Revision 1
December 2014
Reactivity Coefficients
Student Guide
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ii
Table of Contents
INTRODUCTION .............................................................................................................................. 1
TLO 1 REACTIVITY, KEFF AND SHUTDOWN MARGIN ...................................................................... 2
Overview .................................................................................................................................. 2
ELO 1.1 Reactivity ................................................................................................................... 2
ELO 1.2 Reactivity Conversions .............................................................................................. 5
ELO 1.3 Excessive Reactivity .................................................................................................. 6
ELO 1.4 Shutdown Margin ...................................................................................................... 9
ELO 1.5 Shutdown Purpose ................................................................................................... 10
ELO 1.6 Sufficient Reactivity Conversions to Calculate Reactor Shutdown Margin ............ 13
TLO 1 Summary ..................................................................................................................... 15
TLO 2 MODERATOR, VOID AND PRESSURE REACTIVITY ............................................................. 16
Overview ................................................................................................................................ 16
ELO 2.1 Reactivity Coefficients ............................................................................................ 16
ELO 2.2 Moderator Temperature Coefficient (MTC) ............................................................ 18
ELO 2.3 Moderator to Fuel Ratio Effects on MTC ................................................................ 20
ELO 2.4 Void and Pressure Reactivity Coefficients .............................................................. 26
TLO 2 Summary ..................................................................................................................... 28
TLO 3 FUEL TEMPERATURE AND POWER COEFFICIENTS ............................................................. 31
Overview ................................................................................................................................ 31
ELO 3.1 Fuel Temperature Reactivity Coefficient ................................................................ 31
ELO 3.2 Doppler and Self-shielding ...................................................................................... 35
ELO 3.3 Moderator Temperature Effects on the Fuel Temperature Coefficient ................... 44
ELO 3.4 Power Reactivity Coefficient ................................................................................... 48
ELO 3.5 Power Defect on Reactor Power Operations Definition .......................................... 50
TLO 3 Summary ..................................................................................................................... 52
TLO 4 REACTIVITY BALANCES AND BORON REACTIVITY ........................................................... 54
Overview ................................................................................................................................ 54
ELO 4.1 Reactivity Balance ................................................................................................... 54
ELO 4.2 Purpose of Boron Reactivity Control....................................................................... 59
ELO 4.3 Changes in Boron Worth with Changes in Boron Concentration ............................ 62
ELO 4.4 Changes in Boron Concentration over Core Life .................................................... 65
TLO 4 Summary ..................................................................................................................... 67
REACTIVITY COEFFICIENTS SUMMARY ........................................................................................ 68
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Reactivity Coefficients
Revision History
Revision
Date
Version
Number
Purpose for Revision
Performed
By
11/5/2014
0
New Module
OGF Team
12/11/2014
1
Added signature of OGF
Working Group Chair
OGF Team
Introduction
This module includes key concepts that will help the operator understand
power operations of reactivity coefficients.
Rev 1
1
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 Learning Objectives (TLOs):
1. Describe reactivity, keff and shutdown margin and their effect on the
reactor operational status.
2. Describe moderator, void and pressure reactivity coefficients and how
they are affected by changing reactor conditions.
3. Describe the fuel temperature and power reactivity coefficients and
describe how they are affected by changing reactor conditions.
4. Describe how a reactivity balance is performed and methods used to
compensate for excess reactivity.
TLO 1 Reactivity, keff and Shutdown Margin
Overview
Previous sections explained keff, the effective multiplication factor, the ratio
of the neutrons produced by fission in one generation to the number of
neutrons lost through absorption and leakage in the preceding generation.
This section introduces reactivity and its relationship to keff.
Objectives
Upon completion of this lesson, you will be able to do the following:
1. Describe the term reactivity and its relationship to keff and criticality.
2. Convert between alternate units of reactivity.
3. Define excess multiplication factor (kexcess) and excess reactivity
(ρexcess).
4. Define shutdown margin.
5. Evaluate plant parameters or design features that affect shutdown
margin.
6. Given sufficient reactivity information, calculate the reactor shutdown
margin.
ELO 1.1 Reactivity
Introduction
Reactivity is a measure of the fractional change in neutron population per
generation. Reactivity is a function of keff, defined as the ratio of the
neutrons produced by fission in one generation to the number of neutrons
lost through absorption and leakage in the preceding generation. Reactivity,
like keff, describes the reactor's deviation from criticality. Reactivity units
measure the reactor’s deviation from criticality.
Reactivity versus keff
It is possible to determine the number of neutrons after a certain number of
generations if you know the original number of neutrons (No) at the start of
the first generation and the value of keff, with keff at a constant value from
generation to generation. We use the formula below for this purpose:
2
Rev 1
𝑁𝑛 = 𝑁𝑜 (𝑘𝑒𝑓𝑓 )
𝑛
Where:
n = number of generations
Nn = number of neutrons in the nth generation
No = number of neutrons at the start of the first generation
Example:
The number of neutrons in the core at time zero is 1,000 and keff = 1.002.
Calculate the number of neutrons after 50 generations.
Solution:
Using:
𝑁𝑛 = 𝑁𝑜 (𝑘𝑒𝑓𝑓 )
𝑛
𝑁50 = 1,000 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 (1.002)50
𝑁50 = 1,105 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠
If there are No neutrons in the preceding generation, then there are No (keff)
neutrons in the present generation. The numerical change in neutron
population is (No keff - No).
We express reactivity (ρ) as a fraction, therefore the count rate expressed as
a fraction is:
𝜌=
𝑁𝑜 𝑘𝑒𝑓𝑓 − 𝑁𝑜
𝑁𝑜 𝑘𝑒𝑓𝑓
Cancelling out the term No from the numerator and denominator, reactivity
relates to keff as:
𝜌=
𝑘𝑒𝑓𝑓 − 1
𝑘𝑒𝑓𝑓
Reactivity, as shown in the above formula, is the fractional change in
neutron population per generation.
Reactivity versus Criticality
Reactivity is the term used when discussing a nuclear reactor's deviation
from criticality.

If the reactor is critical (keff = 1) then reactivity = 0 for any reactor
power level.
 Reactivity is a positive value >0 for a supercritical reactor (keff > 1).
 Reactivity is negative value <0 for a subcritical reactor (keff < 1).
From the reactivity equation below, ρ may be positive, zero, or negative,
depending upon the value of keff.
𝜌=
𝑘𝑒𝑓𝑓 − 1
𝑘𝑒𝑓𝑓
Rev 1
3
The larger the absolute value of reactivity in the reactor core, the further the
reactor is from criticality.
Example:
Calculate the reactivity in the reactor core when keff is equal to 1.002 and
0.998. For each value of keff, state whether the reactor is critical,
supercritical, or subcritical.
Solution:
The reactivity for each case is determined by substituting the value of keff
into the formula for reactivity:
𝜌=
𝑘𝑒𝑓𝑓 − 1
𝑘𝑒𝑓𝑓
𝜌=
1.002 − 1
1.002
𝜌=
𝜌=
𝑘𝑒𝑓𝑓 − 1
𝑘𝑒𝑓𝑓
0.998 − 1
0.998
𝜌 = 0.001996
𝜌 = −0.0020
Reactivity is positive, therefore the
reactor is supercritical
Reactivity is negative, therefore the
reactor is subcritical
You can determine keff by transforming the equation to solve for keff in
terms of the reactivity if you do not know keff and you know reactivity. The
result is:
𝑘𝑒𝑓𝑓 =
1
1−𝜌
Example:
Given a reactivity of -20.0 x 10-4 ∆k/k, calculate keff.
Solution:
𝑘𝑒𝑓𝑓 =
1
1−𝜌
𝑘𝑒𝑓𝑓 =
1
1 − (−20.0 × 10−4 )
𝑘𝑒𝑓𝑓 = 0.998
4
Rev 1
Knowledge Check
Reactivity is defined mathematically as the fractional
change in _______________.
A.
reactor power per second
B.
neutron population per second
C.
reactor period from criticality
D.
the effective multiplication factor from criticality
Knowledge Check
Given a reactivity of -150.0 x 10-4 ∆k/k, calculate keff to
the nearest thousandth.
A.
1.015
B.
0.985
C.
1.015 x 10-4 ∆k/k
D.
0.985 x 10-4 ∆k/k
ELO 1.2 Reactivity Conversions
Introduction
Reactivity is a dimensionless number. It is simply a ratio of two quantities,
expressed either as a ratio, or in percent (such as ρ).
Reactivity conversions Step-by-Step Table
The value of reactivity is often a small decimal value, often expressed in
special units to make this value easier to express. The value for reactivity
that results directly from the calculation of keff is in units of ∆k/k by
definition. Alternative units for reactivity are percent ∆k/k and pcm
(percent millirho). The table below shows conversions between these units
of reactivity:
Step
Action
1.
Determine the unit of reactivity to be used.
2.
Convert using appropriate step below.
3.
𝜌 = ∆𝑘/𝑘
4.
∆𝑘/𝑘 = 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 ∆𝑘/𝑘/100 or 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 ∆𝑘/𝑘 = ∆𝑘/𝑘(100)
5.
1 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 ∆𝑘/𝑘 = 1,000 𝑝𝑐𝑚
6.
1 𝑝𝑐𝑚 = 10−5 ∆𝑘/𝑘
Rev 1
5
Reactivity Conversion Demonstration
Example:
Convert the values of reactivity listed below to the indicated units.
a. 0.000421 ∆k/k = ____pcm
b. 0.0085 ∆k/k = ____ percent ∆k/k
c. 16 x 10-4 ∆k/k = ____∆k/k
Solution:
a. 42.1 pcm
b. 0.85 percent ∆k/k
c. 0.0016 ∆k/k
Knowledge Check
Convert the values of reactivity listed below to the
indicated units.
A.
1.45 x 10-4 ∆k/k = pcm
B.
350 x 10-4 ∆k/k = %∆k/k
C.
2,500 pcm = ∆k/k
ELO 1.3 Excessive Reactivity
Introduction
Excess reactivity (kexcess) is the reactivity from excess fuel loaded into a
nuclear reactor core beyond the minimum amount necessary to achieve
criticality at the beginning of core life. This is necessary to provide for
longer operational periods between refueling. Excess positive reactivity
must be available to compensate for the following:




Fuel burnup
Fission product poisons (xenon and samarium)
Increases in resonance capture from plutonium-240 buildup
Raising temperature and power to their normal full power values
Excess Reactivity
A critical reactor has a keff = 1. In order to maintain a value of 1 throughout
core life, we must add excess reactivity (extra fuel) to the core at the
beginning of a fuel cycle. This means that we must also add negative
reactivity to the core to counter the positive reactivity from the "excess" fuel
loading. Operators cancel (offset) the excess reactivity from the fuel using
control rods, soluble boron, and fixed burnable poison rods that provide
negative reactivity.
Excess Multiplication Factor
The excess multiplication factor (kexcess) is the amount of excess fuel
loading that causes keff to exceed 1.0. The equation below shows the
mathematical expression:
6
Rev 1
𝑘𝑒𝑥𝑐𝑒𝑠𝑠 = 𝑘𝑒𝑓𝑓 − 1
We express excess reactivity (ρexcess) in terms of kexcess by the following
formula:
𝜌𝑒𝑥𝑐𝑒𝑠𝑠 =
𝑘𝑒𝑥𝑐𝑒𝑠𝑠
𝑘𝑒𝑓𝑓
Example:
Consider the refueling of a reactor. The refueling of the total core increases
keff to a value of 1.5. What is the value of the excess multiplication factor
(kexcess) and excess reactivity (ρexcess) after refueling?
Solution:
Solve for kexcess using the following equation:
𝑘𝑒𝑥𝑐𝑒𝑠𝑠 = 𝑘𝑒𝑓𝑓 − 1
𝑘𝑒𝑥𝑐𝑒𝑠𝑠 = 1.5 − 1
𝑘𝑒𝑥𝑐𝑒𝑠𝑠 = 0.5
Then solve for ρexcess:
𝜌𝑒𝑥𝑐𝑒𝑠𝑠 =
𝑘𝑒𝑥𝑐𝑒𝑠𝑠 0.5
=
𝑘𝑒𝑓𝑓
1.5
𝜌𝑒𝑥𝑐𝑒𝑠𝑠 = 0.333 ∆𝑘/𝑘
Or, we express ρ as:
𝜌𝑒𝑥𝑐𝑒𝑠𝑠 = 33.3% ∆𝑘/𝑘
𝜌𝑒𝑥𝑐𝑒𝑠𝑠 = 33,300 𝑝𝑐𝑚
We generally define excess multiplication factor (kexcess) and excess
reactivity (ρexcess) for specific reactor conditions. Commonly used
conditions are:



Cold, xenon-free, no control rods
Hot, xenon-free, no control rods
Hot, rated power, equilibrium fission product poisons (xenon and
samarium)
Changes in Excess Multiplication Factor over Core Life
The value of kexcess varies over core life due to changing neutron poison
concentrations in the reactor core and fuel burnout. The following figure is
a generic example kexcess over core life.
Rev 1
7
Figure: Core Age versus keff
1. At the beginning of core life, kexcess decreases due to the buildup of
xenon and samarium (fission product poisons) in the reactor (A to B
in the figure above). For core fuel loads that include burnable poison
rods, this reduction would be less significant (removal of negative
reactivity).
2. Toward the middle of core life, kexcess increases to a maximum value
because of the depletion of burnable poisons (B to C in the figure
above). Depending on fuel load /burnable poisons, this peak may be
lower.
3. From middle of core life to end of core life, kexcess decreases due to
fuel burnout, until kexcess is eventually exhausted (C to D in the figure
above). Core coastdown begins at point D to maintain a reduced
power level.
Knowledge Check
After a core reloading pexcess has been calculated to equal
37,500 pcm. What is kexcess equal to?
8
A.
0.8
B.
1.6
C.
0.6
D.
2.66
Rev 1
ELO 1.4 Shutdown Margin
Introduction
Shutdown margin (SDM) is the instantaneous amount of reactivity by
which the reactor is subcritical or would be subcritical from its present
condition, assuming complete insertion of all full-length rod cluster
assemblies (shutdown and control) and the most reactive control rod fully
withdrawn from the core at any time during the core cycle.
The shutdown value (SDV) is the reactivity amount by which nuclear
reactor core is subcritical; or SDV is the additional amount of reactivity that
would make a reactor subcritical from its present condition. These two
terms are closely related. However, most importantly each commercial
nuclear plant has specific SDM requirements required by their operating
license.
Shutdown Value Determination
The SDV is the reactivity amount by which nuclear reactor core is
subcritical or the additional amount of reactivity that would make a reactor
subcritical from its present condition.
We calculate the SDV by using the following equation:
𝑆𝐷𝑉 =
1 − 𝑘𝑒𝑓𝑓
𝑘𝑒𝑓𝑓
The SDV is simply the actual reactivity value by which the reactor is
subcritical or the amount needed to make it subcritical. For example if keff
is 0.99, the SDV is equal to 0.0010 Δk/k.
𝑆𝐷𝑉 =
1 − 𝑘𝑒𝑓𝑓
𝑘𝑒𝑓𝑓
1 − 0.99
0.99
𝑆𝐷𝑉 = 0.0010 ∆𝑘/𝑘
𝑆𝐷𝑉 =
Control rod position, moderator temperature, poisons, boron concentration,
etc. affect SDV.
Shutdown Margin Determination
The plant's technical specifications specify SDM requirements. The SDM
is the instantaneous amount of reactivity by which a nuclear reactor core is
subcritical, or would be subcritical from its present condition with the most
reactive control rod fully withdrawn from the core. Notice from the
definition that SDM exists if the reactor is operating at 100 percent power,
or is shut down.
Nuclear reactor technical specifications require reactors to maintain a
specific minimum SDM, assuming the most reactive rod fully withdrawn
from core. A typical value ranges from 1.0 to 1.7 percent Δk/k. The
required value for SDM will change with core life.
Rev 1
9
The SDM is calculated using same equation as used for SDV:
𝑆𝐷𝑀 =
1 − 𝑘𝑒𝑓𝑓
𝑘𝑒𝑓𝑓
Example:
Calculate SDM of shutdown reactor with a core reactivity value of -0.0055
Δk/k.
Solution:
First, find keff:
𝑘𝑒𝑓𝑓 =
1
1
=
= 0.99453
1 − 𝜌 1 − (−0.0055)
Then, use the SDM equation:
𝑆𝐷𝑀 =
1 − 𝑘𝑒𝑓𝑓 1 − 0.99456
=
= 0.005 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝛥𝑘/𝑘
𝑘𝑒𝑓𝑓
0.99453
Since SDM and SDV have units of reactivity (Δk/k or percent Δk/k) the
value can be determined directly from the -0.0055 Δk/k given in the
problem - just change it to a positive value.
𝑆𝐷𝑀 = 0.55 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝛥𝑘/𝑘
Question: If the plant requires an SMD of 1.0 percent Δk/k, is the above
SDM sufficient?
Answer: No.
Knowledge Check
Calculate shutdown margin of shutdown reactor in Δk/k
with a keff of 0.9.
A.
100 Δk/k
B.
10 Δk/k
C.
1 Δk/k
D.
0.1 Δk/k
ELO 1.5 Shutdown Purpose
Introduction
The time in core life, control rod position, reactivity poison, boron
concentration, and other reactivity related core conditions determine the
amount of reactivity that actually shuts a reactor down, and therefore
determines the SDM.
10
Rev 1
Shutdown Margin Definition
The SDM is the instantaneous amount of reactivity by which a nuclear
reactor core is subcritical, or would be subcritical from its present condition
with the most reactive control rod fully withdrawn from the core.
Understanding this definition is key to understanding how reactivity
conditions in the core can affect its actual value.
The following parameters or design features will affect SDM:








