(麻省理工)电力推进—等离子体(以及电弧)引擎概述

value ce for all electrons. But this is for all electrons colliding with one neutral. We are interested in the reverse (all neutrals, one electron), so the part of νe due to neutrals
The
current
density
is the flux of charge due to motion of all charges. If only the electron motion is
counted (it dominates in this case)
G
JG
j ≡ -ene Ve
(10)
A normal gas at T <∼ 3000K is a good electrical insulator, because there are almost no free electrons in it. For pressure >∼ 0.1 atm, collision among molecule and other particles are frequent enough that we can assume local Thermodynamic Equilibrium, and in particular, ionization-recombination reactions are governed by the Law of Mass Action. Consider neutral atoms (n) which ionize singly to ions (i) and electrons (e):
with some other particle (neutral or ion). Collisions with other electrons are not
counted, because the momentum transfer is in that case internal to the electron
ne = -S +
S2 + Sn = 1+
n
1
+
S
n
(T
)
(6)
Since S increases very rapidly with T, the limits of (6) are
( ) ne ⎯⎯T⎯→0⎯⎯→
Sn → 0 T →0
(Weak ionization)
ne
⎯⎯T⎯→∞⎯⎯→
n 2
(Full ionization)
G OG nce an electron population exists, an electric field E will drive a current density j through the plasma. To understand this quantitatively, consider the momentum
Lecture 10 Page 2 of 12
GG FE + FFriction = 0,
or
JG
G
me Veνe = -e E
JG Ve
=
-
⎛ ⎜ ⎝
e me ν e
⎞G ⎟E ⎠
(9)
The
group
µe
=e meνe
is
the
electron
“mobility”
((m/s)/
(volt/m)).
Lecture 10 Page 1 of 12
Given T, this relates ne to nn . A second relation is needed and very often it is a specification of the overall pressure
P = (ne + ni + nn ) kT = (2ne + nn ) kT
balance of a “typical” electron. It sees an electrostatic force
G
G
FE = -eE
(7)
It also sees a “frictional” force due to transfer of momentum each time it collides
( ) t)
on
the
three
components JJG
of
ce .
In
an
equilibrium
situation all directions are equally likely, so fe = fe ce = fe (ce ) only, and one can
show that the form is Maxwellian.
k = Boltzmann constant = 1.38 ×10-23 J/K
(Note: k = R/Avogadro’s number) h = Plank’s constant = 6.62x10-34 J.s. Vi = Ionization potential of the atom (volts) (Vi = 13.6 V for H)
momentum is fully given up. Suppose there are νe of these collisions per second
( νe
=collision
JG
frequency JJG
per
electron).
The
electron
loses
momentum
at
a
rate
(5)
Combining (3’) and (5),
n2e
=
S
(
T
)
⎛ ⎜⎝
P kT
-
2ne
⎞ ⎟⎠
=
S
(T ) (n
-
2ne
)
Where n = P is the total member density of all particles. kT
We then have
n2e + 2Sne - Sn = 0
fe
= ne
⎛ ⎜ ⎝
me 2πkTe
3
⎞2 ⎟ ⎠
- mec2e
e 2kTe
;
( ) c2e
=
c2ex
+ c2ey
+
c2
ez
(14)
With the normalization
∫ ∫ ∫ fed3ce = ne .
The mean velocity is then
∫ ∫ ∫ ce ≡
ce fed3ce
cancelling remainder is Ve . Think of a swarm of bees moving furiously to and fro, but
moving (as a swarm) slowly.
G The number of electrons per unit volume that have a velocity vector c ending in a
e
“box”
dcex dcey dcez ≡ d3ce in velocity space is defined as
( )G
fe ce d3ce
( ) G JG
Where
fe
ce ,x
is the Distribution function of the JG
electrons which depends (for a Ggiven location x and time
should be nn ceQen . Adding the part due to ions,
νe = nn ceQen + ni ceQei
(13)
16.522, Space Propulsion Prof. Manuel Martinez-Sanchez
Lecture 10 Page 3 of 12
16.522, Space Propulsion Prof. Manuel Martinez-Sanchez Lecture 10: Electric Propulsion - Some Generalities on Plasma (and Arcjet
Engines)
Ionization and Conduction in a High-pressure Plasma
Except for very narrow “sheaths” near walls, plasmas are quasi-neutral:
ne = ni
(4)
So that
n2e = S (T )
(3’)
nn
can be used.
16.522, Space Propulsion Prof. Manuel Martinez-Sanchez
Electrons moving at random with (thermal) velocity ce intercept the area Qen at a rate equal to their flux neceQen . Since a whole range of
speeds ce exists, we use the average
-me V eνe , where Ve =mean directed velocity of electrons, and so
G
JG
FFriction = -me V eνe
(8)
On average,
16.522, Space Propulsion Prof. Manuel Martinez-Sanchez
and from (9),
G j
=
⎛ ⎜⎜⎝
e2ne meνe
⎞G ⎟⎟⎠
E
(11)
The group
σ = e2νe meνe
(12)
is the conductivity of the plasma (Si/m).
Let us consider the collision frequency. Suppose a neutral is regarded as a sphere with a cross-section area Qen.
n R e+i
(1)
One form of the Law of Mass Action (in terms of number densities nj = Pj kT , where T is the same for all species) is
neni = S (T )
(2)
nn
Where the “Saha function” S is given (according to Statistical Mechanics) as
NOTE:
c is e
very
different
(usually
much
larger)
than
Ve .
Most
of
the
thermal
motion
is
fast,
but in random directions, so that on average it nearly cancels out. The non-
( ) S
T
=
2
qi qn
⎛ ⎝⎜
2πmekT h2
3
⎞2 ⎟⎠
-
e
eVi kT
(3)
qi = Ground state degeneracy of ion (= 1 for H+) qn = Ground state degeneracy of neutral (= 2 for H) me = mass of electron = 0.91 × 10-30 Kg
and direct calculation gives
ce =
8 kTe π me
For Hydrogen atoms,
(15)
ce = 6210 Te
(m/s, with Te in K)
(16)
NOTE:
If there is current, the distribution cannot be strictly Maxwellian (or even isotropic). But since Ve << ce, the mean thermal velocity is very close to Equation (15) anyway.
Βιβλιοθήκη Baidu
species. The ions and neutrals are almost at rest compared to the fast-moving
electrons, and we define an effective collision as one in which the electron’s directed