Lectures on Applied Reactor Technology and Nuclear Power Safety
Lecture No 1
Title:
Neutron Life Cycle
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology
KTH
Spring 2005
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 1
Outline of the Lecture
• Infinite Multiplication Factor, k∞
• Four Factor Formula
–
–
–
–
Fast Fission Factor, ε
Resonance Escape Probability, p
Thermal Utilization Factor, f
Reproduction Factor, η
• Effective Multiplication Factor, keff
– Fast Non-Leakage Probability, Pfnl
– Thermal Non-Leakage Probability, Ptnl
• Six Factor Formula
• Neutron Life Cycle of a Fast Reactor
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 2
Introduction
• We start with a short overview of some introductory
topics, which are related to today’s lecture:
–
–
–
–
–
Neutron reactions
Cross-section for neutron reactions
Neutron absorption
Nuclear fission
Neutron moderation (or slowing down)
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 3
Neutron reactions (1)
• 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
– 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
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 4
Neutron reactions (2)
• Inelastic scattering
– When a neutron undergoes inelastic scattering, it is first captured
by the target nucleus to for an excited state of the compound
nucleus
– A neutron of lower kinetic energy is then emitted, leaving the
target nucleus in an excited state
– This excess energy is subsequently emitted as one or two
photons of gamma radiation
• In general, inelastic scattering is limited to some high or
low (depending on target mass) neutron energy ranges
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 5
Neutron reactions (3)
• Elastic scattering
– In elastic scattering the kinetic energy is conserved that is the
total kinetic energy of neutron and the target nucleus is
unchanged due to collision
• There are two types of elastic scattering:
– Potential scattering – resembling collision of two billiard balls
– Resonance or compound nucleus scattering (when target
nucleus is larger) – where neutron is first absorbed and next
expelled from nucleus, leaving the target nucleus in the ground
state
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 6
Neutron reactions (4)
• In considering absorption reactions it is convenient to
distinguish between reactions 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:
–
–
–
–
The emission of gamma radiation (n,γ) – radiative capture
The ejection of an alpha particle (n,α)
The ejection of an proton (n,p)
Fission (n,f)
• Of these the radiative capture is most common, whereas
others are limited to few species.
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 7
Cross-section for neutron reactions (1)
• 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
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 8
Cross-section for neutron reactions (2)
•Suppose a uniform, parallel beam of I
mono-energetic 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 to the right)
II
•The nuclear cross section for a specified
reaction is then defined as
NR
σ=
m 2 / nucleus
(Nδx )I
δx
Cross-section unit: 1 b (barn) =
10-28 m2/nucleus
•Where NR is the number of reactions
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 9
Cross-section for neutron reactions (3)
Total cross section of uranium-238
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 10
Cross-section for neutron reactions (4)
Total cross section of uranium-235
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 11
Cross-section for neutron reactions (5)
Fission cross section of uranium-235
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 12
Cross-section for neutron reactions (6)
• 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
Σ = Nσ m −1
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 13
Cross-section for neutron reactions (7)
• 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
10 3 ρN A
N=
A
• The macroscopic cross section is thus
10 3 ρN A
σ
Σ=
A
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 14
Cross-section for neutron reactions (8)
• 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
10 3 ρN A
Ni =
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 …
and for compound …
10 3 ρN A
Σ i = N iσ i =
ν iσ i
M
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
103 ρN A
(ν 1σ 1 +ν 2σ 2 + L)
Σ=
M
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 15
Nuclear fission (1)
• Fission 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
• 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
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 16
Nuclear fission (2)
• Example of a nuclear fission of U-235
• Each fission of U-235 is followed by a release of 2 or 3
neutrons (2.42 on average for fission of U-235 with
thermal neutrons)
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 17
Nuclear fission (3)
•
Chain reaction will
sustain if there is
enough neutrons to
cause fissions in
coming generations
•
If fission is caused by
slow (thermal)
neutrons, they have to
be moderated (slowed
down) before the next
fission
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 18
Nuclear fission (4)
• Approximate distribution of energy per fission of 235U:
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
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 19
Nuclear fission (5)
•
The neutrons released in fission are of two categories:
– Prompt neutrons (over 99% of all neutrons are prompt) released within
10-14 s
