Lecture 20 Reactor Theory-V

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1 Objectives In this lecture you will learn the following We will discuss the criticality condition and then introduce the concept of k eff.. We then will introduce the four factor formula and two group equations. Finally we will discuss the criticality condition for two groups.

2 Introduction In the last lecture we showed that the Criticality Condition is B 2 = B m 2. From our earlier lecture we know that Thus, we can write One group criticality equation.

3 Interpretation of Criticality Equation To appreciate the physical interpretation of the above equation, We shall go back to diffusion theory. In the above the term volume per unit volume. This would imply that control volume by diffusion. represents the net number of neutrons diffusing into the control would represent the net number of neutrons leaking out of the Since the reactor satisfies the equation, we can state that DB 2 φ would represent the net number of neutrons leaking out of the control volume by diffusion. The neutrons in a finite sized reactor can either get absorbed or can leak out. The total number of neutrons that can leak out of the reactor is Similarly, the total number of neutrons absorbed in the system is Thus, the fraction leaking out is For a homogeneous system, the leaking fraction will be Hence the non-leaking fraction called, the non-leakage probability, P L can be written as

4 Thus the one-group criticality equation can be interpreted as follows k eff is the multiplication constant that includes leakage.

5 Two-Group Approach The calculation performed by one group approach is approximate due to the fact that it ignores the processes that fast neutron undergoes before slowing and eventual absorption. It should be understood that most of the absorption takes place in thermal energies. As the neutron slows down, some of the fast neutrons can escape. During the slowing down process, some neutrons can cause fast fissions. Some of the neutrons can be captured in the resonances in the absorption cross section. All of this requires some modifications to the theory.

6 Four Factor Formula In two group approach we shall have two groups of neutrons. One will be called fast and the other thermal. We had discussed in our previous lectures that the thermal group will have energy upto 5kT, while the fast group will have energies above 5kT. Such a division will eliminate up-scattering in thermal group. Let us now trace the history of a neutron which is thermal. Thus the cycle has been completed. By the definition of the infinite multiplication constant, we can write The above formula is called four factor formula. The definition of each of the term is summarized. η is the number of second generation neutrons generated per neutron absorbed in the fuel. ε, called the fast fission factor is the amplification in the number of fast neutrons due to fast

7 fissions. p, called the resonance escape probability, is the fraction of fast neutrons that succeed in slowing down. Finally, f, the fraction of neutrons that absorb in fuel is called the thermal utilization. It may be noted that η ~ 2.5, ε > 1, p < 1, f < 1. Before we write the governing diffusion equation for two groups, it should be understood that we will write equation for both fast and thermal fluxes. The reactor equation for obtaining the fundamental Eigen values are same as before and the boundary conditions for both the fluxes are also same. Hence, the value of B obtained for different geometry continues to be the same as tabulated in the last lecture.

8 Two-Group Equations The neutron balance equation for both the groups are as follows The third term is the source term for fast neutron. Note that while slowing down only fraction p succeed in slowing down. Since the reactor equation is valid for both the groups, we can rewrite the above equations as Note that we have replaced 2 φ by -B 2 φ. In matrix form the equations can be written as For non-trivial solution

9 Criticality Condition where In the above equation, P LF and P LT are the fast and thermal non leakage probability.