FFluid Mechanics of Liquid Metal Batteries
Douglas H. Kelley
Department of Mechanical EngineeringUniversity of RochesterRochester, NY 14627Email: [email protected]
Tom Weier
Institute of Fluid DynamicsHelmholtz-Zentrum Dresden - RossendorfBautzner Landstr. 40001328 Dresden, GermanyEmail: [email protected]
The design and performance of liquid metal batteries, a newtechnology for grid-scale energy storage, depend on fluidmechanics because the battery electrodes and electrolytesare entirely liquid. Here we review prior and current re-search on the fluid mechanics of liquid metal batteries, point-ing out opportunities for future studies. Because the tech-nology in its present form is just a few years old, only asmall number of publications have so far considered liquidmetal batteries specifically. We hope to encourage collab-oration and conversation by referencing as many of thosepublications as possible here. Much can also be learnedby linking to extensive prior literature considering phenom-ena observed or expected in liquid metal batteries, includingthermal convection, magnetoconvection, Marangoni flow, in-terface instabilities, the Tayler instability, and electro-vortexflow. We focus on phenomena, materials, length scales, andcurrent densities relevant to the liquid metal battery designscurrently being commercialized. We try to point out break-throughs that could lead to design improvements or makenew mechanisms important.
The story of fluid mechanics research in liquid metalbatteries (LMBs) begins with one very important applica-tion: grid-scale storage. Electrical grids have almost no en-ergy storage capacity, and adding storage will make themmore robust and more resilient even as they incorporate in-creasing amounts of intermittent and unpredictable wind andsolar generation. Liquid metal batteries have unique advan-tages as a grid-scale storage technology, but their unique-ness also means that designers must consider chemical andphysical mechanisms — including fluid mechanisms — thatare relevant to few other battery technologies, and in manycases not yet well-understood. We will review the fluid me-chanics of liquid metal batteries, focusing on studies under-taken with that technology in mind, and also drawing exten-sively from prior work considering similar mechanisms in other contexts. In the interest of promoting dialogue acrossthis new field, we have endeavored to include the work ofmany different researchers, though inevitably some will haveeluded our search, and we ask for the reader’s sympathy forregrettable omissions. Our story will be guided by techno-logical application, focusing on mechanisms most relevantto liquid metal batteries as built for grid-scale storage. Wewill consider electrochemistry and theoretical fluid mechan-ics only briefly because excellent reviews of both topics arealready available in the literature. In § §
2, connecting to the thermally regen-erative electrochemical cells developed in the middle of thetwentieth century. Continuing, we consider the fluid mecha-nisms that are most relevant to liquid metal batteries: thermalconvection and magnetoconvection in §
3, Marangoni flow in §
4, interface instabilities in §
5, the Tayler instability in § §
7. We conclude with a summaryand reflection on future directions in § A typical electrical grid spans a country or a continent,serving millions of consumers by linking them to an intricatenetwork of hundreds or thousands of large generators. A gridcan be understood as a single, gigantic machine, because allof its rotating generators must spin in synchrony, and withina fraction of a percent of their design speed, in order for thegrid to function properly. Changes to any one part of the gridaffect all parts of the grid. The implications of this intercon-nectedness are made more profound by the fact that today’sgrids have nearly zero storage capacity. When more electric- a r X i v : . [ phy s i c s . f l u - dyn ] O c t ty is being consumed than generated, conservation of energyrequires that the kinetic energy of the spinning generatorsdrop, so they slow down, quickly losing synchrony, damag-ing equipment, and causing brownouts or blackouts if leftunchecked. Conversely, when more electricity is being gen-erated than consumed, generators speed up, risking all thesame problems. Fluctuations in demand are as old as elec-trical utilities, and have historically been managed by con-tinually adjusting supply by turning generators on and off.Now, grids must also accommodate fluctuations in supply,as intermittent wind and solar generation expand rapidly be-cause of their plummeting costs and the long-term imperativethat humankind generate a significant share of our energyusing renewable sources [1]. Large-scale storage on electri-cal grids would enable widespread deployment of renewablegeneration [2, 3] while maintaining stability [4]. Many tech-nologies for grid-scale storage have been proposed, includ-ing pumped hydro (which accounts for the vast majority ofexisting storage), pressurized air, thermal storage, flywheels,power-to-gas and batteries. Liquid metal batteries are a par-ticular grid-scale storage technology that comes with inter-esting fluid mechanical challenges.Like any battery, a liquid metal battery discharges by al-lowing an energetically-favorable chemical reaction to pro-ceed in a controlled way. Control is maintained by separat-ing the two reactants (the electrodes) with an electrolyte thatprevents electrode materials from passing if they are neutral,but allows them to pass if they are ionized. Thus the reactionproceeds only if some other path passes matching electrons,which then recombine with the ions and go on to react. Theother path is the external circuit where useful work is done,thanks to the energy of the flowing electrons. The batterycan later be recharged by driving electrons in the oppositedirection, so that matching ions come along as well.Battery electrodes can be made from a wide variety ofmaterials, including liquid metals. For example, a liquidsodium negative electrode (anode) can be paired with a sul-fur positive electrode (cathode) and a solid β -alumina elec-trolyte. (Here and throughout, we assign the names “an-ode” and “cathode” according to the roles played during dis-charge.) Na || S batteries operate at about 300 ◦ C and havebeen deployed for grid-scale storage. ZEBRA batteries [5,6],named for the Zeolite Battery Research Africa Project thatdeveloped them, use a NaAlCl negative electrode that al-lows them to operate at temperatures as low as 245 ◦ C. Anelectrolyte composed of Na-doped β -alumina conducts Na + ions. Lower operating temperatures are possible with bat-teries in which a Na negative electrode is combined witha NiCl positive electrode and a NaAlCl electrolyte, sep-arated from the negative electrode with β -alumina to preventcorrosion. Alloying Cs with the Na can substantially im-prove wetting to β -alumina, allowing battery operation atstill lower temperatures [7]. Sumitomo has recently doc-umented a battery design using a eutectic mix of potas-sium and sodium bis(fluorosupfonyl)amide salts along withelectrodes made from unspecified sodium compounds [8, 9].These battery designs and others like them involve liquidmetals but require a solid separator between the layers. MM(N) M z+ loadelectrolyte electrolyte ρ z I anode(alkaline metal)cathode(alloy) e - η a,a η a,c η c,a η c,c
12 IR E a) b) Fig. 1. Sketch of a liquid metal cell with discharge current and den-sity profile for fully charged state and isothermal conditions (a) andschematic discharge process (b) from [12].Fig. 2. Cross-sections of prototype liquid metal batteries. Both areenclosed in a stainless steel casing that also serves as the positivecurrent collector, and both have a foam negative current collectorattached to a copper conductor that exits the top of the battery. In thedischarged state (left), the foam is nearly filled with electrolyte, and adark Li-Bi intermetallic layer is visible at bottom. In the charged state(right), lithium metal is visible in the foam, and the positive electrodeat bottom has been restored to nearly pure bismuth. Because thesephotographs were taken at room temperature, the electrolyte doesnot fill the volume between the electrodes, but during operation, itwould. The space above the negative current collector is filled withinert gas during operation. Adapted from [13], with permission.
As discovered at Argonne National Laboratory in the1960s [10] and rediscovered at MIT recently [11], batteriescan also be designed with liquid metal electrodes and moltensalt electrolytes, requiring no separator at all. We shall usethe term “liquid metal batteries” to refer to those designsspecifically. An example is sketched in Fig. 1a, and cross-sections of two laboratory prototypes are shown in Fig. 2.The internal structure of the battery is maintained by gravity,since negative electrode materials typically have lower den-sity than electrolyte materials, which have lower density thanpositive electrode materials. A solid metal positive currentcollector contacts the positive electrode, and usually servesas the container as well. A solid metal negative current col-lector connects to the negative electrode and is electricallyinsulated from the positive current collector.Because the negative electrode is liquid and the posi-tive current collector is also the battery vessel, some care isrequired to prevent shorts between them. It is possible toelectrically insulate the positive current collector by liningit with a ceramic, but ceramic sleeves are too expensive forgrid-scale applications and are prone to cracking. Instead,ypical designs separate the liquid metal negative electrodefrom vessel walls with a metal foam, as shown in Fig. 6.The high surface tension of the liquid metal provides suffi-cient capillary forces to keep it contained in the pores of thefoam. The solid foam also inhibits flow in the negative elec-trode, which is likely negligible at length scales larger thanthe pore size. In many designs, the foam is held in place bya rigid conductor, as shown, so that its height stays constant.However, as the battery discharges and the positive electrodebecomes a pool of two-part alloy, it swells. If the positiveelectrode swells enough to contact the foam, a short occurs,so the foam height must be carefully chosen, taking into ac-count the thickness of the positive electrode and the densitychange it will undergo during discharge.Most liquid metal cells are concentration cells. Theiropen circuit voltage (OCV) is solely given by the activity ofthe alkaline metal in the cathode alloy. The equations for thetransfer reactions at the two interfaces (see Fig. 1b) read:M → M z + + z e − (1)M z + + z e − → M(N) (2)for the anode/electrolyte interface (1) and the elec-trolyte/cathode interface (2) during discharge. M denotesan alkali ( z =
1) or earth-alkali ( z =
2) metal of the neg-ative electrode, and N refers to the heavy or half metalof the positive electrode. A variety of chemistries havebeen demonstrated, including Mg || Sb [14], Li || Pb-Sb [15],Li || Bi [13], Na | NaCl-CaCl | Zn [16, 17] and Ca-Mg || Bi [18,19]. (See [20] for a review.) The Li || Pb-Sb chemistry hasbeen studied most, and is typically paired with a triple-eutectic LiF-LiCl-LiI electrolyte because of its relatively lowmelting temperature (about 341 ◦ C [10,21]). The equilibriumpotentials ϕ of both half-cells can be written as ϕ ( ) = ϕ + RTzF ln a M z + a M (3) ϕ ( ) = ϕ + RTzF ln a M z + a M(N) (4)with the standard potential ϕ , the universal gas constant R ,the temperature T , the Faraday constant F and the activity a of the metal in the pure (M), ionic (M z + ) and the alloyed(M(N)) state. The difference of the two electrode potentials ϕ ( ) and ϕ ( ) is the cell’s OCV E OC = − RTzF ln a M(N) . (5)Only the activity of the alkali metal in the alloy a M(N) de-termines the OCV since the standard potentials of both halfcells are identical and the activity of the pure anode is oneby definition. Under current flow, only the terminal volt-age E is available. It is the difference of OCV and severalterms describing voltage losses, i.e., polarizations (cf. [22] and Fig. 1b) occurring under current ( I ) flow: E = E OC − IR E − η c , a − η c , c − η a , a − η a , c . (6)These voltage losses are due to the electrolyte resistance R E ,the concentration polarizations at the anode η c , a and cath-ode η c , c and the corresponding activation potentials η a , a and η a , c . Typically, ohmic losses dominate activation and con-centration polarizations by far, but mass transfer limitationsmay nevertheless sometimes occur in the cathodic alloy.Liquid metal batteries have advantages for grid-scalestorage. Eliminating solid separators reduces cost and elimi-nates the possibility of failure from a cracked separator. Per-haps more importantly, solid separators typically allow muchslower mass transport than liquids, so eliminating solids al-lows faster charge and discharge with smaller voltage losses.Liquid electrodes improve battery life, because the life ofLi-ion and other more traditional batteries is limited whentheir solid electrodes are destroyed due to repeated shrink-ing and swelling during charge and discharge. Projectionsfrom experimental measurements predict Li || Pb-Sb batter-ies will retain 85% of their capacity after daily dischargefor ten years [15]. The Li || Pb-Sb chemistry is composed ofEarth-abundant elements available in quantities large enoughto provide many GWh of storage. Low cost is also critical ifa technology is to be deployed widely [23], and liquid metalbatteries are forecast to have costs near the $100/kWh targetset by the US Advanced Research Projects Agency-Energy(ARPA-e). Their energy and power density are moderate,and substantially below the Li-ion batteries that are ubiqui-tous in portable electronics, but density is less essential thancost in stationary grid-scale storage. Li-ion batteries todaycost substantially more than $100/kWh, but their costs havedropped continually over time and will likely drop substan-tially more as the Tesla GigaFactory 1, the world’s largestLi-ion battery plant, continues to increase its production. Theenergy efficiency of liquid metal batteries varies widely withcurrent density, but at a typical design value of 275 mA/cm is 73% [15], similar to pumped hydro storage.Liquid metal batteries also present challenges. Duringdischarge, Li || Pb-Sb batteries provide only about 0.8 V [15].Despite variation with battery chemistry, all conventional liq-uid metal batteries have voltage significantly less than Li-ion batteries. Lacking solid separators, liquid metal batter-ies are not suitable for portable applications in which dis-turbing the fluid layers could rupture the electrolyte layer,causing electrical shorts between the positive and negativeelectrodes and destroying the battery. Rupture might alsoresult from vigorous fluid flows even if the battery is station-ary, such as the Tayler instability ( § § § § ◦ C forLi || Pb-Sb). Little energy is wasted heating large batteriesbecause Joule heating (losses to electrical resistance) pro-vides more than enough energy to maintain the temperature.till, high temperatures promote corrosion and make air-tightmechanical seals difficult. Finally, poor mixing during dis-charge can cause local regions of a liquid metal electrodeto form unintended intermetallic solids that can eventuallyspan from the positive to the negative electrode, destroyingthe battery. Solid formation may well be the leading cause offailure in liquid metal batteries.
The central idea at the heart of LMBs is the three-layerarrangement of liquid electrodes and electrolyte. This seem-ingly simple idea (in fact so apparently simple that it is some-times [24] questioned if it deserves to be patented at all) didnot originate with LMBs. Instead, using a stable stratificationof two liquid metals interspaced with a molten salt for elec-trochemical purposes was first proposed 1905 by Betts [25]in the context of aluminium purification (see Fig. 3b). How-ever, Betts was not able to commercialize his process. In-stead Hoopes, who had a more complicated arrangement us-ing a second internal vessel for aluminium electrorefiningpatented in 1901 [26] (Fig. 3a) developed later a water cooledthree-layer cell [27] (Fig. 3c) that could be successfully op-erated. According to Frary [28] Hoopes as well thought ofusing a three-layer cell around 1900. It can be seen fromTable 1 that even if the idea to use three liquid layers werea trivial one, its realization and transformation to a workingprocess was highly non-trivial indeed.The submerged vessel containing the negative electrode,initially suggested by Hoopes [26] (Fig. 3a) is filled withmolten impure aluminium and surrounded by a bath of fusedcryolite. Cryolite is less dense than the pure or impure Al.In the presence of flow, Al dissolves into the cryolite and de-posits at the carbon walls of the outer vessel, and pure Al canbe collected at the bottom of the outer vessel. However, thecurrent density is distributed very inhomogeneously, concen-trating around the opening of the inner vessel. This implieslarge energy losses and strong local heating rendering a sta-ble operation over longer times impossible.Betts [25, 29] (Fig. 3b) alloyed the impure Al with Cuand added BaF to the cryolite to increase the density of thesalt mixture and to enable the purified Al to float on the fusedsalt. This three-layer arrangement guaranteed the short-est possible current paths and enabled homogeneous currentdensity distributions. Additionally the evaporation of theelectrolyte was drastically reduced by the Al top layer. How-ever, under the high operating temperatures the cell wallsbecame electrically conducting, got covered with metal thatshort-circuited the negative and positive electrodes and thusprevented successful operation of the cell [29].Only Hoopes’ sophisticated construction [27, 28](Fig. 3c) was finally able to operate for longer times. Akey element of Hoopes’ construction is the division of thecell into two electrically insulated sections. The joint be-tween them is water cooled and thereby covered by a crustof frozen electrolyte providing electrical as well as thermalinsulation [30]. A similar idea was later applied to Na || Bi galvanic cells by Shimotake and Hesson [31]. Additionally,instead of using a single electric contact to the purified Al atthe cells’ side as did Betts, Hoopes arranged several graphitecurrent collectors along the Al surface that provided a moreevenly distributed current. However, the electrolyte used byHoopes (see Table 1) had a relatively high melting tempera-ture and a tendency to creep to the surface between the cellwalls and the purified Al [32, 28]. According to Beljajewet al. [34] (see as well Gadeau [35]), the complicated de-sign of Hoopes’ cell, especially the water cooled walls, pre-vented continuous use in production. It was not until 1934that super-purity aluminium became widely available withGadeau’s [36] three-layer refining process that used a dif-ferent electrolyte (see Table 1) according to a patent filed in1932. Its lower melting point allowed for considerably de-creased operating temperature. Gadeau’s cell was lined withmagnesite that could withstand the electrolyte attack withoutthe need of water cooling. However, the BaCl used in theelectrolyte mixture decomposed partially, so the electrolytecomposition had to be monitored and adjusted during celloperation when necessary. This difficulty was overcome byusing the purely fluoride based electrolyte composition sug-gested by Hurter [37] (see Table 1, S.A.I.A. Process).Aluminum refining cells can tolerate larger voltagedrops than LMBs, so the electrolyte layer is often muchthicker. Values quoted are between 8 cm [39, 34] that shouldbe a good estimate for current practice [40], 10 cm [32],20 cm [41] and 25 cm [38]. These large values are on theone hand due to the need for heat production. On the otherhand a large distance between the negative and positive elec-trodes is necessary to prevent flow induced inter-mixing ofthe electrode metals that would nullify refinement. It is oftenmentioned [28, 42, 38, 32] that strong electromagnetic forcestrigger those flows. Unlike aluminium electrolysis cells, re-finement cells have been optimized little, and the technol-ogy would certainly gain from new research [41]. Yan andFray [41] directly invoke the low density differences as acause for the instability of the interfaces, discussed here in §
5. They attribute the limited application of fused salt elec-trorefining to the present design of refining cells that doesnot take advantage of the high electrical conductivity and thevery low thermodynamic potential required for the process.Coupling optimized electrorefining to carbon-free generationof electricity should, according to Yan and Fray [41], resultin “greener” metallurgy.The application of three-layer processes was also pro-posed for electronic scrap reclamation [43], removal ofMg from scrap Al [44, 45, 46], and electrorefining of Si[47,48,49]. Research on the fluid mechanics of current bear-ing three-layer systems can therefore potentially be usefulbeyond LMBs.
