Arc Modeling Challenges

Rümpler Ch. et al.: Arc Modeling Challenges Arc Modeling Challenges Rümpler Ch.1, Narayanan V.R.T.2 1Eaton, 1000 Cherrington Pkwy, Moon Township, PA 15108, USA 2Eaton, Bořivojova 2380, 252 63 Roztoky, Czech Republic ChristianRuempler@Eaton.com Modeling of arcing phenomena has evolved towards becoming a state-of-the-art tool, supporting the design process of power distribution equipment in low-, medium-, and high-voltage applications. Modeling provides a better understanding of the physical processes within the devices which is needed in order to enhance product performance and mitigate risks in the development cycle. In this contribution, modeling challenges related to some of these applications are discussed: a) the calculation of thermodynamic and transport properties, b) the modeling approach for contact arm motion during arc interruption in low- voltage molded case circuit breakers (MCCB’s), c) the model approach for arc flash events in medium- voltage (MV) switchgear. Keywords: arc simulation, plasma transport properties, circuit breaker, switchgear 1 INTRODUCTION eling approach for arc flash in medium- Arcing phenomena are crucial in various ap- voltage switchgear in section 5. plications in the power distribution system. Arcs are used either as a switching element, 2 SIMULATION APPROACH e.g. in circuit breakers with electromechanical A modeling approach has been developed that contact systems or arcs occur as a fault event, covers the highly nonlinear physical processes e.g. as arc flash in low-voltage or medium- during high-current arcing [2]. voltage switchgear. Due to the complex nonlinear processes associated with arcing phenomena, modeling has several useful aspects. A major factor for increased modeling activi- ties in this field is the drive for miniaturization and cost reduction, which leads to increased power density and performance requirements. Here the model driven approach can help to optimize the design towards increased performance. Another aspect is the expensive testing done during development and certification. Having better confidence about the design helps to reduce expensive design-test cycles. In this paper we discuss some of the challeng- ing aspects related to the modeling of arc interruption and arc flash phenomena. Basis for any modeling approach are the necessary input data. In our MHD (magneto-hydrodynamics, [1]) based approach which is introduced in section 2, the thermodynamic and transport properties of the plasma are crucial input data. Fig. 1: Simulation system The calculation method and some results are discussed in the 3rd section. Furthermore, two The set of partial differential equations (PDEs) application specific problems will be dis- that describes the fluid flow, electromagnetic cussed, the contact arm motion modeling ap- field, and radiation transport, respectively, are proach for low-voltage molded case circuit solved in a code coupling approach which is breakers (MCCB’s) in section 4 and the mod- based on customized commercial solvers. Plasma Physics and Technology 2015, 2, 3, 261-270 Rümpler Ch. et al.: Arc Modeling Challenges Splitting the calculation problem into separate used alloy for metallic electrodes. The LTE- tasks for fluid flow and electromagnetics ena- based transport properties are calculated fol- bles a high performance solution. The basic lowing the works of Devoto [6, 7], in which setup of our simulation system is shown in fig. the Chapman-Enskog formulation [8] for pure 1. gases was extended to consider partially ion- The mass, momentum, and energy balance ized and fully ionized gases. In this formula- equations as well as radiation equations are tion, all the transport properties of interest can solved using a finite volume approach (FVM) be obtained as a ratio of determinants of two by means of the ANSYS/Fluent solver [3]. matrices and the size of the matrices depends Several sub models have been implemented as on the desired degree of accuracy (ζ) of the enhancements of the fluid flow (CFD) code, properties. using the user programming interface the (푙,푠) Calculation of collision integrals 훺푖푗 is an in- solver offers: tegral part of the Chapman-Enskog method, An ordinary differential equation (ODE) can which requires the solution of equations (1)- be solved in order to describe the interaction (3). with the electric network that is connected to ∗ 푘푇 ∞ 2 the modeled device. (푙,푠) 푖푗 −훾푖푗 2푠+3 (푙) 훺푖푗 = ( ) ∫ 푒 훾푖푗 푄푖푗 푑훾푖푗 (1) A model for the erosion of electrode material 2휋휇푖푗 0 is used to calculate the mass of metal vapor Here the superscripts (l,s) are related to coeffi- that is ejected into the fluid region. cients of the Sonine polynomials. The integral Another sub model predicts the ablation of th plastic materials [4]. collision cross section of l degree, associated In case of MCCB modeling an ODE can be with the collisional dynamics between “i-j” solved that predicts the rotational or transla- pair of species and dependent on the relative tional motion of the contact arm, driven by speed of collision g, is expressed as magnetic forces. ∞ 푄(푙)(푔) = 2휋 ∫ 휎(푔, 휒)(1 − 푐표푠푙휒) 푠푖푛휒푑휒 To describe the electro-magnetic processes, 푖푗 the finite element approach (FEM) is used. 0 ∞ 푙 = 2휋 (1 − 푐표푠 휒) 푏푑푏 , (2) Therefore the ANSYS/Emag [3] solver is cus- ∫0 tomized, dividing the solution process into where 휎(푔, 휒) is the differential collision two steps, the solution of the electric and the cross section, b is the impact parameter, and χ magnetic equations. For the description of the is the angle of deflection given as arc attachment, a user defined element has ∞ 푑푟 been implemented that is able to represent the 휒 = 휋 − 2푏 ∫ . (3) 푟푚푖푛 2 훷 plasma sheath voltage drop [2]. 