ver-1dd

Permeation Problem without Trapping

Test Description

This verification problem is taken from Ambrosek and Longhurst (2008) and builds on the capabilities verified in ver-1d and ver-1dc. The configuration and modeling parameters are the same as in ver-1d, except that, in the current case, there are no traps in the membrane. This case is simulated in (test/tests/ver-1dd/ver-1dd.i).

This problem models permeation through a membrane with a constant source in which no trap presented. We solve

$(1)var element = document.getElementById("moose-equation-9620105d-b914-4d10-9825-e2f3cd03a9ea");katex.render(" \\frac{dC_M}{dt} = \\nabla D \\nabla C_M ,", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-1a73ceea-631a-433d-911d-b429a6eb68ac");katex.render("C_M", element, {displayMode:false,throwOnError:false});$ is the concentrations of the mobile, $var element = document.getElementById("moose-equation-1cf03032-9b9f-4bad-a088-667c2b48bf6f");katex.render("D", element, {displayMode:false,throwOnError:false});$ is the diffusivity of the mobile species, and $var element = document.getElementById("moose-equation-8177a28f-1faf-465f-b4c8-59a408dd55ac");katex.render("t", element, {displayMode:false,throwOnError:false});$ is the time.

Analytical solution

Ambrosek and Longhurst (2008) provides the analytical equations for the permeation transient as

$(2)var element = document.getElementById("moose-equation-226daabd-6d93-48b1-b4f8-4b8c502ed91c");katex.render(" J_p = \\frac{C_0 D}{l} \\left\\{ 1 + 2 \\sum_{m=1}^{\\infty} \\left[ (-1)^m \\exp \\left( -m^2 \\frac{t}{2 \\; \\tau_{b_e}} \\right) \\right] \\right\\},", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-d36d50cf-4614-4747-ba31-0cef835f29f4");katex.render("C_0", element, {displayMode:false,throwOnError:false});$ is the steady dissolved gas concentration at the upstream (x = 0) side, $var element = document.getElementById("moose-equation-e5e06a87-5790-42ff-81db-2fc7ec5d98c7");katex.render("l", element, {displayMode:false,throwOnError:false});$ is the thickness of the slab, $var element = document.getElementById("moose-equation-a52fe3a6-1fe0-457d-bb46-27b48c867b6b");katex.render("D", element, {displayMode:false,throwOnError:false});$ is the diffusivity of the gas through the material, and $var element = document.getElementById("moose-equation-dc4fe672-ac27-4446-9dd9-84b9d7cf138b");katex.render("\\tau_{b_e}", element, {displayMode:false,throwOnError:false});$ , the breakthrough time, is defined as

$(3)var element = document.getElementById("moose-equation-17c209ef-e54a-4026-8091-e31c123b93b0");katex.render(" \\tau_{b_e} = \\frac{l^2}{2 \\; \\pi^2 \\; D_{eff}},", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-151af82e-ee26-4226-93f4-9d87ede2365b");katex.render("D_{eff}", element, {displayMode:false,throwOnError:false});$ , the effective diffusivity, is equal to $var element = document.getElementById("moose-equation-87b5a646-a400-4d4d-aa0a-90fbb2d46eb3");katex.render("D", element, {displayMode:false,throwOnError:false});$ due to the absence of traps in the membrane.

Results and comparison against analytical solution

The analytical solution for the permeation transient is compared with TMAP8 results in Figure 1. The graphs for the theoretical flux and the calculated flux are in good agreement, with root mean square percentage errors (RMSPE) of RMSPE = 0.14% for $var element = document.getElementById("moose-equation-f56df77f-3891-413f-93ee-e25c1be1de24");katex.render("t \\geq 0.01", element, {displayMode:false,throwOnError:false});$ s. The breakthrough time calculated from the analytical solution provided by Eq. (3) is 0.05 s, and the breakthrough time from TMAP8 is 0.05 s, which matches.

Figure 1: Comparison of TMAP8 calculation with the analytical solution for the permeation transient without traps from Ambrosek and Longhurst (2008).

