Applied Reactor Technology Henryk Anglart

Applied Reactor Technology  Henryk Anglart
AppliedReactor
Technology
Henryk Anglart
Applied Reactor Technology
 2011 Henryk Anglart
All rights reserved
i
Preface
T
he main goal of this textbook is to give an introduction to nuclear engineering
and reactor technology for students of energy engineering and engineering
sciences as well as for professionals working in the nuclear field. The basic
aspects of nuclear reactor engineering are
I C O N K E Y
presented with focus on how to perform analysis
Note Corner
and design of nuclear systems.
Examples
The textbook is organized into seven chapters
Computer Program
devoted to the description of nuclear power
More Reading
plants, to the nuclear reactor theory and analysis,
as well as to the environmental and economical aspects of the nuclear power. Parts in
the book of special interest are designed with icons, as indicated in the table above.
“Note Corner” contains additional information, not directly related to the topics
covered by the book. All examples are marked with the pen icon. Special icons are
used to mark sections containing computer programs and suggestions for additional
reading.
The first chapter of the textbook is concerned with various introductory topics in
nuclear reactor physics. This includes a description of the atomic structure as well as
various nuclear reactions and their cross sections. Neutron transport, distributions and
life cycles are described using the one-group diffusion approximation only. The
intention is to provide an introduction to several important issues in nuclear reactor
physics avoiding at the same time the full complexity of the underlying theory.
Additional literature is suggested to those readers who are interested in a more detailed
theoretical background. The second chapter contains description of nuclear power
plants, including their schematics, major components, as well as the principles of
operation. The rudimentary reactor theory is addressed in chapter three. That chapter
contains such topics as the neutron diffusion and neutron distributions in critical
stationary reactors. It also includes descriptions of the time-dependent reactor behavior
due to such processes as the fuel burnup, the reactivity insertions and changes of the
concentration of reactor poisons. The principles of thermal-hydraulic analyses are
presented in chapter four, whereas chapter five contains a discussion of topics related
to the mechanics of structures and to the selection of materials in nuclear applications.
The principles of reactor design are outlined in chapter six. Finally, in chapter seven a
short presentation of the environmental and economic issues of nuclear power is
given.
i
Table of Contents
PREFACE
I
1
5
INTRODUCTION
1.1 Basics of Atomic and Nuclear Physics
1.1.1 Atomic Structure
1.1.2 Isotopes
1.1.3 Nuclear Binding Energy
5
5
6
7
1.2 Radioactivity
1.2.1 Radioactive Decay
1.2.2 Radioactivity Units
9
9
11
1.3 Neutron Reactions
1.3.1 Cross Sections for Neutron Reactions
1.3.2 Neutron Absorption
1.3.3 Nuclear Fission
1.3.4 Prompt and Delayed Neutrons
1.3.5 Slowing Down of Neutrons
12
12
15
15
16
18
2
NUCLEAR POWER PLANTS
21
2.1 Plant Components and Systems
2.1.1 Primary System
2.1.2 Secondary System
2.1.3 Auxiliary Systems Connected to the Primary System
2.1.4 Plant Auxiliary Systems
2.1.5 Safety Systems
21
21
24
25
25
26
2.2 Nuclear Reactors
2.2.1 Principles of Operation
2.2.2 Reactor Types
2.2.3 Selected Current Technologies
26
27
27
29
2.3 Nuclear Reactor Components
2.3.1 Reactor Pressure Vessel
2.3.2 Reactor Core and Fuel Assemblies
2.3.3 Control Rods
36
36
38
38
2.4 Plant Operation
2.4.1 Plant Startup to Full Power
2.4.2 Plant Shutdown
40
40
41
2.5 Plant Analysis
2.5.1 Steady State Conditions
2.5.2 Transient Conditions
41
41
41
i
2.5.3
3
Computer Simulation of Nuclear Power Plants
NUCLEAR REACTOR THEORY
43
45
3.1 Neutron Diffusion
3.1.1 Neutron Flux and Current
3.1.2 Fick’s Law
3.1.3 Neutron Balance Equation
3.1.4 Theory of a Homogeneous Critical Reactor
45
45
46
47
49
3.2 Neutron Flux in Critical Reactors
3.2.1 Finite-Cylinder Bare Reactor
3.2.2 A Spherical Reactor with Reflector
53
54
57
3.3 Neutron Life Cycle
3.3.1 Four-Factor Formula
3.3.2 Six-Factor Formula
59
60
64
3.4 Nuclear Reactor Transients
3.4.1 Nuclear Fuel Depletion
3.4.2 Fuel Poisoning
3.4.3 Nuclear Reactor Kinetics
3.4.4 Nuclear Reactor Dynamics
3.4.5 Nuclear Reactor Instabilities
3.4.6 Control Rod Analysis
65
65
66
72
74
80
83
4
HEAT GENERATION AND REMOVAL
4.1 Energy from Nuclear Fission
4.1.1 Thermal Power of Nuclear Reactor
4.1.2 Fission Yield
4.1.3 Decay Heat
4.1.4 Spatial Distribution of Heat Sources
89
89
89
90
91
93
4.2 Coolant Flow and Heat Transfer in Rod Bundles
4.2.1 Enthalpy Distribution in Heated Channels
4.2.2 Temperature Distribution in Channels with Single Phase Flow
4.2.3 Heat Conduction in Fuel Elements
4.2.4 Axial Temperature Distribution in Fuel Rods
95
97
97
100
104
4.3 Void Fraction in Boiling Channels
4.3.1 Homogeneous Equilibrium Model
4.3.2 Drift-Flux Model
4.3.3 Subcooled Boiling Region
108
108
109
110
4.4 Heat Transfer to Coolants
4.4.1 Single-phase flow
4.4.2 Two-phase boiling flow
4.4.3 Liquid metal flow
4.4.4 Supercritical water flow
111
111
113
114
115
4.5 Pressure Drops
4.5.1 Single-phase flows
4.5.2 Two-phase flows
117
117
119
4.6 Critical Heat Flux
4.6.1 Departure from Nucleate Boiling
4.6.2 Dryout
119
120
123
ii
5
MATERIALS AND MECHANICS OF STRUCTURES
127
5.1 Structural Materials
5.1.1 Stainless Steels
5.1.2 Low-alloy Carbon Steels
5.1.3 Properties of Selected Steel Materials
127
127
128
128
5.2 Cladding Materials
5.2.1 Zirconium
5.2.2 Nickel Alloys
129
129
129
5.3 Coolant, Moderator and Reflector Materials
5.3.1 Coolant Materials
5.3.2 Moderator and Reflector Materials
5.3.3 Selection of Materials
129
129
131
131
5.4 Mechanical Properies of Materials
5.4.1 Hooke’s Law
5.4.2 Stress-Strain Relationships
5.4.3 Ductile and Brittle Behaviour
5.4.4 Creep
133
133
135
135
136
5.5 Strength of Materials and Stress Analysis
5.5.1 Yield Criteria
5.5.2 Stress Analysis in Pipes and Pressure Vessels
5.5.3 Thermal Stresses
136
136
137
138
5.6 Material Deterioration, Fatigue and Ageing
5.6.1 Radiation Effects in Materials
5.6.2 Corrosion of Metals
5.6.3 Chemical Environment
5.6.4 Material Fatigue
5.6.5 Thermal Fatigue
5.6.6 Ageing
138
138
140
140
141
142
142
6
PRINCIPLES OF REACTOR DESIGN
143
6.1 Nuclear Design
6.1.1 Enrichment design
6.1.2 Burnable absorbers
6.1.3 Refueling
143
145
146
146
6.2 Thermal-Hydraulic Design
6.2.1 Thermal-Hydraulic Constraints
6.2.2 Hot Channel Factors
6.2.3 Safety Margins
6.2.4 Heat Flux Limitations
6.2.5 Core-Size to Power Relationship
6.2.6 Probabilistic Assessment of CHF
6.2.7 Profiling of Coolant Flow through Reactor Core
146
147
147
150
151
154
155
159
6.3 Mechanical Design
6.3.1 Design Criteria and Definitions
6.3.2 Stress Intensity
6.3.3 Piping Design
6.3.4 Vessels Design
161
162
162
163
163
iii
7 ENVIRONMENTAL AND ECONOMIC ASPECTS OF NUCLEAR
POWER
165
7.1 Nuclear Fuel Resources and Demand
7.1.1 Uranium Resources
7.1.2 Thorium Fuel
7.1.3 Nuclear Fuel Demand
165
165
168
168
7.2 Fuel Cycles
7.2.1 Open Fuel Cycle
7.2.2 Closed Fuel Cycle
169
170
170
7.3 Front-End of Nuclear Fuel Cycle
7.3.1 Mining and Milling of Uranium Ore
7.3.2 Uranium Separation and Enrichment
7.3.3 Fuel Fabrication
171
171
171
176
7.4 Back-End of Nuclear Fuel Cycle
7.4.1 Fuel Burnup
7.4.2 Repository
7.4.3 Reprocessing
7.4.4 Partitioning and Transmutation of Nuclear Wastes
7.4.5 Safeguards on Uranium Movement
176
176
179
179
180
181
7.5
Fuel Utilization and Breeding
182
7.6
Environmental Effects of Nuclear Power
186
7.7
Economic Aspects of Nuclear Power
188
APPENDIX A – BESSEL FUNCTIONS……………………..…………...191
APPENDIX B – SELECTED NUCLEAR DATA ……………………….193
APPENDIX C – CUMULATIVE STANDARD NORMAL
DISTRIBUTION
……………………………………………………….195
INDEX ………………………………..………………….............................197
iv
Chapter
1
1
Introduction
N
unclear engineering has a relatively short history. The first nuclear reactor
was brought to operation on December 2, 1942 at the University of
Chicago, by a group of researches led by Enrico Fermi. However, the
history of nuclear energy probably started in year 1895, when Wilhelm
Röntgen discovered X-rays. In December 1938 Otto Hahn and Fritz Strassman found
traces of barium in a uranium sample bombarded with neutrons. Lise Meitner and her
nephew Otto Robert Frisch correctly interpreted the phenomenon as the nuclear
fission. Next year, Hans Halban, Frederic Joliot-Curie and Lew Kowarski
demonstrated that fission can cause a chain reaction and they took a first patent on the
production of energy. The first nuclear power plants became operational in 1954. Fifty
years later nuclear power produced about 16% of the world’s electricity from 442
commercial reactors in 31 countries. At present (2011) the nuclear industry experiences
its renaissance after a decade or so of slowing down in the wakes of two major
accidents that occurred in Three-Mile Island and Chernobyl nuclear power plants.
As an introduction to this textbook, the present Chapter describes the fundamentals of
nuclear energy and explains its principles. The topics which are discussed include the
atomic structure of the matter, the origin of the binding energy in nuclei and the ways
in which that energy can be released.
1.1
Basics of Atomic and Nuclear Physics
1.1.1
Atomic Structure
Each atom consists of a positively charged nucleus surrounded by negatively charged
electrons. The atomic nucleus consists of two kinds of fundamental particles called
nucleons: namely a positively charged proton and an electrically neutral neutron. Mass
of a single proton is equal to 1.007277 atomic mass units (abbreviated as u), where 1
u is exactly one-twelfth of the mass of the 12C atom, equal to 1.661•10-27 kg. Mass of a
single neutron is equal to 1.008665 u and mass of a single electron is 0.000548 u. The
radius of a nucleus is approximately equal to 10-15 m and the radius of an atom is about
10-10 m.
5
C H A P T E R
1
–
I N T R O D U C T I O N
Positively charged
nucleus
-15
~10 m
-10
~10 m
Negatively charged
electrons
FIGURE 1-1: Typical structure and dimensions of atoms.
The number of protons in the atomic nucleus of a given element is called the atomic
number of the element and is represented by the letter Z. The total number of
nucleons in an atomic nucleus is called the mass number of the element and is
denoted with the letter A.
Neutrons, discovered by Chadwick in 1932, are particles of particular interest in
nuclear reactor physics since they are causing fission reactions of uranium nuclei and
facilitate a sustained chain reaction. Both these reactions will be discussed later in a
more detail. Neutrons are unstable particles with mean life-time equal to 1013 s. They
undergo the beta decay according to the following scheme,
~
(1-1)
n → p + e − + ν e + energy .
~
Here p is the proton, e- is the electron and ν e is the electronic antineutrino.
1.1.2
MORE READING: Atomic structure and other topics from atomic and nuclear
physics are presented here in a very simplified form just to serve the purpose of the
textbook. However, for readers that are interested in more thorough treatment of
the subject it is recommended to consult any modern book in physics, e.g.
Kenneth S. Krane, Modern Physics, John Wiley & Sons. Inc., 1996.
Isotopes
Many elements have nuclei with the same number of protons (same atomic number Z)
but different numbers of neutrons. Such atoms have the same chemical properties but
different nuclear properties and are called isotopes. The most important in nuclear
engineering are the isotopes of uranium: 233U, 235U and 238U. Only the two last isotopes
exist in nature in significant quantities. Natural uranium contains 0.72% of 235U and
99.274% of 238U.
A particular isotope of a given element is identified by including the mass number A
and the atomic number Z with the name of the element: ZA X . For example, the
6
C H A P T E R
1
–
I N T R O D U C T I O N
common isotope of oxygen, which has the mass number 16, is represented as 168 O .
Often the atomic number is dropped and the isotope is denoted as 16 O .
1.1.3
Nuclear Binding Energy
The atomic nuclei stability results from a balance between two kinds of forces acting
between nucleons. First, there are attractive forces of approximately equal magnitude
among the nucleons, i.e., protons attract other protons and neutrons as well as
neutrons attract other neutrons and protons. These characteristic intranuclear forces
are operative on a very short distance on the order of 10-15 m only. In addition to the
short-range, attractive forces, there are the conventional, coulomb repulsive forces
between the positively charged protons, which are capable of acting over relatively
large distances.
The direct determination of nuclear masses, by means of spectrograph and in other
ways, has shown that the actual mass is always less than the sum of the masses of the
constituent nucleons. The difference, called the mass defect, which is related to the
energy binding the nucleons, can be determined as follows:
Total mass of protons =
Z ⋅ mp
Total mass of electrons =
Z ⋅ me
Total mass of neutrons =
( A − Z ) ⋅ mn
If the measured mass of the atom is M, the mass defect ∆M is found as,
(1-2)
∆M = Z ⋅ (m p + me ) + ( A − Z ) ⋅ mn − M .
Based on the concept of equivalence of mass and energy, the mass defect is a measure
of the energy which would be released if the individual Z protons and (A-Z) neutrons
combined to form a nucleus (neglecting electron contribution, which is small). The
energy equivalent of the mass effect is called the binding energy of the nucleus. The
Einstein equation for the energy equivalent E of a particle moving with a speed v is as
follows,
(1-3)
E=
m0 c 2
2
1− v c
2
= mc 2 .
Here m0 is the rest mass of the particle (i.e. its mass at v ≈ 0 ), c is the speed of light and
m is the effective (or relativistic) mass of the moving particle.
The speeds of particles of interest in nuclear reactors are almost invariably small in
comparison with the speed of light and Eq. (1-3) can be written as,
(1-4)
E = mc 2
where E is the energy change equivalent to a change m in the conventional mass in a
particular process.
7
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1
–
I N T R O D U C T I O N
EXAMPLE 1-1. Calculate the energy equivalent to a conventional mass equal to
1u.
SOLUTION: Since c = 2.998 • 108 m/s and u = 1.661 • 10-27 kg then E = 1.661
• 10-27 x (2.998 • 108)2 kg m2/s2 = 1.492 • 10-10 J.
EXAMPLE 1-2. Calculate the energy as in EXAMPLE 1-1 using MeV as units.
SOLUTION: One electron volt ( 1 eV) is the energy acquired by a unit charge
which has been accelerated through a potential of 1 volt. The electronic (unit)
charge is 1.602 • 10-19 coulomb hence 1 eV is equivalent to 1.602 • 10-19 J and 1 MeV = 1.602 • 10-13 J.
Finally, E = 1.492 • 10-10 / 1.602 • 10-13 MeV = 931.3 MeV.
NOTE CORNER:
Unit of mass - atomic mass unit:
Unit of energy - electron volt:
Conversion:
1 u = 1.661 • 10-27 kg
1 eV = 1.602 • 10-19 J
1 u is equivalent to 931.3 MeV energy
EXAMPLE 1-3. Calculate the mass defect and the binding energy for a nucleus of
an isotope of tin 120Sn (atomic mass M = 119.9022 u) and for an isotope of
uranium 235U (atomic mass M = 235.0439).
SOLUTION: Using Eq. (1-2) and knowing that A = 120 and Z = 50 for tin and
correspondingly A = 235 and Z = 92 for uranium, one gets:
∆M = 50 ⋅ 1.007825 + 70 ⋅ 1.008665 − 119.9022 = 1.0956 u = 1020.3323 MeV for
tin and correspondingly
∆M = 92 ⋅ 1.007825 + 143 ⋅ 1.008665 − 235.0439 = 1.915095 u = 1783.528 MeV It is interesting to
calculate the binding energy per nucleon in each of the nuclei. For tin one gets eB = EB/A =
1020.3323/120 = 8.502769 MeV and for uranium eB = 1783.528/235 = 7.589481 MeV.
EXAMPLE 1-3 highlights one of the most interesting aspects of the nature. It shows
that the binding energy per nucleon in nuclei of various atoms differ from each other.
In fact, if the calculations performed in EXAMPLE 1-3 are repeated for all elements
existing in the nature, a diagram – as shown in FIGURE 1-2 – is obtained. Sometimes
this diagram is referred to as the “most important diagram in the Universe”. And in
fact, it is difficult to overestimate the importance of that curve.
Assume that one uranium nucleus breaks up into two lighter nuclei. For the time being
it assumed that this is possible (this process is called nuclear fission and later on it will
be discussed how it can be done). From EXAMPLE 1-3 it is clear that the total
binding energy for uranium nucleus is ~235 x 7.59 = 1783.7 MeV. Total binding
energy of fission products (assuming that both have approximately the same eB as
obtained for tin) 235 x 8.5 = 1997.5 MeV. The difference is equal to 213.8 MeV and
this is the energy that will be released after fission of a single 235U nucleus.
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C H A P T E R
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I N T R O D U C T I O N
FIGURE 1-2: Variation of binding energy per nucleon with mass number (from Wikimedia Commons).
The total binding energy can be calculated from a semi-empirical equation,
(A − Z )
E = 15.75 A − 94.8 2
2
(1-5)
A
− 17.8 A2 3 − 0.71Z 2 A−1 3 + 34δA−3 4 ,
where δ accounts for a particular stability of the even-even nuclei, for which δ = 1 and
instability of the odd-odd nuclei, for which δ = -1. This equation is very useful since it
approximates the binding energy for over 300 stable and non-stable nuclei, but it is
applicable for nuclei with large mass numbers only.
1.2
Radioactivity
Isotopes of heavy elements, starting with the atomic number Z = 84 (polonium)
through Z = 92 (uranium) exist in nature, but they are unstable and exhibit the
phenomenon of radioactivity. In addition the elements with Z = 81 (thallium), Z = 82
(lead) and Z = 83 (bismuth) exist in nature largely as stable isotopes, but also to some
extend as radioactive species.
1.2.1
Radioactive Decay
Radioactive nuclide emits a characteristic particle (alpha or beta) or radiation (gamma)
and is therefore transformed into a different nucleus, which may or may not be also
radioactive.
Nuclides with high mass numbers emit either positively charged alpha particles
(equivalent to helium nuclei and consist of two protons and two neutrons) or
negatively charged beta particles (ordinary electrons).
In many cases (but not always) radioactive decay is associated with an emission of
gamma rays, in addition to an alpha or beta particle. Gamma rays are electromagnetic
radiations with high energy, essentially identical with x-rays. The difference between
the two is that gamma rays originate from an atomic nucleus and x-rays are produced
from processes outside of the nucleus.
9
C H A P T E R
1
–
I N T R O D U C T I O N
The radioactive decay of nuclei has a stochastic character and the probability of decay
is typically described by the decay constant λ . Thus, if N is the number of the
particular radioactive nuclei present at any time t, the number of nuclei ∆N that will
decay during a period of time ∆t is determined as,
(1-6)
∆N = −λN∆t ,
which gives the following differential equation for N,
(1-7)
dN
= − λN .
dt
Integration of Eq. (1-7) yields,
(1-8)
N = N 0 e − λt ,
where N0 is the number of radioactive nuclei at time t = 0.
The reciprocal of the decay constant is called the
mean life of the radioactive
species (tm), thus,
(1-9)
tm =
1
λ
.
The most widely used method for representing the rate of radioactive decay is by
means of the half-life of the radioactive species. It is defined as the time required for
the number of radioactive nuclei of a given kind to decay to half its initial value. If N is
set equal to N0/2 in Eq. (1-8), the corresponding half-life time t1/2 is given by,
(1-10)
ln
1
1
= −λt1/ 2 ⇒ t1/ 2 = ln 2 ⋅ = ln 2 ⋅ tm .
2
λ
The half-life is thus inversely proportional to the decay constant or directly
proportional to the mean life.
The half-lives of a number of substances of interest in the nuclear energy field are
given in TABLE 1.1.
TABLE 1.1. Radioactive elements.
Naturally occurring
Artificial
Species
Activity
Half-Life
Species
Activity
Half-Life
Thorium-232
Alpha
1.4•1010 yr
Thorium-233
Beta
22.2 min
Uranium-238
Alpha
4.47•109 yr
Protactinium-233
Beta
27.0 days
Uranium-235
Alpha
7.04•108 yr
Uranium-233
Alpha
1.58•105 yr
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I N T R O D U C T I O N
Uranium-239
Beta
23.5 min
Neptunium-239
Beta
2.35 days
Plutonium-239
Alpha
2.44•104 yr
EXAMPLE 1-4. Calculate the decay constant, mean life and half-life of a
radioactive isotope which radioactivity after 100 days is reduced 1.07 times.
SOLUTION: Equation (1-8) can be transformed as follows: λ = ln(N 0 N ) t .
Substituting
and
yields
t = 100 ⋅ 24 ⋅ 3600 = 8.64 ⋅ 10 6 s
N 0 N = 1.07
−9 −1
λ = 7.83 ⋅ 10 s . The mean life is found from Eq. (1-9) as tm = 1 λ ≈ 4.05 years
and the half-life from Eq. (1-10) t1 2 = ln 2 ⋅ tm ≈ 2.81 years.
NOTE CORNER: Radioactive isotopes are useful to evaluate age of earth and age
of various object created during earth history. In fact, since radioactive isotopes still
exist in nature, it can be concluded that the age of earth is finite. Since the isotopes
are not created now, it is reasonable to assume that at the moment of their creation
the conditions existing in nature were different. For instance, it is reasonable to
assume that at the moment of creation of uranium, both U-238 and U-235 were
created in the same amount. Knowing their present relative abundance (U-238/U235 = 138.5) and half-lives, the time of the creation of uranium (and probably the earth) can be found as:
N 8 N 5 = N 0 e − λ t N 0 e − λ t = e (λ −λ )t = 138.5 . Substituting decay constants of U-235 and U-238 from
8
5
5
8
TABLE 1.1, the age of earth is obtained as t ≈ 5 ⋅ 109 years. In archeology the age of objects is
determined by evaluation of the content of the radioactive isotope C-14. Comparing the content of C-14
at present time with the estimated content at the time of creation of the object gives an indication of the
object’s age. For example, if in a piece of wood the content of C-14 corresponds to 60% of the content in
the freshly cut tree, its age can be found as t = − ln (0.6) λ = t1 2 ⋅ ln (1.667 ) ln 2 ≈ 4000 years (assuming
t1 2 = 5400 years for C-14)
1.2.2
Radioactivity Units
A sample which decays with 1 disintegration per second is defined to have an activity
of 1 becquerel (1 Bq). An old unit 1 curie (1 Ci) is equivalent to an activity of 1 gram
of radium-226. Thus activity of 1 Ci is equivalent to 3.7 1010 Bq.
Other related units of radioactivity are reflecting the influence of the radioactivity on
human body. First such unit was roentgen, which is defined as the quantity of gamma
or x-ray radiation that can produce negative charge of 2.58 10-4 coulomb in 1 kg of dry
air.
One rad (radiation absorbed dose) is defined as the amount of radiation that leads to
the deposition of 10-2 J energy per kilogram of the absorbing material. This unit is
applicable to all kinds of ionizing radiation. For x-rays and gamma rays of average
energy of about 1 MeV, an exposure of one roentgen results in the deposition of 0.96
10-2 J /kg of soft body tissue. In other words the exposure in roentgens and the
absorbed dose in soft tissue in rads are roughly equal numerically.
The SI unit of absorbed dose is 1 gray (Gy) defined as the absorption of 1 J of energy
per kilogram of material, that is 1 Gy = 100 rad.
The biological effects of ionizing radiation depend not only on the amount of energy
absorbed but also on other factors. The effect of a given dose is expressed in terms of
11
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I N T R O D U C T I O N
the dose equivalent for which the unit is rem (radiation equivalent in men). If D is the
absorbed dose in rads, the dose equivalent (DE) in rems is defined by,
DE ( rems ) = D ( rads ) × QF × MF
where QF is the quality factor for the given radiation and MF represents other factors.
Both these factors depend on the kind of radiation and the volume of body tissue
within which various radiations deposit their energy. In SI units the above equation
defines the dose equivalent in Siverts (Sv) with reference to absorbed dose in grays.
Thus, 1 Sv is equivalent to 100 rems.
1.3 Neutron Reactions
As already mentioned, neutrons play a very important role in nuclear reactor
operations and their interactions with matter must be studied in details.
Reaction of neutron with nuclei fall into two broad classes: scattering and absorption.
In scattering reactions, the final result is an exchange of energy between the colliding
particles, and neutron remains free after the interaction. In absorption, however,
neutron is retained by the nucleus and new particles are formed. Further details of
neutron reactions are given below.
Neutrons can be obtained by the action of alpha particles on some light elements, e.g.
beryllium, boron or lithium. The reaction can be represented as,
(1-11)
9
4
Be + 24He→126C + 01n .
The reaction can be written in a short form as 9Be( α ,n)12C indicating that a 9Be
nucleus, called the target nucleus, interacts with an incident alpha particle ( α ); a
neutron (n) is ejected and a 12C nucleus, referred to as the recoil nucleus, remains. As
alpha-particle emitters are used polonium-210, radium-226, plutonium-239 and
americium-341.
1.3.1
Cross Sections for Neutron Reactions
To quantify the probability of a certain reaction of a neutron with matter it is
convenient to utilize the concept of cross-sections. The cross-section of a target
nucleus for any given reaction is thus a measure of the probability of a particular
neutron-nucleus interaction and is a property of the nucleus and of the energy of the
incident neutron.
Suppose a uniform, parallel beam of I monoenergetic neutrons per m2 impinges
perpendicularly, for a given time, on a thin layer δx m in thickness, of a target material
containing N atoms per m3, so that Nδx is the number of target nuclei per m2, see
FIGURE 1-3.
12
C H A P T E R
1
–
I N T R O D U C T I O N
I
δx
FIGURE 1-3: Beam of neutrons impinging a target material.
Let NR be the number of individual reactions occurring per m2. The nuclear cross
section σ for a specified reaction is then defined as the averaged number of reactions
occurring per target nucleus per incident neutron in the beam, thus,
(1-12)
σ=
NR
m 2 / nucleus .
(Nδx )I
Because nuclear cross sections are frequently in the range of 10-26 to 10-30 m2 per
nucleus, it has been the practice to express them in terms of a unit of 10-28 m2 per
nucleus, called a barn (abbreviated by the letter b).
Equation (1-12) can be rearranged as follows,
(1-13)
(Nδx )σ
=
NR
.
I
The right-hand-side of Eq. (1-13) represents the fraction of the incident neutrons
which succeed in reacting with the target nuclei. Thus (Nδx )σ may be regarded as the
fraction of the surface capable of undergoing the given reaction. In other words of 1
m2 of target surface (Nδx )σ m2 is effective. Since 1 m2 of the surface contains
(Nδx ) nuclei, the quantity σ m2 is the effective area per single nucleus for the given
reaction.
The cross section σ for a given reaction applies to a single nucleus and is frequently
called the microscopic cross section. Since N is the number of target nuclei per m3,
the product Nσ represents the total cross section of the nuclei per m3. Thus, the
macroscopic cross section Σ is introduced as,
(1-14)
Σ = Nσ m −1 .
13
C H A P T E R
1
–
I N T R O D U C T I O N
If a target material is an element of atomic weight A, 1 mole has a mass of 10-3 A kg
and contains the Avogadro number (NA = 6.02•1023) of atoms. If the element density
is ρ kg/m3, the number of atoms per m3 N is given as,
(1-15)
N=
103 ρN A
.
A
The macroscopic cross section can now be calculated as,
(1-16)
Σ=
103 ρN A
σ.
A
For a compound of molecular weight M and density ρ kg/m3, the number Ni of
atoms of the ith kind per m3 is given by the following equation (modified Eq. (1-15)),
(1-17)
Ni =
10 3 ρN A
νi ,
M
where ν i is the number of atoms of the kind i in a molecule of the compound. The
macroscopic cross section for this element in the given target material is then,
(1-18)
Σ i = N iσ i =
103 ρN A
ν iσ i .
M
Here σ i is the corresponding microscopic cross section. For the compound, the
macroscopic cross section is expressed as,
(1-19)
Σ = N 1σ 1 + N 2σ 2 + L + N iσ i + L =
103 ρN A
(ν 1σ 1 + ν 2σ 2 + L) .
M
EXAMPLE 1-5. The microscopic cross section for the capture of thermal
neutrons by hydrogen is 0.33 b and for oxygen 2 • 10-4 b. Calculate the
macroscopic capture cross section of the water molecule for thermal neutrons.
SOLUTION: The molecular weight M of water is 18 and the density is 1000
kg/m3. The molecule contains 2 atoms of hydrogen and 1 of oxygen. Equation
3
(1-19) yields, Σ = 10 1000 N A (2 ⋅ 0.33 + 1 ⋅ 2 ⋅ 10 −4 )⋅ 10 −28 ≈ 2.2 m −1
H O
2
18
As a rough approximation, the potential scattering cross section for neutrons of
intermediate energy may be found as,
(1-20)
σ s ≈ 4πR 2 ,
where R is the radius of the nucleus.
At high neutron energies (higher than few MeV) the total cross section (e.g. for various
reactions together) approaches the geometrical cross section of the nucleus,
14
C H A P T E R
(1-21)
1
–
I N T R O D U C T I O N
σ t ≈ σ absorption+inelastic scattering + σ elastic scattering ≈ πR 2 + πR 2 = 2πR 2 .
It has been found that the radii of atomic nuclei (except those with very low mass
number) may be approximated with the following expression,
(1-22)
R ≈ 1.3 ⋅ 10 −15 A1/ 3 m ,
where A is the mass number of the nucleus. The total microscopic cross section is
given by,
(1-23)
σ t ≈ 0.11A1/ 3 b .
In general, the total microscopic cross section is equal to a sum of the scattering (both
elastic and inelastic) and absorption cross sections,
(1-24)
σt = σs + σa .
The microscopic cross section for absorption is further classified into several
categories, as discussed below.
1.3.2
Neutron Absorption
It is convenient to distinguish between absorption of slow neutrons and of fast
neutrons. There are four main kinds of slow-neutron reactions: these involve capture
of the neutron by the target followed by either:
1. The emission of gamma radiation – or the radiative capture- (n, γ )
2. The ejection of an alpha particle (n, α )
3. The ejection of a proton (n,p)
4. Fission (n,f)
Total cross section for absorption is thus as follows,
(1-25)
σ a = σ γ + σ n ,α + σ n , p + σ f + L
One of the most important reactions in nuclear engineering is the nuclear fission,
which is described in a more detail in the following subsections.
1.3.3
Nuclear Fission
Relatively few reactions of fast neutrons with atomic nuclei other then scattering and
fission, are important for the study of nuclear reactors. There are many such fastneutron reactions, but their probabilities are so small that they have little effect on
reactor operation.
is caused by the absorption of neutron by a certain nuclei of high atomic
number. When fission takes place the nucleus breaks up into two lighter nuclei: fission
fragments.
Fission
15
C H A P T E R
1
–
I N T R O D U C T I O N
Only three nuclides, having sufficient stability to permit storage over a long period of
time, namely uranium-233, uranium-235 and plutonium-239, are fissionable by
neutrons of all energies. Of these nuclides, only uranium-235 occurs in nature. The
other two are produced artificially from thorium-232 and uranium-238, respectively.
In addition to the nuclides that are fissionable by neutrons of all energies, there are
some that require fast neutrons to cause fission. Thorium-232 and uranium-238 are
fissionable for neutrons with energy higher than 1 MeV. In distinction, uranium-233,
uranium-235 and plutonium-239, which will undergo fission with neutrons of any
energy, are referred to as fissile nuclides.
Since thorium-232 and uranium-238 can be converted into the fissile species, they are
also called fertile nuclides.
The amount of energy released when a nucleus undergoes fission can be calculated
from the net decrease in mass (mass defect) and utilizing the Einstein’s mass-energy
relationship. The total mean energy released per a single fission of uranium-235 nuclei
is circa 200 MeV. Most of this energy is in a form of a kinetic energy of fission
fragments (84%). The rest is in a form of radiation.
The fission cross sections of the fissile nuclides, uranium-233, uranium-235, and
plutonium-239, depend on neutron energy. At low neutron energies there is 1/v region
(that is, the cross section is inversely proportional to neutron speed) followed by
resonance region with many well defined resonance peaks, where cross section get a
large values. At energies higher than a few keV the fission cross section decreases with
increasing neutron energy. FIGURE 1-4 shows uranium-235 cross section.
100000
Total
Cross section, barns
10000
1000
100
10
Fission
1
0,1
0,00001
0,001
0,1
10
1000
100000
10000000
Neutron energy, eV
FIGURE 1-4: Total and fission cross section of uranium-235 as a function of neutron energy.
1.3.4
Prompt and Delayed Neutrons
The neutrons released in fission can be divided into two categories: prompt neutrons
and delayed neutrons. More than 99% of neutrons are released within 10-14 s and are
the prompt neutrons. The delayed neutrons continue to be emitted from the fission
16
C H A P T E R
1
–
I N T R O D U C T I O N
fragments during several minutes after the fission, but their intensity fall rapidly with
the time.
The average number of neutrons liberated in a fission is designed ν and it varies for
different fissile materials and it also depends on the neutron energy. For uranium-235
ν = 2.42 (for thermal neutrons) and ν = 2.51 (for fast neutrons).
All prompt neutrons released after fission do not have the same energy. Typical
energy spectrum of prompt neutrons is shown in FIGURE 1-5.
0.4
0.35
0.3
X(E)
0.25
0.2
0.15
0.1
0.05
0
0
2
4
6
8
E, MeV
FIGURE 1-5: Energy spectrum of prompt neutrons, Eq. (1-26).
As can be seen, most neutrons have energies between 1 and 2 MeV, but there are also
neutrons with energies in excess of 10 MeV. The energy spectrum of prompt neutrons
is well approximated with the following function,
(1-26)
Χ(E ) = 0.453e −1.036 E sinh 2.29 E ,
where E is the neutron energy expressed in MeV and X(E)dE is the fraction of
prompt neutrons with energies between E and E+dE.
Even though less then 1% of neutrons belong to the delayed group of neutrons, they
are very important for the operation of nuclear reactors. It has been established that
the delayed neutrons can be divided into six groups, each characterized by a definite
exponential decay rate (with associated a specific half-life with each group).
The delayed neutrons arise from a beta decay of fission products, when the “daughter”
is produced in an excited state with sufficient energy to emit a neutron. The
characteristic half-life of the delayed neutron is determined by the parent, or precursor,
17
C H A P T E R
1
–
I N T R O D U C T I O N
of the actual neutron emitter. This topic will be discussed in more detail in sections
devoted to the nuclear reactor kinetics.
1.3.5
Slowing Down of Neutrons
After fission, neutrons move chaotically in all directions with speed up to 50000 km/s.
Neutrons can not move a longer time with such high speeds. Due to collisions with
nuclei the speed goes successively down. This process is called scattering. After a short
period of time the velocity of neutrons goes down to the equilibrium velocity, which in
temperature equal to 20 C is 2200 m/s.
Neutron scattering can be either elastic or inelastic. Classical laws of dynamics are used
to describe the elastic scattering process. Consider a collision of a neutron moving with
velocity V1 and a stationary nucleus with mass number A.
V2
Nucleus
before
V1
Neutron
after
ψ
Neutron
before
V2
Centrer of mass
θ
V1-vm
Neutron
before
A
Nucleus
after
Nucleus
after
Neutron
after
vm
Nucleus
before
FIGURE 1-6: Scattering of a neutron in laboratory (to the left) and center-of-mass (to the right) systems.
It can be shown that after collision, the minimum value of energy that neutron can be
reduced to is αE1 , where E1 is the neutron energy before the collision, and,
2
(1-27)
 A −1
 .
 A +1
α =
The maximum energy of neutron after collision is E1 (neutron doesn’t loose any
energy).
The average cosine of the scattering angle ψ in the laboratory system describes the
preferred direction of the neutron after collision and is often used in the analyses of
neutron slowing down. It can be calculated as follows,
(1-28)
∫
cosψ ≡ µ 0 =
4π
0
cosψdΩ
∫
4π
0
4π
=
2π ∫ cosψ sin θdθ
0
4π
2π ∫ sin θdθ
dΩ
0
since, as can be shown,
(1-29)
cosψ =
A cos θ + 1
A2 + 2 A cos θ + 1
.
18
=
2
,
3A
C H A P T E R
1
–
I N T R O D U C T I O N
EXAMPLE 1-6. Calculate the minimum energy that a neutron with energy 1 MeV
can be reduced to after collision with (a) nucleus of hydrogen and (b) nucleus of
carbon. SOLUTION: For hydrogen A = 1 and α = 0 . For carbon A = 12 and
α = 0.716 . Thus, the neutron can be stationary after the collision with the
hydrogen nucleus, and can be reduced to energy E = 716 keV after collision with
the carbon nucleus.
A useful quantity in the study of the slowing down of neutrons is the average value of
the decrease in the natural logarithm of the neutron energy per collision, or the
average logarithmic energy decrement per collision. This is the average of all
collisions of lnE1 – lnE2 = ln(E1/E2), where E1 is the energy of the neutron before and
E2 is that after collision,
1
E
∫ ln E d (cosθ )
1
(1-30)
E1
=
E2
ξ ≡ ln
−1
2
d (cos θ )
.
Here θ is a collision angle in the centre-of-mass system. Integration means averaging
over all possible collision angles.
Analyzing energy change in scattering, the ratio E1/E2 can be expressed in terms of
mass number A and the cosine of the collision angle cos θ . Substituting this to the
equation above yields,
(1-31)
ξ = 1+
( A − 1)2 ln A − 1 .
2A
A+1
If the moderator is not a single element but a compound containing different nuclei,
the effective or mean-weighted logarithmic energy decrement is given by,
(1-32)
ξ =
ν 1σ s1ξ1 + ν 2σ s 2ξ 2 + ... + ν nσ snξ n
ν 1σ s1 + ν 2σ s 2 + ... + ν nσ sn
.
where n is the number of different nuclei in the compound and ν i is the number of
nuclei of i-th type in the compound. For example, for water (H2O) it yields,
(1-33)
ξH O =
2σ s ( H )ξ H + σ s ( O ) ξ (O )
2σ s ( H ) + σ s (O )
2
.
An interesting application of the logarithmic energy decrement per collision is to
compute the average number of collisions necessary to thermalize a fission neutron. It
can be shown that this number is given as,
(1-34)
NC =
14.4
ξ
.
The moderating power or slowing down power of a material is defined as,
19
C H A P T E R
1
–
I N T R O D U C T I O N
M P = ξΣ s .
(1-35
The moderating power is not sufficient to describe how good a given material is as a
moderator, since one also wishes the moderator to be a week absorber of neutrons. A
better figure of merit is thus the following expression, called the moderating ratio,
(1-36)
MR =
ξΣ s
Σa
.
R E F E R E N C E S
[1-1]
Krane, K.S. Modern Physics, John Wiley & Sons. Inc., 1996
[1-2]
Duderstadt, J.J. and Hamilton, L.J., Nuclear Reactor Analysis, John Wiley & Sons, 1976
[1-3]
Glasstone, S. and Sesonske, A., Nuclear Reactor Engineering, Van Nostrand Reinhold Compant,
1981, ISBN 0-442-20057-9.
[1-4]
Stacey, W.M., Nuclear Reactor Physics, Wiley-VCH, 2004
E X E R C I S E S
EXERCISE 1-1: Disregarding uranium-234, the natural uranium may be taken to be a homogeneous
mixture of 99.28 %w (weight percent) of uranium-238 and 0.72 %w of uranium-235. The density of
natural uranium metal is 19.0 103 kg m-3. Determine the total macroscopic and microscopic absorption
cross sections of this material. The microscopic absorption cross sections for uranium-238 and uranium235 are 2.7 b and 681 b, respectively. Hint: first find mass of uranium-235 and uranium-238 per unit
volume of mixture and then number of nuclei per cubic meter of both isotopes.
EXERCISE 1-2: Calculate the moderating power and the moderating ratio for H2O (density 1000
kg/m3) and carbon (density 1600 kg/m3). The macroscopic cross sections are as follows:
Isotope
σ a , [b]
σ s , [b]
Hydrogen
0.332
38
Oxygen
2.7 10-4
3.76
Carbon
3.4 10-3
4.75
EXERCISE 1-3: A neutron with energy 1 MeV scatters elastically with nucleus of 12C. The scattering
angle in the centre of mass system is 60°. Find the energy of the neutron and the scattering angle in the
laboratory system. What can be the minimum energy of the neutron after collision? Ans. E = 0.929MeV,
θ = 56° , Emin = 0.716 MeV.
EXERCISE 1-4: Assuming that in each collision with the nucleus of 12C neutron loses the maximum
possible energy, calculate the number of collisions after which the neutron energy drops down from
1MeV to 0.025 eV. Ans: 52.
EXERCISE 1-5: Calculate the average cosine of the scattering angle in the laboratory system for 12C and
Ans: 0.0555 and 0.028, resp.
238U.
20
Chapter
2
2 Nuclear Power Plants
N
uclear Power Plants (NPP) are complex systems that transform the fission
energy into electricity on a commercial scale. The complexity of plants stems
from the fact that they have to be both efficient and safe, which requires
that several parallel systems are provided. The central part of a nuclear
power plant consists of a system that ensures a continuous transport of the fission heat
energy out of the nuclear reactor core. Such system is called the primary system.
Equally important are so-called secondary systems, whose main goal is to transform
the thermal energy released from the primary system into electricity (or any other final
form of energy that is required). If the system is based on the steam thermodynamic
cycle, it consists of steam lines, turbine sets with generators, condensers, regeneration
heat exchangers and pumps. In some cases gas turbines are used and the systems then
in addition contain compressors, generators and heat exchangers.
Occasionally the primary and the secondary systems are connected through an
additional intermediate system. This feature is characteristic for sodium-cooled reactor,
where an intermediate sodium loop is used to prevent an accidental leakage of
radioactive material from the primary to the secondary system.
If steam is used as the carrier of the thermal energy, the system is called the Nuclear
Steam Supply System (NSSS). Such systems are typical for nuclear power plants which
are using steam turbines to convert the thermal energy into the kinetic energy.
In addition to the above-mentioned process systems, NPPs contain various safety and
auxiliary systems which are vital for over-all performance and reliability of the plants.
The schematics and principles of operation of such systems are described in the first
section of this chapter. In the following section the focus is on nuclear reactors and
their components. Finally, the last section contains an introduction to plant analysis
using computer simulations.
2.1 Plant Components and Systems
In this section the major systems that exist in NPPs are discussed. To focus the
attention, systems typical to pressurized and boiling water reactors are chosen.
2.1.1
Primary System
The primary system (called also the primary loop) of a nuclear power plant with PWR
is schematically shown in FIGURE 2-1. The main components of the system are as
follows:
•
reactor pressure vessel
21
C H A P T E R
2
–
N U C L E A R
P O W E R
P L A N T S
•
pressurizer
•
steam generator
•
main circulation pipe
•
hot leg (piping connecting the outlet nozzle of the reactor pressure vessel with
the steam generator)
•
cold leg (piping connecting the steam generator with the inlet nozzle of the
reactor pressure vessel)
Pressurizer
Steam
generator
Hot leg
Cold leg
Reactor
pressure
vessel
Main circulation pump
FIGURE 2-1: Primary system of a nuclear power plant with PWR.
Due to a limited power of main circulation pumps, the primary systems of PWRs
consist of several parallel loops. In French PWRs with 910 MWe power there are three
loops, whereas in American reactors with power in range 1100÷1300 MWe there are 2,
3 or 4 parallel loops. In multi-loop systems the pressurizer is present only in one of the
loops.
Typical parameters of the primary loop of PWR with 900 MWe power are given in
TABLE 2.1.
TABLE 2.1. Typical parameters of a primary system of PWR with 900 MWe power.
Parameter
Value
Reactor rated thermal power
2785 MW
Coolant mass flow rate
13245 kg/s
Coolant volume at rated power
263.2 m3
Reactor Pressure Vessel (RPV) rated pressure
15.5 MPa
RPV pressure drop
0.234 MPa
22
C H A P T E R
2
–
N U C L E A R
P O W E R
RPV coolant inlet temperature
286.0 °C
RPV coolant outlet temperature
323.2 °C
Number of circulation loops
3
Steam Generator (SG) inlet coolant temperature
323.2 °C
SG outlet coolant temperature
286.0 °C
SG inlet coolant pressure
15.5 MPa
SG coolant pressure drop
0.236 MPa
SG total heat transfer area
4751 m2
Inside diameter of hot leg
736 mm
Inside diameter of cold leg
698 mm
Main Circulation Pump (MCP) speed
1485 rpm
MCP developed head
91 m
MCP rated flow rate
21250 m3/h
MCP electrical power at cold condition
7200 kW
MCP electrical power at hot conditions
5400 kW
P L A N T S
Nuclear power plants with BWRs are single-loop systems, in which NSSS and the
turbine sets are combined into a single circulation loop. Typical schematic of such loop
is shown in FIGURE 2-2.
Steam
dome
High pressure
turbine
Moisture
separator
reheater
Safety and relief
valve
Main steam
isolation valve
Steam dryer
Downcomer
Steam separator
Core
Lower
plenum
Low pressure
turbine
Bypass valve
Turbine control
and stop valve
Preheater
Recirculation
pump
Feedwater
pump
Reactor pressure
vessel
Condensate
pump
Preheater
FIGURE 2-2: Schematic of a BWR system.
23
Condenser
C H A P T E R
2
–
N U C L E A R
P O W E R
P L A N T S
Typical process parameters for BWR system are given in TABLE 2.2.
TABLE 2.2. Typical process parameters in BWR system.
Parameter
Value
Reactor thermal power
3020 MWt
Generator output (electrical)
1100 MWe
Steam pressure in reactor dome
7 MPa
Steam pressure at inlet to HP turbine
6 MPa
Steam pressure at inlet to LP turbine
0.8 MPa
Pressure in condenser
4 kPa
Fraction of steam flow from reactor to HP turbine
91%
Fraction of steam flow to MSR
9%
Fraction of steam flow to high-pressure preheaters
15%
Fraction of steam flow to low-pressure preheaters
11%
Fraction of steam flow to condenser
54%
Water/steam temperature in upper plenum
286 °C
Feedwater temperature at inlet to RPV
215 °C
Feedwater temperature at inlet to feedwater pump
170 °C
Feedwater temperature at outlet from condensate pump
30 °C
Cooling water temperature at condenser inlet
7 °C (mean)
2.1.2
Secondary System
The secondary system of a nuclear power plant with the PWR is shown in FIGURE
2-3. The main parts of the system are as follows:
•
steam lines
•
turbine set
•
moisture-separator reheater
•
condenser
•
preheaters
24
C H A P T E R
•
condensate and feedwater pumps
•
feedwater piping
Steam
generator
2
N U C L E A R
High pressure
turbine
Moisture
separator
reheater
Safety and relief
valve
Main steam
isolation valve
Steam dryer
–
P O W E R
Low pressure
turbine
Bypass valve
Turbine control
and stop valve
Steam separator
Feedwater
pump
Preheater
Condensate
pump
Condenser
Preheater
FIGURE 2-3: Secondary system in PWR nuclear power plant.
2.1.3
Auxiliary Systems Connected to the Primary System
The following systems are connected to the primary system,
•
chemical and volume control system
•
safety injection system
•
residual heat removal system
•
containment spray system
Other nuclear auxiliary systems,
•
component cooling system
•
reactor cavity and spent fuel pit cooling system
•
auxiliary feedwater system.
2.1.4
Plant Auxiliary Systems
Main auxiliary systems are as follows,
•
ventilation and air-conditioning system
•
compressed air system
•
fire protection system
25
P L A N T S
C H A P T E R
2.1.5
2
–
N U C L E A R
P O W E R
P L A N T S
Safety Systems
The major safety system is the Emergency Core Cooling System (ECCS). It usually
consists of several subsystems as listed below.
ECCS in PWRs
ECCS in PWRs consists of the following subsystems:
•
High-Pressure Injection System (HPIS)
•
Low-Pressure Injection System (LPIS)
•
Accumulators
ECCS in BWRs
ECCS in BWRs consists of:
•
High-Pressure Core-spray System (HPCS)
•
Low-Pressure Core-spray System (LPCS)
•
Low-Pressure Injection System (LPIS)
2.2
Nuclear Reactors
Nuclear reactors are designed to transform heat released from nuclear fissions into
enthalpy of a working fluid, which serves as a coolant of the nuclear fuel. The heat
generated in the nuclear fuel would cause its damage and melting if not proper cooling
was provided. Thus one of the most important safety aspects of nuclear reactors is to
provide sufficient cooling of nuclear fuel under all possible circumstances. In some
reactors it is enough to submerge nuclear fuel in a pool of liquid (or a compartment of
gaseous) coolant, which provides sufficient cooling due to natural convection heat
transfer. Such reactors are called to have passive cooling systems. Such systems are
very advantageous from the safety point of view and are considered in future designs
of nuclear reactors. The difficulty of such designs stems from the fact that the systems
are prone to thermal-hydraulic instabilities.
In the majority of current power reactors a forced convection and boiling heat transfer
is employed to retrieve the heat from the fuel elements. The systems are optimized to
produce electricity by means of the Rankine cycle, in the similar manner as it is done in
conventional power plants. The principles of operation, as well as basic classification of
various reactor types are described in the following sections.
A recommended source of additional information and of the knowledge base on
nuclear reactors is the web site supported by IAEA
(www.iaea.org/inisnkm/nkm/aws/reactors.html).
26
C H A P T E R
2.2.1
2
–
N U C L E A R
P O W E R
P L A N T S
Principles of Operation
The principle of operation of a thermal nuclear reactor is shown in FIGURE 2-4. In
fact, the first nuclear reactor was created by the Nature some 2 billion years ago[2-2], not
by scientists and engineers. Uranium-235 will sustain a chain reaction using normal
water as neutron moderator and reflector. Such conditions can occur if uranium with
3% enrichment will be surrounded or penetrated by water. Due to neutron moderation
by water, self-sustain chain reaction will occur. The released heat will cause water
evaporation, effectively reducing the neutron moderation, and thus the power obtained
from the process is self-controlled. Current reactors are utilizing the same principle,
where self-sustained chain reaction is controlled by either inherent mechanisms (such
as the above-mentioned water evaporation effect) or by deliberately designed systems
that are controlling the distribution and level of the neutron flux in the reactor core.
Fuel elements with
fissile material
Control rods
(neutron absorbers)
Moderator
Coolant inlet (low
temperature)
Coolant outlet (high
temperature)
Radiation
protection
Heat removal
FIGURE 2-4: Principle of operation of a thermal nuclear reactor.
2.2.2
Reactor Types
There are numerous reactor types that have been either constructed or developed
conceptually since the beginning of the nuclear era. The classification of reactors can
be performed using various criteria, such as the type of nuclear fission reaction, type of
coolant or type of moderator. The commonly used classification is given below.
Classification by type of nuclear reaction
•
are such reactors that use slow (thermal) neutrons in selfsustained chain reaction.
•
Fast reactors
Thermal reactors
are such reactors that use fast neutrons (typically average
neutron energies higher than 100 keV) in self-sustained chain reaction.
Classification by moderator material
•
Water-moderated reactors are divided into two different types:
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Light Water Reactors
P L A N T S
(LWR) which are using ordinary water (H2O)
as the moderator.
o
Heavy Water Reactors,
which are using heavy water (D2O) as the
moderator.
•
are using graphite as the moderating material.
Such reactors need additional working fluid as a coolant. They can be further
divided into the following types:
Graphite-moderated reactors
o
Gas-cooled reactors ( for example Magnox and Advanced Gascooled Reactor – AGR)
o Water-cooled reactors (for example Chernobyl-type reactor RBMK)
o
•
High Temperature Gas-cooled Reactors (HTGR), such as
developed in the past AVR, Peach Bottom and Fort St. Vrain, or
currently under development, Pebble Bed Reactor and Prismatic Fuel
Reactor.
are such reactors where either lithium or
beryllium is used as the moderator material. Two types of such reactors are
considered:
Light-element moderated Reactors
o Molten Salt Reactor (MSR) – in which light element (either lithium or
beryllium) is used in combination with the fuel dissolved in the molten
fluoride salt coolant.
o Liquid-metal cooled reactors – in which BeO can be used as moderator
and mixture of lead and bismuth serves as coolant.
•
Organically Moderated Reactors,
in which either biphenyl or terphenyl is
used as the moderating material.
Classification by coolant
Water-cooled reactors are divided into two types: Pressurized Water Reactors
(PWR), which use pressurized water (single-phase water typically at 15.5 MPa pressure)
as coolant and Boiling Water Reactors (BWR), which use boiling water (two-phase
mixture typically at 7 MPa pressure) .
•
•
use liquid metals, such as sodium, NaK (an
alloy of sodium and potassium), lead, lead-bismuth eutectic, or (in earlier stages
of development), mercury, as coolant.
Liquid-metal cooled reactors
Gas-cooled reactors
employ helium, nitrogen or carbon dioxide (CO2) as
coolant.
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Classification by generation
Since early 1950s the reactor designs have been improved on the regular basis, bringing
about various generations of reactors. A typical evolution of reactor generation from
Generation-I through Generation-IV is shown in FIGURE 2-5.
FIGURE 2-5: Evolution of reactor generations (from Wikimedia Commons).
2.2.3
Selected Current Technologies
Not all types of reactors mentioned in the previous section have received commercial
maturity. Actually, most of the currently existing power reactors belong to the LWR
category (in 2005 there were 214 PWRs, 53 WWERs and 90 BWRs out of 443 reactors
in total). Full list of currently operating nuclear reactor types is given in TABLE 2.3.
Some of the most popular reactor designs are described in more detail below.
TABLE 2.3 Reactor types (as of 31 Dec. 2005, source IAEA)
Type
Code
Full Name
Operational
Construction/
shutdown
ABWR
Advanced Boiling Light-Water-Cooled
and Moderated Reactor
4
2/0
AGR
Advanced Gas-Cooled, GraphiteModerated Reactor
14
0/1
BWR
Boiling Light-Water-Cooled and
Moderated Reactor
90
0/20
FBR
Fast Breeder Reactor
3
1/6
GCR
Gas-Cooled Graphite-Moderated
Reactor
8
0/29
HTGR
High-Temperature Gas-Cooled,
0
0/4
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Graphite-Moderated Reactor
HWGCR
Heavy-Water-Moderated, Gas-Cooled
Reactor
0
0/3
HWLWR
Heavy-Water-Moderated, Boiling
Light-Water-Cooled
0
0/2
LWGR
Light-Water-Cooled, GraphiteModerated Reactor
16
1/8
PHWR
Pressurized Heavy-Water-Moderated
and Cooled Reactor
41
7/9
PWR
Pressurized Light-Water-Moderated
and Cooled Reactor
214
4
WWER
Water Cooled Water Moderated Power 53
Reactor
SGHWR
Steam-Generating Heavy-Water
Reactor
Total
443
12
27/110
Pressurized Water Reactor (PWR)
A schematic of a nuclear power plant with the pressurized water-cooled reactor is
shown in FIGURE 2-6. The plant contains two circulation loops: the primary and the
secondary one. The primary circulation loop, in which single-phase water is circulated
between the reactor pressure vessel and the steam generator, is located inside a sealed
containment. The secondary loop circulates steam, which is generated in the steam
generator to the turbine.
FIGURE 2-6: PWR nuclear power plant (from Wikimedia Commons).
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Boiling Water Reactor (BWR)
A nuclear power plant with the boiling water reactor is schematically shown in
FIGURE 2-7. The major difference between BWR and PWR is the direct generation
of steam in the pressure vessel of BWR, which removes the need for steam generators
and for the existence of two separate circulation loops. This particular feature greatly
simplifies the over-all plant structure and allows for reduction of the containment size,
which is much smaller for BWRs than for PWRs.
FIGURE 2-7: BWR nuclear power plant (from Wikimedia Commons).
Pressurized Heavy Water Reactor (PHWR)
The advantage of using heavy water (D2O) as the moderator stems from the fact that,
thanks to lower absorption of neutrons in D2O as compared to H2O, the natural
uranium may be used as the nuclear fuel. Due to that the nuclear fuel is cheaper since
the uranium enrichment in U-235 is not needed. This advantage is partly removed by
the higher costs of the heavy water, which must be obtained in an artificial way.
An example of PHWR is the CANDU (CANada Deuterium Uranium) reactor, which
uses the heavy water as both moderator and coolant, even though the two are
completely separated. A schematic of the CANDU reactor is shown in FIGURE 2-8.
This reactor can also operate with light water coolant. Due to higher neutron
absorption in such systems, the uranium fuel must be slightly enriched.
High Power Channel Reactor (RBMK)
(shown in FIGURE 2-9) is an acronym for the Russian Reaktor Bolshoy
Moshchnosti Kanalniy (High-Power Channel Type Reactor). This type of reactor employs
light water as the coolant and graphite as the moderator. The reactor core consists of
vertical pressure tubes running through the moderator. Fuel is low-enriched uranium
oxide made up into 3.65 m long fuel assemblies. Since the moderator is solid, it is not
expelled from the reactor core with increasing temperature. Since the water coolant is
boiling, the reduction in neutron absorption causes a large positive void coefficient.
Due to this feature the system is inherently unsafe, as it was exposed during the
Chernobyl disaster.
RBMK
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This type of reactor was designed and built in the former Soviet Union. Currently all
units (from 1 to 6) in Chernobyl, Ukraine, are shutdown. Still one unit (Ignalina-2, with
total power of 1500 MWe) is operational in Lithuania. Several units (4 in Kursk, 4 in
Sosnovy Bor, 80 km to the west from St Petersburg; and 3 in Smolensk) are
operational in Russia.
Since the Chernobyl disaster this reactor type underwent a number of updates,
including a new control-rod design, increased number of control rods and increased
enrichment of uranium fro 2 to 2.4%.
FIGURE 2-8: Canadian Heavy Water Reactor, CANDU (from Wikimedia Commons): 1- Fuel bundle, 2
– Calandria, 3 – Adjuster rods, 4 – Heavy water pressure reservoir, 5 – Steam generator, 6 – Light water
pump, 7 – Heavy water pump, 8 – Fueling machines, 9 – Heavy water moderator, 10 – Pressure tube, 11
– Steam to steam turbine, 12 – cold water from condenser, 13 – containment.
FIGURE 2-9: High Power Channel Reactor, RBMK (from Wikimedia Commons).
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Advanced Gas Cooled Reactor (AGR)
Advanced Gas-Cooled Reactors (AGRs) have been developed in United Kingdom as
a second generation of nuclear reactors following the Magnox nuclear power reactor.
On the commercial scale the reactors became operational in 1976 and the estimated
closure dates for 7 units in UK vary from 2014 to 2023. A schematic of AGR is shown
in FIGURE 2-10.
AGRs have high thermal efficiency (up to 41%; to be compared with modern PWRs,
which have the efficiency of 34%) thanks to the high temperature of the CO2 coolant
at the core exit (typically 913 K, or 640 °C and pressure 4 MPa). The benefit of the
high efficiency is however hampered by relatively low fuel burnup ratio. Additional
disadvantage of AGR is that its size must be much larger as compared to PWR of the
same power output.
FIGURE 2-10: Advanced Gas-cooled Reactor, AGR: 1 – Charge tubes, 2 – Control rods, 3 – Graphite
moderator, 4 – Fuel assembly, 5 – Concrete pressure vessel and radiation shielding, 6 – Gas circulator, 7 –
Water, 8 – Water pump, 9 – Heat exchanger, 10 – Steam (from Wikimedia Commons).
Liquid Metal Fast Breeder Reactor (LMFBR)
There are two types of the LMFBR:
•
loop type, in which coolant is circulated through the reactor core and an
intermediate heat exchanger
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pool type (shown in FIGURE 2-11), in which the core and the intermediate
exchangers are submerged in the liquid metal coolant, which is contained in a
pool
The primary goal of the development of LMFBR is to improve the utilization of
natural resources of uranium and to breed fuel by transmuting U-238 into Pu-239, Pu240 and Pu-242, which all are fissile materials.
The major difficulty in the development of LMFBRs lies in the transferring heat from
the liquid metal to other heat carriers (typically water and steam) that can be directly
used in turbines the generate mechanical energy. In addition LMFBR is quite costly
and is economically motivated when fuel prices are high (which has not been the case
in the past decades).
FIGURE 2-11: Liquid Metal cooled Fast Breeder Reactor, LMFBR: pool design to the left; loop design
to the right (from Wikimedia Commons).
High Temperature Gas Cooled Reactor (HTGR)
The specific feature of the HRGR is the ability to operate at high temperatures, up to
1123 K (850 °C) for the pebble bed reactor and 998 K (725 °C) for the General
Atomic’s design. Due to the high temperatures, the over-all thermal efficiency of
HTGR nuclear power plants is very high and comparable to the efficiencies of modern
fossil-fuel plants. In addition, the high-temperature heat generated by HTGRs can be
used in various heat-demanding industrial processes, such as steel manufacture, the
conversion of coal into liquid and gaseous fuels and the steam cracking to produce
hydrogen. Even though currently there are no such reactors under operation or
construction, they have several important advantages and in various forms are
considered as technological options within Generation-IV International Forum
research.
A knowledge base for HTGR is supported by IAEA web site
(www.iaea.org/inisnkm/nkm/aws/htgr/).
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Several engineering designs of HTGRs have been proposed. Two of them are of
particular interest:
•
HTGR designed by General Atomic Co (USA)
•
Pebble Bed Modular Reactor PBMR (Germany and South Africa)
The core of the HTGR (see FIGURE 2-12) consists of stacks of hexagonal graphite
prisms with holes for fuel rods, for coolant (helium) flow and, if any, boron carbide
control rods. The fuel rods are made of graphite containing coated particles (0.6÷0.9
mm in size) of highly enriched uranium and thorium dioxides. The oxide fuel is
sometimes replaced with the carbide fuel. The coolant is not in contact with fuel
elements and the heat is conducted through the graphite prisms. Thanks to the
negative temperature coefficient and virtually no void coefficient (helium is virtually
transparent to neutrons) the reactor possesses good inherent safety features.
PBMR was invented and build in Germany, where the first demonstration reactor
(AVR in Jülich; see FIGURE 2-13) operated from 1967 to 1988. The fuel is similar to
that used in HTGR, but fuel elements are in a shape of spheres with 60 mm diameter.
The spheres are coated with pyrolytic carbon (or sometimes with silicon carbon). Thus
a single fuel ball constitutes a “mini-reactor” containing fuel material, moderator and
cladding. The fuel balls can be added and removed from a reactor on the continuous
basis. A very important feature of PBMR is that it is inherently safe. In absence of
cooling the reactor goes into idle stage (the power is reduced to high negative
temperature coefficient of reactivity) and the remaining heat is removed by natural
convection. The power will increase only when cooling is provided.
There is some criticism of the HTGR design, including the following concerns:
•
hazards due to high content of potentially combustible graphite,
•
lack of containments in some designs,
•
relatively low experience with building HTGRs compared to LWRs,
•
greater volume of the radioactive wastes,
•
potential for jammed pebble damage (it occurred in AVR in 1986, partly
contribution to discontinuation of the PBMR development),
•
potential for contamination of the cooling circuit with metallic fission products
(Sr-90, Cs-137),
•
inaccurate temperature prediction in the core (in AVR the core temperature
was underestimated with about 200 K).
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FIGURE 2-12: HTGR reactor and fuel (www.nukeworker.com).
FIGURE 2-13: Schematics of the AVR; a) reactor, b) fuel, c-d) fueling facility (R. Moormann, Jül 4275).
2.3 Nuclear Reactor Components
Nuclear reactor contains the fuel material, which generates heat. The generated heat
must be removed from the fuel material in an efficient manner to prevent the fuel
melting. Due to that a nuclear reactor has a quite complex internal structure and
contains several sub-systems. The most important ones are the reactor pressure vessel,
the fuel assemblies and the control rods. These systems are discussed in a more detail
in the following sections.
2.3.1
Reactor Pressure Vessel
Tepical reactor pressure vessels are shown in FIGURE 2-14.
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1 Control rod
drive
2 Control rod
3 Inlet from
reactor coolant
pump
4 Outlet to steam
generator
5 Reactor vessel
6 Fuel assembly
7 Thermal
shielding
8 Lower core
plate
1 inlet for head
cooling
2 Steam outlet
3 Moisture
separator
4 Reactor Vessel
5 Control rod
6 Fuel assembly
7 Control rod
guide tube
8 outlet to main
circulation
pump
9 Inlet from main
circulation
pump
FIGURE 2-14: Reactor pressure vessels for BWRs (left) and PWRs (right).
The primary function of the Reactor Pressure Vessel (RPV) is to contain the core
and the auxiliary systems, such as core support plates, shroud and internal pumps (if
any) inside a sealed and pressurized volume. Due to their large volumes and thick
walls, RPVs are made of thick bended steel plates that are welded together. The
preferred material is a special fine-grained low alloy ferritic steel, well suited for
welding. The inside is lined with austenitic steel cladding to protect against corrosion.
Typical pressure vessel of PWR (with the power in the range of 1300 MWe) is 12÷13
m high, has the internal diameter about 5 m and the wall thickness about 20÷30 cm.
The vessel is designed for a pressure of 17.5 MPa and the temperature of 623 K (350
°C).
The pressure vessels in BWRs have large dimensions, however, since the operating
pressure is lower, the wall thickness can be kept in the range of 15÷20 cm.
The internals of the reactor pressure vessel must withstand the loads which are created
during reactor operation and resulting from coolant flow and temperature gradients.
Additional loads that are taken into account include vibrations caused by earthquakes
and flow-structure interactions. The internals are made of hyper-quenched austenitic
stainless steel. The main components of the PWR pressure vessel are as follows:
•
the lower core support structure which consists of,
o the core barrel (a cylinder surrounding the core)
o the thermal shield, made of four pads attached to the outside of the
core barrel by screws
o the lower core plate, supporting the core
o the baffle assembly inside the core barrel, which limits the bypass flow
of the core
o the secondary core support with the instrumentation port columns;
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the upper core support structure which consists of,
o the upper support plate located directly on the fuel assemblies
o the control rod guide tubes and the thermocouple columns
o the guide tube support held in place by the vessel head, which
contributes to support the core barrel
o the support columns for connection between the upper support plate
and the guide tube support
Some of the vessel internals are removable and replaceable for example to allow for
refueling or for reactor vessel inspection.
2.3.2
Reactor Core and Fuel Assemblies
Reactor core consists of a number of fuel assemblies fixed in the pressure vessel
between the lower and the upper core support plates. Fuel assemblies typical for PWRs
and BWRs are shown in FIGURE 2-15.
1 Top tie plate
2 Fuel rod
3 Box
4 Water cross
5 Spacer grid
6 Bottom tie plate
1 Control rod
2 Top tie plate
3 Control rod
thimble tube
4 Fuel rod
5 Spacer grid
6 Bottom tie plate
FIGURE 2-15: Typical fuel assemblies used in BWRs (left) and PWRs (right).
2.3.3 Control Rods
Control rods are movable
elements that can be inserted into or withdrawn from a
reactor to change the reactivity. They are manufactured from materials that are strong
absorbers of neutrons. These are usually special alloys, for example 80 % silver, 15%
indium and 5% cadmium. Generally, the material selected should have a good
absorption cross section for neutrons and have a long lifetime as an absorber (not burn
out rapidly).
The ability of a control rod to absorb neutrons can be adjusted during manufacture. A
control rod that is referred to as a black absorber absorbs essentially all incident
neutrons. A grey absorber absorbs only a part of them. While it takes more grey rods
than black rods for a given reactivity effect, the grey rods are often preferred because
they cause smaller depressions in the neutron flux and power in the vicinity of the rod.
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This leads to a flatter neutron flux profile and more even power distribution in the
core.
If grey rods are desired, the amount of material with a high absorption cross section
that is loaded in the rod is limited. Material with a very high absorption cross section
may not be desired for use in a control rod, because it will burn out rapidly due to its
high absorption cross section. The same amount of reactivity worth can be achieved by
manufacturing the control rod from material with a slightly lower cross section and by
loading more of the material. This also results in a rod that does not burn out as
rapidly.
Another factor in control rod material selection is that materials that resonantly absorb
neutrons are often preferred to those that merely have high thermal neutron
absorption cross sections. Resonance neutron absorbers absorb neutrons in the
epithermal energy range. The path length traveled by the epithermal neutrons in a
reactor is greater than the path length traveled by thermal neutrons. Therefore, a
resonance absorber absorbs neutrons that have their last collision farther (on the
average) from the control rod than a thermal absorber. This has the effect of making
the area of influence around a resonance absorber larger than around a thermal
absorber and is useful in maintaining a flatter flux profile.
There are several ways to classify the types of control rods. One classification method
is by the purpose of the control rods. Three purposes of control rods are as follows,
1.
2.
- used for coarse control and/or to remove reactivity in relatively
large amounts
Shim rods
Regulating rods
- used for fine adjustments and to maintain desired power or
temperature
3.
Safety rods - provide a means for very fast shutdown in the event of an
unsafe condition. Addition of a large amount of negative reactivity by rapidly
inserting the safety rods is referred to as a "scram" or "trip"
Not all reactors have different control rods to serve the purposes mentioned above.
Depending upon the type of reactor and the controls necessary, it is possible to use
dual-purpose or even triple-purpose rods. For example, consider a set of control rods
that can insert enough reactivity to be used as shim rods. If the same rods can be
operated at slow speeds, they will function as regulating rods. Additionally, these same
rods can be designed for rapid insertion, or scram. These rods serve a triple function
yet meet other specifications such as precise control, range of control, and efficiency.
Examples of control rod designs for PWR and BWR are shown in FIGURE 2-16 .
X
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FIGURE 2-16: PWR and BWR control rods.
An important issue is to determine the change of reactivity caused by an insertion of
control rods into a nuclear reactor. Since control rods are strong absorbers of
neutrons, the neutron flux is strongly affected in their neighborhood and the neutron
transport equation has to be solved. Clearly, the diffusion approximation is not able to
replace fully the results obtained from the transport equations, but due to its relative
simplicity it can be used to demonstrate some qualitative results.
2.4 Plant Operation
During the lifetime of a nuclear power plants it is necessary to start-up and stop the
plant operation at various occasions. The typical procedures, which are followed in
such situations, are described in this chapter.
2.4.1
Plant Startup to Full Power
The power of BWR during startup is controlled with the control rods and with
changing the recirculation flow of coolant through the reactor core. With main
circulation pumps kept at low speed (typically at 300 rpm), the control rods are
withdrawn to the level that the reactor becomes critical. After that, further withdrawal
of control rods causes release of power, which is initially used to warm up the reactor.
The warming process after the reactor revision or refueling (when it was previously
cooled down) may take up to 24 hours. After that time the reactor is brought to the
hot standby condition, in which the pressure in the reactor is about 6.5 MPa, but the
steam is not passed to turbines, since the main steam isolation valves and the steam
dump valves are closed.
The first stage of increasing the reactor power from the hot standby to about 65% of
the full power is achieved by further withdrawing the control rods. At that stage the
recirculation flow through the reactor core increases from the natural-circulation level
(roughly about 3000 kg/s for a 1000 MWe reactor) to about 4000 kg/s at power level
of 65-70%. With increasing power the steam is directed to the steam lines, which are
initially warmed up using a dedicated system. Main steam isolation valves open when
the steam lines are warmed up and the steam is directed to the condenser through the
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dump valve, to keep constant pressure in the reactor equal to 7 MPa. The turbine is
started when the power is at about 30% level. At that stage the turbine is put in-phase
with the external grid and rotates at 1500 rpm (50 Hz). After that the reactor power is
increased steadily to 65-70% power.
In the meantime, the feed water system is operating, first at low-load mode and then in
the high-load mode. After passing 40% power, the feedwater system operates at full
capacity.
Power is increased from 65-70% to 100% in steps every 4 hours with magnitude of
2.5%. The reason for such a low paste of the power increase is to protect the fuel
against excessive temperature-induced deformations which might result in so-called
Pellet-Clad Interactions (PCI).
After reaching 100% of the power, the control rods need to be withdrawn additionally
to compensate for the reactivity lost due to xenon build up. The xenon reactivity loss
saturate to approximately 2.6% after 40-50 hours. Total time required for the reactor
startup after refueling is about 3 to 4 days.
2.4.2
Plant Shutdown
The plant is shutdown to perform planned revisions and refueling of the nuclear
reactor. The fission process is stopped by insertion of control rods into the reactor
core. The reactor power, however, does not immediately drop to zero; instead it
initially is at the level that corresponds to the β- and γ-radiation emitted from the
fissions products. After reactor shutdown, the accumulated fission products continue
to decay and generate heat within nuclear fuel elements. This heat is removed from the
reactor using the Residual Heat Removal System (RHRS). In addition, before
removing the reactor upper head, the reactor pressure vessel is cooled down by a
dedicated spray system. After few hours the temperature in the reactor pressure vessel
drops below the level that allows its opening.
2.5 Plant Analysis
The purpose of plant analysis is to optimize the plant operation under rated conditions
and to predict the plant behavior under anticipated transient conditions. The major
aspects of plant analyses are presented in the sub-sections below.
2.5.1
Steady State Conditions
The plant analysis at steady-state conditions can be performed at various power levels.
The most important analysis is that performed at the rated conditions, since the plant is
supposed to operate at such conditions at the most of its lifetime. The purpose of such
analysis is to predict the plant efficiency and the safety margins. Often the steady-state
analysis is performed at off-rated conditions, such as plant operation at the fraction of
the rated power. The reason for such analyses is to predict the safety margins and overall plant behavior, rather than plant efficiency, since plant is not expected to operate at
such conditions for a long time.
2.5.2
Transient Conditions
Even though nuclear power plants are expected to operate at steady-state conditions
during most of their lifetime, they have to be designed in such a way that safety will not
be compromised under various anticipated transient conditions. Some of the transient
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conditions are expected to occur during the normal and upset plant operation. To this
category belong:
•
heat up and cool down of the nuclear steam supply system
•
load variation with prescribed steps (typically 5 to 10% of the rated power)
•
house loading
•
refueling
•
turbine trip
•
inactive loop start up
•
loss of electrical power
•
partial loss of reactor coolant flow
•
reactor trip
•
inadvertent reactor coolant pressure decrease
•
control rod cluster drop
In addition to above-mentioned normal and upset conditions, various less frequent
emergency conditions have to be analyzed. The following situations are considered:
•
small reactor coolant pipe break
•
small steam pipe break
•
total loss of reactor coolant flow
•
total loss of steam flow
Finally, highly improbable but severe faulted conditions are analyzed:
•
reactor coolant pipe break (loss-of-coolant accident – LOCA)
•
steam pipe break
•
feedwater pipe break
•
reactor coolant pump rotor locking
•
control rod cluster ejection
•
steam generator tube break
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Computer Simulation of Nuclear Power Plants
Nuclear power plants are simulated with specialized computer codes, which employ
the lumped-parameter approach. In this approach particular components are
approximated with one or several computational volumes. The codes are particularly
suited for analyses of whole systems and are often termed as system codes. Examples
of such codes are TRACE, RELAP, CATHARE and ATHLET. The system codes
typically require a very extensive description of input data, containing usually the
following main groups:
•
options (e.g. steady-state vs. transient, etc)
•
nodalization (description of computational volumes and their connections)
•
trip signals (usually used for actions of the control system and for the
description of boundary conditions)
•
initial conditions
•
expected output data format and volume
Proper preparation of the input decks to the system codes is very time consuming and
requires extensive system knowledge and experience with code simulations. Usually
system code developers provide user guides, which should be carefully addressed by
potential code users prior starting any simulation. An important part of the input deck
preparation consists of the nodalization of the system. A rule of a thumb is that the
particular nodes should follow the structure of the system under consideration and that
the nodes should be well balanced in size. For example one should avoid placing a very
large node in the direct vicinity to a very small node. An example of a nodalization of
the BWR pressure vessel is shown in FIGURE 2-17.
Steam dome
Steam separator
Upper plenum
Downcomer 2
Bypass
channel
Fuel
Core channel
Downcomer 1
Lower plenum
FIGURE 2-17: Example of nodalization of BWR pressure vessel.
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Plant simulation training can be obtained through an IAEA initiative on the
development of plant simulators for educational purposes, with a goal to provide
insight and understanding of the general design and operational characteristics of
various power reactor systems (www.iaea.org/NuclearPower/Education/Simulators/).
Currently simulators for the following systems are provided:
•
Advanced PWR
•
Advanced BWR
•
2-loop PWR
•
CANDU-9
•
ACR-700
•
WWER-1000
Since 1997 user workshops have been organized to facilitate the exchange of
information and experience on using the plant simulators.
R E F E R E N C E S
[2-1]
IAEA Nuclear Reactors Knowledge Base, www.iaea.org/inisnkm/nkm/aws/reactors.html
[2-2]
New Scientest, Oklo reactor and fine-structure value, June 30, 2004,
http://www.newscientist.com/article/dn6092
[2-3]
Glasstone, S. and Sesonske, A., Nuclear Reactor Engineering, Van Nostrand Reinhold Compant,
1981, ISBN 0-442-20057-9.
[2-4]
IAEA Knowledge Base for HTGR, www.iaea.org/inisnkm/nkm/aws/htgr/
[2-5]
Moormann, R., A safety re-evaluation of the AVR pebble bed reactor operation and its
consequences for future HTR concepts, Forschungszentrum Jülich, report 4275, ISSN 09442952.
[2-6]
IAEA Collection of PC-based Simulators for Education,
www.iaea.org/NuclearPower/Education/Simulators/
E X E R C I S E S
EXERCISE 2-1: Taking typical PWR data given in TABLE 2.1 estimate the pumping power needed to
ensure coolant flow through the reactor core. Compare with the total pump power given in the Table.
EXERCISE 2-2: Estimate the heat transfer coefficient in the steam generator using data from TABLE
2.1.
EXERCISE 2-3: For a BWR plant with parameters given in TABLE 2.2 derive the expression for the
over-all plant efficiency. Plot the process in the T-s diagram.
44
Chapter
3
3
Nuclear Reactor Theory
T
he knowledge of neutron distribution in the reactor core is very important for
prediction of its over-all features like, power level and power distribution. In
general, exact description of positions and velocities of all neutrons is neither
possible nor necessary. Even though the general form of equations are known
(integro-differential Boltzmann equations), their simplified form known as diffusion
theory approximation is employed. Further, it can be assumed that all neutrons have
the same speed – which leads to the 1-group diffusion theory approximation. This
kind of approximation leads to a quite crude model, nevertheless due to its simplicity it
can be used for first-step analyses of nuclear reactors. As a most straightforward
continuation, one can consider multi-group diffusion theory from which two-group
approximation (where only fast and slow or thermal neutrons are considered) gained a
considerable popularity and is often employed for practical analysis of nuclear reactors.
3.1 Neutron Diffusion
3.1.1
Neutron Flux and Current
Let n be a number of neutrons per unit volume and v the mean velocity of neutrons.
The neutron flux is defined as follows,
(3-1)
φ = n⋅v
[neutron/m3 × m/s = neutron/(m2s)].
Inspecting units of neutron flux it can be concluded that nv is the number of neutrons
falling on 1 m2 of target material per second. Since σ [m2] is the effective area per single
nucleus for a given reaction and neutron energy, then Σ [m-1] is the effective area of all
the nuclei per m3 of target and the product Σnv gives the number of interactions
between neutrons and nuclei per m3 of target material and per unit time. In particular,
if Σf is the macroscopic cross section for fission with monoenergetic neutrons with
velocity v, the product Σfnv gives the rate of fissions per unit volume and unit time.
Knowing the energy released per fission, the product determines the spatial power
density distribution in material. Thus in general,
Rate of neutron interactions = Σnv interactions/m 3 ⋅ s
The neutron current density Jn in n direction is defined as a number of neutrons that
cross a unit surface area dA in the n direction per unit time. The number of neutrons
crossing the area dA per unit time is thus equal to Jn dA.
45
C H A P T E R
3
–
N U C L E A R
R E A C T O R
T H E O R Y
dA
Jn
FIGURE 3-1: Neutron current.
3.1.2
Fick’s Law
Neutron concentration is not uniform in a nuclear reactor, even though it would be
most desirable to have this kind of distribution in many practical situations. In the
same manner as for other species it is observed that neutrons will move from regions
of high concentration to regions with low concentration due to a process known as
diffusion. According to the Fick’s law of diffusion, the neutron will flow from high to
low concentration region with a rate proportional to the spatial gradient of
concentration. Using the neutron flux φ as a measure of the neutron concentration,
the Fick’s law states that
(3-2)
J = − D∇φ .
Here D is the neutron diffusion coefficient, ∇φ is the gradient vector of the neutron
flux and J is the vector of neutron current density.
It can be shown by a rigorous analysis that the neutron diffusion coefficient is given as,
(3-3)
D=
1
1
=
,
3[Σ t − µ 0 Σ s ] 3Σ tr
where
(3-4)
Σ tr = Σ t − µ0 Σ s ,
is the macroscopic transport cross section, Σ t is the total macroscopic cross section,
Σ s is the macroscopic scattering cross section and µ0 is the average value of the
cosine of the scattering angle of neutrons ( µ0 = 2 (3 A) , where A is the atomic mass
number of the scattering material). The reciprocal of the macroscopic transport cross
section is called the transport mean free path λtr ; thus,
(3-5)
λtr =
1
1
=
.
Σtr Σt − µ0Σ s
For weakly absorbing materials Σ a is small and Σt may be replaced with Σ s ; hence,
46
C H A P T E R
(3-6)
λtr =
3
–
N U C L E A R
R E A C T O R
T H E O R Y
1
λs
=
,
Σ s (1 − µ0 ) (1 − µ0 )
where λs = Σ −s 1 is the scattering mean free path.
3.1.3
Neutron Balance Equation
Balance equation for neutrons is derived in a same way as it is done for any conserved
property. Consider an arbitrary volume V filled with neutrons. The balance equation
can be expressed as,
{Time rate of change of neutrons in the volume V} =
{the net rate at which neutrons flow in – or out – of V across its surface A (leakage)} +
{rate at which neutrons are produced in volume V}
+ {rate at which neutrons are destroyed (absorbed) in volume V}.
The time rate of change of neutrons in the volume V is given by the following integral,
(3-7)
∫
V
∂n
dV .
∂t
The rate of production of neutrons will be designed by the symbol S(r,t), representing
the production of neutrons per unit volume at location r and at time t.
The rate of absorption of neutrons in volume V is given as,
(3-8)
∫ Σ φdV .
V
a
Finally, the rate of neutron leakage from volume V is given as,
(3-9)
∫ J ⋅ ndA = −∫
A
A
D∇φ ⋅ ndA = − ∫ D∇ 2φdV ,
V
where in the last equation the Fick’s law and the Gauss theorem have been employed.
Thus, the 1-group time dependent neutron diffusion equation (since an arbitrary
volume is considered, integration over the volume is dropped) can be written as,
(3-10)
∂n 1 ∂φ
=
= D∇ 2φ − Σ a φ + S .
∂t v ∂t
The steady-state neutron diffusion equation is obtained by allowing the time derivative
on the left-hand-side to be equal to zero,
(3-11)
D∇ 2φ − Σ a φ + S = 0 .
Equation (3-11) is known as the Helmholtz equation and is very familiar type of
equation in mathematical physics. It can be further transformed as follows,
47
C H A P T E R
(3-12)
3
∇ 2φ −
–
N U C L E A R
R E A C T O R
T H E O R Y
1
S
φ=− ,
2
L
D
where
(3-13)
L ≡ D Σa ,
is, by definition, the neutron diffusion length. It can be shown that L is a measure of
how far a neutron will diffuse from a source before it is absorbed.
To solve the neutron diffusion equation, proper initial and boundary conditions must
be specified. As an initial condition (essential for Eq. (3-10)), it is enough to know the
flux distribution at the initial time,
(3-14)
φ (r,0) = φ0 (r ) .
At a boundary between two media A and B the neutron flux will be the same,
(3-15)
φ A b = φB
b
,
and the neutron current will be continuous,
(3-16)
 dϕ 
 dϕ 
− D A  A  = − DB  B  .
 dx  b
 dx  b
At a boundary between a diffusion medium and vacuum a special condition is applied.
At such boundary the neutron flux gradient is such that the linear extrapolation would
lead to the flux vanishing at a certain distance beyond the boundary, see FIGURE 3-2.
φ
φ0
MEDIUM
VACUUM
x
d
FIGURE 3-2: Extrapolation length.
More detailed transport theory studies indicate that taking,
(3-17)
d ≅ 0.7104λtr ,
48
C H A P T E R
3
–
N U C L E A R
R E A C T O R
T H E O R Y
will result in good prediction of the neutron flux in the medium by the diffusion
theory. Thus the boundary condition for the diffusion equation can be written as,
(3-18)
φ (~rs , t ) = 0 .
where ~
rs ≡ rs + d is the extrapolated boundary.
The source term S in the neutron diffusion equation is the rate of neutron production
per unit volume by fission. One can introduce here a definition of the infinite
multiplication factor (a factor that will be described in more detail later),
k∞ =
Rate of neutron production
,
Rate of neutron absorption
which states that the rate of neutron production is equal to infinite multiplication factor
times the rate of neutron absorption. Since the latter is equal to Σ a ⋅ φ , the source S is
obtained as,
(3-19)
S = k∞ ⋅ Σa ⋅φ .
For a critical reactor the neutron diffusion equation becomes,
(3-20)
D∇ 2φ − Σ aφ + k ∞ Σ aφ = 0 ,
or
(3-21)
 (k − 1)Σ a 
∇ 2φ +  ∞
φ = 0 .
D


