Encyclopedia of physics, vol. 8-1. Fluid mechanics II

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The internal structure of the battery is maintained by gravity, since the negative electrode materials typically have lower density than electrolyte materials, which have lower density than positive electrode materials. A solid metal positive current collector contacts the positive electrode and usually serves as the container as well.

A solid metal negative current collector connects to the negative electrode and is electrically insulated from the positive current collector. Because the negative electrode is liquid and the positive current collector is also the battery vessel, some care is required to prevent shorts between them. It is possible to electrically insulate the positive current collector by lining it with a ceramic, but ceramic sleeves are too expensive for grid-scale applications and are prone to cracking. Instead, typical designs separate the liquid metal negative electrode from vessel walls with a metal foam, as shown in Fig.

The high surface tension of the liquid metal provides sufficient capillary forces to keep it contained in the pores of the foam. The solid foam also inhibits flow in the negative electrode, which is likely negligible at length scales larger than the pore size.

In many designs, the foam is held in place by a rigid conductor, as shown, so that its height stays constant. However, as the battery discharges and the positive electrode becomes a pool of two-part alloy, it swells. If the positive electrode swells enough to contact the foam, a short occurs, so the foam height must be carefully chosen, taking into account the thickness of the positive electrode and the density change it will undergo during discharge. Typically, ohmic losses dominate activation and concentration polarizations by far, but mass transfer limitations may nevertheless sometimes occur in the cathodic alloy.

Liquid metal batteries have advantages for grid-scale storage. Eliminating solid separators reduces cost and eliminates the possibility of failure from a cracked separator. Perhaps more importantly, solid separators typically allow much slower mass transport than liquids, so eliminating solids allows faster charge and discharge with smaller voltage losses. Liquid electrodes improve battery life, because the life of Li-ion and other more traditional batteries is limited when their solid electrodes are destroyed due to repeated shrinking and swelling during charge and discharge.

Hagen–Poiseuille equation

Their energy and power density are moderate, and substantially below the Li-ion batteries that are ubiquitous in portable electronics, but density is less essential than cost in stationary grid-scale storage. Liquid metal batteries also present challenges. Despite variation with battery chemistry, all conventional liquid metal batteries have voltage significantly less than Li-ion batteries. Lacking solid separators, liquid metal batteries are not suitable for portable applications in which disturbing the fluid layers could rupture the electrolyte layer, causing electrical shorts between the positive and negative electrodes and destroying the battery.

Rupture might also result from vigorous fluid flows even if the battery is stationary, such as the Tayler instability Sec. Flow mechanisms may also interact, triggering instabilities more readily. Little energy is wasted heating large batteries because Joule heating losses to electrical resistance provides more than enough energy to maintain the temperature. Still, high temperatures promote corrosion and make air-tight mechanical seals difficult. Finally, poor mixing during discharge can cause local regions of a liquid metal electrode to form unintended intermetallic solids that can eventually span from the positive to the negative electrode, destroying the battery.

Arne Johansson

Solid formation may well be the leading cause of failure in liquid metal batteries. The central idea at the heart of LMBs is the three-layer arrangement of liquid electrodes and electrolyte. This seemingly simple idea in fact so apparently simple that it is sometimes [ 24 ] questioned if it deserves to be patented at all did not originate with LMBs. Instead, using a stable stratification of two liquid metals interspaced with a molten salt for electrochemical purposes was first proposed by Betts [ 25 ] in the context of aluminum purification see Fig.

However, Betts was not able to commercialize his process. Instead Hoopes, who had a more complicated arrangement using a second internal vessel for aluminum electrorefining patented in [ 26 ] Fig. According to Frary [ 28 ], Hoopes as well thought of using a three-layer cell around It can be seen from Table 1 that even if the idea to use three liquid layers were a trivial one, its realization and transformation to a working process was highly nontrivial indeed.

The submerged vessel containing the negative electrode, initially suggested by Hoopes [ 26 ] Fig. Cryolite is less dense than the pure or impure Al. In the presence of flow, Al dissolves into the cryolite and deposits at the carbon walls of the outer vessel, and pure Al can be collected at the bottom of the outer vessel. However, the current density is distributed very inhomogeneously, concentrating around the opening of the inner vessel.

This implies large energy losses and strong local heating rendering a stable operation over longer times impossible. Betts [ 25 , 29 ] Fig. This three-layer arrangement guaranteed the shortest possible current paths and enabled homogeneous current density distributions.

Additionally, the evaporation of the electrolyte was drastically reduced by the Al top layer. However, under the high operating temperatures the cell walls became electrically conducting, got covered with metal that short-circuited the negative and positive electrodes, and thus, prevented successful operation of the cell [ 29 ]. Only Hoopes' sophisticated construction [ 27 , 28 ] Fig. A key element of Hoopes' construction is the division of the cell into two electrically insulated sections. The joint between them is water cooled and thereby covered by a crust of frozen electrolyte providing electrical as well as thermal insulation [ 30 ].

