ver-1kc-1

Sieverts’ Law Boundaries with No Volumetric Source

General Case Description

Two enclosures are separated by a membrane that allows diffusion according to Sievert's law, with no volumetric source present. Enclosure 2 has twice the volume of Enclosure 1.

Case Set Up

This verification problem is taken from Ambrosek and Longhurst (2008).

This setup describes a diffusion system in which tritium T $var element = document.getElementById("moose-equation-464e3736-e363-450a-89ca-dd11d2d832f2");katex.render("_2", element, {displayMode:false,throwOnError:false});$ is modeled across a one-dimensional domain split into two enclosures. The total system length is $var element = document.getElementById("moose-equation-20d4a375-d814-4841-a2cc-af333316f5be");katex.render("2.5 \\times 10^{-4}", element, {displayMode:false,throwOnError:false});$ m, divided into 100 segments. The system operates at a constant temperature of 500 Kelvin. Initial tritium pressures are specified as $var element = document.getElementById("moose-equation-f124e43a-f161-49b2-b2e0-546750eca925");katex.render("10^{5}", element, {displayMode:false,throwOnError:false});$ Pa for Enclosure 1 and $var element = document.getElementById("moose-equation-41dc039f-2a4f-41b2-8f5f-5f84323d5e9e");katex.render("10^{-10}", element, {displayMode:false,throwOnError:false});$ Pa for Enclosure 2.

The diffusion process in each of the two enclosures can be described by

$var element = document.getElementById("moose-equation-475058a6-182f-44e1-bd29-8dd13b6e644d");katex.render("\\frac{\\partial C_1}{\\partial t} = \\nabla D \\nabla C_1,", element, {displayMode:true,throwOnError:false});$ and $var element = document.getElementById("moose-equation-7274bf43-e4ca-4817-b79c-e4fc0a88dbba");katex.render("\\frac{\\partial C_2}{\\partial t} = \\nabla D \\nabla C_2,", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-ecb87fb1-177c-42e9-b063-cf1f8b32a9fa");katex.render("C_1", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-08d740cc-ea9d-4d59-b2ad-f2085e6efdf2");katex.render("C_2", element, {displayMode:false,throwOnError:false});$ represent the concentration fields in enclosures 1 and 2 respectively, $var element = document.getElementById("moose-equation-5af92e80-bdf5-4df6-a478-30b8dca55094");katex.render("t", element, {displayMode:false,throwOnError:false});$ is the time, and $var element = document.getElementById("moose-equation-9d46fe1b-f503-4bd5-9dda-d450fc1e3cc7");katex.render("D", element, {displayMode:false,throwOnError:false});$ denotes the diffusivity.

This case is similar to the ver-1kb case, with the key difference being that sorption here follows Sieverts' law instead of Henry's law. The concentration in Enclosure 1 is related to the partial pressure and concentration in Enclosure 2 via the interface sorption law:

$var element = document.getElementById("moose-equation-4b47d777-ed26-4d18-9c1e-7773827571c5");katex.render("C_1 = K P_2^n = K \\left( C_2 RT \\right)^n", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-615b5120-5199-4fe6-83a9-5f147748ecd8");katex.render("R", element, {displayMode:false,throwOnError:false});$ is the ideal gas constant in J/mol/K, $var element = document.getElementById("moose-equation-7bfb1ab6-1a29-4435-a716-3945b7bab9bc");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature in K, $var element = document.getElementById("moose-equation-45b6d599-2c67-4bc3-9f81-fb6819fe8a13");katex.render("K", element, {displayMode:false,throwOnError:false});$ is the solubility, and $var element = document.getElementById("moose-equation-29c1ed27-e3d1-46e9-96d3-39ea211d1d00");katex.render("n", element, {displayMode:false,throwOnError:false});$ is the exponent of the sorption law. For Sieverts' law, $var element = document.getElementById("moose-equation-1c34b169-666c-41be-a898-10b8ba6d85b0");katex.render("n=0.5", element, {displayMode:false,throwOnError:false});$ .

