GrainRadiusMechanistic

Mechanistic model for calculating grain growth in nuclear fuel.

Description

This auxkernel mechanistically predicts the growth of fuel grains in the presence of a lognormal distribution of mobile fission gas bubbles and sintering pores. Grain growth is modeled using the following expression (Tonks et al., 2021):

$var element = document.getElementById("moose-equation-56542385-596d-4175-b1a7-a4451b044450");katex.render(" \\frac{\\partial\\bar{D}}{\\partial t} = \\alpha \\bar{M} \\frac{\\bar{\\gamma}}{\\bar{D}} - \\alpha^{\\frac{1}{2}} \\sum_p\\left[\\bar{M} - \\frac{M_p}{N_p}\\right]^+ \\bar{P}_p,", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-ad5809c2-fe33-4861-839f-95766a8baaf6");katex.render("\\bar{D}", element, {displayMode:false,throwOnError:false});$ is the average grain diameter, $var element = document.getElementById("moose-equation-9c4ff9e5-dc59-401c-8f2b-f26668a321dd");katex.render("t", element, {displayMode:false,throwOnError:false});$ is the time, $var element = document.getElementById("moose-equation-08343416-751e-4eea-95ba-32b169636926");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ is a geometric constant, $var element = document.getElementById("moose-equation-8a45ef32-a975-40f2-81c9-6a219e56ed0d");katex.render("\\bar{M}", element, {displayMode:false,throwOnError:false});$ is the average grain boundary mobility, $var element = document.getElementById("moose-equation-74a733c7-a6bf-4a93-83c0-62a59575ea77");katex.render("\\bar{\\gamma}", element, {displayMode:false,throwOnError:false});$ is the average grain boundary energy, $var element = document.getElementById("moose-equation-9d1f135f-e549-4e86-a8d9-13241ddc7bee");katex.render("p", element, {displayMode:false,throwOnError:false});$ is the porosity type (fission gas bubbles and sintering pores), $var element = document.getElementById("moose-equation-76cfaac0-0ded-4849-b12f-996d7940e8aa");katex.render("M_p", element, {displayMode:false,throwOnError:false});$ is the mobility of porosity type $var element = document.getElementById("moose-equation-f1a1f598-b34c-4ada-87bf-e8172c0ac2f8");katex.render("p", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-a4fa495e-d2ee-4eaa-a57f-be3b0ab8a093");katex.render("N_p", element, {displayMode:false,throwOnError:false});$ is the number of pores per unit grain boundary area, and $var element = document.getElementById("moose-equation-64c4b8b5-8c83-4091-acae-f282d44375e2");katex.render("\\bar{P}_p", element, {displayMode:false,throwOnError:false});$ is the average pinning force associated with porosity type $var element = document.getElementById("moose-equation-7d7f1994-870e-4482-8316-4502d867ef62");katex.render("p", element, {displayMode:false,throwOnError:false});$ . Here, the $var element = document.getElementById("moose-equation-a88fbe7c-7db5-4871-b5ad-8682f79e1430");katex.render("[x]^+", element, {displayMode:false,throwOnError:false});$ operator equals zero when $var element = document.getElementById("moose-equation-6eadadad-c457-4c5f-9699-1684dfeb3ebf");katex.render("x \\le 0", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-0c9b7b23-b523-4450-926d-9ac9cc65cfc7");katex.render("x", element, {displayMode:false,throwOnError:false});$ otherwise (Tonks et al., 2021).

The data necessary to define the above parameters for UO $var element = document.getElementById("moose-equation-1ab9f2e4-4b17-46e4-95fe-91668256bc6d");katex.render("_2", element, {displayMode:false,throwOnError:false});$ or U $var element = document.getElementById("moose-equation-23adbe67-737a-464c-8ef4-da8c78a4dddc");katex.render("_3", element, {displayMode:false,throwOnError:false});$ Si $var element = document.getElementById("moose-equation-95a2122f-6f74-497c-9210-56fea82d34bf");katex.render("_2", element, {displayMode:false,throwOnError:false});$ (Maiya, 1971; Cheniour et al., 2020; Andersson et al., 2019) can be populated by specifying the fuel_type input parameter. This auxkernel should be used in conjunction with a fission gas model such as UO2Sifgrs to generate the material properties necessary to describe the fuel's microstructure: the fractional coverage of fission gas bubbles on grain boundaries, the radius of the fission gas bubbles, and the fractional reduction in sintering pores due to densification.

Users can specify use_current_values = true to use the current values of the material properties or use_current_values = false to use the values of the material properties at the previous time step. The latter option improves numerical performance at the expense of accuracy.

Note that while the expression above is formulated in terms of the average grain diameter, this auxkernel returns the average grain radius for consistency with GrainRadiusAux.