Additionally, instead of using a single electric contact to the purified Al at the cells' side as did Betts, Hoopes arranged several graphite current collectors along the Al surface that provided a more evenly distributed current. However, the electrolyte used by Hoopes see Table 1 had a relatively high melting temperature and a tendency to creep to the surface between the cell walls and the purified Al [ 28 , 32 ]. According to Eger [ 33 ] and Beljajew et al. It was not until that super-purity aluminum became widely available with Gadeau's [ 36 ] three-layer refining process that used a different electrolyte see Table 1 according to a patent filed in Its lower melting point allowed for considerably decreased operating temperature.

Gadeau's cell was lined with magnesite that could withstand the electrolyte attack without the need of water cooling. However, the BaCl 2 used in the electrolyte mixture decomposed partially, so the electrolyte composition had to be monitored and adjusted during cell operation when necessary. This difficulty was overcome by using the purely fluoride-based electrolyte composition suggested by Hurter [ 37 ] see Table 1 , S.

Aluminum refining cells can tolerate larger voltage drops than LMBs, so the electrolyte layer is often much thicker. These large values are, on the one hand, due to the need for heat production. On the other hand, a large distance between the negative and positive electrodes is necessary to prevent flow induced intermixing of the electrode metals that would nullify refinement.

It is often mentioned [ 28 , 32 , 38 , 42 ] that strong electromagnetic forces trigger those flows. Unlike aluminum electrolysis cells ACEs , refinement cells have been optimized little, and the technology would certainly gain from new research [ 41 ]. Yan and Fray [ 41 ] directly invoke the low density differences as a cause for the instability of the interfaces, discussed here in Sec.

They attribute the limited application of fused salt electrorefining to the present design of refining cells that does not take advantage of the high electrical conductivity and the very low thermodynamic potential required for the process. The application of three-layer processes was also proposed for electronic scrap reclamation [ 43 ], removal of Mg from scrap Al [ 44 — 46 ], and electrorefining of Si [ 47 — 49 ].

Research on the fluid mechanics of current bearing three-layer systems can, therefore, potentially be useful beyond LMBs. After three-layer liquid metal systems were put to use for Al refining, a few decades passed before they were used to generate electricity. TRES combine an electricity delivering cell with a regeneration unit as sketched in Fig. Thermal regeneration implies that the whole system efficiency is Carnot limited [ 55 , 56 ]. A variety of such systems were investigated in the U. Later, Chum and Osteryoung classified the published material on this topic according to system type and thoroughly reviewed it in retrospect [ 53 , 54 ].

LiH-based cells were building blocks of what were probably the first , [ 53 ] experimentally realized thermally regenerative high-temperature systems [ 57 — 59 ], which continue to be of interest today [ 60 , 61 ]. Almost at the same time a patent was filed in by Agruss [ 62 ], bimetallic cells were suggested for the electricity delivering part of TRES.

Henderson et al. Although unmentioned in Ref. Governmental sponsored research on bimetallic cells followed soon after at Argonne National Laboratory , [ 65 ] and at General Motors [ 64 , 66 ]. Research was initially focused on the application of bimetallic cell based TRES on space power applications [ 67 ], namely, systems using nuclear reactors as heat sources. Hydrodynamics naturally plays a vital role in the operation of TRES due to the necessity to transport products and reactants between the electricity producing and the thermal regeneration parts of the system.

However, hydrodynamics of the transport between the cell and the regenerator is mainly concerned with the task of pumping [ 69 , 71 ] and the subtleties of keeping a liquid metal flow through—while preventing electrical contact between—different cells [ 67 ]. Velocities typical for TRES are much lower than those found in conventional heat engines [ 72 ] and could even be achieved using natural circulation driven by heat [ 10 ].

Agruss et al. Publications covering detailed investigations of cell specific fluid mechanics are unknown to the present authors, but the LMB pioneers were obviously aware of its importance as can be seen by a variety of pertinent notes. Cell construction determines to a large extent the influence hydrodynamics can have on cell operation. This is most likely the purest embodiment of an LMB: inside the cell, there are only the three fluid layers that are floating on top of each other according to density. Early on, the vital role of stable density stratification was clearly identified [ 54 , 69 , 73 ].

Cell performance depended on the flow distribution, the volume flux, and, vicariously, on the temperature of the incoming Hg. Cairns et al. Restraining one or more of the liquid phases in a porous ceramic matrix is a straightforward means to guarantee mechanical stability of the interfaces [ 22 ]. A direct mechanical separation of anodic and cathodic compartment is a necessity for space applications that could not rely on gravity to keep the layers apart. Since both matrix and powders had to be electric insulators, an overall conductivity reduction by a factor of about two to four [ 74 , 75 ] resulted even for the better paste electrolytes.

Obviously, using mechanically separated electrode compartments is a prerequisite for any mobile application of LMBs. Equally, for cells used as components in complete TRES, the constant flow to and from the regenerator and through the cell necessitates in almost all cases a mechanical division of positive and negative electrodes. A different purpose was pursued by encasing the negative electrode material into a retainer [ 77 ] made from stainless steel fibers [ 10 ], felt metal [ 78 ], or later, foam [ 13 , 15 , 19 , 79 ] as sketched in Fig. Those retainers allow electrical insulation of the negative electrode from the rest of the cell without resorting to ceramics and restrict fluid mechanics to that in a porous body.

