Model UO2(5%)+Th+U233 Fuel in MCNPX for SCWR

[Aly_19f]

TL;DR Summary

How do I calculate the number density of each element in an enriched fuel?

Hey there, I'm working on an MCNPX modelling for SCWR using different clads and fuels, the first fuel was UO2(5%) and I have calculated the number density correctly since there was only one vector U.
But now I don't know how top deal with the Th+U233 due to the existence of Thorium.
Can anyone help me or guide me to a similar solved problem (Uranium + another heavy fuel)?

 

[Astronuc]

Summary:: How do I calculate the number density of each element in an enriched fuel?

But now I don't know how top deal with the Th+U233 due to the existence of Thorium.

Well, one is asking how to model a 'mixed-oxide' fuel, which could be (U,Pu)O₂, (Th,U)O₂, (Th,Pu)O₂, or even (U,Th,Pu)O₂; they all form MO₂, where M = Th, U, and/or Pu. The enrichments are given in percent by weight (wt%). It's relatively easy in UO₂, since the lattice is the same regardless of U-isotope, but it is fairly straightforward in MOX fuel. One has to do slight adjustments for oxide density of mixtures.

The 3.66% U235 or U233 would imply that the remainder of the metal (Th) is 96.34%. Note the densities to the mixed-oxide and are less than the theoretical density (due to porosity in the ceramic material) and vary slightly due to the differences in mass of the fissile nuclides.

 

[Aly_19f]

Well, one is asking how to model a 'mixed-oxide' fuel, which could be (U,Pu)O₂, (Th,U)O₂, (Th,Pu)O₂, or even (U,Th,Pu)O₂; they all form MO₂, where M = Th, U, and/or Pu. The enrichments are given in percent by weight (wt%). It's relatively easy in UO₂, since the lattice is the same regardless of U-isotope, but it is fairly straightforward in MOX fuel. One has to do slight adjustments for oxide density of mixtures.

The 3.66% U235 or U233 would imply that the remainder of the metal (Th) is 96.34%. Note the densities to the mixed-oxide and are less than the theoretical density (due to porosity in the ceramic material) and vary slightly due to the differences in mass of the fissile nuclides.

I still don't know how to deal with the problem, in UO₂ the sum of the Uranium atom fractions is equal one whatever the enrichment is.
But by applying the same method in the MOX it is always >1.
Is there any textbook or tutorial about burnup calculations?

 

[Astronuc]

The 3.66% U235 or U233 would imply that the remainder of the metal (Th) is 96.34%. Note the densities to the mixed-oxide and are less than the theoretical density (due to porosity in the ceramic material) and vary slightly due to the differences in mass of the fissile nuclides.

Of course, I'm referring to (U,Th)O₂.

But by applying the same method in the MOX it is always >1.

Probably the method assumes a common density, which is a reasonable assumption for UO₂, but not MOX, but of course, the atom fractions should add to one.

Is there any textbook or tutorial about burnup calculations?

I would think most textbooks for Introductory Nuclear Engineering or Nuclear Reactor Theory would have some background or examples. This is essentially a basic chemistry problem.

The atom density, N, is given by N = ρ Av / M, where ρ is the density, Av is Avogadro's number and M is the molecular mass.

When calculating atomic densities of mixed oxides, MO₂, one may apply the rule of mixtures, but must understand the relationship of mass, density and volume, and it can get complicated when each heavy metal element has multiple isotopes, e.g., ₂₃₃U, ²³⁵U, ²³⁸U, and there may be ²³⁴U and ²³⁶U, and with Pu, ²³⁸Pu, ²³⁹Pu, ²⁴⁰Pu, ²⁴¹Pu and ²⁴²Pu.

Rule of mixtures - https://en.wikipedia.org/wiki/Rule_of_mixtures
https://www.sciencedirect.com/topics/engineering/rule-of-mixture-equation

With mixed oxides, one has to remember that there is one M (Th,U,Pu) for each O₂.
So if we want to calculate atomic densities of M, then we need to keep in mind that we would first calculate the molecular density of MO₂ from the mass density, and then multiply by the mass ratio M/MO².

Also, one must remember the physical density is less than the theoretical density, because ceramics do not sinter to 100% of theoretical density. There is some inherent porosity, which is process specific.

The theoretical densities of the compounds interest are:

ThO₂, 10.00 g/cm³
UO₂, 10.96 g/cm³, although some use 10.97 or 10.98, which might indicate an influence of the proportions of ²³⁵U and ²³⁸U in the UO₂; getting a pure amount of ²³⁵U or ²³⁸U is exceedingly difficult, or there could be some porosity or other impurity.
PuO₂, 11.46 g/cm³

One should note the ratio of M to MO₂. For example, for UO₂, one could approximate it by 238/(238+2*16) = 238/270 = 0.8815. If one wishes to be more precise, then one would have to account for the proportions of ²³⁵U and ²³⁸U. For example, for 5% enrichment, the ratio is (0.05*235+0.95*238)/(0.05*235+0.95*238+2*16) = (237.85)/(237.85+32) = 0.8814. So, it's a small difference, since the atomic masses of the isotopes are very close.

For densities of mixtures of oxides, one can use the atom% of the metals, which are close to the wt%. For example, the theoretical density of (U_0.8, Pu_0.2)O₂ is approximately 11.08 g/cm³. Using the densities above for UO₂ = 10.96 and PuO₂ = 11.46, then multiplying by the atomic fractions 0.8 and 0.2, respectively, one obtains 10.96 g/cm³ * 0.8 + 11.46 g/cm³ * 0.2 = 11.06 g/cm³.

In the above example, given the 3.66% ²³³U enrichment in (U,Th)O₂, and densities of 10.96 g/cm³ and 10 g/cm³ for UO₂ and ThO₂, respectively, then the theoretical density of the mixture/compound is 0.0366*10.96 g/cm³ + 0.9634 * 10 g/cm³ = 10.04 g/cm³. So, the theoretical density (TD) is slighly greater than that of ThO₂. The table one provided has the density at 9.5 g/cm³, from which one would infer that the mixed oxide ceramic has a density of ~0.95 of theoretical, which is a typical target of oxide ceramic nuclear fuel. In the distant past, ceramic densities were lower, ~0.87-0.94 TD.

 

[nuclearsneke]

Summary:: How do I calculate the number density of each element in an enriched fuel?

Hey there, I'm working on an MCNPX modelling for SCWR using different clads and fuels, the first fuel was UO2(5%) and I have calculated the number density correctly since there was only one vector U.
But now I don't know how top deal with the Th+U233 due to the existence of Thorium.
Can anyone help me or guide me to a similar solved problem (Uranium + another heavy fuel)?

N/v for fuel is (rho_fuel*N_a/M(th232*(1-0.0366)+u233*(0.0366)))
N/v for u233 is n/v_fuel*0.0366
N/v for th232 is n/v_fuel*(1-0.0366)

These quantities will be in cm^-3

 

[Astronuc]

TABLE 3.2-ROOM TEMPERATURE LATTICE CONSTANTS AND THEORETICAL DENSITIES OF ThO₂-UO₂ SOLID SOLUTIONS
https://inis.iaea.org/collection/NCLCollectionStore/_Public/16/071/16071971.pdf