Single-stage EHD thruster response to several simulation conditions in nitrogen gas
Victor H. Granados, Mario J. Pinheiro, and Paulo A. Sá
Citation: Physics of Plasmas24, 093508 (2017); doi: 10.1063/1.4986219 View online: https://doi.org/10.1063/1.4986219
View Table of Contents: http://aip.scitation.org/toc/php/24/9
Published by the American Institute of Physics
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Single-stage EHD thruster response to several simulation conditions
in nitrogen gas
Victor H.Granados,1,a),b)Mario J.Pinheiro,2and Paulo A.Sa1,b) 1
Departamento de Engenharia Fısica, Faculdade de Engenharia, Universidade do Porto, Rua Doutor Roberto Frias, s/n, 4200-465 Porto, Portugal
2
Departamento de Fısica, Instituto Superior Tecnico - IST, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
(Received 2 June 2017; accepted 10 August 2017; published online 28 August 2017)
We use a numerical model to investigate the influence of pressure from 0.5 Torr (66.7 Pa) to 100 Torr (13.3 kPa) and temperature (190–400 K) on the performance (thrust, fluid velocity, and thrust-to-power-ratio) of a single stage electrohydrodynamic thruster made of a rod anode and funnel-like cathode geometry, using nitrogen as the working gas. The model includes the following nitrogen species: N, Nþ, N2, Nþ2, and N
þ
4. Additional factors are investigated: (i) the ballast
resis-tance, (ii) the secondary electron emission from the cathode (in the range of 105–10), and (iii) the influence of the gap between electrodes on the discharge. As expected, higher pressures increase the net thrust, thrust efficiency, and peak gas velocity; however, with increasing tempera-tures, the trend reverses. We notice that gas flow velocity diminishes for the increasing values of the secondary emission coefficient, and it is possible to identify two working regimes presenting different behaviors: in the first region, for values of the secondary electron emission coefficient between 105 and 102, thrust was not affected, and in the second region, between 102 and 1, a clear decrease in thrust is observed, accompanied by an increase in the discharge current, an unde-sired effect for the purpose of thrust production.Published by AIP Publishing.
[http://dx.doi.org/10.1063/1.4986219]
I. INTRODUCTION
Electrically powered spacecraft propulsion systems make use of electrical energy to accelerate a propellant by different electrical and/or magnetic means. This phenomenon is known as electrical propulsion, and it relies on the momentum con-servation law: a force, called the thrust, is exerted on a space vehicle through the ejection of, usually, a working gas gener-ally containing high kinetic energy. Electrical propulsion engines can be much more fuel-efficient than chemical rock-ets or jet engines, which convert the energy released by chem-ical reactions between the fuel and an oxidizer into kinetic energy. Since electric propulsion systems require very little mass to accelerate a spacecraft, they are mostly limited by the electric-power on-board, and therefore, an electric propulsion system is suitable to low thrust outputs (micro and milli-newton levels). An electrical engine, such as electrohydrody-namic (EHD) thruster, with a partially ionized plasma as the working gas, is fuel efficient but at the expense of electrical power in order to sustain the plasma formation and gas acceleration.
