(麻省理工)电力推进—等离子体(以及电弧)引擎概述

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
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)
cancelling remainder is Ve . Think of a swarm of bees moving furiously to swarm) slowly.
G The number of electrons per unit volume that have a velocity vector c ending in a
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
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)
(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
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):
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-
(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
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.
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
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
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
( ) 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.
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
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
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
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
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.
Lecture 10 Page 2 of 12
GG FE + FFriction = 0,