COMPUTATIONAL MODELING OF ANODIC CURRENT DISTRIBUTION ANDANODE SHAPE CHANGE IN ALUMINIUM REDUCTION CELLS

Y. Xu, J. Li *, H. Zhang, Y. Lai

School of Metallurgy and Environment, Central South University, Changsha, Hunan, China

(Received 23 February 2014; accepted 17 December 2014)Abstract

In aluminium reduction cells, the profile of a new carbon anode changes with time before reaching a steady state shape,since the anode consumption rate, depending on the current density normal to anode surfaces, varies from one region toanother. In this paper, a two-dimension model based on Laplace equation and Tafel equation was built up to calculate thesecondary current distribution, and the shift of anode shape with time was simulated with arbitrary Lagrangian-Eulerianmethod. The time it takes to reach the steady shape for the anode increases with the enlargement of the width of the channelsbetween the anodes or between the anode and the sidewall. This time can be shortened by making a sloped bottom or cuttingoff the lower corners of the new anode. Forming two slots in the bottom surface increases the anodic current density at theunderside of the anode, but leads to the enlargement of the current at the side of the anode.

Keywords: aluminium reduction cell, secondary current distribution, anode consumption, anode shape

* Corresponding author: sqjsxu@gmail.com

Journal of Mining and Metal lurgy,Section B: Metal lurgy

DOI:10.2298/JMMB140223006X

1. Introduction

Aluminum is produced using Hall–Héroultprocess by reducing alumina in aluminium reductioncells, the schematic geometry of which is illustrated infig. 1. In the production, the intense electrolysiscurrent flows downwards from prebaked carbonanodes at top, passing through a molten electrolyte(bath) layer and a liquid aluminium (metal) pad beforebeing collected at the cathode lining at bottom.Aluminium is deposited on the bath-metal interfaceand CO2 is evolved from anode surfaces. The primaryreaction in this process is

The anode surfaces adjacent to the electrolyteinclude the horizontal underside surfaces and thevertical side surfaces. Most current passes through theunderside of anodes, which are normally defined asworking face, while, some current also passed throughthe vertical sides. The current distribution on theanode surfaces, especially the fraction of side current,has important influence on the cell voltage and currentefficiency [1], which are the two key technicalparameters for the industrial production. In addition,the carbon anode is consumed by the anodic reaction.After a new cold anode is inserted into the cell, itslower surface is immediately covered by the insulatedfrozen electrolyte, which slowly melts away in thefirst 12~18 hours [2]. When the anode surface isexposed to the molten electrolyte, the anodic reactiontakes place and the shape of the anode begins totransform. The shift of anode shape causes the

variation of the electromagnetic fields in the cell,which can lead to instability of the working conditionof the cell. Consequently, the anodic currentdistribution and the shift of the anode shape withrespect to time need to be analyzed precisely.

The current distribution in aluminium reductioncells has been widely studied. Different approacheshave been adopted to study the two approximations ofanodic current distribution, primary currentdistribution [1, 3-5] and secondary currentdistribution [6-8]. The former defines the anodicovervoltage as a constant value, but the latterintroduces the overvoltage depending on the localcurrent density. In reference [9, 10], the secondcurrent distribution for a 150 kA prebaked cell wasstudied, and the shift of the anode shape wascalculated with the methods of incremental time stepsand “near steady state shape” condition. The studyshowed it normally took 6 to 8 days until a steadyshape anode was obtained, depending on the width ofthe channels between the two anodes or between theanode and the sidewall. In reference [11], a similarstudy for the anode with chamfers was carried out,and the results indicated it took less time to reach thesteady shape for the anode whose lower corner werecut off by 5 cm (diagonally), as compared to the anodewith rectangular corners. In these studies, the currentdensity at the underside of the anode was assumed asa constant value (0.75 A·cm-2), and it was ignored thatthe local current density at the underside varied withdistance from the edge, which could result in an

J. Min. Metall. Sect. B-Metall. 51 (1) B (2015) 7 - 15

2Al O +3C=4Al+3CO2 3 2

Y. Xu et al. / JMM 51 (1) B (2015) 7 - 15

inaccuracy that the current density varies smoothlynear the corner.

In this paper, a transient calculation for thesecondary current distribution and the change ofanode shape was carried out with a new model, andthe Arbitrary Lagrangian-Eulerian (ALE) based finiteelement method was used in solving this moving-boundary problem. In addition to the normal anodewith rectangular corners, the anodes with specificinitial shape, including a sloped bottom, chamfers andslots, which are widely adopted in aluminiumelectrolysis industry, were taken into account, and theprocesses of shape transformations for the differentanodes have been revealed.

