ver-1n

Voltage-Assisted Transport with Constant Source Boundary Condition

Case Description

This verification case considers one-dimensional deuterium diffusion under an applied voltage through a semi-infinite proton-conducting ceramic (PCC) layer with a constant source at one boundary. PCC materials selectively transport hydrogen isotopes (protium, deuterium, and tritium) at temperatures around 600 $var element = document.getElementById("moose-equation-b4242476-ea53-457b-a7b3-d56f29adcf0a");katex.render("^\\circ", element, {displayMode:false,throwOnError:false});$ C through ionic conduction of hydroxyl defects. This proton-hopping mechanism can be substantially enhanced by applying an electric field, enabling active pumping of hydrogen isotopes across the membrane, even without pressure gradients.

The purpose of this case is to isolate and verify the voltage-assisted migration term in the Nernst–Planck equation. To simplify the comparison with the analytical solution, trapping is excluded, and Sieverts's boundary conditions are imposed on upstream and downstream surfaces.

Case Set Up

This verification case models a one-dimensional PCC membrane with a thickness of 10 mm. The upstream deuterium pressure is held constant, and the corresponding boundary concentration is described by Sieverts' law,

$(1)var element = document.getElementById("moose-equation-cd9d7f12-094c-4401-a24a-4bb0c130b575");katex.render(" C_0 = K_s \\sqrt{P},", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-96537898-ac7a-4b9e-95ef-1b79e7acb0d3");katex.render("C_0", element, {displayMode:false,throwOnError:false});$ is the concentration on the upstream side, $var element = document.getElementById("moose-equation-25f2ab38-8973-4674-ad28-c450141fe9f9");katex.render("K_s", element, {displayMode:false,throwOnError:false});$ is the Sieverts' solubility, and $var element = document.getElementById("moose-equation-3bd3676c-2a7b-4fc5-8852-96255b39a8c9");katex.render("P", element, {displayMode:false,throwOnError:false});$ is the upstream deuterium pressure. The downstream concentration is set to 0.

In the PCC membrane, deuterium occupy charged hydroxyl defects. For the voltage-assisted transport verification, the transported deuterium species is treated as a positively charged mobile species with charge number $var element = document.getElementById("moose-equation-3b3e4f8a-33ed-4c4f-b0a6-6e32c9e7e441");katex.render("z=1", element, {displayMode:false,throwOnError:false});$ . The Nernst–Planck governing equation is

$(2)var element = document.getElementById("moose-equation-a3265a48-37a0-4925-9576-650e782c12c8");katex.render(" \\frac{\\partial C}{\\partial t} = \\nabla \\cdot \\left(D\\nabla C\\right) + \\nabla \\cdot \\left(\\frac{CDF}{RT}\\nabla \\phi\\right),", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-f7389956-26c7-4843-b7bf-8fc0a7f36063");katex.render("C", element, {displayMode:false,throwOnError:false});$ is the concentration of deuterium in the sample, $var element = document.getElementById("moose-equation-82faa3bf-e970-4bee-b71d-7a2ef3c340f2");katex.render("D", element, {displayMode:false,throwOnError:false});$ is the deuterium diffusivity, $var element = document.getElementById("moose-equation-cdaded04-4470-4d79-80f3-265c321e72fe");katex.render("F", element, {displayMode:false,throwOnError:false});$ is the Faraday constant, $var element = document.getElementById("moose-equation-e7028105-c870-4950-98cc-0026917d1926");katex.render("R", element, {displayMode:false,throwOnError:false});$ is the ideal gas constant, $var element = document.getElementById("moose-equation-e3608ef1-7575-4f71-9c71-ff8f754b4a7e");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature, $var element = document.getElementById("moose-equation-00dda54f-aee2-4c61-aa70-0a3d1e9c6535");katex.render("\\phi", element, {displayMode:false,throwOnError:false});$ is the electric potential applied across the sample, and $var element = document.getElementById("moose-equation-0bfcd171-6d93-49d7-89bf-39e55b104c4f");katex.render("t", element, {displayMode:false,throwOnError:false});$ is time. In this verification case, the equation is solved in one dimension, with $var element = document.getElementById("moose-equation-9b404afb-c6f5-4344-8e77-ba298b19c290");katex.render("x", element, {displayMode:false,throwOnError:false});$ representing the distance from the source boundary.

The model parameters used in the verification case are shown in Table 1. The deuterium solubility and diffusivity are taken from Hossain et al. (2020). The applied voltage is 20 V across the 10 mm membrane.

