Skip to main content

Advertisement

Log in

Laboratory model of electrovortex flow with thermal gradients for liquid metal batteries

  • Research Article
  • Published:
Experiments in Fluids Aims and scope Submit manuscript

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.

Graphical abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Ahlers G, Grossmann S, Lohse D (2009) Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection. Rev Mod Phys 81:503–537

    Article  Google Scholar 

  • Ashour RF, Kelley DH, Salas A, Starace M, Weber N, Weier T (2018) Competing forces in liquid metal electrodes and batteries. J Power Sources 378:301–310

    Article  Google Scholar 

  • Aurnou JM, Bertin V, Grannan AM, Horn S, Vogt T (2018) Rotating thermal convection in liquid gallium: multi-modal flow, absent steady columns. J Fluid Mech 846:846–876

    Article  MathSciNet  MATH  Google Scholar 

  • Beltrán A (2017) MHD natural convection flow in a liquid metal electrode. Appl Therm Eng 114:1203–1212

    Article  Google Scholar 

  • Bojarevičs V, Freibergs Y, Shilova EI, Shcherbinin EV (1989) Electrically induced vortical flows. Kluwer Academic Publishers, Dordrecht, The Netherlands

    Book  Google Scholar 

  • Bradwell DJ, Kim H, Sirk AHC, Sadoway DR (2012) Magnesium–antimony liquid metal battery for stationary energy storage. J Am Chem Soc 134(4):1895–1897

    Article  Google Scholar 

  • Brito D, Nataf HC, Cardin P, Aubert J, Masson JP (2001) Ultrasonic Doppler velocimetry in liquid gallium. Exp Fluids 31(6):653–663

    Article  Google Scholar 

  • Chandrasekhar S (1961) Hydrodynamic and hydromagnetic stability, 1st edn. Oxford University Press

  • Cioni S, Ciliberto S, Sommeria J (1997) Strongly turbulent Rayleigh-Bénard convection in mercury: comparison with results at moderate Prandtl number. J Fluid Mech 335:111–140

    Article  MathSciNet  Google Scholar 

  • Dai T, Zhao Y, Ning XH, Narayan RL, Li J, Zw Shan (2018) Capacity extended bismuth-antimony cathode for high-performance liquid metal battery. J Power Sources 381:38–45

    Article  Google Scholar 

  • Davidson HW (1968) Compilation of thermophysical properties of liquid lithium. Report No. NASA TN-D-4650

  • Davidson PA (2001) An introduction to magnetohydrodynamics, 1st edn. Cambridge Texts in Applied Mathematics

  • Eckert S, Cramer A, Gerbeth G (2007) Velocity measurement techniques for liquid metal flows. In: Magnetohydrodynamics. Springer, pp 275–294

  • Fazio C, Sobolev VP, Aerts A, Gavrilov S, Lambrinou K, Schuurmans P, Gessi A, Agostini P, Ciampichetti A, Martinelli L et al (2015) Handbook on lead-bismuth eutectic alloy and lead properties, materials compatibility, thermal-hydraulics and technologies, 2015 edn. Tech. rep, Organisation for Economic Co-Operation and Development

  • Ginter G, Gasser JG, Kleim R (1986) The electrical resistivity of liquid bismuth, gallium and bismuth–gallium alloys. Philos Mag B 54(6):543–552

    Article  Google Scholar 

  • Glazier JA, Segawa T, Naert A, Sano M (1999) Evidence against ‘ultrahard’ thermal turbulence at very high Rayleigh numbers. Nature 398:307–310

    Article  Google Scholar 

  • Gong Q, Ding W, Bonk A, Li H, Wang K, Jianu A, Weisenburger A, Bund A, Bauer T (2020) Molten iodide salt electrolyte for low-temperature low-cost sodium-based liquid metal battery. J Power Sources 475:228674

    Article  Google Scholar 

  • Grossmann S, Lohse D (2000) Scaling in thermal convection: a unifying theory. J Fluid Mech 407:27–56

    Article  MathSciNet  MATH  Google Scholar 

  • Herreman W, Bénard S, Nore C, Personnettaz P, Cappanera L, Guermond JL (2020) Solutal buoyancy and electrovortex flow in liquid metal batteries. Phys Rev Fluids 5(7):074501

    Article  Google Scholar 

  • Herreman W, Nore C, Ramos PZ, Cappanera L, Guermond JL, Weber N (2019) Numerical simulation of electrovortex flows in cylindrical fluid layers and liquid metal batteries. Phys Rev Fluids 4(11):113702