The probably simplest retainer used was an Armco iron ring [ 10 , 65 ] that encased the alkaline metal, a configuration more akin to the differential density cell than to a porous body. Arrangements similar to the iron ring are sometimes used as well in molten salt electrolysis cells [ 27 , 33 , 80 , 81 ] to keep a patch of molten metal floating on top of fused salt while preventing contact with the rest of the cell.

In the case of poorly conducting materials especially Te and Se , the positive electrode had to be equipped with additional electronically conducting components to improve current collection [ 22 , 76 ]. With view on the low overall efficiencies of TRES due to Carnot cycle limitations as well as problems of pumping, plumbing, and separation, research on thermally regenerative systems ceased after [ 54 ] and later LMB work at Argonne concentrated on Li-based systems with chalcogen positive electrodes, namely, Se and Te.

The high strength of the bonds in those systems makes them unsuitable for thermal regeneration [ 22 ]. However, in their review, Chum and Osteryoung [ 54 ] deemed it worthwhile to reinvestigate TRES based on alloy cells once a solar-derived, high temperature source was identified. Using bimetallic cells as secondary elements for off-peak electricity storage was already a topic in the s [ 10 , 83 ].

The initial design conceived in the fall of by Sadoway and Ceder and presented by Bradwell [ 88 ] was one that combined Mg and Sb with a Mg 3 Sb 2 -containing electrolyte, decomposing the Mg 3 Sb 2 on charge and forming it on discharge. Thermodynamic data for the system are available from Ref. Right from the start, research at MIT focused on the deployment of LMBs for large-scale energy storage [ 14 ] concentrating on different practical and economical aspects of utilizing abundant and cheap materials [ 18 , 23 ]. Those cells would have a favorable volume to surface ratio translating into a small amount of construction material per active material and potentially decreasing total costs.

Introduction to Computational Fluid Dynamics

In addition, in large cells, Joule heating in the electrolyte could be sufficient to keep the components molten [ 14 ]. The differential density cells employed for the initial MIT investigations gave later way to cells that used metal foam immersed in the electrolyte to contain the negative electrode [ 13 , 15 , 19 , 79 , 94 ]. Thus, state-of-the-art cells are moderately sized, but the quest for large-scale cells is ongoing.

Recently, Bojarevics and coworkers [ 96 , 97 ] suggested to retrofit old aluminum electrolysis potlines into large-scale LMB installations ending up with cells of 8 m by 3. It should be stressed that LMBs form a whole category of battery systems comprising a variety of material combinations. Consequently, depending on the active materials and the electrolyte selected, different flow situations may arise even under identical geometrical settings.

Because almost every known fluid expands when heated, spatial variations gradients in temperature cause gradients in density. In the presence of gravity, if those gradients are large enough, denser fluid sinks and lighter fluid floats, causing thermal convection.

Usually, large-scale convection rolls characterize the flow shape. Being ubiquitous and fundamental in engineering and natural systems, convection has been studied extensively, and many reviews of thermal convection are available [ 98 — ]. Here, we will give a brief introduction to convection, then focus on the particular characteristics of convection in liquid metal batteries.

Joule heating drives convection in some parts of a liquid metal battery, but inhibits it in others. Broad, thin layers are common, and convection in liquid metal batteries differs from aqueous fluids because metals are excellent thermal conductors that is, they have low Prandtl number. Convection driven by buoyancy also competes with Marangoni flow driven by surface tension, as discussed in Sec.

We close this section with a discussion of magnetoconvection in which the presence of magnetic fields alters convective flow. Usually, both boundaries are held at steady, uniform temperatures, or subjected to steady, uniform heat flux. Convection also occurs in many other geometries, for example, lateral heating.

Heating the fluid from above, however, produces a stably stratified situation in which flow is hindered. Compositional convection is one mechanism by which reaction drives flow; entropic heating, discussed earlier, is another. Thus, compositional convection is likely much stronger than thermal convection. Compositional convection is unlikely during discharge because the less-dense negative electrode material e.

During charge, however, less-dense material is removed from the top of the positive electrode, leaving the remaining material more dense and likely to drive compositional convection by sinking. A ratio of momentum diffusivity kinematic viscosity to thermal diffusivity, the Prandtl number, is a material property that can be understood as a comparison of the rates at which thermal motions spread momentum and heat.

Table 2 lists the Prandtl number of a few relevant fluids. Air and water are very often the fluids of choice for convection studies, since so many industrial and natural systems involve them. We therefore expect thermal convection in liquid metals and molten salts to differ substantially from convection in water or air.

The Prandtl number plays a leading role in the well-known scaling theory characterizing turbulent convection, developed by Grossmann and Lohse [ ]. In fact, the scaling theory expresses the outputs Re and Nu in terms of the inputs Ra and Pr. Boundary layers occur near walls, and transport through them proceeds to a good approximation by diffusion alone.

