(119882 119901gas minus 119901)


(15)

After the germination of gas bubbles and their detachmentfrom the undersurface of the anodes the gas bubbles stillexpand in the gas-liquid mixture and are assumed to bespherical They rise to the surface and start to reduce theirinteraction between them At the surface of the liquid theyoccupy a small percentage of the total volume to a structureof mixture gas and liquid with deformed gas bubbles usingmore than 85 of the total volume

32 Micro-Macro-Coupling The micro-macro-coupling inthe aluminum electrolysis cell is carried out by closing (15)via the expansion rate 119889119882119889119905 (see (14)) One proposesto determine this expansion rate from simulations usinga microscopic model This model takes into account theexpansion of a representative volume during a moment ofthe macroscopic flowThis model takes as input variables theinitial configuration of gas as well as its initial pressure andit intervenes as a variable of macroscopic order 119901ext in theambient pressure at a considered point of the macroscopicflow The expansion rate 119889119882119889119905 is expressed first when theinteractions of gas bubbles are not taken into account In thisstudy a volume representative in expansion of volume |Ω|contains a gaseous part Ωgas of volume |Ωgas| and a liquidpart Ωliq of a constant |Ωliq| over time The gas rate 119882

119888Ω

is defined by 119882119888= |Ωgas||Ω| One recovers the following

relation between |Ωgas| and119882119888

10038161003816100381610038161003816Ωgas

10038161003816100381610038161003816=

119882119888

1 minus119882119888

10038161003816100381610038161003816Ωliq

10038161003816100381610038161003816 (16)

It is assumed that the gas in the cryolite is composed of119899 spherical bubbles of volume |Ωgas119894| radius 119877119894 and thepressure 119875

119892119894 Whereas the cryolite is regarded as Newtonian

and of consistency 120578119900 The pressure is assumed to be constant

in the liquid and equal to the external pressure that representsthe constraint applied on thewall 120597Ω by the externalmediumThus gas bubbles grow by difference of pressure and theirinteractions are neglected here Consequently one finds that

10038161003816100381610038161003816Ωgas

10038161003816100381610038161003816=

119899

sum

119894=0

10038161003816100381610038161003816Ω119892119894

10038161003816100381610038161003816=4120587

3

119899

sum

119894=1

1198773

119894 (17)


So that an algebraic transformation will give

110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

sum119899

119894=11198773

119894

119899

sum

119894=1

1198772

119894 (18)

Without the interactions between bubbles one applies overeach bubble the expression of the velocity (119906(119903) = 11987721199032)pressure (119875(119903) = 119875ext) and 119877 = Δ1199014120578119900 Finally it gives

110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

4120578119900sum119899

119894=11198773

119894

119899

sum

119894=1

(119901gas119894

minus 119901ext) 1198773

119894 (19)

The above equation now is simplified because the bubblesare taken with the same pressure in the entire volume of thedomainΩ One obtains

110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

41205780

(119901gas minus 119901ext) (20)

The differential of (16) and its combinationwith (20) in func-tion of pressurersquos difference writes

119889119882119888

119889119905=

3

41205780

(119901gas minus 119901ext)119882119888 (1 minus 119882119888) (21)

Finally one expresses the pressure of gas 119875gas in functionof 119882119888 119875119892is supposed to be homogeneous in the gaseous

domain by using the ideal gas law for each bubble and bysumming over all bubbles are obtained that 119901gas|Ωgas| =

119901gas0|Ωgas0| Using (16) one can write that

119901gas =1 minus119882

119888

119882119888

1198821198880

1 minus1198821198880

119875gas0 (22)

Thus by neglecting the interactions between bubbles andby assuming a homogeneity of the pressure the gas rateexpansion in the representative volume gives the expression119889119882119888119889119905 in function of119882

119888and 119875ext Now following a macro-

scopic approach the expansion rate of the gas 119882 is a fieldOne obtains then

120597119882

120597119905+ V sdot nabla119882 =

3

41205780

[(1 minus119882)2 1198820

1 minus1198820

119875gas0

minus119875119882(1 minus119882) ] in Ωgas

(23)

To solve numerically (23) its domain of validity must beextended in all the domain Ω in a manner to retrieve itssecond term null in the liquid domainΩliq

4 Numerical Implementation

Reviewing the physicalmodel from anumerical point of viewthe most challenging aspect is the parallel implementationof the large complex and dynamically moving interfacesbetween the cryolite and the gas bubble Among the largevariety of numerical methods the lattice Boltzmann method

(LBM) [20ndash23] is a promising candidate for numerical inves-tigations in aluminum electrolysis cell Suitable free surfaceboundary conditions need to be newly developed for thelattice Boltzmann method In addition the other features ofthe physical model for example the diffusion equation andthe disjoining pressure need to be included into the LBM

At first the focus remains on the fluid dynamics Sincethe beginning of the last century it is known that the kineticBoltzmann equation can approximate the Navier-Stokesequation By numerically solving the kinetic Boltzmannequation the lattice Boltzmann model supplies an approxi-mate solution for theNavier-StokesTheLBM is consequentlya kinetic scheme based on particle dynamics In detail thespace is divided into a 2D regular lattice On each latticesite single-particle distribution functions 119891

119894( 119905) represent

the number of particles at lattice site 119909 and time 119905 in state 119894The distinct states 119894 are defined by a set of nine discrete

velocities

120585119894=Δ119909

Δ119905

(0 0) 119894 = 0

(plusmn1 0) (0 plusmn1) 119894 = 1 4

(plusmn1 plusmn1) 119894 = 5 8

(24)

In each time step Δ119905 the distribution functions are advectedalong their corresponding velocity and subsequently theyundergo a collision operation locally at each time site Inthe simplest case the collision step consists of a relaxationof the function 119891

119894( 119909119905) towards an equilibrium function

119891(0)

119894( 119905) Both steps are graphically displayed in Figure 8 and

combined in the lattice Boltzmann equation (LBE) [24]

119891119894( + 120585

119894 119905 + Δ119905) minus 119891

119894 ( 119905)

= minusΔ119905

120591(119891119894( 119905) minus 119891

(0)

119894( 119905)) + Δ119905119865

119894

(25)

Themacroscopic variables 120588 and are obtained by the zerothand first moments

120588 ( 119905) = sum

119894

119891119894( 119905)

120588 ( 119905) = sum

119894

120585119894119891119894( 119905)

(26)

The collision operator is constructed to fulfill mass momen-tum and energy conservation at each lattice site and to ensurethe correct symmetries of the Navier-Stokes

119891(0)

119894= 119908119894120588(1 +

120585119894120572119906120572

1198882119904

+(120585119894120572119906120572)2

21198884119904

minus119906120572119906120572

21198882119904

) (27)

119908119894=

4

9119894 = 0

1

9119894 = 1 4

1

36119894 = 5 8

(28)

and 1198882119904= 13 is the sound speed squared


The force term includes the gravity g and electromagneticforce [23 25 26]

119865119894= 119908119894120588[

120585119894120572minus 119906120572

1198882119904

+(120585119894120573119906120573) 120585119894120572

1198884119904

] (119892120572+ 119895 and ) (29)

By Chapman-Enskog analysis the LBM is proved to approx-imate the Navier-Stokes in the incompressibility 119906 ≪ 119888

2

119904 and

for small Reynolds numbers The Chapman-Enskog analysisstarts with a separation of the time scales and expansion ofthe distribution functions

119891119894= 119891(0)

119894+ isin 119891

(1)

119894+ isin2119891(2)

119894+ 119874 (isin

3)

119865119894= isin 119865(1)

119894+ 119874 (isin

2)

120597119905= isin 120597(1)

119905+ 1205762120597(2)

119905+ 119874 (isin

2)

120597120572= isin 120597(1)

120572+ 119874 (isin

2)

(30)

where isin is a small number The following constraints areimposed on the expanded distribution functions

sum

119894

[1

120585119894

]119891(0)

119894= 120588 [

1

]

sum

119894

[1

120585119894

]119891(119899)

119894= 120588 [

0

0] for 119899 gt 0

(31)

Further the left hand side of the LBE is expanded by a secondorder Taylor series and the resulting terms in (25) are substi-tuted by the above series expansions An equation is derivedcontaining various orders of epsilon Evaluating the first twomoments of this equation results in the hydrodynamic equa-tions The first order of epsilon provides the Euler equationsIncluding the second order of epsilon the NSE is derived

120597120588

120597119905+120597120588V120572

120597120572= 0

120597120588V120572

120597119905+120597120588V120573V120572

120597120573= minus

120597119875

120597120572+ V120588

120597

120597120573

120597

120597120573V120572+ 120588119892120572+ 119895120572otimes 119861120573

(32)

where V = (1198882119904Δ1199052)(2120591Δ119905 minus 1)

The presented LBM is applied in the fluid region [27ndash29] In the gas bubbles system gas regions and the outer celldesigns are present as well In the regular lattice one maydefine the distinct regions by distinct cells Fluid 119862

119865 bubble

119862119861 interface 119862

119865 and wall cells 119862

119882 The LBM is executed

on the fluid cells In the interface cells the free surfaceboundary conditions in (5) and need to be implemented in(6) After the advection step and before the collision takesplace the interface cells obtain known distribution functionsfrom liquid regions and unknown ones from the gas regionsThe missing populations are reconstructed by a first orderChapman-Enskog approximation see (30) where in 119891(1)

