# Calculate delayed neutron fraction

In summary, the delayed neutron fraction is a measure of the number of neutrons that are released from a fission reaction after a small delay. It is a crucial factor in nuclear reactor design and control, as it affects the overall reactivity and stability of the reactor. The delayed neutron fraction is calculated by dividing the total number of delayed neutrons produced by the total number of neutrons released, and it varies depending on the type of nuclear fuel used. Accurate determination and control of the delayed neutron fraction is essential in ensuring safe and efficient operation of nuclear reactors.
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?

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.

mfb said:
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...

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?

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.

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).

Astronuc said:
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

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.

## 1. What is a delayed neutron fraction?

The delayed neutron fraction (DNF) is a measure of the proportion of neutrons released from a nuclear fission reaction that are delayed, meaning they are released several seconds to minutes after the initial fission event. These delayed neutrons play a crucial role in controlling the rate of a nuclear chain reaction.

## 2. How is the delayed neutron fraction calculated?

The delayed neutron fraction is calculated by dividing the total number of delayed neutrons released in a fission reaction by the total number of neutrons released in the fission reaction (including both prompt and delayed neutrons). This calculation is typically done using experimental data from nuclear reactors or through computer simulations.

## 3. Why is the delayed neutron fraction important?

The delayed neutron fraction is important because it helps to control the rate of a nuclear chain reaction. Without delayed neutrons, a nuclear reactor would rapidly reach criticality and potentially lead to a nuclear meltdown. The delayed neutron fraction also plays a role in reactor safety and design, as it affects the amount of control rods and other safety mechanisms needed to maintain a stable reaction.

## 4. How does the delayed neutron fraction vary between different types of nuclear fuels?

The delayed neutron fraction can vary between different types of nuclear fuels, as it is dependent on the specific isotopes present in the fuel. For example, uranium-235 has a higher delayed neutron fraction than plutonium-239. This variation can impact the stability and efficiency of a nuclear reactor, and is an important consideration in fuel selection and design.

## 5. Can the delayed neutron fraction be changed?

The delayed neutron fraction is an inherent property of a specific nuclear fuel and cannot be changed. However, it can be controlled to some extent through the use of different types of fuel and reactor designs. Additionally, changes in the delayed neutron fraction can occur due to changes in reactor conditions, such as temperature and fuel composition.

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