MOXOxygenDiffusion

MOX oxygen diffusion kernel.

Description

Oxygen redistribution in MOX fuel is modeled by applying the concept of Fickian and Soret diffusion. The model found in Lassmann (1987) is implemented here. Oxygen redistribution is governed by $var element = document.getElementById("moose-equation-f471d849-6869-4768-b1b2-bb340bfc86b3");katex.render(" \\frac{\\partial c}{\\partial t} + \\nabla \\cdot N D \\left( \\nabla c + c \\frac{Q}{RT^2} \\nabla T \\right) = 0", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-814c4aec-e3b9-483a-b1a7-fb11686a32d8");katex.render("c", element, {displayMode:false,throwOnError:false});$ is the atomic fraction of interstitial atoms or oxygen vacancies, $var element = document.getElementById("moose-equation-653fede3-6096-4bc5-a8ec-6d29c2812383");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature, $var element = document.getElementById("moose-equation-cda505b2-bf4f-4b19-9c2d-775eea9fa7c5");katex.render("R", element, {displayMode:false,throwOnError:false});$ is the gas constant, $var element = document.getElementById("moose-equation-3d8ce1ad-084e-44fc-932d-43670f78ecca");katex.render("N", element, {displayMode:false,throwOnError:false});$ is the total number of oxygen atoms per unit volume, $var element = document.getElementById("moose-equation-9f0f1d69-af83-4aba-a366-03869ec851ca");katex.render("D", element, {displayMode:false,throwOnError:false});$ is the diffusion coefficient and $var element = document.getElementById("moose-equation-d9fa59bd-acdd-4cce-b501-ace88690ced1");katex.render("Q", element, {displayMode:false,throwOnError:false});$ is the molar effective heat of transport. By default, $var element = document.getElementById("moose-equation-39c2a163-35b4-4d4c-bb8a-6e7143b4ac76");katex.render("D", element, {displayMode:false,throwOnError:false});$ is defined as $var element = document.getElementById("moose-equation-61a672b0-3728-4cc5-8307-3e22c625015e");katex.render("\\begin{aligned} D &= 1.39 \\times 10^{-6} \\exp{\\left( \\frac{-75900}{RT} \\right)} \\left(\\text{m}^2/\\text{s}\\right).\\\\ \\end{aligned}", element, {displayMode:true,throwOnError:false});$

$var element = document.getElementById("moose-equation-179fb988-acab-4213-9aa1-ddb65f1c1b57");katex.render("Q", element, {displayMode:false,throwOnError:false});$ depends on the materials hyper- and hypo-stoichiometry. The two cases are described below:

Hyper-stoichiometric case

The addition of $var element = document.getElementById("moose-equation-8d412198-95bf-4bff-8d97-6715f41fefe4");katex.render("O^{-2}", element, {displayMode:false,throwOnError:false});$ ions in $var element = document.getElementById("moose-equation-4154a327-737f-4278-ad1d-cb342325ca05");katex.render("(U,Pu)O_{2+x}", element, {displayMode:false,throwOnError:false});$ , when x > 0, requires that $var element = document.getElementById("moose-equation-6d6abdd0-03b8-4533-917f-5bf4c53b94a5");katex.render("U^{+4}", element, {displayMode:false,throwOnError:false});$ ions are converted to $var element = document.getElementById("moose-equation-81d3e721-040f-4ff5-9ae9-dea2a35a8506");katex.render("U^{+5}", element, {displayMode:false,throwOnError:false});$ . The fraction of the total uranium in the compound $var element = document.getElementById("moose-equation-089a193d-c7b2-48ca-84de-5508c3a0b5f2");katex.render("UO_{2+x}", element, {displayMode:false,throwOnError:false});$ in the $var element = document.getElementById("moose-equation-5ee64459-94b5-479a-8545-4c1cb956b73b");katex.render("5^{+}", element, {displayMode:false,throwOnError:false});$ state is $var element = document.getElementById("moose-equation-be41f8c8-550c-405a-bb8f-461708051cb0");katex.render("2\\cdot x", element, {displayMode:false,throwOnError:false});$ ; therefore, $var element = document.getElementById("moose-equation-4140f7e6-f626-4c9e-8b1e-2eefecadad65");katex.render("\\begin{aligned} x &= \\frac{O}{U+Pu} - 2 = \\frac{O}{M} - 2 = c\\\\ Q &= -3.5 \\times 10^{34} \\exp{\\left(-17 V_U\\right)} \\text { (J/mol)} \\end{aligned}", element, {displayMode:true,throwOnError:false});$ with $var element = document.getElementById("moose-equation-38ba6734-9a70-4178-9143-bdd5f8cd33e2");katex.render("\\begin{aligned} V_{U} &= 4 + 2\\left( \\frac{x}{1-q} \\right) = 4 + 2 \\left( \\frac{c}{1 - q} \\right) \\\\ q &= \\frac{q_{0}}{1 - Bu} \\text{ ,} \\\\ \\end{aligned}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-e6d0e79a-7c42-4ed0-946c-d48d3a8d90fd");katex.render("V_{U}", element, {displayMode:false,throwOnError:false});$ is the average valence of uranium ions, $var element = document.getElementById("moose-equation-8e1356ff-a832-4165-a630-b3a57b74f3e5");katex.render("q", element, {displayMode:false,throwOnError:false});$ is the plutonium to heavy metal atoms ratio, $var element = document.getElementById("moose-equation-0026ba19-0042-4b24-a96d-8a6a88405f2f");katex.render("Bu", element, {displayMode:false,throwOnError:false});$ is the burnup (FIMA) and $var element = document.getElementById("moose-equation-25c7fb60-7ee9-438b-82b0-ab026c578ac8");katex.render("q_{0}", element, {displayMode:false,throwOnError:false});$ is the initial plutonium to heavy metal atoms ratio.

