1 Introduction. PACS: Kq, y, Kj, a, Cv, Fv DOI: / /14/11/07
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1 Plasma Science and Technology, Vol.14, No.11, Nov Three-Dimensional Simulation of Plasma Deformation During Contact Opening in a Circuit Breaker, Including the Analysis of Kink Instability and Sausage Instability Vahid ABBASI 1, Ahmad GHOLAMI 1, Kaveh NIAYESH 2 1 High Voltage and Magnetic Material Research Center, Electrical Engineering Department, Iran University of Science and Technology, Narmak, Tehran , Iran 2 School of Electrical and Computer Engineering, University of Tehran, IR Tehran, Iran Abstract A three-dimensional (3-D) transient model has been developed to investigate plasma deformation driven by a magnetic field and its influence on arc stability in a circuit breaker. The 3-D distribution of electric current density is obtained from a current continuity equation along with the generalized Ohm s law; while the magnetic field induced by the current flowing through the arc column is calculated by the magnetic vector potential equation. When gas interacts with an arc column, fundamental factors, such as Ampere s law, Ohm s law, the turbulence model, transport equations of mass, momentum and energy of plasma flow, have to be coupled for analyzing the phenomenon. The coupled interactions between arc and plasma flow are described in the framework of time-dependent magnetohydrodynamic (MHD) equations in conjunction with a K -ε turbulence model. Simulations have been focused on sausage and kink instabilities in plasma (these phenomena are related to pinch effects and electromagnetic fields). The 3-D simulation reveals the relation between plasma deformation and instability phenomena, which affect arc stability during circuit breaker operation. Plasma deformation is the consequence of coupled interactions between the electromagnetic force and plasma flow described in simulations. Keywords: magnetohydrodynamic (MHD), plasma deformation, electromagnetic force, kink instability, sausage instability PACS: Kq, y, Kj, a, Cv, Fv DOI: / /14/11/07 1 Introduction Owing to the excellent dielectric strength and interruption performance of SF 6, it is widely used in gas circuit breakers (GCB) and gas insulated switchgear (GIS), especially for the high voltage class above 110 kv. Therefore, it is very necessary to investigate the SF 6 arc interruption process in a high voltage circuit breaker in order to aid the design of products with higher performances and lower developmental costs [1,2]. During the past 20 years, many researchers of SF 6 high voltage circuit breaker have developed arc models based on magnetohydrodynamics (MHD) [2 4]. In order to investigate an arc and its influences on the characteristics of thermal plasma, a comprehensive three-dimensional (3-D) model is required with a realistic time-dependent description of an arc plasma column with an inherent 3-D and dynamic nature in the presence of a magnetic field. Until now, a great deal of research has been reported on numerical studies on plasma flow and heat transfer inside the plasma [5 10]. However, for simplifying the numerical problem and reducing the computational complexity, two-dimensional (2-D) axisymmetric and steady thermal models have been employed in most previous studies [5 9]. The importance of 3-D simulation becomes pertinent in investigating plasma deformation which is the result of interactions between electromagnetic force and plasma flow. In arc plasma, the flow, electric and magnetic fields interact with each other. The nonlinear transport coefficients of arc plasma, such as electric conductivity, specific heat, dynamic viscosity and thermal conductivity, are dependent on the arc temperature and the pressure of the flow field [11]. The electric conductivity of the arc plasma defines the distribution of the electric potential and current density. Also, the arc current involves the magnetic field and also Lorentz force. Joule heating serves as a heat source and the Lorentz force serves as a momentum source. Therefore, the electric and magnetic fields play an important role in arc behavior and deformation. This paper contributes mainly to the calculation of electromagnetic field and its influence on temperature and electromagnetic force during steps of operation. The simulation results of the paper obtain a view of physical phenomena that makes the nature of an arc more visible.