119894all

terms of 119874(isin 1199062 isin2) are neglected

119891(1)

119894= 119908119894120591(120597120572(120588119906120572) minus

1

1198882119904

120585119894120572120585119894120573120597120572(120588119906120573)) (33)

Interface

FluidBubble

120575

n

t

Figure 3 Interface cell after advection stepwith unknowns function(dotted line) and known function (full line)

In119891(0)119894

(27)119891(1)119894

in (33)Themacroscopic quantities 120588 119906 and120597120572119906120573are unknown The free surface boundary conditions

(5) and (6) are utilised to eliminate the unknowns 120597119899119906119899 120597119905119906119899

and 120597119899119906119905 Therefore the scalar products are expressed in the

local coordinate system at the interfacial consisting of thenormal 119899 and the vector tangential 119905 as depicted in Figure 3Furthermore the quadratic velocity terms in 119891

(0)

119894are

neglected in order to obtain linear system of equationsThen

119891119894= 119908119894(120588 +

120585120572120588119906120572

1198882119904

+ 120591((1 minus1205852

119894119899

1198882119904

)

1198882

119904120588 minus 119875119868

119895

2V)

+ (1 minus1205852

119894119905

1198882119904

)120588120597119905119906119905)

(34)

Here 120585119894119899= 120585119894120572119899120572 120585119894119905= 120585119894120572119905120572 119875119868119895is the pressure at the interface

of the considered gas bubble 119895 including surface tension anddisjoining pressure

After reconstruction the usual collision operation takesplace Yet the interface advection remains to be discussedFor this reason interface cells contain additional informationwhich indicates the amount of fluid inside each cell Thisliquid volume fraction Λ is updated according to the in- andoutflow of the interface cell Formore accuracy the differencebetween the incoming and the outgoing distribution func-tions is calculated on each link and weighted by factor Φwhich depends on the position of the interface

ΔΛ119894 () = Φ (119891119894 ( + 119890

119894) minus 119891119894 ()) (35)

With

Φ =

1 + 119890119894isin 119862119865

Φ () + Φ ( + 119890119894)

2 + 119890119894isin 119862119868

0 + 119890119894isin 119862119861

(36)


Φ is the area fraction obtained through Φ = Λ120588 119890119894= 120585119894Δ119905

and 119894 is related to 119894 via 119890119894= minus 119890119894 The total flux in the interface

cell 119909 is obtained by summing ΔΛ119894over all 119894

Empty or full interface cells are detected when Λ exceedsits limits 0 lt Λ lt 120588 They are changed to gas or fluidcells respectively When cells are transformed the closenessof the interface needs to be guaranteed that is unknowndistribution functions must be advected into interface cellsThis demands sometimes transformations of cells in theclose neighborhood of the initially transformed interface cellUnknowndistribution functions in these cells are obtained bylinear extrapolations The simplest wall boundary conditionsare utilized in the wall cells which represent the cell design

Bubble cells surrounded by interface cells build con-nected and closed areas of cells which represent the bubblesThe volume and amount of gas in the bubbles need to bedetermined in each time step Initially the bubble volume isknownThen it is sufficient to calculate the change of volumeper time step It is given by the change of the volume fractionsΛ in the interface cells 119862

119868119895of the bubble 119895

Δ119881119895= minus sum

119909isin119862119868119895

ΔΛ () (37)

Similarly only the oxide carbon diffusion across the interfaceis recorded in order to obtain the amount of substance in thebubbles In the lattice Boltzmann formalism the flux is givenby the exchange of distribution functions across the interfacewhich is approximated by the cell borders between fluid andinterface cells

5 Numerical Validation

These numerical implementations need to be validated Forthis comparisons between numerical and analytical resultsgive information on the validity range of the parametersand the accuracy of this scheme This test will merely focuson newly developed implementation issues and on singlerelaxation time whereas the interface advection is tested bystreaming a gas bubble in uniform flow field Distortionsof the round bubble shape and the small deviations fromanalytically predetermined way of the bubble center appearafter the advection Overall the advection appears to be verypromising results

The same holds for the curvature algorithm The resultsof template sphere methods are better than those of standardfinite difference methods (FDM) often applied to volume offluid methods (VOF)

The Laplace law has been used for several parameters byconsidering bubble in liquid flow In Figure 4 the numericalresult depicts a small deviation from the theoretical expecta-tion marked by the dotted lines During these investigationssmall spurious currents occur For the tested variables com-prised between 0001 lt 120590 lt 005 003 lt V lt 043 and 120588 = 1for the fluid are used all over the computations

They remain small overall the experiment Dynamicallythe free surface boundary conditions are investigated bystanding capillary and external forces waves such as gravityand the electromagnetic force These waves are damped by

the viscosity of the liquid Frequency and damping rateobtained numerically are in good agreement with theoreticalexpectations Good agreement is also depicted by a numericalinvestigation on the free surface boundary condition asshown in (5) at the interface of a rising gas bubble In additionthe gas bubble shapes and their rising velocity are observedThe later seems to decrease very fast below expectations witha shrinking effect when the surface tension is much largerthan the effects of the external forces impact small or negli-gible In conclusion one can summarise that the numericalexperiment exhibits good agreements with theoretical expec-tations in regard to parameter ranges of our numerical tests

6 Results and Discussion

Having provided and tested the physical model and itsnumerical implementation the simulations of gas bubblesexpansion in aluminum cell process are finally performedThe tests carried out show that the consistency of the liquidintervenes as multiplicative factor in the expression of theviscosity in the system bubbles liquid 120578gas In this studyone considers a system of 119899 bubbles from 84 to 1500 thatis having different densities of gas bubbles The choice ofthe structure of bubbles does not influence the viscosity ofbubbles liquid and helps to keep the integrality of bubblesin the computational domain during the bubbles expansionThus the effects of edges are restricted In Figure 5 oneplotted the values of the ratio 120582 = 120578gas1205780 in function ofgas rate computed during the expansion of bubbles For anumber of bubbles over 84 we have uniquely representedthe obtained values before the effect of edges intervenes(rate lt70) whose description of bubbles-liquid structureis good Comparing the curve of Figure 5 with the right1-W plotted on the same figure one observes a nonlineardependence of the viscosity to the gaseous fraction119882 Onedistinguishes three different areas in the curves of Figure 5On one hand for a gas rate inferior to 20 the more thedensity of bubbles is important the smaller the bubbles areThey offer less resistance to shearing This explains the factthat the equivalent gas rate is high As long as the interactionbetween bubbles is negligible the gas fraction remains small

Finally when the gas fraction tends towards 1 the viscos-ity also tends asymptotically towards zero Thus we have putin evidence on one hand the nonlinear dependence of theviscosity of mixture and on the other hand the dependenceof the viscosity on the bubbles density One remarks that thedensity of bubbles is consequent when the viscosity tends toan asymptotic curve

61 Dependence of Viscosity on Bubbles-Liquid MixtureThere are several numerical simulations of bubbles-liquidstructure by varying the bubbles density The initial gasfraction is the same for each simulation 119882

1198880 as well and

the number of bubbles considered is important more thesizes of whose given at same gas rate are small One used inthis study an ordinated structure (bubbles dispersed regularlyon the grid) and bubbles having different sizes and sameinitial pressure in a manner to conserve the integrality of thestructure in the computational domain on another hand to


1E minus 01

1E minus 02

1E minus 03

1E minus 04

1E minus 05001 01 1

Curvature K

120575 = 005

120575 = 001

120575 = 0005

120575 = 001

(Pga

sminus

Pliq

)

Figure 4 Numerical tests of Laplace lawThe dashed lines representthe theoretical equation

Gas rate0 01 02 03 04 05 06 07 08 09 1

0

01

02

03

04

05

06

07

08

09

1

Visc

osity

of m

ixtu

rec

onsis

tenc

y of

liqu

id

84 bubbles520 bubbles789 bubbles


Figure 5 Function 120582 = 120578gas1205780 obtained by numerical simulation infunction of the number of gas bubbles

satisfy the hypothesis of the model Because the interactionamong bubbles of same pressure is absent the gas rate 119882

119888

model predicts an accurate profile for 84 520 789 1000 and1500 bubbles comparing with the analytical expression of gasrate The results show that the gas rate and the expansionrate are strongly influenced by the density of bubbles Theformer will decrease with the increase of latter This reflectsthe interactions between bubbles and the constraints exertedby the ones to the others when the expansion of 84 bubblesis considered As long as the gas fraction does not exceed45 its evolution remains similar to that obtained withoutinteractionThe divergence shown between the two fractionsbeyond 50 explains the fact that the gas goes out of thecomputational volume causing a decrease of the amount ofliquid present in this volume Consequently beyond a certainlimit the expansion rate of gas fraction is overestimatedcompared to the reality This approach is applicable for eachsimulation

0

01

02

03

04

05

06

07

08

09

1



Gas

rate

0 100 200 300 400 500 600 700

Time (s)

Figure 6 Evolution of gas rate over simulations of volume expan-sion of bubbles liquid with 84 to 1500 bubbles