    Article  Google Scholar 

  • Hodge T (2020) Hourly electricity consumption varies throughout the day and across seasons: today in Energy—U.S. Energy Information Administration (EIA). https://www.eia.gov/todayinenergy/detail.php?id=42915

  • Horstmann GM, Weber N, Weier T (2018) Coupling and stability of interfacial waves in liquid metal batteries. J Fluid Mech 845:1–35

    Article  MathSciNet  MATH  Google Scholar 

  • Iida T, Guthrie RIL (2015a) The thermophysical properties of metallic liquids, vol 1. Fundamentals. Oxford University Press, Oxford

  • Iida T, Guthrie RIL (2015b) The thermophysical properties of metallic liquids, vol 2. Predictive models. Oxford University Press, Oxford

  • Janz GJ, Dampier FW, Lakshminarayanan GR, Lorenz PK, Tomkins RPT (1968) Molten salts, vol 1. Electrical conductance, density, and viscosity data. National Bureau of Standards

  • Kelley DH, Sadoway DR (2014) Mixing in a liquid metal electrode. Phys Fluids 26(5):057102

    Article  Google Scholar 

  • Kelley DH, Weier T (2018) Fluid mechanics of liquid metal batteries. Appl Mech Rev 70(2):020801

    Article  Google Scholar 

  • Keogh DF, Timchenko V, Reizes J, Menictas C (2021) Modelling Rayleigh-Bénard convection coupled with electro-vortex flow in liquid metal batteries. J Power Sources 501:229988

    Article  Google Scholar 

  • Kim H, Boysen DA, Newhouse JM, Spatocco BL, Chung B, Burke PJ, Bradwell DJ, Jiang K, Tomaszowska AA, Wang K, Wei W, Ortiz LA, Barriga SA, Poizeau SM, Sadoway DR (2013a) Liquid metal batteries: past, present, and future. Chem Rev 113(3):2075–2099

    Article  Google Scholar 

  • Kim H, Boysen DA, Ouchi T, Sadoway DR (2013b) Calcium-bismuth electrodes for large-scale energy storage (liquid metal batteries). J Power Sources 241:239–248

    Article  Google Scholar 

  • Köllner T, Boeck T, Schumacher J (2017) Thermal Rayleigh–Marangoni convection in a three-layer liquid-metal-battery model. Phys Rev E 95(5):053114

    Article  Google Scholar 

  • Li H, Yin H, Wang K, Cheng S, Jiang K, Sadoway DR (2016) Liquid metal electrodes for energy storage batteries. Adv Energy Mater 6(14):1600483

    Article  Google Scholar 

  • Lundquist S (1969) On the hydromagnetic viscous flow generated by a diverging electric current. Ark Fys 40(5):89–95

    Google Scholar 

  • Ning X, Phadke S, Chung B, Yin H, Burke P, Sadoway DR (2015) Self-healing li-bi liquid metal battery for grid-scale energy storage. J Power Sources 275:370–376

    Article  Google Scholar 

  • Okada K, Ozoe H (1992) Experimental heat transfer rates of natural convection of molten gallium suppressed under an external magnetic field in either the X, Y, or Z direction. J Heat Transfer 114(1):107–114

    Article  Google Scholar 

  • Ozer N, Tepehan F (1993) Structure and optical properties of electrochromic copper oxide films prepared by reactive and conventional evaporation techniques. Sol Energy Mater Sol Cells 30(1):13–26

    Article  Google Scholar 

  • Perez A, Kelley DH (2015) Ultrasound velocity measurement in a liquid metal electrode. J Vis Exp 102:e52622

    Google Scholar 

  • Personnettaz P, Beckstein P, Landgraf S, Köllner T, Nimtz M, Weber N, Weier T (2018) Thermally driven convection in Li\(\Vert\)Bi liquid metal batteries. J Power Sources 401:362–374

    Article  Google Scholar 

  • Personnettaz P, Landgraf S, Nimtz M, Weber N, Weier T (2019) Mass transport induced asymmetry in charge/discharge behavior of liquid metal batteries. Electrochem Commun 105:106496

    Article  Google Scholar 

  • Poizeau S, Kim H, Newhouse JM, Spatocco BL, Sadoway DR (2012) Determination and modeling of the thermodynamic properties of liquid calcium-antimony alloys. Electrochim Acta 76:8–15

    Article  Google Scholar 

  • Scheel JD, Schumacher J (2017) Predicting transition ranges to fully turbulent viscous boundary layers in low Prandtl number convection flows. Phys Rev Fluids 2(12):123501

    Article  Google Scholar 

  • Sele T (1977) Instabilities of the metal surface in electrolytic alumina reduction cells. Metall Trans B 8(3):613–618

