Numerical simulation of electrovortex flows in cylindrical fluid layers and liquid metal batteries

We use the multiphase magnetohydrodynamic code SFEMaNS to study the generation of electrovortex flows in liquid metal batteries. We first reproduce some well known results in a single-phase liquid metal column and then we characterize the electrovortex flow in layered multiphase fluid systems. A simple energy density balance argument accurately estimates the typical interface deformation caused by the electrovortex flow. When applied to Mg-Sb liquid metal batteries, we find that the electrovortex flows may have the capacity to cause short circuits even in moderate size batteries with radii in the range $[10,20]\mathrm{cm}$.

Figure 1

A liquid metal battery (LMB) is composed of three layers of superposed conducting fluids with different densities. During discharge (a), electrons pass from the $-$ to the $+$ pole in the external circuit. Inside the battery, the top metal $A$ dissolves as ions $A^{+}$ migrate downward in the electrolyte, and ultimately alloy with the bottom metal $B\left(A\right)$. (b) The shape of the current collectors causes the field lines of the electrical current $\mathit{j}$ to spread out as they enter the top and the bottom layers. This causes an azimuthal magnetic field $\mathit{b}$ and an inward Lorentz force $\mathit{j}\times \mathit{b}$. As a result an electrovortex flow $\mathit{u}$ is created in both the top and the bottom layers, and this flow can deform the interfaces (c).

Figure 2

Computational domains used to study the EVF in a liquid metal column. On top of the fluid domain, we suppose a homogeneous current density ${\mathit{J}}_{\text{top}}$. Three different setups are compared. In setup (a), the fluid is connected electrically to a solid wire which is fed from below with a homogeneous current density ${\mathit{J}}_{\text{bot}}$. In setup (b), there is no wire and a homogeneous current density ${\mathit{J}}_{\text{bot}}$ directly enters the fluid domain. In setup (c), the wire is modeled by using a particular Millere current density profile [see Eq. (4)] over the section where the wire should be connected to the liquid bath.

Figure 3

Steady-state fields in the wire setup for $S=250$, $P_m=0.2$, $\widehat{\sigma }=1$. (a) Electrovortex flow and intensity in color code. (b) Field lines and magnitude of the electrical current density $\mathit{j}$. (c) Lorentz force $\mathit{j}\times \mathit{b}$ and intensity. Notice the concentration near $(r,z)=(0.2,0)$. (d) Part of the Lorentz force $-(b_{\theta }^2/r){\mathit{e}}_r$ that effectively drives the flow. Vectors suggest magnitude and direction but are not at scale. (e) Numerical grid used in the computations.

Figure 4

Vertical velocity on the axis $r=0$ for setups (a), (b), and (c) of Fig. 2, all obtained with $S=25$, $P_m=0.2$. (a) At low wire conductivity ($\widehat{\sigma }\ll 1$), the uniform current inlet model is well-adapted and the EVF the most intense. At high wire conductivity ($\widehat{\sigma }\gg 1$), Millere's boundary condition is adequate and the EVF is the least intense. For $\widehat{\sigma }\simeq 1$ the flow intensity ranges between those given by models without wire (uniform and Millere cases). (b) Convergence study for the Millere setup on uniform meshes with varying mesh size.

Figure 5

Current density $j_z$ and azimuthal magnetic field $b_{\theta }$ at the bottom electrical contact $z=0$ with or without wire ($S=25$, $P_m=0.2$). (a) The current density $j_z$ strongly depends on the conductivity of the wire. The (singular) Millere current density profile is adequate for high $\widehat{\sigma }\gg 1$, the uniform current density is adequate for low $\widehat{\sigma }\ll 1$. (b) The magnetic field $b_{\theta }$ is only weakly affected by $\widehat{\sigma }$. The differences between the results obtained with the Millere and the uniform boundary conditions are small.

Figure 6

Reynolds number of the electrovortex flow based on the maximal speed, as a function of $S$ and for various $P_m$. When $S$ is small, we observe the scaling law $\text{Re}\sim S$ for all $P_m$. When $S$ is large, we observe $\text{Re}\sim S^{1/2}$ for low values of $P_m$. Millere's boundary conditions are used at the current inlet.

Figure 7

Rescaled vertical velocity profiles along the axis $r=0$ for various values of $S$ and $P_m\in \{{10}^{-6},0.2\}$ and using Millere boundary conditions. (a) In the low $S$ range: velocity profiles rescale well as $u_z/S$ for both values of $P_m$. (b) In the high $S$ range: velocity profiles rescale well as $u_z/\sqrt{S}$.

Figure 8

Transition from viscous to inertial regime as $S$ increases in axisymmetric computations. We show streamlines of the flow in meridional planes and the colors code the intensity. Millere boundary conditions, $P_m={10}^{-6}$ and $r_w=0.2$.

