Tungsten Thermal Properties Material

Computes tungsten thermal properties as a function of temperature

This class provides tungsten's thermal conductivity, specific heat, and density as material properties as a function of temperature based on the models listed in Milner et al. (2024).

Thermal conductivity

For temperature range: 1 ≤ T < 55 K, the thermal conductivity is defined as

$(1)var element = document.getElementById("moose-equation-34d84c31-2aaa-44c8-af5e-87bd6b02e511");katex.render(" k(T) = \\frac{A0 \\cdot \\left( \\frac{T}{1000} \\right)^N}{1 + A1 \\cdot \\left( \\frac{T}{1000} \\right) + A2 \\cdot \\left( \\frac{T}{1000} \\right)^2 + A3 \\cdot \\left( \\frac{T}{1000} \\right)^3},", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where:

$var element = document.getElementById("moose-equation-eda2f96f-79b7-49d2-b425-061b216511ca");katex.render("k(T)", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the thermal conductivity [W/m.K],
$var element = document.getElementById("moose-equation-6f63893c-e4ac-4242-a57e-229125ad71f5");katex.render("T", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the temperature [K],

and the model parameters values provided in Table 1.

Table 1: Parameters values for the thermal conductivity model from Eq. (1).

Constant	Value	Unit
N	8.740E-01	`[-]`
A0	7.348E+05	`[W/m.K]`
A1	2.544E+01	`[-]`
A2	-8.304E+03	`[-]`
A3	1.180E+06	`[-]`

For temperature range 55 ≤ $var element = document.getElementById("moose-equation-db7671c5-41ab-4d58-92cc-82a299bd287b");katex.render("T", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ≤ 3653 K, the thermal conductivity is defined as

$(2)var element = document.getElementById("moose-equation-f45a3e24-4797-41f7-98bb-2de6e846532a");katex.render(" k(T) = \\frac{B0 + B1 \\cdot \\left( \\frac{T}{1000} \\right) + B2 \\cdot \\left( \\frac{T}{1000} \\right)^2 + B3 \\cdot \\left( \\frac{T}{1000} \\right)^3}{C0 + C1 \\cdot \\left( \\frac{T}{1000} \\right) + \\left( \\frac{T}{1000} \\right)^2},", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where:

$var element = document.getElementById("moose-equation-d64f76aa-a91e-44a3-8490-5b36d829b97f");katex.render("k(T)", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the thermal conductivity [W/m.K],
$var element = document.getElementById("moose-equation-7c8575b1-d1db-4e5b-a300-2d034b7873e3");katex.render("T", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the temperature [K],

and the model parameters values provided in Table 2.

Table 2: Parameters values for the thermal conductivity model from Eq. (2).

Constant	Value	Unit
B0	-3.679E+00	`[W/m.K]`
B1	1.181E+02	`[W/m.K]`
B2	5.879E+01	`[W/m.K]`
B3	2.867E+00	`[W/m.K]`
C0	-2.052E-02	`[-]`
C1	4.741E-01	`[-]`

Specific Heat

For temperature range: 11 ≤ $var element = document.getElementById("moose-equation-a9e5db99-599d-45c2-8e73-b71e1f055cd5");katex.render("T", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ < 293 K, the specific heat is described as:

$(3)var element = document.getElementById("moose-equation-d75028a8-76a2-415e-9bbe-e5c88607940f");katex.render(" C_p(T) =\\frac{A0 \\cdot \\left( \\frac{T}{1000} \\right)^N}{1 + A1 \\cdot \\left( \\frac{T}{1000} \\right) + A2 \\cdot \\left( \\frac{T}{1000} \\right)^2 + A3 \\cdot \\left( \\frac{T}{1000} \\right)^3},", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where:

$var element = document.getElementById("moose-equation-2a40b576-b6da-4527-912b-1d14184e1b75");katex.render("C_p", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ (T)$ is the specific heat [J/g.K],
$var element = document.getElementById("moose-equation-f5754313-429d-408d-bb4d-e43417d55d0a");katex.render("T", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the temperature [K],

and the model parameters values provided in Table 3.

Table 3: Parameters values for the specific heat model from Eq. (3).

