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Magnetic nozzle efficiency in a low power inductive plasma sourceTo cite this article: T A Collard and B A Jorns 2019 Plasma Sources Sci. Technol. 28 105019
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Magnetic nozzle efficiency in a low powerinductive plasma source
T A Collard and B A Jorns
University of Michigan, Ann Arbor, MI 48109, United States of America
E-mail: [email protected]
Received 23 February 2019, revised 4 June 2019Accepted for publication 27 June 2019Published 29 October 2019
AbstractThe nozzle efficiency and performance of a magnetic nozzle operating at low power (<200W)are experimentally and analytically characterized. A suite of diagnostics including Langmuirprobes, emissive probes, Faraday probes, and laser induced fluorescence are employed to mapthe spatial distribution of the plasma properties in the near-field of a nozzle operated with xenonand peak magnetic field strengths ranging from 100 to 600 G. The nozzle efficiency is found tobe <10% with plasma thrust contributions <120 μN and specific impulse <35 s. Theseperformance measurements are compared with predictions from quasi-1D idealized nozzletheory and found to be 50%–70% of the model predictions. It is shown that the reason for thisdiscrepancy stems from the fact that the underlying assumption of the idealized model—that ionsare sonic at the throat of the nozzle—is violated at low power operation. By correcting for theshifting location of the sonic transition point, the model and experiment are made to agree. Thephysical mechanism by which the sonic line moves with respect to the nozzle geometry isattributed to non-ideal behavior at low power. In particular, it is posited that the low ionizationfraction at these low operational powers gives rise to neutral-collisional effects in the near-fieldof the device that can impede ion acceleration. The roles of ionization, enhanced electronresistivity, and charge exchange collisions are all examined. It is found that the ion sonictransition location is most correlated with the ratio of the charge exchange mean free path to thecharacteristic electrostatic acceleration length giving rise to an effective drag term on the ions.
Keywords: magnetic nozzle, electric propulsion, low temperature plasma, Detachment
1. Introduction
The increasing demand for new forms of in-space propulsionfor small spacecraft has given rise in the past two decades to agrowing interest in low power (<200W) magnetic nozzlethrusters [1–16]. This interest stems largely from the numberof advantages magnetic nozzles, a form of electric propulsionthat employs a diverging magnetic field to accelerate a heatedplasma, can offer compared to state-of-the-art technology.Due to the absence of a plasma-wetted electrode, thesedevices can have a longer lifetime compared to other forms ofelectric propulsion. They can operate on a wider range ofpropellants more easily stored for small spacecraft [17–22].From a systems perspective, magnetic nozzles can be simplerthan state-of-the-art thrusters (e.g. Hall effect and ion thrus-ters) as they do not require a separate, dedicated neutralizingelectron source. Despite these apparent advantages, the
demonstrated performance of state-of-the-art magnetic noz-zles to date has been low: ∼8% total efficiency at ∼2 kW [2]with decreasing total efficiency observed at lower power. Thislow performance has limited the viability of magnetic nozzleelectric propulsion as an alternative to more mature forms ofelectric thruster technology. The Hall effect thruster, forexample, exhibits �65% total efficiency when operating at>2 kW power levels [23].
In an effort to explain and potentially improve the lowperformance of magnetic nozzles, a number of previousanalytical and numerical investigations have focused ondeveloping models for their performance scaling [7, 8, 10, 24].These models have shown that the low overall device per-formance largely can be attributed to high ion costs and radialwall losses. Subsequent development efforts (see [9, 12, 25])have focused on exploring new confinement techniquesand methods for ionizing the plasma. Yet, while these
Plasma Sources Science and Technology
Plasma Sources Sci. Technol. 28 (2019) 105019 (18pp) https://doi.org/10.1088/1361-6595/ab2d7d
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first-principles models have been successful in illustrating thechallenges with nozzles at mid- to high-power operation(�200W), the predicted performance from these models inmany cases has not matched measured values at lower powers[10, 16]. This suggests that there may be other effects, beyondthe two dominant ones cited above, that adversely impactmagnetic nozzle performance at low power. For example,Correyero et al [16] were able to match measured ionacceleration and the acceleration predicted by a quasi-1Dkinetic model of a low-power electron-cyclotron resonancethruster, but noted that the ion sonic condition did not coin-cide with the location of peak magnetic field—the locationtraditionally defined as the nozzle throat. Correyero et al positthat this downstream movement of the throat is due to dif-fusion and/or ionization processes, but note that their simu-lation work does not yet include these processes [16]. Inparallel, the potential role of neutral collisions in the near-field plume at low operating powers has been explored as apotential adverse mechanism; indeed, as was shown in [12],the ionization fractions at low power can be less than 10%,thus giving rise to a large fraction of neutrals. Most models todate, however, have neglected the role of this population[3, 5, 7, 10, 24] and its impact on nozzle performance isunclear. Given the fact that low power operation is of part-icular interest for the new paradigm of microsats whereavailable power on orbit is limited [26], there is a pressingneed to understand and mitigate low power effects like these.To this end, the goal of this paper is to quantify and explainthe discrepancies in the predictions from standard magneticnozzle models and actual nozzle performance when operatingat low power. In particular, we focus on the nozzle efficiency—the ability of the nozzle to convert thermal energy todirected kinetic energy—in a low-power, inductively-coupledtest article.
This paper is organized in the following way. Insection 2, we review quasi-1D analysis as applied to magneticnozzles and derive explicit forms for the efficiency and per-formance terms In section 3, we describe an experimentalinductively-coupled magnetic nozzle source and the diag-nostics we employed to measure its plasma properties. These
spatial plasma measurements and the inferred nozzle perfor-mance characteristics are presented in section 4 and comparedto predictions from existing magnetic nozzle models. Finally,in section 5, we discuss the discrepancies between our mea-surements at low power and model predictions, and propose apossible explanation related to the low ionization fraction (theratio of plasma density and neutral density) that these devicesexhibit at low power.
2. Theoretical description of nozzle efficiency
In this work we examine the diverging part of the magneticnozzle, i.e. the region where the plasma thermal energy isconverted into directed kinetic energy. In order to quantify thediscrepancy between model predictions and observed per-formance in this region we focus on one key parameter: thenozzle efficiency. In this section, we introduce a definition forthis parameter that can be measured directly and predictedusing an idealized standard model for this efficiency. Thislatter analysis draws from a number of previous studies onnozzle dynamics [3–6, 8, 10, 24].
2.1. Geometry
Figure 1 shows the canonical, axisymmetric geometry of amagnetic nozzle thruster we employ for this analysis. The‘upstream’ region in this figure is characterized by a gas inlet,a confining liner, and concentric magnets. The plasma iscreated here, and electrons are heated by interactions with theapplied RF field. This heated plasma transits through the‘throat,’ typically assumed to be coincident with the locationof peak magnetic field magnitude. Downstream of the throat,the magnetic field lines diverge and the magnitude of the fielddecreases. In the idealized case, electrons follow the divergingmagnetic field lines and expand. The resulting pressure gra-dients give rise to diamagnetic currents that interact with thethruster magnets to generate a net force on the thruster [4, 6,27–29]. In practice, the diverging magnetic field lines ulti-mately turn back onto themselves, and if unchecked, the
Figure 1. (a) A notional magnetic nozzle thruster. The subscripts denote key locations in the plasma. (b) The normalized magnitude of thecenterline magnetic field as a function of axial position (solid curve) with the source exit plane marked (dashed line).