Hypo-stoichiometric case

Electric neutrality in plutonium oxides, when x < 0, is maintained by conversion of some $var element = document.getElementById("moose-equation-8bde8653-70f6-4c9a-9b2f-146eb0db7373");katex.render("Pu^{+4}", element, {displayMode:false,throwOnError:false});$ to $var element = document.getElementById("moose-equation-ac9f8288-5dfe-4733-b5d3-ff57e4422804");katex.render("Pu^{+3}", element, {displayMode:false,throwOnError:false});$ . The fraction of plutonium in $var element = document.getElementById("moose-equation-781a80f3-cb33-4504-aedd-307e79c85db6");katex.render("3^{+}", element, {displayMode:false,throwOnError:false});$ valence state is $var element = document.getElementById("moose-equation-deaa8f7c-d6fd-47d6-958f-73303dfc2560");katex.render("2\\cdot x", element, {displayMode:false,throwOnError:false});$ ; therefore, $var element = document.getElementById("moose-equation-f81272d6-4650-4166-935c-23945e0c515e");katex.render("\\begin{aligned} x &= -2c\\\\ V_{Pu} &= 4 + 2 \\left( \\frac{x}{q} \\right) = 4 - 4 \\left( \\frac{c}{q} \\right)\\\\ \\end{aligned}", element, {displayMode:true,throwOnError:false});$ $var element = document.getElementById("moose-equation-cca4d1fd-4036-4542-8ae0-72fe209366ef");katex.render(" Q\\text{ (J/mol)}= \\begin{cases} -8.12 \\times 10^{-4} \\exp{\\left(4.85 V_{Pu}\\right)}, & \\text{for $V_{Pu} < 3.3$}\\\\ &\\text{ }\\\\ -3.96 \\times 10^{6} + 2.37 \\times 10^{6} V_{Pu} - 3.6 \\times 10^{5} V_{Pu}^{2}, & \\text{for $3.3 \\leq V_{Pu} < 4$} \\end{cases}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-01f4bdd1-9466-44d5-a4ca-e85da4674bd3");katex.render("V_{Pu}", element, {displayMode:false,throwOnError:false});$ is the average valence of plutonium ions. $var element = document.getElementById("moose-equation-1db853d1-2e11-426e-8071-4e8234e49395");katex.render("\\begin{aligned} N &= \\text{density of UO}_2 \\times \\text{Avagadro's number} \\times \\frac{1}{\\text{molar mass}}\\\\ \\text{ }\\\\ N &= \\text{10,970} \\frac{\\text{kg}}{\\text{m}^3} \\times \\text{6.022140857} \\times \\text{10}^{23} \\frac{\\text{UO}_2 \\text{ molecules}}{\\text{mol}} \\times \\frac{\\text{1 mol}}{\\text{0.270 kg}} \\times 2 \\frac{\\text{oxygen atoms}}{\\text{UO}_2 \\text{ molecules}}\\\\ N &= 4.89 \\times 10^{28} \\frac{\\text{oxygen atoms}}{\\text{m}^3} \\end{aligned}", element, {displayMode:true,throwOnError:false});$