2 푏 푖푗 푟 √(1− 2−2 2) The necessary data exchange and interpolation 푟 휇푖푗푔 between the two mesh based codes is provided by the coupling server MpCCI [5]. The most important parameter in equation (3) representing the collision dynamics is the in- 3 PLASMA TRANSPORT PROPER- teraction potential Φij of the collision, which TIES depends on the interaction partners i and j. In In order to numerically model the arc dynam- this work, while the elastic neutral-neutral and ics within complex circuit breaker and switch- ion-neutral interactions have been character- gear geometries, accurate thermodynamic, ized by the phenomenological Lennard-Jones transport and radiation properties of relevant potential [9], the shielded-Coulomb potential gas mixtures are required as input. In this has been utilized for charge-charge collisions. work, we primarily focus on calculating the Additionally, the contribution from inelastic thermodynamic and transport properties of air- charge-exchange interactions has been includ- metal vapor mixtures. The results will be pro- ed for ion-parent neutral collisions using the vided for air-copper and air-tungsten mixtures, approach suggested by Devoto. Finally, elec- since copper-tungsten (Cu-W) is a commonly tron-impact momentum transfer cross sections 262 Rümpler Ch. et al.: Arc Modeling Challenges data are obtained from relevant databases for are lower than those for 50% and 100% W re- the electron-neutral interactions. The reduced spectively. ∗ mass 휇푖푗, reduced temperature 푇푖푗 and dimen- sionless kinetic energy 훾푖푗 mentioned in equations (1)-(3) are given by Increasing %Cu, %W 푚푖푚푗 휇푖푗 = 푚푖+푚푗 −1 ∗ 1 푚푖 푚푗 푇푖푗 = [ ( + )] 푚푖+푚푗 푇푗 푇푖 1 2 휇푖푗푔 2 Increasing %Cu, %W 훾푖푗 = ( ∗ ) . (4) 2푘푇푖푗 In this section, we provide the transport prop- Fig. 2: Plot of electrical conductivity vs. tempera- erties results for air-copper and air-tungsten ture for different mole fractions of either Cu or W mixtures at atmospheric pressure, for different in air atmosphere proportions of copper and tungsten vapors. The results for air-copper mixtures have al- ready been validated in a previous work [10], while published data does not exist for air- tungsten mixtures to the best of the authors’ knowledge. Fig. 2 depicts the variation of the electrical Increasing %Cu, %W conductivity σ with temperature for different mole fractions of copper and tungsten in air at 1 bar pressure. The following observations can be made: i. The addition of Cu or W results in a sig- nificant increase in σ at temperatures Fig. 3: Plot of viscosity vs. temperature for differ- lower than 15 kK compared to pure air ent mole fractions of Cu and W in air (the red 0% Cu curve). The viscosity peaks are associated with the ii. Above 15000 K increasing content of onset of ionization and subsequent creation of copper and tungsten decrease σ compared electrons and ions, thereby increasing the to air. charge-charge cross section and reducing the iii. Tungsten content intensifies the behavior mean free path. The importance of dynamic in both temperature ranges resulting in viscosity in arc simulations appears through lower conductivity than copper for high Reynolds number (Re), which is the ratio of temperatures and higher σ for tempera- inertial and viscous forces. For a given geome- tures below approximately 15000 K. try, a critical Reynolds number (Recrit) can be The observation (i) is of concern, since the in- determined based on simplifying assumptions. crease in σ at lower temperatures due to metal If the Reynolds number exceeds the critical vapor increases the joule heating effect and limit (Re > Recrit), a transition from laminar to thereby can contribute to a thermal breakdown turbulent flow occurs. Turbulent flows are close to current zero in a circuit breaker. well-known to drastically increase heat and The variation of dynamic viscosity with tem- mass transfer, and hence can efficiently cool perature for different copper and tungsten con- and extinguish the arc. tent is shown in fig. 3. The viscosity peaks for Finally, the variation of total thermal conduc- metal vapor mixtures are lower than that for tivity with temperature for different copper pure air, while those for 50% and 100% Cu and tungsten fractions is presented in fig.

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