*2.1. Model Assumptions*

Though lots of the published investigations on MHD simulations for arcs in low-voltage miniature circuit breakers (MCB) and medium or HV SF6-CBs are mentioned, there is almost no investigation on MHD simulation of FEAs because of some issues and simulation di fficulties [42].

Increasing the fluid velocity, *U*, increases the ratio of advection to thermal di ffusion (Peclet number). For the heat equation, this condition necessitates the use of numerical stabilization and a finer mesh. Reaching convergence on simultaneous solving of heat transfer and laminar flow (LF) equations, especially at high velocities and for gases with low viscosity, is a big challenge, as these two physical concepts are acting against each other. Sometimes, time step reduction can help, but it becomes more complicated if the generated heat increases by increasing the arc current.

Tiny meshes in the presence of a moving part result in distorted mesh elements in a few intervals. To overcome the mentioned issues and trade-o ff the accuracy, time, and complexity, 2-D arc simulations are used. When there are no strong vortices inside the flow, 2-D simulations are accurate enough [43]. This assumption has some e ffects on the gas flow that will be discussed later. In local thermodynamic equilibrium (LTE), which implies local chemical as well as thermal equilibrium [44], the plasma can be considered as a conductive fluid mixture and, thus, be modeled using the single-flow MHD equations instead of two separate flows of electron and ions. The very thin-state plasma flow can be modeled as laminar and the e ffects of symmetry assumption on asymmetric turbulent flows (TF) are ignored due to smaller length scales in Reynolds number (Re).

### *2.2. Experiment Setup and the Numerical Geometry*

Figure 1a shows the elements of the prototype FS. The electromagnetic actuator of CB used in this study consists of a spiral coil in multi-layer formation, which is connected to an L-C current source through a fast-closing switch. The energy storage device consists of a capacitor bank and is connected to the current discharging coil.

**Figure 1.** (**a**) Elements of prototype fast switch (FS) and Thomson coil, (**b**) 2-D geometry, andinitial conditions.

Moving contact (MC) is made from aluminum (Al), while the fixed contacts are made of copper (Cu). The initial gap between the electrodes is 2 mm. The wall of the chamber is made of transparent Poly-methyl-methacrylate (PMMA), known as plexiglass. The length of the arc chamber is 11 cm, and there are two pairs of open hatches at the bottom and the top of the chamber, as shown in Figure 1a. The moving contact is accelerated by the repulsive force generated by a Thomson drive. An appropriate two-dimensional geometry of FS, including the initial conditions, is shown in Figure 1b. The gravity acts in the negative direction of the X-axis. The sharpened edges of the contacts are bent. Avoiding sharp edges is essential in solving the flow and the heat transfer equations because the sharp

points generate a very high current density in terms of numerical resolution, which results in very high-temperature spots, leading to an unrealistic heat and fluid flow. The gaps between fixed and moving contact and areas around contacts are the points with intense changes in the mesh shape, as well as in the value of the heat and fluid variables. Therefore, these areas are modified by two parallel lines, and rectangular boundaries are defined around the fixed and moving contacts as hypothetical boundaries, as shown in Figure 1b. These hypothetical boundaries allow for a smaller mesh in these areas, and secondly, help to solve the moving mesh by path definition. To calculate the arc parameters in the FS model, predefined variables must be calculated in the specific regions of each arc. Therefore, another hypothetical axial boundary is defined in the center of the model (y = 0), as shown in Figure 1b, which outlines the scope of the definition of variables for two series arcs in FS. Studies have shown that 2-D axisymmetric cannot be applied, even with significant simplifications because of two reasons:


### *2.3. Properties of the Material*

### 2.3.1. Gas (Air-Al Plasma)

Thermal plasma properties of air-Aluminium vapor mixtures in [45] show that even a small amount of metal vapor at atmospheric pressure has an appreciable influence on the radiation and the electric conductivity (σ), but a negligible effect on the other features. The effect of metal vapors on σ is dominant in low-current arcs while the impact on radiation dominates at high currents [46] due to higher temperatures (T).

Plasma conductivity in the presence of 1–3% metal vapor in 11,000–13,000 K, is about 20–30% higher than of a pure air plasma [47,48], which is modeled here. When the current is lower than 1 kA, metal vapor resides only in two small regions (1–3 mm) in front of the two electrodes because of limited electrode vaporization [46,49]. Consequently, the arc temperature drops significantly in these small regions near the cathode [50,51]. Then, arc voltage (*Varc*) in low current mode will not be sensitive to the presence of metal vapor [49,51].

