Multiscale simulation of bubble behavior in aluminum reduction cell

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Thermodynamics, Transport, and Fluid Mechanics

Multiscale simulation of bubble behavior in aluminum reduction cell using a combined DBM-VOF-MHD method Meijia Sun, Baokuan Li, and LINMIN LI Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b05109 • Publication Date (Web): 18 Jan 2019 Downloaded from http://pubs.acs.org on January 20, 2019

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Industrial & Engineering Chemistry Research

Multiscale Simulation of Bubble Behavior in Aluminum Reduction Cell Using a Combined DBM-VOF-MHD Method MEIJIA SUN1, BAOKUAN LI1*, LINMIN LI2 1School

of Metallurgy, Northeastern University, Shenyang, Liaoning 110819, China.

2College

of Energy and Electrical Engineering, Hohai University, Nanjing, Jiangsu 210098, China.

* Corresponding Author, Email: [email protected]; Tel: +86-24-83672216; Fax: +86-2423906316

ABSTRACT The physics of aluminum electrolysis process involves many spatial scales, generating a wide variety of bubbles of different sizes in the magnetohydrodynamical (MHD) flow. To capture the dynamics of bubble nucleation, growth, coalescence and the interactions among the bubble-bath-metal, each scale needs to be resolved with the appropriate method. The present study proposes a three-dimensional (3D) multiscale multiphase flow model for bubble behaviors and bath-metal MHD flow, where the largescale interfaces of bubble-bath and bath-metal are resolved by the volume of fluid (VOF) approach and the dispersed micro bubbles by Lagrangian discrete bubble model (DBM). A discrete-continuum transition model is proposed to bridge different scales and handle the multiscale bubbles coexisting at the anode bottom. The predicted gas coverage and bubble thickness are validated by the experimental data in the literature. The numerical results indicate that bath-metal MHD flow accelerates the bubble motion and decreases gas coverage. Keywords: aluminum reduction cell; bubble behaviors; multi-scale; mathematical model; discretecontinuum transition

1

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Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1．INTRODUCTION The bubbles are generated beneath the anode as the reaction occurs between the dissolved alumina and carbon anode in the high-temperature aluminum reduction cell. Figure 1 shows the schematic representations of bubble behaviors. The tiny bubbles nucleate underneath the anode, and then grow and coalesce to large ones. When moving to the anode edge, bubbles will detach from the anode bottom. The large bubble deformation is caused by the unbalanced forces. The released bubble can induce the circulation in the surrounding liquid. The complex phenomenon plays an important role in the magnetohydrodynamic (MHD) instability and current efficiency.

Figure 1. Schematic representations of bubble behaviors underneath the anode surface. (a) micro-bubble nucleation, (b) bubble coalescence, (c) macro-bubble swallowing, (d) bubble detachment.

The bubble dynamics are difficult to measure due to the harsh operating conditions in the real 2


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aluminum reduction cell. Some researchers1-7 developed the air-water and low-temperature reaction models to study bubble behaviors. Fortin et al. 1 explored the relationship between the current density and bubble motions. Vekony et al. 2 found the Fortin bubbles using the inclined anodes. The maximum bubble thickness is about 2cm . Alam et al. 3 investigated the bubble motion beneath the anode in the CuSO4 solution. In their model, the bubbles are generated by the electrochemical reaction instead of by injecting air under the anode. Perron et al.

4, 5

studied the single bubble shape considering the anode

surface inclination and liquid properties. Das et al.

6, 7

studied the bubble detachment and sliding

behavior and analyzed the bubble thickness and the bubble residence time considering various gasliquid surface tensions. However, the bubble dynamics in the air-water systems cannot represent the actual electrolytic features in CO2-cryolite systems. Much research

8-14

have been carried out on the bubble formation, growth, coalescence, and

detachment in the small-scale high-temperature electrolytic model. Qiu et al.

8, 9

established a two-

compartment see-through quartz cell. They reported the visual observations of several molten salt electrolysis processes and described the anode and cathode reactions. Xue and Oye

10

studied bubble

generation, coalescence, growth, and detachment in a transparent cell. They coupled bubble behavior with the cell voltage oscillation. Cassayre et al.

11

compared the wettability between the oxygen-

evolving anode and graphite anode. They found that bubbles detach easily form the anode bottom with a good anode wettability. Zhao et al.