0 200 400 600 800 1000 1200 1400

Number of bubbles in function of parameter

Para

met

er11

1

09

08

07

06

05

04

03

02

Figure 7 Evolution of gas rate during expansion of bubbles-liquidstructure The points correspond to the obtained values by compu-tation of the value of parameter The curves are the approximationsestimated from (23)

Figure 6 shows that the more the number of consideredbubbles is important the more these effects of edges arenegligible compared to whole bubble remaining inside thesystem We choose to express the interaction of bubbles inthe expression of gas rate by introducing simply in (23) aparameterΥdetermined from the curves in Figure 6 Its valueis in function of bubbles density (Figure 7) and then dependsalso on the radius of these bubblesThe curves correspondingto (23) related to the parameter Υ are plotted in Figure 7Finally amicro-macro-coupling is used to express the expan-sion rate of the mixture bubbles liquid in which the liquid isNewtonian and of consistency 120578

0

62 Free Expansion of Gas Bubbles in Cell Thecomputationaldomain Ω is regarded as an open system that is the gaseousmatter can freely go out of the boundary part 120597Ω located ata certain height on which one applies a constraint null suchas (120590 119899 = 0) as indicated in Figure 8 depicting the aluminiumelectrolysis cell We have chosen on Figure 8 the evolution


09

08

07

06

05

04

03

02

01

00 05 1 15 2 25 3 35 4 45 5

Gas

frac

tion

Time (s)

Site ASite B

Site CSite D

(a)

09

08

07

06

05

04

03

02

010 05 1

1

15 2 25 3 35 4 45 5

Time (s)

Site ASite B

Site CSite D

Visc

osity

of m

ixtu

rec

onsis

tenc

e of l

iqui

d

(b)

Velo

city

(mm

s)

Site ASite B

Site CSite D

0 05 1 15 2 25 3 35 4 45 5

Time (s)

35

30

25

20

15

10

5

0

(c)

Pres

sure

(Pa)

0 05 1 15 2 25 3 35 4 45 5

Time (s)

001

0008

0006

0004

0002

0

minus0002

Site ASite B

Site CSite D

(d)

Figure 8 Evolution of gas fraction viscosity velocity and of the pressure at different points in the cell during the gas expansion

of the gas fraction viscosity and of the pressure at differentpoints A B C and D The pressures lower than the previousone in Figure 8(d) particularly at D due to the outgoing faceon which 120590 119899 = 0 the velocity at the end of the expansion isitself higher than the previous one in Figure 8(c)

The micro-macro-coupling used in this study is basedon the fact that the simulation of the expansion of therepresentative volume needs a macroscopic variable that isthe ambient pressure The expansion happens through thefluctuations of pressure Δ119901 in the mixture bubbles-liquidThus heterogeneities of gas fraction and then the size ofbubble (or the density of bubbles) appear in the mixture Theexpansion of the mixture depends on its structure via theviscosity and the volume mass

7 Conclusion

In this study a new scheme has been developed to simulatethe expansion of gas bubbles in aluminum electrolysis cellin order to describe the evolution of the viscosity and the

volume mass of a representative volume of bubbles-liquidmixture during the expansion of the gas It is found that thedependence of these two parameters is on the bubbles densityin the mixture The influence of gas bubbles interactions onthese parameters has been revealed From these results wehave carried out a micro-macro-coupling to express the evo-lution of the mixture volumemass during expansion in func-tion of pressures fluctuation in the mixture This approachhelps to take into account the electrolysis process and alsothe influence of the mixture structure during the simulationsof expansion These simulations provide the variations of gasfraction in the cell The simulations also show that closeto the cooled edges the gas expansion is slower and evenstopped Thus a fine surface skin is created on the cryolite

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


Acknowledgments

The authors would like to acknowledge the financial supportof the Natural Sciences and Engineering Research Council ofCanada and Alcoa A part of the research presented in thispaper was financed by the Fonds de Recherche du Quebec-Nature et Technologies (FRQ-NT) by the intermediary of theAluminium Research Center-REGAL The authors wish alsoto thank Dr Donald Ziegler from the Alcoa Canada PrimaryMetals for the scientific discussions

References

[1] R Keunings ldquoAdvances in the computermodeling of the flow ofpolymeric liquidsrdquo Computational Fluid Dynamics Journal vol9 pp 449ndash458 2001

[2] L Lefebvre and R Keunings ldquoNumerical simulation of chemi-cally-reacting polymer flowsrdquo in Proceedings of the 1st EuropeanComputational Fluid Dynamics Conference vol 2 pp 1133ndash1138Brussels Belgium September 1992

[3] A Arefmanesh and S G Advani ldquoDiffusion-induced growth ofa gas bubble in a viscoelastic fluidrdquo Rheologica Acta vol 30 no3 pp 274ndash283 1991

[4] D F Baldwin C B Park and N P Suh ldquoMicrocellular sheetextrusion system process design models for shaping and cellgrowth controlrdquo Polymer Engineering amp Science vol 38 no 4pp 674ndash688 1998

[5] R Benzi S Succi and M Vergassola ldquoThe lattice Boltzmannequation theory and applicationsrdquo Physics Report vol 222 no3 pp 145ndash197 1992

[6] P L Bhatnagar E P Gross andMKrook ldquoAmodel for collisionprocesses in gases I Small amplitude processes in charged andneutral one-component systemsrdquo Physical Review vol 94 no3 pp 511ndash525 1954

[7] J H Han and C D Han ldquoBubble nucleation in polymeric liq-uids I Bubble nucleation in concentrated polymer solutionsrdquoJournal of Polymer Science B vol 28 no 5 pp 711ndash741 1990

[8] J H Han and C D Han ldquoBubble nucleation in polymeric liq-uids II Theoretical considerationsrdquo Journal of Polymer ScienceB vol 28 no 5 pp 743ndash761 1990

[9] S Hutzler D Weaire and F Bolton ldquoThe effects of Plateauborders in the two-dimensional soap froth III Further resultsrdquoPhilosophical Magazine B vol 71 no 3 pp 277ndash289 1995

[10] C IsenbergThe Science of Films and Soap Bubbles Dover NewYork NY USA 1992

[11] DWeaire and R Phelan ldquoA counter-example to Kelvinrsquos conjec-ture on minimal surfacesrdquo Philosophical Magazine Letters vol69 no 2 pp 107ndash110 1994

[12] J U Brackbill D B Kothe andC Zemach ldquoA continuummeth-od for modeling surface tensionrdquo Journal of ComputationalPhysics vol 100 no 2 pp 335ndash354 1992

[13] N S Ramesh D H Rasmussen and G A Campbell ldquoNumer-ical and experimental studies of bubble growth during themicrocellular foaming processrdquo Polymer Engineering amp Sciencevol 31 no 23 pp 1657ndash1664 1991

[14] L E Scriven ldquoOn the dynamics of phase growthrdquo ChemicalEngineering Science vol 10 no 1-2 pp 1ndash13 1959

[15] D L Weaire and S HutzlerThe Physics of Foams Oxford Uni-versity Press New York NY USA 1999

[16] R Clift J R Grace andMWeber Bubbles Drops and ParticlesAcademic Press New York NY USA 2002

[17] F Bolton and D Weaire ldquoThe effects of Plateau borders in thetwo-dimensional soap froth I Decoration lemma and diffusiontheoremrdquo Philosophical Magazine B vol 63 no 4 pp 795ndash8091991

[18] J M Buick Lattice Boltzmannmethods in interfacial wave mod-elling [PhD thesis] University of Edinburgh Edinburgh Scot-land 1997

[19] J W Bullard E J Garboczi W C Carter and E R FullerJr ldquoNumerical methods for computing interfacial mean curva-turerdquoComputational Materials Science vol 4 no 2 pp 103ndash1161995

[20] N Cao S Chen S Jin and D Martinez ldquoPhysical symmetryand lattice symmetry in the lattice Boltzmannmethodrdquo PhysicalReview E vol 55 no 1 pp R21ndashR28 1997

[21] X He and L-S Luo ldquoA priori derivation of the lattice Boltz-mann equationrdquo Physical Review E vol 55 no 6 pp R6333ndashR6336 1997

[22] X He and L-S Luo ldquoTheory of the lattice Boltzmann methodfrom the Boltzmann equation to the lattice Boltzmann equa-tionrdquo Physical Review E vol 56 no 6 pp 6811ndash6817 1997

[23] Z Guo C Zheng and B Shi ldquoDiscrete lattice effects on theforcing term in the lattice Boltzmann methodrdquo Physical ReviewE vol 65 no 4 Article ID 046308 6 pages 2002

[24] J M Buick and C A Created ldquoGravity in a lattice Boltzmannmodelrdquo Physical Review E vol 61 no 5 pp 5307ndash5320 2000

[25] S Chen D Martınez and R Mei ldquoOn boundary conditions inlattice Boltzmann methodsrdquo Physics of Fluids vol 8 no 9 pp2527ndash2536 1996

[26] M R Swift E Orlandini W R Osborn and J M YeomansldquoLattice Boltzmann simulations of liquid-gas and binary fluidsystemsrdquo Physical Review E vol 54 no 5 pp 5041ndash5052 1996

[27] A K Gunstensen D H Rothman S Zaleski and G ZanettildquoLattice Boltzmannmodel of immiscible fluidsrdquoPhysical ReviewA vol 43 no 8 pp 4320ndash4327 1991

