Victor H. Granados, Mario J. Pinheiro, and Paulo A. Sá
Citation: Physics of Plasmas 23, 073514 (2016); doi: 10.1063/1.4958815
View online: https://doi.org/10.1063/1.4958815
View Table of Contents: http://aip.scitation.org/toc/php/23/7
Published by the American Institute of Physics
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Electrostatic propulsion device for aerodynamics applications
Victor H.Granados,1,a)Mario J.Pinheiro,2and Paulo A.Sa1
1
Departamento de Engenharia Fısica, Faculdade de Engenharia da 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 5 May 2016; accepted 30 June 2016; published online 21 July 2016)
A self-consistent model of single-stage electrohydrodynamic thrusters is proposed in order to com-pare and study their performances in terms of net thrust production and thrust-to-power efficiency. Simulations of three thruster’s cathode configurations (conical, cylindrical, and funnel-like) at a working pressure of’66:7 Pa (0.5 Torr) were conducted. Three working gases were employed: ar-gon (Ar, Ar*, and Arþ), nitrogen (N, Nþ, N2;Nþ2, and N
þ
4), and oxygen (O, O
þ, O;O
2; O
þ
2, and
O2). We found the funnel-like cathode configuration to produce the highest amount of thrust com-paring with the other studied cathode geometries. Additionally, nitrogen gas presented the highest net thrust of 5.2 nN with a thrust-to-power ratio of 0.94lN/W. Although the thrust obtained for ox-ygen is more than one order of magnitude lower than nitrogen’s, the thrust-to-power ratio obtained is more than three times greater.Published by AIP Publishing.
[http://dx.doi.org/10.1063/1.4958815]
I. INTRODUCTION
The study of the flow of an electrically charged fluid un-der the action of an electric field is the subject of electrohy-drodynamics (EHD), an effect referred to in the first place in
1709 by Hauksbee1with regard to sensing a weak air
blow-ing near a charged tube. In 1899, Chattock2 investigated
EHD as a source of pressure difference. Today, thrust, boundary layer enhancement, fluid pumping, and heat trans-fer improvement are specific applications of EHD effects.3
An EHD thruster is a device which, by promoting the ion-ization of a gas, induces the oriented migration of a charged ion cloud, due to the applied electric field, that transfers mo-mentum to the neutral heavy species in the gas. This coupling between charged and neutral particles converts the energy of the charged particles into fluid flow, commonly referred to as ionic wind. The EHD thrust will move in the opposite direc-tion of the ion swarm. By producing the thrust along its own axis, ‘the lifter’ is such a device.3,4In its simplest form, an EHD thruster is made of two electrodes, the cathode or emitter electrode with a sharp edge to facilitate the ionization of the gas, and the anode, or the collector electrode, which when powered by a high dc voltage produces the thrust.
In 1964, de Seversky5patented the first EHD based pro-pulsion system. Still in the 1960s of the last century, Christenson and Moller,6and also Robinson,7,8whose work has been under wraps for a long time, conducted theoretical and experimental research on the EHD thruster. Using coro-na discharge in EHD blowers, both groups came to similar conclusions, namely, that the conversion efficiency of EHD-thrusters, i.e., the electrical to kinetic energy conversion effi-ciency, is very low, approximately 1%. Later, Bondar and Bastien9showed that an increase of fluid velocity leads to a
higher thruster efficiency. More recently, in 2005 Singhal
and Garimella,10 with a 3D coupled EHD/Navier-Stokes
code, conclude that significant increases in conversion effi-ciency are possible by increasing the incoming velocity into
an EHD-pump. At almost the same time, Rickard et al.11
showed that the exhaust velocity can be increased by adding multi stages of EHD-thrusters in series: they tested up to 7 stages in series but with a conversion efficiency very low (1%). In 2009, Wilsonet al.,12although yielding a maximum velocity of over 7 m=s, claimed that the EHD thrusters did not appear to have any practical use for atmospheric applica-tions. These authors reported that parametric experiments to compare two electrode geometries, optimizing their number, distance, and size of the electrodes, model the electrical parameters of the thrusters and obtained mathematical rela-tionships of the optimized designs.