To fulfill the future needs for space exploration, electric propulsion appears as a solid candidate for deep-space mis-sions since it is possible to obtain energy on-board through different mechanisms, e.g., solar panels and nuclear reactors, but there are not, by now, means to resupply spacecrafts with fuel during missions. Research and development programs
about electrical propulsion technologies and new concepts are still unfolding in Europe, Russia, United States, and Japan.1
Plasma thrusters are classically grouped into three main categories according to their thrust generation processes: elec-trothermal (the acceleration mechanism is based on the pres-sure gradient, which is the same acceleration mechanism as the one employed in chemical thrusters but using electric energy to heat the propellant in a chamber, where expansion of the hot neutral or ionized gas through a suitable nozzle allows conversion of the thermal energy into kinetic energy, imparting momentum to the spacecraft); electrostatic (after the ionization of the propellant, an electric field is directly employed to accelerate ions out of the thruster); and electro-magnetic (devices combining both electric and electro-magnetic fields that are employed in order to ionize, accelerate, and guide the propellant out of the thruster).1,2The three groups, along the associated plasma discharge and energy transfer mechanisms, include long-standing technologies such as arc-jet thrusters, magnetoplasmadynamic thrusters, and ion engines, as well as Hall thrusters and variants.3,4We refer the reader to several textbooks and review articles about electric propulsion.3–10
EHD concepts are relatively mature, and the associated research has been pursued since the 18th century. The first instance of ion-wind created by the ionization and acceleration of air molecules between the electrodes of a high-voltage source was observed and described by Francis Hauskbee. Stuhlinger11and Tsiolkovskiy12were the first to propose elec-trostatic thrusters, which were started being used in the 1960s. The first in-space demonstration of electric propulsion occurred
a)Electronic mail: [email protected]
b)Also at Centro de Estudos de Fenomenos de Transporte (CEFT),
Universidade do Porto, Porto, Portugal.
in 1964 with the Space Electric Rocket Test (SERT-1), and this was achieved with an ion engine aboard the Soviet Zond-2 satellite.3About 20% of the commercial satellites use electrohydrodynamic (EHD) mechanisms for attitude and alti-tude control,13but in fact, more research focused on the opti-mization of EHD processes is needed. EHD effects use only electrostatic forces and the transfer of momentum by multiple collisions with a working gas. The electric field accelerates the electrons that subsequently gain sufficient kinetic energy that permits the ionization of the propellant gas. Researchers have explored the use of ion-wind for air filtration devices, solid-fluid boundary layer modification, cooling of integrated circuits, EHD pumping, particle removal in electrostatic pre-cipitators (ESPs), EHD jet printing, electroacoustics (EHD speakers), and propulsion applications. Recent research in this field produced growing interest in utilizing the thrust resulting from a high dc voltage across an asymmetric capacitor as a mean of propulsion.4,14
The present work widens the scope of single-stage EHD thruster research by analyzing the effect of several key condi-tions on the performance of the EHD thrusters in order to optimize their construction and their adequate operation con-ditions. We study the efficiency of several parameters (thrust, T/P ratio, discharge current, peak velocity, spatial velocity profile, electron density, among others) that characterize an EHD thruster under the influence of gas pressure (in the range of 0.5–100 Torr) and gas temperature (190 K–400 K). To avoid overcharging of supplied information, we selected only nitrogen and the funnel-like cathode geometry (because it pro-duced the highest thrust for nitrogen). Besides pressure and temperature studies, we analyzed the role of the discharge current and how it is related to the ballast resistor for the same range of gas pressure. We studied the influence of the secondary electron emission from the cathode and the dis-tance between electrodes on the performance parameters over the extended range of values. All the previously mentioned tasks were not accomplished in the previous works.
In Sec.II, we may find a description of the mathematical model for the EHD thruster. Section IIIintroduces detailed data and analysis of the impact of several key parameters on performance for the EHD thruster, i.e., gas pressure, gas tem-perature, ballast resistance, secondary electron yield, and electrode gap. In Sec.IV, we may find the main conclusions of the investigation along with possible lines for future work in order to improve the current results.
II. MODELLING OF AN EHD THRUSTER
In the near-space region (altitudes of 20–100 km), pressure ranges from roughly 40 Torr (5.33 kPa) to 0.25 Torr (33.3 Pa) and temperature between 190 K and 270 K. Due to this varia-tion, propulsion systems must be designed to operate in these environments. We developed a numerical model to study the performance (thrust, fluid velocity, and thrust-to-power ratio) of a single-stage EHD thruster with a rod anode and funnel-like cathode geometry. Simulations swept over pressures of 0.5–100 Torr and temperatures in the range of 190–400 K.
The details of the model were explained in a previous paper by Granadoset al.,15and the model consists of three
modules: (i) a heavy-species kinetic model for the main ground-states and ionized species of nitrogen gas; (ii) a plasma model obtaining the self-consistent electric field and space charge density using an external electric circuit to power the discharge; and (iii) a fluid model solving the Navier-Stokes equation under the assumption of laminar flow. The electron number density and the electron kinetic energy distribution are determined in the plasma model and used to calculate the electron-impact reaction rates.
