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 Post subject: Cross Section ProportionsPosted: Oct 31, 2015 9:30 am

Joined: Jan 16, 2012 7:15 am
Posts: 94
I hope you folks can confirm some simple assumptions about the equivalence of fission and absorption cross sections.

Data (from BNL Atlas database, for thermal neutrons only)
235-U fission X-section is 582.6.
232-Th absorption X-section is 7.35.

Assumption 1:
If a reactor using pure 235-U provides enough neutrons for a 1.00 breeding ratio, you would need 582.6/7.35 = 80 times as much Thorium in the blanket as Uranium in the core. (Seems high to me!)
Given that 5% enriched Uranium is more likely, you would need about four times as much Thorium as Uranium. This ignores any 239-Pu bred from 238-U in this fuel mix.

Assumption 2:
The absorption X-section for 235-U is 98.8. With its fission X-section (582.6) the total is 681.4. So, fission efficiency is 582.6/681.4 = 85.5%. 14.5% of the 235-U becomes 236-U.
I’ve heard the estimate that a 1 GW reactor will consume a ton of fissile per year. Such a reactor will thus produce 145 kg of 236-U per year. Again, 235-U must be the only fissile. Any 239-Pu contribution would produce 240-Pu “waste” in lieu of 236-U.

Assumption 3:
238-U absorption cross section is 2.68. If the 1 GW plant above uses 5% enrichment, (i.e., 19000 kg 238-U) then 239-Pu production is 19*2.68 = 51 kg per year. (Seems low) This assumes no absorption by 239-U.

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 Post subject: Re: Cross Section ProportionsPosted: Nov 06, 2015 8:53 am

Joined: Jan 16, 2012 7:15 am
Posts: 94
Perhaps I should have repeated at the bottom of the post:

Please confirm my assumptions - or reject them and explain why.

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 Post subject: Re: Cross Section ProportionsPosted: Nov 07, 2015 7:51 pm

Joined: Jul 14, 2011 9:22 pm
Posts: 31
Location: Forest Park, GA
I'm just a student, but it seems that everything hinges on the first assumption of 100% U-235 giving a 1.00 breeding ratio. Where did that assumption come from?

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 Post subject: Re: Cross Section ProportionsPosted: Nov 08, 2015 11:03 am

Joined: Jan 16, 2012 7:15 am
Posts: 94
Vince,
The 100% 235-U assumption allows me to use the cross sections for just 235-U, without worrying how to include the contribution from any other fuel. I intend to refine that assumption as my analysis progresses. For instance, according to assumption 3, after a year there should be 51 kg of 239-Pu in the fuel mix. I would assume that ~5% of the energy from my notional reactor comes from 239-Pu.

That would be my second overarching assumption of linearity. The first assumes that the products of neutron absorption - fission products and heavier actinides - are proportional to the fission and absorption cross-sections. The second assumes that results of fission - energy and fission products - are proportional to the ratio of cross sections between two different fuels. (The simplifying assumptions behind the 5% is that whether the fuel is 235-U or 239-Pu, there are no differences in energy produced or fission product distribution. Again, these assumptions can be removed in later analysis.)

The 1.00 breeding ratio merely simplifies the calculation for Thorium requirements. My notional reactor has no blanket yet. In fact, I'm still working to get it running for more than one second.

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 Post subject: Re: Cross Section ProportionsPosted: Nov 17, 2015 2:20 pm

Joined: May 24, 2015 9:43 pm
Posts: 21
SteveMoniz wrote:
I hope you folks can confirm some simple assumptions about the equivalence of fission and absorption cross sections.

Assumption 1:
If a reactor using pure 235-U provides enough neutrons for a 1.00 breeding ratio, you would need 582.6/7.35 = 80 times as much Thorium in the blanket as Uranium in the core.

Neutron flux is typically not evenly distributed in practical reactors.

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 Post subject: Re: Cross Section ProportionsPosted: Nov 18, 2015 12:27 pm

Joined: Jan 16, 2012 7:15 am
Posts: 94
Quote:
Neutron flux is typically not evenly distributed in practical reactors.

