Environmental Engineering Reference
In-Depth Information
Electrons are in thermal equilibrium with
K
¼
12.93 keV on average. Thus,
R
¼
(2
mK
)
1/2
/
qB
m, and we see that thermal electrons are controlled even more
tightly, to a smaller radius, and move closely along the toroidally oriented magnetic
field lines. So, the charged particle system, closed on itself, is like an in
nite 1D
system. The electron thermal velocity
v
in the plasma is of interest, and is
v
¼
76.7
m
¼
(
k
B
T
/
m
e
)
1/2
10
7
m/s, about 0.16 c. The motion of these electrons is not
much affected by the theory of special relativity, because the relevant factor
[1
¼
4.77
(
v
/
c
)
2
]
1/2
1.012 is close to 1, indicating Newtons laws govern the motion.
We want to learn about the electrical conductivity of the plasma, which is limited by
the mean free path for scattering of the electrons by the ions. Assume the same
parameter
r
2
as was used in Chapter 2 (111.3 f at 12930 eV) determines the collision
cross section. So for the electron mean free path, we
nd
¼
2
l
e
¼
1
=½
N
D
pð
r
2
Þ
¼
257 km
ð
for electrons
Þ:
ð
4
:
32
Þ
This is an amazingly long mean free path, indicating that the electrons orbit
around the torus 6600 times between collisions. (We will see below that there is a
correction that reduces this by a factor 21.1.) The mean time between collisions,
t ¼l
e
/
v
¼
5.38ms. Now, we can
find the mobility
10
8
m
2
m ¼
et=
m
¼
9
:
47
=
Vs
ð
4
:
33
Þ
and the electrical conductivity
10
10
s ¼
ne
m ¼
1
:
52
:
ð
4
:
34
Þ
10
8
K and
n
10
20
deuterons/m
3
is thus
The resistivity of the plasma at 1.5
¼
10
11
r ¼
1
=s ¼
6
:
6
V
m
:
ð
4
:
35
Þ
We can neglect the contribution of the ions to the conductivity because their
mobility is at least 1835 times smaller due to the mass increase between proton and
electron.
We can now
find the resistance
R
for the torus plasma at its operating point as
¼r
2
p
6.2/
p
2
2
R
¼
3.1
r
ohms,
R
¼
0
:
204 n
V:
ð
4
:
36
Þ
Suppose, we want to heat this plasma at power 10MW. Let us
nd the needed
V
2
/
R
(
RP
)
1/2
, which will
voltage
V
to make
P
¼
¼
10MW. The needed voltage is
V
¼
be (10
7
10
9
)
1/2
0.045 V. This voltage is induced by the changing mag-
netic
field through the hole of torus.
Using the Faraday law, EMF
0.204
¼
is magnetic
flux in
Webers, so that we need 0.045 Webers/s. Suppose, we have a superconducting coil
with radius 1m, a bit bigger than used in a typical magnetic resonance imaging
apparatus, cutting through the donut hole of the torus. Thus, we have 0.045
¼
Voltage
¼
d
K
/d
t
, where
K
¼p
d
B
/d
t
,
or d
B
/d
t
14.3mT/s, to provide 10MWheating of the plasma. If the coil will support
5 T, then the time of the ramping can extend to 348 s. In this time, the energy
¼