The reactor is schematically shown in Fig. 1, depicting the axisymmetric computational domain and the thruster geometry. For the general boundary conditions, we consider the fluid to have zero velocity at the surface of both electro-des and the simulation edge parallel to the axis of symmetry in order to simulate a wind tunnel. In the fluid entrance at the bottom of the domain, we consider a pressure pp0, with
p0the average gas pressure, so we solve for both the velocity
and pressure at the boundary but guarantee to avoid any back flow. The top domain boundary is an open end with the same pressure as the gas, allowing it to flow and solving for the velocity.
We designate the distance between electrodes, d, as the distance between two parallel planes tangent to both electro-des according to Fig.1since the cathode is a hollow struc-ture, and this definition simplifies the notion of distances for the purposes of the study.
FIG. 1. Boundary conditions and dimensions of the simulation domain.
We consider a RC electric circuit of discharge initiation between the voltage source and the electrodes. The resis-tance is often called the ballast resisresis-tance (Rb), and it plays
an important role in the stability of the discharge, as well as the onset electric field between electrodes. The circuit is described in detail by Granadoset al.15
The buffer gas is nitrogen, and the species included in the model are N, Nþ, N2, Nþ2, and N
þ
4 (see Table I). After
applying a voltageVinto the RC circuit and into the
electro-des, a plasma is formed consisting of electrons and positive nitrogen (atomic and molecular) ions. Then, the flow of the gas is described by using the Navier-Stokes equation, neglecting turbulence. The inlet flow velocity is assumed axial with a pressure condition higher or equal to the work-ing pressure, so the velocity will be determined by the drag of neutrals between electrodes.
In order to simulate the model, we use a finite-element method, creating a mesh of points in the simulation domain where the differential equations are solved, while consider-ing the appropriate boundary conditions for the problem. We use COMSOL MultiphysicsVR
software with a time-dependent solver in order to achieve steady-state solutions for each case study. At each time step, the electric field,E, is determined by solving the Poisson equation
r E¼qc 0
: (1)
The space charge density,qc, is computed consistently as
qc¼e
XN
j¼1
Zjnjne
0 @
1
A; (2)
whereZjis the ion charge number,njthe ion density, andne
the electron density for speciesj.
We solve the continuity equation for the electron den-sity,ne, and the electron energy density,n, respectively.
@ne
@t þ r Ce¼Re ðu rÞne; (3)
@n
@t þ r CþECe¼R ðu rÞn; (4)
whereCe is the electron flux in the drift-diffusion
approxi-mation, which is given as
Ce¼ ðleEÞne rðDeneÞ (5)
andCis the electron energy flux, which is given as
C¼ ðlEÞn rðDnÞ: (6)
Re andR are the electron density and electron energy
den-sity sources, respectively.De is the electron diffusivity ten-sor, and D is the electron energy diffusivity tensor.
Additionally,leandlrepresent the tensors for the electron
and electron energy mobility, respectively.
The electron source term,Re, is calculated as the sum of
electron impact reaction rates of all the M considered reactions
Re¼X
M
j¼1
xjkjN0ne; (7)
wherexj is the mole fraction of the target species and kj is
the rate coefficient for each reactionj.N0is the total neutral
number density.
We considered an initial mole fraction for each species and restrained the sum of all to be the unit by relaxing the mole fraction ofN2since it is the most abundant species in
the reactions.
The reaction rate coefficients of some electron-impact reactions are computed for electron collisions using a set of cross-sectional data (see TableI) and assuming, for simplic-ity, a Maxwellian distribution function, FMðÞ, from the equation
kj¼ 2e
me
1=2ð1
0
rjðÞFMðÞd; (8)
whereis the electron energy (in eV) andrjðÞare the
elas-tic, excitation, and ionization electron cross sections consid-ered for each reaction.