In 2011, Pekker and Young13developed a 1-D model of
an ideal EHD thruster that provides reasonable estimates of the thrust and efficiency of EHD lifters, as well as their prin-cipal performance limits. Among other conclusions, it was shown that the performance of EHD thrusters drops very fast
at high altitudes. Experiments conducted by Masuyama14in
2012 involved the study of two EHD thruster geometries: wire-to-cylinder and wire-to-rod-to-cylinder; such experi-ments showed that in order to generate thrust between the in-termediate and collector electrodes, it was necessary a voltage of a few thousand volts and that a reverse corona was formed at the intermediate cylinder electrode.
Although there is not still a commonly accepted view on whether EHD-thrusters are worthwhile or not, what level of thrust and thrust efficiency can be obtained from these devi-ces, etc., there are surely some primary limitations of EHD based propulsion systems, because, among others causes, they are typically inefficient in terms of thrust efficiency. This is the cause for only very few researchers tried assess-ing the performance of EHD thrusters, developassess-ing new a)Also at Centro de Estudos de Fenomenos de Transporte (CEFT),
Universidade do Porto, Porto, Portugal. Electronic mail: [email protected]
methods, and studying EHD thrusters’ electrode efficiency, for applications in the earth atmosphere. EHD thrusters are considered one of the most promising solutions for long term space travel as they achieve much higher values of specific impulse over chemical thrusters, which explains why the re-search on these devices is mostly being performed for use in space, even when they still are in their infancy.
In this article, our goal is to contribute to the advancement and the improvement of the performance of EHD thrusters for space missions, especially in what concerns the control of the geometry of the electrodes and the employed gas and its effi-ciency. We use numerical techniques and tools (Comsol Multiphysics15) for the calculation of fluid and electrical parameters crucial to the understanding of our EHD thrusters. At a low pressure (0.5 Torr), we explore three electrode geom-etries (cylindrical, conical and funnel-like cathodes) and com-pare the performance of three different gases, one atomic (argon), other electronegative (oxygen), and another that is the main component of air (nitrogen). The investigation of other gases or mixtures of gases (e.g., with higher mass such as N2–O2, Kr, or Xe) will be performed in the future work.
II. MODEL OF THE EHD THRUSTER
The electrohydrodynamic processes embody interlock-ing aspects of non-compressible gas dynamics (Navier-Stokes equation), ionized gas physics, self-consistent accel-erating electric field adequately described by Poisson’s equa-tion, and migration of charged particles in an electric field in the drift-diffusion approximation.
We considered self-consistent kinetic models for each gas, which includes electron-impact reactions (elastic, exci-tation and ionization), and chemical reactions in order to model the behavior of the gas during the discharge. The reac-tion rates of the kinetic models are detailed in TablesI–III.
A. Determination of the self-consistent electric field
The electrostatic field in the presence of a space-charge is computed using the Poisson equation
r2
V¼ qc
; (1)
where is the plasma permittivity (¼0r), VðrÞ is the
electric potential, andqcðrÞis the space charge density com-puted taking into consideration the plasma chemistry by means of the equation
qc¼e
XN
j¼1
Zjnjne
!
; (2)
whereZjis the charge number of ions andnjandneare the ions and electron densities, respectively. Then, the electric field is computed as
E¼ rV: (3)
B. Species governing equations
In order to emulate the behavior of the EHD fluid, we consider the continuity equation for the electron density,ne
@ne
@t þ r Ce¼Reðu rÞne; (4)
and the continuity equation for the electron energy density, n
TABLE III. Electron-impact and chemical reactions with rate coefficients for oxygen discharge.
Reaction Rate coefficienta[m3/s or m6/s] References
eþO!eþO fðrÞ 22 eþO2!eþO2 fðrÞ 22 eþO2!eþ2O fðrÞ 22 eþO2!2eþOþ2 fðrÞ 22 eþO!2eþOþ fðrÞ 22 eþO2!O2 fðrÞ 22
2eþOþ2 !eþO2 11031ð300=TeÞ4:5 27
eþOþ2 þO2!2O2 61039ð300=TeÞ1:5 27
eþOþ2 !OþO 21013ð300=T
eÞ 27
eþOþO2!OþO2 11043 27 eþOþO2!OþO2 11043 27 OþþOþO
2!Oþ2 þO2 11041 27 OþþO2!Oþ2 þO 3:310
17 27
expð0:00169TgÞ
O2 þO!OþO2 3:31016 28 OþO!O
2þe 1:41016 28 O2 þO2!2O2þe 2:71016ðTg=300Þ0:5 28
expð5590=TgÞ
af
ðrÞsymbolizes that electron-impact reactions rates are functions of cross-section data.