[28] D Grunau S Chen and K Eggert ldquoA lattice Boltzmann modelfor multiphase fluid flowsrdquo Physics of Fluids A vol 5 no 10 pp2557ndash2562 1993

[29] M Do-Quang E Aurell and M Vergassola ldquoAn inventory oflattice Boltzmann models of multiphase flowsrdquo Tech Rep Par-allel and Scientific Computing Institute Department of Scien-tific Computing Uppsala University Uppsala Sweden 2000

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of



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CompositesJournal of

NanoparticlesJournal of



International Journal of

Biomaterials


NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials


which the bubbles clearly belong with its large dynamicallymoving gasliquid interfaces [5 6] Advantages of themethodare the computational locality and parallelism the relativeease of implementation the possibility for the incorporationof additional features and the strong physical backgroundas a kinetic scheme These advantages allowed developing acomplete simulation tool within the scope of this work [7ndash12]Themain part of thiswork is concernedwith the developmentand adaptation of the new method called LBM to specify thealuminum electrolysis process As a main goal the complexgas bubbles evolution in aluminumelectrolysis cells is studiedwith an efficient method that is able to overcome difficultiesin the treatment of complex interfaces [13ndash19] For capturingthe interface novel-free surface boundary conditions aredeveloped In consistency with the hydrodynamical problemthe LBM is utilized for description of the diffusive processesof the propellant Furthermore additionalmodel features canbe incorporated within the framework of the LBM to tackle amultiphysical system of millions of degrees of freedoms

2 Physics of Gas Bubbles

This section gives a detailed introduction to the technologicalaspects and the physical phenomena of the gas bubblesprocess by in situ gas generation In particular the focus reliesupon the anodic chemical reaction of carbon

The properties of the alumina particle size and distribu-tion atomization atmosphere purity and heat treatment arecrucial for the aluminum electrolysis process The aluminaand the cryolite assumed are mixed homogeneously Thesubsequent nucleation method needs to assure a high den-sification in order to suppress gas loss by percolation at thebeginning of the process Optionally the compacted materialcan be cut into near net-shape forms to study the phenomenaof channelization and penetration of fluids in fracture

The carbon anode considered as compacted materialis constituted of one or several pieces of the compactedprecursor In the smelter there are flows from cell to cell inaluminum busbars while in each cell the current flow (I) isset to 25 KA and run downwards through the cell composedof 8 anodes cryolite molten metalpad and a cathode carbonblock

In a matter of minutes after the beginning of the elec-trolysis process gas bubbles nucleate and expand underneaththe anodes in the cryolite and the process is interrupted byremoving the gas out of the cell in order to stabilise thesystemThe final product consists of a closed cell with a cyclicformation of gas bubbles generated from the periodical reac-tion of alumina in the cryolite from feeders and surroundedby a fully dense skin called the top crust The density in themultiphase fluid flows ranges from 04 to 08 gcm3 whichcorresponds to a relative density 120588

119903of 15ndash30

Focusing on the actual electrolysis process five expansionstages dependent on the temperature are identified inFigure 1 thermal expansion of the precursor (0) bubblenucleation under surface of anodes (I) expansion in thesemisolid and liquid range (II) expansion in the liquid tem-perature range (III) and finally the gas bubbles collapse (IV)

0

300 400 500 600 700 800

Expansion of

Oxide carbonrelease from

Expa

nsio

n fa

ctor

IV

III

II

I

Temperature (∘C)

Figure 1 Schematic expansion of gas divided into five regimesPhase (0) depicts the thermal expansion of the precursor phase (I)denotes the bubbles nucleation and expansion under surface ofanodes the expansion in the semisolid and liquid range is in phase(II) phase (III) is the expansion in the liquid temperature range andfinally phase (IV) is the gas bubbles collapse

The beginning of the gas formation regime 0ndashII inFigure 1 is marked by decomposition of the alumina meltingthe precursor and nucleation The decomposition of about6wtup to 960∘C releases approximately a gas volumewhichexceeds 10 times the precursor volume The decompositionbegins while the aluminamatrix is still solid (at about 950∘C)Partial pressures in the order of thematrix yield strengthmayarise and tear open the powder compact precursor

The crack-like openings in the carbon anode are problem-atic because the CO

2can percolate through these channels

to the ambience being lost for the multiphase system Inaddition to these cracks the small pores over the anode serveas nuclei when the alumina melts In the melt heterogeneousnucleationmay occur at nucleation sites such as impurities orpropellant particles

The major gas bubbles expansion takes place while thecryolite is liquid regime III-IV in Figure 1 The strongincrease in expansion relies on the substantial oxide carbonrelease during the considered temperature intervalThe oxidecarbon dissolves and diffuses into numerous nuclei whichconsequently expand At a certain point the gas bubblesimpinge upon each other They must give up their favorableshapeThin fluid films separate the bubblesThe variations ofgas bubble sizes and thus interior pressure aswell as precedingfilm ruptures lead to bent and corrugated cell walls

The stabilization of the gas bubbles films is essen-tial Without stable cell walls bubbles would immediatelyundergo coalescencewhen they contact each other By coales-cencewith the atmosphere enforced by buoyancy the bubblesand consequently massive gas portions from alumina arelost from the system A typical gas bubbles structure cannotevolve The gas bubbles may owe stable films to the presenceof additives such as oxides Since the alumina possesses oxidethat oxide naturally resides as a network in the precursorThe repulsive force is termed disjoining pressure in thefollowing Additionally network fragments make thin filmsless penetrable to the melt so that the cryolite remains in the









119903



3 Physical Model




120597119888

120597119905+ 119906120572

120597119888

120597120572minus120597119863

120597120572

120597119888

120597120572= 119878 (119905) (1)



119888 = 119896119878radic119901119892119894

(2)


119904is


(120597119909

120597119910

) 119888 prop 119899 (3)



Gas-liquidmixture

Cathodeblock

Anodes

contacts

Feeder

xD

xA

xB

xC





120597119906120572

120597120572= 0

120597119906120572

120597119905+120597119906120573

120597120573

120597119906120572

120597120572= minus

120597119875120597120572

120588+ ]1205972120573119906120572+ 119892120572

(4)


119901 minus 2120588]120597119906119899

120597119899= 119901119894minus 120590120581 minus Π (5)

120597119906119905

120597119899+120597119906119899

120597119905= 0 (6)


119894


119901119894=119899119894119877119879

119881119894

(7)






119889119881119894

119889119905= minusint120597Ω119894

119906119899119889119908 at 120597Ω

119894 (8)


119889119899119894

119889119905= minusint120597Ω119894

119863120597119888

120597119899119889119908 (9)






0 119889 ge 119889max(10)











120588gas

119889120588gas

119889119905(119882)


(11)




1

120588gas

119889120588gas

119889119905= minus

1

1 minus119882

119889119882

119889119905 (13)


119889119882

119889119905=120597119882







1 minus119882119891(119882 119901gas minus 119901)


(15)



119888Ω



10038161003816100381610038161003816Ωgas

10038161003816100381610038161003816=

119882119888

1 minus119882119888

10038161003816100381610038161003816Ωliq

10038161003816100381610038161003816 (16)





10038161003816100381610038161003816Ωgas

10038161003816100381610038161003816=

119899

sum

119894=0

10038161003816100381610038161003816Ω119892119894

10038161003816100381610038161003816=4120587

3

119899

sum

119894=1

1198773

119894 (17)



110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

sum119899

119894=11198773

119894

119899

sum

119894=1

1198772

119894 (18)


110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

4120578119900sum119899

119894=11198773

119894

119899

sum

119894=1

(119901gas119894

minus 119901ext) 1198773

119894 (19)


110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

41205780

(119901gas minus 119901ext) (20)


119889119882119888

119889119905=

3

41205780





119901gas =1 minus119882

119888

119882119888

1198821198880

1 minus1198821198880

119875gas0 (22)




120597119882

120597119905+ V sdot nabla119882 =

3

41205780

[(1 minus119882)2 1198820

1 minus1198820

119875gas0


(23)






119894( 119905) represent


velocities

120585119894=Δ119909

Δ119905

(0 0) 119894 = 0

(plusmn1 0) (0 plusmn1) 119894 = 1 4


(24)



119891(0)



119891119894( + 120585

119894 119905 + Δ119905) minus 119891

119894 ( 119905)

= minusΔ119905

120591(119891119894( 119905) minus 119891

(0)

119894( 119905)) + Δ119905119865

119894

(25)


120588 ( 119905) = sum

119894

119891119894( 119905)

120588 ( 119905) = sum

119894

120585119894119891119894( 119905)

(26)


119891(0)

119894= 119908119894120588(1 +

120585119894120572119906120572

1198882119904

+(120585119894120572119906120572)2

21198884119904

minus119906120572119906120572

21198882119904

) (27)

119908119894=

4

9119894 = 0

1

9119894 = 1 4

1

36119894 = 5 8

(28)




119865119894= 119908119894120588[

120585119894120572minus 119906120572

1198882119904

+(120585119894120573119906120573) 120585119894120572

1198884119904

] (119892120572+ 119895 and ) (29)


2

119904 and


119891119894= 119891(0)

119894+ isin 119891

(1)

119894+ isin2119891(2)

