REP/

- page 1 -

PWR Nuclear Reactor Core Design

Power and Reactivity Elements on Reactor Kinetics

and Residual Power

G.B. Bruna

FRAMATOME ANP

REP/

- page 2 -

Foreword

Neutron Balance Equation of a multiplying system at time t:

ttv

ttttStS

tk

tFtAtH

tSttH

effE

eff

1

REP/

- page 3 -

Foreword

Steady State Conditions At any time t :

ttk

tFtS

tt

ttt

effE

)(

0

0

REP/

- page 4 -

Foreword

Steady State ConditionsAt any time t :

The number of neutron in any generation equals the number of neutrons in the previous and following generation;

The prompt-neutron lifetime equals exactly the generation-time .

0.1,

,*

ttA

ttFtk

LtN

tN

tL

tLeff

REP/

- page 5 -

Foreword

Steady State Conditions At any time t, any explicit dependence on the

variable time can be dropped out :

0H0.1effk

REP/

- page 6 -

Foreword

Steady State Conditions

In steady state conditions, the neutron balance of the system changes

Very slightly due to:

Xenon oscillations,

Fuel burn-out,

With a time-constant which is quite long against observation-time.

REP/

- page 7 -

Power and Reactivity

Main Parameters in Reactor Core DesignPower

It is a physical observable which measures the energy released under different forms (kinetic energy of fission fragments, kinetic energy of fission neutrons, gamma) within the system by neutron fission, capture and slowing-down.

ReactivityIt is not a real physical observable because it measures

the reset that is to be applied to the fission operator to restore criticality of a given multiplying system, generally not critical after any perturbation (change of the state Boltzmann operator).

REP/

- page 8 -

Power and Reactivity

Power

Total Fiss Power

Total Power

Local Fiss Power

dEdVNPn

fissnn

fiss

dENP kjin

kjifissnnkji ,,,,,,

dwnslcaptfiss PPPP

REP/

- page 9 -

Power and Reactivity

Power

Power Peak

Axial Offset

P

PMaxPowPeak kji ,,

LH

LH

FF

FFAO

,,

,,

REP/

- page 10 -

Power and Reactivity

Reactivity

FAH

F

H

keff

0

*,0

*,0

0

1

11

REP/

- page 11 -

Power and Reactivity

Control of PowerPower distribution within the reactor core is not flat

because of : Neutron gradient (leakage), Short-life fission-product poisoning, Burn-up and breeding effects, Reflector gain, Fuel and moderator temperature feed-back, Control rod effect; ...

REP/

- page 12 -

Power and Reactivity

Control of PowerPower distribution can be controlled both

At the design stage (assembly and core layout, burnable poisons, reflector, reloading strategy),

In operation (mainly by control rods positioning);

Several strategies of control rod management can be adopted (e.g., in French PWRs : A mode, G mode, X mode).

REP/

- page 13 -

Power and Reactivity

Control of PowerCore design and operation : Typical MOX reloading

strategyIN / O UT

C G G G

G G G

G G G

G G G G

G G GG

G

8

9

10

11

12

13

14

15

H G F E D C B A

HYBRIDE

C8

9

10

11

12

13

14

15

H G F E D C B AOUT / IN

C8

9

10

11

12

13

14

15

H G F E D C B A

1ST CYCLE 2ND CYCLE 3RD CYCLE

4TH CYCLEBP (Gd2O3)G C

REP/

- page 14 -

Power and Reactivity

Control of PowerCore design and operation : X mode operating

Control of AO

Control of Temperature

REP/

- page 15 -

Power and Reactivity

Control of ReactivityReactivity of the core is sensitive to :

Reactor life:Fuel burn-up,Breeding process,Fission-product and actinides build-up,Burnable poison burn-out,

Short-lifetime fission-product poisoning,

Power and temperature feed-back.

REP/

- page 16 -

Power and Reactivity

Control of ReactivityCore reactivity is also sensitive to any external

perturbation of Boltzmann operator :Soluble boron concentration change,

Position of control banks,

Power output,

Any incident and/or reactivity accident.

REP/

- page 17 -

Power and Reactivity

Control of ReactivityIn normal operation, reactivity is to be kept

constant (no measurable reactivity change);

To guarantee respect of this condition, reactivity NEEDS (sources of reactivity changes and design margins) must be compensated exactly by reactivity AVAILABILITIES (worth of control devices).

