Model for arc cathode region in a wide pressure range

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS J. Phys. D: Appl. Phys. 3 (2001) PII: S (01)222-X Model for arc cathode region in a wide pressure range Torbjörn Nielsen 1,, Ahmed Kaddani 2 and Mikhail S Benilov 3 1 FaxénLaboratoriet, KTH, S-0 Stockholm, Sweden 2 ABB Corporate Research, S-21 8 Västerås, Sweden 3 Departemento de Física, Universidade da Madeira, Portugal Received March 2001 Abstract A one-dimensional model for the near-cathode region of electric arcs is applied to define the current, heat and mass transfer mechanisms in the electron temperature range of 0 kk and the pressure range of Pa. The model considers details of the space charge zone, the ionization zone and evaporation of the cathode material. A copper tungsten cathode is investigated, and it is found that evaporation is an important cooling mechanism of the cathode and should not be neglected. The plasma near the cathode is constituted of evaporated contact material in the main part of the investigated pressure temperature domain. The results show that Schottky-enhanced thermoionic emission, together with the present model for the near-cathode layer, explains the current transfer mechanism for a wide range of pressures and temperatures and for refractory and non-refractory cathode materials. 1. Introduction Electric arcs play a dominant role in a wide range of industrial applications, such as different types of circuit breaker, arc welding, plasma spraying, plasma cutting and high-intensity discharge (HID) lamps. Accordingly, extensive research has been performed on modelling of electric arcs and related phenomena. The main differences between these applications are the arc current and the pressure at which the arc operates. The cathode region is considered to be the most active region in arcs. It is governed by both properties of the arc column and the cathode material. A fundamental problem is to explain the emission mechanism of electrons from the cathode surface. One possibility is thermoionic emission, which assumes a well defined surface and a sufficiently high temperature for electrons to be emitted due to temperature alone. This mechanism is not believed to be sufficient for cathode materials with low melting points, i.e. copper or iron, as sufficient temperatures cannot be reached. Therefore, models considering both temperature and the electric field, thermoionic field emission, have been proposed (see, e.g., [1, 2]). In the case of vacuum arcs, theories exist that consider the current transfer from the metal as an ongoing process of Ectons or micro-explosions [3, ]. Micro-protrusions on the surface create concentrations of the electric field and field Present address: ABB Corporate Research, S-21 8 Västerås, Sweden. emission of electrons. The current due to field emission heats the metal locally, causing an explosive emission of electrons, vapour and molten metal. This process has a typical time scale of 9 8 s. The total current is transferred by numerous Ectons appearing and disappearing continuously. It has also been suggested that Ectons are active in atmospheric-pressure arcs []. The emission mechanism seems to be a source of disagreement even to this date. Current transfer near the cathode is not only determined by the emission mechanism, it is also determined by the space charge zone, see figure 1, which is a collisionless layer at the vicinity of the surface. This zone has a positive voltage drop as the plasma is not neutral. Thus, ions are accelerated to the surface, transferring their kinetic energy and ionization energy there. Emitted electrons are accelerated away from the surface and electrons in the arc column may have enough energy to overcome the field and reach the surface. The production of ions and the transition from the highly non-equilibrium plasma in the sheath to the equilibrium plasma in the arc is ensured by the ionization zone, located between the sheath and the equilibrium plasma. Several works considering the near-cathode region exist. Morrow and Lowke [] proposed a non-equilibrium, onedimensional model assuming thermoionic emission and neglecting the space charge zone and evaporation of cathode material. This work included back-diffused electrons due to /01/ $ IOP Publishing Ltd Printed in the UK 201

2 Model for arc cathode region CATHODE J e e J vap J i J e d U = UD A + e > A + + 2e U = U D + U i THERMAL RELAXATION T e = T h and n e = n i T e = T h and n e = n i T e = T h and n e = n i U = 0 SPACE CHARGE IONIZATION J ARC Figure 1. Details of the near-cathode region. J e e, J i, J d e and J vap are fluxes of emitted electrons, ions, back-diffused electrons and evaporated particles, respectively. J is the total current density per unit charge, T e is the electron temperature, T h is the heavy particle temperature and U is the voltage drop. n e and n i are the number densities of electrons and ions, respectively. ambipolar diffusion. Back-diffused electrons were found to be the main reason for the different behaviours of cathode materials with different work functions. It was also found that radiation gives a minor contribution to the energy balance at the cathode