Embed Size (px)
Citation preview
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
1
CH.V: POINT NUCLEAR REACTOR KINETICS
POINT KINETICS EQUATIONS
• INTRODUCTION• INTUITIVE DEDUCTION OF THE POINT KINETICS EQUATIONS• POINT REACTOR MODEL • SOLUTION OF THE POINT KINETICS EQUATIONS FOR A
REACTIVITY STEP
APPROXIMATED SOLUTIONS FOR A TIME-DEPENDENT INSERTED REACTIVITY
• PROBLEM STATEMENT• PROMPT JUMP APPROXIMATION• PROMPT APPROXIMATION
TRANSFER FUNCTION OF THE REACTOR• TRANSFER FUNCTION WITHOUT FEEDBACK• TRANSFER FUNCTION WITH FEEDBACK
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
2
REACTOR DYNAMICS OF POWER TRIPS – FAST TRANSIENTS
• SYSTEM OF EQUATIONS• SOLUTION OF THE EQUATIONS OF THE DYNAMICS
APPENDIX: CORRECT DEDUCTION OF THE POINT KINETICS EQUATIONS
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
3
V.1 POINT KINETICS EQUATIONS
INTRODUCTIONNumerical solution of the time-dependent Boltzmann eq.
complex simplification if flux factorization possible:
Flux shape unchanged and
amplitude factor T alone accounts for time-dependent variations
Problem separable?
Possible only in steady-state regime and with operators J, K constant unrealistic
But acceptable for perturbations little affecting the flux shape around criticality
),,,().(),,,( tvrtTtvr
),,().(),,,( vrtTtvr
T : amplitude factor, fasttime-dependent variations
: shape factor, spatial and slow time-dependent variations
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
4
Reasoning
First Use of the following partial factorization
(+ normalization condition on the factors)in the Boltzmann equation with delayed n (see Appendix)
Secondly Impact of the hypothesis of exact factorization of the time-
dependent part
Deduction of a time-dependent model for the reactor evolution (seen as one point point kinetics) subject to perturbations w.r.t. the critical steady-state regime
Exact and approximated solutions depending on the perturbation type
),,().(),,,( vrtTtvr
),,,().(),,,( tvrtTtvr
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
5
INTUITIVE DEDUCTION OF THE POINT KINETICS EQUATIONS
Evolution of the n population without delayed n and sources (see chap.I):
where : expected lifetime of a n absorbed in the fuel / cycle Considering the amplitude function (TN), delayed n and an
independent source:
where ci(t): concentration of precursors of group i
NB: factorization of in T. to be made dimensionally consistent with ci
)(1)(
tNk
dt
tdN eff
)()()(1)1()( 6
1
tqtctTk
dt
tdTii
i
eff
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
6
Introduce s.t.
and : reactivity, relative distance to criticality
Then:
CommentsPrompt-critical threshold? Criticality obtained only with prompt nkeff = (1 - )-1
= Expressed in % or in pcm, or in $ (1$ if = )
Interpretation of the characteristic times in the one speed case?
Proba of an absorption/u.t.: v.a : destruction time
Close to criticality in an media : keff = J/K = f/a
: production time of the n criticality iff
eff
eff
k
k 1
av1
)()()()( 6
1
tqtctTdt
tdTii
i
effk.
fv1
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
7
POINT REACTOR MODEL
Amplitude function:
Precursor concentration in group i:
Criticality?Steady-state situation with = 0 and q(t) = 0
Comment
Variations of T(t) variation of , hence of the power, hence of temperature
= f(to) point kinetics eq. linear system usually
Neglecting this feedback, (t) = problem data = external reactivity inserted in the reactor
Exact deduction of the point kinetics equation: see appendix
)()()()( 6
1
tqtctTdt
tdTii
i
)()()(
tctTdt
tdcii
ii
(t)
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
8
SOLUTION OF THE POINT KINETICS EQUATIONS FOR A REACTIVITY STEP
Problem
Prompt move of the control rods in an initially steady and critical reactor without source
0,)(
0,0)(
tt
tt
i
ii p
cpTpc
i
)0()()(
i
i
i
ii
pp
i
pc
i
p
TpT
))0(()(
)0(
)0()0( TcCI i
ii
)](1[
1
)0(
)(
pGpT
pT
i
i
i pppG
11)]([ 1
Laplace
>0
<0
/ G(p)
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
9
Inversion of T(p)
Identification of the poles, hence of the roots of
Rem: p = 0 : not a pole (without source)!
