BurnupFunction

Description

The power density in a fuel pellet varies radially as a function of geometry, initial fuel characteristics, and irradiation history. At the beginning of irradiation (low burnup), the concentration of fissile material is uniform, which means that the radial power has a relatively small variation across the radius. With increasing burnup, the Pu concentration markedly increases near the fuel surface due to the capture of epithermal neutrons in the resonances of U. Consequently, the concentration of fissile material and the power density profile are steeper near the pellet surface. These aspects need to be captured in order to calculate the heat generation and temperature distribution in the pellet accurately.

For this purpose, BISON uses the DIONISIO model from Soba et al. (2013), Soba et al. (2014), and Soba and Denis (2017). This model computes the evolution of the concentrations of various heavy metal isotopes (specifically, U, U, U, Pu, Pu, Pu, Pu), in addition to Gd dopping for Gd and Gd (Sweet et al., 2023), locally across the radius of the fuel pellet and the coupled evolution of the radial power and burnup distributions. The model uses one-group cross sections that are function of radial position, U enrichment and radially averaged burnup, and are fitted to the results of extensive neutronics calculations performed with the CONDOR and HUEMUL codes. Neutronics calculations covered LWR-UO fuel rods with initial U enrichments from 0.7% to 12% and average burnups ranging from fresh fuel to 120 MWd/kgU (Soba et al., 2013; Soba et al., 2014; Soba and Denis, 2017).

The rate equations describing the evolution of the concentrations of each heavy metal isotope and gadolinia are where is the local concentration (number of atoms per unit volume), is the absorption cross section, is the capture cross section, and is the neutron flux. Note that decay terms are neglected in the BISON implementation.

Figure 1: Example of calculated radial power form factors (normalized) at various radial average burnups.

The neutron flux, , is calculated in an approximate way using one-group diffusion theory. The neutron diffusion equation is solved in 1D cylindrical coordinates by finite difference.

The local power density at radial position , , which is needed for the thermal analysis is proportional to the neutron flux and the macroscopic cross section for fission, (1)

where is the fission cross section for isotope and is the concentration of isotope . The model computes the radial power profile in the fuel pellet based on Eq. (1). The radial power from factor at radial position is

(2)

with (3)

where and are the inner and outer fuel radii, respectively. The local burnup is computed based on the time integral of the local power density.

An example of model results is given in Figure 1, which illustrates radial power factor distributions at various burnup levels for a UO fuel pellet irradiated under typical PWR conditions.

Example Input Syntax

The burnup function is often created by the Burnup action, as shown below:

[Burnup<<<{"href": "../../syntax/Burnup/index.html"}>>>]
  [burnup2]
    block<<<{"description": "The blocks where radial power factor should be computed."}>>> = 2
    base_name<<<{"description": "Base name for the AuxVariables."}>>> = action_block2
    rod_ave_lin_pow<<<{"description": "Rod average linear power function."}>>> = power_profile
    axial_power_profile<<<{"description": "Axial power peaking function."}>>> = axial_peaking_factors
    num_radial<<<{"description": "Number of radial divisions in secondary grid used to compute radial power profile."}>>> = 80
    num_axial<<<{"description": "Number of axial divisions in secondary grid used to compute radial power profile."}>>> = 20
    a_upper<<<{"description": "The upper axial coordinate of the fuel stack. Required if fuel_pin_geometry is not specified."}>>> = 0.0205
    a_lower<<<{"description": "The lower axial coordinate of the fuel stack. Required if fuel_pin_geometry is not specified."}>>> = 0.0105
    fuel_inner_radius<<<{"description": "The inner radius of the fuel."}>>> = 0.0
    fuel_outer_radius<<<{"description": "The outer radius of the fuel."}>>> = 0.01
    fuel_volume_ratio<<<{"description": "Reduction factor for deviation from right circular cylinder fuel.  The ratio of actual volume to right circular cylinder volume."}>>> = 1.0
  []
[]

It is also possible to directly use the BurnupFunction in an input file:

[Functions<<<{"href": "../../syntax/Functions/index.html"}>>>]
  [burnup2]
    type = BurnupFunction<<<{"description": "Computes burnup and radial power factor.  Built by an Action.", "href": "BurnupFunction.html"}>>>
    rod_ave_lin_pow<<<{"description": "Rod average linear power function."}>>> = power_profile
    axial_power_profile<<<{"description": "Axial power peaking function."}>>> = axial_peaking_factors
    num_radial<<<{"description": "Number of radial divisions in secondary grid used to compute radial power profile."}>>> = 80
    num_axial<<<{"description": "Number of axial divisions in secondary grid used to compute radial power profile."}>>> = 20
    a_upper<<<{"description": "The upper axial coordinate of the fuel stack. Required if fuel_pin_geometry is not specified."}>>> = 0.0205
    a_lower<<<{"description": "The lower axial coordinate of the fuel stack. Required if fuel_pin_geometry is not specified."}>>> = 0.0105
    fuel_inner_radius<<<{"description": "The inner radius of the fuel."}>>> = 0.0
    fuel_outer_radius<<<{"description": "The outer radius of the fuel."}>>> = 0.01
    fuel_volume_ratio<<<{"description": "Reduction factor for deviation from right circular cylinder fuel.  The ratio of actual volume to right circular cylinder volume."}>>> = 1.0
  []
[]

Initial enrichment defaults to U235=5% and U238=95%. These isotopes and values can be customized through the use of isotopes and isotope_fractions. The complete list of available isotopes is U235, U238, Pu239, Pu240, Pu241, and Pu242.

[Burnup<<<{"href": "../../syntax/Burnup/index.html"}>>>]
  [burnup]
    block<<<{"description": "The blocks where radial power factor should be computed."}>>> = 0
    rod_ave_lin_pow<<<{"description": "Rod average linear power function."}>>> = power_profile
    axial_power_profile<<<{"description": "Axial power peaking function."}>>> = axial_peaking_factors
    num_radial<<<{"description": "Number of radial divisions in secondary grid used to compute radial power profile."}>>> = 80
    num_axial<<<{"description": "Number of axial divisions in secondary grid used to compute radial power profile."}>>> = 5
    a_upper<<<{"description": "The upper axial coordinate of the fuel stack. Required if fuel_pin_geometry is not specified."}>>> = 0.0082
    a_lower<<<{"description": "The lower axial coordinate of the fuel stack. Required if fuel_pin_geometry is not specified."}>>> = 0.0
    fuel_inner_radius<<<{"description": "The inner radius of the fuel."}>>> = 0
    fuel_outer_radius<<<{"description": "The outer radius of the fuel."}>>> = 0.0041
    fuel_volume_ratio<<<{"description": "Reduction factor for deviation from right circular cylinder fuel.  The ratio of actual volume to right circular cylinder volume."}>>> = 1
    isotopes<<<{"description": "Fuel isotopes: Gd155 Gd157 U235 U238 Pu239 Pu240 Pu241 Pu242.  Number of entries must match number of entries in isotope_fractions."}>>> = 'U235 U238'
    isotope_fractions<<<{"description": "The isotope fractions associated with the 'isotopes' input line.  Must sum to 1.0."}>>> = '0.045 0.955'
    RPF<<<{"description": "Specifies that the radial power factor is required."}>>> = RPF
  []
[]

Input Parameters

Input Files

• (test/tests/burnup_action/burnup_with_actions.i)
• (test/tests/burnup_action/burnup_without_actions.i)

References