Abstract
The thermal balance of aluminum electrolysis cells (AEC) have to be rigorously controlled in order to improve the efficiency and sustainability of this industrial process. A new modeling strategy is developed to consider the displacements of solid bodies and moving boundaries in finite element models. The transient thermalelectric modeling of the AEC demonstrates the effect of an increase in operating voltage on both the anode cover and the side ledge. With higher heat generation, the anode cover deteriorates and the side ledge thickness decreases. Since the anode cover is characterized by irreversible transformations, the top heat dissipation remains higher even when the operating voltage comes back to its typical value. For the first time, the transient temperature and electric fields throughout the anode life are simulated and validated by industrial measurements. The modeling predictions have been validated from instrumented anodes and manual measurements, all performed on operating AEC.
This is a preview of subscription content, access via your institution.
Abbreviations
 A :

Area (m^{2})
 c _{ p } :

Specific heat capacity (J/kg K)
 E :

Electric field (V/m)
 F :

View factor
 H :

Height (cm or m)
 h :

Convection heat transfer coefficient (W/m^{2} K)
 J :

Current density (A/m^{2})
 k :

Thermal conductivity (W/m K)
 L :

Length (cm or m)
 q :

Heat transfer rate (W)
 q″ :

Heat flux (W/m^{2})
 \( \dot{q} \) :

Rate of energy generation per unit volume (W/m^{3})
 t :

Time (s, h or day)
 T :

Temperature (K or °C)
 V :

Electric potential or voltage (V)
 δ _{ ij } :

Kronecker delta
 \( \nabla \) :

Gradient vector field
 ε :

Emissivity
 ρ :

Density (kg/m^{3})
 σ :

Electrical conductivity (S/m); Stefan–Boltzmann constant (W/m^{2} K^{4})
 ACD:

Anode–cathode distance
 ACM:

Anode cover material
 AEC:

Aluminum electrolysis cell
 BMI:

Bath–metal interface
 CAD:

Computeraided design
 CC:

Center channel
 CFD:

Computational fluid dynamics
 CR:

Cryolite ratio
 HFS:

Heat flux sensor
 MARE:

Mean absolute relative error
 OA:

On the anode
 SC:

Side channel
 bold :

A vector
 [brackets]:

A matrix or a concentration of a chemical species
References
 1.
J. Thonstad, P. Fellner, G.M. Haarberg, J. Hives, H. Kvande, A. Sterten: Aluminium Electrolysis: Fundamentals of the HallHeroult Process, 3rd ed., AluminiumVerlag, Düsseldorf, Germany, 2001, pp. 18.
 2.
A.E. Gheribi, S. Poncsák, S. Guérard, J.F. Bilodeau, L. Kiss, P. Chartrand: J. Chem. Phys., 2017, vol. 146, pp. 110. https://doi.org/10.1063/1.4978235.
 3.
S. Poncsák, L. Kiss, A. Belley, S. Guérard, and J.F. Bilodeau: Light Met. 2015, Proc. Int. Symp., 2015, pp. 655–59.
 4.
S. Poncsák, L. Kiss, R. StPierre, S. Guérard, and J.F. Bilodeau: Light Met. 2014, Proc. Int. Symp., 2014, pp. 585–89.
 5.
S. Poncsák, L. Kiss, S. Guérard, J.F. Bilodeau: Metals, 2017, vol. 7, pp. 110. https://doi.org/10.3390/met7030097.
 6.
F. Allard, G. Soucy, L. Rivoaland, M. Désilets: J. Therm. Anal. Calorim., 2015, vol. 119, pp. 13031314.
 7.
A. FallahMehrjardi, P.C. Hayes, E. Jak: Metall. Mater. Trans. B, 2014, vol. 45B, pp. 12321247.
 8.
J. Liu, A. FallahMehrjardi, D. Shishin, E. Jak, M. Dorreen, M. Taylor: Metall. Mater. Trans. B, 2017, vol. 48B, pp. 31853195.
 9.
F. Allard, M. Désilets, A. Blais: Thermochim. Acta, 2019, vol. 671, pp. 89102.
 10.
F. Allard, M. Désilets, M. LeBreux, A. Blais: Int. J. Heat Mass Transfer, 2019, vol. 132, pp. 12621276.
 11.
Q. Zhang, M.P. Taylor, J.J.J. Chen: Metall. Mater. Trans. B, 2015, vol. 46B, pp. 15201534.
 12.
F. Allard, M. Désilets, M. LeBreux, and A. Blais: Light Met. 2015, Proc. Int. Symp., 2015, pp. 565–70.
 13.
X. Liu, M. Taylor, and S. George: Light Met. 1992, Proc. Int. Symp., 1992, pp. 489–94.
 14.
L.N. Less: Metall. Trans. B., 1977, vol. 8B, pp. 219–225.
 15.
F. Allard, M. Désilets, M. LeBreux, and A. Blais: Light Met. 2016, Proc. Int. Symp., 2016, pp. 289–94.
 16.
H. Wijayaratne, M. Hyland, M. Taylor, A. Grama, and T. Groutso: Light Met. 2011, Proc. Int. Symp., 2011, pp. 399–404.
 17.
X. Shen: PhD thesis, University of Auckland, New Zealand, 2006.
 18.
K.A. Rye, J. Thonstad, and X. Liu: Light Met. 1995, Proc. Int. Symp., 1995, pp. 441–49.
 19.
M.A. Llavona, L.F. Verdeja, R. Zapico, F. Alvarez, and J.P. Sancho: Light Met. 1990, Proc. Int. Symp., 1990, pp. 429–37.
 20.
G. Hatem, M. Llavona, T. Log, J.P. Sancho, and T. Ostvold: Light Met. 1989, Proc. Int. Symp., 1989, pp. 365–70.
 21.
M.A. Llavona, R. Zapico, P. García, J.P. Sancho, and L.F. Verdeja: Light Met. 1988, Proc. Int. Symp., 1988, pp. 201–06.
 22.
K.E. Einarsrud, I. Eick, W. Bai, Y. Feng, J. Hua, P.J. Witt: Appl. Math. Modell., 2017, vol. 44, pp. 3–24.
 23.
B. Bardet, T. Foetisch, S. Renaudier, J. Rappaz, M. Flueck, and M. Picasso: Light Met. 2016, Proc. Int. Symp., 2016, pp. 315–19.
 24.
S. Langlois, J. Rappaz, O. Martin, Y. Caratini, M. Flueck, A. Masserey, and G. Steiner: Light Met. 2015, Proc. Int. Symp., 2015, pp. 771–75.
 25.
M. Ariana, M. Désilets, P. Proulx: Can. J. Chem. Eng., 2014, vol. 92, pp. 19511964.
 26.
M. Blais, M. Désilets, M. Lacroix: Appl. Therm. Eng., 2013, vol. 58, pp. 439446.
 27.
D. Marceau, S. Pilote, M. Désilets, J.F. Bilodeau, L. Hacini, and Y. Caratini: Light Met. 2011, Proc. Int. Symp., 2011, pp. 1041–46.
 28.
Y. Safa, M. Flueck, J. Rappaz: Appl. Math. Modell., 2009, vol. 33, pp. 14791492.
 29.
M. Dupuis and V. Bojarevics: Light Met. 2005, Proc. Int. Symp., 2005, pp. 449–54.
 30.
T.X. Hou, Q. Jiao, E. Chin, W. Crowell, and C. Celik: Light Met. 1995, Proc. Int. Symp., 1995, pp. 755–61.
 31.
A.T. Brimmo, M.I. Hassan, Y. Shatilla: Appl. Therm. Eng., 2014, vol. 73, pp. 116127.
 32.
M. Désilets, D. Marceau, and M. Fafard: Light Met. 2003, Proc. Int. Symp., 2003, pp. 247–54.
 33.
M. LeBreux, M. Désilets, F. Allard, and A. Blais: Numer. Heat Transf. A, 2016, vol. 69A, pp. 128145.
 34.
Q. Wang, L. Gosselin, M. Fafard, J. Peng, B. Li: Metall. Mater. Trans. B, 2016, vol. 47B, pp. 12281236.
 35.