REP/

- page 18 -

Power and Reactivity

Control of Reactivity (NEEDS)Reactivity NEEDS (normal operation) :

Respect of safety criteria,

Respect of margins,

Compensation of fuel burn-up and breeding,

Compensation of burnable poison burn-out,

Compensation of Xenon and Samarium build-up,

Compensation of power and temperature effect.

REP/

- page 19 -

Power and Reactivity

Control of Reactivity (NEEDS)Criteria and margins

The main objective of a nuclear is producing a cheap energy in safest way;

In order to achieve this goal, design and exploitation of the plant must :Guarantee respect of the safety criteria at any time,Maximize energy release from the fuel, according to a given

exploitation strategy.

REP/

- page 20 -

Power and Reactivity

Control of Reactivity (NEEDS)Criteria and margins

In order to guarantee respect of the maximum allowed values (criteria), uncertainty is affected to design parameters;

Uncertainty must account for:Computational precision (base-data, qualification, ..),Technology of the fuel (fabrication tolerance, …),Measurement device precision,Alea (power tilt, ...);

Margins can also be enforced to account for future changes of loading strategies and new fuel features.

REP/

- page 21 -

Power and Reactivity

Control of Reactivity (NEEDS)Fuel burn-up and breeding

Fissile isotopes burn-out,

Plutonium build-up,

Minor Actinides build-up,

Fission-Products build-up.

REP/

- page 22 -

Power and Reactivity

Control of Reactivity (NEEDS)Burnable poison burn-out

Burnable poisons contribute to :Compensate reactivity,Flatten core power;

When they disappear :Fuel reactivity can increase,Power pick can appear.

REP/

- page 23 -

Power and Reactivity

Control of Reactivity (NEEDS)Xenon and Samarium build-up

Short-lifetime Fission Products as Xenon and Samarium build-up as a consequence of production of power,

Any power change engenders a variation of their concentrations which affects the reactivity of the system,

Local power variation engender spatial discontinuities in concentration which produce power tilt.

REP/

- page 24 -

Power and Reactivity

Control of Reactivity (NEEDS)Power and temperature effect :

Doppler broadening of wide epi-thermal resonances:Fissile isotopes do not contribute significantly to Doppler effect owing to

compensation among capture and fission reaction-rates,Fertile isotopes (mainly U238 and Pu 240) have major contribution to

the effect;Moderator effect :

When moderator density varies, neutron spectrum either hardens-up or soften-down and reactivity changes;

Soluble boron poisoning effect :When moderator density varies, amount of boron atoms per unit

volume is modified.

REP/

- page 25 -

Power and Reactivity

Control of Reactivity (NEEDS)Power and temperature effect : Doppler

broadeningBroadening of epi-thermal resonances of heavy isotopes

(at first order, only even ones contribute),

Very fast action (sensitive to temperature changes inside the pellet),

About -3°pcm (1 pcm = 1 E-5) par degree Celsius.

REP/

- page 26 -

Power and Reactivity

Control of Reactivity (NEEDS)Power and temperature effect : Moderator

effectVariation of the water moderating power (neutron

spectrum changes),

Long term action (sensitive to the coolant temperature),

Worth sensitive to isotopic composition of the fuel (stronger for MOX).

REP/

- page 27 -

Power and Reactivity

Control of Reactivity (NEEDS)Power and temperature effect: Moderator

effect

MOX

UOX

Reactivity

Void rate

0100

REP/

- page 28 -

Power and Reactivity

Control of Reactivity (AVAILABILITIES)Reactivity AVAILABILITIES (normal operation)

:Soluble boron,

Control and scram clusters :Black rods,Gray rods,

Burnable poisons :Fixed,Extractable

Extractable poisons.

REP/

- page 29 -

Power and Reactivity

Control of Reactivity (AVAILABILITIES)Soluble boron

Soluble boron is mainly used to compensate the fuel burn-up,

Power shape is quite insensitive to soluble boron poisoning the primary leg,

Soluble boron worth is very sensitive to fuel nature (ranging from 10 pcm/ppm to 4 pcm/ppm and less),

Concentration of boric acid in primary leg is limited by :Crystallization (clad rupture), Moderator density dependence of poisoning effect.

