ver-1ka

Simple Volumetric Source

General Case Description

This problem involves two enclosures connected by a diffusive membrane that follows Sieverts law for diffusion. Both enclosures contain hydrogen (H $var element = document.getElementById("moose-equation-97d75bce-1b40-4add-8948-c75ff2066473");katex.render("_2", element, {displayMode:false,throwOnError:false});$ ), tritium (T $var element = document.getElementById("moose-equation-fb7e6d13-8f69-4aa3-84e4-32c72dfffecd");katex.render("_2", element, {displayMode:false,throwOnError:false});$ ), and tritium gas (HT). In the first enclosure, there is an initial inventory of only hydrogen (H $var element = document.getElementById("moose-equation-9619d632-6898-4ed4-9d0d-37f6b87640f6");katex.render("_2", element, {displayMode:false,throwOnError:false});$ ) along with a constant volumetric source rate of tritium (T $var element = document.getElementById("moose-equation-6511ea02-0ea0-4f6a-b6f1-d1d1e5db376a");katex.render("_2", element, {displayMode:false,throwOnError:false});$ ). The second enclosure starts out empty.

Case Set Up

This verification problem is taken from Ambrosek and Longhurst (2008). The rise in pressure of T $var element = document.getElementById("moose-equation-4b70e979-2ab5-4090-bf67-a660f0c6e38d");katex.render("_2", element, {displayMode:false,throwOnError:false});$ molecules in the first enclosure can be monitored by not enabling T $var element = document.getElementById("moose-equation-b40bf230-bc0d-43e1-9cac-2b6dd6a691b7");katex.render("_2", element, {displayMode:false,throwOnError:false});$ molecules to traverse the membrane between the two enclosures (no tritium flux). Consequently, the rate of pressure increase in the first enclosure can be expressed as:

$(1)var element = document.getElementById("moose-equation-a1107483-e2d8-4ce3-a072-28635f2557e7");katex.render(" \\frac{dP_{T_2}}{dt} = \\frac{S}{V} kT,", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-e1f1ea09-c6e4-4993-bb77-9d0348d9460e");katex.render("S", element, {displayMode:false,throwOnError:false});$ represents the volumetric T $var element = document.getElementById("moose-equation-ace7fb60-8f55-4ba8-be88-b954949e76a3");katex.render("_2", element, {displayMode:false,throwOnError:false});$ source rate, $var element = document.getElementById("moose-equation-97f9bcb1-5e52-48ac-adc9-367f02515ff2");katex.render("V", element, {displayMode:false,throwOnError:false});$ is the volume of the enclosure, $var element = document.getElementById("moose-equation-3efd16b9-a657-4413-99eb-b242f15e4317");katex.render("k", element, {displayMode:false,throwOnError:false});$ is the Boltzmann constant, and $var element = document.getElementById("moose-equation-6c003866-fc5c-439d-927e-25d73db5d29c");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature of the enclosure. In this case, $var element = document.getElementById("moose-equation-f559d0e0-f852-4c6f-ac08-2748cfedf4e5");katex.render("S", element, {displayMode:false,throwOnError:false});$ is set to 10 $var element = document.getElementById("moose-equation-a79fb541-d155-4430-befe-7f0c2a49d110");katex.render("^{20}", element, {displayMode:false,throwOnError:false});$ molecules/m $var element = document.getElementById("moose-equation-97d9a203-bade-4012-9781-8d2cbfebb817");katex.render("^{-3}", element, {displayMode:false,throwOnError:false});$ /s, $var element = document.getElementById("moose-equation-fdfc8d86-352d-4b93-90d9-0dd6b46a9c85");katex.render("V = 1", element, {displayMode:false,throwOnError:false});$ m $var element = document.getElementById("moose-equation-60c844ea-3069-41a6-9df2-202b372b3509");katex.render("^3", element, {displayMode:false,throwOnError:false});$ , and the temperature of the enclosure is constant at $var element = document.getElementById("moose-equation-bead04e5-c6a0-4e70-a7d7-03a23f451fad");katex.render("T = 500", element, {displayMode:false,throwOnError:false});$ K.

Analytical Solution

The analytical solution for Eq. (1) is simply

$var element = document.getElementById("moose-equation-925fa3bb-d1d9-490d-aa1d-312fb69c3580");katex.render("P_{T_2}(t) = \\frac{S}{V} kT t.", element, {displayMode:true,throwOnError:false});$

Results

Comparison of the TMAP8 results and the analytical solution is shown in Figure 1 as a function of time. The TMAP8 code predictions match very well with the analytical solution with a root mean squared percentage error of RMSPE $var element = document.getElementById("moose-equation-da5e0547-b632-41b0-8aee-eed61533f3c7");katex.render("= 0.00", element, {displayMode:false,throwOnError:false});$ %.

Figure 1: Comparison of T $var element = document.getElementById("moose-equation-3b7799e9-4d8a-4be8-ac69-924d85e9fca0");katex.render("_2", element, {displayMode:false,throwOnError:false});$ partial pressure in an enclosure with no loss pathways as function of time calculated through TMAP8 and analytically. TMAP8 matches the analytical solution.

Input files

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

initial_pressure = ${units 0 Pa} # initial internal pressure
kb = ${units 1.380649e-23 J/K} # Boltzmann constant J/K - from PhysicalConstants.h
T = ${units 500 K} # Temperature
S = ${units 1e20 1/m^3/s} # Source term
V = ${units 1 m^3} # Volume
end_time = ${units 10000 s}
time_step = ${units 500 s}

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
  xmax = 1
  ymax = 1
[]

[Variables]
  [pressure]
    family = SCALAR
    order = FIRST
    initial_condition = '${fparse initial_pressure}'
  []
[]

[ScalarKernels]
  [time]
    type = ODETimeDerivative
    variable = pressure
  []
  [source]
    type = ParsedODEKernel
    variable = pressure
    expression = '${fparse - S/V * kb * T}'
  []
[]

[Executioner]
  type = Transient
  dt = ${time_step}
  end_time = ${end_time}
  scheme = 'bdf2'
[]

[Outputs]
  file_base = 'ver-1ka_out'
  csv = true
[]

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

[Tests]
  design = 'ODETimeDerivative.md ParsedODEKernel.md'
  issues = '#12'
  verification = 'ver-1ka.md'
  [ver-1ka_csv]
    type = CSVDiff
    input = ver-1ka.i
    csvdiff = ver-1ka_out.csv
    requirement = 'The system shall be able to model a tritium volumetric source in one enclosure'
  []
  [ver-1ka_comparison]
    type = RunCommand
    command = 'python3 comparison_ver-1ka.py'
    requirement = 'The system shall be able to generate comparison plots between the analytical solution and simulated solution of verification case 1ka, modeling a tritium volumetric source in one enclosure.'
    required_python_packages = 'matplotlib numpy pandas os git'
  []
[]

General Case Description
Case Set Up
Analytical Solution
Results
Input files
References