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Transmission probabilities of rarefied flows in the application of atmosphere-breathing electric propulsion

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Atmosphere-Breathing Electric Propulsion systems (ABEP) are currently investigated to utilize the residual atmosphere as propellant for drag-compensating thrusters on spacecraft in (very) low orbits. The key concept for an efficient intake of such a system is to feed a large fraction of the incoming flow to the thruster by a high transmission probability Θ for the inflow while Θ for the backflow should be as low as possible. This is the case for rarefied flows through tube-like structures of arbitrary cross section when assuming diffuse wall reflections inside and after these ducts, and entrance velocities u larger than thermal velocities v t h ∝ k B T / m . The theory of transmission for free molecular flow through cylinders is well known for u = 0, but less research results are available for u > 0. In this paper, the desired theoretical characteristics of intakes for ABEP are pointed out, a short review of transmission probabilities is given, and results of Monte Carlo simulations concerning Θ are presented. Based on simple algebraic relations, an intake can be optimized in terms of collection efficiency by choosing optimal ducts. It is shown that Θ depends only on non-dimensional values of the duct geometry combined with v th and u. The simulation results of a complete exemplary ABEP configuration illustrate the influence of modeling quality in terms of inflow conditions and inter-particle collisions.
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Transmission Probabilities of Rarefied Flows in the
Application of Atmosphere-Breathing Electric Propulsion
T. Binder, P.C. Boldini, F. Romano, G. Herdrich and S. Fasoulas
Institute of Space Systems, University of Stuttgart, Pfaenwaldring 29, 70569 Stuttgart, Germany
Abstract. Atmosphere-Breathing Electric Propulsion systems (ABEP) are currently investigated to utilize the residual atmosphere
as propellant for drag-compensating thrusters on spacecraft in (very) low orbits. The key concept for an ecient intake of such a
system is to feed a large fraction of the incoming flow to the thruster by a high transmission probability Θfor the inflow while
Θfor the backflow should be as low as possible. This is the case for rarefied flows through tube-like structures of arbitrary cross
section when assuming diuse wall reflections inside and after these ducts, and entrance velocities ularger than thermal velocities
vth kBT/m. The theory of transmission for free molecular flow through cylinders is well known for u=0, but less research
results are available for u>0.
In this paper, the desired theoretical characteristics of intakes for ABEP are pointed out, a short review of transmission proba-
bilities is given, and results of Monte Carlo simulations concerning Θare presented. Based on simple algebraic relations, an intake
can be optimized in terms of collection eciency by choosing optimal ducts. It is shown that Θdepends only on non-dimensional
values of the duct geometry combined with vth and u. The simulation results of a complete exemplary ABEP configuration illustrate
the influence of modeling quality in terms of inflow conditions and inter-particle collisions.
INTRODUCTION
Very low Earth orbits (i.e. below 250 km) are of great interest for many scientific, civil, and military purposes. Higher
accuracy for Earth observations and a persistent signal for Earth communications can be achieved, and manufactur-
ing and launching costs can be reduced. Furthermore, equivalent low orbits are also contemplated for future orbiters
around Mars. The drawback of lower altitudes, however, is the higher density of the residual atmosphere. The con-
siderably increased aerodynamic drag dictates the required amount of propellant on-board which is the main limiting
lifetime factor for such a mission. Even with modern Electric Propulsion at most 2 years of drag compensation can be
accomplished for the majority of missions in Earth orbits below 250 km [1].
Atmosphere-Breathing Electric Propulsion (ABEP) theoretically solves this issue by using the residual atmo-
sphere as propellant. This will decrease, ideally nullify, the on-board propellant requirement and will generate thrust
to partially or fully compensate the drag. A conceptual scheme of a spacecraft with ABEP is shown in Fig. 1a. The
collection process inside the intake is characterized by the highly rarefied (mean free path in the order of 0.1 m–1 km
for ABEP-applicable Mars and Earth orbits) and very directed (u>vth) inflow. A first intuitive configuration based
on continuum flow theories might be a very simple and straightforward funnel-shaped design: a small cylindrical inlet
followed by a cone converging to the thruster. In such a design, though, almost all particles passing through the inlet
section of the intake would be reflected back into flight direction, since nearly no inter-molecular collisions occur and
the solid angle towards the outlet as seen from the reflection positions along the cone is very small.
Studies from ESA [1], BUSEK [2], JAXA [3], and LIP [4] are one of the most recent and detailed examples
dealing with the design of ABEP. Their main common feature regarding the intake is the implementation of small
ducts (e.g. in form of a honeycomb structure) inside the very front part of the inlet, as illustrated in Fig. 1b for the
aforementioned funnel-design. The basic principle of inlet-ducts is that of a molecular trap. The transmission through
the ducts is in case of an axially directed inflow still high, whereas the backflow is significantly reduced when assuming
diuse wall reflections. That backflow corresponds exactly to the theory of transmission as analytically described by
Clausing [5]. For the inflow with u>0, however, less data is available as necessary for a detailed design study.
In this paper, findings regarding those transmission probabilities of rarefied flow in the application of an intake
Inflow
Flight Direction
Solar Array
Solar Array
Intake
Exhaust
S/C Core
(a) Scheme of spacecraft
Inflow
Flight Direction
Thruster
(b) Funnel-like intake with inlet-ducts
FIGURE 1. Concept of Atmosphere-Breathing Electric Propulsion
for an Atmosphere-Breathing Electric Propulsion system are presented. First, the desired theoretical characteristics of
such intakes are pointed out with the help of simple relations derived from the balance of mass flows. Afterwards, a
short review of transmission theory is given and, furthermore, the results of Monte Carlo simulations are presented.
Finally, simulation results of a complete exemplary intake are shown which illustrate the influence of modeling quality
of the ducts on the one hand, and the accuracy of the balance model on the other hand.
BALANCE MODEL
In order to understand the dependencies of the ABEP performance on its characteristics such as the transmittances of
the inlet, we previously derived an analytical model for an ABEP intake [6]. The generic design consists of an inlet
section and a chamber section. For the particles inside the chamber it is assumed that (ideally) every single one has
already performed fully diuse wall reflections and, therefore, proceeds only with thermal movement. The resultant,
in principle non-directional, particle flows are the backflow through the inlet, and the flow through the outlet. The
outlet can represent acceleration grids, an injection device towards the thruster or a further stage of compression. By
balancing all flows, the conditions in the separate sections can be estimated. Figure 2 illustrates the used nomenclature.
