Study of the Magnetohydrodynamic Instability and a New Suppression Method in Liquid Metal Batteries

Abstract

As a strong candidate for energy storage applications, Liquid Metal Batteries (LMBs) have the advantages of higher current density, longer cycle life, and simpler manufacturing of large-scale storage systems. Owing to the all-liquid construction, various kinds of Magnetohydrodynamic instabilities (MHDIs) are present in LMBs. In this paper, an in-depth study of the evolution process of MHDIs within LMBs has been conducted. By analyzing the characteristic velocity, the growth rate of instabilities γ has been defined so that the critical Hartmann number at which the instability occurs can be ascertained. A new critical parameter, mixed Reynolds number Re_mix, has been introduced to determine the duration of stable battery operation across varying charging/discharging currents, including those that may surpass the prescribed safe limits. Finally, a method for mitigating magnetohydrodynamic instability in LMBs through the configuration of busbar current is proposed, which can be seamlessly integrated with parallel battery packs. A comparative analysis of LMBs operation with/without bus current configuration reveals that when bus current is appropriately configured, the magnetic field strength within the battery undergoes a notable reduction of 40%, leading to a significant suppression of instability. The conclusions offer theoretical underpinnings for the application of LMBs in large-scale grid-level energy storage systems.

1. Introduction

As pivotal advancements in wind and solar energy technologies continue to fuel global innovation and infrastructure modernization, the urgent imperative to develop robust, grid-scale energy storage systems has become paramount [1,2]. Such technological breakthroughs are not only pivotal for overcoming the intermittency challenges inherent in renewable energy integration but also represent a cornerstone strategy for accelerating global decarbonization efforts and achieving net-zero carbon emissions targets. Among the diverse energy storage solutions, batteries, particularly liquid metal batteries (LMBs) [3,4,5,6], have garnered considerable attention owing to their cost-effectiveness and adaptable deployment capabilities.

Liquid metal batteries consist of three distinct, immiscible layers of liquid metals or molten salt electrolytes, which naturally stratify due to their varying densities. Typically, the upper negative electrode layer comprises alkali metals or alkaline earth metals characterized by low electronegativity and low density, examples of which include Li, K, Na, Ca, Mg, etc. The bottom positive electrode predominantly utilizes metals or alloys characterized by their high electronegativity and density, exemplary instances being Bi, Pb, Sn, Hg, Se, Te, Sb, and others. Positioned in between, the middle layer comprises a molten salt electrolyte, which doubles as an effective separator. Common LMBs, such as Li||LiCl-KCl||Bi, Li||LiCl-CaCl₂||CaBi [7,8] and Li||LiF-LiCl-LiI||Sb-Pb [3], exhibit remarkable energy conversion efficiencies, reaching as high as 70%. However, the very nature of its all-liquid structure gives rise to various magnetohydrodynamic instabilities (MHDIs), especially under the influence of a magnetic field [9], which includes Taylor instability (TI), sloshing instability (SI, also known as the metal pad roll instability), and Rayleigh–Bénard instability (RBI). In addition, in proximity to the positive and negative current collectors, the abrupt change in the diameter of the electrical path leads to interactions between the current and the self-induced magnetic field, generating electro-vortex flows (EVFs). The combined influence of these factors predisposes the fluid flow in LMBs to instability, potentially causing short circuits between the positive and negative electrodes. Among the myriad instabilities, TI stands out as the most pivotal factor contributing to battery instability and, consequently, is the focus of this paper. For insights into research concerning RBI under the influence of high temperature or magnetic fields, as well as EVFs, please refer to our previous work [10,11,12,13,14].

