HALL THRUSTERS:FLUID MODEL

OF THEDISCHARGE

Electric Space Propulsion, E. Ahedo 3.B.1

Types of models

• Kinetic: based on Boltzmann eq.; unaffordable except for particular aspects of the problem

• Fluid: familiar formulation but important difficulties arising from 1. Weak collisionality (Kn large)2. Wall interaction 3. Curved magnetic topology4. 2D subsonic/supersonic ion flows

• Particle In Cell & MonteCarlo methods (PIC/MCC): good for weak collisionality; simple to implement, but subject to ‘numerical effects’; important difficulties in dealing with disparate scales of electron and ions. MCC model collisions statistically.

• Hybrid (PIC/MCC for heavy species & fluid for electrons): best compromise today; allows 2D (geometrical and magnetic) effects; avoids small electron scales and admits quasineutrality

Electric Space Propulsion, E. Ahedo 3.B.2

Simulation of the plasma discharge• 2D, axisymmetric model• Quasineutral plasma except for sheaths around walls.• Plasma wall interaction treated in separate sheath models.• Boundaries: 1) anode + gas injector, 2) cathode surface,

3) lateral walls• 3 or 4 species: neutrals, ions (+, ++), and electrons• Ion dynamics: unmagnetized, near collisionless, internal

regular sonic transitions, singular sonic transitions at sheath edges.

• Electrons: magnetized, diffusive motion, and weakly collisional (local thermodynamic equilibrium is not assured)

• Fluid modelling: complex and uncertain.• There is no fully 2D model yet. Two existing options:

a) Approximate 1D (axial) fluid modelb) Near 2D hybrid model: fluid eqs. for electrons;

particle model for ions & neutrals

Electric Space Propulsion, E. Ahedo3.B.3

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2D fluid model Basic hypotheses: 1) azimuthal symmetry: 0, 0, 0

2) Quasineutrality: 2 boundary conditions at sheath edges

3) Simplified treatment of pressure tensors

Con

i n

e i i

u u

n n n

θ θθ

+ ++

∂• = = =

∂= + ⇒

•

( )

tinuity equations: ( ) ionization

,

Electron momentum:

ee e ion ion e n ion e

i ni i ion n n ion

e e e

n n u S S n n R Tt

n nn u S n u St t

m n ut

∂+ ∇ ⋅ = ≈ ←

∂

∂ ∂+∇ ⋅ = + ∇ ⋅ = −

∂ ∂

∂•

∂ e e e em n u u+ ∇ ⋅

( )

( ) ( )

( ) e-n momentum transfer

Ion momentum: ( )

e e e e e

e e n en e e e

i i i i i i i i i

n T en E u B M

M n n R T m u

m n u m n u u n Tt

= −∇ − + × +

≈ − ←

∂• + ∇ ⋅ = −∇

∂

( ) ionization + charge-exchange mom. transf.

i i

i ion i n cx i i n

en E M

M S m u S m u u

+ +

≈ − − ←

(here only)e in n=

Electric Space Propulsion, E. Ahedo 3.B.4

2D fluid model

2 2

Electron (total) energy equation :

3 1 5 1 2 2 2 2

( ) ionization ( 12.1eV, 1.5 2.5

e e e e e e e e e e e e e

e e n ion e ion ion ion ion

T m u n T m u n u q en u E Qt

Q n n R T E Eα α

∂ ⎡ ⎤⎛ ⎞ ⎛ ⎞+ + ∇ ⋅ + + = − ⋅ +⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ⎝ ⎠ ⎝ ⎠⎣ ⎦

≈ − ← = −

i

∼

2 2

for Xe) Internal energy = total energy mechanical energy

3 3 ' 2 2

1 1 '2 2

e e e e e e e e e e

e e e e e e e e n en ion e e ion ion

T n T n u q n T u Qt

Q Q M u m u S n n R R m u R E

−

∂ ⎛ ⎞ ⎡ ⎤+ ∇ ⋅ + = − ∇ ⋅ +⎜ ⎟ ⎢ ⎥∂ ⎝ ⎠ ⎣ ⎦

⎛ ⎞≡ − ⋅ + ≈ + −⎜ ⎟⎝ ⎠

i

Heat flux (diffusive model): (Bittencourt)