Table 1: Values of model properties for the Nernst–Planck verification problem.

Parameter	Description	Value	Units	Reference
$var element = document.getElementById("moose-equation-2d1d0a49-c8a8-4790-b3e2-3f3c28c226c5");katex.render("R", element, {displayMode:false,throwOnError:false});$	gas constant	8.31446261815324	J/mol/K	PhysicalConstants.h
$var element = document.getElementById("moose-equation-8d8c23b8-8fbf-4843-8bd6-3c8c6420a02f");katex.render("T", element, {displayMode:false,throwOnError:false});$	temperature	773	K	–
$var element = document.getElementById("moose-equation-d448a6e7-1dd7-41c3-8f73-055845d48c96");katex.render("K_{s}", element, {displayMode:false,throwOnError:false});$	deuterium solubility in PCC	6.38 $var element = document.getElementById("moose-equation-b397644c-3c9a-4250-8c85-653c19dc97a7");katex.render("\\times 10^{23} \\exp(-7726.21 / RT)", element, {displayMode:false,throwOnError:false});$	atom/m $var element = document.getElementById("moose-equation-03867091-b28d-42e4-9c24-5a12f3de2969");katex.render("^3", element, {displayMode:false,throwOnError:false});$ /Pa $var element = document.getElementById("moose-equation-803dfec1-979a-4e77-a428-23e8fd96fa94");katex.render("^{0.5}", element, {displayMode:false,throwOnError:false});$	Hossain et al. (2020)
$var element = document.getElementById("moose-equation-ef2546c3-5b76-42fc-ac59-e92524e08e6a");katex.render("P", element, {displayMode:false,throwOnError:false});$	upstream pressure	100	Pa	–
$var element = document.getElementById("moose-equation-1ac3fed9-f7fe-403d-96f8-bd30726e52cf");katex.render("D", element, {displayMode:false,throwOnError:false});$	deuterium diffusivity in PCC	2.44 $var element = document.getElementById("moose-equation-b4909409-8c9f-4cc2-9a10-510c74665772");katex.render("\\times 10^{-6} \\exp(-71399.15 / RT)", element, {displayMode:false,throwOnError:false});$	m $var element = document.getElementById("moose-equation-167c7bce-7768-4a41-93b3-74e18db80c32");katex.render("^2", element, {displayMode:false,throwOnError:false});$ /s	Hossain et al. (2020)
$var element = document.getElementById("moose-equation-c96ab3d3-de56-4e71-9f7f-68106d56e28b");katex.render("l", element, {displayMode:false,throwOnError:false});$	thickness of PCC sample	10 $var element = document.getElementById("moose-equation-9736f02b-4d32-44b3-821a-b2e591c3ebc5");katex.render("\\times 10^{-3}", element, {displayMode:false,throwOnError:false});$	m	Hossain et al. (2020)
$var element = document.getElementById("moose-equation-69542062-98b5-42f5-87c5-66fd58ef4328");katex.render("F", element, {displayMode:false,throwOnError:false});$	Faraday constant	96485.33	C/mol	–
$var element = document.getElementById("moose-equation-5145ed5c-dae0-4483-b8af-15572caa1516");katex.render("\\phi", element, {displayMode:false,throwOnError:false});$	voltage applied across PCC sample	20	V	–

The verification focuses on two aspects of the solution: (1) the temporal evolution of deuterium concentration at fixed locations, and (2) the spatial concentration profile at fixed times.

Analytical Solution

Luping and Nilsson (1993) provides the analytical solution for a semi-infinite slab as:

$(3)var element = document.getElementById("moose-equation-b2ab2903-e4b0-4bf3-be94-0e63ccf893c2");katex.render(" C = \\frac{C_0}{2}\\left[ \\exp(ax)\\,\\text{erfc}\\left(\\frac{x + aDt}{2\\sqrt{Dt}}\\right) + \\text{erfc}\\left(\\frac{x - aDt}{2\\sqrt{Dt}}\\right) \\right],", element, {displayMode:true,throwOnError:false});$

where

$(4)var element = document.getElementById("moose-equation-963e5fe1-c8b8-4d04-af12-9941d440e0f2");katex.render(" a = \\frac{zF}{RT}\\frac{\\partial \\phi}{\\partial x}.", element, {displayMode:true,throwOnError:false});$

Here, $var element = document.getElementById("moose-equation-eb2140e3-f45b-46bf-ab48-b8e201178e9d");katex.render("z=1", element, {displayMode:false,throwOnError:false});$ is the charge number of the mobile hydroxyl defect carrying the hydrogen isotope. The semi-infinite approximation is valid over the simulated time range because the characteristic diffusion length $var element = document.getElementById("moose-equation-1a9de5b1-6a58-4f44-abca-878e8e0e7a9b");katex.render("\\sqrt{Dt}\\approx 0.14", element, {displayMode:false,throwOnError:false});$ mm remains much smaller than the 10 mm membrane thickness.

