Impact of magnetohydrodynamic and bubbles driving forces on the alumina concentration in the bath of an Hall-Heroult cell Текст научной статьи по специальности «Математика»

Journal of Siberian Federal University. Engineering & Technologies 3 (2013 6) 257-267

УДК 669.713.17:537.84

Impact of Magnetohydrodynamic and Bubbles Driving Forces on the Alumina Concentration in the Bath of an Hall-Heroult Cell

Rene von Kaenela, Jacques Antillea, Michel V. Romeriob and Olivier Bessonb

a KAN-NAK Ltd. 35 B Route de Sion, 3960 Sierre, Switzerland b University of Neuchatel, 11 Emile Argand, 2000 Neuchatel, Switzerland

Received 15.05.2013, received in revised form 22.05.2013, accepted 31.05.2013

The alumina concentration in the bath plays a fundamental role on cell operation. Local depletion may lead to an anode effect when using carbon anodes. A mathematical model describing the alumina convection-diffusion process in the bath coupled to the cell magneto-hydrodynamic (MHD) in the presence of small bubbles is presented. Small bubbles may be assumed when slotted anodes are used. The relative importance of the velocity felds generated by the magnetic effects and/or the small bubbles on the alumina concentration in the bath is discussed.

Keywords: Alumina concentration modelling, gaz bubble interaction, MHD interaction

Introduction

The aluminum industry is continuously increasing the productivity of electrolysis cells by increasing the line current. In order to keep an acceptable anode current density, the anode length is almost systematically increased. As a result the central channel (distance between the anodes in the center of the cell) and the side channels (distance between the anodes to the side lining) are reduced. The channel geometry, Lorentz force fields and bubbles have an important impact on the bath velocity field. We will see that the velocity field plays a key role on the alumina distribution. In order to keep an acceptable energy input when increasing the current, the anode to cathode distance (ACD) is reduced as much as possible before reaching the Magneto-Hydrodynamic constraints. This means a further bath volume reduction. The increase of current imposes an increase of alumina feeding rate simultaneously with a reduction of bath volume. Therefore, the question of dissolution, diffusion and alumina transport becomes an important element for avoiding underfeeding leading to an increase of anode effects (AE) frequency. Alumina dissolution is a very complex phenomena in which the bath chemical composition, bath temperature, alumina temperature and alumina properties play an important role [1, 2, 3]. In this

© Siberian Federal University. All rights reserved * Corresponding author E-mail address: [email protected]

paper we assume that the diss solution is instantaneous when the alumina reaches the bath surface and concentrate the study on the diffusion and transport processes. The purpose ofthe study is to optimize the feeding quantitier (Oeedmgfsequency), aOumina feedtrs location and the number of feeders to minimize the nomber of AE and aeoid sludge.

Bnth velocity field in presence of bubbles and Lorentz force field: Theora

When the numbs r of bubbles producen, per mrr2 add per sfcond, is too latge, a numerical approach describ ing tfe motion oaeffh bubble sepasately sdould be disreaabeled.

Theoe are esoentsa.no two standard wayr to overcome this difficulty. The first cansirts inja^rforming somse kind of averaging over the equations and over the coraerponding iields. The second bypasses the averaging ard directly potrulahes ^huer flow oqpaiions iss each phase.

One oir tht mam dtificulties encountered wlleIt ptrforming ^rt averaging process is retated to the possible jumps that fields can suffer at the boundaries between the two phases. One way to overcomn this problem, see foe exampie [4], consisUs in extending the domain of definition oa each mosioh equation to the domaia occupter. by tre two phases. This if achieved by multiplying each equation by the cMracteriseic function coteeoponding to its domain of definitioa. Derivotives ere tOeii perfosmedin the rense of d^tributions allowing to keep track of these discontinuities in the averaging process.

Whateves choice we make, the resulting equations will contain terms which reflect the interaction between the two phares. The exacd shape of ehese teams are rat known; ghey have to be defined through constitutive equalionf.

