1. Introduction
Magnetoplasmadynamic thrusters (MPDTs) have been one of the most competitive accelerators for space propulsion application such as deep space exploration, orbit transfer, etc. However, improvement of the self-field MPDT’s performance is seriously restricted by low efficiency—about 30% [
1,
2,
3]. A typical self-field MPDT consists of a central cathode with a coaxial annular anode, shown in
Figure 1. The characteristic feature of the self-field MPDT is that the magnetic field B
sf is induced by the discharge current j. The electromagnetic force F
j×B is generated by coupling the discharge current and induced magnetic field in the azimuthal direction. The gas propellant is heated, ionized, and accelerated by the effect of electrothermal or electromagnetic exhaust as high-enthalpy plasma, approximately tens of kilometers per second eventually. It is extremely difficult to improve thruster performance without a profound insight into such a complicated process.
In order to obtain a comprehensive understanding of the self-field MPDT, many models have been developed for the analysis of discharge plasma. Itsuro et al. [
4] calculated the current distribution on the electrodes of self-field MPDTs by non-dimensional Ohm’s law and Maxwell’s equations, with the assumption of small magnetic Reynolds number. The results showed that the current density near the cathode tip has a large value than other positions because of the cathode spot discharge mode. But the assumption of uniform distribution of discharge plasma limits its application, especially in finding the distribution properties of plasma. To predict the performance of self-field steady-state MPDTs, Michael [
5] developed a fully two-dimensional magnetohydrodynamics (MHD) code. The predictions of the model were in a good agreement with experimental results with different current and mass flow rates. Yet, a single fluid and single temperature approximation in this model did not capture the entire details of all the particles in the discharge plasma. Edgar [
6] showed the spatial distribution of the current density at high and low current levels over the electrode surfaces through simple relations for the scaling of performance parameters. In order to simulate the flow field of the self-field MPDT, Sankaran et al. [
7] presented an advanced code based on two-temperature MHD equations. The calculated results matched with the experimental values within 2%. However, the absence of the electrode voltage fall in the simulation caused an incomplete calculation results to compare with the measured total voltage directly. Jorg [
8] described the plasma flow by a series of conservation equations. The effort by Kubota [
9] was aimed at modeling and simulating the processes of self-field MPDT, which analyzed the ionization and acceleration processed in the wide operation range. Xisto et al. [
10,
11] developed a two-dimensional model based on the MHD equations for the simulation of self-field MPDT. This model is used to detect the influence of electrodes on the discharge current or efficiency of the thrusters. Nevertheless, such a set of complicated MHD equations are based on the hypothesis of continuum theory or some empirical coefficients of plasma, which would inevitably cause discrepancies with actual results.
Besides, these several codes based on the MHD approach cannot capture the kinetic description and the interactions of plasma particles, and unless additional boundary conditions [
5] of electrodes’ sheath are added, few of these models can describe the distribution properties of discharge plasma near the electrodes. Fortunately, the particle-in-cell (PIC) [
12,
13] method has been widely used to solve the above issues. Tang et al. [
14,
15,
16] introduced a particle-in-cell with Monte Carlo collision (PIC/MCC) model to simulate the physical processes, acceleration mechanisms, and energy conversion of applied-field MPDT. Yet, until now, there were few published particle models for the simulation of the discharge processes in self-field MPDT.
In this paper, a PIC/MCC simulation model is developed to investigate the characteristics of the discharge plasma in self-field MPDT. In
Section 2, the basic theory of the model is described.
Section 3 details the results and discussions of discharge plasma in self-field MPDTs. Finally,
Section 4 presents the conclusions of this article.
3. Results and Discussion
The simulated MPDT in the present model reported by Gallimore [
1] is a quasi-steady self-field cylinder axisymmetric geometry using an argon propellant at a power level of approximately hundreds of kilowatts. For a description of the thruster in the experiments, it consists of a 1-cm radius central cathode with an inner radius of 7.5 cm and an outer radius of 9.5 cm coaxial annular anode. The argon propellant was ionized through the interaction with the discharge current in the chamber after injection from an insulating plate. Because the reduction factor was 0.01, all of the thruster dimensions were scaled using the scaling schemes in
Table 3. The calculated area was a square region with axial and radial values of 1.2 mm, shown in
Figure 2a. The number of grids in both directions were 120 × 120. That is to say, the grid size Δz = Δr = 0.01 mm. The time-step of the simulation was Δt = 1.4 × 10
−14s, which satisfied the constraints in
Section 2.4. The whole simulation time was 1.4 × 10
−8s, including 10
6 steps.