Input Parameters

burnupCoupled Burnup
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled Burnup
temperatureCoupled Temperature
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled Temperature
variableThe name of the variable that this residual object operates on
C++ Type:NonlinearVariableName
Unit:(no unit assumed)
Controllable:No
Description:The name of the variable that this residual object operates on

Required Parameters

O_M_initial1.98Initial oxygen to metal ratio
Default:1.98
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Initial oxygen to metal ratio
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
displacementsThe displacements
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The displacements
oxygen_to_metal_ratiooxygen_to_metal_ratiooxygen_to_metal_ratio
Default:oxygen_to_metal_ratio
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:oxygen_to_metal_ratio
q00.25Initial plutonium to heavy metal ratio
Default:0.25
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Initial plutonium to heavy metal ratio

Optional Parameters

absolute_value_vector_tagsThe tags for the vectors this residual object should fill with the absolute value of the residual contribution
C++ Type:std::vector<TagName>
Controllable:No
Description:The tags for the vectors this residual object should fill with the absolute value of the residual contribution
extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the matrices this Kernel should fill
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the vectors this Kernel should fill
matrix_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Options:nontime, system
Controllable:No
Description:The tag for the matrices this Kernel should fill
vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
Options:nontime, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Contribution To Tagged Field Data 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.
diag_save_inThe name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Unit:(no unit assumed)
Controllable:No
Description:The name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
save_inThe name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Unit:(no unit assumed)
Controllable:No
Description:The name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
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

(assessment/MOX/JOYO/B14/PTM001/analysis/b14_ptm001_1D_sample3_ox_noAm.i)
(assessment/MOX/JOYO/B14/PTM001/analysis/b14_ptm001_1D_sample3_ox.i)
(test/tests/mox_oxygen_transport/mox_oxygen_transport_hyper.i)
(test/tests/mox_oxygen_transport/mox_oxygen_transport_hypo_V_Pu_low3.3.i)
(test/tests/mox_oxygen_transport/mox_oxygen_transport_hypo_V_Pu_up3.3.i)

References

K. Lassmann. The OXIRED model for redistribution of oxygen in nonstoichiometric uranium-plutonium oxides. Journal of Nuclear Materials, 150:10–16, 1987.

@ARTICLE{lassmann_1987_oxired,
    author = "Lassmann, K.",
    title = "The {OXIRED} model for redistribution of oxygen in nonstoichiometric uranium-plutonium oxides",
    journal = "Journal of Nuclear Materials",
    year = "1987",
    volume = "150",
    pages = "10-16"
}

(assessment/MOX/JOYO/B14/PTM001/analysis/b14_ptm001_1D_sample3_ox_noAm.i)

