Transcript Document
More on Moderators
HW 14
Calculate the moderating power and ratio for pure
D2O as well as for D2O contaminated with a) 0.25%
and b) 1% H2O.
Comment on the results.
In CANDU systems there is a need for heavy water
upgradors.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
1
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slowing down in large mass
number material
u
0
slowing down in hydrogeneous
material
u
continuous slowing-down model
1
2 3
4
5 6
7 n
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
0
1
2
n
3
2
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( A 1) 2 A 1
E
u ln \ 1
ln
2A
A 1
E av
Continuous slowing down model or Fermi model.
• The scattering of neutrons is isotropic in the CM
system, thus is independent on neutron energy.
also represents the average increase in lethargy per
collision, i.e. after n collisions the neutron lethargy
will be increased by n units.
• Materials of low mass number is large Fermi
model is inapplicable.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
3
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Moderator-to-fuel ratio Nm/Nu.
• Ratio leakage a of the moderator f .
• Ratio slowing down time p leakage .
• Water
moderated
reactors, for
example, should
be under
moderated.
• T ratio (why).
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
4
One-Speed Interactions
• Particular general.
Recall:
• Neutrons don’t have a chance to interact with each
other (review test!) Simultaneous beams, different
intensities, same energy:
Ft = t (IA + IB + IC + …) = t (nA + nB + nC + …)v
• In a reactor, if neutrons are moving in all directions
n = nA + n B + nC + …
Rt = t nv = t
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
5
One-Speed Interactions
n(r , )d Neutrons per cm3 at r
whose velocity vector
lies within d about .
d
n ( r ) n ( r , ) d
4
r
• Same argument as before
dI (r , ) n(r , )vd
dF (r , ) t dI (r , )
R(r ) F (r ) dF (r , ) t vn(r , )d v t n(r ) t (r )
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
4
where
(r ) vn(r , )d
4
6
Multiple Energy Interactions
• Generalize to include energy
n(r , E, )dEd Neutrons per cm3 at r with energy
interval (E, E+dE) whose velocity
vector lies within d about .
n( r , E ) dE n( r , E , ) ddE n(r )
n(r , E , )ddE
4
04
R(r , E)dE t ( E)n(r , E)v( E)dE t ( E) (r , E)dE
R(r ) t ( E ) (r , E )dE
0
Scalar
Thus knowing the material properties t and the neutron flux as a
function of space and energy, we can calculate the interaction rate
throughout the reactor.
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
7
Neutron Current
• Similarly RS (r ) S ( E ) (r , E )dE
and so on …
0
Scalar
• Redefine dI (r , ) n(r , )vd as dI (r , ) n(r , )vd
(r ) vn(r , )d
4
J v n ( r , ) d
4
Neutron current density
• From larger flux to smaller flux!
• Neutrons are not pushed!
• More scattering in one direction
than in the other.
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
J
J xˆ J x
8
Equation of Continuity
Net flow of neutrons per second per unit area normal
to the x direction:
J xˆ J x n( r , )v cos x d
In general: J nˆ J n
4
Equation of Continuity
n(r , t )d S (r , t )d a (r ) (r , t )d J (r , t ) nˆdA
t
A
Rate of change in
neutron density
Production
rate
Volume
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
Absorption
rate
Source
distribution
function
“Leakage
in/out” rate
Normal
Surface
to A
area
9
bounding
Equation of Continuity
3
Using Gauss’ Divergence Theorem B dA Bd r
S
J (r , t ) nˆdA J (r , t )d
A
V
n(r , t )d S (r , t )d a (r ) (r , t )d J (r , t ) nˆdA
t
A
1
(r , t ) S (r , t ) a (r ) (r , t ) J (r , t )
v t
Equation of Continuity
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
10
Equation of Continuity
For steady state operation
J (r ) a (r ) (r ) S (r ) 0
For non-spacial dependence
n(t ) S (t ) a (t )
t
Delayed sources?
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
11
Fick’s Law
Assumptions:
1. The medium is infinite.
2. The medium is uniform not (r )
3. There are no neutron sources in the medium.
4. Scattering is isotropic in the lab. coordinate system.
5. The neutron flux is a slowly varying function of
position.
6. The neutron flux is not a function of time.
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
12
Fick’s Law
Current J x
(x)
• Diffusion: random walk of
an ensemble of particles
from region dC/dx
of high
“concentration” to region of
small “concentration”.
• Flow is proportional to the
negative gradient of the
“concentration”.
x
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
Negative Flux Gradient
Current Jx
High flux
More collisions
Low flux
Less collisions
x
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Fick’s Law
Number of neutrons scattered per
second from d at r and going
through dAz
z
dAz
r
cos dAz t r
s (r )
e d
2
4r
y
s not s (r )
x
Removed
(assuming no
buildup)
Slowly varying
Isotropic
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
14
Fick’s Law
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
15
Fick’s Law
HW 15
s dAz
J dAz
4
z
2 / 2
t r
(r )e cos sin drdd
0 0 r 0
z
J dAz ?
s
and show that J z J J 2
3t z 0
s
and generalize J D
D 2
3t
z
z
D
1
3 s
Diffusion
coefficient
The current density is proportional to the negative of the gradient
of the neutron flux.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
16