5 2

Energy equations for ions and neutrals.

ion

e e e e e e e e eo p T eq B m q q T

α

ν

⎡ ⎤⎢ ⎥⎣ ⎦

= − ∇ − ∧ − → = −Κ ⋅∇

i

i

Electric Space Propulsion, E. Ahedo 3.B.5

From 2D to 1D model

Electric Space Propulsion, E. Ahedo 3.B.6

2 2int

1Electron continuity equation:

The 1D axial model works with values averaged over each ('doughnut') radial section

2 1 0

e e ze e ree i

e e ze e ree i

ext

n n u rn u nt z r r

n n u rn u nr r t z r r

ν

ν

∂ ∂ ∂+ + =

∂ ∂ ∂

∂ ∂ ∂⎡= + + −− ∂ ∂ ∂⎣

i

i

int

int2 2

int

( )

( ) |with ( ),..., and 'source' 2 evaluating losses at lateral walls.

An auxiliary radial model (at each ) is needed to

extr e e zee i wr

exte re

e e e wext

n n u nt z

rn un n z nr r

z

ν ν

ν

∂ ∂⎤ → + = −⎢ ⎥ ∂ ∂⎦

≡ =−

∫

i

1 compute & determine radial profiles:

This is a variable-separation type of solution; ( ) is an eigenvalue of the radial model.

We proceed similarly with the rest

e rew e w

w

rn u nr r

z

ν ν

ν

∂≈

∂i

i of fluid equations.

1D axial model Neglect jet divergence, doubly-charged ions, , ,...Wall interaction terms appear as source terms (instead of BCs)

Conservation of species flows: ( ) ( )

: ionization

i n

e zi e i w

i

p p

d n u ndz

ν ν

ν

= −

ii

i

2

frequency : recombination frequency

const /

const /

Ion axial momentum equation:

( ) ( )

Neutral axial momentum

w

n zn e zi A i

e zi e ze d

i e zi e i e i zn w zi

n u n u m Am

n u n u I Ae

d dm n u en m n u udz dz

ν

φ ν ν

+ = =

− = =

= − + −

i

i

2

equation

( ) ( )

velocity of neutrals from recombination ( )

i n zn i e wn znw i zn

znw zi

d m n u m n u udz

u u

ν ν= −

≠

Electric Space Propulsion, E. Ahedo 3.B.7

Electric Space Propulsion, E. Ahedo 3.B.8

1D axial model

2

1Electron Ohm´s law: ( )

: effective collision frequency ( due to wall interaction, due to plasma turb

e eze e e e ze

e e e e

e en ei wm turb

wm turb

d du e n T u um dz n dz θν ωφω ν

ν ν ν ν νν ν

⎡ ⎤− = − = −⎢ ⎥

⎣ ⎦

= + + +

i

2

ulence) Electron internal energy:

( )5 2

( 6 100: factor for energy losses at lateral walls)

Heat conducti

e ee e ze ze ze e e e e i e ion ion e w e e

e

d n Td n T u q u n m u n E n Tdz dz

ν ν α β ν

β

⎛ ⎞+ = + − −⎜ ⎟⎝ ⎠

−

i

∼

i 2

2

55on: 02 2

Normalized set of equations: ( ) ( )

( , , , , , , , ) vector of 8 plasma variables

Singular (sonic) point

e e e ee e e e e e ze

e e

e i zi

e n zi ze zn e ze

n T dTp T eq B m q qm dz

dYT m u F Ydz

Y n n u u u T q

νν

ω

φ

= − ∇ − ∧ − → = −

− =

=

i

i s of the equations at : 1/zi

e i

uMT m

≡ = ±

Electric Space Propulsion, E. Ahedo 3.B.9

1D axial model2

22

lnFor instance:

2 ln 1 (2 )5

A singular transition exists at anode sheath edge (point B):

zii zi zi

e i zi

e zee ze i i zi zn i zi

e e e ze

du d A Gu udz dz T m u

q d AG m u m u u m un T u dz

ν

ων

ν

= − −−

⎛ ⎞= − − − − +⎜ ⎟

⎝ ⎠

i

i 1 &

A regular subsonic/supersonic transition exists inside the channel (point S) with

1 & ( ) 0, i.e. 0 Point S is equivalent to the critical s

B B

S S S

M G

M F Y G

= − = ∞

= = =

i

i2

ection of a Laval nozzle, where 0= ln / corresponds to / 0. In a Hall thruster, =0 corresponds to a certain balance between ionization and electron diffusion.