After three-layer liquid metal systems were put to usefor Al refining, a few decades passed before they were usedto generate electricity. In the meantime, related technolo-gies were developed, including “closed cycle battery sys- ) b) c) frozen electrolyte(crust)water cooledjointpure AlAl-Cu alloy (impure)electrolyteHoopes (1901) Betts (1905) Hoopes (1925)
Fig. 3. Aluminium refinement cells adapted from Hoopes (1901, a), Betts (1905, b), and Hoopes (1925, c).Table 1. Characteristics of different three-layer aluminium refining processes (approximate values, adapted from Table I of [38] and Table 6of [34])
Hoopes Process Gadeau Process S.A.I.A. Process top layer pure Al pure Al pure Aldensity / kgm -3 ◦ C 660 660 660 electrolyte
AlF -NaF-BaF AlF -NaF-BaCl -NaCl AlF -NaF-BaF -CaF composition (mass%) 0.34-0.28-0.38 0.15-0.17-0.6-0.08 0.48-0.18-0.18-0.16density / kgm -3 ◦ C 900 700 670 bottom layer
Al-Cu Al-Cu-Other Al-Cucomposition (mass%) 0.75-0.25 0.6-0.28-0.12 0.7-0.3density / kgm -3 ◦ C 550 unspecified 590 operating temperature / ◦ C 950 800 750tems” (Yeager in [50]), “thermally regenerative fuel cells” or“(thermally) regenerative electrochemical systems (TRES)”as they were later subsumed by Liebhafsky [51], McCullyet al. [52], and Chum and Osteryoung [53, 54]. TRES com-bine an electricity delivering cell with a regeneration unit assketched in Fig. 4: reactants are combined at the low celltemperature T , and then the product is thermally decom-posed at the higher regenerator temperature T . Thermal re-generation implies that the whole system efficiency is Carnotlimited [55, 56].A variety of such systems were investigated in the USduring the period of 1958-1968 [53]. Later, Chum and Os-teryoung classified the published material on this topic ac-cording to system type and thoroughly reviewed it in retro-spect [53, 54]. LiH based cells were building blocks of whatwere probably the first (1958, [53]) experimentally realizedthermally regenerative high-temperature systems [57,58,59],which continue to be of interest today [60, 61]. Almost at the same time a patent was filed in 1960 by Agruss [62],bimetallic cells were suggested for the electricity deliver-ing part of TRES. Henderson et al. [63] concluded their sur-vey of some 900 inorganic compounds for use in thermallyregenerative fuel cell systems with the recommendation toconcentrate on minimizing electrochemical losses, i.e., po-larization and resistance losses, in order to increase overallefficiency. Although unmentioned in [63], bimetallic cellswith liquid metal electrodes and fused salt electrolytes weredeemed most suitable to fulfill those requirements [64]. Gov-ernmental sponsored research on bimetallic cells followedsoon after at Argonne National Laboratory (1961, [65]) andat General Motors (1962 [64, 66]). Research was initiallyfocused on the application of bimetallic cell based TRES onspace power applications [67], namely systems using nuclearreactors as heat sources. Several studies explored the param-eters of concrete designs developed in frame of the “Sys-tems Nuclear Auxiliary Power Program” (SNAP), SNAP- T REGENERATIONBATTERY OPERATION P R O DUC T S R EA C T A N T S Fig. 4. Closed cycle battery system suggested by Yeager in 1957,adapted from Roberts [50]. | KOH-KBr-KI | Hg system were stable enough to allow for a mild Hgfeed of a few milliliters per minute [67]. Cell performancedepended on the flow distribution, the volume flux, and, vi-cariously, on the temperature of the incoming Hg. Cairns etal. [10] presented a conceptual design for a battery of threeNa | NaF-NaCl-NaI | Bi differential density cells, stressed thatthe density differences are large enough to clearly separatethe phases, and mentioned in the same breath that “hydrody-namic stability of the liquid streams must be carefully estab-lished.”Restraining one or more of the liquid phases in a porousceramic matrix is a straightforward means to guarantee me-chanical stability of the interfaces [22]. A direct mechani-cal separation of anodic and cathodic compartment is a ne-cessity for space applications that could not rely on gravityto keep the layers apart. Besides the solid matrix, anothermeans to immobilize electrolytes was to intermix them with
Fig. 5. Sketch of a differential density cell. ceramic powders, so called “fillers”, that resulted in pasteelectrolytes. Since both matrix and powders had to be elec-tric insulators, an overall conductivity reduction by a factorof about two to four [74,75] resulted even for the better pasteelectrolytes. Obviously, using mechanically separated elec-trode compartments is a prerequisite for any mobile appli-cation of LMBs. Equally, for cells used as components incomplete TRES, the constant flow to and from the regenera-tor and through the cell necessitates in almost all cases a me-chanical division of positive and negative electrodes. Exam-ples include the flow through cell with sandwich matrix byAgruss and Karas [69], the earlier single cup cell of “flowingtype” using an electrolyte impregnated alumina thimble [68],and the paste electrolyte cells developed at Argonne NationalLaboratory [76, 74].A different purpose was pursued by encasing the neg-ative electrode material into a retainer [77] made fromstainless steel fibers [10], felt metal [78], or later, foam[15, 13, 79, 19] as sketched in Fig. 6. Those retainers al-low electrical insulation of the negative electrode from therest of the cell without resorting to ceramics and restrict fluidmechanics to that in a porous body. The probably simplestretainer used was an Armco iron ring [65, 10] that encasedthe alkaline metal, a configuration more akin to the differen-tial density cell than to a porous body. Arrangements similarto the iron ring are sometimes used as well in molten saltelectrolysis cells [80, 24, 81] to keep a patch of molten metalfloating on top of fused salt while preventing contact withthe rest of the cell. In the case of poorly conducting mate-rials (especially Te and Se), the positive electrode had to beequipped with additional electronically conducting compo-nents to improve current collection [76, 22].With view on the low overall efficiencies of TRES dueto Carnot cycle limitations as well as problems of pumping,plumbing and separation, research on thermally regenera-tive systems ceased after 1968 [54] and later LMB work atArgonne concentrated on Li-based systems with chalcogenpositive electrodes, namely Se and Te. The high strengthof the bonds in those systems makes them unsuitable forthermal regeneration [22]. However, in their review, Chumand Osteryoung [54] deemed it worthwhile to reinvestigateTRES based on alloy cells once a solar-derived, high tem-perature source was identified. Just recently, a Na || S basedapproaches for solar electricity generation using thermal re- ig. 6. Sketch of a liquid metal cell featuring a retainer (metal foam)to contain the negative electrode. generation was suggested [82].
Using bimetallic cells as secondary elements for off-peak electricity storage was already a topic in the 1960s[10,83]. The very powerful Li || Se and Li || Te cells [75] men-tioned above are, however, unsuitable for wide scale use be-cause of the scarcity of the both chalcogens [22, 85]. In thelate 1960s and early 1970s, research at Argonne moved on toLi || S [86,22,85,87], thus leaving the area of bimetallic cells.LMB activities were reinvigorated at MIT in the firstyears of the 21 st century by Donald Sadoway. The initialdesign conceived in the fall of 2005 by Sadoway and Cederand presented by Bradwell [88] was one that combined Mgand Sb with a Mg Sb containing electrolyte, decomposingthe Mg Sb on charge and forming it on discharge. Thisnew cell design was later termed “Type A” or “ambipolarelectrolysis LMB” [89] and not followed up later. Instead,bimetallic alloying cells ‘(“Type B”) were investigated us-ing different material combinations whose selection was nothampered by the need of thermal regeneration. The Mg || Sbsystem was among the first alloying systems studied at MIT.With MgCl -NaCl-KCl (50:30:20 mol%) it used a standardelectrolyte for Mg electrolysis [90]. Thermodynamic datafor the system are available from [91] and cell performancedata were later reported also by Leung et al. [92].Right from the start research at MIT focused on the de-ployment of LMBs for large-scale energy storage [14] con-centrating on different practical and economical aspects ofutilizing abundant and cheap materials [18, 23]. Initially,large cells with volumes of a few cubic meters [93] and cross-sections of ( > × Because almost every known fluid expands when heated,spatial variations (gradients) in temperature cause gradientsin density. In the presence of gravity, if those gradients arelarge enough, denser fluid sinks and lighter fluid floats, caus-ing thermal convection. Usually large-scale convection rollscharacterize the flow shape. Being ubiquitous and funda-mental in engineering and natural systems, convection hasbeen studied extensively, and many reviews of thermal con-vection are available [98, 99, 100, 101]. Here we will give abrief introduction to convection, then focus on the particularcharacteristics of convection in liquid metal batteries. Jouleheating drives convection in some parts of a liquid metal bat-tery, but inhibits it in others. Broad, thin layers are common,and convection in liquid metal batteries differs from aque-ous fluids because metals are excellent thermal conductors(that is, they have low Prandtl number). Convection drivenby buoyancy also competes with Marangoni flow driven bysurface tension, discussed in §
4. We close this section witha discussion of magnetoconvection, in which the presence ofmagnetic fields alters convective flow.
Thermal convection is driven by gravity, temperaturegradients, and thermal expansion, but hindered by viscosityand thermal diffusion. Convection also occurs more readilyin thicker fluid layers. Combining these physical parametersproduces the dimensionless Rayleigh number Ra = g α T ∆ T L νκ , where g is the acceleration due to gravity, α T is the co-efficient of volumetric expansion, ∆ T is the characteristictemperature difference, L is the vertical thickness, ν is thekinematic viscosity, and κ is the thermal diffusivity. TheRayleigh number can be understood as a dimensionless tem-perature difference and a control parameter; for a givenfluid and vessel shape, convection typically begins at a crit-ical value Ra > Ra crit >
0, and subsequent instabilities thatchange the flow are also typically governed by Ra .f the Rayleigh number is understood as a control param-eter, then the results of changing Ra can also be expressed interms of dimensionless quantities. The Reynolds number Re = UL ν , (7)where U is a characteristic flow velocity, can be understoodas a dimensionless flow speed. The Nusselt number Nu = − QL κ∆ T , where Q is the total heat flux through the fluid, can be under-stood as a dimensionless heat flux.The canonical and best-studied context in which con-vection occurs is the Rayleigh-B´enard case, in which a fluidis contained between upper and lower rigid, no-slip bound-aries, with the lower boundary heated and the upper bound-ary cooled. Usually both boundaries are held at steady, uni-form temperatures, or subjected to steady, uniform heat flux.Convection also occurs in many other geometries, for exam-ple lateral heating. Heating the fluid from above, however,produces a stably-stratified situation in which flow is hin-dered. Temperature is not the only parameter that affects fluiddensity. Chemical reactions, for example, can also changethe local density such that buoyancy drives flow. That pro-cess is known as compositional convection, and the corre-sponding control parameter is the compositional Rayleighnumber Ra X = g α X ∆ XL ν D , where α X is the coefficient of volumetric expansion withconcentration changes, ∆ X is the characteristic concentrationdifference, and D is the material diffusivity. (Compositionalconvection is one mechanism by which reaction drives flow;entropic heating, discussed above, is another.) For liquidmetal batteries, the electrode materials have densities thatdiffer by more than an order of magnitude (see Table 2),and ∆ X ∼
30 mol%, so we expect compositional convectionto cause substantial flow. For comparison, we can considerthermal convection in bismuth at 475 ◦ C, for which the co-efficient of thermal expansion is α T = . × − /K [102].Making the order-of-magnitude estimate ∆ T ∼ α X ∆ X (cid:29) α T ∆ T . For the Na || Bi system at anoperating temperature of 475 ◦ C, the compositional Rayleighnumber exceeds the thermal one by six orders of magnitude.Thus compositional convection is likely much stronger thanthermal convection. Compositional convection is unlikelyduring discharge because the less-dense negative electrode material (e.g., Li) is added to the top of the more-dense posi-tive electrode, producing a stable density stratification. Dur-ing charge, however, less-dense material is removed from thetop of the positive electrode, leaving the remaining materialmore dense and likely to drive compositional convection bysinking.
Liquid metal batteries as sketched in Fig. 8 are a morecomplicated and interesting case than a single layer system.Commercially viable liquid metal battery chemistries involvematerials that are solid at room temperature; to operate, theymust be heated to 475 ◦ C [15]. External heaters produce ther-mal convection in almost any arrangement, especially themost efficient one in which heaters are installed below thebatteries, producing the Rayleigh-B´enard case. During op-eration, however, external heaters are often unnecessary be-cause the electrical resistance of the battery components con-verts electrical energy to heat in a process known as Jouleheating or ohmic heating. If the battery current is largeenough and the environmental heat loss is small enough,batteries can maintain temperature without additional heat-ing [107]. (In fact, cooling may sometimes be necessary.) Inthis case, the primary heat source lies not below the battery,but within it. As Table 3 shows, molten salts have electri-cal conductivity typically four orders of magnitude smallerthan liquid metals, so that essentially all of the Joule heatingoccurs in the electrolyte layer, as shown in Fig. 7. The pos-itive electrode, located below the electrolyte, is then heatedfrom above and becomes stably stratified; its thermal pro-file actually hinders flow. Some flow may be induced bythe horizontal motion of the bottom of the electrolyte layer,which slides against the top of the positive electrode and ap-plies viscous shear stresses, but simulations of Boussinesqflow show the effect to be weak [108, 109]. The electrolyteitself, which experiences substantial bulk heating during bat-tery charge and discharge, is subject to thermal convection,especially in its upper half [109, 110]. One simulation of aninternally-heated electrolyte layer showed it to be character-ized by small, round, descending plumes [111]. Experimentshave raised concern that thermal convection could bring in-termetallic materials from the electrolyte to contaminate thenegative electrode [112]. Convection due to internal heatinghas also been studied in detail in other contexts [113].The negative electrode, located above the electrolyte, isheated from below and is subject to thermal convection. In anegative electrode composed of bulk liquid metal, we wouldexpect both unstable thermal stratification and viscous cou-pling to the adjacent electrolyte to drive flow. Simulationsshow that in parameter regimes typical of liquid metal bat-teries, it is viscous coupling that dominates; flow due to heatflux is negligible [108]. Therefore, in the case of a thickelectrolyte layer, mixing in the electrolyte is stronger thanmixing in the negative electrode above; in the case of a thinelectrolyte layer, the roles are reversed [108]. However, neg-ative electrodes may also be held in the pores of a rigid metalfoam by capillary forces, which prevents the negative elec- able 2. Properties of common electrode materials. Values for metals at respective melting temperature taken from [103, 104] except Liconductivity from [105], Pb-Bi eutectic data from Sobolev [106], and Pb data from [102].
Material ν / − (m /s) κ / − (m /s) ρ (kg/m ) α T / − (K -1 ) σ E / (S/m) Pr Pm / − Negative electrode
Li 1.162 2.050 518 1.9 3.994 0.0567 5.8315Mg 0.7862 3.4751 1590 1.6 3.634 0.0226 3.5907Na 0.7497 6.9824 927 2.54 10.42 0.0107 9.8195
Positive electrode
Bi 0.1582 1.1658 10050 1.17 0.768 0.0136 0.1527Pb 0.253 1.000 10673 1.199 1.050 0.0253 0.334Sb 0.2221 1.3047 6483 1.3 0.890 0.0170 0.2485Zn 0.5323 1.5688 6575 1.5 2.67 0.0339 1.7860eutectic Pb-Sb ν κ ρ α T σ E Pr Pm eutectic Pb-Bi 0.3114 0.5982 10550 1.22 0.909 0.052 0.3557
Table 3. Properties of common electrolyte materials, from Janz et al. [114, 115] Todreas et al. [116], Kim et al. [11], and Masset etal. [117, 118].
