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358 Chapter 7 kTe kTe 1/2 e /kT vi P w = n e e A e 2 m e s e + 2 ( E s ) , ( 7 . 3 - 5 2 ) where A is the total area of the inner and outer channel walls in contact with the high temperature plasma region, vi is the ion velocity toward the wall, and the sheath potential s is given by Eq. (7.3-44). Equation (7.3-52) shows the wall power varies linearly with density but with the electron temperature to the 3/2 power. This is why the dominant wall losses occur in the region of the highest electron temperature. The power into the anode, from Eq. (7.3-47), can be written as Pa = 2IdTeV(anode) . (7.3-53) where the electron temperature in this case is evaluated near the anode. The power radiated is PR =none *ve V, (7.3-54) where the excitation reaction rate coefficient is given in Appendix E as a function of the electron temperature, and V is the volume of the high- temperature plasma region in the channel, which can be taken to be the channel cross-sectional area times the axial thickness L. Equations (7.3-52) and (7.3-54) require knowledge of the plasma density in the high-temperature region in the channel. This can be found to first order from the beam current ne= Ib bId , (7.3-55) evbAc eAc 2 beVd M where Ac is the area of the channel exit. Finally, the power to produce the ions in the thruster is the sum of the beam current and the ion current to the walls times the ionization potential: P i o n = ( I b + I i w ) U + = b + I e w ( 1 ) I d U + , ( 7 . 3 - 5 6 ) where Iiw is given by Eq. (7.3-28) and Iew is given by the left-hand side of Eq. (7.3-52) divided by 2Te (because the electron energy hitting the wall is already included in this equation).PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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