2. Theory2.1 Assumption

To simplify the computational model, someassumptions were made firstly. The currentdistribution on the anode is significantly affected bythe width of the adjacent channel where the bathconducts a small part of the current from the anode.The channels and their typical widths are shown inFig.2. Because the sidewall of the cell is alwayscovered by a ledge of frozen bath, the real widths ofthe side and end channels are about 12~18 cm. In thiswork, a 15 cm wide channel and a 2 cm wide channelwere chosen to represent the side channel and half theinter-anode channel, respectively. Considering thesymmetry of electric field, it is feasible to use half thewidth of the inter-anode channel when modelling asingle anode at one side of the inter-anode channel.The real shape of the ledge was not taken intoaccount.

The new cold anode is covered with the ledge that

melts slowly, and the different parts of anode surfaceare not exposed to the bath at the same time. There issmall current passing through some parts of thesurfaces during the heating-up period, which wasignored in this work. The initial time for thecalculation was taken as the time when the entireanode is clear of the ledge. The gas bubbles generatedby the anodic reaction usually form a thin gas filmunderneath the anode, while, the resistance of the gasfilm was not taken into account in the calculation. Theinfluence of the wave of the bath-metal interface wasalso neglected.

The electrical conductivities of liquid aluminum,typical carbon anode and molten electrolyte are3.5×106 s·m-1 [12], 2.1×104 s·m-1 [13] and 2.2×102s·m-1 [12], respectively. According to the hugedisparity of the three conductivities, it can be assumedthat the anode surface and the bath-metal interface areapproximately equipotential surfaces [9, 14].

When the corners of the anode become roundowing to the consumption, the explicit line dividingthe side surface and bottom surface disappears. To be

8

Figure 2. The channels and their typical widths in thealuminium reduction cell

Figure 1. Schematic geometry of a modern aluminum reduction cell

convenient for presenting the results, the anodesurface was artificially divided as shown in fig. 3. Theanode-cathode distance (so-called ACD) was taken tobe 4.5 cm in this work. The part of the anode curvegoing from point A, the surface of the bath, to point Bwas defined as the side of the anode, and the rest partof the curve going from point B to point C wasregarded as the underside of the anode. Obviously,there is a one-to-one correspondence between thearbitrary point D on the anode curve and the angle θmade by the line OB and OD. The positive angle θcorresponds to the underside of the anode, while, thenegative angle θ corresponds to the side of the anode.

2.2 Computational model

The distribution of the electric potential in theinterelectrode space can be described by the Laplaceequation:

(1)

The current density, J, at any point was determinedfrom the gradient of local potential, , according toOhm’s Law:

(2)

Where is the electric conductivity of the moltenelectrolyte.

The model for the second current distribution canbe established by characterizing the boundaryconditions. The potential at the anode surface wasdefined to be zero, so the inner potential of theelectrolyte at the anode surface can be given byTafel equation:

(3)

Where the current density J is in A·m-2, thepotential is in V, n denotes the unit vector normal tothe anode surface, and J0=100 A·m-2, bc=0.109 V [9].

In this work, the current density at the underside of theanode was not assumed to be a constant value, and themagnitude of the current entering into the system wascontrolled by the equation that described the innerpotential of the electrolyte at the metal-bath interface

:

(4)

Where I is the total current flowing across themetal-bath interface, and the value of I wasdetermined by the average current density at theunderside of the anode, which was chosen to be 0.78A·cm-2 in this work. The other boundaries for theelectric field were defined to be insulated.

When the anode is consumed, the boundaries ofthe model move. The moving velocity of the anodesurface is equal to the anode consumption rate, whichis proportional to the normal current density to theanode surface Jn. Defining the anode contour asy=G(x), the movement of a point on the anode surfacecan be described as

(5)

With

(6)

Here sn(x, G(x)) is the normal moving distance ofthe point (x, G(x)), t is the time of electrolysis, k is acoefficient for describing the relationship betweennormal current density and movement velocity for apoint on anode surface, F is Faraday constant, is theanode density, and is a factor to describe theoverconsumption of the anode with respect to theFaraday’s law. It is widely recognized that theoverconsumption is caused by dusting and theBoudouard reaction [15]. The anode density andfactor were taken to be 1.55 g/cm3 and 1.15,respectively, and then k is 8.251×10-8 m3A-1h-1.

With the consumption of the anode, the moltenaluminium is produced at the top surface of the metaland the metal-bath interface moves upwardsimultaneously. The metal-bath interface commonlymoves up a little faster than the anode bottom surface,while, to maintain a steady cell voltage (a constantACD), the vertical location of the anodes is adjustedfrequently in industry. In this study, the velocity of themetal-bath interface was set following the principlethat the average ACD didn’t change with time.

