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- Detailed numerical simulation of cathode spots in vacuum arcs: Interplay of different mechanisms and ejection of dropletsPublication . Kaufmann, H. T. C.; Cunha, M. D.; Benilov, M. S.; Hartmann, W.; Wenzel, N.A model of cathode spots in high-current vacuum arcs is developed with account of all the poten tially relevant mechanisms: the bombardment of the cathode surface by ions coming from a pre existing plasma cloud; vaporization of the cathode material in the spot, its ionization, and the inter action of the produced plasma with the cathode; the Joule heat generation in the cathode body; melting of the cathode material and motion of the melt under the effect of the plasma pressure and the Lorentz force and related phenomena. After the spot has been ignited by the action of the cloud (which takes a few nanoseconds), the metal in the spot is melted and accelerated toward the periph ery of the spot, with the main driving force being the pressure due to incident ions. Electron emis sion cooling and convective heat transfer are dominant mechanisms of cooling in the spot, limiting the maximum temperature of the cathode to approximately 4700–4800 K. A crater is formed on the cathode surface in this way. After the plasma cloud has been extinguished, a liquid-metal jet is formed and a droplet is ejected. No explosions have been observed. The modeling results conform to estimates of different mechanisms of cathode erosion derived from the experimental data on the net and ion erosion of copper cathodes.
- Numerical investigation of the stability of stationary solutions in the theory of cathode spots in arcs in vacuum and ambient gasPublication . Benilov, M. S.; Cunha, M. D.; Hartmann, W.; Wenzel, N.The stability of stationary spots on cathodes of arcs in vacuum and ambient gas is investigated by means of the simulation of the temporal evolution of perturbations imposed over steady-state solutions. Two cases of loading conditions are considered, namely, spots operating at a fixed current (the case typical of small-scale experiments) and spots operating at a fixed voltage (the case typical of high-power circuit breakers). Results are reported on spots on large copper cathodes of vacuum arcs and on spots on tungsten cathodes of high-pressure argon arcs. It is shown, in particular, that if the ballast resistance in small-scale laboratory experiments with a high-current arc is insufficient, the potential consequence may be a thermal explosion of a spot, if the arc burns in vacuum, and massive melting of the cathode surface, if the arc burns in ambient gas. This conclusion conforms to trends observed in the experiment.
- Simulating changes in shape of thermionic cathodes during operation of high-pressure arc dischargesPublication . Cunha, M. D.; Kaufmann, H. T. C.; Santos, D. F. N.; Benilov, M. S.A numerical model of current transfer to thermionic cathodes of high-pressure arc discharges is developed with account of deviations from local thermodynamic equilibrium occurring near the cathode surface, in particular, of the near-cathode space-charge sheath, melting of the cathode, and motion of the molten metal under the effect of the plasma pressure, the Lorentz force, gravity, and surface tension. Modelling results are reported for a tungsten cathode of an atmospheric-pressure argon arc and the computed changes in the shape of the cathode closely resemble those observed in the experiment. The modelling has shown that the time scale of change of the cathode shape during arc operation is very sensitive to the temperature attained by the cathode. The fact that the computed time scales conform to those observed in the experiment indicate that the model of non-equilibrium near-cathode layers in high pressure arc discharges, employed in this work, predicts the cathode temperature for a given arc current with adequate accuracy. In contrast, modelling based on the assumption of local thermodynamic equilibrium in the whole arc plasma computation domain up to the cathode surface could hardly produce a similar agreement.
- Stability of stationary solutions in the theory of cathode spots in arcs in vacuum and ambient gasPublication . Benilov, M. S.; Cunha, M. D.; Hartmann, W.; Wenzel, N.Stability of stationary spots on cathodes of arcs in vacuum and ambient gas has been investigated by means of simulation of development in time of perturbations imposed over steady-state solutions. Two cases of loading conditions have been considered, namely, spots operating at fixed current (the case typical of small-scale experiments) and spots operating at fixed voltage (the case typical of high-power circuit breakers). Results are reported on spots on large copper cathodes of vacuum arcs and on spots on tungsten cathodes of high-pressure argon arcs. It is shown, in particular, that if the ballast resistance in a small-scale laboratory experiment with a high-current arc is insufficient, potential consequence may be a thermal explosion of a spot, if the arc burns in vacuum, and massive melting of the cathode surface, if the arc burns in ambient gas. This conclusion conforms to trends observed in the experiment.
- Computing DC discharges in a wide range of currents with COMSOL MultiPhysics: time-dependent solvers vs. stationary solversPublication . Almeida, P. G. C.; Benilov, M. S.; Bieniek, M. S.; Cunha, M. D.; Gomes, J. G. L.; Kaufmann, H. T. CostaThe benefits of the usage of stationary over time-dependent solvers of COMSOL Multiphysics in the modelling of DC discharges are explored and demonstrated using as examples glow and high pressure arc discharges; in particular, it is investigated whether time-dependent solvers can be used for a systematic computation of different modes of these discharges. It has been found that most modes of both glow and high-pressure arc discharges cannot be computed in the whole range of their existence by a time-dependent solver. Further, time-dependent solvers are unsuitable for a computation of all the states belonging to the retrograde sections of the current-voltage characteristics of the modes, so the discharge manifests hysteresis, which, in principle, can be observed in the experiment.