TI represents a type of twisted instability, characterized by its non-axisymmetric nature, which closely resembles the z-pinch phenomenon observed in plasma physics. In a liquid metal with a cylindrical structure, the presence of a circular magnetic field determines whether TI occurs, and this is contingent upon the radial distribution of the magnetic field. Consequently, when the charging/discharging currents in LMBs surpass a specific critical threshold, TI may manifest under the influence of the magnetic field induced by these currents themselves [15,16]. Tayler [17] has derived a reference standard, namely ${\partial }_r\left(r{B}_{\phi }^2\left(r\right)\right)>0$. However, it is important to note that this equation remains invariant in the context of LMBs, thereby limiting its application solely to that of a reference point. Consequently, a larger value of ${\partial }_r\left(r{B}_{\phi }^2\left(r\right)\right)$ indicates an increased likelihood of TI occurring. Weier et al. [18] clearly delineated the definition and conditions conducive to the occurrence of TI within LMBs. Stefani et al. [19] employed a simplified model and investigated the disruption of spiral symmetry in TI. Their findings revealed that during the fully matured stage of TI, merely a very small external perturbation suffices to provoke spiral oscillations. In the meantime, these spiral oscillations primarily serve to interchange energy between left-handed and right-handed TI modes. Furthermore, Weber et al. [20] utilized the solver detailed in the reference [21] to examine the process of helical chiral symmetry breaking within TI during both their exponential growth phase and fully developed stage. It has been concluded that the TI will not manifest when the Lundquist number S ($S=HaP{m}^{1/2}$ with $Pm=\nu {\mu }_0\sigma$ denoting the magnetic Prandtl number) is substantially less than 1.

To date, a diverse array of methodologies aimed at mitigating TI has been put forth. Rüdiger et al. [15,22] conducted an investigation into the non-axisymmetric TI induced by the circular magnetic field generated by an axial current flowing between coaxial cylinders devoid of end plates. Their findings revealed a positive correlation between the diameter of the insulating cylinder and its capacity to suppress TI, specifically, as the diameter of the insulating cylinder increased, so did the significance of its suppressive effect. Building upon this foundation, Stefani et al. [23,24] and Weber et al. [20,25,26] proposed analogous approaches to numerically simulate the fluid flow dynamics within an infinitely long cylinder that contains a coaxial insulating cylinder. They demonstrated that as the diameter of the insulating cylinder expands, both the critical current and Hartmann number, Ha, of TI exhibit a corresponding increase. Nonetheless, this increase in insulating cylinder diameter leads to a reduction in the volume of the liquid metal electrode, subsequently decreasing the battery’s capacity. Furthermore, the inclusion of a central ceramic insulating cylinder may induce metal adhesion, potentially causing short circuits. Moreover, Stefani et al. [23] introduced a methodology utilizing coaxial insulated cylinders equipped with charging busbars, demonstrating its efficacy in mitigating the emergence of TI under particular current conditions. Nonetheless, the necessity for incorporating external current results in a decrease in energy conversion efficiency. Liu et al. [4] suggested a method to suppress MHDIs by incorporating a ceramic grid structure, which effectively quells various instabilities in LMBs, including TI, by directing and regulating the flow of magnetic fluids. Notably, the suppression efficacy becomes more significant as the number of grids (or partitions) increases. Weber et al. [25,26] and Herreman et al. [27,28,29] introduced a methodology aimed at mitigating MHDI by augmenting the aspect ratio (D/H). However, the increase in aspect ratio heightens the risk of SI incidents and diminishes battery capacity, falling short of the economic demands for energy storage in power grid applications. In addition to the aforementioned methods, certain scholars have also suggested employing rotating fluids as a means to suppress TI [30,31]. While rotating the fluid at a specific angular velocity can indeed effectively elevate the critical current threshold for TI, this approach remains impractical for stationary LMBs utilized in energy storage applications.

Furthermore, SI can also exert a substantial influence on the operational performance of LMBs. The instability of a sloshing long-wave interface has been investigated by Norbert et al. [32], with a particular focus on the effects of cell current, layer thickness, density, viscosity, conductivity, and the ambient magnetic field. By approximating the top and bottom liquid electrodes as metallic pads, Zikanov et al. [33,34] demonstrated that wave dynamics are governed by the ratio of density discontinuities at the two interfaces. A similar simplification has also been adopted by Norbert et al. [35], to present the oscillation periods of the instability. Due to the difficulty in suppressing SI, it is imperative to maintain a sufficiently large thickness of the electrolyte layer.