I expect the neutron flux is considerably less in the blanket, simply from the extra distance from the core. My assumption specified enough neutrons to compensate for this. ORNL expected a 1.06 breeding ratio for an MSR optimized for breeding, and 0.8 (if I recall correctly) for the non-optimized DMSR. So, the assumption behind my assumption is that the engineers figure out a way to achieve a 1.00 breeding ratio. This is a mental exercise, so I don't need to specify how.

Specific to the first assumption, does a varying neutron flux affect the calculated ratio of 80:1? Is that estimate too high or too low?

More generally, is that estimate meaningful? Remember the overall question is whether or not I can compare barns directly. The second assumption compares fission and absorption barns, from the viewpoint of a neutron approaching a nucleus. It "sees" a cross section of 582 barns for fission, and 99 for absorption. I assume that is where the 85% "fission efficiency" for 235-U comes from. (And with it, 75% for 239-Pu and 92% for 233-U.) Are these calculations over-simple?

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 Post subject: Re: Cross Section ProportionsPosted: Nov 18, 2015 11:39 pm

Joined: Nov 23, 2010 6:51 pm
Posts: 123
The fissile:fertile cross-section ratio will provide an estimate of the minimum blanket to core mass ratio, but there is no maximum blanket size - just diminishing returns after it becomes large enough to absorb the majority of the neutrons that leak from the core. You also want to use the total (fission + capture) cross section for the fissile isotope (total neutron absorption in fissile should equal total absorption in fertile for a breeding ratio of 1.00).

Standard cross section tables are for fully thermalized neutrons. Unless a reactor is strongly over-moderated, there will be a substantial population of higher energy neutrons that have not yet been fully thermalized, which will result in different average cross sections as well as different cross section ratios between nuclides. Generally, cross section ratios become closer to 1 as neutron energy increases. Also, fissile isotope capture resonances in the epithermal region tend to lower the fission probability somewhat compared to either fully thermal or fast neutrons. Common estimates for LWR spectrum are ~75-80% fission probability for 235U and 65-70% for 239Pu.

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 Post subject: Re: Cross Section ProportionsPosted: Nov 20, 2015 9:27 am

Joined: Jan 16, 2012 7:15 am
Posts: 94
Titan,
Quote:
use the total (fission + capture) cross section

I think I get your drift. The 235-U required to produce new 233-U is (582 + 99)/582. You need to add 17% more 235-U because some is “wasted”. What do you call the 17%? Failed fuel? Future fuel?

Quote:
cross section ratios get closer to one as neutron energy increases

If the capture percentage climbs with faster neutrons, then fast reactors would seem to have an inherent inefficiency for burning spent nuclear fuel. Fast reactors may be the only choice for this purpose, but they would be less efficient that I would have assumed before I read your comment.

Quote:
standard tables use fully thermalized neutrons

So where might I get fast cross section data? I can use 100% thermal neutrons as a start, but want to include fast neutrons eventually.

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 Post subject: Re: Cross Section ProportionsPosted: Nov 20, 2015 4:51 pm

Joined: Nov 23, 2010 6:51 pm
Posts: 123
Cross-section ratios between isotopes generally get closer to 1 as neutron energy increases. This is why fast reactors require higher concentrations of fissile isotopes, and also why they are tolerant of higher levels of fission products. The low concentrations of fissile isotopes in thermal reactors are enabled by the very large cross sections of the fissile isotopes at low neutron energy.

Fission to capture ratios behave differently - for most isotopes, they are a bit lower in the resonance region than for thermal neutrons, but rise again when neutron energies are above the capture resonances. In a fast spectrum, the fission to capture ratio is substantially higher than in thermal spectrum for 235U and all Pu isotopes. This is why breeding in the U / Pu cycle requires fast spectrum - there is too much fissionless capture in 239Pu at lower neutron energies. Basically, fast neutrons are good, very slow neutrons are OK, and certain energies in the middle are bad.

Lots of cross section data available at http://www.nndc.bnl.gov/sigma/

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 Post subject: Re: Cross Section ProportionsPosted: Nov 23, 2015 10:13 am

Joined: Jan 16, 2012 7:15 am
Posts: 94
Yes, I used the Sigma database for fission yields. But I could not see any relationship between the (n, total fission) interpreted data and the plot. Looking at the plot again, (thanks) I noticed a link to interpreted data and a text link that leads to a much better looking data set.