- Modeling spots on composite copper-chromium contacts of vacuum arcs and their stabilityPublication . Benilov, Mikhail S.; Cunha, Mário D.; Hartmann, Werner; Kosse, Sylvio; Lawall, Andreas; Wenzel, NorbertCathode spots on copper–chromium contacts of vacuum interrupters are simulated by means of a self-consistent space-resolved numerical model of cathode spots in vacuum arcs developed on the basis of the COMSOL Multiphysics software. Attention is focused on spots attached to Cr grains in the Cu matrix in a wide range of values of the ratio of the grain radius to the radius of the spot. In the case where this ratio is close to unity, parameters of spot are strongly different from those operating on both pure-copper and pure-chromium cathodes; in particular, the spot is maintained by Joule heat generation in the cathode body and the net energy flux is directed from the cathode to the plasma and not the other way round. An investigation of stability has shown that stationary spots are stable if current controlled. However, under conditions of high power circuit breakers, where the near-cathode voltage is not affected by ignition or extinction of separate spots, the spots are unstable and end up either in explosive-like behavior or in destruction by thermal conduction. On the other hand, spots live significantly longer-up to one order of magnitude-if the spot and grain sizes are close; else, typical spot lifetimes are of the order of 10 µs. This result is very interesting theoretically and may explain the changes in grain size occurring in the beginning of the lifetime of contacts of high-power current breakers. A sensitivity study has shown that variations in different aspects of the simulation model produce quantitative changes but do not affect the results qualitatively.
- Space-resolved modeling of stationary spots on copper vacuum arc cathodes and on composite CuCr cathodes with large grainsPublication . Benilov, Mikhail S.; Cunha, Mário D.; Hartmann, Werner; Kosse, Sylvio; Lawall, Andreas; Wenzel, NorbertA self-consistent space-resolved numerical model of cathode spots in vacuum arcs is realized on the computational platform COMSOL Multiphysics. The model is applied to the investigation of stationary spots on planar cathodes made of copper or composite CuCr material with large ( 20 µm) chromium grains. The modeling results reveal a well defined spot with a structure, which is in agreement with the general theory of stationary cathode arc spots and similar to that of spots on cathodes of arcs in ambient gas. In the case of CuCr contacts with large chromium grains, spots with currents of the order of tens of amperes on copper coexist with spots on chromium with currents of the order of one or few amperes. The main effect of change of the cathode material from copper to chromium is a reduction of thermal conductivity of the cathode material, which causes a reduction of the radius of the spot and a corresponding reduction of the spot current.
- Computing Different Modes on Cathodes of DC Glow and High-Pressure Arc Discharges: Time-Dependent Versus Stationary SolversPublication . Almeida, Pedro G. C; Benilov, Mikhail S.; Cunha, Mário D.; Gomes, José G. L.Complex behavior can appear in the modeling of gas discharges even in apparently simple steady-state situations. Time-dependent solvers may fail to deliver essential information in such cases. One of such cases considered in this work is the 1D DC discharge. The other case is represented by multiple multidimensional solutions existing in the theory of DC discharges and describing modes of current transfer with different patterns of spots on the cathodes. It is shown that, although some of the solutions, including those describing beautiful self organized patterns, can be computed by means of a time-dependent solver, in most examples results of time-dependent modeling are at best incomplete. In most examples, numerical stability of the time dependent solver was not equivalent to physical stability.
- Phenomenological approach to simulation of propagation of spots over cathodes of high-power vacuum circuit breakersPublication . Cunha, M. D.; Wenzel, N.; Benilov, M. S.; Hartmann, W.A phenomenological description of an ensemble of a large number of spots on negative contacts of high-power vacuum circuit breakers is developed by means of generalization of the concept of random walk of a single cathode spot in low-current vacuum arcs. The model is formulated in terms of a convection-diffusion equation governing the evolution of the number of spots per unit area, taking into account the variation of the number of spots with the arc current and the “retrograde repulsion” between spots. The approach is applied to description of the distribution of cathode spots during the initial expansion process after arc ignition in conditions of two independent experiments simulating high-power switches. A reasonably good agreement between the theory and the experiment is found. The developed model can be used as a module of global numerical models of the interruption process in high-power vacuum circuit breakers.
- A simple model of distribution of current over cathodes of vacuum circuit breakersPublication . Cunha, Mário D.; Wenzel, Norbert; Almeida, Pedro G. C.; Hartmann, Werner; Benilov, Mikhail S.There are several hundreds of spots operating simultaneously on cathodes of vacuum arcs in high-power vac uum circuit breakers. In this work, the spot distribution along the contact surface is simulated by means of an approach that is based on the concept of surface density of spots and represents a natural alternative to tracing individual spots. An equation governing the evolution of the surface density of the spots or, equivalently, the distribution of macroscopic (averaged over individual spots) current density over the cathode is obtained by generalizing the concept of random walk of a single cathode spot in low-current vacuum arcs. The model relies on empirical parameters characterizing individual spots (the diffusion coefficient of the random motion of cathode spots and the velocity of drift superimposed over the random motion), which may be taken from experiments with low-current arcs, and does not involve adjustable parameters. The model is simple and physically transparent and correctly reproduces the trends observed in the experiments under conditions where the cathode arc attachment is diffuse. The distribution of the macroscopic current density on the cathode, given by the model, represents the boundary condition that is required for existing numerical models of vacuum arcs in high-power vacuum circuit breakers.