In this paper, an in-depth examination of the evolution process of MHDI within LMBs, especially the flow dynamics and characteristics of magnetic fluid, has been conducted. Rather than analyzing the multiple instabilities in isolation, their coupled effects have been comprehensively accounted for. By meticulously analyzing the characteristic velocity, the growth rate of these instabilities γ has been defined, allowing for the determination of the critical Hartmann number Ha_cr, at which instability occurs. Additionally, a novel critical parameter, the mixed Reynolds number Remix, has been introduced, which serves to ascertain the duration of stable battery operation across a range of charging and discharging currents, including those that may exceed predefined safe limits. Furthermore, this paper proposes a method for mitigating magnetohydrodynamic instability in LMBs through the strategic configuration of busbar current. This approach is uniquely suited for integration with parallel battery packs, enhancing its practical applicability. Numerical simulations were conducted to compare the operation of LMBs with and without the bus current configuration. The results reveal that, when the bus current is appropriately configured, the magnetic field strength within the battery undergoes a notable reduction of 40%, leading to a significant suppression of instability. The conclusions provide a robust theoretical foundation for the utilization of LMBs in large-scale grid-level energy storage systems.

The organization of this article is outlined as follows: In Section 2, we introduce the numerical model employed in this study, the numerical methods utilized for its solution, and the validation procedures conducted to ensure the model’s accuracy. Section 3 delves into the progression of MHDI within LMBs and the relevant critical parameters. Section 4 discusses the approach adopted for mitigating instability, based on the optimization of bus current configuration. Lastly, Section 5 summarizes the conclusions drawn from our work and outlines potential avenues for future research.

2. Mathematical Formulations

2.1. Numerical Model

In order to study the instabilities in the LMBs, a square section container with an aspect ratio $\Gamma =h/l_x=h/l_y=0.45 \mathrm{m}/0.1 \mathrm{m}$ is considered here, as shown in Figure 1, where h, l_x, and l_y are the height, length, and width, respectively. The Li||Pb-Bi combination has been chosen as the negative and positive electrodes, respectively, and the molten salt LiCl (59 mol%)-KCl (41 mol%) is employed as the electrolyte, which is the same as in Refs. [36,37]. The electrodes and electrolyte can self-segregate by density into three distinct layers because of the immiscibility of the contiguous metal and salt phases, and the thickness ratio between them is 4:1:4. The detail physical properties at the working temperature of 500 °C (which significantly exceeds the melting point of the three materials ensuring the system remains in the fully liquid state) have been listed in

Table 1. The physical properties of a three-layer battery system Li||LiCl-KCl||Pb-Bi at the working temperature of 500 °C.

Physical Quantity	Symbol	Value	Unit
Mass density	${\rho }^{(1)}$	1.0065 × 104	kg·m−3
	${\rho }^{(2)}$	1597.9	kg·m−3
	${\rho }^{(3)}$	484.7	kg·m−3
Kinematic viscosity	${\nu }^{(1)}$	1.29 × 10−7	m2·s−1
	${\nu }^{(2)}$	1.38 × 10−6	m2·s−1
	${\nu }^{(3)}$	6.64 × 10−7	m2·s−1
Electrical conductivity	${\sigma }^{(1)}$	7.81 × 10−7	kg−1·m−3·s3·A2
	${\sigma }^{(2)}$	187.1	kg−1·m−3·s3·A2
	${\sigma }^{(3)}$	3.0 × 106	kg−1·m−3·s3·A2
Magnetic permeability	${\mu }_{\mathrm{mag}}^{(1,2,3)}$	4π × 10−7	kg−1·m−3·s3·A2

, where superscripts (1), (2) and (3) denotes the properties of negative electrode, electrolyte and positive electrode, respectively.

Before the numerical simulation, some basic assumptions need to be introduced first:

• Both the viscous dissipation and Joule heating are neglected, which means the system temperature is fixed at 500 °C and the physical properties of all the materials remain constant;
• At all solid–liquid interfaces, the no-slip condition is adopted, and a uniform contact angle is maintained (90°);
• The surface tension at all liquid–liquid interfaces is the same and remains constant (0.07 N∙m⁻¹);
• The side walls of LMBs are electrically insulated, while the top and bottom walls are both perfectly conducting;
• All the walls are supposed to be magnetically insulated for the single battery case (Section 3), while only the top and bottom walls are magnetically insulated for the battery pack case (Section 4);
• The thickness of all walls is ignored.