Using the definition of the diffusion length L, the equation reads,
(3-22)
∇ 2φ +
(k ∞ − 1) φ = 0 .
L2
Introducing another important parameter called material buckling Bm, and defined as,
(3-23)
Bm2 =
k∞ − 1
,
L2
Eq. (3-22) becomes,
(3-24)
∇ 2φ + Bm2 φ = 0 .
3.1.4
Theory of a Homogeneous Critical Reactor
In a homogeneous reactor nuclear fuel and other materials form a homogeneous
mixture. It is a convenient approximation of nuclear reactors, which allows for a
simplified analysis of processes that take place in real, heterogeneous reactors. As an
example, a homogeneous spherical reactor will be considered.
49
C H A P T E R
3
–
N U C L E A R
R E A C T O R
T H E O R Y
As derived in the previous subsection, the neutron source is set as S = k∞ Σaφ (r, t ) .
Assuming that this is the only source of neutrons in the reactor, the one-group
diffusion approximation equation for the reactor becomes,
(3-25)
1 ∂φ
= D∇2φ − Σ aφ + k∞Σ aφ ,
v ∂t
with the initial condition given by Eq. (3-14) and the boundary condition given by Eq.
~
(3-18). For a spherical homogeneous reactor with the extrapolated radius R , the
equations become,
(3-26)
 ∂ 2φ 2 ∂φ 
1 ∂φ
 − Σ aφ + k∞Σ aφ ,
= D 2 +
v ∂t
∂
r
r
∂
r


(3-27)
φ (r,0) = φ0 (r ) ,
(3-28)
φ (R , t ) = 0 .
~
Equation (3-26) can be rewritten as follows,
(3-29)
1 ∂φ ∂ 2φ 2 ∂φ k∞ − 1
φ=
=
+
+
vD ∂t ∂r 2 r ∂r D Σa
∂ 2φ 2 ∂φ k∞ − 1
+
+ 2 φ
∂r 2 r ∂r
L
.
To solve the above equation, it is assumed that the solution can be expressed as,
(3-30)
φ (r, t ) = R(r )T (t ) .
Substituting Eq. (3-30) into (3-29) yields,
 d 2 R 2 dR  k∞ − 1
1 dT
 + 2 RT ,
R
= T  2 +
vD dt
r dr 
L
 dr
and dividing both sides with RT yields,
1 1 dT 1  d 2 R 2 dR  k∞ − 1
+ 2
= 
+
vD T dt R  dr 2 r dr 
L
or
(3-31)
1 1 dT 1
1  d 2 R 2 dR 
.
− (k∞ − 1) =  2 +
vD T dt L2
R  dr
r dr 
50
C H A P T E R
3
–
N U C L E A R
R E A C T O R
T H E O R Y
Since the left-hand-side of Eq. (3-31) is a function of t only, and the right-hand side is a
function of r, the equation can be satisfied if both sides are equal to the same constant,
say − B 2 . Thus two ordinary differential equations are obtained,
(3-32)
1 dT
1

= vD  2 (k∞ − 1) − B 2  ,
T dt
L

(3-33)
d 2 R 2 dR
+
+ B2R = 0 .
2
dr
r dr
The solution of Eq. (3-32) is as follows,
(3-34)
 1

T = C ⋅ exp vD  2 (k∞ − 1) − B 2 t  ,

 L
where C is a constant.
Applying substitution R (r ) = u(r ) r in Eq. (3-33) yields,
d 2u
+ B 2u = 0 .
2
dr
A general solution of this equation is as follows,
(3-35)
u (r ) = E1 sin Br + E2 cos Br ,
thus the solution of Eq. (3-33) is as follows,
(3-36)
R (r ) = E1
sin Br
cos Br
+ E2
.
r
r
Since R(r) can not be infinity when r = 0, then E2 must be equal to zero, and the
solution becomes,
(3-37)
R (r ) = E1
sin Br
.
r
This equation must satisfy the boundary condition given by Eq. (3-28), which now
takes the form,
(3-38)
~
~
φ (R , t ) = R (R )⋅ T (t ) = 0
⇒
~
R R =0
( )
for any value of t, thus,
(3-39)
~
sin BR
~
~
~
R R = E1 ~ = 0 ⇒ sin BR = 0 ⇒ Bn R = nπ , n = 1,2,... .
R
( )
As can be seen, Eq. (3-37) is satisfied by all functions of the form,
51
C H A P T E R
(3-40)
3
–
N U C L E A R
Rn (r ) = En
sin Bn r
,
r
R E A C T O R
T H E O R Y
nπ
Bn = ~ .
R
Numbers Bn are called eigenvalues and functions Rn eigenfunctions of the boundary
value problem given by Eq. (3-33) together with the boundary condition given by Eq.
(3-38).
Combining Eq. (3-30) with Eq. (3-34) and Eq. (3-40) yields a solution of Eq. (3-26) as
follows,
(3-41)
∞

n =1

1
(k∞ − 1) − Bn2 t  sin Bn r .
2
L
 r
φ (r, t ) = ∑ Cn ⋅ exp vD 
Coefficients Cn can be found from the initial condition given by Eq. (3-27).
Equation (3-41) can be transformed as follows,
∞
(3-42)


 ln

k − 1 sin Bn r
t
φ (r, t ) = ∑ C n ⋅ exp  n
,
n =1
r
where
(3-43)
kn =
k∞
,
1 + Bn2 L2
(3-44)
ln =
l0
,
1 + Bn2 L2
(3-45)
l0 =
λ
1
= a.
vΣ a
v
Parameter l0 is the mean life-time of neutrons in an infinite media. Since eigenvalues Bn
are increasing with increasing n, values of kn, according to Eq. (3-43), must decrease. It
means that if k1 = 1, then all other values for higher n are less than one. This allows the
following formulation of Eq. (3-42) for k1 = 1,
(3-46)
φ (r, t ) = C1
 k − 1  sin Bn r
sin B1r ∞
+ ∑ Cn ⋅ exp  n
t
.
r
n=2
 ln  r
Now it can be seen that if t → ∞ , then, φ (r , t ) → φ (r ) = C1 sin B1r r . The
conclusion is that if k1 = 1 then the reactor attains steady-state, in which the source of
neutrons comes from the fission of nuclei of the nuclear fuel. This is the condition
when a self-sustained chain reaction exists in the reactor. This is the so-called critical
condition of the reactor.
Taking in the following k1 = k and B1 = B, the criticality condition of a reactor reads
as follows,
52
C H A P T E R
(3-47)
k=
3
–
N U C L E A R
R E A C T O R
T H E O R Y
k∞
=1,
1 + B 2 L2
where
2
(3-48)
π 
B =  ~ ,
R
2
and the neutron flux distribution is given by the following equation,
(3-49)
φ (r, t ) = C1
sin Br
.
r
The criticality condition given by Eq. (3-47) can be derived in another way, by solving
the steady-state diffusion equation (that is Eq. (3-11)) and seeking a positive solution
that satisfies given boundary conditions. This approach will be demonstrated in the
next sections using a cylindrical geometry of the reactor under consideration.
Equation (3-47) can be transformed as,
(3-50)
B2 =
k∞ − 1
.
L2
The parameter B2 as given in Eq. (3-50) is a function of material property only and is
called the material buckling. This is to distinguish from the parameter given by Eq.
(3-48) which is called the geometric buckling. As can be seen, for critical reactor the
material and the geometric buckling are equal to each other, that is,
(3-51)
Bm2 = Bg2 .
This equation plays a central role in the nuclear reactor design. Typically a nuclear
reactor designer starts from evaluation of the geometric buckling to satisfy the
requirements like the required reactor power and the maximum allowed fuel
temperature. The geometric buckling can be found if the shape and the size of the
reactor is known. Eq. (3-51) is then use to evaluate material properties that are required
to make the reactor critical.
For a reactor with known material properties Eqs. (3-48) and (3-50) can be used to
determine the critical size of the reactor. For a reactor with the spherical shape, one can
find its size as follows,
(3-52)
~
R=
Lπ
.
k∞ − 1
3.2 Neutron Flux in Critical Reactors
Distributions of neutron flux can be found by solving Eq. (3-10) (or (3-11) for steadystate situations) with proper boundary (Eq. (3-15) or (3-16)) and, if needed, initial
(3-14) conditions. The most typical geometry - which is a finite cylinder - will be
studied in this section. First a bare reactor, that is a reactor without a reflector will be
53
C H A P T E R
3
–
N U C L E A R
R E A C T O R
T H E O R Y
discussed. Next the influence of the reflector on the neutron distribution will be
shown.
3.2.1
Finite-Cylinder Bare Reactor
As already mentioned, to find the criticality condition in a nuclear reactor one can seek
positive solutions of a steady-state diffusion equation that satisfy proper boundary
conditions.
In the method of variable separation it is assumed that the neutron flux in the finite
cylinder is a function of two variables (r,z) and can be expressed as follows,
z
~z = H / 2 + d
o
r
~
H = H + 2d
~
zi = − H / 2 − d
~
R = R+d
FIGURE 3-3: A finite-cylinder bare reactor with physical dimensions R and H.
(3-53)
φ (r, z ) = R( r ) ⋅ Z ( z ) .
In the cylindrical coordinate system, the Laplacian is as follows,
(3-54)
1 ∂  ∂φ  ∂ 2φ
∇ φ ( r, z ) =
.
r  +
r ∂r  ∂r  ∂z 2
2
Substituting Eqs. (3-53) and (3-54) into (3-24) yields,
(3-55)
1 d  dR( r ) 
d 2 Z (r)
R( z ) + Bm2 R( r ) Z ( z ) = 0 .
r
Z ( z) +
2
r dr  dr 
dz
Dividing both sides of Eq. (3-55) by product R(r)Z(z) gives,
(3-56)
1 d  dR( r )  1
d 2 Z ( z) 1
+
+ Bm2 = 0 .
r

