Fundamentals of Plasma Acceleration
Mariano Andrenucci
Professor, Department of Aerospace Engineering, University of Pisa, Italy
Chairman and CEO, Alta S.p.A, Via A. Gherardesca 5, 56121 Ospedaletto, Pisa, Italy
e-mail: m.andrenucci@alta-space.com
Advanced Course
”Electric Propulsion Concepts and Systems”
ESA/ESTEC, Noordwijk, September 15-19, 2008
Fundamentals of Plasma Acceleration
Plasma Thrusters
•
Unified approach based on acknowledgement that different types of
electric thruster can all be described as Plasma Thrusters
Plasma?
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Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.2
Fundamentals of Plasma Acceleration
Plasma Thrusters
•
Unified approach based on acknowledgement that different types of
electric thruster can all be described as Plasma Thrusters
12
• small Debye length
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 0 k B Te 
 D  
2   L
 ne e 
Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.3
Fundamentals of Plasma Acceleration
Plasma Thrusters
•
Unified approach based on acknowledgement that different types of
electric thruster can all be described as Plasma Thrusters
12
• low neutral collisionality
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 ne 2 
 en  pe   en 
  1
m

 e 0 
Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.4
Fundamentals of Plasma Acceleration
Plasma Thrusters
•
Unified approach based on acknowledgement that different types of
electric thruster can all be described as Plasma Thrusters
• plasma parameter large
ESA/ESTEC, Noordwijk,
September 15-19, 2008
4
   3D ne  1
3
Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.5
Fundamentals of Plasma Acceleration
Plasma Thrusters
•
Unified approach based on acknowledgement that different types of
electric thruster can all be described as Plasma Thrusters
• Main implication: quasi-neutrality assumption
ne  ni  ne  ni  n
• We shall call Plasma Thrusters all devices in which the working fluid
remains quasi-neutral throughout all phases of the process
• Hall Thrusters, Self-field MPD Thrusters and Applied Field MPD
Thrusters belong in this cathegory
• This definition leaves out ion thrusters, which inherently involve
charge separation as a basic feature of the acceleration process
ESA/ESTEC, Noordwijk,
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Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.6
Fundamentals of Plasma Acceleration
Momentum Equation
• To generate thrust we must transfer momentum to a working fluid. How
can momentum be transfered to a plasma?
• Under very general assumptions we can obtain the following
Momentum Equation for the generic species
u

m n   u  u  qnE  u  B     P  Pcol l
 t

collisions
change of momentum
electromotive force
Lorentz emf
Interaction with particles
of other types
pressure, viscosity
interactions with particles
of the same type
ESA/ESTEC, Noordwijk,
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Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.7
Fundamentals of Plasma Acceleration
Two-fluid Model
•
By considering only the electronic and ionic components of the plasma
the following two-fluid model is obtained
u i

mi n 
 u i   u i   e n E  u i  B  pi   p i  Pie
  t





ue

me n 
 ue ue  e n E  ue  B pe   pe  Pei
t


Isotropic pressure terms
non-isotropic (viscous) terms
small m, negligible
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Pi e  Pe i 
m e n e u e  ui 
 ie
Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
friction between
the two fluids
Slide 2.8
Fundamentals of Plasma Acceleration
Two-fluid Model
•
By neglecting the electron inertial term in the second equation and with the substitution
j
ue  ui 
ne
we finally obtain
mi n
d ui
dt


 ne E  u i  B  p i 
ne

j


1
ne
0  ne E  u i  B 
j  B pe 
j

ne


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Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.9
Fundamentals of Plasma Acceleration
Generalized Ohm’s Law
•
With the further useful substitutions
n e2

 ei
m
e  eB m
  e  e
from the electron equation we obtain the Generalized Ohm’s Law
j   (E  v  B 
1

pe )  j  B .
ne
B
electric field
back emf
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Hall’s emf
thermionic emf
Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.10
Fundamentals of Plasma Acceleration
Electric Field
By rearranging the generalized Ohm’s law we obtain the expression
for the self-consistent electric field in the quasi-neutral plasma
E
Ohmic term
j

 ui  B 
Back emf
Resistive heating is
exploited in arcjets
1
1
j B
p
ne
ne e
Hall’s emf
Thermionic emf
This is exploited in
different ways in
MPD and HET
thrusters
Usually small
No useful contribution
in the velocity direction
ESA/ESTEC, Noordwijk,
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Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.11
Fundamentals of Plasma Acceleration
General Vector Diagram
Going back to the two-fluid model, let us visualize
the vector diagram of fields and currents (neglect the
pressure gradient contributions)
Momentum increase
of the ion fluid
mi n
d ui
dt

Electric field
contribution

Collisional
contribution

 n e E  u i  B  p i 

0  n e E  u i  B  j  B  pe 
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ne

ne

B
j
j
Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.12
Fundamentals of Plasma Acceleration
General Vector Diagram
Going back to the two-fluid model, let us visualize
the vector diagram of fields and currents (neglect the
pressure gradient contributions)
Momentum transfer
to the ions
mi n
d ui
dt
Electric field
contribution