119894+ 119874 (isin

3)

119865119894= isin 119865(1)

119894+ 119874 (isin

2)

120597119905= isin 120597(1)

119905+ 1205762120597(2)

119905+ 119874 (isin

2)

120597120572= isin 120597(1)

120572+ 119874 (isin

2)

(30)


sum

119894

[1

120585119894

]119891(0)

119894= 120588 [

1

]

sum

119894

[1

120585119894

]119891(119899)

119894= 120588 [

0

0] for 119899 gt 0

(31)


120597120588

120597119905+120597120588V120572

120597120572= 0

120597120588V120572

120597119905+120597120588V120573V120572

120597120573= minus

120597119875

120597120572+ V120588

120597

120597120573

120597

120597120573V120572+ 120588119892120572+ 119895120572otimes 119861120573

(32)

where V = (1198882119904Δ1199052)(2120591Δ119905 minus 1)


119865 bubble

119862119861 interface 119862




119894all


119891(1)

119894= 119908119894120591(120597120572(120588119906120572) minus

1

1198882119904

120585119894120572120585119894120573120597120572(120588119906120573)) (33)

Interface

FluidBubble

120575

n

t


In119891(0)119894

(27)119891(1)119894





(0)

119894are


119891119894= 119908119894(120588 +

120585120572120588119906120572

1198882119904

+ 120591((1 minus1205852

119894119899

1198882119904

)

1198882

119904120588 minus 119875119868

119895

2V)

+ (1 minus1205852

119894119905

1198882119904

)120588120597119905119906119905)

(34)




ΔΛ119894 () = Φ (119891119894 ( + 119890

119894) minus 119891119894 ()) (35)

With

Φ =

1 + 119890119894isin 119862119865

Φ () + Φ ( + 119890119894)

2 + 119890119894isin 119862119868

0 + 119890119894isin 119862119861

(36)







119868119895of the bubble 119895

Δ119881119895= minus sum

119909isin119862119868119895

ΔΛ () (37)












1198880 as well and



1E minus 01

1E minus 02

1E minus 03

1E minus 04

1E minus 05001 01 1

Curvature K

120575 = 005

120575 = 001

120575 = 0005

120575 = 001

(Pga

sminus

Pliq

)


Gas rate0 01 02 03 04 05 06 07 08 09 1

0

01

02

03

04

05

06

07

08

09

1

Visc

osity

of m

ixtu

rec

onsis

tenc

y of

liqu

id





119888


0

01

02

03

04

05

06

07

08

09

1



Gas

rate

0 100 200 300 400 500 600 700

Time (s)


0 200 400 600 800 1000 1200 1400


Para

met

er11

1

09

08

07

06

05

04

03

02



0



09

08

07

06

05

04

03

02

01

00 05 1 15 2 25 3 35 4 45 5

Gas

frac

tion

Time (s)

Site ASite B

Site CSite D

(a)

09

08

07

06

05

04

03

02

010 05 1

1

15 2 25 3 35 4 45 5

Time (s)

Site ASite B

Site CSite D

Visc

osity

of m

ixtu

rec

onsis

tenc

e of l

iqui

d

(b)

Velo

city

(mm

s)

Site ASite B

Site CSite D

0 05 1 15 2 25 3 35 4 45 5

Time (s)

35

30

25

20

15

10

5

0

(c)

Pres

sure

(Pa)

0 05 1 15 2 25 3 35 4 45 5

Time (s)

001

0008

0006

0004

0002

0

minus0002

Site ASite B

Site CSite D

(d)




7 Conclusion






Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials











119903



3 Physical Model




120597119888

120597119905+ 119906120572

120597119888

120597120572minus120597119863

120597120572

120597119888

120597120572= 119878 (119905) (1)



119888 = 119896119878radic119901119892119894

(2)


119904is


(120597119909

120597119910

) 119888 prop 119899 (3)



Gas-liquidmixture

Cathodeblock

Anodes

contacts

Feeder

xD

xA

xB

xC





120597119906120572

120597120572= 0

120597119906120572

120597119905+120597119906120573

120597120573

120597119906120572

120597120572= minus

120597119875120597120572

120588+ ]1205972120573119906120572+ 119892120572

(4)


119901 minus 2120588]120597119906119899

120597119899= 119901119894minus 120590120581 minus Π (5)

120597119906119905

120597119899+120597119906119899

120597119905= 0 (6)


119894


119901119894=119899119894119877119879

119881119894

(7)






119889119881119894

119889119905= minusint120597Ω119894

119906119899119889119908 at 120597Ω

119894 (8)


119889119899119894

119889119905= minusint120597Ω119894

119863120597119888

120597119899119889119908 (9)






0 119889 ge 119889max(10)











120588gas

119889120588gas

119889119905(119882)


(11)




1

120588gas

119889120588gas

119889119905= minus

1

1 minus119882

119889119882

119889119905 (13)


119889119882

119889119905=120597119882







1 minus119882119891(119882 119901gas minus 119901)


(15)



119888Ω



10038161003816100381610038161003816Ωgas

10038161003816100381610038161003816=

119882119888

1 minus119882119888

10038161003816100381610038161003816Ωliq

10038161003816100381610038161003816 (16)





10038161003816100381610038161003816Ωgas

10038161003816100381610038161003816=

119899

sum

119894=0

10038161003816100381610038161003816Ω119892119894

10038161003816100381610038161003816=4120587

3

119899

sum

119894=1

1198773

119894 (17)



110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

sum119899

119894=11198773

119894

119899

sum

119894=1

1198772

119894 (18)


110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

4120578119900sum119899

119894=11198773

119894

119899

sum

119894=1

(119901gas119894

minus 119901ext) 1198773

119894 (19)


110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

41205780

(119901gas minus 119901ext) (20)


119889119882119888

119889119905=

3

41205780





119901gas =1 minus119882

119888

119882119888

1198821198880

1 minus1198821198880

119875gas0 (22)




120597119882

120597119905+ V sdot nabla119882 =

3

41205780

[(1 minus119882)2 1198820

1 minus1198820

119875gas0


(23)






119894( 119905) represent


velocities

120585119894=Δ119909

Δ119905

(0 0) 119894 = 0

(plusmn1 0) (0 plusmn1) 119894 = 1 4


(24)



119891(0)



119891119894( + 120585

119894 119905 + Δ119905) minus 119891

119894 ( 119905)

= minusΔ119905

120591(119891119894( 119905) minus 119891

(0)

119894( 119905)) + Δ119905119865

119894

(25)


120588 ( 119905) = sum

119894

119891119894( 119905)

120588 ( 119905) = sum

119894

120585119894119891119894( 119905)

(26)


119891(0)

119894= 119908119894120588(1 +

120585119894120572119906120572

1198882119904

+(120585119894120572119906120572)2

21198884119904

minus119906120572119906120572

21198882119904

) (27)

119908119894=

4

9119894 = 0

1

9119894 = 1 4

1

36119894 = 5 8

(28)




119865119894= 119908119894120588[

120585119894120572minus 119906120572

1198882119904

+(120585119894120573119906120573) 120585119894120572

1198884119904

] (119892120572+ 119895 and ) (29)


2

119904 and


119891119894= 119891(0)

119894+ isin 119891

(1)

119894+ isin2119891(2)

119894+ 119874 (isin

3)

119865119894= isin 119865(1)

119894+ 119874 (isin

2)

120597119905= isin 120597(1)

119905+ 1205762120597(2)

119905+ 119874 (isin

2)

120597120572= isin 120597(1)

120572+ 119874 (isin

2)

(30)


sum

119894

[1

120585119894

]119891(0)

119894= 120588 [

1

]

sum

119894

[1

120585119894

]119891(119899)

119894= 120588 [

0

0] for 119899 gt 0

(31)


120597120588

120597119905+120597120588V120572

120597120572= 0

120597120588V120572

120597119905+120597120588V120573V120572

120597120573= minus

120597119875

120597120572+ V120588

120597

120597120573

120597

120597120573V120572+ 120588119892120572+ 119895120572otimes 119861120573

(32)

where V = (1198882119904Δ1199052)(2120591Δ119905 minus 1)


119865 bubble

119862119861 interface 119862




119894all


119891(1)

119894= 119908119894120591(120597120572(120588119906120572) minus

1

1198882119904

120585119894120572120585119894120573120597120572(120588119906120573)) (33)

Interface

FluidBubble

120575

n

t


In119891(0)119894

(27)119891(1)119894





(0)

119894are


119891119894= 119908119894(120588 +

120585120572120588119906120572

1198882119904

+ 120591((1 minus1205852

119894119899

1198882119904

)

1198882

119904120588 minus 119875119868

119895

2V)

+ (1 minus1205852

119894119905

1198882119904

)120588120597119905119906119905)

(34)




ΔΛ119894 () = Φ (119891119894 ( + 119890

119894) minus 119891119894 ()) (35)

With

Φ =

1 + 119890119894isin 119862119865

Φ () + Φ ( + 119890119894)

2 + 119890119894isin 119862119868

0 + 119890119894isin 119862119861

(36)







119868119895of the bubble 119895

Δ119881119895= minus sum

119909isin119862119868119895

ΔΛ () (37)