TABLE II. Electron-impact and chemical reactions with rate coefficients for nitrogen discharge.
Reaction Rate coefficienta[m3/s or m6/s] References
eþN2!eþN2 fðrÞ 25 eþN2!2eþN2þ fðrÞ 25 eþN2!eþ2N fðrÞ 26 eþNþ2 !2N 2:81013ð300=T
eÞ0:5 27
eþNþ2 þN2!2N2 2:61039ð300=TeÞ1:5 27
eþNþ2 !N2 11025ðTg=TeÞ4:5 27
eþNþ4 !N2þN2 21012ðTg=TeÞ0:5 27
N2þþN!NþþN2 7:21019expðTg=300Þ 27
N4þþN2!Nþ2 þ2N2 2:11022expðTg=121Þ 27
N2þþ2N2!Nþ4 þN2 5:01041 27
a
fðrÞsymbolizes that electron-impact reactions rates are functions of cross-section data.
TABLE I. Electron-impact and chemical reactions with rate coefficients for argon discharge.
Reaction Rate coefficienta[m3/s or m6/s] References
eþAr!eþAr fðrÞ 22 eþAr!eþAr fðrÞ 22 eþAr!2eþArþ fðrÞ 22 eþAr!2eþArþ fðrÞ 22 ArþAr!eþArþArþ 6:21016 23,24 ArþAr!ArþAr 3:01021 23,24
af
@n
@t þ r CþECe¼Rðu rÞn: (5)
In Eq.(4),Ce is the electron flux in the drift-diffusion approximation
Ce¼ ðleEÞne rðDeneÞ; (6)
and in Eq.(5),Cis the electron energy flux
C¼ ðlEÞn rðDnÞ: (7)
In the set of Eqs. (4)–(7), Ce is the electron flux,C is the electron energy flux,uis the fluid velocity,Eis the electric field,Reis the electron density source,Ris the electron
en-ergy density source,Deis the electron diffusivity tensor,D
is the electron energy diffusivity tensor, andleandl
repre-sent the tensors for the electron and electron energy mobility, respectively.
On the left side of Eq.(4), the first term stands for the temporal variation of the electron density, while the second term denotes the divergence of the electron flux. On the right side, the last term stands for the convection of electrons. On the right side of Eq.(6), the first term represents the migra-tion of electrons due to the applied electric field, while the second represents the diffusion of electrons from high to low electron density regions.
Eq.(5)is analog to Eq.(4)but now in terms of electron energy density, with the inclusion of the third term on the left side of Eq.(5)that represents the heating of the electrons due to the applied electric field. This term is a heat source or sink depending on whether the electrons are moving in the same or in the opposite direction of the external electric field.
Also, using Einstein’s relation for a Maxwellian electron energy distribution function (EEDF), we can derive from the electron mobility (le), the electron diffusivity (De), energy
diffusivity (D), and electron energy mobility (l)
De¼leTe; (8)
D¼lTe; (9)
l¼5
3le; (10)
whereTeis the electron temperature, defined as a function of the mean electron energy,
Te¼
2
3: (11)
The electron mobility tensor,le, is simplified into a con-stant (le) as we consider the mobility to be isotropic. Thus, the electron diffusivity, the energy diffusivity, and the elec-tron energy mobility are the functions of the elecelec-tron temperature.
The electron source term (Re) is the sum of electron im-pact reaction rates of all theMconsidered reactions:
Re¼
XM
j¼1
xjkjN0ne; (12)
wherexjis the mole fraction of the target species for the re-actionj,kjis the rate coefficient of reactionj, andN0is the
total neutral number density. The electron energy density source (R) is the sum of electron impact reaction rates
mul-tiplied by the energy loss/gain corresponding to each of the Preactions taken into account:
R¼
XP
j¼1
xjkjN0neDj; (13)
whereDj is the energy loss/gain from reaction j. Since we
consider several heavy species for each gas, we need the re-action rate coefficients of their collisional processes, which can be computed for electron collisions using a set of cross section data and assuming, for simplicity, a Maxwellian dis-tribution function,FMðÞ, from the equation
kj¼
2e me
1=2ð1
0
rjð Þ FMð Þ d; (14)
whereis the electron energy (eV) andrjðÞrepresents the elastic and inelastic (excitation and ionization) electron cross sections considered for each reaction.