12-14

studied the bubble motion with different anodes and

investigated bubble rising morphology affected by anode wettability. They pointed out the difference of contact angle in air-water and CO2-cryolite systems. Although the small-scale anode cannot fully reveal the bubble behaviors in the industrial anode, the detailed of bubble motion are well presented. The numerical simulation is a widely used method to predict some complex flows in the aluminum electrolysis process. However, for nowadays computational capacities, the direct numerical simulation is unsuitable for calculating all phenomena in the engineering process. In terms of modeling, the volume of fluid (VOF) 14-20, Euler-Euler 21-23 and Euler- Lagrange 24-25 approaches are employed to predict the bubble motion. The VOF 14-20 has the advantages of capturing the bubble-bath interface under the anode surface. However, the drawback is that it requires fine enough mesh to track small bubble surfaces 3



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accurately, resulting in increasing computation burden. Hence it is not suitable for tracing the small bubble surfaces in the industrial cell. The Euler-Euler method21-23 was applied to simulate the bubble distribution which is collectively modeled using a concentration field. However, the dispersed bubbles are not tracked individually. Then a model developed for tracking the trajectory of single bubble motion is called the discrete bubble model (DBM) 24-25. The predicted bubble thickness under the anode is larger than that observed in some experiment. The large bubbles deformation cannot be predicted because the bubbles’ surfaces cannot be resolved. However, numerical modeling of multiscale bubble motions is a challenge in the industrial aluminum electrolysis process, as in the nucleation stage, the small bubbles are in the micro-scale, after growing up and coalescing, the large bubbles are in the macro-scale. Some research

26-28

has been done to

investigate the complex multiscale flow in chemical, environmental and process engineering. To resolving this multiscale challenge, Ling et al. 26 combined the VOF method and the Lagrangian pointparticle (LPP) model to resolve the large-scale interfaces and trace the small droplets, respectively. In particular, they achieved the two-way conversion between LPP and resolved droplet. G. Pozzetti et al.27 developed a multiscale DEM-VOF method considering the bulk and fluid fine scales. They used a dualgrid method to solve the multiphase flow in different length scales. Davide Zuzio et al. 28 used the LevelSet/Volume-of-Fluid method (CLSVOF) and the adaptive mesh refinement technique to capture the primary atomization. The particle tracking algorithm is employed to track the small droplet trajectory. The Eulerian-Lagrangian transition is achieved to improve the study of multiscale dynamics. Some studies

29-31

are based on a bath-metal two-phase MHD model by ignoring the effect of gas

bubbles. However, the gas bubble, as a driving force, can fluctuate the bath-metal interface, lead a high local current density, and causes the MHD instability. The interactions among the multiphase flow of bubble, bath, and metal are needed to be further studied. In the present work, we explore the multiscale gas bubble motions and study the gas-bath-metal multiphase flow in the small-scale laboratory cell. A three-dimensional (3D) multi-scale mathematic model is used to study the transient gas-bath-metal flows using the MHD model. A discrete-continuum transition algorithm is proposed to the one-way conversion form micro bubble to large one. The single 4


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bubble trajectory is tracked by DPM model. The large bubble surface and bath-metal interface are resolved by the VOF method. The two-way coupling is used to calculate the momentum exchange between dispersed bubbles and continuous fluid. 2．MODEL FORMULATION In the present study, we perform multiscale numerical simulations by combining a VOF method with a DBM model to solve large bubbles’ (macro-scale, ~cm) surface and track the tiny dispersed bubbles’ (micro-scale, ~um or ~mm) trajectory. In particular, a discrete-continuum transition algorithm is developed to bridge different scales of the Eulerian–Lagrangian model and handle the multiscale bubbles coexisting at the anode bottom. To maintain a reasonable computational time, some assumptions adopted in the mathematical model for bubble behaviors coupled with MHD flow are: (a) Bubbles nucleate uniformly beneath the anode. The nucleation sites of bubbles are not analyzed. (b) The temperature field is not considered. (c) The electrochemical reaction in the molten bath with high temperature is disregarded. (d) The bubble, bath and metal are incompressible Newtonian fluids. (e) The contact angle is not explored. 2.1. Discrete Bubble Model Small bubbles formed beneath the anode present spherical pattern. The bubble’s motion is calculated based on the DBM model

24-25, 32-33.