– Delayed neutrons emitted during several minutes after fission
•
The average number of neutrons liberated in fission is designed
as . Typical values are shown below
ν
Uranium-233
Uranium-235
Plutonium-239
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
thermal neutrons
2.49
2.42
2.93
Fast neutrons
2.58
2.51
3.04
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 20
Nuclear fission (6)
• Neutrons released after
fission do not have the
same energy
• Typical energy
spectrum is shown in
figure
• Most neutrons have
energy between 1 and
2 MeV
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 21
Nuclear fission (7)
• Fission of uranium235 can end up with
80 different primary
fission products
• The range of mass
numbers of products
is from 72 to 161
• Figure shows fission
yield vs mass number
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 22
Nuclear fission (8)
• The fission cross section of uranium-235 is very low for
fast (high-energy) neutrons
• To increase the fission rate, it is necessary to slow-down
neutrons
• This subject is treated in the section that follows
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 23
Slowing down of neutrons (1)
• After fission, neutrons move in all directions with speed
up to 50000 km/s
• Neutrons can not move a longer time with such high
speeds, which successively goes down due to collisions
with nuclei; this process is called scattering
• After short period of time the neutron velocities approach
the equilibrium velocity, which is about 2200 m/s at
20 °C
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 24
Slowing down of neutrons (2)
Collision in laboratory system
• Neutron scattering can be
either:
– Elastic (in most cases)
– Inelastic
• Classical dynamics law
are used to describe the
elestic scattering process
• Laboratory or Center-ofmass systems are used
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
V2
Nucleus
before
V1
Neutron
after
ψ
Neutron
before
A
Nucleus
after
Collision in a center-of-mass system
V2
Centrer of mass
θ
V1-v m
Neutron
before
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Nucleus
after
Neutron
after
vm
Nucleus
before
Slide No 25
Slowing down of neutrons (3)
• It can be shown that after collision, the minimum value
of energy to which neutron can be reduced is
αE1
– Where E1 is the neutron energy before the collision, and
 A −1
α =

 A +1
2
• The maximum energy of neutron after collision is E1
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 26
Slowing down of neutrons (4)
• For hydrogen A = 1 and so
α =0
• And it is possible for neutron to lose all energy in one
collision
• For carbon A = 12 and
α = 0.716
• The minimum energy of neutron after collision will be
equal to 71.6% of that before the collision
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 27
Slowing down of neutrons (5)
• 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
• It 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
Here θ is a collision angle in
E1
ln d (cos θ )
the C system; integration
∫
E1 −1 E2
means averaging over all
ξ ≡ ln
=
possible collision angles
E2
d (cos θ )
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 28
Slowing down of neutrons (6)
• Analyzing energy change in scattering, the ratio E1/E2
can be expressed in terms of mass number A and
cosine of the collision angle cosθ
• Using this in the equation in the previous slide, the
following is obtained
2
(
A − 1)
A −1
ln
ξ = 1+
2A
A +1
• It can be seen that for Hydrogen (A=1) ξ=1 and for
uranium-238 ξ=0.0083
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 29
Slowing down of neutrons (7)
• If the moderator is not a single element, but a
compound containing n different nuclei, the effective or
mean (weighted) value of ξ is given by
σ s1ξ1 + σ s 2ξ 2 + ... + σ snξn
ξ =
σ s1 + σ s 2 + ... + σ sn
• For example for H2O we get
ξH O =
2
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
2σ s ( H )ξ H + σ s ( O )ξ(O )
2σ s ( H ) + σ s ( O )
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 30
Slowing down of neutrons (8)
•
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 = 14.4/ξ
One also defines the moderating or slowing down power of a
material as: ξΣs
•
However, this parameter is not sufficient to describe how good a
given material is as a moderator, since one also whishes the
moderator to be a week absorber of neutrons
•
That is why one use moderating ratio = ξΣs/ Σa as a figure of merit
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 31
Infinite Multiplication Factor (1)
• Not all neutrons produced by fission will cause new
fission:
– Some will be absorbed by non-fissionable material
– Some will be absorbed parasitically in fissionable material
– Others 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
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 32
Infinite Multiplication Factor (2)
• The condition of a self-sustaining 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
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 33
Infinite Multiplication Factor (3)
• Any reactor of a finite size will have neutrons leak out of
it
• Generally, the larger the reactor the lower the fraction
the neutron leakage
• For simplicity consider a reactor that is infinitely large,
and therefore has no neutron leakage
• A measure of the increase or decrease in neutron flux in
an infinite reactor is the infinite multiplication factor k∞
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 34
Infinite Multiplication Factor (4)
• The infinite multiplication factor is the ratio of the