Material ν / − (m /s) ρ (kg/m ) σ E (S/m)LiF 1.228 1799 860LiCl 1.067 1490 586LiI 0.702 3.0928 396.68NaCl 0.892 1547 363CaCl ig. 7. Thermal convection in a three-layer liquid metal battery. Avertical cross-section through the center of the battery (a) shows thatthe temperature is much higher in the electrolyte than in either elec-trode. A horizontal cross-section above the electrolyte (b) shows vig-orous flow. (Here u z is the vertical velocity component.) Adaptedfrom [108], with permission. T , ρ z g Iu Fig. 8. Sketch of a liquid metal cell with thermal convection tery cross-section, concentrating electrical current near thecenter and reducing it near the sidewall. The fact that currentcan exit the positive electrodes through the sidewalls as wellas the bottom wall allows further deviation from uniform,axial current. Nonuniform current density causes Joule heat-ing that is also nonuniform — in fact, it varies more sharply,since the rate of heating is proportional to the square of thecurrent density. This gradient provides another source ofconvection-driven flow. Putting more current density near the central axis of the battery creates more heat there andcauses flows that rise along the central axis. Interestinglyelectro-vortex flow (considered in detail in §
7) tends to causethe opposite motion: descent along the central axis. Sim-ulations have shown that negative current collector geome-try and conductivity substantially affect flow in liquid metalbatteries [121]. Other geometric details can also create tem-perature gradients and drive convection. For example, sharpedges on a current collector concentrate current and causeintense local heating. The resulting local convection rollsare small but can nonetheless alter the global topology offlow and mixing. Also, if solid intermetallic alloys form,they affect the boundary conditions that drive thermal con-vection. Intermetallics are typically less dense than the sur-rounding melt, so they float to the interface between the pos-itive electrode and electrolyte. Intermetallics typically havelower thermal and electrical conductivity than the melt, sowhere they gather, both heating and heat flux are inhibited,changing convection in non-trivial ways.
In addition to the Rayleigh number, a second dimension-less parameter specifies the state of a convecting system, thePrandtl number Pr = νκ . A ratio of momentum diffusivity (kinematic viscosity) tothermal diffusivity, the Prandtl number is a material prop-erty that can be understood as a comparison of the rates atwhich thermal motions spread momentum and heat. Table 2lists the Prandtl number of a few relevant fluids. Air and wa-ter are very often the fluids of choice for convection studies,since so many industrial and natural systems involve them.But air and water have Prandtl numbers that differ from liq-uid metals and molten salts by orders of magnitude: Pr = Pr = . Re and Nu in terms of theinputs Ra and Pr . To begin, every possible Rayleigh-B´enardexperiment is categorized according to the role of boundarylayers in transporting momentum and heat. Boundary layersoccur near walls, and transport through them proceeds (to agood approximation) by diffusion alone. On the other hand,in the bulk region far from walls, transport proceeds primar-ily by the fast and disordered motions typical in turbulentflow. Any particular Rayleigh-B´enard experiment can be as-signed to one of eight regimes, depending on three questions:Is momentum transport slower through the boundary layer orthe bulk? Is heat transport slower through the boundary layeror the bulk? And, which boundary layer — viscous or ther-mal — is thicker and therefore dominant? Answering thosehree questions makes it possible to estimate the exponentsthat characterize the dependence of Re and Nu on Ra and Pr .According to the theory [122], the Nusselt number can de-pend on the Prandtl number as weakly as Nu ∝ Pr − / oras strongly as Nu ∝ Pr / , and the Reynolds number can de-pend on the Prandtl number as weakly as Re ∝ Pr − / or asstrongly as Re ∝ Pr − / . Again, convection in liquid metalsand molten salts differs starkly from convection in water orair: changing Pr by orders of magnitude causes Re and Nu tochange by orders of magnitude as well. Experiments study-ing convection in sodium ( Pr = . Nu ) for a given temperature difference ( Ra ) isindeed smaller than for fluids with larger Pr [123]. Exper-iments have also shown that at low Pr , more of the flow’skinetic energy is concentrated in large-scale structures, es-pecially large convection rolls. In a thin convecting layerwith a cylindrical sidewall resembling the positive electrodeof a liquid metal battery, slowly fluctuating concentric ring-shaped rolls often dominate [123]. Those rolls may interactvia flywheel effects [124]. Convection in liquid metal batteries proceeds in thepresence of — and can be substantially altered by — electriccurrents and magnetic fields. Introductions and overviews ofthe topic of magnetoconvection have been provided in textsdedicated to the subject [125] as well as texts on the moregeneral topic of magnetohydrodynamics [126]. The strengthof the magnetic field can be represented in dimensionlessform using the Hartmann number Ha = BL (cid:114) σ E ρν , (8)which is the ratio of electromagnetic force to viscous force.Here B is the characteristic magnetic field strength, σ E isthe electrical conductivity, and ρ is the density. (Magneticfield strength is also sometimes expressed using the Chan-drasekhar number, which is the square of the Hartmann num-ber.) When Ha (cid:29)
1, magnetic fields tend to strongly alterconvection, though the particular effects depend on geome-try. When an electrically conducting fluid flows in the pres-ence of a magnetic field, electrical currents are induced, andthose currents in turn produce magnetic fields. Accordingto Lenz’s law, the direction of any induced current is suchthat it opposes change to the magnetic field. Accordingly,conductive fluids flow most easily in directions perpendicu-lar to the local magnetic field. The simplest such flows formpaths that circulate around magnetic fields; in the presence ofmagnetic fields, convection rolls tend to align with magneticfield lines. That phenomenon is analogous to the tendencyof charged particles in plasmas to orbit magnetic field lines.Other motions, such as helical paths, are also possible.If convection occurs in the presence of a vertical mag-netic field, alignment is impossible, since convection rollsare necessarily horizontal. Accordingly, vertical magnetic fields tend to damp convection [127, 128, 129]. The crit-ical Rayleigh number at which convection begins scalesas Ra crit ∝ Ha [127], as has been verified experimen-tally [130]. The Rayleigh number of oscillatory instabilityof convection rolls is also increased by the presence of a ver-tical magnetic field [128]. Common sources of vertical mag-netic fields in liquid metal batteries include the Earth’s field(though it is relatively weak) and fields produced by wirescarrying current to and from the battery.Just as the Grossmann and Lohse scaling theory [122]considers the dependence of Re and Nu on the inputs Ra and Pr in convection without magnetic fields, a recent scalingtheory by Schumacher and colleagues [131] considers thedependence of Re and Nu on the inputs Ra , Pr — and also Ha — in the presence of a vertical magnetic field. The rea-soning is analogous: the scaling depends on whether trans-port time is dominated by the boundary layer or the bulk, andwhich boundary layer is thickest. However, the situation ismade more complex by the need to consider magnetic fieldtransport in addition to momentum and temperature trans-port, and the possibility that the Hartmann (magnetic) bound-ary layer might be thickest. Altogether, 24 regimes are pos-sible. To reduce the number of free parameters, the authorsconsidered the case in which Pr (cid:28) Pm (cid:28)
1, where Pm = ν µ σ E is the magnetic Prandtl number. (Here µ is themagnetic permeability.) That special case applies to mate-rials common in liquid metal batteries, and still spans fourregimes of magnetoconvection. Categorization depends onwhether the magnetic field is strong ( Ha (cid:29) Ra (cid:29) Re ) and heat flux ( Nu ) in-crease [132]. Moreover, since magnetic fields of any ori-entation damp turbulence [133], convection in the presenceof horizontal magnetic fields tends to be more ordered, spa-tially, than convection in the Ha = Ra increases,waves develop on the horizontal convection rolls [128, 123].In liquid metal batteries, internal electrical currents runprimarily vertically and induce toroidal horizontal magneticfields. Poloidal convection rolls are therefore common, sincetheir flow is aligned and circulates around the magneticfield lines. If the sidewall is cylindrical, boundary condi-tions further encourage poloidal convection rolls. Such rollshave been observed in liquid metal battery experiments, andthe characteristic mass transport time decreases as Ha in-creases [134, 135]. Simulations have shown similar results,with the number of convection rolls decreasing as the currentincreases [136]. Other simulations, however, have suggestedthat electromagnetic effects are negligible for liquid metalbatteries with radius less than 1.3 m [108,109]. Further studymay refine our understanding. In batteries with a rectangularcross-section, we would expect horizontal convection rollscirculating around cores that are nearly circular near the cen-ral axis of the battery, where the magnetic field is strong andthe sidewall is remote. Closer to the wall, we would expectrolls circulating around cores that are more nearly rectangu-lar, due to boundary influence. The molecules of a stable fluid are typically attractedmore strongly to each other than to other materials. The re-sult is the surface tension (or surface energy) σ , which canbe understood as an energy per unit area (or a force per unitlength) of interface between two materials. The surface ten-sions of liquid metals and molten salts are among the highestof any known materials, so it is natural to expect surface ten-sion to play a role in liquid metal batteries. This section willconsider that role.If the surface tension varies spatially, regions of highersurface tension pull fluid along the interface from regionsof lower surface tension. Viscosity couples that motion tothe interior fluid, causing “Marangoni flow”, sketched inFig. 9. Surface tension can vary spatially because it dependson temperature, chemical composition, and other quantities.For most fluids, surface tension decreases with temperature: ∂σ / ∂ T <
0, and flow driven by the variation of surface ten-sion with temperature is called “thermocapillary flow” and isdescribed in existing reviews [137, 138]. Flow driven by thevariation of surface tension with composition, called “solutalMarangoni flow”, has also been considered [139, 140, 141],especially in the context of thin films [142, 143]. In this sec-tion we will focus on Marangoni flow phenomena that arerelevant to liquid metal batteries, focusing on similarities anddifferences to Marangoni flows studied in the past. We willestimate which phenomena are likely to arise, drawing in-sight from one pioneering study that has considered the roleof thermocapillary flow in liquid metal batteries [110].
Thermocapillary flow is driven by temperature gradi-ents but hindered by viscosity, thermal diffusion, and den-sity (which provides inertia). Thermocapillary flow also oc-curs more readily in thicker fluid layers. Combining thesephysical parameters produces the dimensionless Marangoninumber Ma = (cid:12)(cid:12)(cid:12) ∂σ∂ T (cid:12)(cid:12)(cid:12) L ∆ T ρνκ . (9)The Marangoni number plays a role analogous to theRayleigh number in thermal convection. Larger values of Ma make thermocapillary flow more likely and more vigor-ous. Because temperature gradients drive both thermocapil-lary flow and thermal convection, the two phenomena oftenoccur simultaneously. We can compare their relative magni- Fig. 9. Marangoni flow occurs when surface tension at a fluid inter-face varies spatially. Variation along the interface (a) always drivesflow. Variation across the interface (b) causes an instability thatdrives flow if the variation is sufficiently large, as quantified by theMarangoni number Ma . tudes via the dynamic Bond number Bo = RaMa = ρα T gL (cid:12)(cid:12)(cid:12) ∂σ∂ T (cid:12)(cid:12)(cid:12) . (10)Thermal convection dominates when Bo (cid:29)
1, whereas ther-mocapillary flow dominates when Bo (cid:28)
1. Because of the L factor, thermal convection tends to dominate in thick lay-ers, whereas thermocapillary flow tends to dominate in thinlayers. Thermocapillary flow, like thermal convection, isqualitatively different for fluids with small Prandtl number(like liquid metals and molten salts) than for fluids with largePrandtl number. Thermocapillary flow phenomena depend on the direc-tion of the thermal gradient with respect to the interface. Sur-face tension varying along the interface always drives flow,as shown in Fig. 9a. We will consider this case in greaterdetail below. However, if temperature (and thus surface ten-sion) varies across the interface, as in Fig. 9b, the situation ismore complicated. Thermocapillary flow is possible only ifheat flows across the interface in the direction of increasingthermal diffusivity [110]. That condition is satisfied at bothinterfaces between electrolyte and electrode in a liquid metalbattery because the molten salt electrolyte has lower thermaldiffusivity than the metals, and typically higher temperatureas well, because of its low electrical conductivity.With the directional condition satisfied, three phenom-ena are possible [137, 138]. First, the fluid can remainstagnant if thermal conduction carries enough heat and theviscosity is large enough to damp flow. Second, short-wavelength thermocapillary flow can arise, in which the sur-face deformations caused by surface tension are damped pri-arily by gravity. Third, long-wavelength thermocapillaryflow can arise, in which the surface deformations caused bysurface tension are damped primarily by diffusion (of bothmomentum and heat). The relative strength of the two damp-ing mechanisms is quantified by the Galileo number G = gd νκ . (11) G (cid:29) G (cid:28) G (cid:29) Ma >
80 [145], as experimental studieshave confirmed [144]. Typically short-wavelength flow ap-pears as an array of hexagons tiled across the interface. For G ≤ Ma > G /
3, instead of short-wavelength flow, long-wavelengthflow arises [146, 144]. The long-wavelength flow has no re-peatable or particular shape, instead depending sensitivelyon boundary conditions. When observed in experiments, thelong-wavelength flow always ruptures the layer in which itoccurs [138], a property particularly alarming for designersof liquid metal batteries. The short-wavelength mode, on theother hand, causes nearly zero surface deformation [138].We can estimate the relevance of thermocapillary flowand the likelihood of short-wavelength and long-wavelengthflow using dimensionless quantities, as long as the necessarymaterial properties are well-characterized. Most difficult toobtain is the rate of change of surface tension with tempera-ture, ∂σ / ∂ T . Its value is well-known for Pb, Bi, and their eu-tectic alloy [102] because of its importance in nuclear powerplants, however. One pioneering study [110] simulated ther-mocapillary flow in a hypothetical three-layer liquid metalbattery with a eutectic PbBi positive electrode, a LiCl-KClelectrolyte, and a Li negative electrode. First consideringthe positive electrode, for a PbBi layer with L =
20 mmand ∆ T = . Ma = <
80. Nor do we expect long-wavelength flowbecause, according to eq. 11, G = × (cid:29) L =
20 mmand ∆ T = Ma = ,
000 implies vigorous thermocap-illary flow, and Bo = G = . × , we expect the short-wavelength mode, notthe long-wavelength mode. Finally, we expect minimal flowin the negative electrode if it is contained in a rigid metalfoam. All of these predictions should be understood as pre-liminary since eqs. 9, 10, and 11 consider a layer in which only one surface is subject to surface tension effects, but theelectrolyte layer in a liquid metal battery is subject to surfacetension effects on both is upper and lower surfaces.In fact, though the long-wavelength mode can readily beobserved in laboratory experiments with silicone oils [147,144,148], liquid metals and molten salts typically have muchsmaller kinematic viscosity and thermal diffusivity, yieldingsmall values of G that make the long-wavelength mode un-likely. Using the Ma > G / µ m or less in either thePbBi or LiCl-KCl layer. Other considerations require boththe electrolyte and the positive electrode to be much thicker,so rupture via the long-wavelength thermocapillary mode isunlikely in a liquid metal battery.We would expect, however, that the short-wavelengththermocapillary mode often arises in liquid metal batteries,especially in the electrolyte layer. Though unlikely to rup-ture the electrolyte, the short-wavelength mode may mix theelectrolyte, promoting mass transport. The short-wavelengthmode might also couple to other phenomena, for example theinterfacial instabilities discussed in § Surface tension that varies along the interface alwaysdrives flow, and we can estimate its speed by consideringthe energy involved. Suppose a thin, rectangular layer offluid occupies the region 0 ≤ x ≤ L x , 0 ≤ y ≤ L y , 0 ≤ z ≤ L z in Cartesian coordinates ( x , y , z ) , with L z (cid:28) L x and L z (cid:28) L y .Suppose thermocapillary forces act on the z = L z surface, andthat temperature varies in the x direction, such that surfacetension drives flow in the x direction. The work done bythermocapillary forces (per unit volume) scales as ∆ T ∂σ∂ T L x L y L x L y L z . If the flow is steady, if pressure variations are negligible,and if inertial and gravitational forces are negligible, thenthe work done by thermocapillary forces must be dissipatedby viscous damping. For an incompressible Newtonian fluid,the viscous damping term (in energy per unit volume) reads µ τ u j (cid:18) ∂ u i ∂ x i ∂ x j + ∂ u j ∂ x i ∂ x j (cid:19) , where we use indicial notation with summation implied, u j is a velocity component, and τ is a characteristic flow time.We can estimate the flow time in terms of a characteristicspeed U and the total circulation distance: τ ∼ ( L x + L z ) / U .If there is no flow in the y direction and no flow variation inthe x direction, we can estimate the gradients in the viscousdamping term as well. Setting the result equal to the work(per unit volume) done by capillary forces and solving for U ,e estimate a characteristic speed U ∼ ∆ T ∂σ∂ T µ L z L z + L x . As expected, the speed increases with ∆ T and ∂σ / ∂ T , whichincrease the thermocapillary force; and increases with L z ,which reduces viscous shear; but decreases with µ , L z , and L x , which increases viscous drag. Again considering themodel of [110], we find U ∼ . U ∼ Ma <
200 [110]. Electrolyte layers thinner than 2 mm exhibit nei-ther thermocapillary flow nor thermal convection for realisticcurrent densities (less than 2000 A/m ). We raise one caveat:if the negative electrode is contained by a metal foam, flowthere would likely be negligible. Solutal Marangoni flow has been studied less than ther-mocapillary flow, and to our knowledge has not yet beenaddressed in the literature for the specific case of liquidmetal batteries. One experimental and numerical study founda cellular flow structure reminiscent of the hexagons char-acteristic of the short-wavelength mode in thermocapillaryflow [140]. A later experimental and numerical study bythe same authors [141] varied the thickness of the fluid layerand its orientation with respect to gravity, finding that a two-dimensional simulation in which flow quantities are averaged
Fig. 10. Marangoni flow in a three-layer liquid metal battery. Tem-perature is indicated in color, and velocity is indicated by arrows.The horizontal top surface of the electrolyte shows Marangoni cells(a), with downwellings where the temperature is highest. A verticalcross-section through the center of the battery (b) also shows down-wellings, and indicates that the temperature is much higher in theelectrolyte than in either electrode. Adapted from [110], with permis-sion. across the layer thickness fails to match experiments withthick layers. The study also found that cells coarsen overtime, perhaps scaling as t / , where t is time.Though studies of solutal Marangoni flow in liquidmetal batteries have not yet been published, the phenomenonis likely, because charge and discharge alter the compositionof the positive electrode. In past work, salt loss in lithium-chalcogen cells has been attributed to Marangoni flow [150].In the case where composition varies across the interface, so-lutal Marangoni flow is possible only if material flows acrossthe interface in the direction of increasing material diffu-sivity. In a liquid metal battery, the material of interest isthe negative electrode material, e.g. Li, and the interface ofinterest is the one between molten salt electrolyte and liq-uid metal positive electrode. The diffusivity of Li in Bi is1 . × − m /s, and the diffusivity of Li in LiBr-KBr hasbeen calculated as 2 . × − m /s [151]. A battery madewith those materials would be prone to solutal Marangoniflow driven by composition varying across the interface dur-ng discharge, but not during charge. Solutal Marangoniflow driven by variations across the interface is likely to oc-cur in both short- and long-wavelength modes, depending onthe appropriate Marangoni and Galileo numbers (analogousto eqs. 9 and 11). However, a two-layer model for solutalMarangoni flow is unstable at any value of the Marangoninumber [137, 152]. Variations of composition along the in-terface will drive solutal Marangoni flow regardless of theirvalues.An estimate of the magnitude of solutal Marangoni flowwould be useful. Even less is known about the rate of changeof surface tension with composition than about the rate ofchange with temperature. Still, we can put an upper boundon the magnitude of solutal Marangoni flow, and compareto thermocapillary flow, by considering extreme cases. Theforce per unit length that drives thermocapillary flow is (cid:12)(cid:12)(cid:12)(cid:12) ∂σ∂ T ∆ T (cid:12)(cid:12)(cid:12)(cid:12) . Again considering the same situation as [110], we find aforce per unit length around 1 . × − N/m. The force perunit length that drives solutal Marangoni flow is (cid:12)(cid:12)(cid:12)(cid:12) ∂σ∂ X ∆ X (cid:12)(cid:12)(cid:12)(cid:12) , Unfortunately, ∂σ / ∂ X is, to our knowledge, unknown in theliterature for materials common to liquid metal batteries. Al-ternatively, we can consider the extreme case in which dif-ferent regions of the interface are composed of different purematerials, so that the force per unit length is simply thedifference between their (known) surface tensions. Using σ PbBi = . σ Li = .
