PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
CH.VI: REACTIVITY BALANCE AND
REACTOR CONTROL
REACTIVITY BALANCE
•
•
•
•
OPERATION AND CONTROL
CHARACTERISTIC TIMES
INTRODUCTION TO PERTURBATION THEORY
NEUTRON IMPORTANCE
REACTIVITY COEFFICIENTS
• DEFINITION
• EXAMPLES
LONG-TERM NEED FOR REACTIVITY CONTROL
• CONTEXT
• ISOTOPE CONCENTRATION EVOLUTION
XENON EFFECT
• XENON POISONING
• XENON OSCILLATIONS
MEANS TO ENSURE CONTROL
• EXTERNAL MEANS
• REACTIVITY EVOLUTION
1
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
VI.1 REACTIVITY BALANCE
OPERATION AND CONTROL
Variation of the reactor parameters  reactivity
 Loss of the neutron cycle equilibrium  transient
 Control
Criticality to maintain/manage in all circumstances: power,
shutdown, cold shutdown, new/used fuel, whatever qty of
fission products…
 Reactivity margins:
 available at any moment
 same magnitude as and opposite sign to the reactivity
change caused by any factor affecting 
 Characteristic time comparable to that on which  occurs
2
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
CHARACTERISTIC TIMES
Some orders of magnitude
Consumption of fissile matter
1000 h
Xenon effect (see below)
10 h
Delayed n
10 s
Circulation of coolant in the primary circuit
10 s
Transit of the coolant in the core
1s
Heat transfer from the fuel element to the
coolant
Asymptotic period at the prompt-critical
threshold for =10-8 s (small fast reactor)
Mean lifetime of the n
0.1 s
10-3 s
10-3 - 10-8 s
3
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
INTRODUCTION TO PERTURBATION THEORY
Necessity to be able to compute   1
1
in all situations
keff
In practice, calculation of  rarely possible because
 Actual reactor geometry  ideal geometry used in the
computations
 Presence of detectors in the core
 Consummation and production of isotopes = non-uniform f(t)
Simple way to estimate : perturbation from a reference
stationary state Koo  J oo
1
 Perturbed state: K 
J
keff
J  ( J  K )
4
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Let : arbitrary weight function
( , ( J  K ) )

 Static reactivity:
( , J )
1st order
Reference state
Perturbed state
Jo
J = Jo + J
Ko
K = Ko + K
o
 = o + 
0
()
 ( , J oo )  ( , ( J o  Ko ) )  ( , (J  K )o )
 ((J o*  Ko* ) ,  )  ( , (J  K )o )
*
* *
* *



If
:
solution
of
the
adjoint
reference
problem
K


J
o
o o
oo
(o* , (J  K )o )

(o* , J oo )
5
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
NEUTRON IMPORTANCE
Physical meaning of the adjoint flux
Introduction of 1 n at point r with velocity v  in a critical
reactor  secondary n and  
Corresponding augmentation of  ?
 The more important the added n, the larger the increase
Consider a reaction rate R   f ( P ) ( P ) dP
with  ( P)   Q( P' )T ( P'  P)dP'    ( P" )C ( P"  P' )T ( P'  P)dP' dP"
 I ( P)    ( P' ) K ( P'  P)dP'
(see chap.2)
Importance H(P) of a n – entering a collision at P – for R?
 Direct contribution due to a collision at point P: f(P)
 Expected contribution due to the next collisions:
 K ( P  P' ) H ( P' )dP '
6
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
 Adjoint equation: H(P)  *(P)
 * ( P)  f ( P)   K ( P  P' ) * ( P' )dP '
Expression of the reaction rate based on importance? n
emitted by the source, then transported to a 1st collision
R   f ( P) ( P)dP   I ( P) * ( P)dP
Adjoint transport problem in differential form
 f (r , v) 
J 
 (v' )  dv' d'


o
4
4

*
K    .  t (r , v)      s (r , v,   v' ,  ' )  dv' d '
*
o
4
+ adjoint BC for a reactor in vacuum: no importance of the
outgoing n through 
r   ,  * (r , v, )  0 if n.  0
s
s
One speed case: if  (r , ) solution of the direct problem on the
volume V of the reactor with BC in vacuum, then  (r ,)
solution of the adjoint problem with adjoint BC in vacuum
7
Adjoint diffusion problem
 f (r , v) 
J 
 (v' )  dv'

o
4
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
*

K   D(r , v)  t (r , v)    s (r , v  v' )  dv'
*
o
with BC at the extrapolated boundary
One speed diffusion
 diffusion operator: self-adjoint  *  
(at a cst)
Ex: impact of a cross section variation:

V
( f (r )   a (r )) 2 (r )dr
2


(
r
)

( r ) dr
f

V
Variation of  weighted by the flux squared
 Application:  of a control rod more important at mid-height
in the core
8
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
VI.2 REACTIVITY COEFFICIENTS
DEFINITION
Reactivity variations calculable by perturbation theory
 Trace back the causes of the variations of J and K ?
 Modification of the isotope density
 Dilatation due to the  of to
 Production/destruction of isotopes
 Void rate (BWR mainly)
 Move of matter (expulsion of coolant outside the core)
 Modification of microscopic cross sections
 Doppler effect (see chap.VIII)
NB: Effects due to variations of power, or of fuel or coolant to
9
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Variation Tc of the fuel to
(r )  
NU (r )
 (r )
Tc (r )
 (r )  NU (r )
Tc
Tc
< 0 (dilatation)
0
> 0 (Doppler)
One speed diffusion model:
 

( f (r )
V
 ln  f
Tc
  a (r )
 ln  a (r )
)Tc (r ) 2 (r )dr
Tc
2


(
r
)

(r )dr
f

V
Let Tc (r )  c f (r ), with c: mean to and f (r ) the spatial
distribution of Tc. If perturbation Tc only affects c:
 V

 c
( f (r )
 ln  f (r )
Tc
  a (r )
 
f
 ln  a (r )
) f (r ) 2 (r )dr
Tc
(r ) 2 (r )dr

: reactivity
 c coefficient
V
10
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
In general
N independent parameters i  N reactivity coef. s.t.

  
 i
 i
i
EXAMPLES
Power coefficient
If i fct of , hence of P:

  i

P
 i P
i
< 0 for stability!
Doppler coefficient

1 keff 1 f 1 p 1  1  1 P
keff .






T keff T
f T p T  T  T P T
Two to to account for: fuel Tc and moderator Tm
1 keff 1 p
1 keff 1 f 1  1  1 P





(Doppler effect) and
keff Tm
f T  T  T P T
keff Tc
p T
Both < 0
Fast variations
Slower variations
11
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
VI.3 LONG-TERM NEED FOR
REACTIVITY CONTROL
CONTEXT
Time-dependent issues considered up to now (see chap.V) on
time scales characteristic of prompt/delayed n generation
Longer-term time-dependent effects to be considered in the
neutron balance: consumption of fissile material, decay of
fission products…
 Interaction: material consumption/production dependent on
, which in turn depends on the material composition of the
reactor
Reaction rates
dN i
 f ( Ni , )
dt
 (Boltzmann)

 f ( , N i )
t
12
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Time scales likely to be different, however
 Usually:
 flux calculations with Ni constant at each time step t of the irradiation
history of the fuel (from ‘begin of cycle’ (BOC) till ‘end of cycle’ (EOC))
 (TBOC  t )  f ( Ni (TBOC ))
 then Ni evolution (via a depletion code) at the end of the time step
with  constant
dN i (t )
 f ( N i ,  (TBOC  t ))
dt
(possibility to do better than an
explicit Euler scheme but calculations
of  are time-consuming)
Irradiation history
TBOC
∆t
TEOC
 Burnup calculations
13
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
ISOTOPE CONCENTRATION EVOLUTION
Source balance for isotope i
Positive sources
• Isotope i as a fission fragment (fraction ji of fissions with j)
• Isotope i as a result of a n capture by isotope ‘i-1’
• Radioactive decay from parent isotopes
dNi
   ji  N j f , j (v) (v)dv   N i 1 c ,i 1 (v) (v)dv    j i N j
dt
j
j
 ( i  j ) N i  N i   a ,i (v) (v)dv
j
(Bateman equations)
Negative sources
• Isotope i absorbing (capture + fission) a n
• Radioactive decay to daughter isotopes
14
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
VI.4 XENON EFFECT
XENON POISONING
a(Xe135) = 2.7 106 barn at 2200 ms-1 (thermal) !!
 Particular role among all fission products
Production?
Fission
I = 0.061
135
Te

