Cladding Hydrides Action

Action to add kernels and materials to model hydride formation in the cladding (does not include hydrogen pickup).

Description

The Cladding Hydrides Action is designed to simplify the input file when modeling hydride evolution in the cladding. Modeling the evolution of hydrides in the cladding requires two variables to track concentration of the hydrogen in solid solution ( $var element = document.getElementById("moose-equation-83f99181-86ca-4bec-9b6a-77bb1226f7ac");katex.render("C_{ss}", element, {displayMode:false,throwOnError:false});$ ) and the hydrogen as precipitated hydride ( $var element = document.getElementById("moose-equation-e2067d77-62ae-4808-9ea3-67a1da32d749");katex.render("C_{prec}", element, {displayMode:false,throwOnError:false});$ ). These concentrations are usually specified in wt.ppm (can also be written as $var element = document.getElementById("moose-equation-7eaca366-e0c1-4753-b7bd-3d0f3cddc915");katex.render("\\mu g/g", element, {displayMode:false,throwOnError:false});$ ). Since the variable for the hydride content may have a steep gradient, it can be important to use the bounds feature from MOOSE to prevent negative concentrations. A good example can be found in the input file for one of the assessment cases described here.

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 ( $var element = document.getElementById("moose-equation-0f6bb8f6-97af-4041-b73d-5a8d6f511391");katex.render("C_{prec}", element, {displayMode:false,throwOnError:false});$ ) are given by

$var element = document.getElementById("moose-equation-87e7a62a-31f4-400c-a9a7-e95562942d60");katex.render(" \\frac{\\partial C_{prec}}{\\partial t} = S", element, {displayMode:true,throwOnError:false});$

and the evolution of the amount of hydrogen in solid solution ( $var element = document.getElementById("moose-equation-e9e77b2a-1f9a-4779-bffc-4ff605b3517c");katex.render("C_{ss}", element, {displayMode:false,throwOnError:false});$ ) is governed by

$var element = document.getElementById("moose-equation-4dbb3880-0050-4bd0-a0e3-394711d69703");katex.render(" \\frac{\\partial C_{ss}}{\\partial t} = -\\nabla ( -D \\nabla C_{ss} - \\frac{DQ^*C_{ss}}{RT^2} \\nabla T) - S,", element, {displayMode:true,throwOnError:false});$

which includes the diffusion due to Fick's Law and Soret effect in addition to the changes due to precipitation and dissolution. The diffusion coefficient of hydrogen in zirconium $var element = document.getElementById("moose-equation-4817b5a5-e4fc-4b5a-9ead-4470f7a4a4c1");katex.render("D", element, {displayMode:false,throwOnError:false});$ is defined as $var element = document.getElementById("moose-equation-abbc3b3b-8235-4f45-be89-826e3f1b1081");katex.render(" D = D_0 \\exp\\left(\\frac{-E_d}{RT}\\right),", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-51b0f836-57f9-49ca-b870-a719a227b3c3");katex.render("D_0 = 1.08 \\times 10^{-6}", element, {displayMode:false,throwOnError:false});$ m $var element = document.getElementById("moose-equation-fb06d591-7aa9-4396-8408-7aa4bf0a74d1");katex.render("^2", element, {displayMode:false,throwOnError:false});$ s $var element = document.getElementById("moose-equation-78a88c68-795f-4455-a7dd-72ebfffb289d");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-1f8f7fd6-b725-4686-b7e2-1eec762b19ce");katex.render("E_d = 4.4\\times 10^{4}", element, {displayMode:false,throwOnError:false});$ J/mol, $var element = document.getElementById("moose-equation-e443d6dd-ad02-494d-b07b-8e6a8d6a0afc");katex.render("R", element, {displayMode:false,throwOnError:false});$ is the gas constant in J/mol/K, and $var element = document.getElementById("moose-equation-de4f17cc-868b-4dbc-94e4-8f0962f9c8e1");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature in Kelvin (Zhang et al., 2017). $var element = document.getElementById("moose-equation-eee0985b-8910-4855-b2b7-b07e2b89dfc0");katex.render("Q^* = 25500", element, {displayMode:false,throwOnError:false});$ J/mol is the heat of transport measured by Kammenzind (Kammenzind et al., 1996). Note that the heat of transport of hydrogen in Zircaloy-4 was more recently measured and a values of $var element = document.getElementById("moose-equation-1cc3d6be-9412-46cd-bb56-09622785d358");katex.render("Q^* = 30000", element, {displayMode:false,throwOnError:false});$ J/mol was deemed more appropriate given the scatter of measurements (Kang et al., 2023). $var element = document.getElementById("moose-equation-a3b0866f-9b55-45cd-bd85-977ef18dfdb0");katex.render("S", element, {displayMode:false,throwOnError:false});$ is the source term due to precipitation and dissolution derived by HydridePrecipitationRate.

