Revision 2
March 2015
Reactor Kinetics
Student Guide
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ii
Table of Contents
INTRODUCTION .............................................................................................................. 1
TLO 1 THEORY OF SUBCRITICAL MULTIPLICATION ...................................................... 2
Overview .................................................................................................................. 2
ELO 1.1 Subcritical Multiplication .......................................................................... 3
ELO 1.2 Subcritical Multiplication Response to keff Changes ................................. 9
ELO 1.3 Delayed Neutrons .................................................................................... 13
ELO 1.4 Subcritical Reactor Reactivity Rules of Thumb ...................................... 19
TLO 1 Summary ..................................................................................................... 21
TLO 2 REACTOR PERIOD AND STARTUP RATE ............................................................ 24
Overview ................................................................................................................ 24
ELO 2.1 Reactor Period and Startup Equations ..................................................... 25
ELO 2.2 Startup Rate and Reactor Period Calculations ......................................... 29
ELO 2.3 Effects of Reactor Trip and Step Insertion of Reactivity ........................ 33
TLO 2 Summary ..................................................................................................... 36
REACTOR KINETICS MODULE SUMMARY .................................................................... 39
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Reactor Kinetics
Revision History
Revision
Date
Version
Number
Purpose for Revision
Performed
By
10/31/2014
0
New Module
OGF Team
12/11/2014
1
Added signature of OGF
Working Group Chair
OGF Team
3/11/2015
2

Corrected PPT slide 36
and IG/SG page 13 to
label list of nuclides as
“fissile or fissionable.”

Aligned PPT slide 47 to
have correct values for
effective decay constant
that are identified in the
IG/SG.

Aligned references of the
power equation to IG/SG
on PPT slide 79, 99, and
100.

Added units to answer
on PPT slide 79.

Corrected misspelling of
reactor on PPT slide 99.
OGF Team
Introduction
This module includes information on reactor kinetics. The lesson covers
neutron sources, subcritical multiplication, importance of delayed neutrons,
the use of 1/M plots, reactor power changes, and startup rate problems.
Rev 1
1
The reactor operator's primary task and legal responsibility is to operate the
reactor within established operating limits to ensure public health and
safety. This module will provide the operator with a basic theoretical
understanding and knowledge of reactor operations.
Objectives
At the completion of this training session, the trainee will demonstrate
mastery of this topic by passing a written exam with a grade of 80 percent
or higher on the following terminal objectives (TLOs):
1. Describe subcritical multiplication for a nuclear reactor and describe
how subcritical multiplication affects reactor operation.
2. Explain the factors that affect reactor period and startup rate and their
effect on reactor control.
TLO 1 Theory of Subcritical Multiplication
Overview
This chapter covers the theory and use of subcritical multiplication for
prediction of critical rod positions, during a reactor startup. The 1/M plots
are explained, with an opportunity for practice. Subcritical multiplication
and associated rules of thumb are important for understanding and
interpreting reactor response during startup. The following enabling
learning objectives (ELOs) are covered in this lesson:
1. Explain the following:
a. Source neutrons
b. Subcritical multiplication
c. Subcritical multiplication factor
d. Subcritical multiplication response on a nuclear reactor startup.
2. For a subcritical reactor, calculate steady state neutron levels for
various values of keff and reactivity additions.
3. Describe the relationship between the delayed neutron fraction,
average delayed neutron fraction, and average effective delayed
neutron fraction.
4. State the rules of thumb for changing neutron count rate during a
reactor startup.
2
Rev 1
ELO 1.1 Subcritical Multiplication
Subcritical multiplication is the phenomenon that accounts for the changes
in neutron flux that take place in a subcritical reactor, due to reactivity
changes. It is important to understand subcritical multiplication, to
understand reactor response to changes in conditions.
Subcritical Multiplication and Source Neutrons
Neutrons from a variety of sources are always present in a reactor core,
even when the reactor is shut down. Some neutrons are produced by
naturally occurring or intrinsic neutron sources, while other neutrons may
be the result of fabricated or installed neutron sources that are incorporated
into the reactor’s design.
The neutrons produced by sources other than neutron-induced fission are
grouped together and classified as source neutrons. Source neutrons are
important because they help initiate the fission process during a nuclear
reactor startup.
Source neutrons ensure that the neutron population remains high enough to
allow a visible indication of neutron level on the most sensitive monitoring
instruments while the reactor is shut down and during the startup sequence
for source range nuclear instruments. This verifies instrument operability
and allows monitoring of neutron population changes. Source neutrons can
be classified as either intrinsic or installed neutron sources.
Intrinsic Neutron Sources
Some neutrons will be produced in the materials present in the reactor due
to a variety of reactions that occur because of the nature of these materials.
Intrinsic neutron sources are those neutron-producing reactions that always
occur in reactor materials. The most important types of intrinsic neutron
reactions for nuclear reactors include the following:



Spontaneous fission
Photo-neutron reactions
Alpha-neutron reactions
Installed Neutron Sources
Because intrinsic neutron sources can be relatively weak or dependent upon
the recent power history of the reactor, many reactors have artificial sources
of neutrons installed. These neutron sources ensure that shutdown neutron
levels are high enough to be detected by the nuclear instruments at all times.
This provides a true picture of reactor conditions and any change in these
conditions. An installed neutron source is an assembly placed in or near the
reactor for the sole purpose of producing source neutrons.
Rev 1
3
Californium-252
One strong source of neutrons is the artificial nuclide californium-252,
which emits neutrons at the rate of about 2 x 1012 neutrons per second (sec)
per gram as the result of spontaneous fission. Californium-252 is not
widely used as an installed neutron source in commercial nuclear reactors
due to its high cost and its short half-life of 2.65 years.
Beryllium Sources
Alpha-Neutron Beryllium Source
Many installed neutron sources use the alpha-neutron reaction with
beryllium (Be). These sources are composed of a mixture of metallic
beryllium (100 percent beryllium-9) with a small quantity of an alpha (α)
particle emitter, such as a compound of radium, polonium, or plutonium.
The reaction that occurs is presented below.
9
4𝐵𝑒
+ 42𝛼 → ( 136𝐶 )∗ →
12
6𝐶
+ 10𝑛
The beryllium is intimately (homogeneously) mixed with the alpha emitter
and is usually enclosed in a stainless steel capsule.
Photo-Neutron Beryllium Source
Another type of installed neutron source that is widely used is a photoneutron source that employs the photo-neutron reaction with beryllium.
Beryllium is used for photo-neutron sources because its stable isotope
beryllium-9 has a weakly attached last neutron with a binding energy of
only 1.66 MeV. Thus, a gamma ray with greater energy than 1.66 MeV can
cause neutrons to be ejected by the photo-neutron reaction as shown below.
𝛾 + 49𝐵𝑒 → 84𝐵𝑒 + 10𝑛
The most common installed source is antimony-beryllium (Sb-Be). Many
startup sources use antimony and beryllium because after activation with
neutrons the radioactive antimony becomes an emitter of high-energy
gammas, as shown in the reactions below. The activated antinomy also
decays with a 60-day half-life to produce a gamma ray of sufficient energy
to interact with the beryllium (as shown in the above reaction) to produce a
neutron.
123
51𝑆𝑏
124
51𝑆𝑏
+ 10𝑛 →
𝛽−
→
124
51𝑆𝑏
124
52𝑇𝑒
+𝛾
+ −10𝑒 + 𝛾
The Sb-Be photo-neutron sources of this type are constructed somewhat
differently from the alpha-neutron types described above. One design
incorporates a capsule of irradiated antimony enclosed in a beryllium sleeve
4
Rev 1
then the entire assembly is encased in a stainless steel cladding. A large
reactor may have several neutron sources of this type installed within the
core.
Subcritical Multiplication
Fission will occur even in a shutdown reactor with the help of these source
neutrons and available fissionable fuel. Subcritical multiplication is the
process where source neutrons add to the neutrons available in each
generation to sustain the chain reactions when the multiplication factor (keff)
is less than one.
In a subcritical reactor the chain reaction is not self-sustaining because
neutron production is less than absorption plus leakage. However, keff will
be maintained at some constant value less than 1.0 due to the addition of
source neutrons to make up for the insufficient production of neutrons in
that generation.
The neutron count rate will increase as the reactor approaches criticality as
demonstrated during reactor startups. As criticality is approached, keff
approaches a value of 1.0, which means that subcritical multiplication can
be related to the value of keff.
The following equation demonstrates this mathematically:
𝐶𝑅 = 𝑆𝑜
1
𝜂
1 − 𝑘𝑒𝑓𝑓
Where:




CR = neutron count rate in counts per second (cps) [source
range nuclear instrumentation]
𝑆o = source strength in cps
k𝑒𝑓𝑓 = effective neutron multiplication factor
𝜂 = detector efficiency
Exercise 1:
Assuming 𝜂 = 0.1, 𝑆o = 100 cps, and keff = 0.5, the CR would = __ cps
If keff is now increased to 0.75 with no other changes, then the CR would =
__ cps
Solution 1:
𝐶𝑅 = 100
Rev 1
1
0.1
1 − 0.5
5
𝐶𝑅 = 20
𝐶𝑅 = 100
1
0.1
1 − 0.75
𝐶𝑅 = 40
Exercise 2:
What if 𝑆o = 0?
Solution 2:
Mathematically the CR would drop to zero (0). This shows the importance
of source neutrons with a subcritical reactor to provide for monitoring of the
reactor's shutdown status.
When changing keff from 0.5 to 0.75 we have gone halfway to criticality (keff
= 1.0) and the indicated count rate has doubled as shown in the examples
above. The reactor would be critical if the same amount of reactivity is
added again. The reactor would actually be slightly supercritical, which
will be covered in another section of this module.
Subcritical Multiplication Factor (M)
The indicated count rate in the subcritical (source) range, by itself, is not a
good representation of neutron activity in the reactor. Count rate
comparisons (or ratios) are more useful in gauging the reactor's response to
reactivity changes and the approach to criticality during reactor startups.
Count rate ratio is a comparison of two count rates (final count rate divided
by initial count rate) and can be expressed as:
𝐶𝑅2 1 − 𝑘𝑒𝑓𝑓1
=
𝐶𝑅1 1 − 𝑘𝑒𝑓𝑓2
Where:


CR1 = count rate at reference time
CR2 = count rate at some time later after making a change to keff by
the addition positive or negative reactivity.
Going back to the example using keff values of 0.5 and 0.75 we would
obtain a count rate ratio of:
6
Rev 1
𝑀=
𝐶𝑅2
𝐶𝑅1
𝐶𝑅2
1 − 0.5
=
𝐶𝑅1 1 − 0.75
𝐶𝑅2
0.5
=
=2
𝐶𝑅1 0.25
Using the keff values of 0.5 and 0.75 we can see that the count rate ratio is
equal to 2. This ratio of CR2/CR1 is also known as the subcritical
multiplication factor (M) which is the fractional change in neutron
population of a subcritical reactor due to the changes in core reactivity.
Subcritical multiplication factor is also expressed using the following count
rate ratios:
𝑀=
𝐶𝑅𝑛
𝐶𝑅𝑜
Where:


CRn = some count rate at a condition n in cps
CRo = initial count rate in cps
Or relating to source neutrons alone, the following relationship exists:
𝑀=
𝐶𝑅𝑛
1
=
𝐶𝑅3 1 − 𝑘𝑒𝑓𝑓
Where:


CRn = some count rate at a condition n
CRs = initial count rate due to source counts alone
This formula shows that for a given value of keff, there is a subcritical
multiplication factor value that relates the level of source neutrons to a
current steady state level. This formula can be used to determine source
strength if the value of keff is known.
To relate the subcritical multiplication factor to reactivity (ρ), recall the
relationship of keff to ρ:
𝑘𝑒𝑓𝑓 =
Rev 1
1
1−𝜌
7
Substituting this into the count rate ratio equation yields:
𝐶𝑅2 𝜌1 (1 − 𝜌2 )
=
𝐶𝑅1 𝜌2 (1 − 𝜌1 )
If keff is approximately 1.0 and 1-ρ ≈ 1.0, then the equation above can be
approximated by:
𝐶𝑅2 𝜌1
≈
𝐶𝑅1 𝜌2
So, subcritical multiplication factor in terms of reactivity changes:
𝑀=
𝐶𝑅𝑛
𝐶𝑅𝑜
Where:


CRn = some count rate at a condition n in cps
CRo = initial count rate in cps
Subcritical Multiplication Factor Response During Nuclear
Reactor Startup
As control rods are withdrawn, count rate comparisons yield everincreasing values of M as the reactor approaches criticality. Accurately
plotting the expected point of criticality using M values is difficult because
the count rate ratio (M) could increase from approximately one (1) to
several million (or infinity) prior to reaching criticality, making the exact
point of expected criticality difficult to identify.
Instead, the inverse of M or 1/M is used because as keff approaches
criticality (1.0), 1/M approaches zero (0), a much easier identified value for
predicting criticality, which will be discussed later in more detail.
Knowledge Check
A reactor startup is in progress. The initial count rate
was 120 cps. After the first rod pull, the count rate
changed to 150 cps. On the fifth rod pull, the count rate
changed to 3,000 cps. Assuming the initial Keff was 0.9
what is the Keff after the fifth rod pull?
8
Rev 1
A.
0.995
B.
0.996
C.
0.92
D.
0.9996
Knowledge Check
What is keff in a reactor core if it would take 500 PCM of
rod worth to make the reactor critical?
A.
1.005
B.
0.95
C.
0.995
D.
Can’t be calculated from given data
ELO 1.2 Subcritical Multiplication Response to keff Changes
Introduction
The neutron level increases as the control rods are withdrawn when the
reactor operator performs a reactor startup. The count rate levels out at a
new higher value each time the control rods are stopped. This is caused by
subcritical multiplication. This section discusses the effect of subcritical
multiplication on neutron population as keff approaches one.
Definition
The following formula is used to determine count rate changes from
changes in keff:
𝐶𝑅2 1 − 𝑘𝑒𝑓𝑓1
=
𝐶𝑅1 1 − 𝑘𝑒𝑓𝑓2
Where:


CR1 = count rate at reference time
CR2 = count rate at some time later after making a change to keff by
the addition of positive or negative reactivity.
Rev 1
9
Keeping in mind that keff = 1.0 in a critical reactor, what if keff changes from
0.95 to 0.96 (approximately 1,100 per cent mille [pcm]), with an initial
count rate of 1,000 cps?
Substituting values for keff:
𝐶𝑅2 1,000 𝑐𝑝𝑠 =
(1 − 0.95)
(1 − 0.96)
𝐶𝑅2 = 1,250 𝑐𝑝𝑠
Neutron count rate increased by 250 cps.
What happens if keff is changed from 0.989 to 0.999, with the same initial
count rate of 1,000 cps (approximately 1,010 pcm)?
𝐶𝑅2
1 − 0.989
=
1,000 1 − 0.999
𝐶𝑅2 = 11,000
This time, the neutron count rate increased by 10,000 cps with the same
change in keff yielding slightly less reactivity.
Using the subcritical multiplication factor (M) formula, ratio of initial
counts to new counts, demonstrates a similar result:
𝑀=
𝐶𝑅2
𝐶𝑅1


In the first example of changing keff from 0.95 to 0.96, M = 1.25
In the second example of changing keff from 0.989 to 0.999, M = 11
The result is a much larger change in the multiplication factor as keff
approaches 1.0.
Example 1
The subcritical multiplication factor (M) and neutron count rate increase is
greater with progressively smaller changes in keff or reactivity, as criticality
approaches. The following table illustrates this behavior, as keff increases
toward 1.0
keff
Multiplication
Factor (M)
Number of
Doublings
Count Rate
0.99
N/A
N/A
100
0.995
2
1
200
10
Rev 1
keff
Multiplication
Factor (M)
Number of
Doublings
Count Rate
0.9975
4
2
400
0.99875
8
3
800
0.999375
16
4
1,600
0.999687
32
5
3,200
0.99984375
64
6
6,400
0.9999218750
128
7
12,800
0.9999609400
256
8
25,600
Example 2
The number of generations required for the neutron level to reach
equilibrium increases as the reactor approaches a keff of 1.0. Since neutron
generation time does not change and more generations are required for
equilibrium, more time is required for the neutron levels to stabilize
(between control rod pulls) when approaching criticality on a reactor
startup. This is why reactor startups are performed in a controlled and
deliberate manner to ensure 1/M plots can be accurately plotted to predict
criticality.
The following trace of control rod withdrawals versus neutron count rate
(for approximately equal reactivity additions) demonstrates this in the
below figure:
Rev 1
11
Figure Startup Trace Shows Time Versus Count Rate Increase
As keff (shown as the increasing count rate), comes closer to 1.0, the count
rate increases significantly more per rod pull. Since count rate changes are
larger, it also takes longer to reach equilibrium.
Knowledge Check
During a nuclear reactor startup, the operator adds 1.0
percent Δk/k of positive reactivity by withdrawing
control rods, thereby increasing equilibrium source range
neutron level from 220 cps to 440 cps. To raise
equilibrium source range neutron level to 880 cps, an
additional ______________ of positive reactivity must
be added.
12
A.
0.5 percent Δk/k
B.
4.0 percent Δk/k
C.
1.0 percent Δk/k
D.
2.0 percent Δk/k
Rev 1
Knowledge Check
A subcritical reactor has a keff of 0.85 with a stable count
rate of 200 counts per second (cps). If positive reactivity
is added to bring keff to 0.975, at what value will the new
count rate stabilize?
A.
1,000 cps
B.
400 cps
C.
800 cps
D.
1,200 cps
ELO 1.3 Delayed Neutrons
Introduction
Delayed neutrons play an important role in the control and stability of
commercial nuclear reactors. Delayed neutrons increase neutron generation
times, resulting in more controlled power increases with reactivity
additions.
Definition
Delayed neutrons are neutrons produced by the beta decay of fission
product daughters and not directly from the fission event. Delayed neutrons
are born from a few milliseconds to a few minutes after the initiating fission
event.
The delayed neutron fraction (β) is the ratio of delayed neutrons to all
neutrons born (prompt and delayed) for a fuel isotope.
The delayed neutron fraction varies depending on the fissile nuclide,
fissionable nuclide, or mixture of nuclides in use. The delayed neutron
fractions (β) for the fissile and fissionable nuclides of most interest are as
follows:




Uranium-233 β = 0.0026
Uranium-235 β = 0.0065
Uranium-238 β = 0.0148
Plutonium-239 β = 0.0021
For comparison, the majority (≈ 99.36 percent) of the neutrons produced are
prompt, with the remaining (0.64 percent) being delayed.
Rev 1
13
Average Delayed Neutron Fraction (Beta-Bar)
The average delayed neuron fraction is the weighted average of all delayed
neutron fractions of the fuel mixture in the reactor. Each total delayed
neutron fraction value for each fuel isotope is weighted by the percent of
total neutrons that the fuel contributes via fission.
As the fuel mixture changes over core life, such as plutonium-239
production from uranium-238, the average delayed neutron fraction also
changes. For the two most significant fuel isotopes, 𝛽̅ for plutonium-239 is
significantly smaller than 𝛽̅ for uranium-235.
Effective Delayed Neutron Fraction (βeff)
The effective delayed neutron fraction is the fraction of all thermal neutrons
that were born delayed. Delayed and prompt neutrons differ in their
contribution to the fission process. Delayed neutrons are born at lower
energies than prompt neutrons (~ 0.5 mega electron volts [MeV], compared
to 2 MeV).
Delayed neutron average energy is less than the minimum required for fast
fission to occur and therefore delayed neutrons have a lower probability
than prompt neutrons of causing fast fissions. This shows as a decrease in
the fast fission factor.
Since delayed neutrons are born at lower energies, delayed neutrons have a
lower probability of leaking out of the core and do not travel as far to
thermalize, which increases the fast non-leakage factor.
The importance factor (I) relates the average delayed neutron fraction to the
effective delayed neutron fraction. The effective delayed neutron fraction is
the product of the average delayed neutron fraction and the importance
factor.
In a large reactor with low enriched fuel, the decrease in the fast fission
factor dominates the increase in the fast non-leakage probability, thus the
importance factor will be less than one, actually about 0.97 for a
commercial pressurized water reactor (PWR), which does not change over
the core’s life.
The effective delayed neutron fraction is the fraction of neutrons at thermal
energies that were born delayed.
Effect On Reactor Period
A neutron generation time of 10-5 to 10-4 seconds for prompt neutrons only
results in very rapid power excursions. Reactor control is not possible
without the delayed neutrons to slow the reaction rate. The average neutron
generation time, and therefore the rate of power increase, is determined
14
Rev 1
largely by the delayed neutron generation time. The following equation
shows this mathematically.
𝑇𝑖𝑚𝑒𝑎𝑣𝑒𝑟𝑎𝑔𝑒 = 𝑇𝑖𝑚𝑒𝑝𝑟𝑜𝑚𝑝𝑡 (1 − 𝛽) + 𝑇𝑖𝑚𝑒𝑑𝑒𝑙𝑎𝑦𝑒𝑑 (𝛽)
Example:
Assume a prompt neutron generation time for a particular reactor of 1 x 10-4
seconds, and a delayed neutron generation time of 12.7 seconds. If β is
0.0065, then calculate the average generation time.
Solution:
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑇𝑖𝑚𝑒
= 𝑇𝑖𝑚𝑒𝑝𝑟𝑜𝑚𝑝𝑡 (1 − 𝛽) + 𝑇𝑖𝑚𝑒𝑑𝑒𝑙𝑎𝑦𝑒𝑑 (𝛽)
= (1 × 10−4 𝑠𝑒𝑐𝑜𝑛𝑑𝑠)(0.9935) + (12.7 𝑠𝑒𝑐𝑜𝑛𝑑𝑠)(0.0065)
= 0.0827 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
This demonstrates the affect delayed neutrons have on neutron generation
time and thus on reactor control.
If a reactor operated using only prompt neutrons, the generation time from
the previous example would be about 1 x 10-4 seconds. However, by
operating the reactor such that a 0.0065 fraction of neutrons are delayed, the
generation lifetime is extended to 0.0827 seconds. While this is still fast, it
does provide time for adequate operator control.
Delayed neutrons are extremely important in the control of a sustained
fission chain reaction even though they make up only a small fraction of the
total neutron population. Therefore, delayed neutrons provide a necessary
safety factor in operating power reactors.
Another way to look at the effect of delayed neutrons on reactor control is
to look at the reactor period equation:
𝜏=
𝛽𝑒𝑓𝑓 − 𝜌
ℓ∗
+
𝜌 − 𝛽𝑒𝑓𝑓 λ𝑒𝑓𝑓 𝜌 + 𝜌̇
Where:
ℓ*= prompt neutron generation time (≈ 10-5 seconds)
βeff = effective delayed neutron fraction
λ eff = effective delayed neutron precursor decay constant
ρ = reactivity
Rev 1
15
𝜌̇ = rate of change of reactivity (example control rod movement)
The first term considers prompt neutrons and their generation time. If the
reactivity (ρ) added is less than βeff this first term becomes a negative value
negating the prompt neutron's effect to reactor period.
Example
𝜏=
ℓ∗
𝜌 − 𝛽𝑒𝑓𝑓
𝜏=
0.00001
0.001 − 0.0060
𝜏 = −0.002
With the same amount of reactivity added the second term results in:
𝜏=
𝛽𝑒𝑓𝑓 − 𝜌
λ𝑒𝑓𝑓 𝜌 + 𝜌̇
Assume reactive change = 0 and λeff = .1
𝜏=
0.0060 − 0.001
(0.1)(0.001) + 0
𝜏 = 50
This is equal to a startup rate of about half of decades per minute (dpm), an
acceptable value.
If the positive reactivity input exceeds βeff, then the prompt neutrons have a
larger effect on reactor period as well as on generation time.
Example
𝜏=
ℓ∗
𝜌 − 𝛽𝑒𝑓𝑓
𝜏=
0.00001
0.01 − 0.006
𝜏 = 0.0025
This represents a startup rate in excess of 15,000 dpm.
This is a condition known as prompt criticality, or one of
attaining criticality on prompt neutrons alone, which is
not allowed in a commercial power reactor.
16
Rev 1
Note
Note
Important Concept — If the reactivity added is smaller
in magnitude than the average effective delayed neutron
̅ ), power can increase no faster than the rate
fraction (𝛽𝑒𝑓𝑓
of the delayed neutron population’s increase.
Effective Delayed Neutron Precursor Decay Constant (λeff)
The effective delayed neutron precursor decay constant, which is the
inverse of the average delayed neutron generation time for example, λeff =
1/ld is a function of the relative fractions of short-lived and long-lived
delayed neutron precursors. When a reactor is operating at a constant
power, all delayed-neutron-precursor concentrations attain equilibrium
values, such that λeff = ~ 0.08 sec-1.
During an up-power transient, however, more short-lived precursors are
born at the higher power level than are longer-lived precursors. Compared
to steady-state conditions, proportionally more of the shorter-lived lived
precursors are decaying. In this case, the shorter-lived precursors are given
more weight by λeff and its value increases because the average delayed
neutron generation time increases.
During a down-power transient, the longer-lived precursors become more
significant, as the short-lived precursors rapidly decrease to the lowerpower condition. The longer-lived precursors are more heavily weighted by
λeff and its value decreases, due to the increase in the average delayed
neutron generation time.
Values Typically Used for λeff
 Steady state = ~ 0.08 sec1
 Power increases = ~ 0.1 sec-1
 Power decreases = ~ 0.05 sec-1
Delayed Neutron Impacts Control of Reactor Power Over Core