X = 0.003
 I
< 0.5 min
135

 Xe
135

Cs
6.7 h
9.2 h
135

 Ba135
2.6
106
(stable)
ans
Let X, I be the atomic densities of Xe135 and I135
dI
(Bateman equations)
  I  f   I I
dt
(a(I135) neglected)
dX
(I = 2.89 10-5 s-1)
  X  f   I I  ( X   aX  ) X
(X = 2.09 10-5 s-1)
dt
Linked to the Linked to the
current 
 before
(Q: other assumptions?) 15
16
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
In stationary regime with constant flux:
 f
( I   X ) f 
and
X 
I   I
 X   aX 
I
Saturation in Xe for  
X
 0.775 1013 n cm2 s 1
 aX
X ,max 
( I   X ) f
 aX
 aX X  ( I   X ) aX 
Let
: ratio of the nb of n absorbed by

 f
X   aX 
Xe over the nb of fission n
 Reactivity (1G diffusion) :  

V
( I   X ) aX  f  (r )
 X   aX  (r )
2
   f  ( r ) dr
 2 ( r ) dr
V
saturation

 
( I   X )

 0.027
(U235)
 Positive reactivity margin to have in store!
17
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Reactor shutdown in asymptotic regime
dI
 I I
dt
dX
 I I   X X
dt
I (r , t )  I  (r )eI t
( I   X ) f o  t  I  f o  t  t
X (r , t ) 
e

(e
e )
X   aX o
I  X
X
X
I
dX
 
 0 iff   X X  4 1011 n cm2 s 1 (U235)
dt o
 aX  I
[Xe] increases due to disintegration of I135 without destruction
by the n flux ([Xe] maximum after  11h), then decreases
We have
If   starting from a stationary regime, [Xe]  first before 
Negative reactivity following the maximum in [Xe]
Other isotope (poison) with similar effects: Sm
18
XENON OSCILLATIONS
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Reactor of large size, i.e. R(radius)/L(diffusion length) >> 1
 Sufficiently distant regions:
 Both critical
 Might be seen as +/- uncoupled
Timing
Accurate
calculation?
Complex (no point
kinetics!)
Zone 1
Zone 2


Production? Mainly due to fission
10h earlier (see I >> X)
Destruction? Present fission
 X
 X
Risks?
Reactivity?
<0
>0
Power peaks, but
long characteristic
time
Swing
increased


 Easily detected
Longer t Conc.Xe?
X
X
Mitigation?
>0
<0


t=0
 starts to swing
Short t
Conc.Xe?
Reactivity?
Swing reversed
…
Differential
insertion of the
control rods
19
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
VI.5 MEANS TO ENSURE CONTROL
EXTERNAL MEANS
Control rods
 Highly absorbing isotopes (e.g. Ag 80%, In 15%, Cd 5%)
 Impenetrable for thermal n
 Decreasing  in their neighborhood
 Reactivity source > or < 0 in normal operation
 Prompt anti-reactivity source if scram
Chemical poisons
Boric acid: uniformly distributed reactivity source  spatial
power distribution unchanged
20
PHYS-H406 – Nuclear Reactor Physics – Academic year 2013-2014
Consumable poisons
(e.g. borate pyrex rods B2O3)
Isotopes with high , initially put inside the reactor and depleted
because of the (n,) reaction
  of a and compensation for:
 of a due to fission products
 of (f - a) due to the depletion of the fissile matter
REACTIVITY EVOLUTION
(PWR with fresh fuel)
 Cold reactor, P = 0, no poisons (Xe, Sm): keff = 1.229
 Reactor in power, poisons in a steady state: keff = 1.126
cause   due to the  of both the moderator and fuel to
Criticality? Obtained by partly inserting the control rods
21