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

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

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

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

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

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

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)

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

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 )

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 )

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

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

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

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 ?

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

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

Order 1 with previously found Let Steady-state progressively reached  we set Moreover if , we have iff  Validity condition for the prompt jump approximation

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

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

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

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

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

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)

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

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

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

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

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

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

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:

 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

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)

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)

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

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

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))

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

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

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

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

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

compensated  does almost not play before all external  is applied 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 (boils down to setting b0 in the dynamics)

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

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

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

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

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:

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)

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)

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

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

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  

  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  