a. dokhane, phys487, ksu, 2008 chapter4- nuclear chain reaction 1 chapter 4 the nuclear chain...

A. Dokhane, PHYS487, KSU, 2008

Chapter4- Nuclear Chain Reaction

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CHAPTER 4

The Nuclear Chain Reaction



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1. Review

2. Neutron Reactions

3. Nuclear Fission4. Thermal Neutrons

5.5. Nuclear Chain ReactionNuclear Chain Reaction6. Neutron Diffusion

7. Critical Equation



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• Introduction

• Neutron cycle and multiplication factor

• The Thermal utilization factor

• Neutron leakage and critical size

• Nuclear reactors and their classification

• Calculation of for homogeneous reactor

Lecture content:

k



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5.1 Introduction

Nuclear chain reaction is a central theme for reactor physics

Question: What is Nuclear Chain Reaction?

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.



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5.2 Neutron Cycle and Multiplication Factor

Question: When a self-sustaining chain reaction is possible?

Answer: if , the number of neutrons released per fission, is sufficiently greater than 1

Why?

to compensate for neutron loss due to a variety of causes



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HoweverHowever, 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 causesto reduce the various causes which are responsible for the loss of neutrons in a given assembly.



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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.



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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”.



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5.2 Neutron Cycle and Multiplication FactorEvents (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



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5.2 Neutron Cycle and Multiplication FactorAssume 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 1 , called the fast fission factor.

The number of neutron after this event is: 0n



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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

pn 0

where p resonance escape probability



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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 thermal utilization factorthermal utilization factor f.

the number of thermal neutrons that actually absorbed by the fuel:

The fraction of thermal neutrons absorbed by the fuel as compared to all thermal neutron absorption in the assembly is called

pfn 0



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the number of thermal neutrons that actually absorbed by the fuel: pfn 0

This number of thermal neutron absorptions will yield a number of fast fission neutronsThat is times as large

Starting with an initial number of fast neutrons n0, a new generation of fast neutrons is obtained whose total number is

pfn0

pfneutronsfast ofnumber Initial

neutronsfast ofnumber Finalreproduction or multiplication factor =



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neutronsfast ofnumber Finalreproduction or multiplication factor =

pfk four-factor formula

The expression

can be interpreted as the ratio of thermal neutrons created per second to the number of thermal neutrons destroyed per second


neutronsfast ofnumber Final



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A schematic picture of the neutron cycle

n0

0n

pn 0

pfn 0

pfn0



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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 which applies to the uranium mixture

For the natural mixture 34.1

Thus if 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

ioinat NNN

NNf

)238()238(0)235()235(0

)238()238(0)235()235(0)(

represents the thermal neutron absorptions by the two uranium isotopes only

the total number of thermal neutron absorptions in the assembly



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5.3 The thermal Utilization Factor

Also we have)235(

)(

)()(

nata

natfnat

)238()238(0)235()235(0

)235()235()235(0

aa

f

NN

N

Multiplying ioi

nat NNN

NNf

)238()238(0)235()235(0

)238()238(0)235()235(0)( and

)238()238(0)235()235(0

)235()235()235(0

aa

f

NN

N

iiaa

fnatnat NNN

Nf

0)238()238(0)235()235(0

)235()235()235(0)()(



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5.3 The thermal Utilization FactorConsidering only the U235 portion of the natural uranium mixture as the fuel, we have

iiaa

a

NNN

Nf

0)238()238(0)235()235(0

)235()235(0)235(

)235()235(

)235()235(

a

f

iiaa

f

NNN

Nf

0)238()238(0)235()235(0

)235()235()235(0)235()235(

)235()235()()( ff natnat



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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 effk will be less than

k by a factor L (L<1), which is determined by the neutron leakage from the system:

Lkkeff



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5.4 Neutron Leakage and Critical Size

thf llL

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

thfeff llkk

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



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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 1effk , the assembly continues to produce more neutrons than it consumes and is

then said to be supercritical.

For 1effk , fewer neutrons are produced than are consumed. Such an assembly is said to

be subcritical.

For 1effk , 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.



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Question: What is the critical size of an assembly?

Answer:

If we start with an assembly for which 1effk , we can decrease effk 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 1effk ,

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.



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The fundamental problem in the design of a nuclear reactor is to obtain an assembly with 1effk . As a first step to this end, k must be calculated and then, by introducing the

finite size and geometry of the reactor, effk 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, 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 is achieved…) which must be obtained from experimental measurements.



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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.



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5.5 Nuclear Reactors and Their ClassificationQuestion: 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

صغير ) عن بحث يزيد عن ال يزيد أعاله( صفحة صفحة 2020ال المذكورة الخصائص على بناءا المفاعالت أنواع حول



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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 chapter4- nuclear chain reaction 1 chapter 4 the nuclear chain...

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