This paper presents a comprehensive study on the metal pad roll instability (MPRI) in liquid metal batteries (LMBs), a key phenomenon driven by magnetohydrodynamic (MHD) forces. LMBs, which are poised to revolutionize grid-scale energy storage due to their low cost, long lifespan, and high current density, face critical challenges due to instabilities that can compromise operational stability. This work characterizes the MPRI in cylindrical reduction cells and offers theoretical formulations, validated against numerical solvers, such as OpenFOAM and SFEMaNS. By analyzing gravity waves across multiple fluid layers and incorporating capillary, viscous, and Joule dissipation effects, this study provides explicit formulas for determining instability thresholds. The findings are particularly relevant for large-scale LMB applications, where instability driven by weak magnetic fields can lead to short circuits, diminishing battery performance. The paper also addresses the impact of dissipation on the growth rate of instability, and how these results align with experimental data. Our results underscore the importance of accurately modeling and mitigating MHD instabilities in LMB systems for future energy storage solutions.

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1. Herreman W, Nore C, Guermond JL, Cappanera L, Weber N, Horstmann GM. Perturbation theory for metal pad roll instability in cylindrical reduction cells. J Fluid Mech. 2019;878:598–646. doi:10.1017/jfm.2019.642.
2. Herreman W, Wierzchalek L, Horstmann GM, Cappanera L, Nore C. Stability theory for metal pad roll in cylindrical liquid metal batteries. J Fluid Mech. 2023;962:A6. doi:10.1017/jfm.2023.238.
3. Weber N, Beckstein P, Galindo V, Herreman W, Nore C, Stefani F, et al. Metal pad roll instability in liquid metal batteries. Magnetohydrodyn. 2017;53(1):129–140.
4. Nore C, Cappanera L, Guermond JL, Weier T, Herreman W. Feasibility of metal pad roll instability experiments at room temperature. Phys Rev Lett. 2021;126(18):184501. doi:10.1103/PhysRevLett.126.184501.
5. Krastins I, Bojarevics A. Metal pad roll instability threshold with magnetic damping in shallow cylindrical cells. Magnetohydrodyn. 2020;56(4):1–8.
6. Mandin P, Wüthrich R, Roustan H. Industrial aluminium production: the Hall-Heroult process modelling. ECS Trans. 2009;19(26):1. doi:1149/1.3247986.
7. Davidson PA. Overview overcoming instabilities in aluminium reduction cells: a route to cheaper aluminium. Mater Sci Technol. 2000;16(5):475–479. doi:10.1179/026708300101508027.
8. Obaidat M, Al-Ghandoor A, Phelan P, Villalobos R, Alkhalidi A. Energy and exergy analyses of different aluminum reduction technologies. Sustain. 2018;10(4):1216. doi:10.3390/su10041216.
9. Roberts PH, Boardman AD. The effect of a vertical magnetic field on the propagation of gravity waves along the plane surface of a semi-infinite viscous, electrically conducting fluid. Astrophys J. 1962;135:552.
10. Golub MV, Boström A. Interface damage modeled by spring boundary conditions for in-plane elastic waves. Wave Motion. 2011;48(2):105–115. doi:10.1016/j.wavemoti.2010.09.003.

This paper presents a comprehensive study on the metal pad roll instability (MPRI) in liquid metal batteries (LMBs), a key phenomenon driven by magnetohydrodynamic (MHD) forces. LMBs, which are poised to revolutionize grid-scale energy storage due to their low cost, long lifespan, and high current density, face critical challenges due to instabilities that can compromise operational stability. This work characterizes the MPRI in cylindrical reduction cells and offers theoretical formulations, validated against numerical solvers, such as OpenFOAM and SFEMaNS. By analyzing gravity waves across multiple fluid layers and incorporating capillary, viscous, and Joule dissipation effects, this study provides explicit formulas for determining instability thresholds. The findings are particularly relevant for large-scale LMB applications, where instability driven by weak magnetic fields can lead to short circuits, diminishing battery performance. The paper also addresses the impact of dissipation on the growth rate of instability, and how these results align with experimental data. Our results underscore the importance of accurately modeling and mitigating MHD instabilities in LMB systems for future energy storage solutions.

🔒 This is a subscription article

Full text is available to subscribers and institutional members. Please choose an option below to access it.

1. Herreman W, Nore C, Guermond JL, Cappanera L, Weber N, Horstmann GM. Perturbation theory for metal pad roll instability in cylindrical reduction cells. J Fluid Mech. 2019;878:598–646. doi:10.1017/jfm.2019.642.
2. Herreman W, Wierzchalek L, Horstmann GM, Cappanera L, Nore C. Stability theory for metal pad roll in cylindrical liquid metal batteries. J Fluid Mech. 2023;962:A6. doi:10.1017/jfm.2023.238.
3. Weber N, Beckstein P, Galindo V, Herreman W, Nore C, Stefani F, et al. Metal pad roll instability in liquid metal batteries. Magnetohydrodyn. 2017;53(1):129–140.
4. Nore C, Cappanera L, Guermond JL, Weier T, Herreman W. Feasibility of metal pad roll instability experiments at room temperature. Phys Rev Lett. 2021;126(18):184501. doi:10.1103/PhysRevLett.126.184501.
5. Krastins I, Bojarevics A. Metal pad roll instability threshold with magnetic damping in shallow cylindrical cells. Magnetohydrodyn. 2020;56(4):1–8.
6. Mandin P, Wüthrich R, Roustan H. Industrial aluminium production: the Hall-Heroult process modelling. ECS Trans. 2009;19(26):1. doi:1149/1.3247986.
7. Davidson PA. Overview overcoming instabilities in aluminium reduction cells: a route to cheaper aluminium. Mater Sci Technol. 2000;16(5):475–479. doi:10.1179/026708300101508027.
8. Obaidat M, Al-Ghandoor A, Phelan P, Villalobos R, Alkhalidi A. Energy and exergy analyses of different aluminum reduction technologies. Sustain. 2018;10(4):1216. doi:10.3390/su10041216.
9. Roberts PH, Boardman AD. The effect of a vertical magnetic field on the propagation of gravity waves along the plane surface of a semi-infinite viscous, electrically conducting fluid. Astrophys J. 1962;135:552.
10. Golub MV, Boström A. Interface damage modeled by spring boundary conditions for in-plane elastic waves. Wave Motion. 2011;48(2):105–115. doi:10.1016/j.wavemoti.2010.09.003.