See figure on previous slide: 7 real poles
< 0 : 7 poles > 0 : 6 negative poles, 1 positive po > 0 > p1 > … > p6
Asymptotic period of the reactor:
Period – reactivity relation:
)( pG
tpo
ttpk
k
ok eTeTtT
6
0
)(
))(')((1
1
)](1[lim
)0( kkk
k
pp
k
pGppGpGp
pp
T
Tk
2)(1
1
11
ik
ii
i
ik
i
i
p
p
with
)/1(
G
||/1 op
(inhour equation )
(measurement of measurement of )
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
10
Limit cases – inhour equation:
1. Large reactivity: > i << 1
Growth of independent of the delayed n above the prompt-critical threshold, i.e. keff(1 - ) = 1
2. Small reactivity: 0 < << i >> 1
mean lifetime of the delayed n weighted by their relative fraction
Growth of governed by the emission of the delayed n
3. Reactivity < 0
Decrease of governed by the lifetime of the delayed n
)(1
1
11 1
i
i
i
i i
i
i /
1/1
(indep. of )
0
(prompt-criticalthreshold)
1
1
11 i
i
i
i
i
i
i
1
11
with
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
11
Point model with one group of delayed n
Poles ? Solution of
p
p1
1
)()()()(
tctTt
dt
tdT
)()()(
tctTdt
tdc
2
4)( 2
2
1
2
1
ttT
tT
expexp
)0()(
and
Transient quicklydamped
Asymptoticbehaviour
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
12
Possible transients and definition of the unique equivalent group
2 << i slow transients case < beforeCharacteristics of the unique group?
= harmonic mean of the i’s (see above):
2 >> i (but still < ) Inhour equation for p 2:
Characteristics of the unique group?
= arithmetic mean of the i’s:
2 >>> i asymptotic trend of the inhour eq. with > and inversion of the two terms of T(t) (previous slide)
Term increasing with period Tp s.t.
Tp = inverse prompt-critical period
i
i
i pp
11 )1()1(2~
pp
pp i
ii
p i
i
i
i
/1
i
ii
p
p1
1et
1
1
pT
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
13
V.2 APPROXIMATED SOLUTIONS FOR A TIME-DEPENDENT INSERTED REACTIVITY
PROBLEM STATEMENT
Exact solution of the point reactor model equations? Possible only for a reactivity step inhour equation
Other cases? Numerical calculation or possible approximations:
Transients developing on characteristic times long compared to the generation time of the prompt n
governed by the delayed n prompt jump approximation
Very fast transients, beyond the prompt-critical threshold
effect of the prompt n only prompt approximation
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
14
PROMPT JUMP APPROXIMATION
Limit case for 0 Characteristic time of the transient >> Transient governed by the delayed n
Dvp of T(t) in series w.r.t. :
Elimination of c(t) in the point reactor model with 1 group of delayed n:
Replacement of T(t) by its dvp and order 0 in :
Rem: if transient governed by delayed n, what is the consequence on the upper limit of ?
)()(0
tTtT kk
k
0))()(())((2
2
Tttdt
dTt
dt
Td
)(
)0(.'
)'(
)'(exp)(
tdt
t
ttT
t
oo
)()(
))()(()(tT
t
tt
dt
tdTo
o
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
15
Ex: reactivity step with : step function
Discontinuity of T(t) at the originAlternative to integrating the Dirac peak:
If steady-state regime before inserting the step:
As c(t) continuous in t = 0:
Prompt jump approximation in the 1-group result
)(.)( tt
)0()0( cT
)0()0(0 cT
)0()0(
TT
t
t
oeTdtTtT )0('exp).0()(
)(t
and
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
16
Ex2: reactivity ramp
Ex3: sawtooth
For t > t1:
aa at
t
atto
o e
T
tT
211 )1()1(
1
)0(
)()1()2(
att )(
)1()1()0(
)(aat
t
o
o e
T
tT
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
17
Order 1
with previously found
Let
Steady-state progressively reached we set
Moreover if , we have
iff
Validity condition for the prompt jump approximation
0)()( 11
T
dt
dTT
dt
dT
dt
do
o
)()(
))()(()(tT
t
tt
dt
tdTo
o
)()()())(()()( 11 tTttTttTtdt
do
)())((
)()(
)( 1 tTt
tTt
tF o
')'()'()()( 1 dttTtFtFt
0)()( 1 TTo
oTT 1 oo TtTtT
)()(
)()(
21
||
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
18
Inverse of the instantaneous period
By definition:
Prompt jump approximation:
T
c
dt
tTdt ii
i
)(ln
)(
)()(
iii
iii
i cc
ct
)(tPJ
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
19
PROMPT APPROXIMATION
Transients beyond the prompt-critical threshold ( superprompt)Delayed n neglected (once (t) > !!)