Twall
Inflow
nin,Tin ,uin
Inlet
(ducts) (back)
”Chamber”
nch,Tch
˙
Nin
Ain
Θinl.1,˙
Ninl.1
Θinl.2,˙
Ninl.2
Θout,˙
Nout
(˙
Naccel.)
Aout
FIGURE 2. Balance model scheme
R
L
A B
Θback
Θindirect
Θdirect
FIGURE 3. Transmission through
cylinder of length Land radius R
The respective cross sections for the inflow and outlet are defined by Ain and Aout. The parameters of the inflow
are known from the atmospheric model, namely individual species number densities nin, temperature Tin, and the
spacecraft velocity uin. The transmission probability Θrefers to a specific direction through a single structure. It is the
ratio between the gas particles entering the entrance plane A and the gas particles leaving the exit plane B, see Fig. 3.
A particle can directly reach B from A, or can be scattered along the wall before reaching B. However, a particle
coming from A can also return back to A. For the balance model, three transmittances are set based on which the
respective resultant particle flows can be defined: The particle flow into the chamber section (˙
Ninl.1) passes with Θinl.1
through the inlet section, the backflow to the atmosphere after having reached the chamber section ( ˙
Ninl.2) passes with
Θinl.2, and ˙
Nout is the collected net outflow with Θout. Main assumptions of the model are:
Free molecular flow (no inter-particle collisions) in thermodynamic equilibrium;
Diuse reflection at walls;
Fixed chamber and wall temperature (Tch =Twall);
Only non-directional, thermal mass flux inside the chamber.
The total particle flow ˙
Nin onto the intake is calculated using free stream conditions and the extruded intake area
Ain. The part reaching the chamber section includes additionally the inflow transmittance, i.e.:
˙
Ninl.1=˙
NinΘinl.1=nin ¯uinAin Θinl.1.(1)
Based on Tch, the thermal mass flux Γis set, resulting in backflow and outflow from the chamber. The actively
extracted particle flow ˙
Naccel.depends on the actual thruster. It is expected that a minimum number density nch inside
the chamber is necessary for ignition. Therefore, the focus is on the situation before ignition ( ˙
Naccel.=0). The
continuity equation ˙
Ninl.1=˙
Ninl.2+˙
Nout can be applied which determines Γand, thus, nch. Knowing the state inside
the chamber, the collection eciency ηcand number density ratio in Eqs. 2 can be calculated (mis the particle mass
of the respective species). The eciency ηcis equivalent to the thrust-to-drag-ratio of the spacecraft when considering
only the drag on Ain and constant inflow conditions and thruster operation. Hence, ηcis the main figure of merit;
however, also a minimum nch and ˙
Nout for the thruster has to be ensured. The simplicity of the balance model makes
it a very convenient tool for the study of ABEP, provided that the individual transmission probabilities Θare known.
ηc=˙
Nout
˙
Nin
=Θinl.1
Ain
Aout
Θinl.2
Θout +1,nch
nin
=m¯uinΘinl.1
Θinl.2+Aout
Ain Θout r2π
mkBTch
(2)
TRANSMISSION PROBABILITIES
The flow of rarefied gases through tubes has been a problem of great importance during the whole last century [7].
After first discussed by Knudsen [8], the transmission probabilities for cylindrical ducts of various aspect ratios was
described by Clausing [5] by analytical integral equations, providing the fundamental basis for subsequent studies.
As a matter of fact, the balance model through cylinders of finite length with fast inflow and scattered backflow
(but without further ”chamber” outflow) is basically the same as the one for the free molecular ”Patterson” probe, as
discussed with similar focus on transmission by Hughes [9]. Therefore, Hughes’ work constitutes a good basis for the
analytical description of transmission probabilities. In the following, the theory is subdivided into three points:
The transmission with u=0 as published by Clausing [5] (used for Θinl.2in the balance model);
Hughes’ [9] extension to u>0 (used for Θinl.1);
The combination of two adjoining Θ’s such as one of an inlet section with ducts and one of a section without.
Clausing case (u=0)
Clausing’s integral equations [5] are the exact analytical description of the transmission through cylinders with an
inflow corresponding to a Maxwell-Boltzmann distribution around u=0 with Tin =Twall and fully diuse wall reflec-
tions. Their solution is only achievable with numerical approaches. However, Clausing could derive an approximation
in form of an explicit Θ = f(L/R) dependency based on the assumption that a wall reflection leads to a remaining
transmission probability which is linearly dependent on the distance from the inlet and his given function α(L/R).
The first values of transmission probabilities with high accuracy were calculated by Cole [10] in 1977. In the
recent decades the numerical methods were further improved resulting in even more precise results, as summarized
together with their own most recent results by Li et al. [11]. An alternative to methods based on Clausing’s equations
are Monte Carlo simulations applicable to any geometry and also published numerously.
A comparison of Clausing’s approximation with high accuracy results (here by Cole) is shown in Fig. 4a for the
transmission probability of cylindrical ducts in the range of L/R=[0.1,100]. As depicted, Clausing’s values have a
relative error smaller than 0.1% until L/R=3, whereas for higher aspect ratios they remain at least below 4%. For
limL/R→∞ and limL/R0they converge to the exact solution.
0.1 0.20.51 2 510 20 50 100
0.01
0.1
1
0
5
10
15
20
25
Aspect ratio L/R, -
Θcylinder, -
rel. deviation, %
Cole
Clausing
Clausing, rel. deviation
(a) Clausing case: Cole [10] and Clausing [5]
0.1 0.20.51 2 510 20 50 100
0.1
0.2
0.3
0.4
0.5
1
Aspect ratio L/R, -
Θcylinder, -
S=0.5
S=1
S=5
S=10
S=15
(b) u>0 for dierent S=u/vth (Hughes’ method [9])
FIGURE 4. Analytical solutions for transmission probabilities through cylinders of dierent L/R
Non-zero entrance velocity (u>0)
In contrast to Clausing’s assumption of u=0, the inflow in the ABEP application corresponds to in-orbit conditions
(u>vth) which motivates to analyze the influence of a non-zero entrance velocity. Hughes [9] analytically described
the transmission probabilities for cylindrical ducts with a speed ratio S>0, see Eq. 3, and even arbitrary angle of
attack. He adopted Clausing’s α(L/R) approximation, which ultimately leads in the case of a parallel inflow to a single
integration for calculating Θat a specific L/Rand S.