2
r dr  dr  R( r )
Z ( z)
dz
Since the first term is a function of r only and the second term a function of z only, to
satisfy the equation, each of them must be equal to a constant,
54
C H A P T E R
(3-57)
3
–
N U C L E A R
R E A C T O R
T H E O R Y
d 2 Z ( z) 1
d 2 Z ( z)
2
=
−
k
⇒
+ k 2 Z ( z) = 0 .
2
2
Z ( z)
dz
dz
A general solution of the above equation is as follows,
(3-58)
Z ( z ) = A1 sin(kz ) + A2 cos(kz ) .
The equation for the radial part reads,
(3-59)
d 2 R( r ) 1 dR ( r )
+
+ α 2 R( r ) = 0 ,
2
dr
r dr
where,
(3-60)
α 2 = Bm2 − k 2
With a general solution as follows,
(3-61)
R( r ) = C ⋅ J 0 (α ⋅ r ) + D ⋅ N 0 (α ⋅ r ) .
Here J0 is the Bessel function of the first kind and order 0 and N0 is the Bessel
function of the second kind and order 0. Noting that N0 becomes infinite for r = 0, the
constant D in Eq. (3-61) must be equal to zero: D = 0.
At the extrapolated radius of the cylinder the neutron flux will vanish,
(3-62)
~
φ (R , z ) = 0 ,
~
C ⋅ J0 α ⋅ R = 0.
(
)
The above equation will be satisfied for,
(3-63)
~
α n R = ξ n , n = 1,2,...
;
ξ
α n = ~n
R
,
where ξ n is n-th root of the Bessel function J0(x). Equation (3-59) and corresponding
boundary conditions will be satisfied by the following functions,
(3-64)
ξ r 
Rn (r ) = Cn J 0  ~n , n = 1,2,...
 R 
It can be easily checked that only for n = 1 ( ξ1 ≅ 2.405 ) solution given by Eq. (3-64)
is positive in the whole reactor domain. From Eq. (3-58) and boundary conditions at
~
z i and ~
z o results the following solution,
(3-65)
 mπz 
Z m ( z ) = A2, m cos ~  ,
 H 
where again only Z1 is positive in the whole reactor domain. Thus the only positive
solution satisfying the boundary conditions has the following form,
55
C H A P T E R
(3-66)
3
–
N U C L E A R
R E A C T O R
T H E O R Y
 2.405r   πz 
~  cos ~  .
 R  H
φ (r, z ) = AJ 0 
From Eq. (3-60) one gets,
2
(3-67)
2
 2.405   π 
B = ~  + ~  ,
 R  H
2
m
and constant A can be found from the total power of the reactor.
Equation (3-67) expresses the equality of the material buckling with a geometric
parameter, described by the right-hand-side of the equation. This parameter is called, as
already mentioned, the geometric buckling characterizing the considered geometry, and
its values for various shapes (which can be obtained in the same manner as for the
finite cylinder) are shown in TABLE 3.1. The table also shows the critical flux profile
that corresponds to each reactor shape.
TABLE 3.1. Geometric buckling and critical flux profiles for selected geometries.
Shape
Geometric buckling
Slab of thickness a
π 
 ~
a 
2
πx
cos ~
a
Sphere with radius R
π 
 ~
R
2
 πr 
sin ~  r
R
Rectangular
parallelepiped with sides
a, b, c
π  π  π 
 ~ + ~ + ~
a  b  c 
Finite cylinder with
radius R and height H
 2.405   π 
 ~  + ~ 
 R  H
2
2
2
2
2
Flux profile
 πx   πy   πz 
cos ~  cos ~  cos ~ 
a  b  c 
 2.405r   πz 
J 0  ~  cos ~ 
 R  H
Equation (3-67) can be rewritten as,
(3-68)
Bm2 = B g2 ,
where Bg2 is the geometric buckling. Depending on the reactor shape it will be equal to
one of the expressions in TABLE 3.1. Equation (3-68) states the criticality condition
for a reactor. In other words, for given material with given Bm and given reactor shape
with certain Bg, Eq. (3-68) must hold for the reactor to be critical.
Nuclear reactor designers are usually given Bg2 (rather than Bm2 ) since the core
dimensions are determined from the thermal-hydraulic core design, which takes into
56
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R E A C T O R
T H E O R Y
account the thermal limitations in the core. The core must be built large enough and
provide enough heat transfer area to avoid excessively high temperatures for a desired
power output. Once the core size is established, the nuclear designer must determine
the fuel concentration or loading (that corresponds to obtaining a required Bm2 ) that
enable the core to operate at given power for a given period of time.
3.2.2
A Spherical Reactor with Reflector
A bare reactor has several drawbacks which cause that almost no reactors of this type
exists in reality. These drawbacks are as follows:
•
Relatively large amount of neutrons is lost due to leakage from a bare reactor.
As a result, such reactors have large critical size and large critical mass, and
utilization of the nuclear fuel is quite poor.
•
The neutron flux distribution in a bare reactor, and consequently the
distribution of heat sources are very uneven. Even that is causing
uneconomical utilization of the nuclear fuel.
The two drawbacks cause that almost all reactors operating in the world are using
reflectors. As the name indicates, the purpose of a reflector is to reflect back neutrons
to the reactor. The reflector usually is surrounding the reactor core and contains the
same material that is used for moderation (that is a material with large cross section for
scattering and small cross section for absorption).
The one-group diffusion equations for a reactor with reflector are as follows,
(3-69)
Dc ∇2φc − Σ a ,cφc + k∞Σ a ,cφc = 0 ,
(3-70)
Dr ∇ 2φr − Σ a ,rφr = 0 ,
where indices c and r indicate core and reflector, respectively.
The boundary conditions on the core-reflector boundary are as follows,
(3-71)
φc = φr ,
(3-72)
Dc
∂φc
∂φ
= Dr r ,
∂n
∂n
and on the extrapolated boundary of the reflector,
(3-73)
φr = 0 .
Solution of the boundary value problem given by Eqs. (3-69) through (3-73)
determines the critical size of the reactor and the distribution of the neutron flux
(power) in the critical reactor.
As an example, a spherical reactor with reflector is considered. For such reactor with
core radius R and reflector thickness T, the boundary value problem is as follows,
57
C H A P T E R
3
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R E A C T O R
(3-74)
d 2φc 2 dφc
+
+ Bc2φc = 0 ,
2
dr
r dr
(3-75)
d 2φr 2 dφr
+
− κ r2φr = 0 ,
2
dr
r dr
(3-76)
φc (R ) = φr (R ) ,
(3-77)
Dc
(3-78)
dφc
dr
= Dr
R
T H E O R Y
dφr
,
dr R
~
φr (R + T ) = 0 .
Here,
(3-79)
Bc2 =
(3-80)
κ r2 =
k∞ − 1
, Lc = Dc Σ a ,c ,
L2c
Σ a ,r
=
Dr
1
.
L2r
A general solution of Eq. (3-74) is as follows,
φc (r ) = A1
sin Bc r
cos Bc r
+ A2
.
r
r
Since the neutron flux is finite at r = 0, A2 must be equal to zero, and,
(3-81)
φc (r ) = A
sin Bc r
,
r
where A is an arbitrary constant. A general solution of Eq. (3-75) is obtained as,
(3-82)
φr (r ) = C1
sinh κ r r
cosh κ r r
+ C2
.
r
r
From the boundary condition given by Eq. (3-78) results,
[ (
)]
[ (
)]
~
~
sinh κ r R + T
cosh κ r R + T
~
φr R + T = C1
+ C2
=0,
~
~
R+T
R +T
(
(3-83)
)
[ (
~
C 2 = −C1 tanh κ r R + T
)]
Using Eq. (3-83) in (3-82) yields,
(3-84)
[ (
~
sinh κ r R + T − r
φ r (r ) = C
r
)]
58
C H A P T E R
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T H E O R Y
where C is an arbitrary constant.
Substituting Eq. (3-81) and (3-84) into (3-76) yields,
(3-85)
[ (
)]
~
sin Bc R
sinh κ r R + T − R
~
A
=C
⇒ A sin (Bc R ) = C sinh κ r T
R
R
( )
Similarly, substituting the equations to the condition given by Eq. (3-77) yields,
(3-86)
~
~
Dc A(Bc R cos Bc R − sin Bc R ) = − Dr C κ r R cosh κ r T + sinh κ r T
(
)
Combining Eqs. (3-85) and (3-86) yields,
(3-87)
cot Bc R =
1  Dr
1 −
Bc R  Dc
 D rκ r
~
 −
coth κ r T .
 Bc Dc
Having a given thickness of the reflector T and a given material buckling Bm (= Bc),
Eq. (3-87) can be solved for the critical radius of the reactor core R. As can be seen,
the radius is smaller than the corresponding critical radius of a reactor core without
~
reflector, Rc = Rc − d = π Bc − d . The difference,
(3-88)
δ = Rc − R
is introduced as a measure of the reflector savings.
The reflector savings increase with the reflector thickness but it is not larger than Lr. It
can be shown that the reflector thickness should be at most equal to 2 till 3 diffusion
lengths Lr, since thicker reflectors do not improve the savings significantly.
It should be noted that reflected reactors have more uniform distribution of the
neutron flux, and thus the power, which is advantageous from the point of view of
heat transfer in the reactor core and the utilization of the nuclear fuel. The reflector
savings can be calculated in the same manner for other reactor shapes and the
conclusions are identical with those obtained for the spherical reactor.
3.3 Neutron Life Cycle
Not all neutrons produced by fission will cause new fission, since some of them:
1. Will be absorbed by non-fissionable material
2. Will be absorbed parasitically in fissionable material
3. Will leak out of the reactor
For the maintenance of a self-sustaining chain reaction it is enough that – on the
average – at least one neutron produced in fission that causes fission of another
nucleus.
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T H E O R Y
The condition of a self-sustained chain reaction is conveniently expressed in terms of a
multiplication factor. The number of neutrons absorbed or leaking out of the reactor
will determine the value of this multiplication factor, and will also determine whether a
new generation of neutrons is larger, smaller or the same size as the preceding
generation.
Any reactor of a finite size will have neutrons leaking out of it. In general, the larger the
reactor the lower the fraction of the neutron leakage. In particular, if the reactor is
infinitely large there will be no leakage.
3.3.1
Four-Factor Formula
The measure of the increase or decrease in neutron flux in an infinite reactor is the
infinite multiplication factor, k∞. This factor is defined as a ratio of the neutrons
produced by fission ion one generation to the number of neutrons lost through
absorption in the preceding generation.
(3-89)
k∞ =
Neutron production from fission in one generation
Neutron absorption in the preceding generation
k∞ =
Rate of neutron production
.
Rate of neutron absorption
or
(3-90)
The condition for criticality, i.e. for a self-sustaining fission chain to be possible, in the
infinite system is that the rate of neutron production should be equal to the rate of
absorption in the absence of extraneous sources. In other words, requirement for
criticality is,
(3-91)
k∞ = 1.
For some thermal reactors the infinite multiplication factor can be evaluated with a fair
degree of accuracy by means of the four factor formula.
The basis of this formula is the assumed division of the neutrons into three categories:
1. Fission neutrons with energies in excess of about 1 MeV which can cause
fission in uranium-238 as well as in uranium-235
2. Neutrons in the resonance regions which may be captured by uranium-238
3. Thermal neutrons which cause nearly all the fission in uranium-235 and
thereby generate fission neutrons
The four factors formula is as follows,
(3-92)
k ∞ = ε ⋅ p ⋅ f ⋅η
60
C H A P T E R
where : ε
=
3
–
N U C L E A R
R E A C T O R
T H E O R Y
Fast fission factor
p = Resonance escape probability
f
=
Thermal utilization factor
η
=
Reproduction factor
The fast fission factor describes the process where fission is caused by fast neutrons.
In thermal reactors using slightly enriched or natural uranium fuel, some neutrons,
before they have been slowed down appreciably, will cause fission of both uranium235 and uranium-238 nuclei.
At neutron energies greater than about 1 MeV, most of the fast neutron fissions will be
of uranium-238, because of its larger proportions in the fuel. Fast fission results in the
net increase in the fast neutron population of the reactor core. The fast neutron
population in one generation is thus increased by the fast fission factor.
The fast fission factor is defined as the ratio of the net number of fast neutrons
produced by all fissions to the number of fast neutrons produced by thermal fissions:
(3-93)
ε=
Number of fast neutrons produced by all fissions
Number of fast neutrons produced by thermal fissions
In order for a neutron to be absorbed by a fuel nucleus as a fast neutron, it must pass
close to a fuel nucleus while it is a fast neutron. The value of ε will be affected by the
arrangement and concentration of the fuel and the moderator. It will be essentially
equal to one for a homogeneous reactor where the fuel atoms are surrounded by
moderator atoms. However, in a heterogeneous reactor all the fuel atoms are packed
closely together in elements such as pins, rods or pellets. Thus neutron emitted from a
single fission can pass close to another fuel nucleus. Various arrangements in
heterogeneous reactor result in ε ~1.02 ÷ 1.08.
After fission neutrons diffuse through the reactor and collide with nuclei of fuel and
non-fuel material loosing part of their energy. While neutrons are slowing down there
is a chance that some of them will be captured by uranium-238 nuclei. Absorption
cross-section of uranium-238 has several resonance peaks for neutron energies
between 6 to 200 eV. The peak values can be as high as 10000 barns, whereas below 6
eV the absorption cross-section is as low as 10 barns. The probability that the neutron
will not be absorbed by a resonance peak is called the resonance escape probability,
p. the resonance escape probability, p is defined as the ratio of the number of neutrons
that reach thermal energies to the number of fast neutrons that start to slow down:
(3-94)
p=
Number of neutrons that reach thermal energy
.
Number of fast neutrons that start to slow down
The value of resonance escape probability is determined largely by the fuel-moderator
arrangement and the amount of enrichment of uranium-235. In a homogeneous
reactor the neutrons slow down in a region close to fuel nuclei and thus the probability
of being absorbed by uranium-238 is high. In the heterogeneous reactor neutrons slow
down in the moderator where there are no atoms of uranium-238 and the probability
of undergoing resonance absorption is low. The value of the resonance escape
probability is not significantly affected by pressure or poison concentration. In water
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moderated, low uranium-235 enrichment reactors, raising the temperature of the fuel
will raise the resonance absorption in uranium-238 due to the Doppler effect (i.e. an
apparent broadening of normally narrow resonance peaks due to thermal motion of
nuclei). The increase in resonance absorption lowers the resonance escape probability.
The resonance escape probability can be found from the following formula:
(3-95)
 N ⋅I
p( E ) ≈ exp  − F  ,
 ξ ⋅ Σs 
where ξ is the weighted average logarithmic energy decrement for both moderator
and absorber, NF is the number of fuel nuclei per unit volume of the system, I is the
effective resonance integral and Σs is the total macroscopic cross section for
scattering in the system.
Experimental measurements of the resonance integral for a system of isolated rods
give the following formula:
(3-96)
I = a+b
A
,
M
where a and b are constants for a given fuel material (equal to 2.95 and 81.5,
respectively, for uranium and 4.45 and 84.5 for uranium dioxide, such that I will be in
barns), A is the surface area (in m2) and M is the mass (in kg) of a fuel rod:
(3-97)
A
[b]; I UO2 = 4.45 + 84.5 A [b]
M
M
I U = 2.95 + 81.5
The integral I depends on temperature as follows,
(3-98)
[
I (T ) = I (300 K ) 1 + β
(
T − 300
)]
Here I(300 K) is the value of the integral at T = 300 K and β is a constant which
depends on the nature of the fuel and the radius of fuel rods in heterogeneous systems.
For UO2 and typical fuel rods used in LWRs β = 6x10-3.
Since neutrons absorbed by resonance capture in uranium-238 are lost and unable to
take part in sustaining the fission chain, most thermal reactors are design to maximize
the resonance escape probability as far as possible. In a homogeneous mixture of
natural uranium fuel and carbon graphite moderator the highest value of k∞ is 0.855 –
hence a fission chain can not possibly be sustained. Heterogeneous arrangement of the
same materials can lead to k∞ as high as 1.08 due to the increase in the resonance
escape probability.
Once thermalized, the neutrons continue to diffuse throughout the reactor and are
subject to absorption by other materials in the reactor as well as the fuel. The thermal
utilization factor f is defined as the ratio of the number of thermal neutrons absorbed
in the fuel to the number of thermal neutrons absorbed in all reactor material.
62
C H A P T E R
(3-99)
f =
3
–
N U C L E A R
R E A C T O R
T H E O R Y
Number of thermal neutrons absorbed in the fuel
.
Number of thermal neutrons absorbed in all reactor materials
The thermal utilization factor can be expressed as follows:
(3-100)
f =
Σ a , F ⋅ φ F ⋅ VF + Σ a , M
Σ a ,F ⋅ φF ⋅ VF
,
⋅ φM ⋅ VM + Σ a ,C ⋅ φC ⋅ VC + Σ a ,P ⋅ φP ⋅ VP
where subscripts F, M, P and C refer to uranium, moderator, poison and construction
material (clad, spacers, etc), respectively. In a heterogeneous reactor the flux will be
different in the fuel region than in the moderator region due to the high absorption
rate by the fuel. In the homogenous reactor the neutron flux seen by the fuel,
moderator, poisons and the construction material will be the same and the equation for
f can be rewritten as,
(3-101)
f =
Σ a ,F
Σ a ,F
.
+ Σ a ,M + Σ a ,P + Σ a ,C
The coefficient f will not in general depend on the temperature. However, in
heterogeneous water moderated reactors the moderator (water) expands with
temperature and number of moderator atoms will decrease – and that results in
increase of thermal utilization. Because of this effect the temperature coefficient for the
thermal utilization factor is positive.
Most of the neutrons absorbed in the fuel cause fission, but some do not. The
reproduction factor is defined as the ratio of the fast neutrons produced by thermal
fission to the number of thermal neutrons absorbed in the fuel.
(3-102)
η=
Number of fast neutrons produced by thermal fission
.
Number of thermal neutrons absorbed in the fuel
The reproduction factor can also be stated as a ratio of rates as shown below. The rate
of production of fast neutrons by thermal fission is equal to fission reaction rate
( Σ f ,F ⋅ φF ) times the average number of neutrons produced per fission (ν ). The rate of
absorption is Σ a ,F ⋅ φF and thus the reproduction factor becomes,
(3-103)
η=
Σ f , F ⋅ φF ⋅ ν
Σ a ,F ⋅ φF
=ν
Σ f ,F
Σ a ,F
.
When fuel contains several fissionable materials and other materials, it is necessary to
account for each material, e.g.
∑ (ν Σ )
,
η=
∑ (Σ )
i
(3-104)
f ,i
i
a, j
j
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R E A C T O R
T H E O R Y
where the numerator is the sum of the νΣ f terms for all the fissile nuclides and the
denominator is the total absorption cross section for all the species present in the fuel.
3.3.2
Six-Factor Formula
The infinite multiplication factor can fully represent only a reactor that is infinitely
large. To completely describe the neutron life cycle in a real, finite reactor, it is
necessary to account for neutrons that leak out. The multiplication factor that takes
leakage into account is the effective multiplication factor keff. This coefficient is
defined as follows,
(3-105)
keff =
Rate of neutron production
.
Rate of neutron absorption + rate of leakage
For critical reactor the neutron population is neither increasing nor decreasing and keff
= 1. If the neutron production is grater than the absorption and leakage, the reactor is
called supercritical; keff > 1. If the neutron production is less than the absorption and
leakage, the reactor is called subcritical; keff < 1. The effective and the infinite
multiplication factors are related to each other as follows,
(3-106)
k eff = k ∞ PFNL PTNL ,
where PFNL is the fast non-leakage probability and PTNL is the thermal non-leakage
probability. Equation (3-106) represents the so-called six-factor formula.
Comparing Eq. (3-47) with (3-106) and noting that k = keff, one can conclude that in
one-group treatment, the non-leakage probability of neutrons of a specific energy from
a critical system can be calculated as,
(3-107)
PNL = PFNL PTNL =
1
.
1 + L2 B 2
FIGURE 3-4 illustrates the six-factor formula on a generation of 1000 neutrons. Due
to fast fission, the number of neutrons is increased to 1040 (since the fast fission factor
is assumed to be 1.04). Assuming next that the probability of fast non-leakage is equal
to 0.865, the remaining number of fast neutrons is 900. That corresponds to the loss of
140 fast neutrons from the system. In addition, 180 neutrons is absorbed during
moderation (resonance absorption), 100 neutrons escape after moderation (leakage of
thermal neutrons) and 125 thermal neutrons are absorbed in non-fuel material. Finally
the remaining 495 thermal neutrons are participating in fissions, which bring about
1000 fast neutrons of new generation. In that way the chain reaction is sustained at a
constant level of neutrons from generation to generation.
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T H E O R Y
FIGURE 3-4: Illustration of six-factor formula.
3.4 Nuclear Reactor Transients
A nuclear reactor is a time dependent system, even if it operates at apparently steady
conditions. This is due to persistent changes in fuel composition resulting from the
fuel burn-up. The time changes of reactor properties due to fuel depletion are very
slow ones and are typically investigated at time scales comparable to the length of the
fuel cycle. This type of behavior is typically treated as a sequence of steady-state
conditions. There are, however, reactor transient occurrences in which the core
properties significantly change over time periods which are much shorter; with order
of magnitude of hours, minutes or even seconds. Depending of the length of the
characteristic time scale, the nuclear reactor transients can be roughly divided into three
basic groups:
•
Slow transients, in which the characteristic time scale is of order of weeks or
months (for example fuel depletion)
•
Moderate transients, in which the characteristic time scale is of order of hours
(for example fuel poisoning with xenon-135 and samarium-149)
•
Fast transients, in which the characteristic time scale is of order of seconds (for
example control rod insertion)
The three types of transients are discussed below in more detail.
3.4.1
Nuclear Fuel Depletion
Analysis of nuclear fuel depletion is concerned with an analysis of the neutron flux
variation over a long time period, comparable to the length of the fuel cycle. Since the
transient is very slow, it is typically analyzed as a sequence of stationary states. Thus, for
each such state, the neutron flux distribution is obtained from a pertinent set of steadystate equations. If the diffusion approximation is employed, the following equation is
solved,
(3-108) Dn∇ 2φn − Σ a ,nφn = λn vΣ f ,nφn , n = 1, 2, … (state index),
65
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T H E O R Y
where λn is an eigenvalue of the problem at state n, which for a steady-state solution
should be equal to 1. Thus, one of the design problems over a fuel cycle is to obtain
the eigenvalues equal to unity.
Burnup calculations constitute other essential parts of the analysis of changing reactor
properties over a fuel cycle. Typically the calculations take into account the following
property changes (assuming enriched uranium as the fuel),
•
consumption of the fissile material, uranium-235
•
transmutation of uranium-238 into plutonium
•
buildup of the fission products that have a significant cross-section for
absorption (poisons)
The above-mentioned analysis open paths to optimization of the in-core fuel design,
with the over-all goal to maximize the fuel burnup (or the efficiency of the fuel
utilization).
3.4.2
Fuel Poisoning
During operation of a reactor the amount of fuel contained in the core constantly
decreases. If the reactor is to operate during long periods, fuel in excess of that needed
for exact criticality must be added. The positive reactivity due to the excess fuel must
be balanced with negative reactivity from neutron-absorbing material.
Moveable control rods containing neutron-absorbing materials are one method used to
offset the excess fuel. However, using control rods alone may be impractical, e.g. there
is physically insufficient room for the control rods and their large mechanisms. To
control large amounts of excess fuel burnable poisons are used. Burnable poisons are
materials that have a high neutron absorption cross section that are converted into
materials of relatively low absorption cross section as a result of neutron absorption.
Due to the burnup of the poison material, the negative reactivity of the poison
decreases over core life. Ideally, these poisons should decrease their negative reactivity
at the same rate the fuel’s excess positive reactivity is depleted. Fixed burnable poisons
are usually used in the form of compounds of boron or gadolinium that are shaped
into separate lattice pins or plates, or introduced as additives to the fuel.
Soluble poisons,
also called chemical shim, produce spatially uniform neutron
absorption when dissolved in the water coolant. The most common soluble poison in
PWRs is boric acid (soluble boron or solbor). The boric acid in the coolant decreases
the thermal utilization factor, causing the decrease in reactivity.
By varying the concentration of boric acid in the coolant (a process referred to as
boration and dilution) the reactivity of the core can be easily varied. If the boron
concentration is increased (boration), the coolant/ moderator absorbs more neutrons,
adding negative reactivity. If the boron concentration is reduced (dilution), positive
reactivity is added.
is one that maintains a constant negative reactivity worth over
the life of the core. While no neutron poison is strictly non-burnable, certain materials
Non-burnable poison
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T H E O R Y
can be treated as non-burnable poisons under certain conditions – for example
hafnium. The removal – by absorption of neutrons – of one isotope of hafnium leads
to the production of another neutron absorber, and continues through a chain of 5
absorbers – resulting in a long-lived burnable poison.
It is possible to make the reactivity of a poison material that is usually a burnable
poison more uniform over core life through the use of self-shielding. In self-shielding
the poison material is thick enough that only the outer layer of the poison is exposed to
the neutron flux. The absorptions that take place in the outer layers reduce the number
of neutrons that penetrate to the inner material. As the outer layers of poison absorb
neutrons and are converted to non-poison materials, the inner layers begin absorbing
more neutrons, and the negative reactivity of the poison is fairly uniform
Fission fragments generated at the time of fission decay to produce a variety of fission
products. Fission products are of concern because:
•
they become parasitic absorbers of neutrons
•
Result in long term source of heat
Xenon-135 and samarium-149 have the most substantial impact on reactor design and
operation. Both these poisons have impact on the thermal utilization factor and thus
keff and reactivity.
The neutron absorption cross section of xenon-135 is equal to 2.6 x 106 barns. It is
produced directly by some fissions, but it is more commonly a product of the
tellurium-135 decay chain:
β−
135
52
Te
(3-109)
→
19 s
β−
135
53
I
β−
→
6.57 hr
135
54
Xe
→
9.10 hr
β−
→135
55 Cs
→
2.3 ⋅ 10 6 yr
135
56
Ba
(stable)
The half-life of Te-135 is so short that it can be assumed that iodine-135 is produced
directly from fission. Iodine-135 is not a strong neutron absorber, but decays to form
the neutron poison xenon-135. 95% of all the xenon-135 comes from the decay of
iodine-135. Therefore, the half-life of iodine-135 plays an important role in the amount
of xenon-135 present.
The rate of change of iodine (dI/dt; I is the concentration of iodine-135) is equal to the
rate of production minus the rate of removal. The rate of production is just equal to
yield from fission = γI Σf φ, here γI = 0.061 is the fission yield. The rate of removal is
equal to the decay rate (λI I; λI is the decay constant) plus the burnup rate (σa,I I φ),
(3-110)
dI
= γ I Σ f φ − λ I I − σ a , I Iφ
dt
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T H E O R Y
Since the microscopic absorption cross section σI is quite small, the equation for the
iodine-135 concentration can be written as follows,
(3-111)
dI
= γ I Σ f φ − λI I
dt
When the rate of production of iodine equals the rate of removal, equilibrium exists –
the iodine concentration remains then constant and equal to I0,
(3-112)
0 = γ I Σ f φ − λI I 0 ⇒ I 0 =
γ IΣ fφ
.
λI
Since the equilibrium iodine concentration is proportional to the neutron flux, φ , it is
also proportional to reactor power level.
The rate of change of the xenon-135 concentration (dX/dt) is equal to:
•
(+) Xenon-135 production from fission
•
(+) iodine-135 decay
•
(-) xenon-135 decay
•
(-) xenon-135 burnup
(3-113)
dX
= γ X Σ f φ + λI I − λ X X − σ a, X φ X .
dt
The xenon burnup term σ a , X φ X refers to neutron absorption by xenon-135 by the
following reaction,
(3-114)
135
54
Xe + 01n →
136
54
Xe + γ .
Xenon-136 is not a significant neutron absorber – therefore neutron absorption by
xenon-135 constitutes removal of poison from the reactor. At equilibrium,
(3-115)
0 = γ X Σ f φ + λI I 0 − λ X X 0 − σ a, X φ X 0 ⇒ X 0 =
Since,
(3-116)
I0 =
γ IΣ fφ
λI
,
the equilibrium concentration of xenon-135 is,
(3-117)
X0 =
(γ I + γ X )Σ f φ
λ X + σ a, X φ
.
68
γ X Σ f φ + λI I 0
λX + σ a, X φ
.
C H A P T E R
3
–
N U C L E A R
R E A C T O R
T H E O R Y
Comparing the equilibrium concentrations of iodine-135 and xenon-135 reveals that
iodine concentration at equilibrium is linearly proportional to the neutron flux, and
thus to the reactor power. Xenon-135 concentration increases with a lower rate than
linear when reactor power increases.
When a reactor is shutdown, the neutron flux is reduced essentially to zero. Therefore,
after shutdown, xenon-135 is no longer produced by fission and is no longer removed
by burnup. The only remaining production mechanism is the decay of the iodine-135
which was in the core at the time of shutdown, and the only removal mechanism for
xenon-135 is decay. Because the decay rate of iodine-135 is faster than the decay rate
of xenon-135, the xenon concentration builds to a peak. Subsequently, the production
from iodine decay is less than the removal of xenon by decay, and the concentration of
xenon-135 decreases. The greater the flux level prior to shutdown, the greater the
concentration of iodine-135 at shutdown; therefore, the greater the peak in xenon-135
concentration after shutdown.
During periods of steady state operation at a constant neutron flux level, the xenon135 concentration builds up to its equilibrium value for that reactor power in about 40
to 50 hours.
After reactor shutdown, the differential equations for the concentration of iodine-135
and xenon-135 are as follows,
(3-118)
dI
= −λ I I ,
dt
(3-119)
dX
= λI I − λ X X .
dt
The solution of Eq. (3-118) can be readily obtained as,
(3-120)
I = I 0 e −λI t .
To solve Eq. (3-119), the iodine concentration given by Eq. (3-120) is used in Eq.
(3-119) as follows,
(3-121)
dX
= λI I 0 e −λI t − λ X X ,
dt
and, after integration, the xenon-135 concentration is found as,
(3-122)
X (t ) =
λI
λI − λ X
(
)
I 0 e − λ X t − e −λ I t + X 0 e − λ X t .
One can calculate a dimensionless relative change of the xenon-125 concentration as
follows,
(3-123)
ξ (t ) =
γ I λ X + σ a , I φ0 − λ X t − λI t
X (t )
=
e
−e
+ e −λ X t .
X0
γ I + γ X λI − λ X
(
69
)
C H A P T E R
3
–
N U C L E A R
R E A C T O R
T H E O R Y
Here φ0 is the neutron flux in the reactor just before shutdown. The function
expressed by Eq. (3-124) is plotted in FIGURE 3-5 for various values of the neutron
flux just before the shutdown. As can be seen, the concentration increases when the
shutdown neutron flux is high enough, and gets a maximum value after approximately
10 hours after the shutdown.
X
X
Rel. xenon concentration
9
8
1.0E17
7
5.0E17
1.0E18
6
2.0E18
5
4
3
2
1
0
0
10
20
30
40
50
60
Time after shutdown, hr
FIGURE 3-5: Relative xenon-135 concentration after shutdown of a reactor for neutron flux in range
from 1017 to 2x1018 neutrons m-2 s-1.
Assuming equilibrium concentrations of iodine and xenon before the reactor
shutdown given by Eqs. (3-116) and (3-117), the xenon-135 concentration after the
reactor shutdown is given by the following expression,
(3-125)
X (t ) =
γ I Σ f φ0 − λ t −λ
(e − e
λ X − λI
I
Xt
) + (γλ ++γσ )Σφφ
I
X
X
0
f
a, X
e −λX t .
0
Large thermal reactors with little flux coupling between regions may experience spatial
power oscillations because of the non-uniform presence of xenon-135. The
mechanism of xenon oscillations is described in the following four steps:
1. An initial lack of symmetry in the core power distribution (for example,
individual control rod movement or misalignment) causes an imbalance in
fission rates within the reactor core, and therefore, in the iodine-135 buildup
and the xenon-135 absorption
2. In the high-flux region, xenon-135 burnout allows the flux to increase further,
while in the low-flux region, the increase in xenon-135 causes a further
reduction in flux. The iodine concentration increases where the flux is high
and decreases where the flux is low
3. As soon as the iodine-135 levels build up sufficiently, decay to xenon reverses
the initial situation. Flux decreases in this area, and the former low-flux region
increases in power
70
C H A P T E R
3
–
N U C L E A R
R E A C T O R
T H E O R Y
4. Repetition of these patterns can lead to xenon oscillations moving about the
core with periods on the order of about 15 hours
With little change in overall power level, these oscillations can change the local power
levels by a factor of three or more. In a reactor system with strongly negative
temperature coefficients, the xenon-135 oscillations are damped quite readily. This is
one reason for designing reactors to have negative moderator-temperature coefficients.
Samarium-149 is the second most important fission-product poison because of its high
thermal neutron absorption cross section of 4.1 x 104 barns. Samarium-149 is
produced from the decay of the neodymium-149 fission fragment as shown in the
decay chain below,
β−
(3-126)
149
60
→
Nd
β−
149
61
→
Pm
1.72 hr
149
62
Sm (stable)
53.1 hr
For the purpose of examining the behavior of samarium-149, the 1.73 hour half-life of
neodymium-149 is sufficiently shorter than the 53.1 hour value for promethium-149
that the promethium-149 may be considered as if it were formed directly from fission.
This assumption, and neglecting the small amount of promethium burnup, allows the
situation to be described as follows:
Rate of change of 149Pm = yield from fission – decay of
(3-127)
149
Pm. Therefore:
dP
= γ P Σ f φ − λP P
dt
At equilibrium,
(3-128)
0 = γ P Σ f φ − λP P0 ⇒ P0 =
γ PΣ f φ
.
λP
As can be seen, the equilibrium concentration of promethium-149 is linearly increasing
with the neutron flux and thus with the reactor power.
The rate of samarium-149 formation is described as follows:
(3-129)
dS
= γ S Σ f φ + λP P − σ a ,S φ S .
dt
Since the fission yield of samarium-149 is nearly zero, therefore the equation becomes:
(3-130)
dS
= λP P − σ a ,S φ S ,
dt
and at equilibrium,
71
C H A P T E R
(3-131)
3
–
N U C L E A R
R E A C T O R
0 = λ P P0 − σ a , S φ S 0 ⇒ S 0 =
T H E O R Y
λP P0 γ P Σ f
=
σ a ,S φ σ a ,S
.
After reactor shut-down, the neutron flux drops to essentially zero, and the rate of
samarium-149 production becomes as follows,
(3-132)
dS
= λP P .
dt
Because samarium-149 is not radioactive and is not removed by decay, it presents
problems somewhat different from those encountered with xenon-135. The
equilibrium concentration and the poisoning effect build to an equilibrium value during
reactor operation. This equilibrium is reached in approximately 20 days (500 hours),
and since samarium-149 is stable, the concentration remains essentially constant during
reactor operation. When the reactor is shutdown, the samarium-149 concentration
builds up as a result of the decay of the accumulated promethium-149.
The buildup of samarium-149 after shutdown depends upon the power level before
shutdown. Samarium-149 does not peak as xenon-135 does, but increases slowly to a
maximum value. After shutdown, if the reactor is then operated at power, samarium149 is burned up and its concentration returns to the equilibrium value. Samarium
poisoning is minor when compared to xenon poisoning. Although samarium-149 has a
constant poisoning effect during long-term sustained operation, its behavior during
initial startup and during post-shutdown and restart periods requires special
considerations in reactor design.
There are numerous other fission products that, as a result of their concentration and
thermal neutron absorption cross section, have a poisoning effect on reactor operation.
Individually, they are of little consequence, but lumped together they have a significant
impact. These are often characterized as lumped fission product poisons and
accumulate at an average rate of 50 barns per fission event in the reactor.
In addition to fission product poisons, other materials in the reactor decay to materials
that act as neutron poisons. An example of this is the decay of tritium to helium-3.
Since tritium has a half-life of 12.3 years, normally this decay does not significantly
affect reactor operations because the rate of decay of tritium is so slow. However, if
tritium is produced in a reactor and then allowed to remain in the reactor during a
prolonged shutdown of several months, a sufficient amount of tritium may decay to
helium-3 to add a significant amount of negative reactivity. Any helium-3 produced in
the reactor during a shutdown period will be removed during subsequent operation by
a neutron-proton reaction.
3.4.3
Nuclear Reactor Kinetics
The objective of reactor kinetics is to investigate the fast transient behavior of a reactor
subject to external perturbations. A point-kinetics model is obtained when the spatial
distribution of the neutron flux is neglected and only the time behavior of the
amplitude is concerned. The point-kinetics model can be derived for any number of
delayed neutrons. Typically the models with one and six groups of delayed neutrons
are considered. The six-group point kinetics model is summarized in TABLE 3.2 .
X
72
C H A P T E R
3
–
N U C L E A R
R E A C T O R
T H E O R Y
TABLE 3.2. Point reactor kinetics model.
Neutron flux equation
Equations for concentration of precursors of
delayed neutron: i = 1,…,6
6
dn ρ − β
=
n + ∑ λi C i + S
dt
Λ
i =1
dC i β i
=
n − λ i C i , i = 1, ... ,6
dt
Λ
The values of decay constants and yields of delayed-neutron precursors are given in
TABLE 3.3.
TABLE 3.3. Decay constants and yields of delayed-neutron precursors in thermal fission of uranium-235.
Decay constants and yields of delayed-neutron precursors in thermal
fission of uranium-235
t1/2, [s]
λ , [s-1]
β
β λ
i
55.7
22.7
6.22
2.30
0.61
0.23
i
0.0124
0.0305
0.111
0.301
1.1
3.0
0.000215
0.00142
0.00127
0.00257
0.00075
0.00027
0.0065
Total
i
i
0.0173
0.0466
0.0114
0.0085
0.0007
0.0001
0.084
Some properties of the point kinetics model can be investigated using a one-group
approximation. The total yield of the one group of delayed neutrons is obtained as a
sum of yields in all groups. For uranium-235 this value is shown in TABLE 3.3 and is
equal to β = 0.0065. The decay constant can be obtained from a proper averaging, e.g.,
X
(3-133)
6
β
β
=∑ i ⇒λ =
λ i =1 λi
β
βi
∑
i =1 λi
6
X
.
Using data from TABLE 3.3 , the equivalent decay constant for one-group assumption
for uranium-235 is obtained as λ = 0.0065 0.084 ≅ 0.08 s-1.
X
The equations in one group approximation of point kinetic equations are as follows,
(3-134)
dn  ρ − β 
=
 n + λC + S ,
dt  Λ 
(3-135)
dC β
= n − λC .
dt Λ
Here C represents the concentration of precursors of all groups of delayed neutrons.
In the next section special cases of the point kinetic model will be considered and their
solutions will be found.
The average neutron generation time Λ can be written in various forms as follows,
73
C H A P T E R
(3-136)
Λ=
3
–
N U C L E A R
R E A C T O R
T H E O R Y
1
l l
1
= = ∞ =
.
vνΣ f
k k ∞ k ∞ vΣ a
Here l is the average neutron lifetime and k is the effective multiplication factor. The
name “generation time” has been chosen since Λ represents the average time between
two birth events in successive neutron generations. Firstly, 1 Σ f is the mean free path
for fission, that is, it is the average distance a neutron travels from its birth to a fission
event. Then, (1 Σ f ) v = ∆t f is the average time between the birth of a neutron and a
fission event it may cause. Since ν -neutrons is released per fission, the averaged time
between the birth of a neutron and the birth of new generation is as follows,
(3-137)
∆t f
ν
=
1
=Λ.
vνΣ f
In a similar manner, the average traveling distance of a neutron between the birth and
the death (absorption or leakage) is 1 (Σ a + DBg2 ) and the average neutron lifetime
can be obtained as,
(3-138)
l=
1
1
1
1
,
=
2
v Σ a + DB g vΣ a 1 + L2 B g2
since the one-group diffusion length is given as L = D Σ a .
Eq. (3-138) yields,
(3-139)
l=
k∞
1
k
=
= k ⋅Λ .
2 2
vΣ a k ∞ 1 + L Bg vΣ a k ∞
Equation (3-139) indicates that the lifetime and the generation time are equal for a
critical reactor. For subcritical reactor (k < 1) the neutron lifetime is shorter than the
generation time and as a consequence, the neutron population will decrease. For
supercritical reactor the lifetime will be longer and the neutron population will increase,
whereas for a critical reactor the population remains constant.
X
3.4.4
Nuclear Reactor Dynamics
When power changes in a nuclear reactor are large enough to influence the value of the
reactivity, the transient behavior of the reactor is termed as the nuclear reactor
dynamics. As can be expected, the influence of power on the reactivity has to be
quantified in order to properly describe the dynamic behaviour of the nuclear reactor.
During operation of a nuclear reactor the energy released due to nuclear fission is
deposited to the coolant. The resulting temperature distribution in the fuel and coolant
(in BWRs even the void fraction distribution) is a subject of the thermal-hydraulic
analysis of the nuclear reactor. The temperature distribution, which in a general case is
a function of both the time and the location, is influencing the values of microscopic
cross-sections for various nuclear reactions caused by neutrons. As a result the
reactivity will depend on the temperature changes.
74
C H A P T E R
3
–
N U C L E A R
R E A C T O R
T H E O R Y
From a practical point of view it is important to know what the influence of
temperature on reactivity is and how it will influence the operation of the nuclear
reactor. In general the following two cases can be considered:
•
Reactivity increases with the temperature: in this case the increasing reactivity
will cause the increase of the reactor power, which, in turn, will cause the
increase of temperature, etc. That means in this case the reactor will be
inherently unstable.
•
Reactivity decreases with temperature: in this case the decreasing reactivity will
cause the decrease of the reactor power which will be followed by the decrease
of the temperature, and so on. Clearly the reactor will be inherently stable in
such a case.
The conclusion is that reactors should be constructed in such a way which assures the
decreasing reactivity in function of temperature. This can be achieved by using proper
materials and a proper nuclear reactor configuration.
The reactivity changes with temperature because the reactivity depends on
macroscopic cross sections, which themselves involve the atomic number densities of
materials in the core,
(3-140)
Σ(r, t ) = N (r, t )σ (r, t ) .
The atomic density N(r,t) depends on the reactor power level because:
a) material densities depend on temperature T,
b) the concentration of certain nuclei is constantly changing due to neutron
interactions (poisons and fuel burnup).
The microscopic cross section is written in Eq. (3-140) as an explicit function of the
spatial location r and the time t. This must be done since the cross sections that appear
in one-speed diffusion model are actually averaged over energy spectrum of neutrons
that appear in the reactor core. Since this spectrum is itself dependent on the
temperature distribution in the core and hence the reactor power level, this
dependence must be taken into account in Eq. (3-140).
Such reactivity variation with the temperature is the principal feedback mechanisms
determining the inherent stability of a nuclear reactor with respect to short-term
fluctuations in power level. In principle, the reactivity feedback could be evaluated by
solving heat transfer equations, both in fuel and coolant regions and predicting the
temperature distribution in the reactor core for a given reactor power. However, such
approach would end up with a very complex system of partial nonlinear differential
equations. Typical simplification is based on the so-called “lumped-parameter” model
in which the major parts of the reactor core are represented by a single averaged value
of temperature, such as an average fuel temperature, moderator temperature and
coolant temperature.
The subject of reactor dynamics analysis is to accommodate the core average
temperatures such as TF (fuel) and TM (moderator) in suitable models of reactivity
75
C H A P T E R
3
–
N U C L E A R
R E A C T O R
T H E O R Y
feedback. To this end one can write the reactivity change as a sum of two
contributions,
(3-141)
ρ (t ) = δρ ext (t ) + δρ f ( P ) .
The reactivity in Eq. (3-141) is measured with respect to the equilibrium power level P0
(for which the reactivity is just equal to zero) and is a superposition of the “externally”
controlled reactivity insertion (such as by the control rod movement) and “internal”
reactivity change due to inherent feedback mechanisms, indicated here as a function of
the power level.
For steady-state power level P0, Eq. (3-141) becomes,
(3-142)
0 = ρ 0 + ρ f ( P0 ) ,
which merely states that to sustain the criticality of the system one has to supply
external reactivity ρ 0 to counteract the negative feedback reactivity ρ f ( P0 ) . The
incremental reactivities appearing in Eq. (3-141) are then defined as,
(3-143)
δρ ext (t ) = ρ ext (t ) − ρ 0 ,
(3-144)
δρ f ( P ) = ρ f ( P ) − ρ f ( P0 ) .
The feedback mechanism in reactivity is schematically shown in FIGURE 3-6.
δρ ext
+
Σ
-
Neutron
kinetics
Incremental power
δρ ext
δρ f
Feedback
mechanisms
FIGURE 3-6: Reactivity feedback mechanisms.
With increasing material temperature, the nuclei will move with increasing speed. Since
the nuclei move in a chaotic manner, their relative velocity against a monoenergetic
neutron flux will no longer be constant and will have some distribution. This is
equivalent to a situation in which neutrons would have a certain energy distribution
when approaching stationary nuclei. This is the so-called Doppler effect, in analogy to
a similar phenomenon known in acoustics and optics.
Due to the Doppler effect, the number of nuclear reactions caused by monoenergetic
neutrons will depend on the temperature of the material. Without going into details it
can be mentioned that with the increasing material temperature the microscopic
absorption cross section will increase (e.g. of U-238) resulting in decreasing reactivity.
Also the microscopic fission cross-section will decrease with the temperature leading to
additional reduction of the reactivity.
76
C H A P T E R
3
–
N U C L E A R
R E A C T O R
T H E O R Y
One can estimate the Doppler effect on reactivity using the expression for the
resonance escape probability as,
(3-145)
 N ⋅I
p = exp  − F  ,
 ξ ⋅ Σs 
where for metallic uranium and uranium dioxide fuel at 300K temperature, the
effective resonance integral I is given as,
(3-146)
I uranium = 2.95 + 81.5
A
[b]; IUO2 = 4.45 + 84.5 A [b].
M
M
The temperature dependence of the integral I is described by the following correlation
obtained from experimental data,
(3-147)
[
I (T ) = I (300 K ) 1 + β
(
)]
T − 300 ,
where, for 238UO2:
(3-148)
 A
.
M 
β = 6.1 × 10 −3 + 4.7 × 10 −3 
A and M in Eqs. (3-146) and (3-148) are area (in m2) and mass (in kg), respectively, of
a fuel rod.
When keff remains constant from one generation of neutrons to another, it is possible
to determine the number of neutrons at any particular generation by knowing the
number of neutrons at “zero” generation, N0, and the value of keff. Thus, after n
n
generations the number of neutrons is equal to N = N 0 (k eff ) .
In particular, if in the preceding generation there are N 0 neutrons, then there are
N 0 k eff neutrons in the present generation. The change of the number of neutrons
expressed as a fraction of the present number of neutrons is referred to as reactivity
and is expressed as,
(3-149)
ρ=
N 0 k eff − N 0
N 0 k eff
=
k eff − 1
k eff
.
As can be seen, reactivity is a dimensionless number; however, since its value is often
rather small, artificial units are defined. From the definition given by Eq. (3-149), the
value of reactivity is in units of ∆k / k . Alternative units are % ∆k / k and pcm
(percent milli-rho). The conversions between these units are as follows,
(3-150)
1 ∆k k = 100% ∆k k = 10 5 pcm ,
(3-151)
1% ∆k k = 0.01 ∆k k = 10 3 pcm ,
77
C H A P T E R
(3-152)
3
–
N U C L E A R
R E A C T O R
T H E O R Y
1pcm = 10 −5 ∆k k = 10 −3 % ∆k k .
EXAMPLE 3-1. Calculate the reactivity in a reactor core when keff is equal to 1.002
and 0.998. SOLUTION: The reactivity is as follows: for keff = 1.002, ρ = (1.0021/1.002 = 0.001996 ∆k/k = 0.1996 % ∆k/k = 199.6 pcm. For keff = 0.998: ρ =
(0.998-1/0.998 = 0.002 ∆k/k = 0.2 % ∆k/k = 200 pcm.
Other units often used in reactor analysis are dollars ($) and cents. By definition, 1$ is
reactivity which is equivalent to the effective delayed neutron fraction, β , and, as can
be expected, one cent (1c) reactivity is equal to one-hundreth of a dollar.
As already mentioned, the dependence of the reactivity on temperature has an
important influence on the reactor dynamics and stability. In particular, a reactor will
be stable when the reactivity is a decreasing function of temperature. Needless to say
that the dependence of the reactivity on various parameters should be known.
However, it is not possible (and not necessary for most practical purposes) to estimate
such functions with all details. Instead a linearized form of the function is applied to
evaluate the reactivity change due to various parameter changes, that is,
(3-153)
δρ (TC , TF , TM ,...) ≅
∂ρ
∂ρ
∂ρ
δTM + ...
δTC +
δTF +
∂TC
∂TF
∂TM
,
= αTCδTC + α TF δTF + α TM δTM + ...
where α TC , α TF , α TM are the coolant, fuel and moderator temperature coefficient of
reactivity, respectively. These coefficients play important role in safe operation of
nuclear reactors.
A single temperature coefficient of reactivity can be defined as a derivative of the core
reactivity with respect to temperature,
(3-154)
αT ≡
∂ρ
∂ρ
→∑
= ∑ α T( j ) .
∂T
j ∂T j
j
Here j indicates that separate temperatures in the reactor (j = C for coolant, j = F for
fuel, etc.) are taken into account.
The two dominant temperature effects in most reactors are the change in resonance
absorption (Doppler effect) due to fuel temperature change and the change in the
neutron energy spectrum due to changing moderator or coolant density (due to
temperature, pressure or void fraction changes).
Noting that,
(3-155)
αT ≡
∂ρ
1 ∂k eff
1 ∂k eff
= 2
≅
,
∂T k eff ∂T
k eff ∂T
one obtains,
78
C H A P T E R
(3-156)
αT =
3
–
N U C L E A R
R E A C T O R
T H E O R Y
1 ∂k eff
1 ∂k eff
+
= α TF + α TM .
k eff ∂TF k eff ∂TM
Now changes in fuel temperature TF do not affect the shape of the thermal neutron
energy spectrum. It should be mentioned, however, that fuel and moderator
temperature effects can not always be separated. In LWRs a change in moderator
temperature will change the moderator density significantly, thereby influencing
slowing down and hence resonance absorption. In spite of such interference, it is
customary to analyze both coefficients separately.
The fuel temperature coefficient has an important influence on reactor safety in case
of a large positive reactivity insertion. A negative fuel temperature coefficient is
generally considered more important than a negative moderator temperature
coefficient. The reason is that the negative fuel coefficient starts adding negative
reactivity immediately, whereas the moderator temperature cannot turn the power rise
for several seconds.
This coefficient is also called the prompt reactivity coefficient or the fuel Doppler
reactivity coefficient. Its value can be readily obtained from Eq. (3-145) as,
(3-157)
α TF =
1 dk eff
1 dp
1 dI
=
= ln p
.
k eff dTF
p dTF
I dTF
Using Eq. (3-147) in (3-157) yields,
X
(3-158)