Figure 9

Time evolution of the kinetic energy measures $K_{\text{tot}},K_{\text{axi}},K_{\text{tor}}$ [see Eqs. (18a)–(18c)] in three-dimensional electrovortex flow simulations with wire, $\widehat{\sigma }=56$, $P_m=1.57\times {10}^{-7}$. Initial state is axisymmetric. (a) Below $S<S_c=7.2\times {10}^4$ the flow remains axisymmetric, above $S>S_c$ the flow becomes three-dimensional. This bifurcation is illustrated using the kinetic energy carried by the azimuthal component $K_{\text{tor}}$ as proxy for three-dimensionality. Notice the decay or amplification for large times. (b) Simulation done with $S={10}^6$. The energy measures show that three-dimensionality comes along with nonstationarity. Notice that the total kinetic energy $K_{\text{tot}}$ is higher in the three-dimensional regime than in the axisymmetric regime (i.e., for $t<0.04$ when nonaxisymmetric perturbations are negligible).

Figure 10

(a) Snapshot showing streamlines of the three-dimensional electrovortex flow ($\widehat{\sigma }=56$, $P_m=1.57\times {10}^{-7}$, $S=3\times {10}^5$). (b) Maximal flow velocity as a function of $S$: comparison of 3D and axisymmetric simulations.

Figure 11

Electrovortex flow in a two-layer setup in a small cell with radius $R=5\mathrm{cm}$, wire radius $R_w=1\mathrm{cm}$, heights $H_1=1\mathrm{cm},H_2=5\mathrm{cm},H_w=5\mathrm{cm}$, current $I=100\mathrm{A}$ and densities ${\rho }_1=3400\mathrm{kg}{\mathrm{m}}^{-3},{\rho }_2=3500\mathrm{kg}{\mathrm{m}}^{-3}$. (a) Sketch of the setup. (b) Snapshot of the electrovortex flow. (c) Density field.

Figure 12

Variation of the maximal flow speed $U_{max}$ (a) and the maximal interface elevation ${\eta }_{max}$ (b) with current $I$ in the two-layer setup with $(R,R_w,H_1,H_2,H_w)=(5,1,1,5,5)\mathrm{cm}$ and using two different density jumps ${\rho }_2-{\rho }_1\in \{100,500\}\mathrm{kg}/{\mathrm{m}}^3$.

Figure 13

Electrovortex flow in a three-layer setup similar to a Mg-Sb liquid metal battery. Small cell with radius $R=5\mathrm{cm}$, wire radius $R_w=1\mathrm{cm}$, heights $(H_1,H_2,H_3,H_w)=(5,1,5,5)\mathrm{cm}$, and densities $({\rho }_1,{\rho }_2,{\rho }_3)=(1577,1715,6270)\mathrm{kg}{\mathrm{m}}^{-3}$. (a) Sketch of the setup. (b) Meridional grid with mesh size $\in [0.25,1]\mathrm{mm}$. (c), (d) Snapshot of the electrovortex flow and density field at $t=2\mathrm{s}$ with current $I=200\mathrm{A}$.

Figure 14

When the electrovortex flow in a Mg-Sb battery becomes too intense, it can make the electrolyte layer to pinch. In the cell considered here, this occurs at $I=300\mathrm{A}$. Starting from rest at $t=0\mathrm{s}$, we show six successive snapshots of the flow intensity and the interfaces at indicated times.

Figure 15

Simulations with realistic values of electrolyte conductivity ${\sigma }_2=2.13{10}^2\mathrm{S}{\mathrm{m}}^{-1}$ versus simulations with relaxed (fake) electrolyte conductivity ${\sigma }_2=2.56{10}^4\mathrm{S}{\mathrm{m}}^{-1}={\sigma }_3/100$ and with intense currents $I=300\mathrm{A}$. (a) The maximal speed and interface deformation are significantly affected at longer times, but the initial maxima are always the highest ones and these values depend little on ${\sigma }_2$ (arrows). (b) At short times $t=1.5\mathrm{s}$, the flow is similar and slightly more intense with higher ${\sigma }_2$.

Figure 16

(a) Maximal flow intensity $U_{max}$ and upper interface deformation ${\eta }_{max}$ as a function of $I$ in a Mg-Sb battery with dimensions $(R,R_w,H_1,H_2,H_3,H_w)=(5,1,5,1,5,5)\mathrm{cm}$. We observe $U_{max}\sim I$ and ${\eta }_{max}\sim I^2$. At the highest current $I=300\mathrm{A}$ we observe a pinch: ${\eta }_{max}\approx H_2$. (b) nondimensional parameters $\mathrm{\Pi }$ and Ri as a function of $S$ using the data in (a) [see Eq. (22)].

Figure 17

Using the principle of similitude we calculate the maximal flow-speed ${\overline{U}}_{max}$ (a) and the maximal interface elevation ${\overline{\eta }}_{max}$ (panel b) in similar Mg-Sb liquid metal batteries as a function of their radius $\overline{R}$ for the current density $\overline{J}\in \{1,3,10\}{\mathrm{kAm}}^{-2}$. Assuming that the dependence with respect to the height of the electrolyte layer, ${\overline{H}}_2$, is small and taking ${\overline{H}}_2\in [1,5]\mathrm{mm}$, we divide the diagram for ${\overline{\eta }}_{max}$ into three regions. No risk of pinch by EVF when ${\overline{\eta }}_{max}<1\mathrm{mm}$ (green zone). Risk of pinch by EVF when ${\overline{\eta }}_{max}\in [1,5]\mathrm{mm}$ (yellow zone). EVF-induced pinch very likely when ${\overline{\eta }}_{max}>5\mathrm{mm}$ (red zone).