neutrons produced by fission in one generation to the
number of neutrons lost through absorption in the
preceding generation:
Neutron production from fission in one generation
k∞ =
Neutron absorption in the preceding generation
or
Rate of neutron production
k∞ =
Rate of neutron absorption
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 35
Infinite Multiplication Factor (5)
• 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 an extraneous source
• In other words, the requirement for criticality is:
k∞ = 1
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 36
Four Factor Formula (1)
• For some thermal reactors, the infinite multiplication
factor k∞ 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:
– Fission neutrons with energies in excess of about 1MeV which
can cause fission in uranium-238 as well as in uranium-235
– Neutrons in the resonance region which may be captured by
uranium-238
– Thermal neutrons which cause nearly all the fission in uranium235 and thereby generate fission neutrons
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 37
Four Factor Formula (2)
• A group of fast neutrons can enter into several reactions
• Some of these reactions reduce the size of the neutron
group while other reactions allow the group to increase
in size or produce a second generation
• There are four factors that give the inherent
multiplication ability of the fuel and moderator materials:
k∞ = ε ⋅ p ⋅ f ⋅η
where : ε
p
f
η
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Fast fission factor
=
= Resonance escape probability
=
=
Thermal utilization factor
Reproduction factor
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 38
Fast Fission Factor (1)
• The first process that the neutrons of one generation
may undergo is fast fission
• Fast fission is fission caused be neutrons that are in the
fast energy range
• In a thermal reactor using slightly enriched or natural
uranium fuel, some neutrons, before they have been
slowed down appreciably, will cause fission of both
uranium-235 and uranium-238 nuclei
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 39
Fast Fission Factor (2)
• 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 cross-section for fast fission in uranium-235 and
uranium-238 is small
• The fast neutron population in one generation is thus
increased by a factor called the fast fission factor
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 40
Fast Fission Factor (3)
• 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
Number of fast neutrons produced by all fissions
ε=
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
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 41
Fast Fission Factor (4)
•
The value of ε will be affected by the arrangement and
concentrations of the fuel and the moderator
•
The value of ε is essentially 1.00 for a homogenous 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 neutrons emitted from one fission can pass close to another
fuel atom
•
The arrangement in heterogeneous reactors results in ε~1.02 – 1.08
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 42
Resonance Escape Probability (1)
• As already discussed, neutrons increase in number as a
result of fast fission
• After that neutrons continue to diffuse through the
reactor
• As the neutrons move they collide with nuclei of fuel and
non-fuel material and moderator in the reactor loosing
part of their energy
• While they are slowing down there is a chance that some
neutrons will be captured by uranium-238 nuclei
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 43
Resonance Escape Probability (2)
• 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
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 44
Resonance Escape Probability (3)
• 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
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
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 45
Resonance Escape Probability (4)
• 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
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 46
Resonance Escape Probability (5)
• The value of the resonance escape probability is not
significantly affected by pressure or poison concentration
• In water moderated, low uranium-235 enrichment
reactors, raising the temperature of the fuel will rise 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
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 47
Resonance Escape Probability (6)
• The resonance escape probability can be found from the
following formula:
 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
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 48
Resonance Escape Probability (7)
• Experimental measurements of the resonance integral
for a system of isolated rods give the following formula:
A
I =a+b
M
where a and b are constants for a given fuel material (=
2.95 and 81.5 resp for uranium and 4.45 and 84.5 for
uranium dioxide, such that I will be in barns), A is the
area (in m2) and M is the mass (in kg) of a fuel rod
I uranium
A
A
[b]; IUO2 = 4.45 + 84.5
[b]
= 2.95 + 81.5
M
M
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 49
Resonance Escape Probability (8)
• The integral I depends on temperature as follows:
[
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
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 50
Resonance Escape Probability (7)
• Example:
– Calculate the resonance integral I for fuel rods containing UO2
(density 10200 kg m-3); the rod diameter is 8 mm.