Considering the axial symmetry of the geometry, two dimensions in space, and three dimensions in velocity (2D3V) PIC/MCC code (written in Python) was developed in the present model neglecting the azimuthal position of the quantities. The simulation region was the discharge chamber, where neutral particles are ionized and accelerated.
Figure 2 shows the comparison of the induced magnetic field distribution between the calculated value and the experimental value.
Figure 2b shows the experimental current contours in Reference [
1]. As described in Reference [
1], these current contours were obtained by measured magnetic fields using Ampere’s law. So, these percentages illustrated in the graph represent the relative proportion of the magnetic fields at different locations. In order to display the magnitude of the magnetic fields, we calculated the value of the magnetic fields directly in
Figure 2a. As can be seen from the figure, the magnetic field on the side of the discharge chamber at the position of the anode lip was the largest. This is because the induced magnetic field generated by the charged particles was in the same direction at this position, which was in the direction perpendicular to the z–r surface. In this case, the discharge current is 400A and the maximum induced magnetic field is 4.5 gauss. Along the anode lip outward, the magnetic field decreases gradually. This indicates that the direction of induced magnetic field was reversed as the position changes, weakening the whole magnetic field at the middle or right position. The magnetic field outside the anode lip was small, indicating that the current was small in this region. The calculated distribution of the induced magnetic field was basically consistent with the experimental value, which verifies the correctness of the model.
The distribution of electrons’ weight, radial velocity, number density, and ions number density are shown in
Figure 3. The input parameters of the simulation are presented as the following. The simulation results are at 5 × 10
−9s in a quasi-steady discharge process. Tthe voltage was set as 100 V and the discharge current as 400 A. The Ar gas mass flow rate was 4 g/s and the particle weight was 10
6. Most variable weight of the electrons range from 1 to 1000, which have 10
6 real electrons in every super-particle. From
Figure 3b, the electrons’ velocities, at about 10
6 m/s in the cathode vicinity, are higher than other regions. It can clearly be seen from the
Figure 3c,d that the number densities of charged particles in the cathode vicinity are far less than that in the bulk plasma.
Considering the distribution of the electrons’ radial velocity, we can guess that the electrons emitted from the cathode were accelerated by the strong electric field in the cathode sheath. With the increase in the velocity, the electrons’ energy gradually reached the ionization threshold of the gas. In this process, electrons transfer kinetic energy to ions. So, the velocity of electrons in the bulk plasma was less than that in the cathode sheath. The main accelerating region of the electrons was the cathode sheath. In the bulk plasma region, the electron and ion number density near the cathode sheath was the highest with an order of magnitude of 1020 m−3.
In order to visualize the particle density intuitively, the distribution curve of electrons’ and ions’ number density at the axial position of the middle anode lip, about 5.5 × 10
−4 m were calculated in
Figure 4a. The radial position of 0 was the axis of the axisymmetric model. The left and right ends of the curve were the cathode and anode, respectively. In the cathode sheath, the number of particles changed significantly, and the actual width of the sheath was about 5 mm. As the distance increased from the cathode surface, the number density of the charged particles gradually decreased.
As shown in
Figure 4b, the Lorenz force was perpendicular to the current and the induced magnetic field, respectively. The left sectional force F1 on the current was the biggest. Because the trend of the force on the left side of the anode lip was downward, the charged particles were constrained in the red triangle region marked between anode lip and cathode. This phenomenon confirms the pinch effect of the self-field magnetic field of quasi-steady MPDT. Although the force F3 had the opposite effect of the pinch effect, it is so small compared to F
1 and can be ignored. Comparing the number density between electrons and ions, the latter was higher than the former by almost all points. This is because the electrons were much faster than the ions at the whole position.