We note that the processes of gain and loss of electron energy are adequately described by the electron Boltzmann equation, but here we use the Maxwellian distribution func-tion which is well known to overestimate the rate coefficients due to the fact that the high energy tail of the electron energy density function (EEDF) is more populated. In this way, the use of the Maxwellian EEDF leads to a deviation from the more representative EEDF, obtained by solving the full Boltzmann equation for electrons (including elastic, inelas-tic, superelasinelas-tic, and electron-electron collisions). We remind the reader that the Maxwellian is the limit of the EEDF when the ionization degree is high (104), i.e., the electron-electron collisions are relevant.19–21
In our study, we are interested in the main impact of the plasma parameters on the thrust performance in order to obtain the electron rate coefficients, the electron transport parameters, and collisional data to determine the populations
TABLE I. Electron-impact and chemical reactions with rate coefficient con-stants for nitrogen as used in the present work.
Rate coefficienta
Reaction (m3/s or m6/s) References
eþN2!eþN2 fðrÞ 16
eþN2!2eþN2þ fðrÞ 16
eþN2!eþ2N fðrÞ 17
eþNþ
2 !2N 2:81013ð300=TeÞ0:5 18
eþN2þþN2!2N2 2:61039ð300=TeÞ1:5 18
eþNþ
2 !N2 11025ðTg=TeÞ4:5 18
eþN4þ!N2þN2 21012ðTg=TeÞ0:5 18
Nþ
2 þN!NþþN2 7:21019expðTg=300Þ 18
N4þþN2!Nþ2 þ2N2 2:11022expðTg=121Þ 18
Nþ
2 þ2N2!Nþ4 þN2 5:01041 18
af
of the various species. Therefore, the Maxwellian electron energy distribution function must be solved simultaneously with a system of balance equations for the various heavy par-ticle species.
Although the use of the Maxwellian EEDF represents a simplification when compared with the more realistic EEDF Boltzmann equation, it provides computationally faster solu-tions in the framework of a two-dimensional finite element model with a dense mesh.
The electric current constituting of electrons and drifting ions impels the bulk fluid that is assumed to be incompress-ible, i.e.,r u¼0, which is well described by the Navier-Stokes equation as follows:
qf @u
@tþqfðu rÞu¼ rpþ r l ruþ rð uÞ T
h i
þf;
(9)
whereqfis the fluid density anduis the velocity vector.
The EHD force is calculated using f¼qcE. The gas
temperature equation is not included in the model, and we expect this assumption to have marginal impacts on our con-clusions since the electric current is quite low, and contrary to the working of the thermal thrusters, where the pressure and buoyancy effects are very important, in this context, the low temperature has no considerable impact on the rate coef-ficients. However, in Sec.III, a parameter study for the gas temperature is included in order to corroborate the latter assumption.
The total thrust given by the fluid passing through the chamber is determined by
T¼2pqf
ðR
0
rv2zdr; (10)
whereris the distance from the center of the thruster sym-metry axis (r¼0) to the cathode’s wall, located atR, andvz
is the velocity component parallel to the axis of symmetry (in thezdirection).
The current flowing between electrodes is calculated as
I¼
ð
ðnJiþnJeÞdS; (11)
wherenJiis the normal ion current density andnJeis the normal electron current density at the surface of the electro-des. The minus sign in Eq.(11)corrects for the normal com-ponents of the current densities pointing outwards on the
surface of the electrodes, and dS represents the elementary area of the electrode surfaces.
When solving self-consistently the Poisson equation, the electric field (and consequently the E/N ratio) is determined in each point of space, showing the characteristic features of the corona discharge, mainly the fast variation of the electric potential near the electrodes and the verification of quasi-neutrality condition in the remaining volume of the thruster.
We present the reader with five studies, spanning key parameters in order to understand their impact on the overall behavior, namely, (i) the gas pressure, p, (ii) gas tempera-ture, Tg, (iii) ballast resistance, Rb, (iv) secondary electron
emission coefficient (SEEC),ci, and (v) gap between
electro-des, d. For each study, the rest of the conditions remained unchanged. Table II summarizes the main parameters applied to each study. We also see Vinas the input voltage
source, and Cb is the blocking capacitance, used to avoid
voltage spikes between electrodes.