Input files

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

cl = 3.1622e18 # atom/m^3
nx_num = 200 # (-)
diffusivity = 1 # m^2/s
simulation_time = 3 # s
interval_time_min = 0.01 # s
interval_time = 0.01 # s

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = ${nx_num}
  xmax = 1
[]

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

[Variables]
  [mobile]
  []
[]

[Kernels]
  [diff]
    type = Diffusion
    variable = mobile
    extra_vector_tags = ref
  []
  [time]
    type = TimeDerivative
    variable = mobile
    extra_vector_tags = ref
  []
[]

[BCs]
  [left]
    type = DirichletBC
    variable = mobile
    value = '${fparse cl / cl}'
    boundary = left
  []
  [right]
    type = DirichletBC
    variable = mobile
    value = 0
    boundary = right
  []
[]

[Postprocessors]
  [outflux]
    type = SideDiffusiveFluxAverage
    boundary = 'right'
    diffusivity = ${diffusivity}
    variable = mobile
  []
  [scaled_outflux]
    type = ScalePostprocessor
    value = outflux
    scaling_factor = ${cl}
  []
[]

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

[Executioner]
  type = Transient
  end_time = ${simulation_time}
  dt = ${interval_time}
  dtmin = ${interval_time_min}
  solve_type = NEWTON
  scheme = BDF2
  nl_abs_tol = 1e-13
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
  automatic_scaling = true
  verbose = true
  compute_scaling_once = false
[]

[Outputs]
  exodus = true
  csv = true
  [dof]
    type = DOFMap
    execute_on = initial
  []
  perf_graph = true
[]

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

cl = 3.1622e18 # atom/m^3
nx_num = 200 # (-)
diffusivity = 1 # m^2/s
simulation_time = 3 # s
interval_time_min = 0.01 # s
interval_time = 0.01 # s

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = ${nx_num}
  xmax = 1
[]

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

[Variables]
  [mobile]
  []
[]

[Kernels]
  [diff]
    type = Diffusion
    variable = mobile
    extra_vector_tags = ref
  []
  [time]
    type = TimeDerivative
    variable = mobile
    extra_vector_tags = ref
  []
[]

[BCs]
  [left]
    type = DirichletBC
    variable = mobile
    value = '${fparse cl / cl}'
    boundary = left
  []
  [right]
    type = DirichletBC
    variable = mobile
    value = 0
    boundary = right
  []
[]

[Postprocessors]
  [outflux]
    type = SideDiffusiveFluxAverage
    boundary = 'right'
    diffusivity = ${diffusivity}
    variable = mobile
  []
  [scaled_outflux]
    type = ScalePostprocessor
    value = outflux
    scaling_factor = ${cl}
  []
[]

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

[Executioner]
  type = Transient
  end_time = ${simulation_time}
  dt = ${interval_time}
  dtmin = ${interval_time_min}
  solve_type = NEWTON
  scheme = BDF2
  nl_abs_tol = 1e-13
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
  automatic_scaling = true
  verbose = true
  compute_scaling_once = false
[]

[Outputs]
  exodus = true
  csv = true
  [dof]
    type = DOFMap
    execute_on = initial
  []
  perf_graph = true
[]

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

[Tests]
  design = 'TrappingNodalKernel.md ReleasingNodalKernel.md'
  issues = '#12'
  verification = 'ver-1dd.md'
  [ver-1dd]
    type = Exodiff
    input = ver-1dd.i
    exodiff = ver-1dd_out.e
    requirement = 'The system shall be able to model a breakthrough problem without traps.'
  []
  [ver-1dd_csvdiff]
    type = CSVDiff
    input = ver-1dd.i
    should_execute = false # this test relies on the output files from ver-1dd, so it shouldn't be run twice
    csvdiff = ver-1dd_out.csv
    prereq = ver-1dd
    requirement = 'The system shall be able to model a breakthrough problem without traps, and generate CSV data for use in comparisons with the analytic solution.'
  []
  [ver-1d_comparison]
    type = RunCommand
    command = 'python3 comparison_ver-1dd.py'
    requirement = 'The system shall be able to generate comparison plots between the analytical solution and simulated solutions of verification cases 1dd, modeling a breakthrough problem without traps.'
    required_python_packages = 'matplotlib numpy pandas os'
  []
[]

Test Description
Analytical solution
Results and comparison against analytical solution
Input files
References