Constant	Value	Unit
N	3.030E+00	`[-]`
A0	3.103E+02	`[J/g.K]`
A1	-8.815E+00	`[-]`
A2	1.295E+02	`[-]`
A3	1.874E+03	`[-]`

For temperature range: 293 ≤ $var element = document.getElementById("moose-equation-4dddc110-8c53-4bc0-bf81-730f16c7b271");katex.render("T", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ < 3700 K, the specific heat is defined as

$(4)var element = document.getElementById("moose-equation-1bc1d799-9528-4321-8418-7e6b7086cf69");katex.render(" C_p(T) = B0 + B1 \\cdot \\left( \\frac{T}{1000} \\right) + B2 \\cdot \\left( \\frac{T}{1000} \\right)^2 + B3 \\cdot \\left( \\frac{T}{1000} \\right)^3 + \\frac{B_{-2}}{\\left( \\frac{T}{1000} \\right)^2}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where:

$var element = document.getElementById("moose-equation-8e384d04-af51-49f8-97a4-9ed0fb6fa699");katex.render("C_p", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ (T)$ is the specific heat [J/g.K],
$var element = document.getElementById("moose-equation-4eed6686-24d3-4d0a-af1e-c7fd4d0259b9");katex.render("T", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the temperature [K],

and the model parameters values provided in Table 4.

Table 4: Parameters values for the specific heat model from Eq. (4).

Constant	Value	Unit
B0	1.301E-01	`[J/g.K]`
B1	2.225E-02	`[J/g.K]`
B2	-7.224E-03	`[J/g.K]`
B3	3.539E-03	`[J/g.K]`
B $var element = document.getElementById("moose-equation-46c2d518-fefe-4c21-bdf9-83b18f36f453");katex.render("_{-2}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$	-3.061E-04	`[J/g.K]`

Density

For the temperature range: 5 ≤ $var element = document.getElementById("moose-equation-3e64d020-e7d8-4234-9230-a593537b35f6");katex.render("T", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ < 3600 K, the density is defined as $(5)var element = document.getElementById("moose-equation-ff5884ce-9620-4213-8490-4a1687e62d9f");katex.render(" \\rho(T) = \\frac{\\rho_{RT}}{\\left(1 + \\frac{dL}{L_0(T)} \\times \\frac{1}{100}\\right)^3},", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where:

$var element = document.getElementById("moose-equation-36cdbb6a-ee4b-4773-9b8f-580a1400f9a8");katex.render("\\rho(T)", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the density [Kg/m $var element = document.getElementById("moose-equation-94a35aa0-fde7-45d8-9a08-04372b91abb3");katex.render("^3", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ],
$var element = document.getElementById("moose-equation-decedff4-ff6a-4bfd-ab2f-19d6695f91bc");katex.render("\\rho_{RT}(T)", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the room temperature density, set to 19250 [Kg/m $var element = document.getElementById("moose-equation-bf6e1abc-2684-4592-9751-37fc9a83e02b");katex.render("^3", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ],
$var element = document.getElementById("moose-equation-295b002e-c4dd-460b-a362-f195dd461f6f");katex.render("T", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the temperature [K],

and the thermal expansion $var element = document.getElementById("moose-equation-c63834bc-7b84-4196-aafa-8d8a7c5b5b94");katex.render("dL/L_0(T)", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ [%] is given by:

$(6)var element = document.getElementById("moose-equation-4c9d12dc-a0ea-4d03-94a3-c034e941750a");katex.render(" \\frac{dL(T)}{L_0} = A0 + A1 \\cdot \\left( \\frac{T}{1000} \\right) + A2 \\cdot \\left( \\frac{T}{1000} \\right)^2 + A3 \\cdot \\left( \\frac{T}{1000} \\right)^3.", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

The values of the model parameters in Eq. (6) are provided in Table 5 for low temperatures (5 ≤ $var element = document.getElementById("moose-equation-7a12482d-fedd-4269-9e4b-7f61ff615f36");katex.render("T", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ < 294 K), and Table 6 for high temperatures (294 ≤ $var element = document.getElementById("moose-equation-7f275c88-9d70-4ef5-bc41-79dc11e6e023");katex.render("T", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ≤ 3600 K).

Table 5: Parameters values for the thermal expansion model from Eq. (6) for 5 ≤ $var element = document.getElementById("moose-equation-f603fec4-bb5a-4d26-aed1-7487bda38d94");katex.render("T", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ < 294 K.

Constant	Value	Unit
A0	-8.529E-02	`[-]`
A1	-9.915E-02	`[-]`
A2	2.257E+00	`[-]`
A3	-3.157E+00	`[-]`

Table 6: Parameters values for the thermal expansion model from Eq. (6) for 294 ≤ $var element = document.getElementById("moose-equation-eeeedfda-7730-4590-8859-0de81d8b6c02");katex.render("T", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ≤ 3600 K.