2 Vahid ABBASI et al.: 3-D Simulation of Plasma Deformation During Contact Opening in a Circuit Breaker 2 Numerical modeling The coupled interactions among an arc current, a magnetic field and plasma flow are described in the framework of the magnetohydrodynamic (MHD) equation in conjunction with the K -ε turbulence model [5,6], where K and ε are the turbulent kinetic energy and its dissipation rate, respectively. The K -ε model has been widely adopted for arc plasma modeling [5,6,9,10] and the recent numerical studies on turbulence effects in arc simulation have shown that the K -ε model predicts the net plasma powers more accurately than the laminar model [9] as compared with the measured value. The 3-D distribution of arc current j is obtainable from the current continuity equation j = 0, along with the generalized Ohm s law j=σ( ϕ + v B), which results in an elliptic equation for electric potential (ϕ): (σ ϕ) = (σv B), (1) where σ and v are the electric conductivity and the plasma velocity, respectively. The self-induced magnetic field is found using the magnetic vector potential A from B arc = A, where the magnetic vector potential is determined from Ampere s law as follows: 2 A = µ 0 j, (2) thermal conductivity k t are defined as: µ t = ρc µ K 2 /ε and k t = µ t c p /P rt, where C µ, P rt and c p are the constant in the turbulence model, the turbulent Prandtl number and the specific heat, respectively [5,6]. The two turbulence variables k and ε are obtained by solving the following two equations, as given in Ref. [5]: a. Turbulent kinetic energy (ρk)/ t+ (ρvk) = G ρε+ [(µ l +µ t /P rk ) K]. (6) b. Dissipation rate of turbulent kinetic energy (ρε)/ t + (ρvε) = ε K (C 1G C 2 ρε) + [(µ l + µ t /P rε ) ε]. (7) P rk, P rε, C 1 and C 2 are the constants suggested by Launder and Spalding. In addition, G is the turbulent generation term expressed as: G = µ t ( v i / x j + v j / x i ) v i / x j. (8) where v i is the ith component of the velocity vector in Cartesian coordinates x i. By the K -ε model, the viscosity and thermal conductivity in the governing equations include both laminar and turbulent components, i.e. µ = µ l + µ t and k = k l + k t. with µ 0 = 4π 10 7 TmA 1. The time dependent equations for the transports of mass, momentum and energy of plasma flow are described by the conservation laws as follows. a. Conservation of mass b. Conservation of momentum ρ/ t + (ρv) = 0. (3) (ρv)/ t + (ρvv) = j B p + τ. (4) c. Conservation of energy (ρh)/ t + (ρvh) = j E Ṙ + (k T ), (5) where ρ, p, τ, h, Ṙ and T are the plasma density, pressure, stress tensor, enthalpy, radiation loss and temperature, respectively. Energy transport by radiation in an arc plasma system is an extremely complex and important phenomenon [12]. The radiation flux at a particular position is not only a function of the local gas properties, but also depends on the temperature and pressure. LIEBERMANN and LOWKE [13] calculated the net emission coefficient of SF 6 at the center of an isothermal and cylindrical arc column under uniform pressure. Most researchers use this data for saving computational cost, except in some specific cases, such as in circuit breakers [14]. The radiation loss Ṙ is taken into account by using the net emission coefficient. In order to take into account the turbulent effects produced by arc gas, a standard K -ε model is employed, in which the turbulent viscosity µ t and the turbulent 3 Solution and results In order to study the effects of electric field and polarization in a SF 6 circuit breaker, detailed data about the circuit breaker structure are necessary. The case study is a medium voltage (33 kv, 1250 A) circuit breaker, as shown in Fig. 1(a). The upper contact is fixed and the lower contact is movable (Fig. 1(b)), when the circuit breaker switches off the lower contact, SF 6 gas is compressed with 0.35 MPa initial pressure and the gas flows between the contacts to eliminate arc. When the contacts open, current follows among the SF 6. The arc current induces a magnetic field and Lorentz force. Joule heating serves as a heat source and the gas temperature rises intensively. Arc transport coefficients such as electric conductivity, specific heat, dynamic viscosity and thermal conductivity vary according to their relationship with temperature and pressure. Based on the equations described in section 2, the calculation is performed by using the finite element method (FEM). The whole calculation includes the stationary and transient course. The stationary results are used as the initial state for the transient solution under the effect of arc current. Since the arc behavior involves an electromagnetic process combined with a hydrodynamic process, a coupled solution method is adopted to solve the above equations. With the physical properties, source term and boundary condition updated, the discrete equations are formed and solved. During the simulation, first, the mass, momentum and energy conservation are solved simultaneously. After that, the 997
