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120 Chapter 4 where m ̇d is the mass flow into the discharge chamber and Ag is the area of the grids. In the event that there is a significant double-ion content in the discharge plasma, the beam current and the discharge chamber mass utilization efficiency must be corrected using these equations. 4.3.6 Neutral and Primary Densities in the Discharge Chamber The ion and excited neutral production rates described by Eqs. (4.3-25) and (4.3-26) require knowledge of the neutral gas density in the discharge chamber. The neutral gas flow that escapes the chamber (the unionized propellant) is simply the gas injected into the discharge chamber minus the gas particles that are ionized and extracted to form the ion beam: Qout = Qin – Ib . (4.3-32) e The neutral gas that leaks through the grid is the neutral flux on the grids (in particles per second) times the grid optical transparency Ta and a conductance reduction term c known as the Clausing factor [31]: Qout = 41 novoAgTa c, (4.3-33) where vo is the neutral gas velocity, Ag is the grid area, and c is the Clausing factor. The Clausing factor represents the reduced conductance of the grids for finite grid thicknesses and results from Clausing’s original work on gas flow restriction in short tubes. For typical grid apertures with small thickness-to-length ratios, the Clausing factor must be calculated using Monte Carlo techniques, an example of which is given in Appendix G. In general, ion thruster grids will have Clausing factors on the order of 0.5. The mass utilization efficiency of the thruster discharge chamber is defined as md = Ib . (4.3-34) Qine Equating Eqs. (4.3-32) and (4.3-33), using Eq. (4.3-34), and solving for the neutral gas density in the discharge chamber gives no = 4Qin (1 – md ) = 4 I B (1 – md ) . (4.3-35) vo AgTa c voeAgTa c mdPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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