ver-1l

Diffusion with Soret Effect (Thermodiffusion)

General Case Description

This verification case considers tritium diffusion through a semi-infinite layer under the influence of a temperature gradient. The two transport driving forces are Fickian diffusion (which was verified in other verification cases, including ver-1b and ver-1dd), and the Soret effect (also known as thermodiffusion or thermophoresis). The Soret effect describes the phenomenon where species migration occurs in response to a temperature gradient. This coupling of thermal and mass transport can be particularly important in fusion applications where significant temperature gradients exist across material structures.

In this problem, tritium diffuses through a material layer subjected to a constant linear temperature gradient. The left boundary maintains a constant tritium concentration, while the right boundary is impermeable. The combined effects of concentration-driven diffusion and temperature-driven thermodiffusion are verified against an analytical solution. For simplicity, no trapping or solubility effects are included in this case.

Case Set up

This verification case models tritium diffusion through a one-dimensional slab of thickness $var element = document.getElementById("moose-equation-20e10c64-269d-405b-b6f3-7ebd936a86db");katex.render("l = 100", element, {displayMode:false,throwOnError:false});$ m with a linear temperature distribution across the domain. The left boundary ( $var element = document.getElementById("moose-equation-43d40f7f-dc34-488b-9e10-345766dc04ca");katex.render("x = 0", element, {displayMode:false,throwOnError:false});$ ) is maintained at a constant concentration $var element = document.getElementById("moose-equation-6d160b32-a827-4334-987e-f05e385f3dae");katex.render("C_{\\text{left}} = 100", element, {displayMode:false,throwOnError:false});$ mol/m $var element = document.getElementById("moose-equation-648eadd9-4c5e-4d5a-95c4-12b75032946d");katex.render("^3", element, {displayMode:false,throwOnError:false});$ , while the right boundary ( $var element = document.getElementById("moose-equation-3fb353f7-e3da-46c9-8e61-9ac2dade75a5");katex.render("x = l", element, {displayMode:false,throwOnError:false});$ ) is impermeable (zero flux). The initial concentration throughout the domain is $var element = document.getElementById("moose-equation-5d06fcfa-43f4-4299-b543-edcce51db3c5");katex.render("C_0 = 0.1", element, {displayMode:false,throwOnError:false});$ mol/m $var element = document.getElementById("moose-equation-82765558-2aaa-49ee-ad39-cc1a77fe5192");katex.render("^3", element, {displayMode:false,throwOnError:false});$ . The temperature varies linearly from $var element = document.getElementById("moose-equation-f961b604-a2c7-4d20-a9f1-0215f49341c5");katex.render("T_{\\text{left}} = 1", element, {displayMode:false,throwOnError:false});$ K at the left boundary to $var element = document.getElementById("moose-equation-69766a39-9240-4990-a044-b9cfbc5787a7");katex.render("T_{\\text{right}} = 0", element, {displayMode:false,throwOnError:false});$ K at the right boundary, creating a constant temperature gradient. Note that these very simple values are selected for the simplicity of the verification case and do not aim to represent a realistic case.

The governing equation for the coupled Fickian diffusion and the Soret effect is described as:

$var element = document.getElementById("moose-equation-01e6be4a-b9f9-48d0-88bb-f18bf42d9297");katex.render(" \\frac{\\partial C}{\\partial t} = \\nabla \\cdot \\left( D \\nabla C + D S_T C \\nabla T \\right),", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-7b670ea3-507f-4dd2-9a4e-b374c5410995");katex.render("C", element, {displayMode:false,throwOnError:false});$ is the concentration of mobile species, $var element = document.getElementById("moose-equation-d798ecc3-48f3-4353-85bb-d7a935879c10");katex.render("D", element, {displayMode:false,throwOnError:false});$ is the diffusivity, and $var element = document.getElementById("moose-equation-14042bab-45b1-48db-8ff3-8139b6405b6d");katex.render("S_T", element, {displayMode:false,throwOnError:false});$ is the Soret coefficient. The material properties and case parameters are provided in Table 1.

