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 

example on dynamic simulation for temperature field in an aluminum reduction

cell. The sixth section will introduce calculation model of the current efficiency.

Some typical simulation results will also be given in the chapter. It must be noted

that hologram simulation of aluminum cells are not perfect, hence unremitting

hard work is still required.

Fig. 5.3 Principle of hologram simulation technology for aluminum reduction cells

5.2



Computation and Analysis of the Electric Field and Magnetic

Field

The aluminum reduction cell is an electrochemistry reactor acted on high direct

  Naijun Zhou

electric current. The current is the original cause of all the phenomena in

aluminum reduction cells. On one hand, high electric current results in strong

magnetic field and the electromagnetic force then causes melt motion and molten

aluminum flowing and influences heat

and mass transfer as well as the

operating conditions. On the other hand,

the electric current directly produces

heat, which keeps cells at an

appropriate temperature.

The distribution of magnetic field

in a cell depends on the electric

current distribution in the cell and

the adjacent cells. The electric

current is usually divided in several

sections such as bus bar, anode, melt

and cathode etc. and can be treated

respectively. Fig.5.4 is a schematic

of the conduction structure in a

typical aluminum reduction cell.

5.2.1



Computation model of electric current in the bus bar

According to Kirchhoff’s law, current that flows through each part of the bus bar

system can be calculated as:

EIR=

∑∑

• (5.1)

I =

∑

(5.2)

where

j is the node number.

In calculation, all sections of the bus bar are replaced with equivalent resistance.

Then on the basis of series-parallel connection relation among conductors of the

bus bar, a network chart of electric circuit is obtained, and electric potential at

each node and current along each section of the bus bar are computed using a

program.

In calculation, because of the complicated relation of the bus bar sections,

computation is quite difficult and fussy. Fig.5.5 shows a typical bus bar calculation

network chart. For details of the computational methods, program and the

examples, please refer to references (Imery, 1989; Vao, 1990).

Fig. 5.4



Conduction structure in an

aluminum reduction cell

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

Fig. 5.5



Computation current meshwork of bus bars

5.2.2



Computational model of electric current in the anode

Based on the calculated current that flows through anode rod into anode block and

appropriate assumption of boundary conditions, electric potential and current of

each control volume (node) can be calculated with Kirchhaff’s law and resistivity

of anode block by using three-dimensional finite difference method or finite

element method. The procedure is the same as that of calculation of the bus bar.

However, it must be noted that, since the resistivity is usually a function of

temperature, the heat of the control element should be computed before working

out the current distribution (electric field). Then the temperature distribution in

anode block can be obtained by solving heat and conduction differential equations

simultaneously.

The control equations are:

a) Electric conduction differential equation:

∇∇=• (5.3)

b) Heat conduction equation:

vol

0Tq

∇∇+ =• (5.4)

where σ is the electric conduction rate of the anode material; λ is the heat

conduction coefficient;

vol

is Joule heat in control volume that is given by:

1,,

,1,

,,1

,,vol

+−+−+−

+++++=⇒

kji

qqqqqqqq (5.5)

where

ΔΔ

1, , 2

,, 1,, ,,

()

ijk

ijk x i jk ijk

qVV

−

=−•• (5.6)

And the rest may be inferred by analogy.

The difference equations of Eq. 5.3 and Eq. 5.4 can be expressed as:

  Naijun Zhou

, , 1 1, , 1, , 2 , 1, , 1,

3,,1 ,,1 123

[( ) ( )

()]/2()

ijk i jk i jk ij k ij k

ijk ijk

VaV V aV V

aV V a a a

−+ −+

−+

=++++

+++

••

•

(5.7)

,, 1 1,, 1,, 2 , 1, , 1,

3,,1 ,,1 ,, 123

[( ) ( )

()]/2()

ijk i jk i jk ij k ij k

ijk ijk ijk

TbT T bT T

bT T q bb b

−+ −+

−+

=++++

++ ++

••

•

(5.8)

where,

123

xyz

zxzx

aaa

zxzx

bbb

σσσ

λλλ

ΔΔ ΔΔ ΔΔ

ΔΔΔ

ΔΔ ΔΔ ΔΔ

ΔΔΔ

===

•••

Eq.5.7 and Eq.5.8 can be solved by Gaussian elimination or iteration. The

boundary conditions are listed as following:

a) The electric potential at molten aluminum surface is evenly distributed.

b) The electric on the cathode carbon surface is evenly distributed.

c) The temperature in the electrolyte is uniform everywhere, and the value can

be obtained by measuring.

d) The temperature at the joint of anode bus bar and rod should be given.

e) The thermal boundary conditions on the carbon surface are defined as the

type of heat convection boundary.