Therefore, the following governing equations for the LMB system can be obtained:

$$\nabla ·{\mathit{u}}^{\left(i\right)}=0,$$

(1)

$${\partial }_t\left(\rho {\mathit{u}}^{\left(i\right)}\right)+\left(\rho {\mathit{u}}^{\left(i\right)}·\nabla \right){\mathit{u}}^{\left(i\right)}=-\nabla p^{\left(i\right)}+\rho \mathit{g}+\rho \nu \Delta {\mathit{u}}^{\left(i\right)}+{{\mathit{F}}_{\mathrm{L}}}^{\left(i\right)},$$

(2)

$${{\mathit{F}}_{\mathrm{L}}}^{\left(i\right)}={\mathit{J}}^{\left(i\right)}\times {\mathit{B}}^{\left(i\right)},$$

(3)

where ρ, u, t, p, g, ν, F_L, J, and B are the density, velocity, time, pressure, gravitational acceleration, kinematic viscosity, Lorentz force, total current density, and induced magnetic field from J. Superscript ($i=1,2,3$) denotes the negative electrode, electrolyte, and positive electrode, respectively.

In order to get the induced magnetic field of the current, the distribution of the corresponding electric potential field needs to be obtained first from the following equation:

$$\sigma \Delta E^{\left(i\right)}={\displaystyle \underset{S}{\int }{\sigma }_{\mathrm{c}\to \mathrm{f}}}{\left({\mathit{u}}^{\left(i\right)}\times {\mathit{B}}^{\left(i\right)}\right)}_{\mathrm{c}\to \mathrm{f}}\cdot \mathit{n}ds,$$

(4)

where σ, E, and n are the electrical conductivity, electric potential field, and the unit normal vector of each cell face. $\mathrm{c}\to \mathrm{f}$ and S stand for interpolation from cell center to cell face center and surface interpolation, respectively.

By solving Equation (4), the magnetic vector potential (MVP) can be obtained from the electric potential, and then the induced magnetic field is obtained by taking the curl of MVP, which are shown as the following equations:

$$\Delta {\mathit{A}}^{\left(i\right)}={\mu }_0\sigma \nabla E^{\left(i\right)},$$

(5)

$${\mathit{B}}^{\left(i\right)}=\nabla \times {\mathit{A}}^{\left(i\right)},$$

(6)

where A is the MVP, and ${\mu }_0$ denotes vacuum permeability. Simultaneously, as both A and B are solenoidal vector fields, they should satisfy the following relations:

$$\nabla \cdot {\mathit{A}}^{\left(i\right)}=0,$$

(7)

$$\nabla \cdot {\mathit{B}}^{\left(i\right)}=0,$$

(8)

In addition, when considering the additional magnetic field, B_add, generated by the, the Biot–Savart law is employed:

$${\mathit{B}}_{\mathrm{add}}={\displaystyle {\int }_L\frac{{\mu }_0}{4\pi }\frac{I_{\mathrm{add}}d\mathit{l}\times {\mathit{e}}_r}{r^2}},$$

(9)

where I_add is the busbar current.

The volume-of-fluid (VOF) method has been applied to track the liquid–liquid interfaces. Therefore, the volume fraction (α) should satisfy the following equation:

$${\partial }_t{\alpha }^{\left(i\right)}+\left({\mathit{u}}^{\left(i\right)}\cdot \nabla \right){\alpha }^{\left(i\right)}=0,$$

(10)

Therefore, the relative physical properties can be calculated and updated by using ${\alpha }^{\left(i\right)}$ to take the weighted average:

$$\rho ={\displaystyle \sum _{i=1}^3\left({\alpha }^{\left(i\right)}{\rho }^{\left(i\right)}\right)},$$

(11)

$$\nu ={\displaystyle \sum _{i=1}^3\left({\alpha }^{\left(i\right)}{\nu }^{\left(i\right)}\right)},$$

(12)

$$\sigma ={\displaystyle \sum _{i=1}^3\left({\alpha }^{\left(i\right)}{\sigma }^{\left(i\right)}\right)},$$

(13)