Results

We assume that $var element = document.getElementById("moose-equation-a046ff1b-95ee-47d7-8c8b-9c3c817e51f9");katex.render("K = \\frac{10}{\\sqrt{RT}}", element, {displayMode:false,throwOnError:false});$ , which is expected to lead to $var element = document.getElementById("moose-equation-11dbac31-2be6-44de-b8af-f3863df54839");katex.render("C_1 = 10 \\sqrt{C_2}", element, {displayMode:false,throwOnError:false});$ at equilibrium. As illustrated in Figure 1, the pressure jump maintains a ratio of $var element = document.getElementById("moose-equation-fc61c869-f8e8-408a-be1f-8b77f7201c11");katex.render("\\frac{C_1}{\\sqrt{C_2}} \\approx 10", element, {displayMode:false,throwOnError:false});$ , which is consistent with the relationship $var element = document.getElementById("moose-equation-cb9faebc-428f-4b08-b7a1-3522c99b5113");katex.render("C_1 = K (RT C_2)^n", element, {displayMode:false,throwOnError:false});$ for $var element = document.getElementById("moose-equation-1b8fc347-09ef-4ee2-bca6-b3f6eb8b6ed8");katex.render("K = \\frac{10}{\\sqrt{RT}}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-5ff00683-6f90-454b-a583-e31cd0ec7836");katex.render("n=0.5", element, {displayMode:false,throwOnError:false});$ The concentration ratio between enclosures 1 and 2 in Figure 2 shows that the results obtained with TMAP8 are consistent with the analytical results derived from the sorption law for $var element = document.getElementById("moose-equation-653d174d-b568-4ff1-93d2-97827e10b9c6");katex.render("K \\sqrt{RT}=10", element, {displayMode:false,throwOnError:false});$ . As shown in Figure 3, mass is conserved between the two enclosures over time, with a variation in mass of only $var element = document.getElementById("moose-equation-044d5a8c-21d9-47a9-af09-6dd93a710604");katex.render("0.4", element, {displayMode:false,throwOnError:false});$ %. This variation in mass can be further minimized by refining the mesh, i.e., increasing the number of segments in the domain.

Figure 1: Evolution of species concentration over time governed by Sieverts' law with $var element = document.getElementById("moose-equation-e2232bed-a14b-4096-9b50-f5c4d550aca3");katex.render("K = \\frac{10}{\\sqrt{RT}}", element, {displayMode:false,throwOnError:false});$ .

Figure 2: Concentrations ratio between enclosures 1 and 2 at the interface for $var element = document.getElementById("moose-equation-b0ad72a7-e479-4c8e-9997-6a9ca9747326");katex.render("K = \\frac{10}{\\sqrt{RT}}", element, {displayMode:false,throwOnError:false});$ . This verifies TMAP8's ability to apply Sieverts' law across the interface.

Figure 3: Total mass conservation across both enclosures over time for $var element = document.getElementById("moose-equation-e5f999e9-ef95-49aa-8b39-2bf7e43e3fb7");katex.render("K = \\frac{10}{\\sqrt{RT}}", element, {displayMode:false,throwOnError:false});$ .

Input files

The input file for this case can be found at (test/tests/ver-1kc-1/ver-1kc-1.i). To limit the computational costs of the test cases, the tests run a version of the file with a coarser mesh and less number of time steps. More information about the changes can be found in the test specification file for this case (test/tests/ver-1kc-1/tests).

References

James Ambrosek and GR Longhurst. Verification and Validation of TMAP7. Technical Report INEEL/EXT-04-01657, Idaho National Engineering and Environmental Laboratory, December 2008.

@techreport{ambrosek2008verification,
    author = "Ambrosek, James and Longhurst, GR",
    title = "{Verification and Validation of TMAP7}",
    year = "2008",
    number = "INEEL/EXT-04-01657",
    month = "December",
    institution = "Idaho National Engineering and Environmental Laboratory"
}

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

nb_segments_TMAP7 = 20
node_size_TMAP7 = '${units 1.25e-5 m}'
long_total = '${units ${fparse nb_segments_TMAP7 * node_size_TMAP7} m}'
nb_segments_TMAP8 = 1e2
simulation_time = '${units 10 s}'
temperature = '${units 500 K}'
R = '${units 8.31446261815324 J/mol/K}' # ideal gas constant from PhysicalConstants.h
initial_pressure_1 = '${units 1e5 Pa}'
initial_pressure_2 = '${units 1e-10 Pa}'
initial_concentration_1 = '${units ${fparse initial_pressure_1 / (R*temperature)} mol/m^3}'
initial_concentration_2 = '${units ${fparse initial_pressure_2 / (R*temperature)} mol/m^3}'
solubility = '${units ${fparse 10/sqrt(R*temperature)} mol/m^3/Pa^(1/2)}' # Sieverts' law solubility
diffusivity = '${units ${fparse 4.31e-6 * exp(-2818/temperature)} m^2/s}'
n_sorption = 0.5 # Sieverts' Law
unit_scale = 1
unit_scale_neighbor = 1