III. RESULTS AND DISCUSSIONS
A. Influence of gas pressure
Figure 2(a)shows in detail the pressure dependency of the thrust production, where an increase of several of orders of magnitudes is found when the gas pressure varied from 0.5 Torr to 20 Torr. As the pressure increases, so does the power spent on sustaining the discharge as seen in Fig.2(b), which is expected since both the gas velocity and thrust increase as well. The electric current flowing between elec-trodes diminishes for the increase in pressure, which leads to an increase in the onset potential due to the loss of energy for collisions with heavy species, strengthening the electric field on the configuration and improving the efficiency of the thruster as seen by the thrust-to-power (T/P) ratio in Fig.2(c)
since an increasing volume of the plasma feeds the gas with ions able to transmit momentum to the neutrals. We may also notice in Fig. 2(d) how the peak velocity magnitude increases with the pressure, although not as regularly as the rest of the considered parameters.
Figure3shows the natural logarithm of the electron den-sity, depicting how the electron cloud sets on the space between electrodes. For every case, the peak of the electron cloud is located at the tip of the anode with the ionization region expanding with pressure. At 0.5 Torr, electrons tend to spread more regularly around the anode, an outcome of the corona effect favoring the increase in the area of the anode, but this mechanism diminishes in the other cases due
TABLE II. Simulation conditions.
Pressure sweep Temperature sweep Resistance sweep SEEC sweep Gap sweep Parameter (0.5–100) Torr (190–400) K (100–1000) MX (105–100) (0–5) cm
Vin 3000 V 3000 V 3000 V 3000 V 3000 V
Rb 800 MX 800 MX … 800 MX 5000 MX
Cb 1 pF 1 pF 1 pF 1 pF 1 pF
Tg 300 K … 300 K 300 K 300 K
d 0.93 cm 0.93 cm 0.93 cm 2 cm …
ci 0.05 0.05 0.05 … 0.05
p … 10 Torr 0.5–100 Torr 10 Torr 10 Torr
to the increased velocity of the gas flowing around the anode, meaning a higher rate of collision with neutrals. For p¼30 Torr, we notice that the electron cloud tends towards the cathode’s entrance and along the axis of symmetry, an elec-tron behavior that reinforces with increasing gas pressure.
In Fig.4, we may see the gas velocity distribution of the thruster for a range of gas pressures. It is clear how the higher pressures have a directly proportional impact on the fluid velocity. The highest values of the velocity are all located at the tip of the anode. This is due to the increase in power [see Fig.2(b)] with pressure, leading to a higher vol-ume of the plasma and an increase in the electron density. However, the rate of power increase is reduced by the loss of energy due to collision with heavy species, leading to a decrease in the current [Fig.2(b)].
The force acting on charged particles, in a simple approach, is proportional to the current density and inversely proportional to the ionic mobility. Hence, with increasing pres-sure, the ion-neutral collision frequency also increases and ionic mobility decreases; with current, thrust tends to increase.
B. Influence of gas temperature
As the temperature of the gas ranged between 190 K and 400 K, we notice a slight decrease in performance. Namely, Fig.5shows the decrease in the produced thrust, peak veloc-ity, and T/P ratio for a constant gas pressure of 10 Torr. Additionally, higher values of the gas temperature translate into less power delivered to the thruster with a higher current between electrodes since power loss varies as ðTeTgÞ
(with the frequency of collision), which means a lower onset potential and consequently a weaker electric field to accelerate charged particles between electrodes.
The previous behavior may seem counter-intuitive since we may expect a hotter gas to flow faster through the nozzle, but we also need to consider that a hotter gas decreases the gas density for a fixed pressure, thus lowering the gas mass flow rate going inside the thruster’s chamber. Younget al.22
pointed to the same behavior regarding the influence of the gas temperature on the thrust and efficiency of the EHD thrusters.
