Device and Plume Model of an Electrothermal Pulsed Plasma Thruster

Progress in combined device/plume modeling is presented for a Teflon-fed, pulsed plasma thruster from plasma generation to the plume far field. In this work we apply a one-dimensional unsteady model for the plasma generation and acceleration process. A new kinetic ablation algorithm is employed to calculate the Teflon ablation rate as a function of plasma parameters. A near cathode sheath model is included to calculate the plasma potential at the thruster exit plane. Results are compared with data for the electrothermal device, PPT-4. Performance characteristics of the PPT such as mass ablation and thrust impulse are calculated. Predicted plasma properties, thruster performance and plasma parameter distribution in the plume are found to be in agreement with available experimental data. introduction Pulsed plasma thrusters (PPT's) have combined advantages of system simplicity, high reliability, low average electric power requirement and high specific impulse1. The PPT is considered as an attractive propulsion option for orbit insertion, drag makeup and attitude control of small satellites. PPT's, however, have very poor performance characteristics and an overall efficiency at the level 2 3 of about 10% . To improve the PPT performance several directions are being considered . Accurate simulation of these devices and plumes is required for the design of PPT's with improved performances and for assessment of spacecraft integration effects. Research Scientist. Department of Aerospace Engineering, Member of AIAA ' Associate Professor, Department of Aerospace Engineering, Senior Member of AIAA (c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. In the present study we concentrate on the pulsed plasma thruster called PPT-4 that was developed recently at the University of Illinois. This is an electrothermal device that derives most of its acceleration from the electrothermal or gasdynamic mechanism. This thruster is axially symmetric and a discharge occurs between the annular cathode at the thruster exit plane and the circular anode located at the far end of a cyiindrica! cavity made of Teflon. The plasma generated inside this cavity is accelerated in a diverging nozzle that is attached to the downstream end of the cavity. The device has a pulse length of about 10 MS, and the overall specific impulse was measured to be 850 s. In a series of previous papers, we describe our efforts to mode! various aspects of this electrothermal PPT. In Ref. 5, a mode! of the Teflon ablation and plasma discharge processes is described. The mode! was calibrated against mass ablation data from the PPT-4. In Ref.6, the charging, heat and flow effects associated with large Teflon particulates in the plasma jet of the electrothermal PPT were considered. Here it was predicted that the small macro-particles are expected to decompose within the plasma jet. Finally, in Ref. 7, the results obtained in Ref. 5, at the thruster nozzle exit were used as boundary conditions to perform a particle-based PIC-DSMC computation of the electrothermal PPT plume. A significant conclusion from Ref. 7 was that almost all of the back-flow to the spacecraft from this device arises from carbon ions due to their high mobility. The main physical processes in this type of PPT occur in the Teflon cavity. Rapid heating of a thin dielectric surface layer leads to decomposition of the material of the wall. As a result of heating, decomposition and partial ionization of the decomposition products, the total number of particles increases in the cavity. The problem of the ablated controlled discharge has a general interest since it can be used for various applications8'9'10. In these devices, the discharge energy is principally dissipated by ablation of wall material, which then forms the main component of the discharge plasma. The ablated vapor increases the pressure within the capillary and the plasma is expelled through the exit. Previously, discharge evolution in the PPT-4 Teflon cavity was studied by Keidar et a/ assuming uniform plasma parameters. However, further understanding of the physical processes involved requires more detailed analyses including spatial variation of (c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. the plasma parameter distribution along the cavity and a more sophisticated ablation mode!. The present model of the capillary discharge employs a recently developed ablation model. These changes allow us to provide more accurate boundary conditions for the piume simulation. The capillary discharge mode! The model presented here describes the physics of the plasma generation and acceleration in a Teflon cavity for a pulsed electrical discharge as shown in Fig. 1. The main features of the model of the electrical discharge in the dielectric cavity include Joule heating of the piasma, heat transfer to the dielectric, Teflon ablation and electrothermal acceleration of the plasma up to the sound speed at the cavity exit. Mechanisms of energy transfer from the plasma column to the wall of the Teflon cavity includes heat transfer by particle convection and by radiation. The Teflon ablation is based on a recently developed kinetic ablation model, it is assumed that ail parameters vary in the axial direction x (see Fig. 1). Since the axial pressure and velocity gradients are much greater than the radial gradients we assume that radial variation of piasma temperature, pressure and velocity are neglegible,. The axial component of the mass and momentum conservation equations read: ap/at + a(pv) = 2r(t,x)/Ra.............................................................................................................. (1) where p is the piasma density, P is the pressure, V is the plasma velocity and T(t,x) is the ablation rate. The energy balance equation can be written in the form: |ne (3T/3t + V^T/Bz) = Qj Qr where Qj is the Joule heat, Qr is the radiation heat and QF is the heat associated with particle fluxes. This equation depends on the coordinate along the cavity. However, our estimation and (c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. previous calculations show14 that the arc temperature varies only slightly with axial position and therefore we further assume 9T/3z=0. The Teflon surface temperature is calculated from the heat transfer equation with boundary conditions that take into account vaporization heat and conductivity. The solution of this equation is considered for two limiting cases of substantial and small ablation rate very similar to that described in Ref. 1 1 . For known pressure and electron temperature one can calculate the chemical plasma composition assuming LTE11'15'16. The Sana equations are supplemented by the conservation of nuclei and quasi-neutrality. Electrostatic sheaths The electrostatic sheath near the cathode provides the current continuity from the cathode to the plasma bulk as shown in Fig. 1. We assume that the cathode emission plays a small rote in the current balance. The total current density in the sheath consists of electron Je and ion J, current densities: J = Jj + Je ................................................................................................_ In the case of a planar sheath in front of the cathode (the Debye radius is much less than the cathode length) Jj is determined by the Bohm relation: Ji = 0.4en(kTe/mO..............................." where n is the plasma density at the plasma-sheath interface (see Fig. 1). The electron current is due to high energy electrons that penetrate the electrostatic barrier: Je = /4 en(8kTe/me)exp(-eA/kTe) ................................................................................................. (6) where A<|> is the potential drop across the near-cathode sheath. For given current density one can calculate the potential drop: A = Te InUnrie/mi)0-5 J/en(kTe/me) .............................................................................................. (7) One can see that the potential drop depends upon current density, plasma density and electron temperature. (c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. in the cavity near the Teflon surface, the electrostatic sheath potential drop is negative in order to repel the excess thermal electrons, so that the electron current Je is equal to the ion current J,. It was concluded previously that during the discharge pulse, a quasi-steady sheath structure is formed and that under typical PPT conditions this sheath is unmagnetized in the self-magnetic field generated during the pulse. Under the conditions mentioned above, the potential drop in the sheath is calculated as: Ud = -T in (.WJj)..........................................................................................................................(8) Where Jeth is the random electron current density and Jj is the ion current density also determined by the Bohm condition. Ablation mode! The ablation model employed here is based on a kinetic model of the Knudsen layer near the ablated surface, which was analyzed using the distribution function moment method''. This method employs an approximation of the distribution function within the non-equilibrium Knudsen iayer as a sum of the distribution functions before and after this layer with a coordinate dependent coefficient. In our problem of evaporation, it is only important to know the parameters on the boundaries and not their variation between the boundaries. This means that the problem is reduced to the integration of the conservation equations of mass, momentum and energy: JVxf(V) dV = const IVxf(V)dV = const.....................................................................................................(9) JVXV f (V)dV = const After integration of equation (9) we obtain a set of equations in which parameters at the external boundary of the Knudsen iayer depend upon velocity at the Knudsen layer edge. Applying mass and momentum conservation between the edges of the hydrodynamic layer, one can find the velocity at the outer boundary at the Knudsen layer. Velocity and density at the outer boundary of (c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. the Knudsen iayer determine the abiation rate. The system of equations is closed if the equilibrium vapor pressure can be specified. In the case of Teflon, the equilibrium pressure formula is used: where P is the equilibrium pressure, Pc and Tc are the. characteristic pressure and temperature, respectively. Nozzle and Plume models The plasma flow through the conical nozzle is modeled by a quasi-one-dimensional continuum approach similar to that used previously (Ref. 7). We have assumed that the main part of the plasma generated in the cavity accelerates in the nozzle by the gasdynamic mechanism. We have considered the quasi-neutral plasma where ions and electrons are assumed to be idea! gases. This model also relies on the assumption that the flow is sourceless and plasma losses to the wall and wall evaporation are small and can be neglected. The plume mode! is based on a hybrid fluid-particle approach similar to that used previously (Refs. 7, 20). in this model, the neutrals and ions are modeled as particles while electrons are treated as a fluid. Elastic (momentum transfer) and non-elastic (charge exchange) collisions are included in the model. The particle collisions are calculated using the direct simulation Monte Carlo (DSMC) method. Momentum exchange cross sections use the model of Dalgarno et al., ?3 while charge exchange processes use the cross sections proposed by Sakabe and Izawa . Acceleration of the charged particles in the self-consistent electric fields is computed using Particie-!n-Cell method (PIC). We have assumed quasi-neutrality that allows determination of the electron density. The plasma potential with respect to the thruster exit plane is calculated using the Boltzmann relation. The plasma potential at the thruster exit plane with respect to the cathode varies with time and is calculated using the near-cathode sheath model. The grids employed in this computation are similar to those used previously (Ref.7). (c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.