Analytical Solution of the Energy Balance Equation for Thermal Induction Plasmas in Argon

摘要

The Heller-Elenbaas equation for electric arcs is applied to the case of a thermal induction plasma in argon at atmospheric pressure. The dissipated electrical energy is considered to be predominantly lost by conduction to the side walls. For obtaining a solution in closed form, the variation of the induced electric field with radial distance is assumed to follow a power law and use is made of the experimental finding that for argon at moderate temperatures the electric conductivity increases approximately in proportion with the Schmitz heat conduction potential. These two steps permit solving the equation by a Bessel function where a transformed radial coordinate accounts for the field nonuniformity. The solution yields the relative distributions of conduction potential and electric conductivity and shows that with decreasing skin depth these distributions tend toward a rectangular shape. The condition that the profiles vanish at the tube wall establishes an eigenvalue for the induced electric field at this position. From this expression, in combination with the induction law and Poynting's theorem, a relationship between the skin depth and the applied rf magnetic field is obtained which permits determination of the values of conductivity and temperature at the tube axis and thus also their absolute distributions from external conditions. These conditions can be collectively expressed by the time rate of change of applied magnetic flux which emerges as the significant independent parameter. The argon plasma could not be maintained below a flux change of 25 V and dissipates energy most efficiently at about 32 V. (Author)