with To s.t. (To) , to be determined from T(0)
Step
To ? Obtained from the model with 1 group of delayed n for very fast transients
')'(
exp.)( dtt
TtTt
oo
)()()(
tTt
dt
tdT
)(.)( tt ,
)0(TTo
tTtT o
exp.)(and
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
20
RampTo ? Fast transient accounting for the delayed n while
neglecting their variation in time:
with and s.t.
Prompt-critical threshold: tp = /a = p
We also have
and
Prompt approximation:
Rem: which paradoxical result does one get when using the prompt approximation on a non-superprompt transient?
att )(
)0()()()(
TtTt
dt
tdT
)0()0( Tcii
i
'"
exp)0(
''
exp)0()('
dtatT
dtat
TtTt
t
t
o
t
o
'2)0(
222 )'()( deeeT ppp
op
ta
2
a
p 2
a
tat
p
22)(
222
))()(()0()(22)(
ppp erferfeeTtT pp
)()0()()(2
ppoppc erfeTbeforeTtTT p
with
''
exp.)( dtat
TtTt
tpcp
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
21
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
22
V.3 TRANSFER FUNCTION OF THE REACTOR
TRANSFER FUNCTION WITHOUT FEEDBACK
Point reactor kinetics model:
if (t) = 0 t 0
Hence
)()(
'))0()'(.(
))0()(()(
)'(6
1
tTt
dtTtTe
TtTdt
tdT
ttt
o
ii
i
i
)()()()( 6
1
tctTt
dt
tdTii
i
)()()(
tctTdt
tdcii
ii
pp
T
p
TpT
i
ii
i
1
)(1
)0()(
)()(
TpG
L
L
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
23
We thus have
with
G(p) rational fct with Bj, Sj = f() known
For limited relative variations
we have
Transfer function of the reactor
Periodic variations of (t) :
Output of the reactor :
tSj
j
jeBtg )(
')'()'(
)'()0()( dttTt
ttgTtTt
o
))(()(1
pGtg
L
1)0(
)0()(
T
TtT
)()0()()0(
)()( pTpG
p
TpTpT
)(
).0(
)()(
pT
pTpW )( pG
i
i
i Sp
B
tjet .Re)(
.).(Re
)0(
)0()( tjejWT
TtT
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
24
Amplitude and phase of W?
Model with 1 group of delayed n
based on the figures giving |W()| and W() for :
The smaller , the better the reactor responds to high-frequency variations of
Negligible influence of as soon as becomes low
])([
)()(
pp
ppW
][
)(
pp
p
p << << / W(p) / p (/).exp(-j/2)
<< p << / W(p) 1 1
/ << p W(p) /(p) /().exp(-j/2)
comme
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
25
Bode diagram of the transfer function
ln|W()|
W()
/
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
26
Validity limits
Insertion of a sinusoidal reactivity:(t) = /100.sin(t)
Transfer fct with 1G and 6G
Point reactor kinetics equations
t
t
T(t)/T(o)
T(t)/T(o)
(differences?)
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
27
Insertion of a sinusoidal reactivity:(t) = 5/100.sin(t)
Transfer fct with 1G and 6G
t
t
T(t)/T(o)
T(t)/T(o)
Point reactor kinetics equations
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
28
Insertion of a sinusoidal reactivity:(t) = 15/100.sin(t)
t
t
T(t)/T(o)
T(t)/T(o)
Point reactor kinetics equations
Transfer fct with 1G and 6G
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
29
Insertion of a sinusoidal reactivity:(t) = 70/100.sin(t)
t
t
T(t)/T(o)
T(t)/T(o)
Transfer fct with 1G and 6G
Point reactor kinetics equations
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
30
Rem: transfer function: applicable only if (t).(T(t)-T(0)) is negligible
Verification: prompt jump approximation for (t) = o + .sin(t)
Then
Oscillating flux but both the expected and the maximum values exponentially increase with a period given by
This period tends to if
Negative constant reactivity to force in order to hinder the flux from drifting
')'(
)'()'(
)0(
)(ln dt
t
tt
T
tT t
o
')'sin(.