Figure 5 shows the contour plots of axial and radial current density. Among them, the axial current was mainly distributed in the bulk plasma and the left side of the anode lip, while the radial current distribution was more extensive, covering the entire surface of the anode lip. Since the radial current is the main component of the discharge current, the radial current density was greater than the axial current density. The average radial current density at the lower anode lip in
Figure 5b is about 100 A/cm
2, which is in good agreement with the experimental value in the literature [
1]. In fact, the current density in the bulk plasma region was greater than that near both the cathode and the anode.
To explore the collision characteristics of discharge plasma, we calculated the event number and the heavy particles energy of elastic and ionization collisions in
Figure 6, respectively. Comparing
Figure 6a,c elastic collisions (event number about 10
11 each cell) occurred one order of magnitude more frequently than ionization collisions (event number about 10
10 each cell). However, the energy transferred to heavy particles through elastic collisions (about 0.1 eV) was 3~4 orders of magnitude lower than that obtained in ionization collisions. In other words, compared with ionization collision, the contribution of elastic collision to thrust can be neglected, which reflects the important role of gas ionization in the thrust generation.
Figure 7 presents a brief description of the electron’s probability density function with velocity and axial position in different operations of the discharge. The mass flow rate of Ar gas is 4 g/s and the particle weight was 10
6 in
Figure 7a,c. The voltage was 100 V and the discharge current was 400 A in
Figure 7b,d. In
Figure 7a, it was clearly that the greater the voltage, the greater the electrons’ velocity. In addition, more electrons were in a high velocity state with voltage 200 V. The probability density value with velocity below 10
6 m/s exceeded 15%, which means that the shielding electric field in the bulk plasma region plays a significant role. With the increase of discharge voltage, the acceleration effect of the electric field is enhanced, and the number of high velocity electrons increases. However, with the same condition, the number of high velocity electrons decreases with the increase of mass flow rate shown in
Figure 7b. Therefore, it can be concluded that the higher the ionization degree of the gas, the more obvious the acceleration of electrons. Besides, the results also show that the electron weight affects the electron velocity distribution. The smaller the weight of the variable electron, the faster the particle, but the total energy of the particle is the same.
In
Figure 7c, most electrons were distributed in the axial coordinate from 0.2 mm to 0.8 mm when the voltage was 30V and 50V, and it appears to be the highest in the middle, gradually decreasing to both sides. As the voltage increases, more electrons were distributed on both sides, which shows more even distribution with the voltages of 100 V and 200 V. In the same way, in
Figure 7d, the electrons’ probability appears more uniform in the axial direction with electron weight at 10
6 rather than that at 10
5. In a word, with the high voltage and particle weight, the distribution of the electrons was more uniform in the axial direction.
Ion velocity is an important parameter for evaluating the performance of the thruster. We calculate the ions’ axial average velocity of the self-field MPDT using the model shown in
Figure 8. As we can see, the value of the velocity was negative in the left part of the discharge chamber. As the position moved to the right, the speed increased. In the plume region, the velocity was positive and continued to increase, which proves the acceleration effect of the electromagnetic field on the ions. As we can see from
Figure 8, the ions showed obviously lower axial average velocity at the axial position of the anode lip, especially on the left side of the anode lip. In addition, we can also find a similar effect in
Figure 2a. The direction of the force on the current was pointing to cathode at the left of anode lip. These phenomena indicate that the anode lip prevents the ions from ejecting the discharge chamber to some extent.
Figure 9 shows the contour plot of the electric field at a quasi steady-state discharge time. It can be seen that the electric field of the cathode sheath was much larger than that of other positions. The sheath extends about 5–10 mm outward from the cathode surface. Compared with the electric field of the sheath, the electric field in the bulk plasma region was almost unchanged, which reflects the shielding effect of the plasma on the electric field. It has to be mentioned that there was a strong change of the electric field within a small distance on the surface of anode lip, which was caused by the anode sheath. Although the grid near the anode was not encrypted in the simulation program, the electric field of the anode sheath could still be simulated, which proves that the model has the ability to solve the behavior of the anode sheath.