Kinetics Without Delayed Neutrons

1. Kinetics Without Delayed Neutrons B. Rouben McMaster University EP 4D03/6D03 Nuclear Reactor Analysis 2008 Sept-Dec

2. The Time-Independent Diffusion Equation • Several weeks agowe derived and studied the time-independent neutron-balance equation in 1 energy group for a finite, homogeneous reactor:(1) • This equation embodies the concept of a balance between sources (neutron-induced fission) and sinks (absorptions and leakage). cont’d

3. The Time-Independent Diffusion Equation • We also learned that there will not always be a balance if we put together just any combination of fuel and moderator. To ensure a balance, we introduced an adjustment factor keff in the equation: (1) • This adjustment factor (“effective multiplication constant”) on the yield cross section told us whether the original cross section: • was just right (if keff was equal to 1) • had to be reduced, and by how much (if keff was > 1), or • had to be increased, and by how much (if keff was < 1). • It is important to remember that it’s the reactor with the adjusted yield cross section, not the reactor with the given yield cross section, which can operate at steady power.

4. Reactor Kinetics • The previous slides describe reactor statics, the study of time-independent reactor configurations., • We then analyzed various geometries of reactors to determine the time-independent neutron flux shape therein. • Now we want to study reactor kinetics, which means determining how the neutron flux will vary in time when there is no balance between sources and sinks.

5. Time-Dependent Diffusion Equation • In reactor kinetics, we no longer use the keff adjustment factor. We are not trying to modify the reactor to make it time-independent. • Therefore we use the original yield cross section as is. • We can still think of keff, or of reactivity, to have an idea of how far off the reactor is from a balance, but we don’t adjust the cross sections. • And now, since we know that things will evolve in time since/if there is no balance, we must have a time-derivative term, so that the equation becomes the time-dependent diffusion equation.

6. Time-Dependent Equation Without Delayed Neutrons • If there is no balance, then it is the neutron numbers (or neutron density, n) which will change with time. • Let’s first consider the time-dependent diffusion equation in 1 energy group, for a homogeneous reactor, without delayed neutrons – all neutrons are born from f: (2) • But remember the definition of flux: neutron density times speed. • i.e., (3) • where is the one-group neutron speed (not to be confused with , the number of neutrons from fission)

7. Time-Dependent Equation Without Delayed Neutrons • Using Eq. (3) in Eq. (2), we get in terms of : (4) • We can also write the equation in terms of n: (5) • Remember that in thestatic case the flux was the solution of the homogeneous equation

8. Time-Dependent Equation Without Delayed Neutrons • As a simplifying assumption to start with, let’s assume that in the time-dependent case also the flux shape satisfies the same equation. [This is equivalent to a point-kinetics assumption, where the reactor is treated as a point, or spatially uniform.] • Then Eqs. (4) and (5) become:

9. Time-Dependent Equation Without Delayed Neutrons • Looking at Eq. (9) for instance, we see that on the right-hand side we can factorize n out of all terms, to get • Now remember that

10. Time-Dependent Equation Without Delayed Neutrons • So Eq. (11) becomes • If we now define •  becomes

11. Time-Dependent Equation Without Delayed Neutrons • Note the units of  are units of time: •  is called the mean neutron generation time, because from its definition it can be interpreted as any of the following: • the average time between two neutron births in successive generations [since (f)-1 is the mean free path between fissions, and the time for a neutron to cross that distance is (f)-1 /speed] • the time it would take to generate the current number of neutrons at the current generation rate, which we can see is equal to • the average “age” of neutrons in the reactor. (Note that this is a time, and not the Fermi age.)

12. Time-Dependent Equation Without Delayed Neutrons • With all these definitions let’s return to Eq. (10): • This equation is easily solved: • We have shown that in the absence of delayed neutrons the neutron density, and therefore also the neutron flux and power, which are proportional to the neutron density, follow an exponential form in time, with the exponent proportional to the reactivity and inversely proportional to the neutron generation time (as we had concluded last time). [However, we will see that delayed neutrons change the form of the equation!]

13. Time-Dependent Equation Without Delayed Neutrons • Before going on to the equation with delayed neutrons, let’s explore a slightly different treatment. Starting with the definition of α in Eq. (11):

14. Time-Dependent Equation Without Delayed Neutrons • Note that l* also has units of time. It is called the prompt-neutron lifetime, and can be interpreted as any of the following: • average time between the birth and death of a neutron • time necessary to lose all the neutrons in the reactor at the current loss rate. • average life expectancy for neutrons in the reactor. • In this alternative treatment, n and  are still exponential in time: and a similar equation for the flux or power • Either treatment, resulting in the exponential forms in Eq. (15) or (18), is correct, in the absence of delayed neutrons.

15. Reactor Period • By comparing Eqs. (14) and (18), we can see the relationship • Incidentally, given an exponential form of the neutron density, the reactor period is defined as the length of time for the neutron density (or flux, or power) to change (increase or decrease by a factor e). Thus

16. END