magnetic nozzle and plasma detachment...
TRANSCRIPT
In collaboration with an experimental team:R.D. Bengtson and J. Meyers (UT Austin)D.G. Chavers, C.C. Dobson, and J.E. Jones (MSFC)B.M. Schuettpelz (University of Alabama, Huntsville)C. Deline (University of Michigan, Ann Arbour)
Research supported in part by Ad Astra Rocket Company
Boris Breizman, Mikhail Tushentsov, and Alex ArefievInstitute for Fusion Studies, UT Austin
Magnetic Nozzleand Plasma Detachment Scenario
November 12-16, 2007, Orlando, Florida 49th APS-DPP Meeting
TALK OUTLINE
• Plasma detachment issue
• MHD detachment concept and theory (motivated by VASIMR project)
• Numerical model for steady-state magnetic nozzle
• Code testing and numerical results
• Experiment (DDEX, Marshall Space Flight Center)
• Summary
1APS - DPP 2007
PLASMA DETACHMENT PROBLEM
Plasma-based propulsion systems generate thrust by ejecting directed plasma flow.
• A strong magnetic field is used to guide the plasma.
• The ejected plasma must break free from the spacecraft to produce thrust.
There are two scenarios for plasma detachment:
• Detachment from the magnetic field(by breaking the frozen-in constraint viarecombination or some other mechanism).
• Detachment with the magnetic field(by stretching the field lines along the flowdue to plasma plasma current).
2
nozzle wall
plasma current
field lines stretchedby the plasma
vacuum field lines
APS - DPP 2007
MHD DETACHMENT CONCEPT
•Magnetic energy decreases downstream faster than plasma kinetic energy.
•An initially sub-Alfvénic flow becomes super-Alfvénic downstream.
•Plasma flow can stretch the magnetic field lines after drops below . B
28!
m
in
iV
�
22
0.6 0.8 1 1.2 1.40
1
2
3
4
.
2 28B S! "
#
2
1
2
i im nV
S!
�
SS (nozzle cross-section)
E.N.Parker, Astrophys. J. 128, 664 (1958)E.B.Hooper, Journal of Propulsion and Power 9, 757 (1993)
3
Magnetic flux (BS) = const
Flow velocity (V||) = const
Plasma flux (niV
||S) = const
Conserved quantities:
APS - DPP 2007
MAGNETIC NOZZLE LAYOUT
Nozzle endNozzle wall
Sub-Alfvénicflow
Transition to super-Alfvénic flow
Super-Alfvénicflow
Plasma-vacuuminterface
Key requirements to magnetic nozzle:
• Provide a smooth transition from sub- to super-Alfvénic plasma flow.
• Ensure efficient plasma detachment.
4APS - DPP 2007
The main part of the flow remains unperturbed after detachment!The main part of the flow remains unperturbed after detachment!
( )
0
,r z!
!
0z r
0r r
Plasma-vacuuminterface
Inward Alfvén-wave
Rarefaction wave
Unperturbedsuper-Alfvénic flow
CharacteristicNozzle end
Nozzle wall
PLASMA DENSITY PROFILE IN THE PLUME
A. Arefiev and B. Breizman, Phys. Plasmas 12, 043504 (2005)
5APS - DPP 2007
HIGHLY SUPER-ALFVÉNIC PLUME (V/VA>>1)
A. Arefiev and B. Breizman, Phys. Plasmas 12, 043504 (2005)
6
• Rarefaction wave stays at the edge of the plume when V /V
A>> 1 / !
0
B(r, z) =
B0
zmin
2
z2
0 < r < rC
B0
9
zmin
2
z2
V2
VA
2
r
z!"
0! 2
VA
V1!
z*
z
#
$%
&
'(
)
*++
,
-..
2
1!z
*
z
#
$%
&
'(
!2
rC< r < r
PV
0 r > rPV
/
0
111
2
111
rC= z!
01"
1
!0
V
VA
1"z
*
z
#
$%&
'()
*++
,
-..
rPV= z!
01+
2
!0
V
VA
1"z
*
z
#
$%&
'()
*++
,
-..