The single bubble trajectory is predicted by integrating forces

balance on it. The buoyancy, drag force, virtual mass force, and pressure gradient force are considered. The momentum equation referenced Newton’s second law is:

d ub mb  F b  F d  F vir  F p dt g ( b   )

(2)

3 C Vb D u  u b (u  u b ) 4 db

(3)

F b  mb

Fd 

(1)

b

1

𝐹𝑣𝑖𝑟 = 2𝜌𝑉𝑏(𝑢𝑏∇𝑢 ―

𝑑𝑢𝑏 𝑑𝑡

5


)

(4)


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(5)

F p  Vb u bu

Fb , F d ,

and

Fp

are the resultant of buoyancy and gravity, the drag force, and the dynamic

pressure gradient force, respectively. The virtual mass force F vir caused by the acceleration difference between bubble and bath, is important because

  b .35-36

CD is the drag coefficient calculated by

the spherical drag law34. 2.2. Micro Bubble Growth and Coalescence The alumina (Al2O3) is dissolved as the electrochemical reaction happens in the molten cryolite of the cell. The reaction formula is:

2 Al2O3  3C  4 Al  3CO2 The amount of CO2 is calculated by the Faraday law and the volume of CO2 is by the ideal-gas equation. Tiny bubbles formed at the anode bottom is spherical 10, 14. The bubble radius37 is:

r

3

3 RTA It 16 N  Fp1

(6)

r  3 It

(7)

The bubble mass related to the bubble volume can be expressed as:

dmb b RTA I  dt 4 NFp1

(8)

The bubble growth model is only applicable to the micro bubble growth because the large bubbles cannot keep the spherical shape due to the deformation affected by the unbalanced forces. Bubbles grow in nucleation sites, and then move along the anode bottom due to the driven force of bath flow. Some of the bubbles will collide and coalesce with each other. The probability of two bubbles’ collision is calculated by O’Rourke’s algorithm38 :

P

(9)

 (r1  r2 ) 2 u rel t Vcell

The Poisson distribution is employed to calculate the actual probability distribution of 6


N c , which

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can be written as:

P  Nc   e where

N c is

N  N1 P 

the

number

of

N1 (r1  r2 ) 2 u rel t , and Vcell

collisions,

Nc

N

N

N Nc !

is

the

(10) expected

collision

number

N1 is the number of the smaller bubbles.

In the coalescence model, x and y, range in [0, 1], are uniform random numbers, which are selected to calculate whether the two bubbles collide and coalesce. If the first random number x>P (0), two bubbles will collide with each other. 39 The actual collision parameter b related to the second random number y is written as:

b   r1  r2  y

(11) 

2.4 f 

f

and the

If the actual collision parameter b is smaller than the critical offset bcrit   r1  r2  min 1.0, We  ,   the result of the collision is coalescence. The function of the bubble size ratio collisional Weber number We are calculated as: 3

2

r  r  r  r  f  1    1   2.4  1   2.7  1   r2   r2   r2   r2  We =

𝜌𝑏𝑢2𝑟𝑒𝑙𝑑 𝜎

(12)

(13)

The bubble mass and velocity after coalescing are determined according to the conservation law. 38

m1'  m1  m2

(14)

'

m1' u b  m1 u b1  m2 u b 2

(15)

If b  bcrit , the two bubbles will rebound. The velocities of rebounding bubbles are changed: '

u b1  '

u b2 

b  bcrit m1 u b1  m2 u b 2 m2 (u b1  u b 2 )  ( ) m1  m2 m1  m2 r1  r2  bcrit

(16)

b  bcrit m1 u b1  m2 u b 2 m1 (u b1  u b 2 )  ( ) m1  m2 m1  m2 r1  r2  bcrit

(17)

2.3. Discrete-continuum Transition 7



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The physics of bubbles in the aluminum reduction cell involves multiscale dynamics. The tiny bubbles generated beneath the anode are usually small and spherical, which can be tracked in the Lagrangian coordinate. The large bubbles are deformed and flatted under downward facing horizontal surface because of the unbalanced forces, which can be resolved in the Eulerian coordinate. Hence, we developed the discrete-continuum transition to investigate the complex multiscale bubble behaviors. 26, 28

