ver-1fa

Heat Conduction with Heat Generation

General Case Description

This heat transfer verification problem is taken from Longhurst et al. (1992) and Ambrosek and Longhurst (2008), and it has been updated and extended in Simon et al. (2025).

This case models a heat conduction problem through a slab with a heat source. The heat conduction in the one-dimensional model is described as:

$(1)var element = document.getElementById("moose-equation-c8c2a32b-8588-425b-a7aa-e3138165a0b7");katex.render(" \\rho C_P \\frac{d T}{d t} = \\nabla k \\nabla T + Q,", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-58af18ac-3b12-426a-8b16-f7f5e9b3cd83");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature, $var element = document.getElementById("moose-equation-b88bce5e-a111-4e26-9aa3-561e4855898c");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$ is the density, $var element = document.getElementById("moose-equation-e62f8c47-94de-4e67-8b46-c6c343fc3c3b");katex.render("C_P", element, {displayMode:false,throwOnError:false});$ is the specific heat, $var element = document.getElementById("moose-equation-22f85431-043b-4505-baf8-1e4cfd0d9a0c");katex.render("k", element, {displayMode:false,throwOnError:false});$ is the thermal conductivity, and $var element = document.getElementById("moose-equation-6c675f2b-1ddf-4c85-b97f-862ccfc52ed7");katex.render("Q", element, {displayMode:false,throwOnError:false});$ is the internal volumetric heat generation rate.

One end of the slab is kept at a constant temperature of 300 K while the other end acts as an adiabatic surface.

This case uses internal volumetric heat generation rate of $var element = document.getElementById("moose-equation-cf514cf8-dde3-4529-bf4e-818f38f7b72c");katex.render("Q = 1 \\times 10^{4}", element, {displayMode:false,throwOnError:false});$ W/m $var element = document.getElementById("moose-equation-75e3e8c3-6ce4-424b-a15f-71061617c711");katex.render("^3", element, {displayMode:false,throwOnError:false});$ in the slab. The thickness of slab, $var element = document.getElementById("moose-equation-9d9e58ac-6bc2-488f-b9de-694fc5e8d85b");katex.render("L", element, {displayMode:false,throwOnError:false});$ , is 1.6 m, and the thermal conductivity is $var element = document.getElementById("moose-equation-a6ff7124-bcb0-489b-a17f-edc5c595aeeb");katex.render("k = 10", element, {displayMode:false,throwOnError:false});$ W/m/K. The surface temperature, $var element = document.getElementById("moose-equation-ffc8fa2b-684a-4819-be10-3ef500905e3a");katex.render("T_s", element, {displayMode:false,throwOnError:false});$ , on the end with a constant temperature, is 300 K. The material density and specific heat are assumed to be $var element = document.getElementById("moose-equation-851b007a-91ee-4a35-9bb2-d3b6e8200295");katex.render("\\rho = 1", element, {displayMode:false,throwOnError:false});$ kg/m $var element = document.getElementById("moose-equation-b3a4dfb0-1f77-4dc9-8adb-1b229d66cd54");katex.render("^3", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-00162176-224e-4656-8a18-d7d1908e145c");katex.render("C_P = 1", element, {displayMode:false,throwOnError:false});$ J/kg/K, respectively.

Analytical solution

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

$(2)var element = document.getElementById("moose-equation-92f72b43-a27e-4b7a-ab81-23a352aa9ea7");katex.render(" T = T_s \\;+\\; \\frac{QL^2}{2k} \\left(1-\\frac{x^2}{L^2}\\right),", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-ceeabc66-857e-40c2-acab-f76d5f3f6a9b");katex.render("L", element, {displayMode:false,throwOnError:false});$ is the thickness of the slab, and $var element = document.getElementById("moose-equation-ecffd547-580a-4cb8-a485-0c37967423dd");katex.render("T_s", element, {displayMode:false,throwOnError:false});$ is the imposed surface temperature.

Results

A comparison of the temperature calculated through TMAP8 and calculated analytically is shown in Figure 1. The TMAP8 calculations are found to be in good agreement with the analytical solution, with a root mean square percentage error (RMSPE) of RMSPE = 0.05 %.

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

Input files

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

# Verification Problem #1fa from TMAP4/TMAP7 V&V document
# Heat conduction with heat generation

# Data used in TMAP4/TMAP7 case
length = '${units 1.6 m}'
initial_temperature = '${units 300 K}'
density = '${units 1 kg/m^3}'
specific_heat = '${units 1 J/kg/K}'
thermal_conductivity = '${units 10 W/m/K}'
volumetric_heat = '${units 1e4 W/m^3}'

[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
  []
  [heatsource]
    type = HeatSource
    function = volumetric_heat
    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 = '${initial_temperature}'
  []
  # The left boundary of the slab is in adiabatic situation
  [left_flux]
    type = NeumannBC
    boundary = left
    variable = temperature
    value = 0
  []
[]

[Materials]
  # The density of the sample slab
  [density]
    type = GenericConstantMaterial
    prop_names = 'density  thermal_conductivity specific_heat'
    prop_values = '${density} ${thermal_conductivity} ${specific_heat}'
  []
[]

[Functions]
  # The heat source in the sample slab
  [volumetric_heat]
    type = ParsedFunction
    expression = '${volumetric_heat}'
  []
[]

[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 = 1
  end_time = 10
  automatic_scaling = true
[]

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

[Outputs]
  exodus = true
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

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

[Tests]
  design = 'HeatConduction.md HeatConductionTimeDerivative.md HeatSource.md'
  issues = '#12'
  verification = 'ver-1fa.md'
  [ver-1fa]
    type = Exodiff
    input = ver-1fa.i
    exodiff = ver-1fa_out.e
    requirement = 'The system shall be able to model heat conduction in a slab that has heat generation'
  []
  [ver-1fa_lineplot]
    type = CSVDiff
    input = ver-1fa.i
    should_execute = False  # this test relies on the output files from ver-1fa, so it shouldn't be run twice
    csvdiff = ver-1fa_csv_line_0011.csv
    requirement = 'The system shall be able to model heat conduction in a slab that has heat generation to generate CSV data for use in comparisons with the analytic solution for the profile concentration.'
    prereq = ver-1fa
  []
  [ver-1fa_comparison]
    type = RunCommand
    command = 'python3 comparison_ver-1fa.py'
    requirement = 'The system shall be able to generate comparison plots between the analytical solution and simulated solution of verification case 1fa, to model heat conduction in a slab that has heat generation.'
    required_python_packages = 'matplotlib numpy pandas scipy os'
  []
[]

General Case Description
Analytical solution
Results
Input files
References