an overview of reactor kinetics
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17/11/2014
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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
Is not directly dependent on the flux level
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
Is not directly dependent on the flux level
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
1234567 42
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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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Controlling the Power
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Controlling the Power
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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