A one-dimensional theory for the electrode sheaths of electric arcs

A non-equilibrium one-dimensional model is proposed for the plasma sheaths near the electrodes of electric arcs where the cathode is a thermionic emitter. Three coupled equations are solved: (a) the charge continuity equation, which accounts for ambipolar diffusion, recombination and ionization, (b) a form of Ohm's law relating the effective electrical conductivity with a local electric field and (c) the energy balance equation, accounting for thermal conduction, Ohmic heating, plasma radiation and the effects which occur at the electrode surface, i.e. thermionic cooling, ion heating, radiative heating from the plasma and radiative cooling by thermal emission. The three equations give distributions of temperature, electric field and charge density within the sheath. Results are given for the sheath at the thoriated-tungsten cathode and anode for a 200 A arc in argon. Calculations have also been made of sheath properties for various electrode materials to determine the critical current density sustainable by a cathode without cathode melting. Critical current densities for anode melting are less than for cathode melting because of the cooling effect at the cathode of thermionic emission. The authors find that the effect of ion heating tends to keep the cathode temperature near to the temperature for which the current density from thermionic emission equals the imposed current density.