Mei C., Zhou J., Peng X. Simulation and Optimization of Furnaces and Kilns for Nonferrous Metallurgical Engineering

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5 Hologram Simulation of Aluminum Reduction Cells 

Fig. 5.10



Program chart of computation of current field in cathode

Fig. 5.11



Calculation result of current field in cathode (Lu, 1986)

point in the magnetic field, to calculate magnetic vectors produced by each

columnar bus bars with the Biot-Savart Law; the magnetic intensity at that the

specified point can thus be obtained.

In Section 2.4.3, the model and method of computation of magnetic field caused

by electrified bus bars have been discussed in details. In previous research, it has

  Naijun Zhou

been found that the effect of iron shell is not neglectable in computation of the

magnetic field distribution. There are several methods to calculate the additional

magnetic field caused by ferro-magnetic materials, as discussed in Section 2.2.4,

of which the magnetic dipole model method (Robl, 1978; Sele, 1974) is

comparatively more popular and accurate. The principle and calculation procedure

of this method are as follows:

a) Using the equivalent column model to calculate the magnetic induction

intensity (components) of one point produced by a current-bearing conductor.

b) To simplify the iron structure into an equivalent magnetic dipole column rod

with spherical ends (rotation ellipsoid), and to restrict the dipole to be along one of

the three axes of the coordinates. Moreover, the length and the cross-sectional area

of the dipole must be the same as that of the iron structure it represents.

c) To calculate the magnetic effect of each end of the dipole at the specified

point. The magnetic induction intensity at this point is then the superposition of

the magnetic effects of the two ends of the dipole.

A representative structure of the magnetic dipole of a shell is given in Fig. 5.12.

Fig.5.12



Structure of magnetic dipole in a steel shell

(a) Main magnetic flux distribution in the shell; (b) Magnetic dipole of the shell

There are also other methods of calculating the magnetic field, such as

differential method, integral method and two scalar potentials method etc. For

details of these methods, please refer to references (Sneyd, 1985; Qiu, 1992; Li,

1993)

It should be pointed out that a simpler method of equivalent potline

current(JLMRI,1980) was used in original computation of the magnetic field at

early time. However, for either the equivalent series currents method or the

equivalent column method, errors are unavoidable because of the rectangular cross

section of the bus bars, and larger errors may be found near the bus bars. To solve

the problem, Chen Shiyu et al. developed a more accurate method (Chen, 1987) to

directly derive the magnetic field with the actual shape of the bus bars. Meanwhile,

many researchers carried out lots of computations of the magnetic field in

aluminum reduction cells(Chen,1987;Zen,1996;Qiu,1992;Liu et al., 1996; Zeng et

al.,1996;Liang et al.,1998;Severo,2005), and their results are all helpful for the

optimization of the bus bars configuration.

5 Hologram Simulation of Aluminum Reduction Cells 

Here some examples are given to illuminate the application of methods

mentioned above. Fig. 5.13 shows the computational result of the magnetic field

in the middle part of the molten aluminum in a 160kA prebaked cell (Zen, 1996).

In the result, the horizon magnetic field was found in a large vortical structure,

leaning to current inlet side, while the vertical magnetic field is in a saddle

shape.

Fig.5.13



Computational result of magnetic field in an aluminum reduction cell

(160kA cell with four risers)(Zen,1996)

(a) Horizontal magnetic field; (b) Vertical magnetic field

Fig. 5.14 is the computational result of the magnetic field in a 280kA test cell

obtained by Wang Ruliang et al.(Wang et al.,1998) of Shenyang Aluminum &

Magnesium Academe. Significant difference can be found in the result whether or

not the magnetic screening action caused by ferro-magnetic material and the effect

from the neighbor cells are taken into account. The magnetic field distribution was

found to be optimized after some adjustment of the bus bars arrangement,

according to the statement in the paper.