In the meantime, to correctly compute and update the Lorentz force in Equation (3), the current density ${\mathit{J}}^{\left(i\right)}$ needs to be updated first in each calculation:

$${\mathit{J}}^{(i)}={\mathit{J}}_c^{(i)}={\displaystyle \frac{1}{{\Omega }_p}}{\displaystyle \sum _{f=1}^{nf}{\mathit{J}}_f^{(i)}}\left({\mathit{r}}_f-{\mathit{r}}_c\right)\cdot {\mathit{S}}_f,$$

(14)

where ${\mathit{J}}_c$ and ${\mathit{J}}_f$ represent the current density at the cell center and face center of each grid cell, respectively. ${\mathit{r}}_c$ and ${\mathit{r}}_f$ are the position vectors of the control volume center and face center being solved, respectively. ${\mathit{\Omega }}_p$ is the cell volume. ${\mathit{S}}_f$ represents the normal vector of the grid face, whose magnitude is equal to the area of the corresponding cell face.

The boundary conditions are shown as follows:

At the top and bottom walls ($z=0,z=-h$): ${\displaystyle \frac{\partial \mathit{A}}{\partial \mathit{n}}}=0,E=\mathrm{constant},{\displaystyle \frac{\partial \alpha }{\partial \mathit{n}}}=0$.

At the side walls ($\left|x\right|\le l_x/2,y=\pm l_y/2$ and $x=\pm l_x/2,\left|y\right|\le l_y/2$): $\mathit{A}=\left(0,0,0\right)$, ${\displaystyle \frac{\partial E}{\partial \mathit{n}}}=0$${\displaystyle \frac{\partial \alpha }{\partial \mathit{n}}}=0$.

For the initial conditions, both the initial velocity and MVP are set as $\left(0,0,0\right)$, and the internal electric potential is initialized as 0.

For the convenience of obtaining and updating the induced magnetic field, the electric potential, instead of charging/discharging current, is set on the electrodes. Actually, the two can be converted to each other for the same medium according to Ohm’s law.

2.2. Numerical Method and Validation

The governing equations are discretized by adopting the finite volume method. The pressure-velocity coupling is handled by the PIMPLE algorithm, which is a combination of the PISO and SIMPLE algorithms. The Euler scheme is used for the transient terms, the first-order upwind scheme is applied to treat the convective terms, and the second-order central difference scheme is adopted for diffusion terms. The convergence residual for the velocity field is set to 10⁻⁸, and that for the pressure field, MVP, and electric potential field is 10⁻⁷.

As shown in Figure 2, non-equidistant meshes are applied. The computational meshes are refined close to all walls, ensuring there are at least 5 grid points to resolve the Hartmann layer. The grid independence test has been carried out at the charging current $I_{\mathrm{ch}}=2 \mathrm{kA}$. The velocity component in the z-direction U_z and $J/\sigma$ (i.e., electric field intensity) for different grids have been displayed in Figure 3 and Figure 4. It turns out that the solution becomes independent of the grid size for $90\times 90\times 205$. Besides, the adjustable time step algorithm has been adopted to strike a balance between capturing system dynamics accurately and maintaining computational efficiency, and the corresponding maximum Courant number is limited to 0.9.

With the aim of validating the solver, the magnetic fluid (magnetic Prandtl number $Pm={10}^{-3}$) flow in a cuboid cavity with the aspect ratio $\Gamma =H/L=1.5 \mathrm{m}/0.1 \mathrm{m}=15$ is simulated. The comparison between the results of the current solver and those in Refs. [20,21,27] has been displayed in our previous work [4] and will not repeat here.

3. Magnetohydrodynamic Instability in LMBs

3.1. Evolution of Magnetohydrodynamic Instability

Figure 5 and Figure 6 show the shapes of the electrolyte and the magnetic field distribution at the XOY plane ($Z=-0.2$, which is the interface between the negative electrode and electrolyte) for $I_{\mathrm{ch}}= 1.132\mathrm{kA}$. It can be seen that, from $t=1 \mathrm{s}$ to $t=60 \mathrm{s}$, the upper and lower phase interfaces of the electrolyte layer remain stable. Although the fluid velocity increases, especially at the negative electrode, the fluids of different phases are not mixed. In the meantime, there is no significant change in the magnetic field distribution either.

