The Finite Reactor in One Energy Group

1. The Finite Reactor in One Energy Group B. Rouben McMaster University EP 4D03/6D03 Nuclear Reactor Analysis 2008 Sept-Dec

2. Contents • We analyze the criticality or degree of non-criticality of a finite reactor in 1 energy group.

3. The Reactor Equation • In the previous presentation we considered the special case of the infinite lattice. But before considering the infinite lattice, we had written down the time-independent neutron-balance equation in 1 energy group for a finite, homogeneous reactor: • Now, if we integrate this equation, the term is the net leakage out of the reactor.

4. Interactive Discussion/Exercise • What must be the sign of this quantity, ? • Why? • Since f and a are uniform, what does this imply regarding their relative magnitude? • Would this conclusion hold in the same way for an inhomogeneous reactor? Explain. • Do not turn the page until you have attempted/done this discussion/exercise.

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6. Signs and Magnitudes • Since is the leakage out of the reactor, it must be a positive quantity. • If it were negative, it would mean that there is net in-leakage into the reactor from outside, which does not make sense. • From Eq. (1), so if the quantity is positive it implies f > a. • This makes sense, since neutron production must be larger than absorption inside the reactor, to allow for neutron loss by leakage. • If f < a, we have no chance of achieving a time-independent reactor. • Note: For an inhomogeneous reactor also, the overall production must be larger than overall absorption, however, it is quite possible that in some regions of the reactor f < a.

7. Introduce B2 • Since f > a and since D is positive, we can conclude that must be negative everywhere. • This means that the flux curvature is negative. • This is opposite to the situation in the source-sink problem! • We have determined therefore that we can write where B2 > 0 and B is a real (i.e., not imaginary) quantity cont’d

8. The Reactor Equation (cont’d) • B2is called the geometric buckling. • It is a geometric quantity because it is a measure (actually, the negative) of the curvature of the flux. • The flux curvature is everywhere negative in the homogeneous reactor, and B2 is a measure of curvature, or “buckling”. • What are the units of B2? From the definition, we can see they are inverse length squared, e.g., cm-2. • We also know that the flux’s curvature (and therefore the magnitude of B2) must bring the flux to 0 at the reactor’s extrapolated boundary. This means that the geometrical buckling will really depend only on the size and shape of the reactor. It is really a geometrical quantity.

9. The Reactor Equation (cont’d) • The reactor equation can now be written • As we did in the case of the infinite lattice, let us study this equation. • Just as was the case for the infinite lattice, the above reactor equation always has the trivial solution  = 0 (no neutrons anywhere). • However, it has a non-trivial solution if and only if the nuclear properties satisfy the special criticality condition

10. Adjusting the Parameters • The finite reactor has a criticality condition, just as did the infinite lattice. • Once again, what this really means is that we cannot have a viable time-independent reactor with just any properties. • If the properties do not satisfy the criticality condition, we do not have a time-independent reactor. • But we can try to see how far we are from a time-independent reactor. • We can try to modify the reactor in 2 ways: • We can adjust the material properties, e.g., change the yield cross section by, say, dividing it by a quantity keff (‘eff’ reminds us that this is not the infinite lattice, but a finite reactor), or • We can adjust the geometrical buckling by modifying the geometry: change the size or shape of the reactor.

11. Criticality Condition • If we assume that in general we may need to adjust the yield cross section, the relationship which permits a non-trivial solution then becomes • Analogously to k , the infinite-lattice multiplication constant, keff is called the reactor multiplication constant. • We can relate the geometrical buckling to the reactor multiplication constant by rewriting Eq. (5) in 2 ways: or • While we don’t know either B2 or keff (yet), Eq. (6) can be considered the criticality condition for the reactor.

12. Eigenvalue and Reactivity • If we put the flux back into Eq. (5) we get

13. The Reactor Equation for  • While Eqs. (5) or (7) give the criticality condition, they do not contain , and therefore say nothing about it. • To solve for the flux distribution in the reactor, we have to return to Eq. (2): • which we can write in final version as the equation to be solved for the flux shape:

14. Summary • Let’s summarize a few things for the homogeneous 1-group reactor. • If we multiply numerator and denominator of Eq. (6) by  and integrate over the reactor, we get the various total reactor rates: • And also consistent with the standard definition of a multiplication constant

15. Summary • As the (homogeneous) reactor increases in size without bound, the leakage tends to 0, and Eq. (6) rightly reduces to the infinite-lattice criticality condition • Note that for the infinite lattice, the criticality condition is simple: it involves only the nuclear properties f and a.

