Environmental Engineering Reference
In-Depth Information
4.3.2
Scaling the Fusion Power Density from that in the Sun
In Chapter 2, the power output from the sun is analyzed from the point of view of
proton
-
proton fusion, starting with Equation 2.21, and Equations 2.25
-
2.28, then
nally
5
N
p
h
P
¼
0
:
vsi
Q
;
ð
2
:
33
Þ
for the power per cubic meter generated, 313W/m
3
in the core of the sun.
Our program here is to scale this number to
find the power density in a D
D
-
10
20
m
3
(see Figure
reaction Tokamak reactor, assuming deuterium density
N
D
¼
10
31
m
3
. This, along
3.18), to be comparedwith the suns proton density
N
p
¼
3.11
10
24
(Equation 2.31) to
with the change in the reaction probability factor
T
from 8
T
0.1. Assume a factor of 10 increase in
T
from 15 million K to 150 million K.
Scaling must include the
r
2
parameter, noting that the mass of the deuteron is twice
that of the proton, and the temperature enters the velocity and also the cross section
for the geometric collision. Finally, if the Tokamak has a volume 500m
3
, how much
power does it release in fusion reactions (assume
Q
¼
¼
3.5MeV for the DD reaction)?
The scaling ratio,
P
0
/
P
, works out to be
P
0
=
10
20
10
31
2
10
3
=
2
T
0
Gamow
=
10
8
10
24
P
¼ð
=
3
:
11
Þ
ð
1
=
Þð
Þð
0
:
1
=
8
Þ
ð
3
:
5 MeV
=
26
:
2 MeV
Þ¼
54
:
01
;
where
T
0
Gamow
10
4
. (The second factor here comes explicitly from the
change in temperature entering
v
and
s
.) The value forT
T
0
Gamow
follows Equation 2.24
with parameters
r
1
¼
9.9
m
p
.
So the predicted Tokamak power density, assuming pure deuterium fuel, is
16.9 kW/m
3
and its power output is 8.45MW.
¼
3f,
r
2
¼
111.3 f,
E
¼
12.93 keV, and
m
r
¼
1
4.3.3
Adapt DD Plasma Analysis to DT Plasma as in ITER
To convert this result to a DTplasma, approximately, we consider two aspects. First, is
the factor 15 increase in the DTcross section over the DD cross section, and, second,
the larger
Q
energy release, 17.6MeV, versus 3.5MeV. Thus, if the plasma were a DT
plasma, the projected Tokamak power output would be increased by 75, to
500 m
3
Q
¼
634 MW
:
ð
DT plasma at 150 million K
;
Þ:
ð
4
:
25
Þ
This estimate is close to those projected for the ITER Tokamak, which has a larger
volume, 880m
3
, by virtue of its larger, heart-shaped, rather than circular, cross
section.
Let us continue to calculate properties of the assumed Tokamak, starting with the
energy
U
needed to heat the 500m
3
deuterium to 1.5
10
8
K, where we know it is
12.93 keVper particle. In thermal equilibrium, the plasma contains equal numbers of