Rarefaction wave
Unperturbed main flow
θ0 zmin z*
Conical flow without vacuum gap between plasma and nozzle wall
rC! inner front of the rarefaction wave
rPV! outer front (plasma-vacuum interface)
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GOVERNING EQUATIONS
• Collisionless MHD with cold electrons and anisotropic ion pressure (from ICRH)
•Steady-state axisymmetric flow flow along magnetic field lines
1
r
!
!r(r"V
r) +
!
!z("V
z) = 0
" Vr
!Vr
!r+V
z
!Vr
!z
#
$%&
'(= )
!
!rp*+
B2
2µ0
#
$%
&
'( + B
r
!
!r+ B
z
!
!z
#
$%&
'(B
r
µ0
+p*
B2
Br
#
$%
&
'(
" Vr
!Vz
!r+V
z
!Vz
!z
#
$%&
'(= )
!
!zp*+
B2
2µ0
#
$%
&
'( + B
r
!
!r+ B
z
!
!z
#
$%&
'(B
z
µ0
+p*
B2
Bz
#
$%
&
'(
!Bz
!z+
1
r
!
!r(rB
r) = 0
continuity equation
equationsof motion
magnetic flux conservation
7
vr
vz
||v
magnetic field linev!
miV!
2
2B= µ = const
miV
||
2
2+ µB = const
p!= µB
"
mi
; p||= 0
Vr
Vz
=B
r
Bz
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PARAXIAL APPROXIMATION
!
!z+
Vr
Vz
!
!r
"
#$
%
&' (V
z( ) = )(V
z
1
r
!
!rr
Vr
Vz
"
#$
%
&'
!
!z+
Vr
Vz
!
!r
"
#$
%
&'V
r= )
1
(Vz
!
!r
Bz
2
2µ0
+ µBz
(
mi
"
#$
%
&'
!
!z+
Vr
Vz
!
!r
"
#$
%
&'V
z=µB
z
miV
z
1
r
!
!rr
Vr
Vz
"
#$
%
&'
!
!z+
Vr
Vz
!
!r
"
#$
%
&' B
z= )B
z
1
r
!
!rr
Vr
Vz
"
#$
%
&'
!
!z+
Vr
Vz
!
!r
"
#$
%
&' µ = 0
2 2
0 0
ˆ
2 2
z
z
i
B BBm
!µ
µ µ= +
Boundary condition at the plasma-vacuum interface (pressure balance)
B̂ ! external magnetic field at the plasma boundary
• Axial scale-length is much greater than radial scale-length
• Efficient detachment implies that radial velocity and radial magnetic field are small
Vr
Vz
=B
r
Bz
<< 1
8
We use Largangian radial coordinate toaccommodate this boundary condition.
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LAGRANGIAN SOLVER
• Method of lines (ODE with adaptive step) (along flow lines)• Staggered grid second order FD scheme (radial direction)• Mapping to the fixed Eulerian grid
• Natural tracking of the moving plasma-vacuum interface
Zv
ZB
1/ 2jr+
3/ 2jr+
jr
1jr+
vR
rr
k!
1k!
+
9
• Non-uniform profile inputs• Treatment of vacuum gap between plasma and nozzle walls
Steps beyond analytic model
!
r(r0,! ) = r
0+
Vr
Vz
!0
!
" d! '
z(r0,! ) = !