8 boundary c

S i zi SG m u d A dz dA dz G≡ =

i onditions are set at points B, S and P: - Discharge voltage (1) - Density and velocity of injected gas at anode (2) - Electron temperature at cathode (1) - Regularity condition at point S (1) - Anode sheath conditions (3)

2e

e

ωυ−

Anode sheath in a Hall thrusterFor Maxwellian-type VDF electron flux to anode / 2 ,In normal conditions this is much larger than

the quasineutral flux in the channel, ( 0)A negative anode sheath AB is formed i

eA e e

ze e ze

n T m

g n u

π•

•≡ >

•

∼

n order to

satisfy = : exp2

This equation determines the sheath potential fall, .

Bohm condition at sheath edge: /

Electron energy flux deposited: a

eBABzeB zeA eB zeB eB

eB e

AB

ziB eB i

Teg g n u nT m

u T m

φπ

φ

⎛ ⎞≈ − −⎜ ⎟

⎝ ⎠

• ≈ −

•

2 3

) at the anode A,1 ( ) 22

b) at the sheath edge B: (2 )

These are 3 boundary conditions for the axial quasineutral model

totzeA e z eB eB zeB

totzeB zeB eB AB

q m w w f w d w T g

q g T eφ

= − ≈

= +

•

∫

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Electric Space Propulsion, E. Ahedo 3.B.10

Axial structure• Acceleration region:

– Presents most of the potential drop & ion acceleration

– For electrons: Joule heating competes with wall cooling

– Heat conduction smooths Te profile– Plasma production = plasma recombination– Plasma density decreases (due to ion

acceleration)• Ionization region:

– Electron cooling due to ionization– Maximum of plasma density inside

• Ion backstreaming region:– Electric force very small – Ion reverse flow is small– Pressure drop (towards anode) dominates

electron diffusive flow– Length depends on ionization rate (i.e. Te)

• Magnetic field is adjusted for each case:- solid lines: 110V, 110G- dashed lines: 600V, 330G

Electric Space Propulsion, E. Ahedo 3.B.11

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Axial structureInfluence of anode mass flow Influence of magnetic field shape

External B-field profiles

- Solid lines: 4 mg/s- Dashed lines: 10 mg/s

Electric Space Propulsion, E. Ahedo 3.B.12

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Electric Space Propulsion, E. Ahedo 3.B.13

Thrust2

, ,

Plasma momentum flow: ( ) ( ), ( ) ( )

Axial momentum equation for the plasma:

( )

( )

p zi e n

ii i e i zn w zi

ni e w znw i zn

ee e r e e

F z F z F z m u T n A

dF dAen Am n u udz dz

dF Am n u udz

dF d dAAen Aj B n Tdz dz dz

α α α α α αα

θ

φ ν ν

ν ν

φ

=

• = = +

•

= − + −

= −

= + +

⇒

∑

20 ( )

2p

e r w i zi znw e e e

dF d d dAAj B A m u u n n Tdz dz dz dzθ

ε φ ν⎛ ⎞= + − − +⎜ ⎟⎝ ⎠

Electric force

Magnetic force

Ionization + wall recombination

Jet divergence

Non-neutral E-field force

Thrust

20

Integrating the preceding equation between anode and far downstream:

Thrust:

Magnetic force:

Electric force: 2

p plume mag pA elec wall

mag e rA

elec

F F D F F F D

F j B Adz

dF Adz

θ

ε φ

∞

∞

•

• = − = + − −

− =

⎡ ⎤⎛ ⎞− = ⎢ ⎜ ⎟⎝ ⎠⎢⎣ ⎦

∫

Ion wall impact drag: ( )

Jet divergence drag:

Therefore, thrust is transmitted to the thruster through the magnetic reaction force of elect

AE

wall w i zi znw eA

plume e eA

D m u u n Adz

dAD n T dzdz

ν

∞

⎥⎥

− = −

− =

•

∫

∫

rons on the thruster magnetic circuit.Notice the contribution of the external magnetic field to thrust.Pressure forces at the anode make a marginal contributionIon energy accommodation at walls act

−−− s as a drag force.