Motion equations

Let Q; and Qg be the domains containing the fluid and the gas respectively, with corresponding characteristic functions x and xg = 1 - X. X satisfies (see [4]),

dt >l + as • V%l = 0, (1)

where v is the velo city field. < f > being the average of an arbitrary field f we set

a(I oe g) = <X(l oe g) > a(l oe g) e(l oe g) = < X(l oe g) e > (2)

and dsaw from (1)) the ave raging equation

or e) +div[a{, or g) % or g) ) = 0. (3)

Let p;, Ti j, g, B atl MiJ = Os-iB'Bv -—B2^ | be respectivrly the fluid ^nd gas densities , the

H-o V 2 )

stress tensor, the electric current density, the gravitation fotce density, the magnetic induction and Maxwell tenson Multiplying the MHD equations by the characteristic functions x and xg and following the derivation used for (3) we get for the fluid and gas motions the equations

(5,a,v1+<V-(v1 ® v, ))> = v(a; + MM, ))+p;a;g - < (Tl +Ml )-VXi >, (4)

P>(+ < v - (v, ® vj> = V. (a/t)+ PVW-TzVx, > (5)

where, for anyf a(; or g) J(lorg) = < %(l

or g)

Approximatipns and modeling

In order to have a tractable model we make the following assumptions.

p2 = 0; the Reynolds tensor is included in the viscous term (ct/n) with a change of the viscous constant which then becomes a dtffusivity, (the nptation will nor be changed); we assumo that (in Th T ~ and (r =

In ort^e^ir to tija^dle the ave=aging appearing in the equations we consider the different terms seprrrately (without the indices). We also assume that Maxwell tenoor is not affected by tha averaging process. Setting

a = v + w, so ghat < x w> = 0, (6)

one gets

v ® v = o=v®v-ot,TRe , (7)

wttere apRe = -< %,w<8> w >. Vetting T = {-pI + t|t(v)), where t(v) = Vv + (Vv) we finalln note tlhElt

< =T > = < x, (- pI e tit(v)) > = ap + r|a)(v) (8)

Taking these approximations ioto accoutit rlie equaaions (4)) and (5) become

pe J9» («h^i) + v • («=1 ® ( ^ = -a, ^ + v • («A ) + °V v • H + Peg- (9)

-•<ti 'vXt >

~(gVpg+V -agug=<T )%g>. (10)

Neglecting the surface tension ef=ect and handling the jump conditions between the two phases in the setting used for the equations we get

- V/ = =1 < (-^1 - x J - V~o = 0' (11)

whhrt vslip lis? a new field which takes into ac counl for the averaging on the jump f onditio ns.

Second approximation

We wow make the following assumptions

V-(a ^2) = 0. (12)

and

div{ai viP) = (0 ■ (13) We mo jeowe r inSroduce the new field fnt defined by

f2„<= ^(-Vx^^vVx, >. (14)

With this assumption (10) bec ome s

-«^+/.„,=0. (15) From (3) and (13) one draws

div vl = 0. (16)

dtag + div(a=vg) = 0 . (17)

Constitutive equations

We will now make the assumption that the field (14), i.e. f is a function of ~slip only. Following [5] we as sume that

, (18) whewe X is? a constant. this expression into (15) yiel=s, since by assumption p = p2,

-0v gVp^XoL^^p. (19)

so that

S^"^ (20)

The coefficient X has to be determined experimentally.

The model

With the above results we are now ready to give the equations on which our model is leaning. Fluid averaged equations

div vj = 0. (21)

Pi (d> («/~i) + + • (ai~ ®'~i)) = -+ V • (a- ~ ) -4- oc 2 • Mj + p ,a,g. (22)

Gas £avi^ir£i|g;e<i equations

dtag + div(a gP2) = 0, (23)

agVpl+agXv=p = 0 . (24)

Jump averaged c onditions

~2 = ~a + slip' (25)

a; + ag = 1. (26)

Boundary conditions on the different fields have to be adde d .

Alumina diffusiTn and convection: Theory

Let QczR3 be the domain representing the cell, £2Abe the bath, and Qa the anodes.

- 2260 -

Current density

The current density distribution in the bath is a function of alumina concentration, and is given

by

(¿where is the potential, and a is the electrical conductivity, which depends on the alumina concentration c.

Since dii)j = 0, we get

div(a(e)V<p) - 0 (28)

with following the boundary conditions (/'□;?) = /,, on the anodic rod, o = 0 on the bath-metal interface,

QlA

and .tiY)' = () elsewhere. <m

The current j = j j2; j3) is then computed in the following way to avoid rough approximations.

[ jt vdx — — f cr(c) vdx Jn /¡j Vx,

<29>

for suitable functions . ■, and 1 = 1; 2; 3.