S=u
2kB·Tin/m,X=L/R
2S(3)
Figure 4b depicts our results for dierent Sand L/Rcalculated with a numerical integration by the adaptive
Gauss-Kronrod quadrature algorithm quadgk of MATLAB R
. The obtained curves show that with a higher Sthe
Θ(L/R) increases clearly and, therefore, particles are less likely to be scattered at walls or they are scattered at larger
distances from the inlet.
0.05 0.1 0.20.51 2 5
0.2
0.4
0.6
0.8
1
102
101
100
101
102
X=L/R
2S, -
Θcylinder, -
ΘΘS=15
ΘS=15
ΘΘS=15
ΘS=15
rel. deviation from S=15, %
Θ(X): S=0.5S=15 dev. from S=15: S=0.5S=1S=5S=10
FIGURE 5. Θcylinder (S,L
R) by Hughes’ method [9] and deviations from S=15 plotted against X
When plotting Θinstead of L/Ragainst a new variable X, combining both L/Rand S(see Eq. 3), the interesting
fact arises that with increasing Sall curves seem to converge to a single one. Figure 5 shows the Θ(X) curves for
S=15 and S=0.5 as black lines. The S=1 curve would lie between, while the ones for higher Swould virtually
lie on top of S=15. For quantifying the actual dierences, the diagram includes also the relative deviations from
the S=15 curve. Due to the logarithmic scale, the dashed parts depict the negative deviation. For small S, e.g. 0.5
and 1, the transmittances are up to 100% greater, while for S=5 and 10 the deviation is until X1 below 1%. For
1<X<5, it increases as far as 10%. However, it still might be possible that this increased deviation is only due
to the underlying α(L/R) assumption, since the range with increasing deviation roughly corresponds to L/R>3 (see
previous subsection). All in all, it is shown that for the high Sof interest (for ABEP relevant orbits around Earth and
Mars, Sis in the regime of 5–20), the transmission probability approximately becomes from a double dependence a
single one. This makes the regime very convenient for analyses, since the Θ(X) curve has just to be calculated once and
can be evaluated for any Sand L/R. However, the influence of Clausing’s α(L/R) assumption is unknown for S>0,
by which the presented method of Hughes can so far only indicate the existence of a direct Θ(X,S>5) relation, but
for obtaining precise values, a dierent approach is necessary which motivates, e.g., Monte Carlo simulations.
Combining two adjoining transmittances
As shown in Fig. 2, the inlet section is thought to be filled with ducts only in its front part. One reason for that
design is a possible optimization in terms of total transmittance. Based on two adjoined tubes with equal radii this
might seem pointless, since independently of the position where an imagined tube is divided into two parts, the
combined geometry, and therefore its transmittance, is always the same. However, one has to bear in mind that a
transmitted flow has gone through a beaming eect. The outflow has now a radially, non-equal particle distribution
with preferred small deflection angles from the axis; while the inflow has an equal distribution and, for the example
of the Clausing case, a velocity distribution according to Lambert’s cosine law. Therefore, the assumption of taking
the ˙
Nof an outflow from a first structure I as inflow at equilibrium conditions for an adjoined structure II can result
in a total transmittance dierent from the one of a continuous structure. For the derivation of the common theory
of transmittance combination let us assume that a beaming eect can be neglected. Oatley [12] first determined the
combined transmission probability ΘI+II of two adjoining tubes I and II in the Clausing case to be:
1/ΘI+II =1/ΘI+1/ΘII 1.(4)
However, in the case of a non-zero entrance velocity it has to be distinguished between an original Θconsisting of
Θdirect + Θindirect and Θfor particles after wall collisions and corresponding to the Clausing case. Thereby, an extended
ΘI+II is derived by separating the transmitted flow into an infinite number of individual parts, as illustrated in Fig. 6.
The sum inside the resultant combined transmittance of Eq. 5 can be formulated as geometric series, as shown in
Eq. 6. Thereby, Eq. 7 follows after some algebraic manipulations, that obviously collapses to Eq. 4 when assuming
Θ = Θ. The assumption of inflow/outflow conditions at equilibrium can be justified when particularly considering
the transition from a section I filled with a large number of small ducts to a section II without ducts, because the local,
radial outflow distributions of the ducts can be neglected when they are combined to a much larger total inflow for II.
I II
ΘI
Θ
I
1Θ
I
Θ
I
1Θ
I
ΘII
1ΘII
Θ
II
1Θ
II
Θ
II
1Θ
II
...
ΘI·ΘII
+
ΘI·(1 ΘII)(1 Θ
I)·Θ
II
+
ΘI·(1 ΘII)(1 Θ
I)(1 Θ
II)(1 Θ
I)·Θ
II
+...
FIGURE 6. Derivation of two combined transmittances
ΘI+II = ΘI·ΘII + ΘI·(1 ΘII)(1 Θ
I)·Θ
II ·
N
X
n=0
[(1 Θ
II)(1 Θ
I)]n(5)
N
X
n=0
qn=1
1q
N
X
n=0
[(1 Θ
II)(1 Θ
I)]n=1
1(1 Θ
II)(1 Θ
I)=1
Θ
I+ Θ
II Θ
IΘ
II
(6)
ΘI+II =ΘIΘII ΘIΘII(1 Θ
II)(1 Θ
I)+ ΘIΘ
II(1 ΘII )(1 Θ
I)
Θ
I+ Θ
II Θ
IΘ
II
(7)
The concrete transmittances used in Eq. 7 for calculating the individual values of ΘI+II through the combined
ABEP inlet are listed in Tab. 1. Please note that the direct and indirect parts of Θinl.1are dierently transmitted
through the II section, while Θinl.2already corresponds to the Clausing case.
TABLE 1. Used ΘIand ΘII for the individual values of ΘI+II through the ABEP inlet
Θinl.1,direct for fast, unscattered II-inflow: ΘI= ΘI,direct ,ΘII = ΘII,direct + ΘII,ind irect ;
Θinl.1,indirect for scattered II-inflow: ΘI= ΘI,indirect ,ΘII = Θ
II;
Θinl.2for scattered ”chamber” backflow: ΘI= Θ
I,ΘII = Θ
II.
