Voltage distributions in an aluminium smelter prebaked anode*

B. F. Bradley D. W. Macmillan

British Aluminium Co.

S. P. Caste11

Ltd. Gerrards Cross, Bucks SL9 OQB, UK

Bremar Holdings Ltd, Bremar House. Sale Place London W2 IPT (Received 12 October 1976; revised 7 March 197n

Introduction

Aluminium smelting is carried out commercially by the H&roult-Hall reduction process. This is an electrolytic method whereby alumina is reduced in molten cryolite to give aluminium and oxygen. Carbon electrodes are used and the oxygen liberated at the anode reacts with the carbon to produce carbon dioxide. Due to this reaction the anode is consumed and must be continually replenished. The carbon consumption is a significant part of the process cost and for every ton of aluminium produced about 0.5 ton of carbon is used.

In a modern smelter the anode arrangement consists of a number of prebaked carbon blocks placed side by side in a cell, usually in two or four rows. At British Aluminium’s Invergordon smelter 18 of these blocks are used, divided into two rows. Each block has a life of 18 days and at the end of this period it must be removed from the cell and replaced by a new one. The lives of the blocks are staggered so that normally it is only necessary to replace one block per cell per day. A cross-section of the Invergordon cell is shown in Figlnrr 1. There are 320 of these units at the smelter; they are connected in series and the line current is 130 kA. The operating voltage of each cell is about 4.5 V.

Current from the anode bus system above the cell passes down through vertical alumunium conductors, called hangers, to each individual anode which carries an average current of 7200A. At the bottom of the hanger. an 8in diameter mild steel stub is welded to the aluminium and it is this stub that enters a specially prepared hole in the top of the carbon block; cast iron is poured into the annular gap to make the electrical and mechanical joint. This operation is carried out before the anode is brought into the cellroom.

* Paper presented at the United Kingdom Simulation Council Conference on Computer Simulation. Windermere. May 1975.

Carbon anode (+I m \

(+I Electrolyte

/Metal layer

Flgure 1 Cross-section of cell

The carbon blocks in the bottom of the cell, which are covered with molten aluminium, form the cathode. Electrical connexion is made to these by steel bars which run through the base of the box and out through the sides where they connect into the cathode bus system.

The work discussed here began after process difficulties had been experienced with many of the anodes in the cells. After the commissioning of the plant it was found that the voltage drop across the anode was higher and more variable than had been expected. Although this high voltage drop meant that increased power losses were being incurred, the increased Ohmic heat generation could also impair the anode performance in the cell. As a result a number of blocks would have to be removed prematurely, involving considerable extra labour.

At the outset it was realised that the poor and variable characteristics of the cast iron joint were the principal causes of trouble and it was decided therefore, to quantify the voltage distributions throughout the anode, using a mathematical model. as a function of the degree of electrical contact at the joint. As a result. a better understanding of the physics

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Voltage distributions in an aluminium smelter prebaked anode: B. F. Bradley et al.

of the system was expected to emerge and from this it would be possible to make constructive criticism of the existing joint-making procedures and to recommend ways of improving them.

To quantify the voltage distributions the anode, which measured 40 x 30 x 21 ins high, was approximated to a uniform solid cylinder of equivalent base area. This enabled it to be simplified to a two- dimensional system with cylindrical symmetry without, it was felt, significantly affecting the validity of the model. Laplace’s equation was used to define the steady-state voltage distributions throughout the system with appropriate constraints placed on the equation at the boundaries. Finite difference methods were used and the model was programmed on an Elliott 4120 digital computer. Actual values of carbon/ cast iron contact resistance, required to define conditions at this boundary, were measured in the laboratory.

Simulations were carried out to reproduce existing conditions on the plant and also to observe the benefits of modifying the joint design. Plant trials were then implemented to verify these findings.

Model construction

The general form of Laplace’s equation in cylindrical coordinates which defines the steady-state voltage distribution at a point within a uniform solid cylinder is given by:

where R is the radial coordinate; Z is the vertical coordinate; 8 is the azimuth angle and I/ is the voltage at the point R, 0, Z. If circular symmetry is assumed i.e. V is independent of 8, then equation (1) reduces to:

2 2

SRC;+;g+g=o (2)

Equation (2) is the general equation that has been used in the model. Because of the nature of the boundary conditions it could not be integrated directly and was expressed in finite difference form to enable a numerical solution to be obtained.

