HydrideGrowthKinetics

Computes the kinetic parameter for the growth 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 growth kinetic parameter $var element = document.getElementById("moose-equation-ab445611-463d-42a7-a940-7760958f7f8e");katex.render("K_G", element, {displayMode:false,throwOnError:false});$ used by HydridePrecipitationRate.

Growth Kinetic Parameter

The growth of existing hydrides can be limited by hydrogen diffusion or the reaction driving force. Each of these components follows an Arrhenius law, modified by a hydride content dependency (Lacroix, 2019; Lacroix et al., 2021), as

$var element = document.getElementById("moose-equation-0b324965-b6f9-4323-9c90-3be1dd86aefa");katex.render(" K_{mob} = K_{mob0} x_\\alpha v_{\\alpha} \\exp \\left({\\frac{-E_g}{R T}} \\right),", element, {displayMode:true,throwOnError:false});$

$var element = document.getElementById("moose-equation-0c454df2-1a77-4d75-b920-234aa9806273");katex.render(" K_{th} = K_{th0} x_\\alpha v_{\\alpha} \\exp \\left({\\frac{-E_{th}}{R T}} \\right),", element, {displayMode:true,throwOnError:false});$ and

$var element = document.getElementById("moose-equation-b7cd0c42-8c50-4088-bf76-5ac2fb01817d");katex.render(" K_G = \\left(\\frac{1}{K_{mob}} + \\frac{1}{K_{th}} \\right)^{-1},", element, {displayMode:true,throwOnError:false});$

with $var element = document.getElementById("moose-equation-6f9ea109-820d-4b1c-ae9a-d9e6faef2c27");katex.render("v_{\\alpha}", element, {displayMode:false,throwOnError:false});$ the volume fraction of zirconium, deduced from the hydride volume fraction computed in HydrideVolumeFraction. The atomic fraction of hydrogen in the $var element = document.getElementById("moose-equation-02382bb0-f074-4f69-a85d-4fcc7100e5f3");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ phase $var element = document.getElementById("moose-equation-3d983115-e287-426e-827c-eea167b81c11");katex.render("x_\\alpha", element, {displayMode:false,throwOnError:false});$ is given by:

$var element = document.getElementById("moose-equation-cc49f0b0-3eca-4b0d-ab24-1514e6cd949f");katex.render(" x_\\alpha = \\frac{x_{tot} - TSS_D^{at}}{x_{\\delta} - TSS_D^{at}}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-3e112f12-37f4-4215-abea-508f4f1b611b");katex.render("x_{tot}", element, {displayMode:false,throwOnError:false});$ is the atomic fraction of hydrogen, $var element = document.getElementById("moose-equation-1ffc21bc-af34-4eb0-a79b-9f169ac92825");katex.render("TSS_D^{at}", element, {displayMode:false,throwOnError:false});$ is the solubility expressed as an atomic fraction, and $var element = document.getElementById("moose-equation-e7855fd1-ff51-4c22-9d51-a1360e9587f5");katex.render("x_{\\delta}", element, {displayMode:false,throwOnError:false});$ is the $var element = document.getElementById("moose-equation-dcc5b933-bddd-422f-8dbf-5193a7e8e6a9");katex.render("\\alpha/\\alpha+\\delta", element, {displayMode:false,throwOnError:false});$ boundary at the given temperature. The hydrogen atomic fraction at the $var element = document.getElementById("moose-equation-282ccfac-c53f-4d56-97b7-20a78be0373e");katex.render("\\alpha / \\alpha+\\delta", element, {displayMode:false,throwOnError:false});$ boundary $var element = document.getElementById("moose-equation-b2494cb7-53d0-499f-90d4-3e4c760f8ebc");katex.render("x_{\\delta}", element, {displayMode:false,throwOnError:false});$ is also computed in HydrideVolumeFraction.

The default values are $var element = document.getElementById("moose-equation-4c8bc3b9-220b-4cf7-aba6-b4167cc358bc");katex.render("K_{mob0} = 5.35 \\times 10^{5}", element, {displayMode:false,throwOnError:false});$ s $var element = document.getElementById("moose-equation-c9592735-eb11-471b-9d46-c4517257a420");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ; $var element = document.getElementById("moose-equation-592774e5-ba64-4cb5-9881-6c68c1c074f1");katex.render("K_{th0} = 1.6 \\times 10^{-5}", element, {displayMode:false,throwOnError:false});$ s $var element = document.getElementById("moose-equation-6415ed1f-ffc7-45d2-b465-b94224a3a14d");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ; $var element = document.getElementById("moose-equation-7810eefe-780d-434b-a648-35dce89d7f19");katex.render("E_g = 86.8\\times 10^{3}", element, {displayMode:false,throwOnError:false});$ J/mol. The formation energy $var element = document.getElementById("moose-equation-7f3bd13b-0734-4ad4-afaa-2b32d9d3b673");katex.render("E_{th}", element, {displayMode:false,throwOnError:false});$ is computed in HydrideFormationEnergy, following a 3rd degree polynomial of temperature.

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"
}

(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
Growth Kinetic Parameter
Example Input Syntax
References