MONTE CARLO SIMULATIONS
In the previous section it was mentioned that for a set of high speed ratio Sand arbitrary L/R, simple analytical
methods such as Hughes’ might not produce precise results for Θ, and more importantly, they are mostly restricted
to axisymmetric geometries. Therefore, we applied PICLas, a 3-D, highly parallelized, coupled code [13] including
modules for DSMC and Particle-In-Cell (PIC). The modules for boundary treatment and particle movement can be
used without any inter-particle collision or PIC-related routines, making PICLas for us also a conveniently available
Monte Carlo method for free molecular flow (FMF) simulations. In the following, an exemplary outcome for Θ(X) of
cylindrical ducts is presented, and finally, the previous results are applied in complete intake simulations.
Determined transmission probabilities for u>0
We conducted simulations on various duct geometries, here however we confine ourselves to cylindrical tubes. The
aforementioned parameter Xwas varied in the range of 0.01–2 by dierent uin,Tin /m, and L/R, all in the regime
applicable for ABEP. Additionally, Twall was altered which showed that it does not aect the individual transmittances.
Each simulation included a single duct open at both ends (but particle counting) with an incoming flow parallel to
the cylinder axis and fully diuse wall reflections, which additionally mark the reflected particles for distinguishing
between Θdirect and Θindirect . Figure 7 shows the results from the simulations.
110
0.01 0.1 1
0.2
0.4
0.6
0.8
1
L/R(in Mars orbit at 110 km, S=15.89), -
X=L/R
2S, -
Θcylinder, -
1.95 2
0.42
0.44
S=5
S=10
S=15
Total transmitted inflow,
Θ = f(X)
Total transmitted inflow
(analytical, cf. Fig. 5)
Directly transmitted
inflow, Θdirect =f(X)
Indirectly transmitted
inflow, Θindirect =f(X)
FIGURE 7. Cylinder transmittances Θ(X) determined by Monte Carlo simulations
The black points are the total transmittance Θ(X) as sum of Θdirect (blue) and Θindirect (red). In addition to X
(bottom), also the corresponding L/R-axis is included (top), based on a speed ratio Sof a representative Mars orbit
for ABEP. As comparison, the analytical solutions from Hughes’ method (see previous section) are depicted which
match the simulation results very well until X=1. For larger X, it was shown that the analytical solutions themselves
deviate between each other significantly for dierent S, however the influence of Clausing’s α(L/R) approximation
was unknown. Therefore, the region of Θ(X) near X=2 is depicted magnified in the same diagram. It can be seen that
the points from the simulations lie only 0.2–0.4 % above the analytical approximation. Thus, Hughes’ method seems
to predict the actual transmittances for S>0 even better than for Clausing’s S=0 (the corresponding aspect ratios
are of L/R>3) and the unique Θ(X) relation for S>5 is expected to be used at least up to X=1 for ensuring an
error below 1 %, see Fig. 5.
Application to complete intake simulations and DSMC
One specific ABEP configuration was analyzed further after an optimization with the balance model for a specific
small EP thruster being developed at our institute. A Mars orbit at 110 km altitude is chosen [14] with the inflow
conditions shown together with the geometrical parameters in Tab. 2. Internal degrees of freedom are neglected as
well as all other species than CO2(90%), resulting in S=15.89. One single case was simulated including inter-
particle collisions with DSMC by VHS cross sections, the remaining cases are FMF. The chosen geometry considers
a square cross section of the inlet converging at the outlet with 45to the circular section of the thruster. By this
inlet, a filling with ducts of likewise square cross section can be achieved whose wall thickness was approximated as
zero. The outlet was set to Θout =1 and vacuum condition for focusing on the maximum achievable mass flow. Two
dierent lengths were simulated for the inlet part after the ducts (”back”, see Fig. 2) and four for the ducts themselves.
The complete simulations were compared to ones starting directly after the ducts from which an outflow at equi-
librium state is assumed consisting of the original atmospheric condition ( ˙
NinΘinl.1,dir ect) and the scattered particles at
wall temperature ( ˙
NinΘinl.1,indir ect). At these ”equilibrium ducts”, backflowing particles were deleted with a probabil-
ity of the respective Clausing-like transmittance, otherwise diusely reflected. The individual transmittances for the
dierent L/R(Ris now the half side length) were determined analogously to the cylinders of the previous subsection.
For comparison with the balance model (BM), also the Θ’s of the back part have to be known (”II” in Tab. 1). Here,
the converging part was neglected and a large square duct with respective L/Rwas assumed.
The results are depicted in Fig. 8 in terms of collection eciency plotted against duct aspect ratio. It can be seen
that for both back lengths the curve shapes of all models match surprisingly well. This verifies the approximation of
the combined intake. The relative deviations between BM and simulations are approximately 5–8% which is expected
to be, on the one hand, due to the particle part directly transmitted through the whole intake (in the BM all chamber
particles are assumed to be scattered); on the other hand due to the continuous transition from inlet to the ”chamber”
section which is not considered in the BM. The comparison of Fig. 8a and Fig. 8b shows that, eectively, there is an
optimization possible for the L/Rdistribution between duct and back part of the inlet, but for this case the absolute
gain is only 1–2%. However, this confirms the assumption of setting the outflow of the ducts back to equilibrium.
The same interpretation follows when comparing equilibrium ducts with complete simulations - the relative error is
with 1.5% to 5% very small. The deviation between the case without and with inter-particle collisions (L/Rback =4,
L/Rducts =16) is only 0.8% and verifies the FMF assumption for this specific geometry and inflow condition.
TABLE 2. Parameters of intake simulation for Mars at 110 km (Ris half side length of square)
nin (CO2)Tin Twall uin Ain/Aout L/Rduct s Rducts L/Rback Rback =Rin
5.028 ·1017 m3128 K 300 K 3495 m/s 10 [3,8,16,20] 0.25 cm [4,16] 5.185 cm
CONCLUSION
ABEP systems are investigated to use residual atmosphere as propellant for drag compensation in low orbits. The
collection eciency ηc(collected part of the flow onto the intake) can be increased by implementing small ducts at
the front part of the intake. The principle is that of a molecular trap, letting most of the fast inflow coming through
while reducing the backflow. For the optimization of the ABEP performance, an analytical balance model (BM) has
been derived based on the balance of mass flows and transmission probabilities Θthrough the individual sections. The
0510 15 20
20
25
30
35
L/Rducts , -
ηc, %
Equilibrium ducts (sim.)
