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268 Chapter 6 While the 0-D and one-dimensional (1-D) models described above can provide insight into the operation of hollow cathodes, to self-consistently calculate the plasma density in the insert region, including the effects near the cathode orifice, requires a two-dimensional model [32]. The insert plasma energy balance in this model can be found from the electron and ion energy equations. The conservation of energy equations was described in Section 3.5.3 of Chapter 3. The steady-state electron energy equation can be written 0= 5J kTe kTe + J2 J nkTe n ̇eU+, (6.4-28) 2ee e eene where Je is the electron current density in the plasma, is the electron thermal conductivity given by Eq. (3.5-29), is the plasma resistivity given by Eq. (3.6-21), and U+ is the ionization potential of the neutral gas. The steady- state ion energy equation is 0= 5JkTi kTi +v (nkT)+nM v2+Q , (6.4-29) 2iene i i iniT where Ji is the ion current density, n is the thermal conductivity for neutrals, and it is assumed that the ions and neutrals are in thermal equilibrium ( Tn = Ti ) in the collisional insert plasma. The energy balance equations are used to close the system of equations describing the plasma in the insert region. These equations also are used to describe the self-heating mechanism characteristic of hollow cathodes due to the particle flux and energy hitting the cathode walls. This effect will be discussed in Sections 6.5 and 6.6 with respect to the cathode thermal models. Writing the steady-state momentum equations from Eq. (3.5-5) for the ions and electrons, 0=enE– pi –Mn ie(vi –ve)+ in(vi –vn) , (6.4-30) 0=–enE– pe –mn ei(ve –vi)+ en(ve –vn) . (6.4-31) Adding these two equations, assuming that the neutrals move slowly compared to the charged particles, and writing the result in terms of the ion and electron fluxes gives Ji = m en Je (nkTi + nkTe ) , (6.4-32) M in(1+ ) M in(1+ )PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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