I don't see a direct link to capture data cross sections. However, there is a “click to expand” link that has a (n, gamma?) link at the bottom of a list of (n, #) reactions. That leads to another list that has 98.8 as the cross section for 235-U at 0.253 eV, which is a popular neutron energy value for some reason. I assume that is what I should use for capture barns. (Please confirm)

Use for what, you might ask...

I'm still working on my salt analysis tool. The waste stream from an MSR will depend on the reactor design, which is a user input. Currently, that consists of two numbers, fast percent and thermal percent. In a thermal reactor you may get 8-10% of your fissions from fast neutrons, so the user input might be 8/92. (I don't know what the ratio of fast/slow “effective” neutrons in a fast reactor might be.)

If I can get epi-thermal data, I might make that three input numbers – a neutron effectiveness distribution. Note that I can't tell how much time a neutron spends in each region. That depends on the design, and the user will have to consider that in developing the input parameters.

With text data for cross sections, I can aggregate the fission and capture barns into three regions: < 1 eV (thermal), 1 to 1000 eV (epi), and >1000 eV (fast).

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 Post subject: Re: Cross Section ProportionsPosted: Nov 24, 2015 9:02 am

Joined: Jan 16, 2012 7:15 am
Posts: 94
In particular, is there is a standard procedure for aggregating barns?

It would be nice if the x-values chosen for these tables reflects the deceleration curve of a neutron. The neutron would spend the same amount of time in the region of each x-value, and the best aggregate strategy would be the simple average of the barns.

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 Post subject: Re: Cross Section ProportionsPosted: Nov 26, 2015 3:10 pm

Joined: Jan 16, 2012 7:15 am
Posts: 94
Koistinen,
Quote:
Neutron flux is typically not evenly distributed in practical reactors.

I now have a simple model of my reactor, including some neutron flux assumptions.

The core is a 1 meter radius sphere, within a 2 meter radius blanket. The neutron flux at the very center is 2.4 “units” - using whatever multiplier is needed to match a notional average of 2.4 freed neutrons per fission. Most of the neutrons are fast, and escape the inner core. 1.0 of them cause fission in the inner core, which I assume reaches 50% of the way to the core/blanket barrier. In the outer core, keff is less than 1.0, dropping to an assumed 0.5 at the barrier. Note that theses two assumptions use the 50% “first cut estimate” in operations research - if you don't the answer, set upper and lower boundaries, and take the midpoint.

I assume the flux is 1.2 at the barrier, another 50% assumption. But this assumption is based on geometry, which is a bit better justified. Consider a nucleus at the very top of the core. It gets hit by neutrons only from below, so it “sees” only half as many neutrons as a nucleus at the center where the flux is 2.4.

I also assume the flux at the outer blanket boundary is not zero and some neutrons are truly wasted on the containment walls. It's tempting to say 0.2 are wasted, since I'm assuming a 1.00 breeding ratio. However, I fear that would be mixing neutrons and neutron flux.

The outer core is 8x the volume of the inner core. The blanket is 8x the volume of the whole core. This is just a coincidence. It comes from assumption 1 in my first post: that we will need 4 times more TH than (5%) U. This is modified by the model I just described, where the flux in the blanket is about half that in the core. Given Titanium-48’s suggestion, 8x may be closer to 9x, and the blanket radius will be just over 2 meters. 2 meters might still be good if Thorium is slightly more soluble in molten salt than Uranium. 2 meters will not be enough if we go with a tube rather than a sphere.

Actually I'm not sure I need this level of design, though it's fun. I'm trying to simulate the waste streams of various MSR designs. The breeding stream (hardly waste!) is a sideshow, mostly included because it is one of the simplest actinide evolutions. For that, I can just assume that each year in 1 GW iso-breeder, 1000 kg of 232-Th evolves to 1000 kg 233-U. I might have some "waste" (232-U) in this stream if I ever get around to incorporating the (n, 2n) reaction in my sim.

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