Results

Figure 1 compares the TMAP8 results and the analytical solution as a function of time at $var element = document.getElementById("moose-equation-7d23ae93-253e-496e-8288-bd8cd29732cf");katex.render("x = 0.1", element, {displayMode:false,throwOnError:false});$ mm and $var element = document.getElementById("moose-equation-27b3f35a-c08a-4221-b81d-d53126ec836b");katex.render("x = 0.5", element, {displayMode:false,throwOnError:false});$ mm. The TMAP8 calculations closely match the analytical solution at both locations, with root mean square percentage error (RMSPE) values of 0.14% and 0.51%, respectively.

Comparison of deuterium concentration as a function of time at $x = 0.1$ mm and $x = 0.5$ mm calculated by TMAP8 and by the analytical solution.

Figure 1: Comparison of deuterium concentration as a function of time at $var element = document.getElementById("moose-equation-999cecf3-f34e-4f68-ae56-e814794ea9dd");katex.render("x = 0.1", element, {displayMode:false,throwOnError:false});$ mm and $var element = document.getElementById("moose-equation-186df210-d8dc-4ba2-8e12-6ffca90df650");katex.render("x = 0.5", element, {displayMode:false,throwOnError:false});$ mm calculated by TMAP8 and by the analytical solution.

Comparison of deuterium concentration as a function of distance from the source at $t = 30$ s and $t = 500$ s calculated by TMAP8 and by the analytical solution.

Figure 2: Comparison of deuterium concentration as a function of distance from the source at $var element = document.getElementById("moose-equation-4d709a4c-30cf-48d5-a431-3ed5d6a233cc");katex.render("t = 30", element, {displayMode:false,throwOnError:false});$ s and $var element = document.getElementById("moose-equation-c92fcc5c-2cde-495a-bcd4-624d69826265");katex.render("t = 500", element, {displayMode:false,throwOnError:false});$ s calculated by TMAP8 and by the analytical solution.

Input Files

The input file for this case can be found at (test/tests/ver-1n/ver-1n.i). More information about how this is used as a TMAP8 test can be found in the test specification file for this case (test/tests/ver-1n/tests).

References

M Khalid Hossain, Kenichi Hashizume, and Yuji Hatano. Evaluation of the hydrogen solubility and diffusivity in proton-conducting oxides by converting the PSL values of a tritium imaging plate. Nuclear Materials and Energy, 25:100875, 2020.

BibTeX

@article{hossain2020evaluation,
    author = "Hossain, M Khalid and Hashizume, Kenichi and Hatano, Yuji",
    title = "Evaluation of the hydrogen solubility and diffusivity in proton-conducting oxides by converting the {PSL} values of a tritium imaging plate",
    journal = "Nuclear Materials and Energy",
    volume = "25",
    pages = "100875",
    year = "2020",
    publisher = "Elsevier"
}

RIS

TY  - JOUR
AU  - Hossain, M Khalid
AU  - Hashizume, Kenichi
AU  - Hatano, Yuji
TI  - Evaluation of the hydrogen solubility and diffusivity in proton-conducting oxides by converting the PSL values of a tritium imaging plate
JO  - Nuclear Materials and Energy
PY  - 2020
VL  - 25
PB  - Elsevier
SP  - 100875
ER  -

Plain Text

M Khalid Hossain, Kenichi Hashizume, and Yuji Hatano. Evaluation of the hydrogen solubility and diffusivity in proton-conducting oxides by converting the PSL values of a tritium imaging plate. Nuclear Materials and Energy, 25:100875, 2020.

Tang Luping and Lars-Olof Nilsson. Rapid determination of the chloride diffusivity in concrete by applying an electric field. Materials Journal, 89(1):49–53, 1993.