[Mesh]
  [generated]
    type = GeneratedMeshGenerator
    dim = 1
    nx = ${nb_segments_TMAP8}
    xmax = ${long_total}
  []
  [enclosure_1]
    type = SubdomainBoundingBoxGenerator
    input = generated
    block_id = 1
    bottom_left = '0 0 0'
    top_right = '${fparse 1/3 * long_total} 0 0'
  []
  [enclosure_2]
    type = SubdomainBoundingBoxGenerator
    input = enclosure_1
    block_id = 2
    bottom_left = '${fparse 1/3 * long_total} 0 0'
    top_right = '${fparse long_total} 0 0'
  []
  [interface]
    type = SideSetsBetweenSubdomainsGenerator
    input = enclosure_2
    primary_block = 1
    paired_block = 2
    new_boundary = interface
  []
  [interface2]
    type = SideSetsBetweenSubdomainsGenerator
    input = interface
    primary_block = 2
    paired_block = 1
    new_boundary = interface2
  []
[]

[Variables]
  [concentration_enclosure_1]
    block = 1
    order = FIRST
    family = LAGRANGE
    initial_condition = '${initial_concentration_1}'
  []
  [concentration_enclosure_2]
    block = 2
    order = FIRST
    family = LAGRANGE
    initial_condition = '${initial_concentration_2}'
  []
[]

[AuxVariables]
  [pressure_enclosure_1]
    order = CONSTANT
    family = MONOMIAL
    block = 1
    initial_condition = '${initial_pressure_1}'
  []
  [pressure_enclosure_2]
    order = CONSTANT
    family = MONOMIAL
    block = 2
    initial_condition = '${initial_pressure_2}'
  []
[]

[Kernels]
  [diffusion_enclosure_1]
    type = MatDiffusion
    variable = concentration_enclosure_1
    diffusivity = ${diffusivity}
    block = '1'
  []
  [time_enclosure_1]
    type = TimeDerivative
    variable = concentration_enclosure_1
    block = '1'
  []
  [diffusion_enclosure_2]
    type = MatDiffusion
    variable = concentration_enclosure_2
    diffusivity = ${diffusivity}
    block = '2'
  []
  [time_enclosure_2]
    type = TimeDerivative
    variable = concentration_enclosure_2
    block = '2'
  []
[]

[AuxKernels]
  [pressure_enclosure_1]
    type = ParsedAux
    variable = pressure_enclosure_1
    coupled_variables = 'concentration_enclosure_1'
    expression = '${fparse R*temperature}*concentration_enclosure_1'
    block = 1
    execute_on = 'initial timestep_end'
  []
  [pressure_enclosure_2]
    type = ParsedAux
    variable = pressure_enclosure_2
    coupled_variables = 'concentration_enclosure_2'
    expression = '${fparse R*temperature}*concentration_enclosure_2'
    block = 2
    execute_on = 'initial timestep_end'
  []
[]

[InterfaceKernels]
  [interface_sorption]
    type = InterfaceSorption
    K0 = ${solubility}
    Ea = 0
    n_sorption = ${n_sorption}
    diffusivity = ${diffusivity}
    unit_scale = ${unit_scale}
    unit_scale_neighbor = ${unit_scale_neighbor}
    temperature = ${temperature}
    variable = concentration_enclosure_1
    neighbor_var = concentration_enclosure_2
    sorption_penalty = 1e1
    boundary = interface
  []
[]

