Chapter4- Nuclear Chain Reaction
CHAPTER 4
The Nuclear Chain Reaction
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
1.
2.
3.
4.
5.
6.
7.
Review
Neutron Reactions
Nuclear Fission
Thermal Neutrons
Nuclear Chain Reaction
Neutron Diffusion
Critical Equation
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
Lecture content:
• Introduction
• Neutroncycleandmultiplicationfactor
• TheThermalutilizationfactor
• Neutronleakageandcriticalsize
• Nuclearreactorsandtheirclassification
k∞
• Calculationofforhomogeneousreactor
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
5.1 Introduction
Nuclear chain reaction is a central theme for reactor physics
Question: What is Nuclear Chain Reaction?
Answer:
Answer self-sustained process that, one started, needs no additional agents to
keep it going
Goals of this chapter : 1. follow the life of a group of Neutrons and develop a numerical
criterion for a nuclear chain reaction to be possible and
2. apply it to various types of nuclear reactors.
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
5.2 Neutron Cycle and Multiplication Factor
Question: When a self-sustaining chain reaction is possible?
Answer:
Answer ifν , the number of neutrons released per fission, is sufficiently greater
than 1
Why?
to compensate for neutron loss due to a variety of causes
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
5.2 Neutron Cycle and Multiplication Factor
However,
However ν is a constant of nature for a given fissionable material and, therefore,
It is beyond human control,
the only alternative is to reduce the various causes which are responsible
for the loss of neutrons in a given assembly.
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Chapter4- Nuclear Chain Reaction
5.2 Neutron Cycle and Multiplication Factor
Consider now, a natural uranium assembly in which some fission reactions have been initiated
Let us follow the life of a typical neutron from the instant of its creation as a fast fission neutron
listing the various possible events that may occur during its lifetime.
1. Neutron may be absorbed by U238 while its energy is still greater than the threshold energy
for U238 fission and it may cause a fission of a U238 nucleus.
2. It may be absorbed by U238 without leading
to a fission (radiative capture).
This most likely to happen for neutron whose energy
has been reduced by elastic collisions to the epithermal
region (from 1000 ev to 5 ev), where U238 has pronounced
absorption resonance.
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
5.2 Neutron Cycle and Multiplication Factor
3. It may be absorbed by a U235 nucleus causing a fission.
4. It may be absorbed by U235 nucleus without causing fission.
5. It may be absorbed by other materials and impurities that are part
of the assembly without causing a fission.
6. It may escape from the assembly and be lost by what is called “leakage”.
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
5.2 Neutron Cycle and Multiplication Factor
Events (1) and (3) are positive contributions to the neutron economy
Create new neutrons
events (2), (4), (5), and (6) are negative contributions
Remove available neutrons from the assembly
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
5.2 Neutron Cycle and Multiplication Factor
Assume we start with n0 fast neutrons which have just been produced in a uranium assembly.
A part causes fissions in U238 (event 1)
consequent increase in the number of fast neutrons
this small increase by means of a factor ε > 1 , called the fast fission factor.
The number of neutron after this event is:
A. Dokhane, PHYS487, KSU, 2008
n0 ⋅ ε
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Chapter4- Nuclear Chain Reaction
5.2 Neutron Cycle and Multiplication Factor
The energy of the fast neutrons is being reduced steadily by collisions with the other nuclei
in the assembly until they eventually enter the epithermal energy region
The epithermal energy region has strong U238 absorption resonances
Some of the neutrons will be absorbed by U238 (event2),
whereas most of them will escape resonance absorption.