Complete geometry (sim.)
Complete, with DSMC
Balance model (analytical)
(a) Inlet part after ducts (back) with L/Rback =4
0510 15 20
20
25
30
35
L/Rducts , -
ηc, %
Equilibrium ducts (sim.)
Complete geometry (sim.)
Balance model (analytical)
(b) Inlet part after ducts (back) with L/Rback =16
FIGURE 8. Collection eciencies of complete intake simulations compared with balance model
transmission of flows without entrance velocity depends only on the aspect ratio (e.g. L/Rof a tube), while for non-
zero velocities, as present in the inflow condition of an ABEP, an additional dependency on the speed ratio Sarises.
However, for the high Sof an ABEP application the double dependence reduces to a single one on X, combining
aspect ratio and S. It was shown that for cylinders the assumption of an unique Θ(X,S>5) relation results in errors
increasing with X, but for X<1 they stay below 1 %. Therefore, just the Θ(X) curve has to be determined and can be
evaluated for any Sand L/R. This might be convenient particularly for ducts with non-circular cross-sections (but still
similar to regular polygons), since they seem to have an analog behavior, but without analytical Θ(X) descriptions.
Together with a derived equation for combining the Θof the duct-including inlet part with the adjoining back part,
determined Θ(X) points for square ducts were applied in a comparison between the BM and 3-D intake simulations.
In terms of ηc, the relative deviations between both approaches are approximately 5–8% which makes the BM well
suited for quick optimizations. Moreover, simulations have been conducted starting directly after the ducts where
corresponding Maxwellian velocities are assumed. The errors compared to the complete simulation are small, showing
that a beaming eect through the ducts is insignificant, since most of the particles are scattered afterwards anyway.
Last but not least the free molecular flow assumption was verified for the specific geometry and inflow condition.
For further analyses more detailed gas-surface interactions will be included. However, the fully diuse wall
reflections are assumed to be already an useful approximation for first evaluations. Since all analytical relations rely
strongly on them, it is not expected that a similarly comprehensive theory could be developed otherwise.
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... The inlet efficiency η in is defined as the ratio of incident mass flux that is transmitted through the inlet to the propulsion system [27]: ...
... Specular collection methods rely on materials which can resist surface roughening and maintain persistent specular reflection. Such materials are currently under investigation [33], but none currently exist, so this work will be limited to diffuse collimatoraccumulator designs [10,27]. Previous investigations have shown that for passive collimator-accumulator type inlets η in varies with inlet geometry and is inversely proporational to compression ratio of the inlet [12,34]. ...
... Higher inlet efficiencies can be achieved at the cost of gas compression but this is generally considered undesirable [12,34]. For this work η in = 0.35 is assumed to be a reasonable estimate for applicable to a wide range of systems [27,29,35]. ...
Preprint
Full-text available
Air breathing electric propulsion (atmosphere-breathing electric propulsion) (ABEP) has attracted significant interest as an enabling technology for long duration space missions in very low Earth orbit (VLEO) altitudes below about 300 km. The ABEP spacecraft and mission analysis model developed allows parametric characterization of key spacecraft geometry and thruster performance parameters such as spacecraft length-to-diameter, the ratio of solar array span to spacecraft diameter, thrust-to-power, effective exhaust velocity, and inlet efficiency. For the missions analyzed ABEP generally outperforms conventional electric propulsion (EP) below 250 km altitude. Using a 6U spacecraft architecture the model shows that below 220 km ABEP is the only viable propulsion option for desirable mission lifetimes. Parametric evaluations of key spacecraft and ABEP characteristics show that the most significant technological improvements to ABEP spacecraft performance and range of applicability for VLEO missions will come from advancements in inlet efficiency, low drag materials, solar array efficiency, and thrust-to-power.
... At the same time, on the base of the theory of similarity of free molecular gas flows, the effective geometry of honeycomb air intakes was substantiated [135], and the influence of the channel cross section shape was investigated. Subsequently, many research associated with the air intake have been revived within the current century [49], [51], [53], [61], [62], [76], [78], [79], [85], [86], [96], [136]. A fairly complete review of various proposals for ABEP air intakes is presented in [18]. ...
... In TsAGI studies [78], [79], [82], a scheme was proposed for extracting gas from the atmosphere or from a free molecular flow in a laboratory setup using a single channel, shown in Fig. 8.1. Estimation of air intake characteristics is based on balancing the incoming and outgoing molecule flows for the thermalizer and IC [53], [61], [62], [82]. When SC moves at the velocity ∞ V , molecules of the atmosphere gas with the number density ∞ n are captured by the intake device and directly or after a series of collisions with the walls enter to the thermalizer. ...
... They are characterized by roughness and the presence of adsorption layers, which significantly affects the scattering indicatrices [137]. In most theoretical works, when studying the gas flow through an intake in the free molecular mode, the scheme of diffuse reflection of molecules from the surface is taken as the boundary conditions, which determines the choice of geometric shapes of air intakes [49], [54], [55], [58], [59], [61], [62], [78], [79], [82], [96], [136]. This assumption is based, among other things, on the estimation of the aerodynamic drag coefficients of SC in VLEO using experimental data [108], [111], [138], [139]. ...
... While in the latter case particles can cross the duct almost without colliding with it, the transmission probability of thermalized particles is the result of a series of successive collisions and decreases with the duct aspect ratio. Approximate expressions of the Clausing factor are available for simple cases (see, e.g., [83]), but no closed-form exact formula exists and, for complex geometries, the transmission probability can only be calculated with Monte-Carlo simulations. ...
... Even if the orbital velocity is one order of magnitude higher then the mean thermal speed, for ducts with a long aspect ratio a fraction of the particles is reflected before reaching the end of the duct. As described in [70,83], given the thruster transmission probability and the required collection efficiency (or, vice versa, the required compression ratio), an optimal duct length can be deduced. ...