On the other hand, in the bulk region far from walls, transport proceeds primarily by the fast and disordered motions typical in turbulent flow. Is heat transport slower through the boundary layer or the bulk? And, which boundary layer—viscous or thermal—is thicker and therefore dominant? Answering those three questions makes it possible to estimate the exponents that characterize the dependence of Re and Nu on Ra and Pr. Again, convection in liquid metals and molten salts differs starkly from convection in water or air: changing Pr by orders of magnitude causes Re and Nu to change by orders of magnitude as well.

Experiments have also shown that at low Pr, more of the flow's kinetic energy is concentrated in large-scale structures, especially large convection rolls. In a thin convecting layer with a cylindrical sidewall resembling the positive electrode of a liquid metal battery, slowly fluctuating concentric ring-shaped rolls often dominate [ ]. Those rolls may interact via flywheel effects [ ]. If convection occurs in the presence of a vertical magnetic field, alignment is impossible, since convection rolls are necessarily horizontal. Accordingly, vertical magnetic fields tend to damp convection [ — ].

The Rayleigh number of oscillatory instability of convection rolls is also increased by the presence of a vertical magnetic field [ ]. Common sources of vertical magnetic fields in liquid metal batteries include the Earth's field though it is relatively weak and fields produced by wires carrying current to and from the battery. Just as the Grossmann and Lohse scaling theory [ ] considers the dependence of Re and Nu on the inputs Ra and Pr in convection without magnetic fields, a recent scaling theory by Schumacher and colleagues [ ] considers the dependence of Re and Nu on the inputs Ra, Pr—and also Ha—in the presence of a vertical magnetic field.

The reasoning is analogous: the scaling depends on whether transport time is dominated by the boundary layer or the bulk, and which boundary layer is thickest. However, the situation is made more complex by the need to consider magnetic field transport in addition to momentum and temperature transport, and the possibility that the Hartmann magnetic boundary layer might be thickest.

Altogether, 24 regimes are possible. That special case applies to materials common in liquid metal batteries and still spans four regimes of magnetoconvection. Scaling laws are proposed for each regime. The theory's fit parameters remain unconstrained in three of the four regimes because appropriate experimental data are unavailable. Experiments to produce those data would substantially advance the field. On the other hand, if a horizontal magnetic field is present, convection rolls are often able to align with it easily. In that case, flow speed Re and heat flux Nu increase [ ].

As Ra increases, waves develop on the horizontal convection rolls [ , , ]. In liquid metal batteries, internal electrical currents run primarily vertically and induce toroidal horizontal magnetic fields. Poloidal convection rolls are, therefore, common, since their flow is aligned and circulates around the magnetic field lines. If the sidewall is cylindrical, boundary conditions further encourage poloidal convection rolls.

Such rolls have been observed in liquid metal battery experiments, and the characteristic speed increases as Ha increases [ , ]. Simulations have shown similar results, with the number of convection rolls decreasing as the current increases [ ]. Other simulations, however, have suggested that electromagnetic effects are negligible for liquid metal batteries with radius less than 1. Further study may refine our understanding. In batteries with a rectangular cross section, we would expect horizontal convection rolls circulating around cores that are nearly circular near the central axis of the battery, where the magnetic field is strong and the sidewall is remote.

Closer to the wall, we would expect rolls circulating around cores that are more nearly rectangular due to boundary influence.


Liquid metal batteries as sketched in Fig. During operation, however, external heaters are often unnecessary because the electrical resistance of the battery components converts electrical energy to heat in a process known as Joule heating or ohmic heating. If the battery current is large enough and the environmental heat loss is small enough, batteries can maintain temperature without additional heating [ ]. In fact, cooling may sometimes be necessary.

In this case, the primary heat source lies not below the battery, but within it. As Table 3 shows, molten salts have electrical conductivity typically four orders of magnitude smaller than liquid metals, so that essentially all of the Joule heating occurs in the electrolyte layer, as shown in Fig.

The positive electrode, located below the electrolyte, is then heated from above and becomes stably stratified; its thermal profile actually hinders flow. Some flow may be induced by the horizontal motion of the bottom of the electrolyte layer, which slides against the top of the positive electrode and applies viscous shear stresses, but simulations of Boussinesq flow show the effect to be weak [ , ].

The electrolyte itself, which experiences substantial bulk heating during battery charge and discharge, is subject to thermal convection, especially in its upper half [ , ]. One simulation of an internally heated electrolyte layer showed it to be characterized by small, round, descending plumes [ ]. Experiments have raised concern that thermal convection could bring intermetallic materials from the electrolyte to contaminate the negative electrode [ ].