The number of neutrons that will pass through this region
without being absorbed
n0εp
where p resonance escape probability
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
5.2 Neutron Cycle and Multiplication Factor
These neutrons will then reach thermal energies
They may be either absorbed by U235 (events 3 and 4) or absorbed in other materials (event 5)
The fraction of thermal neutrons absorbed by the fuel as compared to all thermal neutron
absorption in the assembly is called
the thermal utilization factor f.
the number of thermal neutrons that actually absorbed by the fuel:
n0εpf
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
5.2 Neutron Cycle and Multiplication Factor
the number of thermal neutrons that actually absorbed by the fuel:
n0εpf
This number of thermal neutron absorptions will yield a number of fast fission neutrons
That is η times as large
Starting with an initial number of fast neutrons n0, a new generation
of fast neutrons is obtained whose total number is
n0 εpfη
reproduction or multiplication factor =
A. Dokhane, PHYS487, KSU, 2008
Final number of fast neutrons
= εpfη
Initial number of fast neutrons
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Chapter4- Nuclear Chain Reaction
5.2 Neutron Cycle and Multiplication Factor
Final number of fast neutrons
= εpfη
reproduction or multiplication factor =
Initial number of fast neutrons
k ∞ = εpfη
The expression
fourfour-factor formula
Final number of fast neutrons
= εpfη
Initial number of fast neutrons
can be interpreted as the ratio of thermal neutrons created per second
to the number of thermal neutrons destroyed per second
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
5.2 Neutron Cycle and Multiplication Factor
n0 εpfη
n0
n0 ⋅ ε
n0εpf
A schematic picture of the neutron cycle
A. Dokhane, PHYS487, KSU, 2008
n0εp
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Chapter4- Nuclear Chain Reaction
5.3 The thermal Utilization Factor
When defining f, we conventionally consider uranium mixture as the fuel,
although the thermal neutrons can produce fission with the U235 component only
We must then also use the numerical value for η which applies to the uranium mixture
For the natural mixture η = 1.34
Thus if σ stands for the thermal absorption cross sections and the suffix i referring to all
non uranium materials and impurities and employing an obvious notation, we can write
for a natural uranium fuel mixture that
f ( nat ) =
N 0 ( 235)σ ( 235) + N 0 ( 238)σ ( 238)
N 0 ( 235)σ ( 235) + N 0 ( 238)σ ( 238) + N oiσ i
the total number of thermal neutron absorptions in the assembly
represents the thermal neutron absorptions by the two uranium isotopes only
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
5.3 The thermal Utilization Factor
Also we have
η ( nat )
σ f ( nat )
=
ν ( 235)
σ a ( nat )
=
Multiplying
f ( nat ) =
N 0 ( 235)σ f ( 235)ν ( 235)
N 0 ( 235)σ a ( 235) + N 0 ( 238)σ a ( 238)
N 0 ( 235)σ ( 235) + N 0 ( 238)σ ( 238)
N 0 ( 235)σ ( 235) + N 0 ( 238)σ ( 238) + N oiσ i
η ( nat ) f ( nat ) =
A. Dokhane, PHYS487, KSU, 2008
and
=
N 0 ( 235)σ f ( 235)ν ( 235)
N 0 ( 235)σ a ( 235) + N 0 ( 238)σ a ( 238)
N 0 ( 235)σ f ( 235)ν ( 235)
N 0 ( 235)σ a ( 235) + N 0 ( 238)σ a ( 238) + N 0iσ i
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Chapter4- Nuclear Chain Reaction
5.3 The thermal Utilization Factor
Considering only the U235 portion of the natural uranium mixture as the fuel, we have
f ( 235) =
η ( 235)
N 0 ( 235)σ a ( 235)
N 0 ( 235)σ a ( 235) + N 0 ( 238)σ a ( 238) + N 0iσ i
σ f ( 235)
=
ν ( 235 )
σ a ( 235)
η ( 235) f ( 235 ) =
N 0 ( 235)σ f ( 235 )ν ( 235)
N 0 ( 235 )σ a ( 235 ) + N 0( 238 )σ a ( 238) + N 0iσ i
η ( nat ) f ( nat ) = η ( 235) f ( 235)
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
5.4 Neutron Leakage and Critical Size
In the preceding derivation of the multiplication factor we omitted completely
the possibility of leakage from the assembly (event 6)
we assumed zero leakage during the neutron cycle
Question: When this assumption is valid?