#
#!=#
#z+
Vr
Vz
#
#r
Transformation to Lagrangian radial coordinate
• Written in MATLAB®
• Runs on single workstation
• Magnetic field at the plasma-vacuum boundary can be precalculated for slowly diverging flow
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NOZZLE CODE VERIFICATION
10
Incoming flow parameters
Ion energy: !||= 250 eV
Plasma density: n = 5.0 "1014 cm-3
Vacuum field linein the presence of plasma
Analytic solution
Numerical solution
Nozzle end
B=const contours
Nozzle wall
The code accurately reproducesthe rarefaction wave at the plasma edge
Magnetic field at the plasma boundary
Rarefaction wave front
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SUB- TO SUPER-ALFVÉNIC TRANSITION
Vacuum field lines(solenoid)
V V
A= 1
11
Alfvén Mach number
R (meter) R (meter)
Den
sity
(101
9 m-3
)
Incoming flow parameters
Ion energy: !||= 10 eV
Plasma density: n = 5.0 "1014 cm-3
Thrust = 114 N
Power = 400 kW ( Ar)
APS - DPP 2007
2µB m
iV
�
2 = 1
Vacuum field lines
Thrust = 16 N
Power = 193 kW ( Ar)
!||=
miV
||
2
2
!"=
miV"
2
2= µB
PLASMA FLOW WITH ION GYROMOTION
12
Incoming flow parameters
Ion gyroenergy: !"= 100 eV
Axial energy: !||= 10 eV
Plasma density: n = 5.0 #1013 cm-3
Plasma radius: Rp= 10 cm
APS - DPP 2007
NOZZLE EFFICIENCY
• There are two factors that affect flow directivity:
• Definition of nozzle efficiency (momentum efficiency):
out
z
Momentum in
z
P
P! "
&
&
out
zP&
in
zP&
divergence of the nozzle itself;radial expansion of the plasma plume at its edge.
= axial momentum flux in the outgoingflow
= axial momentum flux in the incomingflow
13
!Momentum
="V
z
22#rdr$
"VVz2#rdr$
!Power
=
"Vz
2
2V
z2#rdr$
"V2
2V
z2#rdr$
Power efficiency of the nozzle: 1! "
Power( ) # 2 1! "
Momentum( )
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NOZZLE EFFICIENCY CALCULATION
14
Incoming flow parameters
Ion energy: !||= 10 eV
Plasma density: n = 5.0 "1014 cm-3
Plasma radius: Rp= 15 cm
Incoming flow parameters
Ion energy: !||= 100 eV
Plasma density: n = 5.0 "1014 cm-3
Plasma radius: Rp= 15 cm
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DETACHMENT DEMONSTRATION EXPERIMENT
D. Chavers et al., “Status of Magnetic Nozzle and Plasma DetachmentExperiment”, CP813, Space Technology and Applications InternationalForum, p. 465 – 473, AIP 2006
• Washer-stack plasma gun (300 kW) • Plasma density ~ 1019 m-3
• Magnetic field ~ 0.1 Tesla• 3 ms pulse at high power levels
15
Z
R
DDEX facility at MSFC
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PLASMA DENSITY MEASUREMENTS IN DDEX
18
Normal setup Reversed current
Upstream interferometer Upstream interferometer
Downstream interferometersDownstream interferometers
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SIMULATION OF PLASMA FLOW IN DDEX
Vacuum field lines
V V
A= 1
19
nmax
= 0.53 !1011
cm-3
FWHM = 1.1 m
Alfvén Mach number
Incoming flow parameters
Ion energy: !||= 5 eV
Maximum plasma density: n = 1.0 "1013 cm-3
Gaussian profile: FWHM = 9.3 cm @ 0.47 m
Downstream density profile
Experiment: Simulation:
nmax
= 1.0 !1011
cm-3
FWHM = 0.8 m
Magnetic field at the plasma boundary
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REPRODUCTION OF PLASMA PROFILE IN DDEX
20
Den
sity
(101
3 cm
-3)
•15GHz Interferometer•Langmuir probe@ z=1.87 meters
Noz
zle
Effic
ienc
y
z (meter)
Incoming flow parameters
Ion energy: !||= 5 eV
Maximum plasma density: n = 1.0 "1013 cm-3
Gaussian profile: FWHM = 9.3 cm @ 0.47 m
Expected downstream density for plasma flow along vacuum field lines
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SUMMARY
• A properly shaped paraxial magnetic nozzle provides smooth transition from sub- to super-Alfvénic plasma flow and efficient detachment.
• Magnetic nozzle can simultaneously convert ion gyro-motion into axial flow to benefit from the ICRH power deposition.
• The MHD detachment concept is particularly relevant to high-power thrusters (VASIMR).
• The developed steady-state Lagrangian code enables broad parameter scan in detachment modeling with modest computational requirements (single workstation).
• Simulation results match the DDEX experimental data within the confidence range.
21APS - DPP 2007