Convection due to internal heating has also been studied in detail in other contexts [ ]. The negative electrode, located above the electrolyte, is heated from below and is subject to thermal convection. In a negative electrode composed of bulk liquid metal, we would expect both unstable thermal stratification and viscous coupling to the adjacent electrolyte to drive flow. Simulations show that in parameter regimes typical of liquid metal batteries, it is viscous coupling that dominates; flow due to heat flux is negligible [ ]. Therefore, in the case of a thick electrolyte layer, mixing in the electrolyte is stronger than mixing in the negative electrode above; in the case of a thin electrolyte layer, the roles are reversed [ ].

However, the negative electrodes may also be held in the pores of a rigid metal foam by capillary forces, which prevents the negative electrode from contacting the battery sidewall [ 15 ]. The foam also substantially hinders flow within the negative electrode. Essentially, the characteristic length scale becomes the pore size of the foam, which is much smaller than the thickness of the negative electrode. Since the Rayleigh number is proportional to the cube of the characteristic length scale Eq. The physics of convection in porous media [ , ] might apply in this case.

If a liquid metal battery is operated with current density that is uniform across horizontal cross section, we expect uniform Joule heating, and therefore, temperatures that vary primarily in the vertical direction aside from thermal edge effects. However, the negative current collector must not make electrical contact with the vessel sidewall, which is part of the positive current collector. For that reason, the metal foam negative current collector that contains the negative electrode is typically designed to be smaller than the battery cross section, concentrating electrical current near the center and reducing it near the sidewall.

The fact that current can exit the positive electrodes through the sidewalls as well as the bottom wall allows further deviation from uniform, axial current. Nonuniform current density causes Joule heating that is also nonuniform—in fact, it varies more sharply, since the rate of heating is proportional to the square of the current density. This gradient provides another source of convection-driven flow.

Putting more current density near the central axis of the battery creates more heat there and causes flows that rise along the central axis. Interestingly, electro-vortex flow considered in detail in Sec. Simulations have shown that negative current collector geometry and conductivity substantially affect flow in liquid metal batteries [ ]. Other geometric details can also create temperature gradients and drive convection. For example, sharp edges on a current collector concentrate current and cause intense local heating.

The resulting local convection rolls are small but can nonetheless alter the global topology of flow and mixing. Also, if solid intermetallic alloys form, they affect the boundary conditions that drive thermal convection. Intermetallics are typically less dense than the surrounding melt, so they float to the interface between the positive electrode and electrolyte. Intermetallics typically have lower thermal and electrical conductivity than the melt, so where they gather, both heating and heat flux are inhibited, changing convection in nontrivial ways.

The molecules of a stable fluid are typically attracted more strongly to each other than to other materials. The surface tensions of liquid metals and molten salts are among the highest of any known materials, so it is natural to expect surface tension to play a role in liquid metal batteries.

This section will consider that role. If the surface tension varies spatially, regions of higher surface tension pull fluid along the interface from regions of lower surface tension. Surface tension can vary spatially because it depends on temperature, chemical composition, and other quantities. Thermocapillary flow in the presence of a magnetic field has also been studied in prior work [ , ], though we will not consider it further here. In this section, we will consider Marangoni flow phenomena that are relevant to liquid metal batteries, focusing on similarities and differences to Marangoni flows studied in the past.

We will estimate which phenomena are likely to arise, drawing insight from one pioneering study that has considered the role of thermocapillary flow in liquid metal batteries [ ]. Because of the L 2 factor, thermal convection tends to dominate in thick layers, whereas thermocapillary flow tends to dominate in thin layers. Thermocapillary flow, like thermal convection, is qualitatively different for fluids with small Prandtl number like liquid metals and molten salts than for fluids with large Prandtl number.

Typically, short-wavelength flow appears as an array of hexagons covering the interface. The long-wavelength flow has no repeatable or particular shape, instead depending sensitively on boundary conditions. When observed in experiments, the long-wavelength flow always ruptures the layer in which it occurs [ ], a property particularly alarming for designers of liquid metal batteries. The short-wavelength mode, on the other hand, causes nearly zero surface deformation [ ]. We can estimate the relevance of thermocapillary flow and the likelihood of short-wavelength and long-wavelength flow using dimensionless quantities, as long as the necessary material properties are well-characterized.

Its value is well-known for Pb, Bi, and their eutectic alloy [ ] because of its importance in nuclear power plants, however. One pioneering study [ ] simulated thermocapillary flow in a hypothetical three-layer liquid metal battery with a eutectic PbBi positive electrode, a LiCl—KCl electrolyte, and a Li negative electrode. Nor do we expect long-wavelength flow because, according to Eq. Finally, we expect minimal flow in the negative electrode if it is contained in a rigid metal foam. All of these predictions should be understood as preliminary since Eqs.

In fact, though the long-wavelength mode can readily be observed in laboratory experiments with silicone oils [ , , ], liquid metals and molten salts typically have much smaller kinematic viscosity and thermal diffusivity, yielding large values of G that make the long-wavelength mode unlikely. Other considerations require both the electrolyte and the positive electrode to be much thicker, so rupture via the long-wavelength thermocapillary mode is unlikely in a liquid metal battery. We would expect, however, that the short-wavelength thermocapillary mode often arises in liquid metal batteries, especially in the electrolyte layer.