 β
1

 p(300 K )  2 TF
α TF = − ln 
.
When the moderator is at the same time used as a coolant (this is the case in LWRs),
the moderator coefficient of reactivity will in principle reflect the influence of the
coolant density changes on the reactivity.
The dominant reactivity effect in water-moderated reactors arises from changes in
moderator density, due to the thermal expansion of the coolant fluid or due to the void
formation. The principal effect is usually the loss of moderation that accompanies a
decrease in moderator density and causes a corresponding increase in resonance
absorption. It can be calculated as follows,
α TM =
(3-159)
1 dk eff
1 dp
=
=
k eff dTM
p dTM
NF
1 d  
exp −
p dTM   ξ N M σ s

 1  dN M
I  = ln 
 p  dTM

.
Since dNM/dTM is negative and may be quite large, particularly if the coolant
temperature is close to the saturation temperature, the reactivity coefficient can also be
large. Typical values of reactivity coefficients are given in TABLE 3.4.
X
79
C H A P T E R
3
–
N U C L E A R
R E A C T O R
T H E O R Y
TABLE 3.4. Reactivity coefficients.
Type of coefficient
BWR
PWR
HTGR
LMFBR
Fuel Doppler (pcm/K)
-4 to -1
-4 to -1
-7
-0.6 to -2.5
-200 to -100
0
-
-12 to +20
Moderator (pcm/K)
-50 to -8
-50 to -8
+1.0
-
Expansion (pcm/K)
~0
~0
~0
-0.92
Coolant void (pcm/%void)
3.4.5
Nuclear Reactor Instabilities
Important aspect of safe reactor operation is the stability of the nuclear reactor in an
equilibrium state (either operating or shut-down reactor). Strictly speaking, at this point
it is necessary to specify what is meant by the stability. Without going into
mathematical details, it is assumed that the stability is defined in the Lapunov sense. It
is assumed that a system is stable in the Lapunov sense if after a small perturbation
from equilibrium the system will move to new conditions which are close to the initial
equilibrium conditions.
There are several different methods to evaluate reactor stability at equilibrium. One
method, which will be discussed in more detail, is based on the linearization of the
system of differential equations and application of the Laplace transform.
The Laplace transform representation is particularly useful in the development of the
important property of a system called the transfer function. In a general sense, the
transfer function is a mathematical expression which describes the effect of a physical
system on the signal transferred through it. For a system shown in FIGURE 3-7 , the
transfer function G(s) is defined as a ratio of the Laplace transform of the output
signal, Y(s) to the Laplace transform of the input signal, U(s); G(s) = Y(s)/U(s).
X
U(s)
G(s)
X
Y(s)
FIGURE 3-7: Open loop system.
The system represents a so-called open-loop system, that is, a system without feedback.
A system with feedback, also called a closed loop system, is shown in FIGURE 3-8.
U(s)
G(s)
Σ
Y(s)
H(s)
FIGURE 3-8: System with feedback.
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G(s) represents the system (forward) transfer function and H(s) is the feedback
transfer function. Performing a summation of signals at the input to the system yields,
(3-160)
[U ( s ) + H ( s ) ⋅ Y ( s )]G( s ) = Y ( s ) .
Thus, the transfer function for the system with feedback (closed loop transfer
function) becomes,
(3-161)
GT ( s ) ≡
Y ( s)
G( s)
.
=
U ( s) 1 − H ( s) ⋅ G( s)
Both the forward and multiple feedback reactor transfer functions have been derived
in previous sections.
Reactor kinetics equations represent a typical example of an open-loop system. The
input signal is then the reactivity and the output signal is the neutron flux. The transfer
function, referred usually as a zero-power reactor transfer function, is defined as,
(3-162)
G( s) =
xˆ ( s )
.
ρˆ ( s )
Using the linearized reactor point kinetics equations (with dropped non-linear terms,
which holds for small perturbations and ρ < 0.1β ), the transfer function of the zero
power reactor is obtained as,
(3-163)
G( s) =
xˆ ( s ) 1
1
= ⋅
.
ρˆ ( s ) Λ 
1 6 βi 

s 1 + ∑
Λ
+
s
λ
i
=
1
i 

The gain G ( jω ) and the phase angle arg (G ( jω ) ) of the transfer function are shown
in FIGURE 3-9 and FIGURE 3-10, respectively.
X
20
Λ=10−3s
−4
Λ=10 s
−6
Λ=10 s
15
10
Gain, decibeles
5
0
−5
−10
−15
−20
−25
−2
10
−1
10
0
10
Frequency, 1/s
1
10
2
10
FIGURE 3-9: Gain plot for various neutron generation time values.
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The gain plot shown in FIGURE 3-9 (often called the Bode diagram) indicates that an
open-loop reactor system tends to be unstable as frequency becomes small, since the
gain becomes infinite when frequency approaches zero. Thus, a reactor without
feedback is expected to be intrinsically unstable.
X
0
−10
Λ=10−3s
−4
Λ=10 s
−6
Λ=10 s
−20
Phase angle, degrees
−30
−40
−50
−60
−70
−80
−90
−2
10
−1
10
0
10
Frequency, 1/s
1
10
2
10
FIGURE 3-10: Phase plot for various neutron generation time values.
Reactor dynamics equations represent a typical example of a closed loop system. This
is due to the presence of multiple feedbacks which exist in this case, as discussed in
previous sections.
It is interesting to consider a special case of the open loop in which,
(3-164)
U ( s) ⋅ G( s) ⋅ H ( s) = U ( s) .
In this case the system will be self-excited if the feedback loop is closed. There will be
no need for any external input signal since the feedback signal is in-phase with the
external input. Equation (3-164) can be written as,
(3-165)
[1 − H ( s ) ⋅ G( s )]U ( s ) = 0 .
Since the input perturbation is arbitrary, the condition for the instability is,
(3-166)
1 − H ( s) ⋅ G( s) = 0 .
Equation (3-166) is called the characteristic equation of the system and is the same as
the denominator of the closed loop transfer function, given by Eq. (3-161).
One can observe that the roots of the characteristic function will be poles of the closed
loop transfer function. If the closed loop transfer function has the form,
(3-167)
F ( s) =
1
,
s−a
the original f(t) of the transfer function F(s) is,
(3-168)
f (t ) = L−1 {F ( s )} = e at .
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Transfer function F(s) has a pole at s = a, which corresponds to the zero value of its
denominator s – a. If now Re(a) > 0 the function f(t) will diverge with time which
indicates unstable system. In other words, roots of the characteristic equation may
imply exponential divergence in the time domain. The limiting case when Re(a) = 0 is
the marginal stability case in which a perturbation causes the system to oscillate
sinusoidally, but not diverge with time. Naturally system is stable when Re(a) < 0, since
then any perturbation will damp out with time.
One can formulate the following stability criterion for a closed loop system:
“The necessary and sufficient condition for the closed loop system to be stable to small perturbations is
that all the roots of the characteristic equation have negative real parts”.
If one can factor the closed loop transfer function using partial fraction expansion, the
roots can be easily determined. This is the case when the closed loop transfer function
has a form of a rational polynomial. In the case of complicated transcendental algebraic
equations direct evaluation of roots is not trivial, however, and an approach using the
Nyquist Criterion has proven to be an efficient way of investigation of the system
stability.
An important parameter used in the evaluation of BWR stability is the decay ratio,
which is defined as the ratio of two consecutive amplitudes in a given signal, as shown
in FIGURE 3-11. The decay ratio can be calculated from the analytical solution as a
ratio of the system response at time t0 + T to the value at t0, where T is the period of
oscillations. More information about reactor dynamics and stability can be found in
[3-1].
A1
DR=A2/ A1
A2
t0
t
t0+T
FIGURE 3-11: Definition of the decay ratio.
3.4.6
Control Rod Analysis
A control rod, with a cylindrical shape and a radius a (see FIGURE 3-12), inserted
partly into a bare cylindrical reactor, will be considered. From the perturbation
theory[3-4] it is obtained that the reactivity change caused by a perturbation in a reactor
can be calculated as,
X
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
∫ φ  − ∇δD∇ + δΣ
∆ρ =
∫ νΣ φ
1
− νδΣ f
k
2
dV
a
(3-169)
f
T H E O R Y

φdV

,
where the integration is over the whole volume of the reactor. Here
δD, δΣ a and δΣ f are perturbations of the diffusion coefficient, the macroscopic
cross section for absorption and the macroscopic cross section for fission, respectively.
For control rods insertion only the macroscopic cross section for absorption is
perturbed, thus,
∫ δΣ φ dV .
∆ρ =
∫ νΣ φ dV
2
(3-170)
a
2
f
In Eq. (3-170) φ is the neutron flux in a not-perturbed critical reactor; that is in a
reactor without the control rod. For finite-cylinder reactor (and using coordinate
system as shown in FIGURE 3-12 ) the neutron flux is given as,
X
X
(3-171)
 2.405r  πz
~  sin ~ .
H
 R 
φ (r, z ) = AJ 0 
2a
x
z
H
R
FIGURE 3-12: Partly inserted control rod.
For a control rod inserted in the core with a distance x, the perturbation of the
macroscopic cross section for absorption can be expressed as,
(3-172)
Σ a ,cr − Σ a ,c
0

δΣ a = 
for
0 ≤ z ≤ x, 0 ≤ r ≤ a
otherwise
,
where Σ a,cr and Σ a,c are the macroscopic cross sections for absorption for the control
rod and the core, respectively.
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Substituting Eqs. (3-171) and (3-172) into (3-170) yields,
(3-173)
∆ρ ( x ) =
a
x
0
0
∫ ∫ (Σ
a ,cr
 2.405r 
 πz 
− Σ a ,c )A2 J 02  ~  sin 2  ~ 2πrdrdz
 R 
H
,
2
∫ νΣ f φ dV
and for fully inserted control rod,
~
H
a
(3-174)
~
∆ρ H =
( )
∫ ∫ (Σ
0
0
a ,cr
 2.405r 
 πz 
− Σ a ,c )A2 J 02  ~  sin 2  ~ 2πrdrdz
 R 
H
.
2
∫ νΣ f φ dV
Dividing Eqs. (3-173) and (3-174) with each other yields,
(3-175)
∆ρ ( x )
~ =
∆ρ H
( )
∫
x
0
~
H
∫
0
πz
sin 2 ~ dz
x
1
2πx
H
= ~−
sin ~ ,
πz
H
sin 2 ~ dz H 2π
H
and finally,
(3-176)
1
2πx 
~ x
∆ρ (x ) = ∆ρ H  ~ −
sin ~  .
H 
 H 2π
( )
The above equation is graphically illustrated in FIGURE 3-13. As can be seen, for
small changes of x around x = 0.5 the relative change of the reactivity is almost a linear
function of x. This observation is used when performing calibration of control rods.
X
Applying similar as above approach based on the perturbation theory, it can be shown
that if the control rod is placed at a distance r0 from the core centerline (see FIGURE
3-14), then the reactivity change can be calculated as,
X
(3-177)
 2.405r 
∆ρ (r0 ) = J 02  ~ 0 ∆ρ (0 ) .
 R 
Here: ∆ρ (r0 ), ∆ρ (0 ) - changes of the reactivity for a control rod located at a distance
r0 from the centerline and a control rod placed centrally, respectively.
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1
0,9
Relative change of reactivity
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
0
0,2
0,4
0,6
0,8
1
Relative rod position
FIGURE 3-13: Relative reactivity change as a function of insertion distance of a centrally placed control
rod.
R
r0
FIGURE 3-14: Control rod located at a distance r0 from the centerline.
R E F E R E N C E S
[3-1]
Anglart, H., Nuclear Reactor Dynamics and Stability, Compendium, KTH, 2009.
[3-2]
Beckjord, E. Ex. Dir. et al., The Future of Nuclear Power. An Interdisciplinary MIT Study, MIT 2003,
ISBN 0-615-12420-8
[3-3]
Glasstone, S. and Sesonske, A., Nuclear Reactor Engineering, Van Nostrand Reinhold Compant,
1981, ISBN 0-442-20057-9.
[3-4]
Duderstadt, J.J. and Hamilton, L.J., Nuclear Reactor Analysis, John Wiley & Sons, Inc.
[3-5]
Stacey, W.M., Nuclear Reactor Physics, Wiley-VCH, 2004
E X E R C I S E S
EXERCISE 3-1: Calculate the thermal utilization factor for a homogenized core composed of (in % by
volume): UO2 35% and H2O 65%. The enrichment of the fuel is 3.2% (by weight). Microscopic cross
sections [b] for absorption are as follows: water 0.66 [b], oxygen O: 2x10-4 [b], U-235: 681 [b], U-238: 2.7
[b]. Density of UO2: 10200 kg/m3, density of water: 800 kg/m3.
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EXERCISE 3-2: Calculate the resonance escape probability for a reactor as in the previous exercise
assuming the fuel temperature T = 1500 K and the effective resonance integral for fuel at T = 300 K
equal to 25 [b]. Microscopic cross sections for scattering are as follows: water 103 [b], oxygen O: 6 [b], U235: 8 [b], U-238: 8.3 [b].
EXERCISE 3-3: Determine the effective multiplication factor and the neutron flux distribution in a
homogeneous spherical reactor with radius R and with a reflector with the extrapolated thickness
~
T using the one-group diffusion approximation.
EXERCISE 3-4: Determine the effective multiplication factor and the neutron flux distribution in a
homogeneous spherical bare reactor with:
λtr ,c = 5 ⋅ 10 −3 m, Σ a ,c = 4m −1 , Σ f ,c = 3.4m −1 , ν = 2.47, R = 0.2m and k∞ = 2.1.
How the effective multiplication factor will change if the reactor will be surrounded with a graphite
reflector, with the following data:
λtr ,r = 2.75 ⋅ 10 −2 m, Σ a ,r = 3.4 ⋅ 10 −2 m −1 , T = 0.6m .
EXERCISE 3-5: Reactor as in EXERCISE 4 is surrounded with a graphite reflector. Investigate how the
reflector thickness influences the effective multiplication factor. Is it possible to suggest an optimal
thickness of the reflector?
EXERCISE 3-6: A homogeneous reactor with a finite-cylinder shape has a radius R and the extrapolated
~
height H~ . The reactor is surrounded with a radial reflector with the extrapolated thickness T and
extrapolated height H~ . Determine the effective multiplication factor and the distribution of the neutron
flux for a critical reactor using the one-group diffusion approximation.
EXERCISE 3-7: Investigate a possibility to construct a homogeneous reactor which would use the
natural uranium as the fuel and graphite as the moderator. For that purpose it is necessary to calculate the
infinite multiplication factor k ∞ as a function of the ratio NM/NF, where NM and NF are concentration
of nuclei of moderator and fuel, respectively. It can be assumed that η = 1.33 and ε = 1 . Use reasonable
approximations.
EXERCISE 3-8: A cylindrical reactor core with dimensions H = 3.7 m and R = 2.2 m has a centrally
inserted control rod to a certain unknown position x0. The operator decided to measure the control rod
worth and removed the rod from the core with dx = 2 cm. The measured increase of reactivity was 11.2
pcm. After the measurement, the control rod was moved back to the initial position x0 and next inserted
into the reactor core with dx = 2cm. The measured decrease of reactivity was 11.7 pcm. All reactivity
changes are given in relation to the insertion length x0. Calculate the initial insertion length of the control
rod x0 and the worth of the rod in the full-inserted position.
EXERCISE 3-9: LWR reactor core with UO2 fuel contains 5.6x1027 nuclei of U-235 and the average
thermal fission microscopic cross-section of the fuel is σf = 344x10-28 m2. Assuming that the reactor
operates with constant thermal power equal to 3000 MW, calculate: (a) average neutron flux in the core,
(b) equilibrium number of iodine nuclei in the core, (c) equilibrium number of xenon nuclei in the core,
(d) the time after shutdown when the number of xenon nuclei will have maximum. Given: iodine half-life
6.7 h, xenon half-life 9.2 h, iodine effective fractional yield from fission 0.061, xenon effective fractional
yield from fission 0.003, microscopic absorption cross-section for xenon 2.6x106 barn.
87
Chapter
4
4
Heat Generation and Removal
T
his chapter presents methods to determine the distribution of heat sources
and temperatures in various components of nuclear reactor. In safety analyses
of nuclear power plants the amount of heat generated in the reactor core must
be known in order to be able to calculate the temperature distributions and
thus, to determine the safety margins. Such analyses have to be performed for all
imaginable conditions, including nominal operation conditions, reactor startup and
shutdown, as well as for removal of the decay heat after reactor shutdown. The first
section presents the methods to predict the heat sources in nuclear reactors at various
conditions. The following sections discuss the prediction of such parameters as coolant
enthalpy, fuel element temperature, void fraction, pressure drop and the occurrence of
the Critical Heat Flux (CHF) in nuclear fuel assemblies.
4.1
Energy from Nuclear Fission
4.1.1
Thermal Power of Nuclear Reactor
Consider a monoenergetic neutron beam in which n is the neutron density (number
of neutrons per m3). If v is neutron speed then nv is the number of neutron falling
on 1 m2 of target material per second. Since σ is the effective area per single nucleus,
for a given reaction and neutron energy, then Σ is the effective area of all the nuclei
per m3 of target. Hence the product Σnv gives the number of interactions of nuclei
and neutrons per m3 of target material per second.
In particular, the fission rate is found as: Σ f nv = Σ f φ , where φ = nv is the neutron
flux (to be discussed later) and Σ f = Nσ f , N being the number of fissile nuclei/m3
and σ f m2/nucleus the fission cross section. In a reactor the neutrons are not
monoenergetic and cover a wide range of energies, with different flux and
corresponding cross section.
In thermal reactor with volume V there will occur V Σ f φ fissions, where Σ f and φ
are the average values of the macroscopic fissions cross section and the neutron flux,
respectively.
To evaluate the reactor power it is necessary to know the average amount of energy
which is released in a single fission. The table below shows typical values for uranium235.
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TABLE 4.1. Approximate distribution of energy per fission of uranium-235
10-12 J = 1 pJ
MeV
Kinetic energy of fission products
26.9
168
Instantaneous gamma-ray energy
1.1
7
Kinetic energy of fission neutrons
0.8
5
Beta particles from fission products
1.1
7
Gamma rays from fission products
1.0
6
Neutrinos
1.6
10
Total fission energy
32
200
As can be seen, the total fission energy is equal to 32 pJ. It means that it is required
~3.1 1010 fissions per second to generate 1 W of the thermal power. Thus, the thermal
power of a reactor can be evaluated as,
(4-1)
P=
VΣf φ
(W) .
3.1 ⋅ 1010
Thus, the thermal power of a nuclear reactor is proportional to the number of fissile
nuclei, N, and the neutron flux φ . Both these parameters vary in a nuclear reactor and
their correct computation is necessary to be able to accurately calculate the reactor
power.
Power density (which is the total power divided by the volume) in nuclear reactors is
much higher than in conventional power plants. Its typical value for PWRs is 75
MW/m3, whereas for a fast breeder reactor cooled with sodium it can be as high as
530 MW/m3.
4.1.2
Fission Yield
Fissions of uranium-235 nucleus can end up with 80 different primary fission products.
The range of mass numbers of products is from 72 (isotope of zinc) to 161 (possibly
an isotope of terbium). The yields of the products of thermal fission of uranium-233,
uranium-235, plutonium-239 and a mixture of uranium and plutonium are shown in
FIGURE 4-1.
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FIGURE 4-1: Fission yield as a function of mass number of the fission product (from Wikimedia
Commons).
As can be seen in all cases there are two groups of fission products: a “light” group
with mass number between 80 and 110 and a “heavy” group with mass numbers
between 125 and 155.
4.1.3
Decay Heat
A large portion of the radioactive fission products emit gamma rays, in addition to beta
particles. The amount and activity of individual fission products and the total fission
product inventory in the reactor fuel during operation and after shut-down are
important for several reasons: namely to evaluate the radiation hazard, and to
determine the decrease of the fission product radioactivity in the spent fuel elements
after removal from the reactor. This information is required to evaluate the length of
the cooling period before the fuel can be reprocessed.
Right after the insertion of a large negative reactivity to the reactor core (for example,
due to an injection of control rods), the neutron flux rapidly decreases according to the
following equation,
(4-2)
λρ
( β − ρ )t 
t
 β
−
ρ

β −ρ
φ (t ) = φ 0 
e
−
e l .
β −ρ
 β − ρ

Here φ (t ) is the neutron flux at time t after reactor shut-down, φ 0 is the neutron flux
during reactor operation at full power, ρ is the step change of reactivity, β is the
fraction of delayed neutrons, l is the prompt neutron lifetime and λ is the mean decay
constant of precursors of delayed neutrons. For LWR with uranium-235 as the fissile
material, typical values are as follows: λ = 0.08 s-1, β = 0.0065 and l = 10-3 s.
Assuming the negative step-change of reactivity ρ = -0.09, the relative neutron flux
change is given as,
(4-3)
φ (t )
= 0.067e −0.075t + 0.933e −96.5t .
φ0
The second term in Eq. (4-3) is negligible already after t = 0.01s and only the first term
has to be taken into account in calculations. As can be seen, the neutron flux (and thus
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the generated power) immediately jumps to ~6.7% of its initial value and then it is
reduced e-fold during period of time T = 1/0.075 = 13.3 s.
After a reactor is shut down and the neutron flux falls to such a small value that it may
be neglected, substantial amounts of heat continue to be generated due to the beta
particles and the gamma rays emitted by the fission products. FIGURE 4-2 shows the
fission product decay heat versus the time after shut down. The curve, which covers a
time range from 1 to 106 years after shut down, refers to a hypothetical pressurized
water-cooled reactor that has operated at a constant power for a period of time during
which the fuel (with initial enrichment 4.5%) has reached 50 GWd/tU burnup and is
then shut down instantaneously. Contributions from various species which are present
in the spent fuel are indicated.
The power density change due to beta and gamma radiation can be calculated from the
following approximate correlation[4-2],
(4-4)
q′′′
−0.2
≅ 0.065 (t − top ) − t −0.2 ,
q0′′′
[
]
Here q 0′′′ is the power density in the reactor at steady state operation before shut
down, q′′′ is the decay power density, t is the time after reactor shut down [s] and top is
the time of reactor operation before shut down [s]. Equation (4-4) is applicable
regardless of whether the fuel containing the fission products remains in the reactor
core or it is removed from it. However, the equation accuracy and applicability is
limited and can be used for cooling periods from approximately 10 s to less than 100
days.
Equation (4-4) can be transformed as follows,
(4-5)




q′′′ 0.065 
1
1  0.065  1
1 
≅ 0.2 
−
= 0.2  0.2 −
.
0.2
0.2 
0.2 
θ
q0′′′
top   t − t 
t
(
)
θ
+
1


op


op
 t  
  t 
t 
 op  
  op 
Here θ = (t − top ) top is the relative time after reactor shut down. Equation (4-5) is
shown in FIGURE 4-3 for the reactor operation time top from 1 month to 1 year.
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FIGURE 4-2: Fission-product decay heat power (in Watts per metric ton of heavy metal) versus time
after shutdown in years, [3-2].
0.06
t_op = 1 m
0.05
t_op = 2 m
Relative power
t_op = 6 m
t_op = 12 m
0.04
0.03
0.02
0.01
0
0.000001
0.0001
0.01
1
Rel. time after shutdown
FIGURE 4-3: Relative decay power versus relative time after reactor shutdown for various operation
periods from 1 month to 1 year.
4.1.4
Spatial Distribution of Heat Sources
The energy released in nuclear fission reaction is distributed among a variety of
reaction products characterized by different range and time delays. Once performing
the thermal design of a reactor core, the energy deposition distributed over the coolant
and structural materials is frequently reassigned to the fuel in order to simplify the
thermal analysis of the core. The volumetric fission heat source in the core can be
found in general case as,
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∞
(4-6)
q′′′(r ) = ∑ w(fi ) N i (r )∫ dEσ (fi ) (E )φ (r, E ) .
0
i
Here w (if ) is the recoverable energy released per fission event of i-th fissile material,
N i (r ) is the number density of i-th fissile material at location r and σ if (E ) is its
microscopic fission cross section for neutrons with energy E. Since the neutron flux
and the number density of the fuel vary across the reactor core, there will be a
corresponding variation in the fission heat source.
The simplest model of fission heat distribution would correspond to a bare,
homogeneous core. One should recall here the one-group flux distribution for such
geometry given as,
(4-7)
 2.405r   πz 
~  cos ~  .
 R  H 
φ (r, z ) = φ0 J 0 
~
~
Here φ0 is the flux at the center of the core and R and H are effective (extrapolated)
core dimensions that include extrapolation lengths as well as an adjustment to account
for a reflected core.
Having a fuel rod located at r = rf distance from the centerline of the core, the
volumetric fission heat source becomes a function of the axial coordinate, z, only,
(4-8)
 2.405rf
q′′′( z ) = w f Σ f φ0 J 0 
~
 R
 π z 
cos ~  .
 H 
There are numerous factors that perturb the power distribution of the reactor core,
and the above equation will not be valid. For example fuel is usually not loaded with
uniform enrichment. At the beginning of core life, higher enrichment fuel is loaded
toward the edge of the core in order to flatten the power distribution. Other factors
include the influence of the control rods and variation of the coolant density.
All these power perturbations will cause a corresponding variation of temperature
distribution in the core. A usual technique to take care of these variations is to estimate
the local working conditions (power level, coolant flow, etc) which are the closest to
the thermal limitations. Such part of the core is called hot channel and the working
conditions are related with so-called hot channel factors.
One common approach to define hot channel is to choose the channel where the core
heat flux and the coolant enthalpy rise is a maximum. Working conditions in the hot
channel are defined by several ratios of local conditions to core-averaged conditions.
These ratios, termed the hot channel factors or power peaking factors will be
considered in more detail in coming Chapters. However, it can be mentioned already
here that the basic initial plant thermal design relay on these factors.
In thermal reactors it is assumed that 90% of the fission total energy is liberated in fuel
elements, whereas the remaining 10% is equally distributed between moderator and
reflector/shields.
94
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4.2 Coolant Flow and Heat Transfer in Rod
Bundles
Rod bundles in nuclear reactors have usually very complex geometry. Due to that a
thorough thermal-hydraulic analysis in rod bundles requires quite sophisticated
computational tools. In general, several levels of approximations can be employed to
perform the analysis,
•
simple one-dimensional analysis of a single subchannel or bundle,
•
analysis of a whole rod bundle applying the subchannel-analysis code,
•
complex three-dimensional analysis using Computational Fluid Dynamics
(CFD) cods.
In this book only the simples approach is considered. In this approach, the single
subchannel or rod bundle is treated as a one-dimensional pipe with a diameter equal to
the hydraulic (equivalent) diameter of the subchannel or bundle. The hydraulic
diameter of a channel of arbitrary shape is defined as,
(4-9)
Dh =
4A
,
Pw
where A is the channel cross-section area and Pw is the channel wetted perimeter.
FIGURE 4-4 shows typical coolant subchannels in infinite rod lattices.
p
p
d
d
Flow subchannel
Triangular lattice
Square lattice
FIGURE 4-4: Typical coolant subchannels in rod bundles.
The hydraulic diameter for the square and triangular lattice can be calculated from Eq.
(4-9) and from principles shown in FIGURE 4-4 as follows. The subchannel flow area
is given as,
 2 πd 2
 p − 4
A=
2
 3 p 2 − πd
 4
8
for square lattice
,
for triangular lattice
95
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and the wetted perimeter (part of the perimeter filled with walls) is given as,
 πd
Pw =  1
 2 πd
for square lattice
for triangular lattice
.
Here p is the lattice pitch and d is the diameter of fuel rods. The hydraulic diameter is
obtained as:
(4-10)
  4  p 2 
 d    − 1
 π  d 

Dh =  
2
d  2 3  p  − 1

  π  d 




for square lattice
for triangular lattice
In case of fuel assemblies in Boiling Water Reactors (BWR), the hydraulic diameter
should be based on the total wetted perimeter and the total cross-section area of the
fuel assembly. Assuming fuel assembly as shown in FIGURE 4-5, the hydraulic
diameter is as follows,
(4-11)
Dh ≡
4 A 4 w2 − Nπd 2
=
Pw
4 w + Nπd
Here N is the number of rods in the assembly, w is the width of the box [m] and d is
the diameter of fuel rods [m].
Box wall
Fuel rods
p
d
w
FIGURE 4-5: Cross-section of a BWR fuel assembly.
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4
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Enthalpy Distribution in Heated Channels
Assume a heated channel with an arbitrary axial distribution of the heat flux, q’’(z),
and an arbitrary, axially-dependent geometry, as shown in FIGURE 4-6. The coolant
flowing in the channel has a constant mass flow rate W.
z
z=H/2
z+dz
il(z)+dil
q’’
z
il(z)
z=-H/2
W, ili
FIGURE 4-6: A heated channel.
Energy balance for a differential channel length between z and z+dz is as follows,
(4-12)
W ⋅ il ( z ) + q′′( z ) ⋅ PH ( z ) ⋅ dz = W ⋅ [il ( z ) + dil ] ,
which leads to the following differential equation for the coolant enthalpy,
(4-13)
dil ( z ) q′′( z ) ⋅ PH ( z )
=
.
dz
W
Here PH(z) is the heated perimeter of the channel. Integration of Eq. (4-13) from the
channel inlet to a certain location z yields,
(4-14)
1
il ( z ) = ili +
W
z
∫ q′′( z) ⋅ P
H
( z )dz
−H / 2
where il(z) is the coolant enthalpy at location z and ili is the coolant enthalpy at the inlet
to the channel (z = -H/2).
4.2.2
Temperature Distribution in Channels with Single Phase Flow
For small temperature and pressure changes the enthalpy of a single-phase (nonboiling) coolant can be expressed as a linear function of the temperature. Assuming a
uniform axial distribution of heat sources and a constant heated perimeter, Eq. (4-14)
yields,
(4-15)
Tlb ( z ) = Tlbi +
q ′′PH (z + H 2)
c pW
Here Tlb(z) is the coolant bulk temperature at location z. The bulk temperature in a
channel cross section is defined in such a way that it can be obtained from the energy
balance over a portion of the channel. For an arbitrary velocity, temperature and fluid
97
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property distribution across the channel cross section, the bulk temperature is found
as,
∫ρ c
l
Tlb =
v T dA
pl l l
A
∫ρ c
l
v dA
pl l
A
Note that Eq. (4-15) is only valid for coolant without phase change, whereas Eq. (4-14)
is applicable for both single-phase and two-phase flows.
As can be seen, the coolant temperature is a linear function of the distance from the
inlet to the channel. Assuming that the total length of the channel is equal to H, the
exit temperature is as follows,
(4-16)
Tlbex = Tlbi +
q′′PH H
.
c pW
The temperature distribution along the channel is shown in FIGURE 4-7.
z=H/2
Tlbex
q’’
W, Tlbi
Tlbi
z=-H/2
Tlb
FIGURE 4-7: Bulk temperature distribution in a uniformly heated channel with a constant heated
perimeter.
In nuclear reactor cores the axial power distribution may have various shapes. The
cosine-shaped power distribution is obtained in cylindrical homogeneous reactors, as
previously derived using the diffusion approximation for the neutron distribution
calculation.
Using Eq. (4-8) and the coordinate system as indicated in FIGURE 4-8, the power
distribution may be expressed as,
(4-17)
 πz 
q′′( z ) = q0′′ ⋅ cos ~  .
H 
Eq. (4-13) then becomes,
(4-18)
dil ( z ) q0′′ ⋅ PH ( z )
dTlb ( z ) q0′′ ⋅ PH ( z )
 πz 
 πz 
=
cos ~ , or
=
cos ~  .
dz
W
dz
W ⋅ cp
H 
H 
98
C H A P T E R
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G E N E R A T I O N
A N D
R E M O V A L
z=H/2
q’’
z=0
z=-H/2
W, Tlbi
FIGURE 4-8: Heated channel with cosine power shape.
After integration, the coolant enthalpy and temperature distributions are as follows,
(4-19)
~
q0′′ ⋅ PH H   πz 
 πH 
il ( z ) =
⋅ sin  ~  + sin  ~  + ili , or
π  H 
W
 2 H 
~
q′′ ⋅ P H   πz 
 πH 
Tlb ( z ) = 0 H ⋅ sin ~  + sin ~  + Tlbi
W ⋅ cp π   H 
 2 H 
The channel exit temperature and enthalpy can be found substituting z = H/2 into Eq.
(4-19) as follows,
(4-20)
~
2q0′′ ⋅ PH ⋅ H
 πH 
ilex = il ( H / 2) =
sin  ~  + ili , or
π ⋅W
 2H 
~
2q′′ ⋅ P ⋅ H
 πH 
Tlbex = Tlb ( H / 2) = 0 H
sin ~  + Tlbi
π ⋅W ⋅ c p
 2H 
The axial distribution of the coolant temperature is shown in FIGURE 4-9.
q’’
z=H/2
Tlb(z)
T
W, Tlbi
z=-H/2
FIGURE 4-9: Distribution of coolant bulk temperature along channel with the cosine heat flux
distribution.
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4
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Heat Conduction in Fuel Elements
Modern nuclear power reactors contain cylindrical fuel elements that are composed of
ceramic fuel pellets located in metallic tubes (so-called cladding). A cross-section over a
square lattice of fuel rods is shown in FIGURE 4-10. For thermal analyses it is
convenient to subdivide the fuel rod assembly into subchannels. The division can be
performed in several ways; however, most obvious choices are so-called coolantcentered subchannels and rod-centered subchannels. Both types of subchannels are
equivalent in terms of major parameters such as the flow cross-section area, the
hydraulic diameter, the wetted perimenter and the heated perimeter. In continuation,
the thermal analysis will be performed for a single subchannel.
The stationary (time independent) heat conduction equation for an infinite cylindrical
fuel pin, in which the axial heat conduction can be ignored is as follows:
(4-21)
−
1 d 
dT 
 λF r F  = q′′′ ,
r dr 
dr 
where TF is the fuel temperature, [K], λF is the thermal conductivity of the fuel
material, [W m-1 K-1], q′′′ is the density of heat sources, [W m-3] and r is the radial
distance. Here the angular dependence of the temperature is omitted due to the
assumed axial symmetry of the temperature distribution.
Coolant-centered
subchannel
Coolant
Gap
Clad
rFo
Fuel
Rod-centered
subchannel
FIGURE 4-10: Cross-section of a square fuel lattice. Equivalent subchannels (coolant-centered and rod
centerd) suitable for thermal analyses.
Assuming further that q′′′ is constant in a cross-section, Eq. (4-21) can be integrated to
obtain:
(4-22)
λF r
dTF
r2
= − q′′′
dr
2
If the fuel conductivity was constant, Eq. (4-22) could be integrated and the
temperature distribution would be obtained. However in typical fuel materials the fuel
thermal conductivity strongly depends on the temperature and this is the reason why
the temperature distribution can not be found from Eq. (4-22) in a general analytical
form. Instead, Eq. (4-22) is transformed and integrated as follows:
100
C H A P T E R
(4-23)
r
2
4
–
H E A T
TFo
G E N E R A T I O N
λF dTF = − q′′′dr ⇒ ∫ λF dTF = −
TFc
A N D
R E M O V A L
q′′′ rFo
rFo2
rdr
=
−
q′′′ ,
2 ∫0
4
where the integration on the left-hand-side is carried out from the temperature at the
centerline, TFc, to the temperature on the fuel pellet surface TFo=TF (rFo). Defining the
average fuel conductivity as,
(4-24)
λF =
1
TFc − TFo
TFc
∫
TFo
λF dTF
,
the temperature drop across the fuel pellet can be found as,
(4-25)
∆TF ≡ TFc − TFo =
q′′′rFo2
.
4 λF
In the thermal analysis of reactor cores, the power is often expressed in terms of the
linear power density, that is, the power generated per unit length of the fuel element,
(4-26)
q′ ≡ π rFo2 q′′′ .
Combining Eqs.(4-25) and (4-26) yields,
(4-27)
∆TF =
q′
4π λ F
.
Equation (4-27) reveals that the fuel temperature drop is a function of the linear power
density and the averaged fuel thermal conductivity.
In a similar manner the temperature drop across the gas gap can be obtained. In
particular, Eq. (4-21) can be used to describe the temperature distribution in the gas
gap, however, unlike for the fuel pellet, the heat source term is equal to zero and the
gas thermal conductivity can be assumed constant, thus,
(4-28)
−
1 d
dT
dT
C
λG r G = 0 ⇒ λG r G = C1 ⇒ TG (r ) = 1 ln r + C2 .
r dr
dr
dr
λG
The integration constant C1 can be found from the condition of the heat flux
continuity at r = rFo,
(4-29)
− λG
dTG
dr
=−
r = rFo
C1
q′
q′
=
⇒ C1 = −
,
2π
rFo 2π rFo
and the temperature drop in the gap is found as,
(4-30)
∆TG = TG (rGi ) − TG (rGo ) =
q′
2π λG
ln
101
rGo
.
rGi
C H A P T E R
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G E N E R A T I O N
A N D
R E M O V A L
Equation (4-30) is applicable to the clad material as well, since the assumptions on the
heat generation and the thermal conductance are valid in this case as well. Substituting
the proper dimensions and property data yields,
(4-31)
∆TC = TC (rCi ) − TC (rCo ) =
q′
2π λC
ln
rCo
,
rCi
where rCo is the outer clad radius and λC is the clad thermal conductivity.
Heat transfer from the clad surface to the coolant is described by the following
equation,
(4-32)
q′′ = h(TCo − Tlb ) ,
where h is the convective heat-transfer coefficient. Taking into account that
q′′ = q′ (2π rCo ) , the temperature drop in the coolant boundary layer is found as,
(4-33)
∆Tl ≡ TCo − Tlb =
q′
.
2π rCo h
The total temperature drop from the fuel centerline to the coolant is now obtained as,
∆T = ∆TF + ∆TG + ∆TC + ∆Tl =
(4-34)
q′
2π
 1
1 rGo 1 rCo
1  .