Electric Space Propulsion, E. Ahedo 3.B.14

Bibliography1. Ahedo, Presheath/sheath model of a plasma with secondary emission from two

parallel walls, Physics of Plasmas, vol. 9, pp. 4340--4347, 2002.

2. Ahedo, Gallardo, Martínez-Sánchez, Effects of the radial-plasma wall interaction on the axial Hall thruster discharge, Physics of Plasmas, vol. 10, pp. 3397--3409, 2003.

3. Ahedo, Escobar, Influence of design and operation parameters on Hall thruster performances, Journal of Applied Physics, vol. 96, pp. 983--992, 2004.

4. Barral, Makowski, Peradzynski, Gascon, Dudeck, Wall material effects in stationary plasma thrusters. II. Near-wall and in-wall conductivity, Phys. Plasmas, vol. 10, pp.4137 -- 4152, 2003.

5. Boeuf, Garrigues, Low frequency oscillations in a stationary plasma thruster, J. Applied Physics, vol. 84, pp. 3541-3554, 1998.

Electric Space Propulsion, E. Ahedo 3.B.18

HALL THRUSTERS:HYBRID 2D

CODE

Electric Space Propulsion, E. Ahedo 3.C.1

Particle-in-cell (PIC) methods• Individual motion of macroparticles, subject to electromagnetic forces, is

followed in a computational grid. (see Birdsall-Langdon) • Cell size, lcell, smaller than plasma gradients• Timestep, ∆τ, such that particles advance less than cell size. • Number of macroparticles per cell, Ncell, depends

on good statistics and small numerical oscillations.• Mass of macroparticle depends on species density• Different species different sets of macroparticles• Example for Hall thruster (only ions and neutrals)

– axisymmetric, 30mm × 15 mm 900 (toroidal) cells lcell~0.5mm– Ncell (per species) ~ 30 27.000 macroparticles/species

109 -1011 atoms/macroparticle– macroparticle mass ~ atom mass × (atom/macroparticle)– ∆τ ~ 10-8 s (ions), 10-7 s (neutrals), 10-10 s (electrons)

Electric Space Propulsion, E. Ahedo 3.C.2

© IEPC-2005-040. All rights reserved. This content isexcluded from our Creative Commons license. For moreinformation, see http://ocw.mit.edu/help/faq-fair-use/.

Video of particle motion

Electric Space Propulsion, E. Ahedo 3.C.7

Courtesy of Professor Ahedo, Universidad Carlos III de Madrid. Used with permission.

Video of plasma dynamics

Courtesy of Professor Ahedo, Universidad Carlos III de Madrid. Used with permission.

Time-averaged 2D behavior

Electric Space Propulsion, E. Ahedo 3.C.9

Figure removed due to copyright restrictions. Please see Figure 11 in Escobar, D., A. Ant·n, and E. Ahedo."Simulation of High-Specific-Impulse and Double-Stage Hall Thrusters." In Proc. 29th International ElectricPropulsion Conference, Princeton, USA. 2005.

Time-averaged 2D behavior

B streamlines

Te=const along magnetic lines

Ion recombination at walls & neutral

re-emission

Electric Space Propulsion, E. Ahedo 3.C.10

Courtesy of Professor Ahedo, Universidad Carlos III de Madrid. Used with permission.

Bibliography1. Escobar, Antón, Ahedo. Simulation of high-specific-impulse and double-

stage Hall thrusters, IEPC-2005-040, 2005. 2. Hagelaar, Bareilles, Garrigues, Boeuf, Two-dimensional model of a

stationary plasma thruster, Journal of Applied Physics, vol. 91, pp. 5592-5598, 2002.

3. Parra, Ahedo, Fife, Martínez-Sánchez, A two-dimensional hybrid model of the Hall thruster discharge, Journal of Applied Physics,

Electric Space Propulsion, E. Ahedo 3.C.13

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