Low-voltage circuit breakers provide essential protection for industrial and residential power installations, by taking advantage of the voltage drop at the electrode–plasma interface to force current zero. This is accomplished by using the magnetic force and unbalanced pressure on the arc as the contacts open to push the arc toward a stack of steel plates that break the arc into subarcs and thereby multiply the number of voltage drops. As the fault current can be high, substantial energy can be dissipated, which results in interactions among the arc and solid counterparts in terms of wall ablation and metal evaporation. In this study, ablation experiments are conducted to demonstrate its great influence on the arc voltage and on the pressure field. Significant progress has been accomplished in the computation of arc dynamics through the coupling of fluid motion with electromagnetics, although an important mechanism in arc breaking simulation, the effect of Stefan flow caused by species generation, has not been considered. We report out a numerical approach for taking into account the effect of Stefan flow, particularly for the breakers with high gasifying wall materials. This approach accounts for the diffusion induced convection due to added-in species from the evaporation surfaces, which will largely influence the flow field and the properties of the plasma mixture. Apart from the voltage drop, this mechanism plays an important role in simulating arc interruption. The ability of conducting Stefan flow computation further enhances the understanding of arc behaviors and improves the design of practically oriented low-voltage circuit breakers.

## I. INTRODUCTION

The function of a low-voltage circuit breaker (LVCB) in the power distribution system is to protect the electrical installations and to control and isolate the power supply in the electricity networks. The circuit breakers are designed to interrupt fault currents typically within half a cycle once the release mechanism is triggered. The contacts of most LVCBs operate in air, and the opening gap between the contacts increases from zero where the arc plasma starts to form. The initialized arc plasma inside the LVCBs will continue the fault current in the circuit, which is a great threat to the power system.

Comprehensive studies aiming to investigate the arc plasma properties,^{1} simulate the arc interacting with solid counterparts,^{2–4} explore the wall ablation and metal evaporation effect on arc dynamics,^{5–13} and reveal the physics of arc spot^{13–16} have been conducted in the past decades. Rong's group^{2,5,11,13,15,17} has extensively studied the arc spot modeling, wall ablation, metal erosion, ferromagnetic effect of steels splitters, and conducted relevant experiments for validation. Rong *et al.*^{11} modeled the electrode erosion by adding the mass source term in the sheath layer, within which the erosion rate was derived from the energy balance. The added-in copper vapor helps to replenish the properties of plasma mixture in the simulation. Jeanvoine and Muecklich^{18} used the finite element method to simulate the heat transfer and metal erosion, and presented the estimation of temperature distribution in the arc spot. This work proposed another approach to estimate the evaporation rate at the arc spot with Langmuir free evaporation theory;^{18,19} however, his calculation was confined within a solid domain. When it comes to wall ablation in LVCBs, Ruchti and Niemeyer^{6} investigated the ablation controlled arcs in cylindrical tubes theoretically and experimentally, concluding that ablation plays a critical role in controlling pressure, and the ablation rate can be scaled by factors such as current, geometry, and materials. Besides, Anheuser and Beckert^{20} treated the mass loss from the ablation wall by balancing the incident energy flux, namely, $m\u02d9hevp=Frad+Fheat$ ($Frad$ is the radiation heat flux and $Fheat$ is the conduction heat flux). In 2009, Ma *et al.*^{5} modeled the arc–wall interaction by applying the ablation at the boundary between the arc and the ablative wall and concluded that polymer vapor can contribute to fast arc motion.

However, for a multiple species system, the mechanism of describing how the species enters into the plasma mixture needs extra investigation for integrating the multiphysics modeling of arc plasma with multispecies transport. In 2004, Zhang^{21} suggested that vapor entering into the arc column with the initial velocity be determined by the vaporization rate and vapor density and proposed the momentum flux be produced by the vapor flux as $m\u02d92/\rho $. From the Stefan flow theory,^{22} another important factor, diffusion coefficient for the species vapor diffusing into the plasma mixture, should better be taken into consideration. To our best knowledge, in the existing literature, neither metal evaporation modeling nor wall ablation modeling has taken into account the effect of Stefan flow.

