HydrideNucleationKinetics

Computes the kinetic parameter for the nucleation of hydrides in Zr cladding (s-1).

note:Often Created by an Action

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

Description

This material computes the nucleation kinetic parameter $var element = document.getElementById("moose-equation-2e283153-9348-4df8-9911-cfa8eaf8e451");katex.render("K_N", element, {displayMode:false,throwOnError:false});$ used by HydridePrecipitationRate.

Nucleation Kinetic Parameter

The nucleation kinetic factor follows an Arrhenius law, modified by a hydride content dependency (Lacroix, 2019; Lacroix et al., 2021; Passelaigue et al., 2021)

$var element = document.getElementById("moose-equation-8bc7f314-fd91-4140-89a5-5d376c07b9f5");katex.render(" K_N = K_{N0} v_\\alpha exp\\left(\\frac{-E_{th}}{R T}\\right),", element, {displayMode:true,throwOnError:false});$

with the pre-exponential factor $var element = document.getElementById("moose-equation-9281dd1c-61eb-4a79-ad2d-21e762b6aa2f");katex.render("K_{N0} = 2.75 \\times 10^{-5} s^{-1}", element, {displayMode:false,throwOnError:false});$ , the hydride formation energy $var element = document.getElementById("moose-equation-abb84266-efbd-4d62-a888-b4c2e149d870");katex.render("E_{th}", element, {displayMode:false,throwOnError:false});$ computed in HydrideFormationEnergy, and the zirconium volume fraction $var element = document.getElementById("moose-equation-a01a5413-eac2-433d-bb5c-23137a917229");katex.render("v_\\alpha", element, {displayMode:false,throwOnError:false});$ deduced from the hydride volume fraction $var element = document.getElementById("moose-equation-3f729ba9-57c8-4561-ab64-878c4629cc30");katex.render("v_\\delta", element, {displayMode:false,throwOnError:false});$ computed in HydrideVolumeFraction as

$var element = document.getElementById("moose-equation-8d7a0184-b1f8-4fde-8577-c97b7e78999a");katex.render(" v_\\alpha = 1 - v_\\delta.", element, {displayMode:true,throwOnError:false});$

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
  []
[]

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"
}

(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
Nucleation Kinetic Parameter
Example Input Syntax
References