Though unlikely to rupture the electrolyte, the short-wavelength mode may mix the electrolyte, promoting mass transport. The short-wavelength mode might also couple to other phenomena, for example, the interfacial instabilities discussed in Sec. Figure 19 in Ref. Simulations can give further insight into thermocapillary flow in liquid metal batteries. As shown in Fig. Using a range of layer thicknesses and current densities, the study found that five modes of thermal flow arise in typical liquid metal batteries.

In order of decreasing typical speed, they are 1 thermal convection in the electrolyte, 2 thermal convection in the negative electrode, 3 thermocapillary flow driven by the top surface of the electrolyte, 4 thermocapillary flow driven by the bottom surface of the electrolyte, and 5 anticonvection [ ] in the positive electrode. The combined effects of buoyant and thermocapillary forces produce flows much like those produced by buoyancy alone, though thermocapillary forces slightly reduce the characteristic length scale of the flow.

That observation is consistent with an earlier observation that thermocapillary and buoyant forces drive flows having different characteristic lengths [ ]. We raise one caveat: if the negative electrode is contained by a metal foam, flow there would likely be negligible. Solutal Marangoni flow has been studied less than thermocapillary flow, and to our knowledge, has not yet been addressed in the literature for the specific case of liquid metal batteries. One experimental and numerical study found a cellular flow structure reminiscent of the hexagons characteristic of the short-wavelength mode in thermocapillary flow [ ].

A later experimental and numerical study by the same authors [ ] varied the thickness of the fluid layer and its orientation with respect to gravity, finding that a two-dimensional simulation in which flow quantities are averaged across the layer thickness fails to match experiments with thick layers.

Though studies of solutal Marangoni flow in liquid metal batteries have not yet been published, the phenomenon is likely, because charge and discharge alter the composition of the positive electrode. In past work, salt loss in lithium-chalcogen cells has been attributed to Marangoni flow [ ]. In the case where composition varies across the interface, solutal Marangoni flow is possible only if material flows across the interface in the direction of increasing material diffusivity. In a liquid metal battery, the material of interest is the negative electrode material, e.

The diffusivity of Li in Bi is 1. A battery made with those materials would be prone to solutal Marangoni flow driven by composition varying across the interface during discharge, but not during charge. Solutal Marangoni flow driven by variations across the interface is likely to occur in both short- and long-wavelength modes, depending on the appropriate Marangoni and Galileo numbers analogous to Eqs. However, a two-layer model for solutal Marangoni flow is unstable at any value of the Marangoni number [ , ].

Variations of composition along the interface will drive solutal Marangoni flow regardless of their values. Alternatively, we can consider the extreme case in which different regions of the interface are composed of different pure materials, so that the force per unit length is simply the difference between their known surface tensions. These estimates are imprecise: considering temperature will change them by a few percent, and considering different battery chemistry will change them more. These estimates are also upper bounds. Nonetheless, these estimates are two to four orders of magnitude larger than the typical force per unit length that drives thermocapillary flow.

If the true solutal forces reach even a few percent of these estimates, solutal Marangoni flow rivals or dominates thermocapillary flow in liquid metal batteries. According to Eq. Davidson and Lindsay [ ] came to a similar conclusion regarding the instability threshold for circular and square cells using both shallow water theory and a mechanical analog. It can be expected that three-layer systems like Al refinement cells cf.

Knowledge on three-layer systems bearing interface normal currents is currently relatively scarce. Sneyd [ ] treated the case while modeling an electric-arc furnace, assuming a density of zero for the upper phase and the semi-infinite upper and lower layers. He took only the azimuthal magnetic field produced by the current into account and did not consider the action of an additional background field. In addition to long-wave instabilities, Sneyd predicted short-wave instabilities of both sausage and kink type. To the best of our knowledge, experimental results on current-driven interface instabilities in three-layer systems have not been reported to date.

The cause of the violent motions reported by several authors [ 28 , 38 , 32 , 42 ], and already mentioned in Sec. Frary [ 28 ] describes the motion as swirling and attributes it to the interaction of the vertical current within the cell and the magnetic fields of the horizontal current leads.

This is costly in terms of energy but tolerable if the cell is operated as an electrolytic cell. In galvanic mode, OCV obviously limits the permissible current and the resistance of thick electrolyte layers is prohibitive. For LMBs to have an acceptable voltage efficiency, the electrolyte thickness must not exceed a few millimeters, so maintaining interface stability is more difficult.

Again, it is evident from Eq. As estimated by Zikanov [ ], for rectangular cells, the instability due to the interaction of the perturbation currents with the azimuthal field of the main current described by criterion 15 appears to be more dangerous than that caused by the action of the background magnetic field on the horizontal compensating currents It should be mentioned that criteria predicting instability onset for any nonvanishing Lorentz force neglect dissipative effects caused by magnetic induction and viscosity as well as the influence of surface tension [ ].