  Naijun Zhou

Fig.5.14



Vertical magnetic field distribution in molten aluminum

with symmetrical bus bars (Wang et al., 1998)

(a) Magnetic screening action is neglected;(b) Magnetic screening action is considered

1üCurrent inlet side (without effect by neighbor cells);

2üCurrent outlet side (without effect by neighbor cells);

3üCurrent inlet side (with effect by neighbor cells);

4üCurrent outlet side (with effect by neighbor cells)

Up to now ,it is still difficult to make an accurate theoretical computation of the

influence of ferro-magnetic material on the magnetic field. Usually, physical

simulation and experiments have to be employed for assistance (Shen et al., 1994).

Fig.5.15 shows the magnetic distribution in the middle part of the molten

aluminum of a 200kA prebaked cell computed with the commercial software

ANSYS by the authors. In the results, two large vortices are found in the

horizontal magnetic field, whilst the vertical magnetic field presents as

anti-symmetric distribution in two sides of x-axis and y-axis which intersect in

center of the cell.

A research group at the Central South University measured the magnetic

induction intensities in the molten aluminum of the same kind reduction cell using

a three-dimensional gaussmeter. The measured values are listed together with the

corresponding computational results in Table 5.1.

5 Hologram Simulation of Aluminum Reduction Cells 

Fig. 5.15



Computational result in a 200kA pre-baked cell

(a)The horizontal magnetic field; (b)The vertical magnetic field

Table 5.1 Comparison of measured values and simulated values of magnetic field of

a 200kA pre-baked anode cell

Measured

oint

Coordinates of

measurement

points/m

Measured values of

magnetic field

/Gs

Simulated values of

magnetic field

/Gs

x y z B

1 1.13 0.25 1.30 107.5 4.4

3.3

105.45 3.17

5.87

2 2.53 0.25 1.30 108.5 5.3

6.4

103.27 5.06

11.86

3 3.93 0.25 1.30 124.4 20.2 4.1 101.22 10.43 6.34

4 5.33 0.25 1.30 131.4 32.2 13.7 113.86 28.83 7.30

5 6.73 0.25 1.30 123.4 23.2 13.0 108.72 11.61 7.13

6 8.13 0.25 1.30 114.5 14.2 12.2 102.60 10.15 13.68

7 9.53 0.25 1.30 106.0 8.2 11.4 92.97 7.53 6.41

8 1.13 3.65 1.30

103.622.3 14.9 102.30 20.25 13.30

9 2.53 3.65 1.30

102.324.0

1.3

98.08 11.63

5.04

10 3.93 3.65 1.30

90.3 6.5

5.5

86.33 3.42

2.16

11 5.33 3.65 1.30

83.8 5.3

9.5

80.03 2.27

5.56

12 6.73 3.65 1.30



74.5



10.9

17.9



75.95



9.54

8.05

13 8.13 3.65 1.30

105.26.5

16.2

98.18 6.73

5.04

  Naijun Zhou

Continues Table 5.1

Measured

oint

Coordinates of

measurement

points/m

Measured values of

magnetic field

/Gs

Simulated values of

magnetic field

/Gs

x y z B

14 9.53 3.65 1.30 114.8 17.2

11.2

109.07 20.33

8.40

15 0.35 1.10 1.30 25.4

14.4 1.1

29.18

10.14 4.46

16 0.35 1.95 1.30

4.7 16.5 28.5 2.33 3.90 24.50

17 0.35 2.70 1.30

66.4 5.3 22.9 54.12

0.82

17.23

18 10.3 1.00 1.30 37.1 17.6 11.7 44.26 11.55 8.57

19 10.3 1.95 1.30

14.4

13.2 10.6

8.34

3.41 6.59

20 10.3 2.70 1.30

68.4

6.4 31.2

57.75

7.93 28.91

As shown in Table 5.1, the values of measurement are in good agreement with

most simulation results, and their changing trends are the same. The maximum

deviations of the magnetic induction intensities are respectively 20Gs in x

direction, 10Gs in y direction and 5Gs in z direction. The reasons for such

deviations mainly are:

a) Inaccurate positioning of the the measurement;

b) Measurement errors;

c) Fluctuation of currents and voltages during of the measurement.