Moderator temperature
Reactor coolant system boron concentration
Fuel temperature (Doppler)
Control rod position
Xenon/samarium and other reactivity poisons concentration
Number of fuel assemblies loaded in core
Time in core life
Reactor power level
Reactivity effects to Shutdown Margin Example
Each of the following reactivity parameters affects the SDM during reactor
shutdown conditions:






Rev 1
Moderator temperature - an increase in moderator temperature adds
negative reactivity. This increases SDM. During a plant cooldown,
the decreased moderator temperature adds considerable positive
reactivity. It is necessary to increase the RCS boron concentration to
compensate to maintain the required SDM.
Boron concentration in the reactor coolant system - increasing boron
concentration causes a decrease in the thermal utilization factor,
which adds negative reactivity; resulting in an increase in SDM.
Fuel temperature (Doppler) - when in a shutdown condition and the
reactor core is cooling, fuel temperature is maintained constant, and
SDM is unaffected. As the RCS cools, fuel temperature will also
decrease, causing the resonance escape probability to increase. This
adds positive reactivity, with a resulting decrease in SDM.
Control rod position – normally during shutdown conditions, the
control and shutdown rods are in the fully inserted position. If they
are withdrawn, this will add positive reactivity, causing the SDM to
decrease.
Xenon/samarium and other reactivity poisons concentration – during
shutdown conditions, fission product poisons such as xenon and
samarium will either peak or decay off, depending on the power
history and length of shutdown. If poisons increase, this adds
negative reactivity causing SDM to increase. SDM will decrease
from the addition of positive reactivity if poisons are decaying off.
Number of fuel assemblies loaded in core – It is possible to maintain
SDM by a minimum boron concentration during refueling and
shutdown verification via the performance of 1/m plots during fuel
loading and unloading. As fuel is loaded to the core, add positive
reactivity since the concentration of the fuel is increasing. Therefore,
as fuel is loaded, SDM will decrease.
11

Time in core life - when the reactor is shut down, there is no change
in core life and therefore no effect on SDM. However, the time in
core life does affect kexcess, requiring a lower boron concentration
(with increasing core life) to meet minimum SDM requirements.
Reactivity Effects to Shutdown Margin During Operating
Conditions
During reactor operations, the second half of the SDM definition applies the instantaneous amount of reactivity by which a nuclear reactor would be
subcritical from its present condition with the most reactive control rod
fully withdrawn from the core. Therefore, for each of the following
reactivity parameters, operators must consider the reactivity change
immediately following the reactor shutdown or trip.








12
Moderator temperature - RCS temperature following the shutdown
will level off at the no-load value (less than full load), adding positive
reactivity, and causing SDM to decrease.
Reactor coolant system boron concentration - immediately following
the trip or shutdown, boron concentration does not change and there is
no effect on SDM.
Fuel temperature (Doppler) - the cooler fuel temperature from the trip
adds positive reactivity. This is a large effect, causing a large
decrease in SDM.
Control rod position - on a reactor trip, personnel insert all control and
shutdown rods into the core and add a very large amount of negative
reactivity. This results in a large increase in SDM. A reactor
shutdown produces a similar effect; however, since the shutdown rods
may not be inserted, the increase in SDM may be less.
During power operation (power dependent), the control rods must be
above a certain minimum height to ensure adequate SDM on a trip.
Xenon/samarium and other reactivity poisons concentration immediately following a trip or shutdown xenon will peak (first 8
hours). This adds negative reactivity causing an increase to the SDM.
The immediate effect from samarium is much less. Following the
xenon peak, xenon decay is greater than production and positive
reactivity will be added, reducing SDM.
Time in core life - the time in core affects control rod worth, fuel
temperature and moderator temperature reactivity, boron worth, and
other factors. Therefore, time in core life will have an effect on SDM
following a trip or shut down.
Reactor power level - as the power level increases; moderator and fuel
temperatures also increase. Therefore, more positive reactivity would
be added on a trip because of the greater fuel and moderator
temperature decrease. The SDM would be lower on a trip or
shutdown from higher power levels.
Rev 1
Knowledge Check
A nuclear power plant is operating at 70 percent power
with manual rod control. Which one of the following
conditions will increase shutdown margin? Assume that
no unspecified operator actions occur and the reactor
does not trip.
A.
The reactor coolant system is diluted by 10 ppm.
B.
A control rod in a shutdown bank (safety group) drops.
C.
Power is decreased to 50 percent using boration.
D.
The plant experiences a 3 percent load rejection.
ELO 1.6 Calculate Reactor Shutdown Margin
Introduction
With a known value of keff, SDM can be determined using the formula:
𝑆𝐷𝑀 =
1 − 𝑘𝑒𝑓𝑓
𝑘𝑒𝑓𝑓
However, during reactor operation we probably do not know the exact value
of keff. We express SDM in units of reactivity so it is possible to determine
the SDM by accounting for all of the positive and negative reactivities
existing in the reactor core at a given time.
Reactivity Conversions
Each plant has a specific procedure for determining the SDM or a method
for establishing adequate SDM. This lesson will use a generic method for
demonstration purposes.
At most plants when the reactor is at power SDM is maintained (and known
to exist) by ensuring that the control rods are above a certain minimum
height. This minimum height is the insertion limit. The insertion limit
ramps higher as reactor power level is increased. Recall that the SDM is
less on a trip or shutdown as power increases, requiring higher insertion
limits to ensure adequate SDM.
When we shut the reactor down, the rods are on the bottom, so how do we
calculate SDM?
Step
Action
1.
Obtain last critical data as a starting point, where reactivity = 0
2.
Determine all reactivity changes from last critical data
3.
Sum the reactivities to determine reactivity and change the value
to a positive number for the SDM.
Rev 1
13
Calculating Shutdown Margin Demonstration
Example:
The following critical conditions exist just prior to a reactor trip:



Power level = 100 percent
Boron concentration = 660 ppm
Power defect = 1,500 pcm (power defect includes reactivity from the
fuel and moderator temperature coefficients – to be discussed in detail
later)
 Control rod fully withdrawn – 5,000 pcm
 Xenon – at equilibrium
 Samarium – at equilibrium
 RCS temperature at full load
 Middle of core life
Given these reactivity parameters, what is the SDM immediately following
a reactor trip?
Solution:
Just prior to the reactor trip, the core reactivity is 0 pcm:







Reactor critical at 100 percent power
Reactivity from power decrease
Reactivity from control rod insertion
Boron, Xe, Sm no change
Core life no change
RCS temperature at no load
SDM = 3.5
Note
In this example, the SDM is determined immediately
following the trip. If the SDM were to be calculated a
day or more later, xenon and samarium would change in
reactivity value, boron concentration may change, and
the plant may be cooled down or in the process of
cooling down.
Knowledge Check
With a nuclear power plant operating at 85 percent power
and rod control in manual, the operator borates the
reactor coolant system an additional 10 ppm. Assuming
reactor power does not change during the boration,
shutdown margin will _______________
14
A.
decrease and stabilize at a lower value.
B.
decrease, then increase to the original value as coolant
temperature changes.
C.
increase and stabilize at a slightly higher value.
D.
increase, then decrease to the original value as coolant
temperature changes.
Rev 1
TLO 1 Summary
1. Reactivity and its relationship to keff and criticality
 If the reactor is critical (keff = 1), then reactivity = 0 for any reactor
power level.
 Reactivity is a positive value >0 for a supercritical reactor (keff > 1).
 Reactivity is negative value <0 for a subcritical reactor (keff < 1).
2. Convert between alternate units of reactivity.
 The value for reactivity that results directly from the calculation of
keff is in units of ∆k/k. Alternative units for reactivity are percent ∆k/k
and pcm (percent millirho).
3. Excess multiplication and (kexcess) and excess reactivity (ρexcess).
 Excess multiplication factor (kexcess) is the amount of excess fuel
loading that causes keff to exceed 1.0.
 Excess reactivity (ρexcess) is = kexcess / keff.
4. Shutdown Margin
 The plant's technical specifications specify SDM requirements.
 SDM is the instantaneous amount of reactivity by which a nuclear
reactor core is subcritical, or would be subcritical from its present
condition with the most reactive control rod fully withdrawn from the
core.
 SDM is expressed in units of reactivity. Determine the SDM by
accounting for all of the positive and negative reactivities existing in
the reactor core at a given time.
5. Plant parameters or design features that affect SDM.
 The following parameters of design features will affect the SDM:
— Moderator temperature
— Reactor coolant system boron concentration
— Fuel temperature (Doppler)
— Control rod position
— Xenon/samarium and other reactivity poisons concentration
— Number of fuel assemblies loaded in core
— Time in core life
— Reactor power level
Summary
Now that you have completed this lesson, you should be able to:
1. Describe the term reactivity and its relationship to keff and criticality.
2. Convert between alternate units of reactivity.
3. Define excess multiplication factor (kexcess) and excess reactivity
(ρexcess).
4. Define shutdown margin.
5. Evaluate plant parameters or design features that affect shutdown
margin.
6. Given sufficient reactivity information, calculate the reactor shutdown
margin.
Rev 1
15
TLO 2 Moderator, Void and Pressure Reactivity
Overview
This session introduces reactivity coefficients. This section will explore
how moderator temperature, pressure, and voids affect reactivity in the core.
The concepts covered in this lesson are some of the most important to your
reactor operation responsibilities.
In particular, the moderator temperature coefficient provides an inherent
safety feature, along with the fuel temperature coefficient of a PWR. This
lesson includes an explanation of the moderator temperature coefficient, as
well as how boron concentration affects the coefficient. We define SDM in
terms of reactivity coefficients.
Objectives
Upon completion of this lesson, you will be able to do the following:
1. Explain differences between reactivity coefficients and reactivity
defects and explain their use to balance reactivity parameters.
2. Describe the moderator temperature coefficient of reactivity.
3. Describe how the magnitude of the moderator temperature coefficient
varies with changes in the following parameters:
a. Overmoderation and undermoderation of the moderator-to-fuel
ratio
b. Moderator temperature
c. Core age
d. Boron concentration
4. Describe the void and pressure coefficients of reactivity.
ELO 2.1 Reactivity Coefficients
Introduction
The amount of reactivity (ρ) in a reactor determines the neutron population
and/or reactor power state. Many factors affect reactivity, such as fuel
depletion, temperature, pressure, or fission product poisons. This section
discusses the factors affecting reactivity and tells how they control or
predict reactor behavior.
Reactivity Coefficients
Reactivity coefficients quantify the effect that a variation in a reactor
parameter (i.e. a change in temperature, control rod position, boron changes,
etc.) has on the overall reactivity of the core. Reactivity coefficients define
the amount of reactivity change for a given change in the parameter (per °F,
per ppm boron, etc.).
As an example, a moderator temperature increase causes a decrease in the
reactivity of the core. The amount of reactivity change per unit increase of
moderator temperature is the moderator temperature coefficient. Units for
the moderator temperature coefficient are pcm/°F.
16
Rev 1
Generally, αx symbolizes reactivity coefficients, where x represents the
reactor parameter affecting reactivity. The equation below shows reactivity
coefficients expressed as a formula:
𝛼𝑥 =
∆𝜌
∆𝑥
Where:
x = reactivity coefficient for plant parameter x
Δρ = change in reactivity (Δk/k)
Δx = a unit increase in plant parameter x
If the parameter x increases resulting in an addition of positive reactivity,
then x is positive. If the parameter x increases resulting in an addition of
negative reactivity, then x is negative.
Reactivity Defects
Reactivity defects are the total reactivity change caused by variation in a
parameter. The term "reactivity defect" (ρx) describes the total amount of
reactivity added, positive, or negative, due to changing a certain nuclear
reactor parameter by a given amount. Reactivity defects are determined by
multiplying the total change in the parameter by its average coefficient
value. The equations below relate reactivity coefficients to reactivity
defects.
𝜌𝑥 = (∆𝑥)(𝛼𝑥 )
𝜌𝑥 = (∆𝑥) (
∆𝜌
)
∆𝑥
Where:
ρx = reactivity defect (Δk/k)
x = specific parameter (fuel temperature, moderator temperature,
etc.)
Δx = change in parameter x
αx = parameter x reactivity coefficient (fuel temperature, moderator
temperature, etc.)
Example:
The moderator temperature coefficient for a reactor is -8.2 pcm/ °F.
Calculate the reactivity defect that results from a temperature decrease of
5°F.
Solution:
𝜌𝑇 = ∆ 𝑇 𝛼𝑥
Rev 1
17
𝜌𝑇 = (−5℉) (−8.2
𝑝𝑐𝑚
)
℉
𝜌𝑇 = 41 𝑝𝑐𝑚
The reactivity addition due to the temperature decrease was a positive 41
pcm because of the negative temperature coefficient.
Knowledge Check
Moderator temperature coefficient is the change in core
reactivity per degree change in _______________.
A.
fuel temperature
B.
fuel clad temperature
C.
reactor vessel temperature
D.
reactor coolant temperature
ELO 2.2 Moderator Temperature Coefficient (MTC)
Introduction
The moderator temperature coefficient (MTC) of reactivity is the change in
reactivity per degree change in moderator temperature. We discussed the
moderator temperature effect on keff with the six-factor formula, and we will
further review it later in this lesson.
Moderator Temperature Coefficient
The reactivity change per degree change in moderator temperature is the
moderator temperature coefficient (MTC) of reactivity. Its magnitude and
sign (+ or -) is primarily a function of the moderator-to-fuel ratio, density of
the moderator, and boron concentration. Commercial PWRs are designed
with an undermoderated moderator-to-fuel ratio that normally provides a
negative moderator temperature coefficient. Early in core life, the MTC
may be positive with the initial high boron concentration.
If a reactor is overmoderated, it will have a positive MTC as the change in
thermal utilization factor overrides the resonance escape probability.
However, a negative MTC is more desirable because of its power level
regulating effect in the power range.
Assuming reactor power is in the power range, a power increase will cause
moderator temperature to increase. If the core is undermoderated (negative
MTC), the temperature increase will insert negative reactivity into the core
and will slow the power rise. Since power is in the power range and steam
demand has not changed, reactor power will level off at the initial value and
moderator temperature will stabilize at a new value depending on the
amount of reactivity that initially caused the power increase.
The MTC responds to a power decrease in the power range in the opposite
manner (adding positive reactivity to slow the power decrease).
The MTC in equation form is:
18
Rev 1
𝜌𝑓𝑖𝑛𝑎𝑙 − 𝜌𝑖𝑛𝑖𝑡𝑖𝑎𝑙
∆𝜌
𝛼𝑚 = (
)=
∆𝑇𝑚𝑜𝑑
𝑇𝑚𝑜𝑑 𝑓𝑖𝑛𝑎𝑙 − 𝑇𝑚𝑜𝑑 𝑖𝑛𝑖𝑡𝑖𝑎𝑙
Where:
m = moderator temperature coefficient (MTC) (Δk/k/°F)
Δρ = change in reactivity associated with change in moderator
temperature (Δk/k)
ΔTmod = change in moderator temperature (°F)
The symbol m as well as the symbol T represent moderator temperature
coefficient. This text uses the symbol m.
Example:
A reactor is operating at 480°F with an effective multiplication factor of
1.000 (keff = 1.0). The moderator temperature increases to 490°F and keff
decreases to 0.999. What is the value of the moderator temperature
coefficient?
Solution:
First, convert keff values to reactivity.
𝜌=
𝑘𝑒𝑓𝑓 − 1
𝑘𝑒𝑓𝑓
1−1
=0
1
0.999 − 1
=
= −1.001 × 10−3
0.999
𝜌𝑖𝑛𝑖𝑡𝑖𝑎𝑙 =
𝜌𝑓𝑖𝑛𝑎𝑙
Then, calculate the value of MTC.
𝜌𝑓𝑖𝑛𝑎𝑙 − 𝜌𝑖𝑛𝑖𝑡𝑖𝑎𝑙
∆𝜌
𝛼𝑚 = (
)=
∆𝑇𝑚𝑜𝑑
𝑇𝑚𝑜𝑑 𝑓𝑖𝑛𝑎𝑙 − 𝑇𝑚𝑜𝑑 𝑖𝑛𝑖𝑡𝑖𝑎𝑙
(−1.001 × 10−3 ) − (0)
490℉ − 480℉
−1.001 × 10−3 ∆𝑘\𝑘
𝛼𝑚 =
10℉
∆𝑘/𝑘
𝑝𝑐𝑚
= −1.001 × 10−4
= −10
℉
℉
𝛼𝑚 =
𝛼𝑚
Value of Moderator Temperature Coefficient
Boron concentration and time in core life affect the value of MTC. A good
approximation of the MTC is -1 x 10-4Δk/k/°F for the normal operating
range of moderator temperatures in a commercial nuclear reactor.
Rev 1
19
Knowledge Check
A reactor is operating at 560°F with keff = 1.0. The
reactor operator borates the reactor an equivalent of 200
pcm (negative reactivity). RCS temperature responds by
dropping 10 degrees. Assuming no other reactivity
effects what is the MTC?
A.
5 x 10-4 Δk/k/°F
B.
20 x 10-4 Δk/k/°F
C.
10 x 10-4 Δk/k/°F
D.
2 x 10-4 Δk/k/°F
ELO 2.3 Moderator to Fuel Ratio Effects on MTC
Introduction
Moderator temperature coefficient (MTC) values are not constant
throughout core life. As we have learned, the moderator-to-fuel ratio has an
effect on whether or not keff increases or decreases with a moderator
temperature change. In terms of a reactivity coefficient, this translates to
either a positive or negative moderator temperature coefficient. This
section discusses how the following parameters affect MTC:




Overmoderation and undermoderation of the moderator-to-fuel ratio
Moderator temperature
Core age
Boron concentration
Moderator to Fuel Ratio Effects on MTC
The moderator-to-fuel ratio (Nm/Nu) is very important in the discussion of
moderators. The reactor designer adjusts the amount of moderator and fuel
in the core (Nm/Nu ratio) to an optimum value that establishes a negative
MTC throughout core life based on this ratio; however, it is possible during
specific core age and core parameters for a positive MTC to exist.
Moderator temperature affects moderator density and causes the moderatorto-fuel ratio to change. Changes in the moderator-to-fuel ratio affect the
thermal utilization factor (f) and the resonance escape probability (p) which
in turn affect keff and reactivity or more precisely the MTC.
It is possible to design the moderator-to-fuel ratio to be either
undermoderated (too little moderator) or overmoderated (too much
moderator). An overmoderated condition leads to a positive MTC
(undesirable) while an undermoderated condition leads to a negative MTC.
The amount of over or under moderation determines the magnitude of the
MTC.
Commercial PWRs are designed to operate in an undermoderated condition
because of the design requirement to have a negative MTC. The following
graphic illustrates this:
20
Rev 1
Figure: Moderator to Fuel Ratio Curves
Undermoderation
The area to the left of the dotted vertical line is the undermoderated region.
Notice also that at the dotted line, the keff curve peaks. In the
undermoderated region, a decrease in the moderator-to-fuel ratio results in a
decrease in keff, equivalent to negative reactivity. Relating this to
temperature, as temperature is increased, the concentration of the moderator
(Nm) decreases, causing Nm/Nu to decrease (move to the left). This is a
negative MTC.
As you previously learned, the changes in thermal utilization factor (f) and
the resonance escape probability (p) are the main causes for the change in
keff. Using the same illustration of a temperature increase with a decreasing
Nm/Nu, we see that the thermal utilization factor increases while the
resonance escape probability decreases.
In this case, the effect from the resonance escape probability overrides the
effect from the thermal utilization factor leading to a negative MTC. It is
the balance of these two factors (the curves have different slopes) that
determines the magnitude of the MTC (while undermoderated) because one
of these is a positive effect and the other is negative. Recall that the nonleakage factors have a small influence on MTC, which also cause it to be
negative.
Operating in the undermoderated region is very important to reactor control.
The moderator temperature will rise, inserting negative reactivity, thereby
limiting the magnitude of the power excursion if reactor power suddenly
increases. Commercial nuclear reactors are designed with a moderator-tofuel ratio such that MTC is negative in the normal operating temperature
range.
Overmoderation
The area to the right of the dotted vertical line is the overmoderated region.
In the overmoderated region, a reduction in moderator density (temperature
increase) has a greater effect on thermal utilization factor than the resonance
Rev 1
21
escape probability. With f greater than p, keff increases, equivalent to
positive reactivity.
If the reactor operates in the overmoderated region, any increase in reactor
power would result in an increase in moderator temperature. This effect
feeds itself; the increase in moderator temperature adds more positive
reactivity, resulting in an additional increase in reactor power, and even
higher temperatures and higher power. Safe control of the reactor and
maintaining operation within the core operating limits is much more
difficult with a positive MTC (also referred to as PTC).
Moderator Temperature Effects on MTC
As illustrated with the explanation of under- and overmoderation, the
moderator density change affects the moderator-to-fuel ratio, not the
moderator temperature. An increase in moderator temperature results in a
decrease in moderator density. Conversely, a decrease in moderator
temperature results in an increase in moderator density. As we know,
commercial reactors (in USA) use light water as both a coolant and a
moderator.
Another feature of water is that at higher temperatures, the density change
per degree F of water is greater. The figure below shows this relationship:
Figure: Water Density Change versus Moderator Temperature
A greater density change at higher moderator temperatures means a larger
change in the moderator-to-fuel ratio leading to a larger value MTC. The
result is larger absolute value of MTC at high temperatures (500°F to
550°F) than at lower temperatures (100°F to 150°F range).
22
Rev 1
Two points of clarification about the lower absolute
value of MTC at lower moderator temperature:
Note
1. The reactor is only made critical at normal
operating temperatures (around 550°F), so lower
MTC values at lower temperatures are of no
concern when critical.
2. A lower MTC at lower temperatures is a good
thing in regards to a steam break accident, where a
rapid cooldown would cause a large insertion of
positive reactivity for a possible reactor restart
accident. However, accident analysis considers
worst case, which would be a higher MTC.
Boron Concentration Effects on MTC
The discussion so far has considered the moderator to be pure water. This
makes the moderator-to-fuel ratio effect on MTC easier to explain.
However, the moderator is not pure water. Commercial PWRs use soluble
boron, referred to as boric acid, added to the moderator to provide a variable
reactivity poison for control of kexcess, maintaining Tavg in the program band
during power changes, compensating for fission product poisons, and
reactivity adjustment to "trim" the control rods fully withdrawn at 100
percent power.
Boron has a high thermal neutron absorption cross section, adding negative
reactivity to the core much as control rods do - the higher the concentration
of boron the more negative reactivity. Boron concentration is decreased
(diluted) adding positive reactivity to compensate for the negative reactivity
from fuel depletion over the life of a reactor core, as fuel depletes.
The presence of boron in the moderator affects the value of the MTC.
Higher boron concentrations have a greater the effect on the MTC. The
presence of boron in the coolant results in a reduction in the value of the
thermal utilization factor (f) since boron is a neutron absorber. Remember
the ratio for f:
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑢𝑒𝑙
𝑓=
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 𝑖𝑛 𝑎𝑙𝑙 𝑟𝑒𝑎𝑐𝑡𝑜𝑟 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙𝑠
From the formula, as boron absorbs more neutrons, the number of thermal
neutrons absorbed in all reactor material increases, causing f to decrease.
Therefore, increasing the soluble boron concentration causes f to decrease,
which, in turn causes keff to decrease, adding negative reactivity.
When soluble boron is added to the moderator, it becomes an integral part
of the moderator and therefore affects the moderator-to-fuel ratio. Consider
the figure below that illustrates the response of the thermal utilization factor
(f) on moderator/coolant boron concentration.
Rev 1
23
Figure: Boron Effect on the Thermal Utilization Factor
On this family of curves, the area of interest is toward the left side. Notice
that as boron concentration is increased, the slope of the curve (change in f)
becomes steeper. This means that at high boron concentrations, for a given
change in Nm/Nu (or Nmod/Nfuel), the thermal utilization factor will have a
greater change in value. When the density of the moderator changes, Nm
changes, and so does NB (boron concentration). Higher boron
concentrations (atoms/cm3) yield a greater change in NB for the same
temperature (density) change.
The thermal utilization factor (f) and resonance escape probability (ρ) are
two factors affected by moderator temperature. They determine both the
magnitude of MTC, and whether the reactivity coefficient is positive or
negative. Recall that f is the positive factor while ρ is the negative factor to
MTC.
The figure above shows that with high boron concentrations, thermal
utilization becomes a bigger factor. In fact, at very high boron
concentrations (possible after refueling), the thermal utilization factor can
override the negative effect from resonance escape probability resulting in a
positive MTC. Boron has minimal effect on the resonance escape
probability since it is predominantly a thermal neutron absorber.
Remember the ratio for ρ:
𝜌=
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑡ℎ𝑎𝑡 𝑟𝑒𝑎𝑐ℎ 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑠𝑡 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑡ℎ𝑎𝑡 𝑠𝑡𝑎𝑟𝑡 𝑡𝑜 𝑠𝑙𝑜𝑤 𝑑𝑜𝑤𝑛
The MTC becoming less negative as the boron concentration of the
moderator/coolant increases means that the boron concentration must be
limited to prevent the MTC from becoming positive during power
operations. Some plants are allowed to operate with a positive MTC up to
24
Rev 1
some designated power level for a short period, but beyond that, a negative
MTC is required for safety considerations.
By the time the unit is at full power (or before), sufficient buildup of fission
product poisons has occurred, requiring the operators to reduce boron
concentration to compensate, and thereby establishing a negative MTC.
For example, at the beginning of core life (BOL), when the boron
concentration is high, the MTC may be +0.1 x 10-4 Δk/k/°F. At the end of
core life (EOL), after significant boron dilution, the MTC is approximately 2.6 x 10-4 Δk/k/°F.
Alternate Explanation
Another way to look at this concept is to consider a moderator temperature
increase of one degree Fahrenheit (1°F). This temperature increase causes
three effects:
The boron concentration (atoms/cm3) decreases, resulting in a positive
reactivity insertion. Thermal utilization factor increases.
 Decreased moderator density, fewer water atoms (Nm) causes the
thermal utilization factor (f) to increase slightly, causing a positive
reactivity insertion. This insertion is smaller than the insertion due to
the boron effects (depending on boron concentration).
— This positive reactivity insertion is a result of fewer water
molecules and boron atoms per cubic centimeter (cm3) available
for absorption reactions within the reactor core.
 The resonance escape probability (ρ) decreases due to fewer
moderator molecules per cm3 being present in reactor core.
Therefore, neutrons travel further and resonance capture is more
likely, resulting in an insertion of negative reactivity.
The processes listed above are three competing effects that take place with a
moderator temperature increase. For higher boron concentrations, MTC
tends to be less negative (or even positive). Conversely, as boron
concentration approaches zero, MTC tends to be more negative. Therefore,
as previously explained, MTC at the beginning of core life (BOL) can be
slightly positive, whereas the MTC at the end of core life (EOL) will be at
its most negative value.