1198880 as well and



1E minus 01

1E minus 02

1E minus 03

1E minus 04

1E minus 05001 01 1

Curvature K

120575 = 005

120575 = 001

120575 = 0005

120575 = 001

(Pga

sminus

Pliq

)


Gas rate0 01 02 03 04 05 06 07 08 09 1

0

01

02

03

04

05

06

07

08

09

1

Visc

osity

of m

ixtu

rec

onsis

tenc

y of

liqu

id





119888


0

01

02

03

04

05

06

07

08

09

1



Gas

rate

0 100 200 300 400 500 600 700

Time (s)


0 200 400 600 800 1000 1200 1400


Para

met

er11

1

09

08

07

06

05

04

03

02



0



09

08

07

06

05

04

03

02

01

00 05 1 15 2 25 3 35 4 45 5

Gas

frac

tion

Time (s)

Site ASite B

Site CSite D

(a)

09

08

07

06

05

04

03

02

010 05 1

1

15 2 25 3 35 4 45 5

Time (s)

Site ASite B

Site CSite D

Visc

osity

of m

ixtu

rec

onsis

tenc

e of l

iqui

d

(b)

Velo

city

(mm

s)

Site ASite B

Site CSite D

0 05 1 15 2 25 3 35 4 45 5

Time (s)

35

30

25

20

15

10

5

0

(c)

Pres

sure

(Pa)

0 05 1 15 2 25 3 35 4 45 5

Time (s)

001

0008

0006

0004

0002

0

minus0002

Site ASite B

Site CSite D

(d)




7 Conclusion






Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





Gas-liquidmixture

Cathodeblock

Anodes

contacts

Feeder

xD

xA

xB

xC





120597119906120572

120597120572= 0

120597119906120572

120597119905+120597119906120573

120597120573

120597119906120572

120597120572= minus

120597119875120597120572

120588+ ]1205972120573119906120572+ 119892120572

(4)


119901 minus 2120588]120597119906119899

120597119899= 119901119894minus 120590120581 minus Π (5)

120597119906119905

120597119899+120597119906119899

120597119905= 0 (6)


119894


119901119894=119899119894119877119879

119881119894

(7)






119889119881119894

119889119905= minusint120597Ω119894

119906119899119889119908 at 120597Ω

119894 (8)


119889119899119894

119889119905= minusint120597Ω119894

119863120597119888

120597119899119889119908 (9)






0 119889 ge 119889max(10)











120588gas

119889120588gas

119889119905(119882)


(11)




1

120588gas

119889120588gas

119889119905= minus

1

1 minus119882

119889119882

119889119905 (13)


119889119882

119889119905=120597119882







1 minus119882119891(119882 119901gas minus 119901)


(15)



119888Ω



10038161003816100381610038161003816Ωgas

10038161003816100381610038161003816=

119882119888

1 minus119882119888

10038161003816100381610038161003816Ωliq

10038161003816100381610038161003816 (16)





10038161003816100381610038161003816Ωgas

10038161003816100381610038161003816=

119899

sum

119894=0

10038161003816100381610038161003816Ω119892119894

10038161003816100381610038161003816=4120587

3

119899

sum

119894=1

1198773

119894 (17)



110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

sum119899

119894=11198773

119894

119899

sum

119894=1

1198772

119894 (18)


110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

4120578119900sum119899

119894=11198773

119894

119899

sum

119894=1

(119901gas119894

minus 119901ext) 1198773

119894 (19)


110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

41205780

(119901gas minus 119901ext) (20)


119889119882119888

119889119905=

3

41205780





119901gas =1 minus119882

119888

119882119888

1198821198880

1 minus1198821198880

119875gas0 (22)




120597119882

120597119905+ V sdot nabla119882 =

3

41205780

[(1 minus119882)2 1198820

1 minus1198820

119875gas0


(23)






119894( 119905) represent


velocities

120585119894=Δ119909

Δ119905

(0 0) 119894 = 0

(plusmn1 0) (0 plusmn1) 119894 = 1 4


(24)



119891(0)



119891119894( + 120585

119894 119905 + Δ119905) minus 119891

119894 ( 119905)

= minusΔ119905

120591(119891119894( 119905) minus 119891

(0)

119894( 119905)) + Δ119905119865

119894

(25)


120588 ( 119905) = sum

119894

119891119894( 119905)

120588 ( 119905) = sum

119894

120585119894119891119894( 119905)

(26)


119891(0)

119894= 119908119894120588(1 +

120585119894120572119906120572

1198882119904

+(120585119894120572119906120572)2

21198884119904

minus119906120572119906120572

21198882119904

) (27)

119908119894=

4

9119894 = 0

1

9119894 = 1 4

1

36119894 = 5 8

(28)




119865119894= 119908119894120588[

120585119894120572minus 119906120572

1198882119904

+(120585119894120573119906120573) 120585119894120572

1198884119904

] (119892120572+ 119895 and ) (29)


2

119904 and


119891119894= 119891(0)

119894+ isin 119891

(1)

119894+ isin2119891(2)

119894+ 119874 (isin

3)

119865119894= isin 119865(1)

119894+ 119874 (isin

2)

120597119905= isin 120597(1)

119905+ 1205762120597(2)

119905+ 119874 (isin

2)

120597120572= isin 120597(1)

120572+ 119874 (isin

2)

(30)


sum

119894

[1

120585119894

]119891(0)

119894= 120588 [

1

]

sum

119894

[1

120585119894

]119891(119899)

119894= 120588 [

0

0] for 119899 gt 0

(31)


120597120588

120597119905+120597120588V120572

120597120572= 0

120597120588V120572

120597119905+120597120588V120573V120572

120597120573= minus

120597119875

120597120572+ V120588

120597

120597120573

120597

120597120573V120572+ 120588119892120572+ 119895120572otimes 119861120573

(32)

where V = (1198882119904Δ1199052)(2120591Δ119905 minus 1)


119865 bubble

119862119861 interface 119862




119894all


119891(1)

119894= 119908119894120591(120597120572(120588119906120572) minus

1

1198882119904

120585119894120572120585119894120573120597120572(120588119906120573)) (33)

Interface

FluidBubble

120575

n

t


In119891(0)119894

(27)119891(1)119894





(0)

119894are


119891119894= 119908119894(120588 +

120585120572120588119906120572

1198882119904

+ 120591((1 minus1205852

119894119899

1198882119904

)

1198882

119904120588 minus 119875119868

119895

2V)

+ (1 minus1205852

119894119905

1198882119904

)120588120597119905119906119905)

(34)




ΔΛ119894 () = Φ (119891119894 ( + 119890

119894) minus 119891119894 ()) (35)

With

Φ =

1 + 119890119894isin 119862119865

Φ () + Φ ( + 119890119894)

2 + 119890119894isin 119862119868

0 + 119890119894isin 119862119861

(36)







119868119895of the bubble 119895

Δ119881119895= minus sum

119909isin119862119868119895

ΔΛ () (37)












1198880 as well and



1E minus 01

1E minus 02

1E minus 03

1E minus 04

1E minus 05001 01 1

Curvature K

120575 = 005

120575 = 001

120575 = 0005

120575 = 001

(Pga

sminus

Pliq

)


Gas rate0 01 02 03 04 05 06 07 08 09 1

0

01

02

03

04

05

06

07

08

09

1

Visc

osity

of m

ixtu

rec

onsis

tenc

y of

liqu

id





119888


0

01

02

03

04

05

06

07

08

09

1



Gas

rate

0 100 200 300 400 500 600 700

Time (s)


0 200 400 600 800 1000 1200 1400


Para

met

er11

1

09

08

07

06

05

04

03

02



0



09

08

07

06

05

04

03

02

01

00 05 1 15 2 25 3 35 4 45 5

Gas

frac

tion

Time (s)

Site ASite B

Site CSite D

(a)

09

08

07

06

05

04

03

02

010 05 1

1

15 2 25 3 35 4 45 5

Time (s)

Site ASite B

Site CSite D

Visc

osity

of m

ixtu

rec

onsis

tenc

e of l

iqui

d

(b)

Velo

city

(mm

s)

Site ASite B

Site CSite D

0 05 1 15 2 25 3 35 4 45 5

Time (s)

35

30

25

20

15

10

5

0

(c)

Pres

sure

(Pa)

0 05 1 15 2 25 3 35 4 45 5

Time (s)

001

0008

0006

0004

0002

0

minus0002

Site ASite B

Site CSite D

(d)




7 Conclusion






Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials












120588gas

119889120588gas

119889119905(119882)


(11)




1

120588gas

119889120588gas

119889119905= minus

1

1 minus119882

119889119882

119889119905 (13)


119889119882

119889119905=120597119882







1 minus119882119891(119882 119901gas minus 119901)


(15)



119888Ω



10038161003816100381610038161003816Ωgas

10038161003816100381610038161003816=

119882119888

1 minus119882119888

10038161003816100381610038161003816Ωliq

10038161003816100381610038161003816 (16)





10038161003816100381610038161003816Ωgas

10038161003816100381610038161003816=

119899

sum

119894=0

10038161003816100381610038161003816Ω119892119894

10038161003816100381610038161003816=4120587

3

119899

sum

119894=1

1198773

119894 (17)



110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

sum119899

119894=11198773

119894

119899

sum

119894=1

1198772

119894 (18)