The use of a Maxwellian distribution function can lead to an overestimation of ionization and the lowering of ki-netic reaction coefficient with lower threshold. At any rate, it will give an overestimated value of the electronic temper-ature. Even if the latter is true, we use the Maxwellian dis-tribution function instead of a Boltzmann disdis-tribution function because in our study, we are interested in the main behavior of the plasma parameters on the thruster perfor-mance, and additionally we aim to speed up the numerical calculations.
C. Surface kinetics
One of the key mechanisms which guarantees the plas-ma discharge stability is the secondary electron emission from the cathode. Ions are driven towards the cathode by the electric field and those that impact the cathode surface with sufficient energy will release secondary electrons into the bulk, which in turn will be driven by the strong electric field near the cathode, gaining enough energy to begin ionization, thus maintaining the discharge.16,17
In the present work, we consider the interaction of ions with the cathode surface using the secondary electron emis-sion coefficient,ci, and specifying the mean energy of emit-ted secondary electrons,i.
We expect, but have not yet included in the present model, that the plasma electron density should increase dra-matically with the use of metal oxides, yielding highci val-ues. The secondary electron emission coefficient affects the discharge, and a parametrization of thecivalue is in progress in order to have a better understanding of its impact on the thruster performance.
nCe¼ 1
2ve;thne
XciðCinÞ; (15)
nC¼ 5
6ve;thn
XciiðCinÞ; (16)
whereve;this the electron thermal speed, the first term on the
right side of Eq.(15)represents the electrons lost in the wall due to random motion within a few mean free paths, the sec-ond term of the right side of Eq.(15)depicts the secondary emission flux, and the second term of right side of Eq.(16) refers to secondary emission energy flux.
The surface reaction is specified in terms of sticking co-efficients,cf, which has a value of 1 over the reactive surfa-ces such as the cathode, and a value of 0 over non-reactive surfaces
Re;wall¼ cf
1cf=2
1
Ctot ð Þm
1 4
ffiffiffiffiffiffiffiffiffi 8RT
pMn
r YQ
k¼1
ck; (17)
whereCtotis the total electron flux incident on the cathode, mis the reaction order minus 1,ckis the concentration of re-actionk,Ris the gas constant,T is the surface temperature,
Mnis the mean molecular weight of the gas mixture, andQ
is the total number of considered reactions.
It is well known that the vibrational levels of ground-state N2(X1Rþg,v) molecules play a central role in nitrogen
discharge.18 The complexity of nitrogen arises from the
strong coupling between different kinetics such as electron, vibrational, chemical, and surface kinetics. Disregarding non-equilibrium vibrational kinetics of nitrogen (or diatomic molecules such as oxygen) will affect the pumping of vibra-tional quantum states through e-V collisions that are
accompanied by molecular dissociation. Since atoms are not as efficient as molecules in the momentum transfer process, we expect that V-V mechanisms could reduce the thrust-to-power (T/P) ratio. Our model, which is still open to improve-ment, depicts only the most significant species since the amount of reactions can grow rapidly and the calculations become unpractical.
D. Fluid governing equations
The fluid is assumed to be incompressible and the flux viscous laminar which is modelled using the Navier-Stokes
equation. An external volume force f¼qcE is included in
the equation in order to couple with electric field forces
qf
@u
@tþqfðu rÞu¼ rpþ r l ruþ rð uÞ
T
h i
þf;
(18)
whereqfis the fluid density, uis the fluid velocity,pis the pressure, andlis the dynamic viscosity.
Eq.(18)is a balance of forces acting on the fluid, where its left hand side represents a sum of inertial forces, namely, temporal variation and convection forces. In addition, the first term on the right hand side of Eq.(18)accounts for pres-sure forces (prespres-sure gradient), the second term represents the viscous forces, and the third term represents the external electrical force acting on the fluid.