Although an increase in the gas temperature lowers the performance of the thruster, it has less impact compared to the pressure variations, e.g., the thrust was reduced five times
(a) (b)
(d) (c)
FIG. 2. (a) Thrust, (b) current and power, (c) T/P ratio, and (d) peak velocity as functions of pressure.
FIG. 3. Log of electron density,lnðneÞ, for several values of gas pressure.
with the temperature [Fig.5(a)] but increased five orders of magnitude with the pressure [Fig.2(a)].
C. Influence of the ballast resistance
We have seen from the pressure variation results that the produced thrust increased and the current decreased, and the same relation can be drawn in the case of the temperature variation, i.e., inverse proportionality. However, according to Masuyama23 and Pekker,24 the delivered current is directly proportional to the produced thrust (for a fixed pres-sure and temperature), so we may not see this relation in the previous studies.
In order to see how current affects the thrust, we modi-fied the ballast resistance in order to obtain different steady-state discharge currents. The current will be controlled by modifying the ballast resistor and maintaining the input volt-age in the RC-series circuit. We found that increasing the ballast resistance lowers the delivered current between elec-trodes as the input voltage,Vin, stays unchanged (see Fig.6).
The latter is expected since the total resistance seen from the voltage source will increase as well.
In Fig.7, we may see the relationship between the net produced thrust and the current flowing between electrodes for several gas pressures, and we may now notice the direct impact of the current on the thrust. In the curves shown in this figure, the thrust increased about one order of magnitude when the current changes from its minimum to its maximum value considered. Although the change in current is rela-tively small, it took a wide range of the ballast resistance to achieve it, i.e., 200–1000 MX. As expected from the previ-ous analysis, the series with higher pressures produced higher thrust values for a given current between electrodes.
Due to the transitory nature of the discharge at the beginning of the simulation, we cannot always decrease the ballast resistance until any desired value is reached without creating a spike in the current and losing convergence of the time-dependent simulation. Such an effect was noted in the series ofp¼100 Torr of Fig.6when no steady-state solution was found for ballast resistances lower than 800 MX.
We may infer that our methodology, searching for con-sistency among variables, allows direct comparison with those of Masuyama23and Pekker.24
D. Influence of the secondary electron emission coefficient,ci
We investigated the effect that the secondary electron emission from the cathode has on the discharge, which is of especial interest since the electrodes are not parallel and have different sizes. The electric field accelerates ions towards the cathode surface, which impinges on its surface. Ions with sufficient energy upon impact with the cathode will release electrons, commonly called secondary electrons, which in turn contribute to the total current between electro-des. According to Donko,25the electron emission yield may vary considerably with discharge operating conditions, in particular with the reduced electric field E/N, which we cal-culate in each point of space over time. We expect to have a different solution from cases where E/N is a constant.
FIG. 5. (a) Thrust, (b) current and power, (c) T/P ratio, and (d) peak velocity as functions of temperature.
FIG. 6. Current dependence on the ballast resistance,Rb, for several pressure values.
FIG. 7. Thrust as a function of current for several pressure values. Various ballast resistance values needed to obtain the current are indicated near the points.
The secondary electron source induces the modification of the electric potential distribution, deviating the ionic flow to the boundaries, instead of the shortest trajectories to the end of the nozzle.
Figure 8shows the thruster velocity profile for several values of the secondary electron emission coefficient,ci. We
may notice the loss of fluid velocity for increasing values of
ci, and the accumulation of stationary fluid near the anode’s
tip, the region typically presenting its highest value for the extreme case whereci¼1. The gas pressure and temperature
were maintained constant at 10 Torr and 300 K, respectively. In Fig.9, we may identify two regions presenting differ-ent behaviors for all considered parameters: (i) 105<ci <102, where the thrust, current, power, T/P ratio, and even peak velocity remain fairly indifferent to the change in the secondary electron yield, and (ii) 102<ci<10
0, where a
clear decrease in the thrust, power, T/P ratio, and peak veloc-ity may be observed due to an unfavorable distortion of streamlines, accompanied by an increase in the discharge current, an undesired effect for the purposes of thrust produc-tion since this effect is accompanied by a power loss. This
case study may shed light on the appropriate choice of elec-trodes to improve the thrust.
