1. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 1 CHAPTER 4 The Nuclear Chain Reaction

2. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 2 1.Review 2.Neutron Reactions 3.Nuclear Fission 4.Thermal Neutrons 5.Nuclear Chain Reaction 6.Neutron Diffusion 7.Critical Equation

3. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 3 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:

4. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 4 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.

5. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 5 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

6. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 6 5.2 Neutron Cycle and Multiplication Factor However However, is a constant of nature for a given fissionable material and, therefore, It is beyond human control, to reduce the various causes the only alternative is to reduce the various causes which are responsible for the loss of neutrons in a given assembly.

7. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 7 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 U 238 while its energy is still greater than the threshold energy for U 238 fission and it may cause a fission of a U 238 nucleus. 2. It may be absorbed by U 238 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.

8. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 8 5.2 Neutron Cycle and Multiplication Factor 3. It may be absorbed by a U 235 nucleus causing a fission. 4. It may be absorbed by U 235 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”.

9. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 9 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

10. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 10 5.2 Neutron Cycle and Multiplication Factor Assume we start with n 0 fast neutrons which have just been produced in a uranium assembly. A part causes fissions in U 238 (event 1) consequent increase in the number of fast neutrons The number of neutron after this event is:

11. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 11 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 U 238 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

12. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 12 5.2 Neutron Cycle and Multiplication Factor  These neutrons will then reach thermal energies  They may be either absorbed by U 235 (events 3 and 4) or absorbed in other materials (event 5) thermal utilization factor the thermal 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

13. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 13 5.2 Neutron Cycle and Multiplication Factor the number of thermal neutrons that actually absorbed by the fuel: This number of thermal neutron absorptions will yield a number of fast fission neutrons That is times as large Starting with an initial number of fast neutrons n 0, a new generation of fast neutrons is obtained whose total number is reproduction or multiplication factor =

14. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 14 5.2 Neutron Cycle and Multiplication Factor reproduction or multiplication factor = 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

15. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 15 5.2 Neutron Cycle and Multiplication Factor A schematic picture of the neutron cycle n0n0

16. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 16 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 U 235 component only We must then also use the numerical value for which applies to the uranium mixture For the natural mixture represents the thermal neutron absorptions by the two uranium isotopes only the total number of thermal neutron absorptions in the assembly

17. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 17 5.3 The thermal Utilization Factor Also we have Multiplying and

18. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 18 5.3 The thermal Utilization Factor Considering only the U 235 portion of the natural uranium mixture as the fuel, we have

19. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 19 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

20. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 20 5.4 Neutron Leakage and Critical Size l f : fast neutron non-leakage factor l th : 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 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

21. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 21 5.4 Neutron Leakage and Critical Size Question : what does the magnitude of k eff 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

22. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 22 5.4 Neutron Leakage and Critical Size Question : What is the critical size of an assembly? Answer : 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.

23. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 23 5.4 Neutron Leakage and Critical Size Although this plan of action is very straightforward, the actual mathematical work involved can be very complex.

24. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 24 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.

25. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 25 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 لا يزيد عن 20 صفحة بحث صغير (لا يزيد عن 20 صفحة) حول أنواع المفاعلات بناءا على الخصائص المذكورة أعلاه

26. A. Dokhane, PHYS487, KSU, 2008 Chapter4- Nuclear Chain Reaction 26 Homework Problems: 4, 5 of Chapter 7 in Text Book, Pages 241 الى اللقاء في الحصة القادمة ان شاء الله