Since the fluid is assumed incompressible for each gas considered in our study, the volume continuity equation is written as
r u¼0: (19)
E. The plasma electric circuit model
The plasma discharge needs to be strong enough, so the gas can be ionized around the anode but not too strong that an arc forms between electrodes. EHD thrusters work best in the glow discharge regime. As depicted in Fig.1, an RC cou-pling circuit is used in series with the direct-current source in order to avoid the arc regime between its electrodes. The bal-last resistor (Rb) is a tuning parameter that changes amongst simulation cases since a low value could favor an unphysical arc formation with the current increasing exponentially and a
FIG. 2. Conical cathode: geometry, and simulation domain with units in m.
FIG. 3. Cylindrical cathode: geometry, and simulation domain with units in m.
high value could reduce the plasma’s onset potential, reduc-ing the electric field strength and extreduc-inguishreduc-ing the plasma discharge. The blocking capacitance (Cb) is used to avoid voltage peaks between electrodes. The relation between the input voltage (Vin) and the electrodes voltage (Vplasma) is the following:
Vplasma ¼VinIpRbRbCb
dVplasma
dt ; (20)
where Ip represents the current between electrodes and
includes the ions and electrons densities at the electrodes walls.
The power spent in order to create and sustain the plas-ma is calculated from the circuit as
P¼VplasmaIp: (21)
Since the ballast resistor limits the total current deliv-ered to the electrodes, we may notice that an immediate comparison must be made with care, since the plasma con-ductivity is, naturally, different for each gas, implying a dif-ferent resistance for every case studied. Controlling the ballast resistance allows controlling the current, which, in turn, allows increasing the net produced thrust up to a value that provides the conditions for arc formation.
III. THRUSTER DESIGN
The conducted simulations are based on single-stage EHD thrusters with three different cathode configurations, namely: conical cathode, cylindrical cathode, and funnel-like cathode. 3D representations of the configurations along side their actual simulation domains are depicted on Figs.2–4. A list of geometrical parameters such as gap between electro-des and cross sectional areas of input and output of the
thruster nozzles can be found in TableIV. The general struc-ture of the EHD thrusters is presented in Fig.5showing the names of important geometrical dimensions.
In order to calculate the total thrust produced by the flux of gas going through the EHD thruster, we consider,
vzðrÞ, the axial component of the exhaust gas velocity,
which is normal to the nozzle’s output cross-section area, to be time invariant, but space dependent on the radial component in cylindrical coordinates. The time invari-ance is a valid approximation since the steady-state is reached within milliseconds after the corona formation. The radial dependency of the gas normal velocity compo-nent allows to write the following expression for the total thrust:
T¼2pqf
ðR
0
rv2zdr; (22)
whereris the distance from the center of the thruster sym-metry axis (r¼0) to the cathode’s wall, located atR.
In order to simulate the EHD phenomena, we need to choose the global conditions such as the gas temperature,
pres-sure, and input voltage, among others. Table Vsummarizes
the general simulation conditions considered on the present work.
Due to the higher number of electron-impact and chemi-cal reactions considered in the case of oxygen, for conichemi-cal
FIG. 4. Funnel cathode: geometry, and simulation domain with units in m.
TABLE IV. Geometrical dimensions.
Cathode geometry
Parameter Conical Cylindrical Funnel
Input area,Ai(cm
2
) 4.52 4.52 8.55
Output area,Ao(cm2) 2.19 4.52 4.52 Gap between electrodes,d(cm) 2.8 2.8 2.8 Cathode axial length,Lc(cm) 2.0 2.2 3.0 Anode axial length,La(cm) 1.25 1.25 1.25
FIG. 5. EHD thruster with the length of anode (La), length of conical cath-ode (Lc), gap distance (d), and input (Ai) and output (Ao) perpendicular areas.
TABLE V. Simulation conditions.
Parameter Value
Gas temperature,Tg 300 K
Pressure,p 0.5 Torr
Electric source voltage,Vin 500 V Ballast resistance,Rb [10–1500] MX
Blocking capacitance,Cb 1 pF
and cylindrical cathode geometries, we used a smaller com-putational grid in order to decrease the calculation time.
IV. RESULTS AND DISCUSSIONS
A. Conical cathode
The potential distribution is a key factor in the design of the thruster, since the conversion of electric to mechanical energy is made primarily by the electric field. Fig.6shows the electric potential along with the electric field vector, indi-cating its relative magnitude and direction for the three working gases considered. It is clear that the electric field is stronger in the regions near the cathodes when the discharge takes place and it is stronger in the case of nitrogen due to a higher potential gradient.