5.3



Computation and Analysis of the Melt Flow Field

The processes in the melt in aluminum reduction cells are very complex, which

include electrochemical reactions, chemical reactions and physical processes such as

dissolution, diffusion, fluid flow and heat transfer. In those processes, the physical

processes are particularly important, having a critical effect on electrolytic reduction,

of which the melt flow plays a decisive role. Therefore, computation and analysis of

the melt flow field is a key part of the “three fields” analysis.

Movement of the melt may be caused by many factors. Firstly, the

electromagnetic force will cause the movement. Secondly, force produced

when gases rising from the anode bottom in the electrolysis process will also

result in the melt movement (Sun et al.,2004). Other factors such as mass

transfer, heat transfer, pole changing, charging, aluminum discharging, edge

processing and the effect treating will cause melt movement as well. The first

two factors are the main reasons for the movement of the melt during the

stable operation period.

5 Hologram Simulation of Aluminum Reduction Cells 

5.3.1



Electromagnetic force in the melt

Current produces magnetic fields, thus an electromagnetic force (i.e. the Laplace

force) will be caused and acted on the conductor in the magnetic field. The

electromagnetic force is vertical to the plane

that determined by vectors of current

I and

magnetic induction intensity

B, and its

direction can be decided by the right-hand

rule.

In the analysis, the horizontal cross-section

of the bath is divided into four parts (i.e.

section I to

Č , as shown in Fig. 5.16).

Components of the magnetic induction

intensity along axis Ox, Oy or Oz are

respectively B

, B

. The Laplace force

produced in the melt along axis Ox and Oy are correspondingly:

zxxz

BIBIyf −=)( (5.28)

yzzy

BIBIxf −=)( (5.29)

The sum of the Laplace force along y direction in section

ĉis:

∫∫

yyfxF

d)(d (5.30)

where with the magnetic field distribution as shown in Fig. 5.16, the F

is to be

against the direction of y axis.

The sum of Laplace force along x direction in section

ĉis:

∫∫

xxfyF

d)(d (5.31)

where its direction is against x axis. The values and directions of the Laplace force

in other three sections can be decided in the same way. As the horizontal

resultant forces in the four sections all point to the origin of the coordinates,

equations of force balance during the stable operation period of a reduction cell

can be obtained:

, , ,

IIIIIIVIIIIVI yyxxyyxx

FFFFFFFF =−−==−=−

According to above equations, the electrolyte-molten aluminum interface

should be in the shape of a curved surface. In each section, because bearing

unbalanced electromagnetic forces, melt, especially molten aluminum, will move

horizontally (Frederic et al., 2002; Wu et al., 2002; Sun et al., 2005).

Fig. 5.16 Synthesis of horizontal

electromagnetic forces in the melt

  Naijun Zhou

5.3.2



Analysis of the molten aluminum movement

Beside the electromagnetic force, there are many factors determining the flow

speed and surface shape of the molten aluminum. Usually, Navier-Stokes equation

and continuous equation are used to describe the movement of the molten

aluminum.

ρμ

=−∇+∇

∇=

•

(5.32)

where pressure gradient can be obtained by summing internal and external forces:

mwra

P∇= + + +FFFF (5.33)

In which:

External force is:

FF F=+

Electromagnetic force is:

FIB=×

Gravity force is:

Fg=

Flow resistance is:

=∇•FU

Flow inertia force is:

=∇•FUU

It can be proved that(Liu et al., 1996):

P∇=∇• F (5.34)

ρμ

⎛⎞

∇× =∇× − ∇

⎜⎟

⎝⎠

(5.35)

The above two equations can be used to solve the pressure field and velocity

field. As the rotation of the external force seldom equals to zero, the movement in

aluminum reduction cells is a rotational flow. Basically, there are four kinds of

flow fields as shown in Fig. 5.17. Of course, more structures of flow fields can be

formed by superposing the four basic structures, as illustrated in Fig. 5.18.

Fig. 5.17



Four basic structures of velocity field in the molten aluminum

5 Hologram Simulation of Aluminum Reduction Cells 

Fig. 5.18



Superposition of basic movements

5.3.3



Analysis of the electrolyte movement

The movement of electrolyte with certain speed and direction is good for the

dissolution of alumina, the uniform concentration of electrolyte and the release of heat

that produced under anode. However, it will also results in fluctuation of the molten

aluminum and increasing of second reaction, thus makes the cells unstable and reduces

the current efficiency. Hence, it is necessary to study the electrolyte movement.