Table 1: Values of material properties and case geometry for the Soret effect verification problem.

Parameter	Description	Value	Units
$var element = document.getElementById("moose-equation-ab75d109-635c-419e-8b41-a55cc5a1ccd4");katex.render("l", element, {displayMode:false,throwOnError:false});$	Thickness	100	m
$var element = document.getElementById("moose-equation-dafcd727-ce48-479b-a52b-796d9e565ce9");katex.render("D", element, {displayMode:false,throwOnError:false});$	Diffusivity	0.1	m $var element = document.getElementById("moose-equation-b49a16f5-1efc-421a-bc93-48dc98ce27b5");katex.render("^2", element, {displayMode:false,throwOnError:false});$ /s
$var element = document.getElementById("moose-equation-39f829c8-b113-40a1-9842-80d9033e112e");katex.render("S_T", element, {displayMode:false,throwOnError:false});$	Soret coefficient	50	1/K
$var element = document.getElementById("moose-equation-8f7c24ee-c017-4b45-be6e-7bf36e7f59a4");katex.render("C_0", element, {displayMode:false,throwOnError:false});$	Initial concentration	0.1	mol/m $var element = document.getElementById("moose-equation-f214ecdd-f1a1-4b7c-99f4-5a2f228b3cde");katex.render("^3", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-3c218679-1fd4-40eb-849a-3b6f0bc70b1e");katex.render("C_{\\text{left}}", element, {displayMode:false,throwOnError:false});$	Concentration on left	100	mol/m $var element = document.getElementById("moose-equation-3c44d23f-edfe-4e43-9fee-a11b7ccf5105");katex.render("^3", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-5d7b7e72-dc07-41b3-a795-0d006d920073");katex.render("T_{\\text{left}}", element, {displayMode:false,throwOnError:false});$	Temperature on left	1	K
$var element = document.getElementById("moose-equation-302d94d0-9a25-4010-880a-9c12b2cc50ae");katex.render("T_{\\text{right}}", element, {displayMode:false,throwOnError:false});$	Temperature on right	0	K

The verification focuses on two aspects of the solution: (1) the temporal evolution of concentration at a specific location ( $var element = document.getElementById("moose-equation-a0a3179e-e282-4367-9155-6242349d560a");katex.render("x = 10", element, {displayMode:false,throwOnError:false});$ m), and (2) the spatial concentration profile at a specific time ( $var element = document.getElementById("moose-equation-0b68982d-4a13-4eea-98d6-5a9b45322322");katex.render("t = 100", element, {displayMode:false,throwOnError:false});$ s).

Analytical solution

For a semi-infinite domain with constant diffusivity and Soret coefficient, subject to a constant temperature gradient $var element = document.getElementById("moose-equation-99f91a30-b3fa-427a-b8e7-50573af7fb5e");katex.render("\\nabla T", element, {displayMode:false,throwOnError:false});$ , the analytical solution provided in Xie et al. (2015) is described as:

$var element = document.getElementById("moose-equation-7e1d926a-7cca-4fdd-b9af-de296c76e5ec");katex.render(" C(x,t) = \\left[ \\frac{1}{2} \\text{erfc} \\left( \\frac{D S_T t \\nabla T + x}{2\\sqrt{D t}} \\right) + \\frac{1}{2} e^{- S_T x \\nabla T} \\text{erfc} \\left( \\frac{- D S_T t \\nabla T + x}{2\\sqrt{D t}} \\right) \\right] (C_{\\text{left}} - C_0) + C_0,", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-44c9df2f-70b3-4e5e-8230-f716286fc6a5");katex.render("\\text{erfc}", element, {displayMode:false,throwOnError:false});$ is the error function.

Results

Verification of concentration at a fixed location as a function of time

Figure 1 shows the comparison of the TMAP8 calculation and the analytical solution for the concentration at location $var element = document.getElementById("moose-equation-b75f02be-be7d-4a2b-abba-3bafbd5cbc91");katex.render("x = 10", element, {displayMode:false,throwOnError:false});$ m as a function of time. The TMAP8 prediction matches the analytical solution with excellent agreement, yielding a root mean square percentage error of RMSPE = 0.87 %.