Constructed Objects

Table 1: Correspondence Among Action Functionality and MOOSE/BISON Objects for the HydrideAction Action

Functionality	Class	Associated Parameters
Hydrogen concentration evolution ( $var element = document.getElementById("moose-equation-099d3cb5-977d-45d1-a860-ed45c96c13a3");katex.render("\\frac{\\partial C_{ss}}{\\partial t}", element, {displayMode:false,throwOnError:false});$ )	TimeDerivative	`hydrogen_name`: a string of the nonlinear variable name used for the concentration of hydrogen in solid solution
Hydride concentration evolution ( $var element = document.getElementById("moose-equation-87abca3d-4c55-40de-adc4-7e199b3ca299");katex.render("\\frac{\\partial C_{Prec}}{\\partial t}", element, {displayMode:false,throwOnError:false});$ )	TimeDerivative	`hydride_name`: a string of the nonlinear variable name used for the concentration of hydrogen in hydride
Fick's Law	MatDiffusion	`hydrogen_name`: a string consisting of the nonlinear variable name used for the concentration of dissolved hydrogen
		`diffusivity` name of the MaterialProperty used to store the hydrogen diffusivity
Soret effect	ThermoDiffusion	`hydrogen_name`: a string consisting of the nonlinear variable name used for the concentration of dissolved hydrogen
		`temperature`: a string of the temperature field variable
Soluble hydrogen precipitation and dissolution ( $var element = document.getElementById("moose-equation-33ccf2aa-8a82-439a-8205-126abb19697d");katex.render("-S", element, {displayMode:false,throwOnError:false});$ )	HydrogenSource	`hydrogen_name`: a string consisting of the nonlinear variable name used for the concentration of dissolved hydrogen
Hydride hydrogen precipitation and dissolution ( $var element = document.getElementById("moose-equation-24533f00-edd2-432d-9d31-69d62a53332a");katex.render("S", element, {displayMode:false,throwOnError:false});$ )	HydrideSource	`hydride_name`: a string consisting of the nonlinear variable name used for the concentration of dissolved hydrogen
Mass diffusivity of hydrogen in the matrix ( $var element = document.getElementById("moose-equation-58eda24c-3e65-4439-95df-412c684b3278");katex.render("D", element, {displayMode:false,throwOnError:false});$ )	ArrheniusMaterialProperty	`temperature`: a string of the temperature field variable
		`frequency_factor`: value of the Arrhenius hydrogen supersolubility frequency factor
		`activation_energy`: value of the Arrhenius hydrogen supersolubility activation energy
Hydrogen solubility ( $var element = document.getElementById("moose-equation-af00c0bb-173b-4ef1-8631-68cb4eed710f");katex.render("TSS_D^{eff}", element, {displayMode:false,throwOnError:false});$ )	HydrogenSolubility	`temperature`: a string of the temperature field variable
		`hydride_volume_fraction_name` a string of the material name used for the volume fraction of hydrides
		`solubility_frequency_factor`: value Arrhenius hydrogen solubility frequency factor
		`solubility_activation_energy`: value Arrhenius hydrogen solubility activation energy
		`solubility_g`: value of g parameter in modifying function
		`solubility_delta`: value of delta parameter in modifying function
Hydrogen supersolubility ( $var element = document.getElementById("moose-equation-0c993fd6-c0bd-4b38-97ec-fe4e9fa30bb4");katex.render("TSS_P^{eff}", element, {displayMode:false,throwOnError:false});$ )	HydrogenSuperSolubility	`temperature`: a string of the temperature field variable
		`solubility_name`: a string of the material name used for the solubility of hydrogen in zirconium
		`super_solubility_frequency_factor`: value of the Arrhenius hydrogen supersolubility frequency factor
		`super_solubility_activation_energy`: value of the Arrhenius hydrogen supersolubility activation energy
		`super_solubility_tau`: value of the characteristic decrease time of the supersolubility during an annealing
Hydride dissolution kinetics ( $var element = document.getElementById("moose-equation-0d46c14d-7a15-45fe-93ca-51da3e6f8a85");katex.render("K_D", element, {displayMode:false,throwOnError:false});$ )	HydrideDissolutionKinetics	`temperature`: a string of the temperature field variable
		`hydrogen_name`: a string consisting of the nonlinear variable name used for the concentration of dissolved hydrogen
		`solubility_name`: a string of the material name used for the solubility of hydrogen in zirconium
		`dissolution_kinetic_frequency_factor` value of the Arrhenius hydride dissolution kinetic frequency factor
		`dissolution_kinetic_activation_energy` value of the Arrhenius hydride dissolution kinetic activation energy