When the charging current increases to 2.264 kA, LMB will gradually become unstable, as depicted in Figure 7 and Figure 8. The fluctuations at the interface of the electrolyte layer gradually intensify over time, until the positive and negative electrodes come into contact and cause a short circuit ($t=4 \mathrm{s}$). Meanwhile, the magnetic field distribution becomes more disordered, which is much different from Figure 6.

3.2. Critical Parameter

In order to precisely predict the occurrence of MHDI in LMBs, some dimensionless critical parameters have been proposed. For example, Herreman et al. [27] and Rüdiger et al. [22] have found that there is a critical Hartmann number Ha_cr, beyond which MHDI will happen. The definition of Ha is as Equation (15). What is more, this critical Ha is related to the aspect ratio. The conclusion is consistent with those obtained by many other researchers [19,20,21].

$$Ha=B_{\varphi }L{\left({\displaystyle \frac{\sigma }{\nu \rho }}\right)}^{{\displaystyle \frac{1}{2}}}={\displaystyle \frac{{\mu }_{0}I}{2\pi }}{\left({\displaystyle \frac{\sigma }{\nu \rho }}\right)}^{{\displaystyle \frac{1}{2}}},$$

(15)

It can be easily understood from Equation (15) that for the same current, the fluid with a smaller density and larger electrical conductivity will have a larger Ha. Therefore, MDHI usually occurs in the negative electrode region, verifying the results obtained in Section 3.

When MHDI happens, the velocity of fluid in LMBs increases over time until the battery short-circuits. Therefore, the growth rate of MHDI ${\gamma }^{\left(i\right)}$ can be defined as Equation (16) by quantifying the change in characteristic velocity $〈{\mathit{u}}^{\left(i\right)}〉$ with time.

$$〈{\mathit{u}}^{\left(i\right)}〉~{\mathrm{e}}^{{\gamma }^{\left(i\right)}t},$$

(16)

where $〈{\mathit{u}}^{\left(i\right)}〉$ is the root mean square velocity.

Figure 9 shows the effect of the charging current on γ. it can be seen that the curves of all three types of fluid will intersect at one point at low current ($\gamma =0$). When $\gamma >0$, $〈\mathit{u}〉$ increases with time, causing the occurrence of MHDI. In other words, the point, $\gamma =0$, corresponds to the critical current at which MDHI occurs. The value of corresponding Ha is 22, which is in agreement with the result of Refs. [15,16,22].

To ascertain the period during which LMBs can function safely, the mixed Reynold number $R{e}_{mix}$ at the interface between the two layers of fluids has been introduced, which is defined as follows:

$$R{e}_{mix}^{\left(i,j\right)}=\sqrt{R{e}^{\left(i\right)}\cdot R{e}^{\left(j\right)}}={\displaystyle \frac{\overline{u}\cdot L}{\sqrt{{\nu }^{\left(i\right)}{\nu }^{\left(j\right)}}}},\hspace{1em}i,j=\left(1,2,3\right),$$

(17)

As mentioned before, the focus is on the $R{e}_{mix}$ at the interface between the negative electrode and electrolyte, i.e., $R{e}_{mix}^{\left(i,j\right)}$.

Figure 10 depicts the development of $\lg \left(R{e}_{mix}^{\left(1,2\right)}\right)$ with time. By observing the state of the interface, it can be deduced that the interface fluctuates when $\lg \left(R{e}_{mix}^{\left(1,2\right)}\right)>2.27$ (the value of 2.27 is derived based on the observation), and subsequently, the negative electrode begins to mix with the electrolyte layer. The fluctuation and mixing effects become more pronounced in proximity to the walls, as shown in the insert plot of Figure 10, where the distribution of magnetic field strength reaches its maximum. When $\lg \left(R{e}_{mix}^{\left(1,2\right)}\right)<2.27$, LMB is in a state of stability.

4. Instability Suppression Based on Busbar Current

4.1. Configuration of Busbar Current

In large-scale energy storage systems, battery packs are usually used to improve energy conversion efficiency. These packs consist of numerous batteries arranged in either parallel or series configurations. In this paper, the case of parallel battery packs is discussed. The surrounding batteries are idealized as infinitely thin wires, with a distance of $a=0.1 \mathrm{m}$ from the center of the studied battery, as shown in Figure 11.

