Investigation of transition phenomena accompanying the evolution of metallic phase of electric arc into gaseous phase is very important for the further progress in such fields as plasma technologies, electrical apparatus, plasmatrons and other technical applications. Some aspects of this transition are considered in presented paper on the base of mathematical model described dynamics of phenomena in the arc column, near-electrode zones, anode and cathode solids. Cathode and anode phenomena such as ion bombardment, thermionic emission, inverse electron flux, evaporation, radiation, heat conduction etc. are considered in dependence on time, current, opening velocity, parameters of the gas and contact materials. The conditions of the arc transition from one phase to another are formulated in terms of above characteristics and increasing of gas ionization level. Special experiments with two contacts materials, and have been carried to verify the mathematical model. The results of calculation and experimental data enables us to conclude that in metallic arc phase (short arc length), which is characterized by material transfer from the anode to the cathode, the erosion of contacts is considerably small than erosion of contacts both for resistive and inductive circuits, while in gaseous arc phase (long arc length) with opposite material transfer the rate of erosion depends on the inductance. If the inductance, then contacts have smaller erosion in comparison with contacts, however for inductive circuits situation is quite different, thus use of contacts in the case of long arcs burning in gaseous phase is more preferable. It was found also that the addition of niobium diselenide (1%) and tantalum (5%) into silver contact material which are sublimating into arc plasma enables to change ionization potential, that leads to decreasing of the arc temperature, arc duration and contact erosion.

The analytical solutions of the one- and two-phase Stefan problems are found in the form of series containing linear combinations of the integral error functions which satisfy a priori the heat equation. The unknown coefficients are defined from the initial and boundary conditions by the comparison of the like power terms of the series using the Faa di Bruno formula. The convergence of the series for the temperature and for the free boundary is proved. The approximate solution is found using the replacement of series by the corresponding finite sums and the collocation method. The presented test examples confirm a good approximation of such approach. This method is applied for the solution of the Stefan problem describing the dynamics of the interaction of the electrical arc with electrodes and corresponding erosion.

On the base of the Holm model, we represent two phase spherical Stefan problem and its analytical solution, which can serve as a mathematical model for diverse thermo-physical phenomena in electrical contacts. Suggested solution is obtained from integral error function and its properties which are represented in the form of series whose coefficients have to be determined. Convergence of solution series is proved.