UPuZrGaseousEigenstrainwithHotPressingPuSwelling

Description

UPuZrGaseousEigenstrainwithHotPressingPuSwelling computes a volumetric strain to account for gaseous swelling and hot-pressing from Fuel Cladding Mechanical Interaction (FCMI) in U-Pu-Zr metal fuel systems. The derivation for the fuel swelling model for U-Pu-Zr used here was originally presented in Olander (1976) and is used within UPuZrGaseousEigenstrain. UPuZrGaseousEigenstrainwithHotPressingPuSwelling adds to this by allowing the U-Pu-Zr fuel to compress from FCMI and thermal stress by adjusting the volumetric strain and gaseous porosity based off correlations in Ogata and Yokoo (1999). The model is able to simplify the gaseous swelling porosity into an analytical solution by making the following assumptions:

• Fission gas diffusivity is infinite so that all fission gas is born in the bubbles;

• The number density of bubbles is constant and their shape is perfectly spherical;

• The gas within the bubbles can be treated as an ideal gas;

• The gas/vacancy ratio in the bubbles is in equilibrium, i.e., the bubbles are not over- or under-pressurized;

• Re-solution of fission gas atoms from the bubbles is neglected;

• The solid is assumed to be un-stressed.

Following these assumptions, the number of gas atoms in each bubble, , at some time, , can be calculated directly from the total number of fissions, (1) where is the yield of gas atoms per fission, is the number density of bubbles (bub/m), is a fixed constant representing the amount of fission gas retained in the solid, and is the average number of fissions between and , where is the fission rate density.

Also included in Eq. (1) is the impact of interconnectivity of the bubbles on the total number of gas atoms per bubble. In the absence of a mechanistically informed interconnection model, the total bubble interconnection can be estimated from the porosity by, (2) where and is the interconnection initiating and terminating porosity respectively, and is the bulk porosity at time . and can be set via the input parameters interconnection_initiating_porosity and interconnection_terminating_porosity. Eq. (2) ensures a linear transition of interconnection from 0 to 1.0, capturing the smooth transition from non-interconnected to fully interconnected porosity.

The bubble radius can be simply calculated as a function of the total number of fissions, where is the Boltzmann constant, is the temperature (K), is the surface tension of the fuel (J/m) (Karahan, 2009), is the gaseous fission product yield, is the fission rate density, is time, and is the number density of bubbles (bubbles/m).

The total eigenstrain from gaseous fission products and hot pressing of the U-Pu-Zr fuel can be calculated as, (3)

where is the volumetric strain from hot-pressing

where is the creep rate inheritied from UPuZrCreepUpdate and uses the effective hydrostatic stress.

Several checks are performed on the value calculated by Eq. (3) ensuring that swelling due to gas bubbles only increases and is not larger than the maximum permissible swelling calculated from , where is some initial porosity, e.g. fabrication porosity.

The porosity due to gas and hot-pressing can be calculated from the swelling eigenstrain as, and the total porosity can then be calculated from the gaseous eigenstrain as,

Finally, the input parameter anisotropic_factor can be optionally specified between 1 and -1 to preferentially apply the volumetric strain in the radial direction when the axial direction is the y-axis. The factor is applied in Cartesian coordinates, so the x- and z-dimensions are equally modified with the axial y-dimension offsetting both. However, default values for the anisotropic factor are dependent on plutonium content and have been determined empirically using EBR-II data. This resulted in a linear regression of 0.72 due to noise in the EBR-II data. The factor is applied such that the trace of the volumetric swelling is the same as purely isotropic swelling, but preferential in the radial dimensions:

where is the plutonium atom fraction.

Example Input Syntax

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  [gas_swelling]
    type = UPuZrGaseousEigenstrainwithHotPressingPuSwelling<<<{"description": "Computes and sums the change in fuel pellet volume due to gaseous fission product buildup  and hot pressing due to hydrostatic stress in UPuZr.", "href": "UPuZrGaseousEigenstrainwithHotPressingPuSwelling.html"}>>>
    eigenstrain_name<<<{"description": "Material property name for the eigenstrain tensor computed by this model. IMPORTANT: The name of this property must also be provided to the strain calculator."}>>> = gas_swelling_eigenstrain
    temperature<<<{"description": "Coupled temperature variable"}>>> = temp
    initial_porosity<<<{"description": "Initial or fabrication porosity"}>>> = 0.03185
    bubble_number_density<<<{"description": "Material property name for the number density of intragranular bubbles, [bubbles/m^3]"}>>> = 5e17
    interconnection_initiating_porosity<<<{"description": "Porosity at which fission gas release starts"}>>> = 0.24812
    interconnection_terminating_porosity<<<{"description": "Porosity at which fission gas release finishes"}>>> = 0.26812
    creep_rate<<<{"description": "Creep rate material property name"}>>> = creep_rate
    hydrostatic_stress<<<{"description": "Hydrostatic stress in fuel"}>>> = hydrostaticstress
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
    output_properties<<<{"description": "List of material properties, from this material, to output (outputs must also be defined to an output type)"}>>> = 'porosity gaseous_porosity hot_pressing'
    hotpress_scalar<<<{"description": "Scale factor to be applied to the pressing strain. Used for calibration and sensitivity studies"}>>> = 1.0
    plenum_pressure<<<{"description": "plenum pressure used as hydrostatic pressure"}>>> = plenum_pressure
  []
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UPuZrGaseousEigenstrainwithHotPressingPuSwelling must be used in conjunction with the solid mechanics quasi action to apply the calculated eigenstrain:

[Physics<<<{"href": "../../../syntax/Physics/index.html"}>>>]
  [SolidMechanics<<<{"href": "../../../syntax/Physics/SolidMechanics/index.html"}>>>]
    [QuasiStatic<<<{"href": "../../../syntax/Physics/SolidMechanics/QuasiStatic/index.html"}>>>]
      [all]
        strain<<<{"description": "Strain formulation"}>>> = FINITE
        add_variables<<<{"description": "Add the displacement variables"}>>> = true
      []
    []
  []
[]

Input Parameters

Input Files

• (test/tests/solid_mechanics/upuzr_eigenstrains/upuzr_gaseous_eigenstrain_with_hot_pressing_pu_swelling/exact.i)
• (examples/metal_fuel/X447_coarse/DP21_test.i)

References