surface. Zhou et al [] proposed a more detailed model which included the space charge zone. Here, thermoionic field emission was assumed and evaporation of the cathode material was included in a simplified manner. The voltage drop in the space charge zone was obtained from the Steenbeck minimum principle and the ion current was estimated from the thermal speed evaluated at the surface temperature. It was found that cathode erosion depends strongly on the work function and the vapour pressure of the material. Kaddani [8] modelled the complete system of cathode, arc and anode at atmospheric pressure. The work included a detailed description of the anode and cathode regions including space charge zones in the near vicinity of the electrodes. The methodology used showed that high-pressure arc properties could be predicted accurately using numerical simulations. Two-dimensional simulations of the interaction between plasma and cathode, with application to HID lamps, were performed in [9, ]. Wendelstorf [9] found that for a current of A and a pressure of bar, heavy particles and electrons are in thermal equilibrium at a distance of approximately 200 µm from the cathode. Thus, there is an important zone of thermal relaxation in front of the ionization zone. Hu et al [] concluded that arc spots occur if the cathode fall is greater than 20 V. Temperature measurements in the cathode and the nearcathode plasma of HID lamps are reported in [11] and [12], respectively. The aim of this work, which uses the model of Benilov and Marotta [13], is to investigate the behaviour of the nearcathode region for a wide range of pressures and temperatures in terms of current, heat and mass transfer. The different contributions to current and heat flux are studied in detail, including evaporation of the cathode material. 2. Cathode region modelling Details of the model for the space charge zone and the ionization zone are given in [13]. Figure 1 illustrates the important features of the near-cathode region. The space charge zone is considered to be collisionless for ions and electrons and the electron emission mechanism used in this work is Schottky-enhanced thermoionic emission. Besides emitted electrons, the current in the space charge zone is carried by back-diffused electrons and ions. The space charge of the plasma is constituted by ions and back-diffused plasma electrons. Emitted electrons are not considered to contribute to the space charge. It is assumed that all ions reaching the cathode are neutralized and that the mean velocity of ions entering the sheath is given by the Bohm criterion. The energy flux to the cathode surface, q w,isgivenby q w = J i [ 1 2 kt e + e(u D + E i eff )]+Je d (2kT e + e eff ) Je e (2kT w + e eff ) (1) where Je e, J e d and J i are the fluxes of emitted electrons, backdiffused plasma electrons and ions, respectively. Furthermore, U D is the voltage drop in the space charge zone, E i is the ionization energy of the gas, k is Boltzmann s constant, T e is the electron temperature and T w is the temperature of the cathode surface. It has been assumed that ions have the same temperature as the surface. The effective work function, eff, represents the energy needed to extract electrons from the cathode. It is a function of the electric field at the cathode surface. On the right-hand side, the first term represents the energy brought to the cathode surface by ions generated in the ionization zone and accelerated in the space charge zone. The second term is the energy flux due to back-diffused electrons and the third term is the energy flux from the cathode surface due to electron emission. From now on they are denoted q i, qe d and qe e, respectively. The temperature at the cathode surface can now be obtained from the energy balance in the cathode ( d λ dt ) = 0 (2) dx dx with the boundary conditions λ dt dx = q w LJ vap m w at x = 0 T = T 0 at x = L. Here λ is the thermal conductivity, L is the latent heat of evaporation, m w is the mass of the evaporated particles, J vap is the net flux of evaporated particles from the surface, x is the axial coordinate, L is the length of the cathode and T 0 is the temperature at x = L. The boundary condition at x = 0 represents the total heat flux conducted into the cathode and is denoted q cnd. The term LJ vap m w represents the heat loss due to the evaporation of the contact material and is denoted q vap. Cooling of the cathode surface by evaporation was neglected in the work of Benilov and Marotta [13]. The net flux of evaporated contact material, J vap, is here taken into account according to Benilov et al [1]. If the ambient pressure is low, evaporation occurs almost as into a vacuum. For higher pressures, the mass flux is reduced by the ambient gas. If the ambient pressure is higher than the vapour pressure the mass flux is driven by diffusion and not by a pressure gradient. This is not accounted for in the present work. 201

