PLASMA ADIABATICITY IN A DIVERGING MAGNETIC NOZZLE J. P. Sheehan and Benjamin W. Longmier University of Michigan Edgar A. Bering University of Houston Christopher S. Olsen, Jared P. Squire, Mark D. Carter, Franklin R. Chang Díaz, Timothy W. Glover, Andrew V. Ilin, and Leonard D. Cassady Ad Astra Rocket Company
ABSTRACT We propose a fluid model for ambipolar ion acceleration in a magnetic nozzle that preserves the adiabaticity of the plasma. This adiabatic theory predicts that the change in average electron energy depends linearly on the change in plasma potential, providing an important design metric for electric propulsion devices which employ magnetic nozzles. The fluid theory predictions were compared to measurements made in the VASIMR VX-200 experiment which has conditions conducive to ambipolar ion acceleration. A planar Langmuir probe was used to measure the plasma potential, electron density, and electron temperature for a range of mass flow rates (50 140 mg / s) and power levels (12 30 kw). The linear relationship between electron temperature and plasma potential was observed as predicted. The adiabatic theory relies on collisions to rethermalize the electrons and establish a temperature gradient. Coulomb collisions cannot account for the high collisionality but an ion acoustic instability may enhance the collision frequency. 2
HELICONS AND MAGNETIC NOZZLES FOR APPLICATIONS IN PROCESSING AND PROPULSION Helicons Radio frequency High ionizing efficiency Electron heating Magnetic nozzle Functions like physical nozzle Accelerates ions Converts thermal energy into directed kinetic energy Applications Materials processing Electric propulsion CHI KUNG VASIMR 3
CURRENT FREE DOUBLE LAYERS IN HELICON EXPERIMENTS Narrow layer (10s of λ d ) of large potential jump (several T e /e) Isothermal Occurs downstream of nozzle Current free, expanding Accelerates ions May be thrust mechanism in helicon thrusters Open question of how/why current free double layers form 4
ION ACCELERATION IN VASIMR VASIMR: < 200 kw helicon + ICH thruster Went looking for double layers, but found none! V p, n e, and T e derivatives coincide Long length scales: 10,000s of λ d λ d ~ 10 μm Corroborated with RPA B. W. Longmier et al., Plasma Sources Sci. Technol. 20, 015007 (2011). 5
PLASMA IN NOZZLE IS ADIABATIC Assume plasma is adiabatic Energy loss to surfaces small Radiation loss small Geometry dictates degrees of freedom (N) Expanding plasma sphere: N = 3 Magnetic nozzle: N = 2 Electrons remain Maxwellian, though temperature can change Relies on collisions Conservation of momentum Relationship between average electron energy loss and potential ( ion energy gain) 6
HELICON EXPERIMENT USING VASIMR HARDWARE B max = 20,000 G P = 15 30 kw ṁ = 50 140 mg/s Only helicon coupler used, no ICH Superconducting magnets generate converging/diverging magnetic field 7
MEASUREMENTS IN PLUME WERE MADE WITH PLANAR LANGMUIR PROBE V p < 20 V T e < 15 ev n e = 10 10 10 12 cm -3 Planar tungsten probe No RF compensation needed Guard ring reduces sheath expansion effects Parameters extracted from I-V traces V p : knee T e : semilog fit n e : saturation current 8
PARAMETRIC STUDY OF OPERATING PARAMETERS Measured at fixed position 50 cm from throat Highest density where probe could survive Lower mass flow rate Higher T e, V p Lower n e Power flow density input energy per ion Optimize energy deposition for given flow rate 9
V p, T e, n e DECAY DOWNSTREAM Axial measurements Lowest and highest mass flow rates are shown Low flow rates high temperature, larger gradients Temperature decay: plasma is not isothermal Length scale: 10,000s of λ d No double layer, but significant density and temperature gradients 10
DATA WERE CONSISTENT WITH ADIABATIC THEORY Plasma potential decays proportionally to electron temperature Only parallel electron temperature was measured with planar probe Some electron energy may be lost to other sinks Collisions Instabilities Radiation J. P. Sheehan et al., Plasma Sources Sci. Technol., (submitted). 11
ALTERNATIVE THEORY: RAREFACTION WAVE THEORY Nonlocal theory Far downstream, rarefaction waves create potential structures which confine electrons Electrons lose energy on wave, cooling in downstream region Two regions Steady state: ambipolar ion acceleration Rarefaction wave: further acceleration Steady-state part Rarefaction wave part Arefiev and Breizman, Phys. Plasmas 16, 055707 (2009). 12
COMPARISON BETWEEN ADIABATIC THEORY AND RAREFACTION WAVE THEORY Fundamental difference: local theory or non-local theory Both predict same relationship between φ and T e Different relationship between φ and n e Suggests some discrepancy in number of degrees of freedom No observed acceleration in lowfield region Factor of 3 increase in ion velocity from rarefaction wave Importance of collisions Rarefaction wave theory Experiment 13
ADIABATIC THEORY RELIES ON COLLISIONS, RETHERMALIZATION VASIMR has >95% ionization fraction Coulomb collisions dominate Electron-electron collisions cause rethermalization of EVDF Collisions necessary to reduce thermal conductivity along field lines Classical coulomb collision are insufficient to explain rethermalization Collisional mean free path only becomes long enough after major temperature gradient 14
PLASMA IS DETACHED FROM MAGNETIC FIELD LINES In a fully attached nozzle, density should follow field lines n e /B > 1 Plasma beam more focused Detachment occurs Possible detachment mechanisms Particle collisions Anomalous transport Demagnetization Frozen flow Electron inertia C. S. Olsen et al., IEEE Trans. Plasma Sci. (accepted, 2014). 15
INSTABILITIES INCREASE THE COLLISION FREQUENCY In the kinetic equation, collisions represented by collision operator which has depends on the collisional kernel Collision frequency affected by both the stable and unstable parts S. D. Baalrud, J. D. Callen and C. C. Hegna, Phys. Rev. Lett. 102, 245005 (2009). 16
AMBIPOLAR ION ACCELERATION SCALES ARE SIMILAR TO PRESHEATH SCALES L E, L B, λ cx >> λ d T i << T e Quasineutral v i ~ c s Δφ ~ T e J = 0 Presheath and sheath Ambipolar acceleration 17
CONVECTIVE ION ACOUSTIC INSTABILITY INCREASES DOWNSTREAM COLLISION FREQUENCY In presheath: T e constant n e ~ constant In magnetic nozzle: dt e /dx ~ T e /L B dn e /dx ~ n e /L B Instability grows as ions propagate Collisions rethermalize electrons Ignoring gradients (see figure) overestimates collision frequency Complicating factors Ion temperature, i-n collisions Magnetic field divergence Electron-electron collision frequency in magnetic nozzle without considering parameter gradients 18
Ambipolar ion acceleration observed in VX-200 experiment Longer length scale than in CFDLs Adiabatic theory describes experimental results Potential drop depends on temperature, not density Local vs. non-local theory unresolved questions Unknown cause of rethermalization Proposed collision mechanism: ion acoustic instability CONCLUSIONS 19
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