Core Age Effects on MTC
The MTC becomes more negative over core life. The primary reason for
this effect is the decrease in RCS boron concentration as discussed
previously.
Rev 1
25
Note
Commercial PWRs are also limited on how negative the
MTC can become. This restriction is required because
of the Main Steam Line Break Accident. During a steam
line break accident, the reactor coolant system (RCS)
will undergo a rapid cooldown because the steam system
begins to act like an infinite heat sink. This rapid
cooldown will result in large positive reactivity insertion
to the reactor core from the MTC. Some plant accident
analyses demonstrate that the reactor could actually be
rendered supercritical with all control rods fully inserted.
An example of such a limit on the MTC is a value such
as -44 pcm/°F (-4.4 x 10-4 Δk/k/°F).
Knowledge Check
As the reactor coolant boron concentration increases, the
moderator temperature coefficient becomes less
negative. This is because a 1°F increase in reactor
coolant temperature at higher boron concentrations
results in a larger increase in the _______________.
A.
fast fission factor
B.
thermal utilization factor
C.
total nonleakage probability
D.
resonance escape probability
ELO 2.4 Void and Pressure Reactivity Coefficients
Introduction
Void (steam bubbles) and pressure coefficients play a very small role in the
reactivity balances for a commercial PWR compared to MTC. Rules of
thumb for pressure are 100 psi is equal to 1°F temperature change and at
full power voids may occupy about 0.5 percent of the total moderator
volume. Any changes in pressure and voiding large enough to make
significant reactivity changes in normal operating bands do not occur.
The pressure coefficient of reactivity is the result of the effect of pressure
on the density of the moderator. The pressure coefficient of reactivity is the
change in reactivity per unit change in pressure (Δk/k/psi). This implies
that for a given pressure change, a certain amount of water density change
occurs, which, causes a change in reactivity (like the moderator temperature
effect on density).
As pressure increases, density increases, increasing the moderator-to-fuel
ratio. In the undermoderated core, the increase in the moderator-to-fuel
ratio results in positive reactivity addition. Therefore, the pressure
coefficient is a positive reactivity coefficient.
26
Rev 1
A 100-psi increase in pressure causes approximately the same reactivity as a
one-degree decrease in temperature relating the pressure coefficient to
MTC. A typical value for the pressure coefficient of reactivity in a
commercial PWR is 1 x 10-6 Δk/k/psi.
For PWRs, the overall reactivity effect of the pressure coefficient is a minor
factor in normal operation because it is much smaller than the MTC.
Void Coefficient
The void coefficient quantifies the effect that the formation of steam voids
in the moderator has on the MTC. The void coefficient is the change in
reactivity per percent change in void volume (Δk/k/percent void). In
commercial PWRs, the amount of voids is very small; however, in boiling
water reactors (BWR) it is very significant. This discussion is limited to
PWRs.
Voiding may occur in a PWR when power increases to higher levels. These
voids displace moderator from the coolant channels within the core. This
reduces the moderator-to-fuel ratio, and in an undermoderated core, results
in a negative reactivity addition limiting further power increase. The void
coefficient is a negative coefficient.
Moderator Density Effects on Void Coefficient
Bulk boiling of the moderator/coolant does not occur in a PWR; however,
steam bubbles will form in the moderator/coolant around the fuel elements
as reactor power increases. The moderator/coolant sweeps these bubbles
into the bulk coolant where they collapse.
Voids have the effect of reducing the moderator density in the area of the
void. The result is similar to an increase in moderator/coolant temperature
that lowers moderator density. A decreased density causes a decrease in the
resonance escape probability (ρ), an increase in the thermal utilization
factor (f), and an overall decrease in keff.
As with MTC, the dominant effect is the decrease in resonance escape
probability making the void coefficient negative. An approximate value in
a commercial PWR reactor is -1 x 10-3 Δk/k/percent void.
Voids occupy about 0.5 percent of the total moderator/coolant volume at
full power, so like the pressure coefficient, total reactivity inserted by the
void fraction is very small compared to MTC.
Example:
Compute the approximate negative reactivity due to voids in a pressurized
water reactor (PWR) at 100 percent reactor power.
Given:
∆𝑘/𝑘
% 𝑣𝑜𝑖𝑑𝑠
Void fraction at 100 percent power = 0.6 percent
𝛼𝑣 = −1 × 10−3
Rev 1
27
Solution:
∆𝜌𝑣𝑜𝑖𝑑𝑠 = −1 × 10−3
∆𝑘/𝑘
× 0.6% 𝑣𝑜𝑖𝑑𝑠
% 𝑣𝑜𝑖𝑑𝑠
∆𝜌𝑣𝑜𝑖𝑑𝑠 = −0.6 × 10−3 ∆𝑘/𝑘
∆𝜌𝑣𝑜𝑖𝑑𝑠 = −60 𝑝𝑐𝑚
Many plants combine and include the pressure and void
coefficients into the power coefficient because their
reactivity effect is relatively small.
Note
Knowledge Check
Concerning the reactivity affects from the void and
pressure coefficients, which one of the following
statements is true?
A.
The pressure and void coefficient are both negative.
B.
The pressure and void coefficients are both positive.
C.
The void coefficient is negative and the pressure
coefficient positive.
D.
The voice coefficient is positive and the pressure
coefficient negative.
TLO 2 Summary
1. Reactivity coefficients and reactivity
 Reactivity coefficients are the amount that the reactivity will change
for a given change in the parameter (per °F, per ppm boron, etc.).
 Generally, αx symbolizes reactivity coefficients, where x represents
some variable reactor parameter that affects reactivity.
 Reactivity defects (∆ρ) are the total reactivity change caused by a
variation in a parameter.
 Reactivity defects are determined by multiplying the total change in
the parameter by its average coefficient value. The equation below
relates reactivity coefficients to reactivity defects.
∆𝜌 = 𝛼𝑥 ∆𝑡
2. Moderator temperature coefficient of reactivity.
 The reactivity change per degree change in moderator temperature is
the moderator temperature coefficient (MTC) of reactivity.
 MTC is primarily a function of the moderator-to-fuel ratio, density of
the moderator, and boron concentration.
 PWRs are designed with an undermoderated moderator-to-fuel ratio
that provides a negative moderator temperature coefficient except
sometimes early in core life.
 Negative MTC is more desirable because of its power level regulating
effect.
28
Rev 1

MTC works by turning power down when a power increase causes
moderator temperature to increase. The increase in moderator
temperature, adds negative reactivity (MTC) causing reactor power to
stop its increase.
 The MTC in equation form is:
∆𝜌
𝛼𝑚 = (
)
∆𝑇𝑚𝑜𝑑
Approximation of the MTC is -1 x 10-4Δk/k/°F.
The reactor designer adjusts the amount of moderator with the fuel in
the core (Nm/Nu ratio) to an optimum value to ensure a negative MTC
throughout core life.
 Changes in the moderator-to-fuel ratio affect the thermal utilization
factor (f) and the resonance escape probability (ρ), in turn affecting
keff and reactivity or more precisely the MTC.
 It is possible to design the moderator-to-fuel ratio to be either
undermoderated (too little moderator) or overmoderated (too much
moderator).
3. Moderator temperature coefficient variations.
 An overmoderated condition leads to a positive MTC (undesirable)
while an undermoderated condition leads to a negative MTC.
 Commercial PWRs are designed to operate in an undermoderated
condition because of the design requirement to have a negative MTC.
 In the undermoderated region, a decrease in the moderator-to-fuel
ratio results in a decrease in keff, equivalent to negative reactivity.
Relating this to temperature, as temperature is increased,
concentration of the moderator (Nm) decreases, causing Nm/Nu to
decrease (move to the left). This is a negative MTC.
— Thermal utilization factor increases while the resonance escape
probability decreases.
— The balance of these two factors (curves have different slopes)
determines the magnitude of the MTC (while undermoderated).
 In the overmoderated region, a reduction in moderator density
(temperature increase) has a greater effect on thermal utilization
factor than the resonance escape probability. With f greater than ρ,
keff increases, equivalent to positive reactivity.
 The density change per degree F of water is greater at higher
temperatures.
 A temperature increase causes three effects on boron in the
moderator:
— The boron concentration (atoms/cm3) decreases, resulting in a
positive reactivity insertion. Thermal utilization factor
increases.
— Decreased moderator density, fewer water atoms (Nm) causes the
thermal utilization factor (f) to increase slightly, causing a
positive reactivity insertion. This insertion is smaller than the
insertion due to the boron effects (depending on boron
concentration).


Rev 1
29
4.
5.
6.
7.
30
— This positive reactivity insertion is a result of fewer water
molecules and boron atoms per cubic centimeter (cm3) available
for absorption reactions within the reactor core.
The resonance escape probability (ρ) decreases due to fewer moderator
molecules per cm3 being present in reactor core, neutrons travel further,
resonance capture is more likely, resulting in an insertion of negative
reactivity.
With the MTC becoming less negative as the boron concentration of the
moderator increases, the boron concentration must be limited to prevent
the MTC from becoming positive during power operations.
Some plants may operate with a positive MTC up to some designated
power level for a short period, but beyond that, a negative MTC is
required for safety considerations.
 By the time the unit is at full power (or before) sufficient buildup of
fission product poisons has occurred, requiring the operators to reduce
boron concentration to compensate, and thereby reestablishing a
negative MTC
 MTC becomes more negative as a nuclear reactor core life increases the primary reason is the decrease in RCS boron concentration.
Void and pressure coefficients of reactivity
 The pressure coefficient of reactivity is the result of the effect of
pressure on the density of the moderator. The pressure coefficient of
reactivity is the change in reactivity per unit change in pressure
(Δk/k/psi). This implies that for a given pressure change, a certain
amount of water density change occurs, which like the moderator
temperature effects to density, causes a change in reactivity.
 As pressure increases, density increases, increasing the moderator-tofuel ratio. In the undermoderated core, this results in positive
reactivity addition. Therefore, the pressure coefficient is a positive
reactivity coefficient.
 A 100-psi increase in pressure causes approximately the same
reactivity as a one-degree decrease in temperature. The pressure
coefficient of reactivity has a typical value of 1 x 106 Δk/k/psi. The
pressure coefficient effect is much smaller than the MTC effect.
 The void coefficient quantifies the effect that the formation of steam
voids in the moderator has on the MTC. The void coefficient is the
change in reactivity per percent change in void volume (Δk/k/percent
void). In commercial PWRs, the amount of voids is very small.
 Voiding (steam bubbles) may occur when power increases to higher
levels. These voids displace moderator from the coolant channels
within the core, reducing the moderator-to-fuel ratio, and in an
undermoderated core, results in a negative reactivity addition.
— An approximate value in a commercial PWR reactor is -1 x 10-3
Δk/k/percent void.
— At full power, voids occupy about 0.5 percent of the total
moderator/coolant volume
— Void and pressure coefficients total reactivity is very small
compared to MTC.
Rev 1
Now that you have completed this lesson, you should be able to:
1. Explain differences between reactivity coefficients and reactivity
defects, and how they are used to balance reactivity parameters.
2. Describe the moderator temperature coefficient of reactivity.
3. Describe how the magnitude of the moderator temperature coefficient
varies with changes in the following parameters:
a. Overmoderation and undermoderation of the moderator-to-fuel
ratio
b. Moderator temperature
c. Core age
d. Boron concentration
4. Describe the void and pressure coefficients of reactivity.
TLO 3 Fuel Temperature and Power Coefficients
Overview
This session discusses the fuel temperature coefficient, otherwise known as
Doppler broadening or Doppler and power coefficient/defect. It is
important to understand all reactivity coefficients and defects for safe
reactor operations.
The MTC provides an inherent safety feature for PWRs; the fuel
temperature coefficient (FTC) is just as much an inherent safety feature in
that it adds negative reactivity on a power/fuel temperature increase and, as
an added benefit, it is fast acting. This lesson explains Doppler functions
and the Doppler effects on reactor operation.
Objectives
Upon completion of this lesson, you will be able to do the following:
1. Describe the fuel temperature coefficient of reactivity.
2. Explain resonance absorption, Doppler broadening, and selfshielding.
3. Describe how the magnitude of the fuel temperature coefficient varies
with changes in the following parameters:
a. Moderator temperature
b. Fuel temperature
c. Core age
4. Describe the components of the power coefficient of reactivity and
the magnitude of their overall effect over core life.
5. Explain how the power defect affects the reactivity balance on reactor
power operations.
ELO 3.1 Fuel Temperature Reactivity Coefficient
Introduction
Another temperature coefficient of reactivity, the FTC, has a large effect on
reactivity. The FTC is the change in reactivity per degree change in fuel
temperature (Δk/k/°F). Usually, the two dominant temperature coefficients in
a reactor are the moderator temperature coefficient and the FTC.
Rev 1
31
This FTC also responds quicker to an increasing power transient than MTC,
because reactor power causes an immediate increase in fuel temperature.
The moderator lags due to the time for the transfer of heat from the fuel to
the moderator. This is also true for decreasing power (fuel temperature
decrease). The exception to this is when a change in steam demand initiates
the power transient by changing moderator temperature and causing
reactivity to be inserted into the core.
A negative FTC is an important safety feature inherent to PWRs, similar to
the MTC. In the event of a large positive reactivity insertion, because of the
delay in the moderator temperature change, MTC cannot slow the reactor
power rise for several seconds, whereas the FTC starts adding negative
reactivity immediately.
Fuel Temperature Reactivity Coefficient
Another name applied to the FTC is the Doppler reactivity coefficient, often
shortened to Doppler. This coefficient was named after the Doppler Effect
or Doppler broadening of the resonance peaks of U-238 and Pu-240.
The phenomenon of Doppler broadening occurs when the fuel temperature
increases and causes the target nucleus to have more energy. As a result,
the relative energy between the target nucleus and the incident neutron
changes and the acceptable neutron energy band that the nucleus will absorb
will widen.
The actual peak for the microscopic cross-section will lower. However, the
dominant effect is that the nucleus will absorb a broader band of neutrons
(off-peak neutrons). This effect is plays a dominant role in low enriched
cores since there is much more U-238 in the core.
Figure: Doppler Broadening
The broadening of the peaks occurs as fuel temperature increases, making
resonance capture more likely. Therefore, the resonance escape probability
decreases, causing keff to decrease due to the addition of negative reactivity.
Uranium-238 and plutonium-240 are the two significant nuclides with large
resonant peaks.
32
Rev 1
Fuel Temperature Coefficient or Doppler Coefficient
The FTC is the change in reactivity per unit change in fuel temperature.
𝜌𝑓𝑖𝑛𝑎𝑙 − 𝜌𝑖𝑛𝑖𝑡𝑖𝑎𝑙
∆𝜌
𝛼𝐷 = (
)=
∆𝑇𝑓𝑢𝑒𝑙
𝑇𝑓𝑢𝑒𝑙 𝑓𝑖𝑛𝑎𝑙 − 𝑇𝑓𝑢𝑒𝑙 𝑖𝑛𝑖𝑡𝑖𝑎𝑙
Where:
D = Doppler coefficient (FTC) (Δk/k/°F)
Δρ = change in reactivity associated with change in fuel temperature
(Δk/k)
ΔTfuel = change in fuel temperature (°F)
In low enrichment reactor fuel (commercial reactors), most of the uranium
found in the fuel rods is uranium-238 (plutonium-240 builds in over core
life). The magnitude of the Doppler coefficient in PWRs is about -1 x 10-5
Δk/k/°F, or -1 pcm/°F.
Doppler Defect
Although the coefficient is small, the defect can be a very high value
because of reactor power level changes from 0 to 100 percent during power
operations. The average fuel temperature at 100 percent reactor power is
about 2,200°F; however, peak fuel temperature in some fuel rods could be
greater than 3,000°F. Because of this, the magnitude of the change in
reactivity due to fuel temperature changes is large.
The figure below shows an example plot of Doppler defect and rated power:
Figure: Doppler Defect vs. Rated Reactor Core Power
Example
A reactor with an effective multiplication factor of 1.009 (keff = 1.009) has a
fuel temperature of 100°F. When fuel temperature is raised to 600°F, keff =
1.000.
What is value of Doppler coefficient?
Rev 1
33
Solution:
First, solve for ρinitial and ρfinal.
(1.009 − 1)
= 8.92 × 10−3 ∆𝑘/𝑘
1.009
1−1
𝜌𝑓𝑖𝑛𝑎𝑙 =
=0
1
Then, use the above equation to solve for D using ρinitial and ρfinal.
𝜌𝑖𝑛𝑖𝑡𝑖𝑎𝑙 =
𝜌𝑓𝑖𝑛𝑎𝑙 − 𝜌𝑖𝑛𝑖𝑡𝑖𝑎𝑙
∆𝜌
𝛼𝐷 = (
)=
∆𝑇𝑓𝑢𝑒𝑙
𝑇𝑓𝑢𝑒𝑙 𝑓𝑖𝑛𝑎𝑙 − 𝑇𝑓𝑢𝑒𝑙 𝑖𝑛𝑖𝑡𝑖𝑎𝑙
(0) − (8.92 × 10−3 ∆𝑘/𝑘
600℉ − 100℉
−8.92 × 10−3 ∆𝑘/𝑘
𝛼𝐷 =
500℉
∆𝑘/𝑘
𝛼𝐷 = −1.78 × 10−5 (
)
℉
𝛼𝐷 =
Doppler Coefficient Mechanism
A fuel temperature increase causes higher vibrational frequency of the fuel
atoms. This increases neutron absorption by uranium-238 and plutonium240 (Doppler). As shown in the previous Doppler Broadening figure, the
movement of uranium-238 atoms relative to incident high velocity neutrons
results in a broadening and flattening of the resonance absorption peaks;
however, the total area under the resonance peak curve will remain
essentially the same.
The overall effect is that the incident neutrons encounter a higher absorption
cross section over a wider range of neutron energies, resulting in more
resonance absorptions and a decrease in keff. Later sections will provide
more detail on this.
Importance of the Doppler Coefficient
The importance of the Doppler coefficient is that fuel temperature
immediately increases following an increase in reactor power. Uranium
oxide (UO2) (the fuel pellets) is a relatively poor conductor of heat and the
cylindrical fuel rods have a small heat transfer surface per unit volume. It
requires a relatively long time for transfer of the heat generated at any
instant to the moderator/coolant. This time may be 7 to 9 seconds.
In the event of a large positive reactivity addition to the reactor, the MTC
will be subject to this time delay, and therefore have a delayed effect in
countering the insertion of positive reactivity.
On the other hand, the Doppler coefficient, because of its direct association
with the fuel itself, responds immediately. This is why some refer to
Doppler coefficient as the "prompt" coefficient, and MTC as the "delayed"
coefficient. With the Doppler coefficient responding first to an accidental,
34
Rev 1
large positive reactivity addition, Doppler is of paramount importance in the
event a rod ejection accident or other rapid positive reactivity insertion.
Knowledge Check
If fuel temperature decreases by 50°F, the area under the
resonance peak curve will ___________ and positive
reactivity will be added to the core because
____________.
A.
decrease; fewer neutrons will be absorbed by uranium238 overall
B.
decrease; fewer 6.7 eV neutrons will be absorbed by
uranium-238 at the resonance energy
C.
remain the same; fewer neutrons will be absorbed by
uranium-238 overall
D.
remain the same; fewer 6.7 eV neutrons will be absorbed
by uranium-238 at the resonance energy
ELO 3.2 Doppler and Self-shielding
Introduction
Doppler is generally associated with the physics of sound and light, but it
also apples to nuclear physics. The Doppler Effect (or Doppler shift) is the
change in frequency of a sound wave for a listener as the source moves. It
is heard when a vehicle sounding a siren or horn approaches, passes, and
recedes from an observer. Compared to the emitted frequency, the received
frequency is higher during the approach, identical at the instant of passing
and lower during the recession.
We use a source of sound waves moving toward the listener to explain this
phenomenon. As the source moves toward the listener, the source emits
each successive sound wave peak from a position closer to the listener than
the previous sound wave. Therefore, each sound wave takes slightly less
time to reach the listener than the previous one, and the time between
successive sound wave peaks deceases. This is the increase in sound
frequency. The opposite is true when the source of sound is moving away.
In nuclear reactor fuel, Doppler Effect explains the probability of resonant
absorption as a function of the fuel's temperature. Assume a stationary
nucleus will absorb only neutrons of a specific energy Eo. If the nucleus is
moving away from the neutron, the velocity (and energy) of the neutron
must be greater than Eo to undergo resonance absorption. If the nucleus is
moving toward the neutron, the neutron needs less energy than Eo to be
absorbed.
Raising the nuclei temperature causes more rapid vibration within their
lattice structures, in effect broadening the energy range of neutrons for
resonance capture, known as Doppler broadening.
Rev 1
35
Doppler Broadening and Resonance Capture
Neutrons give up energy incrementally via collisions with the nuclei of
materials present in the reactor; this is the purpose of the moderator. The
microscopic cross section for absorption (σa) for uranium-238 is 5,500 barns
for neutrons at 21 eV.
However, the microscopic cross-section for absorption is only 15 to 20
barns for a neutron with an energy level of 20 or 22 eV; either side of 21
eV. These "resonance" peaks, where absorption is most likely to occur, are
where the neutron losses occur from resonance capture or resonance
absorption. The resonance escape probability is the probability that a
neutron will pass through these energy levels without capture.
The figure below shows the U-238 resonance capture cross sections as a
function of neutron energy for two different fuel temperature conditions,
room temperature vs. reactor operating conditions.
Figure: Uranium-238 Cross-Section for Absorption Curve
The relative motion between the incident neutron and the target nucleus
(Doppler Effect) influences the resonance capture cross section for
uranium-238. The average kinetic energy of the uranium-238 nucleus
increases as the temperature increases.
The cross section peak decreases, but the energy spectrum broadens with
increasing temperature. Overall, the likelihood of a neutron capture
increases. This is the Doppler Effect. The motion (KE) or vibration of the
nucleus has a direct impact on its magnitude of capture cross section.
36
Rev 1
To demonstrate this Doppler Effect with different neutron and nucleus
energies, consider the three neutron reactions depicted in the following
figure.
Figure: Doppler Effect in Uranium-238 Resonance Capture
Suppose an incident neutron having 21 eV of kinetic energy impinges on a
target nucleus at room temperature (roughly 0.025 eV), as shown in a. in the
previous figure. The microscopic cross section for absorption for uranium238 at 21 eV is 5,500 barns and the neutron is likely to be absorbed.
Next, consider a 20 eV neutron interacting on a nucleus that is vibrating
toward it with kinetic energy of 1eV, shown in b. in the previous figure.
The relative energy between the incident neutron and target uranium-238
nucleus is, once again 21 eV. The effective absorption cross section is
about 5,500 barns and the neutron is likely to be absorbed as with the
previous example.
In the last example, c. above, the incident neutron possesses KE of about 22
eV, and the target uranium-238 nucleus is vibrating away from the neutron
with KE of 1 eV. The relative energy between the incident neutron and the
target uranium-238 nucleus is, once again 21 eV. The effective absorption
Rev 1
37
cross section is about 5,500 barns and the neutron is likely to be absorbed as
with the previous examples.
These examples depict the Doppler Effect. The KE of the fuel atoms
increases, resulting in neutrons of both higher and lower KE (than required
at room temperature) having an equal probability of resonance absorption
by the fuel atoms as fuel temperature increases. The figure below provides
another illustration of Doppler Effect.
Figure: Resonance Capture in Nucleus Vibrating at 5 eV
The figure above illustrates the effect of heat energy applied to a nucleus.
Upon adding 5 eV of heat energy to the nucleus, the nucleus vibrates
rapidly in all directions. The nucleus still prefers a 21 eV neutron, and has a
high cross section only for neutrons of 21 eV. The nucleus now absorbs
any neutron within the KE range of 16 eV to 26 eV (+ or - 5 eV), depending
upon the neutrons' angle of approach to the nucleus because of the relative
motion between the nucleus and the surrounding neutrons. The motion
between the neutron and the nucleus must be sufficient for a neutron to
"appear" to the nucleus as a 21 eV neutron.
Its speed and area of motion due to vibration increases; however, because it
is vibrating faster, it now spends less time at any given energy within its KE
range if more heat energy is added to the nucleus. The nucleus now has the
capability of capturing "off-resonance" neutrons of 16 eV and 26 eV
respectively. The probability for capturing a 21 eV "resonance" neutron has
decreased, but the probability of capturing neutrons in the 16 eV to 26 eV
range has increased.
38
Rev 1
The net result of heating nuclear fuel is to "broaden" and flatten the
uranium-238 resonance capture cross-section curve. This shift in resonant
capture cross section peaks for uranium-238 is Doppler broadening.
The effects of Doppler broadening result in a modified capture cross section
curve, as shown in a previous figure of the uranium-238 cross-section for
absorption curve. The area under both the original and the broadened curve
is theoretically the same. Therefore, you might assume that the overall
capture of neutrons by uranium-238 would not change significantly.
However, research proves that broadening of the uranium-238 capture cross
section curve increases the resonant neutron capture in uranium oxide (UO2)
fuel pellets. We consider the effects of self-shielding within the fuel pellet
to explain this.
Self-Shielding
The fuel in a commercial nuclear reactor is constructed of ceramic pellets
that are housed in a helium gas-filled, Zircaloytm-clad, cylindrical fuel pin.
The surrounding moderator slows down neutrons (thermalizes). Highenergy neutrons pass through the fuel pellets and the surrounding cladding
into the moderator. The moderator slows the neutrons down into the
epithermal (intermediate) and thermal energy range.
A neutron entering the fuel pellet with the exact resonant energy has a very
high probability of absorption at low fuel temperatures, most likely in the
outer edge of fuel pellet. Epithermal neutrons of other than resonant
energies are more likely to pass directly through the pellet without being
absorbed. The outer fuel atoms tend to shield the inner fuel atoms from
resonant energy neutrons. This is termed self-shielding.
Consider two uranium oxide fuel pellets, one at room temperature and
another at operating reactor fuel temperature, to further explain selfshielding. Refer to the figure below:
Figure: UO2 Fuel Pellet at Room and Operating Reactor Temperature
At room temperature (part a), only resonance neutrons would be captured,
as shown by the 21 eV resonance neutron with the UO2 fuel pellet. OffRev 1
39
resonance neutrons would pass right through and not be "seen" by the UO2
fuel pellet. The inner region of the pellet is termed "self-shielded" by the
outer periphery because the resonance neutron is captured immediately as it
enters the fuel pellet and off-resonance neutrons are not captured.
Part b of the previous figure illustrates what happens when the fuel pellet is
at an elevated temperature. The uranium-238 nuclei tend to capture both
resonance and off-resonance neutrons because of increased vibration due to
increased heat energy (Doppler Effect). The central portion of the fuel
pellet now tends to capture both off-resonance and resonance neutrons
because there is a reduction in fuel pellet self-shielding with the higher
temperatures.
We must consider two issues to determine the amount of self-shielding that
occurs:


Physical size of the fuel pellet
Design characteristics of the fuel pellet
The combination of these two effects determines the overall effect of fuel
temperature on resonance capture within a nuclear reactor core.
Physical Size of Fuel Pellets
The physical size of the fuel pellets and the average distance that a neutron
can travel into a pellet prior to resonance absorption determines if a neutron
will pass through the pellet without absorption. Recall that the mean free
path (Σ) is the average distance that a neutron travels before being absorbed.
The equation below gives the mean free path for absorption:
Σ𝑎 =
1
𝑁𝜎𝑎
Where:
Σa = mean free path (cm)
N = atomic density (atoms/cm3)
σa = microscopic cross section for absorption (barns)
The atomic density (N) is approximately 2 x 1022 atoms/cm3 for the
uranium-238 contained in a fuel pellet. For this discussion, assume that
every neutron is absorbed in three (3) mean free paths.
If 100 neutrons, all at 21 eV, enter the fuel pellet, then all neutrons are
absorbed if the fuel pellet is three mean free paths wide. (At 21 eV,
uranium-238 has a resonance peak of 5,500 barns).
Recall that 1 barn = 10-24 cm2. Therefore:
Σ𝑎 =
(2 ×
1
𝑏𝑎𝑟𝑛𝑠)(1 × 10−24 𝑐𝑚2 /𝑏𝑎𝑟𝑛)
1022 𝑎𝑡𝑜𝑚𝑠/𝑐𝑚3 )(5,500
Σ𝑎 = 0.009 𝑐𝑚
40
Rev 1
Since the average fuel pellet is 1.0 cm in diameter, all 100 neutrons at 21 eV
entering the fuel pellet will be absorbed (0.009 cm x 3 = 0.027 cm < 1 cm).
For neutrons that are not at an energy level of a resonance peak for
uranium-238, the microscopic cross section for absorption is about 15
barns.
This makes the mean free path for these neutrons 3.33 cm.
Σ𝑎 =
(2 ×
1022 𝑎𝑡𝑜𝑚𝑠/𝑐𝑚3 )(15
1
𝑏𝑎𝑟𝑛𝑠)(1 × 10−24 𝑐𝑚2 /𝑏𝑎𝑟𝑛)
Σ𝑎 = 3.33 𝑐𝑚
The fuel pellet would have to be about 10 cm (3 x 3.33 cm) in order for all
of these neutrons (not at 21 eV) to be absorbed in the uranium-238, or
approximately 4.0 inches in diameter. The uranium-238 in the fuel pellet
will absorb very few of the off-resonance neutrons.
Assume that 100 neutrons enter the fuel pellet at 22 eV and two of these are
absorbed in the pellet. The uranium-238 fuel pellet absorbs 102 of the 200
neutrons (we add the two absorptions to the 100-21 eV neutrons).
Now consider an increase in the fuel temperature. The microscopic crosssection for absorption of neutrons at energy levels equal to uranium-238
resonance peaks decreases, but the absorption cross section for neutrons
with energy levels near the resonance peaks increases due to Doppler
broadening. This means that for the 1.0 cm fuel pellet there are still 102
neutrons absorbed within the pellet.
However, now not all of the neutrons at an energy level corresponding to
the resonance peak (21 eV) are absorbed and more of the neutrons not at
resonance peak energy are absorbed. For this example, assume that at
600°F fuel temperature, 99 of the resonant energy (21 eV) neutrons are
absorbed and 3 off-resonance energy neutrons are absorbed. The total
number of neutrons absorbed is the same (102) but the number of resonant
and non-resonant energy neutrons absorbed has changed.
The microscopic cross section for absorption has decreased for the 21 eV
neutrons and increased for the 22 eV neutrons at this higher temperature.
Therefore, there is now a slight possibility that some of 21 eV-neutrons will
escape the fuel pellet without capture.
This decreasing of the microscopic cross section for absorption has the
effect of decreasing the self-shielding occurring within the fuel pellet. A 21
eV-neutron is likely to travel farther into the fuel pellet prior to capture, and
some may pass completely through the pellet without capture.
The off-resonance neutrons that normally would have passed completely
through the pellet now have an increased probability of capture by uranium238 within the fuel pellet at this higher temperature. At lower temperatures,
the average fuel pellet has a diameter smaller than the three mean free paths
needed for total neutron absorption and the internal portion of the fuel pin
does not see neutron flux from neutrons at resonance peak(s) energy.
Rev 1
41
If the fuel temperature is increased, the mean free path increases due to
decreased microscopic cross section (Doppler broadening) and more of the
fuel pellet now experiences resonance neutron flux energy levels. In other
words, as fuel temperature increases, self-shielding decreases.
If the diameter of the fuel pellet is sufficiently large compared to the mean
free path, the effect of self-shielding can be quite pronounced. Not all paths
that a neutron can take will lead through the center of the fuel pellet even
though the diameter of the fuel pellet may be 1 cm. Not all neutrons
entering a fuel pellet have the opportunity to travel 1 cm through the pellet.
In fact, the average straight-line distance a neutron travels through a fuel
pellet is about 0.625 cm.
Using this information, three mean free paths at 0.625 cm would equal a
distance of 0.625 cm divided by three or 0.21 cm of travel for one mean free
path. Using the mean free path equation, this yields a value of
approximately 238 barns as the microscopic cross section for absorption
with a 0.21 cm mean free path, as shown below.
𝜎𝑎 =
𝜎𝑎 =
1
𝑁Σ𝑎
1
(2 ×
1022 𝑎𝑡𝑜𝑚𝑠/𝑐𝑚2 )(0.21
𝑐𝑚)
𝜎𝑎 = 238 𝑏𝑎𝑟𝑛𝑠
For a real fuel pellet, any neutron at an energy level equal to a microscopic
cross section of greater than 238 barns will appear as a resonant energy
neutron and be absorbed in the fuel pellet.
Looking at the figure: Uranium-238 Cross Section for Absorption Curve,
for the energy levels with cross sections for absorption above 238 barns, if
the temperature of the fuel were to increase to 600°F, as in our example, the
energy levels for resonant neutron absorption in uranium-238 with cross
sections above 238 barns are greatly expanded. Therefore, the Doppler
Effect, when combined with the decrease in self-shielding, results in an
increased resonance absorption by uranium-238 at higher fuel temperatures.
The above examples discuss uranium-238; however, all resonant absorbers
found in a nuclear reactor exhibit similar behavior as uranium-238.
Fuel Pellet Design Characteristics
The characteristics of fuel pellet design are a second issue that affects selfshielding. To understand this effect, we must investigate the temperaturedependent characteristics of the fuel pellets.
Manufacturers produce nuclear reactor fuel pellets as ceramic pellets
(uranium oxide). Like other ceramic materials, fuel pellets are poor
conductors of heat. This results in large temperature gradients from the
center to the outer surface of the pellet. This is a major contributor to the
reduction in self-shielding as the fuel temperature is increased.
42
Rev 1
The figure below shows temperature gradients encountered for fuel pellets
located in low and high power regions of the core.
Figure: Fuel Pellet Temperature Profile
Consider the two gradient curves for high and low temperature conditions
as shown in the figure above. The change in temperature across the fuel
pellet increases as well as the center temperatures. For fuel pellets in high
power regions of the core, the fuel centerline temperatures may be above
3,000°F, while temperatures near the fuel pellet surface are closer to
1,000°F. The centerline temperature may be 1,500°F, whereas the
temperature at the surface of the pellet is closer to 700°F for fuel pellets in
lower power regions of the core.
The next figure shows the effect of the increasing temperature gradient on
self-shielding.
Figure: Fuel Pellet Shielded Areas
An epithermal neutron that is not at resonance energy, as it penetrates
deeper into a pellet may appear as a resonance energy neutron in a low
power region of the core. The off-resonance energy neutron may pass
completely through the pellet and not be captured because the temperature
gradient is not as large as that found in a pellet located in a higher power
region of the core.
However, the same neutron entering a fuel pellet in a high power region of
the core would have a higher probability of appearing as a resonance energy
neutron upon entering the pellet and as it penetrates deeper into the pellet.
The result is that as the fuel temperature increases, the effective resonance
capture area for epithermal neutrons also increases. Only a very small
Rev 1
43
fraction of epithermal neutrons escape resonance capture in the fuel pellet at
higher temperatures.
Overall Effect of Temperature on Self Shielding
Increasing the fuel temperature results in a greater fraction of neutrons in
the core being captured in the resonance region. The combination of
Doppler broadening and fuel design (size and operating temperature) does
cause a significant decrease to the resonance escape probability as power
(and fuel temperature) is increased, even though Doppler broadening of the
resonance peaks does not by itself increase the chances of resonance
capture.
Fuel design provides a large volume of resonance absorbers together in a
very dense area, making it difficult for any one neutron to escape resonance
capture. As fuel temperature increases, Doppler broadening results in a
larger fraction of neutrons available for capture. More neutrons are lost
from the neutron life cycle (captured) because more are available for
capture although the probability for capture remains the same.
Knowledge Check
True or False: At higher fuel temperatures, more of the
resonance energy neutrons (peaks) will be captured by
urainium-238.
A.
True
B.
False
ELO 3.3 Fuel Temperature Coefficient Variations
Introduction
The FTC values are not constant throughout core life. Various core
parameters affect the reactivity worth of the FTC. This section discusses
how the following parameters affect FTC:



Moderator temperature
Fuel temperature
Core age
Moderator Temperature Effects on the Fuel Temperature
Moderator temperature/density changes affect the value of the Doppler
coefficient. If moderator density is high (low temperatures), the travel
length, and time for slowing down neutrons are very short.
Resonance capture decreases with less time and exposure available.
Therefore, changes in resonant absorption peaks (Doppler) will cause a
relatively smaller effect on the Doppler coefficient when compared to the
effects at lower moderator density (high temperature).
Slowing down length and time for neutrons increases when the moderator is
hot (less dense) or contains voids. Changes in resonance absorption peaks
(Doppler) will now be more significant since neutrons are spending longer
44
Rev 1
periods in the resonance energy range. This means that the Doppler
coefficient (FTC) is more negative at high moderator temperatures and is
most negative at high void fractions.
Fuel Temperature Effects on the Fuel Temperature Coefficient
The resonance peaks for absorption broaden as the fuel temperature
increases in a nuclear reactor, allowing fuel to capture neutrons resonantly
over a larger range of energy levels. The effects are as follows:

Resonance escape probability decreases, providing a negative effect
on the neutron life cycle.
 Energy of thermal neutrons in fuel increases (higher temperatures),
the absorption cross section of the fuel decreases (fissions).
 Thermal utilization factor decreases, providing a small negative effect
on the neutron life cycle.
At low fuel temperatures, the resonance absorption peaks for uranium-238
and plutonium-240 are very narrow, and only a small fraction of the
neutrons passing through the resonance energy spectrum are absorbed.
Thermal neutron energy is relatively low at low fuel temperatures, and a
sizeable fraction of the neutrons is absorbed in the fuel by uranium-235. A
small increase in fuel temperature causes a significant increase in the
number of neutrons resonantly absorbed in the fuel by uranium-238 and
plutonium-240.
Additionally, uranium-235 absorbs a slightly lower number of thermal
neutrons due to the slightly higher energy thermal neutrons. This results in
the effect from the Doppler coefficient being larger at low fuel temperatures
(greater change).
At high fuel temperatures, the resonance absorption peaks for uranium-238
are broad, and a large fraction of the neutrons slowed down in the core is
resonantly captured. A small increase in temperature results in a small
fractional increase in the number of neutrons resonantly absorbed, and a
small decrease in the number of thermal neutrons absorbed in fuel by
uranium-235. This results in the effect from the Doppler coefficient being
smaller at higher fuel temperatures (smaller change). The figure below
illustrates this effect:
Rev 1
45
Figure: Magnitude Change of Doppler Coefficient versus Fuel Temperature
This figure illustrates that a 1°F change from 1,000°F to 1,001°F results in a
larger magnitude of change for the Doppler coefficient (D) than a 1°F
change from 3,000°F to 3,001°F. This happens because the magnitude of
the Doppler broadening change for uranium-238 (and plutonium-240) target
nuclei is greater at lower fuel temperatures.
Note that while the FTC magnitude is smaller at higher
fuel temperatures, the coefficient is always negative.
Note
Core Age Effects on the Fuel Temperature Coefficient
At the beginning of a fuel cycle, the fuel in the reactor is predominantly
uranium-238 and uranium-235 with some plutonium isotopes from reused
fuel. These fuels cause a significant amount of resonance capture. At the
end of the fuel cycle (EOL), approximately the same amount of uranium238 remains in the fuel and uranium-235 is about 60 percent of its original
concentration. Plutonium-239 and plutonium-240 are also now present in
greater amounts from the following reactions:
238
1
239
𝑈+ 𝑛→
𝑈
92
0 𝜎𝑦 =277𝑏 92
239 𝛽−,𝛾 239
𝑈→
𝑁𝑝
92
93
𝑡1/2 = 23.5 𝑚
𝛽− ,𝛾 239
239
𝑁𝑝 →
𝑃𝑢
93
94
𝑡1/2 = 2.355 𝑑
239
1
240
𝑃𝑢 + 𝑛 →
𝑃𝑢
94
0 𝜎𝑦 =200𝑏 94
Plutomium-239 produces plutonium-240 from neutron capture
approximately 27 percent of the time. Therefore, 73 percent of the time,
fission occurs. The figure below shows the total cross section for
plutonium-240. The capture cross section represents the largest component
of the total cross section for plutonium-240.
46
Rev 1
Figure: Total Cross Section for Plutonium-240
A result of plutonium-240 production over core life is that the Doppler
coefficient will become more negative because plutonium has a very high
capture cross section for 1 eV neutrons (approximately 1 x 105 barns).
Therefore, as plutonium-240 builds up in the reactor core, the value for FTC
becomes more negative later in core life.
There are increasing amounts of fission products present in the core that
resonantly capture neutrons; this leads to a large fractional increase in the
number of neutrons that undergo resonance capture in the core as a result.
Therefore, the Doppler coefficient (FTC) is more negative at EOL than
BOL.
Figure: Value of Doppler Coefficient vs. Temperature over Core Life
Typical values for the Doppler coefficient in a nuclear reactor over core life
are:


Rev 1
-1 x 10-5 Δk/k/°F at BOL
-1.5 x 10-5 Δk/k/°F at EOL
47
Knowledge Check
Concerning the Fuel Temperature Coefficient (FTC),
which one on the following statements is true?
A.
At lower moderator temperatures, FTC is more negative.
B.
At lower fuel temperatures, FTC is less negative.
C.
As the core ages, FTC comes less negative.
D.
None of the above.
Knowledge Check
Which one of the following pairs of isotopes is
responsible for the negative reactivity associated with a
fuel temperature increase near the end of core life?
A.
Uranium-235 and plutonium-239
B.
Uranium-235 and plutonim-240
C.
Uranium-238 and plutonium-240
D.
Uranium-238 and plutonium-239
ELO 3.4 Power Reactivity Coefficient
Introduction
A single coefficient called the power reactivity coefficient combines related
coefficients. The power coefficient of reactivity (αPower) combines the FTC
(or Doppler Coefficient) and MTC. The void coefficient may also be
included for some plants. The power coefficient allows the operator to
determine easily reactivity adjustments for changes in power. Adjustments
would require considering MTC and FTC separately without the power
reactivity coefficient. Additionally, it is much easier to measure reactor
power than quantities such as fuel temperature or percent voids in the
coolant.
Power Reactivity Coefficient
The equation for the power coefficient of reactivity (αPower) is similar to the
equations for other reactivity coefficients:
𝛼𝑃𝑜𝑤𝑒𝑟 =
∆𝜌
∆% 𝑃𝑜𝑤𝑒𝑟
Where:
αPower = Power coefficient of reactivity (Δk/k/°F)
Δρ = change in reactivity associated with change in power (Δk/k)
Δ% power = change in reactor power (%)
48
Rev 1
For practical purposes, the only reactivity coefficients that we need to
consider when calculating the reactivity impact on reactor power are the
MTC and FTC or Doppler coefficient. The amount of voiding in the core
does not change significantly in a PWR. We maintain the reactor coolant
system in a tight pressure band, so its reactivity effects are negligible.
Based on this, we can rewrite the power coefficient equation as:
𝛼𝐷 ∆𝑇𝑓𝑢𝑒𝑙 + 𝛼𝑀 ∆𝑇𝑚𝑜𝑑 + 𝛼𝑉 ∆𝑇𝑣𝑜𝑖𝑑
∆% 𝑃𝑜𝑤𝑒𝑟
Typical values for the power coefficient are:
𝛼𝑃𝑜𝑤𝑒𝑟 =


-1.5 x 10-4 Δk/k/percent power (-15 pcm/percent power) at BOL
-2.2 x 10-4 Δk/k/percent power (-22 pcm/percent power) at EOL
MTC and FTC Effects on the Power Coefficient
The MTC is slow acting because the fuel must first heat up and then
transfer heat to the moderator/coolant. Moderator heating begins at the fuel
cladding surface and transfers heat throughout the bulk of the
moderator/coolant. The FTC is the quickest acting reactivity coefficient
because an increase in power results in an immediate change in fuel
temperature on the other hand.
It is essential that both the MTC and FTC be negative in reactor design.
The resultant increase in fuel temperature and moderator temperature add
negative reactivity to the reactor, which, in turn, will limit or turn the power
increase if power increases due to a positive reactivity insertion. This
makes the reactor inherently stable due to the negative reactivity feedback
from increasing moderator and fuel temperature.
If the moderator and FTCs were positive, any increase in temperature would
add positive reactivity, causing reactor power to increase further. This
increases reactor temperature and results in additional positive reactivity to
the reactor. The industry sometimes refers this condition as a "run-away"
reactor transient, which is very dangerous.
As discussed over core life:
MTC changes from:
+0.1 ( 10-4 Δk/k/°F BOL worst case with a positive MTC
-2.6 x 10-4 Δk/k/°F EOL (26 PCM/°F)
FTC changes from:


-1 x 10-5 Δk/k/°F at BOL
-1.5 x 10-5 Δk/k/°F at EOL (1.5 PCM/°F)
These changes result in power coefficient changes of:




Rev 1
-1.5 x 10-4 (k/k/percent power (-15 pcm/percent power) at BOL
-2.2 x 10-4 (k/k/percent power (-22 pcm/percent power) at EOL
49
Knowledge Check
Which one of the following groups contains parameters
that, if varied, will each have a direct effect on the power
coefficient?
A.
Control rod position, reactor power, moderator voids
B.
Moderator temperature, RCS pressure, xenon
concentration
C.
Fuel temperature, xenon concentration, control rod
position
D.
Moderator voids, fuel temperature, moderator
temperature
ELO 3.5 Power Defect on Reactor Power Operations Definition
Introduction
The power coefficient and defect add to the inherent safety features of a
commercial PWR. However, the power defect, because it adds large
amounts of negative reactivity on a power increase, requires large amounts
of positive reactivity addition to counter its affect. Power decreases have an
equal and opposite effect. This is very different from a highly enriched
reactor where a power defect has little effect. This section will discuss the
operational constraints with changes in reactor power caused by the power
defect.
Power Defect on Reactor Power Operations Definition
In order to raise reactor power from 0 to 100 percent equilibrium reactor
power, compensation must be made for the amount of power defect
involved. Remember that the power coefficient consists of both MTC and
FTC, and the power defect is equal to power coefficient times the delta
power change.
A commercial PWR has control rods and soluble poisons available for
compensating the power defect. Control rods are required to be withdrawn
to certain minimum positions and fully out upon reaching 100 percent
power. Therefore, soluble boron is important for reactivity adjustment to
compensate for the large amount of reactivity caused by the power defect.
The coefficient values include:
MTC changes from:
+0.1 x 10-4 Δk/k/°F BOL worst case with a positive MTC
-2.6 x 10-4 Δk/k/°F EOL (26 PCM/°F)
FTC changes from:


-1 x 10-5 Δk/k/°F at BOL
-1.5 x 10-5 Δk/k/°F at EOL (1.5 PCM/°F)
These changes result in power coefficient changes of:


50
Rev 1
-1.5 x 10-4 (k/k/percent power (-15 pcm/% power) at BOL
-2.2 x 10-4 (k/k/percent power (-22 pcm/% power) at EOL
For comparison at EOL conditions, the following defects apply for a power
increase of 0 to 100 percent.


Note
Note



The moderator temperature defect assumes a 25° to 30°
F increase in RCS average temperature from 0 to 100
percent power; we assume a fuel temperature rise of
1,000°F.
𝑀𝑜𝑑𝑒𝑟𝑎𝑡𝑜𝑟 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑑𝑒𝑓𝑒𝑐𝑡 = −26
𝐹𝑢𝑒𝑙 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑑𝑒𝑓𝑒𝑐𝑡 = −1.5
𝑃𝐶𝑀
𝑝𝑐𝑚
℉
𝑃𝐶𝑀
℉
× 30℉ = −780 𝑃𝐶𝑀
× 1,000℉ = −1,500 𝑃𝐶𝑀
𝑃𝑜𝑤𝑒𝑟 𝑑𝑒𝑓𝑒𝑐𝑡 = −22.8 % 𝑝𝑜𝑤𝑒𝑟 × 100 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 = −2,280 𝑃𝐶𝑀
These values provide an idea of how much reactivity that we must
compensate for during power changes.
Power Defect on Reactor Power Operations Example
Example:
Given the requirement to increase power from 20 percent to 80 percent and
the following reactivity values, describe the boron concentration change
needed. Control rods are at 150 steps and are to be withdrawn to 190 steps.