BibTeX

@article{luping1993rapid,
    author = "Luping, Tang and Nilsson, Lars-Olof",
    title = "Rapid determination of the chloride diffusivity in concrete by applying an electric field",
    journal = "Materials Journal",
    volume = "89",
    number = "1",
    pages = "49--53",
    year = "1993"
}

RIS

TY  - JOUR
AU  - Luping, Tang
AU  - Nilsson, Lars-Olof
TI  - Rapid determination of the chloride diffusivity in concrete by applying an electric field
JO  - Materials Journal
PY  - 1993
VL  - 89
IS  - 1
SP  - 49
EP  - 53
ER  -

Plain Text

Tang Luping and Lars-Olof Nilsson. Rapid determination of the chloride diffusivity in concrete by applying an electric field. Materials Journal, 89(1):49–53, 1993.

(test/tests/ver-1n/ver-1n.i)

# Verification Problem #1n for voltage-assisted transport
# Diffusion verification under an applied voltage with constant source boundary condition

# Physical constants
R = '${units 8.31446261815324 J/mol/K}' # ideal gas constant based on number used in include/utils/PhysicalConstants.h
eV_to_J = '${units 1.602176634e-19 eV/J}'
N_a = '${units 6.02214076e23 at/mol}'
q = '${units 1.602176634e-19 C}'
F = '${fparse N_a * q}'

# Pressure conditions
pressure_high = '${units 100 Pa}'
pressure_low = '${units 1e-6 Pa}'

# Temperature conditions
temperature = '${units 773 K}' # 500 C

# PCC Materials properties
diffusion_pre_PCC = '${units ${fparse sqrt(3) * 1.41e-6} m^2/s -> mum^2/s}' # Deuterium
diffusion_energy_PCC = '${units ${fparse 0.74 * eV_to_J * N_a} J/mol}'
solubility_pre_PCC = '${units ${fparse 1.06 * N_a / 1e18} at/mum^3/Pa^0.5}' # at/m^3/Pa^0.5 -> at/mum^3/Pa^0.5
solubility_energy_PCC = '${units 7726.21 J/mol}'
solubility_order = .5 # Here, we use Sievert's law
charge_number = 1
V_current = '${units 20 V}'

# Important times
end_time = '${units 500 s}'
dt_max = '${units 10 s}'
dt_start = '${units 1 s}'

# Geometry and mesh
length = '${units 10 mm -> mum}'
num_nodes = 1000

[Mesh]
  [cmg]
    type = CartesianMeshGenerator
    dim = 1
    dx = '${fparse length}'
    ix = '${fparse num_nodes}'
  []
[]

[Variables]
  [deuterium_concentration_PCC] # (atoms/microns^3)
  []
[]

[AuxVariables]
  [voltage_phi]
  []
[]

[AuxKernels]
  [phi_auxkernel]
    type = FunctionAux
    variable = voltage_phi
    function = '${V_current} * (${length} - x) / ${length}'
    execute_on = 'INITIAL TIMESTEP_END'
  []
[]

[Problem]
  type = ReferenceResidualProblem
  extra_tag_vectors = 'ref'
  reference_vector = 'ref'
[]

[Kernels]
  [time_PCC]
    type = TimeDerivative
    variable = deuterium_concentration_PCC
    extra_vector_tags = ref
  []
  [diffusion_PCC]
    type = ADMatDiffusion
    variable = deuterium_concentration_PCC
    diffusivity = diffusivity_PCC
    extra_vector_tags = ref
  []
  [diffusion_phi_PCC]
    type = ADMatDiffusion
    variable = deuterium_concentration_PCC
    v = voltage_phi
    diffusivity = conductivity
    extra_vector_tags = ref
  []
[]

[BCs]
  [left_flux]
    type = EquilibriumBC
    Ko = '${solubility_pre_PCC}'
    activation_energy = '${solubility_energy_PCC}'
    boundary = left
    enclosure_var = '${pressure_high}'
    temperature = '${temperature}'
    variable = deuterium_concentration_PCC
    p = ${solubility_order}
  []
  [right_flux]
    type = EquilibriumBC
    Ko = '${solubility_pre_PCC}'
    activation_energy = '${solubility_energy_PCC}'
    boundary = right
    enclosure_var = '${pressure_low}'
    temperature = '${temperature}'
    variable = deuterium_concentration_PCC
    p = ${solubility_order}
  []
[]

[Functions]
  [diffusivity_PCC_func]
    type = ParsedFunction
    expression = '${diffusion_pre_PCC} * exp(-${diffusion_energy_PCC} / ${R} / ${temperature})'
  []
  [solubility_PCC_func]
    type = ParsedFunction
    expression = '${solubility_pre_PCC} * exp(-${solubility_energy_PCC} / ${R} / ${temperature})'
  []
[]

