HydridePrecipitationRate

Computes the preciptation or dissolution rate of hydrogen to ZrHx in Zr cladding (wt.ppm/s).

note:Often Created by an Action

This object can be set up automatically by using the CladdingHydrides action.

Description

A material, HydridePrecipitationRate, is used to calculate the precipitation or dissolution rate $var element = document.getElementById("moose-equation-b5b761bc-360a-4161-9800-526e95b16abd");katex.render("S", element, {displayMode:false,throwOnError:false});$ . This is not a material property per se; it is just used this way for convenience as it is used by both HydrogenSource and HydrideSource.

Hydrogen behavior

In the Hydrogen Nucleation-Growth-Dissolution (HNGD) model (Lacroix, 2019; Lacroix et al., 2021; Passelaigue et al., 2021; Passelaigue et al., 2022), the changes in hydride content ( $var element = document.getElementById("moose-equation-bf7caf30-31c6-40d4-8487-59fec2e62844");katex.render("C_{prec}", element, {displayMode:false,throwOnError:false});$ ) are given by

$var element = document.getElementById("moose-equation-1630b83a-31e0-4132-b3e9-5eab39a81a8b");katex.render(" \\frac{\\partial C_{prec}}{\\partial t} = S", element, {displayMode:true,throwOnError:false});$

and the evolution of the amount of hydrogen in solid solution ( $var element = document.getElementById("moose-equation-fea9fda7-baf5-4121-8978-14b2a24e5abe");katex.render("C_{ss}", element, {displayMode:false,throwOnError:false});$ ) is governed by

$var element = document.getElementById("moose-equation-55524ba5-20f8-4f70-8bdf-39a8dc0ea592");katex.render(" \\frac{\\partial C_{ss}}{\\partial t} = -\\nabla ( -D \\nabla C_{ss} - \\frac{DQ^*C_{ss}}{RT^2} \\nabla T) - S,", element, {displayMode:true,throwOnError:false});$

which includes the diffusion due to Fick's Law and Soret effect in addition to the changes due to precipitation and dissolution.

The rate $var element = document.getElementById("moose-equation-f341582e-a51e-4bfc-80b6-cd1b08af1bb9");katex.render("S", element, {displayMode:false,throwOnError:false});$ accounts for the precipitation, which includes both nucleation and growth, and dissolution. The rate $var element = document.getElementById("moose-equation-0e35c68f-67ab-4373-a3ab-dec1395db7b1");katex.render("S", element, {displayMode:false,throwOnError:false});$ is thus defined as $var element = document.getElementById("moose-equation-236c234f-eb3d-4334-bad2-82596d18cbd9");katex.render(" S = S_N + S_G + S_D", element, {displayMode:true,throwOnError:false});$ with $var element = document.getElementById("moose-equation-05e0e542-0a5d-44fb-bf72-efad3ec460cf");katex.render("S_N", element, {displayMode:false,throwOnError:false});$ the nucleation rate, $var element = document.getElementById("moose-equation-c8e69b0d-fe01-4057-ad8a-e9bcb3afce01");katex.render("S_G", element, {displayMode:false,throwOnError:false});$ the growth rate, and $var element = document.getElementById("moose-equation-c98456e6-e5c2-42d7-aed4-ecb670dba358");katex.render("S_D", element, {displayMode:false,throwOnError:false});$ the dissolution rate.

Precipitation Rate Model

In the HNGD model (Lacroix, 2019; Lacroix et al., 2021; Passelaigue et al., 2021; Passelaigue et al., 2022), nucleation can only happen if the hydrogen content in solid solution ( $var element = document.getElementById("moose-equation-3690b944-9255-45a8-b618-e9dd32fc4c88");katex.render("C_{ss}", element, {displayMode:false,throwOnError:false});$ ) is above the supersolubility ( $var element = document.getElementById("moose-equation-b115649a-1842-4381-8197-d912e07fadef");katex.render("TSS_P^{eff}", element, {displayMode:false,throwOnError:false});$ ), and growth happens if the solid solution content is above the solubility ( $var element = document.getElementById("moose-equation-3cc4b581-21be-42cb-8e98-a1c39f15e113");katex.render("TSS_D^{eff}", element, {displayMode:false,throwOnError:false});$ ) and hydrides already exist.