)'cos())'sin(.(dt
t
tt
o
ot
o
1
22
]1
)()1(
1.[1
o
0)(11 2
o
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
31
If o = 0 and (/) << 1, then
Reminder: prompt jump approximation valid iff
TRANSFER FUNCTION WITH FEEDBACK
Reactivity: depends on parameters i (fuel to, moderator to, void rate…)
Steady-state: i = 0 : variations with respect to the steady-state values
i : solution of evolution equations linking these variables to the reactor power
After a possible linearization:
)()( tt iii
2
2
1
o
ii
)]0()(.[ TtTBAdt
d
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
32
Therefore
reactivity :
Let ext(t) : external reactivity (i.e. controlled by the plant pilot: control rods, poisons…) total reactivity:
Yet
Effective transfer function associated to a feedback effect
dTTBet Att
o)]0()([)( )(
T
i
......1
,
dTTBe Att
o)]0()([, )(
)()(,)( 1 pTBApIp
)()( pTpR
)()()( ttt ext
)().(
)0(
)( ppW
T
pT
)(.)()()0(1
)(
)0(
)( p
pRpWT
pW
T
pT ext
avec
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
33
Representation of the feedback in the case of a fuel temperature reactivity coefficient (see chapter 8) and a void rate reactivity coefficient
Schematic of the feedback
+ KineticsThermal model
of the fuel
Hydrodynamic model
Void ratecoefficient
Fuel temperaturecoefficient
power
W(p) Heatdelivered tothe coolant
ext
+ W(p) T / T(0)ext /
R(p)T(0)
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
34
V.4 REACTOR DYNAMICS OF POWER TRIPS – FAST TRANSIENTS
Particular class of kinetics problems: accidental insertion of a > (prompt-critical threshold)
Delayed n have quasi no effect
Power very fast + fast apparition of a compensated that mitigates the transient
Need to characterize these transients to evaluate the damage induced
In these cases, modification of flux shape: limited in a 1st approximation point kinetics
Amplitude T(t) directly linked to the power P(t)
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
35
SYSTEM OF EQUATIONS
with : reactor “power”
Thermal conduction negligible if transient fast E fct of temperature T only (adapted if compensated due to Doppler effect (see chap.VIII) and to dilatation/expulsion of moderator)
Compensated reactivity? Detailed core calculations with all thermal exchanges,
including possible ebullition
semi-empirical correlation:
with b, n > 0 and : delay (conduction effect or delay till boiling onset)
dt
tdEtP
)()(
)()(]))(()(
[)( 6
1
tctPtEt
dt
tdPii
i
ext
)()(
)(tctP
dt
tdcii
ii
)(.)( tEbE n
et
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
36
Comments
Analytical solution: impossible, even without delayed n Numerical solution: feasible, but unobvious link with
experimental results to fit correlation parameters et : sometimes time-dependent Peak power: up to 106 x nominal power realistic
estimation instead of accuracy
Elimination of the precursors
with
First problem: n = 1 and = 0 Initial time at the prompt critical level = o(0) = 0
')'()()()( )'(
6
1
dttPetPtdt
tdP ttti
ii
i
)()(
)(tEbt
tn
ext )()( tbEt no
dt
tdEtP
)()( and
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
37
SOLUTION OF THE DYNAMICS EQUATIONS
Without delayed n:
with
Hence
If the accidentally inserted reactivity writes as follows:
We have for t > 0 :
)()()(
tPtdt
tdP
)()()( tbPtt o
bP
tPt
d
tdP
o
)()()(
attHt oo )()(
)( bPaP
dPd
dt
PdPtPb
P
tPat o
oo
ln))((2
)(ln2)( 2
(switch from + to – at the maximum of P(t))
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
38
Reactivity step: a = 0
Hence
Solving with respect to E:
Replacing in the expression of P(t):
.
))((2)( 22oo PtPbt
2))()(( tbEto
ooo Pdt
dEtEtE
bPtP )()(
2)( 2
Rt
o
o
Rto
eR
Re
b
RtE
1
1.)( oo bPR 22
2
2
1
..2
)(
Rt
o
o
Rt
o
o
eR
R
e
R
R
b
RtP
2
)(cosh.
22
2mttR
b
R
2
2ln.