D. Picard, J. Tessier, G. Gauvin, D. Ziegler, H. Alamdari, and M. Fafard: Metals, 2017, vol. 7, pp. 1–9. https://doi.org/10.3390/met7090374.
 36.
H. Abbas, M.P. Taylor, M. Farid, and J.J. Chen: Light Met. 2009, Proc. Int. Symp., 2009, pp. 551–56.
 37.
R. Zhao, L. Gosselin, A. Ousegui, M. Fafard, D.P. Ziegler: Numer. Heat Transfer, Part A, 2013, vol. 64A, pp. 317338.
 38.
R. Zhao, L. Gosselin, M. Fafard, J. Tessier, D.P. Ziegler: Int. J. Therm. Sci., 2017, vol. 112, pp. 395407.
 39.
K. Stein, T.E. Tezduyar, R. Benney: Comput. Methods Appl. Mech. Eng., 2004, vol. 193, pp. 20192032.
 40.
A.A. Johnson and T.E. Tezduyar: Comput. Methods Appl. Mech. Eng., 1994, vol. 119, pp. 7394.
 41.
J.N. Reddy and D.K. Gartling: The Finite Element Method in Heat Transfer and Fluid Dynamics, 3rd ed., CRC Press, Boca Raton, Florida, 2010, pp. 229235.
 42.
G. Vidalain, L. Gosselin, M. Lacroix: Int. J. Heat Mass Transfer, 2009, vol. 52, pp. 17531760.
 43.
ANSYS Inc.: ANSYS Mechanical APDL Theory Reference, Canonsburg, Pennsylvania, 2017, pp. 203–18.
 44.
M.F. Cohen and D.P. Greenberg: Comput. Graphics, 1985, vol. 19, pp. 3140.
 45.
Y.S. Touloukian and D.P. DeWitt: Thermophysical Properties of Matter  The TPRC Data Series  Volume 8, Plenum Publishing Corporation, New York, NY, 1972, pp. 873.
 46.
C.F. Windisch, B.B. Brenden, O.H. Koski, and R.E. Williford: Final report on the PNL program to develop an alumina sensor, U.S. Department of Energy, Pacific Northwest Laboratory, United States, 1992, pp. 28–29.
 47.
Hukseflux Thermal Sensors: HF01 High Temperature Heat Flux Sensor (version 1211), Delft, Netherlands, 2003, pp. 1516.
Acknowledgments
This study was supported by Rio Tinto Aluminium, the “Conseil de Recherches en Sciences Naturelles et en Génie du Canada” (CRSNG) and the “Fonds de Recherche du Québec  Nature et Technologies” (FRQNT). The authors wish to thank the staff at Rio Tinto GrandeBaie smelter and Arvida Research & Development Center (ARDC), especially Mr. JeanFrançois Bilodeau and Mr. Sébastien Guérard from ARDC, for the support provided during the realization of this work.
Conflict of interest
The authors declare that they have no competing interests.
Author information
Affiliations
Corresponding author
Additional information
Manuscript submitted September 9, 2018.
Electronic supplementary material
Below is the link to the electronic supplementary material.
Supplementary material 2 (MP4 39624 kb)
Supplementary material 3 (MP4 46052 kb)
Appendices
Appendix A
A modeling method for the geometry and mesh update algorithm followed by the nodal interpolation is detailed in this appendix. This method is applied to the transient thermalelectric modeling of the industrial AEC in order to consider the melting of the anode crust by a moving boundary.
A.1 Python Scripting in ANSYS Workbench
The moving geometry and mesh update algorithms are automated with a Python script interpreted by the ANSYS Workbench software. A while loop is programmed to solve each analysis step of the transient model and the results are saved automatically. The principal steps of this method are described by the scheme of Figure A1. A previously calculated or a new crust–cavity profile (moving boundary) is input in the parameter set to modify the geometry of the model. Accordingly, the geometry is updated using a Python command and a new mesh in the ANSYS Mechanical model is generated.
A.2 APDL Scripting in ANSYS Mechanical