Constant	Value	Unit
A0	-1.400E-01	`[-]`
A1	4.869E-01	`[-]`
A2	-3.056E-02	`[-]`
A3	2.234E-02	`[-]`

Input Parameters

temperatureTemperature
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Temperature

Required Parameters

blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
boundaryThe list of boundaries (ids or names) from the mesh where this object applies
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The list of boundaries (ids or names) from the mesh where this object applies
computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
Default:True
C++ Type:bool
Controllable:No
Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
constant_onNONEWhen ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
Default:NONE
C++ Type:MooseEnum
Options:NONE, ELEMENT, SUBDOMAIN
Controllable:No
Description:When ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
declare_suffixAn optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
densitydensityName to be used for the density
Default:density
C++ Type:std::string
Controllable:No
Description:Name to be used for the density
specific_heatspecific_heatName to be used for the specific heat
Default:specific_heat
C++ Type:std::string
Controllable:No
Description:Name to be used for the specific heat
thermal_conductivitythermal_conductivityName to be used for the thermal conductivity
Default:thermal_conductivity
C++ Type:std::string
Controllable:No
Description:Name to be used for the thermal conductivity

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Controllable:No
Description:The seed for the master random number generator
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Controllable:No
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)
C++ Type:std::vector<std::string>
Controllable:No
Description:List of material properties, from this material, to output (outputs must also be defined to an output type)
outputsnone Vector of output names where you would like to restrict the output of variables(s) associated with this object
Default:none
C++ Type:std::vector<OutputName>
Controllable:No
Description:Vector of output names where you would like to restrict the output of variables(s) associated with this object

Outputs Parameters

prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
Default:False
C++ Type:bool
Controllable:No
Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

Material Property Retrieval Parameters

Input Files

(modules/solid_properties/test/tests/materials/tungsten/tungsten_thermal_properties.i)

References

Justin L Milner, Peter Karkos, and Jessica Jane Bowers. Space nuclear propulsion (snp) material property handbook. Technical Report, National Aeronautics and Space Administration, 2024.

@techreport{milner2024space,
    author = "Milner, Justin L and Karkos, Peter and Bowers, Jessica Jane",
    title = "Space Nuclear Propulsion (SNP) Material Property Handbook",
    year = "2024",
    institution = "National Aeronautics and Space Administration"
}

(modules/solid_properties/test/tests/materials/tungsten/tungsten_thermal_properties.i)

# Analytic formulas are taken from Milner, J. L., Karkos, P., & Bowers, J. J. (2024).
# Space Nuclear Propulsion (SNP) Material Property Handbook (No. SNP-HDBK-0008).
# National Aeronautics and Space Administration (NASA). https://ntrs.nasa.gov/citations/20240004217


[Mesh]
  [f]
    type = GeneratedMeshGenerator
    dim = 1
    xmin = 11
    xmax = 3600
    nx = 100
  []
[]

[Materials]
  [tungsten]
    type = TungstenThermalPropertiesMaterial
    temperature = temperature
    outputs = 'all'
  []
[]

[AuxVariables]
  [temperature]
    [AuxKernel]
      type = FunctionAux
      function = linear_x
      execute_on = 'INITIAL'
    []
  []
  [k_from_functions]
    family = MONOMIAL
    order = CONSTANT
    [AuxKernel]
      type = FunctionAux
      function = 'k_parsed_combination'
    []
  []
  [c_p_from_functions]
    family = MONOMIAL
    order = CONSTANT
    [AuxKernel]
      type = FunctionAux
      function = 'c_p_parsed_combination'
    []
  []
  [rho_from_functions]
    family = MONOMIAL
    order = CONSTANT
    [AuxKernel]
      type = FunctionAux
      function = 'density_analytic'
    []
  []
  [k_diff]
    family = MONOMIAL
    order = CONSTANT
    [AuxKernel]
      type = ParsedAux
      expression = 'thermal_conductivity - k_from_functions'
      coupled_variables = 'thermal_conductivity k_from_functions'
    []
  []
  [c_p_diff]
    family = MONOMIAL
    order = CONSTANT
    [AuxKernel]
      type = ParsedAux
      expression = 'specific_heat - c_p_from_functions'
      coupled_variables = 'specific_heat c_p_from_functions'
    []
  []
  [rho_diff]
    family = MONOMIAL
    order = CONSTANT
    [AuxKernel]
      type = ParsedAux
      expression = 'density - rho_from_functions'
      coupled_variables = 'density rho_from_functions'
    []
  []
[]