The program chart of the computation is showed in Fig.5.6. This procedure was

used to compute and analysis the electric field of anode, as introduced in

references (Gao, 1991; JLMRI, 1980; Zen, 1996; Zen, 2004). Fig.5.7 gives a

computation result.

5.2.3



Computation and analysis of electric field in the melt

The melt in aluminum reduction cells is composed of electrolyte and molten

aluminum. Because of the great difference of their electrical resistivity, the current

distribution in the melt is evident. For instance, since the electrical resistivity in

electrolyte is larger than in the molten aluminum, the current density in the

electrolyte below the anode bottom is almost uniform, and pointing vertically

downwards; however in the electrolyte at the side face of anode, the current

density is much smaller. Haupin had proposed a sector coefficient method to

approximate the current in these parts(Haupin,1990). Of course it can also be

accurately computed by finite difference method or finite element method with

nonlinear boundary conditions.

The current distribution in the molten aluminum is complicated. The molten

aluminum is a good conductor. Moreover, because of the existence of frozen ledge

and deposits, there is usually large horizontal current found in the molten aluminum,

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

Fig. 5.6



Computation procedure of current field for the anode system

which results in the fluctuation and flow of the melt, and causes some unwanted

influences on the operation. Therefore, the current distribution in the melt must be

carefully studied and analyzed.

To compute the current distribution in the melt, there are many kinds of

methods, such as finite difference method, finite element method, network graph

theory, and boundary element method etc. (Arita, 1983; Tarapore, 1982; Richand,

1976; Chen, 1986). The basic control equations are Eq. 5.3 and Eq.5.4. However,

  Naijun Zhou

Fig. 5.7



Computational result of electric distribution in anode (Gao, 1991)

different from that in last section, because of the symmetrical layout of the current

carriers, the current distribution in the melt is symmetrical along the longitudinal

axis in aluminum reduction cells, which therefore can be treated as a

two-dimensional problem. The computation is to be illuminated with method of

network graph theory as below.

The electrical potential drop occurs mainly in the electrolyte. In computation of

the electric field in the electrolyte, the anode and molten aluminum can be

regarded as equipotential volumes. In this case, the electric field in the electrolyte

can be studied as a potential boundary value problem with given electric current.

The potential drop is very small, so the potential difference at the electrolyte-metal

interface and the metal-cathode blocks interface must be taken into account. With

the current density at the interface of metal being determined, the electric field in

it can be studied as a potential field problem with the second boundary condition.

For an electric current field as shown in Fig. 5.8, the medium is homogeneously

isotropic and the thickness is

zΔ . The potential of node A is

, and the

conductance of branch

AB is:

ΔΔ

• (5.9)

Current of branch AB is:

,,1,

()

ij ij i j

IVVG

−

=− • (5.10)

According to Kirchhoff’s junction current law:

0I =

∑

(5.11)

For the inner node

()

ij:

,1, ,,1 1,, ,1,

[( )( )( )( )]0

ij i j ij ij i j ij ij ij

GV V V V V V V V

−−++

−+−−−−−=•

(5.12)

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

Fig. 5.8



Mesh nodes of current field

Namely:

,1,,11,,1

=−+++

+−−+ jijijijiji

VVVVV

(5.13)

Similarly, for the edge node

()

jiB ,:

010343

,1,,11,

=−++

−++ jijijiji

VVVV (5.14)

Potential equations of other edge nodes can be obtained in the same way. For

the boundary problem, the boundary condition of the electrolyte can be expressed

as a matrix after discretization as:

VB (5.15)

For the node that is of the second boundary condition problems such as

()

jiC , ,

if the effluent current is

()

jiI ,

, then:

,1, ,,1 1,, 0

()()()(,)0

[]

ij i j ij ij i j ij

GVV VV V V Iij

++−

−+−− −+ =• (5.16)

That is:

1, , 1 1, , 0

24(,)

i j ij i j ij

VVVV Iij

++−

++−=• (5.17)

A matrix can then be obtained with the potential equations at every boundary

nodes as:

*,7 = (5.18)

After the relative potential has been determined (for example, set the potential

at the electrolyte-metal interface to be zero, i.e.

()

=jiV ), the potential of

each node can be solved with matrix 5.15 by using elimination method or

over-relaxation method, and the current density at the upper interface of the

molten aluminum

()

jiI ,

can be obtained as well. The current density at lower

interface

()

jiI ,

can be obtained in the same way. Substituting the current

densities into matrix 5.18, the potential at each node of the molten aluminum can

be solved using over-relaxation method, and the horizontal and vertical currents

can thus be decided. A computed typical current distribution in the melt is shown

in Fig. 5.9.