Life
Thermal reactors convert a substantial amount of uranium-238 into
plutonium-239, during the fission process, and the power contribution from
the fission of plutonium-239 near the end of core life is significant.
The delayed neutron fraction (β) for uranium-235 is 0.0064; for plutonium239, it is 0.0021. As core age and plutonium-239 concentration increase,
the effective delayed neutron fraction (βeff) for the overall fuel decreases
(typically, from ~ 0.007 to ~ 0.0054). The amount of reactivity insertion
Rev 1
17
needed to produce a given reactor period decreases with the decreasing
value of βeff.
Note
Important Concept — Over the core’s life, βeff
decreases; therefore, for a given amount of added
reactivity, the reactor period decreased and the startup
reactor period (SUR) increases.
Prompt Drop
The prompt neutron population is immediately affected on a reactor trip or
large negative-reactivity insertion, which results in a rapid decrease in the
prompt-neutron population.
Prompt Jump
The term prompt jump describes the reactor's immediate or prompt response
to a positive-reactivity addition.
Prompt Criticality
A reactor is considered prompt critical if it attains criticality via prompt
neutrons alone without any contribution from delayed neutrons. Prompt
critical is not a safe condition, and occurs when the reactivity added in units
of Δk/k equals or exceeds the magnitude of the average effective delayed
̅ ).
neutron fraction (when 𝜌 > 𝛽𝑒𝑓𝑓
Knowledge Check – NRC Exam Bank
Over core life, plutonium isotopes are produced with
delayed neutron fractions that are ______________ than
uranium delayed neutron fractions, thereby causing
reactor power transients to be ______________ near the
end of core life.
A.
larger; slower
B.
larger; faster
C.
smaller; slower
D.
smaller; faster
Knowledge Check – NRC Exam Bank
Following a reactor trip, when does the startup rate
initially stabilize at –1/3 dpm?
18
Rev 1
A.
When decay gamma heating starts adding negative
reactivity
B.
When the long-lived delayed neutron precursors have
decayed away
C.
When the installed neutron source contribution to the
total neutron flux becomes significant
D.
When the short-lived delayed neutron precursors have
decayed away
ELO 1.4 Subcritical Reactor Reactivity Rules of Thumb
Introduction
For shutdown reactors rules of thumb are available for changes in reactivity,
count rate, and keff. These same rules of thumb may be useful during
reactor startups to verify reactor response to reactivity additions.
Detailed Rules of Thumb
A rule of thumb is a principle with broad application, not intended to be
perfectly accurate or 100 percent reliable. Rules of thumb are normally
easy to remember and can be applied for approximating outcomes that do
not always need precise solutions. They are appropriate for use when quick
analysis of a situation is required.
1. Doubling the count rate (subcritical reactor) implies that enough
reactivity was added to take the reactor half way to criticality (keff is
halfway to 1.0).
2. If enough reactivity was added to double count rate and the same
amount of reactivity is added to the reactor again, the reactor will be
approximately critical (actually, slightly supercritical).
3. Using the thumb rule that states with each doubling the distance to
criticality is halved; the count rate doubles again if reactivity equal to
half the original amount is added.
4. Many plants consider that with five (5) to seven (7) count rate
doublings, the reactor should be critical.
Although some plants use count rate doubling to determine 1/M data points,
this fourth rule should be implemented carefully because it is only an
approximation.
Rule of Thumb Proof
Using:
Rev 1
19
𝑘𝑒𝑓𝑓 =
1
1−𝜌
If keff = 0.95, ρ = (-) 0.05263
If keff = 0.975, ρ = (-) 0.02564
Using:
𝐶𝑅2 1 − 𝑘𝑒𝑓𝑓1
=
𝐶𝑅1 1 − 𝑘𝑒𝑓𝑓2
𝐶𝑅2
1 − 0.95
=
𝐶𝑅1 1 − 0.975
𝐶𝑅2
0.05
=
=2
𝐶𝑅1 0.025
To double the count rate, the Δρ= 0.02699. For example, 0.05263 - 0.02564
If 0.02699 Δk/k (2,699 pcm) is added again:
−0.02564
∆𝑘
∆𝑘
∆𝑘
+ 0.02699
= 0.00135
𝑘
𝑘
𝑘
𝑘𝑒𝑓𝑓 =
1
1−𝜌
𝑘𝑒𝑓𝑓 =
1
1 − 0.00135
𝑘𝑒𝑓𝑓 = 1.0013
The reactor is slightly supercritical (keff > 1.0).
Knowledge Check
A reactor operator is performing a reactor startup. The
operator refers to the thumb rule that states it takes five
(5) doublings to attain critically. He has just stopped rod
pull on the fifth doubling. The neutron count rate is
steady with a zero (0) decade per minute (dpm) startup
rate. Should he call the reactor critical?
A.
20
Yes, reactor is likely critical, delayed neutrons take time
to build up.
Rev 1
B.
Yes, it is probably critical or close to critical.
C.
No, it is not possible to tell if the reactor is critical yet.
D.
No, this is only a thumb rule and not exact.
TLO 1 Summary
1. Explanation of key terms
a. Source neurons ensure that the neutron population remains high
enough to allow a visible indication of neutron level while the
reactor is shut down and during the startup. This verifies
instrument operability and allows monitoring of neutron
population changes. Source neutrons are classified as either
intrinsic or installed neutron sources.
b. Subcritical multiplication is the process where source neutrons add
to the neutrons available in each generation for absorption to
sustain the chain reactions in a reactor with a multiplication factor
(keff) of less than one.
c. Subcritical multiplication factor (M)
The subcritical multiplication factor (M), which is equivalent to
CR2 divided by CR1, is the fractional change in neutron population
of a subcritical reactor due to the changes in core reactivity.
𝑀=
𝐶𝑅𝑛
𝐶𝑅𝑜
𝑀=
𝐶𝑅𝑛
1
=
𝐶𝑅3 1 − 𝑘𝑒𝑓𝑓
d. Subcritical multiplication response on a reactor startup
As control rods withdrawn, count rate comparisons yield
increasing values of M as the reactor approaches criticality.
Accurately plotting the expected point of criticality using M
values is difficult because the count rate ratio (M) could increase
from approximately one (1) to several million (or infinity) prior to
reaching criticality, making the exact point of expected criticality
difficult to identify.
The inverse of M or 1/M is used because as keff approaches
criticality (1.0), 1/M approaches zero (0), a much easier identified
value for predicting criticality.
Rev 1
21
2. Calculate neutron levels for Keff and reactivity
For a subcritical reactor calculate steady-state neutron levels for
various values of keff and reactivity additions.
Use these formulas:
𝐶𝑅2 1 − 𝑘𝑒𝑓𝑓1
=
𝐶𝑅1 1 − 𝑘𝑒𝑓𝑓2
𝑘𝑒𝑓𝑓 =
1
1−𝜌
𝐶𝑅2 𝜌1
≈
𝐶𝑅1 𝜌2
𝐶𝑅2 𝜌1 (1 − 𝜌2 )
=
𝐶𝑅1 𝜌2 (1 − 𝜌1 )
Note:
𝜌=
∆𝑘
= 𝑝𝑐𝑚
𝑘
3. Delayed neutron relationships
Delayed neutron fraction — the delayed neutron fraction (β) is the
ratio of delayed neutrons to all neutrons born whether prompt and
delayed for a given isotope, which is the fraction of all neutrons that
began their lives as delayed neutrons.