Beside the electromagnetic force, the bubble movement from chemical reaction

is the other motive force of the electrolyte movement, which thus makes the

process even more complex. Since the force from thermal heat pressure gradient is

relatively less important, it is usually neglected.

The movement caused by the electromagnetic force can be analyzed in the

similar way to that of the molten aluminum. Actually, the horizontal current in the

electrolyte can be neglected, so there is only horizontal electromagnetic force to

be considered.

The movement caused by bubbles is primarily horizontal under the bottom of

the anode but vertical along the side of the anode. This movement are determined

by the generation and motion of bubbles. The motion of bubbles can be divided

into three stages:

a) Bubbles are generated under the anode, and then growing up till depart from

the surface of the anode.

b) Under the anode, the bubbles movement are primarily caused by the pressure

gradient and fluid viscousity.

c) Buoyancy is the main force for the bubbles movement along the side of anode.

The movements of the bubbles will cause the convection, or even turbulence in

the electrolyte, thus leads to oscillation of the free surface. The bubbles can also

produce electromagnetic effect. Because the gas is not a conductor, the effect will

influence the distribution of current and magnetic field, and it may cut off the

current and cause anode effect at worst (Xue et al., 2006).

  Naijun Zhou

The superposition of the motions that result from electromagnetic force and

bubbles is the electrolyte movement. It is difficult to theoretically analyze the

electrolyte movement. Many researchers have engaged in the study with physical

models (Zhou, 2006). For example, Derneded et al. proposed firstly to simulate

the flow of the electrolyte and molten aluminum in the possessing section at the

sidepiece of the anode using water and organic liquid in a cold model. The average

vertical velocity of the electrolyte at sidepiece of anode (Derneded, 1975) they

determined is:

0.044 0.165

0.458 0.467 0.258

0.173 ( ) (cm/s)ugQhl

μσ

ρρ

−−

−

⎛⎞ ⎛⎞

⎟⎟

⎜⎜

⎟⎟

⎜⎜

⎟⎟

⎜⎜

⎟⎟

⎜⎜

⎝⎠ ⎝⎠

••••• (5.36)

And the velocity of the electrolyte at the surface of the ledge is

0.25

⎛

⎞

⎜

⎟

⎠

⎝

• (5.37)

where

is gravity, g=981cm/s

; Q is gas velocity per unit anode perimeter, cm/s;

h is the depth of anode immersed in the electrolyte, cm; l is the distance between

the anode and the ledge, cm;

μρ

is the kinematical viscosity of the electrolyte,

/s;

is surface tension, N/cm;

is density, g/cm

5.3.4



Computation of the melt velocity field

The movement of the melt in reduction cells is a three-dimensional turbulent motion.

However, for the melt under the anode, because its motion is primarily forced by the

horizontal electromagnetic force and the vertical component of the movement is far

less than horizontal one, the computational model is simplified as a two-dimensional

problem(Sun and Mei, 1989; Huang et al., 1994; Liang et al., 1998). The k-

turbulent model is employed in the computation, in which only electromagnetic

force is considered whilst effect of the bubble phase is neglected, and the two

layers of the molten aluminum and electrolyte are considered as immiscible. To

describe the movement of the melt, the Navier-Stocks equations is averaged along

the depth of the layers. Because of the restriction of bottom surface, assumption

of a “rigid cover” has to be made, and the control equations can be set up as

following using square friction resistance:

∂∂

(5.38)

eff eff eff

221/2

eff

( )

uuuuv p u u v

xx x

uv u F

yx xH

ννν

⎛⎞⎛⎞

∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

⎛⎞

++=− + + + +

⎜⎟⎜⎟

⎜⎟

∂∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂

⎝⎠

⎝⎠⎝⎠

∂∂ ∂

⎛⎞

−− + +

⎜⎟

∂∂ ∂

⎝⎠

(5.39)