ver-1ja

Two problems (ver-1ja and ver-1jb) demonstrate tritium decay, though any other isotope could have been chosen. The first (ver-1ja) models simple decay of mobile species in a slab. The second (ver-1jb) models decay of trapped atoms in a similar slab but with a distributed trap concentration. This page presents ver-1ja.

Radioactive Decay of Mobile Tritium in a Slab

General Case Description

This verification case tests the first order radioactive decay capabilities of TMAP8 and is based on the case published in the TMAP7 V&V suite (Ambrosek and Longhurst, 2008). The model assumes pre-charging of a slab with tritium. The tritium (T) is uniformly distributed over the thickness of the slab with an initial concentration of $var element = document.getElementById("moose-equation-b7ade432-d49e-4c08-bea4-da36fd99f574");katex.render("C_T^0 = 1.5 \\times 10^{5}", element, {displayMode:false,throwOnError:false});$ atoms/m $var element = document.getElementById("moose-equation-25ce98b4-eeae-4456-b18d-aa3885faeced");katex.render("^3", element, {displayMode:false,throwOnError:false});$ . The tritium decays to $var element = document.getElementById("moose-equation-8cfc4412-6a5d-4626-aed2-2afcf1fee52a");katex.render("^3\\text{He}", element, {displayMode:false,throwOnError:false});$ with a half-life of $var element = document.getElementById("moose-equation-4dcf04de-9166-4a12-825d-e3286c037ba2");katex.render("t_{1/2} = 12.3232", element, {displayMode:false,throwOnError:false});$ years. The concentrations of the two species are calculated.

The evolution of the tritium and helium concentration, $var element = document.getElementById("moose-equation-c5d71ba0-cda7-413e-8d71-4bf609b3fa21");katex.render("C_T", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-5c566c17-5cf4-4e25-96fe-75c786eac1f8");katex.render("C_{He}", element, {displayMode:false,throwOnError:false});$ , respectively, are governed by

$var element = document.getElementById("moose-equation-808178d8-0407-471f-8a51-a3882336c0f9");katex.render(" \\frac{d C_T}{dt} = -k C_T,", element, {displayMode:true,throwOnError:false});$ and $var element = document.getElementById("moose-equation-8a257c6b-5819-4533-837e-429e13756ea0");katex.render(" \\frac{d C_{He}}{dt} = k C_T,", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-7736c659-fa75-4ff6-a6d0-7d8767895f7f");katex.render("t", element, {displayMode:false,throwOnError:false});$ is the time in seconds, concentrations are in atoms/m $var element = document.getElementById("moose-equation-46cf461c-5172-4bad-808f-563e0c47af6a");katex.render("^3", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-a692be76-b952-4fdc-b4d7-6f926f6261da");katex.render("k= 0.693/t_{1/2}", element, {displayMode:false,throwOnError:false});$ is the decay rate constant in 1/s.

warning:TMAP8 uses different model parameters than TMAP7

The initial tritium concentration in TMAP7 was defined as $var element = document.getElementById("moose-equation-eda832fb-b062-4304-83f1-7bc703bd1132");katex.render("C_T^0 = 1.5", element, {displayMode:false,throwOnError:false});$ atoms/m $var element = document.getElementById("moose-equation-7ab0482e-50ff-41b3-8c9a-d05cbda2596f");katex.render("^3", element, {displayMode:false,throwOnError:false});$ . To use a more realistic values, TMAP8 uses $var element = document.getElementById("moose-equation-652a1004-760a-4b11-b9fe-bcc22939506b");katex.render("C_T^0 = 1.5 \\times 10^{5}", element, {displayMode:false,throwOnError:false});$ atoms/m $var element = document.getElementById("moose-equation-c41e3489-746b-4285-8c3d-4cc7c15533ef");katex.render("^3", element, {displayMode:false,throwOnError:false});$ . Moreover, $var element = document.getElementById("moose-equation-86a0f8b3-0a4e-4eb8-95e0-c5c97f7221d0");katex.render("k", element, {displayMode:false,throwOnError:false});$ is defined as $var element = document.getElementById("moose-equation-a8e5fb21-6b17-41a8-931a-9222ca4af9f3");katex.render("k=0.693/t_{1/2} \\approx 1.78199 \\times 10^{-9}", element, {displayMode:false,throwOnError:false});$ 1/s instead of $var element = document.getElementById("moose-equation-c94eddfd-6d61-4e2d-8879-51a780566481");katex.render("1.78241 \\times 10^{-9}", element, {displayMode:false,throwOnError:false});$ 1/s to be fully consistent with the half-life value (assuming 365.25 days in a year).