Uranium-235 β = 0.0065
Plutonium-239 β = 0.0021
For comparison, the majority (~ 99.36 percent) of the neutrons
produced are prompt, with the remaining (0.64 percent) being delayed.
Average delayed neutron fraction — the term 𝛽̅ (beta-bar) is the
average delayed neutron fraction. The value of 𝛽̅ is the weighted
average of all delayed neutron fractions of the fuel or fuel mixture in
the reactor.
Importance factor — delayed neutron are born at lower energies:
Lower fast fission factor than prompt neutrons, increase to the fast
non-leakage factor., which is about 0.97 for a commercial PWR.
22
Rev 1
𝛽𝑒𝑓𝑓 = 𝛽𝑥 𝐼𝑚𝑝𝑜𝑟𝑡𝑎𝑛𝑐𝑒 𝐹𝑎𝑐𝑡𝑜𝑟
Effective delayed neutron fraction — the effective delayed neutron
fraction is defined as the fraction of thermal neutrons that were born
delayed. It is also the fraction of all fissions that are induced by
neutrons that began their lives as delayed neutrons.
How delayed neutrons affect reactor period:
a. Prompt neutron generation time is in the range of 10-4 to 10-5
seconds.
b. Delayed neutron generation time is ~12.7 seconds.
c. Approximately 0.64 percent of all neutrons are delayed, which
causes the average neutron generation time to increase to ~
0.0827 seconds.
ℓ∗
Delayed Neutron Reaction𝜏 = 𝜌−𝛽
𝑒𝑓𝑓
Note
𝛽𝑒𝑓𝑓 −𝜌
+λ
𝑒𝑓𝑓
𝜌+𝜌̇
Important Concept — If the reactivity added is smaller
in magnitude than the average effective delayed neutron
̅ ), power can increase no faster than the rate
fraction (𝛽𝑒𝑓𝑓
of the delayed neutron population’s increase.
Delayed neutron effect on control of reactor power over core life —
near the end of the core’s life, the power contribution from the fission
of plutonium-239 will be significant.
Note
Important Concept — Over the core’s life, βeff
decreases; therefore, for a given amount of reactivity
addition, the reactor period decreases, and the SUR
increases.
Prompt drop — using the example of a reactor trip, since the prompt
neutrons make up about 99.4 percent of the total neutron population,
there will be a rapid drop in reactor power of about two decades,
known as a prompt drop, until neutron level is at the level of
production of delayed neutrons.
Prompt jump — when positive reactivity is added, the prompt neutron
population immediately increases (10-5 seconds) but the neutron
population change due to the delayed neutron generation time of 12.7
seconds is delayed until the delayed neutron precursor levels have
increased from the reactivity increase.
Prompt criticality — a reactor is considered prompt critical if it is
critical without any contribution from delayed neutrons.
Rev 1
23
𝑘𝑒𝑓𝑓 = 1 + 𝛽𝑒𝑓𝑓
4. Rules of thumb for changing neutron count rate
1. Doubling the count rate (subcritical reactor) implies that enough
reactivity was added to take the reactor half way to criticality (keff
is halfway to 1.0).
2. If enough reactivity was added to double count rate and the same
amount of reactivity is added to the reactor again, the reactor will
be supercritical.
3. Using the thumb rule that states with each doubling the distance to
criticality is halved, the count rate will double again if reactivity
equal to half the original amount is added.
4. Many plants consider that with five (5) doublings the reactor should
be critical.
TLO 2 Reactor Period and Startup Rate
Overview
Calculating power changes, rates of power change, and the effect of prompt
and delayed neutrons on reactor control is an important aspect that should
be understood by operators. This chapter explains prompt and delayed
neutron effects on reactor response, reactor period, and startup rates.
Prompt drop, prompt jump, and prompt criticality are also explained.
Understanding effects of reactivity additions, positive or negative, and their
effect on power and the rate of change in power are important to safe
reactor operations. To ensure an understanding of these concepts the
following ELOs are covered in this lesson:
1. Describe the following equations and associated terms for:
a. Reactor period
b. Doubling time
c. Reactor startup rate
2. Given necessary reactivity variables, calculate the SUR or reactor
period and other variables in the power equations.
3. Describe prompt critical, prompt jump, prompt drop and how reactor
power is affected by a reactor trip and stepped insertion of reactivity.
24
Rev 1
ELO 2.1 Reactor Period and Startup Equations
Introduction
The reactor operator must understand how the reactor will respond when
reactivity is added for safe operation of a nuclear plant. Knowledge of
reactor period assists the operator in understanding the reactor's response to
a reactivity insertion.
Reactor Period and Startup Rate Equations
Reactor period is defined as the length of time in seconds required for
reactor power to change by a factor of e. Where e, the base of the natural
logarithm, is ~ 2.718.
The relationship between reactor power changes and reactor period (τ) is
shown below:
𝑃 = 𝑃𝑜 𝑒 𝑡/𝜏
Where:
P = transient reactor power
Po = original reactor power
τ = reactor period
t = transient time between P and Po (seconds)
From the equation, as reactor period (τ) decreases, the rate of power change
increases.
An equation for transforming the reactor period yields:
𝜏=
𝑡
𝑃
ln
𝑃𝑜
Consider the case of doubling reactor power, (P/Po) = 2. With a reactor
period = 1.44 seconds, the transient time between P and Po, the time it takes
to double power, referred to as the doubling time) is:
𝑡 = 𝜏(ln 𝑃/𝑃𝑜 )
𝑡 = (1.44 𝑠𝑒𝑐)(ln 2/1)
𝑡 = (1.44 𝑠𝑒𝑐)(ln 2)
Rev 1
25
𝑡 = (1.44 𝑠𝑒𝑐)(0.693)
𝑡 = 0.998 𝑠𝑒𝑐 (𝑜𝑟 ~ 1 𝑠𝑒𝑐)
With a reactor period of 1.44 seconds, it takes ~ 1 (one) second to double
reactor power.
Changing reactor period to half of the previous value, or 0.72 seconds:
𝑡 = (0.72 𝑠𝑒𝑐)(0.693)
𝑡 = 0.498 𝑠𝑒𝑐 (𝑜𝑟 ~ 0.5 𝑠𝑒𝑐)
Decreasing the reactor period by half also decreases the time it takes to
double reactor power by half.
What are the factors that make up reactor period itself?
The following equation mathematically expresses reactor period:
Where:
𝜏=
𝛽𝑒𝑓𝑓 − 𝜌
ℓ∗
+
𝜌 − 𝛽𝑒𝑓𝑓 λ𝑒𝑓𝑓 𝜌 + 𝜌̇
ℓ* = prompt neutron generation time (≈ 10-5 seconds)
βeff = effective delayed neutron fraction
λ eff = effective delayed neutron precursor decay constant
ρ = reactivity
𝜌̇ = rate of change of reactivity (example control rod movement)
The above formula factors in both prompt neutrons (the first component)
with a generation time of ~10-5 seconds and delayed neutrons (the second
component) with an effective generation time of ~12.7 seconds. The λeff
adjusts reactor period by the weighted fraction of short-lived and long-lived
delayed neutron precursors. Reactivity added, positive or negative, also
affects reactor period.
Delayed neutrons are what makes the reactor more controllable by