This two-dimension transient model was solvedwith finite element method by discretizing thecomputational domain with quadrate elements. To trackthe moving boundary but avoid the reduction of themesh quality, Arbitrary Lagrangian-Eulerian (ALE)method was adopted. In the calculation, the meshes onthe boundaries move along with the boundaries, and themeshes inside the domain move arbitrarily to optimizethe shapes of elements. Considering that the speed of the

Y. Xu et al. / JMM 51 (1) B (2015) 7 - 15 9

Figure 3. Schematic cross section of a prebaked anode inthe aluminium reduction cell

a

a

2 0

J

0n J J ba

c

exp( )

mn

I

nn

ds x G xdt

k J x G x

k

( , ( )) ( , ( ))

3

kF3

m

anode consumption is quite slow (about 1.5 cm per day),the time step of the transient computation was taken tobe 2 hours, and a sufficient number of steps wereexecuted (more than 120 steps). The calculation startedwith the initial shape of an anode, and the steady stateshape of the anode and the secondary currentdistribution were achieved after the calculation.

3. Results and discussion3.1 The influence of the width of the channel

The distributions of the current densities andequipotential lines in the interelectrode space atdifferent time during the electrolysis for the 15 cmchannel and the 2 cm channel were shown in fig.4 andfig.5, respectively, in which the shift of anodecontours also can be seen. The current densityconcentrates at the corner of the anode at the initialtime, which makes the consumption rate at the cornerlarger than in other regions. In the first 1 to 2 days therectangular corners become round, and the radian ofthe anode curves increases with the time going. Aconstant anode shape was obtained after 8 days for the15cm channel, however, the shape of the anodereaches a steady state after 7 days for the 2 cmchannel. It takes more time to arrive a constant shape

for the anode adjacent to a wider channel, whichagrees with the conclusions reported in reference [9].

The distributions of the anodic current densities forthe two cases are presented in fig.6 and fig.7,respectively. The anode curves are divided by the lineθ=0, the right of which corresponds to the undersides ofthe anodes. From fig.6 or fig.7, one can see that thesharp peak of the anodic current density lies at therectangular corner (θ=0) at the initial time. The peakmoves towards the right and becomes gentle with theprocess of the electrolysis, and it disappears when thesteady shape of the anode is achieved. Thetransformation of the anode shape decreases the anodiccurrent density nearby the corner, increases the anodiccurrent densities at both the bottom and side surfaces,and makes the anodic current distribute more uniformly.As the anode changes from the initial shape to thesteady-state shape, the ratio of the average currentdensity at the underside of the anode to that at the sideof the anode decreases from 2.82 to 2.55 for the 15 cmchannel, while, this ratio decreases from 10.92 to 8.36for the 2 cm channel. It is indicated that the proportionof the current passing through the side of the anode inthe total current increases during the change of theanode shape and this proportion is amplified byenlarging the width of the channel.

Y. Xu et al. / JMM 51 (1) B (2015) 7 - 15 10

Figure 4. The anode shapes and secondary current distributions for the 15cm channel at (a) the initial time, after (b) 1 day,(c) 2 days, (d) 4 days, (e) 6 days and (f) 8 days.

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Figure 5. The anode shapes and secondary current distributions for the 2cm channel at (a) the initial time, after (b) 1 day,(c) 2 days, (d) 4 days, (e) 6 days and (f) 7 days.

Figure 6. The distributions of the anodic current densitiesat different time for the 15cm channel.

Figure 7. The distributions of the anodic current densitiesat different time for the 2cm channel.

3.2 The influence of the initial shape of anode

In order to reduce the thickness of the gas layerunderneath the anode, the anodes are designed tofacilitate the release of gas bubbles. The anodes withchamfers, slots or sloped bottoms are widely used inindustry, and the shape transformations during theelectrolysis for these specific anodes were alsoanalyzed in this work.

The anode with chamfers is commonly producedby cutting off the corners between the bottomsurface and the larger side surface of a rectangularanode. Fig. 8 shows the calculated results for theanode whose lower corners were cut off by 4 cm(diagonally). It can be found that the bevel turnsinto a round corner in the first two days, and thesteady shape of the anode is obtained after 6 days,which is shorter than the time it takes for the normalanode. The anodic current densities for the normalanode and the anode with chamfers both reachmaximums nearby the corners, and the maximumsare compared in fig. 9. It can be seen from fig. 9 thatthe maximum anodic current density for the anodewith chamfers keeps smaller than that for thenormal anode from one day after the electrolysisbegan to the time when the same constant shapes areachieved. The maximum anodic current density forthe anode with chamfers is quite larger at the initialtime, but it decreases very fast in the first day.