3 T Nielsen et al 3. 8 α Figure 2. Benilov s α for copper vapour and air as functions of electron temperature. The pressure is 1 bar and the heavy particle temperature is 3000 K: denotes copper vapour and ---- denotes air. The cathode material is taken to be a mixture of 20% copper and 80% tungsten. This material is very often used in high-voltage circuit breakers. The vapour pressure and the latent heat of evaporation are obtained from linear interpolation between the respective values for copper and tungsten. The vapour pressures are given by [8] p v,cu = 133.3T 1.2 w ( /Tw) (3) ( ) 0.38 p v,w = exp and L is approximated with L = L 0 (T C T )/(T C T B ) if the cathode surface temperature is higher than the boiling point, otherwise with L = L 0. T C is the critical temperature, T B is the boiling point and L 0 is the latent heat at the boiling point, which is L 0,Cu =.8 MJ kg 1 for copper and L 0,W =.82 MJ kg 1 for tungsten. The work function for copper is. ev and for tungsten. ev, the value used in the calculation is. ev. To study the influence of the work function results for 0 = 2. ev are also presented. This value corresponds to the work function of thoriated tungsten. Due to the high boiling point of tungsten, its contribution to evaporation is negligible and the properties of copper are used for the plasma near the cathode. Furthermore, only singly charged ions are considered. The property α characterizing the ionization zone is calculated according to [1] as T w () ( ) 1/ ( ) 1kTi 2 Q 1/2 α =. () m i 3k i Here T i is the ion temperature, k i is the ionization rate coefficient, m i is the mass of ions and Q is the mean cross section for momentum transfer in elastic ion neutral collisions. α is a non-dimensional number comparing the ionization length to the mean free path of ions and neutrals. Benilov and Naidis [1] calculated α for air and the same methodology has been applied for copper vapour in this work. The parameters needed to evaluate the ion neutral cross section, Q, are found in [20]. α is shown in figure 2 for air and copper vapour. Note that in [1] it was pointed out that α depends on the evaporation of the contact, but this effect was not included Figure 3. Isolines of the voltage drop, U D [V], in the space charge zone as a function of ambient pressure and temperature: denotes 0 =.evand----denotes 0 = 2. ev. The ionization rate coefficient in air is obtained from the conventional expression for direct ionization ( ) 1/2 ( ) 8kTe Ei k i = C (E i +2kT e ) exp () πm e kt e where C is the derivative of the ionization cross section with respect to the electron energy, evaluated at the threshold. It is obtained from [18]. For copper, the ionization rate coefficient was calculated taking into account both direct and stepwise ionization. Data such as statistical weights and ionization energy are obtained from [19]. The input parameters are the temperature and pressure in the equilibrium bulk plasma, and the material properties. Equation (2) is solved iteratively for the cathode surface temperature assuming constant material properties, L = 200 mm and T 0 = 300 K. The voltage drop in the space charge zone is then obtained from the energy balance of electrons in the ionization zone, i.e equation (18) in [13]. The total current density is given by the sum of the contributions from backdiffused electrons, emitted electrons and ions at the surface, and the quasi-equilibrium ion number density is obtained from the Saha equation for a two-temperature plasma. Additional output data are all local parameters of the near-cathode layer, in particular the local fluxes of all species and components of the heat flux to the cathode. 3. Results The voltage drop in the space charge zone as a function of pressure and electron temperature is shown in figure 3 for two different work functions. For 0 =. ev, the voltage drop decreases with pressure and increases with electron temperature. The behaviour is significantly different for 0 = 2. ev. In this case, the pressure dependency is very weak and the voltage drop increases with temperature. Figure shows the total current density as a function of pressure and electron temperature for two values of the work function. It is mainly pressure dependant except at low electron temperatures where it increases significantly. The increase of the work function results in decreased total current density. Figure shows the contribution from emitted electrons and ions to the total current density as a function of pressure and electron temperature. The current of back-diffused electrons is the difference between these two values. It can be seen that 2018

4 Model for arc cathode region...e Figure. Isolines of the logarithm of the total current density, j [A m 2 ], as a function of ambient pressure and electron temperature: denotes 0 =.evand----denotes 0 = 2. ev Figure. Isolines of the cathode surface temperature T w [K]asa function of the ambient pressure and the electron temperature: denotes 0 =.evand----denotes 0 = 2. ev Figure. Isolines of the contribution of electrons and ions to the total current density, j, as a function of the ambient pressure and the electron temperature: denotes J e e /J and denotes J i/j. different current transfer mechanisms are active at different pressures and temperatures. For low pressure and electron temperature, the current transfer is dominated by ions and emitted electrons. Increasing the electron temperature or the pressure has the effect of