Analytical Solution

The concentration of T at any given time is given by

$var element = document.getElementById("moose-equation-90dc293d-9739-47b8-936c-0cc63127dfd6");katex.render(" C_T = C_T^0 \\exp(-kt),", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-524c8fef-aedb-4af2-a21a-23a00411d7a1");katex.render("t", element, {displayMode:false,throwOnError:false});$ is the time in seconds and $var element = document.getElementById("moose-equation-bb801c76-4d82-489f-8ccc-7ac682cea86f");katex.render("C_T^0 = 1.5 \\times 10^{5}", element, {displayMode:false,throwOnError:false});$ atoms/m $var element = document.getElementById("moose-equation-24f07bff-7cd5-4257-9cdf-d6397a44eff0");katex.render("^3", element, {displayMode:false,throwOnError:false});$ is the initial concentration of tritium. Applying a mass balance over the system, the time evolution of helium concentration is given by $var element = document.getElementById("moose-equation-b7df1e79-b716-427d-9d99-8f74117753cc");katex.render(" C_{He} = C_T^0 \\left[1- \\exp(-kt) \\right].", element, {displayMode:true,throwOnError:false});$

Results

Figure 1 shows the TMAP8 predictions and how they compare to the analytical solution for the decay of tritium and associated growth of $var element = document.getElementById("moose-equation-dc72ebaf-f288-4e79-8382-903c7ed00011");katex.render("^3\\text{He}", element, {displayMode:false,throwOnError:false});$ in a diffusion segment. The TMAP8 predictions match the analytical solution, with root mean square percentage errors (RMSPE) of 0.79% and 0.17% for the $var element = document.getElementById("moose-equation-33827d66-231e-4a36-9293-6b8727514d40");katex.render("C_T", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-2b06c513-1a95-4924-8e4f-53557b2a4d2b");katex.render("C_{He}", element, {displayMode:false,throwOnError:false});$ concentration curves respectively.

Figure 1: Comparison of TMAP8 predictions against the analytical solution for the decay of tritium and associated growth of $var element = document.getElementById("moose-equation-559a5801-6b90-42be-9868-3c94faff62ea");katex.render("^3\\text{He}", element, {displayMode:false,throwOnError:false});$ in a diffusion segment. The RMSPE is very low for both species.

Input file

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

# Verification Problem #1ja from TMAP7 V&V document
# Radioactive Decay of Mobile Tritium in a Slab

# Case and model parameters (adapted from TMAP7)
tritium_concentration_initial = ${units 1.5e5 atoms/m3}
half_life = ${units 12.3232 year -> s}
decay_rate_constant = ${fparse 0.693/half_life} # 1/s

# Simulation parameters
end_time = ${units 100 year -> s}
dt_start = ${fparse end_time/250} # s

[Mesh]
  type = GeneratedMesh
  dim = 1
[]

[Variables]
  # tritium concentration in atoms/m^3
  [tritium_concentration]
    initial_condition = ${tritium_concentration_initial}
  []
  # helium concentration in atoms/m^3
  [helium_concentration]
  []
[]

[Kernels]
  # kernels for the tritium concentration equation
  [time_tritium]
    type = TimeDerivative
    variable = tritium_concentration
  []
  [decay_tritium]
    type = MatReaction
    variable = tritium_concentration
    v = tritium_concentration
    reaction_rate = '${fparse -decay_rate_constant}'
  []
  # kernels for the helium concentration equation
  [time_helium]
    type = TimeDerivative
    variable = helium_concentration
  []
  [decay_helium]
    type = MatReaction
    variable = helium_concentration
    v = tritium_concentration
    reaction_rate = '${decay_rate_constant}'
  []
[]

[Postprocessors]
  # Average concentration of tritium in the sample in atoms/m^3
  [tritium_concentration]
    type = ElementAverageValue
    variable = tritium_concentration
    execute_on = 'INITIAL TIMESTEP_END'
  []
  # Average concentration of helium in the sample in atoms/m^3
  [helium_concentration]
    type = ElementAverageValue
    variable = helium_concentration
    execute_on = 'INITIAL TIMESTEP_END'
  []
[]

[Executioner]
  type = Transient
  dt = ${dt_start}
  end_time = ${end_time}
  solve_type = PJFNK
  scheme = 'bdf2'
  dtmin = 1
  petsc_options = '-snes_converged_reason'
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]

[Outputs]
  perf_graph = true
  csv = true
[]

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

[Tests]
  design = 'ver-1ja.md MatReaction.md'
  verification = 'ver-1ja.md'
  issues = '#145 #12'
  [ver-1ja_csvdiff]
    type = CSVDiff
    input = ver-1ja.i
    csvdiff = ver-1ja_out.csv
    requirement = 'The system shall be able to model decay of tritium and associated growth of helium in a diffusion segment and generate CSV data for use in comparisons with the analytic solution.'
  []
  [ver-1ja_comparison]
    type = RunCommand
    command = 'python3 comparison_ver-1ja.py'
    requirement = 'The system shall be able to generate comparison plots between the analytical solution and simulated solution of verification case 1ja, which models decay of tritium and associated growth of helium in a diffusion segment.'
    required_python_packages = 'matplotlib numpy pandas os'
  []
[]

General Case Description
Analytical Solution
References