The same computation for the anode with a slopedbottom was carried out. The tilted angle of the bottomsurface of the initial anode was chosen to be 1.7degrees, and the calculated anode shapes andsecondary current distributions at different timeduring the electrolysis are shown in fig. 10.Comparing fig. 10 with fig. 4, one can see that thesloped bottom facilitates the transformation of theanode from its initial shape to the steady shape. Ittakes only 5.5 days to reach the constant shape for thisanode, comparative to 8 days for the normal anode. As

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Figure 9. The changes of the maximum anodic currentdensities with respect to time for the normalanode and the anode with chamfers.

Figure 8. The anode shapes and secondary current distributions for the anode with chamfers (cutting off the lower cornersby 4cm) at (a) the initial time, after (b) 1 day, (c) 2 days and (d) 6 days.

shown in fig. 11, the maximum anodic current densityfor the anode with a sloped bottom has maintainedsmaller than that for the normal anode until the samesteady shape of the anodes is obtained, which, to someextent, indicates the tilted bottom makes the anodiccurrent density distributes more uniformity in theearly period of the electrolysis.

The anode with slots was also taken into considerin our work. It is common to form two slots in thebottom surface of the new anode. The slot is thesame length as the anode, and its typical width is nomore than 2 cm. The calculated secondary currentdistributions and profiles of half the anode withslots, whose width is taken to be 1cm, are shown infig. 12. The rectangular corner formed by the slotalso gradually turns into a round one after theelectrolysis starts, and it takes 7 days for this anodeto reach the steady shape. Obviously, the slot is anequivalent channel that is narrower than otherchannels, and the shape transformation of the corneradjacent to the slot completes faster, hence, theexistence of the slot doesn’t extend the time it needsto arrive the constant shape for the anode. The slotreduces the area of the working face of the anode anddivides the bottom surface into two different regions,which results in the redistribution of the anodiccurrent. The average anodic current densities in theregions of the underside of the anode with slots arecompared to that at the underside of the normalanode in fig. 13, where region A indicates the bottomsurface between the two slots of the anode with slots,region B indicates the bottom surface of the anodewith slots except the region A, and region Cindicates the entire bottom surface of the normal

anode without slots. The average anodic currentdensity in region A is smaller than that in region B atthe initial time, and the latter maintains larger thanthe former until the shape of the anode doesn’tchange any more. After the first day of theelectrolysis, the average anodic current density inregion A becomes larger than that in region C, whichis smaller than that in region B throughout theelectrolysis. It is proved that the reduction of the areaof the bottom surface enlarges the anodic currentdensity at the underside. When the steady shapes ofthe anodes are reached, the average anodic currentdensities at the side surfaces of the anode with slotsand the normal anode are, respectively, 1020 A·m-2and 960 A·m-2, which manifests the slot also enlargesthe anodic current at the side.

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Figure 11. The changes of the maximum anodic currentdensities with respect to time for the normalanode and the anode with a sloped bottom.

Figure 10. The anode shapes and secondary current distributions for the anode with a sloped bottom (the tilted angle of1.7 degrees) at (a) the initial time, after (b) 1 day, (c) 2 days and (d) 5.5 days.

4. Conclusions

The transient calculations for the transformationsof the anode shape and the secondary currentdistribution during electrolysis in aluminiumreduction cells were implemented in this study. Theresults show that the width of the channel and theinitial shape of the prebaked anode have importanteffect on the distribution of anodic current density andthe process of anode shape change. It takes more timeuntil the steady state shape is obtained for the anode

adjacent to the wider channel, and the fraction of thecurrent passing through the side of the anodeincreases with the enlargement of the channel width.Cutting off the lower corners of the anode by 4 cm(diagonally) can reduce the time from the initial shapeto the constant shape of the anode by 1 day, andmaking a sloped bottom with the tilted angle of 1.7degrees can reduce this time by 2.5 days, while,forming two slots in the bottom surface doesn’tchange this time. The presence of the slots, reducingthe area of the working face, increases the anodiccurrent density at the underside of the anode, but leadsto the enlargement of the current at the side of theanode to some degree.

Acknowledgement

The authors are grateful for the financial supportof the National Natural Science Foundation of China(51274241) and the Foundation for InnovativeResearch Groups of the National Natural ScienceFoundation of China (61321003).

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Figure 12. The anode shapes and secondary current distributions for the anode with slots (1cm width) at (a) the initial time,after (b) 1 day, (c) 4 days and (d) 7 days.

Figure 13. The changes of the average anodic currentdensities with respect to time in the differentregions of the undersides of the anodes.

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