16.55 Ionized Gases
Problem Set #4
Problem 1
~
Combine the total energy equation for a species s in terms of (~us , Ts0 , P~s0 , ~qs0 ), as derived in
~
the notes, with the momentum equation in terms of (~us , P~s0 ) to obtain the internal energy
equation. You should obtain,
X
∂ 3
3 0
~
0
0
ns kTs + ∇ · ns~us kTs + ~qs + P~s0 : ∇~us =
Er0 s
(1)
∂t 2
2
r
with,
~ rs
Er0 s = Ers − ~us · M
Notice the absence of any electric or magnetic terms on the LHS.
Problem 2
~ and
Consider a plasma that is homogeneous, steady and stationary, subject to uniform E
~ fields, which produce an electron current density ~je . The gas may be undergoing steady
B
ionization, at a rate ṅe = Ri (Te0 )nn ne per unit volume, where Ri = hce Qioniz ie . Each ionization event absorbs from the electron gas an energy αi Ei , where αi ≈ 2 to 3 accounts for
excitation losses that happen simultaneously. Only one species of ions and of their parent
neutrals are present, and it can be assumed that electron collisions are much more frequent
with the neutrals than with the ions.
(a) For this situation, formulate Eq. (1) of Problem 1, as the equation that will supply the
0
electron temperature. The dissipative term in Ers
should reduce to je2 /σ; eliminate ~je
using the Ohm’s law, and show that the equation reduces to,
!2
hce Qioniz ie
1
eE
me
p
k (Te0 − Tn0 ) +
αi Ei
(2)
= 3
2
me
mn
c̄e Q∗en
1 + βe νen
(b) A model for the ionization rate is given by,
kTe0 −Ei /kTe0
Ri ≈ σ0 c̄e 1 + 2
e
Ei
(3)
Derive an approximate criterion, in the form of a critical value of kTe0 /Ei , to test
whether the ionization loss term or the elastic energy transfer term in Eq. (2) are
dominant. Note that the exponential term in (3) is the controlling factor.
(c) For conditions of low magnetization, βe 1, show that Te0 is a function of Tn0 and
of the “reduced field” E/nn . Conversely, for high magnetization, show that Te0 is a
function of Tn0 and of the drift velocity E/B. As a numerical example, consider a Hall
thruster plasma, with βe 1, where (in the ionization region) E is about 100V /cm
and B is 200Gauss. The gas is Xenon, with Vi =12.1 eV, σ0 = 3.6 × 10−20 m2 and
Q∗ ≈ 3 × 10−19 m2 . Take αi = 2.5.
1
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16.55 Ionized Gases
Fall 2014
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