As we may infer from the previous results, whenever we encounter with an increase in current and a decrease in the delivered power, we notice a degradation of the thrust and T/ P ratio, which occurs for values ofci>102. The secondary
electron stream released from the cathode adds up to the total current between electrodes, so higher SEEC values will diminish the performance of the thruster. ci is a value that
depends on both the impinging ions and the material of the cathode’s surface, which must be considered when building the thrusters to avoid the undesired operation region.
E. Influence of the gap between electrodes, d
An additional parameter with the impact on the dis-charge profile response is the distance between electrodes. We found that the flow speed increases with the gap dis-tance, even when the velocity around the anode gets dimin-ished. As illustrated in Fig. 10, with the increasing gap, the highest speeds tend to occur near the cathode. This is due to the increase in the reduced electric field and deposited power that favors the formation of streamers which tends to increase in length and number with the applied voltage and gap distance.
The net produced thrust increases with the electrodes’s gap for the considered values, as seen in Fig.11(a). Since the input voltage, Vin, is higher than the onset potential of the
anode due to the voltage drop on the ballast resistance, when the distance between electrodes increases, so will the total plasma resistance. This effect, therefore, decreases the electric current flowing between electrodes but strengthens the onset potential and with it the electric field. The combination of those factors driving thrust explains the thrust increase, even when the current decays.
There is, presumably, a gap value at which the electric field becomes weak enough so that this tendency breaks
FIG. 8. Velocity profile (cm/s) for several values of SEEC (peak values on top).
FIG. 9. (a) Thrust, (b) current and power, (c) T/P ratio, and (d) peak velocity as functions of SEEC.
down; such distance must lower the current until the voltage between electrodes becomes close to the input voltage, and after that, the electric field should start decreasing its strength because the potential will not grow any longer as the distance will. This presumed gap limit was not reached due to computational constraints and will be addressed in a future work, but it was pointed out before by Masuyama.23
As shown in Fig.11(c), the thrust-to-power ratio, T/P, also increases, an evidence of the optimization of momentum transfer and absorption of energy. Although the peak veloc-ity of the distribution was found higher for a gap of 3 cm [see Fig. 11(d)] while decreasing for longer electrodes’s gaps, the T/P ratio kept increasing for all the distances con-sidered in the study.
IV. CONCLUSIONS AND FUTURE WORK
An axisymmetric 2-D simulation of a one-stage DC-powered EHD thruster with a funnel-like cathode that runs with nitrogen gas shows that the net produced thrust, thrust efficiency, and the peak gas velocity vary proportionally with pressure. The systematic study carried out in this work has shown that gas pressure plays an important role in the performance of the thruster since it is the variable that impacts the most the fluid between electrodes.
The EHD thruster performed better at lower gas tempera-tures, suggesting that cold neutral gas flowing between elec-trodes is the target flow for improving the thruster’s operation. This model also has shown that the thrust increases with the electric current flowing between electrodes, and this vari-able can be optimized by appropriate control of the ballast resistance for several gas pressure values.
The secondary electron emission is often considered a necessary phenomenon for sustaining DC-discharges; we show how the performance of the discharge decreases when the SEEC,ci, is greater than 10
2
for the proposed geometry at 10 Torr and 300 K.
The distance between electrodes influences the onset potential and the total current. When increased sufficiently,
the gas velocity peak transits from the anode’s tip to the inte-rior of the thruster’s chamber. More research is needed for finding the limits to which the distance can be extended while maintaining the discharge and its performance.
The model proposed in this work may be improved by increasing the number of species considered, along with their excited and vibrational states. The latter will obvi-ously increase the simulation times and computational resources needed to obtain the results, so a compromise has to be reached. Using the two-term approximation Boltzmann equation for solving the electron energy density function (EEDF) from the cross-sectional data of the e-collision reactions will presumably improve the chemical kinetic module of the model instead of the hereby used Maxwellian approximation, prone to errors at higher elec-tron energies.
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