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Plasma Sources Sci. Technol. 28 (2019) 105019 T A Collard and B A Jorns
plasma also will follow these lines and be re-directed back tothe nozzle. It is necessary for net thrust generation that theplasma ‘detach’ from the field lines (denoted as the detach-ment plane in figure 1) at some point downstream. The pro-blem of how and where detachment occurs is an on-goingsubject of research [3–6, 24, 30–34], though a number ofempirical and semi-analytical criteria have been proposed.We reserve a discussion of these for later.
Functionally, although thrust generation happens throughthe mechanisms of plasma pressure and diamagnetic currents,magnetic nozzles can also be analyzed through the lens ofconventional quasi-1D nozzle theory [8, 10, 35]. In thiscontext, the momentum exchange in the nozzle is viewedfrom the perspective of the heavier species—the ions. Thoughthese are too massive to be magnetized, the ions do experi-ence electrostatic acceleration from the potential gradientsthat result from the expansion of the electrons. This processdraws direct parallels to conventional gasdynamic nozzles inwhich the thermal energy of the propellant (as stored in theelectrons) is converted to directed acceleration of the pro-pellant (as represented by the ions). In this case, the ions areaccelerated until the detachment plane, which can be equatedto the effective exit plane of a conventional nozzle. Sig-nificantly, the analogy carries beyond a qualitative analysis,and it has been shown that these nozzles lend themselves toquasi-1D nozzle expansion theory (see [36]) in which thethrust is related to the cross-section of the nozzle. This is theformalism and physical interpretation we apply here in whichwe discuss the expansion and acceleration of the exhaustthrough the throat with cross-sectional area, A0, and the exitplane with cross-sectional area, Ad. In the absence of walls,the radius of the nozzle geometry at each location (seefigure 1) is defined from the centerline to the vacuum inter-face line—the magnetic field line that grazes the edge of thesource at the exit plane.
2.2. Measurable expression for nozzle efficiency
For our analysis, we first begin by introducing a generalizeddefinition of overall thruster efficiency, i.e. directed exhaustpower to input power (see [37]):
h =T
mP2, 1
2
0˙( )
where m denotes the flow rate of the propellant, P0 is theinput power to the system, and T is the thrust. For magneticnozzles, this efficiency is a product of a number of con-tributions, h h h h h= ,rf m loss noz where ηrfdenotes the effi-ciency of conversion of RF input power to thermal energyinside the plasma source, ηm is the mass utilization efficiency,ηloss is the efficiency of the source dictated by losses to thesource walls, and ηnoz is the efficiency of conversion ofthermal energy passing through the nozzle throat to energyemployed for thrust [7, 10, 24, 34, 38]. The first three com-ponents of the efficiency are ‘source-related,’ i.e. they dictatehow well the RF source ionizes and couples thermal energyinto the propellant before it is accelerated through the nozzle[7, 10, 24, 34, 38]. Our analysis, however, focuses on the
downstream contributions to the efficiency from the nozzle,which can be written as
h =T
m P2, 2noz
2
ion noz˙( )
where Pnoz is the power entering the nozzle from the sourceregion, mion˙ is the mass flow rate of ions, and we haveassumed that the plasma is singly ionized. This latterassumption is plausible given the typically low electrontemperatures in these devices [10, 12, 25, 34]. Physically, thisexpression represents the fraction of thermal energy flowinginto the nozzle that is converted to useful directed kineticenergy for thrust.
We can find explicit expressions in terms of plasmaplume properties for both the inlet energy to the nozzle aswell as the thrust by employing quasi-1D nozzle analysis tothe geometry shown in figure 1. For the thrust, we apply acontrol volume analysis coupled with the assumption that theplasma has a quasi-1D expansion (the plasma properties areapproximately constant across each axial slice) to find
= +T m u p A , 3d e dion˙ ¯ ( )
where Ad is defined as before, pe is the average plasmapressure across the nozzle section at the detachment location,and ud is the axial component of ion velocity at the detach-ment location. To evaluate the power flow into the nozzle in(2), we again consider a quasi-1D flow and follow previouswork in assuming the electron pressure expansion in thenozzle is governed by polytropic cooling [9, 39–43]. We thuscan write (see appendix)
gg
= + +--
+
´
P m u qT qT q
n u A
1
2
5
2
3
2
5 3
1
,
4
i e e c
e
noz 02
,0 ,0
,0 0 0
⎡⎣⎢
⎛⎝⎜
⎞⎠⎟
⎤⎦⎥
( )
where mi is the ion mass, òc is the ion cost in eV, ne,0 is theaverage density in the throat, Te,0 is the electron temperaturein the throat, A0 is the cross-sectional area of the throat, q isfundamental charge, u0 is the ion drift velocity in the throat,and γ is the polytropic index. This expression represents thepower flowing into the nozzle with contributions from ionkinetic energy (first term), electron pressure (second term),heat conduction (third term), and ion creation losses (lastterm). This last term historically has been dominant inmagnetic nozzle sources [9, 10, 34] and represents an effec-tive ‘frozen flow’ loss for the creation and maintenance of theplasma. The ‘frozen flow’ losses depend on the ionization andexcitation states of the ions and the electron temperature (seeappendix for further discussion).
Armed with (2)–(4), we thus have the ability to evaluatethe nozzle efficiency experimentally provided we know thelocation of the detachment plane and have measurements ofkey plasma properties in this plane. Given the challenges withmeasuring the low thrusts generated from these devicesdirectly, this type of assessment of performance from plasmameasurements has formed the basis for a significant fraction
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Plasma Sources Sci. Technol. 28 (2019) 105019 T A Collard and B A Jorns
of the studies performed on these devices to date[2, 34, 40, 41, 44].
2.3. Model for nozzle efficiency
There have a been a number of modeling efforts on magneticnozzles to date—ranging from analytical [8, 10, 35, 38] to 2Dnumerical codes [3–7, 24, 39]—to predict key performanceparameters like the the overall efficiency. In order to providerepresentative results for the predictions of these models incomparison to our experimental measurements for a lowpower system, we consider a model informed by the inter-pretation of the magnetic nozzle in terms of a traditionalnozzle analysis. In particular, it is possible to relate the overallperformance and efficiency to the inlet conditions at the throatand the expansion ratio of the nozzle (Ad/A0). The form of thisrelation varies depending on the assumptions made concerningthe expansion process. For our work, we leverage insight fromprevious analytical studies [3–10, 24, 30, 31, 34, 40] to makefour key assumptions:
• The neutral species can be neglected• The ions are cold• The ions are sonic at the throat (location of peakmagnitude field), i.e. M=u0/cs=1, where M denotesthe ion Mach number and cs is the local Bohmspeed ( g=c qT ms e i,0 )
• The plasma expansion is polytropic and governed byan equation of state =g-T n constante e
1( ) , where 1�γ�5/3.
This last assumption of polytropic cooling has beensupported by a number of experiments that have been per-formed to date on magnetic nozzles. These have shownempirical values between 1.1 and 1.4 [9, 42, 43]. Physically,the value of the polytropic index for xenon is bounded by twolimits, representing infinite heat conduction along field linesfrom the source (isothermal, γ=1) and no electron thermalconduction (adiabatic, γ=5/3 for noble gases). We chooseto leave the model with an arbitrary, but constant polytropicindex as we have the ability to measure it directly in oursystem (section 4).