Figure 2 shows micro dispersed bubbles converting to macro resolved ones and macro resolved bubbles engilfing the micro dispersed ones. The coalescence between the micro bubbles can lead to a larger bubble. A criterion is that when the bubble volume is equal to or larger than the cell volume, the dispersed bubble converts to the resolved one. Moreover, the bubble is further solved by the VOF approach by coupling a user-defined function (UDF), as shown in Figure 2(a). If the detection criterion is satisfied, the cell is obscured by discrete bubble volume. The mass source on the right hand in the continuity equation is updated: n

S1   bVbi

(i  1,2,3,..., n)

(18)

i 1

u gi  ubi

(19)

In Figure 2 (b), the large bubble resolved in the Eulerian coordinate spans several grids, while the micro bubble smaller than the grid size is handled by the Lagrangian coordinate system. In the present model, a criterion is implemented to take into account a possible collision between bubbles and the bubble-bath interface. When the dispersed bubble is getting to attach the bubble-bath interface where 0.5   g  1 , it will be removed from the Lagrangian coordinate and recreated in the VOF model. The

dispersed bubble mass is updated in the source term of the mass equation. The fluid velocity is also updated as: n

(i  1,2,3,..., n)

S 2   bVbi i 1

u g '  u gi  ubi

Vbi Vcell

8


(20) (21)

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Figure 2. Schematic diagram of dispersed bubble converting to resolved one. (a) Vb

 Vcell

(b) 0.5  

g

1

2.4. Magnetohydrodynamic Model The current density is solved by electric charge conservation and Ohm’s law: ∇∙𝐽=0

(22)

𝐽 = ― 𝜎𝐸

(23)

The electric potential method is used to calculate the electromagnetic field. 32 𝐸=―

∂𝐴 ∂𝑡

―∇∙𝜑

(24)

In the present study, the influence of the busbar on the magnetic field is ignored. The magnetic field is calculated from the current flowing inside the cell. The magnetic potential vector is employed to calculate the magnetic field by: 40 𝐵=∇×𝐴

(25)

The electromagnetic force is introduced into the fluid momentum equation as a source term using the user-defined functions (UDF), which is given by: 𝐹𝑒 = 𝐽 × 𝐵 9


(26)


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2.5. VOF Model The VOF method is used to trace the interfaces of bubble-bath and bath-metal. 41 The volume fraction of each fluid is solved. ∂𝛼𝑖 ∂𝑡

+ ∇ ∙ (𝛼𝑖𝑢) = 0 3

 i 1

i

1

(27) (28)

The governing equations of the continuity and momentum conservations 42 for the bubble-bath-metal three-phase flow are expressed as:     (  u )  S1  S 2 t ∂(𝜌𝑢) ∂𝑡

where

i

+∇ ∙ (𝜌𝑢𝑢) = ―∇𝑝 +∇ ∙ [𝜇𝑒𝑓𝑓(∇𝑢 + ∇𝑢𝑇)] + 𝐹𝑒 + 𝐹𝑚𝑒 + 𝐹𝑠

(29) (30)

represents the volume fraction of bubble, bath and metal.  is the mixture density and eff

is the mixture effective viscosity composed of the molecular and turbulent viscosities. In the present work, The Lagrangian bubbles are coupled to the Eulerian fluid with a two-way coupling through interchange terms F me such as drag force, virtual mass force and pressure gradient force in the respective momentum equations.

3  C 1 d F me   Vb D u  u b (u  u b )  Vb (u  u b )  Vb u b u t db 2 dt 4 

(31)

The continuum-surface-force (CSF) model 43 is applied to calculated the surface tension F s . Figure 3 displays the solution schematic of the current model. In the DBM-VOF-MHD model, the large bubbles, molten bath, and metal are treated as continuous fluids, while the small bubbles nucleate at anode bottom considered as the dispersed particle. The discrete-continuum transition model is employed to bridge different scales of the hybrid Euler-Lagrangian model and handle the multiscale bubbles coexisting at the anode bottom. Two-way coupling is employed to achieve the exchange momentum between dispersed bubbles and continuous phase, allowing that the continuous flow and dispersed bubbles affect with each other. The electromagnetic force solved by MHD module, as a source term, is added into the momentum conservation equation. Then the multi-physical fields could be solved 10


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simultaneously. Given the characteristic of fluid flow in the process, the RNG k-ε turbulence model 44 is used to calculate multiscale flow dynamics with lower Reynolds number with the appropriate treatment of the near-wall region. An enhanced wall function is employed in the RNG k-ε turbulence model.

