In-pile, short-lived isotopes

commentnote:More Information Available

The material presented here is a subset of the information found in Toptan et al. (2023). See the paper for more information.

A solid sphere of radius is subject to following boundary conditions:

  1. A radial symmetry is considered at : for all .

  2. Concentration is kept at zero at the outer surface: for all .

The in-pile conditions for a decaying fission product () are considered with the following settings: the initial concentration is zero in the sphere ( for ), and a non-zero source (). In this study, we assume a constant source generation, . The mass diffusion solution (or conservation of fission products), one of the Bison's governing equations, is solved in the spherical coordinates. The analytical expressions (Nabielek et al., 1974; Toptan et al., 2023) are given below for concentration, release rate, and total fractional release.

Concentration

The concentration profile, is expressed as (1) and as , Eq. (1) reduces to (2) with the following dimensionless numbers:

  • is the dimensionless radius (unitless)

  • is the diffusion time (unitless)

  • (unitless)

where (s) is the decay constant, and (s) is the reduced diffusivity coefficient.

The concentration, , at a specific point is computed in the code via

[Postprocessors<<<{"href": "../../syntax/Postprocessors/index.html"}>>>]
  [conc_point_25]
    type = PointValue<<<{"description": "Compute the value of a variable at a specified location", "href": "../../source/postprocessors/PointValue.html"}>>>
    variable<<<{"description": "The name of the variable that this postprocessor operates on."}>>> = conc
    point<<<{"description": "The physical point where the solution will be evaluated."}>>> = '${fparse 0.25*kernel_radius} 0 0'
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'initial timestep_end'
  []
[]
(assessment/verification/TRISO_diffusion/inpile/inpile.i)

Figure 1 compares analytical and computed concentration, , at four different radial locations (0.25, 0.50, 0.75, and 0.95) as a function of dimensionless diffusion time, (arbitrarily chosen ).

Figure 1: Comparison of analytical and computed concentration, at four different radial locations (0.25, 0.50, 0.75, and 0.95) as a function of dimensionless diffusion time, (arbitrarily chosen ).

Release-rate over birth-rate (R/B)

The release-rate over birth-rate (R/B) is defined as in terms of the ratio of release rate , the source , and the sphere volume . The release rate, is computed in the code via

[Postprocessors<<<{"href": "../../syntax/Postprocessors/index.html"}>>>]
  [release_inc]
    type = SideIntegralMassFlux<<<{"description": "Computes the integrated mass flux over a given surface.", "href": "../../source/postprocessors/SideIntegralMassFlux.html"}>>>
    variable<<<{"description": "The name of the variable which this postprocessor integrates"}>>> = conc
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = exterior
    arrhenius_prpty_name<<<{"description": "Specify the name of the material property"}>>> = diffusion_coef
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'initial timestep_end'
  []
[]
(assessment/verification/TRISO_diffusion/inpile/inpile.i)

Thus, the (R/B) ratio for a long-lived (or stable; with ) fission product under the in-pile conditions becomes

Figure 2 compares analytical and computed release-rate over birth-rate, (R/B) as a function of the diffusion time, .

Figure 2: Comparison of analytical and computed release-rate over birth-rate (arbitrarily chosen 0.1).

Fractional release

The analytical expression for the fractional release, is expressed as:

The fractional release is computed in the code via

[Postprocessors<<<{"href": "../../syntax/Postprocessors/index.html"}>>>]
  [x_released]
    type = FractionalRelease<<<{"description": "Compute the fractional release of fission products.", "href": "../../source/postprocessors/FractionalRelease.html"}>>>
    released<<<{"description": "Postprocessor to be used as the released value."}>>> = released
    retained<<<{"description": "Postprocessor to be used as the retained value."}>>> = retained
  []
[]
(assessment/verification/TRISO_diffusion/inpile/inpile.i)

Figure 3 compares analytical and computed release fractions, as a function of dimensionless diffusion time, .

Figure 3: Comparison of analytical and computed total release fractions (arbitrarily chosen 0.1).

warningwarning:Fractional release for larger values

The fractional release results do not agree with the expected analytical results for larger values. This behavior is only observed during the in-pile condition for short-lived isotopes (). The source of this mismatch is identified and the relevant post-processors are to be corrected.

Summary

The Bison predictions agree well with the analytical results for concentration profiles, release rate, release-rate over birth-rate (R/B), and fractional release out of a solid sphere under in-pile conditions for a decaying fission product.

References

  1. H. Nabielek, H. Hick, M. Wagner-Lรถffler, and E. H. Voice. Performance limits of coated particle fuel, Part III: Fission product migration in HTR fuel. Technical Report DP-Report-828(Pt.3), O.E.C.D High Temperature Reactor Project Dragon, 6 1974.[BibTeX]
  2. A. Toptan, W. Jiang, J. D. Hales, B. W. Spencer, and S. Novascone. Verification of Bison fission product species conservation under TRISO reactor conditions. Journal of Nuclear Materials, 573:154105, 2023. doi:10.1016/j.jnucmat.2022.154105.[BibTeX]