The model will take intno account the analysis of the parameter cj as a function of the bath composition.

Alumina distribution

In this study, ito is assumed that the alumina feeding is known, ant that the dissolution is instantaneous. The alumina distribution in thebath £2b is given by the following partial differential equation

-^feil _ div(a(x,t))Ve(x,t) + (v(x,i)|Vc(x,i)) = 0, (30)

3t

w here

-c is She alumina concentration in mol/m3,

- <x is the anisotropic diffusion coefficient. A value of 0,5 m2/s was determining the Reynolds mean tensor. St also ltads to an average alununa concentration inflecting Industrial cells.

- v is vhe aefocity field in ¡ab induced by the bubble motion, and the MHD. The boundary and initial conditions have the following foam.

- The concantvstion a ss given on the feeder.

- The concentration flux on the anode and the bath-aluminum inaerface is a—+ p((|m) = 3; the coe

dn

cient (3 verify P =—fc] = molt m3 ,with z is the valence, and 17 is the Faraday constant, in our- case z = 6.

zF

- a— = 0 elsewhere,

dn

- c is given at time / = 0.

Weak formulation

Let us use the following notations.

- rA is uhe botto m ef" the cell (the metal-bath interface),

- ra is the top of the anodic rod (the entrance of the current),

- I}is the fehdee domata,

- rai is Üie ¡interface between tho enode and the Oath. With trhtiessiss notafiont, set

V - {t; € ffl(Q), -0 - 0 on rt} f (31)

equipped with the norm |v| = It'll-' = Li ,,■

Ii - {u G ff1!^), v = Ü On I ,}. (32)

equipped with the norm = Mir = Wlli j, TCf weak formulation oO probtem (28) is: Find o e V such that, t'i/j el'.

I dx = / iavdfr-

Ja Jr.

(33)

Then the computation of the current density j = (j\: j2: /,) is obtained from equation (29) W' e L2(ii). Finally, the weak formulation for the alumina concentration is: Find ce/;(().'/';//'(QJ) with

c = CfOnFf, suchthat € ü -

J fit V™ / Jth 'rn„ur,,

(34)

Numerical methods

Let us dmcompose the domains Q, and Qt into classical tetrahedral finite element mesh. For the numerical simulations of problems (33), (29), and (34), the following algorithm is used.

- IaOiitliztiioe

An initial alumina concentration distribution c0 is given at time t = 0. Let t be the time step.

- ttermtions

For time tm = m- x, if ch is the concentration at time tm

a) Comjfaite the electrisrl conductivity « =

b) Find oh € Vh suchthat, W' e Vh.

j fcrV^lVV') dj- = f joûitu. Jù /r„

(35)

c) The electric current jh = {j,;, 1; j,;,2; jh:3) is obtained as, V/ = 1; 2; 3, and Yv e z>,.

[ jh.tV = - f ■hi Ja

d) A BDF scheme oO order 2 [6] is uand for the time discretization of the concentration equation (34). Moreover a Petrov-Galerkin streamline diffusion method is applied for the advection term [7, 8]. We get the following equation (8 small parameter).

Find c"+1, solution of

f (vlV^K^ + itV^lv))^--& f (37)

for all i-e ii*

Alumina diffusion and convection: Industrial cell

In this section some numerical result for the computation of the alumina distribution in the bath are presented. On the feeders, the alumina concentration is set to 5 % of the bath weight. As a stationary solution is presented, continuous feeding is assumed. The impact of dump feeding could easily be analysed.

Figure 1 correspond to the statio nary alumina distributio n, when the velocity is neglected. The concentration is shown usdet the anndes. The two feeders locations appear clearly in the figure. The asymmetry of the diffusion pattern reflects the larger channel width at the feeders. Figure 2 shows the alumina concentration under the same conditions at metalbath interface level. Away from the feeders, at a distance larger than about one anrde width, the concewtaation is close tn 2,55 %. The vertical variation of thg alumina concentration is 0:5 % under tOe feeders. It is negligiWe awey frrm the feeders.

In fig. 3, the velocity streamlines induced by the MHD are presented.

The impact of this velocity field is shown in fig. 4.