decreasing the contribution of ions to the current. At high pressures and temperatures back-diffused electrons become more significant than ions, and the emitted current becomes higher than the total current. Figure shows the temperature of the cathode surface as a function of pressure and electron temperature for different work functions. As can be seen in this figure, the surface temperature increases with pressure and electron temperature. The increase of the work function leads to a higher surface temperature. As a comparison, Zhou et al [] obtained a surface temperature of 300 K and a current density of. Am 2 for atmospheric pressure and an electron temperature of 20 kk. The relative importance of the two heating mechanisms, ions and back-diffused electrons, as functions of pressure and electron temperature is presented in figure. When the arc pressure or temperature is increased the heating mechanism of the surface switches from ion-dominated heating to heating dominated by back-diffused electrons. For an electron temperature of 20 kk, heating by ions and heating by electrons becomes equally important at approximately 2 bar. In figure 8, the contributions of conduction and evaporation to the cooling of the surface, as functions of pressure and electron temperature, are shown. These Figure. Isolines of the contribution to the total surface heating, q i + qe d, from ions as a function of the ambient pressure and the electron temperature for 0 =. ev Figure 8. Isolines of the contribution to the total surface cooling, q cnd + q vap + qe e, from conduction and evaporation as a function of the ambient pressure and the electron temperature for 0 =. ev: denotes q cnd /(q cnd + q vap + qe e) and----denotes q vap /(q cnd + q vap + qe e). are mainly dependent on pressure except at low electron temperature and high pressure. Below approximately bar the cooling mechanism is mainly evaporation, above this the cathode is mainly cooled by emission of electrons. Conduction in the cathode is not significant. It is found that evaporation is an important mechanism for cooling the cathode especially at pressures below bar. Figure 9 shows the cathode evaporation rate as a function of pressure and electron temperature. As can be seen, evaporation increases with both pressure and electron temperature. This behaviour is mainly due to the increase in current density and therefore the total heat flux to the cathode. When the partial 2019

5 T Nielsen et al C B Figure 9. Isolines of the vapour mass flux from the cathode, J vap m w [kg s 1 m 2 ], as a function of the ambient pressure and the electron temperature A Figure 11. Dominant current transfer mechanisms for different values of the work function, 0. Region A, J e e /J i < 1; region B, J e e /J i > 1 and J d e /J i < 1; and region C, J d e /J i > 1. denotes 0 =.ev,----denotes 0 =. evand denotes 0 =.0 ev Figure. Isolines of the ratio of the vapour pressure to the ambient pressure as a function of the ambient pressure and the electron temperature: denotes 0 =. evand----denotes 0 = 2.eV. pressure of metal vapour approaches the ambient pressure, evaporation is reduced as expected. The ratio of vapour pressure to ambient pressure is shown in figure as a function of pressure and electron temperature for two different work functions. For a low value of the work function, heating by the current becomes less efficient and the ratio falls below unity. This implies that the assumption of 0% metal vapour near the cathode can be questioned. The mechanisms of current and heat transfer are summarized in the maps shown in figure 11 and figure 12, respectively. Figure 11 shows three different regions. Region A corresponds to Je e/j i < 1, region B to Je e/j i > 1 and Je d/j i < 1, and region C to Je e/j i > 1 and Je d/j i > 1. At low pressure (A) the current is carried mainly by ions and also by emitted electrons. As the pressure increases (B) the influence of back-diffused electrons is no longer negligible. The main current transfer mechanism is emission of electrons. At very high pressures (C), electrons dominate the process. The current due to emitted electrons is significantly higher than the total current as the back-diffused electrons play a major role. The contribution from ions is of less importance. Increasing the electron temperature has a similar effect. In the case of 0 = 2. ev, both Je e/j i and Je d/j i are greater than unity, corresponding to region C in figure 11. Figure 12 also shows three different regions. Region A corresponds to q vap /qe e > 1, region B to q vap /qe e < 1 and qe d/q i < 1, and region C to q vap /qe e < 1 and qd e i > 1. At low pressure (A) the cathode is heated by ions and cooled by 0.1. B.. A Figure 12. Dominant heat transfer mechanisms for different values of the work function, 0. Region A, q vap /q e e > 1; region B, q vap /q e e < 1 and qd e /q i < 1; and region C, q d e /q i > 1. denotes 0 =.ev,----denotes 0 =. ev, denotes 0 =.0 ev and denotes 0 = 2. ev. evaporation. As the pressure increases (B), cooling by the emission of electrons and heating by back-diffused electrons becomes more important. At very high pressure (C), cooling by evaporation decreases and the dominant cooling