    Article  Google Scholar 

  • Shen Y, Zikanov O (2016) Thermal convection in a liquid metal battery. Theor Comput Fluid Dyn 30(4):275–294

    Article  Google Scholar 

  • Shercliff JA (1970) Fluid motions due to an electric current source. J Fluid Mech 40(2):241–250

    Article  MATH  Google Scholar 

  • Shishkina O, Grossmann S, Lohse D (2016) Heat and momentum transport scalings in horizontal convection. Geophys Res Lett 43(3):1219–1225

    Article  Google Scholar 

  • Takeda Y (1987) Measurement of velocity profile of mercury flow by ultrasound Doppler shift method. Nucl Technol 79(1):120–124

    Article  Google Scholar 

  • Takeshita T, Segawa T, Glazier JA, Sano M (1996) Thermal turbulence in mercury. Phys Rev Lett 76(9):1465

    Article  Google Scholar 

  • USe (2021) Electricity explained: electricity in the United States. https://www.eia.gov/energyexplained/electricity/electricity-in-the-us.php

  • Vogt T, Horn S, Grannan AM, Aurnou JM (2018) Jump rope vortex in liquid metal convection. Proc Natl Acad Sci USA 115(50):12674–12679

    Article  Google Scholar 

  • Wagner S, Shishkina O (2013) Aspect-ratio dependency of Rayleigh-Bénard convection in box-shaped containers. Phys Fluids 25(8):085110

    Article  Google Scholar 

  • Wang B, Kelley DH (2021) Microscale mechanisms of ultrasound velocity measurement in metal melts. Flow Meas Instrum 81:102010

    Article  Google Scholar 

  • Wang K, Jiang K, Chung B, Ouchi T, Burke PJ, Boysen DA, Bradwell DJ, Kim H, Muecke U, Sadoway DR (2014) Lithium-antimony-lead liquid metal battery for grid-level energy storage. Nature 514(7522):348–350

    Article  Google Scholar 

  • Wang Q, Verzicco R, Lohse D, Shishkina O (2020) Multiple states in turbulent large-aspect-ratio thermal convection: What determines the number of convection rolls? Phys Rev Lett 125(7):074501

    Article  Google Scholar 

  • Weber N, Nimtz M, Personnettaz P, Salas A, Weier T (2018) Electromagnetically driven convection suitable for mass transfer enhancement in liquid metal batteries. Appl Therm Eng 143:293–301

    Article  Google Scholar 

  • Weber N, Nimtz M, Personnettaz P, Weier T, Sadoway D (2020) Numerical simulation of mass transfer enhancement in liquid metal batteries by means of electro-vortex flow. J Power Sources Adv 1:100004

    Article  Google Scholar 

  • Zhang H, Charmchi M, Veilleux D, Faghri M (2007) Numerical and experimental investigation of melting in the presence of a magnetic field: Simulation of low-gravity environment. J Heat Transfer 129(4):568–576

    Article  Google Scholar 

  • Zikanov O (2015) Metal pad instabilities in liquid metal batteries. Phys Rev E 92(6):063021

    Article  MathSciNet  Google Scholar 

  • Zürner T, Schindler F, Vogt T, Eckert S, Schumacher J (2019) Combined measurement of velocity and temperature in liquid metal convection. J Fluid Mech 876:1108–1128. https://doi.org/10.1017/jfm.2019.556

    Article  MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Contributions

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.

Corresponding author

Correspondence to J. S. Cheng.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file 1 (avi 28764 KB)

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 (xz) 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:

$$\begin{aligned} \nabla ^{2}T(x,z) = 0. \end{aligned}$$
(7)

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:

$$\begin{aligned} \frac{\partial T}{\partial x}(x = -\frac{D}{2};\frac{D}{2},z) = 0. \end{aligned}$$
(8)

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:

$$\begin{aligned} \frac{\partial T}{\partial z}(x,z=0) = {\frac{-{\textrm{Bi}}}{H_\textrm{plate}} } (T(x,z=0) - T_{\textrm{Ga}}), \end{aligned}$$
(9)

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:

$$\begin{aligned} T(x,z= _\textrm{plate}) = T_\textrm{set} + \delta T \cos \left( \frac{2\pi x }{\lambda }\right) , \end{aligned}$$
(10)

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).

Fig. 9
figure 9

a Radial cross-section of the copper plate attached to heat exchanger. b Steady state temperature distribution inside the plate for \(\varDelta T = 12\). c The temperature variation at the plate-gallium boundary for different \(\varDelta T\) values where the plate thickness is 0.5 cm. As shown, the variation is \(< 2\%\) for all relevant \(\varDelta T\) cases

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

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

Navigation