Diffusion in 2 Energy Groups

1. Diffusion in 2 Energy Groups B. Rouben McMaster University Course EP 6D03 – Nuclear Reactor Analysis (Reactor Physics) 2009 Jan.-Apr.

2. Contents • We study the diffusion equation in two energy groups of neutrons, which is the methodology most often used for full-core diffusion calculations (but not for lattice calculations). • We treat first the infinite lattice, then move on to the finite reactor.

3. The Diffusion Equation in 2 Groups • In multigroup theory it is conventional to number groups from the highest energy range to the lowest energy range. • Following this convention, in the 2-group model: • Group 1 is the “fast” (or “slowing-down”) group • Group 2 is the “thermal” group • The energy boundary separating the 2 groups is typically taken as 0.625 eV. • Subscripts 1 and 2 refer to groups 1 and 2 respectively, e.g., we will have quantities 1,  2,a1, a2, D1, D2, etc. cont’d

4. The Diffusion Equation in 2 Groups • For now we will make the following simplifying assumptions: • All fission neutrons are born in the fast group • All fissions are lumped as if created by the thermal group, i.e. we have a quantity f2, but no f1 (i.e., f2 is assumed to be renormalized to account for the few percent of fast fissions) • We also have a “downscattering” or “moderation” (i.e., group-1-to-group-2) cross section, which we shall denote 12, but we make the assumption that “upscattering” is negligible, i.e. we take 21  0, and ignore it.

5. The Equation for the Infinite Lattice • As we did in the 1-group treatment, we will start the discussion with the infinite lattice, for which there are no leakage terms. • With the simplifying assumptions in the previous slide, then, the time-independent diffusion equation for the infinite lattice in 2 groups is: • Eq. (1) [a set of 2 equations] expresses the neutron balance: the absorption (loss) terms are balanced by the production terms, in each energy group. cont’d

6. Infinite-Lattice Equation (cont’d) • However, this is a set of 2 homogeneous linear equations in 2 unknowns. Just as in the 1-group case, there is not always a non-trivial solution! • In order that there be a non-trivial solution, the determinant of the equation set has to be 0, i.e. • However, we cannot expect this to be the case in general: physically speaking, we cannot throw together just any kind of material and expect it to constitute a static reactor (except one with 0 neutron flux everywhere)! cont’d

7. Infinite-Lattice Equation (cont’d) • Therefore, in order to ensure a solution, we divide f2 by an adjustable factor k, which will represent the infinite-lattice’s multiplication constant. • The equation set then becomes: • The criticality criterion then becomes: cont’d

8. Infinite-Lattice Equation (cont’d) • i.e., • Eq. (5) can be interpreted according to the neutron cycle (in the same way as we derived the 4-factor formula) as follows: • Suppose Nfast neutrons are born from thermal fissions • These neutrons can be absorbed or be moderated (downscattered to the thermal group). The number which succeed in being moderated is given by the ratio • These surviving neutrons will be absorbed according to cross section a2, but some of these absorptions (if in fuel) will result in fissions and new numbers of neutrons, according to the yield cross section f2. • The ratio of number of neutrons in successive generations is then seen to be exactly k, as given by Eq. (5).

9. Infinite-Lattice Equation (cont’d) • In summary: In the 2-group formalism, the infinite-lattice multiplication constant is given by • which is to be compared to the 1-group result • It is easy to see that the ratio is really the resonance escape probability p, since it is the fraction of fast neutrons which survive to moderation: cont’d

10. Infinite-Lattice Equation (cont’d) • Then, by comparing the rewritten Eq. (5) with the 4-factor formula: (7) we see that we must have (8) • This is actually easy to understand: If we move a2 to the right-hand side of Eq. (8), then we see that the total neutron production (f2) is equal to thermal absortions (a2) times the ratio of thermal absorptions which occur in the fuel(f) times thenumber of fission neutrons born per thermal absorptionin fuel () times the fas-fission factor () [Remember that in our casewe have from the start included fast fissions in f2.] • Therefore we have a consistent and understandable result!