110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

4120578119900sum119899

119894=11198773

119894

119899

sum

119894=1

(119901gas119894

minus 119901ext) 1198773

119894 (19)


110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

41205780

(119901gas minus 119901ext) (20)


119889119882119888

119889119905=

3

41205780





119901gas =1 minus119882

119888

119882119888

1198821198880

1 minus1198821198880

119875gas0 (22)




120597119882

120597119905+ V sdot nabla119882 =

3

41205780

[(1 minus119882)2 1198820

1 minus1198820

119875gas0


(23)






119894( 119905) represent


velocities

120585119894=Δ119909

Δ119905

(0 0) 119894 = 0

(plusmn1 0) (0 plusmn1) 119894 = 1 4


(24)



119891(0)



119891119894( + 120585

119894 119905 + Δ119905) minus 119891

119894 ( 119905)

= minusΔ119905

120591(119891119894( 119905) minus 119891

(0)

119894( 119905)) + Δ119905119865

119894

(25)


120588 ( 119905) = sum

119894

119891119894( 119905)

120588 ( 119905) = sum

119894

120585119894119891119894( 119905)

(26)


119891(0)

119894= 119908119894120588(1 +

120585119894120572119906120572

1198882119904

+(120585119894120572119906120572)2

21198884119904

minus119906120572119906120572

21198882119904

) (27)

119908119894=

4

9119894 = 0

1

9119894 = 1 4

1

36119894 = 5 8

(28)




119865119894= 119908119894120588[

120585119894120572minus 119906120572

1198882119904

+(120585119894120573119906120573) 120585119894120572

1198884119904

] (119892120572+ 119895 and ) (29)


2

119904 and


119891119894= 119891(0)

119894+ isin 119891

(1)

119894+ isin2119891(2)

119894+ 119874 (isin

3)

119865119894= isin 119865(1)

119894+ 119874 (isin

2)

120597119905= isin 120597(1)

119905+ 1205762120597(2)

119905+ 119874 (isin

2)

120597120572= isin 120597(1)

120572+ 119874 (isin

2)

(30)


sum

119894

[1

120585119894

]119891(0)

119894= 120588 [

1

]

sum

119894

[1

120585119894

]119891(119899)

119894= 120588 [

0

0] for 119899 gt 0

(31)


120597120588

120597119905+120597120588V120572

120597120572= 0

120597120588V120572

120597119905+120597120588V120573V120572

120597120573= minus

120597119875

120597120572+ V120588

120597

120597120573

120597

120597120573V120572+ 120588119892120572+ 119895120572otimes 119861120573

(32)

where V = (1198882119904Δ1199052)(2120591Δ119905 minus 1)


119865 bubble

119862119861 interface 119862




119894all


119891(1)

119894= 119908119894120591(120597120572(120588119906120572) minus

1

1198882119904

120585119894120572120585119894120573120597120572(120588119906120573)) (33)

Interface

FluidBubble

120575

n

t


In119891(0)119894

(27)119891(1)119894





(0)

119894are


119891119894= 119908119894(120588 +

120585120572120588119906120572

1198882119904

+ 120591((1 minus1205852

119894119899

1198882119904

)

1198882

119904120588 minus 119875119868

119895

2V)

+ (1 minus1205852

119894119905

1198882119904

)120588120597119905119906119905)

(34)




ΔΛ119894 () = Φ (119891119894 ( + 119890

119894) minus 119891119894 ()) (35)

With

Φ =

1 + 119890119894isin 119862119865

Φ () + Φ ( + 119890119894)

2 + 119890119894isin 119862119868

0 + 119890119894isin 119862119861

(36)







119868119895of the bubble 119895

Δ119881119895= minus sum

119909isin119862119868119895

ΔΛ () (37)












1198880 as well and



1E minus 01

1E minus 02

1E minus 03

1E minus 04

1E minus 05001 01 1

Curvature K

120575 = 005

120575 = 001

120575 = 0005

120575 = 001

(Pga

sminus

Pliq

)


Gas rate0 01 02 03 04 05 06 07 08 09 1

0

01

02

03

04

05

06

07

08

09

1

Visc

osity

of m

ixtu

rec

onsis

tenc

y of

liqu

id





119888


0

01

02

03

04

05

06

07

08

09

1



Gas

rate

0 100 200 300 400 500 600 700

Time (s)


0 200 400 600 800 1000 1200 1400


Para

met

er11

1

09

08

07

06

05

04

03

02



0



09

08

07

06

05

04

03

02

01

00 05 1 15 2 25 3 35 4 45 5

Gas

frac

tion

Time (s)

Site ASite B

Site CSite D

(a)

09

08

07

06

05

04

03

02

010 05 1

1

15 2 25 3 35 4 45 5

Time (s)

Site ASite B

Site CSite D

Visc

osity

of m

ixtu

rec

onsis

tenc

e of l

iqui

d

(b)

Velo

city

(mm

s)

Site ASite B

Site CSite D

0 05 1 15 2 25 3 35 4 45 5

Time (s)

35

30

25

20

15

10

5

0

(c)

Pres

sure

(Pa)

0 05 1 15 2 25 3 35 4 45 5

Time (s)

001

0008

0006

0004

0002

0

minus0002

Site ASite B

Site CSite D

(d)




7 Conclusion






Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

sum119899

119894=11198773

119894

119899

sum

119894=1

1198772

119894 (18)


110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

4120578119900sum119899

119894=11198773

119894

119899

sum

119894=1

(119901gas119894

minus 119901ext) 1198773

119894 (19)


110038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

11988910038161003816100381610038161003816Ωgas

10038161003816100381610038161003816

119889119905=

3

41205780

(119901gas minus 119901ext) (20)


119889119882119888

119889119905=

3

41205780





119901gas =1 minus119882

119888

119882119888

1198821198880

1 minus1198821198880

119875gas0 (22)




120597119882

120597119905+ V sdot nabla119882 =

3

41205780

[(1 minus119882)2 1198820

1 minus1198820

119875gas0


(23)






119894( 119905) represent


velocities

120585119894=Δ119909

Δ119905

(0 0) 119894 = 0

(plusmn1 0) (0 plusmn1) 119894 = 1 4


(24)



119891(0)



119891119894( + 120585

119894 119905 + Δ119905) minus 119891

119894 ( 119905)

= minusΔ119905

120591(119891119894( 119905) minus 119891

(0)

119894( 119905)) + Δ119905119865

119894

(25)


120588 ( 119905) = sum

119894

119891119894( 119905)

120588 ( 119905) = sum

119894

120585119894119891119894( 119905)

(26)


119891(0)

119894= 119908119894120588(1 +

120585119894120572119906120572

1198882119904

+(120585119894120572119906120572)2

21198884119904

minus119906120572119906120572

21198882119904

) (27)

119908119894=

4

9119894 = 0

1

9119894 = 1 4

1

36119894 = 5 8

(28)




119865119894= 119908119894120588[

120585119894120572minus 119906120572

1198882119904

+(120585119894120573119906120573) 120585119894120572

1198884119904

] (119892120572+ 119895 and ) (29)


2

119904 and


119891119894= 119891(0)

119894+ isin 119891

(1)

119894+ isin2119891(2)

119894+ 119874 (isin

3)

119865119894= isin 119865(1)

119894+ 119874 (isin

2)

120597119905= isin 120597(1)

119905+ 1205762120597(2)

119905+ 119874 (isin

2)

120597120572= isin 120597(1)

120572+ 119874 (isin

2)

(30)


sum

119894

[1

120585119894

]119891(0)

119894= 120588 [

1

]

sum

119894

[1

120585119894

]119891(119899)

119894= 120588 [

0

0] for 119899 gt 0

(31)


120597120588

120597119905+120597120588V120572

120597120572= 0

120597120588V120572

120597119905+120597120588V120573V120572

120597120573= minus

120597119875

120597120572+ V120588

120597

120597120573

120597

120597120573V120572+ 120588119892120572+ 119895120572otimes 119861120573

(32)

where V = (1198882119904Δ1199052)(2120591Δ119905 minus 1)


119865 bubble

119862119861 interface 119862




119894all


119891(1)

119894= 119908119894120591(120597120572(120588119906120572) minus

1

1198882119904

120585119894120572120585119894120573120597120572(120588119906120573)) (33)

Interface

FluidBubble

120575

n

t


In119891(0)119894

(27)119891(1)119894





(0)

119894are


119891119894= 119908119894(120588 +

120585120572120588119906120572

1198882119904

+ 120591((1 minus1205852

119894119899

1198882119904

)

1198882

119904120588 minus 119875119868

119895

2V)

+ (1 minus1205852

119894119905

1198882119904

)120588120597119905119906119905)

(34)




ΔΛ119894 () = Φ (119891119894 ( + 119890

119894) minus 119891119894 ()) (35)

With

Φ =

1 + 119890119894isin 119862119865

Φ () + Φ ( + 119890119894)

2 + 119890119894isin 119862119868

0 + 119890119894isin 119862119861

(36)







119868119895of the bubble 119895

Δ119881119895= minus sum

119909isin119862119868119895

ΔΛ () (37)












1198880 as well and



1E minus 01

1E minus 02

1E minus 03

1E minus 04

1E minus 05001 01 1

Curvature K

120575 = 005

120575 = 001

120575 = 0005

120575 = 001

(Pga

sminus

Pliq

)