Weber et al. In agreement with the results of Weber et al. The difference is explained by density differentials: the electrolyte typically has density closer to that of the negative electrode than the positive electrode. However, even in the latter case, the voltage drop in the electrolyte is still found to be excessive with 0. Thus, for practical cases, A g is the control parameter that determines how strongly both interfaces interact.

Wave onset is described by Sele-like parameters extended by interface tension terms for both interfaces. The expressions reduce to the Sele criterion 16 in the limit of large LMBs considered here. The waves at both interfaces can be considered as coupled in the range 0. The threshold values are empirical.

In the weakly coupled regime, the interface waves are antiphase see Fig. Dynamics in the strongly coupled regime are more complex. These strongly coupled instabilities may not occur in cells that are not circular, however. This happened always at the upper interface for the examples discussed earlier. Thus, Eq. Gesing et al. Zikanov [ ] used the St. Venant shallow water equations complemented by electromagnetic force terms to model the rolling pad instability in LMBs with rectangular cross sections. In accordance with Horstmann et al. If the density jump at one interface is much smaller than at the other, only the former develops waves, and the situation is very similar to that in AECs.

This strong effect can be explained by the fact that the aspect ratio determines the set of available natural gravitational wave modes and the strength of the electromagnetic field that is needed to transform them into a pair with complex-conjugate eigenvalues [ , ]. For comparable density jumps at the interfaces, Zikanov's [ ] results again agree with those of Horstmann et al.

The system behavior becomes more complex and is different from that found in AECs. The waves of both interfaces can couple either in phase or antiphase. Zikanov [ ] found examples where the presence of the second interface stabilizes the system, which was not predicted by his two-slab model [ ], whose simplifications are probably too strong to capture this part of the dynamics. In stars, the Tayler instability can in theory overcome gravitational stratification to drive dynamo action, and has therefore been proposed as a source of stellar magnetic fields [ ]. In fusion plasma devices, the kink instability [ ] can disrupt the magnetic fields that prevent plasma from escaping and must therefore be avoided.

First encountered in z-pinch experiments in the s, the kink instability has been studied extensively and reviewed in the plasma physics literature e. Soon after liquid metal batteries were proposed for grid-scale storage, Stefani et al. Their observation prompted a series of studies considering methods to avoid the Tayler instability in liquid metal batteries and the likelihood of it causing rupture.

The Tayler instability can be avoided or damped using a variety of techniques. First, in a real fluid with nonzero viscosity and imperfect electrical conductivity, the onset criterion given by Eq. Second, the instability can be avoided by cleverly routing the battery current to prevent the condition expressed by Eq. Instead of building a cylindrical liquid metal battery carrying an axial current, one can build a battery that is a cylindrical annulus, with an empty central bore. Even better is to route the battery current back through the bore in the opposite direction, which prevents the Tayler instability altogether, since Eq.

However, ohmic losses in the wire would reduce the available voltage. Third, shearing the fluid azimuthally can damp the Tayler instability [ ], though imposing shear is more practical in plasma devices than in liquid metal batteries. Fourth, imposing external magnetic fields—either axial or transverse—damps the Tayler instability [ ], in a process often compared to the damping of thermal convection by vertical magnetic fields [ , ].

Finally, rotation also damps the Tayler instability [ ], though imposing rotation may be impractical for liquid metal batteries. A variety of engineering solutions to avoid or damp the Tayler instability in liquid metal batteries are now known. The characteristics of the Tayler instability, and the particular situations in which it arises, have also been studied extensively in recent work. The instability was observed directly in a laboratory experiment: when the axial current applied to a cylindrical volume of GaInSn alloy exceeded a critical value, the induced axial magnetic field was observed to grow as the square root of the current [ ].

Below the critical current, which nearly matched earlier numerical predictions [ ], no axial field was induced. A central bore was added to the vessel, and as expected, the critical current grew with bore size, ranging from about A to about A. Flow due to the Tayler instability was found to compete with thermal convection caused by Joule heating see Sec. A series of numerical studies have also considered the Tayler instability, with the benefit of being able to characterize its properties in detail. Read copy of the Article. Current adaptation practices: case studies in France, Portugal and Greece.

Extended version in arXiv: Advancements in identifying biomechanical determinants for abdominal aortic aneurysm rupture Vascular, 23 1 , pp. Primary biliary cirrhosis: From bench to bedside. World J Gastrointest Pharmacol Ther. Alexandrakis, E. Spanakis, K. Marias, N. Kampanis, Development of an early warning and coordination system for addressing and managing natural disasters, SafeChania , Chania, Crete, Greece, June AAA risk assessment-integrating morphologic, biomechanic, molecular and clinical risk factors to improve decision making in the management of abdominal aortic aneurysm disease.

Summer Biomechanics, Bioengineering, and Biotransport Conference. The influence of intraluminal thrombus on noninvasive abdominal aortic aneurysm wall distensibility measurement Medical and Biological Engineering and Computing, 53 4 , pp. Tamoxifen induces a pluripotency signature in breast cancer cells and human tumors. Mol Oncol. Epub Jun 5. A case study in Plaka beach, E. Hydrodynamic, neotectonic and climatic control of the evolution of a barrier beach in the microtidal environment of the NE Ionian Sea eastern Mediterranean Geo-Marine Letters, , V 35, Issue 1, pp 37— Botti, D.