Comparison of TMAP8 calculation with the analytical solution for the concentration at $x = 10$ m as a function of time.

Figure 1: Comparison of TMAP8 calculation with the analytical solution for the concentration at $var element = document.getElementById("moose-equation-0df1c21a-2acc-4b3d-8bbb-ee9603fdf2b9");katex.render("x = 10", element, {displayMode:false,throwOnError:false});$ m as a function of time.

Verification of concentration profile as a function of position at a fixed time

Figure 2 shows the comparison of the TMAP8 calculation and the analytical solution for the concentration profile at time $var element = document.getElementById("moose-equation-1044c299-19e7-493c-be5f-b033b5be83e7");katex.render("t = 100", element, {displayMode:false,throwOnError:false});$ s. The concentration profile exhibits a characteristic shape that differs from pure Fickian diffusion due to the thermodiffusion contribution. The TMAP8 prediction is in good agreement with the analytical solution, with a root mean square percentage error of RMSPE = 0.21 %.

Figure 2: Comparison of TMAP8 calculation with the analytical solution for the concentration profile at $var element = document.getElementById("moose-equation-062aa3c4-802d-410e-8a1a-3d38f913cb64");katex.render("t = 100", element, {displayMode:false,throwOnError:false});$ s as a function of position.

Input files

The input file for this case can be found at (tmap8/test/tests/ver-1l/ver-1l.i), which is also used as test in TMAP8 at (tmap8/test/tests/ver-1l/tests).

References

Haijian Xie, Chunhua Zhang, Majid Sedighi, Hywel R Thomas, and Yunmin Chen. An analytical model for diffusion of chemicals under thermal effects in semi-infinite porous media. Computers and Geotechnics, 69:329–337, 2015. doi:https://doi.org/10.1016/j.compgeo.2015.06.012.

@article{xie2015analytical,
    author = "Xie, Haijian and Zhang, Chunhua and Sedighi, Majid and Thomas, Hywel R and Chen, Yunmin",
    title = "An analytical model for diffusion of chemicals under thermal effects in semi-infinite porous media",
    journal = "Computers and Geotechnics",
    volume = "69",
    pages = "329--337",
    year = "2015",
    doi = "https://doi.org/10.1016/j.compgeo.2015.06.012",
    publisher = "Elsevier"
}

(tmap8/test/tests/ver-1l/ver-1l.i)

# Verification Problem #1l for Soret effect
# Tritium diffusion through semi-infinity layer under heat gradient

# Physical Constants
# This is to be consistent with PhysicalConstant.h
# kb = '${units 1.380649e-23 J/K}' # Boltzmann constant
# R = '${units 8.31446261815324 J/mol/K}' # Gas constant

# Material properties
point_location = '${units 10 m}'
thickness = '${units 100 m}'
temperature_left = '${units 1 K}'
temperature_right = '${units 0 K}'
diffusivity = '${units 0.1 m^2/s}'
soret_coefficient = '${units 50 1/K}'
initial_concentration = '${units 0.1 mol/m^3}'
concentration_left = '${units 100 mol/m^3}'

simulation_time = '${units 100 s}'


[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 160
  xmax = '${thickness}'
[]

[AuxVariables]
  [temperature]
  []
[]

[AuxKernels]
  [temperature_kernel]
    type = ParsedAux
    use_xyzt = true
    variable = temperature
    expression = '(${temperature_right} - ${temperature_left}) / ${thickness} * x + ${temperature_left}'
    execute_on = 'initial timestep_end'
  []
[]

[Variables]
  # concentration in mol/m^3
  [concentration]
    initial_condition = '${initial_concentration}'
  []
[]