increasing generation time. This is the βeff component to the reactor period
and affects the rate of power change. More information on delayed
neutrons and their effect on reactor period over core life will be discussed in
the next section.
26
Rev 1
Startup Rate
Startup rate is a more commonly used term among reactor operators for
power changes, because most operators find it easier to use than reactor
period for defining power change. Reactor period is defined as the time in
seconds required for reactor power to change by a factor of e. Monitoring
the power change occurring in one minute is easier than monitoring power
changes by a factor of e.
Reactor period units are seconds and SUR units are decades per minute
(dpm):
𝑃 = 𝑃𝑜 𝑒 𝑡/𝜏
and
𝑃 = 𝑃𝑜 10𝑆𝑈𝑅(𝑡)
Setting these two equations equal to each other:
𝑃𝑜 𝑒𝑡/𝑟 = 𝑃𝑜 10𝑆𝑈𝑅(𝑡)
Working through the math:
𝑆𝑈𝑅 =
26.06
𝜏
Dividing 26.06 by the reactor period in seconds gives us the SUR in dpm.
Going back to the previous example, with 𝜏 = 1.44:
26.06
1.44
𝑆𝑈𝑅 = 18 𝑑𝑝𝑚
𝑆𝑈𝑅 =
PWRs are usually limited to a SUR of one (1) dpm or less; therefore, 18
dpm is too high. What is the reactor-period equivalent to one (1) dpm?
𝑆𝑈𝑅 =
26.06
𝜏
𝜏=
26.06
𝑆𝑈𝑅
𝜏=
26.06
1
If the limit for SUR is one (1) dpm, then the maximum reactor period is
26.06 seconds. The larger the value of the reactor period, the slower the
power increase.
Example:
Rev 1
27
A reactor has a λeff of 0.10 sec-1 and an effective delayed neutron fraction of
0.0070. If keff is equal to 1.0025, what are the stable reactor period and the
SUR?
Solution:
1. First solve for reactivity:
𝜌=
𝑘𝑒𝑓𝑓 − 1
𝑘𝑒𝑓𝑓
𝜌=
1.0025 − 1
1.0025
𝜌 = 0.00249
∆𝑘
𝑘
2. Use this value of reactivity to calculate reactor period:
𝜏=
̅ − 𝜌)
(𝛽𝑒𝑓𝑓
𝜌𝜆̅
𝜏=
(0.0070) − (0.00249)
(0.10 𝑠𝑒𝑐 −1 )(0.00249)
𝜏 = 18.1 𝑠𝑒𝑐
3. The startup rate can then be calculated from the reactor period:
𝑆𝑈𝑅 =
26.06
𝜏
𝑆𝑈𝑅 =
26.06
18.1 𝑠𝑒𝑐
𝑆𝑈𝑅 = 1.44 𝑑𝑝𝑚
Knowledge Check
Which one of the following is a characteristic of
subcritical multiplication?
28
Rev 1
A.
The subcritical neutron level is directly proportional to
the neutron source strength.
B.
Doubling the indicated count rate by reactivity additions
will reduce the margin to criticality by approximately
one quarter.
C.
For equal reactivity additions, it takes less time for the
new equilibrium source range count rate to be reached as
keff approaches unity.
D.
An incremental withdrawal of a given control rod will
produce an equivalent equilibrium count rate increase,
whether keff is 0.88 or 0.92.
ELO 2.2 Startup Rate and Reactor Period Calculations
Introduction
Use of the SUR and reactor period equations to calculate power changes
and rates of power change provides the operator with the knowledge and
understanding necessary to anticipate transient reactor responses.
Startup Rate and Reactor Period Calculations Step-by-Step
Step Action
1.
Determine the unknown variable and identify the applicable
equation(s) to solve for the unknown.
2.
Solve for the unknown variable in the applicable equation(s).
Startup Rate and Reactor Period Calculations Demonstration
The following equations are used to solve for the unknowns in powerchange calculations.
Solving for Power Changes Using the Reactor Period Equation
𝑃 = 𝑃𝑜 𝑒 𝑡/𝜏
Where:
P = transient reactor power
Po = original reactor power
𝜏 = reactor period
Rev 1
29
t = transient time between P and Po in seconds
Solving for reactor period:
First, transform the reactor period equation, and remember that e is the base
of the natural logarithm:
𝜏=
𝑡
ln 𝑃/𝑃𝑜
Solving for time in seconds, for the power change:
𝑡 = 𝜏 (ln
𝑃
)
𝑃𝑜
Solving for startup rate, if the reactor period is known:
𝑆𝑈𝑅 =
26.06
𝜏
Solving for new power level, if both the SUR and time are known:
𝑃 = 𝑃𝑜 10𝑆𝑈𝑅(𝑡)
Solving for both SUR and time, for the power change:
𝑆𝑈𝑅 =
𝑡=
log 𝑃⁄𝑃
𝑜
𝑡
log 𝑃⁄𝑃
𝑜
𝑆𝑈𝑅
Example 1
For a SUR of .two (2) dpm, how long will it take to increase power from 30
percent to 90 percent at a constant rate?
𝑡=
𝑡=
log 𝑃⁄𝑃
𝑜
𝑆𝑈𝑅
log 90⁄30
0.2
𝑡=
log 3
0.2
𝑡=
0.477
= 2.39 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
0.2
30
Rev 1
Example 2
With a constant reactor period of 100 seconds how long will it take to
change power from 7 percent to 100 percent?
𝑡 = 𝜏 (ln
𝑃
)
𝑃𝑜
𝑡 = 100 𝑠𝑒𝑐 (ln
100
)
7
𝑡 = 100 𝑠𝑒𝑐 (ln 14.29)
𝑡 = 100 𝑠𝑒𝑐(2.65) = 265 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 𝑜𝑟 4 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 𝑎𝑛𝑑 25 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
Example 3
A reactor period of 100 seconds equates to a SUR of ______?
𝑆𝑈𝑅 =
26.06
𝜏
𝑆𝑈𝑅 =
26.06
100
𝑆𝑈𝑅 = 0.26 𝑑𝑝𝑚
Example 4
With a nuclear reactor on a constant period of 30 minutes, which one of the
following power changes requires the least time to occur?
A. 1 percent power to 6 percent power
B. 10 percent power to 20 percent power
C. 20 percent power to 35 percent power
D. 40 percent power to 60 percent power
Using:
𝑃 = 𝑃𝑜 𝑒 𝑡/𝜏
Solving for time:
𝑡 = 𝜏 (ln
Rev 1
𝑃
)
𝑃𝑜
31
With a constant reactor preiod the answer to the question is to find the
lowest ration of P/Po. The value of 40 percent/60 percent = 1.5 percent.
The correct choice is D.
Example 5
What is the doubling time for a SUR of 0.25 dpm?
𝑡=
𝑃
log 𝑃
𝑜
𝑆𝑈𝑅
𝑡=
log 2
0.25
𝑡=
0.3
= 1.2 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
0.25
Knowledge Check
A small amount of positive reactivity is added to a
critical reactor in the source/startup range. The amount
of reactivity added is much less than the effective
delayed neutron fraction. Which one of the following
will have a significant effect on the magnitude of the
stable reactor period achieved for this reactivity addition?
A.
Moderator temperature coefficient
B.
Fuel temperature coefficient
C.
Prompt neutron lifetime
D.
Effective decay constant
Knowledge Check – NRC Bank
A reactor is being started for the first time following a
refueling outage. Reactor Engineering has determined
that during the upcoming fuel cycle, β̅eff will range from
a maximum of 0.007 to a minimum of 0.005.
Once the reactor becomes critical, control rods are
withdrawn to increase reactivity by 0.1 % ΔK/K.
Assuming no other reactivity additions, what will the
stable reactor period be for this reactor until the point of
adding heat is reached?
32
Rev 1
A.
20 seconds
B.
40 seconds
C.
60 seconds
D.
80 seconds
ELO 2.3 Effects of Reactor Trip and Step Insertion of Reactivity
Introduction
Prompt jump and prompt drop are the terms used to illustrate the immediate
response of a nuclear reactor to either negative or positive reactivity
insertions. Delayed neutrons are important for safe reactor control, as they
serve to increase the overall neutron generation time; however, prompt