  Naijun Zhou

Fig. 5.9 Computational result of current distribution in the melt

(a) Potential distribution; (b) Current distribution

5.2.4



Computation and analysis of electric field in the cathode

The computation and analysis methods stated in the above two sections are also

applicable for that of the cathode. Because of the existence of collector bar, the

model of cathode is usually three-dimensional (or a mix of two dimensions and

three dimensions). When taking the change of side ledge shape and deformation of

the molten aluminum height into accounts, computation will be more convenient

by using the boundary element method.

The Boundary element method is to solve a group of integral equations with

boundary conditions(Zhou,1983). The procedure is:

a) To discretize the boundary into units with certain shapes.

b) To construct interpolating function in each unit.

c) To convert the boundary integral equations into a group of algebraic equations.

d) To solve the equations.

In Section 2.4.2, the integral equation and fundamental solution of liner

constant current field boundary problem have been introduced. Here the solving

process is to be discussed. Assuming

V is the potential function and

V is the

fundamental solution, Eq. 2.190 and Eq. 2.191 can then be expressed as:

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

()

CV V V S

CiS

⎧

∂∂

=−

⎪

∂∂

⎪

∈Ω⎪

⎧

⎨

⎪

=∈

⎨

⎪

∉Ω

⎪

⎩

∫

(5.19)

For three-dimensional isotropic medium:

π4

= (5.20)

where

is the distance from a field point to the source point.

For two-dimensional field:

π2

= (5.21)

For a two-dimensional field,

S can be divided into N units using liner

interpolation function:

⎪

⎩

⎪

⎨

⎧

121

qqq

VVV

φφ

(5.22)

where

)1(

ζφ

−= , )1(

ζφ

+= , and

is the local coordinate. Eq. 5.19 can

then be discretized as:

∑∑

−=

jijjijii

VHqGVC

(5.23)

where:

HSS

φφ

−

∂∂

∫∫

GVSVS

φφ

−

∫∫

For

i =1, 2, …, N ( Si ∈ ), Eq. 5.23 can be written into a matrix as:

(2)7 = (5.24)

For the Dirichlet problem,

V is a known variable, then:

7)(2

1−

= (5.25)

Note that for the Neumann problem,

V cannot be uniquely determined. Hence,

a reference point must be decided before to solve

V . For the hybrid problem, the

unknown variables in Eq. 5.24 can be moved to the left of equation:

F= (5.26)

where

x is the column vector of V and q , the unknown variables at the

boundary. After solving the unknown variables in

S , the potential at each points

Ω can be obtained by solving Eq. 5.23. Then, with the following equation

(Eq.5.27),current density J

and J

of each points in Ω can be solved.

  Naijun Zhou

∫

∂

−

∂

−

∂

)(

(5.27)

where

∂

;

is conductivity.

In the model, there are following a ssumptions:

a) The electric current filed is approximated to be two-dimension.

b) The side ledge is an insulator.

c) The outlet of the cathode block is isopotential.

d) The cathode-electrolyte interface is isopotential.

e) Temperature inside the cathode blocks and collector bar are uniform, and every

subregions have the same resistivity.

f) The cross sections of the cathode rod are the same, an equivalent height of the

rod is used instead of the real value.

It must be pointed out that, because the cathode carbon block is orthogonal

anisotropic, the computation of the electric field in the cathode is a problem to

solving a multi-media electric field in orthogonal anisotropic material. Lu Jiayu

had proposed a method for this kind of medium(Chen et al., 1986).

The

computation procedure is given in Fig. 5.10, and a typical result is shown in Fig.

5.11.

5.2.5



Computation and analysis of the magnetic field

The operation of aluminum reduction cells is ensured by high DC current.

According to the electromagnetics, when current runs through a conductor, a

magnetic field will be produced in the space around, and an electromagnetic force

field will be produced under the interaction of the magnetic field and the current. It

is the electromagnetic force that results in the melt movement, which has a negative

effect on the process of aluminum production. Once an aluminum reduction cell is

designed and installed, its magnetic field is basically fixed. Therefore optimization

of bus bar collocation in design process is very important, so as to predict the

magnetic field in the operation and to minimize the negative effect.

With the results of the electrical fields as introduced in Section 2.4, computation

and analysis of magnetic field can be carried out. In computation and analysis of

the magnetic field in the cells. An equivalent mathematical model is usually

employed. That is, to replace the rectangular bus bars (including the melt) with

columnar bus bars that is of the same length and cross-sectional area, then for one