An Overview of Reactor Kinetics

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An Overview of Reactor Kinetics

Eleodor NichitaFaculty of Energy Systems and Nuclear Science

University of Ontario Institute of Technology

Outline

Introduction

Simple point Kinetics without Delayed Neutrons

Simple Point Kinetics With Delayed Neutrons

Solution of the PKE - Inhour Equation

Controlling the Power

Reactivity Effects

References and Further Reading

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Introduction

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What is reactor kinetics?

Study of time-dependent (transient) behaviour of nuclear reactors.

Steady State: The fission reaction rate (as well as other parameters) is constant over time.

Transient: The fission reaction rate (as well as other parameters) changes over time.

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If nuclear reactors were cars…

Statics Analyzes how a car works when it moves on the

highway in a perfectly straight line, at perfectly constant speed.

Kinetics Analyzes what happens when you start the car,

accelerate, slow down, stop, turn, drive off a cliff, hit a tree, etc.

Reality There is no such thing as moving in a perfectly

straight line at a perfectly constant speed.

There is no such thing as a perfect steady state.5

Reactor Safety Design

Operating Limit

Trip Limit

Safety Limit

Operating MarginSafety Margin

Operating Domain

Operating Trajectory Design Center

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Reactor Kinetics and Reactor Safety Reactor kinetics allows the study of reactor

operation and control, ensuring operating limits are not exceeded.

Accidents involve transients (usually power increase).

Reactor kinetics allows the study of accidents, their progression and termination.

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Time Scale of Time-Dependent Phenomena Short Time Phenomena (ms, s)

accidents

experiments

startup/shutdown

Medium Time Phenomena (hrs, days) fission product poisoning

Xe

Sm

Long Time Phenomena (months, years) fuel burnup with consequent change in

composition

kinetics

dynamics

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Transients

Without Feedback (approximation) Changes in flux level do not induce changes in

the absorption or production properties of the reactor.

With Feedback Changes in flux level do induce changes in the

absorption or production properties of the reactor.

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Simple Point Kinetics without Delayed Neutrons

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Kinetics is all about balance

productions losses

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Time-dependent neutron balance equation Productions

Fission (all fission neutrons assumed prompt)

Losses Absorption

Leakage

For a homogeneous reactor:

coregcoreacorefcore VDBVVVdt

nd 2

Leakage rateAbsorption rateProduction ratePopulation rate of change

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Notations

Total neutron population

Core-integrated flux

Connection between neutron population and core-integrated flux.

dVtrnVtntncoreV

core ),()()(ˆ

dVtrVttcoreV

core ),()()(ˆ

v)(ˆ)(ˆv)()( tnttnt 13

Neutron balance equation with new notations(all fission neutrons assumed prompt)

)(ˆ)(ˆ)(ˆ)(ˆ 2 tDBttdt

tndgaf

Leakage rateAbsorption rateProduction ratePopulation rate of change

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Kinetics parameters: generation time

Is not directly dependent on the flux level

Interpretations: Time necessary to generate the current number of

neutrons at the current generation rate.

Average “age” of neutrons in the reactor. (Note that this is a time, and not the Fermi age).

v

1

vˆ

ˆˆ

ˆ

rate production

populationneutron

fff n

nn

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Kinetics parameters: lifetime


Interpretations: Time necessary to lose all the neutrons in the

reactor at the current loss rate.

Average life expectancy of neutrons in the reactor.

v1

vˆ

ˆˆ

ˆ

rate loss

populationneutron 222 DBnDB

n

DB

n

aaa

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Generation time and lifetime

For a critical reactor, the lifetime is equal to the generation time.

For a subcritical reactor, the lifetime is shorter than the generation time.

For a supercritical reactor, the lifetime is longer than the generation time.

For an infinite reactor (no leakage):

eff

fa

f

a

kDBDB v

1

v

122

v

1

a

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Kinetics parameters: reactivity

Is a measure of the relative imbalance between productions and losses


efff

ga

f

gaf

f

gaf

k

DBDB

DB

11

rate production

rate loss11

ˆ

ˆˆˆ

rate production

rate loss - rate production

22

2

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Derivation of the Point Kinetics equation w/o delayed neutrons

)(ˆ)(ˆ)(ˆ)(ˆ 2 tDBttdt

tndgaf

nnDB

dt

ndf

f

gaf ˆˆvˆ 2

vˆvˆvˆˆ 2nDBnn

dt

ndgaf

)(ˆ)(ˆ

tndt

tnd

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An Interesting Perspective

vˆvˆvˆˆ 2nDBnn

dt

ndgaf

2ˆˆ ˆv vf a g

dnn DB n

dt

ˆ 1 1ˆ ˆ

dnn n

dt

ˆ 1 1ˆ

dnn

dt

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Point-kinetics without delayed neutrons (all fission neutrons assumed prompt)

Called point kinetics because spatial (as well as energy) dependence is ignored. The reactor is thus reduced to a “point”. Analogous to the concept of point mass.