Using the above assumptions we can find an expressionfor the theoretical nozzle efficiency:
hg
gh=
+ + +gg
--
M T
T5 3, 5t
e d
T e
noz,
2,
5 3
1
2,0
divc
e,0( )[ ]( )
where g=M u qT md e d i, denotes the ion Mach number atthe detachment plane and ηdiv is a divergence efficiency thatcorrects for the 2D nature of the plume expansion. We canfind this latter parameter by geometric arguments or directmeasurements of the plume expansion. Conversely, we canfind the Mach number at the detachment plane provided weknow the detachment location and its associated cross-sec-tional area. In this case, the Mach number is related to the
expansion ratio and inlet ion velocity in the following way:
=g-
T Tu A
u A6e d e
d d, ,0
0 01⎛
⎝⎜⎞⎠⎟ ( )
gg
= +-
-g-
u uqm T u A
u A
2
11 . 7d
i e
d d
202 ,0 0 0
1⎡⎣⎢⎢
⎛⎝⎜
⎞⎠⎟
⎤⎦⎥⎥ ( )
These stem from the equation of state for electron energy andion continuity under the assumption of a collisionless plasma.Provided that the inlet conditions to the nozzle and theexpansion ratio are known (5)–(7), allow for the evaluation ofthe nozzle efficiency and by extension the thrust via (3).Physically, these results show that as the expansion ratioincreases, the plasma can expand more leading to a greateracceleration of the plasma. This is consistent with traditionalde Laval nozzle theory [36].
2.4. Location of detachment plane
Both the experimental and analytical expressions outlined inthe preceding require an estimate for the detachment point ofthe plasma from the magnetic field. While there are a numberof proposed theories for this location, both analytical andsemi-empirical [3–6, 24, 30–34], to date there has not been aconsensus about its location. We introduce for this study anexperimentally-driven criterion that we apply both for theexperimental evaluation of performance and the analyticalmodel. We conjecture that when the ions have stoppedaccelerating, i.e. the ion velocity as measured on the nozzlecenterline is constant, the plasma has effectively becomedetached from the nozzle. This criterion stems from theassumption that if the ion velocity is constant, the ions are nolonger being accelerated by the nozzle dynamics. As wediscuss in the following section, we are able to infer thetransition to this point by employing a combination of ionvelocimetry and current density diagnostics.
3. Experimental apparatus
3.1. Plasma source and vacuum chamber
To quantify the performance of the experimental low-powermagnetic nozzle device using the above generalized perfor-mance framework requires initial and boundary conditionsthat must be informed by experimental measurements. For ourinvestigation we employed a radiofrequency (RF) plasmasource 13 cm in diameter with an integrated 2.5 cm diameterby 1.9 cm long (R0= 12.5 cm, L0=1.9 cm) monolithicquartz plasma liner (figure 2). Neutral xenon propellant wasinjected into the source tube through a central hole in the backof the quartz liner with a fixed flow rate of 3 mg s−1. Theneutral propellant was excited by a 3-turn solenoidal antennawrapped around the quartz liner. The antenna was constructedof 3 mm hollow copper tubing enabling water cooling tominimize impedance variations due to thermal loading. It wasconnected to a transmission line comprised of a bi-directionalcoupler, an L-type automatic matching network, and an RF
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Plasma Sources Sci. Technol. 28 (2019) 105019 T A Collard and B A Jorns
amplifier operating at 13.56 MHz. The directional couplerprovided in situ forward and reflected power telemetry whilethe automatic matching network improved power coupling byreducing the reflected power, typically to below 20% of theforward power.
The converging-diverging magnetic nozzle was producedby an electromagnet constructed of 10 AWG square enamelcoated copper magnet wire. The current through the electro-magnet was controlled to produce peak nozzle throatstrengths up to 900G. To mitigate thermal loading andmaintain steady-state operating temperatures below themaximum 240 °C wire threshold the electromagnet spool waswater cooled.
The integrated plasma source was mounted on supportstructure within the Junior Test Facility at the University ofMichigan. Junior is a 3 m long by 1 m diameter stainless steelclad vacuum chamber backed by a combination of rotary,turbomolecular, and cryogenic pumps. This facility wasselected due to the small size of the device (13 cm in dia-meter, overall) to minimize plasma-swall interactions. Theeffective pumping speed for the experiments described hereinwas ∼15 000 l s−1 on xenon. With this pumping capacity thebase pressure of the Junior facility is ∼1×10−8 Torr xenonand the background pressure was ∼4×10−5 Torr xenonduring full source operation.
3.2. Diagnostics
To fully characterize our nozzle source and the plasmaparameters of interest (section 2), we employed probes tomeasure the magnetic field, ion velocity, neutral density,plasma density, electron temperature, plasma potential, andion current density. The axial and radial ion velocity profileswere measured using a 2D time-averaged laser inducedfluorescence (LIF) setup [45] with an effective spatial reso-lution of ∼1 mm. The non-resonant LIF scheme we usedtargeted the d F5 2
7 2 to p D6 25 20 metastable line of xenon ions
and yielded an ion velocity distribution function (IVDF). The
plasma parameters were measured with a suite of electrostaticprobes, including an unguarded nude Faraday probe (ionsaturation current), a double Langmuir probe (plasma densityand electron temperature), and an emissive probe (plasmapotential). To accommodate the small size of the plasmasource the tungsten collection electrodes of the Faraday probeand double Langmuir probe were 1.6 mm and 0.5 mm indiameter, respectively. Due to the small characteristicdimension of the probes and the sparse plasma density themeasurements were corrected for sheath expansion effectsfollowing Sheridan’s method [46]. This sheath expansioncorrection was included in both the Faraday and doubleLangmuir probe analyses. The thoriated tungsten hairpin loopemissive probe was operated in the limit of high emission,with measurements in both the cold and saturated emissionstates allowing for estimation of the plasma potential byincluding the electron temperature factor correction of the hotfloating potential [47]. The magnetic field topography wasmeasured with a Lakeshore Model 460 3-axis Gaussmeter. AStabil Ion Gauge with an integrated Pitot tube measured theneutral pressure throughout the plume, allowing for estima-tion of the neutral density.
To preserve optical alignment the plasma source wasattached to a pair of translation stages while the laser opticswere fixed. The plasma source then was moved to interrogatemultiple points within the plume. This LIF setup is shown infigure 3. For all other probes the plasma source was mountedat a fixed location within the vacuum chamber and thediagnostics were translated via the motion stages.
4. Results
In this section we present the measurements of the plasmaproperties necessary to evaluate the measured and modelpredictions for nozzle performance. These data were takenwith the nozzle operating at ∼170W combined depositedpower into the transmission line and plasma source and
Figure 2. A (a) front and (b) rear isometric view of an as-built 3D model of the plasma source. Note that the front section of the Faradayshield is removed to allow for the internal components to be viewed. This shield is installed for the experiments discussed herein.
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Plasma Sources Sci. Technol. 28 (2019) 105019 T A Collard and B A Jorns
3 mg s−1 xenon propellant flow rate. We took measurementsfor magnetic field strengths ranging from 100 to 600 G at100 G intervals. These operating conditions were selected dueto the source stability and repeatability over a wide range ofmagnetic field conditions—the source was operated for �12 hper run with <1% change in the source telemetry and exitplane plasma measurements, and between each run the tele-metry and exit plane measurements matched within 1% uponrestart. The ability to tune the magnetic field strength whilekeeping the propellant flow rate and input power constantassists in isolating the underlying mechanisms impactingperformance.