Example Input Syntax

[AuxKernels<<<{"href": "../../syntax/AuxKernels/index.html"}>>>]
  [grain_radius]
    type = GrainRadiusMechanistic<<<{"description": "Mechanistic model for calculating grain growth in nuclear fuel.", "href": "GrainRadiusMechanistic.html"}>>>
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = grain_radius
    temperature<<<{"description": "Coupled temperature."}>>> = temperature
    fuel_type<<<{"description": "Fuel type. The choices are: UO2 U3Si2."}>>> = UO2
    use_current_values<<<{"description": "Whether to use current or old values of variables and material properties in the calculation (use of old values may improve convergence but degrades accuracy)."}>>> = true
  []
[]

Input Parameters

temperatureCoupled temperature.
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled temperature.
variableThe name of the variable that this object applies to
C++ Type:AuxVariableName
Unit:(no unit assumed)
Controllable:No
Description:The name of the variable that this object applies to

Required Parameters

blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
boundaryThe list of boundaries (ids or names) from the mesh where this object applies
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The list of boundaries (ids or names) from the mesh where this object applies
check_boundary_restrictedTrueWhether to check for multiple element sides on the boundary in the case of a boundary restricted, element aux variable. Setting this to false will allow contribution to a single element's elemental value(s) from multiple boundary sides on the same element (example: when the restricted boundary exists on two or more sides of an element, such as at a corner of a mesh
Default:True
C++ Type:bool
Controllable:No
Description:Whether to check for multiple element sides on the boundary in the case of a boundary restricted, element aux variable. Setting this to false will allow contribution to a single element's elemental value(s) from multiple boundary sides on the same element (example: when the restricted boundary exists on two or more sides of an element, such as at a corner of a mesh
execute_onLINEAR TIMESTEP_ENDThe 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.
Default:LINEAR TIMESTEP_END
C++ Type:ExecFlagEnum
Options:XFEM_MARK, NONE, INITIAL, LINEAR, NONLINEAR_CONVERGENCE, NONLINEAR, POSTCHECK, TIMESTEP_END, TIMESTEP_BEGIN, MULTIAPP_FIXED_POINT_END, MULTIAPP_FIXED_POINT_BEGIN, FINAL, CUSTOM, PRE_DISPLACE
Controllable:No
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.
fuel_typeUO2Fuel type. The choices are: UO2 U3Si2.
Default:UO2
C++ Type:MooseEnum
Options:UO2, U3Si2
Controllable:No
Description:Fuel type. The choices are: UO2 U3Si2.
initial_porosity0Initial fuel porosity.
Default:0
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Initial fuel porosity.
use_current_valuesTrueWhether to use current or old values of variables and material properties in the calculation (use of old values may improve convergence but degrades accuracy).
Default:True
C++ Type:bool
Controllable:No
Description:Whether to use current or old values of variables and material properties in the calculation (use of old values may improve convergence but degrades accuracy).

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Controllable:No
Description:The seed for the master random number generator
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Controllable:No
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
Default:False
C++ Type:bool
Controllable:No
Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

Material Property Retrieval Parameters

Input Files

(test/tests/grainRadiusMechanistic/test.i)

References

D. A. Andersson, X. Y. Liu, B. Beeler, S. C. Middleburgh, A. Claisse, and C. R. Stanek. Density functional theory calculations of self- and Xe diffusion in U3Si2. Journal of Nuclear Materials, 515:312–325, 2019. URL: https://doi.org/10.1016/j.jnucmat.2018.12.021, doi:10.1016/j.jnucmat.2018.12.021.

@article{andersson2019,
    author = "Andersson, D. A. and Liu, X. Y. and Beeler, B. and Middleburgh, S. C. and Claisse, A. and Stanek, C. R.",
    doi = "10.1016/j.jnucmat.2018.12.021",
    issn = "00223115",
    journal = "Journal of Nuclear Materials",
    pages = "312-325",
    publisher = "Elsevier B.V.",
    title = "{Density functional theory calculations of self- and Xe diffusion in U3Si2}",
    url = "https://doi.org/10.1016/j.jnucmat.2018.12.021",
    volume = "515",
    year = "2019"
}

Amani Cheniour, Michael R. Tonks, Bowen Gong, Tiankai Yao, Lingfeng He, Jason M. Harp, Benjamin Beeler, Yongfeng Zhang, and Jie Lian. Development of a grain growth model for U3Si2 using experimental data, phase field simulation and molecular dynamics. Journal of Nuclear Materials, 2020. URL: https://doi.org/10.1016/j.jnucmat.2020.152069, doi:10.1016/j.jnucmat.2020.152069.

