Kinetics

1. Kinetics B. Rouben McMaster University EP 4D03/6D03 Nuclear Reactor Analysis 2008 Sept-Dec

2. Kinetics: Time-Dependent Phenomena Kinetics refers to time-dependent phenomena. 3 categories: • On short time scale (ms, s) • Disturbances/accidents • Short experiments • Reactor startup • On medium time scale (hrs, days) • Channel refuellings • Saturating-fission-product transients • Xe-135 • Other saturating fission products (e.g., Sm) • On long time scale (months, years) • fuel burnup & depletion, consequent change in composition

3. Feedback or No Feedback Time-dependent phenomena can be analyzed with or without feedback on fuel/lattice properties during the time evolution of the event. • The phenomena of saturating-fission-product transients, channel refuellings, and fuel depletion most definitely involve changes in fuel properties  should be analyzed with feedback • Some (but not all) of the short-term time-dependent phenomena do not affect the lattice properties (at least not right away), and can therefore be studied without feedback.This is what we will study first. • [Note: when the analysis includes feedback on properties, it is often labelled a “dynamics” study, rather than a kinetics study.]

4. Fast Kinetics & Delayed Neutrons • “Fast” kinetics, i.e., the study of time-dependent phenomena on a very short time scale (seconds), must account for delayed neutrons. • There are 2 categories of neutrons in fission reactors: • prompt neutrons, which are emitted during the fission event itself, and • delayed neutrons, which are emitted later, in the radioactive decay of certain fission products (the delayed-neutron precursors, which have half-lives ranging from fractions of a second to tens or hundreds of seconds) • Delayed neutrons represent only ~ 0.6% of all neutrons. • Despite their very small number, delayed neutrons play a crucial role in fast kinetics, on account of their delayed appearance. • More on delayed neutrons a bit later.

5. Reactor Without Delayed Neutrons Let’s first look at a simple kinetics treatment for a reactor without delayed neutrons. Define • “Average generation time”  average time between the birth of two fission neutrons in successive generations • N(t) = neutron population at time t. • Now keff is the reactor multiplication constant. This can be regarded as the ratio of the neutron populations in successive generations. cont’d

6. Reactor Without Delayed Neutrons (cont’d) From the definitions of andkeff, we can then write If we subtract N(t)from both sides, and divide by: If we identify  as t, or as dt in the limit, we can write

7.  For Reactor Without Delayed Neutrons For a reactor without delayed neutrons, • the average generation time is composed of an average slowing-down time and an “average diffusion time”, td • Actually the slowing down-time is very much shorter than the diffusion time, therefore we can write td • Typical values for various moderators are: • H2O: td 2.1*10-4 s • D2O: td 0.9*10-3 s • Be: td 3.9*10-3 s • Graphite: td 1.7*10-2 s

8. Reactor Without Delayed Neutrons (cont’d) If we take as a numerical example • keff = 1.001 (i.e., a reactivity  1 mk) • and  = 1 ms = 10-3 s, then Eq. (3) gives Thus, the neutron population (and also the power) would multiply • by a factor exp(1) = 2.718 in 1 s • by a factor exp(2) = 2.7182 = 7.389 in 2 s • by a factor exp(3) = 2.7183 = 20.1 in 3 s! This is a very fast rate of increase in the fission power, and it would be impossible to control such a fast power increase with mechanical shutdown systems. This is where delayed neutrons will have a very important effect!

9. Delayed-Neutron Data • In the next 2 slides, Table 1 gives data for 6 delayed-neutron-precursor groups, for fissions from the 4 most important fissionable nuclides: U-235, U-238, Pu-239, Pu-241. • Actually there are many more than 6 delayed-neutron precursors, with various decay constants, but for simplicity they are usually collapsed onto 6 groups. • The decay constants of the 6 groups are selected so as to best fit the net time behaviour of the appearance of delayed neutrons from all precursor groups.

10. x Table 1 (Part 1)Typical 6-Precursor-Group Data for Direct Delayed Neutrons

11. x Table 1 (Part 2)Typical 6-Precursor-Group Data for Direct Delayed Neutrons

12. Delayed-Neutron Data • Of note in Table 1 is the fact that the decay constants of the 6 groups vary by a factor of ~300! • Another important note is that the delayed-neutron fraction from Pu-239 is quite a bit smaller (by a factor of ~3) than that U-235. The delayed fraction from U-238 is the highest, but only ~5% of all fissions in CANDU are U-238 fissions. • For an actual reactor core, one could keep 24 delayed-neutron groups (6 for each fissionable nuclide), or, more usually, collapse the 24 groups to a new “average” 6. • In any case, a core with higher burnup will inevitably have a lower net delayed fraction, because of the smaller delayed fraction in Pu-239! • An equilibrium CANDU core has   0.0058 (~ 6 mk).

13. Photoneutrons • Table 1 gives data for “direct” delayed neutrons, i.e., those born “in the fuel”. • But in CANDU there is another group of delayed neutrons: the photoneutrons. • Photoneutrons are born in the heavy water (moderator, mostly) when a high-energy gamma ray (energy > 2.2 MeV, the binding energy of D) breaks up a deuteron into a proton and a neutron. • There are several precursors which can emit such energetic gammas. • Tables 2 and 3 (next 2 slides) gives typical data for 11 groups of photoneutrons.

14. Table 2Group Yields and Half-Lives for 11 Groups of Photoneutrons

15. x Table 3Photoneutron Delayed Yield and Fraction in a CANDU Lattice

16. Photoneutrons • Photoneutrons represent only ~5% of all delayed neutrons in CANDU. • However, they are important because of the very long half-lives (small decay constants) of their precursors. • When a CANDU is shut down, there is still a measurable source of photoneutrons many months later. • These make it easier to restart the reactor – you don’t need special detectors as in a “new” core.

17. Effect of Delayed Neutrons on Generation Time Now let’s repeat the previous numerical example, but taking into account the delay time in the delayed neutrons’ appearance • The average generation time will now have to be obtained as a weighted average: • For 99.4% of neutrons, take average generation time = 10-3 s • For the other 0.6% (the delayed neutrons) take 10-3 s + (say) the half-life of the precursor group (this requires a 6-term or 17-term sum) If we carry out this weighted average, we find that the average generation time is dominated by the delayed neutrons, because of the very long time for their appearance, and in spite of their small number. We find typically   0.1 s (100 times the previous value!)

18. Effect of Delayed Neutrons on Generation Time Using again keff =1.001, we now find in the exponential equation Thus, the neutron population (and also the power) would multiply • by a factor exp(0.01)  1.01 in 1 s • by a factor exp(0.02)  1.02 in 2 s • by a factor exp(0.03)  1.03 in 3 s! The delayed neutrons have reduced the rate of increase of fission power dramatically. It is now very achievable to control the power transient with mechanical shutdown systems.

19. Improved Formalism Needed However, the “shortcut” formalism that we have used, i.e., determining a “weighted average” generation time, is not rigorous mathematically. We need to derive a formalism which properly takes into account the exact evolution with time of each delayed-neutron-precursor-group’s concentration. This is the subject of future inquiry.

20. END