Generally, the intrinsic magnetic force and the uneven pressure field dominate the arc motion. Meanwhile, some heuristic techniques have been raised for accelerating the arc interruption, such as external magnetic field or wall ablation. For example, Shmelev and co-workers^{23,24} modeled the high-current vacuum arc under the transverse magnetic field, which strengthened the intrinsic magnetic force and accelerated the stepwise movements of arc. If the housing of the LVCB is partly made up of polymers that are easily gasified in high temperature environment, large volume of polymer vapor generation will tremendously increase the local pressure and alter the flow field which, consequently, accelerate the arc motion and expedite arc extinction.^{5,6,25} This technique is also expected to speed up the arc splitting and shorten the duration of arcing. All of these necessitate the implementation of accurate Stefan flow modeling in arc simulation.

To date, not much research related to the ablative material development and its effect on arc interruption^{5,6} has been conducted. As a supplement to this topic, we propose to use the surface reaction model with the effect of Stefan flow^{22} for simulating the evaporation process. Stefan flow is internally generated and can be presented in the absence of any externally imposed flow. As the polymer vapor continuously and increasingly diffuses from the reaction surface to the ambient gas, such diffusion will result in a net transport of mass, a convection process, which can be significant when intense vaporization takes place in hot environment,^{22} i.e., polymer wall ablation resulted from arc burning.

In this paper, the ablation tests are conducted to quantify the ablation effect on arc behaviors. Then, we fully consider the effect of Stefan flow for the species generation and transport in the LVCB model. Section II presents the model and methodology for ablation tests and simulation. The results of ablation test and corresponding simulation are compared and discussed in Sec. III. In Sec. IV, a simplified LVCB model is set up to illustrate the effect of Stefan flow on arc running. The concluding remarks are given in Sec. V.

## II. MODEL AND METHODOLOGY

To illustrate the wall ablation effect, we set up relevant tests to demonstrate the great influence of wall ablatives on arc behaviors by comparing the inner pressure and arc voltage with the results from nonablation materials. Besides, the test results are used to calibrate the simulation methods. For the simulation part, we postulate that the arc plasma is in localized thermal equilibrium (LTE),^{1,4} and calculations are based on continuum physics. The plasma properties are treated as a predefined function of the local environmental variables, such as temperature, pressure, and species concentration. Based on the experimental observations, Lindmayer *et al*.^{26} proposed that a voltage hump (ignition voltage) should be considered for the voltage drop at the electrode–plasma interface. Later, Mutzke *et al*.^{15,27} suggested a 0.1 mm thick layer of elements characterized by a current density dependent electrical resistance to ensure a certain voltage drop. The voltage drop is applied across the sheath as a function of local current density as shown in Fig. 2(a). In the stable stage in Fig. 2(a), the voltage drops for cathode and anode are 10 V and 3.2 V, respectively, and the peak (ignition) voltages for cathode and anode are 22.3 V and 7.16 V, respectively.^{15} The species evaporation is assumed to be gasified directly, without considering the transitional liquid state.

### A. Model

Here, we consider a wall ablation test setup composed of a support base, electrodes, a polymer tube, and a pressure sensor, as shown in Fig. 1. A damped L-C resonant circuit is designed to generate high current discharges. As shown in Fig. 1(a), the energy stored in a bank of 1 mF, high voltage (HV) capacitors, connected in variable serial and parallel configurations, is discharged through a 1 mH HV inductor by triggering a HV SCR switch (S38, Applied Pulsed Power with 4.7 kV peak off-state voltage, 14 kV peak nonrepetitive current) for producing discharge current waveform with a peak current of up to 2 kA and a half cycle time of 10 ms. The arc voltage and arc current are measured with a Texas DP3035 HV differential probe and a Tektronix A622 current probe, respectively. A piezoelectric transducer is used for pressure sensing.

When the sustained current increases to about 10 A, the fuse wire connecting the electrodes will melt and vaporize, resulting in the instant arc formation. Subject to intense heat radiation, the wall of the housing tube starts to ablate. To quantify the ablation effect, we compare the ablation test data between two different tubes, with one made of nylon polymer and another made of alumina (no ablation). The test results will be elaborated in Sec. III. Since the arc plasma typically extinguishes within several milliseconds once generated, it is extremely difficult to observe or measure the temperature, the Lorentz force, and the current density. In order to have a better understanding of arc behaviors, a 1:1 scale numerical model is set up for the purpose of validation, as shown in Fig. 1(c).