The coupled thermalelectric analysis is performed using Mechanical APDL solver in the ANSYS Mechanical software. APDL commands are integrated to read the load data stored in a CSV file, like the thermal and electric properties or the boundary conditions. The new properties or conditions are applied by APDL commands to the current analysis. The previously calculated temperatures are mapped to the new mesh with an interpolation function (*MOPER).[43] This function performs matrix operations by comparing arrays, which contain the previous nodal coordinates and temperatures, and also the new mesh information. A 3D interpolation is applied to get the nodal temperatures on the new mesh. In that simulation case, a similar interpolation method is used to apply the previous material conditions to the newly generated elements. This step allows to consider the irreversible phase transformation in the anode cover (ACM → anode crust). With the previous temperatures and material conditions applied to the new mesh, the transient model is solved for a specified analysis time, before starting the postprocessing. Subsequently, the temperatures in the anode crust are read and a new crust–cavity profile is calculated in order to reach the melting isotherm. The new profile is written in a CSV file to be read by Python commands in order to update the geometric parameter and so on. The nodal coordinates and temperatures are also exported to a CSV file to be read in the next iteration. The procedure in Figure A1 is repeated up to the final time of the analysis.
Appendix B
A moving geometry, mesh update, and nodal interpolation method is developed to reflect the anode consumption and its downward displacement in the AEC. The modeling approach is automated with Python scripting and the results are interpolated from one step to the next one with APDL commands. A moving interface for the melting of the anode crust is integrated in this modeling method. This second case additionally integrates an algorithm to consider the timedependent geometry related to the anode consumption. The motion of the nodes that are located inside the moving parts is governed by the velocity of anode assembly, a downward movement in this case. The node coordinates are modified to follow the anode displacement. Since the geometry is moved and the mesh is automatically updated, the new mesh conforms to the quality criterium initially imposed. The previously calculated nodal temperatures are then applied as initial conditions before the new analysis starts. The complete scheme of resolution is detailed in Figure B1.
B.1 Python Scripting in ANSYS Workbench
A new aspect of this second model is the displacement of the body parts with automatic updates of the CAD geometry by Python scripting. For each analysis time, a new position is calculated and applied to the moving parts. Since the anode is consumed by the process, its size is imposed before each resolution step. The python script also applies the new location of the moving boundary (anode crust melting) according to the postprocessing results.
B.2 APDL Scripting in ANSYS Mechanical
In the ANSYS Mechanical interface, a new mesh is generated for each resolution step. The previously moved nodal and element properties are interpolated and then applied as initial conditions for the thermalelectric model. The boundary conditions and constraints for the analysis are imposed with APDL commands. Thereafter, the model is solved and the postprocessing is performed. During postprocessing, a new crust–cavity profile is calculated only if the anode crust has melted. The nodes in the moving geometry are selected and their coordinated are displaced using matrix operations to account for the downward movement. Consequently, the nodal temperatures and element properties are saved to CSV files to be imported in the next resolution step.
Rights and permissions
About this article
Cite this article
Allard, F., Désilets, M. & Blais, A. A Modeling Approach for TimeDependent Geometry Applied to Transient Heat Transfer of Aluminum Electrolysis Cells. Metall Mater Trans B 50, 958–980 (2019). https://doi.org/10.1007/s11663019015106
Received:
Published:
Issue Date: