2.3.2. Solids

The necessary parameters for the solid parts, including Al [63] and Cu contacts, as well as PMMA [64–66] enclosure to be considered in the numerical model, are presented in Table 1.


**Table 1.** Required parameters for modeling of solid parts.

### *2.4. Equations, Initial, and Boundary Conditions*

### 2.4.1. MHD Equations and Sheath Model

In general, the MHD method is time-consuming but with less data acquisition costs, while many rapid mathematical models do not care about the data acquisition costs. So, some rapid models are full of coefficients, the application of which is inevitably condemned to death after an ad hoc implementation. Therefore, it is essential to define a model description as much as possible on the data flow and information, which is the case of the MHD simulation where a mathematical model that stands on such a solid basis will have a high probability of being applied, and so tested and improved, in a continuous way [67]. That is the reason for the vast utilization of the MHD tools for the design and performance evaluation of switchgear [68]. Four groups of equations used in this study are the current conservation equations Ampere's law and Lorentz force, HT, and Navier-Stokes equations (NSEs) for an incompressible fluid, including conservation of momentum and mass. Details of equations were already presented for the fixed contact simulation [51]. In Newtonian fluids (all gases and most non-viscous liquids [69]), if the Mach number (*Ma*) is more than 0.3 or the *T* inside the fluid changes significantly and rapidly, the fluid is considered compressible. In our study, although the *T* rises

sharply, the changes are not in a vast range, and calculation shows that *Ma* is not more than 0.3. Also, the Re is relatively small, so the flow is laminar in each layer [51]. The *T* in the core is in the range of 9–15k K and densities of ions and electrons at this temperature are high enough to consider that NSEs are valid for highly ionized air plasma [51]. This assumption is justified by the high plasma pressure and the large estimated electron densities in the order of 10<sup>18</sup> cm<sup>−</sup><sup>3</sup> [44]. Here, the sinusoidal 50 Hz 200 A current source *Jn*(*t*) is applied to the terminals, as shown in Figure 1b.

$$J\_n(t) = \pm 5 \times 10^5 \cos(0.314 \times t \text{[ms}^{-1}]) \left[ \text{A/m}^2 \right] \tag{1}$$

The initial conditions for these equations are zero potential at moving contact and zero current density at the outer wall of ACH, as shown in Figure 1b. Due to the current injection direction in the quarter-cycle simulation, the lower side of the moving contact and the upper fixed contact act as an anode, while the lower fixed contact and the upper side of the moving contact are the cathodes. The required parameters to model the e ffect of the Al and Cu cathode and anode are obtained from [66,70]. To solve the heat transfer equations together with the magnetic field equations in a 2-D domain, the thickness should be considered, which is added as *dz* = 10 mm compared with 2-D axisymmetric equations based on arc diameter.

$$
\rho \mathbf{C}\_p \frac{\partial T}{\partial t} + \rho \mathbf{C}\_p \boldsymbol{\mu} \cdot \nabla T - \nabla \cdot (k \nabla T) = Q\_{tot} \tag{2}
$$

As it is shown in Figure 1b, the top and bottom of the enclosure, interior arc chamber, as well as the outer surface of the walls are modeled with the ambient temperature T0, which meets the actual status. It is also assumed that heat transfer from the open points to the outside is negligible. Therefore, these hatches are modeled as thermally insulated surfaces ( → *<sup>n</sup>*·*q* = <sup>0</sup>).

As the time steps and solution methods to solve the MHD and electrodynamic drive are di fferent [71], an appropriate approach is an asynchronous solution. Measurements, and electrodynamic drive simulations, show that moving contact reaches a constant speed at the start of the arc [24]. So, the simulated speed from the electromechanical solution is imported into the fluid flow model as → *uc*. The influence of thermionic electron emission (3–7) is not significant on the current density (*J*), as at the highest melting point of Cu cathode 4.16 × 10−<sup>5</sup> A/m<sup>2</sup> at *T* = 1356 K, while *J* at the cathode is in the order of 9.3 × 10<sup>8</sup> A/m2. Increasing *T* does not ensure observable electron emission since Cu will melt and then evaporates or decomposes, but it a ffects the contact temperature by keeping the continuity in heat transfer equations. Although the first studies on liquid metals [72] show an explosive electron emission from liquid cathodes in special conditions and for a limited time, as far as we know thermionic emission data exist only for polycrystalline solid-state copper emitters [73] and Cu is not the ideal thermionic emitter due to its relatively low melting point (1356 K), which limits the working temperature region to 800–1100 K or so [74]. The cathode temperature does not reach the melting point in the first cycle, so thermionic emission from melted Cu is ignored.

$$
\stackrel{\rightarrow}{J}\_{R} = A\_{R}T^{2} \exp\left(-\frac{q\Phi cff}{k\_{B}T}\right) \stackrel{\rightarrow}{n} \tag{3}
$$

$$J\_{\rm ion} + J\_{\rm clcc} = \left| \overrightarrow{J} \cdot \overrightarrow{n} \right|\tag{4}$$

$$J\_{\rm elec} = \left\{ \begin{array}{c} \left| \stackrel{\rightarrow}{J} \cdot \stackrel{\rightarrow}{\vec{n}} \right| , \left| \stackrel{\rightarrow}{J} \cdot \stackrel{\rightarrow}{\vec{n}} \right| \le J\_R \\\\ J\_{R\prime} \left| \stackrel{\rightarrow}{J} \cdot \stackrel{\rightarrow}{\vec{n}} \right| > J\_R \end{array} \tag{5}$$