The arc survives about 10 ms. The measured arc current is plotted in Fig. 2(b). In order to compare the experimental results with the simulation, the measured current data are filtered for using as the current boundary condition for the simulation.

Based on the melting temperature of fuse wire, the arc is initialized by a predefined temperature and pressure fields. After that, the properties of the plasma mixture are updated at each time step by the local environmental variables.

### B. Mathematical methods

Modeling the arc in LVCB incorporates the fluid motion, radiation heat transfer, electromagnetic field solving, governed by the Navier–Stokes equations, radiative transfer equation (RTE), and electromagnetic equations, respectively. For LVCBs with evaporation, surface chemical reaction and species transport equations will also be considered and solved. In general, the simulation results are obtained by numerically solving the following equations:

- Mass conservation equation:where $\rho $ is the mass density and $V\u2192$ is the flow velocity. This equation needs to be satisfied in the whole computation domain, and there is no explicit source term needed. The generated species are added into the system by applying the species source (or species flux boundary).(1)$\u2202\rho \u2202t+\u2207\u22c5(\rho V\u2192)=0,$
- The species transport equations:where $Yi$ is the species concentration, $Di$ is the diffusion coefficient for species(2)$\u2202(\rho Yi)\u2202t+\u2207\u22c5(\rho YiV\u2192)=\u2207\u22c5(Di\rho \u22c5\u2207Yi)+Si,$
*i*, $Si$ is the species source term (please note that $Di\rho \u22c5\u2207Yi$ here does not invoke the summation convention). - Momentum conservation equations:where $vi$ is the velocity component, $xi$ is the coordinate component,(3)$\u2202(\rho vi)\u2202t+\u2207\u22c5(\rho viV\u2192)=\u2212\u2202P\u2202xi+\u2207\u22c5(\eta \u2207vi)+Smi,$
*P*is the local pressure, $\eta $ is the viscosity, $Smi$ is the momentum source term, including the Lorentz force and gravity. - The energy conservation equation:where $h0$ is the total enthalpy defined as $h0=i+P/\rho +12(vx2+vy2+vz2)$, in which(4)$\u2202(\rho h0)\u2202t+\u2207\u22c5(\rho h0V\u2192)=\u2207\u22c5(k\u2207T)+\u2202P\u2202t+Sh,$
*i*is the internal energy,*k*is the thermal conductivity,*T*is the temperature, $Sh$ is the general source term including the dissipation work done against viscous forces, radiation energy source, Joule heating, etc.^{28} - Ideal gas state equation, assuming high occurring temperatures in the fluid regime and none severe pressure increase:where(5)$P\u22c51\rho =R\u22c5T,$
*R*is the ideal gas constant.

So far, the aforementioned equations are in closed forms and can be solved numerically by the finite volume method. The following radiative heat transfer equation and electromagnetic equations will contribute to the additional energy source term and momentum source term, respectively, while they can be solved independently.

As mentioned before, radiation heat transfer will give rise to an energy source term to mitigate the temperature field, particularly for the system with intense burning like the arc plasma enduring high current. Thus, the radiative transfer equation needs to be accurately solved beforehand to decide the energy source term.

- The radiative transfer equation (RTE):where(6)$dI(S,\Omega )dS+(a+\sigma s)I(S,\Omega )=an2\sigma T4\pi +\sigma s4\pi \u222b04\pi I(S,\Omega i)\Phi (\Omega ,\Omega i)d\Omega i,$
*I*is the radiation intensity,*a*is the absorption coefficient, $\sigma s$ is the scattering coefficient, $\sigma $ is the Stefan–Boltzmann constant, and $\Phi (\Omega ,\Omega i)$ is the phase function.^{29}As the radiation dominates the heat transfer for the high current arc,^{4}the P-1 and discrete ordinates method (DOM) are suggested for accurately solving the radiation heat transfer.^{30} - As the time frequency of the input current is relatively low, the electromagnetic Maxwell's equations are solved under quasistatic condition, i.e.,$\u2207\u22c5(\sigma \u2207\Phi )=0,$$\u22072A\u2192=\u2212\mu j\u2192,$$E\u2192=\u2212\u2207\Phi ,$where $\Phi $ is the electric scalar potential, $A\u2192$ is the magnetic vector potential, and $B\u2192$ is the magnetic flux density, $E\u2192$ is the electric field intensity, $\sigma $ is the electrical conductivity, $\mu $ is the magnetic permeability (assumed as the vacuum permeability).(7)$B\u2192=\u2207\xd7A\u2192,$