It should be noticed that, in the case of parallel battery packs, the side walls of each individual cell have been modified to non-magnetic insulating walls. Utilizing Biot-Savart’s law and the principle of vector addition, it can be observed that the superimposed induced magnetic field produced by the surrounding currents is oriented in the opposite direction to the induced magnetic field generated by the current in the studied battery.

4.2. Inhibition Effect of Busbar Current on MHDI

Figure 12 and Figure 13 show the shapes of the electrolyte and the magnetic field distribution at the XOY plane ($Z=0.25$) for $I_{\mathrm{ch}}=2.264 \mathrm{kA}$. Similarly to the situation without busbar currents, the battery will exhibit instability for high charging current. However, the stable operation time of the battery has been extended nearly twice, indicating that the busbar currents have a suppressive effect on MHDI. Furthermore, in the presence of busbar currents, the region where the fluid attains its peak velocity is situated in proximity to the current collectors. Consequently, this minimizes the likelihood of the negative electrode intermixing with the electrolyte.

A comparison of the magnetic field distribution with and without considering busbar current, as well as the induced magnetic field generated by bus current, has been demonstrated in Figure 13. It is evident that the presence of the magnetic field induced by the bus current, which opposes the direction of the magnetic field generated by the battery’s own current, results in a reduction in the magnetic field strength within the battery. Consequently, the Lorentz force directed towards the axis has been significantly reduced, which in turn has decreased the vortex shear velocity at the electrode-electrolyte interface, and the MHDI is suppressed.

4.3. Critical Parameters for Batteries with Busbar Currents

Figure 14 shows the variation in the MHDI growth rate ${\gamma }^{\left(i\right)}$ under different charging currents, considering the influence of the busbar currents. Compared with Figure 9, there is a noticeable suppression of MHDI across all three layers of LMBs, with a particularly prominent reduction in the growth rate ${\gamma }^{\left(i\right)}$ observed at the negative electrode. This is due to the induced magnetic field generated by the bus, which causes the vortices in two different directions in the negative layer fluid that were originally close to the negative electrolyte interface to migrate to the vicinity of the collector. This migration makes the maximum velocity of the vortex transfer to the symmetric center of the vortex center, as shown in Figure 15 ($I_{\mathrm{ch}}=2.264 \mathrm{kA}$). What is more, the critical current of the battery has experienced an increase of approximately 12%, suggesting that it can sustain stable operation at elevated currents when the influence of the busbar is taken into account. This is highly advantageous for achieving rapid charging and discharging of the battery while simultaneously ensuring its stable operation.

The variation of $R{e}_{mix}$ over time has been illustrated in Figure 16, demonstrating a substantial increase in the battery’s stable operation time. The effect becomes more pronounced as the charging current increases.

5. Conclusions and Outlook

In summary, the MHD characteristics within a three-dimensional square cross-sectional LMB have been investigated through comprehensive numerical simulations and rigorous theoretical analysis. The coupled effects of multiple instabilities have been comprehensively accounted for, rather than being analyzed in isolation. According to the results, the following conclusions can be succinctly summarized:

• Through the investigation of the instability evolution process in LMBs subjected to varying charging and discharging currents, it is found that there exists a threshold current value. Beyond this critical threshold, the electrolyte layer interface experiences significant and violent fluctuations, ultimately leading to short circuits due to the contact between the positive and negative electrodes of the battery.
• The instability growth rate γ is defined in terms of the temporal evolution of the characteristic velocity. The critical Hartmann number, at which instability occurs, can be deduced from the extrapolation point where $\gamma =0$. Furthermore, a novel critical parameter, termed the mixed Reynolds number Remix, has been introduced to assess the stable operation duration of batteries under various charging and discharging currents, even when these currents exceed the recommended safe limits.
• A method has been proposed to mitigate MHDI in LMBs by strategically configuring the bus current. A comparison of LMB operation with and without this bus current configuration reveals that, when the bus current is appropriately configured, the magnetic field strength within the battery is markedly decreased by approximately 40%, thereby effectively suppressing the instability.

The conclusions offer a theoretical underpinning for the application of LMBs in large-scale grid-level energy storage systems. Future research endeavors can delve deeper into exploring the impact of varying bus distances and busbar current magnitudes on the system.