# Sample at midplane

initial_fuel_density = 11026.4

[GlobalParams]
  density = ${initial_fuel_density}
  family = LAGRANGE
[]
[Problem]
  type = ReferenceResidualProblem
  extra_tag_vectors = 'ref'
  reference_vector = 'ref'
[]
[Mesh]
  coord_type = RZ
  [layered1D_mesh]
    type = Layered1DMeshGenerator
    fuel_height = 10e-3
    pellet_outer_radius = 0.0027
    pellet_bottom_coor = 0.0
    pellet_mesh_density = customize
    nx_p = 200
    elem_type = EDGE2
    slices_per_block = 1
    include_plenum = false
    include_clad = false
  []
[]
[UserObjects]
  [pin_geometry]
    type = Layered1DFuelPinGeometry
    include_clad = false
    mesh_generator = layered1D_mesh
  []
[]
[Variables]
  [oxygen]
    initial_condition = 0.009 #nominal value
    scaling = 1e-20
  []
[]
[AuxVariables]
  [temp]
  []
  [fission_rate]
  []
  [burnup]
  []
  [oxygen_to_metal_ratio]
    order = CONSTANT
    family = MONOMIAL
    initial_condition = 1.982
  []
  [saved_t]
  []
  [saved_o]
  []
[]
[Functions]
  [power_history]
    type = PiecewiseLinear
    x = '-200 0  72000  158040  160200  246600  248400  249000.012  251280'
    y = '0 0  39456.16 39456.16  43890.71  43890.71  53442.06  53442.06  0'
  []
[]
[Kernels]
  [oxygen_time_derivative]
    type = TimeDerivative
    variable = oxygen
  []
  [oxygen]
    type = MOXOxygenDiffusion
    variable = oxygen
    temperature = temp
    burnup = burnup
    oxygen_to_metal_ratio = oxygen_to_metal_ratio
    O_M_initial = 1.982
    q0 = 0.3
    extra_vector_tags = 'ref'
  []
[]
[AuxKernels]
  [fission_rate]
    type = FissionRateGeneral
    fission_rate_formulation = MOX
    variable = fission_rate
    block = fuel
    initial_porosity = 0.1372
    rod_ave_lin_pow = power_history
    pellet_diameter = 0.0054
    execute_on = timestep_begin
    energy_per_fission = 3.2e-11
  []
  [burnup]
    type = BurnupAux
    block = fuel
    fission_rate = fission_rate
    variable = burnup
    execute_on = timestep_begin
  []
  [OM]
    type = MaterialRealAux
    variable = oxygen_to_metal_ratio
    property = oxygen_to_metal_ratio
    execute_on = timestep_end
  []
[]
[Materials]
  [fuel_thermal]
    type = MAMOXThermal
    temperature = temp
    porosity = 0.1372
    block = fuel
    Am_content = 0.0
    oxy_to_metal_ratio = 1.982
  []
  [fuel_density]
    type = ParsedMaterial
    block = fuel
    property_name = density
    expression = ${initial_fuel_density}
  []
  [O_M_ratio]
    type = MOXOxygenToMetalRatio
    oxygen = oxygen
    output_properties = 'oxygen_to_metal_ratio'
    O_M_initial = 1.982
  []
[]
[Preconditioning]
  [SMP]
    type = SMP
    full = true
  []
[]
[Dampers]
  [bound]
    type = BoundingValueNodalDamper
    max_value = 1
    min_value = 0
    variable = oxygen
  []
[]
[Executioner]
  type = Transient
  solve_type = 'PJFNK'
  petsc_options = '-snes_ksp_ew'
  petsc_options_iname = '-pc_type -sub_pc_type'
  petsc_options_value = 'asm lu'
  line_search = 'none'
  l_max_its = 50
  l_tol = 8e-3
  nl_max_its = 50
  nl_rel_tol = 1e-5
  nl_abs_tol = 1e-5
  start_time = -200
  n_startup_steps = 1
  end_time = 251280
  dtmax = 10000
  dtmin = 0.25
  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 1e1
    optimal_iterations = 15
    iteration_window = 2
    linear_iteration_ratio = 100
    growth_factor = 2
    cutback_factor = .5
    force_step_every_function_point = true
    timestep_limiting_function = power_history
  []
[]

[PerformanceMetricOutputs]
[]

[Outputs]
  perf_graph = true
  exodus = false
  color = true
  csv = false
  [console]
    type = Console
    max_rows = 25
  []
  [line_plot]
    type = CSV
    execute_on = 'TIMESTEP_END'
    time_step_interval =  1
    file_base = 1d
    create_final_symlink = true
  []
[]

(assessment/MOX/JOYO/B14/PTM001/analysis/b14_ptm001_1D_sample3_ox.i)