Figure 3. Solution schematic of the current model

3．NUMERICAL DETAILS Bubble motion and MHD flow are calculated by the commercial CFD package Fluent-15.0 coupling with the UDF. The computation domain with structured grids is shown in Figure 4. The electrical boundary conditions are shown in Figure 5. In the calculation, we assume that the micro bubbles nucleate uniformly at the anode bottom. We set the initial bubble size of 0.5 mm according to some researchers’ observation

10, 12, 13.

The bubble

injecting velocity is used at the inlet boundary of anode bottom. The outflow boundary is applied at the cell top surface. The no-slip condition is imposed at all boundaries expect the cell top surface, where the zero shear stress is employed.

11



Figure 4. Mesh for simulation

Figure 5. Electrical boundary conditions

Mesh sensitivity study is performed on the bubble thickness and bath-metal interface through the consideration of 587k, 330k, and 182k cells, respectively. The comparison of bubble shape and bathmetal interface at 25s along the centerline of X coordinate is given in Figure 6a. The bubble thickness is about 4.0 mm, as seen in Figure 6b, consistent with some researchers’ observations. 6, 13-14, 45 The enlarged bath-metal interface is presented in Figure 6c. The comparisons deviation of predicted bubble thickness and bath-metal interface are listed in Table 1. Table 1. Comparison deviation of predicted bubble thickness and bath-metal interface

The result reveals that no appreciable difference is found for the predicted bubble shape and bathmetal interface between the 587k and 330k cells. In the view of computational resources and times, a total amount of 330k cells is used to simulate the multiscale bubble motion and bubble-bath-metal multiphase flow field. The grid size is 0.8mm.

12


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Figure 6. Mesh sensitivity. (a) Bubble morphology and bath metal interface at 25s along the centerline of X coordinate (b) Local bubble morphology (c) Local bath-metal interface.

The velocity-pressure coupling of the gas-bath-metal flow is handled through the SIMPLE procedure, and all the governing equations are discretized by a second-order upwind method. A fixed physical time step of 0.0001s is adopted for all transient simulations. The geometrical parameters and material properties are listed in Table 2. Table 2. Geometrical and physical parameters

4.1 Model Validation 13



Figure 7. Comparisons between experimental data and numerical results. (a) bubble coverage; (b) bubble thickness

Figure 7 shows experimental13 and numerical results of bubble coverage and bubble thickness at current density of 9000 A/m2. The effects of micro and macro bubbles are considered in the calculation of average bubble coverage and bubble thickness. Both the predicted bubble coverage and bubble thickness in bubble-bath flow agree well with the bubble-bath experiment data of Zhao et al.,13 which is also conducted on the bubble-bath two-phase flow, indicating that the proposed model could be suitable for predicting bubble-bath flows in aluminum reduction cells. But a small frequency difference between the experimental and numerical results exists due to the different contact angle between the anode and the bath. In the present study, the anode-bath contact angle is assumed to be 90°. But, Zhao et al. found the anode-bath contact angle is 100-120° in their experiments. They pointed out a good anode wettability is due to a small contact angle, which affects the bubble motion under the anode. The peak values of both predicated bubble coverage and bubble thickness in bubble-bath-metal three-phase flow are smaller than that in the bubble-bath two-phase flow, while the fluctuation frequency calculated in the bubble-bath-metal three-phase flow is larger than that in the bubble-bath two-phase flow. The reason is that the bath-metal interface fluctuation, caused by the EMFs, accelerates bubbles’ movement and reduces the residence time of bubbles at the anode bottom. 4.2 Bubble Behaviors

14


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Figure 8. Bubble coalescence and discrete-continuum transition processes. (a) micro-micro (b) micro-macro transition (c) macro-micro (d) macro-macro

