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Ion and Hall Thruster Plumes 407 M Dvi = e . (8.3-32) Dt Since the drift velocity of the ions is much greater than their thermal velocity, the high-velocity ions are modeled as a fluid, with a velocity of vi . The governing equations, solved in two-dimensional (2-D) (R–Z) geometry, are conservation of mass and momentum: nvi = 0 Mvi nvi = en . (8.3-33) The accuracy of the algorithm has been confirmed by comparisons of analytical solutions with model problems in one and two dimensions [8]. The Lagrangian modeling approach leads to reduced numerical noise as compared with PIC algorithms. However, unlike PIC algorithms, the fluid technique ignores spreading of the beam due to ion temperature and, in the case of ion thrusters, the angular distribution coming out of each grid aperture. 8.3.2 Neutral Gas Plumes The neutral gas density in a laboratory vacuum chamber has three components: gas from the thruster, gas from the neutralizer hollow cathode, and the background chamber density. To model the neutral gas density, the gas from ion thrusters can be approximated by isotropic emission from a disk with the diameter of the grid: na ~ cos . (8.3-34) r2 For Hall thrusters, the neutral gas density can be approximated using an annular anode gas flow model with isotropic emission from the channel. This is done by calculating emissions from two disks, one large and one smaller, and subtracting the smaller from the larger. The neutral density drop-off with r and z from a disk emitting a Maxwellian distribution is calculated using an approximate view factor. Energetic charge-exchange (CEX) neutrals are negligible compared to the total neutral density and therefore are not included when modeling the neutral gas density. For plume models, the neutral gas from the neutralizer hollow cathode is usually assumed to be from isotropic emission at a constant temperature equal to the neutralizer cathode orifice temperature. While the neutralizer is offset from the thruster axis of symmetry, in cylindrical 2-D codes there are an equal number of points from the thruster axis closer to and farther from thePDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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