396 N/m at453 K [153], and σ LiCl − KCl = .
122 N/m at 823 K [154],we find σ PbBi − σ Li = . × − N/m and σ Li − σ LiCl − KCl = . × − N/m. These estimates are imprecise: consider-ing temperature will change them by a few percent, and con-sidering different battery chemistry will change them more.These estimate are also upper bounds. Nonetheless, theseestimates are two to four orders of magnitude larger than thetypical force per unit length that drives thermocapillary flow.If the true solutal forces reach even a few percent of theseestimates, solutal Marangoni flow rivals or dominates ther-mocapillary flow in liquid metal batteries. Better constraintson the magnitude of solutal Marangoni flow — beginningwith estimates of ∂σ / ∂ X —would be a valuable contributionfor future work. It is a well known phenomenon in Hall-H´eroult, i.e., alu-minium electrolysis cells (AECs) that long wave instabilitiescan develop at the interface of the cryolite and the liquid alu-minium [155, 156, 157, 158]. Those instabilities are knownas “sloshing” or “metal pad roll instability”. Not only be-cause AECs gave the inspiration for the initial LMB concept
FF I h I h B σ ≫ σ el σ ≫ σ el σ el Fig. 11. Sketch of a liquid metal cell undergoing an interfacial in-stability at MIT [88] it is worthwhile to have a closer look at the roleof interface instabilities in LMBs. If the interface between agood electric conductor (metal, σ = O ( ) S/m) and a poorone (electrolyte, σ el = O ( ) S/m) is slightly inclined withrespect to the horizontal plane, the current distribution insidethe layers changes. In the metal layer(s), horizontal perturba-tion currents ( I h ) arise as sketched in Fig. 11. Those horizon-tal currents interact with the vertical component of a back-ground magnetic field generated, e.g., by the current supplylines, generating Lorentz forces that set the metal layer intomotion. This mechanism was first explained by Sele [155]for AECs. As a consequence, gravity waves with a character-istic length of the vertical cell size develop and culminate in asloshing motion of the aluminium. Wave amplitudes may be-come large enough to reach the graphite negative electrodesand short-circuit the cell, thereby terminating the reductionprocess. In order to prevent the waves from contacting thenegative electrodes for a cell current of about 350 kA, con-sidered as an upper limit for modern cells [159], a cryolitelayer at least 4.5 cm thick is required [156]. These boundaryconditions mean that nearly half of the cell voltage is spentovercoming the electrolyte resistance, and the correspond-ing electric energy is converted to heat [156]. Reducing theelectrolyte layer thickness by even a few millimeters wouldresult in large cost savings, but is made impossible by thesloshing instabilities. Admittedly, Joule heating is not en-tirely wasted, because it maintains the high cell temperatureand to permits the strong wall cooling that allows the for-mation of the side-wall protecting ledge [157]. Metal padrolling in AECs, which typically have a rectangular cross-section, occurs if the parameter β = JB z g ∆ρ CE · L x H E · L y H C (12)exceeds a critical value β cr . Here J and B z denote the ab-solute values of the cell’s current density and of the ver-ical component of the background magnetic field, respec-tively, ∆ρ CE is the density difference between cryolite andaluminium, and H E , H C , L x , L y refer to the layer heights andthe lateral dimensions of the AEC, respectively. See Fig. 12(left) for reference. The first factor in Eq. (12) is the ratio ofLorentz force to gravity force, and the latter ones are ratiosof layer height to lateral cell dimension.Bojareviˇcs and Romerio [160] obtained an expressionfor β cr depending on wave numbers of gravity waves m , n in x , y direction developing in rectangular cells: β cr = π (cid:12)(cid:12)(cid:12)(cid:12) m L y L x − n L x L y (cid:12)(cid:12)(cid:12)(cid:12) . (13)According to Eq. (13) cells with square or circular cross-section are always unstable because their lateral dimensionsare equal and thus β cr =
0. Davidson and Lindsay [161] cameto a similar conclusion regarding the instability threshold forcircular and square cells using both shallow water theory anda mechanical analogue.It can be expected that three-layer systems like Al re-finement cells (cf. § § xyz EC L x L y ρ E ρ C H E H C z rAEC ρ A ρ E ρ C D H A H E H C Fig. 12. Characteristic dimensions and notations for an aluminiumelectrolysis cell (left) and a liquid metal battery (right). dynamic problem by a system possessing only four degreesof freedom associated with the two-dimensional oscillationsof each pendulum. The Lorentz force due to the interactionof the vertical background magnetic field and the horizontalcurrents can cause an instability if C A JB z L x g ρ A H E H A + C C JB z L x g ρ C H E H C > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ω x ω y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (14)Here C A and C C are constants of order one that accountfor geometry [163], and ρ A and ρ C denote the densities of thenegative and positive electrodes, respectively. The pendulaoscillate with their natural gravitational frequencies ω x and ω y . Again, it is evident from Eq. (14) that circular or squarecross-sections are predicted to be always unstable.Zikanov [163, 109] discussed an additional instabilitythat may arise even in the absence of a background magneticfield due to the interaction of J -generated azimuthal mag-netic field B ϕ with the current perturbations. He finds thesystem to be unstable if µ J D g (cid:18) D ρ A H A H v + D ρ C H C H + ρ A H E − ρ C H E (cid:19) > . (15)As estimated by Zikanov [163], for rectangular cells theinstability due to the interaction of the perturbation currentswith the azimuthal field of the main current described by cri-terion (15) appears to be more dangerous than that caused bythe action of the background magnetic field on the horizontalcompensating currents (14).It should be mentioned that criteria predicting instabilityonset for any non-vanishing Lorentz force neglect dissipativeeffects caused by magnetic induction and viscosity as well asthe influence of surface tension [164].Weber et al. [165,164] investigated the metal pad roll in-stability in an LMB using a volume-of-fluid method adaptedfrom the finite volume code OpenFOAM [166] and supple-mented by electromagnetic field calculations to solve the fullNavier-Stokes equations. The material properties correspond ig. 13. Minimum electrolyte layer height h min depending on β ac-cording to Eq. (16) for the Mg | KCl-MgCl -NaCl | Sb system adaptedfrom Weber et al. [164]. For each curve, only the parame-ter named in the legend is varied, the other ones stay constant( j = A/cm , B z = mT, H A = . cm, H E = cm, ρ A = kg/m ). ∆ρ EA = ρ E − ρ A , the inset shows a snapshot ofthe anode/electrolyte interface for β = . . to the special case of the Mg | KCl-MgCl -NaCl | Sb system(see Table 4 for an overview of typical systems).As expected, if one interface is set in motion and theother remains nearly at rest, a criterion similar to the Selecriterion (12) can be formulated: β = JB z D g ( ρ E − ρ A ) H A H E > β cr , (16)using the density differences between negative electrode andelectrolyte and the respective layer heights. Sloshing in cir-cular cells sets in above a relatively well defined β cr, sloshing = .
44. Short-circuiting needs more intense Lorentz forces andhappens above β cr, short-circuit ≈ .
5. The validity of both val-ues is limited to the Mg || Sb system and to the aspect ra-tios H A / D = .
45 and H E = . || Sb system as well, but used a shallow water approxi-mation combined with the electromagnetic field equations.They considered a cell with a 8 × cross-section andMg and Sb layers both 20 cm in height divided by a 5 or8 cm thick electrolyte. In agreement with the results of We-ber et al. [165,164], Bojareviˇcs et al. [96,97] found the inter-face between negative electrode and electrolyte to be muchmore sensitive to the instability than the interface betweenthe electrolyte and positive electrode. The difference is ex-plained by density differentials: the electrolyte typically hasdensity closer to that of the negative electrode than the pos-itive electrode. Bojareviˇcs and Tucs [97] further optimized the magnetic field distribution around the LMB by re-usinga commercial Trimet 180 kA cell series in their simulation.While the unoptimized cell could only be stabilized for the8 cm thick electrolyte, the optimized cell was able to operatewith 2.5 cm electrolyte height. However, even in the lattercase the voltage drop in the electrolyte is still found to beexcessive with 0.49 V at a current of 100 kA.Horstmann et al. [167] investigated the wave couplingdynamics of both interfaces by applying potential theory aswell as direct numerical simulations to LMBs with circularcross-section. While interface tension should be taken intoaccount for (very) small cells and large wave numbers, it isnegligible in the limit of large-scale LMBs. There, the wavesare purely gravitational ones and the strength of their cou-pling depends only on the ratio of the density differences A g = ρ C − ρ E ρ E − ρ A . (17)Thus, for practical cases, A g is the control parameter thatdetermines how strongly both interfaces interact. Wave onsetis described by Sele-like parameters extended by interfacetension terms for both interfaces. The expressions reduces tothe Sele criterion (16) in the limit of large LMBs consideredhere.At the same time A g describes for thin electrolyte lay-ers ( H E →
0) the ratios of amplitudes and frequencies of the(anti-symmetric) waves (cid:12)(cid:12)(cid:12)(cid:12) ˆ η mn AE ˆ η mn EC (cid:12)(cid:12)(cid:12)(cid:12) = ω ω = A g . (18)Here ˆ η mn AE , ˆ η mn EC denote the amplitudes of the of the waves atthe AE and EC interfaces, respectively, with the azimuthalwave number m and radial wave number n . ω AE and ω EC arethe corresponding frequencies.The waves at both interfaces can be considered as cou-pled in the range 0 . < A g <
10. If the metal layer withdensity more similar to the electrolyte is thinner, the lim-iting values have to be corrected by the metal layer heightratio H C / H A . The coupled regime can be further dividedinto “weakly coupled” ( 0 . < A g (cid:46) .
7, 2 (cid:46) A g <
10) and“strongly coupled” (0 . (cid:46) A g (cid:46)
2) regimes. The thresholdvalues are empirical. In the weakly coupled regime the inter-faces are anti-symmetrically displaced and co-rotate in thedirection determined by the more prominent wave. Dynam-ics in the strongly coupled regime are more complex. Formoderate β AE ≈ . β AE ( ≈ .