 2 λ + λ ln r + λ ln r + r h 
F
G
Gi
C
Ci
Co 

The total temperature drop in a fuel rod cross-section is illustrated in FIGURE 4-11.
TFc
FIGURE 4-11. Temperature drops in a fuel rod.
102
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R E M O V A L
EXAMPLE 4-1: During normal operation of a nuclear reactor, the highest power
density in fuel pellets is equal to 900 MW m-3. Find an increase of the maximum
fuel and clad temperature if suddenly the heat transfer coefficient at the clad
surface drops due to CHF from 50000 to 2000 W m-2 K-1. The fuel rod geometry
is shown in FIGURE 4-12. SOLUTION: From Eq. (4-34) it is clear that the only
temperature drop that is affected by a change in the heat transfer coefficient is
given by Eq. (4-33). The temperature drops in the gas gap, as well as in the fuel
pellet and in the clad wall will remain the same, since the power density, and thus the linear power, as well
as material properties and dimensions remain the same. Thus the temperature increase will be the same in
all parts of the fuel rod and will be equal to ∆T = ∆Tl1 − ∆Tl1 = q′ (2πrCo h1 ) − q′ (2πrCo h2 ) , where h1 is the
Fuel
Cladding
Gap
heat transfer coefficient before the occurrence of CHF and h2 is the heat transfer coefficient after the
occurrence of CHF. From Eq. (4-26) the linear power density is found as q′ = πrFo2 q′′′ ≅ 49.876 kW/m.
Thus ∆T = q′ (2πrCo )(1 h1 − 1 h2 ) ≅ 793.8 K. It can be seen that with this increase of the clad temperature
the safety limit value (which is typically about 900 K) will be exceeded and the clad damage may occur.
r
rFo=4.2 mm
rGo=4.25 mm
rCo=4.8 mm
FIGURE 4-12: Fuel element used for calculations in EXAMPLE 4-1.
EXAMPLE 4-2: Calculate temperature drops in a fuel pellet, gas gap, clad and the
thermal boundary layer using the following typical data for PWR:
dimensions – pellet outer diameter dFo = 8.25 mm, gas gap outer diameter dGo =
8.43 mm, clad outer diameter dGo = 9.70 mm; thermal conductivity – clad λC = 11
W/m.K, gas gap λG = 0.6 W/m.K, fuel (UO2) λF = 2.5 W/m.K; heat transfer
coefficient h = 45000 W/m2.K and linear power density q’ = 41 kW/m. Calculate
the maximum allowed linear power density if the fuel temperature shouldn’t
exceed the melting temperature (3073 K) and the coolant temperature is 600 K. SOLUTION:
The temperature drops are obtained as follows,
∆TF =
q′
41000
=
= 1305.07 K
4π λF
4π ⋅ 2.5
∆TG =
 r  41000  8.43 
q′′′rFo2  rGo 
q′
ln  =
ln Go  =
ln
 = 234.73 K
2λG  rFo  2πλG  rFo  2π ⋅ 0.6  8.25 
∆TC =
∆Tl =
 r  41000  9.7 
q′
ln Co  =
ln
 = 83.25 K
2πλC  rGo  2π ⋅11  8.43 
q ′′′rFo2
q′
41000
=
=
= 29.9 K
2rCo h 2πrCo h π 0.0097 ⋅ 45000
Thus the total temperature drop from the pellet center to the fluid bulk is as follows,
∆T = ∆TF + ∆TG + ∆TC + ∆Tl = 1652.95 K .
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The maximum allowed linear power can be found from the expression for the total temperature drop in a
fuel element,
∆T =
q′
4π
1
2  rGo  2  rCo 
2 .
ln  +
ln  +
 +

 λF λG  rFo  λC  rGo  rCo h 
Thus, the maximum linear power is obtained as,
q′max =
4.2.4
4π (Tmelt − Tcool )
W
= 61340.7
m
1
2  rGo  2  rCo 
2 
ln  +
ln  +
 +

r
r
r
h
λ
λ
λ
G
C
 Fo 
 Go  Co 
 F
Axial Temperature Distribution in Fuel Rods
In the previous section expressions for the axial distribution of coolant temperature
have been derived. It has been shown that the axial distribution of coolant temperature
varies with the shape of the axial heat flux distribution.
In particular, substituting Eqs. (4-17) and (4-19) into (4-32) yields the following
expression for the temperature of the clad outer surface,
(4-35)
~
q0′′ ⋅ PH ⋅ H
TCo ( z ) =
π ⋅W ⋅ c p
  πz 
 πH  q ′′
 πz 
⋅ sin  ~  + sin ~  + 0 ⋅ cos ~  + Tlbi .
 2 H  h
H
 H
FIGURE 4-13 shows the temperature of the clad outer surface as a function of the
axial distance.
q’’
z=H/2
zCo,max
TCo,max
TCo(z)
W, Tli z=-H/2 T
FIGURE 4-13: Axial distribution of the temperature of the clad outer surface with cosine axial power
distribution in a channel.
It should be noted that the temperature of the clad outer surface gets its maximum
value TCo,max at a certain location zCo,max. This location can be found from Eq. (4-35)
using the following condition,
(4-36)
dTCo ( z )
= 0.
dz z = zCo ,max
It is convenient to represent the clad outer temperature as,
104
C H A P T E R
(4-37)
4
–
H E A T
G E N E R A T I O N
A N D
R E M O V A L
 πz 
 πz 
TCo ( z ) = A + B sin  ~  + CCo ⋅ cos ~  ,
H 
H 
where,
(4-38)
~
q′′ ⋅ P ⋅ H
q′′
 πH 
, CCo = 0 .
A = B ⋅ sin  ~  + Tlbi , B = 0 H
π ⋅W ⋅ c p
h
 2H 
Using Eq. (4-37) in Eq. (4-36) yields,
 πz
B cos Co~,max
 H

 πz
 − CCo sin  Co~, max

 H

 = 0 ,

which is equivalent to the following equation,
 πz
tan  Co~,max
 H

B
 =
.
 CCo
Thus,
(4-39)
zCo ,max =
~
H
 B 
 .
arctan
π
 CCo 
It should be noted that a physically meaningful solution of the above equation should
be positive and less than H.
Noting that,
 πzCo ,max
sin 

~
H
 πz

tan  Co~,max 

 H 
 = ±

 πz
1 + tan 2  Co~,max
 H



=±
B
CCo
 B 

1 + 
 CCo 
2
,
and
 πz
cos Co~,max
 H

 = ±

1
 πz
1 + tan 2  Co~,max
 H



=±
1
 B 

1 + 
 CCo 
2
,
the maximum temperature of the clad outer surface becomes (taking only + sign),
(4-40)
2
TCo,max = A + B 2 + CCo
.
Using constants A, B and CCo given by Eq. (4-38), the maximum clad outer
temperature is obtained as,
105
C H A P T E R
(4-41)
4
TCo ,max
–
H E A T
G E N E R A T I O N
A N D
~
q′′ ⋅ P ⋅ H
 πH 
= 0 H
⋅ sin  ~  + Tlbi +
π ⋅W ⋅ c p
 2H 
R E M O V A L
~
2
 q0′′ ⋅ PH ⋅ H
′′

 +  q0  ,
 π ⋅W ⋅ c   h 
p 

2
or,
(4-42)
π ⋅W ⋅ c p (TCo ,max − Tlbi )
~
q0′′ ⋅ PH ⋅ H
2
 π ⋅W ⋅ c p 
 πH 
= sin ~  + 1 + 
~  .
 2H 
 PH ⋅ H ⋅ h 
Since the clad maximum temperature is located on the inner surface, it is of interest to
find it as well. The axial distribution of the clad inner temperature can be obtained
from Eqs. (4-31) and (4-35) as,
(4-43)
TCi ( z ) = ∆TC + TCo ( z ) =
~
q′
rCo q0′′ ⋅ PH ⋅ H   πz 
 πH  q′′
 πz 
ln
+
⋅ sin  ~  + sin  ~  + 0 ⋅ cos ~  + Tlbi =
2π λC rCi π ⋅ W ⋅ c p   H 
 2 H  h
H 
~
r
1   πz 
q0′′ ⋅ PH ⋅ H   πz 
r
 πH 
⋅ sin  ~  + sin  ~  + q0′′ Co ln Co +  cos ~  + Tlbi
π ⋅W ⋅ c p   H 
 2 H 
 λC rCi h   H 
Equation (4-43) can be expressed in a short form as,
(4-44)
 πz 
 πz 
TCi ( z ) = A + B sin  ~  + CCi cos ~  .
H 
H
Here A and B are given by Eq. (4-38) and,
(4-45)
r
1
r
CCi = q0′′ Co ln Co +  .
 λC rCi h 
Using the same approach as in the case of the clad outer temperature, the location of
the maximum temperature on the clad inner surface is found as,
(4-46)
zCi ,max =
~
H
 B 
 ,
arctan
π
 CCi 
and the corresponding maximum temperature is,
TCi ,max =
(4-47)
~
q0′′ ⋅ PH ⋅ H
 πH 
⋅ sin  ~  + Tlbi +
π ⋅W ⋅ c p
 2H 
~ 
 q0′′ ⋅ PH ⋅ H



 + q′′ rCo ln rCo + 1 

 π ⋅W ⋅ c   0  λ
p 
  C rCi h 

2
2
In a similar manner the fuel maximum temperature at the centerline can be found as,
106
C H A P T E R
(4-48)
4
–
H E A T
G E N E R A T I O N
A N D
R E M O V A L
 πz 
 πz 
TFc ( z ) = A + B sin  ~  + C Fc cos ~ 
H 
H
where
(4-49)
r
r
r
r
r
1
C Fc = q0′′ Co ln Co + Co ln Go + Co + 
h
 λC rCi λG rGi 2 λF
The maximum fuel temperature is located at,
z Fc ,max =
(4-50)
~
H
π
arctan
B
,
C Fc
and its value is found as,
TFc ,max
(4-51)
~
q0′′ ⋅ PH ⋅ H
 πH 
=
⋅ sin ~  + Tlbi +
π ⋅W ⋅ c p
 2H 
~  
 q0′′ ⋅ PH ⋅ H


 + q0′′ rCo ln rCo + rCo ln rGo + rCo + 1 

 π ⋅W ⋅ c 
h 
p 
  λC rCi λG rGi 2 λF

2
2
.
EXAMPLE 4-3. Calculate the axial temperature distribution of coolant, clad and
fuel in a subchannel of a PWR fuel assembly. The fuel pellets with 8.2 mm
diameter are clad with zircaloy 0.56 mm thick. The outer diameter of the clad rod is
9.6 mm. The coolant bulk inlet temperature is 588 K, mass flow rate per unit
subchannel is 0.4 kg/s, the pressure is 155 bar, and the axial heat flux distribution
on the clad outer surface is q′′(z ) = 2 ⋅ 105 cos(πz H~ ) [W ] , where H = 3.7 m and the
extrapolation length is 0.1 m. Use the same material properties as given in
EXAMPLE 4-2 and assume lattice pitch-to-diameter ratio equal to 1.2. Calculate the heat transfer
coefficient from the Dittus-Boelter correlation, using fluid properties at inlet temperature and pressure.
SOLUTION: The flow cross-section of the subchannel is obtained as Axs = 6.03x10-5 m2 and the mass
flux is obtained as G = W/Axs = 6630 kg/m2.s. Using the Dittue-Boelter correlation, the heat transfer
coefficient is obtained as h = 65347 W/m2.K. Now the constants A, B and CCo can be found as
B=
~
q0′′ ⋅ PH ⋅ H
≅ 3.159 K
π ⋅W ⋅ cp
 πH 
A = B ⋅ sin  ~  + Tlbi ≅ 591.149 K
 2H 
q0′′
CCo =
≅ 3.061 K
h
Thus, the clad outer temperature is given as,
 πz 
 πz 
TCo ( z ) = A + B ⋅ sin ~  + CCo ⋅ cos ~  = 591.149 + 3.159 ⋅ sin (0.8055z ) + 3.061 ⋅ cos(0.8055z )
H
H 
The maximum temperature of the clad outer surface is found as,
2
TCo ,max = A + B 2 + CCo
≅ 595.6 K
and it is located at
zCo, max =
~
H
π
 B 
 ≅ 0.995 m
arctan
 CCo 
Similar calculations are performed for the clad inner surface and for the fuel centerline. The calculated
temperature distributions are shown in FIGURE 4-14.
107
C H A P T E R
4
–
H E A T
G E N E R A T I O N
A N D
R E M O V A L
FIGURE 4-14: Axial distribution of temperature in fuel rod described in EXAMPLE 4-3.
4.3 Void Fraction in Boiling Channels
The characteristic feature of boiling channels is the presence of two phases: the liquid
and the vapor phase. Clearly, the presence of two phases changes the fluid flow and
heat transfer processes as compared to the non-boiling channels. In addition, the
density changes of coolant are more significant in boiling channels due to the dramatic
change of density once liquid transforms into vapor. Thus, to be able to predict the
local value of the coolant density it is required to determine the local volume fraction
of both phases. Typically, the void fraction (that is the volume fraction of the vapor
phase) is determined using various models, as described below.
The various two-phase flow and heat transfer regimes in a boiling channel, such as
BWR fuel assembly, is shown in FIGURE 4-15.
In the simplest two-phase flow model it is assumed that both phases are in the
thermodynamic equilibrium and that they move with the same velocity. These
assumptions are the basis of the Homogeneous Equilibrium Model (HEM), in which
the local, channel-average void fraction is determined from the corresponding local
value of the equilibrium thermodynamic quality.
4.3.1
Homogeneous Equilibrium Model
The HEM expression for the void fraction takes the following form,


0

1

(4-52) α = 
1 + ρ g ⋅  1 − xe 

ρ f  xe 

1

for
xe ≤ 0
for 0 < xe < 1 .
for
xe ≥ 1
108
C H A P T E R
4
–
H E A T
G E N E R A T I O N
A N D
R E M O V A L
Here xe is the equilibrium thermodynamic quality, which is determined from the
energy balance of the coolant in the heated channel.
Temperature
Subcooled boiling
heat transfer
Saturation
temperature
ONB
Saturated boiling
heat transfer
Wall temperature
Wall superheat
Coolant temp.
∆T
OSV
Bubbly flow
Dryout
OAF
Slug flow
Annular flow
FIGURE 4-15: Two-phase flow and heat transfer regimes in a boiling channel: ONB – Onset of Boiling,
OSV – Onset of Significant Void, OAF – Onset of Annular Flow.
Equation (4-52) strongly over-predicts the coolant density (that is it gives a higher value
than the actual one) in the region of sub-cooled boiling, since it assumes only liquid,
whereas in reality both the liquid and the vapor co-exist in that region.
4.3.2
Drift-Flux Model
Once applying the Drift-Flux Model, the void fraction is found as,
(4-53)
α=
Jv
C0 J + U vj
Equation (4-53) expresses the cross-section mean void fraction α in terms of channel
mean superficial velocity of gas, Jv, total superficial velocity, J, and two parameters, C0
and Uvj. The first parameter is the so-called drift-flux distribution parameter and is
simply a covariance coefficient for cross-section distributions of void fraction and total
superficial velocity. The second coefficient is the so-called drift velocity and can be
interpreted as cross-section-averaged difference between the gas velocity and the
superficial velocity, using local the void fraction as a weighting function. The drift-flux
parameters are not constant and depend on flow conditions. TABLE 4.2 gives
expressions for drift-flux parameters, which are valid in a wide range of flow
conditions.
109
C H A P T E R
4
–
H E A T
G E N E R A T I O N
A N D
R E M O V A L
TABLE 4.2. Distribution parameter and drift velocity used drift-flux model.
Flow pattern
Bubbly
0 < α ≤ 0.25
Distribution parameter
Drift velocity
1 − 0.5 p pcr
Dh ≥ 0.05 m 1)


1.2
p pcr < 0.5
C0 = 
Dh < 0.05 m
1.4 − 0.4 p pcr p pcr ≥ 0.5
 σg (ρ l − ρ v ) 

U vj = 1.41
ρ l2


Slug/churn
C 0 = 1.15
 gD (ρ − ρ v ) 

U vj = 0.35 h l
ρl


C 0 = 1.05
 µ j 
U vj = 23 l l 
 ρ v Dh 
0.25 < α ≤ 0.75
Annular
0.75 < α ≤ 0.95
Mist
C0 = 1.0
2)
0.5
(ρ l − ρ v )
ρl
 σg (ρ l − ρ v ) 

U vj = 1.53
ρ v2


0.95 < α < 1
1)
0.5
0.25
0.25
2)
pcr – critical pressure, 2) σ - surface tension
4.3.3
Subcooled Boiling Region
It is commonly accepted that a significant void fraction in a boiling channel appears at
locations where bubbles depart from heated walls. The void fraction between that
point, referred often as the Onset of Significant Void fraction (OSV) point, and the
Onset of Nucleate Boiling (ONB) point is very small and can be neglected.
To establish the location of the OSV point it is recommended to use a correlation
proposed by Saha and Zuber (1974), which states that OSV point is located at such
position in a channel, where the local equilibrium quality is as follows,
(4-54)
xe ,OSV
q′′ ⋅ Dh ⋅ c pf

− 0.0022 i ⋅ λ

fg
f
=
′′
q

− 154
G ⋅ i fg

for
Pe < 70000
.
for
Pe ≥ 70000
Here Pe is the Peclet number defined as,
(4-55)
Pe =
G ⋅ Dh ⋅ c pf
λf
.
For a uniform heat flux distribution, the location of the OSV point is found from the
energy balance as,
(4-56)
zOSV = (xe ,OSV − xei )
W ⋅ i fg
.
q′′ ⋅ PH
Several models have been proposed to predict the flow quality downstream of the
OSV point. Levy (1966) proposed a fitting relationship, which satisfy a condition at z =
zNVG, where x = 0 and also which will predict the flow quality to approach the
equilibrium quality when z is increasing downstream of the OSV point. The Levy’s
relationship is as follows,
110
C H A P T E R
(4-57)
4
–
xa (z ) = xe (z ) − xe (zOSV ) ⋅ e
H E A T
xe ( z )
−1
xe ( zOSV )
G E N E R A T I O N
A N D
R E M O V A L
.
Having the flow quality given by Eq. (4-57), one can apply the general drift flux model
to calculate the void fraction distribution. The recommended expression for the
distribution parameter for subcooled boiling is as follows,
(4-58)
  1 b 
C0 = β 1 +    ,
  β  
where,
1
,
ρ g 1 − xa ( z )
1+
ρ f xa ( z )
(4-59)
β=
(4-60)
 ρg
b=
ρ
 f
0.1

 .


The recommended by Lahey and Moody (1977) drift velocity is as follows,
(4-61)
 σg (ρ f − ρ g ) 

U vj = 2.9
2


ρ
f


0.25
.
This is similar to the expression recommended for bubbly flow (see TABLE 4.2) but a
different constant should be used.
4.4 Heat Transfer to Coolants
4.4.1
Single-phase flow
The heat transfer coefficient h for coolant flow in a rod bundle is calculated from the
Nusselt number Nu as follows,
(4-62)
h=
Nu ⋅ λ
,
Dh
where λ is the fluid thermal conductivity and Dh is the bundle hydraulic diameter.
For laminar flow far from the inlet to a channel, the Nusselt number is as follows,
(4-63)
Nu = 4.364 .
In the inlet region of the channel the following expression is valid,
(4-64)
Nu = 1.31
(1 + 2ς ) , (0 < ς < 0.04)
3
ς
111
C H A P T E R
4
–
H E A T
G E N E R A T I O N
A N D
R E M O V A L
where
(4-65)
ς=
z
Pe
Dh
and Pe is the Peclet dimensionless number given as,
(4-66)
Pe =
UDh
.
a
For turbulent flow in a pipe the Nusselt number can be calculated from the
Dittus-
Boelter correlation,
(4-67)
Nu = 0.023 Re 0.8 Pr n .
Here Pr is the Prandtl number; Pr = ν/a, and ν is the kinematic viscosity of liquid. The
formula is valid for Re > 104 and 0.7 < Pr < 100, n = 0.4 for fluid heating and n = 0.3
for fluid cooling.
Petukhov [4-6] proposed the following semi-empirical expression for the Nusselt
number for turbulent flow in pipes,
(4-68)
Nu =
(C
f ,p
2 ) Re Pr
1 + 13.6C f , p + (11.7 + 1.8 Pr −1 3 ) C f , p 2 (Pr 2 3 − 1)
,
where Cf,p is given by Eq. (4-92).
For rod bundles with triangular lattice and with 1.1 < p/d < 1.8 Ushakov (presented in
[4-7]) proposed the following correlation,
(4-69)
0.15


0.91  p  
0.8
0.4
Nu = 0.0165 + 0.02 1 −
  ⋅ Re Pr ,
2 
d

 ( p / d )   
where the correlation is valid for 5·103 < Re < 5·105 and 0.7 < Pr < 20.
Similar correlation was derived by Weissman[4-14],
(4-70)
Nu = C Re 0.8 Pr 1 3 ,
where:
C = 0.026(p/d) – 0.024 for triangular lattices with 1.1 < p/d < 1.5,
C = 0.042(p/d) – 0.024 for square lattices with 1.1 < p/d < 1.3.
Subbotin et al. recommended for heat transfer to liquids flowing in a bundle with
triangular lattice the following correlation,
(4-71)
Nu = A Re 0.8 Pr 0.4 ,
112
C H A P T E R
4
–
H E A T
G E N E R A T I O N
A N D
R E M O V A L
where,
(4-72)
0.15

0.91  p 
A = 0.0165 + 0.02 1 −
 .
2 
d
(
)
p
d


 
The correlation is valid for 1.1 < p/d < 1.8, 1.0 < Pr < 20 and 5.103 < Re < 5.105.
For gas flow in tight rod bundles Ajn and Putjkov give,
(4-73)
Nu bundle
= 1.184 + 0.351 ⋅ lg( p d − 1) .
Nu DB
The correlation is valid for 1.03 < p/d < 2.4 and Nu DB is found from the DittusBoelter correlation given by Eq. (4-67).
Markoczy performed a study of experimental data obtained in 63 rod bundles with
different geometry details and proposed the following relationship,
(4-74)
Nu bundle
= 1 + 0.91 Re −0.1 Pr 0.4 1 − 2e − B ,
Nu DB
(
)
where,
(4-75)
 2 3  p 2
triangular lattice
 

 π d
B=
.
2
 4  p  − 1
square lattice
π  d 
Here again Nu DB is found from the Dittus-Boelter correlation given by Eq. (4-67).
The correlation is applicable in the following range of parameters:
3·103 < Re < 106
0.66 < Pr < 5
1.02 < p/d < 2.5.
Another approach was proposed by Osmachkin [4-4], who recommended to calculate
the Nusselt number from correlations which are valid for pipes, replacing however the
hydraulic diameter with the “effective” one given by Eq. (4-90).
4.4.2
Two-phase boiling flow
Heat transfer coefficient for two-phase boiling flow can be predicted from various
correlations, for example from the Jens Lottes (subcooled boiling) and the Chen
(saturated boiling) correlations, described in [4-1].
A simple estimation of the boiling heat transfer coefficient can be obtained from a
correlation proposed by Rasohin [4-8],
113
C H A P T E R
(4-76)
4
–
H E A T
G E N E R A T I O N
 5.5 p 0.25 (q′′)2 3
h=
23
1.33
0.577 p (q′′)
A N D
for 0.1 < p ≤ 8
for
8 < p < 20
R E M O V A L
,
where h is heat transfer coefficient [W/m2K], p is pressure [MPa] and q’’ is heat flux
[W/m2].
4.4.3
Liquid metal flow
Due to high thermal conductivity of liquid metals, the correlations for heat transfer
coefficients differ from those for water and gases.
Comprehensive studies have been made on heat transfer to various liquid metals (Hg,
Na, NaK, PbBi, Li, etc). Experiments performed in tubes indicate that a purity of the
liquid metal is of importance. In highly pure liquid metals heat transfer is more
intensive and can be described with the following correlation,
(4-77)
Nu = 5.0 + 0.025 ⋅ Pe 0.8 ,
where Pe is the Peclet number defined as,
(4-78)
Pe = Re⋅ Pr =
DhUρc p
λ
.
If no special measures are undertaken to purify the liquid metal, the heat transfer is
slightly deteriorated due to a deposition of oxide layer on the solid surface. Heat
transfer coefficient is then obtained from the following equation,
(4-79)
Nu = 3.3 + 0.014 ⋅ Pe 0.8 .
There are several correlations which are applicable to fuel rod bundles. The following
correlation has been proposed by Dwyer,
(4-80)
Nu = 6.66 + 3.126 s + 1.184 s 2 + 0.0155(Ψ ⋅ Pe )
0.86
,
where,
Ψ = 1−
0.942s 1.4
(
Pr Re 10 3
)
1.281
,
and s = p/d. Here Nu is the Nusselt number, p and d are lattice pitch and rod diameter,
respectively; Pr is the Prandtl number and Pe is the Peclet number defined by Eq.
(4-78). Equation (4-80) is valid for s > 1.35.
For tightly packed square lattices, the following correlation is applicable,
(4-81)
Nu = 0.48 + 0.0133Pe 0.70 .
For triangular lattice, the following correlations have been proposed:
Borishanski et al.:
114
C H A P T E R
(4-82)
4
–
H E A T
G E N E R A T I O N
[
]
[
A N D
R E M O V A L
]
Nu = 24.151 ⋅ lg − 8.12 + 12.76 s − 3.65s 2 + 0.0174 1 − e −6(s −1) (Pe − 200 ) ,
0.9
Graber:
(4-83)
Nu = 0.25 + 6.2 s + (0.32 s − 0.07 ) ⋅ Pe (0.8-0.024s ) ,
Calamai et al.:
(4-84)
5
Nu = 4 + 0.16 s + 0.33s
3.8
 Pe 
⋅

 100 
0.86
.
For square lattice, the following correlation has been proposed by Zhukov:
(4-85)
Nu = 7.55s − 14 s −5 + A ⋅ Pe (0.64+ 0.264s ) ,
where A = 0.09 for rod bundles with spacers and A = 0.07 for rod bundles without
spacers.
It has been demonstrated that the Nusselt number for the square lattice is lower than
that for triangular lattice bundles, assuming that all other parameters are the same.
4.4.4 Supercritical water flow
Supercritical water is used as the
working fluid in conventional coal-fired power
plants and is considered as coolant in one of the six reactor system concepts envisaged
to deployment after year 2025 (so-called Generation IV reactors). Supercritical water is
an attractive coolant, since the occurrence of the boiling crisis is eliminated. However,
under certain conditions so-called heat transfer deterioration may occur, which require
application of proper correlations to correctly predict temperatures of heated walls.
Critical parameters for water are 22.1 MPa and 374.1 °C. Typical parameters
considered for reactors cooled with critical water are 25 MPa and temperature in a
range from 280 to 580 °C. In this range of parameters, the water property exhibit
anomaly at the pseudo-critical temperature (which for pressure 25 MPa is 384.86 °C).
FIGURE 4-16 shows the temperature dependence of selected properties.
The significant change of physical properties around the pseudo-critical point is
accompanied with anomalous heat transfer results observed in experiments, e.g. [4-10]
and [4-11]. The current understanding of the governing phenomena indicates that the
change of properties is not enough to explain the heat transfer anomalies. It is believed
that the onset of the heat transfer anomaly depends both on fluid property variation as
well as on the flow acceleration and buoyancy effects. However, the mechanism is still
not fully understood.
The heat transfer anomaly manifests itself as either enhancement or deterioration of
heat transfer coefficient. The latter is of particular interest, since it brings about an
increase of the heated wall temperature.
Unlike the Critical Heat Flux (CHF), which occurs only for sub-critical pressures and
which brings about a violent increase of the wall temperature, the Heat Transfer
Deterioration (HTD) is connected to a rather mild wall temperature increase. Such
115
C H A P T E R
4
–
H E A T
G E N E R A T I O N
A N D
R E M O V A L
behavior creates a difficulty once trying to set a criterion for the onset of the HTD
occurrence.
Definitions for HTD used in the literature are quite ambiguous, resulting probably
from the mild character of the governing phenomena. Yamagata et al. Error!
Reference source not found.] define the HTD condition as such, when the
measured heat transfer coefficient is ‘significantly’ lower than that predicted from their
own correlation, valid for low heat flux conditions. Based on that criterion, they
proposed that the limit heat flux for the onset of HTD is as follows
(4-86)
″
q c = 0.2 ⋅ G 1.2 ,
″
where qc is heat flux in [kW/m2] and G is mass flux in [kg/m2.s].
1.0
1.e4*Viscosity
0.9
8*Ther_Exp
0.8
1.e-3*Density
Property
0.7
1.e-5*Cp
0.6
Conductivity
0.5
0.4
0.3
0.2
0.1
0.0
250
300
350
400
450
500
550
Temperature, degC
FIGURE 4-16: Property variation of supercritical water at 25 MPa pressure.
There are many correlations proposed for prediction of the heat transfer coefficient
under deterioration conditions. One of the correlations that shows probably the best
agreement with experiments was proposed by Jackson [4-12] and is as follows,
(4-87)
ρ 
Nu b = 0.0183 ⋅ Re b0.82 Prb0.5  w 
 ρb 
0.3
n
 cp 
 ,

 c pb 


where the exponent n is given as:
For Tb < Tw < T pc and 1.2T pc < Tb < Tw , n = 0.4.
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T

For Tb < T pc < Tw , n = 0.4 + 0.2 w − 1 .
T

 pc

T

T

For T pc < Tb < 1.2T pc and Tb < Tw , n = 0.4 + 0.2 w − 1 1 − 5 b − 1 .
T

T

 pc
 
 pc

Here T is temperature in Kelvin, ρ is density and cp is the specific heat. Indices b , pc
and w in Eq. (4-87) refer to the bulk, pseudo-critical and wall temperature,
respectively. The equation contains a modified specific heat calculated as,
cp =
iw − i
.
Tw − Tb
It should be noted that the application of the correlation requires iterations, since in
order to calculate the Nusselt number, the wall temperature has to be known.
EXAMPLE 4-4. Calculate the wall temperature at the exit from a pipe with
length L = 4.2 m and internal diameter D = 10 mm. The pipe is uniformly heated
with total heat q = 150 kW and cooled with supercritical water at pressure p = 250
bars, inlet temperature 280 °C and inlet mass flux G = 450 kg/m2.s. Apply both
the Dittus-Boelter and the Jackson correlation and compare the results.
SOLUTION: the inlet enthalpy is found from property functions for given
pressure and temperature and is equal to iin = 1.23 MJ/kg. The outlet enthalpy is
found from the energy balance as follows: iout = iin + 4q/(G·π·D2) = 2.291 MJ/kg. From tables, the
corresponding water temperature is found to be equal to 387.2 °C. This is the bulk temperature of the
supercritical water at the outlet from the pipe. The dynamic viscosity of the water at the exit is found
from tables as 3.42x10-5 Pa.s and the Prandtl number is 5.815. The heat transfer coefficient is found from
the Dittus-Boelter correlation as hDB = 16199.3 W/m2.K. Since the heat flux is q’’ = q/(L· π·D) = 1136.8
kW/m2, the wall temperature is found as Tw = Tb + q’’/hDB = 457.4 °C. Similar calculations performed
with the Jackson correlation (note that iterations are needed to find the modified specific heat) give the
following heat transfer coefficient: hJ = 4587 W/m2.K. This gives the wall temperature equal to Tw = Tb
+ q’’/hJ = 635 °C. As can be seen much higher wall temperature is obtained as compared to the
temperature calculated with the Dittus-Boelter correlation.
4.5 Pressure Drops
Calculation of pressure drops in a reactor core is important since they influence the
flow distribution in subchannels and thus affect the local thermal margins. In addition,
the total pressure drop over the coolant circulation loop has to be known in order to
determine the needed pumping power.
4.5.1
Single-phase flows
One can identify several mechanisms that will cause a pressure drop along the fuel
assembly:
1. Friction losses from the fuel rod bundle
2. Local loses from the spacer grids
3. Local loses at the core inlet and exit (contraction and expansion)
4. Elevation pressure drop
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The total pressure drop in a channel with a constant cross-section area can be
calculated from the following equation:
(4-88)
 4C f L
 G2
− ∆ptot = − ∆p fric − ∆ploc − ∆pelev = 
+ ∑ ξ i 
+ Lρ g sin ϕ .
D
2
ρ
i
 h

Here Cf is the Fanning friction coefficient, L is the length of the channel, G is the mass
flux, Dh is the channel hydraulic diameter and ρ is the coolant density.
The friction coefficient for laminar flow can be written in a general form as,
(4-89)
C f = a ⋅ Re − b ,
where a and b are constants, which for the laminar flow in a pipe are equal to 16 and 1, respectively. For laminar flow in rod bundles, Osmachkin proposed to use Eqs.
(4-88) and (4-89), where hydraulic diameter Dh is replaced with an “effective” diameter
given as,
(4-90)
Deff =
2ε
(1 − ε )2
 ε 3 ln ε 
 − −
 Dh ,
 2 2 1−ε 
where ε is the fraction of the cross-section of the bundle which is occupied by rods.
The formula is applicable for rod bundles with triangular lattice and for p/d > 1.3.
For turbulent flow the coefficients in Eq. (4-89) are obtained experimentally. For flow
in a rod bundle with triangular lattice (see FIGURE 4-4) and 1.0 < p/d < 1.5, the
Fanning friction coefficient can be calculated as,
(4-91)
p


C f ,b =  0.96 + 0.63 C f , p ,
d


where Cf,p is the friction factor proposed by Filonenko [4-7], which is valid for tubes
and annuli for Re > 4000:
(4-92)
C f , p = 0.25(1.82 lg10 Re− 1.64 ) .
−2
For rod assemblies Aljoshin et al. [4-5] proposed a general correlation as follows,
(4-93)
P
C f = A w,ch
Pw ,r
m
 Ach 

 Re −0.25 ,
 Ar 
where Pw,ch and Pw,r – are the wetted perimeters of the channel and rods, respectively;
Ach and Ar – are the cross-section areas of the channel and rods, respectively. The
formula is valid for rod bundles with triangular lattice, for which A = 0.47, m = 0.35
and 4·103 < Re < 105, and for rectangular lattice, for which A = 0.38, m = 0.45 and
103 < Re < 5·105.
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Additional pressure losses are associated with spacer grids, the coolant inlet and exit of
the bundle as well as the sudden area changes of the bundle cross-section area. Such
losses are classified as the local pressure losses and are calculated according to the
following general expression,
(4-94)
∆ploc = ξ loc
G2
,
2ρ
where ξ loc is the local pressure loss coefficient.
The local loss coefficient for grid spacers is in general dependent on the spacer
geometry and is usually determined in an experimental way. Typical spacer loss
coefficient is expressed as,
(4-95)
ξ spac = a + b ⋅ Re − c ,
where a, b and c a are constants determined experimentally.
For sudden enlargement and contraction of the channel, the local pressure losses can
be calculated according procedures described in [4-1].
4.5.2
Two-phase flows
Pressure drop in fuel assemblies with two-phase flow can be calculated according to
the procedures described in [4-1], using the hydraulic diameter as described by Eqs.
(4-9) and (4-10) with some modifications appropriate to the fuel assembly design. As
shown in [4-1], the total two-phase flow pressure drop in a channel with a constant
cross-section area can be calculated as,
(4-96)
− ∆p = r3C f ,lo
4L G 2
G2  N 2  G2
+ r4 Lρ l g sin ϕ + r2
+  ∑φ ξ 
.
Dh 2 ρ l
ρ l  i=1 lo ,i i  2 ρ l
Here r2, r3 and r4 are two-phase pressure drop multipliers (acceleration, friction and
gravitation, respectively) and φlo2 ,i , ξi are local loss multiplier and local pressure loss
coefficient, respectively, at location i in the channel.
4.6 Critical Heat Flux
The conditions at which the wall temperature rises and the heat transfer decreases
sharply due to a change in the heat transfer mechanism are termed as the Critical Heat
Flux (CHF) conditions. The nature of CHF, and thus the change of heat transfer
mechanism, varies with the enthalpy of the flow. At subcooled conditions and low
qualities this transition corresponds to a change in boiling mechanism from nucleate to
film boiling. For this reason the CHF condition for these circumstances is usually
referred to as the Departure from Nucleate Boiling (DNB).
At saturated conditions, with moderate and high qualities, the flow pattern is almost
invariably in an annular configuration. In these conditions the change of the heat
transfer mechanism is associated with the evaporation and disappearance of the liquid
film and the transition mechanism is termed as dryout. Once dryout occurs, the flow
pattern changes to the liquid-deficient region, with a mixture of vapor and entrained
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droplets. It is worth noting that due to high vapor velocity the heat transport from
heated wall to vapor and droplets is quite efficient, and the associated increase of wall
temperature is not as dramatic as in the case of DNB.
The mechanisms responsible for the occurrence of CHF (DNB- and dryout-type) are
not fully understood, even though a lot of effort has been devoted to this topic. Since
no consistent theory of CHF is available, the predictions of CHF occurrence relay on
correlations obtained from specific experimental data. LWR fuel vendors perform their
own measurements of CHF in full-scale mock-ups of fuel assemblies. Based on the
measured data, proprietary CHF correlations are developed. As a rule, such
correlations are limited to the same geometry and the same working conditions as used
in experiments.
Most research on CHF published in the open literature has been performed for
upward flow boiling of water in uniformly heated tubes. The overall experimental
effort in obtaining CHF data is enormous. It is estimated that several hundred
thousand CHF data points have been obtained in different labs around the world.
More that 200 correlations have been developed in order to correlate the data.
Discussion of all such correlations is not possible; however, some examples will be
described in this section.
4.6.1
Departure from Nucleate Boiling
The usual form of a DNB correlation is as follows,
(4-97)
qcr′′ = qcr′′ (G, p, Dh , L,...) ,
which means that the main parameters that influence the occurrence of DNB are mass
flux, G, pressure, p, as well as the hydraulic diameter, Dh and length L of the heated
channel.
For upflow boiling of water in vertical 8-mm tubes with constant heat flux, Levitan
and Lantsman recommended the following correlation for DNB:
(4-98)
q cr′′
8 mm
2
1.2{[0.25 ( p − 98 ) / 98 ]− x e }

p
 p   G 
= 10.3 − 7.8 + 1.6  
e −1.5 xe .

98
98
1000

  

Here qcr′′ is the critical heat flux [MW m-2], p is the pressure in [bar], G is the mass flux
in [kg m-2 s-1]. The correlation is valid in ranges 29.4 < p < 196 [bar] and 750 < G <
5000 [kg m-2 s-1] and is accurate to ±15%.
The correlation can be applied to channels with arbitrary diameters if the following
correction factor is applied,
(4-99)
8
qcr′′ = qcr′′ 8mm ⋅  
D
0.5
,
where D is the tube diameter in [mm] and qcr′′ 8mm is the critical heat flux obtained
from Eq. (4-98).
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One of the earliest correlations for DNB applicable to fuel rod bundles was given by
Bowring, who proposed the following expression,
(4-100)
qcr′′ =
A + D ⋅ G∆isubi / 4
,
C+L
where,
(4-101)
A=
0.579 FB1D ⋅ G ⋅ i fg
,
1 + 0.0143FB 2 D1 2G
(4-102)
C=
0.077 FB 3 D ⋅ G
,
n
1 + 0.347 FB 4 (G 1356)
Here D is the hydraulic diameter in [m], G is the mass flux in [kg m-2 s-1], ifg is the latent
heat in [J kg-1], p is the pressure in [Pa], ∆isubi is the inlet subcooling [J kg-1] and L is the
tube length [m]. The correlation parameters n, FB1, FB2, FB3 and FB4 are functions of
pressure and are as follows,
(4-103)
n = 2.0 − 0.5 pR ,
(4-104)
pR =
p
,
6.895 ⋅ 106
 18.942
 pR exp[20.8(1 − pR )] + 0.917

(4-105) FB1 = 
1.917

 pR− 0.368 exp[0.648(1 − pR )]
(4-106)
 1.316
p
exp[2.444(1 − pR )] + 0.309
FB1  R
=
1.309
FB 2 

pR− 0.448 exp[0.245(1 − pR )]
 17.023
 pR exp[16.658(1 − pR )] + 0.667

(4-107) FB 3 = 
1.667


pR− 0.219
(4-108)
pR ≤ 1
,
pR > 1
pR ≤ 1
,
pR > 1
pR ≤ 1
,
pR > 1
FB 4
= p1R.649 .
FB 3
The correlation is based on a fit to data in the ranges 136 < G < 18600 [kg m-2 s-1], 2 <
p < 190 [bar], 2 < D < 45 [mm] and 0.15 < L < 3.7 [m].
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For PWR conditions, when heat flux is uniformly distributed along the channel, the
General Electric Company (GE) correlation is widely applied (Jansen and Levy, [4-13]),
(4-109)
qcr′′ = qcr′′ 70 + 6.2 ⋅103 (70 − p ) ,
where,
(4-110)
qcr′′ 70
(
)

106 2.24 + 0.55 ⋅10 −3 G

= 106 5.16 − 0.63 ⋅10 −3 G − 14.85 x
10 6 1.91 − 0.383 ⋅10 −3 G − 2.06 x

(
(
)
)
if
x < x1
if
if
x1 ≤ x < x2 .
x2 ≤ x
Here,
x1 = 0.197 − 0.08 ⋅10 −3 G , x2 = 0.254 − 0.019 ⋅10 −3 G ,
qcr′′
- critical heat flux, [W/m2]
x
- equilibrium quality,
G
- mass flux, [kg/m2.s],
p
- pressure, [bar].
The correlation is applicable in the following range of parameters: pressure p =
42÷102 bars, mass flux G = 540÷8100 kg/m2.s, quality x = 0.0÷0.45, hydraulic
diameter Dh = 6.2÷32 mm, channel length L = 0.74÷2.8 m.
Westinghouse Company developed similar correlation for uniformly heated bundles,
W-3 (Tong et al.),
(4-111)
{
}
qcr′′ ,U = A 2.022 − 0.0004302 p R + (0.1722 − 0.0000984 pR )e[(18.177−0.004129 pR )x ] ⋅
[(0.1484 − 1.596 x + 0.1729 x x )G + 1.037](1.157 − 0.869 x )⋅
(0.2664 + 0.8357e
)(0.8258 + 0.000784∆i )
R
−3.151 DE
R
where,
pR =
p
, p – pressure [Pa]
6.8947 ⋅103
GR =
G
, G – mass flux [kg/m2.s]
−3
1.3562 ⋅10
A = 3.1695 ⋅105
x
- quality
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C H A P T E R
i f − ili
∆iR =
2326
4
–
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G E N E R A T I O N
A N D
R E M O V A L
, if, ili – saturated and inlet enthalpy, respectively [J/kg].
For non-uniform power distributions, the following correction factor has to be used,
(4-112)
Fc ≡
qcr′′ ,U ( zcr )
C
=
′
′
qcr , NU ( zcr ) 1 − e −C⋅zcr
zcr
q′′( z )
∫ q′′(z ) e
0
−C ( zcr − z )
dz ,
cr
where qcr′′ ,U ( zcr ) is the value of the critical heat flux calculated from Eq. (4-109), zcr is
the axial coordinate of DNB point, q’’(z) is the axial distribution of the heat flux and C
is an empirical constant.
4.6.2
Dryout
The usual form of a dryout correlation is as follows,
(4-113)
xcr = xcr (G, p, Dh , LB ,...) ,
which means that the main parameters that influence the occurrence of dryout are
mass flux, G, pressure, p, hydraulic diameter Dh, boiling length (that it the distance
from the beginning of saturated flow to the dryout point), LB, and possibly other.
For dryout predictios in 8-mm pipes Levitan and Lantsman recommended the
following expression:
(4-114)
2
3
−0.5

p
 p
 p   G 
xcr 8mm = 0.39 + 1.57 − 2.04  + 0.68  
 .
98
 98 
 98   1000 

Here xcr is the critical quality, p is the pressure in [bar] and G is the mass flux in [kg m-2
s-1]. The application region of the correlation is 9.8 < p < 166.6 [bar] and 750 < G <
3000 [kg m-2 s-1] and the accuracy of xcr is ±0.05.
The critical quality given by Eq. (4-114) can be used for other tube diameters with the
following correction factor:
(4-115)
8
xcr = xcr 8mm ⋅  
D
0.15
.
Here xcr 8mm is the critical quality obtained from Eq. (4-114) and D is the tube
diameter in [mm].
For fuel rod bundles the following correlation was proposed by General Electric:
(4-116)
x cr =
A ⋅ L*B
B + L*B
 1.24 