If absorption is ignored (thin layer) then metal vapor increases the NEC, but if absorption is considered then the presence of Al vapor despite Cu reduces the NEC in the temperature range of 10–15k K, if the Al mixture ratio is less than 5% [45] and arc radius is less than 5 mm, which applies to our study. The NEC decreases by an increase in the arc radius [52]. The reported results are utilized to obtain the thermodynamic properties of hot air, including heat capacity, viscosity, density, thermal and electrical conductivity, as well as NEC for the temperature range up to 25,000 K at constant atmospheric pressure [53,54] and is calculated for higher pressure (20 bar). Although the exact formulation of the energy transport by radiation is very complicated, the three mostly used methods are P1, method of partial characteristics, and NEC.

A comparison between these methods is discussed in [55]. The NEC has acceptable accuracy and less entanglement in the range of the optical thicknesses and arc temperatures, (10–15k K and 2–6 mm) [56]. The Air-Al mixture and NEC absorption effect in the presented model will be explained here.

The core diameter varies along with its length at different times. This variation is shown in Figure 2a.

**Figure 2.** (**a**) Variable core diameter along with its length, (**b**) Maximum of core radius, (**c**) Distribution of *T* and metal vapor along the core center, and (**d**) net emission coefficient NECair-Al in arc chamber for 200 *Apeak*.

As it is shown in Figure 2b, the maximum radius of the core varies between 1 mm and 3 mm. So, the radiation absorption radius, *Rp*, for the NEC model is assumed 2 mm. The ablated mass for Cu electrode in low currents is calculated around 100–300 μg/C [57] and in our stationary study was around 316 μg/C [51]. Although the melting point of Al is half of Cu, its latent heat of evaporation is twice the Cu. Assuming a similar ablation rate will result in 150–300 × 10−<sup>3</sup> cm<sup>3</sup> Al vapor. By utilizing the particle tracing method [51], the ratio of Air-Al is calculated and it is shown along with the core center in Figure 2c, which complies with other studies [49]. According to [58], the influence of changing the arc radius from 1 mm to 10 mm on NEC for air plasma in the temperature range of 7–15k K is not significant, and its gradient by increasing the arc radius is negative. So, finding an average radius for the arc gives an accurate estimation of NEC. But according to [45], the influence of metal vapor percentage is enormous at this temperature range. So, metal vapor has been considered, and the arc radius was averaged. Based on the reported NEC of air-aluminium mixture [45] and temperature distribution in Figure 2c, NEC in the arc chamber is calculated through recursive studies to minimize the relative error.

The result is shown in Figure 2d. The volumetric proportion of Al-vapour in a 2-mm gap between electrodes is more than 90% at the start of the arc. Particles have Maxwellian speed distribution, and contact is moving, which results in FEA. So, particles disperse along the arc length, and the mixture ratio decreases as time elapses. Most of the Cu particles remain near the fixed cathode while Al particles are dispersed by moving cathode displacement, as velocity inside the core is around 50 m/s at 0.5 ms but falls to 5 m/s at 1.5 ms. So, the mixture ratio of Al falls to below 5% in an arc core.

The main parameters of the utilized model are the transport and thermodynamic properties of gas mixtures, which would be obtained from the articles or could be calculated through chemical physics [59,60] for pressures and mixtures other than what is reported in the articles. Software packages handle these chemical physics calculations. Here, we used the PLASIMO® package [61], and the electrical conductivity, total heat conductivity, mass density, NEC, viscosity, and heat capacity of pure air and air-Al mixture at 1 and 20 bar at different mixture ratios and pressures are presented in Figure 3. The legends for all sub-figures are the same as Figure 3a.

**Figure 3.** (**a**) Electrical conductivity, (**b**) Heat conductivity, (**c**) Mass density, (**d**) NEC, (**e**) Viscosity, and (**f**) Heat capacity of pure air and air-Al mixture at 1 and 20 bar at different mixture ratios.

The changes in the pressure are very small in this study so its effect on vaporizing temperature is ignorable, but it could be considered as it was explained in the above through assuming a two-phase thermodynamic system and utilizing the method explained in [62].