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Estimation of the climatic change impact to beach tourism usingjoined vulnerability analysis and econometric modelling. An holistic approach to beach erosion vulnerabilityassessment. Petrakis, G. Ghionis, N. Kampanis, S. Akrivis et al. Impact of head rotation on the individualized common carotid flow and carotid bifurcation hemodynamics IEEE Journal of Biomedical and Health Informatics, 18 3 , art. ISBN 21— The case study of the Alfios River delta. Karditsa, D. Sifnioti, M. Vousdoukas, O. Andreadis, S.

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Petrakis, Poulos, S. Velegrakis, N. Kampanis, M. Poster Presentation. Abstract P2E1. Ghionis, M. Vousdoukas, A. Karditsa, O. Petrakis, A. Alexandrakis, D. Sifnioti, I. Monioudi, T. Hasiotis, S. Poulos, A. Thrombus morphology may be an indicator for aneurysm expansion Journal of Cardiovascular Surgery, 55 2 , pp. Value of volume measurements in evaluating abdominal aortic aneurysms growth rate and need for surgical treatment.

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Fluid Mechanics of Liquid Metal Batteries | Applied Mechanics Reviews | ASME Digital Collection

Sifnioti, Th. Hasiotis, M. Lipakis, Kampanis, N. Fluids ; — DOI: Recent and future trends of beach zone evolution in relation to its physical characteristics: the case of the Almiros bay island of Crete, South Aegean Sea. Global Nest 16, - Processes affecting recent and future morphological evolution of the Xylokastro beach zone Gulf of Corinth, Greece. Kazakidi, D. Ekaterinaris A nonlinear dynamic finite element approach for simulating muscular hydrostats, Computer Methods in Biomechanics and Biomedical Engineering, , , DOI: Ghionis G.

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Journal of Endovascular Therapy. Kozyrakis G. V, Katsamouris A. Studying the expansion of small abdominal aortic aneurysms: is there a role for peak wall stress? Angiology, ; 30 5 Yalciner A. C, and. Synolakis C. A novel approach in assessing the effects on hemodynamics of topology preserving shape changes of image based arterial structures. Experimental unsteady flow study in a patient specific abdominal aortic aneurysm model. Experiments in Fluids, ; 50 6 Kalligeris, E.

police-risk-management.com/order/use/car-tasto-accensione-iphone.php Flouri, G. The influence of temperature on rheologocal properties of blood mixtures with different volume expanders - implication in numerical modeling of arterial hemodynamics. Rheologica Acta, ; 50 4 The development of a vulnerability index for beach erosion, based on a sediment budget approach.

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Kamarianakis, Papaharilaou Y. Kostas, E.

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Manousaki, A. The role of geometric parameters in the prediction of abdominal aortic aneurysm wall stress. European Journal of Vascular and Endovascular Surgery, , 39 1 , pp. Bressan, G. Chen, M. Lewicka and D. Wang, eds. Mathioudakis, N. Kampanis, J. Giaouzaki, N. Kampanis, P. Agouridakis and A. Matzarakis, H. Mayer, F. Chmielewski Eds. Papadokostakis, N.

Kampanis, G. Sapkas, S. Papadakis and P. The Association of Lumbar Lordosis with spinal osteoarthritis. BioMed Central Musculoskeletal Disorders , doi Geometric tools for shape mapping in topologically similar arterial structures. Baxevanis, On the effect of fiber creep-compliance in the high-temperature deformation of continuous fiber-reinforced ceramic matrix composites, Intern. Mathioulakis and A. An experimental and numerical flow study in an axisymmetric aneurysm model. Experimental Thermal and Fluid Science, , 34 pp.

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Encyclopedia of physics, vol. 8-1. Fluid mechanics II Encyclopedia of physics, vol. 8-1. Fluid mechanics II
Encyclopedia of physics, vol. 8-1. Fluid mechanics II Encyclopedia of physics, vol. 8-1. Fluid mechanics II
Encyclopedia of physics, vol. 8-1. Fluid mechanics II Encyclopedia of physics, vol. 8-1. Fluid mechanics II
Encyclopedia of physics, vol. 8-1. Fluid mechanics II Encyclopedia of physics, vol. 8-1. Fluid mechanics II
Encyclopedia of physics, vol. 8-1. Fluid mechanics II Encyclopedia of physics, vol. 8-1. Fluid mechanics II
Encyclopedia of physics, vol. 8-1. Fluid mechanics II Encyclopedia of physics, vol. 8-1. Fluid mechanics II
Encyclopedia of physics, vol. 8-1. Fluid mechanics II Encyclopedia of physics, vol. 8-1. Fluid mechanics II
Encyclopedia of physics, vol. 8-1. Fluid mechanics II Encyclopedia of physics, vol. 8-1. Fluid mechanics II
Encyclopedia of physics, vol. 8-1. Fluid mechanics II

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