An approximate relation between the gas dynamic parameters and the electric field, pþ0
2E 2¼
constant, pre-dicts that in regions of higher electric field, the gas pressure is lower than in adjacent regions.19The production of nega-tive ions in oxygen results in a tendency to neutralize the charge near the walls through ion-recombination processes, while in electro-positive plasma environment, the walls tend
to be more strongly charged, with an overall favorable result in the case of nitrogen.
It is clear that, among the three different gases investi-gated, nitrogen gives a streamline more favorable to the gas acceleration as seen in Fig. 7, which shows the fluid speed distributions. Nitrogen presents a peak velocity magnitude of 19.3 cm/s compared to the smaller values of 5.2 cm/s and 3.42 cm/s for argon and oxygen, respectively. Additionally, the fluid speed increases at the nozzle’s output, regardless of the gas, due to the narrower transverse area of the cone’s end.
Fig.8shows the fluid velocity components (axial, radial, and total) at the output of the thruster. As depicted, the radial component of the fluid speed is negligible compared to the axial component, while the axial component contributes to the major work propelling the gas along the central channel of the thruster. As expected from the boundary conditions of the fluid dynamics, the magnitude of all velocities reaches zero at the electrodes walls.
Having argon as working gas creates a counteracting electric field at the outside of the thruster, reducing its effi-ciency. Comparing oxygen with nitrogen, we may notice that
FIG. 6. Electric potential distributions (V), conical configuration for (a) argon, (b) nitrogen and (c) oxygen gases.
in nitrogen, the electric potential has a more steep descent, favoring momentum transfer to Nþ4 and Nþ2 ions. At 0.5 Torr, the concentration and volume production of the heavier Nþ4 ions exceeds slightly the other ions.
B. Cylindrical cathode
The cylindrical cathode geometry is presented as two parallel plates in the two-dimensional plane shown, and due to the parallelism of the walls, the electric field inside of the chamber points mostly radially as seen in Fig.9for the three gases.
Clearly, the cylindrical cathode is not favorable in the case of argon, since the ions are mainly attracted to the walls by the electric field instead of axially along the chamber. The fact that electric potential spreads through the chamber reaching its end induces an opposing electric field at the exit of the nozzle that ultimately produces an undesired
station-ary vortex as seen in Fig. 10. Although laminar flux
equations do not model complex vortex formations, in in-compressible fluids, vortices are created during the process of the separation of boundary layers. We are interested in the cases presenting laminar behavior and the vortex creation shows the limit of our approximation in this respect. At this
pressure of 0.5 Torr, this is the only vortex that appeared once with oxygen or nitrogen as the working gases they were not formed.
As seen in Fig.10(b), with nitrogen as the working gas, the descending plateau seen for the potential favors accelera-tion of ions, with the axial speed attaining about 8 cm/s at the nozzle’s output. The nitrogen speed distribution shows the gas increasing its value at the entrance of the chamber ac-companied by a compression of the fluid, followed by an ex-pansion towards the walls and then gradually slowing down on its departure, a phenomenon that seems to point out that a geometry with a more axial electric field distribution on the chamber could avoid the fluid rushing towards the wall.
In Fig.11, we can observe the components of the veloci-ties of the three gases for the cylindrical cathode. Argon gas shows a prominent negative axial component which corre-sponds to the stationary vortex at the output. The axial com-ponent of the nitrogen gas velocity shows a dip at the center of the thrust chamber due to a natural depletion of ions run-ning from the center to the walls. The oxygen’s axial compo-nent of the fluid velocity attains a maximum at the center, presumably due to the constriction of the discharge in elec-tronegative gases.
FIG. 8. Fluid velocity components at output (cm/s), conical configuration for (a) argon, (b) nitrogen and (c) oxygen gases.
C. Funnel cathode
Fig.12shows the electric potential of the three gases for a funnel-like cathode. The nitrogen gas case showed a more axially distributed electric field with a larger potential gradi-ent at the gradi-entrance of the chamber than the previous cathode configurations, mostly due to the bigger cross sectional area of impact for ions at the entrance of the chamber accompa-nied by a sliding surface for the neutrals to follow their path into the chamber. On the other hand, the funnel geometry does not seem favorable for neither argon nor oxygen, due to the previously observed tendency of the electric potential distribution not having steep descents and being too radially divergent.