... As described in Intake section, the simulation of the atmospheric flow collection typically relies on a TPMC (for free molecular flow) or DSMC approach [147]. A first exception is the work of U. Stuttgart IRS [81,83]. Based on analytical approximations of the transmission probability, the authors derived a simple balance model to assess the performance of an intake with a ducted-inlet. ...
Article
Full-text available
Air-breathing electric propulsion (ABEP) allows for lowering the altitude of spacecraft operations below 250 km, in the so-called Very Low Earth Orbits (VLEOs). Operations in VLEOs will give radical advantages in terms of orbit accessibility, payload performance, protection from radiations, and end-of-life disposal. ABEP combines an intake to collect the residual atmosphere in front of the spacecraft and an electric thruster to ionize and accelerate the atmospheric particles. Such residual gas can be exploited as a renewable resource not only to keep the spacecraft on a VLEO, but also to remove the main limiting factor of spacecraft lifetime, i.e., the amount of stored propellant. Several realizations of the ABEP concept have been proposed, but the few end-to-end experimental campaigns highlighted the need to improve the concept functional design and the representativeness of simulated atmospheric flows. The difficulty in recreating the VLEO environment in a laboratory limits the data available to validate scaling laws and modelling efforts. This paper presents a comprehensive review of the main research and development efforts on the ABEP technology.
... The transmission probability for cylindrical sections with > 0 is plotted in Fig. 30 in Appendix A based on the results obtained by Pond [56] and de Leeuw and Rothe [57], revealing the dependency of the transmittance on the speed ratio and the aspect ratio ∕ of the geometry. However, it has been pointed out by Binder in [59] that both independent quantities can be expressed by a single variable . This approximates the double dependence on the aspect ratio and speed ratio to a single one: ...
... The generic configuration of the domain and its boundary conditions are shown in Fig. 5. The surface interactions are set as diffuse and the binary intermolecular collisions are disabled, in accordance with the BM's basic hypotheses [59]. ...
... • An outer circular cross section = 4.3 ⋅ 10 −3 m 2 of radius = 37 mm with either circular or hexagonal ducts, as shown in Fig. 10 • An outer square cross section = 4.3 ⋅ 10 −3 m 2 of radius ≈ 46.4 mm with either square or circular or hexagonal ducts, as shown in Fig. 11 The maximum packing densities for each configuration are summarised in Table 6. A final consideration must be made on the transmission probability of the flow since the flow transmitted through the inlet has undergone a beaming effect, therefore causing the outflow to have a radially, non-uniform particle distribution with directional angles from the axis of scatter [59]. It is, therefore, necessary to distinguish between the Clausing transmittance and transmittance with macroscopic velocity for both the chamber and the honeycomb sections. ...
Article
Full-text available
In order to extend the orbital lifetime of spacecraft operating in Very-Low Earth Orbit (VLEO), Atmosphere-Breathing Electric Propulsion (ABEP) can be employed for atmospheric drag compensation. The concept is based on the ingestion of rarefied atmospheric particles to be used as propellant for an electric thruster, thereby removing the need for onboard propellant. The present paper aims to design and analyse a passive ABEP intake optimised for the RF Helicon-based Plasma Thruster (IPT), which is selected as the most promising system due to its electrodeless design, as it removes the issue of thruster corrosion, and reduced susceptibility to atmospheric variations. This is achieved by implementing the analytical model, referred to as the Balancing Model (BM), along with its main performance parameters. The Direct Simulation Monte Carlo (DSMC) solver dsmcFoam+ is thus employed to simulate the transmission probability of cylinders and hexagonal prisms with non-zero entrance velocity, yielding a mean percentage error of 2% when compared to independent DSMC results. In addition, a novel DSMC transmittance investigation of square prisms is performed. Various design configurations are optimised for circular and hexagonal intakes based on the functional requirements of the thruster, leading to a maximum collection efficiency of 45% for a cylindrical intake with a hexagonal honeycomb. The optimal intake is hence simulated via DSMC, showing a good accuracy with the BM with or without intermolecular collisions, yielding a percentage error of 0.22% in the former case and 5.53% in the latter one. The variation of incidence flow angle, accommodation coefficient and thruster transmission probability is finally investigated, and the results are validated by the agreement shown with values retrieved from the literature.
... Ṅ s is dependent on thruster geometry and plasma conditions inside the thruster, and accurately computing Ṅ s requires a reacting plasma code which is outside the scope of this work. This analysis will assume frozen flow throughout the thruster such that Ṅ collected s =Ṅ exhaust s The particle flow rate effectively captured by the inlet can be expressed as, The inlet efficiency η in is defined as the ratio of incident mass flux that is transmitted through the inlet to the propulsion system [27]: ...
... Specular collection methods rely on materials which can resist surface roughening and maintain persistent specular reflection. Such materials are currently under investigation [33], but none currently exist, so this work will be limited to diffuse collimator-accumulator designs [10,27]. Previous investigations have shown that for passive collimator-accumulator type inlets η in varies with inlet geometry and is inversely proporational to compression ratio of the inlet [12,34]. ...
... Higher inlet efficiencies can be achieved at the cost of gas compression but this is generally considered undesirable [12,34]. For this work η in =0.35 is assumed to be a reasonable estimate for applicable to a wide range of systems [27,29,35]. ...
Article
Full-text available
Air breathing electric propulsion (atmosphere-breathing electric propulsion) (ABEP) has attracted significant interest as an enabling technology for long duration space missions in very low Earth orbit (VLEO) altitudes below about 300 km. The ABEP spacecraft and mission analysis model developed allows parametric characterization of key spacecraft geometry and thruster performance parameters such as spacecraft length-to-diameter, the ratio of solar array span to spacecraft diameter, thrust-to-power, effective exhaust velocity, and inlet efficiency. For the missions analyzed ABEP generally outperforms conventional electric propulsion (EP) below 250 km altitude. Using a 6U spacecraft architecture the model shows that below 220 km ABEP is the only viable propulsion option for desirable mission lifetimes. Parametric evaluations of key spacecraft and ABEP characteristics show that the most significant technological improvements to ABEP spacecraft performance and range of applicability for VLEO missions will come from advancements in inlet efficiency, low drag materials, solar array efficiency, and thrust-to-power.
