Marc Dupuis, Jonquière
In 2000, before the announcement of the AP50 cell technology, the author presented the thermo-electric design of a 400 kA cell . In 2003, that work was extended in order to present the thermo-electric design of a 500 kA cell .
Those papers were trying to demonstrate that with the support of reliable mathematical modeling tools, there is no limit to the size of cells that could be designed as far as the thermal balance aspect of the cell design is concerned.
Yet, in both occasions, the author has been challenged to design an even bigger cell! This time, the thermo-electric design of a 26.2 meters long 740 kA cell is being presented. In addition, a basic geometric surface area to volume ratio analysis reveals that if the same lining design would be extended to a 85.8 meters long cell, it would be in 'comfortable' thermal balance at around 2380 kA.
Base case 300 kA cell design
The assumed base case 300 kA cell presented in this study is not an existing cell design but it has been strongly inspired by the prototype cell design presented by VAW in the early 90's .
The 350 kA cell design
The author has already presented in  a proposed lining design retrofit that permits to increase the amperage to 350 kA in a way similar to the AP30 cell technology being retrofitted into the AP35 cell technology .
The 265 kA cell design
Alternatively, the author had also demonstrated in  that the same base case 300 kA cell can be retrofitted into a very energy efficient but not so productive 265 kA cell operating at 11.94 kWh/kg.
Considering that cell prototypes operations between 12.2 and 12.6 kWh/kg have been reported in Japan in the early 80's [6,7], it is only a reminder that it is technically possible to operate very low heat dissipating cells.
One potshell size, but three very different thermal balances
At this point, it is important to point out that depending on the cell lining design selected, cells sharing the same 14.4 m X 4.35 m potshell can be operated in 'comfortable' thermal balance, dissipating as low as 422 kW or as high as 712 kW. This is about a ± 25% variation around the average value, clearly demonstrating that the cell thermal balance related design constraints are not terribly limiting!
The 400 kA cell design
From the 14.4 m X 4.35 m base case potshell size 350 kA cell lining design, a 400 kA cell has been produced by extending the potshell length to 16.1 meters . This proposed change has been analyzed using both the very fast lump parameters model (see Table I) and the more elaborated ANSYS® based 3D full cell slice model (see Table II).
Although there is no substitute for performing retrofit studies using reliable mathematical modeling tools, it is worth noticing that a very similar prediction can be made using only a basic geometric surface area to volume ratio analysis .
The initial potshell size being 1.48 m X 14.4 m X 4.35 m, its surface area to volume ratio is 1.95 m2/m3. 32 1.7 m X 0.8 m anodes can be fitted in that cell. In order to be able to add 4 extra anodes, the length of the potshell must be increased to 16.1 meters. At that new length, the surface area to volume ratio of the potshell is now 1.935 m2/m3. Using only that information, the predicted amperage of the extended cell is:
The prediction is not perfectly accurate because, in order to be able to fix 2 extra cathode blocks in the new potshell, the width of all the cathode blocks have been decreased, which in turn decreased the average current density in the collector bars. Hence, the 400 kA cell lining design is not a pure geometric extension of the 350 kA cell design.
Extrapolation to a 1650 kA cell design
Adding 1.7 m to the potshell length only changed its geometric surface area to volume ratio by 1.5%, but a more drastic extension will have a more significant impact.
If instead of adding only 4 extra anodes, the number of anodes is multiplied by 5, the potshell required to fit those 160 anodes must be 68.8 meters long. For that potshell length, the surface area to volume ratio drops to 1.84 m2/m3. This time, the change of aspect ratio will have a sensitive impact on the 'comfortable' cell amperage (see Curve 1 of Figure 1):
This loss of 100 kA or 5.7% due to the change of aspect ratio is not negligible, but remains quite small when compared with the ± 25% due to quite simple lining design changes.
The 500 kA cell design
The 500 kA cell design presented in  was obtained using only the lump parameters model (see Table I). The same cell design analysis study has now been repeated using the more elaborate ANSYS® based 3D full cell slice model (see Table II and Figure 2). This design is using a wider 4.85 meters potshell in order to accommodate 40 longer 1.95 meters anodes with bigger studs diameter. For that reason, the basic geometric surface area to volume ratio analysis cannot be used this time.
The 740 kA cell design
But this will not be the case for the 740 kA cell design as it is purely a 50% extension of the above 500 kA cell. This means that this design uses 60 1.95 m X 0.8 m anodes in a 26.2 m X 4.85 m potshell. There are now 36 cathode blocks and this time there was no need to change their width.
The new design has been analyzed using both types of models (see Tables I and II) and both models predict an increase of cell superheat and a loss of ledge thickness indicating that 740 kA is a bit too much current. 735 kA would have been a more 'comfortable' choice, as we will see below.
The better choice of 735 kA for the new cell design amperage is the result of the basic geometric surface area to volume ratio analysis. The aspect ratio of the 1.48 m X 17.8 m X 4.85 m potshell is 1.876 m2/m3 while the one of the 1.48 m X 26.2 m X 4.85 m potshell is 1.84 m2/m3, which gives:
Extrapolation to a 2380 kA cell design
Pushing once again the basic geometric surface area to volume ratio analysis to the limit, a 85.8 m potshell is required in order to be able to put 200 anodes in a single cell. The aspect ratio of that potshell would be 1.787 m2/m3, which gives for the predicted 'comfortable' amperage (see Curve 2 of Figure 1):
A new generation thermo-electric model
Very recently, a new generation thermo-electric model has been developed: the ANSYS® based 3D full cell and external busbar model  (see Figure 3). When interfaced with an MHD model, this kind of very elaborate thermo-electric model can be used to predict the evolution of the ledge thickness along the perimeter of a cell. While it was not built for that purpose, it is nice to know that it can be used to confirm the heat balance predicted by the two conventional thermo-electric models.
The above analysis clearly demonstrates that if a forced air sidewall cooling system is required in order to be able to operate very high amperage cells , it is clearly not because of the impact of the little reduction of the surface area to volume ratio on the cell heat balance. As far as only the cell heat balance design issue is concerned, there is no limit to the size of cells that can be designed.
Of course, as the cell gets longer, it will be more difficult for the fresh air coming from the potroom walls openings to reach the center of the cell. This impact can be analyzed in potroom ventilation models like the one presented in . It would no doubt be a very serious issue for a 85 meters long cell, but that impact on even a 18 m long cell when compared to a 14 m long one should not be that significant.
This leaves only mechanical design considerations to justify the need of forced air sidewall cooling system on a 500 kA cell design.
Dr. Marc Dupuis is a consultant specialized in the applications of mathematical modeling for the aluminium industry since 1994, the year when he founded his own consulting company GeniSim Inc (www.genisim.com). Before that, he graduated with a Ph.D. in chemical engineering from Laval University in Quebec City in 1984, and then worked 10 years as a research engineer for Alcan International. His main research interests are the development of mathematical models of the Hall-Héroult cell dealing with the thermo-electric, thermo-mechanic, electro-magnetic and hydrodynamic aspects of the problem. He was also involved in the design of experimental high amperage cells and the retrofit of many existing cell technologies.
Table I : Results using Dyna/Marc 1.8 modeling tool
Table II : Results using an ANSYS® based 3D full cell slice modeling tool
Figure 1: Extrapolation of the cell amperage vs the cell aspect ratio
for the 350 kA and the 500 kA cell designs
Figure 2: ANSYS® based 3D full cell slice models
Figure 3: ANSYS® based 3D full cell and external busbar model