2) leads to synchronously ro-tating metal pads (“synchronous tilting instability”). Thesestrongly coupled instabilities may not occur in cells that arenot circular, however.While the two strongly coupled LMB types (Li || Te andLi || Se) have limited practical relevance due to the scarcity of able 4. Coupling parameter A g calculated for different possible working material combinations. The densities are reported at workingtemperature T op , adapted from [167] Electrodes Electrolyte T op ρ A ρ E ρ C A g ( ◦ C) (kg m -3 ) s t r ong l y c oup l e d Li || Se LiCl-LiF-LiI 375 497 2690 3814 0.51Al || Al-Cu ∗ AlF -NaF-CaCl -NaCl 800 2300 2700 3140 1.1Li || Te LiCl-LiF-LiI 475 489 2690 5782 1.41 w ea k l y c oup l e d Na || Sn NaCl-NaI 625 801 2420 6740 2.67Li || Bi LiCl-LiF-LiI 485 488 2690 9800 3.22Na || Bi NaCl-NaI-NaF 550 831 2549 9720 4.18K || Hg KBr-KI-KOH 250 640 2400 12992 6.02 no t c oup l e d Ca || Sb CaCl -LiCl 700 1401 1742 6270 13.28Ca || Bi CaCl -LiCl 550 1434 1803 9720 21.43Mg || Sb KCl-MgCl -NaCl 700 1577 1715 6270 33.06their positive electrode materials, three-layer refinement cellsare almost always strongly coupled. Gesing et al [45, 46]formulate it as a characteristic of their Mg-electrorefinementmethod that the electrolyte has to have a density halfway be-tween that of Al and Mg, i.e., they require a coupling param-eter A g = L x / L y on the critical value of the Sele criterion is quitestrong and resembles the situation in ARCs. This strong ef-fect can be explained by the fact that the aspect ratio deter-mines the set of available natural gravitational wave modesand the strength of the electromagnetic field that is needed totransform them into a pair with complex-conjugate eigenval-ues [168, 161].For comparable density jumps at the interfaces,Zikanov’s [168] results again agree with those of Horstmannet al. [167] in that both interfaces are significantly deformed.The system behavior becomes more complex and is differ-ent from that found in AECs. The waves of both inter-faces can couple either symmetrically or anti-symmetrically.Zikanov [168] found examples where the presence of the sec-ond interface stabilizes the system, which was not predictedby his two-slab model [163], whose simplifications are prob-ably too strong to capture this part of the dynamics. Electric currents induce magnetic fields and interactwith those fields, sometimes bringing unexpected conse-quences. Suppose a current runs axially and has azimuthallysymmetric current density
JJJ , as sketched in Fig. 14. Then,by the right-hand rule, it induces a magnetic field
BBB ϕ thatis purely in the azimuthal direction, and interacts with thatfield to cause a Lorentz force per unit volume FFF
LLL = JJJ × BBB ϕ directed radially inward. That force can be understood asa magnetic pressure. If the current flows through a fluidthat is incompressible, we might expect the magnetic pres-sure to have no effect. However, Tayler [169, 170] and Van-dakurov [171] showed that if the fluid is inviscid ( ν =
0) anda perfect conductor ( σ E = ∞ ), and the induced magnetic fieldsatisfies ∂∂ r (cid:0) rB ϕ (cid:1) < , (19)then the stagnant system is unstable. Given an infinitesimalperturbation, the current drives fluid flow, initially with az-imuthal wave number m =
1. That phenomenon, known asthe Tayler instability in astrophysics and as the kink insta-bility in plasma physics, must also be considered for liquidmetal batteries, which support large axial currents.In stars, the Tayler instability can in theory overcomegravitational stratification to drive dynamo action, and hastherefore been proposed as a source of stellar magneticfields [172]. In fusion plasma devices, the kink instabil-ity [173] can disrupt the magnetic fields that prevent plasmafrom escaping and must therefore be avoided. First encoun-tered in z-pinch experiments in the 1950s, the kink instabil-ity has been studied extensively and reviewed in the plasmaphysics literature (e.g., [174]).Soon after liquid metal batteries were proposed for grid-scale storage, Stefani et al. [175] observed that the technol-
BF u
Fig. 14. Sketch of a liquid metal cell susceptible to the Tayler insta-bility.Fig. 15. Critical current for the Tayler instability depending on theaspect ratio of a cuboid with square cross section filled with Na. Theinsets show contours of the induced vertical magnetic field compo-nent. Adapted from [176]. ogy would be susceptible to the Tayler instability, and thatif the resulting flow were strong enough, it could cause rup-ture the electrolyte layer, destroying the battery. Their obser-vation prompted a series of studies considering methods toavoid the Tayler instability in liquid metal batteries and thelikelihood of it causing rupture.The Tayler instability can be avoided or damped usinga variety of techniques. First, in a real fluid with nonzeroviscosity and imperfect electrical conductivity, the onset cri-terion given by Eq. 19 is no longer strictly correct, becauseviscosity and resistance damp the instability, so that it oc-curs only if the total current (or Hartmann number) exceeds anonzero critical value. Second, the instability can be avoidedby cleverly routing the battery current to prevent the condi-tion expressed by Eq. 19. Instead of building a cylindricalliquid metal battery carrying an axial current, one can builda battery that is a cylindrical annulus, with an empty centralbore. Carrying no current, the bore does not contribute to theinduced magnetic field
BBB ϕ , so that a larger current is required before onset [175]. Even better is to route the battery currentback through the bore in the opposite direction, which pre-vents the Tayler instability altogether, since Eq. 19 is neversatisfied [175,177]. However, ohmic losses in the wire wouldreduce the available voltage. Third, shearing the fluid az-imuthally can damp the Tayler instability [178], though im-posing shear is more practical in plasma devices than inliquid metal batteries. Fourth, imposing external magneticfields — either axial or transverse — damps the Tayler insta-bility [177], in a process often compared to the damping ofthermal convection by vertical magnetic fields [127,129]. Fi-nally, rotation also damps the Tayler instability [176], thoughimposing rotation may be impractical for liquid metal batter-ies. A variety of engineering solutions to avoid or damp theTayler instability in liquid metal batteries are now known.The characteristics of the Tayler instability, and the particu-lar situations in which it arises, have also been studied exten-sively in recent work. The instability was observed directlyin a laboratory experiment: when the axial current appliedto a cylindrical volume of GaInSn alloy exceeded a criti-cal value, the induced axial magnetic field was observed togrow as the square root of the current [179]. Below the crit-ical current, which nearly matched earlier numerical predic-tions [180], no axial field was induced. A central bore wasadded to the vessel, and as expected, the critical current grewwith bore size, ranging from about 2500 A to about 6000 A.Flow due to the Tayler instability was found to compete withthermal convection caused by Joule heating (see § Ha ∼
20 [176, 181, 182].(The Hartmann number (see Eq. 8) is known to control on-set [180].) Using parameters similar to experiments [179],the simulations found similar critical currents. The simula-tions also demonstrated the importance of aspect ratio: liquidmetal layers with high aspect ratio (narrow and tall) are moresusceptible to the Tayler instability [182], see Fig. 15. Simu-lations have also shown the effectiveness of engineering so-lutions for avoiding or damping the Tayler instability, includ-ing adding a central bore, routing current through the bore inthe opposite direction, and applying axial or transverse mag-netic fields [177]. The Tayler instability breaks chiral sym-metry during its growth [181] and might therefore provide alink between planetary tides and the solar cycle [183]. Byallowing for current that is not purely axial, and boundariesthat are not perfect conductors, a later simulation [121] incor-porated more realistic current collectors and considered theireffects. Current collectors with lower conductivity damp theTayler instability, as do fluids with lower conductivity [121].That study also found, however, that electro-vortex flows(see §
7) may play a larger role in the fluid mechanics of liq-uid metal batteries than the Tayler instability. A numericallinear stability analysis of the Tayler instability found criti-cal currents consistent with prior work and described the in-stability as an edge effect, governed by curvature of the mag-etic field [184]. Finally, another simulation study found thatthe Tayler instability occurs in Mg-based liquid metal batter-ies with current density J =
300 mA/cm (a value typical forbatteries being commercialized) if the battery radius exceeds0.43 m, and causes rupture if the radius exceeds 3 m [185].Though there are economic advantages to increasing batterysize, we are unaware of prototypes that large.To summarize: The Tayler instability is a magnetohy-drodynamic phenomenon that drives flow when a large axialcurrent passes through liquid metal. Because of the mag-nitude of the currents involved, and because today’s liquidmetal battery designs have large aspect ratio, the Tayler in-stability may not yet affect the technology. For larger, next-generation batteries, however, the Tayler instability shouldbe considered carefully. A variety of engineering solutionsto avoid or damp the instability are now known. The Tayler instability discussed in § FFF : ∂ uuu ∂ t + ( uuu · ∇∇∇ ) uuu = − ρ ∇∇∇ p + ν∇ uuu + FFF . Here uuu is the fluid velocity. It follows that stagnant fluid( uuu =
0) conserves momentum if and only if1 ρ ∇∇∇ p = FFF . (20)One example, of course, is the gravitational force FFF = − g ˆ zzz ,where we take ˆ zzz as vertical. In that case, Eq. 20 may be inte-grated directly, yielding the familiar fact that the hydrostaticpressure a distance h below the fluid surface is p = ρ gh . Onthe other hand, if ∇∇∇ × FFF (cid:54) =
0, Eq. 20 has no solution, since thecurl of a gradient is always zero. In that case, we concludethat our assumption uuu = FFF for which ∇∇∇ × FFF (cid:54) =
0, the fluid must flow.If we are interested in the Lorentz force per unit volume
FFF = JJJ × BBB due to interaction of an electrical current with its own magnetic field, and under the simplifying assump-tion that all quantities are azimuthally symmetric, it can beshown [186] that ∇∇∇ × FFF (cid:54) = ∂ B ϕ ∂ z (cid:54) = . The magnetic field B ϕ depends, via the Biot-Savart law, onthe current density JJJ . So in azimuthally symmetric systemslike cylindrical liquid metal batteries, electro-vortex flow oc-curs when
JJJ varies axially. In fact, it can be shown that anydivergent current density in an axisymmetric system causeselectro-vortex flow. One canonical example is flow driven bya current from the center of a hemisphere to the hemisphere’ssurface [186]. Taking the additional assumption that the re-sulting flow is irrotational allows analytical solution of manyother cases as well. However, many are similarity solutionsfor which boundary conditions are evaluated at infinity, hin-dering their application to technological applications like liq-uid metal batteries.The details of electro-vortex flow are typically stud-ied via simulation or experiment. Flows typically convergewhere the current density is highest, causing the largestmagnetic pressure. In azimuthally symmetric situations,the result is a poloidal circulation. Current density can beshaped by choosing electrode geometry [188] or by addinga nearby ferromagnetic object, which concentrates magneticfield lines [189]. Electro-vortex flow can cause substantialpressure and was therefore recognized as a promising mech-anism for pumping liquid metals in technological applica-tions including metals processing and nuclear cooling [190].Subsequent efforts produced pumps that drive liquid metalin flat channels with corners [191], through Y- and Ψ -shapedjunctions [192], in a winding-free pump composed of a pairof junctions [193], and centrifugally [194]. Electro-vortexflow also occurs in cylindrical steel furnaces [195].Electro-vortex flow almost certainly occurs in typicalliquid metal batteries because current diverges from the neg-ative current collector to the casing, which serves as posi-tive current collector. Changing the size and aspect ratio ofthe current collectors and electrodes can impede or promoteelectro-vortex flow in simulations [121]. Because electro-vortex flow is not an instability, there are no critical dimen-sionless parameters below which it disappears. Even smallliquid metal batteries supporting gentle currents are suscep-tible to finite electro-vortex flow, which may make it a moreimportant engineering consideration than the Tayler instabil-ity. In fact, an experimental device initially designed to pro-duce the Tayler instability also drove measurable motion atcurrents too small for instability [196] after holes for UDVprobes were drilled in the current collectors. The motionstopped immediately when current was turned off, a behav-ior inconsistent with thermal convection; probably electro-vortex flow was the cause. In fact, electro-vortex flow seemsto suppress the Tayler instability in simulations for thin cur-rent collectors (curve h CC / D = ig. 16. Reynolds number based on the mean velocity in a cylindri-cal liquid metal column vs. time in viscous units. The applied currentdensity corresponds to Ha = , column diameter and height are( D = m, H = . m). The material parameters correspond to Naat 580 ◦ C. h CC denotes the height of the current collectors, their con-ductivity is five times that of sodium. The centered area with fixedpotential has a diameter d = . D . Insets show velocity snapshotsin a meridional plane. Adapted from [121]. flow, see the h CC / D = χ = I πκ (cid:18) µ ρ (cid:19) / (cid:18) κ g α T L ∆ T (cid:19) / , where I is the current. When χ < .
4, thermal convectiondominates, and fluid rises along the central axis, as expected.When χ > .
4, electro-vortex flow dominates, and fluid de-scends along the central axis. At intermediate values of χ ,the two mechanisms have similar strength, and their com-petition produces more complicated flow patterns involvingmultiple circulating rolls. Though liquid metal batteries typ-ically involve different sizes, geometries, and materials thanremelters, χ can probably predict the relative importanceof thermal convection and electro-vortex flow nonetheless,though the values χ < . χ > . χ is written in terms of a fixed temperaturedifferential ∆ T , it may be a better predictor for convectiondriven by temperature differences at the boundaries than forconvection driven by internal Joule heating.Like other motions, electro-vortex flow can enhance bat-tery performance by keeping each layer more chemicallyuniform, but can destroy a battery by rupturing the elec-trolyte layer. A good predictor of rupture is the Richardsonnumber Ri = ( ρ / ρ − ) gL (cid:104) u (cid:105) , a ratio of the gravitational energy of stratification to the ki-netic energy of flow [185, 182]. Here ρ is the density ofthe electrolyte and ρ is the density of the electrode. When Ri <
1, rupture is likely.
In this section, we conclude with a brief summary ofthe fluid mechanics of liquid metal batteries, especially ther-mal convection, compositional convection, Marangoni flow,and electro-vortex flow, including a few words about the im-portance of aspect ratio. Finally, we close with a discus-sion of open questions and future research directions, includ-ing richer experimental measurements, more realistic simu-lations, applications to battery design, size effects, tempera-ture effects, and the role of solid separators.
We have considered the fluid mechanics of liquid metalbatteries, which are made from two liquid metal electrodesand a molten salt electrolyte, without solid separators. In-tended primarily for grid-scale storage, liquid metal batteriesinvolve similar phenomena to those in technologies that wereinvented earlier, including aluminum smelters, aluminum re-finement cells, and thermally regenerative electrochemicalsystems. Fluid flow may occur in the electrolyte or the pos-itive electrode, and affects mass transfer in both layers, butis likely negligible in the negative electrode, where a metalfoam hinders motion. Flow can destroy liquid metal batter-ies if it becomes vigorous enough to rupture the electrolytelayer. However, flow can also be beneficial. In parame-ter regimes common in today’s liquid metal batteries, flows likely important for preventing formation of solid inter-metallic phases, which can swell and cause electrical shortsbetween the positive and negative electrodes, destroying bat-teries. The extent of swelling depends on the density differ-ence between intermetallic and liquid metal. Intermetallicphases are most likely to form during rapid discharge, butcannot form during charge, so rapid charging poses no dan-ger. In current liquid metal battery designs, the primarymechanisms driving flow include thermal convection, com-positional convection, Marangoni flow, and electro-vortexflow. Because the resistivity of the electrolyte layer is fourorders of magnitude higher than the electrode layers, nearlyall Joule heating occurs in the electrolyte, and thermal con-vection is likely strongest there. The positive electrode, be-low the electrolyte, is usually subject to stable temperaturestratification that may actually hinder flow. Thermal convec-tion in the electrolyte layer is constrained — but slowed littleor none — by the magnetic fields produced by battery cur-rents. The low Prandtl number of the molten salt electrolytecauses its convection to have different properties than a high- Pr fluid like air.The magnitude of Marangoni flow in liquid metal bat-teries is difficult to estimate because the surface tension be-tween molten salts and liquid metals — and its variation withtemperature and composition — has rarely been character-ized. That said, because the electrolyte layer is thin, becausethe layer is subject to intense Joule heating, and because liq-uid metals have unusually high surface tension in vacuum,Marangoni flow along electrolyte surfaces is probably sub-stantial.Electro-vortex flow, driven by an electrical current inter-acting with its own magnetic field, is present but probablyweaker than thermal convection or Marangoni flow in mostliquid metal batteries. However, electro-vortex flow may in-teract with other flow mechanisms can cause important ef-fects. Current can also drive flow by interacting with ex-ternal magnetic fields, though that effect is also relativelyweak. Interface instabilities and the Tayler instability are un-likely in today’s liquid metal battery designs, but would be-come crucial in larger batteries (as discussed further below).The aspect ratio of liquid metal batteries plays an importantrole. Given two batteries with the same capacity (and there-fore same volume), the shallower and broader battery is moreprone to interface instabilities, whereas the deeper and nar-rower battery is more prone to the Tayler instability. Inter-actions among mechanisms driving flow are also likely, andcould trigger