The nucleation rate of new hydrides is described by

$var element = document.getElementById("moose-equation-8f00a10d-bbcf-4823-9281-a6e3ce886fde");katex.render(" \\left\\{ \\begin{array}{lll} S_N = K_N ( C_{ss} - TSS_P^{eff} )\\ \\ if\\ C_{ss} > TSS_P^{eff} \\\\ \\\\ S_N = 0\\ \\ else \\end{array} \\right.", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-91c24832-d90e-4235-bc5f-9e750ce82eff");katex.render("K_N", element, {displayMode:false,throwOnError:false});$ is the nucleation kinetic parameter, which is computed in a separate Material. See HydrideNucleationKinetics for more details.

The growth rate of existing hydrides is given by

$var element = document.getElementById("moose-equation-4fbc7e1f-01b4-4e26-af3a-849af11b8f46");katex.render(" \\left\\{ \\begin{array}{lll} S_G = K_G ( C_{tot} - TSS_D^{eff} ) p ( 1 - x ) ( -ln( 1 - x ))^{1 - \\frac{1}{p}}\\ \\ if\\ C_{ss} > TSS_D^{eff} \\ and\\ C_{Prec}>0 \\\\ \\\\ S_G = 0\\ \\ else \\end{array} \\right.", element, {displayMode:true,throwOnError:false});$

with $var element = document.getElementById("moose-equation-005655c2-1dad-47b1-a178-ed2cecd15faa");katex.render("x", element, {displayMode:false,throwOnError:false});$ a measure of the advancement of the the precipitation reaction:

$var element = document.getElementById("moose-equation-4d40685d-050d-4bc7-b775-a2aa68b1b166");katex.render(" x = \\frac{C_{tot} - C_{ss}}{C_{tot} - TSS_D^{eff}}", element, {displayMode:true,throwOnError:false});$

The kinetic parameter $var element = document.getElementById("moose-equation-e2e58caa-0a77-4ca4-a2bc-7e311f5f800a");katex.render("K_G", element, {displayMode:false,throwOnError:false});$ is computed in a separate Material. See HydrideGrowthKinetics for more details.

Dissolution Rate

Dissolution happens if there are hydrides while the solid solution content is under the solubility ( $var element = document.getElementById("moose-equation-a6d3e931-8e1c-4afa-98ec-8eacc50d5d70");katex.render("TSS_D^{eff}", element, {displayMode:false,throwOnError:false});$ ). It is described by a equation similar to the nucleation:

$var element = document.getElementById("moose-equation-95ce5da3-8c3f-46c8-9e61-bebd8494ee86");katex.render(" \\left\\{ \\begin{array}{lll} S_D = K_D ( C_{ss} - TSS_D^{eff} )\\ \\ if\\ C_{ss} < TSS_D^{eff}\\ and\\ C_{Prec}>0 \\\\ \\\\ S_D = 0\\ \\ else \\end{array} \\right.", element, {displayMode:true,throwOnError:false});$

The kinetic parameter $var element = document.getElementById("moose-equation-ef8d4bd8-830f-4315-bfb1-593b040e3e8f");katex.render("K_D", element, {displayMode:false,throwOnError:false});$ is computed in a separated Material. See HydrideDissolutionKinetics for more details.