Figure 8 displays the bubble coalescence and the discrete-continuum transition processes. Micro bubbles are always spherical under the action of surface tension in the initial bubble growth and coalescence stages. In the current model, the micro bubbles are smaller than the grid size and handled in the Lagrange coordinate system. The two micro bubbles collide with each other and coalesce to a larger bubble when moving at the anode bottom, as shown in Figure 8a. Then, since the bubble satisfies the criteria of discrete-continuum transition that the bubble volume is equal to or larger than the volume of the cell, the micro dispersed bubble will convert to the macro resolved one, as shown in Figure 8b. Figure 8c presents the coexistence of macro and micro bubbles. When the micro bubble is getting close to the macro bubble surface where the 0.5   g  1 in the same cell, the macro bubble will swallow up the micro one. Additionally, the macro bubble mass and velocity are updated in the continuity and the momentum equations, respectively. Two macro bubbles coalescence process is resolved in Eulerian coordinates, as presented in Figure 8d. The macro bubble volume is much larger than the cell volume. The bubbles are deformed because of the unbalanced forces (𝐹𝑏, 𝐹𝑠, and 𝐹𝑝). 15



Figure 9. Bottom view of the bubble morphologies. (a) t=4.5s, bubble-bath two-phase flow of numerical simulation; (b) t=4.5s, bubble-bath-metal three-phase flow of numerical simulation.

Figure 9 shows anode bottom view of the bubble morphologies. The small spherical and lager deformed bubbles are distributed nonuniformly on the anode bottom because of the unbalanced forces as shown in Figure 9a. Comparing the experiment result conducted by Zhao et al. 13 and our simulation work, we find the same bubble behaviors which are similar in shape but different in position. Figure 9b displays the predicted bubble distribution in the bubble-bath-metal three-phase flow. A bubble film is formed near the anode edge and some small bubbles close to the anode bottom center. The bath-metal flow promotes the horizontal movement of bubbles along the anode bottom and facilitates the discretecontinuum transition from the micro bubbles to macro ones.

16


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Figure 10. Current line at X=0.05 m in (a) bath-metal flow and in (b) bubble-bath-metal flow and current density of bath-metal interface in (c) bath-metal flow and in (d) bubble-bath-metal flow

In Figure 10a, the current flows vertically from the anode bottom into the cell without considering the bubble’s motion. The change of current density in the bath-metal flow is consistent with the bathmetal interface fluctuation which is driven by EMFs, as seen in Figure 10c. The current density nearby the bubble surface is higher than the apparent current density, as presented in Figure 10b. The nonconductive bubbles adhered to the anode bottom block the current and change its direction. 46-47 Then the current flows around the bubble surface into the electrolyte. The bubble motion as the driven force, combining with EMFs causes a different fluctuation pattern of the bath-metal interface in the bubblebath-metal flow, as seen in Figure 10d. We also find a location match between the peak current density in the bath-metal interface and that in the bubble surface along the X direction. Moreover, the nonconductive bubbles reduce the anode-bath electric contact area, thereby increasing the resistance and local current density in the cell. 4.3 Influence of Bubbles on Cell Flow 17



Figure 11 gives the predicted velocity at the metal plane of Z= 0.004m with different driving forces. Four vortices formed at the cell around are approximately symmetrical under the EMFs. The velocity near the cell side is larger than that in the center. The combined effect of bubbles and EMFs in the metal flow region shows various non-uniform vortices, moreover, the velocity is larger than that with the EMFs. The results show that the combined bubble-EMFs flow increases the momentum exchange among bubble, bath, and metal, which is more effective than the EMFs-induced motion.

Figure 11. Velocity at the metal plane of Z=0.004m with (a) EMFs and with (b) bubbles and EMFs.

Figure12. Bath-metal interface with different driving forces at t=30s. (a) EMFs; (b) Combined bubbles and EMFs

Figure 12 shows the bath-metal interface deformations with different driving forces. In Figure 12a the bath-metal interface with EMFs is smooth and stable. The predicted interface fluctuation under EMFs is high in the middle and low on all sides, which is consistent with some researchers’ observation. 30, 31, 48

In Figure 12b, the bath-metal interface instabilities are developing due to the combined bubble-

EMFs driving flow. The upward amplitude of the maximum interface fluctuation under the EMFs and the combined forces is 0.4mm and 1.3mm, respectively. Figure 13a shows that the dispersed micro 18


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bubbles cause a small-scale disturbance perturbation wave in the bath-metal interface.15 As seen from Figure 13b, 13c and 13d, the bath-metal interface fluctuation waves change consistently with the macro bubble shapes, thereby causing the bubble shadow in the bath-metal interface. The motion of large bubbles increases the fluctuation amplitude of the bath-metal interface, which is easier to cause the MHD instability.

