## Abstract

We present a novel laboratory setup for studying the fluid dynamics in liquid metal batteries (LMBs). LMBs are a promising technology suited for grid-scale energy storage, but flows remain a confounding factor in determining their viability. Two important drivers of flow are thermal gradients, caused by internal heating during operation, and electrovortex flow (EVF), induced by diverging current densities. Our setup explores, for the first time, electrovortex flow combined with both adverse and stabilizing thermal gradients in a cylindrical layer of liquid gallium, simulating the behavior in a single layer of an LMB. In this work, we discuss the design principles underlying our choices of materials, thermal control, and current control. We also detail our diagnostic tools—thermocouple measurements for temperature and Ultrasonic Doppler Velocimetry probes for velocities—and the design principles which go into choosing their placement on the setup. We also include a discussion of our post-processing tools for quantifying and visualizing the flow. Finally, we validate convection and EVF in our setup: we show that scaling relationships between the nondimensional parameters produced by our data agree well with theory and previous studies.

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## Acknowledgements

The authors wish to thank Mike Pomerantz and Jim Alkins for design consultation and construction of the setup. The authors also thank Gerrit M. Horstmann for productive discussions and the idea of writing this paper, as well as Norbert Weber and Tobias Vogt for fruitful scientific discussions. We thank Alex M. Grannan for inspiring the 2D model described in the Appendix, which was built off of his PhD work. This work was supported by the National Science Foundation under award number CBET-1552182.

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All authors contributed to the design and construction of the experiment. JSC, IM, and BW collected data. JSC, IM, BW, and JMF contributed data and error analyses. BW performed the copper oxide coating procedure. IM and DHK wrote the postprocessing scripts. IM developed the theoretical model detailed in the Appendix and wrote the Appendix. JSC wrote the manuscript with feedback from all authors. The second and third authors, IM and BW, contributed equally to this work.

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## Appendix: Theoretical model for heat transfer inside heat exchanger plate

### Appendix: Theoretical model for heat transfer inside heat exchanger plate

We consider a two-dimensional radial cross-section of the copper plate placed in an (*x*, *z*) Cartesian coordinate system with *z* pointing upwards, *x* along the diameter of the plate, and the origin at the midpoint of the plate’s bottom (Fig. 9a). Since we are interested in the steady-state temperature distribution of the plate, the governing equation is the Laplace equation:

The side walls of the plate are in contact with a heat insulator, and we assume that no heat escapes through the sides, i.e., the heat flux through the side walls is zero, which is asserted as the Neumann boundary condition:

At the plate-gallium interface heating (or cooling) convection exists between the plate boundary and the liquid gallium, and the heat transfer through the boundary will be proportional to the temperature difference between the gallium and the boundary. We use Newton’s law of cooling and find the Robin boundary condition:

where \(T_{Ga}\) is the gallium’s mean temperature.

The coil is in contact with the last plate boundary and we represented its thermal effect as an evenly spaced distribution of hot and cold spots added to a set temperature, which can be stated as the Dirichlet boundary condition:

where \(T_\textrm{set}\) is the set temperature of the plate, \(\delta T\) is the temperature difference between a hot spot and a cold spot, and \(\lambda\) is the distance between two hot or two cold spots (Fig. 9b).

We solve Eq. 7 with boundary conditions given by Eqs. 10, 8, and 9 numerically using MATLAB PDE-Model with a mesh of \(\sim\)4800 two-dimensional quadratic triangular elements. We use the parameters \({\textrm{Bi}} = 0.005\), \(\lambda = 2.5\) cm, \(D = 10\) cm, \(T_{{\textrm{Ga}}} = 43~^{\circ } C\), and a range of plate thicknesses \(< 0.6\) cm with \(T_\textrm{set}\) values equivalent to the ones used in experiment. We set \(\delta T\) to 5% of the thermal gradient applied between top and bottom \(\varDelta T\). We quantify the temperature variation at the boundary in contact with gallium as the standard deviation of the temperature there, reported as a percentage of \(\varDelta T\) (Fig. 9c).

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Cheng, J.S., Mohammad, I., Wang, B. *et al.* Laboratory model of electrovortex flow with thermal gradients for liquid metal batteries.
*Exp Fluids* **63**, 178 (2022). https://doi.org/10.1007/s00348-022-03525-3

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DOI: https://doi.org/10.1007/s00348-022-03525-3