[Postprocessors]
  [pressure_enclosure_1]
    type = ElementAverageValue
    variable = pressure_enclosure_1
    block = 1
    execute_on = 'initial timestep_end'
  []
  [pressure_enclosure_2]
    type = ElementAverageValue
    variable = pressure_enclosure_2
    block = 2
    execute_on = 'initial timestep_end'
  []
  [concentration_enclosure_1_at_interface]
    type = SideAverageValue
    boundary = interface
    variable = concentration_enclosure_1
    outputs = 'csv console'
    execute_on = 'initial timestep_end'
  []
  [concentration_enclosure_2_at_interface]
    type = SideAverageValue
    boundary = interface2
    variable = concentration_enclosure_2
    outputs = 'csv console'
    execute_on = 'initial timestep_end'
  []
  [concentration_ratio]
    type = ParsedPostprocessor
    expression = 'concentration_enclosure_1_at_interface / sqrt(concentration_enclosure_2_at_interface)'
    pp_names = 'concentration_enclosure_1_at_interface concentration_enclosure_2_at_interface'
    execute_on = 'initial timestep_end'
    outputs = 'csv console'
  []
  [solubility]
    type = ParsedPostprocessor
    expression = 'concentration_enclosure_1_at_interface / pressure_enclosure_2_at_interface'
    pp_names = 'concentration_enclosure_1_at_interface pressure_enclosure_2_at_interface'
    execute_on = 'initial timestep_end'
    outputs = 'csv console'
  []
  [pressure_enclosure_1_at_interface]
    type = SideAverageValue
    boundary = interface
    variable = pressure_enclosure_1
    outputs = 'csv console'
    execute_on = 'initial timestep_end'
  []
  [pressure_enclosure_2_at_interface]
    type = SideAverageValue
    boundary = interface2
    variable = pressure_enclosure_2
    outputs = 'csv console'
    execute_on = 'initial timestep_end'
  []
  [concentration_encl_1_inventory]
    type = ElementIntegralVariablePostprocessor
    variable = concentration_enclosure_1
    block = 1
    execute_on = 'initial timestep_end'
  []
  [concentration_encl_2_inventory]
    type = ElementIntegralVariablePostprocessor
    variable = concentration_enclosure_2
    block = 2
    execute_on = 'initial timestep_end'
  []
  [mass_conservation_sum_encl1_encl2]
    type = LinearCombinationPostprocessor
    pp_names = 'concentration_encl_1_inventory concentration_encl_2_inventory'
    pp_coefs = '1            1'
    execute_on = 'initial timestep_end'
  []
[]

[Executioner]
  type = Transient
  end_time = ${simulation_time}
  dtmax = 10
  nl_abs_tol = 1e-8
  nl_rel_tol = 1e-6
  scheme = 'bdf2'
  solve_type = NEWTON
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
  nl_max_its = 6
  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 0.01
    optimal_iterations = 5
    iteration_window = 3
    growth_factor = 1.05
    cutback_factor_at_failure = 0.8
  []
[]

[Outputs]
  file_base = 'ver-1kc-1_out_k10'
  csv = true
  exodus = true
  execute_on = 'initial timestep_end'
[]

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

[Tests]
  design = 'InterfaceSorption.md MatDiffusion.md TimeDerivative.md'
  issues = '#12'
  verification = 'ver-1kc-1.md'
  [ver-1kc-1_csv]
    type = CSVDiff
    input = ver-1kc-1.i
    cli_args = "nb_segments_TMAP8=1e1
                simulation_time=4
                Executioner/nl_abs_tol=1e-5
                Executioner/nl_rel_tol=1e-4
                Outputs/exodus=false
                Outputs/file_base=ver-1kc-1_out_k10_light"
    csvdiff = ver-1kc-1_out_k10_light.csv
    requirement = 'The system shall be able to model the diffusion of T2 across a membrane separating two enclosures in accordance with Sieverts’ law with a concentration jump at the interface.'
  []
  [ver-1kc-1_csv_heavy]
    type = CSVDiff
    heavy = true
    input = ver-1kc-1.i
    csvdiff = ver-1kc-1_out_k10.csv
    requirement = 'The system shall be able to model the diffusion of T2 across a membrane separating two enclosures in accordance with Sieverts’ law with a concentration jump at the interface with a fine mesh and tight tolerances for higher accuracy.'
  []
  [ver-1kc-1_exodus_heavy]
    type = Exodiff
    heavy = true
    input = ver-1kc-1.i
    exodiff = ver-1kc-1_out_k10.e
    prereq = ver-1kc-1_csv_heavy
    should_execute = false # this test relies on the output files from ver-1kc-1_csv_concentration_jump, so it shouldn't be run twice
    requirement = 'The system shall be able to model the diffusion of T2 across a membrane separating two enclosures in accordance with Sieverts’ law with a concentration jump at the interface with a fine mesh and tight tolerances for higher accuracy and generate an exodus file.'
  []
  [ver-1kc-1_comparison]
    type = RunCommand
    command = 'python3 comparison_ver-1kc-1.py'
    requirement = 'The system shall be able to generate comparison plots between the analytical solution and simulated solution of verification case 1kc-1, modeling a diffusion across a membrane separating two enclosures in accordance with Sieverts’ law.'
    required_python_packages = 'matplotlib numpy pandas os git'
  []
[]

General Case Description
Case Set Up
Results
Input files
References