1
RR
R
Rt o
o
om
with
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
39
If then
Hence with
and
Max of P(t) ? (t) = 0
bP oo 2
2
o
oo
bPR
2
2
)1(
)(Rt
Rt
o ze
ez
P
tP
o
o
bPz
22
)(0)( mom tbEt
bPtP o
m 2)(
2
max
bEtE o
m
max)(
bE o2
maxmax ETP
o
T2
Rt
Rt
o
o
ze
ez
P
tE
1
)1()(
Equivalent half line widthof the power trip : T s.t.
ze mRt with
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
40
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
41
Case n 1
We obtain:
with
Energy
ntn
ntn
o
o
ne
nePtP
/11
/11
max )1(
)1(.)(
n
no
o bn
nPP
/1
/11
max 1
n
no
bE
/1
/1
max
max/1)1( EnE n
asymmetric impulsionfor n 1
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
42
Compensated with a delay and n 1
If >> 0 : feedback due to values of E corresponding to a time interval where the feedback does not yet apply
if o >> 1
P max at t = T t.q.
o
t
ott
oo
t
o
o
oe
PdtePdttPtE
1
'')'()( '
ntno
no
ooe
bP
dt
tPd]1[
)(ln )(
)(
tn
no
no
ooe
bP
o
ntn
no
no
oo n
eebPt
P
tP oo
][)(ln
)(
)(
Tn
no
no
ooe
bP
]1
)(exp[.)( )(/1
1
Ttnon
o o
nn
en
Ttb
tP
]1
exp[./1max
1
nbP on
onn
and
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
43
Reactivity ramp: o = 0
Max of P(t) ? (t) = 0
Energy:
Impulsion of P keeps symmetric :
Let t = ½ -line width of the transient at half peak:
)1)(
(2)(
ln2)(ln
o
oo P
tPbP
P
tPa
dt
tPd
)(0)( mmm tbEatt
)( mm tEa
bt
Ea
btm 2
a
bP
PP
PPqttPP o
o
om
1/
)/ln(..)(
max
maxmax
oo
o
o
P
P
PP
PPPP
PP
P
dP
at
max
max
2/
ln1/1/
ln2
1 max
max
Equivalent to astep o = atm
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
44
It is shown that t/2tm varies from 0.125 to 0.027 for Pmax/Po ranging from 103 to 1014
Power trip width: very small compared to the time necessary to reach Pmax
compensated does almost not play before all external is applied
Thus
Yet the reactor instantaneous period is minimum at tM s.t.
Then
It can also be shown that b.Ef .t = Cst
compensated x power peak width = Cst
o
m
P
Pat max2
ln2
oo
M
o bP
a
P
tP
P
Pln
)(lnln max
)(0)( MM tbPat
b
aP max
(boils down to setting b0 in the dynamics)
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
45
Comparison with experiments on test reactors
Different n on the rising and descending sides of the peak
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
46
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
47
Influence of the delayed n
Point kinetics equation expressed in energy:
Time origin at max power
1st approx. of E without delayed n
''
)'()()]()([
)( )'(6
12
2
dtdt
tdEe
dt
tdEtbEt
dt
tEd ttti
ii
oi
')'(')0(')]0()0([0 '06
1
dttEeEbE tii
io
i
2cosh)0('
2cosh
2)(' 22
2 RtE
Rt
b
RtE
'2
'cosh)0(')0(')]0()0([ 2'06
1
dtRt
eEEbE tii
io
i
'2
'cosh)0(
2cosh)0(' 2'06
1
20dt
Rtedt
RtbE ti
ii
oi
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
48
As and :
we get:
Effect of delayed n? P does not go down to the initial power level
1o
ii
R
1cosh 2
uduo
)2(2
1)0(' 2
max ii
iob
EP
uduR
EEEo
m2cosh
2)0(')0(
)2(1 2
ii
iobR
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
49
APPENDIX: CORRECT DEDUCTION OF THE POINT KINETICS EQUATIONS
Reminder: Transport equation with delayed n
Let
),,,(),(4
)(),,,(])1[(
),,,(1 6
1
tvrQtrCv
tvrKJt
tvr
v iii
io
t
trCi ),( ),( trCii
dvdtvrvrfo),,,(),(
4
i
po JJ )1(
ifoii Jdvdvrv
),(4
)(
4
),,,(),(4
)(),,,()]()([