[Kernels]
  [time]
    type = ADTimeDerivative
    variable = concentration
  []
  [diffusion]
    type = ADMatDiffusion
    variable = concentration
    diffusivity = '${diffusivity}'
  []
  # Computes thermodiffusive flux: -D * S_T * C * grad(T)
  [thermodiffusion]
    type = ADThermoDiffusion
    variable = concentration
    temperature = 'temperature'
    soret_coefficient = 'thermodiffusion_prefactor'
  []
[]

[Materials]
  # Computes thermodiffusive prefactor: -D * S_T * C
  [thermodiffusion_prefactor]
    type = ADParsedMaterial
    property_name = 'thermodiffusion_prefactor'
    coupled_variables = 'concentration'
    expression = '${soret_coefficient} * ${diffusivity} * concentration'
  []
[]

[BCs]
  # The concentration of tritium on left is assumed constant
  [left]
    type = ADDirichletBC
    variable = concentration
    value = '${concentration_left}'
    boundary = 'left'
  []
  # The concentration of tritium on right is assumed impermeable
  [right]
    type = ADNeumannBC
    variable = concentration
    value = 0
    boundary = 'right'
  []
[]

[Postprocessors]
  # flux of tritium through the left surface
  [flux_surface_left]
    type = SideDiffusiveFluxIntegral
    variable = concentration
    diffusivity = '${diffusivity}'
    boundary = 'left'
    execute_on = 'initial timestep_end'
    outputs = 'console csv exodus'
  []
  # flux of tritium through the right surface
  [flux_surface_right]
    type = SideDiffusiveFluxIntegral
    variable = concentration
    diffusivity = '${diffusivity}'
    boundary = 'right'
    execute_on = 'initial timestep_end'
    outputs = 'console csv exodus'
  []
  # concentration at location
  [concentration_point]
    type = PointValue
    variable = concentration
    point = '${point_location} 0 0'
    outputs = 'csv'
  []
[]

[VectorPostprocessors]
  [line]
    type = LineValueSampler
    start_point = '0 0 0'
    end_point = '${thickness} 0 0'
    num_points = 101
    sort_by = 'x'
    variable = concentration
    outputs = 'vector_postproc'
  []
[]

[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-10
  end_time = '${simulation_time}'
  dtmax = 1
  automatic_scaling = true
  [TimeStepper]
    type = IterationAdaptiveDT
    optimal_iterations = 12
    iteration_window = 1
    growth_factor = 1.1
    dt = 1e-3
    cutback_factor = 0.9
    cutback_factor_at_failure = 0.9
  []
[]

[Outputs]
  exodus = true
  print_linear_residuals = false
  perf_graph = true
  [csv]
    type = CSV
    execute_on = 'initial timestep_end'
  []
  [console]
    type = Console
    time_step_interval = 10
  []
  [vector_postproc]
    type = CSV
    sync_times = '${simulation_time}'
    sync_only = true
  []
[]

(tmap8/test/tests/ver-1l/tests)

[Tests]
  design = 'ADThermoDiffusion.md'
  verification = 'ver-1l.md'
  issues = '#351'
  [ver-1l]
    type = Exodiff
    input = ver-1l.i
    exodiff = ver-1l_out.e
    requirement = 'The system shall be able to model species Fickian and Soret diffusion through a semi-infinity layer.'
  []
  [ver-1l_csvdiff]
    type = CSVDiff
    input = ver-1l.i
    should_execute = False # this test relies on the output files from ver-1l, so it shouldn't be run twice
    csvdiff = ver-1l_csv.csv
    requirement = 'The system shall be able to model species Fickian and Soret diffusion through a semi-infinity layer to generate CSV data for use in comparisons with the analytic solution.'
    prereq = ver-1l
  []
  [ver-1l_comparison]
    type = RunCommand
    command = 'python3 comparison_ver-1l.py'
    requirement = 'The system shall be able to generate comparison plots between the analytical solution and simulated solution of verification case 1l, modeling species Fickian and Soret diffusion through a semi-infinity layer.'
    required_python_packages = 'matplotlib numpy pandas scipy os'
  []
[]

General Case Description
Case Set up
Analytical solution
Results
Input files
References