16. Summary (cont’d) • In the finite reactor, the situation is not as simple: • The flux will not be the same everywhere - we know that it must go to 0 outside the reactor’s extrapolation boundary • The leakage can be written as DB2 (with a negative flux curvature and a positive B2) • The equation which the flux must satisfy is • The criticality condition is a relationship between geometrical and material quantities: cont’d

17. Summary (cont’d) • To make the flux go to 0 at the (extrapolated) boundary, the buckling B2 must necessarily (for given nuclear properties) be larger for a smaller reactor. B2 is a geometrical quantity. • Thus, for criticality, we can adjust either the material properties or the geometry of a reactor design: • We can adjust the relative values of the yield an absorption cross sections (e.g., by changing the fuel composition) • We can adjust the size (or shape) of the reactor, thereby changing the leakage • We can say this in another way: there is a specific minimum critical size (or critical mass) for any planned reactor design with a given composition of fuel and other materials

18. Comparison of Criticality Conditions • For the infinite lattice: • For the finite reactor with the same material properties: • Comparing these equations, we can immediately see that we necessarily always have keff < k (20) • A reactor cannot achieve criticality if the corresponding infinite lattice is subcritical.

19. Interactive Discussion/Exercise • Consider a reactor and its corresponding infinite lattice. • Does it make physical sense that we must have keff < k , and that the reactor cannot achieve criticality if its corresponding infinite lattice is subcritical? Why? • Do not turn the page until you have attempted/done this discussion/exercise.

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21. Reactor & Infinite Lattice • Since the reactor suffers leakage of neutrons but the infinite lattice does not, it does make physical sense that we must have keff < k . • This means that if we know that a certain infinite lattice is subcritical, we cannot ever hope to get criticality if we reduce the size to that of a finite reactor. • We would have to make some other change to get a critical reactor from a subcritical infinite lattice.

22. Some More Definitions and Equations • (Not new) aandf are the overall absorption and fission cross sections for the homogeneous reactor. • If the reactor composition is a mixture of materials (at the very least, of fuel and coolant), then let aF be the absorption cross section of the fuel component. • If is defined as the number of neutrons emitted per absorption in the fuel (not per fission!), then we can write the fission-source density asaF (another expression of course is f ). • The corresponding infinite-lattice multiplication constant (in the 1-group analysis) can then also be written as (21) where we have defined the fuel utilization (22) cont’d

23. Some More Equations (cont’d) • For a critical (keff=1) homogeneous reactor, we can write the buckling in terms of the 1-group infinite-lattice multiplication constant: where is the 1-group diffusion area.

24. Interactive Discussion/Exercise • Some reactors have a fuelled core surrounded on at least some sides by a reflector. • A “bare reactor” is one without a reflector. • Can a bare reactor be considered a “homogeneous reactor”. Why? • Would the equations we have derived apply to a bare reactor? How would you understand the term “reflector savings”? • Do not turn the page until you have attempted/done this discussion/exercise.

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26. Reactor & Infinite Lattice • A “bare reactor” cannot be considered to be a “homogeneous reactor”, since it does not have uniform properties. • This is so even if the fuelled portion of the core is uniform or homogeneous on its own. The presence of a reflector makes the reactor inhomogeneous. • This means that the equations we derived here cannot apply exactly to a bare reactor. They would at most apply as an approximation. • Since a reflector reduces leakage, the term “reflector savings” applies to the gain in reactivity which the reflector provides.

27. Some More Equations (cont’d) • The right-hand side of Eq. (23), has been given a label (as shown) and a name of its own. • It is called the “material buckling” • This is a strange name; the quantity is a function of material properties only, therefore it is not really a “buckling” in the sense of curvature. • Nonetheless, the criticality condition for the 1-group reactor is very often expressed as Geometrical Buckling = Material Buckling (25), a concept worth remembering in that it reminds us that for a reactor to be viable there must be a suitable correspondence between its properties and its size.

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