# Sample at midplane

initial_fuel_density = 11057.75

[GlobalParams]
  density = ${initial_fuel_density}
  family = LAGRANGE
[]
[Problem]
  type = ReferenceResidualProblem
  extra_tag_vectors = 'ref'
  reference_vector = 'ref'
[]
[Mesh]
  coord_type = RZ
  [layered1D_mesh]
    type = Layered1DMeshGenerator
    fuel_height = 10e-3
    pellet_outer_radius = 0.0027
    pellet_bottom_coor = 0.0
    pellet_mesh_density = customize
    nx_p = 200
    elem_type = EDGE2
    slices_per_block = 1
    include_plenum = false
    include_clad = false
  []
[]
[UserObjects]
  [pin_geometry]
    type = Layered1DFuelPinGeometry
    include_clad = false
    mesh_generator = layered1D_mesh
  []
[]
[Variables]
  [oxygen]
    initial_condition = 0.009 #nominal value
    scaling = 1e-20
  []
[]
[AuxVariables]
  [temp]
  []
  [fission_rate]
  []
  [burnup]
  []
  [oxygen_to_metal_ratio]
    order = CONSTANT
    family = MONOMIAL
    initial_condition = 1.982
  []
  [saved_t]
  []
  [saved_o]
  []
[]
[Functions]
  [power_history]
    type = PiecewiseLinear
    x = '-200 0  72000  158040  160200  246600  248400  249000.012  251280'
    y = '0 0  39456.16 39456.16  43890.71  43890.71  53442.06  53442.06  0'
  []
[]
[Kernels]
  [oxygen_time_derivative]
    type = TimeDerivative
    variable = oxygen
  []
  [oxygen]
    type = MOXOxygenDiffusion
    variable = oxygen
    temperature = temp
    burnup = burnup
    oxygen_to_metal_ratio = oxygen_to_metal_ratio
    O_M_initial = 1.982
    q0 = 0.31
    extra_vector_tags = 'ref'
  []
[]
[AuxKernels]
  [fission_rate]
    type = FissionRateGeneral
    fission_rate_formulation = MOX
    variable = fission_rate
    block = fuel
    initial_porosity = 0.1372
    rod_ave_lin_pow = power_history
    pellet_diameter = 0.0054
    execute_on = timestep_begin
    energy_per_fission = 3.2e-11
  []
  [burnup]
    type = BurnupAux
    block = fuel
    fission_rate = fission_rate
    variable = burnup
    execute_on = timestep_begin
  []
  [OM]
    type = MaterialRealAux
    variable = oxygen_to_metal_ratio
    property = oxygen_to_metal_ratio
    execute_on = timestep_end
  []
[]
[Materials]
  [fuel_thermal]
    type = MAMOXThermal
    temperature = temp
    porosity = 0.1372
    block = fuel
    Am_content = 0.0237
    oxy_to_metal_ratio = 1.982
  []
  [fuel_density]
    type = ParsedMaterial
    block = fuel
    property_name = density
    expression = ${initial_fuel_density}
  []
  [O_M_ratio]
    type = MOXOxygenToMetalRatio
    oxygen = oxygen
    output_properties = 'oxygen_to_metal_ratio'
    O_M_initial = 1.982
  []
[]
[Preconditioning]
  [SMP]
    type = SMP
    full = true
  []
[]
[Dampers]
  [bound]
    type = BoundingValueNodalDamper
    max_value = 1
    min_value = 0
    variable = oxygen
  []
[]
[Executioner]
  type = Transient
  solve_type = 'PJFNK'
  petsc_options = '-snes_ksp_ew'
  petsc_options_iname = '-pc_type -sub_pc_type'
  petsc_options_value = 'asm lu'
  line_search = 'none'
  l_max_its = 50
  l_tol = 8e-3
  nl_max_its = 50
  nl_rel_tol = 1e-5
  nl_abs_tol = 1e-5
  start_time = -200
  n_startup_steps = 1
  end_time = 251280
  dtmax = 10000
  dtmin = 0.25
  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 1e1
    optimal_iterations = 15
    iteration_window = 2
    linear_iteration_ratio = 100
    growth_factor = 2
    cutback_factor = .5
    force_step_every_function_point = true
    timestep_limiting_function = power_history
  []
[]

[PerformanceMetricOutputs]
[]

[Outputs]
  perf_graph = true
  exodus = false
  color = true
  csv = false
  [console]
    type = Console
    max_rows = 25
  []
  [line_plot]
    type = CSV
    execute_on = 'TIMESTEP_END'
    time_step_interval =  1
    file_base = 1d
    create_final_symlink = true
  []
[]

(test/tests/mox_oxygen_transport/mox_oxygen_transport_hyper.i)

# This test is designed to exercise the MOXOxygen diffuion kernel. The kernel contains Fickian
# and Soret diffusion terms and a diffusion coefficient that is a function of temperature.
# There doesn't seem to be a simple analytical solution to this pde, so only general observations
# can be made as to whether or not the result is reasonable. The solution should be a form of
# exponential. The solution's shape agrees with Fig.11.20 from "Fundamental ascpects of nuclear reactor fuel elements", by Olander.
# HYPERstochiometric case
[GlobalParams]
  density = 10431.0
  energy_per_fission = 3.2e-11 # J/fission
[]

[Problem]
  type = FEProblem
[]

[Mesh]
  coord_type = RZ
  [mesh]
    type = GeneratedMeshGenerator
    dim = 1
    xmax = 2.794e-3 #pellet radius
    nx = 100
    elem_type = EDGE
  []
[]

[Variables]
  [temp]
    initial_condition = 1400.0
  []
  [oxygen]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.02
    scaling = 1e-20
  []
[]
[AuxVariables]
  [fission_rate]
  []
  [burnup]
  []
[]
[Functions]
  [power_history]
    type = PiecewiseLinear
    x = '0 10000'
    y = '0 50000'
  []
[]
[Kernels]
  [heat]
    type = HeatConduction
    variable = temp
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temp
  []
  [heat_source]
    type = NeutronHeatSource
    variable = temp
    fission_rate = fission_rate
  []
  [oxygen_time_derivative]
    type = TimeDerivative
    variable = oxygen
  []
  [oxygen]
    type = MOXOxygenDiffusion
    variable = oxygen
    temperature = temp
    burnup = burnup
    O_M_initial = 2.02
  []
[]