Collisional
contribution

 n e E  u i  B  p i 


0  n e E  u i  B  j  B  pe 
Electric field effect on
the electron fluid in the
ion comoving frame
ne

ne

B
j
j
Collisional
momentum loss
Electric field effect
due to electrons
relative velocity
ESA/ESTEC, Noordwijk,
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Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.13
Fundamentals of Plasma Acceleration
General Vector Diagram
Thus, the vector diagram of fields and currents for the
two-fluid model (neglecting the pressure gradient
contributions) can be visualized as shown here
-
un B
ne

mi n
d ui
dt


ne

j


1
ne
0  ne E  u i  B 
j  B pe 
j

ne


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September 15-19, 2008

j

jy
ne



E*
1  2
Y
uE

2 ne
uE
1  2 
j 
jx


E*
1  2
1
jB
ne
By combining and posing
we finally have
un  

 ne E  u i  B  p i 
p  pe  pi
un  B
B
 Mn
2
E*
1  2
du
= p + j  B Lorentz force
dt
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+
E*
  tan   j y j x
X
Slide 2.14
Fundamentals of Plasma Acceleration
Energy Equations
The increase in the flow directed kinetic energy can be obtained by taking the
dot product of the momentum equations for the two species by ui and u
e
respectively:
ui  
dui
ne
  pi  ui  ne E  ui 
j ui
dt

0   pe  ue  ne E  ue 
ne

j  ue
which can be rewritten as
d  ui 2 

   pi  ui  n e E ui
d t 
 2 

ne
0   pe ue  ne E  ue 
ne


j ui
j ui

Collisional terms
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j2

Joule heating
Slide 2.15
Fundamentals of Plasma Acceleration
Energy Equations
Adding up the two equations above the collisional terms cancel out, and we
are left with
2
2
d  ui 
j

   pi ui  pe  ue E  j 
d t  2 

Once again we can explicitly highlight the role of the overall Lorentz force.
With a few passages we would obtain
2
d  u 

= p u +  j  B u
d t  2 
but it should be remembered that the increase in the ion fluid kinetic energy is
either drawn from the energy transferred by the electrons through collisions,
or from direct action of the electric field on the ions
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September 15-19, 2008
Advanced Course: Electric Propulsion Concepts and Systems
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Slide 2.16
Fundamentals of Plasma Acceleration
Power transfer efficiency
Thus we see that - neglecting again the pressure gradient terms - the useful
energy transfered to the plasma can utimately be computed in terms of power
delivered by the electric field minus power dissipated as Ohmic heating. We
are thus prompted to define a power transfer efficiency as
E j
2
  1 j 
E j
1.0
uB
cos
E*
0.9
and remembering that
j   E * cos
we finally obtain
u B / E *cos
P 
cos  u B / E *cos
 being is the angle formed by the
Conversion Efficiency
P 
E j 
j2
0.8
0.7
0.1
0.5
1
2
5
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
Lorentz force with the local flow direction
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1.0
2.0
3.0
4.0
Hallparameter
Parameter
Hall
Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.17
Fundamentals of Plasma Acceleration
Example I : large Hall parameter
B
ne

un  
j


j 
Y
1
jB E*
ne
X
ESA/ESTEC, Noordwijk,
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Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.18
Fundamentals of Plasma Acceleration
Example I : large Hall parameter
B
ne

un  
j


.60
.65
j 
Y
.70
c
.75
.80
1
jB E*
ne
.85
X
ESA/ESTEC, Noordwijk,
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Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.19
Fundamentals of Plasma Acceleration
Example I : large Hall parameter
B
ne

un  
j


.60
.65
Y
j 
.70
c
.75
.80
uB
1
jB E*
ne
uB
E
.85
X
ESA/ESTEC, Noordwijk,
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Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.20
Fundamentals of Plasma Acceleration
Example I : large Hall parameter
B
ne

un  
j


.60
.65
Y
j 
.70
c
.75
uB
.80
1
jB E*
ne
.85
X
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uB
E
Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.21
Fundamentals of Plasma Acceleration
ne

un  
j

Example II : Hall parameter ~ 1
B
Y

1
jB
ne
j 
E*
X
ESA/ESTEC, Noordwijk,
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Advanced Course: Electric Propulsion Concepts and Systems
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Slide 2.22
Fundamentals of Plasma Acceleration
ne

c
un  
j

.30
.40
Example II : Hall parameter ~ 1
B
.50
Y
.60
.70

1
jB
ne
j 
E*
X
ESA/ESTEC, Noordwijk,
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Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.23
Fundamentals of Plasma Acceleration
ne

c
.60
.70
un  
j

.30
.40
Example II : Hall parameter ~ 1
B
.50
Y
uB

1
jB
ne
j 
E*
uB
X
E
ESA/ESTEC, Noordwijk,
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Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.24
Fundamentals of Plasma Acceleration
ne

c
un  
j
Example II : Hall parameter ~ 1

B
.30
.40
.50
uB
Y
.60
.70

1
jB
ne
j 
E*
uB
X
E
ESA/ESTEC, Noordwijk,
September 15-19, 2008
Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.25
Fundamentals of Plasma Acceleration
Hall-effect Thrusters
z
z
y
x
j
B
-j/ 
y
jxB
Y
E*
Z
X
x
B
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Slide 2.26
Fundamentals of Plasma Acceleration
Self-field MPD Thrusters
z
y
j
x
B
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Slide 2.27
Fundamentals of Plasma Acceleration
Applied-field MPD Thrusters
z
z
y
B
x
j
y
Y
-j/ 
Z
jxB
E*
B
x
X
ESA/ESTEC, Noordwijk,
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Advanced Course: Electric Propulsion Concepts and Systems
M. Andrenucci
Slide 2.28
Fundamentals of Plasma Acceleration
Applied-field MPD Thrusters
Including effect
of self-field
z
z
y
B
x
j
y
B
Y
-j/ 
Z
E*
jxB
ESA/ESTEC, Noordwijk,
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x
X
Slide 2.29