instabilities, but little is yet known about thedetails of their interactions. Much remains unknown about the fluid mechanics ofliquid metal batteries, leaving many open research opportu-nities. First, richer experimental measurements would sub-stantially advance the field. In experiments, measuring theflow of an opaque, high-temperature fluid is unavoidably dif-ficult. Ultrasound Doppler velocimetry (UDV) [197] pro- vides richer measurements than most other methods. Eachtransducer typically measures one velocity component ata few hundred locations along the probe’s line of sight.Though a line of measurements gives much more insightinto flow shape than point measurements, ultrasound datais nonetheless sparser than the two- and three-dimensionalmeasurements commonly available via optical techniques influids that are transparent. (A recent review [198] of veloc-ity measurement methods for liquid metals is also available.)Experimentalists could contribute substantially to the fieldby using multiple probes, either to make single-componentmeasurements along multiple (carefully chosen) lines [196],or to make multi-component measurements along a sin-gle line. Recent work demonstrating two-component, two-dimensional velocity measurement using phased arrays ofultrasound transducers shows great promise [199, 200, 201,202]. Also, no ultrasound measurements in a working, three-layer battery have been published. Instead, all measure-ments to date were made in a single liquid metal layer, with-out electrolyte or a second metal layer [134, 135]. How-ever, single-layer experimental models capture only a sub-set of the fluid mechanics of liquid metal batteries, includ-ing Joule heating but not heating via entropy change or heatof formation, which may have significant effects [22]; andincluding thermal convection but not compositional convec-tion. Directly measuring the flow in the positive electrodeand the electrolyte of a functioning battery would addressmany open questions and is a worthy goal for experimen-talists. Finally, ultrasound measurements have so far beenrestricted to temperatures below 230 ◦ C, though the batteriesbeing commercialized are solid at those temperatures. High-temperature ultrasound probes with waveguides have beendemonstrated [203] and could be applied to batteries.Second, simulation results incorporating more of the rel-evant physics would substantially advance the field. Sim-ulations to date have sometimes constrained themselves toa single layer [185] and have often constrained themselvesto a subset of possible flow mechanisms, for example, ther-mal convection [108], electro-vortex flow [121], or the com-bination of thermal convection and Marangoni flow [110].These simplifications are important first steps to validatecodes and build intuition for physical mechanisms at playin liquid metal batteries. Nor are the simulations undertakentrivial — low- Pr fluids are tricky and expensive to simulate,and essential material properties have been unavailable. Still,interaction among flow mechanisms are all but certain, sodiscovering the dynamics of liquid metal batteries and accu-rately forecasting their behavior will require simulations in-corporating multiple mechanisms. Measuring material prop-erties, especially the surface tensions between liquid metalsand molten salts, along with their dependence on tempera-ture and composition, would enable better-constrained simu-lations. Simulating three-layer liquid metal batteries includ-ing multiple flow mechanisms would be a great step forward.Among the long list of mechanisms, it seems that thermalconvection, compositional convection, Marangoni flow, andelectro-vortex flow are most relevant for today’s liquid metalbatteries. Heat of formation and entropy change may alsoe significant [22]. In simulations of larger batteries, sur-face instabilities, the Tayler instability, and forces caused byexternal magnetic fields may have stronger effects.Third, richer experimental measurements and simula-tions incorporating more of the relevant physics would allowimproved battery design. Liquid metal batteries involve alarge number of design parameters (electrode materials, elec-trolyte material, size, shape, temperature, current density,etc.) and a large number of engineering metrics (cost, volt-age, temperature, cycle life, etc.). So far, materials choiceshave been considered most carefully, and cost has been iden-tified as the primary engineering metric. Still, trade-offs arecommon, and the technology is new enough that the long-term consequences of design choices are not always clear.For example, Li || PbSb batteries achieve higher voltage andlower cost per kWh than Li || Bi batteries [15,13], but if Li || Bibatteries last much longer because of their ability to elim-inate intermetallic growth electrochemically, their lifetimecost may be lower. Improved fluid mechanical experimentsand simulations might be accurate enough to predict the rateof intermetallic growth and answer the question of materialchoice.Battery size is a particularly interesting design param-eter for fluid mechanical investigation. Aluminum smeltersare typically large, with horizontal dimensions on the orderof 3 m ×
10 m, which lowers the cost of aluminum. Sim-ilarly, liquid metal batteries might provide storage at lowercost if they were larger. For the same energy (or power),larger batteries require less container material and less powerelectronics, offering substantial cost reductions. However,larger batteries are more susceptible to electrolyte ruptureby surface instabilities or the Tayler instability (though as-pect ratio also plays an important role). Preliminary exper-iments also show them to be more susceptible to shorts dueto intermetallic growth. Today’s prototypes are kept small(200 mm) to avoid those drawbacks, but improved under-standing of the fluid mechanics might overcome them andallow larger, lower-cost batteries. In aluminum production,surface instabilities are damped by thickening the electrolyteto about 4 cm [156], at the cost of requiring more voltage —and therefore more energy — to drive current. In batter-ies, however, that mitigation strategy is unavailable, sincethe available voltage is fundamentally limited by electrodematerials, and typically (cid:46) ◦ C, mainly because the melting point of theirmolten salt electrolyte is nearly that high. Batteries stackedin large arrays can maintain that temperature by Joule heating if they are used regularly, but when unused, they require extraheat, adding cost. Reducing the operating temperature wouldreduce that cost and probably have greater effects on othercosts. Lower temperatures would cause slower chemical ki-netics of side reactions that reduce battery efficiency and cor-rode containers and current collectors. Lower temperatureswould ease the design of seals. Much lower temperatureswould allow less expensive container materials, like plasticsinstead of stainless steel. Thus substantial effort is being ded-icated to the search for viable low-temperature liquid metalbattery chemistries [79, 23, 204, 205, 206]. If found, theywould have significantly different material properties thantoday’s liquid metal batteries and therefore give rise to sig-nificantly different fluid mechanics. For example, the lowest-temperature systems use electrolytes composed of room tem-perature ionic liquids instead of molten salts [205,206]. Ionicliquids, however, have typical conductivities orders of mag-nitude smaller than those of molten salts. Liquid metal bat-teries made with ionic liquids, therefore, would require muchthinner electrolyte layers to produce the same voltage, andJoule heating would be still more isolated to the electrolytelayer. The relative magnitude of flow mechanisms (thermalconvection, Marangoni flow, electro-vortex flow, instabili-ties, etc.) may also be different in low-temperature batteries.If low-temperature liquid metal batteries become possible,many practical and interesting questions of fluid mechanicswill arise.Fifth and finally, though we have focused on liquid metalbatteries without solid separators, there are existing tech-nologies [5, 6] as well as proposed next-generation batter-ies [16, 17] that include them. The presence of a solid sepa-rator radically changes the fluid mechanics and design of bat-teries. Without any interfaces between two different liquids,neither Marangoni flow nor surface instabilities are possi-ble. Instead, both electrodes are subject to no-slip boundaryconditions at the separator. Solid separators eliminate thepossibility of rupture due to fluid mechanics and allow elec-trodes to be positioned side-by-side instead of being stackedvertically. Solid separators have electrical conductivity sub-stantially lower than molten salt — about six orders of mag-nitude lower than liquid metals — so heat production wouldbe even more concentrated. High-temperature batteries us-ing solid separators come with their own set of interestingand practical fluid mechanical questions.The fluid mechanics of liquid metal batteries is an excit-ing topic, involving an increasing number of researchers anda large number of open questions. Enabling large-scale stor-age to make electrical grids more robust while incorporatingmore wind and solar generation would make a tremendoussocial impact. Interactions among mass transport, heat trans-port, multiphase flow, magnetohydrodynamics, and chemicalreaction make batteries complicated and interesting. We urgeand encourage researchers to focus on problems that are bothpractical for enabling battery technology and interesting forbroadening human knowledge. cknowledgements
The authors are grateful to F. Stefani for comments onan early draft of the manuscript. This work was supportedby the National Science Foundation under award numberCBET-1552182 and by Helmholtz-Gemeinschaft DeutscherForschungszentren (HGF) in frame of the Helmholtz Al-liance “Liquid metal technologies” (LIMTECH). Fruitfuldiscussions with Valdis Bojareviˇcs, Wietze Herreman, Ger-rit Horstmann, Caroline Nore, Takanari Ouchi, Donald Sad-oway, Norbert Weber, and Oleg Zikanov are gratefully ac-knowledged.
Nomenclature a M z + activity of the cation of M a M activity of M a M ( N ) activity of M alloyed with N A g interface interaction parameter B characteristic magnetic field strength B ϕ azimuthal component of magnetic field Bo dynamic Bond number X characteristic concentration D material diffusivity E voltage E OC open circuit voltage F Faraday constant
FFF force per unit mass
FFF
LLL
Lorentz force per unit volume G Galileo number g acceleration due to gravity Ha Hartmann number H C cryolite layer thickness H E aluminum layer thickness I current I h horizontal perturbation current J current density L vertical thickness L x x-direction size L y y-direction size L z z-direction size m azimuthal wave numberM alkali or earth alkali metal Ma thermal Marangoni numberN heavy or half metal Nu Nusselt number Pm magnetic Prandtl number Pr Prandtl number Q total heat flux R gas constant R textE ohmic resistance electrolyte Ra thermal Rayleigh number Ra crit critical Rayleigh number for flow onset Ra X compositional Rayleigh number Re Reynolds number Ri Richardson number t time T temperature t time uuu velocity u j velocity component, using indicial notation U characteristic flow velocity X concentration of negative electrode material z valency α T thermal coefficient of volumetric expansion α X solutal coefficient of volumetric expansion β Sele criterion for metal pad instability β cr Bojareviˇcs-Romero criterion for metal pad instability ∆ T characteristic temperature difference ∆ X characteristic concentration difference ∆ρ CE characteristic density difference ∆σ characteristic surface tension difference η mn interface wave amplitude η a , a activation polarization at the anode η a , c activation polarization at the cathode η c , a concentration polarization at the anode η c , c concentration polarization at the cathode κ thermal diffusivity µ magnetic permeability ν kinematic viscosity ρ density σ surface tension σ E electrical conductivity τ characteristic flow time ϕ standard half-cell potential ϕ half-cell potential ω oscillation frequency References [1] Kassakian, J. G., and Schmalensee, R., 2011. Thefuture of the electric grid: An interdisciplinary MITstudy. Tech. rep., Cambridge, MA.[2] Whittingham, M. S., 2012. “History, evolution, andfuture status of energy storage”.
Proc. IEEE, ,pp. 1518–1534.[3] Backhaus, S., and Chertkov, M., 2013. “Getting a gripon the electrical grid”.
Phys. Today, (5), pp. 42–578.[4] Nardelli, P. H. J., Rubido, N., Wang, C., Baptista,M. S., Pomalaza-Raez, C., Cardieri, P., and Latva-aho,M., 2014. “Models for the modern power grid”. Eur.Phys. J. Spec. Top., , pp. 2423–2437.[5] Bones, R. J., Teagle, D. A., Brooker, S. D., andCullen, F. L., 1989. “Development of a Ni,NiCI Pos-itive Electrode for a Liquid Sodium (ZEBRA) BatteryCell”.
J. Electrochem. Soc., (5), pp. 1274–1277.[6] Sudworth, J. L., 2001. “The sodium/nickel chloride(ZEBRA) battery”.
J. Power Sources, , Nov.,pp. 149–163.[7] Lu, X., Li, G., Kim, J. Y., Mei, D., Lemmon, J. P.,Sprenkle, V. L., and Liu, J., 2014. “Liquid-metalelectrode to enable ultra-low temperature sodium-betaalumina batteries for renewable energy storage”.
Nat.Commun., , p. 4578.8] Fukunaga, A., Nohira, T., Kozawa, Y., Hagiwara,R., Sakai, S., Nitta, K., and Inazawa, S., 2012.“Intermediate-temperature ionic liquid NaFSA-KFSAand its application to sodium secondary batteries”. J.Power Sources, (0), pp. 52–56.[9] Nitta, K., Inazawa, S., Sakai, S., Fukunaga, A.,Itani, E., Numata, K., Hagiwara, R., and Nohira, T.,2013. Development of Molten Salt Electrolyte Bat-tery. Tech. Rep. 76, Sumitomo Electric Group, Apr.[10] Cairns, E. J., Crouthamel, C. E., Fischer, A. K., Fos-ter, M. S., Hesson, J. C., Johnson, C. E., Shimotake,H., and Tevebaugh, A. D., 1967. Galvanic cells withfused-salt electrolytes. Tech. Rep. ANL-7316, Ar-gonne National Laboratory.[11] Kim, H., Boysen, D. A., Newhouse, J. M. Spatcco,B. L., Chung, B., Burke, P. J., Bradwell, D. J., Jiang,K., Tomaszowska, A. A., Wang, K., Wei, W., Ortiz,L. A., Barriga, S. A., Poizeau, S. M., and Sadoway,D. R., 2013. “Liquid metal batteries: Past, present,and future”.
Chem. Rev., , pp. 2075–2099.[12] Weier, T., Bund, A., El-Mofid, W., Horstmann, G. M.,Lalau, C.-C., Landgraf, S., Nimtz, M., Starace, M.,Stefani, F., and Weber, N., 2017. “Liquid metal bat-teries - materials selection and fluid dynamics”.
IOPConference Series: Materials Science and Engineer-ing, (nil), p. 012013.[13] Ning, X., Phadke, S., Chung, B., Yin, H., Burke, P. J.,and Sadoway, D. R., 2015. “Self-healing Li–Bi liquidmetal battery for grid-scale energy storage”.
J. PowerSources, , pp. 370–376.[14] Bradwell, D. J., Kim, H., Sirk, A. H. C., and Sadoway,D. R., 2012. “Magnesium-antimony liquid metal bat-tery for stationary energy storage”.
J. Am. Chem. Soc., (4), pp. 1895–1897.[15] Wang, K., Jiang, K., Chung, B., Ouchi, T., Burke, P. J.,Boysen, D. A. Bradwell, D. J., Kim, H., Muecke, U.,and Sadoway, D. R., 2014. “Lithium-antimony-leadliquid metal battery for grid-level storage”.
Nature, , 16 October, pp. 348–350.[16] Xu, J., Kjos, O. S., Osen, K. S., Martinez, A. M.,Kongstein, O. E., and Haarberg, G. M., 2016. “Na-Znliquid metal battery”.
J. Power Sources, , pp. 274–280.[17] Xu, J., Martinez, A. M., Osen, K. S., Kjos, O. S.,Kongstein, O. E., and Haarberg, G. M., 2017.“Electrode behaviors of Na-Zn liquid metal battery”.
J. Electrochem. Soc., (12), pp. A2335–A2340.[18] Kim, H., Boysen, D. A., Ouchi, T., and Sadoway,D. R., 2013. “Calcium-bismuth electrodes for large-scale energy storage (liquid metal batteries)”.
J. PowerSources, , pp. 239–248.[19] Ouchi, T., Kim, H., Spatocco, B. L., and Sadoway,D. R., 2016. “Calcium-based multi-element chemistryfor grid-scale electrochemical energy storage”.
Nat.Commun., , Mar., pp. 1–5.[20] Li, H., Yin, H., Wang, K., Cheng, S., Jiang, K.,and Sadoway, D. R., 2016. “Liquid metal electrodesfor energy storage batteries”. Adv. Energy Mater., , p. 1600483.[21] Johnson, C. E., and Hathaway, E. J., 1971. “Solid-liquid phase equilibria for the ternary systemsLi(F,Cl,I) and Na(F,Cl,I)”. J. Electrochem. Soc., (4), pp. 631–634.[22] Swinkels, D. A. J., 1971. “Molten salt batteriesand fuel cells”. In
Advances in Molten Salt Chem-istry , J. Braunstein, G. Mamantov, and G. Smith, eds.,Vol. 1. Plenum Press, New York, ch. 4, pp. 165–223.[23] Spatocco, B. L., and Sadoway, D. R., 2015. “Cost-based discovery for engineering solutions”. In
Ad-vances in Electrochemical Science and Engineering:Electrochemical Engineering Across Scales: FromMolecules to Processes , R. Alikre, P. Bartlett, andJ. Lipkowsi, eds., Vol. 15. Wiley-VCH, ch. 7.[24] Drossbach, P., 1952.
Grundriß der allgemeinen tech-nischen Elektrochemie . Gebr¨uder Borntraeger, Berlin-Nikolassee.[25] Betts, A. G., 1905. Making aluminium. US Patent795,886, Aug. 1.[26] Hoopes, W., 1901. Process of the purification of alu-minium. US Patent 673,364, Apr. 30.[27] Hoopes, W., 1925. Electrolytically refined aluminumand articles made therefrom. US Patent 1,534,315,Apr. 21.[28] Frary, F. C., 1925. “The electrolytic refining of alu-minum”.
Transactions of the American Electrochemi-cal Society, , pp. 275–286.[29] M¨uller, R., 1932. Allgemeine und technische Elektro-chemie . Springer, Vienna.[30] Hoopes, W., Edwards, J. D., and Horsfield, B. T.,1925. Electrolytic cell and method of lining the same.US Patent 1,534,322, Apr. 21.[31] Shimotake, H., and Hesson, J. C., 1968. New bimetal-lic EMF cell shows promise in direct energy con-version. Tech. Rep. Brief 68-10415, Atomic EnergyCommission/NASA.[32] von Zeerleder, A., 1955. “Aluminium”. In Eger [33],pp. 56–364.[33] Eger, G., ed., 1955.
Die technische Elektrolyse imSchmelzfluss , Vol. 3 of
Handbuch der technischenElektrochemie . Akademische VerlagsgesellschaftGeest & Portig K.-G., Leipzig.[34] Beljajew, A. I., Rapoport, M. B., and Firsanowa,L. A., 1957.
Metallurgie des Aluminiums , Vol. 2. VEBVerlag Technik, Berlin.[35] Gadeau, R. A., 1939. “L’aluminium raffin´e”. In
Reine Metalle: Herstellung, Eigenschaften, Verwen-dung , A. E. van Arkel, ed. Springer, Berlin, ch. 32,pp. 145–167.[36] Gadeau, R. A., 1936. Refining of aluminum. USPatent 2,034,339, Mar. 17.[37] Hurter, H., 1937. Improvements in or relating tothe electrolytic refining of aluminium. GB Patent469,361, Jul. 23.[38] Pearson, T. G., and Phillips, H. W. L., 1957. “Theproduction and properties of super-purity aluminium”.
Metall. Rev., (8), pp. 305–360.39] Dube, M. C., 1954. “Extraction and refining of alu-minium”. In Proc. Symp. Non-ferrous metal industryin India, B. Nijhawan and A. Chatterjee, eds., NationalMetallurgical Laboratory, pp. 127–138. Symposiumdate: Feb 1.-3., 1954; published 1957.[40] Wolstenholme, G. A., 1982. “Aluminum extrac-tion”. In Molten salt technology , D. G. Lovering, ed.Springer, New York, ch. 2, pp. 13–55.[41] Yan, X. Y., and Fray, D. J., 2010. “Molten salt elec-trolysis for sustainable metals extraction and materialsprocessing - a review”. In
Electrolysis: Theory, Typesand Applications , S. Kuai and J. Meng, eds. Nova Sci-ence, New York.[42] Edwards, J. D., Frary, F. C., and Jeffries, Z., 1930.
Aluminum and its production . McGraw-Hill, NewYork and London.[43] Singleton, E. L., and Sullivan, T. A., 1973. “Electronicscrap reclamation”.