neutrons are both first on the scene and off the scene. Prompt jump is
shown below in the figure.
Prompt Jump
Figure: Prompt Jump
The term prompt jump describes the immediate effect a positive reactivity
addition will have on the neutron population of either a critical or shutdown
reactor (shutdown reactor needs to be close to a keff of 1.0). When positive
reactivity is added, the prompt neutron population immediately increases.
Rev 1
33
However, neutron population change due to delayed neutrons delays or
waits until the delayed neutron precursor levels have increased from the
reactivity increase. The delayed neutron generation time of 12.7 seconds
delays the time to restore the ratio of delayed neutrons to prompt neutrons
to steady state values.
Prompt Drop
In the case where negative reactivity is added to the core, there will be a
prompt drop in reactor power. The prompt drop, shown below, is the small
immediate decrease in reactor power caused by the negative reactivity
addition. After the prompt drop, the rate of change of power slows and
approaches the rate determined by the delayed term of the reactor period
equation.
Figure: Prompt Drop
Reactor Trip
The prompt neutron population is immediately affected on a reactor trip or
large negative reactivity insertion. The prompt neutrons are gone
immediately since the prompt generation time is short. The neutrons that
remain are those from the delayed neutron precursors.
The short-lived decay first while the longest lived decay at a slower rate.
The longest-lived neutrons account for the negative 80 second period
following the trip (-1/3 dpm) until they have fully decayed, leaving the
source neutrons and subcritical multiplication to maintain an equilibrium
neutron count rate.
34
Rev 1
The following figure illustrates the prompt drop on a reactor trip and
continuing response until subcritical multiplication is reached.
Figure: Reactor Trip Power Decay Response
Knowledge Check – NRC Bank
Delayed neutrons contribute more to reactor stability
than prompt neutrons because they __________ the
average neutron generation time and are born at a
__________ kinetic energy.
Rev 1
A.
increase; lower
B.
increase; higher
C.
decrease; lower
D.
decrease; higher
35
TLO 2 Summary
1. Describe the equations and associated terms for the following:
Reactor period — the time (expressed in seconds) required for reactor
power to change by a factor of e, where e is the base of the natural
logarithm, ~ 2.718.
𝑃 = 𝑃𝑜 𝑒 𝑡/𝜏
Doubling time — the time required for reactor power to change
(increase or decrease) by a factor of two (2).
𝑡 = 𝜏 (ln
𝑃
)
𝑃𝑜
Startup rate (also known as SUR) — the number of factors of ten that
reactor power changes in one minute . SUR is measured in decades
per minute (dpm).
𝑃 = 𝑃𝑜 10𝑆𝑈𝑅(𝑡)
26.06
𝜏
2. Calculate SUR and reactor period
𝑆𝑈𝑅 =
The following equations are used to solve for the unknowns in reactor
power changes:
For power changes: 𝑃 = 𝑃𝑜 𝑒 𝑡/𝜏
𝑃
For time in seconds: 𝑡 = 𝜏 (ln 𝑃 )
𝑜
For reactor period: 𝜏 =
𝑡
𝑃
𝑃𝑜
ln
For SUR, if the period is known: 𝑆𝑈𝑅 =
26.06
𝜏
For new power level, if SUR and time are known: 𝑃 = 𝑃𝑜 10𝑆𝑈𝑅(𝑡)
For times for a power change: 𝑡 =
𝑃
𝑃𝑜
log
𝑆𝑈𝑅
3. Describe prompt critical, prompt jump, prompt drop and how reactor
power is affected by a reactor trip and stepped insertion of reactivity.
Prompt drop — Using the example of a reactor trip, since the prompt
neutrons make up about 99.4 percent of the total neutron population,
there will be a rapid drop in reactor power of about two decades,
36
Rev 1
known as a prompt drop, until neutron level is at the level of
production of delayed neutrons, shown below in the picture.
Figure: Prompt Drop
Prompt jump — when positive reactivity is added, the prompt neutron
population immediately increases (10-5 seconds) but the neutron
population change due to the delayed neutron generation time of 12.7
seconds is delayed until the delayed neutron precursor levels have
increased from the reactivity increase.
Figure: Prompt Jump
Prompt criticality — A reactor is considered prompt critical if it is
critical without any contribution from delayed neutrons.
𝑘𝑒𝑓𝑓 = 1 + 𝛽𝑒𝑓𝑓
Reactor trip — On a reactor trip or large negative reactivity insertion,
the prompt neutron population is immediately affected. The prompt
Rev 1
37
neutrons are gone almost immediately, because the prompt neutron
generation time is short. The neutrons that remain are those from the
delayed neutron precursors, the delayed neutrons. The short-lived
precursors decay off first, while the longer-lived precursors decay at a
slower rate. The longer-lived precursors account for the negative 80second period (-1/3 dpm SUR) following the trip, until a subcritical
neutron count rate is attained.
Figure: Reactor Trip Power Decay Response
Knowledge Check
Two reactors are critical at the same power level, well
below the point of adding heat. The reactors are
identical, except that reactor A is at the beginning of life
and reactor B is near the end of a fuel cycle. If a step
addition of positive 0.001 Δk/k is added to each reactor,
the size of the prompt jump in power level observed in
reactor B near end of cycle (EOC) will be __________
than in reactor A. Given a large reactivity step insertion,
reactor B would go prompt critical with
__________reactivity than in reactor A. (Assume the
power level in each reactor remains below the point of
adding heat.)
38
A.
larger; less
B.
smaller; less
C.
larger; more
D.
smaller; more
Rev 1
Now that you have completed this lesson, you should be able to do the
following:
1. Describe the following equations and associated terms for:
a. Reactor period
b. Doubling time
c. Reactor startup rate
2. Given the necessary reactivity variables, calculate the SUR or reactor
period, and other variables in the power equations.
3. Describe prompt critical, prompt jump, prompt drop and how reactor
power is affected by a reactor trip and stepped insertion of reactivity.
Reactor Kinetics Module Summary
This module covered reactor kinetics and subcritical reactor operation.
Calculating power changes, rates of power changes, and understanding the
effect of prompt and delayed neutrons on reactor control are important
aspects of a reactor operator’s responsibilities.
Now that we have completed this topic, you should be able to demonstrate
mastery of this topic by passing a written exam with a grade of 80 percent
or higher on the following TLOs:
1. Describe subcritical multiplication for a nuclear reactor and state how
subcritical multiplication affects reactor operation.
2. Explain the factors that affect reactor period and start-up rate as well
as their effect on reactor control.
Rev 1
39