• Solution:
– For 1 m long fuel rod its surface area is equal to A =1* πD =
0.0251 m2 and its mass is M = 1* πD2/4*ρ = π*0.0082/4*10200 =
0.513 kg. Thus the integral I is obtained as
0.0251
I = 4.45 + 84.5
= 23.14... b
0.513
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 51
Resonance Escape Probability (8)
• 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
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 52
Thermal Utilization Factor (1)
• 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:
Number of thermal neutrons absorbed in the fuel
f =
Number of thermal neutrons absorbed in all reactor materials
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 53
Thermal Utilization Factor (2)
•
The thermal utilization factor can be expressed as follows:
ΣUa ⋅ φ U ⋅ V U
f = U U U
Σ a ⋅ φ ⋅ V + Σ ma ⋅ φ m ⋅ V m + Σ ca ⋅ φ c ⋅ V c + Σ ap ⋅ φ p ⋅ V p
where superscripts U, 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
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 54
Thermal Utilization Factor (3)
• 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
ΣUa
f = U
Σ a + Σ ma + Σ ap + Σ ca
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 55
Thermal Utilization Factor (4)
• 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 this
results in increase of thermal utilization
• Because of this effect the temperature coefficient for the
thermal utilization factor is positive
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 56
Reproduction Factor (1)
• 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
Number of fast neutrons produced by thermal fission
η=
Number of thermal neutrons absorbed in the fuel
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 57
Reproduction Factor (2)
• 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 =
fission reaction rate ( ΣUf ⋅ φ U ) Χ the average number of neutrons
produced per fission ( ν )
– The rate of absorption of thermal neutrons by the fuel ( ΣUa ⋅ φ U )
η=
ΣUf ⋅ φ U ⋅ν
Σ ⋅φ
U
a
U
=ν
ΣUf
ΣUa
• When fuel contains several fissionable materials, it is
necessary to account for each material, e.g.
η=
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
N U −235 ⋅ σ Uf −235 ⋅ν
N U −235 ⋅ σ aU −235 + N U −238 ⋅ σ aU −238
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 58
Effective Multiplication Factor (1)
• 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
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 59
Effective Multiplication Factor (2)
• 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
• keff = k∞ x PFNL x PTNL, where PFNL is the fast nonleakage probability and PFNL is the thermal nonleakage probability
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 60
Fast Non-Leakage Probability (1)
• In a realistic reactor of finite size some of the fast
neutrons leak out of the boundaries of the reactor core
before they begin the slowing down process
• The fast non-leakage probability PFNL is defined as the
ratio of the number of fast neutrons that do not leak from
the reactor to the number of fast neutrons produced by
all fissions
PFNL
Number of fast neutrons that do not leak from reactor
=
Number of fast neutrons produced by all fissions
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 61
Thermal Non-Leakage Probability (1)
• Neutrons can also leak out of a finite reactor after they
reach thermal energies
• The thermal non-leakage probability is defined as the
ratio of the number of thermal neutrons that do not leak
from the reactor core to the number of neutrons that
reach thermal energies
PTNL
Number of thermal neutrons that do not leak from reactor
=
Number of neutrons that reach the thermal energies
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 62
Six Factor Formula (1)
• keff = k∞ x PFNL x PTNL
• Inclusion of expression for k∞ (four-factor formula) yields
keff = ε ⋅ PFNL ⋅ p ⋅ PTNL ⋅ f ⋅η
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 63