The chemical reaction rate *R* of the wall ablation is calculated by $R=qflux/hpolymer$, where $qflux$ is the radiation flux plus conduction flux to the ablative wall. For copper evaporation, $R=qvap/hcu$, where $qvap$ is the heat allocated for copper evaporation calculated based on Rong's suggestions.^{11}

### C. Arc plasma property and Stefan flow

The thermodynamic properties of plasma at the given temperature and species concentration are calculated under LTE by minimizing Gibbs free energy.^{11,31,32} In this study, there are two kinds of species considered: Nylon 66 polymer and copper vapor. Here, we plot thermal conductivity and electrical conductivity for plasmas of interest to illustrate the influence of the species concentration on the properties of the plasma mixture.

Generally, the species distribution in the computation domain is decided by the generation rate and species transport. Another important factor on the species distribution is the diffusion induced convection, namely, the Stefan flow. In any species transport system, depending on the operating condition, the species vapor transport can be dominated by either diffusion or convection. As is known, the species convection is usually driven by the externally applied pressure gradient. Another type of convection, called Stefan flow,^{22} is internally induced and can present in the absence of the externally imposed gas flow. When generated from the substrate, the species diffusion produces a net mass flow, a convective flux. The magnitude of this convection flux depends on the generation rate and diffusion coefficient. Species generation rate is affected by the local pressure, temperature, or energy input, etc. The diffusion coefficient $Di$ in Eq. (2) can be complicated when the thermal, multicomponent diffusion is considered. For the full multicomponent diffusion modeling, the binary mass diffusion coefficient $Dij$ should be used, which specifies species *i* diffusing in species *j*. In this work, we adopt a constant diffusion coefficient $Di$ for simplicity.

If the species diffusion coefficient and concentration gradient are known, the so-called Stefan flow velocity at the reaction surface can be analytically derived from the species transport equation, expressed as^{22}

where $Di$ is the diffusion coefficient for species *i* and $V\u2192n$ is the velocity normal to the chemical reaction (or evaporation) surface. This corrected Stefan flow velocity will be implemented in the reaction boundary for wall ablation computation (as shown in Fig. 9).^{22}

The velocity given by the Stefan flow for surface evaporation is self-contained when applying the surface flux instead of the source term, as shown in Fig. 9. After the calculation of the species generation rate, species flux can be deduced and applied to the evaporation surfaces with the equivalent velocity given by Eq. (8).^{22}

## III. RESULT ANALYSIS

In this section, the results of ablation tests and simulation will be discussed to illustrate the ablation effects. For both nylon and alumina tubes, the ablative tests were conducted many times. The test data with good repeatability are used to minimize testing variation.

### A. Experimental result analysis

As an important factor in the circuit systems, the arc voltages are measured for both tests, as plotted in Fig. 4. The measured arc current for both cases is roughly the same as shown in Fig. 2(b). It should be noted that the testing conditions are identical for the two cases except for the wall materials, e.g., nylon or alumina.

Figure 4 shows that the wall ablation has a huge influence on the arc voltage and inner pressure. The measured arc voltage with ablation (nylon tube) is generally higher than that without ablation (alumina tube), and the ablation leads to higher arc voltage until arc extinction. These phenomena can be explained by the temperature-dependent properties of the arc plasma. As shown in Fig. 3(b), the electrical conductivity decreases as the temperature drops. This does happen when wall material ablates and absorbs heat intensively. As mentioned before, the arc current is determined partially by the source impedance of the external circuit. Therefore, if the arc resistance increases as a result of temperature decrease, the arc voltage that equals resistance times current will also increase. In Fig. 4(a), as the ablation starts, the arc voltage jumps to a much large value, probably because of the threshold for wall ablation to blast (for instance, the polymer wall will not ablate until its surface temperature increases to the decomposition temperature). Moreover, it takes some time for polymer vapor to diffuse into the arc column.

Figure 4(b) clearly shows that the nylon ablation increases the inner pressure by about 0.8 bar in comparison with the nonablation case. For nonablation tests (blue line), the measured inner pressure also increases to a large value at the initial stage and then quickly fades away. This fleeting high pressure is mainly caused by the high temperature of arc formation. Later as the temperature field becomes stable, the pressure will gradually become balanced by the environmental pressure. For the ablation tests, the generation of polymer vapor increasingly and continuously contributes to the inner pressure until the arc extinguishes. This also demonstrates the great amounts of polymer vapor generated in ablation and necessitates the application of Stefan flow in simulating species transport.

