Environmental Engineering Reference
In-Depth Information
lifetime of a proton (against forming a deuteron) in the sun is about 10
10
years [26].
Yet, it is precisely these rare decays that provide the 170 petawatts heating the
earth over billions of years. This is an example of a situation where rounding off
8.0
10
24
to zero is a serious error!
This factor
T¼
8.0
10
24
will not apply to reactions more likely to be used in
terrestrial fusion reactors such as the D
-
DandD
-
Treactions. Fusion on earth, for
this reason, should be a lot easier (by a factor of 10
24
)thanonthesunbecausewe
can start with deuterons mined from the ocean and do not have to assemble them
from protons as is done on the sun. Deuterons are a necessary step on the way to
making helium from hydrogen. (Again, as on the sun, there are no free neutrons
that could fuse together or with protons with no Coulomb barrier, neutrons are
unstable).
It is useful to consider this reaction in a broader context. In general terms, we
consider a reaction of A and B tomake C, in the present case A and B are both protons.
In general, the rate
R
at which fusion product C is formed is [7]
R ¼ N
A
N
B
ð
1
þd
AB
Þ
1
hvsi
AB
:
ð
2
:
32
Þ
Here
d
AB
¼
0 unless A
¼
B, inwhich case it equals 1.0. The units of
R
are 1/(m
3
s).
If the energy release in the reaction is
Q
, the power density is
P¼RQ
, which, for
N
A
¼N
B
¼N
P
,is
5
N
p
hvsiQ
P ¼
0
:
;
ð
2
:
33
Þ
with units watts/m
3
if
Q
is expressed in Joules.
We
hvsi¼
1.54
10
49
m
3
/s,
P
/
m
3
¼
1.54
10
49
nd,
taking
0.5
(3.106
10
31
)
2
¼
313Ws/m
3
for the core of the sun. This
value is quite close to a published value 276.5W/m
3
at the center of the sun [27]. It is
known that the power density falls off rapidly with increasing radius, and is 19.5W/m
3
at 0.2
R
S
. We can check this value using the total power and assuming it is generated in
the core,
RR
S
/4. So
P
/volume
¼
3.82
10
26
/((4/3)
p
(
R
S
/4)
3
)
¼
17.3W/m
3
.
We will adopt 313W/m
3
as a reasonable basis for scaling to a Tokamak reactor
situation, in Chapter 4, using our approximate analysis summarized in Equa-
tion 2.28 by fusion rate per proton
f
fus
¼
T
Tf
coll
¼
T
T
(
v
/
26.2MeV
1.6
10
19
). The working values
for the center of the sun at 15 million K (1.293 KeV) are
T
T(Gamow factor)
¼
10
8
,
T
(reaction probability in p
-
preaction)
¼
8
10
24
, thermal velocity of proton
v ¼
0.498
10
6
m/s.
This value, about 300W/m
3
, is a small power density, smaller than human
metabolism, in agreement with other estimates. It shows that the large power output
from the sun derives from its immense size. A fusion reactor on earth could be made
to operate at a much higher power density, remember after all that a hydrogen bomb
is a fusion reactor of a sort. For comparison, the power densities available in
commercial processing tools such as
gas tungsten arc welding
(10
8
W/ m
3
) and
plasma
torches
(10
8
L
10
10
W/ m
3
) are much higher [28].
We will return in Chapter 4 to analysis of a DD fusion reactor, which we can
approach by adaptation of our analysis of the situation at the center of the sun.