Six Factor Formula (2)
40 neutrons net
increase
140 fast
neutrons leak
N 0ε = 1040
Fast fission
ε = 1.04
Fast nonleakage
escape
PFNL = 0.865
N0=1000
neutrons at
start of
generation
N 0εPFNL p = 720
Thermal nonleakage
Thermal
utilization
reproduction
η = 2.02
N 0εPFNL pPTNL f = 495
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
p = 0.80
125 neutrons
absorbed in
non-fuel
N 0εPFNL pPTNL fη = 1000
505 neutrons
net increase
N 0εPFNL = 900 Resonance
f = 0.799
180
neutrons
absorbed
PTNL = 0.861
N 0εPFNL pPTNL = 620
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
100
neutrons
leak
Slide No 64
Neutron Life Cycle of a Fast Reactor (1)
• Neutron life cycle in a fast reactor is markedly different
than that for a thermal reactor
• In a fast reactor, care is taken during the reactor design
to minimize thermalization of neutrons
• Virtually all fissions taking place in fast reactor are
caused by fast neutrons
• Resonance escape probability is not significant since
very few neutrons exist at energies where it is significant
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 65
Exercises (1)
• 10:15-10:30 - Divide into groups of 4-5 students and
solve the following problem:
• Exercise 1: 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
assuming that water density is 1000 kg/m3
– Hint: use the formula for compounds:
103 ρN A
(ν 1σ 1 +ν 2σ 2 + L)
Σ=
M
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 66
Exercises (2)
• 10:30-10:45 - solve the following problem:
• Exercise 2: Disregarding the uranium-234, the natural
uranium may be taken to be a homogeneous mixture of
99.28 weight percent of uranium-238 (absorption cross
section 2.7 b) and 0.72 weight percent of uranium-235
(absorption cross section 681 b). The density of natural
uranium metal is 19.0 x 103 kg m-3. Determine the total
macroscopic and microscopic absorption cross
sections of this material
– Hint: first find mass of U-235 and U-238 per unit volume of
mixture and then number of nuclei per cubic meter of U-235 and
U-238
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 67
Exercises (3)
• 10:45-11:00 – solve the following problem:
• Exercise 3: 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], U235: 681 [b], U-238: 2.7 [b].
Density of UO2: 10200 kg/m3
Density of water: 800 kg/m3
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 68
Exercises (4)
• Exercise 4: 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 given below:
Isotope
microscopic cross sections [b]
absorption
scattering
Hydrogen
Oxygen
Carbon
0.332
27x10-5
0.0034
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
38
3.76
4.75
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 69
Exercises (5)
• Exercise 5: Calculate the resonance escape probability
for a reactor as in Exercise 3 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], U-235: 8 [b], U-238: 8.3 [b].
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 70
Home Assignment #1 due 05-01-24
•
Description and data for problems 1 and 2
– A homogenized core has the following composition (in % by volume):
UO2 - 32%, Zr – 10%, H2O 58%. The enrichment of the fuel is 3.5%
(weight). The material data are given in the Table below.
Component density
kg m-3
H2O
Zr-91.2
UO2
U-235
U-238
O
800
6500
10200
18900
18900
-
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
σa
σf
σs
barn
barn
barn
0.66
0.18
-
103
8
681
2.7
2x10-4
580
-
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
ν
8
8.3
6
2.47
Slide No 71
Home Assignment #1 due 05-01-24
• Problem 1 (5 points):
– Calculate the thermal utilization factor in such a core.
• Problem 2 (5 points):
– Calculate k∞ for an identical core as in Problem 1 assuming that
the resonance escape probability p is known and equal to 0.69
and the fast fission factor is 1.04
Applied Reactor Technology and
Nuclear Power Safety – Lecture 1
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 72