Figure 13. Bubble evolution and bath-metal interface fluctuation at (a) t=5s, (b) t=15s, (c) t=25s and (d) t=35s.

4.4 Bubble Coverage and Bubble Thickness

19



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Figure 14. Evolution of gas coverage with different current densities.

Figure 15. Evolution of gas coverage with micro and macro bubbles. (a) Overall diagram; (b) partial magnification

Figure 14 illustrates the evolution of gas coverage with different current densities from 0s to 30s. In the present work, the gas coverage is calculated by tracking micro bubbles’ trajectory and macro bubbles’ surface.

 

A   i’ 1 Ab ,i i 1 g ,i N’

N

A

(32)

The gas coverage curves present sawtooth patterns that are related to the bubble nucleation, growth, coalescence, and detachment. 10 It can be observed that a large current density can increase bubble release frequency. High current accelerates the electrochemical reaction to increase bubble nucleation and growth rates. High current also generates a large EMF to facilitate the bubble motion in the MHD flow. It will help decrease bubble residence time in the cell, leading to a low gas coverage under the 20


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anode. Figure 15 gives the evolution of gas coverage with micro and macro bubbles at 9000A/m2. In the initial stage of bubble formation, gas coverage increases slightly due to the dispersed micro bubble’s growth and coalescence, leading to a small gas coverage at the anode bottom. Once the discretecontinuum transition happens, the gas coverage increases rapidly due to the fast swallowing micro bubbles by macro bubbles. Moreover, the gas coverage of macro resolved bubbles is much larger than that of dispersed micro ones. When bubbles detach from the anode bottom, the gas coverage decreases sharply. Then, the electrical contact area between anode and bath increases so that the micro bubbles can be regenerated at the anode bottom and the gas coverage of dispersed micro bubbles is increased, as shown in Figure 15b.

Figure 16. Evolution of bubble thickness with different current densities.

Figure 16 shows the evolution of bubble thickness with different current densities. Changes in the bubble thickness follow a cyclical pattern. A large current density can slightly reduce the maximum bubble thickness to promote the reduction of ACD. 4.5 Influence of Bubbles on Bath-metal Fluctuation

21



Figure 17. Bath-metal interface fluctuation with (a) 5000 A/m2, (b) 7000 A/m2 and (c) 9000 A/m2

Figure 17 demonstrates the bath-metal interface fluctuation with different current densities. The maximum interface fluctuation occurs around the anode edge because the bubble detaches from the anode bottom and causes a strong disturbance on the interface. Additionally, the MHD flow with bubbleEMF driving forces can facilitate homogenization through efficient mixing. The high current contributes a large EMF and causes the MHD instability. The upward amplitudes of the maximum interface fluctuation under the combined the EMFs and bubble driving forces are about 0.7mm, 1.2mm and 1.8mm with the current density of 5000 A/m2, 7000 A/m2 and 9000 A/m2, respectively. Figure 18 shows the average fluctuation amplitude of bath-metal interface with EMFs and with combined EMFs-bubble driving force. The interface perturbation shows significant damping of the oscillations and tends to be stable after 40 seconds under only EMFs. While the interface perturbation with combined EMFs-bubble driving force presents a sawtooth pattern, revealing that interface fluctuation periodically changes with the bubbles’ motion and increases with increasing current density.

22


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Figure 18. Fluctuation amplitude of bath-metal interface.

5．CONCLUSIONS A multiscale multiphase flow model is used to explore the bubble behaviors and MHD flow in the small-scale cells. The large interfaces are resolved by the VOF approach. The micro bubbles formed at anode bottom are tracked by DBM. The algorithm of discrete-continuum transition is proposed to bridge different scales of the Eulerian–Lagrangian model and handle the multiscale bubbles coexisting at the anode bottom. The numerical results are validated by experimental data in the literature. The conclusions are as follow: 1.