The neutron population increases exponentially for positive reactivity and decreases exponentially for negative reactivity.

tentn

0ˆ)(ˆ

ndt

ndˆ

ˆ

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Comments on power

Power is given by:

Hence:

It follows that the power has the same time dependence as the neutron population

MeVnP f 200vˆ

nP ˆ

tePtP

0)(

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Simple Point Kinetics with Delayed Neutrons

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Delayed Neutrons

In reality, not all neutrons are born prompt. A small fraction (called the delayed neutron fraction) are born indirectly through the decay of fission products called precursors.

There is exactly one precursor for each delayed neutron.

For each fission, we there are neutrons emitted promptly and precursors created, to emit neutrons later, according to the radioactive decay law, that is delayed neutrons per second. is the precursor population.

)1(

CC

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Point Kinetics Equations (PKE) with One Group of Delayed Neutrons

Neutron balance:

Precursor balance:

CDBdt

ndaf

ˆˆˆˆ1ˆ 2

Cdt

Cdf

ˆˆˆ

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Point Kinetics Equations with One Group of Delayed Neutrons Processing of the neutron balance equation:

Cndt

nd

Cnndt

nd

Cnndt

nd

CDBdt

nd

CDBdt

nd

f

faf

af

ˆˆˆ

ˆˆˆˆ

ˆˆvˆˆ

ˆˆˆˆˆˆ

ˆˆˆˆ)1(ˆ

2

2

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Point Kinetics Equations with One Group of Delayed Neutrons Processing of the precursor balance equation:

Point kinetics equations:

Cndt

Cd

Cndt

Cd

Cdt

Cd

f

f

ˆˆˆ

ˆˆvˆ

ˆˆˆ

Cndt

nd ˆˆˆ

Cndt

Cd ˆˆˆ

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Solution of PKE with One Delayed-Neutron Group The solution is a combination of two

exponentials: one fast-varying and one slow-varying.

The reciprocal of the larger (in an algebraic sense) exponent is called the reactor period.

reactivity theassign same;0;

)()(

2121

2121

tttt BeAetPbeaetn

2

1

T

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Effect of Delayed Neutrons on Transients (slower transients)

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Delayed-neutron groups

Different precursors have different half-lives (decay constants).

Precursors with similar half-lives are grouped together into “delayed-neutron groups”.

Delayed neutron fractions, , are defined for each group, k. They add up to the total delayed neutron fraction:

6 delayed-neutron groups are customary.

In heavy-water reactors, photo-neutrons are also important.

There are 11 photon-neutron groups.

K

kk

1

nHH 11

21

k

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PKE with multiple delayed neutron groups

)(ˆ)(ˆ)(ˆ max

1

tCtndt

tnd k

kkk

max...1)(ˆ)(ˆ)(ˆ

kktCtndt

tCdkk

kk

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Kinetics parameters values

0.0001 0.001s s 0.003 0.006

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PKE in the presence of an external source independent of the flux

General Equations

Steady-state (no time dependence)

01

)(ˆ)(ˆ)(ˆ max

StCtndt

tnd k

kkk

max...1)(ˆ)(ˆ)(ˆ

kktCtndt

tCdkk

kk

01

max

ˆˆ0 SCnk

kstationarykkstationary

max...1ˆˆ0 kkCn stationarykkstationaryk

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Solution of steady-state PKE in the presence of an external source independent of the flux

Substitution of the second equation into the first:

Neutron population:

solution exists only for negative reactivity

001

ˆ0ˆ0 SnSnn stationarystationary

K

k

kstationary

0ˆ Snstationary

01

ˆˆ0 SCnK

kstationarykkstationary

max...1ˆˆ0 kkCn stationarykkstationaryk

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Reactor Startup – Approach to Critical

Photo-neutrons act as external source, independent of flux.

The reactivity can be calculated from flux measurements (The flux is proportional to the neutron population.)

is plotted as a function of “poison” concentration or moderator height. Intersection with x axis shows when criticality is expected.

measuredstationarystationary n

SSn

1

ˆ1

ˆ 00

measured

1

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Solution of the PKE - Inhour Equation (case of 6 delayed-neutron groups)

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PKE System of Equations

The reactivity, as well as other parameters, can vary with time but will consider the case when all parameters are constant.