Hydride nucleation kinetics ( $var element = document.getElementById("moose-equation-ad143a96-13e7-4e23-a102-a8eea2373cad");katex.render("K_N", element, {displayMode:false,throwOnError:false});$ )	HydrideNucleationKinetics	`temperature`: a string of the temperature field variable
		`hydrogen_name`: a string consisting of the nonlinear variable name used for the concentration of dissolved hydrogen
		`super_solubility_name`: a string of the material name used for the supersolubility of hydrogen in zirconium
		`nucleation_kinetic_frequency_factor` value of the Arrhenius hydride dissolution kinetic frequency factor
		`hydride_volume_fraction_name` a string of the material name used for the volume fraction of hydrides
		`hydride_formation_energy_name` a string of the material name used for the hydride formation energy
Hydride growth kinetics ( $var element = document.getElementById("moose-equation-92d671d1-f606-4524-a053-8ba99876316a");katex.render("K_G", element, {displayMode:false,throwOnError:false});$ )	HydrideGrowthKinetics	`temperature`: a string of the temperature field variable
		`hydrogen_name`: a string consisting of the nonlinear variable name used for the concentration of dissolved hydrogen
		`hydride_name`: a string of the nonlinear variable name used for the concentration of hydrogen in hydride
		`solubility_name`: a string of the material name used for the solubility of hydrogen in zirconium
		`growth_diffusion_kinetic_frequency_factor` value of the Arrhenius hydride diffusion-driven growth kinetic frequency factor
		`growth_diffusion_kinetic_activation_energy` value of the Arrhenius hydride diffusion-driven growth kinetic activation energy
		`growth_reaction_kinetic_frequency_factor` value of the Arrhenius hydride reaction-driven growth kinetic frequency factor
		`hydride_volume_fraction_name` a string of the material name used for the volume fraction of hydrides
		`hydride_formation_energy_name` a string of the material name used for the hydride formation energy
Heat of transport for hydrogen ( $var element = document.getElementById("moose-equation-8e000ea2-c168-4558-b153-f28b9e15065d");katex.render("Q^*", element, {displayMode:false,throwOnError:false});$ )	GenericConstantMaterial	`heat_of_transport`: value of the energy of hydrogen transport in the cladding
Precipitation and dissolution rate ( $var element = document.getElementById("moose-equation-6b46dc0e-3b0a-433d-893f-88143b775c72");katex.render("S", element, {displayMode:false,throwOnError:false});$ )	HydridePrecipitationRate	`temperature`: a string of the temperature field variable
		`hydrogen_name`: a string consisting of the nonlinear variable name used for the concentration of dissolved hydrogen
		`hydride_name`: a string of the nonlinear variable name used for the concentration of hydrogen in hydride
		`solubility_name`: a string of the material name used for the solubility of hydrogen in zirconium
		`super_solubility_name`: a string of the material name used for the supersolubility of hydrogen in zirconium
		`dissolution_kinetic_name`: a string of the material name used for the kinetic coefficient for hydride dissolution
		`nucleation_kinetic_name`: a string of the material name used for the kinetic coefficient for hydride nucleation
		`growth_kinetic_name`: a string of the material name used for the kinetic coefficient for hydride growth
Hydride formation energy ( $var element = document.getElementById("moose-equation-8b471f5e-175c-4c7f-88de-0417abddd062");katex.render("E_{th}", element, {displayMode:false,throwOnError:false});$ )	HydrideFormationEnergy	`temperature`: a string of the temperature field variable
		`E0, E1, E2, E3`: coefficients of temperature fit for $var element = document.getElementById("moose-equation-e61d16c7-3cb7-400e-aeb3-25b1053d3aee");katex.render("E_{th}", element, {displayMode:false,throwOnError:false});$
Hydride volume fraction ( $var element = document.getElementById("moose-equation-a471a857-95fe-435e-90f1-3508cab137c4");katex.render("f_\\delta", element, {displayMode:false,throwOnError:false});$ )	HydrideVolumeFraction	`temperature`: a string of the temperature field variable
		`x0, x1, x2, x3`: coefficients of temperature fit for $var element = document.getElementById("moose-equation-1a7618af-0e9f-4d6c-ae8e-b7a2b1c023ed");katex.render("x_\\delta", element, {displayMode:false,throwOnError:false});$

If hydrogen pickup is of interest, a corrosion model and pickup boundary condition must be added to the input file (see HydrogenPickup).