### B. Simulation result analysis

In order to fully understand the arc–wall interaction, we set up an identical model with the wall ablation (nylon tube) for the simulation. The simulated arc voltage and gauge pressure are plotted in Figs. 5 and 6, respectively, showing roughly matched results to the test data. For the arc voltage, the simulation results are much smoother than the test. The simulated arc voltage soars after initialization, partly due to the fact that the arc column is initialized by a predefined temperature and pressure fields. As the overall temperature increases due to Joule heating, except for the ablation wall and metal evaporation spot, the electrical resistance decreases in the arc column, followed by the decrease of the arc voltage. After the initial stage, the simulated arc voltage somewhat fluctuates, i.e., going up at 0.002 s then going down at 0.005 s. The possible reason is that the generated species diffusing into and out of the arc column will change the properties of arc plasma. At the last stage, the arc voltage decreases as a result of the decreasing input current (the resistance increase cannot compensate for the current decrease, leading to the overall voltage decrease). Figures 5(b) and 5(c) show that the voltage (electric potential) contour, where the bottom is the cathode, which is earth grounded, and the top anode has the maximum calculated electrical potential. Also, the voltage gradient indicates the path that the current will go through.

Although the simulation model is axially symmetrical, the results deviate to nonsymmetrical after hundreds of time steps (about several milliseconds), probably due to symmetry breaking under fluid turbulence that a small perturbation happened will cause nonsymmetric results, which is also found in Rau's simulation.^{33}

The simulated and measured inner pressures are compared and plotted in Fig. 6(a). Perhaps due to the assumed initial conditions, the simulated inner pressure soars to a very large value after initialization (about 2 bar). Since the ideal gas law $(P/\rho =RT)$ defines the implicit relation between pressure and temperature, the initial condition gives rise to the pressure soar (for the predefined arc, how to give a perfect initial condition is challenging and remains to be solved in the future). The high initial pressure gradually recedes as the pressurized gas escapes from the top and bottom openings.

Figures 6(b) and 6(c) indicate that the inner pressure is primarily influenced by the polymer ablation as the maximum pressure is located near the ablation site. When the polymer vapor is evaporated and injected into the arc column with the so-called Stefan flow, the local fluid flow will be dramatically altered by the species mass flow. Figure 7 illustrates the phenomenon of Stefan flow for both wall ablation and metal evaporation. The projected flow velocity shown in Figs. 7(b) and 7(c) clearly demonstrates the initial velocity given by Stefan flow, as the species comes out from the reaction interface.

When the temperature increases to or above 15 000 K, the electrical conductivity of the plasma gas concentrated by polymer vapor is higher than that concentrated by copper vapor, as shown in Fig. 3(b). The generated species vapor will directly inflate the adjacent space. For example, the space adjacent to the polymer wall will be filled by polymer vapor and thereby has lower electrical resistance. Thus, the current will inevitably bend to the polymer wall, which will further intensify the ablation and lower the local resistance and then let current bend more. At a critical point, the local wall ablation rate will increase large enough to have a strong mass flow departing normally away from the ablation wall, which will blow the arc back to the axis.

The wall ablation provides mass flow of species along the direction as indicated by the velocity arrow in Fig. 7. Similarly, the mass flow of the copper vapor will cause the mass flow normal to the electrode surface and transport the copper vapor into the arc column. Both phenomena will update the plasma properties continuously as the species concentration changes.

## IV. SIMULATION RESULTS OF A SIMPLIFIED LVCB MODEL

As demonstrated by both the wall ablation experiments and the simulations, a great amount of polymer vapor evaporated from the ablation wall has a remarkable influence on the pressure field. With the help of Stefan flow theory describing how polymer vapor enters into the complex system, the flow velocity adjacent to the ablation wall can be accurately simulated. Here, in order to illustrate the Stefan flow effect on arc running, we simulate the arc motion in a classical low voltage circuit breaker model that composed of copper runners, steel splitters, ablation wall, and air chamber as shown in Fig. 8(a). The metal–gas interface is treated as a very thin layer within which the voltage drops are applied. The mesh is densified adjacent to sheath layers to capture the large gradients of electric potential and temperature field, so as to avoid numerical instability.

