Resumen de Fluid modeling and simulation of the electron population in hall effect thrusters with complex magnetic topologies
As a human activity, “Space” has become a major sector in the Global Economy (one worth more than 330,000m$[1]), and an unrelenting contributor to the pursuit of knowledge through scientific and technological achievements. The Space sector today underpins many of the technologies we use in our daily lives and encompasses a breadth of uses, from Commercial and Civil to Defense, and a far greater number of applications: communication, navigation, Earth observation, scientific exploration, and others. The future of this sector is intimately tied to the development of new technologies, which will allow us to expand our reach and capabilities in the Space environment, paving the way for bold missions such as NASA’s Deep Space Gateway[2, 3] or ESA’s Bepi-Colombo[4], which hold the promise of continued study of the solar system under international cooperation.
One of these technologies is Electric Propulsion (EP)[5, 6], a versatile, well established rocket technology which, under certain conditions, can represent a game-changer in the way Space exploration is carried out. Electric thrusters are used for non-atmospheric spacecraft maneuvering and have been present in numerous applications in the last decades: from orbit correction and orientation of commercial satellites[7] to interplanetary-transfers in deep space exploration missions[8]. The advantages of these devices are tied to the basic principles conservation of momentum; at the turn of the 20th century, Tsiolkovsky obtained the momentum conservation equation for the motion of a body of variable mass; integrating said equation yields the change in velocity achievable between two instants of time by ejecting mass at a constant velocity, when no external forces are present. Tsiolkovsky’s equation, which can be considered the beginning of modern rocketry, implies that the total mass required as propellant changes exponentially with its exhaust velocity, which is the effect of having to accelerate the propellant mass alongside the rocket before it is ejected.
In EP thrusters, energy is stored within the spacecraft and transferred to the propellant via heaters, radio-antenna or electro-magnetic fields. Most electric thrusters utilize a propellant in an electrically conducting state, known as a plasma, where electrons are separated from the nuclei of the propellant atoms. Through the acceleration imparted by electric and magnetic fields over the propellant, these devices are capable of achieving the highest exhaust velocities of all existing operational rockets. The exhaust velocity in EP thrusters presents values of 10 − 80Km/s; for comparison, in the more traditional chemical rockets this value is capped at 2−5Km/s. The large difference between both figures of merit make EP thrusters attractive due to the much lower costs in terms of propellant mass. Of course, this technology also presents its share of challenges and limitations: they are unsuitable for orbital launches, due to the very small forces that they impart on the spacecraft and cannot operate immersed in a planet’s atmosphere. Furthermore, because of these very small forces, they must operate for extended periods of time (thousands of hours), which implies that the reliability of the system and its robustness must be kept in mind when designing the propulsion system. Furthermore, they are limited by the state of the art of the electrical power sources aboard the spacecraft (solar-electric, nuclear, etc.).
Of the many types of EP rockets, this thesis focuses on the Hall Effect thruster; in these devices the propellant is accelerated through an electric field, while a magnetic field is employed to trap electrons, which transform the, initially neutral, propellant gas into an electrically conducting plasma. These thrusters have become the main technological solution for near-earth EP, due to their high efficiency over a wide throttle range and larger forces per input power than other types of EP thrusters.
The development of Electric thrusters has faced more barriers than other types of propulsion technologies, such as chemical rockets, due to the complexity of the physical interactions through which these devices function, and the difficulties associated with experimental campaigns, which require producing conditions similar to those of the vacuum of Space.
Over the past two decades, computer simulations were introduced as a novel tool in the characterization of these devices. While computer-aided-design, which is used today in many other engineering disciplines, is not yet a reality, the advent of affordable and scalable computational resources and power transformed a nascent field into one capable of providing us with a virtual laboratory, where different models and interactions may be tested in an attempt to both understand the physical response of a thruster and to predict its capabilities.
The advantages of numerical simulation are apparent in terms of savings in cost and time: the review of different designs and configurations, in an effort to characterize the response of a thruster, has little material cost and can potentially be done in the order of days (in comparison to much larger periods for testing campaigns). Additionally, certain predictive results are of importance in order to understand the performance of the device throughout its lifetime.