A half section of the simplified anode geometry is shown in Figure 2. In the model the cavity was assumed to have a smooth vertical side, although, in practice, it is fluted and slightly tapered. This simplification was necessary so that a practical equation lattice could be constructed to define the system. Part of the lattice construction is illustrated in the bottom left hand corner of Figure 2. Each lattice point that did not coincide with a boundary was defined by equation (2) in numerical form. Along the boundaries, however, a number of constraints had to be met.

The carbon/cast iron interface regions are shown along lines E and G and a carbon/electrolyte interface region was postulated along line K. Current was assumed to flow normally across any interface area and its direction is shown by the arrows. For clarity, the interfaces are shown to have a finite thickness, but in practice they are infinitesimally thin. The dimensions of the anode were considered as variables

Carbon/cast iron R

interface, Oriain 1

z

Carbon/ electrolyte interface, h

4 Anode

D

/Lattice (0.5 in spacing) /

Figure 2 Half section of simulated anode. Arrows show current direction

in the model so that different geometries could be explored.

Boundary conditions

The current was assumed to flow from one uniform potential surface, in the steel stub, through the cast iron to carbon contact and the anode, out through the bottom face of the anode and through the electrolyte layer to a second uniform potential layer, in the metal pool. No allowance was made for any current leaving the side of the anode or for spread in the electrolyte.

Across the boundary line A no current was assumed to flow. This condition was satisfied if aV/aR = 0 for mesh points along the boundary. Equation (2) was then replaced by:

g+E&o (3)

for points on this boundary. Along the boundary line D the same condition applied but as R + 0 it can be shown that in the limit:

1 av a2T/ ---__ R aR aR=

so that along D, equation (2) was replaced by:

2a2v a2v --+E=o dR2

(4)

Along boundary line C, for no current flow across it the condition aV/aZ = 0 must hold. This boundary condition does not alter the form of equation (2) but must be satisfied in conjunction with it.

Along the remaining boundaries E, G and K, and especially at the corner points where boundary A meets boundary C, boundary E meets boundary G, etc., conditions of greater complexity were imposed of more general non-Dirichlet form involving known functions F of the indicated arguments:

av av a9 a2v -,-,p ~ p,J,,J2,J3 aR ?Z aR2’azz’

p being here the resistivity of the carbon anode and Ji, i = 1, 2, 3 variable current densities. The Ji in turn were expressed in terms of the potential of the stub

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Voltage distributions in an aluminium smelter prebaked anode: B. F. Bradley et al.

and the metal pool and the specific resistance of the interfacial materials.

Contact resistance

The concept of contact resistance was used to define the interfacial boundary conditions. It is derived from Ohm’s Law, which is valid across an interface since it is assumed that the current flow across it undergoes no divergence. Across an interface:

V = I x resistance

=I!& A

= $PL) where I is the current through the interface: p is the specific resistance of the interface ‘material’, L is the thickness of the interface ‘material’ and A is the area of the interface. Since the interface is considered to be an infinitesimally thin layer, neither p nor L can be measured independently. The product p x L, however can be determined by measuring the remaining variables; it is this product that is defined as the contact resistance; it is somewhat analogous to a heat transfer coefficient in heat transfer theory. Measurements were carried out in the laboratory to measure the contact resistance between carbon and cast iron as a function of pressure of contact. The actual value used for good contact was 3.6 x 1O-3 ohms in2; which corresponded to a contact pressure of 900 psig.