Example Input Syntax

[CladdingHydrides<<<{"href": "../../syntax/CladdingHydrides/index.html"}>>>]
  [hydrides]
    block<<<{"description": "The list of block ids for the cladding."}>>> = 0
    temperature<<<{"description": "Temperature (K)."}>>> = temp
    hydrogen_in_solution_ppm<<<{"description": "Concentration of dissolved hydrogen (wt.ppm)."}>>> = Css
    hydrogen_as_hydride_ppm<<<{"description": "Concentration of hydrogen as hydride (wt.ppm)."}>>> = Cprec
    solubility_frequency_factor<<<{"description": "Frequency factor for Arrhenius TSSd (s-1)."}>>> = 67116
    solubility_activation_energy<<<{"description": "Activation energy for Arrhenius TSSd (J/mol)."}>>> = 32294
    # solubility_g = 0
    # solubility_delta = 1
  []
[]

Input Parameters

hydrogen_as_hydride_ppmConcentration of hydrogen as hydride (wt.ppm).
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Concentration of hydrogen as hydride (wt.ppm).
hydrogen_in_solution_ppmConcentration of dissolved hydrogen (wt.ppm).
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Concentration of dissolved hydrogen (wt.ppm).

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
computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
Default:True
C++ Type:bool
Controllable:No
Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
compute_growthTrueIs growth computed ?
Default:True
C++ Type:bool
Controllable:No
Description:Is growth computed ?
constant_onNONEWhen ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
Default:NONE
C++ Type:MooseEnum
Options:NONE, ELEMENT, SUBDOMAIN
Controllable:No
Description:When ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
declare_suffixAn optional suffix parameter that can be appended to any declared 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 declared properties. The suffix will be prepended with a '_' character.

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.
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
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

output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)
C++ Type:std::vector<std::string>
Controllable:No
Description:List of material properties, from this material, to output (outputs must also be defined to an output type)
outputsnone Vector of output names where you would like to restrict the output of variables(s) associated with this object
Default:none
C++ Type:std::vector<OutputName>
Controllable:No
Description:Vector of output names where you would like to restrict the output of variables(s) associated with this object

Outputs 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

References

E. Lacroix. Modeling Zirconium Hydride Precipitation and Dissolution in Zirconium Alloys. PhD thesis, The Pennsylvania State University, 2019.

@phdthesis{lacroixPhd,
    Author = "Lacroix, E.",
    Title = "Modeling Zirconium Hydride Precipitation and Dissolution in Zirconium Alloys",
    Year = "2019",
    School = "The Pennsylvania State University"
}

E. Lacroix, P.-C. A. Simon, A. T. Motta, and J.D. Almer. Zirconium hydride precipitation and dissolution kinetics in the hysteresis region in zirconium alloys. Zirconium in the Nuclear Industry: 19th International Symposium, ASTM STP 1597, pages 67–91, 2021.

@article{Lacroix2019,
    author = "Lacroix, E. and Simon, P.-C. A. and Motta, A. T. and Almer, J.D.",
    title = "{Zirconium hydride precipitation and dissolution kinetics in the hysteresis region in zirconium alloys}",
    journal = "Zirconium in the Nuclear Industry: 19th International Symposium, ASTM STP 1597",
    year = "2021",
    pages = "67-91"
}

F. Passelaigue, E. Lacroix, G. Pastore, and A.T. Motta. Implementation and Validation of the Hydride Nucleation-Growth-Dissolution (HNGD) model in BISON. Journal of Nuclear Materials, 544:152683, 2021.

@article{Passelaigue2021,
    author = "Passelaigue, F. and Lacroix, E. and Pastore, G. and Motta, A.T.",
    journal = "Journal of Nuclear Materials",
    title = "{Implementation and Validation of the Hydride Nucleation-Growth-Dissolution (HNGD) model in BISON}",
    volume = "544",
    year = "2021",
    pages = "152683"
}

F. Passelaigue, P.-C. A. Simon, and A.T. Motta. Predicting the hydride rim by improving the solubility limits in the Hydride Nucleation-Growth-Dissolution (HNGD) model. Journal of Nuclear Materials, 558:153363, 2022.