[Materials]
  [diffusion_solubility]
    type = ADGenericFunctionMaterial
    prop_names = 'diffusivity_PCC solubility_PCC'
    prop_values = 'diffusivity_PCC_func solubility_PCC_func'
    outputs = all
  []
  [conductivity]
    type = ADParsedMaterial
    property_name = 'conductivity'
    coupled_variables = 'deuterium_concentration_PCC'
    functor_names = 'diffusivity_PCC_func'
    functor_symbols = 'diffusivity'
    expression = 'diffusivity * ${charge_number} * ${F} * deuterium_concentration_PCC / ${R} / ${temperature}'
    outputs = all
  []
[]

[Postprocessors]
  [concentration_point1]
    type = PointValue
    variable = deuterium_concentration_PCC
    point = '100 0 0'
    outputs = 'csv'
  []
  [concentration_point2]
    type = PointValue
    variable = deuterium_concentration_PCC
    point = '500 0 0'
    outputs = 'csv'
  []
[]

[VectorPostprocessors]
  [line]
    type = LineValueSampler
    start_point = '0 0 0'
    end_point = '1000 0 0'
    num_points = 1001
    sort_by = 'x'
    variable = deuterium_concentration_PCC
    outputs = 'vector_postproc_30s vector_postproc_500s'
  []
[]

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

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = NEWTON
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-8
  end_time = ${end_time}
  automatic_scaling = true
  compute_scaling_once = true
  # nl_max_its = 7
  dtmax = ${dt_max}
  [TimeStepper]
    type = IterationAdaptiveDT
    dt = ${dt_start}
    optimal_iterations = 5
    growth_factor = 1.1
    cutback_factor = 0.9
    cutback_factor_at_failure = 0.9
  []
[]

[Debug]
  show_var_residual_norms = true
[]

[Outputs]
  csv = true
  [exodus]
    type = Exodus
    output_material_properties = true
    time_step_interval = 5
  []
  [vector_postproc_30s]
    type = CSV
    sync_times = '30'
    sync_only = true
  []
  [vector_postproc_500s]
    type = CSV
    sync_times = '500'
    sync_only = true
  []
[]

(test/tests/ver-1n/tests)

[Tests]
  design = 'EquilibriumBC.md'
  verification = 'ver-1n.md'
  issues = '#426'
  [ver-1n]
    type = Exodiff
    input = ver-1n.i
    exodiff = ver-1n_exodus.e
    requirement = 'The system shall be able to model deuterium diffusion under applied voltage through a slab with a constant concentration boundary condition as the species source.'
  []
  [ver-1n_csvdiff]
    type = CSVDiff
    input = ver-1n.i
    should_execute = False  # this test relies on the output files from ver-1n, so it shouldn't be run twice
    csvdiff = ver-1n_out.csv
    requirement = 'The system shall be able to model deuterium diffusion under applied voltage through a slab with a constant concentration boundary condition as the species source to generate CSV data for use in comparisons with the analytic solution over time.'
    prereq = ver-1n
  []
  [ver-1n_lineplot1]
    type = CSVDiff
    input = ver-1n.i
    should_execute = False  # this test relies on the output files from ver-1n, so it shouldn't be run twice
    csvdiff = ver-1n_vector_postproc_30s_line_0015.csv
    requirement = 'The system shall be able to model deuterium diffusion under applied voltage through a slab with a constant concentration boundary condition as the species source to generate concentration CSV data of point 1 for use in comparisons with the analytic solution over depth.'
    prereq = ver-1n
  []
  [ver-1n_lineplot2]
    type = CSVDiff
    input = ver-1n.i
    should_execute = False  # this test relies on the output files from ver-1n, so it shouldn't be run twice
    csvdiff = ver-1n_vector_postproc_500s_line_0066.csv
    requirement = 'The system shall be able to model deuterium diffusion under applied voltage through a slab with a constant concentration boundary condition as the species source to generate concentration CSV data of point 2 for use in comparisons with the analytic solution over depth.'
    prereq = ver-1n
  []
  [ver-1n_comparison]
    type = RunCommand
    command = 'python3 comparison_ver-1n.py'
    requirement = 'The system shall be able to generate comparison plots between the analytical solution and simulated solution of verification case 1n, modeling deuterium diffusion under applied voltage through a slab with a constant concentration boundary condition as the species source.'
    required_python_packages = 'matplotlib numpy pandas scipy os'
  []
[]

Case Description
Case Set Up
Analytical Solution
Results
Input Files
References