3 Plasma Science and Technology, Vol.14, No.11, Nov electric and magnetic fields are computed according to the temperature distribution obtained. The calculation is carried out with a sinusoidal arc current equal to A (the worst case). 1 Fix contact, 2 Nozzle, 3 Movable contact, 4 Gas inlet (a) Structure with complete details, (b) Simplified structure Fig.1 Structure of a typical 33 kv circuit breaker (color online) 3.1 Temperature distribution As the primary parameter of the arc plasma, the temperature is used to define the physical properties of the arc plasma including the electric conductivity. Thus, the temperature distribution is significant for the calculation of electric and magnetic fields. Fig. 2 shows the temperature distribution in an arc chamber. The temperature in the arc core is higher than other areas. The gas temperature rises as the result of current flowing in the arc core. When the arc column moves, the arc current and temperature change correspondingly. As the primary parameter which decides other fields, the temperature distribution in the arc chamber at different steps of the simulation is illustrated in Fig. 3. Fig.2 Temperature distribution in the arc chamber at 0.5 ms (color online) (a) At 3 ms, (b) At 7 ms Fig.3 Temperature distribution in the arc chamber at different steps of movement (color online) 3.2 Temperature distribution Electric potential is function of temperature, electric conductivity, arc current and the distance between two electrodes. Thus, it changes at different steps of movement, as shown in Fig Magnetic field and current distribution Current is injected into the gas from the contacts with the frequency of 50 Hz. The current distribution in the gas depends on temperature and electrical conductivity (Fig. 5(a)). Most of the current distributes in the Z direction, which defines the main arc behavior. However, the amount of current in the radial (r) direction can be neglected; its effects on plasma deformation and electromagnetic force direction in some parts of the chamber are noticeable. Because of the high current magnitude in the Z direction, the higher magnetic field distribution is in the θ direction (Fig. 5(b)). The effect of above phenomenon exhibits itself as electromagnetic force in various part of the chamber (Fig. 6). Electromagnetic force defines the topology of the arc current distribution and arc formation in the next steps. It is the main cause of kink and sausage instabilities, which are described in the following part. 998
4 Vahid ABBASI et al.: 3-D Simulation of Plasma Deformation During Contact Opening in a Circuit Breaker Fig.6 Electromagnetic force vectors between electrodes during operation (color online) 4 (a) At 1 ms, (b) At 7 ms Fig.4 Potential distribution at different steps of movement (color online) Plasma deformation The current that flows in the arc channel is associated with magnetic forces, which lead to an internal overpressure in the arc channel. The phenomenon is often referred to as the pinch effect. The total current in the column may be imagined as consisting of a number of parallel current filaments. Each separate filament will be attracted to other filaments by a magnetic force. The resulting overall force on each filament will be directed towards the center of the column. The resulting overpressure in the center of the arc column may be calculated under the assumption that the current density j is constant over the whole cross section of the arc. With radius R the total current is I =πr2 j. At radius r, the magnetic field H (r ) can be calculated by using Ampere s law: H (r )= 2r j. Consider now a ring segment located at radius r with thickness dr. The current that flows in this ring, in interaction with the magnetic flux, will result in a mechanical force applied at each point of the ring with its direction towards the center. The total current in the ring is: I ring =2πrdrj. The total force (per unit of length of the ring) is Fring = µ0 H(r)Iring = µ0 πr2 drj. (9) The force per unit of area is: F (r) = Fring r = µ0 drj 2, 2πr 2 (10) and the corresponding magnetic pressure gradient is F (r) r dp = = µ0 j 2. dr dr 2 (a) Current distribution, (b) Magnetic field distribution around anode and between electrodes Fig.5 A sample of current and magnetic field distribution (color online) (11) The magnetic pressure in the arc channel is not uniform and causes plasma deformations. Kink and sausage instabilities are two kinds of deformations which occur in particular conditions. a) Kink instability The axial magnetic field B z combined with the azimuthal magnetic field B θ is produced by the plasma current. This helical magnetic field is susceptible to the helical kink instabilities outlined in Fig
5 Plasma Science and Technology, Vol.14, No.11, Nov ity, because it results in plasma resembling a string of sausage [15]. Fig.7 Kink instability may occur when arc current has different directions or when the magnetic field is non-uniform around the plasma (two parameters which affect electromagnetic force vectors) At the concave parts of the plasma surface there is a concentration of the azimuthal field resulting in a magnetic pressure that increases the concavity. Electromagnetic force vectors and consequently the magnetic pressure are stronger in these regions. Similarly, at the convex portions of the surface, the azimuthal field is weaker so that the convex bulge will tend to increase. This is just the hoop force tending to increase the major radius of a curved current [15]. b) Sausage instability Bennett pinch (z-pinch) is axially uniform cylindrical plasma with an axial current and an associated azimuthal magnetic field. The pressure is assumed to be uniform in the interior region r<a where a is the plasma radius and it becomes zero in the exterior region of r>a. In equilibrium, the radial inward J B pinch force balances the radial outward force associated with the pressure gradient so that J z0 B θ0 = P 0 / r where both sides of this equation are finite only in an infinitesimal surface at r = a. We now suppose that thermal noise causes the incompressible plasma to develop axially periodic constrictions and bulges, as shown in Fig. 8. At the constriction positions, the azimuthal magnetic field B θ = µ 0 I/2πr becomes larger than that in an equilibrium state because r<a at a constriction while I is fixed. Thus, at the constriction