@article{cheniour2020,
    author = "Cheniour, Amani and Tonks, Michael R. and Gong, Bowen and Yao, Tiankai and He, Lingfeng and Harp, Jason M. and Beeler, Benjamin and Zhang, Yongfeng and Lian, Jie",
    doi = "10.1016/j.jnucmat.2020.152069",
    issn = "00223115",
    journal = "Journal of Nuclear Materials",
    publisher = "Elsevier B.V",
    title = "{Development of a grain growth model for U3Si2 using experimental data, phase field simulation and molecular dynamics}",
    url = "https://doi.org/10.1016/j.jnucmat.2020.152069",
    volume = "532",
    year = "2020"
}

P. S. Maiya. Surface diffusion, surface free energy, and grain-boundary free energy of uranium dioxide. Journal of Nuclear Materials, 40(1):57–65, 1971. doi:10.1016/0022-3115(71)90116-4.

@article{maiya1971,
    author = "Maiya, P. S.",
    doi = "10.1016/0022-3115(71)90116-4",
    issn = "00223115",
    journal = "Journal of Nuclear Materials",
    number = "1",
    pages = "57-65",
    title = "{Surface diffusion, surface free energy, and grain-boundary free energy of uranium dioxide}",
    volume = "40",
    year = "1971"
}

Michael R. Tonks, Pierre-Clément A. Simon, and Jacob Hirschhorn. Mechanistic grain growth model for fresh and irradiated UO2 nuclear fuel. Journal of Nuclear Materials, 543:152576, 2021. doi:10.1016/j.jnucmat.2020.152576.

@article{TONKS2021152576,
    author = "Tonks, Michael R. and Simon, Pierre-Clément A. and Hirschhorn, Jacob",
    title = "{Mechanistic grain growth model for fresh and irradiated UO2 nuclear fuel}",
    journal = "Journal of Nuclear Materials",
    volume = "543",
    pages = "152576",
    year = "2021",
    issn = "0022-3115",
    doi = "10.1016/j.jnucmat.2020.152576"
}

(test/tests/grainRadiusMechanistic/test.i)

# This template sets up tests using GrainRadiusMechanistic to model grain growth
# in different fuel types.  Piecewise functions are used to specify the
# fractional coverage of fission gas bubbles on grain boundaries (GBCoverage)
# and the fractional reduction in sintering pores due to densification
# (sintering_porosity), which would normally be calculated using a fission gas
# model such as Sifgrs.  Changes in these quantities produce different grain
# growth regimes throughout the simulation:
# Time Regime Number Regime Description
# 1e6  1             The fission gas bubbles move faster than the grain
#                    boundary, and motion is dominated by the grain boundary
# 2e6  3             The fission gas bubbles move faster than the grain
#                    boundary, and sintering pores and grain boundaries both
#                    contribute
# 3e6  2             The fission gas bubbles move faster than the grain
#                    boundary, and the grain boundary is fully pinned by
#                    sintering pores
# 4e6  5             The sintering pores move faster than the grain boundary,
#                    and fission gas bubbles and grain boundaries both
#                    contribute
# 5e6  4             The sintering pores move faster than the grain boundary,
#                    and the grain boundary is fully pinned by fission gas
#                    bubbles
# 6e6  7             Sintering pores, fission gas bubbles, and grain boundaries
#                    all contribute
# 7e6  6             The grain boundary is fully pinned by fission gas bubbles
#                    and sintering pores

[Mesh]
  [generated_mesh]
    type = GeneratedMeshGenerator
    dim = 1
  []
[]

[Problem]
  solve = false
[]

[AuxVariables]
  [temperature]
    initial_condition = 1500
  []
  [GBCoverage_auxvariable]
    order = CONSTANT
    family = MONOMIAL
  []
  [rad_bbl_bdr_auxvariable]
    order = CONSTANT
    family = MONOMIAL
    initial_condition = 2.3e-6
  []
  [sintering_porosity_auxvariable]
    order = CONSTANT
    family = MONOMIAL
  []
  [grain_radius]
    order = CONSTANT
    family = MONOMIAL
    initial_condition = 1e-5
  []
[]

[AuxKernels]
  [GBCoverage_auxkernel]
    type = FunctionAux
    variable = GBCoverage_auxvariable
    function = GBCoverage_function
  []
  [sintering_porosity_auxkernel]
    type = FunctionAux
    variable = sintering_porosity_auxvariable
    function = sintering_porosity_function
  []
  [grain_radius]
    type = GrainRadiusMechanistic
    variable = grain_radius
    temperature = temperature
    fuel_type = UO2
    use_current_values = true
  []
[]

[Functions]
  [GBCoverage_function]
    type = PiecewiseLinear
    x = '0 1e6 2e6 3e6 4e6 5e6 6e6 7e6'
    y = '0 0 0 0.1 0.2 0.05 0.2 0.2'
  []
  [sintering_porosity_function]
    type = PiecewiseLinear
    x = '0 1e6 2e6 3e6 4e6 5e6 6e6 7e6'
    y = '0 0.02 0.04 0 0 0.01 0.04 0.04'
  []
[]