4.1. Plasma properties
4.1.1. Plume properties. We measured plasma density andelectron temperature by using a double Langmuir probe andfitting functions to different parts of the IV curve, followingthe procedures outlined by Brockhaus et al [48]. A sampleraw trace—for the 300 G operating condition and at thecenterline, exit plane location—is shown in figure 4. This
analysis relies on the assumption of a Maxwellian plasma.Following this prescription, we were able to measure theseplasma parameters over a range of positions. In figure 5 weshow contour maps of the measured plasma density, plasmapotential, and electron temperature in the nozzle plume. Theelectron temperature map in figure 5(a) shows cooling as theplasma expands downstream. This indicates that the plasma isnot isothermal and is losing thermal energy. In turn, thiscooling is correlated with the spatial distribution of theplasma potential, which figure 5(b) shows is highest at thenozzle inlet and decreases with distance from this plane. Theelectrostatic field in the plume that results from this potentialstructure is the mechanism for accelerating ions. The plasmadensity plot (figure 5(c)) shows the evident expansion of theplasma in the axial and radial directions with the highestdensity concentrated on centerline at the throat. This isconsistent with the plasma pressure, as represented by theelectron pressure, decreasing as it expands. The convolutionof the ion acceleration and the density profile is captured inthe axial ion current density plot in figure 5(d). However,there is a notable feature in the plasma density and currentdensity profiles: for slices at constant axial location beyondthe (Z/R0)=1 plane, the plasma density exhibits a radiallynon-monotonic dependence. This is characterized by aneffective ‘clustering’ near the vacuum interface line. This typeof structure has been observed before [25] and may beexplained in part by a combination of spatially non-uniformheating within the plasma liner [49, 50] and the presence of acorresponding well in plasma potential along this boundary(figure 5(b)). This potential well structure has been noted in anumber of magnetically confined plasmas [34, 51–53] andcan be interpreted as the consequence of charge separationthat results from ions with sufficient transverse energyovershooting the attached electron fluid. The potential wellforms to counter this finite transverse ion inertia and deflectthe ion streamlines back toward the field-aligned electronstreamlines [34, 51].
4.1.2. Ion acceleration. Taken together, the plasmameasurements in figure 5 show that the conditions areappropriate for electrostatic ion acceleration. We measuredthis acceleration directly in the near-field with the LIF systemby characterizing the axial and radial IVDFs simultaneously(see figure 6(a)). The mean velocity in each direction iscalculated by taking the moment of these distributions afterapplying a sums-of-gaussians fit. Due to the presence of themagnetic nozzle, we assessed the potential impact of Zeemansplitting on our results. Following the approaches of Jornset al [54] and Huang et al [55], and assuming that the ionvelocity distribution is Maxwellian, we estimate that Zeemansplitting adds �5% to the uncertainty in the ion velocitymeasurements at the exit plane, in both the cross-field andfield-aligned directions. Due to the rapid decay in themagnetic field strength with increasing distance from theexit plane, the uncertainty due to Zeeman splitting decreasesdownstream. This low uncertainty contribution allows us toneglect this effect in our analysis of the results. The resulting
Figure 3. The experimental setup used to measure the spatialevolution of the ion velocity distribution throughout the expandingmagnetic nozzle using LIF.
Figure 4. A characteristic I–V trace measured with the doubleLangmuir probe. This specific trace was measured at the centerline,exit plane of the source operating with ∼170W input power,3mg s−1 of xenon, and a field strength of 300G.
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Plasma Sources Sci. Technol. 28 (2019) 105019 T A Collard and B A Jorns
vector map exhibits a subsonic ion speed at the source exitplane (cs∼1.5 km s−1 for this 400 G condition) and largeradial velocity components throughout the plume. The errorin these measurements is estimated as the 95% confidenceinterval after bootstrapping the IVDFs 10 000 times; thelargest error is at (Z/R0, R/R0)=(5/4, 3/2)mm with ±12%error in speed and ±4° in angle. The trajectories of the ionsare consistent with our measurements of the plasma potentialin figure 5. Indeed, following these contours, it is evident thatthe ion trajectories appear to be driven by electrostaticacceleration resulting from the potential structure. This isconsistent with our physical interpretation of the nozzledynamics outlined in section 3. Moreover, we note that ourresults are consistent with previous nozzle studies in twosignficant ways. First, as shown in figure 6(b), the ions are infact accelerated, albeit not prodigiously, in the axial directionas they transit downstream. This is an indication that the ionsare gaining directed kinetic energy as they transit the nozzle.Second, the ions appear to exhibit an ‘outward separation’ inso much as their trajectories diverge more quickly than thenozzle streamlines. This type of ion motion has been seenboth experimentally [34] and predicted numerically [5].
Due to low signal-to-noise ratio in the downstreamregion where the plasma becomes more sparse, we were onlyable to map the ion trajectories using LIF in the region shown
in figure 6. To supplement the near-field data and provide amore complete depiction of the ion dynamics, we alsomapped the ion current density in the far-field (seefigure 5(d)). This spatial map, like all the electrostatic probemaps, extends from the near-field into the plume far-field. Theoverlapping of the ion current density map with the LIFmeasurements allows for the inference of the far-field ionvelocity when coupled with the plasma density measure-ments, assuming that the plasma is singly ionized ( j=qneu).This map reveals a new detail for the ion trajectories: asshown in figure 6(c) the centerline ion velocity plateauswithin (Z/R0)∼5. Consistent with our definition in section 2,thus we are able to empirically designate this plane as thedetachment location.
4.1.3. Mach number. In figure 6(d) we show a plot of thecalculated ion Mach number. This result reveals a notablefeature that is inconsistent with the physical assumptions weoutlined for the model described in section 2: it is evident thatat the inlet plane of the nozzle, which is characterized by thepeak magnetic field magnitude, the ions have not becomesonic. Instead, the sonic transition occurs farther downstreamat location (Z/R0)∼0.5, on average across the nozzle. Thisresult is consistent with recent work that has been performedon a low power electron-cyclotron resonance thruster [16] and
Figure 5. (a) A spatial map of the electron temperature, (b) plasma potential, (c) plasma density and (d) axial ion current density (contour andsolid curves) with an overlay of the magnetic nozzle field lines (dashed curves) for the 400 G, ∼170 W net deposited power, and 3 mg s−1
xenon propellant flow rate operating condition. Note the potential well near the vacuum interface line and that the axial position is referencedto the throat location.
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Plasma Sources Sci. Technol. 28 (2019) 105019 T A Collard and B A Jorns
seems to be evidence that the idealized nozzle expansion mayhave some limitations in describing this low power regime ofoperation. We revisit this in the following section.
4.1.4. Polytropic cooling. With the plasma measurementsfrom the preceding sections, we have nearly all of theinformation required to evaluate both the experimental andanalytical predictions for nozzle conversion efficiency. Theremaining key element is the polytropic cooling index. Wecan estimate this value by noting that df/dTe=γ/(γ−1)and using linear regression of the plasma potential versuselectron temperature [40, 42]. The resulting best fit linearregression to the data along centerline is shown in figure 7(a)for the six magnetic field settings that we investigated in thiswork. The error is estimated by using the lines fit to theextremes of the data and its associated error, resulting in thesteepest and shallowest line slopes (see figure 7(b)). Forcomparison, we also show the polytropic index for adiabaticand isothermal cooling. In all six cases, it is evident thatγ∼1.1, which is within the range of measured values forelectric propulsion devices operating on xenon [9, 41–43],
and between the adiabatic (γ=5/3) and isothermal (γ=1)limits.
Summarizing the above results, we see that this lowpower nozzle does exhibit the type of behavior we expect forthese devices. Electron thermal energy evidently is convertedto ion kinetic energy with the magnetic field acting as amediating factor. This acceleration then occurs until a discretedownstream ‘exit plane.’ In the next section, we apply theformalism outlined in section 2 to quantify the efficacy of thisnozzle conversion and compare it to the analyticalpredictions.