By expressing equation (2) and all boundary conditions in finite difference form, these reduced to a set of linear simultaneous equations: the whole region and its boundaries was covered with a lattice of 0.5 in mesh points such that the point (R.Z) became the lattice point (i, j) where i and j are integers defined in a finite range 0 < i < P, 0 <j < Q, the corner where boundary A meets boundary K being the point (P, Q). Then, the partial first- and second-order derivatives of the potential function V with respect to R and Z were defined in the discrete space by use of Taylor’s theorem and neglecting third-order and higher terms, so that Laplace’s equation (2) took the familiar central- difference form:

Kj = 1/4((1 - l/2i)F_,j + (1 + 1/2i)l$+,j+ xj_,

+Vj+lJ

while, for example, along boundary A, the finite difference equation corresponding to equation (3) was:

l/pj = 1/4[2Vp-,j+ l/pi- 1 + Vpj+l)

with similar equations for all other boundary points. When taken together, these equations constituted a

set of (P + l)(Q + 1) linear simultaneous algebraic equations which, written in matrix form, may be expressed as :

AV=b

with A a sparse.matrix of (P + l)(Q + 1) constant elements, Va column vector with elements:

Kj,i = O,...P j = O,...Q

and h a constant column vector.

A computer program was written (in ICL 4100 ALGOL with NEAT segments) to solve these equations by the well-known interative successive over- relaxation (SOR) method, the advantages of this method in this application being that it gave a simple and compact program textually and used relatively little computer store.

The stub region was assumed to be at uniform potential. To run the program it was first necessary to specify the appropriate anode geometry, carbon resistivity, current and contact resistance values; thus specifying the matrix A and column vector b. It was also necessary to Specify an arbitrary starting potential drop. A distribution Vwas then calculated by solving the system of equations. At the end of the program run this distribution was adjusted to the extent to which the calculated current which emerged along the bottom of the anode (obtained by summing all the currents between the baseline mesh points) compared with the known current through the anode. As a check on the accuracy of the solution, the input current was calculated in the same manner from the adjusted distribution.

The presence of boundaries E, G and K and, in particular, the re-entrant corner point where E meets G. caused the set of equations to be ill-conditioned so that, as is usual with the SOR algorithm in these circumstances, much effort was devoted to manipulating and finding the optimum SOR factor during solution of the equations, iterations being carried out to a residual error of, at most, 0.0001 mV before a stable solution for Vwas achieved.

Simulations

Two main simulations have been carried out. The first was based on the assumption that good electrical contact existed down the vertical sides of the cavity (region E in Figure 2), but poor electrical contact was assumed to exist in the base region G. There was evidence to show that this was fairly typical of the situation in practice, brought about by the procedure for making the cast iron joint. When the stub is placed in the anode cavity it rests on the base, leaving a negligible gap. Consequently, there is little likelihood of there being much penetration of the cast iron beneath the stub and poor electrical contact will result. Even though the stub is in direct contact with the carbon, this was thought unlikely to produce good electrical contact, because of the relative unevenness between the two surfaces.

In the second simulation it was assumed that, in addition to good side contact, good base contact existed; and that this could be obtained by the penetration of the cast iron beneath the stub. In practice, this could be achieved either by lifting the stub slightly off the base or by redesigning the cavity.

The results of the first simulation are shown in Figure 3. Iso-potential lines have been drawn in 25 mV steps and current flow at any point is normal to these lines. It can be seen from the high concentration of lines at the side of the cavity that current predominately enters the block in this region. It then diverges to give a fairly uniform distribution at the bottom of the anode. The mean anode voltage drop (A V) was found to be 0.496 V. The mean baseline

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Voltage distributions in an aluminium smelter prebaked anode: 8. F. Bradley et al.

each of the 320 cells then the total power saving is appreciable. This is seen from the following simple calculation :

1.925

I 875

Figure 3 Predicted voltage distributions I” anode, (good side, poor base contact in cavity). Source voltage = 2.367 V; AV = 496 mV and mean stink voltage = 1.871 V

voltage of 1.871 V represented the voltage drop through the electrolyte.

Results from the second simulation are shown in Figure 4. The benefit of improving the electrical contact all round the stub can be seen by the much more uniform current flow into the anode. The voltage drop across the anode in this case was 0.41OV, representing a saving of 86mV compared with the first simulation. This saving in itself does not appear large, but when it is realised that 130 kA is passing through

:

Figure 4 Predicted voltage distributions In anode, (good side, good base contact m cawty). Source voltage = 2.283 V; A\/ = 410 mV and mean sink voltage = 1.873 V

Total kW = Al/per cell x amps x no. cells

lo3

0.086 x 130 x IO” x 320

lo3

= 3600kW

This figure represented a significant benefit that could be obtained by relatively modest changes to the joint making procedure. Further simulations were also carried out on the basis of more drastic changes in design, namely to introduce a twin stub into the anode. The results indicated that further power saving benefits could be achieved, but since major engineering modifications would be involved, this would offset much of the advantage.

