Out-of-pile, a decaying fission product

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 out-of-pile conditions for a decaying fission product () are considered with the following settings: the initial concentration is non-zero in the sphere ( for ), and a zero source (). 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 with the following dimensionless numbers:

  • is the dimensionless radius (unitless)

  • is the diffusion time (unitless)

  • (unitless)

where (s) is the decay constant, is the initial concentration, 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/outofpile/outofpile.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 ). As , the dimensionless concentration profile approaches zero.

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

The out-of-pile release rate, is expressed as:

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/outofpile/outofpile.i)

Figure 2 compares analytical and computed release rate, as a function of the diffusion time, .

Figure 2: Comparison of analytical and computed release rate as a function of dimensionless diffusion time, (arbitrarily chosen ).

Fractional release

The analytical expression for the fractional release, is given by:

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/outofpile/outofpile.i)

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

Figure 3: Comparison of analytical and computed release fractions.

Summary

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