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
More on Moderators
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).
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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
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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 ,  ) ddE n(r ) 
n(r , E ,  )ddE
 
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).
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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
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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
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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).
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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
4r
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 drdd 
 
0 0 r 0

z
J dAz  ?
  s   
and show that J z  J  J   2 

 3t  z 0

s
and generalize J   D
D 2
3t

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).
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