4.2. Nozzle performance
4.2.1. Efficiency. Leveraging the results from the previoussection, we can employ the formulations from section 2—equations (2)–(5)—to evaluate the nozzle efficiency. To thisend, we estimated the location of the detachment plane as thelocation where the ion velocity plateaus, as inferred from theFaraday probe, yielding Ad. We estimated the axialcomponent of the ion velocity at this exit plane from the
Figure 6. (a) An example axial IVDF at (Z/R0, R/R0)=(0, 1/4) with a sums-of-Gaussian fit (solid curves) to the raw data (circles). (b) Theresulting ion velocity vectors (arrows) and corresponding ion streamlines (dotted curves). (c) The mean axial ion velocity on centerline.(d) The inferred local ion Mach number (contour and solid curves) with an overlay of the magnetic nozzle field lines (dashed curves). All datais taken from the 400 G, ∼170 W net deposited power, and 3 mg s−1 xenon propellant flow rate operating condition. Note that themeasurement locations are indicated by an ‘x’ and that the axial position is referenced to the throat location.
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Plasma Sources Sci. Technol. 28 (2019) 105019 T A Collard and B A Jorns
Faraday probe data, yielding a value for ud. We approximatedthe ion flow rate by using the measured density (doubleLangmuir probe), ion velocity (LIF), and source geometry atthe throat ( =m m n u Ai eion ,0 0 0˙ ). We determined the electrontemperature at the throat using the double Langmuir probemeasurements to yield Te,0 and we used the polytropic indexof ∼1.1 from figure 7. And finally, we estimated the ionenergy cost using the analysis outlined in Lieberman andLichtenberg [56] to yield òc. Combining these measurementsand inferred values at the throat yielded a calculated powerflowing into the nozzle and an estimate for thrust. With these
values, we ultimately were able to use (2)–(4) to generate aplot of the nozzle efficiency, as shown in figure 8(a).
This result shows that the nozzle contribution to theoverall efficiency is markedly low, <10%. In other words, thislow power nozzle is only capable of converting 10% of theavailable energy at the inlet into directed exhaust. The majordriver for this loss in efficiency is illustrated by looking at thedifferent contributions to the nozzle power in (4), shown infigure 9. It is evident from this figure that the largest term isthe ion cost, scaling with òc. This finding is consistent withprevious works [7, 8, 10, 24] and underscores the dominantrole of frozen flow losses associated with ion production. Forillustative purposes and to capture the magnitude of the impactof the ion cost term, consider the nozzle efficiency if thefrozen flow losses could be neglected (h¢noz); for the 400 G
Figure 7. (a) The value of the polytropic index inferred using linear regression of the plasma potential versus the electron temperature for thesource operating conditions (squares). Note that value of the polytropic index is bounded by the adiabatic (γ=5/3, dashed line) andisothermal (γ=1, dotted line) limits. (b) The measured centerline plasma potential versus the electron temperature (circles) for the 400 Goperating condition and the best-fit regression line (solid line). The error in the polytropic index is calculated by using the lines fit to the data,and its associated error, such that the minimum and maximum slopes are obtained (dashed and dotted lines). Across all magnetic fields thedevice is operated at ∼170W net deposited power and 3 mg s−1 xenon propellant flow rate.
Figure 8. (a) The inferred nozzle efficiency from measurements ofthe plasma source (squares) compared to the efficiency predicted bythe ideal model (circles) and the model using the downstream throatproperties (triangles). (b) The divergence efficiency of the plasmasource (squares) compared to the divergence efficiency of a fullyattached, ideal nozzle that detaches at the same location. In allmagnetic field conditions the source is operated with a net depositedpower of ∼170W and propellant flow rate of 3mg s−1 xenon.
Figure 9. The fraction of all modes to the total power flowing intothe nozzle. Note that at each magnetic field strength the net depositedpower is ∼170W and xenon propellant flow rate is 3 mg s−1.
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Plasma Sources Sci. Technol. 28 (2019) 105019 T A Collard and B A Jorns
operating condition h¢ ~ 24%noz . This is ∼5× higher than theactual nozzle efficiency, demonstrating the impact of thefrozen flow losses on the nozzle efficiency.
On the other hand, we note that the nozzle efficiencydoes monotonically improve with magnetic field strength.Physically, this trend is expected because our measuredelectron temperature at the throat increases with increasingmagnetic field, and as previous work has shown, increasedelectron temperature can translate to improved efficiency[9, 10, 34]. We do note that the trend in electron temperaturewith magnetic field is not predicted by classical 0D globalpower balance models [56]; this power balance predicts thatelectron temperature decreases with increasing magnetic fieldstrength due to the correspoding decrease in plasma diffusionto the radial walls. This discrepancy does not impact ouranalysis here as we focus on the downstream dynamics of thesource. However, we note that there are a number of possibleexplanations for this result. For example, as Kinder andKushner proposed, the increased electron temperature athigher magnetic field conditions may be explained byspatially non-uniform power deposition; a result that isnumerically predicted for solenoidal antenna power couplingto a plasma in the presence of a magnetic field [49, 50].
In addition to the various energy terms entering thenozzle, we can observe the energy transfer between thegasdynamic modes throughout the nozzle by using the plasmameasurements in figure 5. Figure 10 illustrates this energytransfer in the nozzle for the 400 G operating condition. Fromthese results the pressure term decreases from ∼20% to∼12% and the heat conduction decreases from ∼79% to∼60%. The energy from these modes is transferred into ionkinetic energy, which increases from ∼1% to ∼28%. It isclear that the decrease in the pressure term is not sufficient toaccount for the ion kinetic energy increase. This suggests thatthe heat conduction is coupled with the ion kinetic energy,and the electron cooling results in a decrease in the heatconduction and a corresponding acceleration of the ions.While the energy transfer between heat conduction and ionkinetic energy is promising, Little and Choueiri [40] showed—through the lens of the Nusselt number, or the ratio of theconvected heat to conducted heat—that efficiency losses may
be incurred due to unrecovered electron thermal energy fordevices in which the power entering the nozzle is dominatedby heat conduction. The dominant energy term in our nozzle,aside from the frozen flow losses, is the heat conduction, sowe next evaluate the potential efficiency penalty incurred byour nozzle by calculating the Nusselt number. For ouroperating conditions the Nusselt number was of the order of∼0.01 to ∼0.1, which suggests that the maximum nozzleefficiency of our device may be limited to ∼50%–80% due toa portion of the electron thermal energy being unrecoverable[40]. This effect is not sufficient to explain the low nozzleefficiency in figure 8(a). A Nusselt number 2–10× smallerthan the values observed in our device would be required tomatch the measured efficiency loss, suggesting that unreco-verable electron thermal energy may be only one of severaleffects limiting the nozzle performance.
For comparison to these measured results, we nextevaluate the analytical predictions for the nozzle efficiency.To this end, we follow the conventional approach used in theformulation in section 2, where we assign the ‘throat’ to bethe exit of the liner. We assume an expansion ratio consistentwith the detachment point measured in figure 6(d)(Ad/A0=6.7) and that the ions are sonic at the throat. Wealso employ the measured electron temperature at the throat(figure 5(a)) and the polytropic index (figure 7). Also, as canbe seen from (5), in order to do a comparison between theanalytical nozzle efficiency and measurements, we need totake into consideration the divergence efficiency of thesystem. We evaluated this correction by using Faraday probemeasurements in the detachment plane and the iterative path-finding method recommended by Brown et al [57] for usewith these near-field Faraday probe measurements. As can beseen in figure 8(b), for all the magnetic field settings thisyielded a divergence efficiency of ηdiv≈68%. Armed withthis result, we are able to plot the analytical predictions fornozzle efficiency in figure 8(a). From the comparison of themodel and source results, it is immediately apparent that theanalytical nozzle efficiency model overpredicts the measuredvalue for all magnetic field settings by a factor of ∼5 at thehighest magnetic field case (600G). This is direct evidencethat these types of idealized nozzle expansions are notrepresentative of our actual system and underscores the pointthat, at low power, similar magnetic nozzles have a morenuanced behavior.