Example Input Syntax

[CladdingHydrides<<<{"href": "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
  []
[]

[Variables<<<{"href": "../Variables/index.html"}>>>]
  [Css]
    scaling<<<{"description": "Specifies a scaling factor to apply to this variable"}>>> = 1e6
  []
[]

[Variables<<<{"href": "../Variables/index.html"}>>>]
  [Cprec]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
    scaling<<<{"description": "Specifies a scaling factor to apply to this variable"}>>> = 1e6
  []
[]

Input Parameters

blockThe list of block ids for the cladding.
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of block ids for the cladding.
hydrogen_as_hydride_ppmConcentration of hydrogen as hydride (wt.ppm).
C++ Type:NonlinearVariableName
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:NonlinearVariableName
Unit:(no unit assumed)
Controllable:No
Description:Concentration of dissolved hydrogen (wt.ppm).
temperatureTemperature (K).
C++ Type:VariableName
Unit:(no unit assumed)
Controllable:No
Description:Temperature (K).

Required Parameters

active__all__ If specified only the blocks named will be visited and made active
Default:__all__
C++ Type:std::vector<std::string>
Controllable:No
Description:If specified only the blocks named will be visited and made active
diffusivity_activation_energy44000Activation energy for Arrhenius hydrogen diffusivity (J/mol).
Default:44000
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Activation energy for Arrhenius hydrogen diffusivity (J/mol).
diffusivity_frequency_factor1.08e-06Frequency factor for Arrhenius hydrogen diffusivity (s-1).
Default:1.08e-06
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Frequency factor for Arrhenius hydrogen diffusivity (s-1).
dissolution_kinetic_frequency_factor1110Constant coefficient for Arrhenius dissolution kinetics (s-1).
Default:1110
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Constant coefficient for Arrhenius dissolution kinetics (s-1).
formation_energy_E0-54543First coefficient of polynomial (J/mol).
Default:-54543
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:First coefficient of polynomial (J/mol).
formation_energy_E138.58Second coefficient of polynomial (J/mol/K).
Default:38.58
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Second coefficient of polynomial (J/mol/K).
formation_energy_E2-0.01929Third coefficient of polynomial (J/mol/K^2).
Default:-0.01929
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Third coefficient of polynomial (J/mol/K^2).
formation_energy_E32.894e-05Fourth coefficient of polynomial (J/mol/K^3).
Default:2.894e-05
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Fourth coefficient of polynomial (J/mol/K^3).
growth_diffusion_kinetic_coef535000Frequency factor for Arrhenius reaction-driven growth kinetics (s-1).
Default:535000
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Frequency factor for Arrhenius reaction-driven growth kinetics (s-1).
growth_diffusion_kinetic_energy86806Activation energy for Arrhenius reaction-driven growth (J/mol).
Default:86806
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Activation energy for Arrhenius reaction-driven growth (J/mol).
growth_reaction_kinetic_coef1.6e-05Frequency factor for Arrhenius diffusion-driven growth kinetics (s-1).
Default:1.6e-05
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Frequency factor for Arrhenius diffusion-driven growth kinetics (s-1).
heat_of_transport25500Heat of transport for hydrogen in cladding (J/mol).
Default:25500
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Heat of transport for hydrogen in cladding (J/mol).
inactiveIf specified blocks matching these identifiers will be skipped.
C++ Type:std::vector<std::string>
Controllable:No
Description:If specified blocks matching these identifiers will be skipped.
nucleation_kinetic_frequency_factor2.75e-05Frequency factor for Arrhenius nucleation kinetics (s-1).
Default:2.75e-05
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Frequency factor for Arrhenius nucleation kinetics (s-1).
solubility_activation_energy35459Activation energy for Arrhenius TSSd (J/mol).
Default:35459
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Activation energy for Arrhenius TSSd (J/mol).
solubility_delta1.05Parameter delta for effective TSSd (-).
Default:1.05
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Parameter delta for effective TSSd (-).
solubility_frequency_factor101999Frequency factor for Arrhenius TSSd (s-1).
Default:101999
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Frequency factor for Arrhenius TSSd (s-1).
solubility_g200Parameter g for effective TSSd (wt.ppm).
Default:200
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Parameter g for effective TSSd (wt.ppm).
super_solubility_activation_energy25249Activation energy for Arrhenius TSSp (J/mol).
Default:25249
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Activation energy for Arrhenius TSSp (J/mol).
super_solubility_frequency_factor30853Frequency factor for Arrhenius TSSp (s-1).
Default:30853
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Frequency factor for Arrhenius TSSp (s-1).
super_solubility_tau10000Characteristic time of supersolubility decrease (s).
Default:10000
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Characteristic time of supersolubility decrease (s).
use_automatic_differentiationFalseFlag to use automatic differentiation (AD) objects when possible
Default:False
C++ Type:bool
Controllable:No
Description:Flag to use automatic differentiation (AD) objects when possible
volume_fraction_x00.623First coefficient of x_delta polynomial (-).
Default:0.623
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:First coefficient of x_delta polynomial (-).
volume_fraction_x1-5.73e-05Second coefficient of x_delta polynomial (K^-1).
Default:-5.73e-05
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Second coefficient of x_delta polynomial (K^-1).
volume_fraction_x28.48e-08Third coefficient of x_delta polynomial (K^-2).
Default:8.48e-08
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Third coefficient of x_delta polynomial (K^-2).
volume_fraction_x3-9.93e-11Fourth coefficient of x_delta polynomial (K^-3).
Default:-9.93e-11
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Fourth coefficient of x_delta polynomial (K^-3).