the pinch force Bθ 2/r is greater than its equilibrium value and so exceeds the outward force from the internally uniform pressure. The resulting net force will be inwards and will cause a radial inwards motion, which will enhance the constriction. Because the configuration is assumed to be incompressible, any plasma squeezed inwards at constrictions must flow into the interspersed bulges, as shown in Fig. 8. The azimuthal magnetic field at the bulge is weaker than its equilibrium value because r>a at the bulge, so the situation is now turned into a competition between outward pressure and inward pinch force. At a bulge the pressure exceeds the weakened pinch force and this force imbalance causes the bulge to increase. Hence any initial perturbed combination of constrictions and bulges will tend to grow and so the system is unstable. This behavior is called the sausage instabil- Fig.8 Sausage instability, current is in the axial direction and the magnetic field is in the azimuthal directions Concaves in kink instability and constrictions in sausage instability lead to arc steadiness in the area where they take place. They occur as the result of strong magnetic pressure and are investigated by simulations to define the possibility of their occurrence. The internal and external surfaces of the arc current have different shapes because of the plasma instabilities. To prove the instability theories, the high current surfaces (current surfaces for current density around 1e7 are chosen) at different times are illustrated in Fig. 9 (the external high current surface of the arc column emerges between the edges of the electrodes, and the internal high current surface emerges between their inner parts). Instabilities are visible on the internal surfaces of the high current column. The shapes of the surfaces are functions of electromagnetic force vectors (Fig. 10). According to the explanation of instabilities and the results of the simulation (Fig. 10), it seems that the possibility of sausage instability occurring in arc column is much higher. At the top and bottom of the current surfaces, the electromagnetic force vectors are in the -r direction, which causes constriction in these areas (the resulting net force is inwards and causes a radial inwards motion, which enhances the constriction). The magnetic pressure in the middle of the surfaces in the Z direction is stronger and the gas pressure causes a radial outwards motion, which enhances the bulge. The electromagnetic force constrictions near the edges of the contacts push the arc current forward into the contacts that are the reason of the arc stability. Similar investigations can be carried out for other current surfaces levels, but the highest levels are more vital because of their higher magnitude of electromagnetic force and their importance in forming the arc column. 1000
6 Vahid ABBASI et al.: 3-D Simulation of Plasma Deformation During Contact Opening in a Circuit Breaker 5 Conclusions A 3-D simulation of the arc in a circuit breaker has been accomplished with a realistic time-dependent description in order to predict the arc deformation and its influence on arc stability. The shrinkage of the arc column near the electrode is visible through the current density distribution and maximum values of electric potential occurring near the contact/plasma interface. Two main criteria, electromagnetic force distribution and its interaction with flow, describe arc deformation and instabilities. By analyzing these criteria, kink and sausage instabilities in high current density plasma are investigated, which is advantageous to improve the efficiency of arc plasma simulation. The results of the simulation demonstrate the higher possibility of sausage instability occurrence in an arc column. During different steps of movement, the arc column consists of two constrictions in the vicinity of the contacts and a bulge in the middle of the column. Constrictions near the contacts can be considered as one of the main cause of arc steadiness, which needs to be reduced in practice. References 1 (a) At 1.5 ms, (b) At 2.5 ms 2 Fig.9 High current column surfaces at different times (current surfaces for current density around 1e7 are chosen) (color online) (a) At 1.5 ms, (b) At 2.5 ms Fig.10 plasma deformation and electromagnetic force vectors on the internal high current surface at different steps (color online) 15 MA Qiang, Rong Mingzhe, Wu Yi, et al. 2008, Plasma Science and Technology, 10: 313 LI Xingwen, Tusongjiang Kari, Chen Degui, et al. 2009, Plasma Science and Technology, 11: 245 Zhang J, Jia S L, Li X W. 2009, Plasma Science and Technology, 11: 152 Liau V K, Lee B Y, Song K D, et al. 2006, J. Phys. D: Appl. Phys, 39: 2114 Jin Myug Park, Keun Su Kim, Tae Hyung Hwang, et al. 2004, IEEE Transaction on Plasma Science, 32: 479 Scott D A, Kovitya P, Haddad G N. 1989, J. Appl. Phys., 66: 5232 Westhoff R, Szekly J. 1991, J. Appl. Phys., 70: 379 Kang K D, Hong S H. 1996, IEEE Transaction on Plasma Science, 24: 89 Hur M, Hong S H. 2002, J. Phys., D: Appl. Phys., 35: 1946 Li H, Pfender E, Chen X. 2003, J. Phys. D: Appl. Phys., 36: 1084 Wu Yi, Rong Mingzhe, Li Jian, et al. 2006, IEEE Transactions on Magnetics, 42: 1007 Lee Jong-chul, Kim Youn J. 2005, Surface Engineering, Surface Instrumentation & Vacuum Technology, 80: 599 Liebermann R W, Lowke J J. 1976, JQSRT, 17: 253 Dixon C M, Yan J D, Fang M T C. 2004, J. Phys. D, Appl. Phys., 37: 3309 Bellan Paul M Fundamentals of Plasma Physics. Cambridge University, Cambridge (Manuscript received 1 June 2011) (Manuscript accepted 9 September 2011) address of Vahid ABBASI: v abbasi@iust.ac.ir 1001
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