Transcript Document

Fuel Depletion
22
3
14
2 1
N ~ 10 cm , ~ 10 cm s
Time scale:
Days and months.
• More depletion  change steady state flux by means

of control rods.
N f (r , t )


f
• For a given fuel isotope
  N f (r , t ) a  (r , t )
t
• For constant flux 0 the solution is





 af 0 ( r ) t
 af  ( r ,t )
N f (r , t )  N f (r ,0)e
 N f (r ,0)e
• For time varying flux


N f (r , t )  N f (r ,0)e
t
 af


 ( r ,t \ ) dt \
0
Neutron fluence


 af  ( r ,t )
 N f (r ,0)e
Solve numerically.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
1
Fuel Depletion
• Constant power.
 




f
P(r , t )  wN f (r , t ) a  (r , t )  P(r ,0)  P0 (r )
Energy
released per
fission
Fission rate




N f (r , t ) (r , t )  N f (r ,0) (r ,0)




 f (r , t ) (r , t )   f (r ,0) (r ,0)
• Power ~ flux only over short time periods during which Nf is constant.

N f (r , t )
t



P0 (r )
f
  N f (r , t ) a  (r , t )  
w
• The solution is obviously


Linear
depletion!



P0 (r )
N f (r , t )  N f (r ,0) 
t
w
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
2
Fuel Depletion
HW 31
Do the
calculations
for different
flux and
power
levels.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
3
Poisoning and Fuel Depletion
Infinite, critical homogeneous reactor.

af (t )
k   f   f
mod erator
poison
control
a (t )  clad


(
t
)


(
t
)


(t )
a
a
a
a





P0 (r )
t
Constant power N f (r , t )  N f (r ,0) 
w



f
 N f (r ,0)  N f (r , t ) a  (r , t )t



N (r ,0) (r ,0)

 (r ,0)



 (r , t ) 



f
N (r , t )
1    (r ,0)t  N f (r ,0)  N f (r ,0) a  (r ,0)t


f
 N f (r ,0) 1   a  (r ,0)t

f 
f 
f
a (r , t )  a (r ,0) 1   a  (r ,0)t
f
f
f
a

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).



4
Poisoning and Fuel Depletion
Xe()
Xe(t ) 
( I   Xe )  f 0
Constant
(1  e
 (  Xe  aXe 0 ) t
 Xe   
 I  f 0
( 

(
e
 Xe  I   aXe0
Xe
a
0
Xe
 aXe 0 ) t
)
 e I t )
Constant


( I   Xe )  f (r ,0) (r ,0)
Xe 
Xe
 a (r , t )   a Xe() 

Xe
  (r , t )
Xe
a



Sm
 (r , t )   a  Sm  f (r ,0) (r ,0)t
Sm
a
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
5
Poisoning and Fuel Depletion
• Now we know all macroscopic cross sections.

af (t )
k   f   f
mod erator
poison
control
a (t )  clad


(
t
)


(
t
)


(t )
a
a
a
a

• When there are no absorbers left to
remove, we need to refuel.
• Absorbers are not only control rods.
• All fuel nuclei should be considered.
• For each species, all sources and
sinks should be taken into account.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).

Until = 0.
Solve for t to get
upper limit for
“core loading
lifetime”
6
Poisoning and Fuel Depletion
• Some poisons are intentionally introduced into
the reactor.
• Fixed burnable poisons.
B, Gd.
More uniform distribution than rods, more
intentionally localized than shim.
• Soluble poisons (chemical shim).
Boric acid (soluble boron, solbor) in coolant.
Boration and dilution.
Emergency shutdown (sodium polyborate or
gadolinium nitrate).
• Non-burnable poisons.
Chain of absorbers or self shielding.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
7
Delayed Precursors
G
G





1 
 g (r , t )   g  g \  fg \ (r ) g \ (r , t )    sg \ g (r ) g \ (r , t )  S gext
v g t
g \ 1
g \ 1





 

  ag (r ) g (r , t )   sg (r ) g (r , t )    Dg (r ) g (r , t )
• For one-group



1 
 (r , t )    f (r ) (r , t )  S ext
v t



  
  a (r ) (r , t )    D(r ) (r , t )
• What about delayed neutrons?
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
8
Delayed Precursors
(s)
< 0.7%
6



1 
 (r , t )  (1   )  f (r ) (r , t )   i Ci  S ext
v t
i 1



  
  a (r ) (r , t )    D(r ) (r , t )




Ci (r , t )
 i Ci (r , t )   i  f (r ) (r , t )
t
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
9
Delayed Precursors
• The multi-group equation now becomes
Different energy spectra
G
6




1 
 g (r , t )   gp (1   )  g \  fg \ (r ) g \ (r , t )  gC  i Ci (r , t )
v g t
i 1
g \ 1


   sg \ g (r ) g \ (r , t )  S gext
G
g \ 1





 

  ag (r ) g (r , t )   sg (r ) g (r , t )    Dg (r ) g (r , t )

G



Ci (r , t )
 i Ci (r , t )  i  g \  fg \ (r )g \ (r , t )
t
g \ 1
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
10
Delayed Precursors
• In steady state
G



i Ci (r , t )  i  g \  fg \ (r )g \ (r , t )
g \ 1
G
0   (1   )  g \
p
g
g \ 1
G




C
 fg \ ( r ) g \ (r )  g   g \  fg \ ( r ) g \ (r )
g \ 1







 

ext
   sg \ g (r ) g \ (r )  S g   ag (r ) g (r )   sg (r ) g (r )    Dg (r ) g ( r )
G
g \ 1
Appearance of 
C
ggg
g

0    (    )
p
g
C
g
p
g

G
g \ 1
g\


 fg \ (r ) g \ (r )
depends on whether G




ext
   sg g (r ) g (r )  S g   ag (r ) g (r )
we have fine or
course energy groups. g 1 


 

\
\
\
  sg (r ) g (r )    Dg (r ) g (r )
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
11