We may notice in Fig. 13 that, with nitrogen
environ-ment, the axial gradient of velocity is more prominent than in the other studied gases, resulting in a higher fluid speed peak (23.7 cm/s), compared to the peak values of argon (1.40 cm/s) and oxygen (3.29 cm/s). Even though nitrogen achieves its peak velocity value at the entrance of the cham-ber, its value does not fall rapidly inside the chamcham-ber, as
seen in Fig. 14 from the peak value of the axial velocity
(’15:5 cm/s) at the output of the nozzle.
Nitrogen ions transferred a higher momentum to the neutral species than both argon and oxygen with the
funnel-like cathode, as reflected on highest axial velocity values, presumably due to the contribution of heavier ions (Nþ2 and Nþ4) as shown in Fig.14.
D. Transient dynamics of various plasma parameters
Although the results of the parameters of interest com-prise of the steady state cases, the simulations are performed as time-dependent, defining an initial state and solving through the evolution of the DC discharge until the steady state is achieved. As depicted in Fig.15, the input voltage, Vin, has a ramp up function in order to give the solver time to set the initial conditions before the discharge takes place. We can also observe the voltage applied between electrodes, commonly known as the onset potential and the complemen-tary ballast resistor voltage which regulates the total current being supplied to the thruster. The role of the ballast is to sta-bilize the plasma, avoiding the creation of a conductive plas-ma bridge (arcing) between electrodes.
Fig. 16 depicts the flux of secondary emission into a
point of the cathode surface showing the transient nature of the discharge, raising its value dramatically during the onset potential transition, before reaching the steady state.
E. Thrust and thrust-to-power ratio
TableVIshows the net thrust produced using the differ-ent gases in the thrusters’ chamber, along with the power consumption and the thrust-to-power (T/P) ratio, a known parameter to determine the amount of thrust obtained for a given power input. The thrust-to-power ratio is calculated as the quotient between the total produced thrust (T) and the
power (P) needed to produce such thrust. According to
Masuyamaet al.:20“EHD thrusters have documented
thrust-to-power ratios as high as 26 mN/W, which is orders of mag-nitude greater than electrostatic propulsion systems used in space,” which refers to the standard atmospheric pressure (760 Torr), implying an applicability in low altitudes. In our simulations, we do not obtain T/P ratio as high as
FIG. 12. Electric potential distributions (V), funnel configuration for (a) argon, (b) nitrogen and (c) oxygen gases.
FIG. 13. Fluid velocity distributions (cm/s), funnel configuration for (a) argon, (b) nitrogen and (c) oxygen gases.
Masuyama’s since the considered pressure is only a fraction (0.5 Torr), which represents the one of a near space altitude. Younget al.21 remark on the direct proportionality that the pressure has on the thrust of EHD devices. In the case of ni-trogen and for the conical cathode configuration, we obtain
the maximum T/P ratio, as 24.59lN/W. Additionally, in all
the cathode geometries analyzed, argon presented the poorest T/P ratio.
Considering the nitrogen gas, the funnel-like cathode presented the highest net thrust (5.181 nN), followed by the
conical (3.440 nN) and cylindrical cases (2.870 nN), respectively.
V. CONCLUSIONS AND FUTURE WORK
In this work, we investigate the effect of the single-stage EHD thruster geometry on the ability to achieve desired ex-haust speeds, and primary economic considerations, as the thrust-power ratio. For stability purposes, the gas discharge is assisted by an electric circuit controlling the glow-to-arc transition. To avoid excessive time-consuming calculations, a reasonable self-consistent set of reactions and proper boundary conditions were considered.
In the funnel geometry, and contrary to the other consid-ered geometries, oxygen has the highest value of T/P ratio, three times higher in magnitude when compared to nitrogen gas.
The most promising cathode geometry is the funnel-like configuration in terms of net produced thrust, specially when considering nitrogen gas, the most abundant gas in the atmosphere.
A follow-up investigation of the EHD thruster, in partic-ular, the role of pressure, temperature, input voltage, and electrodes gap, may further clarify their influence on the pro-pulsive process, namely, the net thrust, consumed power, and T/P ratio.
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configuration Gas Thrust, T (nN) Power, P (mW) T/P ratio (lN=W)
Conical Ar 0.446 0.107 4.168
N2 3.440 0.140 24.590
O2 0.129 0.012 10.670
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