),,,(1 6
1
tvrQtrCv
tvrtKtJt
tvr
v iii
ip
t
trCv ii ),(
4
)(
),,,()(4
)(),( tvrtJ
vtrC i
iii
iip JJJ
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
50
Deducing the kinetics equationsNormalization of the flux factorization:
After replacing by T, dividing by T, multiplying by * and integrating on all variables but t:
dvdrdtvrv
vro
),,,(1.),,(*
4
1),,,(1,),,(* tvrv
vr
),()(
1)
4
)(,(
)(
1))]()([,(
)(
)(
1 **6
1
* QtT
Cv
tTtKtJ
dt
tdT
tT iii
ip
))(,()4
)(,(
)(
1)
4
)(,(
)(
1 ***
tJv
CtT
Cv
dt
d
tT ii
iiii
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
51
Let
Definitions
Generation time:
One-speed case:
Inverse of the expected nb of fission n produced /u.t., induced by 1 n of velocity v
Effective fractions of delayed n
)4
)(,()( *
ii
i Cv
tc
),()( * Qtq
)]()([)( tKtJtA p )()()())(,(
)( 6
1
* tqtctTtAdt
tdTii
i
)()())(,()( * tctTtJ
dt
tdciii
i
),,,()(),,,(
),,,(1,),,(
)(*
*
tvrtJvr
tvrv
vrt
)(
1)(
tvt
f
),,,()(),,,(
),,,()(,),,()(
*
*
tvrtJvr
tvrtJvrt i
i
),,,()(,),,()( * tvrtJvrt i
)()( tt ii
(PWR : ~ 10-5s)
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
52
Reactivity
Dynamic reactivity:
Since
we have
Static reactivity:
with keff = eigenvalue of the stationary problem:
= relative distance to criticalityExpressed in % or in pcm, or in $ (1$ = : prompt-critical
state)
),,,()(),,,(
),,,())()((,),,()(
*
*
tvrtJvr
tvrtKtJvrt
)(),,,()(,),,()( * ttvrtAvrt ii
),,,()(),,,(
),,,()(,),,()(
*
*
tvrtKvr
tvrtJvrtkeff
)(
1)()(
tk
tkt
eff
eff
eff
eff
k
k 1
),,(),,(1
vrvrJkeff
),,(),,( vrvrK
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
53
Equations for the amplitude factor
Choice of normalization ?
s.t. time-dependent fluctuations ofminimal
Weight fct = sol. of the adjoint syst. of the problem stationary
Towards the point reactor model
Up to now, NO approximationIf time-dependent variations in the shape factor neglected:( exact flux factorization in its time-dependent and spatio-energetic parts)
)()()()(
)()()( 6
1
tqtctTt
tt
dt
tdTii
i
)()()(
)()(tctT
t
t
dt
tdcii
ii
),,(* vr ),,,( tvr
)()()()()( 6
1
tqtctTt
dt
tdTii
i
)()(
)(tctT
dt
tdcii
ii
and
(interpretation?)
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
54
Alternative expression of the point kinetics model
Let
One-speed case:
: destruction time of the n
)(
1)(
tvt
a
)()()(1)1)(()( 6
1
tqtctTtk
dt
tdTii
i
eff
)()()(
tctTkdt
tdcii
eff
ii
),,,()(),,,(
),,,(1,),,(
)(*
*
tvrtKvr
tvrv
vrt
)().( tkt eff
Inverse of the expected nb ofn absorbed /u.t. per emitted n
)(t
Criticality iff
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
55
CH.V: POINT NUCLEAR REACTOR KINETICS
POINT KINETICS EQUATIONS
• INTRODUCTION• INTUITIVE DEDUCTION OF THE POINT KINETICS EQUATIONS• POINT REACTOR MODEL • SOLUTION OF THE POINT KINETICS EQUATIONS FOR A
REACTIVITY STEP
APPROXIMATED SOLUTIONS FOR A TIME-DEPENDENT INSERTED REACTIVITY
• PROBLEM STATEMENT• PROMPT JUMP APPROXIMATION• PROMPT APPROXIMATION
TRANSFER FUNCTION OF THE REACTOR• TRANSFER FUNCTION WITHOUT FEEDBACK• TRANSFER FUNCTION WITH FEEDBACK
PH
YS
-H4
06
– N
ucl
ea
r R
ea
cto
r P
hys
ics
– A
cad
em
ic y
ea
r 2
01
4-2
01
5
56
REACTOR DYNAMICS OF POWER TRIPS – FAST TRANSIENTS
• SYSTEM OF EQUATIONS• SOLUTION OF THE EQUATIONS OF THE DYNAMICS
APPENDIX: CORRECT DEDUCTION OF THE POINT KINETICS EQUATIONS