@article{Passelaigue2022,
    author = "Passelaigue, F. and Simon, P.-C. A. and Motta, A.T.",
    journal = "Journal of Nuclear Materials",
    title = "{Predicting the hydride rim by improving the solubility limits in the Hydride Nucleation-Growth-Dissolution (HNGD) model}",
    volume = "558",
    year = "2022",
    pages = "153363"
}

(test/tests/hydrogen/benchmark_transients.i)

# Benchmark : Precipitation and dissolution kinetics (E. Lacroix, PhD Dissertation (2019), 5.2.3)
# In this simulation, the sample is subjected to a uniform thermal treatment that shows the kinetics
# of Nucleation, Growth and Dissolution of hydrides in a zircaloy cladding.

[GlobalParams]
  order = FIRST
  family = LAGRANGE
[]

[Mesh]
  [generated_mesh]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 25.4e-5
    ymin = 0
    ymax = 25e-5
    nx = 1
    ny = 1
    bias_x = 1
  []
[]

[Variables]
  [temp]
  []

  [Css]
    scaling=1e6
  []

  [Cprec]
    order = FIRST
    family = LAGRANGE
    scaling=1e6
  []
[]

[AuxVariables]
  # Used to prevent negative concentrations
  [bounds_dummy]
    order = FIRST
    family = LAGRANGE
  []
[]

[Bounds]
  # To prevent negative concentrations
  [Cprec_lower_bound]
    type = ConstantBounds
    variable = bounds_dummy
    bounded_variable = Cprec
    bound_type = lower
    bound_value = 0.
  []
[]

[ICs]
  [temp_init]
    variable=temp
    value=300
    type=ConstantIC
 []
 [Css_init]
   variable=Css
   value=0.06
   type=ConstantIC
 []
 [Cprec_init]
   variable=Cprec
   value=253.94
   type=ConstantIC
 []
[]

[Kernels]
  [heatflow]
    type = CoefDiffusion
    variable = temp
    coef = 1e-4
  []
  [dTdt]
    type = TimeDerivative
    variable = temp
  []
[]

[CladdingHydrides]
  [hydrides]
    block=0
    temperature = temp
    hydrogen_in_solution_ppm = Css
    hydrogen_as_hydride_ppm = Cprec
    solubility_frequency_factor = 67116
    solubility_activation_energy = 32294
    # solubility_g = 0
    # solubility_delta = 1
  []
[]

[Functions]
  [fct_temp]
    type = PiecewiseLinear
    x = '0    1969  2594    3877  4411  5499  6096  6609  7705  8419  8988  1.02E+04 1.03E+04 1.04E+04 1.05E+04 1.06E+04 1.07E+04'
    y = '300  697.4 697.4   442.7 442.7 656.6 656.6 557.7 558.1 697.4 697.8 451.2    406.4    374.2    350.89   330.3    323.13'
  []
[]

[BCs]
  [T]
    boundary= '0'
    type=FunctionDirichletBC
    variable=temp
    function = fct_temp
  []
[]

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

[Executioner]

  type = Transient

  solve_type = 'PJFNK'

  petsc_options_iname = '-pc_type -snes_type'
  petsc_options_value = 'lu vinewtonrsls' # This petsc option help prevent negative concentrations

  line_search = 'none'

  l_max_its = 100
  l_tol = 8e-3

  nl_max_its = 15
  nl_rel_tol = 1e-6
  nl_abs_tol = 1e-10

  start_time = 0
  end_time = 1.07E+04
  n_startup_steps = 1

  dtmax = 20
  dtmin = 1

  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 20
    timestep_limiting_function = fct_temp
    force_step_every_function_point = true
    optimal_iterations = 60
    iteration_window = 0
    linear_iteration_ratio = 100
  []
[]

[Postprocessors]
  # [temp]
  #   type = ElementAverageValue
  #   variable = 'temp'
  #   block = 0
  # []
  [Css]
    type = ElementAverageValue
    variable = 'Css'
    block = 0
  []
[]

[Outputs]
  # exodus = true
  [console]
    type = Console
    max_rows = 25
  []
  execute_on = 'timestep_end'
  csv = true
[]

Description
Hydrogen behavior
Precipitation Rate Model
Dissolution Rate
Example Input Syntax
Input Parameters
References