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.

Advanced Parameters

Associated Actions

Available Actions

Bison App
HydrideActionAction to add kernels and materials to model hydride formation in the cladding (does not include hydrogen pickup).

References

BG Kammenzind, David G Franklin, H Richard Peters, and Walter J Duffin. Hydrogen pickup and redistribution in alpha-annealed Zircaloy-4. ASTM Special Technical Publication, 1295:338–369, 1996.

@article{kammenzind1996,
    author = "Kammenzind, BG and Franklin, David G and Peters, H Richard and Duffin, Walter J",
    title = "Hydrogen pickup and redistribution in alpha-annealed {Zircaloy-4}",
    journal = "{ASTM} Special Technical Publication",
    volume = "1295",
    pages = "338-369",
    year = "1996"
}

S. Kang, P. Hsun Huang, V. Petrov, A. Manera, T. Ahn, B. Kammenzind, and A. T. Motta. Determination of the hydrogen heat of transport in zircaloy-4. Journal of Nuclear Materials, 573:154122, 1 2023. doi:10.1016/J.JNUCMAT.2022.154122.

@article{Kang2023154122,
    author = "Kang, S. and Huang, P. Hsun and Petrov, V. and Manera, A. and Ahn, T. and Kammenzind, B. and Motta, A. T.",
    doi = "10.1016/J.JNUCMAT.2022.154122",
    issn = "0022-3115",
    journal = "Journal of Nuclear Materials",
    month = "1",
    pages = "154122",
    publisher = "North-Holland",
    title = "Determination of the hydrogen heat of transport in Zircaloy-4",
    volume = "573",
    year = "2023"
}

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"
}

Yongfeng Zhang, Chao Jiang, and Xianming Bai. Anisotropic hydrogen diffusion in α-zr and zircaloy predicted by accelerated kinetic monte carlo simulations. Scientific Reports, 7:1–13, 1 2017. URL: www.nature.com/scientificreports, doi:10.1038/srep41033.

@ARTICLE{zhang2017,
    author = "Zhang, Yongfeng and Jiang, Chao and Bai, Xianming",
    doi = "10.1038/srep41033",
    issn = "20452322",
    issue = "1",
    journal = "Scientific Reports",
    keywords = "Atomistic models,Condensed,Metals and alloys,matter physics",
    month = "1",
    pages = "1-13",
    pmid = "28106154",
    publisher = "Nature Publishing Group",
    title = "Anisotropic hydrogen diffusion in α-Zr and Zircaloy predicted by accelerated kinetic Monte Carlo simulations",
    volume = "7",
    url = "www.nature.com/scientificreports",
    year = "2017"
}

(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
[]

(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
[]

(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
Constructed Objects
Example Input Syntax
Input Parameters
Associated Actions
Available Actions
References