For this LVCB model, an empirical arc is initialized by giving the predefined temperature and pressure fields as shown in Fig. 8(b), instead of opening contacts to extract an arc (which requires the application of dynamic meshing). The initial pressure field is used to balance the magnetic pinching force, which is determined by the initial input current. Therefore, initial pressure is adjusted according to the initial input current.

As the peak of input current reaches thousands of amperes, the results show that the maximum current density can be ∼10^{8} A/m^{2}, and the volumetric heat source given by Joule heating could be larger than 10^{12} J/m^{3}. Under this circumstance, radiation will dominate the heat transfer.

### A. Effect of wall ablation on flow velocity

The species concentration and projected flow velocity are plotted in Fig. 9, which shows that the species generation does amplify the velocity magnitude along the species concentration gradient.

From Fig. 9, the location of maximum copper vapor concentration implies the arc spot on the electrode surface where solid copper first melts and then gasifies into vapor. As the fault current can be very high, the evaporation rate of copper can be very intense in real situations. In addition, the ablation wall can be or designed to be (a potential design strategy) of large surface area. As Stefan flow enables the generated vapor species to obtain initial flow velocity, the generated species vapor can dominate the formation of local flow field, which assists to carry and push the arc toward the steel splitters for arc quenching and interrupting. From another perspective, the ideal LVCB design will fulfill the safe, thorough, and robust arc interruption repeatedly. Wall ablation absorbs considerable amount of heat and generates a great amount of polymer vapor which increases the local pressure and, subsequently, drives the arc movement, thereby expediting the arc extinction and preventing further damage of stationary arc burning.

On the other hand, the obtained initial flow velocity accelerates the species dispersion into ambient gas. The mixture properties will be correspondingly updated, which in turn impacts the arc dynamics and interruption. For example, the dispersed copper vapor will dramatically alter the plasma properties, such as electrical conductivity and absorption coefficient. Specifically, the copper vapor will enhance the electrical conductivity at relatively low temperatures, thereby increasing the local current density and strengthening the Lorentz force proportionally to drive a faster arc motion, as illustrated in Fig. 10(b). Similarly, polymer vapor concentration will greatly influence the thermal conductivity. Generally, the back-striking phenomena of arc will be less likely when a great amount of polymer vapor is generated.^{5} Therefore, precise computation of the species distribution stands a central role in high fidelity arc modeling.

### B. Driving force for arc motion

At the initial stage, as the input current is relatively low (≤500 A), the pressure is the dominant driving force. The unbalanced pressure field stirs up a shock wave that propagates to the chamber wall and outlets. The outlets are open to air and will mute the incoming pressure wave. At the chamber wall, the incoming shock wave will be bounced back. Consequently, the shock wave leads the arc column to move slightly back and forth. This motion pattern subsequently disappears once the Lorentz force increases to be the dominant driving force as the input current increases to a large value. As seen from Fig. 10, both the pressure field and Lorentz force (the component along the X axis) will drive the arc movement to the splitters.

As the current increases, both the pressure and Lorentz force will increase accordingly and push the arc toward splitters as expected. The wall ablation effect is illustrated in Fig. 10(a) by the relatively large pressure field where wall ablates. The red and blue colors in Fig. 10(b) bound the arc column indicate the magnetic self-pinching force, and the resultant of magnetic force is positive to drive arc movement to the right. By virtue of the nonlinear electrical conductivity in the sheath layer, the cross section of the arc column is gradually constricted to a much smaller spot. As the result of ignition voltage drop applied at the frontier of arc spot, the arc presents stepwise movement along the electrode surfaces, as shown in Fig. 11(a) with two arc spots with reduced cross section coexist on the cathode surface. The maximum current density is above 10^{8} A/m^{2} at the cathode arc spot. The voltage drop is indicated by the color change at the electrode surface on the electrical potential contour, as shown in Fig. 11(b).