,
 R 
 f 
where,
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L*B = L B 0.0254 , LB - boiling length in [m],
R f - radial peaking factor,
2
 p − 600 
2
3
A = 1.055 − 0.013 R
 − 1.233GR + 0.907GR − 0.285GR
 400 
B = 17.98 + 78.873GR − 35.464GR2 ,
GR = G 1356.23 , G – mass flux in [kg m-2 s-1],
pR = p 6894.757 , p – pressure in [Pa].
The correlation is valid for 7x7 bundles. It can also be applied to 8x8 bundles once
replacing B with B/1.12.
R E F E R E N C E S
[4-1]
Anglart, H., Thermal-Hydraulics in Nuclear Systems, Compendium KTH, 2010.
[4-2]
Glasstone, S. and Sesonske, A., Nuclear Reactor Engineering, Van Nostrand Reinhold Compant,
1981, ISBN 0-442-20057-9.
[4-3]
Idelchik, Handbook of flow pressure losses
[4-4]
Osmachkin, V.S., Investigation of the thermal-hydraulic characteristics of mockups of fuel rod
assemblies in Kurchatov Institute (Russian), 1974.
[4-5]
Aljoshin, V.S., Kuznetsov, N.M., Sarkisov, A.A., Submarine nuclear reactors (Russian), 1968.
[4-6]
Petukhov, B.S., Genin, L.G., Kovalev, S.A., Heat transfer in nuclear power plants (Russian),
Atomizdat, 1974.
[4-7]
Subbotin, V.I., et al, Hydrodynamics and heat transfer in nuclear power plants (principles of
analysis) (Russian), Atomizdat, 1975.
[4-8]
Rasohin, N.G. et al., ‘Prediction of boiling heat transfer,’ Teploenergetika, 9, 1970.
[4-9]
Tong, L., Boiling crisis and critical heat flux, 1969.
[4-10]
Shitsman, M.E., “Impairment of the heat transmission at supercritical pressures,” High
Temperatures, 1, 2, 237-244 (1963).
[4-11]
Yamagata, K, Nishikawa, K., Hasegawa, S., Fujii, T. and Yoshida, S., “Forced Convective Heat
Transfer to Supercritical Water Flowing in Tubes,” Int. J. Heat Mass Transfer, 15, 2575-2593
(1972).
[4-12]
Jackson, J.D., “Consideration of the Heat Transfer Properties of Supercritical Pressure Water in
Connection of with the Cooling of Advanced Nuclear Reactors,” Proc. 13th Pacific Basin Nuclear
Conference, Shenzen City, China, 21-25 (2002).
[4-13]
Janssen, E. and Levy, S., General Electric Company Report APED-3892, 1962.
[4-14]
Weissman, J. “Heat Transfer to Water Flowing Parallel to Tube Bundles,” Nucl. Sci. and Eng.,
6:78, 1978.
124
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H E A T
G E N E R A T I O N
A N D
R E M O V A L
E X E R C I S E S
EXERCISE 4-1: A nuclear power plant operated during 11 month with constant power 1200 MWe and
with thermal efficiency 34% before the reactor was shutdown for refueling. Calculate the decay power of
the reactor at time equal to 24 hours after the shutdown. Calculate the total decay heat (in MWh)
generated by the core during the refueling outage assuming that it started 1 day after the reactor shutdown
and it lasted for 5 days
EXERCISE 4-2: BWR fuel assembly has 24 rods with diameter dr = 9.63 mm and heated length L =
3.77 m. The cross-section area of the fuel assembly is A = 2.35 10-3 m2, and the hydraulic diameter dh =
9.6 mm. The axial distribution of heat flux is described with the following function:
πz 
1 π
q ′′( z ) = q ′′  + sin  . The bundle is electrically heated and cooled with water flowing vertically
0
2
4
L
upward. The power is distributed uniformly between the rods. The bundle is tested in a thermal-hydraulic
laboratory under the following conditions: inlet mass flux G = 1200 kg/m2.s, inlet water temperature Tli
= 483 K, system pressure p = 7 MPa and the total power applied to the bundle q = 900 kW. Calculate:
(a) total enthalpy increase in the bundle, (b) the maximum value of the heat flux in the bundle, (c) the
maximum clad outer temperature and its axial location (d) the total pressure drop in the rod bundle.
EXERCISE 4-3: Fuel rod, with heat density in fuel pellet q’’’ = 950 MW/m3, has the following radial
dimensions: fuel pellet outer radius: rF = 4.2 mm, gas gap outer radius: rG = 4.25 mm, clad outer radius rC
= 4.82 mm. Calculate: (a) temperature rise in fuel, gas gap and cladding if the thermal conductivities are
4.82 W/m.K, 0.28 W/m.K, and 12 W/m.K, respectively; (b) temperature rise in fuel if the fuel
conductivity is given as the following function of temperature: λ = 4000 + 0.34 ⋅ 10−12 T 4 (T in K),
F
130 + T
assuming that the maximum fuel temperature is equal to 2000 K.
EXERCISE 4-4: A vertical BWR fuel assembly has 100 uniformly heated rods with linear power q’ = 25
kW m-1 in each rod. The bundle is cooled with water flowing upward at pressure 7.0 MPa, inlet
subcooling 10 K and inlet mass flux 1500 kg m-2 s-1. The assembly is 3.65 m long, has a quadratic crosssection with box width equal to 140 mm and the rod outer diameter equal to 10 mm. Given: saturation
temperature: 559 K, latent heat: 1505 kJ kg-1, liquid specific heat: 5205 J kg-1 K-1 , liquid saturation
enthalpy: 1268 kJ kg-1 , density: liquid 740 kg m-3, vapor 35 kg m-3 , liquid thermal conductivity: 0.57 W m1 K-1 , liquid dynamic viscosity: 9.12x10-5 Pa s. Using the Levitan and Lantsman correlation for dryout,
calculate the critical power of the fuel assembly and using HEM, find the total pressure drop.
125
Chapter
5
5
Materials and Mechanics of
Structures
T
he choice of materials for nuclear applications has to be based on
consideration of such properties as the cross-section for neutron absorption,
the resistance to corrosion, the resistance to radiation damages and the
resistance to high temperatures and mechanical loads. Clearly, the prevailing
conditions during the life-time of the nuclear power plant will determine which
materials are suitable for the desired purposes. Consequently, materials which are
acceptable in thermal reactors are not necessarily the best choice for the fast reactors
and better alternatives should be always sought. This chapter contains the general
aspects of material properties and the analyses of their behavior in nuclear applications.
5.1
Structural Materials
Structural materials are such materials that should withstand the desing loads during
the operation of the nuclear power plant. Due to that, the materials should
demonstrate a high enough strength to prevent any kind of failure that would
jeopardize the system integrity and/or safety.
5.1.1
Stainless Steels
Stainless steel is a steel alloy
with a minimum of 11% of chromium content by mass.
Due to this content of chromium stainless steels have a higher resistance to corrosion,
but in general, are not stain-proof.
have austenite (also known as the gamma phase of iron)
as their primary phase. These alloys contain mainly chromium and nickel, but
sometime also other elements. Austenitic stainless steels are corrosion resistant and
have good mechanical properties, but they suffer some degradation as a result of
exposure to fast neutrons. Selected austenitic stainless steels used in reactor
applications are listed in TABLE 5.1.
Austenitic stainless steels
TABLE 5.1. Composition of selected austenitic stainless steels for reactor applications.
SS, Swedish
AISI (USA)
Carbon, % Chromium, %
Nickel, %
Other
Elements
2333
304
0.08
18 to 20
8 to 11
-
2352
304L
0.03
18 to 20
8 to 11
-
2343
316
0.10
16 to 18
10 to 14
Mo (2-3%)
127
C H A P T E R
5
2352
316L
5.1.2
–
M A T E R I A L S
A N D
0.03
M E C H A N I C S
16 to 18
O F
S T R U C T U R E S
10 to 14
Mo (1.752.5%)
Low-alloy Carbon Steels
Low-allow carbon steels are steels in which the main alloying constituent is carbon,
with additional alloying elements such as manganese, nickel, chromium, molybdenum
and others, which are added to improve the alloy mechanical properties. Thees steels
are mainly used in pressure vessels and other components where the corrosion
resistance is not required but the ability to withstand thermal stresses is desirable. Such
steel is used for PWR pressure vessel, where the inside of the vessel is usually clad with
a thin layer of stainless steel or Inconel to provide the corrosion resistance.
5.1.3
Properties of Selected Steel Materials
The physical properties of selected steel materials are given in TABLE 5.2.
TABLE 5.2. Physical properties of selected steel materials
Material
Temper
ature, K
Density,
kg/m3
Young’s
Modulus,
GPa
Poisson’s
Ratio
Yield
Strength,
MPa
Carbon steel
300
7860
207
0.28
340
(A 533-B)
600
-
182
-
280
750
-
172
-
240
800
-
169
-
200
Stainless steel
300
7950
193
0.27
205
(type 347)
500
7860
173
0.30
-
700
7710
166
0.31
150
800
-
157
0.32
-
300
6560
95
0.43
300
500
-
90
0.38
170
600
-
78
-
117
Zircaloy-2
128
C H A P T E R
5
–
M A T E R I A L S
Austenitic steel
300
A N D
8000
M E C H A N I C S
193
O F
S T R U C T U R E S
0.27 - 0.30
205 *)
(type 304 & 316)
*)
0.2% offset yield point
5.2 Cladding Materials
Cladding materials should be resistant to corrosion and they should have good
mechanical properties at high temperatures. Fuel cladding shouldn’t in addition interact
with neutrons.
5.2.1
Zirconium
Zirconium has a small capture cross section for thermal neutrons and furthermore it is
resistant to corrosion by water at the operating temperature of water-cooled reactors.
Due to that, zirconium alloys have found extensive use in the fuel cladding for such
reactors.
Zirconium ores contain 0.5 to 3% of hafnium, which has a large cross section for
thermal neutron capture. This puts high requirements on removal of hafnium from
zirconium to make it suitable for reactor applications. Two alloys are used as cladding
materials: zircaloy-2 (1.5% Zn, 0.15% Fe, 0.1% Cr and 0.05% Ni) and zircaloy-4 (1.5%
Zn, 0.2% Fe, 0.1% Cr and 0.007% Ni).
5.2.2
Nickel Alloys
Nickel alloys demonstrate a resistance to corrosion at high temperatures. Inconels
(containing about 15% chromium, 7% iron and smaller amounts of other elements)
are sometimes used instead of stainless steel because they are less subject to stresscorrosion cracking. However the alloy should not be used inside the reactor core due
to the presence of cobalt. Instead it can be used as pressure-vessel lining or in steam
generators.
5.3 Coolant, Moderator and Reflector
Materials
Coolant, moderator and reflector materials used in nuclear applications have to possess
low cross section for absorption of neutrons in order to improve the neutron economy
in the nuclear reactors. In addition, coolants should have good thermal-hydraulic
properties to effectively cool the nuclear fuel elements.
5.3.1
Coolant Materials
Various coolant materials have been applied in nuclear reactors. Most thermal reactors
are cooled with ordinary water. This type of coolant has several advantages such as low
price, good thermal-hydraulic properties and reasonably good nuclear properties (such
as low cross section for neutron absorption). Heavy water is used when the nuclear
fuel has low enrichment and improved neutron economy is needed. In fast reactors
various gas or liquid metal coolants are employed such as helium, CO2, sodium, lead
and lead-bismuth eutectic. The main properties of selected coolants are given in
TABLE 5.3.
129
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M E C H A N I C S
O F
S T R U C T U R E S
TABLE 5.3. Properties of selected reactor coolants
Fluid
Temerature
Density
K
Kg/m3
Specific
heat
Viscosity,
Pa.s · 105
Thermal
conductivity
Prandtl
number
W/m/K
J/kg/K
Helium
293
0.178
5200
1.86
0.141
0.68
p=
500
0.0973
5200
2.80
0.211
0.69
105 Pa
700
0.0703
5200
3.48
0.278
0.65
900
0.0529
5200
4.14
0.335
0.64
1100
0.0432
5200
4.6
0.389
0.61
558.83
36.524
5354
1.896
0.0643
1.577
p=
600
30.487
3436
2.099
0.0592
1.219
7 MPa
700
23.655
2570
2.559
0.0646
1.018
800
19.913
2403
2.989
0.0755
0.952
900
17.356
2375
3.400
0.0879
0.919
Water
300
999.63
4162
85.27
0.6140
5.780
p=
400
940.91
4241
22.04
0.6886
1.357
7 MPa
450
894.27
4368
15.45
0.6806
0.991
500
835.35
4622
11.88
0.6467
0.849
558.83
739.72
5400
9.125
0.5687
0.866
Sodium
373
925
1382.3
68.19
84.92
0.0111
p=
473
904
1343.3
45.24
81.02
0.0075
105 Pa
573
881
1309.4
34.16
77.12
0.0058
673
856
1282.6
27.96
73.19
0.0049
773
832
1263.9
23.57
69.28
0.0043
873
808
1253.5
20.86
65.37
0.0040
603
10670
147.30
254.7
15.83
0.0237
Steam
Lead,
130
C H A P T E R
5
p = 105
Pa
–
M A T E R I A L S
A N D
M E C H A N I C S
O F
S T R U C T U R E S
673
10580
147.30
213.86
16.58
0.0190
773
10460
147.30
184.63
17.66
0.0154
873
10340
147.30
156.48
18.74
0.0123
973
10210
147.30
139.94
19.82
0.0104
1073
10090
147.30
131.96
20.90
0.0093
[5-2]
5.3.2
Moderator and Reflector Materials
The moderator and reflector materials should have small absorption cross section and
possibly large scattering cross section. Several low mass number materials such as
ordinary water, heavy water, beryllium, carbon (graphite) and zirconium hydride
possess these types of properties. Nuclear properties of typical moderator materials are
given in TABLE 5.4.
TABLE 5.4. Properties of selected moderator materials at 293 K.
Material
Ordinary
water
Heavy
water
Beryllium
Graphite
18
20
9
12
Density, kg/m3
1000
1100
1850
1700
Nr atoms or molecules/m3 x 10-28
3.34
3.32
12.4
8.55
σ a , barns
0.66
0.003
0.0092
0.0034
σ s , barns
103
13.6
6.1
4.8
Σ a , 1/m
2.2
0.0085
0.114
0.029
Σ s , 1/m
345
45
76
41
Atomic or molecular weight
5.3.3
Selection of Materials
The main criteria for material selection in nuclear power plants are safety, reliability,
life-length and economy. One of the important aspects in selection of materials is the
previous experience in using a certain material in equivalent conditions.
To satisfy the criteria the following principles are followed:
•
these parts that are not affected by corrosion should be made of carbon steels
or low alloy steels,
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•
5
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M E C H A N I C S
O F
S T R U C T U R E S
other primary reactor systems should be made of stainless steels or nickel
alloys.
TABLE 5.5 contains a list of materials that are typically selected for various primary
components in nuclear power plants.
TABLE 5.5. Typical material selection for various components in nuclear power plants.
Component
Reactor
vessel
Function
Material requirement
pressure To contain and
support the reactor
core and reactor
internal parts
Material used
High strength material Carbon steel
with as little as possible A 533-B
deterioration with time.
Due to neutrons, low
contents of cobalt,
copper and phosphor is
required.
Reactor inner parts
To support, form
and contain the
core; to direct
coolant (water and
steam) flow; to
control
reactor
power and to
separate steam and
water.
Materials for inner parts
must be resistant to
corrosion.
Materials
should not contain
cobolt-60 isotop and
should be resistant to
high neutron fluxes.
Steam generator
To transfer the Materials
must
be Carbon steel
heat from the resistant to corrosion.
A 533-B for
primary circuit and
pressure
to generate steam.
vessel. Heat
exchanger
tubes
are
made
of
Inconel-600
or Incoloy800.
Turbine set
To transfer the
thermal energy of
steam
into
a
mechanical energy
of rotation. To
condens
steam
after leaving the
turbine.
In nuclear power plants
with secondary circuits
there are no additional
requirements
as
compared with fossilpowered plants. In BWR
materials should not
contain
cobolt-60
isotopes.
Austenitic
stainless steels
(SS 2333 or
AISI 304) or
Inconel-600.
If
high
strength
is
required:
SS2570
(A286)
Turbine rotor
is made of
ASTM A-470
cl. 5.
Condenser
tubes
are
made
of
Titanium.
Pipes
132
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M E C H A N I C S
O F
S T R U C T U R E S
valves
are
made
of
carbon steels.
5.4 Mechanical Properies of Materials
5.4.1
Hooke’s Law
Hooke’s law is the constitutive equation for the linear elastic and one-dimensional
systems is as follows,
(5-1)
ε=
σ
E
,
where ε = ∆l l is the strain, defined as a ratio of the length increase to the initial
length of a material sample, σ = F A is the stress, defined as a ratio of the normal
force F to the sample cross section area A, and E is the Young’s modulus (know also
as modulus of elasticity). Elongation of a material sample into one direction, in which
the load is acting, causes a contraction of the sample in the normal direction. The
contraction is given as,
(5-2)
ε n = −ν
σ
E
,
where ν is the Poisson’s ratio.
Both the Young’s modulus and the Poisson’s ratio are material properties.
EXAMPLE 5-1. Consider a simple bar (shown in FIGURE 5-1) fixed at one end,
at x = 0 and subject to a concentrated force at the other end (x = 2 m). Calculate
the displacement and stress distribution in the bar. SOLUTION: The stress
distribution in the bar can be calculated as: σ = F A = 1000 1e − 4 = 10 MPa . The
corresponding strain is found as ε = σ E = 107 2.1 ⋅ 1011 = 4.76 ⋅ 10 −5 . The
displacement will depend on the distance from the fixed wall and in general can be
given as u = ε ⋅ x . For x = 2 m (whole length of the bar), the displacement is found
as u = 4.76·10-5 x 2·103 mm = 0.0952 mm.
A = 1.E-4 m2, E = 2.1E5 MPa
F=1000 N
2m
FIGURE 5-1. Bar subjected to a concentrated end load.
In the three dimensional state of stress, the Hooke’s law for isotropic materials is as
follows,
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A N D
M E C H A N I C S
O F
S T R U C T U R E S
1
[σ x − ν (σ y + σ z )]
E
1
ε y = [σ y − ν (σ x + σ z )]
E
1
ε z = [σ z − ν (σ x + σ y )]
E
εx =
(5-3)
ε xy =
ε xz =
ε yz =
σ xy
2G
σ xz
2G
σ yz
2G
where x, y and z are Cartesian coordinates, σ x , σ y and σ z are normal stresses acting
along these coordinates, σ xy , σ xz and σ yz are shear stresses and G is the shear
modulus.
For isotropic materials, the relation between the shear modulus G and the Young’s
modulus E is as follows,
(5-4)
G=
E
.
2(1 + ν )
In a two-dimensional state of stress, it can be easily demonstrated that there are two
orthogonal planes in which the shear stress vanishes and only normal stresses are
present. These planes are called the principal planes and the stresses the principal
stresses. There is a graphical representation of the two-dimensional stress state and the
relation between local stresses and the principle stresses known as the Mohr’s circle.
The principle stresses can be also derived for a general three-dimensional stress state,
however, the derivation is more complex. In this case the principle stresses are found
as roots of the following characteristic equation,
(5-5)
σ 3 − I 1σ 2 + I 2σ − I 3 = 0 ,
where,
(5-6)
I1 = σ x + σ y + σ z
(5-7)
I 2 = σ x σ y + σ y σ z + σ z σ x − σ xy2 − σ yz2 − σ xz2
(5-8)
I 3 = σ x σ y σ z + 2σ xy σ yz σ xz − σ x σ xy2 − σ y σ yz2 − σ z σ xz2
are three invariants of the three dimensional stress tensor.
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5.4.2
5
–
M A T E R I A L S
A N D
M E C H A N I C S
O F
S T R U C T U R E S
Stress-Strain Relationships
A typical stress-strain graph for low-carbon steel is shown in FIGURE 5-2. Initially the
stress increases linearly with the strain and the Hooke’s law prevails. At point 2 the
curve levels off and plastic deformation occurs. This point on the stress-strain curve is
called the yield point and the corresponding stress value the yield stress.
The yield point is not always as easily defined as shown in FIGURE 5-2. For example,
for high strength steels and aluminium alloys an arbitrary offset yield point is defined.
The value of this is usually defined at 0.2% of the strain.
Stress
1
2
3
Strain
FIGURE 5-2. Stress versus strain curve: 1 – ultimate strength point, 2 – yield strength point, 3- rupture
point.
The peak stress on the strain-stress curve is known as the ultimate tensile strength
(point 1 in FIGURE 5-2). The region between the yield strength and the ultimate
tensile strength is termed as the strain hardening region. Brittle materials (for example
concrete) do not have a yield point and do not strain-harden which means that the
ultimate strength and breaking strength are the same.
5.4.3
Ductile and Brittle Behaviour
Ductility is a property of a material that describes to what extend the material can be
plastically deformed without fracture. FIGURE 5-3 shows typical examples of the
ductile and brittle fracture.
FIGURE 5-3. Ductile versus brittle of round metal bars after tensile testing; (a) brittle fracture, (b) ductile
fracture, (c) purely ductile fracture.
135
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5.4.4
5
–
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A N D
M E C H A N I C S
O F
S T R U C T U R E S
Creep
Creep is a tendency of a material to the slow plastic deformation under a constant
mechanical and/or thermal load. Generally the creep exhibits three stages: (1) primary
creep with a high rate of the strain increase with time, (2) secondary creep, in which the
strain stays constant or only slightly increases with time, (3) accelerating creep, in which
the strain increases rapidely and becomes so large as to produce failure.
Creep is important in nuclear applications, but its effect may be two-fold:
•
creep can be advantageous as to relief the stresses,
•
creep can be disadvantageous due to significant shape changes of a structure;
for example excessive creep of cladding may lead to fuel damage.
5.5 Strength of Materials and Stress Analysis
The purpose of the strength of materials and stress analysis is to estimate the over-all
resistance of given structure to external loads.
Stress analysis is a discipline which has as a goal to evaluate stresses in a given structure
due to applied loads. The most common way to perform a stress analysis of any system
is to employ computer codes that calculate the distribution of stresses and strains due
to given external or internal loads. Such codes are based on the Finite Element
Methods (FEM), which employ a division of the computational domain into small
objects connected through nodes. The unknown parameters, such as strains and
stresses are determined at nodes, whereas their values inside finite elements are
computed from so called shape functions.
In a simplified analysis the stresses are determined from known dimensions and
applied loads to a specific structure. In many cases known analytical solutions for stress
distributions can be employed.
To evaluate the mechanical integrity of a structure and to provide a proper margin to
failure, the maximum stresses must be limited by certain pre-determined criteria.
Typical approach is to compare the maximum stress with the allowable stress obtained
from a proper yield criterion.
5.5.1
Yield Criteria
One of the objectives of the stress analysis is to determine the conditions when plastic
deformation, or yielding, is initiated in the body under consideration. For that purpose
a special scalar function of the stress tensor is derived. Several such functions have
been proposed, but the most widely used is the von Mises yield criterion based on the
Maxwell-Huber-Hencky-von-Mises maximum strain energy theory. In threedimensional stress state, the Mises stress can be expressed as,
(5-9)
σv =
(σ 1 − σ 2 )2 + (σ 2 − σ 3 )2 + (σ 3 − σ 1 )2
2
.
Von Mises yield criterion states that yielding in 3-D occurs when the distortion strain
energy reaches that required for uniaxial loading. Thus, it can be written as,
136
C H A P T E R
(5-10)
5
–
M A T E R I A L S
A N D
M E C H A N I C S
O F
S T R U C T U R E S
2
2σ max
≥ (σ 1 − σ 2 ) + (σ 2 − σ 3 ) + (σ 3 − σ 1 ) .
2
2
2
Another failure criterion used in practical application is due to Tresca. The criterion
specifies that a material would flow plastically if,
(5-11)
σ tresca = σ 1 − σ 3 ≥ σ max .
5.5.2
Stress Analysis in Pipes and Pressure Vessels
For a pipe or elongated vessel with inside (or outside) pressure p, inner diameter D
(radius R) and wall thickness h, the stresses are calculated as,
pD pR
=
,
2h
h
(5-12)
σ1 =
(5-13)
σ 2 = σ1 =
1
2
pR
,
2h
where σ 1 is the circumferential stress and σ 2 is the longitudinal stress.
The corresponding strain rates are as follows,
(5-14)
ε1 =
1
(σ 1 −νσ 2 ) = pR (2 −ν ) ,
E
2 Eh
(5-15)
ε2 =
1
(σ 2 −νσ 1 ) = pR (1 − 2ν ) .
E
2 Eh
The enlargement of the vessel radius is given as,
(5-16)
∆R = Rε 1 =
pR 2
(2 − ν ) ,
2 Eh
and the elongation of the vessel is,
(5-17)
∆ l = lε 2 =
pRl
(1 − 2ν ) .
2 Eh
Here R and l are the internal radius and the length of the vessel, respectively.
For pipes and pressure vessels with thick walls, the tangential (hoop) stress is given as,
(5-18)
σt = −
p a − pb a 2 b 2 p a a 2 − pb b 2
−
,
a 2 − b2 r 2
a 2 − b2
and the radial stresses as,
(5-19)
pa − pb a 2 b 2 pa a 2 − pb b 2
σr = 2
−
.
a − b2 r 2
a 2 − b2
137
C H A P T E R
5
–
M A T E R I A L S
A N D
M E C H A N I C S
O F
S T R U C T U R E S
Here pa and pb are the external and internal pressure, respectively, a and b are the
external and internal radii of the vessel, and r is the radial coordinate.
In case when only the internal pressure is acting, the stresses are the highest on the
internal surface of the wall and become,
p b (a 2 + b 2 )
,
a 2 − b2
(5-20)
σt =
(5-21)
σ r = − pb .
EXAMPLE 5-2. The design internal pressure in a pipe is 17.5 MPa and its outer
diameter is 0.5 m. The pipe is made of steel with allowable stress 190 MPa at the
design temperature. Evaluate the minimum permissible thickness of the pipe wall
assuming that the circumferential stress should be lower than the allowable stress
and that a thin wall model can be applied. SOLUTION: from Eq. (5-12) the
circumferential stress is obtained as σ 1 = p(Do − 2h ) 2h where Do is the outer
pipe
diameter.
The
pipe
thickness
can
be
found
as
h = pDo [2(σ 1 + p )] = 0.5 ⋅ 17.5 ⋅ 10 6 ⋅ 0.5 (190 ⋅ 106 + 17.5 ⋅ 106 ) ≅ 0.021 m .
5.5.3
Thermal Stresses
The stresses in construction materials are not only created by mechanical loads, but
may result from temperature differences. Typically various systems in nuclear power
plant are assembled at the ambient temperature. During the plant operation, the
temperature is elevated to a new value which corresponds to the local conditions. Due
to that the elements are elongated as a result of the termal dilatation. Thermal stresses
will be developed if the element is not allowed to freely move. Another type of thermal
stresses will develop in solid materials in which exists a strong temperature gradient. In
such case the stresses will depend on the material property (the termal expansion
coefficient) and the temperature gradient. Both types of thermal stresses should be
analysed on the design stage and proper design solutions should be adopted to keep
the stresses below the safety limits.
5.6 Material Deterioration, Fatigue and Ageing
During a nuclear power plant operation, construction materials are constantly changing
their properties. Taking into account the long life-time of a plant, these changes can be
very significant and their over-all influence on the plant components safety and their
integrity must be taken into consideration.
5.6.1
Radiation Effects in Materials
One of the specific conditions that need to be considered for nuclear power plants is
the presence of radiation. There are four types radiation. Alpha and beta radiation is
short-range and typically not influencing the choice of material. Neutron and gamma
radiation, however, have significant range and is therefore influencing the behavior of
construction materials.
Neutrons that participate in the chain reaction in the reactor core have a wide range of
energy: from thermal (~0.025 eV) to fast fission neutrons (~106 – 107 eV). In the
central part of the core the neutron flux can be as high as 1018 n/m2.s. There are no
materials that are not affected (sooner or later) by this high radiation. Typically what
138
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A N D
M E C H A N I C S
O F
S T R U C T U R E S
happens is that atoms are displaced from their positions due to elastic collisions with
neutrons. The total number of atom displacements is a measure of the radiation
damage of a material.
The rate of production of displacements by neutrons may be expressed as,
n& = Nσ d (E )φ (E ) ,
where n& is the rate of displacements, N is the atom density of the target material,
σ d (E ) is the microscopic cross section for displacement and φ (E ) is the flux of
neutrons. In general, both the microscopic cross section for displacement and the
neutron flux are functions of the energy of neutrons, E . It can be shown that,
σ d (E ) ≈ σ s ( E )
E
.
AEd
Here σ s (E ) is the total elastic scattering cross section of the target material for
neutrons of energy E , Ed is the displacement energy (typically its value is equal to 25
to 30 eV for metals) and A is the mass number of the target nucleus. The fraction of
displaced atoms during a period of time ∆t , assuming a constant neutron flux, may be
found as,
(5-22)
n&
∫ N dt ≡
∆t
n( E )
σ (E )E
= φ (E )∆t s
.
N
AEd
The product φ (E )∆t is expressed in units of neutrons/m2 and is called the
The fraction n(E ) N is known as the displacements per atom (dpa).
fluence.
In boiling water reactors only the components inside the reactor core (fuel assemblies,
control rods, etc) and limited parts of the reactor pressure vessel receive high enough
radiation to be affected. The effect of radiation on reactor pressure vessel is larger in
the pressurized water reactors since the vessel is smaller and thus the distance between
the vessel wall and the reactor core is smaller as well.
The changes of material properties due to radiation are monitored through placement
of material samples at various selected places in the nuclear reactor core. The samples
are investigated on a regular basis and in this way it is possible to predict the expected
changes of material properties during the lifetime of the reactor.
Certain atoms may absorb neutrons. Obviously, construction materials should not
contain these types of atoms, since such materials should have the lowest possible
cross-section for absorption. The absorption of neutrons in construction materials
influences the required enrichment of fuel and is thus of highest economic importance.
Finally, neutrons can activate the construction material through creation of radiactive
isotopes. For example Zr-94 after activation by a neutron becomes Zr-95 with halftime equal to 65 days. It is clear that if a construction material contains isotopes that
are activated by neutrons, it will introduce severe limitations on the material
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O F
S T R U C T U R E S
management and possibility to repair the damaged parts. In that respect coblt-59 is
particularly undesirable insertion into a construction material, since, when activated by
neutrons, it creates cobolt-60 which has half-time as long as 5.3 years. Thus
construction materials for nuclear applications have to be carefully analyzed as far as
the component isotopes are concerned and particularly undesirable isotopes (such as
cobolt-60) must be kept at as low level as possible.
5.6.2
EXAMPLE 5-3. Calculate the fraction of iron atoms that are displaced in a sample
of steel, which is exposed to a fluence of 1024 neutrons/m2 with an average energy
of 0.11 MeV. Assume Ed = 25 eV. SOLUTION: The fraction of displayed atoms
is obtained from Eq. (5-22), where for iron A = 56 and the macroscopic cross
section for scattering is obtained from Eq. (1-20) as
2
σ s ≈ 4πR 2 ≈ 4π (1.3 ⋅ 10 −15 A1 3 ) ≈ 3.1 ⋅ 10 −28 m2. Thus
n(E ) / N ≈ 10 24 ⋅ 3.1 ⋅ 10 −28 ⋅ 1.1 ⋅ 105 / 56 / 25 ≅ 0.0244 .
Corrosion of Metals
Corrosion is the disintegration of a material into its constituent atoms due to chemical
reactions with its surroundings. Most often it means electrochemical oxidation of
metals in reaction with oxidants, such as oxygen. Corrosion of metals in aqueous
media generally results from heterogeneities encountered in the solid material. Such
heterogeneities may exist due to impurity inclusions, differences in grain sizes and
composition and differences in stresses. The corrosion might then occur at anodic
regions, where the metal dissolves. At cathodic regions hydrogen is liberated or
oxidized to water.
(IGC) occurs at grain boundaries, which are anodic with
respect to the grains themselves. Intergranular corrosion can occur in certain stainless
steels (e.g. type 304) which have been improperly heat treated. It is sometime observed
on each side of heat-affected zone of a weld in stainless steel.
Intergranular corrosion
Corrosion of metals is significantly influenced by the chemical environment that the
materials are interfacing with. Corrosion may also interfere with other phenomena
such as mechanical or thermal stresses. This may lead to certain undesirable effects
such as the stress corrosion or environmentally accelerated fatigue.
(SCC) is a sudden failure of normally ductile metal
subjected to a tensile stress in a corrosive environment. For example, austenitic steels
may crack in the presence of chlorides. This type of failure is particularly dangerous in
nuclear applications because it is difficult to detect and it usually progresses very fast.
There are three basic factors that influence the stress corrosion cracking: (1) a
susceptible material, (2) a corrosive environment, and (3) tensile stresses above a
threshold. To prevent the occurrence of the stress corrosion cracking is is necessary to
remove at least on of these three factors.
Stress corrosion cracking
5.6.3
Chemical Environment
In BWRs the reference pressure and the temperature are equal to 7 MPa and 559 K
(saturation temperature of water at the reference pressure), respectively. The core inlet
parts are typically affected by 10-15 K lower temperature, whereas the coolant
temperature at the inlet to the reactor pressure vessel is about 100 K below the
saturation temperature. The concentration of oxygen and hydrogen in the reactor
coolant is strictly controlled. The standard concentrations have been 200-400 ppb
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A N D
M E C H A N I C S
O F
S T R U C T U R E S
(part-per-bilion, 10-9) O2 and about 10 ppb H2. In the alternative water chemistry
(AWC) approach, the concentrations are 5-10 ppb O2 and 50-150 ppb H2. To reduce
the corrosion, the water electrical conductivity must be kept as low as possible. This is
achieved by continuous water cleaning. Typical conductivity of water in BWR is about
10 –20 µS/m (at 298 K).
In PWRs the reference pressure and the temperature are equal to 15.5 MPa and 593 K,
respectively. The main difference of water chemistry in PWRs and BWRs is presence
of the boric acid (H3BO3) in the former. The concentration of the boron in water
varies during the fuel cycle. Typical values are 1800 ppm (part-per-milion, 10-6) B at the
beginning of cycle and 10 ppm B at the end of cycle. To mitigate the pH-lowering
effect of the boric acid, various forms of alkali have to be added. In western PWRs
7
LiOH is used, whereas in VVERs KOH/NH3 is employed. The concentration of
alkali is such as to yield pH at the level from 7.5 to 8.5. The water conductivity in the
primary loop depends on the concentrations of the boron and alkali. Typical value for
PWRs is 2000 µS/m (at 298 K).
5.6.4
Material Fatigue
In case of cyclic loads a progressive and localized failure of material may occur even
though the nominal maximum stress values are less than the ultimate tensile stress limit
or even less than the yield stress limit of the material. The starting point of any fatigue
analysis is the Wöhler (or S-N) curve, in which the magnitude of a cyclic stress, S, is
plotted against the logarithmic scale of cycles to failure, N. Typical Wöhler curves for
steels and aluminium alloys are shown in FIGURE 5-4. As can be seen, the peak
stresss decreases with increasing number of cycles. For mild steels the cycles can be
continued indefinitely provided the peak stress is below the endurance (fatigue
strength) limit value.
400
Peak stress, MPa
Mild steel
300
Endurance limit
(fatigue strength)
200
Aluminium alloy
100
0
104
105
106
107
Cycles to failure
108
109
FIGURE 5-4. Typical Wöhler curves for steels and aluminium alloys.
For aluminium alloys the Wöhler curve monotonically decreases in the whole range of
cycles and there is no clear endurance limit value for peak stresses. In this case the
endurance limit is taken to be the stress when the number of cycles to failure is 107 or
108.
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5.6.5
5
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A N D
M E C H A N I C S
O F
S T R U C T U R E S
Thermal Fatigue
Thermal fatigue arises when cyclic stresses result from cyclic temperature changes in a
solid material. Temperature changes may occur in various places in nuclear power
plants. For example in PWRs the temperature will vary in the pressurizer due to
pressure controlling by both water heating and steam spray cooling. Temperature
changes are expected to occur in the whole primary system (reactor pressure vessel,
piping, etc) due to power changes, reactor shutdown and startup. These types of
temperature fluctuations can be forseen on the design stage of the plant and the safe
peak stresses can be evaluated for expected number of cycles. Typically the peak
stresses in the affected areas should be below the endurance limit to withstand
unlimited number of cycles.
Thermal fatigue may lead to severe damages if it occurs in unexpected places. A typical
unfavorable situation can arise if a valve, which is normally closed during standard
plant operation and separates waters at different temperatures, starts to leak. The
mixing of water streams with different temperatures may lead to oscillation of wall
temperature, which may lead to failure due to the thermal fatigue. This process can be
additionally accelerated due to a presence of welds or discontinuity stresses.
5.6.6
Ageing
Managing ageing of structures, system and components of nuclear power plants is
important from the safety point of view. In general, this term is used both to take into
account the physical process of material degradadation that takes place due to the plant
operation, and to express the obsolescence of particular systems as compared to
current standards. The aging process of a nuclear power plant should be monitored
and adequate precautions should be taken as described in Safety Guides (e.g. IAEA
Safety Standard on “Ageing Management for Nuclear Power Plants”).
R E F E R E N C E S
[5-1]
ASME Pressure Vessel Code, Part III.
[5-2]
IAEA Safety Standard: Aeging Management for Nuclear Power Plants.
E X E R C I S E S
EXERCISE 5-1: Calculate the minimum thickness of a pipe with external diameter equal to 550 mm
carrying water at pressure 7 MPa and temperature 300 K. The pipe is made of stainless steel type 347.
EXERCISE 5-2: The design internal pressure in the PWR pressure vessel is 17.5 MPa and the outer
radius is 2.5 m. The vessel is made of carbon steel with allowable stress 190 MPa at the design
temperature. Evaluate the minimum permissible thickness of the reactor pressure vessel wall.
142
Chapter
6
6 Principles of Reactor Design
T
he design of nuclear reactors includes several aspects. Firstly, the design must
reflect the purpose for which the reactor is to be used. Clearly, different design
criteria will be applied for submarine propulsion reactors and for commercial
power reactors. Irrespective of the reactor type, the initial design is typically
based on the previous experience in the field. For example, very little has been changed
in the design of nuclear fuel assemblies starting from generation I through generation
IV of nuclear reactors. The evolutionary – rather than revolutionary – changes in
reactor designs stems from the fact that reactor design is a very costly process and all
possible design flaws must be avoided at earliest possible stage.
This chapter provides principles of the nuclear reactor design. The goal is to indicate
the necessary steps that have to be followed during the design process, however,
without providing in-depth description of the involved methods.
The first section contains a short description of the nuclear design process, in which
the spatial power distribution in a reactor core is evaluated. It also contains such topics
as the enrichment, burnable absorbers and refueling design. The second section is
dealing with the thermal-hydraulic design in which the distribution of temperature in
the reactor core is derermined. Finally, the third section contains a description of the
principles of the mechanical design of nuclear power systems.
6.1 Nuclear Design
A nuclear design of a reactor core is performed at various stages of nuclear power
plant design, construction and operation. Such designs usually have different levels of
accuracy and may be performed as an evaluation, a variant study, or as a verification
calculation. An evaluation design employs simplified methods and is based on various
simplifying assumptions. A variant study has as a goal to demonstrate the advantages
and disadvantages of various design options. Such study is usually performed to
support a choice of the optimal reactor design. Finally, the verification calculation has
as an objective to perform the nuclear design with the highest possible accuracy.
The main goals of the nuclear desing are to determine:
•
the excess reactivity and its effects at any operation condition of the reactor,
•
the composition of the nuclear fuel as a function of burnup,
•
efficiency of control rods and of safety systems,
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power density distribution as a function of space coordinates and time.
A nuclear design is a computation-intensive process and usually engages several
computational programs. Such programs are improved on a regular basis as new
methods, which are both faster and more accurate, are developed.
In general the most optimal reactor design is such that has the minimum capital and
fuel cost components. The fuel cost depends mainly on the initial enrichment and on
the fuel burnup. With increasing fuel enrichment, the fuel burnup increases, but so
does the cost of the fuel loaded into the core.
Another parameter that influences the fuel burnup is the ratio of the moderator to the
fuel. This ratio, however, is not a free parameter and is typically determined based on
safety considerations as to provide a negative moderator reactivity coefficient.
There are various algorithms that may be applied to the nuclear desing of a reactor.
Here a simplified approach is shown in order to make the whole process more
transparent. In real applications the design process might be much more complex.
It is assumed that the reactor has a cylindrical shape and is divided into L axial zones.
Often two zones (L = 2) are assumed. In this case the zones are big enough and their
mutual influence on each other can be neglected while deriving the constants for
analysis of the reactor as a whole. In the same manner the reflector is divided into
several radial zones to include not only water, but also construction materials. In
calculations it is assumed that the reflector consists of a homogeneous mixture of
water and steel.
Nuclear calculations are performed in a equivalent homogeneous cell containing fuel,
moderator and construction material. In calculations only few groups of neutrons are
considered. More detailed analyses indicate that the best results are obtained with four
groups, as shown in
TABLE 6.1.
TABLE 6.1. Four-group model for cell calculations.
Group
High Energy
Low Energy
Comment
1
10 MeV
0.821 MeV
The
low
energy
limit
corresponds to the threshold
energy for fission of 238U
2
821 keV
5.53 keV
The
low
energy
limit
corresponds to the lowest
energy of fission neutrons
3
5530 eV
0.625 eV
Neutrons in the resonance
region
4
0.625 eV
0
Thermal neutrons
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The calculation algorithm is as follows:
1. Determine constants for a unit cell. Perform homogenization of the equivalent
cell.
2. Determine the neutron spectrum and calculate the constants for one- or twogroup diffusion approximation of the whole reactor with the reflector.
3. For given reactor dimensions (they are usually determined on other bases)
calculate the effective multiplication factor, keff.
4. Repeat calculations for various conditions such as: cold reactor, hot reactor,
hot reactor with poisons, etc.
5. Evaluate the effect of the control rods.
6. Calculate the power density distribution.
7. In coupled thermal-hydraulic calculations, determine temperature and density
distributions of all materials in the reactor. If new values significantly differ
from the previous values, repeat calculations from Step 1.
8. Calculate the fuel burnup and the fuel isotopic composition.
6.1.1
Enrichment design
In LWRs light water (H2O) is used as moderator (and coolant). Since H2O is not as
good moderator as e.g. heavy water (D2O), the fuel must be enriched in U-235. The
required enrichment depends on whether the fuel will be used as initial fuel or as
replacement fuel, and also on the required burnup and the length of the fuel cycle.
For initial fuel (first loaded to reactor) the required enrichment is about 2% of U-235.
For replacement fuel (used for refueling) the enrichment should increase to 2.5 – 3.5%
of U-235. The required enrichment will increase with the length of the fuel cycle and
3.5% is a typical value for a 2-year-long fuel cycle.
The neutron flux over a single fuel assembly is not evenly distributed, which is caused
by the non-uniform distribution of the coolant in the assembly. To obtain as flat as
possible neutron flux distribution in a fuel assembly, the enrichment is varied in
different fuel rods. Thus, in rods close to water volumes is the enrichment reduced.
Nearer the bundle center the enrichment is increased, since less moderator is present
there compared to the periphery.
The main principle is that the optimal design of fuel assembly yields as uniform power
distribution as possible, keeping as high reactivity as possible.
The quality of the power distribution is expressed with the internal power distribution
factor, popularly called Fint. The parameter is defined as a ratio of maximum rod power
to mean in the assembly. Normally it should be from 1.06 to 1.16, depending on the
fuel assembly design.
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D E S I G N
Burnable absorbers
The purpose of burnable absorbers is to reduce the excess reactivity at the beginning
of a cycle. The burnable absorbers (called also burnable poisons) are mixed with fuel,
in that way their influence on the neutron flux can be easily controlled in space, since
they are located in such positions where they are needed. The usual material used for
the burnable absorbers include Bor (B) and Gadolinium (Gd). The former one is used
mainly in PWRs, whereas the latter one is used in BWRs.
6.1.3
Refueling
During operation of a nuclear reactor the excess reactivity must be large enough to
allow for the reactor operation at full power. Since this excess reactivity decreases
during reactor operation, it is necessary to refuel the reactor core on regular basis. Fuel
is used in a best way when it is replaced successively, with 20 to 40% at each refueling.
The placement of fresh fuel in a reactor core is governed by requirement of maximum
fuel utilization and preservation of the required thermal margins. It means that fuel
assemblies are reshuffled during each refueling to optimally satisfy the abovementioned requirements.
The length of a fuel cycle varies and can be from 12 to 20 months. The length of the
fuel cycle depends on several factors, like the required fuel burnup at the following
refueling. In some countries (e.g. Sweden) refueling is typically scheduled for the
summer period when the demand on electricity (and thus the price of electricity) is low.
This enforces a 12-month-long fuel cycle.
6.2 Thermal-Hydraulic Design
The primary goals of the thermal core design include achieving a high power density
(to minimize core size), a high specific power (to minimize fuel inventory) and high
coolant exit temperatures (to maximize thermodynamic efficiency). These goals should
be achieved with preserved safety criteria that apply to the nuclear power plant systems
and components.
The thermal-hydraulic design of a nuclear reactor consists typically of the following
steps:
1. Determine the thermal power of individual fuel assemblies using as the starting
point the spatial distribution of power in the reactor core, which is known
from the nuclear core design.
2. Determine the total mass flow rate throough the core, Wc, resulting from the
stationary momentum conservation in the primary loop.
3. Determine the distribution of coolant flow through the fuel assemblies in the
reactor core that satisfy the required principle of the flow profiling.
4. Determine the actual distribution of coolant flow through fuel assemblies that
satisfy the mass and momentum conservation over the reactor core.
5. Determine the pressure losses in fuel assemblies in the reactor core.
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6. Determine the axial distribution of coolant enthalpy, temperature, pressure,
quality and void fraction as well as the heat flux from the fuel elements to the
coolant.
7. Determine the pressure loss coefficients.
8. Determine the critical values of heat flux or critical power of fuel assemblies
with respect of the occurrence of the boiling crisis.
9. Determine heat transfer coefficients.
10. Determine the axial and radial temperature distribution in nuclear fuel rods.
Step 3 is required when a non-uniform distribution of coolant flow through the reactor
core is to be achieved by introduction of the variable throttling at the inlets to fuel
assemblies. The determined throttling distribution is then used in step 4 to calculate the
actual flow distribution in the reactor core.
6.2.1
Thermal-Hydraulic Constraints
The goals of the core design are subject to several important constraints. The first
important constraint is that the core temperatures remain below the melting points of
core components. This is particularly important for the fuel and the clad materials.
There are also limits on heat transfer rate between the fuel elements and coolant, since
if this heat transfer rate becomes too large, critical heat flux may be approached leading
to boiling transition. This, in turn, will result in a rapid increase of the clad temperature.
The coolant pressure drop across the core must be kept low to minimize pumping
requirements as well as hydraulic stresses on core components.
Above mentioned constraints must be analyzed over core live, since as the power
distribution in the core changes due to fuel burnup or core reloading, the temperature
distribution will similarly change.
Furthermore since the cross sections governing the neutronics of the core are strongly
temperature- and density-dependent, there will be a strong coupling between the
thermal-hydraulic and neutronic behavior of the reactor core.
6.2.2
Hot Channel Factors
One of the main goals of the thermal-hydraulic core analysis is to ensure that the
thermal limitations on the core behavior are not exceeded. So far two such limitations
have been discussed shortly:
To exclude melting of the fuel, the linear power density must be limited,
(6-1)
′ .
q ′(r ) < q max
Another limitation is dictated by the requirement that the surface heat flux always
remains below the critical limit,
(6-2)
′′ .
q ′′(r ) < q CHF
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Here r is any location in the nuclear core.
In addition to the two limitations, other issues like thermal and fission gas stresses on
the clad can limit the core performance and thus can limit power generation.
Furthermore, the core stability consideration may be another limiting circumstance.
A thorough thermal-hydraulic analysis of the core requires a detailed, threedimensional calculation of the core power distribution, including the effects of fuel
burnup, fission product buildup, control distributions, and moderator density
variations over core life. This information is next used to determine the coolant flow
and temperature distribution throughout the core. Even though such types of
calculations are performed nowadays, they are quite expensive and time consuming.
Especially for fast transient applications they are prohibitively expensive and are
avoided.
To make the thermal-hydraulic core analysis more practical, a common approach is to
investigate how closely the hot channel in the core approaches the operating
limitations. Then if one can ensure that the thermal conditions of this channel remain
below the core limitations, the remaining channels will presumably fall within design
limitations. One usually defines the hot channel in the core as that coolant channel in
which the core heat flux and enthalpy rise is the largest. Associated with this channel
are various hot channel or hot spot factors relating the performance of this channel to
the average behavior of the core.
The fuel assembly having the maximum power output is defined as the hot assembly.
The hot spot in the core is the point of maximum heat flux or linear power density,
while the hot channel is defined as the coolant (sub-) channel in which the hot spot
occurs or along which the maximum coolant enthalpy increase occurs.
The nuclear hot channel is defined to take into account the variation of the neutron
flux and fuel distribution within the core.
The radial nuclear hot channel factor is defined as,
H /2
(6-3)
∫ q′′(rHC )dz
average heat flux of the hot channel
−H / 2
F =
=
H /2
average heat flux of the channels in core
1 NC
∑ q′′(ri )dz
N C i=1 − H∫/ 2
N
R
.
Here NC is the total number of channels in core. In a similar manner, the axial nuclear
hot channel factor is defined as,
(6-4)
FZN =
max[q′′(rHC )]
maximum heat flux of the hot channel
z
=
.
H /2
average heat flux of the hot channel
1
q′′(rHC )dz
H − H∫/ 2
The total nuclear hot channel factor or nuclear heat flux factor is then,
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FqN =
(6-5)
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– R E A C T O R
D E S I G N
maximum heat flux in the core
= FRN FZN .
average heat flux in the core
EXAMPLE 6-1: The power distribution in a homogeneous, bare cylindrical core is
~
~
described by the following expression (assuming for simplicity R ≅ R and H ≅ H ,
 2.405rf
q′′′( z ) = w f Σ f φ0 J 0 
 R
 π z .
cos 
 H 
The radial factor is then
N
R
F =
J 0 ( 0) ∫
H /2
−H / 2
1
πR 2
∫
R
0
 πz 
cos dz
H
H /2
 2.405r 
 πz 
J0
2πrdr ∫− H / 2 cos dz
 R 
H
≈ 2.32
and the axial factor is,
FZN =
J 0 (0) cos(0)
≈ 1.57
1 H /2
 πz 
J 0 (0 ) ∫
cos dz
H −H / 2  H 
This implies the overall hot channel factor: FqN ≈ 2.32 ⋅ 1.57 ≈ 3.642 .
This is a quite conservative (e.g. high) estimation of the factor. A zone-loaded PWR will typically have a
nuclear hot channel factor of FqN ≈ 2.6 .
The nuclear heat flux hot channel factor is defined assuming nominal fuel pellet and
rod parameters. In reality, however, there will be local variation in fuel pellet density,
enrichment and diameter, surface area of fuel rod and eccentricity of the fuel-clad gap
due to manufacturing tolerances and operating conditions. The more general heat flux
hot channel factor or total power peaking factor Fq is defined as the maximum heat
flux in the hot channel divided by the average heat flux in the core (allowing for abovementioned variability). Fq and FqN are related by defining an engineering heat flux
FqE .
hot-channel factor
(6-6)
FqE =
Fq
FqN
.
Typically FqE is close to unity reflecting the fact that manufacturing tolerances are
quite small (in modern PWRs, FqE ≈ 1.03 ).
One can also define an enthalpy-rise hot channel factor,
(6-7)
F∆H =
maximum coolant enthalpy rise
.
average coolant enthalpy rise
This factor is a function of both variations in the power distribution and coolant flow.
For example, some 3-10% of the coolant flow bypasses the fuel assemblies, due to
leaks or the presence of other core components. This factor accounts for
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manufacturing tolerances and also structure displacement (box bow, rod bow, etc)
caused by the operation conditions.
6.2.3
Safety Margins
Practically all systems have some limits of safe operation. For example, elevators have a
certain limit value of the load that they can carry. Safety limit for engineering systems
can usually be determined with a reasonably good approximation, but never exactly.
This is due to the fact that the limit value, when the system breaks, depends on many
uncertain factors, such as the manufacturing tolerances, material properties, load
characteristics, etc. Thus, for safety reasons the allowed value of the critical parameter
of the considered system must be lower than the known limit value, when the system
breaks. The difference between the actual and the limit value of the critical parameter is
called the safety margin.
are expressed in the same physical units as the critical parameters. For
example, if the critical parameter is a load (as in case of elevators), the safety margin is
expressed in Newtons (N).
Safety margins
Safety limits in nuclear power plants are provided in the technical specifications of the
systems. In each country the safety limits are provided by a proper authority (SSM in
Sweden). Such safety limits are called acceptance criteria and include such parameters
as the reactor coolant system pressure, linear heat generation of fuel, fuel temperature,
fuel clad temperature, percentage of fuel failure, departure from nuclear boiling ratio
and critical power ratio. Thus safety margins can be express as differences between
acceptance criteria and actual values of plant parameters. The various definitions given
above are illustrated in FIGURE 6-1.
FIGURE 6-1: Relationship between the actual value, the safety limit and the safety margin.
The actual value of any parameter in nuclear power plant is not known exactly, since
the calculation procedures are always connected to uncertainties. These uncertainties
result from the modeling approximations, numerical errors and errors imbedded in
correlations that are used in calculations.
There are two principle ways to predict the actual values of parameters:
•
conservative predictions, in which all parameters are assumed to have the most
unfavorable value that could be expected at any foreseen circumstances
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best-estimate predictions, in which all parameters are assumed to have
statistically-mean value from the set of possible values at any foreseen
circumstances
In the best-estimate approach it is necessary to evaluate the final uncertainty of the
parameter under consideration.
Safety limits are set as international standards and are accepted by national regulatory
bodies. Selected safety limits are as follows:
•
Peak Clad Temperature (PCT) – 1478 K (1204 °C),
•
maximum clad oxidation – 17% of clad thickness,
•
maximum hydrogen generation – not to exceed deflagration or detonation
limits for containment integrity.
One of the objectives of the thermal-hydraulic core design is to determine the thermal
margins as accurately as possible. On one hand the margins should be large enough to
ensure safe operation of the nuclear power plant. On the other hand the plant should
operate at conditions that ensure high performance and efficiency. The safety margins
in nuclear power plants contain several contributing factors, as indicated in FIGURE
6-2.
FIGURE 6-2: Various components of safety margins.
6.2.4
Heat Flux Limitations
If the limit of critical heat flux is exceeded in any place in nuclear reactor core, the clad
and fuel temperature will significantly increased due to deteriorated heat transfer to
coolant. The DNB-type limitation is particularly important in PWRs, however, it can
not be ruled out even in BWR, in cases when the axial power distribution is inletskewed. The dryout-type limitation is a concern in BWR exclusively.
Typical way to express the heat flux limitation condition is by means of a ratio of the
critical heat flux to the actual one. For PWR applications the following ratio is used,
(6-8)
DNBR( z ) =
′ (z )
q ′DNB
,
q ′′( z )
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where:
DNBR(z)
- Departure from Nucleate Boiling Ratio at location z,
′ (z )
q ′DNB
- value of the critical flux at location z,
q ′′ ( z )
- value of the actual heat flux at location z.
FIGURE 6-3 illustrates the typical distributions of DNBR, the critical heat flux and the
actual heat flux in a heated channel.
For BWRs the following ratio is used,
(6-9)
CPR =
qcr
,
qac
CPR
- Critical Power Ratio
qcr
- critical power in a fuel assembly (that is the power at which the
dryout occurs),
qac
- actual power of the fuel assembly.
where,
FIGURE 6-3: Distributions of DNBR, critical heat flux and actual heat flux in a heated channel.
EXAMPLE 6-2: A vertical PWR fuel rod bundle with hydraulic diameter Dh = 10
mm and length H = 3.66 m is heated with non-uniform heat flux distribution given
as q ′′( z ) = q0′′{1 + cos[π (z H − 0.5)]} and cooled with water coolant at 15.5 MPa
pressure, 30 K inlet subcooling and mass flux G = 2000 kg/m2.s. Use the
Lntsman&Levitan DNB correlation and find the MDNBR (minimum DNBR)
and its location in the channel. SOLUTION: the actual and the DNB heat flux
distributions are plotted along the channel and shown in FIGURE 6-4. The
MDNBR is found as z = 2.745 m from the inlet and its value is 1.393.
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A script to perform calculations in EXAMPLE 6-2 and to plot the DNBR distribution
along the heated channel is given below.
COMPUTER PROGRAM: Program to plot curves shown in FIGURE 6-4.
// Calculate MDNBR in a channel and plot
// axial distributions of quality, critical
// heat flux, actual heat flux and DNBR
//
//
Channel geometry
D
= 0.01;
// Hydraulic diameter
L
= 3.66;
// Heated length
//------------------// Working conditions
p = 155.;
// Pressure
dTsubi = 30;
// Inlet subcooling
G = 2000;
// Inlet mass flux
q2p0 = 0.54e6;
// Heat flux in the center
//-------------------// properties
i_f = hlsat(p);
// Liquid saturation enthalpy
i_g = hvsat(p);
// Vapor saturation enthalpy
i_lin = hliq(p, tsat(p)-dTsubi); // Inlet liquid enthalpy
i_fg = i_g - i_f; // Latent heat
x_in = (i_lin-i_f)/i_fg; // Inlet quality
//
qcon = (10.3 - 7.8*p/98. + 1.6*(p/98)^2); // Quality-independent part
qcon = qcon*sqrt(0.008/D);
// Diameter correction
//---------------------// Calculations
z = linspace(0,L,25);
q2p = q2p0*(1+cos(%pi*(z/L-0.5)));
x = x_in + (4*q2p0/(G*D*i_fg)) * (z+L/%pi* (sin(%pi*(z/L-0.5)) +1));
q2pcr = 1.e6*qcon*(G/1000.).^(1.2*((0.25*(p-98)/98)-x)).*exp(-1.5*x);
dnbr = q2pcr./q2p;
[MDNBR,k] = min(dnbr);
// --------------------// Plot the axial distributions of heat flux, critical heat flux
// DNBR and quality
f1=scf(1);
af=get("current_axes");
// Get axes
af.font_size = 3;
// Axes font size
af.font_style = 3;
// Axes font style
af.x_label.font_size = 3; // x-axis labels font size
af.y_label.font_size = 3; // y-axis labels font size
tf=af.title;
// Titles
tf.font_size = 3;
// Title font size
xgrid(2);
xtitle(" ","Distance, [m]","Heat Flux, [MW/m^2], Quality [-]");
f=get("current_figure");
fignum=f.figure_id;
fignam="L9Ex2";
plot(z,q2p/1e6,'o-',z,q2pcr/1e6,'^-.',z,dnbr,'v--',z,x,'k*')
legend(['Heat Flux','CHF','DNBR','Quality'],a=1);
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7
Heat Flux
Heat Flux, [MW/m^2], Quality [-]
6
CHF
DNBR
5
Quality
4
3
2
1
0
-1
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Distance, [m]
FIGURE 6-4: Distributions of the actual and DNB heat flux in a heated channel from EXAMPLE 6-2.
6.2.5
Core-Size to Power Relationship
The averaged linear power density in fuel rods is given as,
(6-10)
q′AVR =
C1qR
,
H ⋅ NR
q ′AVR
- core-averaged linear power density [W/m],
C1
- fraction of the core power generated in the fuel (~0.95÷0.97),
qR
- total thermal reactor power [W],
H
- core height [m],
NR
- number of fuel rods.
where:
The average core power density is equal to the core power qR divided by the core
volume VR, that is,
(6-11)
HN R q′AVR
4q′ N
q
C1
′′ ≡ R =
= AVR 2 R .
q′AVR
2
πDC
VR
C1πDC
H
4
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Here DC is the core diameter. On a purely geometrical consideration, the relationship
between the averaged linear and volumetric power densities for square rod lattices is as
follows,
(6-12)
′′ =
q ′AVR
q ′AVR
,
p2
where p is the lattice pitch.
Combining Eqs. (6-10) through (6-12), the following expression for the required core
diameter is obtained,
(6-13)
DC = 2 p
qR
.
Hq′AVRπ
For a target reactor height H and power qR, the core diameter can be decreased by
decreasing the lattice pitch or by increasing the linear power density. Obviously, the
linear power density in the channel with the highest peaking factor has to be lower
than the maximum limit value, that is,
(6-14)
′ = q ′AVR FqN ,
q ′(z ) < q MAX
and the required minimum core diameter is obtained as,
(6-15)
DC ,MIN
qR FqN
= 2p
.
Hq′MAX π
The maximum allowable linear heat density q ′MAX is obtained from a proper DNB or
dryout correlation, assuming the required safety factor DNBR or CPR.
Equation (6-15) reveals the importance of the total nuclear heat flux factor FqN :
keeping this factor low allows to reduce the minimum required core diameter.
6.2.6
Probabilistic Assessment of CHF
Once determining the local value of the critical heat flux, it may be calculated only with
some limited accuracy. Correlation developers usually specify the correlation accuracy
by giving the standard deviation of the predicted value. Typically it is assumed that
CHF is a stochastic variable which has the normal distribution (also called the Gauss
distribution). Such distribution is shown in FIGURE 6-5 (probabilistic density
function) and FIGURE 6-6 (cumulative distribution function for the normal
distribution.
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0.9
0.8
m=0, sig=0.5
m=0, sig=1.0
m=0, sig=2.0
m=-1, sig=1
0.7
0.6
pdf
0.5
0.4
0.3
0.2
0.1
0
-5
-4
-3
-2
-1
0
x
1
2
3
4
5
FIGURE 6-5: Probability density function (pdf) for the normal distribution.
1.2
1
Cumulative
0.8
0.6
m=0, sig=0.5
m=0, sig=1.0
m=0, sig=2.0
m=-1, sig=1
0.4
0.2
0
-5
-4
-3
-2
-1
0
x
1
2
3
4
5
FIGURE 6-6: Cumulative distribution function for the normal distribution.
If the probability distribution function of CHF for a single rod is as shown in
FIGURE 6-7 and the actual value of the heat flux is given by the vertical line AB, then
the shaded area corresponds to the probability that the rod will experience CHF.
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Critical heat flux distribution
Actual heat flux value
′′ )
f (qCHF
B
Probability
of CHF
A
Heat flux
′
q′AB
FIGURE 6-7: Graphical illustration of the probability of CHF with known value of the actual heat flux.
′ can be calculated as,
The probability of CHF with known actual value of heat flux, q′AB
(6-16)
′′
pCHF ≡ p(qCHF
1
′ )=
< q′AB
σ 2π
′
q′AB
∫e
−
2
′′ −qCHF
′′ )
(qCHF
2σ 2
′′ .
dqCHF
0
′′ is the mean value of the critical heat flux and σ is its standard deviation.
Here qCHF
Note that the integration is carried out from 0 rather than from − ∞ since the heat
flux can not be negative. In practical calculations the cumulative distribution function
of a standard normal variable is used. This function, defined as,
(6-17)
Φ ( z ) = p (Z < z ) =
1
2π
z
∫e
−ξ 2
2
dξ ,
−∞
is given in table forms in handbooks dealing with statistics. The function is usually
given for z-values between 0 and 5. To obtain its value for -5 < z < 0, the following
relationship is used,
(6-18)
Φ(− z ) = 1 − Φ ( z ) .
Thus, the CHF probability can be calculated using the standard function as follows,
(6-19)
 q′′ − q′AB
′ 
 .
pCHF = 1 − Φ CHF
σ