There could be a number of reasons why this discrepancyoccurs, but the explanation we explore here stems fromobservations outlined in the previous section and informed bythe work of Correyero et al [16]: the ions are not actuallysonic at the expected throat location. Critically, one of the keyassumptions underpinning the physical model has beenviolated. With this in mind, we can explore the validity ofthe analytical model if we re-define the throat to not becoincident with the location of peak magnetic field, but, inkeeping with the definition of classical nozzles, where theions become sonic—location Z/R0=0.5, as shown infigure 6. By using this A0 and the plasma parameters at thisrevised inlet plane, but preserving the same detachmentlocation, we recover the experimentally inferred nozzle
Figure 10. The axial evolution of the fraction of the total gasdynamicenergy entering the nozzle—the first three terms in (figure 4—for the400G, ∼170W, 3mg s−1 of xenon operating condition.
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Plasma Sources Sci. Technol. 28 (2019) 105019 T A Collard and B A Jorns
efficiency results shown in figure 8(a). This result shows thatby adjusting the throat to this new location the predicted andmeasured efficiency are directly coincident. The physicalreason for this reduced performance is that we are effectivelylowering the expansion ratio by moving the nozzle throatfurther downstream. This translates to a lower overallexpansion through the nozzle and reduced recovery ofthermal energy. This is strong correlational evidence thatthe model is still applicable, provided we track more carefullythe sonic condition, as found by Correyero et al [16].
Moreover, the increasing discrepancy between the modeland source efficiency with increasing nozzle strength can beexplained by tracking the location of the sonic condition as afunction of magnetic field setting. This is shown infigure 11(a) along with the location of the exit plane inferredfrom the procedure outlined above. Notably, the detachmentplane does not shift, but with increasing magnetic field, thelocation of the throat—and therefore effective area of thethroat—moves downstream. As shown in figure 11(b), thistranslates to a lower expansion ratio with increasing magneticfield strength compared to the expansion ratio that would beassumed if we ascribed the throat simply to the source exitplane. This explains why the model and actual efficiencyresults diverge more with increasing magnetic field strength.
4.2.2. Thrust and specific impulse. Physically, the reasonwhy the expected throat and actual sonic line are notcoincident at this low power is not immediately apparent.Before we discuss this in the following section, however, webriefly present here an analysis of other key performanceparameters that can be inferred from these measurements: thethrust and specific impulse. In the experimental plasma sourcethe plasma contribution to thrust is calculated using (3) and isshown in figure 12(a). This thrust is compared to the plasma-generated thrust for the idealized model in section 2. For thislatter calculation, we inferred thrust from the same equation
but used the results from the model—equations (5)–(7) withγ=1.1, Ad/A0=6.7, and measurement-inferred throatparameters—to determine the various detachment planeparameters (ud, Te,d, ne,d). From these results theexperimental source achieves thrust that is ∼50%–70% ofthe expected plasma thrust with a very low overall value. Thisis consistent with the efficiency measurements and can beexplained similarly by the changing location of the throat. Tocalculate the specific impulse the plasma-generated thrustmust be added to the cold gas thrust, which we calculate to be∼850 μN, from = - +F m m v qT m v1g p g g i gion
2( ˙ ˙ ) [ ( )] where
=v qT m5 3g g i( ) [10]. The neutral gas temperature isassumed to be room temperature (Tg=0.026 eV). Due to thedominance of the cold gas thrust at these low powerconditions the specific impulse is limited to several 10s ofseconds, as shown in figure 12(b). The two significantimplications from this discussion of performance are (1) thatthe overall performance of this device is low with thrust in themicronewton range and specific impulse levels little betterthan cold gas, and (2) the analytical model overpredicts theperformance. This type of low performance is consistent withmost low power magnetic nozzles to date [11, 14, 15]. Wealso emphasize that, although we show efficiency values infigure 8 on the order of 10%, these are only nozzleefficiencies. In practice, taking into consideration RFcoupling and wall losses in the liner, the overall efficiencyof this system is less than 1%, as is also consistent with mostlow power studies.
In either case, despite the low overall performance, oneof the major findings from this work is that in low poweroperation analytical predictions that rest on ascribing thethroat location simply to be coincident with the peak magneticfield are insufficient. The correct performance can only berecovered if the location of the throat is moved to the actualion sonic line. We discuss in the next section a possiblephysical explanation for why this result occurs at low power.
Figure 11. (a) The downstream axial location of the effective nozzlethroat (squares) and the detachment plane location (dashed line). (b)The area expansion ratio of the nozzle after the throat is shifteddownstream (squares) compared to the ideal nozzle expansion ration(dashed line). In all magnetic field conditions the source is operatedwith a net deposited power of ∼170 W and propellant flow rate of3mg s−1 xenon.
Figure 12. (a) The inferred plasma contribution to thrust frommeasurements of the plasma source (squares) compared to the thrustpredicted by the ideal model (circles) and the model using thedownstream throat properties (triangles). (b) The inferred specificimpulse. In all magnetic field conditions the source is operated with anet deposited power of ∼170W and xenon propellant flow rate of3mg s−1.
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Plasma Sources Sci. Technol. 28 (2019) 105019 T A Collard and B A Jorns
5. Discussion
In this section, we examine potential physical mechanismsthat may drive the shift in the location of the sonic point in thenozzle from the location of peak magnetic field intensity.Informed by our previous work [12], the hypotheses outlinedin Correyero et al [16], and studies on these low temperaturedevices [3, 5, 7, 10, 24], the conjecture we explore here is thatthe throat may be driven downstream by collisional orionization processes. In particular, this plasma has a lowionization fraction which results in a high neutral densitywithin the nozzle exhaust, leading to effects not accounted forin an ideal analysis of magnetic nozzle performance. We lookat three neutral-dependent mechanisms in particular:enhanced resistivity, charge exchange collisions, and ioniz-ation within the plume.
5.1. Electron-neutral collisions
In order for the nozzle to accelerate the ions up to sonic speed,the electrons must be influenced by the magnetic field. Suf-ficiently upstream of the throat and within the source theplasma is dominated by the neutral propellant gas. In thisregion electron-neutral collisions act to de-magnetize theelectrons. At some point within the source tube in a suffi-ciently ionized device, the degree of ionization becomes highenough that neutral collisions no longer dominate and theplasma is guided by the converging-diverging magnetic fieldlines. However, we conjecture that due to the very lowionization fraction (∼0.1%–1%) the plasma remains domi-nated by collisions in the near-field plume. This precludes theaccelerating action of the magnetic field until the neutral gasbecomes sufficiently sparse that the electrons becomemagnetized.
To evaluate this mechanism empirically, we consider theelectron-neutral Hall parameter:
nW =
qB
m, 8
e en( )
where B is the local magnetic field strength, me is the electronmass, and νen is the electron-neutral collision frequency. Thiselectron-neutral collision frequency is calculated using themeasured electron temperature and neutral density, and thecross section data outlined by Katz et al [58]. Physically, ifthe Hall parameter is large (Ω?1) an electron undergoesmany gyrorotations between collisions with a neutral. Con-versely, if the Hall parameter is small, an electron undergoeseither partial or few gyrorotations between collisions with aneutral. During each collision with a neutral an electron canmove to an adjacent nozzle streamline; a large collision fre-quency results in enhanced cross-field electron transport. Thisbehavior is inconsistent with the idealized, collisionlessmodels outlined above, suggesting that the models may not besuitable for predicting device performance.
