Calculate delayed neutron fraction

1. Nov 28, 2016

Hi guys, I have a question that I can't seem to wrap my head around.

This is the question:
A power reactor is fueled with slightly enriched uranium. At the end of core life (i.e., when it is about to be batch refueled), 30% of the power comes from the fissioning of the built up Pu-239. Calculate the effective value of β at the beginning and end of life; determine the percent increase or decrease.

I don't really know how to go about solving it but I assumed that the total number of neutrons born in fission is equal to those born from Uranium (U-233 and U-238) plus those born from Plutonium (Pu-239). So my fraction at the beginning is equal to β(Pu-239) + β(U-233) + β(U-235) = 0.0021 + 0.0026 + 0.0065.
But the how does power relate to how I would calculate β anyway and how would I use that to find my fraction at the end of life?

2. Nov 28, 2016

Staff: Mentor

I think you have to assume that the reactor starts without Pu-239.

You cannot add β-values like that. As a sanity check: if you could add them, adding more isotopes could give a fraction larger than 1, which is clearly wrong.

3. Nov 28, 2016

That is my problem, I have no idea on how to start.......

4. Nov 28, 2016

Hmmmm.......I think I may have found something.
I've found that "β = ∑(Pi*βi)" where Pi is the fraction of power produced by isotope i.

If this equation is true, then would the β at beginning be equal to 100% β(Uranium) and β at end equal to 70% β(Uranium) + 30% β(Plutonium)??
But then which β do I use for Uranium?

5. Nov 28, 2016

Astronuc

Staff Emeritus
One would need the initial enrichment (at BOC) for the U-235 and U-238, and then the isotopic vector for (U-235, U-238, Pu-239) that would give 30% power generated in the fuel from Pu-239.

One can assume that the neutron flux experienced by the U and Pu is the same, but Pu-239 has a greater (n,f) cross-section in thermal range.

What type of power reactor is this? Most commercial power reactors are LWRs (PWR or BWR) or HWRs (CANDU), but some are graphite-moderated gas-cooled thermal/epithermal reactors or sodium-cooled fast reactors.

What is the initial enrichment?

Unless the reactor fuel includes Th-232, there should be no U-233 in the system, unless it is added into the U.

6. Nov 28, 2016

Astronuc

Staff Emeritus
This is going in the right direction. One must consider the initial enrichment and apply the same principle. Commercial reactors use enrichments in U-235 from natural (0.7%) to 4.95%. The balance is U-238, with some impurities of U-234 and perhaps U-236 (if recycled).

7. Nov 28, 2016

Thanks for the quick reply Astronuc!

The question is actually from the textbook: "Fundamentals of Nuclear Reactor Physics" By Elmer E. Lewis
There is no information on the enrichment nor are there any given β values for U-238

8. Nov 29, 2016

Astronuc

Staff Emeritus
Is there any context in the book regarding 'slightly enriched'? I would think there is an appendix with β values for U-238 and other fissile and fertile species.

U-233, U-235, and Pu-239/-241 are fissile, i.e., they fission readily with thermal neutrons. They also fission with fast (fission) neutrons, but in that energy range, the cross-sections are very low compared to thermal energies. Slightly enriched could mean ~1% U-235. For such a problem, one could construct a plot of β as function of enrichment and burnup.

See Figure 2-5 in this book - https://www.nap.edu/catalog/9263/radiochemistry-in-nuclear-power-reactors

The figure shows the proportion of fissions in commercial reactor fuel for a 2.5% enrichment. Note the fast fissions of U-238 in relation to U-235 and Pu-239.

At the time Lewis wrote the original, commercial LWRs were on annual cycles with exposures typically to three cycles, so enrichments were on the order of 2 to 3%. First cores typically had enrichments in the range of 1 to 2%, since some fuel would be discharged after one cycle or two cycles.