J. Metals, (6), pp. 31–34.[44] Tiwari, B. L., and Sharma, R. A., 1984. “Electrolyticremoval of magnesium from scrap aluminum”. J. Met-als, (7), pp. 41–43.[45] Gesing, A. J., Das, S. K., and Loutfy, R. O.,2016. “Production of magnesium and aluminum-magnesium alloys from recycled secondary aluminumscrap melts”. JOM, (2), pp. 585–592.[46] Gesing, A. J., and Das, S. K., 2017. “Use of ther-modynamic modeling for selection of electrolyte forelectrorefining of magnesium from aluminum alloymelts”. Metall. Mater. Trans. B, , pp. 132–145.[47] Olsen, E., and Rolseth, S., 2010. “Three-layer elec-trorefining of silicon”.
Metall. Mater. Trans. B, ,April, pp. 295–302.[48] Olsen, E., Rolseth, S., and Thonstad, J., 2014. “Elec-trorefining of silicon by the three-layer principle in aCaF -based electrolyte”. In Molten Salts in Chemistryand Technology , M. Gaune-Escard and G. M. Haar-berg, eds. Wiley, ch. 7.6, pp. 569–576.[49] Oishi, T., Koyama, K., and Tanaka, M., 2016. “Elec-trorefining of silicon using molten salt and liquid allyelectrodes”.
J. Electrochem. Soc., (14), pp. E385–E389.[50] Roberts, R., 1958. “The fuel cell round table”.
J. Elec-trochem. Soc., (7), pp. 428–432.[51] Liebhafsky, H. A., 1967. “Regenerative electrochem-ical systems: An introduction”. In Crouthamel andRecht [72], ch. 1, pp. 1–10.[52] McCully, R. C., Rymarz, T. M., and Nicholson, S. B.,1967. “Regenerative chloride systems for conversionof heat to electrical energy”. In Crouthamel and Recht[72], ch. 1, pp. 198–212.[53] Chum, H. L., and Osteryoung, R. A., 1980. Review ofthermally regenerative electrochemical systems vol-ume 1: Synopsis and executive summary. Tech. Rep.SERI/TR-332-416, Solar Energy Research Institute.[54] Chum, H. L., and Osteryoung, R. A., 1981. Review ofthermally regenerative electrochemical systems vol-ume 2. Tech. Rep. SERI/TR-332-416, Solar EnergyResearch Institute. [55] Yeager, E., 1958. “Fuel cells: Basic considerations”.In Proc. 12 th Ann. Battery Research and DevelopmentConf., Power Sources Division, U.S. Army Signal Re-search & Development Laboratory, pp. 2–4.[56] Liebhafsky, H. A., 1959. “The fuel cell and the Carnotcycle”.
J. Electrochem. Soc., (12), pp. 1068–1071.[57] Shearer, R. E., and Werner, R. C., 1958. “Thermallyregenerative ionic hydride galvanic cell”.
J. Elec-trochem. Soc., (11), p. 593.[58] Ciarlariello, T. A., McDonough, J. B., and Shearer,R. E., 1961. Study of energy conversion devices - re-port no. 7. Tech. Rep. MSAR 61-99, MSA ResearchCorporation, Callery, Pennsylvania, Sept. 14.[59] Lawroski, S., Vogel, R. C., and Munnecke, V. H.,1961. Chemical engineering division summary report.Tech. Rep. ANL-6379, Argonne National Laboratory.[60] Roy, P., Salamah, A., Maldonado, J., and Narkiewicz,R. S., 1993. “HYTEC - A thermally regenerative fuelcell”.
AIP Conf. Proc., , pp. 913–921.[61] Wietelmann, U., 2014. “Applications of lithium-containing hydrides for energy storage and conver-sion”.
Chem. Ing. Tech., (12), pp. 2190–2194.[62] Agruss, B., 1966. Regenerative battery. US Patent3,245,836, Apr 12.[63] Henderson, R. E., Agruss, B., and Caple, W. G., 1961.“Resume of thermally regenerative fuel cell systems”.In Energy Conversion for Space Power , N. W. Snyder,ed., Vol. 3 of
Progress in Astronautics and Aeronau-tics . Academic Press, New York, London, pp. 411–423.[64] Agruss, B., and Karas, H. R., 1962. First quarterlytechnical progress report on design and developmentof a liquid metal cell for the period 1 january 1962 - 31march 1962. Tech. Rep. EDR 2678, Allison Divisionof General Motors Corporation, Apr 15.[65] Lawroski, S., Vogel, R. C., and Munnecke, V. H.,1962. Chemical engineering division summary report.Tech. Rep. ANL-6477, Argonne National Laboratory.[66] Austin, L. G., 1967. Fuel cells - a review ofgovernment-sponsored research, 1950-1964. Tech.Rep. NASA-SP-120, NASA, Washington, DC.[67] Agruss, B., Karas, H. R., and Decker, V. L., 1962. De-sign and development of a liquid metal fuel cell. Tech.Rep. ASD-TDR-62-1045, Aeronautical Systems Di-vision, Dir/Aeromechanics, Flight Accessoire Lab,Wright-Patterson AFB.[68] Agruss, B., 1963. “Nuclear liquid metal cell for spacepower”. In Proc. 17 th Annual Power Sources Conf.,pp. 100–103.[69] Agruss, B., and Karas, H. R., 1967. “The thermally re-generative liquid metal concentration cell”. In
Regen-erative EMF cells , R. Gold, ed., Vol. 64 of
Advancesin Chemistry . American Chemical Society, Washing-ton, D.C., pp. 62–81.[70] Kerr, R. L., 1967. “Regenerative fuel cells”. In Perfor-mance Forecast of Selected Static Energy ConversionDevices, 29 th Meeting of AGARD Propulsion and En-ergetics Panel, G. W. Sherman and L. Devol, eds.,GARD, Air Force Aero Propulsion Laboratory andAerospace Research Laboratories Department of theAir Force, pp. 658–715.[71] Groce, I. J., and Oldenkamp, R. D., 1967. “Develop-ment of a thermally regenerative sodium-mercury gal-vanic system part II. design, construction, and testingof a thermally regenerative sodium-mercury galvanicsystem”. In Crouthamel and Recht [72], ch. 5, pp. 43–52.[72] Crouthamel, C. E., and Recht, H. L., eds., 1967.
Re-generative EMF Cells , Vol. 64 of
Advances in Chem-istry . American Chemical Society, Washington, D.C.[73] Weaver, R. D., Smith, S. W., and Willmann, N. L.,1962. “The sodium | tin liquid-metal cell”. J. Elec-trochem. Soc., (8), pp. 653–657.[74] Shimotake, H., Rogers, G. L., and Cairns, E. J., 1969.“Secondary cells with lithium anodes and immobi-lized fused-salt electrolytes”.
Ind. Eng. Chem. ProcessDes. Dev., (1), pp. 51–56.[75] Cairns, E. J., and Shimotake, H., 1969. “High-temperature batteries”. Science, (3886), pp. 1347–1355.[76] Vogel, R. C., Proud, E. R., and Royal, J., 1968. Chem-ical engineering division semiannual report. Tech.Rep. ANL-7425, Argonne National Laboratory.[77] Lawroski, S., Vogel, R. C., Levenson, M., and Mun-necke, V. H., 1963. Chemical engineering divisionresearch highlights. Tech. Rep. ANL-6766, ArgonneNational Laboratory.[78] Vogel, R. C., Burris, L., Tevebaugh, A. D., Webster,D. S., Proud, E. R., and Royal, J., 1971. Chemicalengineering division research highlights. Tech. Rep.ANL-7850, Argonne National Laboratory.[79] Spatocco, B. L., Ouchi, T., Lambotte, G., Burke, P. J.,and Sadoway, D. R., 2015. “Low-temperature moltensalt electrolytes for membrane-free sodium metal bat-teries”.
J. Electrochem. Soc., (14), pp. A2729–A2736.[80] Grube, G., 1930.
Grundz¨uge der theoretischenund angewandten Elektrochemie , 2 nd ed. TheodorSteinkopff, Dresden, Leipzig.[81] Gossrau, G., 1955. “Calcium, Strontium, Barium”. InEger [33], pp. 424–464.[82] Wenger, E., Epstein, M., and Kribus, A., 2017.“Thermo-electro-chemical storage (TECS) of solarenergy”. Appl. Energ., , pp. 788–799.[83] Steunenberg, R. K., and Burris, L., 2000. From testtube to pilot plant: A 50 year history of the chemi-cal technology division at argonne national laboratory.Tech. Rep. ANL-00/16, Argonne National Laboratory.[84] Bockris, J. O., ed., 1972.
Electrochemistry of CleanerEnvironments . Plenum Press, New York.[85] Hietbrink, E. H., McBree, J., Selis, S. M., Trickle-bank, S. B., and Witherspoon, R. R., 1972. “Electro-chemical power sources for vehicle propulsion”. InBockris [84], ch. 3, pp. 47–97.[86] Vogel, R. C., Levenson, M., Proud, E. R., and Royal,J., 1968. Chemical engineering division research highlights. Tech. Rep. ANL-7550, Argonne NationalLaboratory.[87] Kyle, M. L., Cairns, E. J., and Webster, D. S., 1973.Lithium/sulfur batteries for off-peak energy storage:A preliminary comparison of energy storage and peakpower generation systems. Tech. Rep. ANL-7958, Ar-gonne National Laboratory.[88] Bradwell, D., 2006. “Technical and economic feasi-bility of a high-temperature self-assembling battery”.Master’s thesis, Massachusetts Institute of Technol-ogy.[89] Bradwell, D., 2011. “Liquid metal batteries: Am-bipolar electrolysis and alkaline earth electroalloyingcells”. PhD thesis, Massachusetts Institute of Tech-nology.[90] Ray, H. S., 2006.
Introduction to Melts - Molten Salts,Slags and Glasses . Allied Publishers Pvt. Limited,New Delhi.[91] Rao, Y. K., and Patil, B. V., 1971. “Thermody-namic study of the Mg-Sb system”.
Metall. Trans., , pp. 1829–1835.[92] Leung, P., Heck, S. C., Amietszajew, T., Mohamed,M. R., Conde, M. B., Dashwood, R. J., and Bha-gat, R., 2015. “Performance and polarization studiesof the magnesium-antimony liquid metal battery withthe use of in-situ reference electrode”. RSC Adv., ,pp. 83096–83105.[93] Sadoway, D., Ceder, G., and Bradwell, D., 2012.High-amperage energy storage device with liquidmetal negative electrode and methods. US Patent8,268,471 B2.[94] Ouchi, T., and Sadoway, D. R., 2017. “Positive currentcollector for Li || Sb-Pb liquid metal battery”.
J. PowerSources, , pp. 158–163.[95] Sadoway, D. R., 2016. “Innovation in stationary elec-tricity storage: The liquid metal battery”. In StanfordEnergy Seminar, Precourt Institute for Energy.[96] Bojarevics, V., Tucs, A., and Pericleous, K., 2016.“MHD model for liquid metal batteries”. In 10thPAMIR Int. Conf. - Fundamental and Applied MHD,DIEE, University of Cagliary, pp. 638–642.[97] Bojarevics, V., and Tucs, A., 2017.
Light Metals2017 . The Minerals, Metals & Materials Series.Springer, ch. MHD of large scale liquid metal batter-ies, pp. 687–692.[98] Chill`a, F., and Schumacher, J., 2012. “New perspec-tives in turbulent Rayleigh-B´enard convection”.
Eur.Phys. J. E, , p. 58.[99] Lohse, D., and Xia, K.-Q., 2010. “Small-Scale Prop-erties of Turbulent Rayleigh-B´enard Convection”. Annu. Rev. Fluid Mech., (1), pp. 335–364.[100] Ahlers, G., 2009. “Turbulent convection”. Physics, ,p. 74.[101] Bodenschatz, E., Pesch, W., and Ahlers, G., 2000.“Recent developments in Rayleigh-B´enard convec-tion”. Annu. Rev. Fluid Mech., , pp. 709–778.[102] Nuclear Energy Agency, 2015. “Handbook on Lead-ismuth Eutectic Alloy and Lead Properties, Materi-als Compatibility, Thermal-hydraulics and Technolo-gies”. pp. 1–950.[103] Iida, T., and Guthrie, R. I. L., 2015. The Thermophys-ical Properties of Metallic Liquids , Vol. 1. OxfordUniversity Press, Oxford.[104] Iida, T., and Guthrie, R. I. L., 2015.
The Thermophys-ical Properties of Metallic Liquids , Vol. 2. OxfordUniversity Press, Oxford.[105] Davidson, H. W., 1968. Compilation of thermophysi-cal properties of liquid lithium. Tech. Rep. NASA TND-4650, NASA.[106] Sobolev, V., 2011. Database of thermophysical prop-erties of liquid metal coolants for GEN-IV. Tech. Rep.SCK • CEN-BLG-1069, SCK • CEN, Dec.[107] Barriga, S. A., 2013. “An electrochemical investi-gation of the chemical diffusivity in liquid metal al-loys”. PhD thesis, Massachusetts Institute of Tech-nology, Cambridge, MA.[108] Shen, Y., and Zikanov, O., 2016. “Thermal convectionin a liquid metal battery”.
Theor. Comput. Fluid Dyn., (4), pp. 275–294.[109] Zikanov, O., and Shen, Y., 2016. “Mechanisms ofinstability in liquid metal batteries”. In 10th PAMIRInt. Conf. - Fundamental and Applied MHD, DIEE,University of Cagliary, pp. 522–526.[110] K¨ollner, T., Boeck, T., and Schumacher, J., 2017.“Thermal Rayleigh-Marangoni convection in a three-layer liquid-metal-battery model”. Phys. Rev. E, ,p. 053114.[111] Xiang, L., and Zikanov, O., 2017. “Subcritical con-vection in an internally heated layer”. Phys. Rev. Flu-ids, , p. 063501.[112] Foster, M. S., 1967. “Laboratory studies of intermetal-lic cells”. In Crouthamel and Recht [72], pp. 136–148.[113] Goluskin, D., 2015. Internally heated convection andRayleigh-B´enard convection . Springer.[114] Janz, G. J., Dampier, F. W., Lakshminarayanan, G. R.,Lorenz, P. K., and Tomkins, R. P. T., 1968. MoltenSalts: Volume 1, Electrical Conductance, Density, andViscosity Data. Tech. rep.[115] Janz, G. J., Allen, C. B., Bansal, N. P., Murphy, R. M.,and Tomkins, R. P. T., 1979. Physical properties datacompilations relevant to energy storage. II. Moltensalts: Data on single and multi-component salt sys-tems. Tech. Rep. NSRDS-NBS 61, Part II.[116] Todreas, N. E., Hejzlar, P., Fong, C. J., Nikiforova, A.,Petroski, R., Shwageraus, E., and Whitman, J., 2008.Flexible conversion ratio fast reactor systems evalu-ation. Tech. Rep. MIT-NFC-PR-101, MassachusettsInstitute of Technology.[117] Masset, P., Henry, A., Poinso, J.-Y., and Poignet, J.-C., 2006. “Ionic conductivity measurements of molteniodide-based electrolytes”.
J. Power Sources, ,pp. 752–757.[118] Masset, P., Schoeffert, S., Poinso, J.-Y., and Poignet,J.-C., 2005. “Retained molten salt electrolytes in ther- mal batteries”.
J. Power Sources, , pp. 356–365.[119] Shattuck, M. D., Behringer, R. P., Johnson, G. A.,and Georgiadis, J. G., 1997. “Convection and flow inporous media. Part 1. Visualization by magnetic reso-nance imaging”.
J. Fluid Mech., , pp. 215–245.[120] Howle, L. E., Behringer, R. P., and Georgiadis, J. G.,1997. “Convection and flow in porous media. Part 2.Visualization by shadowgraph”.
J. Fluid Mech., ,pp. 247–262.[121] Weber, N., Galindo, V., Priede, J., Stefani, F., andWeier, T., 2015. “The influence of current collectorson Tayler instability and electro-vortex flows in liquidmetal batteries”.
Phys. Fluids, , p. 014103.[122] Grossmann, S., and Lohse, D., 2000. “Scaling in ther-mal convection: A unifying theory”. J. Fluid Mech., , pp. 27–56.[123] Horanyi, S., Krebs, L., and M¨uller, U., 1999. “Tur-bulent Rayleigh–B´enard convection in low Prandtl–number fluids”.
Int. J. Heat Mass Tran., (21),pp. 3983–4003.[124] Jones, C. A., Moore, D. R., and Weiss, N. O., 1976.“Axisymmetric convection in a cylinder”. J. FluidMech., , Jan., pp. 353–388.[125] Weiss, N. O., and Proctor, M. R. E., 2014. Magneto-convection . Cambridge University Press, Cambridge,England, Oct.[126] Davidson, P. A., 2001.
An introduction to magnetohy-drodynamics . Cambridge University Press, London.[127] Chandrasekhar, S., 1954. “On the inhibition of con-vection by a magnetic field: II”.