Embedded within the electron-neutral collision frequencyis a dependency on neutral density. In order to evaluate thisparameter, we mapped the neutral density of the device. Tothis end, we show in figure 13 the neutral density as inferred
from pressure measurements made using a Stabil Ion Gaugewith an integrated pitot tube. Note that the measurements aretaken without power to the RF antenna, but the measuredplasma density is typically ∼3 orders of magnitude lower thanthe neutral density at the source exit plane. This low ionfraction suggests that the neutral profile shape does not sig-nificantly change with plasma ignition.
We show in figure 14 a comparison of the electron-neutral collision frequency with the local magnetic nozzlefield strength via an electron Hall parameter analysis for threemagnetic field settings. We also show on these plots a dottedline coincident with the experimentally determined soniccondition for the ions, i.e. the so-called sonic transition.Significantly, we see that with increasing magnetic fieldstrength this line shifts further downstream and coincides withHall parameter values of 70–200—values that are indicativeof a magnetized electron fluid. Fundamentally, these plotsshow opposite trends between Hall parameter and the throatlocation; as the nozzle field strength increases the Hall para-meter increases within the near-field while the ion sonictransition moves further downstream. For clarity, this trend isillustrated in figure 15, where the centerline Hall parameter iscompared to the centerline ion Mach number. Physically, thistrend states that as the magnetic field increases the electronsin the near-field plume are becoming more magnetized closerto the source exit plane. The fact that this trend is counter tothe downstream movement of the throat suggests that, whileelectron de-magnetization may play a role in the delayed ionacceleration, it is likely not a dominant mechanism.
While electron-neutral collisions may not be a dominantmechanism to explain the downstream throat it is interestingto note that the electrons appear to be magnetized upstream ofthe ion sonic line. This is consistent with the analogy of a deLaval nozzle where the gas is accelerated through a conver-ging section to sonic velocity—the electrons should beeffectively magnetized at the sonic point. In keeping with thisanalogy the electrons must be magnetized for some extentupstream of the sonic line to be compressed and accelerated tothe sonic condition at the throat; the Hall parameter values of70–200 provides evidence that this criterion is met.
Figure 13. A spatial map of the neutral density (contour and solidblack curves) with an overlay of the magnetic nozzle field lines(dashed black curves) for the 400G, ∼170W net deposited power,and 3mg s−1 xenon propellant flow rate source operating condition.
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Plasma Sources Sci. Technol. 28 (2019) 105019 T A Collard and B A Jorns
5.2. Ion-neutral collisions
A high density neutral population in the near-field region mayalso lead to frequent ion-neutral charge exchange collisions(CEX). These collisions act as a drag term on the ion fluid,preventing the ions from reaching sonic speeds until down-stream of the high neutral pressure region, resulting in thethroat shifting downstream into the plume. To assess thepossible impact of this drag term the CEX mean free path(λCEX) can be compared to the characteristic length scale ofion acceleration. Since the ion acceleration appears to begoverned by electrostatic acceleration (see section 4) therelevant characteristic length scale for acceleration can bewritten as (−∇f/f)−1. The ratio of CEX mean free path toacceleration length is then −λCEX∇f/f. If the CEX meanfree path is shorter than the acceleration length scale( l f f- < 1CEX ) the ions are experiencing a dragforce through frequent CEX collisions. Conversely, if−λCEX∇f/f>1 the ion acceleration is not significantlyimpeded by these collisions. To quantify the role of CEXcollisions within our low power test article we track thecenterline evolution of this ratio across four magnetic fieldconditions (see figure 15). By tracking this parameter we seethat as the magnetic field increases the axial location at whichthe CEX mean free path becomes equal to the accelerationlength scale (−λCEX∇f/f=1) shifts downstream. This
trend is influenced by two primary factors: (1) in our device,as the magnetic field increased the measured plasma densitydecreased. This trend is opposite of the improved confinementwith increasing magnetic field observed in other magneticnozzle devices [2, 5, 6, 10, 34, 44]. This deviation from theexpected may be attributed to the quality of the plasma-antenna power coupling, which is shown by modeling resultsof similarly configured magnetically enhanced inductivelycoupled plasmas used in plasma processing applications todegrade when a critical magnetic field threshold is achieved[49, 50]. While the critical magnetic field is dependent onplasma species, source and antenna geometry, magnetictopology, and dominant power coupling mechanisms, it candrop into the 10–100 s of Gauss range [49, 50], which over-laps with our magnetic operating conditions. Estimation ofthe skin depth—a small region into which the bulk of thepower is deposited—for these operating conditions show thatthe skin depth decreases from ∼1 cm to ∼1 mm with ourincreasing magnetic field conditions, further suggesting thatthis trend arises from the power coupling. This means that,with a constant propellant flow rate across all device operat-ing conditions, this trend necessitates a corresponding slightincrease in the un-ionized neutral density in the near-fieldwith increasing magnetic field. This acts to decrease the CEXmean free path. (2) As the magnetic field increased weobserved a decrease in the overall potential drop from source
Figure 14. The electron-neutral Hall parameter (contour and solid curves) overlain with the magnetic nozzle (dashed curves) and ion sonicline (dotted curves) for the (a) 200G, (b) 300G, and (c) 400G source operating condition. For all conditions the net deposited power is∼170 W and the propellant flow rate is 3mg s−1 xenon.
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Plasma Sources Sci. Technol. 28 (2019) 105019 T A Collard and B A Jorns
exit plane to the nozzle detachment plane. This acts toincrease the characteristic acceleration length. Both of theseeffects act to shift the location of −λCEX∇f/f= 1 down-stream. Significantly, this trend in −λCEX∇f/f qualitativelymatches the downstream movement of the ion sonic transitionline with increasing magnetic field. This correlational evi-dence suggests that ion-neutral CEX collisions may play animportant role in shifting the effective nozzle throat down-stream. The collision-induced drag on the ions may delay theacceleration of the subsonic ions, resulting in the sonic con-dition being met at the downstream throat location.
5.3. Ionization within the plume
In addition to the collisional processes discussed above,ionization may also play a role in pushing the ion sonic linedownstream. The volumetric rate of ionization within thenear-field plume region can be estimated from a controlvolume continuity analysis. Leveraging axisymmetry,assuming steady flow, and bounding the control volume to the
region with LIF measurements (refer to figure 6), this takesthe form
p p pp
=G + G - G
KRℓ R R
n n R ℓ
2, 9iz p
r a
g e,
20 0
2
2¯ ¯( )
where R is the radius of the cylindrical control volume, ℓ isthe length of the control volume, and Γr, Γa, and Γ0 are theparticle fluxes through the radial, axial, and source exit planeboundaries, respectively. This volumetric ionization rate canbe compared to the estimated ionization rate within the sourcetube—the location where the RF power is intended to beapplied to the propellant. As shown in figure 16, in themagnetic field conditions that exhibit a downstream throat thevolumetric ionization rate within the near-field plume regionis 15%–35% of the ionization rate within the source tube.This indicates that the RF power may be leaking into the near-field plume region or Ohmic heating may be causing sig-nificant ionization. This behavior is consistent with numerical
Figure 15. The centerline evolution of the (a) electron-neutral Hall parameter, (b) the ratio of the charge exchange mean free path andcharacteristic ion acceleration length scale, and (c) the ion Mach number for the 300–600G operating conditions. For all operating conditionsthe net deposited power is ∼170W and propellant flow rate is 3mg s−1 xenon.