Gas rate0 01 02 03 04 05 06 07 08 09 1

0

01

02

03

04

05

06

07

08

09

1

Visc

osity

of m

ixtu

rec

onsis

tenc

y of

liqu

id





119888


0

01

02

03

04

05

06

07

08

09

1



Gas

rate

0 100 200 300 400 500 600 700

Time (s)


0 200 400 600 800 1000 1200 1400


Para

met

er11

1

09

08

07

06

05

04

03

02



0



09

08

07

06

05

04

03

02

01

00 05 1 15 2 25 3 35 4 45 5

Gas

frac

tion

Time (s)

Site ASite B

Site CSite D

(a)

09

08

07

06

05

04

03

02

010 05 1

1

15 2 25 3 35 4 45 5

Time (s)

Site ASite B

Site CSite D

Visc

osity

of m

ixtu

rec

onsis

tenc

e of l

iqui

d

(b)

Velo

city

(mm

s)

Site ASite B

Site CSite D

0 05 1 15 2 25 3 35 4 45 5

Time (s)

35

30

25

20

15

10

5

0

(c)

Pres

sure

(Pa)

0 05 1 15 2 25 3 35 4 45 5

Time (s)

001

0008

0006

0004

0002

0

minus0002

Site ASite B

Site CSite D

(d)




7 Conclusion






Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





119865119894= 119908119894120588[

120585119894120572minus 119906120572

1198882119904

+(120585119894120573119906120573) 120585119894120572

1198884119904

] (119892120572+ 119895 and ) (29)


2

119904 and


119891119894= 119891(0)

119894+ isin 119891

(1)

119894+ isin2119891(2)

119894+ 119874 (isin

3)

119865119894= isin 119865(1)

119894+ 119874 (isin

2)

120597119905= isin 120597(1)

119905+ 1205762120597(2)

119905+ 119874 (isin

2)

120597120572= isin 120597(1)

120572+ 119874 (isin

2)

(30)


sum

119894

[1

120585119894

]119891(0)

119894= 120588 [

1

]

sum

119894

[1

120585119894

]119891(119899)

119894= 120588 [

0

0] for 119899 gt 0

(31)


120597120588

120597119905+120597120588V120572

120597120572= 0

120597120588V120572

120597119905+120597120588V120573V120572

120597120573= minus

120597119875

120597120572+ V120588

120597

120597120573

120597

120597120573V120572+ 120588119892120572+ 119895120572otimes 119861120573

(32)

where V = (1198882119904Δ1199052)(2120591Δ119905 minus 1)


119865 bubble

119862119861 interface 119862




119894all


119891(1)

119894= 119908119894120591(120597120572(120588119906120572) minus

1

1198882119904

120585119894120572120585119894120573120597120572(120588119906120573)) (33)

Interface

FluidBubble

120575

n

t


In119891(0)119894

(27)119891(1)119894





(0)

119894are


119891119894= 119908119894(120588 +

120585120572120588119906120572

1198882119904

+ 120591((1 minus1205852

119894119899

1198882119904

)

1198882

119904120588 minus 119875119868

119895

2V)

+ (1 minus1205852

119894119905

1198882119904

)120588120597119905119906119905)

(34)




ΔΛ119894 () = Φ (119891119894 ( + 119890

119894) minus 119891119894 ()) (35)

With

Φ =

1 + 119890119894isin 119862119865

Φ () + Φ ( + 119890119894)

2 + 119890119894isin 119862119868

0 + 119890119894isin 119862119861

(36)







119868119895of the bubble 119895

Δ119881119895= minus sum

119909isin119862119868119895

ΔΛ () (37)












1198880 as well and



1E minus 01

1E minus 02

1E minus 03

1E minus 04

1E minus 05001 01 1

Curvature K

120575 = 005

120575 = 001

120575 = 0005

120575 = 001

(Pga

sminus

Pliq

)


Gas rate0 01 02 03 04 05 06 07 08 09 1

0

01

02

03

04

05

06

07

08

09

1

Visc

osity

of m

ixtu

rec

onsis

tenc

y of

liqu

id





119888


0

01

02

03

04

05

06

07

08

09

1



Gas

rate

0 100 200 300 400 500 600 700

Time (s)


0 200 400 600 800 1000 1200 1400


Para

met

er11

1

09

08

07

06

05

04

03

02



0



09

08

07

06

05

04

03

02

01

00 05 1 15 2 25 3 35 4 45 5

Gas

frac

tion

Time (s)

Site ASite B

Site CSite D

(a)

09

08

07

06

05

04

03

02

010 05 1

1

15 2 25 3 35 4 45 5

Time (s)

Site ASite B

Site CSite D

Visc

osity

of m

ixtu

rec

onsis

tenc

e of l

iqui

d

(b)

Velo

city

(mm

s)

Site ASite B

Site CSite D

0 05 1 15 2 25 3 35 4 45 5

Time (s)

35

30

25

20

15

10

5

0

(c)

Pres

sure

(Pa)

0 05 1 15 2 25 3 35 4 45 5

Time (s)

001

0008

0006

0004

0002

0

minus0002

Site ASite B

Site CSite D

(d)




7 Conclusion






Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials









119868119895of the bubble 119895

Δ119881119895= minus sum

119909isin119862119868119895

ΔΛ () (37)












1198880 as well and



1E minus 01

1E minus 02

1E minus 03

1E minus 04

1E minus 05001 01 1

Curvature K

120575 = 005

120575 = 001

120575 = 0005

120575 = 001

(Pga

sminus

Pliq

)


Gas rate0 01 02 03 04 05 06 07 08 09 1

0

01

02

03

04

05

06

07

08

09

1

Visc

osity

of m

ixtu

rec

onsis

tenc

y of

liqu

id





119888


0

01

02

03

04

05

06

07

08

09

1



Gas

rate

0 100 200 300 400 500 600 700

Time (s)


0 200 400 600 800 1000 1200 1400


Para

met

er11

1

09

08

07

06

05

04

03

02



0



09

08

07

06

05

04

03

02

01

00 05 1 15 2 25 3 35 4 45 5

Gas

frac

tion

Time (s)

Site ASite B

Site CSite D

(a)

09

08

07

06

05

04

03

02

010 05 1

1

15 2 25 3 35 4 45 5

Time (s)

Site ASite B

Site CSite D

Visc

osity

of m

ixtu

rec

onsis

tenc

e of l

iqui

d

(b)

Velo

city

(mm

s)

Site ASite B

Site CSite D

0 05 1 15 2 25 3 35 4 45 5

Time (s)

35

30

25

20

15

10

5

0

(c)

Pres

sure

(Pa)

0 05 1 15 2 25 3 35 4 45 5

Time (s)

001

0008

0006

0004

0002

0

minus0002

Site ASite B

Site CSite D

(d)




7 Conclusion






Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




1E minus 01

1E minus 02

1E minus 03

1E minus 04

1E minus 05001 01 1

Curvature K

120575 = 005

120575 = 001

120575 = 0005

120575 = 001

(Pga

sminus

Pliq

)


Gas rate0 01 02 03 04 05 06 07 08 09 1

0

01

02

03

04

05

06

07

08

09

1

Visc

osity

of m

ixtu

rec

onsis

tenc

y of

liqu

id





119888


0

01

02

03

04

05

06

07

08

09

1



Gas

rate

0 100 200 300 400 500 600 700

Time (s)


0 200 400 600 800 1000 1200 1400


Para

met

er11

1

09

08

07

06

05

04

03

02



0



09

08

07

06

05

04

03

02

01

00 05 1 15 2 25 3 35 4 45 5

Gas

frac

tion

Time (s)

Site ASite B

Site CSite D

(a)

09

08

07

06

05

04

03

02

010 05 1

1

15 2 25 3 35 4 45 5

Time (s)

Site ASite B

Site CSite D

Visc

osity

of m

ixtu

rec

onsis

tenc

e of l

iqui

d

(b)

Velo

city

(mm

s)

Site ASite B

Site CSite D

0 05 1 15 2 25 3 35 4 45 5

Time (s)

35

30

25

20

15

10

5

0

(c)

Pres

sure

(Pa)

0 05 1 15 2 25 3 35 4 45 5

Time (s)

001

0008

0006

0004

0002

0

minus0002

Site ASite B

Site CSite D

(d)




7 Conclusion






Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




09

08

07

06

05

04

03

02

01

00 05 1 15 2 25 3 35 4 45 5

Gas

frac

tion

Time (s)

Site ASite B

Site CSite D

(a)

09

08

07

06

05

04

03

02

010 05 1

1

15 2 25 3 35 4 45 5

Time (s)

Site ASite B

Site CSite D

Visc

osity

of m

ixtu

rec

onsis

tenc

e of l

iqui

d

(b)

Velo

city

(mm

s)

Site ASite B

Site CSite D

0 05 1 15 2 25 3 35 4 45 5

Time (s)

35

30

25

20

15

10

5

0

(c)

Pres

sure

(Pa)

0 05 1 15 2 25 3 35 4 45 5

Time (s)

001

0008

0006

0004

0002

0

minus0002

Site ASite B

Site CSite D

(d)




7 Conclusion






Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Acknowledgments


References





































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



Documents

Research Article Gas Bubbles Expansion and Physical