Phil. Mag., (370),pp. 1177–1191.[128] Busse, F. H., and Clever, R. M., 1982. “Stability ofconvection rolls in the presence of a vertical magneticfield”. Phys. Fluids, (6), pp. 931–935.[129] Burr, U., and M¨uller, U., 2001. “Rayleigh-B´enardconvection in liquid metal layers under the influenceof a vertical magnetic field”. Phys. Fluids, (11),pp. 3247–3257.[130] Aurnou, J. M., and Olson, P. L., 2001. “Experimentson Rayleigh-B´enard convection, magnetoconvectionand rotating magnetoconvection in liquid gallium”. J.Fluid Mech., , pp. 283–307.[131] Z¨urner, T., Liu, W., Krasnov, D., and Schumacher, J.,2016. “Heat and momentum transfer for magnetocon-vection in a vertical external magnetic field”.
Phys.Rev. E, (4), Oct., pp. 043108 EP –.[132] Burr, U., and M¨uller, U., 2002. “Rayleigh-B´enardconvection in liquid metal layers under the influenceof a horizontal magnetic field”. J. Fluid Mech., ,pp. 345–369.[133] Moffatt, H. K., 1967. “On the suppression of turbu-lence by a uniform magnetic field”.
J. Fluid Mech., (3), pp. 571–592.[134] Kelley, D. H., and Sadoway, D. R., 2014. “Mix-ing in a liquid metal electrode”. Phys. Fluids, (5),p. 057102.[135] Perez, A., and Kelley, D. H., 2015. “Ultrasound veloc-ity measurement in a liquid metal electrode”. J. Vis.xp. (102), p. e52622.[136] B´eltran, A., 2017. “MHD natural convection flow ina liquid metal electrode”.
Appl. Therm. Eng., ,pp. 1203–1212.[137] Davis, S. H., 1987. “Thermocapillary instabilities”.
Annu. Rev. Fluid Mech., , pp. 403–435.[138] Schatz, M. F., and Neitzel, G. P., 2001. “Experimentson thermocapillary instabilities”. Annu. Rev. FluidMech., , pp. 93–127.[139] Bratsun, D. A., and De Wit, A., 2004. “On Marangoniconvective patterns driven by an exothermic chemicalreaction in two-layer systems”. Phys. Fluids, (4),Apr., pp. 1082–1096.[140] K¨ollner, T., Schwarzenberger, K., Eckert, K., andBoeck, T., 2013. “Multiscale structures in solu-tal Marangoni convection: Three-dimensional simu-lations and supporting experiments”. Phys. Fluids, (9), Sept., pp. 092109–32.[141] K¨ollner, T., Schwarzenberger, K., Eckert, K., andBoeck, T., 2015. “Solutal Marangoni convection ina Hele–Shaw geometry: Impact of orientation andgap width”. Eur. Phys. J.-Spec. Top., (2), Apr.,pp. 261–276.[142] Jensen, K. F., Einset, E. O., and Fotiadis, D. I., 1991.“Flow phenomena in chemical vapor deposition ofthin films”.
Annu. Rev. Fluid Mech., , pp. 197–232.[143] Craster, R. V., and Matar, O. K., 2009. “Dynamics andstability of thin liquid films”. Rev. Mod. Phys., (3),Aug., pp. 1131–1198.[144] VanHook, S. J., Schatz, M. F., McCormick, W. D.,Swift, J. B., and Swinney, H. L., 1995. “Long-Wavelength Instability in Surface-Tension-DrivenB´enard Convection”. Phys. Rev. Lett., , Dec.,pp. 4397–4400.[145] Pearson, J. R. A., 1958. “On convection cells inducedby surface tension”. J. Fluid Mech., (5), pp. 489–500.[146] Smith, K. A., 1966. “On convective instability in-duced by surface-tension gradients”. J. Fluid Mech., (2), pp. 401–414.[147] Koschmieder, E. L., and Switzer, D. W., 1992. “Thewavenumbers of supercritical surface-tension-drivenB´enard convection”. J. Fluid Mech., , pp. 533–548.[148] VanHook, S. J., Schatz, M. F., Swift, J. B., Mc-Cormick, W. D., and Swinney, H. L., 1997. “Long-wavelength surface-tension-driven B´enard convec-tion: experiment and theory”.
J. Fluid Mech., ,pp. 45–78.[149] Welander, P., 1964. “Convective instability in a two-layer fluid heated uniformly from above”.
Tellus, (3), Dec., pp. 349–358.[150] Walsh, W. J., Gay, E. C., Arntzen, J. D., Kinci-nas, J. E., Cairns, E. J., and Webster, D. S., 1973.Lithium/chalcogen secondary cells for componentsin electric vehicular-propulsion generating systems.Tech. Rep. ANL-7999, Argonne National Laboratory.[151] Newhouse, J. M., 2014. “Modeling the operating volt- age of liquid metal battery cells”. PhD thesis, Mas-sachusetts Institute of Technology.[152] Scriven, L. E., and Sternling, C. V., 1964. “On cellu-lar convection driven by surface-tension gradients: ef-fects of mean surface tension and surface viscosity”. J. Fluid Mech., (3), pp. 321–340.[153] Davison, H. W., 1968. Compilation of thermophysicalproperties of liquid lithium. Tech. Rep. TN-D-4650,NASA, July.[154] Janz, G. J., Tomkins, R. P. T., Allen, C. B.,Downey Jr., J. R., Garner, G. L., Krebs, U., andSinger, S. K., 1975. “Molten salts: Volume 4, part 2,chlorides and mixtures—electrical conductance, den-sity, viscosity, and surface tension data”. Journal ofPhysical and Chemical Reference Data, (4), Oct.,pp. 871–1178.[155] Sele, T., 1977. “Instabilities of the metal surface inelectrolytic alumina reduction cells”. Metall. Trans.B, , pp. 613–618.[156] Davidson, P. A., 2000. “Overcoming instabilities inaluminium reduction cells: a route to cheaper alu-minium”. Mater. Sci. Technol., , pp. 475–479.[157] Evans, J. W., and Ziegler, D. P., 2007. “The elec-trolytic production of aluminum”. In Electrochem-ical Engineering , A. Bard and M. Stratmann, eds.,Vol. 5 of
Encyclopedia of Electrochemistry . Wiley-VCH, Weinheim, pp. 224–265. Volume editors: Mac-donald, D.D. and Schmuki, P.[158] Molokov, S., El, G., and Lukyanov, A., 2011. “Classi-fication of instability modes in a model of aluminiumreduction cells with a uniform magnetic field”.
Theor.Comput. Fluid Dyn., , pp. 261–279.[159] Øye, H. A., Mason, N., Peterson, R. D., Richards,N. E., Rooy, E. L., Stevens McFadden, F. J.,Zabreznik, R. D., Williams, F. S., and Wagstaff, R. B.,1999. “Aluminum: Approaching the new millen-nium”. JOM-J Min Met Mat S, (2), pp. 29–42.[160] Bojareviˇcs, V., and Romerio, V., 1994. “Long wavesinstability of liquid metal-electrolyte interface in alu-minum electrolysis cells: a generalization of Sele’scriterion”. Eur. J. of Mech. B, (1), pp. 33–56.[161] Davidson, P. A., and Lindsay, R. I., 1998. “Stabilityof interfacial waves in aluminium reduction cells”. J.Fluid Mech., , pp. 273–295.[162] Sneyd, A. D., 1985. “Stability of fluid layers carry-ing a normal electric current”.
J. Fluid Mech., ,pp. 223–236.[163] Zikanov, O., 2015. “Metal pad instabilities in liquidmetal batteries”.
Phys. Rev. E, (063021).[164] Weber, N., Beckstein, P., Herreman, W., Horstmann,G. M., Nore, C., Stefani, F., and Weier, T., 2017.“Sloshing instability and electrolyte layer rupture inliquid metal batteries”. Phys. Fluids, , p. 054101.[165] Weber, N., Beckstein, P., Galindo, V., Herreman, W.,Nore, C., Stefani, F., and Weier, T., 2017. “Metal padrole instability in liquid metal batteries”. Magnetohy-drodynamics, (1), pp. 129–140.[166] Weller, H. G., Tabor, G., Jasak, H., and Fureby, C.,998. “A tensorial approach to computational con-tinuum mechanics using object-oriented techniques”. Comput. Phys., (6), pp. 620–631.[167] Horstmann, G. M., Weber, N., and Weier, T., 2017.“Coupling and stability of interfacial waves in liquidmetal batteries”. CoRR .[168] Zikanov, O., 2017. “Shallow water modeling ofrolling pad instability in liquid metal batteries”.
CoRR .[169] Tayler, R. J., 1957. “Hydromagnetic Instabilities ofan Ideally Conducting Fluid”.
P. Phys. Soc. Lond. B, (1), p. 31.[170] Tayler, R. J., 1973. “The adiabatic stability of starscontaining magnetic fields-I.Toroidal fields”. Mon.Not. R. Astron. Soc., , p. 365.[171] Vandakurov, Y. V., 1972. “Theory for the Stability ofa Star with a Toroidal Magnetic Field”.
Sov Astron, (2), Oct., pp. 265–272.[172] Spruit, H. C., 2002. “Dynamo action by differentialrotation in a stably stratified stellar interior”. Astron.Astrophys., (3), Jan., pp. 923–932.[173] Rosenbluth, M. N., 1973. “Nonlinear properties of theinternal m=1 kink instability in the cylindrical toka-mak”.
Phys. Fluids, (11), pp. 1894–1902.[174] Freidberg, J. P., 1982. “Ideal magnetohydrodynamictheory of magnetic fusion systems”. Rev. Mod. Phys., , July, pp. 801–902.[175] Stefani, F., Weier, T., Gundrum, T., and Gerbeth, G.,2011. “How to circumvent the size limitation of liquidmetal batteries due to the Tayler instability”. Energ.Convers. Manage., , pp. 2982 – 2986.[176] Weber, N., Galindo, V., Stefani, F., Weier, T., andWondrak, T., 2013. “Numerical simulation of theTayler instability in liquid metals”. New. J. Phys., ,p. 043034.[177] Weber, N., Galindo, V., Stefani, F., and Weier, T.,2014. “Current-driven flow instabilities in large-scaleliquid metal batteries, and how to tame them”. J.Power Sources, , pp. 166–173.[178] Shumlak, U., and Hartman, C. W., 1995. “ShearedFlow Stabilization of the m =1 Kink Mode in Z Pinches”.
Phys. Rev. Lett., , Oct., pp. 3285–3288.[179] Seilmayer, M., Stefani, F., Gundrum, T., Weier, T.,Gerbeth, G., Gellert, M., and R¨udiger, G., 2012. “Ex-perimental evidence for a transient Tayler instabilityin a cylindrical liquid-metal column”. Phys. Rev. Lett., , p. 244501.[180] R¨udiger, G., Schultz, M., and Gellert, M., 2011.“The Tayler instability of toroidal magnetic fieldsin a columnar gallium experiment”.
Astron. Nachr., (1), pp. 17–23.[181] Weber, N., Galindo, V., Stefani, F., and Weier, T.,2015. “The Tayler instability at low magnetic Prandtlnumbers: between chiral symmetry breaking and he-licity oscillations”.
New. J. Phys. , p. 113013.[182] Stefani, F., Galindo, V., Kasprzyk, C., Landgraf, S.,Seilmayer, M., Starace, M., Weber, N., and Weier, T.,2016. “Magnetohydrodynamic effects in liquid metal batteries”.
IOP Conf. Ser.: Mater. Sci. Eng., ,p. 012024.[183] Stefani, F., Giesecke, A., Weber, N., and Weier, T.,2016. “Synchronized helicity oscillations: A link be-tween planetary tides and the solar cycle?”.
SolarPhys., (8), pp. 2197–2212.[184] Priede, J., 2016. “Electromagnetic pinch-type insta-bilities in liquid metal batteries”. In 10th PAMIR Int.Conf. - Fundamental and Applied MHD, DIEE, Uni-versity of Cagliary, pp. 268–273.[185] Herreman, W., Nore, C., Cappanera, L., and Guer-mond, J.-L., 2015. “Tayler instability in liquid metalcolumns and liquid metal batteries”.
J. Fluid Mech., , pp. 79–114.[186] Bojareviˇcs, V., Freibergs, Y., Shilova, E. I., andShcherbinin, E. V., 1989.
Electrically induced vor-tical flows . Kluwer Academic Publishers, Dordrecht.[187] Davidson, P. A., 1999. “Magnetohydrodynamics inmaterials processing”.
Annu. Rev. Fluid Mech., (1),pp. 273–300.[188] Kolesnichenko, I., and Khripchenko, S., 2002. “Math-ematical simulation of hydrodynamic processes in thecentrifugal MHD-pump”. Magnetohydrodynamics, (4), pp. 391–398.[189] Kolesnichenko, I., Khripchenko, S., Buchenau, D.,and Gerbeth, G., 2005. “Electro-vortex flows in asquare layer of liquid metal”. Magnetohydrodynam-ics, , Mar., pp. 39–51.[190] Denisov, S., Dolgikh, V., Mann, M. ´E.., andKhripchenko, S., 1999. “Electrical vortex generationof transit flows across plane MHD channels”. Magne-tohydrodynamics, (1), pp. 52–58.[191] Khripchenko, S., Kolesnichenko, I., Dolgikh, V., andDenisov, S., 2008. “Pumping effect in a flat MHDchannel with an electrovortex flow”. Magnetohydro-dynamics, , Sept., pp. 303–314.[192] Denisov, S., Dolgikh, V., Khalilov, R., Kolesnichenko,I., and Khripchenko, S., 2012. “Pumping effect in Y-and Ψ -shaped channels with ∏ -shaped cores”. Mag-netohydrodynamics, (1), Apr., pp. 197–202.[193] Dolgikh, V., and Khalilov, R., 2014. “Investigationof a model of the winding-free MHD pump with liq-uid metal electrodes”. Magnetohydrodynamics, (2),Aug., pp. 187–192.[194] Denisov, S., Dolgikh, V., Khripchenko, S., andKolesnichenko, I., 2016. “The electrovortex centrifu-gal pump”. Magnetohydrodynamics, (1-2), Nov.,pp. 25–33.[195] Kazak, O. V., and Semko, A. N., 2011. “Electrovortexmotion of a melt in dc furnaces with a bottom elec-trode”. Journal of Engineering Physics and Thermo-physics, (1), pp. 223–231.[196] Starace, M., Weber, N., Seilmayer, M., Kasprzyk, C.,Weier, T., Stefani, F., and Eckert, S., 2015. “Ultra-sound Doppler flow measurements in a liquid metalcolumn under the influence of a strong axial electricfield”. Magnetohydrodynamics, (2), pp. 249–256.197] Takeda, Y., 1995. “Velocity profile measurement byultrasonic Doppler method”. Exp. Therm. Fluid Sci., (4), pp. 444–453.[198] Eckert, S., Cramer, A., and Gerbeth, G., 2007.“Velocity Measurement Techniques for Liquid MetalFlows”. In Magnetohydrodynamics . Springer Nether-lands, pp. 275–294.[199] B¨uttner, L., Nauber, R., Burger, M., R¨abiger, D.,Franke, S., Eckert, S., and Czarske, J., 2013. “Dual-plane ultrasound flow measurements in liquid met-als”.
Measurement Science and Technology, (5),p. 055302.[200] R¨abiger, D., Zhang, Y., Galindo, V., Franke, S.,Willers, B., and Eckert, S., 2014. “The relevance ofmelt convection to grain refinement in Al-Si alloys so-lidified under the impact of electric currents”. ActaMaterialia, , pp. 327–338.[201] Franke, S., R¨abiger, D., Galindo, V., Zhang, Y., andEckert, S., 2016. “Investigations of electrically drivenliquid metal flows using an ultrasound Doppler flowmapping system”. Flow Measurement and Instrumen-tation, , pp. 64–73.[202] Nauber, R., Beyer, H., M¨ader, K., Kupsch, C.,Thieme, N., B¨uttner, L., and Czarske, J., 2016. “Mod-ular ultrasound velocimeter for adaptive flow mappingin liquid metals”. In 2016 IEEE International Ultra-sonics Symposium (IUS), pp. 1–4.[203] Eckert, S., Gerbeth, G., and Melnikov, V. I., 2003.“Velocity measurements at high temperatures by ul-trasound Doppler velocimetry using an acoustic waveguide”. Exp. Fluids, , pp. 381–388.[204] Ashour, R. F., Yin, H., Ouchi, T., Kelley, D. H., andSadoway, D. R., 2017. “Molten amide-hydroxide-iodide electrolyte for a low-temperature sodium-basedliquid metal battery”. J. Electrochem. Soc., (2),pp. A535–A537.[205] Lalau, C.-C., Ispas, A., Weier, T., and Bund, A.,2015. “Sodium-bismuth-lead low temperature liquidmetal battery”.
J. Electrochem. Plating Techn. , June,p. 4808.[206] Lalau, C.-C., Dimitrova, A., Himmerlich, M., Ispas,A., Weier, T., Krischok, S., and Bund, A., 2016. “Anelectrochemical and photoelectron spectroscopy studyof a low temperature liquid metal battery based onan ionic liquid electrolyte”.
J. Electrochem. Soc.,163