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Plasma Sources Sci. Technol. 28 (2019) 105019 T A Collard and B A Jorns
results for magnetically enhanced inductively coupled plasmaprocessing units—devices with similar power couplingarchitectures and magnetic topographies—that also predictpower coupling downstream of the primary ionization andheating region [49, 50]. This result is also consistent withparticle-in-cell modeling results of a plasma thruster plume byLi et al [59], who show that a significant fraction of the inputpower may leak into the thruster plume. While there is noclear trend in ionization rate within the plume compared to themagnetic field operating condition, the source conditions inwhich we observe a downstream effective throat locationexhibit significant levels of ionization. This correlation sug-gests that it may play an important role in the near-fieldregion, and, subsequently, the ion acceleration in that region.The ions that are born in the near-field plume region fallthrough a smaller potential drop than the ions born in thesource tube—except, possibly, those ions that have under-gone CEX collisions. This newly-ionized population maysignificantly reduce the mean ion speed within the fluid,thereby acting as an effective drag term on the ions. Thisionization-induced drag description is consistent with thefindings of Hooper, in a similar device architecture [60].
The ultimate implication from the above discussion isthat the low ionization fraction and corresponding high neu-tral density seems like a plausible explanation for the move-ment in the effective throat of the magnetic nozzle. This isevidence in support of the idea that modeling treatments ofthese systems at low power must be more nuanced, takinginto consideration more than singly-charged, fully-ionizedplasma. With that said, we recognize that three processesdiscussed here may not be the only or dominant drivingfactors, and their interactions with the plume expansion mayvary between devices. Other near-field detachment processeshave been recently identified [13] which may also have aneffect here. The relationships we see, particularly with thecharge exchange process, are compelling but correlational atthis point.
6. Conclusions
In summary, the 2D plasma properties of an experimental lowpower magnetic nozzle plasma source operating at variousmagnetic fields have been measured using electrostatic probesand LIF. In all of the operating conditions a high neutraldensity region is observed in the near-field. The ions initiallyleave the source region subsonically and the axial ion accel-eration region is pushed downstream in high magnetic fieldcases. The impact of this downstream movement of the throaton thrust performance is estimated using spatial plasmaproperty maps to predict the thrust and specific impulse per-formance of the source. The measurement-inferred perfor-mance is compared to the expected performance of an ideal,fully attached nozzle operating at the same conditions; theperformance of the low power device is ∼50%–70% of theideal performance. This measured performance deficit isattributed to the reduced ion acceleration due to the effectiveshortening of the magnetic nozzle and the throat being pusheddownstream. By adjusting the model initial conditions tomatch the empirical downstream throat properties the pre-dicted performance agrees with the results inferred fromplume measurements. This finding is consistent with therecent work performed by Correyero et al [16], though for alow power ECR source compared to our low power inductivesource.
Three potential mechanisms that drive the ion sonictransition downstream are explored: electron-neutral colli-sions, ion-neutral collisions, and ionization within the plume.The electron-neutral collision mechanism is evaluated using aHall parameter analysis. This analysis indicates that theelectrons become more magnetized closer to the source exitplane as the magnetic field strength increases. This runscounter to the measurement-inferred trend of the effectivenozzle throat moving further downstream with increasingmagnetic field, suggesting that this mechanism may not bedirectly responsible for the shift in the throat location. Theimpact of ion-neutral collisions, effectively a drag term on theion fluid, is explored by comparing the CEX mean free pathto the characteristic electrostatic acceleration length scale. Ondevice centerline the trend in the downstream movement ofthe throat location and this parameter are positively corre-lated, suggesting that this mechanism plays an important rolein the near-field expansion and ion acceleration. Finally, inthe magnetic field cases that exhibit a downstream throat thevolumetric ionization rate in the near-field plume region isestimated to be 15%–35% of the ionization rate within thesource. This presence of ionization within the plume is cor-related with the downstream shift in the effective nozzlethroat. This suggests that ionization in the near-field may alsoplay a role in the plasma-nozzle interaction. Overall, it isobserved that in a low power magnetic nozzle test article amechanism exists that shifts the nozzle throat downstream,effectively reducing thrust performance compared to idealmodel predictions. As the research and development trends inthe field shift toward developing low power versions of thesedevices, this finding suggests that it is necessary to include
Figure 16. The ratio of the volumetric ionization rate in the near-fieldplume region (as bounded by the LIF measurements) and thevolumetric ionization rate within the source region. For all magneticfield conditions the net deposited power is ∼170W and propellantflow rate is 3mg s−1 xenon.
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Plasma Sources Sci. Technol. 28 (2019) 105019 T A Collard and B A Jorns
these low-ionization fraction effects in future models anditerations of these devices.
Acknowledgments
This research has been reproduced from [61]. This material isbased upon work supported by the National Science Foun-dation Graduate Research Fellowship Program under GrantNo. DGE 1256260. The authors would like to acknowledgethe members of the Plasmadynamics and Electric PropulsionLaboratory for their insightful discussion concerning thisresearch.
Appendix
When evaluating the denominator of (2), the power enteringthe nozzle is modeled as the sum of the power stored in theflowing ions, in the plasma pressure, and the electron heatconduction. Assuming the plasma profiles are axisymmetric ateach axial slice of the nozzle, the power stored within theflowing ions can be inferred from plasma measurements using
ò p=P m nu r rd , A.1v
R
i0
ion3s
( )
where mi is the ion mass, n is the plasma density, r is theradial coordinate, and Rs is the source tube radius at the exitplane. The power stored within the plasma pressure is
ò p=P qnT u r r5 d . A.2p
R
e0
ions
( )
Assuming that the plasma within the plume is globallygoverned by polytropic cooling the heat conduction term [39]can be written as
ò pg
g=
--
P qnT u r r35 3
1d . A.3Q
R
e0
ions ⎛
⎝⎜⎞⎠⎟ ( )
Physically, this is the energy flowing through the electronfluid along the nozzle field lines to maintain the plasmagradients present within the plume.
The power consumed in creating the ions can beaccounted for by
ò p= P qn u r r2 d , A.4c
R
c0
ions
( )
where òc is the ion cost in eV. The ion cost can be interpretedas the net energy required to create an ion-electron pair. Asdemonstrated in the analysis of ion cost presented by Lie-berman and Lichtenberg [56]:
å= + +=
¥
K T K T K Tm
mK T
3;
A.5
iz e c iz e izj
j e je
iel e
1exc, exc,( ) ( ) ( )
( )
the ion cost exceeds the ionization energy due to the presenceof excited states that absorb a portion of the energy. Note thatin the above expression Kiz is the volumetric ionization rate,
Kexc is the volumetric excitation rate, Kel is the volumetricpolarization scattering rate, òiz is the ionization energy, òexc isthe excitation energy, and j is the state index. Both theionization and excitation rates are functions of electrontemperature. Note that the ion cost exceeds the ionizationenergy due to the presence of excited states and collisionswith neutrals that absorb a portion of the absorbed energy.This expression captures the electron energy loss due to thedominant loss mechanisms for weakly ionized monatomicelectropositive plasmas [56].
The power entering the nozzle is then the sum of(A.1)–(A.4), or
= + + +P P P P P . A.6v p Q cnoz ( )
ORCID iDs
T A Collard https://orcid.org/0000-0002-0723-4868B A Jorns https://orcid.org/0000-0001-9296-2044
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