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360 Chapter 7 where Iiw is the ion flux to the wall. Following Hobbs and Wesson, the modification to the Bohm criterion is small and E Te /2 from the Bohm criterion. From Eq. (7.3-44), the sheath potential for xenon and BNSiO2 walls in the SPT-100 thruster, assuming an average electron temperature along the channel wall of 25 eV, is about –54 V. Plugging these values into Eq. (7.3-57) gives Pw = 45.8IiwTeV + 2.65IiwTeV = 48.5IiwTeV. (7.3-58) The first term on the right-hand side is again the electron power loss to the wall (written in terms of the ion current to the dielectric surface), and the second term is the ion power loss. The power loss to the channel wall due to the electron loss term is an order of magnitude larger than the power loss due to ions. It is convenient in evaluating the efficiency of the thruster to relate the ion current to the wall in Eq. (7.3-58) to the beam current. In the plasma, there is an electric field toward the wall due to the pre-sheath of approximately TeV /2r = Te /w . There is also the axial electric field of Vb /L producing the beam energy. It is common in Hall thrusters to find that the electron temperature is about one-tenth the beam voltage [35], and the channel width is usually approximately L [4,20]. Therefore, the axial electric field is on the order of 10 times the radial electric field. On average, then, the ion current to the channel walls will be about 10% of the beam current. This very simple argument agrees with the SPT-100 example results given in the previous section and the results of Baranov [40]. Using Eq. (7.3-58) with the above estimates for the ion current and electron temperature, the power loss to the insulator walls is Pw = 48.5IiwTeV = 48.5(0.1Ib )(0.1Vb ) = 0.49IbVb . (7.3-59) The power loss to the anode is due to the plasma electrons overcoming the sheath potential at the anode surface. From Eq. (7.2-24), the anode electron current is Iea = Id + Iia. (7.3-60) Neglecting the ion current to the anode as small (due to the mass ratio), and realizing that each electron deposits 2kTe/e to the anode for positive plasma potentials (from Appendix C), the power to the anode isPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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