The previous cases dig not Sake t°e butoles into gctount. It is well known that they have ae important effect on the velocity field. Moceover the considered cell lias slotted anodes. This also has an

Fig. 1. Alumina concentration in the bath when the velocity is zero, under the anodes

Fig. 2. Alumina concentration in the bath when the velocity is zero, bath-metal interface

Fig. 5. Alumina concentration in the bath, velocity induced by MHD, bubbles, and slots

cone. Ctiff.

r

-0.020

-O.&IO -00W -00B0

-o.oaa

Fig. 6. Alumina concentration variaeion due to MHD [ %]

t^ns. dirt.

r

-Q.QJO

|-ao4o

-C.Cfii

impact on the velocity. Figure 5 considers the case when the velocity field consists of the effects of the MDH, the bubbles, and the slots in the anodes.

From tne diffeeent figures, the alumina concentration field appears as sltghtly modified by the velocity field. However, when considering the concertration enolution, thr time needed for reacbing tae stationary stare is eeduced bn a fector 2 in any aituation wien the velocity field is acting. Therefore the velocity field plays an important role in the feeding process (alumina dumps).

To highlight the role of the velocity field, fig. 6 and 7 show the difference between the alumina concentration field, due to the diifusion only and in presence of MHD velocity, aesp. total velocity

The highest diaaerences are onserved at tiie ends of ihe celt, due etsentially to the MHD effecats. High negative values relate ta high alumina concentration difference. The effect; of bubbles tnd slnts generate iutt>ulento, homogenieing the (concentration distribufion.

Conclusions

A new model for the velocity field in presence of MHD, and small bubbles is developed. This velocity field is used to determine the evolution of the alumina concentration using a non-stationary convection-diffusion equation. This equation takes into account the feeding, and the Faraday law at the anodes and cathode.

The application to an existing cell with two point feeders demonstrate the following:

- 265 -

- The alumina concentration can vary up to 2,5 %. Typically a variation 1 % can be expected between anodes.

- The time needed to reach the stationary state due to the diffusion process only is twice the one for the case with MHD and bubbles effects velocity fields. It was found around two minutes.

- The velocity field has an important effect for the alumina distribution under the anodes. It helps to homogenize the alumina concentration.

- Bubbles and slots modify the velocity field which generate turbulences leading to increased homogenizing effects.

References

[1] Kobbeltvedt O., Rolseth S., Thonstad J. // Department of Electrochemistry, Norwegian Institute of Technology, N-7034 Trondheim, Norway SINTEF Materials Technology, N7034 Trondheim, Norway.

[2] Haverkamp R.G. // PhD Thesis, University of Auckland (1992).

[3] Kobbeltvedt O., Rolseth S., Thonstad J. // Light Metals, TMS, 1996.

[4] Drew D.A. and Passman S.L. Theory of Multicomponent Fluids. Spinger 1999.

[5] Panescu F.R.G. // Inria Sophia: Antipolis, 2006.

[6] Hairer E., Nrstett S.P., Wanner G. Solving Ordinary Differential Equations I. Springer-Verlag,

1987.

[7] Johnson C., Saranen J. // Math. of Comp. 47. (1986). P. 118.

[8] Franca L.P., Hauke G., Masud A. // Comput. Methods Appl. Mech. Engrg. 195. (2006). P. 15601572.

Влияние магнитогидродинамической силы и силы, обусловленной движением пузырьков, на концентрацию глинозёма в ванне Эру-Холла

Rene von Kaenel8, Jacques Antillea, Michel V. Romerio5, Olivier Besson5

aKAN-NAK Ltd. Route de Sion 35 B, 3960 Sierre, Switzerland 6University of Neuchatel, Emile Argand 11, 2000 Neuchatel, Switzerland

Концентрация глинозёма в алюминиевом электролизере играет ключевуюроль в технологических операциях. Локальное истощение может привести к анодному эффекту при использовании углеродных анодов. Представлена математическая модель, описывающая конвективно-диффузионный перенос глинозёма, связанный с магнитогидродинамикой (МГД) в ванне при малых размерах пузырьков. Маленькие пузырьки можно получить при использовании анодов с пазами. Обсуждена относительная значимость полей скоростей, создаваемых магнитными эффектами и/или маленькими пузырями, на концентрацию глинозёма в электролизере.

Ключевые слова: моделирование концентрации глинозёма, взаимодействие газовых пузырьков, магнитогидродинамическое взаимодействие.