Answer: valid with an infinite size for the assembly,
hence there will be no leakage of neutrons from the system
That is why we denoted the multiplication factor by
k∞
For an assembly of finite size, the effective reproduction constant k eff will be less than
k ∞ by a factor L (L<1), which is determined by the neutron leakage from the system:
k eff = k ∞ L
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Chapter4- Nuclear Chain Reaction
5.4 Neutron Leakage and Critical Size
L = l f l th
lf : fast neutron non-leakage factor
lth : thermal neutron non-leakage factor
This separation is suggested by the diffusion theory, which treats
the diffusion of fast neutrons and that of thermal neutrons separately
k eff = k ∞ l f l th
keff is the ratio of the number of neutrons in successive generation
This self-multiplication of neutrons is the essential feature of a nuclear chain reaction
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
5.4 Neutron Leakage and Critical Size
Question: what does the magnitude of keff represent?
Answer: the speed with which the number of neutrons builds up and
the rate at which nuclear fissions occur in the assembly
Example: in a nuclear bomb type assembly, this build-up must take place very rapidly
in industrial and research reactors this self-multiplication must be slow enough
to allow the fission rate to remain always under the control of the operator
For k eff > 1 , the assembly continues to produce more neutrons than it consumes and is
then said to be supercritical.
For k eff < 1 , fewer neutrons are produced than are consumed. Such an assembly is said to
be subcritical.
For k eff = 1 , the rate of neutron production is exactly balanced by the rate of neutron
consumption and, in this case, the assembly is called a critical one.
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
5.4 Neutron Leakage and Critical Size
Question: What is the critical size of an assembly?
Answer:
If we start with an assembly for which k eff > 1 , we can decrease k eff by progressively
reducing the reactor size, thereby increasing the neutron loss through leakage from the
assembly. If this reduction in the dimensions of the assembly is continued until k eff = 1 ,
the reactor size of the assembly at that point is called its critical size
the critical size of an assembly is that size for which the rate of neutron loss due to all causes
is exactly equal to the rate of neutron production in the assembly.
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
5.4 Neutron Leakage and Critical Size
The fundamental problem in the design of a nuclear reactor is to obtain an assembly with
k eff > 1 . As a first step to this end, k ∞ must be calculated and then, by introducing the
finite size and geometry of the reactor, k eff can be calculated from the layout and
dimensions of the reactor.
Although this plan of action is very straightforward, the actual mathematical work
involved can be very complex.
Of the four factors, η is a constant of nature for a given nuclear fuel (by changing the
composition of the fuel, i.e. increasing the degree of enrichment, a higher value of η is
achieved…) which must be obtained from experimental measurements.
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
5.4 Neutron Leakage and Critical Size
The other three factors allow the nuclear engineer and designer some “choice”,
as they depend on the physical properties of the fuel as well as on the size of
the reactor, its geometry, fuel arrangement, moderator, as well as on the other
materials incorporated in the reactor assembly.
Question: is the neutron leakage desirable??
Answer: Although neutron leakage is generally not welcomed by the nuclear engineer,
it is important to realize that the critical size requirement for a chain reaction
to become possible is a consequence of neutron leakage.
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
5.5 Nuclear Reactors and Their Classification
Question: How are the reactor classified?
Answer: Reactors are classified based on a variety of characteristic features :
1) Type of fuel used
2) Average neutron energy at which the greater part of all fissions occur
3) Moderator materials used
4) Arrangement and spatial disposition of fuel and moderator
5) Purpose of the reactor
‫ صفحة( حول أنواع المفاعالت بناءا على الخصائص المذكورة أعاله‬20 ‫)ال يزيد عن‬
‫بحث صغير ال‬
A. Dokhane, PHYS487, KSU, 2008
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Chapter4- Nuclear Chain Reaction
Homework
• Problems: 4, 5 of Chapter 7 in Text Book,
Pages 241
• ‫اﱃ اﻟﻠﻘﺎء ﰲ اﳊﺼﺔ اﻟﻘﺎدﻣﺔ ان ﺷﺎء ﷲ‬
A. Dokhane, PHYS487, KSU, 2008
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