Calculate delayed neutron fraction

[Oxlade]

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?

 

[mfb]

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.

 

[Oxlade]

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.

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

 

[Oxlade]

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?

 

[Astronuc]

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.

 

[Astronuc]

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?

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).

 

[Oxlade]

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).

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

 

[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

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.