As a detriment to numerical simulations, we can say that the fidelity of the results and the nature of the physical response of the simulated plasma tend to be questioned. Indeed, the breadth of physical effects which are present in the plasma have a direct impact on the response of the plasma discharge and the performances that may be recovered from the simulation of a thruster. More importantly, these physical effects may be modeled through a variety of approaches, which greatly influences the scales (both temporal and spatial) that need to be resolved in the simulation and the computational resources needed to do so.
The main work of this thesis has been instrumental in the development of a new versatile simulation platform which builds upon the considerable experience in simulation of space propulsion plasmas of the Electric Propulsion & Plasmas (EP2) group at Universidad Carlos III de Madrid (UC3M). This platform, which we have dubbed HYbrid Plasma-thruster Holistic-simulation ENvironment (HYPHEN), hosts a suite of numerical tools focused on solving 2D(r-z) axisymmetric plasma discharges under the influence of magnetic fields; the effects over the discharge of particular magnetic topologies have been given special attention. The plasma regime in the simulations is characterized by the weak collisionality of the heavy-species populations, the magnetization of the electron population and a negligible self-field production by the plasma current. This allows HYPHEN to simulate the response of Hall Effect Thrusters (HETs) and derivative types, and, potentially, other types of thruster architectures, such as High Efficiency Multi-stage Plasma Thrusters (HEMPTs) or applied-field Magneto-Plasma-Dynamic Thrusters (MPDTs). The code is not presently prepared to simulate thrusters that operate in the collisional Magneto Hydro Dynamic (MHD) regime, like the Pulsed Plasma Thruster (PPT) or the self-field MPDT.
The core simulation unit is a hybrid Particle-In-Cell (PIC)-Fluid code: hybrid codes are a popular approach in EP simulation[9, 10, 11] and receive their name because they employ a combination of different models for the description of the various populations in the plasma. In this case, the well known PIC approach was used for the heavy species in the simulation (neutrals and ions), while a fluid approach was been used for the electron population. Under this hybrid approach, ions and neutrals are allowed to present kinetic effects, i.e., those that separate the distribution of particles from the Maxwellian equilibrium.
On the other hand, while plenty of sources for non-Maxwellianity have been found for the electron population, its modeling as a fluid is advantageous mainly from the perspective of computational resources, while retaining the capacity to describe the physics of the “bulk” population. This approach contrasts with full-PIC methods, in which electrons are also treated as particles and are still out of reach of present computational advancements, if representative simulation domains and times are sought.
The work carried out in this thesis has been oriented mainly toward the development of the fluid model for the electron population, its integration alongside the PIC segment in the general structure of the code, and the correct interaction between them. The PIC module itself was developed in its entirety by Domínguez-Vázquez[12] and validated through the simulation of plasma plumes in a Gridded Ion Thruster (GIT). Present efforts are being centered on the simulation of the complex HET plasma discharges.
The electron module was first tackled from the perspective of the requirements needed to solve a fluid approach. The fluid equations can be derived by taking velocity moments of the well known Boltzmann equation, obtaining the transport evolution equations for macroscopic quantities such as electron density,momentum, energy and heat-flow. Under the presence of a confining magnetic field, these transport equations present a large anisotropicity with respect to the perpendicular and parallel directions to the magnetic field.
This lead to proposing that the electron fluid model be resolved in a 2D-Magnetic Field Aligned Mesh (MFAM). The rationale for the use of this mesh was based on an analysis on numerical diffusion in Cartesian meshes for a classic anisotropic diffusion problem: that of a fully magnetized quasineutral plasma for an infinite 2D slab with fully conducting walls and infinitely straight magnetic field lines[13]. The analysis showed that, in order to avoid excessive numerical diffusion, the resolution of the transport equations should be done in a numerical mesh aligned with the preferential directions of the problem, imposed by the magnetic field.
