ver-1fb

Thermal Transient in a Slab

General Case Description

This heat transfer verification problem is taken from Longhurst et al. (1992) and Ambrosek and Longhurst (2008) and builds on the capabilities verified in ver-1fa. It has been updated and extended in Simon et al. (2025). The configuration is the same as in ver-1fa, except that, in the current case, there are no heat source in the slab and both ends with fixed temperature. This case is simulated in (test/tests/ver-1fb/ver-1fb.i).

The heat conduction in the one-dimensional model is described as:

$(1)var element = document.getElementById("moose-equation-8c27d8fa-0be5-47f9-af58-1fb54aa9af6c");katex.render(" \\rho C_P \\frac{d T}{d t} = \\nabla k \\nabla T,", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-fdd872f4-a49d-432f-a0f0-90aeb451bbf7");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature, $var element = document.getElementById("moose-equation-4965333d-f778-48c1-936d-baf88ea8ca7a");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$ is the density, $var element = document.getElementById("moose-equation-72f019f5-2f43-4995-9922-6443042f5385");katex.render("C_P", element, {displayMode:false,throwOnError:false});$ is the specific heat, and $var element = document.getElementById("moose-equation-40679715-7eb5-4dd4-b973-449e4540034e");katex.render("k", element, {displayMode:false,throwOnError:false});$ is the thermal conductivity.

The ends of a slab are kept fixed at different temperatures. The temperature distribution in the slab evolves from an initial state to steady-state.

In this case, the thickness of slab, $var element = document.getElementById("moose-equation-ee8cb52e-95ac-40b6-ba3f-da28a6496d64");katex.render("L", element, {displayMode:false,throwOnError:false});$ , is 4.0 m, the thermal conductivity is $var element = document.getElementById("moose-equation-2495d3bd-0a56-42c1-97e6-01a6529e1743");katex.render("k =10", element, {displayMode:false,throwOnError:false});$ W/m/K, the material density is assumed to be $var element = document.getElementById("moose-equation-5dc7d821-f054-460f-87ab-28a70a649e15");katex.render("\\rho = 1", element, {displayMode:false,throwOnError:false});$ kg/m $var element = document.getElementById("moose-equation-9cec1145-1fdb-4838-a62a-3b6c7b92d34d");katex.render("^3", element, {displayMode:false,throwOnError:false});$ , and the specific heat is assumed to be $var element = document.getElementById("moose-equation-9986109e-3b96-4a9a-a5a5-4039427213d3");katex.render("C_P = 10", element, {displayMode:false,throwOnError:false});$ J/kg/K. The fixed surface temperature, $var element = document.getElementById("moose-equation-5c48db1a-0c30-41c0-a4ee-27bf7e183867");katex.render("T_0", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-71e52b23-36b5-4577-878e-c2ad0a27ebe3");katex.render("T_1", element, {displayMode:false,throwOnError:false});$ , on both ends are defined as 400 K and 300 K, respectively.

Analytical solution

Incropera and DeWitt (2002) provides the analytical solution for the temperature of this case as:

$(2)var element = document.getElementById("moose-equation-089e815d-14d6-4b26-8038-f58c91ec38bc");katex.render(" T(x,t) = T_0 \\;+\\; (T_1-T_0)\\left\\{1-\\frac{x}{L}-\\frac{2}{L}\\sum_{m=1}^{\\infty} \\left(\\frac{1}{\\lambda_m} \\sin(\\lambda_m x) \\exp(-\\alpha \\lambda_m^2 t) \\right)\\right\\},", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-41b4c1c6-1ebc-463d-a7a4-97d2426bddd4");katex.render("x", element, {displayMode:false,throwOnError:false});$ is the distance across the slab, $var element = document.getElementById("moose-equation-00184711-b2f9-4030-8237-69f0a9577d31");katex.render("t", element, {displayMode:false,throwOnError:false});$ is the time, $var element = document.getElementById("moose-equation-4f78628d-9a9e-47c3-abee-3981cdfa014a");katex.render("\\lambda_m", element, {displayMode:false,throwOnError:false});$ is a coefficient of $var element = document.getElementById("moose-equation-496266b0-d72f-4952-aef3-ed55bcc5491d");katex.render("\\frac{m\\pi}{L}", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-507bf948-bd5e-4325-bde6-e9d98fdfdf85");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ is the thermal diffusivity, which is defined as:

$(3)var element = document.getElementById("moose-equation-38e22df0-49dc-42cb-a0f6-814acf45f842");katex.render(" \\alpha = \\frac{k}{\\rho C_p}.", element, {displayMode:true,throwOnError:false});$

warning:Typo in analytical solution from Longhurst et al. (1992)

Both TMAP4 (Longhurst et al., 1992) and TMAP7 (Ambrosek and Longhurst, 2008) provide analytical solutions, but they use different equations. At the initial time and steady state, the solution from TMAP7 matches the real thermal distribution, whereas the solution from TMAP4 does not. Thus, TMAP8 selects the solution from TMAP7 as the analytical solution.

Results

A comparison of the temperature distribution in the slab, computed through TMAP8 and calculated analytically, is shown in Figure 1. The TMAP8 code predictions match very well with the analytical solution with the root mean square percentage errors of RMSPE = 0.09 % at $var element = document.getElementById("moose-equation-e29d7a70-58e3-40d4-896b-1799db2e368b");katex.render("t = 0.1", element, {displayMode:false,throwOnError:false});$ s, RMSPE = 0.03 % at $var element = document.getElementById("moose-equation-cb1921a0-3aaa-410f-99f6-3fb66383c794");katex.render("t = 0.5", element, {displayMode:false,throwOnError:false});$ s, RMSPE = 0.02 % at $var element = document.getElementById("moose-equation-7e4424c1-a6c1-4e6e-9bdd-4dca11f92a2b");katex.render("t = 1", element, {displayMode:false,throwOnError:false});$ s, and RMSPE = 0.00 % at $var element = document.getElementById("moose-equation-c065856f-0fde-4d2f-af32-cb6d40c34baf");katex.render("t = 5", element, {displayMode:false,throwOnError:false});$ s, respectively.

Figure 1: Comparison of temperature distribution in the slab calculated through TMAP8 and analytically

Input files

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

F. P. Incropera and D. P. DeWitt. Fundamentals of Heat and Mass Transfer. John Wiley & Sons, 5th edition, 2002.

@book{Incropera2002,
    author = "Incropera, F. P. and DeWitt, D. P.",
    city = "New York",
    edition = "5th",
    publisher = "John Wiley \& Sons",
    title = "Fundamentals of Heat and Mass Transfer",
    year = "2002"
}

GR Longhurst, SL Harms, ES Marwil, and BG Miller. Verification and Validation of TMAP4. Technical Report EGG-FSP-10347, Idaho National Engineering Laboratory, Idaho Falls, ID (United States), 1992.

@techreport{longhurst1992verification,
    author = "Longhurst, GR and Harms, SL and Marwil, ES and Miller, BG",
    title = "{Verification and Validation of TMAP4}",
    year = "1992",
    institution = "Idaho National Engineering Laboratory, Idaho Falls, ID (United States)",
    number = "EGG-FSP-10347"
}

Pierre-Clément A. Simon, Casey T. Icenhour, Gyanender Singh, Alexander D Lindsay, Chaitanya Vivek Bhave, Lin Yang, Adriaan Anthony Riet, Yifeng Che, Paul Humrickhouse, Masashi Shimada, and Pattrick Calderoni. MOOSE-based tritium migration analysis program, version 8 (TMAP8) for advanced open-source tritium transport and fuel cycle modeling. Fusion Engineering and Design, 214:114874, May 2025. doi:10.1016/j.fusengdes.2025.114874.