The arc column continuously moves forward and slants toward the front electrode surface and then heats up surface mixture gas. Finally, with the improved electricity conductivity in front of the old arc spot, the new current path is created along with the new arc spot, as indicated by Fig. 11(a). Besides, multiple current spots can be observed on both anode and cathode surfaces, which is consistent with the conclusion given by Mutzke.^{27}

### C. Arc voltage validation

The arc voltage and arc current measured from a real LVCB, when a fault current happens, are plotted in Fig. 12(a). The measured current curve is roughly a half sinusoidal curve, which starts to plummet from its peak once the measured arc voltage reaches its maximum value. Meanwhile, the measured arc voltage enters a plateau stage. This plateau stage can be partly interpreted by the fact that, when the arc enters, suffuses, and bypasses the steel plates, the voltage drop will develop and diminish on splitter surfaces simultaneously. Thus, the overall voltage drop remains approximately constant. For example, for a LVCB with a total of 7 steel splitters, ideally the arc column should be cut into 8 subarcs, which subsequently introduces 7 additional voltage drops for both the cathode and the anode. However, seldom will the arc current go through the 7 splitters concurrently. Instead, some splitters may be bypassed by the arc plasma. This phenomenon probably results from several factors: (1) the voltage drop gives rise to an effective resistance at arc-splitter interface, which prevents the current going through the splitters, because the current will go through the path with the lowest resistance if it has multiple choices (such as going through the arc plasma or short-cutting splitters). (2) The temperature field in front of the splitters may be high enough that the gas therein becomes electrically conductive and let the current bypass the splitters. And the local temperature is influenced by the conjugate heat conduction, flow motion, and radiation. (3) Splitter geometry may influence the arc splitting. If the splitter size is too small, it will probably be bypassed by the arc due to the thermal gas diffusion and transport. (4) The outlet boundary condition has a certain influence on the flow field and further affects the temperature field and species distribution.

These factors may be coupled together to make the process of arc interruption complicated. The simulation methods of arc dynamics remain to be complemented and improved in the future. Once the current decreases to zero, the arc extinguishes, and the arc voltage suddenly drops to zero, as shown in Fig. 12(b).

The measured current is smoothed and is used as the boundary condition for the simulating electromagnetic fields. Here, we suggest starting the simulation with relatively low input current, as shown in Fig. 12(a). The initial input current will increase to match the measured current after 1 ms. The computed arc voltage roughly matches the measured arc voltage as a validation of the simulation method. The fluctuation of the simulated arc voltage requires further improvement of the simulation method.

## V. CONCLUSION

In this study, ablation experiments have been designed and conducted to illustrate the wall ablation effect on arc behaviors. The comparison between polymer tube and alumina tube demonstrates that ablation has a great influence on the arc voltage and on the inner pressure. Great heat consumption by ablation lowers the average temperature and contributes to a higher arc voltage. Besides, wall ablation generates a great amount of polymer vapor that amplifies the local flow field and blows the arc away from the species generation site. A CFD/MHD model is set up for capturing the essence of Stefan flow, which not only renders good agreement with the experimental results but also illustrates the important role of diffusion induced convection from the species generation interface. The development and application of high-ablative materials can offer a novel design strategy of LVCBs for fast arc interruption.

Later, in order to illustrate wall ablation on an arc running, a simplified LVCB model is set up with a fully implemented module to incorporate the effect of Stefan flow. The results suggest that the species generation by polymer ablation and/or metal evaporation has a profound influence on arc running and interruption. The intensified flow field enhances the dispersion of species, and subsequently alters the properties of the plasma mixture. Therefore, it is crucial to properly and accurately incorporate the generation and transport of species in arc simulation. In addition, the great amount of species generated will influence the local pressure field. The unbalanced pressure field acts as the initial driving force for arc motion. Later, as the fault current reaches thousands of amperes, the Lorentz force can be outstanding and dominant, so as to pinch and drive the arc toward the splitters.

Thus, incorporating the Stefan flow effect can help to accurately predict the duration of arc interruption in the LVCB. In short, effects of Stefan flow on the arc simulation are embodied in the following aspects: one is to update the composition of plasma mixture dynamically and then affect the arc behaviors; another one is to accelerate the arc motion toward extinction. Both factors are expected to expedite arc to a complete interruption and help to improve the understanding of arc dynamics and arc–solid interactions.

## SUPPLEMENTARY MATERIAL

See the supplementary material for the detailed model configurations and additional simulation results.

## ACKNOWLEDGMENTS

This work was partially supported by the National Science Foundation (NSF) under Grant No. 1650544 and by the GE’s Industrial Solutions Business Unit under Grant No. 6205520. The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of Industrial Solutions or UConn.

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