mechanism is emission. It is worth noting that the two cooling mechanisms, emission of electrons and evaporation, are at least two orders of magnitude higher than the heat flux due to conduction in the cathode. At low pressures, the heat flux components become smaller and the conductive heat flux more significant. The behaviour of the near-cathode region can be explained as follows. When the pressure is increased, the current density increases due to the higher density of the plasma, resulting in more efficient heating of the cathode surface and therefore a higher surface temperature and higher emission current density. In addition, a lower voltage drop is required to maintain the electron energy balance in the ionization zone. Therefore, the current of back-diffused electrons increases and the ion current decreases. Increasing the electron temperature will also enhance the heating of the surface due to a higher current density of ions and a higher temperature of back-diffused electrons, resulting in a higher surface temperature. The voltage drop must now increase for the same reason as above. When the work function is decreased, heating by ions and back-diffused electrons becomes less efficient and the surface C 2020

6 Model for arc cathode region temperature decreases. This results in lower emission and therefore lower current density; the voltage drop decreases accordingly. Evaporation of the cathode surface plays an important role. First, it efficiently keeps the ambient gas away from the near-cathode region. The heating of the cathode is reduced as metal vapour has less ionization energy than, for example, argon. Second, cooling by evaporation also helps to keep the surface temperature down. Therefore, the surface temperature stays well below the critical temperature but above the boiling point of the cathode material. For the pressure and temperature intervals studied in this work, the model can always adjust the current density and the voltage drop such that current transfer is assured.. Summary and conclusions A model for the near-cathode region of electric arcs has been applied to define the current, heat and mass transfer mechanisms for a wide range of pressures and temperatures. The model considers details of the space charge zone, the ionization zone and evaporation of the contact material. The studied parameters are pressure, electron temperature and work function. They are all important in determining which mechanisms are important for current and heat transfer to the cathode. It was found that the current density varies between and Am 2 and the voltage drop in the space charge zone varies between and 3 V, in the pressure range of Pa and the electron temperature range of 0 kk. A high voltage drop and a low current density are obtained for low pressures. It was found that, for a copper tungsten cathode, evaporation is an important cooling mechanism of the cathode and should not be neglected. Except at very low electron temperatures and very high pressures, the vapour pressure is always higher than the background pressure. This means that the plasma near the cathode is, as assumed in this work, constituted of evaporated contact material. As the ionization energy is much lower and the mass of metal atoms is substantially higher than those of the surrounding gas, this has a significant influence on the heat flux to the cathode. In the case of 0 = 2. ev, corresponding to thoriated tungsten, evaporation is negligible and the properties of the bulk plasma as well as evaporated contact material should be considered together with an evaporation model based on diffusion. Thermal conduction inside the cathode was found to be at least two orders of magnitude lower than cooling by electrons or evaporation. This term may thus be neglected and the assumption of constant thermal diffusivity in the cathode does not influence the results. The value of the material work function has a marked influence on the current and heat transfer behaviour of the cathode. Only small changes, which could easily correspond to discrepancies between values found in literature, can influence considerably the cathode region properties. The results of the present work show that Schottkyenhanced thermoionic emission, together with the present model for the near-cathode layer, explain the current transfer mechanism for a wide range of pressure and temperature. The methodology was shown to be feasible for both refractory and non-refractory cathode materials. The total current intensity at the spot is not given by the present model as it is one dimensional and the result is the current density. Acknowledgments The authors would like to acknowledge the contributions of Professor G V Naidis regarding the plasma properties and Mr Staffan Jacobsson regarding vaporization of contacts. This work was financially supported by ABB Switchgear. The work at Universidade da Madeira was partially supported by the project Theory and modelling of plasma cathode interaction in high-pressure arc discharges of the programme POCTI of Fundação para a Ciência e a Tecnologia and FEDER. References [1] Murphy E L and Good R H Phys. 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