[AuxKernels]
  [fission_rate]
    type = FissionRateGeneral
    fission_rate_formulation = MOX
    variable = fission_rate
    rod_ave_lin_pow = power_history
    execute_on = timestep_begin
    pellet_diameter = 0.005588
  []
  [burnup]
    type = BurnupAux
    fission_rate = fission_rate
    variable = burnup
    execute_on = timestep_begin
  []
[]

[BCs]
  [temp_outside]
    type = DirichletBC
    variable = temp
    boundary = right
    value = 1400
  []
[]

[Materials]
  [fuel_thermal]
    type = MAMOXThermal
    temperature = temp
    oxy_to_metal_ratio = 2.02
    Am_content = 0.0
    Np_content = 0.0
    output_properties = 'thermal_conductivity'
  []
  [fuel_density]
    type = ParsedMaterial
    property_name = density
    expression = 10431.0
  []
  [O_M_ratio]
    type = MOXOxygenToMetalRatio
    oxygen = oxygen
    output_properties = 'oxygen_to_metal_ratio'
    O_M_initial = 2.02
    outputs = all
  []
[]

[Preconditioning]
  [SMP]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Transient

  solve_type = 'PJFNK'

  # The -snes_converged_reason option is used here to diagnose some
  # intermittent solver failures we have been seeing with this test
  petsc_options = '-snes_ksp_ew -snes_converged_reason'
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
  petsc_options_value = 'lu       superlu_dist'

  line_search = 'none'

  l_tol = 8e-3
  nl_abs_tol = 1e-9
  nl_rel_tol = 1e-9
  n_startup_steps = 1
  end_time = 10000
  dt = 200
  dtmin = 200
[]

[Postprocessors] #Oxygen to metal ratio is an easier variable to look at for the user
  [ave_ox]
    type = ElementAverageValue
    variable = oxygen
  []
  [max_ox]
    type = NodalExtremeValue
    value_type = max
    variable = oxygen
  []
  [min_ox]
    type = NodalExtremeValue
    value_type = min
    variable = oxygen
  []
  [ave_om_ratio]
    type = ElementAverageValue
    variable = oxygen_to_metal_ratio
  []
  [max_om_ratio]
    type = ElementExtremeValue
    value_type = max
    variable = oxygen_to_metal_ratio
  []
  [min_om_ratio]
    type = ElementExtremeValue
    value_type = min
    variable = oxygen_to_metal_ratio
  []
[]
[VectorPostprocessors]
  [radial_oxygen]
    type = LineValueSampler
    variable = oxygen
    start_point = '0.0 0.0 0.0'
    end_point = '2.794e-3 0.0 0.0'
    num_points = 200
    execute_on = final
    sort_by = x
    outputs = line_plot
  []
  [radial_oxygen-to-metal-ratio]
    type = LineValueSampler
    variable = oxygen_to_metal_ratio
    start_point = '0.0 0.0 0.0'
    end_point = '2.794e-3 0.0 0.0'
    num_points = 200
    execute_on = final
    sort_by = x
    outputs = line_plot
  []
[]
[Outputs]
  exodus = false
  csv = true
  [console]
    type = Console
  []
  [line_plot]
    type = CSV
    execute_on = 'FINAL'
    time_step_interval =  1
  []
[]

[Debug]
  show_var_residual_norms = true
[]

(test/tests/mox_oxygen_transport/mox_oxygen_transport_hypo_V_Pu_low3.3.i)

# This test is designed to exercise the MOXOxygen diffuion kernel. The kernel contains Fickian
# and Soret diffusion terms and a diffusion coefficient that is a function of temperature.
# There doesn't seem to be a simple analytical solution to this pde, so only general observations
# can be made as to whether or not the result is reasonable. The solution should be a form of
# exponential. The solution's shape agrees with Fig.11.21 from "Fundamental ascpects of nuclear reactor fuel elements", by Olander.
# HYPOstochiometric case with a Plutonium Valence lower than 3.3
[GlobalParams]
  density = 10431.0
  energy_per_fission = 3.2e-11 # J/fission
[]

[Mesh]
  coord_type = RZ
  [mesh]
    type = GeneratedMeshGenerator
    dim = 1
    xmax = 2.794e-3 #pellet radius
    nx = 100
    elem_type = EDGE
  []
[]