The predicted bubbles are randomly distributed in the bubble-bath two-phase flow. The bubble film is formed near the anode edge. Some small bubbles are distributed in the anode bottom center in the bubble-bath-metal three-phase flow.

2.

In the initial stage of micro bubble growth and coalescence, dispersed micro bubbles cause a smallscale disturbance perturbation wave in the bath-metal interface. The deformable macro bubbles lead to a bubble shadow in the bath-metal interface. The bath-metal interface perturbation tends to be stable with EMFs and presents sawtooth pattern with combined EMFs-bubble driving force. Moreover, the interface fluctuation periodically changes with the bubbles’ motion and increases with increasing current density.

3.

The gas coverage of resolved macro bubbles increases rapidly due to the macro bubbles fast swallowing micro bubbles. The macro bubbles account for a much larger proportion of gas coverage compared to the micro ones. The gas coverage decreases sharply after large bubbles 23



detach from the anode bottom. Then, the electrical contact area between anode and bath increases so that the micro bubbles can be regenerated at the anode bottom. The present simulation is performed based on a small-scale single-anode laboratory cell model which costs reasonable computing resources. It is true that the present simulation cannot reveal the flow characteristics of the industrial cell. In the future, we will keep our focus on improving the multiscale mathematical models for describing the multiphase bubble-bath-metal flow mechanisms with the external magnetic field in the whole cell. ACKNOWLEDGEMENTS The authors wish to thank the financial support of the National Natural Science Foundation of China (grant numbers 51474065, 51574083).

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NOMENCLATURE

GREEK SYMBOLS

25



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Table Caption Table 1 Comparison deviation of predicted bubble thickness and bath-metal interface Table 2 Geometrical and physical parameters Figures Caption Figure 1. Schematic representations of bubble behaviors underneath the anode surface. (a) microbubble nucleation, (b) bubble coalescence, (c) macro-bubble swallowing, (d) bubble detachment. Figure 2. Schematic diagram of dispersed bubble converting to resolved one. (a) Vb  Vcell (b) 0.5   g  1 Figure 3. Solution schematic of the current model Figure 4. Mesh for simulation Figure 5. Electrical boundary conditions Figure 6. Mesh sensitivity. (a) Bubble morphology and bath metal interface at 25s along the centerline of X coordinate (b) Local bubble morphology (c) Local bath-metal interface. Figure 7. Comparisons between experimental data and numerical results. (a) bubble coverage; (b) bubble thickness Figure 8. Bubble coalescence and discrete-continuum transition processes. (a) micro-micro (b) micromacro transition (c) macro-micro (d) macro-macro Figure 9. Bottom view of the bubble morphologies. (a) t=4.5s, bubble-bath two-phase flow of numerical simulation; (b) t=4.5s, bubble-bath-metal three-phase flow of numerical simulation. Figure 10. Current line at X=0.05m in (a) bath-metal flow and in (b) bubble-bath-metal flow and current density of bath-metal interface in (c) bath-metal flow and in (d) bubble-bath-metal flow Figure 11. Velocity at the metal plane of Z=0.004m with (a) EMFs and with (b) bubbles and EMFs. Figure 12. Bath-metal interface with different driving forces at t=30s. (a) EMFs; (b) Combined bubbles and EMFs. Figure 13. Bubble evolution and bath-metal interface fluctuation at (a) t=5s, (b) t=15s, (c) t=25s and (d) t=35s. Figure 14. Evolution of gas coverage with different current densities. Figure 15. Evolution of gas coverage with micro and macro bubbles. (a) Overall diagram; (b) partial magnification 30


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Figure 16. Evolution of bubble thickness with different current densities. Figure 17. Bath-metal interface fluctuation with (a) 5000 A/m2, (b) 7000 A/m2 and (c) 9000 A/m2 Figure 18. Fluctuation amplitude of bath-metal interface.

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83x119mm (300 x 300 DPI)


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Schematic representations of bubble behaviors underneath the anode surface. (a) micro-bubble nucleation, (b) bubble coalescence, (c) macro-bubble swallowing, (d) bubble detachment. 150x75mm (300 x 300 DPI)



Schematic diagram of dispersed bubble converting to resolved one. (a)Vb>Vcell (b) 0.5

Multiscale simulation of bubble behavior in aluminum reduction cell

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