11. Equation in Operator Form • Note that our original 2-group equation for the infinite lattice, Eq. (1), can be written in matrix form as: • This can be rewritten in operator form as:

12. Equation in Operator Form (cont’d) • With the introduction of keff to ensure a non-trivial solution, the final equation can be written in more conventional eigenvalue-problem form as • Note that if we integrate Eq. (14), we get as we had shown previously

13. Interactive Discussion/Exercise • If we remove the simplifications of assuming that: 1) fast fissions are included in thermal fissions, and 2) there is no upscattering, write the equation for the absorption and scattering operator M and for the fission-source operator F for the infinite lattice. • Note: The advantage of the operator notation is that the basic equation (14) will be the same no matter what exact model we use, i.e., M and F may include different things, but the equation remains the same.

14. Include Fast Fission & Upscattering The operators in the more general case:

15. The Finite Reactor • We now move to the case of the finite homogeneous reactor, and for now retain the simplifying assumptions we had assumed for the infinite lattice • For the finite lattice, we will have to change k to keff and add the leakage terms. • The leakage out of the reactor in each group is • (This must be a positive number for each group – why?) cont’d

16. Finite-Reactor Equation in 2 Groups • The equation for the finite homogeneous reactor, with leakage, is: • [Note: This has to be understood as applying at every point in the reactor.] • As we did in the one-energy-group formalism, let’s write the term as DB2 for each group, with the same buckling (B2) in the 2 groups. [This means that the flux has the same shape in the two groups.]

17. Criticality Criterion for Finite Reactor in 2 Groups • The criterion for having a non-trivial solution to this set of homogeneous equations is (as usual) that the determinant have a zero value: • This reduces to

18. Significance of keff • The significance of Eq. (19) is that the expression on the right-hand side of Eq. (19), which is an expression in the values of the material properties of the reactor, must have the value 1 if the reactor is to be critical (and allowed, therefore, to be a time-independent reactor). • If the right-hand side of Eq. (19) is not zero, then he properties of the reactor do not allow a time-independent solution. However, keff tells us how far off we are, and it also tells us that if we change the reactor to have a fission cross section f2/keff, then that reactor will be critical.

19. Criticality Criterion for Finite Reactor in 2 Groups • If we make use of the following definitions for the thermal diffusion area and the fast diffusion area [or neutron age ] we get for the criticality criterion

20. Relationship Between k and keff • If we compare this to the criticality criterion for the infinite lattice, which we obtained previously: • we see that we have the relationship • This is an important relationship, and can be interpreted as follows. • Since the physical difference between kand keff is the leakage, we can identify the factors linking them:

21. 6-factor formula for keff • The factor represents the non-leakage probability of thermal neutrons from the reactor. • Similarly, the factor represents the non-leakage probability of “fast” (group-1) neutrons. • i.e., the relationship betweenkand keff becomes • Incidentally, if we substitute into Eq. (26) the 4-factor formula for k: • we get the 6-factor formula for keff : another important formula

22. Summary for 2-Group Model • The criticality criterion in 2 energy groups for a finite homogeneous reactor relates the buckling B2 to the material properties: • Regarding the flux shape in the reactor, this is given (for both groups) by the same eigenvalue equation as before: except that in 2 groups the value of the geometrical buckling B2 must be related to the material properties as per Eq. (21). (cont’d)

23. Interactive Discussion/Exercise • Derive the relative amplitudes of the fast and thermal fluxes.

24. Relative Amplitudes of Fluxes The operators in the more general case: Either Eq. (3b) or, just as easily, Eq. (3a), can be used to relate the fluxes as above.

25. Modified 1-Group Criticality Criterion • A modified, approximate criticality criterion is sometimes invoked, by starting from the 2-group criterion, Eq. (21), and neglecting the terms which would involve B2*B2 (since the leakage terms are usually small – a few % - in real reactors, this approximation is usually valid): where is called the migration area.

26. Another Exercise • For the finite homogeneous reactor in 3 groups: • write the equations for  and the operators M and F,if: • all fission neutrons are born in the fast group, but from fissions in any group, and • there is scattering from any group to any other group.

27. Interactive Discussion/Exercise • The usual mathematical intervention (which has been used above) for ensuring that the reactor equation has a solution, is to include an eigenvalue,  (= 1/keff), in front of the yield operator F. • Relate/contrast this intervention with the physical, concrete steps which are or can be used to ensure criticality of real reactors.

28. END