APPENDIX C contains a table with values of the above function.
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EXAMPLE 6-3: A single fuel rod operates at a constant and uniform heat flux
equal to 1 MW/m2. With the current cooling conditions, the calculated critical heat
flux is 1.1 MW/m2. The correlation developer specified that the standard deviation
for the correlation is equal to 5%. Assuming the normal distribution of the CHF
probability density function, calculate the probability of the CHF occurrence for
the rod. SOLUTION: The CHF standard deviation is equal to 0.05*1.1 = 0.055
MW/m2. The argument of the standard cumulative function is equal to (1.11)/0.055 = 1.81818… ~= 1.82. From the table in APPENDIX C, the probability of CHF is found equal
to 0.03438.
In a similar way the probability of CHF can be calculated when both the CHF and the
actual heat flux have known probabilistic density function distributions. In such cases,
the probability of dryout is estimated as illustrated in FIGURE 6-8.
Actual heat flux distribution,
Critical heat flux distribution
f (q ′a′ )
′′ )
f (qCHF
F1
A
′′
qa′′ q′′ A qCHF
F2
Heat flux q’’
FIGURE 6-8: Graphical illustration of the probability of CHF when both the actual and the critical value
of the heat flux are stochastic functions with known distributions.
The figure shows the probability density functions for the actual, q′a′ , and the critical
′′ . Both these functions have in general different shapes, but it is
heat flux, qCHF
assumed that they follow the normal distribution. The curves cross each other at point
A. The shaded areas are given as,
′′ A
qCHF
(6-20)
F1 =
∫ f (q′′ )dq′′
CHF
CHF
,
0
∞
(6-21)
F2 =
∫ f (q′′ )dq′′ .
a
a
q ′a′ A
′′ < q′′ A , that is, the probability that the critical heat flux is
F1 is the probability that qCHF
less then the heat flux at point A. Similarly, F2 is the probability that q′a′ > q′′ A , that is,
the probability that the actual heat flux is larger than the heat flux at point A. Thus the
′′ < q′′ A and q′a′ > q′′ A , in
product F1F2 is the probability that simultaneously qCHF
which case the CHF will occur.
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However, the product F1F2 is not equal to the probability of CHF occurrence, since
′′ ∩ qCHF
′′ > qCHF
′′ A or
such stochastic events corresponding to CHF as qa′′ > qCHF
′′ ∩ q′a′ < q′a′ A are excluded. Thus, the product F1F2 is the lower estimate of
qa′′ > qCHF
the CHF probability, that is,
(6-22)
pCHF > F1 F2 ,
′′ .
for any values of qa′′ > qCHF
′′ > qCHF
′′ A is given by the product (1The probability that both qa′′ < qa′′ A and qCHF
F1)(1-F2) and corresponds to the situations when CHF will not occur. However, this
product is not equal to the probability of the non-occurrence of CHF, since such
′′ ∩ qCHF
′′ > qCHF
′′ A are
stochastic events corresponding to non-CHF cases as qa′′ < qCHF
not included. Thus, the product (1-F1)(1-F2) is the lower estimate of the no-CHF
probability, that is,
(6-23)
pno−CHF > (1 − F1 )(1 − F2 ) .
Since,
pCHF + pno−CHF = 1 ,
then, the CHF probability is determined by the following interval,
(6-24)
F1F2 < pCHF < F1 + F2 − F1 F2 .
In the computational procedure, it is first necessary to find the heat flux at point A and
next to find areas F1 and F2 using the standard cumulative function, as shown in the
example above.
6.2.7
Profiling of Coolant Flow through Reactor Core
A nuclear reactor core consists of many parallel connected heated channels through
which the total flow is distributed according to the distribution of local pressure losses.
Since all channels have common inlet and outlet plena, the total pressure drop in each
channel is the same and equal to the total pressure drop over the whole reactor core. In
a design of nuclear reactor cores it has to be considered how to distribute the flow
through the heated channels. Such process is called flow profiling through the reactor
core.
The flow profiling is performed with two types of general assumptions:
•
profiling with constant distribution of heat sources, in which it is assumed that during
the whole period of the reactor operation the power distribution will have the
same spatial shape,
•
profiling with variable distribution of heat sources, in which a variation of the heat
source distribution during the whole period of the reactor operation is taken
into consideration.
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W
Fuel elements
Flow channels
∆pc=const
Inlet orificing to
be adjusted
WN
W1
W
FIGURE 6-9: Flow distribution through parallel channels in a reactor core.
The following constraints have to be satisfied once performing the flow profiling:
N
(6-25)
∑W
i
=W
i =1
(6-26)
∆pi = ∆pc = const , i = 1, …, N
Using the above equations, the inlet orificing should be selected in such a way that:
•
the total pressure drop over the reactor core is the lowest possible,
•
the total reactor power is the highest possible with given limiting values and
required safety margins for all safety parameters,
•
coolant flow through the reactor core remains stable in all channels under all
anticipated conditions.
A complete and thorough flow profiling that satisfies all three conditions is a complex
process and its description is beyond the scope of the present book. Here a simple
system (see EXAMPLE 6-4 and FIGURE 6-10), is analyzed to demonstrate some
particular features of the flow profiling analysis.
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Total power q
Power fraction f2=1-f1
Power fraction f1
Inlet loss coefficient ξ1
Inlet loss coefficient ξ2
W2
W1
W
FIGURE 6-10: Flow profiling in a two-channel system with non-uniform power distribution.
EXAMPLE 6-4: Two identical heated channels have common inlet and outlet
plena, as shown in FIGURE 6-10. Determine the required inlet loss coefficients to
obtain the maximum total power of both channels q and the minimum required
pumping power if the power fractions in channels are known and equal to f1 and
f2. Neglect the hydrodynamic flow stability considerations. Assume that the limiting
thermal parameter is the heat flux in the channels. Use the Levitan&Lantsman
correlation to determine the DNB heat flux. SOLUTION: Clearly, the solution is
trivial if both power fractions are equal to each other. In this situation the flows should be equal in both
channels. To get the minimum required pumping power, the pressure losses should be as low as possible,
thus no additional orificing should be introduced (the flow stability is not discussed in this example); this
leads to the following solution: W1 = W2 = W/2 and ξ1 = ξ1 = 0. Thus, in the continuation it is assumed
that f1 and f2 are not equal. The critical heat flux in each of the channels is obtained as (see Eqs. (4-98) and
(4-99)),
1.2{[0.25( p −98 ) / 98 ]− xe ,1 }
0 .5
2
p
 p   W A 
8 
−1.5 x
ϕ ( p,x )
q cr′′ ,1 =   10.3 − 7.8 + 1.6   1 
e e ,1 = ϕ1 ( p, D, A) ⋅ ϕ 2 (xe ,1 ) ⋅ W1 3 e ,1
D
98
98
1000
  
  

Here φ1, φ2 and φ3 denote functions of given parameters. Since the channels are identical and assuming
that the reference pressure in the DNB correlation is the same for both channels, the critical heat flux in
the second channel is given as,
ϕ ( p , xe , 2 )
q cr′′ , 2 = ϕ1 ( p, D, A) ⋅ ϕ 2 (xe , 2 ) ⋅ W2 3
The maximum power in the system will be obtained when the critical power ratio are equal in both
channels. That is,
ϕ ( p , xe ,1 )
CPR1 =
qcr′′ ,1 ϕ1 ( p, D, A) ⋅ ϕ 2 (xe,1 )⋅ W1 3
=
q′′f1
q′′f1
ϕ3 ( p , xe , 2 )
= CPR2 =
ϕ1 ( p, D, A) ⋅ ϕ 2 (xe, 2 )⋅W2
q′′(1 − f1 )
The above equation must be solved together with the mass conservation equation W1 + W2 = W to
obtain the required mass flow rate distribution, W1 and W2. In the next step, the pressure drop equations
for both channels must be solved to obtain the required orificing to realize the required mass flux
distribution.
6.3 Mechanical Design
The purpose of the mechanical design is to establish the construction details of plant
components to withstand loads during the operation of the nuclear power plant.
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6.3.1
6
– R E A C T O R
D E S I G N
Design Criteria and Definitions
Design criteria are specified in the boiler and pressure vessel codes such as the ASME
Code section III[6-1].
The basic design criterion stipulates that the nuclear power plant design should be such
that stress intensities will not exceed the specified limits.
The failure theory is based on the maximum shear stress theory. The maximum shear
stress at a point is equal to one-half the difference between the algebraically largest and
the algebraically smallest of the three principal stresses.
A primary stress is any normal stress or a shear stress developed by an imposed
loading which is necessary to satisfy the laws of equilibrium of external and internal
forces and moments. The main characteristic of a primary stress is that it is not selflimiting. Primary stresses that considerably exceed the yield strength will result in
failure. A thermal stress is not classified as a primary stress. Example of primary stress
is a general membrane stress in a circular cylindrical or a spherical shell due to internal
pressure or to distributed live loads.
A secondary stress is a normal stress or a shear stress developed by the constraint of
adjacent material or by self-constraint of the structure. A general thermal stress is an
example of a secondary stress.
A thermal stress is a self-balancing stress produced by a nonuniform distribution of
temperature or by differing thermal coefficients of expansion.
6.3.2
Stress Intensity
The algebraic difference between the largest and (algebraic) smallest of the principal
stresses at a given location is called the equivalent intensity of combined stresses. This term is
commonly abbreviated to stress intensity and is equivalent to twice the maximum
shear stress.
Once determining stress intensities, the following steps should be followed:
1. At the point on the component which is being investigated, choose an
orthogonal set of coordinates, such as tangential, radial and longitudinal, and
designate them with subscripts t, r and l. The stress components in these
directions are then designated σ t , σ r and σ l for direct stresses and τ lt , τ lr
and τ rt for shearing stresses.
2. Calculate the stress components for each type of loading to which the part will
be subjected and assign each set of stress to one or a group of the following
categories:
a. General primary-membrane stress, Pm,
b. Local primary-membrane stress, PL,
c. Primary bending stress, Pb,
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– R E A C T O R
D E S I G N
d. Expansion stress, Pe,
e. Secondary stress, Q,
f. Peak stress, F.
3. For each category, calculate the algebraic sum of the σ t ’s that results from the
different types of loadings and similary for the other five stress components.
Certain combinations of these categories must also be considered.
4. Translate the stress components for the t, r and l directions into principal
stresses, σ 1 , σ 2 and σ 3 .
5. Calculate the stress differences S12, S23, S31 from the relations,
S12 = σ 1 − σ 2 , S 23 = σ 2 − σ 3 , S31 = σ 3 − σ 1 . The stress intensity S is found as
the largest absolute value of S12, S23, S31.
6.3.3
Piping Design
For a straight pipe under internal pressure, the minimum thickness of a pipe wall
required for design pressure shall be determined from one of the following
formulas[6-1],
(6-27)
tm =
pDo
pD + 2a (S m + y ⋅ p )
+ a or tm =
,
2( S m + y ⋅ p )
2( S m + y ⋅ p − p )
where,
tm
- the minimum required wall thickness, m
p
- internal design pressure, MPa
Sm
- maximum allowable stress at design temperature, MPa
D
- inside pipe diameter, m
Do
- outside pipe diameter, m.
a
- an additional thickness (due to corrosion, material removal, etc), m
y = 0 .4 .
Additional thickness should be added to pipe bends, pre-manufactured elbows, branch
connections, openings, closures, etc. The required additional thickness is specified in
more detailed pipe design codes.
6.3.4
Vessels Design
A tentative thickness of cylindrical shells and spherical shells can be found as[6-1],
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– R E A C T O R
D E S I G N
(6-28)
tCS =
pR
pRo
or tCS =
,
S m − 0 .5 p
S m + 0 .5 p
(6-29)
tSS =
pR
pR
or tSS = o .
2 Sm − p
2 Sm
Here,
tCS
- thickness of cylindrical shell, m
tSS
- thickness of spherical shell, m
p
- internal pressure, MPa
Sm
- maximum allowable stress at design temperature, MPa
R
- inside radius of shell, m
Ro
- outside radius of shell, m.
Once a tentative wall thickness in a vessel is establish, additional calculations are
needed to account for local stress values due to openings, nozzles, reinforcements and
welding joints. The recommendations given in a pertinent pressure vessel code should
be followed.
R E F E R E N C E S
[6-1]
ASME Pressure Vessel Code. Part III.
E X E R C I S E S
EXERCISE 6-1: A certain reactor contains 50000 identical fuel rods subject to exactly the same operating
conditions. The safety authority requires that no single fuel rod should be under CHF during normal
operating conditions. Assume a normal distribution of the probability of CHF: (a) What should be the
limit probability of CHF for a single fuel rod to satisfy the requirement of the safety authority? (b) Find
the minimum CPR to satisfy the requirement assuming that the operating conditions are known exactly
and the standard deviation of the CHF correlation is 3.5%, (c) How the reactor power should change to
satisfy the same requirement if the standard deviation of the CHF correlation was 7% instead of 3.5%?.
164
Chapter
7
7 Environmental and Economic
Aspects of Nuclear Power
E
nergy in general can be considered in two categories – primary and secondary.
Primary energy is energy in the form of natural resources, such as wood, coal,
oil, natural gas, natural uranium, wind, hydro power and sunlight. Secondary
energy is in the more usable forms which primary energy may be converted
to, such as electricity, petrol and hydrogen. Primary energy can be either renewable
(solar, wind, wave, biomass, geothermal energy and hydro power) or non-renewable
(fossil fuels – coal, oil and natural gas – and uranium). Each kind of energy
transformation from the primary to the secondary form has some environmental
effects, as well as it is connected with specific costs.
Uranium is abundant in nature and technologies exist that can extend its use 60-fold if
demand requires it. World mine production is about 35000 tons per year, but a lot of
the market is being supplied from secondary sources such as stockpiles, including
material from dismantled nuclear weapons. Practically all of it is used for electricity.
Uranium has the highest heat value of all known fuels and that makes it a very
attractive source of energy.
Due to fission processes that take place in nuclear reactors highly radioactive fission
products are produced. These wastes must be carefully taken care of to not pollute the
environment. All this basic aspects of nuclear fuel cycle are discussed in this Chapter.
7.1 Nuclear Fuel Resources and Demand
7.1.1
Uranium Resources
Identified resources of uranium consist of Reasonably Assured Resources (RAR)
and Inferred Resources which are recoverable at a cost of less than $130/kgU. The
identified resources increased significantly between 2003 and 2005 from 4588000 tU to
4743000 tU (increase with 155000 tU). Some of the increase is due to new discoveries
resulting from increased exploration, but the major part of the increase result from
higher prices of uranium[7-5].
Undiscovered Resources include Prognosticated Resources (PR) and Speculative
Resources (SR). PR refers to uranium resources that are expected to occur in well
defined geological trends of known deposits. SR refers to uranium resources that are
thought to exist in geologically favourable, yet unexplored areas, thus they are assigned
a lower degree of confidence than PR. In total PR are estimated to 2518800 tU and SR
to 7535900 tU, with cost lower than $130/kgU[7-5].
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7
–
E N V I R O N M E N T A L
A N D
E C O N O M I C
A S P E C T S
Other resources include unconventional uranium resources (in which uranium exists
at very low grades) and other potential nuclear fuel materials (e.g. thorium). Most of
the unconventional uranium resources are associated with uranium in phosphates
(about 22 million tons), but other potential sources exist, e.g. seawater and black shale.
Thorium is abundant and widely dispersed, and its resources are estimated to more
than 4.5 million tons. This estimate is considered conservative, since data from China,
Central and Eastern Europe and the Former Soviet Union are not includes[7-5].
Uranium production increased from 36050 tU in 2002 to 40263 tU in 2004 and is
estimated to 41250 tU in 2005. Uranium is mainly produced using open-cut (26.6%)
and underground mining (40.1%) techniques followed by conventional uranium
milling. Other mining methods include in situ leaching (ISL) (19.2%). Total potential
production capability is predicted to rapidly increase to 83370 tU/year in 2010 and
then increase gradually to 86900 tU/year in 2025[7-5].
There are several factors that affect nuclear power capacity and related uranium
requirements. One of the factors is the nuclear energy availability. In 2004, the average
world nuclear energy availability factor was 83.2% compared with 71% in 1990. In
addition, many power plants are undergoing life extension and power uprates leading
to even more increased capacities. All these factors make forecasts of the uranium
requirements rather uncertain.
World annual uranium requirements were about 67320 tU in 2004 and 66840 tU in
2005[7-5]. Current (May 2008) number of operating nuclear reactors worldwide is 439.
They have the total power output of 372.2 GWe and produced during year 2007
2617.9 billion kWh (TWh), which corresponds to 16% of the total electricity
production world-wide. The required uranium to support the current production
corresponds to 65500 tU.
The number of reactors (shown in TABLE 7.1) increases steadily, but not dramatically.
At present there are 28 reactors under construction, with total planned power output
of 22645 MWe; 62 reactors are planned, with total power output of 68021 MWe; and
161 reactors are proposed, with expected total power output of 120625 MWe[7-6]. The
installed nuclear capacity is projected to grow to about 449 GWe net (low estimate –
grow of 22%) or 533 GWe net (high estimate – grow of 44%) by the year 2025. World
reactor-related uranium requirements by the year 2025 are projected to increase to
between 82275 tU (low estimate) and 100760 tU (high estimate). The requirements in
the North America and the Western Europe region are expected to either remain fairly
constant or decline slightly, whereas they will increase in the rest of the World[7-5].
TABLE 7.1. World nuclear power generation and capacity (Source: IAEA and WNA)).
As of May 2008
Country
Argentina
2007
Number
Nuclear
of
Capacity (MW)
Nuclear
Units
2
935
Nuclear
Generation
(BkWh)
Nuclear
Fuel
Share
(Percent)
6.7
.2
Armenia
1
376
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3.5
Belgium
7
5,824
45.9
Brazil
2
1,795
12.4
4.0
.8
Bulgaria
2
1,906
13.7
2.1
Canada
18
96.5
11
12,58
9
8,572
China
6
3,619
24.6
6.0
62.6
.9
Czech RP
0.2
Finland
4
2,696
22.5
8.9
France
59
Germany
17
Hungary
4
63,26
0
20,47
0
1,829
418.
6
133.
2
13.9
6.8
5.9
6.8
India
17
3,782
15.9
.5
Japan
55
Korea Rep.
20
Lithuania
1
47,58
7
17,45
1
1,185
266.
4
136.
6
9.1
7.5
5.3
4.4
Mexico
2
1,360
10.4
.6
Netherlands
1
482
4.0
Pakistan
2
425
2.3
Romania
2
1,300
7.1
.1
.3
3.0
Russia
31
Slovakia
5
21,74
3
2,034
147.
8
14.2
Slovenia
1
666
5.4
South
Africa
Spain
2
1,800
12.6
6.0
4.3
1.6
.5
8
7,450
52.3
7.4
Sweden
10
9,014
64.4
5
3,220
26.3
6.1
Switzerland
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0.0
Taiwan,
China
U.K.
4,921
39.0
19
10,22
2
100,5
82
13,10
7
372,2
02
57.5
9.3
U.S.
10
4
15
Ukraine
Total
7.1.2
6
43
9
5.1
806.
5
87.2
9.4
8.1
2,61
7.9
Thorium Fuel
Thorium is much more abundant in nature than uranium. It is a naturally occurring,
slightly radioactive metal discovered in 1828 by the Swedish chemist Jons Jakob
Berzelius. Thorium occurs in several minerals, such as the rare-earth-thoriumphosphate mineral, monazite, which contains up to about 12% thorium oxide.
TABLE 7.2. Thorium reserves in the world.
Country
Reserves (tons)
Australia
300000
India
290000
Norway
170000
USA
160000
Canada
100000
South Africa
35000
Brazil
16000
Other countries
95000
World total
1200000
Thorium is not fissile itself, but Th-232 isotope, when absorbing slow neutrons, is
producing uranium-233, which is fissile. U-233 has higher neutron yield per neutron
absorbed than both U-235 and Pu-239. This make possible to use thorium in a
breeding cycle.
7.1.3
Nuclear Fuel Demand
In current light water reactors only slightly above 0.5% of natural uranium is used for
energy production. The uranium which is left after enrichment (so-called depleted
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uranium) and in spent fuel is treated as wastes. Some part of uranium is converted in
reactors into plutonium.
With the present fleet of nuclear reactors (439 units with total electricity generation
2.6x1012 kWh per year) would the currently known uranium resources suffice for about
85 years. The same uranium resources would suffice for over 5000 years if fuel
breeding from uranium-238 is employed. Thus breeder reactors should be used if
nuclear power is to play a major role in the energy production in the future.
Evaluation of nuclear fuel demand is difficult, because it depends on many factors
which are changing with time. If the installed electric capacity of a nuclear power plant
is Pe, [GWe] then the annual thermal energy output of the plant is given as,
(7-1)
Q=
Pe C F 365
η th
,
where Q is the annual thermal energy output [GWd], CF is the capacity factor (which
indicates the fraction of the capacity which is actually used. Typical values are 0.8 to
0.9) and η th is the thermal efficiency of the power plant.
The mass of enriched fuel needed per year depends on the generated annual thermal
energy Q and on the fuel burnup Bd, which indicates the amount of thermal energy
obtained from a unit mass (usually metric ton) of enriched fuel. The required mass is
thus obtained as,
(7-2)
M =
Q
,
Bd
where M is the mass of enriched fuel loaded per year [tU/year] and Bd is the fuel
burnup [GWd/tU]. Substituting Eq. (7-1) into(7-2) gives,
(7-3)
M =
Pe C F 365
.
η th Bd
Equation (7-3) gives the required mass of enriched uranium for a given nuclear power
plant. The required mass of the natural uranium depends on the enrichment of fuel
and tails.
7.2 Fuel Cycles
The nuclear fuel cycle consists of the steps required to produce nuclear power,
including the input of fissile material, the processes that convert raw material to useful
forms, the output of energy, and the treatment and/or disposition of spent fuel and
various waste streams. The steps are shown in FIGURE 7-1.
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Step 1
Step 2
Step 3
Step 4
Mining and
Milling
Enrichment
Fabrication
Reactor and
Turbine/Generator
Plutonium
oxide
Mining
tailings
Enrichment
tailings
Step 6
Step 5
Reprocessing
Repository
Reprocessing waste
FIGURE 7-1: Nuclear fuel cycle diagram.
7.2.1
Open Fuel Cycle
In the open fuel cycle fuel is not reprocessed and after usage in reactor is placed in
the repository. It appears as steps 1 through 5 in FIGURE 7-1. This is most often used
system in most countries using the common kinds of reactors. It requires uranium ore
as input, milling and purification of natural uranium, conversion of the uranium to a
chemical form suitable for enrichment, enrichment of the U-235 isotope, fuel
fabrication, loading of uranium fuel assemblies in a reactor and then the reactor
operation. At the end of useful life, spent fuel is removed from the reactor, stored in a
pool of water for cooling and shielding of radioactivity, then removed and placed in
air-cooled casks at the reactor sites for interim storage, and finally removed to geologic
waste storage.
7.2.2
Closed Fuel Cycle
Plutonium production in the open-fuel cycle represents a significant energy resource,
but requires reprocessing of spent fuel to recover the plutonium and to fabricate new
fuel. Recycling of fuel can be done in thermal reactors and in fast reactors.
for thermal reactors is shown in FIGURE 7-1 as steps 1-2-3-4-6-3.
As can be seen it adds another process in comparison with the open fuel cycle, i.e., fuel
reprocessing. Several countries (France, Japan, Russia and the UK) have reprocessing
plants in operation. Spent fuel reprocessing is very costly and, in some countries,
thermal recycle is not considered as an economic choice.
Closed fuel cycle
Close fuel cycle for fast breeder reactors is similar to that of thermal reactors, however,
with some important differences. A fast breeder reactor is capable by design of
producing more fissile isotopes than it consumes, thus making it possible to provide a
growing resource of energy that does not require a continuing supply of U-235 or Pu239 after an initial investment of fissile fuel at the beginning of its life. Fuel enrichment
for fast reactors is higher (15-20%) than it is in LWRs. Breeder reactor cores typically
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have two regions: a ”seed” region on the inside of the core, and a “blanket” region
surrounding the “seed”. Seed fuel assemblies consist of fissile fuel, 15%-20% fissile
plutonium, and this region provides power and fission neutrons to maintain criticality,
while blanket assemblies contain fertile fuel, U-238, for breeding of new plutonium.
7.3 Front-End of Nuclear Fuel Cycle
The “Front-End” of the nuclear fuel cycle consists of several processes which include
mining, milling, conversion, enrichment and fabrication.
7.3.1
Mining and Milling of Uranium Ore
Uranium is usually mined by either surface (open cut) or underground mining
techniques, depending on the depth at which the ore body is found. For example, in
Australia the Ranger mine in the Northern Territory is open cut, while Olympic Dam
in South Australia is an underground mine (which also produces copper, with some
gold and silver). On the contrary, most Canadian uranium mines are underground. Ore
grades vary at different places and are usually less than 0.5% U3O8.
The mined uranium ore (i.e. rock containing economically recoverable concentrations
of uranium) is sent to a mill which is usually located close to the mine. At the mill the
ore is crushed and ground to a fine slurry which is leached in sulfuric acid to allow the
separation of uranium from the waste rock. It is then recovered from solution and
precipitated as uranium oxide (U3O8) concentrate. (Sometimes this is known as
yellowcake, though it is finally khaki in color.) After high-temperature drying, the
uranium oxide, now khaki in color, is packed into 200-litre drums for shipment. The
radiation level one meter from such a drum of freshly processed U3O8 is about half
that (from cosmic rays) received by a person on a commercial jet flight. The solids
remaining after the uranium is extracted are pumped as a slurry to the tailings dam,
which is engineered to retain them securely. Tailings contain most of the radioactive
material in the ore, such as radium.
Some new mines use in situ leaching (ISL) to extract the uranium from the ore body
underground and bring it to the surface in solution. It is recovered in the same fashion.
Uranium minerals are always associated with other elements such as radium and radon
in radioactive decay series. Therefore, although uranium itself is barely radioactive, the
ore which is mined must be regarded as potentially hazardous, especially if it is highgrade ore. The radiation hazards involved, however, are virtually all due to the
associated elements and are similar to those in many mineral sands operations. Any
underground uranium mine is ventilated with powerful fans.
7.3.2
Uranium Separation and Enrichment
The vast majority of all nuclear power reactors in operation and under construction
require 'enriched' uranium fuel in which the content of the U-235 isotope has been
raised from the natural level of 0.72 atom percent (a/o) to about 3.5 weight percent
(w/o) or slightly more. The enrichment process removes 85w/o of the U-238 by
separating gaseous uranium hexafluoride (UF6) into two streams: One stream is
enriched to the required level and then passes to the next stage of the fuel cycle. The
other stream is depleted in U-235 and is called 'tails'. It is mostly U-238.
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So little U-235 remains in the tails (usually less than 0.3w/o) that it is of no further use
for energy, though such 'depleted uranium' is used in metal form in yacht keels, as
counterweights, and as radiation shielding, since it is 1.7 times denser than lead.
NOTE CORNER: It must be noted that although isotopic abundances are always
given in atom percent (a/o), sometime it is preferred to specify enrichments in
weight percent (w/o).
Separation of uranium isotopes is a very difficult task since the two isotopes, U-235
and U-238 have very nearly the same atomic weight. Chemically they can not be
separated, of course, as they have exactly the same chemical properties. Separation is
thus accomplished by physical means, such as either gaseous diffusion method or gascentrifuge method.
Separation process is schematically shown in FIGURE 7-2, where the feed material
with mass MF and weight enrichment xF is separated into the final product with mass
MP and the weight enrichment xP, and the tails with mass MT and enrichment xT.
Product
Feed
MF, xF
Isotope
separation
process
MP, xP
Tails
MT, xT
FIGURE 7-2: Schematics of a separation process.
Since there are virtually no losses of uranium, its total mass is conserved as follows,
M F = M P + MT .
Similarly, the mass of U-235 is the same before and after enrichment,
xF M F = xP M P + xT M T .
Eliminating MF from the last two equations yields,
(7-4)
MF =
xP − xT
MP .
xF − xT
This equation gives the amount of feed material that is required to obtain a specified
amount of enriched product.
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Equation (7-4) indicates that the required mass of natural uranium increases with the
product enrichment. For PWRs, the required enrichment for a given burnup can be
approximated using the following correlation, valid for enrichments up to 20%, [7-1]:
2
(7-5)
 n +1 
 n +1 
x p = 0.41201 + 0.11508
Bd  + .0.00023937
Bd  ,
 2n

 2n

where n is the number of fuel batches, i.e. the fraction of the core refueled per cycle is
1/n.
The isotope separation is a costly process that requires a considerable amount of work.
A so-called value function has been developed on the basis of the theory of the
gaseous diffusion cascade. It can be shown that the value of uranium as a function of
its enrichment in weight percent, x, can be given as,
(7-6)
V ( x) = (1 − 2 x )ln
1− x
.
x
The function is shown in FIGURE 7-3.
The separative work associated with the production of a given amount of enriched
uranium is defined as the increase in total value of the uranium after separation
process. This work is calculated as,
SWU = M PV ( x P ) + M T V ( xT ) − M F V ( x F ) ,
where SWU is the number of separative work units. Since,
MT = M F − M P ,
it can be expressed as,
SWU = M P [V ( x P ) − V ( xT )] − M F [V ( xF ) − V ( xT )].
V(x)
(7-7)
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
x
FIGURE 7-3: The separation value function versus enrichment x in weight percent.
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The separative work can be expressed per unit mass of the product as,
(7-8)
M
SWU
= V ( x P ) − V ( xT ) − F [V ( x F ) − V ( xT )] .
MP
MP
It should be noted that SWU has the same units as mass, which is kg.
EXAMPLE 7-1. Calculate the amount of natural uranium and SWU that are
needed to produce 100000 kg of 3.5w% enriched uranium, assuming tails with
0.2w%. SOLUTION: Note that for natural uranium xF = 0.711w%. Eq. (7-6)
yields: V(xP) = 3.085, V(xF) = 4.869, V(xT) = 6.188. For xT = 0.002, Eq. (7-4)
yields: MF = (0.035-0.002)/(0.00711-0.002)*100000 = 6.458 105 kg. From Eq. (7-7)
the number of separative work units is as follows: SWU = 105*(3.085-6.188)- 6.458
105 *(4.869-6.188) = 5.415 105 kg.
The cost per separative work unit is determined from the enrichment plant operating
costs and the cost of the capital invested in the plant. If CS is the cost of a separative
work unit (in €/kg or $/kg), then
(7-9)
M P C P = SWU ⋅ C S + M F C F − M T CT ,
and
(7-10)
CP =
SWU
M
M
⋅ C S + F C F − T CT .
MP
MP
MP
Here CP, CF and CT are the costs per kg of product, feed and tails, respectively.
According to Eq. (7-9) the cost of the product is equal to the cost of the enrichment
operation plus the cost of the feed, less credit for the value of the tails (which is usually
neglected).
Several methods for enrichment of uranium have been developed, of which two are of
special interest, namely the gaseous-diffusion and gas-centrifuge methods.
The gaseous-diffusion method is based on the different rates at which gases of
different molecular weights diffuse through a porous barrier. The only suitable
compound of uranium for use in the gaseous-diffusion process is the hexafluoride. It
is solid at room temperature but it sublimes above 56.4 ºC.
The basic principle of the separation of isotopes by gaseous diffusion is that, in a
mixture of gases at a certain temperature, the average molecular kinetic energy is the
same for each gas. As a consequence, the lighter molecules will have higher average
speeds than the heavier molecules, with the velocity ratio,
vL
MH
=
.
vH
ML
This ratio is called the theoretical separation factor α * :
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352
MH
=
≈ 1.0043 .
349
ML
α* =
As can be seen this factor is very close to unity and thus, the degree of enrichment in
uranium-235 as a result of diffusion through a single barrier is very small. A “cascade”
of barriers must be employed in order to achieve substantial enrichment.
The stage separation factor α is defined as,
(7-11)
α=
xe (1 − xe )
xd (1 − xd )
where xe and xd are the weight fractions of uranium-235 in the enriched and the
depleted streams leaving the stage. α , which is the actual measure of the uranium-235
enrichment, is found to be less than the theoretical separation factor α * , and
approximately equal to 1.003 as compared to 1.0043.
It can be shown that to obtain the final product and the tails with weight fractions of
uranium-235 xP and xT, respectively, the required number of stages in cascade, Ns-c is
given as,
(7-12)
N s −c =
2
x (1 − xP )
ln P
− 1.
α − 1 xT (1 − xT )
For example, with xP = 0.03, xT = 0.003 and α = 1.003, Ns-c = 1553.
The gas-centrifuge method is based on the principle that if a gas containing molecular
species with different masses is centrifuged, the heavier molecules will move towards
the periphery of the centrifuge whereas the lighter ones will remain in the center. It
can be shown that the equilibrium stage separation factor is,
(7-13)
 (M L − M H )v 2 
,
2 RT


α = exp
where ML and MH are the molecular weights (kg/mol) of the light and the heavy
molecules, R is the universal gas constant (8.31 J/mol/K), T is the absolute
temperature of the gas an v is the linear velocity of gas.
For example, if a cylindrical centrifugal bowl with diameter 0.2 m, rotating with 30000
rpm (500 rotations per second), contains UF6 at temperature T = 330 K, the
equilibrium separation factor is,
 (0.238 − 0.235)(2π ⋅ 0.1 ⋅ 500 )2 
 ≈ 1.055 .
2
⋅
8
.
31
⋅
330