The required parameters of Cu contacts are φ*eff* = 4.94 V, and *AR* = 120.2 A/(<sup>K</sup>· cm) 2 [66,70]. According to (6), the cathode temperature change is the result of the energy absorption and electron emission from the cathode, i.e., overcoming the space charge energy level, and the potential of the plasma ionization layer of positive ions (*Vion* = 15.5 V) near cathode regions [75].

$$-\overrightarrow{n}\cdot(-k\nabla T) = -\overrightarrow{f}\_{\text{elec}}\Phi\_{\text{eff}} + \overrightarrow{f}\_{\text{ion}}\cdot V\_{\text{ion}} + f\_{\text{e}}\cdot\Phi\_{\text{cathode}}\left(\overrightarrow{f}\_{\text{mar}\text{C}}^{\text{max}}\right) \tag{6}$$

According to (7) at the anode surface, the temperature change in the anode is a result of the absorption of electrons at the anode, i.e., overwhelming the energy level because of the accumulation of electrons near the anode.

$$-\overrightarrow{n}\cdot(-k\nabla T) = \left|\overrightarrow{f}\cdot\overrightarrow{n}\right|\cdot\Phi\_{eff}\left(\overrightarrow{f}\right) + \overrightarrow{f}\_{\text{el}}\cdot\Phi\_{\text{anndc}}\left(\overrightarrow{f}\_{\text{narr}A}^{\text{max}}\right) \tag{7}$$

<sup>Φ</sup>*cathode* <sup>→</sup>*<sup>J</sup>* max *nearA* and <sup>Φ</sup>*anode* <sup>→</sup>*<sup>J</sup>* max *nearC* are voltage drops (*Vd*) as a function of *J* over the fluid–contact interface without taking any physical aspect of the sheath into account [76].

### 2.4.2. Moving Mesh and Mesh Validating

As already noted, another simulation difficulty is fast movement. To consider the mechanical motion, Plexiglas walls and fixed contacts are modeled with fixed mesh, but the rest of the interior of FS is defined as a region with free deformation, and a mathematical lema named moving mesh [77] is used to model contact motion. The mesh around the contact moves with *uc*. The maximum and minimum sizes of mesh elements were selected as 1.06 mm and 15.2 μm, respectively, but as it is shown in Figure 4b, boundary layers of fixed contacts, as well as the edges of the fixed and moving contacts, are modeled by a fine mesh with a minimum/maximum size of 0.76/70 μm suitable for CFD.

**Figure 4.** (**a**) Moving mesh with 119,734 elements in 43.48 cm<sup>2</sup> at first and 161,164 elements at the end and (**b**) Extremely fine mesh defined around contacts and gap.

The areas between the sidewalls and the bottom of FS, as well as the linear boundaries parallel to the walls, are modeled by a fine triangular mesh with a maximum size of 150 μm.

To perform a more realistic fluid flow simulation next to the walls, a boundary layer mesh on all internal walls is defined. The final mesh has around 120,000 elements, as shown in Figure 4a. The mesh element quality is shown in Figure 4a is a dimensionless quantity between zero and one, which is an important aspect when validating a model. This measure is based on the equiangular skew. It penalizes elements with large or small angles as compared to the angles in an ideal element. Poor quality elements are considered for quality below 0.1 [78]. Contact movement causes the meshes to be squeezed and drawn, resulting in element quality reduction. Massive compression and stretch cause the model to be unstable and unsolvable. If the mesh overlapping reaches a predefined level, the solution is stopped, and the mesh is repeated considering the previous solution as the initial condition for the new meshed geometry.

In this way, an automatic mesh is generated almost every 120 μs up to 44 times in the course of the simulation, so that the output of mesh number 45 according to Figure 4a contains about 161,000 elements in 43 cm<sup>2</sup> with the same quality as the first mesh. High mesh density below and around moving contact, as well as next to the walls, is visible. Taking into account the equations modeled with this number of mesh elements leads to 460,000 degrees of freedom (DoF) in equations for the primary mesh, which finally reaches more than 635,000 DoF by increasing the number of meshes. The minimum mesh quality that has been ensured for each of the models that have been solved is 0.55, with an element area ratio of about 5.3 × 10−4. In general, the triangular mesh is a quick and straightforward way to obtain meshes of high element quality but comes with a di ffusion cost. However, the extra di ffusion can sometimes be desired, since it becomes easier to achieve convergence. A precisely defined triangular mesh specifying the distribution of mesh elements along an edge, in addition to Refinement of the Corner to decrease the element size at sharp corners, is perfect for our model, where the mesh serves as a starting mesh for mesh adaption. The obtained solution is then used to refine the mesh based on some indicator function. Finally, the adapted mesh is used to simulate the time interval again. After examining di fferent methods, the parallel sparse direct and multi-recursive iterative linear solver (PARDISO) [79], as a stable time resolution method, was used while the Jacobian matrix was updated at each step. Absolute tolerance of 5 × 10−<sup>4</sup> and at least 1 ps time steps at the start of the solution and maximum 2 μs is vital for modeling of *Varc* in the last 20 μs of the arc cycle accurately.