[Materials]
  [GBCoverage]
    type = ParsedMaterial
    coupled_variables = GBCoverage_auxvariable
    property_name = GBCoverage
    expression = GBCoverage_auxvariable
  []
  [rad_bbl_bdr]
    type = ParsedMaterial
    coupled_variables = rad_bbl_bdr_auxvariable
    property_name = bubble_radius_GB
    expression = rad_bbl_bdr_auxvariable
  []
  [sintering_porosity]
    type = ParsedMaterial
    coupled_variables = sintering_porosity_auxvariable
    property_name = sintering_porosity
    expression = sintering_porosity_auxvariable
  []
[]

[Executioner]
  type = Transient
  dt = 1e6
  end_time = 7e6
[]

[Outputs]
  exodus = true
[]

(test/tests/grainRadiusMechanistic/test.i)

# This template sets up tests using GrainRadiusMechanistic to model grain growth
# in different fuel types.  Piecewise functions are used to specify the
# fractional coverage of fission gas bubbles on grain boundaries (GBCoverage)
# and the fractional reduction in sintering pores due to densification
# (sintering_porosity), which would normally be calculated using a fission gas
# model such as Sifgrs.  Changes in these quantities produce different grain
# growth regimes throughout the simulation:
# Time Regime Number Regime Description
# 1e6  1             The fission gas bubbles move faster than the grain
#                    boundary, and motion is dominated by the grain boundary
# 2e6  3             The fission gas bubbles move faster than the grain
#                    boundary, and sintering pores and grain boundaries both
#                    contribute
# 3e6  2             The fission gas bubbles move faster than the grain
#                    boundary, and the grain boundary is fully pinned by
#                    sintering pores
# 4e6  5             The sintering pores move faster than the grain boundary,
#                    and fission gas bubbles and grain boundaries both
#                    contribute
# 5e6  4             The sintering pores move faster than the grain boundary,
#                    and the grain boundary is fully pinned by fission gas
#                    bubbles
# 6e6  7             Sintering pores, fission gas bubbles, and grain boundaries
#                    all contribute
# 7e6  6             The grain boundary is fully pinned by fission gas bubbles
#                    and sintering pores

[Mesh]
  [generated_mesh]
    type = GeneratedMeshGenerator
    dim = 1
  []
[]

[Problem]
  solve = false
[]

[AuxVariables]
  [temperature]
    initial_condition = 1500
  []
  [GBCoverage_auxvariable]
    order = CONSTANT
    family = MONOMIAL
  []
  [rad_bbl_bdr_auxvariable]
    order = CONSTANT
    family = MONOMIAL
    initial_condition = 2.3e-6
  []
  [sintering_porosity_auxvariable]
    order = CONSTANT
    family = MONOMIAL
  []
  [grain_radius]
    order = CONSTANT
    family = MONOMIAL
    initial_condition = 1e-5
  []
[]

[AuxKernels]
  [GBCoverage_auxkernel]
    type = FunctionAux
    variable = GBCoverage_auxvariable
    function = GBCoverage_function
  []
  [sintering_porosity_auxkernel]
    type = FunctionAux
    variable = sintering_porosity_auxvariable
    function = sintering_porosity_function
  []
  [grain_radius]
    type = GrainRadiusMechanistic
    variable = grain_radius
    temperature = temperature
    fuel_type = UO2
    use_current_values = true
  []
[]

[Functions]
  [GBCoverage_function]
    type = PiecewiseLinear
    x = '0 1e6 2e6 3e6 4e6 5e6 6e6 7e6'
    y = '0 0 0 0.1 0.2 0.05 0.2 0.2'
  []
  [sintering_porosity_function]
    type = PiecewiseLinear
    x = '0 1e6 2e6 3e6 4e6 5e6 6e6 7e6'
    y = '0 0.02 0.04 0 0 0.01 0.04 0.04'
  []
[]

[Materials]
  [GBCoverage]
    type = ParsedMaterial
    coupled_variables = GBCoverage_auxvariable
    property_name = GBCoverage
    expression = GBCoverage_auxvariable
  []
  [rad_bbl_bdr]
    type = ParsedMaterial
    coupled_variables = rad_bbl_bdr_auxvariable
    property_name = bubble_radius_GB
    expression = rad_bbl_bdr_auxvariable
  []
  [sintering_porosity]
    type = ParsedMaterial
    coupled_variables = sintering_porosity_auxvariable
    property_name = sintering_porosity
    expression = sintering_porosity_auxvariable
  []
[]

[Executioner]
  type = Transient
  dt = 1e6
  end_time = 7e6
[]

[Outputs]
  exodus = true
[]

Description
Example Input Syntax
Input Parameters
Input Files
References