... The latter can be analytically estimated within a certain range and are divided in those for the particles moving only with thermal diffusion, the Clausing case, and those which have velocities much higher than the thermal velocity. For the Clausing case, Θ is calculated, namely Θ inl.2 and Θ out , assuming α = 1, using the analytical expression of Clausing [64] which results are shown in Fig. 4.2a for the given aspect ratios L/R and for cylindrical structures [63]. For fast particles, Θ inl.1 , see Fig. 4.2b, is the sum of three components: ...
... = Θ indirect , and Θ back = Θ Clausing . In Fig. 4.3 and Fig. 4.4, the horizontal axis is the dimensionless inflow parameter defined as X = (L/R)/( √ 2S), where S = v in / 2k B T in /m p is defined as the molecular speed ratio ranging between 7 < S < 20 for average solar activity in VLEO, being S > 5 the hyperthermal case [66], in which 4. Intake (a) Clausing Θ analytical solutions [63]. ...
... (b) Definition of Θ through a cylinder [63]. ...
Book
This dissertation deals with the development of Atmosphere-Breathing Electric Propulsion (ABEP) technology, that can enable propellant-less continuous orbiting in very low Earth orbits (VLEO). It uses an intake in front of the spacecraft to collect the residual atmosphere and deliver it to an electric thruster as propellant, finally utilizing the cause of aerodynamic drag as source of thrust. A literature review is presented to give the ABEP state-of-the-art of the technology and the most relevant performance parameters are highlighted. The application of ABEP in VLEO is investigated by applying analytical equations based on atmospheric models and intake efficiencies based on the outcome of this work, and available state-of-the-art thruster efficiencies. Such analysis derives the collectible propellant flow, the aerodynamic drag, and the power required to fully compensate the drag. The case of GOCE using an ABEP system is presented, as well as its application in very low Mars orbit (VLMO). The intake and the thruster are investigated and designed within this dissertation. Three ABEP intakes designs are hereby presented, based on gas-surface-interaction prop- erties. Two are based on fully diffuse reflections, delivering collection efficiencies ηc < 0.5 and one based on fully specular reflections of ηc < 0.95. Their sensitivity to misalignment with the flow is analysed as well highlighting the specular design of being more robust compared to the diffuse one by maintaining relatively high ηc even for large angles. The ABEP thruster is based on contactless technology: there is no component in direct contact with the plasma, and a quasi-neutral plasma jet is produced. This enables operation with multiple propellant species (also aggressive such as atomic oxygen in VLEO) and densities, and does not require a neutraliser. The thruster is based helicon plasma discharges to provide higher efficiency compared to inductive ones.
... In an ideal geometry, all molecules will exit in the intended direction of thrust 1 and that all exiting molecules would have the same temperature as the wall. The ratio of the number of particles entering a system from the entrance plane to the number of particles exiting the system from the exit plane is known as transmission coefficient or transmission probability (Mancas, Cervone, and Zandbergen 2014;Binder et al. 2016). Analytically, Daduí C. Guerrieri et al. 2018 showed that for the same geometrical aspect ratio (depth of microchannel/smaller cross-sectional dimension), a rectangular and circular cross-sectional slot had a transmission coefficient of 0.36 and 0.19 respectively. ...
Thesis
Full-text available
A Low-Pressure Micro-resistojet (LPM) is a type of in-space electrothermal propulsion system for satellites that works by heating low-pressure (50 to 300 Pa) fluid flowing through microchannels/slots (typically <1 mm diameter) using resistive heating elements. This thesis delineates a response surface based method (RSM) to explore and optimize the cross-sectional design of microchannels, which, as documented in literature, have so far only been either rectangular or circular in shape. The following steps were performed as part of the RSM: Design sampling using Sobol Sequences, Fluid simulation using Direct Simulation Monte Carlo (DSMC) method with Sparc Industries' VSTRAP software, building surrogate model using Convolutional Neural Network (CNN) and optimization using Genetic Algorithm (GA). The optimization was performed for Argon as the propellant, inlet pressure of 50 Pa, inlet temperature of 300 K and microchannel wall temperature of 600 K. The best-performing design was found to have a thrust efficiency of 0.7019.
... The intake device developed by University of Stuttgart [61,62] are based on the transmission theory of rarefied flows [63,64]. The proposed intake design consists of honeycomb type ducts and a long slender cylindrical tube (called Enhanced Funnel Design, EFD) [65]. ...
Article
Increasing interest in development of very low Earth orbit (VLEO) has attracted more and more researchers to study atmosphere-breathing electric propulsion (ABEP) system in past several decades. This system can use rarefied atmospheric particles as the propellant of electric thrusters, and maintain a long lifetime mission without carrying any propellant from ground. As the key component of system, intake device can realize the collection and compression of atmospheric particles within limited frontal area, which determines the performance of whole ABEP system. This review summarizes the previous studies to develop intake devices, evaluates the corresponding performance and understands the model involved, including atmosphere model, flow physic model and so on. In addition, several continued researches for intake device are also presented, including ground experiment technologies, intake surface material development, space compressor and liquefaction technology. Wherever possible, comments have been provided to provide useful reference to researchers engaged in intake device for ABEP system.
... In the last decade, a number of intake concepts for air-breathing EP application were investigated by several research groups [10,[20][21][22][23][24]. Most of these concepts proposed compact intake geometries featuring an inlet composed of several elongated ducts having a circular, rectangular, or hexagonal section and act as a molecular trap for the collected flow, ideally increasing the collected particles residence time inside the propulsion system, hence the achievable intake compression. ...
Article
Full-text available
Air-breathing electric propulsion has the potential to enable space missions at very low altitudes. This study introduces to a 0D hybrid formulation for describing the coupled intake and thruster physics of an air-breathing electric propulsion prototype. Model derivation is then used to formally derive main system’s key performance indicators and estimate the figure of merit for the design of rarefied flow air intakes. Achievable performance by conical intake shapes are defined and evaluated by Monte Carlo simulations. Influence of inlet flow variation is assessed by dedicated sensitivity analyses. The set of requirements and optimality conditions derived for the downstream plasma thruster suggest concept feasibility within an achievable performance range.
... In 2016, the team studied the transmission probability of rarefied flows in low Earth orbit [39]. The results show that the collection efficiency (collected part of the flow into the air inlet) can be improved by implementing small ducts at the front of the intake. ...