REP/

- page 30 -

Power and Reactivity

Control of Reactivity (AVAILABILITIES)Control and scram clusters

Control clusters can be either homogeneous (AIC) or mixed (axially heterogeneous B4C - AIC),

If needed, boron in boron carbide can be enriched in B10,

The mixed clusters can be more effective then AIC ones, but they posses the inconvenient to bow-up under pressure of He gas produced by B10 neutron capture.

REP/

- page 31 -

Power and Reactivity

Control of Reactivity (AVAILABILITIES)Control and scram clusters

Control clusters are used to Finely adjusting the primary leg output temperature,Controlling Xe oscillations,Maintaining AO inside the operating range;

When inserted into the core control clusters cannot must respect a threshold to avoid prompt criticality in presence of a rod-ejection reactivity accident.

They can (partially) contribute to scam.

REP/

- page 32 -

Power and Reactivity

Control of Reactivity (AVAILABILITIES)Burnable poisons

Burnable poisons are used to :Compensate fuel burn-out,Contribute to power flattening;

Burnable poisons can be : Introduced into guide tubes of some unclustered assemblies (Pyrex),Integrated to the fuel (Gadolinium Oxide)

Thy engender a spectrum hardening,

REP/

- page 33 -

Power and Reactivity

Control of Reactivity (AVAILABILITIES)Extractable poisons

Extractable poisons are introduced at beginning of cycle into guide tubes of assemblies not receiving control and safety clusters,

Their position is not axially adjustable (they can be either OUT or IN),

They engender a spectrum hardening,

When they are dropped out, a spectral-shift is produced.

REP/

- page 34 -

Reactor Kinetics

Neutron Balance Equation of a multiplying system at time t[inhomogeneous equation]:

ttv

ttttStS

tk

tFtAtH

tSttH

effE

eff

1

REP/

- page 35 -

Reactor Kinetics

Lifecycle inside a reactor system (recall)

Production

Neutrons

DiffusionSlowing-down

Capture

Leakage

REP/

- page 36 -

Reactor Kinetics

Lifetime and generation-time (recall)

During transients prompt-neutron lifetime differs from generation time.

0.1,

,*

ttA

ttFtk

LtN

tN

tL

tLeff

REP/

- page 37 -

Reactor Kinetics

Lifetime and generation-timeTypical values for L / L*

Vacuum 20 mm (L* )PWR (UOX) 25 s (L* same)PWR (UOX - MOX) 10 s "PWR (MOX) 7 s "FBR (MOX) 5 s "Critical sphere (U) 6 ns "Critical sphere (Pu) 3 ns "

REP/

- page 38 -

Reactor Kinetics

Point Kinetics Heuristic approach

Reactor is homogenized in space and collapsed to a space-point system (no explicit dependence of variables on space),

Neutrons are collapsed in energy to one group (no explicit dependence of neutrons on energy),

Simplified statistical approach:The number of neutron in the system is quite large,The behavior of the system is described by averaged values of

reaction-rates.

REP/

- page 39 -

Reactor Kinetics

Point Kinetics Heuristic approach : Principle

A quite simple demography problem where, every generation-time L, the neutron population is multiplied by a factor

In a conventional PWR there are about 40 neutron generations per millisecond, i.e. 40 000 per second.

Time Neutron population0 N0L N0 * 2L N0 * *3L N0 * * * tkeff

tkeff

tkeff tkeff

tkeff tkeff

tkeff

REP/

- page 40 -

Reactor Kinetics

Point Kinetics Heuristic approach : Application

Keff = 1.00010 L= 25sNeutron generation per second = 1/25E-6 = 40 000

Time (s) Neutron population 0 N0 1 N0*E+40 000 = N0*55 2 N0*E+80 000 = N0*2980 3 N0*E+120 000 = N0*162 000

Simple but catastrophic scenario!

REP/

- page 41 -

Reactor Kinetics

Point Kinetics Heuristic approach : Application

Keff = 0.900 L= 25sNeutron generation per millisecond = 1E-3/25E-6 = 40

Time (ms) Neutron population 0 N0 1 N0*E+40 = N0*0.0150 2 N0*E+80 = N0*0.0002 3 N0*E+120 = N0*0.000003

Simple but catastrophic scenario!