α = exp
As can be seen the stage separation factor for the gas-centrifugal method is higher than
for the gas diffusion method. The required number of stages is thus reduced. For
example, with xP = 0.03, xT = 0.003 and α = 1.055, Eq. (7-12) yields Ns-c = 84.
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Fuel Fabrication
Enriched UF6 is transported to a fuel fabrication plant where it is converted to
uranium dioxide (UO2) powder and pressed into small pellets. These pellets are
inserted into thin tubes, usually of a zirconium alloy (zircalloy) or stainless steel, to
form fuel rods. The rods are then sealed and assembled in clusters to form fuel
elements or assemblies for use in the core of the nuclear reactor.
7.4 Back-End of Nuclear Fuel Cycle
The Back-End of the nuclear fuel
wastes in environment-friendly way.
cycles
consists of several processes to handle
Despite its demonstrable safety record over half a century, one of the most
controversial aspects of the nuclear fuel cycle today is the question of management and
disposal of radioactive wastes. The most difficult of these are the high-level wastes,
and there are two alternative strategies for managing them:
• reprocessing spent fuel to separate them (followed by vitrification and disposal),
• direct disposal of the fuel containing high levels of radioactivity as waste .
The principal nuclear wastes remain locked up securely in the ceramic reactor fuel.
“Burning” the fuel of the reactor core produces fission products such as various
isotopes of barium, strontium, cesium, iodine, krypton and xenon (Ba, Sr, Cs, I, Kr,
and Xe). Many of the isotopes formed as fission products within the fuel are highly
radioactive, and correspondingly short -lived.
As well as these smaller atoms formed from the fissile portion of the fuel, various
transuranic isotopes are formed by neutron capture. These include Pu-239, Pu-240 and
Pu-241, as well as others arising from some of the U-238 in the reactor core by
neutron capture and subsequent beta decay. All are radioactive and apart from the
fissile plutonium which is partly "burned", they remain within the spent fuel when it is
removed from the reactor. The transuranic isotopes and other actinides form most of
the long-lived portion of high-level waste.
While the civil nuclear fuel cycle generates various wastes, these do not become
“pollution”, since virtually all are contained and managed, otherwise they would be
dangerous. In fact, nuclear power is the only energy-producing industry which takes
full responsibility for all its wastes and fully costs this into the product. Furthermore,
the expertise developed in managing civil wastes is now starting.
7.4.1
Fuel Burnup
During fission process of uranium-235 in a reactor core, fission products are produced
and accumulated. Some of the fission products have large absorption cross-sections
for neutrons and significantly reduce the reactivity of the reactor. When the excess
reactivity is too small to operate the reactor, a portion or whole fuel must be exchange
in the core, even though there is still a large number of uranium-235 atoms that have
not fissioned.
The period of reactor operation with the same fuel batch is called a fuel cycle. The
length of a fuel cycle depends on the available excess reactivity and the increasing
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probability of fuel damage. Some of the fission products are released from fuel pellets
as gases and accumulate in the gas gap and plenum of a fuel rod, leading to an
increased pressure inside the cladding.
The allowable fuel burnup determines the amount of fission products that can be
produced without leading to fuel damages. Higher fuel burnup means longer fuel
cycle. Fuel burnup is expressed in the amount of thermal energy (usually in GWd –
giga-watt-days) that can be obtained from one ton of enriched uranium (1 tU). A
typical value of burnup in current LWRs is 50 GWd/tU. Typical isotopic composition
of PWR fuel with burnup equal to 33 GWd/tU is shown in FIGURE 7-4.
95.5% uranium
3.5% fiss. p.
1.0% TRU
33 GWd/tU
100% U
2.44%
3.2% U-235
0.76% U-235
0.44%
0.44% U-236
2%
3.5% fission
products
0.9% Pu
0.1% Np,Am,Cm
1.5%
96.8% U-238
94.3% U-238
1%
-2.5%
FIGURE 7-4: Change in isotope composition in nuclear fuel with burnup 33 GWd/tU.
As can be seen, uranium-238 (94.3%) remains the major component of the spent fuel.
The new components are as follows: fission products (3.5%), plutonium (0.9%) and
minor actinides (0.1%). Minor actinides are all elements created in reactor with atomic
number higher than uranium, except for plutonium. Minor actinides together with
plutonium are termed as transuranic (TRU) elements. The change in isotopic
composition of spent fuel depends on burnup, as shown in table below.
TABLE 7.3. Isotopic composition of spent fuel, [7-1].
33 GWd/tU
50 GWd/tU
100 GWd/tU
Uranium
95.5%
93.4%
87.43%
Fission products
3.5%
5.15%
10.30%
Plutonium
0.9%
1.33%
1.97%
Minor actinides
0.1%
0.12%
0.30%
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Spent fuel remains radioactive during a long period of time, as shown in FIGURE 7-5.
The curves shown in the figure have been obtained for PWR reactor, with 50
GWd/tU burnup and initial enrichment 4.5%.
Another measure of risks posed by the spent fuel is so called radiotoxicity. Strictly
speaking, radiotoxicity RT at a given time t is calculated as,
(7-14)
RT (t ) =
 λi N i (t )

water
all radionuclides i  MPC i
∑

 ,

where λi N i (t ) is the quantity of radioisotope i present in 1 metric ton of waste at time
t (in Bq/tU) and MPCiwater is the maximum permissible concentration of isotope i in
water (in Bq/m3). The calculation of the maximum permissible concentration for each
radionuclide is based on the assumption that an adult would ingest water containing
the radionuclide at a constant rate of 2 liters per day during one year. The
concentration limit is determined by imposing the requirement that the individual
should receive a committed effective dose no greater than 50 millirems from this
source. The calculated radiotoxicity for PWR spent fuel with 50 GWd/tU and initial
enrichment 4.5% is shown in FIGURE 7-6.
FIGURE 7-5: Radioactivity profile of spent fuel (curie/tU) as a function of time after discharge from
reactor (years), [7-1].
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FIGURE 7-6: Radiotoxicity of PWR spent fuel (in m3 of water) versus time after discharge (in years),
[7-1].
7.4.2
Repository
Spent fuel assemblies taken from the reactor core are highly radioactive and give off a
lot of heat. They are therefore stored in special ponds which are usually located at the
reactor site, to allow both their heat and radioactivity to decrease. The water in the
ponds serves the dual purpose of acting as a barrier against radiation and dispersing the
heat from the spent fuel.
Spent fuel can be stored safely in these ponds for long periods. It can also be dry
stored in engineered facilities. However, both kinds of storage are intended only as an
interim step before the spent fuel is either reprocessed or sent to final disposal. The
longer it is stored, the easier it is to handle, due to decay of radioactivity.
7.4.3
Reprocessing
Spent fuel still contains approximately 96% of its original uranium, of which the
fissionable U-235 content has been reduced to less than 1%. About 3% of spent fuel
comprises waste products and the remaining 1% is plutonium (Pu) produced while the
fuel was in the reactor and not "burned" then.
Reprocessing separates uranium and plutonium from waste products (and from the
fuel assembly cladding) by chopping up the fuel rods and dissolving them in acid to
separate the various materials. Recovered uranium can be returned to the conversion
plant for conversion to uranium hexafluoride and subsequent re-enrichment. The
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reactor-grade plutonium can be blended with enriched uranium to produce a
oxide (MOX) fuel, in a fuel fabrication plant.
mixed
MOX fuel fabrication occurs at five facilities in Belgium, France, Germany and UK,
with two more under construction. There have been 25 years of experience in this, and
the first large-scale plant, Melox, commenced operation in France in 1995. Across
Europe about 30 reactors are licensed to load 20-50% of their cores with MOX fuel,
and Japan plans to have one third of its 53 reactors using MOX by 2010.
The remaining 3% of high-level radioactive wastes (some 750 kg per year from a 1000
MWe reactor) can be stored in liquid form and subsequently solidified.
Reprocessing of spent fuel occurs at seven facilities in Europe with a capacity of over
5000 tons per year and cumulative civilian experience of 55,000 tons over 35 years.
After reprocessing the liquid high-level waste can be calcined (heated strongly) to
produce a dry powder which is incorporated into borosilicate (Pyrex) glass to
immobilize the waste. The glass is then poured into stainless steel canisters, each
holding 400 kg of glass. A year's waste from a 1000 MWe reactor is contained in 5 tons
of such glass, or about 12 canisters 1.3 meters high and 0.4 meters in diameter. These
can be readily transported and stored, with appropriate shielding.
This is as far as the nuclear fuel cycle goes at present. The final disposal of vitrified
high-level wastes, or the final disposal of spent fuel which has not been reprocessed
spent fuel, has not yet taken place. FIGURE 7-7 shows illustratively the amount of
vitrified waste arising from nuclear electricity generation for one person through a
lifetime.
The waste forms envisaged for disposal are vitrified high-level wastes sealed into
stainless steel canisters, or spent fuel rods encapsulated in corrosion-resistant metals
such as copper or stainless steel. The most widely accepted plans are for these to be
buried in stable rock structures deep underground. Many geological formations such as
granite, volcanic tuff, salt or shale will be suitable. The first permanent disposal is
expected to occur about 2010.
Most countries intend to introduce final disposal sometime after about 2010, when the
quantities to be disposed of will be sufficient to make it economically justifiable.
FIGURE 7-7: Borosilicate glass from the first waste vitrification plant in UK in the 1960s. This block
contains material chemically identical to high-level waste from reprocessing spent fuel. A piece this size
from modern vitrification plants would contain the total high-level waste arising from nuclear electricity
generation for one person throughout a lifetime.
7.4.4
Partitioning and Transmutation of Nuclear Wastes
From FIGURE 7-5 and FIGURE 7-6 it is clear that various species have different
properties in terms of radioactivity and radiotoxicity over time. One possible approach
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to optimize the treatment of nuclear wastes would be thus to partition the wastes into
different parts, each containing species with a similar properties. For example, fission
products strontium-90 and cesium-137 with half-lives of about 30 years each, accounts
for the bulk of radioactivity in spent fuel during first several decades. Extracting these
fission products from the spent fuel and storing them separately would allow for
reduction of the required repository space.
Another strategy would be to extract the uranium and TRU elements from the spent
fuel. These would lead to increased storage capacity of a given repository as well.
An option which is thoroughly investigated includes both partitioning and
transmutation of nuclear wastes. There are three principal motivations for this strategy:
•
If the long-lived isotopes could be extracted and destroyed, many more
locations would be considered suitable for hosting repositories, since the
period of time during which the wastes would be dangerous would be shorten
dramatically (from about 300000 years to about 1000 years).
•
The thermal load would be reduced leading to an increased capacity of a given
repository.
•
It would eliminate the risk that plutonium could later be recovered from a
repository and used for weapons.
7.4.5
Safeguards on Uranium Movement
Uranium may only be exported to countries which have bilateral safeguards
agreements, in addition to their acceptance of International Atomic Energy Agency
(IAEA) safeguards under the multilateral Nuclear Non-Proliferation Treaty (NPT).
Australia, for instance, has a network of 19 such bilateral agreements covering almost
30 countries.
Safeguards apply to all exports and subsequent transfers of uranium and to its possible
processing and subsequent re-use. They are based on customer countries being parties
to the NPT.
No uranium can be exported without the government first approving the terms and
conditions of the sale contract.
Since Canada and Australia are major uranium exporters, examples will be given how
uranium movement is handled by these countries.
The Canadian federal nuclear regulatory agency is the Canadian Nuclear Safety
Commission. The CNSC administers the agreement between Canada and the IAEA
for the application of safeguards in Canada and it assists the IAEA by allowing access
to Canadian nuclear facilities and arranging for the installation of safeguards equipment
at Canadian sites. It also reports regularly to the IAEA on nuclear materials held in
Canada.
The Australian Safeguards & Non-proliferation Office (ASNO), which is part of the
Department of Foreign Affairs and Trade, administers Australia's bilateral safeguards
agreements. In addition, ASNO keeps account of nuclear material and associated
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items in Australia through its administration of the Nuclear Non-Proliferation
(Safeguards) Act 1987. It provides information to the IAEA on all nuclear material in
Australia which is subject to safeguards, as well as on uranium exports, as required by
Australia's NPT agreement with the IAEA.
Both countries have in place an accounting system that follows uranium from the time
it is produced and packed for export, to the time it is reprocessed or stored as nuclear
waste, anywhere in the world. It also includes plutonium which is in the spent fuel.
For instance, all documentation relating to Australian-obligated nuclear material
(AONM) is carefully monitored and any apparent discrepancies are taken up with the
country concerned. There have been no unreconciled differences in accounting for
AONM.
These systems operate in addition to safeguards applied by the IAEA which keep track
of the movement of nuclear materials through overseas facilities and which verify
inventories.
A typical contract for the sale of Australian or Canadian uranium oxide concentrate to
an electricity generating utility in say Germany, could first entail shipment to the USA
for conversion to uranium hexafluoride. The equivalent quantity of uranium
hexafluoride might then be sent from USA to the UK for enrichment, and then on to
a fuel fabrication plant in Germany to be turned into uranium dioxide, before going
into the core of a reactor owned by the utility with whom the sale was originally
contracted. Later, the spent fuel from the reactor may go to the UK or France for
reprocessing.
When uranium goes through a continuous process such as conversion or enrichment,
it is not possible to distinguish Australian- or Canadian-origin atoms of uranium from
atoms of uranium supplied by other countries. The only way to track the quantity of
uranium is to use accounting principles, so ensuring that there is no loss of nuclear
material during transportation and processing.
In the 1990s uranium mines gained a competitor, in many ways very welcome, as
military uranium came on to the civil market. Weapons-grade uranium has been
enriched to more than 90% U-235 and must be diluted about 1:25 or 1:30 with
depleted uranium (about 0.3% U-235). This means that progressively, Russian and
other stockpiles of weapons material are used to produce electricity.
Weapons-grade plutonium may also be diluted and used to make mixed oxide (MOX)
fuel for use in ordinary reactors or in special reactors designed to 'burn' it for
electricity.
7.5 Fuel Utilization and Breeding
In thermal reactors not only uranium-235 is fissioned but also some fertile uranium238 is converted into fissile plutonium-239, part of which is fissioned during operation.
However, the total number of plutonium-239 nuclei is less than the number of
uranium-235 nuclei consumed. The efficiency with which fuel is being utilized is
expressed as,
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CR =
7
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A N D
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A S P E C T S
Number of fissile nuclei produced
.
Number of fissile nuclei destroyed
The ratio CR is called the conversion ratio. If the ratio is larger than one, it is called
the breeding ratio (BR).
Both CR and BR are space and time dependent and could be in principle calculated in
the same manner as fuel burnup. In practice, however, some approximations are used
and the ratio is determined at a specific time rather than the global or net value. Thus
the ratio at a given time can be defined as,
CR (or BR) =
Rate of formation of fissile nuclei
,
Rate of destruction of fissile nuclei
where the rates formation and destruction refer to a given time and CR (or BR) in
general vary with time during reactor operation.
The conversion ratio applies to thermal reactors with natural or slightly enriched
uranium as fuel. In such reactors plutonium -239 is produced as a result of the capture
by uranium-238 of thermal and resonant neutrons. The rate of production of fission
neutrons from a given fissile species in a thermal neutron flux φ is φNσ aηε , where N
is the concentration of fissile nuclei, σ a is the thermal-neutron absorption cross
section, η is the reproduction factor (number of fast neutrons produced per neutron
absorbed in fissile nuclei) and ε is the fast fission factor.
The rate of formation of plutonium-239 contains two terms:
•
formation due to thermal neutron capture - φN 238σ c238 ,
•
formation due to resonance neutron capture - φεPFNP (1 − p )
∑ ( Nσ η ) .
a
all fissile species
Here PFNP is the nonleakage probability in slowing down of fast neutrons into the
resonance region; p is the resonance escape probability (thus 1-p is the fraction of the
neutrons in the resonance region that is captured by uranium-238 to form plutonium239).
Initially after reactor startup there is no plutonium-239 and uranium-235 is the only
fissile material. For such conditions, the initial conversion ratio is found as,
(7-15)
CR =
N 238σ c238 + εPFNP (1 − p )N 235σ a235η 235 N 238σ c238
= 235 235 + εPFNP (1 − p )η 235 .
235 235
N σa
N σa
During reactor operation plutonium-239 is produced which contributes in both
capture and generation of neutrons. As a result the conversion ratio is decreasing.
Nevertheless a large initial conversion ratio is desirable, since it extends fuel burnup. In
commercial water-cooled reactors its value is approximately 0.6. However, only slightly
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above 50% of the generated plutonium-239 is fissioned, one-sixth is lost by neutron
capture and the rest remains in the spent fuel.
The breeding ratio, even though in principle equivalent to the conversion ratio, is
calculated in a slightly different manner. This is due to the fact that a breeder usually
consists of a core, containing both fissile and fertile species, surrounded by a blanket
containing only the fertile species. Thus, the breeding ratio at a given time is given as,
∫ φΣ
(7-16)
BR =
core
fertile
c
∫ φ (Σ
f
∫ φΣ dV
.
) dV
fertile
c
dV +
blanket
fissile
c
+Σ
core
Here Σ f and Σ c are the macroscopic fission and capture cross sections, respectively.
The breeding ratio can be divided into two parts as follows.
The external (or blanket) breeding ratio,
EBR =
core leakage
,
fissile
(
)
φ
Σ
+
Σ
dV
∫ f c
core
which can be shown to approximately be equal to,
(7-17)
EBR ≈
Σ core
f
Σ fissile
f
(η − 1) .
The internal (or core) breeding ratio,
∫ φΣ
(7-18)
fertile
c
dV
Σ core
η
f
.
IBR =
≈ fissile
fissile
Σ
ν
(
)
Σ
+
Σ
dV
φ
f
f
c
∫
core
core
Clearly, the total (internal plus external) breeding ratio depends on the neutron
spectrum (since e.g. η depends on the neutron energy) and macroscopic cross
sections. For sodium-cooled fast reactors with dioxides of plutonium-239 (fissile
species) and uranium-238 (fertile species) the total breeding ratio is estimated to be
about 1.2. With carbide (UC and PuC) fuel and blanket material the breeding ratio
should be even larger.
Another figure of merit used to describe the breeding potential is the doubling time,
which is defined as the operating time of a breeder reactor required to produce excess
fissile material equal to the initial quantity in the fuel cycle both inside and outside of
the reactor. The doubling time can be calculated from the following expression[7-3]:
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(7-19)
Td ≈
7
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A S P E C T S
103 M (1 + γ )
days ,
GP(1 + α )(1 − F )
where M is the initial mass of fissile material in the reactor, G is the breeding gain equal
to BR -1, P is the thermal power of the breeder rector in megawatts, F is the fraction
of fissions in fertile nuclides, α is an appropriate average value of Σ c Σ f for all the
fissile species present in the reactor and γM is the mass of fissile material outside of
the reactor during equilibrium operation. In the above formula it is assumed that
fission of 1 kg of any fissile material generates approximately 103 MW of thermal
power.
Td ≈
EXAMPLE 7-2. A 1000 MWe power fast breeder power station operates at a
thermal efficiency of 40% and an average plant capacity factor of 0.75. The initial
reactor inventory is 2600 kg of fissile plutonium and the outside inventory is half
of this amount. Estimate the doubling time assuming BR = 1.2, α = 0.25 and F
= 0.2. SOLUTION: The average thermal power is found as P = 1000*0.75/0.4 =
1.9x103 MW. The doubling time is obtained as
10 3 2.6 ⋅10 3 (1 + 0.5)
days = 28 years .
0.2 ⋅1.9 ⋅10 3 (1 + 0.25)(1 − 0.2)
Due to the desirable breeding potential of fast neutron reactors, a world-wide effort is
undertaken to develop them for commercial use. Starting from 1950 until now about
20 fast neutron reactors have been operating and some have supplied electricity
commercially. This corresponds to over 300 reactor-years of operating experience.
Several fast reactors are in operation, including Phoenix in France (since 1973), FBTR
in India (since 1985), Jojo in Japan (since 1978), BR5/10, BOR 60 and BN600 in
Russia (the last one since 1980). The largest fast reactor for production of electricity Superphenix 1 with 1240 MWe power - was build in France. It operated with major
technical difficulties during 1985-98[7-7].
There are two major types of fast neutron reactors: “burners” – which are net
consumers of plutonium and “breeders” – which produce more plutonium than they
consume. Fast breeding reactors remain the main goal of fast reactor development to
secure long term fuel supply. They can extend the world’s uranium resources by factor
of about 60. This fact makes it economically feasible to utilize ores with very low
uranium concentrations and potentially even uranium found in the oceans[7-4].
Renewed interest in fast breeding reactors has several reasons. In the longer term,
beyond 50 years, uranium resource availability will become a limiting factor unless
breakthrough occurs in mining or extraction technologies. On the one hand the
existing known and speculative economic uranium resources are sufficient to support
thermal reactors with a once-through cycle only until mid-century. Additional limiting
factor facing an essential role for nuclear energy based such reactors is the availability
of the spent-fuel repository space worldwide. On the other hand in the most advanced
fuel cycles using fast spectrum reactors and extensive recycling, it may be possible to
reduce the radiotoxicity of all wastes such that the isolation requirement can be
reduced by several orders of magnitude (e.g. from 300000 to 1000 years) after
discharge from the reactor[7-8].
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The above mentioned issues are addressed in the new development program of
Generation IV reactors. Five out of six Generation IV systems have actinide
management as a mission to address the issues of the deposition of spent nuclear fuel
and high level wastes. These are: Gas-Cooled Fast Reactor (GFR), Lead-Cooled Fast
Reactor (LFR), Molten Salt Reactor (MSR), Supercritical-Water-Cooled Reactor
(SCWR) and Sodium-Cooled Fast Reactor (SFR). Very-High-Temperature Reactor
(VHTR) uses thermal neutron spectrum and once-through system, and is designed for
high-temperature process heat applications, such as the coal gasification and the
thermo-chemical hydrogen production[7-8].
A standing working group within a framework of IAEA – Technical Working Group
on Fast Reactors (TWG-FR) – provides a forum for non-commercial scientific and
technical information and development programs on advances in fast neutron
reactors, [7-9]. The fast reactor database is available on the IAEA website and is
maintained by the International Working Group on Fast Reactors (WGFR)[7-10].
7.6 Environmental Effects of Nuclear Power
The heat values of various fuels are shown in the table below.
TABLE 7.4. Energy conversion: typical heat values of various fuels.
Type of fuel
Heat Value,
MJ/kg
Firewood
16
Brown coal
9
Black coal (low quality)
13-20
Black coal
24-30
Natural gas
39
Crude oil
45-46
Natural uranium in light water
reactor
500 000
The difference in the heat value of uranium compared with coal and other fuels is
important since it directly affects the amount of wastes that each fuel produces. For
instance, 1000 MWe power station consumes about 3.1 million tons of black coal each
year and produces about 7 million tons of waste. A nuclear power plant of equal
power uses 24 tons of uranium UO2 enriched to 4% (this requires mining of over 200
tons of natural uranium or 25000-100000 tons uranium ore) and produces about 700
kg of high-level radioactive waste (after reprocessing of 97% of fuel).
Due to the low rate of wastes produced by nuclear power stations, nuclear industry is
able to take care of all wastes resulting from the electricity production. Using uranium
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as fuel helps to cope with one of the major environmental problems called greenhouse
effect.
In recent years attention has been focused on the climate change effects of burning
fossil fuels, especially coal, due to the carbon dioxide which this releases into the
atmosphere.
Carbon dioxide contributes about half of the human-induced increase in the
greenhouse effect. Electricity generation is one of the major sources of this carbon
dioxide, giving rise to about one quarter of it, or some 9% of the human-induced
greenhouse increase.
Coal-fired electricity generation gives rise to nearly twice as much carbon dioxide as
natural gas per unit of power, but hydro and nuclear do not directly contribute any. If
the entire world's nuclear power were replaced by coal-fired power, emissions from
electricity generation would rise by a third; that is 2400 million tons of carbon dioxide
would be released into the atmosphere. This can be compared with the target of a 5%
reduction – or 600 million tons per year – in carbon dioxide by the year 2010, as
agreed in 1997 at Kyoto just for the developed countries. FIGURE 7-8 shows
greenhouse gas emissions for various fuel types used for the electricity production.
FIGURE 7-8: Greenhouse gas emissions from electricity production.
Conversely, there is scope for reducing coal's carbon dioxide contribution to the
greenhouse effect by substituting natural gas or nuclear power, and by increasing the
efficiency of coal-fired generation itself, a process which is well under way. Nuclear
power is well suited to meeting the demand for continuous, reliable supply on a large
scale (i.e. base-load electricity), the major part of demand.
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7.7 Economic Aspects of Nuclear Power
Estimation of costs is one of the most important tasks while considering a
construction of a new nuclear power plant. A legitimate question is if the electrical
energy obtained from a nuclear power plant is cheaper than that obtained from other
energy sources. The answer to this question is neither clear nor simple. One of the
reasons for this difficulty is the fact that the costs of energy production vary from
country to country and depend on many specific conditions. Due to that the aboveposed question does not have a general answer and the costs of electricity must be
estimated on the case-by-case basis.
In evaluation of total costs, the following part costs must be taken into account:
•
investment cost (project and construction)
•
operation and maintenance costs
•
fuel costs
•
plant decommissioning costs
For any country in the world, a construction of a nuclear power plant is a serious
undertaking. This is particularly true for countries that invest into the first nuclear
power plant. Typically such undertaking engages many parties: governments,
authorities, industries, universities, research centers and – last but not least – the
general public. To carry out such a complex process and evaluate their costs some
standards must be followed. IAEA published a technical report on Economic Evaluation
of Bids for Nuclear Power Plants (Technical Report Series No. 396, IAEA, Vienna, 2000).
The purpose of this guidebook is to facilitate economic and financial bid evaluation.
Thus, many economic aspects of nuclear power are treated there in a great detail.
When calculating costs of the electricity production, it is necessary to take into account
the time aspect, since both the costs and the produced (and sold) electricity are
distributed over time. Simplified cost estimation can be obtained from the following
expression,
∑
(7-20)
C=
I t + M t + Ft + Dt
(1 + d )t
t
Et
∑ (1 + d )
,
t
t
where:
C
- discounted electricity costs in €/kWe
It
- investment cost at time t in €
Mt
- operation and maintenance cost at time t in €
Ft
- fuel cost at time t in €
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Dt
- decommissioning cost at time t in €
Et
- energy production at time t in kWe
d
- discount rate.
E C O N O M I C
A S P E C T S
The costs obtained from Eq. (7-20) for various energy sources can be compared to
each other. Usually the comparisons are performed with two reference values of the
discount rate: 5 and 10%. Typical comparisons are done for nuclear power plants
against coal and natural gas fired power plants. In general, the higher discount rate is in
favor of fossil-fired power plants, since their investment costs are substantially lower
than those of nuclear power plants. Nevertheless, the influence of local conditions
(country) seems to be more significant than the influence of the discount rate. TABLE
7.5 shows the estimated cost ratio of electricity produced in nuclear power plant versus
fossil-fired plants for various countries.
TABLE 7.5. Comparison of unit costs of electricity production in NPP compared with fossil-fired plants
in selected countries.
Country /
Region
Electricity cost from NPP Electricity cost from NPP
(100% - coal-fired plant)
(100% - natural gas-fired
plant)
d = 5%
d = 10%
d = 5%
d = 10%
Belgium
88%
110%
90%
118%
Czech Republic
& Slovakia
88%
98%
79%
100%
Finland
80%
109%
80%
120%
France
60%
80%
60%
80%
Hungary
70%
85%
80%
110%
Japan
85%
95%
70%
95%
UK
100%
130%
110%
170%
Central Canada
85%
120%
60%
92%
Middle West
USA
95%
100%
90%
120%
As can be seen, the cost of electricity from nuclear power plants varies from 60 to
170%. For some countries (France, Japan, Czech Republic and Slovakia) the nuclear
power is the only economically motivated choice. Only for UK the fossil-fired plants
can deliver cheaper electricity.
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It should be mentioned that in the comparison presented in TABLE 7.5 the external
costs of the production of electricity are not taken into account. Such costs (for
example the cost of the emission of greenhouse gases) are significant and their
inclusion into the comparison will make the nuclear power even more profitable
source of electricity.
R E F E R E N C E S
[7-1]
Beckjord, E. Ex. Dir. et al., The Future of Nuclear Power. An Interdisciplinary MIT Study, MIT 2003,
ISBN 0-615-12420-8
[7-2]
Duderstadt, J.J. and Hamilton, L.J., Nuclear Reactor Analysis, John Wiley & Sons, Inc.
[7-3]
Glasstone, S. and Sesonske, A., Nuclear Reactor Engineering, Van Nostrand Reinhold Compant,
1981, ISBN 0-442-20057-9.
[7-4]
Seko, N. “Aquaculture of Uranium in Seawater by a Fabric-Adsorbent Submerged System,”
Nuclear Technology, 144, 274 (Nov. 2003).
[7-5]
Uranium 2005: Resources, Production and Demand, A Joint Report by the OECD Nuclear
Energy Agency and the International Atomic Energy Agency (“Red Book”, 21st edition).
[7-6]
http://www.world-nuclear.org/info/reactors.htm
[7-7]
http://www.world-nuclear.org/info/inf98.htm
[7-8]
http://gif.inel.gov/roadmap/pdfs/gen_iv_roadmap.pdf
[7-9]
http://www.iaea.org/inis/aws/fnss/twgfr/index.html
[7-10]
http://www-frdb.iaea.org/index.html
E X E R C I S E S
EXERCISE 7-1: A core of a pressurized water reactor has a cylindrical shape with height H = 3.66 m
and diameter D = 3.37 m. The mean neutron flux in the core is equal to 2.87x1017 neutron/m2.s and the
mean macroscopic cross section for fission is equal to 10.88 1/m. Calculate the required initial weight
enrichment of the UO2 fuel if the designed burnup is 33 GWd/tU and 1/3 of the core will be refueled
each year.
EXERCISE 7-2: For the same reactor as described in previous exercise calculated the needed mass of
the enriched and the natural uranium assuming that the plant capacity factor is 0.9.
EXERCISE 7-3: For the same reactor as described in EXERCISE 7-1 calculate SWU and the cost of
enrichment assuming xT = 0.3w/o and a unit cost of SWU equal to 45€.
190
Appendix
A
Appendix A - Bessel
Functions
The Bessel differential equation is as follows:
x2
d2y
dy
+x
+ (x 2 − n 2 )y = 0 , n>= 0.
2
dx
dx
(A-1)
Its general solution can be found as,
y ( x ) = C1 J n ( x ) + C 2 N n (x )
(A-2)
where Jn(x): is known as the Bessel function of the first kind:
2k
 x

n ∞ (− 1) 
2
x

J n (x ) =   ∑
 2  k =1 k! (n + k )!
k
(A-3)
and Nn(x) is the Bessel function of the second kind (called also the Neumann function)
given as,
N n (x ) =
J n (x ) cos nπ − J −n (x )
sin nπ
(A-4)
For small x, the functions have an asymptotic values described with:
J n ( x) ≈
1 n
x ,
2 n!
n
N n (x ) ≈ 2 n −1 (n − 1)! x − n ,
n=/ 0
(A-5)
For large x they are approximately described with:
J n ( x) ≈
2
π nπ 

cos x − −
,
πx
4
2 

N n (x ) =
π
2x
e−x
(A-6)
The plot of the Bessel function of the first kind and of the zero order (n = 0) is shown
in the Figure below.
191
A
J0(x)
A P P E N D I X
1,1
1
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
-0,1
-0,2
-0,3
-0,4
-0,5
0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 8,5 9 9,5 10
x
FIG. A-1. Plot of J0(x) function.
Derivatives of the Bessel functions are calculated as follows,
dZ n (αx )
n
= αZ n −1 (αx ) − Z n (αx ), Z = J , N
dx
x
(A-7)
Selected useful integrals of the Bessel functions are as follows,
∫ xZ (x )dx = xZ (x ), ∫ Z (x )dx = − Z (x ) .
0
1
1
0
192
(A-8)
Appendix
B
Appendix B - Selected
Nuclear Data
Data for fissionable and fertile isotopes and for thermal neutrons with v0 = 2200 m/s
Isotope
σ a [b]
σ f [b]
σ s [b]
ν
α
η
T1 2
233U
581
527
10
2.492
0.0899
2.287
1.6 105 y
235U
694
582
10
2.418
0.169
2.068
7.1 108 y
238U
2.71
0.0005
10
-
-
-
4.5 109 y
239U
-
15
-
-
-
-
23.5 m
U-nat
7.68
4.18
8.3
2.418
0.811
1.335
-
232Th
7.65
0.0002
12.6
-
-
-
1.45 1010 y
237Np
170
0.020
239Pu
1026
746
9.6
2.871
0.362
2.108
2.44 104 y
241Pu
1377
1009
-
2.927
0.365
2.145
13.2 y
2.2 106 y
193
Appendix
C
Appendix C – Cumulative
Standard Normal
Distribution
Values of 1 − Φ( z ) where Φ( z ) = p(Z ≤ z ) =
z
1
2π
z
∫e
−ξ 2
2
dξ
−∞
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.0
0.500000
0.496011
0.492022
0.488034
0.484047
0.480061
0.476078
0.472097
0.468119
0.464144
0.1
0.460172
0.456205
0.452242
0.448283
0.444330
0.440382
0.436441
0.432505
0.428576
0.424655
0.2
0.420740
0.416834
0.412936
0.409046
0.405165
0.401294
0.397432
0.393580
0.389739
0.385908
0.3
0.382089
0.378280
0.374484
0.370700
0.366928
0.363169
0.359424
0.355691
0.351973
0.348268
0.4
0.344578
0.340903
0.337243
0.333598
0.329969
0.326355
0.322758
0.319178
0.315614
0.312067
0.5
0.308538
0.305026
0.301532
0.298056
0.294599
0.291160
0.287740
0.284339
0.280957
0.277595
0.6
0.274253
0.270931
0.267629
0.264347
0.261086
0.257846
0.254627
0.251429
0.248252
0.245097
0.7
0.241964
0.238852
0.235762
0.232695
0.229650
0.226627
0.223627
0.220650
0.217695
0.214764
0.8
0.211855
0.208970
0.206108
0.203269
0.200454
0.197663
0.194895
0.192150
0.189430
0.186733
0.9
0.184060
0.181411
0.178786
0.176186
0.173609
0.171056
0.168528
0.166023
0.163543
0.161087
1.0
0.158655
0.156248
0.153864
0.151505
0.149170
0.146859
0.144572
0.142310
0.140071
0.137857
1.1
0.135666
0.133500
0.131357
0.129238
0.127143
0.125072
0.123024
0.121000
0.119000
0.117023
1.2
0.115070
0.113139
0.111232
0.109349
0.107488
0.105650
0.103835
0.102042
0.100273
0.098525
1.3
0.096800
0.095098
0.093418
0.091759
0.090123
0.088508
0.086915
0.085343
0.083793
0.082264
1.4
0.080757
0.079270
0.077804
0.076359
0.074934
0.073529
0.072145
0.070781
0.069437
0.068112
1.5
0.066807
0.065522
0.064255
0.063008
0.061780
0.060571
0.059380
0.058208
0.057053
0.055917
1.6
0.054799
0.053699
0.052616
0.051551
0.050503
0.049471
0.048457
0.047460
0.046479
0.045514
1.7
0.044565
0.043633
0.042716
0.041815
0.040930
0.040059
0.039204
0.038364
0.037538
0.036727
1.8
0.035930
0.035148
0.034380
0.033625
0.032884
0.032157
0.031443
0.030742
0.030054
0.029379
1.9
0.028717
0.028067
0.027429
0.026803
0.026190
0.025588
0.024998
0.024419
0.023852
0.023295
2.0
0.022750
0.022216
0.021692
0.021178
0.020675
0.020182
0.019699
0.019226
0.018763
0.018309
2.1
0.017864
0.017429
0.017003
0.016586
0.016177
0.015778
0.015386
0.015003
0.014629
0.014262
2.2
0.013903
0.013553
0.013209
0.012874
0.012545
0.012224
0.011911
0.011604
0.011304
0.011011
2.3
0.010724
0.010444
0.01017
0.009903
0.009642
0.009387
0.009137
0.008894
0.008656
0.008424
2.4
0.008198
0.007976
0.007760
0.007549
0.007344
0.007143
0.006947
0.006756
0.006569
0.006387
2.5
0.006210
0.006037
0.005868
0.005703
0.005543
0.005386
0.005234
0.005085
0.00494
0.004799
2.6
0.004661
0.004527
0.004396
0.004269
0.004145
0.004025
0.003907
0.003793
0.003681
0.003573
2.7
0.003467
0.003364
0.003264
0.003167
0.003072
0.002980
0.002890
0.002803
0.002718
0.002635
2.8
0.002555
0.002477
0.002401
0.002327
0.002256
0.002186
0.002118
0.002052
0.001988
0.001926
2.9
0.001866
0.001807
0.001750
0.001695
0.001641
0.001589
0.001538
0.001489
0.001441
0.001395
3.0
0.001350
0.001306
0.001264
0.001223
0.001183
0.001144
0.001107
0.00107
0.001035
0.001001
3.1
0.000968
0.000935
0.000904
0.000874
0.000845
0.000816
0.000789
0.000762
0.000736
0.000711
195
A P P E N D I X
C
3.2
0.000687
0.000664
0.000641
0.000619
0.000598
0.000577
0.000557
0.000538
0.000519
0.000501
3.3
0.000483
0.000466
0.000450
0.000434
0.000419
0.000404
0.000390
0.000376
0.000362
0.000349
3.4
0.000337
0.000325
0.000313
0.000302
0.000291
0.000280
0.000270
0.000260
0.000251
0.000242
3.5
0.000233
0.000224
0.000216
0.000208
0.000200
0.000193
0.000185
0.000178
0.000172
0.000165
3.6
0.000159
0.000153
0.000147
0.000142
0.000136
0.000131
0.000126
0.000121
0.000117
0.000112
3.7
0.000108
0.000104
9.96E-05
9.57E-05
9.20E-05
8.84E-05
8.50E-05
8.16E-05
7.84E-05
7.53E-05
3.8
7.23E-05
6.95E-05
6.67E-05
6.41E-05
6.15E-05
5.91E-05
5.67E-05
5.44E-05
5.22E-05
5.01E-05
3.9
4.81E-05
4.61E-05
4.43E-05
4.25E-05
4.07E-05
3.91E-05
3.75E-05
3.59E-05
3.45E-05
3.30E-05
4.0
3.17E-05
3.04E-05
2.91E-05
2.79E-05
2.67E-05
2.56E-05
2.45E-05
2.35E-05
2.25E-05
2.16E-05
4.1
2.07E-05
1.98E-05
1.89E-05
1.81E-05
1.74E-05
1.66E-05
1.59E-05
1.52E-05
1.46E-05
1.39E-05
4.2
1.33E-05
1.28E-05
1.22E-05
1.17E-05
1.12E-05
1.07E-05
1.02E-05
9.77E-06
9.34E-06
8.93E-06
4.3
8.54E-06
8.16E-06
7.80E-06
7.46E-06
7.12E-06
6.81E-06
6.50E-06
6.21E-06
5.93E-06
5.67E-06
4.4
5.41E-06
5.17E-06
4.94E-06
4.71E-06
4.50E-06
4.29E-06
4.10E-06
3.91E-06
3.73E-06
3.56E-06
4.5
3.40E-06
3.24E-06
3.09E-06
2.95E-06
2.81E-06
2.68E-06
2.56E-06
2.44E-06
2.32E-06
2.22E-06
4.6
2.11E-06
2.01E-06
1.92E-06
1.83E-06
1.74E-06
1.66E-06
1.58E-06
1.51E-06
1.43E-06
1.37E-06
4.7
1.30E-06
1.24E-06
1.18E-06
1.12E-06
1.07E-06
1.02E-06
9.68E-07
9.21E-07
8.76E-07
8.34E-07
4.8
7.93E-07
7.55E-07
7.18E-07
6.83E-07
6.49E-07
6.17E-07
5.87E-07
5.58E-07
5.30E-07
5.04E-07
4.9
4.79E-07
4.55E-07
4.33E-07
4.11E-07
3.91E-07
3.71E-07
3.52E-07
3.35E-07
3.18E-07
3.02E-07
196
INDEX
Advanced Gas-Cooled Reactor .............. 33
Energy spectrum of prompt neutrons.... 17
Alpha particles......................................... 9
Engineering heat flux hot-channel factor
Atomic mass units ................................... 5
............................................... 150
Atomic number ........................................ 6
Enthalpy-rise hot channel factor ......... 150
Austenitic stainless steels .................. 127
External (or blanket) breeding ratio .... 185
Average cosine of the scattering angle 18
Extrapolated boundary .......................... 49
Average logarithmic energy decrement 19
Fast fission factor.................................. 61
Avogadro number................................... 14
Fast non-leakage probability ................. 64
Axial nuclear hot channel factor ......... 149
Fast reactors ......................................... 27
Barn
13
Fertile nuclides ...................................... 16
Becquerel ............................................... 11
Fick’s law of diffusion............................ 46
Beta particles .......................................... 9
Fissile nuclides ...................................... 16
Binding energy ......................................... 7
Fission
Black absorber....................................... 38
Fluence
Blanket assemblies ............................. 172
Four factor formula ................................ 60
boiling length ....................................... 123
Fuel assembly ........................................ 38
Boiling Water Reactor............................ 31
Fuel Doppler reactivity coefficient........ 79
Boiling Water Reactors .......................... 28
Fuel temperature coefficient................. 79
Boundary value problem ........................ 52
Gamma rays ............................................. 9
Breeding ratio ...................................... 185
Gas-cooled reactors .............................. 28
Bulk temperature ................................... 97
Geometric buckling ............................... 53
Burnable poisons ................................... 66
Geometrical cross section of the nucleus
15
139
CANDU reactor ...................................... 31
................................................. 14
Chemical shim ....................................... 66
Graphite-moderated reactors ................ 28
Closed fuel cycle ................................. 171
Gray
Control rod ............................................. 38
Grey absorber ........................................ 38
Conversion ratio .................................. 184
Half-life of the radioactive species ....... 10
Critical Heat Flux................................. 119
Heat flux hot channel factor ............... 149
Critical Power Ratio............................. 152
Heat Transfer Deterioration ................ 115
Criticality condition ............................... 52
Heavy Water Reactors ........................... 28
Cross-section ......................................... 12
High Temperature Gas Cooled Reactor 34
Curie
11
High Temperature Gas-cooled Reactors 28
Decay constant ...................................... 10
Homogeneous Equilibrium Model ........ 108
Decay heat ............................................. 92
Hot assembly ....................................... 148
Decay ratio............................................. 83
Hot channel.......................................... 148
Delayed neutrons ................................... 16
Hot spot 148
Departure from Nucleate Boiling ......... 119
Inferred Uranium Resources ............... 166
Departure from Nucleate Boiling Ratio 152
Infinite multiplication factor ................. 60
Displacements per atom...................... 139
Intergranular corrosion ....................... 140
Dittus-Boelter correlation .................... 112
Internal (or core) breeding ratio .......... 185
DNB correlation ................................... 120
Isotopes
Doppler effect .................................. 62, 76
Light Water Reactors............................. 28
Dpa
Light-element moderated Reactors ...... 28
See displacements per atom
11
6
Drift velocity ........................................ 109
Liquid Metal Fast Breeder Reactor ....... 33
Drift-flux distribution parameter.......... 109
Liquid-metal cooled reactors ................ 28
Drift-Flux Model ................................... 109
Lumped fission product poisons ........... 72
Dryout
119
Macroscopic cross section ................... 13
Dryout correlation................................ 123
Mass defect ............................................. 7
Effective multiplication factor .............. 64
Mass number............................................ 6
Effective resonance integral ................. 62
Material buckling ..............................49, 53
Eigenfunctions ....................................... 52
Mean life of the radioactive species ..... 10
Eigenvalues............................................ 52
Mean-weighted logarithmic energy
Electron volt ............................................ 8
decrement ............................... 19
Emergency Core Cooling System .......... 26
Microscopic cross section .................... 13
197
I N D E X
Mixed oxide (MOX) fuel ........................ 181
Regulating rods ...................................... 39
Moderating power .................................. 19
Rem
Moderating ratio .................................... 20
Reproduction factor ............................... 63
Neutron
12
5
Resonance escape probability .............. 61
Neutron current density......................... 45
Roentgen ................................................ 11
Neutron diffusion coefficient ................. 46
Safety limit ........................................... 150
Neutron diffusion equation .................... 47
Safety margin ....................................... 150
Neutron diffusion length ........................ 48
Safety rods ............................................. 39
Neutron flux ........................................... 45
Scattering mean free path ..................... 47
Neutron generation time........................ 73
Secondary stress ................................. 162
Neutron lifetime ..................................... 74
Seed fuel assemblies ........................... 172
Non-burnable poison .............................. 66
Separative work units.......................... 174
Nuclear fission ......................................... 8
Shim rods ............................................... 39
Nuclear hot channel ............................ 148
Sivert
Nucleus
5
Six-factor formula .................................. 64
Offset yield point ................................. 135
Soluble poisons ...................................... 66
Onset of Significant Void fraction ....... 110
Stainless steel ..................................... 127
Open fuel cycle .................................... 171
Stress corrosion cracking ................... 140
Organically Moderated Reactors ........... 28
Stress intensity .................................... 162
Peak Clad Temperature ....................... 151
Supercritical water heat transfer ........ 115
Pebble Bed Modular Reactor ................. 35
Target nucleus ....................................... 12
Percent milli rho - pcm........................... 77
Thermal non-leakage probability ........... 64
Pressurized Heavy Water Reactor ......... 31
Thermal power of a reactor ................... 90
Pressurized water-cooled reactor ......... 30
Thermal reactors ................................... 27
Primary stress ...................................... 162
Thermal stress ..................................... 162
Prompt neutrons .................................... 16
Thermal utilization factor ...................... 62
Prompt reactivity coefficient................. 79
Total nuclear hot channel factor ......... 149
Proton
5
Total power peaking factor ................. 149
11
Transport cross section ........................ 46
Radial nuclear hot channel factor ....... 148
Transport mean free path ...................... 46
RBMK reactor......................................... 31
Ultimate tensile strength .................... 135
Reactor Pressure Vessel ....................... 37
Unconventional uranium resources .... 167
Rad
12
Reasonably Assured Resources ............ 66
Water-moderated reactors..................... 27
Reasonably Assured Uranium Resources
Xenon oscillations ................................. 70
............................................... 166
X-rays
9
Recoil nucleus ....................................... 12
Yellowcake .......................................... 172
Reflector savings ................................... 59
Yield stress .......................................... 135
198
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