Conference Paper
吸气式电推进(Atmosphere-Breathing Electric Propulsion,ABEP)系统在过去十年中对研究人员越来越具有吸引力。该系统可以摄取低轨道环境的稀薄大气分子作为推进工质来提供推力补偿,从而延长飞行器的工作寿命。该综述首先回顾了先前研究人员关于该系统开发作出的努力,然后总结了不同类型吸气式电推力器的主要性能,讨论了各类型推力器的优缺点,最后提出了将来进一步研究的可能途径。结果表明,利用低轨道稀薄大气分子作为推进工质进行推力补偿的潜力很大。但是,先前研究显示出的各种局限性,使得吸气式电推进系统目前还难以实际应用于空间任务。该系统的发展需要解决一些问题,例如进气效率,电离功率,电极腐蚀等。
Article
In this study, a vacuum air-intake device with an inlet diameter of 500 mm is designed for collecting space gas as the propellant of the air-breathing electric thruster, and it is comprised of a multi-hole plate, a big turbo, a small turbomolecular pump and a miniature scroll pump in series. The space gas collection efficiency of the vacuum air-intake device is mainly determined by the performances of the multi-hole plate and the big turbo, which are investigated by Monte Carlo method; the storage of the collected gas with high pressure is mainly achieved by the small turbomolecular pump and the miniature scroll pump, which are analyzed experimentally. The computation results of direct simulation Monte Carlo (DSMC) show that, the collection efficiencies of the high and the low rotational speed cases are 56.47%–57.85% and 41.67%–42.60% at the altitudes of 150–240 km, respectively; the turbo's powers of the high and the low rotational speed cases for gas drag compensation are no more than 52.6 W and 12.3 W, respectively. The experimental results indicate that, the small turbomolecular pump (weight: 5.7 kg) and the miniature scroll pump (weight: 0.84 kg) can quite efficiently compress the collected gas, and get an atmospheric pressure (8.5 × 104 Pa, in Lanzhou); the total power of the two pumps is 27.1–150.3 W at the gas flux of 0–50 sccm.
Article
The air-breathing ion engine (ABIE) is a new type of electric propulsion system to be used to compensate the aerodynamic drag of the satellite orbiting at extremely low altitudes. To save the propellant mass for a long operation lifetime, it inhales the low-density atmosphere surrounding the satellite and use it as the propellant of ion engines. Since feasibility and performance of the ABIE depend strongly on the compression ratio and the air-intake efficiency, numerical analysis has been performed by means of the direct-simulation Monte-Carlo method to clarify the characteristics of the air-intake performance in highly rarefied flows. Influences of the flight altitude, the aspect-ratio of the air-intake duct, and the angle of attack are investigated.
Article
Plasma flows with high Knudsen numbers cannot be treated with classic continuum methods, as represented for example by the Navier–Stokes or the magnetohydrodynamic equations. Instead, the more fundamental Boltzmann equation has to be solved, which is done here approximately by particle based methods that also allow for thermal and chemical non-equilibrium. The Particle-In-Cell method is used to treat the collisionless Vlasov–Maxwell system, while neutral reactive flows are treated by the Direct Simulation Monte Carlo method. In this article, a combined approach is presented that allows the simulation of reactive, partially or fully ionized plasma flows. Both particle methods are briefly outlined and the coupling and parallelization strategies are described. As an example, the results of a streamer discharge simulation are presented and discussed in order to demonstrate the capabilities of the coupled method.
Article
Accurate numerical calculations of molecular flow transmission probabilities and density distributions of tubes are important to the benchmark problems of Monte Carlo solutions, dynamic expansion vacuum gauge calibration systems, and molecular beam formation studies. Although Nawyn and Meyer [published by van Essen and Heerens. J Vac Sci Technol 1976; 13:1183] have solved cylindrical tube problems by using the numerical method based on Clausing's equations, perhaps the calculated results still lack sufficient accuracy. In this paper, we propose a modified method that could achieve calculation accuracies of transmission probabilities as high as 10−11–10−13 in the range of L′/R ≤ 100 (where L′ is the tube length, R is the tube radius), which are more accurate than the results recently reported by Mohan et al. [J Vac Sci Technol A 2007; 25:758] and Gómez-Goñi et al. [J Vac Sci Technol A 2003; 21:1452].
Article
A collisionless gas flows through the interior of a tube of circular cross-section, having both an entrance and an exit and which reflects molecules diffusely. Suitable complementary variational principles are used to obtain upper and lower bounds for the transmission probability of the tube. The numerical results, believed to be the most accurate to date, are compared with those of other authors.
Article
Following a brief historical introduction an overview is given relating the most recent studies of rarefied gas flow to the early work of Knudsen. The first paper submitted in October 1908 (published in 1909) initiated a period of intense activity by Knudsen, Smoluchowski (1910) and, a little later, by Gaede (1913) and Langmuir (1912). This also covered the transition to the already well established hydrodynamic flow expressed in terms of the ratio of mean free path to critical apparatus dimension: which is now referred to as the Knudsen number. The desorption, evaporation and scattering of molecules from surfaces was described in terms of the Knudsen cosine law of scattering. The Knudsen effusion method for determining vapour pressure, also introduced in 1909, has become the main tool for studies of the related problem of the dissociation, chemical bonding and the vaporisation process itself. Clausing (1926) developed, as an alternative to conductances, the concept of transmission probability, still referred to as the Clausing factor, and provided a procedure for their more accurate evaluation in long and short tubes. A number of misconceptions of these early efforts have found their way into the literature and current books on vacuum science and technology. However, detailed studies have clarified the problem of gas-surface interactions; the gas flow in tubes has been tackled with Clausing-type integral equations and by statistical computation techniques based on Monte Carlo analysis procedures adaptable to more complex systems. Results have been confirmed experimentally using molecular-impact pressure probe measuring techniques.
  • F Romano
  • T Binder
  • G Herdrich
  • S Fasoulas
  • T Schönherr
F. Romano, T. Binder, G. Herdrich, S. Fasoulas, and T. Schönherr, 34 th IEPC, Kobe, Japan (2015). [7] W. Steckelmacher, Reports on Progress in Physics 49, 1083–1107 (1986).