[Variables]
  [temp]
    initial_condition = 1400.0
  []
  [oxygen]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.044
    scaling = 1e-20
  []
[]
[AuxVariables]
  [fission_rate]
  []
  [burnup]
  []
[]
[Functions]
  [power_history]
    type = PiecewiseLinear
    x = '0 10000'
    y = '0 50000'
  []
[]
[Kernels]
  [heat]
    type = HeatConduction
    variable = temp
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temp
  []
  [heat_source]
    type = NeutronHeatSource
    variable = temp
    fission_rate = fission_rate
  []
  [oxygen_time_derivative]
    type = TimeDerivative
    variable = oxygen
  []
  [oxygen]
    type = MOXOxygenDiffusion
    variable = oxygen
    temperature = temp
    burnup = burnup
    oxygen_to_metal_ratio = oxygen_to_metal_ratio
    O_M_initial = 1.912
  []
[]

[AuxKernels]
  [fission_rate]
    type = FissionRateGeneral
    fission_rate_formulation = MOX
    variable = fission_rate
    rod_ave_lin_pow = power_history
    execute_on = timestep_begin
    pellet_diameter = 0.005588
  []
  [burnup]
    type = BurnupAux
    fission_rate = fission_rate
    variable = burnup
    execute_on = timestep_begin
  []
[]

[BCs]
  [temp_outside]
    type = DirichletBC
    variable = temp
    boundary = right
    value = 1400
  []
[]

[Materials]
  [fuel_thermal]
    type = MAMOXThermal
    temperature = temp
    oxy_to_metal_ratio = 1.912
    Am_content = 0.0
    Np_content = 0.0
    output_properties = 'thermal_conductivity'
  []
  [fuel_density]
    type = ParsedMaterial
    property_name = density
    expression = 10431.0
  []
  [O_M_ratio]
    type = MOXOxygenToMetalRatio
    oxygen = oxygen
    output_properties = 'oxygen_to_metal_ratio'
    O_M_initial = 1.912
    outputs = all
  []
[]

[Preconditioning]
  [SMP]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Transient

  solve_type = 'PJFNK'

  petsc_options = '-snes_ksp_ew'
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
  petsc_options_value = 'lu       superlu_dist'

  line_search = 'none'

  l_tol = 8e-3
  nl_abs_tol = 1e-9
  nl_rel_tol = 1e-9
  n_startup_steps = 1
  end_time = 10000
  dt = 200
  dtmin = 200
[]

[Postprocessors] #Oxygen to metal ratio is an easier variable to look at for the user
  [ave_ox]
    type = ElementAverageValue
    variable = oxygen
  []
  [max_ox]
    type = NodalExtremeValue
    value_type = max
    variable = oxygen
  []
  [min_ox]
    type = NodalExtremeValue
    value_type = min
    variable = oxygen
  []
  [ave_om_ratio]
    type = ElementAverageValue
    variable = oxygen_to_metal_ratio
  []
  [max_om_ratio]
    type = ElementExtremeValue
    value_type = max
    variable = oxygen_to_metal_ratio
  []
  [min_om_ratio]
    type = ElementExtremeValue
    value_type = min
    variable = oxygen_to_metal_ratio
  []
[]
[VectorPostprocessors]
  [radial_oxygen]
    type = LineValueSampler
    variable = oxygen
    start_point = '0.0 0.0 0.0'
    end_point = '2.794e-3 0.0 0.0'
    num_points = 200
    execute_on = final
    sort_by = x
    outputs = line_plot
  []
  [radial_oxygen-to-metal-ratio]
    type = LineValueSampler
    variable = oxygen_to_metal_ratio
    start_point = '0.0 0.0 0.0'
    end_point = '2.794e-3 0.0 0.0'
    num_points = 200
    execute_on = final
    sort_by = x
    outputs = line_plot
  []
[]
[Outputs]
  exodus = false
  csv = true
  [console]
    type = Console
  []
  [line_plot]
    type = CSV
    execute_on = 'FINAL'
    time_step_interval =  1
  []
[]

[Debug]
  show_var_residual_norms = true
[]

(test/tests/mox_oxygen_transport/mox_oxygen_transport_hypo_V_Pu_up3.3.i)

# This test is designed to exercise the MOXOxygen diffuion kernel. The kernel contains Fickian
# and Soret diffusion terms and a diffusion coefficient that is a function of temperature.
# There doesn't seem to be a simple analytical solution to this pde, so only general observations
# can be made as to whether or not the result is reasonable. The solution should be a form of
# exponential. The solution's shape agrees with Fig.11.21 from "Fundamental ascpects of nuclear reactor fuel elements", by Olander.
# HYPOstochiometric case with a Plutonium Valence between 3.3 and 4
[GlobalParams]
  density = 10431.0
  energy_per_fission = 3.2e-11 # J/fission
[]