@article{Simon2025,
    author = "Simon, Pierre-Clément A. and Icenhour, Casey T. and Singh, Gyanender and Lindsay, Alexander D and Bhave, Chaitanya Vivek and Yang, Lin and Riet, Adriaan Anthony and Che, Yifeng and Humrickhouse, Paul and Shimada, Masashi and Calderoni, Pattrick",
    title = "{MOOSE}-based Tritium Migration Analysis Program, Version 8 ({TMAP8}) for Advanced Open-Source Tritium Transport and Fuel Cycle Modeling",
    journal = "Fusion Engineering and Design",
    publisher = "Elsevier",
    volume = "214",
    pages = "114874",
    year = "2025",
    month = "May",
    issn = "0920-3796",
    doi = "10.1016/j.fusengdes.2025.114874"
}

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

# Verification Problem #1fb from TMAP4/TMAP7 V&V document
# Thermal transient in a slab whitout heat source

# Data used in TMAP4/TMAP7 case
length = '${units 4.0 m}'
initial_temperature = '${units 300 K}'
T_0 = '${units 300 K}'
T_1 = '${units 400 K}'
density = '${units 1 kg/m^3}'
specific_heat = '${units 10 J/kg/K}'
thermal_conductivity = '${units 10 W/m/K}'

[Mesh]
  type = GeneratedMesh
  dim = 1
  xmax = '${length}'
  nx = 20
[]

[Variables]
  # temperature parameter in the slab in K
  [temperature]
    initial_condition = '${initial_temperature}'
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = temperature
  []
  [HeatTdot]
    type = HeatConductionTimeDerivative
    variable = temperature
  []
[]

[BCs]
  # The temperature on the right boundary of the slab is kept constant
  [right_temp]
    type = DirichletBC
    boundary = right
    variable = temperature
    value = '${T_0}'
  []
  # The temperature on the right boundary of the slab is kept constant
  [left_temp]
    type = DirichletBC
    boundary = left
    variable = temperature
    value = '${T_1}'
  []
[]

[Materials]
  # The diffusivity of the sample
  [diffusivity]
    type = GenericConstantMaterial
    prop_names = 'density  thermal_conductivity specific_heat'
    prop_values = '${density} ${thermal_conductivity} ${specific_heat}' # arbitrary values for diffusivity (=k/rho-Cp) to be 1.0
  []
[]

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

[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
  l_tol = 1e-4
  dt = 0.01
  end_time = 10
  automatic_scaling = true
[]

[VectorPostprocessors]
  # The temperature distribution on the sample at coresponding time
  [line]
    type = LineValueSampler
    start_point = '0 0 0'
    end_point = '${length} 0 0'
    num_points = 40
    sort_by = 'x'
    variable = temperature
    outputs = vector_postproc
  []
[]

[Outputs]
  exodus = true
  [vector_postproc]
    type = CSV
    sync_times = '0.1 0.5 1 5'
    sync_only = true
    file_base = 'ver-1fb_u_vs_x'
  []
[]

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

# Verification Problem #1fb from TMAP4/TMAP7 V&V document
# Thermal transient in a slab whitout heat source

# Data used in TMAP4/TMAP7 case
length = '${units 4.0 m}'
initial_temperature = '${units 300 K}'
T_0 = '${units 300 K}'
T_1 = '${units 400 K}'
density = '${units 1 kg/m^3}'
specific_heat = '${units 10 J/kg/K}'
thermal_conductivity = '${units 10 W/m/K}'

[Mesh]
  type = GeneratedMesh
  dim = 1
  xmax = '${length}'
  nx = 20
[]

[Variables]
  # temperature parameter in the slab in K
  [temperature]
    initial_condition = '${initial_temperature}'
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = temperature
  []
  [HeatTdot]
    type = HeatConductionTimeDerivative
    variable = temperature
  []
[]

