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Hall Thrusters 371 The ionization and excitation reaction rate coefficients in Eqs. (7.4-23) and (7.4-24) are given in Appendix E. Finally, the electron energy density is given by convection in the plasma as = 52 nekTe. (7.4-25) 7.4.1.4 Ion Current. Several methods have been used to describe the ion generation and transport in the Hall thruster models. First, the ions have been modeled as a fluid using continuity equations [54,60], where the axial motion is due to the electric field along the channel and the radial motion to the wall is determined by the ion-neutral scattering frequency. The ion current to the wall is then Iw = nino invi AwL , (7.4-26) where Aw is the wall area, L is the plasma length, in is the ion-neutral collision cross section for 90-deg scattering including elastic and charge- exchange collisions, and the velocity of the neutrals is neglected relative to the ion velocity. In PIC numerical codes, this represents the radial flux to the cell boundary where AwL becomes the cell volume. Fife [30] modeled the ion motion using a 2-D PIC code that assumed the ions and neutrals acted as discrete macro-particles in each cell. The time step in the ion-PIC code, in this case, was adjusted (to typically three orders of magnitude slower than the electron model time step) to handle the ion-motion time scales without invoking excessive computational time. Finally, the ion Vlasov equation has been used to solve for the ion generation and motion [52,53]. This has primarily been applied for investigating low- frequency oscillations on the order of the ion-characteristic time scales. In one dimension, this can be written as f+vx f+eE f =neno ive (vx vo), (7.4-27) t tM vx where f is the ion distribution function and (vx vo ) is the Dirac delta function evaluated for the ion velocity relative to the neutral velocity. The ion density is then found from ni = f(x,vx,t)dvx. (7.4-28)PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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