The MFAM was explored in depth, starting with various strategies for mesh generation and correction. The quality of the mesh was also assessed, both from the perspective geometric quality indicators as well as in a representative numerical problem. An investigation into methods for spatial discretization and gradient reconstruction in the proposed mesh was also carried out. An ad-hoc gradient reconstruction method was delivered, based on the well known Weighted Least Squares (WLSQR) approach. The Order-of-Accuracy of the method was checked for a representative mesh with various refinement levels and different analytical trial functions, and was found to be between first and second order, which is in line with results from the literature and is acceptable for the problem presented here.
The electron fluid model was discretized in the 2D(r-z)-axisymmetric numerical MFAM, utilizing a two-temperature bi-Maxwellian closure for the mass, momentum and energy equations, and a collisional closure for the heat transport equations. The model was developed under a number of assumptions: the plasma is quasi-neutral, spatial scales in the order of the Debye length are neglected, viscous stress tensor is neglected and the thermal energy is assumed to be much larger than the kinetic energy of the bulk electron motion. Furthermore, a “free” parameter, known as the anomalous collision frequency, has been used to describe the effects over electron transport associated to the azimuthal direction, wave-like phenomena and near-wall scattering, which all fall under the description of non-classical plasma phenomena[14, 15, 16, 17, 18].
The main difference to previous models found in the literature is the anisotropic two-temperature approach, which permits taking into account the effects of magnetic field non-uniformity in electron transport and preferential heating. While the latter effect may not be of such importance in HETs, it is paramount in the simulation of other types of devices, which we hope to tackle in the near future. The model can also be solved for the isotropic case, which has been the focus of the results obtained in this work.
The fluid model for the electron population was solved using the well known Finite Volume Method (FVM), which poses the transport equations in the strong integral form and is recommended for unstructured meshes in conservation-law type problems. Two temporal schemes were implemented: a forward Euler method and a semi-implicit scheme based both on the forward and backward Euler methods. The stability and convergence of both methods were investigated, concluding that only the semi-implicit scheme is capable of delivering coherent solutions with a reasonable computational toll.
The fluid model was completed with two ancillary physical models: first, a 1D-fluid model for plasma boundary layers (plasma sheaths) in material walls, which has been developed based on a first principles approach to the sheath’s physical mechanisms, tailored to include anisotropicity in the primary electron population and generalized for arbitrary magnetic angles at the wall. Second, a collision rate and collision energy yield model based on various collision frequency models found in the literature and experimental data from well known data repositories, for Xenon and Argon propellants.
The electron fluid module was tested against a solution for a HET discharge obtained from previous codes[10, 19, 20]. The sensitivity of the numerical scheme was evaluated for initial conditions, time-step values and various simulation conditions, including different control schemes for the virtual Power Processing Unit (PPU). The analysis was done from the perspective of energy balances and residuals of the plasma quantities represented by the electron population. The tests performed were oriented toward providing sufficient confidence in the electron population segment, before trialing it alongside the PIC module.
For the complete hybrid simulation (allowing the fluid and particle modules to interact and advance in time) we carried out a parametric investigation for various parameters in the simulation: mainly, the anomalous collision frequency factor, the electron-neutral collision models and others.
The backdrop chosen for this investigation was the well studied SPT-100 HET; this allowed the results to be compared to the experimental performance[21] of the thruster and the reported plasma quantities of the plasma discharge[22, 23, 24, 25]. The reference simulation performance values fall within 10% of the nominal operation performances of the thruster; the recovered plasma profiles are also comparable to the ones measured and found in the literature. Known oscillations in the discharge, associated to the Hall thruster “breathing mode” and ion transit times were also recovered in the results. The current continuity and the energy balances for the electron population and the plasma, show that the simulation conserves the main transport quantities, providing further confidence in the results.
Finally, from the perspective of code development, this work has helped cement a new development methodology for plasma discharge simulation codes in our research group, which is shared with other codes such as EP2-PLUS[26, 27]. The methodology based on test-driven design, modularity, flexibility and a focus on industry and developer community standards and well known libraries. Commonality in different program modules has been encouraged in order to reduce development times and spur reusability. Furthermore sufficient documentation for developers/users has been produced in an effort to ensure continued development of simulation codes in the future; this includes a clear understanding of algorithmic flow and interaction among the different modules.
[