REP/

- page 42 -

Reactor Kinetics

Point Kinetics Heuristic approach : Sub-critical system with

external source Gain amplifying factor

Time Neutron population L S 2L S(1+ ) 3L S(1+ + * ) 4L ………

tkeff1

1

tkeff

tkeff tkeff tkeff

tk

S

eff1

REP/

- page 43 -

Reactor Kinetics

Delayed neutronsStability of the nucleus :

The Electromagnetic field inside nucleus :Effect on protons,

The Nuclear Force field :Contribution of neutrons to nucleus stability,

The Fission process :Compound activated nucleus,Production of fission fragments (Fission Products)Neutron emission.

REP/

- page 44 -

Reactor Kinetics

Delayed neutronsDelayed neutron fraction per fission (UOX fuel) :

U235 0.65%U238 1.48%Pu239 0.21%

Delayed neutron emission time : Br87 -> Kr87 -> Kr86+n 80.6 sI137 -> Xe137 > Xe136+n 32.8 s

REP/

- page 45 -

Reactor Kinetics

Back to the Fission process

Incident neutron

Fission

Delayed neutrons Prompt neutrons

Bv (1-B)v

Diffusion &slowing-down

Delay >0.3 secDelay>03 sec.

REP/

- page 46 -

Reactor Kinetics

Point KineticsThermal feed-back : Power and temperature effect

(recall):Doppler broadening :

Fissile isotopes do not contribute significantly to Doppler effect,Fertile isotopes (mainly U238 and Pu 240) have major contribution to

the effect;Moderator effect :

When moderator changes, neutron spectrum is affected;Soluble boron poisoning effect :

When moderator density varies, amount of boron atoms per unit volume is modified.

REP/

- page 47 -

Residual Power

Time-dependenceAfter shut-down, power does not go immediately to

zero:The system undergoes a fast transient during which

power decrease is driven by decay of residual neutron precursors (Fission Products) [kinetics],

Afterwards, power goes-on decreasing very slowly [activity, residual power].

REP/

- page 48 -

Residual Power

Sources of activityRadioactive decay of:

Fission Products (B+y),

U239, Np239 and daughters (B+y),

Minor Actinides (a),

Other Activation Products (B+y),

Spontaneous Fission,

Induced neutron emission.

REP/

- page 49 -

Residual Power

Sources of activityIn order to explain origin of different contributions

to the activity, several items must analyzed :The fuel burn-up breeding process described by Heavy-

Isotope Depletion Chain,

The decay process of nuclei described in Base-Data Libraries.

REP/

- page 50 -

Residual Power

Sources of activityActivity is also due to the multiplication in sub-

critical conditions of the inherent neutron source :

Spontaneous Fission,Neutron emission by Oxygen 18 :

Actinide decay produces a particles,Free neutrons are generated by stripping by a particles on O18.

k

SCP

eff

1

REP/

- page 51 -

Residual Power

ComputationIn calculation, contributions to residual power are

packed into three groups :Term A includes contribution from residual Neutron

Source,

Term B includes contribution from decay of U239, Np239 and daughters,

Term C includes contribution from decay of :Fission Products and and Activation Products others than U239, Np239

and daughters,Minor Actinides.

REP/

- page 52 -

Residual Power

ComputationSeveral solutions can be adopted to account for

burn-up:Using fuel burn-up averaged values,

Maximize burn-up of the fuel via infinite irradiation.

Uncertainty Can be accounted for in different ways depending on

computational procedure and / or data - library adopted.

REP/

- page 53 -

Keyword Survey

Neutron Balance Equation

Steady state Conditions

Power and Reactivity PowerReactivityControl of PowerControl of Reactivity

REP/

- page 54 -

Keyword Survey

Reactivity NEEDSCriteria and MarginsFuel Burn-up and BreedingBurnable Poison Burn-outXenon and Samarium Build-upPower and Temperature Effect

Doppler BroadeningModerator Effect

Reactivity AVAILABILITIESSoluble BoronControl and Safety ClustersBurnable PoisonsExtractable Poisons

REP/

- page 55 -

Keyword Survey

Reactor KineticsNeutron Balance EquationLifecycle Lifetime and generation-timePoint Kinetics

Heuristic approachApplication

Delayed neutronsPoint kinetics

Heuristic approach,Thermal feed-back,

REP/

- page 56 -

Keyword Survey

Residual PowerTime dependence

Sources of activityRadioactive decay

Actinide depletion Fission Products

Sub-critical conditions

Computation