Plant trial

The trial was carried out over an 18 day period to cover the full anode life in a cell. Two groups of 22 anodes were prepared. The first group consisted of normal production anodes and was used as a control group for comparison purposes. Anodes in the second group had the base of the cavity modified as shown in Figure 5. Three concentric trenches were cut in the carbon using a special tool. In addition, the base of the stub was raised off the carbon by drilling a hole at its centre and dropping a short bolt into it; the stub then rested on the bolt head. With this arrangement, not only would the cast iron penetrate beneath the stub but the effective area of contact had been increased by the trench walls. All the anodes were suitably labelled and introduced into the cells in the normal way. During the trial, carbon/cast iron voltage drops were monitored for each anode. on a daily basis, and an average voltagecs time plot was obtained for each group.

Figure 6 shows the carbon/cast iron voltage history for about the first g-days life of the control group of anodes. The high drop at the start was because the anode was introduced into the cell cold and it is only

I Stub I

Figure 5 Diagram of modified anode cawty

342 Appt. Math. Modelling, 1977, Vol 1, September

Voltage distributions in an aluminium smelter prebaked anode: B. F. Bradley et al.

200 .

0 20 40 60 80 100 120 140 160 180

Time (h)

Figure 6 Voltage histories for control group anodes. Mean AV over 18d = 387mV

when it has heated up and there has been differential expansion of the carbon and steel that good electrical contact is established. The electrolyte temperature is about 960°C and it takes 2-3 days for equilibrium temperature levels to be reached. No data are given for the latter half of the anodes life because, for a number of reasons, measuring difficulties generally occur over this period. Other measurements, however, together with predictions from further simulations, indicate that once the voltage curve has levelled out it is maintained at this level for the remaining life of the anode. The curve drawn through the points was obtained using a curve fitting computer program.

The corresponding voltage history for the modified group of anodes is shown in Figure 7. The effect of the cast iron penetrating beneath the stub has been to lower the average curve and also the scatter of points is less. This latter effect is significant in that production of a more consistent joint can improve the overall cell performance although the actual benefits are more difficult to quantify.

Integrated over the full l&day cycle, the difference in voltage drop between the two groups of anodes was 360mV. This figure is less than the predicted value of 86mV but in the simulations it was assumed that no cast iron penetrated beneath the stubs of the normal anodes. In practice a varying degree of penetration can occur depending on the condition of the stub and this will tend to decrease the difference. This is also thought to be the reason for the greater scatter in the points of the normal anodes. On the basis of a real reduction in voltage drop of 60 mV using modified

l2OOj

0 20 40 bo 80 100 120 140 160 18

Time (h)

C

Figure 7 Voltage histones for modified cavity anodes. Mean Al/ over 18d = 326mV

anodes, the actual power saving to be achieved is corrected to a figure of 2500 kW.

Conclusions

The mathematical model which has been constructed has enabled the voltage distributions likely to occur in the Invergordon smelter prebaked anode to be quantified. The simulations have indicated that appreciable power savings could be achieved by improving the electrical contact at the base of the stub cavity. A plant trial at the smelter has confirmed these

predictions. By modifying the shape of the cavity to ensure that cast iron penetrated beneath the stub, it was demonstrated that power savings across the whole plant of about 2500 kW could be achieved. As a result of the trial. plans have been drawn up to modify existing plant procedures for making the joint.

It is considered that this work gives a good example of how the basic scientific approach of theory and experiment has convincingly analysed a problem. It has also indicated that computer simulation can be a powerful research tool which provides a clear insight into how a process works and what the effects will be of making changes in the system.

Acknowledgements

The authors are grateful to the British Aluminium Co. Ltd. for permission to publish this paper.

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