[Mesh]
  coord_type = RZ
  [mesh]
    type = GeneratedMeshGenerator
    dim = 1
    xmax = 2.794e-3 #pellet radius
    nx = 100
    elem_type = EDGE
  []
[]

[Variables]
  [temp]
    initial_condition = 1400.0
  []
  [oxygen]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.01
    scaling = 1e-20
  []
[]
[AuxVariables]
  [fission_rate]
  []
  [burnup]
  []
[]
[Functions]
  [power_history]
    type = PiecewiseLinear
    x = '0 10000'
    y = '0 50000'
  []
[]
[Kernels]
  [heat]
    type = HeatConduction
    variable = temp
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temp
  []
  [heat_source]
    type = NeutronHeatSource
    variable = temp
    fission_rate = fission_rate
  []
  [oxygen_time_derivative]
    type = TimeDerivative
    variable = oxygen
  []
  [oxygen]
    type = MOXOxygenDiffusion
    variable = oxygen
    temperature = temp
    burnup = burnup
    oxygen_to_metal_ratio = oxygen_to_metal_ratio
  []
[]

[AuxKernels]
  [fission_rate]
    type = FissionRateGeneral
    fission_rate_formulation = MOX
    variable = fission_rate
    rod_ave_lin_pow = power_history
    execute_on = timestep_begin
    pellet_diameter = 0.005588
  []
  [burnup]
    type = BurnupAux
    fission_rate = fission_rate
    variable = burnup
    execute_on = timestep_begin
  []
[]

[BCs]
  [temp_outside]
    type = DirichletBC
    variable = temp
    boundary = right
    value = 1400
  []
[]

[Materials]
  [fuel_thermal]
    type = MAMOXThermal
    temperature = temp
    oxy_to_metal_ratio = 1.98
    Am_content = 0.0
    Np_content = 0.0
    output_properties = 'thermal_conductivity'
  []
  [fuel_density]
    type = ParsedMaterial
    property_name = density
    expression = 10431.0
  []
  [O_M_ratio]
    type = MOXOxygenToMetalRatio
    oxygen = oxygen
    output_properties = 'oxygen_to_metal_ratio'
    outputs = all
  []
[]

[Preconditioning]
  [SMP]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Transient

  solve_type = 'PJFNK'

  petsc_options = '-snes_ksp_ew'
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
  petsc_options_value = 'lu       superlu_dist'

  line_search = 'none'

  l_tol = 8e-3
  nl_abs_tol = 1e-9
  nl_rel_tol = 1e-9
  n_startup_steps = 1
  end_time = 10000
  dt = 200
  dtmin = 200
[]

[Postprocessors] #Oxygen to metal ratio is an easier variable to look at for the user
  [ave_ox]
    type = ElementAverageValue
    variable = oxygen
  []
  [max_ox]
    type = NodalExtremeValue
    value_type = max
    variable = oxygen
  []
  [min_ox]
    type = NodalExtremeValue
    value_type = min
    variable = oxygen
  []
  [ave_om_ratio]
    type = ElementAverageValue
    variable = oxygen_to_metal_ratio
  []
  [max_om_ratio]
    type = ElementExtremeValue
    value_type = max
    variable = oxygen_to_metal_ratio
  []
  [min_om_ratio]
    type = ElementExtremeValue
    value_type = min
    variable = oxygen_to_metal_ratio
  []
[]
[VectorPostprocessors]
  [radial_oxygen]
    type = LineValueSampler
    variable = oxygen
    start_point = '0.0 0.0 0.0'
    end_point = '2.794e-3 0.0 0.0'
    num_points = 200
    execute_on = final
    sort_by = x
    outputs = line_plot
  []
  [radial_oxygen-to-metal-ratio]
    type = LineValueSampler
    variable = oxygen_to_metal_ratio
    start_point = '0.0 0.0 0.0'
    end_point = '2.794e-3 0.0 0.0'
    num_points = 200
    execute_on = final
    sort_by = x
    outputs = line_plot
  []
[]
[Outputs]
  exodus = false
  csv = true
  [console]
    type = Console
  []
  [line_plot]
    type = CSV
    execute_on = 'FINAL'
    time_step_interval =  1
  []
[]

[Debug]
  show_var_residual_norms = true
[]

Description
Input Parameters
Input Files
References