PKE – System of seven differential equations with constant coefficients.

)(ˆ)(ˆ)(ˆ 6

1

tCtndt

tnd

kkk

6...1)(ˆ)(ˆ)(ˆ

ktCtndt

tCdkk

kk

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General solution – Linear combination of Seven Fundamental Solutions

Fundamental solution

t

t

t

t

e

c

c

n

ec

ec

ne

6

1

6

1

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Substitution of Fundamental Solution Initial substitution

Expressing ck from the second eq:

Substituting back into the first eq.:

6

1kkkcnn

6...1,

kcnc kkk

6...1,

knck

kk

6

1k k

kk nnn

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Inhour Equation

Start with:

Express the reactivity:

Inhour Equation

6

1k k

kk nnn

6

1k k

kk

6

1k k

kk

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Graphical Representation of the Inhour Equation

Reactor period:

12 3 4 5 6

max

1

T

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Solutions of the Inhour Equation

For large (positive or negative) values of we obtain the asymptotic behaviour:

so the slope is very small

Because of the very small slope, there is one omega that is much smaller than all the others:

k k

kk

s001.0

max1234567min

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Solutions of the Inhour Equation

All solutions, except for the first one (the largest one in an algebraic sense) are negative.

The largest solution has the same sign as the reactivity.

The reciprocal of the largest solution is called the reactor period.

1max

11

T

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Importance of the Reactor Period

After a long period of time, the neutron population (as well as power) varies as a single exponential:

Proof:

T

t

entn 01ˆ)(ˆ

T

tt

ti

ti

t

i

ti

t

i

ti

enenenen

eneentn

i

ii

0101 largefor

7

2001

7

10

7

10

ˆˆˆˆ

ˆˆ)(ˆ

111

11

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Power is controlled by controlling the reactivity

Reactivity is controlled using Reactivity Devices consisting of neutron absorbers.

By inserting or removing reactivity devices from the core, the absorption rate is varied, hence the reactivity can be varied and the power can be increased, decreased, or the reactor can be completely shut down.

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CANDU Reactivity Devices

Reactor Regulating System (RRS): 14 liquid-zone-control compartments (H2O

filled)

21 adjuster rods

4 mechanical control absorbers

moderator “poison” (poison = element with very large neutron capture cross section, and zero fission cross section)

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CANDU Reactivity Devices

Shutdown systems (SDS): SDS-1, consisting of 28 cadmium shutoff rods

which fall into the core from above

SDS-2, consisting of high-pressure poison injection into the moderator through 6 horizontally oriented nozzles.

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Reactivity Device Worth

Reactivity worth is the difference in reactivity between the core w/o the device and the core w/ the device

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CANDU Shutdown Systems

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Reactivity Effects

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Reactivity Effects

Macroscopic cross sections can change as a consequence of different parameters and, in turn, induce a change in keff and hence in reactivity.

The usual parameters that influence the reactivity are: Fuel Temperature Coolant Temperature Moderator Temperature Coolant Density

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Definition of Reactivity Coefficients

Consider we keep all the reactor parameters constant, with the exception of one, say the fuel temperature.

This is not always possible, as a variation in fuel temperature will induce a variation in coolant temperature, but let us assume we can do it.

Consider we plot the reactivity as a function of the varying parameter (in our case, the fuel temperature).

We can also plot the reactivity change where Tf0 is the reference fuel temperature. This is called the reactivity effect of fuel temperature. We can also calculate and plot

This is called the reactivity coefficient of the fuel temperature.

)()()( 0fff TTT

f

fT dT

Tdf

)(

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Mathematical Expressions of Reactivity Coefficients Let p be the parameter that is being varied,

all others being kept constant.

We define the reactivity coefficient of parameter P as:

Equivalent definition:

dp

pdP

)(

dp

pdk

pkpkdp

d

dp

pdP

)(

)(

1

)(

11

)(2

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CANDU Fuel Temperature Effect

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CANDU Coolant Temperature Effect

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CANDU Moderator Temperature Effect

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CANDU Coolant Density Effect

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Kinetics Codes

Treat as many as 17 delayed neutron groups.

Treat space-dependence of flux as well.

Code examples: CERBERUS (IQS)

SMOKIN (modal expansion)

Module in DONJON (IQS)

NESTLE (direct solution)

others

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References and further reading

Rozon, D., 1998. Nuclear Reactor Kinetics. Polytechnic International Press, Montreal, QC, Canada

Ott, K.O., Neuhold, R.J., 1985. Nuclear Reactor Dynamics. Am. Nucl. Soc., Lagrange Park, IL.

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Documents

An Overview of Reactor Kinetics