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

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

3. V.1 POINT KINETICS EQUATIONS
INTRODUCTION
Numerical 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
T : amplitude factor, fast
time-dependent variations
 : shape factor, spatial and
slow time-dependent variations

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

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 (TN), 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

6. Introduce  s.t.
and : reactivity, relative distance to criticality
Then:
Comments
Prompt-critical threshold? Criticality obtained only with prompt n
keff = (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

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
(t)

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
Laplace
 / G(p)
>0
<0

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:
with
(inhour equation )
(measurement of   measurement of )

10. Limit cases – inhour equation:
Large reactivity:  >   i << 1
Growth of  independent of the delayed n above the prompt-critical threshold, i.e. keff(1 - ) = 1
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
Reactivity  < 0
 Decrease of  governed by the lifetime of the delayed n  
(prompt-
critical
threshold)
with
(indep. of )

11. Point model with one group of delayed n
Poles ? Solution of
and
Asymptotic
behaviour
Transient
quickly
damped

12. Possible transients and definition of the unique equivalent group
2 << i  slow transients   case  <  before
Characteristics of the unique group?
 = harmonic mean of the i’s (see above):
2 >> i (but  still <  )
Inhour equation for p  2:
 = 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 et

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

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 ?

15. Ex: reactivity step with : step function
Discontinuity of T(t) at the origin
Alternative 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
and

16. Ex2: reactivity ramp
Ex3: sawtooth
For t > t1:

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

18. Inverse of the instantaneous period
By definition:
Prompt jump approximation:

19. PROMPT APPROXIMATION Step
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
and

20. Ramp
To ? 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?
with

22. V.3 TRANSFER FUNCTION OF THE REACTOR
TRANSFER FUNCTION WITHOUT FEEDBACK
Point reactor kinetics model:
if (t) = 0 t  0
Hence L L

23. L 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 : L

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
comme
p <<  << /
W(p)   / p
 (/).exp(-j/2)
 << p << /
W(p)  1 1
/ << p
W(p)  /(p)
 /().exp(-j/2)

25. Bode diagram of the transfer function
ln|W()| 
/
W()

26. T(t)/T(o)
Validity limits
Insertion of a sinusoidal reactivity:
(t) = /100.sin(t)
Transfer fct with 1G and 6G
(differences?) t
T(t)/T(o)
Point reactor kinetics equations t

27. T(t)/T(o)
Insertion of a sinusoidal reactivity:
(t) = 5/100.sin(t)
Transfer fct with 1G and 6G t
T(t)/T(o)
Point reactor kinetics equations t

28. T(t)/T(o)
Insertion of a sinusoidal reactivity:
(t) = 15/100.sin(t)
Transfer fct with 1G and 6G t
T(t)/T(o)
Point reactor kinetics equations t

29. T(t)/T(o)
Insertion of a sinusoidal reactivity:
(t) = 70/100.sin(t)
Transfer fct with 1G and 6G t
T(t)/T(o)
Point reactor kinetics equations t

30. Rem: transfer function: applicable only if (t)
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

31. TRANSFER FUNCTION WITH FEEDBACK
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:

32.  Effective transfer function associated to a feedback effect
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
avec

33. Schematic of the feedback
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
ext
power
Kinetics
Thermal model
of the fuel +
W(p)
Heat
delivered to
the coolant
Void rate
coefficient
Hydrodynamic
model
Fuel temperature
coefficient
W(p)
ext /  +
T / T(0)
R(p)T(0)

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)

35. Compensated reactivity?
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) et

36. Elimination of the precursors
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
and

37. SOLUTION OF THE DYNAMICS EQUATIONS
Without delayed n:
with
Hence
If the accidentally inserted reactivity writes as follows:
We have for t > 0 :
(switch from + to – at the maximum of P(t))

38. Reactivity step: a = 0 Hence Solving with respect to E:
Replacing in the expression of P(t): .
with

39. If then
Hence with
and
Max of P(t) ?  (t) = 0
with
Equivalent half line width
of the power trip : T s.t.

41. Case n  1
We obtain:
with
Energy
asymmetric impulsion
for n  1

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.
and

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:
Equivalent to a
step o = atm

44. compensated  does almost not play before all external  is applied
It is shown that t/2tm varies from to 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
(boils down to setting b0 in the dynamics)

45. Comparison with experiments on test reactors
Different n on the rising and
descending sides of the peak

47. Influence of the delayed n
Point kinetics equation expressed in energy:
Time origin at max power
1st approx. of E without delayed n

48. As and :
we get:
Effect of delayed n? P does not go down to the initial power level

49. APPENDIX: CORRECT DEDUCTION OF THE POINT KINETICS EQUATIONS
Reminder: Transport equation with delayed n
Let

50. Deducing the kinetics equations
Normalization of the flux factorization:
After replacing  by T, dividing by T, multiplying by * and integrating on all variables but t:

51. Definitions Let 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
(PWR :  ~ 10-5s)

52. Reactivity Dynamic reactivity: Since we have Static reactivity:
with keff = eigenvalue of the stationary problem:
 = relative distance to criticality
Expressed in % or in pcm, or in $ (1$   =  : prompt-critical state)

53. Equations for the amplitude factor
Choice of normalization ?
s.t. time-dependent fluctuations of minimal
Weight fct = sol. of the adjoint syst. of the problem stationary
Towards the point reactor model
Up to now, NO approximation
If time-dependent variations in the shape factor neglected:
( exact flux factorization in its time-dependent and spatio-energetic parts)
(interpretation?)
and

54. Alternative expression of the point kinetics model
Let
One-speed case:
 : destruction time of the n
Inverse of the expected nb of
n absorbed /u.t. per emitted n
 Criticality iff

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  

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  