Power coefficient = -22 pcm/percent.
Control rod worth = 5 PCM/step
Boron worth = 7 pcm/ppm
Solution:
Step 1: To increase power from 20 to 80 percent =
𝑝𝑐𝑚
−22
× 60% 𝑝𝑜𝑤𝑒𝑟 = 1,320 𝑝𝑐𝑚
% 𝑝𝑜𝑤𝑒𝑟
Step 2: Rod withdrawal from 150 to 190 steps =
5
𝑃𝐶𝑀
× 40 𝑠𝑡𝑒𝑝𝑠 = 200 𝑝𝑐𝑚
𝑠𝑡𝑒𝑝
Step 3: Reactivity required by diluting boron =
1,320 𝑝𝑐𝑚 − 200 𝑝𝑐𝑚 = 1,120 𝑝𝑐𝑚
Step 4: Boron concentration change to add 1,120 pcm =
1,120 𝑝𝑐𝑚
𝑝𝑐𝑚 = 160 𝑝𝑝𝑚
7 𝑝𝑝𝑚
Boron concentration change needed is to dilute 160 ppm
𝑅𝑒𝑎𝑐𝑡𝑖𝑣𝑖𝑡𝑦 = −1,320 𝑝𝑐𝑚 + 200𝑝𝑐𝑚 + 1,120 𝑝𝑐𝑚 = 0
Rev 1
51
The next section discusses reactivity balances and boron coefficients
further. However, this example shows that the power defect (60 percent
power change) requires a considerable amount of boron dilution.
Knowledge Check
True or False. When increasing power to 100 percent,
the control rods have sufficient positive reactivity to
override negative reactivity from the moderator and fuel
temperature defects.
A.
True
B.
False
TLO 3 Summary
1. Fuel temperature coefficient (FTC) is the change in reactivity per degree
change in fuel temperature (Δk/k/°F). Its effect is large because the
change in fuel temperature from 0 to 100 percent power is large.
 MTC is slow acting, whereas the FTC (Doppler coefficient) is the
quickest acting of all of the reactivity coefficients because an increase
in power results in an immediate change in fuel temperature.
 The broadening of the peaks occurs as fuel temperature increases,
makes resonance capture more likely, therefore resonance escape
probability decreases, causing keff to decrease. The effect is added
negative reactivity.
 Doppler coefficient in PWRs is about -1 to -1.5 pcm/°F - always
negative.
 Uranium-238 and plutonium-240 are two nuclides present in some
reactor fuels that have large resonance absorption peaks.
2. The Doppler broadening of resonance peaks occurs because the nuclei
may be moving either toward or away from the neutron at the time of
interaction.
 Neutrons may actually have either slightly more or slightly less than
the resonant energy, but still appear to be at resonant energy relative
to the nucleus.
 Self-shielding - The outer fuel atoms tend to shield the inner fuel
atoms from resonant energy neutrons.
 At low fuel temperatures, a neutron entering a fuel pellet with exact
resonant energy has a very high probability of absorption, most likely
in the outer edge of fuel pellet.
 Epithermal neutrons of other than resonant energies are more likely to
pass directly through the pellet without being absorbed.
 Pressure coefficient is the change in reactivity per unit change in
pressure.
 Pressure coefficient of negligible in reactors moderated by subcooled
liquids because density does not change significantly within the
operating pressure range.
 Void coefficient of is the change in reactivity per unit change in void
volume.
52
Rev 1

Void coefficient becomes significant in a reactor in which the
moderator is at or near saturated conditions.
 Lowering the moderator density as voids and bubbles are created
leads to a decrease in the resonance escape probability (ρ) and an
increase in the thermal utilization factor (f).
3. FTC Variations
 The magnitude of the Doppler coefficient is smaller at higher fuel
temperatures.
 The value for D becomes more negative later in core life.
 The Doppler coefficient is more negative at higher moderator
temperatures.
4. Power coefficient of reactivity
 The Power coefficient of reactivity is:
𝛼𝐷 ∆𝑇𝑓𝑢𝑒𝑙 + 𝛼𝑀 ∆𝑇𝑚𝑜𝑑
𝛼𝑃𝑜𝑤𝑒𝑟 =
∆% 𝑃𝑜𝑤𝑒𝑟
5. Negative reactivity insertion as a function of power increase is the
power defect.
 To raise reactor power from 0 to 100%, requires large amount of +
Δk/k
 Soluble boron is important for reactivity adjustments to compensate
for large amounts of reactivity caused by the power defect.
 Power Defect = Moderator Temperature defect + Fuel Temperature
Defect
Summary
Now that you have completed this lesson, you should be able to:
1. Describe the fuel temperature coefficient of reactivity.
2. Explain resonance absorption, Doppler broadening, and selfshielding.
3. Describe how the magnitude of the fuel temperature coefficient varies
with changes in the following parameters:
a. Moderator temperature
b. Fuel temperature
c. Core age
4. Describe the components of the power coefficient of reactivity and
the magnitude of their overall effect over core life.
5. Explain how the power defect affects the reactivity balance on reactor
power operations.
Rev 1
53
TLO 4 Reactivity Balances and Boron Reactivity
Overview
This session focuses on how all the various reactivities come into play to
take a reactor from cold conditions, with no fission product poisons and a
brand new core, to 100 percent power and keep it there for the entire period
of a fuel cycle. This chapter also includes the purpose of soluble boron,
including its worth and limitations for use in reactivity control.
Objectives
Upon completion of this lesson, you will be able to do the following:
1. Explain a reactivity balance including approximate amounts of
reactivity required to compensate for the following:
a. Reactor heatup
b. Reactor power increase
c. Fission product poison buildup
d. Core life
2. Explain how and why boron is used to control excess reactivity in a
nuclear reactor.
3. Describe how boron reactivity worth changes with the following:
a. Boron concentration
b. Moderator temperature
4. Explain the change in reactivity addition rate resulting from changing
boron concentration over core life.
ELO 4.1 Reactivity Balance
Introduction
A reactivity balance provides a method for summarizing a reactor's
criticality state or overall value of reactivity. These balances are
numerically less than perfect, but they are useful for providing a rough
indicator of the total reactivity in a nuclear reactor.
Reactivity balances assume cold criticality as a starting point (i.e. cold,
clean, keff = 1.0).
Note
"Cold", as used here, refers to a temperature of 68°F.
"Clean" means that there are no fission product
poisons, such as xenon and samarium, etc., present in
the reactor. After initial core criticality and after power
operation, the reactor is no longer considered clean; the
reactor is then referred to as "xenon-free" rather than
"clean".
Enough fuel (positive reactivity) must be in the core to form a critical mass
at 68°F, since the starting point for this reactivity balance is from a cold,
clean, critical condition. We must add additional fuel to the critical mass in
order to be able to achieve 100 percent power equilibrium conditions in the
reactor and sustain power levels throughout core life. This additional
54
Rev 1
reactivity above the amount required for critical mass under cold, clean
conditions is excess reactivity (ρexcess).
Reactivity Balance
The effective multiplication factor (keff) associated with this excess
reactivity is kexcess. The excess multiplication factor (kexcess) is the amount
of excess fuel loading that causes keff to exceed 1.0. The following equation
gives this value:
kexcess = keff - 1 = kmax
The maximum effective multiplication factor (kmax) is the maximum amount
of keff available under reactor cold, clean conditions with no control rods
inserted. The value of kmax is the installed value of keff at core BOL
conditions. Use the formula below to find the excess reactivity value when
you know keff:
𝜌=
𝑘𝑒𝑓𝑓 − 1
𝑘𝑒𝑓𝑓
keff is designated kmax and reactivity (ρ) is designated excess reactivity. This
results in the following relationship:
𝜌𝑒𝑥 =
𝑘𝑚𝑎𝑥 − 1
𝑘𝑚𝑎𝑥
In order to determine the amount of reactivity present in the core, the
amount of positive reactivity due to excess fuel in the core above critical
mass must be determined. This reactivity (excess fuel) is necessary to allow
the reactor to achieve 100 percent power at equilibrium conditions. It can
be determined by considering the processes that occur in order to take the
reactor from a cold, clean, critical condition to a 100 percent power
equilibrium condition.
Reactivity Required to Account for Reactor Heatup
To achieve 100 percent equilibrium reactor power, it is necessary to
increase the reactor temperature to hot operating conditions (545°F) from
cold shutdown conditions (68°F). Reactor heatup results in negative
reactivity added to the core by the moderator and FTCs.
The equations below show calculation of the reactivity defect associated
with the moderator temperature increase:
Assume average MTC = -1 x 10-4 Δk/k/°F
∆𝑇 = 545 ℉ − 68 ℉ = 477 ℉
𝜌𝑚 = (−1 × 10−4
∆𝑘/𝑘
) (477 ℉)
℉𝑚𝑜𝑑
𝜌𝑚 = −4.77 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 ∆𝑘/𝑘 = 4,770 𝑝𝑐𝑚
The reactivity defect associated with the fuel temperature increase is:
Assume:
Rev 1
55
𝐹𝑇𝐶 = −1 × 10−5
∆𝑘/𝑘
℉𝑓𝑢𝑒𝑙
𝜌𝐷 = 𝛼𝐷 ∆𝑇
𝜌𝐷 = (−1 × 10−5
∆𝑘/𝑘
) (477 ℉)
℉𝑓𝑢𝑒𝑙
𝜌𝐷 = −0.477 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 ∆𝑘/𝑘 = 477 𝑝𝑐𝑚
As temperature increases, the negative reactivity added by MTC and FTC
would cause the reactor to go subcritical (keff < 1.0) if the core contains only
enough fuel to achieve cold, clean, critical mass. Positive reactivity must be
added to keep the reactor critical (keff = 1.0).
We add positive reactivity to the core at the beginning of life in the form of
excess fuel. The amount of positive reactivity required to keep the reactor
critical at 545°F, according to our reactivity balance, must be equal to the
negative reactivity added by MTC and FTC as temperature increases. We
must add +5.247 percent Δk/k (+4.77 percent Δk/k plus +0.477 percent
Δk/k) in order to make the reactor critical at 545°F.
Reactivity Required to Account for Power Increase
We will have to take the reactor from the hot, clean, critical condition
(545°F, no fission product poisons) to a 100 percent power, clean, critical
condition in order to achieve 100 percent equilibrium reactor power. We
must load some additional amount of fuel (positive reactivity) and reactivity
from additional fuel is required. The required calculations are:

We must increase fuel and moderator temperature to take the reactor
from a hot, clean, critical condition to a 100 percent power, clean,
critical condition. Negative reactivity is added to the core via MTC
and FTC as a result.
 We calculate the reactivity defect associated with the fuel temperature
increase as follows, assuming a Doppler coefficient of -1 x 10-5
Δk/k/°Ffuel. A typical fuel temperature at 100 percent power is
approximately 1,400°F.
∆𝑇𝑓𝑢𝑒𝑙 = 1,400°F − 545°F = 855℉
∆𝑘/𝑘
𝜌𝐷 = (−1 × 10−5
) (855℉)
℉𝑓𝑢𝑒𝑙
𝜌𝐷 = −0.855 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 ∆𝑘/𝑘 = 855 𝑝𝑐𝑚
 Assuming a value of -1 x 10-4 Δk/k/°Fmod , we calculate the reactivity
defect associated with the moderator temperature increase as shown
below. Assume the change in coolant temperature as the reactor
power increases is 35°F.
𝜌𝑚 = 𝛼𝑚 (∆𝑇)
∆𝑘/𝑘
𝜌𝑚 = (−1 × 10−4
) (35℉)
℉𝑚𝑜𝑑
𝜌𝑚 = −0.35 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 ∆𝑘/𝑘 = 350 𝑝𝑐𝑚
Based on the above calculation, the process of going to a 100 percent power
clean, critical condition adds a total of -1.205 percent Δk/k (-0.855 percent
Δk/k plus -0.350 percent Δk/k) to the core due to the associated fuel and
56
Rev 1
moderator temperature increase. We must add an equal amount of positive
reactivity (in the form of fuel) equal to +1.205 percent Δk/k, for the reactor
to remain critical at 100 percent power, clean critical conditions.
Reactivity Required to Account for Fission Product Poisons
The process of fission results in the buildup of fission fragments. Some of
these fission fragments are nucleons, which readily absorb neutrons. We
refer to these types of nucleons as fission product poisons because they
remove neutrons from the neutron life cycle. The most significant fission
product poisons are xenon and samarium.
In an operating nuclear reactor, the reactivity associated with the xenon and
samarium present in the core is very important. Increasing the
concentration of fission product poisons results in the addition of negative
reactivity, whereas their removal (through decay or burnout) results in
positive reactivity added to the core.
We must maintain equilibrium of fission product poisons at 100 percent
power to continue the reactivity balance. The term equilibrium refers to
equilibrium xenon (Xe) and samarium (Sm) concentrations within the core.
The values below list typical reactivities associated with these equilibrium
values:


𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑠𝑎𝑚𝑎𝑟𝑖𝑢𝑚 = −1.0 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 ∆𝑘/𝑘
𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑥𝑒𝑛𝑜𝑛 = −3.0 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 ∆𝑘/𝑘
We must add additional fuel (positive reactivity) to the core to account for
these fission product poisons (negative reactivity). Sustained criticality is
not possible with the buildup of fission product poisons without additional
reactivity from more fuel.
The approximate value of negative reactivity added by equilibrium
samarium and xenon is -4 percent Δk/k. To compensate, we add additional
fuel equal to +4 percent Δk/k.
Reactivity Required to Account for Core Life
The reactivity balance accounts for enough fuel to operate at 100 percent
power equilibrium conditions. However, reactor operation depletes the
fuel. This fuel depletion adds negative reactivity that will eventually cause
the reactor to become subcritical unless additional fuel is loaded. Reactor
power will decrease causing fuel temperature and the moderator
temperature to decrease, adding some positive reactivity, keeping the
reactor critical but at progressively lower and lower power levels. This is
core coast down.
Since commercial nuclear power plants generate electrical power (revenue),
this scenario is undesirable. To allow reactor operation at 100 percent
power for a specified period, we add additional fuel to the core. This
specified period is the fuel cycle. An 18-month fuel cycle requires
approximately +15 percent Δk/k (15,000 pcm).
Rev 1
57
Total Reactivity Required for Reactor Operation
The total amount of excess reactivity required to operate the nuclear reactor
through an 18-month fuel cycle is determined by summing all of the
reactivities in the reactivity balance. The table below shows these values.
Positive Reactivity Required (Δk/k)
Reactivity Balance
4.770 percent due to MTC
Heatup from 68°F to 545°F
0.477 percent due to FTC
Heatup from 68°F to 545°F
0.855 percent due to FTC
Heatup to 100 percent power,
545°F to 1,400°F
0.350 percent due to MTC
Heatup to 100 percent power
545°F to 580°F
1.000 percent due to samarium
Equilibrium samarium
3.000 percent due to xenon
Equilibrium xenon
15.00 percent for 18 month cycle
Core life
25.45 percent excess reactivity
Reactivity Total
From this information, kexcess can be determined.
Example:
What is kexcess for a reactor with ex = 25.5 percent k/k?
Solution:
𝑘𝑒𝑥𝑐𝑒𝑠𝑠 = 𝑘𝑒𝑓𝑓 − 1 = 𝑘𝑚𝑎𝑥 − 1 (kmax is installed keff in the reactor)
𝑘𝑚𝑎𝑥 =
1
1 − 𝜌𝑒𝑥
1
1 − 0.255
= 1.34
𝑘𝑚𝑎𝑥 =
𝑘𝑚𝑎𝑥
𝑘𝑒𝑥𝑐𝑒𝑠𝑠 = 1.34 − 1 = 0.34
𝑘𝑒𝑥𝑐𝑒𝑠𝑠 = 0.34 ∆𝑘/𝑘
As shown, an 18-month fuel cycle for a PWR requires about 15 percent (k/k
of excess reactivity to account for fuel burnup alone (dependent on MWth
output). An installed keff or kmax of about 1.34 accounts for the cumulative
effects of temperature, fission product poisons, and fuel depletion.
58
Rev 1
PWR Fuel Cycle
(Time)
ρex Required to Account
for Fuel Burnup
keff installed at BOL
12 months
10 percent Δk/k
≈1.26
18 months
15 percent Δk/k
≈1.34
24 months
20 percent Δk/k
≈1.44
Knowledge Check
When performing a reactivity balance for determining
kexcess following refueling, which one of the following
reactivities is not used?
A.
Control rod worth
B.
Fuel worth
C.
Fuel temperature
D.
All are used
ELO 4.2 Purpose of Boron Reactivity Control
Introduction
Operators normally add boron to the moderator/coolant in a commercial
PWR as a method of countering the excess reactivity present in the core
from increased fuel loading. With soluble boron in the RCS, adjusting the
boron concentration is termed a "chemical shim". This term originates from
the movement of control rods to control reactivity.
Inserting and withdrawing (shimming) a reactor's control rods varies the
reactivity present in the core. Adjusting the concentration of boron in the
coolant (chemical shim) affects the amount of reactivity present in the
reactor.
Purpose of Boron Reactivity Control
The industry commonly refers to soluble boron as boric acid or acid for
short. Operators adjust the boric acid principally to control the effects of
slower reactivity changes. The boric acid also serves to reduce the overall
requirements for control rod reactivity in a PWR, allowing for increased
fuel loading, and allowing optimum positioning of control rods.
Soluble boron combined with multi-region fuel loading and programming
of the control rods serve to reduce peak to average power density in a PWR.
In most PWRs, the use of chemical shim allows operation of the reactor
with the control rods fully withdrawn to produce design radial and axial
power distribution within the core. It is possible to minimize the possibility
of excessively high heat flux in any one or more fuel rods by "flattening"
the power distribution throughout the core.
Rev 1
59
Mechanism for Reactivity Control with Boron
Soluble boron added to the reactor coolant system circulates through the
reactor via the moderator/coolant, thoroughly mixing with the coolant in the
process. Operators base the initial required moderator/coolant boron
concentration on the amount of excess reactivity (ρexcess, kexcess) present at
BOL to permit the reactor to operate for a specified period at 100 percent
equilibrium power. Refer back to the lesson on reactivity balances.
The effect of the circulating boron is to increase the macroscopic cross
section for absorption of the moderator/coolant throughout the core. As a
result, there is a decrease in the thermal utilization factor (f), reducing keff
and the amount of reactivity in the core. The reactivity in the core
decreases as power operation depletes the fuel. Dilution with pure water
reduces boron concentration to add positive reactivity (f is increased). This
rebalances reactivity to maintain keff at 1.0.
Chemical shim is a solution of boric acid (H3BO3) and water. Chemical
shim concentration is measured in parts per million of boron by weight.
Calculate the boron concentration (CB) as follows:
𝐶𝐵 =
𝑔𝑟𝑎𝑚𝑠 𝑜𝑓 𝑏𝑜𝑟𝑜𝑛
𝑔𝑟𝑎𝑚𝑠 𝑜𝑓 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛
When 1,000 ppm of boron is present in solution, this ratio is:
1,000 𝑝𝑝𝑚 =
0.001 𝑔𝑟𝑎𝑚𝑠 𝑜𝑓 𝑏𝑜𝑟𝑜𝑛
1.0 𝑔𝑟𝑎𝑚𝑠 𝑜𝑓 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛
Since concentrations are highly dilute, it is accurate to assume no water
displacement upon adding H3BO3 to the reactor coolant system. A 1,000ppm concentration of boron also means:
0.001 grams of boron/cm3 of water using a water density of 1 g/cm3
0.001 grams of boron/ml of water using 1 cm3 of water = 1 ml of
water
Example:


Given the following information, calculate the macroscopic cross section
(Σa) for a 1,000 ppm solution of H3BO3 and H2O.





Atomic weight of boron = 10.81 amu
1,000 ppm contains 0.001 g/cm3 of boron
σa of boron is 765 barns
σa of boric acid (H3BO3) is approximately 765 barns
Σa of pure water is 0.022 cm-1
Solution:
Find the total number of atoms of boron/cm3 in a 1,000-ppm solution:
𝑁𝑜
𝜌
𝐴
Where:
𝑁=
N = atoms/cm3
No = Avogadro's number
60
Rev 1
A = atomic mass of boron or number of grams per GAW for boron
ρ = density of boron
𝑵=(
𝒂𝒕𝒐𝒎𝒔
𝟔. 𝟎𝟐 × 𝟏𝟎𝟐𝟑 𝒈 𝒂𝒕𝒐𝒎𝒔
𝒈
𝟏𝟎. 𝟖𝟏 𝒈 𝒂𝒕𝒐𝒎𝒔
𝑁 = 5.57 ×
𝛴𝑎𝐵
) × (𝟎. 𝟎𝟎𝟏
𝒈
)
𝒄𝒎𝟑
1019 𝑎𝑡𝑜𝑚𝑠
𝑐𝑚3
1019 𝑎𝑡𝑜𝑚
𝑐𝑚2
−22
= (5.57 ×
) × (7.65 × 10
)
𝑐𝑚3
𝑎𝑡𝑜𝑚
𝛴𝑎𝐵 = 0.042 𝑐𝑚−1
For a 1,000-ppm boron solution in water:
Σ𝑎 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 = Σ𝑎 𝑊𝑎𝑡𝑒𝑟 + Σ𝑎𝐵
Σ𝑎 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 = 0.022 𝑐𝑚−1 + 0.043 𝑐𝑚−1
Σ𝑎 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 = 0.064 𝑐𝑚−1
Changing Core Reactivity with Chemical Shim
Changing the concentration of the soluble boron dissolved in the
moderator/coolant in a nuclear reactor is a slow process. The maximum
rate of change in reactivity that can be attained using chemical shim is
approximately 3 pcm/second.
Normally, operators use chemical shim to compensate for slowly changing
reactivity parameters such as fuel depletion and changes in the
concentration of fission product poisons. However, operators also use
boron addition and dilution during transient conditions to maintain control
rod position/axial flux in the desired range.
In such cases, operators must anticipate these boron changes to allow
sufficient time for their desired reactivity effects to take place. Sometimes
operators need to shut down the reactor from 100 percent. They need to add
boron in addition to inserting the control rods to counter the effects of
positive reactivity added from the power defect as power decreases.
Control rods do not provided sufficient negative reactivity since they are
still required to be above their insertion limits for SDM requirements;
therefore, operators must also add boron.
Knowledge Check
Which of the following is a good reason for use of
soluble boron to control kexcess?
A.
Rev 1
Boron worth is constant over core life.
61
B.
Boron allows for control rod positioning to flatten flux
distribution.
C.
Boron has no significant effect on MTC or FTC.
D.
Boron provides a means to change reactivity rapidly.
ELO 4.3 Changes in Boron Worth with Changes in Boron
Concentration
Introduction
Changes in boron concentration and moderator temperature affect the boron
reactivity worth (pcm/ppm or Δk/k/ppm). With the boron concentration
approaching 2,000 ppm at BOL and less than 50 ppm at EOL, the
concentration differences are large for an operating PWR. This section
includes discussion of the effects of these large concentration differences.
The variation of the macroscopic cross section for absorption (Σa) as a
function of boron concentration is shown in the table below:
CB (ppm)
Σa Boron (cm-1)
Σa B and H2O (cm-1)
0
0
0.022
500
0.021
0.043
1,000
0.042
0.064
1,500
0.063
0.085
2,000
0.084
0.106
2,500
0.105
0.127
This table shows that macroscopic absorption cross-section of boron
dissolved in water increases with boron concentration (ppm). The cross
section varies linearly with boric acid concentration.
Differential and Integral Boron Worth
Differential boron worth refers to the reactivity effect of each incremental
increase of dissolved boron added to the core (the coefficient). Integral
boron worth refers to the total reactivity effect on the reactor coolant system
for a specified boron concentration (defect).
Typically, units of pcm/ppm describe the differential boron worth for
PWRs.
62
Rev 1
Figure: Differential Boron Worth
The highest differential (most negative in magnitude) boron worth occurs
for low boron concentrations. This is because of competition. At lower
concentrations, boron atoms present in the moderator are not competing
with the number of boron atoms present at higher concentrations, so their
worth is higher. However, as moderator boron concentration increases,
individual boron atoms are in greater competition for absorbing neutrons, so
their differential worth decreases.
Additionally, as boron concentration becomes greater, there are so many
boron atoms in the moderator that self-shielding between boron atoms
occurs. This decreases the probability that an individual boron atom will
absorb a given neutron, also resulting in a decrease in differential boron
worth for higher boron concentrations.
The differential boron worth figure above also shows that differential boron
worth at 578°F is a lower value (less negative) than at 78°F. This is
because fewer boron atoms are actually in reactor core due to lower
moderator density at higher temperature.
Changes in Boron Worth with Changes in Moderator
Temperature
The figure below shows that the reactivity worth of boron is a function of
the moderator temperature. Differential boron worth curves, like this one,
are useful when making small reactivity changes.
Rev 1
63
Figure: Reactivity Worth of Boron versus Moderator Temperature
At higher temperatures, the reactor core contains a smaller mass of water
(not volume) due to the expansion of water at a constant pressure. The
smaller water mass likewise means a smaller mass of boron in the core for a
given concentration of boron (ppm). This causes a lower boron density in
the core, resulting in lower boron differential worth, as seen in the figure
above.
Note that the concentration (ppm) of boron in the moderator/coolant does
not change as the temperature of the reactor is increased. However, the
actual mass of the water and boron in the system decreases while the
volume remains constant (density decreases).
Knowledge Check
Differential boron reactivity worth will become _______
negative as moderator temperature increases because, at
higher moderator temperatures, a 1-ppm increase in
reactor coolant system boron concentration will add
_______ boron atoms to the core.
64
A.
less; fewer
B.
more; fewer
C.
more; more
D.
less; more
Rev 1
ELO 4.4 Changes in Boron Concentration over Core Life
Introduction
This section describes how the soluble boron concentration changes over
core life and why. Because of the large boron concentration differences
from BOL to EOL, positive reactivity additions via dilution are not the
same throughout core life. However, negative reactivity insertions from
boron addition are the same throughout core life.
Changes in Boron Concentration over Core Life
Operators add boric acid to the reactor coolant system to help control excess
reactivity (kexcess) from fuel loading and to accomplish slow reactivity
changes needed for control rod positioning and compensation of fission
product poisons.
The amount of reactivity controlled by boron is about 20 percent Δk/k or
about 20,000 pcm, with the reactor at cold shutdown and borated to about
2,000 ppm at the beginning of life (BOL). The differential boron worth
curve below shows that the differential boron worth at BOL (and cold) is
about -10 to -11 pcm/ppm.
Figure: Differential Boron Worth
We calculate the negative reactivity effect on the reactor due to a 2,000ppm concentration of boron assuming a value of -10 pcm/ppm:
∆𝜌
−10 𝑝𝑐𝑚
=
∆𝐶𝐵
𝑝𝑝𝑚
∆𝜌(𝑝𝑐𝑚) =
−10 𝑝𝑐𝑚
× 2,000 𝑝𝑝𝑚
𝑝𝑝𝑚
∆𝜌(𝑝𝑐𝑚) = −20,000 𝑝𝑐𝑚
The reactor is taken critical and reactor power increased to 100 percent, the
reactor's control rods must be withdrawn and the boron reduced by dilution
as the reactor coolant system is heated up. The next figure illustrates how
Rev 1
65
boron concentration decreases gradually over core lifetime with fuel
depletion:
Figure: Critical Boron Concentration over Core Life
Beginning of Core Life Boron Concentration
At BOL, boron concentration is about 1,200 ppm. The sharp drop in boron
concentration at BOL is due to the buildup of fission product poisons
having large macroscopic cross sections for absorption. A later module
discusses these further. The buildup of these fission product poisons
requires the insertion of significant positive reactivity in order to maintain
reactor criticality. There is a large initial drop in RCS boron concentration.
End of Core Life Boron Concentration
The figure above shows boron concentration gradually decreasing over core
life until EOL. The flat portion of this curve, around 200 EFPH is a result
of fuel depletion in the core and depletion of burnable poisons (rods or
installed fixed poisons) in core.
The burnup of these burnable poisons is plant specific and aids the soluble
boron in compensating for fuel burnup. Boron concentration then drops in a
nearly linear manner over remainder of core life due to fuel burnup.
Boron Dilution over Core Life
A given amount of boric acid will produce the same ppm change in boron
concentration at any time in core life because the volume of the reactor
coolant system and the concentration of the boric acid used for boron
addition are constant over core life.
Dilutions are a different matter. The concentration of boron dissolved in the
coolant will be much lower at core EOL when compared to core BOL. This
means that every gallon of borated water removed from the core through
dilution will carry with it less boron at core EOL than at core BOL. Boron
concentration decreases by a factor greater than 10 from BOL to EOL. This
means that at core EOL, personnel must remove 10 times as much water to
66
Rev 1
have the same ppm decrease in boron concentration as at BOL. Therefore,
for positive reactivity additions via dilution, much more water is required,
and more time (gallons per minute) is necessary to effect the same positive
reactivity addition.
Knowledge Check
The amount of pure water required to decrease the
reactor coolant boron concentration by 20 ppm at the end
of core life (100 ppm) is approximately ______________
the amount of pure water required to decrease reactor
coolant boron concentration by 20 ppm at the beginning
of core life (1,000 ppm).
A.
one-tenth
B.
the same as
C.
10 times
D.
100 times
TLO 4 Summary
1. A reactivity balance for an operating nuclear reactor involves the
reactivities shown below:
Positive Reactivity Required (k/k)
Reactivity Balance
4.770 percent due to αm
Heatup from 68°F to 545°F
0.477 percent due to αD
Heatup from 68°F to 545°F
0.855 percent due to αD
Heatup to 100 percent power,
545°F to 1,400°F
0.350 percent due to αm
Heatup to 100 percent power
545°F to 580°F
1.000 percent due to samarium
Equilibrium samarium
3.000 percent due to xenon
Equilibrium xenon
15.000 percent for 18 month cycle
Core life
25.452 percent excess reactivity
Reactivity total
2. Boron Reactivity Control
 Operators add boron to the moderator/coolant of a nuclear reactor to
control excess reactivity. This is termed a chemical shim.
 By removing some of the boron, decreasing boron concentration in
the core, reactivity increases. Thermal utilization also increases.
3. Born Reactivity Worth Changes
Rev 1
67

The macroscopic absorption cross section of boron dissolved in water
increases with boron concentration (ppm). The cross section varies
linearly with boric acid concentration.
 The highest differential boron worth occurs for low boron
concentrations.
 At higher temperatures, the reactor contains a smaller mass of water
due to expansion of the water at constant pressure. A smaller mass of
water results in a smaller mass of boron in the core for a given boron
concentration (ppm). This causes a lower boron density in the core,
resulting in a lower boron differential worth.
4. Changes in Boron Concentration over Core Life
 Due to large boron concentration differences from BOL to EOL,
positive reactivity additions via dilution are not the same throughout
core life.
 At BOL, boron concentration is about 1,200 ppm. There is an initial
sharp drop in boron concentration at BOL due to the buildup of
fission product poisons.
 At EOL, more than 10 times as much water must be removed to cause
the same boron concentration decrease as would be needed at BOL.
 Every gallon of borated water removed from the core through dilution
will carry with it less boron at core EOL than at core BOL.
Now that you have completed this lesson, you should be able to:
1. Explain a reactivity balance including approximate amounts of
reactivity required to compensate for the following:
a. Reactor heatup
b. Reactor power increase
c. Fission product poison buildup
d. Core life
2. Explain how and why operators use boron to control excess reactivity
in a nuclear reactor.
3. Describe how boron reactivity worth changes with the following:
a. Boron concentration
b. Moderator temperature
4. Explain the change in reactivity addition rate resulting from changing
boron concentration over core life.
Reactivity Coefficients Summary
Now that you have completed this module, 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 reactivity, keff and shutdown margin and their effect on the
reactor operational status.
2. Describe moderator, void and pressure reactivity coefficients and how
they are affected by changing reactor conditions.
3. Describe the fuel temperature and power reactivity coefficients and
how they are affected by changing reactor conditions.
4. Discuss how a reactivity balance is performed to summarize a
reactor's state of criticality and reactivities considered.
68
Rev 1