[Functions]
  [linear_x]
      type = ParsedFunction
      expression = 'x'
  []
  [x_coord]
    type = ParsedFunction
    expression = 'x'
  []
  [k_analytic_T_less_than_55]
      type = ParsedFunction
      symbol_names = 'T'
      symbol_values = 'x_coord'
      expression = '7.348e5*(T/1000)^0.874/(1+25.44*T/1000-8.304e3*(T/1000)^2+1.18e6*(T/1000)^3)'
  []
  [k_analytic_T_between_55_and_3653]
      type = ParsedFunction
      symbol_names = 'T'
      symbol_values = 'x_coord'
      expression = '(-3.679+1.181e2*T/1000+58.79*(T/1000)*(T/1000)+2.867*(T/1000)*(T/1000)*(T/1000))/(-2.052e-2+4.741e-1*T/1000+(T/1000)*(T/1000))'
  []
  [k_parsed_combination]
    type = ParsedFunction
    expression = 'if(x < 55, k_analytic_T_less_than_55, k_analytic_T_between_55_and_3653)'
    symbol_names = 'k_analytic_T_less_than_55 k_analytic_T_between_55_and_3653'
    symbol_values = 'k_analytic_T_less_than_55 k_analytic_T_between_55_and_3653'
  []
  [c_p_analytic_T_less_than_293]
      type = ParsedFunction
      symbol_names = 'T'
      symbol_values = 'x_coord'
      expression = '3.103e2*(T/1000)^3.03/(1-8.815*T/1000+1.295e2*(T/1000)^2+1.874e3*(T/1000)^3)'
  []
  [c_p_analytic_T_between_293_and_3700]
      type = ParsedFunction
      symbol_names = 'T'
      symbol_values = 'x_coord'
      expression = '1.301e-1+2.225e-2*T/1000-7.224e-3*(T/1000)^2+3.539e-3*(T/1000)^3-3.061e-4/(T/1000)^2'
  []
  [c_p_parsed_combination]
    type = ParsedFunction
    expression = 'if(x < 293, c_p_analytic_T_less_than_293, c_p_analytic_T_between_293_and_3700)'
    symbol_names = 'c_p_analytic_T_less_than_293 c_p_analytic_T_between_293_and_3700'
    symbol_values = 'c_p_analytic_T_less_than_293 c_p_analytic_T_between_293_and_3700'
  []
  [dL_L0_analytic_T_less_than_294]
      type = ParsedFunction
      symbol_names = 'T'
      symbol_values = 'x_coord'
      expression = '-8.529e-2 - 9.915e-2 * T/1000 + 2.257 * (T/1000)^2 - 3.157 * (T/1000)^3'
  []
  [dL_L0_analytic_T_between_294_and_3600]
      type = ParsedFunction
      symbol_names = 'T'
      symbol_values = 'x_coord'
      expression = '-1.4e-1 + 4.869e-1 * T/1000 -3.056e-2 * (T/1000)^2 + 2.234e-2 * (T/1000)^3'
  []
  [dL_L0_parsed_combination]
    type = ParsedFunction
    expression = 'if(x < 294, dL_L0_analytic_T_less_than_294, dL_L0_analytic_T_between_294_and_3600)'
    symbol_names = 'dL_L0_analytic_T_less_than_294 dL_L0_analytic_T_between_294_and_3600'
    symbol_values = 'dL_L0_analytic_T_less_than_294 dL_L0_analytic_T_between_294_and_3600'
  []
  [density_analytic]
      type = ParsedFunction
      symbol_names = 'dL_L0'
      symbol_values = 'dL_L0_parsed_combination'
      expression = '19250 / (1 + dL_L0 / 100)^3'
  []
[]

[Problem]
  solve = false
[]

[Postprocessors]
  [thermal_conductivity_avg]
    type = ElementAverageMaterialProperty
    mat_prop = thermal_conductivity
  []
  [specific_heat_avg]
    type = ElementAverageMaterialProperty
    mat_prop = specific_heat
  []
  [density_avg]
    type = ElementAverageMaterialProperty
    mat_prop = density
  []
[]

[VectorPostprocessors]
  [props_and_diffs]
    type = LineValueSampler
    variable = 'thermal_conductivity specific_heat density k_diff c_p_diff rho_diff'
    start_point = '11 0 0'
    end_point = '3600 0 0'
    num_points = 100
    sort_by = x
  []
[]

[Executioner]
  type = Steady
[]

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

Input Parameters
Input Files
References