The arc root constitute the continuity of the current by the existence of several emitting sites called "spots". The little dimension (<0.1 mm) of this zone and also of the associate sheath make very difficult the experimental studies. This difficulty about the analyse is increased by the very important density of energy in a little volume where the intense radiation forbids a good spectroscopy measurement...
Taking account of the thermal phenomenon at the cathode root and of the material vaporisation the electrode model is solved par a Problem of Stefan with ablation. This model is first with only one dimension and after with two (2D) to obtain a better representation of the thermal conduction in the cathode wire. Our goal is to obtain estimation about the power used for the vaporisation of cathodic material versus the incident energy flux. The numeric work, using the finite element method, with a fixed incident heating gives :
This model describes the heating of a strip of homogenous material whose length is d along z axis an infinite along x and y, the heat flux arrives at the origin (z=0). The solid under the action of this flux overheats locally , then it becomes liquid and vapour. There is apparition of an ablated zone and the excess of energy spreads into the liquid and solid sheets of the electrode. Our goal is to know the instants of appearance of liquid and vapour phases and the values of the corresponding incoming flux.
|Energy flux (W/m²)||10E9||10E10||10E11||10E12||5.10E12||10E13|
|Instant of liquid appearance||0.9ms||12.6µs||0.15µs||1ns||0.05ns||2ps|
|Instant of vapour appearance||3.33 ms||65.7µs||1.08µs||5ns||0.2ns||50ps|
Instants of appearance of liquid and vapour versus values of the incoming energy flux in the case of copper.
As it is shown on the upper table, the instants of the phase changes are function of the values of the heat flux , this give us an information on the values of the necessary energy flux to obtain the existing of a cathodic spot. In vacuum the time life of a cathodic spot is situated between 10 and 100 ns. According to the values found, we can situate the energy flux around 10E12 W/m². For higher values , the situation is this of the case where the liquid and the vapour appear in too small time and the domain is quickly eroded; furthermore we can conclude that the time life of the spot is considered to be equal to 20 ns and that there is no thermal conduction in liquid and in solid. In this case the entire part of the energy is used for the vapour.
When the gap for the variation of the values of the heat flux is determined, we can study the energy repartition in the three phases and the thickness of the liquid and the part corresponding to the metal that is became vapour (ablated length). By example, with an incoming flux of 10E12 W/m², and with a duration of 5 ns , 50% of the energy is used by the liquefaction. After 20 ns, the major part of the energy is used by the vaporisation.
Repartition versus time heating of the energies with an incoming flux equal to 10E12 W/m² in the case of copper.
The apparition of the vapour phase decrease the value of the energy absorbed by the liquid. This fact gives an increasing of the ablated thickness (vapour) and a decrease of the thickness of the liquid phase.
Variation versus time heating , of the liquid thickness in the case of copper and three values of the incoming flux.
Variation versus time heating, of the ablated thickness in the case of copper and three values of the incoming flux.
Liquid depth time evolution at the cathode
This first one dimension model is not very realist, and it gives more vaporisation and less conduction. To correct the influence of the thermal conduction on the ratio between the energy flux used to vaporisation and the energy flux used in arc root model, we used a three dimensions model...
At the end of a tip, the intense electric field create a spot. The evolution versus time of the repartition of the enthalpy with a tip of four micrometers in diameter of a copper tip with an energy flux of 10E12 W/m² is given:
Isotherms for a 1 µm-radius arc and for an incoming flux equal to 1E12 W/m²
Axial temperature for time varying from 10 to 100 ns
Radial temperature on the surface for time varying from 10 to 100 ns