[BCs]
  # The temperature on the right boundary of the slab is kept constant
  [right_temp]
    type = DirichletBC
    boundary = right
    variable = temperature
    value = '${T_0}'
  []
  # The temperature on the right boundary of the slab is kept constant
  [left_temp]
    type = DirichletBC
    boundary = left
    variable = temperature
    value = '${T_1}'
  []
[]

[Materials]
  # The diffusivity of the sample
  [diffusivity]
    type = GenericConstantMaterial
    prop_names = 'density  thermal_conductivity specific_heat'
    prop_values = '${density} ${thermal_conductivity} ${specific_heat}' # arbitrary values for diffusivity (=k/rho-Cp) to be 1.0
  []
[]

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

[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
  l_tol = 1e-4
  dt = 0.01
  end_time = 10
  automatic_scaling = true
[]

[VectorPostprocessors]
  # The temperature distribution on the sample at coresponding time
  [line]
    type = LineValueSampler
    start_point = '0 0 0'
    end_point = '${length} 0 0'
    num_points = 40
    sort_by = 'x'
    variable = temperature
    outputs = vector_postproc
  []
[]

[Outputs]
  exodus = true
  [vector_postproc]
    type = CSV
    sync_times = '0.1 0.5 1 5'
    sync_only = true
    file_base = 'ver-1fb_u_vs_x'
  []
[]

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

[Tests]
  design = 'HeatConduction.md HeatConductionTimeDerivative.md'
  issues = '#12'
  verification = 'ver-1fb.md'
  [ver-1fb]
    type = Exodiff
    input = ver-1fb.i
    exodiff = ver-1fb_out.e
    requirement = 'The system shall be able to model thermal transient in a slab that has temperatures fixed at both the ends'
  []
  [ver-1fb_csvdiff_0pt1sec]
    type = CSVDiff
    input = ver-1fb.i
    csvdiff = ver-1fb_u_vs_x_line_0010.csv
    should_execute = False  # this test relies on the output files from ver-1fb, so it shouldn't be run twice
    requirement = 'The system shall be able to model thermal transient in a slab that has temperatures fixed at both the ends to generate CSV data at time of 0.1 s for use in comparison with analytical solution.'
    prereq = ver-1fb
  []
  [ver-1fb_csvdiff_0pt5sec]
    type = CSVDiff
    input = ver-1fb.i
    csvdiff = ver-1fb_u_vs_x_line_0050.csv
    should_execute = False  # this test relies on the output files from ver-1fb, so it shouldn't be run twice
    requirement = 'The system shall be able to model thermal transient in a slab that has temperatures fixed at both the ends to generate CSV data at time of 0.5 s for use in comparison with analytical solution.'
    prereq = ver-1fb
  []
  [ver-1fb_csvdiff_1pt0sec]
    type = CSVDiff
    input = ver-1fb.i
    csvdiff = ver-1fb_u_vs_x_line_0100.csv
    should_execute = False  # this test relies on the output files from ver-1fb, so it shouldn't be run twice
    requirement = 'The system shall be able to model thermal transient in a slab that has temperatures fixed at both the ends to generate CSV data at time of 1.0 s for use in comparison with analytical solution.'
    prereq = ver-1fb
  []
  [ver-1fb_csvdiff_5pt0sec]
    type = CSVDiff
    input = ver-1fb.i
    csvdiff = ver-1fb_u_vs_x_line_0500.csv
    should_execute = False  # this test relies on the output files from ver-1fb, so it shouldn't be run twice
    requirement = 'The system shall be able to model thermal transient in a slab that has temperatures fixed at both the ends to generate CSV data at time of 5.0 s for use in comparison with analytical solution.'
    prereq = ver-1fb
  []
  [ver-1fb_comparison]
    type = RunCommand
    command = 'python3 comparison_ver-1fb.py'
    requirement = 'The system shall be able to generate comparison plots between the analytical solution and simulated solution of verification case 1fb, modeling thermal transient in a slab with fixed temperatures at both the ends.'
    required_python_packages = 'matplotlib numpy pandas os'
  []
[]

General Case Description
Analytical solution
Results
Input files
References