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Ion and Hall Thruster Plumes 405 where is given by 2 vB t. (8.3-27) R These expressions describe the beam expansion for the case of no initial radial velocity or for an initial radial velocity distribution that is proportional to the radius. Examples of schematic beam profiles as a function of distance from the thruster were given in Fig. 8-8. For the case of an initial radial velocity profile, the integral in Eq. (8.3-25) is h v2 1/2 h 0t = 1+2 B lnx dx, (8.3-28) 1 R 2 h 02 where the integral has to be calculated numerically. Park’s model has been extended by Ashkenazy and Fruchtman [12] to include thermal gradient and two-dimensional effects. The Park’s formula is very similar to an empirical formula developed earlier by Randolph for Hall thrusters [13]. Randolph’s formula has two Gaussians but does not have the curved trajectories of the Park’s formula. The four parameters, k0 through k3, in Randolph’s formula are chosen to fit plume measurements: While the analytical expressions above are invaluable for estimating plume interactions, multi-dimensional computer models are normally used for detailed calculations. There is general agreement on the physics that control the expansion of the main ion beam from ion and Hall thrusters, but there are differences in the numerical algorithms used to calculate the expansion. Several researchers [3,4] employ particle-in-cell (PIC) algorithms, where the beam is modeled as a collection of macro-particles with each particle representing a large number of ions. The velocity and acceleration of each particle are followed in the self-consistently calculated electric field. Another approach, which is much less computationally intensive, is to model the thruster beam as a drifting fluid of cold ions and warm electrons. In this method, the expansion of the fluid-like ion beam is calculated using a Lagrangian algorithm [5,6]. The ion beam profile for the Nuclear Electric 2 2 2 j=R k0exp (sin ) +k2exp . (8.3-29) r2 k12 k32PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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