Solution of Orifice Hollow Cathode Plasma Model Equations by Means of Particle Swarm Optimization
Abstract
:1. Introduction
2. Plasma Model of Hollow Cathode
2.1. Cathode Architecture
2.2. Definition and Assumptions
- The heavy particles (ions and neutrals) are in a thermal equilibrium between each other and their temperature is assumed to be equal to the temperature of the wall, i.e., ;
- A choked gas flow model is used in the orifice regions of both the cathode tube and the keeper;
- A double sheath potential drop is recorded at the orifice entrance region; thus, a planar double sheath is modelled (Figure 1).
2.3. Plasma Model Equations
2.3.1. Keeper
2.3.2. Orifice
2.3.3. Insert
3. Particle Swarm Optimization Methodology
3.1. Overview
3.2. PSO General Model
- = RHS − LHS of Equation (14), (current);
- = RHS − LHS of Equation (15) (emitter power);
- = RHS − LHS of Equation (13) (emitter ions);
- = RHS − LHS of Equation (16) (emitter pressure);
- = RHS − LHS of Equation (5) (orifice power);
- = RHS − LHS of Equation (4) (orifice ions);
- = RHS − LHS of Equation (6) (orifice pressure).
3.3. PSO-Code Based Solver
4. Results
- The confidence bounds range between 2 V and 5 V for the current span examined;
- The discharge current initially decreases with the mass flow rate, as expected, but it increases at the highest mass flow rates; this could be attributed to the absence of a second double sheath at the cathode orifice exit that could be negative because the plasma density decreases passing from the orifice region to the keeper region (see Figure 4a and Figure 5a);
- The shaded zone perfectly covers the experimental measurements obtained for a current of 2A and it is slightly higher at 3A; at 1A and atowest mass flow rates, the current is up to 40%ower, at its maximum, than the reference data, ikely because the system of equations does not adequately describe the plasma physics in those conditions.
Computational Effort
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
OHCs | Orifice Hollow Cathodes |
EP | Electric propulsion |
HETs | Hall Effect Thrusters |
GITs | Gridded Ion Thrusters |
HEMPTs | High-Efficiency Multistage Plasma Thrusters |
PSO | Particle Swarm Optimization |
HPTs | Helicon Plasma Thrusters |
CIRA | Italian Aerospace Research Centre |
PlasMCat | Plasma Model for Cathodes |
RHS | Right-Hand Side |
LHS | Left-Hand Side |
PB | Personal Best |
GB | Global Best |
Symbols | |
A | Area |
L | Length |
r | Radius |
d | Diameter |
q | Elementary charge |
Boltzmann’s constant | |
Vacuum permittivity | |
Residual | |
Electric field at the cathode sheath | |
Discharge current | |
j | Current density |
Ionization rate coefficient | |
Electron mass | |
Ion mass | |
Mass flow rate | |
n | Density |
Ion rate | |
p | Pressure |
P | Power |
R | Resistance |
T | Temperature |
V | Potential |
Work function | |
Degree of ionization | |
Plasma resistivity | |
Collision frequency | |
Cross section, standard deviation | |
Specific heat ratio | |
M | Molecular Weight |
Viscosity | |
Coulomb Logarithm | |
D | Material-specific Richardson-Dushman constant |
Average ionization energy (12.12 eV for xenon) | |
Subscripts | |
n | Neutral |
p | Plasma |
e | Emitter, electron |
o | Orifice |
k | Keeper |
Electron-neutral | |
Electron-ion | |
i | Ion moving at Bohm speed |
Ion or ionization | |
Electron recombination | |
Thermionic emission | |
Thermal | |
Effective | |
s | Emitter surface |
q | Optimization iterations |
z | Swarm individual |
w | Wall |
Double sheath |
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Values |
Variable | Min | Max | Unit |
---|---|---|---|
m−3 | |||
m−3 | |||
5 | 50 | V | |
1 | 4 | eV | |
m−3 | |||
m−3 | |||
1 | 4 | eV | |
1700 | 1950 | K | |
m | |||
1700 | 1950 | K |
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Coppola, G.; Panelli, M.; Battista, F. Solution of Orifice Hollow Cathode Plasma Model Equations by Means of Particle Swarm Optimization. Appl. Sci. 2024, 14, 5831. https://doi.org/10.3390/app14135831
Coppola G, Panelli M, Battista F. Solution of Orifice Hollow Cathode Plasma Model Equations by Means of Particle Swarm Optimization. Applied Sciences. 2024; 14(13):5831. https://doi.org/10.3390/app14135831
Chicago/Turabian StyleCoppola, Giovanni, Mario Panelli, and Francesco Battista. 2024. "Solution of Orifice Hollow Cathode Plasma Model Equations by Means of Particle Swarm Optimization" Applied Sciences 14, no. 13: 5831. https://doi.org/10.3390/app14135831
APA StyleCoppola, G., Panelli, M., & Battista, F. (2024). Solution of Orifice Hollow Cathode Plasma Model Equations by Means of Particle Swarm Optimization. Applied Sciences, 14(13), 5831. https://doi.org/10.3390/app14135831